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Accounting for Incompleteness due to Transit Multiplicity in *Kepler* Planet Occurrence Rates ============================================================================================= Jon K. Zink**¹ , Jessie L. Christiansen**², and Bradley M. S. Hansen**¹ **¹Mani L. Bhaumik Institute for Theoretical Physics, Department of Physics and Astronomy, University of California, Los Angeles, CA 90095 **²NASA Exoplanet Science Institute, California Institute of Technology, Pasadena, CA 91106 E-mail: \\hrefmailto:jzink@astro.ucla.edujzink@astro.ucla.edu Last updated 2018 December 18 ###### Abstract We investigate the role that planet detection order plays in the *Kepler* planet detection pipeline. The *Kepler* pipeline typically detects planets in order of descending signal strength (MES). We find that the detectability of transits experiences an additional 5.5% and 15.9% efficiency loss, for periods <200 days and >200 days respectively, when detected after the strongest signal transit in a multiple-planet system. We provide a method for determining the transit probability for multiple-planet systems by marginalizing over the empirical *Kepler* dataset. Furthermore, because detection efficiency appears to be a function of detection order, we discuss the sorting statistics that affect the radius and period distributions of each detection order. Our occurrence rate dataset includes radius measurement updates from the California Kepler Survey (CKS), *Gaia* DR2, and asteroseismology. Our population model is consistent with the results of Burke et al. ([2015](#bib.bib7)), but now includes an improved estimate of the multiplicity distribution. From our obtained model parameters, we find that only 4.0 ± 4.6% of solar-like GK dwarfs harbor one planet. This excess is smaller than prior studies and can be well modeled with a modified Poisson distribution, suggesting that the *Kepler* Dichotomy can be accounted for by including the effects of multiplicity on detection efficiency. Using our modified Poisson model we expect the average number of planets is 5.86 ± 0.18 planets per GK dwarf within the radius and period parameter space of *Kepler*. ###### keywords: methods: data analysis – planets and satellites: fundamental parameters ††pubyear: 2018††pagerange: Accounting for Incompleteness due to Transit Multiplicity in *Kepler* Planet Occurrence Rates–[8](#A0.F8 "Figure 8 ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates")\\patchcmd\\@combinedblfloats 1 Introduction -------------- The *Kepler* mission has revolutionized our understanding of the frequencies and properties of planets around Sun-like stars. With the final data release DR25, providing all of the data up until the failure of two reaction wheels (Mathur et al., [2017](#bib.bib37)), the primary phase of the project has officially concluded. Within this span, *Kepler* has provided evidence for ≈4, 500 transiting exoplanets.¹¹\\urlhttps://exoplanetarchive.ipac.caltech.edu Nearly 50% of these candidates have been confirmed or validated (Rowe et al., [2014](#bib.bib51); Morton et al., [2016](#bib.bib40)), demonstrating that planets are common and widespread in the Milky Way. There have been many attempts to quantify the frequency of planetary systems and the properties (radius and orbital period) of the planets themselves (Borucki et al., [2011](#bib.bib6); Catanzarite & Shao, [2011](#bib.bib11); Youdin, [2011](#bib.bib64); Howard et al., [2012](#bib.bib30); Batalha et al., [2013](#bib.bib2); Fressin et al., [2013](#bib.bib25); Petigura et al., [2013a](#bib.bib46); Dong & Zhu, [2013](#bib.bib21); Dressing & Charbonneau, [2013](#bib.bib19); Mullally et al., [2015](#bib.bib42); Dressing & Charbonneau, [2015](#bib.bib20); Burke et al., [2015](#bib.bib7); Mulders, Pascucci & Apai, [2015](#bib.bib43); Silburt et al., [2015](#bib.bib54)), with a special attention given to attempting to characterize the frequency of planets with Earth-like properties. One of the most challenging aspects of estimating these occurrence rates is understanding the completeness of the known exoplanet sample. The automation provided by the *Kepler* pipeline has produced a systematic method of detecting transiting exoplanets and thus offers the prospect of a rigorous determination of the survey completeness. With  3.5 years of nearly continuous light curves of  200,000 stars, it is possible to investigate period ranges out to 500 days. Furthermore, the high precision of the *Kepler* light detector has permitted the discovery of planets with radii *r* < 1*r*⊕. Since the completion of the *Kepler* survey, several studies have used this data set to extract population parameters. Petigura et al. ([2013a](#bib.bib46)), using their own *TERRA* pipeline, implemented an *Inverse Detection Method*, where the population CDF (Cumulative Distribution Function) is divided by the detection efficiency. This study also introduced the idea of synthetic planet injections into the *Kepler* light curves to map completeness. Here, artificial transits were injected into the *Kepler* light curves, and the recovery fraction in the *TERRA* pipeline was used to understand the *Kepler* detection efficiency. To avoid confusion from multiple planet transits the Petigura et al. ([2013a](#bib.bib46)) occurrence rate calculation only included the highest SNR (Signal to Noise Ratio) planet in each system, ignoring any multiplicity. To characterize the official *Kepler* completeness, Christiansen et al. ([2015](#bib.bib15)) performed a pixel-level transit injection test to empirically measure how well the pipeline would detect various types of planets. This is discussed in more detail in Section [4](#S4 "4 Injection Recovery ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates"). The results of this study were then used by Burke et al. ([2015](#bib.bib7)) to perform a *Poisson Process Analysis*, where a Bayesian framework is implemented to determine the best population model parameters. The current work employs a similar method. Planet multiplicity introduces detection biases above and beyond those to which single transit systems are subject. When faced with a system of multiple transiting planets, the *Kepler* pipeline will typically find the largest MES (Multiple Event Statistic; comparable to SNR) signal, fit the transit function, and then discard the corresponding data points. The width of discarded data is 3× the transit duration, with 1.5× removed on each side of the transit center. Very few TTVs (Transit-Timing Variations) are large enough to escape this window. Such deletion is necessary to avoid confusion when looking for additional planets, but introduces data gaps into the light curve as noted by Schmitt et al. ([2017](#bib.bib52)). These gaps becomes more invasive in higher multiplicity systems where significant data is being discarded. With each planet removed, the available data set shrinks. This effect creates ‘‘swiss cheese’’-like holes in the light curves, where the number of holes increases with each detected planet. Beyond possible gaps in the light curve, the *Kepler* pipeline fails to detect some short-period planets because of a harmonic fitting function (Christiansen et al., [2013](#bib.bib14)). Here the pipeline attempts to remove sinusoidal variations in the light curve caused by stellar activity, but in doing so, the procedure can overfit a true planet signal and make low SNR planets difficult to detect. To clarify, the baseline wobble from the dataset is removed using a spline smoothing function. The harmonic fitting function is specifically looking for sinusoidal variations in the light curve. This function may or may not be applied, depending on whether the pipeline is able to detect such periodic variation in the light curve. In multiple-planet systems, the harmonic fitter can also overfit the periodic variations caused by transits and remove true signals. Because the pipeline follows these procedures, the order of planet detection can affect it’s detectability. Our goal in this paper is to assess the effect of planet multiplicity and detection order on the completeness of the Kepler results. In Section [2](#S2 "2 Stellar Selection ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates") and [3](#S3 "3 Planet Selection ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates") we describe our methods of stellar and planet selection. In Section [4](#S4 "4 Injection Recovery ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates") we show that detection order affects the detection efficiency for a given planet. In Section [5](#S5 "5 Effects of mutual inclination ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates") we describe how we account for mutual inclination within this study. In Section [6](#S6 "6 Detection Efficiency Grid ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates"), we lay out our process of accounting for overall detection efficiency. In Section [7](#S7 "7 The Likelihood Function ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates") we present our expanded likelihood function used to calculate the posterior for the population parameters. In Section [8](#S8 "8 Discussion ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates"), we discuss the results of our fitting method and the implications of our multiplicity parameters. We provide concluding remarks in Section [9](#S9 "9 Conclusion ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates").  2 Stellar Selection ------------------- Using the final release of *Kepler* data (DR25) which includes Q1-Q17, we select a stellar sample for use in creating a detection efficiency map that accounts for *Kepler* completeness. We use the stellar parameters provided by Mathur et al. ([2017](#bib.bib37)) with improved radius values derived from *Gaia* DR2 (Berger et al., [2018](#bib.bib4)). The updates from *Gaia* DR2 have yet to provide updated corresponding mass values. Thus we must still utilize the *Kepler* DR25 stellar mass parameters (200,038 stars in total). To focus on the occurrence of planets around solar-like GK dwarfs, we only include stars with *T**e**f**f* > 4200, *K* and *T**e**f**f* < 6100*K* (135,494 stars remain). It is also important for completeness mapping that each star has a stellar radius and mass measurement available. ‘‘Null’’ values for either of these fields result in omission (133,056 stars remain). To avoid the inclusion of giants we limit the sample to *l**o**g*(*g*) ≥ 4 and *R*⋆ ≤ 2*R* (96,167 stars remain). We also place requirements on the duty cycle (*f**d**u**t**y*) and the time length of the light curve (*d**a**t**a**s**p**a**n*). These are *f**d**u**t**y* > 0.6 and *d**a**t**a**s**p**a**n* > 2 years are made (86,679 stars remain). The *f**d**u**t**y* limit requires that 60% of *d**a**t**a**s**p**a**n* has been collected. This ensures that a significant portion of the light curve is filled, while still including stars lost in the Q4 CCD loss (Batalha et al., [2013](#bib.bib2)). Time-varying noise measurements have been provided in the DR25 dataset through a value known as CDPP (Combined Differential Photometric Precision; Christiansen et al. [2012](#bib.bib13)). This parameter has been calculated for every field star over 14 different time periods: 1.5, 2.0, 2.5, 3.0, 3.5, 4.5, 5.0, 6.0, 7.5, 9.0, 10.5, 12.0, 12.5, and 15.0 hours (Mathur et al., [2017](#bib.bib37)). These values correspond to the amount of noise a planet signal will need to exceed, given a transit duration, to generate a 1*σ* detection. By requiring stars to have a *C**D**P**P*7.5*h* < 1000 ppm, we minimize the inclusion of stellar and instrumental fluctuations (74 stars exceed this limit). From this we produce a stellar sample of 86,605 solar-like stars. | | | |-------|-------| | ![]() | ![]() | Figure 1: The smoothed recovery fraction at each MES bin. The vertical lines (light blue and red) represent the uncertainty in each bin under the assumption of a binary distribution. The bin values are plotted at the center of each bin. The solid lines (dark blue and red) represent the *Γ**C**D**F* distribution fit. The parameters of this model were fit using a *χ*² minimization. 3 Planet Selection ------------------ When available we utilize the updated planetary parameters provided by the California *Kepler* Survey (CKS) (Petigura et al., [2017](#bib.bib48); Johnson et al., [2017](#bib.bib34)) and the asteroseismic updates provided by Van Eylen et al. ([2018a](#bib.bib59)). One of the main advantages for the inclusion of these updates is the improved planet radius measurements. Since our study, like others, does not account for parameter uncertainty, such improvements are essential for accurate occurrence rates. Where CKS and asteroseismic data are unavailable, the measurements provided by the *Kepler* DR25 catalog (Thompson et al., [2018](#bib.bib56)), in conjunction with the *Gaia* DR2 radius updates (Berger et al., [2018](#bib.bib4)), are implemented. Through private communication, it was indicated that this early release of *Gaia* data may contain some planet radius outliers. To combat this issue, we test the radius values against the *Kepler* DR25 catalog. When the updated *Gaia* measurements differ from the *Kepler* DR25 data by >3*σ*, we utilize the *Kepler* DR25 radius measurements. Overall, 19 planets exceed this outlier limit (statistically, we would expect only 8). All period measurements are drawn from the light curves; thus, improved measurements from *Gaia* and CKS have no effect on the inferred period measurements. We use the periods provided in the *Kepler* DR25 catalogs. Both the CKS and *Kepler* DR25 provide flags for false positives. We include data from both CONFIRMED and CANDIDATE planets in DR25 and *C**K**S**f**p* = *F**A**L**S**E* in the CKS update. To further avoid contamination from false positives, we only include planets with periods .5 < *p* < 500 days and radii .5 < *r* < 16*r*⊕. Periods beyond 500 days have been noted to be highly contaminated by false positives because they barely meet the three transit limit of the pipeline (Mullally et al., [2015](#bib.bib42)). Our period and radii range exceeds the conservative cutoffs adopted by many previous studies, but is necessary when exploring the effects of multiplicity. Often planetary systems span the entire range of the *Kepler* parameter space, thus the inclusion of nearly all the planets is needed for an accurate calculation. There exist 3 multi-planet systems (KIC: 3231341, 11122894, and 11709124) where one planet within the system fall beyond the range of this study. We only select the planets from these system that lie within our radius and period cuts. The inclusion of these planets is useful in providing a stronger statistical argument. Although some of the known planets, in these 3 systems, extend beyond the bounds of this study, we expect many other systems within the dataset to contain planets beyond the range of our selection bounds. Furthermore, if we include the planets that lay beyond our radius and period cuts, our analysis we will artificially inflate the number of inferred planets within this range. The accuracy of the *Kepler* detection order (‘‘TCE Planet Number’’) can be affected by systems with existing false positives. When removing these data points, we manually ensure that the detection order only reflects the order in which valid KOIs (*Kepler* Objects of Interest) are detected. For example, a system with 5 ‘‘real’’ KOIs and 1 false positive would have detection orders ranging from 1-5 regardless of order at which the false positive was detected. It should be noted that these false positives do create cuts in the data, similar to that of a planet and therefore affect the detection order. However, without reordering these systems we artificially inflate our multiplicity calculation in Section [7](#S7 "7 The Likelihood Function ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates"). Higher multiplicities are especially sensitive to mild increases as their detection probabilities are very low. Further discussion in Section [4](#S4 "4 Injection Recovery ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates") shows that we use the same detection efficiency for all planets found after the first detected planet, thus only planets artificially being re-assigned to 1 are of concern. Since most false positives provide relatively weak signals, only 14 systems experience this artificial re-ordering. After making the discussed cuts we find that the highest detection order existing in the parameter space is 7. This means that the highest system multiplicity we consider in this study is a 7 planet system. We find 3062 KOIs meet the indicated period and radius requirements. It has been suggested that gas giants eject companion planets while migrating inward (Beaugé & Nesvorný, [2012](#bib.bib3)). Their large Hill radius forces the planets to become unstable as the Hill radius ratio falls below 10. These hot Jupiters create an independent population of single planet systems (Steffen et al., [2012](#bib.bib55)). If it forms via a distinct channel, this population has the ability to skew the inferred distribution of the model for the generic underlying population. To minimize such contamination, we remove all single planet systems with *r* > 6.7*r*⊕ as indicated by Steffen et al. ([2012](#bib.bib55)). Further evidence of this independent population was discussed by Johansen et al. ([2012](#bib.bib33)), who showed that multi-planet systems with one planet of mass >0.1 Jupiter mass are dynamically unstable on short timescales. This 0.1 Jupiter mass limit roughly corresponds to the *r* = 6.7*r*⊕ limit used here. We find that 120 of these single hot Jupiters exist in the dataset, leaving us with 2942 KOIs that fit all the parameter requirements described. Our final catalog of planets and their corresponding parameters can be found online.²²\\urlhttps://github.com/jonzink/ExoMult Table 1: The *Γ* function parameters used to fit the recovery CDF displayed in Figure [1](#S2.F1 "Figure 1 ‣ 2 Stellar Selection ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates"). | Period Range | Maximum Detection (*c*) | Shape (*a*) | Scale (*b*) | Offset (*x*₀) | |----------------------------|-------------------------|-------------|-------------|---------------| | *Γ**C**D**F**m* = 1 | | | | | | .5 < *p* < 200 days | 0.9825 | 29.3363 | 0.2856 | 0.0102 | | 200 < *p* < 500 days | 0.9051 | 18.4119 | 0.3959 | 1.0984 | | *Γ**C**D**F**m* ≥ 2 | | | | | | .5 < *p* < 200 days | 0.9276 | 21.3265 | 0.4203 | 0.0093 | | 200 < *p* < 500 days | 0.7456 | 5.5213 | 1.2307 | 2.9774 | 4 Injection Recovery -------------------- Here we shall discuss how we can account for the detection efficiency as a function of detection order. Christiansen ([2017](#bib.bib16)) injected artificial planet signals into the calibrated pixels of each of the *Kepler* field stars and processed the altered light curves with the standard detection pipeline. This allows the recovery fraction to be assessed, producing a probability function based on transit MES (Multiple Event Statistic; a detailed description of MES can be found in equation [14](#S6.E14 "(14) ‣ 6.1 Probability of Detection for = m 1 ‣ 6 Detection Efficiency Grid ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates")). A *Γ**C**D**F* (Cumulative Distribution Function) was fit to the empirical probability of recovery, of the form: $${\\Gamma\_{CDF}{({MES})}} = {\\frac{c}{b^{a}{{({a - 1})}!}}{\\int\_{0}^{MES}{{({x - x\_{0}})}^{a - 1}e^{\\frac{- {({x - x\_{0}})}}{b}}{dx}}}}$$ (1) The purpose of this test was to establish an average detection efficiency function for the *Kepler* pipeline as determined by the properties of the target star sample. Therefore planet detection order was not considered. However, many of the target stars are known to host real KOIs, and these signals will remain in the Christiansen ([2017](#bib.bib16)) analysis. This provides an opportunity to consider the effects of detection order on recovery. Here we define detection order by the variable *m*, where *m*=1 indicates the first planet discovered in the system (i.e. highest MES). Likewise, planet *m*=2 and *m*=3 corresponds to the second and third planets found by the *Kepler* pipeline. The highest detection order existing in the parameter space is 7, thus we shall work in the range of *m* = 1 : 7.  We split the data from Christiansen ([2017](#bib.bib16)) into injection with a .5 < *p* < 200 days or 200 < *p* < 500 days. The break at 200 days was selected by testing different values. Beyond 200 days, we find that the distributions begin to change significantly. To focus on the relevant parameter space of our study, we remove all injections with periods beyond 500 days and only consider stars within 4200*K* < *T**e**f**f* < 6100*K* and *l**o**g*(*g*) ≥ 4. Because the goal for the original Christiansen ([2017](#bib.bib16)) experiment was to find an overall detection probability, only one artificial signal was injected into each light curve with a radius and period uniformly sampled from .25 − 7*r*⊕ and .25 − 500 days respectively. To understand the effect of multiple planets we therefore need to investigate systems with existing transit signals in the light curve. Over 30,000 unique signals are found within the *Kepler* data pipeline. Although most of these were later deemed false positives by external checks, the pipeline treats them no differently than an actual planet. It is even very likely that some of them are in fact ‘‘real’’ planets. Therefore, injections in these systems will be subject to the same systematic issues as that of an actual multiple-planet system. This offers a far greater number of *m* ≥ 2 injections than those provided by the KOI list alone. For the system with injected signals, for 200 < *p* < 500 days we find 2,099 *m* ≥ 2 systems and 1,579 *m* ≥ 2 systems for .5 < *p* < 200 days. We also separated the injections into *m* ≥ 3, but this data is extremely limited and cannot produce meaningful results without further injections. Thus, we shall focus only on m=1 and *m* ≥ 2 systems. The data are then binned in MES and the recovery fraction is determined at each binned region of MES space. Because the available data is relatively small compared to the original number of primary injections (31,302 for 200 < *p* < 500 days and 29,083 for .5 < *p* < 200 days), a smoothing technique is utilized. The bin width is set to 2 MES, but instead of moving each bin by steps of width 2, the bins were recalculated at steps of 0.01 MES. This produced 800 data points across a parameter space of 0-16 MES. Utilizing this technique avoids artifacts produced when binning smaller data samples. One issue that can arise from such smoothing is an artificial distribution skew. In acknowledging this possibility, we have tested various bin widths while smoothing and find little deviation from the results with the adopted binning. Since each injection within a bin can have two possible outcome, a detection or a failed detection, the distribution within each bin will follow a binomial model. Here the number of trials corresponds to the number of injections within the bin. Thus, the uncertainty for each bin is calculated assuming a binomial distribution. The recovery CDF is then fit with a 4-parameter *Γ* distribution using a *χ*² minimization. The results of the fit can be seen in Table [1](#S3.T1 "Table 1 ‣ 3 Planet Selection ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates") and Figure [1](#S2.F1 "Figure 1 ‣ 2 Stellar Selection ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates"). One of the main motivations for creating these additional detection efficiency curves was a preliminary search of the results of the Christiansen ([2017](#bib.bib16)) injection test. This showed that 61 previously detected KOIs were lost when the injection of additional planets was made. Thirty-nine of these KOI planets had a low ‘‘Disposition Score’’, indicating that small perturbation to the light curve could easily disrupt their detectability. One KOI was lost because of transit interference, where a higher MES injection with overlapping transits caused some of the transits of the weaker signal planet to be missed. Twenty-one systems had indicated that the harmonic fitting function was triggered when the injection was made, likely overfitting to the transits themselves. Ten of these light curves had no detections of planets at all. Both the injection and the KOI were missed when the artificial planet was placed into the system. This indicates that multiple-planet systems are constrained by additional detection biases not experienced by single planet systems. 5 Effects of mutual inclination ------------------------------- Here, we shall discuss how the effects of mutual inclination are handled within our model. The initial recovery study (Christiansen, [2017](#bib.bib16)) was performed without consideration of higher multiplicity planets. Thus, there was no accounting for mutual inclination. The artificial planets were injected with a random impact parameter (b) from 0 to 1. To understand the effects of mutual inclination on detection efficiency we look at the difference of impact parameters (*Δ**b*) for recovered planet systems. *Δ**b* is calculated by taking the difference of the artificial planet and the largest MES KOI impact parameter in each system. Since an existing KOI is required for this test, we only look at systems with known planets. We find that the detected planets do not significantly differ in *Δ**b* than the difference of two randomly drawn populations of *b* values. Because the artificial planets were injected with uniformly drawn impact parameters, we conclude that the *Δ**b*, and therefore mutual inclination, plays an insignificant role in detection efficiency. However, larger mutual inclinations can cause certain planets to geometrically avoid transit completely. ### 5.1 Transit Probability Analytic models of transit probability have been found for double transit systems as a function of mutual inclination (Ragozzine & Holman, [2010](#bib.bib49)). However, larger multiplicity systems are more difficult and require semi-analytic models (Brakensiek & Ragozzine, [2016](#bib.bib10)). In order to simplify our calculation, we simulate various semi-major axis to stellar radius ratios (*a**p*/*R*⋆) and look at 10⁶ lines of sight to predict the probability of transit. To determine the period population we need a function for *m* transit probability at some semi-major axis value (*a**p*). In order to create a function for probability of transit in addition to *m* − 1 other transits, it is essential that we know the distributions of exoplanet periods. Clearly, this argument is circular in nature. We deal with this issue by using a non-uniform method of sampling from the empirical period population. This is performed for detection order *m* = 2 : 7, since the analytic probability (*R*⋆/*a**p*) is sufficient for m=1. To establish the desired detection order, the required number of planets are drawn from the empirical *Kepler* period data. For example, when looking at the case of m=3, (*a**p*/*R*⋆) is selected and then the two additional planets are drawn from the known *Kepler* period sample. The periods of the additional two planets are redrawn at each line of sight. This is the same as saying we marginalized the additional two planets over the *Kepler* period population. In order to properly account for the transit probability of higher detection orders, we need to know the unbiased underlying populations of periods. To approximate this, we sample the empirical distribution of *Kepler* planet periods, but weighted with a probability ∝*p*2/3. This is done to account for the geometric bias against the detection of longer period planets. To account for the mutual inclination between orbits, we follow the *σ**σ* distribution provided by Fang & Margot ([2012](#bib.bib22)). This mild distribution (⟨*σ*⟩ = 1.6*o*) was found by looking at the impact parameter ratios within *Kepler* systems. Once all orbits have been selected, the number of lines of sight where all planets transit is divided by 10⁶ to establish the transit probability. To determine whether a planet is transiting this equation must be satisfied: $$\\begin{array}{r} {{{{{cos{(i)}}\*c}os{(\\omega)}} - {{{sin{(i)}}\*s}in{(\\omega)}}} \\geq {R\_{\\star}/a\_{p}}} \\\\ {{{{{\\text{or~}sin{(i)}}\*s}in{(\\omega)}} - {{{cos{(i)}}\*c}os{(\\omega)}}} \\leq {R\_{\\star}/a\_{p}}} \\\\ \\end{array}$$ (2) where *i* is the inclination of the of the system and *ω* is the ascending node. Each line of sight is drawn uniformly over *s**i**n*(*i*) and the nodes of each orbit are also drawn uniformly over *s**i**n*(*ω*). For nodes between planets within the same system, we sample uniformly over *s**i**n*(*Δ**ω*). We note that Equation [2](#S5.E2 "(2) ‣ 5.1 Transit Probability ‣ 5 Effects of mutual inclination ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates") is only valid for circular orbits. Consideration of eccentric orbits is presented in Section [8.5](#S8.SS5 "8.5 Considering Eccentricity ‣ 8 Discussion ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates"). To avoid the creation of unstable systems, we check the planet separations (|*a**p*2 − *a**p*1|). If any separation is <10% the semi-major axis of the outer planet we resample the entire system. This process is repeated until no separations fall below the 10% threshold. Although mutual Hill radius would provide a better measure of stability, our metric requires no assumptions about the mass of the planets. Furthermore, we find that changing (or removing) this threshold makes little difference to the probabilities calculated, indicating that stability accounting has little effect statistically. The results of this simulation can be seen in Figure [2](#S6.F2 "Figure 2 ‣ 6 Detection Efficiency Grid ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates").  It is worth noting that equation [2](#S5.E2 "(2) ‣ 5.1 Transit Probability ‣ 5 Effects of mutual inclination ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates") does not account for grazing transits. To properly account for this, *R*⋆ must becomes *R*⋆ ± *r*, where *r* is the radius of the transiting planet. Using a uniform distribution of *r* values from 0.5*r*⊕ to 16*r*⊕, we find that grazing transits provide an increase of 0.2% to the overall transit probability. However, this uniform distribution is weighted far more heavily towards large planets than the underlying planet radius distribution, thus we expect the true correction to be much smaller. To properly account for grazing transit one must have some understanding of the underlying radius population. Any attempt to do so here would add more uncertainty to the calculation and provide a very minimal correction. Thus, we ignore such complications here. 6 Detection Efficiency Grid --------------------------- To represent the *Kepler* survey detection efficiency a grid is created in period and radius space. Both *l**o**g*₁₀*p* and *l**o**g*₁₀*r* are divided into 100 bins, creating 10,000 regions of the parameter space. For every region we uniformly sampled in log space for period and radius, all 86,605 stars are assigned *m* planets based on the detection order of interest. For example, in the detection grid for the first transiting planet (m=1), the probability of detecting at least one planet is calculated at each bin. Similarly for m=2, the probability of detecting at least one planet at each bin in addition to finding another planet in some other arbitrary bin. The average detection probability for each region is calculated using these planetary assignments and the procedures provided in the next Sections ([6.1](#S6.SS1 "6.1 Probability of Detection for = m 1 ‣ 6 Detection Efficiency Grid ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates"); [6.2](#S6.SS2 "6.2 Probability of Detection for ≥ m 2 ‣ 6 Detection Efficiency Grid ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates")). This process is then repeated for each of the 10,000 regions. We calculated 7 detection efficiency grids: first planet probability (m=1), second planet probability (m=2),…, and the seventh planet probability (m=7). This procedure is similar to that of Burke et al. ([2015](#bib.bib7)) and Traub ([2016](#bib.bib57)), but now with 7 different detection order grids. ![]() Figure 2: The probability for transit of high multiplicity systems using the Fang & Margot ([2012](#bib.bib22)) mutual inclination model. The solid black line represents the probability function used for an *m* = 1 planet transit (*R*⋆/*a**p*). A machine-readable version of this data is available \\hrefhttps://github.com/jonzink/ExoMultonline. ![]() ![]() ![]() ![]() ![]() ![]() Figure 3: The detection efficiency maps for m=1:4 exoplanet discovery orders. The color map is representative of *l**o**g*₁₀(Detection Probability). The fading of color across detection order (m) shows the decreasing detection probability. A machine-readable version of this data is available \\hrefhttps://github.com/jonzink/ExoMultonline. ### 6.1 Probability of Detection for *m* = 1 We shall begin with the formula for the detection of the first planet and then discuss the modifications made for the detection of higher order systems. In our base model we assume all planets have perfectly circular orbits and consider the effects of eccentricity in Section [8.5](#S8.SS5 "8.5 Considering Eccentricity ‣ 8 Discussion ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates"). This assumption of little or no eccentricity is reasonable for the typical multiple systems sampled by *Kepler*, where non-circular orbits would result in unstable system architecture. To account for the geometric probability of transit we use: $$P\_{tr} = \\frac{R\_{\\star}}{a\_{p}}$$ (3) where *R*⋆ is the radius of the star and *a**p* is the semi-major axis of the planet orbit. The chord at which the planet transits across the stellar host is given by $$f\_{tr} = \\sqrt{1 - b^{2}}$$ (4) where *b* is the impact parameter of the planet transit. *b* is assigned by uniformly sampling between 0 and 1 for each planet. The duration of the transit can be calculated as $$t\_{dur} = {{{\\frac{R\_{\\star}\*f\_{tr}}{a\_{p}\*\\pi}{(\\frac{p}{1\\text{day}})}}\*24}\\text{hr}}$$ (5) where *p* is the orbital period of the planet. The expected number of transits can be found with $$n\_{tr} = \\frac{data\_{span}}{p}$$ (6) where *d**a**t**a**s**p**a**n* is the span of the data within the *Kepler* survey. Because of various shut downs and data downloads throughout the *Kepler* mission, it is possible that some of the transits may have been missed. To account for the probability of the transit occurring in the window of the *Kepler* mission we adopt the window function provided by Burke et al. ([2015](#bib.bib7)). $$j = \\frac{data\_{span}}{p}$$ (7) $$\\begin{array}{r} {P\_{win} = {1 - {({1 - {duty}})}^{j} - {{j\*d}uty{({1 - {duty}})}^{j - 1}}}} \\\\ {- \\frac{j{({j - 1})}duty^{2}{({1 - {duty}})}^{j - 2}}{2}} \\\\ \\end{array}$$ (8) where *d**u**t**y* is the duty fraction of the targeted stellar source. The *Kepler* pipeline requires at minimum 3 transits for candidate consideration; *P**w**i**n* is the probability that at least 3 transits will be detected by the available *Kepler* data. Since most targets have a *d**u**t**y* = .95, short period transits (*j* ≫ 3) produce a *P**w**i**n* nearly 1 and approach 0 as *j* < 3. Almost all of our sample have data throughout the full data set span of 1458.931 days. The mean *d**a**t**a**s**p**a**n* for this study is 1427.445 days. Other studies have used various way to account for the effects of limb darkening such as that of Claret & Bloemen ([2011](#bib.bib17)). We attempt to mimic the pipeline by looking at the empirical limb darkening values chosen for existing KOIs (with the same stellar parameters discussed in Section [2](#S2 "2 Stellar Selection ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates")). We find that the two limb darkening parameters (*u*₁, *u*₂) used to fit planet transits within the pipeline are strongly correlated to stellar temperature (*T**e**f**f*). The best fit line to this correlation is as follows: $$\\begin{aligned} u\_{1} & {= {{- {1.93\*10^{- 4}\*T\_{eff}}} + 1.5169}} \\\\ u\_{2} & {= {{1.25\*10^{- 4}\*T\_{eff}} - 0.4601}} \\\\ \\end{aligned}$$ (9) We warn that these correlations mimic the choice of the pipeline rather than the true stellar features and should not be used for more evolved stars with *l**o**g*(*g*) < 4. With the given calculated parameters, it is now possible to calculated the expected MES of the *Kepler* pipeline as presented by Burke & Catanzarite ([2017a](#bib.bib8)). $$k\_{rp} = \\frac{r}{R\_{\\star}}$$ (10) *c*₀ = 1 − (*u*₁ + *u*₂) (11) $$\\omega = {{\\frac{c\_{0}}{4} + \\frac{u\_{1} + {2\*u\_{2}}}{6}} - \\frac{u\_{2}}{8}}$$ (12) $$\\begin{array}{r} {depth = 1 - {(\\frac{c\_{0}}{4} + \\frac{{({u\_{1} + {2\*u\_{2}}})}\*{({1 - k\_{rp}^{2}})}^{\\frac{3}{2}}}{6}}} \\\\ {- \\frac{u\_{2}{({1 - k\_{rp}^{2}})}}{8})\\omega^{- 1}} \\\\ \\end{array}$$ (13) $${MES} = {\\frac{depth}{{CDPP}\*10^{6}}\*1.003\*n\_{tr}^{\\frac{1}{2}}}$$ (14) where *C**D**P**P* is in ppm from the *Kepler* stellar catalog, interpolated by the transit duration. Finally, we account for the systematic detection efficiency using the Gamma distribution CDF described in Section [4](#S4 "4 Injection Recovery ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates"). *P**t**i**p**m* = 1 = *Γ**C**D**F**m* = 1(*M**E**S*) (15) where the parameters for *Γ**C**D**F* are the given in Table [1](#S3.T1 "Table 1 ‣ 3 Planet Selection ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates"). Combining all of the discussed probabilities provides an estimate of the detection likelihood of the highest MES planet within the system. This probability is given as follows: *P**d**e**t**m* = 1 = *P**t**r* \* *P**w**i**n* \* *P**t**i**p**m* = 1 (16) This equation provides a metric for understanding the bias of the highest MES planet. This probability is dependent on detection order and we shall now discuss in the next section how higher multiplicity planets (*m* ≥ 2) can be accounted for. ### 6.2 Probability of Detection for *m* ≥ 2 For *m* ≥ 2 planets we follow much of what is described in the previous Section ([6.1](#S6.SS1 "6.1 Probability of Detection for = m 1 ‣ 6 Detection Efficiency Grid ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates")), with a few mild changes to better model the differences in detection probability. We change the transit probability to reflect the probability of *m* planets transiting, accounting for the probability of finding this planet with at minimum *m* − 1 other planets. To best capture the probabilities of our simulation in Section [5.1](#S5.SS1 "5.1 Transit Probability ‣ 5 Effects of mutual inclination ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates"), we interpolate between simulated data points for the transit probability. $$P\_{tr}^{m} = {\\text{Linear\\ Interpolate}{(m,\\frac{a\_{p}}{R\_{\\star}})}}$$ (17) For example, if we are looking at a planet with m=3 (the third planet detected) with *a**p*/*R*⋆ = 32, we would expect a transit probability of ∼0.008. This can be clearly seen in the data provided by Figure [2](#S6.F2 "Figure 2 ‣ 6 Detection Efficiency Grid ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates"). Since no such simulated value exist at this exact point, we interpolate between the the two neighboring estimations to establish this value. Here we use the new detection efficiency for higher m planets. *P**t**i**p**m* ≥ 2 = *Γ**C**D**F**m* ≥ 2(*M**E**S*) (18) *P**d**e**t**m* = *P**t**r**m* \* *P**w**i**n* \* *P**t**i**p**m* ≥ 2 (19) where equation [8](#S6.E8 "(8) ‣ 6.1 Probability of Detection for = m 1 ‣ 6 Detection Efficiency Grid ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates") is again used for *P**w**i**n*. In reality, there are differing window functions for each detection order; when tested, we find that ≈0.4% of the light curve is lost with the addition of each planet. One can see that varying the *d**u**t**y* parameter of equation [8](#S6.E8 "(8) ‣ 6.1 Probability of Detection for = m 1 ‣ 6 Detection Efficiency Grid ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates") by even 3% has negligible effects on the *P**w**i**n* value. Because the detection efficiency is the same for *m* ≥ 2, the only difference between the *m* = 2 : 7 probability maps is the transit probability. This now produced 7 distinct detection grids (*m* = 1 : 7). The first four grids can be seen in Figure [3](#S6.F3 "Figure 3 ‣ 6 Detection Efficiency Grid ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates"). The detection order of the exoplanet in question will dictate which grid is most appropriate for application. To summarize, we have described how the recovery probabilities (CDF) are a function of detection order (m). We use this to create 7 different detection efficiency maps (Figure [3](#S6.F3 "Figure 3 ‣ 6 Detection Efficiency Grid ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates")). In order to create a map for m=1 planets, we sample across planet period and radius space. Doing so, we calculate the probability of detection and averaged over all stars within the *Kepler* stellar sample. We expand upon this idea, creating a map for m=2 planets. Here the new recovery CDF is implemented to account for the additional loss of planets at higher detection orders. Furthermore, we account for the probability of two planets within the system transiting using a mild mutual inclination model (Figure [2](#S6.F2 "Figure 2 ‣ 6 Detection Efficiency Grid ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates")). Jumping from m=1 to m=2 we lose an additional 5.5% and 15.9% of the planets for periods <200 days and periods >200 days respectively. This is due to properties of the pipeline when fitting multiple transit systems. This procedure is repeated for m=3:7 each accounting for the appropriate number of transiting planets according to the data in Figure [2](#S6.F2 "Figure 2 ‣ 6 Detection Efficiency Grid ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates") (3-7 respectively). There is an additional loss of nearly 70% at each respective discovery order due to the unlikely event of multiple orbital alignment with our line of sight. It is clear that these two factors, geometric transit likelihood and pipeline recovery, have a significant effect on the multiplicity extracted from the *Kepler* data set. ![]() Figure 4: The sorting simulation for m=1 and m=2. The solid blue line represents the Beta distribution fit to the respective data set. The boxes are a histogram of the simulated data after being sorted. It is apparent that sorting has a more dramatic effect on radius than period. This is expected as *M**E**S* ∝ *r*²/*p*1/3. A mild deviations from the model is noted in the radius skew. This discrepancy dissolves as we move into higher detection orders. Furthermore, the effects of these deviations are insignificant, given the cuts on duty cycle, data span, and stellar type already made. Table 2: The parameters found in testing the sorting effects of MES. These parameters correspond to a Beta distribution skew expected for the CDF of each multiplicity population. | | m=1 | m=2 | m=3 | m=4 | m=5 | m=6 | m=7 | |--------------|-------|-------|-------|-------|-------|-------|-------| | *a**r**a**d* | 1.095 | 1.030 | 1.028 | 1.013 | 0.998 | 1.065 | 0.951 | | *b**r**a**d* | 0.923 | 1.470 | 2.206 | 3.063 | 4.013 | 4.898 | 6.614 | | *a**p**e**r* | 0.957 | 1.152 | 1.172 | 1.184 | 1.183 | 1.166 | 1.234 | | *b**p**e**r* | 1.004 | 1.010 | 1.000 | 0.999 | 0.997 | 1.006 | 0.994 | 7 The Likelihood Function ------------------------- Using the efficiency grids derived in the previous section, we can infer properties of the underlying planetary population. Here we will discuss the likelihood function required to implement Bayes theorem and extract these population parameters. We adopt the approach of previous studies (e.g. Youdin [2011](#bib.bib64); Petigura et al. [2013a](#bib.bib46); Burke et al. [2015](#bib.bib7)), modeling the underlying population as characterized by independent power-law distributions in period and radius. We also make explicit the assumption that there is a single planetary population – assuming that systems which show only one transit are drawn from the same underlying distribution as those which show multiple transits. We will examine the validity of this assumption in Section [8.1](#S8.SS1 "8.1 Forward Modeling the Results ‣ 8 Discussion ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates"). Our focus on multiple systems also means that we include more of the *Kepler* parameter space than was used in most previous papers. The population of exoplanets is modeled as follows: $$\\frac{d^{2}N}{dpdr} = {fg{(p)}q{(r)}}$$ (20) $${g{(p)}} = \\left\\{ \\begin{array}{ll} {C\_{p1}p^{\\beta\_{1}}} & {\\text{if~}{p < p\_{br}}} \\\\ {C\_{p2}p^{\\beta\_{2}}} & {\\text{if~}{p \\geq p\_{br}}} \\\\ \\end{array} \\right.$$ (21) $${q{(r)}} = \\left\\{ \\begin{array}{ll} {C\_{r1}r^{\\alpha\_{1}}} & {\\text{if~}{r < r\_{br}}} \\\\ {C\_{r2}r^{\\alpha\_{2}}} & {\\text{if~}{r \\geq r\_{br}}} \\\\ \\end{array} \\right.$$ (22) where *f*, *α*₁, *α*₂, *β*₁, *β*₂, *p**b**r*, and *r**b**r* are all fit parameters. We require continuity at *r**b**r* and *p**b**r* through the normalization constants for *q*(*r*) and *g*(*p*). Our method expands on the Poisson process likelihood used by Youdin ([2011](#bib.bib64)). The main difference is the separation of planets by detection order (*m*). In doing so, we require different occurrence factors (*f*) for each *m*, increasing the required number of parameters. Previous studies such as Burke et al. ([2015](#bib.bib7)) have used a single occurrence value, providing an average occurrence factor. By separating the occurrence factor as a function of detection order, we can allow for differences in detection efficiency while simultaneously fitting for the occurrence of planet multiplicity. $${Likelihood} = {\\prod\\limits\_{m = 1}^{7}{{\\lbrack{\\prod\\limits\_{i = 1}^{n\_{m}}{f\_{m}\\eta\_{m}{(p\_{i},r\_{i})}g{(p\_{i})}q{(r\_{i})}}}\\rbrack}e^{- N\_{m}}}}$$ (23) $$\\begin{array}{ll} & {N\_{m} =} \\\\ & {86,{605f\_{m}{\\int\_{.5\\text{days}}^{500\\text{days}}{\\int\_{.5r\_{\\oplus}}^{16r\_{\\oplus}}{\\eta\_{m}{(p,r)}O\_{m}{(p\_{i},r\_{i})}g{(p)}q{(r)}{dr}{dp}}}}}} \\\\ \\end{array}$$ (24) where *N**m* represents the expected number of planets detected for each discovery order (*m*) and *f**m* is an occurrence factor for each *m*. This value provides information on the occurrence of each *m* multiplicity. However, to find meaningful information from these values, they must be disentangled from each other as discussed in Section [7.0.3](#S7.SS2.SSS3 "7.0.3 Occurrence Factor ‣ 7 The Likelihood Function ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates"). The 86,605 accounts for the number of stars in our test sample and *η**m*(*p*, *r*) is the detection probability at the given detection order. The function *O**m*(*p**i*, *r**i*) is the sorting order correction for the (PDF) Probability Distribution Function. This function is necessary to account for the bias in the PDF introduced by our sorting in terms of detection order (discussed further in [7.0.2](#S7.SS2.SSS2 "7.0.2 Sorting Order ‣ 7 The Likelihood Function ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates")). It is often more useful to consider the natural log of the likelihood, which can be simplified: $${ln{({Likelihood})}} \\propto {{\\sum\\limits\_{m = 1}^{7}{\\lbrack{\\sum\\limits\_{i = 1}^{n\_{m}}{ln{({f\_{m}g{(p\_{i})}q{(r\_{i})}})}}}\\rbrack}} - N\_{m}}$$ (25) Using the *l**n*(*L**i**k**e**l**i**h**o**o**d*) is common practice with fitting algorithms, where the ratio of likelihoods are compared to determine the best fit (maximum likelihood). Since *η**m*(*p*, *r*) is not dependent on the fitting parameters it can be considered constant. #### 7.0.1 Calculating *N**m* To find *η**m*(*p*, *r*) we use the detection maps found in Section [6.1](#S6.SS1 "6.1 Probability of Detection for = m 1 ‣ 6 Detection Efficiency Grid ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates") and [6.2](#S6.SS2 "6.2 Probability of Detection for ≥ m 2 ‣ 6 Detection Efficiency Grid ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates"). Here we are assuming an average probability of detection over the stellar population. To properly treat this integral, one would have to compute the detection probability for each star. Such a procedure would be computationally expensive and provide a minimal increase in precision. Table 3: The mixture probabilities for each detection order. For example, this accounts for the possibility that two and three planet systems may only be found with a single planet. These values were found using our transit probability model described in Section [5.1](#S5.SS1 "5.1 Transit Probability ‣ 5 Effects of mutual inclination ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates"). | | m=1 | m=2 | m=3 | m=4 | m=5 | m=6 | |--------------------------------------------|------|------|------|------|------|------| | $\\frac{P{(2|\\overline{(m,1)})}}{P{(m)}}$ | 0.67 | - | - | - | - | - | | $\\frac{P{(3|\\overline{(m,2)})}}{P{(m)}}$ | 0.68 | 0.50 | - | - | - | - | | $\\frac{P{(4|\\overline{(m,3)})}}{P{(m)}}$ | 0.53 | 1.05 | 0.50 | - | - | - | | $\\frac{P{(5|\\overline{(m,4)})}}{P{(m)}}$ | 0.53 | 1.12 | 1.52 | 0.46 | - | - | | $\\frac{P{(6|\\overline{(m,5)})}}{P{(m)}}$ | 0.37 | 1.07 | 1.85 | 1.69 | 1.22 | - | | $\\frac{P{(7|\\overline{(m,6)})}}{P{(m)}}$ | 0.33 | 0.71 | 1.64 | 1.90 | 1.25 | 1.22 | #### 7.0.2 Sorting Order Here we will provide a brief overview of order statistics and why it is an important feature of this model. As mentioned previously, the *Kepler* pipeline finds planets in order of decreasing MES. Such ordering will skew the distribution of planets found in each *m*. Larger, short period, planets will tend to be found in order *m* = 1 or *m* = 2, because there are more transits and deeper transit depths. Smaller, long-period planets will tend towards orders *m* = 6 or *m* = 7. To account for such a skew, a joint distribution model (*P**m*(*x*)) can be utilized (David & Nagaraja, [2003](#bib.bib18)). *P**m*(*x*) ∝ *P*₀(*x*)*C*₀(*x*)*a**m* − 1(1 − *C*₀(*x*))*b**m* − 1 (26) Here, *P*₀(*x*) is the true underlying probability distribution function and *C*₀(*x*) is the true cumulative distribution function. *a**m* and *b**m* can range from (0,inf ) and forces the skew of the distribution. Essentially, the PDF of the distribution is skewed by a Beta distribution of the CDF. In the case of *a**m* = *b**m* = 1 the sorting skew returns the original PDF (*P*₀(*x*)). The parameters *a**m* and *b**m* can be found analytically for equally sampled orders, but becomes far more complex in the decreasing case at hand (each *m* has fewer planets than the last). To determine the best values for this case, we choose to simulate this sorting mechanism on a uniform distribution, where the skew can be clearly isolated and extracted. In doing so, we force the ratio of each *m* sample to mimic that of the empirical population. Each system is then sorted by *r*²/*p*1/3, imitating *Kepler’s* MES sorting. For example, if a system of (r=1.2,p=25), (3.5,20), (4.1,150) were randomly drawn into *m* = 1, 2, 3 detection orders, they would be re-sorted as (3.5,20), (4.1,150), (1.2,25) corresponding to *m* = 1, 2, 3. As we can see, the highest MES will always rise to *m* = 1. This is then repeated for 10⁷ systems. Figure [4](#S6.F4 "Figure 4 ‣ 6.2 Probability of Detection for ≥ m 2 ‣ 6 Detection Efficiency Grid ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates") shows how the first two detection orders are skewed by this procedure. If sorting were not an issue, these distributions would maintain the uniform flat appearance. Fitting a Beta distribution to this skew, we can determine the best *a**m* and *b**m* parameters for our sample. These parameters are provided in Table [2](#S6.T2 "Table 2 ‣ 6.2 Probability of Detection for ≥ m 2 ‣ 6 Detection Efficiency Grid ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates"). Since the this joint distribution is separable, we define the skew portion of the distribution as *O**m*(*p*, *r*). $$\\begin{array}{ll} {{O\_{m}{(p,r)}} = N} & {\*{C\_{r}{(r)}^{a\_{m,r} - 1}{\\lbrack{1 - {C\_{r}{(r)}}}\\rbrack}^{b\_{m,r} - 1}}} \\\\ & {\*{C\_{p}{(p)}^{a\_{m,p} - 1}{\\lbrack{1 - {C\_{p}{(p)}}}\\rbrack}^{b\_{m,p} - 1}}} \\\\ \\end{array}$$ (27) where *C**r*(*r*) and *C**p*(*p*) represents the CDF of the radius and period distributions respectively and *N* represents a normalization factor that we find numerically within the MCMC. #### 7.0.3 Occurrence Factor As noted, the value *f**m* is an integrated occurrence factor. In order to extract meaningful values, we realize that many *m* = 6, 7 planet systems will only provide detectable transits for one or two planets within the system. This will lead to an increased contribution to lower detection orders. Thus we adopt the following method for disentangling the true occurrence factors (*F**m*): $$f\_{m} = {F\_{m} + {\\sum\\limits\_{n = {m + 1}}^{7}{F\_{n}\\frac{P{(n|\\overline{({m:{n-1}})})}}{P{(m)}}}}}$$ (28) Here $P{(n|\\overline{({m:{n-1}})})}$ represents the probability of finding planet *n* given that planets (*m* : *n* − 1) are not found and *P*(*m*) is the probability of finding planet *m*. This ratio accounts for the dependence between occurrence factors. If the mutual inclination is purely isotropic and planets are truly independent this ratio would be one. We use our transit simulation from Section [5.1](#S5.SS1 "5.1 Transit Probability ‣ 5 Effects of mutual inclination ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates") to extract these marginalized probabilities. Table [3](#S7.T3 "Table 3 ‣ 7.0.1 Calculating N m ‣ 7 The Likelihood Function ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates") contains the results from this simulation. This model indicates that each multi-planet system will have more than one opportunity to find an *f*₁ planet. The physical interpretation of the *F**m* values is the fraction of stars that have at least *m* planets. ### 7.1 Fitting the Data We employ EMCEE (Foreman-Mackey et al., [2013](#bib.bib23)), an affine-invariant ensemble sampler (Goodman & Weare, [2010](#bib.bib27)), to explore the parameter space of our study. To better constrain the 13 fit parameters, a Bayesian framework is implemented. Linear space uniform priors are used for all parameters. For *α*₁, *α*₂, *β*₁, and *β*₂ the priors range from -30 to +30. For *r**b**r* and *p**b**r* the priors range from *r**m**i**n* and *p**m**i**n* to *r**m**a**x* and *p**m**a**x* of our planet sample respectively. One unique restriction for our prior is that *F**m* must be larger than *F**m* + 1. It is not possible to have a higher occurrence of *m* + 1 than *m* planets. To avoid truncation bias and maintain order, all *F**m* priors range from 0 to *F**m* − 1. In the special case of *m* = 1, the prior ranges from 0 to 1. It is important to remember that *F**m* represents the fraction of the population containing *m* planets. Therefore, this cascading prior still allows for larger multiplicity systems to be more common than smaller multiplicity systems.  8 Discussion ------------ In this section, we now apply the formalism we have developed to infer the revised occurrence rate parameters for planets orbiting GK dwarfs. This sample includes data from the final *Kepler* release DR25 and updated planet radius measurements from the CKS and *Gaia* DR2. Beyond these recent data improvements, we now include a corrected detection efficiency for multiple-planet systems. Given that many multiple-planet systems span much of the *Kepler* Parameter space, we include planets within .5 < *r* < 16*r*⊕ and .5 < *p* < 500 days. In implementing two detection efficiencies, this study expands on the Poisson process likelihood function used by other authors, allowing for the treatment of planet multiplicity. This Bayesian framework is fit using an MCMC, where 20,000 steps are used to model the posterior of each parameter. The resulting posteriors are presented in Figure [8](#A0.F8 "Figure 8 ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates"). From this model we infer best fit power-law values of $\\alpha\_{1} = {{- 1.65} \\pm\_{0.06}^{0.05}}$, *α*₂ = −4.35 ± 0.12, *β*₁ = 0.76 ± 0.05, and *β*₂ = −0.64 ± 0.02. The breaks in our best fit model occur at $p\_{br} = {7.08 \\pm\_{0.31}^{0.32}}$ days and *r**b**r* = 2.66 ± 0.06*r*⊕. One novel feature of our fitting method is the ability to extract exoplanet multiplicity. This information is provided through the *F**m* parameters. These values indicate the probability of a system having at least *m* planets. We find the following value best fit our model: $F\_{1} = {0.72 \\pm\_{0.03}^{0.04}}$, *F*₂ = 0.68 ± 0.03, *F*₃ = 0.66 ± 0.03, *F*₄ = 0.63 ± 0.03, *F*₅ = 0.60 ± 0.04, $F\_{6} = {0.54 \\pm\_{0.05}^{0.04}}$, and $F\_{7} = {0.39 \\pm\_{0.09}^{0.07}}$. | | | |-------|-------| | ![]() | ![]() | Figure 5: A plot of the forward modeled population derived by our Bayesian analysis. The red x marks symbolize the model values with their corresponding 68.3% confidence intervals. To find this interval the model is sampled 50 times using the posterior parameter distributions, the uncertainty reflects the fluctuations we find from these trials. The black points show the *Kepler* data with poisson uncertainty. For m=5:7 many of the bins have 1 or 0 planets, where small number statistics cause significant variations. In order to minimize this variations we present the resulting combination of *m* ≥ 4. However, it should be noted that our forward model does differentiate between these detection orders. Left: Forward model of multiple and single planet systems. Right: Forward model of only multiple-planet systems. This model was produced by only fitting to the data of multiple-planet systems. ### 8.1 Forward Modeling the Results Thus far, we have accounted for various parameter and population dependencies. To ensure that this process yields meaningful results, we choose to sample the extracted population and subject it to the detection constraints described in Section [6](#S6 "6 Detection Efficiency Grid ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates"). Here we present the \\hrefhttps://github.com/jonzink/ExoMultExoMult forward modeling software. This code, developed in *R*, simulates these detection effects and produces a population of detected planets. Using this program, we can make far fewer assumptions and directly recover the expected population. For example, the probability of transit for all 7 planets can be directly accounted for by sampling system inclination, mutual inclination and the argument of periapsis directly. Furthermore, the detection probability will not be marginalized over all stars, but rather reviewed for each system independently. The first step in our forward model is drawing each system of planets according to the population parameters given in Figure [8](#A0.F8 "Figure 8 ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates"). Each system is randomly oriented with mutual inclinations drawn from a Rayleigh distribution. For planets with detectable impact parameters (*b* < 1), the planets within each system are sorted in decreasing MES. The probability of recovery is assigned to each planet according to the procedure laid out in Sections [6.1](#S6.SS1 "6.1 Probability of Detection for = m 1 ‣ 6 Detection Efficiency Grid ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates") and [6.2](#S6.SS2 "6.2 Probability of Detection for ≥ m 2 ‣ 6 Detection Efficiency Grid ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates"). Based on the calculated probability of detection, the planet is either detected or lost by drawing from a random number generator. Figure [5](#S8.F5 "Figure 5 ‣ 8 Discussion ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates") shows the best fit model to the observed population obtained with this forward model. It is clear the our Bayesian method provides a reasonable model, where nearly all data points are within a 1*σ* deviation of the observed distribution. Fulton et al. ([2017](#bib.bib26)) and Berger et al. ([2018](#bib.bib4)) have provided evidence for a dip in the radius population around 1.5 − 2*r*⊕. This gap is apparent in the m=1 case of Figure [5](#S8.F5 "Figure 5 ‣ 8 Discussion ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates"). While the deviation from a broken power-law is mild, we explore the effects here. When we remove the single planet systems from the data set, this gap is no longer apparent. One plausible explanation for this gap is a unique population of single planet systems (Although evidence from Weiss et al. ([2018](#bib.bib62)) shows that a weak gap can be seen in the multi-planet systems when aggregated). To explore this theory, we isolate the multi-planet systems and run our fitting procedure again. We find a mild difference in the extracted *α* or *β* power law values (*α*₁ = −1.98 ± 0.08 ; *α*₂ = −3.90 ± 0.16 ; *β*₁ = 0.96 ± 0.08 ; *β*₂ = −0.79 ± 0.03). This indicates that if a separate population does exist, the population parameters are weakly affected by their inclusion in our dataset. The resulting forward model of this fit is presented in Figure [5](#S8.F5 "Figure 5 ‣ 8 Discussion ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates"). Furthermore, the increase in uncertainty seen in these parameters is due to the reduced samples used for fitting (1305 multiple system candidates vs. 2942 total candidates). It is notable that the empirical *Kepler* data set is sharply peaked, while the model does not provide a similar sharpness for the *m* = 1 radius population (Figure [5](#S8.F5 "Figure 5 ‣ 8 Discussion ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates") Right). This could be due to the existence of the mentioned radius gap. Furthermore, it is possible that a true accounting for planet period and radius covariance could produce such a peak. Millholland et al. ([2017](#bib.bib38)) and Weiss et al. ([2017](#bib.bib61)) show that the planets within multiple systems tend to have similar mass and radius components. Although these features are not properly accounted for here, Figure [5](#S8.F5 "Figure 5 ‣ 8 Discussion ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates") (Left) shows that these mild population characteristics remains small and do not deviate greatly from a simple broken power-law model. We hope to include such features in the next iteration of this software. It is possible that future studies may use this forward modeling technique to directly determine the population parameters. Unfortunately, it remains computationally expensive to properly account for all detection features. Traub ([2016](#bib.bib57)) overcame this cost by ignoring multiplicity. ### 8.2 Comparison with Prior Work We use a Bayesian method to infer population parameters for the *Kepler* exoplanet population, following much of the procedure presented in Youdin ([2011](#bib.bib64)). However, we build upon this method to extract information about the population multiplicity. Using a broken power-law distribution we find that population parameters of $\\alpha\_{1} = {{- 1.65} \\pm\_{0.06}^{0.05}}$, *α*₂ = −4.35 ± 0.12, *β*₁ = 0.76 ± 0.05, and *β*₂ = −0.64 ± 0.02 provide the best replication of the empirical population. The best fit breaks in these distributions are as follows: $p\_{br} = {7.08 \\pm\_{0.31}^{0.32}}$ days and *r**b**r* = 2.66 ± 0.06*r*⊕. Many prior studies have examined the occurrence of planets as determined by *Kepler*. Youdin ([2011](#bib.bib64)) provided an early estimate of the occurrence rate using a Poisson process likelihood, finding that the PDF exhibited a power law break at periods ∼7 days, with *α* = −2.44 and *β* = 3.23 at short periods, and *α* = −2.93 and *β* = −0.37 at longer periods (we have converted his numbers into the definitions of *α* and *β* adopted here). These suggest a steep rise towards smaller radius planets at all periods, and a sharp rise with increasing periods to the break, followed by a gradual decline to longer periods. This is consistent with other analysis at the same time (Catanzarite & Shao, [2011](#bib.bib11); Howard et al., [2012](#bib.bib30); Dong & Zhu, [2013](#bib.bib21)). With the accumulation of additional data and more detailed treatment of selection effects, subsequent analyses favored a flatter distribution extending to smaller radii (Fressin et al., [2013](#bib.bib25); Petigura et al., [2013b](#bib.bib47); Silburt et al., [2015](#bib.bib54); Traub, [2016](#bib.bib57)), and a distribution falling off inversely with period (*β* ∼ −1) at longer periods (Petigura et al., [2013a](#bib.bib46); Silburt et al., [2015](#bib.bib54)). The plateau at small radii is also found around lower mass hosts (Dressing & Charbonneau, [2013](#bib.bib19), [2015](#bib.bib20); Mulders, Pascucci & Apai, [2015](#bib.bib43)). Burke et al. ([2015](#bib.bib7)) have presented an extensive discussion of planet occurrence using the Q1-Q16 Kepler sample. For their baseline model, they found corresponding values of *α*₁ = −1.54 ± 0.50 and *β*₂ = −0.68 ± 0.17, with only weak evidence for a break in radius and assuming no break in period (they considered only periods >50 days and radii <2.5*r*⊕). This is perhaps the most directly comparable to our analysis, as it uses the completeness estimates from Christiansen et al. ([2015](#bib.bib15)); where this study uses the updated Christiansen ([2017](#bib.bib16)) completeness data. We find very similar values ($\\alpha\_{1} = {{- 1.65} \\pm\_{0.06}^{0.05}}$; *β*₂ = −0.64 ± 0.02) in a comparable regime. In particular, we note that both of these studies find an increasing occurrence of small radius planets down to the detection threshold, a result also supported by another Bayesian methods estimate in Hsu et al. ([2018](#bib.bib32)). Previous studies have used more limited parameter ranges to avoid issues of parameter covariance and susceptibility to completeness mapping. We approach the problem with a rigorous treatment of completeness mapping and a larger parameter space, recovering a similar power-law distribution. This congruity is an encouraging sign as it shows that the inclusion of a larger parameter space does not largely effect the model being inferred. Our inclusion of a broader range of periods and radius allow us to constrain the power-law uncertainty for radius and period to 3.8% and 5.4% respectively. We find breaks in our period and radius distributions occur at $p\_{br} = {7.08 \\pm\_{0.31}^{0.32}}$ days and *r**b**r* = 2.66 ± 0.06*r*⊕. These results are consistent with those found by prior authors. Table 4: A representation of the expected empirical multiplicity as a function of selection effects. Each column shows the expected population using the best fit model from this study (see Figure [8](#A0.F8 "Figure 8 ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates")). Starting from the left, moving right, each effect is adding in addition to all previous effects. The Multiple Detection Efficiency is broken into two columns. The Data column directly used the multiplicity values shown in Figure [6](#S8.F6 "Figure 6 ‣ 8.3 Survival Function ‣ 8 Discussion ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates"). In contrast, the Model column uses the modified Poisson distribution inferred from the multiplicity data (*λ* = 8.40 ± 0.31 and *κ* = 0.70 ± 0.01). Geometric Mutual Inclination Single Detection Multiple Detection Multiple Detection Real *Kepler* Efficiency Efficiency (Data) Efficiency (Model) Data *S**i**n**g**l**e**s* 1870 1910 1558 1649 ± 71 1629 ± 61 1637 *D**o**u**b**l**e**s* 686 816 397 374 ± 29 375 ± 33 346 *T**r**i**p**l**e**s* 354 483 115 103 ± 15 113 ± 15 119 *Q**u**a**d**r**u**p**l**e**s* 207 282 30 26 ± 6 25 ± 6 43 *Q**u**i**n**t**u**p**l**e**s* 127 159 8 5 ± 3 5 ± 3 13 *S**e**x**t**u**p**l**e**s* 132 77 1 1 ± 1 1 ± 1 2 *S**e**p**t**u**p**l**e**s* 167 28 0 0 ± 1 0 ± 1 1 ### 8.3 Survival Function ![]() Figure 6: A plot of the modified Poisson survival function and the system fraction provided by our Bayesian analysis. This model is fit using a likelihood maximization technique, with the assumption of Gaussian uncertainty (essentially, a *χ*² minimization). The posterior distribution for the model is plotted by sampling 5000 models from the parameter posterior distributions in red. The dark red represents the 1*σ* range and the light red indicates the extent of the 2*σ* range. We have included the models provided by Hansen & Murray ([2013](#bib.bib29)) (gold △ makers) and Fang & Margot ([2012](#bib.bib22)) (olive green + markers) for comparison. Both Hansen & Murray ([2013](#bib.bib29)) and Fang & Margot ([2012](#bib.bib22)) models have been renormalized by our best fit *κ* value. Within this study, we only use planets provided by the *Kepler* pipeline. The highest multiplicity seen is m=7 for a GK type star. This is certainly not the actual highest multiplicity within this parameter space. Shallue & Vanderburg ([2018](#bib.bib53)) use a deep convolutional neural network to extract an 8th planet from the *Kepler*-90 light curve, proving this assertion to be true. Using a Poisson survival function we can extrapolate the probability of existence for these higher multiplicity systems. The *F**m* values found by this study represent the fraction of stars with at minimum *m* planets. This lends itself well to a survival function, where the probability of existing up to a certain value (or multiplicity) is obtained. Survival functions (*S*(*x*)) can simply be written as: *S*(*x*) = 1 − *C**D**F*(*x*) (29) where *C**D**F*(*x*) is the cumulative distribution function of the model. In this case we use a modified Poisson distribution to model multiplicity. Poisson distributions are ideal for planet multiplicity as these distributions are used for counting statistics. The modification is that the distribution is not truly normalized, but rather some fraction *κ* of one. Now that that distribution is no longer normalized the survival function must be modified slightly (*S*(*x*) = *κ* − *C**D**F*(*x*)). This modification allows for an excess or scarcity of zero planet systems. We are only interested in stars that do harbor planets, thus this modification is necessary. The CDF for this modified function is given as: $${CDF{(m)}} = {\\sum\\limits\_{n = 1}^{m}{\\kappa\\frac{\\lambda^{n}e^{- \\lambda}}{{(n)}!}}}$$ (30) where *κ* and *λ* are both fit parameters. Further discussion of this modified Poisson distribution can be found in Section 2.3 of Fang & Margot ([2012](#bib.bib22)). The results of this fit are presented in Figure [6](#S8.F6 "Figure 6 ‣ 8.3 Survival Function ‣ 8 Discussion ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates"). We find that *λ* = 8.40 ± 0.31 and *κ* = 0.70 ± 0.01 provide the best match for this distribution. This large *λ* value incorporates a non-negligible fraction of systems with *m* > 10. Since Equation [30](#S8.E30 "(30) ‣ 8.3 Survival Function ‣ 8 Discussion ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates") allows for an inflated number of star without planet, the global average for GK dwarfs in the *Kepler* parameter space (denoted as ⟨*N**p**l*⟩) can be found by multiplying *λ*, an estimate of the average number of planets a planet harboring system will contain, by *κ*, the fraction of stars that do harbor planets. We find that ⟨*N**p**l*⟩ = 5.86 ± 0.18 planets per star. This is likely a lower bound as we have excluded the single Jupiter sized planets that have cleared their systems through migration. Since these stars are currently assumed to have zero planets by this paper, inclusion of these additional planets would increase the *κ* value. However, we would expect our *λ* parameter to slightly decrease, with the inclusion of these additional singles, as this value only considers systems that do harbor planets. Overall the increase in *κ* will dominate, leading to an overall increase in ⟨*N**p**l*⟩. Previous studies have averaged over multiplicity and inferred the ⟨*N**p**l*⟩ value alone. These values are more difficult to compare as they are strongly dependent on the range of planet radius and period include in each study. Looking at short period (*p* < 50 days) planets, Youdin ([2011](#bib.bib64)) found ⟨*N**p**l*⟩ = 1.36. Using our population parameters and making similar cuts we find a comparable value (⟨*N**p**l*⟩ = 1.34 ± .06). Turning the focus towards small planets (.75 < *r* < 2.5*r*⊕) and long periods (50 < *p* < 300 days), Burke et al. ([2015](#bib.bib7)) found ${\\langle N\_{pl}\\rangle} = {0.73 \\pm\_{.07}^{.19}}$. When we apply these same bounds to our model we again find a slightly larger value (⟨*N**p**l*⟩ = 1.15 ± .03). The most comparable parameter space to our study is that of Traub ([2016](#bib.bib57)), who finds ⟨*N**p**l*⟩ = 5.04 ± .23 using a nearly identical parameter range. While there appears to be a mild tension with this value, we note that Traub ([2016](#bib.bib57)) includes a much broader stellar temperature range and pre-*Gaia* radius measurements, likely leading to this deflated ⟨*N**p**l*⟩ value. With this function in hand, we can extrapolate to higher multiplicity. For example, our model suggests that 32.3 ± 2.7% of GK stars will harbor at least 8 planets within the *Kepler* parameter space. In the parameter space of the *Kepler* survey, our solar system has two planets (Venus and Earth). The radius of Mercury is slightly smaller (.387*r*⊕) than our range allows. Since we find ⟨*N**p**l*⟩ = 5.86 ± 0.18 planets per solar-like star in this range, it appears that our system is more underpopulated than most other systems within *p* < 500 days. We would expect 30 ± 1% systems harbor zero planets, 4.0 ± 4.6% harbor just one planet, and 2.0 ± 4.2% harbor only two planets within the range of this study. This lack of multiplicity in our solar system could be important for habitability, but such claims still lack strong evidence. ### 8.4 *Kepler* Dichotomy Analysis of the statistics of the *Kepler* multiple planet systems (Lissauer et al., [2011](#bib.bib36); Fang & Margot, [2012](#bib.bib22); Hansen & Murray, [2013](#bib.bib29); Ballard & Johnson, [2016](#bib.bib1)) suggest that the underlying planetary population requires a two component model. One component is composed of systems with high planet multiplicity and a low inclination dispersion, while the other requires either low intrinsic multiplicity or a large inclination dispersion to reduce the frequency of transits by multiple planets. This has been termed the *Kepler* dichotomy. Lissauer et al. ([2011](#bib.bib36)) inferred that the two populations had roughly equal frequencies and subsequent analyses confirmed this. There have been several models proposed to explain this on dynamical grounds (Johansen et al., [2012](#bib.bib33); Moriarty & Ballard, [2016](#bib.bib39); Hansen, [2017](#bib.bib28)). The simplest solution is to consider a single population of planets in which some fraction have experienced excitation of their mutual inclinations. However, to meet the requirements of the transit statistics, the excitation is sufficiently large that dynamical stability is hard to maintain (Hansen, [2017](#bib.bib28)). Thus, the *Kepler* results seem to imply the existence of a low multiplicity population of planetary systems, whether due to formation or later dynamical instability. However, this finding rests on the relative frequencies of systems with single transiting planets versus multiple transiting planets. If the completeness is a function of the detection order, this may weaken the claim for a *Kepler* dichotomy. In Figure [6](#S8.F6 "Figure 6 ‣ 8.3 Survival Function ‣ 8 Discussion ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates") we show that a single Poisson distribution can account for the multiplicity probabilities (*F**m*) extracted from our analysis. We find a much smaller fraction of intrinsically single systems than Fang & Margot ([2012](#bib.bib22)) and find a distribution broadly similar to the model for a single, dynamically motivated population described in Hansen & Murray ([2013](#bib.bib29)). However, we still find ∼6% of stars harbor intrinsically single or double planet systems. To test the robustness of this low multiplicity contribution we forward model the inferred population using the Poisson multiplicity model. In Table [4](#S8.T4 "Table 4 ‣ 8.2 Comparison with Prior Work ‣ 8 Discussion ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates") under the label ‘‘Multiple Detection Efficiency (Model)’’ we present the multiplicity results of this model. We can see that almost all of the empirical population fall within 1*σ* of the multiplicity model. This indicated that that apparent deviations in our infer *F**m* values can be described by statistical fluctuations in population. Additionally, our *F**m* are very dependent on the choice of mixture values displayed in Table [3](#S7.T3 "Table 3 ‣ 7.0.1 Calculating N m ‣ 7 The Likelihood Function ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates"). A proper accounting of these values would require distribution dependence. Averaging over these parameters, as done here, can cause mild deviations in the inferred *F**m* values. In extracting the population *F**m* values, we have only employed a mild Rayleigh distribution to account for mutual inclination of each system as directed by Fang & Margot ([2012](#bib.bib22)) and have no larger inclination component. It appears that accounting for systematic loss of planets at higher multiplicity substantially reduces the low multiplicity population inferred as per the *Kepler* Dichotomy. We shall now discuss how this works. Using the forward model presented in Section [8.1](#S8.SS1 "8.1 Forward Modeling the Results ‣ 8 Discussion ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates"), we look at how the inclusion of detection efficiency affected the gap seen between systems with one transiting planet and those with two transiting planets. The population provided by the parameters in Figure [8](#A0.F8 "Figure 8 ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates") is modeled 20 times and the median from each group is recorded in Table [4](#S8.T4 "Table 4 ‣ 8.2 Comparison with Prior Work ‣ 8 Discussion ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates"). Using our population parameters and a mild mutual inclination model show that this anomaly is largely due to *Kepler* detection efficiency. Table [4](#S8.T4 "Table 4 ‣ 8.2 Comparison with Prior Work ‣ 8 Discussion ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates") shows how the frequency of detected systems of different transit multiplicity changes as we include different systematic effects. In the first column, we include only the correction of the probability of transit due to geometric alignment. For a simple numerical comparison, this results in a ratio of double transit to single transit systems of 0.37, to be compared to the observed value of 0.21 (the rightmost column). The inclusion of a small mutual inclination dispersion, comparable to that of Fang & Margot ([2012](#bib.bib22)), does not improve the ratio (second column). In the third column, we show the model in which we include the completeness corrections from Christiansen ([2017](#bib.bib16)) without the multiplicity treatment discussed here. This results in a partial improvement of the ratio to 0.25. It is also notable that the number of expected high transit multiplicity systems also drops significantly with the inclusion of this effect. Finally, in the fourth and fifth column, we show the expected numbers including the full, multiplicity-dependent completeness correction discussed here (Section [6.2](#S6.SS2 "6.2 Probability of Detection for ≥ m 2 ‣ 6 Detection Efficiency Grid ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates")). We find that the expected number of different transit multiplicities are now very well matched to the observed numbers, substantially weakening the need for an additional population to explain the observations. The ultimate reason for this is that high transit multiplicity systems usually contain several planets that lie in the low MES region of parameter space, so that the incompleteness (especially when including the detection order effects) knocks planets down the multiplicity scale, resulting in many single transit systems that, in an ideal world, would show two or three transiting planets. Furthermore, the improved stellar radius measurements from *Gaia* suggests that many stars have larger radii than previously believed (Berger et al., [2018](#bib.bib4)). Increasing the stellar radius of system will decreases the probability of detection for an exoplanet. This correction will overall increase the inferred occurrence measurements.  It is important to remember that our dataset does not include single hot Jupiter planets as discussed in Section [3](#S3 "3 Planet Selection ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates"). This observed population of 120 planets does not follow our power-law trend and appears to be uniquely single (Steffen et al., [2012](#bib.bib55)). While these outliers do provide some type of population dichotomy, their presence is not the most prominent cause of the excess of singles. Our extracted population parameters *F*₁ and *F*₂ indicate that 4.0 ± 4.6% of the underlying population does have only one planet, and that this contribution can be described by the modified Poisson distribution used to fit the higher multiplicity systems. There is dynamical evidence that single transiting systems are more dynamically excited than multiple systems (Morton & Winn, [2014](#bib.bib41); Xie et al., [2016](#bib.bib63); Van Eylen et al., [2018b](#bib.bib60)) and this is consistent with the notion that some fraction of compact planetary systems are dynamically perturbed by the existence of giant planets on larger scales. Previously, Hansen ([2017](#bib.bib28)) found that explaining the original excess of single transits required a frequency of giant planets on large scales that was roughly double that found by radial velocity surveys. The reduction found here substantially alleviates that discrepancy. Other recent work also supports the notion that single transiting systems are drawn from the same underlying planetary population as multiple harbor systems. Weiss et al. ([2018](#bib.bib62)) find that both populations share essentially the same stellar and planetary properties, while Zhu et al. ([2018](#bib.bib65)) use transit timing variations to infer that there is a strong correlation between multiplicity and dynamical excitation. They reject the notion that this is driven by giant planet excitation because they see no correlation with the metallicity of the host star, but such a correlation would be difficult to see at the level of 4% as found here. This is further supported by Munoz Romero & Kempton ([2018](#bib.bib44)), who find no metallicity difference between hosts of single and multiple transiting systems, but could easily accommodate mixtures at the 50% level. ### 8.5 Considering Eccentricity ![]() Figure 7: A CDF showing the retrieved eccentricities from our forward modeling pipeline. The red line illustrates the eccentricities used to draw the underlying the Beta distribution (Kipping, [2013](#bib.bib35)). The black line represents the empirical CDF of the detected single planet systems and the blue line represents the eccentricities of the detected multi-planet systems. Including eccentricity into our model increases the number of detected planets. We find that the best fit multiplicity parameters are as follows: *F*₁ = 0.72 ± 0.05, *F*₂ = 0.66 ± 0.03, *F*₃ = 0.63 ± 0.03, *F*₄ = 0.60 ± 0.03, *F*₅ = 0.56 ± 0.03, *F*₆ = 0.51 ± 0.04, and *F*₇ = 0.43 ± 0.07. These parameters are fit using an analog to the Hansen & Murray ([2013](#bib.bib29)) eccentricity model. The original modified Gamma distribution (scale=0.055) is unique to Hansen & Murray ([2013](#bib.bib29)). We map this model to a Beta distribution (*a* = 1.80 and *b* = 14.46), widely used among recent authors, for consistency. This model was inferred by simulating in situ gravitational assembly of planetary embryos and observing the resulting eccentricity population of the fully formed planets. Although derived within a specific scenario, this distribution matches well with a model in which planets explore the full range of available phase space subject to the constraint of dynamical stability (Tremaine, [2015](#bib.bib58)). As such, it represents a plausible description of the level of eccentricity to be expected in such systems. The average eccentricity of this population is ⟨*e*⟩ = 0.11. Comparing these values to those of our base model, we find that eccentricity flattens the CDF of planet multiplicity, slightly decreasing ⟨*N**p**l*⟩ to 5.69 ± 0.17 planets. Recently, Van Eylen et al. ([2018b](#bib.bib60)) provided evidence for two distinct populations of eccentricity (multi-planet systems and single planet systems). Using our forward modeling software (\\hrefhttps://github.com/jonzink/ExoMultExoMult), we test the strength of this hypothesis. Implementing only one true underlying eccentricity model, we inspect the detected eccentricity populations from both the single and multi-planet populations. When tested with the Hansen & Murray ([2013](#bib.bib29)) model (⟨*e*⟩ = 0.11), we find no significant difference between the the observed eccentricities of multi-planet and single planet systems. This indicates that the differences noted by Van Eylen et al. ([2018b](#bib.bib60)) may be real. However, Van Eylen et al. ([2018b](#bib.bib60)) suggests a Beta distribution for single planet systems with ⟨*e*⟩ = 0.26. This is a significantly larger average eccentricity than expected by the Hansen & Murray ([2013](#bib.bib29)) model. When larger eccentricities are tested we do find observable differences between the single and multi-planet systems. The Kipping ([2013](#bib.bib35)) model (*a* = 0.867 and *b* = 3.03) was calculated using radial velocity discoveries and contains a significant fraction of massive planets. This distribution is probably too eccentric (⟨*e*⟩ = 0.22) for the tightly packed model discussed here, but illustrates the effects of detection bias on the eccentricity population. In Figure [7](#S8.F7 "Figure 7 ‣ 8.5 Considering Eccentricity ‣ 8 Discussion ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates") we present the results of our test on the Kipping ([2013](#bib.bib35)) model. We find that multi-planet systems tend to produce more low eccentricity detections than single planet detections despite being drawn from the same underlying population. Analyzing the statistical difference with an *Anderson-Darling* test produces a P-value of 10−7, suggesting these differences would appear statistically significant. Furthermore, we can see that neither of the detected populations closely mimic the true Beta distribution, highlighting the importance of detection efficiency consideration when performing eccentricity occurrence measurements. This effect is caused by the increased transit duration for higher eccentricity transits. Increasing the transit duration improves the planet MES, making the signal easier to detect. Since the highest MES planets are the most likely to be detected, this biases the empirical population toward higher eccentricity. The sorting order in combination with the multiplicity detection efficiency of the *Kepler* pipeline will further exaggerate this bias in the single planet systems. It is clear that low eccentricity distributions are less affected by this bias. Manually tuning the Beta distribution we find that models with ⟨*e*⟩ ≥ 0.18 will produce statistically significant (P-value≤0.001) differences between the empirical eccentricity population of singles and multiple planet systems. Since Van Eylen et al. ([2018b](#bib.bib60)) suggests a ⟨*e*⟩ = 0.26 model for the singles and a ⟨*e*⟩ = 0.05 model for the multi-planet systems, it is difficult to determine the effect of detection bias on their eccentricity model. At this point we cannot rule out that two distinct populations of eccentricity exist between the single and multi-planet systems, but propose that such claims require further evidence. ### 8.6 Extrapolation to Longer Periods As mentioned above, our general populations parameters do not differ greatly from those of previous studies. The quantity *Γ*⊕ is often quoted to avoid any need for understanding the habitable zone or habitable radius range. $$\\frac{dN}{d\\text{ln}p\_{\\oplus}\\ d\\text{ln}r\_{\\oplus}} = \\Gamma\_{\\oplus}$$ (31) We find *Γ*⊕ = 1.31 ± 0.07, consistent with the previous value of Burke et al. ([2015](#bib.bib7)) (*Γ*⊕ = 0.6 with a range of 0.04 to 11.5). Youdin ([2011](#bib.bib64)) found a much higher value of *Γ*⊕ = 2.75 ± 0.3, when extrapolating from periods <50 days. The lack of long period planets provided a weaker power-law, producing the inflated *Γ*⊕ value. Furthermore, we find tension with Foreman-Mackey et al. ([2014](#bib.bib24)) (with $\\Gamma\_{\\oplus} = {0.019 \\pm\_{0.010}^{0.019}}$). Foreman-Mackey et al. ([2014](#bib.bib24)) avoid the assumption of a particular functional form for the extrapolation to longer periods, by using a Gaussian process regression to determine the shape of the distribution. However, they use the results of the *TERRA* pipeline in it’s original form, in which it only reported the highest signal to noise candidate around each star. Although they back out an estimate of the detection efficiency from the results of Petigura et al. ([2013b](#bib.bib47)), we have shown in Section [7](#S7 "7 The Likelihood Function ‣ Accounting for Incompleteness due to Transit Multiplicity in Kepler Planet Occurrence Rates") that detection order can bias the results. In particular, we expect Foreman-Mackey et al. ([2014](#bib.bib24)) to undercount small planets and long period periods. Both of these biases will lower the *Γ*⊕ value and we should regard the Foreman-Mackey et al. ([2014](#bib.bib24)) result as a lower limit. For the occurrence of habitable planets we follow the procedure provided by Burke et al. ([2015](#bib.bib7)). This *ζ*⊕ value is found by integrating the population distribution by 20% of *r*⊕ and *p*⊕ in both directions. We find *ζ*⊕ = 0.217 ± 0.014 using our inferred population parameters, similar to the *ζ*⊕ = 0.10 (with a range of 0.01 to 2) found in Burke et al. ([2015](#bib.bib7)). 9 Conclusion ------------ We present a new method for determining the frequency of exoplanet multiplicity within the *Kepler* dataset. In doing so we provide the following new fitting features and conclusions: 1.Previous studies have discussed and provided methods for calculating high multiplicity transit probabilities (Ragozzine & Holman ([2010](#bib.bib49)); Brakensiek & Ragozzine ([2016](#bib.bib10)); Read et al. ([2017](#bib.bib50))). For occurrence calculations these procedures are often too complex and computationally expensive to carry out. We provide a new method which marginalizes over mutual inclination and the empirical *Kepler* period set to determine the transit probabilities for *Kepler* multi-planet systems. Using this, we provide the transit probabilities for multiple systems containing up to 7 planets. This simplification is important and useful when trying to fit multiplicity parameters via MCMC or some other fitting method that requires 10⁴ calculations. Our method does make some simplification assumptions in the interests of speed. We assume the measurements of planet radius and period are perfect. The uncertainty in period is negligible, however the radius measurements retain significant uncertainty and the present dispersion may yet mask finer features in the distribution. In accounting for mutual inclination, we adopt the model provided by Fang & Margot ([2012](#bib.bib22)). This is derived using a different multiplicity model than that found here. All orbits are assumed to be circular in our base model. Because many of the systems are very compact, circular orbits are required for any type of stability. Tidal circularization will also force many of these planets into circular orbits. However, it is possible that some portion of the population, investigated here, contains varying amounts of eccentricity. We show that any amount of eccentricity will increases our the overall multiplicity values, but decreases the fraction of systems with planets. We have assumed the appropriate model for exoplanet occurrence is a broken power-law. Furthermore, we assume period and radius and uncorrelated. It has been shown by Owen & Wu ([2013](#bib.bib45)) and Weiss et al. ([2017](#bib.bib61)) that a mild correlation exist between period and radius at short periods where photoevaporation can take effect. Nevertheless, the fact that our forward modeling matches the data inspires confidence that the model provides a coherent description of the data. 2.In systems with more than one detected planet, we find that detection efficiency decreases for higher detection order planets. This conclusion was achieved by re-visiting the Christiansen ([2017](#bib.bib16)) injections and looking at systems with pre-existing planets. Multiple planets systems experience an additional loss, for lower MES planets within each system, of at least 5.5% and 15.9% for periods <200 days and >200 days respectively. This type of increased selection effects indicates that a larger fraction of the population is being missed. Being able to infer a larger population of multiple exoplanet systems significantly decreases the gap between single and double planet systems. The initial motivation for additional detection efficiencies for multi-planet systems, was the 61 known KOIs lost during the Christiansen ([2017](#bib.bib16)) injections. When testing our additional selection effects, for multiples, we expect 41 ± 7 planets should be lost due to a similar type of injection test. Because we find that 61 KOIs are lost (rather than 41) we suspect higher order detection efficiencies may be necessary for an accurate accounting of the true underlying populations. 3.Using Bayesian statistics, we expand the Poisson process likelihood to account for variations in detection order. Furthermore, we are able to infer population multiplicity from this fitting process. The results from this fit match that of Burke et al. ([2015](#bib.bib7)), but provide an improved measurement with reduced uncertainty from *Gaia*, CKS, and asteroseismology (Petigura et al., [2017](#bib.bib48); Johnson et al., [2017](#bib.bib34); Berger et al., [2018](#bib.bib4); Van Eylen et al., [2018a](#bib.bib59)). Furthermore, by looking at the occurrence of single and double-planet systems, we only find a 0.9*σ* difference between these two populations (4.0 ± 4.6%). This disparity can be explained by a modified Poisson distribution with *λ* = 8.40 ± 0.31 and *κ* = 0.70 ± 0.01, indicating that the *Kepler* Dichotomy (discussed by Lissauer et al. ([2011](#bib.bib36)); Fang & Margot ([2012](#bib.bib22)); Hansen & Murray ([2013](#bib.bib29)); Ballard & Johnson ([2016](#bib.bib1))) may largely be an artifact of detection efficiency and statistical fluctuation. Using a Poisson process likelihood requires that each planet is drawn independently, which is clearly not the case for planets in multiple systems. Much of the work in this study is accounting for these dependencies. Ignoring the independence requirement of Poisson process could be suspect, but is again justified by the success of our forward model, where this assumption is not necessary. The independence of radius between planets within a system has also not been accounted for within this study. 4.Given our inferred multiplicity model we can extrapolate to higher multi-planet systems. We find that 32.3 ± 2.7% of solar-like stars should contain at least 8 planets within 500 days. The existence of a single 7 planet system and a single 8 planet system (Kepler 90) indicates these systems should be rare but still detectable. We would expect to find <1 eight planet systems within the constraints of this study. 5.We introduce (\\hrefhttps://github.com/jonzink/ExoMultExoMult) and demonstrate that forward modeling a broken power-law distribution can still provide a reasonable model for the exoplanet population, despite growing evidence for a gap in 1.5 − 2*r*⊕ range (Fulton et al., [2017](#bib.bib26); Berger et al., [2018](#bib.bib4); Weiss et al., [2018](#bib.bib62)). We find that our fitting model also produces similar populations of multiplicity to that of the empirical *Kepler* data set, indicating the success of this method. 6.Using the the eccentricity model of Hansen & Murray ([2013](#bib.bib29)), we show that eccentricity can affect the multiplicity occurrence by slightly decreasing the expected number of planets around each star. We also find that for eccentricity models with ⟨*e*⟩ ≥ 0.18 the *Kepler* pipeline will significantly skew the empirical population of eccentricity for single transiting systems, suggesting that differences seen between the single and multiple planet systems may be artificial. ### 9.1 Future Goals As mentioned previously, the uncertainties in the radius measurement are still quite large. Using a Bayesian hierarchical model, this uncertainty can be incorporated when fitting for population parameters (see Foreman-Mackey et al. [2014](#bib.bib24)). We hope to include this feature into our next generation of occurrence fitting. The multiplicity parameters derived here can be use in determining an Eta Earth measurement. The importance of neighboring planets could be essential for the long term stability of an Earth analog (Horner et al., [2017](#bib.bib31)), thus it is important to understand the likelihood of this Earth analog within a multiple system. The new detection efficiency is limited to *m* ≥ 2. Ideally, we would want the detection efficiency for each detection order. To do so one would need to perform an alternative injection experiment, where numerous planets are injected into each system and the recovery of each order can be better sampled. It would also be useful to understand the effects of resonance on detection efficiency. Looking at a select group of stars and injecting many planets at various period ranges could provide an understanding of these features (as performed by Burke & Catanzarite [2017b](#bib.bib9)). With the loss of *Kepler* and the upcoming release of *TESS* it will be essential to combine data across missions to calculate a more robust occurrence measurement. Doing so will require accounting for differing detection efficiencies across each mission. The method described here may provide a unique way of incorporating these different selection effects while producing a uniform population distribution. Acknowledgement --------------- We would like to thank the anonymous referee for useful feedback. The simulations described here were performed on the UCLA Hoffman2 shared computing cluster and using the resources provided by the Bhaumik Institute. This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. References ---------- - Ballard & Johnson (2016) Ballard, S. & Johnson, J. A. 2016, \\apj, 816, 66 - Batalha et al. (2013) Batalha, N. M., Rowe, J. F., Bryson, S. T., et al. 2013, \\apjs,204, 24 - Beaugé & Nesvorný (2012) Beaugé, C. & Nesvorný, D. 2012, \\apj, 751, 2 - Berger et al. (2018) Berger, T. A., Huber, D., Gaido, E., Van Saders, J. L. 2018, submitted, arXiv:1805.00231 - Borucki et al. (2010) Borucki, W. J., et al. 2010, \\sci, 327, 977 - Borucki et al. (2011) Borucki, W. 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--- abstract: 'A possibility is explored to account for the light curve and the low expansion velocity of the supernova SN 2011ht, a member of group of three objects showing signatures of both IIn and IIP supernovae. It is argued that the radiated energy and the expansion velocity are consistent with the low energy explosion ($\approx6\times10^{49}$ erg) and $\leq2~M_{\odot}$ ejecta interacting with the circumstellar envelope of $6-8~M_{\odot}$ and the radius of $\sim2\times10^{14}$ cm. The test of this scenario is proposed.' --- [**SN 2011ht: WEAK EXPLOSION IN MASSIVE EXTENDED\ ENVELOPE**]{} N. N. Chugai [*Institute of Astronomy of Russian Academy of Sciences, Moscow*]{}\ Introduction ============ Among type IIn supernovae (SN IIn, with “n” standing for “narrow lines”) that are commonly associated with the presence of a dense circumstellar medium there is a unique variety composed of SN 1994W (Sollerman et al. 1998), SN 2009kn (Kankare et al. 2012), and SN 2011ht (Roming et al. 2012; Mauerhan et al. 2013). Their bolometric light curve has $\sim120$ days plateau reminiscent of SN IIP. The plateau ends up with the luminosity drop by a factor of ten and a subsequent exit to the tail somewhat similar to the radioactive tail of SN IIP but probably of different origin. Maximum with $\sim-18$ mag is attained at about day 40. The spectrum is smooth continuum with strong emission lines of H$\alpha$ and H$\beta$ characterized by the narrow core (FWHM$\sim 700-800$ km s$^{-1}$) and broad wings $\sim \pm5000$ km s$^{-1}$. Apart from hydrogen lines the spectrum shows narrow metal lines, mostly FeII, with velocity of absorption minima of $\sim-(500...~700)$ km s$^{-1}$. The similarity of light curves and spectra of the mentioned supernovae justifies their selection into a special group designated SN IIn-P (Mauerhan et al. 2013); the notation emphasises their resemblance with SN IIn and SN IIP first mentioned for SN 1994W (Sollerman et al. 1998). The model of SN 1994W proposed earlier suggested the explosion of the red supergiant with $7~M_{\odot}$ ejecta and the kinetic energy of $\sim10^{51}$ erg (Chugai et al. 2004). According to this scenario (dubbed as scenario A) the supernova interacts with the dense extended circumstellar (CS) envelope; narrow lines form in the CS envelope expanded at $\sim10^3$ km s$^{-1}$; broad wings are produced by the scattering of line photons on thermal electrons of the same CS envelope. This scenario, however, faces a serious problem, because it is becoming clear that at the late stage ($t>120$ d) supernovae SN IIn-P do not show signatures of the high-velocity material ($\sim4000$ km s$^{-1}$) that is predicted by this scenario. One might suggest that this gas is not seen because the cool dense shell (CDS) at the contact surface between supernova and CS material is very opaque. At the late stage this situation might occur, if the dust forms in the CDS. Yet the case of SN 1998S where the dust indeed seem to form in the CDS (Pozzo et al. 2004) broad emission lines are seen, possibly because of the mixing of the fragments of the CDS with the hot gas of the forward shock. In the alternative scenario (call it B) proposed by Dessart et al. (2009) the spectrum of SN 1994W, including the continuum and lines, forms in a massive envelope with low expansion velocity ($\sim1000$ km s$^{-1}$) implied by narrow lines. In fact, authors have demonstrated that the expanding atmosphere with a steep density gradient, the effective temperature of $\sim7000$ K, and photosphere radius of $\sim10^{15}$ cm reproduces the observed spectrum fairly well. The success of the straightforward scenario in the modelling of the non-trivial spectrum makes this scenario very attractive. Noteworthy, the scanario B does not contain high velocity gas unlike the scenario A. The scenario B, however, leaves open a question, whether the energy requirements are consistent with the low expansion velocity. The present paper is focused on this issue. To this end a model is developed to describe the phenomenon of SN 2011ht for which most complete observations are available compared to other two SN IIn-P. The model is based on the thin shell approximation that is commonly used for the analysis of SN IIn. Here, however, the model includes diffusion of the trapped radiation in the optically thick envelope. The section 2 describes the model, while the section 3 presents results of the light curve modelling. The modelling of line profiles of H$\alpha$ and H$\gamma$ is presented in section 4. Note, the simultaneous description of these lines in the framework of the unified model of the emission and Thomson scattering in CS envelope turned out problematic in the former scenario of SN 1994W (Chugai et al. 2004). The discovery on 2011 September 29 (JD=2455834) caught SN 2011ht during the rapid flux rise (Roming et al. 2012; Mauerhan et al. 2013). It is reasonable to admit, therefore, that the explosion took place a few days before the discovery. Here the explosion date JD=2455830 is adopted. General considerations and model ================================ The velocity at the photosphere of SN 2011ht fixed by absorption minima, e.g., H$\alpha$ is about 600 km s$^{-1}$ (Mauerhan et al. 2013); this value remains constant through the spectral observations ($t>30$ d) at the plateau stage. The latter indicates that the velocity dispersion and the relative thickness of the shell are rather small. Furthermore, the velocity persistence also sugggests that the shell acceleration phase is brief, $t_a\leq30$ d, which means in turn that the external radius of the CS envelope is $R_{cs}\sim vt_a\leq 2\times10^{14}$ cm. The main stage of the radiative cooling ($\sim120$ d) therefore should be considered as the result of slow diffusion of the trapped radiation generated at the early phase $t\leq 30$ d. In this respect SN 2011ht is similar to SN IIP. The difference is that the bulk of SN IIP matter is distributed in a wide range of velocities which is manifested in the significant decrease of the photospheric velocity at the plateau in contrast to SN IIn-P. The diffusion time characterized by the plateau stage is several times greater than the acceleration time, so a significant fraction of the internal energy is spent on the pressure work at the plateau stage. The strong raiation-dominated shock in the uniform medium deposits 80% of the energy in the internal energy (i.e., radiation) and 20% in the kinetic energy (Chevalier 1976). Assuming that the initial internal energy is equally shared between the work on the expansion and the escaped radiation, we conclude that roughly 2/3 of the explosion energy is spent on the kinetic energy of the accelerated shell, while 1/3 is escaped radiation. Given the radiated energy of SN 2011ht of $\approx 2\times10^{49}$ erg (Mauerhan et al. 2013) we thus conclude that the kinetic energy is $\sim 4\times10^{49}$ erg and the explosion energy is $E\sim6\times10^{49}$ erg. Taking into account the expansion velocity of $v\approx600$ km s$^{-1}$ and the estimated kinetic energy we infer the total mass of the expanding shell as $M\sim10~M_{\odot}$. \[t-par\] ------- --------------- ----------------- ----------------- -------------- --- --------------- Model $E$ $M_{sn}$ $M_{cs}$ $R_{cs}$ s $E_r$ $10^{50}$ erg [$M_{\odot}$]{} [$M_{\odot}$]{} $10^{14}$ cm $10^{50}$ erg m1 0.6 0.01 10 2 0 1.9 m2 0.6 2 8 2 0 1.5 m3 0.57 2 6.5 2.5 0 1.7 m3f 0.57 2 6.5 2.5 0 2.5 m4 0.6 2 7 3.5 2 2.0 m4f 0.6 2 7 3.5 2 3.0 ------- --------------- ----------------- ----------------- -------------- --- --------------- The small relative thickness of the shell and the brief acceleration phase prompts us a simple model based on the thin shell approximation (Giuliani 1982). We consider geometrically thin, but optically thick, shell with the interior filled by the radiation. The radiation energy is determined by the dissipation of the kinetic energy of the supernova ejecta, by the work of the radiation pressure, and the radiation escape via diffusion. The equation of motion for this shell is (cf. Giuliani 1982) $$M\frac{dv}{dt} = 4\pi R^2\left[\rho_{sn}\left(\frac{R}{t}-v\right)^2+ p-\rho_{cs}v^2\right]\,, \label{eq-mom}$$ where $M$ is the mass of the shell with the radius $R$, $v$ is the shell expansion velocity, $\rho_{sn}$ is the supernova density at the radius $R$, $\rho_{cs}$ is the CS density at the radius $R$, and $p$ is the radiation pressure that is assumed to be uniform in the cavity. The expansion velocity of undisturbed CS matter is presumably negligibly small. Undisturbed supernova ejecta are assumed to expand homologously, ($v=r/t$) with the density distribution $\rho \propto (v_0/v)/(1+(v/v_0)^7)$, i.e., $\rho \propto v^{-1}$ in the inner zone ($v<v_0$) and $\propto v^{-8}$ in the outer layers. The radiation pressure in the cavity is $p=E_r/(4\pi R^3)$, where $E_r$ is the radiation energy described by the equation $$\frac{dE_r}{dt} = 2\pi R^2\rho_{sn}\left(\frac{R}{t}-v\right)^3 -E_r\frac{v}{R}-\frac{E_r}{t_c}\,. \label{eq-ener}$$ The first term in the right hand side is the rate of the internal energy generation due to the ejecta collision with the thin shell, the second term is the work of the radiation pressure, and the last term is the luminosity due to the radiation diffusion. The luminosity is determined as $L=E_r/t_d$, where the diffusion time is $$t_d=\xi\frac{R}{c}\tau\,. \label{eq-tcool}$$ Here $\tau\gg1$ is the shell optical depth, $c$ is the speed of light, and $\xi$ is a factor of order unity related to the geometry. For the central source in the uniform sphere $\xi=0.5$ (Sunyaev and Titarchuk 1980); a similar value one obtains for the geometrically thin shell filled by the isotropic radiation. We adopt $\xi=0.5$. The shell optical depth is calculated using Rosseland opacity (Alexander 1975). The temperature distribution in the shell is determined iteratively on the bases of the Eddington solution for the plane slab, $T^4=(3/4)T_e^4(2/3+\tau)$, where $T_e$ is the effective temperature. The shell density is assumed to be equal $\rho_s=7\rho_{cs}$ in line with the density jump in the strong radiation-dominated shock. After the shock break out of the CS envelope ($R>R_{cs}$) the shell density is set to be $\rho_s\propto (R_{cs}/R)^3$ implied by the free expansion. The system of equations of motion, energy, and mass conservation is solved by Runge-Kutta of 4-th order. In every case the energy is conserved with the accuracy of 1%. To test the model we calculated the result of a central explosive release of $10^{51}$ erg in the uniform envelope of $4.2~M_{\odot}$ and the radius of $5\times10^{13}$ cm. The explosion is simulated by the kinetic energy of $0.01~M_{\odot}$ shell. In this formulation the modelling in the framework of the radiation hydrodynamic is available (Chevalier 1976, model A). The light curve in our model slightly differs from that of the model A but the length of the plateau in both models (57 days) coinside within one day. Despite its simplicity our model thus catches the essence of the acceleration dynamics and the light curve produced by the explosion in an extended envelope. ![ Model bolometric light curve (panels [*a*]{} and [*c*]{}), compared to observations of SN 2011ht ([*crosses*]{}, Mauerhan et al. 2013; Roming et al. 2012), and the model thin shell velocity (panels [*b*]{} and [*d*]{}). Model m1 (cf. Table) is ploted in panels [*a*]{} and [*b*]{}, model m2 is in panels [*c*]{} and [*d*]{}. The vertical [*solid*]{} line in panels [*b*]{} and [*d*]{} corresponds to the epoch of the first spectrum (day 30), while the [*dotted*]{} line shows the moment when the thin shell radius is equal to $R_{cs}$. []{data-label="f-lc1"}](fig1.eps){width="90.00000%"} Modelling results ================= SN 2011ht parameter estimates recovered above are used here for two illustrative models, m1 and m2 (cf. Table and Fig.1). The Table contains the explosion energy and mass of SN ejecta, the mass of the CS envelope, its radius $R_{cs}$, the power index $s$ of the density distribution, $\rho\propto r^{-s}$, in the range of $r<R_{cs}$, and the radiated energy. In both models $s=0$ while $\rho\propto r^{-6}$ for $r>R_{cs}$. The model m1 with the ejecta mass of $0.01~M_{\odot}$ in fact simulates the central explosion since the kinetic energy of the low mass ejecta is rapidly ($t<1$ d) thermalized. The aggregated mass of the supernova ejecta and CS envelope is $10~M_{\odot}$ in both models. The models sensibly reproduce the total radiated energy (cf. Table) and the observed expansion velocity of SN 2011ht after day 30. Both models however have apparent drawbacks: the plateau is uacceptably long and the shape of the light curve is unlike the observed one. Particularly, the model does not show the hump at about day 50. Note, the late hump of the model light curve at the plateau end is related to the sharp drop of the opacity with the temperature decrease around $10^4$ K. The comparison of our simplified model with hydrodynamic simulation in the previous section suggests that the strong disagrement between the model and observed plateau duration is unlikely, although cannot be rulled out completely. The problem with the light curve description might arise because some relevant physics is not included in our model. We admit that the missing factor is the fragmentation of the swept-up shell as a result of either the Rayleigh-Taylor instability arising from the rapid deceleration of the shell (Fig.1), or the thin shell instability in the case of the radiative forward shock (Vishniac 1983). The outcome of the fragmentation is the decrease of the shell effective optical depth and, as a result, rapid radiation diffusion and larger luminosity at the early epoch. The fragmentation effect in the light curve can be implemented using the following description. A homogeneous spherical layer with the optical depth $\tau$ breaks down into spherical fragments of a radius $a$ with $N$ to be the number density of fragments. The gas density in fragments is assumed to be the same as in the smooth shell, while the intercloud density to be negligibly small. The average number of clouds along the shell radius is then $\tau_{oc}=\pi a^2N\Delta r$, where $\Delta r$ is the shell thickness. Using $\tau_{oc}$ one can write the expression for the effective optical depth of the cloudy shell (cf. Chugai & Chevalier 2005) as $$\tau_{eff}=\tau_{oc}[1-\exp{(-\tau/\tau_{oc})}]\,. \label{eq-tauef}$$ This expression deviates from the exact one (Utrobin & Chugai 2015) by less than 3%. In the limit of $\tau_{oc}\gg\tau$ the relation (4) reproduces the optical depth of the smooth shell, $\tau_{eff}=\tau$, while for $\tau_{oc}\ll\tau$ the effective optical depth is reduced to average number of clouds along the shell radius, $\tau_{eff}=\tau_{oc}$. The fragmentation evolution is described via the time dependence of $\tau_{oc}$ $$\tau_{oc}=\tau_{oc,2}+\tau_{oc,1}/[1+(t/t_f)^6]\,.$$ The value of $\tau_{oc,1}$ is set to meet the requirement $\tau_{oc,1}\gg\tau_0$, where $\tau_0$ is the initial optical depth of the smooth shell. We assume $\tau_{oc,1}=5\tau_0$ and $\tau_{oc,2}=200$: the choice guaratees that at the early epoch $\tau_{eff}$ is equal to the optical depth of the smooth shell. This description suggests that the fragmentation becomes significant at the stage $t\geq t_f$. ![ The same as Fig.1, but for models with the fragmentation (m3f and m4f, cf. Table). [*Thin*]{} line shows the model without fragmentation. [*Insets*]{} in panels [*a*]{} and [*c*]{} show evolution of the Rosseland optical depth in the model without fragmentation ([*thin*]{} line) and with fragmentation. []{data-label="f-lc2"}](fig2.eps){width="90.00000%"} The fragmentation effect in the light curve is demonstrated by models m3f and m4f (Table, Fig.2) in comparison with models m3 and m4 without fragmentation. Models m3f and m4f differ by the intitial density distribution of the CS envelope: in the model m3f the density is uniform ($s=0$), while in the model m4f $s=2$. The fragmentation time is $t_f=13$ d in the model m3f and 10 days in the model m4f. Model light curves describe better principal features of the observed light curve: maximum at about day 50 and the plateau duration. The shell expansion velocity in both models is close to 600 km s$^{-1}$, in accord with observations. Note that the velocity exit on the constant regime in the model m4f occurs later than both in the model m3f and observations; we therefore conclude that the model m3f is preferred. Exercises with different ejecta mass confirm an obvious guess: a model with larger mass exits to the constant velocity later because of the larger momentum. The model m3f exits to the constant velocity at about day 30, and thus demonstrates that the ejecta mass cannot exceed significantly $2~M_{\odot}$. At the luminosity maximum ($\approx 50$ d) the effective temperature in the model m3f is $10^4$ K in agreement with the temperature inferred from the spectral energy distribution (Mauerhan et al. 2013). This additionally lends credibility that the model reflects basic physics of SN 2011ht phenomenon. For the model verification of great interest is the early stage $t<t_a$ that preceeds complete sweeping of the CS envelope, i.e., when the thin shell radius $R<R_{cs}$. In the model m3f this stage corresponds to $t<t_a=21.6$ d (Fig.2). Our scenario predicts that at $t<t_a$ the photosphere radius is constant and equal to $R_{cs}$, while the velocity at the photosphere should coinside with the velocity of the undisturbed CS envelope, which is presumably small. We expect in this case that the spectrum at $t<t_a$ with the resolution $>100$ km s$^{-1}$ will not reveal absorption lines, while core of emission hydrogen lines will be narrower than at the late time, $t>t_a$. Summing up, the radiated energy and the low expansion velocity of SN 2011ht are consistent with the explosion of supernova of low energy ($\approx6\times10^{49}$ erg) in the CS envelope of the radius $\sim2\times10^{14}$ cm with the total mass of the swept-up shell of $\approx8-9~M_{\odot}$. Hydrogen line profiles ====================== In the proposed scenario the shock wave sweeps up the CS envelope in the initial 20-30 days. After that the shell expands freely with the velocity of $\approx600$ km s$^{-1}$ and kinematics of $v=r/t$. The hydrogen line emission produced by the shell with the large Thomson optical depth, and the hydrogen absorption arising from the external rarefied layer can provide us with an additional test of the SN 2011ht model. The Monte Carlo technique is used below to model hydrogen line profiles. Compared to the similar modelling of H$\alpha$ formed in the cocoon of SN 1998S with the Thomson optical depth of $\tau_T \approx 3$ (Chugai 2001), in the case of SN 2011ht the optical depth of the emitting shell is tremendous ($\tau_T> 10^2$), so one has to take into account the true absorption of quanta between scatterings. This process is modelled by introducing the absorption probability $p=k_a/(k_a+k_T)$, where $k_a$ and $k_T$ are the coefficients of absorption and Thomson scattering respectively. It is assumed that the shell consists of two components: optically thick ($\tau_T >10^2$) spherical layer in the velocity range of $v_1<v<v_2$ and the optically thin ($\tau_T =0.5$) external layer in the velocity range $v_2<v<v_3$ responsible for the absorption component. The ratio of line and continuum emission coefficients is assumed to be constant in the envelope; the electron temperature is set to be $10^4$ K. As an example we consider the spectrum of SN 2011ht on day 37 (Mauerhan et al. 2013). According to the model m3f at this stage the shell optical depth is $\tau_T \approx 10^3$. We adopt $\tau_T = 10^3$ and note that the variation of this value in the range of factor three does not affect the result significantly. The resonance optical depth in lines is assumed to be very large in the range of $v_1<v<v_3$. Computations of the H$\alpha$ profile for the extended set of parameters led us to the optimal choice $v_1=550$ km s$^{-1}$, $v_2=650$ km s$^{-1}$, $v_3=850$ km s$^{-1}$, and $p=0.09$ (Fig.3). To model H$\gamma$ one has to take into account that the absorption probability should be smaller than in the H$\alpha$ band because the absorption is determined primarily by the Paschen continuum. If one takes into account only this absorption mechanism then for the H$\gamma$ band one gets $p=0.026$. However, the H$\gamma$ calculated with this value shows very strong broad wings due to large number of scatterings on thermal electrons. The best agreement with the observed spectrum is found for $p=0.054$. The absorption coefficient is larger than the Paschen value possibly because of the contribution of numerous metall lines in this band. With the correction for the uncertainty of $p$ for H$\gamma$, we find that both H$\alpha$ and H$\gamma$ are well described by the unified model of the SN 2011ht consistent with the light curve model as regards principal parameters (velocity, temperature, and optical depth). This success demonstrates advantage of the scenario B over scenario A: in the latter H$\gamma$ profile could not be reproduced solely by the emission and scattering in the undisturbed CS envelope (Chugai et al. 2004). ![ Model profiles of H$\alpha$ and H$\gamma$ ([*thick*]{} line) compared to the observed spectrum (Mauerhan et al. 2013). Small excess of the observed flux in the red wing of H$\alpha$ at about 5000 km s$^{-1}$ is caused by the presence of weak HeI 6678 Å line. []{data-label="f-sp"}](fig3.eps){width="90.00000%"} Discussion and Conclusions ========================== The goal of the paper was the answer to the question, whether the light curve and spectra of supernovae IIn-P were consistent with the scenario B prompted by the idea of Dessart et al. (2009) that the spectrum of SN 1994W formed in the slowly expanding envelope ($<1000$ km s$^{-1}$). In the thin shell approximation with radiative diffusion a simple model was developed to compute the luminosity and dynamics of SN 2011ht. The modelling demonstrates that the light curve and the low expansion velocity are consistent with the low energy explosion ($\approx 6\times10^{49}$ erg) and ejected mass $\leq2~M_{\odot}$ occured in the CS envelope with the radius of $\sim2\times10^{14}$ cm and the mass of $6-8~M_{\odot}$. In this scenario a better agreement with the observed light curve is achieved, if one admits the shell fragmentation. The issue of instabilities that give rise to the fragmentation is beyond the scope of the present paper. It should be emphasised also that we cannot rule out that in the framework of the radiation hydrodynamics one will be able to reproduce all the observations without invoking fragmentation. The scenario B of the low energy explosion applied to SN 2011ht notably differ from the scenario A proposed for SN 1994W (Chugai et al. 2004). Major differencies of the new scenario from the old one are: (i) factor ten lower energy and, as a result, the lower expansion velocity ($<1000$ km s$^{-1}$), (ii) factor ten smaller radius of the CS envelope, and last but not least (iii) the line emitting region in the new scenario is the massive ($\sim 8~M_{\odot}$) shell accelerated by supernova explosion, while in the old scenario it was the undisturbed CS envelope with the mass of $\sim0.5~M_{\odot}$. The scenario B is favourable by two reasons. First, late time spectra ($t>120$ d) of SN IIn-P do not show high expansion velocities which is a serious problem for the scenario A. Second, the emission of hydrogen lines by the low velocity optically thick shell permits one to describe all the hydrogen lines as demonstrated by the modelling of the H$\alpha$ and H$\gamma$ lines. In contrast, in the scenario A the H$\gamma$ and H$\alpha$ lines cannot be reproduced simultaneously in the model of the emitting undisturbed CS envelope. The concept of the low energy explosion for SN IIn-P events has been proposed earlier by Smith (2013); he attributes this subclass along with the SN 1054 (Crab) to the electron-capture supernovae. Remarkably, the scenario B admits an observational test. It is based on the prediction that at the early stage preceeding the total acceleration of the CS envelope, i.e., at $t<t_a\sim 20$ days, the radial velocities of line absorptions should be equal to the expansion velocity of the undisturbed CS envelope, while the core of hydrogen emission lines should be significantly narrower than at the later epoch ($t>t_a$). The genesis of SN IIn-P is an open issue. Sollerman et al. (1998) have mentioned two possibilities for SN 1994W: star with the initial mass from the range of $8-10~M_{\odot}$ with a core collapsing to the neutron star, or massive star ($M\geq 25~M_{\odot}$) leaving behind the black hole. Both scenario account for the absence of large amount of ejected $^{56}$Ni. From the point of view of producing a close massive CS envelope the progenitor with the mass of $\sim10~M_{\odot}$ is preferred. Indeed, several years prior to the SN outburst the explosive flash of degenerate neon may result in the ejection all the pre-SN envelope (Woosley et al. 2002); for the massive star $>25~M_{\odot}$ that heavy mass loss prior to the collapse is unlikely because the final nuclear burning occurs in non-degenerate fashion. In the case of $\sim10~M_{\odot}$ progenitor the CS envelope with the radius of $2\times10^{14}$ cm forms by 3 yr prior to collapse provided the mass outflow velocity is 20 km s$^{-1}$. Notably, the low explosion energy of SN 2011ht ($\sim 6\times10^{49}$ erg) is consistent with the prediction of neutrino mechanism for $\approx10~M_{\odot}$ progenitor (Kitaura et al. 2006). Yet, it is noteworthy, that the collapse of massive star ($>25~M_{\odot}$) also can produce weak explosion (Woosley et al. 2002). I am grateful to Jon Mauerhan for the spectra of SN 2011ht. Alexander D. R., Astrophys. J. Suppl. [**29**]{}, 363 (1975) Chevalier R. A., Astrophys. J. 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--- abstract: | Let $V$ be a vector space over some field $F$ and let $\rho_{T,S}:T(V)\to S(V)$ be the projection map, given by $x_1\otimes\cdots\otimes x_n\mapsto x_1\cdots x_n$. In this paper we give a descrption of $\ker\rho_{S,T}$ in terms of generators and relations. Namely, we will define a $\ZZ_{\geq 2}$-graded $T(V)$-bimodule $M(V)$, which is a quotient of the $T(V)$-bimodule $T(V)\otimes\Lambda^2(V)\otimes T(V)$, and a morphism of $T(V)$-bimodules $\rho_{M,T}:M(V)\to T(V)$, such that the sequence $$0\to M(V)\xrightarrow{\rho_{M,T}}T(V)\xrightarrow{\rho_{T,S}}S(V)\to 0$$ is exact. In a related result, we define the algebra $S'(V)$ as $T(V)$ factorised by the bilateral ideal generated by $x\otimes y\otimes z-y\otimes z\otimes x$, with $x,y,z\in V$, and we prove that there is a short exact sequence, $$0\to\Lambda^{\geq 2}(V)\xrightarrow{\rho_{\Lambda^{\geq 2},S'}}S'(V)\xrightarrow{\rho_{S',S}}S(V)\to 0.$$ When considering the homogeneous components of degree $2$, we have $M^2(V)=\Lambda^2(V)$ and $S'^2(V)=T^2(V)$ so in both cases we get the well known exact sequence $$0\to\Lambda^2(V)\to T^2(V)\to S^2(V)\to 0.$$ author: - 'Constantin-Nicolae Beli' title: 'On the kernel of the projection map $T(V)\to S(V)$' --- The bimodule $M(V)$ =================== Let $V$ a vector space over a field $F$ and let $(v_i)_{i\in I}$ be a basis, where $(I,\leq )$ is a totally ordered set. Let $\rho_{T,S}:T(V)\to S(V)$ be the canonical projection and, for $n\geq 0$, let $\rho_{T^n,S^n}:T^n(V)\to S^n(V)$ be its homogeneous component of degree $n$. We denote by $[\cdot,\cdot ]:T(V)\times T(V)\to T(V)$ the commutator map, $[\xi,\eta ]=\xi\otimes\eta -\eta\otimes\xi$. Then $\ker\rho_{T,S}$ is the bilateral ideal generated by $[x,y]$, with $x,y\in V$. We want to describe $\ker\rho_{T,S}$ in terms of generators and relations. On homogeneous components, for $n=0,1$ we have $T^n(V)=S^n(V)$ so $\ker\rho_{T^n,S^n}=0$. The first interesting case is $n=2$. The map $V^2\to T^2(V)$, given by $(x,y)\mapsto [x,y]$, is bilinear and alternating, so it induces a linear map $\rho_{\Lambda^2,T^2}:\Lambda^2(V)\to T^2(V)$, given by $x\wedge y\mapsto [x,y]$. Then we have the following well known and elementary result. We have an exact sequence $$0\to\Lambda^2(V)\xrightarrow{\rho_{\Lambda^2,T^2}} T^2(V)\xrightarrow{\rho_{T^2,S^2}} S^2(V)\to 0.$$ We now consider the $T(V)$-bimodule $T(V)\otimes\Lambda^2(V)\otimes T(V)$, generated by $\Lambda^2(V)$. It is $\ZZ_{\geq 2}$ graded, where for every $n\geq 2$ homogeneous component of degree $n$ is $$(T(V)\otimes\Lambda^2(V)\otimes T(V))^n=\bigoplus_{i+j=n-2}T^i(V)\otimes\Lambda^2(V)\otimes T^j(V).$$ Note that we can define the commutator $[\cdot,\cdot ]$, by the same formula, $[\xi,\eta ]=\xi\otimes\eta -\eta\otimes\xi$, also as $[\cdot,\cdot ]:T(V)\times (T(V)\otimes\Lambda^2(V)\otimes T(V))\to T(V)\otimes\Lambda^2(V)\otimes T(V)$, or as $[\cdot,\cdot ]:(T(V)\otimes\Lambda^2(V)\otimes T(V))\times T(V)\to T(V)\otimes\Lambda^2(V)\otimes T(V)$. We consider the map $1\otimes\rho_{\Lambda^2,T^2}\otimes 1:T(V)\otimes\Lambda^2(V)\otimes T(V)\to T(V)$. Note that $1\otimes\rho_{\Lambda^2,T^2}\otimes 1$ is a morphism of graded $T(V)$-bimodules. Now $T(V)\otimes\Lambda^2(V)\otimes T(V)$ is spanned by $\xi\otimes x\wedge y\otimes\eta$, with $x,y\in V$ and $\xi,\eta\in T(V)$, so $\operatorname{Im}(1\otimes\rho_{\Lambda^2,T^2}\otimes 1)$ is spanned by $(1\otimes\rho_{\Lambda^2,T^2}\otimes 1)(\xi\otimes x\wedge y\otimes\eta)=\xi\otimes [x,y]\otimes\eta$, i.e. it is the ideal of $T(V)$ generated by $[x,y]$, with $x,y\in V$. Hence $\operatorname{Im}(1\otimes\rho_{\Lambda^2,T^2}\otimes 1)=\ker\rho_{T,S}$ and we have the exact sequence $$T(V)\otimes\Lambda^2(V)\otimes T(V)\xrightarrow{1\otimes\rho_{\Lambda^2,T^2}\otimes 1}T(V)\xrightarrow{\rho_{T,S}}S(V)\to 0.$$ It follows that $\ker\rho_{T,S}\cong\frac{T(V)\otimes\Lambda^2(V)\otimes T(V)}{\ker (1\otimes\rho_{\Lambda^2,T^2}\otimes 1)}$. The following elements of $T(V)\otimes\Lambda^2(V)\otimes T(V)$ belong to\ $\ker (1\otimes\rho_{\Lambda^2,T^2}\otimes 1)$. \(i) $[x,y]\otimes\xi\otimes z\wedge t-x\wedge y\otimes\xi\otimes [z,t]$, with $x,y,z,t\in V$, $\xi\in T(V)$. \(ii) $[x,y\wedge z]+[y,z\wedge x]+[z,x\wedge y]$, with $x,y,z\in V$. \(i) We have $$\begin{gathered} (1\otimes\rho_{\Lambda^2,T^2}\otimes 1)([x,y]\otimes\xi\otimes z\wedge t-x\wedge y\otimes\xi\otimes [z,t])\\ =[x,y]\otimes\xi\otimes [z,t]-[x,y]\otimes\xi\otimes [z,t]=0.\end{gathered}$$ \(ii) By the Jacobi identity we have $$(1\otimes\rho_{\Lambda^2,T^2}\otimes 1)([x,y\wedge z]+[y,z\wedge x]+[z,x\wedge y])=[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0.$$ Let $M(V)=(T(V)\otimes\Lambda^2(V)\otimes T(V))/W_M(V)$, where $W_M(V)$ is the subbimodule of $T(V)\otimes\Lambda^2(V)\otimes T(V)$ generated by $[x,y]\otimes\xi\otimes z\wedge t-x\wedge y\otimes\xi\otimes [z,t]$, with $x,y,z,t\in V$ and $\xi\in T(V)$, and $[x,y\wedge z]+[y,z\wedge x]+[z,x\wedge y]$, with $x,y,z\in V$. If $\eta\in T(V)\otimes\Lambda^2(V)\otimes T(V)$ then we denote by $[\eta ]$ its class in $M(V)$. On $M(V)$ we keep the notation $\otimes$ for the left and right multiplication from $T(V)\otimes\Lambda^2(V)\otimes T(V)$. That is $\xi\otimes [\eta ]\otimes\xi':=[\xi\otimes\eta\otimes\xi']$ $\forall\xi,\xi'\in T(V)$, $[\eta ]\in M(V)$. Note that $W_M(V)$ is generated by homogeneous elements so it is homogeneous. (In the formula $[x,y]\otimes\xi\otimes z\wedge t-x\wedge y\otimes\xi\otimes [z,t]$ we may restrict ourselves to $\xi\in T^k(V)$, with $k\geq 0$, which makes it homogeneous of degree $k+4$.) Therefore $M(V)$ inherits from $T(V)\otimes\Lambda^2(V)\otimes T(V)$ the property of being a $\ZZ_{\geq 2}$-graded $T(V)$-bimodule. By Lemma 1.2, we have $W_M(V)\sbq\ker (1\otimes\rho_{\Lambda^2,T^2}\otimes 1)$. It follows that\ $1\otimes\rho_{\Lambda^2,T^2}\otimes 1:T(V)\otimes\Lambda^2(V)\otimes T(V)\to T(V)$ induces a morphism of graded bimodules $\rho_{M,T}:M(V)\to T(V)$, given by $[\xi\otimes x\wedge y\otimes\eta ]\mapsto\xi\otimes [x,y]\otimes\eta$ $\forall x,y\in V$ and $\xi,\eta\in T(V)$. Moreover we have the exact sequence $$M(V)\xrightarrow{\rho_{M,T}}T(V)\xrightarrow{\rho_{T,S}}S(V)\to 0.$$ Since $\rho_{M,T}$ is a morphism of graded bimodules, we may consider its homogenous components, $\rho_{M^n,T^n}:M^n(V)\to T^n(V)$ We have $W_M(V)=\ker (1\otimes\rho_{\Lambda^2,T^2}\otimes 1)$, i.e. $\rho_{M,T}$ is injective and we have the exact sequence $$0\to M(V)\xrightarrow{\rho_{M,T}}T(V)\xrightarrow{\rho_{T,S}}S(V)\to 0.$$ We use induction on $n$ to prove that $\rho_{M^n,T^n}$ is injective. If $n=0,1$ then $(T(V)\otimes\Lambda^2(V)\otimes T(V))^n=0$ so $M^n(V)=0$, so there is nothing to prove. Before proving the induction step, we need some preliminary results. We have an action of the symmetric group $S_n$ on $T^n(V)$, given by $$\tau (x_1\otimes\cdots\otimes x_n)=x_{\tau^{-1}(1)}\otimes\cdots\otimes x_{\tau^{-1}(n)}.$$ For $1\leq i\leq n-1$ we denote by $\tau_i$ the transposition $(i,i+1)\in S_n$ and we denote by $f_i:T^n(V)\to M^n(V)$ the linear map given by $x_1\otimes\cdots\otimes x_n\mapsto [x_1\otimes\cdots\otimes x_i\wedge x_{i+1}\otimes\cdots\otimes x_n]$. (Here we repaced the $\otimes$ sign between $x_i$ and $x_{i+1}$ by $\wedge$.) On $T^n(V)$ we have $\rho_{M^n,T^n}f_i=1-\tau_i$. We verify this relation on generators $\xi =x_1\otimes\cdots\otimes x_n$ of $T^n(V)$. We have $$\begin{aligned}\rho_{M^n,T^n}f_i(\xi )&=\rho_{M^n,T^n}([x_1\otimes\cdots\otimes x_i\wedge x_{i+1}\otimes\cdots\otimes x_n])\\ {}&=x_1\otimes\cdots\otimes [x_i,x_{i+1}]\otimes\cdots\otimes x_n\\ {}&=x_1\otimes\cdots\otimes (x_i\otimes x_{i+1}-x_{i+1}\otimes x_i)\otimes\cdots\otimes x_n\\ {}&=x_1\otimes\cdots\otimes x_n-x_1\otimes\cdots\otimes x_{i+1}\otimes x_i\cdots\otimes x_n\\ {}&=x_1\otimes\cdots\otimes x_n-x_{\tau_i(1)}\otimes\cdots\otimes x_{\tau_i(n)}=\xi -\tau_i(\xi ).\qquad\square\end{aligned}$$ For every $1\leq i_1,\ldots,i_s\leq n-1$ in $T(V)$ we have $$1-\tau_{i_s}\cdots\tau_{i_1}=\rho_{M^n,T^n} \sum_{k=1}^sf_{i_k}\tau_{i_{k-1}}\cdots\tau_{i_1}.$$ By Lemma 1.4, for $1\leq k\leq s$ in we have $\rho_{M^n,T^n}f_{i_k}=1-\tau_{i_k}$ so $\rho_{M^n,T^n}f_{i_k}\tau_{i_{k-1}}\cdots\tau_{i_1} =(1-\tau_{i_k})\tau_{i_{k-1}}\cdots\tau_{i_1}$. Hence $\rho_{M^n,T^n} \sum_{k=1}^sf_{i_k}\tau_{i_{k-1}}\cdots\tau_{i_1}$ is equal to the telescoping sum $\sum_{k=1}^s(1-\tau_{i_k})\tau_{i_{k-1}}\cdots\tau_{i_1} =1-\tau_{i_s}\cdots\tau_{i_1}$. \(i) If $\tau\in S_n$ then there is a map $h_\tau :T(V)\to M(V)$ with $h_\tau=\sum_{k=1}^sf_{i_k}\tau_{i_{k-1}}\cdots\tau_{i_1}$ whenever $\tau =\tau_{i_s}\cdots\tau_{i_1}$. In particular, $h_1=0$ and $h_{\tau_i}=f_i$. \(ii) $h_{\sigma\tau}=h_\tau +h_\sigma\tau$ $\forall\sigma,\tau\in S_n$. \(iii) On $T^n(V)$ we have $\rho_{M^k,T^k}h_\tau =1-\tau$ $\forall\tau\in S_n$. (i) We use the fact that $S_n$ is generated by $\tau_1,\ldots,\tau_{n-1}$, with the relations $\tau_i^2=1$, $\tau_i\tau_j=\tau_j\tau_i$ if $j-i\geq 2$ and $(\tau_i\tau_{i+1})^3=1$. (See, e.g., \[KT, Theorem 4.1, pag. 152\].) We consider the set of symbols $A=\{\sigma_1,\ldots,\sigma_{n-1}\}$. Then $S_n$ is isomorphic to the free monoid $(A\cup A^{-1})^*$ factored by the equivalence relation $\sim$, generated by $\alpha\beta\sim\alpha\gamma\beta$ for every $\alpha,\beta\in (A\cup A^{-1})^*$ and $\gamma$ of the form $\gamma =\sigma_i\sigma_i^{-1}$ or $\sigma_i^{-1}\sigma_i$, $\gamma =\sigma_i^2$, $\gamma=(\sigma_i\sigma_{i+1})^3$ or $\gamma =\sigma_i\sigma_j\sigma_i^{-1}\sigma_j^{-1}$, with $j-i\geq 2$. For any $\sigma\in (A\cup A^{-1})^*$ we denote by $[\sigma ]$ its class in $(A\cup A^{-1})^*_{/\sim}$. If $\psi :(A\cup A^{-1})^*\to S_n$ is the morphism of monoids given by $\sigma_i\mapsto\tau_i$ and $\sigma_i^{-1}\mapsto\tau_i^{-1}=\tau_i$ then $\psi$ induces an isomorphism $\tilde\psi :(A\cup A^{-1})^*_{/\sim}\to S_n$, given by $\tilde\psi ([\sigma ])=\psi (\sigma )$ $\forall\sigma\in (A\cup A^{-1})^*$. For every $\sigma =\sigma_s^{\pm 1}\cdots\sigma_1^{\pm 1}\in (A\cup A^{-1})^*$ define the map $g_\sigma =\sum_{k=1}^sf_{i_k}\tau_{i_{k-1}}\cdots\tau_{i_1}$. (If $\sigma =1$ then $s=0$ so $g_1:=0$.) Note that if $\alpha=\sigma_s^{\pm 1}\cdots\sigma_{t+1}^{\pm 1}$ and $\beta =\sigma_t^{\pm 1}\cdots\sigma_1^{\pm 1}$ then $\psi (\beta )=\tau_t\cdots\tau_1$ so $$\begin{aligned}g_{\alpha\beta}& =\sum_{k=1}^sf_{i_k}\tau_{i_{k-1}}\cdots\tau_{i_1} =\sum_{k=1}^tf_{i_k}\tau_{i_{k-1}}\cdots\tau_{i_1} +\left(\sum_{k=t+1}^sf_{i_k}\tau_{i_{k-1}}\cdots\tau_{i_{t+1}}\right) \tau_{i_t}\cdots\tau_{i_1}\\ &=g_\beta+g_\alpha\psi (\beta ).\end{aligned}$$ We now prove that if $\sigma\sim\sigma'$ then $g_\sigma =g_{\sigma'}$. It suffices to take the case when $\sigma=\alpha\gamma\beta$ and $\sigma'=\alpha\beta$, with $\alpha,\beta\in (A\cup A^{-1})^*$ and $\gamma$ is of the form $\sigma_i\sigma_i^{-1}$, $\sigma_i^{-1}\sigma_i$, $\sigma_i^2$, $(\sigma_i\sigma_{i+1})^3$ or $\sigma_i\sigma_j\sigma_i^{-1}\sigma_j^{-1}$, with $j-i\geq 2$. Note that in all these cases we have $\psi (\gamma )=1$. (We have $\tau_i^2=1$, $(\tau_i\tau_{i+1})^3=1$ and, if $j-i\geq 2$, then $\tau_i\tau_j\tau_i\tau_j=\tau_i^2\tau_j^2=1$.) The relation $g_{\alpha\beta}=g_{\alpha\gamma\beta}$ writes as $g_\beta +g_\alpha\psi (\beta )=g_\beta +g_{\alpha\gamma}\psi (\beta )$ so it suffices to prove that $g_\alpha=g_{\alpha\gamma}$. But $\psi (\gamma )=1$ so $g_{\alpha\gamma}=g_\gamma +g_\alpha\psi (\gamma )=g_\gamma +g_\alpha$. Hence we must prove that $g_\gamma =0$. We prove that $g_\gamma (\eta )=0$ for $\eta =x_1\otimes\cdots\otimes x_n$. If $\gamma =\sigma_i^{\pm 1}\sigma_i^{\pm 1}$, which includes the cases $\gamma =\sigma_i\sigma_i^{-1}$, $\sigma_i^{-1}\sigma_i$ and $\sigma_i^2$, we have $$\begin{aligned}g_\gamma (\eta )&=f_i(\eta )+f_i(\tau_i(\eta ))=f_i(x_1\otimes\cdots\otimes x_n)+f_i(x_1\otimes\cdots\otimes x_{i+1}\otimes x_i\otimes\cdots\otimes x_n)\\ &=x_1\otimes\cdots\otimes x_i\wedge x_{i+1}\otimes\cdots\otimes x_n+x_1\otimes\cdots\otimes x_{i+1}\wedge x_i\otimes\cdots\otimes x_n=0.\end{aligned}$$ Let now $\gamma =(\tau_i\tau_{i+1})^3=\tau_i\tau_{i+1}\tau_i\tau_{i+1}\tau_i\tau_{i+1}$. We have $\eta=\eta'\otimes x\otimes y\otimes z\otimes\eta''$, where $\eta'=x_1\otimes\cdots\otimes x_{i-1}$, $\eta''=x_{i+3}\otimes\cdots\otimes x_n$ and $(x,y,z)=(x_i,x_{i+1},x_{i+2})$. Note that when we apply succesively the transpositions $\tau_i=(i,i+1)$ and $\tau_{i+1}=(i+1,i+2)$ to $\eta$ only the factors $x,y,z$ of $\eta$ are permuted, while the factors $\eta'$ and $\eta''$ are unchanged. The factors on the positions $i$, $i+1$ and $i+2$ in $\eta$ are $x,y,z$; in $\tau_{i+1}(\eta )$ they are $x,z,y$; in $\tau_i\tau_{i+1}(\eta )$ they are $z,x,y$; in $\tau_{i+1}\tau_i\tau_{i+1}(\eta )$ they are $z,y,x$; in $\tau_i\tau_{i+1}\tau_i\tau_{i+1}(\eta )$ they are $y,z,x$; and in $\tau_{i+1}\tau_i\tau_{i+1}\tau_i\tau_{i+1}(\eta )$ they are $y,x,z$. Therefore $$\begin{aligned}g_\gamma (\eta )&=f_{i+1}(\eta )+f_i\tau_{i+1}(\eta )+f_{i+1}\tau_i\tau_{i+1}(\eta )+f_i\tau_{i+1}\tau_i\tau_{i+1}(\eta )\\ &\qquad\qquad\qquad +f_{i+1}\tau_i\tau_{i+1}\tau_i\tau_{i+1}(\eta )+f_i\tau_{i+1}\tau_i\tau_{i+1}\tau_i\tau_{i+1}(\eta )\\ &=f_{i+1}(\eta'\otimes x\otimes y\otimes z\otimes\eta'')+f_i(\eta'\otimes x\otimes z\otimes y\otimes\eta'')\\ &\qquad\qquad\qquad +f_{i+1}(\eta'\otimes z\otimes x\otimes y\otimes\eta'')+f_i(\eta'\otimes z\otimes y\otimes x\otimes\eta'')\\ &\qquad\qquad\qquad +f_{i+1}(\eta'\otimes y\otimes z\otimes x\otimes\eta'')+f_i(\eta'\otimes y\otimes x\otimes z\otimes\eta'')\\ &= [\eta'\otimes x\otimes y\wedge z\otimes\eta'']+[\eta'\otimes x\wedge z\otimes y\otimes\eta'']+[\eta'\otimes z\otimes x\wedge y\otimes\eta'']\\ &\qquad +[\eta'\otimes z\wedge y\otimes x\otimes\eta'']+[\eta'\otimes y\otimes z\wedge x\otimes\eta'']+[\eta'\otimes y\wedge x\otimes z\otimes\eta''].\end{aligned}$$ Thus $g_\gamma (\eta )=[\eta'\otimes\xi\otimes\eta'']$, where $$\begin{aligned}\xi&=x\otimes y\wedge z+x\wedge z\otimes y+z\otimes x\wedge y+z\wedge y\otimes x+y\otimes z\wedge x+y\wedge x\otimes z\\ & =x\otimes y\wedge z-z\wedge x\otimes y+z\otimes x\wedge y-y\wedge z\otimes x+y\otimes z\wedge x-x\wedge y\otimes z\\ &=[x,y\wedge z]+[y,z\wedge x]+[z,x\wedge y].\end{aligned}$$ We have $\xi\in W_M(V)$ so $\eta'\otimes\xi\otimes\eta''\in W_M(V)$ and so $g_\gamma (\eta )=[\eta'\otimes\xi\otimes\eta'']=0$. Let now $\gamma =\sigma_i\sigma_j\sigma_i\sigma_j$. We have $\eta =\eta'\otimes x\otimes y\otimes\xi\otimes z\otimes t\otimes\eta''$, where $\eta'=x_1\otimes\cdots\otimes x_{i-1}$, $\xi=x_{i+2}\otimes\cdots\otimes x_{j-1}$, $\eta''=x_{j+2}\otimes\cdots\otimes x_n$, $(x,y)=(x_i,x_{i+1})$ and $(z,t)=(x_j,x_{j+1})$. Note that $\tau_i$ permutes the factors $x$ and $y$ of $\eta$ and leaves all the other factors unchanged, while $\tau_j$ permutes $z$ and $t$ and leaves all the other factors unchanged. We get $\tau_j(\eta )=\eta'\otimes x\otimes y\otimes\xi\otimes t\otimes z\otimes\eta''$, $\tau_i\tau_j(\eta )=\eta'\otimes y\otimes x\otimes\xi\otimes t\otimes z\otimes\eta''$ and $\tau_j\tau_i\tau_j(\eta )=\eta'\otimes y\otimes x\otimes\xi\otimes z\otimes t\otimes\eta''$. Then $$\begin{aligned}g_\gamma (\eta )&=f_j(\eta )+f_i\tau_j(\eta )+f_j\tau_i\tau_j(\eta )+f_i\tau_j\tau_i\tau_j(\eta )\\ &=f_j(\eta'\otimes x\otimes y\otimes\xi\otimes z\otimes t\otimes\eta'')+f_i(\eta'\otimes x\otimes y\otimes\xi\otimes t\otimes z\otimes\eta'')\\ &\qquad\qquad +f_j(\eta'\otimes y\otimes x\otimes\xi\otimes t\otimes z\otimes\eta'')+f_i(\eta'\otimes y\otimes x\otimes\xi\otimes z\otimes t\otimes\eta'')\\ &=[\eta'\otimes x\otimes y\otimes\xi\otimes z\wedge t\otimes\eta'']+[\eta'\otimes x\wedge y\otimes\xi\otimes t\otimes z\otimes\eta'']\\ &\qquad\qquad +[\eta'\otimes y\otimes x\otimes\xi\otimes t\wedge z\otimes\eta'']+[\eta'\otimes y\wedge x\otimes\xi\otimes z\otimes t\otimes\eta''].\end{aligned}$$ Thus $g_\gamma (\eta )=[\eta'\otimes\xi'\otimes\eta'']$, where $$\begin{aligned}\xi'&=x\otimes y\otimes\xi\otimes z\wedge t+x\wedge y\otimes\xi\otimes t\otimes z+y\otimes x\otimes\xi\otimes t\wedge z+y\wedge x\otimes\xi\otimes z\otimes t\\ &=x\otimes y\otimes\xi\otimes z\wedge t+x\wedge y\otimes\xi\otimes t\otimes z-y\otimes x\otimes\xi\otimes z\wedge t-x\wedge y\otimes\xi\otimes z\otimes t\\ &=[x,y]\otimes\xi\otimes z\wedge t-x\wedge y\otimes\xi\otimes [z,t].\end{aligned}$$ We have $\xi'\in W_M(V)$ so $\eta'\otimes\xi'\otimes\eta''\in W_M(V)$ and $g_\gamma (\eta )=[\eta'\otimes\xi'\otimes\eta'']=0$. Since the map $\sigma\mapsto g_\sigma$, defined on $(A\cup A^{-1})^*$, is invariant to the equivalence relation $\sim$, it induces a map defined on $(A\cup A^{-1})^*_{/\sim}$, given by $[\sigma ]\mapsto g_\sigma$. Since $\bar\psi :(A\cup A^{-1})^*_{/\sim}\to S_n$ is an isomorphism we get a map $\tau\mapsto h_\tau$, where $h_\tau=g_\sigma$ for any $\sigma\in (A\cup A^{-1})^*$ such that $\tau =\bar\psi ([\sigma ])=\psi (\sigma )$. If $\tau =\tau_{i_s}\cdots\tau_{i_1}$ then $\tau =\psi (\sigma )$, with $\sigma =\sigma_{i_s}\cdots\sigma_{i_1}$. Hence $h_\tau=g_\sigma =\sum_{k=1}^sf_k\tau_{k-1}\cdots\tau_1$, as claimed. \(ii) Let $\alpha,\beta\in (A\cup A^{-1})^*$ with $\sigma =\psi (\alpha )$ and $\tau =\psi (\beta )$ so that $\sigma\tau =\psi (\alpha\beta )$. Then $h_\sigma =g_\alpha$, $h_\tau =g_\beta$ and $h_{\sigma\tau}=g_{\alpha\beta}=g_\beta +g_\alpha\psi (\beta )=h_\tau +h_\sigma\tau$. \(iii) If we write $\tau=\tau_{i_s}\cdots\tau_1$ then our result is just Corollary 1.5. [**Proof of the induction step.**]{} We must prove that if $[\eta ]\in\ker\rho_{M^n,T^n}$ then $[\eta ]=0$. Note that $\eta$ is a finite linear combination of products of the form\ $v_{i_1}\otimes\cdots\otimes v_{i_k}\wedge v_{i_{k+1}}\otimes\cdots\otimes v_{i_n}$, with $i_1,\ldots,i_n\in I$ and $i_k<i_{k+1}$. We denote by $J=\{ j_1,\ldots,j_m\}$ with $j_1,\ldots,j_m\in I$, $j_1<\cdots <j_m$, the set of all indices $i\in I$ such that $v_i$ is one of the factors $v_{i_h}$ from one of the products in the linear combination that gives $\eta$. We will prove our result by induction on $m$. If $m=0$ there is nothing to be proved. Suppose that $m\geq 1$. Let $J'=\{ j_1,\ldots,j_{m-1}\}$. We have $\eta\in W$, where $W\sbq (T(V)\otimes\Lambda^2(V)\otimes T(V))^n$ is spanned by $v_{i_1}\otimes\cdots\otimes v_{i_k}\wedge v_{i_{k+1}}\otimes\cdots\otimes v_{i_n}$, with $(i_1,\ldots,i_n,;k)\in A$, where $A$ is the set of all $(i_1,\ldots,i_n,;k)$ with $i_1,\ldots,i_n\in J$, $1\leq k\leq n-1$ and $i_k<i_{k+1}$. We also denote by $U\sbq T^n(V)$ the space generated by $v_{i_1}\otimes\cdots\otimes v_{i_n}$, with $(i_1,\ldots,i_n)\in B:=J^n$. We have $A=A_1\sqcup A_2$ where $A_1$ is the set of all $(i_1,\ldots,i_n;k)\in A$ with $i_1,\ldots,i_n\in J'$ and $A_2$ is the set of those where at least one of $i_1,\ldots,i_n$ is $j_m$. Similarly, $B=B_1\sqcup B_2$ where $B_1=J'^n$ and $B_2=J^n\setminus J'^n$, i.e. $B_2$ is the set of all $(i_1,\ldots,i_n)\in J^n$ such that at least one of $i_1,\ldots,i_n$ is $j_m$. For $\alpha =1,2$ we denote by $W_\alpha$ the subspace of $W$ spanned by $v_{i_1}\otimes\cdots\otimes v_{i_k}\wedge v_{i_{k+1}}\otimes\cdots\otimes v_{i_n}$ with $(i_1,\ldots,i_n,;k)\in A_\alpha$ and by $U_\alpha$ the subspace of $U$ spanned by $v_{i_1}\otimes\cdots\otimes v_{i_n}$ with $(i_1,\ldots,i_n)\in B_\alpha$. Then from $A=A_1\sqcup A_2$ and $B=B_1\sqcup B_2$ we deduce that $W=W_1\oplus W_2$ and $U=U_1\oplus U_2$. We have $(1\otimes\rho_{\Lambda^2,T^2}\otimes 1)(v_{i_1}\otimes\cdots\otimes v_{i_k}\wedge v_{i_{k+1}}\otimes\cdots\otimes v_{i_n})=v_{i_1}\otimes\cdots\otimes v_{i_n}-v_{i_1}\otimes\cdots\otimes v_{i_{k+1}}\otimes v_{i_k}\otimes\cdots\otimes v_{i_n}$. If $(i_1,\ldots,i_n;k)\in A$, $A_1$ or $A_2$ then both $(i_1,\ldots,i_n)$ and $(i_1,\ldots,i_{k+1},i_k,\ldots,i_n)$ belong to $B$, $B_1$ or $B_2$ and so $(1\otimes\rho_{\Lambda^2,T^2}\otimes 1)(v_{i_1}\otimes\cdots\otimes v_{i_k}\wedge v_{i_{k+1}}\otimes\cdots\otimes v_{i_n})\in U$, $U_1$ or $U_2$, respectively. It follows that $(1\otimes\rho_{\Lambda^2,T^2}\otimes 1)(W)\sbq U$ and $(1\otimes\rho_{\Lambda^2,T^2}\otimes 1)(W_\alpha )\sbq U_\alpha$ for $\alpha =1,2$. Equivalently, if $\xi\in W$, $W_1$ or $W_2$ then $\rho_{M^n,T^n}([\xi ])=(1\otimes\rho_{\Lambda^2,T^2}\otimes 1)(\xi )\in U$, $U_1$ or $U_2$, respectively. We define $\psi :U_2\to M^n(V)$ on elements on the basis as follows. If $\xi =v_{i_1}\otimes\cdots\otimes v_{i_n}$ with $(i_1,\ldots,i_n)\in B_2$ and $l$ is the smallest index vith $i_l=j_m$ then $\psi (\xi ):=h_{\sigma_l}(\xi )$, where $\sigma_l\in S_n$ is the cyclic permutation $(1,2,\ldots,l)$. After these preliminaries, we start our proof of the induction step. Since $W=W_1\oplus W_2$ we have $\eta =\eta_1+\eta_2$, with $\eta_\alpha\in W_\alpha$. Then $\rho_{M^n,T^n}([\eta_\alpha ])\in U_\alpha$. Since $\rho_{M^n,T^n}([\eta_1])+\rho_{M^n,T^n}([\eta_2])=\rho_{M^n,T^n}([\eta ])=0$ and the sum $U_1+U_2$ is direct, this implies that $\rho_{M^n,T^n}([\eta_1])=\rho_{M^n,T^n}([\eta_2])=0$. Since $\eta_1\in U_1$, it can be written in terms of only $v_i$ with $i\in J'$. Since $|J'|=m-1$, by the induction hypothesis, $\rho_{M^n,T^n}([\eta_1])=0$ implies $[\eta_1]=0$ so $[\eta ]=[\eta_2]$. So we have reduced to the case when $\eta\in U_2$. Then $\eta$ writes as $$\eta =\sum a_{i_1,\ldots,i_n;k}v_{i_1}\otimes\cdots\otimes v_{i_k}\wedge v_{i_{k+1}}\otimes\cdots\otimes v_{i_n},$$ where the sum is take over $(i_1,\ldots,i_n;k)\in A_2$ and $a_{i_1,\ldots,i_n;k}\in F$. Since $[v_{i_1}\otimes\cdots\otimes v_{i_k}\wedge v_{i_{k+1}}\otimes\cdots\otimes v_{i_n}]=f_k(v_{i_1}\otimes\cdots\otimes v_{i_n})$ we have $$[\eta ]=\sum a_{i_1,\ldots,i_n;k}f_k(v_{i_1}\otimes\cdots\otimes v_{i_n}).$$ By Lemma 1.4, on $T^n(V)$ we have $\rho_{M^n,T^n}f_k=1-\tau_k$. It follows that $$0=\rho_{M^n,T^n}[\eta ]=\sum a_{i_1,\ldots,i_n;k}(v_{i_1}\otimes\cdots\otimes v_{i_n}-\tau_k(v_{i_1}\otimes\cdots\otimes v_{i_n})).$$ But for every $(i_1,\ldots,i_n;k)\in A_2$ we have $(i_1,\ldots,i_n)\in B_2$, which implies that also $(i_{\tau_k(1)},\ldots,i_{\tau_k(n)})\in B_2$. Thus both $v_{i_1}\otimes\cdots\otimes v_{i_n}$ and $\tau_k(v_{i_1}\otimes\cdots\otimes v_{i_n})$ belong to $U_2$ so we can apply $\psi$ to the formula above. We get $$0=\sum a_{i_1,\ldots,i_n;k}(\psi (v_{i_1}\otimes\cdots\otimes v_{i_n})-\psi\tau_k(v_{i_1}\otimes\cdots\otimes v_{i_n})).$$ Let $(i_1,\ldots,i_n;k)\in A_2$. We have $\psi (v_{i_1}\otimes\cdots\otimes v_{i_n})=h_{\sigma_l}(v_{i_1}\otimes\cdots\otimes v_{i_n})$ and $\psi\tau_k(v_{i_1}\otimes\cdots\otimes v_{i_n})=h_{\sigma_{l'}}\tau_k(v_{i_1}\otimes\cdots\otimes v_{i_n})$, where $l$ is the smallest index with $i_l=j_m$ and $l'$ is the smallest index with $i_{\tau_k(l')}=j_m$. Since $i_k<i_{k+1}\leq j_m$, we cannot have $l=k$. Since $(i_{\tau_k(1)},\ldots,i_{\tau_k(n)})= (i_1,\ldots,i_{k-1},i_{k+1},i_k,i_{k+2},\ldots,i_n)$, if $l\leq k-1$ or $l\geq k+2$ then $l'=l$. If $l=k+1$ then $l'=k$. Let $\tau =\sigma_{l'}\tau_k\sigma_l^{-1}$ so that $\tau\sigma_l=\sigma_{l'}\tau_k$. Then, by Lemma 1.6(ii), the relation $h_{\tau\sigma_l}=h_{\sigma_{l'}\tau_k}$ writes as $$h_{\sigma_l}+h_\tau\sigma_l =h_{\tau_k}+h_{\sigma_{l'}}\tau_k =f_k+h_{\sigma_{l'}}\tau_k$$ so $f_k=h_\tau\sigma_l+(h_{\sigma_l}-h_{\sigma_{l'}}\tau_k)$. Since $h_{\sigma_l}(v_{i_1}\otimes\cdots\otimes v_{i_n})=\psi (v_{i_1}\otimes\cdots\otimes v_{i_n})$ and $h_{\sigma_{l'}}\tau_k(v_{i_1}\otimes\cdots\otimes v_{i_n})=\psi\tau_k(v_{i_1}\otimes\cdots\otimes v_{i_n})$, this implies that $$f_k(v_{i_1}\otimes\cdots\otimes v_{i_n})=[\zeta_{i_1,\ldots,i_n;k}]+(\psi (v_{i_1}\otimes\cdots\otimes v_{i_n})-\psi\tau_k(v_{i_1}\otimes\cdots\otimes v_{i_n})),$$ where $[\zeta_{i_1,\ldots,i_n;k}] =h_\tau\sigma_l(v_{i_1}\otimes\cdots\otimes v_{i_n})$. It follows that $$\begin{aligned} \left[\eta \right] &=\sum a_{i_1,\ldots,i_n;k}f_k(v_{i_1}\otimes\cdots\otimes v_{i_n})\\ {}&=\sum a_{i_1,\ldots,i_n;k}[\zeta_{i_1,\ldots,i_n;k}]+\sum a_{i_1,\ldots,i_n;k}(\psi (v_{i_1}\otimes\cdots\otimes v_{i_n})-\psi\tau_k(v_{i_1}\otimes\cdots\otimes v_{i_n}))\\ {}&=\sum a_{i_1,\ldots,i_n;k}[\zeta_{i_1,\ldots,i_n;k}].\end{aligned}$$ We now claim that if $(i_1,\ldots,i_n;k)\in A_2$ then $[\zeta_{i_1,\ldots,i_n;k}]=v_{j_m}\otimes [\zeta'_{i_1,\ldots,i_n;k}]$ for some $\zeta'_{i_1,\ldots,i_n;k}\in (T(V)\otimes\Lambda^2(V)\otimes T(V))^{n-1}$. If $l=k+1$ then $l'=k$ so $\tau =\sigma_k\tau_k\sigma_{k+1}^{-1}$. But in $S_n$ we have $(1,2,\ldots,k)(k,k+1)=(1,2,\ldots,k+1)$, i.e. $\sigma_k\tau_k=\sigma_{k+1}$, so $\tau =1$, which implies $h_\tau =0$, so $[\zeta_{i_1,\ldots,i_n;k}] =h_\tau\sigma_l(v_{i_1}\otimes\cdots\otimes v_{i_n})=0$ and we may take $\zeta'_{i_1,\ldots,i_n;k}=0$. Suppose now that $l\leq k-1$ or $l\geq k+2$, so that $l'=l$. We have $\sigma_l(v_{i_1}\otimes\cdots\otimes v_{i_n})=v_{i'_1}\otimes\cdots\otimes v_{i'_n}$, with $i'_h=i_{\sigma_l^{-1}(h)}$. But $\sigma_l(l)=1$ so $i'_1=i_{\sigma_l^{-1}(1)}=i_l=j_m$. So $\sigma_l(v_{i_1}\otimes\cdots\otimes v_{i_n})=v_{j_m}\otimes v_{i'_2}\otimes\cdots\otimes v_{i'_n}$. Since $l'=l$ we have $\tau =\sigma_l\tau_k\sigma_l^{-1} =\sigma_l(k,k+1)\sigma_l^{-1} =(\sigma_l(k),\sigma_l(k+1))$. If $l\leq k-1$ then $\sigma_l(k)=k$ and $\sigma_l(k+1)=k+1$ so $\tau =(k,k+1)=\tau_k$, so $h_\tau =f_k$. Thus $[\zeta_{i_1,\ldots,i_n;k}]=h_\tau\sigma_l(v_{i_1}\otimes\cdots\otimes v_{i_n})=f_k(v_{j_m}\otimes v_{i'_1}\otimes\cdots\otimes v_{i'_n})=[v_{j_m}\otimes\zeta'_{i_1,\ldots,i_n;k}]$, where $\zeta'_{i_1,\ldots,i_n;k}=v_{i'_2}\otimes\cdots\otimes v_{i'_k}\wedge v_{i'_{k+1}}\otimes\cdots\otimes v_{i'_n}$. (Note that $l\leq k-1$ implies $k\geq 2$.) If $l\geq k+2$ then $\sigma_l(k)=k+1$ and $\sigma_l(k+1)=k+2$ so $\tau =(k+1,k+2)=\tau_{k+1}$, so $h_\tau =f_{k+1}$. Then, by the same reasoning from the case $l\leq k-1$, we get $[\zeta_{i_1,\ldots,i_n;k}]=[v_{j_m}\otimes\zeta'_{i_1,\ldots,i_n;k}]$, where $\zeta'_{i_1,\ldots,i_n;k}=v_{i'_2}\otimes\cdots\otimes v_{i'_{k+1}}\wedge v_{i'_{k+2}}\otimes\cdots\otimes v_{i'_n}$. Since $[\zeta_{i_1,\ldots,i_n;k}]=v_{j_m}\otimes [\zeta'_{i_1,\ldots,i_n;k}]$ we have $[\eta ]=\sum a_{i_1,\ldots,i_n;k}[\zeta_{i_1,\ldots,i_n;k}]=v_{j_m}\otimes [\eta']$, where $\eta'=\sum a_{i_1,\ldots,i_n;k}\zeta'_{i_1,\ldots,i_n;k}$. Then $0=\rho_{M^n,T^n}([\eta ])=\rho_{M^n,T^n}(v_{j_m}\otimes [\eta'])= v_{j_m}\otimes\rho_{M^{n-1},T^{n-1}}([\eta'])$. It follows that $\rho_{M^{n-1},T^{n-1}}([\eta'])=0$. But, by the induction hypothesis, $\rho_{M^{n-1},T^{n-1}}$ is injective so we have $[\eta']=0$. It follows that $[\eta ]=v_{j_m}\otimes [\eta']=0$. [**Remark**]{} When $n\geq 2$ we have $(T(V)\otimes\Lambda^2(V)\otimes T(V))^2=\Lambda^2(V)$ and $W_M^2(V)=0$. (All generators of $W_M(V)$ have degrees $\geq 3$.) Thus $M^2(V)=\Lambda^2(V)$ and $\rho_{M^2,T^2}$ coincides with $\rho_{\Lambda^2,T^2}$. Hence $0\to M^2(V)\xrightarrow{\rho_{M^2,T^2}} T^2(V)\xrightarrow{\rho_{T^2,S^2}} S^2(V)\to 0$ coincides with the short exact sequence from Proposition 1.1. The algebra $S'(V)$ =================== We define the algebra $S'(V)$ as $S'(V)=T(V)/W_{S'}(V)$, where $W_{S'}(V)$ is the bilateral ideal of $T(V)$ generated by $x\otimes y\otimes z-y\otimes z\otimes x$, with $x,y,z\in V$. If $x_1\ldots,x_n\in V$ then we denote by $x_1\odot\cdots\odot x_n$ the class of $x_1\otimes\cdots\otimes x_n$ in $S'(V)$. So $(S'(V),+,\odot )$ is an algebra. Since $W_{S'}(V)$ is a homogeneous ideal, $S'(V)$ is a graded algebra. For $n\geq 0$ we denote by $S'^n(V)$ the homogeneous component of degree $n$ of $S'(V)$. We have $S'^n(V)=T^n(V)/W^n_{S'}(V)$, where $W^n_{S'}(V)=W_{S'}(V)\cap T^n(V)$. Since the generators of $W_{S'}(V)$ have degree $3$, for $n\leq 2$ we have $W^n_{S'}(V)=0$ so $S'^n(V)=T^n(V)$. Note that $\rho_{T,S}(x\otimes y\otimes z-y\otimes z\otimes x)=xyz-yzx=0$ so $x\otimes y\otimes z-y\otimes z\otimes x\in\ker\rho_{T,S}$ $\forall x,y,z\in V$. It follows that $W_{S'}(V)\sbq\ker\rho_{S,T}$. Therefore $\rho_{T,S}$ induces a surjective morphism of algebras defined on $T(V)/W_{S'}(V)=S'(V)$. Namely, we have: There is a surjective morphism of algebras $\rho_{S',S}:S'(V)\to S(V)$ given by $x_1\odot\cdots\odot x_n\mapsto x_1\cdots x_n$. The subspace $W_{S'}^n(V)$ of $T(V)$ is spanned by $x_1\otimes\cdots\otimes x_n-x_{\sigma (1)}\otimes\cdots\otimes x_{\sigma (n)}$, with $x_1,\ldots,x_n\in V$ and $\sigma\in A_n$. The bilateral ideal $W_{S'}(V)$ of $T(V)$ is generated by $f(x,y,z)$, where $f:V^3\to T^3(V)$ is given by $(x,y,z)\mapsto x\otimes y\otimes z-y\otimes z\otimes x$. Therefore $W_{S'}^n(V)$ is spanned by $$\begin{gathered} x_1\otimes\cdots\otimes x_{i-1}\otimes f(x_i,x_{i+1},x_{i+2})\otimes x_{i+3}\otimes\cdots\otimes x_n\\ =x_1\otimes\cdots\otimes x_n-x_1\otimes\cdots\otimes x_{i+1}\otimes x_{i+2}\otimes x_i\otimes\cdots\otimes x_n\\ =x_1\otimes\cdots\otimes x_n-x_{\sigma_i(1)}\otimes\cdots\otimes x_{\sigma_i(n)},\end{gathered}$$ with $x_1,\ldots x_n\in V$ and $1\leq i\leq n-2$, where $\sigma_i\in S_n$ is the cycle $(i,i+1,i+2)$. Hence in $S'^n(V)$ we have $x_1\odot\cdots\odot x_n=x_{\sigma_i(1)}\odot\cdots\odot x_{\sigma_i(n)}$ $\forall x_1,\ldots,x_n\in V$ and $1\leq i\leq n-2$. But the cycles $\sigma_i$ generate the the alternating group $A_n$ so in $S'^n(V)$ we have $x_1\odot\cdots\odot x_n=x_{\sigma (1)}\odot\cdots\odot x_{\sigma (n)}$ $\forall\sigma\in A_n$. Hence $W_{S'}^n(V)$ contains $x_1\otimes\cdots\otimes x_n-x_{\sigma (1)}\otimes\cdots\otimes x_{\sigma (n)}$ $\forall x_1,\ldots,x_n\in V$ and $\sigma\in A_n$, which are more general than the original generators, where $\sigma =\sigma_i$ for some $1\leq i\leq n-2$. \(i) If $n\geq 2$ then we have a linear map $c:S'^n(V)\to S'^n(V)$ given by $x_1\odot\cdots\odot x_n\mapsto x_2\odot x_1\odot x_3\odot\cdots\odot x_n$, $\forall x_1,\ldots,x_n\in V$. \(ii) We have $c^2=1$ and $\rho_{S',S}c=\rho_{S',S}$ \(iii) If $x_1,\ldots,x_n\in V$ and $\sigma\in S_n$ then $$x_{\sigma (1)}\odot\cdots\odot x_{\sigma (n)}=\begin{cases}x_1\odot\cdots\odot x_n&\text{ if }\sigma\in A_n\\ c(x_1\odot\cdots\odot x_n)&\text{ if }\sigma\in S_n\setminus A_n\end{cases}.$$ \(iv) If there are $i<j$ with $x_i=x_j$ then $c(x_1\odot\cdots\odot x_n)=x_1\odot\cdots\odot x_n$. Consequently, $x_{\sigma (1)}\odot\cdots\odot x_{\sigma (n)}=x_1\odot\cdots\odot x_n$ holds regardless of the parity of $\sigma$. (i) We define $\bar c:T^n(V)\to T^n(V)$ by $x_1\otimes\cdots\otimes x_n\mapsto x_2\otimes x_1\otimes x_3\otimes\cdots\otimes x_n=x_{\tau (1)}\otimes\cdots\otimes x_{\tau (n)}$, where $\tau\in S_n$ is the transposition $(1,2)$. To prove that $\bar c$ induces the morphism $c:S'^n(V)\to S'^n(V)$ given by $x_1\odot\cdots\odot x_n\mapsto x_2\odot x_1\odot x_3\odot\cdots\odot x_n$, one must prove that $\bar c(W_{S'}^n(V))\sbq W_{S'}^n(V)$. Let $\xi =x_1\otimes\cdots\otimes x_n-x_{\sigma (1)}\otimes\cdots\otimes x_{\sigma (n)}$ be a generator of $W_{S'}^n$, with $x_1,\ldots,x_n\in V$ and $\sigma\in A_n$. Then $\bar c(\xi )=x_{\tau (1)}\otimes\cdots\otimes x_{\tau (n)}-x_{\sigma\tau (1)}\otimes\cdots\otimes x_{\sigma\tau (n)}$. If we denote $y_i=x_{\tau (i)}$, so that $x_i=y_{\tau^{-1}(i)}=y_{\tau (i)}$, then $x_{\sigma\tau (i)}=y_{\tau\sigma\tau (i)}$. Hence $\bar c(\xi )=y_1\otimes\cdots\otimes y_n-y_{\sigma'(1)}\otimes\cdots\otimes y_{\sigma'(n)}$, where $\sigma'=\tau\sigma\tau$. But $\sigma\in A_n$, which implies that $\sigma'\in A_n$ and so $\bar c(\xi )\in W_{S'}^n(V)$. \(ii) Let $\xi =x_1\odot\cdots\odot x_n$. Applying twice $c$ to $\xi$ permutes the first two factors of $\xi$ twice so we have $c^2(\xi )=\xi$. We have $\rho_{S',S}c(\xi )=x_2x_1x_3\cdots x_n=x_1\cdots x_n=\rho_{S',S}(\xi )$. \(iii) We have $c(x_1\odot\cdots\odot x_n)=y_1\odot\cdots\odot y_n$, where $y_i=x_{\tau (i)}$, with $\tau =(1,2)\in S_n$. Then $x_i=y_{\tau (i)}$. If $\sigma\in A_n$ then $x_{\sigma (1)}\odot\cdots\odot x_{\sigma (n)}=x_1\odot\cdots\odot x_n$ follows from Proposition 2.2. If $\sigma\notin A_n$ then note that $x_{\sigma (i)}=y_{\tau\sigma (i)}$ and, since $\tau,\sigma\notin A_n$, we have $\tau\sigma\in A_n$. Therefore $x_{\sigma (1)}\odot\cdots\odot x_{\sigma (n)}=y_{\tau\sigma (1)}\odot\cdots\odot y_{\tau\sigma (n)}=y_1\odot\cdots\odot y_n=c(x_1\odot\cdots\odot x_n)$. \(iv) Let $\tau\in S_n$, $\tau =(i,j)$. Since $x_i=x_j$, permuting the factors $x_i$ and $x_j$ has no effect on te product $x_1\odot\cdots\odot x_n$. Hence $x_{\tau (1)}\odot\cdots\odot x_{\tau (n)}=x_1\odot\cdots\odot x_n$. But $\tau\in S_n\setminus A_n$ so, by (iii), $x_{\tau (1)}\odot\cdots\odot x_{\tau (n)}=c(x_1\odot\cdots\odot x_n)$. Hence the conclusion. We now produce a basis for $S'^n(V)$. For this purpose we need the following elementary result. Let $U$ be a vector space with the basis $(u_\alpha )_{\alpha\in A}$. Let $\sim$ be an equivalence relation on $A$ and let $B$ be a set of representatives for $A_{/\sim}$. Let $W\sbq U$ be the subspace generated by all $u_\alpha -u_\beta$, with $\alpha,\beta\in A$ such that $\alpha\sim\beta$. For every $u\in U$ we denote by $\bar u$ its class in $U/W$. Then $(\bar u_\alpha )_{\alpha\in B}$ is a basis for $U/W$. Let $U'\sbq U$ be the subspace generated by $u_\alpha$, with $\alpha\in B$. Let $f:U\to U'$ be the linear function given by $u_\alpha\mapsto u_\beta$, where $\beta$ is the unique element of $B$ such that $\alpha\sim\beta$. If $u_\alpha -u_\beta$, with $\alpha\sim\beta$, is a generator of $W$ and $\gamma\in B$ such that $\alpha\sim\gamma$ then we also have $\beta\sim\gamma$ and so $f(u_\alpha )=f(u_\beta )=u_\gamma$. It follows that $u_\alpha -u_\beta\in\ker f$ and so $W\sbq\ker f$. Therefore $f$ induces a linear map $\bar f:U/W\to U'$, given by $\bar u_\alpha\mapsto u_\beta$, where $\beta\in B$ such that $\alpha\sim\beta$. We now define $g:U'\to U/W$, given by $u_\alpha\mapsto\bar u_\alpha$ $\forall\alpha\in B$. For every $\alpha\in B$ we have $\bar fg(u_\alpha )=\bar f(\bar u_\alpha )=u_\alpha$. (We have $\alpha\in B$ and $\alpha\sim\alpha$.) Thus $\bar fg=1_{U'}$. If $\alpha\in A$ and $\beta\in B$ such that $\alpha\sim\beta$ then $g\bar f(\bar u_\alpha )=g(u_\beta )=\bar u_\beta =\bar u_\alpha$. (We have $\alpha\sim\beta$ so $u_\alpha -u_\beta\in W$ so $\bar u_\alpha =\bar u_\beta$.) Thus $g\bar f=1_{U/W}$. Thus $g:G'\to G/W$ is an isomorphism and $\bar f$ its inverse. Since $(\bar u_\alpha )_{\alpha\in B}$ is the image with respect to $g$ of the basis $(u_\alpha)_{\alpha\in B}$ of $G'$, it will be a basis for $U/W$. Let $J=\{ (i_1,\ldots,i_n)\in I^n\mid i_1\leq\cdots\leq i_n\}$, $J_1=\{ (i_1,\ldots,i_n)\in I^n\mid i_1<\cdots <i_n\}$ and $J_2= J\setminus J_1$. Then $$\{ v_{i_1}\odot\ldots\odot v_{i_n}\,\mid\, (i_1,\ldots,i_n)\in J\}\cup \{ c(v_{i_1}\odot\ldots\odot v_{i_n})\,\mid\, (i_1,\ldots,i_n)\in J_1\}$$ is a basis of $S'^n(V)$. We use Lemma 2.4 for $U=T^n(V)$, with the basis $(u_\alpha )_{\alpha\in A}$, where $A=I^n$ and $u_{i_1,\ldots,i_n}=v_{i_1}\otimes\cdots\otimes v_{i_n}$ $\forall (i_1,\ldots,i_n)\in A$. The equivalence relation $\sim$ on $A$ is given by $(i_1,\ldots,i_n)\sim (j_1,\ldots,j_n)$ if $(j_1,\ldots,j_n)=(i_{\sigma (1)},\ldots,i_{\sigma (n)})$ for some $\sigma\in A_n$ and $W\sbq U$ is generated by $u_\alpha -u_\beta$, with $\alpha,\beta\in A$, $\alpha\sim\beta$. We claim that $B=J\cup\{ (i_2,i_1,i_3\ldots,i_n)\mid (i_1,\ldots,i_n)\in J_1\}$ is a set of representatives for $A_{/\sim}$. First we show if $\alpha,\beta\in B$, $\alpha\neq\beta$, then $\alpha\not\sim\beta$. We note that if $(i_1,\ldots,i_n)\sim (j_1,\ldots,j_n)$ then then the sequences $i_1,\ldots,i_n$ and $j_1,\ldots,j_n$ written in increasing order are the same. Therefore, if $(i_1,\ldots,i_n),(j_1,\ldots,j_n)\in J$ with $(i_1,\ldots,i_n)\neq (j_1,\ldots,j_n)$ then $(i_1,\ldots,i_n)$ or $(i_2,i_1,i_3,\ldots,i_n)$ cannot be in the relation $\sim$ with $(j_1,\ldots,j_n)$ or $(j_2,j_1,j_3,\ldots,j_n)$. This proves that if $\alpha,\beta\in B$, with $\alpha\neq\beta$ then $\alpha\not\sim\beta$ unless $\alpha =(i_1,\ldots,i_n)$ and $\beta =(i_2,i_1,i_3,\ldots,i_n)$ (or viceversa) for some $(i_1,\ldots,i_n)\in J_1$, i.e. with $i_1<\cdots <i_n$. But in this case the only $\sigma\in S_n$ such that $(i_2,i_1,i_3,\ldots,i_n)=(i_{\sigma (1)},\ldots,i_{\sigma (n)})$ is $\sigma =(1,2)$, which is odd. Hence, again, $\alpha\not\sim\beta$. Next we prove that if $\alpha =(i_1,\ldots,i_n)\in A$ then there is $\beta\in B$ with $\alpha\sim\beta$. We write the sequnce $i_1,\ldots,i_n$ in increasing order as $j_1,\ldots,j_n$. Then $(j_1,\ldots,j_n)\in J\sbq B$ and we have $(j_1,\ldots,j_n)=(i_{\sigma (1)},\ldots,i_{\sigma (n)})$ for some $\sigma\in S_n$. If $\sigma\in A_n$ then $(i_1,\ldots,i_n)\sim (j_1,\ldots,j_n)$ so we may take $\beta =(j_1,\ldots,j_n)$. Suppose now that $\sigma\in S_n\setminus A_n$. If $(j_1,\ldots,j_n)\in J_2$ then $j_k=j_{k+1}$ for some $1\leq k\leq n-1$. Hence if $\tau =(k,k+1)\in S_n$ then $(j_1,\ldots,j_n)=(j_{\tau (1)},\ldots,j_{\tau (n)})=(i_{\sigma\tau (1)},\ldots,i_{\sigma\tau (n)})$. Since $\sigma,\tau\in S_n\setminus A_n$ we have $\sigma\tau\in A_n$ so $(i_1,\ldots,i_n)\sim (j_1,\ldots,j_n)$. So again we may take $\beta =(j_1,\ldots,j_n)$. If $(j_1,\ldots,j_n)\in J_1$ the we also have $(j_2,j_1,j_3\ldots,j_n)\in B$. If $\tau =(1,2)\in S_n$ then $(j_2,j_1,j_3,\ldots,j_n)=(j_{\tau (1)},\ldots,j_{\tau (n)})=(j_{\sigma\tau (1)},\ldots,i_{\sigma\tau (n)})$. Since $\sigma,\tau\in S_n\setminus A_n$ we have $\sigma\tau\in A_n$ so $(i_1,\ldots,i_n)\sim (j_2,j_1,j_3,\ldots,j_n)$. So this time we may take $\beta =(j_2,j_1,j_3,\ldots,j_n)$. The subspace $W_{S'}^n(V)$ of $T^n(V)$ is generated by $f_\sigma (x_1,\ldots,x_n)$ with $x_1,\ldots,x_n$ and $\sigma\in A_n$, where $f_\sigma :V^n\to T^n(V)$ is given by $(x_1,\ldots,x_n)\mapsto x_1\otimes\cdots\otimes x_n-x_{\sigma (1)}\otimes\cdots\otimes x_{\sigma(n)}$. But $f_\sigma$ is multilinear so we may restrict ourselves to the case when $x_1,\ldots,x_n$ belong to the basis $v_i$, with $i\in I$, of $V$. Then $W_{S'}^n(V)$ is generated by $f_\sigma (v_{i_1},\ldots,v_{i_n})=v_{i_1}\otimes\cdots\otimes v_{i_n}-v_{i_{\sigma (1)}}\otimes\cdots\otimes v_{i_{\sigma(n)}}=u_{i_1,\ldots,i_n}-u_{i_{\sigma (1)},\ldots,i_{\sigma (n)}}$, with $i_1,\ldots,i_n\in I$ and $\sigma\in A_n$. Equivalently, $W_{S'}^n(V)$ is generated by $u_\alpha -u_\beta$, with $\alpha,\beta\in A$, $\alpha\sim\beta$, i.e. $W_{S'}^n(V)=W$. It follows that $U/W=T^n(V)/W_{S'}^n(V)=S'^n(V)$. Also if $u=x_1\otimes\cdots\otimes x_n\in U=T^n(V)$ then its class in $U/W=S'^n(V)$ is $\bar u=x_1\odot\cdots\odot x_n$. In particular, $\bar u_{i_1,\ldots,i_n}=v_{i_1}\odot\cdots\odot v_{i_n}$. By Lemma 2.4, $\bar u_\alpha$ with $\alpha\in B$ are a basis of $U/W=S'^n(V)$. If $\alpha =(i_1,\ldots,i_n)\in J$ then $\bar u_\alpha =v_{i_1}\odot\cdots\odot v_{i_n}$. If $\alpha =(i_2,i_1,i_3,\ldots,i_n)$ for some $(i_1,\ldots,i_n)\in J_1$ then $\bar u_\alpha =v_{i_2}\odot v_{i_1}\odot v_{i_3}\odot\cdots\odot v_{i_n}=c(v_{i_1}\odot\cdots\odot v_{i_n})$. Hence the conclusion. For $n\geq 2$ we have the exact sequence $$0\to\Lambda^n(V)\xrightarrow{\rho_{\Lambda^n,S'^n}} S'^n(V)\xrightarrow{\rho_{S'^n,S^n}} S^n(V)\to 0,$$ where $\rho_{\Lambda^n,S'^n}$ is given by $x_1\wedge\cdots\wedge x_n\mapsto x_1\odot\cdots\odot x_n- c(x_1\odot\cdots\odot x_n)$. We already know that $\rho_{S'^n,S^n}$ is surjective. The map $(x_1,\ldots,x_n)\mapsto x_1\odot\cdots\odot x_n-c(x_1\odot\cdots\odot x_n)$ is linear in each variable and anti-symmetric. (If $x_i=x_j$ for some $i\neq j$ then, by Proposition 2.3(iv), we have $c(x_1\odot\cdots\odot x_n)=x_1\odot\cdots\odot x_n$.) Hence the map $\rho_{\Lambda^n,S'^n}$, given by $x_1\wedge\cdots\wedge x_n\mapsto x_1\odot\cdots\odot x_n-c(x_1\odot\cdots\odot x_n)$, is well defined. We prove the injectivity of $\rho_{\Lambda^n,S'^n}$. We use the notations of Propostion 2.5. The set $\{ v_{i_1}\wedge\cdots\wedge v_{i_n}\,\mid\, (i_1,\ldots,i_n)\in J_1\}$ is a basis of $\Lambda^n(V)$. Let $\alpha\in\ker\rho_{\Lambda^n,S'^n}$. We write $\alpha =\sum_{(i_1,\ldots,i_n)\in J_1} a_{i_1,\ldots,i_n}v_{i_1}\wedge\cdots\wedge v_{i_n}$, with $a_{i_1,\ldots,i_n}\in F$. Then $$0=\rho_{\Lambda^n,S'^n}(\alpha )=\sum_{(i_1,\ldots,i_k)\in J_1}a_{i_1,\ldots,i_k}(v_{i_1}\odot\cdots\odot v_{i_n}-c(v_{i_1}\odot\cdots\odot v_{i_k})).$$ Since $v_{i_1}\odot\cdots\odot v_{i_n}$ and $c(v_{i_1}\odot\cdots\odot v_{i_n})$, with $(i_1,\ldots,i_n)\in J_1$, are part of the basis of $S'^n(V)$ form Proposition 2.5, this implies that $a_{i_1,\ldots,i_n}=0$ $\forall (i_1,\ldots,i_n)\in J_1$ so $\alpha =0$. Hence $\rho_{\Lambda^n,S'^n}$ is injective. For the exactness in the second term we use the formulas $\rho_{S'^n,S^n}(x_1\odot\cdots\odot x_n)=\rho_{S'^n,S^n}c(x_1\odot\cdots\odot x_n)=x_1\cdots x_n$. (See Propostion 2.3 (ii).) The map $\rho_{S'^n,S^n}\rho_{\Lambda^n,S'^n}$ is given by $x_1\wedge\cdots\wedge x_k\mapsto\rho_{S'^n,S^n}(x_1\odot\cdots\odot x_n-\\ c(x_1\odot\cdots\odot x_n))=x_1\cdots x_n-x_1\cdots x_n=0$. Hence $\rho_{S'^n,S^n}\rho_{\Lambda^n,S'^n}=0$ so $\ker\rho_{S'^n,S^n}\spq\operatorname{Im}\rho_{\Lambda^n,S'^n}$. For the reverse inclusion let $\alpha\in\ker\rho_{S'^n,S^n}$. By Proposition 2.5, $\alpha$ writes as $$\alpha =\sum_{(i_1,\ldots,i_n)\in J}a_{i_1,\ldots,i_n}v_{i_1}\odot\cdots\odot v_{i_n}+\sum_{(i_1,\ldots,i_n)\in J_1}b_{i_1,\ldots,i_n}c(v_{i_1}\odot\cdots\odot v_{i_n}),$$ where $a_{i_1,\ldots,i_n},b_{i_1,\ldots,i_n}\in F$. Then we have $$\begin{aligned} 0&=\rho_{S'^n,S^n}(\alpha )=\sum_{(i_1,\ldots,i_n)\in J}a_{i_1,\ldots,i_n}v_{i_1}\cdots v_{i_n}+\sum_{(i_1,\ldots,i_n)\in J_1}b_{i_1,\ldots,i_n}v_{i_1}\cdots v_{i_n}\\ &=\sum_{(i_1,\ldots,i_n)\in J_1}(a_{i_1,\ldots,i_n}+b_{i_1,\ldots,i_n})v_{i_1}\cdots v_{i_n}+\sum_{(i_1,\ldots,i_n)\in J_2}a_{i_1,\ldots,i_n}v_{i_1}\cdots v_{i_n}. \end{aligned}$$ Since $v_{i_1}\cdots v_{i_n}$ with $(i_1,\ldots,i_n)\in J=J_1\sqcup J_2$ are a basis of $S^n(V)$, we get $a_{i_1,\ldots,i_n}=0$ $\forall (i_1,\ldots,i_n)\in J_2$ and $a_{i_1,\ldots,i_n}+b_{i_1,\ldots,i_n}=0$, so $b_{i_1,\ldots,i_n}=-a_{i_1,\ldots,i_n}$, $\forall (i_1,\ldots,i_n)\in J_2$. It follows that $$\begin{aligned} \alpha &=\sum_{(i_1,\ldots,i_n)\in J_1}a_{i_1,\ldots,i_n}v_{i_1}\odot\cdots\odot v_{i_n}+\sum_{(i_1,\ldots,i_n)\in J_1}-a_{i_1,\ldots,i_n}c(v_{i_1}\odot\cdots\odot v_{i_n})\\ &=\sum_{(i_1,\ldots,i_n)\in J_1}a_{i_1,\ldots,i_n}(v_{i_1}\odot\cdots\odot v_{i_n}-c(v_{i_1}\odot\cdots\odot v_{i_n}))=\rho_{\Lambda^n,S'^3}(\beta ),\end{aligned}$$ where $\beta=\sum_{(i_1,\ldots,i_n)\in J_1}a_{i_1,\ldots,i_n}v_{i_1}\wedge\cdots\wedge v_{i_n}$. Thus $\alpha\in\operatorname{Im}\rho_{\Lambda^n,S'^3}$. Recall that if $n\leq 2$ then $S'^n(V)=T^n(V)$. If $n=0,1$ then $S^n(V)=S'^n(V)=T^n(V)$ and $\rho_{S'^n,S^n}$ is the identity map so we have the short exact sequence $0\to 0\to S'^n(V)\xrightarrow{\rho_{S'^n,S^n}}S^n(V)\to 0$. By putting together these two sequences with those for $n\geq 2$ from Theorem 2.6, we get: We have a short exact sequence, $$0\to\Lambda^{\geq 2}(V)\xrightarrow{\rho_{\Lambda^{\geq 2},S'}}S'(V) \xrightarrow{\rho_{S',S}}S(V)\to 0.$$ [**Remark**]{} The maps $\rho_{\Lambda^2,S'^2}$ and $\rho_{S'^2,S^2}$ are given by $x\wedge y\mapsto x\odot y-y\odot x$ and $x\odot y\mapsto xy$, respectively. But when identify $S'^2(V)$ with $T^2(V)$ they write as $x\wedge y\mapsto x\otimes y-y\otimes x=[x,y]$ and $x\otimes y\mapsto xy$ so they coincide with $\rho_{\Lambda^2,T^2}$ and $\rho_{T^2,S^2}$. Therefore the short exact sequences form Theorem 2.6, in the case $n=2$, and from Proposition 1.1 are the same. [**References**]{} \[KT\] Christian Kassel and Vladimir Turaev. “Braid groups”, volume 247 of Graduate Texts in Mathematics. Springer, New York, 2008. With the graphical assistance of Olivier Dodane.

--- abstract: 'Due to the combined effect of anisotropic interactions and activity, Janus swimmers are capable to self-assemble in a wide variety of structures, many more than their equilibrium counterpart. This might lead to the development of novel active materials capable of performing tasks without any central control. Their potential application in designing such materials endows trying to understand the fundamental mechanism in which these swimmers self-assemble. In the present work, we study a quasi-two dimensional semi-dilute suspensions of two classes of amphiphilic spherical swimmers whose direction of motion can be tuned: either swimmers propelling in the direction of the hydrophobic patch (WP), or swimmers propelling in the opposite direction (towards the hydrophilic side) (AP). In both systems we have systematically tuned swimmers’ hydrophobic strength and signature, and observed that the anisotropic interactions, characterized by the angular attractive potential and its interaction range, in competition with the active stress, pointing towards or against the attractive patch gives rise to a rich aggregation phenomenology.' author: - Francisco Alarcon - 'Eloy Navarro-Argemí' - Chantal Valeriani - Ignacio Pagonabarraga title: 'Orientational order and morphology of clusters of self-assembled Janus swimmers' --- \[Intro\]Introduction ===================== In the last decade, the pioneer experiments by Dombrowski and Cisneros [@Dombrowski; @Cisneros] inspired the numerical work aimed at understanding the influence of hydrodynamics in the formation of coherent structures of micro-organisms, simulated as spherical squirmers [@Ishikawa2006] in 3D [@Ishikawa2007a; @Ishikawa2008JFM] and quasi-2D [@Ishikawa2008PRL]. The features of a 3D squirmer suspension have been numerically characterized using the smoothed profile method [@molinaSoftMatt13; @molina_PRE16; @JPS_yamamoto] and Lattice Boltzmann (LB), either in bulk [@alarcon2013] or in response to a steady Couette flow [@LlopisSoftMatter13]. While Evans et al. [@evansPoF] used Stokesian Dynamics to detect the instability of the isotropic state in a 3D suspension of weak pullers, Delmotte and coworkers [@Delmotte] applied a force-coupling method to investigate, by large-scale simulations, the emergence of a polar order in a 3D squirmer suspension. More recently, experiments by Ref. [@SoftMatter_35; @SoftMatter_33; @PRL_bacteriaFastest; @Cecille; @Thutupalli] have inspired numerical simulations of 2D and quasi-2D suspensions of squirmers interacting via a repulsive potential. When a suspension (simulated by means of multi-particle collision dynamics) was confined between parallel walls (quasi-2D geometry) [@Zottl_prl], active particles were forming clusters whose size depended on the system’s concentration with the largest being made of pullers. The suspension prepared in such confined geometry would undergo phase separation into a dilute and a dense phase [@Zottl_softmatter16], differently to the 2D suspension of disk-like squirmers [@PRE_Ricard], where hydrodynamics suppressed phase separation. Inspired by the self-assembly observed in experiments with either bacteria [@PRL_bacteriaFastest] or active colloids [@Cecille] and by numerical works on 2D dilute suspensions of active brownian particles (ABP) interacting via isotropic short-range attractions [@SoftMatter_26; @SoftMatter_27; @SoftMatter_28; @SoftMatter_43], we have studied a semi-diluted quasi 2D suspensions of attractive squirmers [@SoftMatter17], to unravel the role played by hydrodynamics in suspensions of attractive active particles. [Most of the nowadays synthetic active colloids are colloids partially coated with Pt. These colloids self-propell by a catalytic reaction of H$_2$O$_2$ and O$_2$ taking place at the Pt surface. This mechanism of colloid’s propulsion is due to phoresis. On the one side, properly modeling the phoretic interactions can be quite complicated [@phoretic39; @phoretic40; @phoretic41; @phoretic42; @phoretic43; @phoretic44], eventhough there are experiments, simulations and theories that consider them relevant for particles’ propulsion. On the other side, which interactions dominate in these systems are still a matter of debate [@JCP_Benno2019].]{} [From the experimental point of view, several studies have focused on understanding the effect of hydrophobicity in catalytic Janus colloids. Modifying the surface of the particles, the authors of Ref. [@Manjare_HydrophobicJanus] found that particles with an hydrophobic patch would move faster than the ones with an hydrophilic patch.]{} [Similarly, Gao et al [@Gao_OTSJanusExp], studied self-assembly of amphiphilic particles made with hydrophobic octadecyltrichlorosilane (OTS)-modified silica microspheres capped with a catalytic Pt hemisphere patch.]{} [Additionally, Yan et al [@Dipolar_Granick] have recently reported different collective states of active dipolar Janus colloids, whose motion is originated by an induced-charge electrophoresis. These colloids self-assemble in different morphologies, ranging from: gas, swarms, chains and clusters, by modifying both the charge imbalance of the colloids and the electric field frequency. Moreover, in [@Dipolar_Granick], the authors were able to numerically reproduce all observed states by means of an overdamped molecular dynamics simulation whose particles were interacting via directed imbalanced interactions.]{} [Motivated by these two experimental works [@Gao_OTSJanusExp; @Dipolar_Granick], simulations of a dilute quasi-2D suspension of ABP interacting via anisotropic amphiphilic Janus interactions have been reported in Ref. [@Stewart_17]. To the best of our knowledge, this is the first numerical attempt to unravel the collective behaviour of amphiphilic active Janus particles, taking into account their anisotropic nature. In Ref. [@Stewart_17] both the effect of interaction directionality and the propulsion speed have been used to control the physical properties of the assembled active aggregates. One might suggest that these two parameters could be easily tuned also in experiments, given that the concentration of a cationic surfactant could be used to reverse the propulsion’s direction and adding pH neutral salts could be used to control the propulsion speed [@Brown_Poon_SaltJanus].]{} In order to establish the relevance of hydrodynamics in this suspension, we study the collective behaviour of quasi 2D suspensions of spherical squirmers interacting via an anisotropic Janus potential, considering different interaction ranges, interaction strengths and hydrodynamic signatures. When clusters appear, we characterise their morphology as a function of the above mentioned features. [The objective of this manuscript is the analysis of minimal models that contain the essential ingredients of activity and steric interactions for heterogeneous particles. The scenarios we identify can be clearly correlated with the competition between self-propulsion, stress generation, and attraction among particles. It is true that phoresis is more complex and can also involve long range interactions among the active elements. Our results serve as a benchmarking scenario if phoresis is controlled by near field effects. In this sense, the systematic analysis we have carried out provides a valuable contribution in this general perspective. In a more specific perspective, this study also helps to understand the effect of the hydrodynamic interactions in the collective dynamics of active Janus colloids, inspired by experiments showing flow fields around a single catalytic Janus sphere [@Campbell_FlowFieldJanus].]{} The manuscript is organised as follows. In section \[model\] we present the anisotropic interaction model, the squirmer model, the lattice Boltzmann methodology and simulations details. In section \[sec:Tools\] we describe the tools used to characterise the Janus squirmers. In section \[sec:Results\] we present all results and discuss our conclusions in section \[sec:Conclusions\]. \[model\]Simulation details =========================== In this work, we have studied a quasi two dimensional dilute suspension of squirmers [@Lighthill; @Blake; @Pedley:2016] interacting via a Janus anisotropic potential similar to the one used in Ref. [@Stewart_17] for Active Brownian Particles (ABP). In order to model the Janus swimmers, we have introduced a pair potential $V(r_{ij},\theta_i,\theta_j)$ which depends on the distance between two squirmers ($r_{ij}$) and their attractive patches’ orientations $$\begin{aligned} \label{eq:JanusGralPotential} %V(r_{ij},\hat{p}_i,\hat{p}_j)= V_{rep}(r_{ij})+V_{att}(r_{ij})\phi(\theta_i,\theta_j)\, . V(r_{ij},\theta_i,\theta_j)= V_{rep}(r_{ij})+V_{att}(r_{ij})\phi(\theta_i,\theta_j)\, \end{aligned}$$ where $\theta_i$ is the angle between the patch unit vector $\vec{p_i}$ and the inter-particle vector $\vec{r}_{ji}=\vec{r}_j - \vec{r}_i$; and $\theta_j$ the angle between $\vec{p_j}$ and $\vec{r}_{ij}=-\vec{r}_{ji}$. $V_{rep}(r_{ij})$ represents a short range soft repulsion [@Allen_Tildesley] (to avoid overlapping) given by $$\begin{aligned} \label{eq:SSPotential} \begin{split} &V_{rep}\left(r_s\right)= \\ &\begin{cases} \epsilon_s \left[ \left(\frac{\sigma_s}{r_{s}} \right)^{\nu}-\left( \frac{\sigma_s}{h_{0}} \right)^{\nu}\left(1-\frac{(r_s-h_0)\nu}{h_0} \right) \right] & 0<r_{s}\leq h_0\\ 0 & otherwise\,,\\ \end{cases} \end{split}\end{aligned}$$ where $\epsilon_s=1.5$ is the energy scale, $\nu=2$, $r_s \equiv r_{ij}-\sigma$ is the inter-particle distance (with $r_s > 0$ to ensure that particles never overlap), $\sigma_s=0.5$ is the characteristic separation length between particles, and $h_0$ the distance at which the potential is shifted to ensure its continuity at $r_s=h_0$. [^1] The attractive potential consists of a radial $V_{att}(r_{ij})$ and an angular $\phi(\theta_i,\theta_j$) contribution. Concerning the radial part of the potential, we have considered two different truncated Lennard-Jones: one long ($r_c =2.5 \sigma$) and another short ($r_c =1.5 \sigma$) range. While the following equation: $$\begin{aligned} \label{eq:VLJmidrange} \begin{split} &V_{att} (r_{ij}) = \\ &\begin{cases} 4 \epsilon \left[ \left( \frac{\sigma}{r_{ij}} \right)^{12} - \left( \frac{\sigma}{r_{ij}} \right)^6 \right] & (2^{1/6}\sigma)<r_{ij}<r_{c}\,,\\ -\epsilon & r_{ij}< (2^{1/6}\sigma)\,, \\ 0 & otherwise\,, \ \end{cases} \end{split}\end{aligned}$$ represents the former, it has been modified for the latter by adding a smooth transition function, so that the potential $V_{att}$ and its derivative vanish continuously as $r_{ij}$ approaches $r_c$ [@pra_smoothfunct] (see appendix \[Appendix:attra\]). Therefore, whenever $(2.2^{1/6}\sigma)<r_{ij}<r_{c}$ $$\begin{aligned} \label{eq:VLJshortrange} V_{att} (r_{ij}) = 4\epsilon \left[ \left( \frac{\sigma}{r_{ij}} \right)^{12} - \left( \frac{\sigma}{r_{ij}} \right)^{6} \right] + 4\epsilon \left[ 6 \left( \frac{\sigma}{r_{c}} \right)^{12} - 3 \left( \frac{\sigma}{r_{c}} \right)^{6} \right] \left( \frac{r_{ij}}{r_{c}} \right)^{2} - 4\epsilon \left[ 7 \left(\frac{\sigma}{r_{c}} \right)^{12} + 4 \left( \frac{\sigma}{r_{c}} \right)^{6} \right]\,, \end{aligned}$$ whereas for $r_{ij} < (2.2^{1/6}\sigma)$ $V_{att}=-0.305 \epsilon$ and $V_{att}=0$ otherwise. Inspired by the work of Hong [*et al.*]{} [@langmuir08] for amphiphilic colloidal spheres, the anisotropic interaction is controlled by the angular contribution of the attractive potential $$\begin{aligned} \label{eq:phi} \begin{split} &\phi(\theta_i,\theta_j)=&\\ &\begin{cases} \cos\theta_i \cos \theta_j &\left( \hat{p}_i \cdot \hat{\mu}_{ji} \right) > 0 \cap \left( \hat{p}_j \cdot \hat{\mu}_{ij} \right) > 0\,,\\ 0 & otherwise\,.\\ \end{cases} \end{split}\end{aligned}$$ where $\hat{\mu}_{ij} = \vec{r}_{ij}/r_{ij}$ is the unit vector joining the squirmers’ center of mass. This anisotropic potential ensures a non-vanishing torque when $\left( \hat{p}_i \cdot \hat{\mu}_{ji} \right) > 0 $ and $ \left( \hat{p}_j \cdot \hat{\mu}_{ij} \right) > 0$ (a more detailed analysis is discussed in appendix \[Appendix:JanusOrient\]). Within our model, each Janus particle self-propels along a direction $\hat{e}_i$ rigidly bounded to the particle. As in Ref. [@Stewart_17], we study two types of active Janus particles depending of the orientation of $\hat{e}_i$ with respect to $\hat{p}_i$. When the propulsion direction ($\hat{e}_i$, red arrow in figure \[fig:JanusSquirmers\]-left panel) is pointing towards the attractive patch $\hat{p}_i$ (red arrow in figure \[fig:JanusSquirmers\]-left panel), swimmers are propelling “with the patch” (WP). When the propulsion direction ($\hat{e}_i$, red arrow in figure \[fig:JanusSquirmers\]-right panel) is pointing against the attractive patch $-\hat{p}_i$ (green arrow in figure \[fig:JanusSquirmers\]-right panel), swimmers are propelling “against the patch” (AP). ![Panel a) WP Janus swimmers. Panel b) AP Janus swimmers. The attractive patch is represented in blue [(dark gray)]{}, and the non-attractive patch is in green [(light gray)]{}. The swimming direction $\hat{e}$ is the red arrow and direction of the attractive patch $\hat{p}$ is the green one. The center-to-center distance between two swimmers $\vec{r_{ij}}$ is the black vector. \[fig:JanusSquirmers\]](WP_pairJanus.pdf "fig:"){width="0.480\linewidth"} ![Panel a) WP Janus swimmers. Panel b) AP Janus swimmers. The attractive patch is represented in blue [(dark gray)]{}, and the non-attractive patch is in green [(light gray)]{}. The swimming direction $\hat{e}$ is the red arrow and direction of the attractive patch $\hat{p}$ is the green one. The center-to-center distance between two swimmers $\vec{r_{ij}}$ is the black vector. \[fig:JanusSquirmers\]](AP_pairJanus.pdf "fig:"){width="0.480\linewidth"} {width=".32\textwidth"}{width=".32\textwidth"}{width=".362\textwidth"} A spherical Janus squirmer of radius $R_p$ is characterized by the tangential surface slip velocity, $$\begin{aligned} \label{eq:surface_veloc} \textbf{u}|_{R_p} = \left[ B_1 \sin \theta + B_2 \sin \theta \cos \theta \right] \boldsymbol{\tau}\, ,\end{aligned}$$ where $\boldsymbol{\tau}$ is a unit vector tangential to the particle surface, $B_1$ quantifies the asymptotic self-propelling speed of an isolated squirmer ($v_s= \frac{2}{3} B_1$) [@Prieve; @Stone] and $B_2$ is proportional to the active stress generated on the surrounding fluid. The ratio between active stress and self-propelling velocity, $\beta=B_2/B_1$ [@Ishikawa2006], quantifies the active state of the squirmer and its interaction with the fluid. A positive value of $\beta$ corresponds to the case when thrust is generated in front of the squirmer’s body (puller); a negative value of $\beta$ to the case when thrust is generated at the back (pusher)  [@Ishikawa2007a] and $\beta=0$ corresponds to a neutral swimmer (see Fig. \[fig:Squirmers\]). In our simulations we disregard thermal fluctuations; thus velocity fluctuations are simply induced by the particles’ activity, where $B_2$ can be understood as an effective temperature. In order to characterize how hydrodynamics affects the assembly of a dilute suspension of Janus swimmers, we model the embedding solvent using a Lattice Boltzmann (LB) method with a D3Q19 three dimensional lattice [@cates_lb; @succi], implemented in a highly parallelized code [@ludwig] that exploit the excellent scalability of LB on supercomputing facilities. Spherical squirmers (with a diameter $\sigma$ of 4.6 lattice nodes [@nguyen]) are individually resolved, imposing a modified bounce-back rule on the one-particle velocity distribution functions to nodes crossing the solid particle’s boundary (including the slip velocity to impose the squirming motion [@llopis_wall; @ricard_epje]). The total force and torque the fluid exerts on the particle is obtained by imposing that the total momentum exchange between the particle and the fluid nodes vanishes (as a result of the modified bounce-back). While accounting for all forces acting on each squirmer allows to update them dynamically, the torque exerted by the fluid determines the change of the squirmer ’s self-propulsion direction. The simulated system consists of a suspension of $N=888$ spherical squirmers at $\phi=0.10$ (where $\phi=\frac{\pi}{4} \rho\sigma^2$ and $\rho=N/L^2$) in a quasi two-dimension geometry with periodic boundary conditions ($L\times L \times k \sigma$, with $k=5$ and $L \approx 83 \sigma$) needed to capture three dimensional hydrodynamics effects, [whereas both colloids’ position and orientation are confined to move in 2D]{}. Additionally, we have carried out bigger simulations with $N=3552$ squirmers at the same packing fraction and $L \approx 167 \sigma$, in order to proof that results are not due to finite size effects, specially cases with large cluster sizes. As in Ref. [@SoftMatter17], we quantify the competition between attractive and self-propelling forces via the dimensionless parameter $$\xi = \frac{F_d}{F_{att}},$$ where $F_d = 6 \pi \eta R_p v_s$ is the friction force associated to the squirmer self-propulsion and $F_{att}$ is the absolute value of the attractive force at its minimum, corresponding to $r=(26/7)^{1/6}\sigma = 1.245\sigma$ for the long-range potential in Eq. \[eq:VLJmidrange\] and to $r=1.239\sigma$ for the short-range potential in Eq. \[eq:VLJshortrange\]. [On the one hand $\xi$ controls the competition between two mechanisms: the self-propulsion and the interaction strength. On the other hand the mechanisms that control the re-orientation of the particles are the active stress given by the $B_2$ squirmer parameter and the orientational-dependence of the potential which is modulated by the range and the strength of the interaction.]{} Given that a squirmer travels its own size in a time $\tau = \frac{\sigma}{v_s}$, we perform simulations from $1450$ up to $3000$ $\tau$. Once the system reaches steady state, in a time window between $1000$ and $2000$ $\tau$, we carry out a systematic analysis of the dynamics of the squirmer suspension, considering $\xi$ from 0.1 to 10 and $\beta$ from -3 up to 3. \[sec:Tools\]Analysis Tools =========================== In order to be able to establish the effect played by the anisotropy of the interaction between swimmers, we follow the procedure in Ref. [@SoftMatter17] and study the degree of aggregation as a function of $\beta$ and $\xi$. Whenever the system forms a steady-state cluster distribution, we explore the clusters morphology by computing their mean size, the cluster-size distribution, their radius of gyration, and both the polar and nematic order parameter. When clusters are present, we identify them following a distance criterion: at each time step, two particles belong to the same cluster of size $s$ whenever their distance is smaller than $r_{cl} = 1.75 \sigma$ (first minimum of the $g(r)$, see appendix \[Appendix:Clusters\]). To calculate the cluster-size distribution $f(s)$ we apply a criterion based on [@chantalbins2012]: (i) We arbitrarily subdivide the range of $s$-values into intervals $\Delta s_i = (s_{i,max} - s_{i,min})$, where $n_i^t$ is the total number of clusters within each interval $\Delta s$; (ii) we assign the value $n_i=n^t_i/\Delta s_i$ to every $s$ within $\Delta s_i$, and compute the fraction of clusters of size $s$ as $f(s)=n_i/N_c$, where $N_c=\sum_i n_i \Delta s_i$ is the total number of clusters. To characterise the clusters’ morphology, we compute the radius of gyration $$\label{eq:Rg} R_g(s) = \sqrt{\sum_{i,j=1}^s \frac{\left(\vec{r}_i-\vec{r}_j\right)^2}{2s}}\,,$$ the average polar order parameter for each cluster size $s$ $$\label{eq:PolarOrderCluster} P(s)=\left\vert\frac{\sum_{i=1}^s \hat{e}_i}{s} \right\vert\,$$ the tensor order parameter (for a $2D$ system [@Barci_2DNematic]) $$\label{eq:lmbdaMtrx} Q_{hk}(t)=\frac{1}{N}\sum_{i=1}^{N}\left(2 e_{ih}(t)e_{ik}(t)-\delta_{hk}\right),$$ (being $h$ and $k$ equal to ${x,y}$ and $N$ the total number of squirmers) and the nematic order parameter $\lambda(t)$ being the largest eigenvalue of $Q_{hk}$ [@Eppenga1984]. We have also computed both global nematic and global polar order parameters, substituting $s$ with $N$ in the global polar order parameter (Eq. \[eq:PolarOrderCluster\]). Therefore, the polar order in steady-state is $P_{\infty} = P(t>>0)$ while the nematic order in steady-state is $\lambda_{\infty} = \lambda(t>>0)$. \[sec:Results\] Results ======================= Mean cluster size ----------------- We start with computing the mean cluster size $\langle s \rangle$ as a function of time for different values of $\beta $ and $\xi$. This allows us to distinguish coarsening from clustering, establishing, in the latter case, when the system reaches a steady state. ![\[fig:nct\_xi0\_1\]Time evolutions of the mean cluster size for squirmers when $\xi=0.1$ (pullers: red [squares]{} and black [circles]{}; neutral: blue [up triangles]{}; pushers: pink [down-triangles]{} and grey [diamonds]{}). (a) AP squirmers with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, (c) WP squirmers with $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. Note the different scale on the y-axis.](ntxi01rc25ttbis.pdf "fig:"){width="0.48\columnwidth"} ![\[fig:nct\_xi0\_1\]Time evolutions of the mean cluster size for squirmers when $\xi=0.1$ (pullers: red [squares]{} and black [circles]{}; neutral: blue [up triangles]{}; pushers: pink [down-triangles]{} and grey [diamonds]{}). (a) AP squirmers with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, (c) WP squirmers with $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. Note the different scale on the y-axis.](ntxi01rc15ttbis.pdf "fig:"){width="0.48\columnwidth"} ![\[fig:nct\_xi0\_1\]Time evolutions of the mean cluster size for squirmers when $\xi=0.1$ (pullers: red [squares]{} and black [circles]{}; neutral: blue [up triangles]{}; pushers: pink [down-triangles]{} and grey [diamonds]{}). (a) AP squirmers with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, (c) WP squirmers with $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. Note the different scale on the y-axis.](ntxi01rc25hhbis.pdf "fig:"){width="0.48\columnwidth"}   ![\[fig:nct\_xi0\_1\]Time evolutions of the mean cluster size for squirmers when $\xi=0.1$ (pullers: red [squares]{} and black [circles]{}; neutral: blue [up triangles]{}; pushers: pink [down-triangles]{} and grey [diamonds]{}). (a) AP squirmers with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, (c) WP squirmers with $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. Note the different scale on the y-axis.](ntxi01rc15hhbis.pdf "fig:"){width="0.48\columnwidth"} Fig. \[fig:nct\_xi0\_1\] represents the mean cluster size as a function of time for suspensions where attraction dominates over propulsion ($\xi=0.1$) of AP squirmers (interacting via long-range (panel-a) or short-range (panel-b) attraction) and of WP squirmers (interacting via long-range (panel-c) or short-range (panel-d) attraction). In all AP suspensions the system coarsen, within the simulated time window as shown by the continuously growing mean cluster size, with a speed dependent on $\beta$. Pullers coarsening faster than pushers, a part from the system with $\beta=-3$ where particles form clusters independently on the interaction range. WP suspensions only coarsen when interactions are long-range (panel-c) with a speed dependent on $\beta$ and pullers coarsening faster than pushers, even though coarsening is much slower than in the AP case. [In panel-c, the coarsening dynamics is so slow that $\langle s \rangle < 10 $ even when $t > 1000 \tau$ (it is worth noting that we have never observed a steady state despite having run up to $6 \times 10^6$ LB-time steps, corresponding to $\sim 8700 \tau$).]{} Comparing panel-c and panel-d, we conclude that the attraction range affects aggregation, given that when the interaction is short-range (panel-d), coarsening is suppressed and $\langle s \rangle$ reaches a steady state at long $t$ for all values of $\beta$. [A more detailed analysis is given in appendix \[Appendix:xi0\_1WPrc1\_5\] for the case of the WP system with $\xi=0.1$. ]{} Fig. \[fig:nct\_xi1\_0\] represents the mean cluster size as a function of time for suspensions where attraction competes with propulsion ($\xi=1$) of AP squirmers (interacting via long-range (panel-a) or short-range (panel-b) attraction) and of WP squirmers (interacting via long-range (panel-c) or short-range (panel-d) attraction). ![\[fig:nct\_xi1\_0\] Time evolutions of the mean cluster size for squirmers when $\xi=1$ (pullers: red [squares]{} and black [circles]{}; neutral: blue [up triangles]{}; pushers: pink [down-triangles]{} and grey [diamonds]{}). (a) AP squirmers with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, (c) WP squirmers with $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. Note the different scale on the y-axis. ](ntxi1rc25ttganso.pdf "fig:"){width="0.48\columnwidth"} ![\[fig:nct\_xi1\_0\] Time evolutions of the mean cluster size for squirmers when $\xi=1$ (pullers: red [squares]{} and black [circles]{}; neutral: blue [up triangles]{}; pushers: pink [down-triangles]{} and grey [diamonds]{}). (a) AP squirmers with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, (c) WP squirmers with $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. Note the different scale on the y-axis. ](ntxi1rc15ttganso.pdf "fig:"){width="0.48\columnwidth"} ![\[fig:nct\_xi1\_0\] Time evolutions of the mean cluster size for squirmers when $\xi=1$ (pullers: red [squares]{} and black [circles]{}; neutral: blue [up triangles]{}; pushers: pink [down-triangles]{} and grey [diamonds]{}). (a) AP squirmers with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, (c) WP squirmers with $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. Note the different scale on the y-axis. ](ntxi1rc25hhganso.pdf "fig:"){width="0.48\columnwidth"}![\[fig:nct\_xi1\_0\] Time evolutions of the mean cluster size for squirmers when $\xi=1$ (pullers: red [squares]{} and black [circles]{}; neutral: blue [up triangles]{}; pushers: pink [down-triangles]{} and grey [diamonds]{}). (a) AP squirmers with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, (c) WP squirmers with $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. Note the different scale on the y-axis. ](ntxi1rc15hhganso.pdf "fig:"){width="0.48\columnwidth"} [ AP squirmers interacting via long-range attractions (panel a) quickly reach a steady state characterized by $\langle s \rangle \approx 2$ except for $\beta=3$ where $\langle s \rangle \approx 10$ (red curve in panel-a). Whereas when dealing with AP squirmers interacting via short-range attractions (panel b), pushers reach a steady state quite quickly with $\langle s \rangle \approx 2$ as well as pullers with $\beta=3$ (with a slightly larger mean cluster size of $\langle s \rangle \approx 2.5$). Interestingly, the mean cluster size for weak pullers (black curve in panel-b) first develops a metastable state at short times ($t \in (10,500) \tau $) and then another one for longer times ($t > 1000 \tau$): the latter due to the formation of dynamic clusters with polar order. Similarly,]{} when $\beta=0$ the decay in $\langle s \rangle $ at long time corresponds to monomers aligning and moving in the same direction. WP squirmers with $\beta <3$ relatively quickly reach steady-state with clusters with an average of 4 particles (panel-c). [Whereas pullers with $\beta=3$ reach a slightly higher mean cluster size $\langle s \rangle \sim 5 $: such increase in the cluster size is observed only for this interaction range, since WP squirmers with short-range attractions (panel d) develop a ]{} mean cluster-size of $\sim 4$ particles, independently on their hydrodynamic signature. Fig. \[fig:nct\_xi10\] represents the mean cluster size as a function of time for suspensions where propulsion dominates over attraction ($\xi=10$) of AP squirmers (interacting via long-range (panel-a) or short-range (panel-b) attraction) and of WP squirmers (interacting via long-range (panel-c) or short-range (panel-d) attraction). ![\[fig:nct\_xi10\] Time evolutions of the mean cluster size for squirmers when $\xi=10$ (pullers: red [squares]{} and black [circles]{}; neutral: blue [up triangles]{}; pushers: pink [down-triangles]{} and grey [diamonds]{}). (a) AP squirmers with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, (c) WP squirmers with $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. Note the different scale on the y-axis.](ntxi10rc25ttbis.pdf "fig:"){width="0.48\columnwidth"} ![\[fig:nct\_xi10\] Time evolutions of the mean cluster size for squirmers when $\xi=10$ (pullers: red [squares]{} and black [circles]{}; neutral: blue [up triangles]{}; pushers: pink [down-triangles]{} and grey [diamonds]{}). (a) AP squirmers with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, (c) WP squirmers with $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. Note the different scale on the y-axis.](ntxi10rc15ttbis.pdf "fig:"){width="0.48\columnwidth"} ![\[fig:nct\_xi10\] Time evolutions of the mean cluster size for squirmers when $\xi=10$ (pullers: red [squares]{} and black [circles]{}; neutral: blue [up triangles]{}; pushers: pink [down-triangles]{} and grey [diamonds]{}). (a) AP squirmers with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, (c) WP squirmers with $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. Note the different scale on the y-axis.](ntxi10rc25hhbis.pdf "fig:"){width="0.48\columnwidth"}![\[fig:nct\_xi10\] Time evolutions of the mean cluster size for squirmers when $\xi=10$ (pullers: red [squares]{} and black [circles]{}; neutral: blue [up triangles]{}; pushers: pink [down-triangles]{} and grey [diamonds]{}). (a) AP squirmers with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, (c) WP squirmers with $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. Note the different scale on the y-axis.](ntxi10rc15hhbis.pdf "fig:"){width="0.48\columnwidth"} When self-propulsion dominates over attraction, squirmers in general form smaller clusters [(never larger than 4 particles on average)]{}. [ In either AP suspensions, the mean cluster size for weak pullers (black circles in panels-a and b) is larger than the one for any other $\beta$, fluctuating around a value of 3 for long times. In the case of neutral squirmers and large $\xi$ and after a long time ($t > 1000 \tau$), particles in the suspension become strongly aligned and most of the clusters are monomers, for both interaction ranges. ]{} [When dealing with WP squirmers and high $\xi$, we have observed a small value of the mean cluster size (around 2), which indicates that particles form dimers on average. A similar behavior is observed for WP squirmers with short-range interactions, with slightly larger fluctuations in the mean cluster size for weak pullers at long times.]{} Having established the conditions needed for clustering to appear, we represent in Fig. \[fig:MeanClustersBetas\] the mean cluster size as a function of $\beta$ for $r_c=2.5 \sigma$ (panel a) and $r_c=1.5 \sigma$ (panel b). [^2] ![\[fig:MeanClustersBetas\]Mean cluster size for suspensions with (a) $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$ in steady-state for WP ($\xi=10$, $\xi=1$ and $\xi=0.1$, square symbols) and AP ($\xi=10$ and $\xi=1$, circular symbols) ](meanclustersize2p5.pdf "fig:"){width="0.48\columnwidth"} ![\[fig:MeanClustersBetas\]Mean cluster size for suspensions with (a) $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$ in steady-state for WP ($\xi=10$, $\xi=1$ and $\xi=0.1$, square symbols) and AP ($\xi=10$ and $\xi=1$, circular symbols) ](meanclustersize1p5.pdf "fig:"){width="0.48\columnwidth"} [As shown in figure \[fig:MeanClustersBetas\], when $\xi=1$ WP swimmers mostly form tetramers and trimers, while WP swimmers with $\xi=10$ form dimers. The clusters observed for WP squirmers have an average size rather insensitive to the squirmer hydrodynamic signature for these two interaction strength $\xi = \{1$,$10$} and the two interaction ranges studied here.]{} [ For lower values of $\xi$, the dependence on $\beta$ becomes more evident for WP squirmers with $\xi=0.1$ (gray squares on panel-b): while for extreme values of $\beta$ (either pushers or pullers) the clusters on average form pentamers, whereas neutral and weak pullers or weak pushers are mostly form tetramers.]{} [AP squirmers with $r_c=2.5 \sigma$ (circular symbols panel-a) show that the mean cluster size increases as $\beta$ increases when $\xi=1$, whereas it does not change significantly when $\xi=10$ or when the interaction range is shorter (circular symbols panel b).]{} As summary, on the one side, comparing the two panels in Fig. \[fig:MeanClustersBetas\], when $\xi=1$ and $10$, the mean cluster size $\langle s \rangle$ for WP squirmers with interaction range of $r_c=1.5 \sigma$ behaves in the same way as the one with a larger cutoff ($r_c=2.5 \sigma$). Therefore, when attraction does not dominate, the mean cluster size for WP squirmers is independent on the interaction range. On the other side, the mean cluster size $\langle s \rangle$ for AP squirmers with $\xi=1$ and $r_c=1.5\sigma$ (red circles in figure \[fig:MeanClustersBetas\]-(b)) does not behave in the same way than when $r_c=2.5\sigma$ (red cirlces in figure \[fig:MeanClustersBetas\]-(a)), the former showing a systematic lower value of $\langle s \rangle$ for all $\beta$ while the latter a monotonic increase of the cluster size for pullers. Finally, AP squirmers with $\xi=10$ (black circles in figure \[fig:MeanClustersBetas\]) [show the same behavior varying $\beta$, independently on the attraction range.]{} With a small peak for weak pullers, and a minimum at $\beta=0$. Coarsening, clustering and aligned states ----------------------------------------- ![\[Glossary\] Snapshots of different types of self-assembly for $r_{c}=2.5 \sigma$. The attractive hemisphere is represented in blue, the repulsive in green and the fixed orientation vector is shown in red. (a), (b) and (c) are AP squirmer suspensions with $\xi=\{0.1,1,10\}$ respectively, while (d), (e) and (f) are WP squirmer suspensions with $\xi=\{0.1,1,10\}$ respectively. (a) Coarsening with $\beta=3$. (b) Clustering with $\beta=3$. (c) Gas system with polar order (polar gas) with $\beta=0$. (d) Chains system with $\beta=0$. (e) Trimers state with $\beta=-3$. (f) Gas case with nematic order (nematic gas) with $\beta=0$.](samplecoarsen.pdf "fig:"){width="0.49\columnwidth"} ![\[Glossary\] Snapshots of different types of self-assembly for $r_{c}=2.5 \sigma$. The attractive hemisphere is represented in blue, the repulsive in green and the fixed orientation vector is shown in red. (a), (b) and (c) are AP squirmer suspensions with $\xi=\{0.1,1,10\}$ respectively, while (d), (e) and (f) are WP squirmer suspensions with $\xi=\{0.1,1,10\}$ respectively. (a) Coarsening with $\beta=3$. (b) Clustering with $\beta=3$. (c) Gas system with polar order (polar gas) with $\beta=0$. (d) Chains system with $\beta=0$. (e) Trimers state with $\beta=-3$. (f) Gas case with nematic order (nematic gas) with $\beta=0$.](samplechains.pdf "fig:"){width="0.49\columnwidth"} ![\[Glossary\] Snapshots of different types of self-assembly for $r_{c}=2.5 \sigma$. The attractive hemisphere is represented in blue, the repulsive in green and the fixed orientation vector is shown in red. (a), (b) and (c) are AP squirmer suspensions with $\xi=\{0.1,1,10\}$ respectively, while (d), (e) and (f) are WP squirmer suspensions with $\xi=\{0.1,1,10\}$ respectively. (a) Coarsening with $\beta=3$. (b) Clustering with $\beta=3$. (c) Gas system with polar order (polar gas) with $\beta=0$. (d) Chains system with $\beta=0$. (e) Trimers state with $\beta=-3$. (f) Gas case with nematic order (nematic gas) with $\beta=0$.](samplecluster.pdf "fig:"){width="0.49\columnwidth"} ![\[Glossary\] Snapshots of different types of self-assembly for $r_{c}=2.5 \sigma$. The attractive hemisphere is represented in blue, the repulsive in green and the fixed orientation vector is shown in red. (a), (b) and (c) are AP squirmer suspensions with $\xi=\{0.1,1,10\}$ respectively, while (d), (e) and (f) are WP squirmer suspensions with $\xi=\{0.1,1,10\}$ respectively. (a) Coarsening with $\beta=3$. (b) Clustering with $\beta=3$. (c) Gas system with polar order (polar gas) with $\beta=0$. (d) Chains system with $\beta=0$. (e) Trimers state with $\beta=-3$. (f) Gas case with nematic order (nematic gas) with $\beta=0$.](sampletrimers.pdf "fig:"){width="0.49\columnwidth"} ![\[Glossary\] Snapshots of different types of self-assembly for $r_{c}=2.5 \sigma$. The attractive hemisphere is represented in blue, the repulsive in green and the fixed orientation vector is shown in red. (a), (b) and (c) are AP squirmer suspensions with $\xi=\{0.1,1,10\}$ respectively, while (d), (e) and (f) are WP squirmer suspensions with $\xi=\{0.1,1,10\}$ respectively. (a) Coarsening with $\beta=3$. (b) Clustering with $\beta=3$. (c) Gas system with polar order (polar gas) with $\beta=0$. (d) Chains system with $\beta=0$. (e) Trimers state with $\beta=-3$. (f) Gas case with nematic order (nematic gas) with $\beta=0$.](samplepolar.pdf "fig:"){width="0.49\columnwidth"} ![\[Glossary\] Snapshots of different types of self-assembly for $r_{c}=2.5 \sigma$. The attractive hemisphere is represented in blue, the repulsive in green and the fixed orientation vector is shown in red. (a), (b) and (c) are AP squirmer suspensions with $\xi=\{0.1,1,10\}$ respectively, while (d), (e) and (f) are WP squirmer suspensions with $\xi=\{0.1,1,10\}$ respectively. (a) Coarsening with $\beta=3$. (b) Clustering with $\beta=3$. (c) Gas system with polar order (polar gas) with $\beta=0$. (d) Chains system with $\beta=0$. (e) Trimers state with $\beta=-3$. (f) Gas case with nematic order (nematic gas) with $\beta=0$.](samplenematic.pdf "fig:"){width="0.49\columnwidth"} Fig. \[Glossary\] displays the different states, types of collective motion or self-assembly that WP and AP Janus squirmers form as $\beta$ and $\xi$ vary. This way to classified our simulations, allow us to observe in a better way all the results. We have identified seven different cases: 3 gas systems, 3 clustering systems and 1 coarsening state. All the different cases, show different types of cluster morphologies and/or particles’ alignment. The emerging structures result to be sensitive to the patch direction (WP or AP), the hydrodynamic signature $\beta$ and the interaction range $r_{c}$ and strength $\xi$. The three gas systems are characterized by the different particles’ alignment: isotropic (with particles randomly swimming), polar (with most of the particles swimming in the same direction) and nematic (with approximately half of the total amount of particles swimming in one direction and the other half in the opposite direction). The three clustering systems can be classified as: finite-size dynamic clusters, chains and trimers. In the coarsening case all particles form a unique macroscopic aggregate. AP Janus squirmers coarsen into a single cluster for $\xi \approx 0.1$, (see Fig. \[Glossary\]-(a)) and self-organise into finite size clusters for $\xi \approx 1$ (see Fig. \[Glossary\]-(b)) (larger values of $\beta$ favoring the formation of larger finite-size clusters, accelerating coarsening). When $\xi=10$ , AP squirmers either form a polar ordered gas (see Fig. \[Glossary\]-(c), for weak positive stresslets, $0<\beta<1$) or an isotropic gas (for AP pushers and pullers with $\beta > 1$). WP Janus squirmers have a strong tendency to form small micelles of three or four particles pointing at each other. For $\xi \approx 0.1$ these micelles coalesce and end up forming chains, which in most of the cases coexist with smaller micelles (see Fig. \[Glossary\]-(d)). At $\xi=1$ the competition with the active stress ($\beta$) gives rise to trimers and tetramers avoiding the formation of larger chains of particles (see Fig. \[Glossary\]-(e)). If activity dominates ($\xi=10$) and $|\beta|<1$, even though a polar order is not favored due to the anisotropic interactions, we still observe a global nematic order: a significant portion of particles move in one direction and the rest in the opposite one (see Fig. \[Glossary\]-(f)). For other values of $\beta$, we observe an isotropic system. [In the Supplemental Material [@Supp] we have included the movies of the six simulations represented in Fig. \[Glossary\], to better capture both the dynamical and morphological features reported in this paper.]{} Cluster size distribution ------------------------- Additionally to the mean cluster size, we calculate the cluster size distribution (CSD) to study the statistics of the different cluster sizes found it for Janus swimmers, computing the CSD for $\xi=1$ and $\xi=10$ and as a function of $\beta$, the direction of the attractive patch and the interaction range. [The case when attraction dominates with respect to self-propulsion ($\xi=0.1$) for AP squirmers is analyzed in appendix \[Appendix:xi0\_1WPrc1\_5\].]{} Results for $\xi=1$ are presented in Fig. \[fig:csd\_xi1\] and for $\xi=10$ in Fig. \[fig:csd\_xi10\]. Following previous studies [@SoftMatter17], where CSD was calculated, we use the same analytical function as a reference. $$\label{eq:csd_cutoff} \frac{f(s)}{f(1)}= A~\frac{\exp(-(s-1)/s_0)}{s^{\gamma_0}}+B~\frac{\exp(-(s-1)/z_0)}{s^{-\gamma_0}},$$ with $\gamma_0$, $s_0$, $z_0$ and $B$ constants such that $A=1-B$. ![\[fig:csd\_xi1\] CSD for $\xi=1$. AP squirmers (a) with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$. WP squirmers (c) with $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. Gray dashed line in (a) and (b) represents the analytical function for a CSD given by Eq. \[eq:csd\_cutoff\] with $B=0$, plotted as guide to the eye, with $s_0=4$, $\gamma_0=5/4$. In (c) and (d) a vertical dashed line is shown as a reference for $s=3$ (trimers). ](pdfxi1rc25ttganso.pdf "fig:"){width="0.48\columnwidth"}![\[fig:csd\_xi1\] CSD for $\xi=1$. AP squirmers (a) with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$. WP squirmers (c) with $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. Gray dashed line in (a) and (b) represents the analytical function for a CSD given by Eq. \[eq:csd\_cutoff\] with $B=0$, plotted as guide to the eye, with $s_0=4$, $\gamma_0=5/4$. In (c) and (d) a vertical dashed line is shown as a reference for $s=3$ (trimers). ](pdfxi1rc15ttganso.pdf "fig:"){width="0.48\columnwidth"} ![\[fig:csd\_xi1\] CSD for $\xi=1$. AP squirmers (a) with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$. WP squirmers (c) with $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. Gray dashed line in (a) and (b) represents the analytical function for a CSD given by Eq. \[eq:csd\_cutoff\] with $B=0$, plotted as guide to the eye, with $s_0=4$, $\gamma_0=5/4$. In (c) and (d) a vertical dashed line is shown as a reference for $s=3$ (trimers). ](pdfxi1rc25hhganso.pdf "fig:"){width="0.48\columnwidth"} ![\[fig:csd\_xi1\] CSD for $\xi=1$. AP squirmers (a) with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$. WP squirmers (c) with $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. Gray dashed line in (a) and (b) represents the analytical function for a CSD given by Eq. \[eq:csd\_cutoff\] with $B=0$, plotted as guide to the eye, with $s_0=4$, $\gamma_0=5/4$. In (c) and (d) a vertical dashed line is shown as a reference for $s=3$ (trimers). ](pdfxi1rc15hhganso.pdf "fig:"){width="0.48\columnwidth"} Fig. \[fig:csd\_xi1\]-(a) represents the CSDs of AP squirmers for $\xi=1$ and $r_c=2.5 \sigma$. The characteristic distribution width grows with $\beta$ until $\beta < 3$. For all the explored parameters, monomers always represent the largest contribution. For large enough values of $\beta$, $\beta = 3$, a second peak emerges for cluster sizes involving a few tens of particles (around 25 for the system parameters explored). Accordingly, CSDs with $\beta < 3$ can be described by Eq. \[eq:csd\_cutoff\], with $B=0$ while the CSD for $\beta=3$ requires a non-vanishing value of $B$ in Eq. \[eq:csd\_cutoff\] to capture the observed maximum. The width of the CSDs is measured by the value of the exponential tail parameter $s_0$, thus pushers CSDs have smaller $s_0$, whereas the corresponding $s_0$ for pullers is larger until $\beta=3$. At larger values of $\beta$ the behavior changes due to the appearance of a second peak in the CSDs. Fig. \[fig:csd\_xi1\]-(b) represents the CSDs of AP squirmers for $\xi=1$ and $r_c=1.5 \sigma$. [CSDs of AP pushers and neutral squirmers follow the analytical behavior of a power law with an exponential tail (Eq. \[eq:csd\_cutoff\] with $B=0$: gray dashed line) with the same width. The squirmers with $\beta=3$ present a CSD wider than the pushers ones, whereas weak pullers CSD follows a power law behavior instead of the power law with an exponential tail (Eq. \[eq:csd\_cutoff\]) which represents clusters of all sizes, such effect is not observed when interaction range is $2.5 \sigma$.]{} Fig. \[fig:csd\_xi1\]-(c) represents the CSDs of WP squirmers for $\xi=1$ and $r_c=2.5 \sigma$. A peak appears between trimers and tetramers, while the characteristic CSD width increases with the active stress $\beta$. Interestingly, when $\beta < 3$ monomers are not observed. Strong WP pushers are characterized by a suspension dominated by trimers, while strong WP pullers form a polydisperse suspension of monomers, dimers, trimers and even chains of tens of particles. Fig. \[fig:csd\_xi1\]-(d) represents the CSDs of WP squirmers for $\xi=1$ and $r_c=1.5 \sigma$, with CSDs presenting a peak between trimers and tetramers. In this case, the CSD width depends on the active stress magnitude rather than in its sign: the higher the value of $|\beta|$, the wider the distribution (with distributions wider than the ones with longer interactions range $r_c=2.5\sigma$, and with a higher number of monomers especially when $\beta=3$ ). Therefore, CSDs present relevant differences depending on whether squirmers are AP or WP-like and on the interaction range of the potential. The CSDs for AP squirmers decay monotonously as a function of cluster size, with a large tail corresponding to cases in which there is the formation of finite-size aggregates. On the contrary, CSDs for WP usually display a peak for trimers. Moreover, the interaction range affects how the CSDs width depends on the hydrodynamic signature: while for large interaction range, strong AP pullers develop a wider distribution than any other $\beta$, for shorter interaction range [weak pullers, develop a power law distribution]{}. In contrast, neither the range of interaction among WP squirmers nor the hydrodynamic signature affect the shape of the CSDs, since CSDs with a peak on trimers is present in all cases, with the widest distribution appearing for the stronger WP or AP pullers. Fig. \[fig:csd\_xi10\]-(a) represents the CSDs of AP squirmers with $\xi=10$ and $r_c=2.5 \sigma$. The corresponding distributions are monotonically decreasing, particularly for $\beta=0.5$ [a power law distribution emerges while in the other cases,]{} the CSD can be approximated by Eq. \[eq:csd\_cutoff\] with $B \neq 0$. ![\[fig:csd\_xi10\] CSD for $\xi=10$. AP squirmers (a) with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$. WP squirmers (c) with $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. Grey dashed line in (a) and (b) represents the analytical function for a CSD given by Eq. \[eq:csd\_cutoff\] with $B=0$, plotted as guide to the eye, with $s_0=4$, $\gamma_0=5/4$.](pdfxi10rc25ttbis.pdf "fig:"){width="0.48\columnwidth"}![\[fig:csd\_xi10\] CSD for $\xi=10$. AP squirmers (a) with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$. WP squirmers (c) with $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. Grey dashed line in (a) and (b) represents the analytical function for a CSD given by Eq. \[eq:csd\_cutoff\] with $B=0$, plotted as guide to the eye, with $s_0=4$, $\gamma_0=5/4$.](pdfxi10rc15ttbis.pdf "fig:"){width="0.48\columnwidth"} ![\[fig:csd\_xi10\] CSD for $\xi=10$. AP squirmers (a) with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$. WP squirmers (c) with $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. Grey dashed line in (a) and (b) represents the analytical function for a CSD given by Eq. \[eq:csd\_cutoff\] with $B=0$, plotted as guide to the eye, with $s_0=4$, $\gamma_0=5/4$.](pdfxi10rc25hhbis.pdf "fig:"){width="0.48\columnwidth"}![\[fig:csd\_xi10\] CSD for $\xi=10$. AP squirmers (a) with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$. WP squirmers (c) with $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. Grey dashed line in (a) and (b) represents the analytical function for a CSD given by Eq. \[eq:csd\_cutoff\] with $B=0$, plotted as guide to the eye, with $s_0=4$, $\gamma_0=5/4$.](pdfxi10rc15hhbis.pdf "fig:"){width="0.48\columnwidth"} Fig. \[fig:csd\_xi10\]-(b) displays AP squirmers’ CSDs for $\xi=10$ and $r_c=1.5 \sigma$. The CSDs are essentially the same as in the previous case with longer interaction range for pushers and $\beta=0$, while for pullers the CSDs are lightly wider. Fig. \[fig:csd\_xi10\]-(c) displays the CSDs for WP squirmers for $\xi=10$ and $r_c=2.5 \sigma$. In this case, hydrodynamics does not significantly affect the CSD, since the CSDs are all monotonically decreasing. Although $\beta=3$ is slightly wider and $\beta=-3$ is slightly narrower than the others, the distributions are similar among them, with approximately the same width. Since activity dominates over the interaction strength, we can infer that this independence on $\beta$ is due to two main effects: on the one hand, the anisotropy of the potential and on the other hand, the range of the interaction. Since in Fig. \[fig:csd\_xi10\]-(d) for WP squirmers and $\xi=10$ and $r_c=1.5 \sigma$, the CSDs have indeed a different width depending on the hydrodynamic signature, corresponding $\beta=0.5$ to the widest distribution. When activity dominates over the interaction strength, we can infer that CSDs depend mainly on hydrodynamics, but the interaction range plays an important role too, mainly for AP and WP pullers. Radius of gyration ------------------ The next feature we compute to study the morphology of the clusters is their radius of gyration $Rg(s)$ (Eq. \[eq:Rg\]), as a function of the cluster size $s$. ![\[fig:Rg\_xi1\] Radius of gyration normalized by the particle radius as a function of the cluster size for suspensions with $\xi=1$. Top panels are AP squirmers (a) with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, whereas bottom panels are WP squirmers (c) with $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. Grey dashed lines in all panels are the function $R_g = K s^{0.64}$.](rgsxi1rc25ttganso.pdf "fig:"){width="0.48\columnwidth"}![\[fig:Rg\_xi1\] Radius of gyration normalized by the particle radius as a function of the cluster size for suspensions with $\xi=1$. Top panels are AP squirmers (a) with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, whereas bottom panels are WP squirmers (c) with $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. Grey dashed lines in all panels are the function $R_g = K s^{0.64}$.](rgsxi1rc15ttganso.pdf "fig:"){width="0.48\columnwidth"} ![\[fig:Rg\_xi1\] Radius of gyration normalized by the particle radius as a function of the cluster size for suspensions with $\xi=1$. Top panels are AP squirmers (a) with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, whereas bottom panels are WP squirmers (c) with $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. Grey dashed lines in all panels are the function $R_g = K s^{0.64}$.](rgsxi1rc25hhganso.pdf "fig:"){width="0.48\columnwidth"}![\[fig:Rg\_xi1\] Radius of gyration normalized by the particle radius as a function of the cluster size for suspensions with $\xi=1$. Top panels are AP squirmers (a) with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, whereas bottom panels are WP squirmers (c) with $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. Grey dashed lines in all panels are the function $R_g = K s^{0.64}$.](rgsxi1rc15hhganso.pdf "fig:"){width="0.48\columnwidth"} Fig. \[fig:Rg\_xi1\]-(a) displays the radius of gyration $R_g(s)$ of AP squirmers with $\xi=1$ and $r_c=2.5\sigma$. We observe that $R_g\sim s^{0.64}$ for $\beta <3$, while for $\beta=3$ (red squares in Fig. \[fig:Rg\_xi1\]-(a)), $R_g$ starts following the same power law before a crossover to a lower value of $R_g$. It is insightful to compare these results with the CSDs, since AP squirmers with $\beta < 3$ have monomodal CSDs, with the width of the distribution depending on the value of $\beta$. Nonetheless, Fig. \[fig:Rg\_xi1\]-(a) shows that clusters morphology is independent of the hydrodynamic signature when $\beta<3$, but once $\beta=3$ the CSD is bimodal and it turns out that the change in $R_g$ coincides with the changes in the CSD. Moreover, dynamic clusters are observed when $\beta < 3$ while less dynamic ones are formed when $\beta=3$: therefore, attractive interaction is enhanced when particles are strong pullers. Similarly, $R_g\sim s^{0.64}$ in Fig. \[fig:Rg\_xi1\]-(b), when the interaction range is short and for small clusters ($s<50$); while for larger clusters of pullers ($s>50$) the $R_g$ power law is smaller than $0.64$. The slower growth of $R_g$ depends on the fact that the clusters’ structures are mostly controlled by hydrodynamics. For WP squirmers, $R_g$ shows a richer behavior: particles first aggregate in trimers and then organize in chain-like structures. As in Fig. \[fig:Rg\_xi1\]-(c), $R_g$ presents a crossover to a different asymptotic regime. Chains are observed only for WP independently of $\beta$; hence, chain formation is driven by the anisotropic interaction and not by hydrodynamics, and is rather insensitive to the range of the potential, as in Fig. \[fig:Rg\_xi1\]-(d). ![\[fig:Rg\_xi10\] Radius of gyration normalized by the particle radius as a function of the cluster size for suspensions with $\xi=10$. Top panels are AP squirmers (a) with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, whereas bottom panels are WP squirmers (c) with $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. Grey dashed lines in all panels are the function $R_g = K s^{0.64}$ and most of the curves fit with this power law.](rgsxi10rc25ttbis.pdf "fig:"){width="0.48\columnwidth"}![\[fig:Rg\_xi10\] Radius of gyration normalized by the particle radius as a function of the cluster size for suspensions with $\xi=10$. Top panels are AP squirmers (a) with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, whereas bottom panels are WP squirmers (c) with $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. Grey dashed lines in all panels are the function $R_g = K s^{0.64}$ and most of the curves fit with this power law.](rgsxi10rc15ttbis.pdf "fig:"){width="0.48\columnwidth"} ![\[fig:Rg\_xi10\] Radius of gyration normalized by the particle radius as a function of the cluster size for suspensions with $\xi=10$. Top panels are AP squirmers (a) with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, whereas bottom panels are WP squirmers (c) with $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. Grey dashed lines in all panels are the function $R_g = K s^{0.64}$ and most of the curves fit with this power law.](rgsxi10rc25hhbis.pdf "fig:"){width="0.48\columnwidth"}![\[fig:Rg\_xi10\] Radius of gyration normalized by the particle radius as a function of the cluster size for suspensions with $\xi=10$. Top panels are AP squirmers (a) with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, whereas bottom panels are WP squirmers (c) with $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. Grey dashed lines in all panels are the function $R_g = K s^{0.64}$ and most of the curves fit with this power law.](rgsxi10rc15hhbis.pdf "fig:"){width="0.48\columnwidth"} When self-propulsion dominates over attraction, $R_g=ks^{0.64}$ for small clusters of AP squirmers independently on $\beta$, while for large clusters of pullers $R_g$ has a smaller exponent Fig. \[fig:Rg\_xi10\]-(a). The same happens when the attraction range is $1.5 \sigma$, Fig. \[fig:Rg\_xi10\]-(b). In Fig. \[fig:Rg\_xi10\]-(c) we report the radius of gyration for the WP squirmers when $r_c=2.5 \sigma$. Given that activity dominates over attraction, particles do not assembly as chains: $R_g$ does not present the cross-over as in Fig. \[fig:Rg\_xi1\]-(c), but rather $R_g \sim s^{0.64}$ for all $s$. Moreover, the clusters’ morphology is the same despite of the active stress $\beta$. However, when the interaction range is reduced to $r_c=1.5 \sigma$, as in Fig. \[fig:Rg\_xi10\]-(d), clusters of pullers are more compact when larger than 20 particles. Local polar and nematic order ----------------------------- We now analyze the degree of alignment as a function of the cluster size, computing in Fig. \[fig:Ps\_xi1\] the polar order $P(s)$ when $\xi=1$ using Eq. \[eq:PolarOrderCluster\]. In general, the polar order inside a cluster usually decreases with cluster size as a power law [@SoftMatter17]. ![\[fig:Ps\_xi1\] Polar order as a function of cluster size for $\xi=1$. Top panels: AP squirmers with (a) $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, whereas bottom panels are WP squirmers with (c) $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. ](psxi1rc25ttganso.pdf "fig:"){width="0.48\columnwidth"}![\[fig:Ps\_xi1\] Polar order as a function of cluster size for $\xi=1$. Top panels: AP squirmers with (a) $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, whereas bottom panels are WP squirmers with (c) $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. ](psxi1rc15ttganso.pdf "fig:"){width="0.48\columnwidth"} ![\[fig:Ps\_xi1\] Polar order as a function of cluster size for $\xi=1$. Top panels: AP squirmers with (a) $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, whereas bottom panels are WP squirmers with (c) $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. ](psxi1rc25hhganso.pdf "fig:"){width="0.48\columnwidth"}![\[fig:Ps\_xi1\] Polar order as a function of cluster size for $\xi=1$. Top panels: AP squirmers with (a) $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, whereas bottom panels are WP squirmers with (c) $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. ](psxi1rc15hhganso.pdf "fig:"){width="0.48\columnwidth"} As reported in Fig. \[fig:Ps\_xi1\]-(a), increasing $|\beta|$ leads to a faster decay of the local polar order for AP squirmers with $\xi=1$ and $r_c=2.5 \sigma$. When decreasing the interaction range, we observe a stronger dependence of the polar order with the hydrodynamic signature: [clusters of neutral squirmers (blue triangles in Fig. \[fig:Ps\_xi1\]-(b)) are practically swimming in the same direction, while clusters of weak-pullers (black circles in Fig. \[fig:Ps\_xi1\]-(b)) show different behavior depending on their size: for small and medium size (up to $100$ particles) the polar order decays with size monotonically; the polar order has a crossover for larger clusters, decreasing faster with the cluster size. In general,]{} weak pullers present higher degree of polar order than pushers and strong pullers behave in the same way as the corresponding pushers, (with $P(s)$ decaying with the cluster-size monotonically). However, the results of the local polar order for WP clusters are quite different. Given that squirmers form elongated clusters from the assembly of trimers, the local order rapidly vanishes as the cluster size grows (Fig. \[fig:Ps\_xi1\]-(c)). This tendency is less marked for short range anisotropic interactions and large $\beta$ (strong pullers in Fig. \[fig:Ps\_xi1\]-(d) (red squares)). In Fig. \[fig:Ps\_xi10\] we report the polar order as a function of the cluster size for both AP and WP squirmers when self-propulsion dominates over attraction ($\xi=10$). ![\[fig:Ps\_xi10\]Polar order as a function of cluster size for $\xi=10$. Top panels: AP squirmers with (a) $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, whereas bottom panels are WP squirmers with (c) $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. ](psxi10rc25ttbis.pdf "fig:"){width="0.48\columnwidth"}![\[fig:Ps\_xi10\]Polar order as a function of cluster size for $\xi=10$. Top panels: AP squirmers with (a) $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, whereas bottom panels are WP squirmers with (c) $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. ](psxi10rc15ttbis.pdf "fig:"){width="0.48\columnwidth"} ![\[fig:Ps\_xi10\]Polar order as a function of cluster size for $\xi=10$. Top panels: AP squirmers with (a) $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, whereas bottom panels are WP squirmers with (c) $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. ](psxi10rc25hhbis.pdf "fig:"){width="0.48\columnwidth"}![\[fig:Ps\_xi10\]Polar order as a function of cluster size for $\xi=10$. Top panels: AP squirmers with (a) $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, whereas bottom panels are WP squirmers with (c) $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. ](psxi10rc15hhbis.pdf "fig:"){width="0.48\columnwidth"} For AP squirmers, $P(s)$ decays faster for pushers than for pullers, independently on the interaction range (panels a and b of Fig. \[fig:Ps\_xi10\]). The degree of polar order is more sensitive to $\beta$ for pullers than for pushers; weak puller’s clusters are more aligned than clusters of stronger pullers, while neutral squirmer’s clusters are the most aligned ones for all cluster sizes. Panels c and d of Fig. \[fig:Ps\_xi10\] report the polar order of clusters of WP squirmers, with a monotonous decay of the polar order with cluster size, weakly dependent on $\beta$ and on the interaction range (except for $\beta=0.5$). In Fig. \[fig:Ns\_xi1\] we report results for the nematic order $\lambda(s)$. ![\[fig:Ns\_xi1\] Nematic order as a function of cluster size when $\xi=1$. Top panels represent results for AP squirmers: (a) with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, whereas bottom panels are WP squirmers (c) with $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. ](nsxi1rc25ttganso.pdf "fig:"){width="0.48\columnwidth"}![\[fig:Ns\_xi1\] Nematic order as a function of cluster size when $\xi=1$. Top panels represent results for AP squirmers: (a) with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, whereas bottom panels are WP squirmers (c) with $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. ](nsxi1rc15ttganso.pdf "fig:"){width="0.48\columnwidth"} ![\[fig:Ns\_xi1\] Nematic order as a function of cluster size when $\xi=1$. Top panels represent results for AP squirmers: (a) with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, whereas bottom panels are WP squirmers (c) with $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. ](nsxi1rc25hhganso.pdf "fig:"){width="0.48\columnwidth"}![\[fig:Ns\_xi1\] Nematic order as a function of cluster size when $\xi=1$. Top panels represent results for AP squirmers: (a) with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, whereas bottom panels are WP squirmers (c) with $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. ](nsxi1rc15hhganso.pdf "fig:"){width="0.48\columnwidth"} The local nematic order for AP squirmers with $\xi=1$ decays with the cluster size $s$ independently on $\beta$ for $\beta < 3$ while presents an even faster decay when $\beta=3$ (see Fig. \[fig:Ns\_xi1\]-(a)). When $r_c=1.5\sigma$, $\lambda(s)$ decays with $s$ in the same way as $P(s)$ does for all $\beta$. Neutral squirmers are completely polarized (as also shown in the nematic order, blue triangles in Fig. \[fig:Ns\_xi1\]-(b)). The high polar order observed in clusters of puller and neutral squirmers is also detected by the nematic order. The behaviour of the local nematic order for WP clusters with $\xi=1$ is quite striking. The reason why $\lambda(1)=\lambda(2)=1$ is that when the interaction range is $2.5\sigma$, once two particles collide they are kept together by the attractive patch. Particles in trimers and tetramers point towards the center of the cluster and are kept together by the attractive patch, reason why they have a small nematic order (all curves in Fig. \[fig:Ns\_xi1\]-(c)). For larger clusters, active stresses become more important: hydrodynamics attract trimers so that particles’ attractive patches reorient in the clusters forming aligned chains. This does not happen for strong pullers, where the nematic order remains low and constant (red squares in Fig. \[fig:Ns\_xi1\]-(c)), due to the fact that trimers form disordered chains (made of trimers formed by hydrodynamics). When the interaction range is $1.5 \sigma$, the local nematic order behaves in the same way as for $2.5 \sigma$, but in this case oriented chains are observed only for strong pushers, whereas the rest of[ squirmers form less oriented chains with a value of $\lambda(s)$ in between.]{} In Fig. \[fig:Ns\_xi10\] we report the local nematic order $\lambda(s)$ for $\xi=10$ and AP squirmers panels a and b. In clusters made of neutral squirmers (blue triangles), particles swim in the same direction independently on the interaction range: $\lambda(s) \approx 1$ for all cluster sizes. The nematic order for all other squirmers decays in the same way as the polar order. ![\[fig:Ns\_xi10\]Nematic order as a function of cluster size for suspensions with $\xi=10$. **(a):** AP squirmers with $r_c=2.5\sigma$. **(b):** AP squirmers with $r_c=1.5\sigma$. **(c):** WP squirmers with $r_c=2.5\sigma$. **(d):** WP squirmers $r_c=1.5\sigma$.](nsxi10rc25ttbis.pdf "fig:"){width="0.48\columnwidth"}![\[fig:Ns\_xi10\]Nematic order as a function of cluster size for suspensions with $\xi=10$. **(a):** AP squirmers with $r_c=2.5\sigma$. **(b):** AP squirmers with $r_c=1.5\sigma$. **(c):** WP squirmers with $r_c=2.5\sigma$. **(d):** WP squirmers $r_c=1.5\sigma$.](nsxi10rc15ttbis.pdf "fig:"){width="0.48\columnwidth"} ![\[fig:Ns\_xi10\]Nematic order as a function of cluster size for suspensions with $\xi=10$. **(a):** AP squirmers with $r_c=2.5\sigma$. **(b):** AP squirmers with $r_c=1.5\sigma$. **(c):** WP squirmers with $r_c=2.5\sigma$. **(d):** WP squirmers $r_c=1.5\sigma$.](nsxi10rc25hhbis.pdf "fig:"){width="0.48\columnwidth"}![\[fig:Ns\_xi10\]Nematic order as a function of cluster size for suspensions with $\xi=10$. **(a):** AP squirmers with $r_c=2.5\sigma$. **(b):** AP squirmers with $r_c=1.5\sigma$. **(c):** WP squirmers with $r_c=2.5\sigma$. **(d):** WP squirmers $r_c=1.5\sigma$.](nsxi10rc15hhbis.pdf "fig:"){width="0.48\columnwidth"} To conclude, in the case of clusters of AP squirmers, alignment is not affected by the interaction range. WP clusters with $\xi=10$ and long range interactions develop a high nematic order for neutral and weak pushers ($\beta=0$ and $-0.5$) as shown in panel c of Fig. \[fig:Ns\_xi10\]. This nematic order is not due to a global polar order, (being very low) but to the competition between the anisotropic potential and the hydrodynamic signature: even though activity dominates over attraction, the interaction range is long enough to allow for the development of a nematic order in the system. Undoubtedly, the interaction range is important to develop clusters with nematic order, since $\lambda(s)$ for WP squirmers with short range [has a different behavior, where the nematic order only shows a non-zero value due to the polar order shown before. ]{} Global polar and nematic order ------------------------------ To study the orientational order of the system, we compute as in [@alarcon2013] the global polar order parameter and the nematic order tensor Eq. \[eq:lmbdaMtrx\] [@Barci_2DNematic], at steady-state at long times. In Fig. \[fig:PolarNematicGlobal\]-(a) we represent the global polar order $P_{\infty}$ for AP squirmers in steady state for $\xi=1$ and $\xi=10$ and both interaction ranges. ![\[fig:PolarNematicGlobal\] Global polar $P_{\infty}$ and nematic $\lambda_{\infty}$ order parameters. Top panels represent results for AP squirmers: (a) polar order parameter $P_{\infty}$, (b) nematic order parameter $\lambda_{\infty}$. Bottom panels represent results for WP squirmers: (c) polar order parameter $P_{\infty}$, (d) nematic order $\lambda_{\infty}$.](globalpolartt.pdf "fig:"){width="0.48\columnwidth"} ![\[fig:PolarNematicGlobal\] Global polar $P_{\infty}$ and nematic $\lambda_{\infty}$ order parameters. Top panels represent results for AP squirmers: (a) polar order parameter $P_{\infty}$, (b) nematic order parameter $\lambda_{\infty}$. Bottom panels represent results for WP squirmers: (c) polar order parameter $P_{\infty}$, (d) nematic order $\lambda_{\infty}$.](globalnematictt.pdf "fig:"){width="0.48\columnwidth"} ![\[fig:PolarNematicGlobal\] Global polar $P_{\infty}$ and nematic $\lambda_{\infty}$ order parameters. Top panels represent results for AP squirmers: (a) polar order parameter $P_{\infty}$, (b) nematic order parameter $\lambda_{\infty}$. Bottom panels represent results for WP squirmers: (c) polar order parameter $P_{\infty}$, (d) nematic order $\lambda_{\infty}$.](globalpolarhh.pdf "fig:"){width="0.48\columnwidth"} ![\[fig:PolarNematicGlobal\] Global polar $P_{\infty}$ and nematic $\lambda_{\infty}$ order parameters. Top panels represent results for AP squirmers: (a) polar order parameter $P_{\infty}$, (b) nematic order parameter $\lambda_{\infty}$. Bottom panels represent results for WP squirmers: (c) polar order parameter $P_{\infty}$, (d) nematic order $\lambda_{\infty}$.](globalnematichh.pdf "fig:"){width="0.48\columnwidth"} When $\xi=10$, activity dominates and the polar order parameter has a non-zero value for weak pullers ($0 \ge \beta \ge 1$), similarly to the 3D case without attractive interactions [@alarcon2013]. When $\xi=1$, the range of interaction affects to the formation of aligned states: for $2.5 \sigma$ the system is completely isotropic despite of the value of $\beta$ (red circles in Fig. \[fig:PolarNematicGlobal\]-(a)), whereas for $r_c=1.5 \sigma$ the system still has polar order for weak pullers (black circles), as with $\xi=10$. The Fig. \[fig:PolarNematicGlobal\]-(b), shows that the nematic order $\lambda_{\infty}$ follows the polar order: for AP squirmers there is not an additional orientation besides the polar order. In Fig. \[fig:PolarNematicGlobal\]-(c) we represent the global polar order $P_{\infty}$ for WP squirmers in steady state for $\xi=1$ and $\xi=10$ and both interaction ranges. For strongly anisotropic interactions ($\xi=1$) the suspension does not develop any significant degree of polar ordering. For weakly interacting WP squirmers ($\xi=10$) and $r_c=1.5 \sigma$, the polar order emerges in the same range of $\beta$’s (black squares) as the AP squirmers. In Fig. \[fig:PolarNematicGlobal\]-(d) we report the nematic order parameter $\lambda_{\infty}$. When $\xi=1$ WP squirmers do not develop any nematic order (red and black circles): either polar or nematic order are absent for WP squirmers with strongly anisotropic interactions. A non-zero nematic order is observed for $\xi=10$, when $r_c=2.5 \sigma$ and $r_c = 1.5 \sigma$: in the latter, the nematic order is due to the polar order (black squares), while in the former mid-range interactions create a new orientational order (red squares). Interestingly, the nematic order parameter for WP squirmers with weak interactions shows a non-zero nematic order when the polar order parameter is zero. \[sec:Conclusions\] Discussion and Conclusions ============================================== In this paper, we report a numerical study of the self-assembly of Janus squirmers in a quasi two dimensional semi-dilute suspension. [Even though we are aware of the fact that experimental catalytic active colloids might present other types of interactions, like phoretic ones, the nature of such interactions is going to depend on the particular system under study [@JCP_Benno2019]. As in the case of active dipolar Janus particles [@Dipolar_Granick] where electrophoresis is effectivelly modelled by the interaction between two imbalanced charges on each particle, in our work we suggest that amphiphilic active Janus colloids [@Gao_OTSJanusExp] can be modelled by means of an effective anisotropic interaction. For this 2D system, we study the effect of the hydrodynamic interactions in the self-assembly of such particles, hopefully shedding some light on experiments with active Janus colloids where the fluid flow field can be measured [@Campbell_FlowFieldJanus] or in experiments where the self-propulsion direction can be tuned [@Brown_Poon_SaltJanus].]{} In a previous work, we have studied aggregation in a semi-dilute quasi two dimensional suspension of isotropically attractive squirmers [@SoftMatter17], demonstrating that both clusters’ morphology and alignment within clusters depended on particles’ interaction strength and hydrodynamic signature. In the current work, by changing the symmetry of the inter-particle interaction (from isotropic to anisotropic) we have observed that the interaction anisotropy, [modulated by its range and strength]{}, compete with the active stress [to re-orient the particles and therefore,]{} giving rise to a rich aggregation phenomenology. [The study of two different interaction ranges was originated from previous results [@SoftMatter17; @SoftMatter_43; @SoftMatter_26; @SoftMatter_27; @Fillion_AABP_3D] where $r_c=2.5 \sigma$ has been used for isotropic potentials, while in [@Stewart_17; @pre09; @Pu_ActiveJanus2017] an interaction range of $r_c=1.5 \sigma$ is used for anisotropic potentials. Both cutoff values are typical in the context of attractive Brownian colloids either passive or active.]{} As shown in all studied cases, the amount of activity versus attraction has a strong effect on the aggregation dynamics, affecting not only the characteristic cluster size $\langle s \rangle$, but also the time scale at which the steady state (if any) is reached. [When the attractive interaction dominates over propulsion, e.g. at $\xi=0.1$, AP squirmers coarsen and the velocity of coarsening will depend on the hydrodynamic stresses for both interaction ranges, similarly to the isotropic case with lower interaction strength ($\xi=0.7$) [@SoftMatter17], where the system coarsens for pullers and weak pullers and form clusters for strong pushers. As opposed to AP, WP squirmers develop more strinking effects, for long range interactions squirmers coarsen very slowly as far as our calculation time is concerned. While for short range, WP squirmers have reached a clustering steady state, where clusters are self-assembly from small chains or trimers. WP cases at $\xi=0.1$ are explained deeply in appendix \[Appendix:xi0\_1WPrc1\_5\]. ]{} [ AP squirmers display a behavior closer to that of squirmers interacting with an isotropic potential when interaction competes with activity and when activity dominates over attraction [@SoftMatter17]. [ The competition between interaction and activity always in terms of $\xi$. ]{} For example, the fluctuation of the mean cluster size for weak pullers observed in panel b of Fig. \[fig:nct\_xi1\_0\] and panels a and b of Fig. \[fig:nct\_xi10\] or the power law behavior of the CSD for weak pullers in panel b of Fig. \[fig:csd\_xi1\] and panels a and b of Fig. \[fig:csd\_xi10\]. In contrast, WP squirmers have a stronger tendency to develop novel collective behavior, from polydisperse suspension of monomers, dimers, trimers and even chains of tens of particles. ]{} [Our interpretation here is that, since two AP squirmers interact when their attractive patches are within their range, this effect is reduced due to their propulsion in opposite directions. Thus the contribution to the torque is mostly given by hydrodynamic stresses rather than by the anisotropic interactions, as in the isotropic case.]{} In order to perform a statistical study of the clusters, we focus on $\xi = 1$ and $\xi=10$ when the system is in steady state. We first compute the cluster size distribution (CSD). On the one side, CSD for AP squirmers can be described by a power law with an exponential tail. When $\xi=1$, CSDs are characterized by a cutoff-algebraic shape except for strong pullers, which develop a bimodal distribution not present for short interaction range ($1.5 \sigma$). In this case, the CSD’s width follows a behaviour reminding that of isotropic squirmers with $\xi>1$, i.e. wider CSDs for weak pullers. When $\xi=10$, the effect of the interaction range on the CSDs is not observed for any value of $\beta$. On the other side, the CSDs for WP squirmers has a peak for trimers when $\xi=1$. The shape of the CSDs is approximately the same independently on the attraction range. When $\xi=10$, even though the CSD is described by a power law with an exponential tail, when $r_c=2.5\sigma$ the larger the value of $\beta$ the wider the distribution; whereas when $r_c=1.5\sigma$ the shape of the CSD is mainly dictated by hydrodynamics, with weak pullers showing a wider distribution. Next, we compute the radius of gyration as a function of the cluster size. When $\xi=1$ and $r_c=2.5\sigma$, the morphology of AP squirmers follows the same behaviour as the one detected in the isotropic case [@SoftMatter17], with more compact clusters for strong pullers; differently from the case when $r_c=1.5\sigma$, where clusters are more compact for weak pullers. When $\xi=10$ the clusters’ $R_g$ does not change, independently on the interaction range or the hydrodynamic signature. When $\xi=1$, the $R_g$ for WP squirmers varies with the clusters sizes since, for both interaction ranges, trimers and tetramers gives rise to the formation of chains, which lead to different exponents. In contrast, when $\xi=10$ and $r_c=2.5 \sigma$, the $R_g$ for WP squirmers is the same despite $\beta$; however, when $r_c=1.5 \sigma$ pullers form more compact clusters. [It is worth mentioning that these chains resemble the chains observed with amphiphilic active brownian particles [@Stewart_17], but clearly their nature is different since WP squirmers form chains due to their stresslet, while the Brownian case depends on the activity and the size of the hydrophobic patch (see appendix \[Appendix:JanusOrient\]). Moreover, it seems that hydrodynamic interactions cancel the formation of chains for AP squirmers, since we have observed chains just for WP squirmers.]{} To establish a potential alignment within clusters, we compute both polar and nematic order. When $\xi=1$, the polar order of the clusters for AP squirmers strongly depends on $r_c$: when $r_c=2.5\sigma$, $P(s)$ quickly decays with the cluster size (quicker for pushers than for pullers); whereas when $r_c=1.5 \sigma$, $P(s)$ corresponds to high polar order for large clusters with $\beta=0$ and weak pullers. The same features can be observed when $\xi=10$, independently on $r_c$. $P(s)$ for WP squirmers vanishes for all cluster sizes, independently of $\beta$ and $r_c$ when $\xi=1$, since the formation of trimers and chains assemble the particles head to head, thus it avoids the polar order. However, when $\xi=10$, the polar order decays with $s$: for $r_c=1.5\sigma$ weak pullers form clusters with high polar order, while for $r_c=2.5\sigma$ $P(s)$ decays faster for pushers than for pullers. In the latter case, the WP squirmers show another type of alignment, with particles with a high nematic order in the range of $\beta=0$ and weak pushers. During aggregation, we have identified seven different cases depending on whether the attractive patch is oriented against the propulsion direction (AP squirmers) or directed towards it (WP squirmers): 3 gas states, 3 clustering cases and 1 coarsening. On the one side, AP squirmers coarsen isotropically if the interaction is strong enough $\xi \approx 0.1$. When the interaction strength competes with self-propulsion ($\xi =1$), dynamic clusters emerge whose mean size depend on hydrodynamic stresses. When activity dominates ($\xi =10$), particles form a gas and depending on the value of $\beta$ this gas can be isotropically oriented or polarly oriented. On the other side, WP squirmers form chains of particles if the interaction is high enough ($\xi \approx 0.1$), or a suspension of trimers and tetramers if the interaction competes with the active propulsion ($\xi =1$). Whenever activity dominates ($\xi =10$), particles form a gas and depending on the value of $\beta$ this gas can be isotropic, polar or even nematically oriented. Therefore, on one hand anisotropy drives the formation of structures of trimers and tetramers for WP squirmers; on the other hand anisotropy modulates the sensitivity of the hydrodynamic signature: while WP squirmers are more sensitive to $r_c$ when $\xi=10$, AP squirmers are more sensitive to the interaction range ($r_c$) when $\xi=1$. To conclude, the rich morphology of the detected squirmers’ aggregates is the result of the anisotropic interactions, characterized by the angular attractive potential and its interaction range, in competition with the active stress, that can be pointing towards or against the attractive patch. Acknowledgments {#acknowledgments .unnumbered} =============== We thank to S.A. Mallory and A. Cacciuto for helpful discussions. This work was possible thanks to the access to MareNostrum Supercomputer at Barcelona Supercomputing Center (BSC). This article is based upon work from COST Action MP1305, supported by COST (European Cooperation in Science and Technology). This work was supported by FIS2016-78847-P of the MINECO and the UCM/ Santander PR26/16-10B-2. IP acknowledges MINECO and DURSI for financial support under projects FIS2015-67837- P and 2017 SGR-884, respectively. FA acknowledges funding from Juan de la Cierva-formación program. [60]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [****,  ()](\doibase 10.1103/PhysRevLett.93.098103) [****,  ()](\doibase 10.1007/s00348-007-0387-y) @noop [****,  ()]{} [****,  ()](\doibase 10.1017/S0022112007007847) [****,  ()](\doibase 10.1017/S0022112008003807) [****,  ()](\doibase 10.1103/PhysRevLett.100.088103) [****,  ()](\doibase 10.1039/C3SM00140G) [****,  ()](\doibase 10.1103/PhysRevE.93.043114) [****,  ()](\doibase 10.7566/JPSJ.86.101008),  @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} [****,  ()](\doibase https://doi.org/10.1016/j.jcp.2015.09.020) [****,  ()](\doibase 10.1126/science.1230020),  [****,  ()](\doibase 10.1073/pnas.1116334109),  [****,  ()](\doibase 10.1103/PhysRevLett.114.158102) [****,  ()](\doibase 10.1103/PhysRevLett.108.268303) [****,  ()](\doibase 10.1088/1367-2630/13/7/073021) [****,  ()](\doibase 10.1103/PhysRevLett.112.118101) [****,  ()](\doibase 10.1039/C6SM02042A) [****,  ()](\doibase 10.1103/PhysRevE.90.032304) [****,  ()](\doibase 10.1103/PhysRevE.88.012305) [****,  ()](\doibase 10.1103/PhysRevLett.111.245702) [****,  ()](\doibase 10.1039/C5SM02350E) [****, ()](\doibase 10.1039/C5SM01061F) @noop [****,  ()]{} [****,  ()](\doibase 10.1103/PhysRevE.89.062316) [****, ()](\doibase 10.1103/PhysRevLett.112.238303) [****,  ()](\doibase 10.1103/PhysRevE.89.022711) [****,  ()](\doibase 10.1103/PhysRevLett.115.258301) [****,  ()](\doibase 10.1039/C6SM01162D) @noop [ ()]{} [****,  ()](\doibase 10.1063/1.5082284) [****,  ()](\doibase 10.1063/1.4863952) [****,  ()](\doibase 10.1021/ja311455k) @noop [****,  ()]{} @noop [****,  ()]{} [****,  ()](\doibase 10.1039/C4SM00340C) @noop [ ()]{} [****,  ()](\doibase 10.1002/cpa.3160050201),  [****,  ()](\doibase 10.1017/S002211207100048X) @noop [****,  ()]{} @noop [**]{} (, ) [****, ()](\doibase 10.1103/PhysRevA.8.1504) [****,  ()](\doibase 10.1021/la7030818), ,  [****,  ()](\doibase 10.1021/la00050a035),  [****,  ()](\doibase 10.1103/PhysRevLett.77.4102) [****,  ()](\doibase 10.1098/rsta.2005.1619) @noop [**]{} (, , ) @noop [****,  ()]{} [****, ()](\doibase 10.1103/PhysRevE.66.046708) [****,  ()](\doibase http://dx.doi.org/10.1016/j.jnnfm.2010.01.023),  [****,  ()](\doibase 10.1140/epje/i2010-10654-7) @noop [****,  ()]{} [****,  ()](\doibase 10.1103/PhysRevB.79.075437) [****,  ()](\doibase 10.1080/00268978400101951) See Supplemental Material at \[URL will be inserted by publisher\] for additional information and videos, @noop [****,  ()](\doibase 10.1039/C5SM00127G) [****,  ()](\doibase 10.1103/PhysRevE.80.021404) [****,  ()](\doibase 10.1039/C7SM00519A) \[Appendix:attra\]Attractive part of the potential ================================================== Fig. \[Fig:Vsmooth\] compares the radial part of the total short range potential given by Eq. \[eq:VLJshortrange\] (red curves) and the classical LJ potential Eq. \[eq:VLJmidrange\] and cut-off distance of $2.5 \sigma$ (blue curves), $\sigma$ is the particles’ diameter. Both curves in Fig \[Fig:Vsmooth\] are the sum of the repulsive soft-sphere potential (Eq. \[eq:SSPotential\]) and the respective attractive potential. \[Appendix:JanusOrient\]Different types of Janus interactions ============================================================= There are different models to simulate the amphiphilic behavior of Janus colloids, in this appendix we have compared two different models for the angular potential $\phi$ in equation \[eq:JanusGralPotential\]. The angular potential used in this manuscript (equation \[eq:phi\]) has been used to simulate spherical particles with one hydrophobic hemisphere and charged on the other [@langmuir08] where both hemispheres have the same size, thus lets call it symmetric potential $\phi_s(\theta_i,\theta_j)$. On the other hand, in reference [@pre09], they developed an asymmetric potential, where the size of the hydrophobic region is tuned, in order to study 3D Janus colloid suspensions in equilibrium and more recently, some of us have used this potential to study the self-assembly of Active amphiphilic Janus particles with attractive patches of different sizes [@Stewart_17]. Analogously to the symmetric potential above, the the orientation depending term is given by $$\begin{aligned} \label{eq:asymetric_orient} \begin{split} &\phi_a \left(\theta_i \right) = \\ &\left\{ \begin{array}{lc} 1 & \mbox{$\theta_i \leq \theta_{max}$}\\ \cos ^2 \left( \frac{\pi \left(\theta_i - \theta_{max} \right)}{2 \theta_{tail}} \right) & \mbox{$\theta_{max} \leq \theta_i \leq \theta_{max} + \theta_{tail} $}\\ 0 & \mbox{otherwise.} \end{array} \right. \end{split}\end{aligned}$$ The orientational part of the potential $\phi_a$, is a a smooth step function that modulates the angular dependence of the potential, the smoothness is modulated by the parameter $\theta_{tail}$, which it is set to $\theta_{tail}=0.2618$ in [@pre09] for 3D Janus colloids while in [@Stewart_17] was tuned in $\theta_{tail}=0.436$ to generate a sufficiently smooth potential at the Janus interface and $\theta_{max}$ tunes the size of the hydrophobic region, for a patch coverage of 50$\%$ then $\theta_{max} = \pi/2$. By the analysis of the curves shown in Fig. \[fig:TwoJanusOrientations\], it is clear that interactions among the Janus particles will depend on the angular function we choose, despite of the fact that $\theta_{max}=\pi/2$ in the asymmetric potential. In terms of the interaction strength, it is clear that $\phi_s$ will give us a lower attraction than $\phi_a$ for any direction of $\theta_i$. An important remark is that $\phi_s=0$ when two particles are parallel while $\phi_a \neq 0$ in the same configuration, therefore $\phi_a$ enhance the emergence of chain-like aggregates [@Stewart_17]. To understand the relevance of the orientational part in the potential and the hydrodynamic interactions, we are working in a model of amphiphilic squirmers using $\phi_a$ in order to compare with the ’dry’ case of Ref. [@Stewart_17], this work is in progress. \[Appendix:Clusters\] Identifying clusters in the suspension ============================================================ In order to determine the criterion to characterize whether a particle belongs or not to a cluster, we computed first the radial distribution function of our swimmer suspensions. {width="0.48\columnwidth"} {width="0.48\columnwidth"} We observe a local peak at $r=1.6\sigma$ for the $g(r)$ corresponding to $\beta=3$ with $r_c=2.5\sigma$ with AP interaction. We took $r_{cl}=1.75\sigma$ for all cases, although for this case it is evident the first minimum cannot be located there. We analyze how the mean cluster size and the cluster size distribution vary for this troublesome case with the choice of $r_{cl}$, comparing $r_{cl}=1.75\sigma$ to a cutoff in the peak $r_{cl}=1.6\sigma$ and a cutoff lying in the first valley $r_{cl}=1.4\sigma$. {width="0.48\columnwidth"} {width="0.48\columnwidth"} We observe that as we reduce $r_{cl}$, the mean cluster size is reduced, being the difference between $r_{cl}=1.75\sigma$ and $r_{cl}=1.4\sigma$ of 0.9 units ($\approx 10\%$). As for the cluster size distribution, we find that all three of them are very similar. Even if the frequencies for the largest observed clusters are reduced with $r_{cl}$, all distributions are cut at the same bin (clusters sized between 240 and 318 particles), with a frequency $7.7 \cdot 10^{-6}$ for $s=279$ and $r_{cl}=1.75\sigma$, and $4.6 \cdot 10^{-6}$ for the same size with $r_{cl}=1.4\sigma$. The monomer frequency goes from 0.680 for $r_{cl}=1.4\sigma$ to 0.656 for $r_{cl}=1.75\sigma$, and all three curves have a very similar shape, sharing even local maxima and minima along the whole profile. \[Appendix:xi0\_1WPrc1\_5\] Clustering analysis for WP Janus with $\xi=0.1$. ============================================================================ Janus squirmers with interaction strength $\xi=0.1$ develop very interesting dynamics, on one hand, for strong pullers with $\beta=3$, particles form mainly trimers and tetramers that eventually aggregate due to hydrodynamic interactions (panel a of Fig. \[Samples\_rc2\_5xi0\_1\] and Fig. \[Samples\_xi0\_1\]). The other case is the chains system, where particles self-assembly in chains of different sizes, such chains are form by two lines of squirmers with the attractive patch pointing towards the other line of squirmers that form the chain (panel b of Fig. \[Samples\_rc2\_5xi0\_1\] and Fig. \[Samples\_xi0\_1\]). The chain-length depends on the value of $\beta$. This chains system has been observed for pushers and weak pullers as well. ![\[Samples\_rc2\_5xi0\_1\] Simulation zoomed snapshots of WP squirmers with $\xi=0.1$ and $r_c=2.5 \sigma$ snapshots are taken at the last timestep of the simulations. The attractive hemisphere is represented in blue, while pure repulsive in green and the fixed orientation vector is shown in red. (a) $\beta=3$ and (b) $\beta=-3$.](fotorc2_5_WPxi0_1_b3.pdf "fig:"){width="0.45\columnwidth"} ![\[Samples\_rc2\_5xi0\_1\] Simulation zoomed snapshots of WP squirmers with $\xi=0.1$ and $r_c=2.5 \sigma$ snapshots are taken at the last timestep of the simulations. The attractive hemisphere is represented in blue, while pure repulsive in green and the fixed orientation vector is shown in red. (a) $\beta=3$ and (b) $\beta=-3$.](fotorc2_5_WPxi0_1_bN3.pdf "fig:"){width="0.45\columnwidth"} When interaction range is $r_c=2.5\sigma$, the clusters either trimers or chains, keep growing slowly, as far as our calculation capacity could reach, we have observed coarsening for all $\beta$. In contrast with WP squirmers with interaction range of $r_c=1.5\sigma$. In this case WP squirmer suspensions reach a clustering state for all hydrodynamic signature $\beta$. We have found two general types of clustering: ![\[Samples\_xi0\_1\] Simulation zoomed snapshots of WP squirmers with $\xi=0.1$ and $r_c=1.5 \sigma$ snapshots are taken once the simulations reach a steady state given by the mean cluster size. The attractive hemisphere is represented in blue, while pure repulsive in green and the fixed orientation vector is shown in red. (a) $\beta=3$ and (b) $\beta=-3$.](foto_WPxi0_1_b3.pdf "fig:"){width="0.45\columnwidth"} ![\[Samples\_xi0\_1\] Simulation zoomed snapshots of WP squirmers with $\xi=0.1$ and $r_c=1.5 \sigma$ snapshots are taken once the simulations reach a steady state given by the mean cluster size. The attractive hemisphere is represented in blue, while pure repulsive in green and the fixed orientation vector is shown in red. (a) $\beta=3$ and (b) $\beta=-3$.](foto_WPxi0_1_bN3.pdf "fig:"){width="0.45\columnwidth"} We have characterized the clusters observed for these WP squirmers. In (panel a in Fig. \[fig:clusters\_xi0\_1WP\]) we show the cluster size distribution $f(s)$. In general, $f(s)$ has two main peaks, one for trimers and another one in the bin of cluster-size around 10 particles for pushers and weak pullers, while strong pullers (red squares) have just one peak for trimers. Moreover, none of these distributions show monomers and just pullers show dimers in their distributions. Another interesting feature of the cluster size distributions are the fact that both strong pushers and pullers have bigger clusters than the rest of squirmers (gray diamonds and red squares respectively) and both distributions follow the analytical shape of equation (\[eq:csd\_cutoff\]) (dotted line) for large clusters. ![\[fig:clusters\_xi0\_1WP\] Cluster analysis of WP squirmers with $\xi=0.1$ and $r_c=1.5\sigma$. (a) Cluster size distribution. (b) Radius of gyration as a function of cluster size. (c) Local polar order and (d) local nematic order parameters. Pullers curves are red and black, while pushers are pink and gray and $\beta=0$ in blue. Note the different scale on the y-axis.](pdfxi01rc15hh.pdf "fig:"){width="0.48\columnwidth"} ![\[fig:clusters\_xi0\_1WP\] Cluster analysis of WP squirmers with $\xi=0.1$ and $r_c=1.5\sigma$. (a) Cluster size distribution. (b) Radius of gyration as a function of cluster size. (c) Local polar order and (d) local nematic order parameters. Pullers curves are red and black, while pushers are pink and gray and $\beta=0$ in blue. Note the different scale on the y-axis.](rgsxi01rc15hh.pdf "fig:"){width="0.48\columnwidth"} ![\[fig:clusters\_xi0\_1WP\] Cluster analysis of WP squirmers with $\xi=0.1$ and $r_c=1.5\sigma$. (a) Cluster size distribution. (b) Radius of gyration as a function of cluster size. (c) Local polar order and (d) local nematic order parameters. Pullers curves are red and black, while pushers are pink and gray and $\beta=0$ in blue. Note the different scale on the y-axis.](psxi01rc15hh.pdf "fig:"){width="0.48\columnwidth"}   ![\[fig:clusters\_xi0\_1WP\] Cluster analysis of WP squirmers with $\xi=0.1$ and $r_c=1.5\sigma$. (a) Cluster size distribution. (b) Radius of gyration as a function of cluster size. (c) Local polar order and (d) local nematic order parameters. Pullers curves are red and black, while pushers are pink and gray and $\beta=0$ in blue. Note the different scale on the y-axis.](nsxi01rc15hh.pdf "fig:"){width="0.48\columnwidth"} Even though that both cluster size distributions are similar for strong pushers and pullers, the morphology of such clusters are not the same. As we can observe in Fig. \[fig:clusters\_xi0\_1WP\]-panel b, where despite that the radius of gyration for small clusters ($s \leq 10$) follows the same behavior for all $\beta$, the large clusters for strong pullers (red squares) are more compact (small slope) than clusters of the rest of squirmers, in fact, clusters of pushers and weak pullers are chains, thus we can observe a greater slope for large clusters (gray diamonds and pink triangles). We have also calculated both polar and nematic order parameters at each cluster, we have observed that clusters have no polar order at any value of $\beta$ (see panel c of Fig. \[fig:clusters\_xi0\_1WP\]), but an alignment, given by a non-zero nematic order is observed (see panel d of Fig. \[fig:clusters\_xi0\_1WP\]). For pullers, all dimers are aligned by a head-to-head configuration and then the nematic order is 1, particles in trimers and tetramers in all cases are pointing out to the center of the clusters therefore, the nematic order is zero. But, in the case of pushers to weak pullers all clusters have a large non-zero nematic order, since particles are forming chains of two lines of particles aligned to the center of the chain giving rise to a nematic order. In contrast, for strong pullers (red squares) large clusters are formed by trimers gather together by hydrodynamic interactions, thus particles are pointing in different directions and nematic order parameter is reduced. In fact, the magnitude of nematic order parameter for large clusters is given by the $\beta$ value. The lower the $\beta$ the greater the nematic order. [^1]: Note that separation $h_0$ is relatively short, $h_0 < \sigma_s$, and its value depends on the location of the minimum of the attractive potential. [^2]: The WP system with $\xi=0.1$, where clustering is observed (panel d of Fig. \[fig:nct\_xi0\_1\]), is studied in more detail in appendix \[Appendix:xi0\_1WPrc1\_5\].

--- abstract: 'Under hard-agar and nutrient-rich conditions, a cell of [*Bacillus subtilis*]{} grows as a single filament owing to the failure of cell separation after each growth and division cycle. The self-elongating filament of cells shows sequential folding processes, and multifold structures extend over an agar plate. We report that the growth process from the exponential phase to the stationary phase is well described by the time evolution of fractal dimensions of the filament configuration. We propose a method of characterizing filament configurations using a set of lengths of multifold parts of a filament. Systems of differential equations are introduced to describe the folding processes that create multifold structures in the early stage of the growth process. We show that the fitting of experimental data to the solutions of equations is excellent, and the parameters involved in our model systems are determined.' author: - 'Ryojiro Honda [^1], Jun-ichi Wakita [^2], and Makoto Katori [^3]' date: 21 September 2015 title: | **Self-Elongation with Sequential Folding\ of a Filament of Bacterial Cells** --- Introduction ============ \[sec:introduction\] Patterns observed in bacterial colonies growing on surfaces of semisolid agar plates realize a variety of fractal and self-affine structures studied in statistical mechanics and fractal physics [@MHKOYM04]. In a series of experimental studies [@RMWSES96; @WIMM97; @IWMM99; @HWKYMM05], it has been clarified that the morphology of growing bacterial colonies at the macroscopic scale does not depend on the biological details of individual organisms, but depends only on environmental conditions controlled by the agar concentration $C_{\rm a}$ and the nutrient concentration $C_{\rm n}$. Cell motility is changed by varying $C_{\rm a}$ and the growth rate is controlled by varying $C_{\rm n}$, and then different patterns appear in different regions in a morphology diagram drawn on the $C_{\rm a}$-$C_{\rm n}$ plane. On this plane, a diffusion-limited aggregation (DLA)-like pattern [@WS81; @Mea86], an Eden-like pattern [@Ede61; @FV85], a concentric-ring pattern, a homogeneously spreading disk like pattern, and a dense branching morphology (DBM) pattern [@MHKOYM04] have been recorded. Since some of these patterns are observed not only in biological systems, but also in chemical and physical systems, such as those in crystal growth, aggregation processes, and viscous fingering [@Vic92], the mechanism of pattern formation could be common in organisms and inorganic substances, and a theoretical study by mathematical modeling and analysis will be useful for understanding the underlying universal principles [@VCBCS95; @VZ12; @BPS11; @PSB13; @BPS14]. In a recent paper [@WTYKY15], we have reported the physical aspects of the collective motion of bacterial cells observed in shallow circular pools prepared on the surface of an agar plate. The diameters of the pools are arranged so as not to be much larger than the length of the bacterial cells swimming in the pools. We used [*Bacillus*]{} ([*B.*]{}) [*subtilis*]{} and found six different types of collective motion, including one-way and two-way rotational motion along the brim of a circular pool and collective oscillatory motion in the entire pool. Analyzing experimental observations in 117 circular pools, we found that these six types of collective motion can be classified using only two parameters: the reduced cell length $\lambda$, which is defined as the ratio of the average cell length in a pool to the pool diameter, and the cell density $\rho$ in the pool. We obtained a phase diagram for the collective motion drawn on the $\lambda$-$\rho$ plane and predicted that simple modeling with the two control parameters will be able to explain the variety of collective motion of bacterial cells. The above results imply that physical considerations are applicable and useful to explain the changes in the morphology of bacterial colonies at the macroscopic scale as well as the dynamical transitions of the collective motion of bacterial cells at the microscopic scale caused by environmental variations. On the basis of them, in this paper, we will report the experimental results and numerical analyses of growth processes starting from a single bacterial cell observed in formation of colony patterns. We tried to analyze the observed growth process by fractal analysis and by using systems of differential equations. Throughout the experiment reported in this paper, we used [*B. subtilis*]{} wild-type strain OG-01. Cells of this strain are rod-shaped (0.5–1.0 $\mu$m in diameter, 2–5 $\mu$m in length) with peritrichous flagella. They swim in a straight line in water by bundling and rotating the flagella. Under unfavorable environmental conditions, such as on a nutrient-poor medium or a dry agar plate, they become spores. When a small number of cells are inoculated on the surface of a semisolid agar plate, they go through a resting period of about 7 h before starting two-dimensional colony expansion. 2.5cm ![Snapshot of the filament configuration of bacterial cells on an agar plate at time $t=40$ min after the first twofold part appeared. The scale bar indicates 20 $\mu$m. []{data-label="fig:picture1"}](67250Fig1.eps "fig:"){width="0.7\linewidth"} As briefly reported in previous papers [@WIMM97; @WKMM10], we observed string like objects in microscopy observations of an inoculation spot of a bacterial suspension in the later stage of the resting phase. In this study, we focus on the growth process that started from a single bacterial cell in the resting period under hard-agar and nutrient-rich conditions. In this case, cell multiplications are repeated with a constant cell cycle (doubling time), but daughter cells fail to separate after each cell cycle, although the cytoplasm has been compartmentalized by septum formation. Then, a long filament is produced, which consists of a chain of cells linked end to end [@Men76]. Such a filament writhes as it elongates on the agar plate, and eventually some segment starts folding and a twofold part of the filament is created. Figure \[fig:picture1\] shows a typical configuration of a filament in which twofold parts have been created. Sooner or later, we will see the appearance of threefold parts, fourfold parts, and so forth, and the filament configuration of linked cells becomes complicated on the agar plate. Mendelson and coworkers have intensively studied the supercoiling processes performed by such cell filaments of [*B. subtilis*]{} [@Men76; @Men78; @MTKL95; @MSL97; @Men99]. They are interested in the situation wherein the filaments twist to make a double-stranded helix. The double-stranded structure itself twists while writhing, eventually comes in contact with itself, and forms a supercoil. By the repetition of such supercoiling processes, macroscopic structures of millimeter length are created, which are called bacterial macrofibers. The interesting motion of macrofibers was reported by Mendelson and coworkers. [@MSWG00; @MST01; @MMT02; @MSRCT03]. Kumada [*et al.*]{} also observed the growth of filaments of cells without separation for [*B. subtilis*]{}, in which the dependence of the morphology on $C_{\rm a}$ was systematically studied [@KIT96]. Such growing filamentous cells were also observed for [*Escherichia*]{} ([*E.*]{}) [*coli*]{} [@TDWW05]. In this paper, we study the simplest situation wherein a filament of bacterial cells does not twist, and hence helical structures are not formulated. In our case, a single self-elongating filament repeatedly folds upon itself and shows a crossover from a one-dimensional structure to a two-dimensional structure. In Sect. \[sec:experimental\], we explain the experimental setup and procedure for recording microscopic snapshots of cell filament configurations for about 6 h. We performed fractal analysis of filament configurations by the box-counting method. In Sect. \[sec:crossover\], we report that the results imply fractal structures in the filament configurations, and the evaluated fractal dimension $D$ shows a crossover from $D=1$ to $2$ as the growth process goes from the exponential phase to the stationary phase. Detailed study of the time evolution of the filament configuration of bacterial cells is discussed in Sects. \[sec:measurement\] and \[sec:analysis\]. In Sect. \[sec:measurement\], we propose a method of characterizing filament configurations on an agar plate by a set of lengths of the simple part and the $k$-fold parts with $k=2,3, 4, \dots$, and the results of experimental measurements of these lengths are shown. To analyze the data given in Sect. \[sec:measurement\], we introduce systems of differential equations in Sect. \[sec:analysis\] and the nonlinear fitting of data to their solutions is performed. Section \[sec:remarks\] is devoted to concluding remarks. Experimental Procedures ======================= \[sec:experimental\] We observe a multiple-fission process from a single bacterial cell in the resting period under hard-agar and nutrient-rich conditions. The experimental setup and procedure are as follows. A solution containing 5 g of sodium chloride (NaCl), 5 g of dipotassium hydrogen-phosphate (K$_{2}$HPO$_{4}$) and 10 g of Bacto-Peptone (Becton, Dickinson and Company, Franklin Lakes, NJ, USA) in 1 L of distilled water is prepared. The environmental parameter $C_{\rm n}$ is given as the concentration of Bacto-Peptone; $C_{\rm n} = 10$ g$\cdot$L$^{-1}$. Then, the solution is adjusted to pH 7.1 by adding 6 N hydrochloric acid (HCl). Moreover, the solution is mixed with 10 g of Bacto-Agar (Becton, Dickinson and Company), which determines the softness of a semisolid agar plate. The environmental parameter $C_{\rm a}$ is given as the concentration of Bacto-Agar; $C_{\rm a} = 10$ g$\cdot$L$^{-1}$. The environmental condition realized by these values of $C_{\rm a}$ and $C_{\rm n}$ gives a typical Eden-like pattern of [*B. subtilis*]{} colonies. The mixture is autoclaved at 121 $^\circ$C for 15 min, and 20 mL of the solution is poured into each sterilized plastic petri dish of 88 mm inner diameter. The thickness of the semisolid agar plates is about 3.2 mm. After solidification at room temperature for 60 min, the semisolid agar plates are dried at 50 $^\circ$C for 90 min. 3 $\mu$L of the bacterial suspension is inoculated at the center of each agar plate surface. 1 $\mu$L of the suspension includes about $10^2$ cells in the spore state. The agar plates are left at room temperature for about 60 min to dry the bacterial suspension droplet. Thereafter, they are cultivated in a stage top incubator at 35 $^\circ$C (INULG2-OTOR-CV, Tokai Hit, Shizuoka), which is attached to the stage of an optical microscope (IX71, Olympus, Tokyo). The spores at the inoculated center spot germinate about 2 h after the inoculation. They repeatedly undergo cell multiplications without splitting or movement, and form a long filament. The filament of cells linked in tandem grows two-dimensionally on the agar plate surface like a self-elongating string. This growth process of a filament of cells is observed through an optical microscope with a 20x objective. A digital camera (DP71, Olympus, Tokyo) is connected to the optical microscope. Microscopy snapshots are recorded every minute for about 6 h using bio-imaging analysis software (Lumina Vision, Mitani, Fukui and Tokyo) from the time when spore germination occurred to the time when the entanglement of a filament started two-dimensional colony expansion. Crossover from One-Dimensional Structure to Two-Dimensional Structure ===================================================================== \[sec:crossover\] In this section, the time after the inoculated spot germinated is denoted by $T$ \[min\]. Let $L(T)$ \[$\mu$m\] be the total length of a filament of bacterial cells observed at time $T$. If the specific growth rate is denoted by $\mu$, it will show an exponential elongation, $$L(T)=L(0) e^{\mu T}, \quad T \geq 0. \label{eqn:elongation}$$ First, we confirmed that the observed $L(T)$ data up to $T=200$ min are well described by the single exponential function Eq. (\[eqn:elongation\]), and the specific growth rate was evaluated to be $$\mu=3.58 \times 10^{-2} \, \mbox{min$^{-1}$}. \label{eqn:alpha_bar}$$ This gives a cycle time (doubling time) of $$\tau= \frac{\ln 2}{\mu}=19.4 \, \mbox{min}. \label{eqn:tau}$$ 2.8cm ![ (Color online) Upper left (lower left) picture: snapshot of the cell filament at time $T=125$ min ($T=245$ min). The upper right (lower right) graph shows the log-log plots of $N_T(\varepsilon)$ versus $\varepsilon$. We find linear regions in the log-log plots, and the fractal dimensions of the filament configurations are evaluated as $D(125)=1.16$ and $D(245)=1.96$, respectively. []{data-label="fig:fractal1"}](67250Fig2.eps "fig:"){width="0.7\linewidth"} 4.5cm ![ Time dependence of fractal dimension $D(T)$ of the filamentous cell configuration on the agar plate. The data are well described by the sigmoid function given by Eq. (\[eqn:sigmoid\]). []{data-label="fig:fractal2"}](67250Fig3.eps "fig:"){width="0.4\linewidth"} The snapshots of filament configurations at $T=125$ and 245 min are shown on the left-hand side in Fig. \[fig:fractal1\] with a resolution of $2040 \times1536$ pixels. They are analyzed by the box-counting method to evaluate the fractal dimensions of the filament configurations of bacterial cells on the agar plate. The procedure is as follows. At each time $T$, the snapshot picture is divided by squares (two-dimensional boxes) of linear size $\varepsilon$, and then is counted the number $N_T(\varepsilon)$ of squares containing the pixels occupied by the filament of bacterial cell. If the configuration has a fractal structure, $N_T(\varepsilon)$ is scaled as $$N_T(\varepsilon) \sim \varepsilon^{-D(T)} \label{eqn:fractal1}$$ with the fractal dimension $D(T)$. We changed the value of $\varepsilon$ from 1 to 1536 pixels (from $2.1 \times 10^{-1}$ to $3.2 \times 10^2$ $\mu$m in the real scale). As shown in the log-log plots given on the right-hand side in Fig. \[fig:fractal1\], the data for $T=125$ min show a power law in the range of 3.5 $\mbox{$\mu$m} < \varepsilon < 5.6 \times 10^1$ in the real scale, and those for $T=245$ min show a power law in the range of 2.5 $\mbox{$\mu$m} < \varepsilon < 2.1 \times 10^1$ in the real scale. The slopes $-1.16$ and $-1.96$ in these log-log plots in Fig. \[fig:fractal1\] provide the fractal dimensions $D(125)=1.16$ and $D(245)=1.96$, respectively. The result implies that the filament configurations have fractal structures. We evaluated the fractal dimensions of the filament at $T=45, 85, 165, 205, 285, 325$, and 365 min and the results are plotted in Fig. \[fig:fractal2\]. We found that the data can be fitted to the following sigmoid function: $$D(T)=c_1 \tanh\bigl[\sigma(T-T_0)\bigr]+c_2 \label{eqn:sigmoid}$$ with $T_0=1.82 \times$ $10^{2}$ min, $\sigma=1.61 \times 10^{-2}$ , $c_1=5.1 \times 10^{-1}$, and $c_2=1.54$. The time evolution of the multiple fission of bacterial cells from the exponential phase to the stationary phase is well described by the time dependence of the fractal dimension of the filament configuration on the agar plate. Folding Processes and Experimental Measurements =============================================== \[sec:measurement\] In this section, we denote the observation time by $t$ \[min\] instead of $T$ \[min\], since we will set $t=0$ at the time when the first folding of a filament occurs as explained below. Let $L(t)$ \[$\mu$m\] be the total length of a filament of bacterial cells at time $t$. Here, we write the specific growth rate as $\alpha$, and then $L(t)$ should obey the following differential equation: $$\frac{d}{dt} L(t)=\alpha L(t). \label{eqn:Lt2}$$ On the semisolid agar plate, the linear growth is unstable, and a filament of cells starts folding after a short time. It repeats the folding processes and the configuration of the filament on the plate becomes complicated very rapidly. Over time, the multifold structure becomes dense and its region spreads over the plates. In this way, the configuration of a filament of cells shows a crossover from the one-dimensional structure to the two-dimensional structure, as shown in the previous section. ![ Illustrations of elementary processes found in the self-elongating process of a filament of bacterial cells in the early stage. \[1\] Elongation of simple segment. \[2\] Creation of a twofold segment by folding a simple segment. Twice the length $r_2$ is added to the length $l_2$. \[3\] Creation of a fourfold segment by elongation of a twofold segment. Four times the length $r_4$ is added to the length $l_4$. \[4\] Creation of a threefold segment by folding of a twofold segment on a simple segment. Three times the length $r_3$ is added to the length $l_3$. []{data-label="fig:processes"}](67250Fig4.eps){width="0.4\linewidth"} In an interval of a filament, if bacterial cells are linked in tandem and form a curved line without folding, the interval is called a simple segment. At each time $t$, we consider the union of all simple segments in a filament and call it the simple part. We represent its length by $l_1(t)$ \[$\mu$m\]. As illustrated by \[1\] in Fig. \[fig:processes\], we see the following process: $$\mbox{Elementary Process [1] : elongation of a simple segment},$$ and $l_1(t)$ rapidly increases with time $t$. We set $t=0$ at the time when the first folding of a filament is observed. In this experiment, $t=T-60$ \[min\]. For $t >0$, the following process occurs: $$\begin{aligned} \mbox{Elementary Process [2]} &:& \mbox{creation of a twofold segment by folding of} \nonumber\\ && \mbox{a simple segment,} \nonumber\end{aligned}$$ as illustrated by \[2\] in Fig. \[fig:processes\]. The twofold part is defined by the union of all twofold segments in the filament, whose length is denoted by $l_2(t)$ \[$\mu$m\]. By definition, $l_1(t)=L(t), l_2(t)=0$ for $t \leq 0$, while $l_1(t)=L(t)-l_2(t), l_2(t)>0$ for $t > 0$. Sooner or later, we will see threefold segments, fourfold segments, and so forth. We call the union of all $k$-fold segments the $k$-fold part, and write the length of the $k$-fold part as $l_k(t)$ \[$\mu$m\], $k =2, 3, 4, \dots$. See the elementary processes \[3\] and \[4\] in Fig. \[fig:processes\], which create a fourfold segment and a threefold segment, respectively. We will attempt to characterize the time evolution of the filament configuration of cells using the set of lengths $(l_1(t), l_2(t), l_3(t), \cdots)$ developing in time $t$. Since $\sum_{k \geq 1} l_k(t)=L(t)$, the data are regarded as a time-dependent ‘partition’ of an exponentially growing length $L(t)$. We found a critical time $t_*$ such that when $0 \leq t \leq t_*$, the whole filament of cells consists of only simple segments and twofold segments, while when $t > t_*$, we observe the appearance of threefold and fourfold segments in a filament of cells and the configuration starts to become complicated. In this experiment, we observed $$t_*=45 \, \mbox{min}. \label{eqn:t_*}$$ It corresponds to the time $T_*=t_*+60 =105$ min after the inoculated spot germinated. Note that it gives the time when the fractal dimensions $D(T)$ of the filament starts to show its rapid increase in Fig. \[fig:fractal2\]. Figure \[fig:picture2\] shows a snapshot of a configuration of bacterial cells at $t=60$ min. We obtained an enlarged photocopy of the snapshot picture and traced the filament segments by hand. Each interval with a number $k$ represents a $k$-fold segment of the filament, where $k=2,3,4$, and 5. The intervals without a number are the simple segments. We used an opisometer, which is an instrument for measuring the length of arbitrary curved lines on a sheet. If the length of a segment indexed $k$ is $r_k$, the length $l_k(t)$ of the $k$-fold part of the filament should be the sum of $k r_k$ over all $k$-fold segments. We measured $(l_k(t))_{k \geq 1}$ up to time $t=90$ min. The results are listed in Table \[tab:data\], where the lengths of parts with $k \geq 3$ are summed and the values of $l_{3_+}(t)=\sum_{k \geq 3} l_{k}(t)$ are given. In the experiment, we fixed the field of vision of the optical microscope. Just before $t=70$ min, a tip of the simple part ran out of the field of vision, and after $t=70$ min, twofold segments may have been created out of our field of vision. Hence, the values of $L$ and $l_1$ at $t=70, 80$, and $90$ and those of $l_2$ at $t=80$ and 90 are not given in Table \[tab:data\]. ![Snapshot of a configuration of the filament of bacterial cells at time $t=60$ min. Each interval with a number $k$ represents a $k$-fold segment of the filament, where $k=2, 3, 4$, and 5. The intervals without a number are the simple segments. The scale bar indicates 20 $\mu$m. []{data-label="fig:picture2"}](67250Fig5.eps){width="0.8\linewidth"} $t$ \[min\] $L$\[$\mu$m\] $l_1$\[$\mu$m\] $l_2$\[$\mu$m\] $l_{3_+}$\[$\mu$m\] ------------- --------------- ----------------- ----------------- --------------------- 0 151 151 0 0 10 228 189 39 0 20 315 247 68 0 30 485 340 145 0 40 733 442 291 0 50 1070 620 362 90 60 1590 810 584 197 70 — — 944 535 80 — — — 1260 90 — — — 2710 : Experimental data.[]{data-label="tab:data"} Analysis by Systems of Differential Equations ============================================= \[sec:analysis\] Exponential growth of total length {#sec:exponential} ---------------------------------- Equation (\[eqn:Lt2\]) is solved as $$L(t)=Ae^{\alpha t}. \label{eqn:L2}$$ Here, $A=L(0)$ is the total length of the filament at $t=0$ when the first folding occurs. As shown in Fig. \[fig:exponential\], the time dependence of $L(t)$ is described by Eq. (\[eqn:L2\]) very well for $0 \leq t \leq 60$ min, and we obtained the following values by the semilog fitting of the data: $$\alpha=3.95 \times 10^{-2} \, \mbox{min$^{-1}$}, \quad A=1.50 \times 10^2 \, \mbox{$\mu$m}. \label{eqn:fitting1}$$ The evaluation of $\alpha$ is consistent with the evaluation given by Eq. (\[eqn:alpha\_bar\]) for the specific growth rate $\mu$ averaged over the longer time period $0 \leq T \leq 200$ min. ![Values of $\ln L(t)$ plotted for $t=0,10, 20, \dots, 60$ min. The linear fitting of Eq. (\[eqn:L2\]) in this semilog plot determines the values of $\alpha=3.95 \times 10^{-2}$ and $A=1.50 \times 10^2$ $\mu$m. []{data-label="fig:exponential"}](67250Fig6.eps){width="0.5\linewidth"} Systems of differential equations and their solutions {#sec:diff_eq} ----------------------------------------------------- In the time interval $0 \leq t \leq t_*$, only elementary processes \[1\] and \[2\] take place. We assume that in the elongation process of a cell filament with specific growth rate $\alpha$, the ratio of the frequency of elementary process \[2\] to that of elementary process \[1\] is given by $\beta/(1-\beta)$ with a constant $0 < \beta < 1$. Then, the time evolution of the lengths $l_1(t)$ and $l_2(t)$ will be described by the following system of linear differential equations: $$\begin{aligned} \frac{d}{dt} l_1(t) &=& \alpha(1-\beta) l_1(t), \nonumber\\ \frac{d}{dt} l_2(t) &=& \alpha l_2(t)+ \alpha \beta l_1(t), \quad 0 \leq t \leq t_*. \label{eqn:diff_eq1}\end{aligned}$$ Note that the first term in the second equation in Eq. (\[eqn:diff\_eq1\]) describes the self-elongation process of the twofold part. Since we have set $t=0$ at the time when the first folding occurs, Eq. (\[eqn:diff\_eq1\]) should be solved under the conditions $$l_1(0)=L(0)=A, \quad l_2(0)=0. \label{eqn:condition1}$$ The solution is then given by $$\begin{aligned} l_1(t) &=& A e^{\alpha(1-\beta) t}, \nonumber\\ l_2(t) &=& A e^{\alpha t} (1-e^{-\alpha \beta t}), \quad 0 \leq t \leq t_*. \label{eqn:solutionA}\end{aligned}$$ For $t > t_*$, we take into account the following elementary processes in addition to processes \[1\] and \[2\]: $$\begin{aligned} \mbox{Elementary Process [3]} &:& \mbox{creation of a fourfold segment by elongation of} \nonumber\\ && \mbox{a twofold segment}, \nonumber\\ \mbox{Elementary Process [4]} &:& \mbox{creation of a threefold segment by folding of} \nonumber\\ && \mbox{a twofold segment on a simple segment}. \nonumber\end{aligned}$$ As illustrated by \[3\] in Fig. \[fig:processes\], a part of the elongating twofold segments becomes a fourfold segment. We assume that the ratio of the frequency of fourfold segment creation to that of simple elongation of the twofold part is given by $\gamma/(1-\gamma)$ with a constant $0 < \gamma < 1$. As illustrated by \[4\] in Fig. \[fig:processes\], the creation of threefold segments can occur only if a twofold segment touches a simple segment and folds on it and if they merge into a threefold segment. Thus, its frequency will be proportional to the product of $l_1(t)$ and $l_2(t)$. We assume that this process reduces the total length of the simple part by $\delta_1 l_1(t) l_2(t)$ and that of the twofold part by $\delta_2 l_1(t) l_2(t)$ with transition rates per unit length $\delta_1>0$ and $\delta_2 > 0$. Then, if we set $l_{3_+}(t)=\sum_{k \geq 3} l_k(t)=l_3(t)+l_4(t)+\cdots$, we will obtain the following system of nonlinear differential equations: $$\begin{aligned} \frac{d}{dt}l_1(t) &=& \alpha(1-\beta)l_1(t)-\delta_1l_1(t)l_2(t), \nonumber\\ \frac{d}{dt}l_2(t) &=& \alpha(1-\gamma)l_2(t)+\alpha \beta l_1(t)-\delta_2l_1(t)l_2(t), \nonumber\\ \frac{d}{dt}l_{3_+}(t) &=& \alpha l_{3_+}(t)+\alpha \gamma l_2(t)+(\delta_1+\delta_2)l_1(t)l_2(t), \quad t \geq t_*. \label{eqn:diff_eq2}\end{aligned}$$ We assume that the parameters $\delta_1$ and $\delta_2$ are sufficiently small and solve the system of nonlinear differential equations given by Eq. (\[eqn:diff\_eq2\]) by perturbation. This assumption will be verified by the data fitting as explained in Sect. \[sec:fittings\]. For $k=1,2$, and $3_+$, we expand $l_k(t)$ as power series of $\delta_1$ and $\delta_2$ as $$\begin{aligned} l_k(t) &=& \sum_{m_1=0}^{\infty} \sum_{m_2=0}^{\infty} \delta_1^{m_1} \delta_2^{m_2} \tilde{l}_k^{(m_1, m_2)}(t) \nonumber\\ &=& \sum_{n=0}^{\infty} \sum_{\substack{m_1 \geq 0, m_2 \geq 0, \cr m_1+m_2=n}} \delta_1^{m_1} \delta_2^{m_2} \tilde{l}_k^{(m_1, m_2)}(t), \label{eqn:expand}\end{aligned}$$ where $\tilde{l}_k^{(m_1, m_2)}(t)$ are time-dependent coefficients of the expansion. The $N$th-order approximate solution, $N=0,1,2, \dots$, is given by $$l_k^{(N)}(t)= \sum_{n=0}^{N} \sum_{\substack{m_1 \geq 0, m_2 \geq 0, \cr m_1+m_2=n}} \delta_1^{m_1} \delta_2^{m_2} \tilde{l}_k^{(m_1, m_2)}(t), \quad k=1,2,3_+. \label{eqn:pth_appr}$$ In the following, we calculate the 0th- and first-order approximate solutions: $$\begin{aligned} l_k^{(0)}(t) &=& \tilde{l}_k^{(0,0)}(t), \nonumber\\ l_k^{(1)}(t) &=& l_k^{(0)}(t)+\delta_1 \tilde{l}_k^{(1,0)}(t) + \delta_2 \tilde{l}_k^{(0,1)}(t), \quad k=1,2,3_+, \quad t \geq t_*. \label{eqn:0_1_appr}\end{aligned}$$ The 0th-order approximate solution, $\{l_k^{(0)}(t) : k=1,2,3_+\}$, solves the system of linear differential equations obtained from Eq. (\[eqn:diff\_eq2\]) by setting $\delta_1=\delta_2=0$. In addition to the initial conditions corresponding to Eq. (\[eqn:condition1\]), $$l_1^{(0)}(0)=A, \quad l_2^{(0)}(0)=0, \label{eqn:condition2}$$ the definition of the critical time $t_*$ gives $$l_{3_+}^{(0)}(t_*)=0. \label{eqn:condition3}$$ Under these conditions, we have the following: $$\begin{aligned} l_1^{(0)}(t) &=& A e^{\alpha(1-\beta) t}, \nonumber\\ l_2^{(0)}(t) &=& \frac{\beta A}{\beta-\gamma} e^{\alpha(1-\gamma) t} (1-e^{-\alpha(\beta-\gamma)t} ), \nonumber\\ l_{3_+}^{(0)}(t) &=& \frac{A}{\beta-\gamma} e^{\alpha t} \Big\{ -\beta(e^{-\alpha \gamma t} - e^{-\alpha \gamma t_*}) +\gamma(e^{-\alpha \beta t}-e^{-\alpha \beta t*}) \Big\}, \quad t \geq t_*. \label{eqn:solution_0th}\end{aligned}$$ To express the first-order approximate solution, we introduce the multiple integrals $$\begin{aligned} I^{(1)}(t; t_*, a_1) &=& \int_{t_*}^t ds \, e^{-a_1 s} l_1^{(0)}(s) l_2^{(0)}(s), \nonumber\\ I^{(2)}(t; t_*, a_2, a_1) &=& \int_{t_*}^t ds \, e^{-a_2 s} I^{(1)}(s; t_*, a_1), \nonumber\\ I^{(3)}(t; t_*, a_3, a_2, a_1) &=& \int_{t_*}^t ds \, e^{-a_3 s} I^{(2)}(s; t_*, a_2, a_1), \label{eqn:integrals}\end{aligned}$$ where $a_i,$ $i=1,2$, and $3$ are constants. By inserting $l_1^{(0)}(t)$ and $l_2^{(0)}(t)$ given in Eq. (\[eqn:solution\_0th\]), they are calculated as $$\begin{aligned} I^{(1)}(t; t_*, a_1) &=& \frac{\beta A^2}{\beta-\gamma} \left\{ \ \frac{e^{(p-a_1)t}-e^{(p-a_1)t_*}}{p-a_1} -\frac{e^{(q-a_1)t}-e^{(q-a_1)t_*}}{q-a_1} \right\}, \nonumber\\ I^{(2)}(t; t_*, a_2, a_1) &=& \frac{\beta A^2}{\beta-\gamma} \left\{ \frac{e^{(p-a_1-a_2)t}-e^{(p-a_1-a_2)t_*}}{(p-a_1)(p-a_1-a_2)} -\frac{e^{(q-a_1-a_2)t}-e^{(q-a_1-a_2)t_*}}{(q-a_1)(q-a_1-a_2)} \right. \nonumber\\ && \quad \left. +\frac{e^{-a_2 t}-e^{-a_2 t_*}}{a_2} \left( \frac{e^{(p-a_1)t_*}}{p-a_1}-\frac{e^{(q-a_1)t_*}}{q-a_1} \right) \right\}, \nonumber\\ I^{(3)}(t; t_*, a_3, a_2, a_1) &=& \frac{\beta A^2}{\beta-\gamma} \left[ \left\{ \frac{e^{(p-a_1-a_2-a_3)t}-e^{(p-a_1-a_2-a_3)t_*}}{(p-a_1)(p-a_1-a_2)(p-a_1-a_2-a_3)} \right. \right. \nonumber\\ && \quad - \frac{e^{(q-a_1-a_2-a_3)t}-e^{(q-a_1-a_2-a_3)t_*}}{(q-a_1)(q-a_1-a_2)(q-a_1-a_2-a_3)} \nonumber\\ && \quad \left. - \frac{e^{-(a_2+a_3)t}-e^{-(a_2+a_3) t_*}}{a_2(a_2+a_3)} \left( \frac{e^{(p-a_1) t_*}}{p-a_1} - \frac{e^{(q-a_1)t_*}}{q-a_1} \right) \right\} \nonumber\\ && \quad +\frac{e^{-a_3 t}-e^{-a_3 t_*}}{a_3} \left\{ \frac{e^{(p-a_1-a_2)t_*}}{(p-a_1)(p-a_1-a_2)} -\frac{e^{(q-a_1-a_2)t_*}}{(q-a_1)(q-a_1-a_2)} \right. \nonumber\\ && \qquad \left. \left. + \frac{e^{-a_2 t_*}}{a_2} \left( \frac{e^{(p-a_1)t_*}}{p-a_1}-\frac{e^{(q-a_1)t_*}}{q-a_1} \right) \right\} \right], \label{eqn:integrals2}\end{aligned}$$ where $p=\alpha(2-\beta-\gamma), q=2 \alpha(1-\beta)$. Then, the first-order approximate solution is given by $$\begin{aligned} l_1^{(1)}(t) &=& l_1^{(0)}(t) -\delta_1 e^{\alpha(1-\beta)t} I^{(1)}(t; t_*, \alpha(1-\beta)), \nonumber\\ l_2^{(1)}(t) &=& l_2^{(0)}(t) - \delta_1 \alpha \beta e^{\alpha(1-\gamma) t} I^{(2)}(t; t_*, -\alpha(\gamma-\beta), \alpha(1-\beta)) \nonumber\\ && \quad \quad \, \, - \delta_2 e^{\alpha(1-\gamma) t} I^{(1)}(t; t_*, \alpha(1-\gamma)), \nonumber\\ l_{3_+}^{(1)}(t) &=& l_{3_+}^{(0)}(t) \nonumber\\ &+& \delta_1 \Big\{ -\alpha^2 \beta \gamma e^{\alpha t} I^{(3)}(t; t_*, \alpha \gamma, -\alpha(\gamma-\beta), \alpha(1-\beta)) +e^{\alpha t} I^{(1)}(t; t_*, \alpha) \Big\} \nonumber\\ &+& \delta_2 \Big\{ -\alpha \gamma e^{\alpha t} I^{(2)}(t; t_*, \alpha, \alpha(1-\gamma)) +e^{\alpha t} I^{(1)}(t; t_*, \alpha) \Big\}, \quad t \geq t_*. \label{eqn:solution_1st}\end{aligned}$$ Note that, owing to the initial condition $l_1^{(1)}(0)=l_1^{(0)}(0)=A$, $l_1^{(1)}(t)$ given in the first line of Eq. (\[eqn:solution\_1st\]) becomes independent of $\delta_2$. Nonlinear fitting {#sec:fittings} ----------------- ![ (Color online) Data of $l_1(t)$, $l_2(t)$, and $l_{3_+}(t)$ plotted by $\bigcirc$, $\bigtriangleup$, and $\diamondsuit$, respectively. The thin curves represent the solution given by Eq. (\[eqn:solutionA\]) for $0 \leq t \leq t_*$ with the parameters given by Eqs. (\[eqn:fitting1\]) and (\[eqn:beta\]). The solution given by Eq. (\[eqn:solution\_1st\]) for $t \geq t_*$ is represented by the thick curves, where the parameters are given by Eq. (\[eqn:fitting2\]). The fitting is excellent. []{data-label="fig:fitting1"}](67250Fig7.eps){width="0.5\linewidth"} ![ (Color online) Data of $l_1(t)$, $l_2(t)$, and $l_{3_+}(t)$ plotted by $\bigcirc$, $\bigtriangleup$, and $\diamondsuit$, respectively. The curves show the 0th-order solution given by Eq. (\[eqn:solution\_0th\]), which ignores the nonlinear terms in Eq. (\[eqn:diff\_eq2\]). We failed to fit $l_{3_+}(t)$. []{data-label="fig:fitting2"}](67250Fig8.eps){width="0.5\linewidth"} First, we used five pairs of data $(l_1(t), l_2(t))$ of $t=0,10, \dots, 40 < t_*=45$ in Table \[tab:data\]. Note that the parameters $\alpha$ and $A$ have already been determined using Eq. (5.2). By fitting to the solution given by Eq. (\[eqn:solutionA\]) of the system given by Eq. (\[eqn:diff\_eq1\]) of linear differential equations for $(l_1(t), l_2(t)), 0 \leq t \leq t_*$, we obtained the value of parameter $$\beta=0.317. \label{eqn:beta}$$ Next, we used the data $l_k(t)$, $k=1,2$, and $3_+$ for $t=50, 60, \dots, 90 > t_*=45$ in Table \[tab:data\] and performed their nonlinear fitting to the first-order approximate solution given by Eq. (\[eqn:solution\_1st\]) of the system of nonlinear differential equations given by Eq. (\[eqn:diff\_eq2\]). Here, $A, \alpha$, and $\beta$ are fixed to be the values given by Eqs. (\[eqn:fitting1\]) and (\[eqn:beta\]), and $\gamma, \delta_1$, and $\delta_2$ are chosen as fitting parameters. They are evaluated as $$\begin{aligned} && \gamma = 0.313, \nonumber\\ && \delta_1=6.24 \times 10^{-7} \, \mbox{min$^{-1}\cdot\mu$m$^{-1}$}, \quad \delta_2=3.75 \times 10^{-8} \, \mbox{min$^{-1}\cdot\mu$m$^{-1}$}. \label{eqn:fitting2}\end{aligned}$$ The fitting is excellent as shown by Fig. \[fig:fitting1\]. The evaluation given by Eq. (\[eqn:fitting2\]) is consistent with the assumption $|\delta_i| \ll 1, i=1,2$, on which we solved the system of nonlinear differential equations given by Eq. (\[eqn:diff\_eq2\]) by perturbation in Sect. \[sec:diff\_eq\]. The nonlinearity is very small but necessary in the fitting. To demonstrate it, we show the 0th-order approximate solution given by Eq. (\[eqn:solution\_0th\]) as curves in Fig. \[fig:fitting2\]. Here, we used the same values of $A$, $\alpha$, and $\beta$ as in Fig. \[fig:fitting1\], but we put $\delta_1=\delta_2=0$. Figure \[fig:fitting2\] shows that if we ignore the nonlinear terms in Eq. (\[eqn:diff\_eq2\]), we fail to fit the data of $l_{3_+}(t), t \geq t_*$. Concluding Remarks ================== \[sec:remarks\] In this paper, we have reported that the growth process of cells of [*B. subtilis*]{} under hard-agar and nutrient-rich conditions allows the realization of the dynamics of a self-elongating filament with sequential folding on a plane. Such a multiple-fission process without cell separation is commonly observed in the early stage of the growth process even if the agar concentration is changed, while the structure and motion of an entangled filament of cells in the later stage depend on environmental conditions [@Men76; @MSL97; @WIMM97; @WKMM10; @KIT96]. Takeuchi [*et al.*]{} reported a study on filamentous cells of [*E.coli*]{} [@TDWW05]. Environmental conditions to realize such self-elongation of cell filaments should be clarified by a systematic study of the early stage of bacterial growth processes. The classification of the morphology and dynamics of a long filament of cells depending on environmental conditions, time periods, spatial and geometrical restrictions, and so forth will be an interesting future problem. Here, we have focused on the simplest situation wherein a filament of cells simply repeats folding processes as it elongates and isotropically spreads over a two-dimensional plate. Note that Mendelson and coworkers have very extensively studied filamentous cell growth in the situation wherein supercoiling processes create helical macrofibers and their chiral self-propulsion motion is observed [@Men76; @Men78; @MTKL95; @MSL97; @Men99; @MSWG00; @MST01; @MMT02; @MSRCT03]. Even in our simple case, it seems to be highly nontrivial to provide a proper description of the filament configuration, which rapidly becomes complicated as it elongates with sequential folding embedded in a plane. In this work, we have proposed describing the global development by the time-dependent fractal dimension $D(T)$ and the local folding processes by the time evolution of partitions $(l_k(t))_{k \geq 1}$ of the exponentially growing total length $L(t)$ of the filament of cells, where $k=1$ for the simple part and $k \geq 2$ for the $k$-fold parts. The analysis discussed in Sect. \[sec:analysis\] could be regarded as a mean-field-type approximation in the following sense. Let us consider a magnetic spin system on a lattice. In the mean-field theory, to describe a phase transition, we consider only the magnetization as an order parameter, which is obtained by averaging over spin configurations. The magnetization per spin $m(T,H)$ at temperature $T$ in an external magnetic field $H$ is calculated by approximating the correlated many-spin system by a single-spin system in a mean field generated by the surrounding spins, which is assumed to be proportional to $m(T, H)$. The proportionality coefficient can be called the effective coupling constant $J_{\rm eff}$. In this way, we obtain the self-consistency equation for $m(T, H)$ with the parameter $J_{\rm eff}$ in addition to the external parameters $T$ and $H$. If we want to compare the experimental data of a magnetization process of some material with the mean-field theory, the parameter $J_{\rm eff}$ should be evaluated by some additional experimental observation. In this analysis of the filament configuration with folding, we summed $l_k(t)$ over $k \geq 3$ to define $l_{3_+}(t)$. By this reduction of variables from the series $(l_k(t))_{k \geq 1}$ to the triplet $(l_1(t), l_2(t), l_{3_+}(t))$, we obtained the finite systems of coupled differential equations given by Eq. (\[eqn:diff\_eq1\]) for $0 \leq t \leq t_*$ and by Eq. (\[eqn:diff\_eq2\]) for $t \geq t_*$. They involve the parameters $\alpha, \beta, \gamma, \delta_1$, and $\delta_2$. We have evaluated these parameters as well as $t_*$ by experimental observations. As shown by Eqs. (5.16) and (5.17), the evaluated $\beta$ and $\gamma$ have almost the same value. We can verify that the quantities given by Eqs. (5.12) and (5.14) have finite values in the limit $\gamma \to \beta$, and then our first-order approximate solution given by Eq. (5.15) is also valid in the case where $\beta=\gamma$. We should note that if $\beta=1/3$ ($\gamma=1/3$), the ratio of the frequency of twofold-segment (fourfold-segment) creation to that of simple elongation of the single (twofold) part is given by $\beta/(1-\beta)=1/2$ ($\gamma/(1-\gamma)=1/2$). We will continue our study to answer the question whether this result, $\beta \simeq \gamma \simeq 1/3$, is universal. To improve the description, we need to carry out further studies on growing elastic filaments. Theoretical investigations can be found in the literature [@PHBC04; @WGP04; @GN06; @MLG13]. We hope that this experimental evaluations of parameters controlling the folding processes, which are described as $l_1 \to l_2$, $l_2 \to l_4$, and $l_1+l_2 \to l_3$ using our variables, will be useful for testing the validity of a possible theoretical consideration in the future. 0.5cm [**Acknowledgments**]{} JW was supported by a Chuo University Grant for Special Research and by a Grant-in-Aid for Exploratory Research (No. 15K13537) from Japan Society for the Promotion of Science. MK was supported in part by a Grant-in-Aid for Scientific Research (C) (No. 26400405) from Japan Society for the Promotion of Science. [9]{} M. Matsushita, F. Hiramatsu, N. Kobayashi, T. Ozawa, Y. Yamazaki, and T. Matsuyama, Biofilms [**1**]{}, 305 (2004). O. Rauprich, M. Matsushita, C. J. Weijer, F. Siegert, S. E. Esipov, and J. A. Shapiro, J. Bacteriol. [**178**]{}, 6525 (1996). J. Wakita, H. Itoh, T. Matsuyama, and M. Matsushita, J. Phys. Soc. Jpn. [**66**]{}, 67 (1997). H. Ito, J. Wakita, T. 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--- abstract: | In this article we consider the moduli space $\mathcal{H}_{g,n}$ of $n$-pointed smooth hyperelliptic curves of genus $g$. In order to get cohomological information we wish to make $\mathbb{S}_n$-equivariant counts of the numbers of points defined over finite fields of this moduli space. We find that there are recursion formulas in the genus that these numbers fulfill. Thus, if we can make $\mathbb{S}_n$-equivariant counts of $\mathcal{H}_{g,n}$ for low genus, then we can do this for every genus. Information about curves of genus zero and one is then found to be sufficient to compute the answers for hyperelliptic curves of all genera and with up to seven points. These results are applied to the moduli spaces of stable curves of genus two with up to seven points, and this gives us the $\mathbb{S}_n$-equivariant Hodge structure of their cohomology. Moreover, we find that the $\mathbb{S}_n$-equivariant counts of $\mathcal{H}_{g,n}$ depend upon whether the characteristic is even or odd, where the first instance of this dependence is for six-pointed curves of genus three. address: 'Department of Mathematics, KTH, S–100 44 Stockholm, Sweden' author: - Jonas Bergström bibliography: - 'cite.bib' title: Equivariant counts of points of the moduli spaces of pointed hyperelliptic curves --- Introduction ============ By virtue of the Lefschetz trace formula, counting points defined over finite fields of a space gives a way of finding information on its cohomology. In this article we wish to count points of the moduli space ${\mathcal H_{{g},{n}}}$ of $n$-pointed smooth hyperelliptic curves of genus $g$. On this space we have an action of the symmetric group ${\mathbb{S}}_n$ by permuting the marked points of the curves. To take this action into account we will make *${\mathbb{S}}_n$-equivariant* counts of the numbers of points of ${\mathcal H_{{g},{n}}}$ defined over finite fields. For every $n$ we will find simple recursive equations in the genus, for the equivariant number of points of ${\mathcal H_{{g},{n}}}$ defined over a finite field. Thus, if we can count these numbers for low genus, we will know the answer for every genus. The hyperelliptic curves will need to be separated according to whether the characteristic is odd or even, see Sections \[sec-repr\] and \[sec-repreven\], and the respective recursion formulas will in some cases be different, see Sections \[sec-u\], \[sec-reca\] and \[sec-ueven\]. When the number of marked points is at most seven, we use the fact that the base cases of the recursions only involve the genus zero case, which is dealt with in Section \[sec-gzero\], and previously known equivariant counts of points of the moduli space of pointed curves of genus one, to get equivariant counts for every genus. These results are presented for odd characteristic in Section \[sec-results\] and for even characteristic in Section \[sec-results2\]. If we consider the odd and even cases separately, then all these counts are polynomials when considered as functions of the number of elements of the finite field. For up to five points these polynomials do not depend upon the characteristic. But for six-pointed hyperelliptic curves there is a dependence, which appears for the first time for genus three. The results of this article are in agreement with the results on the ordinary Euler characteristic and the conjectures on the motivic Euler characteristic of some natural local systems on the moduli space of hyperelliptic curves of genus $3$, which are found in the article [@Bini-Geer] by Bini-van der Geer. There is also an agreement with the results of Getzler, in the article [@G-euler], on the ordinary Euler characteristic of these local systems when the genus is two. The connection is that computing the trace of Frobenius on all the local systems of at most some fixed weight is equivalent to making equivariant counts of the number of points of all the moduli spaces of pointed hyperelliptic curves with at most as many marked points as the weight. Finally, there is an agreement with the results of Tommasi in [@Orsolathesis], where the cohomology of ${\mathcal H_{{g},{2}}}$ is computed for all $g \geq 2$. Consider the moduli space of stable $n$-pointed curves of genus $g$. If the ${\mathbb{S}}_n$-equivariant count of points of this space, when considered as a function of the number of elements of the finite field, gives a polynomial, then we can determine the Hodge structure of its individual cohomology groups (see Theorem 3.4 in [@Mbar4] which is based on a result of van den Bogaart-Edixhoven in [@EB]). All curves of genus two are hyperelliptic and hence in Section \[sec-g2\] we can apply this theorem to the moduli space of stable $n$-pointed curves of genus $2$ for all $n$ up to seven. This computation also gives the Hodge Euler characteristic of the moduli space of smooth $n$-pointed curves of genus $2$ for all $n$ up to seven. These results on genus $2$ curves are all in agreement with the ones of Faber-van der Geer in [@Faber-Geer1] and [@Faber-Geer2]. Moreover, for $n$ up to three they were previously known by the work of Getzler in [@G-2] (see also [@Orsolathesis] mentioned above). Acknowledgements {#acknowledgements .unnumbered} ================ The method I shall use to count points of the moduli space of pointed hyperelliptic curves follows a suggestion by Nicholas M. Katz. I thank Bradley Brock for letting me read an early version of the article [@Brock]. I would also like to thank Carel Faber for all help. Equivariant counts {#sec-equiv} ================== The object of our concern is the moduli space ${\mathcal M_{{g},{n}}}$ of smooth $n$-pointed curves of genus $g$, and in particular the closed subset ${\mathcal H_{{g},{n}}}$ of ${\mathcal M_{{g},{n}}}$ corresponding to the hyperelliptic curves. These are smooth stacks defined over the integers on which we have an action of ${\mathbb{S}}_n$ by permuting the marked points on the curves. Recall that the $n$ points are ordered and distinct. Let $k$ be a finite field with $q$ elements and denote by $k_m$ a degree $m$ extension. Define $H_{g,n}$ to be the coarse moduli space of ${\mathcal H_{{g},{n}}} \otimes \bar k$ and let $F$ be the geometric Frobenius morphism. The purpose of this article is to make ${\mathbb{S}}_n$-equivariant counts of the number of points defined over $k$ of $H_{g,n}$. With this we mean a count, for each element $\sigma$ in ${\mathbb{S}}_n$, of the number of fixed points of $F \cdot \sigma$ acting on $H_{g,n}$. Note that these numbers only depend upon the cycle type $c(\sigma)$ of the permutation $\sigma$. Define $\mathcal{R}_{\sigma}$ to be the category of hyperelliptic curves of genus $g$ that are defined over $k$ together with marked points $(p_1,\ldots,p_n)$ defined over $\bar{k}$ such that $(F \cdot \sigma) (p_i)=p_i$ for all $i$. Points of $H_{g,n}$ are isomorphism classes of $n$-pointed hyperelliptic curves of genus $g$ defined over $\bar k$. For any pointed curve $X$ that is a representative of a point in $H_{g,n}^{F \cdot \sigma}$, there is an isomorphism from $X$ to the pointed curve $(F\cdot \sigma) X$. Using this isomorphism we can descend to an element of $\mathcal{R}_{\sigma}$ (see Lemma 10.7.5 in [@Katz-Sarnak]). Therefore, the number of $\bar k$-isomorphism classes of the category $\mathcal{R}_{\sigma}$ is equal to ${\lvertH_{g,n}^{F \cdot \sigma}\rvert}$. Fix an element $(C,p_1,\ldots,p_n)$ in $\mathcal{R}_{\sigma}$. The sum, over all $k$-isomorphism classes of $n$-pointed curves $(D,q_1,\ldots,q_n)$ that are $\bar{k}$-isomorphic to $(C,p_1,\ldots,p_n)$, of the reciprocal of the number of $k$-automorphisms of $(D,q_1,\ldots,q_n)$, is equal to one (see [@Geer] or [@Katz-Sarnak]). This enables us to go from $\bar k$-isomorphism classes to $k$-isomorphism classes: $${\lvertH_{g,n}^{F \cdot \sigma}\rvert} = \sum_{[Y] \in \mathcal{R}_{\sigma}/\cong_{\bar{k}}} 1 = \sum_{[Y] \in \mathcal{R}_{\sigma}/\cong_{\bar{k}}} \sum_{\substack{[X] \in \mathcal{R}_{\sigma}/\cong{k} \\ X\cong_{\bar{k}}Y}} \frac{1}{{\lvert{\mathrm{Aut}}_{k}(X)\rvert}} = \sum_{[X] \in \mathcal{R}_{\sigma}/\cong_{k}} \frac{1}{{\lvert{\mathrm{Aut}}_{k}(X)\rvert}}.$$ For any curve $C$ over $k$, define $C\bigl(c(\sigma)\bigr)$ to be the set of $n$-tuples of points $(p_1,\ldots,p_n)$ in $C(\bar{k})$ that fulfill $(F \cdot \sigma) (p_i)=p_i$. If $\sigma \in {\mathbb{S}}_n$ has $R_i$ cycles of length $N_i$ for $1 \leq i \leq M$, we can express the number of such $n$-tuples as $$\label{eq-sigma} {\lvertC\bigl(c(\sigma)\bigr)\rvert}=\prod_{i=1}^M \prod_{j=0}^{R_i-1} \Biggl(\sum_{d | N_i} \mu\biggl(\frac{N_i}{d}\biggr)\cdot{\lvertC(k_d)\rvert} -j \cdot N_i \Biggr),$$ where $\mu$ is the Möbius function. Fix a curve $C$ over $k$ and let $X_1, \ldots, X_m$ be representatives of the distinct $k$-isomorphism classes of the subcategory of $\mathcal{R}_{\sigma}$ of elements $(D,q_1,\ldots,q_n)$ where $D \cong_k C$. For each $X_i$ we can act with ${\mathrm{Aut}}_k(C)$ which gives an orbit lying in $\mathcal{R}_{\sigma}$ and where the stabilizer of $X_i$ is equal to ${\mathrm{Aut}}_k(X_i)$. Together the orbits of $X_1, \ldots, X_m$ will contain ${\lvertC\bigl(c(\sigma)\bigr)\rvert}$ elements and hence we obtain $$\label{eq-equiv} {\lvertH_{g,n}^{F \cdot \sigma}\rvert} = \sum_{[X] \in \mathcal{R}_{\sigma}/\cong_{k}} \frac{1}{{\lvert{\mathrm{Aut}}_{k}(X)\rvert}} = \sum_{[C] \in {\mathcal H_{{g}}}(k)/\cong_k} \frac{{\lvertC\bigl(c(\sigma)\bigr)\rvert}}{{\lvert{\mathrm{Aut}}_k(C)\rvert}}.$$ We will compute slightly different numbers than ${\lvertH_{g,n}^{F \cdot \sigma}\rvert}$, but which contain equivalent information. Let $C$ be a curve defined over $k$. The Lefschetz trace formula tells us that for all $m \geq 1$ $$\label{eq-fixC} {\lvertC(k_m)\rvert}={\lvertC_{\bar{k}}^{F^m}\rvert}=1+q^m-a_m(C) \;\; \text{where} \;\; a_m(C) = \mathrm{Tr} \bigl(F^m,H^1(C_{\bar{k}},{\mathbf{Q}_{\ell}}) \bigr).$$ If we consider equations and in view of equation we find that $${\lvertH_{g,n}^{F \cdot \sigma}\rvert} = \sum_{[C] \in {\mathcal H_{{g}}}(k)/\cong_k} \frac{1}{{\lvert\mathrm{Aut}_k(C)\rvert}} \cdot f_{\sigma}(q,a_1(C),\ldots,a_n(C))$$ where $f_{\sigma}(x_0,\ldots,x_n)$ is a polynomial with coefficients in ${\mathbf{Z}}$. Give the variable $x_i$ degree $i$. Then there is a unique monomial in $f_{\sigma}$ of highest degree, namely $x_{N_1}^{R_1} \cdots x_{N_M}^{R_M}$. The numbers which we will pursue will be the following. \[dfn-a\] For $g \geq 2$ define $$\label{eq-a} a_{N_1}^{R_1} \ldots a_{N_M}^{R_M}|_g := \sum_{[C] \in {\mathcal H_{{g}}}(k)/\cong_k} \frac{1}{{\lvert\mathrm{Aut}_k(C)\rvert}} \cdot \prod_{i=1}^M a_{N_i}(C)^{R_i}.$$ Here the $R_i$ and $N_i$ are positive integers. This expression will be said to have weight $\sum_{i=1}^M R_i N_i$. Define also $$a_0|_g := \sum_{C \in {\mathcal H_{{g}}}(k)/\cong_k} \frac{1}{{\lvert\mathrm{Aut}_k(C)\rvert}},$$ an expression of weight zero. If we have already computed for all weights less than $n$, then computing $a_{N_1}^{R_1} \ldots a_{N_M}^{R_M}|_g$ is equivalent to computing ${\lvertH_{g,n}^{F \cdot \sigma}\rvert}$. Representatives of hyperelliptic curves in odd characteristic {#sec-repr} ============================================================= Assume that the finite field $k$ has an odd number of elements. The hyperelliptic curves of genus $g \geq 2$ are the ones endowed with a degree two morphism to ${\mathbf{P}}^1$. This morphism induces a degree two extension of the function field of ${\mathbf{P}}^1$. If we consider hyperelliptic curves defined over the finite field $k$ and choose an affine coordinate $x$ on ${\mathbf{P}}^1$ then we can write this extension in the form $y^2=f(x)$, where $f$ is a square-free polynomial with coefficients in $k$ of degree $2g+1$ or $2g+2$. At infinity, we can describe the curve given by the polynomial $f$ in the coordinate $t=1/x$ by $y^2=t^{2g+2}\cdot f(1/t)$. \[dfn-Pg\] Let $P_g$ be the set of square-free polynomials with coefficients in $k$ and of degree $2g+1$ or $2g+2$. Write $C_f$ for the curve corresponding to the element $f$ in $P_g$. By construction, there exists for each $k$-isomorphism class of objects in ${\mathcal H_{{g}}}(k)$ an $f$ in $P_g$ such that $C_f$ is a representative. Moreover, the $k$-isomorphisms between curves corresponding to elements of $P_g$ are given by $k$-isomorphisms of their function fields. By the uniqueness of the linear system $g^1_2$ on a hyperelliptic curve, these isomorphisms must respect the inclusion of the function field of ${\mathbf{P}}^1$. The $k$-isomorphisms are therefore precisely the ones induced by elements of the group $G:=\mathrm{GL}_2(k) \times k^*/D$ where $$D:=\{(\Bigl(\begin{array}{cc} a & 0 \\ 0 & a \end{array} \Bigr),a^{g+1}) : a \in k^* \} \subset \mathrm{GL}_2(k) \times k^*$$ and where an element $$\gamma= [(\Bigl( \begin{array}{cc} a & b \\ c & d \end{array} \Bigr),e)] \in G$$ gives the isomorphism $$(x,y) \mapsto \left(\frac{ax+b}{cx+d},\frac{ey}{(cx+d)^{g+1}}\right).$$ This defines an action of $G$ on $P_g$, where $g\in G$ takes $f \in P_g$ to $\tilde f \in P_g$, with $$\tilde f(x)=\frac{(cx+d)^{2g+2}}{e^2} \cdot f (\frac{ax+b}{cx+d}).$$ Let us then put $$I:=1/{\lvertG\rvert}=(q^3-q)^{-1}(q-1)^{-1}.$$ Let $\chi_{2,m}$ be the quadratic character on $k_m$. Recall that it is the function that takes $\alpha \in k_m$ to $1$ if it is a square, to $-1$ if it is a nonsquare and to $0$ if it is $0$. With a square or a nonsquare we will always mean a nonzero element. If $C_f$ is the hyperelliptic curve corresponding to $f \in P_g$ then $$a_m(C_f) = -\sum_{\alpha \in {\mathbf{P}}^1(k_m)} \chi_{2,m} \bigl( f(\alpha) \bigr).$$ This follows directly from equation . We will now rephrase equation in terms of the elements of $P_g$. By what was said above, the stabilizer of an element $f$ in $P_g$ under the action of $G$ is equal to ${\mathrm{Aut}}_k(C_f)$ and hence $$\begin{gathered} \label{eq-afp} a_{N_1}^{R_1} \ldots a_{N_M}^{R_M}|_g = \sum_{[f] \in P_g/G} \frac{1}{{\lvert\mathrm{Stab}_G(f)\rvert}} \cdot \prod_{i=1}^M a_{N_i}(C_f)^{R_i} =\\ = \frac{1}{{\lvertG\rvert}} \cdot\sum_{f \in P_g} \prod_{i=1}^M a_{N_i}(C_f)^{R_i} = I \cdot \sum_{f \in P_g} \prod_{i=1}^M \Bigl(-\sum_{\alpha \in {\mathbf{P}}^1(k_{N_i})} \chi_{2,N_i} \bigl( f(\alpha) \bigr) \Bigr)^{R_i}.\end{gathered}$$ This can up to sign be rewritten as $$\label{eq-af} I \cdot \sum_{f \in P_g} \sum_{(\alpha_{1,1}, \ldots,\alpha_{M,R_M}) \in S} \prod_{i=1}^M \prod_{j=1}^{R_i} \chi_{2,N_i} \bigl( f(\alpha_{i,j}) \bigr),$$ where $S$ consists of all tuples $(\alpha_{1,1}, \ldots, \alpha_{1,R_1}, \alpha_{2,1}, \ldots, \alpha_{M,R_M})$ of points in ${\mathbf{P}}^1$ such that $\alpha_{ij} \in {\mathbf{P}}^1(k_{N_i})$. We will split the sum into parts, since for these parts we will be able to find recursive equations in $g$ (see Section \[sec-u\]). Begin by considering for a fixed tuple of points $(\alpha_{1,1},\ldots, \alpha_{M,R_M})$ in $S$. Say that $\alpha \in {\mathbf{P}}^1(k_{s})$, then if $\tilde s/s$ is even we have $\chi_{2,\tilde s} \bigl( f(\alpha) \bigr) = \chi_{2,s} \bigl( f(\alpha) \bigr)^2$ and if $\tilde s / s$ is odd we have $\chi_{2,\tilde s} \bigl( f(\alpha) \bigr) = \chi_{2,s} \bigl( f(\alpha) \bigr)$. Thus, if we introduce exponents, we can assume that for all $i$ and $j$, the smallest field of definition of $\alpha_{i,j}$ is equal to $k_{N_i}$. If for any $\alpha, \beta \in {\mathbf{P}}^1$ we have $F^s(\alpha)=\beta$ for some $s$, then $\chi_{2,i} \bigl( f(\alpha) \bigr) = \chi_{2,i} \bigl( f(\beta) \bigr)$ for all $i$. Hence, by changing the exponents if necessary, we can remove points from the given tuple $(\alpha_{1,1}, \ldots, \alpha_{M,R_M})$ such that the remaining points belong to distinct Frobenius orbits. Finally, we can assume that the exponents, are either one or two because for any $\alpha \in {\mathbf{P}}^1$ and any $s$ we have $\chi_{2,s} \bigl( f(\alpha) \bigr)^r = \chi_{2,s} \bigl( f(\alpha) \bigr)^2$ if $r$ is even and $\chi_{2,s} \bigl( f(\alpha) \bigr)^r = \chi_{2,s} \bigl( f(\alpha) \bigr)$ if $r$ is odd. We end up with the following definition. \[def-ug\] For any $g \geq -1$, any formal tuple $(n_1^{r_1},\ldots,n_m^{r_m})$ where $r_i$ is either $1$ or $2$ for all $i$, and any $\alpha=(\alpha_1,\ldots,\alpha_m)$ in the set $$A(n_1, \ldots, n_m):=\{(\beta_1, \ldots, \beta_m) : \beta_i \in {\mathbf{P}}^1(k_{n_i}), \, F^s(\beta_i)=\beta_j \implies i=j, \, n_i|s \},$$ put $$u_{g,\alpha}^{(n_1^{r_1},\ldots,n_m^{r_m})} := I \cdot\sum_{f \in P_g} \prod_{i=1}^{m} \chi_{2,n_i} \bigl( f(\alpha_{i}) \bigr)^{r_i}$$ and define $$u_{g}^{(n_1^{r_1},\ldots,n_m^{r_m})} := \sum_{\alpha \in A(n_1, \ldots, n_m)} u_{g,\alpha}^{(n_1^{r_1},\ldots,n_m^{r_m})}.$$ The argument above shows that $a_{N_1}^{R_1} \ldots a_{N_M}^{R_M}|_g$ can be decomposed in terms of $u_g$’s, for a finite number of tuples $(n_1^{r_1},\ldots,n_m^{r_m})$. \[dfn-gen\] In the decomposition of $a_{N_1}^{R_1} \ldots a_{N_M}^{R_M}|_g$ the case that $$\begin{cases} m=\sum_{l=1}^M R_l, \\ n_i=N_j \;\; \text{for}\;\; 1 \leq j \leq M \;\; \text{and} \;\; 1+\sum_{l=1}^{j-1} R_l \leq i \leq \sum_{l=1}^j R_l, \\ r_i=1 \;\; \text{for}\;\; 1 \leq i \leq m, \end{cases}$$ will be called the general case and all other cases will be called degenerations of the general one. \[dfn-n\] The number $n:=\sum_{i=1}^m n_i$ will be called the degree of the expression $u_{g}^{(n_1^{r_1},\ldots,n_m^{r_m})}$. \[rmk-decomp\] The general case is the only case in the decomposition which has degree equal to the weight of $a_{N_1}^{R_1} \ldots a_{N_M}^{R_M}|_g$. Note also that the decomposition does not depend upon the finite field $k$. \[lem-decr\] If $u_g^{(n_1^{r_1},\ldots,n_m^{r_m})}$ appears in the decomposition of $a_{N_1}^{R_1} \ldots a_{N_M}^{R_M}|_g$ then $$\sum_{i=1}^m r_i n_i \leq \sum_{i=1}^M R_i N_i,$$ and these two numbers have the same parity. Clear from construction. Let us decompose $a_2^2|_g$ starting with the general case: $$\begin{gathered} a_2^2|_g = I \cdot \sum_{f \in P_g} \Bigl(-\sum_{\alpha \in {\mathbf{P}}^1(k_2)} \chi_{2,2}\bigl(f(\alpha) \bigr) \Bigr)^2 = I \cdot \sum_{f \in P_g} \sum_{\alpha,\beta \in {\mathbf{P}}^1(k_2)} \chi_{2,2}\bigl(f(\alpha) f(\beta) \bigr) =\\ = u_g^{(2^1,2^1)}+2u_g^{(2^1,1^2)}+2u_g^{(2^2)}+u_g^{(1^2,1^2)}+u_g^{(1^2)}.\end{gathered}$$ \[exa-dec\] The decomposition of $a_1^4a_2|_g$, starting with the general case, equals $$\begin{gathered} a_1^4a_2|_g=-u_g^{(2^1,1^1,1^1,1^1,1^1)} - 6u_g^{(2^1,1^2,1^1,1^1)} - 3u_g^{(2^1,1^2,1^2)} - 4u_g^{(2^1,1^1,1^1)} - u_g^{(2^1,1^2)} \\- u_g^{(1^2,1^1,1^1,1^1,1^1)} - 6u_g^{(1^2,1^2,1^1,1^1)} - 4u_g^{(1^1,1^1,1^1,1^1)} - 3u_g^{(1^2,1^2,1^2)} \\ - 22u_g^{(1^2,1^1,1^1)} - 7u_g^{(1^2,1^2)} - 8u_g^{(1^1,1^1)} - u_g^{(1^2)}.\end{gathered}$$ The cases of genus $0$ and $1$ {#sec-g01} ------------------------------ We would like to have an equality of the same kind as in equation , but for curves of genus $0$ and $1$. Every curve of genus $0$ or $1$ has a morphism to ${\mathbf{P}}^1$ of degree two and in the same way as for larger genera, it then follows that every $k$-isomorphism class of curves of genus $0$ or $1$ has a representative among the curves coming from polynomials in $P_0$ and $P_1$ respectively. But there is a difference, compared to the larger genera, in that for curves of genus $0$ or $1$ the $g^1_2$ is not unique. In fact, not all $k$-isomorphisms between curves corresponding to elements of $P_0$ and $P_1$ are induced by elements of the group $G$. Let us, for all $r \geq 0$, define the category $\mathcal{A}_r$ consisting of tuples $(C,Q_0,\ldots,Q_r)$ where $C$ is a curve of genus $1$ defined over $k$ and the $Q_i$ are, not necessarily distinct, points on $C$ defined over $k$. The morphisms of $\mathcal{A}_r$ are, as expected, isomorphisms of the underlying curves that fix the marked points. Note that $\mathcal{A}_0$ is isomorphic to the category ${\mathcal M_{{1},{1}}}(k)$. We also define, for all $r \geq 0$, the category $\mathcal{B}_r$ consisting of tuples $(C,L,Q_1,\ldots,Q_r)$ of the same kind as above, but where $L$ is a $g^1_2$. A morphism of $\mathcal{B}_r$ is an isomorphism $\phi$ of the underlying curves that fixes the marked points, and such that there is an isomorphism $\tau$ making the following diagram commute: $$\begin{CD} C @>\phi>> C'\\ @VLVV @VVL'V\\ {\mathbf{P}}^1 @>\tau>> {\mathbf{P}}^1. \end{CD}$$ Consider $P_1$ as a category, where the morphisms are given by the elements of $G$. Then $P_1$ is naturally a full subcategory of $\mathcal{B}_0$. Moreover, we can identify $\mathcal{B}_0/\cong_k$ with $P_1/G$. For all $r \geq 1$ there are isomorphisms of the categories $\mathcal{A}_r$ and $\mathcal{B}_r$ given by $$(C,Q_0,\ldots,Q_r) \mapsto (C,|Q_0+Q_1|,Q_1,\ldots,Q_r),$$ with inverse $$(C,L,Q_1,\ldots,Q_r) \mapsto (C,|L-Q_1|,Q_1,\ldots,Q_r).$$ We therefore have the equality $$\sum_{[X]\in \mathcal{A}_r/\cong_k}\frac{1}{{\lvert{\mathrm{Aut}}_k(X)\rvert}}\cdot \prod_{i=1}^M a_{N_i}(C)^{R_i} = \sum_{[Y]\in \mathcal{B}_r/\cong_k}\frac{1}{{\lvert{\mathrm{Aut}}_k(Y)\rvert}}\cdot \prod_{i=1}^M a_{N_i}(C)^{R_i}.$$ Every genus $1$ curve has a point defined over $k$, and hence there is a number $s$ such that $1 \leq {\lvertC(k)\rvert} \leq s$ for all genus $1$ curves $C$. As in the argument preceding equation we can take a representative $(C,Q_0,\ldots,Q_r)$ for each element of $\mathcal{A}_r/\cong_k$ and act with ${\mathrm{Aut}}_k(C,Q_0)$, respectively for each representative $(C,L,Q_1,\ldots,Q_r)$ of $\mathcal{B}_0/\cong_k$ act with ${\mathrm{Aut}}_k(C,L)$, and by considering the orbits and stabilizers we get $$\begin{gathered} \sum_{j=1}^s j^r \cdot \sum_{\substack{[X]\in \mathcal{A}_0/\cong_k \\{\lvertC(k)\rvert}=j }}\frac{1}{{\lvert{\mathrm{Aut}}_k(X)\rvert}}\cdot \prod_{i=1}^M a_{N_i}(C)^{R_i} = \\ = \sum_{j=1}^s j^r \cdot\sum_{\substack{[Y]\in \mathcal{B}_0/\cong_k \\ {\lvertC(k)\rvert}=j}}\frac{1}{{\lvert{\mathrm{Aut}}_k(Y)\rvert}}\cdot \prod_{i=1}^M a_{N_i}(C)^{R_i}.\end{gathered}$$ Since this holds for all $r \geq 1$ we can, by a Vandermonde argument, conclude that we have an equality as above for each fixed $j$. We can therefore extend Definition \[dfn-a\] to genus $1$ in the following way $$\begin{gathered} \label{eq-g1a} a_{N_1}^{R_1} \ldots a_{N_M}^{R_M}|_1:=\sum_{\substack{[(C,Q_0)] \in \\{\mathcal M_{{1},{1}}}(k)/\cong_k}} \frac{1}{{\lvert\mathrm{Aut}_k(C,Q_0)\rvert}} \cdot \prod_{i=1}^M a_{N_i}(C)^{R_i} = \\ =\sum_{[f] \in P_1/G} \frac{1}{{\lvert\mathrm{Stab}_G(f)\rvert}} \cdot \prod_{i=1}^M a_{N_i}(C_f)^{R_i} = I \cdot \sum_{f \in P_{1}} \prod_{i=1}^M a_{N_i}(C_f)^{R_i},\end{gathered}$$ which gives an agreement with equation . All curves of genus $0$ are isomorphic to ${\mathbf{P}}^1$ and $a_r({\mathbf{P}}^1)=0$ for all $r \geq 1$. In this trivial case we just let equation  be the definition of $a_{N_1}^{R_1} \ldots a_{N_M}^{R_M}|_0$. Recursive equations for $u_g$ in odd characteristic {#sec-u} =================================================== This section will be devoted to finding, for a fixed tuple $(n_1^{r_1},\ldots,n_m^{r_m})$, a recursive equation for $u_g$. Fix a nonsquare $t$ in $k$ and an $\alpha=(\alpha_1,\ldots,\alpha_m)$ in $A(n_1, \ldots, n_m)$. Multiplying with the element $t$ gives a fixed point free action on the set $P_g$ and therefore $$\begin{gathered} u_{g,\alpha} = I \cdot \sum_{f \in P_g} \prod_{i=1}^m \chi_{2,n_i} \bigl( f(\alpha_{i})\bigr)^{r_{i}} = I \cdot \sum_{f \in P_g} \prod_{i=1}^m \chi_{2,n_i} \bigl( t \cdot f(\alpha_{i})\bigr)^{r_{i}} =\\ = I \cdot \sum_{f \in P_g} \prod_{i=1}^m \chi_{2,n_i}(t)^{r_{i}} \cdot \chi_{2,n_i} \bigl(f(\alpha_{i}) \bigr)^{r_{i}} = (-1)^{\sum_{i=1}^m r_i n_i} \cdot u_{g,\alpha}. \end{gathered}$$ From this computation and Lemma \[lem-decr\] we directly get the following lemma. \[lem-odd\] For any $\alpha \in A(n_1, \ldots, n_m)$ and $g \geq -1$, if $\sum_{i=1}^m r_i n_i$ is odd then $u_{g,\alpha}=0$. Consequently $a_{N_1}^{R_1} \ldots a_{N_M}^{R_M}|_g$ is equal to zero if it has odd weight. The last statement of Lemma \[lem-odd\] can also be found as a consequence of the existence of the hyperelliptic involution. We also see from the computation above that $$\label{eq-monic} u_{g,\alpha} = (q-1) \cdot I \cdot \sum_{f \in P'_g} \prod_{i=1}^m \chi_{2,n_i} \bigl( f(\alpha_{i})\bigr)^{r_{i}} \;\;\; \text{if $\sum_{i=1}^m r_in_i$ is even},$$ where $P'_g$ is the subset of $P_g$ consisting of monic polynomials. From now on we assume that $\sum_{i=1}^m r_in_i$ is even. \[dfn-r\] Define $r$ to be $0$ if all $r_i$ are equal to $2$, and $1$ if not. Recall also from Definition \[dfn-n\] that $n:=\sum_{i=1}^m n_i$. \[def-Ug\] Let $Q_g$ be the set of all polynomials with coefficients in $k$ and of degree $2g+1$ or $2g+2$. For a polynomial $h \in Q_g$ we let $h(\infty)$ be the coefficient of the term of degree $2g+2$. Then with the same conditions as in Definition \[def-ug\] put $$U_{g,\alpha}^{(n_1^{r_1},\ldots,n_m^{r_m})} := I \cdot \sum_{h \in Q_g} \prod_{i=1}^{m} \chi_{2,n_i} \bigl( h(\alpha_{i}) \bigr)^{r_i}$$ and define $$U_g^{(n_1^{r_1},\ldots,n_m^{r_m})} := \sum_{\alpha \in A(n_1, \ldots, n_m)} U_{g,\alpha}^{(n_1^{r_1},\ldots,n_m^{r_m})}.$$ We will find a relation between $U_g$ and $u_{i}$ for all $-1 \leq i \leq g$. The benefit of this relation is that for $g$ large enough we will be able to compute $U_g$. We will divide into two cases. The case that $n_i \geq 2$ for all $i$ {#sec-n2} -------------------------------------- Fix an element $\alpha$ in $A(n_1,\ldots,n_m)$. Since $n_i \geq 2$ for all $i$, we know that none of the points of the tuple $\alpha=(\alpha_1,\ldots,\alpha_m)$ is equal to infinity. Any nonzero polynomial $h$ can be written uniquely in the form $h=f \cdot l^2$ where $f$ is a square-free polynomial and $l$ is a monic polynomial. Moreover, for any $\beta$ in ${\mathbf{A}}^1(k_s)$, if $l(\beta)$ is nonzero, then $\chi_{2,s}\bigl(h(\beta)\bigr)=\chi_{2,s}\bigl(f(\beta)\bigr)$. \[dfn-b\] Let $b_j$ be the number of monic polynomials $l$ of degree $j$ such that $l(\alpha_i)$ is nonzero for all $i$. Let us also put ${\hat{b}}_j:=\sum_{i=0}^j b_i$. The relation above then translates into the equality $$\label{eq-Uua} U_{i,\alpha} = \sum_{j=0}^{i+1} b_j u_{i-j,\alpha}.$$ The numbers $b_j$ can be computed by the sieve method, which gives the result $$\label{eq-b} b_j=q^j+\sum_{i=1}^j (-1)^{i} \cdot \sum_{\substack{1 \leq m_1 < \ldots < m_i \leq m \\ \sum_{l=1}^i n_{m_l} \leq j}} q^{j-\sum_{l=1}^i n_{m_l}}$$ where the choice of $1 \leq m_1 < \ldots < m_i \leq m$ corresponds to demanding the polynomial to be zero in the points $\alpha_{m_1}, \ldots, \alpha_{m_i}$. From this formula it follows that $b_j$ does not depend upon the choice of $\alpha$ in $A(n_1,\ldots,n_m)$. Thus, summing equation over all $\alpha$ in $A(n_1,\ldots,n_m)$ and then over all $i$ between $-1$ and $g$ gives $$\label{eq-UUu} {\hat{U}}_g = \sum_{j=0}^{g+1} {\hat{b}}_j u_{g-j},$$ where ${\hat{U}}_g:=\sum_{i=-1}^g U_i$. Let us try to determine ${\hat{U}}_{g}$. If $r=0$ we can compute ${\hat{U}}_g$ directly: $$\label{eq-UUr0} {\hat{U}}_g= J \cdot {\hat{b}}_{2g+2} \quad \text{for $g\geq -1$ and $r=0$,}$$ where $J:=I \cdot (q-1)\cdot {\lvertA(n_1,\ldots,n_m)\rvert}$. Hence, if $r=0$ we can compute $u_g$ for all $g$, using equations , and . For the case $r=1$, fix again an element $\alpha=(\alpha_1,\ldots,\alpha_m)$ in $A(n_1,\ldots,n_m)$. Using the chinese remainder theorem we find that there is a one to one correspondence between the $n$ coefficients, of a polynomial $f$, of the terms of lowest degree and the tuple of values $(f(\alpha_1),\ldots,f(\alpha_m))$ in $k_{n_1} \times\ldots\times k_{n_m}$. In ${\hat{U}}_{g,\alpha}$ we are summing over all polynomials of degree less than or equal to $2g+2$. Hence if $2g+2 \geq n-1$ we get in the sum of ${\hat{U}}_{g,\alpha}$ an equal contribution from each tuple of values $(f(\alpha_1),\ldots,f(\alpha_m))$. Since for any $j$, half of the nonzero elements in $k_j$ are squares and half are nonsquares, we conclude that $$\label{eq-UUr1} {\hat{U}}_g={\hat{U}}_{g,\alpha}=0 \quad \text{for $g \geq (n-3)/2$ and $r=1$.}$$ Hence, if $r=1$ we can compute $u_g$ for all $g$, if we know $u_g$ for all $g < (n-3)/2$. The case when $n_i=1$ for some $i$ {#sec-n1} ---------------------------------- We can assume that $n_1=1$. As in the former case we will first fix an element $\alpha=(\alpha_1,\ldots,\alpha_m)$ in $A(n_1,\ldots,n_m)$. Since $n_1=1$ we can, using a projective transformation defined over $k$, put $\alpha_1$ at infinity. As long as the number of $i$ such that $n_i=1$ is not equal to the number of elements of ${\mathbf{P}}^1(k)$, we can use a projective transformation defined over $k$ to make sure that none of the points $\alpha_1,\ldots,\alpha_m$ is equal to infinity. In this case, we could directly apply the results of Section \[sec-n2\]. None of the points of the tuple $\tilde \alpha:=(\alpha_2,\ldots,\alpha_m)$ will be equal to infinity and thus we can apply equation which gives $$\label{eq-InfUub} U_{g,\tilde \alpha} = \sum_{j=0}^{g+1} b_j^{(n_2,\ldots,n_m)} u_{g-j,\tilde \alpha}.$$ Recall that for a polynomial $h$ in $Q_g$ we have defined $h(\infty)$ to be equal to the coefficient of the term of degree $2g+2$. Looking at equation we see that $u_{g,\alpha}$ is equal to the part of $u_{g,\tilde \alpha}$ coming from polynomials of even degree. But from the argument in the previous section we find that equation also holds for the ’even’ part of $u_{g,\tilde \alpha}$ and hence $$\label{eq-InfUua} U_{g,\alpha} = \sum_{j=0}^{g+1} b_j^{(n_2,\ldots,n_m)} u_{g-j,\alpha}.$$ Applying a projective transformation to the tuple of points $\alpha$ does not change the value of $u_{g,\alpha}$. Hence, summing equation over all $\alpha$ in $A(n_1,\ldots,n_m)$ gives $$\label{eq-InfUu} U_{g} = \sum_{j=0}^{g+1} b_j^{(n_2,\ldots,n_m)} u_{g-j}.$$ Let us determine $U_{g}$. If $r=0$ we directly get $$\label{eq-InfUr0} U_g= J \cdot b^{(n_2,\ldots,n_m)}_{2g+2} \quad \text{for $g\geq -1$, $r=0$}.$$ If $r=1$ we reduce the sum in $U_g$ in the same way as in equation to be over monic polynomials of degree $2g+2$. As in Section \[sec-n2\], we can then use the one to one correspondence between the $n-1$ coefficients, of a polynomial $f$, of the terms of lowest degree and the tuple of values $(f(\alpha_2),\ldots,f(\alpha_m))$ to conclude that $$\label{eq-InfUr1} U_g=U_{g,\alpha}=0 \quad \text{for $g \geq (n-3)/2$, $r=1$}.$$ The two cases joined {#sec-join} -------------------- In this section we will investigate the numbers $b_j$ further. In particular, this will enable us to connect the results of Section \[sec-n2\] and Section \[sec-n1\]. \[lem-b\] If $n_1=1$ then ${\hat{b}}^{(n_1,\ldots,n_m)}_j=b^{(n_2,\ldots,n_m)}_j$. Fix a tuple $(n_1,\ldots,n_m)$. If we let $t_i=q^{n_i}$ in the formula $$\prod_{i=1}^m(t_i-1)=t_1 \cdots t_m + \sum_{i=1}^m (-1)^{i} \cdot \sum_{1 \leq m_1 < \ldots < m_i \leq m} t_1 \cdots t_m \cdot \frac{1}{t_{m_1}} \cdots \frac{1}{t_{m_i}}$$ we see from equation that $$\label{eq-bn} \prod_{i=1}^m(q^{n_i}-1)=b_n.$$ Again, from equation we find that the coefficient of $q^i$ in $b_j$ is equal to the coefficient of $q^{n+i-j}$ in $b_n$. Hence the coefficient of $q^i$ in ${\hat{b}}_j$ is equal to the sum, over $k$ from $0$ to $j$, of the coefficients of $q^{n+i-k}$ in $b_n$. By equation we know that $q-1$ divides $b_n$. Hence the coefficient of $q^i$ in ${\hat{b}}_j$ is equal to the coefficient of $q^{n-1+i-j}$ in the polynomial $b_n/(q-1)$. Lemma \[lem-b\] shows that we can join equations , and with equations , and . \[thm-rec1\] For any tuple $(n_1^{r_1},\ldots,n_m^{r_m})$, $$\label{eq-rec1} \sum_{j=0}^{g+1} {\hat{b}}_j u_{g-j} = \begin{cases} J \cdot {\hat{b}}_{2g+2} & \text{if $r=0$, $g \geq -1;$} \\ 0 & \text{if $r=1$, $g \geq \frac{n-3}{2}$.} \end{cases}$$ From the description of the coefficients of ${\hat{b}}_j$ above, it follows directly that ${\hat{b}}_j-q{\hat{b}}_{j-1}$ is equal to the coefficient of $q^{n-1-j}$ in $b_n/(q-1)$. Taking formula for $g$, minus $q$ times the same formula but for $g-1$, therefore leaves us with the following theorem. \[thm-rec2\] For any tuple $(n_1^{r_1},\ldots,n_m^{r_m})$, $$\label{eq-rec2} \sum_{j=0}^{\min(n-1,g+1)} ({\hat{b}}_j-q {\hat{b}}_{j-1}) u_{g-j} = \begin{cases} J \cdot ({\hat{b}}_{2g+2}-q{\hat{b}}_{2g}) & \text{if $ r=0$, $g \geq 0;$} \\ 0 & \text{if $r=1$, $g \geq \frac{n-1}{2}.$}\end{cases}$$ For $g \geq (n-1)/2$, Theorem \[thm-rec2\] thus presents us with a linear recursion equation for $u_g$ which has coefficients that are independent of the finite field $k$. \[exa-u1\] Applying Theorem \[thm-rec2\] to $u_g^{(2^1,1^2,1^1,1^1)}$ we get for $g \geq 3$ $$u_g-2 u_{g-1}+2 u_{g-3} -u_{g-4}=0.$$ \[exa-u2\] Let us compute $u_g^{(1^2,1^2,1^2)}$ for all $g \geq -1$. We have that $u_{-1} = J = 1$ and since $r=0$, equation gives the equality $u_0 = 2 u_{-1} + J(q^2-3q+1) = q^2-3q+3$. Applying equation again, we get $$u_g-2 u_{g-1} + u_{g-2} = q^{2g-1}(q-1)^3 \quad \text{for $g \geq 1$}.$$ Solving this recursive equation gives $$u_g^{(1^2,1^2,1^2)}= \frac{q^{2g+3}(q-1)-(2g+2)(q^2-1)+3q+1}{(q+1)^2} \quad \text{for $g \geq -1$}.$$ Linear recursions for $a_{N_1}^{R_1} \ldots a_{N_M}^{R_M}|_g$ {#sec-reca} ============================================================= In the two previous sections we saw that we can decompose $a_{N_1}^{R_1} \ldots a_{N_M}^{R_M}|_g$ into parts for which we, by Theorem \[thm-rec2\], have linear recursive equations. We can put these recursions together to form one for the whole of $a_{N_1}^{R_1} \ldots a_{N_M}^{R_M}|_g$. In this section we will explicitly describe its homogenuous part. Note that the inhomogenities of the recursive equation for $a_{N_1}^{R_1} \ldots a_{N_M}^{R_M}|_g$ all come from $u_g$’s for which $r=0$, and that these $u_g$’s always can be computed using Theorem \[thm-rec1\]. \[thm-chareq\] The characteristic polynomial of the linear recursion we get for $a_{N_1}^{R_1} \ldots a_{N_M}^{R_M}|_g$ from repeated application of Theorem \[thm-rec2\], is equal to $$\label{eq-chareq} \frac{1}{\lambda-1} \prod_{i=1}^M(\lambda^{N_i}-1)^{R_i}.$$ As we saw in Section \[sec-join\], ${\hat{b}}_j-q{\hat{b}}_{j-1}$ is equal to the coefficient of $q^{n-1-j}$ in $b_n/(q-1)$. But if $g \geq n-2$, these numbers are also the coefficients in the recursion of Theorem \[thm-rec2\]. Hence it follows from equation that for the general case in the decomposition of $a_{N_1}^{R_1} \ldots a_{N_M}^{R_M}|_g$, the characteristic polynomial of the linear recursion is equal to . From equation we also find that the characteristic polynomials of all the degenerate cases divide the characteristic polynomial for the general case. Hence, the recursion we get for $a_{N_1}^{R_1} \ldots a_{N_M}^{R_M}|_g$ also has as characteristic polynomial. This tells us that if we can compute $a_{N_1}^{R_1} \ldots a_{N_M}^{R_M}|_g$ when $g+2$ is less than its weight, then we can compute it for every $g$. Note that we will often be able to do much better, for example in Section \[sec-results\] we will compute $a_{N_1}^{R_1} \ldots a_{N_M}^{R_M}|_g$ for every weight up to six using only information from curves of genus $0$ and $1$. In the case of $a_1^4a_2$ we get the characteristic polynomial $(\lambda-1)^4(\lambda+1)$ and hence, using Example \[exa-dec\] to separate the cases where $r=0$, $$a_1^4a_2|_g = -3u_g^{(1^2,1^2,1^2)}-7u_g^{(1^2,1^2)}-u_g^{(1^2)}+A_3 g^3+A_2 g^2+A_1 g+A_0+B_0 (-1)^g$$ where $A_0$, $A_1$, $A_2$, $A_3$ and $B_0$ do not depend upon $g$. Computing $u_0$ {#sec-gzero} =============== In the case of genus $0$ we will find that we can compute $u_g$ for any choice of tuple $(n_1^{r_1},\ldots,n_m^{r_m})$. This is due to the fact that if $C$ is a curve of genus $0$, then for all $n$, ${\lvertC(k_n)\rvert}=1+q^n$ or equivalently $a_n(C)=0$. Using this we will see that we can write $u_0$ for any tuple $(n_1^{r_1},\ldots,n_m^{r_m})$ as a sum of different $u_0$’s for which $r=0$. All these latter cases can then be computed using Theorem \[thm-rec1\]. Let us use induction over the number $n:=\sum_{i=1}^m n_i$ where the base case $n=0$ is trivial. By reordering we can assume that $r_m=1$, because otherwise we would already have $r=0$. Let $N$ be the set of numbers $j \neq m$ such that $n_j = n_m$ and let $\hat{{\mathbf{P}}}^1(k_{n_m})$ be the set of all points in ${\mathbf{P}}^1(k_{n_m})$ that are defined over a proper subfield of $k_{n_m}$. We then have $$\begin{gathered} \label{eq-g0} I \cdot \sum_{\substack{(\alpha_1,\ldots,\alpha_m) \\ \in A(n_1, \ldots, n_m)}} \prod_{i=1}^m \chi_{2,n_i} \bigl( f(\alpha_{i}) \bigr)^{r_{i}} = I \cdot \sum_{\substack{(\alpha_1,\ldots,\alpha_{m-1}) \\ \in A(n_1, \ldots, n_{m-1})}} \prod_{i=1}^{m-1} \chi_{2,n_i} \bigl( f(\alpha_{i}) \bigr)^{r_{i}} \cdot \\ \cdot \Bigl(-a_{n_m}(C_f) -\sum_{\alpha \in \hat{{\mathbf{P}}}^1(k_{n_m})} \chi_{2,n_m} \bigl( f(\alpha) \bigr) - n_m \cdot \sum_{j \in N} \chi_{2,n_m} \bigl( f(\alpha_{j}) \bigr) \Bigr).\end{gathered}$$ Summing the result of the computation over polynomials $f \in P_0$ we get on the left hand side $u_0$ for the tuple $(n_1^{r_1},\ldots,n_m^{r_m})$. The right hand side splits into a finite sum of $u_0$’s and if $(\tilde{n}_1^{\tilde{r}_1},\ldots,\tilde{n}_l^{\tilde{r}_l})$ is one of the tuples appearing then $\sum_{i=1}^l \tilde{n}_i < n$. By induction we are therefore done. Let us compute the first step in the reduction of $u_0^{(6^1,6^1,3^2,1^2,1^2)}$ using equation : $$\begin{gathered} u_0^{(6^1,6^1,3^2,1^2,1^2)}=u_0^{(6^1,3^2,1^2,1^2,6^1)} = -u_0^{(6^1,3^2,3^2,1^2,1^2)}-u_0^{(6^1,3^2,2^1,1^2,1^2)}\\-u_0^{(6^1,3^2,1^2,1^2,1^2)}-5u_0^{(6^1,3^2,1^2,1^2)}-6u_0^{(6^2,3^2,1^2,1^2)}.\end{gathered}$$ Let us fully reduce $u_0^{(4^1,1^2,1^1,1^1)}$ using equation : $$\begin{gathered} u_0^{(4^1,1^2,1^1,1^1)}=-u_0^{(4^1,1^2,1^2)}-u_0^{(4^1,1^1,1^1)} = u_0^{(2^2,1^2,1^2)}+u_0^{(1^2,1^2,1^2)}+2u_0^{(1^2,1^2)} +u_0^{(4^1,1^2)} \\= u_0^{(2^2,1^2,1^2)}+u_0^{(1^2,1^2,1^2)}+2u_0^{(1^2,1^2)}-u_0^{(2^2,1^2)}-u_0^{(1^2,1^2)}-u_0^{(1^2)} \\= u_0^{(2^2,1^2,1^2)}+u_0^{(1^2,1^2,1^2)}+u_0^{(1^2,1^2)}-u_0^{(2^2,1^2)}-u_0^{(1^2)}.\end{gathered}$$ Results for weight up to seven in odd characteristic {#sec-results} ==================================================== In this section we will find that we can compute all $a_{N_1}^{R_1} \ldots a_{N_M}^{R_M}|_g$ of weight at most seven. This will be achieved by decomposing $a_{N_1}^{R_1} \ldots a_{N_M}^{R_M}|_g$, in the manner described in Section \[sec-repr\], into a sum of $u_g$’s for tuples $(n_1,\ldots,n_m)$ and $(r_1,\ldots,r_m)$ with $\sum_{i=1}^m r_i n_i \leq \sum_{i=1}^M R_i N_i$. Then we will compute all the individual $u_g$’s that appear in the decomposition by employing the recursion formula of Theorem \[thm-rec1\]. This involves finding the necessary base cases for the recursions and that will be possible with the help of results on genus $0$ curves obtained in Section \[sec-gzero\] and on genus $1$ curves obtained below. We will write $a_{N_1}^{R_1} \ldots a_{N_M}^{R_M}|_{g,odd}$ and $u_{g,odd}^{(n_1^{r_1},\ldots,n_m^{r_m})}$ to stress that all results are in the case of odd characteristic. See Section \[sec-results2\] for results in the case of even characteristic. \[exa-a0\] Even if the degree is zero, Theorem \[thm-rec1\] is applicable, with $r=0$ and ${\hat{b}}_j=\sum_{i=0}^j q^i$. We then get from equation that $a_0|_{0,odd}=Jq^2=q/(q^2-1)$ and again from equation that $$a_0|_{g,odd}=J(q^{2g+2}-q^{2g})=q^{2g-1} \quad \text{for $g \geq 1$.}$$ This result can also be found in [@Brock]. Degree at most three {#sec-dthree} -------------------- When the degree is at most three we find using Theorem \[thm-rec1\] that we do not need any base cases to compute $u_{g}$ for every $g$. \[exa-w2\] Let us consider $u^{(2^1)}_{g,odd}$. We have $u_{-1}=J= 1/(q+1)$ and using equation we get $u_{0} = -(q+1) u_{-1}= -1$. Since equation tells us that $u_g=-u_{g-1}$ for $g \geq 1$ we conclude that $$u^{(2^1)}_{g,odd} =(-1)^{g+1} \quad \text{for $g \geq 0$}.$$ The result for $a_2|_{g,odd}$ is $$a_2|_{g,odd} = -u^{(2^1)}_g - u^{(1^2)}_g = (-1)^g - q^{2g} \quad \text{for $g \geq 0$}.$$ The result for $a_1^2|_{g,odd}$ is $$a_1^2|_{g,odd} = u^{(1^1,1^1)}_g + u^{(1^2)}_g = -1 + q^{2g} \quad \text{for $g \geq 0$}.$$ The result for $(q^2+1) \cdot a_0|_{g,odd}-a_2|_{g,odd}$ can be found in lecture notes by Bradley Brock and Andrew Granville from 28 July 2003. Consider the case $u_{g,odd}^{(1^2,1^1,1^1)}$. We have $u_{-1}=J=1$ and from equation we get $u_0=-(q-2)u_{-1}=-q+2$. Then gives the recursion equation $u_{g}=2u_{g-1}-u_{g-2}$ for $g \geq 1$ and hence $$u_{g,odd}^{(1^2,1^1,1^1)}=g(-q+1)-q+2.$$ Degree four or five {#sec-dfive} ------------------- From Theorem \[thm-rec1\] we find that when the degree is four or five we need the base case of genus $0$. But the genus $0$ case is always computable following Section \[sec-gzero\], and hence the same is true for $u_g$. For $u_{g,odd}^{(2^1,1^1,1^1)}$ we have $u_{-1}=J=q$ and from Section \[sec-gzero\] it follows that $$u_0^{(2^1,1^1,1^1)}=-u_0^{(2^1,1^2)}=u_0^{(1^2,1^2)}+u_0^{(1^2)}=q.$$ Using equation we get $u_1=-(q-1)u_0-(q^2-q-1)u_{-1}=-q^3+2q$. Solving the recursion equation $u_{g}=u_{g-1}-u_{g-2}-u_{g-3}$ for $g \geq 2$, coming from , gives $$u_{g,odd}^{(2^1,1^1,1^1)}=1/4 \cdot (q^3-q)(-2g+(-1)^g-1)+ q.$$ The result for $a_1^2a_2|_{g,odd}$ is $$\begin{gathered} a_1^2a_2|_{g,odd} = - u_g^{(2^1,1^1,1^1)}-u_g^{(2^1,1^2)}-u_g^{(1^2,1^1,1^1)}- u_g^{(1^2,1^2)} - 2u_g^{(1^1,1^1)}-u_g^{(1^2)} =\\=-\frac{q^{2g+2}-1}{q+1} -q^{2g} +1/2\cdot g(q^3+q-2) +1/2\cdot \begin{cases} 2q & \text{if $g \equiv 0$ mod $2$}\\q^3-q-2 & \text{if $g \equiv 1$ mod $2$}\end{cases}\end{gathered}$$ Weight six {#sec-wsix} ---------- We will not be able to compute all $u_{g}$’s of degree six. But we will find the $u_g$’s that are general cases in the decompositions of $a_{N_1}^{R_1} \ldots a_{N_m}^{R_m}|_g$’s of weight six. This will be sufficient to compute all $a_{N_1}^{R_1} \ldots a_{N_m}^{R_m}|_g$ of weight six, because we saw in Remark \[rmk-decomp\] that only the general case will have degree six and therefore all degenerate cases are covered in Sections \[sec-dthree\] and \[sec-dfive\]. Let $u_g$ be the general case in the decomposition of $a_{N_1}^{R_1} \ldots a_{N_m}^{R_m}|_g$. When the degree is equal to six we see from Theorem \[thm-rec1\] that we need the base cases of genus $0$ and $1$ to compute $u_g$ for all $g$. As we know, we can always compute $u_0$ following Section \[sec-gzero\]. For genus $1$, the numbers $a_{N_1}^{R_1} \ldots a_{N_m}^{R_m}|_1$ have been computed for weight up to six by the author. This was done by embedding every genus $1$ curve with a given point as a plane cubic curve, see [@Jonas2]. Then, since we know all the degenerate cases in the decomposition of $a_{N_1}^{R_1} \ldots a_{N_m}^{R_m}|_1$ we can compute the general case $u_1$. Let us deal with $u_{g,odd}^{(6^1)}$, for which we have $u_{-1}=J=q^3+q-1$. Then, by the use of Section \[sec-gzero\], $u_0^{(6^1)}=-u_0^{(3^2)}-u_0^{(2^1)}-u_0^{(1^2)}=-u_0^{(3^2)}=-q^2$. We know that $a_6|_1=q-1$, see above, and decomposing gives $a_6|_1=-u_1^{(6^1)}-u_1^{(3^2)}-u_1^{(2^1)}-u_1^{(1^2)}$. Thus using Example \[exa-w2\] we get $u_1=-(q-1)-(q^4-q^2-q-1)-1-q^2=-q^4+1$. We can now apply equation which gives $u_2=-(q+1)u_1-(q^2+q+1)u_0-(q^3+q^2+q+1)u_{-1}=-q^6+q^2-q$, $u_3=-u_2-u_1-u_0-u_{-1}=q^6+q^4-q^3$ and $u_4=-u_3-u_2-u_1-u_0-u_{-1}=0$. If we then multiply the characteristic polynomial for the linear recursion of $u_g$ by $\lambda-1$ we get $u_{g}=u_{g-6}$ for all $g \geq 5$. The result for $a_6|_{g,odd}$ is $$\begin{gathered} a_6|_{g,odd} =-u^{(6^1)}_g-u^{(3^2)}_g-u^{(2^1)}_g-u^{(1^2)}_g= -q^{2g}-\frac{q^{2g+3}(q-1)}{q^2-q+1}+\\+\frac{1}{q^2-q+1} \cdot \begin{cases}q^2 & \text{if $g \equiv 0$ mod $3$}\\-q^2-1 & \text{if $g \equiv 1$ mod $3$}\\1 & \text{if $g \equiv 2$ mod $3$}\end{cases} +\begin{cases} q^2+1 & \text{if $g \equiv 0$ mod $6$}\\q^4-2 & \text{if $g \equiv 1$ mod $6$} \\ q^6-q^2+q+1 & \text{if $g \equiv 2$ mod $6$} \\-q^6-q^4+q^3-1 & \text{if $g \equiv 3$ mod $6$} \\1 & \text{if $g \equiv 4$ mod $6$} \\-q^3-q & \text{if $g \equiv 5$ mod $6$} \end{cases}\end{gathered}$$ All $a_{N_1}^{R_1} \ldots a_{N_M}^{R_M}|_{g,odd}$ of weight at most seven are found to be polynomials when considered as functions of $q$. Representatives of hyperelliptic curves in even characteristic {#sec-repreven} ============================================================== Let $k$ be a finite field with an even number of elements. We will again describe the hyperelliptic curves of genus $g \geq 2$ defined over $k$ by their degree two morphism to ${\mathbf{P}}^1$. If we choose an affine coordinate $x$ on ${\mathbf{P}}^1$ we can write the induced degree two extension of the function field of ${\mathbf{P}}^1$ in the form $y^2+h(x)y+f(x)=0$, where $h$ and $f$ are polynomials defined over $k$ that fulfill the following conditions: $$\begin{gathered} \label{eq-deg} 2g+1 \leq \max\bigl(2 \deg(h),\deg(f)\bigr) \leq 2g+2; \\ \label{eq-nons} \gcd(h,f'^2+fh'^2) = 1; \\ \label{eq-inf} t \nmid \gcd(h_{\infty},f_{\infty}'^2+f_{\infty}h_{\infty}'^2). \end{gathered}$$ The last condition comes from the nonsingularity of the point(s) in infinity, around which the curve can be described in the variable $t=1/x$ as $y^2+h_{\infty}(t)y+f_{\infty}(t)=0$ where $h_{\infty}:=t^{g+1}h(1/t)$ and $f_{\infty}:=t^{2g+2}f(1/t)$. Let $P_g$ be the set of pairs $(h,f)$ of polynomials defined over $k$, where $h$ is nonzero, that fulfill all three conditions , and . Write $C_{(h,f)}$ for the curve corresponding to the element $(h,f)$ in $P_g$. To each $k$-isomorphism class of objects in ${\mathcal H_{{g}}}(k)$ there is a pair $(h,f)$ in $P_g$ such that $C_{(h,f)}$ is a representative. All $k$-isomorphisms between the curves represented by elements of $P_g$ are given by $k$-isomorphisms of their function fields, and since the $g^1_2$ of a hyperelliptic curve is unique the $k$-isomorphisms must respect the inclusion of the function field of ${\mathbf{P}}^1$. The $k$-isomorphisms are therefore precisely the ones induced by elements of the group $G_g:=\mathrm{GL}_2(k) \times k^* \times k^{g+2}/D$ where $$D:=\{(\Bigl(\begin{array}{cc} a & 0 \\ 0 & a \end{array} \Bigr),a^{g+1},0) : a \in k^* \} \subset \mathrm{GL}_2(k) \times k^* \times k^{g+2}$$ and where an element $$\gamma= [(\Bigl( \begin{array}{cc} a & b \\ c & d \end{array} \Bigr),e,l(x))] \in G_g$$ gives the isomorphism $$(x,y) \mapsto \left(\frac{ax+b}{cx+d},\frac{ey+l(x)}{(cx+d)^{g+1}}\right).$$ This defines an action of $G_g$ on $P_g$. Let $\tau_m$ be the function that takes $(a,b) \in k_m^2$ to $1$ if the equation $y^2+ay+b$ has two roots defined over $k_m$, $0$ if it has one root and $-1$ if it has none. If $C_{(h,f)}$ is the hyperelliptic curve corresponding to $(h,f) \in P_g$ then $$a_m(C_{(h,f)})=-\sum_{\alpha \in {\mathbf{P}}^1(k_m)} \tau_m \bigl(h(\alpha),f(\alpha) \bigr).$$ This follows directly from equation . Let us define $I_g:=1/{\lvertG_g\rvert}$. In the same way as in the case of odd characteristic we get the equality $$a_{N_1}^{R_1} \ldots a_{N_M}^{R_M}|_g = I_g \cdot \sum_{(h,f) \in P_g} \prod_{i=1}^M \Bigl(-\sum_{\alpha \in {\mathbf{P}}^1(k_{N_i})} \tau_{N_i} \bigl(h(\alpha),f(\alpha) \bigr) \Bigr)^{R_i}.$$ All results of Section \[sec-g01\] are independent of the characteristic and hence we extend the definition of $a_{N_1}^{R_1} \ldots a_{N_M}^{R_M}|_g$ to genus zero and one, in the same way as in that section. \[def-ugeven\] For any $g \geq -1$, any formal tuple $(n_1^{r_1},\ldots,n_m^{r_m})$ such that $r_i$ is either $1$ or $2$ for all $i$, and any $\alpha=(\alpha_1,\ldots,\alpha_m)$ in $A(n_1, \ldots, n_m)$ put $$u_{g,\alpha}^{(n_1^{r_1},\ldots,n_m^{r_m})} := I_g \cdot \sum_{(h,f) \in P_g} \prod_{i=1}^m \tau_{n_i} \bigl(h(\alpha_i),f(\alpha_{i}) \bigr)^{r_{i}}$$ and define $$u_{g}^{(n_1^{r_1},\ldots,n_m^{r_m})}:= \sum_{\alpha \in A(n_1, \ldots, n_m)} u_{g,\alpha}^{(n_1^{r_1},\ldots,n_m^{r_m})}.$$ With this definition we get the same decomposition of $a_{N_1}^{R_1} \ldots a_{N_M}^{R_M}|_g$ into $u_g$’s for a finite number of tuples $(n_1^{r_1},\ldots,n_m^{r_m})$ as in the odd case. Recursive equations for $u_g$ in even characteristic {#sec-ueven} ==================================================== This section will, analogously to Section \[sec-u\], be devoted to finding for a fixed tuple $(n_1^{r_1},\ldots,n_m^{r_m})$ a recursive equation for $u_g$. Fix an $s$ in $k$ which does not lie in the set $\{r^2+r: r \in k \}$, that is, such that $\tau_1(1,s)=-1$. We define an involution on $P_g$ sending $(h,f)$ to $(h,f+s \cdot h^2)$. This involution is fixed point free and hence $$\begin{gathered} u_{g,\alpha} = I_g \cdot \sum_{(h,f) \in P_g} \prod_{i=1}^m \tau_{n_i} \bigl( h(\alpha_i), f(\alpha_{i})\bigr)^{r_{i}} = \\ = I_g \cdot \sum_{(h,f) \in P_g} \prod_{i=1}^m \tau_{n_i} \bigl( h(\alpha_i), f(\alpha_{i}) + s \cdot h^2(\alpha_i)\bigr)^{r_{i}} =(-1)^{\sum_{i=1}^m r_i n_i} \cdot u_{g,\alpha}. \end{gathered}$$ Thus, Lemma \[lem-odd\] also holds in the case of even characteristic. Let $Q_g$ be the set of pairs $(h,f)$ of polynomials over $k$, where $h$ is nonzero and $h$, $f$ are of degree at most $g+1$, $2g+2$ respectively. For a pair $(h,f) \in Q_g$ let $h(\infty)$ and $f(\infty)$ be equal to the degree $g+1$ and $2g+2$ coefficient respectively. Then with the same conditions as in Definition \[def-ugeven\] put $${\hat{U}}_{g,\alpha}^{(n_1^{r_1},\ldots,n_m^{r_m})} := I_g \cdot \sum_{(h,f) \in Q_g} \prod_{i=1}^m \tau_{n_i} \bigl(h(\alpha_i),f(\alpha_{i}) \bigr)^{r_{i}}$$ and define $${\hat{U}}_{g}^{(n_1^{r_1},\ldots,n_m^{r_m})} :=\sum_{\alpha \in A(n_1, \ldots, n_m)}{\hat{U}}_{g,\alpha}^{(n_1^{r_1},\ldots,n_m^{r_m})}.$$ From now on we assume that $\sum_{i=1}^m r_i n_i$ is even. We wish to express the sum ${\hat{U}}_g$ in terms of the sums $u_i$ for $i$ between $-1$ and $g$, because for high enough values of $g$ we will be able to determine ${\hat{U}}_g$. The connection between the sets $Q_g$ and $P_g$ which we will present below is due to Brock and Granville and can be found in an early version of [@Brock]. There the connection is used to count the number of hyperelliptic curves in even characteristic, which is $a_0|_{g,even}$ in our terminology. \[lem-reform\] Let $h$ and $f$ be polynomials over $k$. For any irreducible polynomial $m$ over $k$, the following two statements are equivalent: - $m|\gcd(h,{f'}^2+f{h'}^2)$; - there is a polynomial $l$ over $k$, such that $m|h$ and $m^2 | f+hl+l^2$. Say that $\alpha \in k_n$ is a root of an irreducible polynomial $m$ and of the polynomial $\gcd(h,{f'}^2+f{h'}^2)$. Let $l$ be equal to $f^{q^n/2}$. Working modulo $(x-\alpha)^2$ we then get $$\begin{gathered} f+hl+l^2 = f+hf^{q^n/2}+f^{q^n} \equiv f(\alpha)+ f'(\alpha)(x-\alpha)+h'(\alpha)f(\alpha)^{q^n/2}(x-\alpha)+f(\alpha)^{q^n} \\\equiv (x-\alpha)(f'(\alpha)+h'(\alpha)f(\alpha)^{q^n/2}) \equiv (x-\alpha)(f'(\alpha)^2+h'(\alpha)^2f(\alpha))^{1/2} = 0,\end{gathered}$$ which tells us that $m^2|f+hl+l^2$. For the other direction, assume that we have an irreducible polynomial $m$ and a polynomial $l$ such that $m|h$ and $m^2|f+hl+l^2$. Differentiating the polynomial $f+hl+l^2$ we get that $m^2|f'+h'l+hl'$ and thus $m|f'+h'l$. Taking squares, we get $m^2|f'^2+h'^2l^2$ and then it follows that $m^2|f'^2+h'^2(f+hl)$ and hence $m|f'^2+h'^2f$. Let $(h,f)$ be an element of $Q_g$. In the first part of the proof of Lemma \[lem-reform\], we may take for $l$ any representative of $f^{q^n/2}$ modulo $h$, because for these $l$ we have $f+hl+l^2 \equiv f+hf^{q^n/2}+f^{q^n}$ modulo $(x-\alpha)^2$. In the second part it does not matter which degree $l$ has. We conclude from this that Lemma \[lem-reform\] also holds if we assume that $l$ is of degree at most $g+1$. Choose $g \geq -1$ and let $(h,f) \in Q_g$. Lemma \[lem-reform\] gives the following alternative formulation of the conditions , and . For all polynomials $l$ of degree at most $g+1$, $$\begin{gathered} \label{eq-nons2} m|h, \; m^2|f+hl+l^2 \implies \deg(m)=0; \\ \label{eq-inf2} \deg(h)=g+1 \quad \text{or} \quad \deg(f+hl+l^2) \geq 2g+1.\end{gathered}$$ Here we use that $t | \gcd(h_{\infty},f_{\infty}'^2+f_{\infty}h_{\infty}'^2)$ if and only if $t|h_{\infty}$ and there exists a polynomial $l_{\infty}$ such that $\deg(l_{\infty})\leq g+1$ and $t^2|f_{\infty}+h_{\infty}l_{\infty}+l_{\infty}^2$. In turn, this happens if and only if $\deg(h) \leq g$ and there exists a polynomial $l$ of degree at most $g+1$ such that $\deg(f+hl+l^2) \leq 2g$, where we connect $l$ and $l_{\infty}$ using the definitions $l:=x^{g+1}l_{\infty}(1/x)$ and $l_{\infty}:=t^{g+1}l(1/t)$. This reformulation leads us to making the following definition. Let $\sim_g$ be the relation on $Q_g$ given by $(h,f)\sim_g(h,f+hl+l^2)$ if $l$ is a polynomial of degree at most $g+1$. This is an equivalence relation and since $(h,f)=(h,f+hl+l^2)$ if and only if $l=0$ or $l=h$, the number of elements of each equivalence class $[(h,f)]_g$ is $q^{g+2}/2$. If $(h,f) \in P_g \subset Q_g$ then $[(h,f)]_g \subset P_g$ and we get an induced equivalence relation on $P_g$ which we also denote $\sim_g$. We will now construct all $\sim_g$ equivalence classes of elements of $Q_g$ in terms of the $\sim_i$ equivalence classes of the elements in $P_i$, where $i$ is between $-1$ and $g$. This is the counterpart of factoring a polynomial into a square-free part and a squared part as we did in the case of odd characteristic. For $z:=[(h,f)]_i \in P_i/\sim_i$ let $V_z$ be the set of all equivalence classes $[(mh,m^2f)]_{g}$ in $Q_g$ for all monic polynomials $m$ of degree at most $g-i$. This is well defined since if $(h_1,f_1) \sim_i (h_2,f_2)$ then $(mh_1,m^2f_1) \sim_g (mh_2,m^2f_2)$. \[lem-disjoint\] The sets $V_z$ for all $z \in P_i/\sim_i$ where $-1 \leq i \leq g$ are disjoint. Say that for some $z_1$ and $z_2$ the intersection $V_{z_1} \cap V_{z_2}$ is nonempty. That is, there exist $(h_1,f_1) \in P_{i_1}$, $(h_2,f_2) \in P_{i_2}$ and monic polynomials $m_1$, $m_2$ such that $m_1 h_1 = m_2 h_2$ and $m_1^2 f_1 = m_2^2f_2+m_2h_2l+l^2$. If for some irreducible polynomial $r$ we have $r|m_1$ but $r \nmid m_2$, it follows that $r|h_2$ and $r^2|m_2^2f_2+m_2h_2l+l^2$. By the equivalence of conditions and , this implies that $r|(m_2^2f_2)'^2+m_2^2f_2(m_2h_2)'^2$ which in turn implies that $r|f_2'^2+f_2h_2'^2$. Since $(h_2,f_2) \in P_{i_2}$ we see that $r$ must be constant. Hence every irreducible factor of $m_1$ is a factor of $m_2$. The situation is symmetric and therefore the converse also holds. So far we have not ruled out the possibility that a factor in $m_1$ appears with higher multiplicity than in $m_2$, or vice versa. Let $m$ be the product of all irreducible factors of $m_1$ and put $\tilde m_1:=m_1/m$, $\tilde m_2:=m_2/m$ and $\tilde l :=l/m$. We are then in the same situation as above, that is $\tilde m_1 h_1 = \tilde m_2 h_2$ and $\tilde m_1^2 f_1 = \tilde m_2^2f_2+\tilde m_2 h_2 \tilde l+\tilde l^2$. Thus, if $r$ is an irreducible polynomial such that $r|\tilde m_1$ but $r \nmid \tilde m_2$ we can argue as above to conclude that $r$ is constant. By a repeated application of this line of reasoning, we can conclude that $m_1$ and $m_2$ must be equal. It now follows that $h_1=h_2$ and that $m_2|l$, thus $(h_1,f_1) \sim_{i_1} (h_2,f_2)$. This tells us that $V_{z_1} \cap V_{z_2}$ is nonempty only when $z_1=z_2$. \[lem-cover\] The sets $V_z$ for all $z \in P_i/\sim_i$ where $-1 \leq i \leq g$ cover $Q_g/\sim_g$. Pick any element $(h_1,f_1) \in Q_{g}$ and put $g_1:=g$. We define a procedure, where at the $i$:th step we ask if there are any polynomials $m_i$ and $l_i$ such that $\deg(m_i) > 0$, $\deg(l_i) \leq g_i+1$, $m_i|h_i$ and $m_i^2|f_i+h_il_i+l_i^2$. If so, take any such polynomials $m_i$, $l_i$ and define $h_{i+1}:=h_i/m_i$, $f_{i+1}:=(f_i+h_il_i+l_i^2)/m_i^2$ and $g_{i+1}:=g_i-\deg(m_i)$. This procedure will certainly stop. Assume that the procedure has been carried out in some way and that it has stopped at the $j$:th step, leaving us with some pair of polynomials $(h_j,f_j)$. Next, we take $(h_j,f_{j+1})$ to be any element of the set $[(h_j,f_j)]_{g_j}$ for which $\deg(f_{j+1})$ is minimal. Say that $f_{j+1}=f_j+h_jl_j+l_j^2$ where $\deg(l_j) \leq g_j+1$ and let us define $g_{j+1}$ to be the number such that $2g_{j+1}+1 \leq \max\bigl(2 \deg(h_j),\deg(f_{j+1})\bigr) \leq 2g_{j+1}+2$. The claim is now that $(h_j,f_{j+1}) \in P_{g_{j+1}}$. By definition, condition holds for $(h_j,f_{j+1})$. If there were polynomials $m_{j+1}$ and $l_{j+1}$ such that $m_{j+1}|h_j$ and $m_{j+1}^2|f_{j+1}+h_jl_{j+1}+l_{j+1}^2$ then the pair of polynomials $m_{j+1}$ and $l_j+l_{j+1}$ would contradict that the process above stopped at the $j$:th step. Hence condition is fulfilled for $(h_j,f_{j+1})$. Condition is fulfilled if $2 \deg(h_j) \geq\deg(f_{j+1})$ because then $\deg(h_j)=g_{j+1}+1$. On the other hand, if $2 \deg(h_j) < \deg(f_{j+1})$ and there were a polynomial $l_{j+1}$ such that $\deg(l_{j+1}) \leq g_{j+1}+1$ and $\deg(f_{j+1}+h_jl_{j+1}+l_{j+1}^2) \leq 2g_{j+1}$ then this would contradict the minimality of $\deg(f_{j+1})$. We conclude that $(h_j,f_{j+1}) \in P_{g_{j+1}}$. Finally, we see that if we put $\hat m_r:=\prod_{i=1}^{r-1} m_{i}$ and $l:=\sum_{i=1}^{j} \hat m_i l_i$, then $\deg(l) \leq g+1$, $h_1=\hat m_j h_j$ and $f_1=\hat m_j^2 f_{j+1}+h_1l+l^2$. This shows that $V_z$ contains $[(h_1,f_1)]_g$ where $z:=[(h_j,f_{j+1})]_{g_{j+1}} \in P_{g_{j+1}}/\sim_{g_{j+1}}$. The case that $n_i \geq 2$ for all $i$ {#sec-n2even} -------------------------------------- As promised, we will see here that Lemmas \[lem-disjoint\] and \[lem-cover\] give us a way of writing ${\hat{U}}_g$ in terms of $u_i$ for $i$ between $-1$ and $g$. We will then be able to determine ${\hat{U}}_g$ for large enough values of $g$. Since $n_i \geq 2$ we have $A(n_1,\ldots,n_m) \subset {\mathbf{A}}^m$. Fix elements $z=[(h_0,f_0)]_i \in P_i/\sim_i$ and $\alpha \in {\mathbf{A}}^1(k_s)$ and define $V'_z$ to be the subset of $V_z$ of classes $[(\tilde m h_0,\tilde m^2f_0)]_g$, where $\tilde m$ is a monic polynomial with $\tilde m(\alpha) \neq 0$. Then $\tau_s(h(\alpha),f(\alpha))$ is constant for all $s$ and $(h,f)$ such that $[(h,f)]_g \in V'_z$. This follows directly from the following two properties of $\tau$. Choose any $s \geq 1$ and $t_1,t_2$ in $k_s$. We then have $$\begin{gathered} \label{eq-tau1} \tau_s(vt_1,v^2t_2)=\tau_s(t_1,t_2) \quad \text{for all $v\neq 0 \in k_s$;} \\ \label{eq-tau2} \tau_s(t_1,t_2+vt_1+v^2)=\tau_s(t_1,t_2) \quad \text{for all $v \in k_s$}.\end{gathered}$$ Recalling Definition \[dfn-b\], we have shown that for any $\alpha$ in $A(n_1,\ldots,n_m)$ $$\label{eq-receven} {\hat{U}}_{g,\alpha} = \sum_{i=0}^{g+1}{\hat{b}}_{i} u_{g-i,\alpha} \cdot q^i \cdot \frac{I_{g}}{I_{g-i}},$$ where we have taken into account that the group of isomorphisms depends upon $g$ and that the numbers of elements of the equivalence classes of the relations $\sim_{g-i}$ and $\sim_g$ differ by a factor $q^{i}$. From the definitions we see that $q^i \cdot I_{g}/I_{g-i}=1$. Let us deal with the numbers ${\hat{U}}_{g,\alpha}$. First, recall Definitions \[dfn-n\] and \[dfn-r\]. For any tuple $(\alpha_1,\ldots,\alpha_m)$ in $A(n_1,\ldots,n_m)$, there is a one to one correspondence between $n$ coefficients of a polynomial $f$ and the tuple of values $(f(\alpha_1),\ldots,f(\alpha_m))$ in $k_{n_1}\times \ldots \times k_{n_m}$. Hence, if we fix any $g$ such that $2g+2 \geq n-1$, any nonzero polynomial $h_0$ of degree at most $g+1$ and any tuple $(\alpha_1,\ldots,\alpha_m)$ in $A(n_1,\ldots,n_m)$, we get $$\sum_{(h_0,f) \in Q_g}\prod_{i=1}^m \tau_{n_i} \bigl(h_0(\alpha_i),f(\alpha_{i}) \bigr)^{r_{i}} = \begin{cases}0 & \text{if $r=1$}; \\0 & \text{if $r=0$, $h_0(\alpha_i)=0$ for some $i$}; \\q^{2g+3} & \text{if $r=0$, $h_0(\alpha_i) \neq 0$ for all $i$}, \end{cases}$$ because for all $a\in k_s$ there are as many $b \in k_s$ for which $\tau_s(a,b)=1$ as there are $b \in k_s$ for which $\tau_s(a,b)=-1$. Summing over all elements in $A(n_1,\ldots,n_m)$, we get $$\label{eq-UUeven} {\hat{U}}_g = \begin{cases} Jq^{g+1} {\hat{b}}_{g+1} & \text{if $r=0$, $g \geq -1$;} \\0 & \text{if $r=1$, $g \geq \frac{n-3}{2}$,} \end{cases}$$ where $J:=I_g \cdot q^{g+2} \cdot (q-1)\cdot {\lvertA(n_1,\ldots,n_m)\rvert}= (q^3-q)^{-1}\cdot {\lvertA(n_1,\ldots,n_m)\rvert}$. The case when $n_i=1$ for some $i$ {#sec-n1even} ---------------------------------- We can assume that $n_1=1$. Fix an $\alpha=(\alpha_1,\ldots,\alpha_m)$ in $A(n_1,\ldots,n_m)$ and use a projective transformation to put $\alpha_1$ at infinity. For any $(h,f) \in Q_g$ it holds that if $\mathrm{deg}(h) < g+1$ then $\tau_s(h(\infty),f(\infty))=0$ for all $s$. Define therefore $P'_g$ and $Q'_g$ to be the subsets of $P_g$ and $Q_g$ respectively, that consist of pairs $(h,f)$ such that $\mathrm{deg}(h)=g+1$. We get an induced relation $\sim_i$ on $P'_i$ and $Q'_i$ and we let $V'_z$ be the set of all equivalence classes $[(mh,m^2f)]_{g}$ in $Q'_g$ for all monic polynomials $m$ of degree $g-i$, where $z:=[(h,f)]_i \in P'_i/\sim_i$. In the same way as in Lemma \[lem-disjoint\] and \[lem-cover\] we see that the sets $V'_z$ for all $z \in P'_i/\sim_i$, where $-1 \leq i \leq g$, are disjoint and cover $Q'_g/\sim_g$. For any element $(h,f)\in P'_i$ and any monic polynomial $m$ of degree $g-i$, we have $$\begin{gathered} \tau_s((mh)(\infty),(m^2f)(\infty))=\tau_s(h(\infty),f(\infty)); \\ \tau_s((mh)(\infty),(f+lh+l^2)(\infty))=\tau_s(h(\infty),f(\infty)).\end{gathered}$$ We have thus shown that $$\label{eq-receveninf} U_{g,\alpha}=\sum_{i=0}^{g+1}b_{i}^{(n_2,\ldots,n_m)} u_{g-i,\alpha} \cdot q^i \cdot \frac{I_{g}}{I_{g-i}}.$$ where $U_{g,\alpha}$ is defined to be the part of ${\hat{U}}_{g,\alpha}$ coming from elements $(h,f) \in Q'_g$. Let us fix $(h,f) \in Q'_g$. Since $f(\infty)$ is equal to the degree $2g+2$ coefficient of $f$, there is a one to one correspondence between $n$ coefficients of $f$ and the tuple of values $(f(\infty),f(\alpha_2),\ldots,f(\alpha_m))$ in $k \times k_{n_2}\times \ldots \times k_{n_m}$. We can then conclude in the same way as in Section \[sec-n2even\] that $$\label{eq-Ueven} U_g := \sum_{\alpha \in A(n_1,\ldots,n_m)} U_{g,\alpha} = \begin{cases} Jq^{g+1} b_{g+1}^{(n_2,\ldots,n_m)} & \text{if $r=0$, $g \geq -1$;} \\0 & \text{if $r=1$, $g \geq \frac{n-3}{2}$.} \end{cases}$$ The two cases joined {#the-two-cases-joined} -------------------- Since $I_{g}/I_{g-i}=q^{-i}$ and ${\hat{b}}_i^{(1,n_2,\ldots,n_m)}=b_i^{(n_2,\ldots,n_m)}$ (by Lemma \[lem-b\]) we can put equations , , and together. \[thm-rec1even\] For any tuple $(n_1^{r_1},\ldots,n_m^{r_m})$, $$\label{eq-rec1even} \sum_{j=0}^{g+1}{\hat{b}}_{j} u_{g-j} =\begin{cases} Jq^{g+1} {\hat{b}}_{g+1} & \text{if $r=0$, $g \geq -1$}; \\0 & \text{if $r=1$, $g \geq \frac{n-3}{2} $}.\end{cases}$$ Note that for $r=1$, Theorem \[thm-rec1even\] is the same as Theorem \[thm-rec1\]. Recall from the proof of Lemma \[lem-b\] that ${\hat{b}}_j-q{\hat{b}}_{j-1}$ is equal to zero if $j \geq n$. Let us then take formula for $g$, minus $q$ times the same formula for $g-1$. \[thm-rec2even\] For any tuple $(n_1^{r_1},\ldots,n_m^{r_m})$, $$\label{eq-rec2even} \sum_{j=0}^{\min(n-1,g+1)}({\hat{b}}_{j}-q{\hat{b}}_{j-1}) u_{g-j} = \begin{cases} Jq^{g+1} ({\hat{b}}_{g+1}-{\hat{b}}_{g}) & \text{if $r=0$, $g \geq 0$}; \\0 & \text{if $r=1$, $g \geq \frac{n-1}{2}$}. \end{cases}$$ Since the left hand side of equation is the same as the left hand side of equation , we get the result corresponding to Theorem \[thm-chareq\]. The characteristic polynomial of the linear recursive equation for $a_{N_1}^{R_1}\ldots a_{N_M}^{R_M}|_{g,even}$ is equal to $$\frac{1}{\lambda-1} \prod_{i=1}^M(\lambda^{N_i}-1)^{R_i}.$$ Results for weight up to seven in even characteristic {#sec-results2} ===================================================== In this section we compute all $a_{N_1}^{R_1}\ldots a_{N_M}^{R_M}|_{g,even}$ of weight at most seven. First we will exploit the similarities of Theorems \[thm-rec1\] and \[thm-rec1even\]. Fix a tuple $(n_1^{r_1},\ldots,n_m^{r_m})$ and recall Definition \[dfn-n\]. From the proof of Lemma \[lem-b\] we saw that ${\hat{b}}_j-q{\hat{b}}_{j-1}$ is the coefficient of $q^{n-1-j}$ in $b_n/(q-1)$ for all $j$, and hence ${\hat{b}}_j=q^{j+1-n}{\hat{b}}_{n-1}$ if $j \geq n-1$. Comparing the formulas of Theorems \[thm-rec1\] and \[thm-rec1even\] in the case $r=0$, we see that they are equal if $g \geq n-2$ because then ${\hat{b}}_{2g+2}=q^{g+1}{\hat{b}}_{g+1}$. In the case $r=1$ we see that the formulas are equal if $g \geq \frac{n-3}{2}$. \[thm-ind\] For weight less than or equal to five, $$a_{N_1}^{R_1}\ldots a_{N_M}^{R_M}|_{g,even} = a_{N_1}^{R_1}\ldots a_{N_M}^{R_M}|_{g,odd}$$ as polynomials in $q$. Consider any $a_{N_1}^{R_1}\ldots a_{N_M}^{R_M}|_g$ with weight less than or equal to five. By Lemma \[lem-decr\], which concerns the decomposition of $a_{N_1}^{R_1}\ldots a_{N_M}^{R_M}|_g$, it suffices to show that $u_g$ is independent of characteristic when $(n_1,\ldots,n_m)$ and $(r_1,\ldots,r_m)$ are such that $\sum_{i=1}^m n_ir_i \leq 5$. Clearly $u_{-1}=J$ is always independent of characteristic. The reduction process we found for genus $0$ in Section \[sec-gzero\] in odd characteristic is easily seen to hold also in even characteristic. We can therefore assume that $r=0$ in the case of genus $0$. But if $r=0$ then $n \leq 2$ and hence, by the arguments preceding the theorem, $u_0$ will be independent of characteristic. This takes care of the base cases of the recursions for $u_g$ when $g \geq 1$, given by Theorems \[thm-rec1\] and \[thm-rec1even\]. By the arguments preceding this theorem we see that when $g \geq 1$ these recursions are the same, both for $r=0$ and $r=1$. We can therefore conclude that $u_g$ is independent of characteristic for all $g$. We will now compute $a_{N_1}^{R_1}\ldots a_{N_M}^{R_M}|_{g,even}$ for weight six in the same way as in Section \[sec-wsix\]. To compute $u_g$ of degree at most five using Theorem \[thm-rec1even\] we need to find the base case $u_0$. The arguments of Section \[sec-gzero\] also hold in even characteristic. Hence, when the genus is zero we can reduce to the case $r=0$, which is always computable using Theorem \[thm-rec1even\]. What is left is the general case of the decomposition of $a_{N_1}^{R_1}\ldots a_{N_M}^{R_M}|_{g,even}$. We then need the base cases of genus $0$ and $1$. Again, the genus $0$ part is no problem. The computation of $a_{N_1}^{R_1}\ldots a_{N_M}^{R_M}|_{1}$ in [@Jonas2] is independent of characteristic. We can therefore conclude the genus $1$ part in the same way as in Section \[sec-wsix\]. For certain $a_{N_1}^{R_1}\ldots a_{N_M}^{R_M}|_g$ of weight six, there is a dependence upon characteristic, occuring for the first time for genus three. \[exa-notind\] We wish to compute $u_g^{(1^2,1^2,1^2)}$. We see that $u_{-1}=1$ and equation gives $u_0=q^2-3q+2$. This result is different from the one in the case of odd characteristic, see Example \[exa-u2\]. Continued use of equation gives $u_1=q^4-3q^3+5q^2-6q+3$ and then equation gives $$u_g=2u_{g-1} - u_{g-2} + q^{2g-1}(q-1)^3 \quad \text{for $g \geq 2$}.$$ Solving this leaves us with $$u_{g}^{(1^2,1^2,1^2)}=\frac{(q-1)(q^{2g+3}+g(q^2-1)-3q-2)}{(q+1)^2}.$$ The result for $a_1^6|_{g,even}$ is $$a_1^6|_{g,even} = a_1^6|_{g,odd}-5/8 \cdot g(g-1)(g-2)\bigl((g-3)(q-1)-4\bigr).$$ The result for $a_1^2a_4|_{g,even}$ is $$a_1^2a_4|_{g,even}=a_1^2a_4|_{g,odd}-1/4\cdot\begin{cases}g(q-1) & \text{if $g\equiv 0$ mod $4$};\\(g-1)(q-1) & \text{if $g\equiv 1$ mod $4$};\\(g-2)(q-1) & \text{if $g\equiv 2$ mod $4$};\\(g-3)(q-1)-4 & \text{if $g\equiv 3$ mod $4$}. \end{cases}$$ Euler characteristics in the case of genus two {#sec-g2} ============================================== Let $k$ be any finite field. We have defined $H_{g,n}$ to be the coarse moduli space of ${\mathcal H_{{g},{n}}} \otimes \bar k$. The connection between ${\lvertH_{g,n}^{F \cdot \sigma}\rvert}$ and the cohomology of ${\mathcal H_{{g},{n}}}\otimes \bar k$ is given by the Behrend-Lefschetz trace formula (see [@Behrthesis] and [@Behrend]): $${\lvertH_{g,n}^{F \cdot \sigma}\rvert}= \sum_i (-1)^i \mathrm{Tr} \bigl(F \cdot \sigma,H_{\acute{e}t,c}^i({\mathcal H_{{g},{n}}}\otimes \bar k,{\mathbf{Q}_{\ell}}) \bigr),$$ where $H_{\acute{e}t,c}^{\bullet}(-,{\mathbf{Q}_{\ell}})$ is the compactly supported $\ell$-adic étale cohomology. If we were considering a smooth and proper DM-stack $\mathcal{X}$ defined over the integers together with an action of ${\mathbb{S}}_n$, for which for almost all finite fields $k$ and all $\sigma \in {\mathbb{S}}_n$ the number of fixed points of $F \cdot \sigma$ acting on the coarse moduli space $X_{\bar k}$ of $\mathcal{X}_{\bar k}$ was a polynomial in ${\lvertk\rvert}$, then the Behrend-Lefschetz trace formula would enable us to determine the ${\mathbb{S}}_n$-equivariant Hodge Euler characteristic of $\mathcal{X}_{{\mathbf{C}}}$ (see [@Mbar4] and [@EB]). We can say this in formulas if we define $H^{\bullet}_{c,\lambda}(-,{\mathbf{Q}})$ to be the part of the Betti cohomology corresponding to the irreducible representation of ${\mathbb{S}}_n$ which is indexed by the partition $\lambda$ and has character $\chi_{\lambda}$. We also define $K_0(\mathsf{MHS}_{{\mathbf{Q}}})$ to be the Grothendieck group of rational mixed Hodge structures, with ${\mathbf L}$ the Tate Hodge structure of weight two, and $\Lambda_n$ to be the ring of symmetric polynomials with the basis $\{s_{\lambda}\}$ of Schur polynomials. Then, if there were a polynomial $P_{\lambda}(t)$ for each partition $\lambda$ of $n$ such that $$P_{\lambda}({\lvertk\rvert})= \frac{1}{n!} \cdot \sum_{\sigma \in {\mathbb{S}}_n} \chi_\lambda(\sigma) \cdot {\lvertX_{\bar k}^{F \cdot \sigma}\rvert},$$ the ${\mathbb{S}}_n$-equivariant Hodge Euler characteristic would be determined through $$\label{eq-euler} \sum_{\lambda \, \vdash n} \frac{1}{\chi_\lambda(id)} \Bigl(\sum_i (-1)^i [H^i_{c,\lambda}(\mathcal{X}_{{\mathbf{C}}},{\mathbf{Q}})]\Bigr) \cdot s_{\lambda} = \sum_{\lambda \, \vdash n} P_{\lambda}({\mathbf L}) \cdot s_{\lambda}\in \mathrm{K}_0(\mathsf{MHS}_{{\mathbf{Q}}}) \otimes \Lambda_n.$$ The equality also holds in the Grothendieck group of $\ell$-adic Galois representations, but then the cohomology should be the compactly supported $\ell$-adic étale cohomology of $\mathcal{X}_{{\mathbf{Q}}}$ and ${\mathbf L}$ should be replaced by the Tate twist ${\mathbf{Q}_{\ell}}(-1)$. The moduli space ${\overline{\mathcal M}_{{g},{n}}}$ of stable $n$-pointed curves of genus $g$ is a smooth and proper DM-stack. Using the stratification of ${\overline{\mathcal M}_{{g},{n}}}$ we can make an ${\mathbb{S}}_n$-equivariant count of its number of points using the ${\mathbb{S}}_n$-equivariant counts of the points of ${\mathcal M_{{\tilde{g}},{\tilde{n}}}}$ for all $\tilde{g} \leq g$ and $\tilde{n}\leq n+2(g-\tilde{g})$ (see [@Mbar4] and [@GK]). Since all curves of genus two are hyperelliptic, ${\mathcal M_{{2},{n}}}$ is equal to ${\mathcal H_{{2},{n}}}$. Above, we have made ${\mathbb{S}}_n$-equivariant counts of ${\mathcal H_{{2},{n}}}$ for $n \leq 7$ and they were all found to be polynomial in ${\lvertk\rvert}=q$. These ${\mathbb{S}}_n$-equivariant counts can now be complemented with ones of ${\mathcal M_{{1},{n}}}$ for $n \leq 9$ (see [@Jonas2]) and of ${\mathcal M_{{0},{n}}}$ for $n \leq 11$ (see [@Lehrer]), which are also found to be polynomial in ${\lvertk\rvert}$. We are therefore in a position to apply equation to conclude the ${\mathbb{S}}_n$-equivariant Hodge structure of ${\overline{\mathcal M}_{{2},{n}}}$ for all $n \leq 7$ (see Theorem 3.4 in [@Mbar4]). The results for $n \leq 3$ were previously known by the work of Getzler (see [@G-2]). The equivariant Hodge Euler characteristic of ${\overline{\mathcal M}_{{2},{4}}}$ is equal to $$\begin{gathered} ({\mathbf L}^7+8{\mathbf L}^6+33{\mathbf L}^5+67{\mathbf L}^4+67{\mathbf L}^3+33{\mathbf L}^2+8{\mathbf L}+\mathbf{1})s_{4}\\ +(4{\mathbf L}^6+26{\mathbf L}^5+60{\mathbf L}^4+60{\mathbf L}^3+26{\mathbf L}^2+4{\mathbf L})s_{31}\\ +(2{\mathbf L}^6+12{\mathbf L}^5+28{\mathbf L}^4+28{\mathbf L}^3+12{\mathbf L}^2+2{\mathbf L})s_{2^2}\\ +(3{\mathbf L}^5+10{\mathbf L}^4+10{\mathbf L}^3+3{\mathbf L}^2)s_{21^2}\end{gathered}$$ The equivariant Hodge Euler characteristic of ${\overline{\mathcal M}_{{2},{5}}}$ is equal to $$\begin{gathered} ({\mathbf L}^8+9{\mathbf L}^7+49{\mathbf L}^6+128{\mathbf L}^5+181{\mathbf L}^4+128{\mathbf L}^3+49{\mathbf L}^2+9{\mathbf L}+\mathbf{1})s_{5}\\ +(6{\mathbf L}^7+48{\mathbf L}^6+156{\mathbf L}^5+227{\mathbf L}^4+156{\mathbf L}^3+48{\mathbf L}^2+6{\mathbf L})s_{41}\\ +(3{\mathbf L}^7+31{\mathbf L}^6+106{\mathbf L}^5+159{\mathbf L}^4+106{\mathbf L}^3+31{\mathbf L}^2+3{\mathbf L})s_{32}\\ +(8{\mathbf L}^6+42{\mathbf L}^5+65{\mathbf L}^4+42{\mathbf L}^3+8{\mathbf L}^2)s_{31^2}\\ +(6{\mathbf L}^6+26{\mathbf L}^5+43{\mathbf L}^4+26{\mathbf L}^3+6{\mathbf L}^2)s_{2^21}\\ +({\mathbf L}^5+3{\mathbf L}^4+{\mathbf L}^3)s_{21^3}\end{gathered}$$ The equivariant Hodge Euler characteristic of ${\overline{\mathcal M}_{{2},{6}}}$ is equal to $$\begin{gathered} ({\mathbf L}^9+11{\mathbf L}^8+68{\mathbf L}^7+229{\mathbf L}^6+420{\mathbf L}^5+420{\mathbf L}^4+229{\mathbf L}^3+68{\mathbf L}^2+11{\mathbf L}+\mathbf{1})s_{6}\\ +(7{\mathbf L}^8+75{\mathbf L}^7+317{\mathbf L}^6+641{\mathbf L}^5+641{\mathbf L}^4+317{\mathbf L}^3+75{\mathbf L}^2+7{\mathbf L})s_{5 1}\\ +(5{\mathbf L}^8+62{\mathbf L}^7+292{\mathbf L}^6+615{\mathbf L}^5+615{\mathbf L}^4+292{\mathbf L}^3+62{\mathbf L}^2+5{\mathbf L})s_{4 2}\\ +({\mathbf L}^8+21{\mathbf L}^7+108{\mathbf L}^6+236{\mathbf L}^5+236{\mathbf L}^4+108{\mathbf L}^3+21{\mathbf L}^2+{\mathbf L})s_{3^2}\\ +(17{\mathbf L}^7+118{\mathbf L}^6+278{\mathbf L}^5+278{\mathbf L}^4+118{\mathbf L}^3+17{\mathbf L}^2)s_{4 1^2}\\ +(16{\mathbf L}^7+115{\mathbf L}^6+277{\mathbf L}^5+277{\mathbf L}^4+115{\mathbf L}^3+16{\mathbf L}^2)s_{3 2 1}\\ +(3{\mathbf L}^7+22{\mathbf L}^6+53{\mathbf L}^5+53{\mathbf L}^4+22{\mathbf L}^3+3{\mathbf L}^2)s_{2^3}\\ +(9{\mathbf L}^6+29{\mathbf L}^5+29{\mathbf L}^4+9{\mathbf L}^3)s_{31^3}\\ +(6{\mathbf L}^6+21{\mathbf L}^5+21{\mathbf L}^4+6{\mathbf L}^3)s_{2^2 1^2}\end{gathered}$$ The equivariant Hodge Euler characteristic of ${\overline{\mathcal M}_{{2},{7}}}$ is equal to $$\begin{gathered} ({\mathbf L}^{10}+12{\mathbf L}^9+90{\mathbf L}^8+363{\mathbf L}^7+854{\mathbf L}^6+1125{\mathbf L}^5+854{\mathbf L}^4+363{\mathbf L}^3+90{\mathbf L}^2+\ldots)s_{7}\\ +(9{\mathbf L}^9+109{\mathbf L}^8+580{\mathbf L}^7+1529{\mathbf L}^6+2109{\mathbf L}^5+1529{\mathbf L}^4+580{\mathbf L}^3+109{\mathbf L}^2+9{\mathbf L})s_{6 1}\\ +(6{\mathbf L}^9+100{\mathbf L}^8+606{\mathbf L}^7+1728{\mathbf L}^6+2430{\mathbf L}^5+1728{\mathbf L}^4+606{\mathbf L}^3+100{\mathbf L}^2+6{\mathbf L})s_{5 2}\\ +(3{\mathbf L}^9+58{\mathbf L}^8+389{\mathbf L}^7+1153{\mathbf L}^6+1647{\mathbf L}^5+1153{\mathbf L}^4+389{\mathbf L}^3+58{\mathbf L}^2+3{\mathbf L})s_{4 3}\\ +(28{\mathbf L}^8+258{\mathbf L}^7+831{\mathbf L}^6+1221{\mathbf L}^5+831{\mathbf L}^4+258{\mathbf L}^3+28{\mathbf L}^2)s_{5 1^2}\\ +(34{\mathbf L}^8+331{\mathbf L}^7+1133{\mathbf L}^6+1675{\mathbf L}^5+1133{\mathbf L}^4+331{\mathbf L}^3+34{\mathbf L}^2)s_{4 2 1}\\ +(12{\mathbf L}^8+140{\mathbf L}^7+489{\mathbf L}^6+738{\mathbf L}^5+489{\mathbf L}^4+140{\mathbf L}^3+12{\mathbf L}^2)s_{3^2 1}\\ +(8{\mathbf L}^8+91{\mathbf L}^7+335{\mathbf L}^6+502{\mathbf L}^5+335{\mathbf L}^4+91{\mathbf L}^3+8{\mathbf L}^2)s_{3 2^2}\\ +(28{\mathbf L}^7+143{\mathbf L}^6+228{\mathbf L}^5+143{\mathbf L}^4+28{\mathbf L}^3)s_{4 1^3}\\ +(34{\mathbf L}^7+170{\mathbf L}^6+275{\mathbf L}^5+170{\mathbf L}^4+34{\mathbf L}^3)s_{3 2 1^2}\\ +(10{\mathbf L}^7+47{\mathbf L}^6+77{\mathbf L}^5+47{\mathbf L}^4+10{\mathbf L}^3)s_{2^3 1}\\ +(4{\mathbf L}^6+7{\mathbf L}^5+4{\mathbf L}^4)s_{3 1^4}\\ +(2{\mathbf L}^6+6{\mathbf L}^5+2{\mathbf L}^4)s_{2^2 1^3}\end{gathered}$$ In Table \[tab-coh\] we present the nonequivariant information, in the form of dimensions of the cohomology groups of ${\overline{\mathcal M}_{{2},{n}}}$ for all $n \leq 7$. We already know that the Hodge structure of the cohomology is trivial, that is, that all of the cohomology is Tate. Notice that the table only contains as many dimensions of cohomology groups as we need to be able to fill in the missing ones using Poincaré duality and the fact that the dimension of ${\overline{\mathcal M}_{{2},{n}}}$ is equal to $3+n$. Note that these results agree with Table 2 of ordinary Euler characteristics for ${\overline{\mathcal M}_{{2},{n}}}$ for $n \leq 6$ found in the article [@Bini-Harer] by Bini-Harer. The theorem used above also gives the ${\mathbb{S}}_n$-equivariant Hodge Euler characteristic of ${\mathcal M_{{2},{n}}}$ for $n \leq 7$. We will formulate these results in a slightly different form. Define the local system $\mathbb{V}:=R^1\pi_{*}({\mathbf{Q}_{\ell}})$ where $\pi : \mathcal{M}_{g+1} \to \mathcal{M}_{g}$ is the universal curve. For every irreducible representation of $\mathrm{GSp}(2g)$ indexed by $\lambda=(\lambda_1 \geq \ldots \geq \lambda_g \geq 0)$ and $n$ there is a corresponding local system $\mathbb{V}_{\lambda}(n)$. We say that $\mathbb{V}_{\lambda}:=\mathbb{V}_{\lambda}(0)$ is a local system of weight $\sum_{i=1}^g \lambda_i$. The information contained in the ${\mathbb{S}}_N$-equivariant Hodge Euler characteristics of ${\mathcal M_{{g},{N}}}$ for all $N \leq n$, is equivalent to that of the Hodge Euler characteristics of every local system $\mathbb{V}_{\lambda}$ on ${\mathcal M_{{g}}}$ of weight at most $n$. The Hodge Euler characteristic of the local system $\mathbb{V}_{\lambda}$ is denoted by ${\mathbf{e}}({\mathcal M_{{g}}},\mathbb{V}_{\lambda})$. For more details and for the results on ${\mathbf{e}}({\mathcal M_{{2}}},\mathbb{V}_{\lambda})$ for all $\lambda$ of weight at most three, see [@G-2]. \[thm-M2\] The Hodge Euler characteristics of the local systems $\mathbb{V}_{\lambda}$ on ${\mathcal M_{{2}}}$ of weight four or six are equal to $$\begin{gathered} {\mathbf{e}}({\mathcal M_{{2}}},\mathbb{V}_{(4, 0)})=\mathbf{0}, \;\; {\mathbf{e}}({\mathcal M_{{2}}},\mathbb{V}_{(3, 1)})={\mathbf L}^2-\mathbf{1}, \;\; {\mathbf{e}}({\mathcal M_{{2}}},\mathbb{V}_{(2, 2)})=-{\mathbf L}^4, \\ {\mathbf{e}}({\mathcal M_{{2}}},\mathbb{V}_{(6, 0)})=-\mathbf{1},\;\; {\mathbf{e}}({\mathcal M_{{2}}},\mathbb{V}_{(5, 1)})={\mathbf L}^2-{\mathbf L}-\mathbf{1}, \\ {\mathbf{e}}({\mathcal M_{{2}}},\mathbb{V}_{(4, 2)})={\mathbf L}^3,\;\; {\mathbf{e}}({\mathcal M_{{2}}},\mathbb{V}_{(3, 3)})=-{\mathbf L}-\mathbf{1}.\end{gathered}$$ The results of Theorem \[thm-M2\] all agree with the ones of Faber-van der Geer in [@Faber-Geer1] and [@Faber-Geer2]. Appendix: Introducing $b_i$, $c_i$ and $r_i$ ============================================ This section will give an interpretation of the information carried by the $u_g$’s. It will be in terms of counts of hyperelliptic curves together with prescribed inverse images of points on ${\mathbf{P}}^1$ under their unique degree two morphism. Let $C_{\varphi}$ be a curve defined over $k$ together with a separable degree two morphism $\varphi$ over $k$ from $C$ to ${\mathbf{P}}^1$. We then define $$\begin{gathered} b_{i}(C_{\varphi}):={\lvert\{ \alpha \in A(i) : {\lvert\varphi^{-1}(\alpha)\rvert}=2, \varphi^{-1}(\alpha) \subseteq C(k_i) \}\rvert}, \\ c_{i}(C_{\varphi}):={\lvert\{ \alpha \in A(i) : {\lvert\varphi^{-1}(\alpha)\rvert}=2, \varphi^{-1}(\alpha) \nsubseteq C(k_{i}) \}\rvert}\end{gathered}$$ and put $r_i(C_{\varphi}):=b_{i}(C_{\varphi})+c_{i}(C_{\varphi})$. The number of ramification points of $f$ that lie in $A(i)$ is then equal to ${\lvertA(i)\rvert}-r_{i}(C_{\varphi}).$ Let $\lambda_i$ denote the partition of $i$ consisting of one element. We then find that $${\lvertC_{\varphi}(\lambda_i)\rvert}={\lvertA(i)\rvert}+b_{i}(C_{\varphi})-c_{i}(C_{\varphi})+\begin{cases} 2c_{i/2}(C_{\varphi}) & \text{if $i$ is even}; \\ 0 & \text{if $i$ is odd}. \end{cases}$$ and thus $$a_n(C_{\varphi})=\sum_{i | n \,:\, 2i \nmid n} \bigl(c_i(C_{\varphi})-b_i(C_{\varphi})\bigr) + \sum_{i:2i | n} \bigl(-b_i(C_{\varphi})-c_i(C_{\varphi})\bigr).$$ \[dfn-bc\] For $g \geq 2$ and odd characteristic, define $$b_{N_1}^{R_1}\ldots b_{N_M}^{R_M} c_{N'_1}^{R'_1} \ldots c_{N'_{M'}}^{R'_{M'}}|_g:= \sum_{[C_f] \in {\mathcal H_{{g}}}(k)/\cong_k} \frac{1}{{\lvert{\mathrm{Aut}}_k(C_f)\rvert}}\cdot \prod_{i=1}^{M} b_{N_i}(C_f)^{R_i}\prod_{j=1}^{M'} c_{N'_j}(C_f)^{R'_j}.$$ The number $\sum_{i=1}^M R_iN_i + \sum_{i=1}^{M'} R'_iN'_i$ will be called the weight of this expression. We can in the obvious way also define $$a_{N_1}^{R_1}\ldots a_{N_M}^{R_M} b_{N'_1}^{R'_1}\ldots b_{N'_{M'}}^{R'_{M'}} c_{N^{''}_1}^{R^{''}_1} \ldots c_{N^{''}_{M''}}^{R^{''}_{M''}}|_g,$$ but from the relation between $a_i(C_f)$, $b_i(C_f)$ and $c_i(C_f)$ we see that this gives no new phenomena. Directly from the definitions we get the following lemma. Let the characteristic be odd and let $f$ be an element of $P_g$. We then have $$b_{i}(C_f)=\frac{1}{2} \cdot \sum_{\alpha \in A(i)} \Bigl(\chi_{2,i}\bigl(f(\alpha)\bigr)^2+\chi_{2,i}\bigl(f(\alpha)\bigr)\Bigr)$$ and $$c_{i}(C_f)=\frac{1}{2} \cdot \sum_{\alpha \in A(i)}\Bigl(\chi_{2,i}\bigl(f(\alpha)\bigr)^2-\chi_{2,i}\bigl(f(\alpha)\bigr)\Bigr).$$ If the characteristic is odd we then use the same arguments as in Section \[sec-repr\] to conclude that $$\begin{gathered} b_{N_1}^{R_1}\ldots b_{N_M}^{R_M} c_{N'_1}^{R'_1} \ldots c_{N'_{M'}}^{R'_{M'}}|_g = I \cdot \sum_{f \in P_g} \prod_{i=1}^{M} \Bigl(\frac{1}{2} \cdot \sum_{\alpha \in A(N_i)}\chi_{2,N_i}\bigl(f(\alpha)\bigr)+\chi_{2,N_i}\bigl(f(\alpha)\bigr)^2\Bigr)^{R_i} \cdot \\ \cdot\prod_{j=1}^{M'} \Bigl(\frac{1}{2} \cdot \sum_{\alpha \in A(N'_j)}\chi_{2,N'_j}\bigl(f(\alpha)\bigr)-\chi_{2,N'_j}\bigl(f(\alpha)\bigr)^2\Bigr)^{R'_j}.\end{gathered}$$ Note that this expression is defined for all $g \geq -1$. It can be decomposed in terms of $u_g$’s for tuples $(n_1^{r_1},\ldots,n_m^{r_m})$ such that $$\label{eq-bc-decomp} \sum_{i=1}^m n_i \leq \sum_{i=1}^M R_iN_i + \sum_{j=1}^{M'} R'_j N'_j.$$ The corresponding results clearly hold for elements $(h,f)$ in $P_g$ in even characteristic and the decomposition into $u_g$’s is independent of characteristic. For each $N$ we have the decomposition $$b_{N}|_g=\frac{1}{2}(u_g^{(N^2)}+u_g^{(N^1)}) \; \; \text{and} \; \; c_{N}|_g=\frac{1}{2}(u_g^{(N^2)}-u_g^{(N^1)}).$$ Let us decompose $b_1^2c_2|_g$ into $u_g$’s: $$b_1^2c_2|_g=\frac{1}{8}(u_g^{(2^2,1^2,1^2)}+u_g^{(2^2,1^1,1^1)}+2u_g^{(2^2,1^2)}-u_g^{(2^1,1^2,1^2)}-u_g^{(2^1,1^1,1^1)}-2u_g^{(2^1,1^2)}).$$ In this expression we have removed the $u_g$’s for which $\sum_{i=1}^m r_i n_i$ is odd, since they are always equal to zero. \[lem-bc-u\] For each $N$, the following information is equivalent: - all $u_g$’s of degree at most $N$; - all $b_{N_1}^{R_1}\ldots b_{N_M}^{R_M} c_{N'_1}^{R'_1} \ldots c_{N'_{M'}}^{R'_{M'}}|_g$ of weight at most $N$. From property of the decomposition of $b_{N_1}^{R_1}\ldots c_{N'_{M'}}^{R'_{M'}}|_g$ into $u_g$’s we directly find that if we know $(1)$ we can compute $(2)$. For the other direction we note on the one hand that $$\label{eq-create} I \cdot \sum_{f \in P_g} \prod_{i=1}^j \bigl(b_i(C_f)-c_i(C_f)\bigr)^{s_i} \bigl(b_i(C_f)+c_i(C_f)\bigr)^{t_i}$$ can be formulated in terms of $b_{N_1}^{R_1}\ldots c_{N'_{M'}}^{R'_{M'}}|_g$’s of weight at most $$S:=\sum_{i=1}^j i \cdot (s_i+t_i).$$ If we on the other hand decompose into $u_g$’s we find that there is a unique $u_g$ of degree $S$. The corresponding tuple $(n_1^{r_1},\ldots,n_m^{r_m})$ contains, for each $i$, precisely $s_i$ entries of the form $i^1$ and $t_i$ entries of the form $i^2$. Every $u_g$ of degree $S$ can be created in this way and hence if we know $(2)$ we can compute $(1)$. From the definitions of $a_i(C_f)$ and $r_i(C_f)$ we see that knowing $(1)$ and $(2)$ in Lemma \[lem-bc-u\] is also equivalent to knowing - all $a_{N_1}^{R_1}\ldots a_{N_M}^{R_M} r_{N'_1}^{R'_1} \ldots r_{N'_{M'}}^{R'_{M'}}|_g$ of weight at most $N$, where $a_{N_1}^{R_1}\ldots r_{N'_{M'}}^{R'_{M'}}|_g$ is defined in the obvious way. Moreover, $a_{N_1}^{R_1}\ldots r_{N'_{M'}}^{R'_{M'}}|_g=0$ if $\sum_{i=1}^M R_i N_i$ is odd.

= 6.6truein = 8.7truein = 0.9 in = -1truein = -.8truein plus 0.2pt minus 0.1pt = 45by plus 0.2pt minus 0.1pt .5cm .5cm plus 1pt CERN-TH-97-17\ DAMTP R-97-09\ SU-ITP-97-05\ hep-th/9702103\ February 1997\ [**BLACK HOLES AND CRITICAL POINTS\ IN MODULI SPACE**]{} 1 cm [**Sergio Ferrara,$^a$  Gary W. Gibbons,$^b$  and Renata Kallosh$^c$** ]{}[^1] $^a$[*Theory Division, CERN, 1211 Geneva 23, Switzerland*]{}\ $^b$[*DAMTP, Cambridge University, Silver Street, Cambridge CB3 9EW, United Kingdom*]{}\ $^c$[*Physics Department, Stanford University, Stanford, CA 94305-4060, USA*]{} 1 cm **ABSTRACT** > We study the stabilization of scalars near a supersymmetric black hole horizon using the equation of motion of a particle moving in a potential and background metric. When the relevant 4-dimensional theory is described by special geometry, the generic properties of the critical points of this potential can be studied. We find that the extremal value of the central charge provides the minimal value of the BPS mass and of the potential under the condition that the moduli space metric is positive at the critical point. This is a property of a regular special geometry. We also study the critical points in all N$\geq $2 supersymmetric theories. We relate these ideas to the Weinhold and Ruppeiner metrics introduced in the geometric approach to thermodynamics and used for study of critical phenomena. Introduction ============ In this paper we intend to tie together some recent (and not so recent) work on 4-dimensional black holes in N=2 ungauged supergravity theories [@gary]-[@KLMS] . These theories have [*two types of geometries*]{}: space-time geometry and moduli space geometry, so called special geometry [@special]-[@Cer1] in the space of the scalar fields of the theory. Various properties of space-time singularities including black holes have been studied for a long time. Much less is known about the singularities of the moduli space. In view of the recent understanding that there is a web of connections between different versions of string theories and supergravities one can view the study of these two type of geometries as a useful tool for clarifying such connections. An interesting example of an interplay between the two type of singularities is provided by the massless black holes [@strom]. Such solutions have been constructed and studied before [@massless; @KL] in the heterotic string theory and have found to have naked singularities [@KL]. More recently these solutions have been reexamined with account taken of the first loop corrections of the heterotic string [@klaus]. These corrections modify the prepotential of N=2 supergravity model from $STU + a U^3$. The massless black holes with some charges negative and some charges positive have the following time components of the metric $$g_{tt}^2 = 4 (h_0 + {q_0 \over r}) \left( (h^1 - {p^1 \over r}) (h^2 - {p^2 \over r}) (h^3 + {p^3 \over r}) + a (h^3 + {p^3 \over r})^3 \right) \ .$$ The classical solution ($a=0$) has some naked singularity which makes the black hole repulsive to all matter [@KL]. In the internal space this singularity corresponds to a vanishing cycle. If one includes the quantum corrections one may remove the naked singularities from the space-time by a proper choice of the parameters, still keeping the mass vanishing. The quantum correction proportional to $a$ seem to act as a regulator which removes some of the singularities of the space-time metric. However by looking into the scalar metric in the moduli space $g_{k\bar k} = \partial_k \partial _{\bar k} K(z, \bar z)$ one can find that some components of the scalar metric given by the second derivative of the Kahler potential become negative, at least for $a={1\over 3}$ which is the actual number coming from the first loop calculation in heterotic string theory. This signals that the moduli space geometry becomes singular as the price for having the space-time geometry singularity free. Note that the change in sign of the scalar metric without a change of the signature of the space-time means that the Lagrangian has the wrong sign for the kinetic term for scalars. Moreover the fact that the sign of the scalar metric changes from the positive to the negative one means that somewhere in the space-time the metric vanishes and the special geometry is singular. In general it may be useful to study the properties of these two geometries together. The purpose of this paper is to set up the relevant connections between space-time and special geometry. In particular this will allow us to establish the properties of the critical points of the central charge in supersymmetric theories. We will find out when it is a minimum, when it is a maximum and whether one should expect the uniqueness or non-uniqueness of the critical points. Without the use of the special geometry in the moduli space one can only try to address these issues on a case by case basis. However special geometry will allow us to have a clear answer to all these problems. The generalization of this study to the higher supersymmetries $N>2$ is also possible using the recently developed new formulation of extended supergravities with build-in symplectic structure [@AAF]. The basic difference between N=2 and $N>2$ is in the structure of the moduli space: for N=2 the scalar manifold in not necessarily a coset manifold whereas for $N>2$ it is a coset manifold. Upon identifying the second derivative of the BPS mass and of the black hole potential with the metric on the moduli space of N=2 theory we have realized the deep connection of this construction to the geometric approach to thermodynamics where the Weinhold and Ruppeiner metrics were used for the study of critical phenomena [@rup]. In Sec. 2 we mainly review some work on black holes and one-dimensional geodesic motion which was introduced earlier [@gary] and adapt this work with the purpose of relating it to the special geometry. In Sec. 3 we focus on extreme and double-extreme configurations and find that equations of motion of a particle in a potential together with regularity requirement are sufficient to explain the attractor mechanism for the scalars near the horizon which was understood before [@FKS; @FK] with the help of supersymmetry. Sec. 4 contains new results: we identify the potential of N=2 theory with the first symplectic invariant of special geometry which is homogeneous and of degree 2 in electric and magnetic charges. The critical points of the central charge are found to be also the critical points of the potential. We find the correlation between the sign of the second derivative of the potential at the critical point and the sign of the scalar metric at a given critical point. This relates the singularities of the moduli space to the critical behavior of the potentials defining the motion in space-time. In Sec. 5 we study the critical points of the BPS mass and of the black holes potential in arbitrary extended supergravity with or without matter and calculate the second derivatives at the critical points. Sec. 6 is devoted to connection to the third geometry used in geometric approach to thermodynamics. Geodesic Action with a Constraint ================================== The class of theories we wish to consider has Lagrangian $$-{R\over 2} +\frac{1}{2} G_{ab} \partial_ \mu \phi ^a \partial_\nu \phi ^b g^{\mu\nu} -\frac{1}{4} \mu_{\Lambda \Sigma} {\cal F}^{\Lambda}_{\mu \nu} {\cal F}^{\Sigma}_{\lambda \rho} g^{\mu \lambda} g^{\nu \rho} - \frac{1}{4} \nu_{\Lambda \Sigma} {\cal F}^{\Lambda}_{\mu \nu}{}^*{\cal F}^{\Sigma}_{\lambda \rho} g^{\mu \lambda} g^{\nu \rho} \ . \label{scalaraction}$$ We restrict attention to static solutions and make the ansatz for the metric in the form $$ds^2 = e^{2U} dt^2 - e^{-2U} \gamma_{mn} dx^m dx^n \ .$$ The effective 3-dimensional Lagrangian from which the field equations can be derived takes the form $${1\over 2}R[\gamma_{mn}] - {1\over 2} \gamma^{mn} \partial_m \hat \phi ^a \partial_n \hat \phi ^b \hat G_{ab} \ ,$$ where the “hatted " scalar fields include in addition to the scalar fields $\phi^a$ of the 4-dimensional theories also the function $U$ defining the metric as well as the electrostatic $\psi^A$ and magnetic static $\chi_A$ potentials $$\hat \phi ^a = ( U, \phi ^a, \psi^\Lambda, \chi_\Lambda ) \ .$$ The metric $\hat G$ of the enlarged scalar manifold is independent of $\psi^\Lambda$ and $\chi_\Lambda$ as required by gauge independence. Now consider spherically symmetric solutions. We specify the ansatz $$\begin{aligned} \gamma_{mn} dx^m dx^n = {c^4 d\tau^2 \over \sinh^4 c\tau} + {c^2 \over \sinh^2 c\tau} (d \theta ^2 + \sin^2 \theta d \varphi ^2) \ ,\end{aligned}$$ and discover that the effective 1-dimensional Lagrangian from which the radial equations may be derived is a pure geodesic action: $$\hat G_{ab} {d \hat \phi ^a \over d\tau } {d \hat \phi ^b \over d\tau } \ ,$$ together with the constraint that $$\hat G_{ab} {d \hat \phi ^a \over d\tau } {d \hat \phi ^b \over d\tau } = c^2 \ .$$ Because $\hat G$ is independent of $\psi^\Lambda$ and $\chi_\Lambda$ due to gauge invariance, we have constants of motion: $$\begin{aligned} p^\Lambda &=&\hat G^{\Lambda \Sigma} {d \hat \chi _\Sigma \over d\tau }\ , \nonumber\\ \nonumber\\ q_\Lambda &=&\hat G_{\Lambda \Sigma} {d \hat \psi^\Sigma \over d\tau }\ .\end{aligned}$$ We may now replace the pure geodesic action by the $$\left ({d U \over d\tau}\right )^2 + G_{ab} {d\phi^a \over d\tau } {d\phi^b \over d\tau } +e^{2U} V(\phi, (p,q))$$ and the constraint by $$\left ({d U \over d\tau}\right )^2 + G_{ab} {d\phi^a \over d\tau } {d\phi^b \over d\tau } - e^{2U} V(\phi, (p,q)) = c^2\ . \label{constr}$$ Here $V(\phi, (p,q))$ is a particular potential function constructed from the scalar dependent positive definite couplings $\mu_{\Lambda \Sigma}$ and $\nu_{\Lambda \Sigma}$ of vector fields and $c^2 = 2ST$, where $S$ is the entropy and $T$ is the temperature of the black hole [@GKK]. Specifically $$V= {1\over 2}(p,q) {\cal M} \left (\matrix{ p\cr q\cr }\right ), \label{pot}$$ where $${\cal M} = \left |\matrix{ \mu+ \nu \mu^{-1} \nu & \nu \mu^{-1} \cr \mu^{-1} \nu & \mu^{-1} \cr }\right |. \label{Matrix}$$ It is clear that the properties of the black holes in theories of this type are governed entirely by the metric $G_{ab}$ on the scalars and the potential function $ V(\phi, (p,q)) $. Extreme and Double-Extreme Holes ================================ We begin by considering extreme holes when $c^2 = 2ST=0$ and $\gamma_{mn} dx^m dx^n $ is given by $$\begin{aligned} \gamma_{mn} dx^m dx^n = {d\tau^2 \over \tau^4} + {1 \over \tau^2} (d \theta ^2 + \sin^2 \theta d \varphi ^2) \ .\end{aligned}$$ The geometry becomes $$ds^2 = - e^{2U} dt^2 + e^{-2U} \left[{d\tau^2 \over \tau^4} + {1 \over \tau^2} (d \theta ^2 + \sin^2 \theta d \varphi ^2)\right] \ .$$ Evidently to obtain finite area solution we must have that $$e^{-2U}\rightarrow \left( {A\over 4\pi}\right) \tau^2 \qquad {\rm as} \quad \tau \rightarrow - \infty \ .$$ We also require that this expression for our solution is not infinite near the horizon, $$G_{ab} {d \phi ^a \over d\tau} {d \phi ^b \over d\tau} e^{2U} \tau^4 < \infty \ .$$ This leads to $$G_{ab} {d \phi ^a \over d\tau} {d \phi ^b \over d\tau} \left( {4\pi \over A} \right) \tau^2 \rightarrow X^2 \qquad {\rm as} \quad \tau \rightarrow - \infty\ .$$ Now we can substitute this into the constraint and we get $${1\over \tau^2} + \left( { X^2 A\over \tau^2 4\pi}\right) - {4\pi \over A} {V(p,q,\phi_h)\over \tau^2}=0 \ .$$ This leads to $$A \leq 4\pi V(p,q,\phi_h)$$ The near horizon geometry becomes equal to $$ds^2 = - {4\pi \over A\tau^2} dt^2 + \left( {A\over 4\pi}\right) \left[{d\tau^2 \over \tau^2} + (d \theta ^2 + \sin^2 \theta d \varphi ^2)\right] \ .$$ It is useful to change the variables as follows: $$\rho= -{1\over \tau}\ , \qquad \omega = \log \rho$$ and bring the near horizon geometry to the form of the product space $AdS_2\times S^2$: $$ds^2 = - {4\pi \over A} e^{2\omega} dt^2 + \left( {A\over 4\pi}\right) d\omega^2 + \left( {A\over 4\pi}\right) (d \theta ^2 + \sin^2 \theta d \varphi ^2) \ .$$ In these coordinates the value of the derivatives of the moduli as the function of $\omega$ enters in the term $G_{ab} \partial_ \mu \phi ^a \partial_\nu \phi ^b g^{\mu\nu} $ as follows $$G_{ab} {d \phi ^a \over d\omega } {d \phi ^b \over d\omega } \left( {4\pi \over A} \right) \rightarrow X^2 \qquad {\rm as} \quad \omega \rightarrow \infty\ .$$ Now it is obvious that only $X^2=0$ is consistent with the requirement that the moduli do not blow up near the horizon. Indeed, if $${d \phi ^a \over d\omega } = {\rm const} \qquad {\rm as} \quad \omega \rightarrow \infty$$ and the moduli near the horizon have to be linear in $\omega$ they would not be finite near the horizon. Thus we have proved that for $c=0$ extremal black holes from the single requirement that the geometry as well as moduli are regular near the horizon in the suitably chosen coordinates it follows that the area of the horizon equals the value of the potential near horizon. $${A\over 4\pi} = V(p,q,\phi_h) \ . \label{equal}$$ This property near the horizon applies both to extremal and double-extremal black holes. Double-extremal black holes [@KSW] have the constant moduli, so for them the term $G_{ab} \partial_ \mu \phi ^a \partial_\nu \phi ^b g^{\mu\nu} $ vanishes everywhere and the equality (\[equal\]) between the area of the horizon and the value of the potential near the horizon follows from the constraint equation (\[constr\]) immediately. Thus we have also shown that the area of the horizon of extreme black holes coincides with the area of the horizon of the double-extreme black holes with the same values of charges and is given by the value of the potential $$A_{\rm extr} =A_{\rm double-extr} = 4\pi V(p,q,\phi_h)\ .$$ This universality was understood before [@FKS; @FK] as a consequence of supersymmetry. Here the universal properties of the area of the horizon of the extremal black holes are deduced only from the requirement of the regularity of the configuration. The equation of motion for $\phi^a$ is $$\begin{aligned} {D\phi^a \over D \tau^2} ={1\over 2} {\partial V \over \partial \phi^a} e^{2U} \ .\end{aligned}$$ Near the horizon, taking into account that ${d \phi ^a \over d\omega } =\tau {d \phi ^a \over d\tau } =0$, we get $$\begin{aligned} {d^2 \phi^a \over d \tau^2} \rightarrow {1\over 2} {\partial V \over \partial \phi^a} \left( {4\pi \over \tau^2A} \right) \ .\end{aligned}$$ The solution of this equation near the horizon is $$\phi^a = \left( {2 \pi \over A} \right) {\partial V \over \partial \phi^a} \log \tau + \phi^a_h \label{critical}$$ Above we have omitted a term linear in $\tau$ since it will lead to the singular dilaton at the horizon. Equation (\[critical\]) shows that unless the derivative of the scalar potential over the scalar field vanishes near the horizon, one can not have a regular value of the scalars near the horizon. Thus $$\left({\partial V \over \partial \phi^a}\right)_h=0 \ .$$ Finally we note that the behavior at infinity, at $\tau \rightarrow 0$, is $U\rightarrow M\tau$ leads to the following constraint between the black hole mass, scalar charges, scalar metric and the potential, valid regardless of whether the hole is extreme or not, $$M^2 + G_{ab} \Sigma ^a \Sigma ^b - V(p,q, \phi^a_{\infty} ) = c^2 \ ,$$ where $\phi^a_\infty$ are the values of the scalars at spatial infinity and the scalar charges are defined via the expansion of the scalars at infinity. Of course in the extreme limit we set $c=0$. The double-extreme black holes have a vanishing scalar charge and the constant fixed value of scalars everywhere: $$\begin{aligned} M^2 &=& V(p,q, \phi^a_{fix}) \\ {A\over 4\pi} &=& V(p,q, \phi^a_{fix})\end{aligned}$$ and the fixed value of the scalars is defined by the extremization of the potential $$\left( {\partial V ( \phi, p,q) \over \partial \phi^a }\right)_{fix}= 0$$ Note that in this description of the extremal and double-extremal black holes there was no use of supergravity and/or supersymmetry. We have used the bosonic field equations of the theory and the requirement that the extremal configuration is regular near the horizon, including the regularity of scalars. In the next section we will specify this study to the case of N=2 supergravity and special geometry. We have arrived at the following picture. We may associate with each critical point $\phi ^a_{ fix}$ of the potential $V(\phi, p,q)$ on the manifold of scalars ${\cal M}_\phi$ a supersymmetric Bertotti-Robinson vacuum state. Extreme black hole solutions correspond to dynamical trajectories in the moduli space ${\cal M}_\phi$ starting from the point $\phi ^a _\infty$ at spatial infinity and ending on a critical point $\phi ^a_{fix}$. Double extreme holes with frozen moduli correspond to trivial point trajectories. One could also consider trajectories running between two different critical points but these would not correspond to asymptotically flat solutions. Thus the extreme solutions may be said to spatially interpolate between different vacua. Finding explicit trajectories which effect this interpolation and which satisfy the second order dynamical equations of motion is in general quite difficult, although a number of solutions are known. However we shall see shortly that using special geometry one is able to reduce this problem to the easier one of finding the solutions of a set of first order differential describing the steepest descent curves of another potential function whose physical significance is that it determines the central charge. Critical Points of the Central Charge and of the Potential in Special Geometry ============================================================================== Now we consider the special case for which the scalars field manifold ${\cal M}_4$ is a Kahler manifold with complex coordinates $z^i$ and Kahler potential $K$ so that $$G_{ab} d \phi^a d \phi^b = {\partial ^2 K \over \partial z^i \partial \bar z^n } dz^i d \bar z^{n}\ .$$ The bosonic part of the action of N=2 supergravity interacting with some number of vector multiplets is [^2] $$-{R\over 2} + G_{i\bar j} \partial_ \mu z^i \partial_\nu \bar z^{\bar j} g^{\mu\nu} + {\rm Im} {\cal N}_{\Lambda \Sigma} {\cal F}^{\Lambda}_{\mu \nu} {\cal F}^{ \Sigma}_{\lambda \rho} g^{\mu \lambda} g^{\nu \rho} + {\rm Re} {\cal N}_{\Lambda \Sigma} {\cal F}^{\Lambda}_{\mu \nu}{}^*{\cal F}^{ \Sigma}_{\lambda \rho} g^{\mu \lambda} g^{\nu \rho} \ , \label{scalaraction2}$$ Here the positive definite metric $G_{i\bar j}$ on the scalar manifold as well as scalar dependent negative definite vector couplings ${\rm Re} {\cal N}_{\Lambda \Sigma}$ and ${\rm Re} {\cal N}_{\Lambda \Sigma}$ can be derived from the prepotential or from the symplectic section which defines a particular N=2 theory. The symplectic invariant $I_1$ of the special geometry constructed in [@Cer1] $$I_1 = |Z(z,p,q)|^2 + |D_i Z(z,p,q)|^2=- {1\over 2}(p,q) \left |\matrix{ {\rm Im} {\cal N} + {\rm Re} {\cal N} {\rm Im} {\cal N} ^{-1} {\rm Re} {\cal N} &-{\rm Re} {\cal N} {\rm Im} {\cal N}^{-1} \cr- {\rm Im} {\cal N}^{-1} {\rm Re} {\cal N} & {\rm Im} {\cal N}^{-1} \cr }\right | \left (\matrix{ p\cr q\cr }\right ),$$ can be identified with the potential $$V(p,q, \phi^a ) = I_1 = |Z(z,p,q)|^2 + |D_i Z(z,p,q)|^2 \ .$$ From eqs. (\[pot\]), (\[Matrix\]) of the present paper and from eq. (56) of [@Cer1] it follows that the identification requires that $$\nu + i \mu = - {\cal N }= - {\rm Re} {\cal N } - i {\rm Im} {\cal N }$$ Here $Z$ is the central charge [@Cer], the charge of the graviphoton in N=2 supergravity and $D_i Z$ is the Kahler covariant derivative of the central charge: $$Z(z, \bar z, q,p) = e^{K(z, \bar z)\over 2} (X^\Lambda(z) q_\Lambda - F_\Lambda(z) \, p^\Lambda)= (L^\Lambda q_\Lambda - M_\Lambda p^\Lambda) \ . \label{central}$$ We will use the abbreviation $$\left| {d z \over d \tau} \right|^2 = G_{ab} {d \phi^a \over d \tau} {d \phi^b \over d\tau} = {\partial ^2 K \over \partial z^i \partial \bar z^n } {dz^i \over d \tau} {d \bar z^{n} \over d \tau}\ .$$ The first symplectic invariant of special geometry $I_1$ is positive definite. Our one-dimensional Lagrangian is $${\cal L} \left (U(\tau) , z^i(\tau) , \bar z^i(\tau) \right )= \left ({d U \over d\tau}\right )^2 + \left |{d z \over d\tau } \right |^2 + e^{2U} \left( |Z(z,p,q)|^2 + |D_i Z(z,p,q)|^2 \right) \ . \label{lagr}$$ The constraint becomes $$\left ({d U \over d\tau}\right )^2 + \left |{d z \over d\tau } \right |^2 - e^{2U} \left( |Z(z,p,q)|^2 + |D_i Z(z,p,q)|^2 \right) =c^2 \ . \label{constr1}$$ At infinity, at $\tau \rightarrow 0$, $U\rightarrow M\tau$ we get $$M ^2 (z_{\infty}, \bar z_{\infty} , p,q ) - |Z(z_{\infty} ,p,q)|^2 =c^2 \ .$$ The BPS configuration has the mass equal to the central charge in supersymmetric theories $$M ^2 (z_{\infty}, \bar z_{\infty} , p,q ) - |Z(z_{\infty} ,p,q)|^2 \ , \qquad c=0 \ \label{BPS}$$ since the second and the fourth term in the l.h.s. of eq. (\[constr1\]) cancel at $\tau \rightarrow 0$ and $U\rightarrow M\tau$. By using some properties of special geometry and of the central charge and its covariant derivatives one can rewrite the action as $${\cal L}= \left ({d U \over d\tau} \pm e^U |Z| \right )^2 + \left |{d z^i \over d\tau } \pm e^U G^{i\bar k} \bar D_{\bar k} \bar Z \right |^2 \pm 2 {d \over d\tau } \left ( e^U |Z| \right ) \ . \label{lagr1}$$ Thus we may solve the second order equations by making the ansatz that the following first order equations hold [^3] $$\begin{aligned} {d U \over d\tau}&=&e^U |Z| \\ {d z^i \over d\tau } &=&e^U G^{i\bar k} \bar D_{\bar k} |Z|\end{aligned}$$ These first order equations immediately give (by evaluating at infinity) $$\begin{aligned} M&=& |Z| (z_{0}, p,q)\ , \\ \Sigma^i &=& G^{i\bar k} \bar D_{\bar k} \bar Z \ .\end{aligned}$$ From the behavior at the horizon $\tau \rightarrow \infty$ we get $$\begin{aligned} \left( {A\over 4\pi}\right)^{1/2} &=& |Z| (z_{h}) \ , \\ D^i Z(z_{h}) &=& 0 \ .\end{aligned}$$ where also $$\partial_i V = \partial_i \left( |Z(z)|^2 + |D_i Z(z)|^2\right)$$ Thus we have recovered and specified in this framework of special geometry the equations obtained earlier. To study the critical points of the potential we need few identities from special geometry [@special]-[@Cer1]. $$\begin{aligned} \label{1} D_i D_j Z &=& i C_{ijk} G^{k\bar k} \bar D_{\bar k} \bar Z \ , \\ \nonumber\\\label{2} D_i \bar D_{\bar j} \bar Z &=& G^{i\bar j} \bar Z \ , \\ \nonumber\\\label{3} \bar D_{\bar m} Z &=& 0 \ .\end{aligned}$$ Here $C_{ijk} $ is a completely symmetric covariantly holomorphic tensor. The critical points of the potential coincide with the critical point of the central charge. Using the identities above one gets $$\partial_i V = 2 (D_i Z) \bar Z + i C_{ijk} G^{j\bar m} G^{k \bar k} \bar D_{\bar m} \bar Z \bar D_{\bar k} \bar Z \ .$$ Thus indeed $$D_i Z = \bar D_{\bar k} \bar Z =0 \qquad \Longrightarrow \qquad \partial_i V = \bar \partial_{\bar i} V=0 \ .$$ In what follows we will address the problem: when is the extremum of the central charge and of the potential a minimum and when it is a maximum. Note that the second covariant derivative of the moduli of the central charge at the critical point coincides with the partial (non-covariant) second derivative. We have to calculate $$D_i D_j |Z|= \partial_i \partial_j |Z| \qquad \bar D_{\bar i} D_j |Z|= \bar \partial_{\bar i }\partial_j |Z| \qquad {\rm at} \qquad D_i |Z|= \partial_i |Z|=0$$ and the conjugate to this. We start with the second derivative of the moduli of the central charge and use some identities of special geometry. $$D_i D_j |Z|= -{1\over 4} |Z|^{-3} (\bar Z)^2 D_i Z D_j Z + {1\over 2} |Z|^{-1} i C_{ijk} G^{k\bar k} \bar D_{\bar k} \bar Z \ .$$ It follows that at the critical point $$\partial_i \partial_j |Z| = \bar \partial_{\bar i} \bar \partial_{\bar j} |Z| =0 \ .$$ The mixed derivatives are $$\bar D_{\bar i} D_j |Z|= {1\over 4} |Z|^{-1} \bar D_{\bar i} \bar Z D_j Z + {1\over 2} G_{\bar i j} |Z| \ .$$ It follows that at the critical point we get $$\left ( \bar \partial_i \partial_j |Z|\right) _{\rm cr}= {1\over 2} G_{\bar i j} |Z|_{\rm cr} \ .$$ This means that whenever the metric is positive in the moduli space at the critical point, the BPS mass reaches its minimum. However if the scalar metric is singular and changes the sign the connection established above only signals that the BPS mass at the critical point reaches the maximum outside the range of validity of regular special geometry. In general when the metric changes the sign we have some sort of a phase transition and there is a breakdown of the effective Lagrangian unless new massless states appear. Let us now proceed with the evaluation of the second derivative of the potential $V$ at the critical point where the value of the potential is given by the square of the central charge at the critical point of the central charge. $$\bar \partial_{\bar i} V= \partial_{ i} V=0 \qquad V_{\rm cr} = |Z_{\rm cr}|^2 \ .$$ Upon some extensive use of the identities of special geometry we conclude that at the critical points of the potential which are simultaneously the critical points of the central charge we get $$D_i D_j V= 0 \qquad {\rm at} \qquad D_i Z= \partial_i |Z|=0$$ and $$\left( \bar D_{\bar i} D_j V\right)_{\rm cr}= \left( \bar \partial_{\bar i} \partial_j V \right)_{\rm cr} = 2 G_{\bar i j} V_{\rm cr} \ .$$ Thus again the sign of the second derivative of the potential is defined by the sign of the metric on the moduli space at the critical point where the central charge and the potential have vanishing derivatives. Now we recall that the extremal condition for the central charge was brought to an equivalent form of stabilization equation in [@FK; @Dieter] under condition that the special geometry is not singular. We found that all moduli from the vector multiplets become functions of electric and magnetic charges. The stabilization equation of special geometry is [@FK; @Dieter] $$\left (\matrix{ p^\Lambda\cr q_\Lambda\cr }\right )={ \rm Re} \left (\matrix{ 2i \bar Z L^\Lambda\cr 2i \bar Z M_\Lambda\cr }\right ) \equiv { \rm Re} \pmatrix{2 i Y^I\cr 2 i F_J(Y)\cr}\ . \label{stab}$$ The fixed values of moduli of special geometry in terms of electric and magnetic charges has been found in many examples before [@FKS; @FK; @KSW; @Dieter]. They correspond to the fixed values to which scalar fields are attracted near the black hole horizon. All known examples solve both the extremality condition of the central charge $\partial_i |Z|=0$ and the stabilization equation (\[stab\]). In fact they have been found mostly by solving the easier equation: the stabilization equation (\[stab\]): $$(z^i)_{\rm cr} = z^i (p,q) \ .$$ For many of these solutions the entropy of N=2 black holes has been understood from the microscopic point of view [@KLMS] via the counting of the string or M-theory states. The issue of the uniqueness of the critical points has not been studied before, we have mainly focussed on the goal of finding critical points. Now that we have studied the second derivatives of the central charge and of the potential at the critical points one can address the issue of the uniqueness of the critical points [^4]. Under the assumption that we limit ourself exclusively to study only the potentials and their critical points in the range of applicability of special geometry (the scalar metric is strictly positive) we may expect that the minimum of the potential, which is also the minimum of the BPS mass, is unique. Indeed we have proved that the second derivative at the critical point is positive when the scalar metric is positive and the critical point is a minimum. This concerns any critical point in this class and therefore it has to be unique at least for the continuous branch of the potential. However if we were to relax the condition that the scalar metric is positive we might find some disconnected branches of the potential exhibiting some maxima and various other phenomena. We refer the reader to the 5d case [@new] where we present particular examples of such situations. In the 4d context the analogous investigation would amount to studying all possible branches of the potentials for various interesting theories and calculating the value of the scalar metric at the critical points where it will become a particular function of charges, according to stabilization the equation (\[stab\]) (in case of negative scalar metric the use of stability equation (\[stab\]) may be questionable, rather one may rely on the direct solutions of extremality condition $\partial_i |Z|=0$ for exhibiting the critical point). Critical Points of Moduli Space of Extended Supergravities =========================================================== Here we will use the recent geometric formulation of extended supergravities in which the duality symmetries of the theory are manifest [@AAF]. Below we will present the minimal amount of information on this which will allow us to describe the critical points of these theories. The reader is referred for the details of the new construction to [@AAF]. All $d=4, N>2$ supergravities have scalar fields whose kinetic Lagrangian is described by sigma models of the form ${G\over H}$. Here $G$ is a non compact group acting as an isometry group on the scalar manifold while $H$ is the isotropy group. For $N\leq 4$ a supergravity can be coupled to matter. For such cases the isotropy group is given by a direct product $H= H_{\rm Aut}\otimes H_{\rm matter} $ of the automorphism group of the supersymmetry algebra $H_{\rm Aut}$ and the isotropy group of the matter multiplets $H_{\rm matter} $. In pure supergravity, when matter is absent $H$ is just $H_{\rm Aut}$. The supergravity theory is completely specified in terms of the geometry of the coset space and in particular in terms of the coset representatives $L$. For the case of our interest which are D=4 $N>2$ theories the group $G$ has to be embedded into $Sp(2n,{\bf R})$ group. The construction therefore presents symplectic sections of a $Sp(2n_v,{\bf R})$ bundle over ${G\over H}$, given by $f=(f^\Lambda_{AB}, h=h_{\Lambda AB})$ and $(f^\Lambda_I, h_{\Lambda_I}$). Here ${AB}$ are indices [^5]. in the antisymmetric representation of $H_{\rm Aut}= SU(N)\times U(1)$ and $I$ is the index of the fundamental representation of $H_{\rm matter} $. The graviphoton central charges $Z_{AB}, \bar Z^{AB}$ and the matter charges $Z_I, \bar Z^I$ are defined as linear combination of quantized electric and magnetic charges and moduli as follows: $$\begin{aligned} Z_{AB} &=&f^\Lambda_{AB}q_\Lambda - h_{\Lambda AB} p^\Lambda \ ,\\ \nonumber\\ Z_{I} &=&f^\Lambda_{I}q_\Lambda - h_{\Lambda I} p^\Lambda \ .\end{aligned}$$ The crucial observation which will make it possible to establish a complete universality of critical phenomena in extended supergravities is the following. The manifestly symplectic form of supergravity supplies a simple and completely general expression for the the black hole potential $V$ presented in eq. (\[pot\]) of this paper: it is given in eq. (3.66) of [@AAF] upon identification between the scalar couplings in (\[scalaraction\]) and in manifestly symplectic form of extended supergravities in [@AAF]. $$V= {1\over 2} Z_{AB} \bar Z^{AB}+Z^I \bar Z_I \ . \label{AAF}$$ The differential relations among charges follow from their definition with the use of a vielbein $P$ of ${G\over H}$. The embedded vielbein of the coset consists of blocks: $${\cal P}= \pmatrix{ P_{ABCD} & P_{ABJ} \cr P_{J AB} & P_{IJ} \cr } \ .$$ The differential equations which we will use for the study of the critical points are $$\begin{aligned} \nabla Z_{AB} &=& {1\over 2} \bar Z^{CD} P_{ABCD} + \bar Z_I P^I_{AB} \ , \\ \nonumber\\ \nabla Z^{I} &=& {1\over 2} \bar Z^{CD} P^I_{CD} + \bar Z_J P^{JI} \ . \label{full}\end{aligned}$$ Now we have sufficient amount of the information on the properties of central charges and moduli space of $N\geq 2$ theories to study the critical points. We will first focus on pure supergravity without matter and afterwards will study supergravities coupled to matter. 1\. Critical points in pure $N\geq 2$ supergravities with $Z_I \equiv 0$. The potential and the differential relation simplify: $$V= {1\over 2} Z_{AB} \bar Z^{AB} \ .$$ $$\nabla Z_{AB} = {1\over 2} Z^{CD} P_{ABCD} \ . \label{dif}$$ Using eq. (\[dif\]) to find the derivatives of the central charge we get $$\partial_i V ={1\over 4} \bar Z^{CD} \bar Z^{AB} P_{ABCD,i} + {1\over 4} Z_{AB} Z_{CD}P^{ABCD}{}_{,i} \ .$$ Since $P_{ABCD}$ is completely antisymmetric the solution to this equation exists where only one eigenvalue of the central charge matrix is non-vanishing and other eigenvalues vanish [@FK]: $$\partial_i V =0 \qquad {\rm at} \qquad Z^{12} \neq 0 \ , \quad Z_{\rm other} = 0\ .$$ The non-vanishing eigenvalue of the central charge matrix is the BPS mass at the critical point. For the second derivative we get $$D_j \partial _i V\mid_{\partial_i V =0} = \partial_j \partial _i V = {1\over 2} Z_{LM} P^{ABLM}{}_{,j} \bar Z^{CD} P_{ABCD,i} \ .$$ At present it is not clear if one can simplify this expression and bring it to the form close to what has been found in N=2 case. However, we expect further developments in this direction. 2\. Matter coupled supergravities $N\geq 2$ with $Z_I \neq 0$. The potential (\[AAF\]) and the differential relation (\[full\]) now have a contribution from the matter charges. The critical point is defined by $$\partial_i V ={1\over 4} \bar Z^{CD} \bar Z^{AB} P_{ABCD,i} + {1\over 4} Z_{AB} Z_{CD}P^{ABCD}{}_{,i} + {1\over 2} \bar Z_I P^I_{AB,i} \bar Z^{AB} + {1\over 2} Z^I P_{I,i} ^{AB} Z_{AB}=0 \ . \label{extreme}$$ The configurations with $$Z^{12} \neq 0 \ , \quad Z_{\rm other} = 0 \quad Z^I= \bar Z_I=0$$ solves the extremization condition (\[extreme\]) for the potential. The evaluation of the second derivative of the potential at the critical point proceeds with the use of differential relations (\[full\]) : $$(\partial_j \partial _i V )_{cr}= Z_{CD} \bar Z^{AB} ( P^{CDPQ}{}_{,j} P_{ABPQ,i} + {1\over 2} P^{CD}_{Ij} P^I_{ABi}) \ .$$ Again, it remains to be seen if by using the coset space geometry one can simplify this to a form analogous to a simple result in N=2 theory. Geometry, Thermodynamics and Critical Points ============================================ In section two we described a general formalism for constructing four-dimensional spherically symmetric solutions. In some ways the most symmetrical formulation is the pure geodesics formulation involving the fields ${\hat \phi}$ comprising the scalars $\phi^a$, the potentials for the vectors $\psi^A, \chi_A$ and the Newtonian potential $U$ all on an equal footing. This naturally arises from dimensional reduction, three dimensions which automatically places $U$ and other Kaluza-Klein scalars and vectors on the same footing. Physically it is more convenient to eliminate the potentials $\psi^A$ and $\chi_A$ in favour of their conjugate conserved charges. This gives rise to the potential $V(p,q, \phi,)$ while we have the scalars $\phi^a$ and the Newtonian potential $U$ remain on the same footing and are described by a simple dynamical model involving motion in the $U,\phi$ space with a potential. As we have seen the essential properties of the extreme black holes, such as the area of the event horizon are given by the values of the potential $V(q,p,\phi)$ thought of as a function on the moduli space of scalars ${\cal M}_\phi$ at its critical points $\phi^a_{fix}$. Now it becomes especially interesting to ask how the mass $M$ and area $A$ depend on the moduli. In particular about their second covariant derivatives. At this point it is worth recalling some standard geometrical ideas in ordinary thermodynamics, see [@rup] for a review. Some time ago Weinhold suggested using as a metric the Hessian of the energy $M$, considered as a function of the $n+1$ extensive variables $N^\mu = (S, N^a)$, where $S$ is the entropy and $N^a$, $a= 1,\dots n$ are conserved numbers. Note in this formulation of an ordinary gas the volume is included as one of the $N^a$’s. In the case of black holes, the $N^a$’s include conserved charges, angular momenta and also (see [@GKK]) the values $\phi^\infty$ of the moduli at infinity. Thus in conventional thermodynamics the Weinhold metric $W_{\mu \nu}$ is given by $$W_{\mu \nu} = {\partial M \over \partial N^ \mu \partial N^\nu } \ .$$ In conventional thermodynamics the Weinhold metric is positive semi-definite because of the fact that the energy is least among equilibrium configurations with given entropy $S$, and total numbers $N^a$. Because $$dM= TdS + \mu_a dN^a,$$ the dual function to $M$ is of course $G$ the Gibbs free energy, which should be thought of as function of the intensive variables $\mu _\mu = (T, \mu _a) $ given by $$G=M-TS-\mu _a N^a$$ whence $$dG = -SdT -N^a d\mu _a \ .$$ Thus the inverse metric $ W^ {\mu \nu}$ is given by $$W^{\mu \nu} = - {\partial G \over \partial \mu _\mu \partial \mu_ \nu }\ .$$ Note that the negative sign arises because of the conventional choice of sign made in the thermodynamics literature when defining the Legendre transformation. Sometime after Weinhold, Ruppeiner focussed attention on the entropy $S$ considered as a function of the extensive variables $M$ and $N^a$. It is convenient to define extensive charges $Q^\mu= (M, N^a)$ and conjugate variables $\beta _\mu= ( { 1\over T}, -{ \mu _a \over T})$. Ruppeiner observed that fluctuations of the system are governed by the Ruppeiner metric $$S_{\mu \nu} = -{\partial S \over \partial Q^ \mu \partial Q^\nu }\ .$$ The inverse metric is given by $$S^{\mu \nu} = {\partial \Gamma \over \partial \beta _\mu \partial \beta _\nu }\ .$$ where $\Gamma$ is the Legendre transform of the entropy $S$, i.e $$\Gamma = { G \over T} = -S + {M \over T} -N^a { \mu _a \over T}\ .$$ More symmetrically $$S+ \Gamma= \beta _\mu Q ^\mu \ .$$ An interesting question to ask is how these two metrics are related. The answer is, perhaps, surprisingly, that they are conformally related and the conformal factor is the temperature, in other words $$W_{\mu \nu} d N^\mu d N^\nu = T S_{\mu \nu } dQ^\mu dQ^\nu \ .$$ To see this note that $$W_{\mu \nu} d N^\mu d N^\nu =dT \otimes _s dS + d\mu_a \otimes _s dN^a$$ while $$-S_{\mu \nu} d Q^\mu d Q^\nu =d {1\over T} \otimes _s dM + d {\mu_a \over T } \otimes _s dQ^a \ .$$ It is interesting to observe that it is the conformal geometry which is physically relevant. Thus ratios of specific heats should be conformal invariants. We shall now relate these geometric thermodynamic ideas to the work of this paper. The first thing to note is that that in general, for non-extreme holes these metrics will not be positive definite because of the fact that non-extreme black holes with have negative specific heats. The heat capacity $C= \left ({\partial M \over \partial T}\right)_{p,q,\phi}$ is related to the second derivative of the mass over the entropy at fixed values of charges. For dilaton non-extreme black holes the change in the sign of the heat capacity has been studied in [@G; @Pat; @US]. It has been shown for generic $U(1)^2$ that in the process of the black hole evaporation the temperature increases, reaches the maximum and rapidly drops to zero when the mass of the black hole reaches the value of the central charge. Specific heat blows up when the temperature reaches the maximum and changes the sign, see Figs. 4, 6, 7 of [@US]. The change of the sign of the heat capacity happens at the non-vanishing temperature and means that the corresponding component of the Weinhold metric undergoes the same type of changes. The second important point is that, as pointed out in [@GKK], when considering black holes with scalars we must augment the usual extensive thermodynamic variables such as the area $A$ and the conserved charges $(q,p)$ with the values of the moduli at infinity $\phi_\infty$. This has the consequence that the thermodynamic configuration space is no longer flat ${ \bf R}^k \equiv (A, q,p)$ ( where $k$ is one plus two times the number of electric charges, but becomes its product with the scalar moduli space ${ \bf R}^k \times {\cal M}_\phi$. It was shown in [@GKK] that the thermodynamic variables conjugate to to the moduli $\phi^a_\infty$ , i.e. the analogue of the chemical potentials $\mu_a$ are minus the scalar charges, i.e. one has the relation $$dM=TdS+\psi^A dq_A + \chi_A dp^A - \Sigma _a d \phi ^a _\infty \ .$$ The fact that the scalar moduli space ${\cal M}_\phi$ is no longer flat complicates, but of course does not invalidate, the usual thermodynamic formalism involving Legendre transformations (technically one should speak of Legendre submanifolds etc.) and in particular one is allowed to extend the definition of the Weinhold and Ruppeiner metrics to the scalar moduli space ${\cal M}_\phi$ provided we replace the ordinary derivative by the covariant derivative with respect to the metric $G_{ab}$ on the moduli space. In the present paper we have been considering extreme black holes for which the temperature $T=0$ and it is the Weinhold metric $W_{ab}$ which seems to be the more appropriate one to consider. This may be defined by $$W_{ab}= \nabla _a \nabla _b M(p,q,\phi) \ .$$ The Ruppeiner metric governs fluctuations and naively diverges (see relevant equation above) if $T\rightarrow 0$. This is in agreement with the arguments presented in [@Pat; @US] that near extreme the thermodynamics breaks down. However one might consider a renormalized definition of the the Ruppeiner metric $$S_{ab} = { 1\over 4} \nabla _a \nabla _b A(p,q, \phi) \ .$$ Note that if the mass $M$ considered as a function of the scalars is at a critical point the first derivative vanishes and the covariant derivative may be replaced by the partial derivative. For a general thermodynamic substance or for a general black hole one expects to be able to say very little about the Weinhold metric. In the case of extreme black holes it is given by $$W_{ab}= { 1\over 2 \sqrt V} \nabla _a \nabla _b V \ .$$ As to the Ruppeiner metric, because the area $A$ of the event horizon depends only on the values of the scalars on the horizon and is independent of their values at infinity it follows that $$S_{ab}=0 \ .$$ By contrast for black hole arising form special geometry we are able to make rather more precise statements about the Weinhold metric. We find the remarkable result that the Weinhold metric is proportional to the metric $G_{ab}$ on the moduli space. Conclusion ========== In conclusion we have found the properties of the critical points of the BPS mass in the range of applicability of the special geometry. Supersymmetric states in the spectrum of N=2 theory have the properties that their mass equals the central charge $M^{BPS} = |Z|$. The central charge $Z$ is defined in a generic point of moduli space to be a particular function of moduli and electric and magnetic charges and therefore the BPS mass in N=2 theory is given by $$M^{BPS} = M ( z, \bar z , p,q) \ .$$ When the derivative of the BPS mass over the moduli at fixed values of electric and magnetic charges vanishes we call this a critical point of the moduli space. $$\left( {\partial\over \partial z^i} M ( z, \bar z , p,q) \right) _{\rm cr} =0 \qquad \Longrightarrow \qquad z_{\rm cr} = z (p,q) \ .$$ The critical value of the BPS mass coincides with the value of the entropy of the black hole with the corresponding charges: $\pi M(p,q)_{\rm cr} = S(p,q)$ . In this paper we have calculated the second derivatives of the BPS mass at the critical point. The result is simple and universal for all possible N=2 supergravities interacting with arbitrary number of vector multiplets. It shows that the second derivative is proportional to the metric in the moduli space and the critical value of the BPS mass. $$\left( {\partial\over \partial z^i } { \partial \over \partial \bar z^j } M ( z, \bar z, p,q) \right)_{\rm cr} = {1\over 2} G_{i\bar j }( z_{\rm cr}, \bar z_{\rm cr} ) M (p,q)_{\rm cr} \ .$$ Thus as long as the values of the mass $M_{\rm cr}$ is positive and the scalar metric on the moduli space $(G_{i\bar j})_{\rm cr} $ is positive definite at the critical point, the BPS mass reaches its unique minimum at the critical point. This fact was already applied to the study of the energy of the bound states of branes [@K] and it was pointed out that the extremum of the central charge describes the bound states with minimal energy. If however, any of these two positivity conditions are violated, the analysis based on regular special geometry on N=2 supersymmetric theories does not apply: one has to include the possibility of vanishing moduli and vanishing BPS mass [@strom] and of various singularities of special geometry, in particular the change in the sign of the metric of the scalar manifold. This will extend the study performed here to the interesting cases relevant to possible “phase transitions" between different vacua in different theories as suggested in [@witten]. The examples of such behavior in the context of the 5-dimensional Calabi-Yau type black holes will be presented in [@new]. An interesting outcome of our analysis is the relation of the metric on the moduli space $ G _{ab}$ with the thermodynamic metric $W_{\mu \nu}$ introduced by Weinhold [@rup]. For a general thermodynamic system it would seem to be very difficult to say much about the Weinhold metric. In the present case we are dealing, quite literally, with special geometry and in the extreme case we have found that they are proportional. It would be of interest to extend this analysis to the non-extreme case and this we plan to do in the future. One motivation for studying the Weinhold metric is that one might imagine that in a more exact quantum theory of gravity in which spacetime geometry may not play the same pre-eminent role that it does in classical and semi-classical general relativity one will still be able to talk about the thermodynamic properties of “black holes” but at a more abstract level. One needs therefore some principle to determine the thermodynamic surface giving the equation of state of the system. The thermodynamic properties are encoded in the the Weinhold metric. 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Trivedi and F. Wilczek, Mod. Phys. Lett. [**A6**]{}, 2353 (1991). R. Kallosh, A. Linde, T. Ortín, A. Peet, and A. Van Proeyen, Phys. Rev. [**D46**]{}, 5278 (1992). R. Kallosh, hep-th/9611162. E. Witten Nucl.Phys.B471:195-216,1996, hep-th/9603150. [^1]: E-mail: FERRARAS@vxcern.cern.ch,  G.W.Gibbons@damtp.cam.ac.uk,\   kallosh@physics.stanford.edu [^2]: Here we follow notation of [@KSW] where the black holes were studied in the context of the special geometry. In this context, as different from eq. (\[scalaraction\]) the vector field strength has additional factor 1/2. [^3]: These first order equations were derived using supersymmetry in [@FKS; @fre]. The derivation here is new and does not use supersymmetry. [^4]: The detailed analysis of the uniqueness issue of the critical points (with particular examples) has been performed in the context of the 5-dimensional very special geometry in [@new]. Here we outline how this analysis may be extended to 4-dimensional special geometry. [^5]: Upper $SU(N)$ indices label objects in the complex conjugate representation of $SU(N)$

--- author: - 'Samuel Abreu,' - 'Harald Ita,' - 'Francesco Moriello,' - 'Ben Page,' - 'Wladimir Tschernow,' - Mao Zeng bibliography: - 'main.bib' title: ' Two-Loop Integrals for Planar Five-Point One-Mass Processes ' --- Acknowledgments {#acknowledgments .unnumbered} =============== We would like to thank V. Del Duca and L. Dixon for inspiring discussions. The work of S.A. is supported by the Fonds de la Recherche Scientifique–FNRS, Belgium. The work of B.P. is supported by the French Agence Nationale pour la Recherche, under grant ANR–17–CE31–0001–01. H.I. thanks the Pauli Center of ETH Zürich and the University of Zürich for hospitality. W.T.’s work is funded by the German Research Foundation (DFG) within the Research Training Group GRK 2044. The authors acknowledge support by the state of Baden-Württemberg through bwHPC.

--- abstract: 'Hayashi has extended a result of Mañé, proving that every diffeomorphism $f$ which has a $C^1$-neighborhood $\mathcal{U}$, where all periodic points of any $g\in\mathcal{U}$ are hyperbolic, it is an Axiom A diffeomorphism. Here, we prove the analogous result in the volume preserving scenario.' address: - 'Instituto de Matemática, Universidade Federal do Rio de Janeiro, P. O. Box 68530, 21945-970 Rio de Janeiro, Brazil.' - 'Instituto de Ciências, Matemática e Computação, Universidade de São Paulo, 16-33739153 São Carlos-SP, Brazil' author: - Alexander Arbieto - Thiago Catalan title: Hyperbolicity in the Volume Preserving Scenario --- [^1] [^2] Introduction and Statement of the Results ========================================= Let $M$ be a $C^{\infty}$ $d$-dimensional Riemannian manifold without boundary and let ${{\operatorname{Diff}}^{1}_{m}(M)}$ denote the set of diffeomorphisms which preserves the Lebesgue measure $m$ induced by the Riemannian metric. We endow this space with the $C^1$-topology. In the theory of dynamical systems, one important question is to know whether robust dynamical properties in the phase space leads to differentiable properties of the system. For instance, one of the most important properties that a system can have is stability. This says that any system close enough to the initial have the same orbit structure of the initial. In other terms, this says that there is a topological conjugacy between this system and the initial one. In a striking article [@M3] Mañé proves that any $C^1$ structurally stable diffeomorphism is an Axiom A diffeomorphism. In [@P2] Palis extended this result to $\Omega$-stable diffeomorphisms. Actually, Mañé believe that a weaker property than $\Omega$-stability should be enough to guarantee the Axiom A property. Let us elaborate on this property. Given a diffeomorphism $f$ over $M$, a periodic point $p$ of $f$ is [*hyperbolic*]{} if $Df^{\tau(p)}$ has eigenvalues with absolute values different of one, where $\tau(p)$ is the period of $p$. In the space of $C^1$ diffeomorphisms over $M$, ${\operatorname{Diff}}^1(M)$, we can define the set $\mathcal{F}^1(M)$ as the set of diffeomorphisms $f\in {\operatorname{Diff}}^1(M)$ which have a $C^1$-neighborhood $\mathcal{U}\subset {\operatorname{Diff}}^1(M)$ such that if $g\in {{\mathcal U}}$ then any periodic point of $g$ is hyperbolic. In [@Hayashi], Hayashi proved that any diffeomorphism in $\mathcal{F}^1(M)$ is Axiom A, which means that the periodic points are dense in the nonwandering set $\Omega(f)$ and the last one is a hyperbolic set. We recall that in dimension two, this was proved by Mané [@Mane]. This was also studied for flows without singularities by Gan and Wen in [@GW]. Observe that in the volume preserving scenario, the Axiom A condition is equivalent to the diffeomorphism be Anosov, since $\Omega(f)=M$ by Poincaré Recurrence Theorem. Hence, it is a natural question if Hayashi’s and Mané’s results still holds in the volume preserving scenario. Actually, it seems that the arguments of Mañé holds in this case, specially the ones related to the perturbations. Moreover, using recent generic results in the volume preserving context much of the arguments of the original proof can be avoided. The purpose of this article is to do this. We want to stress out that we follow the same lines of Mañé’s original argument once we show that periodic points must have the same index. We observe also that Bessa and Rocha also have analogous results in [@BR] and together with Ferreira in [@BFR], for the context of incompressible and Hamiltonian flows, but in lower dimensions (three and four respectively). We will also discuss what kind of results our arguments could prove in the contex of incompressible flows in any dimension. We define the set ${{\mathcal F}^1_m(M)}$ as the set of diffeomorphisms $f\in {{\operatorname{Diff}}^{1}_{m}(M)}$ which have a $C^1$-neighborhood $\mathcal{U}\subset {{\operatorname{Diff}}^{1}_{m}(M)}$ such that if $g\in {{\mathcal U}}$ then any periodic point of $g$ is hyperbolic. If $f\in {\operatorname{Diff}}^1(M)$ then an $f-$invariant compact set $\Lambda$ of $M$ is called a [*hyperbolic set*]{} if there is a continuous and $Df$-invariant splitting $T_{\Lambda}M=E^s\oplus E^u$ such that there are constants $0<\lambda<1$ and $C>0$, satisfying $$\|Df_x^k| E^s(x)\|\leq C\lambda^k \quad \text{and}\quad \|Df_x^{-k}| E^u(x)\|\leq C\lambda^k,$$ for every $x\in \Lambda$ and $n>0$. We say that $f$ is an [*Anosov diffeomorphism*]{}, if $M$ is a hyperbolic set for $f$. The main result of this article is the following, Any diffeomorphism in ${{\mathcal F}^1_m(M)}$ is Anosov. \[maintheorem\] If the manifold is symplectic, i.e. it possess a non-degenerated closed 2-form $\omega$, then we can define the set ${{\mathcal F}^{1}_{\omega}(M)}$ using only symplectic diffeomorphisms, i.e. which preserves $\omega$, as we did using volume preserving diffeomorphisms. We denote by ${{\operatorname{Diff}}^{1}_{\omega}(M)}$ the space of symplectic diffeomorphisms. It was proved by Newhouse in [@Newhouse] that any element of ${{\mathcal F}^{1}_{\omega}(M)}$ is Anosov. We observe that, since the neighborhoods of the diffeomorphisms are taken in the respectively spaces ${{\operatorname{Diff}}^{1}_{m}(M)}$, ${{\operatorname{Diff}}^{1}_{\omega}(M)}$, or ${\operatorname{Diff}}^1(M)$ we could take no relation between ${{\mathcal F}^1_m(M)}$, ${{\mathcal F}^{1}_{\omega}(M)}$ and $\mathcal{F}^1(M)$ direct from definition. But, as a corollary of the previous theorem we obtain, ${{\mathcal F}^{1}_{\omega}(M)}\subset {{\mathcal F}^1_m(M)}\subset \mathcal{F}^1(M)$. Since the arguments to prove the main theorem involves heterodimensional cycles, it is natural to try to relate it with the well known Palis conjecture [@Palis] in the volume preserving scenario. In fact the arguments needed to do this are due to Crovisier and they were outlined in [@CROVISIER]. Since it is related to our main theorem, we write it here just for sake of completeness. If $f\in{{\operatorname{Diff}}^{1}_{m}(M)}$ is not an Anosov diffeomorphism then it can be approximated by one diffeomorphism, either exhibiting an heterodimensional cycle if the dimension of $M$ is greater than two or exhibiting an homoclinic tangency if the dimension of $M$ is two. \[palis conj\] Actually, the statement for surfaces was proved by Newhouse in [@Newhouse]. Since for surfaces any volume preserving diffeomorphism is symplectic. This paper is organized as follows. In section 2, we recall the Franks lemma in the volume preserving scenario, which is one of the main tools used in the proof. In section 3, we prove that the index for hyperbolic periodic points is constant in a neighborhood of any $f\in {{\mathcal F}^1_m(M)}$. In section 4, we give a proof of our main theorem. In section 5, we recall Crovisier’s arguments about Palis conjecture. In section 6, we point out how the arguments in the previous sections could be use to prove some analogous results of Gan and Wen [@GW], Toyoshiba [@Toyo] and see how this extends a corollary of Bessa and Rocha, in [@BR] to higher dimensions. Finally, in the appendix we describe the adaptations to the volume preserving case of an argument by Mañé. Franks-type Lemma ================= One of the most useful and basic perturbation lemmas is the Franks lemma [@FRANKS]. This lemma enable us to perform non-linear perturbations along a finite piece of an orbit simply performing arguments from Linear Algebra. In what follows, we will recall this lemma in the volume-preserving scenario, which is contained in the work of Bonatti-Diaz-Pujals [@BDP], proposition 7.4. See also [@LLS]. \[l.franks\] Let $f\in {{\operatorname{Diff}}^{1}_{m}(M)}$ and ${{\mathcal U}}$ be a $C^1$-neighborhood of $f$ in ${{\operatorname{Diff}}^{1}_{m}(M)}$. Then, there exist a neighborhood ${{\mathcal U}}_0\subset {{\mathcal U}}$ of $f$ and $\delta>0$ such that if $g\in {{\mathcal U}}_0(f)$, $S=\{p_1,\dots,p_m\}\subset M$ and $\{L_i:T_{p_i}M\to T_{p_{i+1}}M\}_{i=1}^m$ are linear maps belonging to $SL(d)$ satisfying $\|L_i-Dg(p_i)\|\leq\delta$ for $i=1,\dots m$ then there exists $h\in{{\mathcal U}}(f)$ such that $h(p_i)=g(p_i)$ and $Dh(p_i)=L_i$. Index of Periodic Orbits ======================== In this section we analyze the index of periodic orbits of diffeomorphisms in ${{\mathcal F}^1_m(M)}$. By definition the index of a hyperbolic periodic orbit is the dimension of its stable space. We will see that in the volume preserving case, the property of have two periodic hyperbolic saddles with different indices cannot happen if $f\in {{\mathcal F}^1_m(M)}$. The main result in this section is the following. \[p.index\] Let $f\in {{\mathcal F}^1_m(M)}$ then there exist a neighborhood ${{\mathcal U}}$ of $f$ in ${{\operatorname{Diff}}^{1}_{m}(M)}$ and an integer $i$ such that for every diffeomorphism $g\in{{\mathcal U}}$ and every hyperbolic periodic orbit $p$ of $g$, the index of $p$ with respect to $g$ is $i$. This can be seen through the creation of heterodimensional cycles, thus the proposition can be seen as the volume-preserving and discrete version of a result by Gan and Wen (see theorem 4.1 of [@GW]). But, we can also apply a volume-preserving version of a result by Abdenur-Bonatti-Crovisier-Diaz-Wen (corollary 2 of [@ABCDW]). In fact, if we define the Lyapunov exponent vector of a hyperbolic periodic point $p$ of period $\tau(p)$ by $$v=(\frac{\log|\mu_1|}{\tau(p)},\dots,\frac{\log|\mu_d|}{\tau(p)}),$$ where $\mu_1,\dots,\mu_d$ are the eigenvalues of $Df^{\tau(p)}(p)$ ordered by their moduli. Then the volume preserving version of corollary 2 of [@ABCDW] would gives us a residual subset of ${{\operatorname{Diff}}^{1}_{m}(M)}$ such that for any homoclinic class of a diffeomorphism belonging to this residual subset, the closure of the set of Lyapunov vectors of the saddles of this homoclinic class is convex. This could be done, using the available perturbation tools in the volume preserving case, the Pasting lemma [@A] and the regularization theorem of Ávila [@Avila]. Moreover, by a result of Bonatti and Crovisier [@BONATTICROVISIER], we can assume that for any difffeomorphism in this residual subset the whole manifold is an homoclinic class. Thus, this would give us a saddle with an eigenvalue with norm close to one and by Franks lemma \[l.franks\] this would create a non-hyperbolic point, after an small perturbation. However, since we want to show the relation of these objects with heterodimensional cycles, we will give a proof based on some results by Mañé and Gan-Wen and we will explain why the arguments of [@ABCDW] still are valid in the volume preserving context. First of all, we will recall some good properties of the periodic set that will be very useful. We say that an $f$-invariant compact set $\Lambda$ has a [*dominated splitting*]{} if there exist a continuous splitting $T_{\Lambda}M=E\oplus F$ and constants $m\in{\mathbb{N}}$, $0<\lambda<1$ such that for every $x\in \Lambda$ we have: $$\|Df_x^m| E(x)\|\; \|Df_{f^m(x)}^{-m}| F(f^m(x))\|\leq \lambda.$$ Now, let $\Lambda_i(f)$ denote the close of the set formed by hyperbolic periodic points of $f$ with index $i$. The following proposition which is the volume preserving version of a result by Mañé, proposition II.1 of [@Mane] (see also the work of Liao [@L]) is essential. It can be deduced from Franks Lemma \[l.franks\] and adapting some arguments of Mañé. We will give the necessarily adaptations in an appendix. \[p.domina\] If $f\in {{\mathcal F}^1_m(M)}$, there exist a neighborhood $\mathcal{U}$ of $f$ in ${{\operatorname{Diff}}^{1}_{m}(M)}$, and constants $K>0$, $m\in{\mathbb{N}}$ and $0<\lambda<1$ such that - For every $g\in\mathcal{U}$ and $p\in Per(g)$ with minimum period $\tau(p)\geq m$ $$\prod_{i=0}^{k-1} \|Dg^m(g^{mi}(p))| E^s_g(g^{mi}(p))\|\leq K\lambda^{k}$$ and $$\prod_{i=0}^{k-1} \|Dg^{-m}(g^{-mi}(p))| E^u_g(g^{-mi}(p))\|\leq K\lambda^{k},$$ where $k=[\tau(p)/m]$. - For all $0<i<dim M$ there exists a continuous splitting $T_{\Lambda_i(g)}M=E_i\oplus F_i$ such that $$\|Dg^m(x)| E_i(x)\|\; \|Dg^{-m}(g^m(x))| F_i(g^m(x))\|\leq \lambda.$$ for all $x\in \Lambda_i(g)$ and $E_i(p)=E^s_g(p)$, $F_i(p)=E^u_{g}(p)$ when $p\in Per(g)$ and $dim E^s_{g}(p)=i$. - For all $p\in Per(g)$ $$\limsup_{n\rightarrow +\infty} \frac{1}{n}\sum_{i=0}^{n-1}\log \|Dg^m(g^{mi}(p))| E^s_g(g^{mi}(p))\|<0$$ and $$\limsup_{n\rightarrow +\infty} \frac{1}{n}\sum_{i=0}^{n-1}\log \|Dg^{-m}(g^{-mi}(p))| E^u_g(g^{-mi}(p))\|<0.$$ \[DS prop\] If proposition \[p.index\] is not true, then there exist two hyperbolic periodic orbits $p$ and $q$ of $f$ with respectively indices $i$ and $i+j$, for some $j>0$. Now, we state a result by Abdenur-Bonatti-Crovisier-Diaz-Wen in [@ABCDW] in the volume preserving case, we slightly modify the statement. For any neighborhood $\mathcal{U}$ of $f\in {{\operatorname{Diff}}^{1}_{m}(M)}$, if there exist $p,q\in Per(f)$ with indices $i$ and $i+j$, respectively, then for any positive integer $\alpha$ between $i$ and $i+j$ there exist $g\in\mathcal{U}$ and a hyperbolic periodic point of $g$ with index $\alpha$. \[propABCDW\] We will explain why this proposition holds in the volume preserving case later. Using this proposition, we can find hyperbolic periodic points $p$ and $q$ of $f$, by some perturbation of $f$, with indices $i$ and $i+1$, respectively. In the sequence, we will see how to perturb $f$ in order to get a heterodimensional cycle between these two hyperbolic periodic points. First, we remember a result by Bonatti-Crovisier [@BONATTICROVISIER]: There exists a residual subset ${{\mathcal R}}$ of ${{\operatorname{Diff}}^{1}_{m}(M)}$ such that if $g\in {{\mathcal R}}$ then $M=H(p,g)$, where $H(p,g)$ is the homoclinic class for a hyperbolic periodic point $p$ of $g$. In particular, $g$ is transitive. \[BC\] Hence, perturbing and using the hyperbolicity of $p$ and $q$, we can suppose that our diffeomorphism $f\in {{\mathcal R}}$ and so it is transitive. Then using the connecting lemma of Hayashi for conservative diffeomorphisms, see [@BONATTICROVISIER], we can create an intersection between $W^u(p)$ and $W^s(q)$, also perturbing if necessary, we can assume that this intersection is transversal. Hence, this intersection is robust, and we can suppose that this new diffeomorphism also belongs to ${{\mathcal R}}$. Using the connecting lemma once more, we can create an intersection between $W^s(p)$ and $W^u(q)$. Thus we create a heterodimensional cycle. We observe that this type of argument appears in [@Ab]. As we comment above, what we want to do now is to find a periodic point with at least one Lyapunov exponent close to zero. We can suppose that $p$ and $q$ are fixed points. Now, let $\mathcal{R}_1$ be the set of volume preserving diffeomorphisms where the homoclinic class are disjoint or coincide. Using a result by Carballo, Morales and Pacifico in [@CMP] we know that $\mathcal{R}_1$ is a residual subset in ${{\operatorname{Diff}}^{1}_{m}(M)}$. So, we can assume that $f\in{{\mathcal R}}\cap \mathcal{R}_1$. Hence, $M=\Lambda_i(f)=\Lambda_{i+1}(f)$, i.e., hyperbolic periodic points with indices $i$ and $i+1$ are dense in $M$. Therefore, using Proposition \[DS prop\] we have a dominated splitting for $f$, $TM=E\oplus C\oplus F$, such that the dimension of $E$ and $C$ are equal to $i$ and one, respectively. And thus, by the continuation of the dominated splitting we still have this one for the perturbation of $f$ that exhibits a heterodimensional cycle between $p$ and $q$. Let $\mathcal{U}$ be some neighborhood of $f$ in ${{\mathcal F}^1_m(M)}$ and $\mathcal{U}_1\subset \mathcal{U}$ such that we still have the previous dominated splitting for every $g\in \mathcal{U}_1$. We recall now a perturbation result of Xia in [@Xia]. Fix $\phi\in {{\operatorname{Diff}}^{1}_{m}(M)}$, there exist constants ${\varepsilon}_0>0$ and $c > 0$, depending only of $\phi$, such that for any $x\in M$, any $\psi\in {{\operatorname{Diff}}^{1}_{m}(M)}$ ${\varepsilon}_0-C^1$ close to $\phi$ and any positive numbers $0<\delta<{\varepsilon}_0$ and $0<{\varepsilon}<{\varepsilon}_0$, we have that if $d(y,x) < c\delta{\varepsilon}$, then there is a $\psi_1\in {{\operatorname{Diff}}^{1}_{m}(M)}$ ${\varepsilon}-C^1$ close to $\psi$ such that $\psi_1(\psi^{-1}(x))=y$ and $\psi_1(z)=z$ for all $z\not\in \psi^{-1}(B_{\delta}(x))$. \[xia lemma\] Then, we fix ${\varepsilon}_0$ and $c>0$ for $f$ according the previous lemma. Now, let $0<{\varepsilon}<{\varepsilon}_0$ be such that if $f_1\in {{\operatorname{Diff}}^{1}_{m}(M)}$ is ${\varepsilon}-C^1$ close to $f$ then $f\in \mathcal{U}_1$. Let $x\in W^s(p)\cap W^u(q)$ and $y\in W^u(p)\cap W^s(q)$. Now, let $B_{p}$ and $B_{q}$ be small balls in $M$ centered at $p$ and $q$, respectively. Moreover, given any $\gamma>0$ we may choose $B_{p}$ such that $\|Df(z)-Df(p)\|\leq \gamma$, for every $z\in B_{p}$. By the choice of $x$ and $y$ we can choose $m_1, m_2, m_3$ and $m_4$ positive integers such that $f^{m_1}(x), f^{-m_3}(y)\in B_{p}$ and $f^{-m_2}(x), f^{m_4}(y)\in B_{q}$. Now, let $0<\delta<{\varepsilon}_0$ such that $$f^{-1}(B_{\delta}(f^{m_1}(x)))\cap B_{\delta}(f^{m_1}(x))=\varnothing,\text{ } f^{-1}(B_{\delta}(f^{-m_2+1}(x)))\cap B_{\delta}(f^{-m_2+1}(x))=\varnothing,$$ and $$f^{-1}(B_{\delta}(f^{m_4}(y)))\cap B_{\delta}(f^{m_4}(y))=\varnothing,\text{ } f^{-1}(B_{\delta}(f^{-m_3+1}(y)))\cap B_{\delta}(f^{-m_3+1}(y))=\varnothing.$$ Using the $\lambda$-Lemma, we can find $z_m$ $c\delta{\varepsilon}-$close to $f^{m_1}(x)$ such that $f^m(z_m)$ is also $c\delta{\varepsilon}-$close to $f^{-m_3}(y)$ and $f^{r}(z_m)\in B_{p}$, $0\leq r\leq m$, for every $m>0$ large enough. Analogously, we can find $\overline{z}_n$ satisfying similar conditions exchanging $p$ for $q$ and the respectively iterates of $x$ and $y$. Hence, the set $$O_{mn}=\{z_m,..., f^{m}(z_m),f^{-m_3}(y),..., f^{m_4}(y), \overline{z}_n,..., f^{n}(\overline{z}_n), f^{-m_2}(x),..., f^{m_1}(x)\}$$ is a pseudo periodic orbit. Using lemma \[xia lemma\], we can perturb $f$ in order to find a periodic orbit $p_{mn}$ that shadows $O_{mn}$. Moreover, note that $\{z_m,..., f^{m-1}(z_m),\newline f^{-m_3}(y),..., f^{m_4-1}(y), \overline{z}_n,..., f^{n-1}(\overline{z}_n), f^{-m_2}(x),..., f^{m_1-1}(x)\}$ is the orbit of the periodic point $p_{mn}$. Moreover, $p_{mn}$ pass $m$ and $n$ times in $B_{p}$ and $B_{q}$, respectively. Furthermore, using the dominated splitting of $f$, we observe that $m$ and $n$ could be chosen such that the index of $p_{mn}$ is either $i+1$ or $i$. Now, fix some large $n$ and choose $m=m(n)$ as the biggest one such that, $p_{mn}$ and $p_{m-1\,n}$ are hyperbolic periodic points of different perturbations of $f$ with indices $i$ and $i+1$, respectively. We will call these perturbations of $f$ by $g$ and $h$, i.e., $p_{mn}\in Per(g)$ and $p_{m-1\,n}\in Per(h)$. We would like to remark that the way to perturb $f$ in order to construct these points gives us $g=h$ outside $B_p$. Finally, taking $n$ large enough if necessary we have that $g,h\in \mathcal{U}_1$. By the previous process we have that the orbit of the hyperbolic periodic points $p_{k n}$, $k=m,\, m-1$, is $$\{z_k,..., f^{k-1}(z_k),f^{-m_3}(y),..., f^{m_4-1}(y), \overline{z}_n,..., f^{n-1}(\overline{z}_n), f^{-m_2}(x),..., f^{m_1-1}(x)\},$$ where $z_k$ and $\overline{z}_n$ can be found by $\lambda-$lemma, depending of $k$, as before. Now, denoting by $\tau$ the period of $p_{mn}=f^{-m_3}(y)$ and taking $K=\|Df(p)|C(f)\|$, we have $$\begin{aligned} 0&<\frac{1}{\tau}\log\|Dg^{\tau}(p_{mn})|_{C(g)}\| =\frac{1}{\tau}\sum_{t=0}^{\tau-1}\log\|Dg(g^t(p_{mn}))|_{C(g)}\| \nonumber\\ &<\frac{1}{\tau}(\sum_{t=0}^{\tau-1}\log\|Df(g^t(p_{mn}))|_{C(f)}\|+\gamma) \nonumber\\ &\leq\frac{1}{\tau}(\sum_{t=0}^{\tau-m-1}\log\|Df(g^t(p_{mn}))|_{C(f)}\|+m(\log\|Df(p)|_{C(f)}\|+\gamma)+\gamma\tau) \nonumber\\ &<\frac{1}{\tau-1}(\sum_{t=0}^{\tau-m-1}\log\|Df(g^t(p_{mn}))|_{C(f)}\|+(m-1)(\log\|Df(p)|_{C(f)}\|))+2\gamma+\frac{K}{\tau}, \label{equ6}\end{aligned}$$ where we use that the central direction $C$ is one-dimensional in the first equality, the continuity of the dominated splitting in the second line and the choice of $B_{p}$ in the third one. On the other hand, using the hyperbolic periodic point $p_{m-1\,n}=f^{-m_3}(y)$ of $h$ and similarly arguments we have the following: $$\begin{aligned} 0&>\frac{1}{\tau-1}\log\|Dh^{\tau-1}(p_{m-1\,n})|_{C(h)}\| =\frac{1}{\tau-1}\sum_{t=0}^{\tau-1}\log\|Dh(h^t(p_{m-1\, n})|_{C(h)}\| \nonumber\\ &>\frac{1}{\tau-1}(\sum_{t=0}^{\tau-1}\log\|Df(h^t(p_{m-1\, n}))|_{C(f)}\|-\gamma) \nonumber\\ &\geq\frac{1}{\tau-1}(\sum_{t=0}^{\tau-m-1}\log\|Df(h^t(p_{m-1\, n}))|_{C(f)}\|+(m-1)(\log\|Df(p)|_{C(f)}\|))-2\gamma. \label{eq7}\end{aligned}$$ Now, since $g=h$ outside $B_p$ and the orbit of $p_{mn}$ and $p_{m-1\, n}$ also coincides outside $B_p$ we have $g^t(p_{mn})=h^t(p_{m-1\,n})$ for $0\leq t\leq \tau-m-1$. Hence, we can replace the inequality (\[eq7\]) in (\[equ6\]), to obtain the following $$\begin{aligned} 0&<\frac{1}{\tau}\log\|Dg^{\tau}(p_{mn})|C(g)\|< 4\gamma +\frac{K}{\tau}.\end{aligned}$$ Therefore, since the period $\tau$ goes to infinity when $n$ goes to infinity, and $\gamma>0$ is arbitrary, it is possible to find a hyperbolic periodic point $p_{mn}$ with a Lyapunov exponent sufficiently close to zero. In the general case, when $p$ and $q$ are hyperbolic periodic points, the difference is that the neighborhoods $B_p$ and $B_q$ must be neighborhoods of the orbits of $p$ and $q$, respectively, and then the numbers $m$ and $n$ will be multiples of the periods of $p$ and $q$, respectively. Hence, by the same arguments as before we can find the periodic point $p_{mn}$ with at least one Lyapunov exponent sufficiently close to zero. Finally, using Franks lemma \[l.franks\] again, we can perturb it once more such that we reduce a little bit the force of the eigenvalue associated to the Lyapunov exponent close to zero in each point of the orbit of $p_{mn}$ such that we can get indeed a zero Lyapunov exponent for the periodic point $p_{mn}$. This means, we have an eigenvalue with absolute value one, and then $p_{mn}$ is not a hyperbolic periodic point. Since all of these perturbations can be done inside $\mathcal{U}\subset \mathcal{F}_m^1(M)$ we have a contradiction. We observe that we could do the same arguments, using only that there are periodic points with different indices, as it is done in [@GW]. But, the proof is slightly simplified after that we found periodic points with indices $i$ and $i+1$. Now, we explain how proposition \[propABCDW\] could be proved in the volume preserving case with the same arguments in the dissipative case. First, we will prove a conservative version of the Proposition 2.3 in [@ABCDW]. There is a residual subset ${{\mathcal R}}_2$ of ${{\operatorname{Diff}}^{1}_{m}(M)}$ consisting of diffeomorphisms $f$ such that $Per_{{\mathbb{R}}}(H(p,f))$, the set of hyperbolic periodic points of $f$ with the same index of $p$ and with only real eigenvalues of multiplicity one, is dense in $H(p,f)$ for every non-trivial homoclinic class $H(p,f)$ of $f$. \[proABCDW\] We recall that a [*periodic linear systems (cocycles)*]{} is a 4-tuple $\mathcal{P}=(\Sigma, f, \mathcal{E},A)$, where $f$ is a diffeomorphism, $\Sigma$ is an infinite set of periodic points of $f$, $\mathcal{E}$ an Euclidian vector bundle defined over $\Sigma$, and $A\in GL(\Sigma, f,\mathcal{E})$ is such that $A(x):\mathcal{E}_x\rightarrow \mathcal{E}_{f(x)}$ is a linear isomorphism for each $x$ ($\mathcal{E}_x$ is the fiber of $\mathcal{E}$ at $x$). For the precise definition we refer the reader to the work of Bonatti-Diaz-Pujals [@BDP]. Let $H(p,f)$ be a non-trivial homoclinic class. Then the derivative $Df$ of $f$ induces a periodic linear system with transitions over $Per_h(H(p,f))$, the set of hyperbolic periodic points homoclinically related with $p$. Using this lemma and Franks lemma \[l.franks\], our problem in Proposition \[proABCDW\] becomes a problem of linear algebra. We say that a periodic linear system with transitions $\mathcal{P}=(\Sigma,\, f,\,\mathcal{E},\, A)$ is [*diagonalizable*]{} at the point $x\in \Sigma$ if the linear map $$M_A(x):\mathcal{E}_x\rightarrow \mathcal{E}_x,\quad M_A(x)=A(f^{\tau(x)-1}(x))\circ\ldots\circ A(f^2(x))\circ A(x),$$ only has positive real eigenvalues of multiplicity one. For every periodic linear system with transitions $\mathcal{P}=(\Sigma,\, f,\,\mathcal{E},\, A)$ and every ${\varepsilon}>0$ there is a dense subset $\Sigma'$ of $\Sigma$ and an ${\varepsilon}-$perturbation $A'$ of $A$ defined on $\Sigma'$ which is diagonalizable, that is, $M_{A'}(x)$ has positive real eigenvalues of multiplicity one for every $x\in \Sigma'$. By Remark 7.2 in [@BDP] we can consider the perturbation $A'$ such that $det A'(x)=1$ for every $x\in\Sigma'$. Then, as we have noted before, we can use Franks Lemma \[l.franks\] and the previous lemmas to show proposition \[proABCDW\]. In fact, after we have done all these observations the proof is identically the proof of proposition 2.3 in [@ABCDW]. Hence, we can suppose $f\in \mathcal{R}_2\cap \mathcal{R}$, where the residual set ${{\mathcal R}}$ is given by Proposition \[BC\], and then we may assume that $p$ and $q$ have only real eigenvalues of multiplicity one for $Df^{\tau(p)}(p)$ and $Df^{\tau(q)}(q)$, respectively. As we did before we can suppose, unless some perturbation, that $f$ exhibits a heterodimensional cycle between $p$ and $q$. Hence the proof of the Proposition \[propABCDW\] follows direct from the next result stated in [@ABCDW]. Let $f$ be a diffeomorphism having a heterodimensional cycle associated to periodic saddles $p$ and $q$, of indices $i$ and $i+j$ with $j>0$, with real eigenvalues. Then, for any $C^1-$neighborhood $\mathcal{U}$ of $f$ and for any integer $\alpha$ with $i\leq \alpha \leq i+j$, there exists $g\in \mathcal{U}$ having a periodic point with index $\alpha$. Although this proposition has been stated there for dissipative diffeomorphisms, all of the used perturbations of $f$ are applications of Franks lemma and Hayashi’s connecting lemma. Then, this proposition is still true in the volume preserving case since these two perturbation tools are available in the conservative setting. Proof of Theorem A ================== In the sequence we shall prove the hyperbolicity of ${\overline}{Per(f)}$ for every $f\in{{\mathcal F}^1_m(M)}$, since we already know that the indices of hyperbolic periodic points is constant, say $s$. Let us fixe $f\in {{\mathcal F}^1_m(M)}$ and a continuous dominated splitting $T_{{\overline}{Per(f)}}M=E\oplus F$ given by the Proposition \[DS prop\]. From now, we also consider $m\in{\mathbb{N}}$, $0<\lambda<1$ and $K>0$ as in the Proposition \[DS prop\]. We will prove that this splitting is hyperbolic. We will follow here similar arguments as in the proof of Mane of Theorem B in [@Mane]. To show this we need prove that we have contraction and expansion in the sub-bundles $E$ and $F$, respectively, unless a certain finite time iterate. Hence, by compactness of ${\overline}{Per(f)}$, we just need to show the following $$\liminf_{n\rightarrow +\infty} \|Df^n(x)|E(x)\|=0 \label{eq 1}$$ and $$\liminf_{n\rightarrow +\infty} \|Df^{-n}(x)|F(x)\|=0,$$ for all $x\in {\overline}{Per(f)}$. Observe it’s enough to prove the first case since the second one can be deduced from the first one replacing $f$ by $f^{-1}$. Suppose now $(\ref{eq 1})$ is not true. Then there exists $x\in M$ such that $$\|Df^{jm}(x)|F(x)\|\geq c>0, \text{ for all } j>0.$$ Defining the following probability measure $$\mu_j=\frac{1}{j}\sum_{i=0}^{j-1}\delta_{f^{mi}(x)},$$ where $\delta$ is the dirac measure, we can find a subsequence $j_n\rightarrow \infty$ such that $\mu_{j_n}$ converges to an $f^m-$invariant probability measure $\mu$ in the weak$^*$ topology and $$\lim_{n\rightarrow +\infty} \frac{1}{j_n}\log \|Df^{mj_n}(x)|E(x)\|\geq 0.$$ Hence, taking the continuous functional $\phi(x)=\log\|Df^{m}(x)|E(x)\|$ over ${\overline}{Per(f)}$, we obtain: $$\begin{aligned} \int_{{\overline}{Per(f)}}\phi \; d\mu &= \lim_{n\rightarrow +\infty}\frac{1}{j_n}\sum_{i=0}^{j_n-1}\log\|Df^{m}(f^{mi}(x))|E(f^{mi}(x))\| \\ &\geq \lim_{n\rightarrow +\infty}\frac{1}{j_n}\log\|Df^{mj_n}(x)|E(x)\|\geq 0.\end{aligned}$$ And so, using Ergodic Birkhoff’s Theorem $$0\leq \int_{{\overline}{Per(f)}}\phi\;d\mu=\int_{{\overline}{Per(f)}}\lim_{n\rightarrow +\infty}\frac{1}{n}\sum_{i=0}^{n-1}\log\|Df^{m}(f^{mi}(x))|E(f^{mi}(x))\|\; d\mu. \label{eq 2}$$ Now let $\Sigma(f)\subset M$ a total probability set given by the Ergodic closing lemma in the volume preserving case, see [@Arnaud]. Hence, denote by $\nu=\frac{1}{m}\sum_{i=0}^{m-1} {f^i}^*\mu$ the $f-$invariant probability measure induced by $\mu$, we have $\nu(\Sigma(f)\cap {\overline}{Per(f)})=1$ since $\nu$ is supported on ${\overline}{Per(f)}$. But now, by the invariance of $\Sigma(f)\cap {\overline}{Per(f)}$ for $f$, it’s easy to see that this is also a total probability set for $\mu$. And so, this together with $(\ref{eq 2})$ imply the existence of a point $y\in\Sigma(f)\cap{\overline}{Per(f)}$ such that: $$\lim_{n\rightarrow +\infty}\frac{1}{n}\sum_{i=0}^{n-1}\log\|Df^{m}(f^{mi}(y))|E(f^{mi}(y))\|\geq 0. \label{eq 3}$$ Observe that part (c) of Proposition \[DS prop\] is an obstruction for $y$ be periodic. Hence $y\not\in Per(f)$. By (\[eq 3\]), we can take $\lambda<\lambda_0<1$ and $n_0>0$ such that: $$\frac{1}{n}\sum_{i=0}^{n-1}\log\|Df^{m}(f^{mi}(y))|E(f^{mi}(y))\|\geq \log \lambda_0, \label{eq 4}$$ when $n\geq n_0$. In the next step we will find a hyperbolic periodic point $p\in Per(g)$ such that its orbit is “close" to the orbit of $y$, for $g$ near $f$, and then we will use Lemma \[l.franks\] to exchange the derivative at the orbit of $p$, such that the inequality (\[eq 4\]) gives us a contradiction with part $(a)$ of proposition \[DS prop\]. Using that $y\in \Sigma(f)$ we can approximate $f$ by diffeomorphisms $g$ such that there exists $p\in Per(g)$ and the distance between $f^j(p)$ and $f^{j}(y)$ is arbitrary small, for $0\leq j\leq n$, where $n$ is the minimum period of $p_g$. Since $y$ is not periodic the period $n$ must goes to infinity when $g$ approaches $f$. Hence we may choose $g$ and $p$ such that: $$n\geq m,$$ $$k\geq n_0,$$ $$K\lambda^k<\lambda_0^{k}$$ and $$\left(\frac{\lambda}{\lambda_0}\right)^k C^m\leq \frac{1}{2},$$ where $k=[n/m]$ and $C=\sup_{x\in M}\|Df^{-1}(x)\|$. These choices together with $(\ref{eq 4})$ and the dominated splitting of $f|{\overline}{Per(f)}$ give us the following $$\begin{aligned} \|Df^{-n}_{f^n(y)}|F(f^n(y))\| &\leq \prod_{i=0}^{k-1}\|Df^{-m}_{f^{n-mi}(y)}|F(f^{n-mi}(y))\|\; \|Df^{-(n-mk)}_{f^{n-mk}(y)}|F(f^{n-mk}(y))\| \nonumber\\ &\leq \lambda^k\, C^m\, \lambda_0^{-k}\leq \frac{1}{2}. \label{eq 5}\end{aligned}$$ Let $U$ be a neighborhood of ${\overline}{Per(f)}$ small enough such that the maximal set in $U$ $$\Lambda_U(f)=\bigcap_{n\in{\mathbb{Z}}} f^n(U)$$ has a dominated splitting and satisfying the thesis of the Proposition \[DS prop\]. Hence, we can chose $\mathcal{U}\subset {{\mathcal F}^1_m(M)}$ a neighborhood of $f$ such that every $h\in\mathcal{U}$ has a dominated splitting in $\Lambda_U(h)$ near of the dominated splitting in $\Lambda_U(f)$. Observe that $E_g(p)=E^s_g(p)$ and $F_g(p)=E^u_g(p)$ since dominated splitting is unique if we fix the dimensions, and the index of periodic points is constant for $g\in\mathcal{U}$. Hence, taking a smaller neighborhood $\mathcal{U}$ if necessary we can suppose $E_g^s(g^i(p))$ and $E^u_g(g^i(p))$ as near as we want of $E(f^i(y))$ and $F(f^i(y))$, $0\leq i \leq n$, respectively. In the sequence we will build some volume preserving isomorphisms $A_i:T_{g^i(p)}M\rightarrow T_{f^{i}(y)}M$, $0\leq i\leq n$, near of identity in local coordinates. Moreover, for future convenience, we will have $A_i(E^s_g(g^i(p)))=E(f^i(y))$ isometrically and $A_i(E^u_g(g^i(p)))=F(f^i(y))$, $0\leq i\leq n$. We show how to construct $A_0$, the other cases are analogous. We choose $$\{e_1(i),\,\ldots,\,e_s(i),\, r_1(i),\,\ldots,\,r_{d-s}(i)\} \,\text{ a basis of } T_i M,\, i=y,p,$$ such that $\{e_j(i),\, 1\leq j\leq s\}$ is an orthonormal basis for $E(y)$ if $i=y$ or for $E^s_g(p)$ if $i=p$, and $\{r_j(i),\, 1\leq j\leq d-s\}$ is an orthonormal basis for $F(y)$ if $i=y$ or for $E^u_g(p)$ if $i=p$. Let $A_0: T_p M\rightarrow T_yM$ be a linear map satisfying $A_0(e_j(p))=e_j(y)$ and $A_0(r_j(p))=r_j(y)$, $1\leq j\leq d$. Therefore, by construction, $A_0$ is a volume preserving linear map. Now, let us back to the proof. Let $L_i:T_{g^i(p)} M\rightarrow T_{g^{i+1}(p)} M$ be volume preserving maps defined as follows $$L_i= A^{-1}_{i+1}\, Df(f^i(y))\, A_i, \, \text{ for } 0\leq i\leq n-1.$$ Hence, taking $n$ large enough if necessary, $L_i$ is as close of $Dg(g^i(p))$ as we want, for all $0\leq i\leq n-1$. Then, using lemma \[l.franks\], we can find $h\in \mathcal{U}$ such that $p\in Per(h)$ and $Dh(h^i(p))=L_i$, $0\leq i\leq d-1$. Observe that $E^s_g(p)$ and $E^u_g(p)$ still are invariants by $Dh^n(p)$, by construction of $L_i'$s. This together with (\[eq 5\]), the proximity of $f$ and $g$, and the dimension of the subspaces give us that $E^u_h(p)=E^u_g(p)$. And so, $E^s_h(p)=E^s_g(p)$ too. Finally, since $A_i|E^s_g(g^i(p))$ is an isometry, we have the following $$\|Dh^{m}(h^{im}(p))| E^s_h(h^{im}(p))\|=\|Df^{m}(f^{im}(y))| E(f^{im}(y))\|, \, \text{ for all }\, i\in{\mathbb{N}}.$$ Therefore, $$\prod^{k-1}_{i=0} \|Dh^{m}(h^{im}(p))| E^s_h(h^{im}(p))\|= \prod^{k-1}_{i=0}\|Df^{m}(f^{im}(y))| E(f^{im}(y))\|\geq \lambda_{0}^{k},$$ what contradicts Proposition \[DS prop\]. Therefore we showed that ${\overline}{Per(f)}$ is hyperbolic if $f\in {{\mathcal F}^1_m(M)}$. Finally to conclude that if $f\in {{\mathcal F}^1_m(M)}$ then $f$ is Anosov, we just need to show that $\Omega(f)={\overline}{Per(f)}$ since $\Omega(f)=M$, by Poincaré’s Recurrence Theorem. This will be a consequence of Pugh’s closing lemma. If $f\in{{\mathcal F}^1_m(M)}$ then there exists some neighborhood $\mathcal{U}$ of $f$ in ${{\operatorname{Diff}}^{1}_{m}(M)}$ such that $\sharp H_n(g)$, the number of hyperbolic periodic points of $g$ with period smaller or equal than $n$, is finite and equal for every $g\in\mathcal{U}$, since all diffeomorphisms in $\mathcal{U}$ has only hyperbolic periodic points. Now, suppose ${\overline}{Per(f)}\subsetneq \Omega(f)$, and let $x\in\Omega(f)\backslash {\overline}{Per(f)}$. By Pugh’s closing lemma we can fix $k\in {\mathbb{N}}$ such that all of the perturbations of $f$, needed to create a hyperbolic periodic point near of $x$, are done in an arbitrary small neighborhood of $$\bigcup_{-k\leq j\leq k} f^j(x).$$ Thus, let $U$ be a neighborhood of ${\overline}{Per(f)}$ such that $ f^j(x)\not \in \overline{U}$, $-k\leq j\leq k$. So, using the closing lemma we can get $g\in \mathcal{U}$ and $p\in Per(g)$ with $p\not\in \overline{U}$. However by choice of $k$ and $U$, $f$ is equal to $g$ in $U$ and since $p\not \in \overline{U}$ we have $H_n(f)\neq H_n(g)$ for some $n\in {\mathbb{N}}$, what contradicts the fact of $g\in\mathcal{U}$. Therefore, we have $\Omega(f)={\overline}{Per(f)}$ and this completes the proof. Palis Conjecture in the volume preserving scenario ================================================== In this section, we elaborate the arguments due to Crovisier in [@CROVISIER] on the Palis conjecture in the volume preserving scenario, and this gives the proof of corollary \[palis conj\]. As we said in the introduction we only need to prove that if the dimension of $M$ is greater than two. Suppose $f\in{{\operatorname{Diff}}^{1}_{m}(M)}$ is not Anosov then $f\not\in {{\mathcal F}^1_m(M)}$. Therefore by theorem A, there exists $g\in {{\operatorname{Diff}}^{1}_{m}(M)}$ close to $f$ with a non hyperbolic periodic point $p$. Thus, using Franks lemma \[l.franks\], we can bifurcate this periodic point and produce two hyperbolic periodic points $q$ and $r$ with indices $i$ and $i+1$ respectively. Thus, as we did before in the proof of Proposition \[p.index\], we can perturb once more and create a heterodimensional cycle between $q$ and $r$. For more details we refer the reader to [@Crovisier2]. Star Flows ========== In this section we observe how we can extend some results in the theory of star flows to a divergence free context. Let ${\mathfrak{X}^1(M)}$ be the set of vector fields on $M$ endowed with the $C^1$ topology. And ${\mathfrak{X}^1_m(M)}\subset {\mathfrak{X}^1(M)}$ the subspace of vector fields which are divergence free. By Liouville’s formula, we know that the flow generated by a divergence free vector field is volume preserving, so they are also called incompressible flows in the literature. The analogous version of ${{\mathcal F}^1_m(M)}$ to the flow case are called incompressible star flows. We say that $X$ generates an incompressible star flow if there exits a neighborhood ${{\mathcal U}}$ of $X$ in ${\mathfrak{X}^1_m(M)}$ such that if $Y\in {\mathfrak{X}^1_m(M)}$ then all of its singularities and periodic orbits are hyperbolic. We denote the set of incompressible star flows by ${\mathfrak{X}^*_m(M)}$. The analogous result by Gan and Wen in [@GW] should be true for incompressible star flows. If $X\in{\mathfrak{X}^*_m(M)}$ has no singularities then $X$ is Anosov. In fact, this could be seen as follows. First of all, proposition \[p.domina\] can be extended to the context of incompressible flows exactly as Hayashi did in [@Hayashi]. And the same calculations that we did on heterodimensional cycles (which are inspired on the calculations of [@GW]) can be done to prove the following proposition. If $X\in {\mathfrak{X}^*_m(M)}$ has no singularities then $X$ has no heterodimensional cycles. Now, we observe that the version of Bonatti-Crovisier’s result lemma \[BC\] to incompressible flows can be found in [@Bessa]. Hence, together with the previous proposition we obtain the following proposition. If $X\in {\mathfrak{X}^*_m(M)}$ has no singularities then there exist a neighborhood ${{\mathcal U}}$ of $X$ in ${\mathfrak{X}^1_m(M)}$ and $i\in {\mathbb{N}}$, such that the index of any periodic orbit of any vector field $Y\in {{\mathcal U}}$ is $i$. Now, we can adapt the arguments of Toyoshiba [@Toyo], in the same way that as we did before, to obtain the following result. If $X\in {\mathfrak{X}^*_m(M)}$ has no singularities and there exist a neighborhood ${{\mathcal U}}$ of $X$ in ${\mathfrak{X}^1_m(M)}$ and $i\in {\mathbb{N}}$, such that the index of any periodic orbit of any vector field $Y\in {{\mathcal U}}$ is $i$. Then $X|_{{\overline}{Per(X)}}$ is hyperbolic and ${\overline}{Per(X)}=\Omega(X)$. Finally, by Poincaré’s recurrence, we know that $\Omega(X)=M$ and this would imply that $X$ is Anosov. If we denote by $KS$ the $C^1$-interior of Kupka-Smale incompressible vector fields then $KS\subset {\mathfrak{X}^*_m(M)}$. Now, suppose that $X\in KS$ and $X$ has a singularity $\sigma$. Then since $\Omega(X)=M$ and there are a finite number of singularities then $\sigma$ is approximated by regular orbits and so, by the connecting lemma, after a perturbation we obtain that $W^u(\sigma)\cap W^s(\sigma)-\{\sigma\}\neq \emptyset$. Thus this would be a non-transversal intersection, but this is a contradiction since $X\in KS$. In particular, the analogous result of Toyoshiba holds for incompressible flows. Let $X\in KS$ then $X$ is Anosov. Finally, we observe that if $X$ is a $C^1$-structurally stable divergence free vector field then $X\in KS$. So the following corollary generalizes a corollary found in [@BR] to any dimension (not only 3). If $X\in {\mathfrak{X}^1_m(M)}$ is a $C^1$-structurally stable divergence free vector field then $X$ is Anosov. [**Acknowledgements:**]{} A.A. wants to thank Prof. Sylvain Crovisier by his remarkable comments and for pointing out a mistake in a previous version of this article. He wants to thank Prof. Jorge Rocha for kindly provide reference [@BR]. He also wants to thank ICMC-USP for the kind hospitality. T.C. wants to thank UFRJ for the kind hospitality during preparation of this work. Appendix ======== In this appendix, we show the necessarily modifications to prove proposition \[p.domina\]. In particular, we will review Mañé’s argument from [@Mane] from page 523 to 540. First we recall some notions introduced by Mañé in our context. Let $GL(d)$ be the group of linear isomorphisms and $SL(d)$ be the subgroup of $GL(d)$ of isomorphisms with determinant equal to one. By a hyperbolic sequence we mean a sequence hyperbolic isomorphisms $\xi:{\mathbb{Z}}\to GL(d)$. The sequence is periodic if there exists $m$ such that $\xi_{j+m}=\xi_j$ for $j\in {\mathbb{Z}}$, the minimal positive $m$ is called the period of the sequence. If the stable space is the whole ${\mathbb{R}}^d$ then we call it a contracting sequence. If the sequence is formed by volume preserving isomorphisms, i.e. $\xi:{\mathbb{Z}}\to SL(d)$ then we call it a vol-hyperbolic sequence. Also. a family of periodic sequences $\{\xi^{\alpha}\}$ will be called a vol-family. A family of periodic sequences $\{\xi^{\alpha}\}$ is vol-hyperbolic, if all of its sequences are vol-hyperbolic and $\sup_{\alpha,j}\{\|\xi_j^{\alpha}\|\}<\infty$. Define a distance $d(\xi,\eta)=\sup_{\alpha,j}\{\|\xi_j^{\alpha}-\eta_j^{\alpha}\|$ between two families $\xi$ and $\eta$, we also say that they are equivalent if for every $\alpha$ the period of $\xi^{\alpha}$ and $\eta^{\alpha}$ coincide. Moreover the family $\xi$ is uniformly vol-hyperbolic if there exists ${\varepsilon}>0$ such that every equivalent vol-family $\eta$ which satisfies $d(\xi,\eta)<{\varepsilon}$ is vol-hyperbolic. The proof of proposition \[p.domina\] goes as the same way as in [@Mane], using Franks lemma in the volume preserving case (lemma \[l.franks\]) and the following proposition, which is analogous to lemma II.3 of [@Mane]. \[l.linear\] If $\{\xi^{\alpha}\}$ is an uniform vol-hyperbolic family then there exist constants $K>0$, $m\in {\mathbb{N}}$ and $0<\lambda<1$ such that 1. If $\xi^{\alpha}$ has period $n\geq m$ and $k=[n/m]$ then $$\prod_{j=0}^{k-1}\|(\prod_{i=0}^{m-1}\xi^{\alpha}_{mj+i})|_{E^s_{mj}}\|\leq K\lambda^k\textrm{ and }\prod_{j=0}^{k-1}\|(\prod_{i=0}^{m-1}\xi^{\alpha}_{mj+i})^{-1}|_{E^u_{m(j+1)}}\|\leq K\lambda^k.$$ 2. For every $\alpha$ and $j\in {\mathbb{Z}}$: $$\|(\prod_{i=0}^{m-1}\xi^{\alpha}_{j+i})|_{E^s_{j}}\|\|(\prod_{i=0}^{m-1}\xi^{\alpha}_{j+i})^{-1}|_{E^u_{j+m}}\|\leq \lambda.$$ 3. For every $\alpha$: $$\limsup_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\log\|(\prod_{i=0}^{m-1}\xi^{\alpha}_{mj+i})|_{E^s_{mj}}\|<0\textrm{ and}$$ $$\limsup_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\log\|(\prod_{i=0}^{m-1}\xi^{\alpha}_{mj+i})^{-1}|_{E^u_{m(j+1)}}\|<0.$$ We only indicate the necessarily adaptations. First, we recall lemma II.7 from [@Mane], that will be used to control the stable and unstable parts of the isomorphisms. Let $\{\xi^{\alpha}\}$ be a uniformly contracting family. There exists $K>0$, $m\in {\mathbb{N}}$ and $0<\lambda<1$ such that 1. If $\xi^{\alpha}$ has period $n\geq m$ and $k=[n/m]$ then $$\prod_{j=0}^{k-1}\|\prod_{i=0}^{m-1}\xi^{\alpha}_{mj+i}\|\leq K\lambda^k$$. 2. For every $\alpha$: $$\limsup_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\log\|\prod_{i=0}^{m-1}\xi^{\alpha}_{mj+i}\|<0.$$ With this we can prove the following lemma If $\{\xi^{\alpha}\}$ is a uniformly vol-hyperbolic family then there exist ${\varepsilon}>0$, $K>0$, $0<\lambda<1$ such that if $\{\eta^{\alpha}\}$ is an equivalent vol-family with $d(\xi,\eta)<{\varepsilon}$ then $\eta$ is vol-hyperbolic. Moreover, if $n$ is the period of $\eta^{\alpha}$ and ${\widetilde}{\eta}_j^{\alpha}:{\mathbb{R}}^n/E^s_j\to{\mathbb{R}}^n/E^s_{j+1}$ is the map induced by $\eta_j^{\alpha}$ then $$\|\prod_{j=0}^{n-1}\eta_j^{\alpha}/E^s_0\|\leq K\lambda^n\textrm{ and }\|(\prod_{j=0}^{n-1}{\overline}{\eta}_j^{\alpha})^{-1}\|\leq K\lambda^n.$$ As Mañé did, if ${\varepsilon}>0$ is small and $0<m<d$ and we take a vol-family $\{\phi^{\beta}\}$ containing every sequence $\phi:{\mathbb{Z}}\to GL(d-m)$ with same period of some $\xi^{\alpha}$ such that $\sup_i\|\phi-({\overline}{\xi}^{\alpha}_i)\|<{\varepsilon}$ and this family is uniformly contracting. Indeed, if not there exist a sequence $\psi:{\mathbb{Z}}\to GL(d-n)$ such that for some $\beta$, $\psi$ and $\phi^{\beta}$ have the same period, $\sup_i\|\phi_i^{\beta}-\psi\|$ is small and $\prod_{j=0}^{n-1}\psi_j$ has an eigenvalue with modulus 1 and determinant close to 1 (since is close to $\phi$). Then construct a sequence $\zeta$ with the same period of $\xi$ such that $\zeta_j=\psi_j^{-1}$ restricted to ${\mathbb{R}}^d/E^s_j$ and equal to $\det(\psi_j)\xi$ restricted to $E^s_j$. Note that $\zeta$ is close to $\xi$. This would contradict the vol-uniform hyperbolicity of $\{\xi^{\alpha}\}$. Now, we can proceed exactly as in the rest of the proof of lemma II.8 of [@Mane]. Thus, as in [@Mane]. We obtain (1) and (3) of proposition \[l.linear\]. Now, we prove (2) of proposition \[l.linear\]. First, we observe that the next lemma about angles (lemma II.9 of [@Mane]) holds in the volume preserving case. If $\{\xi^{\alpha}\}$ is an uniformly vol-family then there exist ${\varepsilon}>0$, $\gamma>0$ and $m\in {\mathbb{N}}$ such that if $\{\eta^{\alpha}\}$ is an equivalent vol-family with $d(\xi,\eta)<{\varepsilon}$ then for every $\eta^{\alpha}$ with period $n\geq m$ the angles between stable and unstable spaces are bounded away from zero, i.e. $$\angle (E^s_0(\eta^{\alpha}),E_0^u(\eta^{\alpha}))>\gamma.$$ Indeed, to obtain a contradiction, Mañé construct the following perturbation $\zeta_j=\eta_j$ if $0<j\leq n-1$ and $$\xi_0=\eta_0\left( \begin{array}{cc} I & C \\ 0 & I\\\end{array}\right).$$ For some suitable $C$, note that this is also a vol-sequence. Finally, if (2) does not holds then Mañé shows that the following sequence has small angles between the stable and unstable spaces. The sequence has the form $\eta^t:{\mathbb{Z}}\to GL(d)$, for some $t$ and has period $n$ large, such that for $1\leq j<n-1$, $$\eta^t_i=(I+T_i)\xi_i^{\alpha}(I+P_t) \textrm{ , }\eta^t_{i+j}=(I+T_{i+j})\xi_{i+j}^{\alpha}\textrm{ and }$$ $$\eta^t_{i+n-1}=(I+S_t)T_{i+n_0-1}\xi^{\alpha}_{i+n-1}.$$ However, the transformations involved satisfy the following estimate for ${\varepsilon}>0$ very small, $$\|P_t\|\leq {\varepsilon}\,,\,\|S_t\|\leq {\varepsilon}\textrm{ and }\|T_j\|\leq {\varepsilon}.$$ Hence $\det(I+T_j)$, $\det(I+S_t)$ and $\det (I+P_t)$ are close to 1, so close as we want. Hence we can divide $\eta$ by these determinants appropriately and now we obtain a vol-sequence, with same invariant spaces. In particular the angles are small and again we obtain a contradiction with the previous lemma. Actually, the same argument could be used to show that if a vol-family is not uniformly hyperbolic then it is not uniformly vol-hyperbolic. Indeed, it would have a not hyperbolic sequence sufficiently close. In particular, the determinant would be almost one. Thus multiplying by the inverse of the determinant in a direction distinct of the non-hyperbolic direction, we obtain a not hyperbolic vol-sequence close to the original. A contradiction. We finish this appendix with some comments about proposition \[p.domina\] in the symplectic case. Of course if $f$ is Anosov the statements of that proposition are obviously true. In fact, such proposition is true by the arguments of Newhouse [@Newhouse], since he proves that if a symplectic diffeomorphism has an homoclinic tangency than it can be approximated by one with an 1-elliptic periodic point, which is in particular a non-hyperbolic point. However, proposition \[p.domina\] could be obtained directly. For instance, the domination part could be obtained using arguments from [@HT] and [@Bochi]. 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--- abstract: 'We show using numerical simulations that slowly driven skyrmions interacting with random pinning move via correlated jumps or avalanches. The avalanches exhibit power law distributions in their duration and size, and the average avalanche shape for different avalanche durations can be scaled to a universal function, in agreement with theoretical predictions for systems in a nonequilibrium critical state. A distinctive feature of skyrmions is the influence of the non-dissipative Magnus term. When we increase the ratio of the Magnus term to the damping term, a change in the universality class of the behavior occurs, the average avalanche shape becomes increasingly asymmetric, and individual avalanches exhibit motion in the direction perpendicular to their own density gradient.' author: - 'S. A. D['' i]{}az$^{1,2}$, C. Reichhardt$^{1}$, D. P. Arovas$^{2}$, A. Saxena$^{1}$ and C. J. O. Reichhardt$^{1}$' title: Avalanches and Criticality in Driven Magnetic Skyrmions --- [*Introduction–*]{} Magnetic skyrmions are nanoscale particlelike spin textures that were first observed in chiral magnets in 2009 [@1; @2] and have since been identified in a growing variety of materials, including several that support skyrmions at room temperature [@3; @4; @5; @6; @7; @8]. Skyrmions can exhibit depinning phenomena under an applied current [@2; @9; @10; @11; @12; @13; @14; @15; @16], and their ability to be set in motion along with their size scale make them promising candidates for a variety of applications [@17; @18]. A key feature of skyrmions that is distinct from other depinning systems [@19] is the strong influence on the skyrmion motion of the non-dissipative Magnus term, which arises from the skyrmion topology [@2]. Strong Magnus terms are also relevant for vortex depinning in neutron star crusts [@34; @35]. In particle-based models of vortices in type-II superconductors or colloidal particles, the motion is dominated by the damping term which aligns the particle velocity with the external forces [@19]. In contrast, the Magnus term aligns the particle velocity perpendicular to the direction of the external forces, causing the skyrmions to move at an angle called the intrinsic skyrmion Hall angle $\theta^{\rm int}_{Sk}$ with respect to the external forces [@2; @10; @12; @13; @14]. As recently shown, the Magnus term strongly affects the overall skyrmion dynamics in the presence of disorder, with the measured skyrmion Hall angle starting at zero or a small value for drives just above depinning and gradually increasing to the intrinsic or pin-free $\theta^{\rm int}_{Sk}$ value as the drive increases and the skyrmions move faster [@12; @13; @14; @15; @16; @20; @21; @22; @23]. In many slowly driven systems with quenched disorder, the motion near depinning takes the form of bursts or avalanches of the type observed in driven magnetic domain walls [@24; @24a; @25], vortices in type-II superconductors [@19; @26], earthquake models [@28], and near yielding transitions in sheared materials [@29; @30]. Avalanches or so-called crackling noise arise in a wide range of collectively interacting driven systems, and scaling properties of the avalanche size distributions as well as the average avalanche shape can be used to determine whether the system is at a nonequilibrium critical point and to identify its universality class [@31; @32; @33]. In many avalanche systems, the dynamics is overdamped, but when non-dissipative effects become important, the statistics of the avalanches can change. In particular, the average avalanche shape becomes asymmetric in the presence of an effective mass or stress overshoots [@25; @32; @33]. An open question is whether skyrmions can exhibit avalanche dynamics and, if so, what impact the Magnus term would have on such dynamics. It is important to understand intermittent skyrmion dynamics near the depinning threshold in order to fully realize applications which require skyrmions to be moved and stopped in a controlled fashion, such as in skyrmion race track memories [@18]. In this work we numerically examine avalanches of slowly driven skyrmions moving over quenched disorder for varied ratios $\alpha_m/\alpha_d$ of the Magnus term to the damping term. When $\alpha_m/\alpha_d \leq 1.73$, corresponding to intrinsic skyrmion Hall angles of $\theta^{\rm int}_{Sk} \leq 60^0$, the skyrmion avalanches are power law distributed in both size and duration, and the average avalanche shape for a fixed duration can be scaled to a universal curve as predicted for systems in a nonequilibrium critical state [@31; @32; @33]. For larger values of the Magnus term, the avalanches develop a characteristic size and the average avalanche shape becomes strongly asymmetric, indicative of an effective negative mass similar to that observed for avalanche distributions in certain domain wall systems [@25]. ![(a) Snapshot of the system showing the skyrmions (solid dots) and pinning sites (open circles). Skyrmions are introduced in the unpinned region on the left side of the sample and removed when they reach the right side of the sample. Once the system reaches a steady state, individual skyrmions are added at a slow rate. (b) A segment of the time series of the net skyrmion velocity, $\bar{v}$, versus time in simulation time steps. Clear skyrmion avalanche events appear. []{data-label="fig:1"}](Fig1.pdf){width="3.5in"} [*Simulation and System—*]{} In Fig. \[fig:1\] we show a snapshot of our 2D system which has periodic boundary conditions only in the $y$-direction and contains $N$ skyrmions interacting with $N_p$ randomly placed pinning sites. The skyrmions are modeled as particles with dynamics governed by the modified Thiele equation, used previously to study skyrmions interacting with random [@12; @16] and periodic [@21; @22] pinning substrates. The equation of motion of a single skyrmion $i$ is: $$\alpha_d {\bf v}_{i} - \alpha_m {{\bf \hat z}} \times {\bf v}_{i} = {\bf F}^{ss}_{i} + {\bf F}^{sp}_{i} .$$ Here ${\bf r}_i$ is the skyrmion position and ${\bf v}_{i} = {d {\bf r}_{i}}/{dt}$ is the skyrmion velocity. The damping constant is $\alpha_d$ while $\alpha_m$ is the strength of the Magnus term. In the absence of pinning, a skyrmion experiencing a uniform external force moves at the intrinsic skyrmion Hall angle of $\theta^{\rm int}_{Sk} = \tan^{-1}(\alpha_{m}/\alpha_{d})$ with respect to the direction of the external force, and in the overdamped limit of $\alpha_{m} = 0$, $\theta^{\rm int}_{Sk} = 0^{\circ}$. The skyrmion-skyrmion repulsive interaction force is given by ${\bf F}^{ss}_{i} = \sum^{N}_{j=1} K_{1}(r_{ij}) \hat{\bf r}_{ij}$ where $r_{ij}=|{\bf r}_i - {\bf r}_j|$, $\hat{\bf r}_{ij}=({\bf r}_i - {\bf r}_j)/r_{ij}$, and $K_{1}$ is a modified Bessel function. The pinning force from the quenched disorder ${\bf F}^{sp}_{i}$ arises from $N_p$ randomly placed non-overlapping harmonic traps with maximum pinning force $F_{p}$ and radius $R_{p} = 0.15$. The system dimensions are $L_{x} = 26$ and $L_{y} = 24$, and there is a pin-free region extending from $x=0$ to $x=4$. An artificial wall of stationary skyrmions is placed to the left of $x=0$ to provide confinement. The skyrmions are driven by a gradient, introduced by slowly dropping skyrmions into the pin-free region and allowing them to move into the pinned region under the force of their mutual repulsion[@myaval]. Skyrmions that reach the right edge of the sample are removed from the simulation. After the system reaches a steady state, which typically requires $2 \times 10^3$ skyrmion drops, we examine individual avalanches by measuring the net skyrmion velocity response $\bar v=N^{-1}\sum_{i=0}^{N}|{\bf v}_i|$ between drops, as illustrated in Fig. \[fig:1\](b). We drop skyrmions at a slow enough rate that the time series $\bar v(t)$ shows well-defined avalanches separated by intervals of no motion. We consider five different intrinsic skyrmion Hall angles $\theta^{\rm int}_{Sk} = 0^{\circ}$, $30^{\circ}$, $45^{\circ}$, $60^{\circ}$, and $80^{\circ}$, where we fix $\alpha_{d} = 1.0$ and vary $\alpha_{m}$. We studied several different pinning densities and strengths, but here we focus on systems with $N_{p} = 3700$ and $F_{p} = 1.0$. Experimentally our system corresponds to skyrmions entering from the edge of a sample or moving from a pin-free to a pinned region of the sample driven by a slowly changing magnetic field or small applied current. ![(a) Average avalanche size $\langle S\rangle$ vs avalanche duration $T$ for $\theta^{\rm int}_{Sk} = 30^\circ$. Dashed line is a fit to $\langle S\rangle \propto T^{1/\sigma\nu z}$ with $1/\sigma\nu z = 1.63$. (b) Distribution of avalanche durations $P(T)$ and (c) distribution of avalanche sizes $P(S)$ for $\theta^{\rm int}_{Sk} = 0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$, and $80^\circ$, from bottom to top. The curves have been shifted vertically for clarity. (d) The scaling exponents $\tau$ (triangles), $\alpha$ (squares), and $1/\sigma\nu z$ (circles) vs $\theta^{\rm int}_{Sk}$. For $\theta^{\rm int}_{Sk} \leq 60^\circ$, $\alpha = 1.5$, $\tau = 1.33$, and $1/\sigma\nu z = 1.63$, while for $\theta^{\rm int}_{Sk} = 80^\circ$, $\alpha = 2.5$, $\tau = 1.6$, and $1/\sigma\nu z = 2.4$. []{data-label="fig:2"}](Fig2.pdf){width="3.5in"} [*Results—*]{} From the time series of the skyrmion velocity ${\bar v}(t)$ we determine the avalanche duration $T$ as the time during which ${\bar v}>v_{th}$, where $v_{th}$ is a threshold velocity. We define the avalanche size $S$ as the time integral $S = \int_{t_0}^{t_0+T} {\bar v}(t) - v_{th}$ over the duration of the avalanche. Near a critical point, various quantities associated with the avalanches are expected to scale as power laws [@31]: the average avalanche size $\langle S\rangle(T) \propto T^{1/\sigma\nu z}$, the distribution of avalanche durations $P(T) \propto T^{-\alpha}$, and the avalanche size distribution $P(S)\propto S^{-\tau}$. In Fig. \[fig:2\](a) we plot $\langle S\rangle$ versus $T$ for a system with $\theta^{\rm int}_{Sk} = 30^\circ$ , while Figs. \[fig:2\](b,c) show the corresponding $P(T)$ and $P(S)$ for $\theta^{\rm int}_{Sk} = 0^\circ$ to $80^{\circ}$. In each case we find a range of power law scaling. In Fig. \[fig:2\](d) we plot the extracted critical exponents $\tau$, $\alpha$, and $1/\sigma\nu z$ versus $\theta^{\rm int}_{Sk}$. The exponents are roughly constant for $\theta^{\rm int}_{Sk} \leq 60^{\circ}$ with $\tau = 1.33$, $\alpha = 1.5$, and $1/\sigma\nu z = 1.63$. For $\theta^{\rm int}_{Sk} = 80^{\circ}$, we find longer avalanches of larger size, as indicated by the changes in $P(T)$ and $P(S)$, while $P(T)$ develops a maximum due to the emergence of a characteristic avalanche size. If we consider only the larger avalanches from the $\theta^{\rm int}_{Sk} = 80^\circ$ sample, we obtain considerably larger exponents of $1/\sigma\nu z = 2.4$, $\alpha \approx 2.5$, and $\tau = 1.6$, as shown in Fig. \[fig:2\](d). The exponents for a system in a critical state are predicted to satisfy the following relation[@31]: $$\frac{\alpha -1}{\tau -1} = \frac{1}{\sigma\nu z} .$$ Samples with $\theta^{\rm int}_{Sk} < 60^{\circ}$ obey this relation, samples with $\theta^{\rm int}_{Sk} = 60^{\circ}$ give $1.55$ for the right hand side and 1.63 for the left hand side, and samples with $\theta = 80^{\circ}$ again obey this relation. ![(a) The time averaged avalanche velocity $\langle V\rangle(t,T)$ in a system with $\theta^{\rm int}_{Sk}=30^\circ$, for avalanches of duration $T$, versus time in simulation time steps. The curves represent the time average over ten logarithmically-spaced bins for $T = 150$, 175, 204, 238, 278, 324, 378, 441, 514, 600, and $700$ simulation time steps, from left to right. (b) Scaling collapse of the data in panel (a) plotted as $T^{1 - 1/\sigma\nu z}\langle V\rangle(t,T)$ vs $t/T$, where $1/\sigma\nu z = 1.63$. The dashed curve indicates the overall average avalanche shape. (c) The average avalanche shapes $g/g_{max}$ vs $t/T$ for $\theta^{\rm int}_{Sk} = 0^\circ$ (red), $30^{\circ}$ (orange), $45^{\circ}$ (green), $60^{\circ}$ (blue), and $80^{\circ}$ (purple). []{data-label="fig:3"}](Fig3.pdf){width="3.5in"} A more stringent test of whether a system is at a nonequilibrium critical point is the prediction that the average avalanche shape can be scaled to a universal curve [@31; @32; @33]. This implies that the average skyrmion velocity for a given avalanche duration should scale as $\langle V\rangle(t,T) \propto T^{1/\sigma\nu z - 1}g(t/T)$, where $g(t/T)$ is a universal function of the avalanche shape that can be extracted from the time series by plotting $T^{1 - 1/\sigma\nu z}\langle V\rangle(t,T)$ versus $t/T$. In Fig. \[fig:3\](a) we plot the average avalanche shape $\langle V\rangle$ for different values of $T$ in the $\theta^{\rm int}_{Sk} = 30^\circ$ system, and in Fig. \[fig:3\](b) we show a scaling collapse of the same data versus $t/T$. The dashed line is a fit to the overall average avalanche shape $g(t/T)$. We performed similar scaling collapses for other values of $\theta^{\rm int}_{Sk}$ and find the same universal function $g(t/T)$ for $\theta^{\rm int}_{Sk} \leq 60^{\circ}$, as shown in Fig. \[fig:3\](c), while for $\theta^{\rm int}_{Sk} = 80^{\circ}$, the average avalanche shape is much more asymmetric. ![Snapshots of the avalanche motion during a single large avalanche. Blue dots indicate skyrmions that did not move during the avalanche event, red dots indicate skyrmions that moved a distance greater than $x$, and lines indicate the net displacement of individual skyrmions during the avalanche. (a) $\theta^{\rm int}_{Sk} = 0^\circ$. (b) $\theta^{\rm int}_{Sk} = 30^\circ$. (c) $\theta^{\rm int}_{Sk}=60^\circ$. (d) $\theta^{\rm int}_{Sk} = 80^\circ$. As the Magnus term increases, the avalanche motion starts to show curvature in the positive $y$ direction. []{data-label="fig:4"}](Fig4.pdf){width="3.5in"} The change in the exponents and the average avalanche shape for large $\theta^{\rm int}_{Sk}$ indicates that when the non-dissipative Magus term is strong, there is a change in the universality class. Mean field predictions give $\tau = 1.5$, $\alpha = 2.0$, $1/\sigma\nu z = 2.0$, and a parabolic universal function for the average avalanche shape [@36; @37]. In our system, $\tau = 1.33$, $\alpha = 1.5$, $1/\sigma\nu z = 1.63$, and the universal function $g(t/T)$ has a parabolic shape with some asymmetry at small $t/T$. Since we are working in two dimensions and the skyrmion interaction range is finite, it may be expected that our system would not match the mean field picture; however, it is clear that when the Magnus term is large, the avalanche dynamics show a pronounced change. The asymmetry in the scaling collapse of the avalanches is similar to that found in many systems including magnetic domain avalanches, where it was argued to result from an effective negative mass [@25]. Inertial effects with positive mass tend to give a leftward asymmetry, while an effective negative mass damps the avalanches at later times and produces a rightward asymmetry [@25]. The Magnus term causes the skyrmions to move in the direction perpendicular to the applied external force, and this could reduce the overall avalanche motion in the forward direction at later times, resulting in the skewed average avalanche shape. In Fig. \[fig:4\](a) we plot the skyrmions and their net displacements during a large avalanche in a sample with $\theta^{\rm int}_{Sk} = 0^\circ$, where the motion strongly follows the density gradient from left to right. At $\theta^{\rm int}_{Sk}=30^\circ$ in Fig. \[fig:4\](b), near the right edge of the sample the avalanche motion shows a tendency to curve in the positive $y$ direction. This tendency is enhanced for $\theta^{\rm int}_{Sk}=60^\circ$ in Fig. \[fig:4\](c) and for $\theta^{\rm int}_{Sk}=80^\circ$ in Fig. \[fig:4\](d), where the entire avalanche moves at an angle with respect to the $x$ axis. We have examined several other pinning landscapes, including samples with the same $N_p=3700$ but a lower $F_p=0.3$, where we find results similar to those of the $F_p=1.0$ system. In the limit of strong dilute pinning with $N_p=600$ and $F_p=3.0$, skyrmions that become pinned generally never depin and we observe a strong channeling effect where the avalanches occur through the motion of interstitial or unpinned skyrmions moving along weak links between pinned skyrmions. In this case, the distribution of avalanche sizes is strongly peaked at the size corresponding to the weak link channel. [*Summary—*]{} We have shown that skyrmions driven by their own gradient in the presence of quenched disorder exhibit avalanche dynamics and show power law avalanche duration and size distributions. The average avalanche shape for different avalanche durations can be scaled by a universal function, in agreement with predictions for systems near a nonequilibrium critical point. Skyrmions are distinct from previously studied avalanche systems due to the strong non-dissipative Magnus term in the skyrmion dynamics. We find that as the Magnus term increases, there is a change in the critical behavior of the avalanches as indicated by the critical exponents, and the average avalanche shape develops a strong asymmetry similar to that found for a negative effective mass in magnetic domain depinning avalanches. This change in behavior results when the Magnus term causes the avalanche motion to shift partially into the direction perpendicular to the skyrmion density gradient. 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--- title: 'FePh: An Annotated Facial Expression Dataset for the RWTH-PHOENIX-Weather 2014 Dataset' --- ------------------------------------------------------------------------ \ [**FePh: An Annotated Facial Expression Dataset for the RWTH-PHOENIX-Weather 2014 Dataset**]{} ------------------------------------------------------------------------ \ Background & Summary {#background-summary .unnumbered} ==================== About $466$ million people worldwide have disabled hearing and by $2050$ this number will increase to $900$ million people–one in every ten [^1]. People with hearing loss often communicate through lip-reading skills, texts, and sign language. Sign language, the natural language of people with severe or profound hearing loss, is unique to each country or region [@cite2] and has its own grammar and structure. Combinations of hand movements, postures, and shapes with facial expressions make sign language very unique and complex [@cite3]. [p[1cm]{}p[3.2cm]{}p[1cm]{}p[1.1cm]{}p[1.1cm]{}p[0.6cm]{}p[1.6cm]{}]{} \  Authors & Name & Language & Gesture Type & Language level & Classes & Data Type\ \ [@cite21] & UCI Australian Auslan Sign Language dataset[^2] & Australian & Dynamic & Alphabets & 95 & Data Glove\ [@cite12] & ASL Finger Spelling A [^3] & American & Static & Alphabets & 24 & Depth Images\ [@cite12] & ASL Finger Spelling B [^4] & American & Static & Alphabets & 24 & Depth Image\ [@cite22] & MSRGesture $3$D [^5] & American & - & Words & 12 & Depth Video\ [@cite23] & CLAP14 & Italian & - & Words & 20 & Depth Video?\ [@cite24] & ChaLearn LAP IsoGD[^6] & ConGD[^7]& - & Static & Dynamic & - & 249 & RGB & Depth Video\ [@cite25] & PSL Kinect 30[^8] & Polish & Dynamic & Words & 30 & Kinect Video\ [@cite14] & ISL[^9] & Indian & static & Alphabets, Numbers, Words & 140 & Depth Images\ Hands are widely used in sign language to convey meaning. Despite the importance of hands in sign language, the directions of eye gazes, eyebrows, eye blinks, and mouths, as part of facial expressions also play an integral role in conveying both emotion and grammar [@cite4]. Facial expressions that support grammatical constructions in sign language help to eliminate the ambiguity of signs [@cite5]. Therefore, sign language recognition systems without facial expression recognition are incomplete [@cite6]. Due to the importance of hand shapes and movements in sign language, hands are widely used in sign language recognition systems. However, the other integral part of sign language, facial expressions, has not yet been well studied. One reason is the lack of an annotated facial expression dataset in the context of sign language. To the best of our knowledge, an annotated vision-based facial expression dataset in the field of sign language is a very rare resource. This limits researchers’ ability to study multi-modal sign language recognition models that consider both facial expressions and hand gestures. Conducted research in sign language recognition systems can be categorized in two main groups: vision-based and hardware-based recognition systems. Hardware-based recognition systems use datasets that are collected utilizing special colored gloves [@cite9; @cite10; @cite11], special sensors, and/or depth cameras (such as Microsoft Kinect and Leap Motion) [@cite5; @cite12; @cite13; @cite14; @cite15; @cite16; @cite17; @cite42] to capture special features of the signer’s gestures. Some well known hardware-based datasets are listed in Table \[hardware-based\]. Although utilizing hardware eases the process of capturing special features, they limit the applicability where such hardwares are not available. Therefore, vision-based sign language recognition systems utilizing datasets collected by regular cameras are proposed [@cite2; @cite7; @cite18; @cite19; @cite20]. Furthermore, the new learning techniques such as machine and deep learning techniques have revolutionized many fields of research such as healthcare [@cite43], security [@cite28], industry [@cite44], computer vision [@cite8], etc. Utilizing these learning techniques coupled with vision-based datasets advances the sign language recognition systems. One well known continuous sign language dataset is the RWTH-PHOENIX-Weather corpus [@cite18]. RWTH-PHOENIX-Weather is a large vocabulary (more than $1200$ signs) image corpus containing weather forecasts recorded from German news. Two years later, its publicly available extension called RWTH-PHOENIX-Weather multisigner 2014 dataset was introduced. We will create our annotated facial expression dataset based upon RWTH-PHOENIX-Weather multisigner 2014 continuous sign language benchmark dataset. [p[1cm]{}p[3.2cm]{}p[1cm]{}p[1.1cm]{}p[1.1cm]{}p[0.6cm]{}p[1.6cm]{}]{} \ Authors & Name & Language & Gesture Type & Language level & Classes & Data Type\ \ [@cite29] & - & American & Static & Alphabets, numbers & 36 & Image\ [@cite12] & ASL Finger Spelling A [^10] & American & Static & Alphabets & 24 & Image\ [@cite30] & HUST-ASL[^11] & American & Static & Alphabets, Numbers & 34 & RGB & Kinect Image\ [@cite19; @cite26] & Purdue RVL-SLLL ASL Database[^12] & American & - & Alphabets, Numbers, Words, Paragraphs & 104 & Image, Video\ [@cite31] & Boston ASLLVD[^13] & American & Dynamic & Words & &gt;3300 & Video\ [@cite20] & ASL-LEX[^14] & American & - & Words & Nearly 1000 & Video\ [@cite2] & MS-ASL [^15] & American & Dynamic & - & 1000 & Video\ [@cite32] & - & Arabic & - & Words & 23 & Video\ \ [@cite18] & RWTH-PHOENIX-Weather 2012[^16] & German & - & Sentence & 1200 & Image\ [@cite7; @cite33] & RWTH-PHOENIX-Weather Multisigner 2014[^17] & German & Dynamic & Sentence & &gt;1000 & Video\ [@cite27] & SIGNUM[^18] & German & - & words, Sentences & 450 Words, 780 Sentence & Video\ [@cite10] & LSA16[^19] & Argentinian & - & Alphabets, Words & 16 & Image\ [@cite10] & LSA64[^20] & Argentinian & - & Words & 64 & Video\ [@cite14] & the ISL dataset[^21] & Indian & static & Alphabets, Numbers, Words & 140 & Image\ [@cite34] & ISL hand shape dataset[^22] & Irish & Static & Dynamic & - & 23 Static & 3 Dynamic & Image Video\ [@cite35] & Japaneese Finger spelling sign language dataset & Japan & - & - & 41 & Image\ Facial expression recognition is a very well established field of research with publicly available databases containing basic universal expressions. CK$^+$ [^23] [@cite36] is a well known facial expression database with $327$ annotated video sequences on seven basic universal facial expressions (“anger”, “contempt”, “disgust”, “fear”, “happiness”, “sadness”, and “surprise”). Some other widely used facial expression database are MMI[^24][@cite37; @cite38], Oulu-CASIA [^25] [@cite39], and FER2013[^26] [@cite40]. FER2013 is an unconstrained large database considering seven emotions (previous six emotions plus “neutral”). Despite the value of these facial expression databases, hardly any one them are in the context of sign language. In this paper, we introduce FePh, an annotated facial expression dataset for the publicly available continuous sign language dataset RWTH-PHOENIX-Weather 2014 [@cite7]. As a matter of continuity image data, FePh is similar to CK+, MMI, and Oulu-CASIA. However, it is more complex than those databases as it contains real-life captured videos with more than one basic facial expression in each sequence, with different head poses, orientations, and movements. In addition, FePh not only contains seven basic facial expressions of the FER2013 database, but it also considers their primary, secondary, and tertiary dyads. It is also noteworthy to mention that in sign language, signers mouth the words or sentences to help their audience better grasp meanings. This is a characteristic that makes facial expression datasets in the context of sign language more challenging. As such, this manuscript provides the following contributions: first, introducing annotated facial expression dataset of the RWTH-PHOENIX-Weather 2014 dataset, second, attributing highly used hand shapes with their associated performed facial expressions, and third, illustrating the relationships between hand shapes and facial expressions in sign language. Methods {#methods .unnumbered} ======= Due to the integral role of facial expressions in conveying emotions and grammar in sign language, it is important to use multi-modal sign language recognition models that consider both hand shapes and facial expressions. Therefore, to create FePh and annotate facial expressions of a sign language dataset with annotated hand shapes, we considered the well-known publicly available continuous RWTH-PHOENIX-Weather 2014 dataset. Since the annotated hand shapes dataset RWTH-PHOENIX-Weather 2014 is publicly available as RWTH-PHOENIX-Weather 2014 MS Handshapes dataset [@cite8], we provide facial expression annotations for the same dataset, which enables researchers to utilize a dataset that has both hand shape and facial expression annotations. As a starting point, we collected the full frame images of RWTH-PHOENIX-Weather 2014 development set that are identical to the RWTH-PHOENIX-Weather 2014 MS Handshapes dataset [@cite8]. Furthermore, in order to create a solid facial expression dataset and avoid the influence of hand shapes on the facial expression annotators, faces of all full frame images are automatically detected, tracked, and cropped using facial recognition techniques. Twelve annotators (six women and six men) between 20 to 40 years old were asked to annotate the data. We asked annotators to answer three questions about each static image: the signer’s emotion, visibility, and gender. In terms of emotion, annotators could choose one or more of the following applicable basic universal facial expressions for each static image: “sad”, “surprise”, “fear”, “angry”, “neutral”, “disgust”, and “happy”. Although more than seven emotions and their primary, secondary, and tertiary dyads exist, considering all of them was not within the scale of this project. Therefore, we offered the eighth class of “None” as well. Annotators were asked to choose the “None” class when none of the aforementioned emotions could describe the facial expression of the image. In addition, since annotators could choose more than one facial expression for each individual image, the combinations of basic universal facial expressions were also considered (interestingly, this did not result in choosing more than two emotions for each image) and shown by a “\_” in between such as surprise\_fear. The sequence of emotions is not important in the secondary and tertiary dyads (i.e., surprise\_fear is not different from fear\_surprise). With regard to the second question, visibility, we asked the annotators to evaluate whether the signer’s face is completely visible. Although the signer’s face was visible in majority of images, this was not always the case. The partial visibility of the face was due to the signer’s head movement, position, hand movement, and transitions from one emotion to another emotion. This helped us to detect and opt out these obscured images in the data. Figure \[face\_not\_visible\] shows some obscured exemplary images. {width="\textwidth" height="2cm" width="8cm"} The last question of signer’s gender was asked to provide statistics of signers’ gender. This statistics enable conducting future researches in the affects of gender in expressing emotions and facial expressions. For our labelling purpose, we took advantage of the Labelbox [@cite41] annotating solution tool through which we defined an annotation project and randomly distributed images to be labeled by the annotators. In addition, due to the complexities of the facial images of the RWTH-PHOENIX-Weather 2014 dataset, we used the auto consensus option of the Labelbox tool. These complexities are listed below: - The ambiguity of images, due to signer’s movement, head position, and transitions from one emotion to another (e.g., eyes are closed and/or the lips are still open). - Low quality (resolution) and blurriness of images. - Mouthed words that confuse facial expression annotators. - Personal differences between signers expressing facial expressions. - The best facial expression that describes the image is not included in the dataset. - Images may not be in facial expression’s top frame. - Large intra-class variance (such as “surprised” emotion with open or closed mouth). - Inter-class similarities. With the usage of auto consensus option of Labelbox, we asked more than one annotator (three) to annotate about $60\%$ percent of the data. For the images with three labels, we chose the most voted emotion as the final label of the facial image. In cases where there was not a most voted emotion, but the image was a part of a sequence of images, we have assigned labels based on the before or after images’ facial expression of the same sequence. On the other hand, if there was not a most voted emotion, and the image was not a part of a sequence of images (i.e., one single image without any sequence), we asked our annotators to relabel the image. In this case, all images needed to be labelled by three different annotators. (190.44,100.19) .. controls (190.44,94.36) and (195.36,89.64) .. (201.44,89.64) .. controls (207.51,89.64) and (212.43,94.36) .. (212.43,100.19) .. controls (212.43,106.02) and (207.51,110.74) .. (201.44,110.74) .. controls (195.36,110.74) and (190.44,106.02) .. (190.44,100.19) – cycle ; (149.95,210.68) .. controls (149.95,204.85) and (154.87,200.12) .. (160.94,200.12) .. controls (167.01,200.12) and (171.93,204.85) .. (171.93,210.68) .. controls (171.93,216.5) and (167.01,221.23) .. 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(201.78,36.7) ; (275.59,165.55) .. controls (281.53,142.2) and (276.13,134.42) .. (273.43,117.31) ; (266,141.85) .. controls (266,136.02) and (270.92,131.29) .. (276.99,131.29) .. controls (283.06,131.29) and (287.98,136.02) .. (287.98,141.85) .. controls (287.98,147.67) and (283.06,152.4) .. (276.99,152.4) .. controls (270.92,152.4) and (266,147.67) .. (266,141.85) – cycle ; (220.52,218.51) .. controls (232.4,215.86) and (246.98,213.79) .. (258.85,196.67) ; (231.48,212.23) .. controls (231.48,206.4) and (236.4,201.68) .. (242.47,201.68) .. controls (248.54,201.68) and (253.46,206.4) .. (253.46,212.23) .. controls (253.46,218.06) and (248.54,222.78) .. (242.47,222.78) .. controls (236.4,222.78) and (231.48,218.06) .. (231.48,212.23) – cycle ; (142.76,198.74) .. controls (148.16,207.04) and (163.82,220.01) .. (183.04,218.51) ; (149.95,210.68) .. controls (149.95,204.85) and (154.87,200.12) .. (160.94,200.12) .. controls (167.01,200.12) and (171.93,204.85) .. (171.93,210.68) .. controls (171.93,216.5) and (167.01,221.23) .. (160.94,221.23) .. controls (154.87,221.23) and (149.95,216.5) .. (149.95,210.68) – cycle ; (126.56,117.82) .. controls (119,125.61) and (116.84,146.87) .. (124.94,165.55) ; (109.95,141.33) .. controls (109.95,135.5) and (114.87,130.78) .. (120.94,130.78) .. controls (127.01,130.78) and (131.93,135.5) .. (131.93,141.33) .. controls (131.93,147.15) and (127.01,151.88) .. (120.94,151.88) .. controls (114.87,151.88) and (109.95,147.15) .. (109.95,141.33) – cycle ; (201.78,18.91) node \[font=\] [$Happy$]{}; (136.44,99.83) node \[font=\] [$Sad$]{}; (271.97,183.34) node \[font=\] [$Surprise$]{}; (266.03,99.83) node \[font=\] [$Fear$]{}; (201.78,218.51) node \[font=\] [$Anger$]{}; (131.94,182.66) node \[font=\] [$Disgust$]{}; (201.78,143.4) node \[font=\] [$Neutral$]{}; Data Records {#data-records .unnumbered} ============ The FePh facial expression dataset produced with the above method, is stored on Harvard Dataverse[^27]. All facial images are stored in a folder named FePhimages. Although the full frame images of the FePh dataset are identical to the RWTH-PHOENIX-Weather 2014 images, the image filenames in the FePh\_images folder are different from the original image. FePh filenames consider both image folder\_name and image\_file\_number. For example, the full frame image of the facial image with filename “*01August\_2011\_Monday\_heute\_default-6.avi\_pid0\_fn000054-0.png*” in the FePh dataset is identical to the full frame image with the directory of “*... / 01August\_2011 \_Monday\_heute\_default-6 / 1 / 01August\_2011 \_Monday\_heute.avi\_pid0\_ fn000054-0.png*” in the RWTH-PHOENIX-Weather 2014 and “*... / 01August\_2011 \_Monday\_heute\_default-6 / 1 /\_.png\_ fn000054-0.png*” image in the RWTH-PHOENIX-Weather 2014 MS Handshapes dataset. This helped us to store all images in one single folder (FePh\_images). The FePh\_labels.csv file contains images’ filenames, facial expression labels, and gender labels. To ease data usability, we stored the facial expression labels as codes. Table \[label\_code\] shows the facial expression labels with their corresponding code numbers. In addition, $0$ and $1$ in the gender column represent the male and female genders, respectively. Technical Validation {#technical-validation .unnumbered} ==================== The FePh dataset is created by manually labelling $3359$ images of the RWTH-PHOENIX-Weather 2014 development set that are identical to the full frame images of the RWTH-PHOENIX-Weather 2014 MS Handshapes dataset. Seven universal basic emotions of “sad”, “surprise”, “fear”, “angry”, “neutral”, “disgust”, and “happy” are considered as facial expression labels. In addition to these basic emotions, we asked annotators to choose all the emotions that may apply to an image. This resulted in secondary and tertiary dyads of seven basic emotions such as fear\_sad, fear\_anger, etc. Interestingly, this did not result in having combinations of three basic emotions. Figure \[FE\_relations\] shows the corresponding graph of seven basic emotions, their primary, secondary, and tertiary dydes presented in FePh dataset. Seven basic emotions are shown by colored circles with the emotion labels written inside them. Other colored circles connecting each two basic emotions illustrate the secondary or tertiary dyads of the basic emotions that are connected to. Emotion “Happy” has only one dyad, which is with “Neutral” emotion, named as “Neutral\_Happy”. Although the FePh dataset presents annotated facial expression for all hand shape classes of the RWTH-PHOENIX-Weather 2014, we analyzed the facial expression labels for the top 14 hand shapes (i.e., classes “1”, “index”, “5”, “f”, “2”, “ital”, “b”, “3”, “b\_thumb”, “s”, “pincet”, “a”, “h”, and “ae”). This is due to the demonstrated distribution of the counts per hand shape classes in [@cite8] that shows the top $14$ hand shape classes represent $90\%$ of the data. Seven universal basic facial expressions and their secondary or tertiary dyads occur with different frequencies in the data. Figure \[FE\_frequency\] shows the distribution counts per facial expression class in the data. As it shows, about $90\%$ of the data is expressed with basic facial expressions. In addition, Figures \[none\_dist\] and \[obscured\_dist\] illustrate the frequency of images with obscured faces and the “None” class in the top 14 hand shape classes, respectively. By analyzing facial expressions per hand shape class, we found out that more than one facial expression class represents each hand shape class. Figure \[heatmaps\] shows the frequency heatmaps of the seven facial expressions and their primary, secondary, and tertiary dyads for the top 14 hand shape classes. Each heatmap illustrates the frequency of facial expressions based on the facial expression graph of seven basic universal emotions and their primary, secondary, and tertiary dyads (shown in Figure \[FE\_relations\]) for one of the top 14 hand shape classes. Each heatmap shows that more than one facial expression is expressed within a single hand shape class. This is due to the complexity of sign language in using facial expressions with hand shapes. Two of these complexities, which affect performing different facial expressions within each hand shape class, are as follows: ![Pareto chart showing the distribution of the facial expression “None” and obscured image counts for top 14 hand shape classes (HSH).[]{data-label="dist"}](Figures/all_FE.png){width="\textwidth" height="5cm" width="11cm"} [.5]{} ![Pareto chart showing the distribution of the facial expression “None” and obscured image counts for top 14 hand shape classes (HSH).[]{data-label="dist"}](Figures/FE_None.png "fig:"){width="\linewidth"} [.5]{} ![Pareto chart showing the distribution of the facial expression “None” and obscured image counts for top 14 hand shape classes (HSH).[]{data-label="dist"}](Figures/FE_Obscured.png "fig:"){width="\linewidth"} [.6]{} ![Exemplary full frame images of the RWTH-PHOENIX-Weather 2014 dataset[]{data-label="exampleimage"}](Figures/intraclass_variance.jpg "fig:"){width="0.9\linewidth"} [.4]{} ![Exemplary full frame images of the RWTH-PHOENIX-Weather 2014 dataset[]{data-label="exampleimage"}](Figures/similarhands.jpg "fig:"){width="0.6\linewidth"} [.2]{} ![Heatmaps showing the frequency distribution of the top 14 hand shape classes. Each heatmap, assigned to one hand shape class (briefly mentioned as HSH), shows the frequency of facial expressions on the assigned hand shape class over the facial expression graph of seven basic universal emotions and their primary, secondary, and tertiary dyads. As they show, more than one facial expressions are expressed within a single hand shape class.[]{data-label="heatmaps"}](Figures/HM_HSH_1.png "fig:"){width="\linewidth"} [.2]{} ![Heatmaps showing the frequency distribution of the top 14 hand shape classes. Each heatmap, assigned to one hand shape class (briefly mentioned as HSH), shows the frequency of facial expressions on the assigned hand shape class over the facial expression graph of seven basic universal emotions and their primary, secondary, and tertiary dyads. As they show, more than one facial expressions are expressed within a single hand shape class.[]{data-label="heatmaps"}](Figures/HM_HSH_index.png "fig:"){width="\linewidth"} [.2]{} ![Heatmaps showing the frequency distribution of the top 14 hand shape classes. Each heatmap, assigned to one hand shape class (briefly mentioned as HSH), shows the frequency of facial expressions on the assigned hand shape class over the facial expression graph of seven basic universal emotions and their primary, secondary, and tertiary dyads. As they show, more than one facial expressions are expressed within a single hand shape class.[]{data-label="heatmaps"}](Figures/HM_HSH_5.png "fig:"){width="\linewidth"} [.2]{} ![Heatmaps showing the frequency distribution of the top 14 hand shape classes. Each heatmap, assigned to one hand shape class (briefly mentioned as HSH), shows the frequency of facial expressions on the assigned hand shape class over the facial expression graph of seven basic universal emotions and their primary, secondary, and tertiary dyads. As they show, more than one facial expressions are expressed within a single hand shape class.[]{data-label="heatmaps"}](Figures/HM_HSH_f.png "fig:"){width="\linewidth"} [.2]{} ![Heatmaps showing the frequency distribution of the top 14 hand shape classes. Each heatmap, assigned to one hand shape class (briefly mentioned as HSH), shows the frequency of facial expressions on the assigned hand shape class over the facial expression graph of seven basic universal emotions and their primary, secondary, and tertiary dyads. As they show, more than one facial expressions are expressed within a single hand shape class.[]{data-label="heatmaps"}](Figures/HM_HSH_2.png "fig:"){width="\linewidth"} [.2]{} ![Heatmaps showing the frequency distribution of the top 14 hand shape classes. Each heatmap, assigned to one hand shape class (briefly mentioned as HSH), shows the frequency of facial expressions on the assigned hand shape class over the facial expression graph of seven basic universal emotions and their primary, secondary, and tertiary dyads. As they show, more than one facial expressions are expressed within a single hand shape class.[]{data-label="heatmaps"}](Figures/HM_HSH_ital.png "fig:"){width="\linewidth"} [.2]{} ![Heatmaps showing the frequency distribution of the top 14 hand shape classes. Each heatmap, assigned to one hand shape class (briefly mentioned as HSH), shows the frequency of facial expressions on the assigned hand shape class over the facial expression graph of seven basic universal emotions and their primary, secondary, and tertiary dyads. As they show, more than one facial expressions are expressed within a single hand shape class.[]{data-label="heatmaps"}](Figures/HM_HSH_b.png "fig:"){width="\linewidth"} [.2]{} ![Heatmaps showing the frequency distribution of the top 14 hand shape classes. Each heatmap, assigned to one hand shape class (briefly mentioned as HSH), shows the frequency of facial expressions on the assigned hand shape class over the facial expression graph of seven basic universal emotions and their primary, secondary, and tertiary dyads. As they show, more than one facial expressions are expressed within a single hand shape class.[]{data-label="heatmaps"}](Figures/HM_HSH_3.png "fig:"){width="\linewidth"} [.2]{} ![Heatmaps showing the frequency distribution of the top 14 hand shape classes. Each heatmap, assigned to one hand shape class (briefly mentioned as HSH), shows the frequency of facial expressions on the assigned hand shape class over the facial expression graph of seven basic universal emotions and their primary, secondary, and tertiary dyads. As they show, more than one facial expressions are expressed within a single hand shape class.[]{data-label="heatmaps"}](Figures/HM_HSH_b_thumb.png "fig:"){width="\linewidth"} [.2]{} ![Heatmaps showing the frequency distribution of the top 14 hand shape classes. Each heatmap, assigned to one hand shape class (briefly mentioned as HSH), shows the frequency of facial expressions on the assigned hand shape class over the facial expression graph of seven basic universal emotions and their primary, secondary, and tertiary dyads. As they show, more than one facial expressions are expressed within a single hand shape class.[]{data-label="heatmaps"}](Figures/HM_HSH_S.png "fig:"){width="\linewidth"} [.2]{} ![Heatmaps showing the frequency distribution of the top 14 hand shape classes. Each heatmap, assigned to one hand shape class (briefly mentioned as HSH), shows the frequency of facial expressions on the assigned hand shape class over the facial expression graph of seven basic universal emotions and their primary, secondary, and tertiary dyads. As they show, more than one facial expressions are expressed within a single hand shape class.[]{data-label="heatmaps"}](Figures/HM_HSH_pincet.png "fig:"){width="\linewidth"} [.2]{} ![Heatmaps showing the frequency distribution of the top 14 hand shape classes. Each heatmap, assigned to one hand shape class (briefly mentioned as HSH), shows the frequency of facial expressions on the assigned hand shape class over the facial expression graph of seven basic universal emotions and their primary, secondary, and tertiary dyads. As they show, more than one facial expressions are expressed within a single hand shape class.[]{data-label="heatmaps"}](Figures/HM_HSH_a.png "fig:"){width="\linewidth"} [.2]{} ![Heatmaps showing the frequency distribution of the top 14 hand shape classes. Each heatmap, assigned to one hand shape class (briefly mentioned as HSH), shows the frequency of facial expressions on the assigned hand shape class over the facial expression graph of seven basic universal emotions and their primary, secondary, and tertiary dyads. As they show, more than one facial expressions are expressed within a single hand shape class.[]{data-label="heatmaps"}](Figures/HM_HSH_h.png "fig:"){width="\linewidth"} [.2]{} ![Heatmaps showing the frequency distribution of the top 14 hand shape classes. Each heatmap, assigned to one hand shape class (briefly mentioned as HSH), shows the frequency of facial expressions on the assigned hand shape class over the facial expression graph of seven basic universal emotions and their primary, secondary, and tertiary dyads. As they show, more than one facial expressions are expressed within a single hand shape class.[]{data-label="heatmaps"}](Figures/HM_HSH_ae.png "fig:"){width="\linewidth"} First, although some meanings are communicated using only one hand (usually the right hand), many sign language meanings are communicated using both hands with different hand poses, orientations, and movements. Figure \[intraclass\] shows some exemplary full frame images of hand shape class “1” of the RWTH-PHOENIX-Weather 2014 dataset with different facial expressions. As the figure illustrates, the usage of right hand shapes have large intra-class variance (i.e., the left hand may not be used or may perform similar or different hand shape from the right hand) that may affect the meanings, and as a result, the facial expressions corresponding to them. The first top row in the figure shows the full frame images with the right hand shape of class “1” and no left hand shape. The second row shows some exemplary full frame images, in which the signer has used both hands. In this row, although both right and left hands demonstrate the same hand shape class (hand shape class “1”), their pose, orientation, and movement can differ, which may affect the corresponding facial emotion. The third row of images in Figure \[intraclass\] illustrates full frame examples of using both hands with different hand shapes and facial expressions. Therefore, although RWTH-PHOENIX-Weather 2014 MS Handshapes is a valuable resource presenting right hand shape labels, it lacks pose, orientation, and movement labels of the right hand along with the left hand shape labels. Adding this information to the data affects the communicated meanings as well as the facial expressions that are expressed. Second, due to the communication of grammar via facial expressions, identical hand shapes may be performed with different facial expressions. Figure \[similarhands\] demonstrates some images of this kind that despite the similarity of hand shapes, the facial expressions are different. This complex usage of hands with large intra-class variance and inter-class similarities help signers to communicate different meanings with similar or different facial expressions. In addition to the above, it is noteworthy that the frequency of facial expressions expressed in each hand shape class shows evidence of a meaningful association between hand shapes and facial expressions in the data. To better illustrate this correlation, we calculated the correlation matrix of each facial expression’s frequency in each hand shape class. Since the correlations between the facial expressions together is not the focus of this manuscript, we cropped the first column of each correlation matrix (i.e., showing the correlation between frequency and each facial expression). Table \[correlation \] illustrates the first columns of facial expressions and their frequencies of occurrence in the top 14 hand shape classes correlation matrices. Monitoring each column in Table \[correlation \] gives the most correlated facial expressions for each hand shape. For example, in the hand shape class “3” column, the positive values of 0.697, 0.383, 0.287, 0.214 and 0.093, that are in intersections with “fear”, “surprise”, “anger”, “surprise\_fear” and “None” respectively, show the high correlation with facial expressions in hand shape class “3”. These highly correlated facial expressions in each hand shape class can also be interpreted from heatmaps. For example, the heatmap of hand shape class “3” indicates that for signers signing hand shape class “3”, the distribution of the facial expression label counts is weighted towards expressing more of the “fear” emotion, which is the most correlated facial expression in hand shape class “3”. Understanding these correlations and associations based on heatmaps and correlation matrices will help in increasing the validity and accuracy of sign language, or in a wider application, gesture recognition models. Usage Notes {#usage-notes .unnumbered} =========== To the best of our knowledge, this dataset is currently the first annotated large-scale continuous publicly available facial expression dataset in the context of sign language. Although the number of facial images is enough for statistical methods, it may not be sufficient for some of the state-of-the-art learning techniques in the field of computer vision. Therefore, for such studies, we suggest users to create matched samples choosing subjects from the dataset. This dataset enables the research community to consider both hand and facial signals for vision-based facial/sign language recognition. In addition to facial and sign language recognition systems, FePh has a wider application in other research areas such as gesture recognition and Human-Computer Interaction (HCI) systems. Acknowledgements {#acknowledgements .unnumbered} ================ We thank Hanin Alhaddad, Nisha Baral, Kayla Brown, Erdais Comete, Sara Hejazi, Jasser Jasser, Aminollah Khormali, Toktam Oghaz, Amirarsalan Rajabi, Mostafa Saeidi, and Milad Talebzadeh for their assistance in annotating the data. Author contributions {#author-contributions .unnumbered} ==================== Marie Alaghband and Ivan Garibay initialized the study. Marie Alaghband collected and analyzed the data and wrote the first draft of the manuscript. Niloofar Yousefi and Marie Alaghband developed the dyads graph for the seven facial expressions of the data. All the authors contributed, reviewed, and approved the final version of the manuscript. 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DeepMalaria: Artificial Intelligence Driven Discovery of Potent Antiplasmodials. Frontiers in Pharmacology, 10, 1526. Khormali, A., & Addeh, J. (2016). A novel approach for recognition of control chart patterns: Type-2 fuzzy clustering optimized support vector machine. ISA transactions, 63, 256-264. [^1]: Based on the most recent report of World health Organization [@cite1] (WHO) on January 2020. [^2]: Stands for Australian Sign Language (Available at: <https://archive.ics.uci.edu/ml/datasets/Australian+Sign+Language+signs+(High+Quality)>) [^3]: Available at: <http://empslocal.ex.ac.uk/people/staff/np331/index.php?section=FingerSpellingDataset> [^4]: Available at: <http://empslocal.ex.ac.uk/people/staff/np331/index.php?section=FingerSpellingDataset> [^5]: Available at: <https://www.uow.edu.au/~wanqing/Datasets> [^6]: Available at: [ http://www.cbsr.ia.ac.cn/users/jwan/database/isogd.html]( http://www.cbsr.ia.ac.cn/users/jwan/database/isogd.html) [^7]: Available at: <http://www.cbsr.ia.ac.cn/users/jwan/database/congd.html> [^8]: Available at: <http://vision.kia.prz.edu.pl/dynamickinect.php> [^9]: Available at: [ https://github.com/zafar142007/Gesture-Recognition-for-Indian-Sign-Language-using-Kinect]( https://github.com/zafar142007/Gesture-Recognition-for-Indian-Sign-Language-using-Kinect) [^10]: Available at: <http://empslocal.ex.ac.uk/people/staff/np331/index.php?section=FingerSpellingDataset> [^11]: Stands for Huazhong University of Science & Technology [^12]: Available at: <http://www2.ece.ohio-state.edu/~aleix/ASLdatabase.htm> and <https://engineering.purdue.edu/RVL/Database/ASL/asl-database-front.htm> [^13]: Stands for American Sign Language Lexicon Video Dataset (Available at: <http://www.bu.edu/av/asllrp/dai-asllvd.html>) [^14]: Available at: <http://asl-lex.org/> [^15]: Available at: <https://www.microsoft.com/en-us/download/details.aspx?id=100121> [^16]: Available at: <https://www-i6.informatik.rwth-aachen.de/~forster/database-rwth-phoenix.php> [^17]: Available at: <https://www-i6.informatik.rwth-aachen.de/~koller/RWTH-PHOENIX/> [^18]: Available at: <http://www.phonetik.uni-muenchen.de/Bas/SIGNUM/> [^19]: Available at:<http://facundoq.github.io/unlp/lsa16/index.html> [^20]: Available at: <http://facundoq.github.io/unlp/lsa64/index.html> [^21]: Available at: [ https://github.com/zafar142007/Gesture-Recognition-for-Indian-Sign-Language-using-Kinect]( https://github.com/zafar142007/Gesture-Recognition-for-Indian-Sign-Language-using-Kinect) [^22]: Available at: <https://github.com/marlondcu/ISL> [^23]: Available at: <https://www.computer.org/csdl/proceedings-article/cvprw/2010/05543262/12OmNzZ5olI> [^24]: Available at: <https://mmifacedb.eu/> [^25]: Available at:<https://www.oulu.fi/cmvs/node/41316> [^26]: Available at: <https://github.com/npinto/fer2013> [^27]: <https://dataverse.harvard.edu/>

--- abstract: 'We present the results of a partial upgrade to the Monte Carlo event generator TAUOLA using Resonance Chiral Theory for the two and three meson final states. These modes account for 88% of total hadronic width of the tau meson. The first results of the model parameters have been obtained using Preliminary BaBar data for 3$\pi$ mode.' address: - 'NSC Kharkov Institute of Physics and Technologyâ, Kharkov UA-61108, Ukraine' - 'IFIC, Universitat de València-CSIC, Apt. Correus 22085, E-46071, València, Spain' - 'Institute of Nuclear Physics, PAN, Kraków, ul. Radzikowskiego 152, Poland' - 'RWTH Aachen University, III. Physikalisches Institut B, Aachen, Germany' - | The Faculty of Physics, Astronomy and Applied Computer Science,\ Jagellonian University, Reymonta 4, 30-059 Cracow, Poland - 'Grup de Física Teòrica, Institut de Física d’Altes Energies, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Barcelona, Spain' - 'CERN PH-TH, CH-1211 Geneva 23, Switzerland' author: - 'O.Shekhovtsova' - 'I. M. Nugent' - 'T. Przedzinski' - 'P. Roig' - 'Z. Was' title: 'RChL currents in Tauola: implementation and fit parameters' --- Tau physics ,Monte Carlo generator ,Resonance Chiral Theory ,TAUOLA Introduction {#sec:intro} ============ The tau lepton is the heaviest of the leptons, and provides a unique opportunity to study low energy QCD and the mechanism of hadronization. With the shut-down of both the B-factory collaborations, BaBar [@babar] and Belle [@belle], this is a critical time to make the results from more than a decade of experimental research available in a useful manner to high energy physics community before the opportunity is lost. To accomplish this, interaction between experimentalist and theorists is required to determine the optimal way for comparing experimental data with the theoretical prediction. The approach used by this work is to compare theory to data by using a Monte Carlo event generator to simulate the theoretical models. The Monte Carlo event generator TAUOLA is a long term project that started in the 90’s with the publication [@Jadach:1993hs]. The generator simulates more than twenty modes, including both the leptonic and hadronic modes. Modeling the hadronic decay modes involves matrix elements that convey the hadronization of the vector and axial-vector currents. At present there is no determination from first principles for those matrix elements since they involve strong interaction effects in the non-perturbative regime. Therefore, one has to rely on models that parameterize the form-factors originating from the hadronization. The hadronic currents implemented in the generator TAUOLA [@Jadach:1993hs] were based on theoretical results presented in [@Kuhn]. In these models, the hadronic form-factors are written as a weighted sum of products of Breit-Wigner functions. This approximation, as it is demonstrated in [@GomezDumm:2003ku], is not able to reproduce the next-to-leading-order $\chi$PT results. Later the experimental collaborations, both Cleo [@cleo; @Coan:2004ep] and Aleph [@aleph], introduced improvements based on the result of their data analysis and in some cases spoiling the theoretical constraints [^1]. As an alternative, it was proposed to apply the methods of the Resonance Chiral Theory [@rcht]. This approach is a better theoretically founded than the approach of weighted products of Breit-Wigner functions. However, it requires comparison both BaBar and Belle experimental data with the results from theoretical models which have to be implemented into TAUOLA. In the next sections, we will present the result on the implementation the RChT hadronic currents for $\pi^-\pi^0$, $K^-K^0$, $(K\pi)^-$, $(\pi\pi\pi)^-$ and $(K K\pi)^-$ modes in TAUOLA, for details see [@Shekhovtsova:2012ra; @web:RChL]. Hadronic currents for two and three meson decay modes {#sec:hadr} ===================================================== For any two meson $\tau$ decay channel, the most general form of hadronic currents ($J^\mu$), which is compatible with Lorentz invariance, is written as $$\begin{aligned} J^\mu & = & N \bigl[ (p_1 - p_2 - \frac{\Delta_{12}}{s}(p_1 +p_2))^\mu F^{V}(s) \bigr. \label{eq:twomes} \\ & + & \bigl. \frac{\Delta_{12}}{s}((p_1 + p_2)^\mu F^{S}(s) \bigr], \nonumber\end{aligned}$$ where $p_1$ and $p_2$ are the momenta of hadrons, $\Delta_{12} = m_1^2 -m_2^2$, $s = (p_1 +p_2)^2$. For a final state of three pseudoscalarsis $$\begin{aligned} &&\!\!\!\!\!\!\!\!\!\!\!\!\!J^\mu = N \bigl\{T^\mu_\nu \bigl[ c_1 (p_2-p_3)^\nu F_1 + c_2 (p_3-p_1)^\nu F_2 \bigr.\bigr. \label{eq:fiveF} \\ && \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\bigl.\bigl. + c_3 (p_1-p_2)^\nu F_3 \bigr] + c_4 q^\mu F_4 -{ i \over 4 \pi^2 F^2} c_5 \epsilon^\mu_{.\ \nu\rho\sigma} p_1^\nu p_2^\rho p_3^\sigma F_5 \bigr\}, \nonumber\end{aligned}$$ where $p_1$, $p_2$ and $p_3$ are the momenta, $T_{\mu\nu} = g_{\mu\nu} - q_\mu q_\nu/q^2$ denotes the transverse projector, $q^\mu=(p_1+p_2+p_3)^\mu$ is the momentum of the hadronic system and $F$ is the pion decay constant in the chiral limit. Among the form factors $F_1$, $F_2$, $F_3$, corresponding to the axial-vector part of the hadronic currents, only two are model-independent. For our convenience we keep all of them in Eq. (\[eq:fiveF\]). The model-dependence in Eqs. (\[eq:twomes\]) and (\[eq:fiveF\]) is included in the hadronic form-factors ($F_V$, $F_S$ expressed in terms of $F_i$, i = 1...5). For three meson decay modes, the hadronic form-factors calculated within R$\chi$T can be written as $$F_I = F_I^\chi + F_I^R + F_I^{RR}$$ where $F_I^{\chi}$ is the chiral contribution, $F_I^R$ is the one resonance contribution and $F_I^{RR}$ is the double-resonance part. The explicit form of the functions $F_i$ for 3$\pi$ and $KK\pi$ modes can be found in [@Shekhovtsova:2012ra], Section 2, and in [@GomezDumm:2003ku; @Dumm]. The corresponding theoretical form-factors are obtained in the isospin limit ($m_{\pi}=(2m_{\pi^+}+m_{\pi^0})/3$, $m_K=(m_{K^+}+m_{K^0})/2$), except for the two pion and two kaons modes. Numerical results. Fit of the three pion mode to BaBar data {#sec:num} =========================================================== To avoid problems with multi-dimensional integration of the $a_1$-meson propagator, which is rapidly-changing as a function of its arguments, we first tabulated the $\Gamma_{a_1}(q^2)$ of Eq. (38) of [@Shekhovtsova:2012ra]. Then we use linear interpolation to obtain the value of the $a_1$ width at the required $q^2$. To check the numerical stability of the generator and the multiple numerical integration, the following tests have been done: - For every channel one dimensional spectrum $d\Gamma/dq^2$ (for the three meson decay modes) and $d\Gamma/ds$ (for the two meson decay modes) produced by the generator has been compared with the semi-analytical results. For the three meson decay modes the Gauss integration method has been applied to integrate the analytical results for the hadronic currents. The results agree within statistical errors except for the first and last bins. - The spectrum obtained by the Gauss integration has been compared with the linear interpolated spectrum from the neighboring points and demonstrated that the fluctuations due to numerical problems of integration are absent; - The total rate for every channel obtained from a Monte Carlo run has been compared with the semi-analytical method and required to agree within statistical error. The results of comparison are presented in Tab.\[tab:bench\]. The tests have been performed in the isospin limit of meson masses, except for the two pion and two kaons modes. For more details about these tests see [@Shekhovtsova:2012ra]. Triple Gaussian integration is used for the analytical calculation and double Gaussian integration is used for the current calculation that enters the matrix elements of the Monte Carlo generation. Thus, a pre-tabulation of the $a_1$ width, $\Gamma_{a_1}(q^2)$, is a convenient tool to significantly increase the generation speed. --------------------------------- ----------------------------------------------- ----------------------------------------- ------------------------------------------ Channel PDG Equal masses Phase space with masses [$ \pi^-\pi^0 \; \;\; \;$]{} [($5.778 \pm 0.35\%)\cdot 10^{-13}$]{} [($5.2283 \pm 0.005\%)\cdot10^{-13}$]{} [$(5.2441\pm 0.005\%)\cdot 10^{-13}$]{} [$ K^-\pi^0 \; \;\; \;$]{} [($9.72\;\pm 3.5\%\;)\cdot 10^{-15}$]{} [($8.3981 \pm 0.005\%)\cdot10^{-15}$]{} [$(8.5810\pm 0.005\%)\cdot 10^{-15}$]{} [$ \pi^-\bar K^0 \; \;\; \;$]{} [($1.9\;\;\; \pm 5\%\;\;\;)\cdot 10^{-14}$]{} [($1.6798 \pm 0.006\%)\cdot10^{-14}$]{} [$(1.6512\pm 0.006\%)\cdot 10^{-14}$]{} [$ K^-K^0 \; \;\; \;$]{} [($3.60\; \pm 10\%\;\;)\cdot 10^{-15}$]{} [($2.6502 \pm 0.007\%)\cdot10^{-15}$]{} [$(2.6502 \pm 0.008\%)\cdot 10^{-15}$]{} [$ \pi^-\pi^-\pi^+$]{} [($2.11\; \pm 0.8\%\;\;)\cdot 10^{-13}$]{} [($ 2.1013\pm 0.016\%)\cdot10^{-13}$]{} [$(2.0800\pm 0.017\%)\cdot 10^{-13}$]{} [$ \pi^0\pi^0\pi^-$]{} [($2.10\; \pm 1.2\%\;\;)\cdot 10^{-13}$]{} [($ 2.1013\pm 0.016\%)\cdot10^{-13}$]{} [$(2.1256\pm 0.017\%)\cdot 10^{-13}$]{} [$ K^-\pi^-K^+$]{} [($3.17\; \pm 4\%\;\;\;)\cdot 10^{-15}$]{} [($3.7379 \pm 0.024\%)\cdot10^{-15}$]{} [$(3.8460\pm 0.024\%)\cdot 10^{-15}$]{} [$ K^0\pi^-\bar{K^0}$]{} [($3.9\;\; \pm 24\%\;\;)\cdot 10^{-15}$]{} [($3.7385 \pm 0.024\%)\cdot10^{-15}$]{} [$(3.5917\pm 0.024\%)\cdot 10^{-15}$]{} [$ K^-\pi^0 K^0$]{} [($3.60\; \pm 12.6\%\;\;)\cdot 10^{-15}$]{} [($2.7367\pm 0.025 \%)\cdot10^{-15}$]{} [$(2.7711 \pm 0.024\%)\cdot 10^{-15}$]{} --------------------------------- ----------------------------------------------- ----------------------------------------- ------------------------------------------ To obtain the proper kinematic configurations, the differences between neutral and charged pion and kaon masses were taken into account, more specifically, the physical values were chosen in the phase space generation. On the other hand, this choice breaks constraints resulting from isospin symmetry in potentially uncontrolled way. This is why we collect the numerical results from the Monte Carlo calculation, shown in Table \[tab:bench\], where the partial widths from Particle Data Group compilation [@Nakamura:2010zzi] are compared with our results obtained with isospin-averaged pseudoscalar masses and with the physical ones. The model parameters, more specifically the masses of the resonances and the coupling constants, were fitted to Aleph data [@Barate:1998uf], requiring correct high-energy behavior of the related form factors. For details see Appendix C in [@Shekhovtsova:2012ra]. From Table \[tab:bench\], one can see that the difference between the TAUOLA results for the $\tau$ decay partial width and the PDG ones is $1.5\% -17\%$ depending on the mode. As expected, the agreement is not good because only minimal attempts to adjusting the model parameters have been applied for the comparison with BaBar and Belle data. \[fig:fit\] $M_{\rho'}$ $\Gamma_{\rho'}$ $M_{a_1}$ $F$ $F_V$ $F_{A}$ $\beta_{\rho'}$ ---------------------------------- ------------- ------------------ ----------- ---------- ---------- --------- ----------------- Min. 1.44 0.32 1.00 0.0920 0.12 0.1 -0.36 Max 1.48 0.39 1.24 0.0924 0.24 0.2 -0.18 Default 1.453 0.4 1.12 0.0924 0.18 0.149 -0.25 Fit , $\chi^2/ndf = 2262.12/132$ 1.4302 0.376061 1.21706 0.092318 0.121938 0.11291 -0.208811 Currently, only the differential spectrum of the two pion modes [@belle] and three pseudoscalar modes [@Nugent:2009zz] are published. We begin with a fit to the spectrum of the $\pi^+\pi^-\pi^-$ mode. The result is presented in Table \[tab:fit\] and Fig.\[fig:fit\]. The fit was done taking into account only the dominant S-wave mechanism (we follow the determination done in [@Shibata:2002uv]). As suggested in [@Shibata:2002uv] the discrepancy in the low mass region could be described adding a contribution from a scalar particle, $P$-wave mechanism. We expect that inclusion of the lowest-lying scalar resonance [@sigma] will improve the value of $\chi^2$. The values in the 5th row of Table \[tab:fit\] are only the preliminary results. They do not necessarily correspond to the minimum of $\chi^2/ndf$ of the final fit. Work is in progress. Some technical aspects of the fitting strategy is given in talk by Z. Was [@was_proceed]. Tau physcis at LHC ================== From the perspective of high energy experiments, such as those at LHC, a good understanding of tau leptons properties contributes important ingredients of new physics signatures. With the discovery of a new particle around the mass of 125-126 GeV [@higgs:2012], tau decays are an important decay mode for determining if this is the Standard Model Higgs. This is especially pertinent since CMS has reported a deficit in the number of fermion decays from the new particle relative to the Standard Model Higgs Prediction. At LHC, at the moment, tau decays are only used for identification and are not used to study their dynamic. However, the dynamics of tau decays are important for both modeling the decays and therefore the reconstruction, identification and for measuring the polarization of tau decays. As a result, an upgrade to the TAUOLA based on the BaBar and Belle measurements of tau decays is of some interest for systematic errors evaluation for LHC measurements. Conclusion {#sec:concl} ========== The theoretical results for the hadronic currents of two and three pseudoscalar modes, namely, $\pi \pi$, $K\pi$, $K K$, $\pi\pi\pi$ and $K K \pi$, in the framework of R$\chi$T have been implemented in TAUOLA. These modes, together with the one-meson decay modes, represent more than 88% of the hadronic width of the tau lepton. R$\chi$T is a more controlled QCD-based model than the usually used Breit-Wigner parameterization. However, before making conclusion about validity of the model the theoretical results have to be confronted with the experimental data. This can be achieved by fitting the model to data. As a consequence of comparisons with the data, some of the theoretical assumptions may need to be reconsidered too. Now that the technical work on current installation is complete, the work on fitting the data is in progress in collaboration with theoreticians and experimentalists. Therefore, we consider this work as a step towards a theoretically rigorous description of hadronic tau decay data. As soon as the theoretical results are able to reproduce Babar and Belle data for the most important for LHC decays into two and three pions the TAUOLA-RChT version will be installed into LHC environment as it is described in [@Davidson:2010rw]. Acknowledgement =============== This research was supported by a Marie Curie Intra European Fellowship within the 7th European Community Framework Programme (O.S.) and by Alexander von Humboldt Foundation (I.N.), by the Spanish Consolider Ingenio 2010 Programme CPAN (CSD2007-00042) and by MEC (Spain) under Grants FPA2007-60323, FPA2011-23778 (O.S. and P.R.) and FPA2011-25948 (P.R.) and in part by the funds of Polish National Science Centre under decision DEC-2012/04/M/ST2/00240 and DEC-2011/03/B/ST2/00107 (O.S., T.P., Z.W.). [9]{} B. Aubert et al., Phys.Rev.Lett. [**100**]{} (2008) 011801; J.P. Lees et al., arXiv: 1109.1527 \[hep-ex\]. M. Fujikawa et al., Phys. Rev. [**D78**]{} (2008) 072006; H. Hayashii, M. Fujikawa, Nucl.Phys.Proc.Suppl. [**198**]{} (2010) 157. S. Jadach, Z. Was, R. Decker and J. H. Kühn, Comput. Phys. Commun. [**76**]{} (1993) 361. J.H. Kühn, E. Mirkes, Z. Phys. [**C56**]{} (1992) 661; J.H. Kühn, A. Santamaría, Z. Phys. [**C48**]{} (1990) 445. D. G. Dumm, A. Pich and J. Portolés, Phys. Rev. [**D69**]{} (2004) 073002. A. Weinstein: [*http://www.cithep.caltech.edu/$\sim$ajw/korb\_doc.html*]{} T.E. Coan et al., Phys.Rev.Lett. [**92**]{} (2004) 232001. B. Bloch, private communications G. Ecker, J. Gasser, H. Leutwyler, A. Pich, E. de Rafael, Phys. Lett. [**B223**]{} (1989) 425; Nucl. Phys. [**B321**]{} (1989) 311. O. Shekhovtsova, T. Przedzinski, P. Roig and Z. Was, arXiv: 1203.3955 \[hep-ph\]. To be published in Phys. Rev. D T. Przedzinski, O. Shekhovtsova and Z. Was, [*http://annapurna.ifj.edu.pl/$\sim$wasm/RChL/RChL.htm*]{}. D.G. Dumm, P. Roig, A. Pich, J. Portolés, Phys. Lett. [**B685**]{} (2010) 158; Phys. Rev. [**D81**]{} (2010) 034031. K. Nakamura, J. Phys. [**G37**]{} (2010) 075021. R. Barate [*et al.*]{}, Eur. Phys. J. C [**4**]{} (1998) 409. I.M. Nugent, SLAC-R-936. E.I. Shibata et al., arXiv: hep-ex/0210039. P. Roig, I. Nugent, T. Przedzinski, O. Shekhovtsova, Z. Was, in preparation. Z. Was, talk at this proceedings. G. Aad [*et al.*]{} \[ATLAS Collaboration\], Phys. Lett. B [**716**]{} (2012) 1; S. Chatrchyan [*et al.*]{} \[CMS Collaboration\], Phys. Lett. B [**716**]{} (2012) 30. N. Davidson, G. Nanava, T. Przedzinski, E. Richter-Was and Z. Was, Comput. Phys. Commun.  [**183**]{} (2012) 821, arXiv:1002.0543 \[hep-ph\]. [^1]: In fact, hadronic currents of Cleo and Aleph versions spoil some theoretical prediction. For example, to reproduce $\tau^-\to (K K\pi)^- \nu_\tau $ data the Cleo collaboration [@Coan:2004ep] introduced two ad-hoc parameters and, as a result, the Wess-Zumino part does not reproduce the QCD normalization.

--- abstract: | The Littlewood Conjecture states that $\liminf_{q\to \infty} q\cdot||q\alpha||\cdot||q\beta||=0$ for all $(\alpha,\beta) \in {\mathbb{R}}^2$. We show that with the additional factor of $\log q\cdot \log\log q$ the statement is false. Indeed, our main result implies that the set of $(\alpha,\beta)$ for which $ \liminf_{q\to\infty} q\cdot\log q\cdot\log\log q\cdot||q\alpha||\cdot||q\beta||>0$ is of full dimension. author: - | Dzmitry Badziahin [^1]\ [York]{} title: On multiplicatively badly approximable numbers --- Introduction ============ The famous Littlewood conjecture (LC) states that for any pair of real numbers $(\alpha,\beta)$ $$\label{little} \liminf_{q\to \infty} q\cdot ||q\alpha||\cdot||q\beta||=0$$ where $\|\cdot\|$ denotes the distance to the nearest integer. Equivalently, the set $$\label{cond_lit} \{(\alpha,\beta)\in {\mathbb{R}}^2\;:\; \liminf_{q\to \infty} q\cdot||q\alpha||\cdot||q\beta||>0\}$$ is empty. This problem was conjectured in 1930’s and it is still open. For recent progress concerning this fundamental problem see [@ELK; @PVL] and references therein. It is easily seen that holds for all $\alpha\in{\mathbb{R}}$ and $\beta\in {\mathbb{R}}$ outside the set ${\mathbf{Bad}}$ of badly approximable numbers defined as follows $${\mathbf{Bad}}:=\{\alpha\in {\mathbb{R}}\;:\; \liminf_{q\to \infty} q||q\alpha||>0\}.$$ In attempt to understand what should be a proper analogue of badly approximable points in multiplicative case several authors investigated the following set (we will follow the notation introduced in [@BV_mix]). For $\lambda{\geqslant}0$ let $${\mathbf{Mad}}^{\lambda}:=\{(\alpha,\beta)\in {\mathbb{R}}^2\;:\; \liminf_{q\to\infty} \,(\log q )^\lambda \cdot q \cdot ||q\alpha||\cdot ||q\beta||>0\}.$$ In other words, ${\mathbf{Mad}}^\lambda$ is a modification of the set in such that the corresponding condition is weakened by $(\log q)^\lambda$. More generally, given a function $f\;:\;{\mathbb{N}}\to {\mathbb{R}}^+$, define the set $$\label{def_mp} {\mathbf{Mad}}(f):= \inf\{(\alpha,\beta)\in{\mathbb{R}}^2\;:\; \liminf_{q\to \infty}f(q)\cdot q \cdot ||q\alpha||\cdot||q\beta||>0\}.$$ In [@BV_mix] the author and Velani conjectured that $$\begin{aligned} {\mathbf{Mad}}^\lambda=\emptyset\quad \text{for any }\lambda<1,\\ \dim({\mathbf{Mad}}^\lambda)=2\quad \text{for any }\lambda{\geqslant}1\;\,\end{aligned}$$ where $\dim(\cdot)$ denotes the Hausdorff dimension. If true this conjecture implies that the proper multiplicative analogue of the set ${\mathbf{Bad}}$ is ${\mathbf{Mad}}^1$. Note that LC is equivalent to the statement that ${\mathbf{Mad}}^0$ is empty. Therefore BV conjecture implies LC. Regarding the first part of BV conjecture all that is known to date is the remarkable result of Einsiedler, Katok and Lindenstrauss [@ELK] which states that $\dim{\mathbf{Mad}}^0=0$. On the other hand according to the second part the best known result is due to Bugeaud and Moschevitin [@BM]. It states that $\dim{\mathbf{Mad}}^2=2$. So we have a gap $ 0{\leqslant}\lambda<2 $ where the behavior of ${\mathbf{Mad}}^\lambda$ is completely unknown. In this paper we will address the second part of the BV conjecture. In particular, we will show that $$\dim{\mathbf{Mad}}(f)=2\quad\text{if }\ f(q)=\log q\cdot \log\log q.$$ It will straightforwardly imply that $\dim({\mathbf{Mad}}^\lambda)=2$ for any $\lambda>1$. It is worth mentioning that the ‘mixed’ analogue of this result was achieved recently by author and Velani. It was proven that the set $${\mathbf{Mad}}_{\mathcal{D}}(f):=\{\alpha\in{\mathbb{R}}\; : \; \liminf_{q\to\infty} f(q)\cdot q\cdot |q|_{\mathcal{D}}||q\alpha||>0\}$$ has full Hausdorff dimension. All the details can be found in [@BV_mix]. Simultaneous and dual variants of ${\mathbf{Mad}}$ -------------------------------------------------- It is well known that Littlewood conjecture has an equivalent formulation in terms of linear forms. In other words, is equivalent to the statement that $$\liminf_{|AB|\to\infty}|A|^*|B|^*\cdot||A\alpha-B\beta||>0$$ where $|x|^*:=\max\{|x|,1\}$. However it is not known if can be reformulated in the same manner. In other words, define the sets $$\label{def_ml} {\mathbf{Mad}}_L(f):= \inf\{(\alpha,\beta)\in{\mathbb{R}}^2\;:\; \liminf_{|AB|\to\infty }f(|A|^*|B|^*) \cdot |A|^*|B|^*||A\alpha-B\beta||>0\}$$ and $${\mathbf{Mad}}_L^\lambda:={\mathbf{Mad}}_L(\log^\lambda q).$$ Then ${\mathbf{Mad}}(f)$ and ${\mathbf{Mad}}_L(f)$ are not necessarily the same. However as it will be shown in the next sections these sets are closely related to each other. For consistency in further discussion we will use the notation ${\mathbf{Mad}}_P^\lambda$ and ${\mathbf{Mad}}_P(f)$ instead of ${\mathbf{Mad}}^\lambda$ and ${\mathbf{Mad}}(f)$ respectively. It will reflect the fact that in one case we deal with points and in another case we deal with lines. It appears that instead of investigating ${\mathbf{Mad}}_P(f)$ and ${\mathbf{Mad}}_L(f)$ independently it is easier to deal with them simultaneously. In particular, we prove the following result: \[th\_main1\] Let $f(q)=\log q\cdot \log\log q$. Then $$\dim({\mathbf{Mad}}_P(f)\cap{\mathbf{Mad}}_L(f))=2.$$ Main result ----------- For convenience, we define the ‘modified logarithm’ function $\log^*\;:\; {\mathbb{R}}\to{\mathbb{R}}$ as follows $$\log^* x:=\left\{\begin{array}{ll}1&\mbox{for }x<e;\\ \log x&\mbox{for }x{\geqslant}e. \end{array}\right.$$ From now on $$f(q):=\log^* q\cdot\log^*\log q.$$ The key to establishing Theorem \[th\_main1\] is to investigate the intersection of the sets ${\mathbf{Mad}}_P(f)$ and ${\mathbf{Mad}}_L(f)$ along fixed vertical lines in the $(x,y)$-plane. With this in mind, let ${{\rm L}}_x$ denote the line parallel to the $y$-axis passing through the point $(x,0)$. The following constitutes our main theorem. \[tsc\] For any $\theta \in {\mathbf{Bad}}$ $$\dim ( {\mathbf{Mad}}_P(f)\cap{\mathbf{Mad}}_L(f)\cap {{\rm L}}_\theta) = 1 \ .$$ Since by Jarník (1928) the Hausdorff dimension of ${\mathbf{Bad}}$ is one, Theorem \[th\_main1\] can be easily derived from Theorem \[tsc\] with the help of the following general result that relates the dimension of a set to the dimensions of parallel sections, enables us to establish the complementary lower bound estimate – see [@falc pg. 99]. Let $F$ be a subset of ${\mathbb{R}}^2$ and let $E$ be a subset of the $x$-axis. If  $\dim (F \cap {{\rm L}}_x) {\geqslant}t $ for all $x \in E$, then $\dim F {\geqslant}t + \dim E$. Indeed, let $ F={\mathbf{Mad}}_P(f)\cap {\mathbf{Mad}}_L(f) $ and $E={\mathbf{Bad}}$. In view of $\dim({\mathbf{Bad}})=1$ and Theorem \[tsc\], one gets $\dim {\mathbf{Mad}}_P(f)\cap {\mathbf{Mad}}_L(f) {\geqslant}2$. Since ${\mathbf{Mad}}_P(f)\cap {\mathbf{Mad}}_L(f) \subset {\mathbb{R}}^2$, the upper bound statement for the dimension is trivial. Therefore the main ingredient in establishing Theorem \[th\_main1\] is Theorem \[tsc\]. Regarding the proof of Theorem \[tsc\] we will use ideas similar to those in [@BV_mix] which firstly appeared in joint work of author, Pollington and Velani [@BPV]. However the technical details in this paper are substantially more complicated than those in [@BV_mix]. Preliminaries ============= Let $S$ be any subset of ${\mathbb{R}}^2$. By $S_{\theta}$ we denote its orthogonal projection onto the line ${{\rm L}}_\theta$. Let $P(p,r,q):=(p/q,r/q)$ be a rational point where $(p,r,q)\in{\mathbb{Z}}^3, \gcd(p,r,q)=1$. Denote by the height of $P$ the value $$H(P):=q^2|q\theta-p|{\geqslant}q^2||q\theta||.$$ Denote by $\Delta(P,\delta)$ the following segment on ${{\rm L}}_\theta$: $$\Delta(P,\delta):=\{\theta\}\times\left(\frac{r}{q}-\frac{\delta}{H(P)},\frac{r}{q}+\frac{\delta}{H(P)}\right).$$ So $|\Delta(P,\delta)|=2\delta H(P)^{-1}$. Given a line with integer coeffitients $$L(A,B,C):=\{(x,y)\in {\mathbb{R}}^2\;:\, Ax-By+C=0\},$$ $$\label{cond_abc} (A,B,C)\in{\mathbb{Z}}^3,\ B\neq 0,\ \gcd(A,B,C)=1$$ denote by the height of $L$ the value $$H(L):=|A|^*B^2.$$ Denote by $\Delta(L,\delta)$ the following segment on ${{\rm L}}_\theta$: $$\Delta(L,\delta):=\{\theta\}\times\left( \frac{A\theta+C}{B}-\frac{\delta}{H(L)},\frac{A\theta+C}{B}+\frac{\delta}{H(L)}\right).$$ So $|\Delta(L,\delta)|=2\delta H(L)^{-1}$. Given constants $c>0$ and $Q>0$ define the auxiliary sets: $${\mathbf{Mad}}_P(f,c,Q):= \left\{(\alpha,\beta)\in {\mathbb{R}}^2 \;:\; f(q)\cdot q\cdot ||q\alpha||\cdot ||q\beta||>c\;\ \forall q\in{\mathbb{N}},\; {\geqslant}Q\right\}$$ and $${\mathbf{Mad}}_L(f,c,Q):= \inf\left\{(\alpha,\beta)\in{\mathbb{R}}^2\;:\begin{array}{l} f(|A|^*|B|^*) \cdot |A|^*|B|^*||A\alpha-B\beta||>c,\\[1ex] \forall (A,B)\in{\mathbb{Z}}^2,\ |A|^*B^2{\geqslant}Q \end{array}\right\}.$$ It is easily verified that ${\mathbf{Mad}}_P(f,c,Q) \subset {\mathbf{Mad}}_P(f),$ ${\mathbf{Mad}}_L(f,c,Q) \subset {\mathbf{Mad}}_L(f)$ and $${\mathbf{Mad}}_P(f)\cap{\mathbf{Mad}}_L(f)\, = \, \bigcup_{c > 0} ({\mathbf{Mad}}_P(f,c,Q)\cap{\mathbf{Mad}}_L(f,c,Q))\ .$$ For convenience we will omit the parameter $Q$ where it is irrelevant and write ${\mathbf{Mad}}_P(f,c)$ and ${\mathbf{Mad}}_L(f,c)$ for ${\mathbf{Mad}}_P(f,c,Q)$ and ${\mathbf{Mad}}_L(f,c,Q)$ respectively. So it suffices to prove that the set ${\mathbf{Mad}}_P(f,c)\cap{\mathbf{Mad}}_L(f,c)\cap {{\rm L}}_\theta$ has full Hausdorff dimension for some positive constant $c$. Geometrically, the set ${\mathbf{Mad}}_P(f,c)$ consists of points that avoid the “neighborhood” of each rational point $P=(p/q,r/q)$ defined by the inequality $$\left|x-\frac{p}{q}\right|\left|y-\frac{r}{q}\right|<\frac{c}{f(q)q^3}.$$ This “neighborhood” of $P$ will remove the interval $\Delta(P,cf(q)^{-1})$ from ${{\rm L}}_\theta$. Without loss of generality we can assume that $|q\theta-p|=||q\theta||$. Otherwise we just replace the point $P$ by $P':=(p'/q,r/q)$ such that $|q\theta-p'|=||q\theta||$. Then $\Delta(P')\supset \Delta(P)$ and the “neighborhood” of $P$ will not remove anything more than one of $P'$. Similarly one can show that the set ${\mathbf{Mad}}_L(f,c)$ consists of points that avoid the “neighborhood” of each line $L(A,B,C)$ defined by $$|Ax-By+C|<\frac{c}{f(|A|^*|B|^*)|A|^*|B|^*}$$ where the coefficients $A,B,C$ satisfy $(A,B)<>(0,0)$ and $\gcd(A,B,C)=1$. For $B=0$ it leads to the following inequality: $$||Ax||<\frac{c}{|A|f(|A|)}.$$ Take $c<\inf\limits_{q\in{\mathbb{N}}} q||q\theta||$. Then this inequality is not true for $x=\theta$, in other words the “neighborhood” of the line do not remove anything from ${{\rm L}}_\theta$. Therefore it is sufficient to consider the lines $L(A,B,C)$ with $B\neq 0$, so the coefficients $(A,B,C)$ will satisfy . Then the “neighborhood” of $L(A,B,C)$ will remove the interval $\Delta(L,cf(|A|^*|B|^*)^{-1})$ from ${{\rm L}}_\theta$. Cantor sets ----------- In the proof we will use the general Cantor framework firstly introduced in [@BV_mix]. Here we reproduce the definitions and facts which will be used in later discussion. For more details we refer to the paper [@BV_mix]. Let $I$ be a closed interval in ${\mathbb{R}}$. Let ${\mathbf{R}}:=(R_n)$ with $n\in {\mathbb{Z}}_{{\geqslant}0}$ be a sequence of natural numbers and ${\mathbf{r}}:=(r_{m,n})$ with $m,n\in {\mathbb{Z}}_{{\geqslant}0},\ m{\leqslant}n $ be a two parameter sequence of non-negative real numbers. [**The construction.**]{} We start by subdividing the interval $I$ into $R_0$ closed intervals $I_1$ of equal length and denote by ${\mathcal{I}}_1$ the collection of such intervals. Thus, $$\#{\mathcal{I}}_1 = R_0 \qquad {\rm and } \qquad |I_1| = R_0^{-1}\, |{\rm I}| \ .$$ Next, we remove at most $r_{0,0}$ intervals $I_1$ from ${\mathcal{I}}_1$ . Note that we do not specify which intervals should be removed but just give an upper bound on the number of intervals to be removed. Denote by ${\mathcal{J}}_1$ the resulting collection. Thus, $$\label{iona1} \#{\mathcal{J}}_1 {\geqslant}\#{\mathcal{I}}_1 - r_{0,0} \, .$$ For obvious reasons, intervals in ${\mathcal{J}}_1$ will be referred to as (level one) survivors. It will be convenient to define ${\mathcal{J}}_0 := \{I\}$. In general, for $n {\geqslant}0$, given a collection ${\mathcal{J}}_n$ we construct a nested collection ${\mathcal{J}}_{n+1}$ of closed intervals $J_{n+1}$ using the following two operations. [*Splitting procedure.*]{} We subdivide each interval $J_n\in {\mathcal{J}}_n$ into $R_n$ closed sub-intervals $I_{n+1}$ of equal length and denote by ${\mathcal{I}}_{n+1}$ the collection of such intervals. Thus, $$\#{\mathcal{I}}_{n+1} = R_n \times \#{\mathcal{J}}_n \qquad {\rm and } \qquad |I_{n+1}| = R_n^{-1}\, |J_n| \ .$$ [*Removing procedure.*]{} For each interval $J_n\in {\mathcal{J}}_n$ we remove at most $r_{n,n}$ intervals $I_{n+1} \in {\mathcal{I}}_{n+1} $ that lie within $J_n$. Note that the number of intervals $I_{n+1}$ removed is allowed to vary amongst the intervals in ${\mathcal{J}}_n$. Next, for each interval $J_{n-1}\in {\mathcal{J}}_{n-1}$ we additionally remove at most $r_{n-1,n}$ intervals $I_{n+1} \in {\mathcal{I}}_{n+1}$ that lie within $ J_{n-1}$. In general, for each interval $J_{n-k}\in {\mathcal{J}}_{n-k}$ $(1 {\leqslant}k {\leqslant}n)$ we additionally remove at most $r_{n-k,n}$ intervals $I_{n+1} \in {\mathcal{I}}_{n+1}$ that lie within $J_{n-k}$. Then the collection ${\mathcal{J}}_{n+1}$ consists of all intervals $I_{n+1}\in{\mathcal{I}}_{n+1}$ that survive after all these removing procedures for $k=1,2,\ldots,n$. Thus, the total number of survivors is at most $$\label{iona2} \#{\mathcal{J}}_{n+1}{\geqslant}R_n\#{\mathcal{J}}_n-\sum_{k=0}^nr_{k,n}\#{\mathcal{J}}_k.$$ Finally, having constructed the nested collections ${\mathcal{J}}_n$ of closed intervals we consider the limit set $${\mathbf{K}}(I,{\mathbf{R}},{\mathbf{r}}) := \bigcap_{n=1}^\infty \bigcup_{J\in {\mathcal{J}}_n} J.$$ Any set ${\mathbf{K}}(I,{\mathbf{R}},{\mathbf{r}})$ which can be achieved by the procedure described will be referred to as a [*$(I,{\mathbf{R}},{\mathbf{r}})$ Cantor set.*]{} Of course in general it can happen that for some choice of parameters ${\mathbf{R}}$ and ${\mathbf{r}}$ and some choice of removed intervals in removing procedure the $(I,{\mathbf{R}},{\mathbf{r}})$ Cantor set becomes empty. However the next result shows that with some additional conditions on the parameters the Hausdorff dimension of this set is bounded below. Given a $({\rm I},{\mathbf{R}},{\mathbf{r}})$ Cantor set ${\mathbf{K}}(I,{\mathbf{R}},{\mathbf{r}}) $, suppose that $R_n{\geqslant}4$ for all $n\in{\mathbb{Z}}_{{\geqslant}0}$ and that $$\label{cond_th2} \sum_{k=0}^n \left(r_{n-k,n}\prod_{i=1}^k \left(\frac{4}{R_{n-i}}\right)\right){\leqslant}\frac{R_n}{4}.$$ Then $$\dim {\mathbf{K}}(I,{\mathbf{R}},{\mathbf{r}}) {\geqslant}\liminf_{n\to \infty}(1-\log_{R_n} 2).$$ Here we use the convention that the product term in is one when $k=0$ and by definition $ \log_{R_n}\!2 := \log2/ \log R_n$. The proof of Theorem BV4 is presented in [@BV_mix Theorem 4]. Duality between points and lines -------------------------------- The next two propositions show that there is a ‘kind’ of duality between rational points $P(p,r,q)$ and lines $L(A,B,C)$. It will play a crucial role in our proof. \[prop1\_p\] Let $P_1(p_1,r_1,q_1), P_2(p_2,r_2,q_2)$ be two different rational points with $p_1/q_1\neq p_2/q_2, r_1/q_1\neq r_2/q_2$ and $0<q_1||q_2\theta||{\leqslant}q_2||q_1\theta||$. Let $L(A,B,C)$ with $(A,B,C)$ satisfying be the line passing through $P_1,P_2$. Assume that $(P_2)_\theta\in \Delta(P_1,\delta)$. Then $$\label{prop1_stat_p} (P_2)_\theta\in \Delta\left(L, \frac{\delta^2|B|}{q_2||q_1\theta||}\cdot\frac{H(P_2)}{H(P_1)}\right)\subset \Delta\left(L,2\delta^2\frac{H(P_2)}{H(P_1)}\right).$$ Moreover, $$\label{prop1_stat2_p} H(L){\leqslant}4\delta H(P_1)\frac{q_2^3}{q_1^3}.$$ \[prop1\_l\] Let $L_1(A_1,B_1,C_1), L_2(A_2,B_2,C_2)$ be two lines with integer coefficients $(A_i,B_i,C_i)$ satisfying and $|A_2B_1|{\leqslant}|A_1B_2|$. Assume that they intersect at a rational point $P(p,r,q)$ and that $L_2\cap {{\rm L}}_\theta\in \Delta(L_1,\delta)$. Then $$\label{prop1_stat_l} L_2\cap{{\rm L}}_\theta\in \Delta\left(P,\frac{\delta^2 q}{|B_2A_1|}\cdot \frac{H(L_2)}{H(L_1)}\right)\subset \Delta\left(P,2\delta^2\frac{H(L_2)}{H(L_1)}\right).$$ Moreover, $$\label{prop1_stat2_l} H(P){\leqslant}4\delta H(L_1)\frac{|B_2|^3}{|B_1|^3}.$$ [**Proof of Proposition \[prop1\_p\].**]{} Since $P_1,P_2\in L$ we have the following system of equations $$\left\{\begin{array}{l} Ap_1-Br_1+Cq_1=0;\\ Ap_2-Br_2+Cq_2=0;\\ A\theta-B\omega+C=0 \end{array}\right.$$ where $\omega:=\frac{A\theta+C}{B}$. Since $p_1/q_1\neq p_2/q_2$ and $r_1/q_1\neq r_2/q_1$ we get that the coefficients $A$ and $B$ are nonzero. Let $A':=A/d, B':=B/d$ where $d:=(A,B)$. Then by $(A,B,C)=1$ we get that $q_1=dq_1'$ and $q_2=dq_2'$. Then the first two equations of the system lead to $$A'(p_1q_2'-p_2q_1')=B'(r_1q_2'-r_2q_1').$$ This together with $(A',B')=1$ implies $|p_1q_2'-p_2q_1'|{\geqslant}|B'|$ and $|r_1q_2'-r_2q_1'|{\geqslant}|A'|$ or $$\left|\frac{p_1}{q_1}-\frac{p_2}{q_2}\right|{\geqslant}\frac{|B|}{q_1q_2},\quad \mbox{and}\quad \left|\frac{r_1}{q_1}-\frac{r_2}{q_2}\right|{\geqslant}\frac{|A|}{q_1q_2}.$$ The system also gives us the following equalities $$|A|\left|\frac{p_1}{q_1}-\theta\right|=|B|\left|\frac{r_1}{q_1}-\omega\right|\quad\mbox{and}\quad |A|\left|\frac{p_2}{q_2}-\theta\right|=|B|\left|\frac{r_2}{q_2}-\omega\right|.$$ The assumption $(P_2)_\theta\in \Delta(P_1,\delta)$ is equivalent to $$\left|\frac{r_1}{q_1}-\frac{r_2}{q_2}\right|<\frac{\delta}{q_1^2||q_1\theta||}.$$ Finally by the triangle inequality we find that $$\left|\frac{p_1}{q_1}-\frac{p_2}{q_2}\right|{\leqslant}2\max\left\{\left|\frac{p_1}{q_1}-\theta\right|,\left|\frac{p_2}{q_2}-\theta\right|\right\}= 2\max\left\{\frac{||q_1\theta||}{q_1},\frac{q_2||\theta||}{q_2}\right\}.$$ By combining all these inequalities together we get that $$|B|{\leqslant}q_1q_2\left|\frac{p_1}{q_1}-\frac{p_2}{q_2}\right|{\leqslant}2\max\{q_2||q_1\theta||,q_1||q_2\theta||\}=2q_2||q_1\theta||;$$ $$|A|{\leqslant}q_1q_2\left|\frac{r_1}{q_1}-\frac{r_2}{q_2}\right|<\frac{\delta q_2}{q_1||q_1\theta||}.$$ Now we are ready to calculate the bound $$\left|\frac{r_2}{q_2}-\omega\right|=\frac{|A|}{|B|}\frac{||q_2\theta||}{q_2}<\frac{1}{|AB|}\cdot\frac{\delta^2 q_2^2\cdot ||q_2\theta||}{q_1^2||q_1\theta||^2\cdot q_2}= \frac{1}{|A|B^2}\cdot\frac{\delta^2 |B|\cdot H(P_1)}{q_2||q_1\theta||\cdot H(P_2)}.$$ Then the first inclusion in follows immediately. For the second one we just use calculated estimate for $|B|$. Also by combining the bounds for $|A|$ and $|B|$ we get an estimate for the height $H(L)$: $$H(L)=|A|B^2{\leqslant}\frac{4\delta q_2^3||q_1\theta||}{q_1}=4\delta H(P_1)\frac{q_2^3}{q_1^3}.$$ This completes the proof of Proposition  \[prop1\_p\]. Before we start the proof of Proposition \[prop1\_l\] let’s establish some basic facts regarding the point of intersection of two lines $L_1(A_1,B_1,C_1), L_2(A_2,B_2,C_2)$ with integer coefficients $(A_i,B_i,C_i)\in{\mathbb{Z}}^3\backslash (\{0\}^2\times{\mathbb{Z}}), (A_i,B_i,C_i)=1$; $i=1,2$. These facts will be of use in further discussion as well. An intersection $L_1\cap L_2$ is a rational point $P(p,r,q)$ which is the solution of the following system of equations $$\left\{\begin{array}{l} A_1p-B_1r+C_1q=0;\\ A_2p-B_2r+C_2q=0 \end{array}\right.$$ which leads to the following equalities $$\frac{p}{q}=\frac{B_1C_2-B_2C_1}{A_1B_2-A_2B_1} \qquad {\rm and } \qquad \frac{r}{q}=\frac{A_1C_2-A_2C_1}{A_1B_2-A_2B_1} \ .$$ Therefore we get that $$\label{def_prq} |B_1C_2-B_2C_1|=dp,\ |A_1C_2-A_2C_1|=dr,\ |A_1B_2-A_2B_1|=dq.$$ where $d:=\gcd(A_1B_2-A_2B_1, B_1C_2-B_2C_1)\in{\mathbb{Z}}$. Let $i\in\{1,2\}$. It is easily verified that $$L_i\cap{{\rm L}}_\theta = \left(\theta, \frac{A_i\theta+C_i}{B_i}\right)=\left(\theta, \frac{r}{q}+\frac{A_i}{B_i}\left(\theta-\frac{p}{q}\right)\right).$$ Therefore $$|L_1\cap{{\rm L}}_\theta - L_2\cap{{\rm L}}_\theta|=\left|\frac{A_1}{B_1}-\frac{A_2}{B_2}\right|\cdot\left|\theta-\frac{p}{q}\right|=\frac{d|q\theta-p|}{|B_1B_2|}.$$ Hence $$\label{ineq_qthet} |q\theta-p|= d^{-1}|B_1B_2|\cdot|L_1\cap{{\rm L}}_\theta - L_2\cap{{\rm L}}_\theta|$$ and $$\label{ineq_qgam} |q\omega-r|=\frac{|A_1|}{|B_1|}|q\theta-p|,\quad\text{where }\ \omega:=\frac{A_1\theta+C_1}{B_1}.$$ [**Proof of Proposition \[prop1\_l\].**]{} By an upper bound for $q$ is given by $$q= d^{-1}|A_1B_2-A_2B_1|{\leqslant}2d^{-1}\max\{|A_1B_2|, |A_2B_1|\}=2d^{-1}|A_1B_2|.$$ An upper bound for $|q\theta-p|$ can be derived from and the assumption $L_2\cap{{\rm L}}_\theta \in \Delta(P,\delta)$: $$|q\theta-p|<\frac{\delta |B_1B_2|}{d|A_1|B_1^2}=\frac{\delta|B_2|}{d|A_1B_1|}$$ Finally we get the required bounds $$|L_2\cap{{\rm L}}_\theta - P_\theta|=\frac{|A_2|}{|B_2|}\cdot \frac{|q\theta-p|}{q}< \frac{|A_2|\cdot \delta^2 |B_2|^2}{|B_2|\cdot d^2 |A_1B_1|^2\cdot q|q\theta-p|}{\leqslant}\frac{1}{q^2||q\theta||}\cdot \frac{\delta^2q\cdot H(L_2)}{|B_2A_1|\cdot H(L_1)}$$ and $$H(P)=q^2|q\theta-p|< 4d^{-2}|A_1B_2|^2\cdot\frac{\delta|B_2|}{d|A_1B_1|}{\leqslant}4\delta H(L_1)\cdot\frac{|B_2|^3}{|B_1|^3}.$$ To get the last inclusion in we just use calculated bound for $q$. This completes the proof of Proposition \[prop1\_l\]. As we will see the duality between points and lines will appear throughout the whole paper. Proof of Theorem \[tsc\] ======================== The idea -------- By definition for $\theta\in{\mathbf{Bad}}$ there exists a quantity $c(\theta)>0$ such that $$\inf_{q\in {\mathbb{N}}}q||q\theta||=c(\theta).$$ In other words, for any positive integer $q$ the following inequality is satisfied $$\label{ineq_cthet} q|q\theta-p|{\geqslant}c(\theta).$$ Let $R {\geqslant}e^9c^{-1}(\theta)$ be an integer. Choose constants $c$ and $c_1$ sufficiently small such that they satisfy the following inequalities $$\label{ineq_c1} 2^{12}c<1,\quad 2c<R^2 c_1 c(\theta), \quad c<c(\theta)$$ and $$\label{ineq_c} 2^6\max\left\{\frac{c}{R^2c_1c(\theta)},2^{11}c\right\} \frac{(\log R+2)^2R^4}{(\log2)^2}+2^{15}c_1\frac{R^3(\log R+2)}{\log 2}<1.$$ Finally choose the parameter $Q:=c(\theta)R^2F(2)$ where $$F(n):=\prod_{k=1}^n k\,[\log^* k]\;\mbox{ for }n{\geqslant}1\quad\mbox{ and } F(n):=1\;\mbox{ for }n{\leqslant}0.$$ The goal is to construct a $(I,{\mathbf{R}},{\mathbf{r}})$ Cantor type set ${\mathbf{K}}_c$ with properly chosen parameters $I,{\mathbf{R}}$ and ${\mathbf{r}}$ so that ${\mathbf{K}}_c$ is a subset of ${\mathbf{Mad}}_P(f,c,Q)\cap{\mathbf{Mad}}_L(f,c,Q)$. Then we use Theorem BV4 to estimate its Hausdorff dimension. Let $I$ be any interval of length $c_1$ contained within the unit interval $\{\theta\}\times [0,1]\subset {{\rm L}}_\theta$. Define $ {\mathcal{J}}_0:=\{I\}$. We are going to construct, by induction on $n$, a collection ${\mathcal{J}}_n$ of closed intervals $J_n$ such that ${\mathcal{J}}_n $ is nested in ${\mathcal{J}}_{n-1}$; that is, each interval $J_n$ in ${\mathcal{J}}_n$ is contained in some interval $J_{n-1}$ in ${\mathcal{J}}_{n-1}$. The length of an interval $J_n$ will be given by $$|J_n| \, := \, c_1 \, R^{-n}F^{-1}(n).$$ Moreover, each interval $J_n $ in ${\mathcal{J}}_n$ will satisfy the conditions that $$\label{cond_p} \begin{array}{cl}J_{n} \, \cap \, \Delta(P, cf^{-1}(q)) \, = \, \emptyset & \forall \ \ P(p,r,q)\in{\mathbb{Q}}^2 \ \ \mbox{with } (p,r,q)=1,\\[1ex] &Q<H(P) < c(\theta)R^{n-1}F(n-1) \end{array}$$ and $$\label{cond_l} \begin{array}{cl}J_{n} \, \cap \, \Delta(L,cf^{-1}(|A|^*|B|^*)) \, = \, \emptyset \; & \forall \ L(A,B,C)\text{ with } (A,B,C)\in{\mathbb{Z}}^3,\ B\neq 0, \\[1ex] &(A,B,C)=1, \ Q<H(L) < c(\theta)R^{n-1}F(n-1) \, \end{array}$$ In particular, we put $${\mathbf{K}}_{c} = \bigcap_{n=1}^\infty \bigcup_{J\in{\mathcal{J}}_n}J \ .$$ By construction, conditions and ensure that $${\mathbf{K}}_c \subset {\mathbf{Mad}}_P(f,c)\cap{\mathbf{Mad}}_L(f,c)\cap{{\rm L}}_\theta \ .$$ The aim of the rest of the paper is to show that ${\mathbf{K}}_c $ is in fact a $(I,{\mathbf{R}},{\mathbf{r}})$ Cantor set with ${\mathbf{R}}=(R_n)$ given by $$\label{def_rn} R_n:=R \, (n+1) \, [\log^*\!(n+1)]$$ and ${\mathbf{r}}=(r_{m,n})$ given by $$\label{def_rrnm} r_{m,n} \, := \, \left\{\begin{array}{ll} 25R\log R\cdot n^4(\log^*\!n)^4 &\mbox{ \ if }\; m=n-3\\[2ex] 0 &\mbox{ \ otherwise. } \end{array}\right.$$ Then Theorem \[tsc\] will follow from Theorem BV4. Indeed for $n<3$ the condition is obviously satisfied. For $n{\geqslant}3$ and $R{\geqslant}2^7$ we have that the $$\begin{aligned} {\rm l.h.s. \ of \ } \eqref{cond_th2} & = & r_{n-3,n}\cdot\frac{4^3}{R_{n-1}R_{n-2}R_{n-3}}\\[2ex] &{\leqslant}& \frac{ 4^3}{R^3} \cdot \frac{25R\log R\cdot n^4(\log^*n)^4 }{n(n-1)(n-2)\log^*n\log^*(n-1)\log^*(n-2)} \\[2ex] & {\leqslant}& \frac{16\cdot25\cdot 4^3\log R}{R^3}\cdot\frac{R(n+1)[\log^*(n+1)]}{4}{\leqslant}\frac{R_n }{4} \ = \ {\rm r.h.s. \ of \ } \eqref{cond_th2} \, .\end{aligned}$$ Therefore Theorem BV4 implies that $$\dim {\mathbf{K}}_c {\geqslant}\liminf_{n\to \infty} (1-\log_{R_n}\!2)=1 \,$$ which completes the proof of Theorem \[tsc\]. Basic construction. Splitting into collections $C_P(n,l,k)$ and $C_L(n,l,k)$ ---------------------------------------------------------------------------- Now we will describe the procedure of constructing the collections ${\mathcal{J}}_n$. For $n=0$, we trivially have that (\[cond\_p\]), are satisfied for the sole interval $I\in {\mathcal{J}}_0$. The point is that by the choice of $Q$ there are neither points nor lines satisfying the height condition $Q<H(P),H(L)<c(\theta)$. Then we construct ${\mathcal{J}}_i, i=1,2,3$ by just subdividing each $J_{i-1}$ in ${\mathcal{J}}_{i-1}$ into $R\cdot i[\log^*i]$ closed intervals of equal length. Again for the same reason the conditions (\[cond\_p\]) and are satisfied for any $J_i\in{\mathcal{J}}_i, i=1,2,3$. Note that $$\#{\mathcal{J}}_i=R^iF(i),\quad i=1,2,3.$$ In general, given ${\mathcal{J}}_n$ satisfying and we wish to construct a nested collection ${\mathcal{J}}_{n+1}$ of intervals $J_{n+1}$ for which (\[cond\_p\]) and are satisfied with $n$ replaced by $n+1$. By definition, any interval $J_n$ in ${\mathcal{J}}_n$ avoids intervals $\Delta(P,cf^{-1}(q))$ and $\Delta(L,cf^{-1}(|A|^*|B|^*))$ arising from points and lines with height bounded above by $c(\theta)R^{n-1}F(n-1)$. Since any ‘new’ interval $J_{n+1}$ is to be nested in some $J_n$, it is enough to show that $J_{n+1}$ avoids intervals $\Delta(P,cf^{-1}(q))$ and $\Delta(L,cf^{-1}(|A|^*|B|^*))$ arising from points and lines with height satisfying $$\label{zeq2} c(\theta)R^{n-1}F(n-1){\leqslant}H(P),H(L)<c(\theta)R^nF(n) \ .$$ Denote by $C_P(n)$ the collection of all rational points satisfying this height condition. Formally $$C_P(n) := \left\{P(p,r,q)\in{\mathbb{Q}}^2 \, : \, P \ \ {\rm satsifies \ (\ref{zeq2}) \, } \right\} $$ and it is precisely this collection of rationals that comes into play when constructing ${\mathcal{J}}_{n+1}$ from ${\mathcal{J}}_{n}$. By analogy for ‘lines’ let $$C_L(n) := \left\{L(A,B,C) \, : \, L \ \ {\rm satsifies \ (\ref{zeq2}) \, } \right\} \ .$$ We now proceed with the construction. Assume that $n{\geqslant}3$. We subdivide each $J_n$ in ${\mathcal{J}}_n$ into $R_n=[R(n+1)\log^*(n+1)]$ closed intervals $I_{n+1}$ of length $$|I_{n+1}|=c_1 R^{-n-1}F^{-1}(n+1).$$ Denote by ${\mathcal{I}}_{n+1}$ the collection of such intervals. In view of the nested requirement, the collection ${\mathcal{J}}_{n+1}$ which we are attempting to construct will be a sub-collection of ${\mathcal{I}}_{n+1}$. In other words, the intervals $I_{n+1}$ represent possible candidates for $J_{n+1}$. The goal now is simple — it is to remove those ‘bad’ intervals $I_{n+1}$ from ${\mathcal{I}}_{n+1}$ for which $$\label{svt_p} I_{n+1} \, \cap \, \Delta(P,cf^{-1}(q)) \, \neq \, \emptyset \ \ \mbox{ for some \ } P(p,r,q) \in C_P(n)$$ or $$\label{svt_l} I_{n+1} \, \cap \, \Delta(L,cf^{-1}(|A|^*|B|^*)) \, \neq \, \emptyset \ \ \mbox{ for some \ } L(A,B,C) \in C_L(n) \ .$$ So we define $${\mathcal{J}}_{n+1}:=\left\{J_{n+1}\in{\mathcal{I}}_{n+1}\;:\;\begin{array}{l}J_{n+1} \, \cap \, \Delta(P,cf^{-1}(q))=\emptyset\ \mbox{ for any }P\in C_P(n)\\[1ex] J_{n+1} \, \cap \, \Delta(L,cf^{-1}(|A|^*|B|^*))=\emptyset\ \mbox{ for any }L\in C_L(n). \end{array}\right\}$$ Consider the rational point $P(p,r,q)\in C_P(n)$. Note that since $q^2{\geqslant}q^2 ||q\theta||=H(q){\geqslant}cR^{n-1}F(n-1)$, we have that $$\label{ineq_f} f(q){\geqslant}\frac12\log^* (cR^{n-1}F(n-1))\log^*\frac12\log (c(\theta)R^{n-1}F(n-1))> \frac12n(\log^*n)^2$$ for sufficiently large $R$. We use Stirling formula to show that for $n{\geqslant}3$, $$c(\theta)R^{n-1}F(n-1){\geqslant}c(\theta)R^{n-1}(n-1)!>(8n)^n\quad\mbox{ for }R{\geqslant}e^9c^{-1}(\theta).$$ Therefore the left hand side of is bigger than $$\frac12n\log(8n)\cdot \log^* (\frac12n\log (8n))>\frac12n\log^{*2}n.$$ Note that for any line $L(A,B,C)\in C_L(n)$ we have the analogous bound $$\label{ineq_fl} f(|A|^*|B|^*){\geqslant}\frac12n(\log^*n)^2.$$ For $l\in {\mathbb{Z}}$ we split $C_P(n)$ into sub-collections $$\label{def_cnkl_p} C_P(n,l):=\left\{P(p,r,q)\in C_P(n): \begin{array}{c}c(\theta)2^lR^{n-1}F(n-1){\leqslant}H(P)\\[1ex] H(P)<c(\theta)2^{l+1}R^{n-1}F(n-1) \end{array}\right\}.$$ In view of we have that $$\label{ineq_el} 2^l<R n\log^*n$$ so $$\label{ineq_l} 0{\leqslant}l{\leqslant}[\log_2 (Rn\log^* n)] < \log_2 R + 2\log_2 n<c_3 \log^* n.$$ where $c_3:=(\log R+2)/\log 2$ is an absolute constant independent on $n$ and $l$. Additionally with $k\in{\mathbb{Z}}$ we split the collection $C_P(n,l)$ into sub-collections $C_P'(n,l,k)$ such that $$\label{def_cnk} C_P'(n,l,k):=\left\{P(p,r,q)\in C_P(n,l)\;:\; c(\theta)2^k{\leqslant}q||q\theta||< c(\theta)2^{k+1}\right\}.$$ Take any $P(p,r,q)\in C'_P(n,l,k)$. In view of the value $k$ should be nonnegative. On the other hand one can get an upper bound for $k$ by : $$\label{ineq_k} 0{\leqslant}k{\leqslant}[\log_2(R^n F(n))]<n\log_2R+n\log_2n+n\log_2\log^* n<c_3 n\log^* n,$$ The upshot is that for fixed $n,l$ the number of classes $C'_P(n,l,k)$ is at most $c_3n\log^* n$. Note that within the collection $C_P'(n,l,k)$ we have very sharp control of the height $H(P)$. Then by and we also have very sharp control on the value $q$ as well, namely $$\label{ineq_q} 2^{l-k-1}R^{n-1}F(n-1)<q<2^{l-k+1}R^{n-1}F(n-1).$$ Concerning the collection $C_L(n)$ we also partition it into sub-collections. Firstly we partition it into sub-collections $C_L(n,l)$ such that $$\label{def_cnkl_l} C_L(n,l):=\left\{L\in C_L(n): \begin{array}{c}c(\theta)2^lR^{n-1}F(n-1){\leqslant}H(L)\\[1ex] H(L)<c(\theta)2^{l+1}R^{n-1}F(n-1) \end{array}\right\}.$$ Then we split $C_L(n,l)$ into sub-collections $C_L'(n,l,k)$ such that $$\label{def_cnk_l} C_L'(n,l,k):=\{L(A,B,C)\in C_L(n,l)\;:\; 2^k{\leqslant}|B|<2^{k+1}\}$$ One can check that $l$ and $k$ satisfy the same conditions and as in the case of points. Note that within each collection we have very good control of all point and line parameters. The procedure of removing “bad” intervals from ${\mathcal{I}}_{n+1}$ will be as follows. We will firstly remove all intervals $I_{n+1}\in{\mathcal{I}}_{n+1}$ such that there exists a point $P\in C_P(n,0)$ which satisfy $I_{n+1}\cap \Delta(P,cf^{-1}(q))\neq\emptyset$ or there exists a line a line $L\in C_L(n,0)$ which satisfy $I_{n+1}\cap \Delta(L,cf^{-1}(|A|^*|B|^*))\neq\emptyset$. Then we repeat this removing procedure for collections $$C_P(n,1)\text{ and}\;C_L(n,1),\ldots,\;C_P(n,c_2\log^* n)\text{ and}\;C_L(n,c_2\log^* n)$$ in exactly this order. We will use lexicographical order for pairs in ${\mathbb{Z}}^2$. That is, we say that $(a,b){\leqslant}(c,d)$ if either $a<c$ or $a=c, b{\leqslant}d$. Consider the point $P(p,r,q)\in C_P'(n,l,k)$. If there exists a pair $(n',l'){\leqslant}(n,l)$ and a line $L(A,B,C)\in C_L(n',l')$ such that $$H(L)<H(P)\quad\text{and}\quad\Delta(P,cf^{-1}(q))\subset\Delta(L,cf^{-1}(|A|^*|B|^*))$$ then such a point will not remove anything more than was removed by a line $L$. Therefore such a point can be ignored. The same is true if there exists a point $P'(p',r',q')\in C_P(n',l')$ such that $$H(P')<H(P)\quad\text{and}\quad\Delta(P,cf^{-1}(q))\subset\Delta(P',cf^{-1}(q')).$$ Therefore instead of collection $C_P'(n,l,k)$ we can work with $$C_P(n,l,k):=\left\{P(p,r,q)\in C_P'(n,l,k)\;\left|\; \begin{array}{l} \forall (n',l')<(n,l),\\ \forall L(A,B,C)\in C_L(n',l')\;\text{with }H(L)<H(P),\\[1ex] \forall P'(p',r',q')\in C_P(n',l')\;\text{with }H(P')<H(P)\\[1ex] \Delta(P,cf^{-1}(q))\not\subset\Delta(L,cf^{-1}(|A|^*|B|^*)),\\[1ex] \Delta(P,cf^{-1}(q))\not\subset\Delta(P',cf^{-1}(q')). \end{array}\right. \right\}$$ By the same procedure we construct the collection $C_L(n,l,k)$ from $C_L'(n,l,k)$. Note that by the construction of $C_P(n,l,k)$ there exists at most one point $P(p,r,q)\in C_P(n,l,k)$ with given second coordinate $r/q$. Blocks of intervals ${\mathbf{B}}_P(J)$ and ${\mathbf{B}}_L(J)$ --------------------------------------------------------------- Take the maximal possible constant $c_2>0$ such that $$\label{ineq_c2} c_2{\leqslant}\frac{1}{2^{10}c(\theta)}\quad\text{and}\quad \frac{R^2c_1}{c_2}\in {\mathbb{Z}}.$$ Fix the triple $(n,l,k)$ and consider an arbitrary interval $J\subset {{\rm L}}_\theta$ of length $|J|=c_22^{-l}R^{-n+1}F^{-1}(n-1)$. Then for any $P(p,r,q)\in C_P(n,l,k)$ we have $|\Delta(P,cf^{-1}(q))|<|J|$. Indeed this is true because $$|J|{\geqslant}|\Delta(P,cf^{-1}(q))| \Leftrightarrow \frac{c_2}{2^lR^{n-1}F(n-1)}{\geqslant}\frac{2c}{f(q)H(P)}$$ $$\stackrel{\eqref{def_cnkl_p}}\Leftarrow \frac{c_2}{2^l R^{n-1}F(n-1)}{\geqslant}\frac{2c}{c(\theta)2^lR^{n-1}F(n-1)\cdot f(q)}.$$ The last inequality is true provided $c_2 c(\theta){\geqslant}2c$ which in turn is true by the second inequality of and . One can easily check that the same fact is true for any $\Delta(L,cf^{-1}(|A|^*|B|^*))$ where $L(A,B,C)\in C_L(n,l,k)$. \[triang\_lem\_p\] Let $J$ be an interval on ${{\rm L}}_\theta$ of length $|J|=c_2 2^{-l} R^{-n+1}F^{-1}(n-1)$. Then all rational points $P(p,r,q)\in C_P(n,l,k)$ such that $\Delta(P,cf^{-1}(q))\cap J\neq\emptyset$ lie on a single line.  Consider an arbitrary point $P(p,r,q)\in C_P(n,l,k)$. Then $$\label{cond_pq} \left|\theta-\frac{p}{q}\right|=\frac{H(P)}{q^3}\;\stackrel{\eqref{def_cnkl_p},\eqref{ineq_q}}<\; \frac{c(\theta)}{2^{2l-3k-4}R^{2(n-1)}F^2(n-1)}.$$ Suppose we have three points $P_i(p_i,r_i,q_i)\in C_P(n,l,k), i=1,2,3$ such that $\Delta(P_i,cf^{-1}(q_i))\cap J\neq\emptyset$ and they do not lie on a single line. Then they form a triangle which has the area at least $${\mathbf{area}}(\triangle P_1P_2P_3){\geqslant}\frac{1}{2q_1q_2q_3}\stackrel{\eqref{ineq_q}}{\geqslant}\frac{1}{2^{3l-3k+4}R^{3(n-1)}F^3(n-1)}.$$ On the other hand the first coordinates $p_i/q_i$ of the points $P_i$ should satisfy and their second coordinates $r_i/q_i$ should lie within the interval of length $|J|+|\Delta(P_i,cf^{-1}(q_i))|{\leqslant}2|J|$. Therefore we have the following upper bound for the area of triangle $\triangle P_1P_2P_3$: $${\mathbf{area}}(\triangle P_1P_2P_3)< \frac{ 2c_22^{-l}R^{-n+1}F^{-1}(n-1)\cdot 2c(\theta)}{2^{2l-3k-4}R^{2(n-1)}F^2(n-1)}$$ $${\leqslant}2^{10}c_2c(\theta)\cdot \frac{1}{2^{3l-3k+4}R^{3(n-1)}F^3(n-1)}.$$ Finally by we get that the last value is bounded above by $$\frac{1}{2^{3l-3k+4}R^{3(n-1)}F^3(n-1)}{\leqslant}{\mathbf{area}}(\triangle P_1P_2P_3)$$ which is impossible. So we get a contradiction. So given interval $J$ of length $c_22^{-l}R^{n-1}F^{-1}(n-1)$ if we have at least two points $P\in C_P(n,l,k)$ as in Lemma \[triang\_lem\_p\] then all the points with such property will lie on a single line $L$. We denote this line by $L_J$. If there is at most one point $P\in C_p(n,l,k)$ as in Lemma \[triang\_lem\_p\] then we just say that $L_J$ is undefined. Note that $L_J$ can not be horizontal because by the construction of $C_P(n,l,k)$ there is only one point $P(p,r,q)\in C_P(n,l,k)$ with given second coordinate $r/q$. $L_J$ can not be vertical too. Otherwise its equation can be written as $x=C/A,$ $\gcd(A,C)=1$. Then by the construction of $\theta$ we have that $$\left|\theta-\frac{p}{q}\right|=\left|\theta-\frac{C}{A}\right|{\geqslant}\frac{c(\theta)}{A^2}$$ which together with gives us $$|A|{\geqslant}2^{l-3/2k-2}R^{n-1}F(n-1).$$ Then by defitnition of $L_J$ there exist two points $P_1(p_1,r_1,q_1), P_2(p_2,r_2,q_2)$ with $|r_1/q_1-r_2/q_2|<2|J|$. However $$\left|\frac{r_1}{q_1}-\frac{r_2}{q_2}\right|{\geqslant}\frac{|A|}{q_1q_2}\stackrel{\eqref{ineq_q}}{\geqslant}2^{-l+k/2-4}R^{-n+1}F^{-1}(n-1)>2|J|.$$ So we get a contradiction. The statement of Lemma \[triang\_lem\_p\] can be strengthened if we have more than two points $P\in C_P(n,l,k)$ such that $\Delta(P,cf^{-1}(q))\cap J\neq\emptyset$. \[lem\_mtriang\_p\] Let $J$ be an interval on ${{\rm L}}_\theta$ of length $|J|=c_2 2^{-l} R^{-n+1}F^{-1}(n-1)$. Assume that there exists a line $L_J$. Consider the sequence of consecutive intervals $M_i\subset{{\rm L}}_\theta$, $i\in{\mathbb{N}}$, $|M_i|=|J|$, $M_1:=J$ and bottom end of $M_i$ coincides with the top end of $M_{i+1}$. Define the set $${\mathcal{P}}(J,m):=\left\{P\in C_P(n,l,k)\;:\; P\in L_J \text{ and } \Delta(P,cf^{-1}(q))\cap \left(\bigcup_{i=1}^m M_i\right)\neq\emptyset\right\}$$ and the value $$m_P(J):=\max\{m\in {\mathbb{N}}\;|\; \# {\mathcal{P}}(J,m){\geqslant}m+1\}.$$ Then all rational points $P\in C_P(n,l,k)$ such that $$\Delta(P,cf^{-1}(q))\cap \left(\bigcup_{i=1}^{m_P(J)} M_i\right)\neq\emptyset$$ lie on a line $L_J$. [*Remark 1.*]{} Since the number of points $P\in C_P(n,l,k), P\in L_J$ is finite, the value $m_P(J)$ is correctly defined. Indeed since by assumption $\#{\mathcal{P}}(J,1){\geqslant}2$, $m+1\to\infty$ and $\#{\mathcal{P}}(J,m)$ is bounded then $m(J)$ exists and is finite. [*Remark 2.*]{} We define the block of intervals $${\mathbf{B}}_P(J):=\bigcup_{i=1}^{m(J)}M_i.$$ We will work with it as with one unit. If for some interval $J$ the line $L_J$ is undefined then we define $m(J):=1$ and ${\mathbf{B}}_P(J):=J$. So now $m(J)$ and ${\mathbf{B}}_P(J)$ are well defined for all intervals $J$ of length $c_22^{-l}R^{-n+1}F^{-1}(n-1)$.  Is similar to the proof of Lemma \[triang\_lem\_p\]. Let $${\mathcal{P}}(J,m(J))=(P_i(p_i,r_i,q_i))_{1{\leqslant}i{\leqslant}m(J)+1}$$ where the sequence $r_i/q_i$ is ordered in ascending order. Assume that there is a point $P(p,r,q)\in C_P(n,l,k)$ such that $P\not\in L_J$ and $\Delta(P,cf^{-1}(q))\cap {\mathbf{B}}_P(J)\neq\emptyset$. Then the triangle $\Delta(PP_1P_{m(J)+1})$ is splitted into $m_P(J)$ disjoint triangles $$\Delta(PP_iP_{i+1}),\quad 1{\leqslant}i{\leqslant}m_P(J)$$ each of which has the area $${\mathbf{area}}(\Delta(PP_iP_{i+1})){\geqslant}\frac{1}{2qq_iq_{i+1}}.$$ On the other hand the first coordinates of the points $P_1,\ldots,P_{m_P(J)+1}$ and $P$ satisfy and their second coordinates lie within the interval of length at most $(m_P(J)+1)|J|$. Therefore we have the following estimate for the area of the triangle $$\frac{m_P(J)}{2^{3l-3k+4}R^{3(n-1)}F^3(n-1)}{\leqslant}{\mathbf{area}}(\triangle PP_1P_{m_P(J)+1}){\leqslant}\frac{2^{9}(m_P(J)+1)c_2c(\theta)}{2^{3l-3k+4}R^{3(n-1)}F^3(n-1)}.$$ which is impossible since the l.h.s of this inequality is bigger than its r.h.s. Lemmas \[triang\_lem\_p\] and \[lem\_mtriang\_p\] have their full analogues for lines $L\in C_L(n,l,k)$. However the proofs areslightly different. We will formulate them in the next two lemmata. \[triang\_lem\_l\] Let $J$ be an interval on ${{\rm L}}_\theta$ of length $|J|=c_2 2^{-l} R^{-n+1}F^{-1}(n-1)$. Then all lines $L(A,B,C)\in C_L(n,l,k)$ such that $\Delta(L,cf^{-1}(|A|^*|B|^*))\cap J\neq\emptyset$ pass through a single rational point $P$.  We will use the following well-known fact. Let us have three planar lines $L_i(A_i,B_i,C_i), i=1,2,3$ defined by equations $A_ix-B_iy+C_i=0$. Then they intersect in one point (probably at infinity) if and only if $$\det\left(\begin{array}{ccc} A_1&B_1&C_1\\ A_2&B_2&C_2\\ A_3&B_3&C_3 \end{array}\right)=0.$$ Suppose that there are three lines $L_1,L_2,L_3\in C_L(n,l,k)$ which do not intersect at one point but their thickenings intersect $J$. Then $$\left|\det\left(\begin{array}{ccc} A_1&B_1&C_1\\ A_2&B_2&C_2\\ A_3&B_3&C_3 \end{array}\right)\right|{\geqslant}1.$$ On the other hand we can make a vertical shifts of $L_1,L_2,L_3$ to the distances $\delta_i<|J|+|\Delta(L_i)|<2|J|$, $i=1,2,3$ such that they will intersect at one point on $J$. By vertically shifting a line to the distance $\epsilon$ we change its $C$-coordinate by the value $B\epsilon$. Therefore we have $$\det\left(\begin{array}{ccc} A_1&B_1&C_1+B_1\delta_1\\ A_2&B_2&C_2+B_2\delta_2\\ A_3&B_3&C_3+B_3\delta_3 \end{array}\right)=0 \quad \Rightarrow \quad \left|\det\left(\begin{array}{ccc} A_1&B_1&B_1\delta_1\\ A_2&B_2&B_2\delta_2\\ A_3&B_3&B_3\delta_3 \end{array}\right)\right|{\geqslant}1.$$ However the latter determinant is bounded above by $$2|J|(|B_1(A_2B_3-A_3B_2)|+|B_2(A_1B_3-A_3B_1)|+|B_3(A_1B_2-A_2B_1)|)$$$$\stackrel{\eqref{def_cnkl_l},\eqref{def_cnk_l}}{\leqslant}2 c_2c(\theta)2^{-l}R^{-n+1}F^{-1}(n-1)\cdot 6\cdot2^{l+3} R^{n-1}F(n-1)\stackrel{\eqref{ineq_c2}}< 1$$ We get a contradiction. So given interval $J$ of length $c_22^{-l}R^{n-1}F^{-1}(n-1)$ if we have at least two lines from $C_L(n,l,k)$ as in Lemma \[triang\_lem\_l\] then all lines with such property will intersect at one rational point $P$. We denote this point by $P_J$. If there is at most one line from $C_L(n,l,k)$ as in Lemma \[triang\_lem\_l\] then we just say that $P_J$ is undefined. The next Lemma is a “line” analogue of Lemma \[lem\_mtriang\_p\]. \[lem\_mtriang\_l\] Let $J$ be an interval on ${{\rm L}}_\theta$ of length $|J|=c_2 2^{-l} R^{-n+1}F^{-1}(n-1)$. Assume that there exists a point $P_J$. Consider the sequence of consecutive intervals $M_i\subset{{\rm L}}_\theta$, $i\in{\mathbb{N}}$, $|M_i|=|J|$, $M_1:=J$ and bottom end of $M_i$ coincides with the top end of $M_{i+1}$. Define the set $${\mathcal{L}}(J,m):=\left\{L\in C_L(n,l,k)\;:\; P_J\in L \text{ and } \Delta(L,cf^{-1}(|A|^*|B|^*))\cap \left(\bigcup_{i=1}^m M_i\right)\neq\emptyset\right\}$$ and the value $$m_L(J):=\max\{m\in {\mathbb{N}}\;|\; \# {\mathcal{L}}(J,m){\geqslant}m+1\}.$$ Then all lines $L\in C_L(n,l,k)$ such that $$\Delta(L,cf^{-1}(|A|^*|B|^*))\cap \left(\bigcup_{i=1}^{m_L(J)} M_i\right)\neq\emptyset$$ intersect at a point $P_J$. By analogy with Remark 1 the value $m_L(J)$ is correctly defined. We define the block of intervals $${\mathbf{B}}_L(J):=\bigcup_{i=1}^{m_L(J)}M_i.$$ We will work with it as with one unit. As in Remark 2 if for some interval $J$ the point $P_J$ is not defined then we define $m_L(J):=1$ and ${\mathbf{B}}_L(J):=J$.  If $m_L(J)=1$ then this is simply the statement of Lemma \[triang\_lem\_l\]. Now assume that $m_L(J)>1$. Let $${\mathcal{L}}(J,m_L(J))=(L_i(A_i,B_i,C_i))_{1{\leqslant}i{\leqslant}m_L(J)+1}.$$ Denote by $$\omega_i:=\frac{A_i\theta+C_i}{B_i},\quad 1{\leqslant}i{\leqslant}m_L(J)+1.$$ Then all the triples $(A_i,B_i,C_i)$ lie inside the figure $F$ defined by the inequalities $$\begin{array}{rcl} |A_i|=\displaystyle\frac{H(L_i)}{|B_i|^2}&\stackrel{\eqref{def_cnkl_l},\eqref{def_cnk_l}}<& c(\theta)2^{l-2k+1}R^{n-1}F(n-1),\\[2ex] |B_i|&\stackrel{\eqref{def_cnk_l}}<&2^{k+1}\quad\text{and}\\[1ex] |A_i\theta-B_i\omega_1+C_i|<|B_i|\cdot|\omega_1-\omega_i|&<&c_2m_L(J)2^{k+2-l}R^{-n+1}F^{-1}(n-1). \end{array}$$ The volume of this figure is $16c_2c(\theta)m_L(J)$ which in view of is smaller than $\frac{1}{6}m_L(J)$. All points $(A_i,B_i,C_i)$ together with $(0,0,0)$ lie on the plane defined by $A_ip-B_ir+C_iq=0$. And since $\gcd(A_i,B_i,C_i)=1$ their convex body contains at least $m_L(J)$ disjoint triangles with vertices in points $(A_i,B_i,C_i)$ and $(0,0,0)$. Now suppose that there is a line $L(A,B,C)\in C_L(n,l,k)$ such that $P_J\not\in L$ and $\Delta(L,cf^{-1}(|A|^*|B|^*))\cap {\mathbf{B}}_L(J)\neq\emptyset$. Then $(A,B,C)\in F$ but now this point doesn’t lie on the same plane with points $(A_i,B_i,C_i)$ and $(0,0,0)$. Then it formes at least $m_L(J)$ disjoint tetrahedrons with them each of which has the volume at least $1/6$. Therefore the volume of $F$ is bounded by $$\frac{1}{6}m_L(J){\leqslant}{\mathbf{vol}}(F)<\frac{1}{6}m_L(J).$$ But the last inequality is impossible. Therefore the line $L$ has to pass through the point $P_J$. Properties of blocks ${\mathbf{B}}_P(J)$, ${\mathbf{B}}_L(j)$ and quantities $m_P, m_L$ {#sec_mp} ---------------------------------------------------------------------------------- Take an arbitrary interval $M$ of length $c_22^{-l}R^{n-1}F(n-1)$ and consider the collection ${\mathcal{P}}_M$ of points $P\in C_P(n,l,k)$ such that $\Delta(P,cF^{-1}(q))\cap {\mathbf{B}}_P(M)\neq\emptyset$. Then one of the following cases should be true. [**Case 1P.**]{} For any interval $J\subset {\mathbf{B}}_P(M)$ such that $|J|=|M|$, $$\#{\mathcal{S}}_J:=\#\{P\in{\mathcal{P}}_M\;|\;\Delta(P,cF^{-1}(q))\cap J\neq\emptyset\}{\leqslant}2^2.$$ [**Case 2P.**]{} There exists $J\subset {\mathbf{B}}_P(M), |J|=|M|$ such that $\#{\mathcal{S}}_J>2^2$. Then the line $L_J$ is correctly defined and therefore $L_M=L_M(A,B,C)$ is correctly defined as well. Let the coefficient $B$ satisfy the condition $$\label{cond_ub} |B|<\frac{c(\theta)2^{k+6}}{1/2c n(\log^*n)^2}$$ [**Case 3P.**]{} There exists $J\subset {\mathbf{B}}_P(M), |J|=|M|$ such that $\#{\mathcal{S}}_J>2^2$ and $$\label{cond_lb} |B|{\geqslant}\frac{c(\theta)2^{k+6}}{1/2c n(\log^*n)^2}.$$ Consider Cases 2P and 3P. Since for any $P\in{\mathcal{S}}_J$ all numbers $P_\theta$ lie inside an interval of length at most $2|J|$ there are at least two points $P_1(p_1,r_1,q_1)$ and $P_2(p_2,r_2,q_2)$ from ${\mathcal{S}}_J$ such that $$\left|\frac{r_1}{q_1}-\frac{r_2}{q_2}\right|<2^{-1}|J|.$$ Without loss of generality assume that $q_2||q_1\theta||>q_1||q_2\theta||$. Then $$\label{incl_p} (P_2)_\theta\in \Delta(P_1, 2^{-1}|J|H(P_1))\stackrel{\eqref{def_cnkl_p}}\subset \Delta(P_1, c(\theta)c_2)\stackrel{\eqref{ineq_c2}}\subset\Delta(P_1,2^{-10}).$$ Since $L_M$ is neither vertical nor horizontal, Proposition \[prop1\_p\] is applicable for $\delta=2^{-10}$. It states that $$(P_2)_\theta\in \Delta\left(L_M, \frac{2^{-20}|B|}{q_2||q_1\theta||}\cdot\frac{H(P_2)}{H(P_1)}\right)$$ and $$H(L_M){\leqslant}2^{-8}H(P_1)\frac{q_2^3}{q_1^3}\stackrel{\eqref{ineq_q}}{\leqslant}\frac12H(P_1).$$ It shows that $L_M$ belongs to the class which within the basic construction had been considered before considering the points from ${\mathcal{P}}_M$. Now let’s consider the Case 2P. By , , and the inclusion implies that $$(P_2)_\theta\in \Delta\left(L_M, \frac{2^{-11}}{1/2c n(\log^*n)^2}\right)\stackrel{\eqref{ineq_fl}}\subset \Delta\left(L_M, \frac{1}{8cf(|A|^*|B|^*)}\right).$$ Now since for any $P(p',r',q')\in C_P(n,l,k)$ the distance $|\theta-p'/q'|$ can differ from $|\theta-p/q|$ by factor at most 4 the same thing is true for the value $|\omega-r'/q'|$. An implication of this is that for all $P\in{\mathcal{P}}_M$, $$P_\theta\in \Delta(L_M,1/2 cf^{-1}(|A|^*|B|^*)).$$ Whence $$\bigcup_{P(p,r,q)\in{\mathcal{P}}_M}\Delta(P,cf^{-1}(q))\subset \Delta(L_M,cf^{-1}(|A|^*|B|^*)).$$ However by the construction of the collection $C_P(n,l,k)$, for all $P\in C_P(n,l,k)$ intervals $\Delta(P,cf^{-1}(q))$ are not contained in any interval $\Delta$ previously considered. Therefore since ${\mathcal{P}}_M\subset C_P(n,l,k)$ then the set ${\mathcal{P}}_M$ in case 2P should be empty — a contradiction. Therefore the Case 2P is impossible. Consider the last Case 3P. Let’s order all the points in ${\mathcal{P}}_M=(P_i(p_i,r_i,q_i))_{1{\leqslant}i{\leqslant}m_L(J)+1}$ in such a way that the sequence $p_i/q_i$ is increasing. Then we have $$\left|\frac{p_1}{q_1}-\frac{p_{m_P(M)+1}}{q_{m_P(M)+1}}\right|{\leqslant}\left|\theta-\frac{p_1}{q_1}\right|+ \left|\theta-\frac{p_{m_P(M)+1}}{q_{m_P(M)+1}}\right|\stackrel{\eqref{cond_pq}}{\leqslant}\frac{c(\theta)}{2^{2l-3k-5}R^{2(n-1)}F^2(n-1)}.$$ On the other hand the smallest possible difference between consecutive numbers $p_i/q_i$ and $p_{i+1}/q_{i+1}$ is bounded below by $$\frac{p_{i+1}}{q_{i+1}}-\frac{p_i}{q_i}{\geqslant}\frac{|B|}{q_iq_{i+1}}.$$ and therefore $$\left|\frac{p_1}{q_1}-\frac{p_{m_P(M)+1}}{q_{m_P(M)+1}}\right| \stackrel{\eqref{ineq_q}}{\geqslant}\frac{|B|m_P(M)}{2^{2l-2k+2} R^{2(n-1)}F^2(n-1)}.$$ By combining the last two inequalities and we finally get an estimate $$\label{ineq_mp} m_P(M){\leqslant}c n(\log^* n)^2.$$ Now for the same interval $M$ define the collection ${\mathcal{L}}_M$ of lines $L(A,B,C)\in C_L(n,l,k)$ such that $\Delta(L,cF^{-1}(|A|^*|B|^*))\cap {\mathbf{B}}_L(M)\neq\emptyset$. Consider three different cases which will be full analogues to cases 1P, 2P and 3P. [**Case 1L.**]{} For any interval $J\subset {\mathbf{B}}_L(M)$ such that $|J|=|M|$, $$\#{\mathcal{S}}_J:=\#\{L(A,B,C)\in{\mathcal{L}}_M\;|\;\Delta(L,cF^{-1}(|A|^*|B|^*))\cap J\neq\emptyset\}{\leqslant}2^2.$$ [**Case 2L.**]{} There exists $J\subset {\mathbf{B}}_L(M)$, $|J|=|M|$ such that $\#{\mathcal{S}}_J>2^2$. Then the point $P_J$ is correctly defined and therefore $P_M=P_M(p,r,q)$ is correctly defined as well. Let the coefficient $q$ satisfy the condition $$\label{cond_uq} q<\frac{c(\theta)2^{l-k+3}R^{n-1}F(n-1)}{1/2c n(\log^*n)^2}.$$ [**Case 3L.**]{} There exists $J\subset {\mathbf{B}}_L(M), |J|=|M|$ such that $\#{\mathcal{S}}>2^2$ and $$\label{cond_lq} q{\geqslant}\frac{c(\theta)2^{l-k+3}R^{n-1}F(n-1)}{1/2c n(\log^*n)^2}.$$ Consider Cases 2L and 3L. The arguments will be essentially the same to that about Cases 2P and 3P. So one can get that there are at least two lines $L_1(A_1,B_1,C_1)$ and $L_2(A_2,B_2,C_2)$ from ${\mathcal{S}}$ such that $$|L_1\cap{{\rm L}}_\theta-L_2\cap{{\rm L}}_\theta|<2^{-1}|J|.$$ Without loss of generality suppose that $|A_2B_1|<|A_1B_2|$. Then $$L_2\cap{{\rm L}}_\theta\in \Delta(L_1, 2^{-10}).$$ and Proposition \[prop1\_l\] is applicable with $\delta=2^{-10}$. Therefore arguments analogous to those used in cases 2P, 3P give us $$L_2\cap{{\rm L}}_\theta\in \left(P_M,\frac{2^{-19}q}{|B_2A_1|}\right)$$ and $$H(P_M){\leqslant}\frac12H(L_1).$$ Therefore the point $P_M$ is from the class which has already been considered before considering lines from ${\mathcal{L}}_M$. Now consider the Case 2L. Then by , and we have that $$L_2\cap{{\rm L}}_\theta\in \Delta\left(P_M, \frac{2^{-14}}{1/2c n(\log^*n)^2}\right)\stackrel{\eqref{ineq_f}}\subset \Delta\left(P_M, \frac{1}{32cf(q)}\right).$$ Note that for any line $L(A,B,C)\in C_L(n,l,k)$ which go through $P_M(p,r,q)$ the distance $$\left|\frac{A\theta+C}{B}-\frac{r}{q}\right|=\frac{|A|}{|B|}\left|\theta-\frac{p}{q}\right|$$ can differ by factor at most 16 from the same distance for line $L_2$. Therefore for all $L\in{\mathcal{L}}_M$, $$L\cap{{\rm L}}_\theta\in \Delta(P_M,1/2 cf^{-1}(q)).$$ Whence $$\bigcup_{L(A,B,C)\in{\mathcal{L}}_M}\Delta(L,cf^{-1}(|A|^*|B|^*))\subset \Delta(P_M,cf^{-1}(q)).$$ However since ${\mathcal{L}}_M\subset C_L(n,l,k)$ we get by the construction of $C_L(n,l,k)$ that ${\mathcal{L}}_M$ has to be empty — a contradiction. Therefore the Case 2L is impossible. Now consider the Case 3L. Let’s order all the lines in ${\mathcal{L}}_M=(L_i(A_i,B_i,C_i))_{1{\leqslant}i{\leqslant}m_L(J)+1}$ in such a way that the sequence of the second coordinates of $L_i\cap{{\rm L}}_\theta$ is increasing. Then we have $$|L_1\cap{{\rm L}}_\theta-L_{m_L(M)+1}\cap{{\rm L}}_\theta|{\leqslant}\left|L_1\cap{{\rm L}}_\theta-\frac{r}{q}\right|+ \left|L_{m_L(J)+1}\cap{{\rm L}}_\theta-\frac{r}{q}\right|$$ $$=\left(\frac{|A_1|}{|B_1|}+\frac{|A_{m_L(M)+1}|}{|B_{m_L(M)+1}|}\right)\cdot\frac{|q\theta-p|}{q}\stackrel{\eqref{def_cnkl_l},\eqref{def_cnk_l}}<c(\theta)2^{l-3k+2}R^{n-1}F(n-1)\cdot\frac{|q\theta-p|}{q}.$$ On the other hand by and the smallest difference between two consecutive $L_i\cap{{\rm L}}_\theta$ and $L_{i+1}\cap{{\rm L}}_\theta$ is at least $$\frac{|q\theta-p|}{|B_iB_{i+1}|}> 2^{-2k-2}|q\theta-p|$$ and therefore $$|L_1\cap{{\rm L}}_\theta-L_{m_L(M)+1}\cap{{\rm L}}_\theta|>m_L(M)2^{-2k-2}|q\theta-p|.$$ By combining the upper and lower bounds for $|L_1\cap{{\rm L}}_\theta-L_{m_L(M)+1}\cap{{\rm L}}_\theta|$ and we finally get an estimate $$\label{ineq_ml} m_L(M){\leqslant}c n(\log^* n)^2.$$ Final step of the proof ----------------------- Let $n{\geqslant}3$. Fix an interval $J_{n-3}\in{\mathcal{J}}_{n-3}$. We will firstly estimate the quantity $$\#\{P(p,r,q)\in C_P(n,l,k)\;:\; \Delta(P,cf^{-1}(q))\cap J_{n-3}\neq\emptyset\}.$$ Split $J_{n-3}$ into $$K:= c_1/c_2\cdot2^lR^2(n-1)(n-2)[\log^*(n-1)][\log^*(n-2)]$$ subintervals $M_1,\ldots,M_K$ of equal length $c_22^{-l}R^{-n+1}F^{-1}(n-1)$ such that the bottom endpoint of $M_i$ coincides with the top endpoint of $M_{i+1}$ ($1{\leqslant}i{\leqslant}K-1$). We start by constructing blocks from intervals $M_1,\ldots, M_K$. Define $B_1:={\mathbf{B}}_P(M_{n_1})$, $B_2:={\mathbf{B}}_P(M_{n_2}),\ldots,$ $B_t:={\mathbf{B}}_P(M_{n_t})$ in such a way that $n_1:=1$ and the bottom endpoint of ${\mathbf{B}}_P(M_{n_i})$ coincides with the top endpoint of ${\mathbf{B}}_P(M_{n_{i+1}})$. By Lemma \[lem\_mtriang\_p\] for any $1{\leqslant}i<t$ we have $$\label{est_bi} \#\{P(p,r,q)\in C_P(n,l,k)\;:\; \Delta(P,cf^{-1}(q))\cap B_i\neq\emptyset\}{\leqslant}m_P(M_{n_i})+1{\leqslant}2m_P(M_{n_i}).$$ Now let’s consider the last block $B_t$. The problem is that this block is not necessarily included in $J_{n-3}$ so we need to treat it independently. As it was discussed in Section \[sec\_mp\], we have two possible cases. In Case 1P we have that for any $i{\geqslant}n_t$ $$\#\{P(p,r,q)\in C_P(n,l,k)\;:\; \Delta(P,cf^{-1}(q))\cap M_i\neq\emptyset\}{\leqslant}2^2.$$ By combining it with we get that $$\label{est_pnlk} \#\{P(p,r,q)\in C_P(n,l,k)\;:\; \Delta(P,cf^{-1}(q))\cap J_{n-3}\neq\emptyset\}{\leqslant}4K$$ In Case 3P we have $$\#\{P(p,r,q)\in C_P(n,l,k)\;:\; \Delta(P,cf^{-1}(q))\cap B_t\neq\emptyset\}\stackrel{\eqref{ineq_mp}}{\leqslant}cn(\log^*n)^2+1\stackrel{\eqref{ineq_c1},\eqref{ineq_c2}}<K.$$ By combining this estimate with we get that $$\#\{P(p,r,q)\in C_P(n,l,k)\;:\; \Delta(P,cf^{-1}(q))\cap J_{n-3}\neq\emptyset\}{\leqslant}3K<4K.$$ Now estimate the number of intervals $I_{n+1}\in{\mathcal{I}}_{n+1}$ which are removed by $\Delta(P,cf^{-1}(q))$ where $P$ is some interval from $C_P(n,l,k)$. $$\begin{aligned} \#\{I_{n+1}\in {\mathcal{I}}_{n+1} \!\!\!\!\! & \!\!\!\!\! : \!\!\!\!\! \; & \!\!\!\!\! I_{n+1}\cap \Delta(P,cf^{-1}(q))\neq\emptyset \} \ {\leqslant}\ \displaystyle\frac{|\Delta(P,cf^{-1}(q))|}{|I_{n+1}|}+2 \nonumber \\[2ex] &=& \displaystyle\frac{2cR^{n+1}F(n+1)}{c_1f(q)H(q)}+2 \nonumber \\[2ex] &{\leqslant}& \frac{2c R^2 n(n+1) \; [\log^*\!n] \; [\log^*\!(n+1)]}{c_1 c(\theta)f(q) 2^l}+2 \nonumber \\[2ex] &\stackrel{\eqref{ineq_f}}{<} & \displaystyle\frac{8cR^2 (n+1)}{c_1c(\theta)2^l}+2 \, . \label{iona}\end{aligned}$$ The upshot of the cardinality estimates and is that $$\begin{aligned} \#\{I_{n+1}\in {\mathcal{I}}_{n+1} \!\!\!\! &\!\!\!\! : \!\!\!\! \;& \!\!\!\! J_{n-3} \cap I_{n+1}\cap \Delta(P,cf^{-1}(q))\neq\emptyset\ \mbox{ for some } P\in C_P(n,l)\} \nonumber \\ \stackrel{\eqref{ineq_k}}{\leqslant}\!\! c_3n\log^*n\cdot\#\{I_{n+1}\in {\mathcal{I}}_{n+1} \!\!\!\! &\!\!\!\!\! : \!\!\!\!\!\! \;& \!\!\!\! J_{n-3} \cap I_{n+1}\cap \Delta(P,cf^{-1}(q))\neq\emptyset\ \mbox{for some } P\in C_P(n,l,k)\} \nonumber \\[2ex] &{\leqslant}& (c_3n\log^*n\cdot 4K)\left(2+\frac{8cR^2(n+1)}{c_1c(\theta)2^l}\right) \nonumber \\[2ex] &{\leqslant}& 8R^2c_3\frac{c_1}{c_2}\cdot 2^ln^3(\log^*n)^3+2^5\frac{c_3c}{c_2c(\theta)}R^4\cdot n^4(\log^*n)^3.\end{aligned}$$ By analogy we get the same estimate for $$\#\{I_{n+1}\in {\mathcal{I}}_{n+1} \; | \; J_{n-3} \cap I_{n+1}\cap \Delta(L,cf^{-1}(|A|^*|B|^*))\neq\emptyset\ \mbox{ for some } L\in C_L(n,l)\}.$$ By taking lines and points together and summing over $l$ satisfying we find that $$\#\left\{I_{n+1}\in {\mathcal{I}}_{n+1} \; \left| \begin{array}{l} J_{n-3} \cap I_{n+1}\cap \Delta(P,cf^{-1}(q))\neq\emptyset\ \mbox{ for some } P\in C_P(n)\text{ or}\\[1ex] J_{n-3} \cap I_{n+1}\cap \Delta(L,cf^{-1}(|A|^*|B|^*))\neq\emptyset\ \mbox{ for some } L\in C_L(n) \end{array}\right. \right\}$$ $${\leqslant}16R^2c_3\frac{c_1}{c_2}n^3(\log^*n)^3\sum_{2^l<Rn\log^*n}2^l + 2^6\frac{c_3^2c}{c_2c(\theta)}R^4n^4(\log^*n)^4.$$ If $(2^{10}c(\theta))^{-1}>R^2c_1$ then in view of we have that $c_2=R^2c_1$. Otherwise we have that $c_2{\geqslant}(2^{11}c(\theta))^{-1}$ and $$\frac{c}{c_2c(\theta)}{\leqslant}2^{11}c;\quad \frac{c_1}{c_2}{\leqslant}2^{11} c_1.$$ In any case the last expression is bounded by $${\leqslant}c_4 n^4(\log^*n)^4$$ where $$c_4:=2^6\max\left\{\frac{c}{R^2c_1c(\theta)},2^{11}c\right\} \frac{(\log R+2)^2R^4}{(\log2)^2}+16\max\{R^{-2},2^{11}c_1\}\frac{R^3(\log R+2)}{\log 2}$$ (recall that $c_3=(\log R+2)/\log 2$). In view of the right hand side of this inequality is bounded by $$25R\log R\cdot n^4(\log^* n)^4=r_{n-3,n}.$$ The upshot is that for any interval $J_{n-3}\in{\mathcal{J}}_{n-3}$ the number of ‘bad’ intervals $I_{n+1}\in{\mathcal{I}}_{n+1}$ which are to be removed is bounded by $r_{n-3,n}$. Therefore the desired set ${\mathbf{K}}_c$ is indeed a $(I,{\mathbf{R}},{\mathbf{r}})$ Cantor type set. The proof is complete. Final remark. ============= In the proof of Theorem \[tsc\] we showed that ${\mathbf{Mad}}_P(f,c)\cap{\mathbf{Mad}}_L(f,c)\cap{{\rm L}}_\theta$ contains $(I,{\mathbf{R}},{\mathbf{r}})$ Cantor type set. It allows us to use Theorem 5 from [@BV_mix]: \[th\_icantor\] For each integer $1{\leqslant}i {\leqslant}k $, suppose we are given a Cantor set ${\mathbf{K}}(I,{\mathbf{R}},{\mathbf{r}}_i) $. Then $$\bigcap_{i=1}^{k} {\mathbf{K}}(I,{\mathbf{R}},{\mathbf{r}}_i) $$ is a $(I,{\mathbf{R}},{\mathbf{r}}) $ Cantor set where $${\mathbf{r}}:=(r_{m,n}) \quad\mbox{with } \quad r_{m,n} :=\sum_{i=1}^k r^{(i)}_{m,n} \, .$$ Regarding the sets of the form ${\mathbf{Mad}}_P(f)\cap{\mathbf{Mad}}_L(f)\cap{{\rm L}}_\theta$, Theorem BV5 enables us to show that for any finite family $\theta_1,\ldots,\theta_n$ of badly approximable numbers one can find $\alpha\in{\mathbb{R}}$ such the following inclusion holds simultaneously for all $1{\leqslant}i{\leqslant}n$: $$(\alpha,\theta_i)\in {\mathbf{Mad}}_P(f)\cap{\mathbf{Mad}}_L(f).$$ Moreover the set of such numbers $\alpha$ is of full Hausdorff dimension. The proof is based on intersecting the corresponding Cantor type sets $K_c(i)$ associated with each set ${\mathbf{Mad}}_P(f,c)\cap{\mathbf{Mad}}_L(f,c)\cap {{\rm L}}_{\theta_i}$ for $c$ sufficiently small and then on applying Theorem BV4 to the intersection. We will leave the details to the enthusiastic reader. We also believe that the same fact will be true for countable collection $\{\theta_i\}$ of badly approximable numbers. However it can not be proven with existing technique. [**Acknowledgements.**]{} I would like to thank Sanju Velani for many discussions during the last years. They finally gave rise to many ideas which made this paper possible. [10]{} D. Badziahin, A. Pollington and S. Velani: On a problem in simultaneously Diophantine approximation: Schmidt’s conjecture. [*Pre-print: arXiv:1001.2694*]{} (2010), 1–43. D. Badziahin ans S. Velani: Multiplicatively badly approximable numbers and generalised Cantor sets. [*Pre-print: arXiv:1007.1848*]{} (2010), 1–27. Y. Bugeaud and N. Moshchevitin: Badly approximable numbers and Littlewood-type problems. [*Pre-print: arXiv:0905.0830v1*]{} (2009), 1–15. M. Einsiedler, A. Katok and E. Lindenstrauss: Invariant measures and the set of exceptions to Littlewood’s conjecture. [*Ann. of Math.*]{} **164** (2006), 513–560 K. Falconer: [*Fractal Geometry: Mathematical Foundations and Applications.*]{} John Wiley & Sons, (1990). A. Pollington and S. Velani: On a problem in simultaneously Diophantine approximation: Littlewood’s conjecture. [*Acta Math.*]{} **66** (2000), 29–40. Dzmitry A. Badziahin: Department of Mathematics, University of York, Heslington, York, YO10 5DD, England. e-mail: db528@york.ac.uk [^1]: Research supported by EPSRC grant EP/E061613/1

--- abstract: | We consider the notion of the De Rham operator on finite-dimensional diffeological spaces such that the diffeological counterpart $\Lambda^1(X)$ of the cotangent bundle, the so-called pseudo-bundle of values of differential 1-forms, has bounded dimension. The operator is defined as the composition of the Levi-Civita connection on the exterior algebra pseudo-bundle $\bigwedge(\Lambda^1(X))$ with the standardly defined Clifford action by $\Lambda^1(X)$; the latter is therefore assumed to admit a pseudo-metric for which there exists a Levi-Civita connection. Under these assumptions, the definition is fully analogous the standard case, and our main conclusion is that this is the only way to define the De Rham operator on a diffeological space, since we show that there is not a straightforward counterpart of the definition of the De Rham operator as the sum $d+d^*$ of the exterior differential with its adjoint. We show along the way that other connected notions do not have full counterparts, in terms of the function they are supposed to fulfill, either; this regards, for instance, volume forms, the Hodge star, and the distinction between the $k$-th exterior degree of $\Lambda^1(X)$ and the pseudo-bundle of differential $k$-forms $\Lambda^k(X)$. MSC (2010): 53C15 (primary), 57R35 (secondary). author: - 'Ekaterina [<span style="font-variant:small-caps;">P</span>ervova]{}' title: Diffeological De Rham operators --- Introduction {#introduction .unnumbered} ============ The concept of a *diffeological space* (introduced in [@So1], [@So2]; see [@iglesiasBook] for a recent and comprehensive treatment, and also [@CSW_Dtopology], [@CWhomotopy], [@wu], [@watts], [@iglOrb] for the development of various specific aspects) is a simple and flexible generalization of the concept of a smooth manifold (see [@St] for a review of other similar directions). Many constructions of differential geometry also generalize, although for some of them there is not (yet) a universal agreement on the choice of the proper counterpart of such-and-such notion; this is the case of the tangent bundle, for which there are many proposed versions; the most accepted one at the moment seems to be that of the *internal tangent bundle* [@CWtangent] (see [@HeTangent] for the earlier construction on which it is partially based). Whereas for the cotangent bundle and higher-order differential forms there is a standard version, see for instance [@iglesiasBook], [@Lau], [@karshon-watts] (as well as [@HeCohomology]). Finally, see [@magnot1], [@magnot2] for a more analytic context. A certain development of other concepts of differential geometry on diffeological spaces appears in [@vincent], where (in particular) the basic concept of the diffeological counterpart of a smooth vector bundle was developed to some extent. However, the notion itself and its peculiarities with respect to the standard one were already investigated in [@iglFibre]; this is where diffeological bundles (which herein we call *pseudo-bundles*) were first introduced. The other concepts follow from there, in particular, those needed to define a Dirac operator, which was done in [@dirac]. Since diffeological versions of some classic instances of Dirac operators were not considered therein, in this paper we try to fill this void, describing the diffeological version of the most classic one of all, the De Rham operator. Our main conclusions in this respect are of two sorts. The first is that there does not appear to be any straightforward way of defining a diffeological counterpart of the classical operator $d+d^*$ in the diffeological context. This starts from the fact that the differential itself, defined on the spaces $\Omega^k(X)$, does not descend to the pseudo-bundles $\Lambda^k(X)$. Furthermore, the exterior product defined between the former vector spaces does not yield an identification between $\bigwedge^k(\Lambda^1(X))$ and $\Lambda^k(X)$, although it gives a natural, possibly surjective, map from the former to the latter. We also show that the dimensions of the fibres of $\Lambda^k(X)$ do not truly correlate with the (diffeological) dimension of the space $X$; if $\dim(X)=n$ then $\Lambda^k(X)$ are indeed trivial for $k>n$, but for $k\leqslant n$ the dimensions of fibres of $\Lambda^k(X)$ can be arbitrarily large. Finally, since $\Lambda^1(X)$ may have fibres of varying dimension, there does not seem to be a straightforward definition of the Hodge star on its exterior degrees. Diffeological spaces form a very wide category, so that a statement applying to them all would necessarily risk being too general so as to be meaningless. This issue we resolve by dedicating significant attention to the diffeological gluing procedure ([@pseudobundle]) applied to pairs of diffeological spaces, that in turn satisfy some additional assumptions. Mostly these assumptions have to do with being able to put a *pseudo-metric* on the corresponding pseudo-bundles $\Lambda^k(X)$, and with the extendability of differential forms, that we define below. Under these assumptions we do describe the behavior of pseudo-bundles $\Lambda^k(X)$ under gluing, as well as that of the De Rham groups. #### Acknowledgments Discussions with Prof. Riccardo Zucchi benefitted significantly this work. Main definitions ================ A recent and comprehensive exposition of the main notions and constructions of diffeology can be found in [@iglesiasBook]; that particularly includes the De Rham cohomology (see also [@HeCohomology]). The homological algebra is discussed in a recent [@wu]. Diffeological spaces, pseudo-bundles, and pseudo-metrics -------------------------------------------------------- The concept of a diffeological space is a natural generalization of that of a smooth manifold; briefly, the two differ in that for a diffeological space the notion of atlas is taken by that of a *diffeological structure* whose charts have domains of definition of varying dimension. Furthermore, a diffeological space is not subject to the same topological requirements, such as paracompactness etc. A **diffeological space** is any set $X$ endowed with a **diffeological structure** (or **diffeology**), which is the set ${{\mathcal D}}$ of maps, called **plots**, ${{\mathcal D}}=\{p:U\to X\}$, for all domains $U\subseteq{{\mathbb{R}}}^n$ and for all $n\in{{\mathbb{N}}}$ that satisfies the *covering* condition of every constant map being a plot, the *smooth compatibility* condition of every pre-compostion $p\circ h$ of any plot $p:U\to X$ with any ordinary smooth map $h:V\to U$, being again a plot, and the following *sheaf condition*: if $U=\cup_iU_i$ is an open cover of a domain $U$ and $p:U\to X$ is a set map such that each restriction $p|_{U_i}:U_i\to X$ is a plot (that is, it belongs to ${{\mathcal D}}$) then $p$ itself is a plot. For two diffeological spaces $X$ and $Y$, a set map $f:X\to Y$ is **smooth** if for every plot $p$ of $X$ the composition $f\circ p$ is a plot of $Y$. If the *vice versa* is always true locally, *i.e.* if, whenever the composition of $f\circ p$ with some set map $p:U\to X$ is a plot of $Y$, the map $p$ is necessarily a plot of $X$, and furthermore $f$ is surjective, then $f$ is called a **subduction**. Said in reverse, if, given two diffeological spaces $X$ and $Y$, there exists a subduction $f$ of $X$ onto $Y$, then the diffeology of $Y$ is said to be the **pushforward** of the diffeology of $X$ by the map $f$. For instance, if $X$ is a diffeological space and $\sim$ is any equivalence relation on $X$ then the **quotient diffeology** on $X/\sim$ is defined by the requirement that the quotient projection $X\to X/\sim$ be a subduction. Notice in particular that, unlike in the case of smooth manifolds, every quotient of a diffeological space is again a diffeological space. The same is true for any subset $Y\subseteq X$ of a diffeological space $X$; it is endowed with the **subset diffeology** that consists of precisely the plots of $X$ whose ranges are contained in $Y$. A smooth manifold is an instance of a diffeological space; the corresponding diffeology is given by the set of all usual smooth maps into it. Standard diffeologies are defined for disjoint unions, direct products, and spaces of smooth maps between two diffeological spaces (see [@iglesiasBook]). For a diffeological space carrying an algebraic structure there is an obvious notion of smoothness of that structure, so there are notions of a diffeological vector space, diffeological group, etc. The diffeological counterpart of a smooth vector bundle, that we call a **diffeological vector pseudo-bundle**, is defined analogously to the standard notion, with the exception that there is no requirement of there being an atlas of local trivializations. The precise definition is as follows. A **diffeological vector pseudo-bundle** is a smooth surjective map $\pi:V\to X$ between two diffeological spaces that satisfies the following requirements: 1) for every $x\in X$ the pre-image $\pi^{-1}(x)$ carries a vector space structure; 2) the induced operations of addition $V\times_X V\to V$ and scalar multiplication ${{\mathbb{R}}}\times V\to V$ are smooth for the subset diffeology on $V\times_X V\subseteq V\times V$, for the product diffeologies on $V\times V$ and ${{\mathbb{R}}}\times V$, and for the standard diffeology on ${{\mathbb{R}}}$; 3) the zero section $X\to V$ is smooth. This notion appeared in [@iglFibre], where it is called *diffeological fibre bundle*, and was considered in [@vincent] under the name of *regular vector bundle* and in [@CWtangent], where it is termed *diffeological vector space over $X$*. Some developments of the notion appear in [@pseudobundle]. For such pseudo-bundles there are suitable counterparts of all the usual operations on vector bundles, such as direct sums, tensor products, and taking dual bundles. It is worth noting that already in the case of (finite-dimensional) diffeological vector spaces the expected notion of duality leads, in general, to different conclusions, specifically the diffeological dual of a vector space may not be isomorphic to the space itself. A long-ranging consequence is that there is no proper analogue of a Riemannian metric on a diffeological vector pseudo-bundle, although there is an obvious substitute ([@pseudobundle-pseudometrics]). Let $\pi:V\to X$ be a diffeological vector pseudo-bundle such that the vector space dimension of each fibre $\pi^{-1}(x)$ is finite. A **pseudo-metric** on it is a smooth map $g:X\to V^*\otimes V^*$ such that for all $x\in X$ the bilinear form $g(x)\in(\pi^{-1}(x))^*\otimes(\pi^{-1}(x))^*$ is symmetric, positive semidefinite, and of rank equal to $\dim((\pi^{-1}(x))^*)$. The reason why this definition is stated as it is, is that in general a finite-dimensional diffeological vector space does not admit a smooth scalar product ([@iglesiasBook]). The maximal rank of a smooth symmetric bilinear form on such a space is the dimension of its diffeological dual, and there is always a smooth symmetric positive semidefinite form that achieves that rank ([@dirac], Section 5). The latter is called a **pseudo-metric** on the vector space in question, and the notion of a pseudo-metric on a pseudo-bundle is an obvious extension of that. Notice that not every finite-dimensional pseudo-bundle admits a pseudo-metric (see [@pseudobundle]). If $\pi:V\to X$ is a finite-dimensional diffeological vector pseudo-bundle and $g$ is a pseudo-metric on it then there is an obvious **pairing map** $$\Phi_g:V\to V^*\,\,\mbox{ given by }\,\,\Phi_g(v)(\cdot)=g(\pi(v))(v,\cdot).$$ This map is a subduction onto $V^*$; it is bijective and a diffeomorphism if and only if the subset diffeology on all fibres of $V$ is the standard one, while in general it has a canonically defined right inverse which, however, is not guaranteed to be smooth. The latter is also the reason why the standard construction of the dual $g^*$ via the identity $$g^*(x)(\Phi_g(v),\Phi_g(w))=g(x)(v,w)\,\,\mbox{ for all }\,\,x\in X,\,v,w\in V,$$ although it yields a well-defined family of pseudo-metrics on fibres of $V^*$, may not itself be a pseudo-metric. Diffeological gluing -------------------- The **diffeological gluing** ([@pseudobundle]) is a procedure that mimics the usual topological gluing. Let $X_1$ and $X_2$ be two diffeological spaces, and let $f:X_1\supseteq Y\to X_2$ be a map smooth for the subset diffeology on $Y$. The result of the diffeological gluing of $X_1$ to $X_2$ along $f$ is the space $$X_1\cup_f X_2:=(X_1\sqcup X_2)/_{\sim},\,\,\mbox{ where }x\sim x'\Leftrightarrow x=x'\mbox{ or }f(x)=f(x')$$ endowed with the quotient diffeology of the disjoint union diffeology on $X_1\sqcup X_2$. In practice, the plots of $X_1\cup_f X_2$ can be characterized as follows. Let us first define the **standard inductions** $i_1:X_1\setminus Y\to X_1\cup_f X_2$ and $i_2:X_2\to X_1\cup_f X_2$ given as the compositions $$i_1:X_1\setminus Y\hookrightarrow X_1\sqcup X_2\to X_1\cup_f X_2,\,\,i_2:X_2\hookrightarrow X_1\sqcup X_2\to X_1\cup_f X_2$$ of the obvious inclusions with the quotient projection. Notice that the images $i_1(X_1\setminus Y)$ and $i_2(X_2)$ form a disjoint cover of $X_1\cup_f X_2$, a property that is used to describe maps from/into $X_1\cup_f X_2$. For instance, the plots of $X_1\cup_f X_2$ can be given the following characterization. A map $p:U\to X_1\cup_f X_2$ defined on a connected domain $U$ is a plot of $X_1\cup_f X_2$ if and only if one of the following is true: either there exists a plot $p_1$ of $X_1$ such that $$p(u)=\left\{\begin{array}{ll} p_1(u) & \mbox{if }u\in p_1^{-1}(X_1\setminus Y), \\ i_2(f(p_1(u))) & \mbox{if }u\in p_1^{-1}(Y), \end{array}\right.$$ or there exists a plot $p_2$ of $X_2$ such that $$p=i_2\circ p_2.$$ The right-hand factor $X_2$ always embeds into $X_1\cup_f X_2$, while $X_1$ in general does not, unless $f$ is a diffeomorphism (which is the case we will mostly treat). If it is one then the map $$\tilde{i}_1:X_1\to X_1\cup_f X_2\,\,\mbox{ given by }\,\,\tilde{i}_1(x_1)=\left\{\begin{array}{ll} i_1(x_1) & \mbox{if }x_1\in X_1\setminus Y, \\ i_2(f(x_1)) & \mbox{if }x_1\in Y \end{array}\right.$$ is also an inclusion. Suppose now that we are given two pseudo-bundles $\pi_1:V_1\to X_1$ and $\pi_2:V_2\to X_2$, a gluing map $f:X_1\supseteq Y\to X_2$, and a smooth lift $\tilde{f}:\pi_1^{-1}(Y)\to V_2$ of $f$, that is linear on each fibre in its domain of definition. Then $\tilde{f}$ defines a gluing of $V_1$ to $V_2$ that preserves the pseudo-bundle structures, and specifically, we obtain in an obvious way a new pseudo-bundle denoted by $$\pi_1\cup_{(\tilde{f},f)}\pi_2:V_1\cup_{\tilde{f}}V_2\to X_1\cup_f X_2.$$ The standard inductions $V_1\setminus\pi_1^{-1}(Y)\to V_1\cup_{\tilde{f}}V_2$ and $V_2\to V_1\cup_{\tilde{f}}V_2$ are denoted by $j_1$ and $j_2$ respectively. The gluing of pseudo-bundles is well-behaved with respect to the operations of direct sum and tensor product, while for dual pseudo-bundles its behavior is more complicated, unless both $\tilde{f}$ and $f$ are diffeomorphisms (see [@pseudobundle] and [@pseudobundle-pseudometrics]). Certain pairs of pseudo-metrics on $V_1$ and $V_2$ allow to obtain a pseudo-metric on $V_1\cup_{\tilde{f}}V_2$. Let $\pi_1:V_1\to X_1$ and $\pi_2:V_2\to X_2$ be two finite-dimensional diffeological vector pseudo-bundles, let $(\tilde{f},f)$ be a gluing between them, and let $g_1$ and $g_2$ be pseudo-metrics on $V_1$ and $V_2$ respectively. The pseudo-metrics $g_1$ and $g_2$ are said to be **compatible** (with the gluing along $(\tilde{f},f)$) if for all $y\in Y$ and for all $v_1,w_1\in\pi_1^{-1}(y)$ we have $$g_1(y)(v_1,w_1)=g_2(f(y))(\tilde{f}(v_1),\tilde{f}(w_1)).$$ If $g_1$ and $g_2$ are compatible then the **induced pseudo-metric $\tilde{g}$ on $V_1\cup_{\tilde{f}}V_2$** is defined by $$\tilde{g}(x)(v,w)=\left\{\begin{array}{ll} g_1(i_1^{-1}(x))(j_1^{-1}(v),j_1^{-1}(w)) & \mbox{if }x\in i_1(X_1\setminus Y), \\ g_2(i_2^{-1}(x))(j_2^{-1}(v),j_2^{-1}(w)) & \mbox{if }x\in i_2(X_2)\end{array}\right.$$ for all $x\in X_1\cup_f X_2$ and for all $v,w\in(\pi_1\cup_{(\tilde{f},f)}\pi_2)^{-1}(x)$. See [@pseudobundle-pseudometrics] for details. Differential forms, diffeological connections, and Levi-Civita connections -------------------------------------------------------------------------- The notion of a diffeological differential form is a rather well-developed one by now, see [@iglesiasBook]; it is defined as a collection of usual differential forms satisfying a certain compatibility condition. Namely, let $X$ be a diffeological space, and let ${{\mathcal D}}$ be its diffeology. A **diffeological differential $k$-form on $X$** is a collection $\omega=\{\omega(p)\}_{p\in{{\mathcal D}}}$, where $p:U\to X$ with $U\subseteq{{\mathbb{R}}}^n$ a domain and $\omega(p)\in C^{\infty}(U,\Lambda^k({{\mathbb{R}}}^n))$, such that for any ordinary smooth map $F:V\to U$ defined on another domain $V$ and with values in $U$ we have that $\omega(p\circ F)=F^*(\omega(p))$. The collection of all such forms for a fixed $k$, denoted by $\Omega^k(X)$, is a real vector space and is endowed with the diffeology given by the following condition: a map $q:V\to\Omega^k(X)$ is a plot of $\Omega^k(X)$ is a plot of $\Omega^k(X)$ if and only if for every plot $p:{{\mathbb{R}}}^n\subseteq U\to X$ the map $$V\times U\ni(v,u)\mapsto q(v)(p(u))\in\Lambda^k({{\mathbb{R}}}^n)$$ is smooth in the usual sense. A specific example of a diffeological differential form on $X$ is the **differential** of a smooth function $h:X\to{{\mathbb{R}}}$, where ${{\mathbb{R}}}$ is considered with the standard diffeology. The differential $dh$ is defined by setting $$dh(p)=d(h\circ p)\,\,\mbox{ for every plot }p:U\to X,$$ where $d(h\circ p)$ is the usual differential of an ordinary smooth function $U\to{{\mathbb{R}}}$. Checking that $dh$ is well-defined as an element of $\Omega^1(X)$ is trivial. The definition of $\Omega^k(X)$ then extends to that of the **pseudo-bundle $\Lambda^k(X)$ of $k$-forms over $X$** (termed the *bundle of values of $k$-forms* on $X$ in [@iglesiasBook]), in the following way. We first define, for every $x\in X$, the space $\Omega_x^k(X)$ of **$k$-forms on $X$ vanishing at $x$**. A form $\omega\in\Omega^k(X)$ **vanishes at $x$** if for every plot $p:U\to X$ of $X$ such that $U\ni 0$ and $p(0)=x$ we have that $\omega(p)(0)=0$, the zero form; the set of all such $k$-forms is the subspace $\Omega_x^k(X)$, which is indeed a vector subspace of $\Omega^k(X)$ and is endowed with the subset diffeology. Consider next the trivial pseudo-bundle $X\times\Omega^k(X)$ over $X$. The union $\bigcup_{x\in X}\{x\}\times\Omega_x^k(X)$ is a sub-bundle of $X\times\Omega^k(X)$ in the sense of diffeological vector pseudo-bundles, so the corresponding quotient pseudo-bundle is again a diffeological vector pseudo-bundle; $\Lambda^k(X)$ is precisely this pseudo-bundle: $$\Lambda^k(X):=(X\times\Omega^k(X))/\left(\bigcup_{x\in X}\{x\}\times\Omega_x^k(X)\right).$$ The quotient projection is denoted by $\pi^{\Omega^k,\Lambda^k}$. In particular, if $k=1$ the pseudo-bundle $\Lambda^1(X)$ acts as a substitute of the usual cotangent bundle. Indeed, if $X$ is a smooth manifold considered as a diffeological space for the standard diffeology of a smooth manifold (see above), $\Lambda^1(X)$ coincides naturally with the cotangent bundle $T^1(X)$. Thus, a **diffeological connection on a pseudo-bundle $\pi:V\to X$** is defined as an operator $$\nabla:C^{\infty}(X,V)\to C^{\infty}(X,\Lambda^1(X)\otimes V),$$ satisfying then the usual properties of linearity and the Leibnitz rule. Let $\pi:V\to X$ be a diffeological vector pseudo-bundle. A **diffeological connection on $V$** is a smooth linear operator $$\nabla:C^{\infty}(X,V)\to C^{\infty}(X,\Lambda^1(X)\otimes V)$$ such that for all $h\in C^{X,{{\mathbb{R}}}}$ and for all $s\in C^{\infty}(X,V)$ we have $$\nabla(hs)=dh\otimes s+h\nabla s,$$ where $dh\in\Lambda^1(X)$ is defined by $dh(x)=\pi^{\Omega,\Lambda}(x,dh)$, where $dh$ on the right-hand side is the already-defined differential $dh\in\Omega^1(X)$. A particular instance of a diffeological connection is the **Levi-Civita connection** on $\Lambda^1(X)$ endowed with a pseudo-metric $g^{\Lambda}$. Two assumptions are implicit in this notion: that $X$ is such that $\Lambda^1(X)$ has finite-dimensional fibres, and that $\Lambda^1(X)$ admits a pseudo-metric. If it does then the following definition ([@connectionsLC]) is well-posed. Let $X$ be a diffeological space such that $\Lambda^1(X)$ admits pseudo-metrics, and let $g^{\Lambda}$ be a pseudo-metric on $\Lambda^1(X)$. A **Levi-Civita connection** on $(\Lambda^1(X),g^{\Lambda})$ is a connection $\nabla$ on $\Lambda^1(X)$ which satisfies the usual two conditions. Specifically, $\nabla$ is **compatible with the pseudo-metric $g^{\Lambda}$**, that is, for any two sections $s,t\in C^{\infty}(X,\Lambda^1(X))$ $$d(g^{\Lambda}(s,t))=g^{\Lambda}(\nabla s,t)+g^{\Lambda}(s,\nabla t),$$ where on the left we have the differential of $g^{\Lambda}(s,t)\in C^{\infty}(X,{{\mathbb{R}}})$ that is an element of $C^{\infty}(X,\Lambda^1(X))$ and $g^{\Lambda}$ is extended to sections of $\Lambda^1(X)\otimes\Lambda^1(X)$ by setting $g^{\Lambda}(\alpha\otimes s,t):=\alpha\cdot g^{\Lambda}(s,t)=g^{\Lambda}(s,\alpha\otimes t)$ for any $\alpha\in C^{\infty}(X,\Lambda^1(X))$. Second, $\nabla$ is **symmetric**, that is, for any $s,t\in C^{\infty}(X,\Lambda^1(X))$ we have $$\nabla_s t-\nabla_t s=[s,t],$$ where $\nabla_s t$ is the **covariant derivative of $t$ along $s$** and $[s,t]$ is the **Lie bracket** of $s$ and $t$, both of which are defined via the pairing map $\Phi_{g^{\Lambda}}$ corresponding to the pseudo-metric $g^{\Lambda}$. The (very few, this is a straightforward extension of the standard notion) details concerning the definitions of covariant derivatives and the Lie bracket can be found in [@dirac], Sections 10.2 and 11.1. It is not quite clear when $X$ admits a Levi-Civita connection, but if it does, it is unique. The pseudo-bundles of differential forms are rather well-behaved with respect to the gluing, provided that certain *extendibility conditions* are satisfied (see [@dirac], Section 8.1, for the case of $k=1$), and as a consequence, the same is true for diffeological connections and the Levi-Civita connections. Specifically, given two connections $\nabla^1$ and $\nabla^2$ on pseudo-bundles $\pi_1:V_1\to X_1$ and $\pi_2:V_2\to X_2$ yield a well-defined connection on $\pi_1\cup_{(\tilde{f},f)}\pi_2:V_1\cup_{\tilde{f}}V_2\to X_1\cup_f X_2$, as long as $\nabla^1$ and $\nabla^2$ satisfy a certain compatibility condition with respect to the gluing along $(\tilde{f},f)$, and $\Lambda^1(X_1)$ and $\Lambda^1(X_2)$ satisfy (one of) the already-mentioned extendibility conditions relative to $f$. Furthermore, if $\Lambda^1(X_1)$ and $\Lambda^1(X_2)$ satisfy the extendibility condition and are endowed each with a connection then under a certain additional condition (this is also called a compatibility condition, but it is a different one from that in the case of $V_1\cup_{\tilde{f}}V_2$, see [@dirac], Section 11.4.1; compare with [@dirac], Section 10.3.1) two connections on $\Lambda^1(X_1)$ and $\Lambda^1(X_2)$ yield a well-defined connection $\nabla^{\Lambda}$ on $\Lambda^1(X_1\cup_f X_2)$. Moreover, if $\Lambda^1(X_1)$ and $\Lambda^1(X_2)$ are endowed with pseudo-metrics $g_1^{\Lambda}$ and $g_2^{\Lambda}$ well-behaved (see [@dirac], Section 8.4.2, for definition) with respect to $f$, and the initial connections on them are the Levi-Civita connections then $\nabla^{\Lambda}$ is the Levi-Civita connection on $\Lambda^1(X_1\cup_f X_2)$ endowed with a certain induced pseudo-metric $g^{\Lambda}$ ([@dirac], Section 8.4.3). Pseudo-bundles of Clifford modules, diffeological Clifford connections, and Dirac operators ------------------------------------------------------------------------------------------- As we have mentioned already, the operations of direct sums, tensor products, and quotienting are defined also for diffeological vector pseudo-bundles; this in particular allows to obtain a well-defined pseudo-bundle $\pi^{{C \kern -0.1em \ell}}:{C \kern -0.1em \ell}(V,g)\to X$ of Clifford algebras starting from a given pseudo-bundle $\pi:V\to X$ endowed with a pseudo-metric $g$. Each fibre of ${C \kern -0.1em \ell}(V,g)$ is the Clifford algebra ${C \kern -0.1em \ell}(\pi^{-1}(x),g(x))$. It then makes sense to speak of another pseudo-bundle $\chi:E\to X$ over the same $X$ being a **pseudo-bundle of Clifford modules** over ${C \kern -0.1em \ell}(V,g)$, in the sense that each fibre $\chi^{-1}(x)$ is a Clifford module over ${C \kern -0.1em \ell}(\pi^{-1}(x),g(x))$ with some Clifford action $c(x)$. For $E$ to be a pseudo-bundle of Clifford modules, it suffices to add the requirement that the total action $c$ be smooth. This condition of smoothness can be stated as follows: for every plot $q:U'\to{C \kern -0.1em \ell}(V,g)$ and for every plot $p:U\to E$ the map $$U'\times U\supseteq\{(u',u)\,|\,\pi^{cl}(q(u'))=\chi(p(u))\}\mapsto c(q(u'))(p(u))\in E$$ is smooth for the subset diffeology on its domain of definition. Given then a diffeological space $X$ such that $\Lambda^1(X)$ admits a pseudo-metric $g^{\Lambda}$ such that there exists the Levi-Civita connection $\nabla^{\Lambda}$ on $(\Lambda^1(X),g^{\Lambda})$, and given a pseudo-bundle of Clifford modules $\chi:E\to X$ over ${C \kern -0.1em \ell}(\Lambda^1(X),g^{\Lambda})$ with Clifford action $c$, the notion of a Clifford connection on $E$ is well-defined (although its existence is not guaranteed). A connection $\nabla$ on $E$ is a **Clifford connection** if for every $s,t\in C^{\infty}(X,\Lambda^1(X))$ and for every $r\in C^{\infty}(X,E)$ we have $$\nabla_t(c(s)r)=c(\nabla^{\Lambda}_t s)(r)+c(s)(\nabla_t r).$$ This is quite the same as the standard notion, just using the diffeological counterparts of all components. Then the composition $c\circ\nabla$ of a given Clifford action with the given Clifford connection is, as usual, a **Dirac operator** on $E$. Let $X$ be a diffeological space such that $\Lambda^1(X)$ admits a pseudo-metric $g^{\Lambda}$ and there exists a Levi-Civita connection on $(\Lambda^1(X),g^{\Lambda})$. Let $\chi:E\to X$ be a pseudo-bundle of Clifford modules over ${C \kern -0.1em \ell}(\Lambda^1(X),g^{\Lambda})$ with Clifford action $c$, and let $\nabla$ be a Clifford connection on $E$. Associated to the data $(X,g^{\Lambda},E,c,\nabla)$ is the **Dirac operator** $D:C^{\infty}(X,E)\to C^{\infty}(X,E)$ given by $D=c\circ\nabla$. All these constructions are well-behaved with respect to gluing, provided that all gluing maps are diffeological diffeomorphisms, and that certain compatibility and extendibility conditions are met. Specifically, given two pseudo-bundles $\chi_1:E_1\to X_1$ and $\chi_2:E_2\to X_2$ of Clifford modules over $(\Lambda^1(X_1),g_1^{\Lambda})$ and $(\Lambda^1(X_2),g_2^{\Lambda})$ with Clifford actions $c_1$ and $c_2$, that are endowed with Clifford connections $\nabla^1$ (on $E_1$) and $\nabla^2$ (on $E_2$), and given a gluing of $E_1$ to $E_2$, along a pair of diffeomorphisms $f:X_1\supseteq Y\to X_2$ and $\tilde{f}':E_1\supseteq\chi_1^{-1}(Y)\to E_2$, we need the following conditions for there being a well-defined Dirac operator on the result of gluing: 1. The map $f$ is such that the following two diffeologies on $\Omega^1(Y)$ coincide: the pushforward ${{\mathcal D}}_1^{\Omega}$ of the standard diffeology on $\Omega^1(X_1)$ by the pullback map $i^*:\Omega^1(X_1)\to\Omega^1(Y)$, where $i:Y\hookrightarrow X_1$ is the natural inclusion, and the pushforward ${{\mathcal D}}_2^{\Omega}$ of the standard diffeology on $\Omega^1(X_2)$ by the pullback map $j^*:\Omega^1(X_2)\to\Omega^1(f(Y))$, where $j:f(Y)\hookrightarrow X_2$ is also the natural inclusion. The equality ${{\mathcal D}}_1^{\Omega}={{\mathcal D}}_2^{\Omega}$ is what we previously called the extendibility condition, and it ensures that $\Lambda^1(X_1\cup_f X_2)$ admits a particularly simple description in terms of $\Lambda^1(X_1)$ and $\Lambda^1(X_2)$ (it is possible to give a description without the extendibility condition, but it is far more cumbersome). See [@dirac], Section 8, for details; 2. The pseudo-metrics $g_1^{\Lambda}$ and $g_2^{\Lambda}$ are **compatible** with the gluing along $f$, that is, for every $y\in Y$ and for every pair $\alpha_1\in\Lambda_y^1(X_1)$, $\alpha_2\in\Lambda_{f(y)}^1(X_2)$ such that $i_{\Lambda}^*(\alpha_1)=(f_{\Lambda}^*j_{\Lambda}^*)(\alpha_2)$ (we say that $\alpha_1$ and $\alpha_2$ are **compatible**), where $i_{\Lambda}^*:\Lambda^1(X_1)\supseteq(\pi_1^{\Lambda})^{-1}(Y)\to\Lambda^1(Y)$, $f_{\Lambda}^*:\Lambda^1(f(Y))\to\Lambda^1(Y)$, $j_{\Lambda}^*:\Lambda^1(X_2)\supseteq(\pi_2^{\Lambda})^{-1}(f(Y))\to\Lambda^1(f(Y))$ are induced by the pullback maps $i^*$, $f^*$, and $j^*$, we have that $$g_1^{\Lambda}(y)(\alpha_1,\alpha_1)=g_2^{\Lambda}(f(y))(\alpha_2,\alpha_2);$$ 3. The actions $c_1$ and $c_2$ are compatible with $(\tilde{f}',f)$, specifically, for every $y\in Y$, for every compatible pair $\alpha_1\in\Lambda_y^1(X_1)$, $\alpha_2\in\Lambda_{f(y)}^1(X_2)$, and for every $e_1\in\chi_1^{-1}(y)$ we have that $$c_2(\alpha_2)(\tilde{f}'(e_1))=\tilde{f}'(c_1(\alpha_1)(e_1));$$ 4. The pseudo-bundles $(\Lambda^1(X_1),g_1^{\Lambda})$ and $(\Lambda^1(X_2),g_2^{\Lambda})$ admit Levi-Civita connections $\nabla^{\Lambda,1}$ and $\nabla^{\Lambda,2}$, and these connections are **compatible** in the following sense: for all $t_1\in C^{\infty}(X_1,\Lambda^1(X_1))$ and $t_2\in C^{\infty}(X_2,\Lambda^1(X_2))$ such that for all $\in Y$ we have that $i_{\Lambda}^*(s_1(y))=(f_{\Lambda}^*j_{\Lambda}^*)(s_2(f(y)))$ (these are **compatible** sections of $\Lambda^1(X_1)$ and $\Lambda^1(X_2)$), the following equality holds at every point $yin Y$: $$(i_{\Lambda}^*\otimes i_{\Lambda}^*)((\nabla^{\Lambda,1}s_1)(y))=((f_{\Lambda}^*j_{\Lambda}^*)\otimes(f_{\Lambda}^*j_{\Lambda}^*))((\nabla^{\Lambda,2}s_2)(f(y)));$$ 5. The connections $\nabla^1$ and $\nabla^2$ are **compatible** with the gluing along $(\tilde{f}',f)$, which means the following: for every pair $r_1\in C^{\infty}(X_1,E_1)$, $r_2\in C^{\infty}(X_2,E_2)$ such that for every $y\in Y$ we have $\tilde{f}'(s_1(y))=s_2(f(y))$, and for all $y\in Y$ there is the equality $$(i_{\Lambda}^*\otimes\tilde{f}')((\nabla^1s_1)(y))=((f_{\Lambda}^*j_{\Lambda}^*)\otimes\mbox{Id}_{E_2})((\nabla^2s_2)(f(y))).$$ The conditions just listed provide us with the following: 1. Conditions 1 and 2 yield an **induced pseudo-metric** $g^{\Lambda}$ on $\Lambda^1(X_1\cup_f X_2)$; 2. Condition 3 yields the Levi-Civita connection $\nabla^{\Lambda}$ on $(\Lambda^1(X_1\cup_f X_2),g^{\Lambda})$; 3. Condition 4 provides an **induced Clifford action** $\tilde{c}$ of ${C \kern -0.1em \ell}(\Lambda^1(X_1\cup_f X_2),g^{\Lambda})$ on $E_1\cup_{\tilde{f}'}E_2$; 4. Condition 5 ensures that there is the induced connection $\nabla^{\cup}$ on $E_1\cup_{\tilde{f}'}E_2$, and that it is a Clifford connection. *([@dirac], Proposition 13.3)* Let $D_1$ and $D_2$ be the Dirac operators associated to the data $(X_1,g_1^{\Lambda},E_1,c_1,\nabla^1)$ and $(X_2,g_2^{\Lambda},E_2,c_2,\nabla^2)$ respectively, and suppose that these data and the gluing pair $(\tilde{f}',f)$ satisfy Conditions 1-5 above. The the Dirac operator $D$ associated to the data $(X_1\cup_f X_2,g^{\Lambda},E_1\cup_{\tilde{f}'}E_2,\tilde{c},\nabla^{\cup})$ satisfies the following: for every $s\in C^{\infty}(X_1\cup_f X_2,E_1\cup_{\tilde{f}'}E_2)$ we have that $$D(s)=D_1(s_1)\cup_{(f,\tilde{f}')}D_2(s_2),$$ where $s_1:=\tilde{j}_1^{-1}\circ s\circ\tilde{i}_1\in C^{\infty}(X_1,E_1)$ and $s_2:=j_2^{-1}\circ s\circ i_2\in C^{\infty}(X_2,E_2)$. The map $\tilde{j}_1$ above is the natural inclusion $E_1\hookrightarrow E_1\cup_{\tilde{f}'}E_2$, the analogue of the inclusion $\tilde{i}_1$ (recall that also $\tilde{f}'$ is assumed to be a diffeomorphism), and the sign $\cup_{(f,\tilde{f}')}$ refers to the gluing of the maps $D_1(s_1):X_1\to E_1$ and $D_2(s_2):X_2\to E_2$ along $(f,\tilde{f}')$; see [@dirac], Section 6.3, for details. Diffeological De Rham cohomology -------------------------------- There is an established notion of the De Rham cohomology for diffeological spaces; a complete exposition can be found in [@iglesiasBook], Section 6.73. The construction mimics the standard one and is as follows. Let $X$ be a diffeological space. The already-defined differential of a smooth function $X\to{{\mathbb{R}}}$ provides us with the **coboundary operator** $$d:\Omega^k(X)\to\Omega^{k+1}(X)\,\,\mbox{ for }\,k\geqslant 0,$$ defined by $d\omega(p)=d(\omega(p))$ for any plot $p$ of $X$. This is well-defined and satisfies the coboundary condition $d\circ d=0$, see [@iglesiasBook]. Define, as usual, the space of **$k$-cocycles** to be $$Z_{dR}^k(X):=\mbox{ker}(d:\Omega^k(X)\to\Omega^{k+1}(X)),$$ and let $$B_{dR}^k(X):= d(\Omega^{k-1}(X))\subseteq Z_{dR}^k(X)\,\,\mbox{ for }k\geqslant 1,\,\,\mbox{ and }\,B_{dR}^0(X)=\{0\}$$ be the space of **$k$-coboundaries**. In particular, every $Z_{dR}^k(X)$ is equipped with the subset diffeology relative to the standard diffeology on the corresponding $\Omega^k(X)$. The **de Rham cohomology groups** are then defined as quotients $$H_{dR}^k(X):=Z_{dR}^k(X)/B_{dR}^k(X).$$ They are equipped with the quotient diffeology, with respect to which they become diffeological vector spaces. The pseudo-bundles $\Lambda^k(X_1\cup_f X_2)$, and the groups $H_{dR}^k(X_1\cup_f X_2)$ ======================================================================================= In this section we consider the behavior of $\Lambda^k(X)$ and $H_{dR}^k(X)$ under gluing. The common prerequisite for considering this is to describe first the behavior of the spaces $\Omega^k(X)$ with respect to gluing (as has already been done for $k=1$, [@dirac], Section 8). The vector spaces $\Omega^k(X_1\cup_f X_2)$ ------------------------------------------- As in the case of $k=1$, the spaces $\Omega^k(X_1\cup_f X_2)$ are subspaces of the direct sum $\Omega^k(X_1)\oplus\Omega^k(X_2)$. They can be described as the images of the pullback map $$\pi^*:\Omega^k(X_1\cup_f X_2)\to\Omega^k(X_1\sqcup X_2)\cong\Omega^k(X_1)\oplus\Omega^k(X_2),$$ where $\pi:X_1\sqcup X_2\to X_1\cup_f X_2$ is the quotient projection that defines $X_1\cup_f X_2$, and also given an explicit description in terms of an appropriate compatibility notion. Doing so does not require any additional assumptions on $f$, which appear when we want to establish the surjectivity of the images of the direct sum projections $\pi_1^{\Omega}:\Omega^k(X_1\cup_f X_2)\to\Omega^k(X_1)$ and $\pi_2^{\Omega}:\Omega^k(X_1\cup_f X_2)\to\Omega^k(X_2)$. ### The diffeomorphism $\Omega^k(X_1\sqcup X_2)\cong\Omega^k(X_1)\oplus\Omega^k(X_2)$ The existence (and the construction) of this diffeomorphism is essentially obvious from the definitions. Let $\hat{i}_1:X_1\hookrightarrow X_1\sqcup X_2$ and $\hat{i}_2:X_2\hookrightarrow X_1\sqcup X_2$ be the obvious inclusions. \[omega:of:disjoint:union:splits:as:direct:sum:thm\] For any two diffeological spaces $X_1$ and $X_2$ and for any $k\geqslant 0$ the map $$\hat{i}_1^*\oplus\hat{i}_2^*:\Omega^k(X_1\sqcup X_2)\to\Omega^k(X_1)\oplus\Omega^k(X_2)$$ acting by $(\hat{i}_1^*\oplus\hat{i}_2^*)(\omega)=\hat{i}_1^*(\omega)\oplus\hat{i}_2^*(\omega)$ is a linear diffeomorphism. It suffices to show that $\hat{i}_1^*\oplus\hat{i}_2^*$ has a smooth linear inverse. This inverse is given by assigning to each pair $\omega_1\oplus\omega_2$, where $\omega_1\in\Omega^k(X_1)$ and $\omega_2\in\Omega^k(X_2)$, the form $\omega$ that is defined as follows. Let $p:U\to X_1\sqcup X_2$ be a plot; then there exists a decomposition $U=U_1\sqcup U_2$ of the domain $U$ as a disjoint union of two domains $U_1$ and $U_2$ such that $p_1:=\hat{i}_1^{-1}\circ p|_{U_1}$ and $p_2:=\hat{i}_2^{-1}\circ p|_{U_2}$. We define $$\omega(p)=(\omega_1(p_1),\omega_2(p_2)),$$ the latter pair being naturally seen as a usual differential $k$-form on the disjoint union $U_1\sqcup U_2=U$. That such assignment defines the inverse of $\hat{i}_1^*\oplus\hat{i}_2^*$, and that this inverse is smooth and linear, is immediate from the construction. ### The subspace $\Omega_f^k(X_1)$ of $f$-invariant $k$-forms Let $f:X_1\supseteq Y\to X_2$. In general, the $k$-forms on $X_1$ which can be carried forward to the glued space $X_1\cup_f X_2$ must satisfy a certain additional condition. Two plots $p_1:U\to X_1$ and $p_1':U'\to X_1$ are called **$f$-equivalent** if $U=U'$, and for every $u\in U$ such that $p_1(u)\neq p_1'(u)$ we have that $p_1(u),p_1'(u)\in Y$ and $f(p_1(u))=f(p_1'(u))$. A $k$-form $\omega_1\in\Omega^k(X_1)$ is said to be **$f$-invariant** if for any two $f$-equivalent plots $p_1$ and $p_1'$ of $X_1$ we have that $$\omega_1(p_1)=\omega_1(p_1').$$ The set of all $f$-invariant $k$-forms on $X_1$ is denoted by $\Omega_f^k(X_1)$. It is trivial to establish the following statement (whose proof we therefore omit). For every diffeological space $X_1$ and for every smooth map $f$ defined on a subset of $X_1$ the set $\Omega_f^k(X_1)$ is a vector subspace of $\Omega^k(X_1)$. ### The inverse of the pullback map $\pi^*$ Using the diffeomorphism of Theorem \[omega:of:disjoint:union:splits:as:direct:sum:thm\], we can now describe the inverse of the ($k$th) pullback map $\pi^*$ as a map on the subspace of $\Omega^k(X_1)\oplus\Omega^k(X_2)$ determined by the following condition. \[compatible:forms:defn\] Let $X_1$ and $X_2$ be two diffeological spaces, let $f:X_1\supseteq Y\to X_2$ be a smooth map, and let $k\geqslant 0$. Two forms $\omega_1\in\Omega^k(X_1)$ and $\omega_2\in\Omega^k(X_2)$ are said to be **compatible** if for every plot $p_1$ of the subset diffeology on $Y$ we have $$\omega_1(p_1)=\omega_2(f\circ p_1).$$ We denote by $$\Omega^k(X_1)\oplus_{comp}\Omega^k(X_2)=\{\omega_1\oplus\omega_2\,|\,\omega_1\mbox{ and }\omega_2\mbox{ are compatible}\}$$ the subset in $\Omega^k(X_1)\oplus\Omega^k(X_2)$ that consists of all pairs of compatible forms. We define next the map $$\mathcal{L}^k:\Omega_f^k(X_1)\oplus_{comp}\Omega^k(X_2)\to\Omega^k(X_1\cup_f X_2)$$ given by setting, for every plot $p:U\to X_1\cup_f X_2$ defined on a connected $U$, $$\mathcal{L}^k(\omega_1\oplus\omega_2)(p)=\left\{\begin{array}{ll} \omega_1(p_1) & \mbox{if }p=\hat{i}_1\circ p_1\mbox{ for some plot }p_1\mbox{ of }X_1,\\ \omega_2(p_2) & \mbox{if }p=i_2\circ p_2\mbox{ for some plot }p_2\mbox{ of }X_2. \end{array}\right.$$ For any two diffeological spaces $X_1$ and $X_2$ and for every smooth map $f:X_1\supseteq Y\to X_2$ the map $\mathcal{L}^k$ is well-defined. We need to show that $\mathcal{L}^k(\omega_1\oplus\omega_2)(p)$ does not depend on the choice of the lift of $p$ to a plot $p_i$ of $X_i$, and that the assignment $p\mapsto\mathcal{L}^k(\omega_1\oplus\omega_2)(p)$ satisfies the smooth compatibility condition. The former of these claims is obvious if $p$ lifts to a plot of $X_2$; indeed, since $i_2$ is injective, such a lift is unique. Let $p_1$ and $p_1'$ be two lifts of $p$ to some plots of $X_1$. Then they are obviously $f$-equivalent. Since $\omega_1$ is $f$-invariant by assumption, we have that $\omega_1(p_1)=\omega_1(p_1')$, which implies that $\mathcal{L}^k(\omega_1\oplus\omega_2)(p)$ is well-defined. Let us now show that $\mathcal{L}^k(\omega_1\oplus\omega_2)$ satisfies a smooth compatibility condition. Let $h:U'\to U$ be an ordinary smooth map; then either $\mathcal{L}^k(\omega_1\oplus\omega_2)(p\circ h)=\omega_1(p_1\circ h)=h^*(\omega_1(p_1))=h^*(\mathcal{L}^k(\omega_1\oplus\omega_2)(p))$ or $\mathcal{L}^k(\omega_1\oplus\omega_2)(p\circ h)=\omega_2(p_2\circ h)=h^*(\omega_2(p_2))=h^*(\mathcal{L}^k(\omega_1\oplus\omega_2)(p))$, and we deduce the smooth compatibility condition for $\mathcal{L}^k(\omega_1\oplus\omega_2)$ from those for $\omega_1$ and $\omega_2$ respectively. The map $\mathcal{L}^k$ is therefore well-defined, and it is quite obvious that it is linear. \[cal:L:is:inverse:of:pullback:pi:thm\] The map $\mathcal{L}^k$ is a smooth inverse of the pullback map $\pi^*:\Omega^k(X_1\cup_f X_2)\to\Omega^k(X_1\sqcup X_2)\cong\Omega^k(X_1)\oplus\Omega^k(X_2)$. Let $\omega_1\oplus\omega_2\in\mbox{Range}(\pi^*)$, and let us show that $\omega_1$ and $\omega_2$ are compatible, and that $\omega_1$ is $f$-invariant. Let $p_1$ be a plot for the subset diffeology on $Y$; it is thus a plot of $X_1$, and $f\circ p_1$ is a plot of $X_2$. To both of them there corresponds a plot $p$ of $X_1\cup_f X_2$ given by $p=i_2\circ f\circ p_1=\hat{i}_1\circ p_1$. Since $\omega_1\oplus\omega_2$ is in the range of $\pi^*$, it is the image $\pi^*\omega$ of some $\omega\in\Omega^k(X_1\cup_f X_2)$. The forms $\omega_1$ and $\omega_2$ are given by $$\omega_1(p_1)=\omega(\hat{i}_1\circ p_1)\,\,\mbox{ and }\,\,\omega_2(p_2)=\omega(i_2\circ p_2)$$ respectively (for any arbitrary plots $p_1$ of $X_1$ and $p_2$ of $X_2$. Thus, in the present case we have $$\omega_1(p_1)=\omega(\hat{i}_1\circ p_1)=\omega(p)=\omega(i_2\circ f\circ p_1)=\omega_2(f\circ p_1),$$ which implies the compatibility of $\omega_1$ and $\omega_2$. Suppose now that $p_1$ and $p_1'$ are two $f$-equivalent plots. Then obviously $\hat{i}_1\circ p_1=\hat{i}_1\circ p_1'$, therefore we have $$\omega_1(p_1)=\omega(\hat{i}_1\circ p_1)=\omega(\hat{i}_1\circ p_1')=\omega_1(p_1'),$$ that is, $\omega_1$ is $f$-invariant. In particular, we conclude that the two compositions $\mathcal{L}^k\circ\pi^*$ and $\pi^*\circ\mathcal{L}^k$ are always defined. That they are inverses of each other, is obvious from the construction of $\mathcal{L}^k$. It remains to check that $\mathcal{L}^k$ is smooth. Let $q$ be a plot of $\Omega_f^k(X_1)\oplus_{comp}\Omega^k(X_2)$, and let $U$ be its domain of definition. Then for all $u\in U$ we have that $q(u)=q(u)_1\oplus q(u)_2$ for some $q(u)_1\in\Omega_f^k(X_1)$ and $q(u)_2\in\Omega^k(X_2)$, and the assignments $u\mapsto q(u)_1$ and $u\mapsto q(u)_2$ are plots of $\Omega_f^k(X_1)$ and of $\Omega^k(X_2)$ respectively. To show that $u\mapsto\mathcal{L}^k(q(u)_1\oplus q(u)_2)$ is a plot of $\Omega^k(X_1\cup_f X_2)$, as is required for showing the smoothness of $\mathcal{L}^k$, we need to consider a plot $p:U'\to X_1\cup_f X_2$ and show that the evaluation map $(u,u')\mapsto\mathcal{L}^k(q(u)_1\oplus q(u)_2)(p)(u')$ is a usual smooth section of $\Lambda^k(U\times U')$. It suffices to assume that $U'$ is connected; then $p$ lifts to either a plot $p_1$ of $X_1$ or to a plot $p_2$ of $X_2$. Depending on these two cases, the evaluation map for $q$ either has form $(u,u')\mapsto q(u)_1(p_1)(u')$ or $(u,u')\mapsto q(u)_2(p_2)(u')$, which in both cases is a smooth section of $\Lambda^k(U\times U')$, because $q(u)_1,q(u)_2$ are plots, whence the claim. Theorem \[cal:L:is:inverse:of:pullback:pi:thm\] trivially implies the following. \[omega:of:glued:cor\] The map $\pi^*$ is a diffeomorphism $\Omega^k(X_1\cup_f X_2)\to\Omega_f^k(X_1)\oplus_{comp}\Omega^k(X_2)$. The differential and gluing --------------------------- We shall consider next the behavior of the differential (the coboundary) $d$ operator under gluing. Let $X_1$ and $X_2$ be two diffeological spaces, and let $f:X_1\supseteq Y\to X_2$ be a smooth map. For every $\omega\in\Omega^k(X_1\cup_f X_2)$ the differential $d\omega\in\Omega^{k+1}(X_1\cup_f X_2)$ is determined by the collection of the usual differentials of standard $k$-forms $\omega(p)$ for all plots $p$ of $X_1\cup_f X_2$. Now, we have just seen that $\omega$ is essentially the union (or the wedge) of a $k$-form on $X_1$ with a $k$-form on $X_2$, and every plot $p$ of $X_1\cup_f X_2$ is in some sense a union of a plot of $X_1$ with a plot of $X_2$ (one of which could be absent if the domain of definition of $p$ is connected), see [@pseudobundle] and Lemma 4.1 in [@pseudobundle-pseudometrics]. The following therefore is an expected statement. \[differential:and:gluing:thm\] Let $X_1$ and $X_2$ be two diffeological spaces, let $f:X_1\supseteq Y\to X_2$ be a smooth map, and let $\omega\in\Omega^k(X_1\cup_f X_2)$ be a $k$-form. Let $\pi^*(\omega)=\omega_1\oplus\omega_2$. Then $$\pi^*(d\omega)=d\omega_1\oplus d\omega_2.$$ Let $p:U\to X_1\sqcup X_2$ be a plot of $X_1\sqcup X_2$. We need to compare $\pi^*(d\omega)(p)$ with $(d\omega_1\oplus d\omega_2)(p)$. It suffices to assume that $U$ is connected; then $p$ essentially coincides with either a plot $p_1$ of $X_1$ or a plot $p_2$ of $X_2$. Suppose it coincides with $p_1$. Then by construction and definition $$\pi^*(d\omega)(p)=d\omega(\pi\circ p)=d(\omega(\pi\circ p))=d(\omega_1(p_1)),$$ $$(d\omega_1\oplus d\omega_2)(p)=(d\omega_1)(p_1)=d(\omega_1(p_1)),$$ so the desired equality is true. Since the case when $p$ is equivalent to a plot of $X_2$ is completely analogous, we obtain the desired claim. The extendibility conditions ${{\mathcal D}}_1^{\Omega^k}={{\mathcal D}}_2^{\Omega^k}$ and the images of $\pi_1^{\Omega},\pi_2^{\Omega}$ ---------------------------------------------------------------------------------------------------------------------------------------- So far we have only assumed that the gluing map $f$ is smooth (which is always required for the gluing construction). Obtaining further claims needs some additional conditions, that we call extendibility conditions and describe in this section. Let $X_1$ and $X_2$ be two diffeological spaces, let $f:X_1\supseteq Y\to X_2$ be a smooth map, and let $i:Y\hookrightarrow X_1$ and $j:f(Y)\hookrightarrow X_2$ be the natural inclusions. We say that $f$ satisfies the **$k$-th extendibility condition** if $$i^*(\Omega^k(X_1))=(f^*j^*)(\Omega^k(X_2)).$$ Denote now by ${{\mathcal D}}_1^{\Omega^k}$ the diffeology on $\Omega^k(Y)$ that is the pushforward of the diffeology of $\Omega^k(X_1)$ by the map $i^*$; likewise, denote by ${{\mathcal D}}_2^{\Omega^k}$ the diffeology on $\Omega^k(Y)$ that is the pushforward of the diffeology of $\Omega^k(X_2)$ by the map $f^*j^*$. We say that $f$ satisfies the **$k$-th smooth extendibility condition** if $${{\mathcal D}}_1^{\Omega^k}={{\mathcal D}}_2^{\Omega^k}.$$ The need for these two conditions is based on the following lemma and is rendered explicit by the corollary that follows it. \[when:two:forms:are:compatible:lem\] Let $\omega_1\in\Omega^k(X_1)$ and $\omega_2\in\Omega^k(X_2)$ be two $k$-forms, and let $f:X_1\supseteq Y\to X_2$ be a smooth map. The forms $\omega_1$ and $\omega_2$ are compatible if and only if $$i^*\omega_1=(f^*j^*)\omega_2.$$ Let $\omega_1$ and $\omega_2$ be compatible, and let $p:U\to Y\subseteq X_1$ be an arbitrary plot of $Y$. Since $(i^*\omega_1)(p)=\omega_1(i\circ p)=\omega_1(p)$ and $(f^*j^*)(\omega_2)(p)=\omega_2(f\circ j\circ p)=\omega_2(f\circ p)$, we obtain the desired equality $i^*\omega_1=(f^*j^*)\omega_2$ by the assumption of compatibility of $\omega_1$ and $\omega_2$. Suppose now that $i^*\omega_1=(f^*j^*)\omega_2$ holds; let us show that $\omega_1$ and $\omega_2$ are compatible. Let again $p$ be any plot of $Y$. Then $$\omega_1(p)=\omega_1(i\circ p)=(i^*\omega_1)(p)=(f^*j^*)(\omega_2)(p)=\omega_2(f\circ j\circ p)=\omega_2(f\circ p),$$ therefore the compatibility condition $\omega_1(p)=\omega_2(f\circ p)$ follows from the assumption. \[when:pi-omega:are:surjective:cor\] Let $\pi_1^{\Omega}:\Omega^k(X_1)\oplus_{comp}\Omega^k(X_2)\to\Omega^k(X_1)$ and $\pi_2^{\Omega}:\Omega^k(X_1)\oplus_{comp}\Omega^k(X_2)\to\Omega^k(X_2)$ be induced by the standard direct sum projections. Then $\pi_1^{\Omega}$ and $\pi_2^{\Omega}$ are both surjective if and only if $f$ satisfies the extendibility condition $i^*(\Omega^k(X_1))=(f^*j^*)(\Omega^k(X_2))$. A form $\omega_1\in\Omega^k(X_1)$ belongs to the range of $\pi_1^{\Omega}$ if and only if there exists a form $\omega_2\in\Omega^k(X_2)$ such that $\omega_1$ and $\omega_2$ are compatible. By Lemma \[when:two:forms:are:compatible:lem\] this is equivalent to $i^*\omega_1\in(f^*j^*)(\Omega^k(X_2))$. Asking for this being true for all $\omega_1\in\Omega^k(X_1)$ is obviously equivalent to the inclusion $i^*(\Omega^k(X_1))\subseteq(f^*j^*)(\Omega^k(X_2))$. Applying exactly the same reasoning to an arbitrary $\omega_2\in\Omega^k(X_2)$, we obtain the claim. As is clear from the proof of Corollary \[when:pi-omega:are:surjective:cor\], the necessary and sufficient condition for only $\pi_1^{\Omega}$ to be surjective is $i^*(\Omega^k(X_1))\subseteq(f^*j^*)(\Omega^k(X_2))$; that for surjectivity of only $\pi_2^{\Omega}$ is $(f^*j^*)(\Omega^k(X_2))\subseteq i^*(\Omega^k(X_1))$. The De Rham groups $H_{dR}^k(X_1\cup_f X_2)$ -------------------------------------------- We shall now consider the De Rham groups of $X_1\cup_f X_2$ as they relate to those of $X_1$ and $X_2$. Their description is based on the straightforward behavior of the differential under gluing (Theorem \[differential:and:gluing:thm\]). #### Cocycles and coboundaries Some observations regarding the complex of the coccyges, and that of the coboundaries, are immediate from Theorem \[differential:and:gluing:thm\]. \[cocycles:and:coboundaries:in:glued:lem\] Let $X_1$ and $X_2$ be two diffeological spaces, and let $f:X_1\supseteq Y\to X_2$ be a diffeomorphism such that ${{\mathcal D}}_1^{\Omega^k}={{\mathcal D}}_2^{\Omega^k}$. Then: $$Z_{dR}^k(X_1\cup_f X_2)\subseteq Z_{dR}^k(X_1)\oplus Z_{dR}^k(X_2),$$ $$B_{dR}^k(X_1\cup_f X_2)\subseteq B_{dR}^k(X_1)\oplus B_{dR}^k(X_2).$$ This follows from Theorem \[differential:and:gluing:thm\], whose essence is that $d\omega$, for any $\omega\in\Omega^k(X_1\cup_f )$, is canonically identified, via an isomorphism, to $d\omega_1\oplus d\omega_2$. It is then obvious that $B_{dR}^k(X_1\cup_f X_2)\subseteq B_{dR}^k(X_1)\oplus B_{dR}^k(X_2)$. Furthermore, $d\omega=0$ if and only if both $d\omega_1=0$ and $d\omega_2=0$, therefore $Z_{dR}^k(X_1\cup_f X_2)\subseteq Z_{dR}^k(X_1)\oplus Z_{dR}^k(X_2)$. #### Compatibility of $d\omega_1$ and $d\omega$ *vs.* compatibility of $\omega_1$ and $\omega_2$ That the latter implies the former, is implicit in Theorem \[differential:and:gluing:thm\]. We shall now discuss why the former implies the latter. \[cocycles:and:coboundaries:split:lem\] The differentials $d\omega_1$ and $d\omega_2$ of two forms $\omega_1\in\Omega^{k-1}(X_1)$ and $\omega_2\in\Omega^{k-1}(X_2)$ are compatible if and only if the forms $\omega_1$ and $\omega_2$ are themselves compatible. In particular, $$Z_{dR}^k(X_1\cup_f X_2)=Z_{dR}^k(X_1)\oplus Z_{dR}^k(X_2)$$ $$B_{dR}^k(X_1\cup_f X_2)=B_{dR}^k(X_1)\oplus B_{dR}^k(X_2).$$ Let $p:{{\mathbb{R}}}^n\supseteq U\to Y\subseteq X_1$ be a plot, and let $\omega_1\in\Omega^{k-1}(X_1)$ and $\omega_2\in\Omega^{k-1}(X_2)$ be two forms such that $d\omega_1$ and $d\omega_2$ are compatible. Thus, $(d\omega_1)(p)=(d\omega_2)(f\circ p)$, that is, $d(\omega_1(p))=d(\omega_2(f\circ p))$, where $\omega_1':=\omega_1(p)$ and $\omega_2':=\omega_2(f\circ p)$ are two usual differential forms in $\Omega^{k-1}(U)$. Furthermore, they are such that $\omega_1'-\omega_2'$ is a cocycle, hence its defines an element of $H^{k-1}(U)$. If $U$ is simply connected, $H^{k-1}(U)$ is trivial, so $\omega_1(p)=\omega_2(f\circ p)$. It remains to recall the locality property for diffeological differential forms ([@iglesiasBook], Section 6.36) to conclude that $\omega_1(p)=\omega_2(f\circ p)$ for all other plots $p$ of $Y$. Thus, if $d\omega_1$ and $d\omega_2$ are compatible, which includes the case when they are both zero, then $\mathcal{L}^{k-1}(\omega_1\oplus\omega_2)$ is well-defined. Since $d\circ\mathcal{L}^{k-1}=\mathcal{L}^k\circ(d\oplus d)$, we obtain the claim. #### The diffeomorphism $H_{dR}^k(X_1\cup_f X_2)\cong H_{dR}^k(X_1)\oplus H_{dR}^k(X_2)$ The following is now a trivial consequence of Lemma \[cocycles:and:coboundaries:split:lem\]. \[de:rham:cohomology:splits:thm\] Let $X_1$ and $X_2$ be two diffeological spaces, and let $f:X_1\supseteq Y\to X_2$ be a diffeomorphism such that ${{\mathcal D}}_1^{\Omega^k}={{\mathcal D}}_2^{\Omega^k}$ for all $k$. Then $$H_{dR}^k(X_1)\oplus H_{dR}^k(X_2)\cong H_{dR}^k(X_1\cup_f X_2)$$ via the isomorphism induced by the chain map $\{\mathcal{L}^k\}$. The pseudo-bundles $\Lambda^k(X_1\cup_f X_2)$ relative to $\Lambda^k(X_1)$ and $\Lambda^k(X_2)$ ----------------------------------------------------------------------------------------------- We now consider the pseudo-bundles $\Lambda^k(X_1\cup_f X_2)$ (see [@forms-gluing] for the case of $k=1$, which is treated in a somewhat more general manner). We only do so under substantial restrictions on $f$. The first of them is that $f$ be a diffeomorphism of its domain with its image, and this is necessary for us (we do not know yet how to treat a more general case); the second restriction is that $f$ satisfy the $k$-th smooth extendibility condition, and this, in some cases, may not be strictly necessary (but the results would get far more cumbersome with it). Notice that due to the assumption that $f$ is a diffeomorphism, the map $\tilde{i}_1$ is invertible, and $\Omega_f^k(X_1)=\Omega^k(X_1)$, that is, every $k$-form on $X_1$ is $f$-invariant. ### The vanishing of forms in $\Omega^k(X_1\cup_f X_2)$ Recall that each fibre of $\Lambda^k(X_1\cup_f X_2)$ is the quotient of form $\Omega^k(X_1\cup_f X_2)/\Omega_x^k(X_1\cup_f X_2)$. \[vanishing:forms:according:to:point:thm\] Let $X_1$ and $X_2$ be two diffeological spaces, let $f:X_1\supseteq Y\to X_2$ be a diffeomorphism satisfying the $k$-th smooth compatibility condition ${{\mathcal D}}_1^{\Omega^k}={{\mathcal D}}_2^{\Omega^k}$, and let $x\in X_1\cup_f X_2$ be a point. The the space $\Omega_x^k(X_1\cup_f X_2)$ of $k$-forms vanishing at $x$ is defined by the following: $$\pi^*(\Omega_x^k(X_1\cup_f X_2))\cong\left\{\begin{array}{ll} \Omega_{\tilde{i}_1^{-1}(x)}^k(X_1)\oplus_{comp}\Omega^k(X_2) & \mbox{if }x\in i_1(X_1\setminus Y), \\ \Omega_{\tilde{i}_1^{-1}(x)}^k(X_1)\oplus_{comp}\Omega_{i_2^{-1}(x)}^k(X_2) & \mbox{if }x\in i_2(f(Y)), \\ \Omega^k(X_1)\oplus_{comp}\Omega_{i_2^{-1}(x)}^k(X_2) & \mbox{if }x\in i_2(X_2\setminus f(Y)). \end{array}\right.$$ Let first $\omega\in\Omega_x^k(X_1\cup_f X_2)$, and let $\pi^*\omega$ be written as $\omega_1\oplus\omega_2$. If $x\in i_1(X_1\setminus Y)$, we need to show that $\omega_1$ vanishes at $\tilde{i}_1^{-1}(x)$. Let $p_1$ be a plot of $X_1$, with connected domain of definition, such that $p_1(0)=\tilde{i}_1^{-1}(x)$. Then $p:=\tilde{i}_1\circ p_1$ is a plot of $X_1\cup_f X_2$ such that $p(0)=x$. We have by construction $\omega(p)=\omega_1(p_1)$, therefore $\omega_1(p_1)(0)=\omega(p)(0)=0$, therefore $\omega_1$ vanishes at $\tilde{i}_1^{-1}(x)$. This proves that $$\pi^*(\Omega_x^k(X_1\cup_f X_2))\subseteq\Omega_{\tilde{i}_1^{-1}(x)}^k(X_1)\oplus_{comp}\Omega^k(X_2).$$ The proof that $$\pi^*(\Omega_x^k(X_1\cup_f X_2))\subseteq\Omega^k(X_1)\oplus_{comp}\Omega_{i_2^{-1}(x)}^k(X_2)$$ is completely analogous. Let thus $x\in i_2(f(Y))$. If $p_2$ is a plot of $X_2$ such that $p_2(0)=i_2^{-1}(x)$, we have, as before, $\omega_2(p_2)=\omega(i_2\circ p_2)$, and $i_2(p_2(0))=x$, so $\omega_2$ vanishes at $i_2^{-1}(x)$. Let $p_1$ be a plot of $X_1$. Again, $\omega_1(p_1)=\omega(\tilde{i}_1\circ p_1)$ and $(\tilde{i}_1\circ p_1)(0)=x$, so $\omega_1$ vanishes at $\tilde{i}_1^{-1}(x)$. Therefore $$\pi^*(\Omega_x^k(X_1\cup_f X_2))\subseteq\Omega_{\tilde{i}_1^{-1}(x)}^k(X_1)\oplus_{comp}\Omega_{i_2^{-1}(x)}^k(X_2).$$ Let us establish the reverse inclusion. Let $\omega_1\in\Omega_{x_1}^k(X_1)$ and $\omega_2\in\Omega^k(X_2)$ be two compatible forms, and let $\omega=\mathcal{L}^k(\omega_1\oplus\omega_2)$. Let $p$ be a plot of $X_1\cup_f X_2$ with connected domain of definition and such that $p(0)=\tilde{i}_1(x_1)=:x$. Then $p_1:=\tilde{i}_1^{-1}\circ p$ is a plot of $X_1$ and $p_1(0)=x_1$. Furthermore, $\omega_1(p_1)=\omega(p)$ by construction. We thus conclude that $\omega(p)(0)=0$, hence $\omega$ vanishes at $x$, and in particular, we obtain the first claim. Analogously, if $\omega_1\in\Omega^k(X_1)$ and $\omega_2\in\Omega_{x_2}^k(X_2)$ are compatible then $\omega=\mathcal{L}^k(\omega_1\oplus\omega_2)$ vanishes at $i_2(X_2)$; this yields the third claim. Finally, since $$\Omega_{x_1}^k(X_1)\oplus_{comp}\Omega_{f(x_1)}^k(X_2)=\left(\Omega_{x_1}^k(X_1)\oplus_{comp}\Omega^k(X_2)\right)\,\cap\,\left(\Omega^k(X_1)\oplus_{comp}\Omega_{f(x_1)}^k(X_2)\right)$$ for any $x\in Y$, we obtain the second claim, and the proof is finished. ### The fibres of $\Lambda^k(X_1\cup_f X_2)$ We first define an appropriate compatibility notion for elements of fibres of form $\Lambda_x^k(X_1)$ and $\Lambda_{f(x)}^k(X_2)$, for $x\in Y$. \[compatible:elements:of:lambda:defn\] Let $x\in Y$, let $\alpha_1=\omega_1+\Omega_x^k(X_1)\in\Lambda_x^k(X_1)$, and let $\alpha_2=\omega_2+\Omega_{f(x)}^k(X_2)\in\Lambda_{f(x)}^k(X_2)$. We say that $\alpha_1$ and $\alpha_2$ are **compatible** if any two forms $\omega_1'\in\alpha_1\subseteq\Omega^k(X_1)$ and $\omega_2'\in\alpha_2\subseteq\Omega^k(X_2)$ are compatible. We denote $$\Lambda_x^k(X_1)\oplus_{comp}\Lambda_{f(x)}^k(X_2):=\{\alpha_1\oplus\alpha_2\,|\,\alpha_1\mbox{ and }\alpha_2\mbox{ are compatible}\}$$ for every $x\in Y$. \[fibres:of:lambda:thm\] Let $X_1$ and $X_2$ be two diffeological spaces, let $f:X_1\supseteq Y\to X_2$ be a diffeomorphism such that ${{\mathcal D}}_1^{\Omega^k}={{\mathcal D}}_2^{\Omega^k}$, and let $x\in X_1\cup_f X_2$. Then: $$\Lambda_x^k(X_1\cup_f X_2)\cong\left\{\begin{array}{ll} \Lambda_{\tilde{i}_1^{-1}(x)}^k(X_1) & \mbox{if }x\in i_1(X_1\setminus Y), \\ \Lambda_{\tilde{i}_1^{-1}}^k(X_1)\oplus_{comp}\Lambda_{i_2^{-1}(x)}^k(X_2) & \mbox{if }x\in i_2(f(Y)), \\ \Lambda_{i_2^{-1}(x)}^k(X_2) & \mbox{if }x\in i_2(X_2\setminus f(Y)). \end{array}\right.$$ This is a simple consequence of Theorem \[vanishing:forms:according:to:point:thm\]. It amounts to checking that $$\left(\Omega^k(X_1)\oplus\Omega^k(X_2)\right)/\left(\Omega_{\tilde{i}_1^{-1}(x)}^k(X_1)\oplus_{comp}\Omega^k(X_2)\right)\cong\Omega^k(X_1)/ \Omega_{\tilde{i}_1^{-1}(x)}^k(X_1),$$ $\left(\Omega^k(X_1)\oplus\Omega^k(X_2)\right)/\left(\Omega_{\tilde{i}_1^{-1}(x)}^k(X_1)\oplus_{comp}\Omega_{i_2^{-1}(x)}^k(X_2)\right)\cong$ $\cong\left(\Omega^k(X_1)/ \Omega_{\tilde{i}_1^{-1}(x)}^k(X_1)\right)\oplus_{comp}\left(\Omega^k(X_2)/ \Omega_{i_2^{-1}(x)}^k(X_2)\right),$ $$\left(\Omega^k(X_1)\oplus\Omega^k(X_2)\right)/\left(\Omega^k(X_1)\oplus_{comp}\Omega_{i_2^{-1}(x)}^k(X_2)\right)\cong\Omega^k(X_2)/ \Omega_{i_2^{-1}(x)}^k(X_2),$$ and this is done by completely standard reasoning, of which we omit the details. ### The characteristic maps $\tilde{\rho}_1^{\Lambda^k}$ and $\tilde{\rho}_2^{\Lambda^k}$ It is worth noting that under the assumption of the gluing map $f$ being a diffeomorphism such that ${{\mathcal D}}_1^{\Omega^k}={{\mathcal D}}_2^{\Omega^k}$, the total space $\Lambda^k(X_1\cup_f X_2)$ is a span of the total spaces $\Lambda^k(X_1)$ and $\Lambda^k(X_2)$: it admits two (surjective partially defined) maps $$\tilde{\rho}_1^{\Lambda^k}:(\pi^{\Lambda^k})^{-1}(\tilde{i}_1(X_1))\to\Lambda^k(X_1),\,\,\,\tilde{\rho}_2^{\Lambda^k}:(\pi^{\Lambda^k})^{-1}(i_2(X_2))\to\Lambda^k(X_2),$$ where $\pi^{\Lambda^k}$ is the pseudo-bundle projection $\Lambda^k(X_1\cup_f X_2)\to X_1\cup_f X_2$. The maps $\tilde{\rho}_1^{\Lambda^k}$ and $\tilde{\rho}_2^{\Lambda^k}$ are induced by the pullback maps $\tilde{i}_1^*$ and $i_2^*$ respectively, and can also be given a more direct description, by representing $\Lambda^k(X_1\cup_f X_2)$ as a quotient of $$(X_1\sqcup X_2)\times(\Omega^k(X_1)\oplus_{comp}\Omega^k(X_2))=X_1\times(\Omega^k(X_1)\oplus_{comp}\Omega^k(X_2))\,\sqcup\,X_2\times(\Omega^k(X_1)\oplus_{comp}\Omega^k(X_2)).$$ The domain of definition of $\tilde{\rho}_1^{\Lambda^k}$ corresponds to $X_1\times(\Omega^k(X_1)\oplus_{comp}\Omega^k(X_2))$, and $\tilde{\rho}_1^{\Lambda^k}$ itself is induced by the projection of $X_1\times(\Omega^k(X_1)\oplus_{comp}\Omega^k(X_2))$ to $X_1\times\Omega^k(X_1)$. The direct construction of $\tilde{\rho}_2^{\Lambda^k}$ is completely analogous. Both of these maps are smooth and linear by construction. Furthermore, the following is true. \[tilde-rho:are:subductions:prop\] Let $X_1$ and $X_2$ be two diffeological spaces, and let $f:X_1\supseteq Y\to X_2$ be a diffeomorphism such that ${{\mathcal D}}_1^{\Omega^k}={{\mathcal D}}_2^{\Omega^k}$. Then the maps $$\tilde{\rho}_1^{\Lambda^k}:(\pi^{\Lambda^k})^{-1}(\tilde{i}_1(X_1))\to\Lambda^k(X_1),\,\,\,\tilde{\rho}_2^{\Lambda^k}:(\pi^{\Lambda^k})^{-1}(i_2(X_2))\to\Lambda^k(X_2),$$ where $(\pi^{\Lambda^k})^{-1}(\tilde{i}_1(X_1))$ and $(\pi^{\Lambda^k})^{-1}(i_2(X_2))$ are considered with the subset diffeologies relative to their inclusions in $\Lambda^k(X_1\cup_f X_2)$, are subductions. The two cases of $\tilde{\rho}_1^{\Lambda^k}$ and $\tilde{\rho}_2^{\Lambda^k}$ are fully analogous, so we only consider the first of them. Let $q_1:U\to\Lambda^k(X_1)$ be a plot of $\Lambda^k(X_1)$ (possibly a constant one). We need to show that (at least up to restricting $U$) there exists a plot $q:U\to\Lambda^k(X_1\cup_f X_2)$ such that $q_1=\tilde{\rho}_1^{\Lambda^k}\circ q$. By definition of the diffeology of any $\Lambda^k(\cdot)$, there exists a (local) lift $p_1:U\to X_1\times\Omega^k(X_1)$ of $q_1$, of form $p_1(u)=(\pi_1^{\Lambda^k}(q_1(u)),p_1^{\Omega^k}(u))$ for $u\in U$, where $p_1^{\Omega^k}:U\to\Omega^k(X_1)$ is a plot of $\Omega^k(X_1)$. By the smooth compatibility condition, there exists a plot $p_2^{\Omega^k}:U\to\Omega^k(X_2)$ of $\Omega^k(X_2)$ such that $$i^*\circ p_1^{\Omega^k}=(f^*j^*)\circ p_2^{\Omega^k}.$$ By Lemma \[when:two:forms:are:compatible:lem\] this means that $p_1^{\Omega^k}(u)$ and $p_2^{\Omega^k}(u)$ are compatible for all $u\in U$. Therefore $p:U\to(X_1\cup_f X_2)\times(\Omega^k(X_1)\oplus_{comp}\Omega^k(X_2))$ given by $$p(u)=(\tilde{i}_1(\pi_1^{\Lambda^k}(q_1(u))),p_1^{\Omega^k}(u)\oplus p_2^{\Omega^k}(u))$$ is well-defined, and by construction it is a plot of $(X_1\cup_f X_2)\times(\Omega^k(X_1)\oplus_{comp}\Omega^k(X_2))$. Therefore its composition $q=\pi^{\Omega,\Lambda}\circ p$ with the defining projection $\pi^{\Omega,\Lambda}$ of $\Lambda^k(X_1\cup_f X_2)$ is a plot of $\Lambda^k(X_1\cup_f X_2)$, and by construction $\tilde{\rho}_1^{\Lambda^k}\circ q=q_1$, which completes the proof. The proof of Proposition \[tilde-rho:are:subductions:prop\] provides a working characterization of the diffeology of $\Lambda^k(X_1\cup_f X_2)$, even without any additional conditions on the gluing map $f$. Namely, any plot of $\Lambda^k(X_1\cup_f X_2)$ locally has a lift of form $u\mapsto(p(u),p_1^{\Omega^k}(u)\oplus p_2^{\Omega^k}(u))$, where $p$ is any plot of $X_1\cup_f X_2$, and $p_1^{\Omega^k}$ and $p_2^{\Omega^k}$ are any two plots of $\Omega^k(X_1)$ and $\Omega^k(X_2)$ respectively such that $i^*\circ p_1^{\Omega^k}=f^*\circ j^*\circ p_2^{\Omega^k}$. The operator $d+d^*$ in general is not defined ============================================== In this section we examine the ingredients that usually go into the construction of the De Rham operator as the operator $d+d^*$, showing (via examples based on the gluing construction) that they do not extend, in any straightforward manner, to the diffeological context; whenever, as in the case of volume forms, a formally defined extension exists, it is not really suitable for the purpose it is meant to achieve. The differential is not well-defined as a map on $\Lambda^k(X)$ --------------------------------------------------------------- Let $\alpha\in\Lambda^k(X)$, and let $x:=\pi^{\Lambda^k}(\alpha)$. *A priori*, if a form $\omega\in\Omega^k(X)$ vanishes at $x$, it is not clear why its differential should vanish at $x$ as well; this condition would be needed to ensure that the differential on $\Lambda^k(X)$ could be defined by $d(\omega+\Omega_x^k(X))=d\omega+\Omega_x^{k+1}(X)$. However, already the case of $k=0$ illustrates that this cannot be done. It suffices to consider, on the standard ${{\mathbb{R}}}$, any smooth function $h$ such that $h(0)=0$ and $h'(0)\neq 0$ (for instance, $h(x)=\sin(x)$). The dimension of a diffeological space and pseudo-bundles $\Lambda^k(X)$ ------------------------------------------------------------------------ Although there exists a notion of dimension for diffeological spaces that is similar to the standard one, its implications for the dimensions of fibres of $\Lambda^k(X)$ are not entirely similar to those in the standard case. Specifically, if $\dim(X)=n$ then all pseudo-bundles $\Lambda^{n+k}(X)$, $k=1,2,\ldots$, are trivial; but the dimensions of $\Lambda^k(X)$ with $k=1,\ldots,n$ are not bounded by $n$ and can in fact be arbitrarily large. ### The dimension of $X_1\cup_f X_2$ The dimension of a diffeological space is an extension of the usual notion. It is based on the fact that, although the diffeology ${{\mathcal D}}$ of any given diffeological space $X$ can be quite large, it is usually determined by a smaller subset $\mathcal{A}$ of it, called a **generating family** of ${{\mathcal D}}$. More specifically, a subset $\mathcal{A}\subseteq{{\mathcal D}}$ is called a generating family of ${{\mathcal D}}$ if for any plot $p:U\to X$ in ${{\mathcal D}}$ and for any $u\in U$ there exists a neighborhood $U'\subseteq U$ of $u$ such that either $p|_{U'}$ is constant or there exists a plot $q:U''\to X$ in $\mathcal{A}$ and an ordinary smooth map $h:U'\to U''$ such that $p|_{U'}=q\circ h$. We can re-state this briefly by saying that locally every $p\in{{\mathcal D}}$ either is constant or filters through a plot in $\mathcal{A}$. Almost always, a diffeology admits many generating families. Let $X$ be a diffeological space, and let ${{\mathcal D}}$ be its diffeology. The **dimension** of any generating family $\mathcal{A}=\{q_{\alpha}:U_{\alpha}\to X\}_{\alpha}$ is the supremum of the dimensions of the domains of definition of all $q_{\alpha}\in\mathcal{A}$, $$\dim(\mathcal{A})=\mbox{sup}\{\dim(U_{\alpha})\}.$$ If no supremum exists, the dimension is said to be infinite. The **dimension of $X$** is the infimum of the dimensions of all generating families of ${{\mathcal D}}$, $$\dim(X)=\mbox{inf}\{\dim(\mathcal{A})\,|\,\mathcal{A}\subseteq{{\mathcal D}}\mbox{ is a generating family of }{{\mathcal D}}\}.$$ If ${{\mathcal D}}$ has no generating family with finite dimension, $X$ is said to have infinite dimension. The following is then a trivial observation. \[dimension:of:glued:lem\] Let $X_1$ and $X_2$ be two diffeological spaces of finite dimensions, and let $f:X_1\supseteq Y\to X_2$ be a smooth map. Then $$\dim(X_1\cup_f X_2)=\mbox{max}\{\dim(X_1),\dim(X_2)\}\,\,\mbox{ if }Y\neq X_1,\,\,\mbox{ and }\dim(X_1\cup_f X_2)=\dim(X_2)\mbox{ otherwise}.$$ In particular, $X_1\cup_f X_2$ has finite dimension if and only if both $X_1$ and $X_2$ have finite dimension. Let $\mathcal{A}$ be a generating family of the gluing diffeology on $X_1\cup_f X_2$. We can assume that all plots in $\mathcal{A}$ have connected domains of definition. If $Y=X_1$ then $X_1\cup_f X_2\cong X_2$, so the second statement is obvious. Assume that $Y$ is properly contained in $X_1$. Let $\mathcal{A}_1\subseteq\mathcal{A}$ be the subset of all plots of $\mathcal{A}$ that have lifts to plots of $X_1$; let $\mathcal{A}_2\subseteq\mathcal{A}$ be the subset of plots with lifts to $X_2$. Then $\mathcal{A}=\mathcal{A}_1\cup\mathcal{A}_2$, and $\mathcal{A}_1$ and $\mathcal{A}_2$ are in a natural correspondence with specific generating families $\mathcal{A}_1'$ and $\mathcal{A}_2'$ of the diffeologies of $X_1$ and $X_2$ respectively, and since $X_1\setminus Y$ is non-empty, $\mathcal{A}\setminus\mathcal{A}_2$ is non-empty as well. Therefore we have the inequality $\dim(X_1\cup_f X_2)\leqslant\mbox{max}\{\dim(X_1),\dim(X_2)\}$. *Vice versa*, any two generating families of the diffeologies on $X_1$ and $X_2$ yield automatically a generating family for the gluing diffeology on $X_1\cup_f X_2$. Therefore we obtain the reverse inequality, and so the final claim. ### The dimension of $X$ and pseudo-bundles $\Lambda^k(X)$ For any diffeological space $X$ and for any differential form $\omega\in\Omega^k(X)$, there is a standard way to associate to $\omega$ a smooth section of $\Lambda^k(X)$. This section is defined as the assignment $$x\mapsto\pi^{\Omega^k,\Lambda^k}(x,\omega),$$ where, recall, $\pi^{\Omega^k,\Lambda^k}:X\times\Omega^k(X)\to\Lambda^k(X)$ is the defining quotient projection of $\Lambda^k(X)$ (this is the tautological $k$-form corresponding to $\omega$, that is mentioned in [@iglesiasBook], p. 160). The following is a known fact (see [@iglesiasBook], Section 6.37), but for completeness we provide a proof. Let $X$ be a diffeological space of finite dimension $n$. Then $\Omega^k(X)$ is trivial for $k>n$. Choose a fixed $k>n$. Let $\mathcal{A}$ be a generating family of plots of the diffeology of $X$ that has dimension $n$ (that is, every plot in $\mathcal{A}$ is defined on a domain in ${{\mathbb{R}}}^m$ with $m\leqslant n$, and at least one plot is defined on a domain in ${{\mathbb{R}}}^n$) and let $\omega\in\Omega^k(X)$ be a form. We need to show that $\omega$ is the zero form. Let first $p\in\mathcal{A}$; by assumption, the (usual) dimension of its domain of definition is strictly less than $k$. Therefore obviously $\omega(p)=0$. Let now $q:U\to X$ be any random plot of $X$. Then for every $u\in U$ there exists a subdomain $U'\subseteq U$ such that $q|_{U'}=p\circ F$ for some ordinary smooth map $F:U'\to U''$ and for some plot $p:U''\to X$ that belongs to $\mathcal{A}$. Therefore $\omega(q|_{U'})=F^*(\omega(p))=0$. Since this is true for any $u\in U$, we conclude that $\omega(q)=0$, whence the claim. The following is then immediately obvious. If $X$ is a diffeological space of dimension $n$ then all pseudo-bundles $\Lambda^k(X)$ for $k>n$ are trivial. Suppose now that there exists a volume form $\omega\in\Omega^n(X)$ on $X$ of dimension $n$. Let $\mathcal{A}$ be a generating family for the diffeology of $X$ that has dimension $n$; let $\mathcal{A}_n\subseteq\mathcal{A}$ be the subset consisting of precisely the plots in $\mathcal{A}$ whose domain of definition has dimension $n$. Obviously, if $p\in\mathcal{A}\setminus\mathcal{A}_n$ then $\omega(p)=0$. On the other hand, there are diffeological spaces such that $\mathcal{A}_n$ contains at least two plots that are not related by a smooth substitution, which implies that the dimension, in the sense of pseudo-bundles, of $\Lambda^n(X)$ can be greater than $n$. In fact, it can be arbitrarily greater, as the following example shows. \[dim:X:and:dim:lambda:are:different:ex\] Let $m\in{{\mathbb{N}}}$, $m\geqslant 2$, be any, and let $X$ be the wedge at the origin of $m$ copies of ${{\mathbb{R}}}$ (each copy endowed with its standard diffeology), endowed with the corresponding gluing diffeology. It is quite clear that $X$ is finite-dimensional, and that its dimension is equal to $1$. However, applying repeatedly ($m-1$ times) Theorem 8.5 of [@dirac] (or Theorem \[fibres:of:lambda:thm\] in the case $k=1$), we obtain that the fibre of $X$ at the wedge point has dimension $m$. The volume forms ---------------- The notion of a **volume form** is well-defined for (a subcategory of) diffeological spaces ([@iglesiasBook], Section 6.44). After recalling the necessary definitions, we consider its behavior under gluing. Let $X$ be a diffeological space of dimension $n$. A volume form on it is then a nowhere vanishing $n$-form on $X$; alternatively, it is a collection of usual volume forms on the domains of definition of plots of $X$. Let $X$ be a diffeological space, and let $n=\dim(X)$. A **volume form** on $X$ is a form $\omega\in\Omega^n(X)$ such that for every $x\in X$ there exists a plot $p:U\to X$ of $X$ such that $p(U)\ni x$ and $\omega(p)$ is a volume form on $U$. An alternative way to define a volume form is to ask that, for any $x\in X$, there be a plot $p$ such that $p(0)=x$ and $\omega(p)(0)\neq x$ (see [@iglesiasBook], p. 158). As in the case of smooth manifolds, volume forms do not always exist (obviously, any non-orientable smooth manifold considered with its standard diffeology is an instance of a diffeological space that does not admit any). A characterization of volume forms on $X_1\cup_f X_2$ follows from the definition and the characterization of the space $\Omega^n(X_1\cup_f X_2)$ given above (Corollary \[omega:of:glued:cor\]). Let $X_1$ and $X_2$ be two diffeological spaces of the same finite dimension $n$, let $f:X_1\supseteq Y\to X_2$ be a smooth map, and suppose that both $X_1$ and $X_2$ admit volume forms and that such forms can be chosen to be compatible. Then $X_1\cup_f X_2$ admits a volume form. By assumption and Lemma \[dimension:of:glued:lem\] we have $\dim(X_1\cup_f X_2)=n$. Let $\omega_1$ and $\omega_2$ be compatible volume forms on $X_1$ and $X_2$ respectively. It is then trivial to check that $\mathcal{L}^n(\omega_1\oplus\omega_2)$ is a volume form on $X_1\cup_f X_2$. Indeed, by Lemma \[dimension:of:glued:lem\] $\dim(X_1\cup_f X_2)=n$. Let $x\in X_1\cup_f X_2$ be an arbitrary point. Then it has a lift to either $X_1$ or $X_2$ (possibly to both). Suppose that it has a lift $x_1\in X_1$; since $\omega_1$ is a volume form on $X_1$, there exists a plot $p_1$ of $X_1$ such that $\mbox{Range}(p_1)\ni x_1$ and $\omega_1(p_1)$ is a volume form on the domain of definition of $p_1$. Then $p=\tilde{i}_1\circ p_1$ is plot of $X_1\cup_f X_2$ such that $\mbox{Range}(p)\ni x$ and $\omega(p)=\omega_1(p_1)$ is a volume form on the domain of $p$. The case when $x$ has a lift to $X_2$ is treated analogously, so we obtain the claim. An instance of a volume form on $X_1\cup_f X_2$ is the form $\mathcal{L}^n(\omega_1\oplus\omega_2)$, where $\omega_1$ and $\omega_2$ are compatible volume forms on $X_1$ and $X_2$ respectively. It is not clear whether the *vice versa* of this statement is always true; we can only obtain it under some rather restrictive assumptions. \[splitting:volume:form:on:glued:prop\] Let $X_1$ and $X_2$ be two diffeological spaces of finite dimension and such that $\dim(X_1)=\dim(X_2)$, let $f:X_1\supseteq Y\to X_2$ be a smooth map such that $i_2(f(Y))$ is D-open in $X_1\cup_f X_2$, and let $\omega$ be a volume form on $X_1\cup_f X_2$ such that $\pi^*(\omega)=\omega_1\oplus\omega_2$. Then $\omega_1$ and $\omega_2$ are volume forms on $X_1$ and $X_2$ respectively. Let $x_1\in X_1$ be any point, and let $x=\pi(x_1)\in X_1\cup_f X_2$. Let $p:U\to X_1\cup_f X_2$ be a plot such that $p(U)\ni x$ and $\omega(p)$ is a volume form on $U$. The assumption that $i_2(f(Y))$ allows us to claim that $p$ has a lift to a plot $p_1$ of $X_1$ (which does not have to be true in the case of $x_1\in Y$). Then $p_1(U)\ni x_1$ and $\omega_1(p_1)$ is a volume form on $U$. Since $x_1$ is an arbitrary point, we conclude that $\omega_1$ is a volume form on $X_1$. The case of $\omega_2$ is treated analogously. In the above proposition, let $n$ be the dimension of $X_1\cup_f X_2$. Recall that $\omega_1=\tilde{i}_1^*(\omega)\in\Omega^n(X_1)$ and $\omega_2=i_2^*(\omega)\in\Omega^n(X_2)$ respectively. The assumption that $\dim(X_1)=\dim(X_1\cup_f X_2)=\dim(X_2)$ was not really used in the proof of Proposition \[splitting:volume:form:on:glued:prop\]; rather, we could obtain this equality as part of the conclusion. Notice also that Example \[dim:X:and:dim:lambda:are:different:ex\] implies that there might be many volume forms on $X_1\cup_f X_2$ that are not proportional. $\bigwedge^k(\Lambda^1(X))$ and $\Lambda^k(X)$ are not diffeomorphic -------------------------------------------------------------------- Let $X$ be a finite-dimensional diffeological space, and let $k\geqslant 2$. We now show that $\Lambda^k(X)$ and $\bigwedge^k(\Lambda^1(X))$ are in general not the same. Let $X$ be the wedge at the origin of two copies of the standard ${{\mathbb{R}}}^2$, endowed with the corresponding gluing diffeology. Then by Theorem \[fibres:of:lambda:thm\] we have that $\Lambda_0^1(X)\cong{{\mathbb{R}}}^2\oplus{{\mathbb{R}}}^2\cong{{\mathbb{R}}}^4$. Since by construction $\bigwedge^2(\Lambda_0^1(X))=\left(\bigwedge^2(\Lambda^1(X))\right)_0$, the fibre of $\bigwedge^2(\Lambda^1(X))$ at the wedge point, this is a space of dimension $6$. However, $\Lambda_0^2(X)\cong\Lambda_0^2({{\mathbb{R}}}^2)\oplus\Lambda_0^2({{\mathbb{R}}}^2)$ has dimension $2$. We conclude from the above example that $\bigwedge^k(\Lambda^1(X))$ is *a priori* a much larger space than $\Lambda^k(X)$. We shall see next whether there is any other natural relation between the two, for instance, whether an element of $\bigwedge^k(\Lambda^1(X))$ determines naturally an element of $\Lambda^k(X)$. Recall first that there is a well-defined notion of the **exterior product** $\omega\wedge\mu\in\Omega^{k+l}(X)$ of any two differential forms $\omega\in\Omega^k(X)$ and $\mu\in\Omega^l(X)$ ([@iglesiasBook], Section 6.35), which is defined by setting $$(\omega\wedge\mu)(p):=\omega(p)\wedge\mu(p)$$ for all plots $p$ of $X$. \[exterior:derivative:descends:to:lambda:lem\] The exterior derivative $\wedge:\Omega^k(X)\times\Omega^l(X)\to\Omega^{k+l}(X)$ induces a well-defined and smooth pseudo-bundle map $$\Lambda^k(X)\times_X\Lambda^l(X)\to\Lambda^{k+l}(X).$$ Let $x\in X$; we need to show that if at least one of $\omega,\mu$ vanishes at $x$ then $\omega\wedge\mu$ vanishes at $x$. Let $p$ be a plot of $X$ such that $p(0)=x$. Obviously, $$(\omega\wedge\mu)(p)(0)=\omega(p)(0)\wedge\mu(p)(0),$$ so if one of $\omega(p)(0)$, $\mu(p)(0)$ is zero then $(\omega\wedge\mu)(p)(0)=0$. The smoothness is immediate from the definitions of the respective diffeologies, so we obtain the claim. We thus can obtain the following. \[exterior:derivative:maps:to:lambda:lem\] The exterior derivative yields a well-defined pseudo-bundle map $$\left(\Lambda^k(X)\right)\bigwedge\left(\Lambda^l(X)\right)\to\Lambda^{k+l}(X).$$ This is immediate from Lemma \[exterior:derivative:descends:to:lambda:lem\] and the construction of the diffeology on the exterior product of pseudo-bundle (based essentially on the properties of the tensor product diffeology, see [@wu]). The obvious consequence of Lemma \[exterior:derivative:maps:to:lambda:lem\] is the following statement. There is a well-defined pseudo-bundle map $$\bigwedge^{1,k}:\bigwedge^k(\Lambda^1(X))\to\Lambda^k(X)$$ induced by the exterior derivative. As follows from the example given in this section, the map $\bigwedge^{1,k}$ is in general not injective. It is not clear whether it is surjective. The Hodge star operator does not take values in $\bigwedge^{n-k}(\Lambda^1(X))$ ------------------------------------------------------------------------------- For a diffeological vector space $V$ of finite dimension $n$, the standard definition of the Hodge star $\star:\bigwedge^kV\to\bigwedge^{n-k}V$ by setting $$e_{\sigma(1)}\wedge\ldots\wedge e_{\sigma(k)}\mapsto\mbox{sgn}(\sigma)e_{\sigma(k+1)}\wedge\ldots\wedge e_{\sigma(n)}$$ for all $\sigma\in\mbox{Symm}(n)$ and for all $k=1,\ldots,n$, where $\{e_i\}_{i=1}^n$ is a fixed basis of $V$ (we avoid the requirement of it being an orthonormal basis), yields a well-defined operator that is smooth for the natural diffeology on $\bigwedge^kV$ (induced by the tensor product diffeology). Thus, if $\pi:V\to X$ is a finite-dimensional diffeological vector pseudo-bundle that is locally trivial in the standard sense (in particular, it admits a local basis of smooth sections) then the operator $\star$ is defined on each $\bigwedge^k(V)$ for $k=1,\ldots,n=\dim(V)$, where $\dim(V)$ is the maximum of the usual vector space dimensions of fibres of $V$. Let now $X$ be a diffeological space of finite dimension such that $\Lambda^1(X)$ is finite-dimensional and admits pseudo-metrics; let $g^{\Lambda}$ be a fixed pseudo-metric on $\Lambda^1(X)$. Then for all $x\in X$ the fibre $\Lambda_x^1(X)$ admits an orthonormal, with respect to $g^{\Lambda}(x)$, basis $\alpha_{1,x},\ldots,\alpha_{n,x}$, with respect to which the map $\star_x$ is obviously defined. The collection of the maps $\star_x$ for all $x\in X$ yields in a usual way the operator $\star$ on $\bigwedge^k(\Lambda^1(X))$. However, it does not take values in $\bigwedge^{n-k}(\Lambda^1(X))$, as the next example shows. \[star:undefined:ex\] Let $X$ be the wedge at the origin of two copies, denoted by $X_1$ and $X_2$, of ${{\mathbb{R}}}^2$, endowed with the gluing diffeology; then $\dim(X)=2$. The fibres of $\bigwedge^1(\Lambda^1(X))=\Lambda^1(X)$ can be described as follows: $$\Lambda_x^1(X)=\left\{\begin{array}{ll} \mbox{Span}(dx_1,dy_1) & \mbox{if }\in\tilde{i}_1(X_1\setminus\{(0,0)\}) \\ \mbox{Span}(dx_1\oplus dx_2,dx_1\oplus dy_2,dy_1\oplus dx_2,dy_1\oplus dy_2) & \mbox{if }x=(0,0) \\ \mbox{Span}(dx_2,dy_2) & \mbox{if }\in i_2(X_2\setminus\{(0,0)\}) \end{array}\right.$$ Notice that endowing each fibre with the scalar product for the basis indicated is orthonormal yields a well-defined pseudo-metric on $\Lambda^1(X)$. Now, applying the standard construction of the Hodge star to each fibre yields a map that does *not* take values in $\bigwedge^2(\Lambda^1(X))$. Indeed, on the fibre at the wedge point $(0,0)$ we would, by formal definition, have $$\star_{(0,0)}(dx_1\oplus dx_2)=(dx_1\oplus dy_2)\wedge(dy_1\oplus dx_2)\wedge(dy_1\oplus dy_2)\in\bigwedge^3(\Lambda_{(0,0)}^1(X)).$$ The example just made indicates that, at a minimum, the Hodge star is not readily defined on exterior degrees of $\Lambda^1(X)$. The De Rham operator on $\bigwedge(\Lambda^1(X))$ ================================================= We have established so far that there is no readily available counterpart of the standard operator $d+d^*$ in the diffeological context. Therefore the De Rham operator on $\bigwedge(\Lambda^1(X))$ (endowed with a pseudo-metric) can only be defined as the composition of the standard Clifford action with the Levi-Civita connection, assuming that the latter exists. Another assumption that is needed is that $\bigwedge(\Lambda^1(X))$ have only a finite number of components (summands of form $\bigwedge^k(\Lambda^1(X))$), that is, that there is a uniform bound on the dimensions of fibres of $\Lambda^1(X)$; as we have seen in the previous section, this is not implied by $X$ having finite dimension. Bounding the dimension of $\Lambda^1(X)$ ---------------------------------------- Let $X$ be a diffeological space of finite dimension. The next example shows that the set of the dimensions of fibres of $\Lambda^1(X)$ may not have a supremum. Consider the following sequence $\{X_n\}_{n\in{{\mathbb{N}}}}$ of diffeological spaces: $X_0$ is the standard ${{\mathbb{R}}}$, and if $X_n$ is already defined then $X_{n+1}$ is obtained as the wedge of $X_n$ at the point $x=n+1\in X_n$ with $n+1$ copies of the standard ${{\mathbb{R}}}$ at zero of each copy; formally, $X_{n+1}$ is the result of a sequence of $n+1$ gluings of $X_{n+1}^{(k)}$ (this is $X_{n+1}$ to which $k$ copies of ${{\mathbb{R}}}$ have already been added) to the standard ${{\mathbb{R}}}$ along the map $\{n\}\to\{0\}$. Each $X_n$ is thus endowed with a well-defined diffeology based on the gluing construction; furthermore, there is a sequence of smooth inclusions $X_0\subset X_1\subset\ldots\subset X_n\subset\ldots$. Let $X=\bigcup_{n\in{{\mathbb{N}}}}X_n$; endow it with the minimal diffeology such that all these inclusions are smooth (the diffeology of $X$ is essentially the union of the diffeologies of all $X_n$ and can be called the inductive limit diffeology). Since all the gluing points are isolated and the differential forms are local, we obtain that $\Lambda_{(n)}^1(X)\cong\Lambda_{(n)}^1(X_n)$ and in particular has dimension $n$, by the reasoning made already. Thus, $\Lambda^1(X)$ admits fibres of arbitrarily large dimension, although the dimension of $X$ itself is equal to $1$. The above example shows that the existence of $\max\{\dim(\Lambda_x^1(X))\}$ should be imposed as a separate assumption. If such a maximum exists, we say that $\Lambda^1(X)$ has **bounded dimension**. The definition of the De Rham operator -------------------------------------- Let $X$ be a diffeological space of finite dimension and such that the following two conditions hold. First, $\Lambda^1(X)$ has bounded dimension; second, $\Lambda^1(X)$ admits a pseudo-metric $g^{\Lambda}$ such that there exists a Levi-Civita connection $\nabla^{\Lambda}$ on $(\Lambda^1(X),g^{\Lambda})$. Let $n=\max_{x\in X}\{\dim(\Lambda_x^1(X))\}$, and consider $$\bigwedge(\Lambda^1(X)):=\bigoplus_{k=0}^n\bigwedge^k(\Lambda^1(X)),$$ which is a diffeological vector pseudo-bundle for its standard diffeology based on the tensor product diffeology. Then the standard Clifford action $c^{\Lambda}$ of ${C \kern -0.1em \ell}(\Lambda^1(X),g^{\Lambda})$ on $\bigwedge(\Lambda^1(X))$ is smooth ([@pseudo-bundles-exterior-algebras]); furthermore, the connection $\nabla^{\Lambda}$ induces (in a completely standard way) a connection $\nabla^{\bigwedge(\Lambda)}$ on $\bigwedge(\Lambda^1(X))$. Let $X$ be a diffeological space satisfying the above conditions. The **diffeological De Rham operator** on $(X,g^{\Lambda})$ is the operator $$D_{dR}:C^{\infty}(X,\bigwedge(\Lambda^1(X)))\to C^{\infty}(X,\bigwedge(\Lambda^1(X)))$$ defined as the composition $$D_{dR}=c^{\Lambda}\circ\nabla^{\bigwedge(\Lambda)}.$$ Let $X$ be a wedge of two copies of the standard ${{\mathbb{R}}}^2$ at the origin; endow $X$ with the corresponding gluing diffeology and denote the two copies of ${{\mathbb{R}}}^2$ by $X_1$ and $X_2$ respectively. Each of the two spaces $\Lambda^1(X_1)$, $\Lambda^1(X_2)$ can thus be standardly identified with $T^*{{\mathbb{R}}}^2$, and every fibre written in the form, $\Lambda_{(x_1,y_1)}^1(X_1)=\mbox{Span}(dx_1,dy_1)$ and $\Lambda_{(x_2,y_2)}^1(X_2)=\mbox{Span}(dx_2,dy_2)$. Let $g_i^{\Lambda}$ be the pseudo-metric on $\Lambda^1(X_i)$ given by $$g_i^{\Lambda}(x_i,y_i)=\frac{\partial}{\partial x_i}\otimes\frac{\partial}{\partial x_i}+e^{x_iy_i}\frac{\partial}{\partial y_i}\otimes\frac{\partial}{\partial y_i}\,\,\mbox{ for }i=1,2,$$ where $\frac{\partial}{\partial x_i}$ is the dual map of $dx_i$ and $\frac{\partial}{\partial y_i}$ is the dual of $dy_i$. Let $g^{\Lambda}$ be the corresponding induced pseudo-metric on $\Lambda^1(X)$. Notice that $g_1^{\Lambda}$ and $g_2^{\Lambda}$ are automatically compatible, since all the compatibility conditions are empty in the case of gluing along a single-point set; in particular, we have $$g^{\Lambda}(0,0)=\frac12(\frac{\partial}{\partial x_1}\otimes\frac{\partial}{\partial x_1}+\frac{\partial}{\partial y_1}\otimes\frac{\partial}{\partial y_2}+ \frac{\partial}{\partial x_2}\otimes\frac{\partial}{\partial x_2}+\frac{\partial}{\partial y_2}\otimes\frac{\partial}{\partial y_2}).$$ Every fibre of $\bigwedge(\Lambda^1(X))$ outside of the wedge point coincides with $\bigwedge({{\mathbb{R}}}^2)$, while at the wedge point it is $\bigwedge({{\mathbb{R}}}^2\oplus{{\mathbb{R}}}^2)$. The Clifford algebra ${C \kern -0.1em \ell}(\Lambda^1(X),g^{\Lambda})$ behaves as ${C \kern -0.1em \ell}(\Lambda^1(X_i),g_i^{\Lambda})$, for the appropriate $i=1,2$, outside of the wedge point. At the wedge point it is equivalent to ${C \kern -0.1em \ell}({{\mathbb{R}}}^4,\langle\cdot,\cdot\rangle)$, where $\langle\cdot,\cdot\rangle$ is the canonical scalar product. The Clifford action $c^{\Lambda}$ is standardly defined; for instance, $$c^{\Lambda}(dx_1)(dy_2)=dx_1\wedge dy_2-\frac12.$$ The sections of $\Lambda^1(X)$ are in one-to-one correspondence with pairs of sections of $\Lambda^1(X_1)$ and $\Lambda^1(X_2)$: if $s\in C^{\infty}(X,\Lambda^1(X))$ then $s_1=\tilde{\rho}_1^{\Lambda}\circ s\circ\tilde{i}_1\in C^{\infty}(X_1,\Lambda^1(X_1))$ and $s_2=\tilde{\rho}_2^{\Lambda}\circ s\circ i_2\in C^{\infty}(X_2,\Lambda^1(X_2))$. *Vice versa*, and this is specific to the present instance, given $s_1\in C^{\infty}(X_1,\Lambda^1(X_1))$ and $s_2\in C^{\infty}(X_2,\Lambda^1(X_2))$, we set $s(0,0)=s_1(0,0)\oplus s_2(0,0)$, while outside the wedge point $s$ is equivalent to either $s_1$ or $s_2$ in the obvious sense. Both $X_1$ and $X_2$ are endowed with the standard Levi-Civita connections $\nabla^{\Lambda,1}$ and $\nabla^{\Lambda,2}$ respectively. These induce the Levi-Civita $\nabla^{\Lambda}$ on $X$ (relative to the induced pseudo-metric $g^{\Lambda}$); for a section $s$ of $\Lambda^1(X)$ determined by a pair of sections $s_1\in C^{\infty}(X_1,\Lambda^1(X_1))$ and $s_2\in C^{\infty}(X_2,\Lambda^1(X_2))$, $\nabla^{\Lambda}s$ coincides (up to appropriate identifications) with either $\nabla^{\Lambda,1}s_1$ or $\nabla^{\Lambda,2}s_2$, while at the wedge point its value is essentially $(\nabla^{\Lambda,1}s_1)(0,0)\oplus(\nabla^{\Lambda,2}s_2)(0,0)$ (a formalization of this construction is available in [@connectionsLC]). A fully standard procedure completes the construction. In a previous section we indicated that one (but not the only one) problem in defining the Hodge star for diffeological spaces is that the standard definition does not, in general, yield a map $\bigwedge^k(\Lambda^1(X))\to\bigwedge^{n-k}(\Lambda^1(X))$ for a fixed $n$ independent of $k$. It follows that there might be a way to define $\star$ as taking values in $\bigwedge(\Lambda^1(X))$ if $\Lambda^1(X)$ has bounded dimension. Since this was not the only difficulty in extending the definition of $d+d^*$ to the diffeological context (recall that already $d$ does not descend to a pseudo-bundle map on $\Lambda^1(X)$), we do not go in that direction for now. Appendix: on the possibility of the De Rham-like operator $d^*+d^{**}$ {#appendix-on-the-possibility-of-the-de-rham-like-operator-dd .unnumbered} ---------------------------------------------------------------------- We briefly consider here the possibility of defining a De Rham-like operator $d^*+d^{**}$, based on the map $d^*:(\Omega^{k+1}(X))^*\to(\Omega^k(X))^*$ dual to the differential. This construction comes from the following observations. One, each dual pseudo-bundle $(\Lambda^k(X))^*$ embeds into the trivial pseudo-bundle $X\times(\Omega^k(X))^*$ via the map $(\pi^{\Omega^k,\Lambda^k})^*$ that is the map dual to the defining projection $\pi^{\Omega^k,\Lambda^k}:X\times\Omega^k(X)\to\Lambda^k(X)$ (the dual map is an embedding simply because $\pi^{\Omega^k,\Lambda^k}$ is surjective, and by definition of the dual pseudo-bundle diffeology). Suppose for the moment that we have $$d^*\left((\pi^{\Omega^{k+1},\Lambda^{k+1}})^*((\Lambda^{k+1}(X))^*)\right)\subseteq(\pi^{\Omega^k,\Lambda^k})^*((\Lambda^k(X))^*);$$ then $d^*$ is well-defined as a map $$d^*:(\Lambda^{k+1}(X))^*\to(\Lambda^k(X))^*.$$ The second observation is that if $\Lambda^k(X)$ has only finite-dimensional fibres, then all of these fibres are standard; so if each $\Lambda^k(X)$ is endowed with a pseudo-metric $g^{\Lambda^k}$ then the corresponding pairing map $\Phi_{g^{\Lambda^k}}$ is a diffeomorphism $\Lambda^k(X)\to(\Lambda^k(X))^*$. Therefore the composition $$(\Phi_{g^{\Lambda^k}})^{-1}\circ d^*\circ\Phi_{g^{\Lambda^{k+1}}},$$ also denoted by $d^*$, is a well-defined smooth operator $\Lambda^{k+1}(X)\to\Lambda^k(X)$. The corresponding dual map $$(d^*)^*:(\Lambda^k(X))^*\to(\Lambda^{k+1}(X))^*$$ likewise provides us with a well-defined operator $$d^{**}:=(\Phi_{g^{\Lambda^{k+1}}})^{-1}\circ(d^*)^*\circ\Phi_{g^{\Lambda^k}}:\Lambda^k(X)\to\Lambda^{k+1}(X).$$ It then suffices to assume that $X$ has finite dimension $n$ to obtain a well-defined De Rham-like operator on $\Lambda(X):=\bigoplus_{k=0}^n\Lambda^k(X)$ which is $$d^*+d^{**}:\Lambda(X)\to\Lambda(X).$$ Let us now consider the potential inclusion $$d^*\left((\pi^{\Omega^{k+1},\Lambda^{k+1}})^*((\Lambda^{k+1}(X))^*)\right)\subseteq(\pi^{\Omega^k,\Lambda^k})^*((\Lambda^k(X))^*).$$ Let $\alpha_{k+1}^*\in(\Lambda^{k+1}(X))^*$; consider $$d^*((\pi^{\Omega^{k+1},\Lambda^{k+1}})^*(\alpha_{k+1}^*))(\omega_k)=\alpha_{k+1}^*(d\omega_k+\Omega_x^{k+1}(X)).$$ Under the assumption that all $\Lambda_x^1k(X)$ are finite-dimensional, it is the image of some $\alpha_k^*\in(\Lambda^k(X))^*$ if and only if $$d\omega_k+\Omega_x^{k+1}(X)\in\mbox{Ker}(\alpha_{k+1}^*)\,\,\mbox{ for all }\,\omega_k\in\Omega_x^k(X).$$ Although this is a less restrictive condition than $d\omega_k\in\Omega_x^{k+1}(X)$, there is no obvious reason for it to hold *a priori*; it essentially requires that each element of $(\Lambda_x^{k+1}(X))^*$ vanish on the image $\pi^{\Omega^{k+1},\Lambda^{k+1}}(B_{dR}^{k+1}(X))$ of the space of the coboundaries. 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--- abstract: 'We show that two evanescently coupled $\chi^{(2)}$ parametric downconverters inside a Fabry-Perot cavity provide a tunable source of quadrature squeezed light, Einstein-Podolsky-Rosen correlations and quantum entanglement. Analysing the operation in the below threshold regime, we show how these properties can be controlled by adjusting the coupling strengths and the cavity detunings. As this can be implemented with integrated optics, it provides a possible route to rugged and stable EPR sources.' author: - 'M. K. Olsen and P. D. Drummond' title: 'Entanglement and the Einstein-Podolsky-Rosen paradox with coupled intracavity optical downconverters' --- Introduction ============ The Einstein, Podolsky and Rosen (EPR) paradox stems from a famous paper published in $1935$ [@EPR], which showed that local realism is not consistent with quantum mechanical completeness. A direct and feasible demonstration of the EPR paradox with continuous variables was first suggested using nondegenerate parametric amplification (also known as the OPA) [@eprquad]. The optical quadrature phase amplitudes used in these proposals have the same mathematical properties as the position and momentum originally used by EPR. Even though the correlations between these are not perfect, they are still entangled sufficiently to allow for an inferred violation of the uncertainty principle, which is equivalent to the EPR paradox [@eprMDR; @rd]. An experimental demonstration of this proposal by Ou *et al.* soon followed, showing a clear agreement with quantum theory [@Ou]. In this work, rather than using the nondegenerate optical parametric oscillator (OPO), we consider an alternative device using two degenerate type I downconverters inside the same optical cavity, and coupled by evanescent overlaps of the intracavity modes within the nonlinear medium. Generally, such a device may be considered as either a single nonlinear crystal pumped by two spatially separated lasers, or two waveguides with a $\chi^{(2)}$ component. We calculate phase-dependent correlations between the two low frequency outputs of the cavity in the below threshold regime, showing that this system exhibits a wide range of behaviour and is potentially an easily tunable source of single-mode squeezing, entangled states and states which exhibit the EPR paradox. The spatial separation of the output modes means that they do not have to be separated by optical devices before measurements can be made, along with the unavoidable losses which would result from this procedure. The entangled beams produced can be degenerate in both frequency and polarisation, unlike those of the nondegenerate OPO, and would exit the cavity at spatially separated locations. These correlations are tunable by controlling some of the operational degrees of freedom of the system, including the evanescent couplings between the two waveguides, the input power and the cavity detunings. The term nonlinear coupler was given to a system of two coupled waveguides without an optical cavity by Perina *et al.* [@coupler]. Generically, the device consists of two parallel optical waveguides which are coupled by an evanescent overlap of the guided modes. The quantum statistical properties of this device when the nonlinearity is of the $\chi^{(3)}$ type have been theoretically investigated, predicting energy transfer between the waveguides [@Ibrahim] and the generation of correlated squeezing [@korea]. Coupled $\chi^{(2)}$ downconversion processes in the travelling wave configuration have also been examined theoretically, predicting that light produced in one of the media can be controlled by light entering the other [@mista], and that such a device can produce entanglement of the output beams [@herec]. The coupler with $\chi^{(2)}$ nonlinearity held inside a pumped Fabry-Perot cavity, and operating in the second harmonic generation (SHG) configuration, was introduced by Bache *et al.* [@dimer], who named it the quantum optical dimer by analogy with various systems that display coupling between discrete sites. They analysed intensity correlations between the modes, predicting noise suppression in both the sum and the difference. As the intracavity $\chi^{(2)}$ downconversion processes have long been appreciated as sources of quantum states of the electromagnetic field (See Martinelli *et al.* [@bigpaul] for an overview), we will combine and extend these previous analyses to consider two coupled downconverters operating inside a Fabry-Perot cavity. The advantage of this proposal is the all-integrated nature of the device, which promises greatly increased robustness. Additional potential advantages are the reductions in threshold pump power and phase noise, relative to current practise. Another potential advantage as compared to the normal type II polarisation nondegenerate OPO lies in the difficulty of stabilising this device at frequency degeneracy [@Fabre1; @Fabre2]. Our proposal should be well stabilised by the linear coupling, without having to add any additional features. The system and equations of motion {#sec:equations} ================================== The physical device we wish to examine differs from that described in Ref. [@dimer] in one important detail. We will analyse it in the downconversion regime, where the cavity pumping is at a frequency $2\omega_{L}\simeq\omega_{b}$. As this device has been described in detail in Ref. [@dimer], we will give a briefer description of the essential features here. The system of interest consists of two coupled nonlinear $\chi^{(2)}$ waveguides inside a driven optical cavity, which may utilise integrated Bragg reflection for compactness. Each waveguide supports two resonant modes at frequencies $\omega_{a},\,\omega_{b}$, where $2\omega_{a}\simeq\omega_{b}$. The higher frequency modes at $\omega_{b}$ are driven coherently with an external laser, while the nonlinear interaction within the waveguides produces pairs of downconverted photons with frequency $\omega_{a}$ . We assume that only the cavity modes at these two frequencies are important and that there is perfect phase matching inside the media. The two waveguides are evanescently coupled as in Ref. [@dimer]. Besides the differences in the pumping frequency, we will be interested in the phase-dependent correlations necessary to demonstrate entanglement and the EPR paradox, rather than the intensity correlations of Ref. [@dimer]. The effective Hamiltonian for the system can be written as $${\mathcal{H}}_{eff}={\mathcal{H}}_{int}+{\mathcal{H}}_{couple}+{\mathcal{H}}_{pump}+{\mathcal{H}}_{res}, \label{eq:Heff}$$ where the nonlinear interaction with the $\chi^{(2)}$ media is described by $${\mathcal{H}}_{int}=i\hbar\frac{\kappa}{2}\left[\hat{a}_{1}^{\dag\;2}\hat{b}_{1}-\hat{a}_{1}^{2}\hat{b}_{1}^{\dag}+ \hat{a}_{2}^{\dag\;2}\hat{b}_{2}-\hat{a}_{2}^{2}\hat{b}_{2}^{\dag}\right]\,\,. \label{eq:Hnl}$$ Here $\kappa$ denotes the effective nonlinearity of the waveguides (we assume that the two are equal), and $\hat{a}_{k},\;\hat{b}_{k}$ are the bosonic annihilation operators for quanta at the frequencies $\omega_{a},\;\omega_{b}$ within the crystal $k\;(=1,2)$. The coupling by evanescent waves is described by $${\mathcal{H}}_{couple}=\hbar J_{a}\left[\hat{a}_{1}\hat{a}_{2}^{\dag}+\hat{a}_{1}^{\dag}\hat{a}_{2}\right]+\hbar J_{b} \left[\hat{b}_{1}\hat{b}_{2}^{\dag}+\hat{b}_{1}^{\dag}\hat{b}_{2}\right], \label{eq:Hcouple}$$ where the $J_{k}$ are the coupling parameters at the two frequencies, as described in Ref. [@dimer]. We note that in that work it is stated that the lower frequency coupling, $J_{a}$, is generally stronger than the higher frequency coupling, $J_{b}$, and also that values of $J_{a}$ as high as $50$ times the lower frequency cavity loss rate were calculated to be physically reasonable. The cavity pumping is described by $${\mathcal{H}}_{pump}=i\hbar\left[\epsilon_{1}\hat{b}_{1}^{\dag}-\epsilon_{1}^{\ast}\hat{b}_{1}+\epsilon_{2}\hat{b}_{2}^{\dag}- \epsilon_{2}^{\ast}\hat{b}_{2}\right], \label{eq:Hpump}$$ where the $\epsilon_{k}$ represent pump fields which we will describe classically. Finally, the cavity damping is described by $${\mathcal{H}}_{res}=\hbar\sum_{k=1}^{2}\left(\Gamma_{a}^{k}\hat{a}_{k}^{\dag}+\Gamma_{b}^{k}\hat{b}_{k}^{\dag}\right)+h.c., \label{eq:Hres}$$ where the $\Gamma^{k}$ represent bath operators at the two frequencies and we have made the usual zero temperature approximation for the reservoirs. With the standard methods [@GardinerQN], and using the operator/c-number correspondences $(\hat{a}_{j}\leftrightarrow\alpha_{j},\hat{b}_{j}\leftrightarrow\beta_{j})$, the Hamiltonian can be mapped onto a Fokker-Planck equation for the Glauber-Sudarshan P-distribution [@Roy; @Sud]. However, as the diffusion matrix of this Fokker-Planck equation is not positive-definite, it cannot be mapped onto a set of stochastic differential equations. Hence we will use the positive-P representation [@plusP] which, by doubling the dimensionality of the phase-space, allows a Fokker-Planck equation with a positive-definite diffusion matrix to be found and thus a mapping onto stochastic differential equations. Making the correspondence between the set of operators $(\hat{a}_{j},\hat{a}_{j}^{\dag},\hat{b}_{j},\hat{b}_{j}^{\dag})$ $(j=1,2)$ and the set of c-number variables $(\alpha_{j},\alpha_{j}^{+},\beta_{j},\beta_{j}^{+})$, we find the following set of equations, $$\begin{aligned} \frac{d\alpha_{1}}{dt} & = & -(\gamma_{a}+i\Delta_{a})\alpha_{1}+\kappa\alpha_{1}^{+}\beta_{1}+iJ_{a}\alpha_{2}+\sqrt{\kappa\beta_{1}}\;\eta_{1}(t),\nonumber \\ \frac{d\alpha_{1}^{+}}{dt} & = & -(\gamma_{a}-i\Delta_{a})\alpha_{1}^{+}+\kappa\alpha_{1}\beta_{1}^{+}-iJ_{a}\alpha_{2}^{+}+ \sqrt{\kappa\beta_{1}^{+}}\;\eta_{2}(t),\nonumber \\ \frac{d\alpha_{2}}{dt} & = & -(\gamma_{a}+i\Delta_{a})\alpha_{2}+\kappa\alpha_{2}^{+}\beta_{2}+iJ_{a}\alpha_{1}+\sqrt{\kappa\beta_{2}}\;\eta_{3}(t),\nonumber \\ \frac{d\alpha_{2}^{+}}{dt} & = & -(\gamma_{a}-i\Delta_{a})\alpha_{2}^{+}+\kappa\alpha_{2}\beta_{2}^{+}-iJ_{a}\alpha_{1}^{+}+ \sqrt{\kappa\beta_{2}^{+}}\;\eta_{4}(t),\nonumber \\ \frac{d\beta_{1}}{dt} & = & \epsilon_{1}-(\gamma_{b}+i\Delta_{b})\beta_{1}-\frac{\kappa}{2}\alpha_{1}^{2}+iJ_{b}\beta_{2},\nonumber \\ \frac{d\beta_{1}^{+}}{dt} & = & \epsilon_{1}^{\ast}-(\gamma_{b}-i\Delta_{b})\beta_{1}^{+}-\frac{\kappa}{2}\alpha_{1}^{+\;2}-iJ_{b}\beta_{2}^{+},\nonumber \\ \frac{d\beta_{2}}{dt} & = & \epsilon_{2}-(\gamma_{b}+i\Delta_{b})\beta_{2}-\frac{\kappa}{2}\alpha_{2}^{2}+iJ_{b}\beta_{1},\nonumber \\ \frac{d\beta_{2}^{+}}{dt} & = & \epsilon_{2}^{\ast}-(\gamma_{b}-i\Delta_{b})\beta_{2}^{+}-\frac{\kappa}{2}\alpha_{2}^{+\;2}-iJ_{b}\beta_{1}^{+}, \label{eq:PPSDE}\end{aligned}$$ where the $\gamma_{k}$ represent cavity damping. We have also added cavity detunings $\Delta_{a,b}$ from the two resonances, so that for a pump laser at angular frequency $2\omega_{L}$, one has $\Delta_{a}=\omega_{a}-\omega_{L}$and $\Delta_{b}=\omega_{b}-2\omega_{L}$. Below, in section \[sec:detune\], we will investigate detuning effects in greater detail. The real Gaussian noise terms have the correlations $\overline{\eta_{j}(t)}=0$ and $\overline{\eta_{j}(t)\eta_{k}(t')}=\delta_{jk}\delta(t-t')$. Note that, due to the independence of the noise sources, $\alpha_{k}\;(\beta_{k})$ and $\alpha_{k}^{+}\;(\beta_{k}^{+})$ are not complex conjugate pairs, except in the mean over a large number of stochastic integrations of the above equations. However, these equations do allow us to calculate the expectation values of any desired time-normally ordered operator moments, exactly as required to calculate spectral correlations. Linearised analysis {#sec:linearise} =================== In an operating regions where it is valid, a linearised fluctuation analysis provides a simple way of calculating both intracavity and output spectra of the system [@DFW; @mjc], by treating it as an Ornstein-Uhlenbeck process [@ornstein]. To perform this analysis we first divide the variables of Eq. \[eq:PPSDE\] into a steady-state mean value and a fluctuation part, e.g. $\alpha_{1}\rightarrow\alpha_{1}^{ss}+\delta\alpha_{1}$ and so on for the other variables. We find the steady state solutions by solving the equations (\[eq:PPSDE\]) without the noise terms (note that in this section we will treat all fields as being at resonance), and write the equations for the fluctuation vector $\delta\tilde{x}=[\delta\alpha_{1},\delta\alpha_{1}^{+},\delta\alpha_{2},\delta\alpha_{2}^{+},\delta\beta_{1},\delta\beta_{1}^{+},\delta\beta_{2},\delta\beta_{2}^{+}]^{T}$, to first order in these fluctuations, as $$d\;\delta\tilde{x}=-A\delta\tilde{x}\; dt+BdW, \label{eq:dAB}$$ where the drift matrix is $$A=\left[\begin{array}{cc} A_{aa} & -A_{ba}^{*}\\ A_{ba} & A_{bb}\end{array}\right],$$ with $$\begin{aligned} A_{aa} & = & \left[\begin{array}{cccc} \gamma_{a} & -\kappa\beta_{1}^{ss} & -iJ_{a} & 0\\ -\kappa\beta_{1}^{ss\ast} & \gamma_{a} & 0 & iJ_{a}\\ -iJ_{a} & 0 & \gamma_{a} & -\kappa\beta_{2}^{ss}\\ 0 & iJ_{a} & -\kappa\beta_{2}^{ss\ast} & \gamma_{a}\end{array}\right],\nonumber \\ A_{ba} & = & \left[\begin{array}{cccc} \kappa\alpha_{1}^{ss} & 0 & 0 & 0\\ 0 & \kappa\alpha_{1}^{ss\ast} & 0 & 0\\ 0 & 0 & \kappa\alpha_{2}^{ss} & 0\\ 0 & 0 & 0 & \kappa\alpha_{2}^{ss\ast}\end{array}\right],\nonumber \\ A_{bb} & = & \left[\begin{array}{cccc} \gamma_{b} & 0 & -iJ_{b} & 0\\ 0 & \gamma_{b} & 0 & iJ_{b}\\ -iJ_{b} & 0 & \gamma_{b} & 0\\ 0 & iJ_{b} & 0 & \gamma_{b}\end{array}\right].\end{aligned}$$ In this equation, $dW$ is a vector of real Wiener increments, and the matrix $B$ is zero except for the first four diagonal elements, which are respectively $\sqrt{\kappa\beta_{1}^{ss}},\;\sqrt{\kappa\beta_{1}^{ss\ast}},\;\sqrt{\kappa\beta_{2}^{ss}},\;\sqrt{\kappa\beta_{2}^{ss\ast}}$. The essential conditions for this expansion to be valid are that moments of the fluctuations be smaller than the equivalent moments of the mean values, and that the fluctuations stay small. In the case of the single optical parametric oscillator (OPO), it is well known that there is a critical operating point around which this condition does not hold. This point is easily found by examination of the eigenvalues of the equivalent fluctuation drift matrix for that system, and this procedure is also valid in the present case. The fluctuations will not tend to grow as long as none of the eigenvalues of the matrix $A$ develop a negative real part. At the point at which this happens the linearised fluctuation analysis is no longer valid, as the fluctuations can then grow exponentially and the necessary conditions for linearisation are no longer fulfilled. In this work we will only be interested in a region where linearisation is valid. To examine the stability of the system, we first need to find the steady state solutions for the amplitudes, by solving for the steady state of Eq. \[eq:PPSDE\] with the noise terms dropped. As in the usual optical parametric oscillator, there is an oscillation threshold below which $\alpha_{j}^{ss}=0$ and only the high frequency mode is populated. In the present case, for a real pump, we find $\beta_{j}^{ss}=\epsilon/(\gamma_{b}-iJ_{b})$. Inserting these solutions in the matrix $A$ allows us to find simple expressions for the eigenvalues, $$\begin{aligned} \lambda_{1,2} & = & \gamma_{b}+iJ_{b},\nonumber \\ \lambda_{3,4} & = & \gamma_{b}-iJ_{b},\nonumber \\ \lambda_{5,6} & = & \gamma_{a}+\sqrt{\left[\kappa^{2}\epsilon^{2}/\widetilde{\gamma}_{b}^{2}-J_{a}^{2}\right]},\nonumber \\ \lambda_{7,8} & = & \gamma_{a}-\sqrt{\left[\kappa^{2}\epsilon^{2}/\widetilde{\gamma}_{b}^{2}-J_{a}^{2}\right]}. \label{eq:autovalores}\end{aligned}$$ Here we have introduced auxiliary variables, $\widetilde{\gamma}{}_{a,b}=\sqrt{\gamma_{a,b}^{2}+J_{a,b}^{2}}$. We immediately see that $\lambda_{7,8}$ can develop negative real parts for a pump amplitude greater than the critical value, $\epsilon_{c}=\widetilde{\gamma}{}_{a}\widetilde{\gamma}{}_{b}/\kappa$. As it must, this expression reduces to the single OPO expression of $\gamma_{a}\gamma_{b}/\kappa$ when the couplings are set to zero. In that case, there is then a stable above threshold solution in which the high frequency mode inside the cavity remains constant, independently of any further increase in the pumping, and the low frequency mode becomes occupied. In the present case, it is not simple to find general expressions for these above threshold solutions analytically, but as we will concentrate our attention on the rich variety of below threshold behaviour which is exhibited, this is not important here. We note here that, unlike the single OPO case with a resonant cavity, the threshold pumping is not a constant for fixed cavity loss rates, but is a function of the coupling strengths between the two waveguides. Using the below threshold solutions, we may calculate any desired time normally-ordered spectral correlations inside the cavity using the simple formula $$S(\omega)=\left(A+i\omega\openone\right)^{-1}BB^{\text{T}}\left(A^{\text{T}}-i\omega\openone\right)^{-1}, \label{eq:inspek}$$ after which we use the standard input-output relations [@mjc] to relate these to quantities which may be measured outside the cavity. Quantum correlations {#sec:correlations} ==================== Single mode squeezing {#subsec:squeezado} --------------------- The first quantities we wish to calculate are the single mode quadrature squeezing spectra, to compare these with the well-known results for the normal uncoupled OPO. Defining the quadrature amplitudes as $$\hat{X}_{j}^{\theta}=\hat{a}_{j}\mbox{e}^{-i\theta}+\hat{a}_{j}^{\dag}\mbox{e}^{i\theta},\label{eq:Xthetageral}$$ (where $j=1,2$), we will use the notation $$\begin{aligned} \hat{X}_{j}^{0} & = & \hat{X}_{j},\nonumber \\ \hat{X}_{j}^{\frac{\pi}{2}} & = & \hat{Y}_{j}.\label{eq:XYgeral}\end{aligned}$$ We note here that the quadrature definitions do not need to specify whether it is mode $a$ or $b$ which is involved, as we do not find any interesting behaviour in the high frequency modes below threshold and hence will only present results for the low frequency modes. With this normalisation the coherent state value for the quadrature variances is one. To simplify our results we will assume that the pumping terms for each crystal are real and equal $(\epsilon_{1}=\epsilon_{2}=\epsilon)$. The expressions for the below threshold low frequency quadrature variances in the single OPO case are well known [@osenhor], being $$\begin{aligned} S_{X}^{\text{out}}(\omega) & = & 1+\frac{4\gamma_{a}\gamma_{b}\kappa\epsilon}{(\gamma_{a}\gamma_{b}-\kappa\epsilon)^{2}+\gamma_{b}^{2}\omega^{2}},\nonumber \\ S_{Y}^{\text{out}}(\omega) & = & 1-\frac{4\gamma_{a}\gamma_{b}\kappa\epsilon}{(\gamma_{a}\gamma_{b}+\kappa\epsilon)^{2}+\gamma_{b}^{2}\omega^{2}}, \label{eq:noncoupledVXVY}\end{aligned}$$ and predicting zero-frequency squeezing which becomes perfect in the $Y$ quadrature as the pump approaches the critical threshold value, $\epsilon=\gamma_{a}\gamma_{b}/\kappa$, although the linearised analysis breaks down near this point. Note that the variances inside and outside the cavity are related by $S_{X}^{\text{out}}=1+2\gamma_{a}V(\hat{X})$. Our coupled system would be expected to exhibit the above values in the limit as $J_{a,b}\rightarrow 0$, which provides a standard for comparison with the analytical results. In the general case, we find that $S_{X_{1}^{\theta}}^{\text{out}}=S_{X_{2}^{\theta}}^{\text{out}}$, as expected. We also find that the coupling means that the intracavity high frequency field is no longer real, but has a phase given by $\Theta_{b}=\tan^{-1}(J_{b}/\gamma_{b})$. This will mean that the optimum correlations will no longer generally be found in the $X_{j}$ and $Y_{j}$ quadratures, but at some other phase angle, as found previously for second harmonic generation in detuned cavities [@granja]. Experimentally, this does not present a problem as the local oscillator phase is normally swept across all angles, which must therefore include the optimum angle. We can find analytical solutions for the angle of maximum single-mode squeezing (and antisqueezing), for example, these differing by $\pi/2$ and being found as $$\theta_{opt}=\tan^{-1}\left\{ \frac{2V(\hat{X},\hat{Y})}{V(\hat{Y})-V(\hat{X})\pm\sqrt{\left[V(\hat{Y})-V(\hat{X})\right]^{2}+ 4\left[V(\hat{X},\hat{Y})\right]^{2}}}\right\} , \label{eq:angulo}$$ where $V(A,B)=\langle AB\rangle-\langle A\rangle\langle B\rangle$. However, as this expression is a complicated function of several variables when written out in full, and will not necessarily give the optimum choices at all frequencies, nor when we consider correlations between the modes, we will present results where the local oscillator angle has been optimised numerically. The $\hat{X}$ and $\hat{Y}$ spectral variances outside the cavity are found as $$\begin{aligned} S_{X_{1,2}}^{\text{out}}(\omega) & = & 1+\frac{4\gamma_{a}\kappa\epsilon\left\{\gamma_{b}\left[\widetilde{\gamma_{b}}^{2}(\omega^{2}-J_{a}^{2})+ J_{b}^{2}\gamma_{a}^{2}+(\gamma_{a}\gamma_{b}+\kappa\epsilon)^{2} \right]+2\gamma_{a}J_{b}^{2}\kappa\epsilon\right\}} {4\gamma_{a}^{2}\widetilde{\gamma_{b}}^{4}\omega^{2}+\left[\widetilde{\gamma_{b}}^{2}(\widetilde{\gamma_{a}}^{2}-\omega^{2})-\kappa^{2}\epsilon^{2}\right]^{2}},\nonumber \\ S_{Y_{1,2}}^{\text{out}}(\omega) & = & 1-\frac{4\gamma_{a}\kappa\epsilon\left\{\gamma_{b}\left[\widetilde{\gamma_{b}}^{2}(\omega^{2}-J_{a}^{2})+ J_{b}^{2}\gamma_{a}^{2}+(\gamma_{a}\gamma_{b}-\kappa\epsilon)^{2} \right]+2\gamma_{a}J_{b}^{2}\kappa\epsilon\right\}} {4\gamma_{a}^{2}\widetilde{\gamma_{b}}^{4}\omega^{2}+\left[\widetilde{\gamma_{b}}^{2}(\widetilde{\gamma_{a}}^{2}-\omega^{2})-\kappa^{2}\epsilon^{2}\right]^{2}}, \label{eq:coupledVXVY}\end{aligned}$$ which, as expected, reduce to the single OPO expressions above (\[eq:noncoupledVXVY\]) when the coupling terms are set to zero. The output covariance is $$V(\hat{X}_{j},\hat{Y}_{j})=\frac{4\gamma_{a}J_{b}\kappa\epsilon\left[\widetilde{\gamma}_{b}^{2}(\gamma_{a}^{2}-J_{a}^{2}+\omega^{2})+\kappa^{2}\epsilon^{2}\right]} {4\gamma_{a}^{2}\widetilde{\gamma_{b}}^{4}\omega^{2}+\left[\widetilde{\gamma_{b}}^{2}(\widetilde{\gamma_{a}}^{2}-\omega^{2})-\kappa^{2}\epsilon^{2}\right]^{2}}, \label{eq:covXY}$$ which will give $\theta_{opt}=0,\pi/2$ for the uncoupled case, where $\hat{Y}$ is the squeezed quadrature and $\hat{X}$ the antisqueezed quadrature. In Fig. \[fig:VXJbconst\] we show the single-mode output spectral quadrature variances for the quadrature of best squeezing as the low-frequency mode coupling strength is varied, beginning with $J_{a}=J_{b}=\gamma_{a}=\gamma_{b}=\gamma$. We note here that the pump values used in all the displayed results, $\epsilon_{j}=0.5\epsilon_{c}$, depend on the couplings as stated above and are therefore different for different combinations of the couplings, but are all the same fraction of the threshold value. We find less single-mode squeezing than in the uncoupled case for the same ratio $\epsilon/\epsilon_{c}$, and also find that changing $J_{b}$ mainly serves to change the angle of maximum squeezing. Changing $J_{a}$ changes the frequency at which the maximum of squeezing is found. We see that this device is not as efficient at producing squeezed single-mode outputs as the normal OPO, but as we are interested in the quantum correlations between the two output modes, we will now examine these. Entanglement and the EPR paradox {#subsec:EPR} -------------------------------- An entanglement criterion for optical quadratures has been outlined by Dechoum *et al.* [@ndturco], following from criteria developed by Duan *et al.* [@Duan] which are based on the inseparability of the system density matrix. A theoretical method to demonstrate the EPR paradox using quadrature amplitudes was developed by Reid [@eprMDR], using the mathematical similarities of the quadrature operators to the original position and momentum operators. We will briefly outline these criteria here and then apply them to our system, using the quadrature operators $\hat{X}_{j}$ and $\hat{Y}_{j}$. Note that even though these quadratures have the same mathematical properties as the canonical position and momentum operators for the harmonic oscillator, they correspond physically to the real and imaginary parts of the electromagnetic field, not its position and momentum. To demonstrate entanglement between the modes, we define the combined quadratures $\hat{X}_{\pm}=\hat{X}_{1}\pm\hat{X}_{2}$ and $\hat{Y}_{\pm}=\hat{Y}_{1}\pm\hat{Y}_{2}$ and calculate the variances in these, which we may do analytically. Optimising the result for arbitrary phase angles is better performed numerically. Following the treatment of Ref. [@ndturco], entanglement is guaranteed provided that $$S_{X_{\pm}}^{\text{out}}+S_{Y_{\mp}}^{\text{out}}<4. \label{eq:inequalityduan}$$ We note here that the combined variance defined in this way has an obvious relationship with the well-known two-mode squeezed states which are produced, for example, by the nondegenerate OPO [@democrat1; @democrat2], but that the quadratures between which we find entanglement here are not the same as those which are entangled in that case, where these are $\hat{X}_{-}$ and $\hat{Y}_{+}$. In the present case, considering only the phase angles $\theta=0$ and $\pi/2$, we find entanglement with $\hat{X}_{+}$ and $\hat{Y}_{-}$. The two individual variances can be written as $$\begin{aligned} S_{X_{\pm}}^{\text{out}} & = & S_{X_{1}}^{\text{out}}+S_{X_{2}}^{\text{out}}\pm2V(\hat{X}_{1},\hat{X}_{2}),\nonumber \\ S_{X_{\pm}}^{\text{out}} & = & S_{X_{1}}^{\text{out}}+S_{X_{2}}^{\text{out}}\pm2V(\hat{Y}_{1},\hat{Y}_{2}). \label{eq:sumdiffvars}\end{aligned}$$ The individual quadrature variances are given above (\[eq:coupledVXVY\]), while for the covariances we find: $$V(\hat{X}_{1},\hat{X}_{2})=\frac{-8J_{a}J_{b}\gamma_{a}^{2}\widetilde{\gamma}_{b}^{2}\kappa\epsilon}{{4\gamma_{a}^{2}\widetilde{\gamma}_{b}^{4}\omega^{2} +\left[\widetilde{\gamma}_{b}^{2}(\widetilde{\gamma_{a}}^{2}-\omega^{2})-\kappa^{2}\epsilon^{2}\right]^{2}}}, \label{eq:VXX}$$ and $V(\hat{Y}_{1},\hat{Y}_{2})=-V(\hat{X}_{1},\hat{X}_{2})$, showing that the $\hat{X}$ quadratures are anticorrelated and the $\hat{Y}$ quadratures are correlated. Although these results allow us to write analytical expressions for the combined variances, these are rather bulky and not very enlightening, so we will not reproduce them here. To optimise the degree of entanglement as a function of the quadrature phase angle, we investigate the output spectral correlation $$S_{\theta}^{\text{out}}(\hat{X}_{-})+S_{\theta}^{\text{out}}(\hat{Y}_{+}), \label{eq:Splusminus}$$ where the $\hat{X}$ quadratures are at the angle $\theta$ and the $\hat{Y}$ quadratures at the angle $\theta+\pi/2$. What we find, as shown in Fig. \[fig:entangleJa\], is that the degree of entanglement and the frequency at which it exists depend on the coupling strength $J_{a}$ while the optimum angle depends on $J_{b}$. When we hold $J_{a}$ constant and increase $J_{b}$, we find that the maximum of entanglement is always found at zero frequency, but that the optimum quadrature angle changes. To examine the utility of the system for the production of states which exhibit the EPR paradox, we follow the approach of Reid [@eprMDR]. We assume that a measurement of the $\hat{X}_{1}$ quadrature, for example, will allow us to infer, with some error, the value of the $\hat{X}_{2}$ quadrature, and similarly for the $\hat{Y}_{j}$ quadratures. This allows us to make linear estimates of the quadrature variances, which are then minimised to give the inferred output variances, $$\begin{aligned} S_{\text{inf}}^{\text{out}}(\hat{X}_{1}) & = & S_{X_{1}}^{\text{out}}-\frac{\left[V(\hat{X}_{1},\hat{X}_{2})\right]^{2}}{S_{X_{2}}^{\text{out}}},\nonumber \\ S_{\text{inf}}^{\text{out}}(\hat{Y}_{1}) & = & S_{Y_{1}}^{\text{out}}-\frac{\left[V(\hat{Y}_{1},\hat{Y}_{2})\right]^{2}}{S_{Y_{2}}^{\text{out}}}. \label{eq:EPROPA}\end{aligned}$$ The inferred variances for the $j=2$ quadratures are simply found by swapping the indices $1$ and $2$. As the $\hat{X}_{j}$ and $\hat{Y}_{j}$ operators do not commute, the products of the variances obey a Heisenberg uncertainty relation, with $S_{X_{j}}^{\text{out}}S_{Y_{j}}^{\text{out}}\geq1$. Hence we find a demonstration of the EPR paradox whenever $$S_{\text{inf}}^{\text{out}}(\hat{X}_{j})S_{\text{inf}}^{\text{out}}(\hat{Y}_{j})\leq1. \label{eq:demonstration}$$ With the expressions for the variances given in Eq. \[eq:coupledVXVY\] and the covariances of Eq. \[eq:VXX\], we have all we need to calculate the EPR correlations. Once again, however, the full expressions are somewhat unwieldy, so we will present the results graphically. In Fig. \[fig:epr1\] we present the results for optimised quadrature phase angles while $J_{b}$ is held constant at a value of $\gamma$ while $J_{a}$ is increased. Note that again the angle $\theta$ refers to the $\hat{X}^{\theta}$ quadratures, while the conjugate quadratures are at an angle of $\theta+\pi/2$. Changing $J_{b}$ serves to change the angle of the maximum violation, without changing the degree of violation, while changing $J_{a}$ changes both the degree and the frequency of the maximum violation. As expected, these results are the same for both outputs of the device. Detuning the cavity {#sec:detune} =================== Often in optical systems the best performance is found when the cavity is resonant for the different modes involved in the interactions. In the present case we find that detuning the cavity by the appropriate amount from the two frequencies allows for some simplification of the theoretical analysis and can actually improve some quantum correlations. With detunings included, the steady state below threshold solutions for the high frequency mode are found as $$\beta_{1}^{ss}=\beta_{2}^{ss}=\beta_{ss}=\frac{\epsilon}{\left[\gamma_{b}-i(J_{b}-\Delta_{b})\right]}, \label{eq:betadelta}$$ so that, setting $\Delta_{b}=J_{b}$, we return to the well-known real solutions for a single OPO. If we then set $\Delta_{a}=J_{a}$, define the new variables $A_{p}=\alpha_{1}+\alpha_{2}$ and $A_{m}=\alpha_{1}-\alpha_{2}$, and eliminate the time dependence of $\beta_{a,b}$, we can write positive-P stochastic equations as $$\begin{aligned} \frac{dA_{p}}{dt} & = & -\gamma_{a}A_{p}+\kappa\beta_{ss}A_{p}^{+}+\sqrt{\kappa\beta_{ss}}\left(\eta_{1}+\eta_{3}\right),\nonumber \\ \frac{dA_{p}^{+}}{dt} & = & -\gamma_{a}A_{p}^{+}+\kappa\beta_{ss}A_{p}+\sqrt{\kappa\beta_{ss}}\left(\eta_{2}+\eta_{4}\right),\nonumber \\ \frac{dA_{m}}{dt} & = & -\left[\gamma_{a}+2iJ_{a}\right]A_{m}+\kappa\beta_{ss}A_{m}^{+}+\sqrt{\kappa\beta_{ss}}\left(\eta_{1}-\eta_{3}\right),\nonumber \\ \frac{dA_{m}^{+}}{dt} & = & -\left[\gamma_{a}-2iJ_{a}\right]A_{m}^{+}+\kappa\beta_{ss}A_{m}+\sqrt{\kappa\beta_{ss}}\left(\eta_{2}-\eta_{4}\right). \label{eq:PPcombine}\end{aligned}$$ In the above, the noise terms are the same as those of Eq. \[eq:PPSDE\]. We note here that, although it is the detuning in the low frequency mode that allows us to write the equations for $A_{p}$ and $A_{p}^{+}$ in a particularly simple form, $\Delta_{b}$ also plays a role in that it allows us to treat $\beta_{ss}$ as real, which will make the interesting quantum correlations in and between the $X$ and $Y$ quadratures, so that we do not have to examine all possible local oscillator angles to find the best performance. Following the same linearisation procedure as in section \[sec:linearise\], we find the corresponding drift and noise matrices, $$A_{pm}=\left[ \begin{array}{cccc} \gamma_{a} & -\kappa\beta_{ss} & 0 & 0\\ -\kappa\beta_{ss} & \gamma_{a} & 0 & 0\\ 0 & 0 & \gamma_{a}+2iJ_{a} & -\kappa\beta_{ss}\\ 0 & 0 & -\kappa\beta_{ss} & \gamma_{a}-2iJ_{a}\end{array}\right], \label{eq:Aplusminus}$$ and $$B_{pm}=\left[ \begin{array}{cccc} \sqrt{\kappa\beta_{ss}} & 0 & \sqrt{\kappa\beta_{ss}} & 0\\ 0 & \sqrt{\kappa\beta_{ss}} & 0 & \sqrt{\kappa\beta_{ss}}\\ \sqrt{\kappa\beta_{ss}} & 0 & -\sqrt{\kappa\beta_{ss}} & 0\\ 0 & \sqrt{\kappa\beta_{ss}} & 0 & -\sqrt{\kappa\beta_{ss}}\end{array}\right]. \label{eq:Bplusminus}$$ In terms of the quadratures used in section \[sec:correlations\], we now define $$\begin{aligned} X_{p} & = & A_{p}+A_{p}^{+}=X_{1}+X_{2},\nonumber \\ X_{m} & = & A_{m}+A_{m}^{+}=X_{1}-X_{2},\nonumber \\ Y_{p} & = & -i\left(A_{p}-A_{p}^{+}\right)=Y_{1}+Y_{2},\nonumber \\ Y_{m} & = & -i\left(A_{m}-A_{m}^{+}\right)=Y_{1}-Y_{2}, \label{eq:quadcombine}\end{aligned}$$ and give expressions for the output spectral variances of these new quadratures. For the $X_{p}$ and $Y_{p}$ quadratures these are particularly simple, $$\begin{aligned} S_{X_{p}}^{\text{out}}(\omega) & = & 2+\frac{8\gamma_{a}\gamma_{b}\kappa\epsilon}{\left(\gamma_{a}\gamma_{b}-\kappa\epsilon\right)^{2}+\gamma_{b}^{2}\omega^{2}},\nonumber \\ S_{Y_{p}}^{\text{out}}(\omega) & = & 2-\frac{8\gamma_{a}\gamma_{b}\kappa\epsilon}{\left(\gamma_{a}\gamma_{b}+\kappa\epsilon\right)^{2}+\gamma_{b}^{2}\omega^{2}}, \label{eq:spekplus}\end{aligned}$$ and are readily seen to be the sum of the variances for two uncoupled OPOs, as given in Eq. \[eq:noncoupledVXVY\]. As in that case, the zero-frequency variance in $Y_{p}$ is predicted to vanish at the critical pump value of $\epsilon_{c}=\gamma_{a}\gamma_{b}/\kappa$, although, as should be well known, a linearised analysis is not valid in that region. However, the degree of squeezing is more than was found to be available in the doubly resonant case considered above. The other two variances do not uncouple and have more complicated expressions, $$\begin{aligned} S_{X_{m}}^{\text{out}}(\omega) & = & 2+\frac{8\gamma_{a}\gamma_{b}\kappa\epsilon\left[(\gamma_{a}\gamma_{b}+\kappa\epsilon)^{2}-\gamma_{b}^{2}(4J_{a}^{2}- \omega^{2})\right]}{\left[\gamma_{b}^{2}(\gamma_{a}^{2}+4J_{a}^{2}-\omega^{2})-\kappa^{2}\epsilon^{2}\right]^{2}+4\gamma_{a}^{2}\gamma_{b}^{4}\omega^{2}},\nonumber \\ S_{Y_{m}}^{\text{out}}(\omega) & = & 2-\frac{8\gamma_{a}\gamma_{b}\kappa\epsilon\left[(\gamma_{a}\gamma_{b}-\kappa\epsilon)^{2}-\gamma_{b}^{2}(4J_{a}^{2} -\omega^{2})\right]}{\left[\gamma_{b}^{2}(\gamma_{a}^{2}+4J_{a}^{2}-\omega^{2})-\kappa^{2}\epsilon^{2}\right]^{2}+4\gamma_{a}^{2}\gamma_{b}^{4}\omega^{2}}.\nonumber \\ \label{eq:spekminus}\end{aligned}$$ Graphical results for the combined quadratures which exhibit squeezing are shown in Fig. \[fig:combvar\], from which it is obvious that by far the best squeezing quadrature is $Y_{p}$, which, for these parameters, shows almost $90\%$ squeezing at zero-frequency. The quadratures $X_{m}$ and $Y_{m}$ show only a very small degree of squeezing far from zero frequency. What this result shows, along with the results for the resonant cavity, is that the low frequency modes want to oscillate at two distinct frequencies, as is normal for coupled systems. The detuning chosen, $\Delta_{a}=J_{a}$, moves the sum mode frequency closer to resonance while the other frequency is further detuned. Along with the choice of $\Delta_{b}$ so as to make the intracavity high frequency amplitude real, this results in maximised single-mode noise supression and entanglement centred on zero frequency. Using these results, we can now investigate the degree of entanglement, as done above for the resonant case. As shown in Fig. \[fig:comentangle\], we find that the quadratures $Y_{p}$ and $X_{m}$ are entangled, exactly as in the single OPO case. As with the squeezing, the detunings have moved the maximum of entanglement to zero frequency. A sign of the out of resonance mode attempting to resonate is seen in the small degree of entanglement apparent around $\omega\approx20\gamma$. We also see that the degree of entanglement is less than in the case with zero detuning, shown previously in Fig. \[fig:entangleJa\], although it must be remembered that the absolute pump powers are not the same, merely the ratios $\epsilon/\epsilon_{c}$. Finding analytical expressions for EPR correlations is not possible using this coupled-mode approach, as, although we can calculate the necessary covariances, for example, $V(\hat{X}_{1},\hat{X}_{2})=[V(X_{p})-V(X_{m}))]/4$, it is not obvious how to separate out the single-mode variances. However, these can still be calculated numerically using the full single-mode equations with the appropriate detunings. That the system clearly demonstrates the EPR paradox is shown in Fig. \[fig:detepr\], although again the maximum inferred violation is less than in the resonant case. We note here that all the quantities shown for the detuned system are actually calculated at a lower absolute pump power than in the resonant case. For positive detunings, the critical pump amplitude is found as $$\epsilon_{c}=\sqrt{[\gamma_{a}^{2}+(J_{a}-\Delta_{a})^{2}][\gamma_{b}^{2}+(J_{b}-\Delta_{b})^{2}]}/\kappa\,\,,$$ so that our choice of detunings means this is no longer a function of the coupling strengths. Therefore a careful choice of detunings has two main advantages in that it fixes the quadratures for which the maxima of quantum features are found, and means that the pumping necessary to a good performance does not vary with the coupling strengths. The choice of detunings shown has the possible disadvantage that, as the effective coupling is now only in the $A_{m}$ mode, which is moved away from resonance, the quantum correlations which depend on both the modes are slightly degraded. This is readily seen from the figures because those correlations which include $X_{m}$ and $Y_{m}$ change more with $J_{a}$ than do the others. Conclusion ========== This system exhibits a wide range of behaviour and is potentially an easily tunable source of single-mode squeezing, entangled states and states which exhibit the EPR paradox. The spatial separation of the output modes means that they do not have to be separated by optical devices before measurements can be made, along with the unavoidable losses which would result from this procedure. The entangled beams produced can be degenerate in both frequency and polarisation, unlike those of the nondegenerate OPO, and would exit the cavity at spatially separated locations. This may be a real operational advantage over the nondegenerate OPO, which is also known to produce nonclassical states. 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--- abstract: 'The parity-nonconserving asymmetry in the deuteron photodisintegration, $\vec{\gamma}+d\rightarrow n+p$, is considered with the photon energy ranged up to 10 MeV above the threshold. The aim is to improve upon a schematic estimate assuming the absence of tensor as well as spin-orbit forces in the nucleon-nucleon interaction. The major contributions are due to the vector-meson exchanges, and the strong suppression of the pion-exchange contribution is confirmed. A simple argument, going beyond the observation of an algebraic cancellation, is presented. Contributions of meson-exchange currents are also considered, but found to be less significant.' author: - 'C.-P. Liu' - 'C. H. Hyun' - 'B. Desplanques' bibliography: - 'QEPV.bib' - 'MEC.bib' - 'AM.bib' - 'npPV.bib' title: Parity Nonconservation in the Photodisintegration of the Deuteron at Low Energy --- Introduction \[sec:intro\] ========================== Some interest, both experimental and theoretical, has recently been shown for the study of parity nonconservation in the deuteron photodisintegration by polarized light. Historically, it was its inverse counterpart: the net polarization in radiative thermal neutron capture by proton, $n+p\rightarrow d+\gamma$, which attracted the first attention [@Danilov:1965]. The experimental study was performed by the Leningrad group, taking advantage of new techniques measuring an integrated current [@Lobashov:1972]. The non-zero polarization obtained, $P_{\gamma}=-(1.3\pm0.45)\times10^{-6}$, motivated many theoretical calculations in the frame of strong and weak interaction models known in the 70’s (see for instance Refs. [@Lassey:1975; @Desplanques:1975; @Craver:1976am]). The theoretical results were consistently within the range $P_{\gamma}=(2\sim5)\times10^{-8}$, which is smaller than the measurement by a factor of 30 or more in magnitude and, moreover, of opposite sign. The difficulty to understand the measurement and, also perhaps, the novelty of the techniques, which have been extensively used later on, led to a special reference to this work as “Lobashov experiment”. Later estimates with modern nucleon-nucleon ($NN$) potentials, both parity-conserving (PC) and parity-nonconserving (PNC), give values of $P_{\gamma}$ roughly within the same theoretical range as above. On the experimental side, new results were reported in the early 80’s by the same Leningrad group, giving $P_{\gamma}\leq5\times10^{-7}$ [@Knyazkov:1983ke] and $P_{\gamma}=(1.8\pm1.8)\times10^{-7}$ [@Knyazkov:1984]. Practically, these results indicate an upper limit of $P_{\gamma}$, which is not very constrictive. Since Leningrad group’s last report, the “Lobashov experiment” has long been forgotten by both experimentalists and theorists. Recent experiments such as elastic $\vec{p}$-$p$ scattering (TRIUMF [@Berdoz:2001nu]) and polarized thermal neutron capture by proton (LANSCE [@Snow:2000az]), which directly address the problem of PNC $NN$ interactions; and quasi-elastic $\vec{e}$-$d$ scattering (MIT-Bates [@SAMPLE00b; @Ito:2003mr]), which indirectly involves these interactions, have however raised a new interest for the study of PNC effects in few-body systems. In what could be a golden age for these studies, the “Lobashov experiment” is again evoked. While it seems that there is not much prospect for performing the “Lobashov experiment” in a near future, the inverse process, on the contrary, could be more promising. In this reaction, $\vec{\gamma}+d\rightarrow n+p$, where a deuteron is disintegrated by absorbing a circularly polarized photon, it is expected that, near threshold, the PNC asymmetry ($A_{\gamma}$) is equal to the polarization in the “Lobashov experiment”. This last one can thus be tested from a different approach. The asymmetry $A_{\gamma}$ in the deuteron photodisintegration was first calculated by Lee [@Lee:1978kh] up to the photon energy $\omega_{\gamma}\simeq3.22$ MeV, which is 1 MeV above the threshold. In this energy domain, where the dominant regular transition is $M1$, the result was within the theoretical range of $P_{\gamma}$. Later on, Oka extended Lee’s work, up to $\omega_{\gamma}\simeq35$ MeV [@Oka:1983sp]. Though the cross section still receives a contribution from the $M1$ transition, the dominant contribution comes from the $E1$ transitions. This offers a pattern of PNC effects different from the one at very low photon energy. It was found that $A_{\gamma}$ shows a great enhancement at $\omega_{\gamma}\gtrsim5$ MeV, mainly due to the PNC $\pi$-exchange contribution. If such an enhancement were observed in the experiment, it would provide an important and unambiguous determination of the weak $\pi NN$ coupling constant $h_{\pi}^{1}$. However, a recent schematic calculation of $A_{\gamma}$ by Khriplovich and Korkin [@Khriplovich:2000mb], partly suggested by one of the present author, showed critical contradiction to Oka’s result, with a huge suppression of $A_{\gamma}$ at the energies $\omega_{\gamma}\gtrsim3$ MeV. On the experimental side, a measurement of the asymmetry $A_{\gamma}$ in $\vec{\gamma}+d\rightarrow n+p$ was considered in the 80’s by E. D. Earle *et al.* [@Earle:1981; @Earle:1988fc] but no sensitive result was reported. However, due to advances in experimental techniques and instrumentation, the measurement of $A_{\gamma}$ becomes more feasible nowadays and several groups at JLab [@jlab-lett00], IASA (Athens), LEGS (BNL), TUNL, and SPring-8 show interest in such a measurement. It is therefore important to understand and improve previous estimates. In this work, we carefully re-examine the $\vec{\gamma}+d\rightarrow n+p$ process with two main purposes: 1. Determine how the enhancement of the $h_{\pi}^{1}$ contribution in Oka’s results will change when the calculation is completed with missing parity-admixed components in the final state, in particular in the $^{3}P_{1}$ channel. The role of this last one was revealed by the schematic estimate of Ref. [@Khriplovich:2000mb]. 2. Determine the uncertainty of Khriplovich and Korkin’s calculation in which very simple wave functions are used. It is straightforward to deal with the point 1. In Ref. [@Khriplovich:2000mb], a nice and simple argument about the cancellation of the $h_{\pi}^{1}$ contribution from the final $^{3}P_{0}$, $^{3}P_{1}$, and $^{3}P_{2}$ states along with their parity-admixed partners was given. However, the argument assumed the absence of tensor as well as spin-orbit forces, which are important components of the $NN$ interaction. In order to address these two points (missing components and simplicity of the wave functions), we elaborate our calculation with the Argonne $v_{18}$ $NN$ interaction model. We thus include the $^{1}S_{0}$, $^{3}P_{0}$, $^{3}P_{1}$, $^{3}P_{2}$–$^{3}F_{2}$ channels, deuteron $D$-state, and all their parity-admixed partners consistently. They represent a minimal set of states that allows one to verify the results of the schematic model as well as to include the effect of the tensor and spin-orbit forces that manifest differently in these various channels. We also include other channels, whose role is less important however. As for the $E1$ operator, we employ the Siegert’s theorem [@Siegert:1937yt], which takes into account the contribution of some PC and PNC two-body currents. The small photon energy considered here ($\omega_{\gamma}\leq12$ MeV) justifies this usage. Since there is no theorem similar to the Siegert one for the $M1$ transition operator, two-body currents have to be considered explicitly for both the PC and PNC parts. Adopting Desplanques, Donoghue, Holstein (DDH) potential of the weak interaction [@Desplanques:1980hn], the asymmetry $A_{\gamma}$ will be expressed in terms of the weak $\pi NN$, $\rho NN$ and $\omega NN$ coupling constants, with corresponding coefficients indicating their relative importance. This paper is organized as follows. In Sect. \[sec:formalism\], we review the basic formalism underlying the calculation, which involves both one- and two-body currents. In Sect. \[sec:results\], we show the results and some discussions follow. A particular attention is given to a comparison with earlier works and to new contributions from PNC two-body currents. A simple argument explaining the suppression of the pion-exchange contribution is also given. Conclusions are given in Sect. \[sec:conclusion\]. An appendix contains expressions of $E1$ and $M1$ transition amplitudes due to the PNC two-body currents considered in the present work. Formalism\[sec:formalism\] ========================== For a photodisintegration of an unpolarized target, the asymmetry factor is defined as $$A_{\gamma}\equiv\frac{\sigma_{+}-\sigma_{-}}{\sigma_{+}+\sigma_{-}}\,,$$ where $\sigma_{+(-)}$ denotes the total cross section using right- (left-) handed polarized light. By spherical multipole expansion, it could be expressed as$$A_{\gamma}=\frac{{\displaystyle 2\,\mathrm{Re}\,\sum_{f,i,J}\,\left[F_{EJ}^{*}\,\tilde{F}_{MJ_{5}}+F_{MJ}^{*}\,\tilde{F}_{EJ_{5}}\right]}}{{\displaystyle \sum_{f,i,J}\,\left[F_{EJ}^{2}+F_{MJ}^{2}\right]}}\,.\label{eq:asym}$$ In this formula, the normal electromagnetic (EM) and PNC-induced EM form factors, $F_{XJ}$ and $\tilde{F}_{XJ_{5}}$, with $X$ and $J$ denoting the type and multipolarity of the transition between a specific initial ($i$) and final ($f$) states, are defined in the same way as Refs. [@Musolf:1994tb; @Liu:2002bq]. They depend on the momentum transfer $q$, which equals to the photon energy $\omega_{\gamma}$ in this current case. The form factors $\tilde{F}_{XJ_{5}}$ (and so does the asymmetry) vanish unless some PNC mechanism induces parity admixtures of wave functions and axial-vector currents. In this work, we consider the photon energy $\omega_{\gamma}=q$ up to 10 MeV above the threshold. As the long wavelength limit, $\langle q\, r\rangle\ll1$, is a good approximation, the inclusion of only dipole transitions, *i.e.* $E1$ and $M1$, is sufficient. This leads to 10 possible exit channels connected to the deuteron state by angular momentum considerations. Among them, $^{1}S_{0}$, via the $M1$ transition, and $^{3}P_{0}$, $^{3}P_{1}$, $^{3}P_{2}$–$^{3}F_{2}$, via the $E1$ transitions, dominate the cross section. The transverse multipole operators assume a full knowledge of nuclear currents. This requires, besides the one-body current $\bm j^{(1)}$ from individual nucleons, a complete set of two-body exchange currents (ECs) $\bm j^{(2)}$ which is consistent with the nucleon-nucleon ($NN$) potential. These ECs are usually the sources of theoretical uncertainties, because the $NN$ dynamics is still not fully understood. While there is no alternative for the evaluation of $F_{MJ}$, the Siegert theorem [@Siegert:1937yt] does allow one to transform the evaluation of $F_{EJ}$ into the one of charge multipole $F_{CJ}$. The fact that the PC $NN$ interaction does not give rise to exchange charges at $O(1)$ removes most of the uncertainties related to exchange effects: knowledge of the one-body charge $\rho^{(1)}$ is sufficient for a calculation good to the order of $1/\mN$. In the framework of impulse approximation and using the Siegert theorem, one gets, for the deuteron photodisintegration ($E_{f}-E_{i}=\omega_{\gamma}=q$ and $J_{i}=1$), $$\begin{aligned} F_{E1}^{(S)}(q)_{f,i} & = & \frac{E_{i}-E_{f}}{q}\,\sqrt{\frac{2}{2J_{i}+1}}\,\bra{J_{f}}|\int\, d^{3}x\,[j_{1}(q\, x)\, Y_{1}(\Omega_{x})]\,\rho^{(1)}(\bm x)|\ket{J_{i}}\nonumber \\ & & +\frac{1}{q}\,\frac{1}{\sqrt{2J_{i}+1}}\,\bra{J_{f}}|\int\, d^{3}x\,\bm\nabla\times[j_{1}(q\, x)\,\bm Y_{111}(\Omega_{x})]\cdot\bm j_{spin}^{(1)}(\bm x)|\ket{J_{i}}\nonumber \\ & \simeq & -\frac{q}{3\sqrt{2\,\pi}}\,\bra{J_{f}}|\sum_{i}\,\hat{e}_{i}\,\bm x_{i}|\ket{J_{i}}\equiv-\frac{q}{2\,\sqrt{6\,\pi}}\,\langle E1^{(1)}\rangle\,,\label{eq:Siegert E1}\\ F_{M1}^{(1)}(q)_{f,i} & = & i\,\frac{1}{\sqrt{2J_{i}+1}}\,\bra{J_{f}}|\int\, d^{3}x\,[j_{1}(q\, x)\,\bm Y_{111}(\Omega_{x})]\cdot\bm j^{(1)}(\bm x)|\ket{J_{i}}\nonumber \\ & \simeq & -\frac{q}{3\sqrt{2\,\pi}}\,\bra{J_{f}}|\sum_{i}\,\frac{1}{2\mN}\,[\hat{e}_{i}\,\bm x_{i}\times\bm p_{i}+\hat{\mu}_{i}\,\bm\sigma_{i}]|\ket{J_{i}}\,\equiv-\frac{q}{2\,\sqrt{6\,\pi}}\,\langle M1^{(1)}\rangle\,,\end{aligned}$$ where $\hat{e}_{i}=e\,(1+\tau_{i}^{z})/2$ and $\hat{\mu}_{i}=e\,(\muS+\muV\tau_{i}^{z})/2$ with $\muS=0.88$ and $\muV=4.70$; $Y$ and $\bm Y$ are the spherical and vector spherical harmonics. In these expressions, the approximated results are obtained by replacing the spherical Bessel function $j_{1}(q\, x)$ with its asymptotic form as $q\rightarrow0$, *i.e.* $q\, x/3$, at the long wavelength limit and keeping terms linear in $q$ (the lowest order); they could be related to the forms of $\langle E1^{(1)}\rangle$ and $\langle M1^{(1)}\rangle$ often adopted in the literature. In our numerical calculation, the identity relations are employed instead. Note that the one-body spin current is conserved by itself and not constrained by current conservation. In Eq. (\[eq:Siegert E1\]), this one-body spin current (2nd line) is of higher order in $q$ compared with the Siegert term (1st line), however, it is kept for completeness. As for the PNC-induced form factors $\tilde{F}_{E1_{5}}^{(S)}$ and $\tilde{F}_{M1_{5}}^{(1)}$, one only has to replace either the initial or final state by its opposite-parity admixture, $\widetilde{{\bra{J_{f}}}}$ or $\widetilde{{\ket{J_{i}}}}$, and add a factor “$i$” for $E1$ or “$-i$” for $M1$ matrix elements (in relation with our conventions). The non-vanishing matrix elements for the five dominant exit channels are thus 1. $^{1}S_{0}$:$$\begin{aligned} \langle M1^{(1)}\rangle & = & -\frac{\muV}{\mN}\,\int\, dr\, U^{*}(^{1}S_{0})\, U_{d}(^{3}S_{1})\,,\label{eq:comp-i}\\ \langle E1_{5}^{(1)}\rangle & = & \frac{i}{3}\,\int\, r\, dr\,\tilde{U}^{*}(^{3}P_{0})\,\left[U_{d}(^{3}S_{1})-\sqrt{2}\, U_{d}(^{3}D_{1})\right]\nonumber \\ & & -\frac{i}{\sqrt{3}}\,\int\, r\, dr\, U^{*}(^{1}S_{0})\,\tilde{U}_{d}(^{1}P_{1})\,.\end{aligned}$$ 2. $^{3}P_{0}$:$$\begin{aligned} \langle E1^{(1)}\rangle & = & \frac{1}{3}\,\int\, r\, dr\, U^{*}(^{3}P_{0})\,\left[U_{d}(^{3}S_{1})-\sqrt{2}\, U_{d}(^{3}D_{1})\right]\,,\\ \langle M1_{5}^{(1)}\rangle & = & i\,\frac{\muV}{\mN}\,\int\, dr\,\left[\tilde{U}^{*}(^{1}S_{0})\, U_{d}(^{3}S_{1})-\frac{1}{\sqrt{3}}\, U^{*}(^{3}P_{0})\,\tilde{U}_{d}(^{1}P_{1})\right]\nonumber \\ & & -\, i\,\sqrt{\frac{2}{3}}\,\frac{\muS-1/2}{\mN}\,\int\, dr\, U^{*}(^{3}P_{0})\,\tilde{U}_{d}(^{3}P_{1})\,.\label{eq:comp-ia}\end{aligned}$$ 3. $^{3}P_{1}$:$$\begin{aligned} \langle E1^{(1)}\rangle & = & -\frac{1}{\sqrt{3}}\,\int\, r\, dr\, U^{*}(^{3}P_{1})\,\left[U_{d}(^{3}S_{1})+\frac{1}{\sqrt{2}}\, U_{d}(^{3}D_{1})\right]\,,\\ \langle M1_{5}^{(1)}\rangle & = & -\, i\,\frac{\muV}{\mN}\,\int\, dr\, U^{*}(^{3}P_{1})\,\tilde{U}_{d}(^{1}P_{1})-i\,\frac{\muS+1/2}{\sqrt{2}\,\mN}\,\int\, dr\, U^{*}(^{3}P_{1})\,\tilde{U}_{d}(^{3}P_{1})\nonumber \\ & & -\, i\,\frac{\sqrt{2}\,\muS}{\mN}\,\int\, dr\,\tilde{U}^{*}(^{3}S_{1})\, U_{d}(^{3}S_{1})+i\,\frac{\muS-3/2}{\sqrt{2}\,\mN}\,\int\, dr\,\tilde{U}^{*}(^{3}D_{1})\, U_{d}(^{3}D_{1})\,.\label{eq:comp-ib}\end{aligned}$$ 4. $^{3}P_{2}$–$^{3}F_{2}$:$$\begin{aligned} \langle E1^{(1)}\rangle & = & \frac{\sqrt{5}}{3}\,\int\, r\, dr\,\bigg\{ U^{*}(^{3}P_{2})\,\left[U_{d}(^{3}S_{1})-\frac{1}{5\,\sqrt{2}}\, U_{d}(^{3}D_{1})\right]\,\nonumber \\ & & +\frac{3\,\sqrt{3}}{5}\, U^{*}(^{3}F_{2})\, U_{d}(^{3}D_{1})\bigg\}\,,\\ \langle M1_{5}^{(1)}\rangle & = & -\, i\,\sqrt{\frac{5}{3}}\,\frac{\muV}{\mN}\,\int\, dr\,\left[U^{*}(^{3}P_{2})\,\tilde{U}_{d}(^{1}P_{1})-\sqrt{\frac{3}{5}}\,\tilde{U}^{*}(^{1}D_{2})\, U_{d}(^{3}D_{1})\right]\nonumber \\ & & +i\,\sqrt{\frac{5}{6}}\,\frac{\muS-1/2}{\mN}\,\int\, dr\,\left[U^{*}(^{3}P_{2})\,\tilde{U}_{d}(^{3}P_{1})+\frac{3}{\sqrt{5}}\,\tilde{U}^{*}(^{3}D_{2})\, U_{d}(^{3}D_{1})\right]\,.\label{eq:comp-f}\end{aligned}$$ Results for the remaining five less important channels ($^{3}S_{1}$–$^{3}D_{1}$, $^{1}P_{1}$, $^{1}D_{2}$, $^{3}D_{2}$) will be included in numerical works. The $r$-weighted radial wave functions for scattering and deuteron states, $U$ and $U_{d}$, along with their parity admixtures, $\tilde{U}$ and $\tilde{U}_{d}$, are obtained by solving the Schrödinger equations. Details could be found in Ref. [@Liu:2002bq]. By taking the square of normal EM form factors (PC response function) or the product of normal and PNC-induced ones (PNC response function), we can directly compare Eqs. (5a–5h) in Ref. [@Oka:1983sp]. After removing factors due to wave-function normalizations, the differences are: 1. The parity admixture of the scattering $^{3}P_{1}$ state is included in our work: The admixtures $\tilde{U}(^{3}S_{1})$ and $\tilde{U}(^{3}D_{1})$ are solved from the inhomogeneous differential equations with the source term modulated by $U(^{3}P_{1})$. They are not orthogonal to the deuteron state and thus should not be ignored. Actually, they are required to ensure the orthogonality of the deuteron and the $^{3}P_{1}$ scattering states once these ones are allowed to contain a parity-nonconserving component. 2. The terms involving the scalar magnetic moment are different: Looking for instance at the $M1$ matrix element between $U(^{3}P_{1})$ and $\tilde{U}_{d}(^{3}P_{1})$, the effective $M1$ operator is proportional to $\muS\,\bm S+\bm L/2$. By the projection theorem, $\langle\bm S\rangle=\langle\bm L\rangle$, the overall factor should be $\muS+1/2$, not $\muS+1$ as in Ref. [@Oka:1983sp].[^1] It looks as if this work ignored the $1/2$ factor in front of the $\bm L$ operator. Both points involve the spin-conserving PNC interaction, which is dominated by the pion exchange. Therefore, how these differences change the sensitivity of $A_{\gamma}$ with respect to $h_{\pi}^{1}$ will be elaborated in next section. Now we discuss, in two steps, extra contributions due to ECs when one tries to go beyond the impulse approximation together with the Siegert-theorem framework. First, when PC ECs are included, their contribution to $M1$ matrix elements, $F_{M1}^{(2)}$, definitely needs to be calculated. On the other hand, as PC exchange charges are higher-order in the nonrelativistic limit, $F_{E1}^{(S)}$ is supposed to take care of most two-body effects, and the remaining contribution $\Delta F_{E1}^{(2)}$ can be safely ignored. This argument also applies for the PNC-induced form factors involving the PC ECs: one needs to consider $\tilde{F}_{M1_{5}}^{(2)}$ but can leave out $\Delta\tilde{F}_{E1_{5}}^{(2)}$. Second, the inclusion of PNC ECs, to the first order in weak interaction, only affects the PNC-induced form factors. The contribution $\tilde{F}_{M1_{5}}^{(2')}$ is calculated by using the $M1$ operator constructed from the PNC ECs and unperturbed wave functions (so we use a prime to remind the difference from parity-admixture contributions). One special feature of PNC ECs is that they do have exchange charges of $O(1)$ [@Liu:2003au]. Therefore, one should include them in $\tilde{F}_{E1_{5}}^{(S')}$. As a last remark, we note one advantage of nuclear PNC experiments in processes like photodisintegration or radiative capture. The real photon is “blind” to the nucleon anapole moment, which could contribute otherwise to PNC observables in virtual photon processes. Because this P-odd T-even nucleon moment is still poorly constrained both theoretically and experimentally, the interpretation of real-photon processes, like the one considered here, is thus comparatively easier. Results and Discussions \[sec:results\] ======================================= For practical purposes, we use the Argonne $v_{18}$ [@Wiringa:1995wb] (A$v_{18}$) and DDH [@Desplanques:1980hn] potentials as the PC and PNC $NN$ interactions, respectively. In comparison with earlier works in the 70’s or the 80’s, a strong interaction model like A$v_{18}$ offers the advantage that the singlet-scattering length is correctly reproduced, due to its charge dependence. Correcting results with this respect is therefore unnecessary. The total cross section is plotted in Fig. \[cap:cross section\] as a function of the photon energy and labeled as “IA+Sieg”. Its separate contributions from $E1$ and $M1$ transitions are also shown on the same plot (labeled accordingly). The $M1$ transition only dominates near the threshold region; as the photon energy reaches about 1 MeV above the threshold, the $E1$ transition overwhelms. Away from the threshold, the calculated results agree well with both experiment and existing potential-model calculations up to 10 MeV [@Arenhovel:1991]. Such a good agreement shows the usefulness of the Siegert theorem, by which most of the two-body effects are included. Compared with the curve labeled by “IA”, the result of impulse approximation, one sees the increasing importance of these two-body contributions as $\omega_{\gamma}$ gets larger. On the contrary, because $M1$ matrix elements are purely one-body, we expect our near-threshold results smaller than experiment by about $10\%$ [@Arenhovel:1991]. This discrepancy, originally found in the radiative capture of thermal neutron by proton (the inverse of deuteron photodisintegration), requires various physics such as exchange currents and isobar configurations, to be fully explained. Here, we qualitatively estimate a $5\%$ error for the calculation of $F_{M1}$ near threshold. When calculating the PNC-induced matrix elements with the DDH potential, we use the strong meson-nucleon coupling constants: $g_{\pi\ssst{NN}}=13.45$, $g_{\rho\ssst{NN}}=2.79$, and $g_{\omega\ssst{NN}}=8.37$, and meson masses (in units of MeV): $m_{\pi}=139.57$, $m_{\rho}=770.00$, and $m_{\omega}=781.94$. The resulting asymmetry is then expressed in terms of six PNC meson-nucleon coupling constants $h$’s as $$A_{\gamma}=c_{1}\, h_{\pi}^{1}+c_{2}\, h_{\rho}^{0}+c_{3}\, h_{\rho}^{1}+c_{4}\, h_{\rho}^{2}+c_{5}\, h_{\omega}^{0}+c_{6}\, h_{\omega}^{1}\,,\label{eq:A exp}$$ where the six energy-dependent coefficients $c_{1...6}$ show the sensitivity to each corresponding coupling. It turns out that, for the energy range considered here, $c_{2},\, c_{4},\, c_{5}\gg c_{1}\gg c_{3},\, c_{6}$. This implies the asymmetry has a larger sensitivity to the isoscalar and isotensor couplings than to the isovector ones. The detailed energy dependences of these “large” and “small” coefficients are shown in Fig. \[cap:coeffs\]. In principle, these results are independent. In practice however, they can be shown to depend on three quantities, reflecting the dominant role of the various $S\leftrightarrow P$ neutron-proton transition amplitudes at low energy. These amplitudes have some energy dependence which is essentially determined by the best known long-range properties of strong interaction models. They can therefore be parametrized by their values at zero energy [@Danilov:1965; @Missimer:1976wb; @Desplanques:1978mt], including at the deuteron pole. To a large extent, they can be used independently of the underlying strong interaction model, quite in the spirit of effective-field theories that they anticipated [@Holstein]. In the case of the A$v_{18}$ model employed here, they are given by: $$\begin{aligned} \mN\,\lambda_{t} & = & -0.043\, h_{\rho}^{0}-0.022\, h_{\omega}^{0}\,,\nonumber \\ \mN\,\lambda_{s} & = & -0.125\, h_{\rho}^{0}-0.109\, h_{\omega}^{0}+0.102\, h_{\rho}^{2}\,,\nonumber \\ \mN\, C & = & 1.023\, h_{\pi}^{1}+0.007\, h_{\rho}^{1}-0.021\, h_{\omega}^{1}\,.\end{aligned}$$ The largest corrections to the above approach occur for the PNC pion-exchange interaction which, due to its long range, produces some extra energy dependence and sizable $P\leftrightarrow D$ transition amplitudes. They can show up when the contribution of the $S\leftrightarrow P$ transition amplitude is suppressed, like in this work. For $\omega_{\gamma}=2.235$ MeV, which is very close to the disintegration threshold, we get the asymmetry $$\begin{aligned} A_{\gamma}^{(th)} & \approx & \left[-8.44\, h_{\rho}^{0}-17.65\, h_{\rho}^{2}+3.63\, h_{\omega}^{0}\right.\nonumber \\ & & \left.\hspace{2cm}+O(c_{1},c_{3},c_{6})\right]\times10^{-3}\,.\label{eq:A-threshold}\end{aligned}$$ Using the DDH “best” values as an estimate, we got $A_{\gamma}^{(th)}\approx2.53\times10^{-8}$. By detailed balancing, one expects that $A_{\gamma}^{(th)}$ equals the circular polarization $P_{\gamma}^{(th)}$ observed in the radiative thermal neutron capture by proton, given the same kinematics. Though our result does not exactly correspond to the same kinematics as the inverse process usually considered (the kinetic energy of thermal neutrons $\sim$0.025 eV), it agrees both in sign and order of magnitude with existing calculations of $P_{\gamma}^{(th)}$ [@Lassey:1975; @Desplanques:1975; @Craver:1976am]. We also performed a similar calculation for the latter case with A$v_{18}$, and the result is $$\begin{aligned} P_{\gamma}^{(th)} & \approx & \left[-8.75\, h_{\rho}^{0}-17.47\, h_{\rho}^{2}+3.39\, h_{\omega}^{0}\right.\nonumber \\ & & \left.\hspace{2cm}+O(c_{1},c_{3},c_{6})\right]\times10^{-3}\,.\end{aligned}$$ This is very close to the result of $A_{\gamma}$ quoted above. It is noticed that our expression of $A_{\gamma}^{(th)}$ at very low energy, and therefore that one for $P_{\gamma}^{(th)}$, contains a contribution from the one-pion exchange (see the low-energy part of the $c_{1}$ coefficient given in Fig. \[cap:coeffs\]). This feature, which apparently contradicts the statement often made in the past that this contribution is absent in $P_{\gamma}^{(th)}$, is due to the incorporation in our work of the spin term in Eq. (\[eq:Siegert E1\]), which represents a higher order term in $q$. This correction also explains the difference in the behavior of the $c_{1}$ coefficient with the Oka’s result [@Oka:1983sp]. We note that, because the $M1$ transition dominates at the threshold and we only use the impulse approximation for its matrix element, there should be approximately a $-5\%$ correction to $A_{\gamma}^{(th)}$ (also $P_{\gamma}^{(th)}$) when two-body effects are included in $F_{M1}$. On the other hand, as $\tilde{F}_{E1_{5}}$ is calculated using the Siegert theorem, it should be reliable up to the correction of $\tilde{F}_{E1_{5}}^{(S')}$ from the PNC exchange charge at $O(1)$. When the photon energy gets larger, one can see immediately that the asymmetry gets smaller. A prediction using the DDH best values is shown in Fig. \[cap:Ag DDH\]. In this figure, as soon as the photon energy reaches 1 MeV above the threshold, the asymmetry drops by one order of magnitude. Moreover, the sign changes around $\omega_{\gamma}=4$ MeV. This implies that a higher sensitivity ($\sim10^{-9}$) is needed for any experiment targeting at the kinematic range away from the threshold. Our calculation is consistent with the work by Khriplovich and Korkin [@Khriplovich:2000mb], but is widely different from the one by Oka [@Oka:1983sp]. In the following, we make a closer comparison with these works and, then, present results for the contribution of various PNC two-body currents considered for the first time. Comparison with Oka’s work \[sub:CompOka\] ------------------------------------------ The major difference comes from the pion sector. In Ref. [@Khriplovich:2000mb], where the scattering wave functions are obtained from the zero-range approximation and the deuteron is purely a $^{3}S_{1}$ state, a simple angular momentum consideration leads to a null contribution from pions. Our result shows that the more complex nuclear dynamics has only small corrections, so the asymmetry is not sensitive to $h_{\pi}^{1}$. However, it is not the case at all in Ref. [@Oka:1983sp]: the pion exchange dominates the asymmetry with the coefficient $c_{1}$ being one or two orders of magnitude larger than our result. This discrepancy could be illustrated by considering a case where $\omega_{\gamma}$ is 10 MeV above the threshold. In the central column of Table \[cap:compare Oka’s\], we list the pertinent PNC responses due to the pion exchange among the 5 dominant exit channels. In the right column, we simulate what the outcome will be if the analytical results of Eqs. (5a–5g) in Ref. [@Oka:1983sp] are used, *i.e.* with different factors involving $\mu_{\ssst{S}}$ and no parity admixture of $^{3}P_{1}$ state as mentioned in Sec. \[sec:formalism\]. Comparing the totals from both columns, one immediately observes the simulated result is bigger by an order of magnitude. More inspection shows that, while the changes of the $\mu_{\ssst{S}}$ factors do alter each response somewhat, the major difference depends on whether the big cancellation from the $^{3}P_{1}$ admixture is included or not. By adding contributions from other sub-leading channels, the total will be further downed by a factor of 2.5. Thus the overall difference is about a factor of 30. Transitions Eqs. (\[eq:comp-i\]–\[eq:comp-f\]) Eqs. (5a–5h) in Ref. [@Oka:1983sp] --------------------------------------------------- ------------------------------------ ------------------------------------ $^{3}P_{0}\leftrightarrow\tilde{\mathcal{D}}$ 0.449 -0.142 $^{3}P_{1}\leftrightarrow\tilde{\mathcal{D}}$ -3.217 -4.383 $\widetilde{^{3}P_{1}}\leftrightarrow\mathcal{D}$ 3.942 not considered $^{3}P_{2}\leftrightarrow\tilde{\mathcal{D}}$ -1.231 0.389 $\widetilde{^{3}P_{2}}\leftrightarrow\mathcal{D}$ -0.142 0.045 $^{3}F_{2}\leftrightarrow\tilde{\mathcal{D}}$ -0.151 0.048 $\widetilde{^{3}F_{2}}\leftrightarrow\mathcal{D}$ -0.019 0.006 Total -0.371 -4.037 : The dominant PNC responses due to the pion exchange for $\omega_{\gamma}$ 10 MeV above the threshold (in units of $10^{-5}\times h_{\pi}^{1}$). The central column is calculated by Eqs. (\[eq:comp-i\]–\[eq:comp-f\]), while the right column by Eqs. (5a–5h) in Ref. [@Oka:1983sp]. The symbol $\mathcal{D}$ denotes the deuteron state. \[cap:compare Oka’s\] **Comparison with Khriplovich and Korkin’s work \[sub:CompKK\]** ---------------------------------------------------------------- The vanishing of the $\pi$-exchange contribution in Khriplovich and Korkin’s work [@Khriplovich:2000mb] supposes that the $E1$ transitions from the deuteron state to the different scattering states, $^{3}P_{0}$, $^{3}P_{1}$ and $^{3}P_{2}$, are the same, which implies that one neglects both the tensor and spin-orbit components of the strong interaction. As these parts of the force have large effects in some cases, it is important to determine how the above vanishing is affected when a more realistic description of the interaction is used. We first notice that the isoscalar magnetic operator, $\muS\,\bm S+\bm L/2$, can be written as $\muS\,\bm J+(1/2-\muS)\bm L$. As the operator $\bm J$ conserves the total angular momentum, it follows that the $E1$ transitions from the deuteron state to the $^{3}P_{0}$ and $^{3}P_{2}$ states will be proportional to $\muS-1/2$, in agreement with Eqs. (\[eq:comp-ia\]) and (\[eq:comp-f\]). A similar result holds for the $^{3}P_{1}$ state. For this transition, one has to take into account that the $\bm J$ operator connects states that are orthogonal to each other, including the case where they contain some parity admixture. This unusual but interesting result was originally suggested by a similar result obtained by Khriplovich and Korkin for the $^{1}S_{0}$ and $^{3}P_{0}$ states [@Khriplovich:2000mb]. They used it later on for the $\pi$-exchange contribution on the suggestion of one the present authors. Taking this property into account, one can check that the different $\muS$-dependent terms in Eq. (\[eq:comp-ib\]) combine so that the quantity, $\muS-1/2$, can be factored out. This explains the cancellation of the two largest contributions in Table \[cap:compare Oka’s\], $3.942$ and $-3.217$, approximately proportional to $2\,\muS=1.76$ and $-(\muS+1/2)=-1.38$. Further cancellation is obtained when one considers the sum of the $\pi$-exchange contributions to the asymmetry $A_{\gamma}$ corresponding to the different $P$ states. Taking into account the remark made in the previous paragraph, it can be checked that contributions from Eqs. (\[eq:comp-ia\]), (\[eq:comp-ib\]) and (\[eq:comp-f\]) are proportional to 2, 3 and $-5$ and 4, $-3$, and $-1$ for the $^{3}S_{1}$ and $^{3}D_{1}$ deuteron components respectively (assuming that the $^{3}P$ wave functions are the same). As can be seen in Table \[cap:compare Oka’s\], the dominant contributions, 0.449, 0.725 $(=3.942-3.217)$ and $-1.231$ are not far from the relative ratios 2, 3 and $-5$, expected for the $^{3}S_{1}$ deuteron component. Possible departures can be ascribed in first place to the $^{3}D_{1}$ deuteron component. The above cancellation calls for an explanation deeper than the one consisting in the verification that the algebraic sum of different contributions cancels. An argument could be the following. In the conditions where the cancellation takes place (same interaction in the $^{3}P$ states in particular), a closure approximation involving spin and angular orbital momentum degrees of freedom can be used to simplify the writing of the PNC part of the response function that appears at the numerator of Eq. (\[eq:asym\]). Keeping only the factors of interest here, the interference term of $E1$ and $M1$ matrix elements can be successively transformed as follows $$\begin{aligned} \delta R & \propto & \sum_{M}\bra{J_{i}}\,\hat{r}^{i}\,(\muS-\tfrac12)\, L^{j}\,\left(\delta^{ij}-\hat{q}^{i}\,\hat{q}^{j}\right)\widetilde{{\ket{J_{i}}}}\nonumber \\ & \propto & \sum_{M}\Big[\bra{^{3}S_{1}}U_{d}(^{3}S_{1})+\frac{U_{d}(^{3}D_{1})}{\sqrt{2}}\,\left(3\,(\bm S\!\cdot\!\hat{r})^{2}-S^{2}\right)\,\Big]\nonumber \\ & & \hspace{1cm}\times\:\hat{r}^{i}\, L^{j}\,\left(\delta^{ij}-\hat{q}^{i}\,\hat{q}^{j}\right)\Big[\bm S\cdot\hat{r}\,\ket{^{3}S_{1}}\Big]\nonumber \\ & \propto & {\textrm{Tr}}\left(\Big[U_{d}(^{3}S_{1})+\frac{U_{d}(^{3}D_{1})}{\sqrt{2}}\,\left(3\,(\bm S\!\cdot\!\hat{r})^{2}-S^{2}\right)\,\Big]\,\bm S\!\cdot\!\hat{r}\right)\nonumber \\ & = & 0\,.\label{eq:cancellation}\end{aligned}$$ The first line stems from retaining the isoscalar part of the magnetic operator proportional to $(\muS-1/2)\bm L$ (it is reminded that the $\bm J$ part does not contribute). The next line is obtained by expressing the PC and PNC parts of the deuteron wave function as some operator acting on a pure $|^{3}S_{1}\rangle$ state. Once this transformation is made, it is possible to replace the summation over the deuteron angular momentum components, $M$, by the spin ones, $m_{s}$, which is accounted for at the third line. The last line then follows from the fact that the trace of the spin operator, $\bm S$, possibly combined with a $\Delta S=2$ one, vanishes. A result similar to the above one can be obtained for some contributions involving MECs. It is however noticed that some corrections involving the spin-orbit force, or spin-dependent terms in the E1 transition operator, which both contain an extra $\bm S$ factor in the above equation, could lead to a non-zero trace and therefore to a relatively large correction. Of course, the above cancellation relies on the fact that no polarization of the initial or final state is considered. Had we looked at an observable involving such a polarization, like the asymmetry in the capture of polarized thermal neutrons by protons, the result will be quite different. As is well known, this observable is dominated by the $\pi$-exchange contribution [@Danilov:1965]. **Contributions of PNC ECs \[sub:PNC-ECs\]** -------------------------------------------- In Section \[sec:formalism\], the contributions of PNC ECs were summarized in two additional PNC-induced form factors, $\tilde{F}_{E1_{5}}^{(S')}$ and $\tilde{F}_{M1_{5}}^{(2')}$. Now, we estimate these contributions by considering only the dominant channels $^{1}S_{0}$, $^{3}P_{0}$, $^{3}P_{1}$ and $^{3}P_{2}$–$^{3}F_{2}$. As $E1_{5}$ connects states of same parity, only $^{1}S_{0}$ is allowed; therefore, $\tilde{F}_{E1_{5}}^{(S')}$ plays a more important role for $A_{\gamma}$ near the threshold. On the other hand, $M1_{5}$ connects states of opposite parity, which requires the other four channels, so $\tilde{F}_{M1_{5}}^{(2')}$ has more impact on $A_{\gamma}$ at higher energies. The full set of PNC ECs which is consistent with the DDH potential was derived in [@Liu:2003au], Eqs. (17–24). The whole evaluation is straightforward, however tedious, so we defer all the analytical expressions in Appendix \[sec:appendix-PNC-ECs\] and only quote the numerical results here. With the same parametrization as Eq. (\[eq:A exp\]), the additional contributions to the asymmetry by PNC ECs, via $E1_{5}$ and $M1_{5}$ respectively, are $$\begin{aligned} A_{\gamma}(\tilde{F}_{E1_{5}}^{(S')}) & = & c_{2}^{(S')}\, h_{\rho}^{0}+c_{4}^{(S')}\, h_{\rho}^{2}\,,\label{eq:E1-PNC}\\ A_{\gamma}(\tilde{F}_{M1_{5}}^{(2')}) & = & c_{1}^{(2')}\, h_{\pi}^{1}+c_{2}^{(2')}\, h_{\rho}^{0}+c_{3}^{(2')}\, h_{\rho}^{1}\nonumber \\ & & +c_{4}^{(2')}\, h_{\rho}^{2}+c_{6}^{(2')}\, h_{\omega}^{1}\,.\label{eq:M1-PNC}\end{aligned}$$ The detailed energy-dependence of each coefficient is shown in Fig. \[fig:pncecs\]. The dominance of $\tilde{F}_{E1_{5}}^{(S')}$ near the threshold and $\tilde{F}_{M1_{5}}^{(2')}$ at higher energies could be readily observed in these plots. We discuss their significances to the total asymmetry in the following. For the case where the photon energy is 0.01 MeV above the threshold, only $c_{2}^{(S')}$ and $c_{4}^{(S')}$ are substantial. The former coefficient is about 20% of $c_{2}$, while the latter one is only 2% of $c_{4}$. By using the DDH best values, these contributions give an asymmetry about 1.4$\times10^{-9}$, which is a 6% correction. This is typically the order of magnitude one could expect from the exchange effects. As the energy gets larger, while the coefficients $c_{2}^{(S')}$ and $c_{4}^{(S')}$ keep stable, the coefficients associated with $M1_{5}$ matrix elements grow linearly, roughly. The fastest growing one is $c_{1}^{(2')}$ because the long-ranged pion-exchange dominates the matrix elements. Comparatively, $c_{2}^{(2')}$ has a smaller slope due to less overlap between the effective ranges pertinent to the deuteron wave function and the $\rho$-exchange. For the case where the photon energy is 10 MeV above the threshold, $c_{1}^{(2')}$, $c_{2}^{(2')}$, and $c_{2}^{(S')}$ are substantial. The first coefficient is about 50% of $c_{1}$, and the latter two combined is about 16% of $c_{2}$. The extremely large correction to $c_{1}$ can be simply explained. The cancellation which affects the single-particle contribution (see Eq. (\[eq:cancellation\])) does not apply to the two-body one. By using the DDH best values, all contributions due to PNC ECs give an asymmetry about 3.7$\times10^{-11}$, which is a 5% correction. The reason why large effects from individual meson exchanges lead to an overall small correction is due to the cancellation between pion and heavy-meson exchanges: the DDH best values have opposite signs for the pion and heavy-meson couplings. This conclusion however depends on the sign we assumed for the $g_{\rho\pi\gamma}$ coupling. Conclusion \[sec:conclusion\] ============================= The present work has been motivated by various aspects of the PNC asymmetry $A_{\gamma}$ in the deuteron photodisintegration, especially in the few-MeV photon-energy range. A first work addressing this energy domain [@Oka:1983sp] showed that the process could provide information on the PNC $\pi NN$ coupling constant, $h_{\pi}^{1}$, which allows one to check results from other processes involving this coupling. A later work [@Khriplovich:2000mb], rather schematic, concluded that this contribution could be largely suppressed. Between these two extreme limits, the question arises of what this contribution could be when a realistic description is made, including in particular the tensor and spin-orbit components of the $NN$ interaction. At the first sight, a sizable PNC $\pi$-exchange contribution could arise if one assumes tensor-force effects of about 15% for each partial contribution and no cancellation. The complete calculation shows that the $\pi$-exchange contribution remains strongly suppressed after improving upon the schematic model. Beyond making this observation, a genuine explanation should therefore be found. When considering the asymmetry $A_{\gamma}$, an average is made over the spins of initial and final states. Terms in the interference effects of electric and magnetic transitions, whose spin dependence averages to a non-zero value, are expected to produce a sizable contribution. This discards the $\pi$-exchange contribution which involves a linear dependence on the spin operator $\bm S$ and tensor-force effects which involve the product of spin operators of order 1 and 2. The argument applies to MECs too. A different conclusion would hold for an observable implying a spin polarization of the initial or final state. It thus appears some similarity between the relative role of various contributions here and that one emphasized by Danilov for the inverse process at thermal energies: the circular polarization of photons $P_{\gamma}$ (equivalent to $A_{\gamma}$ here) is mainly dependent on the PNC isoscalar and isotensor contributions while the asymmetry of the photon emission with respect to the neutron polarization depends on the $\pi$-exchange contribution. As the $\pi$-exchange contribution to the asymmetry $A_{\gamma}$ turns out to have a minor role, we can concentrate on the vector-meson ones. At the low energies considered here, it is expected that these contributions depend on two combinations of parameters entering the description of the PNC (and PC) $NN$ interaction. They are the zero-energy neutron-proton scattering amplitudes in the $T=0$ and $T=1$ channels, $\lambda_{t}$ and $\lambda_{s}$. In terms of these quantities introduced by Danilov [@Danilov:1965] (see also later works by Missimer [@Missimer:1976wb], Desplanques and Missimer [@Desplanques:1978mt], Holstein [@Holstein]) the discussion could be simpler. The asymmetry is found to vary between $$A_{\gamma}=0.70\,\mN\,\lambda_{t}-0.17\,\mN\,\lambda_{s}\;\;{\textrm{at \; threshold}}$$ and $$A_{\gamma}=-0.037\,\mN\,\lambda_{t}+0.022\,\mN\,\lambda_{s}\;\;{\textrm{at}}\;\omega_{\gamma}=12{\textrm{ MeV}},$$ thus evidencing a change in sign which occurs around $\omega_{\gamma}=5.5\;{\textrm{MeV}}$ for both amplitudes. Depending on low-energy properties and, thus, on the best known properties of the strong interaction, the place where the cancellation of $A_{\gamma}$ occurs sounds to be well established. It roughly agrees with what can be inferred from the analytic work by Khriplovich and Korkin [@Khriplovich:2000mb]. Not much sensitivity to PNC ECs is found. An experiment should therefore aim at a measurement at energies significantly different, either below or above. The goal for studying PNC effects is to get information on the hadronic physics entering the PNC $NN$ interaction. This supposes that one can disentangle the different contributions to each process. We notice that the combination of parameters $\lambda_{t}$ and $\lambda_{s}$ appearing in the expression of $A_{\gamma}$ is orthogonal to that one determining PNC effects in most other processes, especially in medium and heavy nuclei. The study of the present process is therefore quite useful. Another observation, which is not totally independent of the previous one, concerns the isotensor contribution. This one is especially favored in the present process while it is generally suppressed in processes involving a roughly equal number of protons and neutrons with either spin [@Desplanques:1998ak]. The present process is therefore among the best ones to get information on the isotensor $\rho NN$ coupling constant. We however stress that this supposes the isoscalar parts could be constrained well by other processes. In a meson-exchange model of the PNC interaction, these ones are represented by the isoscalar $\rho NN$ and $\omega NN$ coupling constants. One could add that the relative sign of these two contributions is the same, in a large range of the photon energy ($\omega_{\gamma}\geq3\;$MeV), as in many other processes. It however differs at small photon energies where the asymmetry involves a combination of the various isoscalar and isotensor couplings that is little constrained by other processes. This explains that expectations of $A_{\gamma}$ up to $10^{-7}$ near threshold could be suggested in recent works on the basis of a phenomenological analysis [@Desplanques:1998ak; @Khriplovich:2000mb; @Schi-int]. Measuring this asymmetry could therefore be quite useful to determine a poorly known component of PNC $NN$ interactions. On the theoretical side, the present work should be completed by the contribution of further parity-conserving exchange currents, but also by higher $1/m_{N}$-order corrections from the single-particle current and, consistently, from both PC and PNC exchange currents [@Friar:1983wd]. Though they are not expected to change the main conclusions reached here, they could be required to obtain from experiment a more accurate information on PNC $NN$ forces. C.-P.L. would like to thank R. Schiavilla, M. Fujiwara, and A.I. Titov for useful disussions. C.H.H. gratefully acknowledges the hospitality of the Laboratoire de Physique Subatomique et de Cosmologie, where part of this work was performed. Work of C.H.H. is partially supported by Korea Research Foundation (Grant No. KRF-2003-070-C00015). Non-vanishing Matrix Elements of PNC ECs for Dominant Transitions \[sec:appendix-PNC-ECs\] ========================================================================================== In this section, we summarize the analytical expressions of the non-zero $\tilde{F}_{E1_{5}}^{(S')}$ and $\tilde{F}_{M1_{5}}^{(2')}$ for the five dominant channels which lead to the numerical results in Section \[sub:PNC-ECs\]. $\tilde{F}_{E1_{5}}^{(S')}$ --------------------------- As discussed in Section \[sec:formalism\], an exchange charge at $O(1)$ should contribute to this form factor. According to Ref. [@Liu:2003au], the $\rho$-exchange does generate one:$$\rho_{mesonic}^{\rho}(\bm x;\bm r_{1},\bm r_{2})=2\, e\, g_{\rho\ssst{NN}}\left(h_{\rho}^{0}-\frac{h_{\rho}^{2}}{2\sqrt{6}}\right)(\bm\tau_{1}\times\bm\tau_{2})^{z}(\bm\sigma_{1}-\bm\sigma_{2})\cdot\bm\nabla_{x}\Big(f_{\rho}(r_{x1})f_{\rho}(r_{x2})\Big)\,,$$ with $f_{\ssst{X}}(r)=\exp(-m_{\ssst{X}}\, r)/(4\,\pi\, r)$ and $r_{xi}=|\bm x-\bm r_{i}|$. The $^{1}S_{0}$ state is the only open exit channel and it gives$$\langle E1_{5}^{(S')}\rangle=8\,\frac{g_{\rho\ssst{NN}}}{m_{\rho}}\left(h_{\rho}^{0}-\frac{h_{\rho}^{2}}{2\sqrt{6}}\right)\bra{^{1}S_{0}}r\, f_{\rho}(r)\ket{^{3}S_{1}}_{d}\,,$$ where $\bra{f}F(r)\ket{i}_{d}$ denotes the radial integral $\int dr\, U^{*}(f)\, F(r)\, U_{d}(i)$ and the subscript “$d$” refers to the deuteron state. $\tilde{F}_{M1_{5}}^{(2')}$ --------------------------- As the four allowed exit channels $^{3}P_{0}$, $^{3}P_{1}$ and $^{3}P_{2}$–$^{3}F_{2}$ are spin- and isospin-triplet, the non-vanishing PNC ECs, which satisfy the spin and isospin selections rules, are $$\begin{aligned} \bm j_{pair}^{\rho}(\bm x;\bm r_{1},\bm r_{2}) & = & \frac{e\, g_{\rho\ssst{NN}}}{4\mN}\, h_{\rho}^{1}\, f_{\rho}(r)(\tau_{1}^{z}-\tau_{2}^{z})(\bm\sigma_{1}+\bm\sigma_{2})\Big((1+\tau_{1}^{z})\delta^{(3)}(\bm x-\bm r_{1})\Big)+(1\leftrightarrow2)\,,\\ \bm j_{pair}^{\omega}(\bm x;\bm r_{1},\bm r_{2}) & = & \frac{-e\, g_{\omega\ssst{NN}}}{4\mN}\, h_{\omega}^{1}\, f_{\omega}(r)(\tau_{1}^{z}-\tau_{2}^{z})(\bm\sigma_{1}+\bm\sigma_{2})\Big((1+\tau_{1}^{z})\delta^{(3)}(\bm x-\bm r_{1})\Big)+(1\leftrightarrow2)\,,\\ \bm j_{mesonic}^{\rho}(\bm x;\bm r_{1},\bm r_{2}) & = & \frac{-e\, g_{\rho\ssst{NN}}}{\mN}\left(h_{\rho}^{0}-\frac{h_{\rho}^{2}}{2\sqrt{6}}\right)(\bm\tau_{1}\times\bm\tau_{2})^{z}\nabla_{x}^{a}\Big(i\left\{ \nabla_{1}^{a}\bm\sigma_{2}+\sigma_{1}^{a}\bm\nabla_{2},\, f_{\rho}(r_{x1})\, f_{\rho}(r_{x2})\right\} \nonumber \\ & & -\muV\left[(\bm\sigma_{1}\times\bm\nabla_{1})^{a}\bm\sigma_{2}+\sigma_{1}^{a}\bm\sigma_{2}\times\bm\nabla_{2},\, f_{\rho}(r_{x1})\, f_{\rho}(r_{x2})\right]\Big)+(1\leftrightarrow2)\,,\\ \bm j_{mesonic}^{\rho\pi}(\bm x;\bm r_{1},\bm r_{2}) & = & \frac{-e\, g_{\rho\ssst{NN}}\, g_{\rho\pi\gamma}}{\sqrt{2}\, m_{\rho}}\, h_{\pi}^{1}(\bm\tau_{1}\times\bm\tau_{2})^{z}(\bm\nabla_{1}\times\bm\nabla_{2})\Big(f_{\rho}(r_{x1})f_{\pi}(r_{x2})\Big)+(1\leftrightarrow2)\,.\label{eq:rhopiMesonic}\end{aligned}$$ Note that one additional strong meson-nucleon coupling constant, $g_{\rho\pi\gamma}$, appears in Eq. (\[eq:rhopiMesonic\]). This could be constrained by the $\rho\rightarrow\pi+\gamma$ data. For the numerical calculation, we quote the number $g_{\rho\pi\gamma}=0.585$ as given in Ref. [@Truhlik:2000yx]. The matrix element $\langle M1_{5}^{(2')}\rangle$ can be written as a sum of the contributions from each EC as$$\langle M1_{5}^{(2')}\rangle=\frac{1}{\mN}\left(g_{\rho\ssst{NN}}\, h_{\rho}^{1}\, X_{1}+g_{\omega\ssst{NN}}\, h_{\omega}^{1}\, X_{2}+g_{\rho\ssst{NN}}\,\left(h_{\rho}^{0}-\frac{h_{\rho}^{2}}{2\sqrt{6}}\right)X_{3}\right)+\frac{1}{m_{\rho}}\,\grhoNN\, g_{\rho\pi\gamma}\, h_{\pi}^{1}\, X_{4}\,,$$ and for each exit channel, the quantities $X_{1,2,3,4}$ are 1\. $^{3}P_{0}$$$\begin{aligned} X_{1} & = & -\frac{2}{3}\left(\bra{^{3}P_{0}}r\, f_{\rho}(r)\ket{^{3}S_{1}}_{d}+\frac{1}{\sqrt{2}}\,\bra{^{3}P_{0}}r\, f_{\rho}(r)\ket{^{3}D_{1}}_{d}\right)\,,\\ X_{2} & = & \frac{2}{3}\left(\bra{^{3}P_{0}}r\, f_{\omega}(r)\ket{^{3}S_{1}}_{d}+\frac{1}{\sqrt{2}}\,\bra{^{3}P_{0}}r\, f_{\omega}(r)\ket{^{3}D_{1}}_{d}\right)\,,\\ X_{3} & = & -\frac{8}{3}\left((1+2\muV)\,\bra{^{3}P_{0}}r\, f_{\rho}(r)\ket{^{3}S_{1}}_{d}+\frac{1}{\sqrt{2}}\,(1-\muV)\,\bra{^{3}P_{0}}r\, f_{\rho}(r)\ket{^{3}D_{1}}_{d}\right.\nonumber \\ & & \left.\hspace{0.7cm}-\frac{2}{m_{\rho}}\,\bra{^{3}P_{0}}r\, f_{\rho}(r)\ket{^{3}S_{1}^{(+)}}_{d}-\frac{\sqrt{2}}{m_{\rho}}\,\bra{^{3}P_{0}}r\, f_{\rho}(r)\ket{^{3}D_{1}^{(-)}}_{d}\right),\\ X_{4} & = & \frac{4\sqrt{2}}{3\,(m_{\rho}^{2}-m_{\pi}^{2})}\left(\bra{^{3}P_{0}}f_{\pi\rho}^{\,'}(r)\ket{^{3}S_{1}}_{d}-\sqrt{2}\,\bra{^{3}P_{0}}f_{\pi\rho}^{\,'}(r)\ket{^{3}D_{1}}_{d}\right)\,;\end{aligned}$$ 2\. $^{3}P_{1}$ $$\begin{aligned} X_{1} & = & \frac{1}{\sqrt{3}}\left(\bra{^{3}P_{1}}r\, f_{\rho}(r)\ket{^{3}S_{1}}_{d}-\sqrt{2}\,\bra{^{3}P_{1}}r\, f_{\rho}(r)\ket{^{3}D_{1}}_{d}\right)\,,\\ X_{2} & = & -\frac{1}{\sqrt{3}}\left(\bra{^{3}P_{1}}r\, f_{\omega}(r)\ket{^{3}S_{1}}_{d}-\sqrt{2}\,\bra{^{3}P_{1}}r\, f_{\omega}(r)\ket{^{3}D_{1}}_{d}\right)\,,\\ X_{3} & = & \frac{4}{\sqrt{3}}\bigg((1-\muV)\,\bra{^{3}P_{1}}r\, f_{\rho}(r)\ket{^{3}S_{1}}_{d}-\sqrt{2}\,(1-\muV)\,\bra{^{3}P_{1}}r\, f_{\rho}(r)\ket{^{3}D_{1}}_{d}\nonumber \\ & & \hspace{0.7cm}-\frac{2}{m_{\rho}}\,\bra{^{3}P_{1}}r\, f_{\rho}(r)\ket{^{3}S_{1}^{(+)}}_{d}+\frac{2\sqrt{2}}{m_{\rho}}\,\bra{^{3}P_{1}}r\, f_{\rho}(r)\ket{^{3}D_{1}^{(-)}}_{d}\bigg),\\ X_{4} & = & -\frac{4\sqrt{2}}{\sqrt{3}\,(m_{\rho}^{2}-m_{\pi}^{2})}\left(\bra{^{3}P_{1}}f_{\pi\rho}^{\,'}(r)\ket{^{3}S_{1}}_{d}+\frac{1}{\sqrt{2}}\,\bra{^{3}P_{1}}f_{\pi\rho}^{\,'}(r)\ket{^{3}D_{1}}_{d}\right)\,;\end{aligned}$$ 3\. $^{3}P_{2}$–$^{3}F_{2}$ $$\begin{aligned} X & = & \frac{\sqrt{5}}{3}\left(\bra{^{3}P_{2}}r\, f_{\rho}(r)\ket{^{3}S_{1}}_{d}-\frac{2\sqrt{2}}{5}\,\bra{^{3}P_{2}}r\, f_{\rho}(r)\ket{^{3}D_{1}}_{d}-\frac{3\sqrt{3}}{5}\,\bra{^{3}F_{2}}r\, f_{\rho}(r)\ket{^{3}D_{1}}_{d}\right)\,,\\ X_{2} & = & -\frac{\sqrt{5}}{3}\left(\bra{^{3}P_{2}}r\, f_{\omega}(r)\ket{^{3}S_{1}}_{d}\,-\frac{2\sqrt{2}}{5}\bra{^{3}P_{2}}r\, f_{\omega}(r)\ket{^{3}D_{1}}_{d}-\frac{3\sqrt{3}}{5}\,\bra{^{3}F_{2}}r\, f_{\omega}(r)\ket{^{3}D_{1}}_{d}\right)\,,\\ X_{3} & = & \frac{4\sqrt{5}}{3}\left((1-\muV)\,\bra{^{3}P_{2}}r\, f_{\rho}(r)\ket{^{3}S_{1}}_{d}-\frac{2\sqrt{2}}{5}\,(1-4\muV)\,\bra{^{3}P_{2}}r\, f_{\rho}(r)\ket{^{3}D_{1}}_{d}\right.\nonumber \\ & & \hspace{1.0cm}-\frac{2}{m_{\rho}}\,\bra{^{3}P_{2}}r\, f_{\rho}(r)\ket{^{3}S_{1}^{(+)}}_{d}+\frac{4\sqrt{2}}{5\, m_{\rho}}\,\bra{^{3}P_{2}}r\, f_{\rho}(r)\ket{^{3}D_{1}^{(-)}}_{d}\nonumber \\ & & \left.\hspace{1.0cm}-\frac{3\sqrt{3}}{5}\,(1+\muV)\,\bra{^{3}F_{2}}r\, f_{\rho}(r)\ket{^{3}D_{1}}_{d}+\frac{6\sqrt{3}}{5\, m_{\rho}}\,\bra{^{3}F_{2}}r\, f_{\rho}(r)\ket{^{3}D_{1}^{(+)}}_{d}\right),\\ X_{4} & = & \frac{4\sqrt{10}}{3\,(m_{\rho}^{2}-m_{\pi}^{2})}\left(\bra{^{3}P_{2}}f_{\pi\rho}^{\,'}(r)\ket{^{3}S_{1}}_{d}-\frac{1}{5\sqrt{2}}\,\bra{^{3}P_{2}}f_{\pi\rho}^{\,'}(r)\ket{^{3}D_{1}}_{d}+\frac{3\sqrt{3}}{5}\,\bra{^{3}F_{2}}f_{\pi\rho}^{\,'}(r)\ket{^{3}D_{1}}_{d}\right)\,,\end{aligned}$$ where $\ket{^{2S+1}L_{J}^{(+)}}\equiv\left(\frac{d}{dr}-\frac{L+1}{r}\right)\ket{^{2S+1}L_{J}}$, $\ket{^{2S+1}L_{J}^{(-)}}\equiv\left(\frac{d}{dr}+\frac{L}{r}\right)\ket{^{2S+1}L_{J}}$, and $f_{\pi\rho}^{\,'}(r)\equiv\frac{d}{dr}\left(f_{\pi}(r)-f_{\rho}(r)\right)$. [^1]: We also note that unlike our notation, $\mu_{\ssst{S,V}}$ is used to denote the anomalous magnetic moments in Ref. [@Oka:1983sp].

--- abstract: 'I describe harmonic-oscillator-based effective theory (HOBET) and explore the extent to which the effects of excluded higher-energy oscillator shells can be represented by a contact-gradient expansion in next-to-next-to-leading order (NNLO). I find the expansion can be very successful provided the energy dependence of the effective interaction, connected with missing long-wavelength physics associated with low-energy breakup channels, is taken into account. I discuss a modification that removes operator mixing from HOBET, simplifying the task of determining the parameters of an NNLO interaction.' address: | Inst. for Nuclear Theory and Dept. of Physics, University of Washington\ Seattle, WA 98195, USA\ E-mail: haxton@phys.washington.edu author: - 'W. C. HAXTON' title: 'HARMONIC-OSCILLATOR-BASED EFFECTIVE THEORY' --- Introduction ============ Often the problem of calculating long-wavelength nuclear observables – binding energies, radii, or responses to low-momentum probes – is formulated in terms of pointlike, nonrelativistic nucleons interacting through a potential. To solve this problem theorists have developed both nuclear models, which are not systematically improvable, and exact numerical techniques, such as fermion Monte Carlo. Because the nuclear many-body problem is so difficult – one must simultaneously deal with anomalously large NN scattering lengths and a potential that has a short-range, strongly repulsive core – exact approaches are numerically challenging, so far limited to the lighter nuclei within the 1s and lower 1p shells. The Argonne theory group has been one of the main developers of such exact methods [@argonne]. However, effective theory (ET) offers an alternative, a method that limits the numerical difficulty of a calculation by restricting it to a finite Hilbert space (the $P$- or “included"-space), while correcting the bare Hamiltonian $H$ (and other operators) for the effects of the $Q$- or “excluded"-space. Calculations using the effective Hamiltonian $H^{eff}$ within $P$ reproduce the results using $H$ within $P+Q$, over the domain of overlap. That is, the effects of $Q$ on $P$-space calculations are absorbed into $P(H^{eff}-H)P$. There may exist some systematic expansion – perhaps having to do with the shorter range of interactions in $Q$ – that simplifies the determination $P(H^{eff}-H)P$ [@weinberg; @savage]. One interesting challenge for ET is the case of a $P$-space basis of harmonic oscillator (HO) Slater determinants. This is a special basis for nuclear physics because of center-of-mass separability: if all Slater determinants containing up to $n$ oscillator quanta are retained, $H^{eff}$ will be translationally invariant (assuming $H$ is). Such bases are also important because of powerful shell-model (SM) techniques that have been developed for interative diagonalization and for evaluating inclusive responses. The larger $P$ can be made, the smaller the effects of $H^{eff}-H$. There are two common approaches to the ET problem. One is the determination of $P(H^{eff}-H)P$ from a given $H$ known throughout $P+Q$, a problem that appears naively to be no less difficult than the original $P+Q$ diagonalization of $H$. However, this may not be the case if $H$ is somehow simpler when acting in $Q$. For example, if $Q$ contains primarily high-momentum (short-distance) interactions, then $H^{eff}-H$ might have a cluster expansion: it becomes increasingly unlikely to have $m$ nucleons in close proximity, as $m$ increases (e.g., a maximum of four nucleons can be in a relative s-state). Thus one could approximate the full scattering series in $Q$ by successive two-body, three-body, etc., terms, with the expectation that this series will converge quickly with increasing nucleon number. This would explain why simple two-nucleon ladder sums – g-matrices – have been somewhat successful as effective interactions [@kuobrown] (however, see Ref. ). The second approach is that usually taken in effective field theories [@savage], determining $H^{eff}$ phenomenologically. This is the “eliminate the middleman" approach: $H$ itself is an effective interaction, parameterized in order to reproduce $NN$ scattering and other data up to some energy. So why go to the extra work of this intermediate stage between QCD and SM-like spaces? This alternative approach begins with $PHP$, the long-range $NN$ interaction that is dominated by pion exchange and constrained by chiral symmetry. The effects of the omitted $Q$-space, $P(H^{eff}-H)P$, might be expressed in some systematic expansion, with the coefficients of that expansion directly determined from data, rather than from any knowledge of $H$ acting outside of $P$. While we explored this second approach some years ago, some subtle issues arose, connected with properties of HO bases. This convinced us that the first step in our program should be solving and thoroughly understanding the effective interactions problem via the first approach, so that we would have answers in hand to test the success of more phenomenological approaches. Thus we proceeded to follow the first approach using a realistic $NN$ potential, $av18$ [@argonne2], generating $H^{eff}$ numerically for the two- and three-body problems in a variety of HO SM spaces. Here I will use these results to show that a properly defined short-range interaction provides an excellent representation of the effective interaction. This is an encouraging result, one that suggests a purely phenomenological treatment of the effective interaction might succeed. The key observation is that HOBET is an expansion around momenta $k \sim 1/b$, and thus differs from EFT approaches that expand around $k \sim 0$. Consequently a HOBET $P$-space lacks both high-momentum components important to short-range $NN$ interactions and long-wavelength components important to minimizing the kinetic energy. While our group has previously discussed some of the consequences of the combined infrared/ultraviolet problem in HOBET, here I identify another: a sharp energy dependence in $H^{eff}$ that must be addressed before any simple representation of $H^{eff}-H$ is possible. This, combined with a trick to remove operator mixing, leads to a simple and successful short-range expansion for $H^{eff}-H$. I conclude by noting how these results may set the stage for a successful determination of the HOBET $H^{eff}$ directly from data. Review of the Bloch-Horowitz Equation ===================================== The basis for the approach described here is the Bloch-Horowitz (BH) equation, which generates a Hermitian, energy-dependent effective Hamiltonian, $H^{eff}$, which operates in a finite Hilbert space from which high-energy HO Slater determinants are omitted: $$\begin{aligned} \label{wh:eq1} H^{eff} = H &+& H {1 \over E - Q H} Q H \nonumber \\ H^{eff} |\Psi_P \rangle = E |\Psi_P \rangle ~~~&&~~~ |\Psi_P \rangle = (1-Q) |\Psi \rangle.\end{aligned}$$ Here $H$ is the bare Hamiltonian and $E$ and $\Psi$ are the exact eigenvalue and wave function (that is, the results of a full solution of the Schroedinger equation for $H$ in $P+Q$). The BH equation must be solved self-consistently, as $H^{eff}$ depends on $E$. If this is done, the model-space calculation reproduces the exact $E$, and the model-space wave function $\Psi_P$ is simply the restriction of $\Psi$ to $P$. If one takes for $P$ a complete set of HO Slater determinants with HO energy $\le \Lambda_P \hbar \omega$, $H^{eff}$ will be translational invariant. $P$ is then defined by two parameters, $\Lambda_P$ and the HO size parameter $b$. The BH equation was solved numerically for the $av18$ potential using two numerical techniques. In work carried out in collaboration with C.-L. Song [@song], calculations were done for the deuteron and $^3$He/$^3$H by directly summing the effects of $av18$ in the $Q$-space. Because this potential has a rather hard core, sums to 140 $\hbar \omega$ were required to achieve $\sim$ 1 keV accuracy in the deuteron binding energy, and 70 $\hbar \omega$ to achieve $\sim$ 10 keV accuracies for $^3$He/$^3$H. In work carried out with T. Luu [@luu], such cutoffs were removed by doing momentum-space integrations over all possible excitations. The results are helpful not only to the goals discussed previously, but also in illustrating general properties of $H^{eff}$ that may not be widely appreciated. For example, Table 1 gives the evolution of the $P$-space $^3$He $av18$ wave function as a function of increasing $\Lambda_P$ for fixed $b$. $\Psi_P$ evolves simply, with each increment of $\Lambda_P$ adding new components to the wave function, while leaving previous components unchanged. One sees that the probability of residing in the model space grows slowly from its 0$\hbar \omega$ value (31%) toward unity. Effective operators are defined by $$\hat{O}^{eff} = (1 + HQ {1 \over E_f - HQ}) \hat{O} (1 + {1 \over E_i - QH}QH)$$ and must be evaluated between wave functions $\Psi_P$ having the nontrivial normalization illustrated in Table 1 and determined by $$1 = \langle \Psi| \Psi \rangle = \langle \Psi_P | \hat{1}^{eff} | \Psi_P \rangle.$$ The importance of this is illustrated in Fig. 1, where the elastic magnetic responses for deuterium and $^3$He are first evaluate with exact wave functions $\Psi_P$ but bare operators, then re-evaluated with the appropriate effective operators. Bare operators prove a disaster even at intermediate momentum transfers of 2-3 f$^{-1}$. By using the effective operator and effective wave function appropriate to $\Lambda_P$, the correct result – the form factor is independent of the choice of $\Lambda_P$ (or $b$) – is obtained, as it must in any correct application of effective theory. {width="9cm"} The choice of a HO basis excludes not only high-momentum components of wave functions connected with the hard core, but also low-momentum components connected with the proper asymptotic fall-off of the tail of the wave functions. This combined ultraviolet/infrared problem was first explored by us in connection with the nonperturbative behavior of $H^{eff}$: the need to simultaneously correct for the missing long- and short-distance behavior of $\Psi_P$ is the reason one cannot define a simple $P$ that makes evaluation of $H^{eff}$ converge rapidly. We also found a solution to this problem, a rewriting of the BH equation in which the relative kinetic energy operator is summed to all orders. This summation can be viewed as a transformation of a subset of the Slater determinants in $P$ to incorporate the correct asymptotic falloff. This soft physics, obtained from an infinite sum of high-energy HO states in $Q$, is the key to making $H^{eff}$ perturbative. It turns out this physics is also central to the issue under discussion here, the existence of a simple representation for $H^{eff} = H + H { 1 \over E-QH} QH$, where $H=T+V$. The reorganized BH $H^{eff}$ is the sum of the three left-hand-side (LHS) terms in Eqs. (\[wh:eq4\]) $$\begin{aligned} \langle \alpha | T + TQ {1 \over E-QT} QT | \beta \rangle&\underset{nonedge}{\longrightarrow}&\langle \alpha | T | \beta \rangle \nonumber \\ \langle \alpha | {E \over E-TQ} V {E \over E-QT} | \beta \rangle&\underset{nonedge}{\longrightarrow}&\langle \alpha | V | \beta \rangle \nonumber \\ \langle \alpha | {E \over E-TQ} V {1 \over E-QH} QV {E \over E-QT} | \beta \rangle&\underset{nonedge}{\longrightarrow}&\langle \alpha | V {1 \over E-QH} QV | \beta \rangle \label{wh:eq4}\end{aligned}$$ The first LHS term is the effective interaction for $T$, the relative kinetic energy. As $QT$ acts as a ladder operator in the HO, $E/E-QT$ is the identity except when it operates on an $|\alpha \rangle$ with energy $\Lambda_P\hbar \omega$ or $(\Lambda_P-1) \hbar \omega$. We will call these Slater determinants the edge states. For nonedge states, this new expression and the BH form given in Eq. (\[wh:eq1\]) both reduce to the expressions on the right of Eqs. (\[wh:eq4\]). Noting that the first LHS term in Eqs. (\[wh:eq4\]) can be rewritten as $$\langle \alpha | {E \over E-TQ} (T-{TQT \over E}) {E \over E-QT} | \beta \rangle,$$ we see that the $QT$ summation can be viewed as a transformation to a new basis for $P$, $E/(E-QT) |\alpha \rangle$, that is orthogonal but not orthonormal. This edge-state basis builds in the proper asymptotic behavior governed by $QT$ (free propagation) and the binding energy $E$. The transformation preserves translational invariance, as $T$ is the relative kinetic energy operator. Viewed in the transformed basis, the appropriate effective interaction in given by the LHS terms in the square brackets in Eqs. (\[wh:eq6\]) below. Alternatively, the results can be viewed as two equivalent expressions for the effective interaction between HO states, but with a different division between “bare" and “rescattering" contributions $$\begin{aligned} \mathrm{bare:}{E \over E-TQ} \left[ H - {TQT \over E} \right] {E \over E-QT}&\Leftrightarrow&H \nonumber \\ \mathrm{rescattering:}{E \over E-TQ} \left[ V {1 \over E-QH} QV \right] {E \over E-QT}&\Leftrightarrow& H {1 \over E-QH} QH \label{wh:eq6}\end{aligned}$$ [*It is this new division that is critical.*]{} The expressions are identical for nonedge states. But for edge states, only the expression on the left isolates a quantity, $VGV$, that is short-range and nonperturbative. We will see that this is the term that can be represented by a simple, systematic expansion. Figure 2 shows the extended tail that is induced by $E/E-QT$ acting on a HO state. Figure 3 is included to emphasize that there are important numerical differences between the two expressions in Eqs. (\[wh:eq6\]). It compares calculations done for the deuteron using the two “bare" interactions: thus in both cases $V$ enters only linearly between low-momentum states, and all multiple scattering of $V$ in $Q$ in ignored. Figure 3 gives the resulting deuteron binding energy as a function of $b$, for several values of $\Lambda_P$. For the standard form of the BH equation, a small model space overestimates the kinetic energy (too confining) and overestimates short-range contributions to $V$ (too little freedom to create the needed wave-function “hole"). Making $b$ larger to lower the kinetic energy exacerbates the short-range problem, and conversely. Thus the best $b$ is a poor compromise that, even in a 10 $\hbar \omega$ bare calculation, fails to bind the deuteron. But the new bare $H$ on the LHS of Eqs. (\[wh:eq6\]) sums $QT$ to give the correct wave-function behavior at large $r$, independent of $b$. Then, for the choice $b \sim$ 0.4-0.5 f, the short-range physics can be absorbed directly into the $P$ space. The result is excellent 0th-order ground-state energy, with the residual effects of multiple scattering through $QV$ being very small and perturbative [@luu]. {width="10cm"} {width="10cm"} Harmonic-Oscillator-Based Effective Theory ========================================== Now I turn to the question of whether (and how) the $Q$-space rescattering contribution to $H^{eff}$ might be expressed through some systematic short-range expansion. There are two steps important in applying such an expansion to HOBET. One has to do with the form of the short-range expansion. A contact-gradient (CG) expansion, constructed to include all possible LO (leading order), NLO (next-to-leading order) , NNLO (next-to-next-to-leading order), ..., interactions is commonly used, $$\begin{aligned} a_{LO}^{ss}(\Lambda_P,b) \delta({\bf r}) + a_{NLO}^{ss}(\Lambda_P,b) (\overleftarrow{\nabla}^2 \delta({\bf r}) + \delta({\bf r}) \overrightarrow{\nabla}^2)+ \nonumber \\ a_{NNLO}^{ss,22}(\Lambda_P,b) \overleftarrow{\nabla}^2 \delta({\bf r}) \overrightarrow{\nabla}^2 + a_{NNLO}^{ss,40} (\Lambda_P,b)(\overleftarrow{\nabla}^4 \delta({\bf r}) + \delta({\bf r}) \overrightarrow{\nabla}^4). \label{wh:eq5}\end{aligned}$$ Because HOBET is an expansion around a typical momentum scale $\sim~1/b$, rather than around $\vec{k}=0$, it is helpful to redefine the derivatives appearing in the CG expansion Noting $$\overrightarrow{\nabla}^n \exp{i \vec{k} \cdot \vec{r}}~\arrowvert_{k=0} = 0, n=1,2,....,$$ we demand by analogy in HOBET $$\overrightarrow{\nabla}^n \psi_{1s}(b) = 0, n=1,2,...$$ This can be accomplished by redefining the operators $\hat{O}$ of Eqs. (\[wh:eq5\]) by $$\hat{O} \rightarrow e^{r^2/2} \hat{O} e^{r^2/2}$$ The gradients in Eq. (\[wh:eq5\]) then act on polynomials in $r$, a choice that removes all operator mixing. That is, if $a_{LO}$ is fixed in LO to the $n=1$ to $n=1$ matrix element, where $n$ is the nodal quantum number, it remains fixed in NLO, NNLO, etc. Furthermore, the expansion is in nodal quantum numbers, e.g., $$\overrightarrow{\nabla}^2 \sim (n-1)~~~~~~~~\overrightarrow{\nabla}^4 \sim (n-1)(n-2)$$ so that matrix elements become trivial to evaluate in any order. It can be shown that the leading order in $n$ contribution agrees with the plane-wave result, and that operator coefficients are a generalization of standard Talmi integrals for nonlocal potentials, e.g., $$a_{NNLO}^{ss,22} \sim \int^\infty_0 \int^\infty_0 e^{-r_1^2} r_1^2 V(r_1,r_2) r_2^2 e^{-r_2^2} r_1^2 r_2^2 dr_1 dr_2$$ The next step is to identify that quantity in the BH equation that should be identified with the CG expansion. This has to do with the two forms of the BH equation discussed previously. Consider the process of progressively integrating out $Q$ in favor of the CG expansion, beginning at $\Lambda >> \Lambda_P$ and progressing to $\Lambda=\Lambda_P$. Using the projection operator $$Q_\Lambda = \sum_{\alpha=\Lambda_P+1}^{\Lambda} | \alpha \rangle \langle \alpha |~~~~ Q_{\Lambda_P} \equiv 0$$ we can isolate the contributions, above some scale $\Lambda$, to the two BH rescattering terms of Eqs. (\[wh:eq6\]) $$\begin{aligned} \Delta(\Lambda)&=&H {1 \over E-QH} QH - H {1 \over E-Q_\Lambda H} Q_\Lambda H \nonumber \\ \Delta_{QT}(\Lambda)&=&{E \over E-TQ} \left[ V {1 \over E-QH} QV - V {1 \over E-Q_\Lambda H} Q_\Lambda V \right] {E \over E-QT}.\end{aligned}$$ The goal of a CG expansion might be successful reproduction of the matrix elements of $\Delta(\Lambda)$ – the $Q$-space rescattering contributions for the standard form of the BH equation – as $\Lambda \rightarrow \Lambda_P$. This would allow us to replace all $Q$-space rescattering by a systematic short-range expansion, opening the door to a purely phenomenological determination of $H^{eff}$ for the SM. The test case will be an 8$\hbar \omega$ $P$-space calculation for the deuteron ($\Lambda_{P}$ = 8, $b$=1.7 f). The running of the 15 independent $^3$S$_1$ matrix elements of $\Delta(\Lambda)$ are plottted in Fig. 4a. Five of these are distinguished because they involve an edge-state bra or ket (or both). The evolution of these contributions with $\Lambda$ is seen to be somewhat less regular than that of nonedge-state matrix elements. The results for $\Lambda=\Lambda_P$ show that rescattering is responsible for typically 12 MeV of binding energy. The CG fit to the results in Fig. 4a were done in LO, NLO, and NNLO as a function of $\Lambda$, using the standard form of the BH equation. The coefficients are fit to the lowest-energy matrix elements. Thus in LO $a_{LO}(\Lambda)$ is fixed by the $1s - 1s$ matrix element, leaving 14 unconstrained matrix elements; the NNLO fit ($1s-1s$, $1s-2s$, $1s-3s$, and $2s-2s$) leaves 11 matrix elements unconstrained. This is easily done, because the operators do not mix; e.g., among these four, only the $1s-3s$ matrix element is influenced by $a_{NNLO}^{ss,40}$. The result is a set of coefficient that run as a function of $\Lambda$ in the usual way, with $a_{LO}$ small and dominant for large $\Lambda$, and with the NLO and NNLO terms turning on as the scale is dropped. Figs. 4b-d show the residuals – the differences between the predicted and calculated matrix elements. For non-edge-state matrix elements the scale at which typical residuals in $\Delta$ are significant, say greater than 100 keV (above $\sim$ 1%), is brought down successively, e.g., from $\sim 100 \hbar \omega$, to $\sim 55 \hbar \omega$ (LO), to $\sim 25 \hbar \omega$ (NLO), and finally to $\sim \Lambda_P \hbar \omega$ (NNLO). But matrix elements involving edge states break this pattern: the improvement is not significant, with noticeable deviations remaining at $\sim 100 \hbar \omega$ even at NNLO. {width="12cm"} This failure could be anticipated: because $QT$ strongly couples nearest shells across the $P-Q$ boundary, $H{1 \over E-QH} QH$ contains long-range physics. The candidate short-range interaction is $V {1 \over E-QH} QV$, not $H{1 \over E-QH} QH$: this is the reason $QT$ should be first summed, putting the BH equation in a form – the LHS of Eqs. (\[wh:eq6\]) – that isolates this quantity. This reorganization affects edge-state matrix elements only, those with the large residuals in Figs. 4b-d. To use the reorganized BH equation, the $QT$ sums appearing in Eqs. (\[wh:eq4\]) must be completed. There are several procedures for doing this, but one convenient method exploits the raising/lowering properties of $T$. The result is a series of continued fractions $\tilde{g}_i(2 E/\hbar \omega, \{\alpha_i\},\{\beta_i\})$, where $\alpha_i =(2n+2i+l-1/2)/2$ and $\beta_i = \sqrt{(n+i)(n+i+l+1/2)}/2$. For any operator $\hat{O}$ (e.g., $V$, $V {1 \over E-QH} QV$, etc.) $$\langle n' l' | {E \over E-TQ} O {E \over E-QT} | n l \rangle = \sum_{i,j=0} \tilde{g}_j(n',l') \tilde{g}_i(n,l) \langle n'+j~ l | O |n+i~ l \rangle$$ It follows that the coefficients of the CG expansion for a HO basis must be redefined for edge states, with a state- and E-dependent renormalization $$\begin{aligned} &&a_{LO} \rightarrow a_{LO}'(E/\hbar \omega ; n',l',n,l) = a_{LO} \sum_{i,j=0} \tilde{g}_j(n',l') \tilde{g}_i(n,l) \nonumber \\ &&\times \left[ {\Gamma(n'+j+1/2) \Gamma(n+i+1/2) \over \Gamma(n'+1/2) \Gamma(n+1/2)} \right]^{1/2} \left[{(n'-1)! (n-1)! \over (n'+j-1)! (n+i-1)!} \right]^{1/2} .\end{aligned}$$ This renormalization, which introduces no new parameters, can be evaluated in a similar way for heavier systems: $T$ remains a raising operator. {width="12cm"} Figs. 5 shows the results: the difficulties encountered for $\Delta(\Lambda)$ do not arise for $\Delta_{QT}(\Lambda)$. The edge-state matrix elements are now well behaved, and the improvement from LO to NLO to NNLO is systematic in all cases. When $\Lambda \rightarrow \Lambda_P$, the CG potential continues to reproduce $H^{eff}$ for the $av18$ potential remarkably well, with an the average error in $^3$S$_1$ NNLO matrix elements of about 100 keV (or 1% accuracy). Other channels we explored behaved even better: the average error for the 15 $^1$S$_0$ matrix elements is about 10 keV (or 0.1% accuracy). Because all matrix elements of $H^{eff}$ are reproduced well, the CG potential preserves spectral properties, not simply properties of the lowest energy states within $P$. The NNLO calculation in the $^3$S$_1$-$^3$D$_1$ channel yields a deuteron ground-state energy of -2.21 MeV. Several points can be made:\ $\bullet$ The net effect of the $QT$ summation is to weaken the CG potential for HO edge states: the resulting, more extended state has a reduced probability at small $r$. Consequently the effects of $QV$ are weaker than in states immune from the effects of $QT$.\ $\bullet$ The very strong $QT$ coupling of the $P$ and $Q$ spaces is clearly problematic for an ET: small changes in energy denominators alter the induced interactions. Thus it is quite reasonable that removal of this coupling leads to a strong energy dependence in the effective interaction between HO states. I believe that proper treatment of this energy dependence will be crucial to a correct description of the bound-state spectrum in the HO SM.\ $\bullet$ This process can also be viewed as a transformation to a new, orthogonal (but not normalized) basis for $P$ in which $|\alpha \rangle \rightarrow {E \over E-QT} |\alpha \rangle$. This yields basis states with the proper asymptotic behavior for each channel. A CG expansion with fixed coefficients can be used between these states, following the reorganized BH equation of Eqs. (\[wh:eq4\]).\ $\bullet$ While our calculations have been limited to the deuteron, this same phenomena must arise in heavier systems treated in HO bases – $QT$ remains the ladder operator. As the issue is extended states that minimize the kinetic energy, it is clear that the relevant parameter must be the Jacobi coordinate associated with the lowest breakup channel. This could be an issue for treatments of $H^{eff}$ based on the Lee-Suzuki transformation, which transforms the interaction into an energy-independent one . In approaches like the no-core shell model [@nocore], the Lee-Suzuki transformation is generally not evaluated exactly, but instead only at the two-body level. If such an approach were applied, for example, to $^6$Li, a system weakly bound (1.475 MeV) to breaking up as $\alpha$+d, it is not obvious that a two-body Lee-Suzuki transformation would treat the relevant Jacobi coordinate responsible for the dominant energy dependence. This should be explored.\ $\bullet$ I believe the conclusions about the CG expansion will apply to other effective interactions. For example, V-low-k [@achim], a soft potential obtained by integrating out high-momentum states, is derived in a plane-wave basis, where $T$ is diagonal. Thus it should be analogous to our CG interaction, requiring a similar renormalization when embedded into a HO SM space. It would be interesting to test this conclusion. Summary ======= These results show that the effective interaction in the HO SM must have a very sharp dependence on the binding energy, defined as the energy of the bound state relative to the first open channel. This is typically 0 to 10 MeV for the bound states of most nuclei (and 2.22 MeV for the deuteron ground state explored here). Once this energy dependence is identified, the set of effective interaction matrix elements can be represented quite well by a CG expansion, and the results for successive LO, NLO, and NNLO calculations improve systematically. This result suggests that the explicit energy dependence of the BH equation is almost entirely due to $QT$ – though this inference, based on the behavior of matrix elements between states with different number of HO quanta, must be tested in a case where multiple bound states exist. We also presented a simple redefinition of the gradients associated with CG expansions, viewing the expansion as one around a momentum scale $\sim 1/b$. This definition removes operator mixing, making NNLO and higher-order fits very simple. The expansion then becomes one in nodal quantum numbers, with the coefficients of the expansion related to Talmi integrals, generalized for nonlocal interactions. While our exploration here has been based on “data" obtained from an exact BH calculation of the effective interaction for the $av18$ potential, this raises the question, is such a potential necessary to the SM? That is, now that the success of an NNLO description of $Q$-space contributions is established, could one start with $PHP$ and determine the coefficients for such a potential directly from data, without knowledge of matrix elements of $H$ outside of $P$? I believe the answer is yes, even in cases (like the deuteron) when insufficient information is available from bound states. It turns out that the techniques described here can be extended into the continuum, so that observables like phase shifts could be combined with bound-state information to determine the coefficients of such an expansion. An effort of this sort is in progress. I thank M. Savage for helpful discussions, and T. Luu and C.-L. Song for enjoyable collaborations. This work was supported by the U.S. Department of Energy Division of Nuclear Physics and by DOE SciDAC grant DE-FG02-00ER-41132. References ========== [9]{} S. C. Pieper and R. B. Wiringa, [*Ann.Rev.Nucl.Part.Sci.*]{} [**51**]{}, 53 (2001). S. Weinberg, [*Phys. Lett.*]{} [**B251**]{}, 288 (1990), [*Nucl. Phys.*]{} [**B363**]{}, 3 (1991), and [*Phys. Lett.*]{} [**295**]{}, 114 (1992). S. R. Beane, P. F. Bedaque, W. C. Haxton, D. R. Philips, and M. J. Savage, in [*At the Frontier of Particle Physics, Vol. 1*]{} (World Scientific, Singapore, 2001), p. 133. T. T. S. Kuo and G. E. Brown, [*Nucl. Phys.*]{} [**A114**]{}, 241 (1968). B. R. 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--- abstract: 'We study bond percolation for a family of infinite hyperbolic graphs. We relate percolation to the appearance of homology in finite versions of these graphs. As a consequence, we derive an upper bound on the critical probabilities of the infinite graphs.' author: - 'Nicolas Delfosse[^1]  and Gilles Zémor[^2]' title: A homological upper bound on critical probabilities for hyperbolic percolation --- Introduction ============ Let ${{\EuScript G}}=({{\EuScript V}},{{\EuScript E}})$ be an infinite connected graph. Every edge is declared to be [ *open*]{} with probability $p$, otherwise it is [*closed*]{}. This endows subsets of edges with a product probability measure by declaring edges to be open or closed independently of the others, and creates a random open subgraph $\varepsilon$. If a given edge $e$ belongs to an infinite connected component of $\varepsilon$, we say that (bond) percolation occurs. If the graph ${{\EuScript G}}$ is edge-transitive, then the probability of percolation does not depend on $e$ and we may denote this probability by $f(p)$. Arguably, the most studied parameter of percolation theory is the [*critical probability*]{} $p_c=p_c({{\EuScript G}})$ which is the supremum of the set of $p$’s for which $f(p)=0$. Ever since the seminal work of Kesten [@K80] percolation was extensively studied on the lattices associated to ${\mathbb{Z}}^d$, for background see [@G89]: in the present paper, we are interested in percolation on regular tilings of the hyperbolic plane. This topic was first introduced by Benjamini and Schramm [@Be96], and further studied in [@Be01; @GZ11; @BMK09] among other papers. Specifically, our focus is on the family of graphs that we shall denote by $G(m)$, for $m\geq 4$, that are regular of degree $m$, planar, and tile the plane by elementary faces of length $m$. For $m=4$, the graph $G(m)$ is exactly the square ${\mathbb{Z}}^2$ lattice. The local structure of the graph $G(5)$ is represented in Figure \[fig:G(5)\]. =\[circle, draw, fill=black!100, inner sep=0pt, minimum width=2pt\] (0:1.3) in [72,144,...,359]{} [ – (:1.3) ]{} – cycle (0:1.3) node (a) (72:1.3) node (b) (144:1.3) node (c) (216:1.3) node (d) (288:1.3) node (e) ; (0:.9) node (a2) (60:.9) node (a3) (300:.9) node (a1) ; (a1)–(a)–(a2) (a)–(a3); (72:.9) node (b2) (132:.9) node (b3) (12:.9) node (b1) ; (b1)–(b)–(b2) (b)–(b3); (144:.9) node (c2) (204:.9) node (c3) (84:.9) node (c1) ; (c1)–(c)–(c2) (c)–(c3); (216:.9) node (d2) (276:.9) node (d3) (156:.9) node (d1) ; (d1)–(d)–(d2) (d)–(d3); (288:.9) node (e2) (348:.9) node (e3) (228:.9) node (e1) ; (e1)–(e)–(e2) (e)–(e3); (36:2.1) node (f) (108:2.1) node (g) (180:2.1) node (h) (252:2.1) node (i) (324:2.1) node (j); (a3)–(f)–(b1) (b3)–(g)–(c1) (c3)–(h)–(d1) (d3)–(i)–(e1) (e3)–(j)–(a1); (70:.4) node (a24) (290:.4) node (a21); (345:.4) node(a31); (15:.4) node (a14); (a2)–(a24)–(a31)–(a3) (a1)–(a14)–(a21)–(a2); (142:.4) node (b24) (2:.4) node (b21); (57:.4) node(b31); (87:.4) node (b14); (b2)–(b24)–(b31)–(b3) (b1)–(b14)–(b21)–(b2); (214:.4) node (c24) (74:.4) node (c21); (129:.4) node(c31); (159:.4) node (c14); (c2)–(c24)–(c31)–(c3) (c1)–(c14)–(c21)–(c2); (286:.4) node (d24) (136:.4) node (d21); (201:.4) node(d31); (231:.4) node (d14); (d2)–(d24)–(d31)–(d3) (d1)–(d14)–(d21)–(d2); (358:.4) node (e24) (218:.4) node (e21); (273:.4) node(e31); (303:.4) node (e14); (e2)–(e24)–(e31)–(e3) (e1)–(e14)–(e21)–(e2); Our goal is to study the critical probabilities of these lattices. The simple lower bound $1/(m-1)\leq p_c$ can be derived since $1/(m-1)$ is the critical probability for the $m$-regular tree, and our main concern here is on dealing with upper bounds. Critical probabilities for hyperbolic tilings were studied numerically by Baek et al. [@BMK09] and also by Gu and Ziff [@GZ11] who obtain a “Monte Carlo” upper bound $p_c<0.34$ for $G(5)$. In previous work by the present authors [@DZ13], the rigorous upper bound $p_c < 0.38$ was obtained for $G(5)$ as a by-product of the study of the erasure-correcting capabilities of a family of quantum error-correcting codes. In the present paper we shall obtain a substantially improved upper bound on critical probabilities that gives $p_c<0.30$ for $G(5)$. We remark that we restrict ourselves to the hyperbolic tilings $G(m)$ because they are self-dual and our method is better suited for this case, but results on the critical probabilities for the self-dual case can lead to results for the general case [@LB12]. Classically, one uses finite portions of the infinite graph ${{\EuScript G}}$ to devise intermediate tools for studying percolation. For example, in the original ${\mathbb{Z}}^2$ setting, the standard (by now) method that leads to the computation $p_c=1/2$ is to consider $n\times n$ finite grids and study the probability of the appearance of an open path linking the south boundary to the north boundary (or east to west) [@G89]. In the hyperbolic setting however, trying to mimic this approach directly quickly leads to serious obstacles: what finite portion of the infinite graph $G(5)$ (say) should one consider, and which parts of the boundary should be matched when looking for the appearance of finite open paths ? We shall overcome this difficulty by appealing to finite graphs $G_t(m)$ that are everywhere locally isomorphic to $G(m)$, meaning that every ball of radius $t$ of $G_t(m)$ is required to be isomorphic to a ball of radius $t$ in the infinite graph $G(m)$. We shall derive an upper bound $p_c\leq p_h$ on the critical probability by defining a quantity $p_h$ such that, when $p>p_h$, then with probability tending to $1$ when $t$ tends to infinity, $G_t(m)$ must contain an open cycle that can not be expressed as a sum modulo $2$ of elementary faces. Our end result will be an expression for the upper bound $p_h$ that involves only the structure of the infinite graph $G(m)$, but the existence of the finite graphs $G_t(m)$ (which is non-obvious) will be crucial to the derivation of $p_h$. #### [**Outline and results:**]{} Sections \[section:siran\_graphs\] and \[section:homology\] are background. In Section \[section:siran\_graphs\] we give a short description of a construction of the graphs $G_t(m)$ due to Širáň. We shall need to consider the cycles of those graphs that are not expressible as sums of faces, i.e. that are homologically non-trivial: we shall therefore need background on homology that is dealt with in Section \[section:homology\]. In Section \[section:rank\] we study the appearance of homology in random subgraphs of the finite graphs $G_t(m)$. We introduce a crucial quantity $D(p)$ that we name the [*rank difference function*]{} and that captures the limiting behaviour of the difference of the dimensions of the homologies of the two random subgraphs of $G_t(m)$ chosen through the parameters $p$ and $1-p$. We then define the quantity $$p_h=\sup\left\{p, p-\frac 2m + D(p)=0\right\}.$$ The main result of this section, Theorem \[theo:Dequation\], is that $p_h$ is an upper bound on the critical probability of $G(m)$. We actually conjecture that for $m\geq 5$ (i.e. the genuinely hyperbolic, or non-amenable, case) this upper bound is also a lower bound, i.e. $p_c=p_h$. This would show that for these graphs the critical probability is local in a sense close to [@Be11]. That $p_c\leq p_h$ was derived in [@DZ13] in a roundabout way, through the study of the erasure-decoding capabilities of quantum codes associated to the tilings $G_t(m)$. The present proof not only removes the reference to quantum coding, it is intrinsically shorter and more direct. Section \[section:D(p)\] is dedicated to finding an explicit expression for the rank difference function $D(p)$, and hence for the upper bound $p_h$. Our main result is Theorem \[theo:D\_graphical\], which expresses $D(p)$ as the series: $$\label{eq:D(p)} D(p) = \frac 2 m \sum_{ C } \left( \frac 1 {|V(C)|} \left(p^{|E(C)|} (1-p)^{|\partial(C)|} - (1-p)^{|E(C)|} p^{|\partial(C)|} \right) \right),$$ where $C$ ranges over all connected subgraphs of $G(m)$ containing a given vertex, where $V(C),E(C)$ denote the vertex and edge set of $C$, and where $\partial (C)$ denotes the set of edges with at least one endpoint in $C$, which are not in $E(C)$. As mentioned, this expression for $D(p)$ does not involve the graphs $G_t(m)$ anymore, but its proof crucially relies on their existence. Section \[section:approximation\] proves that replacing $D(p)$ in by a truncated series continues to yield an upper bound on the critical probability $p_c$ of $G(m)$ (Theorem \[theo:bound\]). This allows us to compute explicit numerical upper bounds on $p_c$. Finally, Section \[section:conclusion\] summarizes the results with Theorem \[theo:final\] and gives some concluding comments. Finite quotient of the regular hyperbolic tilings {#section:siran_graphs} ================================================= We are unaware of any method for constructing the required finite versions of $G(m)$ that does not involve a fair amount of algebra. In this section, we briefly recall Širáň’s method to construct such finite versions of the regular hyperbolic tiling $G(m)$. The first step is to construct $G(m)$ from a group of matrices over a ring of algebraic integers. Then this group is reduced modulo a prime number to yield the desired finite graph. Denote by $P_k(X) = 2\cos(k\arccos(X/2))$ the $k$-th normalized Chebychev polynomial and let $\xi = 2\cos(\pi/m^2)$. Let $m \geq 5$ and consider the group $T(m)$ generated by the two following matrices of $SL_3({\mathbb{Z}}[\xi])$. $$a = \left( \begin{array}{ccc} P_m(\xi)^2-1 & 0 & P_m(\xi)\\ P_m(\xi) & 1 & 0\\ -P_m(\xi) & 0 & -1 \end{array} \right) \quad \text{and} \quad b = \left( \begin{array}{ccc} -1 & -P_m(\xi) & 0\\ P_m(\xi) & P_m(\xi)^2-1 & 0\\ P_m(\xi) & P_m(\xi)^2 & 1 \end{array} \right).$$ The group $T(m)$ admits the presentation $$\label{eq:presentation} T(m) = {\langle { a, b \ | \ a^m = b^m = (ab)^2 = 1 }\rangle}.$$ With this group we associate its *coset graph*. The coset graph associated with is defined to be the infinite planar tiling whose vertex set, respectively edge set and face set, is the set of left cosets of the subgroup ${\langle {a}\rangle}$, respectively the set of left cosets of the subgroup ${\langle {ab}\rangle}$ and the subgroup ${\langle {b}\rangle}$. A vertex and an edge, or an edge and a face, are incident if and only if the corresponding cosets have a non-empty intersection. For example, the coset ${\langle {a}\rangle} = \{1, a, a^2, \dots, a^{m-1} \}$ defines a vertex of the graph $G(m)$ and is incident to the $m$ edges represented by the cosets $${\langle {ab}\rangle},\ a{\langle {ab}\rangle},\ a^2{\langle {ab}\rangle},\ \dots,\ a^{m-1} {\langle {ab}\rangle}.$$ We can see that the coset graph is $m$-regular and that its faces contain $m$ edges. It is straightforward to check that the coset graph associated with is the infinite planar graph $G(m)$ [@Si00]. The basic idea to derive a finite version of this tiling is to reduce the matrices defining the group $T(m)$ modulo a prime number. We can reduce the coefficients of the matrices of $T(m)$ thanks to the ring isomorphism ${\mathbb{Z}}[\xi] \simeq {\mathbb{Z}}[X]/h(X)$, where $h(X) \in {\mathbb{Z}}[X]$ is the minimal polynomial of the algebraic number $\xi$. This induces a ring morphism $\pi_p: SL_3({\mathbb{Z}}[\xi]) \rightarrow SL_3({\mathbb{F}}_p[X]/\bar h(X))$ where $\bar h(X)$ is the reduction modulo $p$ of the polynomial $h(X)$. Denote by $\bar T^p(m)$ the image of the group $T(m)$ by the morphism $\pi_p$. The coset graph associated with the group $\bar T^p(m)$ is defined from the cosets of $\bar T^p(m)$, exactly like the coset graph of $T(m)$. Širáň proved that for a well chosen family of prime numbers $p$, this construction provides a sequence of finite tilings $(G_t(m))_t$ which is locally isomorphic to the infinite tiling $G(m)$ [@Si00]. Precisely: \[theo:siran\] For every integer $m \geq 5$, there exists a family of finite tilings $(G_t(m))_{t \geq m}$ and some constant $K$ such that every ball of radius $t$ of $G_t(m)$ is isomorphic to every ball of radius $t$ in $G(m)$. Furthermore, the number of vertices of $G_t(m)$ is at most $K^t$. By construction, the graphs $G_t(m)$ are vertex transitive. Indeed, each element of the group $\bar T(m)$ induces a graph automorphism of the coset graph by left multiplication. An automorphism which sends a vertex $x {\langle {a}\rangle}$ onto the vertex $y {\langle {a}\rangle}$ is given by the left multiplication by $yx^{-1}$ of the cosets representing the vertices. For the same reason, $G_t(m)$ is also edge-transitive and face-transitive. To be sure that the faces of the graph $G_t(m)$ are not degenerate, we require $t \geq m$. We will also use the fact that $G_t$ is a self-dual graph. This is a consequence of the local structure of the graph: every vertex has degree $m$ and every face has length $m$. Background on homology {#section:homology} ====================== Homology of a tiling of surface {#subsection:homology} ------------------------------- A *tiling of a surface* is a graph cellularly embedded in a smooth surface. For us only the combinatorial structure of the surface plays a role, therefore a face of the tiling is represented as the set of edges on its boundary. We denote by $G=(V, E, F)$ such a tiling, where $F$ is the set of faces that, as far as homology is concerned, can be thought of simply as a privileged set of cycles of the graph $(V,E)$. With a tiling of a surface, we associate a *dual tiling* $G^*=(V^*,E^*,F^*)$. The vertices of this dual tiling are given by the faces of $G$. Two vertices of $G^*$ are joined by an edge if the corresponding faces of $G$ share an edge. Since every edge of $E$ belongs to exactly two faces of $F$, there is a one-to-one correspondence between edges of $G$ and edges of $G^*$. Finally, for every vertex $v$ of $V$ the set of edges of $E$ incident to $v$ defines a face of $F^*$ through the above correspondence between $E$ and $E^*$. We assume the graph and its dual have neither multiple edges nor loops. We shall also use $G$ to refer indifferently to the graph $(V,E)$ and to the associated tiling $(V,E,F)$. In the remainder of this section, we consider only finite tilings, and we order the three sets $V, E$ and $F$ by $V = \{v_1, v_2, \dots, v_{|V|}\}$, $E = \{e_1, e_2, \dots, e_{|E|}\}$ and $F = \{f_1, f_2, \dots, f_{|F|}\}$. The *incidence matrix* of the graph $(V,E)$ is defined to be the matrix $B(G) = (b_{ij})_{i, j}$ of ${\mathcal{M}}_{|V|, |E|}({\mathbb{F}}_2)$ such that $b_{ij} = 1 $ if the vertex $v_i$ is incident to the edge $e_j$, and $b_{ij} = 0$ otherwise. To emphasize the ${\mathbb{F}}_2$-linear structure of some subsets of $V$, $E$ and $F$, we introduce the spaces of *$i$-chains* $C_i$: $$C_0 = \bigoplus_{v \in V} {\mathbb{F}}_2 v, \quad C_1 = \bigoplus_{e \in E} {\mathbb{F}}_2 e, \quad C_2 = \bigoplus_{f \in F} {\mathbb{F}}_2 f.$$ In other words, the space $C_0 = \{\sum_v \lambda_v v \ | \ \lambda_v \in {\mathbb{F}}_2\}$ is the set of formal sums of vertices. The sets $C_1$ and $C_2$ are defined similarly. These chain spaces are equipped with two ${\mathbb{F}}_2$-linear mappings $ \partial_2: C_2 \rightarrow C_1 $ and $ \partial_1: C_1 \rightarrow C_0 $ defined by $\partial_2(f) = \sum_{e \in f} e$ and $\partial_1(e) = \sum_{u \in e} u$. These mappings are called *boundary maps*. A subset of the vertex set, respectively the edge set or the face set, can be regarded as its indicator vector in $C_0$, respectively $C_1$ or $C_2$. This yields one-to-one correspondences between subsets and vectors, which allow us to interpret geometrically the boundary maps. In subset language, the map $\partial_2$ sends a subset of faces onto the set of edges on its boundary in the standard sense, and the map $\partial_1$ sends a subset of edges onto its “endpoints” which should be understood modulo $2$, i.e. the set of vertices incident to an odd number of edges in the subset. The singletons $\{v_i\}$, respectively $\{e_i\}$ and $\{f_i\}$, form a basis of the space $C_0$, respectively $C_1$ and $C_2$. The matrix of the map $\partial_1$ in these singleton bases is equal to the incidence matrix $B(G)$ of the graph $(V, E)$ and the matrix of the map $\partial_2$ is equal to the transpose of the incidence matrix $B(G^*)$ of $(V^*,E^*)$. We can easily prove that the composition of these applications is $\partial_1 \circ \partial_2 = 0$, implying the inclusion $\operatorname{Im}\partial_2 \subset \operatorname{Ker}\partial_1$. We can now introduce the ${\mathbb{F}}_2$-homology of tilings of surfaces. The *first homology group* of a finite tiling of a surface $G$, denoted $H_1(G)$, is the quotient space $$H_1(G) = \operatorname{Ker}\partial_1 / \operatorname{Im}\partial_2.$$ Note that $H_1(G)$ is also an ${\mathbb{F}}_2$-vector space. The vectors of $\ker \partial_1$ are called *cycles*. They correspond to the subsets of edges that meet every vertex an even number of times. The set $\ker \partial_1$ of cycles of a graph is an ${\mathbb{F}}_2$-linear space that we refer to as the *cycle code* of the graph. The vectors of $\operatorname{Im}\partial_2$ are called *boundaries* or sums of faces and they describe the sets of edges on the boundary of a subset of $F$. In what follows, we shall study the dimension of the homology group of different tilings of surfaces. The following well known property (see e.g. [@Berge73] for a proof) is used repeatedly . \[lemma:Dcycle\] The dimension of the cycle code of a graph $G=(V, E)$ composed of $\kappa$ connected components, is $ |E| - |V| + \kappa. $ Figure \[fig:torus\](a) represents a square lattice of the torus. A cycle of trivial homology is drawn on Figure \[fig:torus\](b). This cycle is clearly a sum of faces. Two examples of cycles with non trivial homology are given in Figure \[fig:torus\](c) and (d). The first homology group of this tiling of the torus is a binary space of dimension 2. It is generated, for example, by an horizontal cycle which wraps around the torus, such as the one in Figure \[fig:torus\](c) and a vertical cycle which wraps around the torus. The cycle of Figure \[fig:torus\](d) is equivalent to the sum of these horizontal and vertical cycles, up to a sum of faces. =\[circle, draw, fill, inner sep=0pt, minimum width=4pt\]; (0,0) grid (5,5); (0,0)–(5,0) (0,5)–(5,5); (0,0)–(0,5) (5,0)–(5,5); in [0,1,...,5]{} in [0,1,...,5]{} (, ) node ; =\[inner sep=0pt, minimum width=4pt\]; at (0.5,4.5) [$(a)$]{}; =\[circle, draw, fill, inner sep=0pt, minimum width=4pt\]; (0,0) grid (5,5); (0,0)–(5,0) (0,5)–(5,5); (0,0)–(0,5) (5,0)–(5,5); in [0,1,...,5]{} in [0,1,...,5]{} (, ) node ; (5,2)–(3,2)–(3, 1)–(4,1)–(4,3)–(5,3) (0,2)–(2,2)–(2,3)–(0,3); =\[inner sep=0pt, minimum width=4pt\]; at (0.5,4.5) [$(b)$]{}; =\[circle, draw, fill, inner sep=0pt, minimum width=4pt\]; (0,0) grid (5,5); (0,0)–(5,0) (0,5)–(5,5); (0,0)–(0,5) (5,0)–(5,5); in [0,1,...,5]{} in [0,1,...,5]{} (, ) node ; (0,2)–(5,2); =\[inner sep=0pt, minimum width=4pt\]; at (0.5,4.5) [$(c)$]{}; =\[circle, draw, fill, inner sep=0pt, minimum width=4pt\]; (0,0) grid (5,5); (0,0)–(5,0) (0,5)–(5,5); (0,0)–(0,5) (5,0)–(5,5); in [0,1,...,5]{} in [0,1,...,5]{} (, ) node ; (0,2)–(3,2)–(3,3)–(4,3)–(4,2)–(5,2) (2,0)–(2,5); =\[inner sep=0pt, minimum width=4pt\]; at (0.5,4.5) [$(d)$]{}; Induced homology of a subtiling {#section:subhomology} ------------------------------- Percolation theory deals with random subgraphs of a given graph. In what follows, we introduce the homology of a subgraph of a given tiling $G$. The subgraphs that we consider are obtained by selecting a subset of edges. Denote by $G = (V, E, F)$ a tiling of surface and let us consider the subgraph $G_{{\varepsilon}}$ of $G$ whose vertex set is exactly $V$ and whose edge set is a given subset ${\varepsilon}$ of $E$. This graph is not immediately endowed with a set of faces and with a homology group. The proper notion of homology for our purpose is obtained by considering the boundaries of the tiling $G$ which are included in the subgraph $G_{{\varepsilon}}$. More precisely, the subset of edges ${\varepsilon}$ defines the subspace $C_0^{\varepsilon}=C_0$, the subspace $C_1^{\varepsilon}$ of $C_1$ made up of all formal sums of edges of ${\varepsilon}$, and the subspace $C_2^{\varepsilon}$ of $C_2$ made up of all those vectors of $C_2$ whose image under $\partial_2$ is included in $C_1^{\varepsilon}$. The mappings $\partial_1^{\varepsilon}$ and $\partial_2^{\varepsilon}$ are defined as the restrictions of $\partial_1$ and $\partial_2$ to $C_1^{\varepsilon}$ and $C_2^{\varepsilon}$. Let $G=(V, E, F)$ be a tiling of a surface and let ${\varepsilon}\subset E$. The *induced homology group* of $G_{{\varepsilon}}$ is the quotient space $$H_1(G_{{\varepsilon}}) = \operatorname{Ker}\partial_1^{{\varepsilon}} / ( \operatorname{Im}\partial_2^{\varepsilon}).$$ For more detailed background on the homology of surfaces and their tilings see [@Ha02; @Gi10]. Appearance of homology in a random subgraph of $G_t$ {#section:rank} ==================================================== Homology of a subgraph ---------------------- This section is devoted to the analysis of the induced homology of a subgraph of $G_t(m)$. To lighten notation we omit the indices $m$ and $t$ and write $G=G_t(m)$. Following the notation of Section \[section:subhomology\], ${\varepsilon}$ denotes a subset of $E$ and $G_{{\varepsilon}}$ denotes the subgraph of $G$ induced by ${\varepsilon}$. The decomposition of the graph $G_{{\varepsilon}}$ into connected components induces a partition of the edges of ${\varepsilon}$: the set ${\varepsilon}$ is the disjoint union of the subsets ${\varepsilon}_i \subset E$, for $i=1, 2, \dots, r$ and where each set ${\varepsilon}_i$ is the edge set of a connected component of $G_{{\varepsilon}}$. The following lemma proves that this decomposition of the graph $G_{{\varepsilon}}$ induces a decomposition of its homology group. \[lemma:H\_decomposition\] Let ${\varepsilon}= \cup_{i=1}^r {\varepsilon}_i$ be the partition of ${\varepsilon}$ derived from the decomposition of the graph $G_{{\varepsilon}}$ into connected components. Then, the dimension of the first homology group of $G_{{\varepsilon}}$ is at most $$\dim H_1(G_{{\varepsilon}}) \leq \sum_{i=1}^r \dim H_1(G_{{\varepsilon}_i}).$$ Remark that the chain space $C_1^{\varepsilon}$ decomposes as $C_1^{\varepsilon}= \oplus_i C_1^{{\varepsilon}_i}$. This leads to a similar decomposition of the cycle code of $G_{\varepsilon}$. $$\operatorname{Ker}\partial_1^{{\varepsilon}} = \bigoplus_{i=1}^r \operatorname{Ker}\partial_1^{{\varepsilon}_i}.$$ However, the image of $\operatorname{Im}\partial_2^{\varepsilon}$ has a slightly different structure. First, the chain space $C_2^{\varepsilon}$ is has no similar decomposition but it still contains the direct sum $\oplus_i C_2^{{\varepsilon}_i}$. Hence, the image of $\operatorname{Im}\partial_2^{\varepsilon}$ contains the direct sum $ \bigoplus_{i=1}^r \operatorname{Im}\partial_2^{{\varepsilon}_i} $ as a subspace. This implies $$\begin{aligned} \dim H_1(G_{{\varepsilon}}) & = \dim \left( \bigoplus_{i=1}^r \operatorname{Ker}\partial_1^{{\varepsilon}_i} / \operatorname{Im}\partial_2^{{\varepsilon}} \right) \\ & \leq \dim \left( \bigoplus_{i=1}^r \operatorname{Ker}\partial_1^{{\varepsilon}_i} / \bigoplus_{i=1}^r \operatorname{Im}\partial_2^{{\varepsilon}_i} \right).\end{aligned}$$ To conclude, notice that this last quotient is exactly the direct sum $\oplus_i H_1(G_{t, {\varepsilon}_i})$. The next lemma proves that if ${\varepsilon}$ is composed of small clusters, then it covers no homology. \[lemma:smallclusters\] Let $G_{{\varepsilon}}$ be a connected subgraph of $G=G_t(m)$. If ${\varepsilon}$ contains at most $t$ edges, then we have $H_1(G_{{\varepsilon}}) = \{0\}$. Since $G_{{\varepsilon}}$ is connected and contains less than $t$ edges, it is included in a ball of radius $t$. From Theorem \[theo:siran\], this ball is isomorphic with a ball of the planar graph $G(m)$. But this ball is itself planar and in a planar graph, every cycle is a boundary. Thus the group $H_1(G_{{\varepsilon}})$ is trivial. The next lemma will allow us to compute the dimension of the induced homology group of every subgraph $G_{{\varepsilon}}$ of $G=G_t(m)$. Since a set ${\varepsilon}\subset E$ can be regarded as a subset of $E^*$, it also defines a subgraph $G^*_{{\varepsilon}}$ of the graph $G^*$. Let us denote by $\operatorname{rank}G_{\varepsilon}$ ($\operatorname{rank}G_{\varepsilon}^*$) the rank of an incidence matrix of $G_{\varepsilon}$ (of $G^*_{{\varepsilon}}$). By Lemma \[lemma:Dcycle\] these ranks do not depend on the choice of the incidence matrix of the graph. The dimension of the induced homology group is given by: \[lemma:dimH\] For every ${\varepsilon}\subset E$, we have $$\dim H_1(G_{{\varepsilon}}) = |{\varepsilon}| - \frac{2}{m} |E| + 1 + \operatorname{rank}G^*_{\bar {\varepsilon}} - \operatorname{rank}G_{{\varepsilon}}.$$ The group $H_1(G_{{\varepsilon}})$ is the quotient of the cycle code of $G_{{\varepsilon}}$ by $\operatorname{Im}\partial_2^{\varepsilon}$, the set of boundaries of $G$ which are included in the subgraph ${\varepsilon}$. By definition, the cycle code of $G_{{\varepsilon}}$ is the kernel of the map $\partial_1^{{\varepsilon}}$. Moreover, the incidence matrix of $G_{{\varepsilon}}$ is a matrix of this linear map. Therefore, the dimension of the cycle code of the subgraph $G_{{\varepsilon}}$ is $$\label{eq:ker} \dim\ker\partial_1^{{\varepsilon}}=|{\varepsilon}| - \operatorname{rank}G_{{\varepsilon}}.$$ The set of boundaries of $G$ is the image of the map $\partial_2$. We noticed in Section \[subsection:homology\] that a matrix of the map $\partial_2$ is given by the transpose of $B(G^*)$, the incidence matrix of $G^*$. This means that the boundaries of $G$ correspond to the sums of rows of $B(G^*)$. These are the vectors of the form $x B(G^*)$, where $x$ is a binary vector. Consider the incidence matrix of $G^*_{\bar {\varepsilon}}$, where $\bar {\varepsilon}$ denotes the complement of ${\varepsilon}$ in $E$. This matrix can be obtained from $B(G^*)$ by selecting the columns indexed by the edges in $\bar {\varepsilon}$. Let us define a map $\phi$ which sends a sum of rows of $B(G^*)$ onto the same sum of rows in the matrix $B(G^*_{\bar {\varepsilon}})$. It is the map $$\begin{aligned} \phi : \operatorname{Im}\partial_2 &\longrightarrow C_1^{\bar {\varepsilon}}\\ x B(G^*) & \longmapsto x_{\bar {\varepsilon}} B(G^*_{\bar {\varepsilon}}),\end{aligned}$$ where $x$ is a row vector of ${\mathbb{F}}_2^{|V|}$ and $x_{\bar {\varepsilon}}$ is its restriction to the columns indexed by the edges of $\bar {\varepsilon}$. Then, the boundaries of $G$ included in ${\varepsilon}$, are exactly the vectors of the kernel of $\phi$. The dimension of this space is $$\label{eq:kerphi} \dim\operatorname{Im}\partial_2^{\varepsilon}=\dim \ker \phi = \dim\operatorname{Im}\partial_2 - \dim\operatorname{Im}\phi = \operatorname{rank}G^* - \operatorname{rank}G^*_{\bar {\varepsilon}}.$$ Now $\operatorname{rank}G^*=\dim\operatorname{Im}\partial_1^* = |E^*|-\dim\ker\partial_1^*$. Applying Lemma \[lemma:Dcycle\] to the dimension of the cycle code $\ker\partial_1^*$ of $G^*$ and the fact that $G=G_t(m)$ is connected, we get $\operatorname{rank}G^* = |F| - 1 = (2/m)|E| -1$. Injecting this last fact into , we obtain, together with , the formula for $\dim H_1(G_{{\varepsilon}})=\dim\ker\partial_1^{\varepsilon}- \dim\operatorname{Im}\partial_2^{\varepsilon}$. The rank difference function ---------------------------- We now consider the probabilistic behaviour of the induced homology of a random subgraph of $G_t=G_t(m)$. To get a distribution which locally coincides with the distribution of percolation events, the subset of edges ${\varepsilon}$ is chosen by selecting each edge of $G_t$ independently with probability $p$. This defines a random subgraph $G_{t, {\varepsilon}}$ of the graph $G_t$. The intuition we follow is that if we are below the critical probability of the graph $G(m)$, then most connected components appearing in the random subgraph $G_{t, {\varepsilon}}$ should be small. Thanks to Lemma \[lemma:smallclusters\], these clusters do not support any non trivial homology. This implies that if $p < p_c(G(m))$ then the dimension of the induced homology of $G_{t, {\varepsilon}}$ must be small. Conversely, if we compute, using Lemma \[lemma:dimH\], the expected dimension of $H_1(G_{t, {\varepsilon}})$ and find it to be large, we know that $p$ must be above the critical probability $p_c$. These considerations lead us to introduce the following quantity. The rank difference function associated with the family of graphs $(G_t)_t$ is defined to be $$D(p) = \limsup_t {{\mathbb E}}_p \left( \frac{\operatorname{rank}G^*_{t, \bar {\varepsilon}} - \operatorname{rank}G_{t, {\varepsilon}}}{|E_t|} \right).$$ The rank difference function satisfies the folowing equation when $p$ is below the critical probability of $G(m)$. \[theo:Dequation\] If $p<p_c(G(m))$ then the rank difference function associated with the family $(G_t)_t$ satifies $$p - \frac 2 m + D(p) = 0.$$ \[cor:phom\] Defining $p_h = \sup\{p,p - \frac 2 m + D(p) = 0\}$ we have $p_c\leq p_h$. Assume that $p<p_c(G(m))$. By definition of the critical probability, for any fixed edge $e$ of the infinite graph $G(m)$, the probability that $e$ is contained in an open connected component $C(e)$ of $G(m)$ of size strictly larger than $t$ vanishes when $t \rightarrow \infty$. The following lemma shows that we observe a similar behaviour in the finite graphs $G_t$. It will be instrumental in proving Theorem \[theo:Dequation\]. \[lemma:leaving\_components\] For every $t \geq 0$, fix an edge $e_t$ of the graph $G_t$ and denote by $C(e_t)$ its (possibly empty) connected component in the random subgraph $G_{t, {\varepsilon}}$. Then, the probability that $C(e_t)$ contains strictly more than $t-2$ edges tends to $0$ when $t$ goes to infinity. The complementary event depends only on what occurs inside the ball of radius $t$ centered on an endpoint of the edge $e_t$. Since this ball is isomorphic to the ball with the same radius in $G(m)$, this event has the same probability in the space $G(m)$ and in $G_t(m)$. Hence the result by the remark preceding the lemma. [Theorem]{}[\[theo:Dequation\]]{} Thanks to Lemma \[lemma:H\_decomposition\], we have the following upper bound on the dimension of the first homology group of $G_{t, {\varepsilon}}$: $$\dim H_1(G_{t, {\varepsilon}}) \leq \sum_{i=1}^r \dim H_1(G_{t, {\varepsilon}_i}).$$ where ${\varepsilon}_i$ is the edge set of the $i$-th connected component of $G_{t, {\varepsilon}}$. From Lemma \[lemma:smallclusters\], all the components ${\varepsilon}_i$ of size smaller than $t$ have a trivial contribution to $H_1(G_{t, {\varepsilon}})$. For the other components, the dimension of $H_1(G_{t, {\varepsilon}_i})$ is bounded by the number of edges in the component ${\varepsilon}_i$. Indeed, the induced homology group of $G_{t, {\varepsilon}_i}$ is a quotient of the cycle code of this graph, whose dimension is at most the number of egdes in ${\varepsilon}_i$. This implies $$\dim H_1(G_{t, {\varepsilon}}) \leq |\{ e \in E_t \ \text{such that} \ |C(e)| > t \}|,$$ where $C(e)$ denotes the connected component in $G_{t, {\varepsilon}}$ of the edge $e$ and $|C(e)|$ is its number of edges. Let us denote by $X_t=X_t(G_{t, {\varepsilon}})$ the cardinality of the set $\{ e \in E_t \ \text{such that} \ |C(e)| > t \}$. To study the expectation of $X_t$, we define a random variable $X_e$, associated with each edge $e \in E_t$, which takes the value $X_e(G_{t, {\varepsilon}}) = 1$ if the size of $C(e)$ is larger than $t$ and which is 0 otherwise. Consequently, we have $$X_t = \sum_{e \in E_t} X_e,$$ and by linearity of expectation, ${{\mathbb E}}(X_t) = \sum_e {{\mathbb E}}(X_e)$. For every edge $e \in E_t$, this expectation of the random variable $X_e$ is ${{\mathbb E}}(X_e) = {{\mathbb P}}(|C(e)| > t)$. By edge-transitivity of the graph $G_t$, this quantity does not depend on the edge $e$, thus ${{\mathbb E}}(X_t) = |E_t| \ {{\mathbb P}}(|C(e_t)| > t)$, for some fixed edge $e_t$ of the graph $G_t$. Moreover, from Lemma \[lemma:leaving\_components\], this probability vanishes when $t$ goes to infinity. This allows us to bound the expected dimension of the induced homology: $${{\mathbb E}}_p \left( \frac{\dim H_1(G_{t, {\varepsilon}})}{|E_t|} \right) \leq {{\mathbb E}}_p \left( \frac{X_t}{|E_t|} \right) = {{\mathbb P}}_p( |C(e_t)| > t) \rightarrow 0.$$ Since the right-hand side tends to 0 when $t$ goes to infinity, taking the superior limit gives exactly 0, *i.e.* $$\limsup_t {{\mathbb E}}_p \left( \frac{\dim H_1(G_{t, {\varepsilon}})}{|E_t|} \right) = 0.$$ To conclude the proof, we determine the expected dimension of the induced homology group with the help of Lemma \[lemma:dimH\]. We find $$\limsup_t {{\mathbb E}}_p \left( \frac{\dim H_1(G_{t, {\varepsilon}})}{|E_t|} \right) = p - \frac 2 m + D(p).$$ Computation of the rank difference function of hyperbolic tilings {#section:D(p)} ================================================================= The behaviour of the function $D(p)$ is difficult to capture directly from its definition. The aim of this section is to provide an explicit combinatorial description of the rank difference function $D(p)$ associated with the finite tilings $(G_t)_t$. The next lemma enables us to replace the rank which appears in the definition of $D(p)$ by a strictly graph-theoretical quantity. \[lemma:kappa\] Let $\kappa_{t, {\varepsilon}}$ denote the number of connected components of the graph $G_{t, {\varepsilon}}$. We have: $$\operatorname{rank}G_{t, {\varepsilon}} = |V_t| - \kappa_{t, {\varepsilon}}.$$ By definition, the rank of the graph $G_{t, {\varepsilon}}$ is the rank of an incidence matrix of this graph. The kernel of this incidence matrix is the cycle code of the graph $G_{t, {\varepsilon}}$, which has dimension $|{\varepsilon}| - |V_t| + \kappa_{t, {\varepsilon}}$ from Lemma \[lemma:Dcycle\]. The result follows. The function $D(p)$ depends on the expected rank of the random submatrix $G_{t, {\varepsilon}}$. This encourages us to examine the expected number of connected components of the random subgraph $G_{t, {\varepsilon}}$. A key ingredient of our study is the following decomposition of the random variable $\kappa_{t, {\varepsilon}}$. \[lemma:kappa\_decomposition\] Let $C$ be a connected subgraph of $G_t$. Denote by $X_C$ the random variable which takes the value 1 if $C$ is a connected component of the random graph $G_{t, {\varepsilon}}$ and 0 otherwise. Then, we have $$\kappa_{t, {\varepsilon}} = \sum_{C \in {\mathcal{C}}_t} X_C$$ where ${\mathcal{C}}_t$ denotes the set of connected subgraphs $C$ of $G_t(m)$. Moreover, we have ${{\mathbb E}}_p(X_C) = p^{|E(C)|}(1-p)^{|\partial(C)|}$ where $\partial(C)$ is the set of edges of $G_t$ which are incident to at least one vertex of $C$, but which do not belong to $E(C)$. The proof of the above lemma is self-evident. Using this decomposition of $\kappa_{t, {\varepsilon}}$, we derive the following exact expression of the rank difference function as a function of the subgraphs of the infinite graph $G(m)$. \[theo:D\_graphical\] For $m\geq 5$ and $0<p\leq 1/2$, The rank difference function associated with the graphs $(G_t(m))_t$ is equal to $$D(p) = \frac 2 m \sum_{ C \in {\mathcal{C}}(v) } \left( \frac 1 {|V(C)|} \left(p^{|E(C)|} (1-p)^{|\partial(C)|} - (1-p)^{|E(C)|} p^{|\partial(C)|} \right) \right),$$ where ${\mathcal{C}}(v)$ denotes the set of connected subgraphs $C$ of $G(m)$ containing a fixed vertex $v$. From Lemma \[lemma:kappa\], the rank difference function can be rewritten $$\begin{aligned} D(p) & = \limsup_t {{\mathbb E}}_p\left( \frac{\kappa_{t, {\varepsilon}} - \kappa_{t, \bar {\varepsilon}}} {|E_t|} \right)\\ &= \limsup_t \left( {{\mathbb E}}_p \left( \frac{\kappa_{t, {\varepsilon}} } {|E_t|} \right) - {{\mathbb E}}_{1-p} \left( \frac{\kappa_{t, {\varepsilon}} } {|E_t|} \right) \right).\end{aligned}$$ where we used the fact that, $\bar {\varepsilon}$ being the complement of ${\varepsilon}$ in $E_t$, we have ${{\mathbb E}}_p(\kappa_{t, \bar {\varepsilon}}) = {{\mathbb E}}_{1-p}(\kappa_{t, {\varepsilon}})$. Then, using the decomposition of $\kappa_{t, {\varepsilon}}$ proposed in Lemma \[lemma:kappa\_decomposition\] and the linearity of expectation, we obtain $$D(p) = \limsup_t \frac 1 {|E_t|} \sum_{ C \in {\mathcal{C}}_t } \left( {{\mathbb E}}_p(X_C) - {{\mathbb E}}_{1-p}(X_C)\right).$$ [*Elimination of the large components—*]{} Now, remark that the main contribution in this sum is given by the small components. To prove this, consider a sequence of integers $(M_t)_t$ such that $M_t \rightarrow +\infty$. Then, we have $$\begin{aligned} \frac 1 {|E_t|} \sum_{\substack{C \in {\mathcal{C}}_t \\ |E(C)| \geq M_t}} \left( {{\mathbb E}}_p(X_C) - {{\mathbb E}}_{1-p}(X_C)\right) & \leq \frac 1 {|E_t|} \sum_{\substack{C \in {\mathcal{C}}_t \\ |E(C)| \geq M_t}} \left( {{\mathbb E}}_p(X_C) + {{\mathbb E}}_{1-p}(X_C)\right)\\ & = \frac 1 {|E_t|} {{\mathbb E}}_p \left( \sum_{\substack{C \in {\mathcal{C}}_t \\ |E(C)| \geq M_t}} X_C \right) + \frac 1 {|E_t|} {{\mathbb E}}_{1-p} \left( \sum_{\substack{C \in {\mathcal{C}}_t \\ |E(C)| \geq M_t}} X_C\right)\\ & \leq \frac 1 {|E_t|} \frac {2 |E_t|}{M_t} = \frac {2}{M_t} \rightarrow 0\end{aligned}$$ To obtain the last inequality, remark that the sum of all the random variables $X_C$ such that $|E(C)| \geq M_t$ counts the number of connected components of the subgraph $G_{t, {\varepsilon}}$ of size larger than $M_t$. Since connected components are disjoint, this number cannot be larger than $|E_t|/M_t$. The previous paragraph proves that, for every sequence $M_t$ going to infinity, the rank difference function is given by $$D(p) = \limsup_t \frac 1 {|E_t|} \sum_{\substack{C \in {\mathcal{C}}_t \\ |E(C)| < M_t}} \left( {{\mathbb E}}_p(X_C) - {{\mathbb E}}_{1-p}(X_C)\right)$$ [*Recentralization—*]{} In order to remove the dependency on $t$, we would like to apply the local isomorphism between $G_t(m)$ and $G(m)$ and to express everything as a function of the infinite graph $G(m)$. First, we have to recenter all the components $C$ around a fixed vertex $v_t$ of the graph $G_t$. To move a connected component $C$ of the graph $G_t$ onto a component which contains the vertex $v=v_t$, we use a family of automorphisms of the graph $G_t(m)$. For every vertex $w$ of the graph $G_t(m)$, select $\sigma_{v, w}$, an automorphism of the graph $G_t(m)$ sending $v$ onto $w$. We take the identity for $\sigma_{v, v}$. Such an automorphism exists because the graph $G_t$ is vertex transitive, as explained in Section \[section:siran\_graphs\]. From this fixed family of automorphisms, we can reach all the connected subgraphs of $G_t$, starting from the subgraphs containing $v$. Stated differently, we have $${\mathcal{C}}_t =\{ C \ | \ C \text{ connected } \} = \bigcup_{w \in V_t} \{ \sigma_{v, w}(C) \ | \ C \text{ connected }, v \in V(C) \}$$ At the right-hand side of this equality, each component $C$ of the graph appears $|V(C)|$ times. Moreover, the contribution ${{\mathbb E}}_p(X_C)$ of the subgraph $C$, computed in Lemma \[lemma:kappa\_decomposition\], depends only on $|E(C)|$ and $|\partial(C)|$, which are both invariant under the application of an automorphism $\sigma_{v, w}$. Hence, $D(p)$ is equal to $$\begin{aligned} D(p) & = \limsup_t \frac 1 {|E_t|} \sum_{\substack{C \in {\mathcal{C}}_t \\ |E(C)| < M_t}} \left( {{\mathbb E}}_p(X_C) - {{\mathbb E}}_{1-p}(X_C)\right)\\ & = \limsup_t \frac 1 {|E_t|} \sum_{\substack{C \in {\mathcal{C}}_t(v) \\ |E(C)| < M_t}} \sum_{w \in V_t} \frac{1}{|V(C)|} \left( {{\mathbb E}}_p(X_{\sigma_{v,w}(C)}) - {{\mathbb E}}_{1-p}(X_{\sigma_{v,w}(C)})\right)\\ & = \limsup_t \frac 1 {|E_t|} \sum_{\substack{C \in {\mathcal{C}}_t(v) \\ |E(C)| < M_t}} \frac {|V_t|} {|V(C)|} \left( {{\mathbb E}}_p(X_C) - {{\mathbb E}}_{1-p}(X_C)\right)\\ & = \limsup_t \frac 2m \sum_{\substack{C \in {\mathcal{C}}_t(v) \\ |E(C)| < M_t}} \frac {1} {|V(C)|} \left( {{\mathbb E}}_p(X_C) - {{\mathbb E}}_{1-p}(X_C)\right)\end{aligned}$$ where we have used $\frac{|V_t|}{|E_t|}=\frac 2m$ since $G_t$ is $m$-regular. [*Application of the local isomorphism—*]{} We now replace the graph $G_t(m)$ by the infinite graph $G(m)$. Since the balls of radius $t$ are isomorphic in $G_t(m)$ and in $G(m)$, we have that every fixed subgraph $C$ inside such a ball has the same probability of being a connected component whether it is of the random subgraph $G_{t,{\varepsilon}}$ or of the open subgraph of $G(m)$. By choosing $M_t=t-1$, we therefore get $$\label{eq:limsup} D(p) = \limsup_t \frac 2 m \sum_{\substack{C \in {\mathcal{C}}(v) \\ |E(C)| < M_t}} \frac {1} {|V(C)|} \left( {{\mathbb E}}_p(X_C) - {{\mathbb E}}_{1-p}(X_C)\right)$$ where ${\mathcal{C}}(v)$ denotes the set of connected subgraphs $C$ of $G(m)$ containing the fixed vertex $v$. We can now conclude the proof. From Lemma \[lemma:kappa\_decomposition\], the quantity $\left( {{\mathbb E}}_p(X_C) - {{\mathbb E}}_{1-p}(X_C)\right)$ is equal to $\left(p^{|E(C)|} (1-p)^{|\partial(C)|} - (1-p)^{|E(C)|} p^{|\partial(C)|} \right)$, which is positive by Lemma \[lemma:series\_positivity\] to be proven just below. Therefore all the terms of the sum in are positive, which means that the $\limsup$ is in fact a limit. Since $M_t \rightarrow +\infty$, we get $$D(p) = \frac 2 m \sum_{ C \in {\mathcal{C}}(v) } \left( \frac 1 {|V(C)|} \left(p^{|E(C)|} (1-p)^{|\partial(C)|} - (1-p)^{|E(C)|} p^{|\partial(C)|} \right) \right).$$ It remains to prove that the series has positive terms. This result relies on an isoperimetric inequality. \[lemma:series\_positivity\] Let $0<p<1/2$. For every connected subgraph $C$ of $G(m)$, we have $$p^{|E(C)|}(1-p)^{|\partial(C)|} - (1-p)^{|E(C)|} p^{|\partial(C)|} > 0.$$ The parameter $p$ is assumed to be smaller than $1/2$. Thus, to prove that this quantity is strictly positive it suffices to show that for every connected subgraph $C$ of $G(m)$, we have $|E(C)| < |\partial(C)|$. This inequality is somewhat analogous to the isoperimetric inequality that we recall now. The isoperimetric constant of the graph $G(m)$ is defined to be $$i_E(G(m)) = \inf \left\{ \frac{|\partial(C)|}{|V(C)|} \right\}$$ with $C$ ranging over all finite subgraphs (that can be assumed connected) of $G(m)$. This number was computed exactly for hyperbolic graphs in [@HJL02]. It is $$\label{eqn:i_E} i_E(G(m)) = (m-2) \sqrt{1-\frac{4}{(m-2)^2}}.$$ In order to apply this to our problem, we write $$\begin{aligned} \label{eqn:isoperimetric} \frac{|\partial(C)|}{|E(C)|} = \frac{|\partial(C)|}{(m/2)|V(C)|-(1/2)|\partial(C)|} \geq \frac {i_E(G(m))}{m/2 - i_E(G(m))/2}\end{aligned}$$ where we have used the fact that the smallest rate $|\partial(C)|/|E(C)|$ is achieved when $\partial(C)$ contains only edges with exactly one endpoint in $C$. In that case, we have $m|V(C)| = 2|E(C)| + |\partial(C)|$. Using Equation (\[eqn:i\_E\]) and (\[eqn:isoperimetric\]), it is then easy to check that, for all $m \geq 5$, we have $$\frac{|\partial(C)|}{|E(C)|} \geq \frac {i_E(G(5))}{5/2 - i_E(G(5))/2} \approx 1.62 > 1.$$ This proves the lemma. Bound on the critical probability of the hyperbolic lattice $G(m)$ {#section:approximation} ================================================================== We showed in Theorem \[theo:Dequation\] that the critical probability of $G(m)$ is bounded from above as $p_c(G(m))\leq p_h$ with $p_h$ defined in Corollary \[cor:phom\]. Theorem \[theo:D\_graphical\] provides an exact formula for the rank difference function $D(p)$ as a sum of a series depending on the connected subgraphs of $G(m)$. This gives a new expression for $p_h$ that does not involve the finite graphs $G_t(m)$ anymore, but it still leaves $p_h$ difficult to compute. We now show that by replacing the series $D(p)$ by its partial sums, we obtain explicit upper bounds on $p_h$ and hence on $p_c$. \[theo:bound\] Let $n \geq 0$ and let $D_n(p)$ be a partial sum of the series $D(p)$ associated with the hyperbolic graph $G(m)$. Then, the solution $p_h(n) \in [0, 1]$ of the equation $$p - 2/m + D_n(p)=0$$ is an upper bound on $p_h$ and hence on $p_c(G(m))$. We have seen in Lemma \[lemma:series\_positivity\] that all the terms of the series $D(p)$ are strictly positive when $p>0$. Thus, every partial sum $D_n(p)$ satisfies $D_n(p) < D(p)$. As a consequence, if $p_h(n)$ is a solution of the equation $p - 2/m + D_n(p)=0$, then we have $p_h(n) - 2/m + D(p_h(n))>0$. This proves that $D(p)$ does not satisfy the criterion of Theorem \[theo:Dequation\] at $p=p_h(n)$. Therefore $p_h(n)$ is an upper bound on $p_h$. As a first application of this theorem, using only the fact that $D_n(p) \geq 0$, we recover the upper bound $p_c(G(m)) \leq 2/m$, proved in [@DZ10]. The first terms of the series, corresponding to the components of small size can be computed easily. For example the number of connected subgraphs of size 0, that is with 0 edges, containing a fixed vertex of $G(m)$ is 1 and this subgraph has a boundary $\partial(C)$ of size $m$. This gives the partial sum $$D_0(p) = \frac 2 m ( (1-p)^m-p^m ).$$ Applying Theorem \[theo:bound\] to $D_0(p)$, we get an upper close to $0.35$. This is already more precise than the upper bound in [@DZ13]. The next partial sum is given by $$D_1(p) = D_0(p) + \frac 2 m \left( \frac m 2 ( p(1-p)^{2(m-1)} - p^{2(m-1)}(1-p) ) \right),$$ since there are $m$ different connected subgraphs of $G(m)$ composed of one edge and containing a fixed vertex. The first terms can be computed easily in this way. In a tree it is possible to get an exact formula for the number of rooted connected subgraphs using the Lagrange inversion threorem. However this enumeration problem becomes extremely difficult when the subgraphs start covering cycles. Moreover, the size of the boundary and the number of vertices of the subgraph do not depend only on its number of edges. We enumerated all the connected subgraphs of $G(5)$ (hyperbolic animals, as in [@MW10]) of size at most 8 by computer. The results are given in Table \[tab:components\]. Using the partial sum $D_8(p)$ that takes into acount all the subgraphs of size at most 8, we get an upper bound on $p_c(G(5))$ which is approximately $0.299973$: $$p_c(G(5)) \leq 0.299973.$$ To the best of our knowledge, the previous best upper bound was close to $0.38$ [@DZ13]. Gu and Ziff proposed a Monte-Carlo estimation of this threshold of $0.265$ [@GZ11] which is coherent with our upper bound. $|E(C)|$ $|V(C)|$ $\partial(C)$ occurrence ---------- ---------- --------------- ------------ 0 1 5 1 1 2 8 5 2 3 11 30 3 4 14 200 4 5 17 1400 4 5 16 25 5 6 20 10146 5 6 19 450 5 5 15 5 6 7 23 75460 6 7 22 5775 6 6 18 90 7 8 26 572720 7 8 25 64200 7 8 24 480 7 7 21 1155 8 9 29 4418190 8 9 28 661950 8 9 27 13005 8 8 24 12840 8 8 23 180 : Enumeration of the rooted subgraphs of $G(5)$ up to size 8.[]{data-label="tab:components"} Concluding comments {#section:conclusion} =================== Summarising Theorems \[theo:Dequation\] and \[theo:D\_graphical\] we have proved : \[theo:final\] For $m\geq 5$ we have $p_c(G(m))\leq p_h$ with $$\begin{aligned} p_h &= \sup \{ p \in [0, 1/2] \ | \ D(p)+p-\frac 2m = 0 \} \ \text{and}\\ D(p) &= \frac 2 m \sum_{ C \in {\mathcal{C}}(v) } \left( \frac 1 {|V(C)|} \left(p^{|E(C)|} (1-p)^{\partial(C)} - (1-p)^{|E(C)|} p^{\partial(C)} \right) \right) \end{aligned}$$ where ${\mathcal{C}}(v)$ denotes the set of connected subgraphs $C$ of $G(m)$ containing a fixed vertex $v$ of the graph $G(m)$. The value $p_h$ can be thought of as a critical value for the appearance of homology in the graph $G(m)$. It captures the following threshold : for $p>p_h$, open subgraphs of large finite versions of $G(m)$ must have a first homology group of dimension that scales linearly with the total number of edges of the finite graph. For $p<p_h$, the dimension of the homology group is sublinear instead. This bound is really meaningful only for the hyperbolic case $m\geq 5$ since for $m=4$ (the square lattice), the dimension of the total homology group of finite versions of the infinite grid (tori) is limited to $2$. A consequence of Theorem \[theo:final\] is that $p_h$ gives an upper bound on the parameters of the quantum erasure channel that hyperbolic surface codes built on the family $G_t(m)$ can sustain [@DZ13]. We conjecture : \[conj:p\_c=p\_h\] For $m\geq 5$, $p_c=p_h$. Recall that in hyperbolic lattices it has been shown that immediately beyond the critical probability, the open subgraph contains infinitely many infinite connected components [@Be96]. The conjecture could be seen as a “finite” (but unbounded) version of this fact. Acknowledgements {#acknowledgements .unnumbered} ================ Nicolas Delfosse was supported by the Lockheed Martin Corporation. The authors wish to thank Robert Ziff for his comments and Russell Lyons for pointing out an inaccuracy in Lemma \[lemma:H\_decomposition\] in a preliminary version of this article. [1]{} S. K. Baek, P. Minnhagen, and B. Jun Kim. Percolation on hyperbolic lattices. , 79:011124, Jan 2009. I. Benjamini, A. Nachmias and Y. Peres. Is the critical percolation probability local ? [*Probability Theory and Related Fields*]{}, 149, pp. 261–269, 2011. I. Benjamini and O. Schramm. Percolation beyond ${Z}^d$, many questions and a few answers. , 1:71–82, 1996. I. Benjamini, O. Schramm. Percolation in the hyperbolic plane. , 29 pp. 487–507, 2001. C. Berge. . Elsevier, 1973. N. Delfosse and G. Zémor. Quantum erasure-correcting codes and percolation on regular tilings of the hyperbolic plane. In [*Proc. of IEEE Information Theory Workshop, ITW 2010*]{}, pages 1–5, 2010. N. Delfosse and G. Z[é]{}mor. Upper bounds on the rate of low density stabilizer codes for the quantum erasure channel. , 13(9-10):793–826, 2013. P. Giblin. . Cambridge University Press, 2010. G. Grimmett. [*Percolation.*]{} Springer-Verlag, New York, 1989. H. Gu and R. M. Ziff. Crossing on hyperbolic lattices. 85:051141, May 2012. O. Haggstrom, J. Jonasson, and R. Lyons. Explicit isoperimetric constants and phase transitions in the random-cluster model. , pages 443–473, 2002. A. Hatcher. . Cambridge University Press, 2002. H. Kesten. The critical probability of bond percolation on the square lattice equals 1/2. , 74:41–59, 1980. J.F. Lee and S.K. Baek. Bounds of percolation thresholds on hyperbolic lattices. [*Phys. Rev. E*]{} 86, 062105 2012. N. Madras and C.C. Wu. Trees, Animals, and Percolation on Hyperbolic Lattices [*Electronic J. of Probability*]{}, Vol. 15, pp. 2019-2040, 2010. J. Širáň. Triangle group representations and constructions of regular maps. , 82(03):513–532, 2000. [^1]: Département de Physique, Université de Sherbrooke, Sherbrooke, Québec, J1K 2R1 Canada, nicolas.delfosse@usherbrooke.ca [^2]: Institut de Mathématiques de Bordeaux, UMR 5251, université de Bordeaux, zemor@math.u-bordeaux.fr

--- bibliography: - 'journals\_apj.bib' - 'psrrefs.bib' - 'modjoeri.bib' - 'modrefs.bib' - 'repeaters.bib' - 'yreferences.bib' date: 'Received date / Accepted date' title: 'Repeating fast radio bursts with WSRT/Apertif' --- [Repeating fast radio bursts (FRBs) present excellent opportunities to identify FRB progenitors and host environments, as well as decipher the underlying emission mechanism. Detailed studies of repeating FRBs might also hold clues to the origin of FRBs as a population.]{} [We aim to detect bursts from the first two repeating FRBs: (R1) and (R2), and characterise their repeat statistics. We also want to significantly improve the sky localisation of R2 and identify its host galaxy.]{} [We use the Westerbork Synthesis Radio Telescope to conduct extensive follow-up of these two repeating FRBs. The new phased-array feed system, Apertif, allows covering the entire sky position uncertainty of R2 with fine spatial resolution in a single pointing. The data were searched for bursts around the known dispersion measures of the two sources. We characterise the energy distribution and the clustering of detected R1 bursts.]{} [We detected 30 bursts from R1. The non-Poissonian nature is clearly evident from the burst arrival times, consistent with earlier claims. When combined with previous reports, our measurements indicate an increase of $2.7(2) {{\ensuremath\mathrm{\,pc\,cm^{-3}}}}\, \mathrm{yr^{-1}}$ in the dispersion measure along the line of sight. Assuming a constant position angle across the burst, we place an upper limit of 8% on the linear polarisation fraction for the brightest burst in our sample. We did not detect any bursts from R2.]{} [A single power-law might not fit the R1 burst energy distribution across the full energy range or widely separated detections. Our observations provide improved constraints on the clustering of R1 bursts. Our stringent upper limits on the linear polarisation fraction imply a significant depolarisation, either intrinsic to the emission mechanism or caused by the intervening medium, at 1400MHz that is not observed at higher frequencies. The non-detection of any bursts from R2, despite nearly 300hrs of observations, imply either a highly clustered nature of the bursts, a steep spectral index, or a combination of both assuming the source is still active. Another possibility is that R2 has turned off completely, either permanently or for an extended period of time.]{} Introduction {#sec:introduction} ============ Fast radio bursts (FRBs) are transient, highly luminous events characterised by their short timescales of typically only a few milliseconds and dispersion measures (DMs) which are generally much larger than those expected from the Galactic electron density. These properties suggest FRBs to have originated from compact, highly energetic extra-galactic sources [@Lorimer07; @Thornton13]. Despite extensive follow-up, the majority of the discovered FRBs have been found to be one-off events [@Petroff2015]. However, to date, 11 FRBs have been reported to exhibit repeat bursts [@spitler16; @R2CHIME19; @CHIME19c; @Patek19]. For a recent review of FRBs, see @phl19. The repeat bursts from some FRBs enable studies of several FRB properties which are otherwise very hard to do for one-off sources. For example, deep follow-up of the repeating FRBs makes it possible to determine their sky positions with extremely high precision, identify the host galaxies and even associated persistent radio, optical or high-energy sources, if present. The localisation precision also helps in following-up any transient, multi-wavelength emission associated with the bursts. The repetition of bursts from the same source constrains FRB theory as well. A distinct constraint is that a cataclysmic event cannot produce repeating FRBs, and the underlying emission process should be able to sustain and/or repeat itself over considerably long periods of at least several years. The repetition also helps in much detailed investigations of the individual bursts, e.g., using coherently dedispersed high time and frequency resolution and/or over a wide frequency span. At the time of observations used in this work, two FRBs were known to repeat — and [hereafter R1 and R2, respectively, @spitler16; @R2CHIME19]. R1 is precisely localised [@Chatterjee17; @Marcote17] to a low-mass, low-metallicity dwarf galaxy [@Tendulkar-2017], which has helped in theoretically exploring the potential progenitors. Detailed radio follow-up has uncovered several intriguing features of R1. One particularly noteworthy feature is the complex time-frequency structures noted in several individual bursts, in the form of nearly 250MHz wide frequency bands (at 1400MHz) drifting towards lower frequencies [@hss+19]. These bands are not caused by interstellar scintillation, and rather likely to be either intrinsic to the emission process or caused by exotic propagation effects like plasma lensing. Similar frequency bands, albeit without the drifting in some cases, have also been observed from a number of galactic neutron stars (the Crab pulsar PSR B0531+21, @Hankins16; the Galactic Center Magnetar PSR J1745$-$2900, @Pearlman18; Magnetar XTE J1810$-$197, @Maan19b). However, any links between these galactic neutron stars and FRBs are as yet unclear, and require further study. In this work we therefore focus on repeating-FRB pulse-energy distribution and repetition statistics, and compare them to those of pulsar giant pulses and bursts from soft gamma ray repeaters (SGRs). The arrival times of R1 bursts are not well-described by a homogeneous Poisson process [@scholz16; @orp18], implying a clustered nature of the bursts. The clustering of the bursts has important implications for accurately determining the repeat rate as well as optimal observing strategies. It might also contain clues about the emission mechanism. Furthermore, R1 bursts exhibit nearly 100% linear polarisation at 4500MHz and an exceptionally large, rapidly varying rotation measure [RM; $1.33-1.46\times10^5{{\ensuremath\mathrm{\,rad\,m^{-2}}}}$; @msh+18]. This indicates that the R1 bursts are emitted in, or propagate through, an extreme and varying magneto-ionic environment. Due to the potential inter-channel depolarisation caused by the exceptionally large RM, the polarisation characteristics at 1400MHz are not fully known. A polarimetric characterisation at this frequency needs observations with reasonably narrow ($\lesssim$100kHz) frequency channels. R2 was discovered using the pre-commissioning data from the CHIME telescope. Some of the bursts from R2 showed strong similarities with those from [R1]{} in terms of the complex time-frequency structures. However, the uncertainties in the sky position [$\pm4'$ and $\pm10'$ in RA and Dec, respectively @R2CHIME19] have limited any more detailed comparisons between the two repeating FRBs as well as extensive studies of the R2 bursts themselves. A precise localisation using an interferometer will enable detailed polarimetric and high resolution studies of the R2 bursts as well as probes of the host galaxy and any associated persistent radio or high-energy source. In this work, we aim to characterise the 1400MHz polarisation and clustering nature of the bursts from R1, particularly for the bursts at the brighter end of the energy distribution, and localise R2 as well as study many of the above-mentioned aspects of both the repeating FRBs. For this purpose, we have utilised primarily the commissioning data from the new Apertif system on the Westerbork Synthesis Radio Telescope (WSRT). Using the large data sets acquired on R1 and R2, we present here the emission statistics of these two FRBs, over the highest pulse energies, and the longest timescales so-far reported. In the following sections, we provide more details on the time-domain observing modes used in this work (Sect. \[sec:apertif\]) as well as observations and data reduction methods (Sect. \[sec:obs\_and\_data\_reduction\]). We present and discuss our results obtained for R1 and R2 in Sects. \[sec:121102\] and \[sec:r2\], respectively. The overall conclusions are summarised in Sect. \[sec:conclusion\]. The Apertif observing modes {#sec:apertif} =========================== Aperture Tile in Focus (Apertif) is the new phased-array feed system installed on WSRT. It increased the field-of-view (FoV) to ${\sim}8.7\,$ square degrees, turning WSRT into an efficient survey instrument (@ovc10 [@2019NatAs...3..188A]; van Capellen et al. in prep). The system operates in the frequency range of $1130-1760\,$MHz with a maximum bandwidth of $300\,$MHz. The frequency resolution depends on the observing mode, as explained in the next subsections. Each of the WSRT dishes beam-forms the 121 receiver elements into up to 40 partly-overlapping beams on the sky (hereafter compound beams), each with a diameter of roughly $35\arcmin$ at $1400\,\mathrm{MHz}$. The data are then sent to the central system, which either operates as a correlator for imaging (cf. Adams et al. in prep), or as a beamformer for time-domain modes [@leeu14]. The time-domain observing mode system exploits the wide FoV to search for new FRBs as well as to localise any poorly localised repeating FRBs. This mode is enabled by a back-end that is capable of detecting such highly dispersed events in . We commissioned this system on pulsars and repeating FRBs (van Leeuwen et al. in prep). The following two time-domain modes were used to obtain the data presented here. Baseband mode {#sec:sub:sc1} ------------- In baseband mode, the central beam of up to ten telescopes is combined to obtain a high-sensitivity beam in one direction. The pulsar backend then either performs real-time pulsar folding, or records the raw voltages with a time resolution of $1.28\mathrm{\,\mu s}$ and frequency resolution of $0.78125\,$MHz. This allows for coherent dedispersion, as well as choosing an optimal trade-off between time and frequency resolution. During commissioning, we were at first limited to a bandwidth of $200\,$MHz and a single polarisation. The total sensitivity of the single-polarisation system is a factor $\sqrt{2}$ lower than that of the full dual-polarisation system. In early 2019 the system was upgraded to dual-polarisation and in March 2019 the bandwidth was increased to $300\,$MHz. Survey modes {#sec:sub:sc4} ------------ In survey mode, the system beamforms all 40 dish beams either coherently or incoherently. In coherent mode, voltage streams are combined across dishes with the appropriate complex weights. The resulting tied-array beams (TABs) are narrow (${\sim} 25\arcsec$) in East-West direction, but retain the dish resolution of ${\sim} 35\arcmin$ in North-South direction. In incoherent mode, intensity data are summed across dishes for all 40 compound beams individually. The incoherent-array beams (IABs) retain the full dish field-of-view, but at a sensitivity loss of $\sqrt{N_\mathrm{dish}}$ compared to the TAB mode. For survey mode data, both polarisations are summed. The resulting Stokes I data are stored to disk in filterbank format with a time resolution of $81.92\,\mathrm{\mu s}$ and frequency resolution of $0.1953125\,$MHz [@2017arXiv170906104M]. Observations and data reduction {#sec:obs_and_data_reduction} =============================== Observations {#sec:sub:obs} ------------ R1 and R2 were observed with Apertif between November 2018 and August 2019. R1 was observed in baseband mode, as well as both the incoherent and coherent survey modes. R2 is not well enough localised to be observed in the baseband mode, which is only suitable if the source is localised to within one Apertif tied-array beam. It was, however, observed with the incoherent and coherent survey modes. In total we spent ${\sim}130$ hrs on R1 and ${\sim}300$ hrs on R2. An overview of the observations is given in Fig. \[fig:obs\]. ![Overview of Apertif observations of R1 (top) and R2 (bottom). Blue diamonds indicate observations without detected bursts, red triangles indicate observation with detected bursts. Along the vertical axis the observing mode is noted. R1 was observed for a total of ${\sim}130$ hours, and R2 for ${\sim}$300 hours.[]{data-label="fig:obs"}](obs_r1 "fig:"){width="\columnwidth"}\ ![Overview of Apertif observations of R1 (top) and R2 (bottom). Blue diamonds indicate observations without detected bursts, red triangles indicate observation with detected bursts. Along the vertical axis the observing mode is noted. R1 was observed for a total of ${\sim}130$ hours, and R2 for ${\sim}$300 hours.[]{data-label="fig:obs"}](obs_r2 "fig:"){width="\columnwidth"} Reduction of R1 baseband data {#sec:sub:analysis_sc1} ----------------------------- The baseband data were coherently dedispersed using the typical R1 DM of $560.5{{\ensuremath\mathrm{\,pc\,cm^{-3}}}}$ [@hss+19] and converted to filterbank using `digifil`. In this process, the time resolution was reduced from $1.28\,\mathrm{\mu s}$ to $51.2\,\mathrm{\mu s}$. This reduces the computation time while ensuring any burst of at least $51.2\,\mathrm{\mu s}$ in duration, less than the narrowest burst thus far reported, is still detectable. The filterbank data were then searched for any bursts with a DM between $520{{\ensuremath\mathrm{\,pc\,cm^{-3}}}}$ and $600{{\ensuremath\mathrm{\,pc\,cm^{-3}}}}$ in steps of $0.1{{\ensuremath\mathrm{\,pc\,cm^{-3}}}}$ with `PRESTO` [@presto], using a threshold signal-to-noise (S/N) of 8. All candidates were visually inspected. The raw data of a 20 second window around each detected burst were saved for further analysis. Reduction of survey data {#sec:sub:analysis_sc4} ------------------------ For both survey modes, the data were analysed in real-time by our GPU pipeline, AMBER[^1] [@slb+16]. AMBER incoherently dedisperses the incoming Stokes I data to DMs between $0{{\ensuremath\mathrm{\,pc\,cm^{-3}}}}$ and $3000{{\ensuremath\mathrm{\,pc\,cm^{-3}}}}$ in steps of $0.2{{\ensuremath\mathrm{\,pc\,cm^{-3}}}}$ below $820{{\ensuremath\mathrm{\,pc\,cm^{-3}}}}$ and steps of $2.5{{\ensuremath\mathrm{\,pc\,cm^{-3}}}}$ above $820{{\ensuremath\mathrm{\,pc\,cm^{-3}}}}$, and writes a list of candidates with $\mathrm{S/N}\ge8$ to disk. These candidates are automatically further analysed by the offline processing mode of the ARTS processing pipeline, DARC[^2]. First, the candidates are clustered in DM and time to identify bursts that were detected at multiple DMs or in multiple beams simultaneously. Of each cluster, only the candidate with the highest S/N is kept. For all remaining candidates, a short chunk of data (typically 2s) surrounding the candidate arrival time is extracted from the filterbank data on disk and dedispersed to the DM given by AMBER. These data are then given to a machine learning classifier [@cl18], which determines whether a candidate is most likely a radio transient or local interference. For all candidates with a probability greater than $50\%$ of being a real transient, inspection plots are generated and e-mailed to the astronomers. Because the pipeline was still in the commissioning phase, the data were also searched with a `PRESTO`-based pipeline as was done for the baseband data. For R1 we used the same DM range as for the baseband data. For R2, the DM range was set to $150 - 230 {{\ensuremath\mathrm{\,pc\,cm^{-3}}}}$, covering a wide range around the source DM of ${\sim}189{{\ensuremath\mathrm{\,pc\,cm^{-3}}}}$ [@R2CHIME19]. Both pipelines yielded identical results. R1 {#sec:121102} == In total, 30 bursts were detected from R1. Of those, 29 were found in targeted baseband mode observations between 12 and 22 November 2018. One burst was found during regular TAB survey observations in August 2019. An overview of the bursts is shown in Fig. \[fig:burst\_overview\]. {width="\textwidth"} Flux calibration ---------------- The S/N of all bursts identified by the pipelines is determined using a matched filter with boxcar widths between 1 and 50 samples ($0.3$ to $16.4\,$ms). The noise is determined in an area around the burst visually confirmed to be free of interference. We used the modified radiometer equation [@CM03; @MA14] to convert the obtained S/N to peak flux density. For an interferometer, the radiometer equation can be written as $$\label{eq:radiometer} S = \frac{\mathrm{S/N}\,T_\mathrm{sys}}{G N_\mathrm{dish}^{\,\beta} \sqrt{N_\mathrm{pol} \Delta\nu W}},$$ where $S$ is the peak flux density, $T_\mathrm{sys}$ is the system temperature, $G$ is the gain of a single dish, $N_\mathrm{dish}$ is the number of dishes used, $\beta$ is the coherence factor, $N_\mathrm{pol}$ is the number of polarisations, $\Delta\nu$ is the bandwidth, and $W$ is the observed pulse width. We cannot readily measure $T_\mathrm{sys}$ and $G$ independently, but we can measure the system-equivalent flux density ($\mathrm{SEFD} = T_\mathrm{sys}/G$) of each dish. In order to do this, we performed drift scans of calibrator sources 3C147 and 3C286. The flux densities of both sources were taken from @pb17. In TAB mode, we typically find an SEFD of $700\,\mathrm{Jy}$ for the central beam of each dish. In addition, $\beta$ was shown to be consistent with $1$ for the TAB mode and with $1/2$ for the IAB mode [@2018PhDT.......155S], as theoretically expected. These values were used to determine the sensitivity for each of the observations. Although no bursts were detected in IAB mode, we can still use the radiometer equation to set an upper limit to the peak flux density of any bursts during those observations. For each detected burst, the peak flux density was converted to fluence by multiplying with the observed pulse widths, where the pulse width is defined as the width of a top-hat pulse with the same peak and integrated flux density as the observed pulse. This method of calculating the fluence is valid as long as the bursts are resolved in time, which all detected bursts are. Additionally we recorded the interval between that burst and the previous burst, or limits on the interval in case the burst was the first or last of an observation. An overview of the burst parameters is given in Table \[tab:121102\_overview\]. [\*7l]{} Burst & Arrival time & $\mathrm{DM_{S/N}}$ & $\mathrm{DM_{struct}}$ & Boxcar width & Fluence & Wait time\ & (barycentric MJD) & ${{\ensuremath\mathrm{\,pc\,cm^{-3}}}}$ & ${{\ensuremath\mathrm{\,pc\,cm^{-3}}}}$ & (ms) & (Jy ms) & (s)\ 1 & 58434.875313559 & 566.2 & 565(2) & 4.6 & 4.5(9) & ${>}638.51$\ 2 & 58434.889509584 & 565.8 & 567(1) & 4.6 & 4.3(9) & 1226.537\ 3 & 58434.894775566 & 567.6 & 569(2) & 2.3 & 3.9(8) & 454.980\ 4 & 58434.947599142 & 567.0 & 564(3) & 2.9 & 6(1) & 4563.957\ 5 & 58434.966276186 & 577.3 & 566(2) & 3.6 & 11(2) & 1613.701\ 6 & 58434.973174288 & 564.1 & 564(2) & 2.3 & 3.1(6) & 595.991\ 7 & 58435.990826444 & 567.0 & 563(1) & 7.5 & 18(4) & $1236.717$ – $87925.148$\ 8 & 58436.040343161 & 569.6 & 566(2) & 5.9 & 3.8(8) & 4278.245\ 9 & 58436.051467039 & 561.6 & 564(3) & 6.9 & 11(2) & 961.100\ 10 & 58436.054576797 & 568.0 & 569(2) & 4.3 & 6(1) & 268.683\ 11 & 58436.107672326 & 567.8 & 569(3) & 3.6 & 6(1) & $3231.773$ – $4587.543$\ 12 & 58436.110830017 & 567.6 & 565(3) & 2.9 & 8(2) & 272.825\ 13 & 58436.121982835 & 565.0 & 571(4) & 3.3 & 3.1(6) & 963.604\ 14 & 58436.123751681 & 565.6 & 565(4) & 4.9 & 10(2) & 152.828\ 15 & 58436.132117748 & 568.2 & 564(6) & 5.9 & 4.6(9) & 722.828\ 16 & 58436.192189901 & 563.8 & 569(2) & 1.6 & 3.9(8) & $2433.671$ – $5190.235$\ 17 & 58436.235117618 & 565.8 & 566(1) & 2.6 & 20(4) & 3708.959\ 18 & 58436.237531175 & 565.8 & 566(3) & 5.2 & 10(2) & 208.529\ 19 & 58436.242119138 & 565.2 & 568(2) & 4.9 & 11(2) & 396.399\ 20 & 58436.919260205 & 566.6 & 565(2) & 5.9 & 6(1) & $3149.344$ – $58504.989$\ 21 & 58436.964309370 & 567.8 & 564(2) & 2.3 & 3.4(7) & 3892.246\ 22 & 58436.991334107 & 566.6 & 566(2) & 2.9 & 10(2) & $1276.105$ – $2334.933$\ 23 & 58437.051643347 & 572.0 & 569(3) & 8.8 & 12(3) & 5210.724\ 24 & 58437.899455547 & 563.9 & 563(2) & 3.3 & 4.3(9) & $1433.866$ – $73250.973$\ 25 & 58437.924610245 & 567.2 & 569(4) & 8.5 & 9(2) & 2173.365\ 26 & 58437.993893047 & 561.2 & 565(3) & 4.9 & 12(2) & $1492.858$ – $5986.033$\ 27 & 58441.030569351 & 566.8 & 569(5) & 4.3 & 6(2) & $4649.438$ – $262368.833$\ 28 & 58450.903309478 & 566.2 & 565(3) & 2.3 & 3.9(8) & $821.270$ – $853004.746$\ 29 & 58450.974554110 & 564.6 & 569(2) & 2.6 & 3.7(7) & 6155.538\ 30 & 58714.255429157 & 564.6 & 569(1) & 3.3 & 3.9(8) & $1233.459$ – $22747467.604$\ Dispersion measure {#sub:dm} ------------------ The dispersion measure as determined by the pipelines ($\mathrm{DM_{S/N}}$) is optimised for S/N. This S/N value was used to determine the burst peak flux densities. The bursts, however, have complex frequency-time structure [@hss+19]. This is clearly visible in our sample as well, for example in bursts 18 and 19 (Fig. \[fig:burst\_overview\]). We thus also calculated a structure-optimised DM ($\mathrm{DM_{struct}}$) for each burst, which were determined using [dm\_phase]{}[^3]. While the S/N-optimised DM best captures the total energy output of the bursts, it is affected by the complex features that mimic dispersive effects. Though it is unclear whether these features are intrinsic or a propagation effect, their narrowband nature clearly distincts them from cold-plasma dispersion, which is described by a simple power law $\tau \propto \nu^{-2}$, where $\tau$ is the time delay at frequency $\nu$. As $\mathrm{DM_{S/N}}$ is based on the invalid assumption that the signal can be completely described by a $\nu^{-2}$ power law, while $\mathrm{DM_{struct}}$ is not, $\mathrm{DM_{struct}}$ is more likely to represent the actual dispersive effect. Assuming the overall DM remains constant for contemporaneous bursts, we averaged the $\mathrm{DM_{struct}}$ of the 29 bursts detected during the same period in November 2018. The data are not consistent with a single mean, but after visual inspection we concluded that the uncertainties given by [dm\_phase]{} are underestimated for our data, perhaps because many bursts have a relatively low S/N ($<$15). Therefore, we scaled the uncertainties such that the reduced $\chi^2$ of fitting a constant value is unity. The same scaling factor was then applied to burst 30, which was detected several months after the first 29 bursts. The average DM$_\mathrm{struct}$ of the first 29 bursts is $566.2(4){{\ensuremath\mathrm{\,pc\,cm^{-3}}}}$. The average DM we found is higher than the previously reported value of DM$_\mathrm{struct} = 560.5{{\ensuremath\mathrm{\,pc\,cm^{-3}}}}$ [@hss+19]. We ran our pipeline on Apertif data of the Crab pulsar taken in November 2018 to verify the frequency labelling of our data. Several giant pulses were detected, all at the expected DM; We are thus confident the higher DM for this source is real. This increased DM of R1 as detected by Apertif is in line with an R1 burst detected by CHIME, with a DM of $563.6(5){{\ensuremath\mathrm{\,pc\,cm^{-3}}}}$ [@CHIMER1], and several bursts in Arecibo data (Seymour et al. in prep). These were all detected in the same week in November 2018 where most of the Apertif bursts were found. In Fig. \[fig:burst\_overview\], where we display the bursts after dedispersion at the old value, this increase is already apparent in the residual dispersion slope. As noted by @CHIMER1, the DM variation is likely to be local to the source, as such large changes are not seen in Galactic pulsars nor expected in the inter-galactic medium. However, it remains unclear whether these variations are stochastic or a secular trend [@hss+19; @CHIMER1]. To investigate this further, we compared the DM$_\mathrm{struct}$ at different epochs (Fig. \[fig:dm\]). We only considered observations at 1400MHz for this analysis, to avoid any frequency-dependent DM effects as predicted by e.g. the decelerating blast wave model [@metzger19]. We do note that the DM as observed by Apertif is indeed slightly higher than the DM of the burst detected by CHIME in the same week, which is consistent with the model presented by @metzger19. The data are consistent with a monotonically increasing DM of ${\sim} 2.7(2){{\ensuremath\mathrm{\,pc\,cm^{-3}}}}\,\mathrm{yr^{-1}}$. We note that even when fitting a straight line through the first two data points, the predicted DMs for the Apertif data points fall within 1$\sigma$ of the observed values. Hence the good fit is not the result of the relatively large error on the first data point. We suggest that the observed increase in DM is indeed a secular trend. The increasing DM, together with decreasing RM, show that R1 is in a highly magnetised, chaotic environment, where the RM and DM are unlikely to arise from the same region. The DM increase might be explained by a high-density filamentary structure moving into our line-of-sight, although several more complex models also predict changes in both DM and RM over time [e.g. @piro18; @metzger19]. However, the rate at which the DM is increasing remains hard to explain. The origin of the rapid DM increase will be an important aspect for future modelling efforts. ![Structure-optimised DM of R1 at 1400MHz as measured at different epochs. The two Arecibo data points are from @scholz16 and @hss+19, respectively. For Apertif, we averaged the DMs of the 29 bursts detected contemporaneously in November 2018, and show the 30th burst detect in August 2019 separately. The error bars indicate 1$\sigma$ uncertainties. A linear fit is shown in black, with a slope of $2.7(2){{\ensuremath\mathrm{\,pc\,cm^{-3}}}}\,\mathrm{yr^{-1}}$.[]{data-label="fig:dm"}](dm_change){width="\columnwidth"} Energy distribution {#sub:energy_dist} ------------------- From the burst parameters, the intrinsic energy can be calculated as $$E = 4\pi\,d_L^2\,f_\mathrm{b}\,F\,\Delta\nu,$$ were $E$ is the burst energy, $d_L$ is the luminosity distance [$972\,$Mpc, @Tendulkar-2017], $f_\mathrm{b}$ is the beaming fraction of the emission, $F$ is the fluence as observed on Earth, and $\Delta\nu$ is the intrinsic emission bandwidth. Following @law17, we assume isotropic emission, $f_\mathrm{b}=1$. To investigate the energy distribution of R1 bursts, we consider the cumulative distribution of the mean burst rate, defined as the number of detected bursts divided by the total observing time *including* observations without any detected bursts, as function of energy. It is known that the bursts show clustering in time (see @orp18 and Sect. \[sub:rates\]). Therefore it is important to consider the time scales probed by each set of observations: Ten observations spread over a year may yield very different results from ten identical observations spread over one week. Our data are supplemented with the data presented in @law17 and @gourdji19, who have performed similar analyses. An overview of the data used is shown in Table \[tab:rates\_data\]. The resulting cumulative energy distributions are shown in Fig. \[fig:energy\_dist\]. The distribution of burst energies has previously been characterised by a power law, $R({>}E) \propto E^{\gamma}$, where $R$ is the burst rate above energy $E$, and $\gamma$ is the power-law slope. @law17 find a typical slope of $-0.7$ for VLA, GBT, and early Arecibo data. However, @gourdji19 find a significantly steeper slope of $-1.8(3)$ using Arecibo data only, and suggest several reasons why the slope may be different. In some cases, the calculated burst energy is a lower limit. Moving some bursts to a higher energy would flatten the distribution. Additionally, the energies probed by the data presented in @gourdji19 are lower than the others, where perhaps the slope is actually different or cannot be described by a power law at all. The slope is also strongly dependent on the chosen completeness threshold. Perhaps the different timescales probed by the different observations are also important. The 2016 Arecibo data used by @gourdji19 (Table \[tab:rates\_data\]) consist of two observations, one day apart, with detections of several bursts in each observation. This data set thus probes a relatively short timescale, which perhaps influences both the burst rate and energy distribution slope due to the clustered nature of the bursts. To estimate the power-law slope of the Apertif burst energy distribution, we used Eq. \[eq:radiometer\] to set a completeness threshold for WSRT, using the threshold S/N of 8 and a typical pulse width of $4\,$ms. The least sensitive observations were using 8 dishes in IAB mode. The corresponding energy threshold is $1.5\times10^{39}\,$erg. We then calculated the power-law slope using a maximum-likelihood estimator. If we include all data points, we find $\gamma=-1.3(3)$. When including only bursts above $1.5\times10^{39}\,$erg, the slope is $\gamma=-1.7(6)$. Although consistent with both $-0.7$ and $-1.8$ at the 2$\sigma$ level, the slope of the Apertif burst energy distribution favours the value found by @gourdji19. Although care must be taken in comparing slopes, this at least suggests that the probed energy range is not the reason for the steeper slope found in the 2016 Arecibo data that contain the lowest energy bursts, as with Apertif we are probing the highest burst energies at 1400MHz so far reported. The most luminous Apertif burst presented here has an isotropic energy of ${\sim}4.5\times10^{39}\,$erg. During early commissioning, we reported the potential detection of a bright burst from R1 [@atel121102]. Its estimated isotropic energy was $1.2\times10^{40}\,$erg. At that time, such bright bursts were not known to exist. The current energy distribution does, however, credibly allow for a burst this bright. Our slope $\gamma = -1.7(6)$ is the same as the power-law indices found for the giant pulses emitted by Crab pulsar at the same observing frequency of 1400MHz (Table \[tab:rates\_data\]). The steepness is basically unchanged at frequencies a decade lower: at 150MHz it is still $-$2.04(3) [@lkk+19]. The similarity between the brightness distribution fall-off seen in both FRBs and giant pulses suggests these could be related. The distribution steepness matches less convincingly with a neutron-star emission mode that has also been put forward as a source model for FRBs: the soft gamma-ray bursts from magnetars [see, e.g., @2019arXiv191006979W]. The brightness distribution seen in SGR 1900+14 bursts is less steep, following a $-$0.66(13) trend [@1999ApJ...526L..93G]. [\*6l]{} Source & Telescope & Bursts & T$_\mathrm{obs}$ (hr) & Date span & Power-law index ($\gamma$)\ *R1* & Arecibo$^{(1,3)}$ & 11 & 4.5 & 2012-11-02 – 2015-06-02 & $-$0.8$^{+0.3}_{-0.5}$\ & GBT$^{(2,3)}$ & 5 & 15.3 & 2015-11-13 – 2016-01-11 & $-$0.8$^{+0.4}_{-0.5}$\ & VLA$^{(3)}$ & 9 & 28.9 & 2016-08-23 – 2016-09-22 & $-$0.6$^{+0.2}_{-0.3}$\ & Arecibo$^{(4)}$ & 41 & 3.2 & 2016-09-13 – 2016-09-14 & $-$1.8(3)\ & WSRT$^{(5)}$ & 30 & 128.4 & 2018-11-12 – 2019-08-30 & $-$1.7(6)\ *Crab pulsar giant pulses* & WSRT$^{(6)}$ & 13,000 & 6 & 2005-12-10 & $-$1.79(1)\ &ATCA$^{(7)}$ & 700 & 3 & 2006-01-31 & $-$1.33(14)\ *SGR 1900+14 bursts* & BATSE+RXTE$^{(8)}$ & 1,000 & ${\sim}$50 & 1998-1999 & $-$0.66(13)\ ![Cumulative distribution of R1 burst energies detected with VLA (3000MHz), GBT (2000MHz), Arecibo (1400MHz), and Apertif (1400MHz). The data used are described in Table \[tab:rates\_data\]. Poissonian errors on the rates are shown for illustrative purposes. The typical power-law value for the early Arecibo, GBT, and VLA data is $-0.7$, while the later Arecibo data suggest a slope of $-1.8(3)$ above a completeness threshold of $2\times10^{37}\,$erg. The Apertif data suggest a slope of $-1.7(6)$ above the completeness threshold of $1.5\times10^{39}\,$erg, which is indicated by the vertical dashed line.[]{data-label="fig:energy_dist"}](rates){width="\columnwidth"} Burst repetition rate {#sub:rates} --------------------- If repeating FRB burst rates follow Poissonian statistics, the distribution of wait times would be an exponential distribution. However, it has been shown that R1 bursts are highly clustered, which is incompatible with Poissonian statistics [@orp18]. A generalisation of the exponential distribution that allows for clustering is the Weibull distribution, defined as $$\label{eq:wb} \mathcal{W}(\delta|k,r) = \frac{k}{\delta} \, \left[\delta \, r\, \Gamma\left(1 + 1/k\right)\right]^k \, \mathrm{e}^{-\left[\delta \, r \, \Gamma\left(1 + 1/k\right)\right]^k},$$ where $\delta$ is the burst interval, $r$ is the mean burst rate, $\Gamma(x)$ is the gamma function, and $k$ is a shape parameter. $k=1$ is equivalent to Poissonian statistics, $k<1$ indicates a preference for short burst intervals, i.e. bursts are clustered in time, and $k\gg1$ indicates a constant burst rate $r$. In order to apply the Weibull formalism to the R1 bursts observed with Apertif (cf. Table \[tab:121102\_overview\]), we need to consider that subsequent observations may have correlated burst rates as some observations occurred in short succession. This requires small modifications to the equations presented by @orp18. We add a maximum burst interval to their equations to allow for correlated observations. A derivation of the modified equations is given in Appendix \[app:weibull\]. We assume a flat prior on both $k$ and $r$, only requiring that both are positive, and calculate the posterior as the product of the likelihoods of all Apertif observations. The posterior distribution is shown in Fig. \[fig:posterior\_r1\_wsrt\]. The best-fit parameters are $r=6.9^{+1.9}_{-1.5}\,\mathrm{day^{-1}}$ and $k=0.49^{+0.05}_{-0.05}$. Although @orp18 used data from different instruments with different sensitivity thresholds, they all have a lower sensitivity threshold than Apertif. Given the negative slope of the energy distribution (Fig. \[fig:energy\_dist\]), we had thus expected to find a lower rate with Apertif than the rate reported by @orp18. Given the uncertainties, the Apertif rate could still be lower, although we note that our best-fit rate and shape are consistent with that of @orp18 at the $2\sigma$ level. A Poissonian burst rate distribution ($k=1$) is excluded at high significance. This is not surprising, given that all bursts except one were detected within the first 30 observing hours out of a total of ${\sim}130$ hrs. The burst *rate*, however, is consistent with the Poissonian estimate of $5.6(1)\,\mathrm{day^{-1}}$, even though Poissonian statistics cannot explain the distribution of burst *intervals*. Thus, at the time scales probed by our data set, the clustering effect is not important in determining the average burst rate, but it does strongly influence the expected number of detected bursts for any single observation. ![Posterior distribution of R1 burst rate and shape parameters. A *lower* $k$ indicates a *higher* degree of clustering. The green and blue areas indicate the result from @orp18 and this work, respectively. The contours indicate 1, 2, and 3$\sigma$ limits on $r$ and $k$. The best-fit parameters are indicated by the cross and plus.[]{data-label="fig:posterior_r1_wsrt"}](posterior_wsrt){width="\columnwidth"} While the Weibull distribution does not fit previously observed R1 burst wait times very well, it is a significant improvement over Poissonian statistics [@orp18]. There are, however, other ways to describe the clustered behaviour R1 shows. For example, the burst rate might be described by several distinct Poisson processes: one (or more) with a high rate (the “active” state), and one (or more) with a low or zero rate (the “inactive” state). If the burst rate follows Poissonian statistics during an active period, i.e. there is only one burst rate during an active state, the wait time distribution is an exponential distribution. Samples of R1 wait times have indeed been shown to be consistent with exponential [@lin19], but also with log-normal [@gourdji19] and power-law [@lin19] distributions, where in some cases it is not possible to distinguish between these distributions. Considering our observations in November 2018, where 29 out of the 30 bursts were detected, as the active state, we looked at the observed wait times during that time frame. In total, 19 wait times were determined (cf. Table \[tab:121102\_overview\]). The resulting wait time distribution is shown in Fig. \[fig:wait\_times\]. The Apertif sample shows a bi-modal wait time distribution. There is a dearth of burst times between ${\sim}2300\,$s and ${\sim}3600\,$s. The sample of wait times below $2300\,$s does not fit an exponential distribution, but can be fit with a power-law with slope $-0.38(2)$. The sample above $3600\,$s *can* be fit by an exponential distribution, but the steep slope requires a Poissonian burst rate of ${>}25\,\mathrm{day^{-1}}$, which is incompatible with the observed rate. However, it can be fit equally well with a power-law with slope $-3.5(1)$. In Fig. \[fig:wait\_times\], the best-fit power law is shown for both samples. Care has to be taken when interpreting these results, as there is a maximum wait time that can be detected in our observations of typically 2 hrs in duration. The chance of not detecting a given wait time increases linearly with the wait time, and any wait time larger than the observation duration is of course not observable. However, this effect cannot explain the bi-modality nor change in power-law index as those are non-linear in wait-time. Our results indicate that during the active state in November 2018, the burst intervals did not follow a stationary Poisson process. This is incompatible with the wait time distribution of Crab giant pulses, which can be described by an exponential distribution [@L95]. However, a non-stationary Poisson process can result in a power-law wait time distribution at long wait times, which flattens towards shorter wait times. This is seen in for example X-ray solar flares [@aschwanden10; @wheatland00]. The power-law slope depends on the exact form of the burst rate as function of time, but is generally flatter if the burst rate varies rapidly [@aschwanden10]. In magnetar SGR 1900+14, the waiting time distribution between bursts follows a log-normal function, also indicative of a self-organized critical system [@1999ApJ...526L..93G]. ![Wait time distribution of R1 bursts as detected by Apertif. Error bars are 1$\sigma$ Poissonian uncertainties. The distribution is bi-modal. The lower part can be fit by a power-law with slope $-0.38(2)$, but does not fit an exponential distribution. The higher part can be fit by either an exponential distribution or a power law with slope $-3.5(1)$. For both parts, the power-law fit is shown.[]{data-label="fig:wait_times"}](wait_time){width="\columnwidth"} Polarisation properties {#sub:pol} ----------------------- Our baseband data came from only a single linear polarisation receptor, which makes it impossible to determine the total polarisation fraction. Even though the bursts from R1 are known to be highly linearly polarised, its high rotation measure [RM &gt; $10^5{{\ensuremath\mathrm{\,rad\,m^{-2}}}}$; @msh+18] implies that the polarisation angle sweeps around multiple times even within one Apertif frequency channel, so we do not expect to miss any bursts because of misalignment of the polarisation angle between a burst and the receiver elements. However, we can only estimate the RM and degree of linear polarisation [@ramkumar99; @Maan15] if the depolarisation within a single channel is sufficiently small, which is clearly not the case at the native frequency resolution of Apertif. Following @msh+18, the intra-channel polarisation angle rotation ($\Delta\theta$) is given by $$\Delta\theta = \frac{RM \mathrm{c}^2 \Delta\nu}{\nu^3},$$ where $c$ is the speed of light, $\Delta \nu$ is the channel width and $\nu$ is the observing frequency. Evidently, $\Delta\theta$ is higher at lower frequencies, hence a much higher frequency resolution is required at 1400MHz than at 4500MHz. @msh+18 find an intra-channel rotation of $~9\degr$, for a depolarisation fraction of $1.6\%$ for their data. At native frequency resolution, the Apertif data would be over $90\%$ depolarised. Therefore, we reprocessed the baseband data around the bursts and increased the number of channels to 4096 over a bandwidth of $200\,$MHz, implying a frequency resolution of ${\sim}49\,$kHz. This decreased the time resolution to $20.48\,\mathrm{\mu s}$, which is still sufficient to resolve the bursts. The resulting depolarisation fraction is $3\%$ for an RM of $10^5{{\ensuremath\mathrm{\,rad\,m^{-2}}}}$. Following the procedure of @Maan15, we performed a discrete Fourier transform on the intensity spectra in the $\lambda^2$-domain, at each of the time samples in the bursts to obtain corresponding Faraday spectra. The Faraday spectrum represents linearly polarised power as a function of RM if there is only one RM component, which is the case [@msh+18]. We did not find any significant linearly polarised power in the RM range $10^4 - 3.4{\times}10^5{{\ensuremath\mathrm{\,rad\,m^{-2}}}}$. However, due to the low S/N of individual samples within the bursts, we were sensitive to only a reasonably high degree of linear polarisation (50% for the brightest burst, but $>$95% for the other bursts). The R1 bursts are known to exhibit a constant polarisation position angle (PA) over the full burst duration [@msh+18]. To probe linearly polarised emission with higher sensitivity, we used the intensity spectra averaged over the entire burst widths which is valid if the PA is constant. We again did not detect any significant linearly polarised emission. At a 5$\sigma$ level detection threshold, our upper limits on the linearly polarised fractions for the 3 brightest bursts in our sample, burst numbers 17, 7 and 5 in Fig. \[fig:burst\_overview\], are 8%, 14% and 16%, respectively. At 4500MHz, the linear polarisation fraction was measured to be close to 100% [@msh+18]. Hence there must be some additional intrinsic or extrinsic depolarisation at 1400MHz to explain our non-detection. R2 {#sec:r2} == Despite several hundred hours of observations with equivalent or better sensitivity than reported in @R2CHIME19, no bursts from R2 were detected by Apertif. This is in contrast to the six bursts detected by CHIME in 23 hrs of R2 transits. Due to difficulty in measuring their time-dependent sensitivity, @R2CHIME19 calculate R2’s repetition rate with three bursts above their fluence completeness threshold of 13Jyms, found in a total of 14 hrs of exposure. The least sensitive observations in our data set were performed with ten dishes in IAB mode, resulting in a fluence completeness threshold of $8.5 \sqrt{\frac{W}{10 \mathrm{\, ms}}}\,\mathrm{Jy\,ms}$, where $W$ is the pulse width. Most observations were more sensitive, meaning the limits we place on our non-detection are conservative. Our ${\sim}$300 hrs of exposure corresponds to several years worth of CHIME transits, and yet Apertif detected no R2 bursts. We offer two possible explanations, which are addressed independently. Temporal clustering ------------------- R2 shows some striking similarities to R1. Beyond repetition, R2 also has distinct time/spectral structure, with a march-down in frequency of adjacent sub-pulses. It may also exhibit non-Poissonian, or clustered, repetition. As has been previously noted, temporal clustering of bursts can drastically increase the probability of zero events being discovered in a given observation, even if the average repetition rate is high [@connor-2016b; @orp18]. Therefore, it is possible that the reason Apertif did not detect R2 is that it is highly clustered. The posterior for the burst rate $r$ and shape parameter $k$ of a Weibull distribution is shown in Fig. \[fig:posterior\_r2\], where small $k$ corresponds to high clustering. The burst rate as observed by CHIME [&gt;2.16 per day above 13 Jy ms @R2CHIME19] is indicated by the shaded yellow region. If we assume the same rate, $r$, of detectable pulses at CHIME and Apertif (i.e. a flat spectral index in repetition rate), then within the Weibull framework, we can constrain the shape parameter, $k$, to be no greater than 0.12 at $3\sigma$. In other words, if R2’s behaviour at 1400MHz and 600MHz are comparable, then the source’s repetition has to be highly clustered, more so even than R1, for us not to detect any repeat bursts in ${\sim}300$ hrs of exposure. There are reasons to be sceptical of clustering as the sole explanation for our non-detection. For instance, if R2’s repetition statistics were well-described by a Weibull distribution, then the values of $k$ allowed by our non-detection imply that CHIME should have seen many bursts in a single transit, because the temporal clustering would be so significant. From a simple Monte Carlo simulation, we find that with $k\lesssim0.3$, half or more transits in which the FRB is seen to repeat should contain more than one repeat burst. Since CHIME saw its six repeat bursts in six distinct transits, $k$ is either not that small, or clustering only happens on longer time scales. Under our assumptions, the upper-bound on clustering set by our non-detection is inconsistent with the lower-bound on R2’s clustering set by CHIME’s observations. We also emphasise here that the Weibull distribution was chosen as a useful generalisation of the Poisson distribution, in order to account for the observed temporal clustering of R1. However, such clustering may not hold on all time scales, and FRB repetition wait times may not easily be described by a simple continuous distribution. Some FRBs may turn off entirely for extended periods, similar to X-ray binaries in quiescence, and then start back up with Poissonian repetition. Indeed, another explanation for our non-detection of R2 is that the source has turned off, either permanently or for a long, extended period. This will either be corroborated or falsified by CHIME’s daily observing of R2 over the past year. Frequency dependence -------------------- If R2 is not significantly clustered, and the bursts follow Poissonian statistics ($k=1$), we set a $3\sigma$ upper limit to the burst rate at 1400MHz of $r<0.12$ per day above a fluence of $8.5\,$Jy ms. This limit is clearly inconsistent with the CHIME rate of $>$2.16 per day above 13Jyms. This indicates the source may be significantly less bright at 1400MHz than at 600MHz. Under the simplified assumption that R2’s pulses were always the same brightness at a given frequency, and were given by a power law across frequency such that $F(\nu)\propto \nu^{-\alpha}$, then $\alpha$ must be greater than 3.6 at the 3$\sigma$ level. However, it has become increasingly clear that bursts from repeating FRBs are given by bottom-heavy distributions, with many more dim events than bright ones [@gourdji19; @CHIME19c]. The combination of frequency-dependence in the brightness of the source, as well as a power-law brightness distribution of repeat bursts (i.e. $F(\nu)\propto \nu^{-\alpha}$ *and* $N(>F)\propto F^{-\gamma}$), results in strong frequency-dependence in the detection rate. As we show in Appendix B, the frequency-dependent detection rate scales as $N(\nu) \propto \nu^{-\alpha\gamma}$, not as $N(\nu) \propto \nu^{-\alpha}$. In other words, if a source has a red spectrum ($\alpha>0$) and a steep brightness function, then it will be difficult to detect at high frequencies. The analysis holds even if individual bursts from repeaters do not have power-law frequency spectra, so long as their average brightness as a function of frequency is a power law. This new effect may explain our non-detection of R2 at Apertif: From the S/N listed in @R2CHIME19, $\gamma$$\approx$$2.2\pm1.3$, so even a moderate red frequency spectrum could result in considerably lower detection rates at 1400MHz vs 600MHz. ![Posterior distribution of R2 burst rate and shape parameters. The contours indicate 1, 2, and 3$\sigma$ upper limits on $r$ and $k$. A *lower* $k$ indicates a *higher* degree of clustering. The yellow region indicates the CHIME rate with Poissonian error bars. The red region is the CHIME rate modified by a spectral index of -3.6, where the CHIME lower limit on the rate matches the Apertif upper limit at $k=1$, i.e. under the assumption of Poissonian statistics.[]{data-label="fig:posterior_r2"}](posterior_r2){width="\columnwidth"} Conclusions {#sec:conclusion} =========== We have detected 30 bursts from R1 with Apertif. Their structure-optimised DM is higher than previously reported, consistent with an overall increase of ${\sim}2.7(2){{\ensuremath\mathrm{\,pc\,cm^{-3}}}}\,\mathrm{yr^{-1}}$. The isotropic energy distribution of the bursts as determined by several instruments cannot be described by a single power law over the three decades of burst energies. The power-law slope as detected by Apertif, $\gamma = -1.7(6)$, is consistent with that of the Crab pulsar giant pulses. Less convincingly it matches the bursts from magnetar SGR1900+14. The repetition rate of the bursts matches with earlier found values, and confirms their highly clustered nature. Even when considering only the observations during an active period of the source, the burst arrival times are inconsistent with a stationary Poisson process and hence inconsistent with the wait time distribution of Crab giant pulses. However, the wait-time distribution can be described by a double power law, similar to solar flares. We place stringent upper limits on the linear polarisation fractions of some of the brightest bursts in our sample. For the brightest burst, the upper limit is $8\%$, assuming a constant polarisation angle across the burst. These limits suggest that there is an additional depolarising effect at 1400MHz that is not present at 4500MHz. No bursts from R2 were detected. This might be because it has turned of either completely or for an extended period of time. If it has not turned off, the non-detection requires a high degree of clustering within the Weibull framework, assuming a flat spectral index. This is inconsistent with CHIME not having detected several bursts during one transit of R2. Alternatively, R2 may not emit in the Apertif band, or its emission may be intrinsically fainter. We find it unlikely that R2’s statistical frequency spectrum can be described by a power-law. If it can, the spectral index has to be at least $\alpha>3.6$ to explain the Apertif non-detection. This work makes use of data from the Apertif system installed at the Westerbork Synthesis Radio Telescope owned by ASTRON. ASTRON, the Netherlands Institute for Radio Astronomy, is an institute of NWO. Burst wait time formalism {#app:weibull} ========================= We describe the FRB wait time distribution by a Weibull distribution, following @orp18. The Weibull distribution is described by two parameters: the burst rate $r$ and clustering parameter $k$. $k=1$ is equivalent to Poissonian statistics, while a value much smaller or great indicate clustering in time and a constant burst rate, respectively. We incorporate that subsequent observations can be correlated. Here we derive the modifications to the equations presented by @orp18. The probability of measuring some set of burst arrival times $t_\mathrm{1}, t_\mathrm{2}, \dots t_\mathrm{N}$ in a single observation of duration $T$ can be split into three parts: 1. The probability of the interval between the start of the observation and the first burst: $P(t_1)$ 2. The probabilities of the intervals between subsequent bursts in a single observation: $P(t_2 \dots t_\mathrm{N}) = \prod_\mathrm{i=1}^\mathrm{N-1} P(t_\mathrm{i+1} - t_\mathrm{i})$ 3. The probability of the interval between the last burst and the end of the observation: $P(T - t_\mathrm{N})$ Assuming different observations are not correlated, points 1) and 3) describe minimum burst intervals. To include that subsequent observations can be correlated, we include a maximum burst interval, which is simply the interval between the last burst of an observation and the arrival time of next observed burst. Only for the intervals before the first detected burst and after the last burst, there is no constraint on the maximum burst interval. The addition of a maximum burst interval ($\delta_\mathrm{max}$) leads to several minor changes in the probability density functions of @orp18. The probability density distribution of the interval between the start of the observation and the first burst [Eq. 13 of @orp18] is given by $$\label{eq:wb_pdf_start} \begin{aligned} &\mathcal{P}(t_1,\delta_\mathrm{max}|k,r) = r \, \int_{t_1}^{\delta_\mathrm{max}} \mathcal{W}(\delta|k,r) \, \mathrm{d}\delta \\ &= r \, \left[\mathrm{CCDF}(t_1|k,r) - \mathrm{CCDF}(\delta_\mathrm{max}|k,r)\right], \end{aligned}$$ where $\delta$ is the interval between the last unobserved burst and the first observed burst, and CCDF is the cumulative complementary distribution function, defined as $$\label{eq:wb_ccdf} \mathrm{CCDF}(\delta|k,r) = \int_\delta^\infty \mathcal{W}(\delta^\prime|k,r) \, \mathrm{d}\delta^\prime = \mathrm{e}^{-\left[\delta \, r \, \Gamma(1+1/k)\right]^k}.$$ The probability density of the intervals between subsequent bursts in a single observation is unchanged by our addition of correlated observations, and simply given by a product of Weibull distributions for the given intervals, $$\label{wb_pdf_intervals} \mathcal{P}(t_1|k,r) = \prod_\mathrm{i=1}^\mathrm{N-1} \mathcal{W}(t_{\mathrm{i}+1} - t_\mathrm{i}).$$ The probability density of the interval between the last burst and the end of the observation is changed in a similar way as Eq. \[eq:wb\_pdf\_start\] and given by $$\label{eq:wb_pdf_end} \begin{aligned} &\mathcal{P}(T-t_\mathrm{N},\delta_\mathrm{max}|k,r) = \int_{T-t_\mathrm{N}}^{\delta_\mathrm{max}} \mathcal{W}(\delta|k,r) \, \mathrm{d}\delta\\ &= \mathrm{CCDF}(T-t_\mathrm{N}|k,r) - \mathrm{CCDF}(\delta_\mathrm{max}|k,r). \end{aligned}$$ Lastly, we need to consider an observation without any detected bursts [Eq. 17 of @orp18]. The probability density distribution of such an observation is given by $$\label{eq:wb_pdf_0} \begin{aligned} &P(N=0,\delta_\mathrm{max}|k,r) = r \int_T^{\delta_\mathrm{max}} \mathrm{CCDF}(t_1|k,r) \, \mathrm{d}t_1 \\ &= \frac{\Gamma_\mathrm{i}{\left(1/k, \left(T\,r\,\Gamma(1+1/k)\right)^k\right)}}{k\,\Gamma{\left(1+1/k\right)}} \, - \, \\ &\frac{\Gamma_\mathrm{i}{\left(1/k, \left(\delta_\mathrm{max}\,r\,\Gamma(1+1/k)\right)^k\right)}}{k\,\Gamma{\left(1+1/k\right)}}, \end{aligned}$$ where $\Gamma_i(x, z)$ is the upper incomplete gamma function. Note that in the limit $\delta_\mathrm{max} \rightarrow \infty$, all modified equations return to their equivalent versions for non-correlated observations. Frequency-dependent detection rate ================================== Suppose an FRB emits broad-band bursts with a power-law in frequency, given by $L(\nu)\propto \left(\frac{\nu}{\nu_0}\right)^{-\alpha}$. If we assume the differential luminosity function of an individual repeater is given by a power-law $N(L)\propto L^{-(1+\gamma)}$, then the number of events above some minimum detectable luminosity is $$N(>\!L_{\mathrm{min}}) \propto \int\displaylimits_{L_{\mathrm{min}}}^{\infty} N(L)\,dL,$$ where $L_\mathrm{min}$ is determined by the detection instrument’s brightness threshold and the source’s distance scale, such that $L_{\mathrm{min}}=4\pi d^2 S_{\mathrm{min}}$. If we then include the fact that the source is $\left(\frac{\nu}{\nu_0}\right)^{-\alpha}$ times brighter at frequency $\nu$ than at $\nu_0$, we find that $L_{\mathrm{min}}$ is decreased by that same factor, so $$N(>\!L_{\mathrm{min}}, \nu) \propto \int\displaylimits_{L_{\mathrm{min}}(\nu)}^{\infty} L^{-(1+\gamma)}\,dL.$$ For $\gamma>0$, $$N(>\!L_{\mathrm{min}}, \nu) \propto \left [ \left(\frac{\nu}{\nu_0}\right)^\alpha L_{\mathrm{min}} \right ]^{-\gamma},\, $$ and we find a strong relationship between observed repeat rate, $N(>\!L_{\mathrm{min}}, \nu)$, the source’s spectral index $\alpha$, and its luminosity function index $\gamma$, such that $$N(>\!L_{\mathrm{min}}, \nu) \propto \nu^{-\gamma\alpha}.$$ This is striking, because it means that if a repeating FRB’s brightness distribution deviates from $\gamma\approx1$, the source’s detectability across frequency is significantly different from its *brightness* across frequency. As an example, if R2 has $\gamma=2$, similar to the Crab, and $L(\nu)\propto \left(\frac{\nu}{\nu_0}\right)^{-2}$, there will be almost 30 times fewer detectable bursts in the middle of the Apertif band vs. the middle of the CHIME band, assuming $S_{\mathrm{min}}$ is the same at both telescopes. [^1]: <https://github.com/AA-ALERT/AMBER> [^2]: <https://github.com/loostrum/darc> [^3]: <https://github.com/danielemichilli/DM_phase>

--- abstract: 'We study neutrino conversions in a recently envisaged source of high energy neutrinos ($E {\buildrel > \over {_{\sim}}} \, 10^{6}$ GeV), that is, in the vicinity of cosmological Gamma-Ray Burst fireballs (GRB). We consider the effects of flavor and spin-flavor conversions and point out that in both situations, a some what higher than estimated high energy tau neutrino flux from GRBs is expected in new km$^{2}$ surface area under water/ice neutrino telescopes.' address: | Department of Physics, Tokyo Metropolitan University, Minami-Osawa 1-1, Hachioji-Shi,\ Tokyo 192-0397, Japan; E-mail: athar@phys.metro-u.ac.jp author: - 'H. Athar' title: 'Neutrino conversions in cosmological gamma-ray burst fireballs' --- Introduction ============ Recently, cosmological fireballs are suggested as a possible production site for gamma-ray bursts as well as the high energy ($E {\buildrel > \over {_{\sim}}} \, 10^{6}$ GeV) neutrinos [@WB]. Although, the origin of these cosmological Gamma-Ray Burst fireballs (GRB) is not yet understood, the observations suggest that generically a very compact source of linear scale $\sim \, 10^{7}$ cm through internal or/and external shock propagation produces these gamma-ray bursts (as well as the burst of high energy neutrinos) [@P]. Typically, this compact source is hypothesized to be formed possibly due to the merging of binary neutron stars or due to collapse of a supermassive star. The main source of high energy tau neutrinos in GRBs is the production and decay of $D^{\pm}_{S}$. The production of $D^{\pm}_{S}$ may be through $p\gamma $ and/or through $pp$ collisions. In [@PX], the $\nu_{e}$ and $\nu_{\mu}$ flux is estimated in $pp$ collisions, whereas in [@WB], the $\nu_{e}$ and $\nu_{\mu}$ flux is estimated in $p\gamma$ collisions for GRBs. In $pp$ collisions, the flux of $\nu_{\tau}$ may be obtained through the main process of $p+p\rightarrow D^{+}_{S}+X$. The $D^{+}_{S}$ decays into $\tau^{+}$ lepton and $\nu_{\tau}$ with a branching ratio of $\sim \, 3\%$. This $\tau^{+}$ lepton further decays into $\nu_{\tau}$. The cross-section for $D^{+}_{S}$ production, which is main source of $\nu_{\tau}$’s, is $\sim $ 4 orders of magnitude lower than that of $\pi^{+}$ and/or $\pi^{-}$ which subsequently produces $\nu_{e}$ and $\nu_{\mu}$. The branching ratio for $\nu_{e}$ and/or $\nu_{\mu}$ production is higher up to an order of magnitude than that for $\nu_{\tau}$ production (through $D^{\pm}_{S}$). These imply that the $\nu_{\tau}$ flux in $pp$ collisions is suppressed up to 5 orders of magnitude relative to corresponding $\nu_{e}$ and/or $\nu_{\mu}$ fluxes. In $p\gamma$ collisions, the main process for the production of $\nu_{\tau}$ may be $p+\gamma \rightarrow D^{+}_{S}+ \Lambda^{0}+\bar{D}^{0}$ with similar relevant branching ratios and corresponding suppression for cross-section values. Here the corresponding main source for $\nu_{e}$ and $\nu_{\mu}$ production is $p+\gamma \, \rightarrow \Delta^{+}\rightarrow \pi^{+}+n$. Therefore, in $p\gamma $ collisions, the $\nu_{\tau}$ flux is also suppressed up to 5 orders of magnitude relative to $\nu_{e}$ and/or $\nu_{\mu}$ flux. Thus, in both type of collisions, including the relevant kinematics, the intrinsic $\nu_{\tau}(\bar{\nu}_{\tau})$ flux is estimated to be rather small relative to $\nu_{e} (\bar{\nu}_{e})$ and/or $\nu_{\mu} (\bar{\nu}_{\mu})$ fluxes from GRBs, typically being, $F^{0}_{\tau} /F^{0}_{e,\mu}\, <10^{-5}$ [@VZA]. In this paper, we consider the possibility of obtaining higher $\nu_{\tau}$ flux, that is, $F^{0}_{\tau}/F^{0}_{e,\mu}\, >\, 10^{-5}$, from GRBs through neutrino conversions as compared to no conversion situation. The present study is particularly useful as the new under ice/water Čerenkov light neutrino telescopes namely AMANDA and Baikal (also the NESTOR and ANTARES) will have energy, angle and flavor resolution for high energy neutrinos originating at cosmological distances [@M]. Recently, there are several discussions concerning the signatures of a possible neutrino burst from GRBs correlated in time and angle [@W]. In particular, there is a suggestion of measuring $\nu_{\tau}$ flux from cosmologically distant sources through a double shower (bang) event [@L] or through a small pile up of up ward going $\mu$-like events near (10$^{4}-10^{5}$) GeV [@HS]. The plan of this paper is as follows. In Sect. II, we briefly describe the matter density and magnetic field in the vicinity of GRBs. In Sect. III, we discuss in some detail, the range of neutrino mixing parameters that may give rise to relatively large precession/conversion probabilities resulting from neutrino flavor/spin-flavor conversions. In Sect. IV, we give estimates for separable but contained double shower event rates induced by high energy $\nu_{\tau}$’s originating from GRBs without/with conversions for km$^{2}$ surface area under water/ice neutrino telescopes for illustrative purposes and finally in Sect. V, we summarize our results. matter density and magnetic field in the vicinity of GRB ======================================================== According to [@WB], the isotropic high energy neutrino production may take place in the vicinity of $r_{p}\, \sim \, \Gamma^{2}c\Delta t$. Here $\Gamma $ is the Lorentz factor (typically $\Gamma \, \sim \, 300$) and $\Delta t$ is the observed GRB variability time scale (typically $\Delta t\, \sim \, 1$ ms). In the vicinity of $r_{p}$, the fireball matter density is $\rho \, \sim \, 10^{-13}$ g cm$^{-3}$ [@WB]. In these models, the typical distance traversed by neutrinos may be taken as, $\Delta r\, {\buildrel < \over {_{\sim}}}\, (10^{-4}-1)$ pc, in the vicinity of GRB, where 1 pc $\sim \, 3\times 10^{18}$ cm. Matter effects on neutrino oscillations are relevant if $G_{F}\rho /m_{N}\, \sim \, \delta m^{2}/2E$. Using $\rho$ from Ref. [@WB], it follows that matter effects are absent for $\delta m^{2}\, {\buildrel > \over {_{\sim}}} \, {\cal O}(10^{-10})$ eV$^{2}$. Matter effects due to coherent forward scattering of neutrinos off the background for high energy neutrinos originating from GRBs are not expected to be important in the neutrino production regions around GRBs and will not be further discuss here. Taking the observed duration of the typical gamma-ray burst as, $\Delta t\, {\buildrel < \over {_{\sim}}} \, 1$ ms, we obtain the mass of the source as, $M_{GRB}\, {\buildrel < \over {_{\sim}}}\, \Delta t/G_{N}$, where $G_{N}$ is the gravitational constant. For the relatively shorter observed duration of gamma-ray burst from a typical GRB, $\Delta t\, \sim \, 0.2 $ ms, implying $M_{GRB}\, \sim \, 40\, M_{\odot}$ (where $M_{\odot}\, \sim \, 2\times 10^{33}$ g is solar mass). We use $M_{GRB}\, \sim \, 2\times 10^{2}\,M_{\odot}$ in our estimates. The magnetic field in the vicinity of a GRB is obtained by considering the equipartition arguments [@WB]. We use the following profile of magnetic field, $B_{GRB}$, as an example, for $r\, > \, r_{p}$ [@mag] $$B_{GRB}(r)\, \simeq \, B_{0}\left(\frac{r_{p}}{r}\right)^{2}, \label{bprofile}$$ where $B_{0}\, \sim \, L^{1/2}c^{-1/2}(r_{p}\Gamma)^{-1}$ with $L$ being the total wind luminosity (typically $L\, \sim \, 10^{51}$ erg s$^{-1}$). Neutrino conversions in GRB =========================== Flavor oscillations ------------------- In the framework of three flavor analysis, the flavor precession probability from $\alpha $ to $\beta $ neutrino flavor is [@book] $$P(\nu_{\alpha}\rightarrow \nu_{\beta}) \equiv P_{\alpha \beta} = \sum^{3}_{i=1}|U_{\alpha i}|^{2}|U_{\beta i}|^{2} +\sum_{i \neq j} U_{\alpha i}U^{\ast}_{\beta i} U^{\ast}_{\alpha j}U_{\beta j} \cos\left(\frac{2\pi L}{l_{ij}}\right), \label{3flvp}$$ where $\alpha, \beta = e, \mu, $ or $\tau $. $U$ is the 3$\times $3 MNS mixing matrix and can be obtained in usual notation through $$U\, \equiv R_{23}(\theta_{1}) \mbox{diag}(e^{-i\delta /2},1,e^{i\delta /2}) R_{31}(\theta_{2}) \mbox{diag}(e^{i\delta /2},1,e^{-i\delta /2}) R_{12}(\theta_{3}), \label{defU}$$ thus coinciding with the standard form given by the Particle Data Group [@pdg]. In Eq. (\[defU\]), $l_{ij}\simeq 4\pi E/ \delta m^{2}_{ij}$ with $\delta m^{2}_{ij} \equiv |m^{2}_{i}-m^{2}_{j}|$ and $L$ is the distance between the source and the detector. For simplicity, we will assume here a vanishing value for $\theta_{31}$ and CP violating phase $\delta $ in $U$. At present, the atmospheric muon and solar electron neutrino deficits can be explained with oscillations among three active neutrinos [@RECENT]. For this, typically, $\delta m^{2} \sim {\cal O}(10^{-3})$ eV$^{2}$ and $\sin^{2}2\theta \sim {\cal O}(1)$ for the explanation of atmospheric muon neutrino deficit, whereas for the explanation of solar electron neutrino deficit, we may have $\delta m^{2} \sim {\cal O}(10^{-10})$ eV$^{2}$ and $\sin^{2}2\theta \sim {\cal O}(1)$ \[just so\] or $\delta m^{2} \sim {\cal O}(10^{-5})$ eV$^{2}$ and $\sin^{2}2\theta \sim {\cal O}(10^{-2})$ \[SMA (MSW)\] or $\delta m^{2} \sim {\cal O}(10^{-5})$ eV$^{2}$ and $\sin^{2}2\theta \sim {\cal O}(1)$ \[LMA (MSW)\]. The present status of data thus permits multiple oscillation solutions to solar neutrino deficit. We intend to discuss here implications of these mixings for high energy cosmic neutrino propagation. In the above explanations, the total range of $\delta m^{2}$ is $10^{-10}\leq \delta m^{2}/$ eV$^{2} \leq 10^{-3}$ irrespective of neutrino flavor. The typical energy span relevant for possible flavor identification for high energy cosmic neutrinos is $2\times 10^{6}\leq E/$GeV$\leq 2\times 10^{7}$ (see Sect. IV). Taking a typical distance between the GRB and our galaxy as $L \sim 1000$ Mpc, we note that $\cos $ term in Eq. (\[3flvp\]) vanishes and so Eq. (\[3flvp\]) reduces to $$P_{\alpha \beta} \simeq \sum^{3}_{i=1}|U_{\alpha i}|^{2}|U_{\beta i}|^{2}. \label{preduced}$$ It is assumed here that no relatively dense objects exist between the GRB and the earth so as to effect significantly this oscillations pattern. Since $P_{\alpha \beta }$ in above Eq. is symmetric under the exchange of $\alpha $ and $\beta $ indices implying that essentially no $T$ (or $CP$) violation effects arise in neutrino vacuum flavor oscillations for high energy cosmic neutrinos [@cabibbo]. Let us denote by $F^{0}_{\alpha }$, the intrinsic neutrino fluxes. From the discussion in the previous Sect., it follows that $F^{0}_{e} : F^{0}_{\mu} :F^{0}_{\tau} = 1 :2 : < 10^{-5}$. For simplicity, we take these ratios as 1 : 2 : 0. In order to estimate the final flux ratios on earth for high energy cosmic neutrinos originating from cosmologically distant GRBs, let us introduce a 3$\times $3 matrix of vacuum flavor precession probabilities such that $$F_{\alpha} = \sum_{\beta}P_{\alpha \beta} F^{0}_{\beta}, \label{falpha}$$ where the unitarity conditions for $ P_{\alpha \beta} $ read as $$\begin{aligned} P_{ee} + P_{e\mu } + P_{e\tau } & = & 1,\nonumber \\ P_{e\mu} + P_{\mu \mu } + P_{\mu \tau }& = & 1,\nonumber \\ P_{e\tau }+ P_{\mu \tau } + P_{\tau \tau } & = & 1. \label{unitarity} \end{aligned}$$ The explicit form for the matrix $P $ in case of just so flavor oscillations as solution to solar neutrino problem along with the solution to atmospheric neutrino deficit in terms of $\nu_{\mu}$ to $\nu_{\tau}$ oscillations with maximal depth is $$P = \left( \begin{array}{ccc} 1/2 & 1/4 & 1/4 \\ 1/4 & 3/8 & 3/8 \\ 1/4 & 3/8 & 3/8 \end{array} \right). \label{justso}$$ Using Eq. (\[justso\]) and Eq. (\[falpha\]), we note that $F_{e}: F_{\mu }: F_{\tau } = 1: 1: 1$ at the level of $F^{0}_{e}$. Also, Eq. (\[unitarity\]) is satisfied. The same final flux ratio is obtained in remaining two cases for which the corresponding $P$ matrices are \[for SMA (MSW)\] $$P = \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1/2 & 1/2 \\ 0 & 1/2 & 1/2 \end{array} \right), \label{sma}$$ whereas in case of LMA (MSW), $$P = \left( \begin{array}{ccc} 5/8 & 3/16 & 3/16 \\ 3/16 & 13/32 & 13/32 \\ 3/16 & 13/32 & 13/32 \end{array} \right). \label{lma}$$ Thus, independent of the oscillation solution for solar neutrino problem, we have $F_{e}: F_{\mu }: F_{\tau } = 1: 1: 1$. Importantly, these ratios do not depend on neutrino energy or $\delta m^{2}$ at least in the relevant energy interval. Summarizing, although intrinsically the high energy cosmic tau neutrino flux is negligibally small however because of vacuum flavor oscillations it becomes comparable to $\nu_{e}$ flux thus providing some prospects for its possible detection. Spin-flavor oscillations ------------------------ We consider here an example of a possibility that may lead to an energy dependence and/or change in the above mentioned final flux ratios. We consider spin-flavor oscillations resulting from an interplay of possible Violation of Equivalence Principle (VEP) parameterized by $\Delta f$ and the magnetic field in the vicinity of GRBs. We obtain the range of neutrino mixing parameters giving $F_{\tau}/F_{e,\mu} > \, 10^{-5}$. The possibility of VEP arises from the realization that different flavors of neutrinos may couple differently to gravity [@G]. Consider a system of two mixed neutrinos ($\bar{\nu}_{e}$ and $\nu_{\tau}$) for simplicity. The difference of diagonal elements of the $2\times 2$ effective Hamiltonian describing the dynamics of the mixed system of these oscillating neutrinos in the basis $\psi^{T}\, =\, (\bar{\nu}_{e}, \nu_{\tau})$ for vanishing vacuum and gravity mixing angles is [@AK] $$\Delta H\, =\, \delta - V_{G}, \label{deltaH}$$ whereas each of the off diagonal elements is $\mu B$ ($\mu $ is magnitude of neutrino magnetic moment). In Eq. (\[deltaH\]), $\delta \, =\, \delta m^{2}/2E$, where $\delta m^{2}\, =\, m^{2} (\nu_{\tau})-m^{2}(\bar{\nu}_{e}) \, >\, 0$. Here $V_{G}$ is the effective potential felt by the neutrinos at a distance $r$ from a gravitational source of mass $M$ due to VEP and is given by [@G] $$V_{G} \, \equiv \, \Delta f \phi(r)E, \label{defVg}$$ where $\Delta f \, =\, f_{3}-f_{1}$ is a measure of the degree of VEP and $\phi(r)\, = \, G_{N}Mr^{-1}$ is the gravitational potential in the Keplerian approximation. In Eq. (\[defVg\]), $f_{3}G_{N}$ and $f_{1}G_{N}$ are respectively the gravitational couplings of $\nu_{\tau}$ and $\bar{\nu}_{e}$, such that $f_{1}\, \neq \, f_{3}$. We assume here same gravitational couplings for $\nu $ and $\bar{\nu}$ for simplicity. This implies that the sign of $V_{G}$ is same for $\bar{\nu}_{e}\rightarrow \nu_{\tau}$ and $\nu_{e}\rightarrow \bar{\nu}_{\tau}$ conversions [@AK]. If this is not the case then the sign of $V_{G}$ will be different for the two conversions and only one of the two conversions can occur. This may lead to a change in expected $\nu_{e}/\bar{\nu}_{e}$ ratio. We briefly comment on the possibility of empirical realization of this situation in next Sect.. There are at least three relevant $\phi (r)$’s that need to be considered [@MS]. The effect of $\phi (r)$ due to supercluster named Great Attractor with $\phi_{SC} (r)$ in the vicinity of GRB; $\phi (r)$ due to GRB itself, which is, $\phi_{GRB} (r)$, in the vicinity of GRB and the galactic gravitational potential, which is, $\phi_{G}(r)$. Therefore, we use, $\phi(r)\, \equiv \, \phi_{SC}(r)+\phi_{GRB}(r)+\phi_{G}(r)$. However, $\phi_{G}(r)\, \ll \, \phi_{SC}(r),\, \phi_{GRB}(r)$ in the vicinity of GRB. Thus, $\phi(r)\, \simeq \, \phi_{SC}(r)+\phi_{GRB}(r)$. If the neutrino production region $r_{p}$ is ${\buildrel < \over {_{\sim}}} 10^{13}$ cm then at $r\, \sim \, r_{p}$, we have $\phi_{GRB}(r)\, >\, \phi_{SC}(r)$. At $r\, \sim \, 6\times 10^{13}$ cm, $\phi_{GRB}(r)\, \sim \, \phi_{SC}(r)$ and for $r\, {\buildrel > \over {_{\sim}}}\, 10^{14}$ cm, $\phi_{GRB}(r)\, <\, \phi_{SC}(r)$. If the supercluster is a fake object then $\phi(r)\, \simeq \, \phi_{GRB}(r)$. Here we assume the smallness of the effect of $\phi (r)$ due to an Active Galactic Nucleus (AGN), if any, nearby to GRB. The possibility of vanishing gravity and vacuum mixing angle in Eq. (\[deltaH\]) allows us to identify the range of $\Delta f$ relevant for the neutrino magnetic moment effects only. Latter in this Sect., we briefly comment on the ranges of relevant neutrino mixing parameters for non vanishing gravity mixing angle with vanishing neutrino magnetic moment. The case of $\bar{\nu}_{\mu}\rightarrow \nu_{\tau}$ can be studied by replacing $\bar{\nu}_{e}$ with $\bar{\nu}_{\mu}$ along with corresponding changes in $V_{G}$ and $\delta m^{2}$. The intrinsic flux of $\bar{\nu}_{\mu}$ may be greater than that of $\bar{\nu}_{e}$ by a factor of $\sim $ 2 [@WB], thus also possibly enhancing the expected $\nu_{\tau}$ flux from GRBs through $\bar{\nu}_{\mu}\rightarrow \nu_{\tau}$. However, we have checked that observationally this possibility leads to quite similar results in terms of event rates and are therefore not discussed here further. We now study in some detail, the various possibilities arising from relative comparison between $\delta $ and $V_{G}$ in Eq. (\[deltaH\]). Let us first ignore the effects of VEP ($\Delta f\, =\, 0$). For constant $B$, the spin-flavor precession probability $P(\bar{\nu}_{e}\rightarrow \nu_{\tau})$ is obtained using Eq. (\[deltaH\]) as $$P(\bar{\nu}_{e}\rightarrow \nu_{\tau})\, =\, \left[\frac{(2\mu B)^{2}}{(2\mu B)^{2}+\delta^{2}}\right] \sin^{2}\left(\sqrt{(2\mu B)^{2}+\delta^{2}} \cdot \frac{\Delta r}{2}\right). \label{Pdeltaf0}$$ We take $\mu \, \sim \, 10^{-12}\, \mu_{B}$ or less, where $\mu_{B}$ is Bohr magneton, which is less than the stringent astrophysical upper bound on $\mu$ based on cooling of red giants [@R]. We here consider the transition magnetic moment, thus allowing the possibility of simultaneously changing the relevant neutrino flavor as well as the helicity. Therefore, the precessed $\nu_{\tau}$ is an active neutrino and interacts weakly. In Eq. (\[Pdeltaf0\]), $\Delta r$ is the width of the region with $B$. If $\delta \, < \, 2\mu B$, then, for $E\, \sim \, 2\times 10^{6}$ GeV and using Eq. (\[bprofile\]), we obtain $\delta m^{2}\, < \, 5\times 10^{-8}$ eV$^{2}$. We take, $\delta m^{2}\, \sim \, 10^{-9}$ eV$^{2}$, as an example and consequently we obtain from Eq. (\[Pdeltaf0\]) an energy independent large ($P\, >\, 1/2$) spin-flavor precession probability for $\mu \, \sim \, 10^{-12}\mu_{B}$ with $10^{-4}\, {\buildrel < \over {_{\sim}}}\, \Delta r/\mbox{pc} {\buildrel < \over {_{\sim}}}\,1$. This relatively small value of $\delta m^{2}$ is also interesting in the context of sun and supernovae [@ss]. Thus, for $\mu $ of the order of $10^{-12}\, \mu_{B}$, the $\nu_{\tau}$ flux may be higher than the expected one from GRBs, that is, $F_{\tau}/F_{e}\, > \, 10^{-5}$ due to neutrino spin-flavor precession effects. The neutrino spin-flavor precession effects are essentially determined by the product $\mu B$ so one may rescale $\mu $ and $B$ to obtain the same results. For $\delta \, \simeq 2\mu B$ and $\delta \, > \, 2 \mu B$, we obtain from Eq. (\[Pdeltaf0\]), an energy dependent $P$ such that $P\, <\, 1/2$. With non vanishing $\Delta f$ ($\Delta f\, \neq \, 0$), a resonant character in neutrino spin-flavor precession can be obtained for a range of values of relevant neutrino mixing parameters[^1]. Two conditions are essential to obtain a resonant character in neutrino spin-flavor precession: the level crossing and the adiabaticity at the level crossing (resonance). The level crossing condition is obtained by taking $\Delta H\, =\, 0$ and is given by: $$\delta m^{2}\, \sim \, 10^{-3} \mbox{eV}^{2} \left(\frac{|\Delta f|}{10^{-28}}\right). \label{levelcrossing}$$ These $\Delta f$ values are well below the relevant upper limits on $\Delta f$ which are typically in the $10^{-20}$ range [@recent]. Conversely speaking, the prospective detection of high energy neutrinos from cosmologically distant GRBs may be sensitive to $\Delta f$ values as low as $\sim 10^{-28}$. The other essential condition, namely, the adiabaticity in the resonance reads [@adiabaticity] $$\kappa \, \equiv \, \frac{2(2\mu B)^{2}}{|\mbox{d}V_{G}/\mbox{d}r|} \, {\buildrel > \over {_{\sim}}}\, 1. \label{adiabaticity}$$ Note that here $\kappa $ depends explicitly on $E$ through $V_{G}$ unlike the case of ordinary neutrino spin-flip induced by the matter effects. A resonant character in neutrino spin-flavor precession is obtained if $\kappa \, {\buildrel > \over {_{\sim}}}\, 1$ such that Eq. (\[levelcrossing\]) is satisfied. We notice that $B_{ad}/B_{GRB}\, {\buildrel <\over {_{\sim}}}\, 1$ for $\mu \, \sim 10^{-12}\mu_{B}$. Here $B_{ad}$ is obtained by setting $\kappa \, \sim \, 1$ in Eq. (\[adiabaticity\]). The general expression for relevant neutrino spin-flavor conversion probability is given by [@ICTP] $$P(\bar{\nu}_{e}\rightarrow \nu_{\tau})\, =\, \frac{1}{2}-\left(\frac{1}{2}-P_{LZ}\right)\cos 2\theta_{f}\cos 2\theta_{i}, \label{PLZ}$$ where $P_{LZ}\, =\, \exp(-\frac{\pi}{4}\kappa)$ and $ \tan 2\theta_{i}\, =\, (2\mu B)/\Delta H$ is being evaluated at the high energy neutrino production site in the vicinity of GRB, whereas $\tan 2\theta_{f}\, =\, (2\mu B)/\delta $ is evaluated at the exit. In Fig. 1, we plot $P(\bar{\nu}_{e}\rightarrow \nu_{\tau})$ given by Eq. (\[PLZ\]) as a function of $\Delta f $ as well as $\delta m^{2}$ with $E \sim 5\times 10^{6}$ GeV. Four equi $P$ contours are also shown in Fig. 1. Note that the resonant spin-flavor precession probability is relatively small ($P < 1/2$) for $\Delta f {\buildrel > \over {_{\sim}}} 10^{-26}$ essentially irrespective of $\delta m^{2}$ values. The expected spectrum $F_{\tau}$ of the high energy tau neutrinos originating from GRBs due to spin-flavor conversions is calculated as [@ICTP] $$F_{\tau}\simeq P(\bar{\nu}_{e}\rightarrow \nu_{\tau})F^{0}_{e}. \label{FTAU}$$ The energy dependence in $F_{\tau}$ is now evident \[as compared to $F_{\tau}$ given by Eq. (\[falpha\])\] when we convolve $P(\bar{\nu}_{e}\rightarrow \nu_{\tau})$ given by Eq. (\[PLZ\]) with $F^{0}_{e}$ taken from Ref. [@WB]. The degree of energy dependence clearly depends on the extent of spin-flavor conversions. With the improved information on either $\Delta f$ and/or $\mu $, one may be able to distinguish between the situations of resonant and non resonant spin-flavor precession induced by an interplay of VEP and $\mu $ in $B_{GRB}$. Let us now consider briefly the effects of non vanishing gravity mixing angle $\theta_{G}$ for vanishing neutrino magnetic moment. In the case of massless or degenerate neutrinos, the corresponding vacuum flavor oscillation analog for $\nu_{e}\rightarrow \nu_{\tau}$ is obtained through $\theta \rightarrow \theta_{G}$ and $\frac{\delta m^{2}}{4E} \rightarrow V_{G}$ in the standard flavor precession probability formula in 2 flavor approxiamtion. For maximal $\theta_{G}$, the sensitivity of $\Delta f$ may be estimated by equating the argument of second $\sin$ factor equal to $\pi/2$ in the corresponding expression for $P$ [@MS]. This implies $\Delta f\, \sim \, 10^{-41}$ with $\phi(r)\, \simeq \, \phi_{SC}(r)$. This value of $\Delta f$ is of the same order of magnitude as that expected for neutrinos originating from AGNs. In case of non zero $\delta m^{2}$, a resonant or/and non resonant flavor conversion between $\nu_{e}$ and $\nu_{\tau}$ in the vicinity of a GRB is also possible due to an interplay of vanishing/non vanishing vacuum and gravity mixing angles. For instance, a resonant flavor conversion between $\nu_{e}$ and $\nu_{\tau}$ may be obtained if $\sin^{2}2\theta_{G}\, \gg \, 0.25$ with $\Delta f \, \sim 10^{-31}$ ($\theta \, \rightarrow \, 0$). Here the relevant level crossing may occur at $r\, \sim \, 0.1 $ pc with corresponding $\delta m^{2}\, \sim \, 10^{-6}$ eV$^{2}$. Signatures of high energy $\nu_{\tau}$ in neutrino telescopes ============================================================= The km$^{2}$ surface area under water/ice high energy neutrino telescopes may be able to obtain first examples of high energy $\nu_{\tau}$, through [*double showers*]{}, originating from GRBs correlated in time and direction with corresponding gamma-ray burst or may at least provide relevant useful upper limits [@L]. The first shower occurs because of deep inelastic charged current interaction of high energy tau neutrinos near/inside the neutrino telescope producing the tau lepton (along with the first shower) and the second shower occurs due to (hadronic) decay of this tau lepton. The calculation of down ward going contained but separable double shower event rate for a km$^{2}$ surface area under ice/water neutrino telescope can be carried out by replacing the muon range expression with the tau one ($\sim E(1-y)\tau c/m_{\tau} c^{2}$) and then subtracting it from the linear size of a typical high energy neutrino telescope in the event rate formula while using the expected $\nu_{\tau}$ flux spectrum given by Eq. (\[falpha\]) and/or by Eq. (\[FTAU\]). Here, $y$ is the fraction of the neutrino energy carried by the hadrons in lab frame. Thus, $(1-y)$ is the fraction of energy transferred to the associated tau lepton having life time $\tau c$ and mass $m_{\tau}c^{2}$. We take here $y \sim 0.25$ [@L]. The condition of containdness of the two showers is obtained by requiring that the separation between the two showers is less than the typical $\sim $ km size of the neutrino telescope. It is obtained by equating the range of tau neutrino induced tau leptons with the linear size of detector implying $E \, {\buildrel < \over {_{\sim}}}\, 2\times 10^{7}$ GeV. The condition of separableness of the two showers is obtained by demanding that the separation between the two showers is larger than the typical spread of the showers such that the amplitude of the second shower is essentially 2 times the first shower. This leads to $E \, {\buildrel > \over {_{\sim}}}\, 2\times 10^{6}$ GeV [@L]. Thus, the two showers may be separated by a $\mu$-like track within these energy limits. To calculate the event rates, we use Martin Roberts Stirling (MRS 96 R$_{1}$) parton distributions [@mrs] and present event rates in units of yr$^{-1}$ sr$^{-1}$. We have checked that other recent parton distributions give quite similar event rates and are therefore not depicted here. Following [@APZ; @apz], we present in Table I, the expected contained but separable double shower event rates for down word going $\nu_{\tau}$ in km$^{2}$ size under water/ice Čerenkov high energy neutrino telescopes for illustrative purposes. In Table I, the vacuum oscillation situation is essentially independent of the choice of the oscillation solution to solar neutrino problem. From Table I, we notice that the event rates for neutrino flavor/spin-flavor precession are up to $\sim $ 5 orders of magnitude higher than that for typical intrinsic (no oscillations) tau neutrino flux. The possibility of measuring the contained but separable double shower events may enable one to distinguish between the high energy tau neutrinos and electron and/or muon neutrinos originating from cosmologically distant GRBs while providing useful information about the relevant energy interval at the same time. The chance of having double shower events induced by electron and/or muon neutrinos is negligibly small for the relevant energies [@L]. Collective information about directionality of the source, rate and energy dependence of neutrino fluxes will be needed to possibly isolate the mechanism of neutrino oscillation. The up ward going tau neutrinos at these energies may lead to a small pile up of up ward going $\mu$-like events near (10$^{4}-10^{5}$) GeV with fairly flat zenith angle dependence [@HS]. We now briefly discuss the potential of the under water/ice high energy neutrino telescopes to possibly determine an observational consequence of neutrino spin-flip in GRB induced by VEP. In the electron neutrino channel, the $\bar{\nu}_{e}$ interaction rate (integrated over all angles) is estimated to be an order of magnitude higher than that of $(\nu_{e}+\bar{\nu}_{e})$ per Megaton year [@APZ]. This an order of magnitude difference in interaction rate of [*down ward going*]{} $\bar{\nu}_{e}$ is due to Glashow resonance encountered by $\bar{\nu}_{e}$ with $E\, {\buildrel > \over {_{\sim}}} \, 10^{6}$ GeV when $\bar{\nu}_{e}$ interact with electrons inside the detector as compared to corresponding deep inelastic scattering. The up ward going $\bar{\nu}_{e}$, on the other hand, while passing through the earth, at these energies, are almost completely absorbed by the earth mainly due to same resonant effect. Thus, for instance, if $E \, \sim \, 6.4\times 10^{6}$ GeV, an energy resolution $\Delta E/E\, \sim \, 2\Gamma_{W}/M_{W}\, \sim \, 1/20$, where $\Gamma_{W}\, \sim $ 2 GeV is the width of Glashow resonance and $M_{W}\, \sim \, $80 GeV, may be needed to empirically differentiate between $\bar{\nu}_{e}$ and $(\nu_{e}+\bar{\nu}_{e})$. The existing/planned high energy neutrino telescopes may thus in principle attempt to measure the $\nu_{e}/\bar{\nu}_{e}$ ratio in addition to identifying ($\nu_{\tau}+\bar{\nu}_{\tau}$) and ($\nu_{\mu}+\bar{\nu}_{\mu}$) events separately. This feature may be utilized, for instance, to explain a situation in which a [*change*]{} in $\nu_{e}/\bar{\nu}_{e}$ ratio is observed as compared to GRB neutrino flux predictions in [@WB]. This situation, if realized obsevationally may be an evidence for the neutrino spin-flip in GRB due to VEP, provided if neutrinos and antineutrinos couple differently to gravity. This follows from the possibility discussed in previous Sect. that an interplay between VEP and neutrino magnetic moment in $B_{GRB}$ may leads to conversions in either $\nu_{e}$ or $\bar{\nu}_{e}$ channel but not in both channels simultaneously. Results and discussion ====================== 1\. Intrinsically, the flux of high energy cosmic tau neutrinos is quite small, relative to non tau neutrino flavor, typically being $F^{0}_{\tau}/F^{0}_{e, \mu}\, <\, 10^{-5}$ (whereas $F^{0}_{e}/F^{0}_{\mu} \sim 1/2$) from cosmologically distant GRBs. 2\. Because of neutrino oscillations, this ratio can be greatly enhanced. In the context of three flavor neutrino mixing scheme which can accommodate the oscillation solutions to solar and atmospheric neutrino deficits in terms of oscillations between three active neutrinos, the final down ward going ratio of fluxes of high energy cosmic neutrinos on earth is $F_{e}\sim F_{\mu}\sim F_{\tau} \sim F^{0}_{e}$, essentially irrespective of the oscillation solution to solar neutrino problem. The (vacuum) flavor oscillations leads to an essentially energy independent flux of high energy neutrinos of all flavors originating from cosmologically distant GRBs at the level of electron neutrino flux, whereas spin-flavor precessions/conversions may lead to an energy dependence or/and change in this situation. The spin-flavor conversions may occur possibly through several mechanisms. We have discussed in some detail mainly the spin-flavor precession/conversion situation induced by a non zero neutrino magnetic moment and by a relatively small VEP as an example to point out the possibility of obtaining some what higher tau neutrino fluxes as compared to no oscillations/conversions scenarios from GRBs. The matter density in the vicinity of GRB is quite small (up to 4$-$5 orders of magnitude) to induce any resonant flavor/spin-flavor neutrino conversion due to normal matter effects. We have pointed out that a resonant character in the neutrino spin-flavor conversions may nevertheless be obtained due to possible VEP. The corresponding degree of VEP may be $\sim \, (10^{-35}-10^{-25}$) depending on $\delta m^{2}$ value for vanishing gravity mixing angle. 3\. This enhancement in high energy cosmic tau neutrino flux may lead to the possibility of its detection in km$^{2}$ surface area high energy neutrino telescopes. For $2\times 10^{6}\leq E$/GeV $\leq 2\times 10^{7}$, the down ward going high energy cosmic tau neutrinos may produce a double shower signature because of charged current deep inelastic scattering followed by a subsequent hadronic decay of associated tau lepton The double shower event rate for intrinsic (no oscillations/conversions) high energy tau neutrinos originating from GRBs turns out to be small as compared to that due to precession/conversion effects up to a factor of $\sim \, 10^{-5}$. Thus, the high energy neutrino telescopes may possibly provide useful upper bounds on intrinsic properties of neutrinos such as mass, mixing and magnetic moment, etc.. The relevant tau neutrino energy range for detection in km$^{2}$ surface area under water/ice neutrino telescopes may be $2\times 10^{6} {\buildrel < \over {_{\sim}}}\, E/\mbox {GeV}\, {\buildrel < \over {_{\sim}}} \, 2\times 10^{7}$ through characteristic contained but separable double shower events. Observationally, the high energy $\nu_{\tau}$ burst from a GRB may possibly be [*correlated*]{} to the corresponding gamma-ray burst/highest energy cosmic rays (if both have common origin) in time and in direction thus raising the possibility of its detection. If the range of neutrino mixing parameters pointed out in this study is realized terrestially/extraterrestially then a relatively large (energy dependent) $\nu_{\tau}$ flux from GRBs is expected as compared to no oscillation/conversion scenario. #### Acknowledgments. {#acknowledgments. .unnumbered} The author thanks Japan Society for the Promotion of Science for financial support. E. Waxman and J. Bahcall, Phys. Rev. Lett. 78 (1997) 2292 ; Phys. Rev. D 59 (1999) 023002. For a recent review, see, E. Waxman, astro-ph/9911395. For a recent review, see, for instance, T. Piran, Nucl. Phys. B (Proc. Suppl.) 70 (1999) 431 and references cited therein. For a latest discussion, see, P. Meszaros, astro-ph/9904038; T. Piran, astro-ph/9907392. B. Paczyński and G. Xu, Ap. J 427 (1994) 708. A somewhat detailed numerical study supports this order of magnitude estimate, H. Athar, R. A. Vázquez and E. Zas (to be submitted). See, for instance, L. Moscoso, in Proc. [*Sixth International Workshop on Topics in Astroparticle and Underground Physics*]{} (TAUP 99), September 1999, Paris, France (to be published, edited by M. Froissart, J. Dumarchez and D. Vignaud) \[preprint DAPNIA-SPP-00-01 (Jan 2000)\]. T. J. Weiler et al., hep-ph/9411432; F. Halzen and G. Jaczko, Phys. Rev. D 54 (1996) 2779. J. G. Learned and S. Pakvasa, Astropart. Phys. 3 (1995) 267. F. Halzen and D. Saltzberg, Phys. Rev. Lett. 81 (1998) 4305. See also, S. Bottai and F. Becattini, in Proc. [*26th International Cosmic Ray Conference*]{}, Salt Lake City, Utah, 17-25 August, 1999, edited by D. Kieda, M. Salamon and B. Dingus, Vol.2, p. 249; S. Iyer, M. H. Reno and I. Sarcevic, hep-ph/9909393. See, for instance, P. Mészáros, P. Laguna and M. J. Rees, Ap. J 415 (1993) 181. Ta-Pei Cheng and Ling-Fong Li, in [*Gauge theory of elementary particle physics*]{} (Claredon press, Oxford, 1984). C. Caso et al., The Euro. Phys. J. C 3 (1998) 103. See, for instance, E. Lisi, in [*New Era in Neutrino Physics*]{}, Proceedings of the Satellite Symposium after Neutrino 98, Tokyo, Japan, edited by H. Minakata and O. Yasuda, p. 153. N. Cabibbo, Phys. Lett. B 72 (1978) 333. M. Gasperini, Phys. Rev. D 38 (1988) 2635; D 39 (1989) 3606. For an independent similar possibility of testing VEP by neutrinos, see, A. Halprin and C. N. Leung, Phys. Rev. Lett. 67 (1991) 1833. See, for instance, E. Kh. Akhmedov, S. T. Petcov and A. Yu. Smirnov, Phys. Rev. D 48 (1993) 2167. For details in the context of AGNs, see, H. Minakata and A. Yu. Smirnov, Phys. Rev. D 54 (1996) 3698. G. G. Raffelt, Phys. Rev. Lett. 64 (1990) 2856. For a recent up date in the context of sun, see, M. M. Guzzo and H. Nunokawa, Astropart. Phys. 12 (1999) 87, whereas in the context of supernovae, see, for instance, Athar Husain, in Proc. [*New Worlds in Astroparticle Physics*]{}, Eds. A. Mourão, M. Pimento and P. Sá, World Scientific Pub., Singapore, p. 244 (hep-ph/9902222). Athar Husain, Nucl. Phys. B (Proc. Suppl.) 76 (1999) 419 and references cited therein. For a recent analysis, see, G. L. Fogli et al., Phys. Rev. D 60 (1999) 053006 and references cited therein. C.-S. Lim and W. J. Marciano, Phys. Rev. D 37 (1988) 1368; E. Kh. Akhmedov, Sov. J. Nucl. Phys. 48 (1988) 382; Phys. Lett. B 213 (1988) 64. See, C. W. Kim and A. Pevsner, in [*Neutrinos in Physics and Astrophysics*]{} (Harwood Academic Publishers, Switzerland, 1993). A. D. Martins, R. G. Roberts and W. J. Stirling, Phys. Lett. B 387 (1996) 419. R. Gandhi, C. Quigg, M. H. Reno and I. Sarcevic, Astropart. Phys. 5 (1996) 81; Phys. Rev. D 58 (1998) 093009. H. Athar, G. Parente and E. Zas (to be submitted). See also scanned transperacies by Athar Husain at URL http://taup99.in2p3.fr/TAUP99/; hep-ph/9912417. ---------------------------------------------------------------- ------------------- ------------------- ------------------------------ [*Energy Interval*]{} [*no osc*]{} [*vac osc*]{} [*spin-flavor precession*]{} $2\times 10^{6}{\buildrel < \over {_{\sim}}}\, E/\mbox {GeV}\, $10^{-6}$ $1\times 10^{-1}$ $0.5\times 10^{-1}$ {\buildrel < \over {_{\sim}}} \, 5\times 10^{6}$ $5\times 10^{6}{\buildrel < \over {_{\sim}}}\, E/\mbox {GeV}\, $2\times 10^{-7}$ $2\times 10^{-2}$ $10^{-2}$ {\buildrel < \over {_{\sim}}} \, 7\times 10^{6}$ $7\times 10^{6}{\buildrel < \over {_{\sim}}}\, E/\mbox {GeV}\, $2\times 10^{-7}$ $2\times 10^{-2}$ $10^{-2}$ {\buildrel < \over {_{\sim}}} \, 1\times 10^{7}$ $1\times 10^{7}{\buildrel < \over {_{\sim}}}\, E/\mbox {GeV}\, $2\times 10^{-7}$ $2\times 10^{-2}$ $10^{-2}$ {\buildrel < \over {_{\sim}}} \, 2\times 10^{7}$ ---------------------------------------------------------------- ------------------- ------------------- ------------------------------ : Event rate (yr$^{-1}$sr$^{-1}$) for down word going high energy tau neutrino induced contained but separable double showers connected by a $\mu-$ like track in various energy bins using MRS 96 R$_{1}$ parton distributions. For spin-flavor precessions, we use $\delta m^{2}\, {\buildrel < \over {_{\sim}}}\, 10^{-9}$ eV$^{2}$ and $\mu \, \sim 10^{-12}\mu_{B}$ \[see Eq. (\[Pdeltaf0\]) in the text\], whereas for vacuum flavor oscillations, we used Eq. (\[falpha\]). [^1]: From above discussion, it follows that $E$ dependent/independent spin-flavor precession may also be obtained for non zero $\Delta f$, however, given the current status of the high energy neutrino detection, for simplicity, we ignore these possibilities which tend to overlap with this case for a certain range of relevant neutrino mixing parameters; for details of these possibilities in the context of AGN, see [@athar].

--- abstract: 'Photoinduced magnetization dynamics is investigated in chemically ordered $(\mathrm{LaMnO}_3)_{2n}/(\mathrm{SrMnO}_3)_n$ superlattices using the time-resolved magneto-optic Kerr effect. A monotonic frequency-field dependence is observed for the $n=1$ superlattice, indicating a single spin population consistent with a homogeneous hole distribution. In contrast, for $n\geq2$ superlattices, a large precession frequency is observed at low fields indicating the presence of an exchange torque in the dynamic regime. We propose a model that ascribes the emergence of exchange torque to the coupling between two spin populations – viscous and fast spins.' author: - 'H. B. Zhao, K. J. Smith, Y. Fan, G. Lüpke' - 'A. Bhattacharya, S. D. Bader' - 'M. Warusawithana, X. Zhai, J. N. Eckstein' date: 'June 12, 2007' title: | Viscous spin exchange torque on precessional magnetization\ in $(\mathrm{LaMnO}_3)_{2n}/(\mathrm{SrMnO}_3)_{n}$ superlattices --- The manganites exhibit a rich variety of electronic and magnetic phases as a function of doping, or substitution of the A-site cation \[*e.g.* $\mathrm{La}^{3+}$ and $\mathrm{Sr}^{2+}$ in $(\mathrm{La}_{1-x}\mathrm{Sr}_x)\mathrm{MnO}_3]$ to allow the Mn sites to have mixed valence [@R1]. An optimized carrier concentration favors the double exchange interaction which tends to globally align spins on Mn sites ferromagnetically  [@R2]. However, in some doping and temperature ranges, random occupancy on the $A$-site may give rise to coexisting regions with locally ferromagnetic (FM) and antiferromagnetic (AF) correlations. In such a regime, a small magnetic field may have dramatic consequences, because it can align the randomly oriented spins and energetically promote ferromagnetism, rendering a global ferromagnetic order out of a mixed phases. The phase complexity in doped manganites makes the investigation of periodic layered structures especially intriguing, given the possibility of engineering charge, spin and orbital ordering in a layer-by-layer manner [@R3; @R4; @R5]. Using ozone assisted oxide-MBE (Molecular Beam Epitaxy), and a strategy called ’Digital Synthesis’, superlattices can be synthesized using integer unit-cell layers of constituent materials that are fully ordered on the $A$-site [@R6]. First, such structures may lead to novel magnetic interactions due to reconstructions involving charge, spin and orbital degrees of freedom across atomically sharp interfaces, which is absent in materials where $3^+$ and $2^+$ cations randomly occupy the $A$-site. Secondly, the coexisting phases would be regularly arranged, allowing characterization with scattering probes that exploit this periodicity [@R7], or via proximity effects. These approaches can be used to investigate the intimate correlation of the magnetic interaction with the ordered electronic structure, and how this influences the phase transitions into the various ordered states. Recently, ferromagnetic order has been realized in superlattices of dissimilar AF insulating manganites, $\mathrm{LaMnO}_3$ (LMO) and $\mathrm{SrMnO}_3$ (SMO), due to charge transfers promoted by the chemical potential difference across the interface [@R4; @R8]. The AF interaction would still be favored in other regions far from the interface, but a region with glassy/frustrated spins may exist between the FM and AF layers. The magnetic coupling between these layers can dramatically affect the magnetization switching and dynamics, which may be of interest in the context of magnetoelectronics [@R9]. Recent studies of the magnetization dynamics in ferromagnetic manganite random-alloy films have shown that the uniform spin precession mode is governed by anisotropy and demagnetizing fields [@R10; @R11; @R12; @R13]. Due to a homogeneous hole distribution in these films, spins on all Mn sites are coupled parallel by strong ferromagnetic interactions. Therefore, a single spin population is sufficient to describe both quasi-static and dynamic magnetization measurements. However, this may not be true in manganite superlattices, in which the density of holes is modulated, thus modulating the magnetic interaction and spin character in both the static and dynamic regime. In this Letter, we report on a study of the uniform spin precession dynamics in $(\textrm{LMO})_{2n}/(\textrm{SMO})_n$ superlattices through ultrafast pump-probe measurements of the time-resolved magneto-optical Kerr effect (TR-MOKE). We show that the one-spin model typically invoked does not adequately explain the observed frequency-field behavior for the case where $n\geq2$. A large precessional frequency observed at low fields indicates the presence of an exchange torque in the dynamic regime. In contrast, the FM superlattice with $n=1$ and the corresponding random-alloy thin film are shown to have similar precession dynamics, which can be described well by a single spin population and its anisotropy fields. We propose a model that ascribes the emergence of exchange torque in the manganite superlattice to the coupling between two spin populations - viscous and fast spins. TR-MOKE measurements are performed with a Ti:sapphire amplifier laser system providing 150-fs pulses at 1-kHz repetition rate. Figure 1(a) depicts the geometry of our TR-MOKE setup. We use 800-nm pump pulses with fluence of 1 mJ/cm$^2$ to induce magnetization precession in our samples, and the time evolution of the out-of-plane magnetization component $\textrm{M}_z$ is monitored with time-delayed 400-nm probe pulses with fluences on the order of 0.1 mJ/cm$^2$. A split-coil superconducting magnet is employed to study the field dependence of the magnetization precession for both in-plane and out-of-plane sample geometries. All data shown were taken at $\textrm{T}=50 K$. Figure 1(c) shows the time evolution of the out-of-plane magnetization component, as measured by TR-MOKE for the $(\textrm{LMO})_2/(\textrm{SMO})_1$ sample with its spin structure depicted in Fig. 1 (b). All spins are coupled parallel due to strong ferromagnetic exchange interactions in this n=1 superlattice. As a result, an intense oscillation due to the uniform precession of FM spins is observed. The solid line in Fig. 1(c) indicates a fit yielding a precession frequency, $f=6.3$ GHz, and a damping rate, $\Gamma=0.0024$ ps$^{-1}$, defined by $\textrm{M}_z \approx \exp(i2\pi f t-\Gamma t)$. The excitation and precession of the magnetization can be explained by laser-induced demagnetization and anisotropy modulation [@R14; @R15]. When the external field is applied normal to the sample plane, the strong pump pulse instantaneously heats up the sample, which alters the equilibrium direction of the magnetization caused by a sudden change of the demagnetization field. The magnetization subsequently rotates towards the direction of a transient field ($\mathbf{H}_{\textrm{tr}}$), and after the sample cools ($\Delta t>50$ ps), precesses around the original effective field $\mathbf{H}_\textrm{eff}$, as indicated in the inset of Fig. 1(c). {width="\columnwidth"} \[fig:fig1\] Figures 2(a) and (b) show the field dependence of the uniform spin precession for the $(\textrm{LMO})_{2n}/(\textrm{SMO})_{n}$ superlattices with $n=1$ and $n=2$, respectively. The precession frequencies are plotted in Fig. 3. The frequency of the $n=1$ superlattice clearly decreases monotonically to about zero with decreasing field. However, for the $n=2$ superlattice, there is a local minimum at $\mathrm{H}\approx0.8$ T, and an extrapolated non-zero value of $\approx 11$ GHz in the limit $\mathrm{H}=0$. The uniform spin precession in homogeneous ferromagnetic materials can be described by a torque equation: $$\frac{d\mathbf{M}}{d t} = -\gamma \mathbf{M}\times\mathbf{H}_{\mathrm{eff}} \label{torque}$$ where $\mathbf{H}_{\mathrm{eff}} = -\frac{\partial E}{\partial \mathbf{M}}$, and $E$ is the magnetic free energy of the system, which can be written as $E=-\mathbf{H}\cdot\mathbf{M}+2\pi\mathrm{M}_z^2+K_a{\mathrm{M}_z^2}/{\mathrm{M}_s^2}$. The corresponding phenomenological fields are the demagnetizing field $\mathrm{H}_d=4\pi\mathrm{M}_s$ and the out-of-plane anisotropy field $H_a={2K_a}/{\textrm{M}_s}$. $\textrm{H}_d$ is determined independently by SQUID magnetometry measurements. No in-plane anisotropy field needs to be included since the samples are isotropic within the surface plane, as shown in Fig. 4 for the $n=2$ superlattice. {width="\columnwidth"} \[fig:fig2\] The solid lines in Fig. 3 represent fits to the precession frequency using Eq. (1). For the $n=1$ superlattice, both calculated and measured frequencies monotonically decrease with decreasing field to very small values ($< 5$ GHz) at fields below 0.4 T, similar to $\mathrm{La}_{0.67}\mathrm{Sr}_{0.33}\mathrm{MnO}_3$ (LSMO) alloy films [@R12]. An easy plane anisotropy $\mathrm{H}_a = -0.19$ T is obtained for the $n=1$ superlattice, which is comparable to the LSMO alloy film grown on the same $\mathrm{SrTiO}_3$ substrate [@R12]. A small deviation of the measured frequency from the calculated prediction appears at low fields. This discrepancy will be discussed following a description of the $n=2$ superlattice. For the $n=2$ superlattice, the calculated and measured frequency-field dependence shows very good agreement for applied fields larger than 0.8 T. However, a large discrepancy is revealed at low magnetic fields. The measured frequencies increase with decreasing fields while the calculated curve indicates an opposite trend. For a uniform precession mode, a large frequency at very low fields can only be observed in thin films that are anisotropic in the sample plane. In such films, the in-plane anisotropy field acts like an effective field and provides the torque to the magnetization  [@R13; @R16]. However, the ($\mathrm{LMO})_{2n}/(\mathrm{SMO})_n$ superlattices exhibit negligible in-plane anisotropy, as mentioned previously. Nevertheless, a similar effective field must be introduced to account for the large finite frequencies at low fields in the $n=2$ superlattice. Since we excluded an anisotropy field, other possible terms contributing to an effective field are bulk dipolar and exchange fields. However dipolar fields can be neglected in our pump-probe TR-MOKE experiments, since dipolar fields only affect spin waves with nonzero wave vector [@R17]. Moreover, exchange torques are not generated for the uniform mode in systems with a single spin population where all coupled spins are parallel and precess with the same amplitude and phase. We thus conclude that a single spin population is not sufficient to describe the field-frequency dependence for the $n=2$ superlattice. In this structure, spins with different characters may emerge due to variations of magnetic interactions within the different layers. {width="\columnwidth"} \[fig:fig3\] We observe a slight reduction in the magnetic moment per Mn atom in the $n=2$ superlattice, indicating the suppression of ferromagnetic ordering. We suggest that frustrated bonds in regions of varying hole concentration and/or charge carrier mobility result in magnetic disorder and consequently a viscous spin population, as depicted by the blue arrows in Fig. 1(b). In the quasi-static regime, this population is not distinguished from the fast ferromagnetic spins, which constitute a large fraction of Mn sites in the $n=2$ superlattice. A remaining ferromagnetic exchange interaction between viscous and fast spins encourages parallel alignment. Thus, viscous spins rotate with the FM magnetization and applied field; no biased pinning force is produced. This is evident by the absence of an exchange-biased field, as determined from hysteric magnetization curves shown in Fig. 4. However, we notice an increase in coercivity of the $n=2$ superlattice compared to the alloy film and $n=1$ superlattice, as shown in Fig. 4(a). This may be caused by the resistance of viscous spins during the magnetization reversal process, analogous to the phenomenon in FM/AF heterostructures, where the drag of uncompensated AF spins enhances the coercive field but keeps FM and AF spins aligned along the same direction [@R18]. In the dynamic regime, the viscous spins may not readily align with the precessing FM magnetization. FM spins excited by the pump pulse quickly rotate away from the original effective field $\mathbf{H}_\mathrm{eff}$, while viscous spins remain along $\mathbf{H}_{\mathrm{eff}}$ within tens of picoseconds. When the frequency of fast spins (manifest in the magnetization) greatly exceeds the inverse relaxation time of the viscous spins, an exchange torque $\gamma A\mathbf{M}\times\mathbf{M}_v$ is exerted on the precessing magnetization, enhancing the precession frequency. Here, $\mathbf{M}_v$ represents the total magnetization of viscous spins. The exchange field $A\mathrm{M}_v$ exerted on to the FM magnetization is calculated to be $\approx0.20$ T to account for the $\approx 11 $GHz precession frequency of the $n=2$ superlattice in the limit $\mathrm{H_{ext}}=0$. For the $n=1$ superlattice under the same condition, we observe a finite but small frequency ($<5$ GHz); similar calculations indicate a much weaker exchange torque ($<0.03$ T) for this system. The viscous spins rotate in the same manner as the FM magnetization, but on a much longer time scale ($>10$ ns), hence the exchange fields must be isotropic within the sample plane. This interaction is equivalent to a rotational anisotropy or an “isotropic anisotropy” field [@R19]. A similar anisotropy was observed in spin-glass materials in which spin correlations lead to an additional field parallel to the applied field. For example, a spin-glass like phase can exist at a ferromagnetic/superconductor interface, giving rise to a shift in the resonance field of the FMR spectra relative to that observed in the bulk ferromagnetic material [@R20]. An isotropic FMR shift was also observed in FM/AFM heterostructures [@R21]. Although the origin of these fields remains uncertain, some features are linked to spin-glass like phases. At zero field, the exchange interaction between viscous spins and fast spins lies within the sample plane. A strong perpendicular field tilts the FM spins out of the sample plane and reduces magnetic disorder as hole hopping is enhanced concurrently. This might generate an ordered FM state within the whole superlattice, as ferromagnetic interaction between Mn spins is promoted. Hence, the exchange torque is diminished at strong applied fields due to a reduction in the number of viscous spins. This process may explain the observed minimum of the precession frequency around 0.8 T for the $n=2$ superlattice. Using this model, we calculated the frequency-field dependence shown by the dashed line in Fig. 3(b). In this calculation, the exchange torque decreases with increasing applied field and vanishes at $\mathrm{H}=0.8$ T. We neglected the field dependence of $\mathrm{H}_d$ and $\mathrm{H}_a$, since the number of viscous spins is much smaller than the number of free spins. This simulation implies a threshold field at which the viscous spins emerge and an inverse linear dependence of viscous spin population on applied field. The TR-MOKE measurements of the $n=3$ superlattice reveal a similar frequency-field dependence at low fields as for the $n=2$ sample - the precessional frequency increases with decreasing magnetic field, as shown in Fig 3 (c). The same exchange torque generated by viscous spins on the FM magnetization may account for this behavior as well. The overall magnitude of the precession frequency is quite different from that observed in superlattices with $n\leq2$ and LSMO alloy previously discussed. This may be caused by a significant reduction of magnetic moment and demagnetizing field compared to the other structures. The out-of plane anisotropy may also be strongly modified, which dramatically affects the magnitude of the frequency as well. The enhanced damping of the precession in the $n=3$ superlattice is further evidence of the emergence of magnetic disorder. In this superlattice, the large variation of the density of holes may give rise to the coexistence of ferromagnetic and antiferromagnetic phases, which further favors the formation of frustrated bonds at the FM/AF interfaces leading to disordered viscous spins. A strong applied field would align more of these spins, therefore reducing the disorder and exchange torque, similar as in the $n=2$ superlattice. {width="\columnwidth"} \[fig:fig4\] We do not observe magnetization precession in the $n=5$ superlattice. This may be due to the difficulties of exciting a uniform mode in this structure, where the ferromagnetic order is strongly modulated by a periodic modulation of hole density, as evidenced by neutron scattering experiments [@R8]. Fig. 4(a) shows a very high coercive field ($>1$ kOe) in this superlattice, which may point to pinning of the spins due to coexisting FM and AF regions. The strong frustration at FM and AF interfaces creates a large distribution of pinning fields, giving rise to a broad hysteresis loop. This may preclude a coherent rotation of the FM spins. Furthermore, the modulation of magnetism may cause strong magnon scattering [@R22], consequently greatly diminishing the dephasing time for the uniform magnetization precession. Thus, we propose that the increased precessional damping is intimately correlated with increasing magnetic disorder, as manifested in large $n$ superlattices. In conclusion, we have observed coherent spin precession in digital superlattices of $(\mathrm{LMO})_{2n}/(\mathrm{SMO})_n$. The frequency-field dependence for the $n=1$ superlattice is nearly identical to that of random-alloy thin films, consistent with a homogeneous spin character. 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--- abstract: 'This paper considers the realizability of quantum gates from the perspective of information complexity. Since the gate is a physical device that must be controlled classically, it is subject to random error. We define the complexity of gate operation in terms of the difference between the entropy of the variables associated with initial and final states of the computation. We argue that the gate operations are irreversible if there is a difference in the accuracy associated with input and output variables. It is shown that under some conditions the gate operation may be associated with unbounded entropy, implying impossibility of implementation.' author: - | Subhash Kak\ Department of Electrical & Computer Engineering\ Louisiana State University, Baton Rouge, LA 70803, USA title: Information Complexity of Quantum Gates --- psfig \#1 Introduction {#introduction .unnumbered} ------------ In this paper, we consider complexity and realizability of quantum gates from the point of view of information theory. A gate is a physical system that is controlled by varying some input variables, which are classical. In principle, such a physical system could implement a variety of operators based on the control variables. The gate functions may be also implemented by a single physical system that operates sequentially on the qubits in the quantum register. The complexity of the gate will be defined in terms of the entropy associated with its control. From a practical point of view, one is interested in asking how easy it is to control a gate. As no analog system can have infinite precision, we investigate what happens if the precision levels at the input and the output are different. The complexity of the gate, defined in terms of entropy, will be examined for the rotation and [cnot]{} gates in certain circuits. Information processing by gate {#information-processing-by-gate .unnumbered} ------------------------------ One aspect of gate performance is its accuracy. Researchers on quantum information science have given much attention to the question of errors and their correction \[1-3\] by drawing upon parallels with classical information. Quantum error-correction coding works like classical error-correction to correct some large errors. But the framework of quantum information is distinct from that of classical information. In the classical case, it is implicitly assumed that there occurs an automatic correction of errors that are smaller than a threshold by means of clipping or by the use of a decision circuit. In the case of quantum information, the input data is nominally discrete, but in reality its precision cannot be absolute in any actual realization. Furthermore, unknown small errors in quantum information cannot be corrected \[4-5\]. Consequently, proposals for error correction and fault tolerance (such as \[6-8\]) remain unrealistic. Classical analog computation and quantum processing do have parallels. In general, fixed errors in gate operation could become irreversible due to actual small nonlinearity of nominally linear elements. Analog computing is not practical to implement because noise cannot be separated from useful signal and it accumulates, degrading the system performance in an uncorrectable manner. If there were no noise, the practicality of analog computing would depend on the feasibility of the gate implementation over the expected input-output range. This feasibility must be checked in the context of the limitations on information processing by the gate. Consider the gate G of Figure 1. It may be assumed that it is a physical system which is controlled by means of some variable. This control is implemented by choosing a setting on an instrument, and this choice is associated with random error. If one views the circuit operations to be implemented by the same device transitioning through various states in sequence, then one can determine the distribution of the control variable states, and compute its entropy. This entropy, when determined for the entire computing circuit, may be taken to represent its complexity. Information is preserved, therefore one can define the following relationship for the entropy expressions for the input $X$, the gate control information $C$, and the output $Y$: $$H(Y) = H(X) + H(C).$$ Although it is assumed that the variable $X$ is discrete, in reality the lack of perfect precision at the state preparation state makes it a continuous variable \[9-11\]. The lack of precision may not affect the measurement variables, but it would introduce continuous phase error. Similarly, the output variable $Y$ has discrete measurement associated with it, but it may come with additional component states and many unknown, continuous phase terms. This has implications for quantum amplitudes and, consequently, with the probabilities associated with the states. As an aside, equation (1) provides an explanation for the no-cloning theorem. A gate cannot clone a state since this would require the gate to supply information equal to that of the unknown state, which, by virtue of its being unknown, is impossible. As the variables $X$ and $Y$ (defined together with associated continuous phase terms) are continuous, the classical variable $C$ must also be continuous. The entropy associated with a continuous variable $Z$ is given by the expression: $$H (Z) = h(Z) - \lim_{\Delta z \rightarrow 0} log_2 \Delta z$$ where $h(Z)$ is the differential entropy: $$h(Z) = \int_{- \infty}^{\infty} f_Z (z) log_2 \left[\frac{1}{f_Z (z)}\right] dz$$ and $\Delta z$ is the precision associated with the variable. If the precision is the same at both input and output, the term $\lim_{\Delta z \rightarrow 0} log_2 \Delta z$ will cancel out and the differential entropies would be a proper measure of the entropy of $X$ and $Y$. In other words, $$H(C) = h(Y) - h(X).$$ The entropy associated with H(C) is the information lost in the computation process and it may be converted to heat according to thermodynamic laws \[12-14\]. If $H(C)$ is non-zero, error-free quantum computation is impossible, since this is associated with loss of information. Multiplication by constant {#multiplication-by-constant .unnumbered} -------------------------- [**Example 1.**]{} Consider a gate which multiples the inputs by a fixed constant $k > 1$. If the input $X$ is distributed uniformly over the interval $(0,a)$, then the output $Y$ is distributed uniformly over $(0,ka)$. The differential entropy values of the input and the output are: $$h(X) = log_2 a$$ $$h(Y) = log_2 ka$$ Assuming the same precision at input and output, the gate needs to supply entropy equal to $ H(C) = log_2 ka -log_2 a = log_2 k$, which become large as $k$ increases. This supply of entropy will have to be done in terms of interpolation or other processing which cannot be perfect. If $k < 1$, then the output entropy is smaller than input entropy and, therefore, $H(C)$ represents loss of information in the output. In effect, the assumption of fixed amplification of a variable with the same absolute precision at the output amounts to a nonlinear, irreversible process. For example, when a picture is compressed, one cannot obtain the original to the earlier precision by amplifying it back. In practical terms, the precision needed for the realization of a universal gate will be unattainable for a variety of reasons: one cannot have perfectly linear behavior in an electrical circuit over an unrestricted range. Unrestricted multiplication of a continuous variable is not implementable if the precision remains unchanged. In quantum computing, problems that somewhat parallel this above example are the implementation of rotation and [cnot]{} gates, two operators that are basic to the computation process \[15\]. The necessarily classical control of the gate is marred by random errors as well as calibration errors. Rotation {#rotation .unnumbered} -------- [**Example 2.**]{} Consider a quantum gate that rotates the input qubit by a fixed angle. Since the input $X$ and the output $Y$ will be distributed uniformly over the same interval $(0,a)$, the entropy associated with this gate will be $0$ (as per equation 4) as is required by the reversible nature of the assumed quantum evolution. But if the precision associated with the measurement and initialization processes at the input and the output is different, then lossless (or, equivalently, error-free) evolution cannot be assumed. [cnot]{} and Hadamard gates {#cnot-and-hadamard-gates .unnumbered} --------------------------- Consider the [cnot]{} gate together with a companion Hadamard gate. The errors in the device implementation of the [cnot]{} gate may make the gate effectively nonlinear and hence nonunitary. The matrix values that the device embodies may be different from the nominal ones below: $$\left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ \end{array} \right]$$ For simplicity, we consider a very straightforward situation which does not affect the [cnot]{} gate, but where its companion Hadamard gate is off the correct value, stuck in the state $$H_s = \left[ \begin{array}{cc} cos \theta & sin \theta \\ sin \theta & -cos \theta \\ \end{array} \right]$$ where $\theta \neq 45^o$. ### Stuck Hadamard gate before a [cnot]{} {#stuck-hadamard-gate-before-a-cnot .unnumbered} [**Example 3.**]{} Consider the arrangement of Figure 2, where the stuck gate $H_s$ ($\theta \neq \pi /4$, but its value is known) is to the left of the [cnot]{} gate; this circuit demonstrates that quantum processing can compute a global property of a function by a single measurement \[1\]. It will be seen that at the output of the [cnot]{} gate, the state is: $ \frac{1}{\sqrt 2}~(cos \theta |0\rangle - sin \theta |1\rangle ) ~ (|0\rangle - |1\rangle)$ The state $|a\rangle = (cos \theta |0\rangle - sin \theta |1\rangle )$, which is in error, may be passed through the gate $ \left[ \begin{array}{cc} 1 & 0 \\ 0 & -1 \\ \end{array} \right]$ followed by another $H_s$ to yield $|0\rangle$, which can be transformed to the correct $|a\rangle= \frac{1}{\sqrt 2} (|0\rangle - |1\rangle)$. In this example, the state $|b\rangle= \frac{1}{\sqrt 2} (|0\rangle - |1\rangle)$ was not affected by the stuck gate $H_s$. When the stuck gate is the lower Hadamard gate, as in Figure 3, the state at the output of the [cnot]{} gate is: $\frac{1}{\sqrt 2} (sin \theta |00\rangle - cos \theta |01\rangle + sin \theta |11\rangle - cos \theta |10\rangle )$ Corresponding to this we have the density function $\rho^{ab}$ given below: $\rho^{ab} = \frac{1}{2} \left[ \begin{array}{cccc} sin^2 \theta & -sin \theta cos \theta & -sin \theta cos \theta & sin^2 \theta \\ -sin \theta cos \theta & cos^2 \theta & \cos^2 \theta & -sin \theta cos \theta \\ -sin \theta cos \theta & \cos^2 \theta & \cos^2 \theta & -sin \theta cos \theta \\ sin^2 \theta & -sin \theta cos \theta & -sin \theta cos \theta & sin^2 \theta \\ \end{array} \right]$ It follows that the reduced density matrix for the state $|a\rangle$ is: $ \rho^a =\frac{1}{2} \left[ \begin{array}{cc} 1 & -sin 2\theta \\ -sin 2\theta & 1 \\ \end{array} \right]$ Therefore, when $\theta \neq \pi /4 $, $\rho^{ab}$ is a mixture, and we cannot perform any local correction to $|a \rangle$ to obtain the correct product state, for a unitary transformation on a mixture will keep it as a mixture. In other words, this error is not locally correctable. Stuck Hadamard gate in the teleportation protocol {#stuck-hadamard-gate-in-the-teleportation-protocol .unnumbered} ------------------------------------------------- [**Example 4.**]{} In the teleportation protocol, an unknown quantum state (of particle $X$) is teleported to a remote location using two entangled particles ($Y$ and $Z$) and classical information. Here, for convenience, we use the variant teleportation protocol \[16\] which requires only one classical bit in its classical information link (Figure 4). But instead of the Hadamard operator, we consider $H_s$ to be the rotation operator with angle $\theta$. We assume that the receiver has a copy of $H_s$ available for local processing, and we would like to estimate what would happen if this copy is not identical to the one used at the transmitting end. The state $X$ is $|\phi\rangle = \alpha |0\rangle + \beta |1\rangle $, where $\alpha$ and $\beta$ are unknown amplitudes, and $Y$ and $Z$ are in the pure entangled state $ \frac{1}{\sqrt 2} (| 00 \rangle~ +~ | 11\rangle )$. The initial state of the three particles is: $\frac{1}{\sqrt 2} (\alpha~ |000\rangle + \beta~ |100\rangle + \alpha~ |011\rangle + \beta~ |111\rangle $ The sequence of steps in Figure 4 is as follows: 1. Apply chained transformations: [cnot]{} on $X$ and $Y$, followed by [cnot]{} on $Y$ and $Z$. 2. Apply $H_s$ on the state of $X$. 3. Measure the state of $X$ and transfer information regarding it. 4. Apply appropriate operator $G$ to complete teleportation of the unknown state. A simple calculation will show that the state before the measurement is: $ \frac{1}{\sqrt 2} |0\rangle ( |0\rangle + |1\rangle ) (\alpha~cos \theta~ |0\rangle~ +~ \beta~sin \theta~ |1\rangle ) +\frac{1}{\sqrt 2} |1\rangle ( |0\rangle~ +~ |1\rangle ) (\alpha~sin \theta~ |0\rangle~ -~ \beta~cos \theta~ |1\rangle )$ Therefore, after the measurement, we get either $ X^+ = \alpha~cos \theta~ |0\rangle + \beta~sin \theta~ |1\rangle$ or $X^- = \alpha~sin \theta~ |0\rangle - \beta~cos \theta~ |1\rangle$ based on whether the measurement was $0$ or $1$. Assuming that the value of $\theta$ is also communicated to it, the receiver can recover the unknown $X$ probabilistically; when the value of $\theta$ is $45^o$, then the inversion is trivially simple. For simplicity, assume that the receiver needs to invert $X^+ $. He will replicate Figure 4 at his end which means that the Hadamard gate that he would use would have identical characteristics (the same precision) to the one used during the earlier operation. He would now obtain either $ X^{++} = \alpha~cos^2 \theta~ |0\rangle + \beta~sin^2 \theta~ |1\rangle$ or $ X^{+-} = \alpha~ |0\rangle + \beta~ |1\rangle$ Similarly, $X^-$ will, in the next iteration, lead to: $ X^{-+} = \alpha~ |0\rangle + \beta~ |1\rangle$ or $ X^{--} = \alpha~sin^2 \theta~ |0\rangle + \beta~cos^2 \theta~ |1\rangle$ This procedure may be extended, and the probability of recovering the unknown state $X$ can be shown to be given by the tree diagram of Figure 5. In the first pass, there is a fifty percent probability of getting the correct state, and this probability reduces in further passes (Figure 5). The probability of recovering the state $X$ is thus: $$\frac{1}{2} + \frac{1}{2}\frac{1}{4} + \frac{1}{2}\frac{3}{4}\frac{1}{6} + \frac{1}{2}\frac{3}{4}\frac{5}{6} \frac{1}{8} +\frac{1}{2}\frac{3}{4}\frac{5}{6} \frac{7}{8} \frac{1}{10} + ...$$ The ability of the receiver to implement the needed transformation will depend on the precision available in its gate control mechanism. If the value of $\theta$ at the sending point is smaller than the precision available to the receiver, then the state $X$ cannot be recovered. It is interesting that as long as the receiver possesses a rotation operator $H_s$ that is identical to the one used at the sending point, there is no need to know the value of $\theta$ and still obtain the unknown state $X$ probabilistically, as in expression (9). Conclusion {#conclusion .unnumbered} ---------- We have considered the problem of gate complexity in quantum systems. The control of the gate – a physical device – is by modifying some classical variable, which is subject to error. Since one cannot assume infinite precision in any control system, the implications of varying accuracy emongst different gates becomes an important problem. We have shown that in certain arrangements a stuck fault cannot be reversed down the circuit stream using a single qubit operator, for it converted a pure state into a mixed state. We considered the case of the teleportation circuit with the rotation gate stuck at $\theta$. When $\theta = 0^o$, the state $X$ collapses to $0$ or $1$. When $\theta \neq 0^o$ or $ 90^o$, one may obtain the unknown state back probabilistically by passing $X^+$ or $X^-$ back through the circuit of Figure 4 iteratively. Consider two parties, $A$ and $B$, who are both presented with the state $X^+$ or $X^-$. 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--- abstract: | Time dependent entropy of constant force motion is investigated. Their joint entropy so called Leipnik’s entropy is obtained. The main purpose of this work is to calculate Leipnik’s entropy by using time dependent wave function which is obtained by the Feynman path integral method. It is found that, in this case, the Leipnik’s entropy increase with time and this result has same behavior free particle case. Keywords: Path integral, joint entropy, constant force motion. author: - Özgür ÖZCAN - Ethem AKTÜRK - Ramazan SEVER title: Time Dependent Entropy of Constant Force Motion --- Introduction ============ The information entropy plays a major role in a stronger formulation of the uncertainty relations [@ekrem]. This relation may be mathematically defined by using the Boltzmann-Shannon information entropy and the von Neumann entropy. In the literature for both open and closed quantum systems, the different information-theoretical entropy measures have been discussed [@Zurek; @Omnes; @Anastopoulos]. In contrast, the joint entropy [@Leipnik; @Dodonov] can also be used to properties the loss of information, related to evolving pure quantum states [@Trigger]. The joint entropy of the physical systems were conjectured by Dunkel and Trigger [@Dunkel] in which their systems named MACS (maximal classical states). The Leibnik entropy of the simple harmonic oscillator was determined not monotonically increase with time [@Garbaczewski]. In this work, we give a uniform description of the complete joint entropy information of system in motion under a constant force. This paper is organized as follows. In section II, we explain fundamental definitions needed for the calculation. In section III, we deal with calculation and results for constant force systems. Moreover, we obtain the analytical solution of Kernel, wave function in both coordinate and momentum space and its joint entropy. We also obtain same quantities for constant magnetic field case. Finally, we present the conclusion in section IV. Fundamental Definitions ======================= We consider a classical system with $d=sN$ degrees of freedom, where N is the particle number and s is number of spatial dimensions [@Dunkel]. Apart from this, let us describe $g(x,p,t)=g(x_1,...,x_n,p_1,...,p_d,t)$ which is non-negative, time dependent phase space density function of system. The density function is assuming to be normalized to unity, $$\int dx dp g(x,p,t)=1.$$ The Gibbs-Shannon entropy is described by $$S(t)=-\frac{1}{N!}\int dx dp g(x,p,t)ln(h^{d} g(x,p,t)),$$ where $h=2\pi\hbar$ is the Planck constant. Schrödinger wave equation with the Born interpretation [@Born] is given by $$i\hbar\frac{\partial\psi}{\partial t}=\hat{H}\psi.$$ The quantum probability densities are defined in position and momentum spaces as $|\psi(x,t)|^2$ and $|\tilde{\psi}(p,t)|^2$, where $|\tilde{\psi}(p,t)|^2$ is given as $$\tilde{\psi}(p,t)=\int\frac{dx e^{-ipx/\hbar}}{(2\pi\hbar)^{d/2}}\psi(x,t).$$ Leipnik proposed the product function as [@Dunkel] $$g_{j}(x,p,t)=|\psi(x,t)|^2|\tilde{\psi}(p,t)|^2\geq0.$$ Substituting Eq. (5) into Eq. (2), we get the joint entropy $S_{j}(t)$ for the pure state $\psi(x,t)$ or equivalently can be written in the following form  [@Dunkel] $$\begin{aligned} S_{j}(t)&=&-\int dx |\psi(x,t)|^{2}\ln|\psi(x,t)|^{2}- \int dp |\tilde{\psi}(p,t)|^2 \ln |\tilde{\psi}(p,t)|^2-\nonumber\\&-&\ln h^{d}.\end{aligned}$$ We find time dependent wave function by means of the Feynman path integral which has form [@Feynman] $$\begin{aligned} K(x'',t'';x',t')&=&\int^{x''=x(t'')}_{x'=x(t')}Dx(t)e^{\frac{i}{\hbar}S[x(t)]} \nonumber\\&=&\int^{x''}_{x'}Dx(t)e^{\frac{i}{\hbar}\int_{t'}^{t''}L[x,\dot{x},t]dt}.\end{aligned}$$ The Feynman kernel can be related to the time dependent Schrödinger’s wave function $$\begin{aligned} K(x'',t'';x',t')=\sum_{n=0}^{\infty}\psi_{n}^{*}(x',t')\psi_{n}(x'',t'').\end{aligned}$$ The propagator in semiclassical approximation reads $$\begin{aligned} K(x'',t'';x',t')=\Big[\frac{i}{2\pi\hbar}\frac{\partial^2}{\partial x'\partial x''}S_{cl}(x'',t'';x',t')\Big]^{1/2}e^{\frac{i}{\hbar}S_{cl}(x'',t'';x',t')}.\end{aligned}$$ The prefactor is often referred to as the Van Vleck-Pauli-Morette determinant  [@Khandekar; @Kleinert]. The $F(x'',t'';x',t')$ is given by $$\begin{aligned} F(x'',t'';x',t')=\Big[\frac{i}{2\pi\hbar}\frac{\partial^2}{\partial x'\partial x''}S_{cl}(x'',t'';x',t')\Big]^{1/2}.\end{aligned}$$ CALCULATION AND RESULTS ======================= [**Constant Force**]{} ---------------------- The Lagrangian for present case is $$L(x,\dot{x},t)=\frac{1}{2}m\dot{x}^2+fx$$ The classical path obeys $$m\ddot{x}_{cl}=f$$ The solution of above equation is $$x_{cl}(\tau)=x_{0}+\Big(\frac{x-x_{0}}{t-t_{0}}-\frac{1}{2}\frac{f}{m}(t-t_{0})\Big)\tau+\frac{1}{2}\frac{f}{m}\tau^2$$ One obtains for classical action integral along the classical path [@Feynman] $$\begin{aligned} S(x_{cl}(\tau))=\frac{1}{2}m\frac{(x-x_{0})^2}{t-t_{0}}+\frac{1}{2}(x+x_{0})f(t-t_{0})-\frac{1}{24}\frac{f^2}{m}(t-t_0)^3\end{aligned}$$ and finally, for the kernel $$\begin{aligned} K(x'',x';T)&=&\Big[\frac{m}{2\pi i\hbar T}\Big]^{1/2} \exp\Big[\frac{im}{2\hbar}\frac{(x-x_{0})^2}{T}+\frac{i}{2\hbar}(x+x_{0})fT-\nonumber\\&-&\frac{i}{24\hbar}\frac{f^2}{m}T^3\Big]\end{aligned}$$ The dependent wave function at time $t>0$ $$\begin{aligned} \Psi(x,t)&=&\Big[\frac{1-i\frac{\hbar t}{m\sigma^2}}{1+i\frac{\hbar t}{m\sigma^2}}\Big]^{1/4}\Big[\frac{1}{\pi\sigma^{2}(1+\frac{\hbar^{2} t^{2}}{m^{2}\sigma^4})}\Big]^{1/4}\exp\Big[-\frac{(x-\frac{p_{0}}{m}t-\frac{ft^2}{2m})^2}{2\sigma^2(1+\frac{\hbar^{2} t^{2}}{m^{2}\sigma^4})}\times\nonumber\\&\times&(1-i\frac{\hbar t}{m\sigma^2})\Big] \exp\Big[\frac{i}{\hbar}(p_{0}+ft)x-\frac{i}{\hbar}\int^{t}_{0}d\tau\frac{(p_{0}+f\tau)^2}{2m}\Big]\end{aligned}$$ where $\sigma$ is width of Gaussian curve. The corresponding probability distribution is $$\begin{aligned} |\Psi(x,t)|^2=\Big[\frac{1}{\pi\sigma^{2}(1+\frac{\hbar^{2} t^{2}}{m^{2}\sigma^4})}\Big]^{1/2}\exp\Big[-\frac{(x-\frac{p_{0}t}{m}-\frac{ft^2}{2m})^2}{\sigma^2(1+\frac{\hbar^{2} t^{2}}{m^{2}\sigma^4})}\Big]\end{aligned}$$ or $$\begin{aligned} |\Psi(x,t)|^2=\Big[\frac{1}{\pi\sigma^{2}(1+\frac{\hbar^{2} t^{2}}{m^{2}\sigma^4})}\Big]^{1/2}\exp\Big[-\frac{(x-\frac{p_{0}t}{m}-\frac{ft^2}{2m})^2}{\sigma^2(1+\frac{\hbar^{2} t^{2}}{m^{2}\sigma^4})}\Big]\end{aligned}$$ The probability density in coordinate space is shown Fig.\[eps1\]. The probability density in momentum space can be written easily $$\begin{aligned} |\Psi(p,t)|^2=\Big[\frac{\sigma^{2}}{\pi \hbar^{2}}\Big]^{1/2}\exp\Big[\frac{-\sigma^{2}}{\hbar^{2}}(p+(p_{0}+f t))^{2}\Big]\end{aligned}$$ The time dependent joint entropy can be obtained from Eq. [6]{} as $$S_{j}(t)=\ln(\frac{e}{2})\sqrt{1+\frac{\hbar^{2}t^{2}}{m^{2}\sigma^{4}}}$$ The joint entropy of this system is shown Fig.\[eps2\] and Fig.\[eps3\]. It is important that Eq. [20]{} is in agreement with following general inequality for the joint entropy: $$S_{j}(t)\geq\ln(\frac{e}{2})$$ originally derived by Leipnik for arbitrary one-dimensional one-particle wave functions. Conclusion ========== We have investigated the joint entropy for constant for motion. We have obtained the time dependent wave function by means of Feynman Path integral technique. In this case, we have found that the joint entropy increase with time and the results harmony prior studied. The joint entropy has same behavior as free particle case. This result indicates that the information entropy is getting increase with time. Acknowledgements ================ This research was partially supported by the Scientific and Technological Research Council of Turkey. [99]{} E. Aydiner, C. Orta and R.Sever, E-print:quant-ph/0602203 N. J. Cerf and C. Adami, Phys. Rev. Lett. **79**, 5194–5197 (1997). W.H. Zurek, Phys. Today **44**(10), 36 (1991). R. Omnes, Rev. Mod. Phys. **64**, 339 (1992). C. Anastopoulos, Ann. Phys. **303**, 275 (2003). R. Leipnik, Inf. Control. **2**, 64 (1959). V.V.Dodonov, J.Opt. B: Quantum Semiclassical Opt. **4**, S98 (2002). S. A. Trigger, Bull. Lebedev Phys. Inst. **9**, 44 (2004). J. Dunkel and S. A. Trigger, Phys. Rev.A**71**, 052102 (2005). P. Garbaczewski, Phys. Rev. A **72**, 056101 (2005). M. Born, Z. Phys. **40**, 167 (1926). R.P. Feynmann, A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, USA (1965). D.C. Khandekar, S.V. Lawande, K.V. Bhagwat, Path-Integral Methods and Their Applications, World Scientific, Singapore (1993). H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, World Scientific, 3rd Edition (2004).

--- abstract: 'We study the space of generalized translation invariant valuations on a finite-dimensional vector space and construct a partial convolution which extends the convolution of smooth translation invariant valuations. Our main theorem is that McMullen’s polytope algebra is a subalgebra of the (partial) convolution algebra of generalized translation invariant valuations. More precisely, we show that the polytope algebra embeds injectively into the space of generalized translation invariant valuations and that for polytopes in general position, the convolution is defined and corresponds to the product in the polytope algebra.' address: - 'Institut für Mathematik, Goethe-Universität Frankfurt, Robert-Mayer-Str. 10, 60054 Frankfurt, Germany' - 'Institut des Hautes Études Scientifiques, 35 Route de Chartres, 91440 Bures-sur-Yvette, France' author: - Andreas Bernig - Dmitry Faifman title: Generalized translation invariant valuations and the polytope algebra --- Introduction ============ Let $V$ be an $n$-dimensional vector space, $V^*$ the dual vector space, $\mathcal{K}(V)$ the set of non-empty compact convex subsets in $V$, endowed with the topology induced by the Hausdorff metric for an arbitrary Euclidean structure on $V$, and $\mathcal{P}(V)$ the set of polytopes in $V$. A valuation is a map $\mu:\mathcal{K}(V) \to \mathbb{C}$ such that $$\mu(K \cup L)+\mu(K \cap L)=\mu(K)+\mu(L)$$ whenever $K,L, K \cup L \in \mathcal{K}(V)$. Continuity of valuations will be with respect to the Hausdorff topology. Examples of valuations are measures, the intrinsic volumes (in particular the Euler characteristic $\chi$) and mixed volumes. Let $\operatorname{Val}(V)$ denote the (Banach-)space of continuous, translation invariant valuations. It was the object of intensive research during the last few years, compare [@alesker_mcullenconj01; @alesker_fourier; @alesker_bernig_schuster; @alesker_faifman; @bernig_aig10; @bernig_broecker07; @bernig_fu06; @bernig_fu_hig; @bernig_hug; @fu_barcelona] and the references therein. Valuations with values in semi-groups other than $\mathbb{C}$ have also attracted a lot of interest. We only mention the recent papers [@abardia12; @abardia_bernig; @bernig_fu_solanes; @haberl10; @hug_schneider_localtensor; @schneider13; @schuster10; @schuster_wannerer; @wannerer_area_measures; @wannerer_unitary_module] to give a flavor on this active research area. Of particular importance is the class of the so-called smooth valuations. The importance of this class stems from the fact that it admits various algebraic structures, which include two bilinear pairings, known as product and convolution, and a Fourier-type duality interchanging them. These algebraic structures are closely related to important notions from convex and integral geometry, such as the Minkowski sum, mixed volumes, and kinematic formulas. This emerging new theory is known as algebraic integral geometry [@bernig_aig10; @fu_barcelona]. A different, more classical type of algebraic object playing an important role in convex geometry is McMullen’s algebra of polytopes. In this paper, we show how McMullen’s algebra fits into the framework of algebraic integral geometry. More precisely, we show that McMullen’s algebra can be embedded as a subalgebra of the space of generalized valuations, which is, roughly speaking, the dual space of smooth valuations. Let us now give the necessary background required to state our main theorems. The group $\operatorname{GL}(V)$ acts in the natural way on $\operatorname{Val}(V)$. The dense subspace of $\operatorname{GL}(V)$-smooth vectors in $\operatorname{Val}(V)$ is denoted by $\operatorname{Val}^\infty(V)$. It carries a Fréchet topology which is finer than the induced topology. In [@bernig_fu06], a convolution product on $\operatorname{Val}^\infty(V) \otimes \operatorname{Dens}(V^*)$ was constructed. Here and in the following, $\operatorname{Dens}(W)$ denotes the $1$-dimensional space of densities on a linear space $W$. Note that $\operatorname{Dens}(V) \otimes \operatorname{Dens}(V^*) \cong \mathbb{C}$: if $\operatorname{vol}$ is any choice of Lebesgue measure on $V$, and $\operatorname{vol}^*$ the corresponding dual measure on $V^*$, then $\operatorname{vol}\otimes \operatorname{vol}^* \in \operatorname{Dens}(V) \otimes \operatorname{Dens}(V^*)$ is independent of the choice of $\operatorname{vol}$. If $\phi_i(K)=\operatorname{vol}(K+A_i) \otimes \operatorname{vol}^*$ with smooth compact strictly convex bodies $A_1,A_2$, then $\phi_1 * \phi_2(K)=\operatorname{vol}(K+A_1+A_2) \otimes \operatorname{vol}^*$. By Alesker’s proof [@alesker_mcullenconj01] of McMullen’s conjecture, linear combinations of such valuations are dense in the space of all smooth valuations. The convolution extends by bilinearity and continuity to $\operatorname{Val}^\infty(V) \otimes \operatorname{Dens}(V^*)$. By [@bernig_fu06], the convolution product is closely related to additive kinematic formulas. It was recently used in the study of unitary kinematic formulas [@bernig_fu_hig], local unitary additive kinematic formulas [@wannerer_area_measures; @wannerer_unitary_module] and kinematic formulas for tensor valuations [@bernig_hug]. In this paper, we will extend the convolution to a (partially defined) convolution on the space of generalized translation invariant valuations. Elements of the space $$\operatorname{Val}^{-\infty}(V):=\operatorname{Val}^\infty(V)^* \otimes \operatorname{Dens}(V)$$ are called generalized translation invariant valuations. By the Alesker-Poincaré duality [@alesker04_product], $\operatorname{Val}^\infty(V)$ embeds in $\operatorname{Val}^{-\infty}(V)$ as a dense subspace. More generally, it follows from [@alesker_fourier Proposition 8.1.2] that $\operatorname{Val}(V)$ embeds in $\operatorname{Val}^{-\infty}(V)$, hence we have the inclusions $$\operatorname{Val}^\infty(V) \subset \operatorname{Val}(V) \subset \operatorname{Val}^{-\infty}(V).$$ Generalized translation invariant valuations were introduced and studied in the recent paper [@alesker_faifman]. Note that another notion of generalized valuation was introduced by Alesker in [@alesker_val_man1; @alesker_val_man2; @alesker_val_man4; @alesker_val_man3]. In the next section, we will construct a natural isomorphism between the space of translation invariant generalized valuations in Alesker’s sense and the space of generalized translation invariant valuations in the sense of the above definition. Given a polytope $P$ in $V$, there is an element $M(P) \in \operatorname{Val}^{-\infty}(V) \otimes \operatorname{Dens}(V^*) \cong \operatorname{Val}^\infty(V)^*$ defined by $$\langle M(P),\phi\rangle=\phi(P).$$ Let $\Pi(V)$ be McMullen’s polytope algebra [@mcmullen_polytope_algebra]. As a vector space, $\Pi(V)$ is generated by all symbols $[P]$, where $P$ is a polytope in $V$, modulo the relations $[P] \equiv [P+v], v \in V$, and $[P \cup Q]+[P \cap Q]=[P] + [Q]$ whenever $P,Q, P \cup Q$ are polytopes in $V$. The product is defined by $[P] \cdot [Q]:=[P+Q]$. The map $M:\mathcal{P}(V) \to \operatorname{Val}^{-\infty}(V) \otimes \operatorname{Dens}(V^*)$ extends to a linear map $$M: \Pi(V) \to \operatorname{Val}^{-\infty}(V) \otimes \operatorname{Dens}(V^*).$$ Our first main theorem shows that McMullen’s polytope algebra is a subset of $\operatorname{Val}^{-\infty}(V)\otimes \operatorname{Dens}(V^*)$. \[mainthm\_injection\_mcmullen\] The map $M:\Pi(V)\to \operatorname{Val}^{-\infty}(V)\otimes \operatorname{Dens}(V^*)$ is injective. Equivalently, the elements of $\operatorname{Val}^\infty(V)$ separate the elements of $\Pi(V)$. In Section \[sec\_partial\_conv\] we will introduce a notion of transversality of generalized translation invariant valuations. Our second main theorem is the following. \[mainthm\_convolution\] There exists a partial convolution product $*$ on $\operatorname{Val}^{-\infty}(V) \otimes \operatorname{Dens}(V^*)$ with the following properties: 1. If $\phi_1,\phi_2 \in \operatorname{Val}^{-\infty}(V) \otimes \operatorname{Dens}(V^*)$ are transversal, then $\phi_1 * \phi_2 \in \operatorname{Val}^{-\infty}(V) \otimes \operatorname{Dens}(V^*)$ is defined. 2. If $\phi_1 * \phi_2$ is defined and $g \in \operatorname{GL}(V)$, then $(g_* \phi_1) * (g_* \phi_2)$ is defined and equals $g_*(\phi_1 * \phi_2)$. 3. Whenever the convolution is defined, it is bilinear, commutative, associative and of degree $-n$. 4. The restriction to the subspace $\operatorname{Val}^\infty(V) \otimes \operatorname{Dens}(V^*)$ is the convolution product from [@bernig_fu06]. 5. If $x,y$ are elements in $\Pi(V)$ in general position, then $M(x),M(y)$ are transversal in $\operatorname{Val}^{-\infty}(V) \otimes \operatorname{Dens}(V^*)$ and $$M(x \cdot y)=M(x) * M(y).$$ Stated otherwise, the maps in the diagram $$\xymatrix{\operatorname{Val}^\infty(V) \otimes \operatorname{Dens}(V^*) \ar@{^{(}->}[r] & \operatorname{Val}^{-\infty}(V) \otimes \operatorname{Dens}(V^*) & \Pi(V) \ar@{_{(}->}[l]}$$ have dense images and are compatible with the (partial) algebra structures. [**Remarks:**]{} 1. In Prop. \[prop\_continuity\] we will show that, under some technical conditions in terms of wave fronts, the convolution on generalized translation invariant valuations from Theorem \[mainthm\_convolution\] is the unique jointly sequentially continuous extension of the convolution product on smooth translation invariant valuations. 2. In [@alesker_bernig], it was shown that the space $\mathcal{V}^{-\infty}(X)$ of generalized valuations on a smooth manifold $X$ admits a partial product structure extending the Alesker product of smooth valuations on $X$. If $X$ is real-analytic, then the space $\mathcal{F}_\mathbb{C}(X)$ of $\mathbb{C}$-valued constructible functions on $X$ embeds densely into $\mathcal{V}^{-\infty}(X)$. It was conjectured that whenever two constructible functions meet transversally, then the product in the sense of generalized valuations exists and equals the generalized valuation corresponding to the product of the two functions. The relevant diagram in this case is $$\xymatrix{\operatorname{Val}^\infty(X) \ar@{^{(}->}[r] & \operatorname{Val}^{-\infty}(X) & \mathcal{F}_\mathbb{C}(X), \ar@{_{(}->}[l]}$$ where both maps are injections with dense images and are (conjecturally) compatible with the partial product structure. Theorem 5 in [@alesker_bernig] gives strong support for this conjecture. 3. The Alesker-Fourier transform from [@alesker_fourier] extends to an isomorphism $\mathbb{F}:\operatorname{Val}^{-\infty}(V^*) \to \operatorname{Val}^{-\infty}(V) \otimes \operatorname{Dens}(V^*)$, compare [@alesker_faifman]. Another natural partially defined convolution on $\operatorname{Val}^{-\infty}(V) \otimes \operatorname{Dens}(V^*)$ would be $$\psi_1 * \psi_2:=\mathbb{F} \left(\mathbb{F}^{-1}\psi_1 \cdot \mathbb{F}^{-1} \psi_2\right),$$ where the dot is the partially defined product on $\operatorname{Val}^{-\infty}(V^*)$ from [@alesker_bernig]. It seems natural to expect that this convolution coincides with the one from Theorem \[mainthm\_convolution\], but we do not have a proof of this fact. Plan of the paper {#plan-of-the-paper .unnumbered} ----------------- In the next section, we introduce and study the space of generalized translation invariant valuations and explore its relation to generalized valuations from Alesker’s theory. In Section \[sec\_embedding\_polytopes\] we show that McMullen’s polytope algebra embeds into the space of generalized translation invariant valuations. A partial convolution structure on this space is constructed in Section \[sec\_partial\_conv\]. In Section \[sec:Appendix\] we construct a certain current on the sphere which is related to the volumes of spherical joins, and can be viewed as a generalization of the Gauss area formula for the plane. This current also plays a major role in the proof of the second main theorem. Its construction is based on geometric measure theory and is independent of the rest of the paper. Finally, Section \[sec\_compatibility\] is devoted to the proof of the fact that the embedding of the polytope algebra is compatible with the two convolution (product) structures. Acknowledgements {#acknowledgements .unnumbered} ---------------- We wish to thank Semyon Alesker for multiple fruitful discussions and Thomas Wannerer for useful remarks on a first draft of this paper. Preliminaries {#sec_prels} ============= In this section $X$ will be an oriented $n$-dimensional smooth manifold and $S^*X$ the cosphere bundle over $X$. It consists of all pairs $(x,[\xi])$ where $x \in X, \xi \in T^*_xX, \xi \neq 0$ and where the equivalence relation is defined by $[\xi]=[\tau]$ if and only if $\xi=\lambda \tau$ for some $\lambda>0$. The projection onto $X$ is denoted by $\pi:S^*X \to X, (x,\xi) \mapsto x$. The antipodal map $s:S^*X \to S^*X$ is defined by $(x,[\xi]) \mapsto (x,[-\xi])$. The push-forward map (also called fiber integration) $\pi_*:\Omega^k(S^*X) \to \Omega^{k-(n-1)}(X)$ satisfies $$\int_{S^*X} \pi^*\gamma \wedge \omega=\int_X \gamma \wedge \pi_*\omega, \quad \gamma \in \Omega_c^{2n-k-1}(X).$$ If $V$ is a vector space, then $S^*V \cong V \times \mathbb P_+(V^*)$, where $\mathbb P_+(V^*):=V^* \setminus \{0\}/ {\mathbb{R}}_+$ is the sphere in $V^*$. Moreover, if $V$ is a Euclidean vector space of dimension $n$, we identify $S^*V$ and $SV=V \times S^{n-1}$ and write $\pi=\pi_1:SV \to V, \pi_2:SV \to S^{n-1}$ for the two projections. Currents -------- Let us recall some terminology from geometric measure theory. We refer to [@federer_book; @morgan_book] for more information. The space of $k$-forms on $X$ is denoted by $\Omega^k(X)$, the space of compactly supported $k$-forms is denoted by $\Omega^k_c(X)$. Elements of the dual space $\mathcal{D}_k(X):=\Omega^k_c(X)^*$ are called [*$k$-currents*]{}. A $0$-current is also called [*distribution*]{}. The boundary of a $k$-current $T \in \mathcal{D}_k(X)$ is defined by $\langle \partial T,\phi\rangle=\langle T,d\phi\rangle, \phi \in \Omega^{k-1}_c(X)$. If $\partial T=0$, $T$ is called a [*cycle*]{}. If $T \in \mathcal{D}_k(X)$ and $\omega \in \Omega^l(X), l \leq k$, then the current $T \llcorner \omega \in \mathcal{D}_{k-l}(X)$ is defined by $\langle T \llcorner \omega,\phi\rangle:=\langle T, \omega \wedge \phi\rangle$. If $f:X \to Y$ is a smooth map between smooth manifolds $X,Y$ and $T \in \mathcal{D}_k(X)$ such that $f|_{\operatorname{spt}T}$ is proper, then the push-forward $f_*T \in \mathcal{D}_k(Y)$ is defined by $\langle f_*T,\phi\rangle:=\langle T,\zeta f^*\phi\rangle$, where $\zeta \in C^\infty_c(X)$ is equal to $1$ in a neighborhood of $\operatorname{spt}T \cap \operatorname{spt}f^*\omega$. It is easily checked that $$\label{eq_boundary_vs_differential} \partial ([[X]] \llcorner \omega)=(-1)^{\deg\omega+1} [[X]] \llcorner (d\omega)$$ and $$\label{eq_push_forward_forms_currents} \pi_*\left([[S^*X]] \llcorner \omega\right)=(-1)^{(n+1)(\deg\omega+1)} [[X]] \llcorner \pi_*\omega.$$ Every oriented submanifold $Y \subset X$ of dimension $k$ induces a $k$-current $[[Y]]$ such that $\langle [[Y]],\phi\rangle=\int_Y \phi$. By Stokes’ theorem, $\partial [[Y]]=[[\partial Y]]$. A [*smooth current*]{} is a current of the form $[[X]] \llcorner \omega \in \mathcal{D}_{n-k}(X)$ with $\omega \in \Omega^k(X)$. If $X$ and $Y$ are smooth manifolds, $T \in \mathcal{D}_k(X), S \in \mathcal{D}_l(Y)$, then there is a unique current $T \times S \in \mathcal{D}_{k+l}(X \times Y)$ such that $\langle T \times S, \pi_1^*\omega \wedge \pi_2^*\phi\rangle=\langle T,\omega\rangle \cdot \langle S,\phi\rangle$, for all $\omega \in \Omega^k(X), \phi \in \Omega^l(Y)$. Here $\pi_1,\pi_2$ are the projections from $X \times Y$ to $X$ and $Y$ respectively. If $T=[[X]] \llcorner \omega, S=[[Y]] \llcorner \phi$, then $$\label{eq_product_smooth_currents} T \times S=(-1)^{(\dim X-\deg \omega)\deg \phi}[[X \times Y]] \llcorner (\omega \wedge \phi).$$ The boundary of the product is given by $$\label{eq_boundary_product} \partial(T \times S)=\partial T \times S+(-1)^k T \times \partial S,$$ compare [@federer_book 4.1.8]. If $X$ is a Riemannian manifold, the [*mass*]{} of a current $T \in \mathcal{D}_k(X)$ is $$\mathbf{M}(T):=\sup \{\langle T,\phi\rangle: \phi \in \Omega^k_c(X), \|\phi(x)\|^* \leq 1, \forall x \in X\},$$ where $\|\cdot\|^*$ denotes the comass norm. Currents of finite mass having a boundary of finite mass are called [*normal currents*]{}. The flat norm of $T$ is defined by $$\mathbf{F}(T):=\sup\{\langle T,\phi\rangle: \phi \in \Omega^k_c(X), \|\phi(x)\|^* \leq 1, \|d\phi(x)\|^* \leq 1, \forall x \in X\}.$$ If $X$ is compact, then the $\mathbf{F}$-closure of the space of normal $k$-currents is the space of [*real flat chains*]{}. Wave fronts ----------- We refer to [@guillemin_sternberg77] and [@hoermander_pde1] for the general theory of wave fronts and its applications. For the reader’s convenience and later reference, we will recall some basic definitions and some fundamental properties of wave fronts, following [@hoermander_pde1]. First let $X$ be a linear space of dimension $n$, $T$ a distribution on $X$. The cone $\Sigma(T) \subset X^*$ is defined as the closure of the complement of the set of all $\eta \in X^*$ such that for all $\xi$ in a conic neighborhood of $\eta$ we have $$\|\hat T(\xi)\| \leq C_N(1+\|\xi\|)^{-N}, \quad N \in \mathbb{N}$$ (with constants $C_N$ only dependent on $N$ and the chosen neighborhood). Here $\hat T$ denotes the Fourier transform of $T$, and the norm is taken with respect to an arbitrary scalar product on $X^*$. Next, for an affine space $X$ and a point $x \in X$, the set $\Sigma_x(T) \subset T_x^* X$ is defined by $$\Sigma_x(T):=\bigcap_\phi \Sigma(\phi T),$$ where $\phi$ ranges over all compactly supported smooth functions on $X$ with $\phi(x) \neq 0$. Note that one uses the canonic identification $T^*_xX=X^*$. The [*wave front*]{} set of $T$ is by definition $$\operatorname{WF}(T):=\{(x,[\xi]) \in S^*X: \xi \in \Sigma_x(T)\}.$$ The set $\operatorname{singsupp}(T):=\pi(\operatorname{WF}(T)) \subset X$ is called [*singular support*]{}. Let $(x_1,\ldots,x_n)$ be coordinates on $X$. Given a current $T \in \mathcal{D}_k(X)$, we may write $$T=\sum_{\substack{I=(i_1,\ldots,i_k)\\1 \leq i_1<i_2<\cdots<i_k \leq n}} T_I \frac{\partial}{\partial x_{i_1}} \wedge \ldots \wedge \frac{\partial}{\partial x_{i_k}}$$ with distributions $T_I$. Then the wave front of $T$ is defined as $\operatorname{WF}(T):=\bigcup_I \operatorname{WF}(T_I)$. A current $T$ is smooth, i.e. given by integration against a smooth differential form, if and only if $\operatorname{WF}(T)=\emptyset$. \[def\_dkgamma\] Let $\Gamma \subset S^*X$ be a closed set. Then we set $$\mathcal{D}_{k,\Gamma}(X):=\{T \in \mathcal{D}_k(X): \operatorname{WF}(T) \subset \Gamma\}.$$ A sequence $T_j \in \mathcal{D}_{k,\Gamma}(X)$ converges to $T \in \mathcal{D}_{k,\Gamma}(X)$ if (writing $T_j, T$ as above) $T_{j,I} \to T_I$ weakly in the sense of distributions and for each compactly supported function $\phi \in C^\infty_c(X)$ and each closed cone $A$ in $X^*$ such that $\Gamma \cap (\operatorname{spt}\phi \times A)=\emptyset$ we have $$\sup_{\xi \in A} |\xi|^N \left|\widehat{\phi T_{I}}(\xi)-\widehat{\phi T_{j,I}}(\xi)\right| \to 0, \quad j \to \infty$$ for all $N \in \mathbb{N}$. \[prop\_approx\_smooth\] Let $T \in \mathcal{D}_{k,\Gamma}(X)$. Then there exists a sequence of compactly supported smooth $k$-forms $\omega_i \in \Omega^{n-k}(X)$ such that $[[X]] \llcorner \omega_i \to T$ in $\mathcal{D}_{k,\Gamma}(X)$. In other words, smooth forms are dense in $\mathcal{D}_{k,\Gamma}(X)$. \[prop\_intersection\_current\] Let $T_1 \in \mathcal{D}_{k_1}(X), T_2 \in \mathcal{D}_{k_2}(X)$ such that the following transversality condition is satisfied: $$\operatorname{WF}(T_1) \cap s \operatorname{WF}(T_2) = \emptyset.$$ Then the intersection current $T_1 \cap T_2 \in \mathcal{D}_{k_1+k_2-n}(X)$ is well-defined. More precisely, if $[[X]] \llcorner \omega_i^j \to T_j$ in $\mathcal{D}_{k_j,\operatorname{WF}(T_j)}(X)$ with $\omega_i^j \in \Omega^{n-k_j}(X)$, $j=1,2$, then $[[X]] \llcorner (\omega_i^1 \wedge \omega_i^2) \to T_1 \cap T_2$ in $\mathcal{D}_{k_1+k_2-n,\Gamma}(X)$, where $$\Gamma:=\operatorname{WF}(T_1) \cup \operatorname{WF}(T_2) \cup \left\{(x,[\xi_1+\xi_2]):(x,[\xi_1]) \in \operatorname{WF}(T_1), (x,[\xi_2]) \in \operatorname{WF}(T_2)\right\}.$$ The boundary of the intersection is given by $$\label{eq_boundary_intersection} \partial (T_1 \cap T_2)=(-1)^{n-k_2} \partial T_1 \cap T_2+T_1 \cap \partial T_2.$$ The wave front of a distribution is defined locally and behaves well under coordinate changes. Using local coordinates, one can define the wave front set $\operatorname{WF}(T) \subset S^*X$ for a distribution $T$ on a smooth manifold $X$. Definition \[def\_dkgamma\] and Propositions \[prop\_approx\_smooth\] and \[prop\_intersection\_current\] remain valid in this greater generality. We will need a special case of these constructions. \[prop\_wf\_submfld\] Let $Y \subset X$ be a compact oriented $k$-dimensional submanifold. Then $$\operatorname{WF}([[Y]])=N_X(Y)=\{(x,[\xi]) \in S^*X|_Y:\xi|_{T_xY}=0\}.$$ An example for Proposition \[prop\_intersection\_current\] is when $T_i=[[Y_i]]$ with oriented submanifolds $Y_1,Y_2 \subset X$ intersecting transversally (in the usual sense). Then Proposition \[prop\_wf\_submfld\] implies that the transversality condition in Proposition \[prop\_intersection\_current\] is satisfied, and $T_1 \cap T_2=[[Y_1 \cap Y_2]]$. It is easily checked using , that for currents $A_1,A_2$ on an $n$-dimensional manifold $X$ and currents $B_1,B_2$ on an $m$-dimensional manifold $Y$, we have $$\label{eq_intersection_product} (A_1 \times B_1) \cap (A_2 \times B_2) = (-1)^{(n-\deg A_1)(m-\deg B_2)} (A_1 \cap A_2) \times (B_1 \cap B_2)$$ whenever both sides are well-defined. Given a differential operator, we have $\operatorname{WF}(PT) \subset \operatorname{WF}(T)$ with equality in case $P$ is elliptic [@hoermander_pde1 (8.1.11) and Corollary 8.3.2]. In particular, it follows that for a current on a manifold $X$, we have $$\label{eq_wavefront_boundary} \operatorname{WF}(\partial T) \subset \operatorname{WF}(T).$$ If $T$ is a current on the sphere $S^{n-1}$ and $\Delta$ the Laplace-Beltrami operator, then $$\label{eq_wavefront_laplacian} \operatorname{WF}(\Delta T)=\operatorname{WF}(T).$$ Valuations ---------- Let us now briefly recall some notions from Alesker’s theory of valuations on manifolds, referring to [@alesker_val_man1; @alesker_val_man2; @alesker_survey07; @alesker_val_man4; @alesker_val_man3] for more details. Let $X$ be a smooth manifold of dimension $n$, which for simplicity we suppose to be oriented. Let $\mathcal{P}(X)$ be the space of all compact differentiable polyhedra on $X$. Given $P \in \mathcal{P}(X)$, the conormal cycle $N(P)$ is a Legendrian cycle in the cosphere bundle $S^*X$ (i.e. $\partial N(P)=0$ and $N(P) \llcorner \alpha=0$ where $\alpha$ is the contact form on $S^*X$). A map of the form $$P \mapsto \int_{N(P)} \omega+ \int_P \gamma, \quad P \in \mathcal{P}(X), \omega \in \Omega^{n-1}(S^*X), \gamma \in \Omega^n(X)$$ is called a [*smooth valuation*]{} on $X$. The space of smooth valuations on $X$ is denoted by $\mathcal{V}^\infty(X)$. It carries a natural Fréchet space topology. The subspace of compactly supported smooth valuations is denoted by $\mathcal{V}_c^\infty(X)$. The valuation defined by the above equation will be denoted by $\nu(\omega,\gamma)$. We remark that, without using an orientation, we can still define $\nu(\omega,\phi)$, where $\omega \in \Omega^{n-1}(S^*X) \otimes \operatorname{or}(X), \phi \in \Omega^n(X) \otimes \operatorname{or}(X)$. Here $\operatorname{or}(X)$ is the orientation bundle over $X$. Elements of the space $$\mathcal{V}^{-\infty}(X):=(\mathcal{V}_c^\infty(X))^*$$ are called [*generalized valuations*]{} [@alesker_val_man4]. Each compact differentiable polyhedron $P$ defines a generalized valuation $\Gamma(P)$ by $$\langle \Gamma(P),\phi\rangle:=\phi(P), \quad \phi \in \mathcal{V}^\infty_c(X).$$ A smooth valuation can be considered as a generalized valuation by Alesker-Poincaré duality . We thus have injections $$\xymatrix{ \mathcal{V}^\infty(X) \ar@{^{(}->}[r] & \mathcal{V}^{-\infty}(X) & \mathcal{P}(X) \ar@{_{(}->}[l]}.$$ By the results in [@bernig_broecker07] and [@alesker_bernig], a generalized valuation $\phi \in \mathcal{V}^{-\infty}(X)$ is uniquely described by a pair of currents $E(\phi)=(T(\phi),C(\phi)) \in \mathcal{D}_{n-1}(S^*X) \times \mathcal{D}_n(X)$ such that $$\label{eq_conditions_t_c} \partial T=0, \pi_*T=\partial C, T \text{ is Legendrian, i.e.} T \llcorner \alpha=0.$$ Note that, in contrast to different uses of the word [*Legendrian*]{} in the literature, $T$ is not assumed to be rectifiable. Given $(T,C)$ satisfying these conditions, we denote by $E^{-1}(T,C)$ the corresponding generalized valuation. If $\mu$ is a compactly supported smooth valuation on $X$, then we may represent $\mu=\nu(\omega,\gamma)$ with compactly supported forms $\omega,\gamma$. If $E(\phi)=(T,C)$, then $$\langle \phi,\mu\rangle=T(\omega)+C(\gamma).$$ In particular, the generalized valuation $\Gamma(P)$ corresponding to $P \in \mathcal{P}(X)$ satisfies $$\label{eq_currents_for_polytope} E(\Gamma(P))=(N(P),[[P]]).$$ If $\phi=\nu(\omega,\gamma)$ is smooth, then $$\begin{aligned} T(\phi) & = [[S^*X]] \llcorner s^*(D\omega+\pi^* \gamma),\label{eq_pair_currents_smooth_a}\\ C(\phi) & = [[X]] \llcorner \pi_*\omega, \label{eq_pair_currents_smooth}\end{aligned}$$ where $D$ is the Rumin operator and $s$ is the involution on $S^*X$ given by $[\xi] \mapsto [-\xi]$ [@alesker_bernig; @rumin94]. Let us specialize to the case where $X=V$ is a finite-dimensional vector space. We denote by $\mathcal{V}^\infty(V)^{tr}$ and $\mathcal{V}^{-\infty}(V)^{tr}$ the spaces of translation invariant elements. Alesker has shown in [@alesker_val_man2] that $$\mathcal{V}^\infty(V)^{tr} \cong \operatorname{Val}^\infty(V).$$ A similar statement for generalized valuations is shown in the next proposition. \[prop\_identification\_vals\] The transpose of the map $$\begin{aligned} F: \mathcal{V}_c^\infty(V) & \to \operatorname{Val}^\infty(V) \otimes \operatorname{Dens}(V^*) \\ \mu & \mapsto \int_V \mu(\bullet + x) d\operatorname{vol}(x) \otimes \operatorname{vol}^*\end{aligned}$$ induces an isomorphism $$F^*: \operatorname{Val}^{-\infty}(V) \stackrel{\cong}{\longrightarrow} \mathcal{V}^{-\infty}(V)^{tr}.$$ The diagram $$\xymatrix{\operatorname{Val}^{-\infty}(V) \ar_\cong^{F^*}[r] & \mathcal{V}^{-\infty}(V)^{tr}\\ \operatorname{Val}^\infty(V) \ar^{\cong}[r] \ar@{^{(}->}[u] & \mathcal{V}^\infty(V)^{tr} \ar@{^{(}->}[u]}$$ commutes and the vertical maps have dense images. First we show that the diagram is commutative, i.e. that the restriction of $F^*$ to $\operatorname{Val}^\infty(V)$ is the identity. Let $\phi\in \operatorname{Val}^\infty(V)$ be a smooth valuation. We will show that for any $\mu \in \mathcal V_c^\infty(V)$ one has $$\langle F^*\phi,\mu\rangle_{\mathcal V^\infty(V)}=\langle \phi,\mu\rangle_{\mathcal V^\infty(V)},$$ or equivalently that $$\langle \phi,F\mu\rangle_{\operatorname{Val}^\infty(V)}=\langle \phi,\mu\rangle_{\mathcal V^\infty(V)}.$$ Fix a Euclidean structure on $V$, which induces canonical identifications $\operatorname{Dens}(V) \cong \mathbb C$ and $S^*V \cong V \times S^{n-1}$. We may assume by linearity that $\phi$ is $k$-homogeneous. Represent $\mu=\nu(\omega,\gamma)$ with some compactly supported forms $\omega \in \Omega_c^{n-1}(S^*V)$, $\gamma \in \Omega_c^n(V)$. We may write $$F\mu=\nu\left(\int_V x^* \omega d\operatorname{vol}(x),\int_V x^*\gamma d\operatorname{vol}(x)\right).$$ If $k<n$, then $\phi=\nu(\beta,0)$ for some form $\beta\in\Omega^{n-1}(S^*V)^{tr}$ by the irreducibility theorem [@alesker_mcullenconj01]. By the product formula from [@alesker_bernig] we have $$\langle \phi,\mu\rangle_{\mathcal V^{\infty}(V)}=\int_{S^*V}\omega\wedge s^{*}D\beta+\int_{V}\gamma\wedge\pi_{*}\beta\label{eq:manifolds_pairing}$$ and $$\langle \phi, F\mu\rangle_{\operatorname{Val}^{\infty}(V)}=\pi_{*}\left(\int_{V}x^{*}\omega d\operatorname{vol}(x)\wedge s^{*}D\beta\right) +\int_{S^{n-1}}\beta\cdot\int_V x^{*}\gamma d\operatorname{vol}(x). \label{eq:valuation_pairing}$$ The second summand in is $$\int_{V}\gamma\wedge\pi_{*}\beta=\int_{S^{n-1}} \beta \cdot \int_V \gamma$$ which coincides with the second summand of . Denoting $\psi:=s^{*}D\beta \in\Omega^{n}(S^*V)^{tr}$ and $\tau:=\omega \wedge \psi \in \Omega_{c}^{2n-1}(S^*V)$, it remains to verify that $$\pi_{*}\left(\int_{V}x^{*}\omega d\operatorname{vol}(x) \wedge\psi\right)=\int_{S^*V}\omega\wedge\psi,$$ which is equivalent to $$\pi_{*}\left(\int_{V} x^{*} \tau d\operatorname{vol}(x)\right)=\left(\int_{S^*V} \tau \right)\operatorname{vol}.$$ Write $\tau=f(y,\theta)d\operatorname{vol}(y)d\theta$ for $y\in V$, $\theta\in S^{n-1}$ and $d\theta$ the volume form on $S^{n-1}$. Then $$\begin{aligned} \pi_{*}\left(\int_{V}x^{*}\tau d\operatorname{vol}(x)\right)(y) & =\left(\int_{S^{n-1}} \left(\int_{V}f(y+x,\theta)d\operatorname{vol}(x)\right) d\theta \right)\operatorname{vol}(y)\\ & =\left(\int_{S^{n-1}}\left(\int_{V}f(x,\theta)d\operatorname{vol}(x)\right) d\theta \right)\operatorname{vol}(y)\\ & = \left(\int_{S^*V} \tau \right)\operatorname{vol}(y),\end{aligned}$$ as required. Now assume $k=n$, so $\phi=\nu(0,\lambda \operatorname{vol})$ is a Lebesgue measure on $V$. Then $$\langle \phi, F\mu\rangle_{\operatorname{Val}^{\infty}(V)}=\lambda \pi_*\left(\int_V x^*\omega d\operatorname{vol}(x)\right)$$ and $$\langle \phi,\mu\rangle_{\mathcal V^{\infty}(V)}=\lambda \int_{V} \pi_{*}\omega \operatorname{vol}.$$ Since the right hand sides coincide, the commutativity of the diagram follows. We proceed to show surjectivity of $F$. Let $\phi$ be a smooth translation invariant valuation on $V$. We fix translation invariant differential forms $\omega \in \Omega^{n-1}(S^*V), \gamma \in \Omega^n(V)$ with $\phi=\nu(\omega,\gamma)$. Let $\operatorname{vol}$ be a density on $V$ and let $\beta$ be a compactly supported smooth function such that $\int_V \beta(x) d \operatorname{vol}(x)=1$. The valuation $\mu:=\nu(\pi^*(\beta) \wedge \omega,\beta \gamma)$ is smooth, compactly supported and satisfies $F(\mu)=\phi \otimes \operatorname{vol}^*$. This shows that $F$ is onto. Thus $F$ induces an isomorphism $$\tilde F: \mathcal{V}_c^\infty(V)/\ker F \stackrel{\cong}{\longrightarrow} \operatorname{Val}^\infty(V) \otimes \operatorname{Dens}(V^*).$$ The transpose of $\tilde F$ is an isomorphism $$\tilde F^*: \operatorname{Val}^{-\infty}(V) \to (\ker F)^\perp.$$ The proof will be finished once we can show $(\ker F)^\perp = \mathcal{V}^{-\infty}(V)^{tr}$. If $\mu \in \mathcal{V}_c^\infty(V)$, then $(t_v)_*\mu-\mu \in \ker F$ for every $v \in V$, where $t_v$ is the translation by $v$. It follows that $(\ker F)^\perp \subset \mathcal{V}^{-\infty}(V)^{tr}$. Let $\phi \in \mathcal{V}^{-\infty}(V)^{tr}$. Fix a compactly supported approximate identity $f_\epsilon$ in $\operatorname{GL}(n)$ and set $\phi_\epsilon:=\phi * f_\epsilon$. Then $T(\phi_\epsilon)=T(\phi) * f_\epsilon$ and $C(\phi_\epsilon)=C(\phi)*f_\epsilon$ are smooth currents. By [@alesker_bernig Lemma 8.1], $\phi_\epsilon \in \operatorname{Val}^\infty(V)$ and $\phi_\epsilon \to \phi$. This shows that $\operatorname{Val}^\infty(V) \cong \mathcal{V}^\infty(V)^{tr}$ is dense in $\mathcal{V}^{-\infty}(V)^{tr}$. Clearly $\operatorname{Im}\left(F^*:\operatorname{Val}^\infty(V) \to \mathcal{V}^{-\infty}(V)^{tr}\right) \subset (\ker F)^\perp$. Since the image is dense in $\mathcal{V}^{-\infty}(V)^{tr}$, it follows that $\mathcal{V}^{-\infty}(V)^{tr} \subset (\ker F)^\perp$. For $\phi \in \operatorname{Val}^{-\infty}(V)$ we set $\operatorname{WF}(\phi):=\operatorname{WF}(T(\phi)) \subset S^*(S^*V)$. Given $\Gamma \subset S^*(S^*V)$ a closed set, we define $$\operatorname{Val}^{-\infty}_\Gamma(V):=\left\{\phi \in \operatorname{Val}^{-\infty}(V): \operatorname{WF}(\phi) \subset \Gamma\right\}.$$ \[lemma\_density\_smooth\] The subspace $\operatorname{Val}^\infty(V) \subset \operatorname{Val}^{-\infty}_\Gamma(V)$ is dense. This follows by [@alesker_bernig Lemma 8.2], noting that in the proof of that lemma, a translation invariant generalized valuation is approximated by translation invariant smooth valuations. Embedding McMullen’s polytope algebra {#sec_embedding_polytopes} ===================================== Let $V$ denote an $n$-dimensional real vector space, and $\Pi(V)$ the McMullen polytope algebra on $V$. It is defined as the abelian group generated by polytopes, with the relations of the inclusion-exclusion principle and translation invariance. The product is defined on generators by $[P] \cdot [Q]:=[P+Q]$. It is almost a graded algebra over $\mathbb{R}$. We refer to [@mcmullen_polytope_algebra] for a detailed study of its properties. For $\lambda \in {\mathbb{R}}$, the dilatation $\Delta(\lambda)$ is defined by $\Delta(\lambda)[P]=[\lambda P]$. The [*$k$-th weight space*]{} is defined by $$\Xi_k:=\{x \in \Pi: \Delta(\lambda)x=\lambda^k x \text{ for some rational } \lambda>0, \lambda \neq 1\}.$$ McMullen has shown ([@mcmullen_polytope_algebra], Lemma 20) that $$\Pi(V)=\bigoplus_{k=0}^n \Xi_k.$$ The aim of this section is to prove Theorem \[mainthm\_injection\_mcmullen\], namely that $\Pi(V)$ embeds into $\operatorname{Val}^{-\infty}(V) \otimes \operatorname{Dens}(V^*)$. Recall that the map $M:\Pi(V) \to \operatorname{Val}^{-\infty}(V) \otimes \operatorname{Dens}(V^*) \cong \operatorname{Val}^{\infty}(V)^*$ is defined by $$\langle M([P]),\phi\rangle=\phi(P), \quad P \in \mathcal{P}(V).$$ We will denote by $M_k:\Pi(V)\to \operatorname{Val}_k(V) \otimes \operatorname{Dens}(V^*)$ the $k$-homogeneous component of the image of $M$, and similarly $[\phi]_{k}$ is the $k$-homogeneous component of a valuation $\phi$. $$M([P])=\operatorname{vol}(\bullet-P)\otimes \operatorname{vol}^*.$$ In particular $\operatorname{Im}(M) \subset \operatorname{Val}(V)\otimes \operatorname{Dens}(V^*)$. We claim that $\langle \operatorname{PD}(\psi),\operatorname{vol}(\bullet - K) \otimes \operatorname{vol}^* \rangle=\psi(K)$ for $\psi \in \operatorname{Val}^\infty(V)$ and $K \in \mathcal{K}(V)$, where $\operatorname{PD}:\operatorname{Val}^\infty(V) \to \operatorname{Val}^{-\infty}(V)$ denotes the natural embedding given by the Poincaré duality of Alesker [@alesker04_product]. Indeed, let first $K$ be smooth with positive curvature. Then, by ([@bernig_aig10], (15)), $$\psi \cdot \operatorname{vol}(\bullet-K)= \int_V \psi((y+K) \cap \bullet) d\operatorname{vol}(y)$$ and hence $$\langle \operatorname{PD}(\psi), \operatorname{vol}(\bullet-K) \otimes \operatorname{vol}^*\rangle=\psi(K).$$ By approximation, this holds for non-smooth $K$ as well, showing the claim. Since by definition $\langle M[P],\psi\rangle=\psi(P)$, the statement follows. \[lemma\_m\_as\_average\] Let $P$ be a polytope and $\Gamma(P) \in \mathcal{V}^{-\infty}(V)$ the associated generalized valuation, i.e. $\langle \Gamma(P), \phi\rangle=\phi(P)$ for $\phi \in \mathcal{V}^\infty_c(V)$. Then $$F^*M([P])=\int_V \Gamma(P+x) d \operatorname{vol}(x) \otimes \operatorname{vol}^*,$$ with $F$ as in Proposition \[prop\_identification\_vals\]. Let $\phi \in \mathcal{V}_c^\infty(V)$. Then $$\begin{aligned} \langle F^* M([P]),\phi\rangle & =\langle M([P]),F\phi\rangle\\ & = F(\phi)(P)\\ & = \int_V \phi(P+x)d\operatorname{vol}(x) \otimes \operatorname{vol}^*\\ & = \int_V \langle \Gamma(P+x),\phi\rangle d\operatorname{vol}(x) \otimes \operatorname{vol}^*\\ & = \left\langle \int_V \Gamma(P+x) d\operatorname{vol}(x) \otimes \operatorname{vol}^*,\phi \right\rangle.\end{aligned}$$ For the following, we fix a Euclidean structure and an orientation on $V$ and denote by $\operatorname{vol}$ the corresponding Lebesgue measure on $V$. Denote by $\mathcal{C}_{j}$ the collection of $j$-dimensional oriented submanifolds $N \subset S^{n-1}$, obtained by intersecting $S^{n-1}$ with a $(j+1)$-dimensional polytopal cone $\hat{N}$ in $V$. Given $v\in \Lambda^k V$, define the current $[v] \in \mathcal{D}_{k}(V)$ by $$\langle [v],\omega\rangle:=\int_V \omega|_x(v) d\operatorname{vol}(x), \omega \in \Omega_c^k(V).$$ Let $\Lambda_s^kV$ denote the cone of simple $k$-vectors in $V$. Given a pair $(v,N) \in \Lambda_s^k V \times \mathcal{C}_{n-k-1}$, we define the current $A_{v,N} = [v] \times [[N]]\in \mathcal{D}_{n-1}(V \times S^{n-1})$, where $[[N]]$ is the current of integration over $N$. Observe that changing the sign of $v$ and the orientation of $N$ simultaneously leaves the current $A_{v,N}$ invariant. Let $Y_k \subset \mathcal{D}_{n-1}(V \times S^{n-1})$ be the $\mathbb{C}$-span of currents of the form $A_{v,N}, v \in \Lambda^k_sV, N \in \mathcal{C}_{n-k-1}$ such that $\operatorname{Span}(v)\oplus \operatorname{Span}(\hat{N})=V$ as oriented spaces. Given a polytope $P \subset V$, we let $\mathcal{F}_k$ be the set of $k$-faces of $P$. Each face is assumed to possess some fixed orientation. If $F$ is a face of $P$ of dimension strictly less than $n$, we let $n(F,P)$ be the normal cone of $F$, and $\check n(F,P):=n(F,P) \cap S^{n-1}$ is oriented so that the linear space parallel to $F$, followed by $\operatorname{Span}(\check n(F,P))$, is positively oriented. Moreover, let $v_F$ be the unique $k$-vector in the linear space parallel to $F$ such that $|v_F|=\operatorname{vol}_k F$, the sign determined by the orientation of $F$. \[lemma\_image\_of\_polytope\] Let $P$ be a polytope. Then $$\begin{aligned} E(M_0([P])) & =\left(0,\operatorname{vol}(P) [[V]]\right),\\ E(M_{n-k}([P])) & =\left(\sum_{F \in \mathcal{F}_k(P)} A_{v_{F},\check{n}(F,P)},0\right), \quad 0\leq k\leq n-1. \end{aligned}$$ It follows from and Lemma \[lemma\_m\_as\_average\] that $$\begin{aligned} T(M([P])) & = \int_{V} T(\Gamma(P+x))d\operatorname{vol}(x)=\sum_{F \in \mathcal{F}_{\leq n-1}(P)} A_{v_F,\check{n}(F,P)}\\ C(M([P])) & = \int_{V} C(\Gamma(P+x))d\operatorname{vol}(x) = \operatorname{vol}(P) [[V]].\end{aligned}$$ From this the statement follows. Let $$T_k:\operatorname{Im}(M_{n-k}) \to Y_k, \quad 0\leq k\leq n-1$$ be the restriction of $T$ to $\operatorname{Im}(M_{n-k})$. For a linear subspace $L\subset V$, we will write $S(L)=S^{n-1} \cap L$. Let $\mathcal{F}(V)$ denote the space of $\mathbb{Z}$-valued constructible functions on $V$, i.e. functions of the form $\sum_{i=1}^N n_i 1_{P_i}$ with $n_i \in \mathbb{Z}, P_i \in \mathcal{P}(V)$ (compare [@alesker_val_man4]). Let $\mathcal{F}_\text{a.e.}(V)$ be the set of congruence classes of constructible functions where $f \sim g$ if $f-g=0$ almost everywhere. \[lemma\_inclusion\_into\_constr\_functions\] Denote by $Z$ the abelian group generated by all formal integral combination of compact convex polytopes in $V$. Let $W \subset Z$ denote the subgroup generated by lower-dimensional polytopes and elements of the form $[P \cup Q]+[P \cap Q]-[P]-[Q]$ where $P \cup Q$ is convex. Then the map $$\begin{aligned} Z/W & \to \mathcal{F}_\text{a.e.}(V)\\ \sum_i n_i [P_i] & \mapsto \sum_i n_i 1_{P_i}, \quad n_i \in \mathbb{Z} \end{aligned}$$ is injective. It is easily checked that the map is well-defined. To prove injectivity, it is enough to prove that $f:=\sum_i 1_{P_i} \sim \sum_j 1_{Q_j}$ implies $\sum_i [P_i] \equiv \sum_j [Q_j]$. Decompose the connected components of $$\left(\cup_i P_i \cup \cup_j Q_j\right) \setminus \left(\cup_i \partial P_i \cup \cup_j \partial Q_j\right)$$ into simplices $\{\Delta\}$, disjoint except at their boundary. Then, by the inclusion-exclusion principle, $$\sum_i [P_i] \equiv \sum_i \sum_{\Delta \subset P_i} [\Delta] \equiv \sum_{k \geq 1} (-1)^{k+1} \sum_{\Delta \subset \bigcup_{i_1<\ldots<i_k} \cap_{j=1}^k P_{i_j}}[\Delta]$$ By examining the superlevel set $\{f \geq k\}$ we see that $$\bigcup_{i_1<\ldots<i_k}(P_{i_1} \cap \ldots \cap P_{i_k}) \sim \bigcup_{j_1<\ldots<j_k}(Q_{j_1} \cap \ldots \cap Q_{j_k}),$$ where $A \sim B$ means the sets $A,B$ coincide up to a set of measure zero. Therefore, $$\sum_i [P_i] \equiv \sum_j [Q_j] \mod W,$$ as claimed. The same claim and proof apply if we replace $Z$ with the free abelian group of polytopal cones with vertex in the origin. Let us recall some notions from [@mcmullen_polytope_algebra]. Let $L$ be a subspace of $V$. The [*cone group*]{} $\hat \Sigma(L)$ is the abelian group with generators $[C]$, where $C$ ranges over all convex polyhedral cones in $L$, and with the relations 1. $[C_1 \cup C_2]+[C_1 \cap C_2]=[C_1]+[C_2]$ whenever $C_1,C_2,C_1 \cup C_2$ are convex polyhedral cones; 2. $[C]=0$ if $\dim C < \dim L$. The [*full cone group*]{} is given by $$\hat \Sigma:=\bigoplus_{L \subset V} \hat \Sigma(L),$$ where the sum extends over all linear subspaces of $L$. \[lemma\_lemma39\] The map $$\begin{aligned} \sigma_k : \Pi(V) & \to \mathbb{C} \otimes_ \mathbb{Z} \hat \Sigma \\ [P] & \mapsto \sum_{F \in \mathcal{F}_k(P)} \operatorname{vol}(F) \otimes n(F,P)\end{aligned}$$ restricts to an injection on $\Xi_k$. For all $0\leq k\leq n-1$, there is a linear map $\Phi_k:Y_k \to\mathbb{C} \otimes_{\mathbb{Z}} \hat \Sigma$ such that $\Phi_k(A_{v,N})=|v| \otimes [\hat{N}]$. The first thing to note is that if $A_{v_1,N_1}=A_{v_2,N_2}$, then either $v_1=v_2$, $N_1=N_2$ or $v_1=-v_2$ and $N_1=\overline{N_2}$, where $\overline{N_j}$ is $N_j$ with reversed orientation. Thus on the generators of $Y_k$, $\Phi_k(A_{v,N}):=|v| \otimes [\hat{N}]$ is well-defined. Now assume that $\sum_j c_j A_{v_j,N_j}=0$. We shall show that $\sum_j c_j |v_j| \otimes [\hat{N}_j]=0$. Note that for all $\rho \in \Omega_c^k(V)$ and $\omega \in \Omega^{n-k-1}(S^{n-1})$, one has $$\sum_j c_j \int_V \rho|_x(v_j)d\operatorname{vol}(x) \int_{N_j} \omega=0.$$ More generally, suppose that we have $$\sum_j \lambda_j \int_{N_j} \omega=0$$ for some coefficients $\lambda_j \in \mathbb{C}$ and for all $\omega\in\Omega^{n-k-1}(S^{n-1})$. Let $L \subset V$ be a linear subspace of dimension $n-k$. Let $U_\epsilon \subset S^{n-1}$ denote the $\epsilon$-neighborhood of $L \cap S^{n-1}$, where $\epsilon$ is sufficiently small. Let $p_\epsilon:U_\epsilon \to L\cap S^{n-1}$ denote the nearest-point projection. Fix some $\beta \in C^{\infty}[0,1]$ such that $\beta(x)=1$ for $0 \leq x \leq \frac13$ and $\beta(x)=0$ for $\frac23 \leq x \leq 1$. Let $ \beta_\epsilon \in C^{\infty}(U_\epsilon)$ be given by $\beta_\epsilon(x)=\beta(\operatorname{dist}(x,p(x))/\epsilon)$. Given a form $\sigma \in \Omega^{n-k-1}(L\cap S^{n-1})$, we set $\omega:=\beta_\epsilon p_\epsilon^* \sigma \in \Omega^{n-k-1}(S^{n-1})$. If $N_j \subset L \cap S^{n-1}$, then $$\int_{N_j} \omega=\int_{N_j} \sigma,$$ while if $N_j$ does not lie in $L \cap S^{n-1}$ then $\int_{N_j} \omega \to 0$ as $\epsilon \to 0$. Thus, letting $\epsilon \to 0$, we obtain $$\sum_{N_j \subset L} \lambda_j \int_{N_j} \sigma=0,$$ where the sum is over all $N_j$ contained in $L$. Now fix an orientation on $L$ and assume without loss of generality that all $N_j \subset L$ have the induced orientation. Going back to the original equation we may write $$\sum_{N_j \subset L} c_j \int_V \rho|_x(v_j)d\operatorname{vol}(x) \int_{N_j}\sigma=0.$$ Let $v_0 \in \Lambda^k L^\perp \subset \Lambda^k V$ be the unique vector with $|v_0|=1$ and $v_0^\perp=L$ (the last equality understood with orientation). Then $v_j=|v_j|v_0$ for all $j$ with $N_j \subset L$. Therefore, the above equation implies that $$\sum_{N_j \subset L} c_j |v_j| \int_{N_j}\sigma=0$$ for all $\sigma \in \Omega^{n-k-1}(L \cap S^{n-1})$ and this implies that $$\sum_{N_j \subset L} c_j |v_j| 1_{N_j}=0$$ almost everywhere. By Lemma \[lemma\_inclusion\_into\_constr\_functions\] we have in $\mathbb{C} \otimes \hat{\Sigma}(L)$ $$\sum_{N_j \subset L} c_j |v_j| \otimes [\hat{N}_j]=0.$$ Since $L$ was arbitrary, we deduce that in $\mathbb{C} \otimes \hat{\Sigma}$ we have $$\sum_j c_j |v_j| \otimes [\hat N_j]=0,$$ as required. By Lemma \[lemma\_lemma39\], the map $$\sigma_k:\Xi_k \to \mathbb{C} \otimes \hat{\Sigma}$$ is injective. For $0 \leq k \leq n-1$ we have $$\sigma_k=\Phi_k \circ T_k \circ M_{n-k},$$ while $\sigma_n$ is the volume functional on $\Pi(V)$ restricted to $\Xi_n$, so identifying the space of Lebesgue measures on $V$ with $\mathbb{C}$ allows us to write $\sigma_n=C \circ M_0$. We conclude that $(M_{n-k})|_{\Xi_k}$ is injective for each $k$, and hence $M$ is injective. Partial convolution product {#sec_partial_conv} =========================== Let us first describe the convolution in the smooth case, see [@bernig_fu06], but using a more intrinsic approach. Let $V$ be an $n$-dimensional vector space. Let $\operatorname{Dens}(V)$ denote the $1$-dimensional $\operatorname{GL}(V)$-module of densities on $V$. The orientation bundle $\operatorname{or}(V)$ is the real $1$-dimensional linear space consisting of all functions $\rho:\Lambda^{top}V \to \mathbb R$ such that $\rho (\lambda \omega)=\mathrm{sign}(\lambda) \rho(\omega)$ for all $\lambda \in\mathbb R$ and $\omega \in \Lambda^{top}V$. Note that there is a canonical isomorphism $\operatorname{or}(V) \cong \operatorname{or}(V^*)$. Let $\mathbb P_+(V)$ be the space of oriented $1$-dimensional subspaces of $V$. Then $ S^*V:=V \times \mathbb P_+(V^*)$ has a natural contact structure. We have a natural non-degenerate pairing $$\Lambda^k V^* \otimes \Lambda^{n-k}V^* \stackrel{\wedge}{\longrightarrow} \Lambda^n V^* \cong \operatorname{Dens}(V) \otimes \operatorname{or}(V),$$ which induces an isomorphism $$*:\Lambda^k V^* \otimes \operatorname{or}(V) \otimes \operatorname{Dens}(V^*) \stackrel{\cong}{\longrightarrow} (\Lambda^{n-k} V^*)^* \cong \Lambda^{n-k} V.$$ Let $$\ast_1:\Omega (S^*V)^{tr} \otimes \operatorname{or}(V) \otimes \operatorname{Dens}(V^*) \stackrel{\cong}{\longrightarrow} \Omega(V^* \times \mathbb P_+(V^*))^{tr}$$ be defined by $$\ast_1(\pi_1^*\gamma_1 \wedge \pi_2^*\gamma_2):=(-1)^{\binom{n-\deg \gamma_1}{2}} \pi_1^*(*\gamma_1) \wedge \pi_2^*\gamma_2$$ for $\gamma_1\in \Omega(V)^{tr} \otimes \operatorname{or}(V) \otimes \operatorname{Dens}(V^*), \gamma_2 \in \Omega(\mathbb P_+(V^*))$. Let $\phi_j=\nu(\omega_j,\gamma_j) \in \operatorname{Val}^\infty(V) \otimes \operatorname{Dens}(V^*),j=1,2$, where $\omega_j \in \Omega^{n-1}(S^*V)^{tr} \otimes \operatorname{or}(V) \otimes \operatorname{Dens}(V^*), \gamma_j \in \Omega^n(V)^{tr} \otimes \operatorname{or}(V) \otimes \operatorname{Dens}(V^*)$. Then the convolution product $\phi_1 * \phi_2$ is defined as $\nu(\omega,\gamma) \in \operatorname{Val}^\infty(V) \otimes \operatorname{Dens}(V^*)$, where $\omega$ and $\gamma$ satisfy $$\begin{aligned} D\omega +\pi^*\gamma & = \ast_1^{-1}\left(\ast_1 (D\omega_1+\pi^*\gamma_1) \wedge \ast_1(D\omega_2+\pi^*\gamma_2)\right)\\ \pi_*\omega & = \pi_* \circ \ast_1^{-1} \left( \ast_1 \kappa_1 \wedge \ast_1 (D\omega_2)\right)+(*\gamma_1)\pi_*\omega_2+(*\gamma_2)\pi_*\omega_1.\end{aligned}$$ Here $\kappa_1 \in \Omega^{n-1}(S^*V)^{tr}$ is any form such that $d\kappa_1=D\omega_1$. The convolution extends to a partially defined convolution on the space $\operatorname{Val}^{-\infty}(V) \otimes \operatorname{Dens}(V^*)$ as follows. The space $\mathcal{D}_*(S^*V)$ admits a bigrading $$\mathcal{D}_*(S^*V)=\bigoplus_{k=0}^n \bigoplus_{l=0}^{n-1} \mathcal{D}_{k,l}(S^*V),$$ and for $T \in\mathcal{D}_*(S^*V)$, we denote by $[T]_{k,l}$ the component of bidegree $(k,l)$. We consider now the $\operatorname{GL}(V)$-module of translation-invariant currents $\mathcal D(S^*V)^{tr}=\mathcal D(V)^{tr} \otimes \mathcal D(\mathbb P_+(V^*))$. One has the natural identification $\mathcal D_k(V)^{tr}=\Omega^{n-k}(V)^{tr} \otimes \operatorname{or}(V)$, and a non-degenerate pairing $\mathcal D_k(V)^{tr} \otimes \Omega^k(V)^{tr}\to \operatorname{Dens}(V)$. We define for $T \in \mathcal{D}_{k,l}(S^*V)^{tr} \otimes \operatorname{or}(V) \otimes \operatorname{Dens}(V^*)$ the element $\ast_1 T \in \mathcal{D}_{n-k,l}(V^*\times \mathbb P_+(V^*))^{tr}$ by $$\langle \ast_1 T,\delta\rangle:=(-1)^{nk+nl+k+{\binom{n}{2}}} \langle T,\ast_{1}^{-1}\delta\rangle \in \operatorname{Dens}(V^*)$$ for all $\delta \in \Omega_c^{n-k,l}(V^*\times \mathbb P_+(V^*))^{tr}$. With this definition of $\ast_1$ the diagram $$\xymatrixcolsep{2pc} \xymatrix{\Omega^{k,l}(S^*V)^{tr} \otimes \operatorname{or}(V) \otimes \operatorname{Dens}(V^*) \ar[r]^-{*_1} \ar@{_{(}->}[d] & \Omega^{n-k,l}(V^*\times \mathbb P_+(V^*))^{tr} \ar@{_{(}->}[d] \\ \mathcal{D}_{n-k,n-l-1}(S^*V)^{tr} \otimes \operatorname{or}(V)\otimes \operatorname{Dens}(V^*)\ar[r]^-{*_1}& \mathcal{D}_{k,n-l-1}(V^*\times \mathbb P_+(V^*))^{tr}}$$ commutes. Equivalently, $$\ast_1 \left([[S^*V]] \llcorner \gamma\right)=[[S^*V]] \llcorner \ast_1 \gamma$$ for all $\gamma \in \Omega(S^*V)^{tr} \otimes \operatorname{or}(V) \otimes \operatorname{Dens}(V^*)$. Clearly we have $\operatorname{WF}(\ast_{1} T)=\operatorname{WF}(T)$ for $T\in \mathcal{D}(S^*V)^{tr}$ Without the choice of an orientation and a Euclidean scalar product, we may intrinsically define $A_{v,N} \in \mathcal D_{k,n-1-k}(S^*V)^{tr} \otimes \operatorname{or}(V) \otimes \operatorname{Dens}(V^*)$ as follows. Let $v \in \Lambda^k_s V$ and let $N \subset \mathbb P_+(V^*)$ be an $(n-1-k)$-dimensional, geodesically convex polytope contained in $v^\perp \cap \mathbb P_+(V^*)$. Then define $$\langle A_{v,N}, \pi_1^*\gamma \wedge \pi_2^* \delta \otimes \sigma \otimes \epsilon \rangle= \int_V \langle x^*\omega, v\rangle d\sigma(x)\int_{N_\epsilon} \delta$$ for $\gamma \in \Omega_c^k(V), \delta \in \Omega^{n-k-1}(\mathbb P_+(V^*)), \epsilon\in\operatorname{or}(V)$ and $\sigma \in \operatorname{Dens}(V)$. Here $N_\epsilon$ equals $N$ with a choice of orientation such that the orientation of the pair $(v,N_\epsilon)$ is induced by $\epsilon$. Given a Euclidean trivialization, this reduces to the previous definition of $A_{v,N}$. \[lemma\_filling\_current\] Given a Legendrian cycle $T\in\mathcal{D}_{n-k,k-1}(S^*V)^{tr}$ with $1\leq k\leq n-1$, there exists $\tilde{T} \in \mathcal{D}_{n-k,k}(S^*V)^{tr}$ with $T=\partial \tilde{T}$ and $\operatorname{WF}(\tilde{T})=\operatorname{WF}(T)$. Let us use a Euclidean scalar product and an orientation on $V$. Then we may identify $\operatorname{Dens}(V) \cong \mathbb{C}, \operatorname{or}(V) \cong \mathbb{C}, S^*V \cong SV=V \times S^{n-1}$. Let $\phi \in \operatorname{Val}_{k}^{-\infty}(V)$ be the valuation represented by $(T,0)$. Choose a sequence $\phi_j \in \operatorname{Val}_k^\infty(V)$ such that $\phi_j \to \phi$. Let $\omega_j \in \Omega^{k,n-k-1}(SV)^{tr}=\Omega^{n-k-1}(S^{n-1}) \otimes \Lambda^kV^*$ be a form representing $\phi_j$ and such that $D\omega_j=d\omega_j$. Note that $\mathcal{D}_{n-k,k-1}(SV)^{tr}=\mathcal{D}_{k-1}(S^{n-1}) \otimes \Lambda^{n-k} V$. Then $d\omega_j \in \Omega^{n-k}(S^{n-1}) \otimes \Lambda^kV^* \subset \mathcal{D}_{k-1}(S^{n-1}) \otimes \Lambda^{n-k}V$ converges weakly to $T$. Let $G:\Omega^*(S^{n-1}) \to \Omega^*(S^{n-1})$ denote the Green operator on $S^{n-1}$ and $\delta:\Omega^*(S^{n-1}) \to \Omega^{*-1}(S^{n-1})$ the codifferential. We define $\beta_j:=G(d\omega_j) \in \Omega^{n-k}(S^{n-1}) \otimes \Lambda^{n-k}V$, i.e. $\Delta \beta_j=d\omega_j$. Then $\Delta d\beta_j=d\Delta \beta_j=0$, hence $d\beta_j$ is harmonic, which implies that $\delta d\beta_j=0$. We define $$\tilde T:=(-1)^{n+k}\lim_{j \to \infty} [[S^{n-1}]] \llcorner \delta\beta_j \in \mathcal{D}_k(S^{n-1}) \otimes \Lambda^{n-k}V \subset \mathcal{D}_{n-k,k}(SV)^{tr}.$$ Then $$\partial \tilde T = \lim_j [[S^{n-1}]] \llcorner d \delta\beta_j = \lim_j [[S^{n-1}]] \llcorner \Delta \beta_j=\lim_j [[S^{n-1}]] \llcorner d\omega_j =T.$$ From $\tilde T=\delta \circ G(T)$ and , we infer that $\operatorname{WF}(\tilde T) \subset \operatorname{WF}(T)$. Conversely, from we deduce that $\operatorname{WF}(T)=\operatorname{WF}(\partial \tilde{T}) \subset \operatorname{WF}(\tilde T)$. Recall that $ s:S^*V \to S^*V$ is the antipodal map. Let $\phi_j \in \operatorname{Val}^{-\infty}(V) \otimes \operatorname{Dens}(V^*), E(\phi_j)=:(T_j,C_j)$, $j=1,2$. We call $\phi_1,\phi_2$ transversal if $$\operatorname{WF}(T_1) \cap s(\operatorname{WF}(T_2)) = \emptyset.$$ \[prop\_def\_convolution\] Let $\phi_j \in \operatorname{Val}^{-\infty}(V) \otimes \operatorname{Dens}(V^*)$, $j=1,2$ be transversal and $(T_j,C_j):=E(\phi_j), j=1,2$. Decompose $T_j=t_j+T'_j$, where $t_j=\alpha_j \pi^*([[V]] \llcorner \operatorname{vol}_n) \otimes \operatorname{vol}_n^* \in \mathcal{D}_{0,n-1}(S^*V)^{tr} \otimes \Lambda^n V^*$ is the corresponding $(0,n-1)$-component, and let $\tilde T_1 \in \mathcal{D}_n(S^*V)^{tr} \otimes \Lambda^n V^*$ be a current such that $\partial \tilde T_1=T'_1$ and $\operatorname{WF}(\tilde T_1)=\operatorname{WF}(T_1)$, guaranteed to exist by Lemma \[lemma\_filling\_current\]. Then the currents $$\begin{aligned} T & :=\ast_1^{-1}\left(\ast_1T_1 \cap \ast_1T_2\right)\\ C & :=\pi_*\left(\ast_1^{-1}\left(\ast_1\tilde{T}_1 \cap \ast_1 T'_2\right)\right)+\alpha_1C_2+\alpha_2C_1\end{aligned}$$ are independent of the choice of $\tilde T_1$ and satisfy the conditions . Note first that $\partial$ commutes (up to sign) with $*_1$ on translation invariant currents. By , $T$ equals (up to sign) the boundary of the $n$-current $S:=\ast_1^{-1}\left(\ast_1 \tilde T_1 \cap \ast_1T_2\right)$. In particular, $T$ is a cycle. Moreover, $\pi_*T= \pm \partial \pi_* S=0$, since $\pi_*S$ is a translation invariant $n$-current, hence a multiple of the integration current on $V$ which has no boundary. For the same reason, $\partial C=0$, hence the condition $\pi_*T=\partial C$ is trivially satisfied. Note that whenever $Q \in \mathcal{D}_n(S^*V)$ is a boundary, then $\pi_*Q=0$. Indeed, let $Q=\partial R$ and $\rho \in \Omega^n_c(V)$. Then $$\label{eq_push_forward_boundary} \langle \pi_*Q,\rho\rangle=\langle Q,\pi^*\rho\rangle=\langle \partial R,\pi^* \rho\rangle=\langle R,\pi^* d\rho\rangle=0.$$ By Lemma \[lemma\_filling\_current\], there exists a translation invariant $n$-current $\tilde T_2$ with $\partial \tilde T_2=T_2'$. Suppose that $\partial \tilde T_1=\partial \hat T_1=T_1'$ for two $n$-currents $\tilde T_1$ and $\hat T_1$. Then the $n$-current $Q:=*_1^{-1} \left(*_1(\tilde T_1-\hat T_1) \cap *_1 T_2'\right)$ is (up to a sign) the boundary of the $(n+1)$-current $R:=*_1^{-1} \left(*_1(\tilde T_1-\hat T_1) \cap *_1 \tilde T_2\right)$. By , it follows that $\pi_*Q=0$, which shows that $C$ is independent of the choice of $\tilde T_1$. Let us finally show that $T$ is Legendrian. Fix sequences $(\phi_j^i)_i$ of smooth and translation invariant valuations converging to $\phi_j, j=1,2$. Let $\phi_j^i$ be represented by the forms $(\omega_j^i, \gamma_j^i)$. Then $E(\phi_j^i)=(T_j^i, C_j^i)$ is given by the formulas , and hence $\phi_1^i * \phi_2^i$ is represented by the current $T^i=[[S^*V]] \llcorner s^*\kappa^i$, with $$\label{eq_conv_smooth} s^*\kappa^i:=\ast_1^{-1}(\ast_1 s^*(D\omega_1^i+\pi^*\gamma_1^i) \wedge \ast_1 s^*(D\omega_2^i+\pi^* \gamma_2^i)).$$ It is easily checked that $\kappa^i$ is a horizontal, closed $n$-form (compare also [@bernig_fu06 Eq. (37)]). It follows that $T^i$ is Legendrian. Note that $ [[S^*V]] \llcorner s^*(D\omega_j^i+\pi^*\gamma_j)$ converges to $T_j$. By the definition of the intersection current, $[[S^*V]] \llcorner s^*\kappa^i$ converges to $T$ and hence $T$ is Legendrian. \[def\_convolution\] In the same situation, the convolution product $\phi_1 * \phi_2 \in \operatorname{Val}^{-\infty}(V) \otimes \operatorname{Dens}(V^*)$ is defined as $\phi_1 * \phi_2:=E^{-1}(T,C)$. If $\phi_1,\phi_2 \in \operatorname{Val}^\infty(V) \otimes \operatorname{Dens}(V^*) \subset \operatorname{Val}^{-\infty}(V) \otimes \operatorname{Dens}(V^*)$, then the convolution of Definition \[def\_convolution\] coincides with the convolution from [@bernig_fu06]. For $\phi_j \in \operatorname{Val}^\infty(V)$ given by the pairs $(\omega_j, \gamma_j)\in \Omega^{n-1}(S^*V)^{tr} \otimes \operatorname{Dens}(V)$, the corresponding currents are $E(\phi_{j})=([[S^*V]] \llcorner s^*(D\omega_j+\pi^*\gamma_j), [[V]] \llcorner \pi_*\omega_j)$. We consider two cases: If $\omega_1=0$ and $\gamma_1=c\cdot \operatorname{vol}$, then $\phi_1=c \cdot \operatorname{vol}$, and $\phi_1 \ast \phi_2=c\phi_2$ by the original definition of convolution. By the new definition, $E(\phi_1)=([[S^*V]] \llcorner \pi^* \gamma_1,0)$, and $$\ast_1 \pi^* \gamma_1=c \in \Omega^0(V\times S^{n-1}).$$ Write $$\phi_1 \ast \phi_2=E^{-1}( [[S^*V]] \llcorner \pi^*\gamma_1,0) \ast E^{-1}(0,C_2)+E^{-1}( [[S^*V]] \llcorner \pi^*\gamma_1,0) \ast E^{-1}(T_2,0).$$ If $E^{-1}(0,C_2)=\lambda\chi$, by definition the first summand equals $c\lambda\chi=cE^{-1}(0,C_2)$. The second summand is $c\cdot E^{-1}(T_2,0)$, and so $\phi_1 \ast \phi_2=c E^{-1}(T_2,C_2)=c \phi_2$, as required. In the remaining case, we may assume $\gamma_1=\gamma_2=0$, so $E(\phi_{j})=( [[S^*V]] \llcorner s^{*}D\omega_{j}, [[V]] \llcorner \pi_{*}\omega_{j})$. Moreover, we may assume that $d\omega_{1},d\omega_{2}$ are vertical. The original definition of convolution gives $E(\phi_{1}\ast\phi_{2})=( [[S^*V]] \llcorner s^* D\omega, [[V]] \llcorner \pi_* \omega)$ with $$\begin{aligned} \omega & =\ast_1^{-1}\left(\ast_1 \omega_1 \wedge \ast_1 D\omega_2\right)\\ D\omega & =\ast_1^{-1}\left(\ast_1 D\omega_1 \wedge \ast_1 D\omega_2\right).\end{aligned}$$ Since $[E^{-1}([[S^*V]] \llcorner s^* D\omega_j,0)]_{n}=[\nu(\omega_{j},0)]_n=0$, it remains to verify that by the new definition, $$E^{-1}([[S^*V]] \llcorner s^* D\omega_1,0) \ast E^{-1}( [[S^*V]] \llcorner s^* D\omega_2,0)=E^{-1}( [[S^*V]] \llcorner s^* D\omega, [[V]] \llcorner \pi_* \omega).$$ By homogeneity, we may assume that $\deg \omega_1=(k,n-1-k)$ and $\deg\omega_{2}=(l,n-1-l)$. If $k+l<n$ then by dimensional considerations $\omega=0$. If $k+l>n$ then $\pi_* \omega=0$, and the new definition of convolution gives $$E(\phi_1 \ast \phi_2) =\left( [[S^*V]] \llcorner \ast_1^{-1}\left((\ast_1s^*D\omega_1) \wedge (\ast_1 s^* D\omega_2)\right),0\right)=\left( [[S^*V]] \llcorner s^* D\omega,0\right)$$ as required. Finally, if $k+l=n$, then $1 \leq k \leq n-1$, $D\omega=0$, and by the new definition $T(\phi_1 \ast \phi_2)=0$. Since $T_1= [[S^*V]] \llcorner s^* d\omega_1$ and $\omega_1\in\Omega^{n-1}(S^*V)$, using one can take $\tilde{T}_1=(-1)^n [[S^*V]] \llcorner s^* \omega_1$. Using , the fact that the operations $s^* $, $\ast_1$ and $[[S^*V]]\llcorner$ commute, while $\pi_* \circ s^*=(-1)^n \pi_*$, we obtain $$\begin{aligned} C(\phi_1 \ast \phi_2) & =(-1)^n \pi_* \left( [[S^*V]] \llcorner \ast_1^{-1}(\ast_1 s^* \omega_1 \wedge \ast_1 s^* d\omega_2)\right)\\ & =(-1)^n \pi_* ([[S^*V]] \llcorner s^*\omega)\\ & =(-1)^n [[V]] \llcorner \pi_* s^*\omega\\ & =[[V]] \llcorner \pi_* \omega,\end{aligned}$$ completing the verification. \[prop\_continuity\] Let $\Gamma_1,\Gamma_2 \subset S^*(S^*V)$ be closed sets with $\Gamma_1 \cap s\Gamma_2 = \emptyset$ and set $$\Gamma:=\Gamma_1 \cup \Gamma_2 \cup \left\{(x,[\xi],[\eta_1+\eta_2]):(x,[\xi]) \in S^*V, (x,[\xi],[\eta_1]) \in \Gamma_1, (x,[\xi],[\eta_2]) \in \Gamma_2 \right\}.$$ Then the convolution is a (jointly sequentially) continuous map $$\operatorname{Val}^{-\infty}_{\Gamma_1}(V) \otimes \operatorname{Dens}(V^*) \times \operatorname{Val}^{-\infty}_{\Gamma_2}(V) \otimes \operatorname{Dens}(V^*) \to \operatorname{Val}^{-\infty}_\Gamma(V) \otimes \operatorname{Dens}(V^*).$$ In the notations of Proposition \[prop\_def\_convolution\], we have $\operatorname{WF}(*_1 \tilde T_1)=\operatorname{WF}(\tilde T_1) \subset \operatorname{WF}(T_1) \subset \Gamma_1$ and $\operatorname{WF}(*_1 T_2')=\operatorname{WF}(T_2') \subset \operatorname{WF}(T_2) \subset \Gamma_2$. Since the intersection of currents is a jointly sequentially continuous map $\mathcal{D}_{*,\Gamma_1}(S^*V) \times \mathcal{D}_{*,\Gamma_2}(S^*V) \to \mathcal{D}_{*,\Gamma}(S^*V)$, the statement follows. Whenever it is defined, the convolution is commutative and associative. Let $\phi_j \in \operatorname{Val}^{-\infty}_{\Gamma_i}(V) \otimes \operatorname{Dens}(V^*), j=1,2$. By Lemma \[lemma\_density\_smooth\] there exist sequences $\phi_j^i \in \operatorname{Val}^\infty(V) \otimes \operatorname{Dens}(V^*), j=1,2$ converging to $\phi_j$ in $\operatorname{Val}^{-\infty}_{\Gamma_j}(V) \otimes \operatorname{Dens}(V^*)$. By Proposition \[prop\_continuity\], $\phi_1^i * \phi_2^i$ converges to $\phi_1 * \phi_2$ in $\operatorname{Val}^{-\infty}_{\Gamma}(V)$, while $\phi_2^i * \phi_1^i$ converges to $\phi_2 * \phi_1$. Since the convolution on smooth valuations is commutative, it follows that $\phi_1 * \phi_2=\phi_2 * \phi_1$. For associativity, let $\Gamma_1,\Gamma_2,\Gamma_3 \subset S^*(S^*V)$ be closed sets such that if $(x,[\xi]) \in S^*V, (x,[\xi],[\eta_i]) \in \Gamma_i, i=1,2,3$, then $$\eta_1+\eta_2 \neq 0, \eta_1+\eta_3 \neq 0, \eta_2+\eta_3 \neq 0, \eta_1+\eta_2+\eta_3 \neq 0.$$ One easily checks that both maps $\operatorname{Val}^{-\infty}_{\Gamma_1}(V) \times \operatorname{Val}^{-\infty}_{\Gamma_2}(V) \times \operatorname{Val}^{-\infty}_{\Gamma_3}(V) \to \operatorname{Val}^{-\infty}(V)$ are well-defined. An approximation argument as above shows that they agree. The volume current on the sphere {#sec:Appendix} ================================ The aim of the section is the construction of a certain family of currents on the sphere, which can be used to compute the volume of the convex hull of two polytopes on the sphere. This can be viewed as a generalization of the Gauss formula for area in the plane. The construction in this section uses tools from geometric measure theory, and is independent of the rest of the paper. It is used in the next section for the proof of the second main theorem. A geodesically convex polytope on $S^{n-1}$ is the intersection of a proper convex closed polyhedral cone in $V$ with $S^{n-1}$. We let $\mathcal{P}(S^{n-1})$ denote the set of oriented geodesically convex polytopes on $S^{n-1}$. For $I \in \mathcal{P}(S^{n-1})$ of dimension $k$, denote by $E_I=\operatorname{Span}_V I\cap S^{n-1}$ the $k$-dimensional equator that it spans. For $I,J \subset S^{n-1}$ geodesically convex polytopes, we let $\operatorname{conv}(I,J) \subset S^{n-1}$ denote the union of all shortest geodesic intervals having an endpoint in $I$ and an endpoint in $J$. If $\dim I+\dim J=n-2$, both $I,J$ are oriented and $E_I \cap E_J=\emptyset$, one has a natural orientation on $\operatorname{conv}(I,J)$, by comparing the orientation of $\operatorname{Span}_V(I) \oplus \operatorname{Span}_V(J)=V$ with the orientation of $V$. The geodesically convex polytope $-I$ is oriented in such a way that the antipodal map $I \mapsto -I$ is orientation preserving. Note that $\partial(-I)=-\partial I$ whenever $\dim I>0$, while when $\dim I=0$ we have $\partial (-I)=\partial I$. Here and in the following, $\partial$ denotes the extended singular boundary operator, i.e. for a positively oriented point $I$ we have $\partial I=1$. If $\epsilon=\pm 1$, we write $\operatorname{conv}(I,\epsilon)=I^\epsilon$, where $I^{-1}$ denotes orientation reversal. We denote $A_{n-1}(I,J):=\operatorname{vol}_{n-1}(\operatorname{conv}(I,J))$ provided that $(-I)\cap J=\emptyset$. Note that whenever the orientation of $\operatorname{conv}(I,J)$ is not well-defined, it is a set of volume zero. Note also that $A_{n-1}$ is a partially-defined bi-valuation in $I,J$, so we may extend $A_{n-1}$ as a partially-defined bilinear functional on chains of polytopes. \[lemma\_cancellation\] Let $J \subset S^{n-1}$ be an oriented geodesically convex polytope of dimension $k$. Suppose it does not intersect $\operatorname{Span}_{{\mathbb{R}}^n}(I) \cap S^{n-1}$ for all $I \in \partial n_{F}$, for all $F \in \mathcal{F}_k(P)$. Let $\omega \in \Omega_c^k(\mathbb{R}^{n})$. Then $$\label{eq:conjecture} \sum_{F \in \mathcal{F}_k(P)} \langle \ [v_F],\omega\rangle A_{n-1}(-\partial n_F,J)=0.$$ Remark: Note that if $-\partial n_F=\sum_j I_j$ is the decomposition of $\partial n_F$ into geodesically convex polytopes, then by definition $$A_{n-1}(-\partial n_F,J)=\sum A_{n-1}(I_{j},J)$$ is the sum of the oriented volumes. Note also that in formula (\[eq:conjecture\]) the orientation of $F$ in each summand appears twice, and so the summands are well-defined. For $k=n-1$, this is the well-known statement that $$\sum_{F\in\mathcal{F}_{n-1}(P)} [v_F]=0,$$ where the orientation is given by fixing the outer normals to $P$. Now for $k<n-1$, fix an arbitrary orientation for all faces of dimension $k$ and $(k+1)$. For a pair of faces $F \in \mathcal{F}_k(P)$, $G \in \mathcal{F}_{k+1}(P)$, s.t. $F \subset \partial G$, define $\mathrm{sign}(F,G)=\pm 1$ according to the orientation. Then $$A_{n-1}(-\partial n_F,J)=\sum_{G\supset F} A_{n-1}(-n_G,J)\mathrm{sign}(F,G),$$ where the sum is over all $G \in \mathcal{F}_{k+1}(P)$ containing $F$ in their boundary, and therefore $$\sum_{F \in \mathcal{F}_k(P)} \langle [v_F],\omega\rangle A_{n-1}(-\partial n_F,J) = \sum_{G \in \mathcal{F}_{k+1}(P)} A_{n-1}(-n_{G},J) \sum_{F \subset G}\mathrm{sign}(F,G)\langle [v_F],\omega\rangle,$$ but the internal sum is obviously zero by the case $k=n-1$. \[lemma\_symmetry\] Let $I,J \subset S^{n-1}$ be oriented, geodesically convex polytopes with $\dim I=k$, $\dim J=n-1-k$, such that $J \cap E_I=\emptyset$. Then $$A_{n-1}(\partial I,J)= (-1)^k A_{n-1}(I,\partial J).$$ Let us consider $I,J$ as singular cycles, such that the singular boundary operator on an oriented point equals its sign. Denote $\partial I=\sum_i I_i$, $\partial J=\sum_j J_j$, where $I_i,J_j$ are geodesically convex. Choose any point $x \in S^{n-1}$ outside $I \cup J$, and let $H=S^{n-1} \setminus\{x\}$. Choose a form $\beta \in \Omega^{n-2}(H)$ such that $d\beta=\operatorname{vol}_{n-1}$. Then $$\partial \operatorname{conv}(I_{i},J)=\operatorname{conv}(\partial I_{i},J)+ (-1)^k\sum_j \operatorname{conv}(I_{i},J_{j}),$$ and since $\partial^{2}=0$, we can write $$\partial\sum_i \operatorname{conv}(I_{i},J)= (-1)^k \sum_{i,j} \operatorname{conv}(I_{i},J_{j}).$$ Similarly, $$\partial \sum_j \operatorname{conv}(I,J_{j})=\sum_{i,j} \operatorname{conv}(I_{i},J_{j}).$$ Therefore $$\begin{aligned} A_{n-1}(\partial I,J) & = \left\langle \operatorname{vol}_{n-1}, \sum_i \operatorname{conv}(I_{i},J)\right\rangle\\ & = \left\langle \beta,\partial \sum_i \operatorname{conv}(I_{i},J)\right\rangle\\ & = (-1)^k\left\langle\beta,\sum_i \sum_j \operatorname{conv}(I_{i},J_{j})\right\rangle\\ & = (-1)^k A_{n-1}(I,\partial J),\end{aligned}$$ concluding the proof. The following proposition is the main result of this section. It shows that in fact $A_{n-1}$ can be uniquely extended as a bilinear functional on all chains. \[prop\_volume\_current\] Given $I \in \mathcal{P}(S^{n-1})$ of dimension $k$, where $0 \leq k \leq n-2$, there exists a unique $L^1$-integrable form $\omega_I \in \Omega^{n-k-2}(S^{n-1} \setminus I)$, such that for any $J \in \mathcal{P}(S^{n-1})$ of dimension $(n-k+2)$ with $J \cap I=\emptyset$ one has $$\label{eq_omegaI} \int_J \omega_I=A_{n-1}(-I,J).$$ The current $T_I \in \mathcal{D}_{k+1}(S^{n-1})$ with $$\langle T_I,\phi\rangle:=\int_{S^{n-1}} \omega_I \wedge \phi, \quad \phi \in \Omega^{k+1}(S^{n-1})$$ has the following properties: 1. $T_I$ is additive (i.e. $T_I=T_{I_1}+T_{I_2}$ whenever $I=I_1 \cup I_2$ with geodesically convex polytopes $I_1,I_2$ such that $I_1 \cap I_2$ is a common face of $I_1$ and $I_2$); 2. the singular support of $T_I$ equals $I$; 3. If $k>0$, then $$\label{eq_boundary_T_I} \partial T_I= (-1)^{nk+1} \operatorname{vol}(S^{n-1})[[I]]+ (-1)^{n+1} T_{\partial I},$$ where $[[I]] \in \mathcal{D}_k(S^{n-1})$ is the $k$-current of integration over $I$; 4. If $k=0$, $$\partial T_I=-\operatorname{vol}(S^{n-1})[[I]]+(-1)^nT_{\partial I},$$ where we adopt the convention $\omega_{\pm 1}:=\mp \operatorname{vol}_{n-1} \in \Omega^{n-1}(S^{n-1}), T_{\pm 1}=\mp [[S^{n-1}]] \llcorner \operatorname{vol}_{n-1} \in \mathcal{D}_0(S^{n-1})$. Then holds also for $I=\pm 1$. *Step 1.* Define for $v\in S^{n-1}$ the hemisphere $H_v:=\{p\in S^{n-1}: \langle p,v\rangle>0\}$. Let $W \subset (S^{n-1})^n$ be the set of $n$-tuples $(p_1,...,p_n)$ belonging to some $H_v$. Define $F: W \to {\mathbb{R}}$ by $F(p_1,...p_n)=\operatorname{vol}_{n-1}(\Delta(p_1,...,p_n))$, the oriented volume of the geodesic simplex $\Delta(p_1,\ldots,p_n)$ with vertices $p_1,\ldots,p_n$. $F$ is well defined and smooth, since all $p_j$ lie in one hemisphere. For two non-antipodal points $q,p \in S^{n-1}$, we define $$\omega_{q,p} \in \Lambda^k T ^*_qS^{n-1} \otimes \Lambda^{n-k-2} T ^*_qS^{n-1}$$ by setting, for $u_1,\ldots,u_k \in T_q S^{n-1}$, $v_1,\ldots,v_{n-k-2}\in T_p S^{n-1}$ $$\begin{gathered} \omega_{q,p}(u_1,\ldots,u_k,v_1,\ldots,v_{n-k-2})\\ :=\left.\frac{d^{n-2}}{ds^k dt^{n-k-2}}\right|_{s,t=0} F(q, \gamma_1(s) ,\ldots,\gamma_k(s),p,\delta_1(t),\ldots,\delta_{n-k-2}(t)),\end{gathered}$$ where $\gamma_i$ resp. $\delta_j$ are any smooth curves through $q$ resp. $p$ such that $\gamma_i'(0)=u_i$, $\delta_j'(0)=v_j$. It is immediate that the definition is independent of the choice of such curves, and that $\omega_{q,p}$ defines a unique element $\omega\in \Omega^{k,n-k-2}(S^{n-1}\times S^{n-1}\setminus \overline\Delta)$, where $ \overline\Delta=\{(q,-q):q\in S^{n-1}\} \subset S^{n-1} \times S^{n-1}$ denotes the skew-diagonal. Note that $$\begin{gathered} F(q,\gamma_1(\epsilon),\ldots,\gamma_k(\epsilon),p,\delta_1(\epsilon),\ldots,\delta_{n-k-2}(\epsilon))\\ =\frac{1}{k!(n-k-2)!}\omega_{q,p}(u_1,\ldots,u_k,v_1,\ldots,v_{n-k-2})\epsilon^{n-2}+o(\epsilon^{n-2}). \end{gathered}$$ Given an oriented geodesic $k$-dimensional polytope $I\subset S^{n-1}$, define $\omega_I \in \Omega^{n-k-2}(S^{n-1}\setminus I)$ by $$\omega_I\big|_p=\int_I\omega_{-q,p}dq.$$ Let us verify that $\int_J \omega_I=A_{n-1}(-I,J)$ for an $(n-k-2)$-dimensional geodesic polytope $J$ such that $J\cap I=\emptyset$. Since both sides are additive in both $I,J$, we may assume that $I,J$ are geodesic simplices. We may further assume that there are vector fields $U_1,\ldots,U_k$ on $-I$ that are orthonormal and tangent to $-I$, and $V_1,\ldots,V_{n-k-2}$ on $J$ orthonormal and tangent to $J$. These vector fields define flow curves on $-I,J$. For $\epsilon>0$ one can use those curves to define a grid on $-I$, resp. $J$ denoted $\{-q_i\}$ resp. $\{p_j\}$, defining parallelograms $-Q_i$ resp. $P_j$ of volumes $\epsilon^k+o(\epsilon^k)$ resp. $\epsilon^{n-k-2}+o(\epsilon^{n-k-2})$. Note that the volume of the convex hull of two $\epsilon$-simplices is equal, up to $o(\epsilon^{n-2})$, to $\frac{1}{k!(n-k-2)!}$ times the volume of the convex hull of the corresponding parallelograms. Thus the total volume is given by $$\begin{aligned} A(-I,J) & =\sum A(-Q_i,P_j)\\ & =\sum _{i,j}\left(\omega_{-q_i,p_j}(U_1,\ldots,U_k,V_1,\ldots,V_{n-k-2})\epsilon^{n-2}+o(\epsilon^{n-2})\right)\\ & =\int_{(-I)\times J}\omega +o(1).\end{aligned}$$ Taking $\epsilon\to 0$, this proves the claim. *Step 2.* Let $J \subset S^{n-1}\setminus I$ be a geodesic polytope of dimension $(n-k-1)$. Then $$\begin{aligned} \int_J d\omega_I & = \int_{\partial J} \omega_I\\ & = A_{n-1}(-I,\partial J) \\ & = (-1)^k A_{n-1}(-\partial I,J) \quad \text{ by Lemma \ref{lemma_symmetry}} \\ & = (-1)^k \int_J \omega_{\partial I}.\end{aligned}$$ It follows that on $S^{n-1} \setminus I$ we have $$d\omega_I= (-1)^k \omega_{\partial I}.$$ *Step 3.* Let us verify that $\omega_I$ is an integrable section of $\Omega^{n-k-2}(S^{n-1} \setminus I)$, and therefore admits a unique extension to all of $S^{n-1}$ as a current of finite mass. Introduce spherical coordinates $$\begin{aligned} \Phi_n:[0,2\pi] \times [0,\pi]^{n-2} & \to S^{n-1}\\ (\theta_0,\theta_1,\ldots,\theta_{n-2}) & \mapsto \Phi_n(\theta_0,\theta_1,\ldots,\theta_{n-2}),\end{aligned}$$ which are inductively defined by $$\begin{aligned} \Phi_2(\theta_0) & :=(\cos \theta_0,\sin \theta_0),\\ \Phi_n(\theta_0,\theta_1,\ldots,\theta_{n-2}) & := \left(\sin \theta_{n-2} \Phi_{n-1}(\theta_0,\theta_1,\ldots,\theta_{n-3}),\cos \theta_{n-2}\right).\end{aligned}$$ Note that $\theta_{n-2}$ is defined on the whole sphere $S^{n-1}$ and smooth outside $\{\theta_{n-2}=0,\pi\}=S^0$, while for $i>0$, $\theta_{n-2-i}$ is undefined in $\{\theta_{n-1-i}=0,\pi\} \cup \{\theta_{n-1-i}\text{ undefined}\}=S^{i}$, and constitutes a coordinate outside $\{\theta_{n-2-i}=0,\pi\}$. The volume form of $S^{n-1}$ is given by $$\operatorname{vol}_{n-1}=\prod_{i=0}^{n-3} \sin^{n-2-i} \theta_{n-2-i} \bigwedge_{i=0}^{n-2} d\theta_{n-2-i}.$$ Define for $0 \leq i\leq n-2$ the vector fields $$X_{n-2-i}=\frac{1}{\prod_{j=0}^{i-1}\sin^{n-2-j}\theta_{n-2-j}}\frac{\partial}{\partial\theta_{n-2-i}}.$$ The vector field $X_{n-2-i}$ is well defined outside the set $\{\theta_{n-2-i}=0,\pi\}$. Whenever two such vector fields are defined, they are pairwise orthonormal. Now $\omega_I$ is integrable if $$\int_{S^{n-1}} |\omega_I(X_{i_1},\ldots,X_{i_{n-k-2}})| \operatorname{vol}_{n-1}<\infty$$ for all $i_1,\ldots,i_{n-k-2}$. Let $j_1,\ldots,j_{k+1}$ be the indices not appearing in $\{i_1,\ldots,i_{n-k-2}\}$. We consider the common level sets $C=C(\theta_{j_1},\ldots,\theta_{j_{k+1}})$, with volume element $\sigma_C$, so that $$\operatorname{vol}_{n-1}=\left(\prod_{l=1}^{k+1}\sin^{j_l}\theta_{j_l}\wedge_{l=1}^{k+1} d\theta_{j_l}\right)\wedge \sigma_C.$$ Then $$\begin{aligned} \int_{S^{n-1}} & |\omega_{I}(X_{i_1},\ldots,X_{i_{n-k-2}})| \operatorname{vol}_{n-1} \\ & =\int_{\theta_{j_1},\ldots,\theta_{j_{k+1}}} \prod_{l=1}^{k+1} \sin^{j_l}\theta_{j_l} \prod_{l=1}^{k+1}d\theta_{j_l} \int_{C(\theta_{j_1},\ldots,\theta_{j_{k+1}})}|\omega_I(X_{i_1},\ldots,X_{i_{n-k-2}})| \sigma_C.\end{aligned}$$ While $C(\theta_{j_{1}},...,\theta_{j_{k+1}})$ is not a geodesic polytope in $S^{n-1}$, it nevertheless holds by the definition of $\omega_I$ that the internal integral is bounded by the total area of the sphere. Thus the entire integral is finite. We can therefore define the current $T_I \in \mathcal{D}_{k+1}(S^{n-1})$ by $$\langle T_I,\phi\rangle := \int_{S^{n-1} \setminus I} \omega_I \wedge \phi, \quad \phi \in \Omega^{n-k-2}(S^{n-1}).$$ *Step 4.* We prove by induction on $k$. For the induction base $k=0$, recall that $T_1=-[[S^{n-1}]] \llcorner \operatorname{vol}_{n-1}, \omega_1=-\operatorname{vol}_{n-1}$. The $0$-dimensional geodesic polytope $I$ is just a point, which we may suppose to be positively oriented. Then $-I$ is the positively oriented antipodal point. Let $S^{n-1}_\epsilon$ be the sphere $S^{n-1}$ minus the geodesic ball of radius $\epsilon$ centered at $I$. For $g \in C^{\infty}(S^{n-1})$, we compute $$\begin{aligned} \langle \partial T_I,g\rangle & = \langle T_I,dg\rangle\\ & = \int_{S^{n-1} \setminus \{I\}} \omega_I \wedge dg\\ & = \lim_{\epsilon \to 0} \int_{S^{n-1}_\epsilon} \omega_I \wedge dg\\ & = \lim_{\epsilon \to 0} \left[ (-1)^{n+1}\int_{S^{n-1}_\epsilon} d\omega_I \wedge g+ (-1)^n\int_{\partial S^{n-1}_\epsilon} \omega_I \wedge g\right]\\ & = (-1)^{n+1} \int_{S^{n-1}} g \operatorname{vol}_{n-1}+ (-1)^n \lim_{\epsilon \to 0} \int_{\partial S^{n-1}_\epsilon} \omega_I \wedge g.\end{aligned}$$ The boundary of $S^{n-1}_\epsilon$ is an $(n-2)$-dimensional geodesic sphere around $I$. Since $\int_{\partial S^{n-1}_\epsilon} \omega_I=A_{n-1}(-I, \partial S_\epsilon^{n-1})=(-1)^{n-1}\operatorname{vol}S^{n-1}_\epsilon$ (note that Lemma (\[lemma\_symmetry\]) does not apply here, as $S_\epsilon$ is not geodesically convex), the second integral tends to $(-1)^{n-1}g(I)$ times the volume of $S^{n-1}$. It follows that $$\partial T_I= -\operatorname{vol}(S^{n-1}) [[I]]+ (-1)^{n} T_1,$$ as claimed. *Step 5.* Suppose now that $k>0$ and that holds for all polytopes of dimension strictly smaller than $k$. Define the current $U_I:= \partial T_I+(-1)^n T_{\partial I} \in \mathcal{D}_k(S^{n-1})$. By step 2 and equation , $U_I$ is supported on $I$. We have to show that $U_I= (-1)^{nk+1} \operatorname{vol}(S^{n-1}) [[I]]$. Choose a family of closed neighborhoods $I_\epsilon$ with smooth boundary such that $I_\epsilon$ converges to $I$ as $\epsilon \to 0$. Define currents $T_{I,\epsilon}, V_{I,\epsilon}, U_{I,\epsilon}$ on $S^{n-1}$ by $$\begin{aligned} \langle T_{I,\epsilon},\phi\rangle & := \int_{S^{n-1} \setminus I_\epsilon} \omega_I \wedge \phi, \quad \phi \in \Omega^{k+1}(S^{n-1}),\\ \langle V_{I,\epsilon},\phi\rangle & := \int_{S^{n-1} \setminus I_\epsilon} \omega_{\partial I} \wedge \phi,\quad \phi \in \Omega^k(S^{n-1}),\\ \langle U_{I,\epsilon},\phi\rangle & := (-1)^{n+k} \int_{\partial (S^{n-1} \setminus I_\epsilon)} \omega_I \wedge \phi, \quad \phi \in \Omega^k(S^{n-1}).\end{aligned}$$ By Step 3, $\mathbf{M}(T_{I,\epsilon}-T_I) \to 0, \mathbf{M}(V_{I,\epsilon}-T_{\partial I})\to 0$ as $\epsilon \to 0$. Since $U_{I,\epsilon}$ is given by integration of a smooth form on a compact smooth manifold, it is a normal current (i.e. its mass and the mass of its boundary are finite). By Stokes’ theorem and Step 2, we have $U_{I,\epsilon}= \partial T_{I,\epsilon} + (-1)^n V_{I,\epsilon} $. Therefore $$\mathbf{F}(U_{I,\epsilon}-U_I)=\mathbf{F}(\partial(T_{I,\epsilon}-T_I)+ (-1)^n(V_{I,\epsilon}-T_{\partial I})) \leq \mathbf{M}(T_{I,\epsilon}-T_I)+\mathbf{M}(V_{I,\epsilon}-T_{\partial I}) \to 0.$$ It follows that $U_I$ is a real flat $k$-chain supported on the $k$-dimensional spherical polytope $I$. By induction, we have $\partial T_{\partial I}= (-1)^{nk+n+1}\operatorname{vol}(S^{n-1}) [[\partial I]]$ and hence $\partial U_I= (-1)^n \partial T_{\partial I}= (-1)^{nk+1} \operatorname{vol}(S^{n-1}) [[\partial I]]$. The constancy theorem [@federer_book 4.1.31], [@morgan_book Proposition 4.9] implies that $U_I= (-1)^{nk+1} \operatorname{vol}(S^{n-1}) [[I]]$, as claimed. For further use, we give the following current-theoretic interpretation of . If $\phi \in \Omega^{k+1}_c(S^{n-1} \setminus I)$, then $T_I \cap \left([[S^{n-1}]] \llcorner \phi\right) = [[S^{n-1}]] \llcorner (\omega_I \wedge \phi)$ and hence $$\langle T_I \cap ([[S^{n-1}]] \llcorner \phi),1\rangle=\int_{S^{n-1}} \omega_I \wedge \phi=(-1)^{(n-k-2)(k+1)} \langle [[S^{n-1}]] \llcorner \phi,\omega_I\rangle.$$ Let $\phi_i \in \Omega^{k+1}_c(S^{n-1} \setminus I)$ be a sequence with $[[S^{n-1}]] \llcorner \phi_i \to [[J]]$ in $\mathcal{D}_{n-k-2,\operatorname{WF}([[J]])}(S^{n-1})$ Since $\operatorname{WF}(T_I) \cap s \operatorname{WF}([[J]])=\emptyset$ by the second item, $\langle T_I \cap ([[S^{n-1}]] \llcorner \phi_i),1\rangle \to \langle T_I \cap [[J]],1\rangle$, while $\langle [[S^{n-1}]] \llcorner \phi_i, \omega_I\rangle \to \int_J \omega_I=A_{n-1}(-I,J)$. Therefore $$\label{eq_intersection_volume_current} \langle T_I \cap [[J]],1\rangle = (-1)^{n(k+1)} A_{n-1}(-I,J).$$ \[prop\_boundary\_tildet\] Let $P$ be a polytope and let $(T,C):=E(M([P]))$ the associated currents. Decompose $T=t+T'$ as in Proposition \[prop\_def\_convolution\], where $t$ is the $(0,n-1)$-component of $T$ and $T'=\sum_{k=1}^{n-1} \sum_{F \in \mathcal{F}_k(P)} A_{v_F,\check{n}(F,P)}$. Let $$\tilde T:= \frac{1}{\operatorname{vol}(S^{n-1})} \sum_{k=1}^{n-1} (-1)^{nk+k+1}\sum_{F \in \mathcal{F}_k(P)} [v_F] \times T_{\check n(F,P)} \in \mathcal{D}_n( SV).$$ Then $$\partial \tilde T=T'.$$ By and Proposition \[prop\_volume\_current\] we have $$\begin{aligned} \partial \tilde{T} & = \frac{1}{\operatorname{vol}(S^{n-1})} \sum_{k=1}^{n-1} (-1)^{nk+k+1} \sum_{F \in \mathcal{F}_k(P)} \partial([v_F] \times T_{\tilde n(F,P)})\\ & = \frac{1}{\operatorname{vol}(S^{n-1})} \sum_{k=1}^{n-1} (-1)^{nk+1} \sum_{F\in \mathcal{F}_k(P)} [v_F] \times \partial T_{\tilde n(F,P)}\\ & =\sum_{k=1}^{n-1} \sum_{F \in \mathcal{F}_k(P)} A_{v_F,\check{n}(F,P)}+\frac{1}{\operatorname{vol}(S^{n-1})} \sum_{k=1}^{n-1} \sum_{F\in \mathcal{F}_k(P)} (-1)^{nk+n} [v_F] \times T_{\partial n_F}.\end{aligned}$$ Let $ 1 \leq k \leq n-1$ be fixed and let $J \subset S^{n-1}$ be a $(k-1)$-dimensional geodesic polytope not intersecting any $\partial n_F$ for $F \in \mathcal{F}_k(P)$. Lemma \[lemma\_cancellation\] implies that $$\sum_{F \in \mathcal{F}_k(P)} \langle [v_F],\omega\rangle T_{\partial n_F} \cap [[J]] =0$$ for all $\omega \in \Omega_c^k({\mathbb{R}}^n)$. It follows that $\sum_{F \in \mathcal{F}_k(P)} [v_F] \times T_{\partial n_F}=0$, and the statement follows. Compatibility of the algebra structures {#sec_compatibility} ======================================= From now on, we fix an orientation and a Euclidean scalar product on $V$ and identify $\operatorname{Dens}(V) \cong \mathbb{C}, \operatorname{or}(V) \cong \mathbb{C}, S^*V \cong SV=V \times S^{n-1}$. It then holds that $$\langle \ast_1 T,\omega\rangle =(-1)^{nk+nl+k}\langle T,\ast_1\omega \rangle$$ for $T\in \mathcal D_{k,l}(S^*V)^{tr}$, $\omega \in \Omega^{k,n-1-l}(S^*V)^{tr}$. \[prop\_star\_product\] Let $v_i \in \Lambda^{k_i}_sV$ and $T_i \in \mathcal{D}_{l_i}(S^{n-1}), i=1,2$ be currents on the sphere such that $T_1 \cap T_2 \in \mathcal{D}_{l_1+l_2-n+1}(S^{n-1})$ is defined. Then $$\ast_1 ([v_1] \times T_1) \cap \ast_1([v_2] \times T_2) = (-1)^{k_1(n-k_2-l_2-1)}\ast_1 \left([v_1 \wedge v_2] \times (T_1 \cap T_2)\right).$$ In particular, for $A_{v_i,N_i} \in Y_{k_i}$ with $N_1$ and $N_2$ being transversal, one has $$\ast_1 A_{v_1,N_1} \cap \ast_1 A_{v_2,N_2}= \ast_1 A_{v_1 \wedge v_2,N_1 \cap N_2}.$$ Let $v \in \Lambda^k_s V, T \in \mathcal{D}_l(S^{n-1})$. We compute for $\gamma_1 \in \Omega_c^{n-k}(V), \gamma_2 \in \Omega_c^l(S^{n-1})$ $$\begin{aligned} \langle \ast_1 ([v] \times T),\pi_1^* \gamma_1 \wedge \pi_2^*\gamma_2\rangle & =(-1)^{nk+nl+k}\langle [v] \times T,\ast_1 (\pi_1^* \gamma_1 \wedge \pi_2^*\gamma_2)\rangle\\ & =(-1)^{nk+nl+k+\binom{k}{2}} \langle [v],\ast \gamma_1\rangle \cdot \langle T,\gamma_2\rangle\\ & =(-1)^{nk+nl+k+\binom{k}{2}+k(n-k)} \langle [\ast v], \gamma_1\rangle \cdot \langle T,\gamma_2\rangle\\ & =(-1)^{\binom{k}{2}+nl} \langle [\ast v] \times T,\pi_1^* \gamma_1 \wedge \pi_2^*\gamma_2\rangle,\end{aligned}$$ i.e. $$\label{eq_star1} \ast_1 ([v] \times T)=(-1)^{\binom{k}{2}+nl} [\ast v] \times T.$$ Let now $v_i \in \Lambda^{k_i}_s V, T_i \in \mathcal{D}_{l_i}(S^{n-1}), i=1,2$. We have $[*v_1] \cap [*v_2]=[*(v_1 \wedge v_2)]$. Using it follows that $$([*v_1] \times T_1) \cap ([*v_2] \times T_2) = (-1)^{k_1(n-1-l_2)} [*(v_1 \wedge v_2)] \times (T_1 \cap T_2).$$ The statement now follows from . Two elements $x,y \in \Pi(V)$ are in general position, if any two normal cones to a pair of faces of $x$ and $y$ are transversal. Given two elements $x,y\in\Pi(V)$ in general position, the convolution $M(x) \ast M(y)$ is well-defined, and $M(x\cdot y)=M(x) \ast M(y)$. By linearity, it suffices to consider $x=[P]$ and $y=[Q]$ for some polytopes $P,Q$. The wavefront of the current $A_{v,N} \in Y_k$ is the conormal bundle to $N$ in $S^{n-1}$. Thus $\operatorname{WF}( A_{v_1,N_1}) \cap \operatorname{WF}( A_{v_2,N_2})=\emptyset$ if and only if for all $x \in N_1 \cap N_2$ we have $$T_x N_1+T_x N_2=T_x S^{n-1},$$ that is, if and only if $N_1$ and $N_2$ are transversal. Recall from Lemma \[lemma\_image\_of\_polytope\] that for $0 \leq k \leq n-1$ $$T(M_{n-k}[P])=\sum_{F \in \mathcal{F}_k(P)} A_{v_F,\check{n}(F,P)}.$$ Thus, given $[P],[Q] \in \Pi(V)$ in general position, the normal cones in $S^{n-1}$ have disjoint wavefronts. Let $F$ be a face of $P$ and $G$ a face of $Q$. If $\dim F+\dim G \geq n$ then $\check{n}(F,P) \cap \check{n}(G,Q)=\emptyset$ by transversality. Thus, by Proposition \[prop\_star\_product\] $$\begin{aligned} \ast_1 T(M([P]) \ast M([Q])) & =\sum_{\dim F+\dim G<n} \ast_1 A_{v_F,\check{n}(F,P)} \cap \ast_1 A_{v_G,\check{n}(G,Q)}\\ & =\sum_{\dim F+\dim G<n} \ast_1 A_{v_F \wedge v_G,\check{n}(F,P) \cap \check{n}(G,Q)}\\ & =\ast_1 \sum_{H \in \mathcal{F}(P+Q)} A_{v_H,\check{n}(H,P+Q)},\end{aligned}$$ and so $$T(M[P] * M[Q]) =\sum_{H\in \mathcal{F}(P+Q)} A_{v_{H},\check{n}(H,P+Q)}=T(M[P+Q]).$$ It remains to verify that $C(M[P] \ast M[Q])=C(M[P+Q])$. We set $$T_1=t_1+T_1':=T(M[P]), T_2=t_2+T_2':=T(M[Q])$$ as in Proposition \[prop\_def\_convolution\]. Then $t_1=\sum_{F \in \mathcal{F}_0(P)} A_{F,n(F,P)}=\pi^*( [[V]] \llcorner \operatorname{vol}_n)$, hence $\alpha_1=1$. Similarly $\alpha_2=1$. Let $F$ be a $k$-dimensional face of $P$ and $G$ an $(n-k)$-dimensional face of $Q$, where $1 \leq k \leq n-1$. By Proposition \[prop\_star\_product\], $$\begin{aligned} \ast_1([v_F] \times T_{\check n(F,P)})\cap \ast _1 (A_{G, \check n(G, Q)}) & = \ast_1([v_F\wedge v_G] \times (T_{\check n(F,P)}\cap [[\check n(G,Q)]]).\end{aligned}$$ Summing over all faces of $P$ and $Q$, using and Proposition \[prop\_boundary\_tildet\], we obtain that $$\begin{aligned} C( & M[P] * M[Q]) = \pi_* *_1^{-1} (*_1 \tilde T_1 \cap *_1 T_2') + \alpha_1 C_2+\alpha_2 C_1\\ & = \sum_{k=1}^{n-1} (-1)^{nk+k+1} \sum_{F \in \mathcal{F}_k(P)} \pi_* *_1^{-1} (*_1 ([v_F] \times T_{\check n(F,P)}) \cap *_1 T_2') \\ & \quad + \alpha_1 C_2+\alpha_2 C_1\\ & =\sum_{k=1}^{n-1} (-1)^{nk+k+1} \sum_{\substack{F \in \mathcal{F}_k(P)\\G \in \mathcal{F}_{n-k}(Q)}} \pi_* \left([v_F\wedge v_G] \times (T_{\check n(F,P)}\cap [[\check n(G,Q)]])\right) \\ & \quad + \operatorname{vol}(P)[[V]]+\operatorname{vol}(Q)[[V]]\\ & =\sum_{k=1}^{n-1} (-1)^{n(n-k)+nk+k+1} \sum_{\substack{F \in \mathcal{F}_k(P)\\G \in \mathcal{F}_{n-k}(Q)}} [v_F\wedge v_G] \operatorname{vol}(\operatorname{conv}(-\check n(F,P),\check n(G,Q))) \\ & \quad + \operatorname{vol}(P)[[V]]+\operatorname{vol}(Q)[[V]]\\ & =\sum_{k=1}^{n-1} (-1)^{n+k+1} \sum_{\substack{F \in \mathcal{F}_k(P)\\G \in \mathcal{F}_{n-k}(Q)}} [v_F\wedge v_G] \operatorname{vol}(\operatorname{conv}(-\check n(F,P),\check n(G,Q))) \\ & \quad + \operatorname{vol}(P)[[V]]+\operatorname{vol}(Q)[[V]].\end{aligned}$$ Note that the pair $(-1)^{n+\dim F+1}(v_F,-\check n(F,P))$ is positively oriented, and coincides with $(v_{-F}, \check n(-F,-P))$. 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--- abstract: 'We study stability of gas accretion in Active Galactic Nuclei (AGN). Our grid based simulations cover a radial range from 0.1 to 200 pc, which may enable to link the galactic/cosmological simulations with small scale black hole accretion models within a few hundreds of Schwarschild radii. Here, as in previous studies by our group, we include gas radiative cooling as well as heating by a sub-Eddington X-ray source near the central supermassive black hole of $10^8 M_{\odot}$. Our theoretical estimates and simulations show that for the X-ray luminosity, $L_X \sim 0.008~L_{Edd}$, the gas is thermally and convectivelly unstable within the computational domain. In the simulations, we observe that very tiny fluctuations in an initially smooth, spherically symmetric, accretion flow, grow first linearly and then non-linearly. Consequently, an initially one-phase flow relatively quickly transitions into a two-phase/cold-hot accretion flow. For $L_X = 0.015~L_{Edd}$ or higher, the cold clouds continue to accrete but in some regions of the hot phase, the gas starts to move outward. For $L_X < 0.015~L_{Edd}$, the cold phase contribution to the total mass accretion rate only moderately dominates over the hot phase contribution. This result might have some consequences for cosmological simulations of the so-called AGN feedback problem. Our simulations confirm the previous results of Barai et al. (2012) who used smoothed particle hydrodynamic (SPH) simulations to tackle the same problem. However here, because we use a grid based code to solve equations in 1-D and 2-D, we are able to follow the gas dynamics at much higher spacial resolution and for longer time in comparison to the 3-D SPH simulations. One of new features revealed by our simulations is that the cold condensations in the accretion flow initially form long filaments, but at the later times, those filaments may break into smaller clouds advected outwards within the hot outflow. Therefore, these simulations may serve as an attractive model for the so-called Narrow Line Region in AGN.' author: - 'M. Mo[ś]{}cibrodzka$^{1,\dagger}$, D. Proga$^{1}$' title: Thermal and dynamical properties of gas accreting onto a supermassive black hole in an AGN --- Introduction ============ Physics within the central parsecs of a galaxy is dominated by the gravitational potential of a compact supermassive object. In a classical theory of spherical accretion by @bondi:1952, Bondi radius $R_B$ determines the zone of the gravitational influence of a central object and it is given by $R_{B} \approx 150 (M_{BH}/10^8 M_{\odot}) (T_{\infty}/10^5 {\rm K})^{-1} \, {\rm pc}$, where $M_{BH}$ is the central object mass, and $T_{\infty}$ is the temperature of the uniform surrounding medium. At radii smaller than the Bondi radius, $R_{B}$, the interstellar medium (ISM) or at least its part is expected to turn into an accretion flow. Physics of any part of a galaxy is complex. However near the Bondi radius, it is particularly so because there, several processes compete to dominate not only the dynamical state of matter but also other states such as thermal and ionization. Therefore studies of the central parsec of a galaxy require incorporation of processes and their interactions that are typically considered separately in specialized areas of astrophysics, e.g., the black hole accretion, physics of ISM and of the galaxy formation and evolution. One of the main goals of studying the central region of a galaxy is to understand various possible connections between a supermassive black hole (SMBH) and its host galaxy. Electromagnetic radiation provides one of such connections. For example, the powerful radiation emitted by an AGN, as it propagates throughout the galaxy, can heat up and ionize the ISM. Subsequently, accretion could be slowed down, stopped or turned into an outflow if the ISM become unbound. Studies of heated accretion flows have a long history. Examples of early and key works include @ostriker:1976, @cowie:1978, @mathews:1978, @stell:1982, @bisnovatyi:1980, @krolik:1983, and @balbus:1989. The accretion flows and their related outflows are very complex phenomena. It is likely that several processes are responsible for driving an outflow, i.e., not just the energy of the radiation, as mentioned above, but also for example, the momentum carried by the radiation. Therefore, our group explored combined effects of the radiation energy and momentum on the accretion flows and on producing outflows (e.g. @proga:2007; @proga:2008; @kurosawa:2008; @kurosawa:2009a; @kurosawa:2009b; @kurosawa:2009c). These papers reported on results from simulations carried out using Eulerian finite difference codes where effects of gas rotation and other complications such non-spherical and non-azimuthal effects were included (see e.g. @janiuk:2008). To identify the key processes determining the gas properties (here, we are mainly concerned with thermal properties) and to establish any code limitations in modeling an accretion flow, in this paper we adopt a relatively simple physical set up. Namely, the modeled system consists of a central SMBH of mass $M_{BH}= 10^8 M_{\odot}$ and a spherical shell of gas inflowing to the center. The simulations focus on regions between 0.1-200 pc from the central object, where the outer boundary is outside of $R_B$. The key difference compared to the Bondi problem is an assumption that the central accretion flow is a point-like X-ray source. The X-rays illuminate the accreting gas and the gas itself is allowed to cool radiatively under optically thin conditions. To keep the problem as simple as possible, the radiation luminosity is kept fixed instead of being computed based on the actual accretion rate for an assumed radiation efficiency (see also @kurosawa:2009a). To model the presented problem, one needs to introduce extra terms into the energy equation to account for energy losses and gains. The physics of an optically thin gas that is radiatively heated and cooled, in particular, its thermal and dynamical stability has been analyzed in a great detail by @field:1965. Therefore, to study thermal properties of accretion flows or dynamical properties of thermally unstable gas, it is worthwhile to combine @bondi:1952 and @field:1965 theories. Notice, that our set-up is very similar that that used in the early works in the 70-ties and 80-ties that we mentioned above. Some kind of complexity and time variability in a heated accretion flow is expected based on the 1-D results from the early work. A dynamical study of the introduced physical problem requires resolving many orders of magnitude of the radial distance from the black hole. Our goal is not only to cover the largest radial span as possible but also to resolve any small scale structure of the infalling gas. This is a challenging goal. To study the dynamics of gas in a relatively well controlled computer experiment, we use the Eulerian finite difference code ZEUS-MP [@hayes:2006]. We address systematically numerical requirements to adequately treat the problem of thermally unstable accretion flows. We introduce an accurate heating-cooling scheme that incorporates all relevant physical processes of X-ray heating and radiative cooling. Low optical thickness is assumed, which decouples fluid and radiation evolution. We resolve three orders of magnitude in the radial range by using a logarithmic grid where the logarithm base is adjusted to the physical conditions. We solve hydrodynamical equations in 1- and 2- spatial dimensions (1-D and 2-D). We follow the flow dynamics for a long time scale in order to investigate the non-linear phase of gas evolution. Notice that most of the earlier work in the 70-ties and 80-ties, focused on linear analysis of stability, early stages of evolution of the solutions, and considered only 1-D cases. As useful our group’s past studies are, we keep in mind that any result should be confirmed by using more than one technique or approach. Therefore, @barai:2012 (see also @barai:2011), began a parallel effort to model accretion flows including the same physics but instead of performing simulations using a grid based code we used the smoothed particle hydrodynamic (SPH) GADGET-3 code @springel:2005. Overall, the 3-D SPH simulations presented by @barai:2012 showed that despite this very simple set up, accretion flows heated by even a relatively weak X-ray source (i.e., with the luminosity around 1% of the Eddington luminosity) can undergo a complex time evolution and can have a very complex structure. However, the exact nature and robustness of these new 3-D results has not been fully established. @barai:2012 mentioned some numerical issues, in particular, artificial viscosity and relatively poor spacial resolution in SPH, because of the usage of linear length scale (as opposed to logarithmic grid in ZEUS-MP), limit ability to perform a stability analysis where one wishes to introduce perturbations to an initially smooth, time independent solution with well controlled amplitude and spatial distribution (SPH simulations have intrinsic limitations in realizing a smooth flow). Therefore, the robustness and stability of the solutions found in the SPH simulations are hard to access due to mixing of physical processes and numerical effects. Here, we aim at clarifying the physics of these flows and measure the role of numerical effects in altering the effects of physical processes. Our ultimate goal is to provide insights that could help to interpret observations of AGN. We explore the conditions under which the two phase, hot and cold, medium near an AGN can form and exist. Such two phase accretion flows can be a hint to explain the modes of accretion observed in galactic nuclei, but also to explain the formation of broad and narrow lines which define AGN. We also measure the so-called covering and filling factors and other quantities in our simulations in order to relate the simulation to the origin of the broad and narrow line regions (BLRs and NLRs, respectively). The connection of this work to the galaxy evolution and cosmology is that we resolve lower spatial scales, and hence can probe what physical processes affect the accretion flow. In our models, we can directly observe where the hot phase of accretion turns into a cold one or where an eventual outflow is launched. In most of the current simulations of galaxies (e.g. @dimatteo:2012 and references therein) these processes are assumed or modeled by simple, so called sub-resolution, approximations because, contrary to our simulations, the resolution is too low to capture the flow properties on adequately small scales. The article is organized as follows. In § \[sec:equation\], we present the basic equations describing the physical problem. In § \[sec:num\_setup\], we show the details of the numerical set up. Results are in § \[sec:results\_1d\] and in § \[sec:results\_2d\]. We summarize the results in § \[sec:discussion\]. Basic Equations {#sec:equation} =============== We solve equations of hydrodynamics: $$\frac{D\rho}{Dt} + \rho {\bf \nabla \cdot v}=0 \label{eq:mass}$$ $$\rho \frac{D{\bf v}}{Dt} = -\nabla P + \rho {\bf g} \label{eq:mom}$$ $$\rho \frac{D}{Dt} (\frac{e}{\rho})= -P {\bf \nabla \cdot v} + \rho {\mathcal L} \label{eq:energy}$$ where $D/Dt$ is Lagrangian derivative and all other symbols have their usual meaning. To close the system of equations we adopt the $P=(\gamma-1)e$ equation of state where $\gamma = 5/3$. Here $g$ is the gravitational acceleration near a point mass object in the center. The equation for the internal energy evolution has an additional term $\rho {\mathcal L}$, which accounts for gas heating and cooling by continuum X-ray radiation produced by an accretion flow near the central SMBH. The heating/cooling function contains four terms which are: (1) Compton heating/cooling ($G_{Compton}$), (2) heating and cooling due to photoionization and recombination ($G_X$), (3) free-free transitions cooling ($L_{b}$) and (4) cooling via line emission ($L_{l}$) and it is given by (@blondin:1994, @proga:2000): $$\rho {\mathcal L} = n^2 (G_{Compton} + G_X - L_b - L_l ) \, \, {\rm [erg \,\, cm^{-3} s^{-1} ]}\label{eq:HC_full}$$ where $$G_{Compton}=\frac{k_b \sigma_{TH}}{4 \pi m_e c^2} \xi T_X \left(1- \frac{4T}{T_X}\right)$$ $$\label{eq:HC_2} G_X= 1.5 \times 10^{-21} \xi^{1/4} T^{-1/2} \left(1-\frac{T}{T_X}\right)$$ $$L_b= \frac{2^5 \pi e^6}{ \sqrt{27} h m_e c^2 } \sqrt{\frac{2\pi k_b T}{m_ec^2}}$$ $$\label{eq:HC_4} L_l= 1.7 \times 10^{-18} \exp\left(- \frac{1.3 \times 10^5}{T}\right) \xi^{-1} T^{-1/2} - 10^{-24}$$ where $T_X$ is the radiative temperature of X-rays and $T$ is the temperature of gas. We adopt a constant value $T_X = 1.16 \times 10^8$ K ($E=10$ keV) at all times. The numerical constants in Equation \[eq:HC\_2\] and \[eq:HC\_4\] are taken from an analytical formula fit to the results from a photoionization code XSTAR [@kallman:2001]. XSTAR calculates the ionization structure and cooling rates of a gas illuminated by X-ray radiation using atomic data. The photoionization parameter $\xi$ is defined as: $$\xi \equiv \frac{4 \pi F_X}{n} = \frac{L_X} {n r^2} = \frac{f_X L_{Edd}} {n r^2} = \frac{f_X L_{Edd} m_p \mu} {\rho r^2} \,\, {\rm [ergs \,\, cm \,\, s^{-1}]}$$ where $F_X$ is the radiation flux, $n=\rho/(\mu m_p)$ is the number density, and $\mu$ is a mean molecular weight. Given $\xi$ definition, notice that $\mathcal L$ is a function of thermodynamic variables but also strongly depends on the distance from the SMBH. The luminosity of the central source $L_X$ is expressed in units of the Eddington luminosities, $f_X \equiv L_X/L_{Edd}$. The reference Eddington luminosity for a supermassive black hole mass considered in this work is $$L_{Edd} \equiv \frac{ 4 \pi G M_{BH} m_p c}{\sigma_{TH}} = 1.25 \times 10^{46} \left(\frac{M }{10^8 M_{\odot}}\right) \,\, {\rm [ergs \,\, s^{-1}]}$$ Method and Initial Setup {#sec:num_setup} ======================== To solve Equations. \[eq:mass\], \[eq:mom\], \[eq:energy\], we use the numerical code ZEUS-MP [@hayes:2006]. We modify the original version of the code in particular, we use a Newton-Raphson method to find roots of Equation \[eq:energy\] numerically at each time step. We have successfully tested the numerical method against an analytical model with heating and cooling. We describe the numerical code tests in the Appendix, showing the thermal instability (TI) development in the uniform medium. We solve equations in spherical-polar coordinates. Our computational domain extends in radius from 0.1 to 200 pc. The useful reference unit is a radius of the innermost stable circular orbit of a central black hole: $r_*= 6 GM_{BH}/c^2$. We assume the fiducial mass of the black hole $M_{BH}=10^8 M_{\odot}$ for which $r_*=8.84 \times 10^{13} {\rm cm}$. The computational domain in these units ranges from $r_i=3484.2 r_*$ to $r_o=6.9683 \times 10^6 r_*$ (or $r_i = 6.6 10^{-4} R_B$ and $r_o=1.3 R_B$, where $R_B = 152$ pc). Since $r_i$ is relatively large in comparison to the BH horizon we cannot model here the compact regions near the black hole where X-ray emission is produced. Instead we parameterize the X-ray luminosity using $f_X$, so that $L_X=f_X L_{Edd}$. We solve equations for five values of $f_X$=0.0005, 0.008, 0.01, 0.015, 0.02 (these numbers correspond to models later labeled as A, B, C, D and E). As initial conditions for the A model (lowest luminosity), we use an adiabatic, semi-analytical solution from @bondi:1952. For higher luminosities the integration of equations starts from last data from a model with one level lower luminosity provided that the lower $f_X$ solution is time-independent. The procedure is adopted in order to increase the luminosity in a gradual manner rather than sudden. [^1] Only for steady state solutions (with assumption that the mass accretion rate is constant from $r_i$ to $r_*$) the efficiency of conversion of gravitational energy into radiation $\eta$ is related to $f_X$ as $$\frac{\eta}{\eta_r} = \frac{f_X}{ \dot{m}}$$ where $\dot{m}$ is a mass accretion rate in Eddington units ($\dot{M}_{Edd}=L_{Edd}/\eta_r c^2$ and $\eta_r=0.1$ is a reference efficiency) and it is measured from the model data. In our steady state models, $\dot{m} \approx 1$, therefore the energy conversion efficiency in these cases is approximately $\eta=0.1f_X$. Our boundary conditions put constrains on a density at $r_o$, it is set to be $\rho_o=10^{-23} {\rm g \, cm^{-3}}$. For other variables we use an outflow type of boundary conditions at the inner and outer radial boundary. In 2-D models our computational domain extends in $\theta \in (0,90^\circ)$. At the symmetry axis and at the equator we use appropriate reflection boundary conditions. The numerical resolution used depends on the number of dimensions i.e.: in 1-D $N_r=256, 512, 1024, 2048, 4096$; in 2-D $(N_r,N_\theta)=(256,64), (512,128), (1024,256)$. The spacing of the radial grid is set as $dr_i/dr_{i+1}$=1.023, 1.01, 1.0048, 1.002, 1.0008 for $N_r$=256, 512, 1024, 2048, and 4096, respectively. The number of grid points in the second dimension are chosen so that the linear size of the grid zone in all directions is similar (i.e., $r_i \Delta \theta_j \approx \Delta r_i$). Results: 1-D models {#sec:results_1d} =================== 1-D Steady Solutions -------------------- We begin with presenting the basic characteristics of 1-D solutions. Table \[tab:1d\] shows a list of all our 1-D simulations. Each simulation was performed until $t_{f}=20$ Myr equivalent to 4.7 dynamical time scales at the outer boundary $r_o=200$pc ($t_{dyn}=t_{ff}=\sqrt{r_o^3/2GM_{BH}}=4.21$ Myr). Only some of the numerical solutions settled down to a time-independent state at $t_{f}$. We focus on analyzing two representative solutions, that are steady-state at $t_{f}$: 1D256C and 1D256D, with the X-ray luminosity of the former $f_X=10^{-2}$, and of the latter $f_X=1.5 \times 10^{-2}$. Note that these solutions were obtained using the lowest resolution. We find these two solutions instructive in showing the thermal properties of the gas. ---------- ---------------------- ------- --------- -------------------------------- ---------------------------- ------------------------------- -------------------- --------- Model ID $f_X$ $N_r$ $t_f$ $\langle\dot{M}\rangle_t$ $\langle\chi\rangle_{r,t}$ $\langle\tau_{X,sc}\rangle_t$ Max($\tau_{X,sc}$) comment \[Myr\] ${\rm [M_{\odot} \, yr^{-1}]}$ 1D256A $5 \times 10^{-4}$ 256 20 2.0 3.9 0.44 0.49 s 1D512A $5 \times 10^{-4}$ 512 20 2.0 6.1 0.45 0.51 s 1D1024A $5 \times 10^{-4}$ 1024 20 2.0 6.8 0.46 0.52 s 1D2048A $5 \times 10^{-4}$ 2048 20 2.0 6.8 0.46 0.53 s 1D4096A $5 \times 10^{-4}$ 4096 20 2.0 6.8 0.46 0.53 s 1D256B $8 \times 10^{-3}$ 256 20 1.8 0 0.1 0.12 s 1D512B $8 \times 10^{-3}$ 512 20 1.8 0 0.1 0.13 s 1D1024B $8 \times 10^{-3}$ 1024 20 1.8 0 0.1 0.13 s 1D2048B $8 \times 10^{-3}$ 2048 20 1.9 0 0.1 1.7 s 1D4096B $8 \times 10^{-3}$ 4096 20 1.95 0.05 0.13 5.8 ns 1D256C $1 \times 10^{-2}$ 256 20 1.7 0 0.09 0.09 s 1D512C $1 \times 10^{-2}$ 512 20 1.8 0 0.09 0.09 s 1D1024C $1 \times 10^{-2}$ 1024 20 1.8 0 0.11 10.4 s 1D2048C $1 \times 10^{-2}$ 2048 20 1.8 0.11 0.13 9.3 ns 1D256D $1.5 \times 10^{-2}$ 256 20 1.5 0.7 0.2 24 s 1D512D $1.5 \times 10^{-2}$ 512 20 1.5 1.9 0.23 28 ns 1D1024D $1.5 \times 10^{-2}$ 1024 20 2.1 5.8 0.55 61 ns 1D256E $2 \times 10^{-2}$ 256 20 1.23 4.1 0.2 30 ns ---------- ---------------------- ------- --------- -------------------------------- ---------------------------- ------------------------------- -------------------- --------- Figure \[fig:st1d\] presents the overall structure of model $1D256~C$ and $D$ (model C and D in the left and right column, respectively). Panels from top to bottom in Figure \[fig:st1d\] display: radial profiles of gas density, gas temperature overplotted with the Mach number (red line with the labels on the right hand side of the panels), the net heating/cooling rate plotted together with contribution from each physical process (see Equation \[eq:HC\_full\]), the entropy S, and the bottom row shows gas temperature as a function of $\xi$. In the bottom panels, the red line indicates the T-$\xi$ relation for radiative equilibrium (i.e. solving ${\mathcal L}(\xi,T)=0$ for each $T$). The green line indicates a $T-\xi$ relation for a gas being adiabatically compressed due to the geometry of the spherical accretion ($T \propto \xi^2$), while the blue line for a constant pressure gas ($T \propto \xi^1$). The 1D256C and D solutions differ mainly in the position of the sonic point and in the fact that the model 1D256D is strongly time dependent for a short period of time during initial evolution (see below). However, in most part the solution share several common properties. In particular, in both solutions, the gas is nearly in radiative equilibrium at large radii whereas, at small radii (below $r\approx2\times10^{19}$cm, where $T>2 \times10^6$K) they depart from the equilibrium quite significantly. In the inner and supersonic parts of the solutions, $T$ scales with $\xi$ as if gas was under constant pressure. At large radii where the solutions are nearly in the radiative equilibrium, the net heating/cooling is not exactly zero. One can identify, four zones where either cooling or heating dominates. In the most inner regions where the gas is supersonic, adiabatic heating is very strong and the dominant radiative process is cooling by free-free emission. At the outer radii, the cooling in lines and heating by photoionization dominates. For models considered in this paper the Compton cooling is the least important. ![ Structure of 1-D accretion flow in run 1D256C (left column, $f_X=1\times10^{-2}$) and 1D256D (right column, $f_X=1.5\times10^{-2}$). Each panel is a snapshot taken at t=20 Myr. Panels from top to bottom show: density, temperature with Mach number (Mach number scale is on the right hand side), heating/cooling rates, and entropy S. The dashed vertical line in top panels marks the position of the sonic point. In panel with heating/cooling rates the black solid line is a net heating/cooling and color lines indicate particular physical process included in the calculations: Compton heating (red line), photoionization heating (green line), bremsstrahlung cooling (magenta-line), cooling through line emission (blue-line). The bottom panels display the gas temperature as a function of photoionization parameter; color lines indicate gas in radiative equilibrium (red), constant pressure conditions (blue) and free-fall compression (green).[]{data-label="fig:st1d"}](fig1_col.eps) Inspecting the bottom panels in Figure \[fig:st1d\], one can suspect the gas is in the middle section of the computational domain to be thermally unstable because the slope of the $T-\xi$ relation (in the log-log scale) is larger than 1. Notice also that in both solutions the entropy is a non-monotonic function of radius. The regions where the entropy decreases with increasing radius correspond to the regions where there is net heating and the Schwarzschild criterion indicates convective instability at these radii. We therefore conclude that both solutions could be unstable. We first check more formally the thermal stability of our solutions. Thermal Stability of Steady Accretion Flows ------------------------------------------- The linear analysis of the growth of thermal modes under the radiative equilibrium conditions (${\mathcal L}(\rho_0,T_0) = 0$) has been examined in detail by @field:1965 (see Appendix for basic definitions). In Figure \[fig:tibv\], in the top panels (left and right column correspond again to model 1D256C and D), we show the radial profiles of various mode timescales. The timescales, $\tau=1/n$, are calculated using definitions \[eq:Np\] and \[eq:Nv\]. The growth timescale of short wavelength, isobaric condensations $\tau_{TI}=-1/N_p$ is positive (thermally unstable zone marked with the dotted line) in a limited radial range between about 10 and 100 pc. The location of the thermally unstable zone depends on the central source luminosity, and it moves outward with increasing $f_X$. The long wavelength, isochoric perturbations are damped, at all radii, on timescales of $\tau_{v}=-1/N_v$ (faster than TI development). The short wavelength nearly adiabatic, acoustic waves are damped as well, and $\tau_{ac}=-2/(N_v-N_p)$. In Figure \[fig:tibv\], the dashed line is the accretion timescale $\tau_{acc}=r/v$. Within the thermally unstable zone, $\tau_{TI}$ is short in comparison to $\tau_{acc}$, in both models. ![ Left and right panels correspond, respectively to runs 1D256 C and D. Top panels show the instability growth rates in comparison to the accretion time scale ($\tau_{acc}=r/v$, dashed line). The time scale for the short wavelength isobaric mode growth is displayed as the dotted line while the damping rate as solid line ($\tau_{N_p}$). Other two lines show the long-wavelength isochoric mode damping rate $\tau_{N_v}$ (heavy line) and the effective acoustic waves damping time scale $\tau_{N_v}$ (light line). Middle panel: The dashed line is $\tau_{acc}$ and solid line is $t_{BV}=1/\omega_{BV}$, where $\omega_{BV}^2>0$ is the ${\rm Brunt-V\ddot{a}is\ddot{a}l\ddot{a}}$ oscillation frequency for a spherical system. Solid lines show the regions which are unstable convectivelly. The dotted line indicates region where $\omega_{BV}^2<0$ and oscillations are possible. Bottom panels: the derivative $d \ln T/ d \ln \xi$ as a function of radius is shown as a solid line. $(d \ln T/ d \ln \xi)_{ad}$ for an adiabatic inflow is marked as dotted line, and dashed line is the same derivative for radiative equilibrium conditions. Horizontal one indicates slope of 1. \[fig:tibv\]](fig2_col.eps) @balbus:1986, @balbus:1989, @mathews:1978, (and also @krolik:1983) extended the analysis by @field:1965 to spherical systems with gravity, in more general case when initially the gas is not in the radiative equilibrium. Their approximate solution gives the formula for linear evolution of the short wavelength, isobaric, radial perturbation as it moves with smooth background accretion flow (Equation 23 in @balbus:1986 or Equation 4.12 in @balbus:1989). Since the two presented solutions are close to radiative equilibrium, the approximate formula for the growth of a comoving perturbation given by @balbus:1989 reduces to $$\delta (r) = \frac{\delta \rho}{\rho} =\delta_s \exp \left( \int_{r_s}^{r_f} - \frac{N_p(r')}{v(r')} dr' \right), \label{eq:amp}$$ where $N_p(r')$ is a locally computed growth rate of a short wavelength, isobaric perturbation as defined in the Appendix or @field:1965, $r'$ is radius where $N_p(r') < 0$, and $\delta_s$ is an initial amplitude of a perturbation at some starting radius $r=r_s$. Using Equation \[eq:amp\], the isobaric perturbation amplification factors are $\delta/\delta_s \approx 10^{10}, 10^{16}, 10^{19}$ and $10^{33}$, for models 1D256A, B, C, and D, respectively. Notice that these amplification factors are calculated for the asymptotic, maximum physically allowed growth rate, $n=-N_p$, which might not be numerically resolved. To quantify the role of TI in our simulations we ought to address the following question. What is the minimum amplitude and wavelength of a perturbation in our computer models? The smallest amplitude variability is due to machine precision errors, $\epsilon_{machine} \approx 10^{-15}$ (for a double float computations). The typical $\lambda$ of these numerical fluctuations are of the order of the numerical resolution, $\Delta r_i$. The discretization of the computational domain affects the TI growth rates in our models in two ways: (1) the numerical grid refinement limits the size of the smallest fragmentation that can be captured; (2) the rate at which the condensation grows in the numerical simulations depends on number of points resolving a condensation. As shown in Appendix the perturbation of a given $\lambda$ has to be resolved by 20, or more, grid points. A wavelength $\lambda_0$ for which $n = - 0.9973 \times N_p$ is shown in Figure \[fig:res\] together with $\Delta r_i $ as a function of radius for models with $N_r$=256, 512, 1024, 2048, and, 4096 grid points. In low resolution models we marginally resolve $\lambda_0$. We therefore expect the TI fragmentations to grow slower than theoretical estimates. Reduction of the growth rate due to these numerical effects even by a factor of a few is enough to suppress variability because of the strong exponential dependence. ![Grid spacing (red lines) in models with $N_r$=256, 512, 1024, 2048, and 4096 points and $\lambda_0$ (black, dotted line) as a function of radius in models 1D256 C (left panel) and D (right panel). \[fig:res\]](fig3_col.eps) Thermal mode evolution depends not only on the numerical effects but also other processes affecting the flow. Figure \[fig:time\_scales\] shows the comparison of time scales of physical processes involved: the compression due to geometry of the inflow and stretching due to accretion dynamics. We expect that any eventual condensation formed from the smooth background which leaves the thermally unstable zone, would accrete with supersonic background velocity. From the continuity equation, the co-moving density evolution is a balance of two terms $(1/\rho) (D\rho/Dt)=-2v/r - \partial v/\partial r$, i.e., the compression and tidal stretching. The amplitude of condensation grows in regions where there is compression due to geometry and decreases in regions where fluid undergoes acceleration - it stretches the perturbation. In the models 1D256D and C interior of the TI zone, the evolution of the perturbation is dominated by compression because the compression time scale is the shortest. ![Time scales in 1-D, stationary models 1D256 C (left panel) and 1D256 D (right panel): accretion time scale ($\tau_{acc}$, dashed line), compression time scale ($\tau_c$, solid line), tidal stretching time scale ($\tau_s$, dotted-dashed line), and condensation growth time scale ($\tau_{TI}$, dotted line). \[fig:time\_scales\]](fig4_col.eps) Convective Stability of Steady Accretion Flows ---------------------------------------------- In this subsection, we examine in more detail convective stability of our solutions. In Figure \[fig:tibv\], (middle panels), we compare the accretion time scale $\tau_{acc}$ and the ${\rm Brunt-V\ddot{a}is\ddot{a}l\ddot{a}}$ time scale $\tau_{BV}=\frac{1}{\omega_{BV}}$ associated with the development of convection. The frequency $\omega_{BV}$ is defined as $\omega_{BV}^2 \equiv (-\frac{1}{\rho} \frac{\partial P}{\partial r}) \frac{\partial \ln S}{dr}$. The convectivelly unstable regions are marked as solid lines ($\omega_{BV}^2 >0$). The convectivelly unstable zones overlap with the thermally unstable zones. Since $\tau_{acc} \ll \tau_{BV}$ convective motions might not develop, at least at the linear stage of the development of TI. In the bottom panels of Figure \[fig:tibv\] we show the logarithmic derivatives of $d \ln T / d \ln \xi$ that could be used to graphically assess the stability of the flow. This can be done by comparing the derivatives (the slopes of the $ln T - \ln \xi$ relation) for three cases: model data (solid line where $T$ and $\xi$ are taken directly from the simulations), purely adiabatic inflow (dotted line, assuming that the velocity profile is same as in the numerical solution), and radiative equilibrium conditions (dashed line). In particular, the regions where the solid line is above the red line correspond to the potentially TI zones. The regions where the dotted line is below the solid line correspond with the zone where the flow is potentially convectivelly unstable. The conclusion regarding the flow stability is consistent with the conclusion reached by analyzing the time scales shown in the top and middle panels of Fig. 2. Other Physical Consequences of Radiative Heating and Cooling - obscuration effects ---------------------------------------------------------------------------------- ![ Fraction of central illuminating source radiative energy intrinsically absorbed (upper panels) and emitted (bottom panels) by gas per second as a function of time. We show the steady state solutions $1D256C$ and $1D256D$ in left and right panels, respectively. Solid lines show the net absorption/emission and dashed lines indicate the intrinsic absorption and emission. \[fig:en1d\]](f5.eps) The growth of the thermal instability leads to the development of a dense cold clouds (shells in 1-D models; e.g. variable phase in model 1D256D). The enhanced absorption in the dense condensations may make them optically thick. Here we check if the time-dependent models are self-consistent with our optically thin assumption. Figure \[fig:en1d\] shows the amount of energy absorbed and emitted by the gas (heating and cooling rates integrated over a volume at each time moment) in comparison to the luminosity of the central source in models 1D256C and D. Solid lines show the net rate of the energy exchange between radiation and matter (cooling function ${\mathcal L}$ integrated over the simulation volume) while dashed lines indicate the intrinsic absorption and emission (heating and cooling terms used in ${\mathcal L}$ are integrated independently). The net heating-cooling rate is mostly much lower than unity reflecting the fact that in the steady state the gas is nearly in a radiative equilibrium. During a variable phase (part of model 1-D256D) the energy absorbed by the accretion flow (black solid line) becomes comparable to the X-ray luminosity of the central black hole. During this variable phase the optical thickness of accreting shells can increase up to $\tau_{X,sc} \approx 20$ where the majority contribution to opacity is due to photoionization absorption. The average optical thickness increases in models with higher resolution indicating that the flow is more variable and condensations are denser. This increase in optical depth is related to shells condensating much faster in runs with higher resolutions. The dense condensations falling towards the center could reduce the radiation flux in the accretion flow at larger distances. It is beyond the scope of the present paper to investigate the dependence of the flow dynamics on the optical thickness effects and we leave it to the future study. Significant X-ray absorption is related also to transfer of momentum from radiation field to the gas. To estimate the importance of the momentum exchange between radiation and matter, one can compute a relative radiation force: $$f_{force} \equiv \frac{\sigma_{sc}+\sigma_{X}}{\sigma_{TH}} f_X \label{eq:rforce}$$ where $\sigma_X$ is the energy averaged X-ray cross-section. The momentum transfer is significant when $f_{force}>1$. Using our expression for the heating function due to X-ray photoionization $\sigma_X/\sigma_{TH}=H_X / n / F_X = 2.85\times 10^4 \xi^{-3/4} T^{-1/2}$ (see § \[sec:equation\]). Even for a dense cold shell $f_{force}$ is at most 0.1 (in case when $\tau_{max}\approx60$). Therefore radiation force is not likely to directly launch an outflow. However, the situation may change when optical effects are taken into account. $\dot{M}$ Evolution ------------------- We end our presentation of 1-D results with a few comments on the time evolution of the mass accretion rate, $\dot{M}$. Figure \[fig:mdot1d\] displays $\dot{M}$ vs time measured for all of our 1-D models. One can divide the solutions into two subcategories: steady and unsteady where $\dot{M}$ varies from small fluctuations to large changes. For a given $f_X$, the time behavior of the solution depends on the resolution, due to effects described above. In the variable models a fraction of the accretion proceeds in a form of a cold phase defined as all gas with $T < 10^5$ K. Column 6 in Table \[tab:1d\] shows the ratio of cold to hot mass accretion rates, $\chi$, computed by averaging $\dot{M}'s$ over the radius and simulation time. We average $\dot{M}'s$ over $r<100$ pc because the cadence of our data dumps is comparable to the dynamical time scale at 100 pc. The larger the luminosity the more matter is accreted via the cold phase. However, the maximum value of $\chi$ is of the order of a few, so the dominance of the cold gas is not too strong. ![ Mass accretion rate in 1-D solutions for $f_X=0.0005, 0.08, 0.01$, and $0.015$. Different colors shows $\dot{M}$ for various number of grid points: $N_r$=256 (blue), 512 (red), 1024 (green), 2048 (magenta), 4096 (black). \[fig:mdot1d\]](f6.eps) ------------- ---------------------- ------- ------------ --------- -------------------------------- ------------------------------ ---------------------------- ---------------------------------------- -------------------- ------------------- Model ID $f_X$ $N_r$ $N_\theta$ $t_f$ $\langle\dot{M}\rangle$ $\langle f_{Vol}\rangle_{t}$ $\langle\chi\rangle_{r,t}$ $\langle\tau_{X,sc}\rangle_{\theta,t}$ Max($\tau_{X,sc}$) final state \[Myr\] ${\rm [M_{\odot} \, yr^{-1}]}$ 2D256x64A $5 \times 10^{-4}$ 256 64 20 2.0 $1\times10^{-6}$ 0 0.45 0.46 smooth 2D256x64B $8 \times 10^{-3}$ 256 64 15.4 2.04 $1\times10^{-6}$ $10^{-6}$ 0.1 0.99 smooth 2D512x128B $8 \times 10^{-3}$ 512 128 20 1.95 $1\times10^{-4}$ 0.02 0.11 4.7 clouds 2D1024x256B $8 \times 10^{-3}$ 1024 256 1.83 1.84 $1.5\times10^{-3}$ 0.39 0.19 24. clouds 2D256x64C $1 \times 10^{-2}$ 256 64 11.8 1.94 $7\times 10{-5}$ 0.15 0.11 2.5 smooth 2D512x128C $1 \times 10^{-2}$ 512 128 20 1.88 $5\times10^{-4}$ 0.09 0.13 17.3 clouds 2D1024x256C $1 \times 10^{-2}$ 1024 256 1.12 1.95 $4\times10^{-3}$ 0.5 0.24 70 clouds 2D256x64D $1.5 \times 10^{-2}$ 256 64 12 1.57 $5\times10^{-3}$ 0.3 0.14 37.8 clouds 2D512x128D $1.5 \times 10^{-2}$ 512 128 11 1.6 $3\times10^{-3}$ 0.43 0.12 13.2 outflow,filaments ------------- ---------------------- ------- ------------ --------- -------------------------------- ------------------------------ ---------------------------- ---------------------------------------- -------------------- ------------------- Results: 2-D models {#sec:results_2d} =================== Seeding the TI -------------- To investigate the growth of instabilities in 2-D, we solve eqs. \[eq:mass\], \[eq:mom\], \[eq:energy\], for the same parameters $M_{BH}$ and $f_X$ as in § \[sec:results\_1d\], but on a 2-D, axisymmetric grid with $\theta$ angle changing from 0 to $90 \deg$. We use three sets of numerical resolutions described in § \[sec:num\_setup\]. To set initial conditions in axisymmetric models, we copy the solutions found in 1-D models onto the 2-D grid. In case of time-independent 1-D models, our starting point, is the data from $t=t_{f}$. We checked, for example, the runs 2D256x64 A, B, C, and D (which are 2-D version of models 1D256A, B, C, and, D) are time-independent at all times as expected. In case of higher resolution models, for which the 1-D, steady state models do not exist we adopt a quasi-stationary data from the 1-D run early evolution (at $t$ of a fraction of a Myr), during which the flow is already relaxed from its initial conditions but the TI fluctuations are not yet developed. Models which are time varying in 1-D, in 2-D develop dynamically evolving spherical shells, as expected, also indicating that our numerical code keeps a symmetry in higher dimensions. To break the symmetry in 2-D models, we perturb the smooth solutions adopted as initial conditions. The perturbation of a smooth flow is seeded everywhere and has a small amplitude randomly chosen from a uniform distribution. The new density at each point is $\rho=\rho_0 (1+ Amp*rand)$, where $rand$ is a random number $rand \in (-1,1)$ and maximum amplitude $Amp=10^{-3}$. To seed the isobaric, divergence free fluctuations other (than $\rho$) hydrodynamical variables are left unchanged. The amplitude magnitude $Amp$ is chosen to be much higher in comparison to $\epsilon_{machine}$, in order to investigate the development and evolution of strongly non-linear TI on relatively short time scales, starting directly from a linear regime. The list of all perturbed, 2-D models is given in Table \[tab:2d\]. Formation of Clouds, Filaments, & Rising Bubbles ------------------------------------------------ For luminosities $L_X < 0.015~L_{Edd}$, the 2-D models show similar properties to the 1-D models. The gas is thermally and convectivelly unstable within the computational domain, and we observe that very tiny fluctuations in an initially smooth, spherically symmetric, accretion flow, grow first linearly and then non-linearly. Since the symmetry is broken the cold phase of accretion forms many small clouds. For $L_X = 0.015~L_{Edd}$ or higher, the cold clouds continue to accrete but in some regions a hot phase of the gas starts to move outward.  In Figure \[fig:st2d\], we show three snapshots of representative 2-D model 2D512x128D at various times (t = 3, 6 and 11.8 Myrs). This model has the best resolution and the highest luminosity for which we are able to start the evolution from nearly steady state conditions. Columns from left to right show density, temperature, and total gas velocity overplotted with the arrows indicating the direction of flow. Initially (at t=3 Myr) the smooth accretion flow fragments into many clouds, which are randomly distributed in space. The cooler, denser regions are embedded in a warm background medium. The colder clouds are stretched in the radial direction and they have varying sizes. This initial phase of the evolution is common for all models in Table \[tab:2d\]. The phase where many cold clouds accrete along with the warm background inflow is transient. At a later stage (t=6 Myr, middle panels), model 2D512x128D shows a systematic outflow in form of rising, hot bubbles. The outflow is caused by the pressure imbalance between the cold and hot matter and buoyancy forces. The hot bubbles expand at speeds of a few hundreds km/s. Despite of the outflow, the accretion is still possible. During the rising bubble phase, the smaller clouds merge and sink towards the inner boundary as streams/filaments. However, even this phase is relatively short-lived. Bottom panels in Figure \[fig:st2d\] show the later phase of evolution when some of the filaments occasionally break into many clouds (this process takes place between 10 and 50 pc). These ’second generation’ clouds occasionally flow out together with a hot bubble. Along the X-axis, we see an inflow of a dense filament. To quantify the properties of clumpy accretion flow, we measure the volume filling factor of a cold gas $f_{vol}$, defined as: $$f_{Vol}=V_{cloud}/V_{tot}$$ where $V_{cloud}$ is the volume occupied by gas of $T<10^{5}$ K, and $V_{tot}$ is the total volume of the computational domain. In model 2D512x128D, the time-averaged $f_{Vol}$ is $\langle f_{Vol}\rangle=3\times10^{-3}$. The time evolution of $f_{vol}$ within 60 pc is shown in Figure \[fig:vol\] (black, solid line). The $f_{Vol}$ is variable and at the moment of the outflow formation, $f_{Vol}$ suddenly decreases by a factor of about 4. For comparison $f_{Vol}$ calculated during run 2D256x64D is also shown (blue, dashed line). Run 2D256x64D has the same physical parameters as 2D512x128D, however, no outflow forms. In the latter case, $f_{vol}$ is less variable and larger. In Table \[tab:2d\], we gather the time averaged $\langle f_{Vol}\rangle$ for all 2-D solutions. Measuring $f_{Vol}$ allows to quantify whether the perturbed accretion flow returns to its original, smooth state. We find that this happens when the $\langle f_{Vol}\rangle\approx 10^{-5}$ or smaller (models 2D256x64A, B, and C). ![Evolution of the volume filling factor $f_{vol}$ in model 2D512x128D (black, solid line) and 2D256x64D (dashed, blue line). \[fig:vol\]](f8.eps) $\dot{M}$ Evolution ------------------- Figure \[fig:mdot2d\] presents $\dot{M}$ through the inner boundary measured as a function of time. In most cases (except model 2D256x64A), $\dot{M}$ becomes stochastic instantly with spikes corresponding to the accretion of colder but denser clouds similar to those in Figure \[fig:st2d\] (upper panels). Similar to 1-D models, one can divide the solution into two types: steady and unsteady state. In the latter, $\dot{M}$ fluctuates on various levels depending on $f_X$. ![Mass accretion rate in 2-D models with initially seeded random perturbations. Various colors code the $\dot{M}$ in models calculated with various grid resolutions: 2D256x64 (blue), 512x128 (red), 1024x256 (green). Results are sensitive to the resolution same as in 1-D models. \[fig:mdot2d\]](f9.eps) Table \[tab:2d\] lists several characteristics of our 2-D simulations, for example, the ratio $\langle\chi\rangle_t$ averaged over time. In 2-D models this variable is smaller in comparison to 1-D due to geometry of the clouds. The maximum value of $\langle\chi\rangle_t$ is less than unity. This indicates that multi-dimensional effects (specifically development of convection) promote hot phases accretions. We plan to investigate this issue in future by carring out 3-D simulations. We find that a large scale outflow forms only in run 2D512x128D. But even in this case, the $\dot{M}$ is not significantly affected by the outflow. Figure \[fig:mdotinout\] shows the mass outflow rate (dashed line), inflow rate (dotted lines) and total mass flow rate (solid line) as a function of radius. The same types of lines show $\dot{M}$ for various times of the simulation (t=3, 6 and 11.8 Myr, green, blue and black lines, respectively) and they are averaged over $\theta$ angle. The rising bubble originates at about 10 pc in this case. We anticipate that large scale outflow are common and significant for high luminosity cases (i.e., for $f_X > 0.02$). ![Model 2D512x128D: In-, out- and total mass flow rate as a function of radius for three time moments shown in Figure \[fig:st2d\] (green, blue, and black correspond to t=3, 6 and 11.8 Myr). The solid lines mark the inflow rates while the dashed line - outflow rate. The dotted line is the total mass flow. \[fig:mdotinout\]](f10.eps) Obscuration Effects ------------------- Here we again check whether the cloud opacity might affect our results. The averaged optical thickness of the filaments and clouds is similar (see Table \[tab:2d\], columns 8 and 9). In Figure \[fig:energy2d\_all\] we show how much energy is absorbed and emitted in run 2D512x128D during the evolution. The figure shows the intrinsic absorption and emission integrated over the entire computational domain. We next calculate $\langle\tau_{X,cs}\rangle$ (optical thickness due to absorption, averaged over angles and times) and maximum value of $\tau_{X,cs}$ that occurred during the evolution. In case of the largest optical depth of $\tau \approx 70$ (in run 2D1024x256C) the radiation force coefficient from Equation \[eq:rforce\]: $f_{force} \approx 0.2$ which, as in 1-D models, is small but might not be negligible. Therefore, we are planning to explore the effects of optical depth in a follow-up paper. ![Fraction of central illuminating source radiative energy intrinsically absorbed (upper panel) and emitted (bottom panel) by gas per second as a function of time in models 2D512x128 (red, solid line) and 2D256x64D (blue, dotted line). \[fig:energy2d\_all\]](f11.eps) Summary and Discussions {#sec:discussion} ======================= In this work we show the evolution of thermal instabilities in gas accreting onto a supermassive black hole in an AGN. A simplified assumptions made in this work, in particular constant X-ray luminosity emitted near the central SMBH regardless of the $\dot{M}$, allows to follow the development of TI from the linear to strongly non-linear and dynamical stage up to luminosities of $L \approx 1.5 \times 10^{-2}~L_{Edd}$. In our 1-D models the TI is seeded by numerical errors which might be non-isobaric and are initially under-resolved. In the initial phase, the TI growth rate is smaller than predicted by theory. The rate is affected by grid resolution which leads to the formation of cold clouds of various sizes and density contrasts. This is reflected in the mass accretion rate fluctuating at different amplitude and rate for the same physical conditions but different resolutions. One cannot avoid dealing with these numerical difficulties in the numerical models. Nevertheless, we find the under-resolved, 1-D models very useful in quick checking where the thermally unstable zone exists and what type of fluctuation could cause the smooth to turn into a two-phase medium. For given physical conditions, Figure \[fig:res\] shows the wavelength $\lambda_0$ and Equation \[eq:amp\] gives the amplitude of an isobaric perturbation required to break the smooth flow into a two-phase, time-dependent model. In 2-D models, although the models depend on the resolution effects same way as in 1-D setup, we can observe an outflow formation. The convectivelly unstable gas buoyantly rises and, as found in this work, controls the later evolution of the two-phase medium and mass accretion rate. Given a simple set up with minimum number of processes included, our models display the three major features needed to explain some of the AGN observations: cold inflow, hot outflow and cold, dense clouds which occasionally escape, advected with the hot wind. We show that an accretion flow at late, non-linear stages, thus most relevant to observations, are dominated by buoyancy instability not TI. This suggests that the numerical resolution might not have to be as high as that needed to capture the small scale TI modes and it is sufficient to capture significantly larger and slower buoyancy modes. We plan to check the consistency of the models with the observations by calculating the synthetic spectra, including emission and absorption lines based on our simulation following an approach like the one in @sim:2012. Here, we only briefly comment on the main outflow properties and compare them some observations of outflows in Seyfert galaxies. Space Telescope Imaging Spectrograph (STIS) on board the [*Hubble Space Telescope*]{} allows us to map the kinematics of the Narrow Line Regions in some nearby Seyfert Galaxies (e.g. for NGC 4151 @das:2005; NGC 1068; @das:2006; Mrk 3, @crenshaw:2010; Mrk 573, @fisher:2010; and Mrk 78, @fisher:2011). Position-dependent spectra in \[O III\] $\lambda$ 5007 and $H_{\alpha}$ $\lambda$ 6563, and the measurements of the outflow velocity profiles show the following general trend: the outflow has a conical geometry and the \[O III\] emitting gas accelerates linearly up to some radius and then decelerates. The velocities typically reach up to about 1000 km/s and a turnover radius is on one hundred to a few hundred parsec scales. To compare our results with the observations, Figure \[fig:vr\_scatter\] shows the radial velocity of hot and cold gas versus the radius at t=11.8 Myr for model 2D512x128D (the data correspond to a snapshot shown in the right panels in Figure \[fig:st2d\]). We reiterate that our model is quite simplified (e.g., no gas rotation) and the outer radius is relatively small (i.e., 200 pc). Therefore, our comparison is only illustrative. We find that the hot outflow originates at around 10 pc and accelerates up to about $v_{max} \approx 200 {\rm km/s}$, which is comparable to the escape velocity from 10 pc, $v_{esc}=314~{\rm km/s}$. At larger distances, r = 100-200 pc, we see a signature of deceleration which is consistent with the observations of Seyferts outflows. We note that the geometry of the simulated flow is affected by our treatment of the boundaries of the computational domain, specifically along the pole and the equator we use reflection boundary conditions (see § \[sec:num\_setup\]). The scatter plot also indicates that the cold clouds appear at about 20-80 pc. Their maximum velocity is about $v_{max}=100$ km/s, which is smaller than the velocity of the hot outflow. The plot does not show a clear indication of a linear acceleration of the outflowing cold gas. However, it is possible that the cold clouds, seen in this snapshot, will continue to be dragged by the hot outflow and eventually will reach higher velocities. We also measured the column density of the hot and cold gas for the same, representative, snapshot at t=11.8 Myrs. The typical column densities vary with the observer inclination, $N_{H}=5 \times 10^{22} - 10^{24} \, {\rm cm^2}$ for gas $T > 10^5$ K, and $N_{H}=10^{20}-10^{23} \, {\rm cm^2}$ for gas with $T<10^5$ K. This is roughly consistent with column densities estimated from observations of AGN (e.g. for NGC 1068 $N_H= 10^{19}-10^{21} \, {\rm cm^2}$, @das:2007 and references therein). Our results are similar in many respects, to the previous findings presented in @barai:2012, i.e. the accretion evolution depends on $f_X$ luminosity; we also observe clouds, filaments and outflow. The outflow appears at $f_X=0.015$ which is consistent with $f_X=0.02$ found by @barai:2012. Here, we are able to calculate models for about 10 times longer in comparison to 3-D SPH models. We confirm the previous results that the cold phase of accretion rate can be only a few times larger in comparison to the hot one. Similar models has been investigated in the past by e.g. @krolik:1983. Our work is on one hand a simplified and on the other hand an extended version of these previous works. The key extention here is that our new results cover the non-linear phase of the evolution. There are two new conclusions added by our analysis to the previous investigations. The 2-D models with outflows are possibly governed by other than TI instabilities mainly convection. Another non-linear effects found in our 1-D and 2-D models is that the fragmentation of the flow makes it optically thick for photoionization. Further investigation of shadowing effects is required. Some sub-resolution models of AGN feedback in galaxy formation (@dimatteo:2008; @dubois:2010; @lusso:2011) assume that BH accretion is dominated by an unresolved cold phase, in order to boost up the accretion rate obtained in simulations. Our results indicate that the cold phase accretion is unlikely dominant as even in well-developed and well- resolved multi-phase cases, the accretion is typically dominated by a hot phase. However, we note that the cold phase of our solution might be an upper branch of some more complicated multi-phase medium (i.e., a mixture of molecular, atomic and dusty gas). This work was intentionally focused on a very limited number of processes and effects. Its results suggest that the future work should include more self-consistent approach not only with shadowing effects but also with the radiation force. Our next step would be to investigate the non-axisymmetric effects via fully 3-D simulations. The latter is challenging and one may not be able to see very fine details of the gas dynamics as in 2-D models due to resolution effects. ![ Scatter plot of radial velocity of hot ($T>10^5 K$, smaller red symbols) and cold ($T < 10^5 K$, larger blue symbols) phase of the flow in model 2D512x128D at t=11.8 Myr (model shown in right panels in Figure \[fig:st2d\]).[]{data-label="fig:vr_scatter"}](f13_reduced.eps) This work was supported by NASA under ATP grant NNX11AI96G and NNX11AF49G. DP thanks J. Ostriker, J. Stone, and S. Balbus for discussions and also Department of Astrophysical Sciences, Princeton University for its hospitality during his sabbatical. DP also acknowledges the UNLV sabbatical assistance. Authors would like to thank Paramita Barai, Ken Nagamine and Ryuichi Kurosawa for their comments on the manuscript. 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Here, we briefly recall his most important, for our analysis, equations. We disregard the thermal conduction effects. The dispersion relation derived from linearized local fluid equations with heating/cooling described by ${\mathcal L}$ function and perturbed by a periodic, small amplitude wave given by $\exp(nt+ikx)$, is: $$n^3 + N_v n^2 + k^2 c_s^2 n + N_p k^2 c_s^2 = 0 \label{eq:cube}$$ where $k$ is the perturbation wave number ($k=2 \pi /\lambda$) and functions $N_p$ and $N_v$ are defined as $$N_p \equiv \left .\frac{1}{c_p} \left(\frac{\partial {\mathcal L} }{\partial T}\right)\right|_P \label{eq:Np}$$ and $$N_v \equiv \left. \frac{1}{c_v} \left(\frac{\partial {\mathcal L}}{\partial T}\right)\right|_\rho \label{eq:Nv}$$ with $c_p$ and $c_v$ being the specific heats under constant pressure and constant volume conditions, respectively, and $T$ is the gas temperature. Vertical line means that the derivative is taken under constant thermodynamical variable condition. Dispersion Equation \[eq:cube\] has three roots. In a short wavelength regime ($\lambda \ll 2\pi N_p / c_s$), two, complex roots correspond to two conjunct nearly adiabatic sound waves and third, real one is an isobaric condensation mode (the gas density and temperature change in anti-phase so that the pressure remains constant). The sign of the real part of the root gives the stability criterion. The sound wave will grow if $\left. \partial{\mathcal L}/\partial T \right|_S < 0$ (known as Parker’s criterion, @parker:1953). The condensation mode will grow if $\left. \partial {\mathcal L}/\partial T \right|_P < 0$ (Field’s criterion). In a short wavelength limit, the growth rates asymptote to $n=-0.5(N_v-N_p)$ (for sound waves) and $n=-N_p$ (for condensation modes). Isochoric modes ($n \rightarrow -N_v$) and effective acoustic waves are eigen modes of long wavelengths perturbations. The perturbation growth/damp time scale is $\tau_{TI}=1/n$. We use the above @field:1965 theory to show that our numerical scheme for solving the modified energy conservation equation (Equation \[eq:energy\]) together with two other fluid dynamics equations is accurate. The test calculations are carried out in 1-D Cartesian coordinates within $x\in(0,L)$ range where L is the size of the computational domain in dimensionless units. The boundary conditions for all variables are periodic. In an unperturbed state, the gas density ($\rho_0=1$) and internal energy density ($e_0=1$) are constant in the entire computational domain. The velocity of gas is set to zero. We assume that the gas is heated by an external source of radiation and cools due to free-free transitions. The test cooling function is simple: $${\mathcal L}= C \rho T^{1/2} \label{testcool} - H$$ The normalization constants $H$ (for heating) and $C$ (for cooling) are set so that in the unperturbed state the gas is in radiative equilibrium i.e. ${\mathcal L}(\rho_0,e_0)=0$. In this test the functions $N_p$ and $N_v$ have explicit, analytical forms $$N_p \equiv \left .\frac{1}{c_p} \left(\frac{\partial {\mathcal L} }{\partial T}\right)\right|_P \equiv \frac{1}{c_p} \left( \left . \frac{\partial{\mathcal L}}{\partial T} \right |_\rho - \frac{\rho}{T} \left . \frac{\partial{\mathcal L}}{\partial \rho} \right |_T \right) =- \frac{1}{2 c_p} C \rho_0 T_0^{-1/2}$$ and $$N_v \equiv \left. \frac{1}{c_v} \left(\frac{\partial {\mathcal L}}{\partial T}\right)\right|_\rho = \frac{1}{2 c_v} C \rho_0 T_0^{-1/2} = -\gamma N_p.$$ The numerical values of limiting growth/damp rates are $N_p=-0.04$, and $N_v=0.067$, while the speed of sound is: $c_s^2=1.11$ ($\gamma=5/3$). The domain sound crossing time is much shorter than the perturbation growth time scale which allows to keep the constant pressure. Our numerical scheme implemented into ZEUS-MP code correctly reproduces the expected growth rates of small amplitude perturbation of the uniform medium. The perturbation is an eigen mode of TI, and its properties depend on the assumed $\lambda$. Eigen modes are realized by first applying a cosine perturbation to the gas density $\rho= \rho_0 + A \rho_0 \cos(k x)$ and calculating profiles of $e$ and $v$ from e.g. Equations 11 and 14 in [@field:1965], for a given $k$ and corresponding theoretical value of $n$ (given by Equation \[eq:cube\]). Next we measure how fast the perturbation grows while it is in the linear regime. Figure \[fig:app\] (left panel), shows the analytical solution of the theoretical dispersion relation $n(\lambda)$ (solid line, third root of Equation \[eq:cube\]), and the numerical growth rates calculated with ZEUS-MP (points). For very short $\lambda$’s the eigen mode of this root is converging to the isobaric condensation mode and grows at $n=-N_p$ rate, as expected. The long $\lambda$ modes grow slower in comparison to the very short $\lambda$ condensations, as predicted by theory. For relatively large $\lambda$, the third root changes into an effective acoustic wave, it becomes complex with the real part negative meaning that the waves are damped (see @shu:1992, Equation 41 in the Problem Set No 3).   In the second test, we measure the growth rate of a condensation mode that has a finite size (i.e. smaller than the domain length). We are interested in how many numerical grid points is required to resolve the correct $n$. We set $\lambda=0.1$ while $L=1$. Figure \[fig:app\] (right panel) shows the same time snapshots of the growing condensation mode density, calculated with various numerical resolutions. Models with lower resolution evolve slower. When $\lambda$ resolved with 16 points it starts converging to the right solution. We conclude that about 20 or more grid points per $\lambda$ is required to resolve the isobaric condensation. [^1]: We also decouple $L_X$ from $\dot{M}$ in order to avoid introducing additional parameters into the equations. While coupling these quantities not only a radiative efficiency of gravitational to radiative energy has to be assumed but one also needs to know how to calculate the mass accretion rate at the very compact region way below $r_i=0.1 pc$. Another reason for decoupling $L_X$ and $\dot{M}$ is that we are interested in caring out a stability analysis and perturb a steady state solutions with all model parameters fixed.

--- abstract: 'We show that the deformation space of complex parallelisable nilmanifolds can be described by polynomial equations but is almost never smooth. This is remarkable since these manifolds have trivial canonical bundle and are holomorphic symplectic in even dimension. We describe the Kuranishi space in detail in several examples and also analyse when small deformations remain complex parallelisable' address: | Sönke Rollenske\ Department of Mathematics\ Imperial College London\ SW7 2AZ London\ United Kingdom author: - Sönke Rollenske title: 'The Kuranishi-space of complex parallelisable nilmanifolds' --- Introduction ============ Left-invariant geometric structures on nilmanifolds, i.e., compact quotients of (real) nilpotent Lie groups, have proved to be both very rich and accessible for an in depth study. Thus many examples and counter-examples in (complex) differential geometry are of this type. In this paper we are concerned with deformations of complex structures for complex parallelisable nilmanifolds, which are the compact quotient of complex nilpotent Lie groups. The study of deformations of complex structures on compact complex manifolds has been an important topic since it was first developed by Kodaira and Spencer in [@Kod-sp58]. A deformation of a given compact complex manifold $X$ is a flat proper map $\pi:\kx\to \kb$ of (connected) complex spaces such that all the fibres are smooth manifolds together with an isomorphism with $X\isom \ky_0=\inverse\pi(0)$ for a point $0\in \kb$. If $\kb$ is a smooth $\pi$ is just a holomorphic submersion. Kodaira and Spencer showed that first order deformations correspond to elements in $H^1(X, \Theta_X)$ where $\Theta_X$ is the sheaf of holomorphic tangent vectors. A key result is now the theorem of Kuranishi which, for a given compact complex manifold $X$, guarantees the existence of a locally complete space of deformations $\kx\to{\mathrm{Kur}}(X)$ which is versal at the point corresponding to $X$. In other word, for every deformation $\ky\to \kb$ of $X$ there is a small neighbourhood $\ku$ of $0$ in $\kb$ yielding a diagram $$\xymatrix{ \ky\restr{\ku}\isom f^*\kx \ar[d]\ar[r] & \kx \ar[d]\\ \ku\ar[r]^f&{\mathrm{Kur}}(X),}$$ and in addition the differential of $f$ at $0$ is unique. The Kuranishi family ${\mathrm{Kur}}(X)$ hence parametrises all sufficiently small deformations of $X$. In general the map $f$ will not be unique which is roughly due to the existence of automorphisms. Another point of view, which we will mainly adopt in this paper, is the following: consider $X$ as a differentiable manifold together with an integrable almost complex structure $(M,J)$, i.e., $J:TM\to TM$, $J^2=-\id_{TM}$ and the Nijenhuis integrability condition holds (see [[(\[nijenhuis\])]{}]{} below). A deformation of $X$ can be viewed as a family of such complex structures $J_t$ depending on some parameter $t\in \kb$ with $J=J_0$. The construction of the Kuranishi space can then be made explicit after the choice of a hermitian metric on $M$. We will go through this construction for our special case of complex parallelisable nilmanifolds in Section \[kurani\]. In general the Kuranishi space can be arbitrarily bad but we can hope for better control over the deformations if we restrict our class of manifolds. If, for example, $X$ is Kähler and has trivial canonical bundle, i.e., $X$ is a Calabi-Yau manifold, then the Tian-Todorov Lemma implies that the Kuranishi space is indeed smooth; we say that $X$ has unobstructed deformations. These manifolds are very important both in physics and in mathematics for example in the context of mirror symmetry. This result fails if we drop the Kähler condition [@ghys95]. The only nilmanifolds which can carry a Kähler structure are tori but it was proved by Cavalcanti and Gualtieri [@caval-gual04] and, independently, by Babaris, Dotti and Verbitsky [@0712.3863v3] that nilmanifolds with left-invariant complex structure always have trivial canonical bundle. In addition, all known examples in the context of left-invariant complex structures on nilmanifolds, e.g., complex tori, the Iwasawa manifold [@nakamura75], Kodaira surfaces [@borcea84], abelian complex structures [@con-fin-poon06; @mpps06] (see Section \[definitions\] for a definition), had unobstructed deformations. Therefore it was speculated if this holds for all left-invariant complex structures. This was supported by results on weak homological mirror symmetry for nilmanifolds [@poon06; @0708.3442v2]. On the other hand Catanese and Frediani observed in their study of deformations of principal holomorphic torus bundles, which are in particular nilmanifolds with left-invariant complex structure, that the Kuranishi space can be singular [@cat-fred06]. In this article we want to study the Kuranishi space of complex parallelisable nilmanifolds. These were very intensively studied by Winkelmann [@winkelmann98] and they enjoy many interesting properties. We will only be concerned with their deformations. We will show in particular, that the Kuranishi space of a complex parallelisable nilmanifold is *almost always* singular thus showing that no analog of the Tian-Todorv theorem can exist for nilmanifolds. Nevertheless, the Kuranishi space can not become too ugly: If $X=\Gamma\backslash G$ is a complex parallelisable nilmanifold and $G$ is $\nu$-step nilpotent, then ${\mathrm{Kur}}(X)$ is cut out by polynomial equations of degree at most $\nu$. In Section \[nonexam\] we will give an example that the bound on the degree does not remain valid for general nilmanifolds but, as far as we know, there could be a larger bound depending on the step-length and the dimension only. We believe that the Lie-algebra ${\ensuremath{\gothg}}$ of $G$ cannot be too far from being free if the Kuranishi space is smooth and all examples that we found were actually free. Unfortunately, the analysis of the obstructions of higher order becomes very complicated but we can at least prove the following: Let $X=\Gamma\backslash G$ be a complex parallelisable nilmanifold and let ${\ensuremath{\gothg}}$ be the Lie-algebra of $G$. If ${\ensuremath{\gothg}}\slash [{\ensuremath{\gothg}},[{\ensuremath{\gothg}},{\ensuremath{\gothg}}]]$ is not isomorphic to a free 2-step nilpotent Lie-algebra then there is a non-vanishing obstruction in degree 2 and the Kuranishi space is singular. In particular, if ${\ensuremath{\gothg}}$ is 2-step nilpotent then ${\mathrm{Kur}}(X)$ is smooth if and only if ${\ensuremath{\gothg}}$ is a free 2-step nilpotent Lie-algebra. It is a natural question which infinitesimal deformations in $H^1(X,\Theta_X)$ integrate to a 1-parameter family of complex parallelisable complex structure and we show in Section \[remainparall\] that this is the case if and only if they are infinitesimally complex parallelisable. The same results holds for abelian complex structures [@con-fin-poon06]. From this we can also deduce that every complex parallelisable nilmanifold which is not a torus has small deformations which are no longer complex parallelisable (Corollary \[nondef\]). On the other hand it is known that small deformations at least remain in the category of nilmanifolds with left-invariant complex structure (see Section \[kurani\] or [@rollenske07b]). In Section \[examples\] we will give several explicit examples, mostly in small dimension. As far as we know, these are the first examples of compact complex manifolds with trivial canonical bundle (or even holomorphic symplectic structure) which have non-reduced Kuranishi-space. ### Acknowledgements {#acknowledgements .unnumbered} This research was carried out at Imperial College London supported by a DFG Forschungsstipendium and I would like to thank the Geometry group there for their hospitality. Fabrizio Catanese, Fritz Grunewald, Andrey Todorv and Jörg Winkelmann made several useful comments during a talk at the University of Bayreuth. Complex parallelisable nilmanifolds and nilmanifolds with left-invariant complex structure {#definitions} ========================================================================================== Let $G$ be a simply connected, complex, nilpotent Lie-group with Lie-algebra ${\ensuremath{\gothg}}$ and $\Gamma\subset G$ a lattice, i.e., a discrete cocompact subgroup. By a theorem of Mal’cev [@malcev51] such a lattice exists if and only if the real Lie-algebra underlying ${\ensuremath{\gothg}}$ can be defined over $\IQ$. The most important invariant attached to a nilpotent Lie-algebra (or Lie-group) is its nilpotency index, also called step length. It is defined as follows: consider the descending central series, inductively defined by $$\kc_0{\ensuremath{\gothg}}:={\ensuremath{\gothg}}, \qquad \kc_{k+1}{\ensuremath{\gothg}}=[\kc_k{\ensuremath{\gothg}},{\ensuremath{\gothg}}].$$ Then ${\ensuremath{\gothg}}$ is nilpotent if and only if there exists a $\nu$ such that $\kc^\nu{\ensuremath{\gothg}}=0$. The smallest such $\nu$ is called the nilpotency index. Since the multiplication in $G$ is holomorphic we can act with elements of $\Gamma$ on the left; the quotient $X:=\Gamma\backslash G$ is a complex parallelisable compact nilmanifold. The nilpotent complex Lie-group $G$ acts transitively on $X$ by multiplication on the right and this is in fact an equivalent characterisation of $\IC$-parallelisable nilmanifolds [@wang54]. As already remarked by Nakamura [@nakamura75] not all deformations of $\IC$-parallelisable nilmanifolds are again $\IC$-parallelisable but, as we will discuss in section \[kurani\], we can describe all deformations in the slightly more general framework of nilmanifolds with left-invariant complex structures which we will now explain. Let $H$ be a simply connected, real, nilpotent Lie-group with Lie-algebra ${\gothh}$ and containing a lattice $\Gamma$. Taking the quotient yields a real nilmanifold $M:=\Gamma\backslash H$. An almost complex structure $J: {\gothh}\to {\gothh}$ defines an almost complex structure on $H$ by left-translation and this almost complex structure is integrable if and only if the Nijenhuis condition $$\label{nijenhuis} [x,y]-[Jx,Jy]+J[Jx,y]+J[x,Jy]=0$$ holds for all $x,y\in {\gothh}$. In this case we call the pair $({\gothh}, J)$ a Lie-algebra with complex structure. The action of $\Gamma$ on the left is then holomorphic and we get an induced complex structure on $M$. We call $(M,J)$ a nilmanifold with left-invariant complex structure. Note that the multiplication in $H$ induces an action on the left on $M$ if and only if $\Gamma$ is normal if and only if $H=\IR^n$ is abelian; there is always an action on the right which is holomorphic if and only if $(H,J)$ is a complex Lie-group. By abuse of notation we will call a tensor, e.g., a vector field, differential form or metric, on $M$ left-invariant if its pullback to the universal cover $H$ is left-invariant. The complexified Lie-algebra ${\gothh}_\IC={\gothh}\tensor_\IR\IC$ decomposes as $${\gothh}_\IC={{{{\gothh}}^{1,0}}}\oplus{{{{\gothh}}^{0,1}}}$$ where ${{{{\gothh}}^{1,0}}}$ is the $i$-eigenspace of $J$ and ${{{{\gothh}}^{0,1}}}=\overline{{{{{\gothh}}^{1,0}}}}$ is the $(-i)$-eigenspace. It is not hard to see that the complex structure is integrable if and only if ${{{{\gothh}}^{1,0}}}$ is a (complex) Lie-subalgebra of ${\gothh}_\IC$. The complex structure $J$ makes $({\gothh},J)$ into a complex Lie-algebra if and only if the bracket is $J$-linear, i.e., for all $x,y\in {\gothh}$ we have $$\label{Clie} [Jx,y]=J[x,y].$$ In this case $H$ is a complex Lie-group and $(M,J)$ is $\IC$-parallelisable as above. The following equivalent characterisation is also well known. \[parallchar\] A Lie-algebra with complex structure $({\gothh},J)$ is a complex Lie-algebra if and only if $[{{{{\gothh}}^{1,0}}}, {{{{\gothh}}^{0,1}}}]=0$. In this case the canonical projection $$\pi: ({\gothh},J)\to {{{{\gothh}}^{1,0}}}, \qquad z\mapsto \frac{1}{2}(z-iJz)$$ is an isomorphism of complex Lie algebras. Let $x,y\in {\gothh}$ and consider $X:=\frac{1}{2}(x-iJx)\in {{{{\gothh}}^{1,0}}}$ and $\bar Y:=\frac{1}{2}(y+iJy)\in {{{{\gothh}}^{0,1}}}$. Then $$\begin{aligned} [X,\bar Y]&= \frac{1}{4}[x-iJx, y+iJy]\\&= \frac{1}{4}([x,y]-i^2[Jx,Jy]-i([Jx,y]-[x,Jy])\\&=\frac{1}{4}([x,y]+[Jx,Jy])-i([Jx,y]-[x,Jy])\end{aligned}$$ and we see that this vanishes if and only if $$[x,y]=-[Jx,Jy] \text{ and } [Jx,y]=[x,Jy].$$ If we combine these two equations with the Nijenhuis tensor [[(\[nijenhuis\])]{}]{} then we get the identity $-2[x,y]=2J[Jx,y]$ which becomes [[(\[Clie\])]{}]{} after applying $J$ to it and dividing by $-2$. On the other hand the equations are certainly fulfilled if [[(\[Clie\])]{}]{} holds and we have shown the claimed equivalence. The second claim is proved by a similar computation: since $\pi$ is an isomorphism of complex vector spaces it remains to show that $\pi$ is a homomorphism of Lie-algebras. Indeed for $x,y\in{\gothh}$ we have using [[(\[Clie\])]{}]{} $$[\pi(x), \pi(y)]= \frac{1}{4}[x-iJx, y-iJy]= \frac{1}{4}([x,y]+i^2[Jx,Jy]-2iJ[x,y])=\pi([x,y]).$$ \[notation\] In order to make our notation more transparent ${\gothh}$, $H$ and $M$ will always denote a real Lie-algebra, Lie-group or nilmanifold, often equipped with a (left-invariant) complex structure $J$. We will only consider integrable complex structures. The notations ${\ensuremath{\gothg}}$, $G$ and $X$ will be reserved for their complex parallelisable counterparts. If we need to access the underlying real object with left-invariant complex structure we will write for example ${\ensuremath{\gothg}}=({\gothh},J)$. By the above Lemma we can then identify $${\ensuremath{\gothg}}_\IC={\gothh}_\IC={\ensuremath{\gothg}}\oplus \bar {\ensuremath{\gothg}}$$ where the bracket on $\bar {\ensuremath{\gothg}}$ is given by $[\bar x, \bar y]=\overline{[x,y]}$ and $[{\ensuremath{\gothg}}, \bar{\ensuremath{\gothg}}]=[{{{{\gothh}}^{1,0}}}, {{{{\gothh}}^{0,1}}}]=0$. Another important class of left-invariant complex structures are so-called abelian complex structures, which are characerised by $[{{{{\gothh}}^{1,0}}},{{{{\gothh}}^{1,0}}}]=0$ or, equivalently, $[Jx,Jy]=[x,y]$ for all $x,y\in {\gothh}$. In some sense this is the opposite condition to being a complex Lie-algebra and their deformations have been studied in [@mpps06; @con-fin-poon06]. As we pointed out in the introduction, deformations behave much more nicely in this case. Dolbeault cohomology ==================== In this section we will describe how the Dolbeault cohomology of a nilmanifold with left-invariant complex structure $(M,J)$ is completely controlled by the Lie-algebra with complex structure $({\gothh}, J)$. This reduces many problems in the study of nilmanifolds to finite dimensional linear algebra. We will soon concentrate on the complex parallelisable case. Let $(M,J)$ be a nilmanifold with left-invariant complex structure and ${\gothh}$ be the Lie-algebra of the corresponding Lie-group. We can identify elements in $$\Lambda ^{p,q}:=\Lambda^{p,q}({\gothh}^*,J)=\Lambda^p{{{{\gothh}^*}^{1,0}}}\tensor\Lambda^q{{{{\gothh}^*}^{0,1}}}$$ with left-invariant differential forms of type $(p,q)$ on $M$. The differential $d=\del+\delbar$ restricts to $$\Lambda^*{\gothh}_\IC^*=\bigoplus\Lambda ^{p,q}$$ and can in fact be defined in terms of the Lie bracket only: for $\alpha \in {\gothh}^*$ and $x,y\in {\gothh}$ considered as differential form and vectorfields we have $$\label{differential} d\alpha(x,y)=x(\alpha(y))-y(\alpha(x))-\alpha([x,y])=-\alpha([x,y])$$ since all left-invariant functions are constant. Let $H^k({\gothh}, \IC)$ be the $k$-th cohomology group of the complex $$\Lambda^*{\gothh}^*_\IC: \quad 0\to \IC\overset{0}{\to} {\gothh}_\IC^*\overset{d}{\to}\Lambda^2{\gothh}_\IC^* \overset{d}{\to}\Lambda^3{\gothh}_\IC^* \overset{d}{\to}\dots$$ and $H^{p,q}({\gothh},J)$ be the $q$-th cohomology group of the complex $$\Lambda^{p,*}:\quad 0\to \Lambda^{p,0}\overset{\delbar}{\to}\Lambda^{p,1}\overset{\delbar}{\to}\Lambda^{p,2}\overset{\delbar}{\to}\dots$$ In fact, the first complex calculates the usual Lie-algebra cohomology with values in the trivial module $\IC$ while the second calculates the cohomology of the Lie-algebra ${{{{\gothh}}^{0,1}}}$ with values in the module $\Lambda^{p,0}$ (see [@rollenske07b]). \[cohom\] Let $M=\Gamma\backslash H$ be a real nilmanifold with Lie-algebra ${\gothh}$. 1. The inclusion of $\Lambda^*{\gothh}^*_\IC$ into the de Rham complex induces an isomorphism $$H_{\mathrm{dR}}^*(M, \IC) \isom H^*({\gothh}, \IC)$$ in cohomology. (Nomizu, [@nomizu54]) 2. The inclusion of $\Lambda^{p,*}$ into the Dolbeault complex induces an inclusion $$\label{iota} \iota_J:H^{p,q}({\gothh},J)\to H^{p,q}(M,J)$$ which is an isomorphism if $(M,J)$ is complex parallelisable (Sakane, [@sakane76]) or if $J$ is abelian (Console and Fino, [@con-fin01]). Moreover, there exists a a dense open subset $U$ of the space of all left-invariant complex structures on $M$ such that $\iota$ is an isomorphism for all $J\in U$ ([@con-fin01]) Other work in this direction was done by Cordero, Fernándes, Gray and Ugarte [@cfgu00]. Conjecturally $\iota$ is an isomorphism for all left-invariant complex structures; in particular no counterexample is known. For further reference we describe some cohomology groups in these terms. Let ${\ensuremath{\gothg}}$ be a complex Lie-algebra. Let us denote by $K^k:=\im(d:\Lambda^{k-1}{\gothh}_\IC^*{\to}\Lambda^{k}{\gothh}_\IC^*)$ the space of $k$-boundaries. Then $$\begin{gathered} H^0({\ensuremath{\gothg}}, \IC)=\IC,\\ H^1({\ensuremath{\gothg}},\IC)=\Ann(\kc_1{\ensuremath{\gothg}})=\Ann([{\ensuremath{\gothg}},{\ensuremath{\gothg}}]),\\ K^2=\Ann(\ker([-,-]:\Lambda^2{\ensuremath{\gothg}}\to {\ensuremath{\gothg}})).\end{gathered}$$ Moreover, $H^{0,1}({\ensuremath{\gothg}})=\overline{H^1({\ensuremath{\gothg}},\IC)}$ and $\im(\delbar:\bar{\ensuremath{\gothg}}^*\to \Lambda^2\bar{\ensuremath{\gothg}}^*)=\bar K^2$. All assertions follow immediately from the fact that the differential $d:{\ensuremath{\gothg}}^*\to\Lambda^2{\ensuremath{\gothg}}^*$ is the dual of the Lie bracket $[-,-]:\Lambda^2{\ensuremath{\gothg}}\to {\ensuremath{\gothg}}$ and from the identification ${\ensuremath{\gothg}}_\IC={\ensuremath{\gothg}}\oplus \bar {\ensuremath{\gothg}}$. Since we are interested in deformations, the cohomology of the holomorphic tangent bundle (resp. tangent sheaf) $\Theta_{(M,J)}$ is of particular interest. It has been calculated in [@rollenske07b] for left-invariant complex structures for which [[(\[iota\])]{}]{} is an isomorphism, generalising results on abelian complex structure in [@mpps06; @con-fin-poon06]. But for a complex parallelisable nilmanifold $X$ we can calculate it directly (as observed by Nakamura [@nakamura75]). Any element of the complex Lie-algebra ${\ensuremath{\gothg}}$ gives rise to a holomorphic vector field. Hence the tangent sheaf is isomorphic to $\ko_X\tensor {\ensuremath{\gothg}}$ and in cohomology we have a natural isomorphism $$H^q(X,\Theta_X)=H^q(X, \ko_X\tensor {\ensuremath{\gothg}})\isom H^q(X, \ko_X)\tensor {\ensuremath{\gothg}}=H^{0,q}(X)\tensor {\ensuremath{\gothg}}\isom H^{0,q}({\ensuremath{\gothg}})\tensor {\ensuremath{\gothg}}.$$ Combining this with the previous results we get \[cohomcalc\] Let $X=\Gamma\backslash G$ be a complex parallelisable nilmanifold. Then the tangent sheaf $\Theta_X\isom \ko_X\tensor {\ensuremath{\gothg}}$ and its cohomology is calculated by the complex $$0\to {\ensuremath{\gothg}}\overset{0}{\to} \bar{{\ensuremath{\gothg}}}^*\tensor {\ensuremath{\gothg}}\overset{\delbar}{\to}\Lambda^{2} \bar{{\ensuremath{\gothg}}}^*\tensor {\ensuremath{\gothg}}\overset{\delbar}{\to}\dots$$ where the differential of $\bar\alpha\tensor X \in \Lambda^{p,0}{\ensuremath{\gothg}}$ is given by $ \delbar (\bar\alpha\tensor X)=(\delbar\bar\alpha) \tensor X$. In particular we have $$\begin{gathered} H^0(X, \Theta)={\ensuremath{\gothg}}\\ H^1(X,\Theta)=H^1(X, \ko_X)\tensor {\ensuremath{\gothg}}=\overline {\Ann([{\ensuremath{\gothg}},{\ensuremath{\gothg}}])}\tensor {\ensuremath{\gothg}}\end{gathered}$$ Kuranishi theory {#kurani} ================ In [@kuranishi62] Kuranishi showed that for every compact complex manifold $X$ there exists a locally complete family of deformations which is versal at $X$. He constructs this family explicitly as a small neighbourhood of zero in the space of harmonic $(0,1)$-forms with values in the holomorphic tangent bundle after choosing some hermitian metric on $X$ (which always exists). We will now apply his construction to complex parallelisable nilmanifolds using the results of the last section. Let $(M,J)=(\Gamma\backslash H,J)$ be the real nilmanifold with left-invariant complex structure underlying a complex parallelisable nilmanifold $X=\Gamma\backslash G$. The complex structure $J:{\gothh}\to {\gothh}$ is uniquely determined by the eigenspace decomposition ${\gothh}_\IC={{{{\gothh}}^{1,0}}}\oplus {{{{\gothh}}^{0,1}}}$. A (sufficiently small) deformation of this decomposition ${\gothh}_\IC=V\oplus \bar V$ can be encoded in a map $\Phi:{{{{\gothh}}^{0,1}}} \to {{{{\gothh}}^{1,0}}}$ such that $\bar V=(\id+\Phi) {{{{\gothh}}^{0,1}}}$, i.e., the graph of $\Phi$ in ${\gothh}_\IC$ is the new space of vectors of type $(0,1)$. This decomposition then determines a unique almost complex structure $J_V$ which is integrable if and only if $[V,V]\subset V$. So far we have only described deformations of $J$ which remain left-invariant; this will be justified in a moment. The integrability condition is most conveniently expressed using the so-called Schouten bracket: for $X,Y\in {{{{\gothh}}^{1,0}}}$ and $(0,1)$-forms $\bar \alpha, \bar\beta\in{{{{\gothh}^*}^{0,1}}}$ we set $$\label{Schouten} [\bar\alpha\tensor X, \bar\beta\tensor Y]:=\bar \beta \wedge L_{Y}\bar \alpha \tensor X+ \bar\alpha \wedge L_{X}\bar\beta\tensor Y+\bar \alpha\wedge \bar \beta \tensor [X,Y]$$ where $L_{X}\bar\beta=i_Xd\bar\beta+d(i_X \bar\beta)$ is the Lie derivative and $i_X$ is the contraction with $X$. One can then show that the new complex structure is integrable if and only if $\Phi$ satisfies the Maurer-Cartan equation $$\label{MC} \delbar \Phi +[\Phi, \Phi]=0$$ and it is well known that infinitesimal deformations, which correspond to first-order solutions, are parametrised by classes in $H^1(X,\Theta_X)$ (see for example [@catanese88] or [@Huybrechts] for an overview). But different solutions may well yield isomorphic deformations. In order to single out a preferred solution we choose a hermitian structure on ${\ensuremath{\gothg}}$ which induces a left-invariant hermitian structure on $X$. Using the Hodge star operator associated to the hermitian metric we can define the formal adjoint $\delbar^*$ to $\delbar$ and the Laplace operator $$\Delta:=\delbar\delbar^*+\delbar^*\delbar.$$ Defining the space of harmonic forms to be $\kh^k=\ker (\Delta:\Lambda^{k} \bar{{\ensuremath{\gothg}}}^*\to \Lambda^{k} \bar{{\ensuremath{\gothg}}}^* )$ there is an orthogonal decomposition $$\Lambda^{k} \bar{{\ensuremath{\gothg}}}^*=B^k\oplus\kh^k\oplus V^k$$ where $B^k=\im(\delbar:\Lambda^{k-1} \bar{{\ensuremath{\gothg}}}^*\to \Lambda^{k} \bar{{\ensuremath{\gothg}}}^*)$ and $V^k=\im(\delbar^*:\Lambda^{k+1} \bar{{\ensuremath{\gothg}}}^*\to \Lambda^{k} \bar{{\ensuremath{\gothg}}}^*)$; this is just the intersection of usual Hodge-decomposition with the subcomplex of left-invariant differential forms. The main point is that all harmonic forms are in left-invariant in our setting. Since $\ker(\delbar)=B^k\oplus \kh^k$ we get an isomorphism $$H^k(X,\Theta_X)\isom H^k(X,\ko_X)\tensor {\ensuremath{\gothg}}\isom \kh^k\tensor {\ensuremath{\gothg}}.$$ We are especially interested in the first two cohomology groups. By Lemma \[cohomcalc\] we have $B^1=0$ which yields a commutative diagram $$\xymatrix{ \bar{\ensuremath{\gothg}}^*\tensor {\ensuremath{\gothg}}\ar[rr]^\delbar \ar@{=}[d]&& \ar@{=}[d]\Lambda^2\bar{\ensuremath{\gothg}}^*\tensor {\ensuremath{\gothg}}\\ (\kh^1\tensor{\ensuremath{\gothg}})\oplus (V^1 \tensor {\ensuremath{\gothg}})\ar[rr]^\delbar \ar[d]^{\mathrm{pr}} && (B^2\tensor {\ensuremath{\gothg}})\oplus( \kh^2\tensor {\ensuremath{\gothg}})\oplus( V^2\tensor {\ensuremath{\gothg}}) \ar[dl]^P\ar[d]_H\\ V^1\tensor {\ensuremath{\gothg}}& \ar[l]^{\delta}_{\isom} B^2\tensor{\ensuremath{\gothg}}&\kh^2\tensor {\ensuremath{\gothg}}. }$$ We denote by $\delta$ the inverse of the isomorphism $P\circ \delbar: V^1\tensor{\ensuremath{\gothg}}\to B^2\tensor{\ensuremath{\gothg}}$. We will now use these operators to describe the Kuranishi space: let $X_1,\dots, X_n$ be a basis of ${\ensuremath{\gothg}}$ and $\bar\omega^1, \dots \bar\omega^m$ be a basis for $\kh^1$. Then $\{\bar\omega^i\tensor X_j\}$ is a basis of $H^1(X,\Theta_X)$ and we define recursively $$\label{Phi} \begin{split} \Phi_1(\underline t)&=\sum_{i,j} t_i^j \bar\omega^i\tensor X_j, \\ \Phi_2(\underline t)&:=-\delta\circ P [\Phi_1(\underline t), \Phi_1(\underline t)],\\ \Phi_k(\underline t)&:=-\delta\circ P \sum_{1\leq i<k} \left[\Phi_i(\underline t), \Phi_{k-i}(\underline t)\right] \quad (k\geq 2), \end{split}$$ obtaining a formal power series $$\Phi(\underline t)=\sum_{k\geq1} \Phi_k(\underline t).$$ We see that $\Phi_k$ is a homogeneous polynomial of degree $k$ in the variables $t_i^j$ and it is easy to verify that $$\delbar\Phi+[\Phi,\Phi]=H[\Phi,\Phi].$$ The map $\Phi$ does not depend on the choice of the basis and we can define the obstruction map $${\mathrm{obs}}:\kh^1\tensor {\ensuremath{\gothg}}\to \kh^2\tensor {\ensuremath{\gothg}},\qquad \mu=\sum_{i,j} t_i^j \bar\omega^i\tensor X_j\mapsto H[\Phi(\underline t),\Phi(\underline t)].$$ We can now formulate Kuranishi’s theorem in our context. The formal powerseries $\Phi(\underline t)$ converges for sufficiently small values of $\underline t$ and there is a versal family of deformations of $X$ over the space $${\mathrm{Kur}}(X):=\{\mu\in \kh^1(\Theta_X)\mid \|\mu\|<\epsilon; {\mathrm{obs}}(\mu)=0\}.$$ where $\kh^1(\Theta_X)=\kh^1\tensor {\ensuremath{\gothg}}$ is the space of harmonic 1-forms with values in $\Theta_X$. ${\mathrm{Kur}}(X)$ is called the Kuranishi space of $X$. By construction $\Phi$ is left-invariant and hence the new complex structure will also be left-invariant. In fact, the new subbundle of tangent vectors of type $(0,1)$ in $TM_\IC$ is obtained by translating the subspace $(\id+\Phi){{{{\ensuremath{\gothg}}}^{0,1}}}\subset {\ensuremath{\gothg}}_\IC$. We have reproved that all sufficiently small deformations of our complex parallelisable nilmanifold carry a left-invariant complex structure. Note that the construction involved the choice of a hermitian structure so ${\mathrm{Kur}}(X)$ is not defined in a canonical way. Nevertheless for different choices of a metric (the germs of) the resulting spaces are (non canonically) isomorphic. The values of $\underline t$ have to be small for two different reasons. First of all we need to ensure the convergence of the formal power series $\Phi(\underline t)$ and secondly $(\id+\Phi)\bar {\ensuremath{\gothg}}$ should be the space of $(0,1)$ vectors for an integrable almost complex structure, in other words we need $(\id+\Phi)\bar {\ensuremath{\gothg}}\oplus\overline{(\id+\Phi)\bar {\ensuremath{\gothg}}}={\ensuremath{\gothg}}_\IC$. We will see that the first issue will not arise in our setting. Usually the terms of the formal power series $\Phi$ are described using Green’s operator, which inverts the Laplacian on the orthogonal complement of harmonic forms, setting $$\Phi_k(\underline t):=-\delbar^*G \sum_{1\leq i<k} \left[\Phi_i(\underline t), \Phi_{k-i}(\underline t)\right]$$ It is straight-forward to check that this agrees with our definition above using the identities $G\circ\Delta+H=\Delta\circ G+H=\id$ and definition of the Laplacian. Our formula involves only $\delta=\inverse \delbar$ and the projection $P$ which will facilitate the computation of examples in Section \[examples\]. Now that we have seen how the Kuranishi space is constructed we want to investigate its structure in detail for complex parallelisable nilmanifolds. The key result is the following: \[schoutennil\] Let $\bar\alpha\tensor X, \bar\beta\tensor Y \in \bar{\ensuremath{\gothg}}^*\tensor {\ensuremath{\gothg}}$. Then their Schouten bracket is $$[\bar\alpha\tensor X, \bar\beta\tensor Y ]=\bar\alpha\wedge \bar \beta\tensor[X,Y].$$ Comparing the expression with the general formula [[(\[Schouten\])]{}]{} it suffices to show that for $X\in {\ensuremath{\gothg}}$ and $\bar\alpha\in\bar{\ensuremath{\gothg}}$ the Lie-derivative $L_X\bar\alpha=i_Xd\bar\alpha+d(i_X \bar\alpha)=0$. But $\bar\alpha$ is of type $(0,1)$ and $d\bar\alpha$ is of type $(0,2)$ (since $[{\ensuremath{\gothg}}, \bar{\ensuremath{\gothg}}]=0$) so both vanish when contracted with a vector of type $(1,0)$. This gives us For $\Phi$ as in the recursive description [[(\[Phi\])]{}]{} we have $[\Phi_k, \Phi_l] \in \Lambda^2\bar{{\ensuremath{\gothg}}}^*\tensor \kc_{k+l-1}{\ensuremath{\gothg}}\subset \Lambda^2\bar{{\ensuremath{\gothg}}}^*\tensor{\ensuremath{\gothg}}$. We prove our claim by induction: for $k=1$ there is nothing to prove since $\kc_0{\ensuremath{\gothg}}={\ensuremath{\gothg}}$. Note that, by the Jacobi identity, $[\kc_k{\ensuremath{\gothg}}, \kc_l{\ensuremath{\gothg}}]\subset \kc_{k+l+1}{\ensuremath{\gothg}}$. Since the Schouten bracket is the Lie bracket on the vector part and the map $\delta=\inverse\delbar$ acts only on the form part our claim follows. We deduce immediately that the Kuranishi space can not be too complicated: \[polynom\] If ${\ensuremath{\gothg}}$ is $\nu$-step nilpotent and $\Phi$ as in [[(\[Phi\])]{}]{} then $${\mathrm{obs}}(\underline t)=\sum_{\stackrel{1\leq i,j< \nu,}{ i+j\leq \nu}}H[\Phi_i, \Phi_j].$$ In particular ${\mathrm{Kur}}(X)$ is cut out by polynomial equations of degree at most $\nu$. Since ${\ensuremath{\gothg}}$ is $\nu$-step nilpotent $\kc_k{\ensuremath{\gothg}}=0$ for $k\geq \nu$. By the previous Lemma this implies that $[\Phi_i, \Phi_j]=0$ whenever $i+j-1\geq \nu$ and hence the only possibly non-vanishing terms of ${\mathrm{obs}}=H[\Phi,\Phi]$ are the ones given above. For further reference we use Lemma \[schoutennil\] to calculate the second order obstructions, i.e., the quadratic term of the obstruction map ${\mathrm{obs}}$: let as before $\bar\omega^1, \dots, \bar\omega^{m}$ be a basis of $\kh^1=\overline {\Ann([{\ensuremath{\gothg}},{\ensuremath{\gothg}}])}$ and $X_1, \dots, X_n$ be a basis of ${\ensuremath{\gothg}}$. Then we can represent any element in $H^1(X, \Theta_X)$ as $$\Phi_1(\underline t)=\sum_{i,j} t_i^j \bar\omega^i\tensor X_j$$ and consequently $$\label{deg2} \begin{split} [\Phi_1(\underline t),\Phi_1(\underline t)]&=[\sum_{i,k} t_i^k \bar\omega^i\tensor X_k, \sum_{j,l} t_j^l \bar\omega^j\tensor X_l]\\ &= \sum_{i,j,k,l}(t_i^k t^l_j)[ \bar\omega^i\tensor X_k, \bar\omega^j\tensor X_l]\\ &= \sum_{i,j,k,l}(t_i^k t^l_j)\bar\omega^i\wedge\bar\omega^j\tensor [X_k, X_l]\\ & = \sum_{1\leq i<j\leq m}\sum_{k,l}(t_i^k t^l_j-t_j^k t^l_i)\bar\omega^i\wedge\bar\omega^j\tensor [X_k, X_l]\\ & = \sum_{1\leq i<j\leq m}\sum_{1\leq k<l\leq n}2(t_i^k t^l_j-t_j^k t^l_i)\bar\omega^i\wedge\bar\omega^j\tensor [X_k, X_l]\\ & = 2 \sum_{1\leq i<j\leq m}\sum_{1\leq k<l\leq n} \det\begin{pmatrix}t_i^k & t_i^l\\t_j^k & t_j^l \end{pmatrix} \bar\omega^i\wedge\bar\omega^j\tensor [X_k, X_l]. \end{split}$$ We deduce from this formula a necessary condition for the Kuranishi space to be smooth: \[lambda2\] If the subspace $\Lambda^2 \kh^1 \subset \Lambda^2 \bar{\ensuremath{\gothg}}$ is not contained in $B^2$ and ${\ensuremath{\gothg}}$ is not abelian then there is a non-vanishing obstructions in degree 2 and the Kuranishi space is singular. Assume that $\Lambda^2 \kh^1 \subset \Lambda^2 \bar{\ensuremath{\gothg}}$ is not contained in $B_2$. Then there is some basis vector $\bar\omega^i\wedge\bar\omega^j$ which is not contained in the image of $\delbar$. Since ${\ensuremath{\gothg}}$ is not abelian there are vectors $X_k, X_l$ such that $[X_k, X_l]\neq 0$. Setting $t_p^q=0$ if $p\neq i,j$ or $q\neq k,l$ and choosing the remaining coefficient such that $\det\begin{pmatrix}t_i^k & t_i^l\\t_j^k & t_j^l \end{pmatrix}\neq 0$ we have found an obstructed element in $H^1(X, \Theta_X)$. The condition that the Kuranishi space be smooth is very strong. To make this more precise we need to recall the definition of the free 2-step Lie-algebra: let $m\geq2$, $V=\IC^m$ and $\gothb_m:=V\oplus \Lambda^2 V$. Then $\gothb_m$ with the Lie bracket $$[ \cdot, \cdot]: \gothb_m\times \gothb_m\to \gothb_m, \qquad [a+b\wedge c, a'+b'\wedge c']:=a\wedge a'$$ is the free 2-step nilpotent Lie-algebra. \[freecond\] If ${\ensuremath{\gothg}}$ is not abelian then there is a non-vanishing obstruction in degree 2 if and only if ${\ensuremath{\gothg}}\slash \kc^2{\ensuremath{\gothg}}$ is not isomorphic to a free 2-step nilpotent Lie-algebra. Hence, if ${\ensuremath{\gothg}}\slash \kc^2{\ensuremath{\gothg}}$ is not free then the Kuranishi space is singular. The vanishing of all obstructions on degree 2 is not a sufficient condition for the Kuranishi space to be smooth. A 4-dimensional example where ${\mathrm{Kur}}(X)$ is cut out by a single cubic equation can be found in Section \[explicit4\]. All examples with smooth Kuranishi space which we could find were actually free Lie algebras and at least in the 2-step nilpotent case there are no other: \[b\_m\] If ${\ensuremath{\gothg}}$ is 2-step nilpotent then the Kuranishi space is smooth if and only if ${\ensuremath{\gothg}}$ is a free 2-step nilpotent Lie-algebra, i.e., ${\ensuremath{\gothg}}\isom \gothb_m$ with $m=h^{0,1}(X)$. This follows immediately from the theorem since for a 2-step nilpotent Lie-algebra we have $\kc_2{\ensuremath{\gothg}}=0$, hence ${\ensuremath{\gothg}}/\kc_2{\ensuremath{\gothg}}\isom{\ensuremath{\gothg}}.$ Note that $\gothb_2$ is the complex Heisenberg algebra, which is the Lie-algebra of the universal cover of the Iwasawa-manifold. So we have reproved the smoothness of the Kuranishi space of the Iwasawa manifold first observed by Nakamura. It is very easy to produce examples with singular Kuranishi space: If ${\ensuremath{\gothg}}\isom {\ensuremath{\gothg}}'\oplus \gotha$ where $\gotha\isom \IC^n$ is an abelian Lie-algebra and ${\ensuremath{\gothg}}'$ is not abelian, then the Kuranishi-space is singular. In particular, if $X$ is any complex parallelisable nilmanifold which is not a torus and $T$ is a complex torus then $X\times T$ has obstructed deformations. We have ${\ensuremath{\gothg}}\slash \kc_2{\ensuremath{\gothg}}={\ensuremath{\gothg}}'\slash\kc_2{\ensuremath{\gothg}}'\oplus \gotha$ which is not free. An application of the theorem proves the assertion. Before we can address the proof of Theorem \[freecond\] we need a technical lemma. Let ${\ensuremath{\gothg}}=\kc_0 {\ensuremath{\gothg}}\supset \kc_1{\ensuremath{\gothg}}\supset \dots \supset \kc_{\nu}{\ensuremath{\gothg}}=0$ be the descending central series and let $\kc^k{\ensuremath{\gothg}}^*=\Ann\kc_k{\ensuremath{\gothg}}$. We get a filtration $$0=\kc^0{\ensuremath{\gothg}}^*\subset \kc^1{\ensuremath{\gothg}}^*=\Ann(\kc_1{\ensuremath{\gothg}})\subset \dots\subset \kc^\nu{\ensuremath{\gothg}}^*={\ensuremath{\gothg}}^*.$$ \[dck\] Setting $$W^k=\langle \alpha\wedge \beta \in \Lambda^2{\ensuremath{\gothg}}^*\mid \alpha \in \kc^i{\ensuremath{\gothg}}^*, \beta\in \kc^j{\ensuremath{\gothg}}^*, i+j\leq k\rangle \subset \Lambda^2{\ensuremath{\gothg}}^*$$ we have $$d\alpha \in W^k \iff \alpha \in \kc^k{\ensuremath{\gothg}}^*.$$ Assume that there is $\alpha \notin \kc^k{\ensuremath{\gothg}}^*$ with $d\alpha \in W^k$. By the Jacobi-identity $\kc_k{\ensuremath{\gothg}}$ is generated by elements of the form $X=[Y,Z]$ where $Y\notin \kc_1{\ensuremath{\gothg}}$ and $Z\in \kc_{k-1}{\ensuremath{\gothg}}$ and hence $\alpha(X)\neq 0$ for one such element. By the definition of $\kc^i{\ensuremath{\gothg}}^*$ we have $\beta(Y,Z)=0$ for all $\beta \in W^k$. On the other hand $$d\alpha(Y,Z)=-\alpha([Y,Z])=-\alpha(X)\neq 0$$ so $d\alpha\notin W_k$ – a contradiction. The other direction is a well known fact for nilpotent Lie algebras. It can be easily seen picking a basis adapted to the descending central series (often called Malcev or Engel basis) and writing $\alpha$ as a linear combination of the elements of the dual basis. *Proof of Theorem \[freecond\].* Let ${\ensuremath{\gothg}}$ be a non-abelian Lie-algebra. By Lemma \[lambda2\] it suffices to show that $\Lambda^2\kh^1\subset B_2$ if and only if ${\ensuremath{\gothg}}\slash \kc^2{\ensuremath{\gothg}}$ is a free 2-step Lie-algebra. Recalling that $\kh^1=\overline{\kc^1{\ensuremath{\gothg}}^*}$ and $B^2=\overline{\im(d)}$ we have to prove that $\Lambda^2\kc^1{\ensuremath{\gothg}}^*=W^1$ is in the image of the differential if and only if ${\ensuremath{\gothg}}\slash \kc^2{\ensuremath{\gothg}}$ is free. The Lie bracket in ${\ensuremath{\gothg}}$ can also be considered as a linear map $$b:\Lambda^2 {\ensuremath{\gothg}}\to \kc_1{\ensuremath{\gothg}},$$ which is, by definition, surjective. Dualising we get (the restriction of) the differential $$d: (\kc_1{\ensuremath{\gothg}})^* \to \Lambda^2 {\ensuremath{\gothg}}^*,$$ which is now injective. Let $A$ be the anullator of $\kc_2{\ensuremath{\gothg}}$ in $(\kc_1{\ensuremath{\gothg}})^*$. Then we infer from Lemma \[dck\] that $d\restr{A}:A\to W^2=\Lambda^2\kc^1{\ensuremath{\gothg}}^*$, in fact, $$dA=\im (d) \cap W^2=\im(d)\cap \Lambda^2\kc^1{\ensuremath{\gothg}}^*.$$ The dual map $$b': \left(\Lambda^2\kc^1{\ensuremath{\gothg}}^*\right)^*=\Lambda^2({\ensuremath{\gothg}}\slash\kc_1{\ensuremath{\gothg}}) \to A^*=\kc_1{\ensuremath{\gothg}}\slash\kc_2{\ensuremath{\gothg}}$$ gives an anti-symmetric bilinear form on ${\ensuremath{\gothg}}\slash\kc_1{\ensuremath{\gothg}}$ with values in $\kc_1{\ensuremath{\gothg}}\slash\kc_2{\ensuremath{\gothg}}$ which is exactly the Lie bracket in the quotient Lie-algebra ${\ensuremath{\gothg}}\slash\kc_2{\ensuremath{\gothg}}$. Hence we see that $\Lambda^2\kc^1{\ensuremath{\gothg}}^*$ is in the image of $d$ if and only if $d: A \to W^2$ is surjective if and only if $b'$ is injective. But $b'$ is by definition surjective so it is injective if and only if it is bijective in which case $\Lambda^2({\ensuremath{\gothg}}\slash\kc_1{\ensuremath{\gothg}})\isom\kc_1{\ensuremath{\gothg}}\slash\kc_2{\ensuremath{\gothg}}$ and the Lie-algebra ${\ensuremath{\gothg}}\slash \kc^2{\ensuremath{\gothg}}$ is indeed free. Deformations remaining complex parallelisable {#remainparall} ============================================= It is a natural question if there are conditions which guarantee that a given small deformation of our complex parallelisable manifold $X$ is again complex parallelisable. So let $\mu \in H^1(X, \Theta_X)=\kh^1\tensor {\ensuremath{\gothg}}$ be a infinitesimal deformation and $\Phi$ the corresponding iterative solution of the Maurer-Cartan equation as in [[(\[Phi\])]{}]{}. The new space of $(0,1)$-vectors is $(\id+\Phi)\bar {\ensuremath{\gothg}}$. (Recall that we identified ${\ensuremath{\gothg}}_\IC={\ensuremath{\gothg}}\tensor\bar{\ensuremath{\gothg}}$.) By Lemma \[parallchar\] the new complex structure is again complex parallelisable if and only if $$[(\id+\bar\Phi) X, (\id+\Phi)\bar Y]=0$$ for all $X,Y\in {\ensuremath{\gothg}}$. Looking at the terms up to first order yields $$[X, \bar Y]+[\bar\mu X,\bar Y] +[ X,\mu\bar Y]=[\bar\mu X,\bar Y] +[ X,\mu\bar Y]=0.$$ The first of these terms is in $\bar{\ensuremath{\gothg}}$ while the second is in ${\ensuremath{\gothg}}$ and they are complex conjugate to each other up to sign and renaming. Thus we call $\mu$ an infinitesimally complex parallelisable deformation if $$\begin{gathered} \forall X,Y \in {\ensuremath{\gothg}}: [ X,\mu\bar Y]=0 \iff \mu \in \kh^1\tensor \kz{\ensuremath{\gothg}}.\end{gathered}$$ Such infinitesimal deformations are always unobstructed: if $\mu \in \kh^1\tensor \kz{\ensuremath{\gothg}}$ then $[\mu,\mu]\in \Lambda^2\bar{\ensuremath{\gothg}}^*\tensor [\kz{\ensuremath{\gothg}}, \kz{\ensuremath{\gothg}}]=0$. Hence in the recursive definition [[(\[Phi\])]{}]{} all higher order terms vanish, $\Phi=\mu$ and ${\mathrm{obs}}(\mu)=0$. We have proved \[remaining\] For an element $\mu \in H^{1}(X, \Theta_X)=H^1(X,\ko_X)\tensor {\ensuremath{\gothg}}$ the following are equivalent: 1. $\mu \in H^1(X, \ko_X)\tensor \kz{\ensuremath{\gothg}}$. 2. $\mu$ defines an infinitesimally complex parallelisable deformation. 3. $t\mu$ induces a 1-parameter family of complex parallelisable manifolds for $t$ small enough, i.e., provided that $(\id+t\mu)\bar{\ensuremath{\gothg}}\oplus(\id+t\bar\mu){\ensuremath{\gothg}}={\ensuremath{\gothg}}_\IC$. Hence the Kuranishi family is (locally) a cylinder over an analytic subset of $H^1(X, \ko_X)\tensor ({\ensuremath{\gothg}}/\kz{\ensuremath{\gothg}})$. Since ${\ensuremath{\gothg}}=\kz{\ensuremath{\gothg}}$ if and only if ${\ensuremath{\gothg}}$ is abelian we deduce: \[nondef\] If ${\ensuremath{\gothg}}$ is not abelian then there are small deformations of $X$ which are not complex parallelisable. Examples ======== We continue to use the notation from Remark \[notation\]. The deformation theory of the complex parallelisable nilmanifold $X$ is completely determined by the Lie-algebra ${\ensuremath{\gothg}}$ and we have already discussed two series of examples where ${\mathrm{Kur}}(X)$ is smooth. - If ${\ensuremath{\gothg}}=\mathfrak a_k$ is the $k$-dimensional abelian Lie-algebra then $X$ is a torus and ${\mathrm{Kur}}(X)$ is smooth of dimension $\frac{k^2(k+1)}{2}$. - If ${\ensuremath{\gothg}}=\mathfrak b_m$ is the free 2-step nilpotent Lie-algebra on $m$ generators, which has dimension $\frac{m(m+3)}{2}$, then ${\mathrm{Kur}}(X)$ is smooth of dimension $\frac{m^2(m+3)}{2}$ (see Corollary \[b\_m\]). Examples in low dimension – overview ------------------------------------ Nilpotent complex Lie algebras are classified up to dimension 7 [@magnin86] and partial results are known in dimension 8 . Starting from dimension 7 there are infinitely many non-isomorphic cases. We will now describe the Kuranishi-space of complex parallelisable nilmanifolds up to dimension 5. There is a convenient way to describe a nilpotent Lie-algebra ${\ensuremath{\gothg}}$ using the differential $d: {\ensuremath{\gothg}}\to \Lambda^2{\ensuremath{\gothg}}$. The expression $${\ensuremath{\gothg}}=(0,0,0,0,12+34)$$ means the following: with respect to a basis $\omega^1, \dots , \omega^5$ the differential is given by $$d\omega^1=d\omega^2=d\omega^3=d\omega^4=0 \text{ and } d\omega^5=\omega^1\wedge\omega^2+\omega^3\wedge\omega^4.$$ This determines the Lie bracket, which is the dual map (see [[(\[differential\])]{}]{}). More precisely, if we denote by $X_1, \dots, X_5$ the dual basis then the only non-zero Lie brackets are $[X_1, X_2]=[X_3,X_4]=-X_5$. Table 1 lists all Lie-algebras up to dimension 5 in this notation together with some information on the Kuranishi space of an associated complex parallelisable nilmanifold. We denote the nilpotency index by $\nu$. Note that all Lie-algebras with smooth Kuranishi space are either free or abelian. One can check that also the free 4-step nilpotent Lie-algebra on 2 generators $(0,0,12,13,23,14,25,24+15)$ has smooth Kuranishi space. \[alle\] $\dim$ Lie-algebra $\nu$ $h^1(\Theta_X)$ smooth irreducible reduced -------- --------------------- ------- ----------------- --------- ------------- --------- 1 $\gotha_1$ 1 1 $\surd$ $\surd$ $\surd$ 2 $\gotha_2$ 1 6 $\surd$ $\surd$ $\surd$ 3 $\gotha_3$ 1 18 $\surd$ $\surd$ $\surd$ 3 $\gothb_1$ 2 6 $\surd$ $\surd$ $\surd$ 4 $\gotha_4$ 1 40 $\surd$ $\surd$ $\surd$ 4 $(0,0,0,12)$ 2 12 $-$ $-$ $\surd$ 4 $(0,0,12,13)$ 3 8 $-$ $-$ $\surd$ 5 $\gotha_5$ 1 75 $\surd$ $\surd$ $\surd$ 5 $(0,0,0,12,13)$ 2 15 $-$ $-$ $\surd$ 5 $(0,0,0,0,12+34)$ 2 20 $-$ $-$ $\surd$ 5 $(0,0,12,13,23)$ 3 10 $\surd$ $\surd$ $\surd$ 5 $(0,0,0,12,13+24)$ 3 15 $-$ $-$ $-$ 5 $(0,0,12,13,14)$ 4 10 $-$ $-$ $-$ 5 $(0,0,12,13,14+23)$ 4 10 $-$ $-$ $-$ : Kuranishi spaces up to dimension 5. Examples in low dimension – explicit descriptions ------------------------------------------------- In this section we will give explicit equations for the Kuranishi space of some examples. In order to avoid cumbersome notation we will only consider the germ of the Kuranishi space at zero which will be denoted by ${\mathrm{Kur}}(X)_0$. Since nothing interesting happens in dimension 1, 2, and 3 we start in dimension 4. ### Computations in dimension 4 {#explicit4} We will now compute the Kuranishi space explicitly for the two singular examples in dimension 4. The structure equations of the considered Lie-algebras are given with respect to the bases $X_1, \dots, X_n$ and $\omega^1, \dots, \omega^n$ as described at the beginning of this section. Thus we will always start the computation of the iterative solution of the Maurer-Cartan equation with the element $$\Phi_1(\underline t)=\sum_{i=1}^{n}\sum_{j=1}^{m} t_i^j \bar\omega^i\tensor X_j$$ where $n=\dim{\ensuremath{\gothg}}$ and ${m}=\codim\kc_1{\ensuremath{\gothg}}=h^{0,1}(X)$. In order to use harmonic forms we equip ${\ensuremath{\gothg}}$ with the unique hermitian metric such that the $X_i$ form an orthonormal basis. In every step of the recursion [[(\[Phi\])]{}]{} we will decompose $[\Phi_k, \Phi_l]=\beta+\chi$ where $\chi$ is harmonic and $\beta$ is exact. Then $\chi$ will contribute to the obstruction map and $\delta(\beta)=\inverse{(\delbar)}\beta$ will, if necessary, be used to compute the next iterative step. #### The Lie-algebra ${\ensuremath{\gothg}}=(0,0,0,12)$ {#the-lie-algebra-ensuremathgothg00012 .unnumbered} Since ${\ensuremath{\gothg}}$ is 2-step nilpotent we only have to look at obstructions in degree 2, i.e., ${\mathrm{obs}}=H[\Phi_1, \Phi_1]$. Since $[X_1,X_2]=-X_4$ is the only non-zero bracket we deduce from [[(\[deg2\])]{}]{} that $$\begin{aligned} [\Phi_1(\underline t),\Phi_1(\underline t)]&= - 2 \sum_{1\leq i<j\leq 3}\det\begin{pmatrix}t_i^1 & t_i^2\\t_j^1 & t_j^2 \end{pmatrix} \bar\omega^i\wedge\bar\omega^j\tensor X_4\\ &=-2\det\begin{pmatrix}t_1^1 & t_1^2\\t_3^1 & t_3^2 \end{pmatrix} \bar\omega^1\wedge\bar\omega^3\tensor X_4-2\det\begin{pmatrix}t_2^1 & t_2^2\\t_3^1 & t_3^2 \end{pmatrix} \bar\omega^2\wedge\bar\omega^3\tensor X_4\\ &\qquad-\delbar\left(2\det\begin{pmatrix}t_1^1 & t_1^2\\t_2^1 & t_2^2 \end{pmatrix} \bar\omega^4\tensor X_4\right).\end{aligned}$$ Hence $$\begin{aligned} {\mathrm{Kur}}(X)_0&=\{ \underline t \in \IC^{12}\mid \det\begin{pmatrix}t_1^1 & t_1^2\\t_3^1 & t_3^2 \end{pmatrix}=\det\begin{pmatrix}t_2^1 & t_2^2\\t_3^1 & t_3^2 \end{pmatrix}=0\}_0\\ &=\left(\IC^6\times Y\right)_0\end{aligned}$$ where $$Y= \{ t_3^1 = t_3^2=0\}\cup \{ \rk\begin{pmatrix}t_1^1 &t_2^1 & t_3^1 t_1^2\\t_1^2 & t_2^2& t_3^2 \end{pmatrix}\leq 1\}.$$ In particular we see that the Kuranishi space is a cylinder over the reducible space $Y$. #### The Lie-algebra ${\ensuremath{\gothg}}=(0,0,12,13)$ {#the-lie-algebra-ensuremathgothg001213 .unnumbered} We infer from [[(\[deg2\])]{}]{} that $$\begin{aligned} [\Phi_1(\underline t),\Phi_1(\underline t)]&= - 2 \det\begin{pmatrix}t_1^1 & t_1^2\\t_2^1 & t_2^2 \end{pmatrix} \bar\omega^1\wedge\bar\omega^2\tensor X_3 - 2 \det\begin{pmatrix}t_1^1 & t_1^3\\t_2^1 & t_2^3 \end{pmatrix} \bar\omega^1\wedge\bar\omega^2\tensor X_4\\ &=-\delbar\left(2\det\begin{pmatrix}t_1^1 & t_1^2\\t_2^1 & t_2^2 \end{pmatrix} \bar\omega^3\tensor X_3 + 2 \det\begin{pmatrix}t_1^1 & t_1^3\\t_2^1 & t_2^3 \end{pmatrix}\bar\omega^3\tensor X_4\right)\end{aligned}$$ and by the recursion formula we set $$\Phi_2:=2\det\begin{pmatrix}t_1^1 & t_1^2\\t_2^1 & t_2^2 \end{pmatrix} \bar\omega^3\tensor X_3 + 2 \det\begin{pmatrix}t_1^1 & t_1^3\\t_2^1 & t_2^3 \end{pmatrix}\bar\omega^3\tensor X_4.$$ We see that there are no obstructions of second order and calculate (noting that $X_4$ is in the centre and that $[X_2, X_3]=0$) $$\begin{aligned} [\Phi_1(\underline t),\Phi_2(\underline t)]&=[t^1_1\bar\omega^1\tensor X_1 + t^1_2 \bar\omega^2\tensor X_1, 2\det\begin{pmatrix}t_1^1 & t_1^2\\t_2^1 & t_2^2 \end{pmatrix} \bar\omega^3\tensor X_3]\\ &= -2 \det\begin{pmatrix}t_1^1 & t_1^2\\t_2^1 & t_2^2 \end{pmatrix} \left( t^1_1\bar\omega^1\wedge \bar\omega^3\tensor X_4 +t^1_2\bar\omega^2\wedge \bar\omega^3\tensor X_4\right)\\ &= -2\det\begin{pmatrix}t_1^1 & t_1^2\\t_2^1 & t_2^2 \end{pmatrix} \left( t^1_2\bar\omega^2\wedge \bar\omega^3\tensor X_4+t^1_1\delbar \bar\omega^4\tensor X_4 \right)\\ &=-2 t^1_2\det\begin{pmatrix}t_1^1 & t_1^2\\t_2^1 & t_2^2 \end{pmatrix} \bar\omega^2\wedge \bar\omega^3\tensor X_4 \mod B_2\end{aligned}$$ Hence we have $${\mathrm{Kur}}(X)_0=\{ \underline t \in \IC^2\tensor \IC^4 =\IC^8\mid t^1_2\det\begin{pmatrix}t_1^1 & t_1^2\\t_2^1 & t_2^2 \end{pmatrix}=0\}_0,$$ in other words, ${\mathrm{Kur}}(X)_0$ is a cylinder over the cone over the union of a plane and a quadric in $\IP^3$. ### Remarks on dimension 5 {#dim5sect} The computations in dimension 5 proceed along the same lines as in dimension 4 but are, as one might imagine, much more involved. Thus, we will only present the results. In the view of Theorem \[remaining\] the Kuranishi space is a cylinder over an analytic subset of the vector space $H^1(\bar {\ensuremath{\gothg}},\IC)\tensor ({\ensuremath{\gothg}}\slash \kz{\ensuremath{\gothg}})$ whose dimension we denote by $d$. The germ of the Kuranishi space at 0 is cut out by polynomial function and we will give the primary decomposition, computed using the program Singular [@GPS05], of the ideal $I$ of all these functions. Different ideals in the decomposition correspond to different irreducible components. If some component is set-theoretically contained in another component we call it an embedded component; this can only happen if the component is not reduced. Non-reduced components occur if there are infinitesimal deformations which can be lifted up to a certain order but not to actual deformations. In all examples the Kuranishi space has several irreducible components. We denote by $k$ be the number of components of the reduced space and by $e$ the number of embedded components. Note that in the case ${\ensuremath{\gothg}}=(0,0,12,13,14+23)$ there are two non-reduced components which are not embedded, both supported on linear subspaces. To simplify the description of the ideals we introduce the notation $$\begin{gathered} \delta_{ij}^{kl}:=\det\begin{pmatrix}t_i^k & t_i^l\\t_j^k & t_j^l \end{pmatrix},\\ \Delta_{ijk}^{lmn}:=\det\begin{pmatrix} t_i^l &t_j^l &t_k^l\\ t_i^m &t_j^m &t_k^m\\ t_i^n &t_j^n &t_k^n \end{pmatrix}.\end{gathered}$$ The results can now be found in Table 2 where we also give the codimension and the degree of the various components. \[dim5\] [ccccccX]{} $\mathbf{{\ensuremath{\gothg}}}$ & $\mathbf d$ & $\mathbf{(k,e)}$ & **Codim.** &**Degree** & **reduced?** & **Ideal (primary decomposition)**\ &\ $(0,0,0,12,13)$ &9&$(2,0)$&$(2,2)$&$(3,1)$& $(\surd,\surd)$&$(\delta_{23}^{23}, \delta_{23}^{13}, \delta_{23}^{12})\cap (t_3^1, t_2^1)$\ &\ $(0,0,0,12,13+24)$ & 12 &$(3,2)$ & $(4, 5, 4);( 5, 5)$ &$(9, 3, 3);(2, 4)$ & $(\surd, \surd, \surd)$& [ $$\begin{aligned} &(\delta^{13}_{23}+\delta^{24}_{23}, \delta_{23}^{12}, \delta_{13}^{14}+\delta_{13}^{24}, \delta_{13}^{12}, \delta_{12}^{13}+\delta_{12}^{24}, \delta^{12}_{12})\\ \cap &( t_3^2, t_3^1,t_1^2, t_2^1t_3^3+t^2_2t_3^4, 2(t^2_2)^2-t_3^3, 2t_2^1t_2^2+t_3^4)\\ \cap&( t_3^2, t_3^1, t_2^1t_3^3+t^2_2t_3^4, t^1_1 + t_1^2t_3^4, \delta_{12}^{12})\\ \cap&(t_3^2, t_1^2, (t_3^3)^2, t_3^1t_3^3, t_2^2t_3^3, (t_3^1)^2, \delta_{23}^{13}+t_2^2t_3^4, t_2^2t_3^1, \delta_{13}^{13}, (t_2^2)^2)\\ \cap&(t_1^2, t_3^1t_3^3+t_3^2t_3^4, (t_3^2)^2, t_3^1t_3^2, t_1^1t_3^2,(t_3^1)^2, \\ &\qquad\delta^{13}_{23}+\delta^{24}_{23}, \delta_{23}^{12}, t_1^1t_3^1,(t_1^1)^2, \delta^{13}_{13}+t_1^4t_3^2+2t_1^1(t_2^2)^2)\end{aligned}$$]{} \ &\ $(0,0,12,13,14)$ & 8 & $(2,1)$ & $(2,1);(2)$ & $(3,1);(2)$ & $(\surd,\surd)$ & $$(\delta_{12}^{23}, \delta_{12}^{13}, \delta_{12}^{12})\cap (t_2^1)\cap (\delta^{12}_{12}, t_1^1t_2^1, (t_1^1)^2, (t_2^1)^2)$$ \ &\ $(0,0,12,13,14+23)$ & 8 & $(4,0)$ & $(2, 2, 2, 2)$ & $(3,2,2,3)$ & $(\surd, \surd,-, -)$ & [$$\begin{aligned} & (\delta_{12}^{23}, \delta_{12}^{13}, \delta^{12}_{12}) \cap (t_2^1, 2(t_1^1)^2-t_2^2)\\ \cap &((t_2^2)^2,t_2^1t_2^2, (t_2^1)^2, 2t_1^1\delta_{12}^{12}-t_2^1t_2^3)\\ \cap& ( (t_2^1)^3, t_2^2\delta_{12}^{12}+t_2^1\delta_{12}^{13}, t_2^1\delta_{12}^{12},\\ & \qquad t_1^1(t_2^1)^2,t_1^2\delta_{12}^{12} +t_1^1\delta_{12}^{13}, t_1^1\delta_{12}^{12}, (t_1^1)^2t_2^1, (t^1_1)^3)\end{aligned}$$]{} \ &\ $(0,0,0,0,12+34)$ & 16 & $(2,0)$ & $(5,5)$ & $(20,12)$ & $(\surd, \surd)$ & [$$\begin{aligned} (&\delta^{12}_{34}+\delta_{34}^{34}, \delta^{12}_{24}+\delta^{34}_{24}, \delta^{12}_{14}+\delta^{34}_{14}, \delta^{12}_{23}+\delta^{34}_{23}, \delta^{12}_{13}+\delta^{34}_{14},\\ & \delta^{12}_{12}+\delta^{34}_{12}, \Delta^{234}_{234}, \Delta^{234}_{134}, \Delta^{234}_{124}, \Delta^{134}_{234}, \Delta^{134}_{134}, \Delta^{134}_{124}, \Delta_{123}^{234}, \Delta^{134}_{123})\\ \cap&(\delta^{12}_{24}+\delta^{34}_{23}, \delta^{12}_{14}+\delta^{34}_{14}, \delta^{12}_{23}+\delta^{34}_{23}, \delta^{12}_{13}+\delta^{34}_{13}, \delta^{12}_{34}-\delta^{34}_{12},\\ &\delta^{24}_{12}+\delta^{24}_{34}, \delta^{23}_{12}+\delta^{23}_{34}, \delta^{14}_{12}+\delta^{14}_{34}, \delta^{13}_{12}+\delta^{13}_{34},\delta^{12}_{12}-\delta^{34}_{34},\\ & t^1_2\delta^{12}_{34}-t^1_4\delta^{34}_{23}+t^1_3\delta^{14}_{34}, t^1_1\delta^{12}_{34}-t^1_4\delta^{34}_{13}+t^1_3\delta^{34}_{14})\end{aligned}$$]{} \ A non-parallelisable example {#nonexam} ---------------------------- The Kuranishi space of a nilmanifold which is neither complex parallelisable nor carries an abelian complex structure can be much more complicated. We will illustrate this fact by describing a 2-step nilpotent Lie-algebra such that the Kuranishi space of an associated nilmanifold is singular but not cut out by quadrics, i.e., there are non-vanishing obstructions of higher order. We use here an alternative way to describe a real Lie-algebra with complex structure: consider the complex vectorspace $V:=\langle X_1, \dots, X_7\rangle_\IC$. There is a natural real vectorspace ${\gothh}\subset V\oplus \bar V$ invariant under complex conjugation such that ${\gothh}_\IC=V\oplus \bar V$. This decomposition defines a complex structure $J$ on ${\gothh}$ via ${{{{\gothh}}^{1,0}}}:=V$. Let $\omega^1, \dots, \omega^7$ be the basis of $V^*$ dual to the $X_i$. Then, by the formula for the differential [[(\[differential\])]{}]{}, a Lie bracket on ${\gothh}$ is uniquely determinded by $$\begin{gathered} d\omega^1=d\omega^2=d\omega^3=d\omega^4=d\omega^5=0,\\ d\omega^6=\omega^1\wedge\omega^2,\\ d\omega^7=\omega^3\wedge\omega^4+ \bar \omega^1\wedge\omega^5,\end{gathered}$$ and the complex conjugate equations. For example, we have $[\bar X_5, X_1]=\bar X_7$. Then $$d {{{{\gothh}^*}^{1,0}}}\subset \Lambda^{2,0}\oplus \Lambda^{1,1}$$ which means $d=\del+\delbar$ and the complex structure is integrable with respect to this Lie bracket. But since the image of $d$ is not contained in one of the components $\Lambda^{1,1}$ and $\Lambda^{2,0}$ neither the complex structure is abelian nor is $({\gothh}, J)$ a complex Lie-algebra. Our Lie-algebra with complex structure $({\gothh},J)$ is defined over $\IQ$ and by the theorem of Mal’cev [@malcev51] there exists a lattice $\Gamma$ in the corresponding real simply connected nilpotent Lie-group $H$. We obtain a nilmanifold with left-invariant complex structure $(M,J)=(\Gamma\backslash H, J)$. Now let $$\mu:= \bar \omega^3\tensor X_1 +\bar \omega^4\tensor X_2.$$ Recall that for $X\in {{{{\gothh}}^{1,0}}}$ and $\bar Y \in {{{{\gothh}}^{0,1}}}$ we have $\delbar X (\bar Y)= {{{ [\bar Y, X]}^{1,0}}}$ where ${{{x}^{1,0}}}$ is the image of $x\in {\gothh}_\IC$ under the projection to the $(1,0)$-part. In particular we see that $\delbar X_1=\delbar X_2=0$. This implies $\delbar\mu=0$ and $\mu$ defines a class in $H^1((M,J), \Theta_{(M,J)})$. Since every left-invariant function is constant and the contraction of a vector of type $(1,0)$ with a form of type $(0,2)$ is zero the Schouten-bracket is given by $$[\bar\alpha\tensor X, \bar\beta\tensor Y]:=\bar\beta\wedge (i_Y\del\bar \alpha) \tensor X+ \bar\alpha \wedge (i_X\del\bar\beta)\tensor Y+\bar\alpha\wedge \bar \beta\tensor [X,Y].$$ We compute the first two steps of the iterative solution $\Phi$ with $\Phi_1=\mu$ of the Maurer-Cartan equation. Since $\del\bar\omega^3=\del\bar\omega^4=0$ we get $$\begin{aligned} [\mu,\mu]&= 2\bar\omega^3\wedge\bar\omega^4\tensor [X_1, X_2]\\ &=-\delbar(2\bar \omega^7\tensor X_6).\end{aligned}$$ We see that the obstruction in degree 2 vanishes. Following the recursion [[(\[Phi\])]{}]{} we set $\Phi_2=2\bar\omega^7\tensor X_6$ and hence $$\begin{aligned} [\Phi_1, \Phi_2]&= [\bar \omega^3\tensor X_1, 2\bar\omega^7\tensor X_6] +[\bar \omega^4\tensor X_2, 2\bar\omega^7\tensor X_6]\\ &= 2\bar\omega^3\wedge(i_{X_1}\del\bar\omega^7)\tensor X_6 +2\bar\omega^4\wedge(i_{X_2} \omega^1\wedge\bar\omega^5)\tensor X_6\\ &= 2\bar\omega^3\wedge\bar\omega^5\tensor X_6.\end{aligned}$$ It is immediate from the equations that this $2$-form with values in the tangent bundle is not exact and hence there is a non-vanishing obstruction in degree three. [CFGU00]{} Maria Laura Barberis, Isabel G. Dotti, and Misha Verbitsky. Canonical bundles of complex nilmanifolds, with applications to hypercomplex geometry, 2007, arXiv:0712.3863v3 \[math.DG\]. Ciprian Borcea. Moduli for [K]{}odaira surfaces. , 52(3):373–380, 1984. F. Catanese. Moduli of algebraic surfaces. 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Differential gerstenhaber algebras associated to nilpotent algebras, 2007, arXiv:0708.3442v2 \[math.AG\]. tienne Ghys. Déformations des structures complexes sur les espaces homogènes de [$\mathrm{ SL}(2,\mathbb C)$]{}. , 468:113–138, 1995. G.-M. Greuel, G. Pfister, and H. Schönemann. 3.0. , Centre for Computer Algebra, University of Kaiserslautern, 2005. . Daniel Huybrechts. . Universitext. Springer-Verlag, Berlin, 2005. K. Kodaira and D. C. Spencer. On deformations of complex analytic structures. [I]{}, [II]{}. , 67:328–466, 1958. M. Kuranishi. On the locally complete families of complex analytic structures. , 75:536–577, 1962. L. Magnin. Sur les algèbres de [L]{}ie nilpotentes de dimension [$\leq 7$]{}. , 3(1):119–144, 1986. A. I. Malcev. On a class of homogeneous spaces. , 1951(39):33, 1951. C. Maclaughlin, H. Pedersen, Y. S. Poon, and S. Salamon. Deformation of 2-step nilmanifolds with abelian complex structures. , 73(1):173–193, 2006. Iku Nakamura. 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[**A proposal for nonabelian (0,2) mirrors**]{} Wei Gu$^1$, Jirui Guo$^2$, Eric Sharpe$^1$ [cc]{} ------------------------- $^{1}$ Dep’t of Physics Virginia Tech 850 West Campus Dr. Blacksburg, VA 24061 ------------------------- & -------------------------------------------- $^2$ Dep’t of Physics Center for Field Theory & Particle Physics Fudan University 220 Handan Road 200433 Shanghai, China -------------------------------------------- [weig8@vt.edu]{}, [jrguo@fudan.edu.cn]{}, [ersharpe@vt.edu]{} $\,$ In this paper we give a proposal for mirrors to (0,2) supersymmetric gauged linear sigma models (GLSMs), for those (0,2) GLSMs which are deformations of (2,2) GLSMs. Specifically, we propose a construction of (0,2) mirrors for (0,2) GLSMs with $E$ terms that are linear and diagonal, reducing to both the Hori-Vafa prescription as well as a recent (2,2) nonabelian mirrors proposal on the (2,2) locus. For the special case of abelian (0,2) GLSMs, two of the authors have previously proposed a systematic construction, which is both simplified and generalized by the proposal here. August 2019 Introduction ============ One of the outstanding problems in heterotic string compactifications is to understand nonperturbative effects due to worldsheet instantons. For type II strings and (2,2) worldsheet theories, these effects are well-understood, and are encoded in quantum cohomology rings and Gromov-Witten theory. In principle, there are analogues of both for more general heterotic theories, but there are comparatively many open questions. For example, in a heterotic $E_8 \times E_8$ compactification on a Calabi-Yau threefold with a rank three bundle, the low-energy theory contains states in the ${\bf 27}$ and ${\bf \overline{27}}$ representations of $E_6$, with cubic couplings appearing as spacetime superpotential terms. On the (2,2) locus (the standard embedding, where the gauge bundle equals the tangent bundle), for the case of the quintic threefold, those couplings have the standard form [@Strominger:1985ks; @Candelas:1990rm] $${\bf \overline{27}}^3 \: = \: 5 \: + \: \sum_{k=1}^{\infty} n_k \frac{k^3 q^k}{1 - q^k} \: = \: 5 \: + \: 2875 \, q \: + \: 4876875 \, q^2 \: + \: \cdots,$$ where the $n_k$ encode the Gromov-Witten invariants. These are computed by three-point functions in the A model topological field theory on the worldsheet. Off the (2,2) locus, for more general gauge bundles, these couplings have a closely analogous form: a classical contribution plus a sum of nonperturbative contributions, without any perturbative loop corrections [@Wen:1985jz; @Dine:1986zy; @Dine:1987bq; @Silverstein:1995re; @Berglund:1995yu; @Beasley:2003fx]. As a result, we know that more general heterotic versions of the Gromov-Witten invariants exist, but they have only ever been computed on the (2,2) locus. Their more general expressions are an unsolved open problem. For heterotic versions of quantum cohomology rings, comparatively more is known. The heterotic analogue is a quantum-corrected ring of sheaf cohomology groups [@Distler:1987ee] of the form $H^{\bullet}(X, \wedge^{\bullet} {\cal E}^*)$, which was introduced in [@Katz:2004nn; @Adams:2005tc; @Sharpe:2006qd; @Guffin:2007mp], and has since been computed for toric varieties [@McOrist:2007kp; @McOrist:2008ji; @Donagi:2011uz; @Donagi:2011va; @Closset:2015ohf], Grassmannians $G(k,n)$ [@Guo:2015caf; @Guo:2016suk], and flag manifolds [@Guo:2018iyr], all for the case that the gauge bundle is a deformation of the tangent bundle. (Cases involving more general gauge bundles are not currently understood.) See for example [@McOrist:2010ae; @Guffin:2011mx; @Melnikov:2012hk; @Melnikov:2019tpl] for reviews. Historically, Gromov-Witten invariants in (2,2) supersymmetric theories were first computed using mirror symmetry, and so one might hope that a (0,2) supersymmetric version of mirror symmetry might aid in such developments. This is one of the motivations to understand (0,2) mirrors (see e.g. [@Blumenhagen:1996vu; @Blumenhagen:1996tv] for some early work). To date, there has been significant progress on understanding (0,2) mirror symmetry, but many results are still limited (and certainly heterotic Gromov-Witten invariants are not yet known). For example, for the case of reflexively-plain polytopes, and bundles that are deformations of the tangent bundle, a generalization of the Batyrev construction of ordinary Calabi-Yau mirrors exists, see [@Melnikov:2010sa; @Bertolini:2018qlc; @Bertolini:2018usi]. In this paper, we shall propose what is ultimately a (0,2) analogue of the Hori-Vafa construction [@Hori:2000kt; @Morrison:1995yh], which is to say, a mirror construction for two-dimensional gauge theories, resulting in a Landau-Ginzburg model. For abelian theories, there has been nontrivial work in this area in the past [@Adams:2003zy; @Chen:2016tdd; @Chen:2017mxp; @Gu:2017nye]. This work has included ansatzes for various special cases of toric varieties [@Chen:2016tdd; @Chen:2017mxp], as well as a more systematic proposal for abelian theories [@Gu:2017nye]. The proposal in this paper will both extend such constructions to nonabelian[^1] (0,2) GLSMs, as well as give a simpler, more straightforward, presentation in abelian cases than that in [@Gu:2017nye]. We do not claim to have a proof of the construction; we only give the proposal and describe some consistency tests. We begin in section \[sect:proposal\] by describing our proposal. As many subtleties of nonabelian mirrors have already been extensively discussed in [@Gu:2018fpm; @Chen:2018wep; @Gu:2019zkw], here we focus solely on the novel aspects introduced by (0,2) supersymmetry. In section \[sec:justification\], we justify this proposal, by giving formal arguments for why it correctly reproduces quantum sheaf cohomology relations and correlation functions in the original theory. We do not, however, claim to have a proof. In section \[sect:specialize\], we specialize to abelian theories. In particular, the ansatz here simplifies and generalizes the ansatz two of the authors previously discussed in [@Gu:2017nye]. In the next several sections, we discuss concrete examples. We begin in section \[sect:pnpm\] by giving a detailed analysis of mirrors to ${\mathbb P}^n \times {\mathbb P}^m$. We verify correlation functions in the original theory, construct lower-energy Landau-Ginzburg theories in the style of (2,2) Toda duals to projective spaces, discussing subtleties that arise in their construction, explicitly verify correlation functions in those lower-energy theories, and also compare to previous (0,2) mirrors for these spaces in [@Chen:2016tdd]. In section \[sect:hirzebruch\] we perform analogous analyses for (0,2) mirrors to Hirzebruch surfaces, constructing lower-energy theories and comparing to results in [@Chen:2017mxp]. In section \[sect:previous\] we compare to the previous systematic proposal for (0,2) mirrors to abelian theories by two of the authors [@Gu:2017nye]. The ansatz presented here is both more general and rather simpler, and we also argue that when we restrict to (0,2) deformations of the form considered in [@Gu:2017nye], our current proposal gives the same results as [@Gu:2017nye]. In section \[sect:grassmannian\], we discuss our first nonabelian examples, GLSMs for (0,2) deformations of Grassmannians $G(k,N)$. These are two-dimensional $U(k)$ gauge theory with matter in copies of the fundamental representation. We construct lower-energy Landau-Ginzburg models, analogues of (2,2) Toda duals, that generalize the Grassmannian mirrors discussed in [@Gu:2018fpm], and explicitly verify that quantum sheaf cohomology rings [@Guo:2015caf; @Guo:2016suk] are reproduced. We also explicitly verify that correlation functions are correctly reproduced in a few tractable examples. In section \[sect:flag\] we briefly discuss (0,2) deformations of flag manifolds, generalizations of Grassmannians that are also described by two-dimensional nonabelian gauge theories. We verify that quantum sheaf cohomology rings [@Guo:2018iyr] are reproduced. Finally, in section \[sect:hypersurfaces\], we briefly discuss (0,2) mirrors to theories with hypersurfaces. The rest of the paper is concerned with mirrors to theories without a (0,2) superpotential; in this section, we discuss how the result is modified to take into account a (0,2) superpotential, and also discuss how the mirror ansatz reproduces some conjectures regarding hypersurface mirrors in [@McOrist:2008ji]. Proposal {#sect:proposal} ======== In this section, we will describe our ansatz for mirrors to (0,2) supersymmetric[^2] GLSMs which are deformations of (2,2) supersymmetric GLSMs. Our ansatz will apply to both abelian and nonabelian theories, but with a restriction on the form of the functions $E = \overline{D}_+ \Psi$, which we shall describe in a moment. For simplicity, in this section we will assume the original gauge theory has no superpotential, and will discuss mirrors to theories with (0,2) superpotentials in section \[sect:hypersurfaces\]. We do not claim a physical proof of this proposal, though in later sections we will provide numerous consistency tests. We will consider (0,2) deformations encoded in $\overline{D}_+ \Psi$ which can be arbitrary holomorphic functions of the chiral superfields – any product compatible with gauge representations is permissible. In our proposal, we make two restrictions on the forms for which we consider mirrors, one more restrictive than the other: - We assume that $\overline{D}_+ \Psi$ is linear in chiral superfields, rather than a more general holomorphic function of chiral superfields. This may sound very restrictive, but in fact, it has been argued that only linear terms contribute to A/2-twisted GLSMs[^3] – nonlinear terms are irrelevant. (This was conjectured in [@McOrist:2008ji]\[section 3.5\], [@Kreuzer:2010ph]\[section A.3\], and rigorously proven in [@Donagi:2011uz; @Donagi:2011va] for abelian GLSMs. It also is a consequence of supersymmetric localization [@Closset:2015ohf], and see in addition [@Donagi:2014koa]\[appendix A\].) - We assume that $\overline{D}_+ \Psi$ is also diagonal, meaning, for theories which are deformations of (2,2) theories, that for any Fermi superfield $\Psi$, $\overline{D}_+ \Psi$ is proportional to the chiral superfield with which it is partnered on the (2,2) locus. On the (2,2) locus, the $\overline{D}_+ \Psi$ are both linear and diagonal, and there exist nontrivial (0,2) deformations which are also linear and diagonal. The constraints above, that $\overline{D}_+ \Psi$ be both linear and diagonal, imply the form $$\overline{D}_+ \Psi_{i} = E_{i}(\sigma) \Phi_{i}.$$ Now that we have stated the restrictions, we give the proposal. Let us consider a (0,2) GLSM with connected[^4] gauge group $G$ of dimension $n$ and rank $r$, chiral fields $\Phi_i$ and Fermi fields $\Psi_i$ in a (possibly reducible) representation $R$ for $i=1, \cdots, N= {\rm dim}\, R$. If $\mathcal{W}$ is the Weyl group of $G$, then the proposed mirror theory is a $\mathcal{W}$-orbifold of a (0,2) Landau-Ginzburg model given by the following matter fields: - $r$ chiral fields $\sigma_a$ and $r$ Fermi fields $\Upsilon_a$, $a=1,\cdots,r$, - chiral fields $Y_{i}$ and Fermi fields $F_{i}$ where $i=1,\cdots,N$, - $n-r$ chiral fields $X_{\tilde{\mu}}$ and $n-r$ Fermi fields $\Lambda_{\tilde{\mu}}$, following the same pattern as the (2,2) nonabelian mirror proposal [@Gu:2018fpm]. For linear and diagonal $\overline{D}_+ \Psi$ as above, the proposed (0,2) superpotential of the mirror Landau-Ginzburg orbifold is $$\label{superpotential} \begin{split} W=&\sum_{a=1}^r \Upsilon_a \left( \sum_{i=1}^N \rho^a_i Y_{i} \: - \: \sum_{\tilde{\mu}=1}^{n-r} \alpha_{\tilde{\mu}}^a \ln X_{\tilde{\mu}} \: - \: t_a \right) \\ & + \sum_{i=1}^N F_{i}\left( E_{i}(\sigma)-\exp(-Y_{i})\right) \: + \: \sum_{\tilde{\mu}=1}^{n-r}\Lambda_{\tilde{\mu}}\left(1 \: - \: \sum_{a=1}^r \sigma_a \alpha_{\tilde{\mu}}^a X_{\tilde{\mu}}^{-1} \right), \end{split}$$ where $\rho_i^a$ is the $a$-th component of the weight $\rho_i$ of representation $R$, and $\alpha_{\tilde{\mu}}$, $\tilde{\mu}=1, \cdots, n-r$ are the roots of $G$. In later sections, we will slightly modify the index structure above, to be more convenient in each case, just as in [@Gu:2018fpm; @Chen:2018wep; @Gu:2019zkw]. For example, if the matter representation $R$ consists of multiple fundamentals, we will break $i$ into separate color and flavor indices. The Weyl orbifold group acts on the superpotential above in essentially the same form as discussed in detail in [@Gu:2018fpm; @Chen:2018wep; @Gu:2019zkw], so we will be brief. In broad brushstrokes, the orbifold group acts by a combination of exchanging fields and multiplying by signs. In the present case, such actions happen on pairs $(Y_i, F_i)$, $(X_{\tilde{\mu}},\Lambda_{\tilde{\mu}})$, $(\sigma_a, \Upsilon_a)$ simultaneously. For example, if $Y_i$ is swapped with $Y_j$, then simultaneously $F_i$ is swapped with $F_j$. If $Y_i$ is multiplied by a sign, then simultaneously $F_i$ is multiplied by a sign. It is then straightforward to show that the superpotential above is invariant under the orbifold group, following the same arguments as in [@Gu:2018fpm; @Chen:2018wep; @Gu:2019zkw]. Furthermore, because the $\Lambda_{\tilde{\mu}}$ terms have the same form as on the (2,2) locus, the part of the excluded locus corresponding to $X_{\tilde{\mu}}$ poles is the same as on the (2,2) locus, and so, for mirrors to connected gauge groups, the fixed points of the Weyl orbifold do not intersect non-excluded critical loci. In passing, another part of the excluded locus is defined by the fact that $\exp(-Y)$ is nonzero for finite $Y$, and that part of the excluded locus will change as the $\exp(-Y)$’s are now determined by the $E$’s. Most of the superpotential above is simply the (0,2) version of the (2,2) mirrors of [@Hori:2000kt; @Gu:2018fpm; @Chen:2018wep; @Gu:2019zkw], with the exception of the $F E$ terms in the second line. For a (2,2) supersymmetric mirror, each of those $E$’s would be $$E_{i}(\sigma) \: = \: \sum_{a=1}^r \rho_i^a \sigma_a.$$ Allowing for more general $E$’s encodes the (0,2) deformation. We should also observe that in the original (0,2) gauge theory, those $E$’s are not in the superpotential; the fact that they appear in the mirror (0,2) superpotential is as one expects for mirror symmetry. Just as in [@Gu:2018fpm], we omit the Kähler potential from our ansatz, partly because it is not pertinent to the tests we will perform. For abelian (0,2) GLSMs, detailed discussions of dualities and corresponding Kähler potentials can be found in [@Adams:2003zy]. The constraints implied by the Fermi fields imply the operator mirror map $$\begin{aligned} \label{mirror map} \exp(-Y_{i}) & = & E_{i}(\sigma), \\ X_{\tilde{\mu}} & = & \sum_{a=1}^r \alpha^a_{\tilde{\mu}} \sigma_a,\end{aligned}$$ as well as the constraints $$\sum_{i=1}^N \rho^a_i Y_{i} \: - \: \sum_{\tilde{\mu}=1}^{n-r} \alpha_{\tilde{\mu}}^a \ln X_{\tilde{\mu}} \: = \: t_a.$$ Exponentiating the constraints and applying the operator mirror map, we get the relations $$\left[ \prod_i E_i(\sigma)^{ \rho_i^a } \right] \left[ \prod_{\tilde{\mu}} X_{\tilde{\mu}}^{ \alpha^a_{\tilde{\mu}} } \right] \: = \: q_a.$$ Just as in the (2,2) case [@Gu:2018fpm section 3.3], and as we will see in more detail in section \[sec:justification\], the factor $$\left[ \prod_{\tilde{\mu}} X_{\tilde{\mu}}^{ \alpha^a_{\tilde{\mu}} } \right]$$ just contributes a phase, so that these relations reduce to $$\label{qsc} \prod_i E_{i}(\sigma)^{\rho^a_i} = \tilde{q}_a,$$ for suitably phase-shifted $\tilde{q}_a \propto q_a$, which are precisely the quantum sheaf cohomology relations for these theories (see [*e.g.*]{} [@Closset:2015ohf]). Justification {#sec:justification} ============= In this section, we will provide a few general tests of the (0,2) mirror proposal of the previous section. Specifically, we will see in greater detail how the quantum sheaf cohomology ring relations arise in these theories, and we will reproduce the one-loop effective (0,2) superpotential of [@McOrist:2008ji] and also argue how correlation functions in these theories reproduce those of the original gauge theories, in cases in which vacua are isolated. Our arguments in this section will be somewhat formal, but in concrete examples in later sections we will verify these properties explicitly. Integrate out $X_{\tilde{\mu}}$, $\Lambda_{\tilde{\mu}}$ -------------------------------------------------------- First, following [@Gu:2018fpm], to better understand the properties of this theory, we integrate out the fields $X_{\tilde{\mu}}$ and $\Lambda_{\tilde{\mu}}$. The Hessian of $X_{\tilde{\mu}}$ is $$H_X \: = \: \prod_{\tilde{\mu}} \left( \sum_{a=1}^r \sigma_a \alpha_\mu^a \right)^{-1},$$ which generates a factor in the path integral measure which vanishes along the excluded locus, exactly the same as in (2,2) mirrors [@Gu:2018fpm]. The equations of motion of $X_{\tilde{\mu}}$ are $$X_{\tilde{\mu}} \: = \: \sum_{a=1}^r \sigma_a \alpha^a_{\tilde{\mu}}.$$ Therefore, integrating out $X_{\tilde{\mu}}$ and $\Lambda_{\tilde{\mu}}$ amounts to eliminating the terms proportional to $\Lambda_{\tilde{\mu}}$ and $\ln X_{\tilde{\mu}}$ in and shifting the FI parameters $t_a$ to $\tilde{t}_a$, just as happens in (2,2) mirrors [@Gu:2018fpm], reproducing a phase discussed in [@Hori:2013ika section 10]. For example, for each $U(k)$ factor of the gauge group, $$\alpha_{\tilde{\mu}}^a \: = \: \alpha_{bc}^a \: = \: \delta_c^a-\delta_b^a$$ for $a,b,c=1,\cdots,k$ and $b \neq c$ and thus $$\sum_{\tilde{\mu}} \alpha_{\tilde{\mu}}^a \ln X_{\tilde{\mu}} \: = \: \sum_{b \neq c} \alpha_{bc}^a \ln\left( \sigma_c - \sigma_b \right) \: = \: \sum_{b \neq a} \ln \left( \frac{\sigma_a-\sigma_b}{\sigma_b-\sigma_a} \right) \: = \: (k-1) \pi i$$ from the equation of motion. Therefore, after integrating out $X_{\tilde{\mu}}$ and $\Lambda_{\tilde{\mu}}$, the superpotential reduces to $$\label{superpotential-} \tilde{W}=\sum_{a=1}^r \Upsilon_a \left( \sum_{i=1}^N \rho^a_i Y_{i} - \tilde{t}_a \right) \: + \: \sum_{i=1}^N F_{i}\left( E_{i}(\sigma)-\exp(-Y_{i})\right)$$ through a redefinition $\tilde{t}_a$ of $t_a$. The equations of motion of $\sigma_a$ and $Y_{i}$ derived from then gives the mirror map and the expected quantum sheaf cohomology relations . Correlation functions --------------------- In this section, we will compare correlation functions in the B/2-twisted Landau-Ginzburg model just defined  with corresponding A/2 model correlation functions, in cases with isolated Coulomb branch vacua, and along the way, recover the one-loop effective (0,2) superpotential of [@McOrist:2008ji] along the Coulomb branch. Now, for a (0,2) superpotential of the form $W = F^i J_i$ with isolated vacua, correlation functions are schematically of the form [@Melnikov:2007xi] $$\langle f \rangle \: = \: \sum_{\rm vacua} \frac{ f }{ \det \partial_i J_j },$$ closely related to formulas for correlation functions in (2,2) Landau-Ginzburg models involving determinants of matrices of second derivatives of the superpotential. Thus, we need to compute some analogues of Hessians. The Hessian of $Y_{i}$ is $$H_Y = \prod_i \exp(-Y_{i}) = \prod_i E_{i}(\sigma),$$ which is nonzero at generic points on the Coulomb branch. From , integrating out $Y_{i}$ and $F_{i}$ reduces  to $$\label{superpotential--} W_{\rm eff} \: = \: \sum_{a=1}^r \Upsilon_a J^a_{\rm eff} \: = \: \sum_{a=1}^r \Upsilon_a \left( -\sum_{i=1}^N \rho^a_i \ln E_{i}(\sigma) - \tilde{t}_a \right),$$ which is the same as the effective superpotential on the Coulomb branch of the original GLSM. Consequently, assuming isolated vacua, for any operator $\mathcal{O}(\sigma)$, the B/2 correlation functions of our proposed Landau-Ginzburg mirror are [@Melnikov:2007xi] $$\begin{aligned} \langle\mathcal{O}(\sigma)\rangle & = & \frac{1}{|\mathcal{W}|} \sum_{J^a_{\rm eff}=0} \frac{\mathcal{O}(\sigma)}{\left( \det_{a,b} \partial_b J^a_{\rm eff} \right) H_X H_Y}, \\ & = & \frac{1}{|\mathcal{W}|} \sum_{J^a_{\rm eff}=0} \frac{\mathcal{O}(\sigma) \prod_{\tilde{\mu}} \left( \sum_{a=1}^r \sigma_a \alpha_{\tilde{\mu}}^a \right)}{\left( \det_{a,b} \partial_b J^a_{\rm eff} \right) \left(\prod_i E_{i}(\sigma)\right)},\end{aligned}$$ which is the same as the A/2 correlation function computed from the original GLSM [@Closset:2015ohf equ’n (3.63)]. (The factor of $1/| {\mathcal W}|$ reflects the Weyl orbifold, which acts freely on the critical locus, as in [@Gu:2018fpm], so that twisted sectors do not enter this computation, at least for mirrors to theories with connected gauge groups.) Specialization to abelian theories {#sect:specialize} ================================== Let’s consider a GLSM with gauge group $U(1)^r$. The chiral field $\Phi_i$ and Fermi field $\Psi_i$ have charge $Q^a_i$ under the $a$-th $U(1)$, for $i=1, \cdots, N$. Assuming linear and diagonal (0,2) deformations, as discussed before, these fields satisfy $$\overline{D}_+ \Psi_i \: = \: \sum_{a=1}^r E_i^a \sigma_a \Phi_i,$$ where $E_i^a=Q_i^a$ on the (2,2) locus. In the abelian case, the fields $X_\mu$ and $\Lambda_\mu$ are absent in the mirror theory. The matter content of the mirror Landau-Ginzburg model thus consists of chiral fields $\sigma_a, Y_i$ and Fermi fields $\Upsilon_a, F_i$, $a=1,\cdots,r, i=1,\cdots,N$. The superpotential is $$W \: = \: \sum_{a=1}^r \Upsilon_a \left( \sum_{i=1}^N Q_i^a Y_i - t^a \right) \: + \: \sum_{i=1}^N F_i \left( \sum_{a=1}^r E_i^a \sigma_a - \exp(-Y_i) \right).$$ Following the argument of section \[sec:justification\], it is easy to see that the mirror map is $$\exp(-Y_i) \: = \: \sum_{a=1}^r E_i^a \sigma_a$$ and the effective superpotential is $$W_{\rm eff} \: = \: \sum_{a=1}^r \Upsilon_a J^a_{\rm eff} \: = \: \sum_{a=1}^r \Upsilon_a \left( -\sum_{i=1}^N Q_i^a \ln \left(\sum_{b=1}^r E_i^b \sigma_b\right) - t^a \right),$$ which reproduces the expected correlation functions $$\langle \mathcal{O}(\sigma) \rangle \: = \: \sum_{J^a_{\rm eff}=0} \frac{\mathcal{O}(\sigma)}{\left( \det_{a,b} \partial_b J^a_{\rm eff} \right) H_Y},$$ where $$H_Y \: = \: \prod_{i=1}^N \left(\sum_{a=1}^r E_i^a \sigma_a \right).$$ Example: ${\mathbb P}^n \times {\mathbb P}^m$ {#sect:pnpm} ============================================= Setup {#sect:pnpm:UV} ----- In this section we will compare to proposals for (0,2) mirrors to ${\mathbb P}^n \times {\mathbb P}^m$ with a deformation of the tangent bundle, as discussed in [@Chen:2016tdd]. In this case, a general deformation of the tangent bundle is described as the cokernel $$0 \: \longrightarrow \: {\cal O}^2 \: \stackrel{E}{\longrightarrow} \: {\cal O}(1,0)^{n+1} \oplus {\cal O}(0,1)^{m+1} \: \longrightarrow \: {\cal E} \: \longrightarrow \: 0,$$ where $$E \: = \: \left[ \begin{array}{cc} Ax & B x \\ C y & D y \end{array} \right],$$ where $x$, $y$ are vectors of homogeneous coordinates on ${\mathbb P}^n$, ${\mathbb P}^m$, respectively, and where $A$, $B$ are constant $(n+1) \times (n+1)$ matrices, and $C$, $D$ are constant $(m+1) \times (m+1)$ matrices. In this language, the (2,2) locus corresponds for example to the case that $A$ and $D$ are identity matrices, and $B = 0 = C$. Physically, in the corresponding (0,2) GLSM, we can write $$\overline{D}_+ \Lambda_i \: = \: \left( A_{ij} \sigma + B_{ij} \tilde{\sigma} \right) x_j, \: \: \: \overline{D}_+ \tilde{\Lambda}_j \: = \: \left( C_{jk} \sigma + D_{jk} \tilde{\sigma} \right) y_k,$$ and so we have $$E_{ij}(\sigma, \tilde{\sigma}) \: = \: (A \sigma + B \tilde{\sigma})_{ij}, \: \: \: \tilde{E}_{jk}(\sigma, \tilde{\sigma}) \: = \: (C \sigma + D \tilde{\sigma})_{jk}.$$ The (0,2) mirror ansatz of this paper is only defined for diagonal $E$’s, so we shall assume the matrices $A$, $B$, $C$, $D$ are diagonal: $$\begin{aligned} A & = & {\rm diag}\left( a_0, \cdots, a_n \right), \\ B & = & {\rm diag}\left( b_0, \cdots, b_n \right), \\ C & = & {\rm diag}\left( c_0, \cdots, c_m \right), \\ D & = & {\rm diag}\left( d_0, \cdots, d_m \right).\end{aligned}$$ We also define $$E_i(\sigma, \tilde{\sigma}) \: = \: a_i \sigma + b_i \tilde{\sigma}, \: \: \: \tilde{E}_i(\sigma, \tilde{\sigma}) \: = \: c_i \sigma + d_i \tilde{\sigma}.$$ Following the (0,2) mirror ansatz given earlier, we take the (0,2) mirror to be defined by the superpotential $$\begin{aligned} W & = & \Upsilon_1 \left( \sum_{i=1}^n Y_i \: - \: t_1 \right) \: + \: \Upsilon_2\left( \sum_{j=0}^m \tilde{Y}_j \: - \: t_2 \right) \nonumber \\ & & \: + \: \sum_{i=0}^n F_i \left( E_i(\sigma, \tilde{\sigma}) \: - \: \exp\left( - Y_i \right) \right) \: + \: \sum_{j=0}^m \tilde{F}_j \left( \tilde{E}_j(\sigma, \tilde{\sigma}) \: - \: \exp\left( - \tilde{Y}_j \right) \right). \label{eq:pnpm:mirror1}\end{aligned}$$ As a first consistency test, let us verify that this produces the quantum sheaf cohomology ring of ${\mathbb P}^n \times {\mathbb P}^m$. First, we integrate out the $\Upsilon_i$, which gives the usual constraints $$\label{eq:pnpm:constr1} \prod_{i=0}^n \exp\left( - Y_i \right) \: = \: q_1, \: \: \: \prod_{j=0}^m \exp\left( - \tilde{Y}_j \right) \: = \: q_2.$$ Integrating out the $F_i$, $\tilde{F}_j$ gives the operator mirror maps $$\exp\left( - Y_i \right) \: = \: E_i(\sigma, \tilde{\sigma}), \: \: \: \exp\left( - \tilde{Y}_j \right) \: = \: \tilde{E}_j(\sigma, \tilde{\sigma}),$$ and combining these with the constraints (\[eq:pnpm:constr1\]), one immediately has $$\det(A \sigma + B \tilde{\sigma}) \: = \: \prod_i E_i(\sigma, \tilde{\sigma}) \: = \: q_1, \: \: \: \det(C \sigma + D \tilde{\sigma}) \: = \: \prod_j \tilde{E}_j(\sigma, \tilde{\sigma}) \: = \: q_2,$$ which are precisely the quantum sheaf cohomology ring relations for this model [@McOrist:2007kp; @McOrist:2008ji; @Donagi:2011uz; @Donagi:2011va; @Closset:2015ohf]. Correlation functions in the UV {#sect:p1p1:2pt:uv} ------------------------------- Before going on to integrate out some of the fields, let us take a moment to explicitly compute two-point B/2-model correlation functions in the case of the mirror to ${\mathbb P}^1 \times {\mathbb P}^1$. (As we already know the chiral ring matches that of the A/2 model, computing the two-point correlation functions suffices to determine all of the B/2-model correlation functions.) Correlation functions for the ${\mathbb P}^1 \times {\mathbb P}^1$ model were computed in [@Closset:2015ohf]\[section 4.2\]. We repeat the highlights here for completeness. The two-point correlation functions have the form $$\langle \sigma \sigma \rangle \: = \: - \frac{\Gamma_1}{\alpha}, \: \: \: \langle \sigma \tilde{\sigma} \rangle \: = \: + \frac{\Delta}{\alpha}, \: \: \: \langle \tilde{\sigma} \tilde{\sigma} \rangle \: = \: - \frac{\Gamma_2}{ \alpha},$$ where $$\begin{array}{lr} \gamma_{AB} \: = \: \det(A+B) - \det A - \det B, & \gamma_{CD} \: = \: \det(C+D) - \det C - \det D, \\ \Gamma_1 \: = \: \gamma_{AB} \det D - \gamma_{CD} \det B, & \Gamma_2 \: = \: \gamma_{CD} \det A - \gamma_{AB} \det C, \\ \Delta \: = \: (\det A)(\det D) - (\det B)(\det C), & \alpha \: = \: \Delta^2 - \Gamma_1 \Gamma_2. \end{array}$$ We can compute correlation functions in the present mirror B/2-twisted Landau-Ginzburg model with superpotential (\[eq:pnpm:mirror1\] using the methods of [@Melnikov:2007xi]. Specializing to $n=m=1$, we have six functions $J_i$, corresponding to the coefficients of $\Upsilon_{1,2}$, $F_{1,2}$, $\tilde{F}_{1,2}$, and six fields $\sigma$, $\tilde{\sigma}$, $Y_{0,1}$, $\tilde{Y}_{0,1}$. The resulting matrix of derivatives $(\partial_i J_j)$ has the form $$(\partial_i J_j) \: = \: \left[ \begin{array}{cccccc} 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 \\ a_0 & b_0 & \exp\left( - Y_0 \right) & 0 & 0 & 0 \\ a_1 & b_1 & 0 & \exp\left( - Y_1 \right) & 0 & 0 \\ c_0 & d_0 & 0 & 0 & \exp\left( - \tilde{Y}_0 \right) & 0 \\ c_1 & d_1 & 0 & 0 & 0 & \exp\left( - \tilde{Y}_1 \right) \end{array} \right],$$ and then correlation functions have the form $$\langle f(\sigma,\tilde{\sigma}) \rangle \: = \: \sum_{ J=0 } \frac{ f(\sigma, \tilde{\sigma}) }{ \det (\partial_i J_j ) },$$ where the sum is over the solutions of $\{ J_i = 0 \}$. It is straightforward to compute that the resulting correlation functions precisely match those listed above from the A/2 model [@Closset:2015ohf]\[section 4.2\]. More nearly standard expressions {#sect:pnpm:IR} -------------------------------- More nearly standard expressions for Landau-Ginzburg mirrors do not involve $\sigma$ fields, so in this section, we shall integrate out these fields to derive expressions for mirrors of a more nearly standard form. We will encounter some interesting subtleties. Specifically, some other expressions for possible (0,2) mirrors to ${\mathbb P}^n \times {\mathbb P}^m$ are in [@Chen:2016tdd; @Gu:2017nye]. Those expressions have precisely $n$ $Y$’s and $m$ $\tilde{Y}$’s, so we first integrate out the $\Upsilon_i$, eliminating $Y_0$, $\tilde{Y}_0$: $$\exp\left( - Y_0 \right) \: = \: q_1 \prod_{i=1}^n \exp\left( + Y_i \right), \: \: \: \exp\left( - \tilde{Y}_0 \right) \: = \: q_2 \prod_{j=1}^m \exp\left( + \tilde{Y}_j \right).$$ Next, we can either integrate out some of the Fermi fields $F_i$, $\tilde{F}_j$, and then integrate out $\sigma$’s, or we can integrate out $\sigma$’s first, and then some of the Fermi fields. This order-of-operations ambiguity does not exist in (2,2) theories. The results are independent of choices, as one should expect, but we illustrate both methods next, to illustrate various subtleties in both the analysis and the normalization of the results. In later analyses in this paper, we will be much more brief. ### First method {#sect:pnpm:first} Having integrating out the $\Upsilon_i$, our strategy in this approach is to next integrate out some $F$, $\tilde{F}$ (as many as $\sigma$’s), and then use the resulting constraints to eliminate $\sigma$’s. The expressions in [@Chen:2016tdd; @Gu:2017nye] have as many $F$’s as $Y$’s, so we need to integrate out one $F$ and one $\tilde{F}$. This will mean solving for $\sigma$ and $\tilde{\sigma}$ in terms of other variables. There are a number of ways to proceed, and indeed, one expects that there will be many equivalent but different-looking expresions for $\sigma$, $\tilde{\sigma}$ in terms of $Y_i$ and $\tilde{Y}_j$. To pick one, we choose an index $i$ and $j$ such that the expressions we get from integrating out the corresponding $F$ and $\tilde{F}$, namely $$\exp\left( - Y_i \right) \: = \: E_i(\sigma, \tilde{\sigma}), \: \: \: \exp\left( - \tilde{Y}_j \right) \: = \: \tilde{E}_j(\sigma, \tilde{\sigma}),$$ can be inverted to solve for $\sigma$, $\tilde{\sigma}$ in terms of $Y_i$, $\tilde{Y}_j$. Put another way, using an index $I$ to denote either $i$ or $j$, and writing, schematically, $$E_I(\sigma,\tilde{\sigma}) \: = \: S_I^{\alpha} \sigma_{\alpha},$$ we pick two indices $I$ such that the resulting $2 \times 2$ matrix $S$ is invertible. (Here we are deliberately making contact with the notation used in [@Gu:2017nye].) Suppose, for example, that the two equations $$\label{eq:pnpm:f0ft0:1} \exp\left( - Y_0 \right) \: = \: E_0(\sigma, \tilde{\sigma}), \: \: \: \exp\left( - \tilde{Y}_0 \right) \: = \: \tilde{E}_0(\sigma, \tilde{\sigma}),$$ can be inverted to solve for $\sigma$, $\tilde{\sigma}$. Let us do this explicitly, and examine the result. From our earlier discussion, $$E_0(\sigma, \tilde{\sigma}) \: = \: a_0 \sigma + b_0 \tilde{\sigma}, \: \: \: \tilde{E}_0(\sigma, \tilde{\sigma}) \: = \: c_0 \sigma + d_0 \tilde{\sigma}.$$ Assuming that $$\Delta_0 \: \equiv \: \det \left[ \begin{array}{cc} a_0 & b_0 \\ c_0 & d_0 \end{array} \right] \: \neq \: 0,$$ we first integrate out $F_0$, $\tilde{F}_0$ to get the constraints (\[eq:pnpm:f0ft0:1\]), and then these equations to find $$\begin{aligned} \sigma & = & \frac{1}{\Delta_0} \left( d_0 \exp\left( - Y_0 \right) \: - \: b_0 \exp\left( - \tilde{Y}_0 \right) \right), \\ \tilde{\sigma} & = & \frac{1}{\Delta_0} \left( - c_0 \exp\left( - Y_0 \right) \: + \: a_0 \exp\left( - \tilde{Y}_0 \right) \right).\end{aligned}$$ Then, after finally integrating out $\sigma$ and $\tilde{\sigma}$, the (0,2) superpotential reduces to $$\begin{aligned} W & = & \sum_{i=1}^n F_i \left( a_i \sigma + b_i \tilde{\sigma} - \exp\left( - Y_i \right) \right) \: + \: \sum_{j=1}^m \tilde{F}_j \left( c_j \sigma + d_j \tilde{\sigma} - \exp\left( - \tilde{Y}_j \right) \right), \\ & = & \sum_{i=1}^n F_i \Biggl[ \frac{ \left( a_i d_0 - b_i c_0 \right) }{\Delta_0} q_1 \prod_{i'=1}^n \exp\left( + Y_{i'} \right) \: + \: \frac{ \left( - a_i b_0 + b_i a_0 \right)}{\Delta_0} q_2 \prod_{j'=1}^m \exp\left( + \tilde{Y}_{j'} \right) \nonumber \\ & & \hspace*{3.0in} \: - \: \exp\left( - Y_i \right) \Biggr] \nonumber \\ & & \: + \: \sum_{j=1}^m \tilde{F}_j \Biggl[ \frac{ \left( c_j d_0 - d_j c_0 \right)}{\Delta_0} q_1 \prod_{i'=1}^n \exp\left( + Y_{i'} \right) \: + \: \frac{ \left( - c_j b_0 + d_j a_0 \right)}{\Delta_0} q_2 \prod_{j'=1}^m \exp\left( + \tilde{Y}_{j'} \right) \nonumber \\ & & \hspace*{3.0in} \: - \: \exp\left( - \tilde{Y}_j \right) \Biggr]. \label{eq:pnpm:first}\end{aligned}$$ Before going on, there is a subtlety we should discuss, that will become important when comparing correlation functions between the UV and lower-energy theories. Specifically, when we integrated out $\sigma$ and $\tilde{\sigma}$, one effect is to multiply the path integral by a constant. Specifically, after integrating out $F_0$ and $\tilde{F}_0$, we had constraints which schematically appear in the B/2 model path integral in the form $$\int d \sigma d \tilde{\sigma} \, \delta\left( a_0 \sigma + b_0 \tilde{\sigma} - \exp\left( - Y_0 \right) \right) \delta\left( c_0 \sigma + d_0 \tilde{\sigma} - \exp\left( - \tilde{Y}_0 \right) \right).$$ Then, integrating over $\sigma$, $\tilde{\sigma}$ generates a factor of $$\frac{1}{a_0 d_0 - b_0 c_0} \: = \: \frac{1}{\Delta_0}$$ from the Jacobian. This will multiply correlation functions in the lower-energy theory, and we will see later in subsection \[sect:pnpm:corrfns:ir\] that this will be required in order for the lower-energy-theory’s correlation functions to match the UV correlation functions. ### Second method As a consistency test, and to illuminate the underlying methods, we will now rederive the same result via a different approach. Having integrated out the $\Upsilon_i$, our strategy in this approach is to next integrate out the $\sigma_a$. This will generate constraints on the $F$, $\tilde{F}$, which we will use to write some in terms of the others. (This is the opposite order of operations from the previous approach.) The result of this method will be an expression for the (0,2) mirror that is not of the form described in [@Chen:2016tdd; @Gu:2017nye], and also does not respect symmetries of the parametrization. We restrict to ${\mathbb P}^1 \times {\mathbb P}^1$ for simplicity. Integrating out $\sigma_a$, we have the constraints $$\begin{aligned} \sum_{i=0}^n a_i F_i \: + \: \sum_{j=0}^m c_j \tilde{F}_j & = & 0, \\ \sum_{i=0}^n b_i F_i \: + \: \sum_{j=0}^m d_j \tilde{F}_j & = & 0.\end{aligned}$$ Solving for $F_0$, $\tilde{F}_0$, we find $$\begin{aligned} F_0 & = & - \frac{1}{\Delta_0} \left[ \sum_{i=1}^n (a_i d_0 - b_i c_0) F_i \: + \: \sum_{j=1}^m (c_j d_0 - c_0 d_j) \tilde{F}_j \right], \\ \tilde{F}_0 & = & - \frac{1}{\Delta_0} \left[ \sum_{i=1}^n (a_0 b_i - b_0 a_i) F_i \: + \: \sum_{j=1}^m (d_j a_0 - b_0 c_j) \tilde{F}_j \right]\end{aligned}$$ where $$\Delta_0 \: = \: a_0 d_0 - b_0 c_0.$$ Plugging this back into the (0,2) superpotential, we have $$\begin{aligned} W & = & - \sum_{i=0}^n F_i \exp\left( - Y_i \right) \: - \: \sum_{j=0}^m \tilde{F}_j \exp\left( - \tilde{Y}_j \right), \\ & = & - \sum_{i=1}^n F_i \Biggl[ \exp\left( - Y_i \right) \: - \: \frac{(a_i d_0 - b_i c_0)}{\Delta_0} q_1 \prod_{k=1}^n \exp\left( + Y_k \right) \nonumber \\ & & \hspace*{1.5in} \: - \: \frac{ (a_0 b_i - b_0 a_i)}{\Delta_0} q_2 \prod_{k=1}^m \exp\left( + \tilde{Y}_k \right) \Biggr] \nonumber \\ & & - \sum_{j=1}^m \tilde{F}_j \Biggl[ \exp\left( - \tilde{Y}_j \right) \: - \: \frac{ (c_j d_0 - c_0 d_j) }{\Delta_0} q_1 \prod_{k=1}^n \exp\left( + Y_k \right) \nonumber \\ & & \hspace*{1.5in} \: - \: \frac{ (d_j a_0 - b_0 c_j}{\Delta_0} q_2 \prod_{k=1}^m \exp\left( + \tilde{Y}_k \right) \Biggr]. \label{eq:p1p1:mirror1}\end{aligned}$$ This precisely matches the superpotential (\[eq:pnpm:first\]) derived from integrating out fields in a different order, as expected. As in the first ordering, there is a subtlety we have glossed over, a multiplicative factor arising when integrating out some of the fields. Here, the factor arises when integrating out $F_0$, $\tilde{F}_0$, for the same reasons as before: schematically, the B/2 model path integral measure contains a factor of the form $$\int d F_0 d \tilde{F}_0 \, \delta( a_0 F_0 + b_0 \tilde{F}_0 + \cdots) \, \delta( c_0 F_0 + d_0 \tilde{F}_0 + \cdots),$$ which again generates a numerical factor[^5] of $\Delta_0^{-1}$ that multiplies correlation functions, and which will be important in subsection \[sect:pnpm:corrfns:ir\]. Correlation functions in the lower-energy theory {#sect:pnpm:corrfns:ir} ------------------------------------------------ Next, we compute correlation functions in the new theory, for the case of ${\mathbb P}^1 \times {\mathbb P}^1$, obtained after integrating out fields, and compare to the results for correlation functions computed in the UV theory, before integrating out fields. We will see an important subtlety. Using the mirror (0,2) superpotential (\[eq:p1p1:mirror1\]), and the operator mirror map $$\begin{aligned} \sigma & = & \frac{1}{\Delta_0} \left( d_0 \exp\left( - Y_0 \right) \: - \: b_0 \exp\left( - \tilde{Y}_0 \right) \right), \\ & = & \frac{1}{\Delta_0} \left( d_0 q_1 \exp\left( + Y_1 \right) \: - \: b_0 q_2 \exp\left( + \tilde{Y}_1 \right) \right), \\ \tilde{\sigma} & = & \frac{1}{\Delta_0} \left( a_0 \exp\left( - \tilde{Y}_0 \right) \: - \: c_0 \exp\left( - Y_0 \right) \right), \\ & = & \frac{1}{\Delta_0} \left( a_0 q_2 \exp\left( + \tilde{Y}_1 \right) \: - \: c_0 q_1 \exp\left( + Y_1 \right) \right),\end{aligned}$$ where $$\Delta_0 \: = \: a_0 d_0 \: - \: b_0 c_0,$$ using the methods of [@Melnikov:2007xi], we find that the two-point functions computed from the mirror above are all $\Delta_0$ times the A/2 model correlation functions in [@Closset:2015ohf]\[section 4.2\], reviewed in section \[sect:p1p1:2pt:uv\], or in other words, $$\langle \sigma \sigma \rangle_{\rm mirror} \: = \: - \Delta_0 \frac{\Gamma_1}{\alpha}, \: \: \: \langle \sigma \tilde{\sigma} \rangle_{\rm mirror} \: = \: + \Delta_0 \frac{\Delta}{\alpha}, \: \: \: \langle \tilde{\sigma} \tilde{\sigma} \rangle_{\rm mirror} \: = \: - \Delta_0 \frac{\Gamma_2}{ \alpha},$$ However, we still need to take into account the subtlety discussed in subsection \[sect:pnpm:IR\]. Specifically, when deriving the (0,2) Landau-Ginzburg model above from the UV presentation, we had to perform changes-of-variables when integrating out fields, with the effect that low-energy correlation functions should be multiplied by factors of $1/\Delta_0$. Taking that subtlety into account, and dividing out the extra $\Delta_0$ factors, we find that the correct two-point functions precisely match both those of the A/2 model [@Closset:2015ohf]\[section 4.2\], as well as those of the original (UV) theory described in subsection \[sect:pnpm:UV\]. It is also straightforward to compute four-point functions. Their values in the A/2 model are given in [@Chen:2016tdd]\[appendix A.1\]. When one computes them in the (lower-energy) Landau-Ginzburg model above, not taking into account the subtlety discussed above, one finds that the Landau-Ginzburg correlation functions are $\Delta_0$ times the A/2 model correlation functions. Taking into account the subtlety above, the overall factor of $1/\Delta_0$ multiplying all correlation functions, fixes the four-point functions also. In any event, once one knows that the two-point functions and the quantum sheaf cohomology relations match, all of the higher-point functions are guaranteed to match. Comparison to other (0,2) mirrors {#sect:pnpm:compare} --------------------------------- Now, let us compare to the (0,2) mirrors in [@Chen:2016tdd; @Gu:2017nye], for brevity just for the case of ${\mathbb P}^1 \times {\mathbb P}^1$. As a matter of principle, these mirrors need not necessarily match – there could be multiple different UV theories describing the same IR physics. Nevertheless, in special families, we will see that there is a match. For example, in [@Chen:2016tdd]\[section 4.2\], it was argued that one (0,2) Landau-Ginzburg model those B/2 correlation functions correctly match those of the corresponding A/2 theory on ${\mathbb P}^1 \times {\mathbb P}^1$ had superpotential $$W \: = \: F_1 J_1 \: + \: \tilde{F}_1 \tilde{J}_1,$$ where $$\begin{aligned} J_1 & = & a X_1 \: - \: \frac{q_1}{X_1} \: + \: b \frac{ \tilde{X}_1^2}{ X_1 } \: + \: \mu \tilde{X}_1, \\ \tilde{J}_1 & = & d \tilde{X}_1 \: - \: \frac{q_2}{\tilde{X}_1} \: + \: c \frac{ X_1^2 }{\tilde{X}_1} \: + \: \nu X_1,\end{aligned}$$ with $$\mu \: = \: \det(A+B) - \det A - \det B, \: \: \: \nu \: = \: \det(C+D) - \det C - \det D,$$ and operator mirror map $$\sigma \: = \: X_1, \: \: \: \tilde{\sigma} \: = \: \tilde{X}_1.$$ These expressions have the good property that they are in terms of determinants of the matrices $A$, $B$, $C$, $D$, and so respect global symmetries of the original theory. For that matter, the A/2 correlation functions only depend upon those determinants, which is explicit in the mirrors constructed in [@Chen:2016tdd]. For purposes of comparison, for ${\mathbb P}^1 \times {\mathbb P}^1$, the superpotential (\[eq:p1p1:mirror1\]) takes the form $$\begin{aligned} W & = & - F_1 \left[ \exp\left( - Y_1 \right) \: - \: q_1 \frac{ ( a_1 d_0 - b_1 c_0 ) }{\Delta_0} \exp\left( + Y_1\right) \: - \: q_2 \frac{ ( b_1 a_0 - a_1 b_0)}{\Delta_0} \exp\left( + \tilde{Y}_1\right) \right] \nonumber \\ & & \: - \: \tilde{F}_1 \left[ \exp\left( - \tilde{Y}_1 \right) \: - \: q_1 \frac{ ( c_1 d_0 - d_1 c_0)}{ \Delta_0} \exp\left( + Y_1 \right) \: - \: q_2 \frac{ ( d_1 a_0 - c_1 b_0 ) }{\Delta_0} \exp\left( + \tilde{Y}_1 \right) \right]. \nonumber \\ & &\end{aligned}$$ On the face of it, this clearly does not match the mirror proposal of [@Chen:2016tdd], and in fact, is not even written in terms of global-symmetry-invariant determinants of $A$, $B$, $C$, $D$. Nevertheless, as we have seen, it does reproduce the same correlation functions. One could imagine using global symmetry transformations to rotate to $a_0 = d_0=1, b_0=c_0 = 0$, the case considered in [@Gu:2017nye]\[section 5.1\], in which case the result above reduces to $$\begin{aligned} W & = & - F_1 \left[ \exp\left( - Y_1\right) \: - \: q_1 a_1 \exp\left( + Y_1 \right) \: - \: q_2 b_1 \exp\left( + \tilde{Y}_1 \right) \right] \nonumber \\ & & \: - \: \tilde{F}_1 \left[ \exp\left( - \tilde{Y}_1\right) \: - \: q_1 c_1 \exp\left( + Y_1 \right) \: - \: q_2 d_1 \exp\left( + \tilde{Y}_1 \right) \right].\end{aligned}$$ In this case, $$a \: = \: a_1, \: \: \: b \: = \: 0 \: = \: c, \: \: \: d \: = \: d_1, \: \: \: \mu \: = \: b_1, \: \: \: \nu \: = \: c_1,$$ with operator mirror map $$\sigma \: = \: q_1 \exp\left( + Y_1 \right), \: \: \: \tilde{\sigma} \: = \: q_2 \exp\left( + \tilde{Y}_1 \right).$$ If we change variables as $$\exp\left( - Y_0 \right) \: = \: q_1 \exp\left( + Y_1 \right), \: \: \: \exp\left( - \tilde{Y}_0 \right) \: = \: q_2 \exp\left( + \tilde{Y}_1 \right),$$ then we can rewrite the superpotential as $$\begin{aligned} W & = & - F_1 \left[ q_1 \exp\left( + Y_0 \right) \: - \: a_1 \exp\left( - Y_0 \right) \: - \: b_1 \exp\left( - \tilde{Y}_0 \right) \right] \nonumber \\ & & \: - \: \tilde{F}_1 \left[ q_2 \exp\left( + \tilde{Y}_0 \right) \: - \: c_1 \exp\left( - Y_0 \right) \: - \: d_1 \exp\left( - \tilde{Y}_0 \right) \right],\end{aligned}$$ which precisely matches the (0,2) mirror in [@Chen:2016tdd] for the case $a_0 = d_0 = 1$, $b_0 = c_0 = 0$. We will return to this case, which also arose in [@Gu:2017nye], in a more systematic analysis in section \[sect:previous\]. Example: Hirzebruch surfaces {#sect:hirzebruch} ============================ In this section we will compare to proposals for (0,2) mirrors to Hirzebruch surfaces with a deformation of the tangent bunde, as discussed in [@Chen:2017mxp]. Our analysis will follow the same form as that for the mirror to ${\mathbb P}^n \times {\mathbb P}^m$, so we will be comparatively brief. A Hirzebruch surface ${\mathbb F}_n$ can be described by a GLSM with gauge group $U(1)^2$ and matter fields $x_0$ $x_1$ $w$ $s$ ---------- ------- ------- ----- ----- $U(1)_1$ $1$ $1$ $n$ $0$ $U(1)_2$ $0$ $0$ $1$ $1$ A deformation ${\cal E}$ of the tangent bundle is described mathematically as the cokernel $$0 \: \longrightarrow \: {\cal O}^2 \: \stackrel{*}{\longrightarrow} \: {\cal O}(1,0)^2 \oplus {\cal O}(n,1) \oplus {\cal O}(0,1) \: \longrightarrow \: {\cal E} \: \longrightarrow \: 0,$$ where $$* \: = \: \left[ \begin{array}{cc} A x & Bx \\ \gamma_1 w & \beta_1 w \\ \gamma_2 s & \beta_2 s \end{array} \right],$$ and $x = [x_0, x_1]^T$. In principle, additional nonlinear deformations are also possible, but as they do not contribute to quantum sheaf cohomology rings (see section \[sect:proposal\]), we omit them here. The (2,2) locus corresponds to the case $A = I$, $B = 0$, $\gamma_1 = n$, $\beta_1 = 1$, $\gamma_2=0$, $\beta_2=1$. For a general (0,2) theory (with linear diagonal deformations), the $E$’s take the form $$\overline{D}_+ \Lambda_{x, i} \: = \: ((\sigma A + \tilde{\sigma} B) x)_i, \: \: \: \overline{D}_+ \Lambda_w \: = \: (\gamma_1 \sigma + \beta_1 \tilde{\sigma}) w, \: \: \: \overline{D}_+ \Lambda_s \: (\gamma_2 \sigma + \beta_2 \tilde{\sigma}) s,$$ where the $\Lambda$’s are the Fermi superfield partners to the bosonic chiral fields. Our mirror construction applies to diagonal deformations, so we only consider the case that $$\begin{array}{cc} \overline{D}_+ \Lambda_{x,0} \: = \: ( a_0 \sigma + b_0 \tilde{\sigma}) x_0, & \overline{D}_+ \Lambda_{x,1} \: = \: ( a_1 \sigma + b_1 \tilde{\sigma}) x_1, \\ \overline{D}_+ \Lambda_w \: = \: (\gamma_1 \sigma + \beta_1 \tilde{\sigma}) w, & \overline{D}_+ \Lambda_s \: = \: (\gamma_2 \sigma + \beta_2 \tilde{\sigma}) s. \end{array}$$ From our ansatz, the mirror Landau-Ginzburg model has fields - $\sigma, \tilde{\sigma}$, - $(Y_{0,1}, F_{0,1})$, corresponding to $(x_{0,1}, \Lambda_{x,0-1})$ of the A/2 model, - $(Y_w, F_w)$, corresponding to $(w, \Lambda_w)$ of the A/2 model, - $(Y_s, F_s)$, corresponding to $(s, \Lambda_s)$ of the A/2 model, and superpotential $$\begin{aligned} W & = & \Upsilon_1 \left( Y_0 + Y_1 + n Y_w - t_1 \right) \: + \: \Upsilon_2 \left( Y_w + Y_s - t_2 \right) \nonumber \\ & & \: + \: F_0 \left( a_0 \sigma + b_0 \tilde{\sigma} - \exp\left( - Y_0 \right) \right) \: + \: F_1 \left( a_1 \sigma + b_1 \tilde{\sigma} - \exp\left( - Y_1 \right) \right) \nonumber \\ & & \: + \: F_w \left( \gamma_1 \sigma + \beta_1 \tilde{\sigma} - \exp\left( - Y_w \right) \right) \: + \: F_s \left( \gamma_2\sigma + \beta_2 \tilde{\sigma} - \exp\left( - Y_s \right) \right).\end{aligned}$$ The operator mirror map is defined by the constraints imposed by the $F$’s: $$\begin{aligned} \exp\left( - Y_0 \right) & = & a_0 \sigma + b_0 \tilde{\sigma}, \\ \exp\left( - Y_1 \right) & = & a_1 \sigma + b_1 \tilde{\sigma}, \\ \exp\left( - Y_w \right) & = & \gamma_1 \sigma + \beta_1 \tilde{\sigma}, \\ \exp\left( - Y_s \right) & = & \gamma_2 \sigma + \beta_2 \tilde{\sigma},\end{aligned}$$ and using the mirror D-term relations imposed by the $\Upsilon$’s, namely $$\exp\left( - Y_0 - Y_1 - n Y_w \right) \: = \: q_1, \: \: \: \exp\left( - Y_w - Y_s \right) \: = \: q_2,$$ we quickly derive the quantum sheaf cohomology (chiral ring) relations $$\left( a_0 \sigma + b_0 \tilde{\sigma} \right) \left( a_1 \sigma + b_1 \tilde{\sigma} \right) \left( \gamma_1 \sigma + \beta_1 \tilde{\sigma} \right)^n \: = \: q_1, \: \: \: \left( \gamma_1 \sigma + \beta_1 \tilde{\sigma} \right) \left( \gamma_2 \sigma + \beta_2 \tilde{\sigma} \right) \: = \: q_2,$$ or equivalently $$\det \left( A \sigma + B \tilde{\sigma} \right) \left( \gamma_1 \sigma + \beta_1 \tilde{\sigma} \right)^n \: = \: q_1, \: \: \: \left( \gamma_1 \sigma + \beta_1 \tilde{\sigma} \right) \left( \gamma_2 \sigma + \beta_2 \tilde{\sigma} \right) \: = \: q_2, \label{eq:fn:qsc}$$ which precisely match the known quantum sheaf cohomology ring relations for this case [@McOrist:2007kp; @McOrist:2008ji; @Donagi:2011uz; @Donagi:2011va]. Next, we integrate out some of the fields to find a lower-energy effective Landau-Ginzburg description of the same physics. If we integrate out $F_0$, $F_w$, we get the constraints $$\begin{aligned} a_0 \sigma \: + \: b_0 \tilde{\sigma} & = & \exp\left( - Y_0 \right), \\ \gamma_1 \sigma \: + \: \beta_1 \tilde{\sigma} & = & \exp\left( - Y_w \right),\end{aligned}$$ which can be solved to give $$\begin{aligned} \sigma & = & \frac{1}{\Delta_0} \left( \beta_1 \exp\left( - Y_0 \right) \: - \: b_0 \exp\left( - Y_w \right) \right), \label{eq:fn:opmirror1} \\ \tilde{\sigma} & = & \frac{1}{\Delta_0} \left( a_0 \exp\left( - Y_w \right) \: - \: \gamma_1 \exp\left( - Y_0 \right) \right), \label{eq:fn:opmirror2}\end{aligned}$$ for $$\Delta_0 \: = \: a_0 \beta_1 - b_0 \gamma_1.$$ Using the $\Upsilon$ constraints to eliminate $Y_0$, $Y_w$, we have $$\begin{aligned} \exp\left( - Y_w \right) & = & q_2 \exp\left( + Y_s \right), \\ \exp\left( - Y_0 \right) & = & q_1 \exp\left( + Y_1 \right) \exp\left( + n Y_w \right) \: = \: (q_1 q_2^{-n} ) \exp\left( + Y_1 \right) \exp\left( - n Y_s \right),\end{aligned}$$ and finally plugging in we get the lower-energy effective superpotential $$\begin{aligned} W & = & F_1 \left( a_1 \sigma + b_1 \tilde{\sigma} - \exp\left( - Y_1 \right) \right) \: + \: F_s \left( \gamma_2 \sigma + \beta_2 \tilde{\sigma} - \exp\left( - Y_s \right) \right), \\ & = & F_1 \left( \frac{ (a_1 \beta_1 - b_1 \gamma_1 ) }{\Delta_0} \exp\left( - Y_0 \right) \: + \: \frac{ (-a_1 b_0 + b_1 a_0 ) }{\Delta_0} \exp\left( - Y_w \right) \: - \: \exp\left( - Y_1 \right) \right) \nonumber \\ & & \: + \: F_s \left( \frac{ ( \gamma_2 \beta_1 - \beta_2 \gamma_1)}{\Delta_0} \exp\left( - Y_0 \right) \: + \: \frac{ ( - \gamma_2 b_0 + \beta_2 a_0 ) }{\Delta_0} \exp\left( - Y_w \right) \: - \: \exp\left( - Y_s \right) \right), \nonumber \\ & = & F_1 \biggl[ \frac{ (a_1 \beta_1 - b_1 \gamma_1 ) }{\Delta_0} (q_1 q_2^{-n} ) \exp\left( + Y_1 \right) \exp\left( - n Y_s \right) \: + \: \frac{ (-a_1 b_0 + b_1 a_0 ) }{\Delta_0} q_2 \exp\left( + Y_s \right) \nonumber \\ & & \hspace*{1.5in} \: - \: \exp\left( - Y_1 \right) \biggr] \nonumber \\ & & \: + \: F_s \biggl[ \frac{ ( \gamma_2 \beta_1 - \beta_2 \gamma_1)}{\Delta_0} (q_1 q_2^{-n} ) \exp\left( + Y_1 \right) \exp\left( - n Y_s \right) \: + \: \frac{ ( - \gamma_2 b_0 + \beta_2 a_0 ) }{\Delta_0} q_2 \exp\left( + Y_s \right) \nonumber \\ & & \hspace*{1.5in} \: - \: \exp\left( - Y_s \right) \biggr]. \label{eq:fn:mirror:ir}\end{aligned}$$ To be clear, because of the change of variables we performed in constraints above, to match A/2 correlation functions, correlation functions in this model must be multiplied by a factor of $1/\Delta_0$, just as in our analysis in subsection \[sect:pnpm:IR\]. As a consistency check, let us quickly verify from the mirror (\[eq:fn:mirror:ir\]) above, plus the operator mirror map (\[eq:fn:opmirror1\]), (\[eq:fn:opmirror2\]), that the quantum sheaf cohomology relations are obeyed. Briefly, $$\begin{aligned} a_0 \sigma + b_0 \tilde{\sigma} & = & (q_1 q_2^{-n}) \exp\left( + Y_1 \right) \exp\left( - n Y_s \right) \: \: \: \mbox{ from the operator mirror map}, \\ a_1 \sigma + b_1 \tilde{\sigma} & = & \exp\left( - Y_1 \right) \: \: \: \mbox{ from the $F_1$ constraint}, \\ \gamma_1 \sigma + \beta_1 \tilde{\sigma} & = & q_2 \exp\left( + Y_s \right) \: \: \: \mbox{ from the operator mirror map}, \\ \gamma_2 \sigma + \beta_2 \tilde{\sigma} & = & \exp\left( - Y_s \right) \: \: \: \mbox{ from the $F_s$ constraint},\end{aligned}$$ hence $$\begin{aligned} \left(a_0 \sigma + b_0 \tilde{\sigma} \right) \left( a_1 \sigma + b_1 \tilde{\sigma} \right) \left( \gamma_1 \sigma + \beta_1 \tilde{\sigma} \right)^n & = & q_1, \\ \left( \gamma_1 \sigma + \beta_1 \tilde{\sigma} \right) \left( \gamma_2 \sigma + \beta_2 \tilde{\sigma} \right) & = & q_2,\end{aligned}$$ which are precisely the quantum sheaf cohomology ring relations (\[eq:fn:qsc\]) for this case. Now, consider the mirror in the special case that $a_0 = 1$, $b_0 = 0$, $\beta_1 = 1$, $\gamma_1 = n$, in other words, that they take their values on the (2,2) locus. In this case, $\Delta_0 = 1$, and the mirror above becomes $$\begin{aligned} W & = & F_1 \left[ \left(a_1 - n b_1\right) (q_1 q_2^{-n}) \exp\left( + Y_1 \right) \exp\left( - n Y_s \right) \: + \: b_1 q_2 \exp\left( + Y_s \right) \: - \: \exp\left( - Y_1 \right) \right] \nonumber \\ & & \: + \: F_s \left[ \left( \gamma_2 - n \beta_2 \right) (q_1 q_2^{-n} ) \exp\left( + Y_1 \right) \exp\left( - n Y_s \right) \: + \: \beta_2 q_2 \exp\left( + Y_s \right) \: - \: \exp\left( - Y_s \right) \right]. \nonumber \\\end{aligned}$$ Using the operator mirror map, we can write this more simply as $$\begin{aligned} W & = & F_1 \left[ a_1 \sigma \: + \: b_1 \tilde{\sigma} \: - \: \exp\left( - Y_1 \right) \right] \nonumber \\ & & \: + \: F_s \left[ \gamma_2 \sigma \: + \: \beta_2 \tilde{\sigma} \: - \: \exp\left( - Y_s \right) \right].\end{aligned}$$ Now, we can perform a change of variables to relate this to the ${\mathbb F}_n$ mirror described in [@Chen:2017mxp]\[section 4.2\], [@Gu:2017nye]\[section 5.2.1\]. To relate to their notation, if we define $X_1$, $X_3$ by $$\begin{aligned} \sigma & = & X_1 \: = \: \exp\left( - Y_0 \right), \\ \tilde{\sigma} & = & X_3 - n X_1 \: = \: \exp\left( - Y_w \right),\end{aligned}$$ then the (0,2) superpotential above becomes $$\begin{aligned} W & = & F_1 \left[ a_1 X_1 \: + \: b_1 \left( X_3 - n X_1 \right) \: - \: \frac{ q_1 }{ X_1 X_3^n } \right] \nonumber \\ & & \: + \: F_s \left[ \gamma_2 X_1 \: + \: \beta_2 \left( X_3 - n X_1 \right) \: - \: \frac{q_2}{X_3} \right].\end{aligned}$$ For the case we are considering ($a_0 = 1$, $b_0 = 0$, $\gamma_1 = n$, $\beta_1 = 1$), $$\begin{aligned} a & = & \det A \: = \: a_1, \\ b & = & \det B \: = \: 0, \\ \mu_{AB} & = & b_1,\end{aligned}$$ the coefficient of $F_1$ can be identified with the $J_1$ in [@Chen:2017mxp]\[section 4.2\], [@Gu:2017nye]\[section 5.2.1\], and their $J_2$ is $n J_1$ plus the coefficient of $F_s$. After a trivial linear rotation of $F_1$, $F_s$, we see that this change of variables identifies, in this case, the (0,2) mirror superpotential to ${\mathbb F}_n$ above, derived from our general ansatz, with that discussed in [@Chen:2017mxp; @Gu:2017nye]. This matching was not necessary – there can be different UV representations of the same IR physics – but it is certainly satisfying. We will discuss a more general form of this construction in section \[sect:previous\]. Comparison to previous abelian proposal {#sect:previous} ======================================= A proposal was made for a systematic mirror construction in abelian (0,2) GLSMs in [@Gu:2017nye]. The proposal of this paper both generalizes and simplifies the proposal given there. In this section, we will explicitly relate our ansatz to that discussed there. (Special cases have already been discussed, in sections \[sect:pnpm:compare\] and \[sect:hirzebruch\].) Briefly, the proposal in [@Gu:2017nye] considered abelian (0,2) GLSMs with $E$’s that are both linear and diagonal, as here, but with two additional restrictions: - To compute the mirror, one picked an invertible submatrix $S$ of the charge matrix, - and the (0,2) deformations vanished for $E$’s corresponding to rows of $S$. The physics of the resulting mirror was independent of choices, but nevertheless this was a more restrictive mirror than that given in this paper. We will outline a derivation of the construction in [@Gu:2017nye] from the mirror in this paper, but first, with the benefit of hindsight, let us outline in general terms how they are related. - In the proposal of this paper, to generate a lower-energy Landau-Ginzburg model, we may for example integrate out a subset of the $F$ Fermi fields, and solve for the $\sigma_a$. This procedure only works if the corresponding submatrix of the $E$’s is invertible, and so, broadly speaking, corresponds to a choice of invertible submatrix. - Assuming that the $E$ submatrix chosen above is the same as on the (2,2) locus removes the necessity of keeping track of overall numerical factors multiplying partition functions and correlation functions, the subtlety discussed in [*e.g.*]{} subsection \[sect:pnpm:first\]. Next, we shall outline a derivation of the ansatz of [@Gu:2017nye] from the proposal of this paper. First, they wrote their linear diagonal $\overline{D}_{+} \Psi_i$ in terms of deformations $B_{ij}$ off the (2,2) locus, as $$E_i \: = \: \sum_j \sum_a \left( \delta_{ij} + B_{ij} \right) Q_i^a \sigma_a.$$ For these $E_i$, our ansatz (\[superpotential\]) can be written as $$\begin{aligned} W & = & \sum_{a=1}^r \Upsilon_a \left( \sum_{i=1}^N Q_i^a \sigma_a \: - \: t_a \right) \nonumber \\ & & \hspace*{0.5in} \: + \: \sum_{i=1}^N F_i \left( \sum_a Q_i^a \sigma_a \: + \: \sum_j B_{ij} Q^a_j \sigma_a \: - \: \exp\left( - Y_i \right) \right).\end{aligned}$$ Now, in the ansatz of [@Gu:2017nye], one picks an invertible submatrix $S$ of the charge matrix, and for $i$ corresponding to a column of $S$, $B_{ij} = 0$. As a result, for those $i$, the $F_i$ terms are simply $$F_i \left( \sum_a Q^a_i \sigma_a \: - \: \exp\left( - Y_i \right) \right),$$ and so we have a constraint that relates, for those $i$, $$\sum_a Q^a_i \sigma_a \: = \: \exp\left( - Y_i \right),$$ or equivalently, in the notation of [@Gu:2017nye], $$\sum_a S^a_{i_S} \sigma_a \: = \: \exp\left( - Y_i \right).$$ Solving for $\sigma_a$, we have $$\sigma_a \: = \: \sum_{i_S} \left( S^{-1} \right)_{a i_S} \exp\left( - Y_{i_S} \right),$$ and plugging back in, our (0,2) superpotential becomes $$\begin{aligned} W & = & \sum_{a=1}^r \Upsilon_a \left( \sum_{i=1}^N Q_i^a \sigma_a \: - \: t_a \right) \nonumber \\ & & \: + \: \sum_{i=1}^N F_i \left( \sum_a Q_i^a \sigma_a \: + \: \sum_{a, j, i_S} B_{ij} Q^a_j \left( S^{-1} \right)_{a i_S} \exp\left( - Y_{i_S} \right) \: - \: \exp\left( - Y_i \right) \right),\end{aligned}$$ which is precisely the (0,2) superpotential of [@Gu:2017nye]. Example: Grassmannians {#sect:grassmannian} ====================== So far all of our examples have involved abelian GLSMs. We next turn to a nonabelian example. The Grassmannian $G(k,N)$ is described by a $U(k)$ GLSM with chirals $\Phi^a_i$ and Fermis $\Psi^a_i$ in N copies of the fundamental representation, $a \in \{1,\cdots,k\}$, $i \in \{1,\cdots,N\}$. For linear and diagonal (0,2) deformations off the (2,2) locus [@Guo:2015caf] $$\overline{D}_+ \Psi^a_i \: = \: \left(\sigma^a_b + B_i^j \left( {\rm Tr} \, \sigma \right) \right) \Phi^b_j,$$ where $B$ is diagonal, $B={\rm diag}(b_1,\cdots,b_N)$. The mirror theory consists of chiral fields $\sigma_a, Y_{ia}, X_{\mu\nu}$ and Fermi fields $\Upsilon_a, F_{ia}, \Lambda_{\mu\nu}$ with $a,\mu,\nu=1,\cdots,k, i=1,\cdots,N$ and $\mu \neq \nu$, in the notation of [@Gu:2018fpm]. For the fundamental representation of $U(k)$, the $a$-th component of the weight associated with $Y_{ib}$ is $$\rho^a_{ib} = \delta^a_b$$ and the roots are given by $$\alpha_{\mu\nu}^a = \delta_\nu^a-\delta_\mu^a,$$ therefore the superpotential reads $$\begin{split} W=&\sum_{a=1}^k \Upsilon_a \left( \sum_{i=1}^N Y_{ia} + \sum_{\mu \neq a}(\ln X_{a\mu} - \ln X_{\mu a}) - t \right) \\ &+ \: \sum_{i=1}^N \sum_{a=1}^k F_{ia} \left( \sigma_a + b_i \left(\sum_b \sigma_b\right) - \exp(-Y_{ia}) \right) \: + \: \sum_{\mu \neq \nu} \Lambda_{\mu\nu} \left( 1+\frac{\sigma_\mu - \sigma_\nu}{X_{\mu\nu}} \right), \end{split} \label{eq:gkn:genlw}$$ which gives the operator mirror map $$\exp(-Y_{ia}) \: = \: \sigma_a + b_i \left(\sum_b \sigma_b \right).$$ Next, we compute the excluded locus. From the $X_{\mu \nu}$ poles, since $X_{\mu \nu} = \sigma_{\nu} - \sigma_{\nu}$ along the critical locus, we have $$\sigma_a \: \neq \: \sigma_b$$ for $a \neq b$. That part is the same as on the (2,2) locus. From the fact that $\exp(-Y) \neq 0$, the $F_{ia}$ coefficients imply that $$\sigma_a \: + \: b_i \left( \sum_c \sigma_c \right) \: \neq \: 0,$$ for all $a$ and $i$, which is a deformation of what one gets on the (2,2) locus. Let us take a moment to examine the second excluded locus condition further. If we sum over $\sigma_a$, we get $$\left( 1 + k b_i \right) \left( \sum_c \sigma_c \right) \: \neq \: 0$$ for all $i$, hence for example $$1 + k b_i \: \neq \: 0$$ for all $i$. This condition is closely related to a constraint that arises on the $B_i^j$ in order for the gauge bundle defined by the $\overline{D}_+ \Psi$ to be a bundle, and not some more general sheaf. Specifically, it was shown in [@Guo:2016suk]\[theorem 3.3\] that the $B$’s define a bundle, and not a sheaf, if and only if there do not exist $k$ eigenvalues of $B$ that sum to $-1$. The excluded locus condition we have just derived on the Coulomb branch implies that none of the $B$ eigenvalues equals $-1/k$, which is closely related. Next, let us recover the A/2 model. Upon integrating out $X_{\mu\nu}$ and $Y_{ia}$, we get $$W_{\rm eff} \: = \: \sum_{a=1}^k \Upsilon_a \left( -\ln \prod_{i=1}^N \left(\sigma_a + b_i \left(\sum_b \sigma_b\right)\right) - t \right)$$ and $$H_X \: = \: \prod_{\mu \neq \nu} (\sigma_\mu - \sigma_\nu)^{-1},$$ $$H_Y \: = \: \prod_{i=1}^N \prod_{a=1}^k \left(\sigma_a + b_i \left(\sum_b \sigma_b \right) \right),$$ which reproduce the A/2 correlation functions of the $U(k)$ GLSM $$\langle\mathcal{O}(\sigma)\rangle \: = \: \frac{1}{k!} \sum_{J^a_{\rm eff}=0} \frac{\mathcal{O}(\sigma)}{\left( \det_{a,b} \partial_b J^a_{\rm eff} \right) H_X H_Y}.$$ Next, we shall integrate out some of the fields to construct a lower-energy Landau-Ginzburg model in the pattern of [@Gu:2018fpm]\[section 4.1\]. Beginning with the (0,2) superpotential (\[eq:gkn:genlw\]), integrating out the $\Upsilon_a$ gives the constraints $$\sum_{i=1}^N Y_{ia} \: + \: \sum_{\mu \neq a} \ln \left( \frac{ X_{a \mu} }{ X_{\mu a} } \right) \: = \: t.$$ Using these to eliminate $Y_{N a}$, we have $$Y_{Na} \: = \: - \sum_{i=1}^{N-1} Y_{ia} \: - \: \sum_{\mu \neq a} \ln \left( \frac{ X_{a \mu} }{ X_{\mu a} } \right) \: = \: t,$$ and so we define $$\begin{aligned} \Pi_a & = & \exp\left( - Y_{N a} \right), \\ & = & q \left[ \prod_{i=1}^{N-1} \exp\left( + Y_{i a} \right) \right] \left[ \prod_{\mu \neq a} \frac{ X_{a \mu} }{ X_{\mu a} } \right],\end{aligned}$$ which happens to match the $\Pi_a$ defined in the (2,2) mirror of $G(k,N)$ in [@Gu:2018fpm]\[section 4.1\]. Next, we integrate out $F_{N a}$, which gives constraints $$\sigma_a \: + \: b_N \left( \sum_c \sigma_c \right) \: = \: \exp\left( - Y_{N a} \right) \: = \: \Pi_a.$$ These equations can be solved to give $$\sigma_a \: = \: \frac{1}{1 + k b_N} \left[ \left(1 + (k-1) b_N \right) \Pi_a \: - \: b_N \sum_{c \neq a} \Pi_c \right].$$ Plugging this back in, we get our expression for a mirror Landau-Ginzburg theory: $$\begin{aligned} W & = & \sum_{i=1}^{N-1} \sum_{a=1}^k F_{i a} \left( \sigma_a + b_i \left( \sum_c \sigma_c \right) \: - \: \exp\left( - Y_{ia} \right) \right) \nonumber \\ & & \: + \: \sum_{\mu \neq \nu} \Lambda_{\mu \nu} \left( 1 \: + \: \frac{ \sigma_{\mu} - \sigma_{\nu} }{ X_{\mu \nu} } \right), \\ & = & \sum_{i=1}^{N-1} \sum_{a=1}^k F_{i a} \left[ \frac{1}{1 + k b_N} \left( \left(1 + (k-1) b_N + b_i \right) \Pi_a \: + \: (b_i - b_N) \sum_{c \neq a} \Pi_c \right) \: - \: \exp\left( - Y_{ia} \right) \right] \nonumber \\ & & \: + \: \sum_{\mu \neq \nu} \Lambda_{\mu \nu} \left( 1 \: + \: \frac{ \Pi_{\mu} - \Pi_{\nu} }{ X_{\mu \nu} } \right).\end{aligned}$$ As in earlier discussions, we have glossed over a subtlety: when integrating out the $F_{N a}$, we omitted a Jacobian factor of $$\det({\rm Jac})^{-1} \: = \: \det\left[ \begin{array}{cccc} 1+b_N & b_N & \cdots & b_N \\ b_N & 1+b_N & \cdots & b_N \\ \vdots & & & \vdots \\ b_N & b_N & \cdots & 1+b_N \end{array} \right]^{-1} \: = \: \frac{1}{1 + k b_N},$$ which should be multiplied into correlation functions in order to match against A/2 results. As a consistency check, when all the $b_i = 0$, the (0,2) superpotential above reduces to $$\begin{aligned} W & = & \sum_{i=1}^{N-1} \sum_{a=1}^k F_{i a} \left( \Pi_a \: - \: \exp\left( - Y_{ia} \right) \right) \: + \: \sum_{\mu \neq \nu} \Lambda_{\mu \nu} \left( 1 \: + \: \frac{ \Pi_{\mu} - \Pi_{\nu} }{X_{\mu \nu} } \right),\end{aligned}$$ which is precisely the (0,2) expansion of the (2,2) mirror superpotential $$W \: = \: \sum_{i=1}^{N-1} \sum_{a=1}^k \exp\left( - Y_{ia} \right) \: + \: \sum_{\mu \neq \nu} X_{\mu \nu} \: + \: \sum_{a=1}^k \Pi_a$$ computed in [@Gu:2018fpm]\[section 4.1\]. Next, we will derive the quantum sheaf cohomology relations from this lower-energy Landau-Ginzburg model. The $\Lambda_{\mu \nu}$ imply the constraints $$X_{\mu \nu} \: = \: \Pi_{\nu} - \Pi_{\mu}$$ along the critical locus, and similarly from the $F_{ia}$, $$\begin{aligned} \exp\left( - Y_{ia} \right) & = & \frac{1}{1 + k b_N} \left( \left(1 + (k-1) b_N + b_i \right) \Pi_a \: + \: (b_i - b_N) \sum_{c \neq a} \Pi_c \right), \\ & = & \sigma_a \: + \: b_i \left( \sum_c \sigma_c \right)\end{aligned}$$ along the critical locus. Plugging into the definition of $\Pi_a$, we have $$\Pi_a \: = \: q \left[ \prod_{i=1}^{N-1} \exp\left( + Y_{ia} \right) \right] (-)^{k-1},$$ hence $$\Pi_a \prod_{i=1}^{N-1} \left[ \sigma_a \: + \: b_i \left( \sum_c \sigma_c \right) \right] \: = \: (-)^{k-1} q,$$ or more simply $$\det\left( I \sigma_a + B ({\rm Tr}\, \sigma) \right) \: = \: \prod_{i=1}^{N} \left[ \sigma_a \: + \: b_i \left( \sum_c \sigma_c \right) \right] \: = \: (-)^{k-1} q,$$ This is precisely the physical description of the quantum sheaf cohomology ring relation in the A/2 model on $G(k,n)$ with the tangent bundle deformation described above [@Guo:2015caf], as expected. Thus, we see this mirror correctly duplicates the quantum sheaf cohomology ring. Now, let us perform some consistency checks by computing correlation functions in the mirror Landau-Ginzburg model above in two simple examples and comparing to known results. Our first example is the special case of $G(1,3) = {\mathbb P}^2$. This has no mathematically nontrivial tangent bundle deformations, but nontrivial parameters can still enter the GLSM and appear in correlation functions, and so it will give a nontrivial test. In this case, the (0,2) superpotential above reduces to $$\begin{aligned} W & = & \sum_{i=1}^2 F_{i} \left[ \frac{1}{1+b_3} \left( 1 + b_i \right) \Pi \: - \: \exp\left( - Y_{i} \right) \right],\end{aligned}$$ with $$\Pi \: = \: q \prod_{i=1}^2 \exp\left( + Y_i \right), \: \: \: \sigma \: = \: \frac{1}{1 + b_3} \Pi.$$ The matrix of derivatives of the superpotential terms is $$(\partial_i J_j) \: = \: \frac{1}{1+b_3} \left[ \begin{array}{cc} (1+b_1) \Pi \: + \: (1 + b_3) \exp\left( - Y_1 \right) & (1+b_1) \Pi \\ (1+b_2) \Pi & (1+b_2) \Pi \: + \: (1 + b_3)\exp\left( - Y_2 \right) \end{array} \right],$$ and using the methods of [@Melnikov:2007xi], we find $$\langle \sigma^2 \rangle \: = \: \frac{1}{(1+b_1) (1+b_2) }, \: \: \: \langle \sigma^5 \rangle \: = \: \frac{q}{(1+b_1)^2 (1+b_2)^2 (1+b_3) }.$$ These are exactly $(1+b_3)$ times the A/2 correlation functions for this model given in [@Guo:2015caf]\[section 4.1\], which are $$\langle \sigma^2 \rangle \: = \: \frac{1}{(1+b_1) (1+b_2)(1+b_3) }, \: \: \: \langle \sigma^5 \rangle \: = \: \frac{q}{(1+b_1)^2 (1+b_2)^2 (1+b_3)^2 }.$$ As predicted, we multiply the (lower-energy) Landau-Ginzburg model correlation functions by $1/(1+b_3)$ to get the A/2 model correlation functions. Next, consider the case of $G(2,3) = {\mathbb P}^2$. This model, mirror to a $U(2)$ gauge theory, again has no mathematically nontrivial tangent bundle deformations, but will also serve as a test of correlation functions, as nontrivial parameters do enter the GLSM and appear in correlation functions. Briefly, one now constructs a matrix of derivatives of the functions multiplying $F_{11}$, $F_{12}$, $F_{21}$, $F_{22}$, $\Lambda_{12}$, $\Lambda_{21}$, with respect to $Y_{11}$, $Y_{12}$, $Y_{21}$, $Y_{22}$, $X_{12}$, $X_{21}$, and using the methods of [@Melnikov:2007xi], we find $$\begin{aligned} \langle \sigma_1^2 \rangle & = & \frac{1 + 2 b_3}{\Delta} \left( -1 - 2 I_2 - 2 I_1 \right), \\ \langle \sigma_1 \sigma_2 \rangle & = & \frac{1 + 2 b_3}{\Delta} \left( 2 + 2 I_2 + 2 I_1 \right), \\ \langle \sigma_2^2 \rangle & = & \frac{1 + 2 b_3}{\Delta} \left( -1 - 2 I_2 - 2 I_1 \right),\end{aligned}$$ where, following the notation of [@Guo:2015caf], $$\begin{aligned} I_1 & = & \sum_i b_i, \\ I_2 & = & \sum_{i < j} b_i b_j, \\ I_3 & = & b_1 b_2 b_3, \\ \Delta & = & 2 \prod_{i < j} \left( 1 + b_i + b_j \right).\end{aligned}$$ The correlation functions above are precisely $(1+2 b_3)$ times the A/2 model correlation functions computed in [@Guo:2015caf], precisely as expected from the normalization subtlety discussed in section \[sect:pnpm:IR\]. Example: Flag manifolds {#sect:flag} ======================= In this section, we will briefly outline mirrors to flag manifolds. The GLSM describing the flag manifold $F(k_1,k_2,\cdots,k_n,N)$ is a quiver gauge theory with gauge group $U(k_1) \times \cdots \times U(k_n)$ [@Donagi:2007hi]. For each $s=1,\cdots,n-1$, there is a chiral multiplet $\Phi_{s,s+1}$ and a Fermi multiplet $\Psi_{s,s+1}$ transforming in the fundamental representation of $U(k_s)$ and in the antifundamental representation of $U(k_{s+1})$. There are also chiral multiplets $\Phi_{n,n+1}^i$ and Fermi multiplets $\Psi_{n,n+1}^i$ transforming in the fundamental representation of $U(k_n)$ for $i=1,\cdots,N$. The $E$-terms of this theory are given by [@Guo:2018iyr] $$\begin{split} &\overline{D}_+ \Psi_{s,s+1} \: = \: \Phi_{s,s+1} \Sigma^{(s)} \: - \: \Sigma^{(s+1)} \Phi_{s,s+1} \: + \: \sum_{t=1}^n u^s_t \left({\rm Tr}\, \Sigma^{(t)} \right) \Phi_{s,s+1}, \\ &\quad s=1,\cdots,n-1, \\ &\overline{D}_+ \Psi_{n,n+1}^i \: = \: \Phi_{n,n+1} \Sigma^{(n)} \: + \: \sum_{t=1}^n \left({\rm Tr}\, \Sigma^{(t)}\right) {A_t}_j^i \Phi_{n,n+1}^j, ~i,j=1,\cdots,N. \end{split}$$ The matrices $A_t$ are assumed to be diagonal in this paper, i.e. $${A_t}_j^i = A_{ti} \delta^i_j.$$ The mirror theory is a Landau-Ginzburg model consisting of chiral fields $$\sigma_{a_s}^{(s)},~ {Y^{(s)}}^{a_s}_{b_s},~ X^{(s)}_{\mu_s\nu_s}$$ and Fermi fields $$\Upsilon_{a_s}^{(s)},~ {F^{(s)}}^{a_s}_{b_s},~ \Lambda^{(s)}_{\mu_s\nu_s}$$ for $s=1,\cdots,n$, $a_s=1,\cdots,k_s$, $b_s=1,\cdots,k_{s+1}$, $\mu_s,\nu_s=1,\cdots,k_s$ and $\mu_s \neq \nu_s$ where $k_{n+1}=N$. The superpotential is $$\begin{split} W=&\sum_{s=1}^n\sum_{a_s=1}^{k_s} \Upsilon^{(s)}_{a_s} \left( \sum_{b_s=1}^{k_{s+1}} {Y^{(s)}}^{a_s}_{b_s} - \sum_{\alpha_s=1}^{k_{s-1}}{Y^{(s-1)}}^{\alpha_s}_{a_s} + \sum_{\mu_s \neq a_s}(\ln X^{(s)}_{a_s\mu_s} - \ln X^{(s)}_{\mu_s a_s}) - t_s \right) \\ &+ \: \sum_{s=1}^n \sum_{a_s=1}^{k_s} \sum_{b_s=1}^{k_{s+1}} {F^{(s)}}^{a_s}_{b_s} \left( {E^{(s)}}^{a_s}_{b_s}(\sigma) - \exp\left(-{Y^{(s)}}^{a_s}_{b_s}\right) \right) \\ & + \: \sum_{s=1}^n \sum_{\mu_s \neq \nu_s} \Lambda^{(s)}_{\mu_s\nu_s} \left( 1+\frac{\sigma^{(s)}_{\mu_s} - \sigma^{(s)}_{\nu_s}}{X^{(s)}_{\mu_s\nu_s}} \right), \end{split}$$ where $k_0=0$, $${E^{(s)}}^{a_s}_{b_s}(\sigma) \: = \: \sigma^{(s)}_{a_s} \: - \: \sigma^{(s+1)}_{b_s} \: + \: \sum_{t=1}^n u^s_t {\rm Tr}\, \sigma^{(t)}$$ for $s=1,\cdots,n$, $a_s=1,\cdots,k_s$, $b_s=1,\cdots,k_{s+1}$ and $${E^{(n)}}^{a_n}_{b_n}(\sigma) \: = \: \sigma^{(n)}_{a_n} \: + \: \sum_{t=1}^n A_{tb_n} {\rm Tr} \, \sigma^{(t)}$$ for $a_n=1,\cdots,k_n$ and $b_n=1,\cdots,N$. Again, integrating out $X^{(s)}_{\mu_s\nu_s}$ and $\Lambda^{(s)}_{\mu_s\nu_s}$ shifts the FI parameters $$t_s \rightarrow t_s + (k_s-1) \pi i.$$ Hypersurfaces {#sect:hypersurfaces} ============= So far, our examples have involved mirrors to GLSMs without a superpotential. One can add a superpotential to the original theory, following the same prescription as [@Gu:2018fpm]; namely, one assigns R-charges to the fields, and then takes the mirrors to fields with nonzero R charges, following the same pattern as in [@Gu:2018fpm]. For example, if a chiral field $\phi$ of the original theory has R-charge $r$, then the fundamental field in the mirror is $$X \: \equiv \: \exp\left( - (r/2) Y \right),$$ and the theory has a ${\mathbb Z}_{2/r}$ orbifold. As a result, the mirror (0,2) theory does not depend upon the details of the original superpotential, only upon R-charges. For (2,2) theories, such statements are standard, but in (0,2) theories, they have come to be believed only somewhat more recently [@McOrist:2008ji], and only as statements about GLSM descriptions. In any event, the point is that our mirror construction implicitly reproduces the conjecture of [@McOrist:2008ji] that A/2-twisted GLSMs are independent of precise superpotential terms, and depend only upon R-charges. Conclusions =========== In this paper we have described an extension of the nonabelian mirror proposal of [@Gu:2018fpm] from two-dimensional (2,2) supersymmetric theories to (0,2) supersymmetric theories. The result is a simple systematic ansatz which both generalizes and simplifies previous approaches to Hori-Vafa-style (0,2) abelian mirrors [@Adams:2003zy; @Chen:2016tdd; @Chen:2017mxp; @Gu:2017nye], and also applies to nonabelian cases [@Gu:2018fpm; @Chen:2018wep; @Gu:2019zkw]. 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Other proposals have appeared in the math community in e.g. [@rietsch1; @teleman], as reviewed in [@Gu:2018fpm]\[section 4.9, appendix A\]. [^2]: For introductions to (0,2) GLSMs and (0,2) Landau-Ginzburg models, we recommend [@Distler:1993mk; @Distler:1995mi]. [^3]: For A/2-twisted nonlinear sigma models, this story is not settled, not least because we know of no simple way to distinguish the UV linear from UV nonlinear deformations in the IR. [^4]: It is very straightforward to extend this proposal to $O(k)$ gauge theories in the same fashion as the (2,2) case, discussed in [@Gu:2019zkw], but we shall not discuss any examples of $O(k)$ (0,2) mirrors in this paper. [^5]: Tracing through this a bit more carefully, the numerical factor arises from the delta functions, which arose from bosonic fields ($\sigma$’s), hence the numerical factor is $\delta_0^{-1}$ instead of $( \Delta_0^{-1} )^{-1} = \Delta_0$ as one might have expected from a fermionic integral.

LU TP 99–31\ hep-ph/9910288\ October 1999 [**QCD Interconnection Effects[^1]**]{}\ [Torbjörn Sjöstrand[^2]]{}\ [*Department of Theoretical Physics,*]{}\ [*Lund University, Lund, Sweden*]{} [**Abstract**]{}\ Heavy objects like the $W$, $Z$ and $t$ are short-lived compared with typical hadronization times. When pairs of such particles are produced, the subsequent hadronic decay systems may therefore become interconnected. We study such potential effects at Linear Collider energies. This talk mainly reports on work done in collaboration with Valery Khoze [@work]. The widths of the $W$, $Z$ and $t$ are all of the order of 2 GeV. A Standard Model Higgs with a mass above 200 GeV, as well as many supersymmetric and other Beyond the Standard Model particles would also have widths in the range. Not far from threshold, the typical decay times $\tau = 1/\Gamma \approx 0.1 \, {\mathrm{fm}} \ll \tau_{\mathrm{had}} \approx 1 \, \mathrm{fm}$. Thus hadronic decay systems overlap, between pairs of resonances ($W^+W^-$, $Z^0Z^0$, $t\bar{t}$, $Z^0H^0$, …), so that the final state may not be just the sum of two independent decays. Pragmatically, one may distinguish three main eras for such interconnection: Perturbative: this is suppressed for gluon energies $\omega > \Gamma$ by propagator/timescale effects; thus only soft gluons may contribute appreciably. Nonperturbative in the hadroformation process: normally modelled by a colour rearrangement between the partons produced in the two resonance decays and in the subsequent parton showers. Nonperturbative in the purely hadronic phase: best exemplified by Bose–Einstein effects. The above topics are deeply related to the unsolved problems of strong interactions: confinement dynamics, $1/N^2_{\mathrm{C}}$ effects, quantum mechanical interferences, etc. Thus they offer an opportunity to study the dynamics of unstable particles, and new ways to probe confinement dynamics in space and time [@GPZ; @ourrec], [*but*]{} they also risk to limit or even spoil precision measurements [@ourrec]. So far, studies have mainly been performed in the context of $W$ mass measurements at LEP2. Perturbative effects are not likely to give any significant contribution to the systematic error, $\langle \delta m_W \rangle {\raisebox{-0.8mm}{\hspace{1mm}$\stackrel{<}{\sim}$\hspace{1mm}}}5$ MeV [@ourrec]. Colour rearrangement is not understood from first principles, but many models have been proposed to model effects [@ourrec; @otherrec; @HR], and a conservative estimate gives $\langle \delta m_W \rangle {\raisebox{-0.8mm}{\hspace{1mm}$\stackrel{<}{\sim}$\hspace{1mm}}}40$ MeV. For Bose–Einstein again there is a wide spread in models, and an even wider one in results, with about the same potential systematic error as above [@ourBE; @otherBE; @HR]. The total QCD interconnection error is thus below $m_{\pi}$ in absolute terms and 0.1% in relative ones, a small number that becomes of interest only because we aim for high accuracy. ------------------------------------------------------------------------ More could be said if some experimental evidence existed, but a problem is that also other manifestations of the interconnection phenomena are likely to be small in magnitude. For instance, near threshold it is expected that colour rearrangement will deplete the rate of low-momentum particle production [@lowmom], Fig. \[figlowmom\]. Even with full LEP2 statistics, we are only speaking of a few sigma effects, however. Bose-Einstein appear more promising to diagnose, but so far experimental results are contradictory [@BEstatus]. One area where a linear collider could contribute would be by allowing a much increased statistics in the LEP2 energy region. A 100 fb$^{-1}$ $W^+W^-$ threshold scan would give a $\sim 6$ MeV accuracy on the $W$ mass [@Wilson], with negligible interconnection uncertainty. This would shift the emphasis from $m_W$ to the understanding of the physics of hadronic cross-talk. A high-statistics run, e.g. 50 fb$^{-1}$ at 175 GeV, would give a comfortable signal for the low-momentum depletion mentioned above, and also allow a set of other tests [@othertest; @lowmom]. Above the $Z^0Z^0$ threshold, the single-$Z^0$ data will provide a unique $Z^0Z^0$ no-reconnection reference. Thus, high-luminosity, LEP2-energy LC (Linear Collider) runs would be excellent to [*establish*]{} a signal. To explore the [*character*]{} of effects, however, a knowledge of the energy dependence could give further leverage. ------------------------------------------------------------------------ In QED, the interconnection rate dampens with increasing energy roughly like $(1 - \beta)^2$, with $\beta$ the velocity of each $W$ in the CM frame [@QED]. By contrast, the nonperturbative QCD models we studied show an interconnection rate dropping more like $(1 - \beta)$ over the LC energy region (with the possibility of a steeper behaviour in the truly asymptotic region), Fig. \[figprob\]. If only the central region of $W$ masses is studied, also the mass shift dampens significantly with energy, Fig. \[figprob\]. However, if also the wings of the mass distribution are included (a difficult experimental proposition, but possible in our toy studies), the average and width of the mass shift distribution do not die out. Thus, with increasing energy, the hadronic cross-talk occurs in fewer events, but the effect in these few is more dramatic. ------------------------------------------------------------------------ The depletion of particle production at low momenta, close to threshold, turns into an enhancement at higher energies [@lowmom]. However, in the inclusive $W^+W^-$ event sample, this and other signals appear too small for reliable detection. One may instead turn to exclusive signals, such as events with many particles at low momenta, or at central rapidities, or at large angles with respect to the event axis, Fig. \[figexclusive\]. Unfortunately, even after such a cut, fluctuations in no-reconnection events as well as ordinary QCD four-jet events (mainly $q\bar{q}gg$ split in $qg + \bar{q}g$ hemispheres, thus with a colour flow between the two) give event rates that overwhelm the expected signal. It could still be possible to observe an excess, but not to identify reconnections on an event-by-event basis. The possibility of some clever combination of several signals still remains open, however. ------------------------------------------------------------------------ Since the $Z^0$ mass and properties are well-known, $Z^0Z^0$ events provide an excellent hunting ground for interconnection. Relative to $W^+W^-$ events, the set of production Feynman graphs and the relative mixture of vector and axial couplings is different, however, and this leads to non-negligible differences in angular distributions, Fig. \[figzzww\]. Furthermore, the higher $Z^0$ mass means that a $Z^0$ is slower than a $W^{\pm}$ at fixed energy, and the larger $Z^0$ width also brings the decay vertices closer. Taken together, at 500 GeV, the reconnection rate in $Z^0Z^0$ hadronic events is likely to be about twice as large as in $W^+W^-$ events, while the cross section is lower by a factor of six. Thus $Z^0Z^0$ events are interesting in their own right, but comparisons with $W^+W^-$ events will be nontrivial. (2880,1728)(0,0) (1836,1049)[(0,0)\[r\][[$\Delta R$]{}]{}]{} (1836,1149)[(0,0)\[r\][[$\langle\delta m_{W}^{4j}\rangle$ $\mathrm{BE}_m'$]{}]{}]{} (1836,1249)[(0,0)\[r\][[$\langle\delta m_{W}^{4j}\rangle$ $\mathrm{BE}_m$]{}]{}]{} (2896,907)[(0,0)\[l\][(fm)]{}]{} (2896,1021)[(0,0)\[l\][$\Delta R$]{}]{} (2741,251)[(0,0)\[l\][0.0]{}]{} (2741,536)[(0,0)\[l\][0.2]{}]{} (2741,821)[(0,0)\[l\][0.4]{}]{} (2741,1107)[(0,0)\[l\][0.6]{}]{} (2741,1392)[(0,0)\[l\][0.8]{}]{} (2741,1677)[(0,0)\[l\][1.0]{}]{} (1648,31)[(0,0)[$E_{\mbox{cm}}$ (GeV)]{}]{} (220,964) (0,0)\[b\] (2587,151)[(0,0)[1000]{}]{} (2145,151)[(0,0)[800]{}]{} (1704,151)[(0,0)[600]{}]{} (1262,151)[(0,0)[400]{}]{} (821,151)[(0,0)[200]{}]{} (540,1677)[(0,0)\[r\][1000]{}]{} (540,1392)[(0,0)\[r\][800]{}]{} (540,1107)[(0,0)\[r\][600]{}]{} (540,821)[(0,0)\[r\][400]{}]{} (540,536)[(0,0)\[r\][200]{}]{} (540,251)[(0,0)\[r\][0]{}]{} ------------------------------------------------------------------------ As noted above, the Bose–Einstein interplay between the hadronic decay systems of a pair of heavy objects is at least as poorly understood as is colour reconnection, and less well studied for higher energies. In some models [@ourBE], the theoretical mass shift increases with energy, when the separation of the $W$ decay vertices is not included, Fig. \[figboei\]. With this separation taken into account, the theoretical shift levels out at around 200 MeV. How this maps onto experimental observables remains to be studied, but experience from LEP2 energies indicates that the mass shift is significantly reduced, and may even switch sign. ------------------------------------------------------------------------ The $t\bar{t}$ system is different from the $W^+W^-$ and $Z^0Z^0$ ones in that the $t$ and $\bar{t}$ always are colour connected. Thus, even when both tops decay semileptonically, $t \to b W^+ \to b \ell^+ \nu_{\ell}$, the system contains nontrivial interconnection effects. For instance, the total hadronic multiplicity, and especially the multiplicity at low momenta, depends on the opening angle between the $b$ and $\bar{b}$ jets: the smaller the angle, the lower the multiplicity [@topmult], Fig. \[figtop\]. On the perturbative level, this can be understood as arising from a dominance of emission from the $b\bar{b}$ colour dipole at small gluon energies [@dipole], on the nonperturbative one, as a consequence of the string effect [@string]. Uncertainties in the modelling of these phenomena imply a systematic error on the top mass of the order of 30 MeV already in the semileptonic top decays. When hadronic $W$ decays are included, the possibilities of interconnection multiply. This kind of configurations have not yet been studied, but realistically we may expect uncertainties in the range around 100 MeV. In summary, LEP2 may clarify the Bose–Einstein situation and provide some hadronic cross-talk hints. A high-luminosity LEP2-energy LC run would be the best way to establish colour rearrangement, however. Both colour rearrangement and BE effects (may) remain significant over the full LC energy range: while the fraction of the (appreciably) affected events goes down with energy, the effect per such event comes up. If the objective is to do electroweak precision tests, it appears feasible to reduce the $WW/ZZ$ “interconnection noise” to harmless levels at high energies, by simple proper cuts. It should also be possible, but not easy, to dig out a colour rearrangement signal at high energies, with some suitably optimized cuts that yet remain to be defined. The $Z^0Z^0$ events should display about twice as large interconnection effects as $W^+W^-$ ones, but cross sections are reduced even more. The availability of a single-$Z^0$ calibration still makes $Z^0Z^0$ events of unique interest. While detailed studies remain to be carried out, it appears that the direct reconstruction of the top mass could be uncertain by maybe 100 MeV. Finally, in all of the studies so far, it has turned out to be very difficult to find a clean handle that would help to distinguish between the different models proposed, both in the reconnection and Bose–Einstein areas. Much work thus remains for the future. [99]{} V.A. Khoze and T. Sjöstrand, LU TP 99-23, hep-ph/9908408, to appear in the Proceedings of the International Workshop on Linear Colliders, Sitges (Barcelona), Spain, April 28 - May 5, 1999 G. Gustafson, U. Pettersson and P. Zerwas, Phys. Lett. [**B209**]{} (1988) 90. T. Sjöstrand and V.A. Khoze, Z. Physik [**C62**]{} (1994) 281, Phys. Rev. Lett. [**72**]{} (1994) 28.. G.Gustafson and J.Häkkinen, Z. Physik [**C64**]{} (1994) 659;\ L. Lönnblad, Z. Physik [**C70**]{} (1996) 107;\ Š. Todorova–Nová, DELPHI Internal Note 96-158 PHYS 651;\ J. Ellis and K. Geiger, Phys. Rev. [**D54**]{} (1996) 1967, Phys. Lett. [**B404**]{} (1997) 230;\ B.R. Webber, J. Phys. [**G24**]{} (1998) 287. J. Häkkinen and M. Ringnér, Eur. Phys. J. [**C5**]{} (1998)275. L. Lönnblad and T. Sjöstrand, Phys. Lett. [**B351**]{} (1995)293, Eur. Phys. J. [**C2**]{} (1998) 165. S. Jadach and K. Zalewski, Acta Phys. Polon. [**B28**]{} (1997) 1363;\ V. Kartvelishvili, R. Kvatadze and R. M[ø]{}ller, Phys. Lett. [**B408**]{} (1997) 331;\ K. Fia[ł]{}kowski and R. Wit, Acta Phys. Polon. [**B28**]{} (1997) 2039, Eur. Phys. J. [**C2**]{} (1998) 691;\ Š. Todorova–Nová and J. Rameš, hep-ph/9710280. V.A. Khoze and T. Sjöstrand, Eur. Phys. J. [**C6**]{} (1999) 271. F. Martin, presented at XXXIV Rencontres de Moriond, France, March 20—27, 1999, preprint LAPP–EXP 99.04. G. Wilson, presented at the International Workshop on Linear Colliders, Sitges (Barcelona), Spain, April 28 – May 5, 1999 E. Norrbin and T. Sjöstrand, Phys. Rev. [**D55**]{} (1997) R5. A.P. Chapovsky and V.A. Khoze, Eur. Phys. J. [**C9**]{} (1999) 449. V.A. Khoze and T. Sjöstrand, Phys. Lett. [**B328**]{} (1994) 466. Ya.I. Azimov, Yu.L. Dokshitzer, V.A. Khoze and S.I. Troyan, Phys. Lett. [**B165**]{} (1985) 147. B. Andersson, G. Gustafson, G. Ingelman and T. Sjöstrand, Phys. Rep. [**97**]{} (1983) 31. [^1]: To appear in the Proceedings of the Workshop on the development of future linear electron-positron colliders for particle physics studies and for research using free electon lasers, Lund, Sweden, 23–26 September 1999 [^2]: torbjorn@thep.lu.se

Introduction {#intro} ============ One-dimensional or quasi-one-dimensional magnetic systems show many fascinating properties which continue to attract an intense theoretical activity. One of these properties is the presence of a spin gap in antiferromagnetic Heisenberg chains with integer spin[@haldane] and in ladders.[@ladders] Another, particularly complex, system which presents a spin gap is the spin-Peierls (SP) system. In this system a Heisenberg chain coupled to the lattice presents an instability at a critical temperature, $T_{SP}$, below which a dimerized lattice pattern appears and a spin gap opens in the excitation spectrum.[@pytte] The interest in the spin-Peierls phenomena was recently revived after the first inorganic SP compound, CuGeO$_3$, was found.[@hase] This inorganic material allows the preparation of better samples than the organic SP compounds and hence several experimental techniques can be applied to characterize the properties of this system.[@regnault] Besides, this compound can be easily doped with magnetic and non-magnetic impurities, leading to a better understanding of its ground state and excitations.[@lussier] Spin-Peierls systems present also a very rich and interesting behavior in the presence of an external magnetic field. Below the spin-Peierls transition temperature, and for magnetic fields $H$ smaller than a critical value $H_{cr}(T)$, the system is in its spin-Peierls phase, characterized by a gapped nonmagnetic ($S^z=0$) ground state with a dimerized pattern or alternating nearest-neighbor (NN) interactions. For $T < T_{tc} < T_{SP}$, at $H=H_{cr}(T)$ a transition occurs from the dimerized phase to a gapless incommensurate (IC) state characterized by a finite magnetization, $S^z > 0$. $T_{tc}$ is the temperature of the point at which the dimerized, incommensurate and uniform phases meet. The dimerized-IC transition was predicted by some theories[@cross2] to be of first order at low temperatures, and this is the behavior found in experimental studies[@hase2; @loosd]. Other theories predict that this transition is a second order one.[@fujita] A simple picture of the dimerized-IC transition can be obtained by mapping the Heisenberg spin chain to a spinless fermion system by a Jordan-Wigner transformation. The effect of the magnetic field favoring a nonzero $S^z$ due to the Zeeman energy can be interpreted as a change in the band filling of the equivalent spinless fermion system. As a result, the momentum of the lattice distortion moves away from $\pi$ as $\tilde{q} = (1-S^z/N) \pi$, where $N$ is the number of sites on the chain. However, since [*umklapp*]{} processes pin the momentum at $\pi$ up to a critical field $H_{cr}(T)$, the lattice distortion will remain a simple dimerization and the magnetic ground state will remain a singlet.[@crossfisher] Theoretical[@fujita; @nakano; @buzdin] and experimental[@kiry; @fagot] studies indicate that the lattice distortion pattern in the IC phase corresponds to an array of solitons. A complementary picture indicating how a soliton lattice could appear as a consequence of the finite magnetization in the IC phase is the following. Let’s assume that the dominant contribution to the magnetic ground state comes from a state of NN singlets or dimers. An up spin replacing a down spin destroys a singlet and gives rise to two domain-walls or solitons separating regions of dimerized order which are shifted in one lattice spacing with respect to each other. Each soliton carries a spin-1/2. Due to the spin-lattice coupling it is expected that the lattice solitons are driven by these magnetic solitons. The soliton formation in spin-Peierls systems has been studied analytically by bosonization techniques applied to the spinless fermion model.[@affleck] The coupling to the lattice is treated usually in the adiabatic approximation. The resulting field-theory formalism has lead to important results, the most remarkable being the relation between the soliton width and the spin-Peierls gap, $\xi \sim \Delta^{-1}$.[@nakano] Although this formalism has been extended to a Heisenberg model with competing NN and next-nearest-neighbor (NNN) antiferromagnetic interactions[@zang2; @dobryriera], it presents some unsatisfactory features. In the first place, there are some recent experimental results[@kiry] for the soliton width in the IC phase in CuGeO$_3$ indicating a disagreement with the theoretical prediction. Although there might be a contribution to the soliton width coming from magnetic[@zang2] or elastic[@dobryriera] interchain couplings which would explain at least partially this disagreement, it is also possible that the differences could be due to several approximations involved in the bosonized field theory. One should take into account that these theories are valid in principle in the long wave-length limit, and the applicability of their results to real materials can not be internally assessed. Then, our first motivation to start a numerical study of the IC phase in spin-Peierls systems is to measure the importance of these approximations in the analytical approach. In the second place, the field theory approach does not provide a detailed dependence of the magnitudes involved in terms of the original parameters of the microscopical models. For example, even for the simplest case[@nakano] the expression obtained for the spin-wave velocity must be replaced by the exact one known from Bethe’s exact solution of the Heisenberg chain. In this sense, numerical studies could give information about how the relevant magnitudes depend on the original parameters without further approximations. With these motivations, in this article we want to initiate the study of the incommensurate phase in SP systems using numerical methods. These methods give essentially exact results for finite clusters, and they can be used to check various approximations required by the analytical approaches and the validity of their predictions. Besides, the numerical simulations provide a detailed information of the dominant magnetic and lattice states. In Section \[lanczos\] we present the model considered and we study several features of the soliton formation in the IC phase using the Lanczos algorithm. In particular we analyze the effect of NNN interactions on the soliton width. In Section \[monte\] we perform Monte Carlo simulations using the world line algorithm –which allows us to study larger chains than the ones accessible to the Lanczos algorithm– in order to reduce finite size effects. Exact diagonalization study {#lanczos} =========================== The one-dimensional model which contains both the antiferromagnetic Heisenberg interactions and the coupling to the lattice is: $$\begin{aligned} {\cal H} &=& J \sum_{i = 1}^N (1 + (u_{i+1}- u_{i}))\; {\bf S}_i \cdot {\bf S}_{i+1} \nonumber \\ &+& J_2 \sum_{i = 1}^N {\bf S}_i \cdot {\bf S}_{i+2} + \frac{K}{2} \sum_{i=1}^N (u_{i+1}- u_{i})^2 \label{hamtot}\end{aligned}$$ where ${\bf S}_i$ are the spin-1/2 operators and $u_i$ is the displacement of magnetic ion $i$ with respect to its equilibrium position. Periodic boundary conditions are imposed. The first term, which corresponds to the nearest neighbor (NN) interactions, contains the spin-lattice coupling in the adiabatic approximation. The second term contains the AF NNN interactions, which were proposed in Refs. \[\] to fit the experimental magnetic susceptibility data in CuGeO$_3$. Several other properties of this material have been reasonably described using this model.[@haas; @rierakoval; @poilblanc] As in Ref. \[\], we assume for simplicity that the lattice distortion does not affect the second neighbor interactions. In principle, the NNN interactions should be corrected by a term proportional to $(u_{i+2}- u_{i})$ which vanishes in the dimerized phase but not necessarily in the incommensurate phase. This correction should be important precisely in the region around a soliton. It is customary to introduce the frustration constant $\alpha= J_2/J$. The estimated value of $\alpha$ in CuGeO$_3$ varies between 0.24 (Ref. \[\]) and 0.36 (Ref. \[\]). In this second case, $\alpha$ is larger than the critical value $\alpha_c \approx 0.2411$ above which in the absence of dimerization a gap opens in the excitation spectrum.[@okamoto] Our purpose is to study numerically Hamiltonian (\[hamtot\]) with exact diagonalization (Lanczos) techniques and by Monte Carlo simulations. In this latter case, in order to avoid the well-known sign problem due to the frustration, we will consider only the diagonal second neighbor interaction $$\begin{aligned} {\cal H}_2^{zz} = J_2^{z} \sum_{i = 1}^N S_i^{z} S_{i+2}^{z}, \label{h2n-zz}\end{aligned}$$ instead of the isotropic NNN interactions (second term of Eq. \[hamtot\]). It is quite apparent that the main numerical difficulty is related to the handling of the set of displacements $\{u_i\}$, which in principle can take arbitrary values to describe the various distortion patterns present in the dimerized and IC phases of the system. These displacements are calculated self-consistently by the following iterative procedure. First, we introduce the bond distortions defined as $\delta_i = (u_{i+1}- u_{i})$. Then, the equilibrium conditions for the phononic degrees of freedom: $$\begin{aligned} \frac{\partial \langle {\cal H} \rangle}{\partial \delta_i} + \lambda = 0 \label{eqcond}\end{aligned}$$ lead to the set of equations: $$\begin{aligned} J \langle {\bf S}_i \cdot {\bf S}_{i+1} \rangle + K \delta_i - {J \over N} \sum_{i=1}^N \langle {\bf S}_i \cdot {\bf S}_{i+1} \rangle = 0 , \label{distor-eqn}\end{aligned}$$ which satifies the constraint $\sum_i \delta_i =0$. This constraint has been included in Eq. (\[eqcond\]) through the corresponding Lagrange multiplier $\lambda$. The expectation values are taken with respect to the ground state of the system. The iterative procedure starts with an initial distortion pattern $\{ \delta_i^{(0)} \}$, which in general we choose at random. At the step $n$, with a distortion pattern $\{ \delta_i^{(n-1)} \}$, we diagonalize Hamiltonian (\[hamtot\]) using the Lanczos algorithm and compute the correlations $\langle {\bf S}_i \cdot {\bf S}_{i+1} \rangle$. We replace these correlations in Eq. (\[distor-eqn\]) and the new set $\{ \delta_i^{(n)} \}$ is obtained. We repeat this iteration until convergence. Essentially the same procedure is followed in the quantum Monte Carlo algorithm, as it is discussed in Section \[monte\]. We have applied this exact diagonalization procedure to determine the distortion patterns in the 20 site chain at $T=0$. In the first place we consider the case of $S^z=0$. As mentioned above, this corresponds to a dimerized lattice, i.e. $\delta_i = (-1)^i \delta_0$. Notice that for this simple case, the equilibrium distortion amplitude $\delta_0$ could be determined in an easier way by computing the energies of the spin part of Hamiltonian for a set of values of $\delta_0$. Then, adding the elastic energy and interpolating one obtains the minimum total energy. We have performed this calculation in order to check our iterative algorithm. The results for $\delta_0$ vs. $K$, for $S^z=0$, are shown in Fig.1 for $\alpha = 0.0$, 0.2 and 0.4, and $J_2^{z} = 0.2$, and 0.4. It can be seen that, as expected, for $\alpha >0$ the dimerized state is more favorable and this leads to a larger $\delta_0$ for a given $K$. To a lesser extent this trend is also present for $J_2^{z} > 0$. The dependence of $\delta_0$ with $K$ can be inferred from the scaling relation between the energy and the dimerization, $E_0(\delta_0)-E_0(0) \sim \delta_0^{2\nu}$ (plus logarithmic corrections) with $\nu=2/3$, in principle valid for $\alpha < \alpha_c$ and small $\delta_0$.[@crossfisher; @spronken; @laukamp] Then, it is easy to obtain $\delta_0 \sim K^{-3/2}$, a relation which is approximately satisfied by our numerical data. The fact that $\delta_0$ vanishes at a finite value $\hat{K}$ of the elastic constant, is just a finite size effect. By diagonalizing chains of $N=12$, 16 and 20 sites, for $\alpha=0$, we have verified that $\hat{K}$ increases with the lattice size, as it can be seen in Fig. \[fig1.5\], and it should eventually diverge in the bulk limit. Once we have determined the equilibrium distortion as a function of $K$, we are able to compute the singlet-triplet spin gap, defined as the following difference of ground state energies: $$\begin{aligned} \Delta = E_{0,dim}(S^z=1)-E_0(S^z=0) \label{spingap}\end{aligned}$$ It is worth to emphasize that $E_{0,dim}$ is the ground state energy of the system for $S^z=1$ with the dimerization obtained for $S^z=0$ and the same set of parameters. The results of this calculation are shown in Fig. \[fig2\]. Consistently with the larger $\delta_0$ shown in Fig. \[fig1\], the gap increases with $\alpha$. The effect of $J_2^{z}$ is much weaker than that of the isotropic second neighbor interaction which is not surprising since the 1D ground state magnetic structure, with a dominant dimerized state, has essentially a quantum (off-diagonal) origin. This small increase in $\Delta$ for a given $K$ is consistent with the small increase in $\delta_0$ shown in Fig. \[fig1\]. The corresponding scaling relation, $\Delta \sim K^{-1}$, obtained from the relation between the singlet-triplet gap and the dimerization, $\Delta \sim \delta_0^{2/3}$, is again reasonably satisfied by our numerical data. . \[fig2\] We now consider the case of $S^z=1$, which corresponds to the incommensurate region just above the dimerized-incommensurate transition. We have determined the distortion pattern for a 20 site chain using the iterative procedure described above. As discussed at the beginning of this section, the two solitons or domain walls separating dimerized regions are clearly distinguishable. (A typical pattern can be seen in Fig. \[soliton\].) The maximum distortion $\delta_0$, shown in Fig. \[fig3\], presents similar behavior as the one shown in Fig. \[fig1\] corresponding to $S^z=0$. In particular, the fact that $\delta_0$ vanishes at a finite $K$ is again due to finite size effects. In order to compute the soliton width, we use the following form to fit the numerically obtained distortion patterns: $$\begin{aligned} \delta_i = (-1)^i \tilde{\delta} \tanh \left(\frac{i-i_0 - {\frac d 2}}{\xi} \right)\tanh \left(\frac{i-i_0+{\frac d 2}}{\xi}\right), \label{fittanh}\end{aligned}$$ which corresponds to modeling each soliton as an hyperbolic tangent, as obtained in the analytical approach to this problem.[@nakano] The amplitude $\tilde{\delta}$, the soliton width $\xi $, and the soliton-antisoliton distance $d$, are the parameters determined by the numerical fitting. The amplitude $\tilde{\delta}$ should be equal to the maximum distortion $\delta_0$ defined above for well separated solitons, i.e. $d \gg \xi$. The main limitation of this calculation arises in the region where, for a given $\alpha$, $K$ is so large that the solitons have a substantial overlap in the 20 site chain, and the fitting function (\[fittanh\]) is no longer appropriate. In this case, the elliptic sine should be used to describe the soliton lattice. This is the region where finite size effects are important, as it was discussed above with respect to Figs. \[fig1\] and \[fig3\]. However, this situation is not directly relevant to experiment since in real materials the solitons are well-separated.[@kiry] We show in Fig. \[fig4\] the soliton width as a function of the gap $\Delta$ for the 20 site chain, for the same values of $\alpha$ and $J_2^{z}$ as before. It can be seen that the there is a linear dependence of the soliton width with the inverse of the gap. This behavior is consistent with the theoretical prediction:[@nakano] $$\begin{aligned} \xi =v_s/\Delta, \label{width-gap}\end{aligned}$$ where $v_s$ is the spin-wave velocity for $\alpha < \alpha_c$. It was recently shown that the relation (\[width-gap\]), originally obtained for the unfrustrated chain,[@nakano] is also valid in the presence of frustration.[@dobryriera] For $\alpha > \alpha_c$, $\Delta$ contains a contribution from the frustration due to the presence of a gap even in the absence of dimerization. A linear fitting of these curves in the region $\xi > 2.5$ gives the slopes 1.87, 1.70 and 1.63 , for $\alpha =0.0,\;0.2$ and $0.4$ respectively. Recently, a numerical study[@fledder] has proposed the law: $v_s=\frac \pi 2 (1-1.12 \alpha )$ in the bulk limit for $\alpha < \alpha_c$, From this law one gets $v_s =$ 1.57, 1.22, for $\alpha =0.0$ and $0.2$ respectively. We can observe that the slopes obtained by fitting the curves shown in Fig. (\[fig4\]) are systematically larger than these values of $v_s$. Besides, the effect of $\alpha$ is weaker in the numerical data than that predicted by Eq. (\[width-gap\]). For $\alpha= 0.4 > \alpha_c \approx 0.2411$, we have estimated $v_s$ by fitting the excitation dispersion relation $\varepsilon(k) = E_{0,dim}(S^z=1,k) - E_0(S^z=0,k=0)$ with the law $\varepsilon(k)^2 = \Delta^2 + v_s^2 k^2 + c k^4$ around $k=0$ and $\delta_i =0$. For $L=20$ we obtained $v_s = 0.707$, a value which is also smaller than the slope of the curve $\xi$ vs. $1/\Delta$ for $\alpha=0.4$ in Fig. (\[fig4\]). This disagreement between the prediction obtained by the continuum bosonized theory and the numerical results could be due to the approximations involved in the former or to finite size effects present in the latter. The study of much larger lattices than those considered in this section will be done in the following section using quantum Monte Carlo simulations. On the other hand, for the case of $J_2^z = 0.4$ the slope is actually [*larger*]{} ($\approx 2.1$) than the value obtained for the Heisenberg chain with NN interactions only. This effect is opposite to that of the isotropic NNN interactions and it will be further discussed in the next section. Monte Carlo simulations {#monte} ======================= In order to treat longer chains than those considered in the Lanczos diagonalization study of the previous section, we have implemented a world-line Monte Carlo algorithm[@WLMC] suited to this problem. The partition function is re-expressed as a functional integral over wordline configurations, where the contribution on each imaginary-time slice is given by the product of the two-site evolution matrix elements, $$\begin{aligned} W_{i,i+1}(\tau )=\langle S_{i,\tau }^zS_{i+1,\tau }^z\left| {\rm e}^{- \Delta \tau J_i {\bf S}_i \cdot {\bf S}_{i+1}}\right| S_{i,\tau +\Delta \tau }^zS_{i+1,\tau +\Delta \tau }^z\rangle \nonumber\end{aligned}$$ where $J_i = J(1 + \delta_i)$. These matrix elements are the Boltzmann weights associated with a bond ($ i,i+1$) in a time step $\Delta \tau =1/mT$ in the Trotter direction, where $ T $ is the temperature and $m$ is the Trotter number. Since the exchange couplings depend on the lattice displacements, these matrix elements are site dependent. We implemented the algorithm with the addition of a dynamic minimization of the free energy with respect to the lattice displacements. Starting from a given initial configuration (random distribution of spins and a dimerized pattern for the lattice displacements) we typically considered $2\times 10^3$ sweeps for thermalization. During the next $4\times 10^3$ sweeps we measured the derivative of the magnetic free energy, which, in the limit of $T \rightarrow 0$, is given by $$\frac{\partial {\cal F}_M}{\partial \delta _i}=J\langle \langle {\bf S}_i {\bf \cdot S}_{i+1}\rangle \rangle _T \ . \label{DF}$$ Leaving 3 sweeps between each measurement for de-correlation this produces $10^3$ independent values to obtain the thermal average. With this free-energy gradient we corrected the displacements according to (\[distor-eqn\]) and repeated the procedure, including the $2\times 10^3$ sweeps for thermalization since the spins have to accommodate to the new lattice distorsions. Once the displacement pattern is stabilized within statistical fluctuations —we typically considered $\sim $150 iterations, see Fig. \[150\]— we performed measurements of several quantities. For this we obtained 100 independent groups of $10^3$ measurements each, following the same procedure as described above, [*i.e.*]{}, i) thermalization, ii) measurements of $\frac{\partial {\cal F}_M}{\partial \delta _i}$ and observables, and iii) correction of the displacement pattern due to statistical fluctuations. In our calculations we considered chains of 64 sites with periodic boundary conditions and a temperature $T=0.05J$. We checked that this value is low enough to study ground-state properties by comparison with measurements at even lower temperatures. On the other hand, at higher temperatures the soliton is not observed and there is no definite pattern of lattice displacements. We took $m=80$ for the Trotter number, which is large enough to reproduce the Lanczos results on smaller chains (see Fig. \[fig1.5\]). For some particular quantities like the energy gap, which require more precision, we considered also $m=160.$ In addition, comparison with results for a longer chain with $N=128$ indicates that in the parameter range of our calculations the Monte Carlo results have no sizeable finite-size effects. In Fig. \[fig1.5\] we show the Monte Carlo results for the homogeneous dimerization of the 64 site chain in the $S^z=0$ subspace as a function of the elastic constant $K$, together with the Lanczos results for smaller chains. Notice that in the parameter range considered the 64 site chain does not have the finite size effects present for smaller chains, namely, the vanishing of $\delta_0$ for finite values of $K$. The inset shows the expected scaling behavior $\delta \propto K^{-3/2}$ discussed in the previous section. As a further check, we have also reproduced the scaling behavior of the energy gain $E_0(\delta_0)-E_0(0)$ and gap with $\delta_0$ with a measured exponent $\nu =2/3$ within statistical errors. =9.00cm The soliton structure in the subspace with $S^z=1$ is given in Fig. \[soliton\], where we plot the displacement envelope $\widetilde{\delta }_i=(-)^i\delta_i$ and the local magnetization $\langle S_i^z\rangle $ , for different values of the elastic constant $K.$ Notice that the displacements are normalized by their maximum values (shown in Fig. \[fig1.5\]) and the local magnetization by the classical value $S=1/2.$ Consequently, the size of lattice distortions in different panels cannot be directly compared. For small values of $K$ there is a well defined soliton-antisoliton structure in the distortion pattern, with the associated local magnetization following a staggered order. There is a net $1/2$ spin density near each domain wall, which makes the excess $S^z=1$. As in the previous section, we fitted a two-soliton solution (\[fittanh\]), with $\tilde{\delta }_0=1$ because of the normalization adopted. The results for the soliton width $\xi$ are shown in Fig. \[xi-k\]. For increasing values of $K$ the soliton width grows until the displacement profile resembles a sine law (see Fig. \[soliton\]). This sinusoidal pattern is typical of the soliton lattice, observed for large values of $S^z$. It can be seen that the scaling $\xi \sim K$ obtained in \[\] is well reproduced in the whole parameter range considered, as indicated by the linear fit to the data (dashed line). This figure shows that the soliton width for $J_2^z = 0.3$ also presents a linear dependence with $K$. These features observed in the 64 site chain are qualitatively similar to those present in the 20 site chain as determined by exact diagonalization. Besides, it can seen in this figure that the reduction of $\xi$ is much stronger when the isotropic NNN is taking into account. We have performed a simple study on the soliton-antisoliton interaction. For this study we fixed the distortion pattern to the law (\[fittanh\]) with the previously fitted value of $\xi ,$ and considered increasing values of $d.$ For small $K$ ($\leq 2J$) we found that the total energy becomes a constant (within statistical fluctuations) when $ d\geq 4\xi ,$ which implies that the soliton-antisoliton pairs shown in the left panels of Fig. 4 are not interacting. This was confirmed by allowing the lattice distortion to evolve starting from a pattern like (\[fittanh\]) with an initial separation larger than $d,$ which produces the same result for $\xi $ and the total energy. Next, we study the behavior of the soliton width $\xi $ with the spin-Peierls gap $\Delta $. That is, we compare the quantity $\xi $ that characterizes the $S^z=1$ soliton state, with the singlet-triplet excitation gap $\Delta $ above the dimerized $S^z=0$ ground state. As shown in Fig. \[xi-gap\], these two quantities are inversely related to each other, as discussed in the previous section. The slope of the linear fit is $1.9$, very close to the value 1.87 obtained by exact diagonalization of the 20 site chain in the previous section. This result confirms the disagreement between the numerical results with the analytical prediction pointed out in Section \[lanczos\]. Also shown in Fig. \[xi-gap\] are the results for $J_2^z = 0.3$. A linear fit to these results leads to a slope $\approx 2.3$ , i.e. larger than the value corresponding to $J_2^z = 0.0$. This increase of the slope between $\xi$ and $\Delta^{-1}$ is consistent with the result obtained for the 20 site lattice by exact diagonalization and $J_2^z = 0.4$. This behavior should be contrasted with the [*reduction*]{} of the slope found for the isotropic NNN interaction. A possible explanation of this behavior could be the following. As discussed in the previous section, the term ${\cal H}_2^{zz}$ leads to a smaller increase of the spin gap than the fully isotropic NNN interaction. On the other hand, the Ising interaction could be more effective in punishing the excess $\langle S^z \rangle$ which appears around a soliton leading to a smaller reduction of the soliton width than the one caused by the isotropic term, as it can be seen in Fig. \[xi-k\]. A more detailed study of the Hamiltonian in the presence of the term of ${\cal H}_2^{zz}$ is clearly necessary to fully understand this behavior. Finally, it is possible to estimate the critical value of the magnetic field at zero temperature. By adding a Zeeman term to the Hamiltonian (\[hamtot\]), $-g \mu_B S^z H$ ($\mu_B$: Bohr’s magneton), $H_{cr}$ may be calculated as: $$\begin{aligned} H_{cr} = E_0(S^z=1)-E_0(S^z=0) \label{hmagcrit}\end{aligned}$$ in units of $g \mu_B$. $E_0(S^z=1)$ is the ground state energy of (\[hamtot\]), and then $H_{cr} < \Delta$, which is the value expected of a gapped system in the absence of magneto-elastic coupling. The behavior of $H_{cr}$ as a function of $\Delta$ is shown in Fig. \[hcritvsgap\] for the 64 site chain, $\alpha=J_2^z=0.0$, and for the 20 site chain, $\alpha=J_2^z=0.4$. It is apparent a linear dependence over all the range studied, which is in agreement with the mean-field prediction[@pytte; @crossfisher], $H_{cr} \approx 0.84 \Delta$. However, we obtain a coefficient considerable smaller, $H_{cr}/\Delta \approx 0.47$, almost independent of $\alpha$. This value is also smaller than twice the soliton formation energy calculated in Ref. \[\]. The finite value at the origin of the curves corresponding to $\alpha=J_2^z=0.4$ is a finite size effect. Conclusions {#conclu} =========== In this article we have analyzed the magnetic soliton lattice in the incommensurate phase of spin-Peierls systems using numerical methods. There is a remarkable agreement between the results obtained by exact diagonalization using the Lanczos algorithm and those obtained by quantum Monte Carlo with the world-line algorithm. The relations among various features of the solitons and magnetic properties of the system have been determined and compared with analytical results. Our starting point is a microscopical model proposed to describe several properties of CuGeO$_3$, consisting of a 1D AF Heisenberg model with nearest and next-nearest neighbor interactions. In the first place we have not detected any crossover in the behavior of the quantities examined as $\alpha$, the ratio of NNN to NN interactions, becomes greater than $\alpha_c$ at least for the small chains considered. That is, there are only smooth changes as $\alpha$ varies between 0.0 and 0.4. The most important effect of the competing NNN interaction is a [*reduction*]{} of the soliton width $\xi$ as a function of the inverse of the singlet-triplet spin gap $\Delta$. Furthermore, the effect of the diagonal term (\[h2n-zz\]) is much less important and in some cases even qualitatively different to that of the isotropic NNN term. Although several functional forms predicted by continuum analytical theories have been confirmed by our numerical data, there are some important quantitative differences. The most important disagreement between our numerical results and the analytical predictions is related to the coefficient in the relation $\xi \sim \Delta^{-1}$, i.e. we have obtained a systematically higher value than the theoretical value which is the spin-wave velocity. The estimated value of $H_{cr}/\Delta$ is also noticeable smaller than the mean-field result and slighly smaller than the prediction of bosonized field theory. The relevance of these numerical results to real SP materials, such as CuGeO$_3$ and the recently discovered NaV$_2$O$_5$,[@navo] has to be determined experimentally. The numerical procedures developed in this article could be applied to the study of several other properties of the incommensurate phase of spin-Peierls systems, such as the static magnetization as a function of the magnetic field (recently measured in CuGeO$_3$ by Fagot-Revurat [*et al.*]{}[@fagot]) and the order of the transition from the dimerized to the incommensurate phases.[@loosd] F. D. 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Numerical calculations indicate that this relation still holds with an exponent $\nu$ close to 2/3, for $\alpha > \alpha_c$, at least for not too small $\delta_0$ and $\alpha < 1/2$; M. Laukamp and J. Riera, (to be published). A. Fledderjohann and C. Gros, cond-mat/9612013. J. E. Hirsch [*et al.*]{}, Phys. Rev. B [**26**]{}, 5033 (1982). M. Isobe and Y. Ueda, J. Phys. Soc. Jpn. [**65**]{}, 1178 (1996); M. Weiden, R. Hauptmann, C. Geibel, F. Steglich, M. Fischer, P. Lemmens and G. Güntherodt, preprint cond-mat/9703052.

--- abstract: 'In this article we explicitly determine the structure of the Weierstrass semigroups $H(P)$ for any point $P$ of the Giulietti–Korchmáros curve $\cX$. We show that as the point varies, exactly three possibilities arise: One for the $\mathbb{F}_{q^2}$-rational points (already known in the literature), one for the $\mathbb{F}_{q^6} \setminus \mathbb{F}_{q^2}$-rational points, and one for all remaining points. As a result, we prove a conjecture concerning the structure of $H(P)$ in case $P$ is a $\mathbb{F}_{q^6} \setminus \mathbb{F}_{q^2}$-rational point. As a corollary we also obtain that the set of Weierstrass points of $\cX$ is exactly its set of $\mathbb{F}_{q^6}$-rational points.' author: - Peter Beelen and Maria Montanucci title: 'Weierstrass semigroups on the Giulietti–Korchmáros curve' --- [^1] [^2] Introduction ============ Let $\cC$ be a nonsingular, projective algebraic curve of genus $g$ defined over a field $\mathbb{F}$. Let $P$ be a rational point on $\cC$. The *Weierstrass semigroup* $H(P)$ is defined as the set of integers $k$ such that there exists a function on $\cC$ having pole divisor exactly $kP$. More generally $H(P)$ can be defined for any point $P$ on $\cC$ by considering $\cC$ as an algebraic curve over the algebraic closure of $\mathbb{F}$. It is clear that $H(P)$ is a subset of natural numbers $\mathbb{N}=\{0,1,2,\ldots\}$. The Weierstrass gap Theorem, see [@Sti Theorem 1.6.8], states that the set $G(P):= \mathbb{N} \setminus H(P)$ contains exactly $g$ elements, which are called *gaps*. The structure of $H(P)$ is not always the same for every point $P$ of $\cC$. However, it is known that for generically the semigroup $H(P)$ is the same, but there can exist finitely many points of $\cC$, called *Weierstrass points*, with a different gap set. These points are of intrinsic interest, for example in Stöhr–Voloch theory [@SV], but in case $\mathbb{F}=\mathbb{F}_q$, the finite field with $q$ elements, they also occur in the study of algebraic geometry (AG) codes [@TV1991]. In this context, a commonly studied class of curves are the so-called *maximal curves*, that is, algebraic curves defined over a finite field $\mathbb{F}_q$ having as many rational points as possible according to the Hasse–Weil bound. More precisely, an algebraic curve $\cC$ with genus $g(\cC)$ and defined over $\mathbb{F}_q$ is said to be an $\mathbb{F}_{q}$-maximal curve if it has $q+1+2g(\cC)\sqrt{q}$ points defined over $\mathbb{F}_q$. Clearly, this can only be the case if the cardinality $q$ of the finite field is a square. An important and well-studied example of an $\mathbb{F}_{q^2}$-maximal curve is given by the Hermitian curve $\mathcal{H}$. For fixed $q$, the curve $\mathcal{H}$ has the largest possible genus $g(\mathcal{H}) =q(q-1)/2$ that an $\mathbb{F}_{q^2}$-maximal curve can have. The Weierstrass points on $\mathcal H$ and the precise structure of the semigroups for $P$ on $\mathcal{H}$ are known; see [@GV]. By a result commonly attributed to Serre, see [@L1987 Proposition 6], any $\mathbb{F}_{q^2}$-rational curve which is covered by an $\mathbb{F}_{q^2}$-maximal curve is also $\mathbb{F}_{q^2}$-maximal. Most of the known maximal curves are subcovers of the Hermitian curve. The first known example of a maximal curve which is not a subcover of the Hermitian curve was constructed by Giulietti and Korchmáros; see [@GK]. This curve is an $\mathbb{F}_{q^6}$-maximal curve and commonly called the Giulietti–Korchmáros (GK) curve. The aim of this paper is to complete the description of the Weierstrass semigroups occurring for this curve. The Weierstrass semigroup for any $\mathbb{F}_{q^2}$-rational point of $\cX$ was computed in [@GK], but the structure of the Weierstrass semigroup $H(P)$ where $P \not\in \cX(\mathbb{F}_{q^2})$ is not known, except for $q \le 9$, [@FG2010; @D2011]. Based on the available data for small $q$, a conjecture concerning the structure of $H(P)$ was stated in [@D2011] for $P \in \cX(\mathbb{F}_{q^6}) \setminus \cX(\mathbb{F}_{q^2})$. For $P \not\in \cX({\mathbb {F}_{q^6}})$ nothing specific is known about $H(P).$ In this article we determine settle the conjecture from [@D2011] and also determine the structure of the generic semigroup for $P$ on $\cX$. More precisely, we show the following theorem. \[mainth\] Let $q$ be a prime power and let $P$ be a point of the Giulietti–Korchmáros curve $\cX$. The Weierstrass semigroup $H(P)$ is given by - $H(P)=\langle q^3 -q^2 +q, q^3, q^3 + 1 \rangle,$  if $P \in \cX(\mathbb{F}_{q^2})$; - $H(P)=\langle q^3-q+1,q^3+1,q^3+i(q^4-q^3-q^2+q-1) \mid i=0,\ldots,q-1\rangle,$  if $P \in \cX(\mathbb{F}_{q^6}) \setminus \cX(\mathbb{F}_{q^2})$; - $H(P)=\mathbb{N} \setminus G,$  if $P \not\in \cX(\mathbb{F}_{q^6})$, where $$G=\left\{iq^3+kq+m(q^2+1)+\sum_{s=1}^{q-2} n_s ((s+1)q^2)+j+1 \mid i,j,k,m,n_1,\ldots,n_{q-2} \in \mathbb{Z}_{\geq0}, \ j \le q-1 \ \makebox{and}\right.$$ $$\left.i+j+k+mq+\sum_{s=1}^{q-2} n_s ((s+1)q-s) \leq q^2-2\right\}.$$ As mentioned above, the case $P \in \cX(\mathbb{F}_{q^2})$ is already known and taken from [@GK]. As a bonus, we will also obtain the set of Weierstrass points of $\cX$. \[mainth2\] Let $W$ denote the set of Weierstrass points of the Giulietti–Korchmáros curve $\cX$. Then $W=\cX(\mathbb{F}_{q^6})$. The paper is organized as follows: In the next section we give the necessary background on the GK curve as well as some results on Weierstrass semigroups and their gaps that we will need later. In section three, we settle the conjecture from [@D2011] concerning $H(P)$ for $P \in \cX(\mathbb{F}_{q^6}) \setminus \cX(\mathbb{F}_{q^2})$, while in section four, we compute the Weierstrass semigroup for $P \not \in \cX(\mathbb{F}_{q^6})$. We finish with some concluding remarks and observations. The Giulietti–Korchmáros curve {#sec2} ============================== Let $q$ be a prime power and $\K=\overline{\mathbb{F}}_q$. The Giulietti–Korchmáros (GK) curve $\cX$ is a non-singular curve in ${\rm PG}(3,\K)$ defined by the affine equations $$\label{eq:GK} \cX: \left\{ \begin{array}{l} Y^{q+1}=X^q+X,\\ Z^{q^2-q+1}=Y^{q^2}-Y.\\ \end{array} \right.$$ This curve has genus $g(\cX)=(q^5-2q^3+q^2)/2$ and $q^8-q^6+q^5+1$ $\mathbb{F}_{q^6}$-rational points. The curve $\cX$ has been introduced in [@GK], where it was proved that $\cX$ is maximal over ${\mathbb {F}_{q^6}}$, that is, the number $|\cX({\mathbb {F}_{q^6}})|$ of ${\mathbb {F}_{q^6}}$-rational points of $\cX$ equals $q^6+1+2gq^3$. Also, for $q>2$, the curve $\cX$ is not ${\mathbb {F}_{q^6}}$-covered by the Hermitian curve maximal over ${\mathbb {F}_{q^6}}$; $\cX$ was the first maximal curve shown to have this property. Note that equation implies that $\cX$ is a cover of the Hermitian curve over ${\mathbb {F}_{q^2}}$ given by the affine equation $Y^{q+1}=X^q+X$. We will denote this curve by $\cH$. The automorphism group ${\rm Aut}(\cX)$ of $\cX$ is defined over ${\mathbb {F}_{q^6}}$ and has order $q^3(q^3+1)(q^2-1)(q^2-q+1)$. Moreover, it has a normal subgroup isomorphic to ${\rm SU(3,q)}$, the automorphism group of the Hermitian curve $\cH$. The set $\cX({\mathbb {F}_{q^6}})$ of the $\mathbb{F}_{q^6}$-rational points of $\cX$ splits into two orbits under the action of ${\rm Aut}(\cX)$: one orbit $\mathcal O_1=\cX({\mathbb {F}_{q^2}})$ of size $q^3+1$, which coincides with the intersection between $\cX$ and the plane $Z=0$; and another orbit $\mathcal O_2=\cX({\mathbb {F}_{q^6}})\setminus\cX({\mathbb {F}_{q^2}})$ of size $q^3(q^3+1)(q^2-1)$; see [@GK Theorem 7]. The orbits $\mathcal O_1$ and $\mathcal O_2$ are the short orbits of ${\rm Aut}(\cX)$, that is, the unique orbits of points of $\cX$ having a non-trivial stabilizer in ${\rm Aut}(\cX)$. Let $x,y,z\in \K(\cX)$ be the coordinate functions of the function field of $\cX$, which satisfy $y^{q+1}=x^q+x$ and $z^{q^2-q+1}=y^{q^2}-y$. Then we denote by $P_{(a,b,c)}$ the affine point of $\cX$ with coordinates $(a,b,c)$ and by $P_\infty$ the unique point at infinity. Similarly, we denote by $Q_{(a,b)}$ the affine point of the Hermitian curve $\cH$ with coordinates $(a,b)$ and by $Q_\infty$ its unique point at infinity. The Weierstrass semigroup at $P_\infty$, and hence at every $\mathbb{F}_{q^2}$-rational point of $\cX$ (since they lie in the same short orbit $\mathcal O_1$ of ${\rm Aut}(\cX)$) was computed in [@GK]. [[@GK Proposition 6.2]]{} The Weierstrass semigroup of $\cX$ at $P_\infty$ is generated by $q^3 -q^2 +q$, $q^3$, $q^3 + 1$. Before describing what is known about $H(P)$ for $P \not \in \cX({\mathbb {F}_{q^2}})$, we introduce several functions on $\cX$ and give their divisors. Some of these functions can be interpreted as functions on $\cH$ as well and therefore have a divisor on $\cH$. To differentiate, we will write $(f)_\cH$ (resp. $(f)_\cX$) for divisors on the Hermitian curve $\cH$ (resp. the GK curve $\cX$). Given a point $P=P_{(a,b,c)}$ on $\cX$, we define the functions $$\label{tilda} \tilde{x}_{P}=-a^q-x+b^qy, \quad \tilde{y}_P=y-b, \quad \tilde{z}_P=-a^{q^3}-x+b^{q^3}y+c^{q^3}z.$$ Then it is not hard to show the following. $$\begin{aligned} (\tilde{x}_{P})_\cX &=q\sum_{\xi^{q^2-q+1}=1}P_{(a,b,\xi c)}+\sum_{\xi^{q^2-q+1}=1}P_{(a^{q^2},b^{q^2},\xi c^{q^2})}-(q^3+1)P_{\infty},\label{eq:divxpX}\\ (\tilde{y}_{P})_\cX & =\sum_{s^q+s=0, \ \xi^{q^2-q+1}=1} P_{(a+s,b,\xi c)}-(q^3-q^2+q)P_\infty,\label{eq:divypX}\\ (\tilde{z}_{P})_\cX & =q^3 P_{(a,b,c)}+{P_{(a^{q^6},b^{q^6},c^{q^6})}}-(q^3+1)P_\infty,\label{eq:divzpX}\\ (z)_\cX & =\sum_{P\in\cX({\mathbb {F}_{q^2}}),P\ne P_\infty} P \, - \, q^3P_{\infty}.\label{eq:divz}\end{aligned}$$ Now let $P=P_{(a,b,c)}$ be a fixed $\mathbb{F}_{q^6}$-rational point of $\cX$ which is not $\mathbb{F}_{q^2}$-rational (implying $c \neq 0$). In this case equation implies: $$\label{eq:divzpX6} (\tilde{z}_{P})_\cX=(q^3+1)(P-P_\infty) \ \makebox{for} \ P=P_{(a,b,c)} \in \cX({\mathbb {F}_{q^6}}).$$ The Weierstrass semigroup $H(P)$ is only completely known in finitely many cases if $P\in \cX({\mathbb {F}_{q^6}})\setminus \cX({\mathbb {F}_{q^2}})$. It was computed for $q=2$ and $q=3$ in [@FG2010] and for $4 \le q \le 9$ in [@D2011]. Also in [@D2011], the following partial information was obtained for general $q$: Equations , and imply that the functions $1/\tilde z_P, \tilde y_P / \tilde z_P, \tilde x_P / \tilde z_P$ have poles only in $P$ of orders $q^3+1$, $q^3$ and $q^3-q+1$ respectively. Hence $$\label{eq:somepoles} \langle q^3-q+1,q^3,q^3+1\rangle \subseteq H(P) \ \makebox{for} \ P \in \cX({\mathbb {F}_{q^6}}) \setminus \cX({\mathbb {F}_{q^2}}).$$ Based on this and the results for $q \le 9$, the following conjecture was stated in [@D2011], which we will prove in the next section. \[conjD\] The Weierstrass semigroup $H(P)$ of $\cX$ at $P \in \cX(\mathbb{F}_{q^6}) \setminus \cX(\mathbb{F}_{q^2})$ is given by $$H(P)=\langle q^3-q+1,q^3+1,q^3+i(q^4-q^3-q^2+q-1) \mid i=0,\ldots,q-1\rangle.$$ Finally, for $P \not \in \cX({\mathbb {F}_{q^6}})$ nothing specific is known about the structure of semigroup $H(P)$. We will completely determine its gap structure, but for now, we finish this section by stating some facts that we will use to achieve this. We start with the following well-known lemma connecting regular differentials (i.e., differential forms having no poles anywhere on $\cX$) and gaps of $H(P)$. [[@VS Corollary 14.2.5]]{}\[prop:holom\] Let $\cX$ be an algebraic curve of genus $g$ defined over $\K$. Let $P$ be a point of $\cX$ and $\omega$ be a regular differential on $\cX$. Then $v_P(\omega)+1$ is a gap at $P$. This proposition has the following, for us very useful, consequence. \[holom\] For any point $P$ on the GK curve $\cX$ distinct from $P_\infty$ and for any $f \in L((2g(\cX)-2)P_\infty),$ we have $v_P(f)+1 \in \mathbb{N} \backslash H(P).$ First note that $(dy)_\cH=(q^2-q-2)Q_\infty$. The set of points that ramify in the covering of $\cX$ by $\cH$ is exactly $\cH({\mathbb {F}_{q^2}})$, the set of ${\mathbb {F}_{q^2}}$-rational points of the Hermitian curve, all with ramification index $q^2-q+1$. Moreover, the points of $\cX$ above $\cH({\mathbb {F}_{q^2}})$ are precisely the ${\mathbb {F}_{q^2}}$-rational points of $\cX$. Therefore, we immediately obtain that $$(dy)_\cX=(q^4-2q^3+q^2-2)P_\infty+(q^2-q)\sum_{P\in\cX({\mathbb {F}_{q^2}}),P\ne P_\infty} P.$$ Thus, from $z^{q^2-q+1}=y^{q^2}-y$ and equation , $$(dz)_\cX=(-dy/z^{q^2-q})_\cX=(q^5-2q^3+q^2-2)P_\infty.$$ In particular a differential $fdz$ is regular if and only if $f \in L((q^5-2q^3+q^2-2)P_\infty)=L((2g(\cX)-2)P_\infty)$. The corollary now follows by applying Proposition \[prop:holom\]. The Weierstrass semigroup $H(P)$ for $P \in \cX(\mathbb{F}_{q^6}) \setminus \cX(\mathbb{F}_{q^2})$ ================================================================================================== This section is devoted to the proof of Conjecture \[conjD\] for any prime power $q$. In particular in this section $P=P_{(a,b,c)}$ will always denote a point in $\cX(\mathbb{F}_{q^6}) \setminus \cX(\mathbb{F}_{q^2})$. Further we define the semigroup $$T:=\langle q^3-q+1,q^3+1, q^3+i(q^4-q^3-q^2+q-1) \mid i=0,\ldots,q-1\rangle.$$ Conjecture \[conjD\] then simply states that $H(P)=T$. Our proof of the conjecture consists of two main steps. In the first step, we will show that $T \subset H(P)$ by showing that the generators of $T$ are in $H(P)$. In the second step, we show that the number of gaps of the semigroup $T$ (also known as the genus of $T$) is exactly equal to the genus of $\cX$. Once this has been established, the equality $H(P)=T$ will follow immediately, proving Conjecture $\ref{conjD}$. $T \subset H(P)$ ---------------- As before we use the function $\tilde x_P$ defined in equation and its divisor in equation . Moreover, for $k \in \mathbb{Z}$, we define the $k$-th Frobenius twist of $\tilde x_P$ as the follows: $$\label{eq:FrobxP} \tilde x_P^{(k)}:=-a^{q^{2k+1}}-x+b^{q^{2k+1}}y \ \makebox{for} \ P=P_{(a,b,c)}.$$ Since we assume that $P \in \cX(\mathbb{F}_{q^6}) \setminus \cX(\mathbb{F}_{q^2}),$ equation implies that $$\begin{aligned} \label{eq:divFrobxP} (\tilde x_P^{(1)})_{\cX} & =q\sum_{\xi^{q^2-q+1}=1}P_{(a^{q^2},b^{q^2},\xi c^{q^2})}+\sum_{\xi^{q^2-q+1}=1}P_{(a^{q^4},b^{q^4},\xi c^{q^4})}-(q^3+1)P_{\infty},\notag\\ (\tilde x_P^{(2)})_{\cX} & =q\sum_{\xi^{q^2-q+1}=1}P_{(a^{q^4},b^{q^4},\xi c^{q^4})}+\sum_{\xi^{q^2-q+1}=1}P_{(a,b,\xi c)}-(q^3+1)P_{\infty}.\end{aligned}$$ \[functions\] Let $P=P_{(a,b,c)} \in \cX(\mathbb{F}_{q^6}) \setminus \cX(\mathbb{F}_{q^2})$ and let $\tilde f_i=f_i / \tilde z_P^{iq-i+1}$ where $$f_i:=\frac{(\tilde x_P)^{qi} \cdot \tilde x_P^{(2)}}{(\tilde x_P^{(1)})^{i}}, \ \makebox{for} \ i=1,\dots,q-1.$$ Then $(\tilde f_i)_{\infty}=(q^3+i(q^4-q^3-q^2+q-1))P$ and in particular $q^3+i(q^4-q^3-q^2+q-1) \in H(P)$ for $i=1,\ldots,q-1$. Using equations and , we directly obtain that $$(f_i)_{\cX}=(iq^2+1)\sum_{\xi^{q^2-q+1}=1}P_{(a,b,\xi c)}+(q-i)\sum_{\xi^{q^2-q+1}=1}P_{(a^{q^4},b^{q^4},\xi c^{q^4})}-(q^3+1)(iq-i+1)P_\infty.$$ Now using the divisor of $\tilde z_P$ given in equation , we find that $$(\tilde f_i)_{\cX}=-(q^3+i(q^4-q^3-q^2+q-1))P+(iq^2+1)\sum_{\substack{\xi^{q^2-q+1}=1, \\ \xi \neq 1}}P_{(a,b,\xi c)}+(q-i)\sum_{\xi^{q^2-q+1}=1}P_{(a^{q^4},b^{q^4},\xi c^{q^4})}.$$ The lemma now follows. Note that the lemma is also true for $i=0$. Considering the corresponding function $\tilde f_0=\tilde x_P^{(2)}/\tilde z_P$, gives a way to show that $q^3 \in H(P)$. However, this is already known, see equation . \[prop:contained\] Let $P \in \cX(\mathbb{F}_{q^6}) \setminus \cX(\mathbb{F}_{q^2})$. Then $T\subset H(P).$ Equation and Lemma \[functions\] imply that $\{ q^3-q+1,q^3+1, q^3+i(q^4-q^3-q^2+q-1) \mid i=0,\ldots,q-1\} \subset H(P)$. Since by definition these numbers generate $T$, the proposition follows. The genus of the numerical semigroup $T$ equals $g(\cX)$ -------------------------------------------------------- We now show that the genus $g(T)$ of the numerical semigroup $T=\langle q^3-q+1,q^3+1,q^3+i(q^4-q^3-q^2+q-1) \mid i=0,\ldots,q-1\rangle$ is equal to $g(\cX)=(q^5-2q^3+q^2)/2$. In this way, since we already know that $T \subseteq H(P_{(a,b,c)})$ from Proposition \[prop:contained\], Conjecture \[conjD\] will be completely proved. We recall that a numerical semigroup is called *telescopic* if it is generated by a telescopic sequence, that is by a sequence $(a_1,\ldots,a_k)$ such that - $\gcd(a_1, \ldots , a_k)=1$; - for each $i=2,\ldots,k$, $a_i/d_i \in \langle a_1/d_{i-1},\ldots, a_{i-1}/d_{i-1}\rangle$, where $d_i=\gcd(a_1,\ldots,a_i)$ and $d_0=0$; see [@KP1995]. From [@HVP1998 Proposition 5.35], the genus of a semigroup $\Gamma$ generated by a telescopic sequence $(a_1,\ldots,a_k)$ is $$\label{gentelescopic} g(\Gamma)=\frac{1}{2} \bigg( 1+ \sum_{i=1}^k \bigg( \frac{d_{i-1}}{d_i}-1\bigg) a_i \bigg).$$ For the semigroup $S$ defined by $S:=\langle q^3-q+1,q^3+1\rangle$ we obtain the following: \[genusS\] The numerical semigroup $S=\langle q^3-q+1,q^3+1\rangle$ is telescopic. Its genus $g(S)$ is given by $$g(S)=\frac{q^3(q^3-q)}{2}.$$ Let $a_1=q^3-q+1$ and $a_2=q^3+1$. Then $gcd(a_1,a_2)=1$ and, using the same notation as above, $d_1=a_1$ and $d_2=1$. Since $a_2/d_2 \in \langle 1 \rangle= \langle a_1/d_1 \rangle$, $S$ is telescopic. Thus from equation , $$g(S)=\frac{1}{2} \bigg( 1-a_1+(a_1-1)a_2 \bigg)=\frac{q^3(q^3-q)}{2}.$$ Now the idea is to compute the number of gaps of $T$ by identifying the elements of $T$ that are gaps of $S$. The following observation is trivial, but will be very useful. \[obs:representationab\] For any integer $n$, there exist unique integers $a$ and $b$ such that $n=a(q^3-q+1)+b(q^3+1)$ and $0 \le b \le q^3-q.$ An integer $n$ is an element of the semigroup $S=\langle q^3-q+1,q^3+1\rangle$ if and only if there exist integers $a$ and $b$ such that $n=a(q^3-q+1)+b(q^3+1)$, $a \ge 0$ and $0 \le b \le q^3-q.$ In the following lemma, we identify several elements of $T \setminus S$ that turn out to play an important role. \[lem:sij\] For any $i=0,\ldots,q-1$ and $j=1,\ldots,q-1$, define the set $$S_{i,j}:=\{(iq-jq^2+k_1)(q^3-q+1)+(jq^2-i+k_2)(q^3+1) \mid k_1=0,\ldots,q-1, \ k_2=0,\ldots,q^3-q-jq^2+i\}.$$ Then we have: 1. $S_{i,j} \subset T \setminus S.$ 2. $S_{i,j} \cap S_{i'j'} = \emptyset$ if $(i',j') \neq (i,j)$, $0 \le i' \le q-1$ and $1 \le j' \le q-1.$ 3. $|S_{i,j}|=q(q^3-q-jq^2+i+1).$ First of all note that $$jq^3+i(q^4-q^3-q^2+q-1)=(-jq^2+iq)(q^3-q+1)+(jq^2-i)(q^3+1).$$ Using this, it is clear from Proposition \[prop:contained\], that $(iq-jq^2+k_1)(q^3-q+1)+(jq^2-i+k_2)(q^3+1) \in T$ for any $i,j,k_1,k_2$ in the given range. To show that these elements are not in $S$, observe that $$\label{eq:ijk} iq-jq^2+k_1 \le (q-1)q-q^2+q-1<0 \ \makebox{and} \ 0 \le jq^2-i+k_2 \le q^3-q.$$ Observation \[obs:representationab\] now implies that $(iq-jq^2+k_1)(q^3-q+1)+(jq^2-i+k_2)(q^3+1) \not \in S.$ This completes the proof of the first item. Now suppose that $S_{i,j} \cap S_{i'j'} \neq \emptyset$. Then there exist integers $k_1,k_1',k_2,k_2'$ satisfying the defining requirements of $S_{i,j}$ and $S_{i'j'}$ such that $$(iq-jq^2+k_1)(q^3-q+1)+(jq^2-i+k_2)(q^3+1)=(i'q-j'q^2+k'_1)(q^3-q+1)+(j'q^2-i'+k'_2)(q^3+1).$$ As above, we have equation as well as the similar equation $$i'q-j'q^2+k'_1 <0 \ \makebox{and} \ 0 \le j'q^2-i'+k'_2 \le q^3-q.$$ Observation \[obs:representationab\] therefore implies that $$iq-jq^2+k_1=i'q-j'q^2+k'_1 \ \makebox{and} \ jq^2-i+k_2=j'q^2-i'+k'_2,$$ and in particular $(i-i')q-(j-j')q^2+(k_1-k_1')=0.$ Considering this equation modulo $q$ and modulo $q^2$, we see that $k_1=k_1'$ and $i=i'$, implying that $j=j'$ as well. Then it is also clear that $k_2=k_2'$. This implies the second item. As for the third item: if $$(iq-jq^2+k_1)(q^3-q+1)+(jq^2-i+k_2)(q^3+1)=(iq-jq^2+k'_1)(q^3-q+1)+(jq^2-i+k'_2)(q^3+1),$$ with integers $k_1,k_1',k_2,k_2'$ satisfying the defining requirements of $S_{i,j}$, then the same reasoning as in above proof of the second item, shows that $k_1=k_1'$ and $k_2=k_2'$. Hence the cardinality of $S_{i,j}$ is simply the number of possibilities for $k_1$ times that for $k_2$. Picture \[fig1\] describes the sets $S_{i,j}$ for $q=3$. In this picture a point of coordinates $(a,b)$ is used to represent the element $a(q^3-q+1)+b(q^3+1)$. Black dots represent elements of the numerical semigroup $S$, while white dots represent the elements contained in $S_{i,j}$ for some $i$ and $j$. \[fig1\] (-2.8,0)–(3,0) node \[right\] [$a$]{}; (0,0)–(0,2.7) node \[above\] [$b$]{}; (-2.8,2.4)–(3,2.4) node \[right\] [$24$]{}; (0,0)–(-2.8,2.4); in [0,0.1,...,2.8]{} in [0,0.1,...,2.5]{} at (,) ; at (-0.3,0.7) ; at (-0.9,0.9) ; at (-0.6,0.8) ; at (-1.2,1.6) ; at (-1.5,1.7) ; at (-1.8,1.8) ; (1.3,2.5) –(1.3,2.5) node \[above, right\][$S$]{}; (-0.3,0.7)–(-0.3,2.5) node \[above, right\][$S_{2,1}$]{}; (-0.9,0.9) –(-0.9,2.5) node \[above, right\][$S_{0,1}$]{}; (-0.6,0.8) –(-0.6,2.5) node \[above, right\][$S_{1,1}$]{}; (-1.2,1.6) –(-1.2,2.5) node \[above, right\][$S_{2,2}$]{}; (-1.5,1.7) –(-1.5,2.5) node \[above, right\][$S_{1,2}$]{}; (-1.8,1.8) –(-1.8,2.5) node \[above, right\][$S_{0,2}$]{}; (-0.3,0.7)–(0,0.7); (-0.9,0.9) –(-0.6,0.9); (-0.6,0.8) –(-0.3,0.8); (-1.2,1.6) –(-0.9,1.6); (-1.5,1.7) –(-1.2,1.7); (-1.8,1.8) –(-1.5,1.8); (-2.8,2.4) –(-2.8,0) node \[below\] [ ]{}; in [-0.1,-0.2,-0.3]{} in [0.7,0.8,...,2.5]{} at (,) ; in [-0.4,-0.5,-0.6]{} in [0.8,0.9,...,2.4]{} at (,) ; in [-0.7,-0.8,-0.9]{} in [0.9,1,...,2.5]{} at (,) ; in [-1,-1.1,-1.2]{} in [1.6,1.7,...,2.4]{} at (,) ; in [-1.3,-1.4,-1.5]{} in [1.7,1.8,...,2.5]{} at (,) ; in [-1.6,-1.7,-1.8]{} in [1.8,1.9,...,2.4]{} at (,) ; (-0.3,0.7) –(-0.3,0) node \[below\] [ ]{}; (-0.6,0.8) –(-0.6,0) node \[below\] [ ]{}; (-0.9,0.9) –(-0.9,0) node \[below\] [ ]{}; (-1.2,1.6) –(-1.2,0) node \[below\] [ ]{}; (-1.5,1.7) –(-1.5,0) node \[below\] [ ]{}; (-1.8,1.8) –(-1.8,0) node \[below\] [ ]{}; (-0.3,0.7) –(-2.8,0.7) node \[left\] ; (-0.6,0.8) –(-2.8,0.8); (-0.9,0.9) –(-2.8,0.9) node \[left\] [ ]{}; (-1.2,1.6) –(-2.8,1.6) node \[left\] [ ]{}; (-1.5,1.7) –(-2.8,1.7); (-1.8,1.8) –(-2.8,1.8) node \[left\] [ ]{}; --------- ----------------------- -- ----------------- $\circ$ Elements in $S_{i,j}$ Elements in $S$ --------- ----------------------- -- ----------------- We are now ready to prove Conjecture \[conjD\]. We have $g(T)=g(\cX)$ and in particular $H(P)=T.$ Proposition \[prop:contained\] implies that $g(T) \ge g(\cX)$. Hence the theorem follows once we show that $g(T) \le g(\cX)$. However, using the first two items of Lemma \[lem:sij\], we see that $$g(T) \le g(S)-\sum_{i=0}^{q-1}\sum_{j=1}^{q-1}|S_{i,j}|.$$ Using Lemma \[genusS\] and item three of Lemma \[lem:sij\] we obtain $$\begin{aligned} g(T) &\leq \frac{q^6-q^4}{2} -\sum_{i=0}^{q-1} \sum_{j=1}^{q-1} q(q^3-q+1-jq^2+i)\\ &= \frac{q^6-q^4}{2} -\sum_{i=0}^{q-1} \sum_{j=1}^{q-1} q(q^3-q+1) +\sum_{i=0}^{q-1} \sum_{j=1}^{q-1}jq^3-\sum_{i=0}^{q-1} \sum_{j=1}^{q-1}iq\\ &= \frac{q^6-q^4}{2}-q^2(q-1)(q^3-q+1)+\frac{q^5(q-1)}{2}-\frac{q^2(q-1)^2}{2}=\frac{q^5-2q^3+q^2}{2}=g(\cX).\end{aligned}$$ A direct consequence of the above theorem is that $H(P)=\left(\bigcup_{i,j}S_{i,j} \right) \cup S$. It is not hard to obtain more information about $H(P)$ from the above calculations. For example, it is clear that the multiplicity of $H(P)$ (i.e., the smallest positive element in $H(P)$) is equal to $q^3-q+1$, while its conductor (i.e., the largest gap) is $2g(\cX)-1$. This means in particular that like $H(P_\infty)$, the semigroup $H(P)$ is symmetric. Since $H(P_\infty)$ has multiplicity $q^3-q^2+q$, we also see that $H(P) \neq H(P_\infty).$ The Weierstrass semigroup $H(P)$ for $P \not\in \cX(\mathbb{F}_{q^6})$ ====================================================================== In this section we determine the Weierstrass semigroup $H(P)$ for $P \not\in \cX(\mathbb{F}_{q^6})$. In particular in this section $P=P_{(a,b,c)}$ will always denote a point on $\cX$ not in $\cX(\mathbb{F}_{q^6})$. For future reference, note that as in the previous section, this means that $c \neq 0$. As we will see, the semigroup $H(P)$ is the same for all $P \not\in \cX(\mathbb{F}_{q^6})$ and hence the ‘generic’ semigroup for a point on $\cX$. Our approach is use Corollary \[holom\] to construct gaps of $H(P)$ by computing the valuation at $P$ of functions $f \in L((2g(\cX)-2)P_\infty).$ It is very easy to find a basis of the Riemann–Roch space $L((2g(\cX)-2)P_\infty)$. For example the functions $x^iy^jz^k$ where $i \ge 0$, $0 \le j \le q$, $0 \le k \le q^2+q$ and $i(q^3+1)+j(q^3-q^2+q)+kq^3\le 2g(\cX)-2$ form a basis. However, this does not settle the matter, since these basis elements all will have valuation $0$ at $P$. Therefore an effort must be made to construct functions in $L((2g(\cX)-2)P_\infty)$ having distinct valuations at $P$. In the next subsection, we construct functions with various valuations at $P$. After that we will combine these functions and obtain a set $G$ of several explicitly described gaps of $H(P)$ using Corollary \[holom\]. The remainder of the section will then be a somewhat lengthy calculation showing that the set $G$ in fact contains $g(\cX)$, and hence all, gaps of $H(P)$. Construction of functions. -------------------------- We start by constructing a function $g_1$ with small, but positive, valuation at $P=P_{(a,b,c)}$. It will be convenient to define $\beta=b^{q^2}-b$. Note that $b^{q^2}-b=c^{q^2-q+1} \neq 0$, since $P \not \in \cX({\mathbb {F}_{q^6}})$ (and therefore a fortiori $P \not \in \cX({\mathbb {F}_{q^2}})$). We define $$g_1:=(\beta^{q^2-1}-1)\tilde x_P^q+\beta^{q^2+q}+\beta^{q}\left((\tilde y_P-\beta)(\tilde x_P+\beta^q(\tilde y_P-\beta))^{q-1}\right).$$ The functions $\tilde x_P$ and $\tilde y_P$ are as in equation . This definition may seen ad hoc, but it arises naturally when constructing functions of low pole order at $P_\infty$ and large vanishing order at $P$. More precisely, we have the following lemma. \[lem:g1\] The function $g_1$ is an element of $L((2g(\cX)-2)P_\infty)$. Moreover $v_{P_\infty}(g_1)\ge-q(q^3+1)$ and $v_P(g_1)=q^2+1$. It is clear that $g_1$ only can have a pole at $P_\infty$. Moreover, from equations and imply that $\tilde x_P$ (resp. $\tilde y_P$) has a pole at $P_\infty$ of order $q^3+1$ (resp. $q^3-q^2+q$). Therefore, the triangle inequality implies that $v_{P_\infty}(g_1) \ge v_{P_\infty}(\tilde x_P^q)=-q(q^3+1),$ which is what we want to show. From equation , we see that the function $\tilde y_P$ is a local parameter for the point $P=P_{(a,b,c)}$. The defining equation for $\mathcal H_q$ directly implies that $\tilde x_P^q+\tilde x_P=\beta \tilde y_P^q-\tilde y_P^{q+1}$. Hence we easily can obtain the power series development of $\tilde x_P$ in terms of $\tilde y_P$. More precisely, we obtain that $$\begin{aligned} \label{eq:powerx} \tilde x_P & = \beta \tilde y_P^q-\tilde y_P^{q+1}-\tilde x_P^q=\beta \tilde y_P^q-\tilde y_P^{q+1}-\beta^q \tilde y_P^{q^2}+\tilde y_P^{q^2+q}+\cdots \notag\\ & = (\tilde y_P-\beta)(-\tilde y_P^q+(\tilde y_P-\beta)^{q-1}\tilde y_P^{q^2})+\cdots\end{aligned}$$ Using this, we also obtain that $$\begin{aligned} \label{eq:powerw} (\tilde y_P-\beta)\left(\tilde x_P+\beta^q(\tilde y_P-\beta)\right)^{q-1}&=(\tilde y_P-\beta)\left((\tilde y_P-\beta)(-\tilde y_P^q+(\tilde y_P-\beta)^{q-1}\tilde y_P^{q^2}) + \beta^q(\tilde y_P-\beta) \right)^{q-1}+\cdots\notag\\ & = (\tilde y_P-\beta)^q\left( -(\tilde y_P-\beta)^q+(\tilde y_P-\beta)^{q-1}\tilde y_P^{q^2}\right)^{q-1}+\cdots\notag\\ &=(\tilde y_P-\beta)^{q^2-q+1}\left( -(\tilde y_P-\beta)+\tilde y_P^{q^2}\right)^{q-1}+\cdots\notag\\ &=(\tilde y_P-\beta)^{q^2}-(\tilde y_P-\beta)^{q^2-1}\tilde y_P^{q^2}+\cdots\notag\\ &=-\beta^{q^2}+(1-\beta^{q^2-1})\tilde y_P^{q^2}+\beta^{q^2-2}\tilde y_P^{q^2+1}+\cdots.\end{aligned}$$ Combining equations and , we see that $$\begin{aligned} g_1&=(\beta^{q^2-1}-1)\beta^q \tilde y_P^{q^2}+\beta^{q^2+q}+\beta^q(-\beta^{q^2}+(1-\beta^{q^2-1})\tilde y_P^{q^2}+\beta^{q^2-2}\tilde y_P^{q^2+1}) + \cdots\\ &=\beta^{q^2+q-2}\tilde y_P^{q^2+1}+\cdots\end{aligned}$$ This implies that $v_P(g_1)=q^2+1$, which is what we wanted to show. The next functions are inspired by the previous section in the sense that we again use the functions $\tilde x_P^{(k)}$ introduced in equation , but now for $P=P_{(a,b,c)} \not\in \cX({\mathbb {F}_{q^6}})$. For $s=1,\ldots,q-2$ we define $$h_s:=\left( \frac{\tilde x_P^q}{\tilde x_P^{(1)}} \right)^{s+1} \cdot \tilde x_P^{(2)}.$$ We have the following lemma about these functions. \[lem:hs\] Let $s=1,\dots,q-2$. The function $h_s$ is an element of $L((2g(\cX)-2)P_\infty)$. Moreover $v_{P_\infty}(h_s)=-(q(s+1)-s)(q^3+1)$ and $v_P(h_s)=(s+1)q^2$. Using equations and , we see that $v_{P_\infty}(h_s)=-(q(s+1)-s)(q^3+1)$ and that $h_s$ has no other poles. Further it is well known that $\mathcal H_q({\mathbb {F}_{q^2}})=\mathcal H_q(\mathbb{F}_{q^4}).$ Since any point in $\mathcal H_q({\mathbb {F}_{q^2}})$ ramifies totally in the cover $\cX \to \cH$, this means that also $\mathcal \cX({\mathbb {F}_{q^2}})=\mathcal \cX(\mathbb{F}_{q^4}).$ Therefore $v_P(\tilde x_P^{(2)})=0$, since $P \not \in \cX({\mathbb {F}_{q^6}})$. This implies that $$v_P(h_s)=(s+1)\left(qv_P(\tilde x_P)-v_P(\tilde x_P^{(1)})\right)=(s+1)q^2,$$ as claimed. Now we able to determine several gaps of $H(P)$. \[gaps\] Let $P \not\in \cX(\mathbb{F}_{q^6})$ be a point on $\cX$. Then $$\begin{gathered} G:=\{iq^3+j+kq+m(q^2+1)+\sum_{s=1}^{q-2} n_s ((s+1)q^2)+1 \mid i,j,k,m,n_1,\ldots,n_{q-2} \in \mathbb{Z}_{\geq 0}, \ \makebox{and}\\ i(q+1)+jq+k(q+1)+mq(q+1)+\sum_{s=1}^{q-2} n_s ((s+1)q-s)(q+1) \leq (q+1)(q^2-2)\},\notag\end{gathered}$$ is a set of gaps at $P$. Let $i,j,k,m,n_1,\ldots,n_{q-2}$ be nonnegative integers and write $f= \tilde z_P^i\tilde y_P^j\tilde x_P^k g_1^m \prod_{s=1}^{q-2} h_s^{n_s}$. Equations , , combined with Lemmas \[lem:g1\] and \[lem:hs\] imply that $f \in L((2g(\cX)-2)P_\infty)$ if $$i(q^3+1)+j(q^3-q^2+q)+k(q^3+1)+m(q^4+q)+\sum_{s=1}^{q-2} n_s ((s+1)q-s)(q^3+1) \leq q^5-2q^3+q^2-2,$$ which is equivalent to $$\label{eq:inquality1} i(q+1)+jq+k(q+1)+mq(q+1)+\sum_{s=1}^{q-2} n_s ((s+1)q-s)(q+1) \leq (q+1)(q^2-2).$$ On the other hand we have $$v_P(f)=iq^3+j+kq+m(q^2+1)+\sum_{s=1}^{q-2} n_s ((s+1)q^2).$$ Hence the claim follows from Lemma \[holom\]. \[obs:largestgapinG\] Inequality implies in particular that $i\leq q^2-2,j \leq q^2+q-3,k \leq q^2-2$, $m \leq q-1$ and $n_s \leq \lfloor (q+1)/(s+1)\rfloor$. This implies directly that the largest gap of $H(P)$ that is contained in $G$ is obtained by putting $i=q^2-2$ and all other remaining variables to $0$. In other words: the largest element in $G$ is $q^5-2q^3+1=2g(\cX)-q^2+1.$ \[obs:jsmall\] If $j\ge q$ and the tuple $(i,j,k,m,n_1,\dots,n_{q-2})$ satisfies inequality , then the tuple $(i,j-q,k+1,m,n_1,\dots,n_s)$ will also satisfy inequality . This implies that when calculating the set $G$, we may assume that $j \le q-1$. Moreover, inequality is equivalent to $$i+j+k+mq+\sum_{s=1}^{q-2} n_s ((s+1)q-s) \leq q^2-2+\frac{j}{q+1},$$ which for $j \le q-1$ is equivalent to $$\label{eq:inquality} i+j+k+mq+\sum_{s=1}^{q-2} n_s ((s+1)q-s) \leq q^2-2,$$ since all variables involved are integers. $|G|=g(\cX)$. ------------- We now prove that $G$ is exactly the set of gaps $G$ at $P=P_{(a,b,c)} \not\in \cX({\mathbb {F}_{q^6}})$, that is $|G|=g(\cX)$. Since we already know that $G$ contains gaps of $H(P)$, it is sufficient to show that $|G| \ge g(\cX)$. This will require a detailed study of the elements of $G$. To this end we consider the following map $$\varphi: \mathbb{Z}_{\geq 0}^{q+2} \rightarrow \mathbb{Z}_{\geq 0}, \quad {\rm with} \quad \varphi(i,j,k,m,n_1,\ldots,n_{q-2}) = iq^3+j+kq+m(q^2+1)+\sum_{s=1}^{q-2} n_s ((s+1)q^2)+1,$$ and consider the set $$\mathcal{G}=\{(i,j,k,m,n_1,\ldots,n_{q-2}) \in \mathbb{Z}_{\geq 0}^{q+2} \mid j \le q-1, \ \makebox{inequality \eqref{eq:inquality} holds}\}.$$ Then by Observation \[obs:jsmall\] we have $G=\varphi(\mathcal{G})$. The main difficulty is that $\varphi_{\big | \mathcal{G}}$, the restriction of the map $\varphi$ to $\mathcal{G}$, is not injective. This makes estimating the cardinality of $G$ somewhat tricky. We proceed by studying the image of $\varphi$ on the following three subsets of $\mathcal G$. $$\begin{aligned} \mathcal G_1&:=\{(i,0,k,m,0,\dots,0) \in \mathcal G\},\\ \mathcal G_2&:=\{(i,j,k,m,0,\dots,0) \in \mathcal G \mid 1 \le j \le q-1, k \le q-1, j+m \le q-1\}\\ \mathcal G_3&:=\{(i,j,k,0,\dots,0,n_s,0,\dots,0) \in \mathcal G \mid k \le q-1,1 \le s \le q-2,n_s=1,i+k+(s+1)q \ge q^2-1\}.\end{aligned}$$ Further, we write $G_1=\varphi(\mathcal G_1)$, $G_2=\varphi(\mathcal G_2)$ and $G_3=\varphi(\mathcal G_3)$. We will show that these sets are mutually disjoint and that their cardinalities add up to $|G|$ in a series of lemmas. \[lem:G1\] Let $\mathcal G_1$ and $G_1=\varphi(\mathcal{G}_1)$ be as above. Then $\varphi$ restricted to $\mathcal G_1$ is injective and $$|G_1|=\frac12 q^2(q-1)\left( \frac13 q^2+\frac56 q + \frac12\right).$$ If $(i,0,k,m,0,\dots,0) \in \mathcal G_1$, then $\varphi(i,0,k,m,0,\dots,0)=iq^3+kq+m(q^2+1)+1$ and by inequality $i+k+mq \le q^2-2.$ This implies in particular that $$0 \le m \le q-1 \ \makebox{and} \ 0 \le kq+m(q^2+1) \le (k+mq)q+q-1 \le (q^2-2)q+q-1 < q^3.$$ Now suppose $(i_1,0,k_1,m_1,0,\dots,0),(i_2,0,k_2,m_2,0,\dots,0) \in \mathcal G_1$ and $$i_1q^3+k_1q+m_1(q^2+1)=i_2q^3+k_2q+m_2(q^2+1).$$ Calculating modulo $q$ and using that $0 \le m_1 \le q-1$ and $0 \le m_2 \le q-1$ (see Observation \[obs:jsmall\]), we see that $m_1=m_2$. Further, since $0 \le k_1q+m_1(q^2+1)<q^3$ and $0 \le k_2q+m_2(q^2+1)<q^3$, we see that $k_1q+m_1(q^2+1)=k_2q+m_2(q^2+1)$ and $i_1q^3=i_2q^3$. Combining these equalities, we see that $(i_1,0,k_1,m_1,0,\dots,0)=(i_2,0,k_2,m_2,0,\dots,0)$, which is what we wanted to show. Now we compute $|G_1|.$ First of all, from the above we see that $|G_1|=|\mathcal G_1|$. Further we have $$\begin{aligned} |\mathcal G_1|&= \sum_{m=0}^{q-1}\sum_{i=0}^{q^2-2-mq}\sum_{k=0}^{q^2-2-mq-i} 1=\sum_{m=0}^{q-1}\sum_{i=0}^{q^2-2-mq} (q^2-1-mq-i)\\ & = \sum_{m=0}^{q-1}\frac{(q^2-1-mq)(q^2-mq)}{2} = \frac{(q^2-1)q^3}{2}+\sum_{m=0}^{q-1}\frac{-2q^3-q^2+q}{2} m+\binom{m+1}{2}q^2\\ & = \frac{(q^2-1)q^3}{2}+\frac{-2q^3-q^2+q}{2}\binom{q}{2}+\binom{q+1}{3}q^2.\end{aligned}$$ In the last equality we used *summation on the upper index* to evaluate the summation $\sum_m\binom{m+1}{2}$; see [@GKP Eqn. (5.10)]. The desired equality for $|G_1|$ now follows. \[lem:G2\] Let $\mathcal G_2$ and $G_2=\varphi(\mathcal{G}_2)$ be as above. Then $\varphi$ restricted to $\mathcal G_2$ is injective and $$|G_2|=\frac12 q^2(q-1)\left( \frac23 q^2-\frac16 q - \frac56\right).$$ If $(i,j,k,m,0,\dots,0) \in \mathcal G_2$, then $\varphi(i,j,k,m,0,\dots,0)=iq^3+j+kq+m(q^2+1)+1$ and by definition we have $1 \le j \le q-1$, $1 \le j+m \le q-1$ and $0 \le k \le q-1$. Moreover, inequality gives that $i+j+k+mq \le q^2-2.$ Similarly as in the previous lemma, we obtain that $$0 \le m \le q-1 \ \makebox{and} \ 0 \le j+kq+m(q^2+1) \le (k+mq)q+q-1 \le (q^2-2)q+q-1 < q^3.$$ Now suppose $(i_1,j_1,k_1,m_1,0,\dots,0),(i_2,j_2,k_2,m_2,0,\dots,0) \in \mathcal G_2$ and $$i_1q^3+j_1+k_1q+m_1(q^2+1)=i_2q^3+j_2+k_2q+m_2(q^2+1).$$ Reasoning exactly as in the previous lemma, we obtain that $j_1+m_1=j_2+m_2$, $j_1+k_1q+m_1(q^2+1)=j_2+k_2q+m_2(q^2+1)$ and $i_1=i_2$. Combining the first two equations, we deduce that $k_1q+m_1q^2=k_2q+m_2q^2$. Since $0\le k_1 \le q-1$ and $0 \le k_2 \le q-1$, we see $k_1=k_2$, which now implies that $(i_1,j_1,k_1,m_1,0,\dots,0)=(i_2,j_2,k_2,m_2,0,\dots,0).$ Now we compute $|G_2|$. First note that $k\le q-1$, but for a given $j$ and $m$, we also have $k \le q^2-2-j-mq$. However, since $j \ge 1$ and $0 \le j+m \le q-1$, we see that $m \le q-2$. Hence $q^2-2-j-mq \ge q^2-2-1-(q-2)q \ge q-1,$ implying that the condition $k \le q^2-2-j-mq$ is trivially satisfied. Hence $$\begin{aligned} |\mathcal G_2|&= \sum_{j=1}^{q-1}\sum_{m=0}^{q-1-j}\sum_{k=0}^{q-1} \sum_{i=0}^{q^2-2-j-k-mq}1=\sum_{j=1}^{q-1}\sum_{m=0}^{q-1-j}\sum_{k=0}^{q-1} (q^2-1-j-k-mq)\\ & = \sum_{j=1}^{q-1}\sum_{m=0}^{q-1-j} (q^2-1-j-mq)q-\binom{q}{2}=\sum_{j=1}^{q-1}\left((q^2-1-j)q-\binom{q}{2}\right)(q-j)-q^2\binom{q-j}{2}\\ & = \sum_{j=1}^{q-1}\left((q^2-q)q-\binom{q}{2}\right)(q-j)-(q^2-2q)\binom{q-j}{2} =\left((q^2-q)q-\binom{q}{2}\right)\binom{q}{2}-(q^2-2q)\binom{q}{3}.\end{aligned}$$ The desired equality now follows. \[lem:G3\] Let $\mathcal G_3$ and $G_3=\varphi(\mathcal{G}_3)$ be as above. Then $\varphi$ restricted to $\mathcal G_3$ is injective and $$|G_3|=\frac12 q^2(q-1)\left( \frac13 q - \frac23\right).$$ If $(i,j,k,0,0,\dots,0,n_s,0,\dots,0) \in \mathcal G_3$, then $\varphi(i,j,k,0,0,\dots,0,n_s,0,\dots,0)=iq^3+j+kq+(s+1)q^2+1$ and by definition we have $n_s=1$, $1 \le s \le q-2$, $0 \le j \le q-1$, $0 \le k \le q-1$ and $i+k+(s+1)q \ge q^2-1$ (that is $i+k+sq \ge q^2-q-1$). Moreover, inequality gives that $i+j+k+s(q-1) \le q^2-q-2.$ Note that the inequalities $i+k+sq \ge q^2-q-1$ and $i+j+k+s(q-1) \le q^2-q-2$ only can be satisfied simultaneously, if $j \le s-1$, so we may assume this as well in the remainder of the proof. Now suppose $(i_1,j_1,k_1,0,0,\dots,0,n_s,0,\dots,0),(i_1,j_1,k,0,0,\dots,0,1,0,\dots,0) \in \mathcal G_3$ and $$i_1q^3+j_1+k_1q+(s_1+1)q^2=i_2q^3+j_2+k_2q+(s_2+1)q^2.$$ Since the $q$-ary expansion of a number is unique, we immediately obtain that $j_1=j_2$, $k_1=k_2$ and $s_1=s_2$, since all variables involved at between $0$ and $q-1$. Hence $i_1=i_2$ as well and the first part of the lemma follows. Now we compute $|G_3|$. Recall that we may assume $j \le s-1$. Hence $$\begin{aligned} |\mathcal G_3|&= \sum_{s=1}^{q-2}\sum_{j=0}^{s-1}\sum_{k=0}^{q-1}\sum_{i=q^2-q-1-k-sq}^{q^2-q-2-j-k-s(q-1)}1= \sum_{s=1}^{q-2}\sum_{j=0}^{s-1}\sum_{k=0}^{q-1}(s-j)\\ & = q\sum_{s=1}^{q-2}\sum_{j=0}^{s-1}(s-j)=q\sum_{s=1}^{q-2}\binom{s+1}{2}=q\binom{q}{3}.\end{aligned}$$ The desired equality now follows. Finally to obtain an estimate for $|G|$, we need to study the intersections of the sets $G_1$, $G_2$ and $G_3$. It turns out that they are disjoint, as we will now show. \[G1G2G3disjoint\] The sets $G_1$, $G_2$ and $G_3$ defined above are mutually disjoint. [**Part 1. $G_1 \cap G_2 = \emptyset$.**]{} Let $(i_1,0,k_1,m_1,0,\dots,0) \in \mathcal G_1$, $(i_2,j_2,k_2,m_2,0,\dots,0) \in \mathcal G_2$ and suppose that $$i_1q^3+k_1q+m_1(q^2+1)=i_2q^3+j_2+k_2q+m_2(q^2+1).$$ Since $0 \le m_1 \le q-1$ and $1 \le j_2+m_2 \le q-1$, we see that $m_1=j_2+m_2$ and hence that $i_1q^2+k_1+m_1q=i_2q^2+k_2+m_2q.$ Note that $m_1-m_2=j_2 \ge 0$, where the inequality follows from the definition of $\mathcal G_2$. Inequality implies that $k_1+m_1q < q^2$ as well as $k_2+m_2q < q^2$. Hence we obtain $i_1=i_2$ and $k_1+m_1q=k_2+m_2q$, whence $(m_1-m_2)q=k_2-k_1$. This implies that $k_1 \equiv k_2 \pmod{q}$, but since $k_1 \ge 0$ and $0 \le k_2 \le q-1$ we can deduce $k_1-k_2 \ge 0$. On the other hand we already have seen that $m_1-m_2=j_2 \ge 1$, but then we arrive at a contradiction, since $0<(m_1-m_2)q=k_2-k_1 \le 0$. [**Part 2. $G_1 \cap G_3 = \emptyset$.**]{} Let $(i_1,0,k_1,m_1,0,\dots,0) \in \mathcal G_1$, $(i_3,j_3,k_3,0,0,\dots,0,1,0,\dots,0) \in \mathcal G_3$ and suppose that $$i_1q^3+k_1q+m_1(q^2+1)=i_3q^3+j_3+k_3q+(s+1)q^2.$$ Similarly as in part 1 above, we obtain that $m_1=j_3$, whence $i_1q^2+k_1+m_1q=i_3q^2+k_3+(s+1)q$, as well as the inequality $k_1+m_1q<q^2$. However, since $k_3 \le q-1$ and $s+1 \le q-1$, we also have $k_3+(s+1)q<q^3$. Therefore we obtain that $i_1=i_3$ as well as $k_1+m_1q=k_3+(s+1)q$. This implies that $$i_3+k_3+(s+1)q=i_1+k_1+m_1q \le q^2-2,$$ where we have used inequality to obtain the inequality. On the other hand $i_3+k_3+(s+1)q \ge q^2-1$ by the definition of $\mathcal G_3$ and we arrive at a contradiction. [**Part 3. $G_2 \cap G_2 = \emptyset$.**]{} Let $(i_2,j_2,k_2,m_2,0,\dots,0) \in \mathcal G_2$, $(i_3,j_3,k_3,0,0,\dots,0,1,0,\dots,0) \in \mathcal G_3$ and suppose that $$i_2q^3+j_2+k_2q+m_2(q^2+1)=i_3q^3+j_3+k_3q+(s+1)q^2.$$ Reasoning very similarly as in Part 1 and Part 2, we obtain $j_2+m_2=j_3$, $i_2=i_3$ and $$i_3+k_3+(s+1)q=i_2+k_2+m_2q \le q^2-2.$$ Again we arrive at a constriction. We are now ready to prove the main theorem of this section. Let $P$ be a point of $\cX$ with $P\not\in \cX(\mathbb{F}_{q^6})$. Then the set of gaps of $H(P)$ is given by, $$G=\{iq^3+kq+m(q^2+1)+\sum_{s=1}^{q-2} n_s ((s+1)q^2)+j+1 \mid i,j,k,m,n_1,\ldots,n_{q-2} \in \mathbb{Z}_{\geq 0}, j \le q-1,\ \makebox{and}$$ $$i+j+k+mq+\sum_{s=1}^{q-2} n_s ((s+1)q-s) \leq q^2-2\}.$$ Moreover, the set of Weierstrass points $W$ on $\cX$ coincides with $\cX(\mathbb{F}_{q^6})$. Combing Lemmas \[lem:G1\], \[lem:G2\], \[lem:G3\], and \[G1G2G3disjoint\] we see that $$|G| \ge |G_1|+|G_2|+|G_3|=\frac12 q^2(q-1)(q^2+q-1)=g(\cX).$$ Since we know that $H(P)$ has exactly $g(\cX)$ gaps, Proposition \[gaps\] then implies that $H(P)=\mathbb{N} \setminus G$. From Observation \[obs:largestgapinG\], we deduce that the largest gap in $H(P)$ is $2g(\cX)-q^2+1$, while we already know that for any $P \in \cX({\mathbb {F}_{q^6}})$, the largest gap is $2g(\cX)-1$. This implies the last statement in the theorem. The proof also shows that the gaps of $H(P)$ are precisely $G_1 \cup G_2 \cup G_3$, which is convenient when checking if a particular number is a gap or not. For example, this allows us to compute the multiplicity (smallest positive element) of $H(P)$ fairly easily. Let $P$ be a point of $\cX$ with $P\not\in \cX(\mathbb{F}_{q^6})$. The multiplicity of $H(P)$ is equal to $q^3-1$. From Stöhr-Voloch Theory we know that $q^3-1$ and $q^3$ are non-gaps at $P$, since $P$ is not a Weierstrass point; see [@HKT Proposition 10.9]. It is also not difficult to verify this directly. On the other hand, let $1 \le a \le q^3-2$ be an integer and write $a-1=c_0+c_1q+c_2q^2$ with $0\le c_t \le q-1$ for $t=1,2,3$. Then we distinguish three cases. [**Case 1. $c_2 \ge c_0$ and $(c_1,c_2) \neq (q-1,q-1)$.**]{} In this case a direct verification shows that $a=\varphi(0,0,c_1+(c_2-c_0)q,c_0,0\dots,0)$ and that $(0,0,c_1+(c_2-c_0)q,c_0,0\dots,0) \in \mathcal G_1$. [**Case 2. $c_2<c_0$.**]{} We have $a=\varphi(0,c_0-c_2,c_1,c_2,0\dots,0)$ and $(0,c_0-c_2,c_1,c_2,0\dots,0) \in \mathcal G_2$ in this case. [**Case 3. $(c_1,c_2) = (q-1,q-1)$**]{} Note that in this case $c_0 \le q-3,$ since $a-1=c_0+(q-1)q+(q-1)q^2 \le q^3-3$. One then checks that $a=\varphi(0,c_0,q-1,0,0,\dots,0,1)$ and that $(0,c_0,q-1,0,0,\dots,0,1) \in \mathcal G_3$. At this point seems to be reasonable to ask for the generators of the Weierstrass semigroup $H(P)$ for $P \not\in \cX(\mathbb{F}_{q^6})$. Their explicit determination seems to be a challenging task as the following examples show. In particular the number of generators of $H(P)$ seems to grow quickly with respect to $q$. - If $q=2$ then $g=10$ and $$G=\{1,2,3,4,5,6,9,10,11,17\}.$$ Clearly $7$ and $8$ must be generators of $H(P)$ and since $12 \not\in \langle 7,8\rangle$ and $13 \not\in \langle 7,8,12 \rangle$ we obtain that also $12$ and $13$ are generators. Note that $\langle 7,8,12,13\rangle \cap \{0, \ldots,20\}=\{7,8,12,13,14,15,16\}$ and hence also $18$ is a generator. In fact $$H(P)=\langle 7,8,12,13,18\rangle.$$ Moreover, if $P \in \cX$ then $$H(P)=\begin{cases} \{{ 0, 6, 8, 9, 12, 14, 15, 16, 17, 18,20,\ldots }\}, \ \makebox{if} \ P \in \cX(\mathbb{F}_{4}), \\ \{ 0, 7, 8, 9, 13, 14, 15, 16, 17, 18,20,\ldots\}, \ \makebox{if} \ P \in \cX(\mathbb{F}_{64}) \setminus \cX(\mathbb{F}_4), \\ \{0,7,8,12,13,14,15,16,18,19,20 \ldots\}, \ \makebox{otherwise}. \end{cases}$$ - If $q=3$ then $g=99$ and $$G=\{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 28, 29, 30, 31, 32, 33, 34,$$ $$35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69,$$ $$70, 71, 73, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 109, 110, 111, 112, 113, 114, 115, 116, 118, 119,$$ $$136, 137, 138, 139, 140, 142, 163, 164, 166, 190 \}.$$ Arguing as for the previous case, one can prove that $$H(P)=\langle 26,27,50,51,72,74,75,96,97,117,120,121,141,145,165\rangle.$$ It is unclear what the number of generators for general $q$ is. For $q=4$ the semigroup turns out to have $28$ generators. Collecting the results in the paper, we have proven Theorem \[mainth\] and Corollary \[mainth2\] from the introduction. We finish by summing up some further facts on the various semigroups on $\cX$ in a table, leaving a question mark for the minimal number of generators in the case $P \not\in \cX({\mathbb {F}_{q^6}})$. Determining this number could be interesting future work.\ $P$ multiplicity conductor number of generators -------------------------------------------------------------------- -------------- ----------------- ---------------------- $P \in \cX({\mathbb {F}_{q^2}})$ $q^3-q^2+q$ $2g(\cX)-1$ $3$ $P \in \cX({\mathbb {F}_{q^6}})\setminus \cX({\mathbb {F}_{q^2}})$ $q^3-q+1$ $2g(\cX)-1$ $q+2$ $P \not\in \cX({\mathbb {F}_{q^6}})$ $q^3-1$ $2g(\cX)-q^2+1$ ? Acknowledgments {#acknowledgments .unnumbered} =============== The first author gratefully acknowledges the support from The Danish Council for Independent Research (Grant No. DFF–4002-00367). The second author would like to thank the Italian Ministry MIUR, Strutture Geometriche, Combinatoria e loro Applicazioni, Prin 2012 prot. 2012XZE22K and GNSAGA of the Italian INDAM. [99]{} I. Duursma, [*Two-Point Coordinate Rings for GK-Curves*]{}, IEEE Trans. Inf. Theory [**57**]{}(2), 593-600 (2011). S. Fanali and M. Giulietti, [*One-Point AG Codes on the GK Maximal Curves*]{}, IEEE Trans. Inf. Theory [**56**]{}(1), 202–210 (2010). A. Garcia and P. Viana, Weierstrass points on certain nonclassical curves, Arch. Math. [**46**]{}(4), 315-322 (1986). M. Giulietti and G. Korchmáros, [*A new family of maximal curves over a finite field*]{}, Math. Ann. [**343**]{}, 229–245 (2009). R. Graham, D. E. Knuth, O. Patashnik: *Concrete mathematics. A foundation for computer science.* Second edition. Addison-Wesley Publishing Company, Reading, MA, (1994), xiv+657 pp. J.W.P. Hirschfeld, G. Korchmáros and F. Torres, *Algebraic Curves over a Finite Field,* *Princeton Series in Applied Mathematics*, Princeton, (2008). T. Høholdt, J. Van Lint, R. Pellikaan, [*Algebraic geometry codes*]{}, in: V.S. Pless, W.C. Huffman (Eds.), Handbook of Coding Theory, North-Holland, 871-961 (1998). C. Kirfel and R. Pellikaan, [*The minimum distance of codes in an array coming from telescopic semigroups*]{}, IEEE Trans. Inf. Theory [**41**]{}, 1720-1732 (1995). G. Lachaud, *Sommes d’Eisenstein et nombre de points de certaines courbes algébriques sur les corps finis*, C.R. Acad. Sci. Paris **305**(Série I) 729-732 (1987). H. Stichtenoth, *Algebraic function fields and codes*, Springer, 2009. K.O. Stöhr, J.F. Voloch: *Weierstrass points and curves over finite fields*, Proc. London Math. Soc. **52**(3), 1-19 (1986). M.A. Tsfasman, G. Vladut, *Algebraic-geometric Codes*, Kluwer, Dordrecht, (1991). G. D. Villa Salvador, *Topics in the theory of algebraic function fields*, Mathematics: Theory & Applications. Birkhäuser Boston, Inc., Boston, MA, (2006). Peter Beelen Technical University of Denmark,\ Department of Applied Mathematics and Computer Science,\ Matematiktorvet 303B,\ 2800 Kgs. Lyngby,\ Denmark,\ pabe@dtu.dk\ Maria Montanucci Universita’ degli Studi della Basilicata,\ Dipartimento di Matematica, Informatica ed Economia,\ Campus di Macchia Romana,\ Viale dell’ Ateneo Lucano 10,\ 85100 Potenza,\ Italy,\ maria.montanucci@unibas.it [^1]: [*Math. Subj. Class.:*]{} Primary: 11G20.   Secondary: 11R58, 14H05, 14H55. [^2]: [*Keywords:*]{} Giulietti–Korchmáros maximal curve, Weierstrass semigroup, Weierstrass points.

--- abstract: 'We report on the successful science verification phase of a new observing mode at the Keck interferometer, which provides a line-spread function width and sampling of 150 km/s at $K''$-band, at a current limiting magnitude of $K''\,\sim\,7$ mag with spatial resolution of $\lambda\,/\,2\,B\,\approx\,2.7\,{\rm mas}$ and a measured differential phase stability of unprecedented precision (3 mrad at $K\,=\,5\,{\rm mag}$, which represents 3 $\mu$as on sky or a centroiding precision of $10^{-3}$). The scientific potential of this mode is demonstrated by the presented observations of the circumstellar disk of the evolved Be-star 48 Lib. In addition to indirect methods such as multi-wavelength spectroscopy and polaritmetry, the here described spectro-interferometric astrometry provides a new tool to directly constrain the radial density structure in the disk. We resolve for the first time several Pfund emission lines, in addition to [$Br\,\gamma$ ]{}, in a single interferometric spectrum, and with adequate spatial and spectral resolution and precision to analyze the radial disk structure in 48 Lib. The data suggest that the continuum and $Pf$-emission originates in significantly more compact regions, inside of the [$Br\,\gamma$ ]{}emission zone. Thus, spectro-interferometric astrometry opens the opportunity to directly connect the different observed line profiles of [$Br\,\gamma$ ]{}and Pfund in the total and correlated flux to different disk radii. The gravitational potential of a rotationally flattened Be star is expected to induce a one-armed density perturbation in the circumstellar disk. Such a slowly rotating disk oscillation has been used to explain the well known periodic V/R spectral profile variability in these stars, as well as the observed V/R cycle phase shifts between different disk emission lines. The differential line properties and linear constraints set by our data are consistent with theoretical models and lend direct support to the existence of a radius-dependent disk density perturbation. The data also shows decreasing gas rotation velocities at increasing stello-centric radii as expected for Keplerian disk rotation, assumed by those models.' author: - 'J.-U. Pott, J. Woillez, S. Ragland, P. L. Wizinowich, J. A. Eisner, J. D. Monnier, R. L. Akeson, A. M. Ghez, J. R. Graham, L. A. Hillenbrand, R. Millan-Gabet, E. Appleby, B. Berkey, M. M. Colavita, A. Cooper, C. Felizardo, J. Herstein, M. Hrynevych, D. Medeiros, D. Morrison, T. Panteleeva, B. Smith, K. Summers, K. Tsubota, C. Tyau, E. Wetherell' title: 'Probing local density inhomogeneities in the circumstellar disk of a Be star using the new spectro-astrometry mode at the Keck interferometer' --- Introduction {#sec:1} ============ The ASTRA (ASTrometric and phase-Referencing Astronomy) upgrade program, funded by the NSF Major Research Instrumentation program, aims at extending the sensitivity and spectral resolution of the Keck interferometer (KI) through phase referencing and to implement a narrow-angle astrometry mode to enable a much broader scientific use of the KI . In this paper we report on results of the successful implementation of the first ASTRA mode, the self phase referencing (SPR) mode. In the SPR mode, 55 % of the light from the on-axis science target is used to provide fringe tracking (i.e., phase referencing) information while 20 % is used to make the science measurement. Due to the stabilized fringes, $>$100x longer integrations can be taken on the second camera. Therefore, it is possible in SPR to increase the spectral resolution of the interferometric measurement by at least an order of magnitude. SPR as currently implemented and provided to the general user, offers a spectral resolution $R\,(\,=\,\lambda/\Delta \lambda)$ of 2000, which is Nyquist sampled at R = 1000 (330 pixels across the K’-band) at K $\lesssim\,7\,{\rm mag}$. Further technical aspects and results from the commissioning are discussed by @2010Woi. Besides simple amplitude spectro-interferometry at this resolution, the here presented data demonstrate that the differential visibility phase can be retrieved at a precision of at least 3 mrad for a bright star. Non-linear phase changes in the source spectrum can be measured, and typically indicate translations of the photo-center over the respective spectral channels on the sky . The operational readiness of KI-SPR was successfully demonstrated with the here discussed observations of on the night of April 25, 2008 (UT). In addition, several young stellar objects were observed successfully [@2010Eis]. We show at the example of 48 Lib the wealth of spatial and spectral information that can be provided by SPR data. 48 Lib is a well studied classical Be-star with circumstellar shell-emission lines. Previous optical-nearinfrared long-baseline interferometric (OLBI) measurements of Be-star-shells demonstrated that 3d-models for H-line and continuum emission often overpredict the true size, which demonstrates the need for direct OLBI measurements . Narrow-band observations with the Mark III interferometer , resulted in Be-star disk diameter estimates ranging in 2.6 to 4.5 mas [@1997ApJ...479..477Q], well suited for the angular resolution of the 85 m baseline of the KI ($\lambda/2B\,=\,2.7~{\rm mas}$). We chose 48 Lib as science demonstration target because recent detection of very narrow satellite absorption features in Fe II shell lines reliably indicate very high inclination angles . This ensures strong features in the differential visibility and phase signals across the emission lines. A large fraction of Be stars show cyclic variations in the ratio between the violet and red flux of the HI emission lines (V/R), an enigmatic phenomenon studied since almost a century [@1925PA.....33..537C]. Early measurements with the GI2T interferometer spatially and timely resolved the $H\,\alpha$ emission of $\gamma$-Cas, showing spatial variation of the line emission region with time @1989Natur.342..520M. The fast rotation of Be-stars [near-critical rotational velocities for cooler stars like 48 Lib, @2005ApJ...634..585C] results in a flattening of the stellar shape, which can be observationally confirmed by OLBI . While the rapid rotation might play a role in the ejection of the circumstellar envelope, the quadrupole moment of the gravitational potential of a flattened Be star is expected to create a prograde one-armed spiral density pattern, slowly precessing in the disk and creating the V/R cycles . The latter authors calculate that the radial extension of such a disk oscillation mode should be confined to a few stellar radii, which can be probed by interferometry. Central properties of this theory have been observed indirectly by the continuous observation of individual line profiles , and the detection of phase-shifts between different emission line cycles pointing towards a spiral density perturbation [@2007ApJ...656L..21W]. Spectro-interferometric observations of the photocenter shift in hydrogen lines of $\zeta$ Tau appear to directly confirm a one-armed density mode in the disk . After summarizing the observation and data reduction (Sect. \[sec:2\]), we will discuss in Sect. \[sec:3\] that our data show that the radial dependence of the hydrogen emission regions can be observed [*directly*]{} with KI-SPR, thanks to resolving several emission lines of the Pfund series together with [$Br\,\gamma$ ]{}at unprecedented differential precision in the $K$-band. Concluding remarks are given in Sect. \[sec:5\]. Observations {#sec:2} ============ [ccccc]{} Name & $V/H/K$$\,^{(a)}$ & UT & BL$_{\rm proj}$& Stellar diam\ & & & & \[mas\] $\,^{(b)}$\ (Tar, B4III$\,^{(c)}$) & 4.9/4.8/4.6 & 13:20:49 & 66.6 m, 39$^\circ$ EoN & $0.23\,\pm\,0.1$\ (Cal, A4V) & 5.4/5.3/5.1 & 13:28:17 & & $0.25\,\pm\,0.04$\ Details of the observations are given in Table \[tab:11\]. Each dataset consists of 155 frames of 0.5 sec integration time, taken on the second fringe camera. The frames were selected to have an absolute group delay of less then 4.5 $\mu$m to ensure high SNR and limited impact of the dispersion introduced by the air-filled delay lines [see Appendix C in @1999ApJ...510..505C]. The remaining 137 (151) frames of target (calibrator) were corrected for group delay (linear) and dispersion (quadratic) wavenumber slopes. The resulting pre-processed frames were stacked by calculating the mean in each wavelength bin, and the standard deviation of this mean is used throughout the paper as error. This statistical error is adequately describing the differential uncertainty of the data. The total flux data were flux normalized before the averaging. This is necessary to avoid that the larger absolute flux variation due to the spatial filtering of the single-mode fibre fed fringe camera dominates the here interesting differential flux variation between adjacent lambda bins. The resulting measurables of the SPR observation (flux, $V^2$, and [$d\phi$ ]{}) are shown in Fig. \[fig:1\]. To achieve the rest wavelength calibration shown in the figure, two steps were performed. First, the observed wavelength calibration needs to be rectified. The slope is measured internally by a Fourier transform based technique. In standard KI observing setup, this slope is applied automatically to the data provided through the NExScI archive. Furthermore, we used telluric absorption lines in the raw spectra to estimate an offset in the wavelength calibration (typically of order of a few tenths of one spectral pixel), and to verify the archived rectification. The fact that our data do not show the full $K'$ band is due to an alignment problem of the early instrument setup used. For current performance, we refer to the KI support webpage [^1]. In a second step, the corrections for barycentric motion of the earth (-13 [${\rm km\,s}^{-1}$ ]{}) and peculiar motion of the star [-6 [${\rm km\,s}^{-1}$ ]{}, @1967IAUS...30...57E] have been applied. Note for [$Br\,\gamma$ ]{}, that the offset between zero [$d\phi$ ]{}and rest wavelength is significantly larger than the accuracy of our wavelength calibration. It is attributed to observing a profile which is significantly asymmetric at the level of the used spectral resolution. The quality of the resulting wavelength calibration is shown by overplotting the HI-recombination line centers in Fig. \[fig:1\] (dotted lines). It is reassuring, that (1) the flux line profiles are centered, that (2) the various Pfund lines show the same line profile, and most importantly, that (3) the [$d\phi$ ]{}of both [$Br\,\gamma$ ]{}and Pfund lines crosses zero close to the transition wavelengths, and with the [*same*]{} sign of the slope. (3) is expected for an inclined Keplerian disk. The target data were calibrated with a subsequently measured unresolved star, observed only 8 min later under equal atmospheric conditions. We applied simple calibration strategies: the raw target flux was divided by the calibrator, and multiplied by a measured spectral template from the IRTF spectral library (A3V, Vacca, personal comm.) containing the HI absorption features of the calibrator of similar spectral type. The spectral template provides a relative flux precision of 1 %, and was folded to $R\,=\,1000$, and divided by a black body continuum fit to the line free regions before application to our data. The resultant spectrum (upper panel in Fig. \[fig:1\]) thus shows in each spectral bin the flux with respect to the calibrator. The $V^2$ was corrected for the system visibility, estimated by observing an unresolved calibrator. We did not apply any flux bias correction in our data calibration. Although the primary fringe tracker at KI has a flux bias of a few percent per magnitude [^2] internal tests have demonstrated that the detector used for these science observations does not display such a bias. The differential phase is calibrated by subtraction of the calibrator [$d\phi$ ]{}. To show the impressive differential stability of the data over the used band, we overplotted in red linear continua, fitted to the line-free regions ($\lambda\,\le\,2.15\,\mu$m and $2.18\,\le\,\lambda\,\le\,2.315\,\mu$m), over flux ratio and $V^2$. The slightly red slope of the linear flux ratio continuum probably originates in the shell-induced NIR excess of 48 Lib, which photosphere is hotter than the calibrator of spectral type A4V. The continuum trend in the visibility is due to resolving the innermost circumstellar gas, and is discussed in Sect. \[sec:31\]. The red line in the [$d\phi$ ]{}panel marks zero phase, and is not a fit. The resulting pixel-to-pixel errors are $\Delta {\rm flux ratio}\,=\,0.003$, $\Delta\,V^2\,=\,0.007$, and $\Delta\,d\phi\,=\,3\,{\rm mrad}$. The higher relative precision of the [$d\phi$ ]{}signal with respect to the precision of the visibility is expected for such spectro-interferometric measurements [@2006dies.conf..291M]. This differential stability at the level of a few $10^{-3}$ is unprecedented for spectrally dispersed OLBI data at the spectral resolution and sensitivity offered by KI-SPR. It is the key to derive the scientific results presented in the next section. A more comprehensive description of the SPR mode and its sensitivity, performance, and technology will be given elsewhere [@2010Woi]. Results {#sec:3} ======= ![[*Top:*]{} Mean calibrated flux ratio between the 48 Lib and the continuum divided calibrator. The red solid line marks a linear continuum fit. The different line profiles of [$Br\,\gamma$ ]{}and $Pf$-emission lines are clearly visible [*Center:*]{} Mean calibrated $V^2$ of 48 Lib showing that both the NIR continuum and the recombination line emission are spatially resolved by the interferometer. A linear continuum (red line) was fitted to the line free regions. [*Bottom:*]{} The calibrated differential phase data. The red line marks zero phase. All plots show the rest wavelength of the target. The vertical dotted lines indicate the rest wavelength of the recombination lines. It is apparent that all lines show the same slope at the line center, as expected for disk emission. \[fig:1\]](plot_48lib_v2_frcont_lin.eps) [lccc]{} $K'_{\rm cont}$ (Gauss-disk only): &$\theta_{\rm cont}^{\rm FWHM}$& 0.94$\pm$0.03 mas&\ $K'_{\rm cont}$ (Gauss-disk + star): &$\theta_{\rm cont,disk}^{\rm FWHM}$& 1.65$\pm$0.05 mas&\ $Br\,\gamma$ & $V_{(-250\,,\,-100\,{\rm km\,s^{-1})}}$ & $R_{(100\,,\,250\,{\rm km\,s^{-1})}}$& $V\,/\,R$\ $F_{\rm line}$ (norm.) & $0.29 \pm 0.01$ & $0.16 \pm 0.01$ & $1.80 \pm 0.2$\ $F_{\rm line}^{\rm corr}$ (norm.) & $0.23 \pm 0.01$ & $0.14 \pm 0.01$ & $1.66 \pm 0.2$\ $\theta_{\rm line}^{\rm FWHM}$ (mas) & $1.7 \pm 0.2$ & $1.4 \pm 0.2$ & $1.2 \pm 0.2$\ [[*PC* ]{}$_{\rm line}^{\rm shift}$ ]{}(mas) & $2.2 \pm 0.2$ & $1.9 \pm 0.2$ & $1.2 \pm 0.2$\ [$Pf_{\rm avg}$ ]{}$\,(5:24..28)$ & $V_{(-325\,,\,-175\,{\rm km\,s^{-1})}}$ & $R_{(175\,,\,325\,{\rm km\,s^{-1})}}$& $V\,/\,R$\ $F_{\rm line}$ (norm.) & $0.029 \pm 0.01$ & $0.028 \pm 0.01$ & $1.0\pm0.1$\ $F_{\rm line}^{\rm corr}$ (norm.) & $0.024 \pm 0.01$ & $0.024 \pm 0.01$ & $1.0\pm0.1$\ $\theta_{\rm line}^{\rm FWHM}$ (mas) & $1.9 \pm 0.3$ & $1.5 \pm 0.2$ & $1.2\pm0.3$\ [[*PC* ]{}$_{\rm line}^{\rm shift}$ ]{}(mas) & $0.9 \pm 0.3$ & $0.9 \pm 0.3$ & $1.1\pm0.3$\ Fig. \[fig:1\] shows that different components have been detected in the spectra of : a clearly resolved continuum emission, and HI recombination line emission of [$Br\,\gamma$ ]{}and various Pfund lines. The visibility data suggest that the continuum and the line emission stem from spatially separated, distinct regions in the circumstellar disk. Also, they derive from different emission processes. Therefore, we discuss in the following the two components separately. Continuum emission {#sec:31} ------------------ The disk continuum emission, emitted by bound-free and free-free thermal emission in the disk, is clearly resolved. To facilitate the comparison of our results with other published disk diameters, we give the size of a disk-only fit to the continuum visibilities ($\theta_{\rm cont}^{\rm FWHM}\,=\,0.95\,\pm\,0.03~\,{\rm mas}$) in Table \[tab:2\]. We model the disk continuum by a maximally inclined Gaussian profile, with a position angle of 50$^\circ$ which is orthogonal to the axis of polarization [@1999PASP..111..494M]. However the measured continuum emission consists of both stellar and disk contributions. Taking into account, that the disk continuum of 48 Lib contributes one third of the $K$-band emission , the resulting disk size scale ($\theta_{\rm cont, disk}^{\rm FWHM}\,=\,1.65\,\pm\,0.05~\,{\rm mas}$) is seven times larger than the stellar photospheric diameter. Source variability may compromise this estimation of $\theta_{\rm cont, disk}^{\rm FWHM}$. But a flux calibration of the KI photometry of 48 Lib, based on HD 145607 is consistent with the [2Mass]{} point source flux of our target [@2006AJ....131.1163S] to better than 10%, which is the precision level of our absolute flux calibration. report an irregular variability in the Hipparcos band at the 5 % level. Therefore, we can expect that variability of the star-disk flux ratio has no strong effect on the $\theta_{\rm cont, disk}^{\rm FWHM}$ value reported above. Note that although important for the estimation of the disk size, we cannot derive the NIR disk excess from the visibility data due to our small $u,v$-coverage. Similarly, two-dimensional physical or phenomenological disk models of the continuum emission are not well constraint by a single baseline measurement. @2007ApJ...654..527G showed for several Be stars, that, even for significantly two-dimensional $u,v$-coverages, physical models, using radiative transfer calculations, and phenomenological models, using Gaussian intensity profiles are difficult to distinguish by the visibility data alone. Physical and phenomenological FWHM estimates of the average disk compactness may differ by 50-100 % [@2007ApJ...654..527G]. This may be used as an accuracy estimate for a physical interpretation of the here derived $\theta_{\rm cont, disk}^{\rm FWHM}$. Since near-infrared disk excess and $\theta_{\rm cont, disk}^{\rm FWHM}$ of 48 Lib are within the typical range of Be star disks, we can expect that a physical modeling of the disk continuum, as done by @2007ApJ...654..527G, would lead to similar densities as described therein. The estimated disk continuum-to-stellar radii ratio matches other interferometrically observed edge-on systems with comparable properties [e.g. $\zeta$ Tau in @2007ApJ...654..527G]. The typically found moderate extent of the $K$-band emission size (5-10 times the stellar radius) is significantly smaller than recently modeled , confirming the need for interferometric constraints to understand the physical conditions in the stellar outflows of Be-stars. There is several indications in our data, that the disk continuum emission zone is inside the [$Br\,\gamma$ ]{}and smaller or equal to the $Pf-$ recombination line emission zones. All fitted linear characteristics of the violet and red part of the [$Br\,\gamma$ ]{}([$Pf_{\rm avg}$ ]{}) line emission zones (that is $\theta_{\rm line}^{\rm FWHM}$ and the photocenter shifts [[*PC* ]{}$_{\rm line}^{\rm shift}$ ]{}, for details see the next section; violet and red labels negative and positve Doppler shifts, following standard Be-star terminology) indicate that the majority of the line emission is created at stellocentric radii larger than (comparable to) the radii dominating the continuum emission. Furthermore, the [$Pf_{\rm avg}$ ]{}line center in total and correlated flux drops below the continuum level (Fig. \[fig:3\]). This points to a significant absorption of the continuum flux by gas [*in front*]{} of the continuum source, that is the continuum emission is embedded and surrounded by the zone dominating the line emission. Therefore, we confirm the findings of @2007ApJ...654..527G [studied $H\,\alpha$ versus continuum size] and that the observed [$Br\,\gamma$ ]{} (and [$Pf_{\rm avg}$ ]{}) hydrogen line emission in Be stellar shells appears to occur at larger (or equal) stellocentric radii than the disk continuum emission. HI recombination lines {#sec:32} ---------------------- ![Zoom into the [$Br\,\gamma$ ]{}line. The flux ratio ([*top*]{}) is normalized to the linear continuum shown in Fig. \[fig:1\]. The [*central*]{} panel shows the correlated flux, normalized to the linear continuum fit. Note that the linear flux ratio continuum fit may overestimate the actually observed continuum emission due to absorption by gas [*in front*]{} of the continuum emission, which would result in underestimated fluxes at the line center. The measured differential phase is plotted in the [*bottom*]{} panel. At the top right of each figure, the integrated violet and red line profile is given. The red area marks the region of integration (100-250 km/s), which excludes the uncertain central continuum absorption. The wavelength-differential precision is indicated by the red errorbar in the left of each panel. \[fig:2\] ](plot_48lib_BrGam_zoom.eps) ![The three panels show the [$Pf_{\rm avg}$ ]{}profile at the flux levels of Pf-(5-24) in the same way as in Fig. \[fig:2\]. The smoother line profile with respect to [$Br\,\gamma$ ]{}derives from the line averaging over the linearly interpolated individual Pfund lines. The hatched area indicates the beginning of the velocity region where the shape of the mean line profile is significantly compromised by the neighboring lines due to the averaging. The profile at large velocities ($\ge\,330\,{\rm km\,s^{-1}}$) are a result of our data processing, and unreal. We show this part here to demonstrate that the central, real line profile is more significant than these relatively large profile changes in the outer velocity-space compromised from the averaging. The statistical error of the mean shapes shown is indicate by the red errorbar in the left of each figure. \[fig:3\]](plot_pfund_avg_frcont_lin.eps) The emission line widths of Be-stars are dominated by large-scale kinematic Doppler broadening (due to the fact that the ionized gas is located in a rotating disk around the star), without significant contributions of non-kinematic scattering processes . The disk of 48 Lib is known to be edge on, which simplifies the geometric and kinematic interpretation of the line profiles . The spectrometer does not fully Nyquist-sample the double peak of the [$Br\,\gamma$ ]{}and Pfund lines, but the central dip in the flux and the shape of the [$d\phi$ ]{} signal show that the violet (V) and red (R) part of the disk are clearly separated by the spectrometer, and that meaningful V/R ratios can be derived. $Pf$-24..34 have been detected for the first time in an interferometric spectrum. The five strongest individual $Pf$-lines are clear detections ($\sim\,5-8\,\sigma$) above the differential precision level of flux ratio, visibility, and [$d\phi$ ]{}. To improve on the SNR for the $Pf-$lines, and to minimize the effects of the limited sampling of the line-profile, we average over the five strongest, detected $Pf-$lines (24..28). To do so, the $Pf-$25..28 lines were scaled to match flux levels of $Pf-$24. This avoids that the final average profile ([$Pf_{\rm avg}$ ]{}) is affected by the individual line fluxes. We checked that none of the features of [$Pf_{\rm avg}$ ]{}discussed below is significantly altered or vanishes when averaging over all 11 detected Pfund lines. However, our choice of averaging over the five strongest lines to create [$Pf_{\rm avg}$ ]{}leads to the highest SNR. In addition, the omitted higher-order Pfund lines are closer in rest wavelength, and therefore, overlapping of adjacent line profiles would become apparent and compromise the profile of [$Pf_{\rm avg}$ ]{}. Similarly, at velocities $\ge 330~$[${\rm km\,s}^{-1}$ ]{}(indicated by the dashed region in Fig. \[fig:3\]), the average profile is affected by adjacent lines. We purposefully show in the figure a large velocity space including these artifacts to demonstrate that [$Pf_{\rm avg}$ ]{}emerges even out of these averaging artifacts. Unless indicated otherwise, we use a linear flux ratio continuum fit. Using a black body instead of a linear shape for the flux ratio continuum leads to a comparably good fit, and slightly increases the off-center line emission of [$Pf_{\rm avg}$ ]{}. We include this uncertainty in the shape of the continuum at the red end of the K’-band in the errors given in Table \[tab:2\]. The statistical precision of the averaged profiles are shown as red errorbars. We compared the individual properties (flux, visibility, and differential phase) of the $Pf-$(24..28) included in [$Pf_{\rm avg}$ ]{}to check if meaningful geometric constraints on the Pfund line emission zone can be derived from [$Pf_{\rm avg}$ ]{}. The decreasing signature in visibility and phase with increasing excitation level is consistent with the $Pf$-n/$Pf-$24 line flux ratios. This means that a similar geometry is encoded in the measured visibility and phase of each line, suggesting a common origin of the lines, and justifying the averaging. The individual line ratios used match the respective Pfund-line ratio trends measured for other Be-star envelopes, such as and [@2010AJ....139.1983G]. Furthermore, these flux ratios vary by no more than up to about 30 % ($Pf$-28/$Pf-$24), which indicates together with a similarly moderate variation of the respective Einstein coefficients that the lines included in [$Pf_{\rm avg}$ ]{}are emitted under similar conditions. Therefore, they most likely originate from similar regions in the disk. The geometric constraints, derived from [$Pf_{\rm avg}$ ]{}, cannot be more accurate than this level, which matches the numbers, given in Table \[tab:2\]. The precision of the $V/R$-estimate of the individual Pfund lines is clearly limited by our spectral resolution and the sampling of the line profile. Apparent ratios of up to 20 % above and below unity for the Pfund lines included in [$Pf_{\rm avg}$ ]{}are regarded as not significantly different from unity. Therefore, the lines show comparable $V/R$. Given the previous arguments for a common origin of the individual lines in the disk, we can expected that averaging does improve the $V/R$-signal for the Pfund emission zone. We concentrate on answering the key question qualitatively: are KI-SPR data suitable to measure directly, and efficiently in a single dataset, core parameters like radial extent and radius-dependence of the often discussed slowly rotating density inhomogeneity in the gas around Be stars? To do so, violet and red velocity bins outside of the line center were defined to derive average V-R properties of the lines, while avoiding the zone of highest absorption and uncertain continuum (see below). The velocity bins enclose the maximum line emission with a width of 150 [${\rm km\,s}^{-1}$ ]{}, which is chosen to be comparable to the spectral resolution (bins are marked as colored background in Fig. \[fig:2\]&\[fig:3\]). The profile averages for each bin are given in the figures. We derived intrinsic line emission sizes and photo-center shifts from the data (Table \[tab:2\]) by assuming the superposition of a linear continuum and additional [$Br\,\gamma$ ]{}disk emission . Note that the intrinsic photocenter shifts are on the order of the fringe spacing ($\lambda /2B$), which requires the use of the exact complex relation between measured and intrinsic phase to derive the correct [[*PC* ]{}$_{\rm line}^{\rm shift}$ ]{}. Table \[tab:2\] reports the mean linear properties fitted to the linearly interpolated data for each velocity bin. For our analysis, we used a linear continuum extrapolation of the continuum flux ratio into the line center to derive the quantitative constraints in Table \[tab:2\]. However, the actual continuum at the [$Br\,\gamma$ ]{}line center may deviate from this linear continuum. The stellar continuum has some photospheric [$Br\,\gamma$ ]{}absorption, and since we observe an edge-on disk in emission with, at least partially, optically thick [$Br\,\gamma$ ]{}emission, additional absorption and emission makes it difficult to estimate the exact continuum shape at the line center. This presents a difficulty for isolating the correlated flux, visibility and intrinsic [$d\phi$ ]{}of the line emission at the line center from the measured data, which contain a flux-weighted average of both continuum and line emission. An absorbed continuum profile (in contrast to the here assumed linear extrapolation of the off-line continuum) at the center of [$Br\,\gamma$ ]{}would increase the (correlated) line flux, and therefore decrease the derived intrinsic photocenter shift. To quantify this, we also applied a B4III [$Br\,\gamma$ ]{}template to the data (see also footnote $^{\rm (c)}$ in Table \[tab:11\]). This would not change the measured V/R asymmetries significantly, but it would increase the given [$\theta_{\rm line}^{\rm FWHM}$]{}by about 40 % and decrease [[*PC* ]{}$_{\rm line}^{\rm shift}$ ]{}by 20%, which gives the order of magnitude of this systematic uncertainty in our quantitative analysis. Radial disk structure\[sec:33\] ------------------------------- ![Visualization of the measured disk properties (not to scale). The shown one-armed over-density pattern would explain the measured [*V/R*]{} and correlated flux profiles of [$Br\,\gamma$ ]{}and [$Pf_{\rm avg}$ ]{}, as predicted by , but the here shown pattern is not based on a model calculation. The exact shape of the pattern is not constrained by a single KI-SPR dataset. The relative location of the optical $H\alpha$ line is added for completeness, matching previous, single telescope velocity and [*V/R*]{} measurements [@2007ApJ...656L..21W]. \[fig:4\]](48lib_disk.eps) [$Br\,\gamma$ ]{}and [$Pf_{\rm avg}$ ]{}profiles differ. Above all, they show different V/R ratios and central absorption levels. The total and correlated [$Br\,\gamma$ ]{}line profiles are with $V/R\,\sim\,1.8$, and 1.6 respectively, significantly asymmetric. In contrast, [$\theta_{\rm line}^{\rm FWHM}$]{}and [[*PC* ]{}$_{\rm line}^{\rm shift}$ ]{}of the violet and red [$Br\,\gamma$ ]{}emitting regions coincide within the uncertainties. resolved the [$Br\,\gamma$ ]{}disk emission of $\zeta$ Tau with the VLTI with a spectral and spatial resolution similar to our data. They found a larger intrinsic line photocenter shift of the brighter wing, emitted, at the time of their observation, from the south-eastern, red part of the disk of $\zeta$ Tau. A non-LTE disk model can explain such a trend in the [$d\phi$ ]{}as well as an extensive multi-wavelength and multi-technique dataset with a global oscillation of a spiral density pattern . Our data might show a similar trend of a slightly larger [[*PC* ]{}$_{\rm line}^{\rm shift}$ ]{}of the brighter side of [$Br\,\gamma$ ]{}. However, this trend is not significant with respect to the estimated uncertainties. We would encourage a similar, complex modeling effort of the disk of 48 Lib, including the new differential constraints from the Pfund emission and based on repeated observations to improve on the significance of the [$d\phi$ ]{}trend of [$Br\,\gamma$ ]{}. Here we concentrate on the overall line emission zone properties. The bulk of the [$Br\,\gamma$ ]{}emission appears to come from similar stello-centric radii, and the enhanced emission on the violet side indicates a locally enhanced density. The observed asymmetry is inline with the $\sim$9 yr periodicity of cyclic changes of V/R($H\,\alpha\,$) . The $H\,\alpha$ spectra of 48 Lib, provided by the [BeSS]{} database [^3] also report a V/R $>\,1$ in 2008. In contrast, the [$Pf_{\rm avg}$ ]{}profile appears to be symmetric in all properties within the errors. Again, [$\theta_{\rm line}^{\rm FWHM}$]{}and [[*PC* ]{}$_{\rm line}^{\rm shift}$ ]{}of [$Pf_{\rm avg}$ ]{}are comparable for both sides, but the smaller [[*PC* ]{}$_{\rm line}^{\rm shift}$ ]{}points to significantly smaller stello-centric radii of the bulk of the [$Pf_{\rm avg}$ ]{}emission, if compared to [$Br\,\gamma$ ]{}. Both measured V/R([$Br\,\gamma$ ]{}) and V/R([$Pf_{\rm avg}$ ]{}) can be reconciled with a one-armed density perturbation, if a radial dependence of the perturbation is allowed, as in the case of a spiral density wave, precessing through the disk. There is several indications for that the Pfund emission emerges from inside the bulk of the [$Br\,\gamma$ ]{}emission. The mean [[*PC* ]{}$_{\rm line}^{\rm shift}$ ]{}([$Br\,\gamma$ ]{}) is with 2.1 mas about twice as large as [[*PC* ]{}$_{\rm line}^{\rm shift}$ ]{}([$Pf_{\rm avg}$ ]{}). Furthermore, the bins are centered at different stello-centric radii and velocities. The peak velocities appear offset ([$Br\,\gamma$ ]{}: $150\,\pm\,50\,$[${\rm km\,s}^{-1}$ ]{}versus [$Pf_{\rm avg}$ ]{}: $230\,\pm\,30\,$[${\rm km\,s}^{-1}$ ]{}). These velocity and [[*PC* ]{}$_{\rm line}^{\rm shift}$ ]{}numbers are consistent with Keplerian gas motion, a typical assumption for Be star disk. The Keplerian rotation assumption is further supported by the measured double-peaked line profiles (in contrast to highly disordered or asymmetric profiles). Since Be-star disks are (at least partially) optically thick in the hydrogen emission lines , it is expected that the stellocentric radius of the respective line anti-correlates with the specific line absorption properties. Comparing the respective Einstein absorption coefficients, we find $B_{5,24}\,<\,B_{4,7}\,<\,B_{2,3}$, which suggests that the [$Pf_{\rm avg}$ ]{}is emitted inside of the [$Br\,\gamma$ ]{}as described by our data. [$Br\,\gamma$ ]{}should be emitted at smaller stellocentric radii than $H_{\alpha}$, as indirectly confirmed by the data of @2007ApJ...656L..21W . Also the smaller $H_{\alpha}$ peak velocities [the spectrum in Fig. 1 of @2007ApJ...656L..21W suggests about 110 km /s] are consistent with larger stellocentric radii compared to [$Br\,\gamma$ ]{}and [$Pf_{\rm avg}$ ]{}. Another indication for the different location of the Pfund line emission are the total and correlated flux profiles. Both show a clear absorption below the continuum emission level. This, together with the symmetric line profile might indicate that the gas dominating the Pfund emission is located [*in front*]{} of the inner disk and the star (responsible for the continuum) . As discussed, we cannot estimate what fraction of this continuum absorption is due to intrinsic photospheric absorption. But a domination of the measured continuum absorption by photospheric absorption is unlikely for a Be star. Having the majority of the [$Br\,\gamma$ ]{}emission at larger stellocentric radii on the violet side, but the Pfund emission at smaller radii in front of the star suggests that the over-density pattern follows a radial, one-armed spiral pattern . We depict the radial structure of the disk emission as measured by the KI in a schematic way in Fig. \[fig:4\]. However, our single dataset obviously cannot distingiush a spiral wave from other radius-dependent perturbations patterns. Our data provide a further quantitative test of Papaloizou’s density wave model. discuss that the temporal prediction of V/R cycles of about 10 yr periods fits the available spectroscopic observations. Interferometric observations directly resolve the expected linear scale of the predicted density waves, as opposed to indirect methods such as polarimetry [as discussed for instance in @2010ApJ...709.1306W]. The mean photocenter shifts result in stellocentric radii of about 18 (8) $R_*$ for [$Br\,\gamma$ ]{}([$Pf_{\rm avg}$ ]{}), which is consistent with both hydrogen emission line modeling of Be stars and with density wave models predicting that the local density perturbation, induced by the rotational flattening of the Be stars, disappears outside a few stellar radii . Conclusions {#sec:5} =========== We presented the first interferometric detection of several Pfund lines at the red end of the $K'$-band. Thanks to the simultaneous detection of [$Br\,\gamma$ ]{}in the spectrum of a single KI-SPR observation, we can derive stello-centric radius dependent properties the gas disk of the classical Be-star 48 Lib at high precision. The geometry and kinematics of the emission zones of [$Br\,\gamma$ ]{}and of eleven higher Pfund lines are clearly detected in the differential visibility and phase signals, and the Pfund emission originates from smaller stello-centric radii than [$Br\,\gamma$ ]{}. The radial separation of the various emission lines is convincingly explained by the physics of optically thick line emission in the Be circumstellar disks. In addition, the $K'$-band disk continuum emission was resolved as well, and shown to be emitted from inside the [$Br\,\gamma$ ]{}(V/R $>1$) hydrogen line zone, at radii comparable to the bulk of the $Pf$-emission (with V/R $\approx\,1$). Our findings match qualitatively and quantitatively theoretical Be-star disk model calculations, predicting a precessing one-armed over-density pattern. The spatial constraints from our interferometric data coincide with the common notion that Be-star disks have a Keplerian-like radial velocity profile. Observations similar to the here presented will shed more light on the enigmas of the creation and properties of circumstellar Be shells at all stages of evolution. In particular time-resolved monitoring of Be circumstellar environments with KI-SPR together with radiative transfer modeling of the disk emission will deliver strong constraints on the actual radial and azimuthal geometry of the disk inhomogeneities likely to be responsible for the HI line emission asymmetries. In addition to the presented scientific results on the disk of 48 Lib, we demonstrate with the detection of the Pfund lines, that even weak line emission can produce a significant signal in the differential phase. KI-SPR data of bright stars can be calibrated to a differential precision of $\Delta {\rm flux ratio}\,=\,0.003$, $\Delta\,V^2\,=\,0.007$, and $\Delta\,d\phi\,=\,3\,{\rm mrad}$. This extreme precision in the differential phase equals an on-sky centroid shift of 3 $\mu$as (a relative precision of 10$^{-3}$), which is an improvement by two orders of magnitude over state-of-the-art single-telescope spectro-astrometry. This improvement is however limited to the relatively small interferometric field of view ($\sim$30 mas) and sensitivity of the current KI-SPR mode. Dual field phase referencing, the second phase of the KI-ASTRA upgrade, is currently under construction, and will allow fainter science targets to be observed. Using a different dispersive element, even higher spectral resolution could be achieved in SPR mode. We are grateful to W. Vacca and A. Seifahrt for many useful discussions and contributions to this work. The excellent support of the KI team at WMKO and NExScI helped making these observations a success. The realization of the KI-ASTRA upgrade is supported by the NSF MRI grant, AST-0619965. The data presented herein were obtained at the W.M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W.M. Keck Foundation. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain. The Keck Interferometer is funded by the National Aeronautics and Space Administration as part of its Exoplanet Exploration program. 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--- abstract: 'This research work presents a new class of non-blind information hiding algorithms that are stego-secure and robust. They are based on some finite domains iterations having the Devaney’s topological chaos property. Thanks to a complete formalization of the approach we prove security against watermark-only attacks of a large class of steganographic algorithms. Finally a complete study of robustness is given in frequency DWT and DCT domains.' author: - | Jacques M. Bahi, Jean-François Couchot, and Christophe Guyeux[^1]\ University of Franche-Comté, Computer Science Laboratory, Belfort, France\ {jacques.bahi, jean-francois.couchot, christophe.guyeux}@univ-fcomte.fr bibliography: - 'abbrev2.bib' - 'mabase.bib' - 'biblioand2.bib' title: 'Steganography: a class of secure and robust algorithms' --- Introduction {#sec:intro} ============ This work focus on non-blind binary information hiding chaotic schemes: the original host is required to extract the binary hidden information. This context is indeed not as restrictive as it could primarily appear. Firstly, it allows to prove the authenticity of a document sent through the Internet (the original document is stored whereas the stego content is sent). Secondly, Alice and Bob can establish an hidden channel into a streaming video (Alice and Bob both have the same movie, and Alice hide information into the frame number $k$ iff the binary digit number $k$ of its hidden message is 1). Thirdly, based on a similar idea, a same given image can be marked several times by using various secret parameters owned both by Alice and Bob. Thus more than one bit can be embedded into a given image by using this work. Lastly, non-blind watermarking is useful in network’s anonymity and intrusion detection [@Houmansadr09], and to protect digital data sending through the Internet [@P1150442004]. Furthermore, enlarging the given payload of a data hiding scheme leads clearly to a degradation of its security: the smallest the number of embedded bits is, the better the security is. Chaos-based approaches are frequently proposed to improve the quality of schemes in information hiding [@Wu2007; @Liu07; @CongJQZ06; @Zhu06]. In these works, the understanding of chaotic systems is almost intuitive: a kind of noise-like spread system with sensitive dependence on initial condition. Practically, some well-known chaotic maps are used either in the data encryption stage [@Liu07; @CongJQZ06], in the embedding into the carrier medium, or in both [@Wu2007; @Wu2007bis]. Methods referenced above are almost based on two fundamental chaotic maps, namely the Chebychev and logistic maps, which range in $\mathbb{R}$. To avoid justifying that functions which are chaotic in $\mathbb{R}$ still remain chaotic in the computing representation (*i.e.*, floating numbers) we argue that functions should be iterated on finite domains. Boolean discrete-time dynamical systems (BS) are thus iterated. Furthermore, previously referenced works often focus on discretion and/or robustness properties, but they do not consider security. As far as we know, stego-security [@Cayre2008] and chaos-security have only been proven on the spread spectrum watermarking [@Cox97securespread], and on the dhCI algorithm [@gfb10:ip], which is notably based on iterating the negation function. We argue that other functions can provide algorithms as secure as the dhCI one. This work generalizes thus this latter algorithm and formalizes all its stages. Due to this formalization, we address the proofs of the two security properties for a large class of steganography approaches. This research work is organized as follows. It firstly introduces the new class of algorithms (Sec. \[sec:formalization\]), which is the first contribution. Next, the Section \[sec:security\] presents a State-of-the-art in information hiding security and shows how secure is our approach. The proof is the second contribution. The chaos-security property is studied in Sec. \[sec:chaossecurity\] and instances of algorithms guaranteeing that desired property are presented. This is the fourth contribution. Applications in frequency domains (namely DWT and DCT embedding) are formalized and corresponding experiments are given in Sec. \[sec:applications\]. This shows the applicability of the whole approach. Finally, conclusive remarks and perspectives are given in Sec. \[sec:concl\]. Information hiding algorithm: formalization {#sec:formalization} =========================================== As far as we know, no result rules that the chaotic behavior of a function that has been established on $\mathbb{R}$ remains on the floating numbers. As stated before, this work presents the alternative to iterate a Boolean map: results that are theoretically obtained in that domain are preserved during implementations. In this section, we first give some recalls on Boolean discrete dynamical Systems (BS). With this material, next sections formalize the information hiding algorithms based on these Boolean iterations. Boolean discrete dynamical systems {#sub:bdds} ---------------------------------- Let us denote by $\llbracket a ; b \rrbracket$ the interval of integers: $\{a, a+1, \hdots, b\}$, where $a \leqslant b$. Let $n$ be a positive integer. A Boolean discrete-time network is a discrete dynamical system defined from a [*Boolean map*]{} $f:{\ensuremath{\mathds{B}}}^n\to{\ensuremath{\mathds{B}}}^n$ s.t. $$x=(x_1,\dots,x_n)\mapsto f(x)=(f_1(x),\dots,f_n(x)),$$ [*and an iteration scheme*]{}: parallel, serial, asynchronous… With the parallel iteration scheme, the dynamics of the system are described by $x^{t+1}=f(x^t)$ where $x^0 \in {\ensuremath{\mathds{B}}}^n$. Let thus $F_f: \llbracket1;n\rrbracket\times {\ensuremath{\mathds{B}}}^{n}$ to ${\ensuremath{\mathds{B}}}^n$ be defined by $$F_f(i,x)=(x_1,\dots,x_{i-1},f_i(x),x_{i+1},\dots,x_n),$$ with the *asynchronous* scheme, the dynamics of the system are described by $x^{t+1}=F_f(s^t,x^t)$ where $x^0\in{\ensuremath{\mathds{B}}}^n$ and $s$ is a [*strategy*]{}, *i.e.*, a sequence in $\llbracket1;n\rrbracket^{\ensuremath{\mathbb{N}}}$. Notice that this scheme only modifies one element at each iteration. Let $G_f$ be the map from $\mathcal{X}= \llbracket1;n\rrbracket^{\ensuremath{\mathbb{N}}}\times{\ensuremath{\mathds{B}}}^n$ to itself s.t. $$G_f(s,x)=(\sigma(s),F_f(s^0,x)),$$ where $\sigma(s)^t=s^{t+1}$ for all $t$ in ${\ensuremath{\mathbb{N}}}$. Notice that parallel iteration of $G_f$ from an initial point $X^0=(s,x^0)$ describes the “same dynamics” as the asynchronous iteration of $f$ induced by the initial point $x^0$ and the strategy $s$. Finally, let $f$ be a map from ${\ensuremath{\mathds{B}}}^n$ to itself. The [*asynchronous iteration graph*]{} associated with $f$ is the directed graph $\Gamma(f)$ defined by: the set of vertices is ${\ensuremath{\mathds{B}}}^n$; for all $x\in{\ensuremath{\mathds{B}}}^n$ and $i\in \llbracket1;n\rrbracket$, $\Gamma(f)$ contains an arc from $x$ to $F_f(i,x)$. We have already established [@GuyeuxThese10] that we can define a distance $d$ on $\mathcal{X}$ such that $G_f$ is a continuous and chaotic function according to Devaney [@Devaney]. The next section focus on the coding step of the steganographic algorithm based on $G_f$ iterations. Coding {#sub:wmcoding} ------ In what follows, $y$ always stands for a digital content we wish to hide into a digital host $x$. The data hiding scheme presented here does not constrain media to have a constant size. It is indeed sufficient to provide a function and a strategy that may be parametrized with the size of the elements to modify. The *mode* and the *strategy-adapter* defined below achieve this goal. \[def:mode\] A map $f$, which associates to any $n \in \mathds{N}$ an application $f_n : \mathds{B}^n \rightarrow \mathds{B}^n$, is called a *mode*. For instance, the *negation mode* is defined by the map that assigns to every integer $n \in \mathds{N}^*$ the function $${\neg}_n:\mathds{B}^n \to \mathds{B}^n, (x_1, \hdots, x_n) \mapsto (\overline{x_1}, \hdots, \overline{x_n}).$$ \[def:strategy-adapter\] A *strategy-adapter* is a function $\mathcal{S}$ from ${\ensuremath{\mathbb{N}}}$ to the set of integer sequences, which associates to $n$ a sequence $S \in \llbracket 1, n\rrbracket^\mathds{N}$. Intuitively, a strategy-adapter aims at generating a strategy $(S^t)^{t \in {\ensuremath{\mathbb{N}}}}$ where each term $S^t$ belongs to $\llbracket 1, n \rrbracket$. Moreover it may be parametrized in order to depend on digital media to embed. For instance, let us define the *Chaotic Iterations with Independent Strategy* (*CIIS*) strategy-adapter. The CIIS strategy-adapter with parameters $(K,y,\alpha,l) \in [0,1]\times [0,1] \times ]0, 0.5[ \times \mathds{N}$ is the function that associates to any $n \in {\ensuremath{\mathbb{N}}}$ the sequence $(S^t)^{t \in \mathds{N}}$ defined by: - $K^0 = \textit{bin}(y) \oplus \textit{bin}(K)$: $K^0$ is the real number whose binary decomposition is equal to the bitwise exclusive or (xor) between the binary decompositions of $y$ and of $K$; - $\forall t \leqslant l, K^{t+1} = F(K^t,\alpha)$; - $\forall t \leqslant l, S^t = \left \lfloor n \times K^t \right \rfloor + 1$; - $\forall t > l, S^t = 0$. where $F$ is the piecewise linear chaotic map [@Shujun1], recalled in what follows: \[def:fonction chaotique linéaire par morceaux\] Let $\alpha \in ]0; 0.5[$ be a control parameter. The *piecewise linear chaotic map* is the map $F$ defined by: $$F(t,\alpha) = \left\{ \begin{array}{cl} \dfrac{t}{\alpha} & t \in [0; \alpha],\\ \dfrac{t-\alpha}{\frac{1}{2}-\alpha} & t \in [\alpha;\frac{1}{2}],\\ F(1-t,\alpha) & t \in [\frac{1}{2}; 1].\\ \end{array} \right.$$ Contrary to the logistic map, the use of this piecewise linear chaotic map is relevant in cryptographic usages [@Arroyo08]. Parameters of CIIS strategy-adapter will be instantiate as follows: $K$ is the secret embedding key, $y$ is the secret message, $\alpha$ is the threshold of the piecewise linear chaotic map, which can be set as $K$ or can act as a second secret key. Lastly, $l$ is for the iteration number bound: enlarging its value improve the chaotic behavior of the scheme, but the time required to achieve the embedding grows too. Another strategy-adapter is the *Chaotic Iterations with Dependent Strategy* (CIDS) with parameters $(l,X) \in \mathds{N}\times \mathds{B}^\mathds{N}$, which is the function that maps any $ n \in \mathds{N}$ to the sequence $\left(S^t\right)^{t \in \mathds{N}}$ defined by: - $\forall t \leqslant l$, if $t \leqslant l$ and $X^t = 1$, then $S^t=t$, else $S^t=1$; - $\forall t > l, S^t = 0$. Let us notice that the terms of $x$ that may be replaced by terms taken from $y$ are less important than other: they could be changed without be perceived as such. More generally, a *signification function* attaches a weight to each term defining a digital media, w.r.t. its position $t$: A *signification function* is a real sequence $(u^k)^{k \in {\ensuremath{\mathbb{N}}}}$. For instance, let us consider a set of grayscale images stored into portable graymap format (P3-PGM): each pixel ranges between 256 gray levels, *i.e.*, is memorized with eight bits. In that context, we consider $u^k = 8 - (k \mod 8)$ to be the $k$-th term of a signification function $(u^k)^{k \in {\ensuremath{\mathbb{N}}}}$. Intuitively, in each group of eight bits (*i.e.*, for each pixel) the first bit has an importance equal to 8, whereas the last bit has an importance equal to 1. This is compliant with the idea that changing the first bit affects more the image than changing the last one. \[def:msc,lsc\] Let $(u^k)^{k \in {\ensuremath{\mathbb{N}}}}$ be a signification function, $m$ and $M$ be two reals s.t. $m < M$. Then the *most significant coefficients (MSCs)* of $x$ is the finite vector $u_M$, the *least significant coefficients (LSCs)* of $x$ is the finite vector $u_m$, and the *passive coefficients* of $x$ is the finite vector $u_p$ such that: $$\begin{aligned} u_M &=& \left( k ~ \big|~ k \in \mathds{N} \textrm{ and } u^k \geqslant M \textrm{ and } k \le \mid x \mid \right) \\ u_m &=& \left( k ~ \big|~ k \in \mathds{N} \textrm{ and } u^k \le m \textrm{ and } k \le \mid x \mid \right) \\ u_p &=& \left( k ~ \big|~ k \in \mathds{N} \textrm{ and } u^k \in ]m;M[ \textrm{ and } k \le \mid x \mid \right)\end{aligned}$$ For a given host content $x$, MSCs are then ranks of $x$ that describe the relevant part of the image, whereas LSCs translate its less significant parts. We are then ready to decompose an host $x$ into its coefficients and then to recompose it. Next definitions formalize these two steps. Let $(u^k)^{k \in {\ensuremath{\mathbb{N}}}}$ be a signification function, $\mathfrak{B}$ the set of finite binary sequences, $\mathfrak{N}$ the set of finite integer sequences, $m$ and $M$ be two reals s.t. $m < M$. Any host $x$ can be decomposed into $$(u_M,u_m,u_p,\phi_{M},\phi_{m},\phi_{p}) \in \mathfrak{N} \times \mathfrak{N} \times \mathfrak{N} \times \mathfrak{B} \times \mathfrak{B} \times \mathfrak{B}$$ where - $u_M$, $u_m$, and $u_p$ are coefficients defined in Definition \[def:msc,lsc\]; - $\phi_{M} = \left( x^{u^1_M}, x^{u^2_M}, \ldots,x^{u^{|u_M|}_M}\right)$; - $\phi_{m} = \left( x^{u^1_m}, x^{u^2_m}, \ldots,x^{u^{|u_m|}_m} \right)$; - $\phi_{p} =\left( x^{u^1_p}, x^{u^2_p}, \ldots,x^{u^{|u_p|}_p}\right) $. The function that associates the decomposed host to any digital host is the *decomposition function*. It is further referred as $\textit{dec}(u,m,M)$ since it is parametrized by $u$, $m$, and $M$. Notice that $u$ is a shortcut for $(u^k)^{k \in {\ensuremath{\mathbb{N}}}}$. Let $(u_M,u_m,u_p,\phi_{M},\phi_{m},\phi_{p}) \in \mathfrak{N} \times \mathfrak{N} \times \mathfrak{N} \times \mathfrak{B} \times \mathfrak{B} \times \mathfrak{B} $ s.t. - the sets of elements in $u_M$, elements in $u_m$, and elements in $u_p$ are a partition of $\llbracket 1, n\rrbracket$; - $|u_M| = |\varphi_M|$, $|u_m| = |\varphi_m|$, and $|u_p| = |\varphi_p|$. One can associate the vector $$x = \sum_{i=1}^{|u_M|} \varphi^i_M . e_{{u^i_M}} + \sum_{i=1}^{|u_m|} \varphi^i_m .e_{{u^i_m}} + \sum_{i=1}^{|u_p|} \varphi^i_p. e_{{u^i_p}}$$ where $(e_i)_{i \in \mathds{N}}$ is the usual basis of the $\mathds{R}-$vectorial space $\left(\mathds{R}^\mathds{N}, +, .\right)$ (that is to say, $e_i^j = \delta_{ij}$, where $\delta_{ij}$ is the Kronecker symbol). The function that associates $x$ to any $(u_M,u_m,u_p,\phi_{M},\phi_{m},\phi_{p})$ following the above constraints is called the *recomposition function*. The embedding consists in the replacement of the values of $\phi_{m}$ of $x$’s LSCs by $y$. It then composes the two decomposition and recomposition functions seen previously. More formally: Let $\textit{dec}(u,m,M)$ be a decomposition function, $x$ be a host content, $(u_M,u_m,u_p,\phi_{M},\phi_{m},\phi_{p})$ be its image by $\textit{dec}(u,m,M)$, and $y$ be a digital media of size $|u_m|$. The digital media $z$ resulting on the embedding of $y$ into $x$ is the image of $(u_M,u_m,u_p,\phi_{M},y,\phi_{p})$ by the recomposition function $\textit{rec}$. Let us then define the dhCI information hiding scheme presented in [@gfb10:ip]: \[def:dhCI\] Let $\textit{dec}(u,m,M)$ be a decomposition function, $f$ be a mode, $\mathcal{S}$ be a strategy adapter, $x$ be an host content,$(u_M,u_m,u_p,\phi_{M},\phi_{m},\phi_{p})$ be its image by $\textit{dec}(u,m,M)$, $q$ be a positive natural number, and $y$ be a digital media of size $l=|u_m|$. The dhCI dissimulation maps any $(x,y)$ to the digital media $z$ resulting on the embedding of $\hat{y}$ into $x$, s.t. - we instantiate the mode $f$ with parameter $l=|u_m|$, leading to the function $f_{l}:{\ensuremath{\mathds{B}}}^{l} \rightarrow {\ensuremath{\mathds{B}}}^{l}$; - we instantiate the strategy adapter $\mathcal{S}$ with parameter $y$ (and possibly some other ones); this instantiation leads to the strategy $S_y \in \llbracket 1;l\rrbracket ^{{\ensuremath{\mathbb{N}}}}$. - we iterate $G_{f_l}$ with initial configuration $(S_y,\phi_{m})$; - $\hat{y}$ is finally the $q$-th term of these iterations. To summarize, iterations are realized on the LSCs of the host content (the mode gives the iterate function, the strategy-adapter gives its strategy), and the last computed configuration is re-injected into the host content, in place of the former LSCs. Notice that in order to preserve the unpredictable behavior of the system, the size of the digital medias is not fixed. This approach is thus self adapted to any media, and more particularly to any size of LSCs. However this flexibility enlarges the complexity of the presentation: we had to give Definitions \[def:mode\] and \[def:strategy-adapter\] respectively of mode and strategy adapter. ![The dhCI dissimulation scheme[]{data-label="fig:organigramme"}](organigramme2.eps){width="8.5cm"} Next section shows how to check whether a media contains a watermark. Decoding {#sub:wmdecoding} -------- Let us firstly show how to formally check whether a given digital media $z$ results from the dissimulation of $y$ into the digital media $x$. Let $\textit{dec}(u,m,M)$ be a decomposition function, $f$ be a mode, $\mathcal{S}$ be a strategy adapter, $q$ be a positive natural number, $y$ be a digital media, and $(u_M,u_m,u_p,\phi_{M},\phi_{m},\phi_{p})$ be the image by $\textit{dec}(u,m,M)$ of a digital media $x$. Then $z$ is *watermarked* with $y$ if the image by $\textit{dec}(u,m,M)$ of $z$ is $(u_M,u_m,u_p,\phi_{M},\hat{y},\phi_{p})$, where $\hat{y}$ is the right member of $G_{f_l}^q(S_y,\phi_{m})$. Various decision strategies are obviously possible to determine whether a given image $z$ is watermarked or not, depending on the eventuality that the considered image may have been attacked. For example, a similarity percentage between $x$ and $z$ can be computed and compared to a given threshold. Other possibilities are the use of ROC curves or the definition of a null hypothesis problem. The next section recalls some security properties and shows how the *dhCI dissimulation* algorithm verifies them. Security analysis {#sec:security} ================= State-of-the-art in information hiding security {#sub:art} ----------------------------------------------- As far as we know, Cachin [@Cachin2004] produces the first fundamental work in information hiding security: in the context of steganography, the attempt of an attacker to distinguish between an innocent image and a stego-content is viewed as an hypothesis testing problem. Mittelholzer [@Mittelholzer99] next proposed the first theoretical framework for analyzing the security of a watermarking scheme. Clarification between robustness and security and classifications of watermarking attacks have been firstly presented by Kalker [@Kalker2001]. This work has been deepened by Furon *et al.* [@Furon2002], who have translated Kerckhoffs’ principle (Alice and Bob shall only rely on some previously shared secret for privacy), from cryptography to data hiding. More recently [@Cayre2005; @Perez06] classified the information hiding attacks into categories, according to the type of information the attacker (Eve) has access to: - in Watermarked Only Attack (WOA) she only knows embedded contents $z$; - in Known Message Attack (KMA) she knows pairs $(z,y)$ of embedded contents and corresponding messages; - in Known Original Attack (KOA) she knows several pairs $(z,x)$ of embedded contents and their corresponding original versions; - in Constant-Message Attack (CMA) she observes several embedded contents $z^1$,…,$z^k$ and only knows that the unknown hidden message $y$ is the same in all contents. To the best of our knowledge, KMA, KOA, and CMA have not already been studied due to the lack of theoretical framework. In the opposite, security of data hiding against WOA can be evaluated, by using a probabilistic approach recalled below. Stego-security {#sub:stegosecurity} -------------- In the Simmons’ prisoner problem [@Simmons83], Alice and Bob are in jail and they want to, possibly, devise an escape plan by exchanging hidden messages in innocent-looking cover contents. These messages are to be conveyed to one another by a common warden named Eve, who eavesdrops all contents and can choose to interrupt the communication if they appear to be stego-contents. Stego-security, defined in this well-known context, is the highest security class in Watermark-Only Attack setup, which occurs when Eve has only access to several marked contents [@Cayre2008]. Let $\mathds{K}$ be the set of embedding keys, $p(X)$ the probabilistic model of $N_0$ initial host contents, and $p(Y|K)$ the probabilistic model of $N_0$ marked contents s.t. each host content has been marked with the same key $K$ and the same embedding function. \[Def:Stego-security\] The embedding function is *stego-secure* if $\forall K \in \mathds{K}, p(Y|K)=p(X)$ is established. Stego-security states that the knowledge of $K$ does not help to make the difference between $p(X)$ and $p(Y)$. This definition implies the following property: $$p(Y|K_1)= \cdots = p(Y|K_{N_k})=p(Y)=p(X)$$ This property is equivalent to a zero Kullback-Leibler divergence, which is the accepted definition of the “perfect secrecy” in steganography [@Cachin2004]. The negation mode is stego-secure --------------------------------- To make this article self-contained, this section recalls theorems and proofs of stego-security for negation mode published in [@gfb10:ip]. *dhCI dissimulation* of Definition \[def:dhCI\] with negation mode and CIIS strategy-adapter is stego-secure, whereas it is not the case when using CIDS strategy-adapter. On the one hand, let us suppose that $X \sim \mathbf{U}\left(\mathbb{B}^n\right)$ when using CIIS$(K,\_,\_,l)$. We prove by a mathematical induction that $\forall t \in \mathds{N}, X^t \sim \mathbf{U}\left(\mathbb{B}^n\right)$. The base case is immediate, as $X^0 = X \sim \mathbf{U}\left(\mathbb{B}^n\right)$. Let us now suppose that the statement $X^t \sim \mathbf{U}\left(\mathbb{B}^n\right)$ holds until for some $t$. Let $e \in \mathbb{B}^n$ and $\mathbf{B}_k=(0,\cdots,0,1,0,\cdots,0) \in \mathbb{B}^n$ (the digit $1$ is in position $k$). So $P\left(X^{t+1}=e\right)=\sum_{k=1}^n P\left(X^t=e\oplus\mathbf{B}_k,S^t=k\right)$ where $\oplus$ is again the bitwise exclusive or. These two events are independent when using CIIS strategy-adapter (contrary to CIDS, CIIS is not built by using $X$), thus: $$P\left(X^{t+1}=e\right)=\sum_{k=1}^n P\left(X^t=e\oplus\mathbf{B}_k\right) \times P\left(S^t=k\right).$$ According to the inductive hypothesis: $P\left(X^{n+1}=e\right)=\frac{1}{2^n} \sum_{k=1}^n P\left(S^t=k\right)$. The set of events $\left \{ S^t=k \right \}$ for $k \in \llbracket 1;n \rrbracket$ is a partition of the universe of possible, so $\sum_{k=1}^n P\left(S^t=k\right)=1$. Finally, $P\left(X^{t+1}=e\right)=\frac{1}{2^n}$, which leads to $X^{t+1} \sim \mathbf{U}\left(\mathbb{B}^n\right)$. This result is true for all $t \in \mathds{N}$ and then for $t=l$. Since $P(Y|K)$ is $P(X^l)$ that is proven to be equal to $P(X)$, we thus have established that, $$\forall K \in [0;1], P(Y|K)=P(X^{l})=P(X).$$ So dhCI dissimulation with CIIS strategy-adapter is stego-secure. On the other hand, due to the definition of CIDS, we have $P(Y=(1,1,\cdots,1)|K)=0$. So there is no uniform repartition for the stego-contents $Y|K$. To sum up, Alice and Bob can counteract Eve’s attacks in WOA setup, when using dhCI dissimulation with CIIS strategy-adapter. To our best knowledge, this is the second time an information hiding scheme has been proven to be stego-secure: the former was the spread-spectrum technique in natural marking configuration with $\eta$ parameter equal to 1 [@Cayre2008]. A new class of $\varepsilon$-stego-secure schemes ------------------------------------------------- Let us prove that, \[th:stego\] Let $\epsilon$ be positive, $l$ be any size of LSCs, $X \sim \mathbf{U}\left(\mathbb{B}^l\right)$, $f_l$ be an image mode s.t. $\Gamma(f_l)$ is strongly connected and the Markov matrix associated to $f_l$ is doubly stochastic. In the instantiated *dhCI dissimulation* algorithm with any uniformly distributed (u.d.) strategy-adapter that is independent from $X$, there exists some positive natural number $q$ s.t. $|p(X^q)- p(X)| < \epsilon$. Let $\textit{deci}$ be the bijection between ${\ensuremath{\mathds{B}}}^{l}$ and $\llbracket 0, 2^l-1 \rrbracket$ that associates the decimal value of any binary number in ${\ensuremath{\mathds{B}}}^{l}$. The probability $p(X^t) = (p(X^t= e_0),\dots,p(X^t= e_{2^l-1}))$ for $e_j \in {\ensuremath{\mathds{B}}}^{l}$ is thus equal to $(p(\textit{deci}(X^t)= 0,\dots,p(\textit{deci}(X^t)= 2^l-1))$ further denoted by $\pi^t$. Let $i \in \llbracket 0, 2^l -1 \rrbracket$, the probability $p(\textit{deci}(X^{t+1})= i)$ is $$\sum\limits^{2^l-1}_{j=0} \sum\limits^{l}_{k=1} p(\textit{deci}(X^{t}) = j , S^t = k , i =_k j , f_k(j) = i_k )$$ where $ i =_k j $ is true iff the binary representations of $i$ and $j$ may only differ for the $k$-th element, and where $i_k$ abusively denotes, in this proof, the $k$-th element of the binary representation of $i$. Next, due to the proposition’s hypotheses on the strategy, $p(\textit{deci}(X^t) = j , S^t = k , i =_k j, f_k(j) = i_k )$ is equal to $\frac{1}{l}.p(\textit{deci}(X^t) = j , i =_k j, f_k(j) = i_k)$. Finally, since $i =_k j$ and $f_k(j) = i_k$ are constant during the iterative process and thus does not depend on $X^t$, we have $$\pi^{t+1}_i = \sum\limits^{2^l-1}_{j=0} \pi^t_j.\frac{1}{l} \sum\limits^{l}_{k=1} p(i =_k j, f_k(j) = i_k ).$$ Since $\frac{1}{l} \sum\limits^{l}_{k=1} p(i =_k j, f_k(j) = i_k ) $ is equal to $M_{ji}$ where $M$ is the Markov matrix associated to $f_l$ we thus have $$\pi^{t+1}_i = \sum\limits^{2^l-1}_{j=0} \pi^t_j. M_{ji} \textrm{ and thus } \pi^{t+1} = \pi^{t} M.$$ First of all, since the graph $\Gamma(f)$ is strongly connected, then for all vertices $i$ and $j$, a path can be found to reach $j$ from $i$ in at most $2^l$ steps. There exists thus $k_{ij} \in \llbracket 1, 2^l \rrbracket$ s.t. ${M}_{ij}^{k_{ij}}>0$. As all the multiples $l \times k_{ij}$ of $k_{ij}$ are such that ${M}_{ij}^{l\times k_{ij}}>0$, we can conclude that, if $k$ is the least common multiple of $\{k_{ij} \big/ i,j \in \llbracket 1, 2^l \rrbracket \}$ thus $\forall i,j \in \llbracket 1, 2^l \rrbracket, {M}_{ij}^{k}>0$ and thus $M$ is a regular stochastic matrix. Let us now recall the following stochastic matrix theorem: If $M$ is a regular stochastic matrix, then $M$ has an unique stationary probability vector $\pi$. Moreover, if $\pi^0$ is any initial probability vector and $\pi^{t+1} = \pi^t.M $ for $t = 0, 1,\dots$ then the Markov chain $\pi^t$ converges to $\pi$ as $t$ tends to infinity. Thanks to this theorem, $M$ has an unique stationary probability vector $\pi$. By hypothesis, since $M$ is doubly stochastic we have $(\frac{1}{2^l},\dots,\frac{1}{2^l}) = (\frac{1}{2^l},\dots,\frac{1}{2^l})M$ and thus $\pi = (\frac{1}{2^l},\dots,\frac{1}{2^l})$. Due to the matrix theorem, there exists some $q$ s.t. $|\pi^q- \pi| < \epsilon$ and the proof is established. Since $p(Y| K)$ is $p(X^q)$ the method is then $\epsilon$-stego-secure provided the strategy-adapter is uniformly distributed. This section has focused on security with regards to probabilistic behaviors. Next section studies it in the perspective of topological ones. Chaos-security {#sec:chaossecurity} ============== To check whether an existing data hiding scheme is chaotic or not, we propose firstly to write it as an iterate process $x^{n+1}=f(x^n)$. It is possible to prove that this formulation can always be done, as follows. Let us consider a given data hiding algorithm. Because it must be computed one day, it is always possible to translate it as a Turing machine, and this last machine can be written as $x^{n+1} = f(x^n)$ in the following way. Let $(w,i,q)$ be the current configuration of the Turing machine (Fig. \[Turing\]), where $w=\sharp^{-\omega} w(0) \hdots w(k)\sharp^{\omega}$ is the paper tape, $i$ is the position of the tape head, $q$ is used for the state of the machine, and $\delta$ is its transition function (the notations used here are well-known and widely used). We define $f$ by: - $f(w(0) \hdots w(k),i,q) = ( w(0) \hdots w(i-1)aw(i+1)w(k),i+1,q')$, if $\delta(q,w(i)) = (q',a,\rightarrow)$; - $f( w(0) \hdots w(k),i,q) = (w(0) \hdots w(i-1)aw(i+1)w(k),i-1,q')$, if $\delta(q,w(i)) = (q',a,\leftarrow)$. Thus the Turing machine can be written as an iterate function $x^{n+1}=f(x^n)$ on a well-defined set $\mathcal{X}$, with $x^0$ as the initial configuration of the machine. We denote by $\mathcal{T}(S)$ the iterative process of a data hiding scheme $S$. ![Turing Machine[]{data-label="Turing"}](Turing.eps){width="8.5cm"} Let us now define the notion of chaos-security. Let $\tau$ be a topology on $\mathcal{X}$. So the behavior of this dynamical system can be studied to know whether or not the data hiding scheme is $\tau-$unpredictable. This leads to the following definition. \[DefinitionChaosSecure\] An information hiding scheme $S$ is said to be chaos-secure on $(\mathcal{X},\tau)$ if its iterative process $\mathcal{T}(S)$ has a chaotic behavior, as defined by Devaney, on this topological space. Theoretically speaking, chaos-security can always be studied, as it only requires that the two following points are satisfied. - Firstly, the data hiding scheme must be written as an iterate function on a set $\mathcal{X}$; As illustrated by the use of the Turing machine, it is always possible to satisfy this requirement; It is established here since we iterate $G_f$ as defined in Sect. (\[sub:bdds\]); - Secondly, a metric or a topology must be defined on $\mathcal{X}$; This is always possible, for example, by taking for instance the most relevant one, that is the order topology. Guyeux has recently shown in [@GuyeuxThese10] that chaotic iterations of $G_f$ with the vectorial negation as iterate function have a chaotic behavior. As a corollary, we deduce that the dhCI dissimulation algorithm with negation mode and CIIS strategy-adapter is chaos-secure. However, all these results suffer from only relying on the vectorial negation function. This problem has been theoretically tackled in [@GuyeuxThese10] which provides the following theorem. \[Th:Caracterisation des IC chaotiques\] Functions $f : \mathds{B}^{n} \to \mathds{B}^{n}$ such that $G_f$ is chaotic according to Devaney, are functions such that the graph $\Gamma(f)$ is strongly connected. We deduce from this theorem that functions whose graph is strongly connected are sufficient to provide new instances of dhCI dissimulation that are chaos-secure. Computing a mode $f$ such that the image of $n$ (*i.e.*, $f_n$) is a function with a strongly connected graph of iterations $\Gamma(f_n)$ has been previously studied (see [@bcgr11:ip] for instance). The next section presents a use of them in our steganography context. Applications to frequential domains {#sec:applications} =================================== We are then left to provide an u.d. strategy-adapter that is independent from the cover, an image mode $f_l$ whose iteration graph $\Gamma(f_l)$ is strongly connected and whose Markov matrix is doubly stochastic. First, the $\textit{CIIS}(K,y,\alpha,l)$ strategy adapter (see Section \[sub:wmcoding\]) has the required properties: it does not depend on the cover and the proof that its outputs are u.d. on $\llbracket 1; l \rrbracket$ is left as an exercise for the reader. In all the experiments parameters $K$ and $\alpha$ are randomly chosen in $\rrbracket 0, 1\llbracket$ and $\rrbracket 0, 0.5\llbracket$ respectively. The number of iteration is set to $4*lm$, where $lm$ is the number of LSCs that depends on the domain. Next, [@bcgr11:ip] has presented an iterative approach to generate image modes $f_l$ such that $\Gamma(f_l)$ is strongly connected. Among these maps, it is obvious to check which verifies or not the doubly stochastic constrain. For instance, in what follows we consider the mode $f_l: {\ensuremath{\mathds{B}}}^l \rightarrow {\ensuremath{\mathds{B}}}^l$ s.t. its $i$th component is defined by $$\label{eq:fqq} {f_l}(x)_i = \left\{ \begin{array}{l} \overline{x_i} \textrm{ if $i$ is odd} \\ x_i \oplus x_{i-1} \textrm{ if $i$ is even} \end{array} \right.$$ Thanks to [@bcgr11:ip Theorem 2] we deduce that its iteration graph $\Gamma(f_l)$ is strongly connected. Next, the Markov chain is stochastic by construction. Let us prove that its Markov chain is doubly stochastic by induction on the length $l$. For $l=1$ and $l=2$ the proof is obvious. Let us consider that the result is established until $l=2k$ for some $k \in {\ensuremath{\mathbb{N}}}$. Let us then firstly prove the doubly stochasticity for $l=2k+1$. Following notations introduced in [@bcgr11:ip], let $\Gamma(f_{2k+1})^0$ and $\Gamma(f_{2k+1})^1$ denote the subgraphs of $\Gamma(f_{2k+1})$ induced by the subset ${\ensuremath{\mathds{B}}}^{2k} \times\{0\}$ and ${\ensuremath{\mathds{B}}}^{2k} \times\{1\}$ of ${\ensuremath{\mathds{B}}}^{2k+1}$ respectively. $\Gamma(f_{2k+1})^0$ and $\Gamma(f_{2k+1})^1$ are isomorphic to $\Gamma(f_{2k})$. Furthermore, these two graphs are linked together only with arcs of the form $(x_1,\dots,x_{2k},0) \to (x_1,\dots,x_{2k},1)$ and $(x_1,\dots,x_{2k},1) \to (x_1,\dots,x_{2k},0)$. In $\Gamma(f_{2k+1})$ the number of arcs whose extremity is $(x_1,\dots,x_{2k},0)$ is the same than the number of arcs whose extremity is $(x_1,\dots,x_{2k})$ augmented with 1, and similarly for $(x_1,\dots,x_{2k},1)$. By induction hypothesis, the Markov chain associated to $\Gamma(f_{2k})$ is doubly stochastic. All the vertices $(x_1,\dots,x_{2k})$ have thus the same number of ingoing arcs and the proof is established for $l$ is $2k+1$. Let us then prove the doubly stochasticity for $l=2k+2$. The map $f_l$ is defined by $f_l(x)= (\overline{x_1},x_2 \oplus x_{1},\dots,\overline{x_{2k+1}},x_{2k+2} \oplus x_{2k+1})$. With previously defined notations, let us focus on $\Gamma(f_{2k+2})^0$ and $\Gamma(f_{2k+2})^1$ which are isomorphic to $\Gamma(f_{2k+1})$. Among configurations of ${\ensuremath{\mathds{B}}}^{2k+2}$, only four suffixes of length 2 can be obviously observed, namely, $00$, $10$, $11$ and $01$. Since $f_{2k+2}(\dots,0,0)_{2k+2}=0$, $f_{2k+2}(\dots,1,0)_{2k+2}=1$, $f_{2k+2}(\dots,1,1)_{2k+2}=0$, and $f_{2k+2}(\dots,0,1)_{2k+2}=1$, the number of arcs whose extremity is - $(x_1,\dots,x_{2k},0,0)$ is the same than the one whose extremity is $(x_1,\dots,x_{2k},0)$ in $\Gamma(f_{2k+1})$ augmented with 1 (loop over configurations $(x_1,\dots,x_{2k},0,0)$); - $(x_1,\dots,x_{2k},1,0)$ is the same than the one whose extremity is $(x_1,\dots,x_{2k},0)$ in $\Gamma(f_{2k+1})$ augmented with 1 (arc from configurations $(x_1,\dots,x_{2k},1,1)$ to configurations $(x_1,\dots,x_{2k},1,0)$); - $(x_1,\dots,x_{2k},0,1)$ is the same than the one whose extremity is $(x_1,\dots,x_{2k},0)$ in $\Gamma(f_{2k+1})$ augmented with 1 (loop over configurations $(x_1,\dots,x_{2k},0,1)$); - $(x_1,\dots,x_{2k},1,1)$ is the same than the one whose extremity is $(x_1,\dots,x_{2k},1)$ in $\Gamma(f_{2k+1})$ augmented with 1 (arc from configurations $(x_1,\dots,x_{2k},1,0)$ to configurations $(x_1,\dots,x_{2k},1,1)$). Thus all the vertices $(x_1,\dots,x_{2k})$ have the same number of ingoing arcs and the proof is established for $l=2k+2$. DWT embedding ------------- Let us now explain how the dhCI dissimulation can be applied in the discrete wavelets domain (DWT). In this paper, the Daubechies family of wavelets is chosen: each DWT decomposition depends on a decomposition level and a coefficient matrix (Figure \[fig:DWTs\]): $\textit{LL}$ means approximation coefficient, when $\textit{HH},\textit{LH},\textit{HL}$ denote respectively diagonal, vertical, and horizontal detail coefficients. For example, the DWT coefficient *HH*2 is the matrix equal to the diagonal detail coefficient of the second level of decomposition of the image. ![Wavelets coefficients.[]{data-label="fig:DWTs"}](DWTs.eps){width="7cm"} The choice of the detail level is motivated by finding a good compromise between robustness and invisibility. Choosing low or high frequencies in DWT domain leads either to a very fragile watermarking without robustness (especially when facing a JPEG2000 compression attack) or to a large degradation of the host content. In order to have a robust but discrete DWT embedding, the second detail level (*i.e.*, $\textit{LH}2,\textit{HL}2,\textit{HH}2$) that corresponds to the middle frequencies, has been retained. Let us consider the Daubechies wavelet coefficients of a third level decomposition as represented in Figure \[fig:DWTs\]. We then translate these float coefficients into their 32-bits values. Let us define the significance function $u$ that associates to any index $k$ in this sequence of bits the following numbers: - $u^k = -1$ if $k$ is one of the three last bits of any index of coefficients in $\textit{LH}2$, $\textit{HL}2$, or in $\textit{HH}2$; - $u^k = 0$ if $k$ is an index of a coefficient in $\textit{LH}1$, $\textit{HL}1$, or in $\textit{HH}1$; - $u^k = 1$ otherwise. According to the definition of significance of coefficients (Def. \[def:msc,lsc\]), if $(m,M)$ is $(-0.5,0.5)$, LSCs are the last three bits of coefficients in $\textit{HL}2$,$\textit{HH}2$, and $\textit{LH}2$. Thus, decomposition and recomposition functions are fully defined and dhCI dissimulation scheme can now be applied. Figure \[fig:DWT\] shows the result of a dhCI dissimulation embedding into DWT domain. The original is the image 5007 of the BOSS contest [@Boss10]. Watermark $y$ is given in Fig. \[(b) Watermark\]. From a random selection of 50 images into the database from the BOSS contest [@Boss10], we have applied the previous algorithm with mode $f_l$ defined in Equation (\[eq:fqq\]) and with the negation mode. DCT embedding ------------- Let us denote by $x$ the original image of size $H \times L$, and by $y$ the hidden message, supposed here to be a binary image of size $H' \times L'$. The image $x$ is transformed from the spatial domain to DCT domain frequency bands, in order to embed $y$ inside it. To do so, the host image is firstly divided into $8 \times 8$ image blocks as given below: $$x = \bigcup_{k=1}^{H/8} \bigcup_{k'=1}^{L/8} x(k,k').$$ Thus, for each image block, a DCT is performed and the coefficients in the frequency bands are obtained as follows: $x_{DCT}(m;n) = DCT(x(m;n))$. To define a discrete but robust scheme, only the four following coefficients of each $8 \times 8$ block in position $(m,n)$ will be possibly modified: $x_{DCT}(m;n)_{(3,1)},$ $x_{DCT}(m;n)_{(2,2)},$ or $x_{DCT}(m;n)_{(1,3)}$. This choice can be reformulated as follows. Coefficients of each DCT matrix are re-indexed by using a southwest/northeast diagonal, such that $i_{DCT}(m,n)_1 = x_{DCT}(m;n)_{(1,1)}$,$i_{DCT}(m,n)_2 = x_{DCT}(m;n)_{(2,1)}$, $i_{DCT}(m,n)_3 = x_{DCT}(m;n)_{(1,2)}$, $i_{DCT}(m,n)_4 = x_{DCT}(m;n)_{(3,1)}$, ..., and $i_{DCT}(m,n)_{64} =$ $ x_{DCT}(m;n)_{(8,8)}$. So the signification function can be defined in this context by: - if $k$ mod $64 \in \{1,2,3\}$ and $k\leqslant H\times L$, then $u^k=1$; - else if $k$ mod $64 \in \{4, 5, 6\}$ and $k\leqslant H\times L$, then $u^k=-1$; - else $u^k = 0$. The significance of coefficients are obtained for instance with $(m,M)=(-0.5,0.5)$ leading to the definitions of MSCs, LSCs, and passive coefficients. Thus, decomposition and recomposition functions are fully defined and dhCI dissimulation scheme can now be applied. Image quality ------------- This section focuses on measuring visual quality of our steganographic method. Traditionally, this is achieved by quantifying the similarity between the modified image and its reference image. The Mean Squared Error (MSE) and the Peak Signal to Noise Ratio (PSNR) are the most widely known tools that provide such a metric. However, both of them do not take into account Human Visual System (HVS) properties. Recent works [@EAPLBC06; @SheikhB06; @PSECAL07; @MB10] have tackled this problem by creating new metrics. Among them, what follows focuses on PSNR-HVS-M [@PSECAL07] and BIQI [@MB10], considered as advanced visual quality metrics. The former efficiently combines PSNR and visual between-coefficient contrast masking of DCT basis functions based on HVS. This metric has been computed here by using the implementation given at [@psnrhvsm11]. The latter allows to get a blind image quality assessment measure, *i.e.*, without any knowledge of the source distortion. Its implementation is available at [@biqi11]. Embedding ------------ ------- ------- ------- ------- Mode $f_l$ neg. $f_l$ neg. PSNR 42.74 42.76 52.68 52.41 PSNR-HVS-M 44.28 43.97 45.30 44.93 BIQI 35.35 32.78 41.59 47.47 : Quality measeures of our steganography approach\[table:quality\] Results of the image quality metrics are summarized into the Table \[table:quality\]. In wavelet domain, the PSNR values obtained here are comparable to other approaches (for instance, PSNR are 44.2 in [@TCL05] and 46.5 in [@DA10]), but a real improvement for the discrete cosine embeddings is obtained (PSNR is 45.17 for [@CFS08], it is always lower than 48 for [@Mohanty:2008:IWB:1413862.1413865], and always lower than 39 for [@MK08]). Among steganography approaches that evaluate PSNR-HVS-M, results of our approach are convincing. Firstly, optimized method developed along [@Randall11] has a PSNR-HVS-M equal to 44.5 whereas our approach, with a similar PSNR-HVS-M, should be easily improved by considering optimized mode. Next, another approach [@Muzzarelli:2010] have higher PSNR-HVS-M, certainly, but this work does not address robustness evaluation whereas our approach is complete. Finally, as far as we know, this work is the first one that evaluates the BIQI metric in the steganography context. With all this material, we are then left to evaluate the robustness of this approach. Robustness ---------- Previous sections have formalized frequential domains embeddings and has focused on the negation mode and $f_l$ defined in Equ. (\[eq:fqq\]). In the robustness given in this continuation, [dwt]{}(neg), [dwt]{}(fl), [dct]{}(neg), [dct]{}(fl) respectively stand for the DWT and DCT embedding with the negation mode and with this instantiated mode. For each experiment, a set of 50 images is randomly extracted from the database taken from the BOSS contest [@Boss10]. Each cover is a $512\times 512$ grayscale digital image and the watermark $y$ is given in Fig \[(b) Watermark\]. Testing the robustness of the approach is achieved by successively applying on watermarked images attacks like cropping, compression, and geometric transformations. Differences between $\hat{y}$ and $\varphi_m(z)$ are computed. Behind a given threshold rate, the image is said to be watermarked. Finally, discussion on metric quality of the approach is given in Sect. \[sub:roc\]. Robustness of the approach is evaluated by applying different percentage of cropping: from 1% to 81%. Results are presented in Fig. \[Fig:atck:dec\]. Fig. \[Fig:atq:dec:img\] gives the cropped image where 36% of the image is removed. Fig. \[Fig:atq:dec:curves\] presents effects of such an attack. From this experiment, one can conclude that all embeddings have similar behaviors. All the percentage differences are so far less than 50% (which is the mean random error) and thus robustness is established. ### Robustness against compression Robustness against compression is addressed by studying both JPEG and JPEG 2000 image compressions. Results are respectively presented in Fig. \[Fig:atq:jpg:curves\] and Fig. \[Fig:atq:jp2:curves\]. Without surprise, DCT embedding which is based on DCT (as JPEG compression algorithm is) is more adapted to JPEG compression than DWT embedding. Furthermore, we have a similar behavior for the JPEG 2000 compression algorithm, which is based on wavelet encoding: DWT embedding naturally outperforms DCT one in that case. ### Robustness against Contrast and Sharpness Attack Contrast and Sharpness adjustment belong to the the classical set of filtering image attacks. Results of such attacks are presented in Fig. \[Fig:atq:fil\] where Fig. \[Fig:atq:cont:curve\] and Fig. \[Fig:atq:sh:curve\] summarize effects of contrast and sharpness adjustment respectively. ### Robustness against Geometric Transformation Among geometric transformations, we focus on rotations, *i.e.*, when two opposite rotations of angle $\theta$ are successively applied around the center of the image. In these geometric transformations, angles range from 2 to 20 degrees. Results are presented in Fig. \[Fig:atq:rot\]: Fig. \[Fig:atq:rot:img\] gives the image of a rotation of 20 degrees whereas Fig. \[Fig:atq:rot:curve\] presents effects of such an attack. It is not a surprise that results are better for DCT embeddings: this approach is based on cosine as rotation is. Evaluation of the Embeddings {#sub:roc} ---------------------------- We are then left to set a convenient threshold that is accurate to determine whether an image is watermarked or not. Starting from a set of 100 images selected among the Boss image Panel, we compute the following three sets: the one with all the watermarked images $W$, the one with all successively watermarked and attacked images $\textit{WA}$, and the one with only the attacked images $A$. Notice that the 100 attacks for each images are selected among these detailed previously. For each threshold $t \in \llbracket 0,55 \rrbracket$ and a given image $x \in \textit{WA} \cup A$, differences on DCT are computed. The image is said to be watermarked if these differences are less than the threshold. In the positive case and if $x$ really belongs to $\textit{WA}$ it is a True Positive (TP) case. In the negative case but if $x$ belongs to $\textit{WA}$ it is a False Negative (FN) case. In the positive case but if $x$ belongs to $\textit{A}$, it is a False Positive (FP) case. Finally, in the negative case and if $x$ belongs to $\textit{A}$, it is a True Negative (TN). The True (resp. False) Positive Rate (TPR) (resp. FPR) is thus computed by dividing the number of TP (resp. FP) by 100. ![ROC Curves for DWT or DCT Embeddings[]{data-label="fig:roc:dwt"}](ROC.eps){width="7cm"} The Figure \[fig:roc:dwt\] is the Receiver Operating Characteristic (ROC) curve. For the DWT, it shows that best results are obtained when the threshold is 45% for the dedicated function (corresponding to the point (0.01, 0.88)) and 46% for the negation function (corresponding to the point (0.04, 0.85)). It allows to conclude that each time LSCs differences between a watermarked image and another given image $i'$ are less than 45%, we can claim that $i'$ is an attacked version of the original watermarked content. For the two DCT embeddings, best results are obtained when the threshold is 44% (corresponding to the points (0.05, 0.18) and (0.05, 0.28)). Let us then give some confidence intervals for all the evaluated attacks. The approach is resistant to: - all the croppings where percentage is less than 85; - compressions where quality ratio is greater than 82 with DWT embedding and where quality ratio is greater than 67 with DCT one; - contrast when strengthening belongs to $[0.76,1.2]$ (resp. $[0.96,1.05]$) in DWT (resp. in DCT) embedding; - all the rotation attacks with DCT embedding and a rotation where angle is less than 13 degrees with DWT one. Conclusion {#sec:concl} ========== This paper has proposed a new class of secure and robust information hiding algorithms. It has been entirely formalized, thus allowing both its theoretical security analysis, and the computation of numerous variants encompassing spatial and frequency domain embedding. After having presented the general algorithm with detail, we have given conditions for choosing mode and strategy-adapter making the whole class stego-secure or $\epsilon$-stego-secure. To our knowledge, this is the first time such a result has been established. Applications in frequency domains (namely DWT and DCT domains) have finally be formalized. Complete experiments have allowed us first to evaluate how invisible is the steganographic method (thanks to the PSNR computation) and next to verify the robustness property against attacks. Furthermore, the use of ROC curves for DWT embedding have revealed very high rates between True positive and False positive results. In future work, our intention is to find the best image mode with respect to the combination between DCT and DWT based steganography algorithm. Such a combination topic has already been addressed (*e.g.*, in [@al2007combined]), but never with objectives we have set. Additionally, we will try to discover new topological properties for the dhCI dissimulation schemes. Consequences of these chaos properties will be drawn in the context of information hiding security. We will especially focus on the links between topological properties and classes of attacks, such as KOA, KMA, EOA, or CMA. Moreover, these algorithms will be compared to other existing ones, among other things by testing whether these algorithms are chaotic or not. Finally we plan to verify the robustness of our approach against statistical steganalysis methods [@GFH06; @ChenS08; @DongT08; @FridrichKHG11a]. [^1]: Authors in alphabetic order

--- abstract: 'We derive a general formulation of the laws of irreversible thermodynamics in the presence of electromagnetism and gravity. For the handling of macroscopic material media, we use as a guide the field equations and the Noether identities of fundamental matter as deduced in the framework of gauge theories of the Poincaré$\otimes U(1)$ group.' author: - Romualdo Tresguerres title: Thermodynamics in dynamical spacetimes --- Introduction ============ The present work is based on our previous paper [@Tresguerres:2007ih]. There we studied jointly gravitation and electrodynamics in the form of a gauge theory of the Poincaré group times the internal group $U(1)$. Following the approach of Hehl et al. to gauge theories of gravity [@Hehl:1974cn]–[@Obukhov:2006ge], we made use of a Lagrangian formalism to get the field equations and the Noether identities associated to the gauge symmetry, devoting special attention to energy conservation. This latter aspect of [@Tresguerres:2007ih], where exchange between different forms of energy plays a central role, strongly suggests to look for a thermodynamic interpretation of the corresponding formulas, although this aim remains unattainable as only single matter particles are involved. For this reason, we are interested in extending similar energetic considerations to macroscopic matter in order to be able to construct an approach to thermodynamics compatible with gauge theories of gravity. In this endeavor, our starting point is provided by the dynamical equations found for a particular form of fundamental matter, namely Dirac matter, with the help of the principle of invariance of the action under local Poincaré$\otimes U(1)$ transformations. Our main hypothesis is that the equations still hold for other forms of matter with the same $U(1)$, translational and Lorentz symmetry properties, and we assume that these are possessed by macroscopic matter. Accordingly, we consider that material media obey equations with a form which is known to us, also when we have to reinterpret several quantities involved in them –in particular the matter sources– in order to give account of macroscopic features which are not present in the original formulation. Moreover, a major alteration of the almost purely geometrical approach to physical reality characteristic for gauge theories occurs with the introduction of thermodynamic variables. Briefly exposed, regarding the latter ones we proceed as follows. From the original gauge theoretically defined matter energy current $\epsilon ^{\rm matt}$, we define a modified matter energy current $\epsilon ^{\rm u}$ with an energy flux component $q$ identified as heat flux, and a further component $\mathfrak{U}$ representing the internal energy content of a volume element. As a requirement of the transition to macroscopic matter [@Callen], we postulate $\mathfrak{U}$ to depend, among others, on a new macroscopic variable $\mathfrak{s}$ with the meaning of the entropy content of an elementary volume. (Contrary to other authors [@Landau:1958]-[@Priou:1991], we do not introduce an additional entropy flow variable.) The definition of temperature as the derivative of $\mathfrak{U}$ with respect to $\mathfrak{s}$ completes the set of fundamental thermal variables. We are going to prove that they satisfy the first and second laws of thermodynamics. In our approach, the energy and entropy forms, as much as the temperature function, are Lorentz invariants, as in Eckart’s pioneering work [@Eckart:1940te]. There, as in our case, the first principle of thermodynamics is derived from the energy-momentum conservation law not as the zero component of this vector equation, but as a scalar equation. The paper is organized as follows. In Sections II and III we present the gauge-theoretically derived field equations and Noether identities. After introducing in IV a necessary spacetime foliation, Section V is devoted to defining total energy and its various constitutive pieces, and to studying the corresponding conservation equations. In VI, explicit Lagrangians for electrodynamics and gravity are considered, while VII deals with some aspects of the energy-momentum of macroscopic matter. In Section VIII we argue on the most suitable way to include the features of material media in the dynamical equations. Lastly, the main results are presented in Section IX, where we deduce the laws of thermodynamics in two different scenarios. The paper ends with several final remarks and the conclusions. Field equations =============== The results of [@Tresguerres:2007ih] relevant for the present paper are summarized in what follows with slight changes needed to replace the fundamental Dirac matter by macroscopic matter. Interested readers are referred to [@Tresguerres:2007ih] for technical details, in particular those concerning the handling of translations. A complementary study of the underlying geometry of dynamical spacetimes of Poincaré gauge theories can be found in Refs. [@Tresguerres:2002uh] and [@Tresguerres:2012nu]. Our point of departure is a Lagrangian density 4-form $$L=L(\,A\,,\vartheta ^\alpha\,,\Gamma ^{\alpha\beta}\,;F\,,T^\alpha\,,\,R^{\alpha\beta}\,;{\rm matter\hskip0.2cm variables}\,)\,,\label{totalLag}$$ invariant under local Poincaré$\otimes U(1)$ symmetry. Its arguments, along with matter fields, are the following. On the one hand, we recognize the connection 1-forms of $U(1)$, of translations and of the Lorentz subgroup respectively: that is, the electromagnetic potential $A$, the (nonlinear) translational connections $\vartheta ^\alpha$ geometrically interpreted as tetrads, and the Lorentz connections $\Gamma ^{\alpha\beta}$ required to guarantee gauge covariance, being antisymmetric in their indices. On the other hand, further arguments are the covariantized derivatives of the preceding connections. The differential of the electromagnetic potential is the familiar electromagnetic field strength $$F:= dA\,,\label{Fdef}$$ and analogously, torsion [@Hehl:1995ue] defined as the covariant differential of tetrads $$T^\alpha := D\,\vartheta ^\alpha = d\,\vartheta ^\alpha + \Gamma _\beta{}^\alpha\wedge\vartheta ^\beta\,,\label{torsiondef}$$ together with the Lorentz curvature $$R^{\alpha\beta} := d\,\Gamma ^{\alpha\beta} + \Gamma _\gamma{}^\beta\wedge \Gamma ^{\alpha\gamma}\,,\label{curvdef}$$ play the role of the field strengths associated respectively to translations and to the Lorentz group. Lorentz indices are raised and lowered with the help of the constant Minkowski metric $o_{\alpha\beta}= diag(-+++)$. The derivatives of (\[totalLag\]) with respect to the connections $A$, $\vartheta ^\alpha $ and $\Gamma ^{\alpha\beta}$ are the electric four-current 3-form $$J :={{\partial L}\over{\partial A}}\,,\label{definition03a}$$ the total energy-momentum 3-form $$\Pi _\alpha :={{\partial L}\over{\partial \vartheta ^\alpha}}\,,\label{definition03b}$$ (including, as we will see, electrodynamic, gravitational and matter contributions), and the spin current[^1] $$\tau _{\alpha\beta} :={{\partial L}\over{\partial \Gamma ^{\alpha\beta}}}\,.\label{definition03c}$$ Finally, derivatives of (\[totalLag\]) with respect to the field strengths (\[Fdef\]), (\[torsiondef\]) and (\[curvdef\]) yield respectively the electromagnetic excitation 2-form $$H:=-{{\partial L}\over{\partial F}}\,,\label{definition01}$$ and its translative and Lorentzian analogs, defined as the excitation 2-forms $$\quad H_\alpha :=-{{\partial L}\over{\partial T^\alpha}}\,,\quad H_{\alpha\beta}:=-\,{{\partial L}\over{\partial R^{\alpha\beta}}}\,.\label{definition02}$$ With these definitions at hand, the principle of extremal action yields the field equations $$\begin{aligned} dH &=&J\,,\label{covfieldeq1} \\ DH_\alpha &=&\Pi _\alpha\,,\label{covfieldeq2}\\ DH_{\alpha\beta} +\vartheta _{[\alpha }\wedge H_{\beta ]}&=&\tau _{\alpha\beta}\,.\label{covfieldeq3}\end{aligned}$$ As we will see below, suitable explicit Lagrangians uncover respectively (\[covfieldeq1\]) as Maxwell’s equations and (\[covfieldeq2\]) as a generalized Einstein equation for gravity, whereas (\[covfieldeq3\]) completes the scheme taking spin currents into account. Notice that Eqs. (\[covfieldeq1\])–(\[covfieldeq3\]) are explicitly Lorentz covariant[^2]. In addition, they are invariant with respect to translations as much as to $U(1)$ as a consequence of the (nonlinear) symmetry realization used in [@Tresguerres:2007ih]. Noether identities ================== Following [@Hehl:1995ue], we separate the total Lagrangian density 4-form (\[totalLag\]) into three different pieces $$L=L^{\rm matt}+L^{\rm em}+L^{\rm gr}\,,\label{Lagrangedecomp}$$ consisting respectively in the matter contribution $$L^{\rm matt} = L^{\rm matt}(\,\vartheta ^\alpha\,;{\rm matter\hskip0.2cm variables}\,)\,,\label{mattLagcontrib}$$ (in the fundamental case, matter variables consisting of matter fields $\psi$ and of their covariant derivatives including connections $A$ and $\Gamma ^{\alpha\beta}$), together with the electromagnetic part $L^{\rm em}(\,\vartheta ^\alpha\,,\,F\,)\,$ and the gravitational Lagrangian $L^{\rm gr}(\,\vartheta ^\alpha\,,\,T^\alpha\,,\,R_\alpha{}^\beta\,)$. According to (\[Lagrangedecomp\]), the energy-momentum 3-form (\[definition03b\]) decomposes as $$\Pi _\alpha =\Sigma ^{\rm matt}_\alpha +\Sigma ^{\rm em}_\alpha +E_\alpha\,,\label{momentdecomp}$$ with the different terms in the right-hand side (rhs) defined respectively as $$\Sigma ^{\rm matt}_\alpha :={{\partial L^{\rm matt}}\over{\partial \vartheta ^\alpha}}\,,\quad \Sigma ^{\rm em}_\alpha :={{\partial L^{\rm em}}\over{\partial \vartheta ^\alpha}}\,,\quad E_\alpha :={{\partial L^{\rm gr}}\over{\partial \vartheta ^\alpha}}\,.\label{momentdecompbis}$$ Starting with the matter Lagrangian part $L^{\rm matt}\,$, let us derive the Noether type conservation equations for the matter currents associated to the different symmetries, that is $$J={{\partial L^{\rm matt}}\over{\partial A}}\,,\quad \Sigma ^{\rm matt}_\alpha = {{\partial L^{\rm matt}}\over{\partial \vartheta ^\alpha }}\,,\quad\tau _{\alpha\beta} = {{\partial L^{\rm matt}}\over{\partial \Gamma ^{\alpha\beta}}}\,.\label{mattcurrdefs}$$ Provided the field equations (\[covfieldeq1\])–(\[covfieldeq3\]) are fulfilled, as much as the Euler-Lagrange equations for matter fields (non explicitly displayed here), from the invariance of $L^{\rm matt}$ under vertical (gauge) Poincaré $\otimes$ $U(1)$ transformations follow the conservation equations for both, the electric current $$dJ =0\,,\label{elcurrcons}$$ and the spin current $$D\,\tau _{\alpha\beta} +\vartheta _{[\alpha}\wedge\Sigma ^{\rm matt}_{\beta ]}=0\,.\label{spincurrconserv}$$ On the other hand, the Lie (lateral) displacement ${\it{l}}_{\bf x} L^{\rm matt}$ of the Lagrangian 4-form along an arbitrary vector field $X$ yields the identity $$D\,\Sigma ^{\rm matt}_\alpha =(\,e_\alpha\rfloor T^\beta )\wedge\Sigma ^{\rm matt}_\beta +(\,e_\alpha\rfloor R^{\beta\gamma}\,)\wedge\tau _{\beta\gamma} +(\,e_\alpha\rfloor F\,)\wedge J\,,\label{sigmamattconserv}$$ with the matter energy-momentum 3-form given by $$\Sigma ^{\rm matt}_\alpha =-(\,e_\alpha\rfloor\overline{D\psi}\,)\,{{\partial L^{\rm matt}}\over{\partial d\overline{\psi}}} +{{\partial L^{\rm matt}}\over{\partial d\psi}}\,(\,e_\alpha\rfloor D\psi\,) + e_\alpha\rfloor L^{\rm matt}\label{sigmamatt}$$ (for Dirac matter, and thus to be modified for the case of macroscopic matter). In the rhs of (\[sigmamattconserv\]) we recognize, besides the proper Lorentz force 4-form in the extreme right, two additional terms with the same structure, built with the field strengths and the matter currents of translational and Lorentz symmetry respectively. Next we apply the same treatment to the remaining constituents of (\[Lagrangedecomp\]). The gauge invariance of the electromagnetic Lagrangian piece implies $$\vartheta _{[\alpha}\wedge\Sigma ^{\rm em}_{\beta ]} =0\,,\label{Symem-emt}$$ while in analogy to (\[sigmamattconserv\]) we find $$D\,\Sigma ^{\rm em}_\alpha =(\,e_\alpha\rfloor T^\beta )\wedge\Sigma ^{\rm em}_\beta -(\,e_\alpha\rfloor F\,)\wedge dH\,,\label{sigmaemconserv}$$ being the electromagnetic energy-momentum $$\Sigma ^{\rm em}_\alpha =(\,e_\alpha\rfloor F\,)\wedge H + e_\alpha\rfloor L^{\rm em}\,.\label{sigmaem}$$ Finally, regarding the gravitational Lagrangian part, its gauge invariance yields $$D\,\Bigl( DH_{\alpha\beta} +\vartheta _{[\alpha }\wedge H_{\beta ]}\,\Bigr) +\vartheta _{[\alpha}\wedge\Bigl( DH_{\beta ]} -E_{\beta ]}\,\Bigr)=0\,,\label{redund}$$ (derivable alternatively from (\[spincurrconserv\]) with (\[covfieldeq2\]), (\[covfieldeq3\]), (\[momentdecomp\]) and (\[Symem-emt\])), and the (\[sigmamattconserv\]) and (\[sigmaemconserv\])– analogous equation reads $$\begin{aligned} &&D\,\Bigl( DH_\alpha -E_\alpha\,\Bigr) -(\,e_\alpha\rfloor T^\beta )\wedge\Bigl( DH_\beta -E_\beta\,\Bigr)\nonumber\\ &&\hskip0.2cm -(\,e_\alpha\rfloor R^{\beta\gamma}\,)\wedge\Bigl( DH_{\beta\gamma}+\vartheta _{[\beta }\wedge H_{\gamma ]}\,\Bigr)=0\,,\label{ealphaconserv}\end{aligned}$$ with the pure gravitational energy-momentum given by $$\begin{aligned} E_\alpha =(\,e_\alpha\rfloor T^\beta )\wedge H_\beta +(\,e_\alpha\rfloor R^{\beta\gamma}\,)\wedge H_{\beta\gamma} +e_\alpha\rfloor L^{\rm gr}\,.\label{ealpha}\end{aligned}$$ Eq.(\[ealphaconserv\]) is also redundant, being derivable from (\[sigmamattconserv\]) and (\[sigmaemconserv\]) together with the field equations (\[covfieldeq1\])–(\[covfieldeq3\]) and (\[momentdecomp\]). Spacetime foliation =================== General formulas ---------------- The definition of energy to be introduced in next section, as much as its subsequent thermodynamic treatment, rests on a foliation of spacetime involving a timelike vector field $u$ defined as follows. (For more details, see [@Tresguerres:2012nu].) The foliation is induced by a 1-form $\omega = d\tau $ trivially satisfying the Frobenius’ foliation condition $\omega\wedge d\omega =0$. The vector field $u$ relates to $d\tau$ through the condition $u\rfloor d\tau =1$ fixing its direction. This association of the vector $u$ with $\tau $, the latter being identified as [*parametric time*]{}, allows one to formalize time evolution of any physical quantity represented by a $p$-form $\alpha$ as its Lie derivative along $u$, that is $${\it{l}}_u\alpha :=\,d\,(u\rfloor\alpha\,) + u\rfloor d\alpha \,.\label{Liederdef}$$ (Notice that the condition $u\rfloor d\tau =1$ itself defining $u$ in terms of $\tau$ means that ${\it l}_u\,\tau := u\rfloor d\tau =1$.) With respect to the direction of the time vector $u$, any $p$-form $\alpha$ decomposes into two constituents [@Hehl-and-Obukhov], longitudinal and transversal to $u$ respectively, as $$\alpha = d\tau\wedge\alpha _{\bot} +\underline{\alpha}\,,\label{foliat1}$$ with the longitudinal piece $$\alpha _{\bot} := u\rfloor\alpha\,,\label{long-part}$$ consisting of the projection of $\alpha$ along $u$, and the transversal component $$\underline{\alpha}:= u\rfloor ( d\tau\wedge\alpha\,)\,,\label{trans-part}$$ orthogonal to the former as a spatial projection. The foliation of exterior derivatives of forms is performed in analogy to (\[foliat1\]) as $$d\,\alpha = d\tau\wedge\bigl(\,{\it{l}}_u\underline{\alpha} -\,\underline{d}\,\alpha _{\bot}\,\bigr) +\underline{d}\,\underline{\alpha }\,,\label{derivfoliat}$$ with the longitudinal part expressed in terms of the Lie derivative (\[Liederdef\]) and of the spatial differential $\underline{d}$. For its part, the Hodge dual (\[dualform\]) of a $p$-form $\alpha$ decomposes as $${}^*\alpha =\,(-1)^p\, d\tau\wedge {}^{\#}\underline{\alpha} - {}^{\#}\alpha _{\bot}\,,\label{foliat2}$$ being $^\#$ the Hodge dual operator in the three-dimensional spatial sheets. Foliation of tetrads -------------------- Let us apply the general formulas (\[Liederdef\])–(\[foliat2\]) to the particular case of tetrads $\vartheta ^\alpha $, which, as universally coupling coframes [@Tresguerres:2007ih], will play a significant role in what follows. Their dual vector basis $\{e_\alpha\}$ is defined by the condition $$e_\alpha\rfloor \vartheta ^\beta = \delta _\alpha ^\beta\,.\label{dualitycond}$$ When applied to tetrads, (\[foliat1\]) reads $$\vartheta ^\alpha = d\tau\,u^\alpha + \underline{\vartheta}^\alpha\,,\label{tetradfoliat}$$ where the longitudinal piece $$u^\alpha := u\rfloor\vartheta ^\alpha\label{fourvel}$$ has the meaning of a four-velocity. In terms of it, the time vector $u$ can be expressed as $u =u^\alpha e_\alpha$, being the requirement for $u$ to be timelike fulfilled as $$u_\alpha u^\alpha = -1\,.\label{form01}$$ In terms of (\[fourvel\]), let us define the projector $$h_\alpha{}^\beta :=\delta _\alpha ^\beta + u_\alpha u^\beta\,.\label{form03}$$ Replacing (\[tetradfoliat\]) in (\[dualitycond\]) and making use of (\[form03\]) we find $$e_\alpha\rfloor \Big(\,d\tau\,u^\beta + \underline{\vartheta}^\beta\,\Bigr) = \delta _\alpha ^\beta =-u_\alpha u^\beta +h_\alpha{}^\beta \,.\label{dualitycondbis}$$ implying $$e_\alpha \rfloor d\tau = -\,u_\alpha\,,\label{form02}$$ and $$e_\alpha\rfloor \underline{\vartheta}^\beta = h_\alpha{}^\beta\,.\label{dualitycondbis}$$ On the other hand, let us generalize the definition (\[Liederdef\]) of Lie derivatives by considering covariant differentials instead of ordinary ones [@Hehl:1995ue]. In particular, we will make extensive use of the covariant Lie derivative of the tetrads, defined as $$\begin{aligned} {\cal \L\/}_u\vartheta ^\alpha &:=& D\left( u\rfloor\vartheta ^\alpha\right) + u\rfloor D\vartheta ^\alpha\nonumber\\ &=& D u^\alpha + T_{\bot}^\alpha \,,\label{thetaLiederiv01}\end{aligned}$$ where $${\cal \L\/}_u\vartheta ^\alpha = {\it{l}}_u\vartheta ^\alpha +{\Gamma _{\bot}}_\beta{}^\alpha\wedge\vartheta ^\beta\,,\label{thetaLiederiv02}$$ with (\[thetaLiederiv01\]) decomposing into the longitudinal and transversal pieces $$\begin{aligned} ({\cal \L\/}_u\vartheta ^\alpha )_{\bot} &=& {\cal \L\/}_u u^\alpha\,,\label{thetaLiederiv03}\\ \underline{{\cal \L\/}_u\vartheta ^\alpha} &=& \underline{D} u^\alpha + T_{\bot}^\alpha\nonumber\\ &=& {\cal \L\/}_u\underline{\vartheta}^\alpha\,.\label{thetaLiederiv04}\end{aligned}$$ For what follows, we also need complementary formulas concerning the foliation of the eta basis. Since they require more space, we introduce them in Appendix A. Definition and conservation of energy ===================================== In Ref.[@Tresguerres:2007ih] we discussed the definition of the total energy current 3-form $$\epsilon := -\left(\,u^\alpha\,\Pi _\alpha + Du^\alpha\wedge H_\alpha\,\right)\,.\label{energycurr}$$ By rewriting it as $$\epsilon =-d\left( u^\alpha H_\alpha\right) + u^\alpha \left( DH_\alpha -\Pi _\alpha \right)\,,\label{exactform01}$$ and making use of (\[covfieldeq2\]), we find that it reduces to an exact form $$\epsilon =-d\left( u^\alpha H_\alpha\right)\,,\label{exactform02}$$ automatically satisfying the continuity equation $$d\,\epsilon =0\,.\label{energyconserv01}$$ The interpretation of (\[energycurr\]) as total energy, and thus of (\[energyconserv01\]) as local conservation of total energy, becomes apparent with the help of (\[momentdecomp\]). The energy (\[energycurr\]) reveals to be the sum of three pieces $$\epsilon =\epsilon ^{\rm matt}+\epsilon ^{\rm em}+\epsilon ^{\rm gr}\,,\label{energydec}$$ defined respectively as $$\begin{aligned} \epsilon ^{\rm matt} &:=& -u^\alpha\,\Sigma ^{\rm matt}_\alpha\,,\label{mattenergy}\\ \epsilon ^{\rm em} &:=& -u^\alpha\,\Sigma ^{\rm em}_\alpha\,,\label{emenergy}\\ \epsilon ^{\rm gr} &:=& -\left(\,u^\alpha\,E_\alpha + D u^\alpha\wedge H_\alpha\,\right)\,.\label{grenergy}\end{aligned}$$ On the other hand, decomposing (\[energycurr\]) into its longitudinal and transversal components $$\epsilon = d\tau\wedge\epsilon _{\bot} +\underline{\epsilon}\,,\label{energyfol01}$$ the foliated form of the local energy conservation equation (\[energyconserv01\]) reads $${\it l}_u\,\underline{\epsilon}-\underline{d}\,\epsilon _{\bot}=0\,,\label{conteq}$$ showing (when integrated) that the rate of increase of the energy $\underline{\epsilon}$ contained in a small volume equals the amount of energy flowing into the volume over its boundary surface as the result of the balance of inflow and outflow of the energy flux $\epsilon _{\bot}$ crossing through the closed surface. Conservation of total energy is the result of exchanges between the different forms of energy. Let us write the continuity equations of the different pieces (\[mattenergy\])–(\[grenergy\]). As we will see immediately, in all these equations, when considered separately, sources and sinks of energy are involved, reflecting the fact that, inside the small volume considered, energy is produced or consumed, wether on account of work or of any other manifestation of energy. These terms only cancel out when all forms of energy are considered together, that is, in (\[energyconserv01\]) with (\[energydec\]). Regarding the matter contribution to energy (\[mattenergy\]), using (\[sigmamattconserv\]) we find $$d\,\epsilon ^{\rm matt} = -{\cal \L\/}_u\,\vartheta ^\alpha\wedge\Sigma ^{\rm matt}_\alpha -R_{\bot}^{\alpha\beta}\wedge\tau _{\alpha\beta} -F_{\bot}\wedge J\,.\label{mattender}$$ The interpretation of this conservation equation when its validity is extended to macroscopic matter constitutes the main task of the present work. Actually, Eq. (\[mattender\]) provides the basis for our approach to thermodynamics. In analogy to (\[mattender\]), definition (\[emenergy\]) of electromagnetic energy with (\[sigmaemconserv\]) yields the Poynting equation $$d\,\epsilon ^{\rm em} = -{\cal \L\/}_u\,\vartheta ^\alpha\wedge\Sigma ^{\rm em}_\alpha + F_{\bot}\wedge dH\,,\label{emender}$$ generalized to take into account spacetime as defined in Poincaré gauge theories. In (\[emender\]), the energy flux (or intensity of flowing energy) is represented by the Poynting 2-form $\epsilon ^{\rm em}_{\bot}$, and the last term in the rhs is related to Joule’s heat. Finally, from the gravitational energy definition (\[grenergy\]) with (\[ealphaconserv\]) we get $$\begin{aligned} d\,\epsilon ^{\rm gr} &:=& -{\cal \L\/}_u\,\vartheta ^\alpha\wedge\left(\,E_\alpha -DH_\alpha\right)\nonumber\\ &&+R_{\bot}^{\alpha\beta}\wedge \left(\,DH_{\alpha\beta} +\vartheta _{[\alpha }\wedge H_{\beta ]}\right)\,.\label{grender}\end{aligned}$$ The field equations (\[covfieldeq1\])–(\[covfieldeq3\]) guarantee that the sum of (\[mattender\]), (\[emender\]) and (\[grender\]) is conserved, in agreement with (\[energyconserv01\]). Electrodynamical and gravitational Lagrangians ============================================== In the present Section we introduce explicit Lagrangian pieces (\[Lagrangedecomp\]) describing electrodynamics and gravity. We do so in order to calculate in particular the excitations defined in (\[definition01\]) and (\[definition02\]), which extend to the macroscopic arena without alterations, as will be discussed in Section VIII. We also derive the electromagnetic and gravitational energy-momentum contributions to (\[momentdecomp\]) as defined in (\[momentdecompbis\]), and the corresponding energies (\[emenergy\]) and (\[grenergy\]). The form found for (\[emenergy\]), namely (\[explemen1\]), and in particular that of its transversal part (\[emendh\]), provides us with a criterion to choose the way to extend the [*microscopic*]{} fundamental equations to macroscopic material media. (See Section VIII.) Electrodynamics --------------- In the context of fundamental matter in vacuum, we consider the Maxwell Lagrangian $$L^{\rm em}=-{1\over 2}\,F\wedge\,^*F\,.\label{emlagrang1}$$ From it follows a field equation of the form (\[covfieldeq1\]) where the excitation (\[definition01\]) is given by the Maxwell-Lorentz electromagnetic spacetime relation $$H={}^*F\,,\label{emmom}$$ involving (\[Fdef\]), which identically satisfies $$dF =0\,.\label{vanfder}$$ Eqs. (\[covfieldeq1\]) and (\[vanfder\]) complete the set of Maxwell’s equations for fundamental matter in vacuum. On the other hand, the electromagnetic part (\[sigmaem\]) of energy-momentum derived from the explicit Lagrangian (\[emlagrang1\]) reads $$\Sigma ^{\rm em}_\alpha = {1\over 2}\,\left[\,\left( e_\alpha\rfloor F\right)\wedge H -F\wedge\left( e_\alpha\rfloor H\right)\,\right]\,,\label{emenergymom}$$ so that (\[emenergy\]) becomes $$\epsilon ^{\rm em} = -{1\over 2}\,\bigl(\,F_{\bot}\wedge H -F\wedge H_{\bot}\,\bigr)\,,\label{explemen1}$$ obeying Eq.(\[emender\]). The transversal component $\underline{\epsilon}^{\rm em}$ of the electromagnetic energy current 3-form (\[explemen1\]) is the energy 3-form representing the amount of electric and magnetic energy contained in a small volume, and the longitudinal part $\epsilon ^{\rm em}_{\bot}$ is the energy flux or Poynting 2-form. Gravity ------- For the gravitational action, we consider a quite general Lagrangian density taken from Ref. [@Obukhov:2006ge], including a Hilbert-Einstein term with cosmological constant, plus additional contributions quadratic in the Lorentz-irreducible pieces of torsion and curvature as established by McCrea [@Hehl:1995ue] [@McCrea:1992wa]. The gravitational Lagrangian reads $$\begin{aligned} L^{\rm gr}&=&{1\over{\kappa}}\,\left(\,\,{a_0\over 2}\,\,R^{\alpha\beta}\wedge\eta_{\alpha\beta} -\Lambda\,\eta\,\right)\nonumber\\ &&-{1\over 2}\,\,T^\alpha\wedge \left(\sum_{I=1}^{3}{{a_{I}}\over{\kappa}}\,\,{}^{*(I)} T_\alpha\right)\nonumber\\ &&-{1\over 2}\,\,R^{\alpha\beta}\wedge\left(\sum_{I=1}^{6}b_{I}\,\, {}^{*(I)}R_{\alpha\beta}\right)\,,\label{gravlagr}\end{aligned}$$ with $\kappa$ as the gravitational constant, and $a_0$, $a_{I}$, $b_{I}$ as dimensionless constants. From (\[gravlagr\]) we calculate the translational and Lorentz excitations (\[definition02\]) to be respectively $$\begin{aligned} H_\alpha &=& \sum_{I=1}^{3}{{a_{I}}\over{\kappa}}\,\,{}^{*(I)} T_\alpha\,,\label{torsmom}\\ H_{\alpha\beta}&=&-{a_0\over{2\kappa}}\,\eta_{\alpha\beta} +\sum_{I=1}^{6}b_{I}\,\, {}^{*(I)}R_{\alpha\beta}\,,\label{curvmom}\end{aligned}$$ and we find the pure gravitational contribution (\[ealpha\]) to the energy-momentum $$\begin{aligned} E_\alpha &=& {a_0\over {4\kappa}}\,e_\alpha\rfloor \left(\,R^{\beta\gamma}\wedge\eta_{\beta\gamma}\,\right)-{\Lambda\over{\kappa}}\,\eta _\alpha\nonumber\\ &&+{1\over 2}\,\left[\,\left( e_\alpha\rfloor T^\beta\right)\wedge H_\beta -T^\beta\wedge\left( e_\alpha\rfloor H_\beta \right)\,\right]\nonumber\\ &&+{1\over 2}\,\left[\,\left( e_\alpha\rfloor R^{\beta\gamma}\right)\wedge H_{\beta\gamma} -R^{\beta\gamma}\wedge\left( e_\alpha\rfloor H_{\beta\gamma}\right)\,\right]\,.\nonumber\\ \label{gravenergymom}\end{aligned}$$ (Notice the resemblance between (\[gravenergymom\]) and (\[emenergymom\]).) The gauge-theoretical equations (\[covfieldeq2\]) with (\[gravenergymom\]) and (\[momentdecomp\]) constitute a generalization of Einstein’s equations. Actually, for $a_0=1\,$, $a_{I}=0\,$, $b_{I}=0\,$ and vanishing torsion, (\[gravenergymom\]) reduces to $$E_\alpha = {1\over{\kappa}}\,\left(\,\,{1\over 2}\,\,R^{\beta\gamma}\wedge\eta_{\beta\gamma\alpha} -\Lambda\,\eta _\alpha\,\right)\,,\label{H-Egravenergymom}$$ which is simply an exterior calculus reformulation of Einstein’s tensor plus a cosmological constant term. Using the general expression (\[gravenergymom\]), we calculate the gravitational energy (\[grenergy\]) to be $$\begin{aligned} \epsilon ^{\rm gr} &=& -{a_0\over {4\kappa}}\,\bigl(\,R^{\alpha\beta}\wedge\eta_{\alpha\beta}\,\bigr)_{\bot}+{\Lambda\over{\kappa}}\,u^\alpha\eta _\alpha\nonumber\\ &&-{1\over 2}\,\bigl(\,T_{\bot}^\alpha\wedge H_\alpha -T^\alpha\wedge H_{{\bot}\alpha}\,\bigr)\nonumber\\ &&-{1\over 2}\,\bigl(\,R_{\bot}^{\alpha\beta}\wedge H_{\alpha\beta} -R^{\alpha\beta}\wedge H_{{\bot}\alpha\beta}\,\bigr)\nonumber\\ &&-D u^\alpha\wedge H_\alpha\,,\label{explgren}\end{aligned}$$ (compare with (\[explemen1\])), obeying Eq.(\[grender\]). Energy-momentum 3-form of macroscopic matter ============================================ Contrarily to the former cases of electromagnetism and gravity, we do not propose a Lagrangian for macroscopic matter. Instead, we focus our attention on the matter energy-momentum 3-form $\Sigma ^{\rm matt}_\alpha $, for which we postulate the dynamical equation (\[sigmamattconserv\]), and any other in which it appears, to hold macroscopically. The energy-momentum (\[sigmamatt\]) found for Dirac matter does not play any role when considering macroscopic systems. The description of each kind of material medium requires the construction of a suitably chosen energy-momentum 3-form adapted to it. In the present Section we merely present a useful decomposition applicable to any $\Sigma ^{\rm matt}_\alpha$, and we consider the form of the simplest of all mechanic energy-momentum contributions, namely that due to pressure, which we explicitly separate from the whole macroscopic matter energy-momentum. By using projectors (\[form03\]) and definition (\[mattenergy\]), we find $$\begin{aligned} \Sigma ^{\rm matt}_\alpha &&\equiv ( -u_\alpha u^\beta + h_\alpha{}^\beta ) \Sigma ^{\rm matt}_\beta\nonumber\\ &&=: u_\alpha\,\epsilon ^{\rm matt} +\widetilde{\Sigma}^{\rm matt}_\alpha\,,\label{enmom02}\end{aligned}$$ making apparent the pure energy content of energy-momentum . On the other hand, to give account of pressure, we separate the pressure term from an energy-momentum 3-form as $$\begin{aligned} \Sigma ^{\rm matt}_\alpha &=& p\,h_\alpha{}^\beta\,\eta _\beta +\Sigma ^{\rm undef}_\alpha\nonumber\\ &=&-d\tau\wedge p\,\overline{\eta}_\alpha +\Sigma ^{\rm undef}_\alpha\,,\label{enmom01}\end{aligned}$$ with $\overline{\eta}_\alpha$ as defined in (\[3deta07\]), while $\Sigma ^{\rm undef}_\alpha $ is left undefined. By decomposing (\[enmom01\]) according to (\[enmom02\]), we get $$\Sigma ^{\rm matt}_\alpha = u_\alpha\,\epsilon ^{\rm matt} -d\tau\wedge p\,\overline{\eta}_\alpha +\widetilde{\Sigma}^{\rm undef}_\alpha\,.\label{enmom03}$$ The piece $\widetilde{\Sigma}^{\rm undef}_\alpha $ present in (\[enmom03\]) after the separation of the energy term can be chosen in different manners to describe, as the case may be, viscosity, elasticity, plasticity, etc. Actually, (\[enmom03\]) resembles the energy-momentum 3-form of a fluid plus additional contributions responsible for different mechanic features. Notice that, being (\[sigmamattconserv\]) a dynamical equation of the form $$D\,\Sigma ^{\rm matt}_\alpha = f_\alpha\,,\label{force01}$$ where the 4-form $f_\alpha$ is a generalized Lorentz force, by replacing (\[enmom03\]) in it, we get (at least formally) an extended Navier-Stokes equation. Electrodynamic equations in material media ========================================== Looking for a general criterion about the most suitable procedure to include phenomenological matter in the fundamental equations, let us examine in particular electromagnetism in order to find out how to generalize (\[covfieldeq1\]) as much as (\[emender\]) in such a manner that they become applicable macroscopically while preserving their form. As a matter of fact, Maxwell’s equations in matter admit two alternative formulations, depending on how the electric and magnetic properties of material media are taken into account [@Hehl-and-Obukhov] [@Obukhov:2003cc]. Actually, polarization and magnetization can be described, in seemingly equivalent ways, either as due to modifications of the electromagnetic excitations $H$ or as the result of the existence inside such materials of generalized currents $J$ including both, free and bound contributions. With the latter approach in mind, we define the total current density $J^{\rm tot}$ as the sum of a current $J^{\rm free}$ of free charge and a matter-bounded contribution $J^{\rm matt}$ characteristic for the medium, that is $$J^{\rm tot} = J^{\rm free} + J^{\rm matt}\,,\label{totcurr01}$$ with the assumption that they are conserved separately as $$dJ^{\rm free}=0\,,\qquad dJ^{\rm matt}=0\,,\label{totcurrconserv}$$ so that, although both types of charge can coexist, no exchange occurs between them. From the second conservation condition in (\[totcurrconserv\]), we infer the existence of an independent excitation 2-form, which we denote as $H^{\rm matt}$, such that $$J^{\rm matt}= -dH^{\rm matt}\,.\label{indepexcits}$$ For the longitudinal and transversal pieces of $H^{\rm matt}$ we introduce the notation $$H^{\rm matt}= -d\tau\wedge M + P\,,\label{matexcit01}$$ where $M$ is the magnetization 1-form and $P$ the polarization 2-form. The extension of Maxwell’s equations (\[covfieldeq1\]) to include the contribution (\[indepexcits\]) of the material medium without altering their form can then be performed in any of the alternative ways mentioned above. Let us define $$H^{\rm bare} :={}^*F\,,\label{macMax05}$$ (where we call [*bare fields*]{} the fields in vacuum) in analogy to the Maxwell-Lorentz spacetime relation (\[emmom\]). Then, according to the first procedure, consisting in considering the electromagnetic effects of the medium as due to a modification of the electromagnetic excitations, the latter ones $H$ as much as $J$ in (\[covfieldeq1\]) are to be understood respectively as $$H = H^{\rm tot} := H^{\rm bare} +H^{\rm matt}\quad{\rm and}\quad J= J^{\rm free}\,,\label{secondcase}$$ while in the second case such effects are characterized in terms of bounded currents, so that the same equation (\[covfieldeq1\]) is to be read taking in it now $$H = H^{\rm bare}\quad{\rm and}\quad J = J^{\rm tot} := J^{\rm free} - dH^{\rm matt}\,.\label{firstcase}$$ Let us show that, despite appearances, both formulations are not trivially interchangeable. Actually, only one of them can be easily adjusted to our program of generalizing the [*microscopic*]{} formulas (\[mattender\]) and (\[emender\]) to include the contributions of the medium. Our main argument to decide in favor of one of both alternatives (in the present context) is that the electromagnetic energy (\[explemen1\]) is different in each case, in such a way that, for arbitrary $P$ and $M$, Eq.(\[emender\]) is compatible with only one of the possible choices. Making use of (\[foliat1\]), we decompose the electromagnetic excitation 2-form $H$, the electromagnetic field strength 2-form $F$ and the current $J$ of Maxwell’s equations (\[covfieldeq1\]) and (\[vanfder\]) as $$\begin{aligned} H &=& d\tau\wedge {\cal H} + {\cal D}\,,\label{Max01}\\ F &=& -d\tau\wedge E + B\,,\label{Max02}\\ J &=& -d\tau\wedge j + \rho\,.\label{Max03}\end{aligned}$$ Accordingly, the foliation of (\[covfieldeq1\]) yields $$\begin{aligned} {\it{l}}_u {\cal D} -\underline{d}\,{\cal H} &=& -j\,,\label{Max07}\\ \underline{d}\,{\cal D}&=& \rho\,.\label{Max08}\end{aligned}$$ and that of (\[vanfder\]) gives rise to $$\begin{aligned} {\it{l}}_u B +\underline{d}\,E &=& 0\,,\label{Max09}\\ \underline{d}\,B &=& 0\,.\label{Max10}\end{aligned}$$ In Eqs. (\[Max07\])–(\[Max10\]) we do not prejudge which of both interpretations is to be given to the different fields. In order to decide, we express (\[macMax05\]) in terms of the Hodge dual (\[foliat2\]) of (\[Max02\]) $$^*F = d\tau\wedge{}^\#B + {}^\#E\,.\label{Max04}$$ So we see that (\[secondcase\]) corresponds to the choice $${\cal D} ={}^\#E +P\,,\quad {\cal H}={}^\#B -M\,,\quad J=J^{\rm free}\,,\label{elmagexcits02}$$ in the Maxwell equations (\[Max07\])–(\[Max10\]), with $$J^{\rm free}= -d\tau\wedge j^{\rm free} + \rho ^{\rm free}\,,\label{freecurr}$$ while (\[firstcase\]) gives rise to $${\cal D} ={}^\#E\,,\quad {\cal H}={}^\#B\,,\quad J=J^{\rm tot}\,,\label{elmagexcits01}$$ being $$J^{\rm tot}= -d\tau\wedge ( j^{\rm free} +{\it{l}}_u P +\underline{d}\,M\,) + (\rho ^{\rm free}-\underline{d}\,P\,)\,,\label{totcurr}$$ as calculated from (\[totcurr01\]) with (\[indepexcits\]) and (\[matexcit01\]). Now, in order to check the compatibility either of (\[elmagexcits02\]) or (\[elmagexcits01\]) with (\[emender\]), we add (\[Max07\]) and (\[Max09\]) to each other, respectively multiplied by $E$ and ${\cal H}$, to get $$E\wedge{\it{l}}_u {\cal D} + {\it{l}}_u B\wedge{\cal H} +\underline{d}\,(E\wedge {\cal H}) = -E\wedge j\,,\label{Poynting01}$$ and on the other hand, we rewrite the transversal part of (\[explemen1\]) as $$\underline{\epsilon}^{\rm em} ={1\over 2}\,( E\wedge{\cal D} + B\wedge {\cal H}\,)\,.\label{emendh}$$ We can see that, in general, for nonspecified $P$ and $M$, the step from (\[Poynting01\]) to (\[emender\]) with $\epsilon ^{\rm em}$ given by (\[emendh\]) is only possible with the choice (\[elmagexcits01\]) for the excitations. Indeed, notice that the first term in the rhs of (\[emender\]) has its origin in the relation $$\begin{aligned} &&{\it{l}}_u \underline{\epsilon}^{\rm em} := {\it{l}}_u\,{1\over 2}\left( E\wedge{}^\#E + B\wedge {}^\#B\,\right)\nonumber\\ &&\hskip1.0cm \equiv E\wedge {\it{l}}_u {}^{\#}E + {\it{l}}_u B\wedge {}^{\#}B -({\cal \L\/}_u\,\vartheta ^\alpha\wedge\Sigma ^{\rm em}_\alpha )_{\bot}\,,\nonumber\\ \label{ident02}\end{aligned}$$ derived with the help of the identities $$\begin{aligned} {\it{l}}_u {}^{\#}E &\equiv &\,{}^{\#}\Bigl(\,{\it{l}}_u E -{\cal \L\/}_u\underline{\vartheta}^\alpha\, e_\alpha\rfloor E\,\Bigr) +{\cal \L\/}_u\underline{\vartheta}^\alpha\wedge\left( e_\alpha\rfloor {}^{\#}E\,\right)\,,\nonumber\\ \label{formula01}\\ {\it{l}}_u {}^{\#}B &\equiv &\,{}^{\#}\Bigl(\,{\it{l}}_u B -{\cal \L\/}_u\underline{\vartheta}^\alpha\wedge e_\alpha\rfloor B\,\Bigr) +{\cal \L\/}_u\underline{\vartheta}^\alpha\wedge\left( e_\alpha\rfloor {}^{\#}B\,\right)\,.\nonumber\\ \label{formula02}\end{aligned}$$ (Compare with (\[dualvar\]).) Thus, although (\[Poynting01\]) holds in both approaches, it only can be brought to the form (\[emender\]) within the scope of choice (\[elmagexcits01\]), or equivalently of (\[firstcase\]), the latter thus revealing to be necessary in order to guarantee the general applicability of the fundamental formulas found for microscopic matter. Accordingly, we choose option (\[firstcase\]), which in practice means that, in order to apply the original formula (\[covfieldeq1\]) of the fundamental approach, we have to keep in it the excitation $H =H^{\rm bare}={}^*F$ built from bare fields, and to include all contributions of the medium in the matter current by replacing $J$ by $J^{\rm tot}=J -dH^{\rm matt}$, where the new $J$ in $J^{\rm tot}$ is understood to be $J^{\rm free}$. In the following, we generalize this criterion of strict separation between bare electromagnetic fields (say radiation in vacuum) and matter, in such a way that it also applies to the gravitational case. So, in all field equations and Noether identities established in Sections II and III, we have to leave untouched the excitations $H$, $H_\alpha$, $H_{\alpha\beta}$ built from bare fields as in Section VI, while modifying the matter currents $J$, $\Sigma ^{\rm matt}_\alpha $, $\tau _{\alpha\beta}$. The matter contributions separated from bare fields will enter $\epsilon ^{\rm matt}$ and thus $\epsilon ^{\rm u}$ as defined in Section IX, so that they will play a role in the thermodynamic relations to be established there. Deduction of the laws of thermodynamics ======================================= First approach, in an electromagnetic medium -------------------------------------------- In view of the discussion of previous section, we identify $H$ with $H^{\rm bare}$ and, in order to adapt Eq.(\[mattender\]) to a macroscopic medium with electromagnetic properties, we replace in it (as everywhere) $J$ by $J^{\rm tot}$, that is $$d\,\epsilon ^{\rm matt} = -{\cal \L\/}_u\,\vartheta ^\alpha\wedge\Sigma ^{\rm matt}_\alpha -R_{\bot}^{\alpha\beta}\wedge\tau _{\alpha\beta} -F_{\bot}\wedge J^{\rm tot}\,.\label{emmattender}$$ Taking into account the explicit form (\[totcurr\]) of $J^{\rm tot}$, we find that (\[emmattender\]) can be rewritten as $$\begin{aligned} &&\mkern-60mu d\,\bigl(\,\epsilon ^{\rm matt} +F\wedge M\,\bigr)\nonumber\\ &&= -{\cal \L\/}_u\,\vartheta ^\alpha\wedge\Sigma ^{\rm matt}_\alpha -R_{\bot}^{\alpha\beta}\wedge\tau _{\alpha\beta}-F_{\bot}\wedge J\nonumber\\ &&\quad + d\tau\wedge\Bigl\{ -F_{\bot}\wedge{\it l}_u P +\underline{F}\wedge{\it l}_u M\Bigr\}\,,\label{diff02}\end{aligned}$$ where we use simply $J$ instead of $J^{\rm free}$. Let us define the modified matter energy current in the left-hand side (lhs) of (\[diff02\]) as $$\epsilon ^{\rm u} := \epsilon ^{\rm matt} + F\wedge M\,.\label{intenergycurr01}$$ Then, from (\[diff02\]) and (\[intenergycurr01\]) and using the notation (\[Max02\]) for $F$, we find the more explicit version of (\[diff02\]) $$\begin{aligned} d\,\epsilon ^{\rm u} &=&d\tau\wedge\Bigl\{{\Sigma _{\alpha}}_{\bot}^{\rm matt}\wedge{\cal \L\/}_u\underline{\vartheta}^\alpha -\underline{\Sigma}^{\rm matt}_\alpha\,{\cal \L\/}_u u^\alpha +R_{\bot}^{\alpha\beta}\wedge{\tau _{\alpha\beta}}_{\bot}\nonumber\\ &&\hskip1.5cm +E\wedge j +E\wedge{\it l}_u P +B\wedge{\it l}_u M \Bigr\}\,,\label{diff01}\end{aligned}$$ where we recognize in the rhs, among other forms of energy, the electric and magnetic work contributions $E\wedge{\it l}_u P$ and $B\wedge{\it l}_u M$ respectively. Let us now decompose (\[intenergycurr01\]) foliating it according to (\[foliat1\]) and introducing a suitable notation for the longitudinal and transversal pieces, namely $$\begin{aligned} \epsilon ^{\rm u} &=& d\tau\wedge\epsilon ^{\rm u}_{\bot} + \underline{\epsilon}^{\rm u}\nonumber\\ &=:& d\tau\wedge q + \mathfrak{U}\,.\label{intenergycurr02}\end{aligned}$$ As we are going to justify in the following (in view of the equations satisfied by these quantities), $q$ will play the role of the heat flux 2-form and $\mathfrak{U}$ that of the internal energy 3-form. From (\[intenergycurr02\]) with (\[derivfoliat\]) we get $$d\,\epsilon ^{\rm u}= d\tau\wedge\left(\,{\it l}_u\,\mathfrak{U} - \underline{d}\,q\,\right)\,.\label{energycurrder01}$$ At this point, we claim as a characteristic of macroscopic matter systems [@Callen] the dependence of the internal energy 3-form $\mathfrak{U}$ on a certain new quantity $\mathfrak{s}$ –the entropy– which we take to be a spatial 3-form (representing the amount of entropy contained in an elementary volume). Eq.(\[secondlaw\]) to be found below confirms [*a posteriori*]{} that $\mathfrak{s}$ actually behaves as expected for entropy. Moreover, the structure of (\[diff01\]) suggests to promote a shift towards a fully phenomenological approach by considering $\mathfrak{U}$ to possess [@Callen] the following general functional dependence $$\mathfrak{U} = \mathfrak{U}\,(\mathfrak{s}\,,P\,,M\,,\underline{\vartheta}^\alpha \,, u^\alpha\,)\,.\label{uargs}$$ In (\[uargs\]), as in the matter Lagrangian piece (\[mattLagcontrib\]), tetrads are still taken as arguments of $\mathfrak{U}$ while new variables replace the fundamental matter fields $\psi$ and their covariant derivatives $D\psi$. Connections involved in the derivatives $D\psi$ are thus excluded together with the fields. Besides the new entropy variable and the polarization and magnetization of the medium (induced by external fields), we find the components (\[tetradfoliat\]) of the tetrads in terms of which the volume 3-form (\[3deta06\]) with (\[3deta09\]) is defined. Accordingly, the Lie derivative of (\[uargs\]) present in (\[energycurrder01\]) takes the form $$\begin{aligned} {\it l}_u\,\mathfrak{U} &=& {{\partial\mathfrak{U}}\over{\partial\mathfrak{s}}}\,{\it l}_u\mathfrak{s} +{{\partial\mathfrak{U}}\over{\partial P}}\wedge{\it l}_u P +{{\partial\mathfrak{U}}\over{\partial M}}\wedge{\it l}_u M\nonumber\\ &&+{{\partial\mathfrak{U}}\over{\partial\underline{\vartheta}^\alpha}}\wedge{\it l}_u\underline{\vartheta}^\alpha +{{\partial\mathfrak{U}}\over{\partial u^\alpha}}\,{\it l}_u u^\alpha\,,\label{uLiederiv01}\end{aligned}$$ where we identify the derivatives [@Callen] as $$\begin{aligned} && {{\partial\mathfrak{U}}\over{\partial\mathfrak{s}}} =T\,,\quad {{\partial\mathfrak{U}}\over{\partial P}} =E\,,\quad {{\partial\mathfrak{U}}\over{\partial M}} =B\,,\label{Uder01}\\ &&{{\partial\mathfrak{U}}\over{\partial\underline{\vartheta}^\alpha}}={\Sigma _{\alpha}}_{\bot}^{\rm matt}\,,\quad {{\partial\mathfrak{U}}\over{\partial u^\alpha}}=-\underline{\Sigma}^{\rm matt}_\alpha\,.\label{Uder02}\end{aligned}$$ Let us call attention to the temperature defined in (\[Uder01\]) as the derivative of the internal energy with respect to the entropy. On the other hand, a plausibility argument to justify the identifications we make in (\[Uder02\]) can be found in Appendix B. Replacing (\[Uder01\])–(\[Uder02\]) in (\[uLiederiv01\]) we get $$\begin{aligned} {\it l}_u\,\mathfrak{U} &=& T\,{\it l}_u\mathfrak{s} +E\wedge{\it l}_u P +B\wedge{\it l}_u M\nonumber\\ &&+{\Sigma _{\alpha}}_{\bot}^{\rm matt}\wedge{\it l}_u\underline{\vartheta}^\alpha -\underline{\Sigma}^{\rm matt}_\alpha\,{\it l}_u u^\alpha\,.\label{uLiederiv02}\end{aligned}$$ In order to rearrange the non explicitly invariant terms in (\[uLiederiv02\]) to get invariant expressions, we replace the ordinary Lie derivatives by covariant Lie derivatives of the form (\[thetaLiederiv02\]), so that the last terms in (\[uLiederiv02\]) become $$\begin{aligned} {\Sigma _{\alpha}}_{\bot}^{\rm matt}\wedge{\it l}_u\underline{\vartheta}^\alpha -\underline{\Sigma}^{\rm matt}_\alpha\,{\it l}_u u^\alpha &\equiv& {\Sigma _{\alpha}}_{\bot}^{\rm matt}\wedge{\cal \L\/}_u\underline{\vartheta}^\alpha -\underline{\Sigma}^{\rm matt}_\alpha\,{\cal \L\/}_u u^\alpha\nonumber\\ &&+\Gamma _{\bot}^{\alpha\beta}\bigl(\,\vartheta _{[\alpha}\wedge\Sigma ^{\rm matt}_{\beta ]}\bigr)_{\bot}\,.\label{identity01}\end{aligned}$$ Replacing (\[identity01\]) in (\[uLiederiv02\]) we finally arrive at $$\begin{aligned} {\it l}_u\,\mathfrak{U} &=& T\,{\it l}_u\mathfrak{s} +E\wedge{\it l}_u P +B\wedge{\it l}_u M\nonumber\\ &&+{\Sigma _{\alpha}}_{\bot}^{\rm matt}\wedge{\cal \L\/}_u\underline{\vartheta}^\alpha -\underline{\Sigma}^{\rm matt}_\alpha\,{\cal \L\/}_u u^\alpha\nonumber\\ &&+\Gamma _{\bot}^{\alpha\beta}\bigl(\,\vartheta _{[\alpha}\wedge\Sigma ^{\rm matt}_{\beta ]}\bigr)_{\bot}\,.\label{uLiederiv03}\end{aligned}$$ In the rhs of (\[uLiederiv03\]), the term containing explicitly the Lorentz connection is obviously noninvariant. Its emergence is due to an inherent limitation of the phenomenological approach, namely the absence of explicit dependence of $\mathfrak{U}$ on fundamental matter fields and their derivatives, together wit connections. Indeed, provided matter fields $\psi$ with derivatives $d\psi$ were present, connections were required to define covariant derivatives preserving local symmetry. However, in the phenomenological case, $\mathfrak{U}$ depends neither on $\psi$ nor on $d\psi$, so that (since $d\psi$ and connections need each other) it cannot give rise to invariant expressions, either one takes it or not to depend on the connections. The noninvariant term in (\[uLiederiv03\]), reflecting the lack of invariance of the terms in the lhs of (\[identity01\]), will be dragged to equations (\[energycurrder02\]) and (\[secondlaw\]) below. (We will find a similar situation in (\[uLiederiv04bis\]) and (\[diff01tot\]).) In any case, let us mention that the invariance is restored in the particular case when the macroscopic free spin current $\tau _{\alpha\beta}$ vanishes. Making use of (\[uLiederiv03\]), Eq.(\[diff01\]) reduces to $$\begin{aligned} d\,\epsilon ^{\rm u} &=& d\tau\wedge\Bigl[\,{\it l}_u\,\mathfrak{U} - T\,{\it l}_u\mathfrak{s} + E\wedge j + R_{\bot}^{\alpha\beta}\wedge{\tau _{\alpha\beta}}_{\bot}\nonumber\\ &&\hskip1.5cm -\Gamma _{\bot}^{\alpha\beta}\bigl(\,\vartheta _{[\alpha}\wedge\Sigma ^{\rm matt}_{\beta ]}\bigr)_{\bot} \,\Bigr]\,,\label{energycurrder02}\end{aligned}$$ and finally, comparison of (\[energycurrder02\]) with (\[energycurrder01\]), making use of (\[spincurrconserv\]), yields $${\it l}_u\mathfrak{s} -{{\underline{d}\,q}\over T} = {1\over T}\,\bigl[\,E\wedge j + R_{\bot}^{\alpha\beta}\wedge{\tau _{\alpha\beta}}_{\bot} +\Gamma _{\bot}^{\alpha\beta}\bigl(\,D\,\tau _{\alpha\beta}\bigr)_{\bot}\,\bigr]\,.\label{secondlaw}$$ In the lhs of (\[secondlaw\]) we find the rate of change of the entropy 3-form combined in a familiar way with heat flux and temperature. The interpretation of the first term in the rhs is facilitated by the fact that, according to Ohm’s law $j=\sigma\,{}^\# E$, it is proportional to $E\wedge j ={1\over\sigma} j\wedge{}^\# j \geq 0$, so that it is responsible for entropy growth. The second term is analogous to the first one. If we suppose that all terms in the rhs of (\[secondlaw\]) are $\geq 0$, or, in any case, for vanishing macroscopic free spin current $\tau _{\alpha\beta}$, we can consider (\[secondlaw\]) to be a particular realization of the second law of thermodynamics. On the other hand, the first law is no other than the conservation equation (\[emmattender\]) for matter energy, rewritten as (\[diff01\]) in terms of the internal energy current 3-form (\[intenergycurr01\]). This reformulation is necessary in order to bring to light the components of $\epsilon ^{\rm u}$ defined in (\[intenergycurr02\]), that is, heat flux and internal energy respectively, thus making possible to compare the first law with the second one (\[secondlaw\]) deduced above. (By the way, notice that the inversion of (\[intenergycurr01\]) to express $\epsilon ^{\rm matt}$ in terms of $\epsilon ^{\rm u}$ suggests to interpret $\epsilon ^{\rm matt}$ as a sort of enthalpy current 3-form.) Making use of (\[energycurrder01\]), the first law (\[diff01\]) can be brought to the more compact form $$\begin{aligned} {\it l}_u\,\mathfrak{U} -\underline{d}\,q &=& -\bigl(\,{\cal \L\/}_u\,\vartheta ^\alpha\wedge\Sigma ^{\rm matt}_\alpha \bigr)_{\bot} +R_{\bot}^{\alpha\beta}\wedge{\tau _{\alpha\beta}}_{\bot}\nonumber\\ &&+E\wedge j +E\wedge{\it l}_u P +B\wedge{\it l}_u M\,.\label{uLiederiv03bis}\end{aligned}$$ The first term in the rhs of (\[uLiederiv03bis\]), that is, the longitudinal part of ${\cal \L\/}_u\,\vartheta ^\alpha\wedge\Sigma ^{\rm matt}_\alpha $, encloses information about mechanic work, whose form depends on the explicit matter energy-momentum 3-form we consider. In particular, by taking it to consist of a pressure term plus an undefined part, as in (\[enmom01\]), we find $$\bigl(\,{\cal \L\/}_u\,\vartheta ^\alpha\wedge\Sigma ^{\rm matt}_\alpha\,\bigr)_{\bot} = \bigl(\,{\cal \L\/}_u\,\vartheta ^\alpha\wedge\Sigma ^{\rm undef}_\alpha\,\bigr)_{\bot} +{\cal \L\/}_u\underline{\vartheta}^\alpha\wedge p\,\overline{\eta}_\alpha\,,\label{presscontrib}$$ where the last term, in view of (\[volLieder\]), results to be $${\cal \L\/}_u\underline{\vartheta}^\alpha\wedge p\,\overline{\eta}_\alpha = p\,{\it l}_u\overline{\eta}\,,\label{pressderiv}$$ being thus identifiable as the ordinary pressure contribution to work as pressure times the derivative of the volume. It is worth remarking that the emergence of this pressure contribution to the first law does not ocur through derivation of $\mathfrak{U}$ with respect to the volume $\overline{\eta}$ (which is not an independent variable by itself, being defined from the tetrads as (\[3deta06\])), but with respect to the tetrad components, as in (\[Uder02\]). Replacing (\[presscontrib\]) with (\[pressderiv\]) in the first law equation (\[uLiederiv03bis\]), we get for it the more explicit formulation $$\begin{aligned} {\it l}_u\,\mathfrak{U} -\underline{d}\,q &=& -\bigl(\,{\cal \L\/}_u\,\vartheta ^\alpha\wedge\Sigma ^{\rm undef}_\alpha \bigr)_{\bot} +R_{\bot}^{\alpha\beta}\wedge{\tau _{\alpha\beta}}_{\bot} +E\wedge j\nonumber\\ &&-p\,{\it l}_u\overline{\eta}+E\wedge{\it l}_u P +B\wedge{\it l}_u M \,,\label{firstlaw01}\end{aligned}$$ where one recognizes the familiar contributions of internal energy, heat flux and work \[including $\bigl(\,{\cal \L\/}_u\,\vartheta ^\alpha\wedge\Sigma ^{\rm undef}_\alpha\,\bigr)_{\bot}$ among the latter ones\], together with additional terms. In particular, $E\wedge j$ and the formally similar quantity $R_{\bot}^{\alpha\beta}\wedge{\tau _{\alpha\beta}}_{\bot}$ are present in (\[firstlaw01\]) due to irreversibility, as read out from (\[secondlaw\]). General approach ---------------- Let us extend the previous results to the most general scenario in which we modify all matter currents in analogy to $J^{\rm (tot)}$ in order to take into account further possible contributions of a medium. In an attempt to expand the electromagnetic model, we introduce –associated to gravitational interactions– translational and Lorentz generalizations of the electromagnetic polarization and magnetization of macroscopic matter. Maybe this constitutes a merely formal exercise. However, it can also be understood as a proposal to look for new properties of material media, since we are going to consider the hypothesis of certain new phenomenological matter contributions to the sources of gravity, acting perhaps as dark matter. Generalizing (\[firstcase\]), we propose to modify the complete set of field equations (\[covfieldeq1\])–(\[covfieldeq3\]) as $$\begin{aligned} dH &=&J^{\rm (tot)}\,,\label{covfieldeq1bis} \\ DH_\alpha &=&\Pi ^{\rm (tot)}_\alpha\,,\label{covfieldeq2bis}\\ DH_{\alpha\beta} +\vartheta _{[\alpha }\wedge H_{\beta ]}&=&\tau ^{\rm (tot)}_{\alpha\beta}\,,\label{covfieldeq3bis}\end{aligned}$$ with bare excitations and total currents consisting of the sum of free and bound contributions, defined respectively as $$\begin{aligned} J^{\rm (tot)} &=& J-dH^{\rm matt}\,,\label{Jtot} \\ \Pi ^{\rm (tot)}_\alpha &=& \Pi _\alpha -DH^{\rm matt}_\alpha \,,\label{Pitot}\\ \tau ^{\rm (tot)}_{\alpha\beta} &=& \tau _{\alpha\beta} - ( DH^{\rm matt}_{\alpha\beta} +\vartheta _{[\alpha }\wedge H^{\rm matt}_{\beta ]})\,,\label{Tautot}\end{aligned}$$ where we introduce generalizations of the electromagnetic polarization and magnetization (\[matexcit01\]) as $$\begin{aligned} H^{\rm matt} &=& -d\tau\wedge M + P\,,\label{matexcit01bis}\\ H_\alpha ^{\rm matt} &=& -d\tau\wedge M_\alpha + P_\alpha \,,\label{matexcit02}\\ H_{\alpha\beta}^{\rm matt} &=& -d\tau\wedge M_{\alpha\beta} + P_{\alpha\beta}\,,\label{matexcit03}\end{aligned}$$ whatever the physical correspondence of these quantities may be. Since, as discussed above, only matter currents are to be modified, we understand (\[Pitot\]) in the sense that only the matter part is altered, that is $$\Pi ^{\rm (tot)}_\alpha = \Sigma ^{\rm matt}_{{\rm (tot)}\alpha } +\Sigma ^{\rm em}_\alpha +E_\alpha\,,\label{totmomentdecomp}$$ being $$\Sigma ^{\rm matt}_{{\rm (tot)}\alpha } = \Sigma ^{\rm mat}_\alpha -DH^{\rm matt}_\alpha\,.\label{totmattmom}$$ In view of (\[totmattmom\]), we extend (\[mattenergy\]) as $$\epsilon _{\rm (tot)}^{\rm matt} := -u^\alpha\,\Sigma ^{\rm matt}_{{\rm (tot)}\alpha } =\epsilon ^{\rm matt} + u^\alpha DH^{\rm matt}_\alpha\,,\label{totmattenergy}$$ and, as a generalization of (\[mattender\]) to include macroscopic matter, we postulate the formally analogous equation $$d\,\epsilon _{\rm (tot)}^{\rm matt} = -{\cal \L\/}_u\,\vartheta ^\alpha\wedge\Sigma ^{\rm matt}_{{\rm (tot)}\alpha } -R_{\bot}^{\alpha\beta}\wedge\tau ^{\rm (tot)}_{\alpha\beta} -F_{\bot}\wedge J^{\rm (tot)}\,,\label{genmattender01}$$ as the law of conservation of total matter energy. Eq.(\[genmattender01\]) can be rearranged as $$\begin{aligned} &&\mkern-60mu d\,\bigl(\,\epsilon ^{\rm matt} +F\wedge M + T^\alpha\wedge M_\alpha + R^{\alpha\beta}\wedge M_{\alpha\beta}\,\bigr)\nonumber\\ &&= -{\cal \L\/}_u\,\vartheta ^\alpha\wedge\Sigma ^{\rm matt}_\alpha -R_{\bot}^{\alpha\beta}\wedge\tau _{\alpha\beta} -F_{\bot}\wedge J\nonumber\\ &&\quad + d\tau\wedge\Bigl\{ -F_{\bot}\wedge{\it l}_u P +\underline{F}\wedge{\it l}_u M\nonumber\\ &&\hskip1.6cm -T_{\bot}^\alpha\wedge{\cal \L\/}_u P_\alpha +\underline{T}^\alpha\wedge{\cal \L\/}_u M_\alpha\nonumber\\ &&\hskip1.6cm -R_{\bot}^{\alpha\beta}\wedge{\cal \L\/}_u P_{\alpha\beta} +\underline{R}^{\alpha\beta}\wedge{\cal \L\/}_u M_{\alpha\beta}\Bigr\}\,.\nonumber\\ \label{diff03bis}\end{aligned}$$ (Compare with (\[diff02\]).) Without going into details, we proceed in analogy to the former case. We define a similar internal energy current 3-form $$\widehat{\epsilon}^u := \epsilon ^{\rm matt} +F\wedge M + T^\alpha\wedge M_\alpha + R^{\alpha\beta}\wedge M_{\alpha\beta}\,,\label{totintenergy}$$ decomposing as $$\widehat{\epsilon}^{\rm u} =: d\tau\wedge \widehat{q} + \widehat{\mathfrak{U}}\,.\label{intenergycurr02bis}$$ Supposing the functional form of $\widehat{\mathfrak{U}}$ to be $$\widehat{\mathfrak{U}} = \widehat{\mathfrak{U}}\,(\widehat{\mathfrak{s}}\,,P\,,M\,,P_\alpha\,,M_\alpha\,,P_{\alpha\beta}\,,M_{\alpha\beta}\,,\underline{\vartheta}^\alpha \,, u^\alpha\,)\,,\label{uargsbis}$$ and with the pertinent definitions analogous to (\[Uder01\]) and (\[Uder02\]), first we get $$\begin{aligned} {\it l}_u\,\widehat{\mathfrak{U}} &=& \widehat{T}\,{\it l}_u\widehat{\mathfrak{s}} +{\Sigma _{\alpha}}_{\bot}^{\rm matt}\wedge{\it l}_u\underline{\vartheta}^\alpha -\underline{\Sigma}^{\rm matt}_\alpha\,{\it l}_u u^\alpha\nonumber\\ &&-F_{\bot}\wedge{\it l}_u P +\underline{F}\wedge{\it l}_u M\nonumber\\ &&-T_{\bot}^\alpha\wedge{\it l}_u P_\alpha +\underline{T}^\alpha\wedge{\it l}_u M_\alpha\nonumber\\ &&-R_{\bot}^{\alpha\beta}\wedge{\it l}_u P_{\alpha\beta} +\underline{R}^{\alpha\beta}\wedge{\it l}_u M_{\alpha\beta}\,,\label{uLiederiv02bis}\end{aligned}$$ and finally, suitably rearranging the noncovariant quantities in (\[uLiederiv02bis\]) into covariant ones defined in analogy to (\[thetaLiederiv01\]), we arrive at $$\begin{aligned} {\it l}_u\,\widehat{\mathfrak{U}} &=& \widehat{T}\,{\it l}_u\widehat{\mathfrak{s}} +{\Sigma _{\alpha}}_{\bot}^{\rm matt}\wedge{\cal \L\/}_u\underline{\vartheta}^\alpha -\underline{\Sigma}^{\rm matt}_\alpha\,{\cal \L\/}_u u^\alpha\nonumber\\ &&-F_{\bot}\wedge{\it l}_u P +\underline{F}\wedge{\it l}_u M\nonumber\\ &&-T_{\bot}^\alpha\wedge{\cal \L\/}_u P_\alpha +\underline{T}^\alpha\wedge{\cal \L\/}_u M_\alpha\nonumber\\ &&-R_{\bot}^{\alpha\beta}\wedge{\cal \L\/}_u P_{\alpha\beta} +\underline{R}^{\alpha\beta}\wedge{\cal \L\/}_u M_{\alpha\beta}\nonumber\\ &&+\Gamma _{\bot}^{\alpha\beta}\Bigl[\,D\,\bigl(\tau ^{\rm (tot)}_{\alpha\beta} -\tau _{\alpha\beta}\bigr) +\vartheta _{[\alpha}\wedge\Sigma ^{\rm matt}_{\beta ]{\rm (tot)}}\Bigr]_{\bot} \,.\label{uLiederiv04bis}\end{aligned}$$ Assuming that the analogous of (\[spincurrconserv\]) holds for generalized matter, that is $$D\,\tau ^{\rm (tot)}_{\alpha\beta} +\vartheta _{[\alpha}\wedge\Sigma ^{\rm matt}_{\beta ]{\rm (tot)}} =0\,,\label{totspinconserv}$$ from (\[diff03bis\]) with (\[totintenergy\]) and (\[uLiederiv04bis\]) follows $$\begin{aligned} d\,\widehat{\epsilon}^{\rm u} &=& d\tau\wedge\Bigl[\,{\it l}_u\,\widehat{\mathfrak{U}} -\widehat{T}\,{\it l}_u\widehat{\mathfrak{s}} -F_{\bot}\wedge j + R_{\bot}^{\alpha\beta}\wedge{\tau _{\alpha\beta}}_{\bot}\nonumber\\ &&\hskip1.5cm +\Gamma _{\bot}^{\alpha\beta}\bigl(\,D\,\tau _{\alpha\beta}\bigr)_{\bot}\,\Bigr]\,,\label{diff01tot}\end{aligned}$$ giving rise, when compared with the differential of (\[intenergycurr02bis\]), to the second law of thermodynamics with exactly the same form as (\[secondlaw\]). Regarding the first law (\[diff03bis\]) with (\[totintenergy\])–(\[uargsbis\]), taking (\[enmom01\]) as before and using the notation (\[Max02\]), it takes the form $$\begin{aligned} {\it l}_u\,\widehat{\mathfrak{U}} -\underline{d}\,\widehat{q} &=& -\bigl(\,{\cal \L\/}_u\,\vartheta ^\alpha\wedge\Sigma ^{\rm undef}_\alpha \bigr)_{\bot} +R_{\bot}^{\alpha\beta}\wedge{\tau _{\alpha\beta}}_{\bot} +E\wedge j\nonumber\\ &&-p\,{\it l}_u\overline{\eta}+E\wedge{\it l}_u P +B\wedge{\it l}_u M \nonumber\\ &&-T_{\bot}^\alpha\wedge{\cal \L\/}_u P_\alpha +\underline{T}^\alpha\wedge{\cal \L\/}_u M_\alpha\nonumber\\ &&-R_{\bot}^{\alpha\beta}\wedge{\cal \L\/}_u P_{\alpha\beta} +\underline{R}^{\alpha\beta}\wedge{\cal \L\/}_u M_{\alpha\beta}\,,\label{firstlaw01bis}\end{aligned}$$ which only differs from (\[firstlaw01\]) in the additional work contributions corresponding to the gravitational generalizations of polarization and magnetization. Final remarks ============= Gravity and conservation of total energy ---------------------------------------- Let us examine the role played by gravity in the conservation of energy. In our approach, the first law of thermodynamics can take alternatively the forms (\[emmattender\]) or (\[diff01\]), being concerned with the matter energy current either in its form $\epsilon ^{\rm matt}$ or $\epsilon ^{\rm u}$. Differentiation of such matter energy currents generates work expressions, the latter ones acting physically by transforming themselves into different forms of energy. So, mechanic work can produce electric effects, etc. However, these subsequent transformations are not explicitly shown by the thermodynamic equation (\[diff01\]). Neither the sum of the matter and electromagnetic energy currents is conserved separately, since the addition of (\[mattender\]) and (\[emender\]) yields $$\begin{aligned} d\,(\epsilon ^{\rm matt}+\epsilon ^{\rm em}) &&= -{\cal \L\/}_u\,\vartheta ^\alpha\wedge (\,\Sigma ^{\rm matt}_\alpha +\Sigma ^{\rm em}_\alpha \,) -R_{\bot}^{\alpha\beta}\wedge\tau _{\alpha\beta}\nonumber\\ &&\neq 0\,.\label{energyconserv02}\end{aligned}$$ Conservation of energy in an absolute sense, with all possible transformations of different forms of energy into each other taken into account, requires to include also the gravitational energy. Indeed, from (\[energyconserv01\]) with (\[energydec\]) we get $$d\,(\epsilon ^{\rm matt}+\epsilon ^{\rm em}+\epsilon ^{\rm gr})=0\,.\label{energyconserv03}$$ This conservation equation, concerned with all forms of energy simultaneously, completes the first law of thermodynamics (\[diff01\]), which concentrates on the behavior of only the matter energy current $\epsilon ^{\rm u}$. The total energy flux $\epsilon _{\bot}$ in (\[energyconserv03\]) includes heat flux, Poynting flux in a strict sense and other Poynting-like contributions. The integrated form (\[exactform02\]) of (\[energyconserv03\]) can be seen as a sort of generalized Bernouilli’s principle. Thermal radiation ----------------- The formalism is not necessarily restricted to gauge theoretically derived forms of energy. It is flexible enough to deal with other thermodynamic approaches, as is the case for thermal radiation, the latter being described not in terms of electromagnetic fields but as a foton gas [@Prigogine] [@Demirel]. A body in thermal equilibrium is modelized as a cavity filled with a gas of thermal photons in continuous inflow and outflow. The number of photons, the internal energy and the entropy contained in the cavity, the pressure of thermal radiation on the walls and the chemical potential are all functions of the temperature, being respectively given by $$\begin{aligned} \mathcal{N} &=& \alpha\,T^3\,\overline{\eta}\,,\label{photgas01}\\ \mathfrak{U} &=& \beta\,T^4\,\overline{\eta}\,,\label{photgas02}\\ T \mathfrak{s} &=& {4\over 3}\,\mathfrak{U}\,,\label{photgas03}\\ p\,\overline{\eta} &=& {1\over 3}\,\mathfrak{U}\,,\label{photgas04}\\ \mu &=& 0\,.\label{photgas05}\end{aligned}$$ The quantities (\[photgas01\])–(\[photgas05\]) automatically satisfy the relation $${\it l}_u\,\mathfrak{U} = T\,{\it l}_u\mathfrak{s} -p\,{\it l}_u \overline{\eta}\,,\label{uLiederiv09}$$ which constitutes a particular case of the thermodynamic equations found above. Indeed, Eq. (\[uLiederiv03\]) with vanishing $P$, $M$ and $\tau _{\alpha\beta}$ reduces to $$\begin{aligned} {\it l}_u\,\mathfrak{U} = T\,{\it l}_u\mathfrak{s} +{\Sigma _{\alpha}}_{\bot}^{\rm matt}\wedge{\cal \L\/}_u\underline{\vartheta}^\alpha -\underline{\Sigma}^{\rm matt}_\alpha\,{\cal \L\/}_u u^\alpha \,.\label{uLiederiv07}\end{aligned}$$ By handling the photon gas as matter, and taking for it an energy-momentum (\[enmom03\]) with $\widetilde{\Sigma}^{\rm undef}_\alpha =0$ as $$\Sigma ^{\rm matt}_\alpha = u_\alpha\,\epsilon ^{\rm matt} -d\tau\wedge p\,\overline{\eta}_\alpha\,,\label{enmom04}$$ replacement of (\[enmom04\]) in (\[uLiederiv07\]) yields $${\it l}_u\,\mathfrak{U} = T\,{\it l}_u\mathfrak{s} -p\,{\it l}_u \overline{\eta} +\epsilon ^{\rm matt}_{\bot}\wedge u_\alpha\,T_{\bot}^\alpha\,,\label{uLiederiv08}$$ from where, for vanishing torsion, (\[uLiederiv09\]) follows. On the other hand, for thermal radiation, the second law (\[secondlaw\]) reduces [@Prigogine] to that of reversible processes $${\it l}_u\mathfrak{s} -{{\underline{d}\,q}\over T} = 0\,,\label{revsecondlaw}$$ and since the number of photons (\[photgas01\]) inside the cavity is in general not constant, we propose for this quantity the continuity equation $${\it l}_u\mathcal{N} +\underline{d} j_{_N} = \sigma _{_N}\,,\label{photnumber}$$ where we introduce $j_{_N}$ as the photon flux and $\sigma _{_N}$ as the rate of photon creation or destruction. Now, from (\[photgas01\])–(\[photgas03\]), replacing the values $$\alpha ={{16\,\pi\,k_B^3\,\zeta (3)}\over{c^3\,h^3}}\,,\qquad \beta ={{8\,\pi ^5\,k_B^4}\over{15\,c^3\,h^3}}\,,\label{alphabeta01}$$ with $\zeta $ as the Riemann zeta function, such that $\zeta (3)\approx 1.202$, and being $k_B$ the Boltzmann constant, we get the relation $$\mathfrak{s} = {4\over 3}\,{\mathfrak{U}\over T} = {{4\beta}\over{3\alpha}}\,\mathcal{N}\approx 3.6\,k_B\,\mathcal{N}\,,\label{alphabeta02}$$ so that (\[revsecondlaw\]) with (\[alphabeta02\]) yields $$\underline{d}\,q = T\,{\it l}_u\mathfrak{s} \approx 3.6\,k_B\,T\,{\it l}_u\mathcal{N}\,.\label{photheatflux}$$ With (\[photnumber\]), Eq.(\[photheatflux\]) transforms into $$\underline{d}\,q \approx 3.6\,k_B\,T\,(\sigma _{_N} -\underline{d} j_{_N})\,.\label{fluxrelat}$$ According to (\[fluxrelat\]), the divergence of the heat flux $q$ of thermal radiation is proportional to the divergence of the photon flux $j_{_N}$ continuously emitted and absorbed by a body, and it also depends on possible additional contributions $\sigma _{_N}$ due to photon production or destruction. Conclusions =========== We propose an approach to thermodynamics compatible with gauge theories of gravity and beyond. Indeed, the formalism developed in the present paper is explicitly covariant under local Lorentz transformations unless for the symmetry breaking terms present in (\[secondlaw\]) and (\[diff01tot\]), (which vanish for $\tau _{\alpha\beta}=0$). Moreover, local translational symmetry as much as local $U(1)$ symmetry are also present in our equations as hidden symmetries, due the particular realization of the Poincaré$\otimes U(1)$ gauge group used to derive the field equations and Noether identities which constituted our starting point [@Tresguerres:2007ih] [@Tresguerres:2002uh] [@Tresguerres:2012nu]. In particular, the thermodynamic equations, concerned with the exchange between different forms of energy, are both Poincaré and $U(1)$ gauge invariant. The laws of thermodynamics deduced by us concentrate on the conservation of the matter energy current $\epsilon ^{\rm matt}$ (or, equivalently, $\epsilon ^{\rm u}$), but in addition we complete the scheme giving account of the conservation of total energy, as discussed in Sec. X. In this way we synthesize the total energy balance in classical physics of material media. Eta basis and its foliation =========================== Four-dimensional formulas ------------------------- The eta basis consists of the Hodge duals of exterior products of tetrads. One defines $$\begin{aligned} \eta &:=&\,^*1 ={1\over{4!}}\,\eta _{\alpha\beta\gamma\delta}\,\vartheta ^\alpha\wedge\vartheta ^\beta\wedge\vartheta ^\gamma\wedge\vartheta ^\delta\,,\label{eta4form}\\ \eta ^\alpha &:=&\,^*\vartheta ^\alpha ={1\over{3!}}\,\eta ^\alpha{}_{\beta\gamma\delta} \,\vartheta ^\beta\wedge\vartheta ^\gamma\wedge\vartheta ^ \delta\,,\label{antisym3form}\\ \eta ^{\alpha\beta}&:=&\,^*(\vartheta ^\alpha\wedge\vartheta ^\beta\,)={1\over{2!}}\,\eta ^{\alpha\beta}{}_{\gamma\delta}\,\vartheta ^\gamma\wedge\vartheta ^\delta\,,\label{antisym2form}\\ \eta ^{\alpha\beta\gamma}&:=&\,^*(\vartheta ^\alpha\wedge\vartheta ^\beta\wedge\vartheta ^\gamma\,)=\,\eta ^{\alpha\beta\gamma}{}_\delta\,\vartheta ^\delta\,,\label{antisym1form}\end{aligned}$$ with $$\eta ^{\alpha\beta\gamma\delta}:=\,^*(\vartheta ^\alpha\wedge\vartheta ^\beta\wedge\vartheta ^\gamma\wedge\vartheta ^ \delta\,)\,,\label{levicivita}$$ as the Levi-Civita antisymmetric object, and where (\[eta4form\]) is the four-dimensional volume element. With tetrads $\vartheta ^\alpha$ chosen to be a basis of the cotangent space, an arbitrary $p$-form $\alpha$ takes the form $$\alpha ={1\over{p\,!}}\,\vartheta ^{\alpha _1}\wedge ...\wedge\vartheta ^{\alpha _p}\,(e_{\alpha _p}\rfloor ... e_{\alpha _1}\rfloor\alpha\,)\,.\label{pform}$$ Its Hodge dual is expressed in terms of the eta basis (\[eta4form\])–(\[levicivita\]) as $$\,{}^*\alpha ={1\over{p\,!}}\,\eta ^{\alpha _1 ... \alpha _p}\,(e_{\alpha _p}\rfloor ... e_{\alpha _1}\rfloor\alpha\,)\,.\label{dualform}$$ Comparison of the variations of (\[pform\]) with those of (\[dualform\]) yields the relation $$\delta \,{}^*\alpha =\,{}^*\delta\alpha -{}^*\left(\delta\vartheta ^\alpha\wedge e_\alpha\rfloor\alpha\,\right) +\delta\vartheta ^\alpha\wedge\left( e_\alpha\rfloor {}^*\alpha\,\right)\,,\label{dualvar}$$ analogous to the three-dimensional identities (\[formula01\]) and (\[formula02\]) used in the main text. Foliated eta basis ------------------ Let us now make use of (\[foliat1\]) and (\[tetradfoliat\]) to calculate $$\begin{aligned} \vartheta ^\alpha &=& d\tau\,u^\alpha + \underline{\vartheta}^\alpha\,,\label{teth01}\\ \vartheta ^\alpha\wedge\vartheta ^\beta &=& d\tau\,\Bigl( u^\alpha\,\underline{\vartheta}^\beta - u^\beta\,\underline{\vartheta}^\alpha \Bigr) + \underline{\vartheta}^\alpha\wedge\underline{\vartheta}^\beta\,,\label{teth02}\end{aligned}$$ etc. Taking then the Hodge duals of (\[teth01\]), (\[teth02\]) etc., we find the foliated version of (\[eta4form\])–(\[levicivita\]), that is $$\begin{aligned} \eta &=& d\tau\wedge\overline{\eta}\,,\label{eta04}\\ \eta ^\alpha &=& -d\tau\wedge\overline{\eta}^\alpha - u^\alpha\,\overline{\eta}\,,\label{eta03}\\ \eta ^{\alpha\beta} &=& d\tau\wedge\overline{\eta}^{\alpha\beta} -\Bigl( u^\alpha\,\overline{\eta}^\beta - u^\beta\,\overline{\eta}^\alpha \Bigr)\,,\label{eta02}\\ \eta ^{\alpha\beta\gamma} &=& -d\tau\,\epsilon ^{\alpha\beta\gamma} -\Bigl( u^\alpha\,\overline{\eta}^{\beta\gamma} + u^\gamma\,\overline{\eta}^{\alpha\beta} + u^\beta\,\overline{\eta}^{\gamma\alpha}\Bigr)\,,\nonumber\\ \label{eta01}\\ \eta ^{\alpha\beta\gamma\delta}&=& -\Bigl( u^\alpha\,\epsilon ^{\beta\gamma\delta}-u^\delta\,\epsilon ^{\alpha\beta\gamma}+ u^\gamma\,\epsilon ^{\delta\alpha\beta}- u^\beta\,\epsilon ^{\gamma\delta\alpha}\Bigr)\,,\nonumber\\ \label{eta00}\end{aligned}$$ where $$\begin{aligned} \overline{\eta}&:=& \Bigl( u\rfloor \eta \Bigr) ={1\over{3!}}\,\epsilon _{\alpha\beta\gamma}\, \underline{\vartheta}^\alpha\wedge\underline{\vartheta}^\beta\wedge\underline{\vartheta}^\gamma ={}^{\#}1 \,,\label{3deta06}\\ \overline{\eta}^\alpha &:=&-\Bigl( u\rfloor \eta ^{\alpha}\Bigr) ={1\over{2!}}\,\epsilon ^\alpha{}_{\beta\gamma}\,\underline{\vartheta}^\beta\wedge\underline{\vartheta}^\gamma ={}^{\#}\underline{\vartheta}^\alpha \,,\label{3deta07}\\ \overline{\eta}^{\alpha\beta}&:=&\Bigl( u\rfloor \eta ^{\alpha\beta}\Bigr) =\,\epsilon ^{\alpha\beta}{}_{\gamma}\,\underline{\vartheta}^\gamma ={}^{\#}(\underline{\vartheta}^\alpha\wedge\underline{\vartheta}^\beta\,)\,,\label{3deta08}\\ \epsilon ^{\alpha\beta\gamma}&:=&-\Bigl( u\rfloor \eta ^{\alpha\beta\gamma}\Bigr) =\,u_\mu\,\eta ^{\mu\alpha\beta\gamma}={}^{\#}(\underline{\vartheta}^\alpha\wedge\underline{\vartheta}^\beta\wedge\underline{\vartheta}^\gamma\,)\,,\nonumber\\ \label{3deta09}\end{aligned}$$ being (\[3deta06\]) the three-dimensional volume element, such that $\overline{\eta} = u^\alpha\,\eta _\alpha$. Making use of (\[thetaLiederiv01\])–(\[thetaLiederiv04\]), (\[3deta06\]) and (\[3deta07\]), one can prove that the Lie derivative of this volume can be decomposed as $$\begin{aligned} {\it l}_u \overline{\eta} ={\cal \L\/}_u\underline{\vartheta}^\alpha\wedge\overline{\eta}_\alpha\,.\label{volLieder}\end{aligned}$$ On the other hand, the contractions between tetrads and eta basis in four dimensions (see for instance [@Hehl:1995ue]), when foliated reduce to $$\begin{aligned} \underline{\vartheta}^\mu\wedge\overline{\eta}_\alpha &=& h^\mu{}_\alpha\,\overline{\eta}\,,\label{rel04bis}\\ \underline{\vartheta}^\mu\wedge\overline{\eta}_{\alpha\beta} &=& -h^\mu{}_\alpha\,\overline{\eta}_\beta +h^\mu{}_\beta\,\overline{\eta}_\alpha\,,\label{rel03bis}\\ \underline{\vartheta}^\mu\,\epsilon _{\alpha\beta\gamma} &=& h^\mu{}_\alpha\,\overline{\eta}_{\beta\gamma} +h^\mu{}_\gamma\,\overline{\eta}_{\alpha\beta} +h^\mu{}_\beta\,\overline{\eta}_{\gamma\alpha}\,,\label{rel02bis}\\ 0 &=& -h^\mu{}_\alpha\,\epsilon _{\beta\gamma\delta} + h^\mu{}_\delta\,\epsilon _{\alpha\beta\gamma} -h^\mu{}_\gamma\,\epsilon _{\delta\alpha\beta} + h^\mu{}_\beta\,\epsilon _{\gamma\delta\alpha}\,.\nonumber\\ \label{rel01bis}\end{aligned}$$ Taking (\[dualitycondbis\]) into account, we also find $$\begin{aligned} e_\alpha\rfloor\overline{\eta}&=&\overline{\eta}_\alpha\,,\label{contract02}\\ e_\alpha\rfloor\overline{\eta}_{\beta}&=&\overline{\eta}_{\beta\alpha}\,,\label{contract03}\\ e_\alpha\rfloor\overline{\eta}_{\beta\gamma}&=&\epsilon _{\beta\gamma\alpha}\,.\label{contract04}\end{aligned}$$ In view of definition (\[3deta09\]), the contraction of all objects (\[3deta07\])-(\[3deta09\]) with $u_\alpha$ vanishes. From (\[3deta07\]) then follows that $0=u_\alpha\,\overline{\eta}^\alpha ={}^{\#}(u_\alpha\,\underline{\vartheta}^\alpha )\,$, thus implying $u_\alpha\,\underline{\vartheta}^\alpha =0\,$. Plausibility argument ===================== Let us argue here against the seemingly [*ad hoc*]{} character of Eqs.(\[Uder02\]), namely $${{\partial\mathfrak{U}}\over{\partial\underline{\vartheta}^\alpha}}={\Sigma _{\alpha}}_{\bot}^{\rm matt}\,,\qquad {{\partial\mathfrak{U}}\over{\partial u^\alpha}}=-\underline{\Sigma}^{\rm matt}_\alpha\,,\label{condit1bbb}$$ showing that, in fact, the internal energy 3-form $\mathfrak{U}$ inherits properties of the original mater Lagrangian, in particular of $L^{\rm matt}_{\bot}$. First we notice that, according to (\[intenergycurr02\]), $\mathfrak{U}$ is the transversal part of the internal energy current $\epsilon ^{\rm u}$ defined in (\[intenergycurr01\]) as proportional to $\epsilon ^{\rm matt}$. On the other hand, from the fundamental matter energy-momentum 3-form (\[mattenergy\]) with (\[sigmamatt\]) follows $$\epsilon ^{\rm matt} =\overline{{\cal \L\/}_u\psi}\,\,{{\partial L}\over{\partial d\overline{\psi}}} -{{\partial L}\over{\partial d\psi}}\,\,{\cal \L\/}_u\psi -L^{\rm matt}_{\bot}\,,\label{expmattenergy}$$ so that, at least for Dirac matter, we get $\mathfrak{U}= -L^{\rm matt}_{\bot} +$ additional terms. According to this relation, Eqs.(\[condit1bbb\]) should resemble the analogous derivatives of $L^{\rm matt}_{\bot}$. In order to calculate them, we make use of the following result proved in [@Tresguerres:2007ih]. When considering the foliated Lagrangian density form $L = d\tau\wedge L_{\bot}\,$, depending on the longitudinal and transversal parts of any dynamical variable $Q = d\tau\wedge Q_{\bot} + \underline{Q}\,$, Eq.(D14) of [@Tresguerres:2007ih] establishes that $${{\partial L}\over{\partial Q}} = (-1)^p\, d\tau\wedge{{\partial L_{\bot}}\over{\partial\underline{Q}}}+{{\partial L_{\bot}}\over{\partial Q_{\bot}}}\,,\label{condit1}$$ with $p$ standing for the degree of the $p$-form $Q$. In view of (\[condit1\]), the matter energy-momentum 3-form defined in (\[momentdecompbis\]) decomposes as $$\begin{aligned} \Sigma ^{\rm matt}_\alpha := {{\partial L^{\rm matt}}\over{\partial \vartheta ^\alpha}} = -d\tau\wedge{{\partial L_{\bot}^{\rm matt}}\over{\partial\underline{\vartheta}^\alpha}}+{{\partial L_{\bot}^{\rm matt}}\over{\partial u^\alpha}}\,,\label{condit1b}\end{aligned}$$ implying $${{\partial L_{\bot}^{\rm matt}}\over{\partial\underline{\vartheta}^\alpha}} = -{\Sigma _{\alpha}}_{\bot}^{\rm matt}\,,\qquad {{\partial L_{\bot}^{\rm matt}}\over{\partial u^\alpha}} = \underline{\Sigma}^{\rm matt}_\alpha\,,\label{condit1bb}$$ which reproduce the form of (\[condit1bbb\]), provided $\mathfrak{U}= -L^{\rm matt}_{\bot}$ as suggested above. R. Tresguerres, Translations and dynamics, Int.J.Geom.Meth.Mod.Phys. [**05**]{} (2008) 905-945, arXiv:gr-qc/0707.0296. F.W. Hehl, G.D. Kerlick and P. Von der Heyde, General relativity with spin and torsion and its deviations from Einstein’s theory, Phys. Rev. [**D10**]{} (1974) 1066-1069. F.W. Hehl, P. Von der Heyde, G.D. Kerlick and J.M. Nester, General Relativity with spin and torsion: Foundations and prospects, Rev. Mod. Phys. [**48**]{} (1976) 393-416. F.W. Hehl, Four lectures on Poincaré gauge field theory, given at 6th Course of Int. School of Cosmology and Gravitation, Erice, Italy, 6-18 May 1979, eds. P.G. Bergmann and V. de Sabbata (New York: Plenum, 1980). F. Gronwald, Metric-affine gauge theory of gravity. I: Fundamental structure and field equations, Int. J. Mod. Phys. [**D 06**]{} (1997) 263-304, gr-qc/9702034. F.W. Hehl, J.D. McCrea and E.W. Mielke and Y. Neeman, Metric affine gauge theory of gravity: Field equations, Noether identities, world spinors, and breaking of dilation invariance, Phys. Rept. [**258**]{} (1995) 1-171, gr-qc/9402012. R.D. Hecht, Conserved quantities in the Poincaré gauge theory of gravitation (in German), Ph.D. Thesis, University of Cologne, 1993. Y.N. Obukhov, Poincaré gauge gravity: Selected topics, Int. J. Geom. Meth. Mod. Phys. [**03**]{} (2006) 95-138, gr-qc/0601090. H.B. Callen, Thermodynamics and an introduction to thermostatistics, (John Wiley $\&$ Sons, New York, Chichester, Brisbane, Toronto, Singapore, 1985). L.D. Landau and E. M. Lifshitz, Fluid Mechanics, Addison Wesley, Reading, Mass. (1958). W. Israel, Nonstationary irreversible thermodynamics: A causal relativistic theory, Annals Phys. [**100**]{} (1976) 310-331. W. Israel and J.M. Stewart, Transient relativistic thermodynamics and kinetic theory, Annals Phys. [**118**]{} (1979) 341-372. B. Carter, Convective variational approach to relativistic thermodynamics of dissipative fluids, Proc. Roy. Soc. Lond. [**A433**]{} (1991) 45. D. Priou, Comparison between variational and traditional approaches to relativistic thermodynamics of dissipative fluids, Phys. Rev. [**D43**]{} (1991) 1223. C. Eckart, The Thermodynamics of irreversible processes, III. Relativistic theory of the simple fluid, Phys.Rev. [**58**]{} (1940) 919-924. R. Tresguerres, Unified description of interactions in terms of composite fiber bundles, Phys. Rev. [**D66**]{} (2002) 064025. R. Tresguerres, Motion in gauge theories of gravity, Int.J.Geom.Meth.Mod.Phys. [**10**]{} (2013) 1250085, arXiv:gr-qc/1202.2569. F.W. Hehl and Y.N. Obukhov, Foundations of Classical Electrodynamics, (Birkhauser Boston, Basel, Berlin, 2003). J.D. McCrea, Irreducible decompositions of non-metricity, torsion, curvature and Bianchi identities in metric-affine spacetimes, Class. Quant. Grav. [**9**]{} (1992) 553-568. F.W. Hehl and Y.N. Obukhov, Electromagnetic energy-momentum and forces in matter, Phys.Lett. [**A311**]{} (2003) 277-284, arXiv:physics/0303097v1 D. Kondepudi and I. Prigogine, Modern thermodynamics, From heat engines to dissipative structures, (John Wiley $\&$ Sons, New York, 1998). Y. Demirel, Nonequilibrium thermodynamics: Transport and rate processes in physical and biological systems, (Elsevier Science $\&$ Technology Books, Amsterdam, 2002). [^1]: The definition of spin current given in Eq.(61) of Reference [@Tresguerres:2007ih] differs from the present one due to the fact that there we considered an internal structure for the tetrads, with a particular dependence on $\Gamma ^{\alpha\beta}$, giving rise to additional terms. The latter ones are not present when the internal structure of the tetrads is ignored, as is the case here. [^2]: The covariant differentials in (\[covfieldeq2\]) and (\[covfieldeq3\]) are defined as $$DH_\alpha := dH_\alpha -\Gamma _\alpha{}^\beta\wedge H_\beta\,,$$ and $$DH_{\alpha\beta} := dH_{\alpha\beta} -\Gamma _\alpha{}^\gamma\wedge H_{\gamma\beta} -\Gamma _\beta{}^\gamma\wedge H_{\alpha\gamma}\,,$$ respectively.

--- abstract: | A C-coloring of a hypergraph $\cH=(X,\cE)$ is a vertex coloring $\vp:X\to\enn$ such that each edge $E\in\cE$ has at least two vertices with a common color. The related parameter $\UU (\cH)$, called the upper chromatic number of $\cH$, is the maximum number of colors can be used in a C-coloring of $\cH$. A hypertree is a hypergraph which has a host tree $T$ such that each edge $E \in \cE$ induces a connected subgraph in $T$. Notations $n$ and $m$ stand for the number of vertices and edges, respectively, in a generic input hypergraph. We establish guaranteed polynomial-time approximation ratios for the difference $n-\overline{\chi}({\cal H})$, which is $2+2 \ln (2m)$ on hypergraphs in general, and $1+ \ln m$ on hypertrees. The latter ratio is essentially tight as we show that $n-\overline{\chi}({\cal H})$ cannot be approximated within $(1-\epsilon) \ln m$ on hypertrees (unless [NP]{}$\subseteq$[DTIME]{}$(n^{\cO(log\;log\; n)})$). Furthermore, $\overline{\chi}({\cal H})$ does not have ${\cal O}(n^{1-\epsilon})$-approximation and cannot be approximated within additive error $o(n)$ on the class of hypertrees (unless ${\sf P}={\sf NP}$). [**Keywords:**]{} approximation ratio, hypergraph, hypertree, C-coloring, upper chromatic number, multiple hitting set. **AMS 2000 Subject Classification:** 05C15, 05C65, 05B40, 68Q17 author: - | Csilla Bujtás $^{1}$   Zsolt Tuza $^{1,2}$\ $^1$ Department of Computer Science and Systems Technology\ University of Pannonia, Veszprém, Hungary\ $^2$ Alfréd Rényi Institute of Mathematics\ Hungarian Academy of Sciences, Budapest, Hungary title: '-1.5cm [   ]{} Approximability of the upper chromatic number of hypergraphs[^1]' --- Introduction ============ In this paper we study a hypergraph coloring invariant, termed upper chromatic number and denoted by $\UU(\cH)$, which was first introduced by Berge (cf. [@B]) in the early 1970’s and later independently by several further authors [@ABN; @Vol2] from different motivations. The present work is the very first one concerning approximation algorithms on it. We also consider the complementary problem of approximating the difference $n-\UU$, the number of vertices minus the upper chromatic number. One of our main tools to prove a guaranteed upper bound on it is an approximation ratio established for the 2-transversal number of hypergraphs. As problems of this type are of interest in their own right, we also prove an approximation ratio in general for the minimum size of multiple transversals, i.e., sets of vertices intersecting each edge in a prescribed number of vertices at least. Earlier results allowed to select a vertex into the set several times; we prove bounds for the more restricted scenario where the set does not include any vertex more than once. Notation and terminology ------------------------ A *hypergraph* $\cH=(X, \cE)$ is a set system, where $X$ denotes the set of vertices and each edge $E_i\in \cE$ is a nonempty subset of $X$. Here we also assume that for each edge $E_i$ the inequality $|E_i|\ge 2$ holds, moreover we use the standard notations $|X|=n$ and $|\cE|=m$. A hypergraph $\cH$ is said to be *$r$-uniform* if $|E_i|=r$ for each $E_i \in \cE$. We shall also consider hypergraphs with restricted structure, where some kind of host graphs are assumed. A hypergraph $\cH=(X,\cE)$ admits a *host graph* $G=(X,E)$ if each edge $E_i \in \cE$ induces a connected subgraph in $G$. The edges of the host graph $G$ will be referred to as *lines*. Particularly, $\cH$ is called *hypertree* or *hyperstar* if it admits a host graph which is a tree or a star, respectively. Note that under our condition, which forbids edges of size 1, $\cH$ is a hyperstar if and only if there exists a fixed vertex $c^*\in X$ (termed the center of the hyperstar) contained in each edge of $\cH$. A *C-coloring* of $\cH$ is an assignment $\vp:X\to\enn$ such that each edge $E\in\cE$ has at least two vertices of a common color (that is, with the same image). The *upper chromatic number* $\UU(\cH)$ of $\cH$ is the maximum number of colors that can be used in a C-coloring of $\cH$. We note that in the literature the value $\UU(\cH)+1$ is also called the ‘cochromatic number’ or ‘heterochromatic number’ of $\cH$ with the terminology of Berge [@B p. 151] and Arocha *et al.* [@ABN], respectively. A C-coloring $\vp$ with $|\vp(X)|=\UU(\cH)$ colors will be referred to as an *optimal coloring* of $\cH$. The *decrement* of $\cH=(X,\cE)$, introduced in [@proj-plane], is defined as $\dec(\cH)=n-\UU(\cH)$. Similarly, the decrement of a C-coloring $\vp:X\to\enn$ is meant as $\dec(\vp)=|X|-|\vp(X)|$. For results on C-coloring see the recent survey [@BT-JGeom]. A *transversal* (also called hitting set or vertex cover) is a subset $T \subseteq X$ which meets each edge of $\cH=(X, \cE)$, and the minimum cardinality of a transversal is the *transversal number* $\tau(\cH)$ of the hypergraph. An *independent set* (or stable set) is a vertex set $I \subseteq X$, which contains no edge of $\cH$ entirely. The maximum size of an independent set in $\cH$ is the *independence number* (or stability number) $\aaa(\cH)$. It is immediate from the definitions that the complement of a transversal is an independent set and vice versa, so the Gallai-type equality $\tau(\cH)+\aaa(\cH)=n$ holds for each hypergraph. Remark that selecting one vertex from each color class of a C-coloring yields an independent set, therefore $\UU(\cH)\le\aaa(\cH)$ and, equivalently, $\dec(\cH) \ge \tau (\cH)$. More generally, a *$k$-transversal* is a set $T\subseteq X$ such that $|E_i \cap T|\ge k$ for every $E_i \in \cE$. A 2-transversal is sometimes called double transversal or strong transversal, and its minimum size is the *2-transversal number* $\tau_2(\cH)$ of the hypergraph. For an optimization problem and a constant $c>1$, an algorithm $\cA$ is called a *$c$-approximation algorithm* if, for every feasible instance $\cI$ of the problem, if the value has to be minimized, then $\cA$ delivers a solution of value at most $c\cdot Opt(\cI)$; if the value has to be maximized, then $\cA$ delivers a solution of value at least $Opt(\cI)/c$. Throughout this paper, an approximation algorithm is always meant to be one with polynomial running time on every instance of the problem. We say that a value has guaranteed approximation ratio $c$ if it has a $c$-approximation algorithm. In the other case, when no $c$-approximation algorithm exists, we say that the value cannot be approximated within ratio $c$. For a function $f(n,m)$, an $f(n,m)$-approximation algorithm and the related notions can be defined similarly. A polynomial-time approximation scheme, abbreviated as PTAS, means an algorithm for every fixed $\eps > 0$ which is a $(1+\eps)$-approximation and whose running time is a polynomial function of the input size (but any function of $1/\eps$ may occur in the exponent). For further terminology and facts we refer to [@B; @BM; @Vaz] in the theory of graphs, hypergraphs, and algorithms, respectively. The notations $\ln x$ and $\log x$ stand for the natural logarithm and for the logarithm in base 2, respectively. Approximability results on multiple transversals ------------------------------------------------ The transversal number $\tau(\cH)$ of a hypergraph can be approximated within ratio $(1+\ln m)$ by the classical greedy algorithm (see e.g. [@Vaz]). On the other hand, Feige [@Fei] proved that $\tau(\cH)$ cannot be approximated within $(1-\epsilon) \ln m$ for any constant $0<\epsilon <1$, unless [NP]{}$\subseteq$[DTIME]{}$(n^{\cO(\log \log n)})$. As relates to the $k$-transversal number, in [@Vaz] a $(1+ \ln m)$-approximation is stated under the less restricted setting which allows multiple selection of vertices in the $k$-transversal. In the context of coloring, however, we cannot allow repetitions of vertices. For this more restricted case, when the $k$-transversal consists of pairwise different vertices, we prove a guaranteed approximation ratio $(1+ \ln (km))$. In fact we consider a more general problem, where the required minimum size of the intersection $E_i \cap T$ can be prescribed independently for each $E_i \in \cE$. \[multiple\] Given a hypergraph $\cH=(X,\cE)$ with $m$ edges $E_1,\dots,E_m$ and positive integers $w_1,\dots,w_m$ associated with the edges, the minimum cardinality of a set $S\sst X$ satisfying $|S\cap E_i|\ge w_i$ for all $1\le i\le m$ can be approximated within $\sum_{i=1}^{W} 1/i < 1+\ln W$, where $W= \sum_{i=1}^m w_i$. This result, proved in the next section, implies a guaranteed approximation ratio $(1+ \ln 2m)$ for $\tau_2(\cH)$. Approximability results on the upper chromatic number ----------------------------------------------------- The problem of determining the upper chromatic number is -hard, already on the class of 3-uniform hyperstars. On the other hand, the problems of determining $\overline{\chi}({\cal H})$ and finding a $\overline{\chi}({\cal H})$-coloring are fixed-parameter tractable in terms of maximum vertex degree on the class of hypertrees [@BT-cejor]. A notion closely related to our present subject was introduced by Voloshin [@Vol93; @Vol2] in 1993. A *mixed hypergraph* is a triple $\cH =(X, \cC, \cD)$ with two families of subsets called $\cC$-edges and $\cD$-edges. By definition, a coloring of a mixed hypergraph is an assignment $\vp:X\to\enn$ such that each $\cC$-edge has two vertices of a common color and each $\cD$-edge has two vertices of distinct colors. Then, the minimum and the maximum possible number of colors, that can occur in a coloring of $\cH$, is termed the lower and the upper chromatic number of $\cH$ and denoted by $\chi(\cH)$ and $\UU(\cH)$, respectively. For detailed results on mixed hypergraphs we refer to the monograph [@Volmon]. Clearly, the of a hypergraph $\cH=(X, \cE)$ are in one-to-one correspondence with the colorings of the mixed hypergraph $\cH'=(X, \cE, \es)$, and also $\UU(\cH)= \UU(\cH')$ holds. The following results are known on the approximation of the upper chromatic number of mixed hypergraphs: For mixed hypergraphs of maximum degree 2, the upper chromatic number has a linear-time $\frac{5}{3}$-approximation and an $O(m^3+n)$-time [@KKV-degree Theorem 14 and Theorem 15] There is no PTAS for the upper chromatic number of mixed hypergraphs of maximum degree 2, unless $=$. [@KKV-degree Theorem 20] There is no $o(n)$-approximation algorithm for the upper chromatic number of mixed hypergraphs, unless $=$. [@K-spect Corollary 5] All these results assume the presence of $\cD$-edges in the input mixed hypergraph. In this paper we investigate how hard it is to estimate $\UU$ for C-colorings of hypergraphs. On the positive side, we prove a guaranteed approximation ratio for the decrement of hypergraphs in general, furthermore we establish a better ratio on the class of hypertrees. \[appr-gen\] The value of $\dec(\cH)$ is $(2+2\ln (2m))$-approximable on the class of all hypergraphs. \[appr-htree\] The value of $\dec(\cH)$ is $(1+\ln m)$-approximable on the class of all hypertrees. These theorems are essentially best possible concerning the ratio of approximation, moreover the upper chromatic number turns out to be inherently non-approximable already on hypertrees with rather restricted host trees, as shown by the next result. \[ratio\] For every $\epsilon > 0$, $\dec(\cH)$ cannot be approximated within $(1-\epsilon)\ln m$ on the class of hyperstars, unless [NP]{}$\subseteq$[DTIME]{}$(n^{\cO(\log \log n)})$. For every $\epsilon > 0$, $\UU(\cH)$ cannot be approximated within $n^{1-\epsilon}$ on the class of $3$-uniform hyperstars, unless [P]{}$=$[NP]{}. As regards the *difference* between a solution determined by a polynomial-time algorithm and the optimum value, the situation is even worse. \[additive\] Unless $=$, neither of the following values can be approximated within additive error $o(n)$ for hypertrees of edge size at most 7: $\UU(\cH)$,  $\dec(\cH)$, $\aaa(\cH)-\UU(\cH)$,  $\tau(\cH)-\dec(\cH)$, $\dec(\cH)-\tau_2(\cH)/2$. The relevance of the last quantity occurs in the context of Proposition \[decr-tau2\] of Section \[decr-transv\]. We prove the positive results with guaranteed approximation ratio in Section 3, and the negative non-approximability results in Section 4. Lemmas on connected colorings of hypertrees ------------------------------------------- Suppose that $\cH$ is a hypergraph over a host graph $G$, and $\vp$ is a C-coloring of $\cH$. We say that $\vp$ is a *connected coloring* if each color class of $\vp$ induces a connected subgraph of $G$. We will use the following two lemmas concerning connected C-colorings of hypertrees, both established in [@BT-cejor]. A line $uv$ of the host tree $G$ is termed *monochromatic line* for a C-coloring $\vp$ if $\vp(u)=\vp(v)$. ([@BT-cejor Proposition 2]) \[conn\] If a hypertree admits a C-coloring with $k$ colors, then it also has a connected C-coloring with $k$ colors over any fixed host tree. ([@BT-cejor Proposition 3]) \[mono-lines\] If $\vp$ is a connected C-coloring of a hypertree $\cH$ over a fixed host tree $G$, then the decrement of $\vp$ equals the number of monochromatic lines in $G$. Multiple transversals ===================== In this section, we describe a variation of the classical greedy algorithm, with the goal to produce a multiple transversal with pairwise different elements. Analyzing the greedy selection we will prove Theorem \[multiple\]. We recall its statement. **Theorem \[multiple\]**. *Given a hypergraph $\cH=(X,\cE)$ with $m$ edges $E_1,\dots,E_m$ and positive integers $w_1,\dots,w_m$ associated with its edges, the minimum cardinality of a set $S\sst X$ satisfying $|S\cap E_i|\ge w_i$ for all $1\le i\le m$ can be approximated within $\sum_{i=1}^{W} 1/i < 1+\ln W$, where $W= \sum_{i=1}^m w_i$.* Denote by $\cS$ the collection of all feasible solutions, that are the sets $S\sst X$ such that $|S\cap E_i|\ge w_i$ holds for all $i=1,\dots,m$. By definition, the optimum of the problem is the integer $$M:=\min_{S\in\cS} |S| .$$ We will show that the greedy selection always yields an $S^*\in\cS$ with $$|S^*| \le M \cdot \left(1 + 1/2 + \dots + 1/W\right).$$ To prove this, for any $Y\sst X$ and any $1\le i\le m$ we define $$w_{i,Y} := \max \left(0, \, w_i - |E_i\cap Y|\right)$$ which means the reduced number of elements to be picked further from $E_i$, once the set $Y$ has already been selected. Moreover, to any vertex $x\in X\smin Y$ we associate its usefulness $$u_{x,Y} := | \{E_i \mid x\in E_i, \ w_{i,Y} > 0 \}|.$$ The greedy algorithm then starts with $Y_0=\es$ and updates $Y_k := Y_{k-1}\cup\{x_k\}$ where $x_k\in X\smin Y_{k-1}$ has maximum usefulness among all values $u_{x,Y_{k-1}}$ in the set $X\smin Y_{k-1}$, as long as this maximum is positive. Reaching $u_{x,Y_t}=0$ for all $x\in X\smin Y_t$ (for some $t$), we set $S^* := Y_t$; we will prove that this $S^*$ satisfies the requirements. It is clear by the definition of $u_{x,Y}$ that $S^*$ meets each $E_i$ in at least $w_i$ elements, i.e. $S^*\in\cS$. We need to prove that $S^*$ is sufficiently small. For this, consider the following auxiliary set of cardinality $W$: $$Z := \{ z(i,j) \mid 1\le i\le m, \ 1\le j\le w_i \}.$$ At the moment when $Y_k$ is constructed by adjoining an element $x_k$ to $Y_{k-1}$, we assign weight $1/u_{x,Y_{k-1}}$ to all elements $z(i,w_{i,Y_{k-1}})$ such that $x_k\in E_i$ and $w_{i,Y_{k-1}}>0$. Note that $w_{i,Y_{k}}=w_{i,Y_{k-1}}-1$ will hold after the selection of $x_k$. Moreover, total weight 1 is assigned in each step, hence the overall weight after finishing the algorithm is exactly $|S^*|$. We put the elements $z(i,j)$ in a sequence $Z^*=(z_1,z_2,\dots,z_W)$ such that the elements of $Z$ occur in the order as they are weighted (i.e., those for $x_1$ first in any order, then the elements weighted for $x_2$, and so on). Just before the selection of $x_k$, the number of elements $z(i,j)$ to which a weight has been assigned is precisely $m_{k-1} := \sum_{\ell=1}^{k-1} u_{x_\ell,Y_{\ell-1}}.$ We are going to prove that $u_{x_k,Y_{k-1}} \ge (W-m_{k-1})/M$. Assuming that this has already been shown, it follows that each $z_q$ in $Z^*$ has weight at most $M/(W+1-q)$ and consequently $|S^*| \le M \cdot \left(1 + 1/2 + \dots + 1/W\right)$ as required. Let now $S_0\in\cS$ be any fixed optimal solution. Consider the bipartite incidence graph $B$ between the sets $E_i$ and the elements of $S_0$. That is, the first vertex class of $B$ has $m$ elements $a_1,\dots,a_m$ representing the sets $E_1,\dots,E_m$ while the second vertex class consists of the elements of $S_0$; we denote the latter vertices by $b_1,\dots,b_M$. There is an edge joining $a_i$ with $b_j$ if and only if $b_j\in E_i$. Since $S_0\in \cS$, each $a_i$ has degree at least $w_i$. Moreover, considering the moment just before $x_k$ is selected, if we remove the vertices of $S_0\cap Y_{k-1}$, in the remaining subgraph still each $a_i$ has degree at least $w_{i,Y_{k-1}}$. We take a subgraph $B'$ of this $B-Y_{k-1}$ (possibly $B$ itself if $Y_{k-1}\cap S_0=\es$) such that each $a_i$ has degree *exactly* $w_{i,Y_{k-1}}$. The number of edges in $B'$ is then equal to $W-m_{k-1}$; hence, some $b_j$ has degree at least $(W-m_{k-1})/M$. It follows that this $b_j$ has usefulness at least $(W-m_{k-1})/M$ at the moment when $x_k$ is selected; but $x_k$ is chosen to have maximum usefulness, hence $u_{x_k,Y_{k-1}} \ge (W-m_{k-1})/M$. This completes the proof. \[k-transv\] For each positive integer $k$, the $k$-transversal number $\tau_k$ has a $(1+\ln (km))$-approximation on the class of all hypergraphs. Guaranteed approximation ratios for the decrement ================================================= In this section we establish a connection between the parameters $\dec(\cH)$ and $\tau_2(\cH)$, and then we prove our positive results stated in Theorems \[appr-gen\] and \[appr-htree\]. Decrement vs. 2-transversal number {#decr-transv} ---------------------------------- First, we give an inequality valid for all hypergraphs without any structural restrictions and then, using this relation, we prove Theorem \[appr-gen\]. \[decr-tau2\] For every hypergraph $\cH$ we have $\tau_2(\cH)/2\le\dec(\cH)\le\tau_2(\cH)-1$, and both bounds are tight. In particular, $\tau_2(\cH)$ is a 2-approximation for $\dec(\cH)$. [*Lower bound:*]{}If $\UU(\cH) \le n/2$, then $\dec(\cH)\ge n/2 \ge \tau_2(\cH)/2$ automatically holds. If $\UU(\cH) > n/2$, then every $\UU$-coloring contains at least $2\UU(\cH)-n$ singleton color classes, therefore the total size of non-singleton classes is at most $n-(2\UU(\cH)-n)= 2(n-\UU(\cH))$. Since the union of the latter meets all edges at least twice, we obtain $2\dec(\cH)\ge\tau_2(\cH)$. [*Upper bound:*]{}If $S$ is a 2-transversal set of cardinality $\tau_2(\cH)$, we can assign the same color to the entire $S$ and a new dedicated color to each $x\in X\smin S$. This is a C-coloring with $n-|S|+1$ colors and with decrement $\tau_2(\cH)-1$. [*Tightness:*]{}The simplest example for equality in the upper bound is the hypergraph in which the vertex set is the only edge, i.e. $\cH=(X,\{X\})$. Many more examples can be given. For instance, we can specify a proper subset $S\sst X$ with $|S|\ge 2$, and take all triples $E\sst X$ such that $|E\cap S|=2$ and $|E\smin S|=1$. If $|S|\le n-2$, then $S$ is the unique smallest 2-transversal set, and every C-coloring with more than two colors makes $S$ monochromatic, hence the unique $\UU$-coloring uses $n-|S|+1$ colors. For the lower bound, we assume that $n=3k+1$. Let $X=\{1,2, \dots, 3k+1\}$ and $$\begin{aligned} \cE&=&\{\{3r+1, 3r+2, 3r+3\}\mid 0\le r\le k-1\} \nonumber \\ & &\cup~ \{\{3r+2, 3r+3, 3r+4\}\mid 0\le r\le k-1\}\} \nonumber\end{aligned}$$ Then $\tau_2(\cH)=2k$ because the $k$ edges in the first line are mutually disjoint and hence need at least $2k$ vertices in any 2-transversal set, while the $2k$-element set $\{3r+2\mid 0\le r\le k-1\}\cup \{3r+3\mid 0\le r\le k-1\}$ meets all edges twice. On the other hand, there exists a unique C-coloring with decrement $k$, obtained by making $\{3r+2,3r+3\}$ a monochromatic pair for $r=0,1,\dots,k-1$ and putting any other vertex in a singleton color class. This verifies equality in the lower bound. Now, we are ready to prove Theorem \[appr-gen\]. Let us recall its statement. **Theorem \[appr-gen\]**. *The value of $\dec(\cH)$ is $(2+2\ln (2m))$-approximable on the class of all hypergraphs.* By Corollary \[k-transv\], we have a $(1+ \ln (2m))$-approximation algorithm $\cA$ for $\tau_2$. Hence, given a hypergraph $\cH=(X,\cE)$, the algorithm $\cA$ outputs a 2-transversal $T$ of size at most $(1+ \ln (2m))\tau_2(\cH)$. Then, assign color 1 to every $x\in T$, and color the $n-|T|$ vertices in $X\setminus T$ pairwise differently with colors $2,3,\dots, n-|T|+1$. As each edge $E_i\in \cE$ contains at least two vertices of color 1, this results in a C-coloring $\vp$ with decrement satisfying $$\dec(\vp) = |T|-1 \le (1+ \ln (2m))\tau_2(\cH) -1 < 2(1+ \ln (2m))\dec(\cH),$$ where the last inequality follows from Proposition \[decr-tau2\]. Therefore, algorithm $\cA$ together with the simple construction of coloring $\vp$ is a $(2+2\ln 2m)$-approximation for $\dec(\cH)$. Guaranteed approximation ratio on hypertrees -------------------------------------------- In this short subsection we prove Theorem \[appr-htree\]. We recall its statement. **Theorem \[appr-htree\]**. *The value of $\dec(\cH)$ is $(1+\ln m)$-approximable on the class of all hypertrees.* Given a hypertree $\cH=(X, \cE)$ and $G=(X,L)$ which is a host tree of $\cH$, construct the auxiliary hypergraph $\cH^*=(L^*, \cE^*)$ such that each vertex $l_i^* \in L^*$ represents a line $l_i$ of the host tree, moreover each edge $E_i^* \in \cE^*$ of the auxiliary hypergraph corresponds to the edge $E_i \in \cE$ in the following way: $$E_i^*=\{l_j^* \mid l_j \subseteq E_i\}.$$ Now, consider any connected C-coloring $\vp$ of $\cH$. This coloring determines the set $S \subseteq L$ of monochromatic lines in the host tree, moreover the corresponding vertex set $S^* \subseteq L^*$ in $\cH^*$. By Lemma \[mono-lines\], $\dec(\vp)=|S|=|S^*|$. As $\vp$ is a connected C-coloring, each edge of $\cH$ contains a monochromatic line and, consequently, $S^*$ is a transversal of size $\dec(\vp)$ in $\cH^*$. Similarly, in the opposite direction, if a transversal $T^*$ of $\cH^*$ is given and the corresponding line-set is $T$ in the host tree, then every edge $E_i$ of $\cH$ contains two vertices, say $u$ and $v$, such that the line $uv$ is contained in $T$. Then, the vertex coloring $\phi$, whose color classes correspond to the components of $(X, T)$, is a connected C-coloring of $\cH$, and in addition $\dec(\phi)=|T|=|T^*|$ holds. By Lemma \[conn\], $\cH$ has a connected C-coloring $\vp$ with $\dec(\vp)=\dec(\cH)$, therefore the correspondence above implies $\dec(\cH)= \tau(\cH^*)$. As $\cH^*$ can be constructed in polynomial time from the hypertree $\cH$, and since a transversal $T^*$ of size at most $(1+ \ln m)\tau(\cH^*)$ can be obtained by greedy selection, a C-coloring $\phi$ of $\cH$ with $$\dec(\phi)=|T^*| \le (1+ \ln m)\tau(\cH^*)=(1+ \ln m)\dec(\cH)$$ can also be constructed in polynomial time. This yields a guaranteed approximation ratio $(1+ \ln m)$ for the decrement on the class of hypertrees. Approximation hardness ====================== The bulk of this section is devoted to the proof of Theorem  \[additive\] on non-approximability for hypertrees. Then, we prove a lemma concerning parameters $\UU(\cH)$ and $\dec(\cH)$ of hyperstars. The section is closed with the proof of Theorem \[ratio\] and with some remarks. Additive linear error --------------------- Our goal in this subsection is to prove Theorem \[additive\]. This needs the following construction, which was introduced in [@perf-htree]. (We note that a similar construction was given already in [@KKPV].) #### Construction of $\cH(\Phi)$. Let $\Phi= C_1 \wedge \cdots \wedge C_m$ be an instance of 3-SAT, with $m$ clauses of size 3 over the set $\{x_1,\dots,x_n\}$ of $n$ variables, such that the three literals in each clause $C_j$ of $\Phi$ correspond to exactly three distinct variables. We construct the hypertree $\cH=\cH(\Phi)$ with the set $$X = \{ c^* \} \cup \{ x'_i,\, t_i,\, f_i \mid 1 \le i \le n \}$$ of $3n+1$ vertices, where the vertices $x'_i,t_i,f_i$ correspond to variable $x_i$. First, we define the host tree $T=(X,E)$ with vertex set $X$ and line-set $$E = \{ c^*x_i',\, x_i't_i,\, x_i'f_i \mid 1\leq i\leq n \}.$$ Hypergraph $\cH$ will have 3-element “variable-edges” $H_i=\{x_i',t_i,f_i\}$ for $i=1,\dots,n$, and 7-element “clause-edges” $F_j$ representing clause $C_j$ for $j=1,\dots,m$. All the latter contain $c^*$ and six further vertices, two for each literal of $C_j$: If $C_j$ contains the positive literal $x_i$, then $F_j$ contains $x_i'$ and $t_i$. If $C_j$ contains the negative literal $\neg x_i$, then $F_j$ contains $x_i'$ and $f_i$. Since $H_1,\dots,H_n$ are disjoint edges, it is clear that $\dec(\cH)\ge n$ and $\UU(\cH)\le 2n+1$. We shall see later that equality holds if and only if $\Phi$ is satisfiable. In addition, since $x_1',\dots,x_n'$ is a transversal set of $\cH$, the equalities $\tau(\cH)=n$ and $\aaa(\cH)=2n+1$ are valid for all $\Phi$, no matter whether satisfiable or not. Also, $\tau_2(\cH)=2n$ for all $\Phi$. #### Optimal colorings of $\cH$. By Lemma \[conn\], we may restrict our attention to colorings where each color class is a subtree in $T$. This makes a coloring irrelevant if it 2-colors a variable-edge in such a way that $\{t_i,f_i\}$ is monochromatic but $x_i'$ has a different color. Hence, at least one of the lines $x_i't_i$ and $x_i'f_i$ is monochromatic (maybe both) for each $i$. Moreover, we may assume the following further simplification: there is no monochromatic line $c^*x_i'$. Indeed, if the entire $H_i$ is monochromatic, then we would lose a color by making the line $c^*x_i'$ monochromatic. On the other hand, if say the monochromatic pair inside $H_i$ is $x_i't_i$, then every clause-edge $F_j$ containing $c^*x_i'$ but avoiding $t_i$ also contains the line $x_i'f_i$, therefore we get a coloring with the same number of colors if we assume that $x_i'f_i$ is monochromatic instead of $c^*x_i'$. Summarizing, we search an optimal coloring $\vp:X\to\enn$ with the following properties for all $i=1,\dots,n$: $\vp(c^*)\ne\vp(x_i')$ $\vp(x_i')=\vp(t_i)$ or $\vp(x_i')=\vp(f_i)$ In the rest of the proof we assume that all vertex colorings occurring satisfy these conditions. #### Truth assignments. Given a coloring $\vp$, we interpret it in the following way for truth assignment and clause deletion: If $H_i$ is monochromatic, delete all clauses from $\Phi$ which contain literal $x_i$ or $\neg x_i$. Otherwise, assign truth value $x_i\mapsto\ttt$ if $\vp(x_i')=\vp(t_i)$, and $x_i\mapsto\fff$ if $\vp(x_i')=\vp(f_i)$. It follows from the definition of $\cH(\Phi)$ that this truth assignment satisfies the modified formula after deletion if and only if $\vp$ properly colors all edges of $\cH$. Also conversely, if $\Phi'$ is obtained from $\Phi$ by deleting all clauses which contain $x_i$ or $\neg x_i$ for a specified index set $I\ssq\{1,\dots,n\}$, then a truth assignment $a:\{x_i \mid i\in \{1,\dots,n\}\smin I \}\to\{\ttt,\fff\}$ satisfies $\Phi'$ if and only if the following specifications for the monochromatic lines yield a proper coloring $\vp$ of $\cH$: If $i\in I$, then $\vp(x_i')=\vp(t_i)=\vp(f_i)$. Otherwise, let $\vp(x_i')=\vp(t_i)$ if $a(x_i)=\ttt$, and $\vp(x_i')=\vp(f_i)$ if $a(x_i)=\fff$. The observations above imply the following statement: For any instance $\Phi$ of [3-SAT]{}, the value of $\dec(\cH(\Phi))$ is equal to the minimum number of variables whose deletion from $\Phi$ makes the formula satisfiable. To complete our preparations for the proof of the theorem, let us quote an earlier result on formulas in which every positive and negative literal occurs in at most four clauses. The problem [Max 3Sat$(4,\overline 4)$]{} requires to maximize the number of satisfied clauses in such formulas. The following assertion states that this optimization problem is hard to approximate, even when the input is restricted to satisfiable formulas. ([@bounded-sat Corollary 5]) \[sat-bounded\] Satisfiable [Max 3Sat$(4,\overline 4)$]{} has no PTAS, unless $=$. Now we are ready to verify Theorem \[additive\], which states: **Theorem \[additive\]**. *Unless $=$, neither of the following values can be approximated within additive error $o(n)$ for hypertrees of edge size at most 7:* $\UU(\cH)$,  $\dec(\cH)$, $\aaa(\cH)-\UU(\cH)$,  $\tau(\cH)-\dec(\cH)$, $\dec(\cH)-\tau_2(\cH)/2$. We apply reduction from Satisfiable [Max 3Sat$(4,\overline 4)$]{}. For each instance $\Phi$ of this problem, we construct the hypergraph $\cH=\cH(\Phi)$. Since $\Phi$ is required to be satisfied, no variables have to be deleted from it to admit a satisfying truth assignment. This means precisely one monochromatic line inside each variable-edge. Hence, the above observations together with Lemma \[conn\] imply that $\dec(\cH)=n$ and $\UU(\cH)=2n+1$. On the other hand, Lemma \[sat-bounded\] implies the existence of a constant $c>0$ such that it is -hard to find a truth assignment that satisfies all but at most $cm$ clauses in a satisfiable instance of [Max 3Sat$(4,\overline 4)$]{} with $m$ clauses. Since each literal occurs in at most four clauses, this may require the cancelation of at least $cm/8\ge c'n$ variables. Thus, for the coloring $\vp$ determined by a polynomial-time algorithm, $\dec(\vp)-\dec(\cH)=\Theta(n)$ may hold, and hence also $\UU(\cH)-|\vp(X)|=\Theta(n)$. No efficient approximation on hyperstars ---------------------------------------- Proposition \[decr-tau2\] established a relation between $\dec(\cH)$ and $\tau_2(\cH)$, valid for all hypergraphs. Here we show that for hyperstars there is a stronger correspondence between the parameters. After that, we prove Theorem \[ratio\] which states non-approximability results on hyperstars. Given a *hyperstar* $\cH=(X,\cE)$, let us denote by $c^*$ the center of the host star. Hence, $c^*\in E$ holds for all $E\in\cE$. We shall use the following notations: $$E^- = E \smin \{c^*\}, \quad \cE^- = \{E^-\mid E\in\cE\}, \quad \cH^- = (X\smin\{c^*\},\cE^-) .$$ \[star-decrem\] If $\cH$ is a hyperstar, then $\dec(\cH) = \tau(\cH^-) = \tau_2(\cH) -1$ and $\UU(\cH)=\aaa(\cH^-)+1$. If a 2-transversal set $S$ does not contain $c^*$, then we can replace any $s\in S$ with $c^*$ and obtain another 2-transversal set of the same cardinality. This implies $\tau(\cH^-) = \tau_2(\cH) -1$. Let us observe next that the equalities $\UU(\cH)=\aaa(\cH^-)+1$ and $\dec(\cH) = \tau(\cH^-)$ are equivalent, due to the Gallai-type equality for $\aaa+\tau$ in $\cH^-$. Now, the particular case of Lemma \[conn\] for hyperstars means that there exists a $\UU$-coloring of $\cH$ such that all color classes but that of $c^*$ are singletons. Those singletons form an independent set in $\cH^-$, because the color of $c^*$ is repeated inside each $E^-$. Thus, we necessarily have $\UU(\cH)\le\aaa(\cH^-)+1$. Conversely, if $S$ is a largest independent set in $\cH^-$, i.e.$|S| = \aaa(\cH^-) = |X|-1-\tau(\cH^-)$ and $E^-\smin S\ne\es$ for all $E^-$, then making $X\smin S$ a color class creates a monochromatic pair inside each $E\in\cE$ because the color of $c^*$ is repeated in each $E^-$. Hence, assigning a new private color to each $x\in S$ we obtain that $\UU(\cH)\ge\aaa(\cH^-)+1$, consequently $\UU(\cH)=\aaa(\cH^-)+1$ and $\dec(\cH)=\tau(\cH^-)$. The following non-approximability results concerning $\UU(\cH)$ and $\dec(\cH)$ are valid already on the class of hyperstars. We recall the statement of Theorem \[ratio\]. **Theorem \[ratio\]**. By Proposition \[star-decrem\], the equalities $\UU(\cH)=\aaa(\cH^-)+1$ and $\dec(\cH)=\tau(\cH^-)$ hold whenever $\cH$ is a hyperstar. If $\cH$ is a generic hyperstar (with no restrictions on its edges), then $\cH^-$ is a generic hypergraph. Thus, approximating $\dec(\cH)$ on hyperstars is equivalent to pproximating $\tau(\cH^-)$ on hypergraphs, which is known to be intractable within ratio $(1-\eps)(\log m)$ unless [ NP]{}$\subseteq$ [DTIME]{}$(n^{\cO(\log \log n)})$, by the result of Feige [@Fei]. If $\cH$ is a generic 3-uniform hyperstar, then $\cH^-$ is a generic graph. Thus, approximating $\UU(\cH)$ on 3-uniform hyperstars is equivalent to approximating $\aaa(\cH^-)+1$ on graphs, which is known to be intractable within ratio $n^{1-\eps}$ unless =, by the result of Zuckerman [@Zuck]. In a similar way, we also obtain the following non-approximability result concerning $\tau_2$. The value $\tau_2(\cH)$ does not have a polynomial-time $((1-\eps)\ln m)$-approximation on hyperstars, unless [NP]{}$\subseteq$[DTIME]{}$(n^{\cO(\log \log n)})$. By Proposition \[star-decrem\], the approximation of $\tau_2(\cH)$ on hyperstars $\cH$ is as hard as that of $\tau(\cH^-)$ on general hypergraphs $\cH^-$. In connection with Theorem \[ratio\] one may observe that, even if we restrict the problem instances to 3-uniform hypergraphs in which each vertex pair is contained in at most three edges, $\UU(\cH)$ does not admit a PTAS. This follows from the fact that the determination of $\aaa(G)$ is -complete on graphs of maximum degree 3, by the theorem of Berman and Fujito [@BF]. Concluding remarks ================== Our results on hyperstars show that $\dec(\cH)$ admits a much better approximation than $\UU(\cH)$ does. In a way this fact is in analogy with the following similar phenomenon in graph theory: The independence number $\aaa(G)$ is not approximable within $n^{1-\eps}$, but $\tau(G)=n-\aaa(G)$ admits a polynomial-time 2-approximation because $\nu(G)\le\tau(G)\le 2\nu(G)$, and the matching number $\nu(G)$ can be determined in polynomial time. In this way, both comparisons $\dec(\cH)$ with $\UU(\cH)$ and $\tau(G)$ with $\aaa(G)$ demonstrate that there can occur substantial difference between the approximability of a graph invariant and its complement. Perhaps hypertrees with not very large edges admit some fairly efficient algorithms: Determine the largest integer $r$ such that there is a PTAS to approximate the value of $\UU(\cH)$ for hypergraphs $\cH$ in which every edge has at most $r$ vertices. Our results imply that $r\le 6$ is necessary. 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[^1]:  Research supported in part by the Hungarian Scientific Research Fund, OTKA grant T-81493, and by the European Union and Hungary, co-financed by the European Social Fund through the project TÁMOP-4.2.2.C-11/1/KONV-2012-0004 – National Research Center for Development and Market Introduction of Advanced Information and Communication Technologies.

--- abstract: 'We study the impact of the capture and annihilation of Weakly Interacting Massive Particles (WIMPs) on the evolution of Pop III stars. With a suitable modification of the Geneva stellar evolution code, we study the evolution of 20 and 200 M$_\odot$ stars in Dark Matter haloes with densities between 10$^{8}$ and $10^{11}$ GeV/cm$^3$ during the core H-burning phase, and, for selected cases, until the end of the core He-burning phase. We find that for WIMP densities higher than 5.3 $10^{10}(\sigma^{SD}_p/10^{-38} \mbox{ cm}^2)^{-1}$ GeV cm$^{-3}$ the core H-burning lifetime of $20 M_{\odot}$ and $200 M_{\odot}$ stars exceeds the age of the Universe, and stars are sustained only by WIMP annihilations. We determine the observational properties of these ‘frozen‘ objects and show that they can be searched for in the local Universe thanks to their anomalous mass-radius relation, which should allow unambiguous discrimination from normal stars.' author: - 'Marco Taoso$^{1,2}$' - 'Gianfranco Bertone$^{2}$' - 'Georges Meynet$^{3}$' - 'Silvia Ekstr$\ddot{\mbox{o}}$m$^{3}$' title: Dark Matter annihilations in Pop III stars --- In the Standard Cosmological Model, the matter density of the Universe is dominated by an unknown component, approximately 5 times more abundant than baryons, dubbed Dark Matter (DM). Among the many DM candidates proposed in the literature, Weakly Interacting Massive Particles (WIMPs), i.e. particles with mass $\cal{O}$$(100)$ GeV and weak interactions, appear particularly promising, also in view of their possible connection with well motivated extensions of the Standard Model of particle physics (see Ref. [@reviews] for recent reviews on particle DM, including a discussion of ongoing direct, indirect and accelerator searches). Despite their weak interactions, WIMPs can lead to macroscopic effects in astrophysical objects, provided that they have a sizeable scattering cross section off baryons. In this case, in fact, DM particles traveling through stars can be captured, and sink at the center of the stars. Direct searches and astrophysical arguments, however, severely constrain the strength of DM-baryons interactions (see e.g. Ref. [@mack] and references therein). Since the capture rate is proportional to the product of the scattering cross section times the local DM density, large effects are thus expected in regions where the DM density is extremely high (this was already noticed in the context of the so called ’cosmions’ [@cosmions]). Recent progress in our understanding of the formation and structure of DM halos has prompted a renewed interest in the consequences of DM capture in stars, in particular in the case of White Dwarfs [@Moska], compact objects [@Bertone:2007ae] and main sequence stars [@Fairbairn] at the Galactic center, where the DM density could be extremely high [@Bertone:2005hw]. Alternatively, one may focus on the first stars, which are thought to form from gas collapsing at the center of $10^6-10^8 M_{\odot}$ DM halos at redshift $z\lesssim 10-30$. In fact, Spolyar, Freese and Gondolo [@Freese] have shown that the energy released by WIMPs annihilations in these mini-halos, during the formation of a proto-star (thus even before DM capture becomes efficient), may exceed any cooling mechanism, thus inhibiting or delaying stellar evolution (see also Ref. [@Freeseb]). The formation of proto-stars with masses between $6 M_{\odot}$ and $600 M_{\odot}$ in DM halos of $10^6 M_{\odot}$ at z=20, can actually be delayed by $\sim 10^3-10^4 $ yrs [@Iocco]. Once the star forms, the scattering of WIMPs off the stellar nuclei becomes more efficient and a large number of WIMPs can be trapped inside the gravitational potential well of the star. The WIMPs luminosity can overwhelm that from nuclear reactions and therefore strongly modify the star evolution [@Iocco2; @Freese2], and the core H-burning phase of Pop III stars, in DM halos of density of $10^{11} \mbox{ GeV cm}^{-3}$, is substantially prolonged, especially for small mass stars ($M_{*}<40 M_{\odot}$) [@Iocco]. In this letter, we perform a detailed study of the impact of DM capture and annihilation on the evolution of Pop. III stars with a suitable modification of the Geneva stellar evolution code [@Ekstrom; @Maynet]. With respect to previous analyses, this already allows us to properly take into account the stellar structure in the calculation of the capture rate, that we compute, following Ref. [@Gould], as $$C= 4 \pi \int_{0}^{R_*} dr r^2 \frac{dC(r)}{dV} \label{eqn:C}$$ with $$\begin{aligned} \frac{dC(r)}{dV} &=& \left(\frac{6}{\pi}\right)^{1/2} \sigma_{\chi,N} \frac{\rho_{i}(r)}{M_i}\frac{\rho_{\chi}}{m_{\chi}} \frac{v^{2}(r)}{\bar{v}^2} \frac{\bar{v}}{2 \eta A^2} \\ \nonumber &\times & \left\{ \left( A_+ A_- -\frac{1}{2}\right) [\chi(-\eta,\eta)-\chi(A_-,A_+) ] \right.\\ \nonumber &+& \left. \frac{1}{2} A_+ e^{-A_-^2} -\frac{1}{2} A_- e^{-A_+^2} -\frac{1}{2} \eta e^{-\eta^2} \right\} \label{eqn:dCdV}\end{aligned}$$ $$A^2=\frac{3 v^2(r)\mu}{2 \bar{v}^2 \mu_-^2} \mbox{, }\hspace{0.5cm} A_{\pm}=A \pm \eta \mbox{,} \hspace{0.5cm}\eta^2=\frac{3v_{*}^2}{2\bar{v}^2}$$ $$\chi(a,b)=\frac{\sqrt{\pi}}{2}[\mbox{Erf}(b)-\mbox{Erf}(a)]=\int_a^bdy e^{-y^2}$$ $$\mu_-=(\mu_i-1)/2 \mbox{,} \hspace{0.5cm} \mu_{i}=m_{\chi}/M_i$$ where $\rho_i(r)$ is the mass density profile of a given chemical element in the interior of the star and $M_i$ refers to its atomic mass, while $\rho_{\chi}$, $m_{\chi}$ and $\bar{v}$ are respectively the WIMP mass and the WIMP density and velocity dispersion at the star position. The velocity of the star with respect to an observer, labeled as $v_*$, is assumed to be equal to $\bar{v}$, giving therefore $\eta=\sqrt{3/2}$. The radial escape velocity profile depends on $M(r)$, i.e. the mass enclosed within a radius $r$, $v^2(r)=2 \int_{r}^{\infty} G M(r^{'})/r^{'2} dr'$. The WIMP scattering cross section off nuclei, $\sigma_{\chi,N}$ is constrained by direct detection experiments and for a WIMP mass of 100 GeV the current upper limits are $\sigma_{SI}=10^{-43} \mbox{ cm}^2$ [@SI] and $\sigma_{SD}=10^{-38} \mbox{ cm}^2 $ [@SD] respectively for spin-independent and spin-dependent WIMP interactions off a proton. We will adopt these reference values throughout the paper, but the capture rate can be easily rescaled for other scattering cross sections by using Eq. \[eqn:dCdV\]. The spin-independent interactions with nucleons inside nuclei add up coherently giving an enhancement factor $A^4$ with respect to the interaction with a single nucleon: $\sigma^{SI}_{\chi,N}=A^4\sigma_{\chi,p}$, where $A$ is the mass number. There is no such enhancement for the spin-dependent interactions. We consider the contribution to the capture rate from WIMP-hydrogen spin dependent interactions and WIMP-helium $^4$He spin-independent interactions, neglecting the presence of other elements because of their very low abundance. The contribution of Helium is found to be negligible with respect to that from hydrogen. Once captured, WIMPs get redistributed in the interior of the star reaching, in a characteristic time $\tau_{th},$ a thermal distribution [@GriestSeckel]: $$n_{\chi}(r)=n_0 e^{\frac{-r^2}{r_w^2}} \mbox{ with } r_{\chi}=\sqrt{\frac{3 k T_c}{2\pi G \rho_c m_{\chi}}} \label{eq:Distribuzione}$$ with $T_c$ and $\rho_c$ referring to the core temperature and density. The distribution results quite concentrated toward the center of the star: e.g. for a $20 M_{\odot}$ star immersed in a WIMP density of $\rho_{\chi}=10^{9} \mbox{ GeV cm}^{-3}$ at the beginning of the core H-burning phase we obtain $r_{\chi}= 2 \times 10^{9} \mbox{cm},$ a value much lower than the radius of the star, $R_*= 10^{11} \mbox{cm}.$ This consideration underlines the importance of an accurate spatial resolution in the core to properly treat the luminosity produce from WIMPs annihilations. We have also checked that regardless the extremely high concentrations of WIMPs obtained at the center of the stars, the gravity due to WIMPs is completely negligible. The number of scattering events needed for DM particles to thermalize with the nuclei in the star is of order $\approx m_{\chi}/M_H$, thus an upper limit on the thermalization time can be obtained as $\tau_{th}=(m_{\chi}/M_H)/(\sigma_{SD}\bar{n}_H \bar{v})$ where $\bar{n_H}$ is the average density on the star. The WIMPs luminosity is simply $L_{\chi}(r)= 4 \pi (\sigma v) m_{\chi} c^2 n_{\chi}^2(r)$. For the annihilation cross section times relative velocity $(\sigma v)$, we assume the value $3 \times10^{-26} \mbox{ cm}^2$, as appropriate for a thermal WIMP, but note that the total WIMP luminosity at equilibrium does not depend on this quantity. After a time $$\tau_{\chi}= \left(\frac{C (\sigma v)}{\pi^{3/2} r_{\chi}^3} \right)^{-1/2}$$ an equilibrium between capture and annihilation is established, and this incidentally allows to determine the normalization constant $n_0$ above. We have checked that the two transients $\tau_{\chi}$ and $\tau_{th}$ remain much smaller, during the evolution of the star, than the Kelvin-Helmotz timescale, $\tau_{KH}$ and the timescale needed for the nuclear reactions to burn an hydrogen fraction $\Delta X_c=0.002$ of the convective core, $\tau_{nucl}$: $$\tau_{KH}=\frac{G M_*^2}{R_* L_*}\hspace{0.5cm} \tau_{nucl}=\frac{q_c \Delta X_c M_* 0.007 c^2}{L_*}$$ where the \* labels quantities relative to the the star and $q_c$ is the core convective mass fraction. This argument justifies the assumption of equilibrium between capture and annihilation and the use of the radial distribution in Eq. \[eq:Distribuzione\]. We assume here an average WIMP velocity $\bar{v}= 10 \mbox{ Km s}^{-1},$ the virial velocity in an halo of $10^{5}-10^6 M_{\odot}$ at z=20. As for the DM density, semi-analytic computations of the adiabatic contraction of DM halos [@Freese2; @Freese3], in agreement with the results extrapolated from simulations of first star formation [@AbelBryan], suggest DM densities of order $10^{12} \mbox{ GeV cm}^{-3}$ or even higher. We have implemented the effects of WIMPs annihilation in the Geneva stellar evolution code (see Ref. [@Ekstrom; @Maynet] for details), and followed the evolution of a $20 M_{\odot}$ and $200 M_{\odot}$ stars for different DM densities. We show in Fig.\[HR\] the evolutionary tracks for the $20 M_{\odot}$ model, and show for comparison (black line) the case of a standard Pop III star without WIMPs. For DM densities smaller than $10^9 \mbox{ GeV cm}^{-3}$ the evolutionary tracks closely follow that of a normal star and they are not shown for simplicity. The position of the star at the beginning of the core H-burning phase (zero-age main sequence, or ZAMS) is obtained when, after a short transient, the luminosity produced at the center of the star equals the total luminosity and the star settles down in a stationary regime. For increasing DM densities the WIMPs luminosity produced at the center overwhelms the luminosity from nuclear reactions and makes the star inflate, producing therefore a substantial decrease of the effective temperature and a moderate decrease of the star luminosity at the ZAMS position, with respect to the standard scenario. For $\rho_{\chi}=10^{10} \mbox{ GeV cm}^{-3}$, the energy produced by WIMPs present in the star at a given time, estimated as $E_{\chi} \simeq L_{\chi} \tau_{KH},$ is, at the ZAMS, $\sim 0.8$ times the gravitational potential energy of the star, and the star therefore starts to contract. In this phase, the core temperature, and consequently also the nuclear reactions, increase. When the latter become comparable with the WIMPs luminosity, the standard situation is recovered and the evolutionary track joins the classical tracks of a star without WIMPs. An important difference from standard evolution is that in the first phase, the nuclear reactions are slowed down and therefore the core H-burning lifetime is prolonged. For Dark Matter densities $\rho_{\chi}\leq1.6$ $10^{10} \mbox{ GeV cm}^{-3},$ the picture is qualitatively the same, and for these models we only show in Fig. \[HR\] the first phases of the evolution. In Fig. \[tcxc\], we show the core temperature as a function of the DM density, at different stages of the core H-burning phase. At high DM densities hydrogen burns at much lower core temperatures than in the usual scenario, till a certain mass fraction is reached, e.g. $X_c=0.3$ for $\rho_{\chi}=10^{10} \mbox{ GeV cm}^{-3},$ and the standard evolutionary track is joined. For increasing DM densities the nuclear reaction rate is more and more delayed till the contraction of the star is inhibited, due to the high DM energy accumulated, and the evolution is frozen. In Fig.\[HR\] for $\rho_{\chi}=2\cdot10^{10} \mbox{ GeV cm}^{-3}$ and $\rho_{\chi}=3\cdot10^{10} \mbox{ GeV cm}^{-3}$ the stars seems to remain indefinitely at the ZAMS position. In Fig. \[den\] we show the core H-burning lifetime as a function of the DM density. In the case of a $20 M_{\odot}$ model, for $\rho\leq10^{10} \mbox{ Gev cm}^{-3}$ the core H-burning phase is prolonged by less then 10 % but the delay increases rapidly for higher DM densities. Extrapolating the curve we determine a critical density, $\rho_c =2.5\cdot10^{10} \mbox{ Gev cm}^{-3}$, beyond which the core H-burning lifetime is longer then the age of the Universe. All the calculations have been repeated for the $200 M_{\odot}$ model and we find that both the $20 M_{\odot}$ and $200 M_{\odot}$ stars evolutions are stopped for DM densities higher than $5.3\cdot10^{10} (\frac{\sigma^{SD}_p}{10^{-38} \mbox{ cm}^2})^{-1} \mbox{ GeV cm}^{-3}.$ We have also verified that the results weakly depend on the WIMP mass, e.g. the core H-lifetime is modified by a factor 0.2% and 5% respectively for $m_{\chi}=10 \mbox{ GeV}$ and $m_{\chi}=100 \mbox{ GeV},$ if $\rho_{\chi}=10^{10}\mbox{ GeV cm}^{-3}$. It is remarkable that under these circumstances, frozen Pop III stars can survive until the present epoch, and can be searched for as an anomalous stellar population. In Fig.\[gt\] we show the effective temperature and gravity acceleration at the surface of these frozen Pop III stars, kept in the H-burning phase, for different DM densities. Frozen stars would thus appear much bigger and with much lower surface temperatures with respect to normal stars with the same mass and metallicity. Our results are qualitatively consistent with the preliminary estimates in [@Iocco2; @Freese2] and the analysis in [@Iocco]. However, for a given DM density, we obtain a somewhat longer core H-burning lifetime with respect to [@Iocco], possibly due to their use of an approximated expression for the capture rate. We have also followed, for selected models, the evolution during the core He-burning phase. During this evolutionary stage, the Dark Matter luminosity is lower than the nuclear reaction luminosity, therefore the impact of DM annihilations is found to be rather weak. For the $20 M_{\odot}$ model and for $\rho_{\chi} = 1.6\cdot10^{10} \mbox{ GeV cm}^{-3}$ the He-lifetime is prolonged by a factor 1.2, rather than a factor 37 found for the H-burning phase for the same DM density. 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--- abstract: 'In classically chaotic systems, small differences in initial conditions are exponentially magnified over time. However, it was observed experimentally that the (necessarily quantum) “branched flow” pattern of electron flux from a quantum point contact (QPC) traveling over a random background potential in two-dimensional electron gases(2DEGs) remains substantially invariant to large changes in initial conditions. Since such a potential is classically chaotic and unstable to changes in initial conditions, it was conjectured that the origin of the observed stability is purely quantum mechanical, with no classical analog. In this paper, we show that the observed stability is a result of the physics of the QPC and the nature of the experiment. We show that the same stability can indeed be reproduced classically, or quantum mechanically. In addition, we explore the stability of the branched flow with regards to changes in the eigenmodes of quantum point contact.' author: - Bo Liu - 'Eric J. Heller' title: Stability of Branched Flow from a Quantum Point Contact --- Branching is a universal phenomenon of wave propagation in a weakly correlated random medium. It is observed in 2DEGs with wavelength on the scale of nanometers[@a2; @a3], in quasi-two-dimensional resonator with microwave[@a12] and used to study sound propagation in oceans with megameter length scales[@a10]. It has significant influence on electron transport in 2DEGs[@a14; @a13] and is found to be implicated in the formation of freak waves in oceans[@a9]. In all these studies, classical trajectory simulations show closely similar branch formation. {width="14cm"} However, the classical interpretation was challenged by a recent experiment on 2DEGs[@a1], where it was observed that the (necessarily quantum) branched flow pattern even far away showed stability of the branches against the changes in the QPC. This stability was conjectured to be of quantum origin[@a1]. To our best knowledge, no insights into this stability have been provided since, and it remains a puzzle in the literature. In this paper, we provide an explanation for the observed stability. Moreover, we provide numerical simulations to show that it can indeed be reproduced by classical trajectories. To proceed, we need to recount what was done in the experiment[@a1]. To create a large change in initial conditions, the QPC was shifted by about one correlation length of the underlying random potential, which is also roughly the width of the QPC. Classically, a one correlation length shift is indeed very significant for the chaotic dynamics, making the trajectories very different, as seen in the classical simulations of reference[@a1]. If one launches two separate quantum wavepackets[@a4] through QPCs differing by this amount, the coherent overlap between the two initial wavepackets is estimated at less than five percent. However, in the experiment it is nonetheless observed that some branches remain at almost exactly the same locations seventy correlation lengths away from the injection points, with the only observed difference being the relative strength of each branch. This lack of sensitivity even at long range was termed “the unexpected features of the branched flow”. The present paper gives an explanation to the observed stability, by taking into account the effect of the change of QPCs. As a Gedanken experiment, consider a pair of side-by-side QPCs differing by a shift. This could not be in the experiment, which had only one QPC, which however was able to be shifted relative to the rest of the device and the branched flow imaged again. In the Gedanken experiment, suppose we put wavepacket A though one QPC and wavepacket B through the other. Can the coherent overlap between the initially nonoverlapping A and B wavepackets increase over time and distance from the QPC’s? The answer is of course no, both classically (considered as overlap in phase space) and quantally. It is elementary to show that the coherent overlap must remain the same over time if the wavepackets are propagated under the same Hamiltonian. This is true whether or not disorder is present. However, in the experiment as performed, *the Hamiltonian of a single QPC and the Hamiltonian with the QPC shifted over are not the same.* Therefore, no theorem constrains the evolution of the coherent overlap between the two different initial wavepackets. As it will become clear, this is exactly what leads to the observed stability. In order to show this, we first consider the ideal case where the QPC is perfectly adiabatic and provide an analytical solution of the coherent overlap between the two wavepackets launched from *two different* QPCs as a function of time. We show the correlation reaches almost one at sufficiently large distance even if the initial coherent overlap is negligible. We then choose a more realistic QPC potential and also add smooth disorder of the type causing the branching into the system. The coherent overlap in this case still reaches $85\%$. Finally, we show that the same mechanism works for classical trajectories. For the classical case, we calculate an overlap of 79$\%$ in the phase space. Both results prove that the stability in the experiment is due to the nature of the experimental QPC shift. In the last part of the paper, we also make a prediction on the stability of the branched flow when the second mode in the QPC is open. For a QPC with harmonic confinement, the Hamiltonian is $$\begin{aligned} H_{o}=\frac{\vec{p}^{2}}{2m}+\frac{1}{2}m\omega^2(y)x^2 \end{aligned} \label{H}$$ where $\omega(y)$ is a slowly varying function of y and decreases monotonically as the QPC opens up. According to the approach developed in [@a4], we can reproduce the experimental results by propagating an initial wavepacket of the following form through the system. $$\begin{aligned} \Psi_{o}(x,y,0)&=\int dE \ e^{i\varphi (E,y_{0})}\sqrt{-\frac{m}{2\pi\hbar^2}\frac{\partial f_{T}(E,E_F)}{\partial E}}\Psi_{1}(x,y,E) \end{aligned} \label{iwave}$$ where $\Psi_{1}(x,y,E)$ is the scattering eigenstate at energy E , $\varphi (E,y_{0})$ is chosen so that it is a compact wavepacket centered at $y=y_{0}$, and $f_{T}(E,E_F)$ is the Fermi distribution with temperature T and Fermi energy $E_F$. This is a so-called “thermal wavepacket” at temperature T. For one mode open in the QPC, the thermal wavepacket gives the correct thermally averaged conductance, by propagating the wavepacket through the scattering region and counting the total flux that passes through a given point. More information about this method can be found in both reference [@a4] and the supplementary material. First we consider a perfect QPC and assume that disorder is absent. Numerical results including both disorder and an imperfect QPC follow. For a perfectly adiabatic QPC satisfying $\frac{\omega'(y)}{\omega^{2}(y)}\ll\frac{m}{\hbar k_{F}}$ and $\frac{\omega''(y)}{\omega^{2}(y)}\ll\frac{m}{\hbar}$, where $k_F$ is the Fermi wavevector, $\Psi_{1}(x,y,E)$ can be approximated by $$\begin{aligned} \Psi_{1}(x,y,E)=\frac{A(y_{0})}{\sqrt{\hbar k(y,E)\sqrt{\pi}\sigma(y)} }e^{i\int_{y_{0}}^{y}k(y',E)dy'-\frac{x^{2}}{2\sigma(y)^{2}}} \end{aligned} \label{eigenchannel}$$ where $\sigma(y)=\sqrt{\frac{\hbar}{m\omega(y)}}$, $\frac{\hbar^2 k^2(y,E)}{2m}+\frac{1}{2}\hbar w(y)=E$ and $A(y_0)$ is the normalizing constant. The effect of shifting the QPC is incorporated in the initial wavepacket as $$\begin{aligned} \Psi_{s}(x,y,0)=\hat{L}(x_{0})\Psi_{o}(x,y,0) \end{aligned}$$ where $\hat{L}(x_{0})=e^{-ix_{0} \hat{p}_{x}/\hbar}$ is the translation operator, $x_0$ is the displacement of the QPC and $\hat{p}_{x}$ is the momentum operator in the x direction. The two wavepackets evolve under the influence of their respective QPC and the coherent overlap between them at a later time t is given by $$\begin{aligned} C_{o,s}(t)=&\left | \int dxdy\ \Psi^{*}_{o}(x,y,t)\Psi_{s}(x,y,t)\right |\\ =&\left | \int dy\ H(y,t) S(y) \right | \end{aligned} \label{eoverlap}$$ and $$\begin{aligned} H(y,t)=&\int dEdE' \ \frac{\left | A(y_{0})\right |^{2}a^{*}(E')a(E)}{\hbar\sqrt{k(y,E)k(y,E')}}e^{-i(E-E')t/\hbar} \\ &\times e^{i(\varphi (E,y_{0})-\varphi (E',y_{0})+\int_{y_{0}}^{y}(k(y',E)-k(y',E'))dy')}\\ S(y)=&\ e^{-\frac{x_{0}^{2}}{4\sigma(y)^{2}}} \end{aligned}$$ {width="17.2cm"} H(y,t) is essentially a function needed for normalization and the integral can be estimated by considering only S(y). When $x_{0}$ is 0, $S(y)=1$ and normalization guarantees that $C_{o,s}(t)=1$. Initially, the wavepackets are centered around $y=y_{0}$ and we could choose an initial displacement $x_{0}\gg \sigma(y_{0})$ such that $S(y_0) \sim 0$ and $C_{o,s}(0) \sim 0$ . As time increases, the wavepacket will move away from the injection point and broaden. At typical experimental temperatures, the broadening is small compared with the distance it travels in y[@a4]. When the centers of the wavepackets reach a region far from the the injection point, $\sigma(y)$ around the new centers will grow to be much larger than $x_0$ and we have $S(y) \sim 1$, $C_{o,s}(t) \sim 1$. In other words, even though we start with two almost nonoverlapping(incoherent) initial wavepackets, the QPCs increase the coherent overlap as the wavepackets move away and this coherent overlap can reach unity in the far region. This overlap is of coherent nature and is different from the trivial spatial overlap one might expect. Spatial overlap is not enough to explain the experimentally observed stability due to the fluctuating phase in chaotic systems. However, our result shows that the overlap is large even if one takes into account the phases of the wavepackets and this coherent overlap can not be destroyed by disorders. This large coherent overlap only exists because the two different QPCs represent two different Hamiltonians. If propagated under the same Hamiltonian, the coherent overlap will always remain small. In the experiment, both the finite size of the QPCs (making them not perfectly adiabatic) and the disorder can degrade the coherent overlap. We numerically estimate the coherent overlap under these conditions. The QPC’s size is estimated from the Scanning Gate Microscopy data in [@a1] and the random potential has a correlation length of $0.9\lambda_F$ and standard deviation of $8\%E_F$, where $\lambda_F$ is the Fermi wavelength and $E_F=7.5meV$ is chosen to match that in the experiment. The random potential is generated to match both the sample mobility and the distance from donors to 2DEGs[@a19]. The numerical results show that $C_{o,s}(t)$= $85\%$ when the QPC is shifted by $\lambda_{F}$ as in the experiment. Thus the degradation of the coherent overlap at long range from the QPC is modest. Starting with two almost nonoverlapping(incoherent) initial wavepackets, and evolving separately under the influence of two different QPCs, their coherent overlap increases with time and distance, increasing fastest close to the QPCs. The coherent overlap eventually saturates to some constant value far from the QPCs. The two wavepackets now evolve effectively under the same Hamiltonian and their coherent overlap cannot be changed by the presence of disorder, for example. This is why we measure 25$\lambda_{F}$ downstream from the injection point, where the potential due to the QPC has died off. *Given the large coherent overlap between the two wavepackets, we should expect the same set of branches far from the injection point even though the flow patterns look different close to the QPC, as shown in our quantum simulations in Fig.\[qflux\]*. Another kind of disorder can reduce the overlap: backscattering from hard impurity scatterers. However, backscattering was suppressed in the original experiment due to the high purity of the samples used[@a1]. Reference[@a1] included both classical and quantum simulations and discussion. Does our explanation of the branch stability also apply to classical simulations? Indeed it does, but the proper classical initial conditions to represent the QPC are subtle and require care. A choice that closely resembles the quantum initial conditions is to use the Wigner quasiprobability distribution[@a7], defined as $$P(\vec{x},\vec{p})=\frac{1}{\pi\hbar}\int_{-\infty}^{\infty}d\vec{s}\ e^{2i\vec{p}\cdot \vec{s}/\hbar}\Psi^{*}(\vec{x}+\vec{s})\Psi(\vec{x}-\vec{s}) \label{eWigner}$$ The advantages are twofold: a) it produces the correct quantum spatial distribution $P(\vec{x})=\int P(\vec{x},\vec{p})d\vec{p}=|\Psi(\vec{x})|^{2}$ and momentum distribution $P(\vec{p})=\int P(\vec{x},\vec{p})d\vec{x}=|\Psi(\vec{p})|^{2}$; b) it properly accounts for the momentum uncertainty due to the confinement of QPC. Keeping y fixed at $y_{0}$, applying (\[eWigner\]) to (\[eigenchannel\]) in x yields $$P(x,p_{x})=\frac{1}{\pi \sigma_{p_{x}} \sigma_{x}} e^{-\frac{p_{x}^2}{\sigma_{p_{x}}^{2}}-\frac{x^{2}}{\sigma_{x}^{2}}} \label{eigenwi}$$ where $\sigma_{p_{x}}^{2}=m\hbar\omega(y_{0})$ and $\sigma_{x}^{2}=\hbar/m\omega(y_{0}) $ When $\omega(y)$ changes sufficiently slowly compared to the motion in y, (\[eigenwi\]) holds approximately true for any $y>y_{0}$, which implies that momentum distributions are highly correlated at any position no matter which QPC the electron originates from. The only difference is the overall probability of arriving at that point. This already hints as to why branches remain at the same positions with a modified strength. As in the quantum case, we use a realistic QPC potential, and weak random potentials in the open regions of the 2DEGs. We sample according to (\[eigenwi\]), with the keeping energy fixed at $E_{F}$ by eliminating trajectories with larger energy in the Wigner distribution, and boosting those with less in $p_y$. These details may be omitted and do not change the conclusions about branch populations and overlap. We propagate the electrons classically. We show Poincare’ surface of section plots[@a6; @a19] in Fig.\[surface\]. In the absence of disorder(Fig.\[surface\]a and b), the adiabaticity of the QPC ensures that when the electrons emerge, most energy is transferred from the x (transverse) direction to the y (longitudinal) direction, which is also expected in the quantum case. The results when disorder is present is shown in Fig.\[surface\]c$\&d$. As can be seen, very similar regions in phase space are occupied, with different relative strengths, when the QPC is shifted. To quantify the overlap in phase space, we define the correlation to be $$C(P_{o},P_{s})=\int dxdp_{x} \ \sqrt{P_{o}(x,p_{x})P_{s}(x,p_{x})}$$ where $P_{o}$ corresponds to the distribution in the original QPC and $P_{s}$ the shifted one. Twenty five wavelengths away from the injection point, it is measured that $C(P_{o},P_{s})$=79$\%$, which is comparable to our quantum result. ![(Color Online) Classical simulations of the total flux in the y direction that passes through a given point. ([**a**]{}) shows the case where the QPC is not shifted and ([**b**]{}) shows that when the QPC is shifted by $\lambda_{F}$ to the right. The white reference grid is at the same location in both images and the plots start at y=$y_{0}+15\lambda_{F}$. In both figures, the branches labeled by the red and yellow arrows are clearly visible. More information about this figure can be found in the supplementary material. []{data-label="cflux"}](cflux.pdf){width="8cm"} The classical approach to branching is based on caustics which develop in coordinate space due to focussing effects, and stable regions in phase space[@a8; @a5; @a16] which persist some distance away from the injection point. (Eventually, stable regions, which form by chance so to speak in the random potential, are also subject to destruction further on in the random potential). Each branch corresponds to a localized region in phase space with its strength determined by the electron density in those regions. After shifting the QPC, similar regions in phase space are occupied with only a changed relative density, which means in coordinate space that the same branches are occupied with a different strength. This explains the observation in the experiment[@a1]. In Fig.\[cflux\], we present our simulations of the total classical flux that passes through a given point, which confirms that classical trajectories can indeed reproduce the observed stability. It is worth noting that the same stable regions could in principle be populated from both QPCs, causing some similarity of branch appearance, but this will not be a generic effect for all random potentials and QPC shifts[@a17]. ![(Color Online) Quantum simulations of total flux in the y direction that passes through a given point when the second mode of the QPC is open. ([**a**]{}) corresponds to the first mode of the QPC while ([**b**]{}) shows contribution from the second mode alone. The starting point and length scale are the same as in Fig.\[qflux\], but the flux strength is presented in log scale instead. In both figures, the branches pointed to by the red, yellow and purple arrows are clearly visible. []{data-label="second"}](second.pdf){width="6cm"} One advantage of a classical interpretation is that it can provide intuition in cases where the quantum dynamics is less intuitive. One example would be to consider what happens when the second mode of QPC is open. According to reference [@a4], we need to independently propagate two wavepackets where one corresponds to the first mode and the other corresponds to the second mode. Their contributions to the flux are then added up incoherently to produce the experimental measurements. Since the contribution from the first mode is added incoherently, it is no surprise that the same set of branches recurs when both modes are open. However, it is interesting to ask what happens if one looks at the contribution from each mode alone. Quantum mechanically, the first and second mode are orthogonal to each other, and, therefore have zero overlap at all time since the Hamiltonian is the same. However, the classical phase space regions corresponding to the second mode alone would still overlap more or less with that due to the first mode[@a18]. As a result, the second mode alone should still produce some similar branches that appear in the first mode with a different strength. In order to see this effect, we take the logarithm of the quantum flux due to each mode alone and present the results in Fig.\[second\]. As we can see, some of the strongest branches are clearly preserved, which verifies our prediction. In conclusion, we have successfully explained the stability of branched flow against large changes in initial conditions using both quantum and classical simulations, which agree on the fact of the stability of branches against shifts of the QPC injection point. This resolves a puzzle raised by a recent experiment[@a1] and shows the role of the QPC in enhancing the stability of branched flow in 2DEGs. Our classical interpretation predicts a further stability of the branched flow that can not be readily inferred form the experiment. The interpretations in this paper can provide useful insights into future applications in the coherent control of electron flow, branch management and probing local random potential. We acknowledge support from Department of Energy under DE-FG02-08ER46513. [99]{} M.A. Topinka, B. J. LeRoy, S. E. J. Shaw, E. J. Heller, R. M. Westervelt, K. D. Maranowski and A. C. Gossard, Science [**289**]{}, 2323 (2000). M. A. Topinka, B. J. LeRoy, R. M. Westervelt, S. E. J. Shaw, R. Fleischmann, E. J. Heller, K. D. Maranowski and A. C. Gossard, Nature [**410**]{}, 183-186 (2001). R. Hohmann, U. Kuhl, H.-J. Stockmann, L. Kaplan and E. J. Heller, Phys. Rev. Lett [**104**]{}, 093901 (2010). M. A. Wolfson and S. Tomsovic, J. Acoust. Soc. Am. [**109**]{}, 2693 (2001). K. E. Aidala, R. E. Parrott, T. Kramer, E. J. Heller, R. M. Westervelt, M. P. Hanson and A. C. Gossard, Nat. Phys. [**3**]{}, 464 (2007). D. Maryenko, F. Ospald, K. V. Klitzing, J. H. Smet, J. J. Metzger, R. Fleischmann, T. Geisel and V. Umansky, Phys. Rev. B [**85**]{}, 195329 (2012). E. J. Heller, L. Kaplan, and A. Dahlen, J. Geophys.Res. [**113**]{}, C09023(2008). M. P. Jura, M. A. Topinka, L. Urban, A. Yazdani, H. Shtrikman, L. N. Pfeiffer, K. W. West and D. Goldhaber-Gordon, Nat. Phys. [**3**]{}, 841-845 (2007). E. J. Heller, K. E. Aidala, B. J. LeRoy, A. C. Bleszynski, A. Kalben, R. M. Westervelt, K. D. Maranowski and A. C. Gossard, Nano Lett. [**5**]{}, 1285 (2005). S. E. J. Shaw’s Thesis, Harvard University, available at http://www.physics.harvard.edu/Thesespdfs/sshaw.pdf (2002). Adel Abbout, Gabriel Lemarie and Jean-Louis Pichard, Phys. Rev. Lett [**106**]{}, 156810 (2011). E. J. Heller, J. Chem. Phys. [**65**]{}, 1289 (1976). L. Kaplan, Phys. Rev. Lett [**89**]{}, 184103 (2002). E. J. Heller and S. E. J. Shaw, International Journal of Modern Physics B [**17**]{}, 3977 (2003). J. J. Metzger, R. Fleischmann and T. Geisel, Phys. Rev. Lett [**105**]{}, 020601 (2010). If we have a stable region in phase space that happens to be cut into halves by the shifting of the QPC, it will produce close branches with similar strength even with zero overlap at all times. However, there are two reasons why this will only have marginal effect in producing the experimentally observed stability. First, the probability of cutting a stable regions in halves such that both regions have appreciable strength is negligible. Therefore, it is not reproducible by experiment. Secondly and most importantly, any classical stable regions will decay with distance. Thus, any close branches, if exist, will separate eventually and contribute only marginally to the experimental stability. This doesn’t depend on the choice of the Wigner quasiprobability distribution. It is simply due to the fact that both modes have to live in the same region in coordinate space and maintain considerable amount of momentum uncertainty due to the confinement. See the supplementary material provided for more information.

--- abstract: | We discuss certain generalization of the Hilbert space of states in noncommutaive quantum mechanics that, as we show, introduces magnetic monopoles into the theory. Such generalization arises very naturally in the considered model, but can be easily reproduced in ordinary quantum mechanics as well. This approach offers a different viewpoint on the Dirac quantization condition and other important relations for magnetic monopoles. We focus mostly on the kinematic structure of the theory, but investigate also a dynamical problem (with the Coulomb potential).\ **Keywords:** Magnetic monopoles, quantum mechanics, noncommutative space author: - Samuel Kováčik - Peter Prešnajder bibliography: - 'your-bib-file.bib' title: Magnetic monopoles in noncommutative quantum mechanics --- (Submitted to JMP ) Introduction ============ Magnetic monopoles are a unique part of physics. Their existence is being considered for more than a century, yet they have never been observed. They appear (in theory) in various areas of physics, persistently throughout different models, always playing a slightly different role. They premiered in the classical theory of electromagnetism. Maxwell equations in vacuum are symmetric under a transformation known as electric-magnetic duality $\left(\textbf{E},\textbf{B}\right) \rightarrow \left( \textbf{B}, -\textbf{E}\right)$. This symmetry is violated in the presence of electric sources $\rho_E$, but can be recovered by introducing (monopole) magnetic sources $\rho_M$. New phenomena appear in such a generalized theory, for example, electromagnetic fields generated by a static system of electric and magnetic monopole have a non-vanishing angular momentum. It is often comfortable to work with electromagnetic potentials $\textbf{A}, \varphi$ instead of electromagnetic fields $\textbf{E}, \textbf{B}$. It might seem that magnetic potentials cannot describe magnetic monopoles, since $\mbox{div rot } \textbf{A} = 0$ seems to follow directly (as $ \partial_{[ i} \partial_{j ]} A_k=0$), resulting in the absence of magnetic monopoles. This, however, holds only for nonsingular potentials $\textbf{A}$ (for which the order of derivatives can be exchanged) and, therefore, monopoles could be described by singular potentials, see e.g. [@zwanziger; @milton]. The following potentials $$\textbf{A} = \frac{g}{4 \pi r} \frac{\textbf{r}\times \textbf{n}}{r-\textbf{r} . \textbf{n}}\ \ \ \ \mbox{or}\ \ \ \ \textbf{A} = - \frac{g}{4 \pi r} \frac{\textbf{r}\times \textbf{n}}{r+\textbf{r} . \textbf{n}}$$ (where $\textbf{n}$ is a unit constant vector) result into Coulomb(-like) magnetic field $$\textbf{B} = \frac{g}{4 \pi} \frac{\textbf{r}}{r^3}.$$ In the quantum theory the description of Yang [@Yang] is preferred. In this framework, one describes monopoles with sections (avoiding the singularity) related by a gauge condition in the overlapping regions. A consistent quantum theory requires the electric and the magnetic charge to satisfy the Dirac quantization condition (in convenient units) $$e g = \frac{n}{2} \, , n \in \mathbb{Z} \,.$$ This condition has an appealing physical consequence - the electric charge has to be quantized, as is observed in nature. The appearance of magnetic monopoles in quantum field theory is (again) slightly different, they appear as topological solutions, in contrast to ordinary particles appearing as quantum excitations. As was shown by Polyakov and ’t Hooft, monopoles are a general consequence of grand unification theories (GUT), appearing when a higher symmetry brakes down into a product containing $U(1)$, [@pol; @hooft]. Mass of the monopoles is, therefore, expected to be on the GUT breaking scale. Monopoles also appear in cosmology, existing as topological defects between domains of different vacua. Cosmology also offers an explanation why we have not observed any magnetic monopoles yet, the process of inflation diluted them remarkably. For the purpose of this paper is the quantum mechanical (QM) description the most convenient one. Let us quote the results of Zwanziger [@zwanziger], in the presence of monopole states is the usual Heisenberg algebra modified as $$\begin{aligned} \label{MMzw1} \left[ \hat{x}_i , \hat{x}_j \right] &=& 0 , \\ {\nonumber}\left[ \hat{\pi}_i , \hat{x} _j \right] &=& - i \delta_{ij} , \\ {\nonumber}\left[\hat{\pi}_i ,\hat{\pi}_j \right] &=& i \mu \varepsilon_{ijk} \frac{\hat{x}_k}{r^3} ,\end{aligned}$$ where $\mu = e g$ and the Dirac quantization condition dictates $\mu \in \mathbb{Z} /2$. In the same paper, a dynamical problem with the Coulomb potential was analyzed. The Coulomb problem in ordinary QM can be solved algebraically, as was first proposed by Pauli, generalizing the classical notion of the Laplace-Runge-Lenz vector $A_i$, [^1]. Components of this vector, together with the components of the angular momentum operator form a representation of either the $so(1,3)$ or the $so(4)$ algebra, depending on the sign of the energy of the system. The algebra closes only on energy eigenstates $\hat{H}\psi = E \psi$ $$\begin{aligned} \label{LRLalg} &\left[ \hat{L}_i, \hat{L}_j \right] =&i \varepsilon_{ijk} \hat{L}^k ,\\ {\nonumber}&\left[ \hat{L}_i, \hat{A}_j \right] =& i\varepsilon_{ijk} \hat{A}^k ,\\ {\nonumber}&\left[ \hat{A}_i, \hat{A}_j \right] =& - 2i \varepsilon_{ijk} \hat{H} \hat{L}^k,\\ {\nonumber}&\left[ \hat{L}_i, \hat{H} \right] =&\left[ \hat{A}_i, \hat{H} \right]= 0 .\end{aligned}$$ These relations are indifferent to the presence of monopole states, however, the Casimir operators, which determine the energy spectrum, are not $$\begin{aligned} \label{Zwanziger} \hat{A}_i \hat{L}_i &=& q \mu , \\ {\nonumber}\hat{A}_i \hat{A}_i - 2 \hat{H} \left( \hat{L}_i \hat{L}_i +1\right) &=& q^2+ 2 \hat{H} \left(- \mu^2 \right) ,\end{aligned}$$ where $q$ is the electric charge from the Coulomb potential. Below we shall investigate magnetic monopoles in the framework of noncommutative quantum mechanic (NC QM), which is a particular application of the ideas of noncommutative geometry to QM, [@Con; @Mad1]. NC QM differs from ordinary QM by having a nonvanishing commutator of the coordinate operators. This results in the impossibility of exact position measurements, which can be motivated by (thought experiments in) quantum theory of gravity, [@DFR]. NC theories are closely related to different candidates for such a theory, the string/M-theory being a prominent example, [@string]. NC QM models that do not possess rotational invariance have been investigated in [@AB], [@CP], however, our problem requires full 3D rotational symmetry. Such a model was proposed in [@Jabbari], the construction used here has been developed in [@GP1; @GP2; @vel; @LRL]. Using the auxiliary bosonic operators approach, the exact solution of NC Coulomb problem was found, both dynamically and algebraically. In this paper, we utilize the same approach, but consider a generalized class of physical states to describe magnetic monopoles. This paper is organized as follows. First, we construct NC QM using auxiliary bosonic operators. In subsection A we analyze general kinematical structures, in subsection B a dynamical one (with the Coulomb potential). In subsection C we briefly present how can the results be reproduced in the context of ordinary QM. Conclusions are followed by the Appendix containing all lengthy and technical calculations. Noncommutative quantum mechanics ================================ The first thing to do is to decide on the RHS of the noncommutativity relation, from which the restriction on position measurements follows. We study a rotationally invariant model described by $$\label{NCrel} [x_i, x_j] = 2 i \lambda \varepsilon_{ijk} x_k ,$$ where $\varepsilon_{ijk}$ is the Levi-Civita symbol and $\lambda$ is a constant describing the NC length scale. It is not fixed, but as an artifact of quantum gravity it could be expected to be approximately the Planck length. Resulting NC space corresponds to an infinite sequence of fuzzy spheres. Let us consider two set of auxiliary bosonic creation and annihilation (c/a) operators satisfying $$\label{aux} [a_\alpha,a^+_\beta]=\delta_{\alpha\beta },\ \ [a_\alpha,a_\beta]=[a^+_\alpha, a^+_\beta]=0,$$ with $ \alpha , \beta = 1,2$, which act in a Fock space $\mathcal{F}$ spanned on normalized vectors $$|n_1,n_2\rangle= \frac{(a^+_1)^{n_1}\,(a^+_2)^{n_2}}{ \sqrt{n_1!\,n_2!}}\ |0\rangle.$$ The NC coordinates satisfying (\[NCrel\]) are constructed using the c/a operators as $$\label{NCx} x_i = \lambda \sigma^i_{\alpha \beta} a^+_\alpha a_\beta ,$$ where $\sigma^i$ are the Pauli matrices. Using the number operator $N=a^+_\alpha a_\alpha$ we can define the radial coordinate operator as $$r = \lambda \left( N+1 \right) = \lambda \left( a^+_\alpha a_\alpha +1 \right) .$$ It can be easily checked that $r^2 = x^2 + \lambda^2$, which differs from the ordinary result but reproduces it in the $\lambda \rightarrow 0$ limit. We refer to such as ’the commutative limit’, since in it (\[NCrel\]) becomes $[x_i,x_j]=0$ as in ordinary QM. In this limit should the results either reproduce the ordinary ones or vanish. We define the Hilbert space $\mathcal{H}_\kappa $ as a completion of the linear space of operators in the auxiliary Fock space spanned by analytic functions $\Psi_\kappa(a^+,a)$ satisfying relation $$\label{states} \Psi_\kappa(e^{-i\tau} a^+,e^{i\tau} a) = e^{-i\tau\kappa} \Psi_\kappa (a^+,a), \ \tau \in {\bf R}, \ \mbox{fixed}\ \kappa \in \mathbb{Z} ,$$ that possesses finite norm $$\label{norm} ||\psi_\kappa||^2 = 4\pi \lambda ^2 Tr [ \psi^+_\kappa \,\hat{r}\, \psi_\kappa] ,$$ where $\hat{r}$ acts as $\hat{r} \psi_\kappa = \frac{1}{2}\left( r \psi_\kappa + \psi_\kappa r \right)$ and has been added to reproduce the ordinary integration $\int d^3x$ in the commutative limit [^2] Because NC coordinates (\[NCx\]) contain equal number of creation and annihilation operators, for any state of the form $\Psi_0(\textbf{x})$ is $\kappa=0$. In this paper we consider a generalized class of states with $\kappa\neq 0$. If $\kappa < 0$ there is $|\kappa|$ more annihilation than creation operators $\# a - \# a^+ = -\kappa = |\kappa|$, if $\kappa > 0 $ it is vice versa $\#a^+ - \#a = \kappa$. Kinematic structures -------------------- We shall now define important physical operators on $\mathcal{H}_\kappa$. To distinguish them from the ones on the auxiliary space $\mathcal{F}$, we denote them with a hat. We are using a lower index to distinguish between left and right multiplication $$\begin{aligned} \hat{X}_{i,L} \Psi_\kappa = x_i \Psi_\kappa ,\ & & \hat{X}_{i,R} \Psi_\kappa = \Psi_\kappa x_i ,\\ \nonumber \hat{r}_{L} \Psi_\kappa = r \Psi_\kappa , \ & & \hat{r}_{R} = \Psi_\kappa r .\end{aligned}$$ The operators $\hat{X}_{i,L}$ and $\hat{X}_{i,R}$ carry the $so_L(3)$ and the $so_R(3)$ Lie algebra representation respectively. Coordinate operators on $\mathcal{H}_\kappa$ are defined as symmetrical combinations $$\label{Xnc} \hat{X}_i = \frac{1}{2} \left( \hat{X}_{i,L} + \hat{X}_{i,R}\right) , \ \hat{r} = \frac{1}{2} \left( \hat{r}_{L} + \hat{r}_{R}\right),$$ while the angular momentum operator satisfying $[\hat{L}_i, \hat{L}_j] = \varepsilon_{ijk} \hat{L}_k$ is an antisymmetrical one $$\hat{L}_i = \frac{1}{2\lambda} \left( \hat{X}_{i,L} - \hat{X}_{i,R}\right).$$ Note that the angular momentum operator acts on $\psi_\kappa$ as $\hat{L}_i \psi_\kappa = \frac{1}{2\lambda} [x_i , \psi_\kappa]$ and that [^3] $$\label{[x,x]} [\hat{X}_i, \hat{X}_j] = \lambda^2 \varepsilon_{ijk}\hat{L}_k.$$ Even thought for generalized states $\hat{r}_L \neq \hat{r}_R \neq \hat{r}$, they are closely related for each $\mathcal{H}_\kappa, \ \kappa \in \mathbb{Z}$ $$\begin{aligned} \hat{r}&=&\hat{r}_L - \frac{\lambda \kappa}{2} = \hat{r}_R + \frac{\lambda \kappa}{2}, \\ {\nonumber}\lambda \kappa &=& \hat{r}_L- \hat{r}_R.\end{aligned}$$ As $\hat{r}$ commutes with the generators $\hat{X}_{i,L}$ and $\hat{X}_{i,R}$, the $so_L(3)$ and the $so_R(3)$ Casimir operators can be expressed in terms of $\hat{r}$ and $\kappa$ as $$\begin{aligned} \hat{X}^2_L &=& \left( \hat{r}+ \frac{\lambda \kappa}{2} \right)^2 - \lambda^2, \\ {\nonumber}\hat{X}^2_R &=& \left( \hat{r}- \frac{\lambda \kappa}{2} \right)^2- \lambda^2 .\end{aligned}$$ For states $\psi_0(\textbf{x})$ with $\kappa=0$ it holds that $\hat{r}_L = \hat{r}_R$ and $\hat{r} = \hat{r}_L$ can be chosen for simplicity, as was done in the aforementioned references. We can use their definitions and results, sharing the same line of reasoning, but have to replace $\hat{r}_L \rightarrow \hat{r}$ and check for possible consequences and modifications. The functions $\psi_\kappa$ with fixed $\kappa$ are mappings $\mathcal{F}_n \rightarrow \mathcal{F}_{n+\kappa}$, they form a representational space for an irreducible $SO(4)$ representation in which it holds that $\hat{r} = \lambda \left( n+1 \right) + \frac{\lambda \kappa}{2}$. The Casimir operators are [^4] $$\begin{aligned} \label{cas4} \hat{c}_1 &=&\hat{L}^2 + \frac{1}{ \lambda^2} \hat{X}^2 = \frac{1}{4 \lambda^2} \left( \hat{X}^2_L + \hat{X}^2_R \right) = \frac{1}{2\lambda^2}\left( \hat{r}^2-\lambda^2+\left( \frac{\lambda \kappa}{2} \right)^2 \right), \\ {\nonumber}\hat{c}_2 &=&\frac{1}{2\lambda} \hat{X}_i\hat{L}_i =\frac{1}{4\lambda^2} \left( \hat{X}^2_L - \hat{X}^2_R \right) = \frac{\kappa}{2\lambda}\hat{r},\end{aligned}$$ Two of the most important physical operators, namely the free Hamiltonian and the velocity operators are defined as $$\begin{aligned} \hat{H}_0 \psi_\kappa &=& \frac{1}{2\lambda \hat{r}} [ a^+_\alpha , [ a_\alpha , \psi_\kappa ]] , \\ {\nonumber}\hat{V}_i \psi_\kappa &=&i [ \hat{H}_0,\hat{X}_i]\psi_\kappa= \frac{i}{2 \hat{r}} \sigma^i_{\alpha \beta} \left(a^+_\alpha \psi_\kappa a_\beta - a_\beta \psi_\kappa a^+_\alpha \right) .\end{aligned}$$ To begin revealing the overall structure let us first combine $\hat{X}_i$ and $\hat{L}_i$ together as $$\hat{L}_{ij} = \varepsilon_{ijk} \hat{L}_k , \ \hat{L}_{k4} = - \hat{L}_{4k} = \lambda^{-1} \hat{X}_k \,$$ to observe an $so(4) \cong su_L(2) \oplus su_R(2)$ Lie algebra structure $$[\hat{L}_{ab},\hat{L}_{cd}] = i \left( \delta_{ac} \hat{L}_{bd}-\delta_{bc} \hat{L}_{ad}-\delta_{ad} \hat{L}_{bc} + \delta_{bd} \hat{L}_{ac} \right) ,$$ where indices go over as $i,j,k,...=1,2,3$ and $a,b,c, ... = 1,...,4$. The central point of ordinary QM is the Heisenberg uncertainty relation, the commutator of $[\hat{V}_i, \hat{X}_j]$. In [@vel] it has been shown that this relation obtains a $\lambda$-correction already for $\psi_0$ states and as it turns out, this correction is of the same for $\psi_\kappa$ states as well $$\label{xV} [\hat{X}_i,\hat{V}_j] = i \delta_{ij} \left(1-\lambda^2 \hat{H}_0\right) \equiv i \lambda \delta^{ij} \hat{V}_4 ,$$ where $\hat{V}_4 \psi_\kappa = \frac{1}{\lambda}- \lambda \hat{H}_0= \frac{1}{2 \hat{r}}\left( a^+_\alpha \psi_\kappa a_\alpha + a_\alpha \psi_\kappa a_\alpha^+ \right)$. Note that $\hat{V}_a$ transforms as an $SO(4)$ vector. Another interesting result of [@vel] is that even though the coordinates do not commute, the velocities do. This, however, fails to be true for $\kappa \neq 0$ states, instead it holds $$\label{F} \left[\hat{V}_i,\hat{V}_j \right] = i\hat{F}_{ij} ,$$ with the magnetic field strength given as $$\hat{F}_{ij} = \varepsilon_{ijk} \frac{- \frac{\kappa}{2}\hat{X}_k}{\hat{r}(\hat{r}^2-\lambda^2)}.$$ This can be generalized into an $SO(4)$ structure by noting that $$\hat{F}_{ab} =-i[\hat{V}_a,\hat{V}_b]= -\frac{\kappa \lambda}{2}\frac{ \varepsilon_{abcd} \hat{L}_{cd}}{\hat{r}\left(\hat{r}^2-\lambda^2\right)}.$$ For its square it holds that $$\label{F42} \frac{1}{2}\hat{F}_{ab}^2=\frac{1}{2}\hat{F}_{ab} \hat{F}_{ab} =\frac{1}{2} \left( \frac{\kappa }{2}\right)^2 \frac{\left(r^2 - \lambda^2 +\left( \frac{\kappa}{2}\right)^2\right)}{r^2 \left( r^2 - \lambda^2 \right)^2} \, .$$ In the aforementioned reference it was noted that the eigenvalues of $\hat{V}_a^2$ lay on a $S^3$ sphere with a radius of $\lambda^{-1}$. This structure is modified for $\kappa \neq 0$ states as well $$\label{V42} \hat{V}_a^2 = \frac{1}{\lambda^2} \left( 1 - \frac{\left( \frac{\kappa \lambda}{2}\right)^2}{\hat{r}^2 - \lambda^2}\right) .$$ In the classical theory such a ($\kappa$ dependent) term arises due to the effective potential of the angular momentum of the fields. Note the similar terms appearing on the RHS of (\[F42\], \[V42\]) and the equations for Casimir operators (\[cas4\]), they allow us to express the squares as $$\hat{V}_a^2 = \frac{1}{\lambda^2} - \frac{\lambda^2 \hat{c}^2_2}{\hat{r}^2 \left( \hat{r}^2 - \lambda^2\right)} , \ \hat{F}^2_{ab} = \frac{\lambda^4}{\hat{r}^4 \left(\hat{r}^2 - \lambda^2\right)^2}\hat{c}_1 \hat{c}_2^2 .$$ We can combine these equations to obtain a single one, generalizing the important $\kappa=0$ result $\hat{V}_a^2 = \lambda^{-2}$ to $$\hat{V}_a^2 + \hat{\varphi}\hat{F}_{ab}^2 = \lambda^{-2} , \,\hat{\varphi}= \frac{\hat{r}^2 \left( \hat{r}^2 - \lambda^2 \right)}{\hat{r}^2 -\lambda^2 + \left(\frac{\lambda \kappa}{2}\right)^2}.$$ Dynamical structure ------------------- Before drawing any conclusion let us take a look at a certain dynamical structure. It is convenient to choose the Coulomb potential $U=\frac{q}{r}, \ q = e^2$, there are two reasons for it. First, the Coulomb problem can be solved algebraically (as was found by Pauli) and second, it has already been analyzed in the framework of NC QM (for $\kappa=0$ states) [@GP1; @GP2; @LRL] . The time independent Schrödinger equation with the Coulomb potential for generalized $\kappa$ states is $$\label{SchrK} \hat{H}\psi_\kappa = \left( \hat{H}_0 - \frac{q}{\hat{r}}\right) \psi_{\kappa} = E \psi_{\kappa} .$$ The following vector is called the Laplace-Runge-Lenz (LRL) vector[^5] and is conserved in the presence of such a potential $$\hat{A}_k = \frac{1}{2} \varepsilon_{ijk} \left( \hat{L}_i \hat{V}_j + \hat{V}_j \hat{L}_i \right) + q \frac{\hat{X}_k}{\hat{r}} .$$ The same is true for the angular momentum operator. We can express it as $$[\hat{H} , \hat{L}_i ] = 0 , \, [\hat{H}, \hat{A}_i] = 0 .$$ Commutators of the angular momentum and the LRL vector are $$\begin{aligned} \label{LRLalgNC} [\hat{L}_i, \hat{L}_j ] &=& i \varepsilon_{ijk} \hat{L}_k , \\ \nonumber [\hat{L}_i, \hat{A}_j] &=& i \varepsilon_{ijk} \hat{A}_k , \\ \nonumber [\hat{A}_i, \hat{A}_j] &=& -2i\hat{H} \left(1 - \lambda^2 \hat{H}\right) \varepsilon_{ijk} \hat{L}_k,\end{aligned}$$ Restricting to energy eigenstates we can take $\hat{H} =E$ and obtain either the $so(3,1)$ or the $so(4)$ Lie algebra, depending on the sign of $E\left(1-\lambda^2 E\right)$. Following from the group theory we know that their Casimir operators are allowed to take discrete values only, from which the discreteness of the spectrum follows. For generalized $\kappa$ states the Casimir operators are $$\begin{aligned} \label{cas} \hat{C}_1 &=& \hat{L}_i \hat{A}_i = -\frac{\kappa}{2}q , \\ \nonumber \hat{C}_2 &=& \hat{A}_i \hat{A}_i + (- 2E +\lambda^2 E^2) (\hat{L}_i \hat{L}_i + 1) \\ {\nonumber}&=& q^2 + \left(\frac{\kappa}{2}\right)^2 (- 2E +\lambda^2 E^2) .\end{aligned}$$ Again, we observe a $\kappa$ correction. Ordinary space -------------- It has been noted earlier that the results can be reproduced in ordinary QM. The starting point is to realize that the isometry group of three-dimensional Euclidean space is locally isomorphic to that of complex $\textbf{C}^2$ plane. Two complex coordinates $z_1, z_2$ of $\textbf{C}^2$ can be mapped into three real $\textbf{R}^3$ coordinates by (a Hopf fibration) $x_i = \bar{z} \sigma^i z$. This relation can be understood using Cayley-Klein parameters $$\begin{aligned} \label{CP} z_1 = \sqrt{r} \cos \left(\theta /2 \right) e^{\frac{i}{2}\left(\varphi+\gamma \right)}, \ && \bar{z}_1 = \sqrt{r} \cos \left(\theta /2\right) e^{-\frac{i}{2}\left(\varphi+\gamma \right)}, \\ {\nonumber}z_2 = \sqrt{r} \sin \left(\theta /2 \right) e^{\frac{i}{2}\left(-\varphi+\gamma \right)}, \ &&\bar{z}_2 =\sqrt{r} \sin \left(\theta /2 \right) e^{-\frac{i}{2}\left(-\varphi+\gamma \right)} ,\end{aligned}$$ which are by $x_i = \bar{z} \sigma^i z$ transformed into spherical coordinates of $\textbf{R}^3$, the angle $\gamma$ is lost in translation. $\textbf{C}^2$ is naturally equipped with a Poisson structure $$\label{Poisson} \{z_\alpha, \bar{z}_\beta\}=i \delta _{\alpha \beta} .$$ The (free) Hamiltoanian is $$\label{cHam} \hat{H}_0 = \frac{1}{2r} \{ \bar{z}_\alpha , \{ z_\alpha , . \} \} , \ H_0 \psi (z,\bar{z}) = - \frac{1}{2r}{\partial}_{\bar{z}_\alpha} {\partial}_{z_\alpha} \psi (z,\bar{z}),$$ where $r=\bar{z}_\alpha z_\alpha$. Using this we can define the velocity operator as $$\label{cV} \hat{V}_i =\{ \hat{X}_i, \hat{H}_0 \}= - \frac{i}{2r}\sigma^i_{\alpha \beta} (\bar{z}_\alpha \partial _{\bar{z}_\beta} + z_\beta \partial_{z_\alpha}),$$ the coordinate operator acting only as a left multiplication now. Quantization of $\textbf{C}^n$ can be carried out by replacing $\bar{z}_\alpha , z_\alpha \rightarrow \sqrt{\lambda}a_\alpha^+, \sqrt{\lambda}a_\alpha$ and derivatives with commutators. Note that our model of NC QM can be reconstructed this way, for example the Hopf relation $x_i = \bar{z} \sigma^i z$ becomes (\[NCx\]). If we restrict the algebra of functions to $\textbf{C}^2$ on only those of the form $\psi_0(\textbf{x})$, the Hamiltonian (\[cHam\]) and velocity operator (\[cV\]) are acting as in ordinary QM $$\hat{H}_0 \psi_0(\textbf{x}) = - \frac{1}{2}\partial_i \partial_i \psi_0(\textbf{x}), \, \hat{V}_i \psi_0(\textbf{x}) = -i \partial_i \psi_0(\textbf{x}) ,$$ as follows from the chain rule for derivatives. This way we can formulate ordinary QM on $\textbf{C}^2$ instead of $\textbf{R}^3$. We can also consider a generalized class of states $$\label{comStates} \psi_\kappa (\textbf{x}, \xi) = \psi_0 (\textbf{x}) \xi , \ \xi = \sum \limits_\kappa{}^{'} C_{\kappa_1 \kappa_2} z_1^{\kappa _1}z_2^{\kappa _2} ,$$ with the sum $ \sum \limits_\kappa{}^{'}$ going over all $\kappa_1, \kappa_2$ such that $\kappa_1 + \kappa_2 = - \kappa$. [^6] This alters the action of (\[cV\]) as $$\label{comVA} \hat{V}^j \psi_\kappa= (- i \partial_j + \mathcal{A}^j)\psi_\kappa , \, \mathcal{A}_j = - \frac{i}{2r\xi} \sigma ^j _{\gamma \delta} z_\delta (\partial _{z_\gamma} \xi) .$$ The gauge potential $\mathcal{A}_j$ satisfies (compare it with the last term in (\[V42\])) $$\label{AA} \frac{1}{2}(\mathcal{A}_j)^+\mathcal{A}_j = \left(\frac{\kappa}{2}\right)^2 \frac{1}{2r^2} ,$$ The commutative limit of the results derived in NC QM can be obtained by considering states (\[comStates\]), for example $$\label{VVcom} [\hat{V}_i, \hat{V}_j] = -\frac{\kappa}{2}i\varepsilon^{ijk} \frac{\hat{X}^k}{r^3}.$$ We are now ready to draw conclusions about our results and their relation to magnetic monopoles. Summary and conclusions ======================= Let us recall the kinematic structure of ordinary QM in the presence of monopole states as was derived in [@zwanziger] (on the left) and compare it with the kinematic structure of NC QM with generalized $\kappa \neq 0$ states (equations (\[\[x,x\]\]), (\[xV\]), (\[F\]) on the right) $$\begin{array}{lcl} \ \left[ \hat{x}_i , \hat{x}_j \right] = 0 & \leftrightarrow & \ [\hat{X}_i, \hat{X}_j ] = \lambda^2 \varepsilon_{ijk} \hat{L}_k, \\ \ \left[ \hat{x}_i, \hat{\pi}_j \right] = i \delta_{ij} &\leftrightarrow & \ [\hat{X}^i,\hat{V}^j] = i \delta^{ij} \left(1-\lambda^2 \hat{H}_0\right), \\ \ \left[\hat{\pi}_i ,\hat{\pi}_j \right] = i \mu \varepsilon_{ijk} \frac{\hat{x}_k}{r^3} & \leftrightarrow & \ \left[\hat{V}_i,\hat{V}_j \right] = i\frac{-\kappa}{2} \varepsilon_{ijk} \frac{\hat{X}_k}{\hat{r}(\hat{r}^2-\lambda^2)}. \end{array}$$ The relations between the angular momentum operators and the LRL vector are the same (in the $\lambda \rightarrow 0$ limit), as one can check comparing (\[LRLalg\]) and (\[LRLalgNC\]). Zwanziger [@zwanziger] derived the Casimir operators for the symmetry algebra of the Coulomb problem in the presence of magnetic monopoles (on the left). Let us compare his results with those for $\kappa$ states in equation (\[cas\]) (on the right) $$\begin{array}{lcl} \hat{C}_1 = -q \mu & \leftrightarrow & \hat{C}_1= \frac{\kappa}{2}q , \\ \hat{C}_2 = q^2 + (\mu)^2(-2E) & \leftrightarrow & \ \hat{C}_2 = q^2 + \left(\frac{\kappa}{2}\right)^2 (-2E+\lambda^2 E^2). \end{array}$$ The results are the same (in the commutative limit) if we set $\mu = -\frac{\kappa}{2}$. We need to check if such identification is possible, since $\mu$ has to obey the Dirac quantization condition $\mu \in \mathbb{Z}/2$. Recall that $\kappa$ counts the difference in the number of creation and annihilation operators and therefore $\kappa/2 \in\mathbb{Z}/2$ as well. The identification is perfect and offers a different viewpoint on the Dirac condition. Therefore $\psi_\kappa$ are to be interpreted as monopole states in NC QM. If we set $\lambda=0$, but keep $\kappa \neq 0$ we obtain ordinary QM with magnetic monopoles. By setting $\kappa = 0,\ \lambda \neq 0$ we obtain NC QM without monopoles. Finally by setting $\kappa = \lambda = 0$ ordinary QM (without monopoles) is recovered. It shall be reminded that for a system of two dyons[^7] are the parameters $q, \mu$ defined (in convenient units) as $$q = - \frac{e_1 e_2 + g_1 g_2}{4\pi} , \, \mu =\frac{e_1 g_2 - g_1 e_2}{4\pi} \, .$$ Therefore, the considered case describes for example an electron orbiting a nucleus with a magnetic monopole in it or an electrically charged magnetic monopole. From (\[comStates\]), it can be understood how do the generalized states describe monopoles. In $\textbf{C}^2$ there are 4 coordinates, but for wavefunctions of the form $\psi_0(\textbf{x})$ one of them, with a topology of $S^1$, vanishes. However, for $\psi_\kappa$ states it persists as a factor $e^{- \frac{i}{2}\kappa \gamma}$ winding around $$\begin{aligned} \psi_{0}&=&\psi_0(\textbf{x})=\Phi(r,\varphi, \theta), \\ \nonumber \psi_{1}&=&\psi_0(\textbf{x})\bar{z}_1=\Phi(r,\varphi, \theta) e^{-\frac{i}{2}\gamma}, \\ \nonumber \psi_{2}&=&\psi_0(\textbf{x})\bar{z}_1\bar{z}_2 =\Phi(r,\varphi, \theta) e^{- i \gamma}, \\ {\nonumber}&...& \\ {\nonumber}\psi_\kappa &=& \Phi(r,\varphi, \theta) e^{- i \frac{\kappa}{2}\gamma}.\end{aligned}$$ Note that $|\psi_\kappa|^2 = \psi_\kappa^\dagger \psi_\kappa$ always contains equal number of creation and annihilation operators (or $\bar{z}$ and $z$). Appendix ======== Strategy is the same for most of the calculations. If we want to prove an equation we express its LHS in terms of c/a operators, shuffle them using (\[aux\]) and recombine them to obtain the RHS. This procedure is often rather straightforward, but sometimes involves a tricky step or two. Writing down everything would be overwhelming (not to mention unnecessary), therefore we gather only the crucial steps here. As was mentioned, the important novelty for generalized $\kappa \neq 0$ states is that the left and the right multiplication by $r$ are unequal $$\begin{aligned} \label{rLrR} \hat{r}_L &=& \hat{r}+\rho, \ \hat{r}_R = \hat{r}-\rho , \ \rho = \frac{\lambda \kappa}{2} , \\ {\nonumber}\hat{r}&=&\frac{1}{2}\left(\hat{r}_L + \hat{r}_R\right), \\ {\nonumber}\lambda \kappa &=& \hat{r}_L -\hat{r}_R=2 \rho .\end{aligned}$$ One needs to go through the same calculations as in [@vel; @LRL], identify where the assumption $\hat{r}_L=\hat{r}_R$ was used and track down the corrections using (\[rLrR\]). It is very useful to use auxiliary operators $$\begin{aligned} \label{ab} &\hat{a}_\alpha \psi = a_\alpha \psi , \, & \hat{a}_\alpha^+ \psi = a_\alpha^+ \psi , \\ \nonumber &\hat{b}_\alpha \psi = \psi a_\alpha , \, & \hat{b}_\alpha^+ \psi = \psi a_\alpha^+ \end{aligned}$$ and their quadratic combinations $$\begin{aligned} \label{auxALG} \hat{w}_{\alpha \beta} = \hat{a}^+_\alpha \hat{b}_\beta -\hat{a}_\beta \hat{b}^+_\alpha ,\ && \hat{\zeta}_{\alpha \beta} = \hat{a}^+_\alpha \hat{b}_\beta + \hat{a}_\beta \hat{b}^+_\alpha , \\ \ \hat{\chi}_{\alpha \beta} = \hat{a}^+_\alpha \hat{a}_\beta + \hat{b}_\beta \hat{b}^+_\alpha \nonumber, \ && \hat{\mathcal{L}}_{\alpha \beta} = \hat{a}^+_\alpha \hat{a}_\beta - \hat{b}_\beta \hat{b}^+_\alpha \nonumber .\end{aligned}$$ Most of the physical operators can be expressed using those either after contracting the indices $\alpha, \beta$ together ($\hat{A}_{\alpha \alpha} = \hat{A}$) or with those of Pauli matrices ($\hat{A}_{\alpha \beta} \sigma^i_{\alpha \beta} = \hat{A}_i$). For example $\hat{L}_i = \frac{1}{2}\hat{\mathcal{L}}_i$, $\hat{X}_i = \frac{\lambda}{2} \hat{\chi}_i$, $\hat{r} = \frac{\lambda}{2}( \hat{\chi}+2 )$, $\hat{V}_i = \frac{i}{2r} \hat{w}_i$, $\hat{H}_0= \frac{1}{2\lambda r}(\chi - \zeta + 2)$. [^8] **The velocity commutator**\ This calculation is almost a carbon copy of the one for $\kappa = 0$ states in [@LRL], the only modification appears right before the final step $$\begin{aligned} \varepsilon_{ijk} [ \hat{V}_i, \hat{V}_j] &=& \left( \mbox{same steps as for $\kappa=0$ states} \right)\\ {\nonumber}&=&\frac{-\frac{i}{2}\sigma^k_{\alpha \delta}}{\hat{r}^2} ( \frac{\lambda}{\hat{r}} (\cancel{\hat{a}^+_\alpha \hat{b}_\beta \hat{a}^+_\beta \hat{b}_\delta} + \hat{a}_\beta \hat{b}^+_\alpha \hat{a}^+_\beta \hat{b}_\delta - \cancel{\hat{a}_\beta \hat{b}_\alpha^+ \hat{a}_\delta \hat{b}^+_\beta} - \hat{a}^+_\alpha \hat{b}_\beta \hat{a}_\delta \hat{b}^+_\beta \\ {\nonumber}&& - \cancel{\hat{a}^+_\beta \hat{b}_\delta \hat{a}^+_\alpha \hat{b}_\beta} - \hat{a}_\delta \hat{b}^+_\beta \hat{a}^+_\alpha \hat{b}_\beta + \hat{a}^+_\beta \hat{b}_\delta \hat{a}_\beta \hat{b}^+_\alpha + \cancel{\hat{a}_\delta \hat{b}^+_\beta \hat{a}_\beta \hat{b}_\alpha ^+} ) \\ {\nonumber}&&+(\cancel{\hat{a}_\alpha^+\hat{b}_\beta \hat{a}^+_\beta \hat{b}_\delta}-\hat{a}^+_\alpha \hat{b}_\beta \hat{a}_\delta \hat{b}^+_\beta - \hat{a}_\beta\hat{b}^+_\alpha \hat{a}^+_\beta \hat{b}_\delta + \cancel{\hat{a}_\beta \hat{b}^+_\alpha \hat{a}_\delta \hat{b}_\beta^+} \\ {\nonumber}&& - \cancel{\hat{a}_\beta^+ \hat{b}_\delta \hat{a}_\alpha^+ \hat{b}_\beta} + \hat{a}_\delta \hat{b}^+_\beta \hat{a}^+_\alpha \hat{b}_\beta - \cancel{\hat{a}_\delta \hat{b}^+_\beta \hat{a}_\beta \hat{b}^+_\alpha} + \hat{a}_\beta^+ \hat{b}_\delta \hat{a}_\beta \hat{b}^+_\alpha )) \\ {\nonumber}&=&\frac{-\frac{i}{2}\sigma^k_{\alpha \delta}}{\hat{r}^2} \left( \frac{\lambda}{\hat{r}}\left( \frac{2\hat{r}_L}{\lambda}\hat{b}^+_\alpha \hat{b}_\delta - \frac{2\hat{r}_R}{\lambda}\hat{a}^+_\alpha \hat{a}_\delta \right)+\hat{a}_\delta \hat{a}^+_\alpha [\hat{b}^+_\beta, \hat{b}_\beta] + \hat{b}_\delta \hat{b}^+_\alpha [\hat{a}^+_\beta , \hat{a}_\beta]\right) \\ {\nonumber}&=&\frac{-i}{\hat{r}^2 - \lambda^2 }\left( \frac{1}{\hat{r}} \frac{\hat{r}_L \hat{X}_{R,k} - \hat{r}_R \hat{X}_{L,k}}{\lambda}+ \frac{\hat{X}_{L,k} - \hat{X}_{R,k}}{\lambda} \right) \\ {\nonumber}&=&\frac{-i}{\hat{r}(\hat{r}^2 - \lambda^2)}\frac{2 \rho}{\lambda} \hat{X}_k ,\end{aligned}$$ which is equal to $$[\hat{V}_i,\hat{V}_j] = \varepsilon_{ijk} \frac{-i \left(\frac{\kappa}{2}\right) \hat{X}_k}{\hat{r}(\hat{r}^2-\lambda^2)}.$$ **Square of the velocity operator and the (free) Hamiltonian**\ For this calculation, it is convenient to express the velocity operator using (\[auxALG\]) (pairs of terms with contracted indices are put into parenthesis as $a^+_\alpha a_\alpha = (a^+a)$) $$\begin{aligned} \hat{V}_i \hat{V}_i &=& - \frac{1}{4 \hat{r}} \sigma^i _{\alpha \beta} \sigma^i _{\gamma \delta} \hat{w}_{\alpha \beta} \frac{1}{\hat{r}} \hat{w}_{\gamma \delta} \\ {\nonumber}&=&-\frac{1}{4\hat{r}}\left( 2 \delta_{\alpha \delta} \delta_{\beta \gamma} - \delta_{\alpha \beta} \delta_{\gamma \delta} \right) \left( \left(\frac{1}{\hat{r}-\lambda}\hat{a}^+_\alpha \hat{b}_\beta - \frac{1}{\hat{r}+\lambda}\hat{a}_\beta \hat{b}^+_\alpha \right)\left( \hat{a}^+_\gamma \hat{b}_\delta - \hat{a}_\delta \hat{b}^+_\gamma \right) \right) \\ {\nonumber}&=&-\frac{1}{4\hat{r}} \frac{1}{\hat{r}-\lambda}\left( 2\hat{a}^+_\alpha \hat{b}_\beta (\hat{a}^+_\beta \hat{b}_\alpha - \hat{a}_\alpha \hat{b}^+_\beta) - \hat{a}^+_\alpha \hat{b}_\alpha (\hat{a}^+_\delta \hat{b}_\delta - \hat{a}_\delta \hat{b}^+_\delta)\right) \\ {\nonumber}&& +\frac{1}{4\hat{r}} \frac{1}{\hat{r}+\lambda}\left(2 \hat{a}_\beta \hat{b}^+_\alpha (\hat{a}^+_\beta \hat{b}_\alpha - \hat{a}_\alpha \hat{b}^+_\beta) - \hat{a}_\alpha \hat{b}^+_\alpha (\hat{a}^+_\delta \hat{b}_\delta - \hat{a}_\delta \hat{b}^+_\delta)\right) \\ {\nonumber}&=&-\frac{1}{4\hat{r}} \frac{1}{\hat{r}-\lambda}\left( \cancel{2} (\hat{a}^+\hat{b})^2 - 2(\hat{a}^+\hat{a})(\hat{b}\hat{b}^+) - \cancel{(\hat{a}^+\hat{b})^2} +(\hat{a}^+\hat{b})(\hat{a}\hat{b}^+) \right) \\ {\nonumber}&&+\frac{1}{4\hat{r}} \frac{1}{\hat{r}+\lambda} \left( 2 (\hat{a}\hat{a}^+)(\hat{b}^+\hat{b}) - (\hat{a}\hat{b}^+)^2 - (\hat{a}\hat{b}^+)(\hat{a}^+\hat{b}) + \cancel{(\hat{a}\hat{b}^+)^2}\right) \\ {\nonumber}&=& - \frac{1}{4\hat{r}} \frac{1}{\hat{r}-\lambda} \left((\hat{a}^+\hat{b})^2 - 2 \frac{\hat{r}_L-\lambda}{\lambda}\frac{\hat{r}_R-\lambda}{\lambda}+(\hat{a}^+\hat{b})(\hat{a}\hat{b}^+)\right) \\ {\nonumber}&& -\frac{1}{4\hat{r}} \frac{1}{\hat{r}+\lambda}\left( (\hat{a}\hat{b}^+)^2 - 2 \frac{\hat{r}_L+\lambda}{\lambda}\frac{\hat{r}_R+\lambda}{\lambda} +(\hat{a}\hat{b}^+)(\hat{a}^+\hat{b})\right) \\ {\nonumber}&=& - \frac{1}{4\hat{r}} \left( \frac{1}{\hat{r}-\lambda} \left((\hat{a}^+\hat{b})^2 + (\hat{a}^+\hat{b})(\hat{a}\hat{b}^+)\right) +\frac{1}{\hat{r}+\lambda}\left( (\hat{a}\hat{b}^+)^2 + (\hat{a}\hat{b}^+)(\hat{a}^+\hat{b})\right)\right) \\ {\nonumber}&& + \frac{1}{2\hat{r}\lambda^2} \left( \frac{(\hat{r}-\lambda+\rho)(\hat{r}-\lambda - \rho)}{\hat{r}-\lambda} + \frac{(\hat{r} + \lambda+\rho)(\hat{r}+\lambda-\rho )}{\hat{r}+\lambda} \right)\\ {\nonumber}&=& - \frac{1}{4\hat{r}}\left( \frac{1}{\hat{r}-\lambda}\left((\hat{a}^+\hat{b})^2 + (\hat{a}^+\hat{b})(\hat{a}\hat{b}^+)\right) + \frac{1}{\hat{r}+\lambda}\left((\hat{a}\hat{b}^+)^2 + (\hat{a}\hat{b}^+)(\hat{a}^+\hat{b})\right)\right) \\ {\nonumber}&&+\frac{1}{\cancel{2\hat{r}}\lambda^2}\cancel{2\hat{r}} \left( 1 - \frac{\rho^2}{\hat{r}^2 - \lambda^2}\right) .\end{aligned}$$ To identify the $(ab)$ terms we first take $$\hat{H}_0 - \frac{1}{\lambda^2} = - \frac{1}{2\lambda \hat{r}} \left((\hat{a}^+\hat{b}) + (\hat{b}^+\hat{a})\right),$$ and square it to $$\begin{aligned} \left(\hat{H}_0 - \frac{1}{\lambda^2} \right)^2 &=& \frac{1}{2\lambda \hat{r}} \left((\hat{a}^+\hat{b}) + (\hat{b}^+\hat{a})\right) \frac{1}{2\lambda \hat{r}} \left((\hat{a}^+\hat{b}) + (\hat{b}^+\hat{a})\right) \\ {\nonumber}&=& \frac{1}{4 \lambda^2 \hat{r}} \left( \frac{1}{\hat{r}-\lambda}\left( (\hat{a}^+\hat{b})^2 +(\hat{a}^+\hat{b})(\hat{b}^+\hat{a})\right) +\frac{1}{\hat{r}+\lambda}\left( (\hat{b}^+\hat{a})^2 + (\hat{b}^+\hat{a})(\hat{a}^+\hat{b})\right)\right).\end{aligned}$$ Comparing these two expressions we obtain $$\lambda^2 \left( \hat{H}_0 - \frac{1}{\lambda^2}\right)^2 = - \hat{V}^2 + \frac{1}{\lambda^2}\left(1- \frac{\rho^2}{\hat{r}^2-\lambda^2}\right) ,$$ or equivalently $$\hat{V}_a^2 = \frac{1}{\lambda^2}\left( 1 -\frac{\rho^2}{\hat{r}^2-\lambda^2} \right),$$ where $a=1,...,4$ (recall that $\hat{V}_4 = \frac{1}{\lambda}- \lambda \hat{H}_0$). **The Coulomb problem**\ Derivation of the Coulomb system spectrum in an algebraic way (developed by Pauli) is done in detail in [@LRL]. There, it was first shown that the Laplace-Runge-Lenz (LRL) vector defined as $\hat{A}_k =\frac{1}{2}\varepsilon_{ijk} (\hat{L}_i \hat{V}_j + \hat{V}_j\hat{L}_i) + q \frac{\hat{X}_k}{\hat{r}} $ can be expressed using (\[auxALG\]) as $\hat{A}_k = -\frac{1}{2\lambda \hat{r}}(\hat{r}\hat{\zeta} _k - \hat{X}_k \hat{\zeta}) + q \frac{\hat{X}_k}{\hat{r}}$. The Schrödinger equation can be, after restricting on energy eigenstates, expressed as $\hat{W}' = 2 \lambda q$, where $\hat{W}'= \eta \hat{r} - \hat{\zeta}$ with $\eta = \frac{2}{\lambda} + \omega $ and $ \omega = -2 \lambda E$. Afterwards, it is shown that the LRL vector together with the angular momentum operator satisfy $$[\hat{A}_i,\hat{A}_j] =\frac{1}{4\lambda^2} [\hat{W}'_i, \hat{W}'_j] = i\frac{\omega}{\lambda}\left(1+\frac{\omega\lambda}{4}\right)\varepsilon_{ijk} \hat{L}_k =i \varepsilon_{ijk} \left(- 2E + \lambda^2 E^2\right) \hat{L}_k ,$$ $$\label{LA} [\hat{L}_i, \hat{L}_j] = i \varepsilon_{ijk} \hat{L}_k, \ [\hat{L}_i,\hat{A}_j] = i \varepsilon_{ijk} \hat{A}_k ,\ [\hat{L}_i, \hat{H}] = [\hat{A}_i, \hat{H}] =0 .$$ Perhaps rather surprisingly this is not affected by considering $\kappa \neq 0$ states at all. The only differences appear for the Casimir operators, which are used to derive the energy spectrum. The first Casimir operator follows easily from $$\begin{aligned} \hat{X}_i \hat{L}_i &=& \frac{1}{4\lambda}(\hat{X}_{L,i} + \hat{X}_{R,i})(\hat{X}_{L,i} - \hat{X}_{R,i}) = \frac{1}{4\lambda}(\hat{X}_L^2 - \hat{X}_R^2) = \frac{1}{4\lambda}(\hat{r}_L^2 - \hat{r}_R^2) \\ {\nonumber}&=&\frac{1}{4\lambda}(\hat{r}_L+\hat{r}_R)(\hat{r}_L-\hat{r}_R) = \frac{1}{2\lambda}\hat{r} (\hat{r}_L-\hat{r}_R) = \frac{\kappa}{2}\hat{r} , \\ {\nonumber}\hat{L}_j\hat{\zeta}_j &=& \frac{1}{2\lambda}\left((\hat{r}_L-\hat{r}_R)\hat{a}^+_\alpha \hat{b}_\alpha + (\hat{r}_L -\hat{r}_R) a_\alpha \hat{b}^+_\alpha \right) = \frac{\kappa}{2}\hat{\zeta}, \end{aligned}$$ as $$\hat{C}_1' = \hat{L}_j \hat{A}_j = \frac{1}{2\lambda} \hat{L}_j (\eta \hat{X}_j - \hat{\zeta}_j)= -\frac{1}{2\lambda}\left(-\frac{\kappa}{2}\right) (\eta \hat{r} - \hat{\zeta}) = \frac{\kappa}{2}q .$$ The $\kappa \neq 0$ correction is apparent. Derivation of the second Casimir operator is considerably more complicated, the RHS of the following equation is a constant and we need to identify its value $$\label{Peq} \hat{C}_2' = \hat{W}'_i \hat{W}'_i +(\eta^2 \lambda^2 - 4)(\hat{L}_i \hat{L}_i+1) \, .$$ Expressing the terms on the RHS we obtain (after a number of auxiliary calculations) $$\begin{aligned} \hat{W}'_i \hat{W}'_i &+&(\eta^2 \lambda^2 - 4)(\hat{L}_i \hat{L}_i+1) \\ {\nonumber}&=& \eta^2 \hat{X}^2 - \eta\{\hat{X}_i, \hat{\zeta}_i \} + \hat{\zeta}^2 +\frac{2}{\lambda^2}\left( \hat{r}_L \hat{r}_R - \hat{X}_{L,i} \hat{X}_{R,i} + \lambda^2 \right) \\ {\nonumber}&&+ \eta^2 \frac{1}{4}(\hat{X}_L^2 +\hat{X}_R^2 - 2 \hat{X}_{L,i} \hat{X}_{R,i} )- \frac{1}{\lambda^2}(\hat{X}_L^2 + \hat{X}_R^2 - 2 \hat{X}_{L,i} \hat{X}_{R,i}) + \eta^2 \lambda^2 - 4 \\ {\nonumber}&=&\left( - \eta\{\hat{r},\hat{\zeta}\} + \hat{\zeta}^2 \right) +\frac{\eta^2}{4}\left(\hat{X}_L^2 +\hat{X}_R^2 +\cancel{2 \hat{X}_{L,i} \hat{X}_{L,i}} \right)+ \frac{2}{\lambda^2}(\hat{r}_L\hat{r}_R - \cancel{\hat{X}_{L,i}\hat{X}_{R,i}} + \lambda^2) \\ {\nonumber}&&+ \eta^2 \frac{1}{4}(\hat{X}_L^2 +\hat{x}_R^2 - \cancel{2 \hat{X}^L_i \hat{X}^R_i}) - \frac{1}{\lambda^2} (\hat{X}_L^2 + \hat{X}_R^2 - \cancel{2 \hat{X}_{L,i} \hat{X}_{R,i}}) + \eta^2 \lambda^2 - 4 \\ {\nonumber}&=&\left( - \eta\{\hat{r},\hat{\zeta}\} + \hat{\zeta}^2 \right) + \frac{\eta^2}{\cancel{2}}\left( \cancel{2} (\hat{r}^2 - \cancel{\lambda^2}) + \cancel{2} \lambda^2 \left(\frac{\kappa}{2}\right)^2\right) - \frac{1}{\lambda^2}\left( 2 (\cancel{\hat{r}^2} - \cancel{\lambda^2}) + 2 \lambda^2 \left(\frac{\kappa}{2}\right)^2 \right) \\ {\nonumber}&&+\frac{2}{\lambda^2}\left(\cancel{\hat{r}^2} - \lambda^2 \left(\frac{\kappa}{2}\right)^2 + \cancel{\lambda^2}\right) + \cancel{\eta^2 \lambda^2} - \cancel{4} \\ {\nonumber}&=& (\hat{W}')^2 + \eta^2 \lambda^2 \left(\frac{\kappa}{2}\right)^2 - 2 \left(\frac{\kappa}{2}\right)^2 - 2 \left(\frac{\kappa}{2}\right)^2 = 4 \lambda^2 q^2 + (\eta^2 \lambda^2 - 4) \left(\frac{\kappa}{2}\right)^2 .\end{aligned}$$ This work was partially supported by COST action MP1405 (QSPACE) and by VEGA project 1/0985/16. 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--- abstract: | Modeling user engagement dynamics on social media has compelling applications in market trend analysis, user-persona detection, and political discourse mining. Most existing approaches depend heavily on knowledge of the underlying user network. However, a large number of discussions happen on platforms that either lack any reliable social network (news portal, blogs, Buzzfeed) or reveal only partially the inter-user ties (Reddit, Stackoverflow). Many approaches require observing a discussion for some considerable period before they can make useful predictions. In real-time streaming scenarios, observations incur costs. Lastly, most models do not capture complex interactions between exogenous events (such as news articles published externally) and in-network effects (such as follow-up discussions on Reddit) to determine engagement levels. To address the three limitations noted above, we propose a novel framework, , which, to our knowledge, is the first that can model and predict user engagement [*without considering the underlying user network*]{}. Given streams of timestamped news articles and discussions, the task is to observe the streams for a short period leading up to a time horizon, then predict *chatter*: the volume of discussions through a specified period after the horizon. processes text from news and discussions using a novel time-evolving recurrent network architecture that captures both temporal properties within news and discussions, as well as influence of news on discussions. We report on extensive experiments using a two-month-long discussion corpus of Reddit, and a contemporaneous corpus of online news articles from the Common Crawl. shows considerable improvements beyond recent state-of-the-art models of engagement prediction. Detailed studies controlling observation and prediction windows, over $43$ different subreddits, yield further useful insights. author: - '$^1$Subhabata Dutta, $^2$Sarah Masud, $^3$Soumen Chakrabarti, $^2$Tanmoy Chakraborty' bibliography: - 'ref.bib' --- Introduction {#sec:intro} ============ The Web is the most popular medium for large-scale public interaction and information propagation. About 4.3 billion people used the Internet in 2019, with $3.53$ billion using at least one social media site[^1]. Unlike radio or television, social media convey information with active participation of users. One can broadly identify two modes of engagement within user communities. In the [*reshare*]{} mode, a user shares some information with a community (friends, followers, groups, etc.), and members of that community recursively propagate the information. This process creates a tree of reshares, where information flows from the root to the leaves. The other is the [*reply*]{} mode, where one user posts some opinion, and other users reply to that post (or to other replies of the post), thus, forming a discussion. Mining the dynamics of these modes can yield useful insights for opinion mining, market research [@lee2015role; @hu2018luxury], political analysis [@jenkins2018any; @ekstrom2018social] and human psychology [@khan2017social]. A growing body of research has focused on modeling the dynamics of such information propagation — both for [*reshare*]{} [@kobayashi2016tideh; @zhao2018attentional; @wang2018retweet] and [*reply*]{} [@nishi2016reply]. There are broadly two approaches. [*Feature-driven models*]{} mainly rely on three types of features for modeling the growth of reply trees, based on the social network among users, the propagated content, and temporal observations. The other approach is to fit [*self-exciting process*]{} models [@zhao2015seismic] (reviewed in Section \[sec:Related\]). In terms of growth prediction, the existing body of literature has a bias for reshare trees; specifically, Twitter retweet trees. #### **Latent/Implicit social networks.** Most reshare models depend heavily on the user network (*follow* on Twitter, *friend* on Facebook), which is available on only a limited number of platforms. However, if we focus on the dynamics of *discussions*, many platforms do not offer explicit user-user social ties. One such influential platform is Reddit. As of , it is used by $430$ million active monthly users[^2]. Users can post [*submissions*]{} to one of the Reddit communities, commonly called [*subreddits*]{}. Other users can then [*comment*]{} in reply to the submission or any earlier comment on the submission. These comments form a discussion tree, with the submission at the root. Though Reddit provides a [*subscribe*]{} option to its users, using which they implement an internal community structure (and not an inter-user link as in Twitter or Facebook), Reddit keeps this information private. HackerNews, IRC, and Slack assert similar constraints. Engagement prediction methods that rely on the social network structure will lose applicability and performance. ![Hourly submission and comment count for the keyword [*Black Mirror*]{} (a popular Web-series). We show the counts in three different communities (subreddits), and across whole of Reddit. We can clearly observe the differences in user reaction towards the same event in different communities.[]{data-label="fig:subreddit_sub_comment"}](Figures/sub-comment_plot.pdf){width="\columnwidth"} ![Hourly submission and comment count for the keyword [*Black Mirror*]{} (a popular Web-series). We show the counts in three different communities (subreddits), and across whole of Reddit. We can clearly observe the differences in user reaction towards the same event in different communities.[]{data-label="fig:subreddit_sub_comment"}](Figures/subreddit_plot.pdf){width="\columnwidth"} #### **Exogenous effects.** A second critical modeling issue is [*exogenous*]{} influence. An influential event (recurring or sporadic) in the real world determines what topic will be the “talk-of-the-town” on the social media. In Figure \[fig:event\_submission\_comment\], we show how the activities in Reddit change over time for four different topics, each corresponding to different events. [*Narcos*]{} and [*Game of Thrones*]{} are popular television series, releasing each season (collection of episodes) periodically. For both of these recurring events, there are sudden spikes in the number of submissions and comments. The last spikes in activities correspond to the final seasons where we can observe a huge gap in the number of submissions and comments, corresponding to more extensive discussions (high comment per submission ratio). The discovery of Higgs Boson (2012) and its decay (2018) corresponds to the two abrupt spikes in the activity plots of [*God Particle*]{}. [*Demonetization*]{} was an event of national importance in India in November 2016. Owing to its long-standing effects and several events triggered by it, we can observe multiple spikes even after the initial one. It was intensely discussed during the parliamentary elections of India in 2019, as reflected in the large spikes with a high comment-to-submission ratio in the first quarter of 2019. [*Demonetization*]{} is a perfect example of the superposition of multiple exogenous events, financial and political. From these examples, it is quite evident that a model that takes these exogenous events into consideration during training may result in better user engagement predictions. #### **Endogenous (within-community) influence.** While external events play a crucial role in determining “what people will talk about”, the degree to which such ‘chatter’ will evolve, and the time it will take to decay, are strongly dependent on the internal or [*endogenous*]{} states of a community. Figure \[fig:subreddit\_sub\_comment\] shows activity levels related to [*Black Mirror*]{} in different subreddits. Whereas the whole of Reddit ([*r/all*]{}) and the dedicated subreddit [*r/blackmirror*]{} follow the recurring pattern of event arrival (release of seasons), this is not the case for [*r/AskReddit*]{} and [*r/conspiracy*]{}. Interestingly, the subreddit [*r/blackmirror*]{} was created at the end of 2016. Flocking of dedicated followers to this subreddit and their intense engagement played a pivotal role in raising the topic’s popularity in other subreddits and in Reddit overall (sharp spikes in Figure \[fig:subreddit\_sub\_comment\] after 2017 compared to the previous releases in 2014–15). #### **.** We present , a system for chatter prediction that handles the combined challenges of unknown influence network structure and exogenous influence. observes news and discussion streams for a limited time window up to a time **horizon**, after which it predicts **chatter**: the intensity of subsequent discussion up to another specified time. (For concreteness, throughout this paper, we will use **news** articles as the prototypical exogenous influence. We will use **submissions**, **comments**, and **discussion** as prototypical social network activity. Note that these are broad model concepts that may be embodied differently in other chatter prediction applications.) achieves our goal using a network architecture inspired by a **unified chatter model**. In this model, each user follows a two step process of [*read and react*]{}. Upon the arrival of a discussion item, a user reads its content (or views images or video). Then, depending on his/her cognitive state and the content features (topic, complexity, opinion), s/he decides whether or not to react and contribute to the discussion. Any contribution, in turn, affects the state of other users. This process, “read and react”, aggregated over uesrs, can be conceptualized as an evolving mapping of content to its virality, conditioned on the dynamics of the exogenous and the endogenous states. Using two months of Reddit discussions on 43 different subreddits, amounting to nine million submissions and comments, along with 3.9 million time-aligned news articles, we show that makes more accurate chatter predictions compared to recent competitive approaches based on Hawkes Processes [@kobayashi2016tideh], cascades [@cheng2014cascadepred], and others. Drilling down into the subreddits and contrastic their dynamics give additional insights. #### **Summary of contributions** Our contributions are four-fold: - Formal specification of a new chatter prediction problem in settings where social network knowledge is absent and exogenous influence is present. - Design and implementation of , a new chatter prediction system that targets the above setting. - Extensive experimental comparison against prior chatter prediction methods, demonstrating the superiority of . - New chatter prediction data set and accompanying code[^3]. {width="\textwidth"} Design of {#sec:method} ========== Guided by the unified chatter model discussed in Section \[sec:intro\], we describe in this section the complete working of . We set up notation in Section \[subsec:Method:Prelim\], and discuss some basic approaches that cannot capture all the signals available in our setting. We describe how combines these signals from news and discussion, in Section \[subsec:influence\_aggregate\]. Then, in Section \[subsec:evolving\_conv\] we motivate why events arriving after the horizon need time-evolving network components, and present a suitable network for processing post-horizon events. We tie these pieces together with a training loss in Section \[subsec:final\_prediction\]. Figure \[fig:whole\_model\] shows a sketch of . Preliminaries and Notation {#subsec:Method:Prelim} -------------------------- Time is quantized into observation intervals of length $\Delta_\text{obs}$ (e.g., $[t_{k-1}, t_k]$ in Figure \[fig:whole\_model\]). Intervals are indexed with $k$. We denote as $\hat{N}:=\langle (n_i, t_i)|\forall i\in \mathbb{Z}^+\rangle$ the stream of news articles $n_i$ with publication timestamps $t_i$. Let $\hat{S}:=\langle (s_j, t_j)|\forall j\in \mathbb{Z}^+\rangle$ be a stream of submission items $s_j$ posted at time $t_j$. Every news item $n_i$ consists of a text digest with headline and body, while every submission item $s_j$ is a triplet $(s^T_j, s^V_j, s^R_j)$, where $s^T_j$ is the text digest of $s_j$, $s^V_j$ is the subreddit to which $s_j$ was posted, and $s^R_j$ is the average commenting activity (number of comments) in the subreddit within the previous interval. For any submission $s_j$ posted at timestamp $t_j$ with $t_k<t_j\leq t_{k+1}$, we define an [**observation window**]{} $[t_j, t_j+m \Delta_\text{obs} ]$ and a [**prediction window**]{} $[t_j+m\Delta_\text{obs}, t_j+\Delta_\text{pred}]$, where $m\in \mathbb{N}$ is an application-driven hyperparameter. Note that, in this setting it is important to differentiate the roles of submissions posted before and after $t_k$, which is the boundary up to which we have the most recent exogenous-endogenous signals defined. So every submission up to $t_k$ contributes to this endogenous signal. The submission posted within $t_k<t_j\le t_{k+1}$ lies between the two arrivals of influence signal, one in the past ($t_k$) and one in the future ($t_{k+1}$). So, every such submission might be under the influence signal at $t_k$. Our goal is to predict **chatter** pertaining to $s_j$, defined as $y_j = \ln (1+C_j)$, where $C_j$ is the total number of comments made about $s_j$ within the prediction window, after observing commenting activity about $s_j$ within the observation window. (Additive error in log-count prediction amounts to count prediction within a multiplicative factor. $C_j$ depends on $\Delta_\text{pred}$ but we elide that for simpler notation.) As early predictions are most beneficial, we specify two settings: Zero-shot: : Empty observation window, with $m=0$. Minimal early observation: : Here $m>0$, but $m\Delta_\text{obs} \ll \Delta_\text{pred}$. Popular time-series models cannot perform the above tasks well, for several reasons. First, any static mapping from the textual features of the submissions to their corresponding future chatter fails to incorporate the dynamic exogenous and endogenous influences that govern chatter. Second, generative models need a substantial degree of early observation which is not available in our setting. Third, the high arrival rates demand fast response. Therefore, when predicting the future chatter of a submission, chatter under its predecessors in the stream remains mostly unobservable. This precludes autoregressive [@kobayashi2016tideh] approaches. Finally, the lack of knowledge of user-user ties inhibits the employment of information diffusion models [@cheng2014cascadepred]. The cumulative aggregate influence of exogenous and endogenous signals from news and submission streams during $[t_{k-1}, t_k]$ will be endowed a deep representation $G_k$, as defined in Section \[subsec:influence\_aggregate\]. $G_k$ will help map submission texts posted within the next interval to a base chatter intensity, which will be aggregated with the activity within the observation window to predict the final chatter. In what follows, every weight and bias matrix (denoted as $W$ and $Q$, respectively with various subscripts) belongs to the trainable model parameter set. Every text segment (news and submission) is mapped to a sequence of low dimensional representations using a shared word embedding layer. Moreover, the subreddit information $s^V_j$ corresponding to each submission $s_j$ is mapped to a **subreddit embedding vector** $\mathbf{U}_j$ using a shared embedding layer. All these embeddings are part of the trainable parameters of . Cumulative Influence Aggregation {#subsec:influence_aggregate} -------------------------------- We define the exogenous knowledge state $\mathcal{G}_\text{X}$ and endogenous knowledge state $\mathcal{G}_\text{E}$ as functions of subsets of $\hat{N}$ and $\hat{S}$, respectively, such that, at any time instance $t$, $$\label{Eq:knowledge_state} \begin{split} \mathcal{G}_\text{X}(t) &= \mathcal{F}_\text{X}(N|t_i\leq t \ \forall (n_i, t_i)\in N\subseteq \hat{N})\\ \mathcal{G}_\text{E}(t) &= \mathcal{F}_\text{E}(S|t_j\leq t \ \forall (s_j, t_j)\in S\subseteq \hat{S}) \end{split}$$ We define two functions that compute exogenous and endogenous influences: $$\begin{aligned} \text{Exogenous:} & \quad \mathcal{F}_\text{X}(N|t_i\leq t \ \forall (n_i, t_i)\in N\subseteq \hat{N})\\ \text{Endogenous:} & \quad \mathcal{F}_\text{E}(S|t_j\leq t \ \forall (s_j, t_j)\in S\subseteq \hat{S})\end{aligned}$$ We model each of these functions $\mathcal{F}_\text{X}$ and $\mathcal{F}_\text{E}$ in two steps. First, we map each text digest to its influence feature map using a shared convolution block (refer to component ($1$) in Figure \[fig:whole\_model\]). Given an input sequence of word vectors of a news (submission text) as $n_i$ ($s^T_j$), we apply successive 1-dimensional convolution and max-pooling operations to produce a feature map $X^n_i$ ($X^s_j$): $$\label{eq:static_conv1d} \begin{split} C^n_i &= \text{ReLU}\bigl(\text{Conv1D}(n_i|W_\text{static}) \bigr)\\ X^n_i &= \text{MaxPool}(C^n_i) \end{split}$$ where $W_\text{static}$ denotes filer kernels and $C^n_i$ ($C^s_j$) is an intermediate representation. We use parallel branches of convolution and pooling operations with different kernel sizes (1, 3, and 5) to capture textual features expressed by contexts of different sizes. The outputs from each branch are then concatenated to produce the final feature representation $f^n_i$ ($f^s_j$) corresponding to $n_i$ ($s_j$) (detailed organization explained in the Appendix, Figure \[fig:text\_cnn\]). The submission feature maps are concatenated with subreddit vector $\mathbf{U}_j$ to differentiate the influences of submissions from different subreddits, as discussed in Section \[sec:intro\]. The final feature map corresponding to $s_j$ is then $f^{sv}_j$. We denote these stages as [*static*]{} because $W_\text{static}$ remains temporally invariant. #### **ConvNet vs. LSTM/BERT** We chose a simple convolutional architecture [@kalchbrenner2014convnetsentence; @kim2014convnetsentence] over a recurrent one for two reasons — i) convolution reduces the size of parameter space, which is essential in handling large data streams, and ii) our task requires efficient understanding of topic-specific keywords to constitute the influence, as opposed to the complex linguistic structures with long-term dependencies; this makes recurrent architectures an overkill. Our experiments with BERT [@DBLP:conf/naacl/DevlinCLT19] instead of convolution to produce text representations (expectedly) gave no significant gain. Next, using the convoluted feature maps $f^n_i$ ($f^{sv}_j$), we compute discrete approximations of the functions $\mathcal{F}_\text{X}$ ($\mathcal{F}_\text{E}$) at the end of every interval $[t_{k-1}, t_k]$ as $G^n_k$ ($G^s_k$), as follows (see component (4) in Figure \[fig:whole\_model\]): $$G^n_k = \text{GRU}\Bigl(h^n_{k-1}, \langle f^n_i|t_{k-1}<t_i\leq t_{k}\rangle | W^n_G\Bigr)$$ where $\text{GRU}$ is a Gated Recurrent Unit, $t_i$ ($t_j$) corresponds to the timestamp associated with $n_i$ ($s_j$), $h^n_{k-1}$ is the hidden state of the GRU from the previous interval, and $W^n_G$ ($W^s_G$) is the parameter set. Stateful propagation of the hidden state ensures the modeling of short-term as well as long-term influence signals. Finally, the cumulative influence $G_k$ is computed as the concatenation of $G^n_k$ and $G^s_k$. We deploy two different GRUs to aggregate the exogenous and endogenous influences over the intervals given the different arrival patterns of news articles over web and submissions over Reddit (news come in sparse bursts, while submissions mostly come in very high rate). Again, our choice of GRU as the recurrent information processing layer for this task is motivated by our experiments confirming LSTMs to be slower with no performance gain compared to GRUs. Time-evolving Convolution (TEC) {#subsec:evolving_conv} ------------------------------- Following the intuitive motivation of the unified chatter model, we may now seek to map any submission $s_j$ posted in the interval $[t_k, t_{k+1}]$ to its potential to invoke future chatter, controlled by the cumulative influence from the previous interval. Formally, this mapping can be defined as, $$\mathcal{B} = \mathcal{F}_G(s_j|G_k, \ t_k<t_j\leq t_{k+1})$$ We again resort to 1-dimensional convolution to learn feature maps from $s_j$, but this time, the filter kernel $W_\text{TEC}$ being a function of $G_k$ and the subreddit vector $\mathbf{U}_j$ corresponding to $s_j$: $$W_\text{TEC} = W_S\odot\gamma (W_G\cdot G_k + W_V\cdot s^V_j)$$ where $\gamma(x)=x$ if $x\ge 0$, and $\alpha x$ of $x < 0$ (standard LeakyReLU activation with parameter $\alpha$, experimentally set to $0.2$). $W_G$ and $W_V$ (corresponding to the two feed-forward layers inside component (3) in Figure \[fig:whole\_model\]) control the contributions of the cumulative influence and the subreddit, respectively. This $\gamma (W_G\cdot G_k + W_V\cdot s^V_j)$ component ‘calibrates’ the static kernel $W_S$ with the element-wise multiplication according to the influence. As $G_k$ evolves over time, so does $W_\text{TEC}$. Equipped with this influence-controlled kernel, the time-evolving convolution and max-pooling on $s_j$ can be defined similar to Eq. \[eq:static\_conv1d\]. Again, we apply parallel branches of successive convolution with different filter sizes and max-pooling, concatenation, and another series of convolutions (complete organization shown in Appendix, Figure \[fig:ada\_cnn\]) to finally map $s_j$ to a non-negative real value $\Tilde{B}_j$, the [**potential chatter intensity**]{} of $s_j$ independent of the subreddit where $s_j$ is posted. Final Prediction {#subsec:final_prediction} ---------------- Chatter levels in different subreddits vary with the number of active users at any time. The average commenting activity $s^R_j$ of the subreddit corresponding to $s_j$ enables us to compute the relative activity signal $r_j \in \mathbb{R}$ such that, $$r_j = \sigma(W_R\cdot s^R_j + Q_R)$$ where $\sigma(x)= (1+e^{-x})^{-1}$ (shown as “FF-$\sigma$” blocks in Figure \[fig:whole\_model\]). We compute the [**base chatter intensity**]{} corresponding to $s_j$ as $B_j = r_j\Tilde{B}_j$ such that $0<r_j<1$ plays the role of a scaling factor to calibrate $B_j$ according to the activity level of the subreddit. Having computed $B_j$, the chatter intensity invoked by $s_j$, under the influence of past history of news and submission arrival and calibrated by the subreddit information, we next observe the commenting activity under $s_j$ within the observation window (see component (4) in Figure \[fig:whole\_model\]). We employ a binning over time intervals to transform the comment arrivals within the observation window $[t_j, t_j+m\Delta_\text{obs}]$ into a sequence $\langle c^1_j, c^2_j, \cdots, c^m_j\rangle$ where each $c^l_j$ is the total number of comments arrived within $[t_j+(l-1)\Delta_\text{obs}, t_j+l\Delta_\text{obs}]$. This sequence serves as a coarse approximation of the rate of comment arrivals over the observation window. We use a single LSTM layer to aggregate this sequence and predict the final chatter $y^m_j$ (superscript corresponds to the length of the observation window). In the zero-shot setting (i.e., $m=0$) this LSTM is not used and we predict chatter $y^0_j = B_j$. Details of  parameters are given in Appendix \[appendix:parameters\]. Cost/Loss Functions {#subsec:cost_function} ------------------- In a realistic setting, there are far too many discussions invoking near-zero chatter along with very small number of those which go viral heavily. As we take both of these types without any filtering (opposed to excluding less viral ones in the cascade prediction tasks like [@kobayashi2016tideh] or [@zhao2015seismic]), the cost function needs to handle skewed ground truth values. To deal with this, we train  by minimizing the mean absolute relative error given by $\sum_j \frac{\lvert y_j - y^m_j \rvert}{y_j+\epsilon}$, where $y_j$ is the chatter ground truth, $y^m_j$ is the predicted chatter, and $\epsilon$ is a small positive real number to avoid division by zero (as implemented in Keras/Tensorflow). [|l|l|]{} & [**Expression**]{}\ Bag-of-words \* & Unigram features with tf-idf\ Complexity \* &\ LIX Score \* &\ Polarity &\ Referral count \* & Number of URLs in the submission\ Size \* &\ Subreddit \* & In which subreddit the submission is posted\ Commenting time &\ ------------------------- Average time difference in first $k/2$ comments ------------------------- : Feature set for the baseline [*CasPred*]{}, where $T$:= set of unique terms in corpus, $tf_t$:= term frequency of term $t\in T$ in the submission, $|w|$:= number of words in the submission, $|cw|$:= number of words in the submission with more than 6 letters, $|s|$:= number of sentences in the submission, $k$:= observable discussion size, $t_i$:= time when $i$-th comment was put, $t_0$:= time of submission. \* signifies a feature not in the original paper but added by us. & $\frac{1}{k/2-1}\sum_{i=1}^{k/2-1}(t_i-t_{i-1})$\ ------------------------- Average time difference in last $k/2$ comments ------------------------- : Feature set for the baseline [*CasPred*]{}, where $T$:= set of unique terms in corpus, $tf_t$:= term frequency of term $t\in T$ in the submission, $|w|$:= number of words in the submission, $|cw|$:= number of words in the submission with more than 6 letters, $|s|$:= number of sentences in the submission, $k$:= observable discussion size, $t_i$:= time when $i$-th comment was put, $t_0$:= time of submission. \* signifies a feature not in the original paper but added by us. & $\frac{1}{k/2-1}\sum_{i=k/2}^{k}(t_i-t_0)$\ \[tab:cascade\_features\] Experiments {#sec:experiment} =========== Dataset {#subsec:data} ------- We collected the discussion data from Pushshift.io[^4], a publicly available dump of Reddit data, stored in monthly order. We used the discussion data of October and November 2018, from 43 different communities (Subreddits). The October data was used for training and development, and the November data was used for testing. In total, we have a collection of 751,866 submissions with 2,604,839 comments in the training data, and 1,334,341 submissions with 4,264,177 comments in the test data. To fetch the news articles published online, we relied on the news-please crawler [@Hamborg2017], which extracts news articles from the Common Crawl archives[^5]. We crawled the news articles published in the same timeline as of the Reddit discussions. We got 1,851,022 articles from $4757$ different news sources for the month of October, and 2,010,985 articles from $5054$ sources for November. It should be noted that this covers all the news articles published in English in this period, as we chose not to use non-English data. Details of preprocessings are given in Appendix \[appendix:preprocess\]. Training and Evaluation Protocols {#subsec:param_select} --------------------------------- For training the model, we use the October data divided into the train-validate split. The news and discussion streams in the period from GMT to GMT are used for training , while the validation is done using the data from GMT to GMT. We set $\Delta_\text{obs}$ and $\Delta_\text{pred}$ to $60$ seconds and $30$ days, respectively. We train multiple variations of  with different observation windows: $15$, $30$, $45$, and $60$ minutes ($m=15, 30, 45, 60$). Additional training detail are given in the Appendix \[appendix:training\]. Baseline Models {#subsec:baseline} --------------- Due to the novelty of the problem setting and the absence of social network information, comparing  with the state-of-the-art is not straightforward. We engage four external baselines for retweet cascade prediction and Reddit user engagement prediction, tailored to our setting. We also implement multiple variants of  for extensive ablation analysis of the different signals. ### **TiDeH** To adapt Time Dependent Hawkes Process [@kobayashi2016tideh] as a baseline in the absence of any knowledge of the underlying user network, we set the follower count of each Reddit poster/commenter as 1. In addition, we set the minimum thread size (minimum number of comments) to 10. ### **CasPred** {#subsubsec:cascade} The cascade prediction approach of @cheng2014cascadepred provides an interesting baseline for  by allowing us to test not only the temporal but also the textual features of our dataset. Due to limitations of Reddit metadata we can make use of only a subset of the features they used. We also include some additional content features fitting to discussions in Reddit [@DBLP:conf/icdm/Dutta0019] (complete feature set in Table \[tab:cascade\_features\]). We implement CasPred-org (original features) and CasPred-full (augmented with additional features) with observable cascade size $k=10$. ### **RGNet** Our third external baseline is an adaptation of the Relativistic Gravitational Network [@DBLP:conf/icdm/Dutta0019], primarily designed to predict user engagement behavior over Reddit. ### **DeepCas** DeepCas [@DBLP:conf/www/LiMGM17] makes use of the global weighted topology of inter-user ties. Since, Reddit does not posses any explicit user-user mapping, we consider an edge between the posters and the commenters of a post (to generate the global network). Also, each post(with its set of poster and commenters) is treated as a cascade. ### **Ablation variants of** {#subsubsec:ablation} To observe the contribution of different components of , we implement the following ablated variants: $\bullet$ [**-N**]{} which uses only the news-side influence signal; $\bullet$ [**-S**]{} which uses only the submission-side influence signal; $\bullet$ [**-Static**]{} which does not use any influence signal; for this, the time-evolving convolution block is replaced by a static convolution block; $\bullet$ [**LSTM-CC**]{} which uses only the LSTM layer aggregating the observed comment arrivals (Section \[subsec:final\_prediction\]). LSTM-CC allows implementation for only the minimal early observation setting; zero-shot is not supported. Other variants are implemented for both of the task settings. [**Model**]{} [**MAPE**]{} $\mathbf{\tau}$ $\mathbf{\rho}$ [**Step-wise**]{} $\mathbf{\tau}$ --------------- ---------------- ----------------- ----------------- ----------------------------------- + 33.142 0.4042 0.4601 0.8781 ++ [**25.893**]{} [**0.4439**]{} [**0.5050**]{} [**0.8980**]{} TiDeH (1hr.) 35.178 0.0715 0.1140 0.5622 CasPred-full - - - 0.4741 CasPred-org - - - 0.3515 RGNet 148.34 0.1871 0.2273 0.5305 DeepCas 163.6 0.2362 0.3309 0.2636 : Evaluation of  and the external baselines (over complete test data). $\tau$ and $\rho$ correspond to Kendall’s $\tau$ and Spearman’s $\rho$, respectively, whereas Step-wise $\tau$ corresponds to Kendall $\tau$ on the sampled ground truth. + and ++ correspond to the zero-shot and minimal early observation of $1$ hour. \[tab:result\_overall\_agg\] [**Model**]{} [**Setting**]{} [**MAPE**]{} $\tau$ --------------- ----------------- -------------- -------- -N -S -Static LSTM-CC OBS 152.313 0.3580 : Performances of different ablation variants of  (see Section \[subsubsec:ablation\]). Except for LSTM-CC, all the variations are tested for zero-shot (ZS) and early observation of 1 hour (OBS). MAPE and $\tau$ are as mentioned in Table \[tab:result\_overall\_agg\]. \[tab:ablation\] [**Model**]{} [**AR\***]{} [**TD\***]{} [**gaming**]{} [**politics**]{} [**technology**]{} [**Music**]{} [**techsupport**]{} [**WITT\***]{} [**news**]{} [**movies**]{} [**RL\***]{} --------------- ---------------- --------------- ------------------- ------------------ -------------------- ----------------- --------------------- ---------------- ---------------- -------------------- --------------- [**26.031**]{} 31.854 24.462 [**23.389**]{} 25.686 21.249 25.13 26.002 [**29.102**]{} 25.076 24.221 -S [**28.411**]{} 34.712 31.011 [**38.105**]{} 36.122 28.310 36.54 27.151 [**35.671**]{} 28.510 27.907 -N [**30.008**]{} 35.003 28.949 [**25.610**]{} 30.991 27.569 29.171 27.711 [**29.342**]{} 26.134 27.886 [**Model**]{} [**Tinder**]{} [**TNF\***]{} [**anime**]{} [**india**]{} [**Jokes**]{} [**soccer**]{} [**FF\***]{} [**NSQ\***]{} [**nfl**]{} [**AS**]{} [**WSB\***]{} 23.861 33.420 26.004 [**27.009**]{} 25.562 [**24.753**]{} 27.419 25.510 28.911 23.70 23.251 -S 25.945 36.138 28.040 [**39.202**]{} 32.787 [**33.402**]{} 29.958 29.875 35.007 26.419 24.36 -N 26.020 36.287 29.011 [**32.784**]{} 34.095 [**30.92**]{} 29.011 31.592 32.019 24.499 24.212 [**Model**]{} [**InNews**]{} [**GO\***]{} [**teenagers**]{} [**POLITIC**]{} [**brasil**]{} [**NBA2k**]{} [**bussiness**]{} [**PF\***]{} [**nba**]{} [**worldnews**]{} [**UPO\***]{} 29.534 24.993 24.771 27.364 24.77 25.333 26.001 23.122 25.251 24.924 28.037 -S 32.119 27.604 26.759 29.990 26.904 30.424 31.213 31.797 28.751 28.117 30.301 -N 30.54 27.601 27.002 32.013 27.591 28.701 28.10 32.107 26.29 25.023 29.213 [**Model**]{} [**EI\***]{} [**AN\***]{} [**NBB\***]{} [**FIFA**]{} [**BN24\***]{} [**BCAll\***]{} [**NBTMT\***]{} [**TTF\***]{} [**PH\***]{} [**NBMARKET\***]{} - 27.001 27.159 24.146 0.754 24.113 28.011 26.112 25.301 24.35 23.109 - -S 28.994 32.571 32.386 29.778 29.292 27.003 29.203 26.906 24.997 27.904 - -N 28.923 28.529 25.183 29.022 28.124 27.091 28.114 28.476 26.870 27.878 - \[tab:subreddit\_aggregate\] Evaluation {#sec:evaluation} ========== We explore multiple evaluation strategies to see how  responds to different challenges of chatter prediction. We use three different evaluation metrics: a) Mean Absolute Percentage Error (MAPE), b) Kendall rank correlation coefficient (Kendall’s $\tau$), and c) Spearman’s $\rho$. As the CasPred model does not predict the exact size of the discussion but gives a binary decision of whether a given submission will reach at least size $l\times k$, $l\in \mathbb{Z}^{+}$ after observing a growth of size $k$, we can not evaluate this with the mentioned three metrics directly. Instead, we map the ground-truth to these $l\times k$ values such that the label of a discussion with size $d$ would be $\left \lfloor{\frac{x}{k}}\right \rfloor $. Then we compute Kendall’s $\tau$ over this values (hereafter called as step-wise $\tau$). We use the same binning (with $k=10$) for rest of the models to evaluate the step-wise $\tau$. Overall Performance ------------------- In Table \[tab:result\_overall\_agg\], we show the evaluation results for  in zero-shot and early observation settings along with all the external baselines. While  exploiting comment arrival within the early observation window outperforms rest of the models by a large margin, it also performs better than the external baselines in the zero-shot setting. An interesting pattern can be observed with TiDeH, RGNet and DeepCas. While TiDeH produces predictions comparable to  in terms of MAPE, it suffers largely in terms of rank correlation. On the other hand, both RGNet and DeepCas follow a completely opposite pattern – better ranking of future chatter compared to predicting the actual value of chatter. The poor performance of CasPred and RGNet can be explained in terms of the difference between their original design context and the way they are deployed in our problem setting. Almost two-third of the feature set originally used for CasPred can not be implemented here. Also, it is evident that our additional feature set actually improves the performance of CasPred, signifying the importance of these features for engagement modeling in Reddit. In case of RGNet, it is built to take into account the dynamics of user engagement over time. But in the absence of a rich feature set and social network information, it is the use of endogenous and exogenous influence which gives  such leverage compared to the baselines. Earlier we highlighted the major challenge of predicting future chatter without delayed observation of chatter evolution.  satisfies this requirement better than other baselines, because they were all designed for much larger early observation windows. *TiDeH takes $24$ hours of observation to outperform  with $1$ hour of observation.* Via time-sensitive combination of exogenous and endogenous signals,  achieves superior performance without network knowledge. Ablation of  Components {#subsec:ablation_results} ----------------------- We justify the complexity of , and show that all its pieces are critical. In Table \[tab:ablation\], we present the performances of the various ablation models described in Section \[subsubsec:ablation\]. It is evident that removal of either exogenous or endogenous signals from  results in a degraded performance. However,  with only endogenous signal slightly outperforms its counterpart with only exogenous signal. This difference does not tell us whether endogenous signals are more important — we need to study their effect for individual subreddits to comment on that. Removing both signals degrades performance heavily, particularly in the zero-shot setting. This is expected, because -Static in the zero-shot regime is simply a static convolution block mapping submission texts to their future chatter – a regression task juxtaposed with simple text classification engine. The decrease in performance is more evident with the MAPE measure ($195.344$ and $149.558$, respectively for zero-shot and early observation). Even in zero-shot setting -Static utilizes the information of average comment arrival in the subreddit to scale the future chatter accordingly and learn at least a possible ranking of submissions with respect to their future chatter. Additionally we removed this operation as well for -Static; kendall $\tau$ for this further ablated model dropped to $0.02$. Comparing the performances of -Static in zero-shot (only submission features) and LSTM-CC (only comment arrival features), one can easily conclude that, when the exogenous and endogenous signals are not taken into account, comment arrival patterns are much powerful indicators of future chatter compared to submission texts. Effect of Observation Window and Size of Discussion --------------------------------------------------- [**Model**]{} [**MAPE**]{} $\tau$ --------------- -------------- -------- -0 33.142 0.4042 -15 31.886 0.4278 -30 28.12 0.4302 -45 26.042 0.4361 -60 25.893 0.4439 LSTM-CC-15 196.128 0.0639 LSTM-CC-30 174.667 0.1307 LSTM-CC-45 167.024 0.2259 LSTM-CC-60 152.313 0.3580 : MAPE and kendal-$\tau$ scores to predict future chatter using  and LSTM-CC, each with varying size of the observation window. [*model\_name-$x$*]{} signifies the model uses an observation window $x$ minutes long. \[tab:obs-window\] Table \[tab:obs-window\] shows the variation of performance for  and LSTM-CC with different sizes of observation window used to aggregate early arrivals of comments. While LSTM-CC shows a steady betterment of performance with increasing observation,  takes a quick leap from zero-shot to $15$ minute early observation and then reaches a nearly-stationary state. As shown in Figure \[fig:observation\_vs\_size\], with longer initial observation,  tends to decrease the error rate for predicting high values of chatter. @cheng2014cascadepred reported increasing uncertainty in predicting larger cascades. We plot the absolute error in prediction vs. ground-truth value in Figure \[fig:observation\_vs\_size\], for different early observation windows. We measure the absolute error to predict the size gain after observation. In all four cases, absolute error varies almost linearly with size. However, with longer observation, the slope drops. With a 60 minutes long early observation, absolute error nearly grazes a zero slope line. However, these plots shows the joint effect of increasing observation and decreasing post-observation gold value. Table \[tab:pred-window\] summarizes how well  predicts future chatter with different prediction windows. Again, longer a discussion persists, harder it becomes to predict the final amount of chatter. Also, as can be expected, the zero-shot system tends to suffer more with longer prediction window. Subreddit-wise Analysis {#subsec:subreddit_analysis} ----------------------- #### **Exogenous vs. Endogenous influence** While Table \[tab:ablation\] provides useful insights about the roles played by components, drilling down from aggregate performance into different subreddits gives additional insight. As discussed in Section \[sec:intro\], endogenous and exogenous influence manifest themselves differently over different subreddits (which is why we used subreddit embeddings $\mathbf{U}_j$ in both components: influence aggregation and time-evolving convolution). In Table \[tab:subreddit\_aggregate\], we present the performances of -N, and -S for each of the 43 subreddits. In some subreddits (e.g., [*r/techsupport, r/india, r/business, r/POLITIC, r/InNews*]{}, etc.),  suffers more with the ablation of exogenous signal compared to the endogenous one. Most of this subreddits are either directly news related (like [*r/news, r/InNews, r/worldnews,*]{} etc.), or very closely governed by what is happening in the real world, like [*r/technology, r/business, r/nfl, r/movies,*]{} etc. Some subreddits are naturally grouped, i.e., they share common topics of discussion, common set of commenting users, etc. Subreddits like [*r/Music, r/movies and r/anime*]{} fall into one such group. This sharing of information facilitates -S to perform better compared to -N as it uses the endogenous knowledge in terms of previous submissions, subreddit embeddings, and comment rates. Also there are some particular subreddits (e.g., [*r/AskReddit, r/teenagers, r/NoStupidQuestions*]{}) where endogenous information becomes more important than the exogenous one. [**Model**]{} AR Anime FF EI UPO --------------- -------- --------- --------- ---------- --------- + 37.184 37.545 39.060 38.219 37.011 ++ 26.031 26.004 27.419 27.001 28.037 LSTM-CC 110.34 108.69 129.078 121.336 115.414 [**Model**]{} india soccer NBB business TNF + 34.323 31.095 31.866 30.519 33.911 ++ 27.009 24.753 24.146 26.001 25.686 LSTM-CC 168.91 177.247 163.077 165.325 162.12 : Effect of early observation on the chatter prediction performance over different subreddits. We take  in zero-shot and early observation setting (+ and ++, respectively) and LSTM-CC as models for comparison; results are reported on 10 of the 43 total subreddits – 5 showing most response towards observation (top half), and 5 with least response (bottom half). Abbreviations of subreddits follow the same definitions in Table \[tab:subreddit\_aggregate\]. \[tab:subreddit\_observe\] [**Model**]{} [**MAPE**]{} $\tau$ --------------- -------------- -------- -$E_1$ 19.020 0.5145 -$E_{10}$ 21.979 0.4610 -$E_{20}$ 25.224 0.4437 -$E_{30}$ 26.042 0.4361 -$Z_1$ 22.198 0.4812 -$Z_{10}$ 25.064 0.4334 -$Z_{20}$ 32.719 0.4098 -$Z_{30}$ 33.142 0.4042 : MAPE and Kendall’s $\tau$ to predict future chatter using  and LSTM-CC, each with varying size of the prediction window. -$E_x$ and -$Z_x$ signify  in $1$ hour early observation and zero-shot setting using a prediction window $x$ days long, respectively. \[tab:pred-window\] #### **Zero-shot vs. Early Observation** Much similar to exogenous and endogenous influence signals, the role of early-observation to predict future chatter differs between subreddits. In Table \[tab:subreddit\_observe\], we explore this phenomena by comparing  in zero-shot and early observation settings. We also gauge the performance of LSTM-CC as this ablation variant depends purely on comment arrival in observation window to predict future chatter. For the subreddits in the top half of Table \[tab:subreddit\_observe\] the performance gain with moving from zero-shot to early observation regime is significantly higher compared to those in the bottom half. LSTM-CC follows the same pattern, with MAPE for each of the top-half subreddits being substantially lower than the global average (see Table \[tab:result\_overall\_agg\]) and higher for the top-half subreddits. Various characteristics of the subreddits might be put as responsible for this: size of the subreddit (large subreddits like AskReddit embody complex dynamics of user interests, resulting in chatter signals that can not be modeled using influence signals alone), small secluded subreddits like EcoInternet with focus of discussion not much available over news or rest of the Reddit, etc. {width="0.7\columnwidth"} . \[fig:observation\_vs\_size\] Related Work {#sec:Related} ============ Most prior work on engagement prediction requires knowledge of the underlying social network. Such systems are mostly based on modeling information or influence diffusion at a microscopic level [@kupavskii2012retweetcascade; @cheng2014cascadepred], recently enhanced with point processes [@zhao2015seismic; @kobayashi2016tideh]. Separation of exogenous and endogenous influences [@de2018endoexo] in the predictive model can increase interpretability and accuracy. Some systems predict if, after observing $k$ instances of influence or transfer along edges, the process will cascade to over $2k$ transfers ‘eventually’ [@cheng2014cascadepred]. Most such systems use a richly designed space of temporal, structural, and contextual features, along with the standard supervised classifiers, to predict the evolving dynamics of the cascades. In contrast, we do not assume any knowledge of an underlying network. Moreover, we are not tracking the diffusion of any uniquely identifiable content like an image or a hashtag. There is no one-to-one mapping between a specific news story and a network community. In fact, the same event may be reported in multiple news stories. While subrredits generally have overlapping interests, the extent of exogenous influence of a news story within a subreddit is topical in nature. Endogenous influence within a community depends on the invisible and possibly transient social links. @guerini2011virality explore the prediction of viral propagation, based solely on the textual features and not the social network structure, which is closer in spirit to  compared to network-assisted prediction. They track the spread of specific identified information items (short ‘stories’). A story can be submitted only once, unlike multiple submissions on a topic in our setting. They propose hardwired definitions of appreciation, discussion, controversiality, and ‘white’ and ‘black’ buzz, then use an SVM classifier to predict such labels successfully. @aswani2017socialbuzz presented similar studies. @shulman2016predictability found early adoption a stronger predictor of later popularity than other content features. @weng2012competition showed how limited attention of individuals causes competition in the evolution of memes. @peng2018emerging seek to predict (as early as possible) emerging discussions about products on social media without information about the social network structure. Like @guerini2011virality, they engineer a variety of rich features, including author diversity, author engagement, competition from other products, and temporal, content, and user features in a conventional classifier. Their task is thus limited to a vertical domain (products), and the prediction task is discrete classification (will a burst of activity emerge or not). In contrast, we seek to predict quantitative levels of chatter. Unlike both @peng2018emerging and @guerini2011virality, we avoid extensive feature engineering and instead focus on the design of a deep network that integrates exogenous and endogenous influences. Chatter intensity or related quantities have been used for predicting other social outcomes as well. @asur2010predicting used a linear regression to predict their box-office success. Other examples such as election outcome and stock movements are surveyed by @yu2012socialprediction. Thus, our work on chatter intensity prediction opens up avenues toward such compelling downstream applications. Chatter intensity or related quantities have been used for predicting other social outcomes as well. @asur2010predicting used a simple linear regression on just the observed count of tweets mentioning movies to predict their box-office success more accurately than other prediction markets. Other examples such as election outcome and stock movements are surveyed by @yu2012socialprediction. Thus, our work on chatter intensity prediction opens up avenues toward such compelling downstream applications. Conclusion {#sec:End} ========== Activity prediction usually depends on knowledge of the underlying social network structure. However, on several important social platforms, the social network is incomplete, not directly observable, or even transient. We introduce the problem of predicting social chatter level without graph information, and present a new deep architecture, , for this setting.  combines deep text representation with a recurrent network that tracks the temporal evolution of the state of a community with latent connectivity. Without knowledge of social network topology,  achieves new state-of-the-art accuracy. Here we have regarded chatter as caused by news events but not vice versa, whereas chatter is having increasing effects on the real world. Modeling such feedback effects may be a natural avenue for future work. Corpus Preprocessing {#appendix:preprocess} ==================== We use same strategy for text cleaning of both news articles and submissions. After tokenization, replacing URLs, and converting numeric values to their textual counterpart, we set a maximum document frequency of 0.8 (fraction of the total number of news articles and submissions) and minimum document frequency of $5$ (absolute count) to exclude stopwords and extremely rare words. We trained the Word2Vec model for $500$ iterations with window size set to $10$ and output dimension $100$. We take the maximum length of texts to be $50$ and $100$ words for submissions and news articles, respectively. Design details of {#appendix:parameters} ================== For each of the $43$ subreddits represented as one-hot vector, the subreddit embedding layer outputs a $32$-dimensional vector. Every weight matrix is randomly initialized using Xavier initialization. All bias matrices are initialized with zeros. Organization of Convolution Blocks {#appendix:conv} ----------------------------------  uses two separate stackings of convolution-maxpool operations: static convolution block and time-evolving convolution block (components $1$ and $3$ in Figure \[fig:whole\_model\]). Internal organizations of the blocks are shown in Figure \[fig:text\_cnn\] (static) and Figure \[fig:ada\_cnn\] (time-evolving). For all the convolution operations in the static block and the branched segments of the time-evolving block we use padding to keep the size of output feature maps to be same as inputs. For the last three convolution operations in the temporal block we do not use any padding (as the kernel size is 1). All the branches of convolution-maxpooling in both the blocks have number of filters $128$, $64$, and $32$, successively. Last three convolution operations in time-evolving block have filter numbers $64$, $32$, and $1$, successively. Parameters of Recurrent Units {#appendix:recurrent} ----------------------------- The news-aggregating and the submission-aggregating GRUs (see component (2) in Figure \[fig:whole\_model\]) both have hidden state size equal to $128$. The LSTM layer aggregating comment arrivals in the early observation window uses hidden state of size of $8$. ![Static convolution of news and submission texts to obtain the latent feature representations. The three parallel branches of convolution-maxpooling is repeated thrice, and then concatenated to produce the feature map. The initial list of the word vectors comes from the word embedding layer.[]{data-label="fig:text_cnn"}](Figures/cnn.pdf){width="\columnwidth"} ![The time-evolving convolution component. $K$ denotes size of convolution filters. Cumulative influence and subreddit vectors corresponding to the input submissions are the control inputs for every convolution layer. []{data-label="fig:ada_cnn"}](Figures/ada_cnn.pdf){width="\columnwidth"} training details {#appendix:training} ================= To initialize the word embedding layers, we train a skip-gram Word2Vec model on the training split of the news and submission data. The resulting word vectors are of size $100$, and they are further trained while training .  is optimized using Adam optimizer [@kingma2014adam], with the learning rate set to $0.00001$. As the model works in an online setting, the batch size is set to one. While training , we reset the states of stateful GRUs after every epoch (after testing on the validation data).  is trained on a Intel Xeon Processor ($16$ cores, $32$ GB RAM) with NVIDIA Quadro K6000 GPU. Each training iteration of full  takes 13 hours and 22 minutes (roughly). We trained all the models for $25$ iterations and save top $5$ best models (on validation loss). All the results reported are averaged over these $5$ models. [^1]: Source: https://www.statista.com/ [^2]: https://www.adweek.com/digital/reddit-reaches-430-million-monthly-active-users-looks-back-at-2019/ [^3]: Code and Sample Data at <https://github.com/LCS2-IIITD/ChatterNet> [^4]: <https://files.pushshift.io/reddit/> [^5]: <https://commoncrawl.org/>

--- abstract: 'We consider initial value problems for differential-algebraic equations in a possibly infinite-dimensional Hilbert space. Assuming a growth condition for the associated operator pencil, we prove existence and uniqueness of solutions for arbitrary initial values in a distributional sense. Moreover, we construct a nested sequence of subspaces for initial values in order to obtain classical solutions.' address: - | Insitut für Analysis\ Fakultät Mathematik\ Technische Universität Dresden\ Germany - | Department of Mathematics and Statistics\ University of Strathclyde\ Glasgow, United Kingdom author: - Sascha Trostorff - Marcus Waurick date: 'October 23, 2017' title: 'On higher index differential-algebraic equations in infinite dimensions' --- Introduction and main results ============================= In this short note, we consider two solution concepts of differential-algebraic equations (DAEs) in infinite dimensions. For this, let $E$ and $A$ be bounded linear operators in some possibly infinite dimensional Hilbert space $H$. We consider the implicit initial value problem $$\tag{{\ensuremath{\ast}}}\begin{cases} Eu'(t)+Au(t)=0, & t>0,\\ u(0+)=u_{0} \end{cases}\label{eq:IVP0}$$ for some given $u_{0}\in H$. In order to talk about a well-defined problem in , we assume that the pair $(E,A)$ is *regular*, that is, $$\begin{aligned} \exists\nu\in\mathbb{R}\colon & \mathbb{C}_{{\operatorname{Re}}>\nu}\subseteq \rho(E,A),\\ \exists C\geq0,k\in\mathbb{N}\:\forall s\in\mathbb{C}_{{\operatorname{Re}}>\nu}\colon & \|\left(sE+A\right)^{-1}\|\leq C|s|^{k},\end{aligned}$$ where $$\rho(E,A)\coloneqq \{s\in \mathbb{C}\,;\, (sE+A)^{-1}\in L(H)\}.$$ We note here that these two conditions are our replacements for regularity in finite dimensions. Indeed, for $H$ finite-dimensional, $(E,A)$ is called regular, if $\det(sE+A)\neq0$ for some $s\in\mathbb{C}$. Thus, $s\mapsto\det(sE+A)$ is a polynomial of degree at most $\dim H$, which is not identically zero. The growth condition is a consequence of the Weierstrass or Jordan normal form theorem valid for finite spatial dimensions, see e.g. [@Berger2012; @Dai1989; @Mehrmann2006]. The smallest possible $k\in\mathbb{N}$ occurring in the resolvent estimate is called the *index of $(E,A)$*: $$\operatorname{ind}(E,A)\coloneqq\min\{k\in\mathbb{N}\,;\,\exists C\geq0\,\forall s\in\mathbb{C}_{{\operatorname{Re}}>\nu}:\|\left(sE+A\right)^{-1}\|\leq C|s|^{k}\}.$$ We shall also define a sequence of (initial value) spaces associated with $(E,A)$: $$\mathrm{IV}_{0} \coloneqq H\mbox{ and } \mathrm{IV}_{k+1} \coloneqq\{x\in H;Ax\in E[\mathrm{IV}_{k}]\}\quad(k\in\mathbb{N}).$$ A first observation is the following. \[prop:stabInde\]Let $k=\operatorname{ind}(E,A)$ and assume that $E[\mathrm{IV}_{k}]\subseteq H$ is closed. Then $\mathrm{IV}_{k+1}=\mathrm{IV}_{k+2}$. Since the sequence of spaces $(\mathrm{IV}_{k})_{k}$ is decreasing (see ), Proposition \[prop:stabInde\] leads to the following question. \[prob:indexProblem\]Assume that $E[\mathrm{IV}_{j}]\subseteq H$ is closed for each $j\in\mathbb{N}$. Do we then have $$\min\{k\in\mathbb{N};\mathrm{IV}_{k+1}=\mathrm{IV}_{k+2}\}=\operatorname{ind}(E,A)?$$ With the spaces $(\mathrm{IV}_{k})_{k}$ at hand, we can present the main theorem of this article. \[thm:mainTh\]Assume that $E[\mathrm{IV}_{\operatorname{ind}(E,A)}]\subseteq H$ is closed, $u_{0}\in\mathrm{IV}_{\operatorname{ind}(E,A)+1}$. Then there exists a unique continuously differentiable function $u\colon\mathbb{R}_{>0}\to H$ with $u(0+)=u_{0}$ such that $$Eu'(t)+Au(t)=0\quad(t>0).$$ With Proposition \[prop:stabInde\] and , it is possible to derive the following consequence. \[cor:mainCor\]Assume that $E[\mathrm{IV}_{j}]\subseteq H$ is closed for each $j\in\mathbb{N}$, $u_0\in H$. Then there exists a continuously differentiable function $u\colon\mathbb{R}_{>0}\to H$ with $u(0+)=u_{0}$ and $$Eu'(t)+Au(t)=0\quad(t>0),$$ if, and only if, $u_{0}\in\mathrm{IV}_{\operatorname{ind}(E,A)+1}$. Corollary \[cor:mainCor\] suggests that the answer to Problem \[prob:indexProblem\] is in the affirmative for $H$ being finite-dimensional. Also in our main result, there is room for improvement: In applications, it is easier to show that $R(E)\subseteq H$ is closed as the $\mathrm{IV}$-spaces are not straightforward to compute. Thus, we ask whether the latter theorem can be improved in the following way. Does $R(E)\subseteq H$ closed imply the closedness of $E[\mathrm{IV}_{\operatorname{ind}(E,A)}]\subseteq H$ or even closedness of $E[\mathrm{IV}_{j}]\subseteq H$ for all $j\in\mathbb{N}$? We shall briefly comment on the organization of this article. In the next section, we introduce the time-derivative operator in a suitably weighted vector-valued $L_{2}$-space. This has been used intensively in the framework of so-called ‘evolutionary equations’, see [@PicPhy]. With this notion, it is possible to obtain a *distributional* solution of such that the differential algebraic equation holds in an integrated sense, where the number of integrations needed corresponds to the index of the DAE. We conclude this article with the proofs of Proposition \[prop:stabInde\], , and Corollary \[cor:mainCor\]. We emphasize that we do not employ any Weierstrass or Jordan normal theory in the proofs of our main results. We address the case of unbounded $A$ to future research. The case of index $0$ is discussed in [@Trostorff2017b], where also exponential stability and dichotomies are studied. The time derivative and weak solutions of DAEs ============================================== Throughout this section, we assume that $H$ is a Hilbert space and that $E,A\in L(H)$ with $(E,A)$ regular. We start out with the definition of the space of (equivalence classes of) vector-valued $L_{2}$ functions: Let $\nu\in\mathbb{R}$. Then we set $$L_{2,\nu}(\mathbb{R};H)\coloneqq\left\{ f:\mathbb{R}\to H\,;\,f\mbox{ measurable,\,}\intop_{\mathbb{R}}|f(t)|_{H}^{2}\exp(-2\nu t)\,\mathrm{d}t<\infty\right\} ,$$ see also [@PicPhy; @KPSTW14_OD; @Picard1989]. Note that $L_{2,0}(\mathbb{R};H)=L_{2}(\mathbb{R};H)$. We define $H_{\nu}^{1}(\mathbb{R};H)$ to be the ($H$-valued) Sobolev space of $L_{2,\nu}(\mathbb{R};H)$-functions with weak derivative representable as $L_{2,\nu}(\mathbb{R};H)$-function. With this, we can define the derivative operator $$\partial_{0,\nu}\colon H_{\nu}^{1}(\mathbb{R};H)\subseteq L_{2,\nu}(\mathbb{R};H)\to L_{2,\nu}(\mathbb{R};H),\phi\mapsto\phi'.$$ In the next theorem we recall some properties of the operator just defined. For this, we introduce the *FourierLaplace transformation* $\mathcal{L}_{\nu}\colon L_{2,\nu}(\mathbb{R};H)\to L_{2}(\mathbb{R};H)$ as being the unitary extension of $$\mathcal{L}_{\nu}\phi(t)\coloneqq\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}\phi(s){\mathrm{e}}^{-\left({\mathrm{i}}t+\nu\right)s}\mathrm{d}s\quad(\phi\in C_{c}(\mathbb{R};H),\,t\in\mathbb{R}),$$ where $C_{c}(\mathbb{R};H)$ denotes the space of compactly supported, continuous $H$-valued functions defined on $\mathbb{R}$. Moreover, let $$\begin{aligned} {\operatorname{m}}\colon\{f\in L_{2}(\mathbb{R};H);(t\mapsto tf(t))\in L_{2}(\mathbb{R};H)\}\subseteq L_{2}(\mathbb{R};H) & \to L_{2}(\mathbb{R};H),\\ f & \mapsto(t\mapsto tf(t))\end{aligned}$$ be the multiplication by the argument operator with maximal domain. \[thm:FLT\]Let $\nu\in\mathbb{R}$. Then $$\partial_{0,\nu}=\mathcal{L}_{\nu}^{\ast}({\mathrm{i}}{\operatorname{m}}+\nu)\mathcal{L}_{\nu}.$$ A direct consequence of is the continuous invertibility of $\partial_{0,\nu}$ if $\nu\ne 0$. \[cor:solOp\]Let $\nu>0$ be such that $\rho(E,A)\supseteq\mathbb{C}_{{\operatorname{Re}}>\nu}$ and $\|\left(sE+A\right)^{-1}\|\leq C|s|^{\operatorname{ind}(E,A)}$ for some $C\geq0$ and all $s\in\mathbb{C}_{{\operatorname{Re}}>\nu}$. Then $$\partial_{0,\nu}^{-k}\left(\partial_{0,\nu}E+A\right)^{-1}\in L(L_{2,\nu}(\mathbb{R};H)),$$ where $k=\operatorname{ind}(E,A)$. Moreover, $\partial_{0,\nu}^{-k}\left(\partial_{0,\nu}E+A\right)^{-1}$ is causal, i.e., for each $f\in L_{2,\nu}(\mathbb{R};H)$ with ${\operatorname{spt}}f\subseteq\mathbb{R}_{\geq a}$ for some $a\in\mathbb{R}$ it follows that $${\operatorname{spt}}\partial_{0,\nu}^{-k}\left(\partial_{0,\nu}E+A\right)^{-1}f\subseteq\mathbb{R}_{\geq a}.$$ By and the unitarity of $\mathcal{L}_{\nu}$, we obtain that the first claim is equivalent to $$\left({\mathrm{i}}{\operatorname{m}}+\nu\right)^{-k}\left(\left({\mathrm{i}}{\operatorname{m}}+\nu\right)E+A\right)^{-1}\in L(L_{2}(\mathbb{R};H)),$$ which, in turn, would be implied by the fact that the function $$t\mapsto\left({\mathrm{i}}t+\nu\right)^{-k}\left(\left({\mathrm{i}}t+\nu\right)E+A\right)^{-1}$$ belongs to $L^\infty(\mathbb{R};L(H))$. This is, however, true by regularity of $(E,A)$. We now show the causality. As the operator $\partial_{0,\nu}^{-k}\left(\partial_{0,\nu}E+A\right)^{-1}$ commutes with translation in time, it suffices to prove the claim for $a=0.$ So let $f\in L_{2,\nu}(\mathbb{R};H)$ with ${\operatorname{spt}}f\subseteq\mathbb{R}_{\geq0}.$ By a Paley-Wiener type result (see e.g. [@rudin1987real 19.2 Theorem]), the latter is equivalent to $$(\mathbb{C}_{{\operatorname{Re}}>\nu}\ni z\mapsto\left(\mathcal{L}_{{\operatorname{Re}}z}f\right)({\operatorname{Im}}z))\in\mathcal{H}^{2}(\mathbb{C}_{{\operatorname{Re}}>\nu};H),$$ where $\mathcal{H}^{2}(\mathbb{C}_{{\operatorname{Re}}>\nu};H)$ denotes the Hardy-space of $H$-valued functions on the half-plane $\mathbb{C}_{{\operatorname{Re}}>\nu}$. As $$\left(\mathcal{L}_{{\operatorname{Re}}z}\partial_{0,\nu}^{-k}\left(\partial_{0,\nu}E+A\right)^{-1}f\right)({\operatorname{Im}}z)=z^{-k}\left(zE+A\right)^{-1}\left(\mathcal{L}_{{\operatorname{Re}}z}f\right)({\operatorname{Im}}z)$$ for each $z\in\mathbb{C}_{{\operatorname{Re}}>\nu},$ we infer that also $$(\mathbb{C}_{{\operatorname{Re}}>\nu}\ni z\mapsto\left(\mathcal{L}_{{\operatorname{Re}}z}\partial_{0,\nu}^{-k}\left(\partial_{0,\nu}E+A\right)^{-1}f\right)({\operatorname{Im}}z))\in\mathcal{H}^{2}(\mathbb{C}_{{\operatorname{Re}}>\nu};H),$$ due to the boundedness and analyticity of $$\left(\mathbb{C}_{{\operatorname{Re}}>\nu}\ni z\mapsto z^{-k}\left(zE+A\right)^{-1}\in L(H)\right).$$ This proves the claim. Corollary \[cor:solOp\] states a particular boundedness property for the solution operator associated with . This can be made more precise by introducing a scale of extrapolation spaces associated with $\partial_{0,\nu}$. Let $k\in\mathbb{N}$, $\nu>0$. Then we define $H_{\nu}^{k}(\mathbb{R};H)\coloneqq D(\partial_{0,\nu}^{k})$ endowed with the scalar product $\langle\phi,\psi\rangle_{k}\coloneqq\langle\partial_{0,\nu}^{k}\phi,\partial_{0,\nu}^{k}\psi\rangle_{0}$. Quite similarly, we define $H_{\nu}^{-k}(\mathbb{R};H)$ as the completion of $L_{2,\nu}(\mathbb{R};H)$ with respect to $\langle\phi,\psi\rangle_{-k}\coloneqq\langle\partial_{0,\nu}^{-k}\phi,\partial_{0,\nu}^{-k}\psi\rangle_{0}$. We observe that the spaces $(H_{\nu}^{k}(\mathbb{R};H))_{k\in\mathbb{Z}}$ are nested in the sense that $j_{k\to\ell}\colon H_{\nu}^{k}(\mathbb{R};H)\hookrightarrow H_{\nu}^{\ell}(\mathbb{R};H),x\mapsto x$, whenever $k\geq\ell$. The operator $\partial_{0,\nu}^{\ell}$ can be considered as a densely defined isometry from $H^{k}$ to $H^{k-\ell}$ with dense range for all $k\in\mathbb{Z}$. The closure of this densely defined isometry will be given the same name. In this way, we can state the boundedness property of the solution operator in Corollary \[cor:solOp\] equivalently as follows: $$\left(\partial_{0,\nu}E+A\right)^{-1}\in L\left(L_{2,\nu}(\mathbb{R};H),H_{\nu}^{-k}(\mathbb{R};H)\right).$$ More generally, as $\left(\partial_{0,\nu}E+A\right)^{-1}$ and $\partial_{0,\nu}^{-1}$ commute, we obtain $$\left(\partial_{0,\nu}E+A\right)^{-1}\in L\left(H_{\nu}^{j}(\mathbb{R};H),H_{\nu}^{j-k}(\mathbb{R};H)\right)$$ for each $j\in\mathbb{Z}$. Note that by the Sobolev embedding theorem (see e.g. [@KPSTW14_OD Lemma 5.2]) the $\delta$-distribution of point evaluation at $0$ is an element of $H_{\nu}^{-1}(\mathbb{R};H)$; in fact it is the derivative of $\chi_{\mathbb{R}_{\geq0}}\in L_{2,\nu}(\mathbb{R};H)=H_{\nu}^{0}(\mathbb{R};H)$. With these preparations at hand, we consider the following implementation of the initial value problem stated in : Let $u_{0}\in H$. Find $u\in H_{\nu}^{-k}(\mathbb{R};H)$ such that $$\left(\partial_{0,\nu}E+A\right)u=\delta\cdot Eu_{0}.\label{eq:IVPdis}$$ \[thm:solthdis\]Let $(E,A)$ be regular. Then for all $u_{0}\in H$ there exists a unique $u\in H_{\nu}^{-k}(\mathbb{R};H)$ such that holds. Moreover, we have $$u=\chi_{\mathbb{R}_{\geq0}}u_{0}-\left(\partial_{0,\nu}E+A\right)^{-1}\chi_{\mathbb{R}_{\geq0}}Au_{0}$$ and $${\operatorname{spt}}\partial_{0,\nu}^{-k}u\subseteq\mathbb{R}_{\geq0}.$$ Note that the unique solution is given by $$u=\left(\partial_{0,\nu}E+A\right)^{-1}\delta\cdot Eu_{0}\in H_{\nu}^{-k-1}(\mathbb{R};H).$$ Hence, $$\begin{aligned} u-\chi_{\mathbb{R}_{\geq0}}u_{0} & =\left(\partial_{0,\nu}E+A\right)^{-1}\left(\delta\cdot Eu_{0}-\left(\partial_{0,\nu}E+A\right)\chi_{\mathbb{R}_{\geq0}}u_{0}\right)\\ & =-\left(\partial_{0,\nu}E+A\right)^{-1}\chi_{\mathbb{R}_{\geq0}}Au_{0},\end{aligned}$$ which shows the desired formula. Since $\chi_{\mathbb{R}_{\geq0}}u_{0}\in L_{2,\nu}(\mathbb{R};H)\hookrightarrow H_{\nu}^{-k}(\mathbb{R};H)$ and $\left(\partial_{0,\nu}E+A\right)^{-1}\chi_{\mathbb{R}_{\geq0}}Au_{0}\in H_{\nu}^{-k}(\mathbb{R};H)$ by Corollary \[cor:solOp\] we obtain the asserted regularity for $u$. The support statement follows from the causality statement in Corollary \[cor:solOp\]. In the concluding section, we will discuss the spaces $\mathrm{IV}_{k}$ in connection to $(E,A)$ and will prove the main results of this paper mentioned in the introduction. Proofs of the main results and initial value spaces =================================================== Again, we assume that $H$ is a Hilbert space, and that $E,A\in L(H)$ with $(E,A)$ regular. At first, we turn to the proof of Proposition \[prop:stabInde\]. For this, we note some elementary consequences of the definition of $\mathrm{IV}_{k}$ and of regularity. \[lem:aux\] (a) For all $k\in\mathbb{N}$, we have $\mathrm{IV}_{k}\supseteq\mathrm{IV}_{k+1}.$ (b) Let $s\in\mathbb{C}\cap\rho(E,A)$. Then $$E(sE+A)^{-1}A=A(sE+A)^{-1}E.$$ (c) Let $k\in\mathbb{N}$, $x\in\mathrm{IV}_{k}$. Then for all $s\in\mathbb{C}\cap\rho(E,A)$ we have $$(sE+A)^{-1}Ex\in\mathrm{IV}_{k+1}.$$ (d) Let $s\in\mathbb{C}\cap\rho(E,A)\setminus\{0\}$. Then $$(sE+A)^{-1}E=\frac{1}{s}-\frac{1}{s}(sE+A)^{-1}A.$$ (e) Let $k\in\mathbb{N}$, $x\in\mathrm{IV}_{k}$. Then for all $s\in\mathbb{C}\cap\rho(E,A)\setminus\{0\}$ we have $$(sE+A)^{-1}Ex=\frac{1}{s}x+\sum_{\ell=1}^{k}\frac{1}{s^{\ell+1}}x_{\ell}+\frac{1}{s^{k+1}}(sE+A)^{-1}Aw.$$ for some $w\in H$, $x_{1},\ldots,x_{k}\in H$. The proof of (a) is an induction argument. The claim is trivial for $k=0$. For the inductive step, we see that the assertion follows using the induction hypothesis by $$\mathrm{IV}_{k+1}=A^{-1}[E[\mathrm{IV}_{k}]]\supseteq A^{-1}[E[\mathrm{IV}_{k+1}]]=\mathrm{IV}_{k+2}.$$ Next, we prove (b). We compute $$\begin{aligned} E(sE+A)^{-1}A= & E(sE+A)^{-1}(sE+A-sE)\\ = & E-E(sE+A)^{-1}sE\\ = & E-\left(sE+A-A\right)(sE+A)^{-1}E\\ = & A(sE+A)^{-1}E.\end{aligned}$$ We prove (c), by induction on $k$. For $k=0$, we let $x\in\mathrm{IV}_{0}=H$ and put $y\coloneqq\left(sE+A\right)^{-1}Ex.$ Then, by (b), we get that $$Ay=A\left(sE+A\right)^{-1}Ex=E\left(sE+A\right)^{-1}Ax\in R(E)=E[\mathrm{IV}_{0}].$$ Hence, $y\in\mathrm{IV}_{1}$. For the inductive step, we assume that the assertion holds for some $k\in\mathbb{N}$. Let $x\in\mathrm{IV}_{k+1}$. We need to show that $y\coloneqq\left(sE+A\right)^{-1}Ex\in\mathrm{IV}_{k+2}$. For this, note that there exists $w\in\mathrm{IV}_{k}$ such that $Ax=Ew$. In particular, by the induction hypothesis, we have $\left(sE+A\right)^{-1}Ew\in\mathrm{IV}_{k+1}$. Then we compute using (b) again, $$\begin{aligned} Ay & =A\left(sE+A\right)^{-1}Ex\\ & =E\left(sE+A\right)^{-1}Ax\\ & =E\left(sE+A\right)^{-1}Ew\in E[\mathrm{IV}_{k+1}].\end{aligned}$$ Hence, $y\in\mathrm{IV}_{k+2}$ and (c) is proved.\ For (d), it suffices to observe $$\begin{aligned} (sE+A)^{-1}E&=\frac{1}{s}(sE+A)^{-1}sE\\ &=\frac{1}{s}(sE+A)^{-1}(sE+A-A)\\ &=\frac{1}{s}-\frac{1}{s}(sE+A)^{-1}A.\end{aligned}$$ In order to prove part (e), we proceed by induction on $k\in\mathbb{N}$. The case $k=0$ has been dealt with in part (d) by choosing $w=-x.$ For the inductive step, we let $x\in\mathrm{IV}_{k+1}$. By definition of $\mathrm{IV}_{k+1}$, we find $y\in\mathrm{IV}_{k}$ such that $Ax=Ey.$ By induction hypothesis, we find $w\in H$ and $x_{1},\ldots,x_{k}\in H$ such that $$(sE+A)^{-1}Ey=\frac{1}{s}y+\sum_{\ell=1}^{k}\frac{1}{s^{\ell+1}}x_{\ell}+\frac{1}{s^{k+1}}(sE+A)^{-1}Aw.$$ With this we compute using (d) $$\begin{aligned} (sE+A)^{-1}Ex & =\frac{1}{s}x-\frac{1}{s}(sE+A)^{-1}Ax\\ & =\frac{1}{s}x-\frac{1}{s}(sE+A)^{-1}Ey\\ & =\frac{1}{s}x-\frac{1}{s}\left(\frac{1}{s}y+\sum_{\ell=1}^{k}\frac{1}{s^{\ell+1}}x_{\ell}+\frac{1}{s^{k+1}}(sE+A)^{-1}Aw\right)\\ & =\frac{1}{s}x+\sum_{\ell=1}^{k+1}\frac{1}{s^{\ell+1}}{\widetilde}{x}_{\ell}+\frac{1}{s^{k+2}}(sE+A)^{-1}A{\widetilde}{w},\end{aligned}$$ with ${\widetilde}{x}_{1}=-y,{\widetilde}{x}_{\ell}=-x_{\ell-1}$ for $\ell\geq2$ and ${\widetilde}{w}=-w$. With (a), we obtain the following reformulation of Proposition \[prop:stabInde\]. \[prop:stabInd0\]Assume that $E[\mathrm{IV}_{\operatorname{ind}(E,A)}]\subseteq H$ is closed. Then $$\mathrm{IV}_{\operatorname{ind}(E,A)+1}\subseteq\mathrm{IV}_{\operatorname{ind}(E,A)+2}.$$ Note that the closedness of $E[\mathrm{IV}_{\operatorname{ind}(E,A)}]$ implies the same for the space $\mathrm{IV}_{\operatorname{ind}(E,A)+1}$ since $A$ is continuous. We set $k\coloneqq\operatorname{ind}(E,A)$. Let $x\in\mathrm{IV}_{k+1}$. Then we need to find $y\in\mathrm{IV}_{k+1}$ with $Ax=Ey$. By definition there exists $x_{0}\in\mathrm{IV}_{k}$ with the property $Ax=Ex_{0}$. For $n\in\mathbb{N}$ large enough we define $y_{n}\coloneqq n\left(nE+A\right)^{-1}Ex_{0}.$ Since, $x_{0}\in\mathrm{IV}_{k}$, we deduce with (c) that $y_{n}\in\mathrm{IV}_{k+1}$. Moreover, by (e), $(y_{n})_{n}$ is bounded. Choosing a suitable subsequence for which we use the same name, we may assume that $(y_{n})_{n}$ is weakly convergent to some $y\in H$. The closedness of $\mathrm{IV}_{k+1}$ implies $y\in\mathrm{IV}_{k+1}$. Then using (e) we find $w\in H$ and $x_{1},\ldots,x_{k+1}\in H$ such that $$\left(nE+A\right)^{-1}Ex_{0}=\sum_{\ell=0}^{k}\frac{1}{n^{\ell+1}}x_{\ell}+\frac{1}{n^{k+1}}(nE+A)^{-1}Aw.$$ Hence, we obtain $$\begin{aligned} Ey & =\operatorname*{w-lim}_{n\to\infty}Ey_{n}\\ & =\operatorname*{w-lim}_{n\to\infty}E\left(nE+A\right)^{-1}nEx_{0}\\ & =\operatorname*{w-lim}_{n\to\infty}nE\left(nE+A\right)^{-1}Ax\\ & =\operatorname*{w-lim}_{n\to\infty}\left(nE+A-A\right)\left(nE+A\right)^{-1}Ax\\ & =Ax-\operatorname*{w-lim}_{n\to\infty}A\left(nE+A\right)^{-1}Ex_{0}\\ & =Ax-A\operatorname*{w-lim}_{n\to\infty}\left(\sum_{\ell=0}^{k}\frac{1}{n^{\ell+1}}x_{\ell}+\frac{1}{n^{k+1}}(nE+A)^{-1}Aw\right)=Ax,\end{aligned}$$ which yields the assertion. With an idea similar to the one in the proof of Proposition \[prop:stabInde\] (Proposition \[prop:stabInd0\]), it is possible to show that $E\colon\mathrm{IV}_{k+1}\to E[\mathrm{IV}_{k}]$ is an isomorphism if $k=\operatorname{ind}(E,A)$ and $E[\mathrm{IV}_{k}]\subseteq H$ is closed. We will need this result also in the proof of our main theorem. \[thm:Eiso\]Let $(E,A)$ be regular and assume that $E[\mathrm{IV}_{k}]\subseteq H$ is closed, $k=\operatorname{ind}(E,A).$ Then $$E\colon\mathrm{IV}_{k+1}\to E[\mathrm{IV}_{k}],x\mapsto Ex$$ is a Banach space isomorphism. Note that by the closed graph theorem, it suffices to show that the operator under consideration is one-to-one and onto. So, for proving injectivity, we let $x\in\mathrm{IV}_{k+1}$ such that $Ex=0.$ By definition, there exists $y\in\mathrm{IV}_{k}$ such that $Ey=Ax=Ax+nEx$ for all $n\in\mathbb{N}$. Hence, for $n\in\mathbb{N}$ large enough, we have $x=\left(nE+A\right)^{-1}Ey$. Thus, from $y\in\mathrm{IV}_{k}$ we deduce with the help of (e) that there exist $w,x_1,\ldots.x_{k}\in H$ such that $$x=\left(nE+A\right)^{-1}Ey=\frac{1}{n}y+\sum_{\ell=1}^{k}\frac{1}{n^{\ell+1}}x_{\ell}+\frac{1}{n^{k+1}}(nE+A)^{-1}Aw\to0\quad(n\to\infty),$$ which shows $x=0$. Next, let $y\in E[\mathrm{IV}_{k}]$. For large enough $n\in\mathbb{N}$ we put $$w_{n}\coloneqq(nE+A)^{-1}ny.$$ By (c), we obtain that $w_{n}\in\mathrm{IV}_{k+1}.$ Let $x\in\mathrm{IV}_{k}$ with $Ex=y$. Then, using (e), we find $w,x_{1},\ldots,x_{k}\in H$ such that $$\begin{aligned} w_{n} & =(nE+A)^{-1}ny\\ & =(nE+A)^{-1}nEx\\ & =x+\sum_{\ell=1}^{k}\frac{1}{n^{\ell}}x_{\ell}+\frac{1}{n^{k}}(nE+A)^{-1}Aw,\end{aligned}$$ proving the boundedness of $(w_{n})_{n}.$ Without loss of generality, we may assume that $(w_{n})_{n}$ weakly converges to $z\in\mathrm{IV}_{k+1}=A^{-1}[E[\mathrm{IV}_{k}]]$. Hence, $$\begin{aligned} Ez&=\operatorname*{w-lim}_{n\to\infty}Ew_{n}\\ &=\operatorname*{w-lim}_{n\to\infty}\frac{1}{n}\left(nE+A\right)w_{n}\\ &=\operatorname*{w-lim}_{n\to\infty}\frac{1}{n}\left(nE+A\right)(nE+A)^{-1}ny\\ &=y\tag*{{\qedhere}}.\end{aligned}$$ Next, we come to the proof of our main result , which we restate here for convenience. \[thm:mainTh-1\]Assume that $E[\mathrm{IV}_{\operatorname{ind}(E,A)}]\subseteq H$ is closed, $u_{0}\in\mathrm{IV}_{\operatorname{ind}(E,A)+1}$. Then has a unique continuously differentiable solution $u:\mathbb{R}_{>0}\to H$, satisfying $u(0+)=u_{0}$ and $$Eu'(t)+Au(t)=0\quad(t>0).\label{eq:dAe}$$ Moreover, the solution coincides with the solution given in . Let $u_{0}\in\mathrm{IV}_{\operatorname{ind}(E,A)+1}$. We denote ${\widetilde}{E}\colon\mathrm{IV}_{k+1}\to E[\mathrm{IV}_{k}],x\mapsto Ex,$ where $k=\operatorname{ind}(E,A)$. By , we have that ${\widetilde}{E}$ is an isomorphism. For $t>0$, we define $$u(t)\coloneqq\exp\left(-t{\widetilde}{E}^{-1}A\right)u_{0}.$$ Then $u(0+)=u_{0}.$ Moreover, $u(t)$ is well-defined. Indeed, if $u_{0}\in\mathrm{IV}_{k+1}$ then $Au_{0}\in E[\mathrm{IV}_{k}]$. Hence, ${\widetilde}{E}^{-1}Au_{0}\in\mathrm{IV}_{k+1}$ is well-defined. Since $E[\mathrm{IV}_{k}]$ is closed, and $A$ is continuous, we infer that $\mathrm{IV}_{k+1}$ is a Hilbert space. Thus, we deduce that $u\colon\mathbb{R}_{>0}\to\mathrm{IV}_{k+1}$ is continuously differentiable. In particular, we obtain $$\mathrm{IV}_{k+1}\ni u'(t)=-{\widetilde}{E}^{-1}Au(t).$$ If we apply ${\widetilde}{E}$ to both sides of the equality, we obtain . If $u:\mathbb{R}_{>0}\to H$ is a continuously differentiable solution of with $u(0+)=u_{0},$ we infer that $u\in L_{2,\nu}(\mathbb{R};H)$ for some $\nu>0$ large enough, where we extend $u$ to $\mathbb{R}_{<0}$ by zero. Hence, $$\partial_{0,\nu}Eu+Au=E\partial_{0,\nu}u+Au=Eu'+Au+\delta\cdot Eu(0+)=\delta\cdot Eu_{0},$$ where we have used that $u$ is differentiable on $\mathbb{R}_{<0}\cup\mathbb{R}_{>0}$ and jumps at $0$. Thus, $u$ is the solution given in , from which we also derive the uniqueness. We conclude with a comment on the proof of Corollary \[cor:mainCor\]. We note that the condition $u_{0}\in\mathrm{IV}_{\operatorname{ind}(E,A)+1}$ arises naturally if we assume that $\mathrm{IV}_{j}$ is closed for each $j\in\mathbb{N}.$ Indeed, if $u:\mathbb{R}_{>0}\to H$ is a continuously differentiable solution of , we infer that $$Au(t)=-Eu'(t)\quad(t>0)$$ and thus $u(t)\in\mathrm{IV}_{1}$ for $t>0$. Since $\mathrm{IV}_{1}$ is closed, we derive $u'(t)\in\mathrm{IV}_{1}$ and hence, inductively $u(t)\in\bigcap_{j\in\mathbb{N}}\mathrm{IV}_{j}$ for each $t>0.$ Since $\bigcap_{j\in\mathbb{N}}\mathrm{IV}_{j}=\mathrm{IV}_{\operatorname{ind}(E,A)+1}$ by Proposition \[prop:stabInd0\], we get $$u_{0}=u(0+)\in\mathrm{IV}_{\operatorname{ind}(E,A)+1}.$$ [1]{} T. Berger, A. Ilchmann, and S. Trenn. The quasi-[W]{}eierstraß form for regular matrix pencils. , 436(10):4052–4069, 2012. L. Dai. . Springer-Verlag New York, Inc., Secaucus, NJ, USA, 1989. A. Kalauch, R. Picard, S. Siegmund, S. Trostorff, and M. M. Waurick. . , 26(2):369–399, 2014. P. [Kunkel]{} and V. [Mehrmann]{}. European Mathematical Society Publishing House, Zürich, 2006. R. Picard. Volume 196 of Pitman Research Notes in Mathematics Series. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989. R. Picard. . , 32:1768–1803, 2009. W. Rudin. . Mathematics series. McGraw-Hill, 1987. S. Trostorff and M. Waurick. . Technical report, TU Dresden, University of Strathclyde, 2017.

--- abstract: 'Let $f \in S_{\kappa}(\Gamma_0(N))$ be a Hecke eigenform at $p$ with eigenvalue $\lambda_f(p)$ for a prime $p \nmid N$. Let $\alpha_p$ and $\beta_p$ be complex numbers satisfying $\alpha_p + \beta_p = \lambda_{f}(p)$ and $\alpha_p \beta_p = p^{\k-1}$. We calculate the norm of $f_{p}^{\alpha_p}(z) = f(z) - \beta_{p} f(pz)$ as well as the norm of $U_p f$, both classically and adelically. We use these results along with some convergence properties of the Euler product defining the symmetric square $L$-function of $f$ to give a ‘local’ factorization of the Petersson norm of $f$.' address: - | $^1$Department of Mathematical Sciences\ Clemson University\ Clemson, SC 29634 - | $^2$Department of Mathematics\ Queens College\ City University of New York\ Flushing, NY 11367 - | $^3$Department of Mathematics\ University of Connecticut\ Storrs, CT 06269 author: - Jim Brown$^1$ - Krzysztof Klosin$^2$ bibliography: - 'mybib.bib' title: 'On the norms of $p$-stabilized elliptic newforms' --- [^1] Introduction ============ Let $\k \geq 2$ and $N \geq 1$ be integers and $p$ an odd prime with $p \nmid N$. Let $f \in S_{\k}(\Gamma_0(N))$ be a newform. It is well-known that the Petersson norm $\langle f,f \rangle$ serves as a natural period for many $L$-functions of $f$ [@HidaInvent81; @ShimuraMathAnn77]. In this paper we focus on related periods $\langle f_p^{\alpha_p}, f_p^{\alpha_p} \rangle$ (defined below) for $\alpha_p$ a Satake parameter of $f$. When $f$ is ordinary at $p$, the forms $f_p^{\alpha_p}$ arise naturally in the context of Iwasawa theory as the objects which can be interpolated into a Hida family. It is in fact in the context of ‘$p$-adic interpolation’ of some automorphic lifting procedures (between two algebraic groups, one of them being $\GL_2$) that these calculations arise (see [@BrownKlosinGSp(4)] for example); however, our results apply in a more general setup as specified below. Let $f \in S_{\k}(\Gamma_0(N))$ be an eigenform for the $T_p$-operator with eigenvalue $\lambda_{f}(p)$. Let $\alpha_p$ and $\beta_p$ be the pair of complex numbers satisfying $\alpha_p + \beta_p = \lambda_f(p)$ and $\alpha_p \beta_p = p^{\k-1}$. We set $f_{p}^{\alpha_{p}}(z) = f(z) - \beta_{p} f(pz)$. In the case that $f$ is ordinary at $p$, we can choose $\alpha_p$ and $\beta_p$ so that $\alpha_{p}$ is a $p$-unit and $\beta_{p}$ is divisible by $p$. In this special case $f_p^{\alpha_p}$ is the $p$-stabilized ordinary newform of tame level $N$ attached to $f$. Since $f_p^{\alpha_p} = p^{1-\kappa}\beta_p (U_p-\beta_p)f$, calculating $\langle f_p^{\alpha_p}, f_p^{\alpha_p} \rangle$ is in fact equivalent to calculating $\langle U_p f, U_p f \rangle$. While computation of any of these inner products does not present any difficulties (see Section \[Relations between the norms\]), it is an accident resulting from the relative simplicity of the Hecke algebra on $\GL_2$, where the $T_p$ and the $U_p$ operators differ by a single term. It turns out that in the higher-rank case it is the calculation of the latter inner product that provides the fastest route to computing the Petersson norm of various $p$-stabilizations. With these future applications in mind we present an alternative approach to calculating $\langle U_p f, U_p f \rangle$, this time working adelically (see Sections \[relations\] and \[sec:adeliccalc\]), as this is the method that generalizes to higher genus most readily (see [@BrownKlosinGSp(4)], where this is done for the group $\GSp_4$). It is well-known that the Petersson norm $\langle f,f \rangle$ is closely related to the value $L(\kappa, \Sym^2 f)$ at $\kappa$ of the symmetric square $L$-function of $f$. The absolutely convergent Euler product defining this $L$-function for ${\operatorname{Re}}(s) >\kappa$ converges (conditionally) to the value $L(s, \Sym^2 f)$ when ${\operatorname{Re}}(s)=\kappa$ (this and in fact a more general result is proved in the appendix by Keith Conrad). On the other hand our computation of $\langle f_p^{\alpha_p}, f_p^{\alpha_p} \rangle$ shows that this inner product differs from $\langle f,f \rangle$ by essentially the $p$-Euler factor of $L(\kappa, \Sym^2 f)$. Combining these facts we exhibit a (conditionally convergent) factorization of $\langle f,f \rangle$ into local components defined via the inner products $\langle f_p^{\alpha_p}, f_p^{\alpha_p} \rangle$ (for details, see Section \[Applications to $L$-values\]). The authors would like to thank Henryk Iwaniec, and Keith Conrad would like to thank Gergely Harcos, for helpful email correspondence. Classical calculation of $\langle f_p, f_p \rangle$ and $\langle U_p f, U_p f \rangle$ {#Relations between the norms} ====================================================================================== Let $N$ be a positive integer. Let $\Gamma_0(N)\subset \SL_2(\bfZ)$ denote the subgroup consisting of matrices whose lower-left entry is divisible by $N$. For a holomorphic function $f$ on the complex upper half-plane $\fh$ and for $\gamma=\bmat a&b \\ c&d \emat \in \GL_2^+(\bfR)$, where $+$ denotes positive determinant, and $\kappa \in \bfZ_+$ we define the slash operator as $$(f|_{\k} \gamma)(z) = \frac{\det(\gamma)^{\kappa/2}}{(cz+d)^\kappa} f\left(\frac{az+b}{cz+d}\right). $$ If $\kappa$ is clear from the context we will simply write $f|\gamma$ instead of $f|_{\kappa}\gamma$. We will write $S_{\kappa}(\Gamma_0(N))$ for the $\bfC$-space of cusp forms of weight $\kappa$ and level $\Gamma_0(N)$ (i.e., functions $f$ as above which satisfy $f|_{\kappa} \gamma = f$ for all $\gamma \in \Gamma_0(N)$ and vanish at the cusps - for details see [@Miyake89]). The space $S_{\kappa}(\Gamma_0(N))$ is endowed with a natural inner product (the *Petersson inner product*) defined by $$\langle f, g \rangle_{N} = \int_{\Gamma_0(N) \backslash \fh} f(z) \overline{g(z)} y^{\k-2} dx dy$$ for $z = x+iy$ with $x,y \in \bfR$ and $y>0$. If $\Gamma \subset \Gamma_0(N)$ is a finite index subgroup we also set $$\langle f, g \rangle_{\Gamma} = \int_{\Gamma \backslash \fh} f(z) \overline{g(z)} y^{\k-2} dx dy.$$ From now on let $p$ be a prime which does not divide $N$. Set $\eta = {\left[ \begin{matrix} p & 0 \\ 0 & 1 \end{matrix} \right]}$. We have the decomposition $$\label{decomp3} \Gamma_0(N) {\left[ \begin{matrix} 1 & 0 \\ 0 & p \end{matrix} \right]} \Gamma_0(N) = \bigsqcup_{j=0}^{p-1} \Gamma_0(N) {\left[ \begin{matrix} 1 & j \\ 0 & p \end{matrix} \right]} \sqcup \Gamma_0(N) \eta.$$ Recall the $p$th Hecke operator acting on $S_{\k}(\Gamma_0(N))$ is given by $$T_p f = p^{\k/2-1}\left( \sum_{j=0}^{p-1} f|_{\k} {\left[ \begin{matrix} 1 & j \\ 0 & p \end{matrix} \right]} + f|_{\k} \eta\right)$$ and the $p$th Hecke operator acting on $S_{\k}(\Gamma_0(Np))$ is given by $$U_pf = p^{\k/2-1}\left( \sum_{j=0}^{p-1} f|_{\k} {\left[ \begin{matrix} 1 & j \\ 0 & p \end{matrix} \right]} \right).$$ As we will be viewing $f \in S_{\k}(\Gamma_0(N))$ as an element of $S_{\k}(\Gamma_0(Np))$, we use $T_p$ and $U_p$ to distinguish the two Hecke operators at $p$ defined above. Let $f \in S_{\k}(\Gamma_0(N))$ be an eigenfunction for $T_p$ with eigenvalue $\lambda_f(p)$. There exist (up to permutation) unique complex numbers $\alpha_p$ and $\beta_p$ satisfying $\lambda_f(p) = \alpha_p + \beta _p$ and $\alpha_p\beta_p=p^{\kappa-1}$. We consider the following two forms: $$\begin{aligned} f_{p}^{\alpha_p}(z) &= f(z) - \beta_{p} p^{-\k/2} (f|_{\kappa}\eta)(z),\\ f_{p}^{\beta_p}(z) &= f(z) - \alpha_{p} p^{-\k/2} (f|_{\kappa}\eta)(z).\end{aligned}$$ One immediately obtains that $f_p^{\alpha_p} \in S_{\k}(\Gamma_0(Np))$ and that $f_p^{\alpha_p}$ is an eigenfunction for the operator $U_p$ with eigenvalue $\alpha_p$. Furthermore, if $f$ is also an eigenform for $T_{\ell}$ for a prime $\ell \neq p$, then so is $f_p^{\alpha_p}$ and it has the same $T_{\ell}$-eigenvalue as $f$. The analogous statements for $f_{p}^{\beta_p}$ hold as well. Note that if $f$ is ordinary at $p$, then one can choose $\alpha_p$ and $\beta_p$ so that $\ord_p(\alpha_p) = 0$ and then $f_p^{\alpha_p}$ is the $p$-stabilized newform associated to $f$, see [@WilesInventMath88] for example. \[classform\] Let $f \in S_{\k}(\Gamma_0(N))$ be defined as above, where $p \nmid N$. We have $$\frac{\langle U_p f, U_p f \rangle_{Np}}{\langle f, f \rangle_{Np}} = p^{\k-2} + \frac{(p-1) \lambda_{f}(p)^2}{p+1}$$ and $$\frac{\langle f_p^{\alpha_p}, f_p^{\alpha_p} \rangle_{Np}}{\langle f, f \rangle_{Np}} = \frac{p}{p+1}\left(1 - \frac{\alpha_p^2}{p^\kappa}\right)\left(1 - \frac{\beta_p^2}{p^\kappa}\right).$$ The definition of $f_{p}^{\alpha_p}$ and the fact that $U_p f = T_{p} f - p^{\k/2-1}f|\eta$ immediately give $$\begin{aligned} \langle f_p^{\alpha_p}, f_p^{\alpha_p} \rangle_{Np} &= (1 + |\beta_{p}|^2 p^{-\k}) \langle f, f \rangle_{Np} - p^{-\k/2}(\beta_{p} \langle f| \eta, f \rangle_{Np} + \ov{\beta_{p} \langle f|\eta, f\rangle}_{Np}) \end{aligned}$$ and $$\begin{aligned} \langle U_p f, U_p f \rangle_{Np} &= (p^{\k-2} + \lambda_{f}(p)^2) \langle f, f \rangle_{Np} - p^{\k/2-1}\lambda_{f}(p) (\langle f|\eta, f \rangle_{Np} + \ov{\langle f|\eta, f\rangle}_{Np}).\end{aligned}$$ Let us now compute $\langle f|\eta, f \rangle_{Np}$. Observe that by the definition of $T_p$ we have $$\langle T_p f, g \rangle_{Np} = p^{\k/2-1}\left( \sum_{j=0}^{p-1} \left \langle f\mid {\left[ \begin{matrix} 1 & j \\ 0 & p \end{matrix} \right]}, g\right \rangle_{Np} + \langle f|\eta, g \rangle_{Np}\right).$$ Using the decomposition (\[decomp3\]) we can find $a_{j}, b_{j} \in \Gamma_0(N)$ so that $a_{j} {\left[ \begin{matrix} 1 & j \\ 0 & p \end{matrix} \right]} b_{j} = {\left[ \begin{matrix} 1 & 0 \\ 0 & p \end{matrix} \right]}$, and $a,b \in \Gamma_0(N)$ so that $a {\left[ \begin{matrix} 1 & 0 \\ 0 & p \end{matrix} \right]} b = {\left[ \begin{matrix} p & 0 \\ 0 & 1 \end{matrix} \right]}.$ Using this and the fact that $f, g \in S_{\k}(\Gamma_0(N))$, we have $$\begin{aligned} p^{1-\k/2} \langle T_p f,g \rangle_{Np} &= \sum_{j=0}^{p-1} \left \langle f\mid a_{j} {\left[ \begin{matrix} 1 & j \\ 0 & p \end{matrix} \right]}, g|b_{j}^{-1} \right \rangle_{Np} + \langle f|\eta, g \rangle_{Np} \\ &=\sum_{j=0}^{p-1} \left \langle f\mid a_{j}{\left[ \begin{matrix} 1 & j \\ 0 & p \end{matrix} \right]} b_{j}, g \right \rangle_{Np} + \langle f|\eta, g \rangle_{Np} \\ &= p \left \langle f\mid {\left[ \begin{matrix} 1 & 0 \\ 0 & p \end{matrix} \right]}, g \right \rangle_{Np} + \langle f|\eta, g \rangle_{Np} \\ &= p \left \langle f\mid a {\left[ \begin{matrix} 1 & 0 \\ 0 & p \end{matrix} \right]}, g|b^{-1} \right \rangle_{Np} + \langle f|\eta, g \rangle_{Np}\\ &= (p+1) \langle f|\eta, g \rangle_{Np}. \end{aligned}$$ Thus, setting $g = f$ we obtain $$\langle f|\eta, f \rangle_{Np} = p^{1-\k/2}\frac{\lambda_{f}(p)}{p+1} \langle f, f \rangle_{Np}.$$ We can now easily conclude that $$\frac{\langle f_p^{\alpha_p}, f_p^{\alpha_p} \rangle_{Np}}{\langle f, f \rangle_{Np}} = 1 + |\beta_{p}|^2 p^{-\k} - p^{1-\k}(\beta_{p}+\ov{\beta_p})\frac{\lambda_{f}(p)}{p+1}$$ and $$\frac{\langle U_p f, U_p f \rangle_{Np}}{\langle f, f \rangle_{Np}} = p^{\k-2} + \frac{(p-1) \lambda_{f}(p)^2}{p+1}.$$ Using the fact that $T_p$ is self-adjoint with respect to the Petersson inner product we have $\alpha_p + \beta_p = \lambda_f(p) \in \bfR$. We note by Lemma \[Ramanujan\] below that $\ov{\alpha_{p}} = \beta_{p}$. Thus $|\alpha_p|^2=|\beta_p|^2 = |\alpha_p||\beta_p|=p^{\kappa-1}$ and $\beta_p + \ov{\beta_p}=\lambda_f(p)$. This allows us to simplify the formula for $ \frac{\langle f_p^{\alpha_p}, f_p^{\alpha_p} \rangle_{Np}}{\langle f, f \rangle_{Np}}$ to $$\frac{\langle f_p^{\alpha_p}, f_p^{\alpha_p} \rangle_{Np}}{\langle f, f \rangle_{Np}} = 1 + \frac{1}{p} - p^{1-\k}\frac{\lambda_{f}(p)^2}{p+1}.$$ Again using that $\lambda_f(p) = \alpha_p+\beta_p$ we obtain $$\begin{aligned} \frac{\langle f_p^{\alpha_p}, f_p^{\alpha_p} \rangle_{Np}}{\langle f, f \rangle_{Np}} &= \frac{1}{p+1}\left(p+ 1+\frac{p+1}{p} - p^{1-\kappa}(\alpha_p^2+\beta_p^2 + 2p^{\kappa-1})\right)\\ &= \frac{p}{p+1}\left(1-\frac{\alpha_p^2}{p^{\kappa}}\right)\left(1-\frac{\beta_p^2 }{p^{\kappa}}\right).\end{aligned}$$ We have $$\lim_{\substack{p\to \infty\\ p\hs\textup{is prime}}} \frac{\langle f_p^{\alpha_p}, f_p^{\alpha_p} \rangle_{Np}}{p+1}=\langle f, f \rangle_N.$$ Using Theorem \[classform\] and the fact that $\langle f,f\rangle_{Np} = (p+1)\langle f, f\rangle_N$, we have for every prime $p \nmid N$ that $$\frac{\langle f_p^{\alpha_p}, f_p^{\alpha_p} \rangle_{Np}}{p+1}= \frac{p}{p+1}\left(1-\frac{\alpha_p^2}{p^{\kappa}}\right)\left(1-\frac{\beta_p^2 }{p^{\kappa}}\right)\langle f, f \rangle_{N}.$$ Since $|\alpha_p|^2=|\beta_p|^2=p^{\kappa-1}$, we see that the first three factors on the right tend to 1 as $p$ tends to infinity. Relation between the classical and adelic inner products {#relations} ======================================================== While the classical calculations for $\langle U_p f, U_p f \rangle$ are rather elementary, it is also useful to note that one can perform these calculations adelically. The problem of calculating $\langle U_p f, U_p f \rangle$ is one that is local in nature, so it lends itself nicely to such an approach. Moreover, in a higher genus setting such as when working with Siegel modular forms, it is the adelic approach that generalizes most readily [@BrownKlosinGSp(4)]. In this section we provide the necessary background relating the adelic and classical inner products that is needed to relate the adelic inner product calculated in Section \[sec:adeliccalc\] to the calculation given in the previous section. In this and the following sections $p$ will denote a prime number and $v$ will denote an arbitrary place of $\bfQ$ including the Archimedean one, which we will denote by $\infty$. Let $G=\GL_2$ and fix $N \geq 1$. By strong approximation (see for example [@BumpCambridgeUniversityPress88 Theorem 3.3.1, p. 293]) we have $$\label{strapp} G(\AQ) = G(\bfQ) G(\bfR) \prod_{p} K_{p},$$ where $K_p$ is a compact subgroup of $G(\bfQ_p)$ such that $\det K_p = \bfZ_p^{\times}$. One example would be to take $K_{p} = K_{0}(N)_p$, where $$K_0(N)_p = \left\{ \bmat a& b \\ c& d \emat \in G(\bfZ_{p}): c \equiv 0 \bmod{N}\right\}.$$ Note that $K_0(N)_p=G(\bfZ_p)$ if $p \nmid N$. We will also set $$K_0(N):= \left\{ \bmat a& b \\ c& d \emat \in G(\hat{\bfZ}): c \equiv 0 \bmod{N}\right\}=\prod_p K_0(N)_p.$$ The decomposition (\[strapp\]) implies that \[2\] G() G() = G\^+() \_[p]{} K\_[p]{}, where $+$ indicates positive determinant. Let $Z \subset G$ denote the center. For every $p$ there is a unique Haar measure $dg_p$ on $G(\bfQ_p)$ normalized so that the volume of any maximal compact subgroup of $G(\bfQ_p)$ is one. We use the standard Haar measure on $G(\bfR)$ as defined in [@BumpCambridgeUniversityPress88 $\S$ 2.1]. Define the adelic analogue of the Petersson inner product: $$\langle \phi_1, \phi_2 \rangle = \int_{Z(\AQ) G(\bfQ) \setminus G(\AQ)} \phi_1(g) \ov{\phi_2(g)} dg,$$ where $\phi_1$ and $\phi_2$ lie in $L^2(Z(\AQ) G(\bfQ) \setminus G(\AQ))$ and have the same central character and $dg$ is the Haar measure on $Z(\AQ) G(\bfQ) \setminus G(\AQ)$ corresponding to our choice of local Haar measures. Let $f \in S_{\kappa}(\Gamma_0(N))$ be an eigenform. For $g=\gamma g_{\infty} k\in G(\AQ)$ with $\gamma \in G(\bfQ)$, $g_{\infty}=\bmat a&b\\c&d\emat \in G^+(\bfR)$ and $k \in K_0(N)$, set \[adelicdef\] \_f(g)=f(g\_).Then $\phi_f$ is an automorphic form on $G(\AQ)$ and it is easy to see (using the bijection in [@GelbartAFAG Equation 5.13]) that one has $$\label{eqn:classicalrelation} \langle \phi_f, \phi_{f} \rangle = \frac{1}{\left[\SL_2(\bfZ):\Gamma_0(N)\right]} \langle f, f \rangle_{N}.$$ Let $\pi_{f} \cong \otimes \pi_{f,v}$ be the automorphic representation generated by $\phi_{f}$. If $f$ is a newform, then we can write $\phi_f= \otimes_v \phi_{f,v}$ for $\phi_{f,v} \in \pi_{f,v}$ and $\phi_{f,v}$ are spherical vectors for all $v \nmid N$, $v \neq \infty$. For every $v$ we can choose a $G(\bfQ_v)$-invariant inner product $\langle\cdot, \cdot\rangle_v$ (and any two such are scalar multiples of each other) so that $\langle \phi_{f,v}, \phi_{f,v}\rangle_v=1$ for all $v \nmid N$, $v \neq \infty$. It follows that there is constant $c$ so that $$\label{eqn:factorization}\langle \phi_{f}, \phi_{f}\rangle = c \prod_{v} \langle \phi_{f,v}, \phi_{f,v} \rangle_{v}.$$ We are now in a position to relate the ratio $\frac{\langle U_p f, U_p f \rangle_{Np}}{\langle f, f \rangle_{Np}}$ to something that can be calculated locally. In fact since we are only interested in this ratio, the precise value of the constant $c$ in (\[eqn:factorization\]) will be irrelevant. Fix $p \nmid N$. As noted above, we normalize our Haar measure so that $\vol(K_0(1)_p) = 1$. For a vector $v_p$ inside the space of $\pi_{f,p}$, we set $$T_p v_p = \int_{K_0(1)_p \bsmat p&0 \\ 0& 1 \esmat K_0(1)_p} \pi_{f,p}(g) v_{p} dg$$ and $$V_p v_{p} = \int_{\bsmat 1 & 0 \\ 0 & p \esmat K_0(1)_p} \pi_{f,p}(g) v_{p} dg.$$ Note that we have the decompositions \[dec\] K\_0(1)\_p p&0\ 0& 1 K\_0(1)\_p = \_[b=0]{}\^[p-1]{} p & b\ 0 & 1 K\_0(1)\_p 1&0\ 0& pK\_0(1)\_p and \[dec1\] K\_0(p)\_p p&0\ 0& 1 K\_0(p)\_p= \_[b=0]{}\^[p-1]{} p & b\ 0 & 1 K\_0(p)\_p. The adelic operator corresponding to the $U_p$-operator acting on classical modular forms as defined in Section \[Relations between the norms\] is given by $$U_p v_{p} := T_p v_{p} - V_p v_{p}.$$ \[classtoadel\] We have $$\frac{\langle U_p^{\rm cl} f, U_p^{\rm cl} f \rangle_{Np}}{\langle f, f \rangle_{Np}} = p^{\k-2}\langle U_p \phi_{f,p}, U_p \phi_{f,p} \rangle_{p}$$ for any local inner product pairing $\langle , \rangle_{p}$ so that $\langle \phi_{f,p}, \phi_{f,p} \rangle_{p} = 1$ and $U_p^{\rm cl}$ is the classical $U_p$-operator as defined in Section \[Relations between the norms\]. If we set $U_p \phi_f := (U_p \phi_{f,p}) \otimes \otimes_{v \neq p} \phi_{f,v}$ then it follows by the same argument as the one in the proof of [@GelbartAFAG Lemma 3.7] that \[adelclas\] \_[U\_p\^[cl]{}f]{}=p\^[/2-1]{}U\_p \_f. The lemma is now immediate from (\[eqn:classicalrelation\]) and (\[eqn:factorization\]). It only remains to calculate $\langle U_p \phi_{f,p}, U_p \phi_{f,p} \rangle$, which is done in the next section. Local calculation of $\langle U_p \phi_{f,p}, U_p \phi_{f,p} \rangle$ {#sec:adeliccalc} ===================================================================== We will now give a calculation that, when combined with the results of the previous section, provides a local way to calculate $\langle U_p f, U_p f \rangle$ in terms of $\langle f, f \rangle$. As in the previous section we fix $f \in S_{k}(\Gamma_0(N))$ a newform and a prime $p$ not dividing $N$. We again let $\pi_{f} = \otimes_{v} \pi_{f,v}$ be the automorphic representation associated to $f$. Note that since $p \nmid N$, we can take the principal series representation $\pi_{p}(\chi_1, \chi_2)$ to be the model for $\pi_{f,p}$ and for functions $\psi, \psi' \in \pi_p(\chi_1, \chi_2)$ define the local inner product by $$\langle \psi, \psi' \rangle_p := \int_{K_0(N)_p} \psi(g) \ov{\psi'(g)}dg,$$ where the Haar measure is normalized so that $\vol(K_0(N)_p)=1$. Then the vector $\phi_{f,p}\in \pi_p$ corresponds to the function (which we will also denote by $\phi_{f,p} \in \pi_p(\chi_1,\chi_2)$) which can be described explicitly as $$\phi_{f,p}\left( \bmat a &* \\ 0&b \emat k\right) = \chi_1(a) \chi_2(b)|ab^{-1}|_p^{1/2},$$ where $|\cdot|_p$ denotes the standard $p$-adic norm ($|p|_p=p^{-1}$) and $k \in K_0(N)_p$. As this section is focused on the calculation of $\langle U_p \phi_{f,p}, U_p \phi_{f,p} \rangle$, we will from now on write $\phi$ for $\phi_{f,p}$ and $K_0(1)$ (resp., $K_0(p)$) for $K_0(1)_p$ (resp., $K_0(p)_p$). We note here that the calculation which follows can also be performed using the MacDonald formula for matrix coefficients (see [@CasselmanCompMath80 $\S$ 4]). However, in the relatively simple case of $\GL_2$ the elementary approach which we present below does not add any computational difficulty and is perhaps more transparent. Set $$\mB:= \left\{ \bmat p & b \\ 0 & 1 \emat : b\in \{0,1, \dots, p-1\}\right\}, \quad \mB':= \mB \cup \left\{ \bmat 1&0 \\ 0&p \emat \right\}.$$ If $g \in K_0(1)$ and $\beta \in \mB'$, there is a permutation $\sigma_{g}$ of $\mB'$ and elements $k(g,\beta) \in K_0(1)$ such that $g \beta = \sigma_{g}(\beta) k(g, \beta)$. Furthermore, note that if $g \in K_0(1) - K_0(p)$, then the corresponding permutation cannot fix $\bmat 1&0 \\ 0& p \emat$. This implies that for such a $g$, there exists $\beta \in \mB$ such that $\sigma_{g}(\beta) = \bmat 1&0 \\ 0& p \emat$. Since in the computation of $U_p\phi$ only matrices in $\mB$ are used, we are interested in the restriction of $\sigma$ to $\mB$. For such a $g$ there are $p-1$ matrices in the image of $\sigma_{g}$ which have $(p,1)$ on the diagonal and one that has $(1,p)$ on the diagonal. Set $\mB_1(g) = \{\beta \in \mB: \sigma_{g}(\beta) \in \mB\}$ and $\mB_2(g) = \{\beta \in \mB: \sigma_{g}(\beta) \in \mB' - \mB\}$. So for $g \in K_0(1) - K_0(p)$ we have (note that our $\phi$ is right-$K_0(1)$-invariant and $\vol(K_0(1))=1$) $$\begin{split} (U_p\phi)(g) &= \vol(K_0(1)) \sum_{\beta \in \mB} \phi(g\beta)\\ &= \sum_{\beta \in \mB} \phi(\sigma_{g}(\beta) k(g,\beta)) \\ &= \sum_{\beta \in \mB_1(g)} \phi(\sigma_{g}(\beta)) + \sum_{\beta \in \mB_2(g)} \phi(\sigma_{g}(\beta))\\ & = (p-1)\chi_1(p) p^{-1/2} + \chi_2(p) p^{1/2}. \end{split}$$ If $g \in K_0(p)$, then the permutation $\sigma$ fixes $\bmat 1&0 \\0 & p \emat$, hence we obtain $$(U_p\phi)(g) = \vol(K_0(1)) \sum_{\beta \in \mB} \phi(g\beta) = \sum_{\beta \in \mB} \phi(\sigma(\beta))=p\chi_1(p)p^{-1/2} = \chi_1(p)p^{1/2}.$$ Now let us compute the integral: $$\left< U_p \phi, U_p \phi \right>_{K_0(1)} = \int_{K_0(p)} U_p\phi(g) \ov{U_p\phi(g)} dg + \int_{K_0(1) - K_0(p)} U_p\phi(g) \ov{U_p\phi(g)} dg.$$ We have \[j1\] \_[K\_0(p)]{} U\_p(g) dg &= \_[K\_0(p)]{} p |\_1(p)|\^2 dh\ &=(K\_0(p)) p |\_1(p)|\^2\ &= and, since $\vol(K_0(1) - K_0(p))=p/(p+1)$, $$\begin{gathered} \label{k1} \int_{K_0(1) - K_0(p)} U_p\phi(g) \ov{U_p\phi(g)} dg\\ = \int_{K_0(1)- K_0(p)} \left[ \frac{(p-1)^2}{p} |\chi_1(p)|^2 + (p-1)\tr(\chi_1(p) \ov{\chi_2(p)})+ p|\chi_2(p)|^2\right] dg \\ = \frac{p}{p+1} \left[ \frac{(p-1)^2}{p} |\chi_1(p)|^2 + (p-1)\tr(\chi_1(p) \ov{\chi_2(p)})+ p|\chi_2(p)|^2\right].\end{gathered}$$ Putting (\[j1\]) and (\[k1\]) together we get \[for32\] &= |\_1(p)|\^2 + |\_2(p)|\^2+ (\_1(p)). \[Ramanujan\] We have $\chi_j(p) = p^{s_j}$ for $j=1,2$, where $s_j$ is a purely imaginary number. In particular, $|\chi_{j}(p)|=1$ for $j=1,2$. Moreover, we have $\ov{\alpha_p} = \beta_{p}$. The first part follows from [@GelbartAFAG p. 92] and is a direct consequence of the fact that cusp forms on $\GL_2$ satisfy the Ramanujan conjecture. Observe that $\alpha_p = p^{(\k-1)/2} \chi_1(p)$ and $\beta_{p} = p^{(\k-1)/2} \chi_2(p)$. Using that $\alpha_p \beta_{p} = p^{\k-1}$, we obtain $\chi_1(p) \chi_2(p) = 1$. This, combined with the fact that $\chi_{j}(p) = p^{s_{j}}$ with $s_{j}$ purely imaginary implies $\chi_1(p) = \ov{\chi_2(p)}$. Thus $\ov{\alpha_p} = \beta_p$. Using Lemma \[Ramanujan\] we can simplify (\[for32\]) to $$\langle U_p \phi, U_p \phi \rangle = \frac{2p^2-p+1}{p+1} + \frac{p^2-p}{p+1} \tr (\chi_1(p)\ov{\chi_2(p)}).$$ Moreover using that $\alpha_p = p^{(\kappa-1)/2}\chi_1(p)$, $\beta_p=p^{(\kappa-1)/2}\chi_2(p)$ and $\chi_1(p) = \ov{\chi_2(p)}$ we have $$\tr (\chi_1(p)\ov{\chi_2(p)}) = (\chi_1(p) + \chi_2(p))^2 - 2\chi_1(p) \chi_2(p) = p^{1-\k}\lambda_f(p)^2 - 2.$$ Thus we obtain $$\begin{aligned} \langle U_p \phi, U_p \phi \rangle &= \frac{p+1}{p+1} + \frac{p(p-1)p^{1-k}\lambda_f(p)^2}{p+1}\\ &= 1+ \frac{(p-1)\lambda_f(p)^2}{p+1}p^{2-k},\end{aligned}$$ hence we see that by Lemma \[classtoadel\] this recovers the classical formula from Theorem \[classform\]. Applications to $L$-values {#Applications to $L$-values} ========================== Let $f \in S_{\kappa}(\Gamma_0(N))$ be a newform. In this section we apply the results of the previous sections to give a ‘local’ decomposition of the Petersson norm of $f$. This depends on showing that value $L^{N}(k, \Sym^2 f)$ obtained by meromorphic continuation of $L(s,\Sym^2 f)$ can be expressed as a conditionally convergent Euler product. Recall that the (partial) *symmetric square $L$-function* of $f$ is defined by the Euler product \[Ep\] L\^N(s, \^2 f) = \_[p N]{}, where $$L_{p}(s, \Sym^2 f):=\left(1-\frac{\alpha_{p}^2}{p^{s}}\right)\left(1-\frac{\alpha_{p} \beta_{p}}{p^{s}}\right)\left(1-\frac{\beta_{p}^2}{p^{s}}\right).$$ The product (\[Ep\]) converges absolutely for ${\operatorname{Re}}s>\kappa$. It is well-known that $L^N(s, \Sym^2 f)$ admits meromorphic continuation to the entire complex plane with possible poles only at $s = \kappa$ and $\kappa - 1$, of order at most one [@ShimuraPLMS75 Theorem 1]. In our case (since $f$ is assumed to have trivial character), the $L$-function does not have a pole at $s=\kappa$ [@ShimuraPLMS75 Theorem 2]. We will continue to denote this extended function by $L^N(s, \Sym^2 f)$. Using that $\alpha_p\beta_p=p^{\kappa-1}$ we conclude that $$\label{form1} \frac{\langle f_p^{\alpha_p}, f_p^{\alpha_p} \rangle_{Np}}{\langle f, f \rangle_{Np}} = \frac{p^2}{p^2-1}\frac{1}{L_p(\kappa,\Sym^2 f)} =\frac{\zeta_p(2)}{L_p(\kappa, \Sym^2 f)},$$ where $\zeta_{p}(s) = 1/(1 - 1/p^s).$ We have $$\langle f_p^{\alpha_p}, f_p^{\alpha_p}\rangle_{Np} = \langle f_p^{\beta_p}, f_p^{\beta_p}\rangle_{Np}.$$ Set $$\langle f,f\rangle^{(p)}_N:= \frac{\langle f, f \rangle_{Np}}{\langle f_p^{\alpha_p}, f_p^{\alpha_p} \rangle_{Np}} = \frac{\langle f, f \rangle_{Np}}{\langle f_p^{\beta_p}, f_p^{\beta_p} \rangle_{Np}}.$$ We will now show that $\langle f,f\rangle^{(p)}_N$ can in some sense be regarded as a ‘local’ (at $p$) period for the symmetric square $L$-function. \[conv1\] The value $L^N(\kappa, \Sym^2 f)$ given by the meromorphic continuation is equal to the conditionally convergent Euler product $$\prod_{p \nmid N} \frac{1}{L_{p}(\kappa, \Sym^2 f)}$$ when we order the factors according to increasing $p$. Let $\phi_f$ be defined as in (\[adelicdef\]) and let $\chi_1(p)=\alpha_p/p^{(\kappa-1)/2}$ and $\chi_2(p)=\beta_p/p^{(\kappa-1)/2}$ be its Satake parameters for $p \nmid N$ as in Section \[sec:adeliccalc\]. For $p \nmid N$ define $$L_{p}(s, \Sym^2 \phi_f):= \left(1-\frac{\chi_1(p)^2}{p^{s}}\right)\left(1-\frac{1}{p^{s}}\right)\left(1-\frac{\chi_2(p)^2}{p^{s}}\right)$$ and note that $L_{p}(s, \Sym^2 \phi_f)=L_{p}(s+\kappa-1, \Sym^2 f)$. Thus the Euler product $$L^N(s, \Sym^2 \phi_f):=\prod_{p \nmid N}\frac{1}{L_{p}(s, \Sym^2 \phi_f)}$$ converges absolutely for ${\operatorname{Re}}s>1$ and inherits all the corresponding properties (in particular the meromorphic continuation and the lack of a pole at $s=1$) from $L^N(s, \Sym^2 f)$. As before we will continue to denote this extended function by $L^N(s, \Sym^2 \phi_f)$. Let $\pi$ be the automorphic representation of $\GL_2(\AQ)$ associated with $\phi_f$. It is known [@GelbartJacquet78 Theorem 9.3] that there exists an automorphic representation $\sigma$ of $\GL_3(\AQ)$ such that the (partial) standard $L$-function $L^N(s, \sigma)$ coincides with $L^N(s, \Sym^2 \pi):=L^N(s, \Sym^2 \phi_f)$. Also $L^N(s, \sigma)$ does not vanish on the line ${\operatorname{Re}}s=1$ by a result of Jacquet and Shalika (see [@JacquetShalika76 Theorem 1]; see also [@KohnenSengupta00]). Finally note that by Lemma \[Ramanujan\], we have $|\chi_1(p)| = |\chi_2(p)| = 1$ if $p \nmid N$. Thus we are in a position to apply Theorem \[ethm\] in the appendix with $K=\bfQ$ and $d=3$ to $L^N(s, \Sym^2 \phi_f)$ and the theorem follows. By Theorem \[conv1\] and (\[form1\]) we have $$\frac{L^N(\kappa, \Sym^2 f)}{\prod_{p \nmid N} \langle f,f\rangle^{(p)}_N} = \zeta^N(2),$$ where the superscript means that we omit the Euler factors at primes dividing $N$, and the product $\prod_{p \nmid N}$ (here and below) is ordered according to increasing $p$. Using [@HidaInvent81 Theorem 5.1] we have $$L^N(\kappa, \Sym^2 f) = \prod_{p\mid N}\left(1-\frac{\lambda_f(p)^2}{p^{\kappa}}\right) \times \frac{2^{2\kappa}\pi^{\kappa+1}}{(\kappa-1)! \delta(N) N \phi(N)}\langle f,f \rangle_N,$$ where $\delta(N)=2$ or 1 according as $N \leq 2$ or not. Using this we obtain the following corollary that can be viewed as a factorization of the ‘global’ period $\langle f,f\rangle_N$ in terms of the ‘local’ periods $ \langle f,f\rangle^{(p)}_N$. We have $$\langle f,f \rangle_N = \frac{(\kappa-1)! \delta(N) N \phi(N)\zeta^N(2)}{2^{2\kappa}\pi^{\kappa+1}} \prod_{p\mid N}\frac{1}{1-\lambda_f(p)^2/p^{\kappa}} \prod_{p \nmid N} \langle f,f\rangle^{(p)}_N.$$ Convergence of Euler products on ${\rm Re}(s) = 1$\ by Keith Conrad$^3$ =================================================== Let $K$ be a number field. A degree $d$ Euler product over $K$ is a product $$L(s) = \prod_{\mathfrak p} \frac{1}{(1 - \alpha_{{\mathfrak p},1}{{\rm N}}{\mathfrak p}^{-s})\cdots (1 - \alpha_{{\mathfrak p},d}{{\rm N}}{\mathfrak p}^{-s})},$$ where $|\alpha_{{\mathfrak p},j}| \leq 1$ for all nonzero prime ideals ${\mathfrak p}$ in the integers of $K$ and $1 \leq j \leq d$. On the half-plane ${\operatorname{Re}}(s) > 1$ this converges absolutely and is nonvanishing. Combining factors at prime ideals lying over a common prime number, $L(s)$ is also an Euler product over ${\mathbf Q}$ of degree $d[K:{\mathbf Q}]$. We want to prove a general theorem about the representability of $L(s)$ by its Euler product on the line ${\operatorname{Re}}(s) = 1$. If $L(s)$ is the $L$-function of a nontrivial Dirichlet character, this is in [@davenport pp. 57–58], [@landau $\S$ 109], and [@mv p. 124] if $s = 1$ and [@landau $\S$ 121] if ${\operatorname{Re}}(s) = 1$. \[ethm\] If $L(s)$ is a degree $d$ Euler product over $K$ and it admits an analytic continuation to ${\operatorname{Re}}(s) = 1$ where it is nonvanishing, then $L(s)$ is equal to its Euler product on ${\operatorname{Re}}(s) = 1$ when factors are ordered according to prime ideals of increasing norm: if ${\operatorname{Re}}(s) = 1$ then $$L(s) = \lim_{x \rightarrow \infty} \prod_{{{\rm N}}{\mathfrak p}\leq x} \frac{1}{(1 - \alpha_{{\mathfrak p},1}{{\rm N}}{\mathfrak p}^{-s})\cdots (1 - \alpha_{{\mathfrak p},d}{{\rm N}}{\mathfrak p}^{-s})}.$$ The proof is based on the following lemma about representability of a Dirichlet series on the line ${\operatorname{Re}}(s) = 1$. \[rthm\] Suppose $g(s) = \sum_{n \geq 1} b_n n^{-s}$ has bounded Dirichlet coefficients. If $g(s)$ admits an analytic continuation from ${\operatorname{Re}}(s) > 1$ to ${\operatorname{Re}}(s) \geq 1$, then $g(s)$ is still represented by its Dirichlet series on the line ${\operatorname{Re}}(s) = 1$. See [@newman]. Here is the proof of Theorem \[ethm\]. We will apply Lemma \[rthm\] to a logarithm of $L(s)$, namely the absolutely convergent Dirichlet series $$(\log L)(s) := \sum_{{\mathfrak p}} \sum_{k \geq 1} \frac{\alpha_{{\mathfrak p},1}^k + \cdots + \alpha_{{\mathfrak p},d}^k}{k{{\rm N}}{\mathfrak p}^{ks}},$$ where ${\operatorname{Re}}(s) > 1$. The coefficient of $1/{{\rm N}}{\mathfrak p}^{ks}$ has absolute value at most $d/k \leq d$, so if we collect terms and write $(\log L)(s)$ as a Dirichlet series indexed by the positive integers, say $\sum_{n \geq 1} c_n/n^s$, then $c_n = 0$ if $n$ is not a prime power and $|c_n| \leq d[K:{\mathbf Q}]$ if $n$ is a prime power. Therefore the coefficients of $(\log L)(s)$ as a Dirichlet series over ${\mathbf Z}^+$ are bounded. Since $L(s)$ is assumed to have an analytic continuation to a nonvanishing function on ${\operatorname{Re}}(s) \geq 1$, $(\log L)(s)$ has an analytic continuation to ${\operatorname{Re}}(s) \geq 1$, so Lemma \[rthm\] implies that $$\label{L1t} (\log L)(s) = \sum_{{\mathfrak p}^k} \frac{\alpha_{{\mathfrak p},1}^k + \cdots + \alpha_{{\mathfrak p},d}^k}{k{{\rm N}}{\mathfrak p}^{ks}}$$ for ${\operatorname{Re}}(s) = 1$, where the terms in the series are collected in order of increasing values of ${{\rm N}}({\mathfrak p}^k)$. Although a rearrangement of terms in a conditionally convergent series can change its value, one particular rearrangement of the series in (\[L1t\]) doesn’t change the sum: $$\label{apt} \sum_{{\mathfrak p}^k} \frac{\alpha_{{\mathfrak p},1}^k + \cdots + \alpha_{{\mathfrak p},d}^k}{k{{\rm N}}{\mathfrak p}^{ks}} = \sum_{{\mathfrak p}}\sum_{k \geq 1} \frac{\alpha_{{\mathfrak p},1}^k + \cdots + \alpha_{{\mathfrak p},d}^k}{k{{\rm N}}{\mathfrak p}^{ks}}$$ when ${\operatorname{Re}}(s) = 1$, where the sum on the left is in order of increasing values of ${{\rm N}}({\mathfrak p}^k)$ and the outer sum on the right is in order of increasing values of ${{\rm N}}({\mathfrak p})$. To prove (\[apt\]), we rewrite it as $$\label{sum2} \sum_{{{\rm N}}({\mathfrak p}^k) \leq x} \frac{\alpha_{{\mathfrak p},1}^k + \cdots + \alpha_{{\mathfrak p},d}^k}{k{{\rm N}}{\mathfrak p}^{ks}} = \sum_{{{\rm N}}({\mathfrak p}) \leq x} \sum_{k \geq 1} \frac{\alpha_{{\mathfrak p},1}^k + \cdots + \alpha_{{\mathfrak p},d}^k}{k{{\rm N}}{\mathfrak p}^{ks}} + o(1)$$ as $x \rightarrow \infty$, and we will prove (\[sum2\]) when ${\operatorname{Re}}(s) > 1/2$, not just ${\operatorname{Re}}(s) = 1$. For ${\operatorname{Re}}(s) = 1$ we can pass to the limit in (\[sum2\]) as $x \rightarrow \infty$ and conclude (\[apt\]). The sum on the right in (\[sum2\]) the sum on the left in (\[sum2\]) is equal to $$\sum_{{{\rm N}}({\mathfrak p}) \leq x} \sum_{\substack{k \geq 2\\ {{\rm N}}({\mathfrak p})^k > x}} \frac{\alpha_{{\mathfrak p},1}^k + \cdots + \alpha_{{\mathfrak p},d}^k}{k{{\rm N}}{\mathfrak p}^{ks}},$$ which is equal to $$\label{bigt} \sum_{\sqrt{x} < {{\rm N}}({\mathfrak p}) \leq x} \sum_{k \geq 2} \frac{\alpha_{{\mathfrak p},1}^k + \cdots + \alpha_{{\mathfrak p},d}^k}{k{{\rm N}}{\mathfrak p}^{ks}} + \sum_{{{\rm N}}({\mathfrak p}) \leq \sqrt{x}} \sum_{\substack{k \geq 3 \\ {{\rm N}}({\mathfrak p})^k > x}} \frac{\alpha_{{\mathfrak p},1}^k + \cdots + \alpha_{{\mathfrak p},d}^k}{k{{\rm N}}{\mathfrak p}^{ks}}.$$ The absolute value of the first sum in (\[bigt\]) is bounded above by $$\begin{aligned} \sum_{\sqrt{x} < {{\rm N}}({\mathfrak p}) \leq x} \sum_{k \geq 2} \left|\frac{\alpha_{{\mathfrak p},1}^k + \cdots + \alpha_{{\mathfrak p},d}^k}{k{{\rm N}}{\mathfrak p}^{ks}}\right| & \leq & \sum_{\sqrt{x} < {{\rm N}}({\mathfrak p}) \leq x} \sum_{k \geq 2} \frac{d}{k{{\rm N}}{\mathfrak p}^{k\sigma}} \ \ \text{ where } \sigma = {\operatorname{Re}}(s) \\ & < & \frac{d}{2}\sum_{\sqrt{x} < {{\rm N}}({\mathfrak p}) \leq x} \sum_{k \geq 2} \frac{1}{{{\rm N}}{\mathfrak p}^{k\sigma}} \\ & = & \frac{d}{2}\sum_{\sqrt{x} < {{\rm N}}({\mathfrak p}) \leq x} \frac{1}{{{\rm N}}{\mathfrak p}^{\sigma}({{\rm N}}{\mathfrak p}^{\sigma} - 1)} \\ & < & \frac{d}{2}\sum_{\sqrt{x} < {{\rm N}}({\mathfrak p}) \leq x} \frac{4}{{{\rm N}}{\mathfrak p}^{2\sigma}} \quad \textup{since ${{\rm N}}(\mathfrak p)^\sigma > \sqrt{2} > \frac{4}{3}$,}\end{aligned}$$ which tends to 0 as $x \rightarrow \infty$ since $\sum_{{\mathfrak p}} 1/{{\rm N}}{\mathfrak p}^{2\sigma}$ converges. The absolute value of the second sum in (\[bigt\]) is bounded above by $$\begin{aligned} \sum_{{{\rm N}}({\mathfrak p}) \leq \sqrt{x}} \sum_{\substack{k \geq 3\\ {{\rm N}}({\mathfrak p})^k > x}} \frac{d}{k{{\rm N}}{\mathfrak p}^{k\sigma}} &< \sum_{{{\rm N}}({\mathfrak p}) \leq \sqrt{x}} \sum_{\substack{k \geq 3\\ {{\rm N}}({\mathfrak p})^k > x}} \frac{d}{\log_{{{\rm N}}{\mathfrak p}}(x){{\rm N}}{\mathfrak p}^{k\sigma}}\\ &= \sum_{{{\rm N}}({\mathfrak p}) \leq \sqrt{x}} \frac{d\log {{\rm N}}({\mathfrak p})}{\log x}\sum_{\substack{k \geq 3 \\ {{\rm N}}({\mathfrak p})^k > x}} \frac{1}{{{\rm N}}{\mathfrak p}^{k\sigma}}.\end{aligned}$$ Letting $n$ be the least integer above $\log_{{{\rm N}}{\mathfrak p}}(x)$, $$\sum_{\substack{k \geq 3 \\ {{\rm N}}({\mathfrak p})^k > x}} \frac{1}{{{\rm N}}{\mathfrak p}^{k\sigma}} = \frac{1/{{\rm N}}({\mathfrak p})^{n\sigma}}{1 - 1/{{\rm N}}({\mathfrak p})^{\sigma}} < \frac{1/x^{\sigma}}{1/4} = \frac{4}{x^{\sigma}},$$ so $$\sum_{{{\rm N}}({\mathfrak p}) \leq \sqrt{x}} \sum_{\substack{k \geq 3 \\ {{\rm N}}({\mathfrak p})^k > x}} \frac{d}{k{{\rm N}}{\mathfrak p}^{k\sigma}} < \sum_{{{\rm N}}({\mathfrak p}) \leq \sqrt{x}} \frac{4d\log {{\rm N}}({\mathfrak p})}{x^{\sigma}\log x} = O\left(\frac{\sqrt{x}}{x^{\sigma}\log x}\right),$$ which tends to 0 as $x \rightarrow \infty$ since $\sigma > 1/2$. Now that we established (\[apt\]), take the exponential of the right side: if ${\operatorname{Re}}(s) = 1$, then $$\prod_{{\mathfrak p}} \exp\left(\sum_{k \geq 1} \frac{\alpha_{{\mathfrak p},1}^k + \cdots + \alpha_{{\mathfrak p},d}^k}{k{{\rm N}}{\mathfrak p}^{ks}}\right) = \prod_{{\mathfrak p}} \frac{1}{(1 - \alpha_{{\mathfrak p},1}{{\rm N}}{\mathfrak p}^{-s})\cdots (1 - \alpha_{{\mathfrak p},d}{{\rm N}}{\mathfrak p}^{-s})},$$ where the products run over ${\mathfrak p}$ in order of increasing norms and the last calculation is justified since $|\alpha_{{\mathfrak p},j}/{{\rm N}}({\mathfrak p})^s| \leq 1/{{\rm N}}({\mathfrak p}) < 1$. Since $L(s) = e^{(\log L)(s)}$ for ${\operatorname{Re}}(s) \geq 1$, by (\[L1t\]) and (\[apt\]) we have $$L(s) = \prod_{{\mathfrak p}} \frac{1}{(1 - \alpha_{{\mathfrak p},1}{{\rm N}}{\mathfrak p}^{-s})\cdots (1 - \alpha_{{\mathfrak p},d}{{\rm N}}{\mathfrak p}^{-s})}$$ for ${\operatorname{Re}}(s) = 1$, where the product is in order of increasing values of ${{\rm N}}{\mathfrak p}$. Let $L(s)$ be the $L$-function of the elliptic curve $y^2 = x^3 - x$ over ${\mathbf Q}$. For ${\operatorname{Re}}(s) > 3/2$ it has an Euler product over the odd primes of the form $$\label{L1p} L(s) = \prod_{p \not= 2} \frac{1}{1 - a_p p^{-s} + p \cdot p^{-2s}} = \prod_{p \not= 2} \frac{1}{(1 - \alpha_{p}p^{-s})(1 - \beta_{p}p^{-s})},$$ where $|\alpha_{p}| = \sqrt{p}$ and $|\beta_{p}| = \sqrt{p}$ for $p \not= 2$. Since $y^2 = x^3 - x$ has CM by ${\mathbf Z}[i]$, $L(s)$ is also the $L$-function of a Hecke character $\chi$ on ${\mathbf Q}(i)$ such that $|\chi((\alpha))| = |\alpha| = |{{\rm N}}(\alpha)|^{1/2}$ for all nonzero $\alpha$ in ${\mathbf Z}[i]$ with odd norm. Therefore $L(s)$ also has an Euler product over the nonzero prime ideals of ${\mathbf Z}[i]$ of odd norm: for ${\operatorname{Re}}(s) > 3/2$, $$\label{L2p} L(s) = \prod_{(\pi) \not= (1+i)} \frac{1}{1 - \chi(\pi)/{{\rm N}}(\pi)^s}.$$ The function $L(s)$ is entire and is nonvanishing on the line ${\operatorname{Re}}(s) = 3/2$, so $L(s + 1/2)$ fits the conditions of Theorem \[ethm\] using $K = {\mathbf Q}$ and $d = 2$ for (\[L1p\]), and $K = {\mathbf Q}(i)$ and $d = 1$ for (\[L2p\]). Therefore (\[L1p\]) and (\[L2p\]) are both true on the line ${\operatorname{Re}}(s) = 3/2$. For instance, $L(3/2) \approx .826348$, the partial Euler product for (\[L1p\]) at $s = 3/2$ over prime numbers up to 100,000 is $\approx .826290$, and the partial Euler product for (\[L2p\]) at $s = 3/2$ over nonzero prime ideals in ${\mathbf Z}[i]$ with norm up to 100,000 is $\approx .826480$. [^1]: The first author was partially supported by the National Security Agency under Grant Number H98230-11-1-0137. The United States Government is authorized to reproduce and distribute reprints not-withstanding any copyright notation herein. The second author was partially supported by a PSC-CUNY Award, jointly funded by The Professional Staff Congress and The City University of New York.

--- abstract: 'This is a summary of how the definition of quantum singularity is extended from static space-times to conformally static space-times. Examples are given.' address: - | Department of Mathematics, U.S. Naval Academy\ Annapolis, Maryland, 21012, USA\ E-mail: dak@usna.edu - | Department of Physics, Harvey Mudd College\ Claremont, California, 91711, USA\ E-mail: helliwell@HMC.edu author: - 'DEBORAH A. KONKOWSKI' - 'THOMAS M. HELLIWELL' bibliography: - 'ws-pro-sample.bib' title: 'A SUMMARY: QUANTUM SINGULARITIES IN STATIC AND CONFORMALLY STATIC SPACETIMES' --- Introduction ============ The question addressed in this review is: What happens if instead of classical particle paths (time-like and null geodesics) one uses quantum mechanical particles to identify singularities in conformally static as well as static spacetimes? A summary of the answer is given together with a couple of example applications. This conference proceeding is based primarily on an article by the authors [@HK1]. Types of Singularities ====================== Classical Singularities ----------------------- A classical singularity is indicated by incomplete geodesics or incomplete paths of bounded acceleration [@HE; @Geroch] in a maximal spacetime. Since, by definition, a spacetime is smooth, all irregular points (singularities) have been excised; a singular point is a boundary point of the spacetime. There are three different types of singularity [@ES]: quasi-regular, non-scalar curvature and scalar curvature. Whereas quasi-regular singularities are topological, curvature singularities are indicated by diverging components of the Riemann tensor when it is evaluated in a parallel-propagated orthonormal frame carried along a causal curve ending at the singularity. Quantum Singularities --------------------- A spacetime is QM (quantum-mechanically) nonsingular if the evolution of a test scalar wave packet, representing the quantum particle, is uniquely determined by the initial wave packet, manifold and metric, without having to put boundary conditions at the singularity[@HM]. Technically, a static ST (spacetime) is QM-singular if the spatial portion of the Klein-Gordon operator is not essentially self-adjoint on $C_{0}^{\infty}(\Sigma)$ in $L^2(\Sigma)$ where $\Sigma$ is a spatial slice. This is tested (see, e.g., Konkowski and Helliwell [@HK1]) using Weyl’s limit point - limit circle criterion [@RS; @Weyl] that involves looking at an effective potential asymptotically at the location of the singularity. Here a limit-circle potential is quantum mechanically singular, while a limit-point potential is quantum mechanically non-singular. This definition of quantum singularity has been utilized in the analysis of several timelike spacetime singularities; three examples by the authors include asymptotically power-law spacetimes [@HK3], spacetimes with diverging higher-order curvature invariants [@HK1], and a two-sphere singularity [@HK5]. Conformally Static Space-Times ============================== The Klein-Gordon with general coupling of a scalar field to the scalar curvature is given by $$(\Box - \xi R)\Phi=M^2\Phi$$ where $M$ is the mass if the scalar particle, $R$ is the scalar curvature, and $\xi$ is the coupling ($\xi=0$ for minimal coupling and $\xi=1/6$ for conformal coupling). Using the natural symmetry of conformally static space-times the radial equation easily separates allowing it to be put into so-called Schrödinger form to identify the potential, allowing easy analysis of the quantum singularity structure. Friedmann-Robertson-Walker Space-Times with Cosmic String ========================================================= A metric modeling a Friedmann-Robertson-Walker cosmology with a cosmic string [@DS] can be written as $$ds^2= a^2(t)( -dt^2 + dr^2 + \beta^2 r^2 d\phi^2 +dz^2)$$ where $\beta=1-4\mu$ and $\mu$ is the mass per unit length of the cosmic string. This metric is conformally static (actually conformally flat). Classically it has a scalar curvature singularity times when $a(t)$ is zero and a quasiregular singularity when $\beta^2\neq1$. We focus on resolving the timelike quasiregular singularity. For the quantum analysis [@HK1], the Klein-Gordon equation with general coupling can be separated into mode solutions [^1] with the radial equation changed to Schrödinger form, $$u'' + (E - V(x))u = 0$$ where $E$ is a constant, $x=r$, and the potential $$V(x) = \frac{m^2 - \beta^2 /4}{\beta^2 x^2}$$ . Near zero one can show that the potential $V(x)$ is limit point if $m^2/\beta^2 \geq 1$. So any modes with sufficiently large $m$ are limit point, but $m=0$ is limit circle and thus generically this conformally static space-time is quantum mechanically singular. Roberts Solution ================ The Roberts metric [@Roberts] $$ds^2 = e^{2t}(-dt^2 + dr^2 + G^2(r) d\Omega^2)$$ where $G^2(r) = 1/4[ 1+ p - (1 -p) e^{-2r}]( e^{2r} - 1)$ is conformally static, self-similar, and spherically symmetric. It has a classical scalar curvature singularity at $r = 0$ for $0 < p < 1$ that is timelike. The massive minimally coupled Klein-Gordon equation can be separated into mode solutions and by changing both dependent and independent variables ($r=x$), we get an appropriate inner product and a one-dimensional Schrödinger equation similar to Eq.(3) where again $E$ is a constant but, here, near zero, $V(x)$ goes like $-1/4x^2 < 3/4x^2$ so the potential is limit circle and there is a quantum singularity [@HK1]. Acknowledgments {#acknowledgments .unnumbered} =============== One of us (DAK) thanks B. Yaptinchay for discussions. [9]{} T.M. Helliwell and D.A. Konkowski, [*Int. J. Mod. Phys. A*]{} [**26**]{}, 3878 (2011). S.W. Hawking and G.F.R. Ellis, [*The Large-Scale Structure of Spacetime*]{} (Cambridge University Press, 1973). R. Geroch, [*Ann. Phys.*]{} [**48**]{}, 526 (1968) G.F.R. Ellis and B.G. Schmidt, [*Gen. Rel. Grav.*]{} [**8**]{}, 915 (1977). G.T. Horowitz and D. Marolf, [*Phys. Rev. D*]{} [**52**]{}, 5670 (1995). M. Reed and B. Simon, [*Functional Analysis*]{} (Academic Press, 1972); M. Reed and B. Simon, [*Fourier Analysis and Self-Adjointness*]{} (Academic Press, 1972). H. Weyl, [*Math. Ann.*]{} [**68**]{}, 220 (1910). T.M. Helliwell and D.A. Konkowski, [*Class. Quantum Grav.*]{} [**24**]{}, 3377 (2007). T.M. Helliwell and D.A. Konkowski, [*Gen. Rel. Grav.*]{} [**43**]{}, 695 (2011). P.C.W. Davies and V. Sahni, [*Class. Quantum Grav.*]{} [**5**]{}, 1 (1988). M.D. Roberts, [*Gen. Rel. Grav.*]{} [**21**]{}, 907 (1989). [^1]: Only the time equation contains the coupling constant $\xi$.

--- abstract: 'Ultrashort electron bunches are useful for applications like ultrafast imaging, coherent radiation production, and the design of compact electron accelerators. Currently, however, the shortest achievable bunches, at attosecond time scales, have only been realized in the single- or very few-electron regimes, limited by Coulomb repulsion and electron energy spread. Using *ab initio* simulations and complementary theoretical analysis, we show that highly-charged bunches are achievable by subjecting relativistic (few MeV-scale) electrons to a superposition of terahertz and optical pulses. We provide two detailed examples that use realistic electron bunches and laser pulse parameters which are within the reach of current compact set-ups: one with bunches of $>$ 240 electrons contained within 20 as durations and 15 $\mathrm{\mu}$m radii, and one with final electron bunches of  1 fC contained within sub-400 as durations and 8 $\mathrm{\mu}$m radii. Our results reveal a route to achieve such extreme combinations of high charge and attosecond pulse durations with existing technology.' author: - 'Jeremy Lim[^1]' - Yidong Chong - Liang Jie Wong title: 'Terahertz-optical intensity grating for creating high-charge, attosecond electron bunches' --- =4 , Introduction ============ Electron bunches of femtosecond-to-attosecond-scale duration are useful tools for studying ultrafast atomic-scale processes, including structural phase transitions in condensed matter [@Siwick2003An-Atomic-Level; @Baum20074D-Visualizatio; @Morrison2014A-photoinduced-; @Gedik2007Nonequilibrium-; @Musumec2010Capturing-ultra; @ScianiMiller2011], sub-cycle changes in oscillating electromagnetic waveforms [@Ryabov2016Electron-micros], and the dynamics of biological structures [@Fitzpatrick2013Exceptional-rig; @Anthony-W.-P.-Fitzpatrick20134D-Cryo-Electro]. High-density electron bunches of sub-femtosecond durations are potentially useful in high-resolution, time-resolved atomic diffraction [@Baum2017NatPhys], as sources of extreme-ultraviolet radiation through inverse Compton scattering [@PhysRevLett.104.234801; @PhysRevSTAB.14.070702; @Kiefer_rel_elec_mirrors; @PhysRevLett.119.254801], and as injection bunches for compact charged-particle accelerators [@RJEngland_DLA; @FerrariE_ACHIP]. Existing schemes for electron bunch compression include the use of electrostatic elements [@WangGedikIEEE2012], time-varying fields within radio-frequency (RF) cavities [@Gao12; @vanOudheusden2010PRL; @Chatelain2012; @Kassier2012; @Gliserin2015Sub-phonon-peri], electromagnetic transients [@Baum2007Attosecond-elec; @Hilbert2009Temporal-lenses; @WLJ2015NJP; @PriebeNatPhoton; @KozakNatPhys2017; @PhysRevLett.120.103203; @KealhoferSci2016; @EhbergerOSA2017], and a combination of optical laser pulses and dielectric membranes [@Baum2017NatPhys]. In all of these schemes, space charge effects and velocity spread enforce a tradeoff between electron bunch charge and pulse duration. Consequently, whereas an electron bunch of pulse duration 0.1-1 ps may contain  250 fC [@vanOudheusden2010PRL], electron bunches of attosecond-scale durations (attobunches) are typically realized with single or very few electrons [@Baum2017NatPhys; @PriebeNatPhoton; @KozakNatPhys2017; @PhysRevLett.120.103203; @KealhoferSci2016; @EhbergerOSA2017; @Zewail187; @Aidelsburger2010PNAS]. Here, we use *ab initio* numerical simulations and complementary analytical theory to show that high-charge electron bunches of attosecond-scale durations can be produced by interfering coherent terahertz and optical pulses. We study two regimes of operation: in the first regime, 5 MeV electrons are compressed into attobunches of about $20$ as duration, each containing $\sim 240$ electrons. In the second regime, 5 MeV electrons are compressed into bunches of $<400$ as duration, each containing $\sim 1$ fC of charge. By comparison, theoretical predictions of electron bunch compression using realistic bunches have so far been limited to about 200 as [@KozakNatPhys2017] in the single-electron regime. Experimentally, the shortest electron bunches produced to date lie in the single-electron regime, with durations of 655 as [@PriebeNatPhoton] and 820 as [@Baum2017NatPhys], and indirect measurements indicating durations as short as 260 as [@PhysRevLett.120.103203]. In addition, we obtain fully closed-form expressions for the dynamics of electrons subject to a general combination of counter-propagating pulses. Given a specific initial electron bunch configuration, these analytical tools enable us to predict various key features of our compression scheme, such as the bunch duration at focus (maximum compression), and the final kinetic energy (KE) spread. Our analytical predictions agree well with our *ab initio* numerical simulation results in regimes where space charge effects are negligible. Our work complements existing theoretical formulations for the behavior of charged particles in counter-propagating electromagnetic fields, which have been confined to the sub-relativistic regime [@Hilbert2009Temporal-lenses; @PhysRevA.98.013407]. In the proposed scheme, shown in figure \[Fig\_01\](a), the counter-propagating terahertz and optical pulses interfere to form an intensity grating, which is velocity-matched to the relativistic (few MeV-scale) electrons by choosing the proper carrier frequency for each pulse. The ponderomotive force, which is proportional to the negative intensity gradient, compresses the electrons into a train of attobunches. Bunch compression schemes based on intensity gratings have previously been studied only in the regime where both electromagnetic pulses are at optical/infrared frequencies for applications like electron acceleration [@Hafizi1997Vacuum-beat-wav; @Kozak2015Electron-accele; @Esarey1995PRE], and the compression of non-relativistic, single and few-electron bunches [@Baum2007Attosecond-elec; @Hilbert2009Temporal-lenses; @KozakNatPhys2017; @PhysRevLett.120.103203]. Here, we show that combining terahertz frequencies with optical frequencies creates an intensity grating that can be used to compress relativistic electron pulses achievable in lab-scale setups [@Maxson2017Direct-Measurem; @fs_time_res_diffraction; @SLAC_MeV_rev_sci_instr; @C4FD00204K] to attosecond scale durations with as much as 1 fC of charge per attobunch. We use counter-propagating terahertz pulses of $<100~\mathrm{\mu}$J and optical pulses of $<100$ mJ, which are readily obtained with today’s technology [@Yeh2007Generation-of-1; @single_cycle_1THz; @OR_organic_crystals; @THz_DFG; @HuangOptLett2013; @Dhillon2017a; @Fulop_THz_OR; @Fulop_mJ_THz; @THz_0.4mJ; @THz_0.9mJ]. The absence of material structures in the interaction region of this scheme removes the possibility of material damage, allowing the intensity of our lasers to be scaled to arbitrarily high values for rapid focusing and strong compression of relativistic electron bunches. Due to the suppression of space charge effects at relativistic energies [@HastingsApplPhysLett2006; @MusumeciApplPhysLett2010], the resulting attobunches can hold substantially higher charge than existing attobunches in the non-relativistic, single-electron regime [@Baum2017NatPhys; @KozakNatPhys2017; @PhysRevLett.120.103203]. Our *ab initio* simulations (as described in the next section) exactly model the interactions of electrons with each other as well as with external laser fields. In particular, our simulations account for both near-field and far-field space charge effects, where near-field refers to fields responsible for the Coulomb force, and far-field refers to fields associated with radiation from the electron. We model the external laser fields using exact, finite-energy, non-paraxial solutions to Maxwell’s equations. This is critical for accuracy since terahertz pulses from compact sources usually operate in the near-single-cycle limit and have beam waists tightly focused down to wavelength-scale dimensions [@single_cycle_1THz] in order to achieve desired on-axis field strengths. Results ======= High-charge attosecond electron bunches --------------------------------------- Figure \[Fig\_01\] presents results from 2 regimes of our study: (i) a regime where $\sim 20$ as electron bunch durations containing 246 electrons are realized and (ii) a regime where $<400$ as electron bunch durations are realized with fC-scale charge per bunch. The durations of the compressed bunches are stated using full width at half maximum (FWHM) values. In all simulation results presented in this section, the optical (co-propagating) and terahertz (counter-propagating) pulses have central wavelengths of $\lambda_{1} = 0.65~\mathrm{\mu m}$ and $\lambda_{2} = 300~\mathrm{\mu{m}}$ respectively, are linearly-polarized in $x$, and propagate in the $\pm z$ direction. The electron bunches have a mean KE of $\langle \mathrm{KE}\rangle =5$ MeV. The velocity of the intensity grating, $v_{gr}$, is matched to the mean velocity of the electrons, $v_{0}$, by choosing wavelengths, $\lambda_{1}$ and $\lambda_{2}$, such that [@Baum2007Attosecond-elec; @Esarey1995PRE]: $$v_{gr} = v_{0} = c\bigg{(} \frac{\lambda_{2} - \lambda_{1}}{\lambda_{1}+\lambda_{2}} \bigg{)} \label{eqn_standing_wave_condition}$$ where $c$ is the speed of light in free space. In the lab frame, the grating period is given by $$\lambda_{gr} = \frac{\lambda_{1}}{2\gamma_{0}^{2}(1 - \beta_{0})} = \frac{\lambda_{2}}{2\gamma_{0}^{2}(1 + \beta_{0})}$$ where $\vec{\beta}=\beta_{0}\hat{z}=(v_{0}/c)\hat{z}$ is the normalized mean velocity of the electron bunch propagating in $\hat{z}$. The corresponding Lorentz factor is $\gamma_{0} = 1/\sqrt{1 - \beta_{0}^{2}}$. The mean KE of the electrons is $\langle \mathrm{KE} \rangle = (\gamma_{0} - 1)m_{e}c^{2}$, where $m_{e}$ is the electron rest mass. (\[eqn\_standing\_wave\_condition\]) shows the necessity of combining very disparate counter-propagating laser wavelengths where relativistic electrons are concerned: for $v_{gr}$ close to the speed of light, $v_{0} \sim c$, $\lambda_{2} \gg \lambda_{1}$ is necessary. The use of relativistic electrons takes us into a regime beyond what has been studied for compressing non-relativistic electrons and gives us an opportunity to leverage the developments of high-intensity terahertz pulses in combination with optical pulses in our scheme. Figures \[Fig\_01\](b) and \[Fig\_01\](d) show the electron density distribution obtained by averaging over 300 sets of *ab initio* simulations using an initial 5 MeV electron bunch containing 2 fC of charge uniformly distributed across 10 grating periods. After the laser-electron interaction, each resulting attobunch has about 1250 electrons contained within each $\lambda_{gr}$, and 246 electrons within the FWHM duration of 20 as. The non-paraxial optical and terahertz pulses have energies of 90.3 mJ and 39.0 $\mu$J respectively. The optical pulse has a duration of 80 fs (intensity FWHM) and a peak on-axis field strength $E_{01} \approx 4.96\times 10^{10}$ V/m. The terahertz pulse has a 1 ps duration (intensity FWHM) and a peak on-axis field strength $E_{02}\approx 2.95\times10^{8}$ V/m. Both laser pulses have the same waist radius, $w_{0}=450~\mathrm{\mu{m}}$. During interaction, the bunch has a radius of about $15~\mathrm{\mu{m}}$. The initial relative KE spread is $\sigma_{\mathrm{KE}}/\langle\mathrm{KE}\rangle = 10^{-3} \%$. While this value is small, relative KE spreads as low as $\sigma_{\mathrm{KE}}/\langle\mathrm{KE}\rangle = 4\times10^{-4} \%$ have been predicted for existing RF gun set-ups [@PhysRevSTAB.18.120102]. The full set of electron bunch and laser pulse parameters, as well as a plot of the non-paraxial terahertz pulse electric field spatial profile is found in Supporting Information Section S.5(iv). The second scenario, shown in figures \[Fig\_01\](c) and \[Fig\_01\](e), involves compressing an electron bunch of $\langle \mathrm{KE} \rangle=5$ MeV, 20 fC (total charge), 16.5 fs FWHM duration, relative energy spread $\sigma_{\mathrm{KE}}/\langle \mathrm{KE} \rangle\approx 0.146 \%$ and $8~\mathrm{\mu{m}}$ bunch radius, into a train of sub-400 as duration, fC-scale electron bunches. The electron density heatmap and distribution are averaged over 200 sets of *ab initio* simulation results. The initial electron bunch was modelled after the bunch experimentally demonstrated in [@Maxson2017Direct-Measurem] (see Supporting Information Section S.5(v)). Both pulsed lasers have the same beam waist: $w_{0} = 200~\mathrm{\mu{m}}$. The optical pulse has a duration of 30 fs (intensity FWHM) and an on-axis peak field strength $E_{01} \approx 5\times10^{10}$ V/m, corresponding to a pulse energy of 6.66 mJ. The terahertz pulse (figure S10 in Supporting Information Section S.5(v)) has a duration of 1 ps (intensity FWHM) and an on-axis peak field strength $E_{02}\approx4.18\times10^8$ V/m, corresponding to a pulse energy of 16.9 $\mu$J. Such optical and terahertz pulses are readily achievable today in a table-top setup [@Yeh2007Generation-of-1; @single_cycle_1THz; @OR_organic_crystals; @THz_DFG; @HuangOptLett2013; @Dhillon2017a; @Fulop_THz_OR; @Fulop_mJ_THz; @THz_0.4mJ; @THz_0.9mJ]. At the focus, we observe the formation of electron bunches with about 1 fC of charge in a FWHM duration of 367 as (figure \[Fig\_01\](e)). Figures \[Fig\_02\](a) and \[Fig\_02\](c) show the transverse dynamics induced by the intensity grating for the case studied in figures \[Fig\_01\](b) and \[Fig\_01\](d) while and those in figures \[Fig\_02\](b) and \[Fig\_02\](d) correspond to the case studied in figures \[Fig\_01\](c) and \[Fig\_01\](e). The evolution of $\sigma_{x}$ and $\sigma_{\gamma\beta_{x}}$ with space-charge effects, but without laser-electron interaction, has been plotted using red dotted lines. It can be seen that the laser interaction imparts large transverse momenta in $x$ only during the time of interaction (vertical dotted line labeled “OL”), but long after interaction, the transverse dynamics are practically indistinguishable from the case with no electron-intensity grating interaction. For the case shown in figure \[Fig\_02\](c), the compression is strong enough such that maximum compression (vertical dashes labeled “MC”) occurs before the grating has completely faded. Thus, the transverse momentum spread is still significant ($\sigma_{\gamma\beta_{x}} = 1.28\times10^{-3}$) compared to the case without the grating ($\sigma_{\gamma\beta_{x}} = 0.11\times10^{-3}$). For the case shown in figure \[Fig\_02\](d), maximum compression is attained just after the intensity grating has faded. Hence, the transverse momentum spread at maximum compression ($\sigma_{\gamma\beta_{x}} = 2.11\times10^{-3}\%$) is similar to the case where there is no electron-grating interaction ($\sigma_{\gamma\beta_{x}} = 2.01\times10^{-3}$). Hence, when low transverse momentum spread and bunch expansion is desired, care should be taken to ensure maximum compression is attained long after the intensity grating has faded. Figures \[Fig\_02\](e) and \[Fig\_02\](f) show the achievable electron bunch duration at the focus and the amount of charge contained within the FWHM duration as a function of initial electron KE spread for a fixed amount of total charge. The laser pulse and electron bunch parameters used (except for the charge amount and initial KE spread) are the same as those used to produce figures \[Fig\_01\](b) and \[Fig\_01\](d). Figure \[Fig\_02\](e) indicates that with initial relative KE spreads on the order of $0.1\%$, which is achievable with the current state-of-the-art few-MeV beamlines [@Maxson2017Direct-Measurem], compressed bunches of durations on the order of hundreds of attoseconds can already be realized. For initial KE spreads on the order of about $10^{-2}\%$, sub-100 as bunches can be attained, and $\lesssim 10^{-3}\%$ initial KE spread yields bunches which have durations of 20 as and below at the focus. It should be noted that the charge contained within the attobunches can be enhanced by increasing the initial charge values without increasing the attobunch durations significantly (even up to 10 fC) due to the relativistic suppresion of space charge effects at few-MeV electron energies. Figure \[Fig\_02\](g) shows the electron density distribution for all attobunches at the time of maximum compression of the attobunch closest to the grating center (red distribution, values used to plot figure \[Fig\_02\](e)) for the 0.2 fC case. Our results indicate that despite each attobunch having differing focal times which depend on their relative distance from the center of the intensity grating, the final bunch durations across the entire macrobunch are similar, and by appropriate selection of laser pulse durations, the focal times for each attobunch can be controlled (Supporting Information Section S.2). Our results show that a combination of terahertz and optical technologies can be enabling concepts for the realization of high-charge electron bunches of sub-fs durations. Theoretical predictions of key bunch parameters ----------------------------------------------- We now present fully closed-form expressions for the behavior of charged particles subject to a pair of counter-propagating electromagnetic pulses. These expressions, which neglect space charge effects, have been used to predict various key properties of our bunch compression scheme – including the focal time (maximum compression), the bunch duration at focus, and the final KE spread – and show excellent agreement with the results of our *ab initio* simulations in regimes where space charge and non-paraxial laser pulse effects are small (see figure \[Fig\_03\]). We start from the Newton-Lorentz equations of motion, which describe the dynamics of electrons in arbitrary electromagnetic fields. Treating the counter-propagating laser pulses as pulsed plane waves and considering an electron moving in an arbitrary direction such that the transverse ($x,y$-direction) momenta are small compared to the longitudinal ($z$-direction) momentum, we obtain the normalized electron velocity long after interaction as (see Supporting Information Sections S.1 to S.3 for detailed derivations): $$\beta_{z,f}' \approx \Bigg{(} \sqrt{\frac{\pi}{\alpha_{a}}}\frac{T_{1}'}{\omega'}\cos(\Delta\theta)\frac{e^{2}E_{01}'E_{02}'}{m_{e}^{2}c^{2}}\sin(2k'z_{OLe}' + \phi_{0})\exp\Bigg{\{} \frac{-4(z_{OLe}' - z_{OL}')^{2}}{c^{2}[T_{1}'^{2}(1+\beta_{z,i}')^{2} + T_{2}'^{2}(1 - \beta_{z,i}')^{2}]} \Bigg{\}} \Bigg{)} + \beta_{z,i}' \label{eqn_betafp}$$ where the primes on the variables indicate that they are evaluated in the frame moving at normalized velocity $\vec{\beta}=\beta_{0}\hat{z}$. We define this to be the primed frame. For our electron bunch compression scheme, we take $\beta_{0}$ as the mean normalized velocity of the electron bunch being compressed. $E_{0j}'$ and $T_{j}'$ respectively refer to the electric field amplitude and pulse duration of the laser pulse labelled by subscript $j$, where $j=1$ ($j=2$) refers to the laser pulse which co-propagates (counter-propagates) with respect to the electron bunch. $\omega' = k'c$ is the central angular frequency of the laser pulses (which have the same frequency in the primed frame), $\Delta\theta$ is the relative angle between the polarization vectors associated with the two laser pulses (which we set to $0$ here for the strongest compression), $\phi_{0}$ is a phase constant that depends on the carrier envelope phase of each laser pulse, and $\beta_{z,i}'$ is the initial normalized electron speed. The intensity peaks of the counter-propagating laser pulses overlap at position $z' = z_{OL}'$ and time $t' = t_{OL}'$, and we define the longitudinal electron position at the time $t' = t_{OL}'$ to be $z_{OLe}'$ in the limit where the laser field strengths go to zero. $\alpha_{a}$ is defined as $$\alpha_{a} \equiv (1 - \beta_{z,i}')^{2} + \frac{T_{1}'^{2}}{T_{2}'^{2}}(1+\beta_{z,i}')^{2}.$$ We also obtain the corresponding electron position long after interaction as $$\begin{split} z_{f}'(t') &= \beta_{z,f}'ct' + z_{OLe}' - \beta_{z,i}'ct_{OL}'\\ &+\Bigg{(} \frac{\alpha_{b}}{\alpha_{a}}\sqrt{\frac{\pi}{\alpha_{a}}}\frac{T_{1}'}{2\omega'}\cos(\Delta\theta)\frac{e^{2}E_{01}'E_{02}'}{m_{e}^{2}c}\sin(2k'z_{OLe}' + \phi_{0}) \exp\Bigg{\{} \frac{-4(z_{OLe}' - z_{OL}')^{2}}{c^{2}[T_{1}'^{2}(1 + \beta_{z,i}')^{2} + T_{2}'^{2}(1 - \beta_{z,i}')^{2}]} \Bigg{\}} \Bigg{)} \end{split} \label{eqn_zfp}$$ where $$\alpha_{b} = \frac{2}{c}\bigg{\{} \frac{T_{1}'^{2}}{T_{2}'^{2}}(1+\beta_{z,i}')[z_{OLe}' - z_{OL}' - (1+\beta_{z,i}')ct_{OL}'] -(1 - \beta_{z,i}')[z_{OLe}' - z_{OL}' + (1 - \beta_{z,i}')ct_{OL}'] \bigg{\}}.$$ When the bunch has vanishing longitudinal velocity spread, i.e. $\beta_{z,i}'=0$, the general expression for the focal time, defined as the time between $t_{OL}'$ and the electrons reaching maximum compression, $t_{comp}'$ , is: $$t_{comp}' - t_{OL}' = \frac{m_{e}}{K_{0}'\sqrt{\pi}}\sqrt{\frac{1}{T_{1}'^{2}} + \frac{1}{T_{2}'^{2}}}\exp\Bigg{[} \frac{4(z_{OLe}' - z_{OL}')^{2}}{c^{2}(T_{1}'^{2} + T_{2}'^{2})} \Bigg{]} + \frac{z_{OLe}' - z_{OL}'}{c}\Bigg{(} \frac{T_{2}'^{2} - T_{1}'^{2}}{T_{1}'^{2}+T_{2}'^{2}} \Bigg{)}. \label{eqn_foc_times}$$ Here, $K_{0}' = (2e^{2}E_{01}'E_{02}'\cos\Delta\theta)/(m_{e}c^{2})$. In the special case where we consider the electrons near the center of the intensity grating ($z_{OLe}' \approx z_{OL}'$) and $T_{1}' = T_{2}' = T'$,$E_{01}' = E_{02}' = E_{0}'$ , (\[eqn\_foc\_times\]) reduces to $$t_{comp}' - t_{OL}' = \Bigg{(} \frac{m_{e}}{K_{0}'}\sqrt{\frac{2}{\pi}}\Bigg{)}\frac{1}{T'}$$ which agrees with the analytical result obtained in [@Hilbert2009Temporal-lenses], modulo a factor of $\sqrt{2/\pi}$ which comes from our choice of a Gaussian pulsed profile. The overlap of the optical and terahertz pulses results in a finite-length intensity grating in which electrons farther from the center of the intensity grating generally experience a weaker compressive force. This effect is taken into account through the exponential factors in (\[eqn\_betafp\]) and (\[eqn\_zfp\]), as well as through $\alpha_{b}$. The results in figure \[Fig\_03\] show the excellent agreement between our analytical predictions (circles) and numerical results when the laser pulses are modelled as pulsed plane waves (crosses). The discrepancy between the plane wave simulations and the exact numerical results using non-paraxial pulses (triangles) shows the importance of taking into account the transverse profiles of the focused optical and terahertz pulses in our simulations. Nevertheless, we also note that these exact results follow the trend predicted by our theory relatively well in the regime considered in figure \[Fig\_03\]. In figure \[Fig\_03\], the 5 MeV, $10~\mathrm{\mu{m}}$-radius electron bunch was modelled using $3.75\times10^{5}$ particles, and has a uniform random distribution in $z$ over a length of $\lambda_{gr}$. The initial bunch is normally-distributed in $x$ and $y$. The initial momentum spread for all cases is normally-distributed in all directions and isotropic: $\sigma_{\gamma\beta_{x}} = \sigma_{\gamma\beta_{y}} = \sigma_{\gamma\beta_{z}}$. We used the following initial relative KE spreads: $\sigma_{\mathrm{KE}}/\langle\mathrm{KE}\rangle= $ 0.02%, 0.06%, 0.10%, and 0.14%. The corresponding momentum spreads are $\sigma_{\gamma\beta_{i}}=1.9615\times10^{-3}$, $5.8848\times10^{-3}$, $9.8075\times10^{-3}$, and $1.3731\times10^{-2}$ respectively ($i\in\{x,y,z\}$). All electron bunch and laser pulse parameters are listed in Supporting Information Section S.5(vi). Figures \[Fig\_03\](a) and \[Fig\_03\](d) show that a larger initial electron bunch KE spread makes it more difficult to compress the bunch unless higher laser field strengths field strengths are used. Figure \[Fig\_03\] thus highlights the importance of low energy spread in realizing attosecond bunches. As seen in figure \[Fig\_03\](a), a change in relative initial KE spread from 0.02% to 0.14% can cause the electron bunch durations at the focus to increase by almost an order of magnitude. In the limit where $E_{01}E_{02}$ is small, we see from figure \[Fig\_03\](a)-(c) that it is possible to obtain fs-scale electron bunches with a very small (practically negligible) change in energy spread, at the cost of a longer focal time. In figure \[Fig\_03\](b), the decrease in the focal time approximately as $1/E_{01}'E_{02}' = 1/E_{01}E_{02}$ agrees with the trend predicted by (\[eqn\_foc\_times\]). Using the formalism described here, the predicted durations and focal times for the cases shown in figures \[Fig\_01\](b)-\[Fig\_01\](e) are also in good agreement with our *ab initio* simulations. For the case shown in figures \[Fig\_01\](b) and \[Fig\_01\](d), the predicted FWHM duration for both the left and right attobunches is $9$ as, which is a good estimate of the numerically computed values of $21$ as and $20$ as; the theoretical time of maximum compression is $0.123$ ns, which is very close to the actual value of $0.127$ ns. For the case shown in figures \[Fig\_01\](c) and \[Fig\_01\](e), the theoretically predicted durations of the left and right attobunches are $352$ as and $338$ as respectively while the numerically computed durations are $391$ as and $367$. The theoretical time of maximum compression is $0.032$ ns, which is very close to the actual value of $0.033$ ns. Discussion ========== Here, we present an overview of the electron kinetic energies which can be matched using sources of coherent light at various wavelengths, as well as a brief comparison between our scheme and existing electron bunch compression schemes. The interest in working with electrons of larger kinetic energies is due to the relativistic suppression of space charge effects, which allows shorter bunch durations to be achieved in this compression scheme. The development of intense, coherent terahertz sources on a table-top scale [@Yeh2007Generation-of-1; @single_cycle_1THz; @OR_organic_crystals; @THz_DFG; @HuangOptLett2013; @Dhillon2017a; @Fulop_THz_OR; @Fulop_mJ_THz; @THz_0.4mJ; @THz_0.9mJ] as figure \[Fig\_04\] shows, unlocks a range of electron kinetic energies spanning 4 orders of magnitude (keV to 10 MeV). By contrast, using only wavelengths falling in the optical to near-infrared regime (0.4 $\mathrm{\mu{m}}$ to 1.4 $\mathrm{\mu{m}}$) would limit us to electron kinetic energies of 100 keV or less. Although the mechanism here can be extended to electron kinetic energies on the order of $10^{2}$ MeV and higher, much larger laser intensities would be involved for effective compression. The study of the use of this mechanism for such ultrarelativistic electrons is beyond the scope of this work. We note that alternative techniques for producing highly-compressed electron bunches of kinetic energies from tens-of-MeV to GeV include the use of compact inverse free electron laser systems [@PRL_hightrapping_efficiency] and undulator modulators [@PRL_single_cycle_XUV] have already been demonstrated and proposed. These methods may be more practical when larger dedicated accelerator facilities are available. However, for compact acceleration schemes such as dielectric laser acceleration [@RJEngland_DLA], the ability to produce fC-scale, few-MeV electron bunches modulated to sub-fs scales as injection sources, like those presented in this work, are of interest. A number of laser-based sources of intense terahertz radiation, suitable for the use in the present compression scheme, as well as other forms of charged particle manipulation, have already been reported in the literature. Single-cycle and quasi-single-cycle terahertz radiation centered at 1 to 2 THz with peak field strengths on the order of 1 MV/cm have been achieved using optical rectification of LiNbO3 with tilted pulse front pumping [@Yeh2007Generation-of-1; @single_cycle_1THz] and optical rectification of organic crystals with high non-linear constants [@OR_organic_crystals]. Semiconductors have also been shown to be a promising alternative for generating high-energy THz pulses using this technique [@Fulop_THz_OR]. Terahertz pulse energies on the order of tens of $\mu$J are already routinely produced [@Dhillon2017a] from these compact sources and energies up to 1 mJ [@THz_0.9mJ; @THz_0.4mJ] have already been demonstrated. With the high field strengths accompanying these high pulse energies, shorter attobunch durations and focusing times can be achieved, which our results in figure \[Fig\_03\] predicts. The development of compact THz sources of higher energies, would alleviate the need for extremely tight-focusing of the THz pulse in order to achieve the desired field strengths. Difference frequency generation (DFG) of optical parameteric amplifiers have been used to produced narrow-band, multi-cycle pulses at mid-infrared frequencies (15-30 terahertz) and higher fields strengths of 100 MV/cm [@THz_DFG]. While ultra-broadband terahertz radiation can be produced using plasma ionization [@THz_plasma_ionization], the field strengths are typically lower than those achieved using optical rectification. However, they could potentially be used for the compression of low-charge or single-electron bunches with small energy spreads over longer focal distances. We note that greater flexibility in our choice of wavelength for matching a given electron kinetic energy can be achieved by tilting the counter-propagating pulses [@Hilbert2009Temporal-lenses; @KozakNatPhys2017; @PhysRevLett.120.103203; @Kozak2015Electron-accele]. In this case, however, too large a tilt angle will lead to restrictions on the transverse size of the electron bunch. Nevertheless, the concept of tilting laser pulses could be implemented in the terahertz-optical scheme to accommodate an even wider range of electron kinetic energies. Dielectric membranes, in combination with an optical laser pulse, have been used to compress non-relativistic (70 keV), single-electron bunches to attosecond-scale durations [@Baum2017NatPhys]. When non-relativistic electrons are considered, the laser field strength required to modulate the bunch remains low enough to avoid material damage. However, relativistic electron bunches require much higher intensities for compression to attosecond time-scales, making material damage more likely. The scheme studied in the present paper allows high-intensity lasers to be used without the risk of material damage. Conclusion ========== We presented a scheme in which counter-propagating terahertz and optical pulses are used to compress relativistic electrons into a train of attosecond-duration bunches. Due to the space-charge suppression at few MeV-scale energies, significant amounts of charge can be contained within each attobunch, compared to previously realized attobunches that have only single or very few electrons. Our *ab initio* simulations take near- and far-field space charge effects (associated with the Coulomb force and the electron radiation respectively) into account, and use exact, non-paraxial pulse profiles to model single-cycle, tightly-focused terahertz pulses; this is a significant advance over previous numerical studies of similar intensity grating compression schemes, which assumed non-interacting electrons and planar or paraxial electromagnetic waves. We presented results for attosecond electron bunch compression in two regimes. The first case involved the compression of a lower-charge electron cloud into attobunches with durations of about $20$ (FWHM), containing about 246 electrons. Such short-duration bunches could be used, for instance, as sources of high-quality coherent radiation through processes like inverse Compton scattering [@Kiefer_rel_elec_mirrors], Smith-Purcell radiation [@Sergeeva2017Smith-Purcell-r], transition radiation [@Zhang2017Transition-radi], and through electron-plasmon scattering [@WLJ_nat_photon_2016; @Rosolen_LightSci_2018]. We find that the realization of this scenario depends on having kinetic energy spreads which are extremely low but feasible [@PhysRevSTAB.18.120102]. In the second, the initial electron bunch contains 20 fC of charge and is comparable to the bunches that can be produced by existing few-MeV scale electron sources. In this case, we showed that the electrons can be compressed into smaller bunches of sub-400 as durations (FWHM), each containing up to 1 fC of charge. Besides electron diffraction applications (e.g. time-resolved atomic diffraction in [@Baum2017NatPhys]), these bunches could potentially serve as pre-accelerated injection sources for compact dielectric laser acceleration (DLA) schemes, in which fC-scale, few-MeV electron bunches are desirable as input [@RJEngland_DLA]. The modulated sub-fs bunches generated by our scheme can fit into the phase space acceleration buckets – typically also of sub-laser wavelength length-scales – which could improve the accelerated beam quality [@RJEngland_DLA]. The sub-micron transverse bunch dimensions required for injection into typical optical DLA schemes can be achieved through the use of electron beam focusing optics. Our results indicate that attosecond-scale electron bunches are not inherently limited to the few-to-single-electron regime, which has been the focus of other studies. We thank the National Supercomputing Center (NSCC) Singapore for the use of their computing resources. LJW acknowledges support from the Science and Engineering Research Council (SERC; grant no. 1426500054) of the Agency for Science, Technology and Research (A\*STAR), Singapore. All authors made critical contributions to this manuscript and declare no competing financial interests. [60]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} [****,  ()](\doibase 10.1063/1.3520283) [****,  ()](http://stacks.iop.org/0034-4885/74/i=9/a=096101) @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} [****,  ()](\doibase 10.1103/PhysRevLett.104.234801) [****,  ()](\doibase 10.1103/PhysRevSTAB.14.070702) [****,  ()](\doibase 10.1038/ncomms2775) [****, ()](\doibase 10.1103/PhysRevLett.119.254801) @noop [****,  ()]{} @noop [****,  ()]{} [****,  ()](\doibase 10.1109/JSTQE.2011.2112339) [****,  ()](\doibase 10.1364/OE.20.012048) [****,  ()](\doibase 10.1103/PhysRevLett.105.264801) [****,  ()](\doibase 10.1063/1.4747155) [****,  ()](\doibase 10.1007/s00340-012-5207-2) @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} [****,  ()](\doibase 10.1103/PhysRevLett.120.103203) @noop [****,  ()]{} in [**](\doibase 10.1364/NLO.2017.NW2A.2) (, ) p.  [****,  ()](\doibase 10.1126/science.1166135) [****,  ()](\doibase 10.1073/pnas.1010165107) [****,  ()](\doibase 10.1103/PhysRevA.98.013407) @noop [****,  ()]{} [****,  ()](http://stacks.iop.org/0953-4075/48/i=19/a=195601) [****,  ()](\doibase 10.1103/PhysRevE.52.5443) [****,  ()](\doibase 10.1103/PhysRevLett.118.154802) [****, ()](http://stacks.iop.org/1367-2630/17/i=6/a=063004) [****,  ()](\doibase 10.1063/1.4926994) [****,  ()](\doibase 10.1039/C4FD00204K) [****,  ()](\doibase 10.1063/1.2734374) [****,  ()](\doibase 10.1063/1.3560062) [****,  ()](\doibase 10.1063/1.3655331) @noop [****,  ()]{} [****,  ()](\doibase 10.1364/OL.38.000796) [****,  ()](http://stacks.iop.org/0022-3727/50/i=4/a=043001) @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} [****,  ()](\doibase 10.1063/1.2372697) @noop [****,  ()]{} [****,  ()](\doibase 10.1103/PhysRevSTAB.18.120102) @noop [****,  ()]{} [****,  ()](\doibase 10.1103/PhysRevLett.113.104801) [****,  ()](\doibase 10.1063/1.2828709) @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} [^1]: Present address: Singapore University of Technology and Design, 8 Somapah road, Singapore 487372, Singapore

--- author: - Andrea Fubini - Tommaso Roscilde - Valerio Tognetti - Matteo Tusa - Paola Verrucchi date: 'Received: date / Revised version: date' title: Reading entanglement in terms of spin configurations in quantum magnets --- [leer.eps]{} gsave 72 31 moveto 72 342 lineto 601 342 lineto 601 31 lineto 72 31 lineto showpage grestore Introduction ============ Entanglement properties have recently entered the tool kit for studying magnetic systems, thanks to the insight they provide on aspects which are not directly accessible through the analysis of standard magnetic observables [@ArnesenBV01; @Osterlohetal02; @OsborneN02; @Ghoshetal03; @Vidaletal03; @Verstraeteetal04; @Roscildeetal04]. The analysis of entanglement properties is particularly indicated whenever purely quantum effects come into play, as in the case of quantum phase transitions. However, in order to gain a deeper insight into quantum criticality, as well as into other phenomena such as field-induced factorization [@Roscildeetal04] and saturation, the connection between magnetic observables and entanglement estimators should be made clearer, a goal we aim at in this paper. On the other hand, most entanglement estimators, as defined for quantum magnetic systems, are expressed in terms of magnetizations and spin correlation functions. It comes therefore natural to wonder where, inside the standard magnetic observables, the information about entanglement is actually stored, and how entanglement estimators can extract it. Quite clearly, by posing this question, one does also address the problem of finding a possible experimental measure of entanglement, which is of crucial relevance in developing possible solid-state devices for quantum computation. In this context a privileged role is played by the concurrence $C$, which measures the entanglement of formation between two q-bits by an expression which is valid not only for pure states but also for mixed ones[@Hilletal97; @Wootters98]. In the framework of interacting spin systems, exploiting different symmetries of such systems the concurrence has been related to spin-spin correlators and to magnetizations.[@Wangetal02; @Syljuasen03; @Glaseretal03; @Amicoetal04] However, $C$ has not yet been given a general interpretation from the magnetic point of view, and a genuinely physical understanding of its expression is still elusive. Scope of this paper is therefore that of giving a simple physical interpretation of bipartite entanglement of formation, building a direct connection between entanglement estimators and occupation probabilities of two-spin states in an interacting spin system. To this purpose we develop a general formalism for analyzing the spin configuration of the system, so as to directly relate it with the expression of the concurrence. The resulting equations are then used to read our data relative to the $S=1/2$ antiferromagnetic Heisenberg model in a uniform magnetic field, both on a chain and on a two-leg ladder. The model is a cornerstone in the study of magnetic systems, extensively investigated and quite understood in the zero-field case. When a uniform magnetic field is applied the behavior of the system is enriched, gradually transforming its ground state and thermodynamic behavior. The analysis of entanglement properties in this model, and in particular that referring to the range of pairwise entanglement as field increases, sheds new light not only on the physical mechanism leading to magnetic saturation in low-dimensional quantum systems, but also on the nature of some $T=0$ transitions observed in bosonic and fermionic systems, such as that of hard-core bosons with Coulomb interaction, and that described by the bond-charge extended Hubbard model, respectively. In the former case, the connection between magnetic and bosonic model is obtained by an exact mapping that allows a straightforward generalization of our results to the discussion of the phase diagram of the strongly interacting boson-Hubbard model [@BruderFS93]. In the more complex case of the the bond-charge extended Hubbard model, a direct connection between the Heisenberg antiferromagnet in a field is not formally available, but a recent work by Anfossi [*et al.*]{}[@Anfossietal05] has shown that some of the $T=0$ transitions observed in the system are characterized by long-ranged pairwise entanglement of the same type we observe in our magnetic model at saturation. Our data result from stochastic series expansion (SSE) quantum Monte Carlo simulations based on the directed-loop algorithm[@SyljuasenS02]. The calculations were carried on a chain with size $L=64$ and on a $L\times 2$ ladder with $L=40$. In order to capture the ground-state behavior we have considered inverse temperatures $\beta=2L$. In Sec.\[s.magtoprob\] we define the magnetic observables we refer to, and develop the formalism which allows us to write them in terms of probabilities for two spins to be in specific states, both at zero and at finite temperature. In Sec. \[s.spinconftoentanglement\] we show how concurrence extracts, out of the above probabilities, the specific information on bipartite entanglement of formation. In Secs. \[s.chain\] and \[s.ladder\] we present our SSE data for the antiferromagnetic Heisenberg model on a chain and on a square ladder respectively, and read them in light of the discussion of Secs. \[s.magtoprob\] and \[s.spinconftoentanglement\]. Conclusions are drawn in Sec. \[s.conclusions\]. From magnetic observables to spin configurations {#s.magtoprob} ================================================ We study a magnetic system made of $N$ spins $S=1/2$ sitting on a lattice. Each spin is described by a quantum operator ${\bm S}_l$, with $[S^\alpha_l,S^\beta_m]= i\delta_{lm}\varepsilon_{\alpha\beta\gamma}S^\gamma_l$, $l$ and $m$ being the site-indexes. The magnetic observables we consider are the local magnetization along the quantization axis: $$M_l^z{\equiv}\left\langle S^z_l\right\rangle \label{e.Mz}~,$$ and the correlation functions between two spins sitting on sites $l$ and $m$: $$g_{lm}^{\alpha\alpha} \equiv \left\langle S^\alpha_l S^\alpha_m\right\rangle \label{e.galphaalpha}~.$$ The averages $\left\langle~\cdot~\right\rangle$ represent expectation values over the ground state for $T=0$, and thermodynamic averages for $T>0$. We now show that the above single-spin and two-spin quantities provide a direct information on the specific quantum state of any two spins of the system. Let us consider the $T=0$ case first: For a lighter notation we drop site-indexes, allowing their appearance whenever needed. After selecting two spins, sitting on sites $l$ and $m$, any pure state of the system may be written as $$|\Psi\rangle=\sum_{\nu\in {\cal S}}|\nu\rangle\sum_{\Gamma\in {\cal R}} c_{\nu_\Gamma}|\Gamma\rangle~, \label{e.Psi}$$ where ${\cal S}$ is an orthonormal basis for the $4$-dimensional Hilbert space of the selected spin pair, while ${\cal R}$ is an orthonormal basis for the $2^{N-2}$-dimensional Hilbert space of the rest of the system. Moreover, in order to simplify the notation, we understand products of kets relative to (operators acting on) different spins as tensor products, meanwhile dropping the corresponding symbol $\otimes$. The quantum probability for the spin pair to be in the state $|\nu\rangle$, being the system in the pure state $|\Psi\rangle$, is $p_{\nu}\equiv\sum_\Gamma|c_{\nu_\Gamma}|^2$, and the normalization condition $\langle\Psi|\Psi\rangle=1$ implies $\sum_\nu p_{\nu}=1$. We consider three particular bases for the spin pair: $$\begin{aligned} {\cal{S}}_1 {\equiv}&\{|u_{_{\rm I}}\rangle,|u_{_{\rm I\!I}}\rangle,|u_{_{\rm I\!I\!I}}\rangle,|u_{_{\rm I\!V}}\rangle\}~, \label{e.bstandard}\\ {\cal{S}}_2 {\equiv}&\{|e_1\rangle,|e_2\rangle,|e_3\rangle,|e_4\rangle\}~, \label{e.bBell}\\ {\cal{S}}_3 {\equiv}&\{|u_{_{\rm I}}\rangle,|u_{_{\rm I\!I}}\rangle,|e_3\rangle,|e_4\rangle\}~, \label{e.bmixed}\end{aligned}$$ with $$\begin{aligned} |u_{_{\rm I}}\rangle\equiv|\uparrow\rangle_l|\uparrow\rangle_m~&,~ |u_{_{\rm I\!I}}\rangle\equiv|\downarrow\rangle_l|\downarrow\rangle_m~,\nonumber\\ |u_{_{\rm I\!I\!I}}\rangle\equiv|\uparrow\rangle_l|\downarrow\rangle_m~&,~ |u_{_{\rm I\!V}}\rangle\equiv|\downarrow\rangle_l|\uparrow\rangle_m~,\nonumber\\ |e_1\rangle={\textstyle\frac{1}{\sqrt{2}}} \left(|u_{_{\rm I}}\rangle+|u_{_{\rm I\!I}}\rangle\right)~&,~ |e_2\rangle={\textstyle\frac{1}{\sqrt{2}}} \left(|u_{_{\rm I}}\rangle-|u_{_{\rm I\!I}}\rangle\right)~,\nonumber\\ |e_3\rangle={\textstyle\frac{1}{\sqrt{2}}} \left(|u_{_{\rm I\!I\!I}}\rangle+|u_{_{\rm I\!V}}\rangle\right)~&,~ |e_4\rangle={\textstyle\frac{1}{\sqrt{2}}} \left(|u_{_{\rm I\!I\!I}}\rangle-|u_{_{\rm I\!V}}\rangle\right)~,\label{e.states}\end{aligned}$$ where $|\uparrow\rangle_{l,m}(|\downarrow\rangle_{l,m})$ are eigenstates of $S^z_{l,m}$ with eigenvalue $+{\textstyle\frac{1}{2}}(-{\textstyle\frac{1}{2}})$. For the coefficients entering Eq. (\[e.Psi\]), and for each state $\Gamma$, the following relations hold $$\begin{aligned} c_{_{1\Gamma}}={\textstyle{\frac{1}{\sqrt{2}}}}\left(c_{{{_{\rm I}}}_{\Gamma}} +c_{{{_{\rm I\!I}}}_{\Gamma}}\right)&,& c_{_{2\Gamma}}={\textstyle{\frac{1}{\sqrt{2}}}}\left(c_{{{_{\rm I}}}_{\Gamma}} -c_{{{_{\rm I\!I}}}_{\Gamma}}\right)~, \label{e.c12}\\ c_{_{3\Gamma}}={\textstyle{\frac{1}{\sqrt{2}}}}\left(c_{{{_{\rm I\!I\!I}}}_{\Gamma}} +c_{{{_{\rm I\!V}}}_{\Gamma}}\right)&,& c_{_{4\Gamma}}={\textstyle{\frac{1}{\sqrt{2}}}}\left(c_{{{_{\rm I\!I\!I}}}_{\Gamma}} -c_{{{_{\rm I\!V}}}_{\Gamma}}\right)~, \label{e.c34}\end{aligned}$$ meaning also $$\begin{aligned} |c_{_{1\Gamma}}|^2+|c_{_{2\Gamma}}|^2&=& |c_{{{_{\rm I}}}_{\Gamma}}|^2+|c_{{{_{\rm I\!I}}}_{\Gamma}}|^2~,\label{e.c12+}\\ |c_{_{1\Gamma}}|^2-|c_{_{2\Gamma}}|^2&=& |c_{{{_{\rm I}}}_{\Gamma}}c_{{{_{\rm I\!I}}}_{\Gamma}}| \cos(\varphi^\Gamma_{{{_{\rm I}}}}-\varphi^\Gamma_{{{_{\rm I\!I}}}})~,\label{e.c12-}\\ |c_{_{3\Gamma}}|^2+|c_{_{4\Gamma}}|^2&=& |c_{{{_{\rm I\!I\!I}}}_{\Gamma}}|^2+|c_{{{_{\rm I\!V}}}_{\Gamma}}|^2\label{e.c34+}~,\\ |c_{_{3\Gamma}}|^2-|c_{_{4\Gamma}}|^2&=& |c_{{{_{\rm I\!I\!I}}}_{\Gamma}}c_{{{_{\rm I\!V}}}_{\Gamma}}| \cos(\varphi^\Gamma_{{{_{\rm I\!I\!I}}}}-\varphi^\Gamma_{{{_{\rm I\!V}}}})~, \label{e.c34-}\end{aligned}$$ where $c_{\nu_\Gamma}\equiv|c_{\nu_\Gamma}|e^{i\varphi^\Gamma_\nu}$. According to the usual nomenclature ${\cal{S}}_1$ and ${\cal{S}}_2$ are the [*standard*]{} and [*Bell*]{} bases, respectively, while ${\cal{S}}_3$ is here called the [*mixed*]{} basis. Such bases are characterized by the fact that states corresponding to parallel and antiparallel spins do not mix with each other. It therefore makes sense to refer to $|u_{_{\rm I}}\rangle,|u_{_{\rm I\!I}}\rangle,|e_1\rangle$, and $|e_2\rangle$ as [*parallel states*]{}, and to $|u_{_{\rm I\!I\!I}}\rangle,|u_{_{\rm I\!V}}\rangle,|e_3\rangle$, and $|e_4\rangle$ as [*antiparallel states*]{}. The probabilities specifically related with the elements of ${\cal{S}}_1$ will be hereafter indicated by $p_{{_{\rm I}}}, p_{{_{\rm I\!I}}}, p_{{_{\rm I\!I\!I}}}$, and $p_{{_{\rm I\!V}}}$ while $p_1,p_2,p_3,$ and $p_4$ will be used for those relative to the elements of ${\cal{S}}_2$. From the normalization conditions $$\begin{aligned} &&p_{{_{\rm I}}}+p_{{_{\rm I\!I}}}+p_{{_{\rm I\!I\!I}}}+p_{{_{\rm I\!V}}}=1 \label{e.normstand}\\ &&p_1+p_2+p_3+p_4=1 \label{e.normBell}\\ &&p_{{_{\rm I}}}+p_{{_{\rm I\!I}}}+p_3+p_4=1~, \label{e.normmix}\end{aligned}$$ or equivalently from Eqs. (\[e.c12+\]) and (\[e.c34+\]), follows $p_{{_{\rm I}}}+p_{{_{\rm I\!I}}}=p_1+p_2$, and $p_{{_{\rm I\!I\!I}}}+p_{{_{\rm I\!V}}}=p_3+p_4$, representing the probability for the two spins to be parallel and antiparallel, respectively. We do also notice that the elements of ${\cal{S}}_1$ are factorized states, while those of ${\cal{S}}_2$ are maximally entangled ones. The above description is easily translated in terms of the two-site reduced density matrix $$\rho=\sum_{\Gamma} \langle\Gamma|\Psi\rangle\langle\Psi|\Gamma\rangle= \sum_{\nu\lambda}|\nu\rangle\langle\lambda| \sum_{\Gamma} c_{\nu_\Gamma}c_{\mu_\Gamma}^*~,$$ whose diagonal elements are the probabilities for the elements of the basis chosen for writing $\rho$. The normalization conditions Eqs. (\[e.normstand\]-\[e.normmix\]) translate into ${\rm Tr}~(\rho)=1$. Thanks to the above parametrization, the magnetic observables (\[e.Mz\]) and (\[e.galphaalpha\]) are directly connected to the probabilities of the two spins being in one of the states . In fact it is $$\begin{aligned} 2(g^{xx}+g^{yy})= \langle\Psi\,|S^+_lS^-_m+S^-_lS^+_m|\Psi\rangle & =\nonumber\\ &=\langle\Psi|S^+_lS^-_m+S^-_lS^+_m|\left( |e_3\rangle\sum_\Gamma c_{_{3\Gamma}}|\Gamma\rangle+ |e_4\rangle\sum_\Gamma c_{_{4\Gamma}}|\Gamma\rangle\right)=\nonumber\\ &=\langle\Psi| \left(|e_3\rangle\sum_\Gamma c_{_{3\Gamma}}|\Gamma\rangle- |e_4\rangle\sum_\Gamma c_{_{4\Gamma}}|\Gamma\rangle\right)\nonumber\\ &=(p_3-p_4)~, \label{e.gxx+gyy.p}\end{aligned}$$ and similarly $$\begin{aligned} &&2(g^{xx}-g^{yy})=(p_1-p_2)~, \label{e.gxx-gyy.p}\\ &&g^{zz}={\textstyle\frac{1}{2}} \left(p_{{_{\rm I}}}+p_{{_{\rm I\!I}}}\right)-{\textstyle\frac{1}{4}}= {\textstyle\frac{1}{2}}\left(p_1+p_2\right)-{\textstyle\frac{1}{4}}~, \label{e.gzz.p}\\ &&M_z\equiv{\textstyle\frac{1}{2}}\left(M^z_l+M^z_m\right)= \left(p_{{_{\rm I}}}-p_{{_{\rm I\!I}}}\right)~, \label{e.Mz.p}\end{aligned}$$ where all ${\cal{S}}_i$ are suitable to calculate $g^{zz}$, while $(g^{xx}\pm g^{yy})$ and $M_z$ specifically require ${\cal{S}}_2$ and ${\cal{S}}_3$, respectively. After Eqs. (\[e.gxx+gyy.p\])-(\[e.Mz.p\]), one finds $$\begin{aligned} p_{{_{\rm I}}}&=&\textstyle{\frac{1}{4}}+g^{zz}+M_z~, \label{e.puu}\\ p_{{_{\rm I\!I}}}&=&\textstyle{\frac{1}{4}}+g^{zz}-M_z~, \label{e.pdd}\\ p_1&=&\textstyle{\frac{1}{4}}+g^{xx}-g^{yy}+g^{zz}~, \label{e.p1}\\ p_2&=&\textstyle{\frac{1}{4}}-g^{xx}+g^{yy}+g^{zz}~, \label{e.p2}\\ p_3&=&\textstyle{\frac{1}{4}}+g^{xx}+g^{yy}-g^{zz}~, \label{e.p3}\\ p_4&=&\textstyle{\frac{1}{4}}-g^{xx}-g^{yy}-g^{zz}~. \label{e.p4}\end{aligned}$$ It is to be noticed that the probabilities relative to the Bell states do not depend on the magnetization. In the the finite temperature case, the generalization is straightforwardly obtained by writing each of the Hamiltonian eigenstates, numbered by the index $n$, in the form (\[e.Psi\]), so that $$\rho(T)= \sum_{\nu\mu}|\nu\rangle\langle\mu| \sum_n e^{-E_n/T}\sum_{\Gamma}c_{\nu_\Gamma\!,n}c_{\mu_\Gamma\!,n}^*~. \label{e.rho(t)}$$ In terms of probabilities the above expression simply means that the purely quantum $p_\mu$ shall be replaced by the quantum statistical probabilities $$p_\mu(T)\equiv\sum_n e^{-E_n/T} \sum_{\Gamma}\left|c_{\nu_\Gamma\!,n}\right|^2~. \label{e.p(t)}$$ Therefore, apart from the further complication of the formalism, the discussion developed for pure states stays substantially unchanged when $T>0$. Equations (\[e.puu\])-(\[e.p4\]) show that magnetic observables allow a certain insight into the spin configuration of the system, as they give, when properly combined, the probabilities for any selected spin pair to be in some specific quantum state. However, the mere knowledge of such probabilities is not sufficient to appreciate the quantum character of the global state, and more specifically to quantify its entanglement properties. From spin configurations to entanglement properties {#s.spinconftoentanglement} =================================================== We here analyze the entanglement of formation[@Bennettetal96; @Hilletal97; @Wootters98] between two spins, quantified by the concurrence $C$. In the simplest case of two isolated spins in the pure state $|\phi\rangle$ the concurrence may be written as $C=|\sum_i\alpha_i^2|$, where $\alpha_i$ are the coefficients entering the decomposition of $|\phi\rangle$ upon the magic basis $\{|e_1\rangle,i|e_2\rangle,i|e_3\rangle,|e_4\rangle\}$. However, if one refers to the notation of the previous section, it is easily shown that $$C(|\phi\rangle) = \left|(c_{_1}^2-c_{_2}^2)-(c^2_{_3}-c^2_{_4})\right|= 2\left|c_{{_{\rm I}}}c_{{_{\rm I\!I}}}-c_{{_{\rm I\!I\!I}}}c_{{_{\rm I\!V}}}\right|~, \label{e.Ctwospins}$$ where Eqs. (\[e.c12\])-(\[e.c34\]) have been used, with index $\Gamma$ obviously suppressed. The above expression shows that $C$ extracts the information about the entanglement between the two spins by combining probabilities and phases relative to specific two-spin state. In fact, one should notice that a finite probability for two spins to be in a maximally entangled state does not guarantee *per se* the existence of entanglement between them, since this probability may be finite even if the two spins are in a separable state.[@esempio] In a system with decaying correlations, at infinite separation all probabilities associated to Bell states attain the value of $1/4$, but this of course tells nothing about the entanglement between them, which is clearly vanishing. It is therefore expected that *differences* between such probabilities, rather than the probabilities themselves give insight in the presence or absence of entanglement. When the many-body case is tackled, the mixed-state concurrence of the selected spin pair has an involved definition in terms of the reduced two-spin density matrix.[@Wootters98] However, possible symmetries of the Hamiltonian ${\cal{H}}$ greatly simplify the problem to the extent that $C$ results a simple function of the probabilities (\[e.puu\])-(\[e.p4\]) only. We here assume that ${\cal{H}}$ is real, has parity symmetry (meaning that either ${\cal{H}}$ leaves the $z$ component of the total magnetic moment unchanged, or changes it in steps of $2$), and is further characterized by translational and site-inversion invariance. The two latter properties implies $M^z_l$ as defined in Eq. (\[e.Mz\]) to coincide with the uniform magnetization $M_z\equiv \sum_l\langle S^z_l\rangle/N$, and the probabilities $p_{{_{\rm I\!I\!I}}}=p_{{_{\rm I\!V}}}$, respectively. Under these assumptions, the concurrence for a given spin pair is[@Amicoetal04] $$\begin{aligned} C_{(r)}&{\equiv}&2\max\{0,C'_{(r)},C''_{(r)}\}~, \label{e.Cr}\\ C'_{(r)}&{\equiv}&|g_{(r)}^{xx}+g_{(r)}^{yy}| - \sqrt{\left({\textstyle{\textstyle\frac{1}{4}}}+g_{(r)}^{zz}\right)^2-M_z^2}~, \label{e.C'}\\ C''_{(r)}&{\equiv}&|g_{(r)}^{xx}-g_{(r)}^{yy}| -{\textstyle\frac{1}{4}}+g_{(r)}^{zz}~, \label{e.C''}\end{aligned}$$ where $r$ is the distance in lattice units between the two selected spins. Despite being simple combinations of magnetic observables, the physical content of the above expressions is not straightforward. However, by using the expression found in Section \[s.magtoprob\], one can write Eqs.  and in terms of the probabilities for the two spins to be in maximally entangled or factorized states, thus finding, in some sense, an expression which is analogous to Eq.  for the case of mixed states. In fact, from Eqs. (\[e.gxx+gyy.p\])-(\[e.gxx-gyy.p\]), it follows $$\begin{aligned} 2C'&=&|p_3-p_4|-2\sqrt{p_{{_{\rm I}}} p_{{_{\rm I\!I}}}}\label{e.C'p}~,\\ 2C''&=&|p_1-p_2|-(1-p_1-p_2)=\nonumber\\ &=&|p_1-p_2|-2\sqrt{p_{{_{\rm I\!I\!I}}} p_{{_{\rm I\!V}}}}\label{e.C''p}~,\end{aligned}$$ where we have used $p_{{_{\rm I\!I\!I}}}{=}p_{{_{\rm I\!V}}}$ and hence $p_3+p_4=2p_{{_{\rm I\!I\!I}}}=2\sqrt{p_{{_{\rm I\!I\!I}}}p_{{_{\rm I\!V}}}}$. The expression for $C''$ may be written in the particularly simple form $$2C''=2{\rm max}\{p_1,p_2\}-1~, \label{e.C''pmax}$$ telling us that, in order for $C''$ to be positive, it must be either $p_1>1/2$ or $p_2>1/2$. This means that one of the two parallel Bell states needs to saturate at least half of the probability, which implies that it is by far the state where the spin pair is most likely to be found. Despite the apparently similar structure of Eqs. (\[e.C’p\]) and (\[e.C”p\]), understanding $C'$ is more involved, due to the fact that $\sqrt{p_{{_{\rm I}}}p_{{_{\rm I\!I}}}}$ cannot be further simplified unless $p_{{_{\rm I}}}=p_{{_{\rm I\!I}}}$. The marked difference between $C'$ and $C''$ reflects the different mechanism through which parallel and antiparallel entanglement is generated when time reversal symmetry is broken, meaning $p_{{_{\rm I}}}\neq p_{{_{\rm I\!I}}}$ and hence $M_z\neq 0$. In fact, in the zero magnetization case, it is $p_{{_{\rm I\!I}}}=p_{{_{\rm I}}}=(p_1+p_2)/2$ and hence $$2 C'=2{\rm max}\{p_3,p_4\}-1~, \label{e.C'p.Mz=0}$$ which is fully analogous to Eq. (\[e.C”pmax\]), so that the above analysis can be repeated by simply replacing $p_1$ and $p_2$ with $p_3$ and $p_4$. For $M_z \neq 0$, the structure of Eq. (\[e.C’p.Mz=0\]) is somehow kept by introducing the quantity $$\Delta^2\equiv(\sqrt{p_{{_{\rm I}}}}-\sqrt{p_{{_{\rm I\!I}}}})^2~, \label{e.Delta2}$$ so that $$2 C'=2{\rm max}\{p_3,p_4\}-(1-\Delta^2)~, \label{e.C'Delta}$$ meaning that the presence of a magnetic field favors bipartite entanglement associated to antiparallel Bell states, $|e_3\rangle$ and $|e_4\rangle$. In fact, when time reversal symmetry is broken the concurrence can be finite even if $p_3,~p_4<1/2$. From Eqs. (\[e.C”pmax\]) and (\[e.C’Delta\]) one can conclude that, depending on $C$ being finite due to $C'$ or $C''$, the entanglement of formation originates from finite probabilities for the two selected spins to be parallel or antiparallel, respectively. In this sense we will speak about [*parallel*]{} and [*antiparallel*]{} entanglement. Moreover, from Eqs. (\[e.C’p\])-(\[e.C”p\]) we notice that, in order for parallel (antiparallel) entanglement to be present in the system, the probabilities for the two parallel (antiparallel) Bell states must be not only finite but also different from each other. Thus, the Bell states $|e_1\rangle$ and $|e_2\rangle$ ($|e_3\rangle$ and $|e_4\rangle$) result mutually exclusive in the formation of entanglement between two spins in the system, the latter being present only if one of the Bell state is more probable than the others. The case $p_1=p_2=1/2$ ($p_3=p_4=1/2$) corresponds in turn to an incoherent mixture of $|e_1\rangle$ and $|e_2\rangle$ ($|e_3\rangle$ and $|e_4\rangle$). In fact, the occurrence of the differences $|p_1-p_2|$ and $|p_3-p_4|$ is intriguing. Let us comment on $|p_1-p_2|$, as the same kind of analysis holds for $|p_3-p_4|$. In the general case the difference $p_1-p_2$ can vanish because of genuine many-body effects which are not directly readable in terms of 2-spin entangled or separable states. It is easier to interpret Eq.  \[Eq. \], if one restricts the possibilities to the case in which the two spins are not entangled with the rest of the system. By using Eq. (\[e.c12-\]), one can select two particular situations all leading to $p_1=p_2$:\ [*(i)*]{} $c_{{{_{\rm I}}}_{\Gamma}}$ or $c_{{{_{\rm I\!I}}}_{\Gamma}}$ vanishes $\forall\Gamma$, meaning that $|\Psi\rangle$ does not contain states where the two selected spins are parallel and entangled;\ [*(ii)*]{} for each $\Gamma$ such that both $|c_{{{_{\rm I}}}_{\Gamma}}|$ and $|c_{{{_{\rm I\!I}}}_{\Gamma}}|$ are non-zero, it is $\varphi_{_{\rm I}}^\Gamma-\varphi_{_{\rm II}}^\Gamma=\pi/2$. Thus, whatever the antiparallel components are, the parallel terms of $|\Psi\rangle$ appear in the form $(\alpha |e_1\rangle+\alpha^*|e_2\rangle)$. The above analysis suggests the first term in $C''$ ($C'$) to distill, out of all possible parallel (antiparallel) spin configurations, those which are specifically related with entangled parallel (antiparallel) states. These characteristics reinforce the meaning of what we have called parallel and antiparallel entanglement. Chain {#s.chain} ===== We consider the isotropic Heisenberg antiferromagnetic chain in a uniform magnetic field, described by $$\frac{\cal{H}}{J}= \sum_i \boldsymbol{S}_i\cdot\boldsymbol{S}_{i+1}-h S^z_i~, \label{e.chain}$$ where the exchange integral $J$ is positive, and the reduced magnetic field $h{\equiv}g\mu_{\rm B}H/J$ is assumed uniform. This model is characterized by the rotational symmetry on the $xy$ plane, as well as by the existence of a saturation field $h_{\rm s}=2$, such that for $h\geq h_{\rm s}$ the ground state is the factorized ferromagnetic one, with all spins aligned along the field direction. Moreover, Eq. (\[e.chain\]) has all the necessary symmetries for Eqs. (\[e.Cr\])-(\[e.C”\]) to hold. Due to the rotational symmetry on the $xy$ plane, it is $g^{xx}=g^{yy}$, meaning $p_1=p_2=\frac14 + g^{zz} \leq 1/2$, according to Eqs. (\[e.p1\]) and (\[e.p2\]), and hence null parallel entanglement ($C'' \leq 0$) between any two spins along the chain, no matter the field, the temperature, and the distance between them. In Fig. \[f.corr.chain\] we show the $T=0$ correlation functions for nearest neighboring (n.n.) and next-nearest neighboring (n.n.n.) spins, together with the uniform magnetization, as the field is varied. Beyond the overall regular behavior, we notice that there exists a value of the magnetic field where one simultaneously observes $g_{(1)}^{zz}=0$ and $M_z=1/4$ (indicated by the dashed lines). According to Eqs. (\[e.gzz.p\]) and (\[e.Mz.p\]) this implies null probability $p_{{{}_{\rm I\!I}}}$ for adjacent spins to be parallel in the direction opposite to the field. This means that the ground-state configuration is a superposition of spin configurations entirely made of stable clusters of spins parallel to the field separated by Néel-like strings. In Fig. \[f.C1prob.chain\] we show the probabilities for n.n. spins to be in the states of the mixed basis, together with the n.n. concurrence: The value of the n.n. concurrence for $h=0$ is in agreement with the exact resut in the thermodynamic limit.[@Glaseretal03] In presence of an external magnetic field, $C_{(1)}$ is found positive $\forall h$, meaning that, no matter the value of the field, the probabilities $p_3$ and $p_4$ for adjacent spins are always different from each other. The probabilities for the triplet states $|e_3\rangle,|u_{{_{\rm I\!I}}}\rangle,$ and $|u_{{_{\rm I}}}\rangle$ are equal for $h=0$ and depart from each other when the field is switched on. The singlet $|e_4\rangle$ evidently dominates the ground state up to a field which roughly corresponds to the value where $p_{{_{\rm I\!I}}}$ vanishes. As for the concurrence, despite the ground-state structure evidently changes as the field increases, $C_{(1)}$ stays substantially constant up to a large value of the field, mainly due to the fact that not only $p_4$ but also $p_3$ decreases with the field. This behavior mimics the one occurring in a spin dimer, whose ground state is the singlet state $|e_4\rangle$ up to $h=1$ where, after a level crossing, $|u_{{_{\rm I}}}\rangle$ becomes energetically favored. However, in a spin chain, many-body effects smear the sharp behavior of the dimer due to the level crossing. We do also notice that $C_{(1)}$ starts to decrease as soon as the total probability for parallel spins ($p_{{_{\rm I}}}+p_{{_{\rm I\!I}}}$) gets larger than that for antiparallel spins ($p_3+p_4$). The further reduction of $C_{(1)}$ is mainly driven by $p_{{_{\rm I}}}$ starting to rapidly increase. In the same field region where a substantial change in the n.n. configuration occurs, the n.n.n. concurrence $C_{(2)}$ switches on. This is seen in Fig. \[f.C2prob.chain\], where the probabilities for n.n.n. spins are shown together with the corresponding concurrence. In fact, when considering the n.n.n. quantities, we notice that both $g_{(2)}^{xx}$ and $g_{(2)}^{zz}$ have a non-monotonic behavior, displaying a maximum and a minimum, respectively, in the field region where $C_{(2)}$ gets positive (as from the comparison between Fig. \[f.corr.chain\] and \[f.C2prob.chain\]). Regarding the probabilities, one finds that, although the most likely state is always $|u_{{_{\rm I}}}\rangle$, $p_3$ is surprisingly large, and almost equal to $p_{{_{\rm I}}}$, as far as $h<1$. Moreover, both $p_3$ and $p_4$ have a non monotonic behavior and increase with $h$ up to the field where we simultaneously observe $g_{(2)}^{xx}$ and $g_{(2)}^{yy}$ attaining their extreme values, $p_{{_{\rm I}}}$ exceeding $1/2$, $p_4$ getting larger than $p_{{_{\rm I\!I}}}$, and $C_{(2)}$ switching on. As observed in the n.n. case, when $p_{{_{\rm I\!I}}}$ for n.n.n. spins vanishes $C_{(3)}$ switches on. Let us further comment upon $C_{(1)}$, $C_{(2)}$, and $C_{(3)}$. Given the fact that only antiparallel entanglement may exist in this chain, it is not surprising that $C_{(1)}>0$ and $C_{(2)}=0$ at low fields, as n.n. spins belong to different sublattices, while n.n.n. spins belong to the same sublattice. However, the fact that $C_{(2)}$ becomes finite indicates a ground-state evolution from the Néel-like to the ferromagnetic state such that the system enters a region where quantum fluctuations increase the total probability for spins belonging to the same sublattice to be antiparallel and entangled. The opposite effect is understood when $C_{(3)}$ is considered: in order to keep $C_{(3)}=0$ almost up to the saturation field, quantum fluctuations must reduce the total probability for spins belonging to different sublattices to be antiparallel and entangled. The above comments upon $C_{(2)}$ and $C_{(3)}$ may be generalized to $C_{(n)}$ with even and odd $n$, respectively. In Fig. \[f.Cn.chain\] we in fact show $C_{n}$ up to $n=5$. The concurrence for increasing distance between the two spins gets finite for a big enough field resembling the phenomenology of finite spin clusters.[@ArnesenBV01] Moreover, combining the exact results of Refs.[@Jinetal04] and [@Hikiharaetal04], we find that the range of the concurrence for the model , namely the distance $R$ such that $C_{(r)}$ vanishes for $r>R$, is $$R=\left|\frac{\rho}{\sqrt{\pi}(2+4M_z)(\frac12 - M_z)^{1/2}} \right|^\theta~, \label{e.Crange}$$ with the constant $\rho=0.924...$. When $h\to h_{\rm s}$, it is $M_z \simeq \frac12 - \frac{\sqrt{2}}{\pi}\sqrt{h_{{\rm s}}-h}$ and $\theta \simeq 2 - \frac{2\sqrt{2}}{\pi}\sqrt{h_{{\rm s}}-h}$, and the range of the concurrence is seen to diverge according to $R\simeq \frac{\rho\sqrt2}{32}(h_{{\rm s}}-h)^{-1/2}$. In other terms, approaching the saturation field, all $C_{(n)}$ become finite of order $O(1/N)$, consistently with the occurrence of a $|W_N\rangle$ state[@Duretal00]. For such state the entanglement is maximally bipartite in the sense of the Coffman-Kundu-Wootters inequality.[@Duretal00; @Coffmanetal00; @Osborne05] This scenario is consistent with our numerical data. As shown in Fig.\[f.Cn.chain\] up to $n=5$, for any $C_{(n)}$ it exists a field $h_n>h_{n-1}$ such that $C_{(n)}$ is positive for $h\in[h_n,2)$, with $h_n\to 2$ for $n\to\infty$. The divergence of the range of the concurrence for $h\to h_{{\rm s}}$ is shown in the inset of Fig. \[f.Cn.chain\]. Although the correct power-law behavior shows up, the precision of the numerical data is not sufficient to get the correct multiplicative constant. In fact, the above expression  is derived from asymptotic exact results, valid only for $r\gg 1$, when $C_{(r)}$ becomes too small to resolve it numerically. The formalism introduced in the previous sections works also in the finite temperature case, where it describes the effects of thermal fluctuations on quantum coherence. In Fig. \[f.temp\], the temperature dependence of probabilities and concurrences, for $h=1.8$, shows how thermal fluctuations progressively drive the system towards an incoherent mixture of states. Increasing $T$ the concurrences (right panel) are progressively suppressed and above $k_{_{\rm B}}T\sim0.8J$ also the n.n. concurrence vanishes. At higher temperatures none of the spin pairs in the system is entangled and quantum coherence is lost. The temperature behavior of the probabilities (left panel) is non monotonic, signaling the relative weight of the different states in the energy spectrum of the system. Eventually, at high $T$ all the probabilities tends to the asymptotic value $p_\nu=1/4$. Two-Leg Ladder {#s.ladder} ============== The above picture further enriches when considering the two-leg isotropic ladder, described by $$\frac{\cal{H}}{J}=\sum_i\sum_{\alpha=0,1} (\boldsymbol{S}_{i,\alpha}\cdot\boldsymbol{S}_{i+1,\alpha}{-} hS^z_{i,\alpha}) +\gamma\boldsymbol{S}_{i,{0}}\cdot\boldsymbol{S}_{i,{1}}~, \label{e.hHeisenberg.ladder}$$ where the index $i$ runs on both the right ($\alpha=0$) and left ($\alpha=1$) leg. The first term is the Heisenberg Hamiltonian for the right and left legs, while the last term describes the exchange interaction between spins of the same rung, whose relative weight is $\gamma$. The model is known to describe cuprate compounds like SrCu$_2$O$_3$ and it has been extensively studied for zero[@Barnesetal93; @Dagottoetal96] and finite field[@Chaboussantetal98]. The system shows a gap $\Delta$ in the excitation spectrum that can be interpreted essentially as due to the energy cost for producing a triplet excitation on a rung [@Barnesetal93]. The system reaches full polarization[@Chaboussantetal98], with all spins aligned along the field direction, for $h > \gamma + 2$. In the following we will specifically consider the isotropic case $\gamma=1$, which is characterized by a gap $\Delta \approx 0.5 J$,[@Barnesetal93] and by a saturation field $h_{\rm s}=3$. As in the chain case, due to the rotational invariance on the $xy$ plane, parallel entanglement cannot develop in the isotropic ladder. On the other hand, antiparallel bipartite entanglement can here develop between spins belonging to the same leg, or to the same rung, or to a different rung and leg. Two-spin quantities will be hereafter pinpointed by the two-component vector $(r_i,r_\alpha)$ joining the two selected spins, the first component referring to the direction of the legs, and the second one to that of the rungs. The indexes $(01)$, $(10)$, $(11)$, $(20)$ will therefore indicate n.n. spins on the same rung, n.n. along one leg, n.n.n. on adjacent rungs, and n.n.n. along the same leg, respectively. Our SSE data in Fig. \[f.corr.ladder\] for the uniform magnetization and the n.n. correlation functions $g^{\alpha\alpha}_{(01)}$, $g^{\alpha\alpha}_{(10)}$ confirm the description given in the previous paragraph: Before the Zeeman interaction fills the energy gap at the critical value $h_{\rm c}\simeq 0.5$, the ground-state configuration is frozen and characterized by the singlet $|e_4\rangle$ being by far the most likely state for each rung. The use of the formalism developed in Sec. \[s.magtoprob\] gives a direct information on the physics of the system: In Fig. \[f.prob.ladder\] we see that the singlet probabilities $p_4$ relative to n.n. spins on a rung and along one leg, as functions of the field, share a similar behavior everywhere but at the critical field $h_c$, where $p_4$ for n.n. spins sitting on the same rung shows up a kink that is not present in the singlet probability along the leg. This qualitatively different behavior clearly reflects the nature of the energy gap that closes at $h_c$. The sharp decrease of $p_4$ in favor of $p_{{_{\rm I}}}$ on the rung just above $h_c$ testifies that, even in the case, here considered, of equal exchange interaction along the legs and on the rungs ($\gamma=1$), the first excitations in the energy spectrum of the ladder are triplet excitations on the rungs. When the Zeeman energy becomes larger than the gap, for $h>h_{\rm c}$, the ground state starts to evolve with the field, whose immediate effect is that of pushing the quantities relative to spins on the rungs and along the legs towards each other: In fact, for $h>1$ n.n. spins along the legs and on the rungs substantially share the same behavior. As for the probabilities, we see that $p_{{_{\rm I\!I}}}$ and $p_3$ keep being equivalent, no matter the value of the field, and slowly vanish as saturation is reached. On the contrary the probability for n.n. spins to be in $|u_{{_{\rm I}}}\rangle$ increases at the expense of the probability relative to the singlet state until, for $h\simeq 1.8$, the two probabilities cross each other. Finally, we apply the formalism of Sec. \[s.spinconftoentanglement\] to extract features of the ground state from the concurrences. Fig. \[f.Cn.ladder\] shows the concurrences $C_{(r_i,r_\alpha)}$ up to the distance $r\equiv r_i+r_\alpha=4$ for spins sitting on the same (upper panel) and on different legs. The bipartite antiparallel entanglement between two spins sitting at a given distance $r$ is in general larger on different legs, even beyond the n.n. case. As expected, the field, after closing the gap, pushes $C_{(01)}$ and $C_{(10)}$ towards each other. Quite unexpectedly, however, this evolution includes a region where n.n. concurrence along the leg, $C_{(10)}$, increases. It is interesting to notice that $C_{(11)}$ switches on at $h\simeq 1.8$, where $p_4$ and $p_{{_{\rm I}}}$ for n.n. spins are seen to cross each other in Fig. \[f.corr.ladder\], signaling the crossover from an antiferromagnetic to a ferromagnetic-like configuration of the n.n. spins. Conclusions {#s.conclusions} =========== In this paper we developed a simple and effective formalism that allows to reconstruct the probability for two spins of a multi-spin system to be in a given quantum state, once the collective state of the system is given. Remarkably, such probabilities are found to be simple combination of standard magnetic observables, Eqs. (\[e.puu\]-\[e.p4\]). Within such formalism it is very natural to understand how concurrence quantifies the amount of entanglement between two spins by comparing the probabilities for those spins to be in different Bell states. In particular the expression for the concurrence clearly separates the case of parallel \[Eq. \] and antiparallel \[Eq. \] spins, leading to the introduction of the concept of *parallel* and *antiparallel* entanglement. The knowledge of the probability distribution for a given set of two-spin states can be a useful tool to study quantum phases dominated by the formation of particular local two-spin states and to investigate the transitions given by the alternation of such states. Within this class of phenomena we can cite the occurrence of short-range valence-bond states in low-dimensional quantum antiferromagnets [@Whiteetal94], and the transition from a dimer-singlet phase to long-range order in systems of weakly coupled dimers under application of a field or by tuning of the inter-dimer coupling [@Matsumotoetal04]. Acknowledgments =============== Fruitful discussions with L. Amico, G. Falci, S. Haas, D. Patanè, J. Siewert, and R. Vaia are gratefully acknowledged. We acknowledge support by SQUBIT2 project EU-IST-2001-390083 (A.F.), and by NSF under grant DMR-0089882 (T.R.). [200]{} M.C. Arnesen, S. Bose, and V. Vedral, Phys. Rev. Lett. [**87**]{}, 017901 (2001). 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--- abstract: 'We present results of the thermal conductivity of $\rm La_2CuO_4$ and $\rm La_{1.8}Eu_{0.2}CuO_4$ single-crystals which represent model systems for the two-dimensional spin-1/2 Heisenberg antiferromagnet on a square lattice. We find large anisotropies of the thermal conductivity which are explained in terms of two-dimensional heat conduction by magnons within the CuO$_2$ planes. Non-magnetic Zn substituted for Cu gradually suppresses this magnon thermal conductivity $\kappa_{\mathrm{mag}}$. A semiclassical analysis of $\kappa_{\mathrm{mag}}$ is shown to yield a magnon mean free path which scales linearly with the reciprocal concentration of Zn-ions.' author: - 'C. Hess' - 'B. Büchner' - 'U. Ammerahl' - 'L. Colonescu' - 'F. Heidrich-Meisner' - 'W. Brenig' - 'A. Revcolevschi' title: 'Magnon Heat Transport in doped $\rm La_2CuO_4$' --- Apart from their intimate relation to the physics of the copper-oxide high-temperature superconductors low dimensional quantum magnets are at the center of many novel phenomena. Among them are magnetic contributions to the thermal conductivity $\kappa$ of an unprecedentedly large magnitude, which have been discovered recently in the quasi one-dimensional (1D) spin ladder compound $\rm (Sr,Ca,La)_{14}Cu_{24}O_{41}$ [@Hess01; @Sologubenko00]. Similar findings of large and quasi 1D magnetic heat conduction have been reported also for cuprate spin-chain compounds [@Sologubenko00a; @Sologubenko01]. In this context it has become a key issue whether thermal transport due to magnetic degrees of freedom in certain classes of spin models may be intrinsically quasi-[*ballistic*]{}, i.e., dissipationless, thereby leading to high magnetic thermal conductivities $\kappa_{\mathrm{mag}}$ [@Zotos97; @Alvarez02]. Of particular interest is the mean free path $l_{\mathrm{mag}}$ of the heat-current carrying magnetic excitations. Based on a kinetic approach it has been shown recently [@Hess01], that $l_{\mathrm{mag}}$ for the spin ladder compound is of the order of 3000 [Å]{} . Direct microscopic evaluation of the mean free path is an open issue yet, with conflicting conclusions published [@Alvarez02]. In this letter we turn to two-dimensional (2D) quantum magnets by investigating the thermal conductivity of $\rm La_2CuO_4$ which realizes the spin-1/2 Heisenberg-antiferromagnet on the square lattice. The magnetic structure consists of Cu$^{2+}$-ions in CuO$_2$-planes extending along the crystallographic $a$- and $b$-directions. A strong antiferromagnetic intraplanar exchange coupling $J/k_B\approx1550$ K [@Hayden91a] is present whereas the interplanar exchange $J_\perp$ is negligible ($J_\perp /J\approx 10^{-5}$) [@Thio88]. Our primary focus will be on a broad peak at high temperatures ($T$) in $\kappa(T)$ which we observe if measured by a thermal current parallel to the CuO$_2$-planes, but which is absent for the perpendicular direction. This is consistent with previous results obtained in Ref. [@Nakamura91]. Several attempts to explain this peak have been made, e.g. scattering processes of acoustic phonons with soft optical phonons [@Cohn95] or magnons [@Morelli89]. It has been speculated also, that the high-$T$-maximum could be related to heat carried by magnetic excitations [@Nakamura91]. However, its origin has never been elucidated unambiguously. Here, and employing the effects of doping by various types of impurities we provide clear evidence in favor of the anisotropic heat conduction to be due to 2D magnons indeed. We extract the mean free path from our data using an approach similar to that for the case of $\rm (Sr,Ca,La)_{14}Cu_{24}O_{41}$ [@Hess01]. It will be shown that $l_{\mathrm{mag}}$ does not only scale linearly with the reciprocal concentration of magnetic impurities which are generated by Zn-ions, but moreover, that it is roughly equal to the mean unidirectional distance between these ions. We have measured $\kappa$ of $\rm La_2CuO_4$ and $\rm La_{1.8}Eu_{0.2}CuO_{4}$ single crystals as well as $\rm La_2Cu_{1-z}Zn_zO_4$ ($z=0$, 0.005, 0.008, 0.01, 0.02, 0.05) polycrystals as a function of $T$. Stoichiometric oxygen contents were achieved by annealing in high vacuum ($p<10^{-4}$ mbar) for 1-3 hours at 800$^\circ$ C. The preparation of the samples [@Hucker02] and the experimental method [@Hess01] have been described elsewhere. ![\[fig1\]Thermal conductivity of $\rm La_2CuO_4$ (bottom) and $\rm La_{1.8}Eu_{0.2}CuO_4$ (top). Full circles: $\kappa_c$. Open circles: $\kappa_{ab}$. Open squares: $\kappa_{\mathrm{mag}}$. Solid line: fit according to Eq. \[fit2d\]. Dashed line: $\kappa_{ab,\mathrm {ph}}$.](fig1.eps){width="\columnwidth"} In the lower panel of Fig. \[fig1\] we present the thermal conductivity of single crystalline $\rm La_2CuO_4$ if measured parallel ($\kappa_{ab}$) and perpendicular ($\kappa_c$) to the CuO$_2$-planes as a function of $T$. The pronounced anisotropy of $\kappa$ strongly resembles that of spin ladder compounds [@Hess01]: $\kappa_c$ exhibits a $T$-dependence which is typical for insulating crystalline materials with pure phononic thermal conductivity $\kappa_{\mathrm {ph}}$. In contrast to this, the low-$T$ peak of $\kappa_{ab}$ is followed by a minimum around 80 K and a strong increase of $\kappa_{ab}$ which eventually develops into a broad peak around 300 K that clearly exceeds the low-$T$ peak. Several conventional mechanisms which may be invoked to explain this unusual $T$-dependence and the resulting anisotropy can be dismissed. First, we can exclude effects of radiative heat transport since the optical properties of $\rm La_2CuO_4$ are almost isotropic in the relevant energy-range below $\rm h\nu\lesssim0.1$ eV [@Uchida91]. Second, electronic contributions to the thermal current can be excluded since the material is insulating. Therefore, phononic heat transport is expected to dominate. However, and third, a scenario solely based on thermal conduction by acoustic phonons is very unlikely since an increase of scattering usually causes a negative slope of $\kappa(T)$ for intermediate and higher $T$, in contrast to our observation. Hence, either an unusual scattering process acts on the acoustic phonons or an additional contribution apart from the conventional phononic background must be present for $\kappa_{ab}$. We now turn to the upper panel of Fig. \[fig1\] where $\kappa_{ab}(T)$ and $\kappa_c(T)$ of $\rm La_{1.8}Eu_{0.2}CuO_{4}$ are shown. Both curves strongly resemble our findings for $\rm La_2CuO_4$, yet exhibiting obvious differences: the low $T$-peaks of $\kappa_{ab}$ and $\kappa_c$ are slightly larger and more sharply shaped than in the case of $\rm La_2CuO_4$. Furthermore, a step-like anomaly is present at $T_{LT}\approx135$ K. Above $T_{LT}$ $\kappa_c$ remains almost constant and stays below the value for the undoped case, while $\kappa_{ab}$ also exhibits a high-$T$ maximum at $T\approx270$ K which is even larger than in $\rm La_2CuO_4$. The difference between $\kappa_c$ of Eu-doped and of pure $\rm La_2CuO_4$ can be attributed to a difference in phononic heat conduction. Upon doping $\rm La_2CuO_4$ with Eu, enhanced scattering of phonons reduces $\kappa_c$ for $T>T_{LT}$, where both compounds have the same structure. The anomaly at $T_{LT}$ ($\rm La_{1.8}Eu_{0.2}CuO_{4}$) signals the transition to a new structural phase for $T<T_{LT}$ where $\kappa_{\mathrm {ph}}$ is enhanced. The structural peculiarities of rare earth doped cuprates are well known [@Buchner94]. Their influence on $\kappa_{\mathrm {ph}}$ will be discussed in detail in a forthcoming paper [@Hess03]. Despite the increase of scattering of phonons due to Eu-impurities in $\rm La_{1.8}Eu_{0.2}CuO_{4}$ the high-temperature peak of $\kappa_{ab}$ is of larger magnitude than in $\rm La_2CuO_4$. This excludes the peak to stem from the previously mentioned heat transport by acoustic phonons. The high-$T$ peaks in $\kappa_{ab}$ of both compounds therefore must originate from a different heat transport channel. In principle, two different kinds of excitation could generate such an additional heat current. One possibility is heat transport by dispersive optical phonons, the other one is thermal current carried by magnetic excitations. Magnetic heat transport is the most likely explanation, since the magnetism is truly two-dimensional and the refore could explain the observed anisotropy. This is not true for dispersive optical phonons, which certainly are present in this material, but are found along all crystal axes [@Pintschovius91]. This result is corroborated by the strong suppression of the high-$T$ peak in hole-doped $\rm La_{2-x}Sr_xCuO_{4}$ [@Nakamura91], even at a very low Sr-content of $x\approx0.01$ [@Hess03]. On the one hand such an effect of doping is not to be expected in the case of a high-$T$ peak originating from optical phonons, since the lattice impurities induced by the Sr-ions are similar to the Eu-impurities discussed above. On the other hand the strong frustration of antiferromagnetism upon doping of mobile holes (cf. Ref. [@Hucker02] and references therein) would provide for a satisfying explanation for the suppression of a peak of magnetic origin. Note in this context that Eu-doping leaves the CuO$_2$-planes and therefore the magnetism almost unaffected. Hence we conclude, that the high-$T$ peak of $\kappa_{ab}$ originates from magnetic excitations which propagate only within the CuO$_2$-planes. $\kappa_{ab}$ therefore consists of a usual phonon background and a magnon contribution $\kappa_{\mathrm{mag}}$ while $\kappa_c$ is purely phononic. Upon applying a magnetic field of 8 T no significant changes of $\kappa$ were detected. This is, however, consistent with a magnetic origin of the high-$T$ peak since the corresponding Zeeman-energy is orders of magnitude smaller than the magnetic exchange coupling $J$. In order to extract $\kappa_{\mathrm{mag}}$ from $\kappa_{ab}$ we make use of the anisotropy of $\kappa$ assuming that the phononic part $\kappa_{ab,\mathrm {ph}}$ of $\kappa_{ab}$ is roughly proportional to $\kappa_c$ which is justified by only weakly anisotropic elastic constants [@Pintschovius91]. Since the magnetic contributions roughly follow a $T^2$-law (see below) they are expected to be negligible in the range of the low-$T$ peak. Therefore, for $\rm La_2CuO_4$ as well as for $\rm La_{1.8}Eu_{0.2}CuO_{4}$ a reasonable estimate of $\kappa_{ab,\mathrm {ph}}$ is achieved by scaling the corresponding data for $\kappa_c$ such as to match its low-$T$ peaks with that of $\kappa_{ab}$. In Fig. \[fig1\] the data thus obtained for $\kappa_{ab,\mathrm {ph}}$ of both compounds are represented by dashed lines. $\kappa_{\mathrm{mag}}$ was extracted by subtracting $\kappa_{ab,\mathrm {ph}}$ from $\kappa_{ab}$ (open squares in Fig. \[fig1\]). This procedure involves significant uncertainties only at low $T$ where $\kappa_{\mathrm{mag}}\ll\kappa_{ab,\mathrm {ph}}$. At intermediate and higher $T$ the relative uncertainties are smaller and rather concern the magnitude of $\kappa_{ab,\mathrm {ph}}$ than its slope. Therefore we assume that in this $T$-range the difference between the extracted and the true $\kappa_{\mathrm{mag}}$ is roughly constant. Following standard linearized Boltzmann-theory we have $\varkappa^i=\frac{1}{2}\frac{1}{(2\pi)^2}\int v_{\bf k}l_{\bf k}\epsilon_{\bf k} \frac{d}{dT}n_{\bf k}d{\bf k}$ for the 2D thermal conductivity of a single magnon dispersion branch (labelled by $i$), where $v_{\bf k}$, $l_{\bf k}$, $\epsilon_{\bf k}$ and $n_{\bf k}$ denote velocity, mean free path, energy and Bose-function of a magnon. Note that $\kappa_{\mathrm{mag}}^i$ of a three-dimensional ensemble of planes, as realized in $\rm La_2CuO_4$, is given by $\kappa_{\mathrm{mag}}^i=\frac{2}{c}\varkappa^i$, where $c=13.2$ [Å]{} is the lattice constant of $\rm La_2CuO_4$ perpendicular to the planes. Then the total $\kappa_{\mathrm{mag}}$ results from summing up $\kappa_{\mathrm{mag}}^i$ of each magnon branch. In order to calculate $\kappa_{\mathrm{mag}}^i$ we approximate the magnon dispersion relation $\epsilon_{\bf k}$ of the two branches $i=1,2$ with the 2D-isotropic expression $\epsilon_{\bf k}=\epsilon_k=\sqrt{\Delta^2_i+(\hbar v_0k)^2}$, which describes the dispersion observed experimentally [@Keimer93; @Coldea01] for small values of $k$. Here, $v_0$ is the spin wave velocity while $\Delta_1$ and $\Delta_2$ denote the spin gaps of each magnon branch. For clarity, we define the characteristic temperature $\Theta_M=(\hbar v_0 \sqrt{\pi}) /(a k_B)$ where $a=3.8$ [Å]{} is the lattice constant of the CuO$_2$-planes [@note1]. Assuming a momentum independent mean free path, i.e. $l_{\bf k}\equiv l_{\mathrm{mag}}$, we obtain for each magnon branch: $$\label{fit2d}{\kappa_{\mathrm{mag}}^i=\frac{v_0k_B \,l_{\mathrm{mag}}}{2a^2c} \frac{T^2}{\Theta_M^2} \int\limits_{x_{\mathrm{0,i}}}^{x_{\mathrm{max}}}x^2\sqrt{x^2-x_{\mathrm{0,i}}^2} \frac{dn(x)}{dx}dx.}$$ Here, the integral is dimensionless but temperature dependent via $x_{\mathrm{0},i}=\Delta_i/(k_B T)$ and $x_{\mathrm{max}}$. The upper boundary $x_{\mathrm{max}}$, however, may be set to infinity without affecting the fit at temperatures $T\lesssim 300$ K. Since from neutron scattering experiments $v_0\approx1.287\cdot10^5$ m/s [@Hayden91a] as well as $\Delta_{1}/k_B\approx26$ K and $\Delta_{2}/k_B\approx58$ K [@Keimer93] are well known quantities, $l_{\mathrm{mag}}$ is the only unknown parameter in Eq. \[fit2d\]. Assuming $l_{\mathrm{mag}}$ to be $T$-independent at low $T$, we can use Eq. \[fit2d\] in order to check the $T$-dependence of our experimental $\kappa_{\mathrm{mag}}$, which for $T\gtrsim\Delta_{1,2}$ should roughly be $\kappa_{\mathrm{mag}}\propto T^2$, as the integral is only weakly $T$-dependent in this $T$-range. Furthermore, $l_{\mathrm{mag}}$ can be extracted from the data. For $T\gtrsim250$ K interactions between the magnons become important [@Keimer92] which requires a renormalization of the spin wave parameters and therefore we restrict the application of Eq. \[fit2d\] with constant $l_{\mathrm{mag}}$ to $T\lesssim250$ K. Fitting Eq. (\[fit2d\]) to the experimental data we allow for an additive shift of the $\kappa_{\mathrm{mag}}$-curve which yields a further free parameter apart from $l_{\mathrm{mag}}$ and accounts for the aforementioned uncertainties in the magnitude of $\kappa_{\mathrm{mag}}$ [@note2]. For both compounds satisfactory fits were obtained at intermediate $T$-ranges. These are listed in Table \[tab1\], together with the resulting values for $l_{\mathrm{mag}}$. The fitting curves are represented by solid lines in Fig. \[fig1\]. While the slight deviations between the fitted and experimental data towards low $T$ are due to the uncertainties in $\kappa_{ab,\mathrm {ph}}$ in this range, the deviations at high $T$ can be understood in terms of the $T$-dependence of $l_{\mathrm{mag}}$ due to enhanced magnon-magnon scattering. ![\[fig2\]Open circles: Thermal conductivity $\kappa$ of $\rm La_2Cu_{1-z}Zn_zO_4$ polycrystals ($z=0$, 0.005, 0.01, 0.05) as a function of $T$. Solid lines: extrapolated $\kappa_{\mathrm {ph}}$.](fig2.eps){width="\columnwidth"} The data are consistent with a constant $l_{\mathrm{mag}}$ for $T$ both, within the fit interval and below, indicating that in this range $T$-dependent scattering processes like magnon-magnon scattering or even magnon-phonon scattering may be discarded. Therefore, relevant processes seem to be sample-boundary scattering or scattering at static magnetic defects. For $\rm La_{1.8}Eu_{0.2}CuO_{4}$ and $\rm La_2CuO_4$ we find $l_{\mathrm{mag}}\approx1160$ [Å]{} and $l_{\mathrm{mag}}\approx560$ [Å]{}, respectively. Since these values are far too small to correspond to the crystal dimensions, magnon-defect scattering is the most likely candidate. This conclusion is consistent with the fact that $\kappa_{\mathrm{mag}}$ and $l_{\mathrm{mag}}$ are quantitatively different for $\rm La_{1.8}Eu_{0.2}CuO_{4}$ and $\rm La_2CuO_4$: since the magnetic properties of both compounds are expected to be identical in essence, unequal $\kappa_{\mathrm{mag}}$ can arise only due to a difference in densities of the magnetic defects that restrict $l_{\mathrm{mag}}$. In order to check our analysis quantitatively it would be desirable to measure the density of static magnetic defects of our crystals. In lack of such methods we have performed measurements of $\kappa$ on samples with a well defined density of magnetic defects. Such defects can be induced in $\rm La_2CuO_4$ by substituting a small amount of Cu$^{2+}$-ions by non-magnetic Zn$^{2+}$-ions. Representative results on our polycrystalline samples of $\rm La_2Cu_{1-z}Zn_zO_4$ are presented in Fig. \[fig2\]. As is expected for doping of static structural and magnetic defects, the Zn-impurities lead to a gradual suppression of both, the phonon as well as the magnon contribution to $\kappa$ [@notesrzn]. Due to the polycrystalline nature of our samples anisotropic information on $\kappa$ is averaged over. We therefore estimate the phonon contributions $\kappa_{\mathrm {ph}}^{\mathrm {poly}}$ by fitting $\kappa$ right of its maximum by $\kappa_{\mathrm {ph}}=\alpha/T+\beta$ and by extrapolating this fit towards high $T$ (solid lines in Fig. \[fig2\]). In turn, $\kappa_{\mathrm{mag}}^{\mathrm {poly}}$ on the polycrystals is obtained by $\kappa_{\mathrm{mag}}^{\mathrm {poly}}=\kappa-\kappa_{\mathrm {ph}}$. ![\[fig3\]Main panel: Magnon thermal conductivity $\kappa_{\mathrm{mag}}$ of $\rm La_2Cu_{1-z}Zn_zO_4$ ($z=0$, 0.005, 0.008, 0.01, 0.02, 0.05) as a function of $T$ (open circles). Solid lines: fits according to Eq. \[fit2d\]. Inset: $l_{\mathrm{mag}}$ as a function of $1/z$ in units of lattice constants $a$. Solid line: fit line through origin.](fig3.eps){width="\columnwidth"} Note, that the measured $\kappa_{\mathrm{mag}}^{\mathrm {poly}}$ is smaller than the intrinsic $\kappa_{\mathrm{mag}}$ of these compounds by the factor of 2/3 due to averaging over all three components of the $\kappa$ tensor [@note3]. As shown in Fig. \[fig3\], $\kappa_{\mathrm{mag}}=(3/2)\kappa_{\mathrm{mag}}^{\mathrm {poly}}$ systematically decreases with increasing Zn-content as it is expected for static defects. Analyzing these data we proceed analogous to the undoped case by using Eq. \[fit2d\]. However, we neglect a slight reduction of the spin wave velocity [@Brenig91] and changes of the spin gaps induced by the Zn-ions. These effects lead to corrections smaller than the experimental error. The fits are represented by the solid lines in Fig. \[fig3\], the resulting $l_{\mathrm{mag}}$ and the fit intervals are reproduced in Table \[tab1\]. For typical transport experiments it is expected that the mean free path of the quasi-particles is proportional to the reciprocal defect concentration, i.e., to $1/z$. We therefore plot $l_{\mathrm{mag}}$ as a function of $1/z$ in the inset of Fig. \[fig3\]. The linear scaling between $l_{\mathrm{mag}}$ and $1/z$ is clearly confirmed. From a linear fit including the origin (solid line) we find that $l_{\mathrm{mag}}\approx 0.74 a/z$. Since $1/z$ is equal to the mean unidirectional distance of Zn-ions, i.e. static magnetic defects, $l_{\mathrm{mag}}$ gives a direct measure of these distances [@note4]. Therefore, this result confirms the above quantitative analysis of $\kappa_{\mathrm{mag}}$ of the single crystals based on Eq. \[fit2d\]. In conclusion, both, qualitatively as well as quantitatively our results strongly suggest, that the high-$T$ peak observed in $\kappa_{ab}$ of doped and undoped $\rm La_2CuO_4$ arises due to magnon heat transport which is confined to the CuO$_2$-planes. We note, that in general a large $\kappa_{\mathrm{mag}}$ is rarely observed at high $T$ and requires a unique synergy of magnon-phonon coupling [@Sanders77], large $l_{\mathrm{mag}}$ and high spin wave velocity $v_0$. Its realization in $\rm La_2CuO_4$ may lead to future use of magnetic heat transport as a tool to study the interactions of magnetic excitations with other quasiparticles like holes or phonons. We acknowledge support by the DFG through SP1073. 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--- abstract: 'As falhas de software estão com frequência associadas a acidentes com graves consequências económicas e/ou humanas, pelo que se torna imperioso investir na validação do software, nomeadamente daquele que é crítico. Este artigo endereça a temática da qualidade do software através de uma análise comparativa da usabilidade e eficácia de quatro ferramentas de análise estática de programas em C/C++. Este estudo permitiu compreender o grande potencial e o elevado impacto que as ferramentas de análise estática podem ter na validação e verificação de software. Como resultado complementar, foram identificados novos erros em programas de código aberto e com elevada popularidade, que foram reportados.' author: - 'Patrícia Monteiro, João Lourenço, e António Ravara' bibliography: - 'mybibliography.bib' title: | Uma análise comparativa de\ ferramentas de análise estática\ para deteção de erros de memória --- Introdução ========== [Edsger Dijkstra, The Humble Programmer, ACM Turing Lecture 1972]{} “*Program testing can be a very effective way to show the presence of bugs, but is hopelessly inadequate for showing their absence*.” A crescente necessidade de desenvolvimento de software cada vez mais complexo, no menor tempo possível e com baixo custo, conduz ao aumento da densidade de erros, sendo o controlo de qualidade frequentemente assegurada com base em testes e revisão manual de código. Estas técnicas, apesar de eficazes, não permitem garantir a ausência total de erros num programa. A presença de defeitos no software pode conduzir a problemas variados, tais como erros funcionais (o programa não cumpre os requisitos), falhas e/ou vulnerabilidades de segurança, baixa performance ou a interrupção da execução do programa. As falhas de software crítico são muitas vezes associadas a desastres com graves consequências económicas e/ou humanas. São exemplos disso casos como a autodestruição do Ariane 5 (1996)[^1] e do Mars Climate Orbiter (1999) [@board_mars_1999], o bloqueio do terminal 5 no Aeroporto de Londres-Heathrow (2008)[^2] e, mais recentemente, o erro encontrado pela Amazon no Jedis (2018)[^3]. No software escrito em C/C++ é comum a existência de erros de memória, tais como desreferências inválidas, acesso a variáveis não inicializadas e fugas e operações inválidas de libertação de memória. Este tipo de erros são de difícil deteção pois normalmente conduzem ao comportamento indefinido do programa [@john_guide_2010], isto é, tanto podem causar a sua falha imediata como permitir que este continue a funcionar de forma silenciosamente defeituosa. A verificação e validação de software é feita, essencialmente, através de três técnicas: revisão manual de código [@Trisha2016], análise automática (dinâmica [@ball_1999] ou estática [@wichmann_industrial_1995]) e semi-automática com provadores de teoremas [@duffy_1991]. As duas últimas são rigorosas e permitem analisar a totalidade do código, verificando propriedades que são definidas matematicamente e identificando todas as situações em que as mesmas são violadas: *os erros*. A análise automática de software exige menos esforço por parte das equipas de desenvolvimento, fator importante no contexto atual da industria. A análise dinâmica implica a execução do programa ou de uma sua representação, enquanto que a estática analisa o programa sem executar o código fonte ou uma sua representação. A utilização precoce de técnicas de análise estática permite a identificação de erros numa fase inicial do desenvolvimento, o que reduz consideravelmente os custos [@patton_software_2006]. Como estas ferramentas utilizam frequentemente um processo de sobre-aproximação das propriedades que pretendem verificar, reportam com frequência *falsos positivos*, i.e., erros que não existem. Os falsos positivos requerem um tratamento adicional que tem custos não negligenciáveis e cuja filtragem automática pode facilmente gerar *falsos negativos*, ou seja, erros reais que não são reportados. Interessa-nos estudar a viabilidade de usar ferramentas automáticas para procurar erros de memória, e o objetivo deste artigo é apresentar uma análise comparativa preliminar de quatro ferramentas de análise estática de código aberto: Cppcheck[^4] [@cppcheck_manual_2017], Clang Static Analyzer[^5], Infer[^6] [@calcagno_infer_2011] e Predator[^7] [@dudka_byte-precise_2013]. O processo e critérios de seleção destas ferramentas encontra-se descrito na Secção \[subsec:tools\]. Através desta análise pretende-se identificar o tipo de padrões de erros de memória que as ferramentas conseguem detetar, identificando aquelas que são funcionalmente equivalentes, por reportarem os mesmos erros, e as que são complementares, por reportarem erros distintos. Neste artigo apresenta-se a análise de dois programas, um de pequena e outro de média dimensão, estando a decorrer a análise de outros dois programas de grande dimensão. Como seria de esperar, verificou-se que havia consideravelmente mais erros de memória reportados no software de média dimensão que no de pequena dimensão. Naturalmente, o número de falsos positivos também aumentou. No entanto, através dos resultados obtidos foi possível construir exemplos mínimos que permitiram perceber de forma clara o tipo de padrões de erros identificados pelas várias ferramentas. Os padrões de erros descobertos permitiram identificar as diferenças e falhas na análise das ferramentas. Além disso, foram descobertos novos erros nos programas escrutinados, sendo que os mesmos foram reportados nos respetivos repositórios. Metodologia =========== ![Etapas seguidas durante a realização da experiência.[]{data-label="fig:processo"}](Figures/process.pdf){width="\linewidth"} A Figura \[fig:processo\] apresenta o processo seguido durante este trabalho. O processo teve início com a identificação e seleção das ferramentas de análise estática a estudar (Secção \[subsec:tools\]) e dos programas alvo (Secção \[subsec:software\]). Concluída a fase de seleção das ferramentas e dos programas, seguiu-se a análise dos mesmos. Inicialmente, foram recolhidas informações sobre o número e tipo de erros de memória já identificados nos programas, pesquisando os repositórios dos mesmos. Os erros reportados como questões de utilizadores (i.e., *issue*) foram verificados manualmente, de maneira a verificar se são erros reais ou falsos positivos e se já estão ou não corrigidos. Por outro lado, os erros reportados como *commit* foram considerados erros reais e classificados como corrigidos. As versões onde foram identificados erros foram selecionadas e analisadas por todas as ferramentas. Uma vez que as ferramentas de análise estática devolvem falsos positivos, seguiu-se a fase de validação dos seus relatórios de resultados. Esta validação, foi feita em duas etapas: comparação com o histórico de erros dos repositórios; e revisão manual. A comparação com o histórico de erros, para além de permitir validar resultados, também permitiu calcular a percentagem de erros identificados corretamente e detetar a existência de novos erros. Estes foram, posteriormente, verificados através da revisão manual do código. Por fim, com as informações recolhidas foi possível identificar padrões de erros reconhecidos pelas ferramentas e construir exemplos mínimos para cada um deles. Escolha de ferramentas {#subsec:tools} ---------------------- A escolha das ferramentas de análise estática a utilizar nesta experiência foi feita com base nos seguintes critérios: analisarem programas em C/C++; serem ferramentas de código aberto; serem ferramentas de projetos ativos; e identificarem pelo menos dois dos seguintes erros: desreferenciação inválida, operação de libertação inválida, fuga de memória e variáveis não inicializadas. Assim, numa primeira seleção foram identificadas 32 ferramentas, que foram dispostas na Tabela \[tab:hla:ferramentas\_analise\_estatica\] (ver Apêndice \[app:ferramentas\_analise\_estatica\]), onde cerca de metade corresponde a ferramentas de código aberto e outra metade a ferramentas comerciais. Ficámos com uma seleção de apenas 10 ferramentas que eram simultaneamente projetos ativos e de código aberto. Posteriormente, estas 10 ferramentas foram classificadas relativamente às suas funcionalidades de verificação de software (Tabela \[tab:hla:funcionalidades\]), tendo sido selecionadas as ferramentas que garantissem pelo menos duas das funcionalidades listadas. [width=0.85]{} **** **** **** **** ------------------- ------ ------ ------ ------ Cppcheck CppLint x x x x Clang Static An. Cobra x x x x Flawfinder x x x x Frama-C x x Infer x Predator x Uno x x VisualCodeGrepper x x x x : Funcionalidades das ferramentas de análise estática.[]{data-label="tab:hla:funcionalidades"} Foram identificadas 6 ferramentas que cumpriam com todos os critérios referidos (cerca de 19`%` da lista de ferramentas inicial), que foram dispostas na Tabela \[tab:hla:caracteristicas\], onde se fez a sua caracterização relativamente à facilidade de instalação, facilidade de utilização, teoria e correção. Nesta última tabela podemos observar que as ferramentas se baseiam em diferentes teorias para realizar a sua análise, sendo que a descrição de cada uma delas pode ser consultada no Apêndice \[app:teorias\]. Além disso, na Tabela \[tab:hla:caracteristicas\], é importante ressaltar que a correção de uma ferramenta é influenciada pelo facto de esta apresentar, ou não, filtragem de falsos negativos. Relativamente à facilidade de instalação, a maioria das ferramentas tem de ser instalada manualmente. Por outro lado, todas as ferramentas estão disponíveis nas três plataformas mais comuns (Linux, macOS e Windows) excepto o Predator, que apenas funciona em Linux. Relativamente à facilidade de utilização, metade das ferramentas selecionadas não precisa de uma função `main` e apenas o Predator requere anotações adicionais no código fonte, que apenas são necessárias para imprimir informação e não para a execução da análise. Por fim, todas as ferramentas permitem a análise de ficheiros isolados e algumas delas (Cppcheck e Frama-C) disponibilizam interface gráfica, o que facilita a sua utilização. [width=0.9]{} ---------- -------- ------ --- --- -- --- --- -- --- **** **** Cppcheck AST x x x CFG x x x Frama-C AST x x Infer SL, AI x x x x Predator SMG x x x Uno CFG x x x x ---------- -------- ------ --- --- -- --- --- -- --- : Características das ferramentas de análise estática.[]{data-label="tab:hla:caracteristicas"} No final do processo de seleção obtiveram-se as seguintes ferramentas: Cppcheck [@cppcheck_manual_2017], Clang Static Analyzer[^8], Frama-C [@frama_c_manual_2017], Infer [@calcagno_infer_2011], Predator [@dudka_byte-precise_2013] e UNO [@holzmann_uno_2002]. Numa primeira análise experimental, não conseguimos que as ferramentas UNO e Frama-C produzissem resultados relevantes, pelo que optámos por excluir estas ferramentas da análise comparativa final. Escolha de projetos a analisar {#subsec:software} ------------------------------ Através de uma pesquisa de software aberto no GitHub, selecionámos 16 projetos escritos nas linguagens C/C++, que foram classificados e ordenados de de acordo com os seguintes critérios: Prioridade: : a nossa avaliação ponderada (1 = mais prioritário) dos demais critérios, tendo em especial consideração a quantidade de operações de manipulação de memória (`malloc`, `calloc`, `realloc`, `free` e utilização de ponteiros) e tipo de impacto que os erros têm . Popularidade: : determinada pela quantidade de estrelas atribuídas pelos utilizadores ao repositório no GitHub. Número de versões: : favorecemos os programas com múltiplas versões, verificando se uma ferramenta identifica os erros presentes numa determinada versão e confirma que os mesmos foram corrigidos nas versões seguintes. Dimensão: : Assumindo que 1kB corresponde aproximadamente a 40 linhas no programa fonte, os programas foram divididos por dimensão (Tabela \[tab:hla:dimensao\]). Agrupámos programas com a mesma dimensão para facilitar a seleção, uma vez que nos interessava testar projetos com dimensões diferentes. [width=0.5]{} --------- ------- -------- ------- -------- Pequeno 50 400 2000 6000 Médio 400 1600 6000 64000 Grande 16000 128000 64000 512000 --------- ------- -------- ------- -------- : Classificação das dimensões dos Programas.[]{data-label="tab:hla:dimensao"} Erros de memória: : número e percentagem de erros de memória reportados nos *commits*. Falha: : impacto dos erros identificados no software, classificado em 3 categorias (por ordem decrescente): falha, performance e segurança. Consideramos mais prioritários programas com maior percentagem de falhas. Questões sobre erros: : submetidas pelos utilizadores e referentes a erros de memória identificados durante a instalação ou utilização do software. Em alguns casos, os programas poderão não ter erros de memória identificados, mas ter uma grande quantidade de questões sobre esse tipo de erros por responder ou resolver. Manipulação de memória: : operações de manipulação de memória e quantidade de ponteiros utilizados. Seguindo os critérios apresentados, os programas foram agrupados por dimensão, e depois ordenados por prioridade. Os com a mesma dimensão e prioridade foram ordenados por popularidade. Na Tabela \[tab:hla:software\] estão listados, de forma já ordenada, todos os programas analisados e respetivas características. Os quatro escolhidos (um de pequena dimensão, um de média dimensão e dois de grande dimensão) são os que se encontram posicionados no topo de cada um grupos, tendo eles prioridade igual a 1 e elevada popularidade: Simple Dynamic String (SDS) : [^9] biblioteca de *strings* dinâmicas, projetada para aumentar as funcionalidades limitadas das *strings* da biblioteca C; Beanstalkd : [^10] gestor de tarefas para aplicações distribuídas; Tmux : [^11] multiplexador de terminal, que permite que uma série de terminais possam ser acedidos e controlados a partir de um único terminal; Memcached : [^12] sistema distribuído de cache em memória, que é frequentemente utilizado para acelerar sites dinâmicos, colocando os dados dos mesmos em cache para reduzir o número de vezes que uma fonte de dados externa precisa de ser acedida. [width=1]{} ---------------- ------ --------- --------------- ------ ------------------- ------------------ ------------------- -------------- ---------------- **** **** **** **** **** **** **** (\# Estrelas) `(%)` total erros `(%)` erros mem. `(%)` total casos \# Operações \# Apontadores 1 Pequena 2215 2 0 0 23 121 71 TinyVM 1 Pequena 1254 1 25 4 25 64 115 Twemcache 1 Pequena 811 8 0 0 21 257 205 GloVe 2 Pequena 2204 2 18 0 28 118 64 Sparkey 2 Pequena 784 2 5 0 0 86 69 Beanstalkd 1 Média 4405 34 6 7 10 93 129 Openwebrtc 1 Média 1519 1 3 1 2 439 323 Redcarpet 2 Média 4169 23 4 0 7 66 44 Http-Parser 2 Média 3816 20 5 0 5 30 96 Leveldb 3 Média 12915 18 4 6 7 37 173 Tmux 1 Grande 9530 20 8 11 3 1116 918 Memcached 1 Grande 7452 73 3 7 9 591 395 The foundation 1 Grande 2810 670 3 6 7 682 1342 Timescaledb 1 Grande 4259 25 3 1 3 59 214 Lwan 2 Grande 4131 1 15 8 21 462 700 Ziparchive 2 Grande 3368 37 1 2 3 67 284 ---------------- ------ --------- --------------- ------ ------------------- ------------------ ------------------- -------------- ---------------- : Características dos programas.[]{data-label="tab:hla:software"} Experimentação e análise ======================== Relatório de resultados {#subsec:reports} ----------------------- Neste artigo apresentamos os resultados referentes a dois programas. O SDS é um software de pequena dimensão com apenas 2 versões no seu repositório. Os respetivos *commits* não reportam erros de memória, mas existem vários erros deste tipo identificados por utilizadores (i.e., *issue*), sendo que alguns desses erros ainda se encontram por corrigir e/ou validar. Portanto, todos os erros reportados no repositório e devolvidos pelas ferramentas foram verificados através de revisão manual de código. O Beanstalkd é um software de dimensão média e possui 34 versões no repositório. As ferramentas foram executadas em 12 versões deste software onde haviam sido identificados erros de memória, permitindo assim comparar os resultados destas com o histórico de erros disponível. Os erros identificados no SDS e no Beanstalkd, quer pelos programadores e utilizadores, quer pelas ferramentas, encontram-se listados nas Tabelas \[tab:hla:erros-sds\] e \[tab:hla:erros-beanstalkd\] (Apêndice \[app:relatorio\_erros\]). No total foram analisados 30 erros no SDS, dos quais 24 foram identificados pelas ferramentas, e 188 erros no Beanstalkd, dos quais 175 foram identificados pelas ferramentas. De notar que, na Tabela \[tab:hla:erros-beanstalkd\] foram omitidos os 135 falsos positivos identificados pelo Predator. Na secção seguinte é feita uma descrição da análise dos relatórios obtidos. Análise dos resultados ---------------------- Os resultados obtidos pela execução das ferramentas de análise estática foram comparados com o histórico de erros dos repositório e os novos erros foram sujeitos a revisão manual de código. Assim, a partir da análise dos resultados foi possível calcular a percentagem de falsos positivos (FP), falsos negativos (FN), verdadeiros positivos (TP) e verdadeiros negativos (TN) identificados por cada uma das ferramentas. Para uma determinada ferramenta a classificação dos erros em cada uma das categorias segue os seguintes critérios: ------------------------- ----------------------------------------------- **Falso positivo** erro não real mas reportado pela ferramenta **Falso negativo** erro real mas não reportado pela ferramenta **Verdadeiro positivo** erro real e reportado pela ferramenta **Verdadeiro negativo** erro não real e não reportado pela ferramenta ------------------------- ----------------------------------------------- Um erro foi considerado real quando está reportado no histórico de erros do repositório ou quando é identificado por uma das ferramentas e validado com revisão manual de código. ### Simple Dynamic String (SDS). {#subsubsec:sds} Na versão 1 foram reportados 7 erros no repositório, sendo um desses um falso positivo identificado por um utilizador. Na versão 2 foram reportados 2 erros, mas um já existia na versão 1. Por outro lado, as ferramentas encontram um total de 24 erros diferentes, sendo que 21 desses não foram reportados. No entanto, apenas 6 dos 21 erros poderão ser classificados como reais. Assim, no SDS foram identificados 26 erros, dos quais 13 são reais. Os erros reais que encontrámos na última versão e que reportámos são: - *sds.c:1123: error: null dereference*, identificado pelo Infer[^13]; - *sds.c:1240: error: dead store*, identificado pelo Clang e pelo Infer[^14]. Na Figura \[fig:per\_error\_sds\] está representado um gráfico com as percentagens dos vários tipos de erros identificados pelas ferramentas no SDS. Este software retorna, intencionalmente, apontadores para o meio de blocos de memória alocados com `malloc`. Portanto, as operações de desreferência ou libertação de memória aplicadas a ponteiros nestas situações são reportadas pelas ferramentas Clang Static Analyzer e Predator como erros, que nós classificámos como falsos positivos. Verificámos que o Clang Static Analyzer apresentou uma percentagem relativamente baixa de falsos positivos, mas verificou-se também 30`%` de falsos negativos. Esta percentagem deve-se ao facto da Clang não identificar um padrão de erros relevante, nomeadamente a verificação dos resultados de operações de alocação de memória (i.e., `malloc`, `calloc` e `realloc`). Por princípio, a ferramenta Predator não permite chamadas de funções externas, a fim de excluir qualquer efeito colateral que possa potencialmente quebrar a segurança da memória. As únicas funções externas permitidas são aquelas que o Predator reconhece como funções integradas e as modela apropriadamente, provando a segurança da memória (`malloc`, `free`, e algumas funções da biblioteca do C tais como `memset`, `memcpy` e `memmove` [@dudka_byte-precise_2013]). Por este motivo, o Predator não foi capaz de identificar nenhum erro real neste software, sendo a ferramenta que devolveu a maior percentagem de falsos positivos e falsos negativos. Apesar de falhar na identificação de alguns erros, o Infer é a ferramenta que devolve uma maior percentagem de verdadeiros positivos e a menor percentagem de falsos negativos neste software. Como a análise realizada pelo Infer consiste numa execução simbólica do código, mantendo uma *heap* simbólica, quando a ferramenta não consegue provar a segurança da memória, pode reportar um erro, se encontrar uma desreferência nula ou uma fuga de memória, ou pode perceber que se encontra num estado inconsistente. Em ambos os casos, a análise é interrompida, porque a tentativa de prova não faz sentido. Outra razão para que o Infer não consiga reportar erros que poderia identificar é a existência de um tempo limite para execução da análise, que é por vezes atingido antes da análise chegar ao fim [@calcagno_infer_2011], e que parece não ser possível de parametrizar. O Cppcheck é a ferramenta que devolve a menor quantidade de erros. Por uma opção de engenharia, esta ferramenta realiza filtragem dos erros para reduzir o número de falsos positivos reportado, mas nesse processo acabar por eliminar erros reais gerando falsos negativos. ### Beanstalkd. {#subsubsec:beanstalkd} Este software tem uma dimensão e complexidade maior que o SDS. Como o Predator não está preparado para analisar programas muito complexos [@dudka_byte-precise_2013], a análise do Beanstalkd revelou-se pouco conclusiva. Numa tentativa de extrair algum tipo de resultado relevante, o Predator foi utilizado para testar cada um dos ficheiros do software individualmente, tendo sido obtidos 135 falsos positivos (cerca de 72`%` do total de erros reportados) que se devem às chamadas de funções externas que são ignoradas. O Infer foi mais uma vez a ferramenta que devolveu uma maior percentagem de verdadeiros positivos. No entanto, foi também a ferramenta com a segunda maior percentagem de falsos positivos. O Cppcheck não devolveu falsos positivos, no entanto, revelou-se pouco eficaz devolvendo a menor percentagem de verdadeiros positivos e a maior percentagem de falsos e verdadeiros negativos. Estes resultados devem-se ao facto desta ferramenta fazer filtragem de falsos positivos. Por fim, o Clang Static Analyzer foi a ferramenta que devolveu uma maior quantidade de verdadeiros negativos (80`%`), juntamente com o Cppcheck. Além disso, esta ferramenta identificou a segunda maior percentagem de verdadeiros positivos no Beanstalkd. Os novos erros identificados e classificados como verdadeiros positivos no repositório do Beanstalkd foram os seguintes: - *prot.c:501: error: null dereference*, identificado pelo Infer[^15]; - *testheap.c:222: error: memory leak*, identificado pelo Cppcheck e Predator[^16]. Estes erros, tal como aconteceu para o software SDS, foram reportados e aguardam *feedback* por parte dos programadores responsáveis pelo repositório. Conclusões ========== Como se pode verificar nas Tabelas \[tab:hla:erros-sds\] e \[tab:hla:erros-beanstalkd\] do Apêndice \[app:sintese\_erros\], é útil usar as 4 ferramentas, pois obtém-se resultados complementares: cada ferramenta identificou erros que nenhuma das outras identificou. Este facto valida a seleção feita. É também importante salientar que as utilização destas ferramentas não pesa significativamente no desenvolvimento de software, pois os tempos que cada uma leva a analisar os softwares escolhidos são muito reduzidos, como se pode ver nas Tabelas \[tab:hla:tempos\_execucao\_sds\] e \[tab:hla:tempos\_execucao\_beanstalkd\] do Apêndice \[app:tempos\_execucao\]: vão de alguns segundos, no caso do SDS, a no máximo pouco mais de um minuto, no caso do Beanstalkd. Note-se que vários dos erros encontrados pelas ferramentas estiveram presentes nos softwares por longos períodos de tempo, como se vê nas Tabelas \[tab:hla:tempos\_permanencia\_sds\] e \[tab:hla:tempos\_permanencia\_beanstalkd\] do Apêndice \[app:tempos\_permanencia\]: o SDS teve erros que foram corrigidos só passados dois anos e tem erros que presentes há cerca de quatro anos; o Beanstalkd teve erros que foram corrigidos só passados quatro anos e tem erros presentes há quase dez anos. A deteção destes erros pelas ferramentas foi no entanto muito rápida (como se referiu acima) e a verificação de que se tratavam de erros reais levou poucas horas. É então muito vantajosa a utilização destas ferramentas de análise estática no processo de desenvolvimento de software para evitar erros de memória. A partir desta experiência foi possível identificar padrões de erros e construir exemplos mínimos capazes de reproduzir os resultados observados nos programas analisados. Na Tabela \[tab:hla:padroes\] (Apêndice \[app:patterns\]) estão representados os padrões identificados para cada tipo de erro. Os exemplos mínimos e respetivos relatórios de resultados podem ser consultados no Apêndice \[app:exemplos\_minimos\]. A ferramenta Infer é a única que identifica a possibilidade da ocorrência de uma desreferência nula quando não é feita a verificação dos resultados das operações de alocação de memória. Se as funções `malloc`, `calloc` e `realloc` falharem, estas retornam o valor `NULL` e, portanto, a desreferência dessa variável poderá gerar um erro. O Infer é também a única ferramenta que identifica que o valor de um endereço não está a ser utilizado (i.e., *dead store*), caso esse valor seja 0 ou *NULL*. A ferramenta Clang Static Analyzer não considera esta situação um erro, uma vez que considera que o valor 0 ou *NULL* ao ser atribuído à variável na sua inicialização não é um valor não utilizado. O Infer (até à versão 0.13.1) não conseguia identificar qualquer tipo de erro de memória relacionado com a utilização de uma variável alocada usando o padrão *sizeof(\*ptr)*, e.g., `ptr = malloc(sizeof(*ptr))`. Na versão mais recente do Infer (versão 0.14.0, lançada no dia 1 de Maio de 2018) este defeito já foi corrigido. No Apêndice \[app:exemplos\_minimos\] deste artigo encontra-se um exemplo mínimo da situação descrita. Apesar de o Infer ser a única ferramenta que reporta a não verificação das chamadas a funções de alocação de memória, observou-se que o também o Cppcheck, em algumas situações, reporta este tipo erros. No entanto, os erros reportados pelo Cppcheck e Infer são diferentes. O Cppcheck identifica uma possível fuga de memória, enquanto que o Infer identifica uma desreferência nula. Isto acontece porque é feita uma atribuição sem que seja verificado o resultado da chamada à função `realloc`, ficando assim a memória anteriormente atribuída inacessível no caso de esta operação falhar. Este tipo de ferramentas está já a ser incluído no processo de desenvolvimento de alguns programas de grande dimensão e relevância, como o LibreOffice[^17]. Atualmente, este software usa duas ferramentas, o Cppcheck e o Coverity [@llaguno_2017]. Segundo o relatório disponibilizado, até ao momento foram analisadas mais de 6 milhões de linhas de código do projeto LibreOffice, onde foram identificados mais de 25 mil erros, dos quais foram corrigidos 99`%` [@coverity_libreoffice]. No caso do Cppcheck, segundo o último relatório disponível (de 27 de Janeiro de 2018), foram identificados através desta ferramenta cerca de 6 mil erros no repositório do LibreOffice [@cppcheck_libreoffice]. O trabalho futuro passa pela análise dos programas de grande dimensão já selecionados (Tmux e Memcached). Os dados recolhidos dessa análise serão depois utilizados para identificar novos padrões de erros e construir exemplos mínimos, tal como foi feito para o SDS e para o Beanstalkd. Desta forma espera-se obter novos dados para o desenvolvimento do estudo comparativo das ferramentas. ### Agradecimentos. {#agradecimentos. .unnumbered} [*Este trabalho foi suportado pela FCT-NOVA e parcialmente financiado pela FCT-MCTES e pelo programa POCI-COMPETE2020 nos projetos UID/CEC/04516/2013 e PTDC/CCI-COM/32456/2017.*]{} [^1]: <https://around.com/ariane.html> [^2]: <http://www.computerweekly.com/news/2240086013/British-Airways-reveals-what-went-wrong-with-Terminal-5> [^3]: <https://github.com/xetorthio/jedis/issues/1747> [^4]: <https://github.com/danmar/cppcheck> [^5]: <http://clang.llvm.org/> [^6]: <https://github.com/facebook/infer> [^7]: <https://github.com/kdudka/predator> [^8]: <http://clang.llvm.org/> [^9]: <https://github.com/antirez/sds> [^10]: <https://github.com/kr/beanstalkd> [^11]: <https://github.com/tmux/tmux> [^12]: <https://github.com/memcached/memcached> [^13]: <https://github.com/antirez/sds/issues/99> [^14]: <https://github.com/antirez/sds/issues/100> [^15]: <https://github.com/kr/beanstalkd/issues/384> [^16]: <https://github.com/kr/beanstalkd/issues/382> [^17]: <https://github.com/LibreOffice/core>

--- abstract: 'The paper studies the suppression of cross-tier inter-cell interference (ICI) generated by a macro base station (MBS) to pico user equipments (PUEs) in heterogeneous networks (HetNets). Different from existing ICI avoidance schemes such as enhanced ICI cancellation (eICIC) and coordinated beamforming, which generally operate at the MBS, we propose a full duplex (FD) assisted ICI cancellation (fICIC) scheme, which can operate at each pico BS (PBS) individually and is transparent to the MBS. The basic idea of the fICIC is to apply FD technique at the PBS such that the PBS can send the desired signals and forward the listened cross-tier ICI simultaneously to PUEs. We first consider the narrowband single-user case, where the MBS serves a single macro UE and each PBS serves a single PUE. We obtain the closed-form solution of the optimal fICIC scheme, and analyze its asymptotical performance in ICI-dominated scenario. We then investigate the general narrowband multi-user case, where both MBS and PBSs serve multiple UEs. We devise a low-complexity algorithm to optimize the fICIC aimed at maximizing the downlink sum rate of the PUEs subject to user fairness constraint. Finally, the generalization of the fICIC to wideband systems is investigated. Simulations validate the analytical results and demonstrate the advantages of the fICIC on mitigating cross-tier ICI.' author: - 'Shengqian Han, Chenyang Yang, and Pan Chen [^1]' bibliography: - 'IEEEabrv.bib' - 'Hanbib\_D2D.bib' title: 'Full Duplex Assisted Inter-cell Interference Cancellation in Heterogeneous Networks' --- Inter-cell interference cancellation, Full duplex, Heterogeneous networks, eICIC, CoMP, OFDM. Introduction ============ Cellular systems are evolving toward heterogeneous networks (HetNets) with universal frequency reuse in order to support the galloping demand of mobile wireless services [@Soret2013]. By deploying low-power nodes such as micro, pico, or femto base stations (BSs) within the coverage of traditional macro BSs (MBSs), HetNets bring the network close to the user equipments (UEs) to obtain the “cell-splitting” gain, and therefore improve the area spectral efficiency. For simplicity, we refer to the low-power nodes as pico BSs (PBSs) hereinafter. In practice, however, straightforwardly deploying PBSs in the coverage of MBSs cannot effectively realize the promised benefits of HetNets because the large difference of the two types of BSs in transmit power makes pico UEs (PUEs) suffer from severe cross-tier inter-cell interference (ICI) generated by the MBS. Efficient ICI cancellation (ICIC) mechanisms are therefore critical to support the HetNet deployment [@Soret2013]. To mitigate the ICI, enhanced ICIC (eICIC) techniques have been developed in Release 10 of Long-term Evolution (LTE)[@Soret2013]. In the time-domain eICIC, the MBS remains silent in the so-called almost blank subframes (ABS), during which the cell-edge PUEs are served without interference. In the frequency-domain eICIC, the MBS and PBSs schedule UEs in orthogonal frequency resources to avoid the ICI. The eICIC methods are of low complexity and easy to implement, but they limit the performance of both macro UEs (MUEs) and PUEs since the UEs can be only served in partial time-frequency resources. Coordinated multi-point (CoMP) transmission is another promising technique for cross-tier ICI suppression, which exploits the antenna resources of the MBS [@Yang2013]. Considering the fact that providing high-performance backhaul from all PBSs to the core network may be cost prohibitive, coordinated beamforming (CB) based CoMP transmission, requiring no data sharing among the BSs, has received wide attention. CoMP-CB can be considered as a sort of spatial-domain eICIC, which enables the PUEs to be served with the whole time-frequency resources. However, the performance of CoMP-CB is limited by the number of antennas at the MBS, which is usually not acceptable when the PBSs are densely deployed in the coverage of the MBS. Different from eICIC and CoMP-CB, both of which control the transmission of the MBS in time, frequency or spatial domain in order to generate an ICI-free environment for the transmission between PBSs and PUEs, in this paper we strive to study the cross-tier ICI suppression scheme without the participation of MBSs, which therefore will not consume the resources of the MBS. Specifically, we consider using full duplex (FD) technique to HetNets. FD communication was long believed impossible in wireless system design due to the severe self-interference within the same transceiver. However, the belief has been overturned recently with the tremendous progress in self-interference cancellation [@Choi2010; @Duarte2012], where the plausibility of FD technique for short-range point-to-point communications was approved. FD techniques have been applied to provide bi-directional communications over the same time and frequency resources, and exhibited noticeable spectral efficiency gains over half-duplex (HD) schemes [@Hong2014; @Everett2014]. FD techniques are also applied in relay systems to improve service coverage, where the usage of FD avoids the waste of resources as in HD relay systems [@Riihonen2009; @Day2012]. To the best of our knowledge, leveraging FD in HetNets to suppress the cross-tier ICI has not been addressed in the literature. $B, K_M, K_P$ Numbers of PBSs, MUEs, and PUEs per pico cell ------------------------------------------------ ---------------------------------------------------------------------------------- $M, N_t$ Numbers of transmit antennas at the MBS and each PBS $N_r, N_t$ Numbers of FD and HD receive antennas and HD $\mathbf{h}_{Pk}, \mathbf{H}_{PP}$ Channel from PBS to ${\text{PUE}}_k$ and self-interference channel at the FD PBS $\bar{\mathbf{H}}_{MP}, \bar{\mathbf{h}}_{Mk}$ Equivalent channels from MBS to PBS and ${\text{PUE}}_k$ $\bar{\mathbf{h}}_{MP}, \bar{h}_{Mk}$ Equivalent channels from MBS to PBS and ${\text{PUE}}_k$ in single-user case $\bar{\mathbf{g}}_{MPn}, \bar{g}_{Mkn}$ Equivalent frequency-domain channels from MBS to PBS and ${\text{PUE}}_k$ $\mathbf{y}'_P, \mathbf{y}_P$ Receive signals of FD PBS before and after self-interference cancellation $\mathbf{z}_x, \mathbf{z}_y$ Transmitter and receiver distortions from hardware impairments $\mathbf{W}_f, L$ Forwarding precoder at PBS and its order in wideband systems $\mathbf{w}_f$ Vectorization of $\mathbf{W}_f^H$ $\mathbf{w}_{d,k}$ Precoder at PBS for desired signals of ${\text{PUE}}_k$ $\bar{\mathbf{W}}_{fn}, \bar{\mathbf{w}}_{dn}$ Frequency-domain precoders for forwarded ICI and desired signals $P_{out}, P_0$ Total and maximal transmit power of PBS $P_{out,n}$ Transmit power on the $n$-th subcarrier of PBS : List of major symbols[]{data-label="T:Acronyms"} In this paper, we consider a HetNet consisting of a MBS and multiple PBSs. We investigate the application of FD technique to mitigate the ICI generated by the MBS to the PUEs. The main contributions are summarized as follows: - We propose a novel ICI cancellation scheme for HetNets, named FD assisted ICI cancellation (fICIC), where the PBSs are supposed to have the FD capability of transmitting and receiving simultaneously. The basic idea of the proposed fICIC is to let the FD PBSs forward the listened interference from the MBS to the PUEs to neutralize the ICI, while at the same time sending the desired signals. Since the ICI is mitigated at PBSs now, the proposed fICIC is transparent to the MBS in the sense that no changes are needed for the transmission of the MBS. Note that a similar usage of the FD technique was presented in [@Zheng2013] for the cooperative cognitive network, where the FD secondary BS forwards the listened primary signals in order to increase the primary spectrum accessing opportunities, which leads to completely different problem and strategy design from ours. - In narrowband single-user case, where the MBS serves a single MUE and each PBS serves a single PUE, we first find the explicit expressions of the optimal fICIC precoders, which maximize the signal-to-interference-plus-noise ratio (SINR) of the PUE in each pico cell. We then analyze the asymptotical performance of the fICIC in ICI-dominated scenario. The results show that under perfect self-interference cancellation for FD, the fICIC can thoroughly eliminate weak ICI, while when the ICI is very strong or the residual self-interference is very large, the fICIC will reduce to the HD scheme. - In narrowband multi-user case, where the MBS serves multiple MUEs and each PBS serves multiple PUEs, we propose a low-complexity algorithm to optimize the fICIC scheme, aimed at maximizing the downlink sum rate of each pico cell under the fairness constraint over the PUEs. Simulations validate the analytical results, and demonstrate a significant performance gain of the fICIC over the HD scheme as well as evident benefits of combining the fICIC with existing eICIC and CoMP techniques. - We generalize the narrowband fICIC to orthogonal frequency-division multiplexing (OFDM) systems, where the optimization problem for the fICIC precoder design aimed at maximizing sum rate over multiple subcarriers is obtained, which can be solved with a gradient based method. Simulations shows the advantages of the fICIC on mitigating wideband ICI. **Notations:** $(\cdot)^T$, $(\cdot)^\ast$, and $(\cdot)^H$ denote transpose, complex conjugate, and conjugate transpose, respectively. $\mathbf{X} \succeq \mathbf{0}$ represents that matrix $\mathbf{X}$ is positive semi-definite, and $\mathbf{X}^{\frac{1}{2}}$ denotes hermitian square root of $\mathbf{X}$. $\mathtt{tr}(\cdot)$ denotes matrix trace, $\mathtt{rank}(\cdot)$ denotes matrix rank, $\mathcal{E}\{\cdot\}$ denotes expectation operator, $\mathtt{vec}(\cdot)$ denotes vectorization operator, $\otimes$ denotes Kronecker product, $\odot$ denotes convolution product, and $\|\cdot\|$ denotes Euclidian norm. $\mathtt{diag}(\mathbf{X})$ denotes the diagonal matrix with the same diagonal elements as $\mathbf{X}$. $\mathbf{I}_N$, $\mathbf{0}_N$ and $\bar{\mathbf{0}}_N$ denote $N\times N$ identity and zero matrices, and $N\times 1$ zero vector, respectively. The major symbols used in the paper are listed in Table \[T:Acronyms\]. System Model {#S:system_model} ============ We first consider the downlink transmission of a narrowband time division duplex (TDD) HetNet, and then generalize the model to wideband systems. Suppose that the HetNet consists of one MBS and $B$ PBSs, where the MBS serves ${K_M}$ single-antenna MUEs and each PBS serves ${K_P}$ single-antenna PUEs. Assume that the MUEs experience negligible interference from PBSs due to the coverage range expansion (CRE) of pico cells, and the pico cells are geographically separated so that each PUE receives much weaker interference from interfering PBSs compared to the interference generated by the MBS, which is treated as noise in the paper. Therefore, we focus on the suppression of the cross-tier ICI generated by the MBS to PUEs, which is commonly recognized as a bottleneck to improve the spectral efficiency in real-world HetNets [@Soret2013]. We consider applying FD technique at each PBS in the downlink transmission, with which the PBS can send the desired signals and forward the listened cross-tier ICI simultaneously. The structure of the FD PBS transceiver is illustrated in Fig. \[F:system\], where uplink and downlink data flows are indicated by dashed and solid lines, respectively. The FD PBS is comprised of traditional baseband (BB) and radio frequency (RF) modules in HD transceiver, taking charge of the transmission and reception of desired signals of PUEs, as well as additional FD modules, dedicated for self-interference cancellation and ICI suppression. The antennas at the PBS can be divided into two parts as shown in Fig. \[F:system\], where the “FD receive antennas” are only used to receive the ICI from the MBS in the downlink, while the “transmit antennas” and the “HD receive antennas” share the same antennas in a TDD manner, which are used to send and receive the signals of PUEs in downlink and uplink, respectively. Therefore, the uplink and downlink channel reciprocity holds between the PBS and the PUE.     Since the proposed fICIC scheme will not affect the performance of MUEs and other-cell PUEs, in the sequel we only consider a reference PBS and focus on the performance of the PUEs served by the reference PBS. The resulting interference environment is demonstrated in Fig. \[F:network\]. Suppose that the MBS has $M$ transmit antennas, $M \geq {K_M}$, and the FD PBS has $N_t$ transmit antennas, $N_t$ HD receive antennas, and $N_r$ FD receive antennas, $N_t \geq {K_P}$. Let $\mathbf{h}_{Mk}\in\mathbb{C}^{M\times 1}$ and $\mathbf{h}_{Pk}\in\mathbb{C}^{N_t\times 1}$ denote the channels from the MBS and the PBS to the $k$-th PUE (denoted by ${\text{PUE}}_k$), $\mathbf{H}_{MP}\in\mathbb{C}^{M\times N_r}$ denote the channel from the MBS to the PBS, and $\mathbf{H}_{PP}\in\mathbb{C}^{N_t\times N_r}$ denote the self-interference channel of the FD PBS. Signals of the FD PBS {#S:system_model_1} --------------------- In the downlink the FD PBS can transmit and receive signals simultaneously. We use the index $t$ to denote time instant. To reflect the impact of hardware impairments of transmitter chains on self-interference cancellation, we can express the receive signal at the PBS before self-interference cancellation based on [@Day2012; @Zheng2013] as $$\label{E:receivesignal} \bar{\mathbf{y}}_p[t] = \mathbf{H}_{MP}^H \mathbf{W}_M \mathbf{s}_M[t] + \mathbf{H}_{PP}^H \big(\mathbf{x}_p[t] + \mathbf{z}_x[t]\big) + \mathbf{n}_p[t],$$ where $\mathbf{W}_M\in\mathbb{C}^{M\times {K_M}}$ is the precoding matrix at the MBS for sending the signals $\mathbf{s}_M\sim\mathcal{CN}(\bar{\mathbf{0}}_{K_M}, \mathbf{I}_{{K_M}})$ to the MUEs, the term $\mathbf{H}_{PP}^H \big(\mathbf{x}_p[t] + \mathbf{z}_x[t]\big)$ is the self-interference, $\mathbf{x}_p\in\mathbb{C}^{N_t\times 1}$ is the transmit signal vector of the FD PBS, $\mathbf{z}_x\sim\mathcal{CN}(\bar{\mathbf{0}}_{N_t}, \mu_x\mathtt{diag}(\boldsymbol{\Phi}_{x}))$ is the transmitter distortion with $\boldsymbol{\Phi}_{x}$ denoting the covariance matrix of $\mathbf{x}_p$, $\mu_x\ll 1$ is a scaling constant, which reflects the combined effects of additive power-amplifier noise, non-linearities in digital-to-analog converter and power amplifier, I/Q imbalance, and oscillator phase noise, and $\mathbf{n}_p\!\sim\!\mathcal{CN}(\bar{\mathbf{0}}_{N_r}, \sigma_n^2\mathbf{I}_{N_r})$ is the additive white Gaussian noise (AWGN), which takes into account both thermal noises and the ICI from other PBSs. Further considering the hardware impairments of receiver chains based on [@Day2012; @Zheng2013], the distorted receive signal can expressed as $$\label{E:distorted-y} \mathbf{y}'_p[t] = \bar{\mathbf{y}}_p[t] + \mathbf{z}_y[t],$$ where $\mathbf{z}_y\sim\mathcal{CN}(\bar{\mathbf{0}}_{N_r}, \mu_y\mathtt{diag}(\boldsymbol{\Phi}_y))$ is the additive distortion caused by adaptive gain control noise, non-linearities in analog-to-digital converter and gain control, I/Q imbalance, and oscillator phase noise in receiver chains, $\mu_y\ll 1$ is a scaling constant, and $\boldsymbol{\Phi}_y$ is the covariance matrix of the undistorted receive signal $\bar{\mathbf{y}}_p$, which can be obtained from (\[E:receivesignal\]) as $$\begin{aligned} \label{E:Phi-y} \boldsymbol{\Phi}_y & = \mathcal{E}_{\mathbf{s}_M, \mathbf{x}_{p}, \mathbf{z}_x,\mathbf{n}_p}\{\bar{\mathbf{y}}_p[t]\bar{\mathbf{y}}_p^H[t]\} = \mathbf{H}_{MP}^H \mathbf{W}_M\mathbf{W}_M^H\mathbf{H}_{MP} \nonumber\\ & + \mathbf{H}_{PP}^H (\boldsymbol{\Phi}_x+\mu_x\mathtt{diag}(\boldsymbol{\Phi}_x)) \mathbf{H}_{PP} + \sigma_n^2\mathbf{I}_{N_r}.\end{aligned}$$ Since the transmit signal $\mathbf{x}_p$ is known at the FD PBS, the self-interference $\mathbf{H}_{PP}^H\mathbf{x}_p$ in (\[E:receivesignal\]) can be cancelled if the self-interference channel $\mathbf{H}_{PP}$ is estimated. During a dedicated training phase, the orthogonal pilot signals $\sqrt{P_{tr}}\mathbf{C}$ are transmitted for self-interference channel estimation, where $P_{tr}$ is the transmit power of pilot signals, and $\mathbf{C}=[\mathbf{c}_1,\dots,\mathbf{c}_{N_t}]\in\mathbb{C}^{N_t\times N_t}$ is unitary with $\mathbf{C}\mathbf{C}^H = \mathbf{I}_{N_t}$. Then, similar to (\[E:receivesignal\]) and (\[E:distorted-y\]), we can obtain the receive pilot signals with hardware impairments of both transmitter and receiver chains as $$\label{E:Training} \mathbf{Y}[t]\!=\! \tilde{\mathbf{Y}}[t] \!+ \!\tilde{\mathbf{Z}}_{y}[t] \!\triangleq\! \mathbf{H}_{PP}^H \big(\sqrt{P_{tr}}\mathbf{C}\!+\!\tilde{\mathbf{Z}}_{x}[t]\big)\! +\! \tilde{\mathbf{N}}[t] \!+\! \tilde{\mathbf{Z}}_{y}[t],$$ where $\tilde{\mathbf{Y}}= [\tilde{\mathbf{y}}_{1},\dots,\tilde{\mathbf{y}}_{N_t}] \in \mathbb{C}^{N_r\times N_t}$ denotes the undistorted receive signal, $\tilde{\mathbf{Z}}_y=[\tilde{\mathbf{z}}_{y,1},\dots,\tilde{\mathbf{z}}_{y,N_t}] \in \mathbb{C}^{N_r\times N_t}$ is the additive distortion of receive signals with $\tilde{\mathbf{z}}_{y,i}\sim\mathcal{CN}(\bar{\mathbf{0}}_{N_r}, \mu_y\mathtt{diag}(\tilde{\boldsymbol{\Phi}}_{y,i}))$, $\tilde{\boldsymbol{\Phi}}_{y,i}$ is the covariance matrix of $\tilde{\mathbf{y}}_{i}$, $\tilde{\mathbf{Z}}_x=[\tilde{\mathbf{z}}_{x,1},\dots,\tilde{\mathbf{z}}_{x,N_t}] \in \mathbb{C}^{N_t\times N_t}$ is the transmitter distortion with $\tilde{\mathbf{z}}_{x,i} \sim \mathcal{CN}(\bar{\mathbf{0}}_{N_t}, \mu_xP_{tr}\mathtt{diag}(\mathbf{c}_i\mathbf{c}_i^H))$, and $\tilde{\mathbf{N}} \in \mathbb{C}^{N_r\times N_t}$ is the AWGN whose columns follow the distribution $\mathcal{CN}(\bar{\mathbf{0}}_{N_r}, \sigma_n^2\mathbf{I}_{N_r})$. It can be obtained from (\[E:Training\]) that $\tilde{\boldsymbol{\Phi}}_{y,i} = P_{tr}\mathbf{H}_{PP}^H \big(\mathbf{c}_i\mathbf{c}_i^H+\mu_x\mathtt{diag}(\mathbf{c}_i\mathbf{c}_i^H)\big)\mathbf{H}_{PP} + \sigma_n^2\mathbf{I}_{N_r}$. With least-squares channel estimator, we can estimate the self-interference channel as $$\begin{aligned} \label{E:CE} \hat{\mathbf{H}}_{PP} = &\frac{1}{\sqrt{P_{tr}}}\mathbf{C}\mathbf{Y}^H[t] = \mathbf{H}_{PP} + \frac{1}{\sqrt{P_{tr}}}\big(\mathbf{C}\tilde{\mathbf{Z}}_x^H[t]\mathbf{H}_{PP} + \nonumber\\ &\mathbf{C}\tilde{\mathbf{N}}^H[t] + \mathbf{C}\tilde{\mathbf{Z}}_y^H[t]\big) \triangleq \mathbf{H}_{PP} + \mathbf{E}_{PP}[t],\end{aligned}$$ where $\mathbf{E}_{PP} = [\mathbf{e}_{PP,1},\dots,\mathbf{e}_{PP,N_t}]^H \in \mathbb{C}^{N_t\times N_r}$ denotes the channel estimation errors. Denoting $\mathbf{c}_i = [c_{i1}, \dots, c_{iN_t}]^T$, we can express $\mathbf{e}_{PP,i}$ as $$\mathbf{e}_{PP,i} = \frac{1}{\sqrt{P_{tr}}} \Big(\mathbf{H}_{PP}^H \sum_{j=1}^{N_t}{c}_{ij}\tilde{\mathbf{z}}_{x,j} + \tilde{\mathbf{N}}\mathbf{c}_i + \sum_{j=1}^{N_t} c_{ij}\tilde{\mathbf{z}}_{y,j}\Big),$$ from which we can obtain that $\mathbf{e}_{PP,i}$ follows the distribution $\mathcal{CN}(\bar{\mathbf{0}}_{N_r}, \tilde{\boldsymbol{\Phi}}_{e,i})$ with $$\begin{aligned} \label{E:Epp-cov} & \tilde{\boldsymbol{\Phi}}_{e,i} \!=\! \mathbf{H}_{PP}^H \sum_{j=1}^{N_t}|c_{ij}|^2\mu_x\mathtt{diag}(\mathbf{c}_j\mathbf{c}_j^H) \mathbf{H}_{PP} \! +\! \frac{(1\! +\! \mu_y)\sigma_n^2}{P_{tr}}\mathbf{I}_{N_r}\! +\! \nonumber\\ & \sum_{j=1}^{N_t} |c_{ij}|^2 \mu_y\mathtt{diag}\big(\mathbf{H}_{PP}^H \big(\mathbf{c}_j\mathbf{c}_j^H\! +\! \mu_x\mathtt{diag}(\mathbf{c}_j\mathbf{c}_j^H)\big)\mathbf{H}_{PP}\big).\end{aligned}$$ With $\hat{\mathbf{H}}_{PP}$, the receive signal at the PBS after self-interference cancellation is $$\begin{aligned} \label{E:yp} \mathbf{y}_p[t] = & \mathbf{y}_p'[t] - \hat{\mathbf{H}}_{PP}^H[t] \mathbf{x}_p[t] \triangleq \bar{\mathbf{H}}_{MP}^H \mathbf{s}_M[t] - \mathbf{E}_{PP}^H[t]\mathbf{x}_p[t] \nonumber\\ & + \mathbf{H}_{PP}^H \mathbf{z}_x[t] + \mathbf{n}_p[t] + \mathbf{z}_y[t],\end{aligned}$$ where $\bar{\mathbf{H}}_{MP} \triangleq \mathbf{W}_M^H\mathbf{H}_{MP}$ is the equivalent channel from the MBS to the FD PBS. The FD PBS then transmits the desired signals of the PUEs together with the self-interference cancelled receive signals $\mathbf{y}_p$. The combined transmit signal of the PBS can be expressed as $$\label{E:x2} \mathbf{x}_p[t] = \mathbf{W}_f \mathbf{y}_p[t-\tau] + \sum_{k=1}^{K_P} \mathbf{w}_{d,k} s_{p,k}[t],$$ where $\tau$ is the processing delay introduced by the FD modules, $\mathbf{W}_f\in\mathbb{C}^{N_t\times N_r}$ is the precoding matrix for the forwarded signals, $\mathbf{w}_{d,k}\in\mathbb{C}^{N_t\times 1}$ is the precoding vector for the desired signal, $s_{p,k}$, of ${\text{PUE}}_k$, and $s_{p,k}\sim\mathcal{CN}(0,1)$. In (\[E:x2\]), the receive signal $\mathbf{y}_p$ by $N_r$ additional FD receive antennas of the PBS are forwarded via $\mathbf{W}_f$. The forwarded signal will occupy a part of transmit power of the PBS, which leads to the reduction of the transmit power for desired signals. As will be clear later, however, the forwarded signal can be used to mitigate the cross-tier ICI at PUEs efficiently with the optimized $\mathbf{W}_f$, and hence result in the improvement of PUEs’ data rate. With (\[E:yp\]) and (\[E:x2\]) we can calculate the transmit power of the FD PBS as $$\label{E:Pout-new} P_{out} \!=\! \mathtt{tr}(\boldsymbol{\Phi}_x) \!=\! \mathtt{tr}\big(\mathcal{E}_{\mathbf{s}_M, s_{p,k}, \mathbf{n}_p, \mathbf{z}_x,\mathbf{z}_y, \mathbf{E}_{PP},\mathbf{H}_{PP}}\{\mathbf{x}_p[t]\mathbf{x}_p^H[t]\}\big),$$ where the expectations are taken over data $\mathbf{s}_M$ and $s_{p,k}$, noises $\mathbf{n}_p$, transmitter and receiver distortions $\mathbf{z}_x$ and $\mathbf{z}_y$, channel estimation errors $\mathbf{E}_{PP}$, and self-interference channel $\mathbf{H}_{PP}$, respectively.[^2] Assume that $\mathbf{H}_{PP}$ follows Rayleigh distribution, i.e., $\mathtt{vec}(\mathbf{H}_{PP})\sim\mathcal{CN}(\bar{\mathbf{0}}_{N_rN_t}, \bar{\alpha}_{PP}\mathbf{I}_{N_rN_t})$ with $\bar{\alpha}_{PP}$ denoting the average channel gain, which is reasonable because the transmit antennas and the FD receive antennas can be well isolated in the considered scenario as will be detailed in Section \[S:SIC\]. Then, we show in Appendix \[A:Pout\] that the transmit power $P_{out}$ can be expressed as $$\begin{aligned} \label{E:x22} P_{out} \approx & \mathtt{tr}\left(\mathbf{W}_f\bar{\mathbf{H}}_{MP}^H\bar{\mathbf{H}}_{MP}\mathbf{W}_f^H\right) + \sum_{k=1}^{K_P} \|\mathbf{w}_{d,k}\|^2\nonumber\\ & + \big(\sigma_n^2 + \sigma_e^2P_{out} \big) \mathtt{tr}(\mathbf{W}_f\mathbf{W}_f^H),\end{aligned}$$ where the approximation follows from $\mu_x\ll 1$ and $\mu_y\ll 1$ as in [@Day2012], and $\sigma^2_e = \frac{\sigma_n^2}{P_{tr}}+2\bar{\alpha}_{PP}(\mu_x+\mu_y)$ reflects the residual self-interference, in which the term $\frac{\sigma_n^2}{P_{tr}}$ comes from imperfect channel estimation and the term $2\bar{\alpha}_{PP}(\mu_x+\mu_y)$ comes from hardware impairments. From the equation with respect to $P_{out}$ given in (\[E:x22\]), we can obtain the transmit power of the FD PBS as $$\begin{aligned} \label{E:Pout} & P_{out} =\\ & \frac{\mathtt{tr}\!\big(\bar{\mathbf{H}}_{MP}\mathbf{W}_f^H\mathbf{W}_f\bar{\mathbf{H}}_{MP}^H\big) \!+\! \sigma_n^2\mathtt{tr}\!\big(\mathbf{W}_f^H\mathbf{W}_f\big) \!+\! \sum_{k=1}^{K_P}\! \|\mathbf{w}_{d,k}\|^2}{1 \!-\! \sigma_e^2\mathtt{tr}\big(\mathbf{W}_f^H\mathbf{W}_f\big)},\nonumber\end{aligned}$$ which shows that the power of the precoding matrix $\mathbf{W}_f$ for ICI forwarding must be limited by $$\label{E:oscillations} \mathtt{tr}\left(\mathbf{W}_f^H\mathbf{W}_f\right) < \frac{1}{\sigma_e^2}.$$ Otherwise, amplifier self-oscillations will occur at the FD PBS [@Riihonen2009]. Let $P_0$ denote the maximal transmit power of the PBS. Then the transmit power $P_{out}$ needs to satisfy $P_{out} \leq P_0$, which can be rewritten based on (\[E:Pout\]) as $$\begin{aligned} \label{E:PBPC} & \mathtt{tr}\left(\bar{\mathbf{H}}_{MP}\mathbf{W}_f^H\mathbf{W}_f\bar{\mathbf{H}}_{MP}^H\right) + (P_0\sigma_e^2+ \sigma_n^2)\mathtt{tr}\left(\mathbf{W}_f^H\mathbf{W}_f\right) \nonumber\\ & \qquad\qquad\qquad\qquad\qquad\quad\quad + \sum_{k=1}^{K_P} \|\mathbf{w}_{d,k}\|^2 \leq P_0.\end{aligned}$$ One can observe that the self-oscillation constraint (\[E:oscillations\]) is implicitly reflected in the maximal power constraint (\[E:PBPC\]). Therefore, in the following only the constraint in (\[E:PBPC\]) is considered. Signals of ${\text{PUE}}_k$ --------------------------- The receive signal of ${\text{PUE}}_k$ can be expressed as $$\label{E:yk} y_k[t] = \mathbf{h}_{Pk}^H\mathbf{x}_p[t] + \mathbf{h}_{Mk}^H\mathbf{W}_M\mathbf{s}_M[t] + n_k[t],$$ where the impact of hardware impairments at the HD ${\text{PUE}}$ is ignored as commonly considered for HD transmission in the literature, and $n_k[t]\sim\mathcal{CN}(0, \sigma_n^2)$ is the AWGN at ${\text{PUE}}_k$, which includes the received ICI from interfering PBSs and thermal noises. By noting the equivalence between time delay and phase shift in narrowband systems, we have $\mathbf{s}_M[t-\tau] = \mathbf{s}_M[t]e^{-j\phi}$, where $\phi = 2\pi f_c\tau$ with $f_c$ denoting the carrier frequency.[^3] Then based on (\[E:yp\]) and (\[E:x2\]), we can rewrite (\[E:yk\]) as $$\begin{aligned} \label{E:yk2} & y_k[t] = \mathbf{h}_{Pk}^H \mathbf{w}_{d,k} s_{p,k}[t] + \underbrace{\sum_{j=1, j\neq k}^{K_P} \mathbf{h}_{Pk}^H \mathbf{w}_{d,j} s_{p,j}[t]}_{\text{Intra-cell interference}} \nonumber\\ & + \mathbf{h}_{Pk}^H \mathbf{W}_f \big(-\mathbf{E}_{PP}^H[t-\tau]\mathbf{x}_p[t-\tau] + \mathbf{H}_{PP}^H \mathbf{z}_x[t-\tau] +\nonumber\\ &\quad\underbrace{\qquad\qquad\qquad\qquad\qquad\qquad\quad \mathbf{n}_p[t-\tau] + \mathbf{z}_y[t-\tau]\big)}_{\text{Forwarded residual self-interference and noises}} \nonumber\\ & + \underbrace{\left( \bar{\mathbf{h}}_{Mk}^H + \mathbf{h}_{Pk}^H \mathbf{W}_f \bar{\mathbf{H}}_{MP}^H e^{-j\phi} \right) \mathbf{s}_M[t]}_{\text{Cross-tier ICI}} + n_k[t],\end{aligned}$$ where $\bar{\mathbf{h}}_{Mk} \triangleq \mathbf{W}_M^H\mathbf{h}_{Mk}$ is the equivalent channel from the MBS to ${\text{PUE}}_k$. To achieve coherent combination between the ICI forwarded by the PBS and that directly received at the PUEs as shown in (\[E:yk2\]), the PBS needs to implement sample-by-sample ICI forwarding in time domain to reduce the processing delay as considered in [@Riihonen2008a; @Riihonen2008] for relay systems, where the time-domain FD amplify-and-forward relay was employed to provide co-phasing combining gain at the destination node. This is different from frequency domain forwarding for FD relay systems, e.g., in[@Riihonen2009] by the same authors as [@Riihonen2008a; @Riihonen2008], where the processing delay will be in symbol-level because a complete symbol needs to be received and demodulated before forwarding in the frequency domain. Similar to the derivations in Appendix \[A:Pout\], we can compute the signal and interference power from (\[E:yk2\]) by taking the expectation with respect to $\mathbf{s}_M$, $s_{p,k}$, $\mathbf{n}_p$, $\mathbf{z}_x$, $\mathbf{z}_y$, $\mathbf{E}_{PP}$, and $\mathbf{H}_{PP}$, and finally obtain the SINR of ${\text{PUE}}_k$ as $$\begin{aligned} \label{E:SINR3} &\mathtt{SINR}_{k,FD}= \nonumber\\ & |\mathbf{h}_{Pk}^H\mathbf{w}_{d,k}|^2 \!/\! \big(\sum_{j\neq k}\! \|\mathbf{h}_{Pk}^H \mathbf{w}_{d,j}\|^2 \!+\! \|\bar{\mathbf{h}}_{Mk}^H \!+\! \mathbf{h}_{Pk}^H \mathbf{W}_f \bar{\mathbf{H}}_{MP}^H e^{-j\phi}\!\|^2 \nonumber\\ &+ \|\mathbf{h}_{Pk}^H \mathbf{W}_f\|^2(P_{out}\sigma_e^2+\sigma_n^2) + \sigma_n^2\big),\end{aligned}$$ where $P_{out}$ is the transmit power of the PBS, which is a function of the precoders $\mathbf{W}_f$ and $\mathbf{w}_{d,k}$ as given in (\[E:Pout\]). Narrowband Single-user Case =========================== In this section, we investigate the single-user case, i.e., ${K_M}={K_P}=1$, to gain insight into the ICI mitigation mechanism of the fICIC scheme. In this case, the equivalent channel $\bar{\mathbf{H}}_{MP}$ is a row vector and $\bar{\mathbf{h}}_{Mk}$ is a scalar, which are denoted by $\bar{\mathbf{h}}_{MP}^H \in \mathbb{C}^{1\times N_r}$ and $\bar{h}_{Mk}\in \mathbb{C}^{1\times 1}$ for clarity, respectively. Moreover, noting that the intra-cell interference does not exist now, the SINR can be simplified as $$\begin{aligned} \label{E:SINR3-1} &\mathtt{SINR}_{k,FD} =\\ &\frac{|\mathbf{h}_{Pk}^H\mathbf{w}_{d,k}|^2}{ |\bar{h}_{Mk}^\ast \!+\! \mathbf{h}_{Pk}^H \mathbf{W}_f \bar{\mathbf{h}}_{MP} e^{-j\phi}|^2 \!+\! \|\mathbf{h}_{Pk}^H \mathbf{W}_f\|^2\!(P_{out}\sigma_e^2\!+\!\sigma_n^2) \!+\! \sigma_n^2}, \nonumber\end{aligned}$$ where $P_{out}$ given in (\[E:Pout\]) can be rewritten as $$\label{E:pout-2} P_{out} = \frac{\|\mathbf{W}_f\bar{\mathbf{h}}_{MP}\|^2 + \sigma_n^2\mathtt{tr}\left(\mathbf{W}_f^H\mathbf{W}_f\right) + \|\mathbf{w}_{d,k}\|^2}{1 - \sigma_e^2\mathtt{tr}\left(\mathbf{W}_f^H\mathbf{W}_f\right)}.$$ When the precoder for forwarding the listened ICI, $\mathbf{W}_f$, in (\[E:SINR3-1\]) is selected as zero, the FD PBS reduces to a HD PBS. It is not hard to obtain the optimal precoder for transmitting the desired signal in HD case that maximizes the SINR of ${\text{PUE}}_k$ as $\mathbf{w}_{d,k} = \frac{\sqrt{P_0}\mathbf{h}_{Pk}}{\|\mathbf{h}_{Pk}\|}$, which is referred to as “the HD scheme” in the sequel. The maximum SINR achieved by the HD scheme can be obtained from (\[E:SINR3-1\]) as $$\label{E:SINR3-HD} \mathtt{SINR}_{k,HD}^\star = \frac{P_0\|\mathbf{h}_{Pk}\|^2}{|\bar{h}_{Mk}|^2 + \sigma_n^2}.$$ Compared to the HD scheme where the ICI power is $|\bar{h}_{Mk}|^2$ as shown in (\[E:SINR3-HD\]), the FD scheme turns the ICI power into $|\bar{h}_{Mk}^\ast + \mathbf{h}_{Pk}^H \mathbf{W}_f \bar{\mathbf{h}}_{MP} e^{-j\phi}|^2$ as shown in (\[E:SINR3-1\]), which can be reduced by optimizing the precoders $\mathbf{W}_f$ and $\mathbf{w}_{d,k}$. The optimization problem, aimed at maximizing the SINR of ${\text{PUE}}_k$ subject to the maximal transmit power constraint, can be formulated as \[E:problem\] $$\begin{aligned} \underset{\mathbf{W}_f, \mathbf{w}_{d,k}}{\max}\ & \mathtt{SINR}_{k,FD} \label{E:Objective}\\ s.\ t.\ & \|\mathbf{W}_f\bar{\mathbf{h}}_{MP}\|^2 + (P_0\sigma_e^2 + \sigma_n^2)\mathtt{tr}\left(\mathbf{W}_f^H\mathbf{W}_f\right) \nonumber\\ &\qquad\qquad\qquad\qquad\qquad+ \|\mathbf{w}_{d,k}\|^2 \leq P_0, \label{E:Constraint} \end{aligned}$$ where the power constraint (\[E:Constraint\]) comes from (\[E:PBPC\]). Optimal fICIC Scheme -------------------- In this subsection, we strive to find the optimal fICIC scheme with explicit expressions for $\mathbf{W}_f$ and $\mathbf{w}_{d,k}$. Given that it is difficult to directly solve problem (\[E:problem\]) due to the complex expression of $\mathtt{SINR}_{k,FD}$ in (\[E:SINR3-1\]), we start with investigating the properties of the optimal solutions to problem (\[E:problem\]) in the following. First, by substituting the expression of $P_{out}$ given in (\[E:pout-2\]) into (\[E:SINR3-1\]), we can find that $\mathtt{SINR}_{k,FD}$, i.e., the objective function of problem (\[E:problem\]), is an increasing function of $\|\mathbf{w}_{d,k}\|^2$. Therefore, for any given $\mathbf{W}_{f}$ and the direction of $\mathbf{w}_{d,k}$, $\frac{\mathbf{w}_{d,k}}{\|\mathbf{w}_{d,k}\|}$, we can always improve the SINR by increasing $\|\mathbf{w}_{d,k}\|^2$ until the PBS transmits with its maximum power. This means that the global optimal solution to problem (\[E:problem\]) is obtained when the power constraint in (\[E:Constraint\]) holds with equality. Based on this result, we can obtain from (\[E:pout-2\]) that $P_{out} = P_0$, with which $\mathtt{SINR}_{k,FD}$ is simplified as $$\begin{aligned} \label{E:SINR3-2} &\mathtt{SINR}_{k,FD} = \\ &\frac{|\mathbf{h}_{Pk}^H\mathbf{w}_{d,k}|^2}{ |\bar{h}_{Mk}^\ast \!+\! \mathbf{h}_{Pk}^H \mathbf{W}_f \bar{\mathbf{h}}_{MP} e^{-j\phi}|^2 \!+\! \|\mathbf{h}_{Pk}^H \mathbf{W}_f\|^2\!(P_0\sigma_e^2\!+\!\sigma_n^2) \!+\! \sigma_n^2}. \nonumber\end{aligned}$$ Second, because the direction of $\mathbf{w}_{d,k}$ affects only the numerator of $\mathtt{SINR}_{k,FD}$, we can obtain that the optimal $\mathbf{w}_{d,k}^\star$ satisfies $$\label{E:w2_direction} \frac{\mathbf{w}_{d,k}^\star}{\|\mathbf{w}_{d,k}^\star\|} = \frac{\mathbf{h}_{Pk}}{\|\mathbf{h}_{Pk}\|}.$$ Third, we can show that the optimal $\mathbf{W}_{f}^\star$ has the following property. ***Lemma 1:** The optimal $\mathbf{W}_{f}^\star$ is of rank 1, which can be decomposed as* $$\label{E:w1} \mathbf{W}_{f}^\star = - \bar{h}_{Mk}^\ast e^{j\phi}\cdot \beta \cdot\mathbf{h}_{Pk}\bar{\mathbf{h}}_{MP}^H,$$ where $\beta$ is a positive scalar. See Appendix \[S:prooflemma2\]. We can see from the expression of the optimal $\mathbf{W}_{f}^\star$ that with the fICIC the listened ICI is first enhanced by the maximal ratio combining with $\bar{\mathbf{h}}_{MP}^H$, and then forwarded by the maximal ratio transmission with $\mathbf{h}_{Pk}$. Based on (\[E:w2\_direction\]) and (\[E:w1\]), we can express the SINR with $\beta$ and $\|\mathbf{w}_{d,k}\|$. Further considering that the optimal solution is obtained when the power constraint in (\[E:Constraint\]) holds with equality, we can replace $\|\mathbf{w}_{d,k}\|$ with $\beta$, and finally convert problem (\[E:problem\]) into the following optimization problem \[E:problem1\] $$\begin{aligned} \underset{\beta}{\max}\ & f_0(\beta) \label{E:Objective1}\\ s.\ t.\ & \left( \|\bar{\mathbf{h}}_{MP}\|^2 + \sigma_I^2+\sigma_n^2 \right)|\bar{h}_{Mk}|^2\|\mathbf{h}_{Pk}\|^2 \|\bar{\mathbf{h}}_{MP}\|^2 \beta^2 \leq P_0 \label{E:Constraint1}\\ \ &\beta \geq 0, \end{aligned}$$ where $f_0(\beta) \triangleq \Big(\|\mathbf{h}_{Pk}^H\|^2\big(P_0 - \big(\|\bar{\mathbf{h}}_{MP}\|^2 + \sigma_I^2+\sigma_n^2 \big) |\bar{h}_{Mk}|^2\|\mathbf{h}_{Pk}\|^2 \|\bar{\mathbf{h}}_{MP}\|^2 \beta^2 \big)\Big) / \Big( \big(|\bar{h}_{Mk}| - \beta|\bar{h}_{Mk}|\|\mathbf{h}_{Pk}\|^2 \|\bar{\mathbf{h}}_{MP}\|^2\big)^2 + \beta^2|\bar{h}_{Mk}|^2\|\mathbf{h}_{Pk}\|^4 \|\bar{\mathbf{h}}_{MP}\|^2(\sigma_I^2+\sigma_n^2) + \sigma_n^2\Big)$, and $\sigma_I^2 \triangleq P_0\sigma_e^2$ denotes the average power of residual self-interference. ***Proposition 1:** The maximum of the objective function in (\[E:Objective1\]), i.e. the maximal SINR of ${\text{PUE}}_k$, can be expressed as* $$\label{E:maxSINR} \mathtt{SINR}_{k,FD}^\star = \frac{1}{\frac{1}{B\beta^\star}-1},$$ *with $\beta^\star = \frac{2\left(A+D-\sqrt{(A+D)^2 - \frac{AC^2}{B}}\right)}{C^2}$ representing the optimal solution of $\beta$,* *where $A = P_0\|\mathbf{h}_{Pk}\|^2$, $B = \|\mathbf{h}_{Pk}\|^2 (\|\bar{\mathbf{h}}_{MP}\|^2 + \sigma_I^2 + \sigma_n^2)$, $C = 2|\bar{h}_{Mk}|\|\mathbf{h}_{Pk}\|\|\bar{\mathbf{h}}_{MP}\|$, and $D = |\bar{h}_{Mk}|^2+\sigma_n^2$.* See Appendix \[S:prooftheorem1\]. With the optimal $\beta^\star$, we can directly obtain the optimal $\mathbf{W}_f^\star$ from (\[E:w1\]). To compute the optimal $\mathbf{w}_{d,k}^\star$, recalling that constraint (\[E:Constraint\]) holds with equality for the optimal solutions, we can obtain the norm of the optimal $\mathbf{w}_{d,k}^\star$ as $$\label{E:w2_norm} \|\mathbf{w}_{d,k}^\star\| = \sqrt{P_0 - \frac{1}{4}\left(\|\bar{\mathbf{h}}_{MP}\|^2 + \sigma_I^2 + \sigma_n^2\right)C^2\beta^{\star 2}}.$$ Then by substituting (\[E:w2\_norm\]) into (\[E:w2\_direction\]), we can obtain $\mathbf{w}_{d,k}^\star$. Asymptotic Performance Analysis {#S:asymptotic} ------------------------------- To understand the behavior of the fICIC scheme, we next consider an ICI-dominated scenario, where the noise power $\sigma_n^2$ approaches to zero. ### Perfect Self-interference Cancellation If the self-interference can be perfectly eliminated, i.e., $\sigma_I^2 = 0$, from the definitions after (\[E:maxSINR\]) we have $B \doteq \|\mathbf{h}_{Pk}\|^2\|\bar{\mathbf{h}}_{MP}\|^2$ and $D \doteq |\bar{h}_{Mk}|^2$, where $\doteq$ denotes asymptotic equality. Then based on Proposition 1, after some manipulations we can obtain the optimal $\beta^\star$ as $$\begin{aligned} \label{E:w1-app} &\beta^\star\doteq \frac{P_0\|\mathbf{h}_{Pk}\|^2 + |\bar{h}_{Mk}|^2 - \left|P_0\|\mathbf{h}_{Pk}\|^2 - |\bar{h}_{Mk}|^2\right|}{2\|\bar{\mathbf{h}}_{MP}\|^2|\bar{h}_{Mk}|^2\|\mathbf{h}_{Pk}\|^2}\\ & = \frac{\min(P_0\|\mathbf{h}_{Pk}\|^2, |\bar{h}_{Mk}|^2)}{\|\bar{\mathbf{h}}_{MP}\|^2|\bar{h}_{Mk}|^2\|\mathbf{h}_{Pk}\|^2} \triangleq \frac{\eta}{\|\bar{\mathbf{h}}_{MP}\|^2|\bar{h}_{Mk}|^2\|\mathbf{h}_{Pk}\|^2}.\nonumber\end{aligned}$$ By substituting (\[E:w1-app\]) into (\[E:Objective1\]), the maximal SINR of ${\text{PUE}}_k$ becomes $$\label{E:maxSINR-app} \mathtt{SINR}_{k,FD}^\star \doteq \frac{P_0|\bar{h}_{Mk}|^2\|\mathbf{h}_{Pk}\|^2 \!-\! \eta^2}{(\eta \!-\! |\bar{h}_{Mk}|^2)^2 \!+\! \left( \frac{\eta^2}{\|\bar{\mathbf{h}}_{MP}\|^2} \!+\! |\bar{h}_{Mk}|^2\right)\sigma_n^2},$$ where $\eta = \min(P_0\|\mathbf{h}_{Pk}\|^2, |\bar{h}_{Mk}|^2)$ as defined in (\[E:w1-app\]), which depends on the strengths of the desired signal, $P_0\|\mathbf{h}_{Pk}|^2$, and the ICI, $|\bar{h}_{Mk}|^2$. In the following two cases are discussed. - **Case 1:** $|\bar{h}_{Mk}|^2 < P_0\|\mathbf{h}_{Pk}\|^2$ This is a case where the ICI is weaker than the desired signal. Then, we have $\eta = |\bar{h}_{Mk}|^2$ and the maximal SINR is $$\label{E:maxSINR-largeS} \overline{\mathtt{SINR}}_{k,FD}^\star \doteq \frac{P_0\|\mathbf{h}_{Pk}\|^2 - |\bar{h}_{Mk}|^2}{\left(\frac{|\bar{h}_{Mk}|^2}{\|\bar{\mathbf{h}}_{MP}\|^2} + 1\right)\sigma_n^2}.$$ It implies that the FD PBS can thoroughly eliminate the ICI generated by the MBS by properly designing the forwarding and transmitting precoders. When compared with the HD scheme, the performance gain of the fICIC can be obtained as $$\label{E:FD-HD-1} \frac{\overline{\mathtt{SINR}}_{k,FD}^\star}{\mathtt{SINR}_{k,HD}^\star} \doteq \frac{ 1 - \frac{|\bar{h}_{Mk}|^2}{P_0\|\mathbf{h}_{Pk}\|^2}}{\left(\frac{1}{\|\bar{\mathbf{h}}_{MP}\|^2} + \frac{1}{|\bar{h}_{Mk}|^2} \right)\sigma_n^2}.$$ - **Case 2:** $|\bar{h}_{Mk}|^2 \geq P_0\|\mathbf{h}_{Pk}\|^2$ In this case where the ICI is stronger, we have $\eta = P_0\|\mathbf{h}_{Pk}\|^2$ and the maximal SINR is $$\label{E:maxSINR-largeI-1} \widehat{\mathtt{SINR}}_{k,FD}^\star \doteq \frac{P_0\|\mathbf{h}_{Pk}\|^2}{|\bar{h}_{Mk}|^2\!- \!P_0\|\mathbf{h}_{Pk}\|^2 \!+ \! \frac{\frac{P_0^2\|\mathbf{h}_{Pk}\|^4}{\|\bar{\mathbf{h}}_{MP}\|^2} \!+\! |\bar{h}_{Mk}|^2}{|\bar{h}_{Mk}|^2\!-\!P_0\|\mathbf{h}_{Pk}\|^2}\sigma_n^2}.$$ For very strong interference, i.e., $|\bar{h}_{Mk}|^2 \gg P_0\|\mathbf{h}_{Pk}\|^2$, $\widehat{\mathtt{SINR}}_{k,FD}^\star$ can be approximated as $$\label{E:maxSINR-largeI} \widehat{\mathtt{SINR}}_{k,FD}^\star \approx \frac{P_0\|\mathbf{h}_{Pk}\|^2}{|\bar{h}_{Mk}|^2} \doteq \mathtt{SINR}_{k,HD}^\star.$$ From the above analysis, we can obtain the following observations. - ***Impact of $\bar{\mathbf{h}}_{MP}$**:* It is shown from (\[E:maxSINR-largeS\]) and (\[E:maxSINR-largeI-1\]) that the SINR of ${\text{PUE}}_k$ increases with the channel gain between MBS and PBS $\|\bar{\mathbf{h}}_{MP}\|$. This is because given the power of the PBS allocated for forwarding the listened ICI, denoted as $$\label{E:Powerforward} P_{out}^{fw} \triangleq \|\mathbf{W}_f\bar{\mathbf{h}}_{MP}\|^2 + (\sigma_I^2 + \sigma_n^2)\mathtt{tr}\left(\mathbf{W}_f^H\mathbf{W}_f\right),$$ a large $\|\bar{\mathbf{h}}_{MP}\|$ will reduce the power used for forwarding the residual self-interference and noises and thus improve the efficiency of power usage. However, when the ICI power $|\bar{h}_{Mk}|^2$ is very large, the impact of $\|\bar{\mathbf{h}}_{MP}\|$ can be neglected as shown in (\[E:maxSINR-largeI\]). - ***Impact of ${\mathbf{h}}_{Pk}$ and $\bar{h}_{Mk}$**:* As shown by (\[E:SINR3-HD\]), (\[E:maxSINR-largeS\]) and (\[E:maxSINR-largeI\]), increasing the desired signal $P_0\|\mathbf{h}_{Pk}\|^2$ can improve the performance of both the FD scheme and the HD scheme.[^4] However, the performance improvement for the FD scheme is more significant because (\[E:FD-HD-1\]) shows that the performance gain of FD over HD increases with $P_0\|\mathbf{h}_{Pk}\|^2$. It can also be seen that the performance gain of FD over HD decreases with the ICI power $|\bar{h}_{Mk}|^2$, until vanishes for very large $|\bar{h}_{Mk}|^2$ as shown by (\[E:maxSINR-largeI\]). It should be pointed out that the extreme case with very strong ICI in (\[E:maxSINR-largeI\]) rarely happens in practice even when the maximum CRE offset of $9$ dB in LTE systems is considered [@CRE2013]. As will be verified in Fig. \[F:placement\], the fICIC still exhibits evident performance gain over the HD scheme when the average power of the ICI is $10.4$ dB stronger than the desired signal. ### Large Residual Self-interference When the self-interference cancellation for FD is imperfect and the residual self-interference, $\sigma_I^2$, is large, the parameter $B$ is large and the term $\frac{AC^2}{B}$ is small. Then by using the first-order taylor expansion, $\sqrt{c-z} \doteq \sqrt{c} - \frac{1}{2\sqrt{c}}z$ for small $z$, we can obtain $\beta^\star \doteq \frac{A}{(A+D)B}$ when $\frac{AC^2}{B}$ approaches to zero, with which we can obtain from (\[E:maxSINR\]) the maximal SINR of ${\text{PUE}}_k$ as $$\label{E:SINR-large-selfI} \widetilde{\mathtt{SINR}}_{k,FD}^\star \doteq \frac{P_0\|\mathbf{h}_{Pk}\|^2}{|\bar{h}_{Mk}|^2 + \sigma_n^2} = \mathtt{SINR}_{k,HD}^\star.$$ Moreover, we can compute the power of the PBS allocated for forwarding the listened ICI from (\[E:Powerforward\]) as $$\begin{aligned} \label{E:Forward-power} & P_{out}^{fw}\doteq \frac{P_0^2\|\bar{\mathbf{h}}_{MP}\|^2|\bar{h}_{Mk}|^2\|\mathbf{h}_{Pk}\|^2}{(|\bar{h}_{Mk }|^2 \!+\! P_0\|\mathbf{h}_{Pk}\|^2 \!+ \! \sigma_n^2)^2(\|\bar{\mathbf{h}}_{MP}\|^2\! +\! \sigma_I^2 \!+\! \sigma_n^2)}.\end{aligned}$$ It can be seen from (\[E:Forward-power\]) that the transmit power of the PBS allocated for forwarding ICI decreases with the growth of residual self-interference $\sigma_I^2$. When $\sigma_I^2$ is very large, the PBS will use all power to transmit desired signals, and therefore the fICIC will reduce to the HD scheme. Narrowband Multi-user Case ========================== In this section, we consider the general multi-user case, where the MBS serves ${K_M}$ MUEs and the PBS serves ${K_P}$ PUEs with ${K_M} \geq 1$ and ${K_P} \geq 1$. We optimize the fICIC scheme, aimed at maximizing the sum rate of ${K_P}$ PUEs served by the reference PBS while guaranteeing the fairness among the PUEs, subject to the maximal transmit power constraint of the PBS. The problem can be formulated as \[E:problem-MU\] $$\begin{aligned} & \underset{\mathbf{W}_f, \{\mathbf{w}_{d,k}\}}{\max}\ R_{sum} \label{E:Objective-MU}\\ & s.\ t.\ \log(1 \!+\! \mathtt{SINR}_{k,FD}) \!=\! \alpha_k R_{sum}, \ k = 1,\dots, K_P \label{E:fairness-constraint}\\ & \ \ \ \ \mathtt{tr}\left(\bar{\mathbf{H}}_{MP}\mathbf{W}_f^H\mathbf{W}_f\bar{\mathbf{H}}_{MP}^H\right) + (\sigma_I^2 + \sigma_n^2)\mathtt{tr}\left(\mathbf{W}_f^H\mathbf{W}_f\right)\nonumber\\ & \qquad\qquad\qquad\qquad\qquad\qquad + \sum_{k=1}^{K_P} \|\mathbf{w}_{d,k}\|^2 \leq P_0, \label{E:Constraint-MU} \end{aligned}$$ where $R_{sum}$ is the sum rate of all PUEs, the constraints in (\[E:fairness-constraint\]) ensure the data rate proportion among the PUEs with predefined fairness factors $\{\alpha_k\}$ as considered in [@Emil2012], $\alpha_k \geq 0$, and $\sum_{k=1}^{K_P} \alpha_k = 1$. The expression of $\mathtt{SINR}_{k,FD}$ in multi-user case is given in (\[E:SINR3\]), which is a function of the transmit power $P_{out}$ as shown in (\[E:Pout\]) and thus is very complicated. To simplify $\mathtt{SINR}_{k,FD}$, we can prove that the optimal solution to problem (\[E:problem-MU\]) is obtained when the power constraint (\[E:Constraint-MU\]) holds with equality.[^5] Based on this result, we have $P_{out} = P_0$ in (\[E:SINR3\]). Problem (\[E:problem-MU\]) is to find the maximal achievable $R_{sum}$, denoted by $R_{sum}^\star$, ensuring all constraints satisfied, which can be solved by bisection methods [@boyd2009convex]. Specifically, for a given $R_{sum}$ in an iteration, denoted by $R_{sum}^0$, if the following optimization problem \[E:problem-MU-bisection\] $$\begin{aligned} \mathtt{Find}\ & \mathbf{W}_f, \{\mathbf{w}_{d,k}\} \label{E:Objective-MU-bisection}\\ s.\ t.\ \ & (\ref{E:Constraint-MU})\nonumber \\ & \mathtt{SINR}_{k,FD} = 2^{\alpha_k R_{sum}^0} - 1, \ \ k = 1,\dots, K_P \label{E:fairness-constraint-bisection} \end{aligned}$$ is feasible, then it follows that $R_{sum}^0$is an achievable sum rate of all ${K_P}$ ${\text{PUE}}$s, i.e., $R_{sum}^0 \leq R_{sum}^\star$, otherwise, $R_{sum}^0 > R_{sum}^\star$. This condition can be used in bisection algorithms to find $R_{sum}^\star$. Now the remaining issue is to find efficient approaches to evaluate the feasibility of problem (\[E:problem-MU-bisection\]). In the following, we show that the feasibility problem can be solved by investigating the optimization problem below \[E:problem-MU-reform-feasibility\] $$\begin{aligned} \underset{\mathbf{W}_f, \{\mathbf{w}_{d,k}\}}{\min}\ & \mathtt{tr}\left(\bar{\mathbf{H}}_{MP}\mathbf{W}_f^H\mathbf{W}_f\bar{\mathbf{H}}_{MP}^H\right) + (\sigma_I^2 + \sigma_n^2)\cdot\nonumber\\ &\qquad\mathtt{tr}\left(\mathbf{W}_f^H\mathbf{W}_f\right) + \sum_{k=1}^{K_P} \|\mathbf{w}_{d,k}\|^2 \label{E:Objective-MU-reform-feasibility}\\ s.\ t.\ \ & \mathtt{SINR}_{k,FD} \geq \gamma_{k}, \ k = 1,\dots, {K_P}, \label{E:constraint-SINR-feasibility} \end{aligned}$$ where the objective function is the left-hand side of constraint (\[E:Constraint-MU\]), and $\gamma_k \triangleq 2^{\alpha_k R_{sum}^0} - 1$. To see this, we note that the optimal solution to problem (\[E:problem-MU-reform-feasibility\]) is obtained when all SINR constraints in (\[E:constraint-SINR-feasibility\]) hold with equality, otherwise, the value of the objective function can be always further reduced by properly decreasing $\|\mathbf{w}_{d,k}\|^2$. It suggests that if the minimum of the objective function (\[E:Objective-MU-reform-feasibility\]) is smaller than $P_0$, then constraints (\[E:Constraint-MU\]) and (\[E:fairness-constraint-bisection\]) in problem (\[E:problem-MU-bisection\]) are all satisfied. This means that problem (\[E:problem-MU-bisection\]) is feasible and $R_{sum}^0$ is an achievable sum rate, otherwise, problem (\[E:problem-MU-bisection\]) will be infeasible. In what follows, we solve problem (\[E:problem-MU-reform-feasibility\]). By defining $\mathbf{w}_f = \mathtt{vec}(\mathbf{W}_f^H)$, we can rewrite problem (\[E:problem-MU-reform-feasibility\]) as \[E:problem-MU-reform-feasibility-reform\] $$\begin{aligned} & \underset{\mathbf{w}_f, \{\mathbf{w}_{d,k}\}}{\min} \|\big(\mathbf{I}_{N_t}\!\otimes\! \bar{\mathbf{H}}_{MP}\big)\! \mathbf{w}_f\|^2\! +\! (\!\sigma_I^2 + \sigma_n^2\!)\|\mathbf{w}_f\|^2 \!+\! \sum_{k=1}^{K_P}\! \|\mathbf{w}_{d,k}\|^2 \label{E:Objective-MU-reform-feasibility-reform}\\ &s.\ t.\ \frac{|\mathbf{h}_{Pk}^H\mathbf{w}_{d,k}|^2}{\Omega_k} \geq \gamma_k, \ \forall k, .\label{E:constraint-SINR-feasibility-reform} \end{aligned}$$ where $\Omega_k \triangleq {\sum_{j\neq k}} \|\mathbf{h}_{Pk}^H \mathbf{w}_{d,j}\|^2 + \|\bar{\mathbf{h}}_{Mk} + e^{-j\phi} \left(\mathbf{h}_{Pk}^T \otimes\bar{\mathbf{H}}_{MP}\right) \mathbf{w}_f\|^2 + \|\left(\mathbf{h}_{Pk}^T\otimes\mathbf{I}_{N_r}\right) \mathbf{w}_f\|^2(\sigma_I^2+\sigma_n^2) + \sigma_n^2$. Problem (\[E:problem-MU-reform-feasibility-reform\]) is non-convex because the constraints in (\[E:constraint-SINR-feasibility-reform\]) are non-convex. A common method to solve such a problem is to convert the non-convex SINR constraints into convex second-order cone constraints [@Yu07]. Specifically, since adding any phase rotation to $\mathbf{w}_{d,k}$ will not affect the SINR of all ${\text{PUE}}$s, we can assume that $\mathbf{h}_{Pk}^H \mathbf{w}_{d,k}$ is real-valued, which does not affect the global optima of problem (\[E:problem-MU-reform-feasibility-reform\]). Then we can convert the non-convex problem (\[E:problem-MU-reform-feasibility-reform\]) into the following second-order cone constrained problem \[E:problem-MU-reform-feasibility-reform-socp\] $$\begin{aligned} \underset{\mathbf{w}_f, \mathbf{w}_{d}}{\min} & \|\big(\mathbf{I}_{N_t}\otimes \bar{\mathbf{H}}_{MP}\big)\! \mathbf{w}_f\|^2 \!+\! (\sigma_I^2 \!+\! \sigma_n^2)\!\|\mathbf{w}_f\|^2 \!+\! \|\mathbf{w}_{d}\|^2 \label{E:Objective-MU-reform-feasibility-reform-socp}\\ s.\ t.\ & \sqrt{1 + \frac{1}{\gamma_k}} \mathbf{h}_{Pk}^H\mathbf{w}_{d,k} \nonumber\\ & \geq \left\| \left[\! \begin{smallmatrix} \left(\mathbf{I}_{N_r}\otimes \mathbf{h}_{Pk}^H\right) \mathbf{w}_d\\ e^{-j\phi} \left(\mathbf{h}_{Pk}^T \otimes\bar{\mathbf{H}}_{MP}\right) \mathbf{w}_f\\ \sqrt{\sigma_I^2+\sigma_n^2}\left(\mathbf{h}_{Pk}^T\otimes\mathbf{I}_{N_r}\right)\mathbf{w}_f\\ \sigma_n \end{smallmatrix} \!\right] \!+\! \left[\! \begin{smallmatrix} \bar{\mathbf{0}}_{N_r}\\ \bar{\mathbf{h}}_{Mk}\\ \bar{\mathbf{0}}_{N_r}\\ 0 \end{smallmatrix} \!\right] \right\|, \forall k, \label{E:constraint-SINR-feasibility-reform-socp} \end{aligned}$$ where $\mathbf{w}_d = \mathtt{vec}\left([\mathbf{w}_{d,1}, \dots, \mathbf{w}_{d,{K_P}}]\right)$. The resultant problem (\[E:problem-MU-reform-feasibility-reform-socp\]) is convex since both the objective function and constraints are convex, which can be solved by standard convex optimization algorithms [@boyd2009convex]. However, the computational complexity of the standard algorithms is still too high and prohibits the practical use of the fICIC scheme, especially when the numbers of MUEs, PUEs, and the transmit and receive antennas at the PBS are large. In the following we strive to propose a low-complexity algorithm to solve problem (\[E:problem-MU-reform-feasibility-reform\]), with which the optimal precoders are obtained with explicit expressions. We begin with the discussion regarding the strong duality of problem (\[E:problem-MU-reform-feasibility-reform\]). Recalling that we have shown the equivalence between problem (\[E:problem-MU-reform-feasibility-reform\]) and problem (\[E:problem-MU-reform-feasibility-reform-socp\]), and also known that strong duality holds for problem (\[E:problem-MU-reform-feasibility-reform-socp\]) because it is a convex problem. Further, along the lines of [@Yu07 App.A], we can show the equivalence between the Lagrangian functions of problem (\[E:problem-MU-reform-feasibility-reform-socp\]) and problem (\[E:problem-MU-reform-feasibility-reform\]). Therefore, strong duality holds for the non-convex optimization problem (\[E:problem-MU-reform-feasibility-reform\]). This means that we can solve problem (\[E:problem-MU-reform-feasibility-reform\]) with the Lagrange dual method. The dual problem to problem (\[E:problem-MU-reform-feasibility-reform\]) can be expressed as \[E:dual\] $$\begin{aligned} \underset{\bar{\lambda}_k}{\max}\ & \underset{\mathbf{w}_f, \{\mathbf{w}_{d,k}\}}{\min} \ J(\bar{\lambda_k}, \mathbf{w}_f, \mathbf{w}_{d,k}) \\ s.t. \ \ & \bar{\lambda}_k \geq 0, \ \forall k,\end{aligned}$$ where $\bar{\lambda}_k$ is the lagrangian multiplier, and $J(\bar{\lambda_k}, \mathbf{w}_f, \mathbf{w}_{d,k})$ is the Lagrangian function of problem (\[E:problem-MU-reform-feasibility-reform\]) with the expression $$\begin{aligned} \label{E:Largrangian-reform} &{J(\bar{\lambda}_k, \mathbf{w}_f, \mathbf{w}_{d,k}) =} {\left\|\left(\mathbf{I}_{N_t}\otimes \bar{\mathbf{H}}_{MP}\right) \mathbf{w}_f\right\|^2 + (\sigma_I^2 + \sigma_n^2)\|\mathbf{w}_f\|^2} \nonumber\\ & { + \sum_{k=1}^{K_P} \bar{\lambda}_k \left(\|\bar{\mathbf{h}}_{Mk} + e^{-j\phi} \left(\mathbf{h}_{Pk}^T \otimes\bar{\mathbf{H}}_{MP}\right) \mathbf{w}_f\|^2 \right. }\nonumber\\ & {\left.+ \|\left(\mathbf{h}_{Pk}^T\otimes\mathbf{I}_{N_r}\right) \mathbf{w}_f\|^2(\sigma_I^2+\sigma_n^2) + \sigma_n^2\right)} \nonumber\\ & { \!+\! \sum_{k=1}^{K_P} \mathbf{w}^H_{d,k}\Big(\mathbf{I}_{N_t} \!+\!\sum_{j\neq k} \bar{\lambda}_j \mathbf{h}_{Pj}\mathbf{h}_{Pj}^H \!- \! \frac{\bar{\lambda}_k}{\gamma_k} \mathbf{h}_{Pk} \mathbf{h}_{Pk}^H\Big)\mathbf{w}_{d,k}.}\end{aligned}$$ Optimal Solution to $\bar{\lambda}_k$ ------------------------------------- It can be seen from (\[E:Largrangian-reform\]) that $\underset{\mathbf{w}_f, \{\mathbf{w}_{d,k}\}}{\min} \ J(\bar{\lambda_k}, \mathbf{w}_f, \mathbf{w}_{d,k}) \to -\infty$ except when $\mathbf{I}_{N_t} +\sum_{j\neq k} \bar{\lambda}_j \mathbf{h}_{Pj}\mathbf{h}_{Pj}^H - \frac{\bar{\lambda}_k}{\gamma_0} \mathbf{h}_{Pk} \mathbf{h}_{Pk}^H \succeq \mathbf{0}_{N_t}$. Hence, the dual problem (\[E:dual\]) is equivalent to \[E:problem-dual\] $$\begin{aligned} \underset{\mathbf{w}_f}{\min}\ & \underset{\bar{\lambda}_k}{\max}\ J(\bar{\lambda}_k, \mathbf{w}_f, \bar{\mathbf{0}}_{N_t}) \label{E:Objective-dual}\\ s.\ t.\ & \mathbf{I}_{N_t} +\sum_{j\neq k} \bar{\lambda}_j \mathbf{h}_{Pj}\mathbf{h}_{Pj}^H - \frac{\bar{\lambda}_k}{\gamma_k} \mathbf{h}_{Pk} \mathbf{h}_{Pk}^H \succeq \mathbf{0}_{N_t}, \ \forall k \label{E:con-lambda} \\ & \bar{\lambda}_k \geq 0, \ \forall k, \end{aligned}$$ where the objective function (\[E:Objective-dual\]) comes from the fact that $J(\bar{\lambda_k}, \mathbf{w}_f, \mathbf{w}_{d,k})$ given in (\[E:Largrangian-reform\]) is minimized when $\mathbf{w}_{d,k} \!=\! \bar{\mathbf{0}}_{N_t}$. It is difficult to directly find the optimal $\bar{\lambda}_k$ from problem (\[E:problem-dual\]) due to the complicated semi-definite positive constraints (\[E:con-lambda\]). To solve the problem, we simplify the constraints as follows. ***Proposition 2:*** The semi-definite positive constraints (\[E:con-lambda\]) can be equivalently expressed as $$\label{E:Con-semidefinite} \bar{\lambda}_k\mathbf{h}_{Pk}^H\Big(\mathbf{I}_{N_t} +\sum_{j\neq k} \bar{\lambda}_j \mathbf{h}_{Pj}\mathbf{h}_{Pj}^H\Big)^{-1} \mathbf{h}_{Pk} \leq \gamma_k, \forall k.$$ See Appendix \[S:prooflemma4\]. Note that for any given $\mathbf{w}_f$, the objective function of problem (\[E:problem-dual\]) is an increasing function of $\bar{\lambda}_k$. Then we can show that the optimal $\bar{\lambda}_k$ maximizing $J(\bar{\lambda}_k, \mathbf{w}_f, \bar{\mathbf{0}}_{N_t})$ is obtained when the constraints in (\[E:Con-semidefinite\]) hold with equality (otherwise, one can always increase $\bar{\lambda}_k$ to improve the value of the objective function). It suggests that the optimal $\bar{\lambda}_k^\star$ should satisfy $$\label{E:opt-lambda} \bar{\lambda}_k^\star = \frac{\gamma_k}{\mathbf{h}_{Pk}^H\big(\mathbf{I}_{N_t} +\sum_{j\neq k} \bar{\lambda}_j^\star \mathbf{h}_{Pj}\mathbf{h}_{Pj}^H\big)^{-1} \mathbf{h}_{Pk}}.$$ (\[E:opt-lambda\]) provides a fixed-point iterative algorithm to find the optimal $\bar{\lambda}_k^\star$, which can be expressed as $$\label{E:opt-lambda11} \bar{\lambda}_k^{\star(n+1)} = \frac{\gamma_k}{\mathbf{h}_{Pk}^H\big(\mathbf{I}_{N_t} +\sum_{j\neq k} \bar{\lambda}_j^{\star(n)} \mathbf{h}_{Pj}\mathbf{h}_{Pj}^H\big)^{-1} \mathbf{h}_{Pk}}, $$ where the superscript $(n)$ denotes the $n$-th iteration. The convergence of the fixed-point iterative algorithm can be proved based on the standard function theory [@Wiesel2006], which shows that the algorithm given in (\[E:opt-lambda11\]) will converge to a unique optimal solution from any initial value $\{\bar{\lambda}_k^{\star(0)}\}$. Optimal Solution to $\mathbf{w}_f$ ---------------------------------- After obtaining the optimal $\bar{\lambda}_k^\star$ for $k = 1,\dots, {K_P}$, we next find the optimal $\mathbf{w}_f^\star$ based on the Karush-Kuhn-Tucker (KKT) condition [@boyd2009convex] associated with the Lagrangian function $J(\bar{\lambda}_k^\star, \mathbf{w}_f, \mathbf{w}_{d,k})$ . We can obtain that the optimal $\mathbf{w}_{f}^\star$ satisfies $$\begin{aligned} \label{E:optimalwf} &\sum_{k=1}^{K_P} e^{j\phi} \bar{\lambda}_k^\star \left(\mathbf{h}_{Pk}^T \otimes\bar{\mathbf{H}}_{MP}\right)^H \bar{\mathbf{h}}_{Mk} + \left(\mathbf{I}_{N_t}\otimes \bar{\mathbf{H}}_{MP}^H\bar{\mathbf{H}}_{MP}\right) \mathbf{w}_f^\star \nonumber\\ & + (\sigma_I^2+\sigma_n^2)\mathbf{w}_f^\star + \sum_{k=1}^{K_P} \bar{\lambda}_k^\star\left( \left((\mathbf{h}_{Pk}\mathbf{h}_{Pk}^H)^T \otimes\bar{\mathbf{H}}_{MP}^H\bar{\mathbf{H}}_{MP}\right)\mathbf{w}_f^\star \right. \nonumber\\ &\left. + (\sigma_I^2+\sigma_n^2) \left((\mathbf{h}_{Pk}\mathbf{h}_{Pk}^H)^T\otimes\mathbf{I}_{N_r}\right)\mathbf{w}_f^\star\right) = \bar{\mathbf{0}}_{N_tN_r}.\end{aligned}$$ By applying the properties of Kronecker product [@meyer2000matrix] and after some manipulations, we can obtain the optimal $\mathbf{w}_f^\star$ as $$\begin{aligned} \label{E:optimalwf2} & \mathbf{w}_f^\star \!=\! \!-\!e^{j\phi} \big( \big(\sum_{k=1}^{K_P} \bar{\lambda}_k^\star(\mathbf{h}_{Pk}\mathbf{h}_{Pk}^H)^T \!+\! \mathbf{I}_{N_t}\big)^{-\!1} \big(\sum_{k=1}^{K_P} \bar{\lambda}_k^\star\bar{\mathbf{h}}_{Mk} \mathbf{h}_{Pk}^H\big)^T \nonumber\\ & \otimes \!\big(\bar{\mathbf{H}}_{MP}^H\bar{\mathbf{H}}_{MP}\!+\! (\sigma_I^2 \!+\! \sigma_n^2)\mathbf{I}_{N_r}\big)^{-\!1}\bar{\mathbf{H}}_{MP}^H\big) \mathtt{vec}(\mathbf{I}_{N_tN_r}).\end{aligned}$$ Then we can recover the optimal $\mathbf{W}_f^\star$ from $\mathbf{w}_f^\star$ as $$\begin{aligned} \label{E:Wf} &{\mathbf{W}_f^\star = } -e^{j\phi}\Big(\sum_{k=1}^{K_P} \bar{\lambda}_k^\star\mathbf{h}_{Pk}\mathbf{h}_{Pk}^H + \mathbf{I}_{N_t}\Big)^{-1} \cdot\\ & \Big(\sum_{k=1}^{K_P} \bar{\lambda}_k^\star{\mathbf{h}}_{Pk} \bar{\mathbf{h}}_{Mk}^H\Big) \bar{\mathbf{H}}_{MP}\big(\bar{\mathbf{H}}_{MP}^H\bar{\mathbf{H}}_{MP}+ (\sigma_I^2+\sigma_n^2)\mathbf{I}_{N_r}\big)^{-1}.\nonumber\end{aligned}$$ Optimal Solution to $\mathbf{w}_{d,k}$ --------------------------------------- With the optimal $\lambda_k^\star$ and $\mathbf{W}_f^\star$, the optimal $\mathbf{w}_{d,k}^\star$ of problem (\[E:problem-MU-reform-feasibility-reform\]) can be solved as follows. According to the KKT condition, the optimal $\mathbf{w}_{d,k}^\star$ satisfies $$\begin{aligned} & \frac{\partial J(\bar{\lambda_k}^\star, \mathbf{w}_f^\star, \mathbf{w}_{d,k})}{\partial \mathbf{w}_{d,k}}\\ &= 2\Big(\mathbf{I}_{N_t} + \sum_{j\neq k} \bar{\lambda}_j^\star \mathbf{h}_{P,j}\mathbf{h}_{P,j}^H - \frac{\bar{\lambda}_k^\star}{\gamma_k} \mathbf{h}_{Pk}\mathbf{h}_{Pk}^H\Big)\mathbf{w}_{d,k} = \bar{\mathbf{0}}_{N_t},\nonumber $$ from which we have $$\begin{aligned} \label{E:opt-wtk} \mathbf{w}_{d,k}^\star = & \frac{\lambda_k^\star\mathbf{h}_{Pk}^H\mathbf{w}_{d,k}^\star}{\gamma_k} \big(\mathbf{I}_{N_t} + \sum_{j\neq k} \bar{\lambda}_j^\star \mathbf{h}_{P,j}\mathbf{h}_{P,j}^H\Big)^{-1} \mathbf{h}_{Pk} \nonumber\\ \triangleq & \sqrt{p_k^\star} \tilde{\mathbf{w}}_{d,k}^\star,\end{aligned}$$ where $\tilde{\mathbf{w}}_{d,k}^\star = \big(\mathbf{I}_{N_t} + \sum_{j\neq k} \bar{\lambda}_j^\star \mathbf{h}_{P,j}\mathbf{h}_{P,j}^H\big)^{-1} \mathbf{h}_{Pk}$, and $p_k^\star$ is a scalar controlling the power allocated to ${\text{PUE}}_k$ for transmitting desired signals. To find the optimal $\{p_k^\star\}$ in (\[E:opt-wtk\]), recalling that the optimal solution to problem (\[E:problem-MU-reform-feasibility-reform\]) is obtained when all SINR constraints in (\[E:constraint-SINR-feasibility-reform\]) hold with equality if problem (\[E:problem-MU-reform-feasibility-reform\]) is feasible for given {$\gamma_k$}, then we can obtain the following equations with respect to $p_k^\star$ $$\begin{aligned} \label{E:opt-p} & \frac{p_k^\star|\mathbf{h}_{Pk}^H\tilde{\mathbf{w}}_{d,k}^\star|^2}{ \sum_{j\neq k} p_j^\star\|\mathbf{h}_{Pk}^H \tilde{\mathbf{w}}_{t,j}^\star\|^2 \!+\! \zeta_k(\mathbf{W}_f^\star)} \!= \!\gamma_k,\ k \!=\! 1,\dots, {K_P},\end{aligned}$$ where $\zeta_k(\mathbf{W}_f^\star) = \|\bar{\mathbf{h}}_{Mk}^H + \mathbf{h}_{Pk}^H \mathbf{W}_f^\star \bar{\mathbf{H}}_{MP}^H e^{-j\phi}\|^2 + |\mathbf{h}_{Pk}^H \mathbf{W}_f^\star|^2(\sigma_I^2+\sigma_n^2) + \sigma_n^2$. From the equations in (\[E:opt-p\]), we can solve the optimal $\mathbf{p}^\star\triangleq [p_1^\star, \dots, p_{K_P}^\star]^T$ as $$\label{E:PA} \mathbf{p}^\star = \mathbf{M}^{-1} \boldsymbol{\zeta},$$ where $\boldsymbol{\zeta} = [\zeta_1(\mathbf{W}_f^\star), \dots, \zeta_{K_P}(\mathbf{W}_f^\star)]^T$, and $\mathbf{M}\in\mathbb{C}^{{K_P}\times {K_P}}$ is defined as $$[\mathbf{M}]_{kj} = \left\{ \begin{array}{ll} \frac{1}{\gamma_k}|\mathbf{h}_{Pk}^H\tilde{\mathbf{w}}_{d,k}^\star|^2 , & k = j,\\ - |\mathbf{h}_{Pk}^H\tilde{\mathbf{w}}_{t,j}^\star|^2, & k \neq j. \end{array} \right.$$ Finally, we summarize the proposed low-complexity algorithm to solve the original optimization problem (\[E:problem-MU\]) in general multiuser case in Table \[tab:Distributed-P-B-allocation\]. ------------------------------------------------------------------------ [aaa]{} **Initialization:** Set $i=0$, $\underline{R_{sum}}=0$, and $\overline{R_{sum}} = \sum_{k=1}^{K_P} \log(1 +\mathtt{SINR}_k^{ub})$, where $\mathtt{SINR}_k^{ub} = \frac{P_0\|\mathbf{h}_{Pk}\|^2}{\sigma_n^2}$ is an upper bound of the SINR of ${\text{PUE}}_k$, which is achieved when the ICI disappears and only ${\text{PUE}}_k$ is served by the PBS.\ **Bisection Iteration:** At the $i$-th iteration, set $i \leftarrow i + 1$. - Compute $R_{sum}^0 = \frac{\underline{R_{sum}} + \overline{R_{sum}}}{2}$. - Given $R_{sum}^0$, compute $\{\bar{\lambda}_k^\star\}$ with the fixed-point iterative algorithm given by (\[E:opt-lambda11\]). - Given $\{\bar{\lambda}_k^\star\}$, compute $\mathbf{W}_f^\star$ based on (\[E:Wf\]). - Given $R_{sum}^0$, $\{\bar{\lambda}_k^\star\}$ and $\mathbf{W}_f^\star$, compute $\{\tilde{\mathbf{w}}_{d,k}^\star\}$ and $\{p_k^\star\}$ based on (\[E:opt-wtk\]) and (\[E:PA\]), respectively, then obtain $\mathbf{w}_{d,k}^\star = \sqrt{p_k^\star}\tilde{\mathbf{w}}_{d,k}^\star$. - Given $\mathbf{W}_f^\star$ and $\{\mathbf{w}_{d,k}^\star\}$, compute the value of the objective function of problem (\[E:problem-MU-reform-feasibility\]), denoted by $P_{0}^{i}$. - **If** $P_{0}^{i} \leq P_0$, let $\underline{R_{sum}} \leftarrow R_{sum}^0$, **otherwise**, let $\overline{R_{sum}} \leftarrow R_{sum}^0$. \ **Repeat:** Iterate step 2 until the required accuracy is reached, i.e., $ \overline{R_{sum}} - \underline{R_{sum}} \leq \varepsilon$, where $\varepsilon$ is a specific threshold. ------------------------------------------------------------------------ Generalization to wideband systems ================================== In previous sections the fICIC is optimized in narrowband systems, where all the channels are flat fading and therefore only single-tap forwarding precoder is designed. When considering wideband systems, frequency-selective fading channels should be taken into account and hence a multi-tap finite impulse response (FIR) forwarding precoder needs to be designed, which makes the design of the fICIC more involved. In this section, we generalize the fICIC to OFDM systems. For the sake of notational simplicity, we consider single-user case, i.e., $K_M=1$ and $K_P=1$, which can be straightforwardly extended to multiuser orthogonal frequency division multi-accessing (OFDMA) systems due to the orthogonality between subcarriers. Following the previous narrowband-system definitions, let $\mathbf{h}_{Mk}(t)$ and $\mathbf{h}_{Pk}(t)$ denote the time-domain channels from the MBS and the PBS to ${\text{PUE}}_k$, $\mathbf{H}_{MP}(t)$ denote the time-domain channel from the MBS to the PBS, and $\mathbf{H}_{PP}(t)$ denote the time-domain self-interference channel of the FD PBS. The corresponding frequency-domain channels on the $n$-th subcarrier of the four channels are denoted by $\mathbf{g}_{Mkn}$, $\mathbf{g}_{Pkn}$, $\mathbf{G}_{MPn}$, and $\mathbf{G}_{PPn}$, respectively. Consider that the PBS employs a FIR precoder $\mathbf{W}_f(t) = \sum_{l=0}^{L-1} \mathbf{W}_{fl} \delta(t-lT_s)$ to forward the listened ICI, where $L$ is the order of the FIR precoder and $T_s$ is the sampling interval. The selection of $L$ needs to ensure that the delay spread of the equivalent channel for the forwarded ICI, $\mathbf{h}_{Pk}(t)\odot\mathbf{W}_f(t)\odot\mathbf{H}_{MP}(t)\odot\delta(t-\tau)$, does not exceed the cyclic prefix (CP) of the OFDM system in order to maintain orthogonality between subcarriers. Let $\bar{\mathbf{W}}_{fn}$ denote the frequency response of $\mathbf{W}_f(t)$ on the $n$-th subcarrier. Then, following the same derivations in Section \[S:system\_model\] for narrowband systems, we can obtain the transmit power of the PBS on the $n$-th subcarrier as $$\begin{aligned} \label{E:Pout-wbn} &P_{out,n} = \\ &\frac{\mathtt{tr}\big(\bar{\mathbf{W}}_{fn} \bar{\mathbf{g}}_{MPn}\bar{\mathbf{g}}_{MPn}^H \bar{\mathbf{W}}_{fn}^H\big) \!+ \! \sigma_n^2\mathtt{tr}\big(\bar{\mathbf{W}}_{fn}\bar{\mathbf{W}}_{fn}^H\big)\! +\! \|\bar{\mathbf{w}}_{dn}\|^2}{1 - \sigma_e^2\mathtt{tr}\big(\bar{\mathbf{W}}_{fn}\bar{\mathbf{W}}_{fn}^H\big)},\nonumber\end{aligned}$$ where $\bar{\mathbf{g}}_{MPn}=\mathbf{G}_{MPn}^H\bar{\mathbf{w}}_{Mn}$ is the equivalent frequency-domain channel from the MBS to the PBS, $\bar{\mathbf{w}}_{Mn}$ is the precoder at the MBS for the MUE on the $n$-th subcarrier, $\bar{\mathbf{w}}_{dn}$ is the precoder at the PBS for the desired signal of ${\text{PUE}}_k$, and $\sigma^2_e = \frac{\sigma_n^2}{P_{tr}} + 2\bar{\alpha}_{PP}(\mu_x+\mu_y)$ is the same as defined in (\[E:x22\]). With (\[E:Pout-wbn\]), the total transmit power constraint for the PBS can be expressed as $$\label{E:PBPC-wb} P_{out} = \sum_{n=1}^N P_{out,n} \leq P_0.$$ Compared to the power constraint in (\[E:Pout\]) for narrowband system, which can be converted into a convex constraint for the precoders as shown in (\[E:PBPC\]), the constraint in (\[E:PBPC-wb\]) for wideband system is much more complicated and is non-convex. Similar to (\[E:SINR3\]), we can obtain the SINR of ${\text{PUE}}_k$ on the $n$-th subcarrier as $$\begin{aligned} \label{E:SINR3-wb} &\mathtt{SINR}_{kn,FD} \nonumber\\ &= |\mathbf{g}_{Pkn}^H\bar{\mathbf{w}}_{dn}|^2/\big(\|\bar{g}_{Mkn}^\ast + \mathbf{g}_{Pkn}^H \bar{\mathbf{W}}_{fn} \bar{\mathbf{g}}_{MPn} e^{-jdn}\|^2 \nonumber\\ &\qquad + \|\mathbf{g}_{Pkn}^H \bar{\mathbf{W}}_{fn}\|^2(P_{out,n}\sigma_e^2+\sigma_n^2) + \sigma_n^2\big),\end{aligned}$$ where $\bar{g}_{Mkn} \triangleq \bar{\mathbf{w}}_{Mn}^H\mathbf{g}_{Mkn}$ is the equivalent channel from the MBS to the ${\text{PUE}}$ on the $n$-th subcarrier, and $P_{out,n}$ is given in (\[E:Pout-wbn\]), which is a function of the precoders $\bar{\mathbf{W}}_{fn}$ and $\bar{\mathbf{w}}_{dn}$. Then, the wideband fICIC precoder optimization problem, aimed at maximizing the sum rate of ${\text{PUE}}_k$ over all subcarrier, can be formulated as \[E:problem-wb\] $$\begin{aligned} & \max_{\mathbf{W}_f(t), \{\bar{\mathbf{w}}_{dn}\}} \sum_{n=1}^N \log(1 +\mathtt{SINR}_{kn,FD}) \label{E:objective-wb}\\ & s.t. \big[[\bar{\mathbf{W}}_{f1}]_{ij}, \!\dots\!, [\bar{\mathbf{W}}_{fN}]_{ij}\big]^T\! =\! \mathbf{F}\big[[\mathbf{W}_{f1}]_{ij}, \!\dots\!, [\mathbf{W}_{fL}]_{ij}\big]^T \label{E:cons-wb0}\\ & \qquad \sum_{n=1}^N P_{out,n} \leq P_0, P_{out,n} \geq 0,\ \forall n, \label{E:cons-wb2} \end{aligned}$$ where constraint (\[E:cons-wb0\]) restricts that the $N$ frequency-domain precoders $\{\bar{\mathbf{W}}_{fn}\}$ are generated from the $L$-tap time-domain FIR precoder $\mathbf{W}_f(t)$, and $\mathbf{F}\in\mathbb{C}^{N\times L}$ is the matrix containing the first $L$ columns of the $N\times N$ fast fourier transformation matrix. Problem (\[E:problem-wb\]) is non-convex, whose global optimal solution is difficult to find. We can obtain a local optimal solution to the problem by using a gradient-based solution (specifically using the function $\mathtt{fmincon}$ of the optimization toolbox of MATLAB). Note that the direction of $\bar{\mathbf{w}}_{dn}$ only affects the power of desired signal in the numerator of $\mathtt{SINR}_{n,FD}$. Therefore, the direction of the optimal $\bar{\mathbf{w}}_{dn}$ can be obtained as $\frac{\bar{\mathbf{w}}_{dn}^\star}{\|\bar{\mathbf{w}}_{dn}^\star\|} = \frac{\mathbf{g}_{Pkn}}{\|\mathbf{g}_{Pkn}\|}$, with which the number of variables in problem (\[E:problem-wb\]) is reduced. Practical Issues ================ By now, we have introduced the concept of fICIC and optimized the associated precoders. In this section, we discuss some practical issues regarding the application of the fICIC. Channel Acquisition ------------------- To apply the fICIC, the PBS needs to have the channels from the MBS to both the PBS and PUEs, i.e., ${\mathbf{H}}_{MP}$ and ${\mathbf{h}}_{Mk}$, as well as the channel from the PBS to PUEs, $\mathbf{h}_{Pk}$, $\forall k$. First, the channel ${\mathbf{H}}_{MP}$ can be directly estimated at the PBS by using the FD receive antennas to receive the downlink training signals sent from the MBS. Second, noting that in TDD systems channel reciprocity holds between the HD transceiver at the PBS and each PUE, the channel $\mathbf{h}_{Pk}$ can be estimated at the PBS by using the HD receive antennas to receive the uplink training signals sent by ${\text{PUE}}_k$. Finally, the channel ${\mathbf{h}}_{Mk}$ can be first estimated by ${\text{PUE}}_k$ and then fed back to the PBS, where digital or analog feedback schemes can be employed [@Marzetta2006]. We will evaluate the performance of the fICIC under imperfect channel estimation and feedback in next section. fICIC [v.s.]{} the HD Scheme ---------------------------- As we analyzed in Section \[S:asymptotic\], with the fICIC, the FD PBS can adaptively switch between FD mode and HD mode by optimizing the precoders for transmitting desired signals and forwarding listened interference. For instance, when the ICI is very strong or the residual self-interference is very large, the fICIC will reduce to the HD scheme, as shown by (\[E:maxSINR-largeI\]) and (\[E:SINR-large-selfI\]). Therefore, with perfect channel information at the PBS, the proposed fICIC will always outperform the HD scheme given the same number of uplink and downlink RF chains and BB modules. Yet, to support the FD technique extra passive antennas (though cheap) and FD modules are required. When imperfect channels are considered at the PBS, the performance of the fICIC will decrease. Nevertheless, simulations in next section show that significant performance gain can be still achieved by the fICIC over the HD scheme with imperfect channels. Self-interference Cancellation {#S:SIC} ------------------------------ Effective self-interference cancellation is crucial for the fICIC, where isolation of the transmit and FD receive antennas is an important approach [@Everett2014]. Considering the transceiver structure of the FD PBS given in Fig. \[F:system\], where the FD receive antennas are only active in the downlink to receive the ICI from the MBS while not receiving the uplink signals from the PUEs, the FD receive antennas can be mounted far away from the transmit antennas, e.g., installing the FD receive antennas outside a building and transmit antennas inside, respectively. In this manner, the self-interference can be largely suppressed in general. Joint Application with Existing ICIC Schemes -------------------------------------------- Existing ICIC schemes including eICIC and CoMP-CB operate at the MBS, while the fICIC can operate at each PBS individually. Therefore, the fICIC can be directly applied together with existing ICIC schemes. As will be shown in next section, a joint application of the fICIC and existing schemes can achieve evident performance improvement. Simulation Results ================== In this section we verify our analytical results and evaluate the performance of the proposed fICIC scheme via simulations. ![Network layout for simulations, where two PBSs are deployed in the macro cell covered by the MBS.[]{data-label="F:layout"}](Makedrop){width="46.00000%"} Simulation Setups ----------------- For all simulations, unless otherwise specified, the parameters in this subsection are used. The considered HetNet layout is shown in Fig. \[F:layout\], where the MBS is located at the center of the macro cell, ${\text{PBS}}_1$ and ${\text{PBS}}_2$ are respectively located at $(d_{P1}, 0)$ and $(d_{P2}, 0)$, the MBS serves two MUEs located at $(d_{M1}, 0)$ and $(d_{M2}, 0)$, and ${\text{PBS}}_k$ serves two PUEs located at $(d_{Pk}, r)$ and $(d_{Pk}, -r)$, respectively. We set the radius of the macro cell $R_M$ as $500$ $m$, $d_{P1} = 60$ $m$, $d_{P2} = 180$ $m$, $r = 40$ $m$, $d_{M1} = 120$ $m$, and $d_{M2}= 240$ $m$. Since we have shown in the analytical results that the performance of the fICIC depends on the strength of the ICI, the positions of the two PBSs, $d_{P1}$ and $d_{P2}$, are selected to reflect the cases of strong and weak ICI, respectively. We will also evaluate the performance of the fICIC with random PBS placements later. The MBS transmits with $M = 4$ antennas and the power of $P_M = 46$ dBm, each PBS has $N_r = 2$ FD receive antennas and transmits with $N_t = 2$ antennas and the maximal power of $P_0 = 30$ dBm, and each UE (including MUE and PUE) has one receive antenna. The path loss is set as $128.1 + 37.6 \log_{10}d$ for macro cell and $140.7 + 36.7 \log_{10} d$ for pico cell, respectively, where $d$ is the distance in $km$ [@TR36.814]. A penetration loss of $20$ dB is considered for the channels to PUEs. We model the interference from the MBSs in adjacent macro cells and the surrounding PBSs as noises. Define the average receive SNR of a MUE located at the cell edge as $\text{SNR}_{\text{edge}}$, then the noise variance $\sigma_n^2$ can be obtained as $\sigma_n^2 = P_M - (128.1 + 37.6 \log_{10} R_M) - \text{SNR}_{\text{edge}}$ in dBm. To evaluate the impact of imperfect self-interference cancellation for FD, we define the signal to self-interference ratio as $\text{SIR}_{\text{self}} = P_0 - \sigma_I^2$ in dB in order to reflect the level of self-interference cancellation. The Rayleigh flat small-scale fading channels are considered. The fairness factors are set as $\alpha_k=\frac{1}{K_P}$ in multi-user case. All the results are averaged over 1000 channel realizations. Narrowband Single-user Case --------------------------- ![Average sum rate versus $\text{SNR}_{\text{edge}}$, where perfect self-interference cancellation and perfect channels are considered, $K_M = 1$, $K_P = 1$, and $N_r = 1$. []{data-label="F:rate"}](Fig1_NtP2){width="46.00000%"} We first simulate the single-user case, where the MBS serves one MUE, each PBS serves one PUE, and each PBS has $N_r = 1$ FD receive antenna. Under the assumption of perfect self-interference cancellation, the average sum rate achieved by the fICIC is depicted in Fig. \[F:rate\]. For comparison, the performance of the HD scheme, given by $\mathcal{E}\{\log(1+\mathtt{SINR}_{k,HD}^\star)\}$, and the performance in ICI free case, given by $\mathcal{E}\{\log(1+\frac{P_0\|\mathbf{h}_{Pk}\|^2}{\sigma_n^2})\}$, are also presented. First, we can see that all the schemes perform closely in low SNR regime, where the system operates in noise-limited scenario. With the increase of SNR, the performance floor appears for the HD scheme, which is caused by the ICI from the MBS. The performance of Cell 2 is better than Cell 1 due to $d_{P2} > d_{P1}$ that leads to weaker ICI. The proposed fICIC exhibits a noticeable performance gain over the HD scheme. For weak ICI case, e.g., Cell 2, as we analyzed, the fICIC can thoroughly eliminate the ICI in a high probability (considering the randomness of small-scale channels), and thus the performance gap between the fICIC and the ICI free case is very small. ![Average sum rate versus $d_{P1}$ for different PBS placements, where perfect self-interference cancellation and perfect channels are considered, $K_M = 1$, $K_P = 1$, and $N_r = 1$. []{data-label="F:placement"}](Fig2_NtP2_new){width="46.00000%"} In Fig. \[F:placement\], we evaluate the impact of the strength of the ICI on the performance of the fICIC by simulating a single PBS with different positions $d_{P1}$. The performance achieved by the fICIC and the HD scheme as well as the performance gain of the fICIC are depicted. It is shown that the performance of the HD scheme increases slowly with the reduction of ICI, i.e., the increase of $d_{P1}$, while the performance of the fICIC first increases fast and then keeps nearly constant. It implies that the fICIC can make a large area of the macro cell, e.g., from $150$ $m$ to $500$ $m$, experience very weak ICI. Moreover, we can see that the fICIC performs close to the HD scheme at large $d_{P1}$ because the ICI is very weak and has negligible impact on the performance in this case. Further recalling that the fICIC will degenerate to the HD scheme when the ICI is very strong as analyzed in Section \[S:asymptotic\], we can understand that the gain of the fICIC over the HD scheme first increases and then decreases with $d_{P1}$. However, we can still observe an evident performance improvement of nearly $300\%$ even when $d_{P1} = 10$ $m$, in which case the average power of the ICI is $10.4$ dB stronger than that of the desired signals. Figure \[F:SIR\] shows the performance of the fICIC as a function of $\text{SIR}_{\text{self}}$, where imperfect self-interference cancellation is considered. First, it can be seen that the fICIC reduces to the HD scheme for low $\text{SIR}_{\text{self}}$, which agrees with our previous analysis. Second, the fICIC outperforms the HD scheme when $\text{SIR}_{\text{self}} \geq 60$ dB for Cell 1 but $75$ dB for Cell 2. This is because ${\text{PBS}}_1$ is closer to the MBS and hence can listen a stronger ICI, which can relax the requirement of self-interference cancellation for FD. Moreover, it is shown that the self-interference cancellation of $110$ dB is sufficient for the fICIC, which is practically possible because on one hand existing work has reported $90$ dB self-interference cancellation even for closely placed transmit and receive antennas [@Everett2014], and on the other hand, as we discussed before, in our case the transmit antennas and FD receive antennas can be separated far apart, leading to further reduction of the self-interference, e.g., with an additional $20$ dB penetration loss. ![Average sum rate versus $\text{SIR}_{\text{self}}$ with imperfect self-interference cancellation, where perfect channels are considered, $\text{SNR}_{\text{edge}} = 20$ dB, $K_M = 1$, $K_P = 1$, and $N_r = 1$.[]{data-label="F:SIR"}](Fig4_NtP2){width="46.00000%"} Narrowband Multi-user Case -------------------------- In this subsection the multi-user case is simulated, where each PBS serves two PUEs. We compare the performance of the fICIC with the HD scheme, a successive interference cancellation (SIC) based non-linear HD scheme (denoted by HD-SIC) [@Wildemeersch2014], the ABS-based eICIC and CoMP-CB, and also evaluate the performance of the combinations of the fICIC with eICIC and CoMP-CB. To study the effectiveness of the seven schemes for cancelling different-strength ICI, we fix the location of Cell 2 and adjust the location of Cell 1 to generate different interference to noise ratio (INR) for PUEs in Cell 1, where $\mathtt{INR}\triangleq\frac{\mathcal{E}\{\|\bar{\mathbf{h}}_{Mk}\|^2\}}{\sigma_n^2}$. To enable the HD-SIC scheme, we consider that the MBS serves a single MUE because each PUE has only one antenna so that only one interference signal can be decoded and cancelled. In the simulations, the proposed low-complexity algorithm given in Table \[tab:Distributed-P-B-allocation\] is employed to obtain the performance of both the fICIC and the HD scheme, where $\mathbf{W}_f$ is set as zero for the HD scheme. For the HD-SIC, since the ICI needs to be decoded first by regarding the desired signal from the PBS as interference, the PBS may not transmit with its maximal power. Given the data rate of the MUE as $4$ bps/Hz, we employ exhaustive searching to find the maximal feasible transmit power of the PBS under the ICI decoding constraint, where for any given transmit power the precoder of the PBS is computed the same as the HD scheme. For the ABS-based eICIC, it is considered that the MBS mutes in half time to provide ICI-free environment to the PUEs. For CoMP-CB, the signal-to-leakage-plus-noise ratio (SLNR) based precoder [@VSINR2009] is employed. The results are depicted in Fig. \[F:eICIC\]. Compared with the fICIC, we can see that the eICIC achieves worse performance when the ICI is not strong, say $\mathrm{INR}< 25$ dB, because the fICIC can effectively cancel the weak-medium level of ICI and allow the PUEs to use all the time-frequency resources, while with the eICIC the PUEs only uses half resources. When the ICI is strong, as we analyzed before, the fICIC is not effective, and the eICIC shows large performance gain. By combining eICIC with fICIC, the two schemes supplement each other and achieve much better performance. ![Average sum rate of PUEs in Cell 1 achieved by relevant schemes versus $\text{INR}$, where perfect self-interference cancellation and perfect channels are considered, $\text{SNR}_{\text{edge}} = 20$ dB, $N_r = 2$, $K_M = 1$, and $K_P = 2$. []{data-label="F:eICIC"}](Fig14_new){width="48.00000%"} For CoMP-CB, we can see that it is superior to the fICIC when the ICI is strong, say $\mathrm{INR} \geq 26$ dB, but inferior for weak-medium ICI. This can be explained as follows. In the simulated case, the MBS has only four antennas, which are not adequate to serve one MUE and suppress the ICI for four PUEs simultaneously. As a result, when the INR of Cell 1 is smaller than Cell 2, the SLNR-based CoMP-CB only suppresses the stronger ICI to the PUEs in Cell 2, while the ICI to PUEs in Cell 1 will be mitigated only for large INR (this also explains why the performance of Cell 1 with pure CoMP-CB or the combination of CoMP-CB and fICIC first deceases and then increases with the growth of INR.). By contrast, the fICIC is effective to suppress weak-medium ICI but not for strong ICI. Therefore, CoMP-CB and fICIC perform better in different ICI cases. Since CoMP-CB has no enough antenna resources to cancel weak-medium ICI in the simulation, the combination of CoMP-CB and fICIC has negligible gain over the pure fICIC for weak-medium INR. However, evident performance improvement is achieved by their combination for strong ICI, because the strong ICI can be first suppressed into weak-medium ICI by CoMP-CB and then further effectively cancelled by the fICIC. For the HD-SIC, we can observe that it is not always feasible when the ICI is not strong, e.g., $\mathrm{INR} < 15$ dB. With the increase of ICI, the interference signal becomes decodable and more power can be transmitted by the PBS, which results in the performance improvement as expected. It can be seen that the HD-SIC outperforms the fICIC for strong ICI, which inspires us to investigate the SIC-based fICIC in future work, where the FD PBS forwards the listened ICI to further enhance the ICI at the PUE so as to extend the feasible region of SIC. ![Average sum rate of PUEs in Cell 1 versus $\text{SNR}_{\text{edge}}$, where ${\text{PBS}}_1$ is uniformly placed within the area suffering from strong ICI with $d_{P1}\sim U([50, 250])$ $m$, $\text{SIR}_{\text{self}} = 110$ dB, $N_r = 1, 2$, $K_M = 1, 2$, and $K_P = 1, 2$. In the legends, “p-CE” and “i-CE” denote perfect and imperfect channel estimation, respectively, and “p-FB” and “i-FB” denote perfect and imperfect feedback, respectively.[]{data-label="F:CE"}](Fig12){width="46.00000%"} Finally, in Fig. \[F:CE\] the performance of the fICIC under imperfect self-interference cancellation, imperfect channel estimation and practical channel feedback is evaluated, where only Cell 1 is considered and ${\text{PBS}}_1$ is randomly placed within the area experiencing strong ICI, specifically with $d_{P1}$ following uniform distribution between $[50, 250]$ $m$. As discussed before, the channel $\bar{\mathbf{H}}_{MP}$ can be directly estimated at the PBS, the channel $\mathbf{h}_{Pk}$ can be obtained at the PBS by estimating the uplink channel based on channel reciprocity, and the channel $\bar{\mathbf{h}}_{Mk}$ can be first estimated at the ${\text{PUE}}_k$ and then fed back to the PBS. In simulations, we employ linear minimum mean-squared error estimator to estimate $\bar{\mathbf{H}}_{MP}$, $\mathbf{h}_{Pk}$ and $\bar{\mathbf{h}}_{Mk}$, and use analog feedback [@Marzetta2006] to send back the estimate of $\bar{\mathbf{h}}_{Mk}$ to the PBS , where the transmit power of ${\text{PUE}}_k$ is set as $23$ dBm. Figs. \[F:CE\](a) and \[F:CE\](b) show the results in single-user and multi-user cases, respectively. In both cases we can see the performance degradation of the fICIC caused by imperfect channel estimation and imperfect feedback. However, compared with the HD scheme, significant performance gain achieved by the fICIC can be still observed, even when imperfect self-interference cancellation, imperfect channel estimation and practical analog feedback are taken into account. Wideband Single-user Case ------------------------- In this subsection we evaluate the performance of the fICIC in wideband systems. We consider a LTE system with $5$ MHz bandwidth, where the sampling interval is $T_s = 0.13$ $\mu$s [@TS36.211]. The small-scale channels are generated based on WINNER II clustered delay line model [@WinnerII]. Specifically, the channels from the MBS to the PBS, $\mathbf{H}_{MP}$, use the typical urban macro-cell line of sight (LoS) model considering that the receive antennas of the PBS can be mounted outside a building as discussed before, the channels from the MBS to PUEs, $\mathbf{h}_{Mk}$, use the typical urban macro-cell non-LoS model, and the channels from the PBS to PUEs, $\mathbf{h}_{Pk}$, use the typical urban micro-cell NLoS model. After sampling the multipath channels with the considered bandwidth, we obtain the maximal delay spread of $\mathbf{H}_{MP}$ and $\mathbf{h}_{Pk}$ as three and six samples, respectively. We consider the processing delay of the FD PBS as $4$ samples, i.e., $\tau = 0.52\ \mu$s. Since the CP of the LTE system is $4.7$ $\mu$s [@TS36.211], i.e., $36$ samples, we can obtain the maximal order of the FIR forwarding precoder $\mathbf{W}_f(t)$ as $23$, i.e., $L\leq23$. Note that the maximum value of $L$ can be even larger if the extended CP with $16.7$ $\mu$s is considered. ![Average sum rate of the PUE over $N=25$ subcarriers in Cell 1 versus $\text{SNR}_{\text{edge}}$, where $\text{SIR}_{\text{self}} = 110$ dB, $K_M = 1$, and $K_P = 1$.[]{data-label="F:WB"}](WB){width="46.00000%"} Considering that the complexity of solving problem (\[E:problem-wb\]) increases rapidly with the number of subcarriers and also considering the channel correlation in frequency domain, in the simulations we treat each resource block (RB) as a subcarrier and then $N=25$ subcarriers are simulated corresponding to the total $25$ RBs of the $5$ MHz LTE system [@TS36.211]. In addition, we select the order of $\mathbf{W}_f$ as $L=1,2,4$ to speed up the simulations. In Fig. \[F:WB\], the average sum rate of Cell 1 with $d_{P1}=60~m$ achieved by the fICIC is depicted, where imperfect self-interference cancellation, imperfect channel estimation and practical channel feedback as considered in Fig. \[F:CE\] are taken into account. The compared HD scheme is obtained from problem (\[E:problem-wb\]) by setting $\mathbf{W}_f$ as zeros. We can see from Fig. \[F:WB\] that the performance of the wideband fICIC improves with the increase of $L$, and an evident performance gain over the HD scheme can be observed when $L=4$. More significant performance gain can be expected when larger $L$ is used for the fICIC, to achieve which more efficient low-complexity algorithms to problem (\[E:problem-wb\]) need to be studied in future work. Conclusions =========== In this paper we proposed to eliminate the cross-tier ICI in HetNets by using FD technique at the PBS. We derived a FD assisted ICI cancellation (fICIC) scheme, with which the ICI can be mitigated by merely designing the precoders at PBSs without relying on the participation of the MBS. We first investigated the narrowband single-user case to gain some insight into the behavior of the fICIC, where we found closed-from solution of the optimal fICIC, and analyzed its asymptotical performance in ICI-dominated scenario. We then studied the general narrowband multi-user case, and devised a low-complexity algorithm to find the optimal fICIC scheme that maximizes the downlink sum rate of PUEs subject to the fairness constraint. Finally, we generalized the fICIC to wideband systems and discussed the practical issues regarding the application of the fICIC. Simulations validated the analytical results. Compared with the traditional HD scheme, the fICIC exhibits significant performance gain even when imperfect self-interference cancellation and imperfect channel information are taken into account. By combining the fICIC with eICIC or CoMP-CB, the ICI with various levels can be effectively eliminated. Derivation of Transmit Power $P_{out}$ {#A:Pout} ====================================== By substituting (\[E:yp\]) into (\[E:x2\]), we can obtain $$\begin{aligned} \label{E:xp-new} &\mathbf{x}_p[t] =\mathbf{W}_f \big(\bar{\mathbf{H}}_{MP}^H \mathbf{s}_M[t\!-\!\tau] - \mathbf{E}_{PP}^H[t\!-\!\tau]\mathbf{x}_p[t\!-\!\tau]+\\ & \mathbf{H}_{PP}^H \mathbf{z}_x[t\!-\!\tau]+ \mathbf{n}_p[t\!-\!\tau] + \mathbf{z}_y[t\!-\!\tau]\big)+ \textstyle{\sum_{k=1}^{K_P}} \mathbf{w}_{d,k} s_{p,k}[t].\nonumber\end{aligned}$$ Since the terms at the right-hand side of (\[E:xp-new\]) are independent, we can obtain $P_{out}$ with (\[E:Pout-new\]) by taking the expectations over each term, yielding $$\begin{aligned} P_{out}& = \mathtt{tr}\left(\mathbf{W}_f\bar{\mathbf{H}}_{MP}^H\bar{\mathbf{H}}_{MP}\mathbf{W}_f^H\right) + \sigma_n^2\mathtt{tr}\left(\mathbf{W}_f\mathbf{W}_f^H\right)\nonumber\\ & + \textstyle{\sum_{k=1}^{K_P}} \|\mathbf{w}_{d,k}\|^2 + \mathtt{tr}\left(\mathbf{W}_f\boldsymbol{\Sigma}_1\mathbf{W}_f^H\right) \nonumber\\ &+ \mathtt{tr}\left(\mathbf{W}_f\boldsymbol{\Sigma}_2\mathbf{W}_f^H\right) +\mathtt{tr}\left(\mathbf{W}_f\boldsymbol{\Sigma}_3\mathbf{W}_f^H\right),\label{E:Pout-0}\end{aligned}$$ where $\boldsymbol{\Sigma}_1 \triangleq \mathcal{E}_{\mathbf{x}_p, \mathbf{E}_{PP},\mathbf{H}_{PP}}\{\mathbf{E}_{PP}^H[t\!-\!\tau]\mathbf{x}_p[t\!-\!\tau]\mathbf{x}_p^H[t\!-\!\tau]\mathbf{E}_{PP}[t\!-\!\tau]\}$, $\boldsymbol{\Sigma}_2 \triangleq \mathcal{E}_{\mathbf{H}_{PP}}\{\mathbf{H}_{PP}^H \mu_x\mathtt{diag}(\boldsymbol{\Phi}_x) \mathbf{H}_{PP}\}$, and $\boldsymbol{\Sigma}_3\triangleq\mathcal{E}_{\mathbf{H}_{PP}}\{\mu_y\mathtt{diag}(\boldsymbol{\Phi}_y)\}$, which can be derived as follows. Denoting $\mathbf{x}_p = [x_{p1},\dots,x_{pN_t}]^T$, we can rewrite $\boldsymbol{\Sigma}_1$ as $$\begin{aligned} \label{E:Sigma-1} &\boldsymbol{\Sigma}_1= \mathcal{E}_{\mathbf{x}_p, \mathbf{E}_{PP},\mathbf{H}_{PP}}\Big\{\textstyle{\sum_{i=1}^{N_t}} |x_{pi}[t-\tau]|^2 \mathbf{e}_{PP,i}\mathbf{e}_{PP,i}^H\Big\} \nonumber\\ &\stackrel{(a)}{=} \mathcal{E}_{\mathbf{H}_{PP}}\Big\{\textstyle{\sum_{i=1}^{N_t}} [\boldsymbol{\Phi}_x]_{ii} \tilde{\boldsymbol{\Phi}}_{e,i}\Big\}\nonumber\\ &\stackrel{(b)}{=} \mathcal{E}_{\mathbf{H}_{PP}}\Big\{\textstyle{\sum_{i=1}^{N_t}} [\boldsymbol{\Phi}_x]_{ii} \Big(\mathbf{H}_{PP}^H \textstyle{\sum_{j=1}^{N_t}}|c_{ij}|^2\mu_x\mathtt{diag}(\mathbf{c}_j\mathbf{c}_j^H) \mathbf{H}_{PP} \nonumber \\ &\quad + \frac{1}{P_{tr}}(1+\mu_y)\sigma_n^2\mathbf{I}_{N_r} + \textstyle{\sum_{j=1}^{N_t}} |c_{ij}|^2 \mu_y\cdot\nonumber\\ &\quad \mathtt{diag}\big(\mathbf{H}_{PP}^H \big(\mathbf{c}_j\mathbf{c}_j^H+\mu_x\mathtt{diag}(\mathbf{c}_j\mathbf{c}_j^H)\big)\mathbf{H}_{PP}\big) \Big) \Big\} \nonumber\\ & \stackrel{(c)}{=} \textstyle{\sum_{i=1}^{N_t}} [\boldsymbol{\Phi}_x]_{ii} \Big(\textstyle{\sum_{j=1}^{N_t}} |c_{ij}^2|\mu_x \bar{\alpha}_{PP} \mathtt{tr}\big(\mathtt{diag}(\mathbf{c}_j\mathbf{c}_j^H)\big)\mathbf{I}_{N_r} +\nonumber\\ &\quad \frac{1}{P_{tr}}(1+\mu_y) \sigma_n^2\mathbf{I}_{N_r} + \textstyle{\sum_{j=1}^{N_t}} |c_{ij}|^2\mu_y \bar{\alpha}_{PP}\big(\mathtt{tr}(\mathbf{c}_j\mathbf{c}_j^H)\mathbf{I}_{N_r} +\nonumber\\ &\quad \mu_x\mathtt{tr}\big(\mathtt{diag}(\mathbf{c}_j\mathbf{c}_j^H)\mathbf{I}_{N_r}\big)\big) \Big)\nonumber\\ &\stackrel{(d)}{=} \big(\frac{(1\!+\!\mu_y)\sigma_n^2}{P_{tr}} \!+\! \bar{\alpha}_{PP}(\mu_x\!+\!\mu_y\!+\!\mu_x\mu_y)\big) \mathtt{tr}(\boldsymbol{\Phi}_x) \mathbf{I}_{N_r},\end{aligned}$$ where the expectations over $\mathbf{x}_p$ and $\mathbf{E}_{PP}$ are taken in step $(a)$, $[\boldsymbol{\Phi}_x]_{ii}$ denotes the $i$-th diagonal element of $\boldsymbol{\Phi}_x$, step $(b)$ comes from (\[E:Epp-cov\]), step $(c)$ takes the expectation over $\mathbf{H}_{PP}$ considering $\mathtt{vec}(\mathbf{H}_{PP})\sim\mathcal{CN}(\bar{\mathbf{0}}_{N_rN_t}, \bar{\alpha}_{PP}\mathbf{I}_{N_rN_t})$, and step $(d)$ comes from the fact that $\mathbf{CC}^H = \mathbf{I}_{N_t}$ such that $\mathtt{tr}(\mathbf{c}_i\mathbf{c}_i^H) = \mathtt{tr}(\mathtt{diag}(\mathbf{c}_i\mathbf{c}_i^H)) = \sum_{j=1}^{N_t} |c_{ij}|^2 = 1$. The term $\boldsymbol{\Sigma}_2$ can be obtained as $$\label{E:Sigma-2} \boldsymbol{\Sigma}_2 = \mathcal{E}_{\mathbf{H}_{PP}}\{\mathbf{H}_{PP}^H \mu_x\mathtt{diag}(\boldsymbol{\Phi}_x) \mathbf{H}_{PP}\} = \bar{\alpha}_{PP}\mu_x\mathtt{tr}(\boldsymbol{\Phi}_x) \mathbf{I}_{N_r}.$$ With (\[E:Phi-y\]), the term $\boldsymbol{\Sigma}_3$ can be obtained as $$\begin{aligned} \label{E:Sigma-3} & \boldsymbol{\Sigma}_3 =\mathcal{E}_{\mathbf{H}_{PP}}\{\mu_y\mathtt{diag}(\boldsymbol{\Phi}_y)\} \nonumber\\ & = \mathcal{E}_{\mathbf{H}_{PP}}\Big\{\mu_y\mathtt{diag}\big( \bar{\mathbf{H}}_{MP}^H \bar{\mathbf{H}}_{MP} +\nonumber\\ &\quad \mathbf{H}_{PP}^H (\boldsymbol{\Phi}_x+\mu_x\mathtt{diag}(\boldsymbol{\Phi}_x)) \mathbf{H}_{PP} + \sigma_n^2\mathbf{I}_{N_r} \big)\Big\}\\ & = \mu_y\Big(\mathtt{diag}(\bar{\mathbf{H}}_{MP}^H \bar{\mathbf{H}}_{MP}) \!+\! \bar{\alpha}_{PP}(1\!+\!\mu_x)\mathtt{tr}(\boldsymbol{\Phi}_x)\mathbf{I}_{N_r} \!+\! \sigma_n^2\mathbf{I}_{N_r} \Big).\nonumber\end{aligned}$$ Substituting (\[E:Sigma-1\]), (\[E:Sigma-2\]) and (\[E:Sigma-3\]) into (\[E:Pout-0\]) and noting that $\mathtt{tr}(\boldsymbol{\Phi}_x) = P_{out}$, we can obtain $$\begin{aligned} \label{E:Pout-final} &P_{out} \!=\! \mathtt{tr}\left(\mathbf{W}_f\big(\bar{\mathbf{H}}_{MP}^H\bar{\mathbf{H}}_{MP} \!+\! \mu_y \mathtt{diag}(\bar{\mathbf{H}}_{MP}^H \bar{\mathbf{H}}_{MP})\big)\mathbf{W}_f^H\right) \!+\! \nonumber\\ & \ \ \textstyle{\sum_{k=1}^{K_P}} \|\mathbf{w}_{d,k}\|^2 \!+\!\Big((1+\mu_y)\sigma_n^2 \!+\! \nonumber\\ & \ \ \big({\frac{(1\!+\!\mu_y)\sigma_n^2}{P_{tr}}} \!+\! 2\bar{\alpha}_{PP}(\mu_x\!+\!\mu_y\!+\!\mu_x\mu_y)) \big) P_{out}\Big) \mathtt{tr}(\mathbf{W}_f\mathbf{W}_f^H) \nonumber\\ &\approx \mathtt{tr}\left(\mathbf{W}_f\bar{\mathbf{H}}_{MP}^H\bar{\mathbf{H}}_{MP}\mathbf{W}_f^H\right) + \textstyle{\sum_{k=1}^{K_P}} \|\mathbf{w}_{d,k}\|^2 +\nonumber\\ &\ \ \big(\sigma_n^2 + (\frac{\sigma_n^2}{P_{tr}}+2\bar{\alpha}_{PP}(\mu_x+\mu_y)) P_{out} \big) \mathtt{tr}(\mathbf{W}_f\mathbf{W}_f^H),\end{aligned}$$ where the approximation follows from $\mu_x\ll 1$ and $\mu_y\ll 1$ as in [@Day2012]. Proof of Lemma 1 {#S:prooflemma2} ================ To prove this lemma, we rewrite problem (\[E:problem\]) by expressing $\mathbf{W}_f$ with its vectorization, denoted by $\bar{\mathbf{w}}_f = \mathtt{vec}(\mathbf{W}_f)$, as follows. \[E:problem1-vec\] $$\begin{aligned} &\underset{\bar{\mathbf{w}}_f, \mathbf{w}_{d,k}}{\max}\ \frac{|\mathbf{h}_{Pk}^H\mathbf{w}_{d,k}|^2}{\Omega'_k } \label{E:Objective-vec}\\ &s.t.\ \|\big(\bar{\mathbf{h}}_{MP}^T\!\otimes\!\mathbf{I}_{N_t}\big)\bar{\mathbf{w}}_f\|^2 \!+\! (P_0\sigma_e^2 \!+\! \sigma_n^2)\|\bar{\mathbf{w}}_f\|^2 \!+\! \|\mathbf{w}_{d,k}\|^2 \!\leq\! P_0, \label{E:Constraint-vec} \end{aligned}$$ where $\Omega'_k \triangleq |\bar{h}_{Mk}^\ast + e^{-j\phi} \big(\bar{\mathbf{h}}_{MP}^T \otimes\mathbf{h}_{Pk}^H \big) \bar{\mathbf{w}}_f|^2 + \|\big(\mathbf{I}_{N_r}\otimes \mathbf{h}_{Pk}^H\big) \bar{\mathbf{w}}_f\|^2(P_0\sigma_e^2+\sigma_n^2) + \sigma_n^2$, and the property $\mathtt{vec}(\mathbf{AXB}) = \left(\mathbf{B}^T\otimes\mathbf{A}\right)\mathtt{vec}(\mathbf{X})$ is used. Based on the KKT condition, we can obtain the optimal solution of $\bar{\mathbf{w}}_{f}$ as $$\begin{aligned} \label{E:optwf} &\bar{\mathbf{w}}_f^\star = - \bar{h}_{Mk}^\ast e^{j\phi}\Big(\frac{\mathtt{SINR}_{k,FD}}{\Delta_1} \big(\bar{\mathbf{h}}_{MP}^T \otimes \mathbf{h}_{Pk}^H \big)^H\big(\bar{\mathbf{h}}_{MP}^T \otimes \mathbf{h}_{Pk}^H\big) \nonumber\\ &\!+\! \frac{\mathtt{SINR}_{k,FD}\Delta_2}{\Delta_1} \!\big(\! \mathbf{I}_{N_r}\! \otimes\! \mathbf{h}_{Pk}^H\!\big)^H\big(\mathbf{I}_{N_r}\!\otimes\! \mathbf{h}_{Pk}^H\big)\!+\! \lambda \big(\bar{\mathbf{h}}_{MP}^T\!\otimes\!\mathbf{I}_{N_t}\big)^H\cdot\nonumber\\ & \big(\bar{\mathbf{h}}_{MP}^T \otimes\mathbf{I}_{N_t}\big) + \lambda\Delta_2 \mathbf{I}_{N_tN_r}\Big)^{-1} \big(\bar{\mathbf{h}}_{MP}^T \otimes\mathbf{h}_{Pk}^H\big)^H,\end{aligned}$$ where $\Delta_1 = |\bar{h}_{Mk}^\ast + e^{-j\phi} \left(\bar{\mathbf{h}}_{MP}^T \otimes \mathbf{h}_{Pk}^H\right) \bar{\mathbf{w}}_f^\star|^2 + \|\left(\mathbf{I}_{N_r}\otimes \mathbf{h}_{Pk}^H\right) \bar{\mathbf{w}}_f^\star\|^2(P_0\sigma_e^2+\sigma_n^2) + \sigma_n^2$, $\Delta_2 = P_0\sigma_e^2+\sigma_n^2$, and $\lambda$ is the lagrangian multiplier. By applying the properties of Kronecker product and after some regular manipulations, we can simplify (\[E:optwf\]) as $$\begin{aligned} \label{E:optwf-sim} \bar{\mathbf{w}}_f^\star = & - \bar{h}_{Mk}^\ast e^{j\phi}\left(\left((\bar{\mathbf{h}}_{MP}\bar{\mathbf{h}}_{MP}^H)^T + \Delta_2\mathbf{I}_{N_r}\right)^{-1} (\bar{\mathbf{h}}_{MP}^H)^T\right)\nonumber\\ &\otimes \left(\left(\frac{\mathtt{SINR}_{k,FD}}{\Delta_1}\mathbf{h}_{Pk}\mathbf{h}_{Pk}^H + \lambda\mathbf{I}_{N_t} \right)^{-1}\mathbf{h}_{Pk}\right).\end{aligned}$$ Then based on the matrix inversion lemma, we can further rewrite (\[E:optwf-sim\]) as $$\begin{aligned} \label{E:optwf2} \bar{\mathbf{w}}_{f}^\star &= \frac{ - \bar{h}_{Mk}^\ast e^{j\phi}\Delta_1(\bar{\mathbf{h}}_{MP}^H)^T\otimes \mathbf{h}_{Pk}}{\left(\mathtt{SINR}_{k,FD}\|\mathbf{h}_{Pk}\|^2 + \lambda\Delta_1\right)\left( \|\mathbf{h}_{MP}\|^2 + \Delta_2\right)} \nonumber\\ &\triangleq - \bar{h}_{Mk}^\ast e^{j\phi}\cdot \beta \cdot (\bar{\mathbf{h}}_{MP}^H)^T\otimes\mathbf{h}_{Pk},\end{aligned}$$ where $\beta = \frac{\Delta_1}{\left(\mathtt{SINR}_{k,FD}\|\mathbf{h}_{Pk}\|^2 + \lambda\Delta_1\right)\left( \|\mathbf{h}_{MP}\|^2 + \Delta_2\right)}$ is a positive scalar. According to (\[E:optwf2\]), we can recover the optimal $\mathbf{W}_f^\star$ from $\bar{\mathbf{w}}_f^\star$ as (\[E:w1\]), which is of rank 1. Proof of Proposition 1 {#S:prooftheorem1} ====================== By defining $\nu \triangleq |\bar{h}_{Mk}|\|\mathbf{h}_{Pk}\|\|\bar{\mathbf{h}}_{MP}\| \beta = \frac{C}{2}\beta$, problem (\[E:problem1\]) can be rewritten as \[E:problem1-A\] $$\begin{aligned} \underset{\nu}{\max}\ & f(\nu) \triangleq \frac{A-B\nu^2}{B\nu^2 - C\nu + D} \label{E:Objective1-A}\\ s.\ t.\ & \nu^2 \leq \frac{A}{B}\label{E:Constraint1-A}\\ \ & \nu \geq 0, \label{E:Constraint2-A} \end{aligned}$$ where the parameters $A$, $B$, $C$ and $D$ are defined in Proposition 1. Since the numerator of the objective function (\[E:Objective1-A\]) is concave, the denominator is convex, and both are differential, we know that the objective function of problem (\[E:problem1-A\]) is quasi-concave [@boyd2009convex]. This suggests that the global optimal solution to problem (\[E:problem1-A\]), denoted by $\bar \nu$, needs to satisfy the following KKT conditions, \[E:KKT-A\] $$\begin{aligned} -\nabla f(\bar{\nu}) + 2u\bar{\nu} - v &= 0 \label{E:KKT-A1}\\ u(\bar{\nu}^2 - \frac{A}{B}) = 0,\ u &\geq 0 \label{E:KKT-A2}\\ v \bar{\nu} = 0,\ v &\geq 0, \label{E:KKT-A3} \end{aligned}$$ where $u$ and $v$ are the lagrangian multipliers. We next show that $u = 0$ and $v = 0$ for problem (\[E:problem1-A\]). First, we find that $\bar{\nu}^2 < \frac{A}{B}$ must hold for the constraint (\[E:Constraint1-A\]), otherwise, when $\bar{\nu}^2 = \frac{A}{B}$, the objective function will be zero that is obviously non-optimal. Therefore, from (\[E:KKT-A2\]) we have $u = 0$. Second, the objective function satisfies $\nabla f(0) > 0$, which means that we can always find a small $\epsilon > 0$ making $f(\epsilon) > f(0)$, i.e., $\bar{\nu} > 0$ must hold. Therefore, based on (\[E:KKT-A3\]) we have $v = 0$. Then the condition (\[E:KKT-A1\]) for $\bar{\nu}$ can be simplified as $$\label{E:Largrange} 2B\bar{\nu} - \frac{(B\bar{\nu}^2-A)(2B\bar{\nu}-C)}{B\bar{\nu}^2 - C\bar{\nu} + D} = 0.$$ From (\[E:Largrange\]), we can derive the optimal value of the objective function as $$f(\bar{\nu}) =\frac{2B\bar{\nu}}{C-2B\bar{\nu}} = \frac{1}{\frac{C}{2B\bar{\nu}} -1}.$$ The condition (\[E:Largrange\]) can be further expressed as a quadratic equation with respect to $\bar{\nu}$, which is $$\label{E:equation} g(\bar{\nu}) \triangleq BC\bar{\nu}^2 - 2B(A+D)\bar{\nu} + AC = 0,$$ whose two solutions can be obtained as $$\begin{aligned} \label{E:two-solutions} \bar{\nu} &= \frac{A+D\pm\sqrt{(A+D)^2 - \frac{AC^2}{B}}}{C}.\end{aligned}$$ The feasibility of the two solutions is examined as follows. Recalling that $0\leq \bar{\nu}\leq \sqrt{\frac{A}{B}}$ from (\[E:Constraint1-A\]) and (\[E:Constraint2-A\]), we can see that $g(0) = AC > 0$ and $g(\small{\sqrt{\frac{A}{B}}})<0$ because $$\begin{aligned} & g(\textstyle{\sqrt{\frac{A}{B}}})= 2AC - 2(A+D)\sqrt{AB} \nonumber\\ & < 4P_0|\bar{h}_{Mk}|\|\bar{\mathbf{h}}_{MP}\|\|\mathbf{h}_{Pk}\|^3 - 2(P_0\|\mathbf{h}_{Pk}\|^2 + |\bar{h}_{Mk}|^2)\cdot\nonumber\\ &\qquad\sqrt{P_0\|\bar{\mathbf{h}}_{MP}\|^2\|\mathbf{h}_{Pk}\|^4}\nonumber\\ & = -2\sqrt{P_0}\|\bar{\mathbf{h}}_{MP}\|\|{\mathbf{h}}_{Pk}\|^2(|\bar{h}_{Mk}| - \sqrt{P_0}\|\mathbf{h}_{Pk}\|)^2,\end{aligned}$$ where the inequality is obtained by setting $\sigma_e^2 = 0$ and $\sigma_n^2 = 0$ in the definitions of $B$ and $D$. The results indicate that equation (\[E:equation\]) has one and only one solution within $[0, \sqrt{\frac{A}{B}}]$, which is the smaller one in (\[E:two-solutions\]). Finally, recalling that $\nu = \frac{C}{2}\beta$, we can obtain the optimal $\beta$ as shown after (\[E:maxSINR\]). Proof of Proposition 2 {#S:prooflemma4} ====================== By defining $\mathbf{A} = \mathbf{I}_{N_t} +\sum_{j\neq k} \bar{\lambda}_j \mathbf{h}_{Pj}\mathbf{h}_{Pj}^H$ and $\mathbf{B} = \mathbf{I}_{N_t} - \frac{\bar{\lambda}_k}{\gamma_k} \mathbf{A}^{-\frac{1}{2}} \mathbf{h}_{Pk} \mathbf{h}_{Pk}^H\mathbf{A}^{-\frac{H}{2}}$, we can express the left-hand side of (\[E:con-lambda\]) as $\mathbf{A}^{\frac{1}{2}}\mathbf{B}\mathbf{A}^{\frac{H}{2}}$. Since $\mathbf{A}$ is positive definite and $\mathbf{B}$ is Hermitian, it is not difficult to show that $\mathbf{A}^{\frac{1}{2}}\mathbf{B}\mathbf{A}^{\frac{H}{2}}\succeq \mathbf{0}_{N_t}$ and $\mathbf{B}\succeq \mathbf{0}_{N_t}$ are equivalent to each other. Then, the semi-definite positive constraints (\[E:con-lambda\]) can be equivalently expressed as $$\begin{aligned} \label{E:semi-definite} & \mathbf{I}_{N_t} - \frac{\bar{\lambda}_k}{\gamma_k}\big(\mathbf{I}_{N_t} +\sum_{j\neq k} \bar{\lambda}_j \mathbf{h}_{Pj}\mathbf{h}_{Pj}^H\big)^{-\frac{1}{2}} \mathbf{h}_{Pk} \mathbf{h}_{Pk}^H\big(\mathbf{I}_{N_t} \nonumber\\ & \quad\qquad+\sum_{j\neq k} \bar{\lambda}_j \mathbf{h}_{Pj}\mathbf{h}_{Pj}^H\big)^{-\frac{H}{2}} \succeq \mathbf{0}_{N_t}, \forall k.\end{aligned}$$ The positive semi-definite constraint in (\[E:semi-definite\]) means that the minimal eigenvalue of the matrix in the left-hand side should be non-negative. Note that the term $\left(\mathbf{I}_{N_t} +\sum_{j\neq k} \bar{\lambda}_j \mathbf{h}_{Pj}\mathbf{h}_{Pj}^H\right)^{-\frac{1}{2}}$ $\mathbf{h}_{Pk} \mathbf{h}_{Pk}^H\left(\mathbf{I}_{N_t} +\sum_{j\neq k} \bar{\lambda}_j \mathbf{h}_{Pj}\mathbf{h}_{Pj}^H\right)^{-\frac{H}{2}}$ is of rank one. It has only one positive eigenvalue, which is $\mathbf{h}_{Pk}^H\left(\mathbf{I}_{N_t} +\sum_{j\neq k} \bar{\lambda}_j \mathbf{h}_{Pj}\mathbf{h}_{Pj}^H\right)^{-1} \mathbf{h}_{Pk}$, and all other eigenvalues are zeros. Therefore, the constraint in (\[E:semi-definite\]) can be further converted into the constraint on the minimal eigenvalue of the matrix in the left-hand side of (\[E:semi-definite\]) as $$1 - \frac{\bar{\lambda}_k}{\gamma_k}\mathbf{h}_{Pk}^H\left(\mathbf{I}_{N_t} +\sum_{j\neq k} \bar{\lambda}_j \mathbf{h}_{Pj}\mathbf{h}_{Pj}^H\right)^{-1} \mathbf{h}_{Pk} \geq 0, \forall k.$$ which can be further rewritten as (\[E:Con-semidefinite\]). [^1]: This work was supported in part by the National Natural Science Foundation of China (No. 61301084) and by the Fundamental Research Funds for the Central Universities. S. Han, C. Yang, and P. Chen are with the School of Electronics and Information Engineering, Beihang University, Beijing, China (e-mail: {sqhan, cyyang, chenpan}@buaa.edu.cn). [^2]: In order to highlight the benefits of the fICIC scheme via forwarding the listened ICI, we restrict ourselves to the case where the precoders $\mathbf{W}_f$ and $\{\mathbf{w}_{d,k}\}$ are designed only for transmitting the listened ICI and desired signals but not for spatial-domain self-interference cancellation (i.e., design $\mathbf{W}_f$ and $\{\mathbf{w}_{d,k}\}$ to pre-null self-interference). Towards this end, we take the expectation over $\mathbf{H}_{PP}$ in (\[E:Pout-new\]), such that the precoders are independent of the instantaneous self-interference channel $\mathbf{H}_{PP}$. [^3]: One can further incorporate the propagation delay difference experienced by the forwarded ICI and the direct ICI into $\tau$, which is not considered here because the PUEs are close to the PBS leading to negligible additional propagation delay. [^4]: Herein, we consider that the increase of the desired signal’s strength comes from reducing the distance between the PUE and its serving PBS, but not from increasing the power of the PBS. As a result, the interference from surrounding PBSs to the PUE will be even weaker, making it reasonable to focus on the dominant cross-tier ICI as we considered. [^5]: Otherwise, suppose that the power constraint holds with inequality with the optimal precoders $\mathbf{W}^\star_{f}$ and $\{\mathbf{w}_{d,k}^\star\}$, then given $\mathbf{W}^\star_{f}$, we can always find new precoders $\{\mathbf{w}'^\star_{d,k}\}$, defined as $\mathbf{w}'^\star_{d,k} = c\mathbf{w}^\star_{d,k}$ for $k = 1, \dots, {K_P}$ with $c > 1$, which can further improve the SINR of all PUEs until the constraint (\[E:Constraint-MU\]) holds with equality.

--- abstract: | We use the two-point correlation function to calculate the clustering properties of the recently completed SSRS2 survey, which probes two well separated regions of the sky, allowing one to evaluate the sensitivity of sample-to-sample variations. Taking advantage of the large number of galaxies in the combined sample, we also investigate the dependence of clustering on the internal properties of galaxies. The redshift space correlation function for the combined magnitude-limited sample of the SSRS2 is given by $\xi(s)=(s/5.85$ 1 Mpc)$^{-1.60}$ for separations between 2 $\leq s \leq$ 11 1 Mpc, while our best estimate for the real space correlation function is $\xi (r) = (r/5.36$ 1 Mpc)$^{-1.86}$. Both are comparable to previous measurements using surveys of optical galaxies over much larger and independent volumes. By comparing the correlation function calculated in redshift and real space we find that the redshift distortion on intermediate scales is small. This result implies that the observed redshift-space distribution of galaxies is close to that in real space, and that $\beta = \Omega^{0.6}/b < 1$, where $\Omega$ is the cosmological density parameter and $b$ is the linear biasing factor for optical galaxies. We have used the SSRS2 sample to study the dependence of $\xi$ on the internal properties of galaxies such as luminosity, morphology and color. We confirm earlier results that luminous galaxies ($L>L^*$) are more clustered than sub-$L^*$ galaxies and that the luminosity segregation is scale-independent. We also find that early types are more clustered than late types. However, in the absence of rich clusters, the relative bias between early and late types in real space, $b_{E+S0}$/$b_S$ $\sim$ 1.2, is not as strong as previously estimated. Furthermore, both morphologies present a luminosity-dependent bias, with the early types showing a slightly stronger dependence on the luminosity. We also find that red galaxies are significantly more clustered than blue ones, with a mean relative bias of $b_R/b_B$ $\sim$ 1.4, stronger than that seen for morphology. Finally, by comparing our results with the measurements obtained from the infrared-selected galaxies we determine that the relative bias between optical and galaxies in real space is $b_o/b_I$ $\sim$ 1.4. author: - 'C. N. A. Willmer' - 'L. Nicolaci da Costa' - 'P. S. Pellegrini' title: 'Southern Sky Redshift Survey: Clustering of Local Galaxies [^1]' --- -1[$h^{-1}$]{} \#1 1[$h^{-1}$]{} 8 [$\sigma_8\>$]{} Introduction ============ Optical surveys using full sampling (CfA2, Geller & Huchra, 1989; SSRS2, da Costa 1994, 1997) now probe over 30% of the sky down to a magnitude limit of $m_B$ = 15.5. These surveys allow delineating various large-scale structures in the different volumes, thanks to their wide angular coverage. The combination of the large number of galaxies, complete sampling and sky coverage makes these surveys extremely useful to study some of the statistical properties of galaxy clustering. Thus, it becomes possible to divide the sample in bins of luminosities, morphologies and colors and study the dependence of clustering on these parameters. The simplest characterization of the galaxy clustering can be expressed in terms of the two-point correlation function $\xi$. This statistic has been widely applied to a variety of samples which include surveys of optical (Davis & Peebles 1983; Davis et al. 1988; de Lapparent et al. 1988; Pellegrini 1990$b$; Santiago & da Costa 1990; Loveday 1995; Marzke 1995; Hermit 1996; Tucker 1997; Guzzo et al. 1997) and galaxies (Davis 1988; Saunders et al. 1992; Strauss et al. 1992; Fisher et al. 1994). The two-point correlation function has also been used to characterize the dependence of the galaxy clustering on the internal properties of galaxies such as morphology (Davis & Geller 1976; Giovanelli, Haynes & Chincarini 1986; Iovino et al. 1993; Loveday et al. 1995; Hermit et al. 1996; Guzzo et al. 1997), color (Tucker et al. 1996), surface brightness (Santiago & da Costa 1990), luminosity (Benoist et al. 1996 and references therein; Valotto & Lambas 1997; Guzzo et al. 1997) and internal dynamics (White, Tully & Davis 1988). However, since most samples analyzed so far have been relatively small, the quantitative results are by and large tentative. In this paper we use the SSRS2 to investigate the correlation properties of galaxies in the nearby Universe, comparing our results with those obtained in other surveys such as the Stromlo-APM (Loveday 1995), Las Campanas Redshift Survey (LCRS, Tucker 1997) and (Fisher 1994). We also measure how sensitive our results are to sample-to-sample variations. With our enlarged sample, we re-examine the luminosity dependence of $\xi$, previously studied by Benoist et al. (1996) who only used the southern galactic cap portion of the SSRS2. Finally, we investigate the dependence of the correlation properties on morphologies and colors. This information is a key ingredient for studies of galaxy biasing. In Section 2 we describe the different catalogs we analyzed, while in Section 3 we describe the method used to compute the two-point correlation function in redshift and real space. In Section 4 we present the results we obtain for magnitude-limited samples. In Section 5 we examine the dependence of clustering on internal properties of galaxies, while in Section 6 we compare the correlation properties of our optical sample with the infrared-selected 1.2 Jy survey (Fisher et al. 1994). A summary is presented in Section 7. Data ==== In this work we use data obtained for the SSRS2, which contains 5512 galaxies in both galactic hemispheres. The data are discussed in greater detail by da Costa (1994; 1997) so only a brief description will be made here. The SSRS2 is extracted from the list of non-stellar objects in the Hubble Space Telescope Guide Star Catalog (Lasker 1990, hereafter GSC). The SSRS2 is a complete catalog which is magnitude-limited at $m_B$ = 15.5. Alonso (1994) have shown that the galaxy magnitudes derived from the GSC are on a uniform isophotal magnitude system ($\sim$ 26 $mag$ $arc$ $sec^{-2}$) with estimated magnitude errors 0.3 $mag$. The SSRS2 south contains 3573 galaxies distributed over 1.13 steradians of the southern galactic cap ($b \leq -40^o$), within the declination range $-40^o \leq \delta \leq -2.5^o$. The SSRS2 north contains 1939 galaxies distributed over a solid angle of 0.57 steradians with $\delta \leq 0^o$ with $b \geq +35^o$. As described by da Costa (1997) the galaxy morphologies in this sample used classifications based on other works (Lauberts & Valentijn 1989) as well as those made by the authors. In this work we also considered a sub-sample of SSRS2 galaxies which have measured colors. For this we used the Lauberts & Valentijn (1989) catalog, so that the subcatalog of galaxies with colors only contains objects south of $\delta$ $\approx$ -17.5$^o$, in both galactic hemispheres. We also imposed a cut at $m_B$ = 14.5 because beyond this magnitude the Lauberts & Valentijn (1989) catalog becomes incomplete, in particular for early type galaxies (Pellegrini 1990$a$). This selection effect is caused by the fact that the Lauberts & Valentijn (1989) catalog is derived from the diameter-limited Lauberts (1982) catalog. The sample limited at $m_B$=14.5 contains 780 galaxies in both galactic hemispheres, of which 694 (89%) have colors. As pointed out by Marzke & da Costa (1997), this sample presents no systematic dependence of the color completeness with magnitude down to the 14.5 limit of this sub-sample. All galaxy heliocentric velocities, $v_\odot$, have been corrected for the Solar motion with respect to the centroid of the Local Group using $v =v_\odot + 300 sin(l) cos(b)$ , where ($l$, $b$) are the galactic coordinates of the galaxy. We also remove from the analyses all galaxies with radial velocities less that 500 , as the redshifts for these objects are likely to be dominated by their peculiar velocities. We use throughout $H_o = 100 h$Mpc$^{-1}$. The Two-Point Correlation Function ================================== Method ------ The two-point correlation function $\xi (r)$ can be computed from the data using the estimator suggested by Hamilton (1993): $$\xi(r) = {DD(r) RR(r) \over [DR(r)]^2} - 1,$$ where $DD(r)$, $RR(r)$ and $DR(r)$ are the number of data–data, random–random and data–random pairs, with separations in the interval between $r$ and $r + dr$. The random catalog is generated using the same selection criteria as the galaxy sample. This estimator has the advantage that it is not too sensitive to uncertainties in the mean density, which is only a second order effect. The counts $DD(r)$, $DR(r)$ and $RR(r)$ can be generalized to include a weight $w$ which is particularly important to correct for selection effects at large distances in magnitude-limited samples: $$\begin{aligned} DD(r) = &{\displaystyle{ \sum_i^{N_{\scriptscriptstyle{gal}}} \sum_j^{N_{\scriptscriptstyle{gal}}}} } & w(s_j, r) w(s_i, r), \\ & \scriptscriptstyle {r- \Delta r \leq |s_i-s_j| \leq r+ \Delta r} \nonumber\end{aligned}$$ where $i$ sums over all objects in the sample and the sum over $j$ includes all particles at a distance $s$ from the origin, which in this work is taken as the centroid of the Local Group, and $r = | {\bf{s}}_i -{\bf{s}}_j|$ is the separation of the pair $(i,j)$. The galaxy-random pairs $DR(r)$ and random-random pairs $RR(r)$ are similarly weighted. The most common weighting schemes are: equally weighted pairs $w(s_i, r)=1$; equally–weighted volumes where $w(s_i, r)=1/\phi(s_i)$ and the minimum variance weighting given by $$w (s_i, r) = {1 \over 1 +4\pi \bar n J_3(r) \phi(s_i)}, \quad J_3 (r) = \int_0^r dr'r'^2 \xi(r'),$$ where $\phi (s_i)$ is the selection function at distance $s_i$ from the origin and $J_3$ is the mean number of excess galaxies out to a distance $r$ around each galaxy. Even though in the last scheme the weights depend on the unknown correlation function, in practice, it is not very sensitive to the exact form of $\xi(r)$ (Loveday 1992; Marzke, Huchra & Geller 1994). In this work we adopt the minimum-variance weighting and take $J_3 (r$ = 30 1 Mpc) $\sim 1100$, obtained by using the real-space correlation function of Davis & Peebles (1983). The mean densities were calculated using the estimator $$\bar n = \sum_{i=1}^{N_{gal}}w_i/\int_{s_{min}}^{s_{max}} dV\phi(s)w(s)$$ where again $\phi(s)$ is the selection function, derived from the luminosity function and $w(s)$ is the weight (Davis & Huchra 1982). The errors for the redshift space correlation function () as well as for the real-space correlation discussed below, were calculated by means of bootstrap resampling (Ling, Frenk & Barrow 1986). For the volume-limited samples the total of bootstraps was 50, while for magnitude-limited samples 25 resamplings were calculated. As shown by Fisher (1994), bootstraping tends to overestimate the true errors, so that the estimate of the latter will in general be rather conservative. Real space ---------- In order to estimate real space correlation functions, we follow Davis & Peebles (1983). For any two galaxies with redshifts [**s$_1$**]{} and [**s$_2$**]{}, we define the separation in redshift space, and the separation perpendicular to the line of sight respectively as $${\bf {s = s_1 - s_2}}, \quad {\bf {l}} = {1 \over 2} \bf{(s_1 + s_2)},$$ in the small angle approximation. From these parameters one can derive $\pi$, the separation between two galaxies parallel to the line of sight and $r_p$, the separation perpendicular to the line of sight using: $$\pi = {{\bf s.l} \over |l|}, \quad r_p = \sqrt{|{\bf{s}}|^2 -\pi^2}.$$ These are then used to compute the statistic $\xi(r_p,\pi)$ estimated from the pair-counts as $$1 + \xi(r_p,\pi) = {DD(r_p,\pi) RR(r_p,\pi) \over [DR(r_p,\pi)]^2}.$$ From we define the projected function: $$\omega(r_p) = 2 \int_0^\infty d\pi \quad \xi (r_p,\pi),$$ which is related to the real space correlation function through $$\omega(r_p) = 2\int_0^\infty dy \quad \xi[(r_p^2 + y^2)^{1/2}].$$ The inverse is the Abel integral: $$\xi(r) = -{1 \over \pi} \int_r^\infty dr_p {\omega^\prime(r_p) \over (r_p^2-r^2)^{1/2}, }$$ where $\omega^\prime(r_p)$ is the first derivative of $\omega (r_p)$. If the real space correlation function is a power-law, the integral for $\omega(r_p)$ can be performed analytically to give $$\omega(r_p) = r_p ({r_o \over r_p})^{\gamma} { \Gamma({1 \over 2}) \Gamma({\gamma -1 \over 2}) \over \Gamma({\gamma \over 2})}.$$ Biasing ------- The variance of galaxy counts measures the clustering amplitude at intermediate scales. It is also a useful quantity to compare models and data. The variance in the counts is defined as $$\langle (N -nV)^2 \rangle = nV +n^2V^2\sigma^2,$$ where nV is the mean number of galaxies in the volume V and $ n^2V^2\sigma^2$ is the mean number of galaxies in excess of random inside a sphere of volume V. It is related to the moment of the correlation function (Peebles 1980) $$\sigma^2 = {1 \over V^2} \int_VdV_1dV_2 \xi(|r_1-r_2|),$$ which can be calculated numerically. For a power law correlation function $\xi(r) = (r/r_o)^\gamma$, and a spherical volume of radius R we get $$\sigma^2(R) = 72(r_o/R)^\gamma/\lbrack 2^\gamma(3-\gamma)(4-\gamma)(6-\gamma)\rbrack.$$ This is the expression we have used to compute 8 , and which is often used to normalize theoretical models. The relative bias between two different samples at a given separation $s$ may be estimated through (Benoist et al. 1996) : $$\frac {b} {b_*}(s) = \sqrt{\frac{\xi(s)}{\xi_*(s)}} = \sqrt{\frac{J_3(s)}{J_3*(s)}},$$ where the starred symbols denote a sample taken as a fiducial. The relative bias of the clustering may also be estimated through $$\frac {b} {b_*}(s) = \sqrt{\frac{\sigma^2(s)}{\sigma_*^2(s)}},$$ where $\sigma^2$ is the variance of counts in cells described above. These are the expressions used in this work to calculate the relative bias between galaxies of different luminosities relative to $L^*$ galaxies, as well as for different morphological types and colors. Magnitude-limited Samples ========================= In order to estimate the effects due to the finite volume we are probing and to estimate the importance of cosmic variance, we compare the clustering properties of the individual SSRS2 south and north samples as well as the combined sample, with previous estimates of $\xi$. In this analysis, we have computed taking into account all galaxies brighter than $M=-13$ in the velocity range $500 < v < 12,000$ . The correlation function was computed using the minimum-variance weighting discussed in Section 3 and a random background catalog of 10,000 points for the individual samples and 20,000 points for the combined sample. In the calculation of the selection function we have used the Schechter parameters determined for the entire SSRS2 survey by da Costa et al. (1997), which are $M^*$ = -19.55 and $\alpha$ = -1.15. These values are virtually identical to those measured by da Costa et al. (1994) for the SSRS2 south. We tested whether our results are affected by the presence of clusters of galaxies. For this we used a list of galaxy clusters with richness R $\geq$ 1 (J. Huchra, private communication). All galaxies whose positions were within one Abell radius of the central position of cluster, and that had radial velocities within 500 of the cluster’s mean radial velocity were culled from the sample. We find that the correlation parameters are virtually unchanged for the vast majority of the samples, and when there are changes, these are within the quoted errors of the complete sample. Therefore, we will not consider the removal of galaxies in clusters in this work. The redshift space correlation function, , for the SSRS2 samples is shown in Fig. 1, where we plot the correlation function out to separations of 30 1 Mpc. For the sake of clarity, in the figure we only show error bars calculated for the combined sample. One can see that beyond $\sim$ 15 1 Mpc, the errors become progressively larger, and sometimes the sample-to-sample variations are larger than the estimated errors. In general, is adequately described by a power-law on small scales. For most cases in this paper, the power-law fits were calculated in the interval $2 < s < 11$ 1 Mpc. The upper-limit was chosen because there is a suggestion of an abrupt change of slope in on scales $s$ $\lsim$ 12 1 Mpc. The lower-limit was chosen to minimize the effects on due to peculiar motions of galaxies in virialized systems. The best power-law fits obtained for each sub-sample of the SSRS2 are represented as lines in Fig. 1, as explained in the caption. The correlation parameters derived from the fits are presented in Table 1, where we list: the sample identification (column 1); the correlation length (column 2) and slope $\gamma_s$ (column 3) obtained from the power-law fits; and in column (4) the rms variance in galaxy counts within spheres 8 1 Mpc in radius, followed in columns (5) through (7) by the same parameters determined for real space, which will be discussed below. An inspection of both Table 1 and Fig. 1 shows that the redshift correlation functions for SSRS2 sub-samples are very similar on small scales ($s < 10$ 1 Mpc). This also demonstrates that the sampling variations are consistent with the error estimates, at least in the range of separations for which the fits are calculated. In Fig. 2 we compare measured in this work for the combined SSRS2 with the measured in other surveys - the sparsely sampled Stromlo-APM survey (Loveday et al. 1995), the Las Campanas Redshift Survey (LCRS) (Tucker 1997), and that measured by Fisher et al. (1994) for the 1.2 Jy survey. The fit parameters calculated in these papers, as well as by other workers can be found in Table 2. Despite small differences in amplitude, the shapes of the three optical surveys are remarkably similar. It is important to note that the volumes of the Stromlo-APM ( 2.5 $\times$ 10$^6$ $h^{-3}$ Mpc$^3$) and the LCRS ( 2.6 $\times$ 10$^6$ $h^{-3}$ Mpc$^3$) are about 5 times larger than that of the SSRS2 (5.2 $\times$ 10$^5$ $h^{-3}$ Mpc$^3$), and probe different regions of space, and thus independent structures. The lower amplitude of the survey compared to the optical samples reflects the relative bias that exists between optically and infrared-selected galaxies, which will be further discussed in Section 6 below. In Fig. 3, we compare our power-law fit parameters with equivalent measurements by other authors (see Table 2). In general, there is a good agreement between our values for the redshift space parameters and those obtained from other optical surveys, specially the Stromlo-APM and LCRS. The effect of redshift distortions on the observed redshift correlation function has also been estimated for the SSRS2. These distortions are caused by the peculiar velocities of galaxies, which on large scales, are due to the infall of galaxies from low-density regions into high-density regions, while on small scales the correlations are smeared out by virial motions of galaxies in groups and clusters (Kaiser 1987). As described in Section 3, these effects may be accounted for by calculating the correlation function as a function of the separations parallel and perpendicular to the line of sight, which can then be used to define the estimator, which is unaffected by redshift distortions. However, one should bear in mind that in general the calculation of the real space correlation function is much more susceptible to noise than that calculated in redshift space. From the correlation functions computed using the minimum-variance weighting scheme, we have obtained . From power-law fits, in the interval 2 $< r_p <$10 1 Mpc, we have derived the correlation parameters listed in Table 1. By comparing the real space fit parameters obtained in this work (Table 1) with previous measures (columns 4 and 5 in Table 2), we find a good agreement with the real space measurements of Davis & Peebles (1983), Loveday et al. (1995) and Marzke et al. (1995). The fit to the real space correlation function for the combined sample is compared in Fig. 4 with the redshift space correlation function. One can see that at intermediate separations the redshift is amplified relative to the real space correlation . The small amplification suggests that the observed redshift distribution is close to the real space distribution. At separations of $\sim$ 10 1 Mpc, the ratio between the real and redshift space correlations is $\sim$ 1.5. In the linear regime, peculiar motions on large scales cause to be amplified by a factor $\sim$ $1 + { 1 \over 2 } {\beta} + { 1 \over 5 } {\beta^2}$ where $\beta = {\Omega^{0.6} \over b}$ and $b$ is the linear biasing factor (Kaiser 1987). Therefore, a rough estimate for $\beta$ is $ \sim 0.6$, on scales of the order of 10 1 Mpc, consistent with that determined by Loveday (1996). The Clustering Dependence on the Internal Properties of Galaxies ================================================================ Luminosity ---------- In this work we use the combined SSRS2, as well as the SSRS2 north and south sub-samples to further explore the clustering dependence on luminosity, as was carried out by Benoist et al. (1996), but who only used the SSRS2 south. Probing independent structures in different volumes we can estimate the impact of cosmic variance. It should also be noted that the absolute magnitude limits considered in this section differ slightly from those of Benoist et al. (1996), and were chosen to compare our results with the volume-limited samples of Fisher (1994), which will be discussed in Section 6 below. The volume-limited samples considered in this section only contain galaxies bright enough that would allow them to be included in the sample when placed at the cutoff distance. We defined samples limited at radial distances of 60, 80, 100 and 120 1 Mpc. The absolute magnitude limits corresponding to these distances are -18.39, -19.01 (both $L < L^*$) , -19.50 ($\sim L^*$) and -19.89 ($L > L^*$), respectively. For all galaxies in these samples, the weighting function is $w(r)=1$ and the volume densities are simply the total number of galaxies divided by the corresponding volume. The correlation functions obtained for the volume-limited sub-samples at the different depths are shown in Fig. 5 for $s \leq 20$ 1 Mpc, where the different symbols represent different volume limits. For reasons of clarity, we only present error bars for the samples volume-limited at 60 1 and 120 1 Mpc. The meaning of these symbols, as well as the indication of the parent sample (SSRS2 south, north or combined) are shown in each panel. The power-law fits are represented by lines in the figure and the parameters are summarized in Table 3 where we list: in column (1) the sample; in column (2) the depth R; in column (3) the number of galaxies N$_g$; in column (4) the mean density; in columns (5) and (6) the power-law fit parameters and formal errors; and in column (7) $\sigma_8$, the rms fluctuation of the number of galaxies in a sphere of radius 81 Mpc. The interval used in the fits is $ 2 < s < 11$ 1 Mpc, the same as that adopted in the previous section. An inspection of Table 3 and Fig. 5 shows that the amplitude of tends to increase with the sample depth, the variation being somewhat larger in the northern and combined samples. We point out that for the SSRS2 north (panel b), is noisier because of the smaller number of galaxies, in most cases about half of those in the southern sample. The correlation length ($s_0$) ranges from 3.8 1 Mpc to 6.8 1 Mpc. However, the slope varies considerably from sample to sample, though with a tendency of becoming steeper as the depth increases. In order to evaluate the cosmic variance, we show in Fig. 6 for each of the volume-limited sub-samples, but now plotting the results for the southern, northern and combined samples in each panel. For the samples in smaller volumes, the differences between the northern and southern samples are larger than the estimated error calculated for the combined sample, and probably reflect the amplitude of the sample to sample variation, with the north being systematically lower. For the larger volumes the samples present similar behavior, and the variations are generally consistent with the estimated errors. To remove the effects of distortions due to motions, which may affect our estimates of the strength of clustering and the relative bias between different samples, we have also computed the real-space correlation function for the volume-limited samples. As above, we have computed for the sub-samples volume-limited at R = 60, 80, 100 and 120 1 Mpc in each galactic hemisphere and for the combined sample. The resulting real space correlation parameters are listed in Table 4. In Fig. 7 we compare measured for each volume limit, denoted by open symbols, with the real-space correlation fits described above, represented as a solid line. For the sake of clarity, we only show the fits we measure for the combined sample, as this will be the one less affected by noise. The smearing due to motions in virialized systems for $r < 3$ 1 Mpc is quite noticeable for all samples, while the effect of peculiar motions is only obvious for the smaller volumes, little evidence being seen in the samples in larger volumes. The dependence of clustering in redshift–space (as measured by $\sigma_8$) with luminosity (as measured by the limiting absolute magnitude of each sample) is shown in Fig. 8(a), where we use as fiducial magnitude the value of $M^*$=-19.55 (see Section 4). The figure shows an overall behavior consistent with that found by Benoist et al. (1996) and which is detected in all samples, further demonstrating that this effect is unlikely to be spurious. This result supports their finding that there is a dependence of clustering on luminosity, as measured in redshift space. To further investigate its reality, we have computed in real space for the same volume limited samples. The results are shown in Fig. 8(b). Here again it is immediately apparent that the clustering amplitude increases with luminosity in the same way as seen in redshift space. On the whole, these results, using a larger sample, confirm in real space the conclusions of Benoist (1996). In Fig. 9 we present the relative bias with scale calculated using equation (15), where we compare the the correlation function for the volume-limited samples at 60, 80 and 120 1 Mpc relative to the 100 1 Mpc sample. From the figure one may see that there are only minor differences between the smaller volumes. In the case of the sample volume-limited at 120 1 Mpc, the relative bias is fairly constant over the range of scales we consider at $\sim$ 1.5. This suggests that the luminosity bias is scale-independent, and that it starts to become important only for galaxies brighter than $\sim$ $L^*$. Morphology ---------- Since all galaxies in the SSRS2 have morphological classifications we can also analyze the clustering dependence on morphology. With this aim, we have calculated and $\xi(r)$ for the SSRS2 for different morphological types, dividing galaxies into broad morphological bins - early types comprising E, S0 and S0-a, and late types containing Sa galaxies and later. In contrast to the results of the Stromlo-APM, the luminosity function parameters used in the selection function for both samples are quite similar to those measured for the SSRS2 as a whole (Marzke et al. 1997). Furthermore, since sample-to-sample variations are within our estimated errors both for the magnitude and volume-limited samples, as shown in Section 4, in the analysis below we only consider the combined sample to improve the statistics. The resulting correlation functions for early and late type galaxies are presented in Figure 10 panel (a) in redshift space and (b) in real space, while the fit parameters are presented in Table 5. For the late type galaxies we find that the correlation function is adequately described by ($s_0$ = 5.40.21 Mpc; $\gamma_s$=1.480.09), while for early types we find $s_0$ = 6.50.2 1 Mpc; $\gamma_s$=1.860.11. Our values for early type galaxies are close to those of Santiago & da Costa for the diameter-limited SSRS ($s_0$ = 6.01.5 1 Mpc, $\gamma_s$=1.69) and Hermit (1996) for the ORS, who measure $s_0$=6.7 and $\gamma_s$=1.52. Our value for late types is somewhat larger than that measured by Santiago & da Costa (1990), while a proper comparison with Hermit (1996) cannot be made, because we have not subdivided spirals into earlier (Sa/Sb) and later (Sc/Sd) types as they did. A comparison between the fit parameters obtained from available redshift space correlation functions, is shown in Fig. 11, where open symbols represent fits for late type galaxies and solid symbols represent early types. Although all works agree that early types are more clustered than late types, as indicated by the larger correlation length, the scatter is large with the Stromlo-APM results yielding very extreme results. This, in turn, implies large uncertainties in the measurement of the relative bias between the two populations. Based on the redshift space information, we estimate the relative bias between morphological types as 1.25. However, for a proper estimate of the dependence of the correlation properties on morphology it is important to take into account the fact that redshift distortions may affect early and late type galaxies in different ways. Therefore a more meaningful comparison must be carried out in real space. The values we measure for the correlation length in real space for early types ($r_0$=6.00.4; $\gamma_r$=1.910.18) show a very good agreement with those of Loveday (1995), ($r_0$=5.90.7; $\gamma_r$=1.850.13). For late types we find ($r_0$=5.30.3; $\gamma_r$=1.890.15), which is somewhat larger than those measured by the same authors ($r_0$=4.40.1; $\gamma_r$=1.640.05). Our value of the correlation length is significantly smaller than that measured by Guzzo et al. (1997) ($r_0$=8.40.8; $\gamma_r$=2.050.09) for early types. We should note that their sample is volume-limited at $M < -19.5$, whereas we consider galaxies down to $M = -13$. In order to compare with these authors we consider a volume-limited sub-sample of SSRS2 galaxies with $M \leq$ -19.5, which corresponds to maximum distance of 100 1 Mpc. Using this sample, for early types we measure ($r_0$=5.70.8; $\gamma_r$=2.090.49) while for late types we find ($r_0$=5.00.5; $\gamma_r$=2.010.28). For both early as well as late types, there are still discrepancies relative to the results of Guzzo et al. (1997), which could reflect the paucity of rich clusters in our sample. By using the variance, we estimate that the relative bias between the different morphologies is $b_{E+S0}/b_S$ = 1.18$\pm$0.15 in a sample where clusters are not important. This value is smaller than the determination derived from the real-space correlations of Loveday et al. (1995) $b_{E+S0}/b_S$ = 1.33 and Guzzo et al. (1997) $b_{E+S0}/b_S$ = 1.68. From these results we may conclude that the relative bias between the two populations range from roughly 1.2 to 1.7, depending on the cluster abundance in the sample, with the former value representing a lower limit. We have also calculated the correlation function for galaxies discriminated by morphological types for volume-limited samples using the same absolute magnitude limits as in Section 5.1. This calculation was carried out both in redshift, as well as real space, and the results are presented in Tables 6 and 7 respectively. In redshift space there is a trend of the correlation function amplitude increasing with luminosity for both morphological classes. The magnitude of this variation is larger for early types than for late types, although the errors are large. The same trend may be inferred from the analysis in real space, as shown in Figure 12(a), where we compare the $\sigma_8$ values obtained for the different sub-samples. Here it may be clearly seen that there is a trend of $\sigma_8$ increasing with luminosity, suggesting that the morphological and luminosity segregations are two separate effects. Using equation (15) we can also examine how the relative bias varies as a function of scale. This is shown in Figure 12(b), using the real space correlation functions. In contrast to the luminosity bias we find that the morphological bias presents a small decrease from $\sim$ 1.4 on small scales to $\sim$ 1.0 on larger scales ($\sim$ 8 1 Mpc). Although the latter value is slightly smaller than that estimated through the $\sigma_8$ values ($b_{E+S0}/b_S$ = 1.18 $\pm$ 0.15), it is still within the estimated error. A similar behavior of the morphological bias changing with scale, was found by Hermit et al. (1996) but using the redshift space correlation function of the ORS, which may not be as meaningful, because of possible biases introduced by virial motions. Taken together, the above results are consistent with the interpretation that luminosity segregation could be a primordial effect, while the morphological segregation could be enhanced by environmental effects (e.g. Loveday 1995). Colors ------ Another internal characteristic available in the present catalog is color. Although morphology and colors are correlated the scatter is large, and galaxies of a given type exhibit a broad range of colors, indicating different star-formation histories. On the other hand, colors are easily measured and are an objective criterion, in particular for samples of distant galaxies, whereas the morphological classification is somewhat subjective and becomes increasingly difficult to carry out as the apparent sizes of galaxies get smaller. A further evidence that morphology and colors have somewhat different distributions comes from the calculation of the luminosity function, which presents significantly different shapes for blue and red galaxies (Marzke & da Costa 1997), while the luminosity function calculated by separating galaxies between early and late types presents similar Schechter parameters (Marzke et al. 1997). The few works calculating the correlation properties of galaxies divided by colors present rather conflicting results for the deep samples. Works by Infante & Pritchet (1993) and Landy, Szalay & Koo (1996) using the angular correlation function show that the correlation of redder galaxies is significantly stronger than for bluer galaxies, except for the very bluest ones (Landy et al. 1996). Carlberg et al. (1996) analyzing a redshift survey of K-band selected galaxies, find that for $0.3 \leq z \leq 0.9$ red galaxies are more correlated than blue galaxies by a factor of five. These results differ from those of Le Fèvre et al. (1996) who find that at $z \geq$ 0.5 blue and red galaxies have the same correlation properties, while for 0.2 $\leq z \leq$ 0.5 blue galaxies are less correlated than red ones. For nearby galaxies, Tucker et al. (1996) have calculated the correlation function and showed that at small scales ($s \leq 10$ 1 Mpc) red galaxies ($[b_J - R]_0 > 1.25$) cluster more strongly than blue ($[b_J - R]_0 < 1.05$) ones, while for larger scales no evidence of color segregation is seen. In order to make an independent estimation of the dependence of on colors, we use the the $m_B$ = 14.5 sample described in Section 2, which contains galaxies in both galactic hemispheres. As mentioned in Section 2, this bright limit was used because of incompleteness in colors, as we are restricted to galaxies with measurements in the Lauberts & Valentijn (1989) catalog. In this work we adopted the restframe color cutoff as $(B_T-R_T)_0$ = 1.3 which is roughly the color of an Sbc galaxy, and was the criterion adopted by Marzke & da Costa (1997) in the determination of the luminosity function by colors. This value is close to the median value of $B_T-R_T$ in our sample which is $B_T-R_T$=1.2. The conversion of observed into restframe colors used the no-evolution models calculated by Bruzual & Charlot (1993), where we assume that the B and R measures in the Lauberts & Valentijn (1989) catalog are on the same system of $b_J$ and $r_F$ used by Bruzual & Charlot (1993). To calculate we used the following Schechter function parameters; for blue galaxies ($B_T-R_T \leq 1.3$), $M^*$ = -19.43, $\alpha$ = -1.46; for red galaxies ($B_T-R_T > 1.3$), $M^*$ = -19.25, $\alpha$ = -0.73, which were obtained by Marzke & da Costa (1997). The sample, which only considers galaxies out to a maximum distance of 8000 , contains 387 blue and 219 red galaxies. The results of the two-point correlation function are shown in Figure 13 (a) for redshift space while the fit parameters may be found in Table 8. Because of the small number of objects, the correlation function is very noisy, yet it is unquestionable that the red galaxies present a systematically higher amplitude at all separations compared to blue galaxies. In order to verify how sensitive the results may be to incompleteness, we re-calculated for the $m_B$=14.2 sample which is 92 % complete in colors. The fit parameters present a similar behavior, although the values differ from those measured for the 14.5 sample. The results we obtain for the samples discriminated in colors present a qualitative agreement with those of Tucker (1996), in the sense that red galaxies are more strongly correlated than blue galaxies. We have also calculated the real-space correlation function for the 14.5 sample and the power-law fit is presented in Fig. 13 (b), together with the redshift space correlation. The figure shows that the slopes of both power law fits are fairly similar ($\gamma_r$=1.99 for blue, $\gamma_r$=2.18 for red galaxies), though the uncertainties are rather large, in particular for the red galaxies. The observed suggests that red galaxies are probably more affected by peculiar motions than blue galaxies. Because of the relatively small size of the sample with colors, we have not been able to investigate the dependence on luminosity, which would be dominated by errors because of the small number of objects assigned to each luminosity bin. The relative bias estimated from $\sigma_8$ in real space is $b_R$/$b_B$ = 1.400.33, and a similar result is obtained if the redshift space results are considered. As in the case of luminosity and morphology, one may calculate the relative bias between galaxies of different colors as a function of scale, which is presented in Fig. 14. Because the observed correlation function is rather noisy, for this plot we used the fits to . Taking the results at face value they would suggest that the relative bias between red and blue galaxies on small scales is comparable to that seen for early and late type galaxies. However, it levels off more rapidly ($\sim $ 4 1 Mpc), remaining constant at $b_R/b_B \sim 1.2$ thereafter. This behavior could be the result of evolution due to environmental effects, where early type galaxies in higher density regions lost their gas more rapidly than bluer galaxies, and thus present a much lower star formation rate. However, because the errors are large, these results should only be considered as tentative. Comparison with galaxies ======================== In this section we compare the correlation properties of both the entire SSRS2 sample as well as for the volume-limited sub-samples described in Section 5.1 with the 1.2 Jy survey (Fisher et al. 1994). In Fig. 2 we presented a comparison between as measured by different surveys. In that figure it is quite apparent that the three optical surveys are in very good agreement, while the survey presents systematically lower amplitude, which reflects the bias that exists between the distribution of optically-selected and infrared-selected galaxies, previously noted by several authors (Davis et al 1988; Babul & Postman 1990; Lahav et al. 1990; Saunders et al. 1992; Strauss et al. 1992; Fisher et al. 1994). The clustering dependence on infrared luminosity was investigated by Fisher et al. (1994), using different sub-samples of the 1.2 Jy survey, who found no evidence for such dependence. This result differs from that presented in Section 5.1 above, and a comparison between the different volume-limited sub-samples of SSRS2 and galaxies is presented in Fig. 15. We point out that as there are no available measures in real space calculated by Fisher (1994), here we use the values obtained in redshift space, which are presented in the last column of Table 3. The inspection of this figure shows that the amplitude and shape of have the best agreement for the 60 1 Mpc sample ($M < -18.39$), while the other SSRS2 sub-samples containing brighter galaxies present $\xi(s)_{SSRS2} > \xi(s)_{IRAS}$. A possible explanation for the different behavior of the optical and relative to luminosity is that the optical luminosity is more strongly related to the mass, while the infrared luminosity of galaxies reflects rather the star-formation rate which is only weakly dependent on the mass (Davis et al. 1988). The relative bias between the different volume-limited samples of the SSRS2 and the 1.2 Jy survey are shown in Fig. 16(a). For this calculation we used the variance calculated in redshift space, shown in column (7) of Table 3, with the values in Table 1 of Fisher et al. (1994). An inspection of the figure shows that the relative bias between both samples increases with luminosity, ranging from $b_o/b_I$ = 0.94 to $b_o/b_I$=1.91. The smallest value is obtained for the sample which includes less luminous galaxies ($M < -18.39$), while the largest value is for the sample with the brightest galaxies ($M < -19.89$). This result suggests that the relative bias between optical and galaxies depends on the luminosity of the objects, the less luminous optical galaxies showing a clustering amplitude comparable to that found for galaxies. The mean relative bias between the optical and samples may be estimated using the $\sigma_8$ values both in redshift and real space. For this we use the $\sigma_8$ for the magnitude-limited SSRS2 sample and the 1.2 Jy sample (Fisher 1994), using for the former, values in the last column of Table 1. By using the value of $\sigma_8$ for the combined sample we find that the relative bias between optical and sample is $\sim$ 1.20 0.07. The bias in real space can be obtained using the $\sigma_8$ value derived from the real space correlation function. For our combined sample we measure 8 = 0.96 $\pm$ 0.06, while Fisher (1994) quote for galaxies 8 = 0.69 $\pm$ 0.04. This result implies in a relative bias $b_o/b_i = 1.39\pm 0.17$, $\sim$ 17 % larger than that estimated in redshift space. This value is consistent with the value of $\sigma_8$ = 1.38$\pm$ 0.12 reported by Fisher (1994). One may also calculate the relative bias with scale in real space by using equation (15) for the combined magnitude-limited optical and flux-limited samples. The results are presented in Fig. 16(b). Notwithstanding the large error bars, the results suggest that relative bias between optical and galaxies decrease with scale varying from about 1.4 on 1 1 Mpc to close to 1 on 10 1 Mpc scales (e.g. Strauss et al. 1992). Summary ======= We have investigated the correlation properties of galaxies in the SSRS2 catalog for which we have considered both volume and magnitude-limited samples. The main results may be summarized as follows: - In spite of the small volume probed relative to the scale of inhomogeneities, we find an excellent agreement between our correlation function and those of other surveys probing volumes more than 5 times larger. This result is in contradiction with the fractal interpretation of the galaxy distribution in the Universe, which predicts that the correlation length increases with volume. - The relatively small differences between redshift and real space correlations on intermediate scales ($s \sim$ 10 1 Mpc) suggest a low value of $\beta = \Omega^{0.6}/b < 1$, indicating that the redshift distribution of galaxies is close to that in real space. - We confirm the existence of a luminosity-dependent bias for super-L\* galaxies that is scale-independent, suggesting that it is of primordial nature. - In contrast, the relative bias between early and late types shows a scale dependence, varying from about 1.4 on small scales to 1 at $\sim$ 8 1 Mpc. The mean relative bias is found to be $b_{E+S0}/b_S \sim$ 1.2. This small value, when compared to previous surveys, probably reflects the paucity of rich clusters in the surveyed region. - Both early and late types show separately a luminosity-dependent bias similar to the sample as a whole further suggesting that the luminosity bias is primordial in nature while the excess clustering of early types relative to spirals on small scales may be caused by environmental effects. - The relative bias between red and blue galaxies is similar to that observed between early and late type galaxies. However, it levels off on smaller scales $\sim$ 4 1 Mpc at a constant value of about 1.2. We find that the mean relative bias of galaxies selected by colors is greater than when selected by morphologies. 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--- abstract: 'We derive general formulæ for counting the number of homomorphisms between dihedral groups using only elementary group theory.' address: 'Department of Mathematics and Computer Science, Penn State Harrisburg, Middletown PA 17057' author: - 'Jeremiah W. Johnson' title: 'The number of group homomorphisms from $D_m$ into $D_n$' --- This note considers the problem of counting the number of group homomorphisms from $D_m$ into $D_n$, where for a positive integer $l$, $D_l$ denotes the finite group generated by two generators $r_l$ and $f_l$ subject to the relations $r_l^l = e = f_l^2$ and $r_lf_l = f_lr_l^{-1}$. We derive some general formulæ using only elementary group theory and a few basic facts about the dihedral groups. We will assume throughout that $\phi$ represents Euler’s totient function. \[THM:ODD\] Let $m$ and $n$ be positive odd integers. The number of group homomorphisms from $D_m$ into $D_n$ is $$1+n\left(\sum_{k|\gcd(m,n)}\phi(k)\right).$$ Suppose that $\rho\colon D_m \to D_n$ is a group homomorphism, where $m$ and $n$ are positive odd integers. We consider all of the places that $\rho$ could send the generators $r_m$ and $f_m$ of $D_m$ which yield group homomorphisms. As $m$ is odd, it must be the case that $\rho(r_m) = r_n^{\alpha}$, where $r_n^{\alpha}$ is an element of $D_n$ whose order divides both $m$ and $n$. Let $k$ represent the order of this element. There are precisely $\phi(k)$ elements of order $k$ in $D_n$. Since $\rho$ can send $r_m$ to any one of these elements, we have $\sum_{k | m, n} \phi(k)$ choices for $\rho(r_m)$. Next, consider our choices for $\rho(f_m)$. Since $|\rho(f_m)|$ divides $|f_m| = 2$, either $\rho(f_m) = r_n^{\beta}f_n$, $0 \leq \beta < n$, or $\rho(f_m) = e_n$. But not all of these choices for $\rho(f_m)$ yield homomorphisms, as can be seen when we consider where $\rho$ sends the remaining elements in $D_m$ of the form $r_m^kf_m$, where $0 < k < m$. If $\rho(f_m) = e_n$ and $\rho(r_m) = r_n^{\alpha}$, where $\alpha \neq 0$ or $n$, then $\rho(r_mf_m) = r_n^{\alpha}e_n = r_n^{\alpha}$, and $|r_n^{\alpha}|$ does not divide $|r_mf_m|$. Therefore, if $\rho(f_m) = e_n$, then $\rho$ must be trivial. Conversely, when $\rho(f_m) = r_n^{\beta}f_n$, $\rho(r_m^kf_m) = r_n^{k\alpha +\beta \mod n}f_n$, and $|r_n^{k\alpha +\beta \mod n}f_n|$ divides $|r_m^kf_m|$. So, given any choice for $r_m$, we have $n$ choices for $f_m$. Including the trivial homomorphism gives the result. When $m$ and $n$ are positive odd integers and $m|n$, it follows from the fact that $\sum_{k|n}\phi(k) = n$ [@WS88] that there are $mn+1$ group homomorphisms from $D_m$ into $D_n$, and furthermore, there are $n^2+1$ group endomorphisms of $D_n$. When $m$ is a positive odd integer and $n$ is a positive even integer, $r_n^{n/2}$ is a possible choice for the image of $f_m$. However, if $f_m$ is sent to $r_n^{n/2}$, then the image of $r_m$ must be $e_n$; otherwise the map fails to be a homomorphism. Again let $\rho\colon D_m\to D_n$ denote the map and suppose that $\rho(r_mf_m) = r_n^{\alpha}r_n^{n/2}$ for some $\alpha \neq 0$ or $n$ This element necessarily has order not equal to 2 or 1; a contradiction. So in this case, we gain a single additional map sending $r_m$ to $e_n$ and $f_m$ to $r_n^{n/2}$. Taking this additional consideration into account, a proof nearly identical to that used for Theorem \[THM:ODD\] yields the following result. \[THM:ODDEVEN\] Let $m$ be a positive odd integer and $n$ a positive even integer. The number of group homomorphisms from $D_m$ into $D_n$ is $$2+n\left(\sum_{k|\gcd(m, n)} \phi(k)\right).$$ When $m$ is a positive even integer, the number of choices that exist for the image of $r_m$ includes all elements of the form $r_n^kf_n$, $0 \leq k < n$. This creates a number of additional possibilities. \[THM:EVEN\] Let $m$ and $n$ be positive even integers. The number of group homomorphisms from $D_m$ into $D_n$ is $$4+4n+n\left(\sum_{k|\gcd(m,n)}\phi(k)\right).$$ Suppose that $\rho\colon D_m \to D_n$ is a group homomorphism, where $m$ and $n$ are positive even integers. When $m$ is even, we have in addition to the $\sum_{k | m, n} \phi(k)$ possible choices for $\rho(r_m)$ that occur when $m$ is odd the possibility of mapping $r_m$ to those elements in $D_n$ of the form $r_n^{\beta}f_n$. As there are $n$ such elements of the latter type, we have $\sum_{k | m, n} \phi(k) + n$ possible choices for $\rho(r_m)$. Next, suppose $\rho(r_m) = r_n^{\alpha}$ and consider $\rho(f_m)$. Since $|\rho(f_m)|$ divides $|f_m| = 2$, it must be the case that either $\rho(f_m) = r_n^{\beta}f_n$, $0 \leq \beta < n$, $\rho(f_m) = r_n^{n/2}$, or $\rho(f_m) = e_n$. If $\alpha = 0$ or $n/2$, any of these $n+2$ choices for $\rho(f_m)$ will yield a homomorphism. If $\alpha \neq 0$ or $n/2$, then $\rho(f_m)$ cannot equal $e_n$ or $r_n^{n/2}$. So, there are $n\left(\sum_{k|\gcd(m,n)}\phi(k)\right)+4$ homomorphisms sending $r_m$ to an element of the form $r_n^{\alpha}$. Assume next that $\rho(r_m) = r_n^{\alpha}f_n$. Since $|\rho(r_m)| = |\rho(f_m)| = 2$, it follows that if $\rho$ is a homomorphism, then the size of the image of $\rho$ is either 2 or 4. There is only one subgroup of each order containing $r_n^{\alpha}f_m$; the cyclic subgroup $\langle r_n^{\alpha}f_m\rangle$, and the subgroup $\langle r_n^{\alpha}f_m, r_n^{\alpha+n/2 \mod n}f_n\rangle$. There are two choices for $f_m$ which result in the first case; namely, $\rho(f_m) = e_n$, or $\rho(f_m) = r_n^{\alpha}$. Similarly, there are two choices for $f_m$ which result in the second case; $\rho(f_m) = r_n^{\alpha+n/2 \mod n}f_n$ or $\rho(f_m) = r^{n/2}$. A brief calculation shows that each of these four possibilities does in fact give a homomorphism, which leads to the conclusion. When $m$ and $n$ are positive even integers and $m|n$, it follows that the number of group homomorphisms from $D_m$ into $D_n$ is $4+4n+mn$, while the number of group endomorphisms of $D_n$ is $(n+2)^2$. The last case to consider is when $m$ is even and $n$ is odd. \[THM:EVENODD\] Let $m$ be a positive even integer and $n$ a positive odd integer. The number of group homomorphisms from $D_m$ into $D_n$ is $$1 + 2n + n\left(\sum_{k|\gcd(m, n)}\phi(k)\right).$$ As in the proof of Theorem \[THM:ODD\], there are $n\left(\sum_{k|\gcd(m,n)}\phi(k)\right)$ homomorphisms in which $r_m$ is sent to an element of the form $r_n^{\alpha}$, $0 < \alpha < n$, plus the trivial homomorphism. In addition, we could send $r_m$ to any of the $n$ elements of the form $r_n^{\alpha}f_n$, $0 \leq \alpha < n$. If $\rho(r_m) = r_n^{\alpha}f_n$, then the image of $\rho$ is a subgroup of order 2, the cyclic subgroup $\langle \rho(r_m)\rangle$. That leaves two choices for $\rho(f_m)$; either $\rho(f_m) = e_n$ or $\rho(f_m) = r_n^{\alpha}f_n$, from which the result follows. When $\gcd(m,n) = 1$, Theorems \[THM:ODDEVEN\] and \[THM:EVENODD\] lead to the succinct formulæ that the number of group homomorphisms from $D_m$ into $D_n$ equals $n+2$ when $m$ is odd and $n$ is even, and $3n+1$ when $m$ is even and $n$ is odd. [99]{} W. Sierpinski, *Elementary Theory of Numbers*, 2nd ed., North-Holland, Amsterdam.

--- abstract: 'We investigate the order parameter of noncentrosymmetric superconductors Li$_2$Pd$_3$B and Li$_2$Pt$_3$B via the behavior of the penetration depth $\lambda(T)$. The low-temperature penetration depth shows BCS-like behavior in Li$_2$Pd$_3$B, while in Li$_2$Pt$_3$B it follows a linear temperature dependence. We propose that broken inversion symmetry and the accompanying antisymmetric spin-orbit coupling, which admix spin-singlet and spin-triplet pairing, are responsible for this behavior. The triplet contribution is weak in Li$_2$Pd$_3$B, leading to a wholly open but anisotropic gap. The significantly larger spin-orbit coupling in Li$_2$Pt$_3$B allows the spin-triplet component to be larger in Li$_2$Pt$_3$B, producing line nodes in the energy gap as evidenced by the linear temperature dependence of $\lambda(T)$. The experimental data are in quantitative agreement with theory.' author: - 'H. Q. Yuan' - 'D. F. Agterberg' - 'N. Hayashi' - 'P. Badica' - 'D. Vandervelde' - 'K. Togano' - 'M. Sigrist' - 'M. B. Salamon' title: 'S-wave/spin-triplet order in superconductors without inversion symmetry: Li$_2$Pd$_3$B and Li$_2$Pt$_3$B' --- The crystal structure of most superconducting materials investigated to date includes a center of inversion. The Pauli principle and parity conservation then dictate that superconducting pairing states with even parity are necessarily spin-singlet, while those with odd parity must be spin-triplet [@MUMM; @2003]. In materials that lack inversion symmetry, the tie between spatial symmetry and the Cooper-pair spin may be broken [@Gorkov; @2001; @Frigeri; @2004; @Frigeri; @2005; @Edelstein; @1989; @Levitov; @1985; @Samokhin; @2004]. The absence of inversion symmetry along with parity-violating antisymmetric spin-orbit coupling (ASOC) allows admixture of spin-singlet and spin-triplet components.  Unconventional behavior, including zeroes in the superconducting gap function, is then possible, even if the pair wavefunction exhibits the full spatial symmetry of the crystal. In this Letter we report the dramatically different electrodynamic behavior of two newly discovered noncentrosymmetric superconductors Li$_{2}$Pd$_{3}$B and Li$% _{2}$Pt$_{3}$B [@Togano; @2004; @Badica; @2005]. The penetration depth $% \lambda (T)$ in the former material has the expected exponential temperature dependence of a fully-gapped superconductor while the latter exhibits a linear temperature dependence over the range $0.05\leq T/T_{c}\leq 0.3.$ Inasmuch as the main difference between these two compounds is the larger spin-orbit coupling strength for Pt ((Z$_{Pt}$/Z$_{Pd}$)$^{2}$ = 2.9), we argue that the unconventional behavior is evidence for admixed singlet and triplet order as a consequence of ASOC. Indeed, we show quantitative agreement between the experimental data of $\lambda(T)$ and the theoretical calculations for mixed singlet and triplet states based on ASOC. Parity-broken superconductivity (SC) was previously discussed in the context of surface superconductors [@Edelstein; @1989] and for dirty bulk materials [@Levitov; @1985]. Recently, the discovery of SC in the magnetic compounds CePt$_{3}$Si [@Bauer; @2004], UIr [@Akazawa; @2004] and CeRhSi$_3$ [@Kimura; @2005](under pressure) has attracted extensive interest in studying SC without inversion symmetry. Unfortunately, in these correlated electron systems the nature of superconductivity is complicated by its coexistence with magnetism, therefore severely restricting the study of parity-broken SC. Li$_{2}$Pd$_{3}$B and Li$_{2}$Pt$_{3}$B crystallize in a perovskite-like cubic structure composed of distorted octahedral units of BPd$_{6}$ and BPt$% _{6}$ [@Eibenstein; @1997]. Unlike CePt$_3$Si, CeRhSi$_3$ and UIr, these materials show no evidence of magnetic order or strong correlated-electron effects [@Togano; @2004; @Badica; @2005; @Nishiyama; @2005; @Takeya; @2005] that could lead to unconventional superconducting behavior. Further, the increased spin-orbit coupling in Pt leads to much larger band-splitting in Li$_{2}$Pt$_{3}$B than in Li$_{2}$Pd$_{3}$B [@Lee; @2005], allowing us to study the dependence of superconductivity on the ASOC strength. Therefore, we argue that Li$_{2}$Pd$_{3}$B and Li$_{2}$Pt$_{3}$B provide a *model system* in which to study SC without inversion symmetry. Polycrystalline samples of Li$_2$Pd$_3$B and Li$_2$Pt$_3$B were prepared by arc melting [@Togano; @2004; @Badica; @2005]. Powder x-ray diffraction and metallography identify them as being single phase. The sharp superconducting transitions with a width less than 10% of $T_c$ observed in either bulk magnetization $M(T)$ (see, e.g., the inset of Fig.1), penetration depth $% \lambda(T)$ or electrical resistivity $\rho(T)$ (not shown) indicate good sample homogeneity. The mean free path [@note], estimated from the rf resistivity $% \rho$ at $T_c$, coherence length $\xi$ and specific heat coefficient $\gamma$ [@Takeya; @2005] ($\rho=20\mu\Omega$cm, $\xi=9.5$nm, $\gamma=9$mJ/molK$^2$ for Li$_2$Pd$_3$B and $\rho=28 \mu\Omega$cm, $\xi=14.5$nm, $\gamma=7$mJ/molK$% ^2$ for Li$_2$Pt$_3$B), is 24 nm for Li$_2$Pd$_3$B and 42 nm for Li$_2$Pt$_3$B, a few times larger than the corresponding coherence length, indicating clean samples. Precise measurements of penetration depth $% \Delta\lambda(T)$ were performed utilizing a tunnel-diode based, self-inductive technique at 21 MHz down to 90 mK in a dilution refrigerator. The change in penetration depth $\Delta\lambda(T)$ is proportional to the resonant frequency shift $\Delta f(T)$, i.e., $\Delta \lambda(T)=G\Delta f(T) $, where the factor $G$ is determined by sample and coil geometries [@Chia; @2003]. Due to the uneven sample surface, the uncertainty of the $% G $-factor can be up to 15%. In this paper, $\Delta\lambda(T)$ is extrapolated to zero at $T=0$, i.e., $\Delta\lambda(T)=\lambda(T)-\lambda_0$. The values of zero temperature penetration depth $\lambda_0$ ($% \lambda_0=190$ nm for Li$_2$Pd$_3$B and $\lambda_0=364$ nm for Li$_2$Pt$_3$B) are taken from Ref. [@Badica; @2005], determined from the magnetic critical field measurements. The difference of $\lambda_0$ in the two compounds might result from their distinct Fermi surfaces due in part to the spin-orbit coupling [@Lee; @2005]. The magnetization $M(T, H)$ was measured using a commercial SQUID magnetometer (MPMS, Quantum Design). ![(Color online). Temperature dependence of the penetration depth $% \triangle\protect\lambda(T)$ for Li$_2$Pd$_3$B (\#2) and Li$_2$Pt$_3$B (\#3), showing distinct low-temperature behavior [@Yuan; @2005]. The inset shows the magnetization $M(T)$ for Li$_2$Pt$_3$B (\#3) measured in zero-field-cooling (ZFC) and field-cooling (FC) in a magnetic field of 5 Oe. The values of $T_c$ ($T_c=6.7$ K for Li$_2$Pd$_3$B (\#2) and $T_c=2.43$ K for Li$_2$Pt$_3$B (\#3)) were determined from the mid-points of magnetization drop at $T_c$.](Fig1.eps){width="0.76\columnwidth"} In Fig. 1 the penetration depth change $\Delta \lambda (T)$ is shown for Li$% _{2}$Pd$_{3}$B and Li$_{2}$Pt$_{3}$B, respectively. The nearly $T$-independence of $\lambda (T)$ at low temperatures for the Pd-compound is characteristic of fully gapped behavior, consistent with NMR experiments [@Nishiyama; @2005] and specific heat measurements [@Takeya; @2005]. However, the penetration depth $\lambda (T)$ of Li$_{2}$Pt$_{3}$B follows a linear temperature dependence [@Yuan; @2005]. Such a $T$-linear behavior of $\lambda(T)$ can be theoretically interpreted by (a) phase fluctuations among Josephson-coupled grains [@Ebner; @1983] and (b) line nodes in the superconducting energy gap. The former one can be ruled out in this context. The importance of phase fluctuations depends inversely on grain size, which is large ($> $100 $\mu $m) in both the Pt- and Pd-samples [@Togano; @2004].If phase fluctuations dominate, the Pd sample, with comparable normal-state resistivity and grain size, should also show a strong linear temperature dependence. Further, the transition temperature $T_c$ is strongly dependent on the normal state resistivity in the phase-fluctuation regime. We find that the $T_c$ varies by less than 10% among samples that have normal-state resistivities that differ by a factor of three or more. Finally we have reanalyzed the specific heat of Li$_{2}$Pt$% _{3}$B reported in Ref.[@Takeya; @2005] and find that $C_{el}/T\sim T$  is a much better representation of those data at low temperature than is an exponential dependence, further supporting the existence of line nodes. Before describing our model, we explore possible non s-wave states that might exhibit line nodes in Li$_2$Pt$_3$B. Weak-coupling theory of SC, justified by the low $% T_{c}$, permits only the following three:\ (i) $\Delta_{+}(\mathbf{k})\simeq\Delta_{-}(\mathbf{k})=(k_x^2-k_y^2)(k_y^2-k_z^2)(k_z^2-k_x^2)$,\ (ii)$\Delta_{+}(\mathbf{k},z)\simeq\Delta_{-}(\mathbf{k},z)=e^{iqz}k_z[k_y(k_y^2-k_z^2)+ik_x(k_z^2-k_x^2)]$,\ (iii) $\Delta_{+}(\mathbf{k},z)\simeq\Delta_{-}(\mathbf{k},z)=e^{iqz}k_z(k_x+ik_y)$.\ In the latter two cases, broken parity and time reversal symmetries combine to destabilize the spatially uniform state, giving rise to the spatial dependence in the gap functions. The former two states are unlikely in any theory that is based on local interactions (like the single band Hubbard model). Since the above three states are not s-wave pairing states, and (as argued below) Li$_{2}$Pd$_{3}$B appears to be s-wave, a phase transition in the pairing state of Li$_{2}$(Pd$_{1-x}$Pt$_{x}$)$_{3}$B with varying $x$ would have to occur for any of these states to exist in Li$_{2}$Pt$_{3}$B. Furthermore, these states should be extremely sensitive to impurities and $% T_{c}$ should be strongly suppressed when $x$ is varied away from 1. These are in contrast with the experimentally observed smooth evolution of $T_{c}$ with $x$ [@Badica; @2005]. Given these arguments against unconventional superconductivity, we attribute the dramatic difference between these two compounds to the variation of ASOC. When parity symmetry is violated, the ASOC that breaks the spin degeneracy of each band takes the form $\alpha\mathbf{g(k)\cdot S(k)}/\hbar$, where $% \alpha$ denotes the spin-orbit coupling strength, $\mathbf{S(k)}$ is the spin of an electron with momentum $\hbar \mathbf{k}$, and $\mathbf{g(k)}$ is a dimensionless vector ($\mathbf{g(-k)=-g(k)}$ to preserve time reversal symmetry). This ASOC leads to an energy splitting of the originally degenerate spin states and results in spin-eigenstates that are polarized parallel or anti-parallel to $\mathbf{g(k)}$. The ASOC plays a crucial role in the determination of the superconducting state. The key point is that if a spin-triplet contribution to the superconducting gap function is to emerge, its characteristic d-vector $\mathbf{d(k)}$ must be parallel to $% \mathbf{g(k)}$ (provided that the ASOC is sufficiently large) [@Frigeri; @2004; @Frigeri; @2005]. This leads to two gap functions $\Delta_{\pm}(\mathbf{k}% )=\psi\pm t \mid\mathbf{g(k)}\mid$, where each gap is defined on one of the two bands formed by the degeneracy lifting of the ASOC; $\psi$ and $t$ are the singlet and triplet order parameters respectively. For a range of values of $\nu=\psi/t$, $\Delta_{-}(\mathbf{k})$ can change sign and nodes may exist in the superconducting gap. Recent band structure calculations for these compounds [@Lee; @2005] provide information about $\mid\mathbf{g(k)}\mid$. These results indicate that $\alpha$ is a large energy scale relative to the bandwidth and that $% \mid\mathbf{g(k)}\mid$ is highly anisotropic. To capture these results in a model we take:$\mathbf{g(k)}=a_1\mathbf{k}-a_2[\widehat{\mathbf{x}}k_x(k_y2+k_z2)+% \widehat{\mathbf{y}}k_y(k_z2+k_x2)+\widehat{\mathbf{z}}k_z(k_x2+k_y2)]$,with $a_2/a_1=3/2$, $\mathbf{k}$, a unit vector, and the spherical average of $\mid\mathbf{g(k)}\mid^2$ equal to unity. This form of $\mid\mathbf{g(k)}% \mid$ is the simplest that is consistent with cubic symmetry and allows for anisotropy on a model, spherical Fermi surface. {width="0.75\columnwidth"} {width="0.75\columnwidth"} We compute the penetration depth (the superfluid density) on the basis of the formula described in [@Hayashi; @2005]. These fits provide estimates for $\nu $ (defined at $T\rightarrow T_{c}$) and $\delta $, the ratio of the relative density of states between the spin-orbit-split bands. The resulting fits are shown in Figs.2(a) and 3(a), respectively. Li$_{2}$Pd$_{3}$B is nearly a pure spin-singlet state, with a large value of $\nu \simeq 4$. We note that the preliminary fit of two-band model in Li$_{2}$Pd$_{3}$B with a fraction of 4% from the small energy gap [@Yuan; @2005] treated data only for $T<0.3T_{c}$, while the present analysis covers the whole temperature range. As argued above, Li$_{2}$Pt$_{3}$B clearly evidences line nodes, meaning that $\Delta _{-}(\mathbf{k})$ changes sign for a range of wavevectors. The best fit for Li$_{2}$Pt$_{3}$B has $\nu =0.6$ and $\delta =0.3$, indicating that the spin-triplet component is dominant. We expect $% \delta $ to be proportional to the strength $\alpha $ of the ASOC, which in turn varies as the square of the atomic number, as above. The obtained value of $\delta (Pt)/\delta (Pd)=0.3/0.1=3$ is consistent with the expectations. In the insets to Figs. 2(a) and 3(a) we show polar plots of $\Delta _{-}(% \mathbf{k})$ for the two compounds; for Li$_{2}$Pt$_{3}$B, the existence of line nodes appears in the form of circular bands. For Li$_{2}$Pd$_{3}$B, both $\Delta _{+}(\mathbf{k})$ and $\Delta _{-}(\mathbf{k})$ are non-zero, but anisotropic, over the entire Fermi surface. It is noted that the gap functions $\Delta _{+}(\mathbf{k})$ and $\Delta _{-}(\mathbf{k})$ possess cubic symmetry (see the insets to Figs. 2(a) and 3(a)) and the pairing states break only gauge invariance symmetry, i.e., an s-wave orbital symmetry for both Li$_{2}$Pd$_{3}$B and Li$_{2}$Pt$_{3}$B. However, $\Delta _{-}(\mathbf{k})$ exhibits a sign change in Li$_{2}$Pt$_{3}$B indicated by dark circles in Fig. 3(a). Figures 2(b) and 3(b) present the superfluid density $\rho _{s}(T)$ obtained from the penetration depth ($\rho _{s}(T)=\lambda _{0}^{2}/\lambda ^{2}(T)$), along with calculated curves. The agreement is satisfactory. One notes that the weak tail in the experimental $\rho _{s}(T)$ as $T\rightarrow T_{c}$ is mainly due to the influence of rf skin depth upon approaching $T_{c}$. The insets to Figs. 2(b) and 3(b) show the calculated temperature dependences of the order parameters $\psi $ and $t$. Obviously, the spin-singlet component is dominant in the order parameter of Li$_{2}$Pd$_{3}$B, but it is not the case in Li$_{2}$Pt$_{3}$B. For the latter compound, the spin-triplet component $t$ is sufficiently large to give rise to the existence of line nodes in the superconducting energy gap. The existence of a spin-triplet state may be stabilized by the inter-parity" coupling (termed $e_{m}$ in Ref.[@Frigeri; @2005]) between singlet and triplet channels as allowed by broken inversion symmetry. This interaction can arise from el.-ph. (and el.-el.) coupling and may dominate in Li$_{2}$Pt$_{3}$B because of the large ASOC [@Lee; @2005]. We note that while our model (spherical Fermi surface and isotropic spin-singlet gap) predicts that spin-triplet component is larger than the spin-singlet component, this needs not be the case in reality. In particular, if the spin-singlet gap is anisotropic, then the Fermi surface average of the magnitude of the spin-triplet component required to produce line nodes can be significantly decreased. In addition to the profound effect on the pairing state in Li$_2$Pt$_3$B, broken parity symmetry has other non-trivial consequences. For example, the cubic symmetry allows for a novel contribution to the Ginzburg Landau (GL) free energy density of the form $\varepsilon\mathbf{B \cdot j_{so}}$, where $% \mathbf{j_{so}}$ is the supercurrent as defined in the usual GL theory and $% \varepsilon$ is a constant. As a consequence, the condensate wavefunction will not be spatially uniform along the direction of the applied magnetic field as it usually is. Near the upper critical field, it will develop a finite center of mass momentum that is parallel to the applied field [Kaur 2005]{}. This helical structure of the order parameter is similar to that of a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconductor [@Fulde; @1964; @Larkin; @1965]. However, in contrast to the FFLO phase, a non-zero center of mass momentum exists at all temperatures. In the vortex state, this coupling term causes the magnetization to develop a transverse component that is parallel to the supercurrent. This may be observable through small angle neutron scattering experiments. In summary, our observations have demonstrated that superconductors lacking inversion symmetry exhibit qualitatively distinct properties from those with an inversion center. The existence of unconventional properties (e.g., line nodes) for an s-wave type superconductor, found in Li$_2$Pt$_3$B, provides an alternative way to study unconventional SC, especially that arising from phonon pairing mechanism. Indeed, the absence of parity symmetry coupled with strong spin-orbit coupling, which results in an admixture of spin-singlet and spin-triplet pairing, requires a complete reconceptualization of Cooper pairs and the nature of the superconducting state. We thank P. Frigeri, R. Kaur, I. Milat and H. Takeya for useful discussions. This work is supported by the National Science Foundation under awards No. NSF-EIA0121568 and NSF-DMR0318665, the Department of Energy under award No. DEFG02-91ER45439, the Swiss National Science Foundation and Petroleum Research Funds. We also acknowledge supports from ICAM (HQY), the 2003-PFRA program of Japan Society for the Promotion of Science (NH) and the CEEX program of Romanian Ministry of Education and Research (PB). 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--- abstract: 'We evaluate background rejection capabilities and physics performance of a detector composed of two diverse elements: a sensitive target (filled with one or two species of liquefied noble gasses) and an active veto (made of Gd-doped ultra-pure water). A GEANT4 simulation shows that for a direct WIMP search, this device can reduce the neutron background to O(1) event per year per tonne of material. Our calculation shows that an exposure of one tonne $\times$ year will suffice to exclude spin-independent WIMP-nucleon cross sections ranging from $10^{-9}$ pb to $10^{-10}$ pb.' address: 'Dpto. de F[í]{}sica Teórica y del Cosmos & C.A.F.P.E., Universidad de Granada, Spain' author: - 'A Bueno, M C Carmona and A J Melgarejo' title: 'Direct WIMP identification: Physics performance of a segmented noble-liquid target immersed in a Gd-doped water veto' --- [*Keywords*]{}: Dark matter, argon, xenon, neutron, background rejection Introduction ============ Evidence for the existence of dark matter is overwhelming. The first was found by F. Zwicky in 1933 while measuring the rotational velocity of the Coma cluster of galaxies [@Zwicky]. Since then astronomers have gathered more relevant data, which likely point in the direction of the existence of a new form of matter [@Trimble]. For instance, the striking optical and X-ray images, obtained by the Chandra telescope, of the so-known bullet cluster (1E0657-558) [@bullet; @bullet2] cannot be explained simply by advocating that Newtonian dynamics is modified at large scales [@Sanders]. Additional support for dark matter comes from the precise measurement of the cosmic microwave background radiation done by the Wilkinson Microwave Anisotropy Probe (WMAP). Together with the Sloan Digital Sky Survey (SDSS) large-scale structure data, it tells us that only a small fraction of the matter content of the Universe ($\Omega_B = 0.042\pm 0.002$) is of baryonic origin, while the rest is made of a totally unknown new form of matter ($\Omega_{DM} = 0.20\pm 0.02$) [@wmap]. The rest of the energy content of the Universe ($\Omega_\Lambda = 0.76\pm 0.02$) is accounted for by a very smooth form of energy called dark energy. So far dark matter has been observed only through its gravitational effects, therefore we know little about its fundamental properties. Massive neutrinos contribute to a small fraction of dark matter [@atm; @solar]. However the most plausible hypothesis for potential candidates (generically known as Weakly Interacting Massive Particles, WIMPs) states that they should be neutral weakly-interacting heavy particles with lifetimes comparable to the age of the Universe. No such thing exists in the Standard Model of Particle Physics. However theories beyond it (like, for example, Supersymmetry, Extra Dimensions, etc.) do have particles that naturally arise as dark matter candidates [@Bertone]. Parallel to model building developments, there is an intense and challenging experimental activity devoted to WIMP detection, as well [@Gaitskell]. DAMA/NaI claims to have found evidence for the presence of WIMPs in the Galactic halo [@Bernabei:2000qi]. Using a direct detection method, they have measured an annual modulation, over seven annual cycles (107,731 kg day total exposure), consistent with expectations from a WIMP signature. The collected DAMA/LIBRA data [@Bernabei:2008yi] supports this claim. Considering the data from both experiments (amounting to an exposure of 0.82 tonnes $\times$ yr), the presence of dark matter particles in the galactic halo is supported at 8.2 $\sigma$ C.L. However, the situation is highly controversial since other direct-search experiments, probing similar regions of the parameter space, have found negative results [@Akerib:2004fq; @Chardin:2003vn; @Ahmed:2003su; @warp; @xenon10]. Indirect detection methods have not found signals that could be attributed to WIMP annihilation [@Ambrosio:1998qj; @Desai:2004pq]. To improve current sensitivities and explore in depth the parameter space of the most favoured dark matter models, there is an indisputable need for more massive detectors with enhanced background rejection capabilities. The use of liquefied noble gasses, as target for WIMP interactions, ranks among the most promising detection techniques [@nobleliquids1; @nobleliquids2; @nobleliquids3]. This technology is easily scalable and allows to build detectors in the range of few tonnes of fiducial mass [@andre]. This paper concentrates on evaluating the physics potential of one of such detectors. An effort has been made to design an experiment that allows to reduce as much as possible the background caused by neutron interactions inside the active target. In addition, we consider several independent target units that can be filled with different noble liquids; in this way we can aim at confirming, with a single experiment, that the event rate and the recoil spectral shape follow the expected dependence on A$^2$ for WIMP signals [@pfsmith]. In the following Sections, we describe the basic detector layout and its foreseen physics performance. Detector description {#sec:det} ==================== Liquid noble elements, used as sensitive medium for direct dark matter searches, are a promising alternative to ionization, solid scintillation and milli-Kelvin cryogenic detectors. When a WIMP particle scatters off a noble element, scintillation photons and ionization electrons are produced due to the interactions of the recoiling nucleus with the neighbouring atoms. The simultaneous detection of primary scintillation photons and ionization charge (or the secondary photons produced when this charge is extracted from the liquid to the gas phase [@rusos]) is a powerful discriminator against backgrounds. Pulse shape provides an additional tool to identify true signals: depending on the nature of the interacting particle, the scintillation light shows a different time dependence [@Hitachi]. In addition, thanks to the high level of purity achieved, these detectors can drift ionization charges for several meters, hence it is conceivable to reach masses of the order of several tonnes. Nowadays XENON [@xenon10], ZEPLIN [@zeplin1; @zeplin2; @zeplin3; @zeplin4], XMASS-DM [@xmass], WARP [@warp] and ArDM [@andre] collaborations use liquid argon or xenon targets to look for WIMPs. Similar detectors can be used to detect the yet unobserved coherent neutrino-nucleus elastic scattering [@betapaper]. Assuming a one tonne detector, we expect an event rate of O(10) events per year of operation for a WIMP-nucleus cross section of 10$^{-10}$ pb. To explore such small cross sections, backgrounds should be reduced to very challenging levels (about 1 event per tonne per year). In case the target is filled with argon, the $^{39}$Ar isotope, which is a beta particle emitter, is a serious source of concern (its activity is approximately 1 Bq per kg of natural argon [@Ar39radio]). However, the most important source of background is due to neutrons produced in detector components or in the rock of the underground cavern. A large fraction of these external neutrons can be rejected using external hydrocarbon shields, active vetoes or a combination of the two [@cline]. High-energy neutrons induced by muon interactions in the rock are a more serious concern. Recently, an innovative neutron multiplicity meter, very similar in concept to the detector discussed in this document, has been proposed to monitor this neutron flux [@akerib]. The flux of internal neutrons can be highly reduced using low activity materials for the inner parts of the detector. However, it is unavoidable that some of them interact with target nuclei mimicking a WIMP signal. Detectors with a large fiducial volume offer the advantage of an increased probability for neutrons to interact several times, before they exit the target. For WIMPs this is highly unlikely given the small cross sections involved. This fact can be used to further reduce neutron backgrounds. In our case, given the reduced dimensions of the sensitive targets (see below), the signal due to multiple interactions cannot be the main tool for background rejection. To reduce neutron contamination, we propose a detector made of two diverse elements: an external detector, acting as an active veto, made of ultra-pure water doped with gadolinium. This will enhance neutron capture and its posterior identification [@gadzooks]. Immersed in this veto, the internal detector, that acts as target, consists of cells filled with a noble element. This configurations allows to operate the sensitive target with two noble liquids simultaneously. If energy deposits occur, within a certain time window, both in the cell and water, the event is tagged as neutron-like provided the external veto records the typical 8 MeV gamma cascade from neutron capture on gadolinium. Noble liquid target {#sec:target} ------------------- We have carried out a full simulation of the detector using GEANT4 [@Allison] (see Figure \[fig:detsim\]). The target is made of 100 low-background metal cylinders (each 40 cm high and 30 cm in diameter). The internal volume, that can be filled with a noble liquid, has 30 cm drift distance and 24 cm in diameter. For our physics studies, the fiducial region corresponds to a cylinder of 6 cm radius and 25 cm high. The fiducial mass amounts up to 0.8 tonnes in case the target is filled with liquid xenon (LXe) and 0.4 tonnes in case liquid argon (LAr) is used. Our device can detect simultaneously the ionization charge and the scintillation light resulting from the scattering of incoming particles off xenon or argon nuclei. Light is read by means of photomultiplier tubes (PMTs) placed at the target bottom. Ionization electrons are drifted to the liquid surface where they are converted into secondary scintillation light that is read by PMTs on top of the cylinders. Charge amplification devices (i.e., GEM, LEM, Micromegas [@Sauli; @Jeanneret; @Giomataris2]) are a possible alternative for charge readout. This configuration of independent cylinders, apart from the fact of being easily scalable, offers a clear experimental advantage: data taking can proceed with two different targets simultaneously. Cylinders can be filled with argon and xenon, for example. In case a WIMP signal is observed with enough statistical relevance, we can confirm in a single experiment that the event rate and the recoil spectral shape follow the expected dependence on A$^2$. ![ Artist’s view of the detector: (Top) A target cell. (Bottom) Noble liquid target plus active veto.[]{data-label="fig:detsim"}](EPS/cylinder "fig:")\ ![ Artist’s view of the detector: (Top) A target cell. (Bottom) Noble liquid target plus active veto.[]{data-label="fig:detsim"}](EPS/det_side_view "fig:")\ Water-Čerenkov neutron detector {#sec:veto} ------------------------------- The active target is immersed in a water tank (1.6 m height, 6.9 m width and 6.9 m long), made of copper or other low background material. The distance between cylinders is 30 cm. The distance to the veto walls is 60 cm. This distance has been optimized to allow for an efficient neutron capture by gadolinium. The veto contains 70 tonnes of ultra-pure water, once we subtract the volume taken by the sensitive targets and the ancillary system. 1500 9" PMTs (40% photo-coverage), mounted at the water-tank walls, are used to detect the photons produced by neutron capture on Gd. They will detect the light produced by penetrating cosmic muons, as well, thus providing an efficient veto against this kind of events. Following the approach discussed in [@gadzooks], we have doped the water-filled parallelepiped with highly-soluble gadolinium trichloride (GdCl$_3$). To avoid the absorption of photons by the cylindrical targets and the supporting system associated to them, we propose a solution similar to the one used in the Pierre Auger Observatory [@pao], namely to cover their external walls with Tyvek (a material that shows a reflectivity higher than 90% to Čerenkov light [@justus]). A particular source of concern is the radio-purity of the additive. According to the estimations given in [@gadzooks] and [@sno], the potential background caused by it, especially the alpha particle decays of $^{152}$Gd, is much smaller than what is expected from the sources considered in Section \[sec:results\]. The amount of gadolinium has being chosen in order to minimize the number of neutrons captured by hydrogen nuclei, since we consider the 2.2 MeV gammas coming from this reaction are extremely hard to detect with the outer veto. A dedicated GEANT4 simulation has been carried out to study which is the optimal Gd concentration. A 1 meter radius sphere filled with Gd-doped water has been simulated and neutrons with energies up to 10 MeV have been shot from the center. Figure \[fig:gdconc\] shows the obtained results. ![Number of absorbed particles as a function of the Gd concentration.[]{data-label="fig:gdconc"}](EPS/percent_gd.eps){width=".7\textwidth"} We observe that while the total number of absorbed particles does not change with Gd concentration, the proportion of Gd-absorbed particles does, saturating at a value $\sim2\%$. Hence, we will use for our calculations a $2\%$ admixture by mass of GdCl$_3$. To evaluate the veto efficiency, we follow the gammas produced in the 8 MeV cascade following neutron capture by Gd. The detectable signal corresponds to Compton electrons above Cerenkov threshold. More than 90$\%$ of these electrons have energies above 3 MeV, with a mean value of about 5 MeV [@sno]. Nearly 50$\%$ of the Cerenkov photons are detected and only 3$\%$ of them are absorbed by the targets and their associated ancillary system. Considering a global detection efficiency of 15$\%$ for the simulated PMTs, we obtain a light yield of 6 photo-electrons/MeV. Assuming a detection threshold of 3 MeV, our trigger efficiency is $>$95$\%$ for this energy and reaches $\sim$ 100$\%$ at 4 MeV [@hosaka]. The overall detection efficiency for the 8 MeV gamma cascade is $>$90$\%$ and the energy resolution is assumed to be 20$\%$. Physics performance {#sec:results} =================== The estimation of the overall background event rate must take into account both internal and external sources of gamma rays and neutrons. We conservatively assume that, due to instrumental limitations, we cannot detect signals below 15 (30) keV of true recoil energy in case we use a xenon (argon) target. On the other hand, we will assume a maximum true recoil energy for WIMP like events of 50 (100) keV for the xenon (argon) target. We note that it has been recently suggested that neutrinos can be a source of background for the next generation of direct-search dark matter experiments [@Monroe:2007]. However, as shown in [@Monroe:2007], for Ar and Xe targets, this background is easily eliminated considering a cut on the true recoil energy above 10 keV. Therefore the cuts we impose throughout this work on the true recoil energy reduce the neutrino contamination to a negligible level. Contamination from radioactive nuclei, xenon and argon isotopes {#sec:isotopes} --------------------------------------------------------------- For a target made of argon, an important source of background comes from the presence of radioactive $^{39}$Ar. This is a beta particle emitter with an activity of about 1 Bq per kg of natural argon, which for a single-volume 1 tonne detector translates into a 1 kHz rate. In our case, since the target is divided into hundred independent units, the event rate due to $^{39}$Ar decays does not represent an issue for the design of the data-acquisition system. In addition, the probability to have a $^{39}$Ar decay overlapping with a different sort of interaction is smaller than in the case of a single-volume large-size detector due to the smaller drift times involved. Regarding the possibility of misidentifying $^{39}$Ar signals as WIMPs, we should note that $\beta$ particles mainly interact with atomic electrons, while nuclear recoils deposit their energy through transfers to screened nuclei [@Lindhard]. This affects the charge generated by an event (for the same energy is around three times bigger for electrons), the charge to light ratio, which is bigger for electrons, and the ratio between the slow and the fast component of the scintillation light of liquid argon (pulse shape discrimination). According to our simulations, the background due to radioactive nuclei can be reduced to a level well below the one expected from neutrons using the ratio of measured scintillation light over ionization and pulse shape discrimination [@pulse; @warp]. For a nuclear recoil acceptance of 50$\%$, the rejection power against backgrounds caused by electromagnetic particles is $\sim 5 \cdot 10^{-7}$ for each individual target. This rejection power agrees with the results quoted in [@warp] and [@Lippincott:2008]. A further reduction of this kind of background will come from the use of underground-extracted argon [@Galbiati:2007]. Its $^{39}$Ar activity has been recently measured for the first time and shown to be $<$5% of the one present in natural argon. These reasons lead us to not consider further this sort of background. In case the target is filled with xenon, $^{136}$Xe is the most important radioactive isotope. It decays through double beta decay and therefore, given the small probability of the process, the resulting count rate, in the energy band of interest, is negligible compared to other sources of background, even before any rejection cut is applied. Krypton and radon are two radioactive nuclides present in commercially available noble gasses and therefore a potential source of background as well. The highest contamination comes from $^{85}$Kr, which $\beta$–decays with an endpoint energy of 678 keV. As has been discussed, impurities of Kr below 10 ppb can be reached [@xenon10], making negligible the contamination produced by those radioactive decays. Neutrons from target components ------------------------------- One of the most important sources of background comes from neutrons produced by radioactive contamination of the materials constituting the detector itself. To minimize their rate, the use of copper for all the vessels is likely to be the best possible choice. The radioactive impurities can be reduced below 0.02 ppb in some copper samples which would bring the neutron contamination to below 1 event per year [@ILIAS]. If we conservatively assume a 0.1 ppb contamination, one obtains a neutron production rate of $4.54\times10^{-11}$ s$^{-1}$cm$^{-3}$. Being each cylinder 6 mm thick, its total volume amounts to 2217 cm$^{3}$. This means a total of one neutron per cylinder per year. The contamination induced by PMTs must be carefully evaluated as well. Main manufacturers continue to optimize the choice of materials used in PMT construction to reduce their radioactivity levels. Typical contamination values for U and Th range from a few tens to several hundreds parts per billion. Among the wide variety of tubes available in the market, it is possible to find out some models specially designed for low background applications where the measured uranium and thorium concentrations in quartz and metal components is of the order of ten or even less ppbs [@Kudryavtsev], giving a yearly production of less than one neutron per PMT. The phototube windows could be coated with Tetra-Phenyl-Butadiene (TPB) to shift the ultra-violet light to the maximum of the phototube spectral response without an increase on contamination. If we assume a rate of 1 neutron emitted per year per PMT and 8 PMTs per cylinder, we expect a total emission of 8 neutrons per cylinder. In total, PMTs and the copper vessel contribute to 9 neutrons emitted per cylinder per year. Although they will not be considered in the present work, there are several possibilities to reduce the rate of neutrons coming from PMTs. One is to set acrylic light-guides between photomultipliers and the active volume [@Kudryavtsev] which can reduce by a factor 2 the rate of neutrons. Another possibility is to substitute the PMTs on the top of the cylinder by charge readout devices, which can be constructed from low radioactivity materials, having a negligible neutron production rate. We have studied the background rate due to detector components using a simulated sample that amounts to 50 years of data taking. The results shown in Tables \[tab:detres\] and \[tab:detresXe\] are normalized to one year of operation. Table \[tab:detres\] corresponds to the configuration where LAr is used. Throughout this work, columns labeled as [*Total*]{} refer to the total number of neutrons per year, while columns labeled as [*Not vetoed*]{} refer to those neutrons not being absorbed in the Gd-doped water tank; likewise by $E_{recoil}$ we mean the equivalent recoil energy inferred from the energy measured in the active target. Neutrons Total Not vetoed ---------------------------------- ------- ------------ Produced in 1 year 900 20 30 keV $<$ E$_{recoil}<$ 100 keV 19 0.3 : LAr target: Neutron background from detector components normalized to one year of data taking. \[tab:detres\] After a simple selection cut based on the nuclear recoil energy, we find a background of 0.3 neutrons per year for a LAr detector with a fiducial mass of 0.4 tonnes. Considering as signal only those neutrons interacting just once inside the active volume, we can get rid of some additional background. However given the small dimensions of the targets, we expect a modest reduction factor from events with multiple interactions. It is important to note that when the active Gd-doped veto is used, the amount of background is reduced by roughly a factor fifty. The results using liquid xenon as target (fiducial mass 0.8 tonnes) are shown in Table \[tab:detresXe\]. The overall background, after the energy cut, amounts to 1 neutron per year. The reduction given by the active veto in this case is only a factor ten. The amount of background for xenon is larger than for argon. The reason comes from the fact that some xenon isotopes like $^{131}$Xe and $^{129}$Xe show a very high cross section for neutron absorption. For the case of a xenon-filled detector, the smaller the dimensions of the target cylinder the better to identify neutrons in the external active veto. Neutrons Total Not vetoed --------------------------------- ------- ------------ Produced in 1 year 900 64 15 keV $<$ E$_{recoil}<$ 50 keV 12 1 : LXe target: Neutron background from detector components normalized to one year of data taking. \[tab:detresXe\] Neutrons and gamma rays from active veto components --------------------------------------------------- Assuming the same contamination levels we used in the previous Section to estimate the neutron flux due to copper walls, PMTs, voltage divider bases, etc., we obtain that the active veto system contributes with approximately $10^4$ emitted neutrons per year. The flux of these neutrons is orders of magnitude smaller than the ones that reach the external walls of the detector, after being produced in the rock of the cavern by natural radioactivity (according to Table \[tab:rockres\] it amounts to O(10$^7$) rock-emitted neutrons per year). Therefore the contribution of the neutron-induced background from veto components is added to the contamination induced by the walls of the cavern and will be treated in the next Section, but it represents a small fraction of the total expected background. Another source of contamination is the gamma ray flux produced by the PMTs of the veto system. They mainly come from the decay of $^{208}$Tl (thorium chain) and $^{214}$Bi (uranium chain). The former produces a 2.6 MeV gamma and the latter emits 2.2 MeV and 2.4 MeV photons. As explained before, we can set a threshold of 3 MeV for the veto system without a significant loss of efficiency. In these conditions, the majority of those gamma rays will fall below threshold. Those reaching the targets can be rejected using the criteria discussed in Section \[sec:isotopes\], and therefore their contribution to the total background will be significantly smaller than the one expected from neutrons. Neutrons from surrounding rock ------------------------------- Neutrons coming from the rock have two possible origins: (1) underground production by cosmic muons (called hereafter “muon–induced neutrons”) and (2) neutrons induced by spontaneous fission and ($\alpha$, n) reactions due to uranium and thorium present in the rock (generically called from now on “radioactive”). The latter have a very soft spectrum (typically energies of few MeV). The energy spectrum from muon–induced neutrons is harder and therefore are more difficult to moderate by the water shielding. Those neutrons may come from larger distances and produce recoils with energies well above the energy threshold set for signal events [@Kudryavtsev]. The active external water veto will efficiently tag crossing muons by Čerenkov light detection. Neutron signals occurring in the noble liquid target in coincidence with water PMT signals will be rejected. There are $\sim10^{7}$ neutron absorptions per year in the water volume. If we assume a 100 $\mu$s veto time per interaction, this will correspond to a total of $\sim20$ min dead time of the detector per year due to neutron interactions. ### Neutrons from radioactivity: Natural radioactivity can produce neutrons either directly from spontaneous fissions or by means of emitted alpha particles through $(\alpha, n)$ reactions. To compute the spectrum and the rate of those neutrons, the program SOURCES-4C [@SOURCES] has been used. Since the original program provides $(\alpha,n)$ reactions up to 6.5 MeV $\alpha$ particle energy, we include the modifications done in [@Kudryavtsev] to obtain a more realistic neutron spectrum. The thorium and uranium contamination has been taken as the average of those given in reference [@Amare], namely 18.8 Bq/kg for $^{238}$U and 42 Bq/kg for $^{232}$Th. Secular equilibrium is considered. Accordingly, the computed rates for neutron production amounts to $8.44\times10^{-8}$ s$^{-1}$cm$^{-3}$ from $(\alpha, n)$ reactions and to $7.38\times10^{-8}$ s$^{-1}$cm$^{-3}$ from spontaneous fission. Neutrons have been generated according to the energy spectrum shown in Figure \[fig:enerock\] and propagated through the rock using the GEANT4 simulation code and the prescriptions given in [@Kudryavtsev]. As a result we get the neutron spectrum in the walls of the laboratory. To get the final number of neutrons impinging in the detector outer walls and their energy, we have simulated a cavern of $15\times12\times40$ m$^3$, similar in dimensions to the experimental main hall at Canfranc underground laboratory [@Canfranc]. Table \[tab:rockres\] shows the number of neutrons that reach the detector and those that produce energy deposits in the same range as WIMP interactions. Normalized to one year of data taking, we show the level of expected background for two different distances between the external vessel wall and the first active cylinder a neutron will encounter. With a 60 cm thick water active veto, the number of interactions inside the liquid argon volume is well below 1 per year. ![Energy spectrum of neutrons produced in the rock by natural radioactivity.[]{data-label="fig:enerock"}](EPS/rock_ene.eps){width=".7\textwidth"} --------------------------------- ----------------- ----------------- ----------------- ----------------- Neutrons Total Not vetoed Total Not vetoed Produced in 1 year $4.2\times10^7$ $1.6\times10^7$ $4.2\times10^7$ $1.6\times10^7$ 30 keV$<$$E_{recoil}$$<$100 keV $175$ $3$ $9$ $<0.1$ --------------------------------- ----------------- ----------------- ----------------- ----------------- : Neutron background from rock radioactivity. We assume a LAr target with a fiducial mass of 0.4 tonnes and one year of data taking. Results are shown for two different configurations of the active water veto.\[tab:rockres\] The study has been repeated considering a liquid xenon target and 60 cm thick water veto. For this configuration, the expected background from rock radioactivity amounts to nearly one event per year (see Table \[tab:rockresXe\]). In accordance with the results got while studying the contamination due to neutrons from detector components, once more the larger cross section for neutron absorption is responsible for having a bigger expected background when xenon is considered as detector target. Neutrons Total Not vetoed -------------------------------- ----------------- ----------------- Produced in 1 year $4.2\times10^7$ $1.6\times10^7$ 15 keV$<$$E_{recoil}$$<$50 keV $5$ 0.7 : Neutron background from rock radioactivity. A LXe target (0.8 tonnes fiducial mass) has been considered together with an active water veto 60 cm thick. Results are shown for one year of data taking\[tab:rockresXe\] ### Muon-induced neutrons: Fast neutrons from cosmic ray muon interactions represent an important background for dark matter searches. Unlike charged particles, they can not be tagged by veto systems, and unlike lower energy neutrons from rock radioactivity, they can not be stopped by a passive shielding. However, as proposed in [@akerib], it is possible to place close to the detector some material in which this fast neutrons produce secondary low energy neutrons that can be detected by the proposed veto system when absorbed by Gd. The total muon-induced neutron flux $\phi_{n}$ as a function of the depth for a site with a flat rock overburden can be estimated as [@Mei:2006]: $$\phi_{n}=P_0\left(\frac{P_1}{h}\right)e^{-h/P_1}$$ where $h$ is the vertical depth in kilometers water equivalent (km.w.e.), $P_0=4.0\times10^{-7}$ cm$^{-2}$ s$^{-1}$ and $P_1=0.86$ km.w.e. If we consider the Canfranc underground laboratory, with a depth of $\sim2500$ m.w.e. [@Canfranc], the total neutron flux is $7.52\times10^{-9}$ cm$^{-2}$ s$^{-1}$. The neutron energy spectrum is given by [@Wang:2001]: $$\frac{dN}{dE_{n}}=A\left(\frac{e^{-7E_{n}}}{E_{n}}+B(E_{\mu})e^{-2E_{n}}\right)$$ $A$ is a normalization constant and $B(E_\mu)=0.52-0.58e^{-0.0099E_\mu}$. The muon energy spectrum can be estimated with the following equation [@Mei:2006]: $$\frac{dN}{dE_{\mu}}=Ce^{-bh(\gamma_\mu-1)} \cdot \left(E_\mu+ \epsilon_\mu(1-e^{-bh}) \right)^{-\gamma_\mu}$$ where $C$ is a normalization constant, $E_\mu$ is the muon energy in GeV, $b=0.4$/km.w.e, $\gamma_\mu=3.77$ and $\epsilon_\mu=693$ GeV. The previous equations give rise to the energy spectrum displayed in Figure \[fig:muonsrock\]. ![Energy spectrum of neutrons produced by muons interacting in the surrounding rock as estimated for the Canfranc lab.[]{data-label="fig:muonsrock"}](EPS/muon_neutron_energy.eps){width=".7\textwidth"} The angular neutron distribution can be expressed as [@Wang:2001]: $$\frac{dN}{dcos\theta}=\frac{A}{(1-cos\theta)^{0.6}+B(E_\mu)}$$ with $B(E_\mu)=0.699E_\mu^{-0.136}$. In order to simulate the fast neutron background, we consider a $10\times10$ m$^2$ surface on the detector from which we simulate neutrons with the specified angular and energy distributions. Together with the detector itself, we simulate a lead block in which neutrons will create secondary particles. We have considered two different configurations (lead block on the top or at the bottom of the detector) and two different thicknesses for the passive lead veto. [|c|cc|cc|]{} Neutrons & &\ Produced in 1 year & &\ \ 30 keV$<$$E_{recoil}$$<$100 keV & &\ Muon veto & &\ \ & & & &\ 30 keV$<$$E_{recoil}$$<$100 keV & 149 & 149 & 8 & 11\ Muon veto & 15 & 15 & 0.7 & 1\ \ & & & &\ 30 keV$<$$E_{recoil}$$<$100 keV & 34 & 9 & 4 & 0.3\ Muon veto & 3 & 1 & 0.4 & $<0.1$\ According to Table \[tab:muonres\], out of the four configurations studied, the best one corresponds to the case where a 60 cm thick lead block is placed on top of the detector. With this passive veto alone, the background is about ten events per year, provided the lead is 60 cm thick. When combined with the Gd-doped water veto, the background drops well below one event per year. The simulations have been repeated considering liquid xenon as the target material. Results are shown in Table \[tab:muonresXe\]. Again, the expected background coming from muon-induced neutrons is $\sim$1 event per year for the whole detector. However, further reduction can be achieved when water itself is considered as an active veto (as shown in Tables \[tab:muonres\] and \[tab:muonresXe\], where we refer to it as Muon veto). It has been demonstrated that by rejecting events in coincidence with a muon, the contamination level decreases by a factor 10 [@Kudryavtsev]. In our case, this means the overall background would be well below 1 event per year per tonne of target material. Neutrons Total Not vetoed -------------------------------- ---------------- ---------------- Produced in 1 year $2.4\cdot10^5$ $1.9\cdot10^5$ 15 keV$<$$E_{recoil}$$<$50 keV 8 0.7 Muon veto 0.8 $<0.1$ : Background events coming from cosmic muon-induced neutrons using LXe as target material for a data taking period of one year. We assume that a 60 cm thick lead block is installed on top of the detector.\[tab:muonresXe\] Discussion ========== As a result of our study, we have seen that the combination of a noble liquid (used as sensitive target) and a Gd-doped active water veto efficiently reduces neutron background. For idealized data taking conditions, if we take as reference value an exposure of one tonne $\times$ year, the total neutron-induced contamination for the case of an argon-filled detector is one event, while two events are expected for the case of xenon (see Table \[tab:summary\]). As shown in Figure \[fig:limit\], in case no statistically significant signal is observed, we can reach sensitivities [@Feldman] for the WIMP-nucleon spin-independent cross section close to 10$^{-10}$ pb. To compute these limits we have assumed a standard dark matter galatic halo [@lewin], an energy resolution that amounts to 25$\%$ for the energy range of interest and 50$\%$ nuclear recoil acceptance. For completeness, we note that the best upper limit to date excludes (at 90$\%$ C.L.) cross sections above $4.5 (4.6)\times 10^{-8}$ pb for a WIMP mass of 30(60) GeV/c$^2$ [@xenon10; @cdmslimit]. --------------------- ----------------------------------- ----------------------------------- Background LAr target LXe target (exposure: 1 tonne $\times$ year) (exposure: 1 tonne $\times$ year) Detector components 0.9 1.2 Rock radioactivity $\!\!\!\!<$0.1 0.9 Muon-induced 0.1 0.1 [**Total**]{} [**1.0**]{} [**2.2**]{} --------------------- ----------------------------------- ----------------------------------- : Total expected backgrounds for two different target configurations. Figures have been normalized to an exposure of one tonne per year.\[tab:summary\] ![Achievable sensitivities for the case sensitive targets are filled either with LAr or LXe. The curves have been computed assuming an exposure of one tonne $\times$ year, 50$\%$ nuclear recoil acceptance. The tool from reference has been used [@DAMNED].[]{data-label="fig:limit"}](EPS/limit.eps){width=".7\textwidth"} Conclusions =========== We have evaluated the performance of a detector devised to carry out a direct search for WIMPs. It is made of two sub-detectors: the active target consists of small cylinders filled with a liquefied noble gas. They are immersed inside an ultra-pure water tank doped with gadolinium, that acts as a veto system against neutrons and cosmic muons. This configuration enhances the probability for neutron capture in water and its identification, thus providing a much improved rejection tool against this kind of background. This technique is scalable and allows the construction of large detectors with masses in the tonne range. We have observed that the use of a Gd-doped veto reduces by about a factor fifty the neutron contamination in case the target is filled with argon and up to a factor ten in case xenon is used. 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--- abstract: 'A collection of $k$ sets is said to form a *$k$-sunflower*, or $\Delta$*-system* if the intersection of any two sets from the collection is the same, and we call a family of sets $\mathcal{F}$ *sunflower-free* if it contains no sunflowers. Following the recent breakthrough of Ellenberg and Gijswijt and Croot, Lev and Pach [@EllenbergGijswijtCapsets; @CrootLevPachZ4] we apply the polynomial method directly to Erdős-Szemerédi sunflower problem [@ErdosSzemeredi1978UpperBoundsForSunflowerFreeSets] and prove that any sunflower-free family $\mathcal{F}$ of subsets of $\{1,2,\dots,n\}$ has size at most $$|\mathcal{F}|\leq3n\sum_{k\leq n/3}\binom{n}{k}\leq\left(\frac{3}{2^{2/3}}\right)^{n(1+o(1))}.$$ We say that a set $A\subset(\mathbb Z/D \mathbb Z)^{n}=\{1,2,\dots,D\}^{n}$ for $D>2$ is sunflower-free if every distinct triple $x,y,z\in A$ there exists a coordinate $i$ where exactly two of $x_{i},y_{i},z_{i}$ are equal. Using a version of the polynomial method with characters $\chi:\mathbb{Z}/D\mathbb{Z}\rightarrow\mathbb{C}$ instead of polynomials, we show that any sunflower-free set $A\subset(\mathbb Z/D \mathbb Z)^{n}$ has size $$|A|\leq c_{D}^{n}$$ where $c_{D}=\frac{3}{2^{2/3}}(D-1)^{2/3}$. This can be seen as making further progress on a possible approach to proving the Erdős-Rado sunflower conjecture [@ErdosRadoTheorem], which by the work of Alon, Sphilka and Umans [@AlonSphilkaUmansSunflowerMatrix Theorem 2.6] is equivalent to proving that $c_{D}\leq C$ for some constant $C$ independent of $D$.' author: - 'Eric Naslund, William F. Sawin' date: May 27th 2016 title: 'Upper Bounds for Sunflower-Free sets' --- Introduction ============ A collection of $k$ sets is said to form a *$k$-sunflower*, or $\Delta$*-system*, if the intersection of any two sets from the collection is the same. A family of sets $\mathcal{F}$ is said to be *$k$-sunflower free* if no $k$ members form a $k$-sunflower, and when $k=3$ we simply say that the collection $\mathcal{F}$ is *sunflower-free.* It is a longstanding conjecture that sunflower-free families must be small, and there are two natural situations in which we may ask this question. The first, and most general case, is when each set in the family has size $m$. Erdős and Rado made the following conjecture which is now known as the *Sunflower Conjecture.* \[conj:Erdos-Rado-Sunflower-Conjecture\](Erdős-Rado Sunflower Conjecture [@ErdosRadoTheorem]) Let $k\geq3$, and suppose that $\mathcal{F}$ is a $k$-sunflower free family of sets, each of size $m$. Then $$|\mathcal{F}|\leq C_{k}^{m}$$ for a constant $C_{k}>0$ depending only on $k$. In their paper, Erdős and Rado [@ErdosRadoTheorem] proved that any $k$-sunflower free family of sets of size $m$ has size at most $m!(k-1)^{m}$, and the conjectured bound of $C_{k}^{m}$ remains out of reach for any $k\geq3$. The second setting for upper bounds for $k$-sunflower-free sets concerns the case where each member of $\mathcal{F}$ is a subset of the same $n$-element set. There can be at most $2^{n}$ such subsets, and the Erdős-Szemerédi sunflower conjecture states that this trivial upper bound can be improved by an exponential factor. \[conj:Erdos-Szemeredi-Sunflower-Conjecture\](Erdős-Szemerédi Sunflower Conjecture [@ErdosSzemeredi1978UpperBoundsForSunflowerFreeSets]) Let $S$ be a $k$-sunflower free collection of subsets of $\left\{ 1,2,\dots,n\right\} $. Then $$|S|<c_{k}^{n}$$ for some constant $c_{k}<2$ depending only on $k$. In Erdős and Szemerédi’s paper [@ErdosSzemeredi1978UpperBoundsForSunflowerFreeSets], they prove that conjecture \[conj:Erdos-Szemeredi-Sunflower-Conjecture\] follows from conjecture \[conj:Erdos-Rado-Sunflower-Conjecture\] (see also [@AlonSphilkaUmansSunflowerMatrix Theorem 2.3]), and so it is a weaker variant of the sunflower problem. Let $F_{k}(n)$ denote the size of the largest $k$-sunflower-free collection $\mathcal{F}$ of subsets of $\{1,2,\dots,n\}$, and define $$\mu_{k}^{S}=\limsup_{n\rightarrow\infty}F_{k}(n)^{1/n}$$ to be the the *Erdős-Szmerédi-$k$-sunflower-free capacity*. The trivial bound is $\mu_{k}^{S}\leq2$, and the Erdős-Szemerédi sunflower conjecture states that $\mu_{k}^{S}<2$ for all $k\geq3$. In this paper we prove new bounds for the sunflower-free capacity $\mu_{3}^{S}$. It is a theorem of Alon, Shpilka and Umans [@AlonSphilkaUmansSunflowerMatrix pp. 7] that the recent work of Ellenberg and Gijswijt [@EllenbergGijswijtCapsets] on progression-free sets in $\mathbb{F}_{3}^{n}$ implies that $\mu_{3}^{S}<2$, and before this, the best upper bound for a sunflower-free collection of $\{1,2,\dots,n\}$ was $2^{n}\exp\left(-c\sqrt{n}\right)$ due to Erdős and Szemerédi [@ErdosSzemeredi1978UpperBoundsForSunflowerFreeSets]. We give a simple proof of a quantitative version of [@AlonSphilkaUmansSunflowerMatrix pp. 7], showing that $\mu_3^S \leq \sqrt{1+C}$ where $C$ is the capset capacity. However, using the ideas from the recent breakthrough of Ellenberg and Gijswijt and Croot, Lev and Pach, [@CrootLevPachZ4; @EllenbergGijswijtCapsets] on progressions in $\mathbb{F}_{3}^{n}$, and from Tao’s version of the argument [@TaosBlogCapsets], we apply the polynomial method directly to this problem, and obtain a stronger result: \[thm:Main-Sunflower-Free-Capacity\]Let $\mathcal{F}$ be a sunflower-free collection of subsets of $\{1,2,\dots,n\}$. Then $$|\mathcal{F}|\leq3(n+1)\sum_{k\leq n/3}\binom{n}{k},$$ and $$\mu_{3}^{S}\leq\frac{3}{2^{2/3}}=1.889881574\dots$$ The best known lower bound for this problem is $\mu_{3}^{S}\geq1.554$ due to the first author [@NaslundLowerBounds], and so there is still a large gap between upper and lower bounds for the sunflower-free capacity $\mu_{3}^{S}$. In section \[sec:Sunflower-Free-Sets-in-Z\_D\] we turn to the sunflower problem in the set $\{1,2,\dots,D\}$, which we will always think of as $\mathbb Z/D \mathbb Z$ . Alon, Shpilka and Umans [@AlonSphilkaUmansSunflowerMatrix Definition 2.5], defined a $k$-sunflower in $(\mathbb Z/D \mathbb Z)^{n}$ for $k\leq D$ to be a collection of $k$ vectors such that in each coordinate they are either all different or all the same. When $k=3$ and $D=3$ this condition is equivalent to being a three-term arithmetic progression in $\mathbb{F}_{3}^{n}$. \[conj:Sunflower-Conjecture-in-Z\_D\](Sunflower-Conjecture in $(\mathbb Z/D \mathbb Z)$) Let $k\leq D$, and let $A\subset(\mathbb Z/D \mathbb Z)^{n}$ be a $k$-sunflower-free set. Then $$|A|\leq b_{k}^{n}$$ for a constant $b_{k}$ depending only on $k$. The motivation for this problem comes from [@AlonSphilkaUmansSunflowerMatrix Theorem 2.6] where they proved that conjecture \[conj:Sunflower-Conjecture-in-Z\_D\] is equivalent to the Erdős-Rado sunflower conjecture. In particular, if there exists a constant $C$ independent of $D$ such that any $3$-sunflower-free set in $(\mathbb Z/D \mathbb Z)^{n}$ has size at most $C^{n}$, then Conjecture \[conj:Erdos-Rado-Sunflower-Conjecture\] holds for $k=3$ with $c_{3}=e\cdot C$. Since a sunflower-free set cannot contain a $3$-term arithmetic progression, the recent result of Ellenberg and Gijswijt [@EllenbergGijswijtCapsets] implies an upper bound for sunflower-free sets $A\subset(\mathbb Z/D \mathbb Z)^{n}$ for $D$ prime of the form $|A|\leq c_D^{n}$, where $c_D = D e^{ -I((D-1)/3)}$ for a function $I$ defined in [@EllenbergGijswijtCapsets] in terms of a certain optimization problem. It’s not too hard to see that $$0 < \lim_{D \to \infty} \frac{c_D}{D} <1$$ Using the characters $\chi:\mathbb{Z}/D\mathbb{Z}\rightarrow\mathbb{C}$ instead of polynomials, we prove the following theorem: \[thm:Main-Z\_D-sunflower-free-capacity\]Let $D\geq3$, and let $A\subset(\mathbb Z/D \mathbb Z)^{n}$ be a sunflower-free set. Then $$|A|\leq c_{D}^{n}$$ where $c_{D}=\frac{3}{2^{2/3}}(D-1)^{2/3}.$ This can be seen as progress towards the Erdős-Rado sunflower conjecture, and we remark that the now resolved Erdős-Szemerédi conjecture for $k=3$ is equivalent to proving that $c_{D}<D^{1-\epsilon}$ for some $\epsilon$ and all $D$ sufficiently large [@AlonSphilkaUmansSunflowerMatrix Theorem 2.7]. To prove Theorem \[thm:Main-Sunflower-Free-Capacity\] and \[thm:Main-Z\_D-sunflower-free-capacity\] we bound the slice rank of a function of three variables $T(x,y,z)$ which is nonvanishing if and only if $x=y=z$ or $x,y,z$ form a sunflower. A function $f:A^{k}\rightarrow\mathbb{F}$, where $A^{k}=A\times A\times\cdots\times A$ denotes the cartesian product and $\mathbb{F}$ is a field, is said to be a *slice* if it can be written in the form $$f(x_{1},\dots,x_{k})=h(x_{i})g(x_{1},\dots,x_{i-1},x_{i+1},\dots,x_{k})$$ where $h:A\rightarrow\mathbb{F}$ and $g:A^{k-1}\rightarrow\mathbb{F}$. The *slice rank* of a general function $f:A^{k}\rightarrow\mathbb{F}$ is the smallest number $m$ such that $f$ is a linear combination of $m$ slices. If $A$ is a sunflower-free set, it follows that, for $x,y,z\in A$, $T(x,y,z)$ is nonzero if and only if $x=y=z$. We then have the following lemma: \[lem:Tao-rank-hyperdiagonal-matrices\](Rank of diagonal hypermatrices [@TaosBlogCapsets Lemma 1]) Let $A$ be a finite set and $\mathbb{F}$ a field. Let $T(x,y,z)$ be a function $A \times A \times A \to \mathbb F$ such that $T(x,y,z)\neq 0$ if and only if $x=y=z$. Then the slice rank of $T$ is equal to $|A|$. Using this lemma, we need only find an upper bound on the slice rank of $T$ to obtain an upper bound on the size of the sunflower-free set. In each case we do this by an explicit decomposition of $T$ into slices found by writing $T$ as either a polynomial or as a sum of characters. We refer the reader to section 4 of [@BlasiakChurchCohnGrochowNaslundSawinUmans2016MatrixMultiplication] for further discussion of the slice rank. This method is the direct analogue of Tao’s interpretation [@TaosBlogCapsets] of the Ellenberg-Gijswijt argument for capsets, and can be thought of as a $3$-tensor generalization of the Haemmer bound [@Haemer1978AnUpperBoundForTheShannonCapacityOfAGraph], which bounds the Sperner capacity of a hypergraph rather than the Shannon capacity of a graph. We stress two differences between our result and several other papers which use the slice rank method [@TaosBlogCapsets] [@BlasiakChurchCohnGrochowNaslundSawinUmans2016MatrixMultiplication], or which have been reinterpreted to use the slice rank [@CrootLevPachZ4] [@EllenbergGijswijtCapsets]. First, these papers study functions valued in finite fields, whose characteristic is chosen for the specific problem and cannot be changed without affecting the bound. Our work uses functions valued in a field of characteristic zero, though we could have done the same thing in any finite field of sufficiently large characteristic. Second, these papers mainly describe functions as low-degree polynomials and use that structure to bound their slice rank. In the proof of Theorem \[thm:Main-Z\_D-sunflower-free-capacity\], we describe functions as sums of characters. One can interpret characters as polynomials restricted to the set of roots of unity, but under this interpretation the degree of the polynomial is not relevant to the proof of Theorem \[thm:Main-Z\_D-sunflower-free-capacity\] - only the number of nontrivial characters is. The proofs of theorems \[thm:Main-Sunflower-Free-Capacity\] and \[thm:Main-Z\_D-sunflower-free-capacity\] can be extended without modification to handle a multicolored version of the problem analogous to multicolored sum-free sets as defined in [@BlasiakChurchCohnGrochowNaslundSawinUmans2016MatrixMultiplication]*.* \[sec:The-Erdos-Szemeredi-Sunflower-Problem\]The Erdős-Szemerédi Sunflower Problem ================================================================================== Any subset of $\{1,2,\dots.n\}$ corresponds to a vector in $\{0,1\}^{n}$ where a $1$ or $0$ in coordinate $i$ denotes whether or not $i$ lies in the subset. A sunflower-free collection of subsets of $\{1,2,\dots,n\}$ gives rise to a set $S\subset\{0,1\}^{n}$ with the property that for any three distinct vectors $x,y,z\in S$, there exists $i$ such that $\{x_{i},y_{i},z_{i}\}=\{0,1,1\}$. Moreover, a sunflower-free collection of subsets of $\{1,2,\dots,n\}$ that also does not contain two subsets with one a proper subset of the other gives rise to a set $S \subset \{0,1\}^{n}$ such that for any $x,y,z\in S$ not all equal, there exists $i$ such that $\{x_{i},y_{i},z_{i}\}=\{0,1,1\}$. This holds because the only new case is when two are equal and the third is not (say $x=y$ and $z$ is distinct), and then because $x \neq z$, $x$ is not a subset of $z$, so there exists some $i$ such that $x_i=y_i=1$ and $z_i=0$. Given a sunflower-free set $S\subset\{0,1\}^{n}$, let $S_{l}$, for $l=1,\dots,n$, denote the elements of $S$ with exactly $l$ ones so that $S=\cup_{l=0}^{n}S_{l}$. Then for each $l$, $S_l$ is a sunflower-free collection of subsets with none a proper subset of another, hence whenever $x,y,z\in S_{l}$ satisfy $x+y+z\notin\{0,1,3\}^{n}$ we must have $x=y=z$. For $x,y,z\in\{0,1\}^{n}$ consider the function $T:\left\{ 0,1\right\} ^{n}\times\left\{ 0,1\right\} ^{n}\times\left\{ 0,1\right\} ^{n}\rightarrow\mathbb{R}$ given by $$T(x,y,z)=\prod_{i=1}^{n}\left(2-(x_{i}+y_{i}+z_{i})\right).$$ The function $T(x,y,z)$ is nonvanishing precisely on triples $x,y,z$ such that there does not exist $i$ where $\{x_i,y_i,z_i\}=\{1,1,0\}$. Hence restricted to $S_l \times S_l \times S_l$, $T(x,y,z)$ is nonzero if and only if $x=y=z$. So by Lemma \[lem:Tao-rank-hyperdiagonal-matrices\], the slice rank of $T$ is at least $|S_{l}|$. Expanding the product form for $T(x,y,z)$, we may write $T(x,y,z)$ as a linear combination of products of three monomials $$x_{1}^{i_{1}}\cdots x_{n}^{i_{n}}y_{1}^{j_{1}}\cdots y_{n}^{j_{n}}z_{1}^{k_{1}}\cdots z_{n}^{k_{n}}$$ where $i_{1},\dots,i_{n},j_{1},\dots,j_{n},k_{1},\dots,k_{n}\in\{0,1\}^{n}$, and $$i_{1}+\cdots+i_{n}+j_{1}+\cdots+j_{n}+k_{1}+\cdots+k_{n}\leq n.$$ For each product of three monomials, at least one of $i_{1}+\cdots+i_{n}$, $j_{1}+\cdots+j_{n}$, $k_{1}+\cdots+k_{n}$ is at most $n/3$. For each term in $T$, choose either $x_{1}^{i_{1}}\cdots x_{n}^{i_{n}}$, $y_{1}^{j_{1}}\cdots y_{n}^{j_{n}}$, or $z_{1}^{k_{1}}\cdots z_{n}^{k_{n}}$, making sure to choose one of total degree at most $n/3$. Divide the expansion of $T$ into, for each possible monomial, the sum of all the terms where we chose that monomial. Because one monomial in each of these sums is fixed, we can express each sum as a product of that monomial (a function of one variable) times the sum of all the other terms (a function of the other variables), hence each of these sums is a slice. The total slice rank is at most the number of slices, which is at most the number of monomials we can choose: $3$ times the number of monomials in $n$ variables of degree at most $1$ in each variable and of total degree at most $k$. The number of such monomials is exactly $\sum_{k\leq n/3}\binom{n}{k}$, so this yields the upper bound $$|S_{l}|\leq3\sum_{k\leq n/3}\binom{n}{k},$$ $$|S| \leq \sum_{l=0}^n |S_l| \leq 3(n+1) \sum_{k\leq n/3}\binom{n}{k}$$ which is the statement of Theorem \[thm:Main-Sunflower-Free-Capacity\]. Capset Capacity --------------- A capset $A$ is a subset of $\mathbb{F}_{3}^{n}$ containing no three-term arithmetic progressions. Let $A_{n}\subset\mathbb{F}_{3}^{n}$ denote the largest capset in dimension in $n$, and define $$C=\limsup_{n\rightarrow\infty}|A_{n}|^{1/n}$$ to be the *capset capacity*. Note that $|A_n|$ is super-multiplicative, that is for $m,n\geq 1$ we have $|A_{mn}|\geq |A_{n}|^m$ since $A_n$ Cartesian-producted with itself $m$ times is a capset in $\mathbb{F}_3^{mn}$. Ellenberg and Gijswijt [@EllenbergGijswijtCapsets] proved that $C\leq2.7552$, and the following theorem is a quantitative version of a result of Alon, Sphlika, and Umans [@AlonSphilkaUmansSunflowerMatrix pp. 7]. We have that $\mu_{3}^{S}\leq\sqrt{1+C}$ where $C$ is the capset capacity and $\mu_{3}^{S}$ is th*e* Erdős-Szmeredi-sunflower-free capacity. We will bound the size of the largest sunflower-free set in $\{0,1\}^{2n}$ by writing each vector in terms of the four vectors in $\{0,1\}^{2}$ $$u_{0}=\left[\begin{array}{c} 0\\ 0 \end{array}\right],\ u_{1}=\left[\begin{array}{c} 1\\ 0 \end{array}\right],\ u_{2}=\left[\begin{array}{c} 0\\ 1 \end{array}\right],\ u_{3}=\left[\begin{array}{c} 1\\ 1 \end{array}\right].$$ Every set $S\subset\{0,1\}^{2n}$ corresponds to a set $\tilde{S}\in\{0,1,2,3\}^{n}$ where we obtain $S$ from $\tilde{S}$ by replacing each symbol $i$ for $i\in\{0,1,2,3\}$ with the vector $u_{i}$. For example, $$\left[\begin{array}{c} 1\\ 0\\ 1\\ 1 \end{array}\right]\longleftrightarrow\left[\begin{array}{c} 1\\ 3 \end{array}\right]\text{ and }\left[\begin{array}{c} 0\\ 1\\ 0\\ 0 \end{array}\right]\longleftrightarrow\left[\begin{array}{c} 2\\ 0 \end{array}\right].$$ For each $x\in\{0,1\}^{n}$ consider $$\tilde{S}_{x}=\left\{ v\in\tilde{S}:\ v_{i}=3\text{ if and only if }x_{i}=1\right\} .$$ We may view elements of $\tilde{S}_x$ as elements of $\{0,1,2\}^{n-x} =\mathbb F_3^{n-x}$ by ignoring the coordinates where $x$ is $1$. If three elements in $\tilde{S}_x$ form an arithmetic progression in $\mathbb F_3^{n-x}$, then in each coordinate the elements of $\tilde{S}_x$ are either all the same or are $0,1,2$ in any order, so the entries of the corresponding vectors in $S$ are either all the same or $u_0,u_1,u_2$ in any order. Because $u_{0},u_{1},u_{2}$ form a sunflower, these three elements of $S$ form a sunflower. Because $S$ is a sunflower-free set, $\tilde{S}_x$ is a capset. Let $w(x)=\sum_{i=1}^{n}x_{i}$ be the weight of the vector $x$, then $$|\tilde{S}_{x}|\leq C^{n-w(x)}$$ where $C$ is the capset capacity. Hence $$|S|\leq\sum_{x}C^{n-w(x)}=\sum_{j=0}^{n}\binom{n}{j}C^{n-j}=(1+C)^n,$$ and we obtain the desired bound. Using the Ellenberg-Gijswijt upper bound on capset capacity, this gives $\mu_3^S \leq 1.938$, which is not as strong a bound as Theorem \[thm:Main-Sunflower-Free-Capacity\]. \[sec:Sunflower-Free-Sets-in-Z\_D\]Sunflower-Free Sets in $(\mathbb Z/D \mathbb Z)^{n}$ ======================================================================================= Consider the $D$ characters $\chi:\mathbb{Z}/D\mathbb{Z}\rightarrow\mathbb{C}^{\times}$. By the orthogonality relations, for any $a,b\in\mathbb{Z}/D\mathbb{Z}$ $$\frac{1}{|D|}\sum_{\chi}\chi(a-b)=\begin{cases} 1 & \text{if}\ a=b\\ 0 & \text{otherwise} \end{cases}.$$ Hence $$\frac{1}{|D|}\sum_{\chi}\left(\chi(a)\overline{\chi(b)}+\chi(b)\overline{\chi(c)}+\chi(a)\overline{\chi(c)}\right)=\begin{cases} 0 & \text{if }a,b,c\text{ are distinct}\\ 1 & \text{if exactly two of }a,b,c\text{ are equal}\\ 3 & \text{if }a=b=c \end{cases}.$$ For $x,y,z\in(\mathbb Z/D \mathbb Z)^{n}$, define the function $T:(\mathbb Z/D \mathbb Z)^{n}\times(\mathbb Z/D \mathbb Z)^{n}\times(\mathbb Z/D \mathbb Z)^{n}\rightarrow\mathbb{C}$ by $$T(x,y,z)=\prod_{j=1}^{n}\left(\frac{1}{|D|}\sum_{\chi}\left(\chi(a)\overline{\chi(b)}+\chi(b)\overline{\chi(c)}+\chi(a)\overline{\chi(c)}\right)-1\right) ,\label{eq:product_for_T}$$ $$= \prod_{j=1}^{n}\left(\frac{1}{|D|}\sum_{\chi}\left(\chi(a)\overline{\chi(b)}1(c)+1(a)\chi(b)\overline{\chi(c)}+\chi(a)1(b)\overline{\chi(c)}\right)-1(a)1(b)1(c)\right)$$ which is non-zero if and only if $x,y,z$ form a $\mathbb Z/D \mathbb Z$-sunflower or all equal. Let $A\subset(\mathbb Z/D \mathbb Z)^{n}$ be a sunflower free set. Then restricted to $A \times A \times A$, $T$ is nonzero if and only if $x=y=z$. Hence by Lemma \[lem:Tao-rank-hyperdiagonal-matrices\] the slice rank of $T$ is at least $|A|$. Expanding the product in (\[eq:product\_for\_T\]), we see that $T$ can be written as a linear combination of terms of the form $$\chi_{1}(x_{1})\cdots\chi_{n}(x_{n})\psi_{1}(y_{1})\cdots\psi_{n}(y_{n})\xi_{1}(z_{1})\cdots\xi_{n}(z_{n})$$ where $\chi_{1},\dots,\chi_{n},\psi_{1},\dots,\psi_{n},\xi_{1},\dots,\xi_{n}$ are characters on $\mathbb{Z}/D\mathbb{Z}$, at most $2n$ of which are nontrivial. For any such term, at least one of the tuples $\chi_1,\dots,\chi_n$, $\psi_1,\dots,\psi_n$, $\xi_1,\dots,\xi_n$ must contain at most $2n/3$ nontrivial characters. Grouping the terms by the one containing the fewest nontrivial characters like this, we are able to upper bound the slice rank of $T$ by $$3\sum_{k\leq2n/3}\binom{n}{k}(D-1)^{k},$$ where the $(D-1)^{k}$ comes from the fact that for each set of $k$ indices, we have $(D-1)^{k}$ possible choices of non-trivial characters. Because $D \geq 3$, $\frac{D-1}{2} \geq 1$, so: $$\sum_{k\leq2n/3}\binom{n}{k}(D-1)^{k} \leq \sum_{k\leq2n/3}\binom{n}{k}(D-1)^{k} \left(\frac{D-1}{2}\right)^{2n/3-k}= \left(\frac{D-1}{2} \right)^{-\frac{n}{3}} \sum_{k\leq2n/3}\binom{n}{k}(D-1)^{k} \left( \frac{D-1}{2} \right)^{n-k}$$$$\leq \left(\frac{D-1}{2} \right)^{-\frac{n}{3}} \sum_{k\leq n}\binom{n}{k}(D-1)^{k} \left( \frac{D-1}{2} \right)^{n-k} = \left(\frac{D-1}{2} \right)^{-\frac{n}{3}} \left( D-1 + \frac{D-1}{2} \right)^n = \left(\frac{3}{2^{2/3}}(D-1)^{2/3}\right)^{n}$$ Let $c_{D}=\frac{3}{2^{2/3}}(D-1)^{2/3}.$ This inequality proves that for $|A|$ a sunflower-free set, $$|A|\leq 3c_{D}^{n}$$ To prove Theorem \[thm:Main-Z\_D-sunflower-free-capacity\] (that $|A| \leq c_D^n$), we can remove the factor of $3$ by a standard amplification argument, as for $A$ a sunflower-free set in $(\mathbb Z/D \mathbb Z)^n$, $A^k$ is a sunflower-free set in $(\mathbb Z/D \mathbb Z)^{nk}$, so $|A| \leq (3c_D^{nk})^{1/k} = 3^{1/k} c_D^n$. Taking $k\rightarrow\infty$, we obtain Theorem \[thm:Main-Z\_D-sunflower-free-capacity\]. [1]{} Noga Alon, Amir Shpilka, and Christopher Umans. On sunflowers and matrix multiplication. , 22(2):219–243, 2013. Jonah Blasiak, Thomas Church, Henry Cohn, Joshua Grochow, Eric Naslund, Will Sawin, and Christopher Umans. On cap sets and the group-theoretic approach to matrix multiplication. 2016. URL:https://arxiv.org/abs/1605.06702/. Ernie Croot, Vsevolod Lev, and Peter Pach. Progression-free sets in $\mathbb Z_4^n$ are exponentially small. 2016. URL:https://arxiv.org/abs/1605.01506/. Jordan Ellenberg and Dion Gijswijt. On large subsets of $\mathbb F_q^n$ with no three-term arithmetic progression. 2016. URL:https://arxiv.org/abs/1605.09223/. P. Erd[ő]{}s and R. Rado. Intersection theorems for systems of sets. , 35:85–90, 1960. P. Erd[ő]{}s and E. Szemer[é]{}di. Combinatorial properties of systems of sets. , 24(3):308–313, 1978. W. Haemers. An upper bound for the [S]{}hannon capacity of a graph. In [*Algebraic methods in graph theory, [V]{}ol. [I]{}, [II]{} ([S]{}zeged, 1978)*]{}, volume 25 of [*Colloq. Math. Soc. János Bolyai*]{}, pages 267–272. North-Holland, Amsterdam-New York, 1981. Eric Naslund. Lower bounds for capsets and sunflower-free sets. Unpublished. Terence Tao. A symmetric formulation of the croot-lev-pach-ellenberg-gijswijt capset bound, 2016. URL:https://terrytao.wordpress.com/2016/05/18/a-symmetric-formulation-of-the-croot-lev-pach-ellenberg-gijswijt-capset-bound/.

--- abstract: 'We investigate the quantum entropy, its power spectrum, and the excitation inversion of a Cooper pair box interacting with a nanomechanical resonator, the first initially prepared in its excited state, the second prepared in a “cat"-state. The method uses the Jaynes-Cummings model with damping, with different decay rates of the Cooper pair box and distinct detuning conditions, including time dependent detunings. Concerning the entropy, it is found that the time dependent detuning turns the entanglement more stable in comparison with previous results in literature. With respect to the Cooper pair box excitation inversion, while the presence of detuning destroys the its collapses and revivals, it is shown that with a convenient time dependent detuning one recovers such events in a nice way.' address: - 'Universidade Paulista, Rod. BR 153, km 7, 74845-090 Goiânia, GO, Brazil.' - 'Universidade Estadual de Goiás, Rod. BR 153, 3105, 75132-903 Anápolis, GO, Brazil.' - 'Instituto de Física, Universidade Federal de Goiás, 74001-970 Goiânia, GO, Brazil.' author: - 'C. Valverde' - 'A.T. Avelar' - 'B. Baseia' title: Controlling Statistical Properties of a Cooper Pair Box Interacting with a Nanomechanical Resonator --- , Quantum Entropy ,Power Spectrum ,Cooper Pair Box ,Nanomecanical Resonator ,Excitation Inversion 65.40.gd ,32. 80. Bx ,42.50.Dv Introduction ============ In the last years there has been a great interest in the production of new nonclassical states of the quantized electromagnetic field, one of the interesting topics of Quantum Optics. Despite the field quantization in 1925, quantum optical effects were observed only seven decades after, the first of them being the antibunching effect, as predicted by Carmichael and Walls in 1976 [@1], experimentally confirmed by Kimble, M. Dagenais and L. Mandel [@2] in 1977. A second nonclassical effect was observed in 1985 by Slusher et al. [@3], theoretically anticipated by Stoler et al. [@4] in 1970. A third one, the oscillations in the photon statistical distribution, was observed in 1987 by Rempe et al. [@5]. Since then, various nonclassical states of the quantized electromagnetic field were studied, including their practical realization in laboratories in different systems - one of them being the famous Schrödinger “cat" state, its generation being sugested by Yurke and Stoler [@6], Davidovich et al. [@7], etc; its first experimental observation was obtained by the group of Haroche [@8]. More recently, the community became aware of the first experimental observation of the decoherence of the Schrödinger “cat" state, in both realms of optical [@8.1] and atomic physics [@8.2], and constituting the first observation of the passage through the frontier that separates the quantum and classical physics. After that, another interesting topic emerged, as the quantum teleportation of states, first suggested by Bernnett et al. [@10], based on the nonlocal character of quantum mechanics and contextualized by the EPR entangled states [@10.5]. This somewhat “bizarre" effect was first observed experimentally in 1997, by the group of Zeilinger [@11], concerning the teleportation of a single photon state; the effect was later extended for atomic states and also for a huge quantity of photons [@12]. Then, several publications in this line appeared in the literature [@13; @13.1; @13.2; @13.3]. Besides the nonclassical effects of light field states, many researchers became interested in the study of new states and new effects they could exhibit, mainly concerning with their potential applications [@cv1]. Then, it became also interesting the study of various schemes for the generation of nonclassical light states [@114; @14; @cv2]. To this end, two lines of study emerged: (i) when the issue is concerned with a state of a stationary field, inside a high-Q microwave cavity; (ii) when concerning with a traveling field, either thoughout the free space or a medium (optical fibers, beam splitters, prisms, etc.). In both cases various proposals appeared in the literature [@15a; @15ab; @15; @15.1]. The extension of these investigations for atomic systems has been also implemented. In this case the system no longer concerns with traveling fields or a field trapped inside a high-Q cavity; instead, it consists of atoms either inside or crossing a cavity, including atoms inside a magneto-optical trap [16,16.1]{}. When focusing either the field or the atomic case the theoretical strategy starts from a Hamiltonian describing the atom-field system, traditionally treated via the Jaynes-Cummings model and the atom-field coupling usually considered as a constant parameter. Comparatively, the number of such works in the literature is very small when one considers the atom-field coupling and/or atomic frequency as a time dependent parameter [@law; @l1; @l22; @l2; @l3], including the case of time dependent amplitude [@abdalla]. Nevertheless, this scenario is also relevant;  for example, the state of two qubits (qubits stand for quantum bits) with a desired degree of entanglement can be generated via a time dependent atom-field coupling [olaya]{}; such coupling can modify the dynamical properties of the atom and the field, with transitions that involve a large number of photons [yang]{}. In general, these studies are simplified by neglecting the atomic decay from an excited level. Theoretical treatments taking into account this complication of the real world also employs the Jaynes-Cummings model. In these case, as expected, one finds decoherence of the state describing the system, since the presence of dissipation destroys the state of a system as time flows. Here, taking advantage of what we have learned on the atom-field interaction, we will study an advantageous system in practice (due to its rapid response and better controllability [@y1]) by considering a nanomechanical resonator (NR) interacting with a Cooper pair box (CPB). This nanodevice has its own interest since its macroscopic nature and peculiar effects of low-frequency noise in the solid-state impose obstacles requiring more careful studies than a mere translation from quantum optics. It has been explored in the study of quantum nondemolition measurements [ak1,e1]{}, in the study of decoherence of nonclassical states, as Fock states and superposition or entangled states describing mesoscopic systems [@e2], etc. The fast advance in the tecnique of fabrication in nanotecnology implied great interest in the study of the NR system in view of its potential modern applications, as a sensor, largely used in various domains, as in biology, astronomy, quantum computation, and more recently in quantum information [@ak2] to implement the quantum qubit [@ak3] and in the production of nonclassical states, e.g.: Fock states [@akk], Schrödinger’s “cat" states [@akk2], squeezed states [@a34], clusters states [@ak], etc. In particular, when accompanied by superconducting charge qubits, the NR has been used to prepare entangled states [@akk1]. Zhou et al.[@a34] have proposed a scheme to prepare squeezed states using a NR coupled to a CPB qubit; in this proposal the NR-CPB coupling is under an external control while the connection between these two interacting subsystems play an important role in quantum computation. Such a control is achieved via convenient change of system parameters, which can set “on" and “off" the interaction between the NR and the CPB, on demand. One of the desired goals in this report is to verify the behavior and properties of an entangled state describing the CPB-NR system, via the Jaynes-Cummings model, by considering the energy dissipation in the CPB during its transitions from an excited level to a ground state. Another target is to verify if, and in which way, the time dependence of the CPB-NR coupling modifies the dynamical properties of the state describing a subsystem. We will also study the time evolution of the quantum entropy and its power spectrum, as well as the CPB excitation inversion. There are some evidences of entropy production, including the fact that the power spectrum of stationary systems and subsystems can be used as dynamical criteria for quantum caos [@Avi; @avi1]. For the entropy power spectrum, such criteria embody those already discussed in the literature concerned with fixed parameters. Then, it seems adequate to look at the various characteristics of the entropy to formulate a reasonable and suficient universal dynamical criterium for the quantum caos. The degree of entanglement, represented by the entropy in certain circumstances, has also shown itself being sensible to the presence of a classical caos [@k2; @k3]. Model hamiltonian for the CPB-NR system ======================================= There exist in the literature a large number of devices using the SQUID-base, where the CPB charge qubit consists of two superconducting Josephson junctions in a loop. In the present model a CPB is coupled to a NR as shown in Fig. (\[cooper\]); the scheme is inspired in the works by Jie-Qiao Liao et al. [@ak3] and Zhou et al. [@a34] where we have substituted each Josephson junction by two of them. This creates a new configuration including a third loop. A superconducting CPB charge qubit is adjusted via a voltage $V_{1}$ at the system input and a capacitance $C_{1}$. We want the scheme ataining an efficient tunneling effect for the Josephson energy. In Fig.(\[cooper\]) we observe three loops: one great loop between two small ones. This makes it easier controlling the external parameters of the system since the control mechanism includes the input voltage $V_{1}$ plus three external fluxes $\Phi (\ell ),$ $\Phi (r)$ and $\Phi _{e}(t)$. In this way one can induce small neighboring loops*.* The great loop contains the NR and its effective area in the center of the apparatus changes as the NR oscillates, which creates an external flux $\Phi _{e}(t)$ that provides the CPB-NR coupling to the system. In this work we will assume the four Josephson junctions being identical, with the same Josephson energy $E_{J}^{0}$, the same being assumed for the external fluxes $\Phi (\ell )$ and $\Phi (r)$, i.e., with same magnitude, but opposite sign: $\Phi (\ell )=-\Phi (r)=\Phi (x)$. In this way, we can write the Hamiltonian describing the entire system as $$\hat{H}=\omega \hat{a}^{\dagger }\hat{a}+4E_{c}\left( N_{g}-\frac{1}{2}\right) \hat{\sigma}_{z}-4E_{J}^{0}\cos \left( \frac{\pi \Phi _{x}}{\Phi _{0}}\right) \cos \left( \frac{\pi \Phi _{e}}{\Phi _{0}}\right) \hat{\sigma}_{x}, \label{a1}$$ where $\hat{a}^{\dagger }(\hat{a})$ is the creation (annihilation) operator for the excitation in the NR, corresponding with the frequency $\omega $ and mass $m$; $E_{J}^{0}$ and $E_{c}$ are respectively the energy of each Josephson junction and the charge energy of a single electron; $C_{1}$ and $C_{J}^{0}$ stand for the input capacitance and the capacitance of each Josephson tunel, respectively. $\Phi _{0}=h/2e$ is the quantum flux and $N_{1}=C_{1}V_{1}/2e$ is the charge number in the input with the input voltage $V_{1}$. We have used the Pauli matrices to describe our system operators, where the states $\left\vert g\right\rangle $ and $\left\vert e\right\rangle $ (or 0 and 1) represent the number of extra Cooper pairs in the superconduting island. We have: $\hat{\sigma}_{z}=\left\vert g\right\rangle \left\langle g\right\vert -\left\vert e\right\rangle \left\langle e\right\vert $, $\hat{\sigma}_{x}=\left\vert g\right\rangle \left\langle e\right\vert -\left\vert e\right\rangle \left\langle g\right\vert $ and $E_{C}=e^{2}/\left( C_{1}+4C_{J}^{0}\right) .$ The magnectic flux can be written as the sum of two terms, $$\Phi _{e}=\Phi _{b}+B\ell \hat{x}\text{ }, \label{a4}$$where the first term $\Phi _{b}$ is the induced flux, corresponding to the equilibrium position of the NR and the second term describes the contribution due to the vibration of the NR; $B$ represents the magnectic field created in the loop. We have assumed the displacement $\hat{x}$ described as $\hat{x}=x_{0}(\hat{a}^{\dagger }+\hat{a})$, where $x_{0}=\sqrt{m\omega /2}$ is the amplitude of the oscillation. Substituting the Eq.(\[a4\]) in Eq.(\[a1\]) and controlling the flux $\Phi _{b}$ we can adjust $\cos \left( \frac{\pi \Phi _{b}}{\Phi _{0}}\right) =0$ to obtain $$\hat{H}=\omega \hat{a}^{\dagger }\hat{a}+4E_{c}\left( N_{g}-\frac{1}{2}\right) \hat{\sigma}_{z}-4E_{J}^{0}\cos \left( \frac{\pi \Phi _{x}}{\Phi _{0}}\right) \sin \left( \frac{\pi B\ell \hat{x}}{\Phi _{0}}\right) \hat{\sigma}_{x} \label{a8}$$and making the approximation $\pi B\ell x/\Phi _{0}<<1$ we find $$\hat{H}=\omega \hat{a}^{\dagger }\hat{a}+\frac{1}{2}\omega _{0}\hat{\sigma}_{z}+\lambda _{0}(\hat{a}^{\dagger }+\hat{a})\hat{\sigma}_{x}, \label{a9}$$where the constant coupling $\lambda _{0}=-4E_{J}^{0}\cos \left( \frac{\pi \Phi _{x}}{\Phi _{0}}\right) \left( \frac{\pi B\ell x_{0}}{\Phi _{0}}\right) $ and the effective energy $\omega _{0}=8E_{c}\left( N_{g}-\frac{1}{2}\right) .$ In the rotating wave approximation the above Hamiltonian results as $$\hat{H}=\omega \hat{a}^{\dagger }\hat{a}+\frac{1}{2}\omega _{0}\hat{\sigma}_{z}+\lambda _{0}(\hat{\sigma}_{+}\hat{a}+\hat{a}^{\dagger }\hat{\sigma}_{-}).$$ Next, we will consider a more general scenario by substituting $\omega \rightarrow \omega (t)=\omega +f\left( t\right) $ and $\lambda _{0}\rightarrow \lambda (t)=\lambda _{0}\left[ 1+f\left( t\right) /\omega \right] $ [@yang; @jf]; in addition, we assume the presence of a constant decay rate $\gamma $ in the CPB, from its excited level to the ground state; $\omega _{0}$ is the transition frequency of the CPB and $\lambda _{0}$ stands for the CPB-NR coupling. $\hat{\sigma}_{\pm }$ and $\hat{\sigma}_{z}$ are the CPB transition and excitation inversion operators, respectively; they act on the Hilbert space of atomic states and satisfy the commutation relations $\left[ \hat{\sigma}_{+},\hat{\sigma}_{-}\right] =\hat{\sigma}_{z}$ and $\left[ \hat{\sigma}_{z},\hat{\sigma}_{\pm }\right] =\pm \hat{\sigma}_{\pm }$. As well known, the coupling parameter $\lambda (t)$ is proportional to $\sqrt{\upsilon \left( t\right) /V\left( t\right) }$, where the time dependent quantization volume $V\left( t\right) $ takes the form $V\left( t\right) =V_{0}/\left[ 1+f\left( t\right) /\omega \right] $ [scully,l2,jf]{}. Accordingly, we obtain the new (*non hermitean*) Hamiltonian $$\hat{H}=\omega (t)\hat{a}^{\dagger }\hat{a}+\frac{1}{2}\omega _{0}\hat{\sigma}_{z}+\lambda (t)(\hat{\sigma}_{+}\hat{a}+\hat{a}^{\dagger }\hat{\sigma}_{-})-i\frac{\gamma }{2}\left\vert e\right\rangle \left\langle e\right\vert . \label{b1}$$ Non hermitean Hamiltonians (NHH) have been largely used in the literature. As some few examples we mention: Ref. [@nh5], where the authors use a NHH and an algorithm to generalize the conventional theory; Ref. [@nh1], using a NHH to get information about entrance and exit channels; Ref. [nh6]{}, using non hermitean techniques to study canonical transformations in quantum mechanics; Ref. [@nh7], solving quantum master equations in terms of NHH; Ref. [@nh3], using a new approach for NHH to study the spectral density of weak H-bonds involving damping; Ref. [@nh8], studing NHH with real eighenvalues; Ref. [@nh4], using a canonical formulation to study dissipative mechanics exhibing complex eigenvalues; Ref. [@nh9], studing NHH in non commutative space, and more recently: Ref. [@nh10], studing the optical realization of relativistic NHH; Ref. [@l2], studing the evolution of entropy of atom-field interation; Ref. [@l22], using a damping JC-Model to study entanglement between two atoms, each one inside distinct cavities Solving the CPB-NR system ========================= Now, the state describing our time dependent system can be written as $$\left\vert \Psi \left( t\right) \right\rangle =\sum\nolimits_{n=0}^{\infty }(C_{g,n}\left( t\right) \left\vert g,n\right\rangle +C_{e,n}\left( t\right) \left\vert e,n\right\rangle ). \label{b2}$$ Taking the CPB initially prepared in its excited state $\left\vert e\right\rangle $ and the NR in a superposition of two coherent states, $\left\vert \beta \right\rangle =\eta (\left\vert \alpha \right\rangle +\left\vert -\alpha \right\rangle )$, and expanding each coherent state component in the Fock’s basis, i.e., $\left\vert \alpha \right\rangle =exp(-|\alpha |^{2}/2)\sum_{n=o}^{\infty }(\alpha ^{n}/\sqrt{n!})|n\rangle $, we have $\left\vert \beta \right\rangle =\sum\nolimits_{n=0}^{\infty }F_{n}\left\vert n\right\rangle ,$where $\eta =[2+2\exp (-2\alpha ^{2})]^{-1/2}$ is the normalization factor. Assuming the NR and CPB decoupled at $t=0$ and the initial conditions $C_{g,n}\left( 0\right) =0$ and $\sum\nolimits_{n=0}^{\infty }\left\vert C_{e,n}\left( 0\right) \right\vert ^{2}=1$ we may write the Eq. (\[b2\]) as $$\left\vert \Psi \left( 0\right) \right\rangle =\sum\nolimits_{n=0}^{\infty }F_{n}\left\vert e,n\right\rangle . \label{b4}$$ The time dependent Schrödinger equation for the present system is $$i\dfrac{d\left\vert \Psi \left( t\right) \right\rangle }{dt}=\hat{H}\left\vert \Psi \left( t\right) \right\rangle , \label{b5}$$ with the Hamiltonian $\hat{H}$ given in Eq. (\[b1\]).* *Substituting Eq.(\[b1\]) in Eq.(\[b5\]) we get the (coupled) equations of motion for the probabilitity amplitudes $C_{e,n}(t)$ and $C_{g,n+1}(t)$:$$\begin{aligned} \frac{\partial C_{e,n}(t)}{\partial t} &=&-in\omega (t)C_{e,n}(t)-\frac{i}{2}\omega _{0}C_{e,n}(t)-i\lambda (t)\sqrt{n+1}C_{g,n+1}(t)-\frac{\gamma }{2}C_{e,n}(t), \label{b8} \\ \frac{\partial C_{g,n+1}(t)}{\partial t} &=&-i(n+1)\omega (t)C_{g,n+1}(t)+\frac{i}{2}\omega _{0}C_{g,n+1}(t)-i\lambda (t)\sqrt{n+1}C_{e,n}(t). \label{b9}\end{aligned}$$ The solutions of the coefficientes $C_{e,n}(t)$, $C_{g,n+1}(t)$ furnish the quantum dynamical properties of the system, including the CPB-NR entanglement. For the cases $f(t)=0$ and $f(t)=$ $const,$ the Eq.(\[b8\]) and Eq.([b9]{}) are exactly soluble. We find, analytically,$$\begin{aligned} C_{g,n+1}(t) &=&(1/\zeta )[-2i\lambda e^{-1/4\delta t}\left( e^{1/4\zeta t}-e^{-1/4\zeta t}\right) \sqrt{n+1}F_{n}]{,} \label{bb1} \\ C_{e,n}(t) &=&\left( 1/2\zeta \right) [ie^{-1/4\delta t}\left( e^{1/4\zeta t}\left( i\gamma +2\omega -i\zeta -2\omega _{0}\right) -e^{-1/4\zeta t}\left( i\gamma +2\omega +i\zeta -2\omega _{0}\right) \right) F_{n}]{,} \label{bb2}\end{aligned}$$where $\delta =\gamma +2i\omega (1+2n)$ and $\zeta =[\gamma (\gamma +4i(\omega _{0}-\omega ))-4({\omega }^{2}+\omega _{0}^{2})-16{\lambda }^{2}(1+n)+8\omega \omega _{0}]^{1/2}.$ However, when the coupling $f(t)$ is time dependent the solution of this system of equations is found only numerically. As well known, in the presence of decay rate $\gamma $ in the CPB the state of the whole CPB-NR system becomes mixed. In this case its description requires the use of the density operator $\hat{\rho}_{CN}$, which describes the entire system. To obtain the reduced  density matrix describing the CPB (NR) sub-system we must trace over variables of the NR (CPB) sub-system. For example, $$\hat{\rho}_{NR}=Tr_{CPB}(\hat{\rho}_{CN})=\sum\limits_{n}\sum\limits_{n^{\prime }}\left[ C_{e,n}(t)C_{e,n^{\prime }}^{\ast }(t)+C_{g,n}(t)C_{g,n^{\prime }}^{\ast }(t)\right] \left\vert n\right\rangle \left\langle n\prime \right\vert . \label{c2}$$ Entropy of sub-systems ====================== Recently, researchers have employed several methods to study the dynamical of entanglement [@l22; @l2; @l3; @k41; @k6]. As proved by Phoenix and Knight [@k4] the von Neumann entropy offers a quantitative measure of disorder of a system and of the purity of a quantum state. Such entropy, defined as $S_{NR(C)}=-Tr(\hat{\rho}_{N(C)}\ln \hat{\rho}_{N(C)})$, is a measure that is sensible to quantum entanglement of two interacting subsystems. The quantum dynamics described by the Eq. (\[b1\]) furnishes the CPB-NR entanglement and we will employ the von Neumann quantum entropy as a measure of the degree of entanglement. The entropy $S$ of a quantum system, when composed of two subsystems, obeys a theorem due to Araki and Lieb, which stablishes that: $\left\vert S_{CPB}-S_{NR}\right\vert \leq S\leq S_{CPB}+S_{NR}$; $S_{CPB}$ and $S_{NR}$ standing for the entropies of the subsystems. $S$ stands for the total entropy of CPB-NR system. One immediate consequence of the above inequality is that, if one prepares the entire system in a pure state at $t=0$, then both components of the whole system have the same entropy for the subsequent time evolution.* *So, when assuming our system initially in a pure and decoupled state the entropies of the CPB and NR become identical, namely, $S_{CPB}(t)=S_{NR}(t).$ Then, one only needs to calculate the quantum entropy of a subsystem to get its entanglement evolution. We obtain, from the Eqs. (\[c2\]) and $S_{NR}=-Tr(\hat{\rho}_{NR}\ln \hat{\rho}_{NR})$, $$S_{NR}(t)=-\left[ \wedge _{NR}^{+}(t)\ln (\wedge _{NR}^{+}(t))+\wedge _{NR}^{-}(t)\ln (\wedge _{NR}^{-}(t))\right] , \label{c3}$$ where, $$\wedge _{NR}^{\pm }(t)=\frac{1}{2}\left( 1\pm \sqrt{(\left\langle R_{1}|R_{1}\right\rangle -\left\langle R_{2}|R_{2}\right\rangle )^{2}+4\left\vert \left\langle R_{1}|R_{2}\right\rangle \right\vert ^{2}}\right) , \label{c4}$$ with  $\left\langle R_{1}|R_{1}\right\rangle =\sum_{n=0}^{\infty }\left\vert C_{e,n}(t)\right\vert ^{2},~$ $\left\langle R_{2}|R_{2}\right\rangle =\sum_{n=0}^{\infty }\left\vert C_{g,n+1}(t)\right\vert ^{2}$ and$\ \left\langle R_{1}|R_{2}\right\rangle =\left\langle R_{2}|R_{1}\right\rangle ^{\ast }=\sum_{n=0}^{\infty }C_{e,n+1}^{\ast }(t)C_{g,n+1}(t).$ We can now look at the time evolution of the NR entropy. We will assume the NR subsystem initially in an even “Schrödinger-cat" state. Firstly we consider the resonant case $(f(t)=0);$ the time evolution of the NR entropy with different decay rates $\gamma $ in the CPB**,** with** **$\omega =\omega _{0}=2000\lambda _{0}$ and the “cat"-state with $\alpha =5$, as shown in Figs. \[s4\](a), \[s4\](b), and \[s4\](c). In an ideal case the CPB decay rate vanishes. As displayed in Fig. \[s4\](a) the maximum value of the entropy of the NR is close to $\ln 2$. Just after the start of the CPB-NR interaction the entropy of the NR stabilizes at this value by a small interval and then recovers the oscillations as time goes on. The CPB sub-system is stable while standing in its ground state $\left\vert g\right\rangle $; but when lying in its excited state $\left\vert e\right\rangle ,$ various factors as spontaneous emission among others, imply its decay to the ground state. For a small decay rate (see Fig. [s4]{}(b) the maximum entanglement becomes significant only for large times. However, the increase of the decay rate produces a drastic change on the entanglement (see Fig. \[s4\](c), with a great reduction in their swings, leading the CPB to its ground state (zero entropy). This effect upon the entropy of the CPB also affects the entropy of the NR (Figs. \[s4\]). Secondly, we modify the previous case by including the presence of a detuning $(f(t)=\Delta \neq 0)$ to verify its influence opon our interacting system. We take the decay rate as $\gamma =0.05\lambda _{0}$, with $f(t)=\Delta =const$ and $\Delta \ll \omega _{0}$, $\omega $. As result the entanglement remains for long time as the value of $\Delta $ increases*,* as we see comparing Fig. \[s5\](a) with Fig. \[s4\](c); this event is accompanied by a diminution of the maximum entropy (see Figs \[s5\](a) and \[s5\](b)). When the detuning increases, the CPB transitions decreases (cf. Figs. \[inv5\]). \ \ Thirdly, we extend the detuning to the time dependent case, assuming $f(t)=c\sin (\omega \prime t)$, where $c$ and $\omega \prime $ are parameters of amplitude and frequency modulation of the NR with the condition $\omega \prime <c\ll \omega _{0}$*,* $\omega $. Comparing the Fig. \[s6\](a) with Fig. \[s5\](b) we see that the sinusoidal modulation does not favor the entanglement for long time. However, the frequency modulation turns the maxima of entanglement greater (see Fig. \[s6\]). Comparing Fig. \[s6\](a) with \[s6\](b) we see that when the frequency $\omega \prime $ grows the oscillations of the entropy decrease. Power Spectrum of the Entropy ============================= To get a better understanding of the entropy we have considered its power spectrum (**PS**). It consists of a frequency-dependent function, being real, positive, and constructed from the following Fourier transform [scully]{}$$PS(\varpi )=\frac{1}{\pi }\int_{0}^{\tau _{\max }}S_{NR}(t)\exp (i\varpi t)dt, \label{c5}$$where $\tau _{\max }=$ $\lambda _{0}t_{\max }$ stands for the maximum interaction interval in the plot $S_{NR}(t)$ versus $\lambda _{0}t$. The entropy PS is obtained from the above equation and plotted in Figs. [s4]{}(a), \[s4\](b) and \[s4\](c). As expected, the amplitude of oscillations of this PS is reduced in the presence of growing decay rates (see Figs. \[ps4\](a), \[ps4\](b) and \[ps4\](c) ); when we add a constant detuning $($ $f(t)=\Delta \neq 0)$ and a decay rate $\gamma =0.05\lambda _{0}$ we see in Figs. \[ps5\](a) and \[ps5\](b) the maximum value of the PS increasing when $\Delta $ also increases. In the case $f(t)=csin(\omega \prime t)$ the frequency of entropy PS is smoothly attenuated, with a peak around $\Omega =0.1,$ as shown in Fig. \[ps6\](a). When the frequency $\omega \prime $ increases the entropy PS is rapidly attenuated (cf. Figs. \[ps6\](a) and \[ps6\](b)). \ \ Excitation Inversion of the CPB =============================== The CPB excitation inversion, $I_{CPB}(t)$, is an important observable of two level systems. It is defined as the difference of the probabilities of finding this system in the excited and in the ground state; for the CPB it reads,$$I_{CPB}(t)=\sum\limits_{n=0}^{\infty }\left[ \left\vert C_{e,n}(t)\right\vert ^{2}-\left\vert C_{g,n+1}(t)\right\vert ^{2}\right] . \label{joao}$$ The Eq. (\[joao\]) allows us to look at the time evolution of the CPB excitation inversion. First, we assume the resonant case $(f(t)=0),$ for different values of the decay rate $\gamma $, with $\alpha =5$ and $\omega =\omega _{0}=2000\lambda _{0}$ as in Fig. \[inv4\]. Figs. \[inv4\](a), (b) and (c) exhibit identical collapse and revival, but with different amplitudes: the higher the decay rate, the lower the amplitude of oscillations of CPB excitation inversion. However, in the presence of a fixed detuning, with $f(t)=\Delta =const$ and $\Delta \ll \omega _{0},$ $\omega $, we see that CPB excitation inversion in Fig. \[inv5\](a) occurs only inside the interval $\tau =\lambda _{0}t\in $ $(15,30)$; differently, in Fig. \[inv5\](b) this event occurs inside the interval $\tau \in (50,75) $, with amplitude smaller than that in Fig. \[inv5\](a); this is the effect caused by a constant detuning upon the CPB excitation inversion. For a time dependent detuning, $f(t)=csin(\omega \prime t)$, the frequency of  the CPB excitation inversion accompanies the frequency $\omega \prime $, as shown in Figs. \[inv6\]. When we compare the Fig. \[inv6\](b) with Fig. \[inv4\](c), we see: if $\omega \prime $ increases, the interval of collapse of the excitation inversion also increases. However, looking at Figs. \[inv6\](a), \[inv6\](b) we see that the increasing of $\omega \prime $ as in Fig. \[inv6\](b) introduces equally spaced collapse intervals, accompanied by revivals modulated by the parameter $\omega \prime $ . Now, we compare the CPB excitation inversion with constant detuning $(f(t)=\Delta =const)$ and with a time dependent detuning ($f(t)=csin(\omega \prime t)$): looking at Figs. \[inv5\] we see the plots of excitation inversion showing neither collapses nor revivals, with exceptions of small regions exhibinting excitation inversion (Fig.\[inv5\](a)); in Fig \[inv5\](b) only a single such region appears. However, when considering a time dependent detuning (Fig. \[inv6\](b)), it nicely restitutes those collapses and revivals that appear in the resonant case (cf. Fig. \[inv4\](c)). \ \ Conclusion ========== We have considered a Hamiltonian model for a CPB-NR interacting system to study Entropy, its Power Spectrum and the CPB Excitation Inversion. These properties characterize the entangled state that describes this coupled system for various values of the parameters involved. We have included dissipation and assumed the NR initially in a Schrödinger “cat"-state and the CPB in excited state. We have also considered the following scenarios: (*i*) both subsystems in resonance (detuning $f=0$); (*ii*) off-resonance, with a constant detuning $(f=\Delta \neq 0)$, and (*iii*) with a time dependent detuning $(f(t)=csin(\omega \prime t))$. The results were discussed in the previous section. Concerning with the entropy we see that when the NR is initially in a Schrödinger “cat"-state, the entropy lasts longer than in an atom-field system, with the field initially in a coherent state (cf. Ref. [@l2]). 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--- abstract: 'The graph isomorphism (GI) problem is the computational problem of finding a permutation of vertices of a given graph $G_1$ that transforms $G_1$ to another given graph $G_2$ and preserves the adjacency. In this work, we propose a quantum algorithm to determine whether there exists such a permutation. To find such a permutation, we introduce isomorphic equivalent graphs of the given graphs to be tested. We proof that the GI problem of the equivalent graphs is equivalent to the GI problem of the given graphs. The idea of the algorithm is to determine whether there exists a permutation can transform the eigenvectors of the adjacency matrix of the equivalent graphs each other. The cost time of the algorithm is polynomial.' author: - Xi Li - Hanwu Chen title: The quantum algorithm for graph isomorphism problem --- [^1] Introduction ============ The GI problem has been heavily studied in computer science[@kobler2012graph]. Although GI problem for many special classes of graphs can be solved in polynomial time, and in practice graph isomorphism can often be solved efficiently[@mckay1981practical], the universal polynomial for GI is still open. The problem is not known to be solvable in polynomial time nor to be NP-complete, and therefore may be in the computational complexity class NP-intermediate. On the 11th of December 2015, L´aszl´o Babai proposed a quasi-polynomial algorithm on classical computer for GI problem other than Johnson graphs[@babai2016graph]. The GI problem is believed to be of comparable computational difficulty as well as integer factorization problem[@arora2009computational]. However, the integer factorization problem can be settled by Shor’s algorithm in polynomial time, but the efficient quantum algorithm for GI is not known. A number of researchers have considered the quantum physics-based algorithms for solving the graph isomorphism problem. In some of these algorithms, quantum systems droved by the Hamiltonians defined by the topology structure of the GI instance are settled[@rudolph2002constructing; @shiau2005s; @gamble2010two; @shiau2003physically]. Evolution results of the quantum system droved by different Hamiltonians imply that whether the two graphs are isomorphic. In the algorithms based on the continuous time quantum walk on graphs, the adjacency matrix of the associated graph is used to define the Hamiltonian $H(G)$ that drive the system. The state of system is changed by the unitary operator $U\left( G \right) = {e^{ - iH\left( G \right)t}}$. For the same initial state and the same sample time, if the measurement values of the systems droved by distinct Hamiltonians are equal, then the algorithm judges that the two graphs are isomorphic. The GI test algorithms based on discrete quantum are similar[@berry2011two]. However, both the continuous time quantum and the discrete time quantum walk on graphs are invalid for distinguishing the pairs of non-isomorphic strongly regular graphs with the same parameters even increasing the number of walkers or adding interacting between walkers[@smith2010k; @berry2011two]. Another kind of quantum algorithm for solving the GI problem are based the adiabatic quantum evolution. In these algorithms, every permutation in the symmetry group $S_n$ is encoded as a binary vector which corresponds a computational basis state[@gaitan2014graph; @hen2012solving; @tamascelli2014quantum]. Defining a time depended Hamiltonian $H(t)$. the $H(0)$ is a proper initial Hamiltonian whose ground state is easily preparing and the $H(T)$ is ending Hamiltonian whose ground state is a valid permutation for GI problem. Via adiabatic quantum evolution $T$ time, if the quantum system ending at a ground state corresponds a permutation, then the given graphs are isomorphic. These algorithms can distinguish non-isomorphism SRG with the same parameters. However, finding the time complexity of the algorithms is intricate since obtaining the energy gap of Hamiltonians is an open problem and the number of permutations needs to encode is $N!$ which is too large. In our scheme, the permutation is not directly checked one by one. We introduce a kind of graphs correspond the original given graphs, the GI problem of such kind of graphs is equivalent to the GI problem of the given graphs, hence we call it isomorphic equivalent graph. The lowest eigenvalue of the Hamiltonian defined by the adjacency matrix of isomorphic equivalent graph is simple, namely the ground state is a non-degenerate state. After we prepare the ground states of distinct Hamiltonians, we check that whether these ground states can transfer to each other by a permutation matrix. If yes, then the two original given graphs are isomorphic, otherwise they are non-isomorphic. We introduce an he altered Grover algorithm, and acquire such a matrix by this algorithm. Since quantum algorithm for the GI problem of pairs of SRGs with the same parameters are hard, we illustrate our algorithm via the instances of SRGs. This paper is organized as follows: the second section presents isomorphic conditions of isomorphic equivalent graphs. In the third section, the procedure of ground state of the equivalent graphs of SRG preparing is given. The fourth section discusses how to determine the transfer matrix between the ground states. The time complexity be presented in the fifth section, and the final section provides conclusions. Equivalent graph of isomorphism =============================== A graph, denoted as $G(V,E)$, consists of a vertex set $V$ and an edge set $E$. The set $E$ is a subset of $V\times V$, which implies the connection relationship between pairs of vertices in $V$. The connection relationship of $G$ is generally represented via the adjacency matrix $A$. It is a $N \times N$ real symmetric matrix, where $A_{jk} = 1$ if vertex $v_j$ and $v_k$ are connected otherwise $A_{jk}= 0$. For graphs with loops, the diagonal entry is the number of loop attached on that vertex and the degree is the sum of the number of neighbors and the diagonal entry. For loop-less graphs, the diagonal entry $A_{jj}$=0 and the number of neighbors of a vertex is known as its degree. For two given loop-less graphs $G_1$ and $G_2$, it is well known that $G_1$ is isomorphic to $G_2$ if and only if there exist a permutation matrix $P$ such that ${A_2} = P{A_1}{P^T}$ , where $A_1$ and $A_2$ are the adjacency matrices of $G_1$ and $G_2$ respectly. Now considering add a loop to every vertex of $G_1$ and $G_2$, the result graphs be denoted as $\widetilde{G_1}$ and $\widetilde{G_2}$. The correspond adjacency matrices are $A_1^ \prime {\rm{ = }}{A_1}{\rm{ + }}I$ and $A_2^\prime {\rm{ = }}{A_2}{\rm{ + }}I$. Adding the equal number of loops to every vertex does not change the adjacency of the original graph. Hence the isomorphism between $\widetilde{G_1}$ and $\widetilde{G_2}$ is equivalent to the isomorphism between $G_1$ and $G_2$. The isomorphism of $\widetilde{G_1}$ and $\widetilde{G_2}$ apparently be implied in the below theorem. Graphs $\widetilde{G_1}$ and $\widetilde{G_2}$are isomorphic if and only if there exists a permutation matrix $P$, such that equation $A_2^\prime = P {A_1^\prime}{P^T}$ satisfies. More operations can be executed on graph $\widetilde{G_1}$ and $\widetilde{G_2}$ such that the isomorphism between the resulting graphs imply the isomorphism between the original given graph $G_1$ and $G_2$. Now, choosing a pair vertices $v \in V( \widetilde{G_1} )$ and $w \in V( {\widetilde{G_2} } )$, deleting the loop of $v$ and $w$ in $\widetilde{G_1}$ and $\widetilde{G_2}$. The resulting spanning subgraph are denoted as $\widetilde{G_1^v}$ and $\widetilde{G_2^w}$ respectively. A similar theorem can be obtained. Graphs $\widetilde{G_1}$ and $\widetilde{G_2}$ are isomorphic if and only if there exist a pair of vertices $v$ and $w$ such that $\widetilde{G_1^v}$ and $\widetilde{G_2^w}$ are isomorphism. If $\widetilde{G_1}$ and $\widetilde{G_2}$ are isomorphic, then there exist a isomorphic mapping $f$ and a pair of vertices $v$ and $w$ such that $f$ maps $v$ to $w$,namely $$f:v \to w$$. Deleting the loop of $v$ and $w$ in$\widetilde{G_1}$ and $\widetilde{G_2}$ to obtain the spanning subgraph $\widetilde{G_1^v}$ and $\widetilde{G_2^w}$. Apprently, the map $f$ is aslo a isomorphic map from $\widetilde{G_1^v}$ to $\widetilde{G_2^w}$. Analogously, if graphs $\widetilde{G_1^v}$ and $\widetilde{G_2^w}$ are isomorphic, then there exists a isomorphic map $f$ such that $f:v \to w$. Adding a loop to $v$ and $w$, one will obtain graphs $\widetilde{G_1}$ and $\widetilde{G_2}$. Then the map $f$ is an isomorphic mapping from $\widetilde{G_2^w}$ to $\widetilde{G_2^w}$. $\widetilde{G_1^v}$ and $\widetilde{G_2^w}$ are isomorphic if and only if there exists a permutation matrix $Q$ such that $A_2^w = Q A_1^v{Q^T}$, where $A_1^v$ and $A_2^w$ are adjacency matrices of $\widetilde{G_2^w}$ and $\widetilde{G_2^w}$ respectively. By proper relabeling, one can give vertices $v$ and $w$ index $1$, then $$A_2^w = A_2^\prime - \left[ {\begin{array}{*{20}{c}} 1&0& \ldots &0\\ 0&0&{}&{}\\ \vdots &{}&{}&{}\\ 0&{}&{}&0 \end{array}} \right],$$ and $$A_1^v = A_1^\prime - \left[ {\begin{array}{*{20}{c}} 1&0& \ldots &0\\ 0&0&{}&{}\\ \vdots &{}&{}&{}\\ 0&{}&{}&0 \end{array}} \right].$$ Based on theorem 2, $\widetilde{G_1^v}$ and $\widetilde{G_2^w}$ are isomorphic then $\widetilde{G_1}$ and $\widetilde{G_2}$ are isomorphic. Hence, there exists a permutation matrix Q such that $A_2^\prime = QA_1^\prime{Q^T}$. this leads to $$\label{eq1} A_2^\prime - \left[ {\begin{array}{*{20}{c}} 1&0& \ldots &0\\ 0&0&{}&{}\\ \vdots &{}&{}&{}\\ 0&{}&{}&0 \end{array}} \right] = QA_1^\prime{Q^T} - Q\left[ {\begin{array}{*{20}{c}} 1&0& \ldots &0\\ 0&0&{}&{}\\ \vdots &{}&{}&{}\\ 0&{}&{}&0 \end{array}} \right]{Q^T}.$$ Analogously, if the $A_2^w = Q A_1^v{Q^T}$ valid, then $\widetilde{G_1}$ and $\widetilde{G_2}$ are isomorphic. by **Theorem 2**, $\widetilde{G_1^v}$ and $\widetilde{G_2^w}$ are isomorphic. Loop-less graphs $G_1$ and $G_2$ are isomorphic if and only if there exist a pair of vertices $v$ and $w$ such that $\widetilde{G_1^v}$ and $\widetilde{G_2^w}$ are isomorphism. From the above theorems, the isomorphism problem between $G_1$ and $G_2$ can be reduced to the isomorphism problem between $\widetilde{G_1^v}$ and $\widetilde{G_2^w}$. Hence, we call the graph $\widetilde{G_1^v}$ and $\widetilde{G_2^w}$ the isomorphic equivalent graphs of $G_1$ and $G_2$ respectively. In next sections, we will see that finding isomorphic permutation matrix between equivalent graphs is more facile than the original given graphs. We give an instance in Fig.(\[iso\_graph\]) The spectrum of isomorphic equivalent SRG ========================================= Since distinguishing non-isomorphic SRGs is hard by quantum algorithm, we will apply the algorithm to the GI problem of SRG at first. First of all, we will introduce the spectrum of equivalent SRG which is significant for the preparing a non-degenerate eigenvector. The connection relationship of a SRG satisfies[@cvetkovic1980spectra] (1). It is neither a complete graph nor an empty graph, (2). Any two adjacent vertices have $a$ common adjacent vertices (3). Any two non-adjacent vertices have $c$ common adjacent vertices If the number and degree of the SRG are $N$ and $k$ respectively, then it is labelled SRG with parameters $(N,k,a,c)$. From conditions (1) to (3), one can obtain the spectrum of the adjacent matrix of SRG, or spectrum of SRG. These are $k$, ${\lambda _1} = \frac{1}{2}\left( {a - c + \sqrt \Delta } \right)$ and ${\lambda _2} = \frac{1}{2}\left( {a - c - \sqrt \Delta } \right)$ with multiplicity 1, ${m_1} = \frac{1}{2}\left( {N - 1 + \frac{{\left( {N - 1} \right)\left( {c - a} \right) - 2k}}{{\sqrt \Delta }}} \right)$ and ${m_2} = \frac{1}{2}\left( {N - 1 - \frac{{\left( {N - 1} \right)\left( {c - a} \right) - 2k}}{{\sqrt \Delta }}} \right)$ respectively[@cvetkovic1980spectra], where $\Delta = {\left( {a - c} \right)^2} + 4\left( {k - c} \right)$. Hence, the SRGs with the same parameters have the same eigenvalues and multiples of the eigenvalues. $G_1$ and $G_2$ are SRGs, for $v \in \widetilde{G_1}$ and $w \in \widetilde{G_2}$, the adjacency matrices of $\widetilde{G_1}$ and $\widetilde{G_2}$ are $A_1^v = {A_1} + I - \left| v \right\rangle \left\langle v \right| $, $A_2^v = {A_2} + I - \left| v \right\rangle \left\langle v \right| $. Where $A_1$ and $A_2$ are adjacency matrices of $G_1$ and $G_2$ respectively, $\left|v\right>$ is a vector with only one non-zero component 1 in the index of vertex of $v$. For instance, the index of $v$ is $1$, then $$\left| v \right\rangle \left\langle v \right| = \left[ {\begin{array}{*{20}{c}} 1&0& \ldots &0\\ 0&0&{}&{}\\ \vdots &{}&{}&{}\\ 0&{}&{}&0 \end{array}} \right]$$. From literature [@cvetkovic1997eigenspaces], the characteristic polynomial of $A_1^v$ is $$\label{eigenp} {P_v}\left( x \right) = P\left( x \right)\left( {1 + \sum\limits_{k = 1}^m {\frac{{\alpha _{kj}^2}}{{x - {\mu _k}}}} } \right).$$ Where the $\alpha_{kj}$, $P(x)$ and $\mu_k=\lambda_k+1$ are the graph angle, characteristic polynomial and eigenvalues of ${A_1}+I$ respectively. For SRG, ${\alpha _{kj}} = \sqrt {\frac{{{m_k}}}{N}}$. From equation (1), the least eigenvalue is simple, it corresponds a unique eigenvector. Oppositely, the least eigenvalue of a SRG is multiple, and it corresponds multiple eigenvectors. The adjacency matrix with simple eigenvalue is critical for our GI algorithm. We consider to define Hamiltonian via the adjacency matrix whose least eigenvalue is simple and to prepare this eigenvector. For general graph, the similar operation still can produce a simple least eigenvalue. This conclusion can be clearly obtained from Eq.\[eigenp\] For a arbitrary graph Graph $G$, the least eigenvalue of graph $\widetilde{G^v}$ is simple. Frame of the isomorphic algorithm ================================= In this section, the frame of the isomorphic algorithm is presented. The sub-procedures in the algorithm are introcudes in next sections. Now, we are given two SRGs $G_1$ and $G_2$ with the same parameters. We fix a vertex $v$ of $G_1$, and let $w$ run over all vertices of $G_2$. If $G_1$ and $G_2$ are isomorphic, then there exists a vertex $w$ in $G_2$ such that ${A_1^v} = Q {A_2^w}{Q^T}$. The ground states of ${A_1^v}$ and ${A_2^w}$ are denoted as $\left| \varphi_1 \right>$ and $\left| \varphi_2 \right>$ , both of them correspond the least eigenvalue $\mu_{min}$. It provides that $${A_2^w}\left| {{\varphi _2}} \right\rangle = {\mu _{\min }}\left| {{\varphi _2}} \right\rangle.$$ If the two give SRGs are isomorphic, then $${A_1^w}{Q^T}\left| {{\varphi _2}} \right\rangle = {\lambda _{\min }}{Q^T}\left| {{\varphi _2}} \right\rangle.$$ Hence, if the two give SRGs are isomorphic, then the eigenvectors of ${A_1^v}$ and $A_2^w$ can be transform by a permutation matrix $Q^T$. If the eigenvectors are degenerate, the Eq.(3) is general not valid in a quantum system. That’s why we must perform our algorithm on isomorphic equivalent graphs. For one turn, namely a specific vertex $w \in G_2$, we illustrate the steps in Fig.1. For the whole algorithm, we may do this this procedure $N$ times at worst. We list the procedures blow and also give Fig.(\[frame\]) to illustrate. **Procedure (1)**. Preparing the ground state of ${A_1^v}$ and $A_2^w$ by adiabatic quantum algorithm. **Procedure (2)**. Finding the linear operator $Q^k$ that can transform the two ground states each other by an altered Grover’s algorithm. **Procedure (3)**. Checking that whether $Q^k$ is a permutation matrix. The eigenstate preparing via adiabatic quantum evolution ======================================================== The adiabatic quantum algorithm can be realized on quantum computer. The adiabatic quantum algorithm is usually used for combinational optimization problem. In this paper, we apply it to prepare the eigenvector of the Hamiltonian defined by the adjacency matrix. The time depend Hamiltonian of adiabatic quantum algorithm has the format[@farhifarhi] $$H\left( t \right) = \left( {1 - \frac{t}{T}} \right){H_i} + \left( {\frac{t}{T}} \right){H_P}$$ $H_i$ is the initial Hamiltonian, $H_P$ is the ending Hamiltonian which is defined relying on specific problems. Here, we define that $${H_P} = {A_1} + I - \xi \left| v \right\rangle \left\langle v \right|,$$ or $${H_P} = {A_2} + I - \xi \left| w \right\rangle \left\langle w \right|,$$ Let $s=\frac{t}{T}$, the eigenvalues of $H(t)$ are $\mu_1(s)<,\ldots,\mu_N(s)$, and the eigenvectors are $\left|\mu_1(s)\right>,\ldots,\left|\mu_1(s)\right>$. The evolution time satisfies $$T\geq\frac{\epsilon}{g_{min}^2}$$ where $${g_{\min }} = \mathop {\min }\limits_{0 \le t \le 1} \left( {\mu \left( 1 \right) - \mu \left( 0 \right)} \right),$$ and $$\varepsilon = \mathop {\max }\limits_{0 \le t \le 1} \left| {\left\langle {\mu \left( 1 \right)} \right|\frac{{dH\left( s \right)}}{{ds}}\left| {\mu \left( 0 \right)} \right\rangle } \right|.$$ When $\xi \ll 1$, $\mu \left( 1 \right) \approx \mathop {\min }\limits_j \left| {{\lambda _j} + 1} \right|$. Choosing a proper initial Hamiltonian that $\mu(0)=c$, such that c is far less than $\mu(1)$. Since $\varepsilon \geq 1$, the evolution time reaches $$T \ge \frac{1}{{{{\left( {\mathop {\min }\limits_j \left| {{\lambda _j} + 1} \right|} \right)}^2}}}.$$ From the above formula, the evolution time is not very long. Hence, in the analysis of time complexity, the preparing time of eigenvectors can be ignored. Note that the initial eigenvector mustn’t be the equal superposition state, since equal superposition state is approximately equal to another eigenvector of $H_P$ when The parameter $\xi$ be taken to a small enough value. Finding permutation via an altered Grover algorithm =================================================== In the previous section, we have illustrated that the procedure of preparing the ground state of adjacency matrix of isomorphic equivalent graph. Now we have such two ground states, how to check whether there exists a permutation matrix that can transform them each other. Our approach isn’t checking every permutation in the symmetric group $S_n$, but directly find what a unitary matrix can transform one eigenvector to another one. We adopt an altered Grover’s algorithm to determine that unitary transform. The Grover’s original algorithm can be found in literature [@grover1996fast], or one can find the algorithm version described by unitary matrix in literature [@williams2010explorations]. we describe the altered Grover’s algorithm by the way of the latter. Now we have the two eigenvectors $\left|\varphi_1\right>$ and $\left|\varphi_2\right>$, no matter if the two graphs are isomorphic $\left<\varphi_1|\varphi_2\right>\neq 0$. By using the altered Grover’s algorithm, $\left|\varphi_1\right>$($\left|\varphi_2\right>$) can be transformed to $\left|\varphi_2\right>$($\left|\varphi_1\right>$). The algorithm procedures are listed below: **Step (1)**. Given two oracles, construct the iterative operator $Q = - U{{\bf{1}}_s}{U^\dag}{{\bf{1}}_t}$, where $\left|s\right>=\left|\varphi_1\right>$ is the initial state, $\left|s\right>=\left|\varphi_2\right>$ is the target state, ${{\bf{1}}_s} = I - 2\left| s \right\rangle \left\langle s \right|$ and ${{\bf{1}}_t} = I - 2\left| t \right\rangle \left\langle t \right|$ are constructed relied on the oracles, $U$ is a unitary operator satisfies that $\left\langle {{\varphi _1}} \right|U\left| {{\varphi _2}} \right\rangle \ne 0$. **Step (2)**. Iterative execution the operator $k = \frac{\pi }{4}\sqrt N$ times, namaly $\left| \phi \right\rangle = {Q^k}U\left| s \right\rangle $. **Step (3)**. Checking that whether $\left\langle {{\varphi _2}} \right.\left| \phi \right\rangle \approx 1$ is valid, if yes, turn to Step (4), otherwise, turn to Step (3). **Step (4)**. Testing whether the operator $Q^k$ is a permutation matrix. If yes, then the given two graphs are isomorphic, otherwise the given graphs are non-isomorphic. Although we have prepared the two ground states in some format Qbit, we don’t know what they are. Hence, we need two oracles in our algorithm and the original Grover’s algorithm only needs one, since we call it the altered Grover’s algorithm. Checking whether a matrix is a permutation can be realized by quantum algorithm just involved the computational basis state. We first need prepare a group of computational basis $\left|j\right>(j=0,...,N-1)$, which can be written is format of vector $$\left| 0 \right\rangle = \left[ {\begin{array}{*{20}{c}} 1\\ 0\\ \vdots \\ 0 \end{array}} \right],\left| 1 \right\rangle = \left[ {\begin{array}{*{20}{c}} 0\\ 1\\ \vdots \\ 0 \end{array}} \right], \ldots ,\left| {N - 1} \right\rangle = \left[ {\begin{array}{*{20}{c}} 0\\ 0\\ \vdots \\ 1 \end{array}} \right]$$. Then, we let the operator $Q^k$ acts on every basis vector and measure the result. If the result vectors are all basis states and there are no one pair of them are equal, then the matrix is a permutation. In this manner, we will spend $N$ time since the number of computational basis vector is $N$. Checking all pairs of basis vectors will cost $O(N^2)$. Time complexity analysis ======================== The algorithm contains three main steps. In the first step, we need prepare $N+1$ ground statesat most. Since the time of preparing one ground state is far less than the time of other steps. In the second step, one need to transform the two ground states by the Grover algorithm, for one round the time is $k = \frac{\pi }{4}\sqrt N$. In the third step, we need check that if the matrix $Q^k$ is a permutation, we need cost $N^2$ time in one turn. The whole procedure, we will do $N$ turn for the worst case that we will check all vertices in the second given graph. So worst time complexity is $O(N^3)$. Conclusion ========== In this work, we put forward a quantum algorithm for GI problem. The time complexity of the algorithm is polynomial. We introduce the isomorphic equivalent graph, and present several theorems for GI test. Via that kind of graph, we transform the GI problem given graphs to GI problem of isomorphic equivalent graphs. By the transformation, the least eigenvalue of the adjacency matrix becomes simple and the corresponding ground state is non-degenerate. That ground state can be efficaciously prepared in a short time by adiabatic quantum evolution. Then, by using the altered Grover algorithm, we can find the transformation matrix between the two ground states. If the given two graph are just co-spectrum but not isomorphic, then that matrix is no longer a permutation matrix. In the original Grover’s algorithm, one needs an oracle, but in the altered Grover’s algorithm we need two oracles. Theoretically, if we can prepare the eigenvector, then the oracle can be made. The work of oracle making is not the main part of our algorithm just as in Grover algorithm. [18]{} \[1\][\#1]{} \[1\][`#1`]{} urlstyle \[1\][doi: \#1]{} Johannes Kobler, Uwe Sch[ö]{}ning, and Jacobo Tor[á]{}n. *The graph isomorphism problem: its structural complexity*. Springer Science &amp; Business Media, 2012. Brendan D McKay et al. Practical graph isomorphism. 1981. L[á]{}szl[ó]{} Babai. Graph isomorphism in quasipolynomial time. In *Proceedings of the forty-eighth annual ACM symposium on Theory of Computing*, pages 684–697. ACM, 2016. Sanjeev Arora and Boaz Barak. *Computational complexity: a modern approach*. Cambridge University Press, 2009. Terry Rudolph. Constructing physically intuitive graph invariants. *arXiv preprint quant-ph/0206068*, 2002. SY Shiau. S.-y. shiau, r. joynt, and sn coppersmith, quantum inf. comput. 5, 492 (2005). *Quantum Inf. Comput.*, 5:0 492, 2005. John King Gamble, Mark Friesen, Dong Zhou, Robert Joynt, and SN Coppersmith. Two-particle quantum walks applied to the graph isomorphism problem. *Physical Review A*, 810 (5):0 052313, 2010. Shiue-yuan Shiau, Robert Joynt, and Susan N Coppersmith. Physically-motivated dynamical algorithms for the graph isomorphism problem. *arXiv preprint quant-ph/0312170*, 2003. Scott D Berry and Jingbo B Wang. Two-particle quantum walks: Entanglement and graph isomorphism testing. *Physical Review A*, 830 (4):0 042317, 2011. Jamie Smith. k-boson quantum walks do not distinguish arbitrary graphs. *arXiv preprint arXiv:1004.0206*, 2010. Frank Gaitan and Lane Clark. Graph isomorphism and adiabatic quantum computing. *Physical Review A*, 890 (2):0 022342, 2014. Itay Hen and AP Young. Solving the graph-isomorphism problem with a quantum annealer. *Physical Review A*, 860 (4):0 042310, 2012. Dario Tamascelli and Luca Zanetti. A quantum-walk-inspired adiabatic algorithm for solving graph isomorphism problems. *Journal of Physics A: Mathematical and Theoretical*, 470 (32):0 325302, 2014. Drago[š]{} M Cvetkovi[ć]{}, Michael Doob, and Horst Sachs. *Spectra of graphs: theory and application*, volume 87. Academic Pr, 1980. Dragos Cvetkovic, Drago[š]{} M Cvetkovi[ć]{}, Peter Rowlinson, and Slobodan Simic. *Eigenspaces of graphs*, volume 66. Cambridge University Press, 1997. E Farhi. E. farhi, j. goldstone, s. gutmann, and m. sipser, quantum computation by adiabatic evolution, arxiv: quant-ph/0001106. *Quantum computation by adiabatic evolution*. Lov K Grover. A fast quantum mechanical algorithm for database search. In *Proceedings of the twenty-eighth annual ACM symposium on Theory of computing*, pages 212–219. ACM, 1996. Colin P Williams. *Explorations in quantum computing*. Springer Science &amp; Business Media, 2010. [^1]: A footnote to the article title

--- abstract: 'In this work, it is presented a characterization of star condition for a $C^1$ vector field based on Lyapunov functions. It is obtained conditions to strong homogeneity for singular sets by using the notion of infinitesimal Lyapunov functions. As an application we obtain some results related to singular hyperbolic sets for flows.' author: - Luciana Salgado title: 'Star Flows: a characterization via Lyapunov functions' --- [^1] Introduction and statement of results {#sec:int-stat} ===================================== Star systems has been studied by many renowned researchers, among them R. Mañé and S. Liao, whom many years ago used it in order to prove the famous *stability conjecture* from Palis and Smale. For more details about star systems, see for instance [@Liao1980],[@Man82],[@Palis87],[@Man88],[@shi-gan-wen2014],[@GanWen2006],[@ArbMo2013]. \[def:star-flow\] A $C^1$ vector field $X$ (or its flow $X_t$) is said to be *star* if it cannot be $C^1$-approximated by ones exhibiting nonhyperbolic periodic orbits. The definition for diffeomorphisms is analogous. In the case of diffeomorphisms, $C^1$ structural stability is equivalent to star condition (and equivalent to Axiom A plus no cycle), see e.g. [@Man88]. However, in the case of flows, the situation is much more complex. In fact, in absence of singularities, Gan and Wen [@GanWen2006] proved that nonsingular star flows satisfy Axiom A and no cycle condition. But, in the singular setting, this is no longer true. One of the most emblematic example that a star flow does not satisfy Axiom A is the geometric Lorenz Attractor [@Guck76; @Lo63]. From this, remained the question about singular star vector fields. \[ex:geomLorenz\] The geometric Lorenz attractor is an emblematic example of star flow. see Figure \[fig:Lorenz\]. ![The geometric Lorenz flow.[]{data-label="fig:Lorenz"}](L3D){width="6cm"} In an attempt to study the behaviour of robust singular attractors like Lorenz ones, Morales, Pacífico and Pujals in [@MPP99] defined the so called *singular hyperbolic systems*. Many researchers have worked about this notion in order to understand it as an extension of the hyperbolic theory for invariant sets for flows which are not (uniformly) hyperbolic, but which have some robust properties, certain kind of weaker hyperbolicity and also admit singularities. The authors in [@MPP99] proved that, $C^1$-generically in dimension three, chain recurrence classes are singular hyperbolic. Without the generic assumption this result does not hold, as we can see in [@BaMo14]. In [@shi-gan-wen2014], Gan, Shi and Wen, proved that if a chain recurrent class of a star flow $X$ has homogeneous index for singularities (the same dimension of stable manifold), then this is a singular hyperbolic set of $X$, in any dimension. Moreover, they proved that, $C^1$-generically in dimension four, the chain recurrent set of a star flow is singular hyperbolic. In [@MPP04], it has been proved that every robustly transitive singular set for a three dimensional flow is a partially hyperbolic attractor or repeller and the singularities in this set must be Lorenz-like. In [@GanLiWen2005], Gan, Li and Wen extended the result in [@MPP04] to higher dimension assuming that the set is also strongly homogeneous. We recall that a vector field $X$ is said to be *strongly homogeneous of index $0 \leq {\operatorname{ind}}(\Lambda) \leq n-1$* over a set $\Lambda$ whether it cannot be $C^1$-approximated by one which has some hyperbolic periodic orbit of index different of ${\operatorname{ind}}(\Lambda)$ in a neighborhood $U$ of $\Lambda$. Here, the index ${\operatorname{ind}}(\cdot)$ means the dimension of the contracting subbundle from the hyperbolic splitting of a hyperbolic periodic orbit . We recall that a compact invariant set $\Lambda$ is *robustly transitive* for a vector field $X$ if there exist a neighborhood $U$ of $\Lambda$ and a neighborhood ${{\mathcal U}}\in {\mathfrak{X}^{1}(M)}$ of $X$ such that, for every $Y \in {{\mathcal U}}$, the maximal invariant set $\Lambda_Y = \cap_{t \in {{\mathbb R}}} X_t(U)$ is contained in the interior of $U$ and is non-trivially (not a single orbit) transitive. Many researchers believed that, at least generically, singular star flows should be singular hyperbolic ones. There are many conjectures about this and various results involving the most varied assumptions, see for instance, [@ArbMo2013; @GanLiWen2005; @shi-gan-wen2014]. Another point of view on the influence of the singularities emerged in the recent work of Bonatti and da Luz [@BodL17], where the authors have defined *multisingular hyperbolicity* which admit singularities of different indexes in the same recurrent class, and have shown that this kind of hyperbolicity implies star condition. It is easy to see that the classical singular hyperbolic flows (central dimension $2$ or the full dimension of central subbundle) are a particular case of the multisingular ones. Moreover, they proved that the converse holds on the $C^1$ generic assumption. In this paper, we prove a characterization of star condition for flows via ${\EuScript{J}}$-algebra of Potapov (infinitesimal Lyapunov functions) [@Pota60; @Pota79; @Wojtk01]. We also show a relation between ${\EuScript{J}}$-algebra and strong homogeneity. Then, we apply this to obtain some results about singular hyperbolic systems. Recall that a pseudo-euclidean space is a real finite-dimensional vector space endowed with a non-degenerate quadratic form (in case of an euclidean space, a positive definite one). The ${\EuScript{J}}$-algebra here means a pseudo-euclidean structure given by a $C^1$ non-degenerate quadratic form ${\EuScript{J}}$, defined on $\Lambda$, which generates positive and negative cones with maximal dimension $p$ and $q$, respectively, with $p + q = \dim (M)$. The maximal dimension of a cone in $T_xM$ is the maximal dimension of the subspaces contained in there. This algebraic/geometric approach has been very useful in the study of weak and uniform hyperbolicity, see [@BurnKatok94], [@lewow80], [@Wojtk01]. In [@ArSal2012], this author jointly with V. Araújo, obtained characterizations of partial and singular/sectional hyperbolicity based on ${\EuScript{J}}$-algebra. In [@ArSal2015], the same authors proved an equivalence between dominated splittings for the flow and dominated splittings for the $k$-th exterior powers of the tangent cocycle. More results relating geometric and algebraic features of singular hyperbolicity can be viewed in [@ArSal2012], [@ArSal2015], [@SalgCoelho2017], for the classical sectional and singular hyperbolicity definitions. Also see [@Salg2016] for singular hyperbolicity in a broad sense involving sectional expansion of intermediate dimensions between two and the full dimension of the central subbundle. Our main result (Theorem A) rounds about of the so called *star flows* and its relation with infinitesimal Lyapunov functions. We recall that a point $p \in M$ is said to be a $C^1$ preperiodic point of $X$ if for any $C^1$ neighborhood $\mathcal{V}$ of $X$ and any neighborhood $U \subset M$ of $p$, there is $g\in \mathcal{V}$ and $q \in U$ such that $q \in {\operatorname{Per}}(g)$. We denote this set by $P_*(X)$, and it is easy to see that it is closed and $X$-invariant. We can assume that the periodic points are hyperbolic and we can define a $C^1 i$-preperiodic point $p$ of $X$, $0 \leq i \leq d$, if there are sequences $X_n$ of flows and $p_n$ of periodic points of $X_n$ with index $i$ such that $$\begin{aligned} \lim\limits_{n \to \infty} X_n = X \ \textrm{and} \ \lim\limits_{n \to \infty} p_n = p.\end{aligned}$$ We denote by $P_*^i(X)$ the set of $C^1 i$-preperiodic of $X$ and, then, $P_*(X) = \cup_{i=0}^{d} P_*^i(X)$. Precisely, it is proved the following. \[cor:star-flow\] A vector field $X \in {\mathfrak{X}^{1}(M)}$ is star if, and only if, satisfies all of the next properties: 1. there is a neighborhood $U$ of $P_*(X)$ and a field of quadratic forms ${\EuScript{J}}$ with index $0 \leq {\operatorname{ind}}\leq \dim(M) - 1$ defined on the preperiodic set $P_*(X)$, $C^1$ along the flow direction over each preperiodic orbit, such that $X$ is strictly ${\EuScript{J}}$-separated on every $p \in P_*(X)$; 2. for every $\sigma \in {\mathrm{Sing}}(X\vert_U)$ and $\forall v \in T_\sigma M, {\EuScript{J}}'(v) > 0$; 3. the linear Poincaré flow $P^t$ associated to each preperiodic orbit $\gamma$ of $X\vert_U$ is strictly ${\EuScript{J}}$-monotone. The definitions concerning the quadratic forms are given in the next section. Let us explain why we are not requiring strict monotonicity in a whole nonsingular neighborhood of $X$. In fact, under certain assumptions, if we have strict monotonicity over any compact invariant nonsingular subset then it is a hyperbolic subset (see [@ArSal2012 Theorem D]). The idea here is to extend the characterization for any star flow, since it is already known the existence of star flows which are not neither uniformly hyperbolic nor singular/sectional hyperbolic (see [@BodL17]). Now, we present the definition of strong homogeneity. \[def:str-hom-set\] We say that a set $\Lambda$ is strongly homogeneous of index ${\operatorname{ind}}$ for a flow $X_t$, if there exist neighborhoods $U$ of $\Lambda$ and ${\EuScript{U}}$ of $X$ such that all periodic orbits in $U$ with respect to any flow in ${\EuScript{U}}$ have index ${\operatorname{ind}}$. The second result guarantees that a compact connected invariant set for a flow is strongly homogeneous under the existence of a field of non-degenerate quadratic forms ${\EuScript{J}}$ defined on a neighborhood of this set such a way that the flow derivative $DX_t$ keeps positive cones $C_+(x) : = \{0\} \cup \{ v \in T_xM ; {\EuScript{J}}(x) v > 0\}$ invariants, i.e., $DX_t(C_+(x)) \subset C_+(X_t(x))$, for all $t>0$, $x \in \Lambda$ and the projected quadratic forms on the normal bundle, with respect to the flow direction, are monotonic functions. \[mthm:strong-homog-equiv\] A compact invariant set $\Lambda$ for $X \in {\mathfrak{X}^{1}(M)}$ is strongly homogeneous with index ${\operatorname{ind}}$ if and only if there is a neighborhood $U\subset M$ of $\Lambda$ and a continuous field of non-degenerate quadratic forms ${\EuScript{J}}$ on $U$ with fixed index ${\operatorname{ind}}({\EuScript{J}}) = {\operatorname{ind}}$ such that the preperiodic set $P_*(X)$ of $X\vert_{U}$ is strictly ${\EuScript{J}}$-separated and the associated linear Poincaré flow $P^t$ is strictly ${\EuScript{J}}$-monotone on $P_*(X\vert_{U})$. If $\Lambda$ is the maximal invariant set of a trapping region $U$ and we require, in addition, that the field direction must be inside the non-positive cone, we obtain an equivalence between the existence of such a quadratic forms and singular hyperbolicity, as in [@ArSal2012 Theorem D]. Note that in the last result monotonicity is only required on preperiodic orbits. If we require it over any nonsingular compact invariant subsets in $\Lambda$, it is possible to evaluate the index of a singularity, once it is accumulated by regular orbits. \[mcor:strong-homog\] Let $\Lambda$ be a maximal invariant set of a neighborhood $U$ for $X \in {\mathfrak{X}^{1}(M)}$. Then, $\Lambda$ is strongly homogeneous with index ${\operatorname{ind}}$ and ${\operatorname{ind}}(\sigma) \geq {\operatorname{ind}}$ for all $\sigma \in {\mathrm{Sing}}(X\vert_{\Lambda})$ if there is a field of non-degenerate $C^1$ quadratic forms ${\EuScript{J}}$ on $U$ with index ${\operatorname{ind}}({\EuScript{J}}) = {\operatorname{ind}}$ such that $X$ is strictly ${\EuScript{J}}$-separated, the associated linear Poincaré flow $P^t$ is strictly ${\EuScript{J}}$-monotone on every compact invariant nonsingular subset $K$ of $U$ and for every $\sigma \in {\mathrm{Sing}}(X\vert_U)$ and $\forall v \in T_\sigma M, {\EuScript{J}}'(v) > 0$. We may ask if the converse of Corollary \[mcor:strong-homog\] is valid. But, just by supposing strongly homogeneity, we could not obtain the field of quadratic forms, because we need some kind of decomposition on the tangent bundle to create the cones. \[mthm:strong-homog-to-quad\] Let $\Lambda$ be a compact invariant set whose singularities are hyperbolic (if any) and accumulated by regular orbits for a $C^1$ vector field $X$, which is strongly homogeneous with index ${\operatorname{ind}}$ and ${\operatorname{ind}}(\sigma) > {\operatorname{ind}}$ for all $\sigma \in {\mathrm{Sing}}(X\vert_{\Lambda})$. Then, there exists a field of non-degenerate quadratic forms ${\EuScript{J}}$ on $\Lambda$ with index ${\operatorname{ind}}({\EuScript{J}}) = {\operatorname{ind}}(\Lambda)$ for which $X$ is ${\EuScript{J}}$-separated and the associated linear Poincaré flow $P^t$ is strictly ${\EuScript{J}}$-monotone on every compact invariant nonsingular subset $\Gamma$ of $\Lambda$. As an application, we obtain some results about partial hyperbolicity for robustly transitive strongly homogeneous singular sets of [@GanLiWen2005] and transitive set of [@ArbMo2013]. The text is organized as follow. In the first section, it is given the main definitions and stated the main results. In second section, it is presented the main tools by using the notion of ${\EuScript{J}}$-algebra of Potapov. In third section, it is given some applications concerning singular hyperbolicity. In fourth section is proved the main theorems. Some definitions and auxiliary results {#sec:prelim-definit} ====================================== Now, we give some definitions. Let $M$ be a connected compact finite $d$-dimensional manifold, $d \geq 3$, without boundary, together with a flow $X_t : M \to M, t \in \mathbb{R}$ generated by a $C^1$ vector field $X: M \to TM$. An *invariant set* $\Lambda$ for the flow of $X$ is a subset of $M$ which satisfies $X_t(\Lambda)=\Lambda$ for all $t\in{{\mathbb R}}$. A *trapping region* $U$ for a flow $X_t$ is an open subset of the manifold $M$ which satisfies: $X_t(U)$ is contained in $U$ for all $t>0$; and there exists $T>0$ such that $\overline{X_t(U)} $ is contained in the interior of $U$ for all $t>T$. The maximal invariant set $\Lambda_X(U):= \cap_{t \geq 0} X_t(U)$ of $U$ is called an *attracting set*. An attracting set for $X$ which is transitive is called an *attractor* for $X$. A *repeller* for $X$ is an attractor for $-X$. We say that a set $\Lambda$ is *Lyapunov stable* if for every neighborhood $U$ of $\Lambda$ there is another one $V \subset U$ such that every point $p \in V$ has its forward orbit contained in $U$. A *singularity* for the vector field $X$ is a point $\sigma\in M$ such that $X(\sigma)=0$ or, equivalently, $X_t(\sigma)=\sigma$ for all $t \in {{\mathbb R}}$. The set formed by singularities is the *singular set of $X$* denoted ${\mathrm{Sing}}(X)$ and ${\operatorname{Per}}(X)$ is the set of periodic points of $X$. We say that a *singularity is hyperbolic* if the eigenvalues of the derivative $DX(\sigma)$ of the vector field at the singularity $\sigma$ have nonzero real part. The set of critical elements of $X$ is the union of the singularities and the periodic orbits of $X$, and will be denoted by ${\operatorname{Crit}}(X)$. We recall that an hyperbolic set $\Lambda$ for a flow $X_t$ is an invariant subset of $M$ with a decomposition $T_\Lambda M= E^s\oplus E^X \oplus E^u$ of the tangent bundle over $\Lambda$ which is a continuous splitting, where $E^X$ is the direction of the vector field, the subbundles are invariant under the derivative $DX_t$ of the flow $$\begin{aligned} DX_t\cdot E^*_x=E^*_{X_t(x)},\quad x\in\Lambda, \quad t\in{{\mathbb R}},\quad *=s,X,u;\end{aligned}$$ $E^s$ is uniformly contracted by $DX_t$ and $E^u$ is uniformly expanded: there are $K,\lambda>0$ so that $$\begin{aligned} \label{eq:def-hyperbolic} \|DX_t\mid_{E^s_x}\|\le K e^{-\lambda t}, \quad \|DX_{-t} \mid_{E^u_x}\|\le K e^{-\lambda t}, \quad x\in\Lambda, \quad t\in{{\mathbb R}}.\end{aligned}$$ We say that a point $x \in M$ is [*nonwandering*]{} for $X$ provided for every neighborhood $U$ of $x$ there is $t > 0$ such that $X_t(x) \cap U \neq \emptyset$, i.e., there exists a point $y \in U$ with $X_t(y) \in U$. We denote by $\Omega(X)$ the non-wandering set of $X$. An ${\varepsilon}$-chain from $x_0$ to $x_l$ for $X$ is a sequence $\{x_0, x_1, \cdots, x_l\}$ such that for all $0 \leq j \leq l$, the distance $d(x_{j-1}, x_j) < {\varepsilon}$. We define the [*chain recurrent set*]{} of $X$ by $R(X) = \{x \in M; \textrm{there \ is \ an} \ {\varepsilon}-\textrm{chain \ from} \ x \ \textrm{to} \ x\}$. We say that two points are [*chain equivalent*]{} provided, given ${\varepsilon}> 0$, there is an ${\varepsilon}$-chain from $x$ to $y$ and from $y$ to $x$. It is known that this is an equivalence relation, the equivalence classes are called [*chain components*]{} of $R(X)$ and, for flows, the components are actually the connected components of $R(X)$. If $X$ admits a single chain component on an invariant set $\Lambda$, we say that $X$ is [*chain transitive*]{} on $\Lambda$. See [@Rob99], for instance. From $C^1$ Pugh’s closing lemma we have $\Omega(X) \subset P_*(X) \subset R(X)$. \[rmk:wen-preperiod\] As proved by Wen [@wen00], $C^1$ preperiodic sets do not explode under $C^1$ perturbations, i.e., for any $0 \leq i \leq d$ and for any neighborhood $U$ of $P_*(X)$, there is a $C^1$ neighborhood $\mathcal{V}$ of $X$ such that $P_*(Y) \subset U$, for any $Y \in \mathcal{V}$. Indeed, for example in [@shub87], we can see that the recurrent set do not explode under $C^0$ perturbations, i.e., if $U$ is a neighborhood of $R(X)$ there is a neighborhood $\mathcal{V}$ of $X$ such that $R(Y) \subset U$ for all $Y \in \mathcal{V}$. Next, we explain the kind of hyperbolicities we are dealing. \[def1\] A *dominated splitting* over a compact invariant set $\Lambda$ of $X$ is a continuous $DX_t$-invariant splitting $T_{\Lambda}M = E \oplus F$ with $E_x \neq \{0\}$, $F_x \neq \{0\}$ for every $x \in \Lambda$ and such that there are positive constants $K, \lambda$ satisfying $$\begin{aligned} \label{eq:def-dom-split} \|DX_t|_{E_x}\|\cdot\|DX_{-t}|_{F_{X_t(x)}}\|<Ke^{-{\lambda}t}, \ \textrm{for all} \ x \in \Lambda, \ \textrm{and all} \,\,t> 0. \end{aligned}$$ A compact invariant set $\Lambda$ is said to be *partially hyperbolic* if it exhibits a dominated splitting $T_{\Lambda}M = E \oplus F$ such that subbundle $E$ is uniformly contracted. In this case $F$ is the *central subbundle* of $\Lambda$. A compact invariant set $\Lambda$ is said to be *singular-hyperbolic* if it is partially hyperbolic and the action of the tangent cocycle expands volume along the central subbundle, i.e., $$\begin{aligned} \label{eq:def-vol-exp} \vert \det (DX_t\vert_{F_x}) \vert > C e^{{\lambda}t}, \forall t>0, \ \forall \ x \in \Lambda. \end{aligned}$$ The following definition was given as a particular case of singular hyperbolicity. \[def:sec-exp\] A *sectional hyperbolic set* is a singular hyperbolic one such that for every two-dimensional linear subspace $L_x \subset F_x$ one has $$\begin{aligned} \label{eq:def-sec-exp} \vert \det (DX_t \vert_{L_x})\vert > C e^{{\lambda}t}, \forall t>0. \end{aligned}$$ In [@Salg2016], this author give another definition of singular hyperbolicity encompassing two previous as follow. \[new-def-singhyp\] A compact invariant set $\Lambda \subset M$ is *$p$-sectional hyperbolic or singular hyperbolic of order $p$* for $X$ if all singularities in $\Lambda$ are hyperbolic, there exists a partially hyperbolic splitting of the tangent bundle on $T_{\Lambda}M = E \oplus F$ and constants $C,\lambda > 0$ such that for every $x \in \Lambda$ and every $t>0$ we have 1. $\Vert DX_t\vert_{E_x} \Vert \leq C e^{- \lambda t}$; 2. $\vert \wedge^p DX_t \vert_{L_x}\vert > C^{-1} e^{{\lambda}t}$, for every $p$-dimensional linear subspace $L_x \subset F_x$. In our applications here, we only deal with two dimensional singular hyperbolic case, but we conjecture that analogous results hold for singular hyperbolic sets of any order $p$, with $2 \leq p \leq \dim{F}$. From now on, we consider $M$ a connected compact finite dimensional riemannian manifold and all singularities of $X$ (if they exist) are hyperbolic. Fields of quadratic forms {#sec:fields-quadrat-forms} ------------------------- From now, we introduce the quadratic forms and its properties. Let ${\EuScript{J}}:E_U\to{{\mathbb R}}$ be a continuous family of quadratic forms ${\EuScript{J}}_x:E_x\to{{\mathbb R}}$ which are non-degenerate and have index $0<q<\dim(E)=n$, where $U\subset M$ is an open set such that $X_t(U) \subset \overline{U}, \forall t \geq 0,$ for a vector field $X$. We also assume that $({\EuScript{J}}_x)_{x\in U}$ is continuously differentiable along the flow. The continuity assumption on ${\EuScript{J}}$ just means that for every continuous section $Z$ of $E_U$ the map $U\to{{\mathbb R}}$ given by $x\mapsto {\EuScript{J}}(Z(x))$ is continuous. The $C^1$ assumption on ${\EuScript{J}}$ along the flow means that the map $x\mapsto {\EuScript{J}}_{X_t(x)} (Z(X_t(x)))$ is continuously differentiable for all $x\in U$ and each $C^1$ section $Z$ of $E_U$. The assumption that $M$ is a compact manifold enables us to globally define an inner product in $E$ with respect to which we can find the an orthonormal basis associated to ${\EuScript{J}}_x$ for each $x$, as follows. Fixing an orthonormal basis on $E_x$ we can define the linear operator $$\begin{aligned} J_x:E_x\to E_x \quad\text{such that}\quad {\EuScript{J}}_x(v)=<J_x v,v> \quad \text{for all}\quad v\in T_xM,\end{aligned}$$ where $<,>=<,>_x$ is the inner product at $E_x$. Since we can always replace $J_x$ by $(J_x+J_x^*)/2$ without changing the last identity, where $J_x^*$ is the adjoint of $J_x$ with respect to $<,>$, we can assume that $J_x$ is self-adjoint without loss of generality. Hence, we represent ${\EuScript{J}}(v)$ by a non-degenerate symmetric bilinear form $<{\EuScript{J}}_x v,v>_x$. Now we use Lagrange’s method to diagonalize this bilinear form, obtaining a base $\{u_1,\dots,u_n\}$ of $E_x$ such that $$\begin{aligned} {\EuScript{J}}_x(\sum_{i}\alpha_iu_i)=\sum_{i=1}^q -\lambda_i\alpha_i^2 + \sum_{j=q+1}^n \lambda_j\alpha_j^2, \quad (\alpha_1,\dots,\alpha_n)\in{{\mathbb R}}^n.\end{aligned}$$ Replacing each element of this base according to $v_i=|\lambda_i|^{1/2}u_i$ we deduce that $$\begin{aligned} {\EuScript{J}}_x(\sum_{i}\alpha_iv_i)=\sum_{i=1}^q -\alpha_i^2 + \sum_{j=q+1}^n \alpha_j^2, \quad (\alpha_1,\dots,\alpha_n)\in{{\mathbb R}}^n.\end{aligned}$$ Finally, we can redefine $<,>$ so that the base $\{v_1,\dots, v_n\}$ is orthonormal. This can be done smoothly in a neighborhood of $x$ in $M$ since we are assuming that the quadratic forms are non-degenerate; the reader can check the method of Lagrange in a standard Linear Algebra textbook and observe that the steps can be performed with small perturbations, for instance in [@Maltsev63]. In this adapted inner product we have that $J_x$ has entries from $\{-1,0,1\}$ only, $J_x^*=J_x$ and also that $J_x^2=J_x$. Having fixed the orthonormal frame as above, the *standard negative subspace* at $x$ is the one spanned by $v_{1},\dots, v_{q}$ and the *standard positive subspace* at $x$ is the one spanned $v_{q+1},\dots,v_n$. ### Positive and negative cones {#sec:positive-negative-co} Let ${\EuScript{C}}_\pm=\{C_\pm(x)\}_{x\in U}$ be the family of positive and negative cones $$\begin{aligned} C_\pm(x):=\{0\}\cup\{v\in E_x: \pm{\EuScript{J}}_x(v)>0\} \quad x\in U\end{aligned}$$ and also let ${\EuScript{C}}_0=\{C_0(x)\}_{x\in U}$ be the corresponding family of zero vectors $C_0(x)={\EuScript{J}}_x^{-1}(\{0\})$ for all $x\in U$. In the adapted coordinates obtained above we have $$\begin{aligned} C_0(x)=\{v=\sum_{i}\alpha_iv_i\in E_x : \sum_{j=q+1}^n \alpha_j^2 = \sum_{i=1}^q \alpha_i^2\}\end{aligned}$$ is the set of *extreme points* of $C_\pm(x)$. The following definitions are fundamental to state our main result. \[def:J-separated\] Given a continuous field of non-degenerate quadratic forms ${\EuScript{J}}$ with constant index on the trapping region $U$ for the flow $X_t$, we say that the flow is - ${\EuScript{J}}$-*separated* if $DX_t(x)(C_+(x))\subset C_+(X_t(x))$, for all $t>0$ and $x\in U$; - *strictly ${\EuScript{J}}$-separated* if $DX_t(x)(C_+(x)\cup C_0(x))\subset C_+(X_t(x))$, for all $t>0$ and $x\in U$; - ${\EuScript{J}}$-*monotone* if ${\EuScript{J}}_{X_t(x)}(DX_t(x)v)\ge {\EuScript{J}}_x(v)$, for each $v\in T_xM\setminus\{0\}$ and $t>0$; - *strictly ${\EuScript{J}}$-monotone* if $\partial_t\big({\EuScript{J}}_{X_t(x)}(DX_t(x)v)\big)\mid_{t=0}>0$, for all $v\in T_xM\setminus\{0\}$, $t>0$ and $x\in U$; - ${\EuScript{J}}$-*isometry* if ${\EuScript{J}}_{X_t(x)}(DX_t(x)v) = {\EuScript{J}}_x(v)$, for each $v\in T_xM$ and $x\in U$. Thus, ${\EuScript{J}}$-separation corresponds to simple cone invariance and strict ${\EuScript{J}}$-separation corresponds to strict cone invariance under the action of $DX_t(x)$. \[rmk:J-separated-C-\] If a flow is strictly ${\EuScript{J}}$-separated, then for $v\in T_xM$ such that ${\EuScript{J}}_x(v)\le0$ we have ${\EuScript{J}}_{X_{-t}(x)}(DX_{-t}(v))<0$ for all $t>0$ and $x$ such that $X_{-s}(x)\in U$ for every $s\in[-t,0]$. Indeed, otherwise ${\EuScript{J}}_{X_{-t}(x)}(DX_{-t}(v))\ge0$ would imply ${\EuScript{J}}_x(v)={\EuScript{J}}_x\big(DX_t(DX_{-t}(v))\big)>0$, contradicting the assumption that $v$ was a non-positive vector. This means that a flow $X_t$ is strictly ${\EuScript{J}}$-separated if, and only if, its time reversal $X_{-t}$ is strictly $(-{\EuScript{J}})$-separated. A vector field $X$ is ${\EuScript{J}}$-*non-negative* on $U$ if ${\EuScript{J}}(X(x))\ge0$ for all $x\in U$, and ${\EuScript{J}}$-*non-positive* on $U$ if ${\EuScript{J}}(X(x))\leq 0$ for all $x\in U$. When the quadratic form used in the context is clear, we will simply say that $X$ is non-negative or non-positive. We apply this notion to the linear Poincaré flow defined on regular orbits of $X_t$ as follows. Suppose that the vector field $X$ is non-negative on $U$. Then, the span $E^X_x$ of $X(x)\neq 0$ is a ${\EuScript{J}}$-non-degenerate subspace. According to item (1) of Proposition \[pr:propbilinear\], we have that $T_xM=E_x^X\oplus N_x$, where $N_x$ is the pseudo-orthogonal complement of $E^X_x$ with respect to the bilinear form ${\EuScript{J}}$, and $N_x$ is also non-degenerate. Moreover, by the definition, the index of ${\EuScript{J}}$ restricted to $N_x$ is the same as the index of ${\EuScript{J}}$. Thus, we can define on $N_x$ the positive and negative cones with core $N_x^+$ and $N_x^-$, respectively. Define the Linear Poincaré Flow $P^{\, t}$ of $X_t$ along the orbit of $x$, by projecting $DX_t$ orthogonally (with respect to ${\EuScript{J}}$) over $N_{X_t(x)}$ for each $t\in{{\mathbb R}}$: $$\begin{aligned} P^{\, t} v := \Pi_{X_t(x)}DX_t v , \quad v\in T_x M, t\in{{\mathbb R}}, X(x)\neq 0,\end{aligned}$$ where $\Pi_{X_t(x)}:T_{X_t(x)}M\to N_{X_t(x)}$ is the projection on $N_{X_t(x)}$ parallel to $X(X_t(x))$. We remark that the definition of $\Pi_x$ depends on $X(x)$ and ${\EuScript{J}}_X$ only. The linear Poincaré flow $P^{\,t}$ is a linear multiplicative cocycle over $X_t$ on the set $U$ with the exclusion of the singularities of $X$. In this setting we can say that the linear Poincaré flow is (strictly) ${\EuScript{J}}$-separated and (strictly) ${\EuScript{J}}$-monotonous using the non-degenerate bilinear form ${\EuScript{J}}$ restricted to $N_x$ for a regular $x\in U$. More precisely: $P^t$ is ${\EuScript{J}}$-monotonous if $\partial_t{\EuScript{J}}(P^tv)\mid_{t=0}\ge0$, for each $x\in U, v\in T_xM\setminus\{0\}$ and $t>0$, and strictly ${\EuScript{J}}$-monotonous if $\partial_t{\EuScript{J}}(P^tv)\mid_{t=0}>0$, for all $v\in T_xM\setminus\{0\}$, $t>0$ and $x\in U$. \[pr:J-separated-spectrum\] Let $L:V\to V$ be a ${\EuScript{J}}$-separated linear operator. Then 1. $L$ can be uniquely represented by $L=RU$, where $U$ is a ${\EuScript{J}}$-isometry and $R$ is ${\EuScript{J}}$-symmetric (or ${\EuScript{J}}$-pseudo-adjoint; see Proposition \[pr:propbilinear\]) with positive spectrum. 2. the operator $R$ can be diagonalized by a ${\EuScript{J}}$-isometry. Moreover the eigenvalues of $R$ satisfy $$\begin{aligned} 0<r_-^q\le\dots\le r_-^1=r_-\le r_+=r_1^+\le\dots\le r_+^p. \end{aligned}$$ 3. the operator $L$ is (strictly) ${\EuScript{J}}$-monotonous if, and only if, $r_-\le (<) 1$ and $r_+\ge (>) 1$. ${\EuScript{J}}$-separated linear maps {#sec:j-separat-linear} -------------------------------------- ### ${\EuScript{J}}$-symmetrical matrixes and ${\EuScript{J}}$-selfadjoint operators {#sec:j-symmetr-matrix} The symmetrical bilinear form defined by $(v,w)=\langle J_x v,w\rangle$, $v,w\in E_x$ for $x\in M$ endows $E_x$ with a pseudo-Euclidean structure. Since ${\EuScript{J}}_x$ is non-degenerate, then the form $(\cdot,\cdot)$ is likewise non-degenerate and many properties of inner products are shared with symmetrical non-degenerate bilinear forms. We state some of them below. We recall that $E^\perp:=\{v\in V: (v,w)=0 \quad\text{for all}\quad w\in E\}$, the pseudo-orthogonal space of $E$, is defined using the bilinear form. \[pr:propbilinear\] Let $(\cdot,\cdot):V\times V \to{{\mathbb R}}$ be a real symmetric non-degenerate bilinear form on the real finite dimensional vector space $V$. 1. $E$ is a subspace of $V$ for which $(\cdot,\cdot)$ is non-degenerate if, and only if, $V=E\oplus E^\perp$. 2. Every base $\{v_1,\dots,v_n\}$ of $V$ can be orthogonalized by the usual Gram-Schmidt process of Euclidean spaces, that is, there are linear combinations of the basis vectors $\{w_1,\dots, w_n\}$ such that they form a basis of $V$ and $(w_i,w_j)=0$ for $i\neq j$. Then this last base can be pseudo-normalized: letting $u_i=|(w_i,w_i)|^{-1/2}w_i$ we get $(u_i,u_j)=\pm\delta_{ij}, i,j=1,\dots,n$. 3. There exists a maximal dimension $p$ for a subspace $P_+$ of ${\EuScript{J}}$-positive vectors and a maximal dimension $q$ for a subspace $P_-$ of ${\EuScript{J}}$-negative vectors; we have $p+q=\dim V$ and $q$ is known as the *index* of ${\EuScript{J}}$. 4. For every linear map $L:V\to{{\mathbb R}}$ there exists a unique $v\in V$ such that $L(w)=(v,w)$ for each $w\in V$. 5. For each $L:V\to V$ linear there exists a unique linear operator $L^+:V\to V$ (the pseudo-adjoint) such that $(L(v),w)=(v,L^+(w))$ for every $v,w\in V$. 6. Every pseudo-self-adjoint $L:V\to V$, that is, such that $L=L^+$, satisfies 1. eigenspaces corresponding to distinct eigenvalues are pseudo-orthogonal; 2. if a subspace $E$ is $L$-invariant, then $E^\perp$ is also $L$-invariant. The proofs are rather standard and can be found in [@Maltsev63]. The following simple result will be very useful in what follows. \[le:kuhne\] Let $V$ be a real finite dimensional vector space endowed with a non-positive definite and non-degenerate quadratic form ${\EuScript{J}}:V\to{{\mathbb R}}$. If a symmetric bilinear form $F:V\times V\to{{\mathbb R}}$ is non-negative on $C_0$ then $$\begin{aligned} r_+=\inf_{v\in C_+} \frac{F(v,v)}{\langle Jv,v\rangle} \ge \sup_{u\in C_-}\frac{F(u,u)}{\langle Ju,u\rangle}=r_-\end{aligned}$$ and for every $r$ in $[r_-,r_+]$ we have $F(v,v)\ge r\langle Jv,v\rangle$ for each vector $v$. In addition, if $F(\cdot,\cdot)$ is positive on $C_0\setminus\{0\}$, then $r_-<r_+$ and $F(v,v)> r\langle Jv,v\rangle$ for all vectors $v$ and $r\in(r_-,r_+)$. \[rmk:Jseparated\] Lemma \[le:kuhne\] shows that if $F(v,w)=\langle \tilde J v,w\rangle$ for some self-adjoint operator $\tilde J$ and $F(v,v)\ge0$ for all $v$ such that $\langle J v, v\rangle=0$, then we can find $a\in{{\mathbb R}}$ such that $\tilde J \ge a J$. This means precisely that $\langle \tilde J v,v\rangle\ge a\langle Jv, v\rangle$ for all $v$. If, in addition, we have $F(v,v)>0$ for all $v$ such that $\langle J v, v\rangle=0$, then we obtain a strict inequality $\tilde J > a J$ for some $a\in{{\mathbb R}}$ since the infimum in the statement of Lemma \[le:kuhne\] is strictly bigger than the supremum. The (longer) proofs of the following results can be found in [@Wojtk01] or in [@Pota79]; see also [@Wojtk09]. For a ${\EuScript{J}}$-separated operator $L:V\to V$ and a $d$-dimensional subspace $F_+\subset C_+$, the subspaces $F_+$ and $L(F_+)\subset C_+$ have an inner product given by ${\EuScript{J}}$. Thus both subspaces are endowed with volume elements. Let $\alpha_d(L;F_+)$ be the rate of expansion of volume of $L\mid_{F_+}$ and $\sigma_d(L)$ be the infimum of $\alpha_d(L;F_+)$ over all $d$-dimensional subspaces $F_+$ of $C_+$. \[pr:product-vol-exp\] We have $\sigma_d(L)=r_+^1 \cdots r_+^d$, where $r^i_+$ are given by Proposition \[pr:J-separated-spectrum\](2). Moreover, if $L_1,L_2$ are ${\EuScript{J}}$-separated, then $\sigma_d(L_1L_2)\ge\sigma_d(L_1)\sigma_d(L_2)$. The following corollary is very useful. \[cor:compos-max-exp\] For ${\EuScript{J}}$-separated operators $L_1,L_2:V\to V$ we have $$\begin{aligned} r_+^1(L_1L_2)\ge r_+^1(L_1) r_+^1(L_2) \quad\text{and}\quad r_-^1(L_1L_2)\le r_-^1(L_1)r_-^1(L_2). \end{aligned}$$ Moreover, if the operators are strictly ${\EuScript{J}}$-separated, then the inequalities are strict. \[rmk:J-mon-spec\] Another important property about the singular values of a ${\EuScript{J}}$-separated operator $L$ is that $$r_+^1 = r_+ \ge 1 (> 1) \quad\text{and}\quad r_-^1 = r_- \le 1 (< 1)$$ if, and only if, $L$ is (strictly) ${\EuScript{J}}$-monotone. This property will be used a lot of times in our proofs. ### Lyapunov exponents By Oseledec’s Ergodic Theorem [@Os68], there exist a full probability set $X$ such that for every $x \in Y$ there is an invariant decomposition $$\begin{aligned} T_xM = \langle X\rangle \oplus E_{1}(x) \oplus \cdots \oplus E_{l(x)}(x)\end{aligned}$$ and numbers $\chi_1 < \cdots < \chi_l$ corresponding to the limits $$\begin{aligned} \chi_j = \lim\limits_{t \to +\infty} \frac{1}{t} \log \Vert DX_t(x) \cdot v\Vert,\end{aligned}$$ for every $v \in E_i(x)\setminus \{0\}, i = 1, \cdots, l(x)$. In this setting, Wojtkowski [@Wojtk01] proved that the logarithm of the pseudo-Euclidean singular values $0 \leq r_q^- \leq \cdots \leq r_1^- \leq r_1^+ \leq \cdots \leq r_p^+$ of $DX_t$ are $\mu$-integrable, and obtained estimates of the Lyapunov exponents related to the singular eigenvalues of strictly ${\EuScript{J}}$-separated maps. [@Wojtk01 Corollary 3.7] \[thm:lyap-exp-sing-val\] For $1 \leq k_1 \leq p$ and $1 \leq k_2 \leq q$ $$\begin{aligned} \chi^-_1 + \cdots + \chi^-_{k_1} \leq \sum_{i=1}^{k_1} \int \log r^-_i d\mu \ \textrm{and} \ \chi^+_1 + \cdots + \chi^+_{k_2} \geq \sum_{i=1}^{k_2} \int \log r^+_i d\mu.\end{aligned}$$ This result will be very useful in proof of Theorem \[mthm:strong-homog-to-quad\]. Some applications ================= In this section, we present some applications of this theory related to some kind of hyperbolicities, as partial and singular ones. In particular, we provide another proof of [@GanLiWen2005 Theorem A]. ### Some results about partial and sectional hyperbolicity from ${\EuScript{J}}$-separation The author, together with V. Araújo, proved in [@ArSal2012] the following useful theorem which relates partial hyperbolicity and ${\EuScript{J}}$-separated sets for a flow. [@ArSal2012 Theorem A] \[mthm:Jseparated-parthyp\] A maximal invariant subset $\Lambda$ of a trapping region $U$ whose singularities are hyperbolic is a partially hyperbolic set for a flow $X_t$ if, and only if, there is a $C^1$ field ${\EuScript{J}}$ of non-degenerate quadratic forms with constant index, equal to the dimension of the stable subspace of $\Lambda$, such that $X_t$ is a non-negative strictly ${\EuScript{J}}$-separated flow on $U$. This result will be useful in our applications of Theorem \[mthm:strong-homog-equiv\]. In the sequence, we can give another proof of next result from [@GanLiWen2005]. [@GanLiWen2005 Theorem A] Let $X \in {\mathfrak{X}^{1}(M)}$, and $\Lambda$ be a robustly transitive singular set of $X$ that is strongly homogeneous of index ${\operatorname{ind}}$. If every singularity $\sigma$ of $X$ is hyperbolic of index ${\operatorname{ind}}(\sigma) > {\operatorname{ind}}$, then $\Lambda$ has a partially hyperbolic splitting of contracting dimension Ind. Likewise, if every singularity $\sigma$ of $X$ is hyperbolic of index ${\operatorname{ind}}(\sigma) \leq {\operatorname{ind}}$, then $\Lambda$ has a partially hyperbolic splitting of expanding dimension $n - 1 - Ind$. We are going to deal with the case ${\operatorname{ind}}(\sigma) > {\operatorname{ind}}$, the other case is analogous. Since $\Lambda$ is strongly homogeneous and ${\operatorname{ind}}(\sigma) > {\operatorname{ind}}$, by [@GanLiWen2005 Lemma 4.1] there is a dominated splitting $T_{\sigma}M = E_{\sigma} \oplus F_{\sigma}$ such that $\dim(E_{\sigma}) = {\operatorname{ind}}$. Hence, Theorem \[mthm:strong-homog-to-quad\] implies that there exists a field of non-degenerate quadratic forms ${\EuScript{J}}$ on $\Lambda$ with index ${\operatorname{ind}}({\EuScript{J}}) = {\operatorname{ind}}(\Lambda)$ for which $X$ is strictly ${\EuScript{J}}$-separated and the associated linear Poincaré flow $P^t$ is strictly ${\EuScript{J}}$-monotone on every compact invariant subset $\gamma$ of $\Lambda^*$. Therefore, Theorem \[mthm:Jseparated-parthyp\] completes the proof. Some immediate results follow from the main theorems. The following consequences of these results follows from the robustness of sectional hyperbolicity and the theory of sectional hyperbolic transitive sets for homogeneous flows from [@MeMor06] and [@ArbMo2013]. \[mcor:2-sec-exp-J-monot\] Let $X\in{\mathfrak{X}^{1}(M)}, \dim(M) \geq 4$ with a nontrivial transitive compact invariant set $\Lambda$ whose singularities, if any, are hyperbolic. Then the following conditions are equivalent: 1. There exists a family ${\EuScript{J}}$ of smooth non-degenerate indefinite quadratic forms with constant index ${\operatorname{ind}}({\EuScript{J}})$ on $\Lambda$ such that $X$ is a non-negative strictly ${\EuScript{J}}$-separated vector field, for which the linear Poincaré flow is strictly ${\EuScript{J}}$-monotonous on every compact invariant set in $\Lambda_X(U)^*=\Lambda_X(U)\setminus {\mathrm{Sing}}(X)$ 2. The set $\Lambda$ is a sectional-hyperbolic subset for $X$ with constant index ${\operatorname{ind}}({\EuScript{O}})={\operatorname{ind}}({\EuScript{J}})$ for all periodic orbits ${\EuScript{O}}$ of $\Lambda$ and ${\operatorname{ind}}(\sigma)=Ind({\EuScript{J}})+1$ for all singularities $\sigma\in\Lambda\cap {\mathrm{Sing}}(X)$. For the next statement, we recall that a hyperbolic singularity $\sigma$ is said to be of codimension one if its index satisfies either ${\operatorname{ind}}(\sigma) = 1$ or ${\operatorname{ind}}(\sigma) = n - 1$, where $n = \dim(M)$. \[obs-lyap-est\] Every attracting set is Lyapunov stable. \[thm:lyap-est-sec-hyp\] Let $\Lambda \subset M^n, n \geq 4$, be a nontrivial transitive set, which is Lyapunov stable for $X$, with singularities all of them hyperbolic of codimension one. Then, the following properties are equivalent: 1. $\Lambda$ is sectional-hyperbolic with $1 \leq dim(E^s) = {\operatorname{ind}}({\EuScript{J}}) \leq n-2$; 2. There exists a field of non-degenerate quadratic forms with constant index $1 \leq {\operatorname{ind}}({\EuScript{J}}) \leq n-2$ such that $X$ is non-negative strictly ${\EuScript{J}}$-separated on $\Lambda$ and every compact invariant subset $\Gamma \subset \Lambda$ is strictly ${\EuScript{J}}$-monotone for linear Poincaré flow associated to $X$. Proof of Corollaries \[mcor:2-sec-exp-J-monot\] and \[thm:lyap-est-sec-hyp\] ---------------------------------------------------------------------------- Indeed, suppose that $(2)$ is true. Then, $X$ is strongly homogeneous on $\Lambda$. By [@ArbMo2013 Corollary 8], this is a sectional hyperbolic set for $X$. To prove the converse statement, we need just use [@ArSal2012 Theorem D]. The next proof needs the following lemma. Let $\Lambda$ be a compact invariant set for a flow $X$ of a $C^1$ vector field $X$ on $M$. [@AraArbSal Lemma 5.1] \[le:flow-center\] Given a continuous splitting $T_\Lambda M = E\oplus F$ such that $E$ is uniformly contracted, then $X(x)\in F_x$ for all $x\in \Lambda$. Suppose that $\Lambda$ is sectional-hyperbolic with decomposition $E \oplus F$. So, it is clearly strongly homogeneous. Once the subbundles are non-trivial and $E$ is uniformly contracting, we must have $1 \leq \dim(E) := {\operatorname{ind}}({\EuScript{J}}) \leq n-2$, because by Lemma \[le:flow-center\], $\langle X \rangle \subset F$. By Theorem \[mthm:Jseparated-parthyp\], there exists a field ${\EuScript{J}}$ of differentiable quadratic forms with constant index equal to the dimension of $E$ with the required properties. Reciprocally, the existence of such a field ${\EuScript{J}}$ implies, by Theorem \[mthm:strong-homog-equiv\], that $\Lambda$ is strongly homogeneous of index ${\operatorname{ind}}({\EuScript{J}})$. Thus, once the singularities are hyperbolic of codimension one, it is enough to use Lemma [@ArbMo2013 Corollary 9]. Proof of Theorems ================= Now, we prove our mains results. To prove the Theorem \[cor:star-flow\] we use the following result from [@ArSal2012]. [@ArSal2012 item 3,Theorem 2.23] \[prop:J-hyperbolic\] Let $\Gamma$ be a compact invariant set for $X$ with a dominated splitting $T_{\Gamma}M = E \oplus F$. Let ${\EuScript{J}}$ be a $C^1$ field of indefinite quadratic forms such that $DX_t$ is strictly ${\EuScript{J}}$-separated. Then, $E \oplus F$ is uniformly hyperbolic if, and only if, there is an equivalent field ${\EuScript{J}}$ of quadratic forms on a neighborhood of $\Gamma$ such that ${\EuScript{J}}'(v) > 0$, for all $v \in T_{\Gamma}M$ and all $x \in \Gamma$. If $X$ is a star flow, then each singular point $\sigma$ is hyperbolic and it is well known that its hyperbolic decomposition $E^s_{\sigma} \oplus E^u_{\sigma}$ is a dominated one. So, by using adapted metrics (see [@Goum07]) we construct the desired quadratic form $J_\sigma$ such that $X$ is strictly separated (see [@ArSal2012]) and, by Proposition \[prop:J-hyperbolic\] ${\EuScript{J}}'(v)>0$ for all $v \in T_{\Gamma}M$. Analogously, for every periodic orbit $\gamma$ of $X$, consider the hyperbolic splitting $T_\gamma M = E^s \oplus E^X \oplus E^u$. Again, considering $E^s \oplus (E^X \oplus E^u)$ as a dominated splitting we obtain a quadratic form ${\EuScript{J}}$ for which $X$ is strictly separated on $\gamma$. By construction of the adapted metrics, we have that ${\EuScript{J}}$ is $C^ 1$ along the flow (see [@Goum07] for details about the construction of such a adapted metric). In addition, the linear Poincaré flow associated to $X$, $P^t$ is hyperbolic and then ${\EuScript{J}}$-monotone on $\gamma$. If $\gamma$ is a sink (respectively, a source) the splitting $E^s \oplus E^X$ (respectively, $E^u \oplus E^X$) is a dominated one and we proceed constructing the cones the same way, however the core of the nonnegative cone is the field direction. Reciprocally, take a small neighborhood $U$ of $Per_*(X)$ such that there is a $C^1$ neighborhood $\mathcal{V}$ of $X$ for which $Per_*(Y) \subset U$ and suppose that such a field of quadratic forms is defined on $U$. By Proposition \[prop:J-hyperbolic\], every singularity $\sigma \in U$ is hyperbolic. The case of periodic orbits is analogous. Shrinking $U$, if necessary, we may suppose that, for each preperiodic orbit and each singularity in $U$ of each $Y \in \mathcal{V}$, we have quadratic forms (still denoted ${\EuScript{J}}$) with the same features as before. Indeed, since the quadratic form on each periodic orbit is $C^1$ along the flow, for any $Y \in \mathcal{V}$, shrinking $\mathcal{V}$ if necessary, we must have that any preperiodic orbit of $X$ present stricly montonicity for the linear Poincaré flow. If, for some $Y \in \mathcal{V}$, another periodic orbit is created, by $C^1$-closeness it is ${\EuScript{J}}$-monotone for the linear Poincaré flow $P^t_Y$ associated to $Y$, since it comes from a preperiodic one of $X$ which is ${\EuScript{J}}$-monotone for the linear Poincaré flow $P^t_X$, by hyphotesis. See [@ArSal2012 Section 2.5.4]. Hence, every periodic orbit for any $Y \in \mathcal{V}$ is hyperbolic. Therefore, $X$ is a star flow. Now, it is proved the second main result. Since the linear Poincaré flow is strictly ${\EuScript{J}}$-monotone on each preperiodic orbit $\gamma \subset \Lambda^*$ implies that, if $\gamma$ is a closed orbit, then it is a hyperbolic subset of $\Lambda$, with a constant index, which we denote ${\operatorname{ind}}({\EuScript{J}})$. Moreover, taking a small enough neighborhood $U$ of $Per_*(X\vert_\Lambda)$ there exists some neighborhood $\mathcal{V}$ of $X$ such that $Per_*(Y\vert_{\Lambda_Y})\subset U$, $\forall Y \in \mathcal{V}$. If some periodic orbit $\gamma_Y$ is created by a small $C^1$ perturbation of $X$, it comes from a preperiodic orbit of $X$. Thus, $\gamma_Y$ is a hyperbolic closed orbit of $Y$, and must have index equal to ${\operatorname{ind}}({\EuScript{J}})$. Hence, ${\operatorname{ind}}({\EuScript{J}})$ does not change by small differentiable perturbations of $X$ on a neighborhood of $\Lambda$, so the index of hyperbolic periodic orbits also does not change. Therefore, $\Lambda$ is strongly homogeneous for $X$. Reciprocally, if $\Lambda$ is strongly homogeneous of index ${\operatorname{ind}}$, then cannot be there a non-hyperbolic periodic orbit. Otherwise, we can create two periodic orbits with different indices, by Frank’s Lemma. Moreover, $X$ is a star flow in a neighborhood of $\Lambda$. Hence, by Theorem \[cor:star-flow\] we can define the desired field of quadratic forms ${\EuScript{J}}$, with fixed index ${\operatorname{ind}}({\EuScript{J}}) = {\operatorname{ind}}$, defined on a neighborhood $U$ of $\Lambda$, where $U$ is the neighborhood for which $Per_*(Y\vert_{\Lambda_Y}) \subset U$ for any $Y$ close enough to $X$. Note that Corollary \[mcor:strong-homog\] follows from Theorem \[mthm:strong-homog-equiv\], just observing now that if any singularity $\sigma$ is accumulated by regular orbits, it cannot present ${\operatorname{ind}}(\sigma) < {\operatorname{ind}}{{\EuScript{J}}}$, once $X \in {\mathfrak{X}^{1}(M)}$, ${\EuScript{J}}$ is a continuous field of quadratic forms and $X$ is ${\EuScript{J}}$- monotonic over any compact invariant nonsingular set $\Gamma$. Now, we prove our last main result. First of all, we recall some definitions which are necessary here. Let $Z$ be a compact metric space and denote $\mathcal{M}(Z)$ the set of probabilities measures on the Borel $\sigma$-algebra of $Z$. If $T: Z \to Z$ is a measurable map, we say that a probability measure $\mu$ is an invariant measure of $T$, if $\mu(T^{-1}(A)) = \mu(A)$, for every measurable set $A \subset Z$. We say that $\mu$ is an invariant measure of $X$ if it is an invariant measure of $X_t$ for every $t \in \mathbb{R}$. We will denote by $\mathcal{M}_X$ the set of all invariant measures of $X$. A subset $Y\subset Z$ has *total probability* if for every $\mu\in \mathcal{M}_X$ we have $\mu(Y)=1$ (see [@Man82]). The support of a measure $\mu$, denoted by $supp(\mu)$, is the set of points for which the measure is non-zero. An invariant measure is said to be *atomic* if its support is either a closed orbit or a singularity. A probability measure $\mu$ is an *ergodic measure* if for every invariant set $A$ we have $\mu(A) = 1$ or $\mu(A) = 0$. Finally, a certain property is said to be valid in *$\mu$-almost every point* if it is valid in the whole Z except, possibly, in a set of null measure. We recall the definition of $\delta$-closable points of [@Man82]. We say that a point $x \in M \setminus Sing(X)$ is $\delta$-closable if, for any $C^1$ neighborhood ${{\mathcal U}}\subset {\mathfrak{X}^{1}(M)}$ of $X$, there exists a vector field $Z \in {{\mathcal U}}$, a point $z \in M$ and $T > 0$ such that: 1. $Z_T(z) = z$, 2. $Z = X$ on $M \setminus B_{\delta} (X_{[0, T]}(x))$ and 3. $dist(Z_t(z), X_t(x)) < \delta, \forall 0 \leq t \leq T$. We denote by $\Sigma(X)$ the set of points of $M$ which are $\delta$-closable for any $\delta$ sufficiently small. If $\Lambda$ is a strongly homogeneous set for $X$ with singularities all of them hyperbolic, then $X$ is a star flow in $\Lambda$. By Ergodic Closing Lemma, the $\delta$-closable set of $X$ has total probability. If $x \in \Lambda$ is a regular $\delta$-closable point, then it is a pre-periodic point of index ${\operatorname{ind}}(\Lambda)$. According the proof of [@GanLiWen2005 Lemma 5.3], we have a dominated splitting $E_x \oplus F_x$ of index ${\operatorname{ind}}(\Lambda)$ in $T_xM$, for all $x$ . By Theorem \[thm:lyap-exp-sing-val\], we have $$\begin{aligned} \chi^-_1 + \cdots + \chi^-_{k_1} \leq \sum_{i=1}^{k_1} \int \log r^-_i d\nu \ \textrm{and} \ \chi^+_1 + \cdots + \chi^+_{k_2} \geq \sum_{i=1}^{k_2} \int \log r^+_i d\nu,\end{aligned}$$ for any $k_1 \leq q, k_2 \leq p$. Also according the proof of [@GanLiWen2005 Lemma 5.3], the ergodic probability measures are not atomic. Now, Birkhoff’s ergodic theorem and Corollary \[cor:compos-max-exp\] imply that the Lyapunov exponents on $E$ are negative and the sectional Lyapunov exponents are positive, in a total probability subset of $\Lambda$. Moreover, for singularities $\sigma \in {\mathrm{Sing}}(\Lambda)$ we have two possibilities: First case: $\sigma$ is accumulated by recurrent orbits (including periodic orbits), then since ${\operatorname{ind}}(\sigma) \geq {\operatorname{ind}}(\Lambda)$, by [@GanLiWen2005 Lemma 4.1] there is a dominated splitting $T_{\sigma}M = E_{\sigma} \oplus F_{\sigma}$, where $dim (E) = {\operatorname{ind}}(\Lambda)$. Second case: Either there exists a dominated splitting on $T_{\sigma}M = E_{\sigma} \oplus F_{\sigma}$ with $dim (E) = {\operatorname{ind}}(\Lambda)$, which guarantees the definition of ${\EuScript{J}}$ such that $X$ is stricly ${\EuScript{J}}$-separated. Or, otherwise, since $\sigma$ is an isolated hyperbolic singularity with ${\operatorname{ind}}(\sigma) \geq {\operatorname{ind}}(\Lambda)$, we have an invariant splitting for which we only guarantee that ${\EuScript{J}}$ such that $X$ is (not strictly) ${\EuScript{J}}$-separated. So, we have an invariant splitting $T_{\Lambda}M = E_{\Lambda} \oplus F_{\Lambda}$ which has uniformly angle bounded away from zero and $T_{\sigma}M = E_{\sigma} \oplus F_{\sigma}$ is dominated for every $\sigma \in {\mathrm{Sing}}(X)$. Now, [@AraArbSal Theorem C] implies that the corresponding decomposition $T_{\Lambda}M = E \oplus F$ is dominated of index ${\operatorname{ind}}(\Lambda)$. By using the adapted metric for dominated splitting [@Goum07], we obtain a field of $C^1$ non-degenerated quadratic forms ${\EuScript{J}}$ such that $X$ strictly ${\EuScript{J}}$-separated over $\Lambda$, as in [@ArSal2012]. Now, to prove the ${\EuScript{J}}$-monotonicity, take a compact invariant set $\Gamma$ in $\Lambda^*$. Since $X$ is a star flow and $\Gamma$ is nonsingular, by [@GanWen2006 Theorem A], this set must be a hyperbolic one. So, by well known results, the linear Poincaré flow associated to $X$ is strictly ${\EuScript{J}}$-monotone on any compact invariant set $\Gamma \in \Lambda^*$. 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--- abstract: 'We introduce the notion of mixed-$\omega$-sheaves and use it for the study of a relative version of Fujita’s freeness conjecture.' address: 'Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan' author: - Osamu Fujino date: '2019/7/29, version 0.40' title: 'On mixed-$\omega$-sheaves' --- Introduction {#z-sec1} ============ Let us recall Fujita’s famous freeness conjecture on adjoint bundles. Note that everything is defined over $\mathbb C$, the complex number field, in this paper. \[z-conj1.1\] Let $X$ be a smooth projective variety with $\dim X=n$ and let $\mathcal L$ be any ample invertible sheaf on $X$. Then $\omega_X\otimes \mathcal L^{\otimes l}$ is generated by global sections for every $l\geq n+1$. Although there have already been many related results, Conjecture \[z-conj1.1\] is still open. As a generalization of Conjecture \[z-conj1.1\], Popa and Schnell proposed the following conjecture, which is a relative version of Fujita’s freeness conjecture. \[z-conj1.2\] Let $f:X\to Y$ be a surjective morphism between smooth projective varieties with $\dim Y=n$. Let $\mathcal L$ be any ample invertible sheaf on $Y$. Then, for every positive integer $k$, the sheaf $$f_*\omega^{\otimes k}_X\otimes \mathcal L^{\otimes l}$$ is generated by global sections for $l\geq k(n+1)$. We can find some interesting results on Conjecture \[z-conj1.2\] in [@denf], [@dutta], [@dutta-murayama], and [@iwai]. In this paper, we do not directly treat Conjecture \[z-conj1.2\]. We propose a new conjecture similar to Conjecture \[z-conj1.2\]. It is also a generalization of Conjecture \[z-conj1.1\]. \[z-conj1.3\] Let $f: X\to Y$ be a surjective morphism between smooth projective varieties with $\dim Y=n$. Let $\mathcal L$ be any ample invertible sheaf on $Y$. Then, for every positive integer $k$, the sheaf $$f_*\omega^{\otimes k}_{X/Y}\otimes \omega_Y\otimes \mathcal L^{\otimes l}$$ is generated by global sections for $l\geq n+1$. If Conjecture \[z-conj1.3\] and Fujita’s original freeness conjecture (see Conjecture \[z-conj1.1\]) hold true, then $f_*\omega^{\otimes k}_X\otimes \mathcal L^{\otimes l}$ is generated by global sections for every $l\geq k(n+1)$ since $$f_*\omega^{\otimes k}_X\otimes \mathcal L^{\otimes l}\simeq f_*\omega^{\otimes k}_{X/Y}\otimes \omega_Y\otimes \mathcal L^{\otimes \left(l-(k-1)(n+1)\right)}\otimes \left(\omega_Y\otimes \mathcal L^{\otimes n+1}\right)^{\otimes k-1}.$$ Therefore, Conjecture \[z-conj1.3\] is sharper than Conjecture \[z-conj1.2\]. It is well known that Conjecture \[z-conj1.3\] holds true when $Y$ is a curve. This means that $f_*\omega^{\otimes k}_{X/Y}\otimes \omega_Y\otimes \mathcal L^{\otimes l}$ is generated by global sections for every $l\geq 2$. More generally, we have: \[z-thm1.4\] Let $f:X\to C$ be a surjective morphism from a smooth projective variety $X$ onto a smooth projective curve $C$. Let $\mathcal H$ be an ample invertible sheaf on $C$ with $\deg \mathcal H\geq 2$ and let $k$ be any positive integer. Then the sheaf $$f_*\mathcal \omega^{\otimes k}_{X/C}\otimes \omega_C\otimes \mathcal H$$ is generated by global sections. Here, we give a detailed proof of Theorem \[z-thm1.4\] in order to explain our idea. If $f_*\omega^{\otimes k}_{X/C}=0$, then there are nothing to prove. So we assume that $f_*\omega^{\otimes k}_{X/C} \ne 0$. We take any closed point $P$. \[z-claim\] $H^1(C, f_*\omega^{\otimes k}_{X/C}\otimes \omega_C \otimes \mathcal H\otimes \mathcal O_C(-P))=0$. By Viehweg’s weak positivity theorem, $f_*\omega^{\otimes k}_{X/C}$ is a nef locally free sheaf since $C$ is a smooth projective curve. Therefore, $\mathcal E:=f_*\omega^{\otimes k}_{X/C}\otimes \mathcal H\otimes \mathcal O_C(-P)$ is ample. If $H^1(C, \mathcal E\otimes \omega_C)\ne 0$, then we get $H^0(C, \mathcal E^*)\ne 0$ by Serre duality. This implies that there is a nontrivial inclusion $0\to \mathcal O_C\to\mathcal E^*$. By taking the dual of this inclusion, we have the following surjection $\mathcal E\to \mathcal O_C\to 0$. This is a contradiction since $\mathcal E$ is ample. Hence we have $H^1(C, \mathcal E\otimes \omega_C)=0$. By Claim, the natural restriction map $$H^0(C, f_*\omega^{\otimes k}_{X/C}\otimes \omega_C\otimes \mathcal H)\to f_*\omega^{\otimes k}_{X/C}\otimes \omega_C\otimes \mathcal H\otimes \mathbb C(P)$$ is surjective. This means that $f_*\omega^{\otimes k}_{X/C}\otimes \omega_C\otimes \mathcal H$ is generated by global sections. The following theorem supports Conjecture \[z-conj1.3\]. \[z-thm1.5\] Let $f:X\to Y$ be a surjective morphism between smooth projective varieties and let $H$ be an ample divisor on $Y$ such that $|H|$ is free. We put $\dim Y=n$. Then $$f_*\omega^{\otimes k}_{X/Y}\otimes \omega_Y\otimes \mathcal O_Y(lH)$$ is generically generated by global sections for all integers $k\geq 1$ and $l\geq n+1$. Moreover, $$\left(\bigotimes ^s (f_*\omega^{\otimes k}_{X/Y})\right)^{**} \otimes \omega_Y\otimes \mathcal O_Y(lH)$$ is generically generated by global sections for all integers $k\geq 1$, $s\geq 1$, and $l\geq n+1$. Let $H^\dag$ be an ample divisor on $Y$ such that $|H^\dag|$ is not necessarily free. Then $$f_*\omega^{\otimes k}_{X/Y}\otimes \omega_Y\otimes \mathcal O_Y(lH^\dag)$$ is generically generated by global sections for all integers $k\geq 1$ and $l\geq n^2+\min\{2, k\}$. Moreover, the sheaf $$\left(\bigotimes ^s (f_*\omega^{\otimes k}_{X/Y})\right)^{**} \otimes \omega_Y\otimes \mathcal O_Y(lH^\dag)$$ is generically generated by global sections for all integers $k\geq 1$, $s\geq 1$, and $l\geq n^2+\min\{2, k\}$. The author learned the following remark from Masataka Iwai. \[y-rem1.6\] Let $f:X\to Y$ and $H^\dag$ be as in Theorem \[z-thm1.5\]. Let $U$ be the largest Zariski open set of $Y$ such that $f$ is smooth over $U$. Then, by the argument in [@iwai], we can check that the natural map $$H^0(Y, f_*\omega^{\otimes k}_{X/Y}\otimes \omega_Y\otimes \mathcal O_Y(lH^\dag))\otimes \mathcal O_Y\to f_*\omega^{\otimes k}_{X/Y}\otimes \omega_Y\otimes \mathcal O_Y(lH^\dag)$$ is surjective on $U$ for all integers $k\geq 1$ and $l\geq \frac{n(n+1)}{2}+1$. Theorem \[z-thm1.5\] is a special case of Theorems \[y-thm1.7\] and \[y-thm1.8\]. In this paper, we will establish Theorems \[y-thm1.7\] and \[y-thm1.8\] by our new theory of mixed-$\omega$-sheaves. \[y-thm1.7\] Let $f:X\to Y$ be a surjective morphism from a normal projective variety $X$ onto a smooth projective variety $Y$ with $\dim Y=n$. Let $\Delta$ be an effective $\mathbb R$-divisor on $X$ such that $K_X+\Delta$ is $\mathbb R$-Cartier and that $(X, \Delta)$ is log canonical over a nonempty Zariski open set of $Y$. Let $L$ be a Cartier divisor on $X$ with $L\sim _{\mathbb R}k(K_{X/Y}+\Delta)$ for some positive integer $k$. Let $H$ be a big Cartier divisor on $Y$ such that $|H|$ is free. Then $$f_*\mathcal O_X(L)\otimes \mathcal O_Y(K_Y+lH)$$ is generically generated by global sections for every $l\geq n+1$. Moreover, $$\left(\bigotimes ^s f_*\mathcal O_X(L)\right)^{**} \otimes \mathcal O_Y(K_Y+lH)$$ is generically generated by global sections for all integers $s\geq 1$ and $l\geq n+1$. \[y-thm1.8\] In Theorem \[y-thm1.7\], we assume that $H^\dag$ is a nef and big Cartier divisor on $Y$ such that $|H^\dag|$ is not necessarily free. Then we have that $$f_*\mathcal O_X(L)\otimes \mathcal O_Y(K_Y+lH^\dag) \quad \text{and}\quad \left(\bigotimes ^s f_*\mathcal O_X(L)\right)^{**} \otimes \mathcal O_Y(K_Y+lH^\dag)$$ are generically generated by global sections for all integers $s\geq 1$ and $l \geq n^2+\min\{2, k\}$. Let us quickly explain the idea of the proof of Theorem \[z-thm1.5\], which is mainly due to Nakayama (see [@nakayama]). Let $f:X\to Y$ be a surjective morphism between smooth projective varieties and let $H$ be an ample Cartier divisor on $Y$ such that $|H|$ is free. We fix a positive integer $k\geq 2$. Then we can construct a surjective morphism $g:Z\to Y$ from a smooth projective variety $Z$ and a direct summand $\mathcal F$ of $g_*\mathcal O_Z(K_Z)$ such that there exists a generically isomorphic injection $$\mathcal F\hookrightarrow f_*\omega^{\otimes k}_{X/Y}\otimes \omega_Y\otimes \mathcal O_Y(H).$$ By Kollár’s vanishing theorem, we have $$H^i(Y, \mathcal F\otimes \mathcal O_Y((n+1-i)H))=0$$ for every $i>0$, where $n=\dim Y$. Therefore, by Castelnuovo–Mumford regularity, $\mathcal F\otimes \mathcal O_Y((n+1)H)$ is generated by global sections. This implies that $$f_*\omega^{\otimes k}_{X/Y}\otimes \omega_Y\otimes \mathcal O_Y((n+2)H)$$ is generically generated by global sections. Note that we do not try to establish any vanishing theorem for $f_*\omega^{\otimes k}_{X/Y}\otimes \omega_Y\otimes \mathcal O_Y(H)$ directly. Anyway, it is natural to consider: \[y-def1.9\] A torsion-free coherent sheaf $\mathcal F$ on a normal quasi-projective variety $W$ is called a [*[mixed-$\omega$-sheaf]{}*]{} if there exist a projective surjective morphism from a smooth quasi-projective variety $V$ and a simple normal crossing divisor $D$ on $V$ such that $\mathcal F$ is a direct summand of $f_*\mathcal O_V(K_V+D)$. When $D=0$, $\mathcal F$ is called a [*[pure-$\omega$-sheaf]{}*]{} on $W$. For the study of klt pairs, the notion of pure-$\omega$-sheaves is sufficient and is essentially due to Nakayama (see [@nakayama]). In this paper, we study some basic properties of mixed-$\omega$-sheaves. They are indispensable for the study of log canonical pairs. Of course, the theory of mixed-$\omega$-sheaves (resp. pure-$\omega$-sheaves) in this paper is based on that of mixed (resp. pure) Hodge structures. Roughly speaking, Nakayama only treats pure-$\omega$-sheaves in [@nakayama Chapter V]. However, his theory of $\omega$-sheaves is more sophisticated and some of his results are much sharper than ours. We do not try to make the framework discussed in this paper supersede Nakayama’s theory of $\omega$-sheaves in [@nakayama Chapter V]. The main purpose of this paper is to make Nakayama’s theory of $\omega$-sheaves more accessible and make it applicable to the study of log canonical pairs. Theorem \[x-thm9.3\] (and Remark \[x-rem9.4\]) is one of the main results of this paper, which we call a fundamental theorem of the theory of mixed-$\omega$-sheaves. \[y-thm1.10\] Let $f:X\to Y$ be a surjective morphism from a normal projective variety $X$ onto a smooth projective variety $Y$. Let $L$ be a Cartier divisor on $X$ and let $\Delta$ be an effective $\mathbb R$-divisor on $X$ such that $K_X+\Delta$ is $\mathbb R$-Cartier. Let $D$ be an $\mathbb R$-divisor on $Y$. Let $k$ be a positive integer with $k\geq 2$. Assume the following conditions: - $(X, \Delta)$ is log canonical [*[(]{}*]{}resp. klt[*[)]{}*]{} over a nonempty Zariski open set of $Y$, and - $L+f^*D-k(K_{X/Y}+\Delta)-f^*A$ is semi-ample for some big $\mathbb R$-divisor $A$ on $Y$. If $f_*\mathcal O_Y(L)\ne 0$, then there exist a mixed-$\omega$-big-sheaf [*[(]{}*]{}resp. pure-$\omega$-big-sheaf[*[)]{}*]{} $\mathcal F$ on $Y$ and a generically isomorphic injection $$\mathcal F\hookrightarrow \mathcal O_Y(K_Y+\lceil D\rceil)\otimes f_*\mathcal O_X(L).$$ For the precise definition of mixed-$\omega$-big-sheaves and pure-$\omega$-big-sheaves, see Definition \[x-def5.3\] below. As an application of Theorem \[y-thm1.10\], we have: \[y-thm1.11\] Let $f:X\to Y$ be a surjective morphism from a normal projective variety $X$ onto a smooth projective variety $Y$ with connected fibers. Let $\Delta$ be an effective $\mathbb R$-divisor on $X$ such that $K_X+\Delta$ is $\mathbb R$-Cartier and that $(X, \Delta)$ is log canonical over a nonempty Zariski open set of $Y$. Let $D$ be an $\mathbb R$-Cartier $\mathbb R$-divisor on $X$ such that $D-(K_{X/Y}+\Delta)$ is nef. Then, for any $\mathbb R$-divisor $Q$ on $Y$, we have $$\kappa _\sigma(X, D+f^*Q)\geq \kappa _\sigma(F, D|_F) +\kappa (Y, Q)$$ and $$\kappa _\sigma(X, D+f^*Q)\geq \kappa (F, D|_F) +\kappa _\sigma(Y, Q)$$ where $F$ is a sufficiently general fiber of $f:X\to Y$. We note that $\kappa_\sigma(X, D)$ and $\kappa (X, D)$ denote Nakayama’s numerical dimension and the Iitaka dimension of $D$, respectively. Theorem \[y-thm1.11\] already played a crucial role in the theory of minimal models. We explain the organization of this paper. In Section \[x-sec2\], we collect some basic definitions. In Section \[x-sec3\], we prepare some useful and important lemmas. They will play a crucial role in this paper. In Section \[x-sec4\], we quickly explain some basic properties of Viehweg’s weakly positive sheaves and big sheaves. In Section \[x-sec5\], we introduce mixed-$\omega$-sheaves and mixed-$\omega$-big-sheaves. In Sections \[x-sec6\] and \[x-sec7\], we prove some basic properties of mixed-$\omega$-sheaves based on the theory of mixed Hodge structures. In Section \[x-sec8\], we treat a very special but interesting case. Section \[x-sec9\] is the main part of this paper. We establish a fundamental theorem of the theory of mixed-$\omega$-sheaves. Section \[x-sec10\] is devoted to the proof of Theorems \[z-thm1.5\], \[y-thm1.7\], and \[y-thm1.8\]. In Section \[x-sec11\], we treat Nakayama’s inequality on $\kappa_\sigma$, which has already played a crucial role in the theory of minimal models, and a slight generalization of the twisted weak positivity theorem. The author was partially supported by JSPS KAKENHI Grant Numbers JP16H03925, JP16H06337. He thanks Masataka Iwai for useful comments. We will work over $\mathbb C$, the complex number field, throughout this paper. We note that a [*[scheme]{}*]{} is a separated scheme of finite type over $\mathbb C$ and a [*[variety]{}*]{} is an integral scheme. Preliminary {#x-sec2} =========== In this section, we collect some basic definitions. For the details, see [@fujino-fundamental], [@fujino-foundations], and [@fujino-iitaka]. Let us start with the definition of canonical sheaves and canonical divisors. \[x-def2.1\] Let $X$ be an equidimensional scheme of dimension $n$ and let $\omega^\bullet_X$ be the dualizing complex of $X$. Then we put $$\omega_X:=h^{-n}(\omega^\bullet_X)$$ and call it the [*[canonical sheaf]{}*]{} of $X$. We further assume that $X$ is normal. Then a [*[canonical divisor]{}*]{} $K_X$ of $X$ is a Weil divisor on $X$ such that $$\mathcal O_{X_{\mathrm{sm}}}(K_X)\simeq \Omega^n_{X_{\mathrm{sm}}}$$ holds, where $X_{\mathrm{sm}}$ is the largest smooth Zariski open set of $X$. It is well known that $$\mathcal O_X(K_X)\simeq \omega_X$$ holds when $X$ is normal. If $f:X\to Y$ is a morphism between Gorenstein schemes, then we put $$\omega_{X/Y}:=\omega_X\otimes f^*\omega^{\otimes -1}_Y.$$ If $f:X\to Y$ is a morphism from a normal scheme $X$ to a normal Gorenstein scheme $Y$, then we put $$K_{X/Y}:=K_X-f^*K_Y.$$ Let us quickly see the definition of singularities of pairs. \[x-def2.2\] Let $X$ be a normal variety and let $\Delta$ be an effective $\mathbb R$-divisor on $X$ such that $K_X+\Delta$ is $\mathbb R$-Cartier. Let $f:Y\to X$ be a resolution of singularities of $X$ such that ${{\operatorname{Exc}}}(f)\cup f^{-1}_*\Delta$ has a simple normal crossing support, where ${{\operatorname{Exc}}}(f)$ is the exceptional locus of $f$ on $Y$ and $f^{-1}_*\Delta$ is the strict transform of $\Delta$ on $Y$. Then we can write $$K_Y=f^*(K_X+\Delta)+\sum _i a_i E_i$$ with $f_*\left(\sum _i a_i E_i\right)=-\Delta$. We say that $(X, \Delta)$ is [*[log canonical]{}*]{} (resp. [*[klt]{}*]{}) if $a_i\geq -1$ (resp. $a_i>-1$) for every $i$. If $(X, \Delta)$ is log canonical and there exist a resolution of singularities $f:Y\to X$ as above and a prime divisor $E_i$ on $Y$ with $a_i=-1$, then $f(E_i)$ is called a [*[log canonical center]{}*]{} of $(X, \Delta)$. \[x-def2.3\] Let $(X, \Delta)$ be a log canonical pair. If there exists a resolution of singularities $f:Y\to X$ such that the exceptional locus ${{\operatorname{Exc}}}(f)$ of $f$ is a divisor on $Y$, ${{\operatorname{Exc}}}(f)\cup f^{-1}_*\Delta$ has a simple normal crossing support, and $$K_Y=f^*(K_X+\Delta)+\sum _i a_i E_i$$ with $a_i>-1$ for every $f$-exceptional divisor $E_i$, then the pair $(X, \Delta)$ is called a [*[dlt]{}*]{} pair. The following definitions are very useful in this paper. \[x-def2.4\] Let $f:X\to Y$ be a dominant morphism between normal varieties and let $D$ be an $\mathbb R$-divisor on $X$. We can write $$D=D_{\mathrm{hor}}+D_{\mathrm{ver}}$$ such that every irreducible component of $D_{\mathrm{hor}}$ (resp.  $D_{\mathrm{ver}}$) is mapped (resp. not mapped) onto $Y$. If $D=D_{\mathrm{hor}}$ (resp. $D=D_{\mathrm{ver}}$), $D$ is said to be [*[horizontal]{}*]{} (resp. [*[vertical]{}*]{}). \[x-def2.5\] Let $D=\sum _i d_i D_i$ be an $\mathbb R$-divisor on a normal variety $X$, where $D_i$ is a prime divisor on $X$ for every $i$, $D_i\ne D_j$ for $i\ne j$, and $d_i \in \mathbb R$ for every $i$. Then we put $$\lfloor D\rfloor =\sum _i \lfloor d_i \rfloor D_i, \quad \{D\}=D-\lfloor D\rfloor, \quad \text{and} \quad\lceil D\rceil =-\lfloor -D\rfloor.$$ Note that $\lfloor d_i \rfloor$ is the integer which satisfies $d_i-1<\lfloor d_i \rfloor \leq d_i$. We also note that $\lfloor D\rfloor$, $\lceil D\rceil$, and $\{D\}$ are called the [*[round-down]{}*]{}, [*[round-up]{}*]{}, and [*[fractional part]{}*]{} of $D$ respectively. If $0\leq d_i \leq 1$ for every $i$, then we say that $D$ is a [*[boundary]{}*]{} $\mathbb R$-divisor on $X$. We note that $\sim _{\mathbb Q}$ (resp. $\sim_{\mathbb R}$) denotes the [*[$\mathbb Q$-linear]{}*]{} (resp. [*[$\mathbb R$-linear]{}*]{}) [*[equivalence]{}*]{} of $\mathbb Q$-divisors (resp. $\mathbb R$-divisors). In this paper, we will repeatedly use the following notation: $$D^{=1}=\sum _{d_i=1} D_i, \quad D^{>1}=\sum _{d_i>1} d_i D_i, \quad \text{and} \quad D^{<0}=\sum _{d_i<0}d_i D_i.$$ We close this section with the definition of exceptional divisors for proper surjective morphisms between normal varieties. \[x-def2.6\] Let $f:X\to Y$ be a proper surjective morphism between normal varieties. Let $E$ be a Weil divisor on $X$. We say that $E$ is [*[$f$-exceptional]{}*]{} if $\mathrm{codim}_Yf({{\operatorname{Supp}}}E)\geq 2$. We note that $f$ is not always assumed to be birational. Preliminary lemmas {#x-sec3} ================== In this section, we collect some useful and important lemmas for the reader’s convenience. They are more or less well known to the experts. Let us start with the following easy lemmas on $\mathbb R$-divisors. We will use them repeatedly in this paper. \[x-lem3.1\] Let $A$ be a Cartier divisor on a normal variety $V$. Let $B$ be an $\mathbb R$-Cartier $\mathbb R$-divisor on $V$ such that $B=\sum _{i\in I} b_iB_i$ where $b_i\in \mathbb R$ and $B_i$ is a prime divisor on $V$ for every $i$ with $B_i\ne B_j$ for $i\ne j$. Assume that $A\sim _{\mathbb R} B$. Then we can take a $\mathbb Q$-Cartier $\mathbb Q$-divisor $C=\sum _{i\in I} c_i B_i$ on $V$ such that - $A\sim _{\mathbb Q} C$, - $c_i=b_i$ holds if $b_i\in \mathbb Q$, and - $|c_i-b_i|\ll 1$ holds for $b_i\in \mathbb R\setminus \mathbb Q$. In particular, ${{\operatorname{Supp}}}C={{\operatorname{Supp}}}B$, $\lfloor C\rfloor =\lfloor B\rfloor$, $\lceil C\rceil =\lceil B\rceil$, and ${{\operatorname{Supp}}}\{C\}={{\operatorname{Supp}}}\{B\}$. It is an easy exercise. For the details, see, for example, the proof of [@fujino-fundamental Lemma 4.15]. \[x-lem3.2\] Let $D=\sum _{i\in I} a_i D_i$ be an $\mathbb R$-divisor on a smooth projective variety $V$, where $a_i\in \mathbb R$ and $D_i$ is a prime divisor on $V$ for every $i$ with $D_i\ne D_j$ for $i\ne j$. Assume that $D$ is semi-ample. Then we can construct a $\mathbb Q$-divisor $D^\dag=\sum _{i\in I} b_iD_i$ such that - $D^\dag$ is semi-ample, - $b_i=a_i$ holds if $a_i \in \mathbb Q$, and - $|b_i-a_i|\ll1$ holds for $a_i\in \mathbb R\setminus \mathbb Q$. Since $D$ is semi-ample, we can write $D=\sum _{j\in J} m_jM_j$ where $m_j\in \mathbb R$ and $M_j$ is a semi-ample Cartier divisor on $V$ for every $j$. As usual, by perturbing $m_j$s suitably, we get a desired semi-ample $\mathbb Q$-divisor $D^\dag$ on $V$. For the details, see, for example, the proof of [@fujino-fundamental Lemma 4.15]. Next, we treat a very useful covering trick, which is essentially due to Yujiro Kawamata. We will use it in the proof of Theorem \[x-thm9.3\]. \[x-lem3.3\] Let $f:V\to W$ be a projective surjective morphism between smooth quasi-projective varieties and let $H$ be a Cartier divisor on $W$. Let $d$ be an arbitrary positive integer. Then we can take a finite flat morphism $\tau:W'\to W$ from a smooth quasi-projective variety $W'$ and a Cartier divisor $H'$ on $W'$ such that $\tau^*H\sim dH'$ and that $V'=V\times _W W'$ is a smooth quasi-projective variety with $\omega_{V'/W'}=\rho^*\omega_{V/W}$, where $\rho:V'\to V$. By construction, we may assume that $\tau:W'\to W$ is Galois. We take general very ample Cartier divisors $D_1$ and $D_2$ with the following properties. - $H\sim D_1-D_2$, - $D_1$, $D_2$, $f^*D_1$, and $f^*D_2$ are smooth, - $D_1$ and $D_2$ have no common components, and - ${{\operatorname{Supp}}}(D_1+D_2)$ and ${{\operatorname{Supp}}}(f^*D_1+f^*D_2)$ are simple normal crossing divisors. We take a finite flat cover due to Kawamata with respect to $W$ and $D_1+D_2$. Then we obtain $\tau:W'\to W$ and $H'$ such that $\tau^*H\sim dH'$. By the construction of the above Kawamata cover $\tau:W'\to W$, we may assume that the ramification locus $\Sigma$ of $\tau$ in $W$ is a general simple normal crossing divisor. This means that $f^*P$ is a smooth divisor for any irreducible component $P$ of $\Sigma$ and that $f^*\Sigma$ is a simple normal crossing divisor on $V$. In this situation, we can easily check that $V'=V\times _WW'$ is a smooth quasi-projective variety. $$\xymatrix{ V' \ar[r]^{\rho} \ar[d]_{f'} & V\ar[d]^{f} \\ W' \ar[r]_{\tau} & W }$$ By construction, we can also easily check that $\omega_{V'/W'}=\rho^*\omega_{V/W}$ by the Hurwitz formula. Let us see the construction of $f':V'\to W'$ more precisely for the reader’s convenience. Let $\mathcal A$ be an ample invertible sheaf on $W$ such that $\mathcal A^{\otimes d}\otimes \mathcal O_W(-D_i)$ is generated by global sections for $i=1, 2$. We put $n=\dim W$. We take smooth divisors $$H^{(1)}_1, \ldots, H^{(1)}_n, H^{(2)}_1, \ldots, H^{(2)}_n$$ on $W$ in general position such that $\mathcal A^{\otimes d}= \mathcal O_W(D_i+H^{(i)}_j)$ for $1\leq j\leq n$ and $i=1, 2$. Let $Z^{(i)}_j$ be the cyclic cover associated to $\mathcal A^{\otimes d}= \mathcal O_W(D_i+H^{(i)}_j)$ for $1\leq j\leq n$ and $i=1, 2$. Then $W'$ is the normalization of $$\left(Z^{(1)}_1\times _W \cdots \times _WZ^{(1)}_n\right) \times _W \left(Z^{(2)}_1\times _W \cdots \times _WZ^{(2)}_n\right).$$ For the details, see, for example, [@esnault-viehweg 3.19. Lemma] and [@viehweg3 Lemma 2.5]. Let $S^{(i)}_j$ be the cyclic cover of $V$ associated to $\left(f^*\mathcal A\right)^{\otimes d} =\mathcal O_V(f^*D_i+f^*H^{(i)}_j)$. Then we define $V'$ as the normalization of $$\left(S^{(1)}_1\times _V \cdots \times _VS^{(1)}_n\right) \times _V \left(S^{(2)}_1\times _V \cdots \times _VS^{(2)}_n\right).$$ Note that $\rho:V'\to V$ is a finite flat morphism between smooth quasi-projective varieties. Since $V\times _WW'\to V$ is finite and flat and $V$ is smooth, $V\times _W W'$ is Cohen–Macaulay (see, for example, [@kollar-mori Corollary 5.5]). By construction, we can easily see that $V\times _W W'$ is smooth in codimension one. Therefore, $V\times _W W'$ is normal. Since $\rho$ factors through $V\times _W W'$ by construction, we see that $V'=V\times _W W'$ by Zariski’s main theorem. By the above construction of $\tau:W'\to W$, we see that $\tau:W'\to W$ is Galois. We give a very important remark on Lemma \[x-lem3.3\]. \[x-rem3.4\] In the proof of Lemma \[x-lem3.3\], let $S$ be any simple normal crossing divisor on $V$. Then we can choose the ramification locus $\Sigma$ of $\tau$ such that $f^*P\not\subset S$ for any irreducible component $P$ of $\Sigma$ and that $f^*\Sigma\cup S$ is a simple normal crossing divisor on $V$. If we choose $\Sigma$ as above, then we obtain that $\rho^*S$ is a simple normal crossing divisor on $V'$. Finally, let us explain Viehweg’s fiber product trick. We include the proof for the benefit of the reader. We will use it in the proof of Theorems \[y-thm1.7\] and \[y-thm1.8\] in Section \[x-sec10\]. \[x-rem3.5\] Let $V$ be a reduced Gorenstein scheme. Note that $V$ may be reducible. We consider $$V'\overset{\delta}\longrightarrow V^\nu\overset{\nu}\longrightarrow V$$ where $\nu:V^\nu\to V$ is the normalization and $\delta:V'\to V^\nu$ is a resolution of singularities. Then, for every positive integer $n$, we have $$\label{x-3.1} \nu_*\mathcal O_{V^\nu}(nK_{V^\nu})\subset \omega^{\otimes n}_V$$ and $$\label{x-3.2} \delta_*\mathcal O_{V'}(nK_{V'}+E)\subset \mathcal O_{V^\nu}(nK_{V^\nu})$$ where $E$ is any $\delta$-exceptional divisor on $V'$. In particular, we have $$\label{x-3.3} (\nu\circ \delta)_*\mathcal O_{V'}(nK_{V'}+E)\subset \omega^{\otimes n}_V$$ for every positive integer $n$. If $U$ is a Zariski open set of $V$ such that $\nu\circ \delta$ is an isomorphism over $U$, then the inclusion is an isomorphism over $U$. In Steps \[x-step3.5.1\] and \[x-step3.5.2\], we will prove and , respectively. \[x-step3.5.1\] By taking the double dual of $\delta_*\mathcal O_{V'}(nK_{V'}+E)$, we obtain $\mathcal O_{V^\nu}(nK_{V^\nu})$. Therefore, we have $$\delta_*\mathcal O_{V'}(nK_{V'}+E)\subset \mathcal O_{V^\nu}(nK_{V^\nu})$$ for every integer $n$. \[x-step3.5.2\] Since $\nu$ is birational, the trace map $\nu_*\mathcal O_{V^\nu}(K_{V^\nu})\to \omega_V$ is a generically isomorphic injection $$\label{x-3.4} \nu_*\mathcal O_{V^\nu}(K_{V^\nu})\hookrightarrow \omega_V.$$ Since $\nu$ is finite, $$\label{x-3.5} \nu^*\nu_*\mathcal O_{V^\nu}(K_{V^\nu})\to \mathcal O_{V^\nu}(K_{V^\nu})$$ is surjective and the kernel of is the torsion part of $\nu^*\nu_*\mathcal O_{V^\nu}(K_{V^\nu})$. Therefore, by , we get an inclusion $$\label{x-3.6} \mathcal O_{V^\nu}(K_{V^\nu})\hookrightarrow \nu^*\omega_V.$$ Let $n$ be a positive integer with $n\geq 2$. Then we have $$\mathcal O_{V^\nu}(nK_{V^\nu})=\mathcal O_{V^\nu} (K_{V^\nu}+(n-1)K_{V^\nu})\hookrightarrow \mathcal O_{V^\nu}(K_{V^\nu})\otimes \nu^*\omega^{\otimes n-1}_V$$ by . Therefore, by taking $\nu_*$, we get $$\nu_*\mathcal O_{V^\nu}(nK_{V^\nu}) \hookrightarrow \nu_*\mathcal O_{V^\nu}(K_{V^\nu})\otimes \omega^{\otimes n-1}_V\hookrightarrow \omega^{\otimes n}_V$$ by . This is what we wanted. By the above construction of and , it is obvious that the inclusion $$(\nu\circ \delta)_*\mathcal O_{V'}(nK_{V'}+E)\subset \omega^{\otimes n}_V$$ is an isomorphism over $U$. \[x-lem3.6\] Let $f:X_0\to Y_0$ be a projective surjective morphism between smooth quasi-projective varieties and let $\Delta_0$ be an effective $\mathbb R$-divisor on $X_0$ such that ${{\operatorname{Supp}}}\Delta_0$ is a simple normal crossing divisor on $X_0$ and $(X_0, \Delta_0)$ is log canonical over a nonempty Zariski open set of $Y_0$. Let $L_0$ be a Cartier divisor on $X_0$ such that $L_0\sim _{\mathbb R} k(K_{X_0/Y_0}+\Delta_0)$ for some positive integer $k$. Assume that $f$ is flat. We consider the $s$-fold fiber product $$X^s_0:=\underbrace{X_0\times _{Y_0} X_0 \times _{Y_0} \cdots \times _{Y_0} X_0}_{s}$$ of $X_0$ over $Y_0$ and let $f^s:X^s_0\to Y_0$ be the induced morphism. We take a resolution of singularities $\rho:X^{(s)}_0\to X^s_0$ which is an isomorphism over a nonempty Zariski open set of $Y_0$. Then we can write $$\mathcal O_{X^{(s)}_0}(K_{X^{(s)}_0})=\rho^*\omega_{X^s_0}\otimes \mathcal O_{X^{(s)}_0}(R)$$ where $R$ is an $(f^s\circ \rho)$-vertical Cartier divisor by construction. Let $p_i:X^s_0\to X_0$ be the $i$-th projection. We put $\pi_i=p_i\circ \rho:X^{(s)}_0\to X_0$. We consider $$L^{(s)}_0:=\sum _{i=1}^s \pi_i^*L_0+kR.$$ We further assume that $f_*\mathcal O_{X_0}(L_0)$ is locally free. Then there exists a generically isomorphic injection $$f^{(s)}_*\mathcal O_{X^{(s)}_0}(L^{(s)}_0)\hookrightarrow \bigotimes _{i=1}^sf_*\mathcal O_{X_0}(L_0)$$ with $f^{(s)}=f^s\circ \rho$. By construction, we have $$L^{(s)}_0 \sim _{\mathbb R}k\left(K_{X^{(s)}_0/Y_0}+\sum _{i=1}^s \pi_i^*\Delta_0\right).$$ Note that $$\left(X^{(s)}_0, \sum _{i=1}^s \pi_i^*\Delta_0\right)$$ is log canonical over a nonempty Zariski open set of $Y_0$. We also note that $X^{(s)}_0$ may be reducible, that is, $X^{(s)}_0$ may be a disjoint union of smooth varieties. By the flat base change theorem, we have $$\omega_{X^s_0/Y_0}=\bigotimes _{i=1}^s p^*_i\omega_{X_0/Y_0}.$$ In particular, $X^s_0$ is Gorenstein. We note that $$\label{x-3.7} \begin{split} L^{(s)}_0&=\sum _{i=1}^s \rho^*p^*_i (kK_{X_0/Y_0}+(L_0-kK_{X_0/Y_0}))+kR \\& \sim kK_{X^{(s)}_0/Y_0}+\sum _{i=1}^s \pi^*_i (L_0-kK_{X_0/Y_0}). \end{split}$$ We have the following isomorphism of locally free sheaves: $$f^s_*\mathcal O_{X^s_0}\left(\sum _{i=1}^s p^*_iL_0\right)\simeq \bigotimes ^s f_*\mathcal O_{X_0}(L_0).$$ We use induction on $s$. If $s=1$, then the statement is obvious. So we assume that $s\geq 2$. We consider the following commutative diagram $$\xymatrix{ X^s_0 \ar[dr]^-{f^s}\ar[d]_-{p_s}\ar[r]^-q& X^{s-1}_0 \ar[d]^-{f^{s-1}}\\ X_0 \ar[r]_-f& Y_0 }$$ where $q=(p_1, \cdots, p_{s-1})$. Then we have $$\label{x-3.8} \mathcal O_{X^s_0}\left(\sum _{i=1}^s p^*_i L_0\right)\simeq \mathcal O_{X^s_0}(p^*_sL_0)\otimes q^*\mathcal O_{X^{s-1}_0}\left(\sum _{i=1}^{s-1}p^*_iL_0\right).$$ Therefore, we obtain $$\begin{split} f^s_*\mathcal O_{X^s_0}\left(\sum _{i=1}^s p^*_iL_0\right) &\simeq f_*{p_s}_*\left(\mathcal O_{X^s_0}(p^*_sL_0)\otimes q^*\mathcal O_{X^{s-1}_0}\left(\sum _{i=1}^{s-1}p^*_iL_0\right)\right) \\ &\simeq f_*\left(\mathcal O_{X_0}(L_0)\otimes {p_s}_*q^*\mathcal O_{X^{s-1}_0}\left( \sum _{i=1}^{s-1}p^*_iL_0\right)\right) \\ &\simeq f_*\left(\mathcal O_{X_0}(L_0)\otimes f^*f^{s-1}_*\mathcal O_{X^{s-1}_0}\left( \sum _{i=1}^{s-1}p^*_iL_0\right)\right) \\ &\simeq f_*\left(\mathcal O_{X_0}(L_0)\otimes f^*\left(\bigotimes^{s-1}f_*\mathcal O_{X_0}(L_0)\right)\right) \\ &\simeq f_*\mathcal O_{X_0}(L_0)\otimes \bigotimes ^{s-1}f_*\mathcal O_{X_0}(L_0) \\ &\simeq \bigotimes ^s f_*\mathcal O_{X_0}(L_0). \end{split}$$ Note that the first isomorphism follows from , the second one is due to the projection formula, the third one is obtained by the flat base change theorem, the fourth one is due to induction on $s$, and the fifth one follows from the projection formula. Anyway, we obtain the desired isomorphism. Let us go back to the proof of Lemma \[x-lem3.6\]. We have an inclusion $$\begin{split} \rho_*\mathcal O_{X^{(s)}_0}(L^{(s)}_0)&\subset \omega^{\otimes k}_{X^s_0/Y_0}\otimes \mathcal O_{X^s_0}\left(\sum _{i=1}^s p^*_i (L_0-kK_{X_0/Y_0})\right)\\ &\simeq \mathcal O_{X^s_0}\left(\sum_{i=1}^s p^*_i L_0\right) \end{split}$$ by and Lemma \[x-rem3.5\], which is an isomorphism over a nonempty Zariski open set of $Y_0$. By taking $f^s_*$, we obtain a generically isomorphic injection $$\begin{split} f^{(s)}_*\mathcal O_{X^{(s)}_0}(L^{(s)}_0)&\subset f^s_*\mathcal O_{X^s_0} \left(\sum_{i=1}^s p^*_i L_0\right)\\ &\simeq \bigotimes _{i=1}^s f_*\mathcal O_{X_0}(L_0), \end{split}$$ where $f^{(s)}=f^s\circ \rho:X^{(s)}_0\to Y_0$. By assumption, $L_0-kK_{X_0/Y_0}\sim_{\mathbb R} k\Delta_0$. Therefore, $$L^{(s)}_0\sim _{\mathbb R} k\left(K_{X^{(s)}_0/Y_0} +\sum _{i=1}^s \pi^*_i \Delta_0\right)$$ by . We can take a nonempty Zariski open set $U$ of $Y_0$ such that $f$ is smooth over $U$, ${{\operatorname{Supp}}}\Delta$ is relatively simple normal crossing over $U$, and $\rho$ is an isomorphism over $U$. Then we see that $\left(X^{(s)}_0, \sum _{i=1}^s \pi^*_i \Delta_0\right)$ is log canonical over $U$. Weakly positive sheaves and big sheaves {#x-sec4} ======================================= We quickly see some basic properties of Viehweg’s weakly positive sheaves and big sheaves. For the details, see [@fujino-iitaka Chapter 3], [@viehweg1], [@viehweg2], and [@viehweg3]. \[x-def4.1\] Let $\mathcal F$ be a torsion-free coherent sheaf on a smooth quasi-projective variety $W$. We say that $\mathcal F$ is [*[weakly positive]{}*]{} if, for every positive integer $\alpha$ and every ample invertible sheaf $\mathcal H$, there exists a positive integer $\beta$ such that $\widehat{S}^{\alpha\beta}(\mathcal F)\otimes \mathcal H^{\otimes \beta}$ is generically generated by global sections. We say that a nonzero torsion-free coherent sheaf $\mathcal F$ is [*[big]{}*]{} (in the sense of Viehweg) if, for every ample invertible sheaf $\mathcal H$, there exists a positive integer $a$ such that $\widehat{S}^a(\mathcal F)\otimes \mathcal H^{\otimes -1}$ is weakly positive. For the reader’s convenience, let us recall the following basic properties of big sheaves without proof. \[x-lem4.2\] Let $\mathcal F$ be a nonzero torsion-free coherent sheaf on a smooth quasi-projective variety $W$. Then the following three conditions are equivalent. - There exist an ample invertible sheaf $\mathcal H$ on $W$, some positive integer $\nu$, and an inclusion $\bigoplus \mathcal H\hookrightarrow \widehat{S}^\nu(\mathcal F)$, which is an isomorphism over a nonempty Zariski open set of $W$. - For every invertible sheaf $\mathcal M$ on $W$, there exists some positive integer $\gamma$ such that $\widehat {S}^\gamma (\mathcal F)\otimes \mathcal M^{\otimes -1}$ is weakly positive. In particular, $\mathcal F$ is a big sheaf. - There exist some positive integer $\gamma$ and an ample invertible sheaf $\mathcal M$ on $W$ such that $\widehat {S}^\gamma (\mathcal F)\otimes \mathcal M^{\otimes -1}$ is weakly positive. We will use the following two easy lemmas on big sheaves in this paper. So we explicitly state them here for the reader’s convenience. \[x-lem4.3\] Let $\mathcal F$ be a weakly positive sheaf and let $\mathcal H$ be an ample invertible sheaf on a smooth quasi-projective variety $W$. Then $\mathcal F\otimes \mathcal H$ is big. We give a proof for the sake of completeness. Since $\mathcal F$ is weakly positive, $\widehat{S}^{2b}(\mathcal F)\otimes \mathcal H^{\otimes b}$ is generically generated by global sections for some positive integer $b$. By replacing $b$ with a multiple, we may assume that $\mathcal H^{\otimes b-1}$ is generated by global sections. Then $\widehat{S}^{2b}(\mathcal F\otimes \mathcal H) \otimes \mathcal H^{\otimes -1}= \widehat{S}^{2b}(\mathcal F)\otimes \mathcal H^{\otimes 2b-1}$ is generically generated by global sections. In particular, $\widehat {S}^{2b}(\mathcal F\otimes \mathcal H) \otimes \mathcal H^{\otimes -1}$ is weakly positive. This implies that $\mathcal F\otimes \mathcal H$ is big by Lemma \[x-lem4.2\]. \[x-lem4.4\] Let $\mathcal F$ be a torsion-free coherent sheaf on a smooth quasi-projective variety $W$ and let $\tau:W'\to W$ be a finite surjective morphism from a smooth quasi-projective variety $W'$. Assume that $\tau^*\mathcal F$ is big. Then $\mathcal F$ is a big sheaf on $W$. We include the proof for the benefit of the reader. We take an ample invertible sheaf $\mathcal H$ on $W$. By replacing $W$ with $W\setminus \Sigma$ for some suitable closed subset $\Sigma$ of codimension $\geq 2$ (see, for example, [@fujino-iitaka Lemma 3.1.12 (i)]), we may assume that $\mathcal F$ is locally free. Since $\tau^*\mathcal F$ is big by assumption, there exists a positive integer $a$ such that $S^a(\tau^*\mathcal F)\otimes \tau^*\mathcal H^{\otimes -1}= \tau^*\left(S^a(\mathcal F)\otimes \mathcal H^{\otimes -1}\right)$ is weakly positive (see Lemma \[x-lem4.2\]). Therefore, $S^a(\mathcal F)\otimes \mathcal H^{\otimes -1}$ is weakly positive since $\tau$ is finite (see, for example, [@fujino-iitaka Lemma 3.1.12 (v)]). This means that $\mathcal F$ is big by Lemma \[x-lem4.2\]. Mixed-$\omega$-sheaves and mixed-$\omega$-big sheaves {#x-sec5} ===================================================== In this section, we introduce mixed-$\omega$-sheaves, mixed-$\omega$-big-sheaves, mixed-$\widehat\omega$-sheaves, and mixed-$\widehat\omega$-big-sheaves. We also treat some important examples in Lemmas \[x-lem5.5\], \[x-lem5.8\], and \[x-lem5.9\]. Let us start with the definition of mixed-$\omega$-sheaves and pure-$\omega$-sheaves. \[x-def5.1\] A torsion-free coherent sheaf $\mathcal F$ on a normal quasi-projective variety $W$ is called a [*[mixed-$\omega$-sheaf]{}*]{} if there exist a projective surjective morphism from a smooth quasi-projective variety $V$ and a simple normal crossing divisor $D$ on $V$ such that $\mathcal F$ is a direct summand of $f_*\mathcal O_V(K_V+D)$. When $D=0$, $\mathcal F$ is called a [*[pure-$\omega$-sheaf]{}*]{} on $W$. We give a very important remark on Definition \[x-def5.1\]. \[x-rem5.2\] The notion of pure-$\omega$-sheaves is essentially the same as that of Nakayama’s $\omega$-sheaves in [@nakayama] when we treat torsion-free coherent sheaves on normal projective varieties. However, the definition of pure-$\omega$-sheaves in Definition \[x-def5.1\] does not coincide with [@nakayama Chapter V, 3.8. Definition]. Our definition seems to be more reasonable than Nakayama’s from the mixed Hodge theoretic viewpoint. For some geometric applications, the notion of mixed-$\omega$-big-sheaves and pure-$\omega$-big-sheaves is very useful. \[x-def5.3\] Let $\mathcal F$ be a torsion-free coherent sheaf on a normal quasi-projective variety $W$. If there exist projective surjective morphisms $f:V\to W$, $p:V\to Z$, and an ample divisor $A$ on $Z$ satisfying the following conditions: - $V$ is a smooth quasi-projective variety, - $Z$ is a normal quasi-projective variety, - $D$ is a simple normal crossing divisor on $V$, - there exists a projective surjective morphism $q:Z\to W$ such that $f=q\circ p$, and - $\mathcal F$ is a direct summand of $f_*\mathcal O_V(K_V+D+P)$, where $P$ is a Cartier divisor on $V$ such that $P\sim _{\mathbb Q}p^*A$, $$\xymatrix{ V \ar[dd]_-f\ar[dr]^-p& \\ & Z \ar[dl]^-q\\ W }$$ then $\mathcal F$ is called a [*[mixed-$\omega$-big-sheaf]{}*]{} on $W$. As in Definition \[x-def5.1\], $\mathcal F$ is called a [*[pure-$\omega$-big-sheaf]{}*]{} on $W$ when $D=0$. \[x-rem5.4\] Of course, we defined mixed-$\omega$-big-sheaves and pure-$\omega$-big-sheaves referring to [@nakayama Chapter V, 3.16. Definition (1)]. However, Nakayama’s definition of $\omega$-bigness is different from ours. Lemma \[x-lem5.5\] gives a very basic example of mixed-$\omega$-sheaves. \[x-lem5.5\] Let $V$ be a smooth quasi-projective variety and let $D$ be a simple normal crossing divisor on $V$. Let $L$ be a semi-ample Cartier divisor on $V$. Then $\mathcal O_V(K_V+D+L)$ is a mixed-$\omega$-sheaf on $V$ and $\mathcal O_V(K_V+L)$ is a pure-$\omega$-sheaf on $V$. Although this lemma is well known, we give a proof for the sake of completeness. Let $m$ be a positive integer such that $|mL|$ is free. We take a general section $s\in H^0(V, \mathcal O_V(mL))$, whose zero divisor is $B$. We may assume that $B$ is a smooth divisor, $B$ and $D$ have no common irreducible components, and ${{\operatorname{Supp}}}(B+D)$ is a simple normal crossing divisor on $V$. The dual of $$s: \mathcal O_V\to \mathcal O_V(mL)$$ defines an $\mathcal O_V$-algebra structure on $$\bigoplus_{i=0}^{m-1}\mathcal O_V(-iL).$$ We put $$\pi:Z:={{\operatorname{Spec}}}_V\bigoplus_{i=0}^{m-1}\mathcal O_V(-iL) \to V.$$ Then $Z$ is a smooth quasi-projective variety and $\pi^*D$ is a simple normal crossing divisor on $Z$ by construction. We can check that $$\pi_*\mathcal O_Z(K_Z+\pi^*D)\simeq \bigoplus _{i=0}^{m-1} \mathcal O_V(K_V+D+iL)$$ since $\pi_*\mathcal O_V=\bigoplus_{i=0}^{m-1} \mathcal O_V(-iL)$. This means that $\mathcal O_V(K_V+D+L)$ is a mixed-$\omega$-sheaf on $V$. We put $D=0$ in the above argument. Then we see that $\mathcal O_V(K_V+L)$ is a pure-$\omega$-sheaf on $V$. We treat two elementary lemmas. \[x-lem5.6\] Let $\mathcal F$ be a mixed-$\omega$-big-sheaf [*[(]{}*]{}resp. pure-$\omega$-big-sheaf[*[)]{}*]{} on a normal quasi-projective variety $W$. Then $\mathcal F$ is a mixed-$\omega$-sheaf [*[(]{}*]{}resp. pure-$\omega$-sheaf[*[)]{}*]{} on $W$. We may assume that $\mathcal F$ is a direct summand of $f_*\mathcal O_V(K_V+D+P)$ as in Definition \[x-def5.3\]. By Lemma \[x-lem5.5\], $\mathcal O_V(K_V+D+P)$ is a mixed-$\omega$-sheaf on $V$. Therefore, we see that $\mathcal F$ is a mixed-$\omega$-sheaf on $W$. If we put $D=0$, then we see that $\mathcal F$ is a pure-$\omega$-sheaf on $W$. \[x-lem5.7\] Let $\mathcal F$ be a mixed-$\omega$-sheaf [*[(]{}*]{}resp. pure-$\omega$-sheaf[*[)]{}*]{} on a normal quasi-projective variety $W$ and let $\mathcal A$ be an ample invertible sheaf on $W$. Then $\mathcal F\otimes \mathcal A$ is a mixed-$\omega$-big-sheaf [*[(]{}*]{}resp. pure-$\omega$-big-sheaf[*[)]{}*]{} on $W$. We may assume that $\mathcal F$ is a direct summand of $f_*\mathcal O_V(K_V+D)$ as in Definition \[x-def5.1\]. We put $Z=W$. Let $A$ be an ample divisor on $W$ such that $\mathcal O_W(A)=\mathcal A$. Then $\mathcal F\otimes \mathcal A$ is a direct summand of $f_*\mathcal O_V(K_V+D+f^*A)$. Therefore, $\mathcal F\otimes \mathcal A$ is a mixed-$\omega$-big-sheaf on $W$. When $D=0$, we see that $\mathcal F\otimes \mathcal A$ is a pure-$\omega$-big-sheaf on $W$. Lemmas \[x-lem5.8\] and \[x-lem5.9\] give many nontrivial important examples of mixed-$\omega$-sheaves and mixed-$\omega$-big-sheaves in the study of higher-dimensional algebraic varieties. \[x-lem5.8\] Let $f: V\to W$ be a projective surjective morphism from a smooth projective variety $V$ onto a normal projective variety $W$. Let $D$ be a simple normal crossing divisor on $V$ and let $M$ be an $\mathbb R$-divisor on $V$ such that $M-f^*H$ is semi-ample for some ample $\mathbb Q$-divisor $H$ on $W$. We assume that $D$ and ${{\operatorname{Supp}}}\{M\}$ have no common irreducible components and ${{\operatorname{Supp}}}(D+\{M\})$ is a simple normal crossing divisor on $V$. Then $f_*\mathcal O_V(K_V+D+\lceil M\rceil)$ is a mixed-$\omega$-big-sheaf on $W$ and $f_*\mathcal O_V(K_V+\lceil M\rceil)$ is a pure-$\omega$-big-sheaf on $W$. By Lemma \[x-lem3.2\], we can construct a $\mathbb Q$-divisor $M^\dag$ on $V$ such that $M^\dag-f^*H$ is semi-ample, ${{\operatorname{Supp}}}\{M^\dag\}={{\operatorname{Supp}}}\{M\}$, and $\lceil M^\dag\rceil =\lceil M\rceil$. Therefore, we may assume that $M$ is a $\mathbb Q$-divisor by replacing $M$ with $M^\dag$. By Kawamata’s covering construction, we can construct a finite Galois cover $\pi:V'\to V$ from a smooth projective variety $V'$ with the following properties: - $\pi^*D$ is a simple normal crossing divisor on $V'$, - $\pi^*\{M\}$ is a $\mathbb Z$-divisor on $V'$, - ${{\operatorname{Supp}}}(\pi^*D+\pi^*\{M\})$ is a simple normal crossing divisor on $V'$, and - $\left(\pi_*\mathcal O_{V'}(K_{V'}+\pi^*D+\pi^*M)\right)^G \simeq \mathcal O_V(K_V+D+\lceil M\rceil)$, where $G$ is the Galois group of $\pi:V'\to V$. By assumption, $\pi^*M$ is semi-ample. Since $\pi^*M-(f\circ \pi)^*H$ is semi-ample, we have the following commutative diagram: $$\xymatrix{ V'\ar[dd]_-{f\circ\pi}\ar[dr]^-p & \\ & Z\ar[dl]^-q\\ W }$$such that - $Z$ is a normal projective variety, and - there is an ample $\mathbb Q$-divisor $A$ on $Z$ with $\pi^*M\sim _{\mathbb Q}p^*A$. Therefore, $f_*\mathcal O_V(K_V+D+\lceil M\rceil)$ is a mixed-$\omega$-big-sheaf on $W$ since it is a direct summand of $(f\circ \pi)_*\mathcal O_{V'}(K_{V'}+ \pi^*D+\pi^*M)$. We put $D=0$ in the above argument. Then $f_*\mathcal O_V(K_V+\lceil M\rceil)$ is a pure-$\omega$-big-sheaf on $W$. \[x-lem5.9\] Let $V$ be a smooth quasi-projective variety and let $D$ be a simple normal crossing divisor on $V$. Let $B$ be a $\mathbb Q$-divisor on $V$ such that $rB\sim 0$ for some positive integer $r$, ${{\operatorname{Supp}}}\{B\}$ and $D$ have no common irreducible components, and ${{\operatorname{Supp}}}(\{B\}+D)$ is a simple normal crossing divisor on $V$. Then there exist a generically finite morphism $\pi:V'\to V$ from a smooth quasi-projective variety $V'$ and a simple normal crossing divisor $D'$ on $V'$ such that $\mathcal O_V(K_V+D+\lceil B\rceil)$ is a direct summand of $\pi_*\mathcal O_{V'}(K_{V'}+D')$. In particular, $\mathcal O_V(K_V+D+\lceil B\rceil)$ is a mixed-$\omega$-sheaf on $V$. When $D=0$, $\mathcal O_V(K_V+\lceil B\rceil)$ is obviously a pure-$\omega$-sheaf on $V$. If $B\sim 0$, then there are nothing to prove. By replacing $r$ suitably, we may assume that $iB\not\sim 0$ for $1\leq i\leq r-1$ and that $r\geq 2$. We consider the following $\mathcal O_V$-algebra $$\mathcal A=\bigoplus _{i=0}^{r-1} \mathcal O_V(\lfloor -iB\rfloor)$$ defined by an isomorphism $\mathcal O_V(-rB)\simeq \mathcal O_V$. Let $Z$ be the normalization of ${{\operatorname{Spec}}}_V\mathcal A$. Then we have $$\tau_*\mathcal O_Z(K_Z+\tau^*D)\simeq \bigoplus _{i=0}^{r-1}(K_V+D+ \lceil iB\rceil)$$ where $\tau:Z\to V$. By construction, we see that $(Z, \tau^*D)$ is dlt. We take a suitable resolution of singularities $\rho:V'\to Z$ and write $$K_{V'}+D'=\rho^*(K_Z+\tau^*D)+E$$ where $D'$ is a reduced simple normal crossing divisor on $V'$ and $E$ is an effective $\rho$-exceptional $\mathbb Q$-divisor on $V'$. We put $\pi:=\tau\circ \rho: V'\to V$. Then $$\begin{split} \pi_*\mathcal O_{V'}(K_{V'}+D')&\simeq \tau_*\mathcal O_Z(K_Z+\tau^*D)\\ &\simeq \bigoplus_{i=0}^{r-1}\mathcal O_V(K_V+D+\lceil iB\rceil). \end{split}$$ Therefore, we have the desired statement. We close this section with the definition of mixed-$\widehat\omega$-sheaves, mixed-$\widehat\omega$-big-sheaves, pure-$\widehat\omega$-sheaves, and pure-$\widehat\omega$-big-sheaves. \[x-def5.10\] A torsion-free coherent sheaf $\mathcal G$ on a normal quasi-projective variety $W$ is called a [*[mixed-$\widehat\omega$-sheaf]{}*]{} (resp. [*[mixed-$\widehat\omega$-big-sheaf]{}*]{}) if there exist a mixed-$\omega$-sheaf (resp. mixed-$\omega$-big-sheaf) $\mathcal F$ on $W$ and a generically isomorphic injection $\mathcal F\hookrightarrow \mathcal G^{**}$ into the double dual $\mathcal G^{**}$ of $\mathcal G$. If $\mathcal F$ is a pure-$\omega$-sheaf (resp. pure-$\omega$-big-sheaf) in the above inclusion $\mathcal F\hookrightarrow \mathcal G^{**}$, then $\mathcal G$ is called a [*[pure-$\widehat\omega$-sheaf]{}*]{} (resp. [*[pure-$\widehat\omega$-big-sheaf]{}*]{}). Let $f:X\to Y$ be a surjective morphism between smooth projective varieties and let $\Delta$ be a simple normal crossing divisor on $X$. Let $k$ be a positive integer with $k\geq 2$ and let $H$ be an ample Cartier divisor on $Y$. Then we will show that $$\mathcal O_Y(K_Y+H)\otimes f_*\mathcal O_X(k(K_{X/Y}+\Delta))$$ is a mixed-$\widehat\omega$-big-sheaf on $Y$. This is a special case of Theorem \[x-thm9.3\], which we call a fundamental theorem of the theory of mixed-$\omega$-sheaves. Basic properties: Part 1 {#x-sec6} ======================== In this section, we treat the weak positivity and the bigness of mixed-$\omega$-sheaves and mixed-$\omega$-big-sheaves, respectively. Let us start with the following weak positivity theorem, which follows from the theory of mixed Hodge structures. \[x-thm6.1\] Let $f:V\to W$ be a projective surjective morphism between smooth quasi-projective varieties. Let $D$ be a simple normal crossing divisor on $V$. Then $f_*\mathcal O_V(K_{V/W}+D)$ is weakly positive. We may assume that $V$ and $W$ are smooth projective varieties by compactifying $f:V\to W$ suitably. Then this result is more or less well known. For the proof based on the theory of variations of mixed Hodge structure (see [@fujino-higher], [@ffs], [@fujino-fujisawa], [@fujisawa], and so on), see [@fujino-weak Theorem 7.8 and Corollary 7.11]. For the proof based on the vanishing theorem, see [@fujino-quasi-alb Theorem 8.4]. As an easy consequence of Theorem \[x-thm6.1\], we have: \[x-thm6.2\] Let $\mathcal F$ be a mixed-$\omega$-sheaf on a smooth quasi-projective variety $W$. Then $\mathcal F\otimes \omega^{\otimes -1}_W$ is weakly positive. We may assume that $\mathcal F$ is a direct summand of $f_*\mathcal O_V(K_V+D)$ as in Definition \[x-def5.1\]. By Theorem \[x-thm6.1\], $f_*\mathcal O_V(K_{V/W}+D)$ is weakly positive. Then $\mathcal F\otimes \omega^{\otimes -1}_W$ is weakly positive since it is a direct summand of $f_*\mathcal O_V(K_{V/W}+D)$. When $\mathcal F$ is a mixed-$\omega$-big-sheaf on $W$ in Theorem \[x-thm6.2\], $\mathcal F\otimes \omega^{\otimes -1}_W$ is not only weakly positive but also big. \[x-thm6.3\] Let $\mathcal F$ be a mixed-$\omega$-big-sheaf on a smooth quasi-projective variety $W$. Then $\mathcal F\otimes \omega^{\otimes -1}_W$ is big. Without loss of generality, we may assume that $\mathcal F$ is a direct summand of $f_*\mathcal O_V(K_V+D+P)$ as in Definition \[x-def5.3\]. It is sufficient to prove that $f_*\mathcal O_V(K_{V/W}+D+P)$ is big. Let $$\xymatrix{ V \ar[dd]_-f\ar[dr]^-p& \\ & Z \ar[dl]^-q\\ W }$$ and $A$ be as in Definition \[x-def5.3\]. Let $H$ be an ample Cartier divisor on $W$. We take a positive integer $m$ such that $mA-q^*H$ is ample. We can take a finite surjective morphism $\tau:W'\to W$ from a smooth quasi-projective variety $W'$ and get the following commutative diagram $$\xymatrix{ V' \ar[d]_-{f'}\ar[r]^-\rho& V \ar[d]^-f\\ W' \ar[r]_-\tau&W }$$ such that $\tau^*H\sim mH'$ for some Cartier divisor $H'$, $V'=V\times _W W'$ is a smooth quasi-projective variety, $\rho^*D$ is a simple normal crossing divisor, and $\rho^*\omega^{\otimes n}_{V/W} =\omega^{\otimes n}_{V/W}$ holds for every integer $n$ (see Lemma \[x-lem3.3\] and Remark \[x-rem3.4\]). By Lemma \[x-lem4.4\], It is sufficient to prove that $$\tau^*f_*\mathcal O_V(K_{V/W}+D+P)\simeq f'_*\mathcal O_{V'}(K_{V'/W'}+ \rho^*D+\rho^*P)$$ is a big sheaf on $W'$. By construction, we see that $\rho^*P-f'^*H'$ is a semi-ample Cartier divisor on $V'$ since it is $\mathbb Q$-linearly equivalent to $\rho^*p^*(A-(1/m)q^*H)$. Therefore, by Lemma \[x-lem5.5\], $\mathcal O_{V'}(K_{V'}+\rho^*D+\rho^*P-f'^*H')$ is a mixed-$\omega$-sheaf on $V'$. Therefore, $\mathcal E:=f'_*\mathcal O_{V'}(K_{V'}+\rho^*D +\rho^*P-f'^*H')$ is a mixed-$\omega$-sheaf on $W'$. We note that $$f'_*\mathcal O_{V'}(K_{V'/W'}+\rho^*D +\rho^*P)\simeq \mathcal E \otimes \omega^{\otimes -1}_{W'}\otimes \mathcal O_{W'}(H').$$ By Theorem \[x-thm6.2\], $\mathcal E\otimes \omega^{\otimes -1}_{W'}$ is weakly positive. By Lemma \[x-lem4.3\], $\mathcal E\otimes \omega^{\otimes -1}_{W'}\otimes \mathcal O_{W'}(H')$ is big since $H'$ is ample. This is what we wanted. We close this section with an obvious corollary. \[x-cor6.4\] Let $\mathcal F$ be a mixed-$\widehat\omega$-sheaf [*[(]{}*]{}resp. mixed-$\widehat\omega$-big-sheaf[*[)]{}*]{} on a smooth quasi-projective variety $W$. Then $\mathcal F\otimes \omega^{\otimes -1}_W$ is weakly positive [*[(]{}*]{}resp. big[*[)]{}*]{}. We note that $\mathcal F\otimes \omega^{\otimes -1}_W$ is weakly positive (resp. big) if and only if so is $\mathcal F^{**}\otimes \omega^{\otimes -1}_W$. Therefore, the desired statement follows from Theorems \[x-thm6.2\] and \[x-thm6.3\]. Basic properties: Part 2 {#x-sec7} ======================== In this section, we discuss some vanishing theorems for mixed-$\omega$-sheaves and several related topics. \[x-lem7.1\] Let $\mathcal F$ be a mixed-$\omega$-big-sheaf on a normal projective variety $W$. Then $H^i(W, \mathcal F\otimes \mathcal N)=0$ for every $i>0$ and every nef invertible sheaf $\mathcal N$ on $W$. We may assume that $\mathcal F$ is a direct summand of $f_*\mathcal O_V(K_V+D+P)$ as in Definition \[x-def5.3\]. Let $N$ be a Cartier divisor on $W$ such that $\mathcal N\simeq \mathcal O_W(N)$. It is sufficient to prove that $H^i(W, f_*\mathcal O_V(K_V+D+P+f^*N))=0$ for every $i>0$. We take an ample $\mathbb Q$-divisor $H$ on $W$ such that $A-q^*H$ is an ample $\mathbb Q$-divisor on $Z$, where $A$ and $q:Z\to W$ are as in Definition \[x-def5.3\]. Then we can take a boundary $\mathbb Q$-divisor $\Delta$ on $V$ such that $\Delta\sim _{\mathbb Q} D+P-f^*H$ and that ${{\operatorname{Supp}}}\Delta$ is a simple normal crossing divisor on $V$. Then we have $$K_V+D+P+f^*N-(K_V+\Delta)\sim _{\mathbb Q} f^*(H+N).$$ We note that $H+N$ is ample. Therefore, by [@fujino-fundamental Theorem 6.3 (ii)] (see also [@fujino-foundations Theorem 3.16.3 (ii) and Theorem 5.6.2 (ii)], and so on), we obtain that $H^i(W, f_*\mathcal O_V(K_V+D+P+f^*N))=0$ for every $i>0$. As an easy consequence of Lemma \[x-lem7.1\], we have: \[x-lem7.2\] Let $\mathcal F$ be a mixed-$\omega$-sheaf [*[(]{}*]{}resp. mixed-$\omega$-big-sheaf[*[)]{}*]{} on a normal projective variety $W$ with $\dim W=n$. Let $\mathcal A$ be an ample invertible sheaf on $W$ such that $|\mathcal A|$ is free. Then $\mathcal F\otimes \mathcal A^{\otimes n+1}$ [*[(]{}*]{}resp. $\mathcal F\otimes \mathcal A^{\otimes n}$[*[)]{}*]{} is generated by global sections. This is a direct consequence of Lemmas \[x-lem5.7\], \[x-lem7.1\], and Castelnuovo–Mumford regularity. Let us recall a vanishing theorem for dlt pairs. \[x-lem7.3\] Let $f:V\to W$ be a surjective morphism from a smooth projective variety $V$ onto a normal projective variety $W$. Let $\Delta$ be an effective $\mathbb R$-divisor on $V$ such that $(V, \Delta)$ is dlt and that every log canonical center of $(V, \Delta)$ is dominant onto $W$. Let $L$ be a Cartier divisor on $V$ such that $L-(K_V+\Delta)\sim _{\mathbb R} f^*H$ for some nef and big $\mathbb R$-divisor $H$ on $W$. Then $H^i(W, R^jf_*\mathcal O_V(L)\otimes \mathcal N)=0$ for $i>0$, $j\geq 0$, and every nef invertible sheaf $\mathcal N$ on $W$. By Kodaira’s lemma, we can write $H\sim _{\mathbb R}A+E$ such that $A$ is an ample $\mathbb R$-divisor on $W$ and $E$ is an effective $\mathbb R$-Cartier $\mathbb R$-divisor on $W$. Since every log canonical center of $(V, \Delta)$ is dominant onto $W$, $(V, \Delta+\varepsilon f^*E)$ is dlt for $0<\varepsilon \ll 1$. Let $N$ be a Cartier divisor on $W$ such that $\mathcal N\simeq \mathcal O_W(N)$. We note that $$L+f^*N-(K_V+\Delta+\varepsilon f^*E)\sim _{\mathbb R} f^*(N+(1-\varepsilon )H+\varepsilon A)$$ and that $N+(1-\varepsilon )H+\varepsilon A$ is ample for $0<\varepsilon \ll 1$. By [@fujino-weak Lemma 7.14], $$H^i(W, R^jf_*\mathcal O_V(L)\otimes \mathcal N)=0$$ for $i>0$ and $j\geq 0$. In Lemmas \[x-lem7.4\] and \[x-lem7.5\], we treat mixed-$\widehat\omega$-big-sheaves on smooth projective curves. \[x-lem7.4\] Let $\mathcal G$ be a mixed-$\widehat\omega$-big-sheaf on a smooth projective curve $C$. Then $H^1(C, \mathcal G\otimes \mathcal N)=0$ holds for every nef invertible sheaf $\mathcal N$ on $C$. We note that $\mathcal G$ is locally free since $C$ is a smooth curve. By definition, we have a mixed-$\omega$-big-sheaf $\mathcal F$ on $C$ and a generically isomorphic injection $\iota: \mathcal F\hookrightarrow \mathcal G$. Note that the cokernel of $\iota$ is a skyscraper sheaf on $C$. By Lemma \[x-lem7.1\], $H^1(C, \mathcal F\otimes \mathcal N)=0$ holds. Therefore, we have $H^1(C, \mathcal G\otimes \mathcal N)=0$ by the surjection $H^1(C, \mathcal F\otimes \mathcal N)\to H^1(C, \mathcal G \otimes \mathcal N)$. \[x-lem7.5\] Let $\mathcal E$ be a locally free sheaf on a smooth projective curve $C$ and let $P$ be a closed point of $C$. If $\mathcal E\otimes \mathcal O_C(-P)\otimes \mathcal N^{\otimes -1}$ is a mixed-$\widehat \omega$-big-sheaf on $C$ for some nef invertible sheaf $\mathcal N$ on $C$, then $\mathcal E$ is generated by global sections at $P$. By Lemma \[x-lem7.4\], $H^1(C, \mathcal E\otimes \mathcal O_C(-P))=0$. This means that the natural restriction map $$H^0(C, \mathcal E)\to \mathcal E\otimes \mathbb C(P)$$ is surjective. Therefore, $\mathcal E$ is generated by global sections at $P$. Let us discuss generically global generations of mixed-$\omega$-big-sheaves. \[x-lem7.6\] Let $\mathcal F$ be a mixed-$\omega$-sheaf on a normal projective variety $W$ with $\dim W=n$. Let $H$ be a big Cartier divisor on $W$ such that $|H|$ is free. Then $\mathcal F\otimes \mathcal O_W((n+1)H)$ is generically generated by global sections. We may assume that $\mathcal F=f_*\mathcal O_V(K_V+D)$ as in Definition \[x-def5.1\]. \[x-step7.6.1\] Let $\mu:\widetilde V\to V$ be a projective birational morphism from a smooth projective variety $\widetilde V$ such that $$K_{\widetilde V}+\widetilde D=\mu^*(K_V+D)+E$$ where $\widetilde D$ and $E$ are effective divisors and have no common irreducible components. Since $\mu_*\mathcal O_{\widetilde V}(K_{\widetilde V} +\widetilde D)\simeq \mathcal O_V(K_V+D)$, we may replace $(V, D)$ and $f:V\to W$ with $(\widetilde V, \widetilde D)$ and $f\circ\mu: \widetilde V\to W$, respectively. By taking $\mu:\widetilde V\to V$ suitably, we may assume that all the log canonical centers of $(V, D_{\mathrm{hor}})$ are dominant onto $V$, where $D_{\mathrm{hor}}$ is the horizontal part of $D$. Since $f_*\mathcal O_V(K_V+D_{\mathrm{hor}}) \hookrightarrow f_*\mathcal O_V(K_V+D)$ is a generically isomorphic injection, we may replace $D$ with $D_{\mathrm{hor}}$. \[x-step7.6.2\] We will prove that $f_*\mathcal O_V(K_V+D)\otimes \mathcal O_V((n+1)H)$ is generically generated by global sections by induction on $n=\dim W$. We take a general member $W'$ of $|H|$. We put $f^{-1}(W')=V'$. Then we have a short exact sequence $$0\to \mathcal O_V(K_V+D)\to \mathcal O_V(K_V+V'+D)\to \mathcal O_{V'}(K_{V'}+D|_{V'})\to 0$$ by adjunction. Since $W'$ is a general member of $|H|$, we get a short exact sequence $$\label{x-7.1} \begin{split} 0&\to f_*\mathcal O_V(K_V+D)\otimes \mathcal O_W(nH)\to f_*\mathcal O_V(K_V+V'+D)\otimes \mathcal O_W(nH)\\&\to f_*\mathcal O_{V'}(K_{V'}+D|_{V'})\otimes \mathcal O_{W'}(nH|_{W'})\to 0. \end{split}$$ By the vanishing theorem (see Lemma \[x-lem7.3\]), we have $$\label{x-7.2} H^1(W, f_*\mathcal O_V(K_V+D)\otimes \mathcal O_W(nH))=0.$$ Therefore, the restriction map $$H^0(W, f_*\mathcal O_V(K_V+D)\otimes \mathcal O_W((n+1)H))\to H^0(W', f_*\mathcal O_{V'}(K_{V'}+D|_{V'})\otimes \mathcal O_{W'}(nH|_{W'}))$$ is surjective by and . By induction on $n$, $f_*\mathcal O_{V'}(K_{V'}+D|_{V'}) \otimes \mathcal O_{W'}(nH|_{W'})$ is generically generated by global sections. This implies that so is $f_*\mathcal O_V(K_V+D)\otimes \mathcal O_W((n+1)H)$. We obtain the desired statement. Lemma \[x-lem7.7\] is similar to Lemma \[x-lem7.6\]. \[x-lem7.7\] Let $\mathcal F$ be a mixed-$\omega$-big-sheaf on a normal projective variety $W$ with $\dim W=n$. Let $H$ be a big Cartier divisor on $W$ such that $|H|$ is free. Then $\mathcal F\otimes \mathcal O_W(nH)$ is generically generated by global sections. The proof of Lemma \[x-lem7.7\] is essentially the same as that of Lemma \[x-lem7.6\]. If $n=0$, then the statement is trivial. If $n=1$, then it follows from Lemma \[x-lem7.5\]. Therefore, we assume that $n\geq 2$. As in the proof of Lemma \[x-lem7.6\], we may assume that $\mathcal F=f_*\mathcal O_V(K_V+D+P)$ as in Definition \[x-def5.3\]. Moreover, we may further assume that $D=D_{\mathrm{hor}}$ and that every log canonical center of $(V, D)$ is dominant onto $W$ (see Step \[x-step7.6.1\] in the proof of Lemma \[x-lem7.6\]). We take a general member $W'$ of $|H|$ and put $V'=f^{-1}(W')$. Then the natural restriction map $$\begin{split} &H^0(W, f_*\mathcal O_V(K_V+D+P)\otimes \mathcal O_W(nH))\\ &\to H^0(W', f_*\mathcal O_{V'}(K_{V'}+D|_{V'}+P|_{V'})\otimes \mathcal O_{W'}((n-1)H|_{W'})) \end{split}$$ is surjective as in Step \[x-step7.6.2\] in the proof of Lemma \[x-lem7.6\]. By induction on dimension, we see that $$f_*\mathcal O_V(K_V+D+P)\otimes \mathcal O_W(nH)$$ is generically generated by global sections. We close this section with the following result, which is due to [@dutta-murayama]. We will use it in the proof of Theorem \[y-thm1.8\]. \[x-lem7.8\] Let $\mathcal F$ be a mixed-$\omega$-sheaf on a normal projective variety $W$ and let $H$ be a nef and big Cartier divisor on $W$. We put $\dim W=n$. Then $\mathcal F\otimes \mathcal O_W(lH) $ is generically generated by global sections for $l\geq n^2+1$. We may assume that $\mathcal F=f_*\mathcal O_V(K_V+D)$ as in Definition \[x-def5.1\]. Then, by [@dutta-murayama Theorems C and 2.20], $\mathcal F\otimes \mathcal O_W(lH) $ is generically generated by global sections for $l\geq n^2+1$. For the details of Lemma \[x-lem7.8\], we recommend the reader to see [@dutta-murayama]. A special case {#x-sec8} ============== In this section, we will freely use the standard notation and some basic results in the theory of minimal models (see, for example, [@fujino-fundamental], [@fujino-foundations], and [@fujino-iitaka]). The reader can skip this section. \[x-thm8.1\] Let $f:X\to Y$ be a surjective morphism from a normal projective variety $X$ onto a smooth projective variety $Y$ with connected fibers. Assume that $f$ is weakly semistable in the sense of Abramovich–Karu and that the geometric generic fiber $X_{\overline \eta}$ of $f: X\to Y$ has a good minimal model. Let $H$ be an ample Cartier divisor on $Y$. Let $k$ be a positive integer such that $k\geq 2$ and $f_*\omega^{\otimes k}_{X/Y}\ne 0$. Then $$f_*\omega^{\otimes k}_{X/Y}\otimes \omega_Y\otimes \mathcal O_Y(H)$$ is locally free and is a pure-$\omega$-big-sheaf on $Y$. In particular, $$H^i(Y, f_*\omega^{\otimes k}_{X/Y}\otimes \omega_Y\otimes \mathcal O_Y(H))=0$$ for every $i>0$. As mentioned above, we will freely use some basic results in the theory of minimal models. By the proof of [@fujino-direct Theorem 1.6] (see also [@fujino-direct-corri]), we have already known that $f_*\omega^{\otimes m}_{X/Y}$ is a nef locally free sheaf on $Y$ for every $m\geq 1$. We consider a relative good minimal model $f':X'\to Y$ of $f:X\to Y$. $$\xymatrix{ X \ar[dr]_-f\ar@{-->}^-\phi[rr]&& X'\ar[dl]^-{f'} \\ & Y& }$$ Since $$f_*\omega^{\otimes m}_{X/Y}\simeq f'_*\mathcal O_{X'}(mK_{X'/Y})$$ holds for every $m\geq 1$, it is sufficient to prove that $$f'_*\mathcal O_{X'}(K_{X'}+(k-1)K_{X'/Y}+f'^*H)$$ is a pure-$\omega$-big-sheaf on $Y$. In this step, we will prove: $K_{X'/Y}$ is nef and $f'$-semi-ample. Since $f': X'\to Y$ is a relative good minimal model of $f:X\to Y$, $K_{X'/Y}$ is $f'$-semi-ample. Therefore, $$f'^*f'_*\mathcal O_{X'}(lK_{X'/Y})\to \mathcal O_{X'}(lK_{X'/Y})$$ is surjective for a sufficiently large and divisible positive integer $l$. Since $f'_*\mathcal O_{X'}(lK_{X'/Y})\simeq f_*\omega^{\otimes l}_{X/Y}$ is a nef locally free sheaf, $K_{X'/Y}$ is nef by the above surjection. Since $K_{X'/Y}$ is nef and $f'$-semi-ample, $(k-1)K_{X'/Y}+af'^*H$ is semi-ample for every positive rational number $a$. We take a birational morphism $\rho:\widetilde X\to X'$ from a smooth projective variety $\widetilde X$ such that the exceptional locus ${{\operatorname{Exc}}}(\rho)$ of $\rho$ is a simple normal crossing divisor on $\widetilde X$. Since $X'$ has only canonical singularities, we see that $$\rho_*\mathcal O_{\widetilde X}(K_{\widetilde X}+\lceil (k-1)\rho^*K_{X'/Y}+\rho^*f'^*H\rceil) \simeq \mathcal O_{X'}(K_{X'}+(k-1)K_{X'/Y}+f'^*H)$$ holds. By Lemma \[x-lem5.8\], $$(f'\circ\rho)_*\mathcal O_{\widetilde X}(K_{\widetilde X}+\lceil (k-1)\rho^*K_{X'/Y}+\rho^*f'^*H\rceil) \simeq f'_*\mathcal O_{X'}(K_{X'}+(k-1)K_{X'/Y}+f'^*H)$$ is a pure-$\omega$-big-sheaf on $Y$ since $$(k-1)\rho^*K_{X'/Y}+\rho^*f'^*H-\frac{1}{2}\rho^*f'^*H$$ is semi-ample. This means that $$f_*\omega^{\otimes k}_{X/Y}\otimes \omega_Y\otimes \mathcal O_Y(H)$$ is locally free and a pure-$\omega$-big-sheaf on $Y$. The vanishing theorem follows from Lemma \[x-lem7.1\]. Theorem \[x-thm8.1\] predicts that $f_*\omega^{\otimes k} _{X/Y}\otimes \omega_Y \otimes \mathcal O_Y(H)$ has good properties. Fundamental theorem {#x-sec9} =================== This section is the main part of this paper. The main result of this section is Theorem \[x-thm9.3\], which we call a fundamental theorem of the theory of mixed-$\omega$-sheaves. Let us start with the following lemma. \[x-lem9.1\] Let $f:X\to Y$ be a surjective morphism from a normal projective variety $X$ onto a smooth projective variety $Y$. Let $L$ be a Cartier divisor on $X$ and let $\Delta$ be an effective $\mathbb R$-divisor on $X$ such that $K_X+\Delta$ is $\mathbb R$-Cartier. Let $k$ be a positive integer with $k\geq 2$. We assume the following conditions: - $(X, \Delta)$ is log canonical over a nonempty Zariski open set of $Y$, and - $L-k(K_{X/Y}+\Delta)$ is nef and $f$-semi-ample. Let $H$ be an ample divisor on $Y$. We assume that $f_*\mathcal O_Y(L)\ne 0$. We take a positive integer $l$ such that $$\mathcal O_Y(lH)\otimes f_*\mathcal O_X(L)$$ is big. Then $$\mathcal O_Y(K_Y+(l-\lfloor l/k\rfloor)H)\otimes f_*\mathcal O_X(L)$$ is a mixed-$\widehat \omega$-big-sheaf on $Y$. Hence we obtain that $$\mathcal O_Y(K_Y+(k-1)H)\otimes f_*\mathcal O_X(L)$$ is always a mixed-$\widehat \omega$-big-sheaf on $Y$. We include all the details although Lemma \[x-lem9.1\] is essentially the same as [@nakayama Chapter V, 3.34. Lemma]. We divide the proof into several small steps. \[x-step9.1\] Let $\mu:\widetilde X\to X$ be a projective birational morphism from a smooth projective variety $\widetilde X$ such that $K_{\widetilde X}+\widetilde \Delta=\mu^*(K_X+\Delta)$ and that ${{\operatorname{Supp}}}\widetilde \Delta$ is a simple normal crossing divisor on $\widetilde X$. We put $E=\lceil -(\widetilde \Delta^{<0})\rceil$. Then $E$ is an effective $\mu$-exceptional divisor on $\widetilde X$, $\widetilde \Delta+E$ is effective, and $(\widetilde X, \widetilde \Delta+E)$ is log canonical over a nonempty Zariski open set of $Y$ by construction. We note that $$\mu^*L+kE-k(K_{\widetilde X/Y}+\widetilde \Delta+E) =\mu^*(L-k(K_{X/Y}+\Delta))$$ and that $\mu_*\mathcal O_{\widetilde X}(\mu^*L+kE)\simeq \mathcal O_X(L)$. Therefore, by replacing $f:X\to Y$, $L$, and $\Delta$ with $f\circ \mu: \widetilde X\to Y$, $\mu^*L+kE$, and $\widetilde \Delta+E$ respectively, we may assume that $X$ is smooth and ${{\operatorname{Supp}}}\Delta$ is a simple normal crossing divisor on $X$. \[x-step9.2\] Since $\mathcal O_Y(lH)\otimes f_*\mathcal O_X(L)$ is big, we can take a positive integer $a$ such that $$\widehat S^a(\mathcal O_Y(lH)\otimes f_*\mathcal O_X(L)) \otimes \mathcal O_Y(-H)=\mathcal O_Y((al-1)H)\otimes \widehat S^a(f_*\mathcal O_X(L))$$ is generically generated by global sections by Lemma \[x-lem4.2\]. \[x-step9.3\] We take an effective $f$-exceptional divisor $E$ on $X$ such that $$\left(f_*\mathcal O_X(bL)\right)^{**}\simeq f_*\mathcal O_X(b(L+E))$$ holds for every $1\leq b\leq a$. By taking a resolution of singularities as in Step \[x-step9.1\], we may assume that ${{\operatorname{Supp}}}(\Delta+E)$ is a simple normal crossing divisor on $X$. Since $$(L+E)-k(K_{X/Y}+\Delta+(1/k)E)=L-k(K_{X/Y}+\Delta),$$ we may replace $L$ and $\Delta$ with $L+E$ and $\Delta+(1/k)E$, respectively. This is because $$\mathcal O_Y(K_Y+(l-\lfloor l/k\rfloor)H)\otimes f_*\mathcal O_X(L)$$ is a mixed-$\widehat\omega$-big-sheaf on $Y$ if and only if so is $$\mathcal O_Y(K_Y+(l-\lfloor l/k\rfloor)H)\otimes \left(f_*\mathcal O_X(L)\right)^{**}.$$ Anyway, we may assume that $f_*\mathcal O_X(bL)$ is reflexive for every $1\leq b\leq a$. \[x-step9.4\] By taking a suitable birational modification of $X$ again (see Step \[x-step9.1\]), we may further assume that the image of the natural map $$f^*f_*\mathcal O_X(L)\to \mathcal O_X(L)$$ is invertible and can be written as $\mathcal O_X(L-B)$ such that ${{\operatorname{Supp}}}(\Delta+B)$ is a simple normal crossing divisor on $X$. By the definition of $B$, we have $f_*\mathcal O_X(L-B)=f_*\mathcal O_X(L)$. \[x-step9.5\] We note that we can take an effective $f$-exceptional divisor $E$ on $X$ such that the map $f^*f_*\mathcal O_X(L)\to \mathcal O_X(L-B)$ induces $$f^*\widehat S^a(f_*\mathcal O_X(L))\to \mathcal O_X(a(L-B)+E).$$ Then we have the following map $$\label{x-9.1} H^0(Y, \mathcal O_Y((al-1)H)\otimes \widehat S^a(f_*\mathcal O_X(L)))\otimes \mathcal O_X\to \mathcal O_X(a(L-B)+E+(al-1)f^*H).$$ By taking a suitable birational modification of $X$ again (see Step \[x-step9.1\]), we may assume that the image of is $$\mathcal O_X(a(L-B)+E-F+(al-1)f^*H)$$ for some effective $f$-vertical divisor $F$ on $X$. We may further assume that ${{\operatorname{Supp}}}(\Delta+B+E+F)$ is a simple normal crossing divisor on $X$. We put $$N:=a(L-B)+E-F+(al-1)f^*H.$$ Then $|N|$ is free by construction. \[x-step9.6\] We take a positive number $\varepsilon$. Then $L-k(K_{X/Y}+\Delta)+\varepsilon f^*H$ is semi-ample because $L-k(K_{X/Y}+\Delta)$ is nef and $f$-semi-ample by assumption. We put $$M:=L-(K_{X/Y}+\Delta)-\frac{k-1}{k}B+\frac{k-1}{ak}(E-F) +\left(l-\left\lfloor \frac{l}{k}\right\rfloor\right)f^*H.$$ Then $$M-\frac{k-1}{ak}N-\frac{1}{k}(L-k(K_{X/Y}+\Delta)+\varepsilon f^*H)=\alpha f^*H$$ for some $\alpha>0$ if $\varepsilon$ is sufficiently small. We note that $$\frac{(al-1)(k-1)}{ak}<\frac{\lceil l(k-1)\rceil}{k}=l-\left\lfloor\frac{l}{k}\right\rfloor.$$ Thus $M$ and $M-\alpha f^*H$ are semi-ample. Without loss of generality, we may assume that $\varepsilon$ and $\alpha$ are rational numbers since $$\left(l-\left\lfloor \frac{l}{k}\right\rfloor\right)-\frac{(al-1)(k-1)}{k} -\frac{\varepsilon}{k}=\alpha.$$ \[x-step9.7\] We consider $$\left\lfloor \frac{k-1}{k}B+\Delta\right\rfloor.$$ We put $$B_0=\max\left\{ T \left| \text{$T$ is a Weil divisor with $0\leq T\leq B$ and $T\leq \left\lfloor \frac{k-1}{k}B+ \Delta\right\rfloor$}\right.\right\}.$$ We write $$\left\lfloor \frac{k-1}{k}B+\Delta\right\rfloor-B_0=\Delta_1+\Delta_2$$ where $\Delta_1$ is the horizontal part and $\Delta_2$ is the vertical part. By assumption (i), $\Delta_1=\Delta^{=1}_1$. By construction, we see that $\Delta_1\subset {{\operatorname{Supp}}}\Delta^{=1}$ and that $\Delta_1$ and ${{\operatorname{Supp}}}\{M\}$ have no common irreducible components. \[x-step9.8\] We have the following generically isomorphic injections: $$\begin{split} f_*\mathcal O_X(K_X+\Delta_1+\lceil M\rceil)&\hookrightarrow \omega_Y((l-\lfloor l/k\rfloor)H)\otimes \left( f_*\mathcal O_X\left(L-\left\lfloor\frac{k-1}{k}B+\Delta\right\rfloor+\Delta_1\right)\right)^{**} \\& =\omega_Y((l-\lfloor l/k\rfloor)H)\otimes \left( f_*\mathcal O_X\left(L-B_0-\Delta_2\right)\right)^{**} \\ &\hookrightarrow \omega_Y((l-\lfloor l/k\rfloor)H)\otimes f_*\mathcal O_X(L). \end{split}$$ We note that $$f_*\mathcal O_X(L)=f_*\mathcal O_X(L-B)\subset f_*\mathcal O_X(L-B_0)\subset f_*\mathcal O_X(L).$$ This implies that $$\mathcal O_Y(K_Y+(l-\lfloor l/k\rfloor)H)\otimes f_*\mathcal O_X(L)$$ is a mixed-$\widehat \omega$-big-sheaf on $Y$ because $f_*\mathcal O_X(K_X+\Delta_1+\lceil M\rceil)$ is a mixed-$\omega$-big-sheaf by Lemma \[x-lem5.8\]. \[x-step9.9\] Let $l_0$ be the minimum positive integer such that $$\mathcal O_Y(K_Y+l_0H)\otimes f_*\mathcal O_X(L)$$ is a mixed-$\widehat\omega$-big-sheaf on $Y$. By Theorem \[x-thm6.3\], $$\mathcal O_Y(l_0H)\otimes f_*\mathcal O_X(L)$$ is big. By the result obtained above, $$\mathcal O_Y(K_Y+(l_0-\lfloor l_0/k\rfloor)H)\otimes f_*\mathcal O_X(L)$$ is a mixed-$\widehat\omega$-big-sheaf on $Y$. This implies that $l_0-\lfloor l_0/k\rfloor\geq l_0$. Thus we get $l_0\leq k-1$. Hence we have $$\mathcal O_Y(K_Y+(k-1)H)\otimes f_*\mathcal O_X(L)$$ is a mixed-$\widehat\omega$-big-sheaf on $Y$. Thus we get the desired statements. \[x-rem9.2\] In Lemma \[x-lem9.1\], we further assume that $(X, \Delta)$ is klt over a nonempty Zariski open set of $Y$. Then we can easily see that $\Delta_1=0$ in Step \[x-step9.7\] in the proof of Lemma \[x-lem9.1\]. Therefore, we obtain that $$\mathcal O_Y(K_Y+(l-\lfloor l/k\rfloor)H)\otimes f_*\mathcal O_X(L)$$ and $$\mathcal O_Y(K_Y+(k-1)H)\otimes f_*\mathcal O_X(L)$$ are pure-$\widehat\omega$-big-sheaves on $Y$. Theorem \[x-thm9.3\] is the most important result in the theory of mixed-$\omega$-sheaves. So we call it a fundamental theorem of the theory of mixed-$\omega$-sheaves. \[x-thm9.3\] Let $f:X\to Y$ be a surjective morphism from a normal projective variety $X$ onto a smooth projective variety $Y$. Let $L$ be a Cartier divisor on $X$ and let $\Delta$ be an effective $\mathbb R$-divisor on $X$ such that $K_X+\Delta$ is $\mathbb R$-Cartier. Let $D$ be an $\mathbb R$-divisor on $Y$. Let $k$ be a positive integer with $k\geq 2$. Assume the following conditions: - $(X, \Delta)$ is log canonical over a nonempty Zariski open set of $Y$, and - $L+f^*D-k(K_{X/Y}+\Delta)-f^*A$ is semi-ample for some big $\mathbb R$-divisor $A$ on $Y$. If $f_*\mathcal O_Y(L)\ne 0$, then $$\mathcal O_Y(K_Y+\lceil D\rceil)\otimes f_*\mathcal O_X(L)$$ is a mixed-$\widehat \omega$-big-sheaf on $Y$. We divide the proof into several small steps. \[x-step9.3.1\] By taking a resolution as in Step \[x-step9.1\] in the proof of Lemma \[x-lem9.1\], we may assume that $X$ is a smooth projective variety and that ${{\operatorname{Supp}}}\Delta$ is a simple normal crossing divisor on $X$. We note that $$L+f^*\lceil D\rceil -k\left(K_{X/Y}+\Delta+\frac{1}{k} f^*\{-D\}\right)-f^*A=L+f^*D-k(K_{X/Y}+\Delta)-f^*A.$$ Therefore, by replacing $L$ and $\Delta$ with $L+f^*\lceil D\rceil$ and $\Delta+\frac{1}{k}f^*\{-D\}$, respectively, we may assume that $D=0$. By Kodaira’s lemma, we have $A\sim_{\mathbb R} A_1+A_2$ such that $A_1$ is an ample $\mathbb R$-divisor and $A_2$ is an effective $\mathbb R$-divisor. By replacing $A$ and $\Delta$ with $A_1$ and $\Delta+\frac{1}{k}f^*A_2$ respectively, we may further assume that $A$ is an ample $\mathbb R$-divisor on $Y$. We take an ample Cartier divisor $H$ on $Y$ and a positive integer $m$ such that $A-\frac{k-1}{m}H$ is ample. Then $$L-k(K_{X/Y}+\Delta)-\frac{k-1}{m}f^*H$$ is semi-ample. Therefore, we may replace $A$ with $\frac{k-1}{m}H$. By taking a resolution as in Step \[x-step9.1\] in the proof of Lemma \[x-lem9.1\] again, we may assume that $X$ is a smooth projective variety and that ${{\operatorname{Supp}}}\Delta$ is a simple normal crossing divisor on $X$. By Lemma \[x-lem3.2\], we may further assume that $\Delta$ is a $\mathbb Q$-divisor. We take an effective $f$-exceptional divisor $E$ and replace $L$ and $\Delta$ with $L+E$ and $\Delta+(1/k)E$ respectively. Then we may assume that $f_*\mathcal O_X(L)$ is reflexive. By taking a birational modification of $X$, we may assume that the image of $$f^*f_*\mathcal O_X(L)\to \mathcal O_X(L)$$ is $\mathcal O_X(L-B)$ for some effective divisor $B$ such that ${{\operatorname{Supp}}}(\Delta+B)$ is a simple normal crossing divisor on $X$. Let $S$ denote the union of all $f$-exceptional divisors on $X$. We may assume that ${{\operatorname{Supp}}}(\Delta+B+S)$ is a simple normal crossing divisor on $X$ by taking a suitable birational modification of $X$ again (see Step \[x-step9.1\] in the proof of Lemma \[x-lem9.1\]). \[x-step9.3.2\] By Lemma \[x-lem3.3\] and Remark \[x-rem3.4\], we take a finite flat Galois cover $\tau:Y'\to Y$ from a smooth projective variety $Y'$ and get the following commutative diagram $$\xymatrix{ X' \ar[d]_-{f'}\ar[r]^-\rho& X\ar[d]^-f \\ Y' \ar[r]_-\tau&Y }$$ such that $X'=X\times _YY'$ is a smooth projective variety, $\tau^*H\sim mH'$ for some ample Cartier divisor $H'$ on $Y'$, and $\rho^*\omega^{\otimes n}_{X/Y}=\omega^{\otimes n}_{X'/Y'}$ for every integer $n$. Let $G$ denote the Galois group of $\tau:Y'\to Y$. By construction (see the proof of Lemma \[x-lem3.3\]), we may assume that $H'$ is $G$-invariant. We put $L'=\rho^*L$, $B'=\rho^*B$, $\Delta'=\rho^*\Delta$, and $S'=\rho^*S$. Without loss of generality, we may assume that ${{\operatorname{Supp}}}(\Delta'+B'+S')$ is a simple normal crossing divisor on $X'$ and that $\rho^*(K_{X/Y}+\Delta)=K_{X'/Y'}+\Delta'$ holds (see Remark \[x-rem3.4\]). We note that $$L'-(k-1)f'^*H'-k(K_{X'/Y'}+\Delta')\sim _{\mathbb Q} \rho^*\left(L-k(K_{X/Y}+\Delta)-\frac{k-1}{m} f^*H\right)$$ by construction. This implies that $\left(L'-(k-1)f'^*H'\right)-k(K_{X'/Y'}+\Delta')$ is semi-ample. \[x-step9.3.3\] We apply Lemma \[x-lem9.1\] to $$\left(L'-(k-1)f'^*H'\right)-k(K_{X'/Y'}+\Delta').$$ Then we obtain that $\mathcal O_{Y'}(K_{Y'})\otimes f'_*\mathcal O_{X'}(L')$ is a mixed-$\widehat\omega$-big-sheaf on $Y'$. Therefore, $f'_*\mathcal O_{X'}(L')$ is a big sheaf on $Y'$ by Theorem \[x-thm6.3\]. Thus we can take a positive integer $a$ such that $\widehat S^a(f'_*\mathcal O_{X'}(L'))$ is generically generated by global sections (see Lemma \[x-lem4.2\]). Then we take an effective $G$-invariant $f'$-exceptional divisor $E'$ on $X'$ such that $$\left(f'_*\mathcal O_{X'}(bL')\right)^{**}\simeq f'_*\mathcal O_{X'}(b(L'+E'))$$ holds for every $1\leq b\leq a$. By replacing $L'$, $\Delta'$, and $B'$ with $L'+E'$, $\Delta'+(1/k)E'$, and $B'+E'$ respectively, we may assume that $f'_*\mathcal O_{X'}(bL')$ is reflexive for every $1\leq b\leq a$. \[x-step9.3.4\] We can take an effective $G$-invariant $f'$-exceptional divisor $E'$ on $X'$ such that the surjective map $$f'^*f'_*\mathcal O_{X'}(L')\to \mathcal O_{X'}(L'-B')$$ induces $$f'^*\widehat S^a(f'_*\mathcal O_{X'}(L')) \to \mathcal O_{X'}(a(L'-B')+E').$$ Then we have the following map $$\label{x-9.2} H^0(Y', \widehat S^a(f'_*\mathcal O_{X'}(L'))) \otimes \mathcal O_{X'}\to \mathcal O_{X'} (a(L'-B')+E').$$ By taking an equivariant resolution of singularities of $X'$, we may assume that the image of is $$\mathcal O_{X'}(a(L'-B')+E'-F')$$ for some effective $G$-invariant $f'$-vertical divisor $F'$ on $X'$. Of course, we may assume that ${{\operatorname{Supp}}}(\Delta'+B'+E'+F')$ is a simple normal crossing divisor on $X'$. We put $$N':=a(L'-B')+E'-F'.$$ Then $|N'|$ is free by construction. We put $$M':=L'-(K_{X'/Y'}+\Delta')-\frac{k-1}{k}B'+\frac{k-1}{ak} (E'-F').$$ Then $$M'-\frac{k-1}{ak}N'-\frac{1}{k}(L'-k(K_{X'/Y'}+\Delta')- (k-1)f'^*H')=\frac{k-1}{k}f'^*H'.$$ In particular, $M'$ and $M'-\frac{k-1}{k}f'^*H'$ are semi-ample. \[x-step9.3.5\] We put $$\left\lfloor \frac{k-1}{k}B'+\Delta'\right\rfloor=B'_0+\Delta'_1+\Delta'_2$$ as in Step \[x-step9.7\] in the proof of Lemma \[x-lem9.1\]. Then $\Delta'_1$ is a $G$-invariant $f'$-horizontal simple normal crossing divisor on $X'$. As before, ${{\operatorname{Supp}}}\{M'\}$ and $\Delta'_1$ have no common irreducible components. Thus, by Lemma \[x-lem5.8\], $f'_*\mathcal O_{X'}(K_{X'}+\Delta'_1+\lceil M'\rceil)$ is a mixed-$\omega$-big-sheaf on $Y'$. Note that the Galois group $G$ acts on $f'_*\mathcal O_{X'}(K_{X'}+\Delta'_1+\lceil M'\rceil)$. \[x-step9.3.6\] Therefore, we get the following generically isomorphic $G$-equivariant embedding: $$\label{x-9.3} f'_*\mathcal O_{X'}(K_{X'}+\Delta'_1+\lceil M'\rceil) \hookrightarrow \mathcal O_{Y'}(K_{Y'}) \otimes f'_*\mathcal O_{X'}(L')$$ as in Step \[x-step9.8\] in the proof of Lemma \[x-lem9.1\]. We note that $f'_*\mathcal O_{X'}(L')\simeq \tau^*f_*\mathcal O_X(L)$ by the flat base change theorem. We take $\tau_*$ of and then take the $G$-invariant parts. Thus, we get a mixed-$\omega$-big-sheaf $$\mathcal F:=\left(\tau_*f'_*\mathcal O_{X'}(K_{X'} +\Delta'_1+\lceil M'\rceil)\right)^G$$ on $Y$ and a generically isomorphic injection $$\mathcal F\hookrightarrow \mathcal O_Y(K_Y)\otimes f_*\mathcal O_X(L).$$ This means that $\mathcal O_Y(K_Y)\otimes f_*\mathcal O_X(L)$ is a mixed-$\widehat\omega$-big-sheaf on $Y$. Anyway, we obtain that $\mathcal O_Y(K_Y+\lceil D\rceil)\otimes f_*\mathcal O_X(L)$ is a mixed-$\widehat\omega$-big-sheaf on $Y$. \[x-rem9.4\] As in Remark \[x-rem9.2\], we further assume that $(X, \Delta)$ is klt over a nonempty Zariski open set of $Y$ in Theorem \[x-thm9.3\]. Then we see that $\Delta'_1=0$ in Step \[x-step9.3.5\] in the proof of Theorem \[x-thm9.3\]. Hence we obtain that $$\mathcal O_Y(K_Y+\lceil D\rceil)\otimes f_*\mathcal O_X(L)$$ is a pure-$\widehat\omega$-big-sheaf on $Y$. As a corollary of Theorem \[x-thm9.3\], we have: \[x-cor9.5\] Let $f:X\to Y$ be a surjective morphism from a normal projective variety $X$ onto a smooth projective variety $Y$ with $\dim Y=n$. Let $L$ be a Cartier divisor on $X$ and let $\Delta$ be an effective $\mathbb R$-divisor on $X$ such that $K_X+\Delta$ is $\mathbb R$-Cartier. Let $D$ be an $\mathbb R$-divisor on $Y$. Let $k$ be a positive integer with $k\geq 2$. Assume the following conditions: - $(X, \Delta)$ is log canonical over a nonempty Zariski open set of $Y$, and - $L+f^*D-k(K_{X/Y}+\Delta)$ is nef and $f$-semi-ample. Let $H$ be a big Cartier divisor on $Y$ such that $|H|$ is free. Then $$\mathcal O_Y(K_Y+\lceil D\rceil+(n+1)H)\otimes f_*\mathcal O_X(L)$$ is generically generated by global sections. Let $H^\dag$ be a nef and big Cartier divisor on $Y$ such that $|H^\dag|$ is not necessarily free. Then the sheaf $$\mathcal O_Y(K_Y+\lceil D\rceil+lH^\dag)\otimes f_*\mathcal O_X(L)$$ is generically generated by global sections for $l\geq n^2+2$. By taking a resolution of singularities as in Step \[x-step9.1\] in the proof of Lemma \[x-lem9.1\], we may assume that $X$ is smooth. By replacing $L$ and $\Delta$ with $L+f^*\lceil D\rceil$ and $\Delta+\frac{1}{k}f^*\{-D\}$ respectively, we may assume that $D=0$. By the flattening theorem, there is a birational morphism $\tau:Y'\to Y$ from a smooth projective variety $Y'$ such that the main component of $X\times _Y Y'$ is flat over $Y'$. Let $X'$ be a resolution of the main component of $X\times _Y Y'$. Then we get the following commutative diagram. $$\xymatrix{ X' \ar[d]_-{f'}\ar[r]^-\rho& X \ar[d]^-f\\ Y' \ar[r]_-\tau&Y }$$ By construction, any $f'$-exceptional divisor is $\rho$-exceptional. We put $K_{X'}+B=\rho^*(K_X+\Delta)$. We may assume that ${{\operatorname{Supp}}}B$ is a simple normal crossing divisor on $X'$. We write $$K_{Y'}=\tau^*K_Y+R$$ where $R$ is an effective $\tau$-exceptional divisor on $Y'$. We put $$L':=\rho^*L+k\lceil -(B^{<0})\rceil -kf'^*R$$ and $\Delta'=B+\lceil -(B^{<0})\rceil$. Note that $\lceil -(B^{<0})\rceil$ is effective and $\rho$-exceptional. Then we have $$L'-k(K_{X'/Y'}+\Delta')=\rho^*(L-k(K_{X/Y}+\Delta)).$$ We take an effective $f'$-exceptional divisor $E$ on $X'$ such that $$\left(f'_*\mathcal O_{X'}(L')\right)^{**} \simeq f'_*\mathcal O_{X'}(L'+E).$$ Note that $E$ is $\rho$-exceptional and that there is a generically isomorphic injection $$\tau_*f'_*\mathcal O_{X'}(L'+E) =f_*\rho_*\mathcal O_{X'}(L'+E)\subset f_*\mathcal O_X(L).$$ Therefore, we have a generically isomorphic injection $$\tau_*\left((f'_*\mathcal O_{X'}(L'))^{**}\right) \subset f_*\mathcal O_X(L).$$ By Kodaira’s lemma, we have $\tau^*H\sim _{\mathbb Q} A+B$ such that $A$ is an ample $\mathbb Q$-divisor and $B$ is an effective $\mathbb Q$-divisor. Note that $$\begin{split} &L'+E+f'^*\tau^*H-k\left(K_{X'/Y'}+\Delta'+\frac{1}{k}E+ \frac{1}{k}f'^*B\right)-\frac{1}{2}f'^*A\\ &=L'-k(K_{X'/Y'}+\Delta')+\frac{1}{2}f'^*A \end{split}$$ is semi-ample. Therefore, by Theorem \[x-thm9.3\], $$\mathcal O_{Y'} (K_{Y'}+\tau^*H)\otimes f'_*\mathcal O_{X'}(L'+E)$$ is a mixed-$\widehat\omega$-big-sheaf on $Y'$. Thus, by Lemma \[x-lem7.7\], $$\mathcal O_{Y'}(K_{Y'}+(n+1)\tau^*H) \otimes (f'_*\mathcal O_{X'}(L'))^{**}$$ is generically generated by global sections. Therefore, so is $\mathcal O_Y(K_Y+(n+1)H) \otimes f_*\mathcal O_X(L)$. By the same argument, we see that $$\mathcal O_{Y'}(K_{Y'}+\tau^*H^\dag)\otimes f'_*\mathcal O_{X'}(L'+E)$$ is a mixed-$\widehat\omega$-big-sheaf on $Y'$. Hence the sheaf $$\mathcal O_{Y'}(K_{Y'}+l\tau^*H^\dag) \otimes (f'_*\mathcal O_{X'}(L'))^{**}$$ is generically generated by global sections for $l\geq n^2+2$ by Lemma \[x-lem7.8\]. Therefore, so is $\mathcal O_Y(K_Y+lH^\dag)\otimes f_*\mathcal O_X(L)$. Anyway, we get the desired statements. We note that [@nakayama Chapter V, 3.37. Corollary] needs the assumption that $(X, \Delta)$ is klt over a nonempty Zariski open set of $Y$. On the other hand, Corollary \[x-cor9.5\] can be applied to log canonical pairs. This is the main difference between [@nakayama Chapter V, 3.37. Corollary] and Corollary \[x-cor9.5\]. Proof of Theorems \[z-thm1.5\], \[y-thm1.7\], and \[y-thm1.8\] {#x-sec10} ============================================================== In this section, we prove Theorems \[z-thm1.5\], \[y-thm1.7\], and \[y-thm1.8\] in Section \[z-sec1\]. Let us first prove Theorem \[y-thm1.7\]. We divide the proof into small steps. \[x-step10-1\] By taking a suitable resolution of singularities of $X$, we may assume that $X$ is a smooth projective variety and ${{\operatorname{Supp}}}\Delta$ is a simple normal crossing divisor on $X$ (see Step \[x-step9.1\] in the proof of Lemma \[x-lem9.1\]). We may further assume that every log canonical center of $(X, \Delta_{\mathrm{hor}})$ is dominant onto $Y$. \[x-step10-2\] In this step, we will prove the generically generation of $f_*\mathcal O_X(L)\otimes \mathcal O_Y(K_Y+lH)$ when $k=1$. By replacing $L$ and $\Delta$ with $L-\lfloor \Delta_{\mathrm{ver}}\rfloor$ and $\Delta- \lfloor \Delta_{\mathrm{ver}}\rfloor$ respectively, we may further assume that $(X, \Delta)$ is dlt and that every log canonical center of $(X, \Delta)$ is dominant onto $Y$. By the arguments in Step \[x-step7.6.2\] in the Proof of Lemma \[x-lem7.6\], we see that $f_*\mathcal O_X(L)\otimes \mathcal O_Y(K_Y+lH)$ is generically generated by global sections. In this step, we will see that $f_*\mathcal O_X(L)\otimes \mathcal O_Y(K_Y+lH)$ is generically generated by global sections when $k\geq 2$. This follows directly from Corollary \[x-cor9.5\]. More precisely, we put $D=0$ and apply Corollary \[x-cor9.5\]. In this final step, we treat the case when $s\geq 2$. We take the $s$-fold fiber product $$X^s:=\underbrace{X\times _Y X\times _Y \cdots \times _Y X}_{s}$$ of $X$ over $Y$. Let $f^s: X^s\to Y$ be the induced morphism. Let $\rho:X^{(s)}\to X^s$ be a resolution of singularities of the dominant components of $X^s$ such that $\rho$ is an isomorphism over a nonempty Zariski open set of $Y$. We put $f^{(s)}=f^s\circ \rho: X^{(s)}\to Y$. We note that $X^{(s)}$ may be reducible, that is, a disjoint union of some smooth projective varieties. We can take a Zariski open set $U$ of $Y$ such that $\mathrm{codim}_Y(Y\setminus U)\geq 2$, $f_*\mathcal O_X(L)$ is locally free on $U$, and $f$ is flat over $U$. By applying Lemma \[x-lem3.6\] to $f^{-1}(U)\to U$, we can construct a Cartier divisor $L^{(s)}$ on $X^{(s)}$ and an effective $\mathbb R$-divisor $\Delta^{(s)}$ on $X^{(s)}$ such that $$L^{(s)}\sim_{\mathbb R} k(K_{X^{(s)}/Y}+\Delta^{(s)}),$$ $(X^{(s)}, \Delta^{(s)})$ is log canonical over a nonempty Zariski open set of $Y$, and there exists a generically isomorphic injection $$\left(f^{(s)}_*\mathcal O_{X^{(s)}}(L^{(s)})\right)^{**}\subset \left(\bigotimes ^s f_*\mathcal O_X(L)\right)^{**}.$$ By Theorem \[x-thm9.3\], $$\mathcal O_Y(K_Y+H)\otimes f^{(s)}_*\mathcal O_{X^{(s)}}(L^{(s)})$$ is a finite direct sum of mixed-$\widehat\omega$-big-sheaves when $k\geq 2$. Note that $X^{(s)}$ may be reducible. Therefore, $$\mathcal O_Y(K_Y+H)\otimes \left(\bigotimes ^s f_*\mathcal O_X(L)\right)^{**}$$ is also a finite direct sum of mixed-$\widehat\omega$-big-sheaves. Thus, by Lemma \[x-lem7.7\], $$\mathcal O_Y(K_Y+lH)\otimes \left(\bigotimes ^s f_*\mathcal O_X(L)\right)^{**}$$ is generically generated by global sections for $l\geq n+1$ when $k\geq 2$. If $k=1$, then we can check that $\mathcal O_Y(K_Y+lH)\otimes f^{(s)}_*\mathcal O_{X^{(s)}}(L^{(s)})$ is generically generated by global sections for $l\geq n+1$ by the arguments in Steps \[x-step10-1\] and \[x-step10-2\]. Therefore, $$\mathcal O_Y(K_Y+lH)\otimes \left(\bigotimes ^s f_*\mathcal O_X(L)\right)^{**}$$ is generically generated by global sections for $l\geq n+1$ when $k=1$. Anyway, we have obtained the desired statements. Next we prove Theorem \[y-thm1.8\]. It is not difficult to modify the proof of Theorem \[y-thm1.7\]. In this step, we will treat the case when $k=1$. As usual, by taking a suitable birational modification of $X$, we may assume that $X$ is smooth and ${{\operatorname{Supp}}}\Delta$ is a simple normal crossing divisor on $X$. By replacing $L$ and $\Delta$ with $L-\lfloor \Delta^{>1}\rfloor$ and $\Delta-\lfloor \Delta^{>1}\rfloor$ respectively, we may assume that $\Delta$ is a boundary $\mathbb R$-divisor on $X$. Note that $\Delta^{>1}$ is $f$-vertical. By perturbing the coefficients of $\Delta$, we may further assume that $\Delta$ is a $\mathbb Q$-divisor with $L\sim _{\mathbb Q} K_{X/Y}+\Delta$. By Lemma \[x-lem5.9\], $$\mathcal O_X(K_X+\lfloor \Delta\rfloor +\lceil L-K_{X/Y}-\Delta\rceil)\simeq \mathcal O_X(L)\otimes f^*\mathcal O_Y(K_Y)$$ is a mixed-$\omega$-sheaf on $X$. Therefore, $f_*\mathcal O_X(L)\otimes \mathcal O_Y(K_Y)$ is a mixed-$\omega$-sheaf on $Y$. Thus, by Lemma \[x-lem7.8\], $f_*\mathcal O_X(L)\otimes \mathcal O_Y(K_Y+lH^\dag)$ is generically generated by global sections for $l\geq n^2+1$. Similarly, we may assume that the sheaf $f^{(s)}_*\mathcal O_{X^{(s)}}(L^{(s)})\otimes \mathcal O_Y(K_Y)$ in the proof of Theorem \[y-thm1.7\] is a finite direct sum of mixed-$\omega$-sheaves on $Y$ when $k=1$. Therefore, $$\left(\bigotimes ^sf_*\mathcal O_X(L)\right)^{**} \otimes \mathcal O_Y(K_Y+lH^\dag)$$ is generically generated by global sections for $l\geq n^2+1$. In this step, we will treat the case when $k\geq 2$. If we use Lemma \[x-lem7.8\] instead of Lemma \[x-lem7.7\], then the proof of Theorem \[y-thm1.7\] implies that $$\mathcal O_Y(K_Y+lH^\dag)\otimes \left(\bigotimes ^sf_*\mathcal O_X(L)\right)^{**}$$ is generically generated by global sections for $l\geq n^2+2$. By Corollary \[x-cor9.5\], $$\mathcal O_Y(K_Y+lH^\dag)\otimes f_*\mathcal O_X(L)$$ is generically generated by global sections for $l\geq n^2+2$. Thus we get the desired statements. Finally, we prove Theorem \[z-thm1.5\]. We put $L=kK_{X/Y}$. Then this theorem directly follows from Theorems \[y-thm1.7\] and \[y-thm1.8\]. We close this section with an easy remark. \[x-rem10.1\] Let $Y$ be a smooth projective variety and let $H$ be an ample Cartier divisor on $Y$. Let $m$ be any positive integer. Then we can construct a finite cover $f:X\to Y$ from a smooth projective variety $X$ such that $\mathcal O_Y(-mH)$ is a direct summand of $f_*\mathcal O_X$. Therefore, we need the condition $k\geq 1$ in Theorems \[z-thm1.5\], \[y-thm1.7\], and \[y-thm1.8\]. Some other applications {#x-sec11} ======================= In this final section, we treat Nakayama’s inequality on $\kappa _\sigma$ and a slight generalization of the twisted weak positivity theorem. Theorem \[x-thm11.3\] and a special case of Theorem \[x-thm11.7\] have already played a crucial role in the theory of minimal models. Let us first recall the definition of $\kappa _\sigma$ for the reader’s convenience. \[x-def11.1\] Let $D$ be a pseudo-effective $\mathbb R$-Cartier divisor on a normal projective variety $X$ and let $A$ be a Cartier divisor on $X$. If $H^0(X, \mathcal O_X(\lfloor mD\rfloor +A))\ne 0$ for infinitely many positive integers $m$, then we set $$\sigma (D; A)=\max \left\{ k\in \mathbb Z_{\geq 0}\, \left|\, {\underset{m\to \infty}{\limsup}} \frac{\dim H^0(X, \mathcal O_X(\lfloor mD\rfloor +A))}{m^k}>0 \right.\right\}.$$ If $H^0(X, \mathcal O_X(\lfloor mD\rfloor +A))\ne 0$ only for finitely many $m\in \mathbb Z_{\geq 0}$, then we set $\sigma (D; A)=-\infty$. We define [*[Nakayama’s numerical dimension $\kappa _{\sigma}$]{}*]{} by $$\kappa _\sigma(X, D)=\max \{\sigma(D; A)\, |\, A \ {\text{is a Cartier divisor on}} \ X\}.$$ It is well known that $\kappa_{\sigma}(X, D)\geq 0$ (see, for example, [@nakayama Chapter V. 2.7. Proposition]). If $D$ is not pseudo-effective, then we put $\kappa_\sigma (X, D)=-\infty$. By this convention, we can define $\kappa_\sigma(X, D)$ for every $\mathbb R$-Cartier divisor $D$ on $X$. It is obvious that $$\kappa_\sigma(X, D)\geq \kappa(X, D)$$ always holds for every $\mathbb R$-Cartier divisor $D$ on $X$ by definition, where $\kappa (X, D)$ denotes the [*[Iitaka dimension]{}*]{} of $D$. For the details of $\kappa _\sigma(X, D)$ and $\kappa (X, D)$, we recommend the reader to see [@nakayama]. The following remark is easy but very useful. \[x-rem11.2\] Let $X$ be a smooth projective variety and let $D$ be an $\mathbb R$-divisor on $X$. We put $$\sigma (D; A)'=\max \left\{ k\in \mathbb Z_{\geq 0}\cup \{-\infty\}\, \left|\, {\underset{m\to \infty}{\limsup}} \frac{\dim H^0(X, \mathcal O_X(\lceil mD\rceil +A))}{m^k}>0 \right.\right\},$$ where $A$ is a divisor on $X$. Then we have the following equality $$\kappa _\sigma(X, D)=\max\{ \sigma(D; A)' \, |\, \text{$A$ is a divisor}\}.$$ We will use this characterization of $\kappa_\sigma$ in the proof of Theorem \[x-thm11.3\] below. We note the following easy but important fact that $\kappa _\sigma(X, lD)=\kappa_\sigma(X, D)$ holds for every positive integer $l$ (see [@fujino-corri Remark 2.2]), which will be useful in the proof of Theorem \[x-thm11.3\] below. The inequalities in Theorem \[x-thm11.3\] are indispensable in the theory of minimal models (see Remarks \[x-rem11.4\] and \[x-rem11.5\]). \[x-thm11.3\] Let $f:X\to Y$ be a surjective morphism from a normal projective variety $X$ onto a smooth projective variety $Y$ with connected fibers. Let $\Delta$ be an effective $\mathbb R$-divisor on $X$ such that $K_X+\Delta$ is $\mathbb R$-Cartier and that $(X, \Delta)$ is log canonical over a nonempty Zariski open set of $Y$. Let $D$ be an $\mathbb R$-Cartier $\mathbb R$-divisor on $X$ such that $D-(K_{X/Y}+\Delta)$ is nef. Then, for any $\mathbb R$-divisor $Q$ on $Y$, we have $$\kappa _\sigma(X, D+f^*Q)\geq \kappa _\sigma(F, D|_F) +\kappa (Y, Q)$$ and $$\kappa _\sigma(X, D+f^*Q)\geq \kappa (F, D|_F) +\kappa _\sigma(Y, Q)$$ where $F$ is a sufficiently general fiber of $f:X\to Y$. Before we prove Theorem \[x-thm11.3\], we give two important remarks. \[x-rem11.4\] We think that one of the most important results of Nakayama’s theory of $\omega$-sheaves is the inequality on $\kappa_\sigma$ in [@nakayama Chapter V, 4.1. Theorem (1)]. However, as we explained in [@fujino-subadd Remark 3.8] and [@fujino-corri Section 3], the proof of [@nakayama Chapter V, 4.1. Theorem (1)] is incomplete. For the details, see, for example, [@fujino-corri Section 1]. So, in Theorem \[x-thm11.3\], we claim two weaker inequalities than Nakayama’s original one (see [@fujino-corri (3.3) and (3.4)]). Anyway, the first inequality in Theorem \[x-thm11.3\] is still sufficiently powerful for some geometric applications (see [@fujino-corri Section 3]). \[x-rem11.5\] The troubles in the proof of [@dhp Remark 2.6] and [@gongyo-lehmann Theorem 4.3] caused by the incompleteness of [@nakayama Chapter V, 4.1. Theorem (1)] can be corrected by using the first inequality in Theorem \[x-thm11.3\]. For the details, we recommend the reader to see [@hashizume-hu Lemma 2.10]. Let us prove Theorem \[x-thm11.3\]. If $Q$ is not pseudo-effective, then the desired inequalities are obviously true. So we may assume that $Q$ is pseudo-effective. Similarly, we may further assume that $D|_F$ is pseudo-effective. As usual (see Step \[x-step9.1\] in the proof of Lemma \[x-lem9.1\]), we may assume that $X$ is smooth and ${{\operatorname{Supp}}}\Delta$ is a simple normal crossing divisor on $X$ by the basic properties of $\kappa _\sigma$ and $\kappa$. We take a sufficiently ample Cartier divisor $A$ on $X$ such that $A+\{-mD\}$ is ample for every integer $m$. Then $$\lceil mD\rceil +A-m(K_{X/Y}+\Delta)=m(D-(K_{X/Y}+\Delta))+A+\{-mD\}$$ is ample for every positive integer $m$. Then we can take an ample Cartier divisor $H$ on $Y$ such that $\mathcal O_Y(H)\otimes f_*\mathcal O_X(\lceil mD\rceil +A)$ is generically generated by global sections for every positive integer $m$ by Corollary \[x-cor9.5\]. Thus there exists a generically isomorphic injection $$\mathcal O_Y^{\oplus r(mD; A)}\hookrightarrow \mathcal O_Y(H)\otimes f_*\mathcal O_X(\lceil mD\rceil +A),$$ where $r(mD; A):={{\operatorname{rank}}}f_*\mathcal O_X(\lceil mD\rceil +A)$. This induces the following injection $$\mathcal O_Y(\lfloor mQ\rfloor +H)^{\oplus r(mD; A)}\hookrightarrow \mathcal O_Y(\lfloor mQ\rfloor +2H)\otimes f_*\mathcal O_X(\lceil mD\rceil +A).$$ Therefore, we have $$\label{eq1.1} \begin{split} &\dim _{\mathbb C} H^0(X, \mathcal O_X(\lceil m(D+f^*Q)\rceil +A+2f^*H) \\&\geq \dim _{\mathbb C} H^0(X, \mathcal O_X(\lceil mD\rceil+f^*(\lfloor mQ\rfloor)+A+2f^*H)) \\ & \geq r(mD; A)\cdot \dim _{\mathbb C} H^0(Y, \mathcal O_Y(\lfloor mQ\rfloor +H)) \end{split}$$ for every positive integer $m$. We can take a positive integer $m_0$ and a positive real number $C_0$ such that $$\label{eq1.2} C_0m^{\kappa (F, D|_F)}\leq r(mm_0D; A)$$ for every large positive integer $m$ (see, for example, ). Thus we have $$\label{eq1.3} \begin{split} &\dim H^0(X, \mathcal O_X(\lceil mm_0(D+f^*Q)\rceil+A+2f^*H))\\ &\geq C_0m^{\kappa(F, D|_F)} \cdot \dim H^0(Y, \mathcal O_Y(\lfloor mm_0Q\rfloor +H)) \end{split}$$ for every large positive integer $m$ by and . We may assume that $H$ is sufficiently ample. Then we get $$\label{eq1.4} \limsup_{m\to \infty} \frac{\dim H^0(X, \mathcal O_X(\lceil mm_0(D+f^*Q)\rceil+A+2f^*H))} {m^{\kappa (F, D|_F)+\kappa _{\sigma}(Y, Q)}}>0$$ by and the definition of $\kappa _\sigma(Y, Q)$. This means that the following inequality $$\label{eq1.5} \kappa _\sigma(X, D+f^*Q)\geq \kappa (F, D|_F)+ \kappa _\sigma(Y, Q)$$ holds. Similarly, we can take a positive integer $m_1$ and a positive real number $C_1$ such that $$\label{eq1.6} \begin{split} C_1m^{\kappa (Y, Q)}&\leq \dim H^0(Y, \mathcal O_Y(\lfloor mm_1Q\rfloor))\\ &\leq \dim H^0(Y, \mathcal O_Y(\lfloor mm_1Q\rfloor+H)) \end{split}$$ for every large positive integer $m$ (see, for example, ) if $H$ is a sufficiently ample Cartier divisor. Then, by and , we have $$\label{eq1.7} \begin{split} &\dim H^0(X, \mathcal O_X(\lceil mm_1 (D+f^*Q)\rceil +A+2f^*H))\\ &\geq C_1m^{\kappa(Y, Q)} \cdot r(mm_1D; A) \end{split}$$ for every large positive integer $m$. Therefore, we get $$\label{eq1.8} \limsup_{m\to \infty} \frac{\dim H^0(X, \mathcal O_X(\lceil mm_1 (D+f^*Q)\rceil +A+2f^*H))} {m^{\kappa _\sigma(F, D|_F)+\kappa(Y, Q)}}>0$$ when $A$ is sufficiently ample. Note that $$\label{eq1.9} \sigma (m_1D|_F; A|_F)'=\max \left\{ k\in \mathbb Z_{\geq 0} \cup \{-\infty\}\, \left|\, \underset{m\to \infty}{\limsup}\frac{r(mm_1D; A)}{m^k}>0\right.\right\}$$ for a sufficiently general fiber $F$ of $f:X\to Y$ and that $$\label{eq1.10} \begin{split} \kappa_\sigma(F, D|_F) &=\kappa_\sigma (F, m_1D|_F) \\&=\max\{\sigma(m_1D|_F; A|_F)'\, |\, {\text{$A$ is very ample}}\}. \end{split}$$ Hence we have the inequality $$\label{eq1.11} \kappa _\sigma(X, D+f^*Q)\geq \kappa_\sigma (F, D|_F)+ \kappa (Y, Q)$$ by . It is highly desirable to solve the following conjecture. As we explained in [@fujino-corri], Nakayama’s original inequality on $\kappa_\sigma$ (see [@nakayama Chapter V, 4.1. Theorem (1)]) follows from Conjecture \[x-conj11.6\] and the argument in the proof of Theorem \[x-thm11.3\]. \[x-conj11.6\] Let $X$ be a smooth projective variety and let $D$ be a pseudo-effective $\mathbb R$-divisor on $X$. Then there exist a positive integer $m_0$, a positive rational number $C$, and an ample Cartier divisor $A$ on $X$ such that $$Cm^{\kappa_\sigma(X, D)}\leq \dim H^0(X, \mathcal O_X (\lfloor mm_0D\rfloor +A))$$ holds for every large positive integer $m$. Finally, we treat a slight generalization of the twisted weak positivity theorem. \[x-thm11.7\] Let $f:X\to Y$ be a surjective morphism from a normal projective variety $X$ onto a smooth projective variety $Y$. Let $\Delta$ be an effective $\mathbb R$-divisor on $X$ such that $K_X+\Delta$ is $\mathbb R$-Cartier and that $(X, \Delta)$ is log canonical over a nonempty Zariski open set of $Y$. 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--- abstract: 'Compressed sensing is a new methodology for constructing sensors which allow sparse signals to be efficiently recovered using only a small number of observations. The recovery problem can often be stated as the one of finding the solution of an underdetermined system of linear equations with the smallest possible support. The most studied relaxation of this hard combinatorial problem is the $l_1$-relaxation consisting of searching for solutions with smallest $l_1$-norm. In this short note, based on the ideas of Lagrangian duality, we introduce an alternating $l_1$ relaxation for the recovery problem enjoying higher recovery rates in practice than the plain $l_1$ relaxation and the recent reweighted $l_1$ method of Candès, Wakin and Boyd.' author: - 'Stéphane Chrétien [^1]' title: An Alternating l1 approach to the compressed sensing problem --- Introduction ============ Compressed Sensing (CS) is a very recent field of fast growing interest and whose impact on concrete applications in coding and image acquisition is already remarkable. Up to date informations on this new topic may be obtained from the website [*http://nuit-blanche.blogspot.com/*]{}. The foundational paper is [@Candes:IEEEIT06] where the main problem considered was the one of reconstructing a signal from a few frequency measurements. Since then, important contributions to the field have appeared; see [@Candes:ICM06] for a survey and references therein. The Compressed Sensing problem ------------------------------ In mathematical terms, the problem can be stated as follows. Let $x$ be a $k$-sparse vector in $\mathbb R^n$, i.e. a vector with no more than $k$ nonzero components. The observations are simply given by $$\label{linsys} y=Ax$$ where $A\in \mathbb R^{m\times n}$ and $m$ small compared to $n$ with ${\rm rank} A=m$, and the goal is to recover $x$ exactly from these observations. The main challenges concern the construction of observation matrices $A$ which allow to recover $x$ with $k$ as large as possible for given values of $n$ and $m$. The problem of compressed sensing can be solved unambiguously if there is no sparser solution to the linear system (\[linsys\]) than $x$. Then, recovery is obtained by simply finding the sparsest solution to (\[linsys\]). If for any $x$ in $\mathbb R^n$ we denote by $\|x\|_0$ the $l_0$-norm of $x$, i.e. the cardinal of the set of indices of nonzero components of $x$, the compressed sensing problem is equivalent to $$\label{l0} \min_{x\in \mathbb R^n} \|x\|_0 \hspace{.3cm} {\rm s.t. } \hspace{.3cm} Ax=y.$$ We denote by $\Delta_0(y)$ the solution of problem (\[l0\]) and $\Delta_0(y)$ is called a decoder [^2]. Thus, the CS problem may be viewed as a combinatorial optimization problem. Moreover, the following lemma is well known. \[algcond\] [(See for instance [@Cohen:Pre06])]{} If $A$ is any $m \times n$ matrix and $2k \leq m$, then the following properties are equivalent: i\. The decoder $\Delta_0$ satisfies $\Delta_0(Ax) = x$, for all $x \in \Sigma_k$, ii\. For any set of indices $T$ with $\#T = 2k$, the matrix $A_T$ has rank $2k$ where $A_T$ stands for the submatrix of $A$ composed of the columns indexed by $T$ only. The $l_1$ relaxation -------------------- The main problem in using the decoder $\Delta_0(y)$ for given observations $y$ is that the optimization problem (\[l0\]) is NP-hard and cannot reasonably be expected to be solved in polynomial time. In order to overcome this difficulty, the original decoder $\Delta_0(y)$ has to be replaced by simpler ones in terms of computational complexity. Assuming that $A$ is given, two methods have been studied for solving the compressed sensing problem. The first one is the orthognal matching pursuit (OMP) and the second one is the $l_1$-relaxation. Both methods are not comparable since OMP is a greedy algorithm with sublinear complexity and the $l_1$-relaxation offers usually better performances in terms of recovery at the price of a computational complexity equivalent to the one of linear programming. More precisely, the $l_1$ relaxation is given by $$\label{l1} \min_{x\in \mathbb R^n} \|x\|_1 \hspace{.3cm} {\rm s.t. } \hspace{.3cm} Ax=y.$$ In the following, we will denote by $\Delta_1(y)$ the solution of the $l_1$-relaxation (\[l1\]). From the computational viewpoint, this relaxation is of great interest since it can be solved in polynomial time. Indeed, (\[l1\]) is equivalent to the linear program $$\nonumber \min_{x\in \mathbb R^n} \sum_{i=1}^n z_i \hspace{.3cm} {\rm s.t. } \hspace{.3cm} -z\leq x\leq z,\hspace{.3cm}{\rm and}\hspace{.3cm} Ax=y.$$ The main subsequent problem induced by this choice of relaxation is to obtain easy to verify sufficient conditions on $A$ for the relaxation to be exact, i.e. to produce the sparsest solution to the underdetermined system (\[linsys\]). A nice condition was given by Candes, Romberg and Tao [@Candes:IEEEIT06] and is called the Restricted Isometry Property. Up to now, this condition could only be proved to hold with great probability in the case where $A$ is a subgaussian random matrix. Several algorithmic approaches have also been recently proposed in order to garantee the exactness of the $l_1$ relaxation such as in [@Juditsky:ArXiv08] and [@Daspremont:ArXiv08]. The goal of our paper is different. Its aim is to present a new method for solving the CS problem generalizing the original $l_1$-relaxation of ([@Candes:IEEEIT06]) and with much better performance in pratice as measured by success rate of recovery versus original sparsity $k$. Lagrangian duality and relaxations ================================== Equivalent formulations of the recovery problem ----------------------------------------------- Recall that the problem of exact reconstruction of sparse signals can be solved using $\Delta_0$ and Lemma \[algcond\]. Let us start by writing down problem (\[l0\]), to which $\Delta_0$ is the solution map, as the following equivalent problem $$\label{quad} \max_{z,\: x\in \mathbb R^n} e^t z$$ subject to $$\nonumber z_ix_i=0, \hspace{.3cm} z_i(z_i-1)=0 \hspace{.3cm} i=1,\ldots,n, \textrm{ and } Ax=y$$ where $e$ denotes the vector of all ones. Here since the sum of the $z_i$’s is maximized, the variable $z$ plays the role of an indicator function for the event that $x_i=0$. This problem is clearly nonconvex due to the quadratic equality constraints $z_ix_i=0, \hspace{.3cm} i=1,\ldots,n$. The standard Semi-Definite Programming (SDP) relaxation scheme {#SDP} -------------------------------------------------------------- A simple way to construct a SDP relaxation is to homogenize the quadratic forms in the formulation at hand using a binary variable $z_0=1$. Indeed, by symmetry, it will suffice to impose $z_0^1=1$ since, if the relaxation turns out to be exact and a solution $(z_0,z,x)$ is recovered with $z_0=-1$, then, as the reader will be able to check at the end of this section, $(-z_0,-z,-x)$ will also solve the relaxed problem. For instance, problem (\[quad\]) can be expressed as $$\label{quadhom} \max_{z,\: x\in \mathbb R^n} e^t zz_0$$ subject to $$\nonumber z_ix_i=0, \hspace{.3cm} z_i(z_i-z_0)=0 \textrm{ and } z_0Ax=y$$ for $i=1,\ldots,n, z_0^2=1$. If we choose to keep explicit all the constraints in problem (\[quadhom\]), the Lagrange function can be easily be written as $$\nonumber \begin{array}{l} L_{SDP}(w,\lambda ,\mu,\nu)=w^tQw+\sum_{i=1}^n \lambda_i w^tC_iw \\ \hspace{1cm}+\sum_{i=1}^n \mu_i w^tE_iw+v_0w^tE_0w\\ \hspace{1cm}+\sum_{j=1}^m \nu_j w^t A_j w-\nu^ty, \end{array}$$ where $w$ is the concatenation of $z_0$, $z$, $x$ into one vector, $\lambda$ (resp. $\mu$ and $\nu$) is the vector of Lagrange multipliers associated to the constraints $z_ix_i=0$, $i=1,\ldots,n$ (resp. $z_i(z_i-z_0)$, $i=1,\ldots,n$, and $z_0a_j^tx=y_j$, $j=1,\ldots,m$) and where all the matrices $Q$, $A_j$, $j=1,\ldots,m$, $E_i$, $i=1,\ldots,n$ and $C_i=1,\ldots,n$ belong to $\mathbb S_{2n+1}$, the set of symmetric $2n+1\times 2n+1$ real matrices and are defined by $$\nonumber Q= \left[ \begin{array}{ccc} 0 & \frac12 e^t & 0_{1,n} \\ \frac12 e & 0_{n,n} & 0_{n,n} \\ 0_{n,1} & 0_{n,n} & 0_{n,n} \end{array} \right] A_j= \left[ \begin{array}{ccc} 0 & 0_{1,n} & \frac12 a_j^t \\ 0_{n,1} & 0_{n,n} & 0_{n,n} \\ \frac12 a_j & 0_{n,n} & 0_{n,n} \end{array} \right]$$ for $j=1,\ldots,m$, where $a_i^t$ is the $j^{th}$ row of $A$, $$\nonumber E_0= \left[ \begin{array}{ccc} 1 & 0_{1,n} & 0_{1,n} \\ 0_{n,1} & 0_{n,n} & 0_{n,n} \\ 0_{n,1} & 0_{n,n} & 0_{n,n} \end{array} \right], E_i= \left[ \begin{array}{ccc} 0 & -e_i^t & 0_{1,n} \\ -e_i & 2D(e_i) & 0_{n,n} \\ 0_{n,1} & 0_{n,n} & 0_{n,n} \end{array} \right]$$ and $$\nonumber C_i= \left[ \begin{array}{ccc} 0 & 0_{1,n} & 0_{1,n} \\ 0_{n,1} & 0_{n,n} & D(e_i) \\ 0_{n,1} & D(e_i) & 0_{n,n} \end{array} \right]$$ for $i=1,\ldots,n$ where $e_i$ is the vector with all components equal to zero except the $i^{th}$ which is set to one, $e$ is the vector of all ones, $D(e_i)$ is the diagonal matrix with diagonal vector $e_i$ and where $0_{k,l}$ denotes the $k\times l$ matrix of all zeros. The dual function is given by $$\nonumber \theta_{SDP}(\lambda,\mu,\nu)=\sup_{w\in \mathbb R^{2n+1}}L(w,\lambda,\mu,\nu),$$ and thus $$\nonumber \theta_{SDP}(\lambda,\mu,\nu)=\begin{cases} -\nu^ty \textrm{ if } Q(\lambda,\mu,\nu)\preceq 0 \\ +\infty \textrm{ otherwise } \end{cases}$$ with $$\nonumber Q(\lambda,\mu,\nu)=w^tQw+\sum_{i=1}^n \lambda_i w^tC_iw+\sum_{i=0}^n \mu_i w^tE_iw+\sum_{j=1}^m \nu_j w^t A_j w$$ and where $\succeq$ is the Löwner ordering ($A\succeq B$ iff $A-B$ is positive semi-definite). Therefore, the dual problem is given by $$\nonumber \inf_{\lambda \in \mathbb R^n, \mu\in \mathbb R^{n+1}, \nu\in \mathbb R^m} \theta_{SDP}(\lambda,\mu,\nu),$$ which is in fact equivalent to the following semi-definite program $$\label{dualsdp} \inf_{\lambda \in \mathbb R^n, \mu\in \mathbb R^{n+1}, \nu\in \mathbb R^m} -y^t \nu,$$ subject to $$Q(\lambda,\mu,\nu)\preceq 0.$$ We can also try and formulate the dual of this semi-definite program which is called the bidual of the initial problem. This bidual problem is easily seen after some computations to be given by $$\label{bidual} \max_{X\in \mathbb S_{2n+1},\: X\succeq 0} {\rm trace} (QX)$$ subject to $$\nonumber {\rm trace} (A_jX)=y_j,\: j=1,\ldots,m,$$ $$\label{z0} {\rm trace}(E_0X)=1,$$ $$\nonumber {\rm trace}(E_iX)=0 \textrm{ and } {\rm trace}(C_iX)=0,\: i=1,\ldots,n.$$ Now, if $X^*$ is an optimal solution with ${\rm rank}(X^*)=1$, then $$\nonumber X^*=\Big(\pm \left[ \begin{array}{c} z_0^* \\ z^* \\ x^* \end{array} \right]\Big) \Big(\pm \left[ \begin{array}{c} z_0^* \\ z^* \\ x^* \end{array} \right]\Big)^t$$ and it can be easily verified that all the constraints in (\[quadhom\]) are satisfied. Moreover, we may additionally impose that $z_0^*=1$ [^3]. However, the following proposition ruins the hopes for the occurance of such an agreable situation. \[optrank\] If non empty, the solution set of the bidual problem (\[bidual\]) is not a singleton and it contains matrices with rank equal to $n-m$. [**Proof**]{}. Consider the subspace $W_0$ of $\mathbb R^{2n+1}$ as the set of vectors whose $n+1$ first coordinates are equal to zero and such that the last $n$ coordinates form a vector in the kernel of $A$. Since we assumed that ${\rm rank} A=m$, we have that ${\rm dim} W_0=n-m$. Assume that there exists a solution $X^*$ to (\[bidual\]) with rank less than or equal to $n-m-1$. Then, it is possible to find a vector $w$ in $W_0$ with $w^t\perp P_{W_0}({\rm Range}(X^*))$. On the other hand, one can easily check that $X^{**}=X^*+ww^t$ satisfies all the bidual constraints and has the same objective value as $X^*$. Thus, $X^{**}$ is also a solution of the bidual problem and ${\rm rank} X^{**}={\rm rank} X^*+1$. Iterating the argument up to matrices of dimension equal to $n-1$, we obtain that the solution set contains matrices with rank equal to $n-m$. To prove non uniqueness of the solution, for any solution matrix $X^*$, set $X^{***}=X^*+ww^t$ for any choice of $w$ in $W_0$ and $X^{***}$ is also a solution of the bidual problem. $\Box$ Comments on the SDP relaxation ------------------------------ Despite the powerfull Lagrangian methodology behind its construction, the SDP relaxation of the problem has three major drawbacks: - as implied by Proposition \[optrank\], the standard SDP relaxation scheme leads to solutions which naturally have rank greater than one which makes it hard to try and recover a nice primal candidate. Moreover, even if the rank problem could be overcome in practice in the case where $x$ is sparse enough, by adding more ad hoc constraints in the SDP, finding the most natural way to do this seemed quite non trivial to us. - in the case where the SDP has a duality gap, proposing a primal suboptimal solution does not seem to be an easy task. - the computational cost of solving Semi-Definite Programs is much greater than the cost of solving our naive relaxation, a fact which may be important in real applications. An utopic relaxation {#Utop} -------------------- In order to overcome the drawbacks of the SDP relaxation, we investigate another scheme which may look utopic at first sight. Notice that one interesting variant of formulation (\[quad\]) could be the following in which the nonconvex complementarity constraints are merged into the unique constraint $\|D(z)x\|_1=0$ $$\label{prel1} \max_{z\in \{0,1\}^n} e^t z \hspace{.3cm} {\rm s.t. } \|D(z)x\|_1=0, \hspace{.3cm} Ax=y.$$ Choosing to keep the constraints $Ax=y$ and $z\in \{0,1\}^n$ implicit in (\[prel1\]), the Lagrangian function is given by $$L(x,z,u)=e^tz-u\|D(z)x\|_1$$ where $D(z)$ is the diagonal matrix with diagonal vector equal to $z$. The dual function (with values in $\mathbb R\cup +\infty$) is defined by $$\label{theta} \theta(u)=\max_{z\in \{0,1\}^n, \: Ax=y} L(x,z,u)$$ and the dual problem is $$\label{dual} \inf_{u\in \mathbb R} \theta(u).$$ The main problem with the dual problem (\[dual\]) is that the solutions to (\[theta\]) are as difficult to obtain as the solution of the original problem (\[prel1\]) because of the nonconvexity of the Lagrangian function $L$. The Alternating $l_1$ method ============================ We now present a generalization of the $l_1$ relaxation which we call the Alternating $l_1$ relaxation with better experimental performances than the standard $l_1$ relaxation and the SDP relaxation. A practical alternative to the utopic relaxation ------------------------------------------------ Due to the difficulty of computing the dual function $\theta$ in the relaxation \[Utop\], the interest of this scheme seems at first to be of pure theoretical nature only. In this section, we propose a suboptimal but simple alternating minimization approach. When we restrict $z$ to the value $z=e$, i.e. the vector of all ones, solving the problem $$\label{partialtheta} x_(u)={\rm argmax}_{z=e, \: x\in \mathbb R^n, \: Ax=y} L(x,z,u)$$ gives exactly the solution $\Delta_1(y)$ of the $l_1$ relaxation. From this remark, and the Lagrangian duality theory above, it may be supected that a better relaxation can be obtained by trying to optimize the Lagrangian even in a suboptimal manner. $u>0$ and $L \in \mathbb N_*$ $z_u^{(0)}=e$ $x_u^{(0)}\in{\rm argmax}_{x\in \mathbb R^n, \: Ax=y} L(x,z^{(0)},u)$ $l=1$ $z_u^{(l)}\in {\rm argmax}_{z\in \{0,1\}^n} L(x_u^{(l)},z,u)$ $x_u^{(l)}\in {\rm argmax}_{x\in \mathbb R^n, \: Ax=y} L(x,z_u^{(l)},u)$ $l \leftarrow l+1$ Output $z_u^{(L)}$ and $x_u^{(L)}$. At each step, knowing the value of $z_u^{(l)}$ implies that optimization with respect to $x\in\mathbb R^n$ can be equivalently restricted to the set of variables $x_i$ which are indexed by the $i$’s associated with the values of $z_u^{(l)}$ which are equal to one. Thus, the choice of $z_u^{(l)}$ corresponds to adaptive support selection for the signal to recover. The following lemma states that $z_u^{(l)}$ is in fact the solution of a simple thresholding procedure. \[01\] For all $x$ in $\mathbb R^n$, any solution $z$ of $$\label{rlxstp1} \max_{z\in [0,1]^n} L(x,z,u)$$ satisfies that $z_i=1$ if $|x_i|< \frac1{u}$, 0 if $|x_i|> \frac1{u}$ and $z_i\in [0,1]$ otherwise. [**Proof**]{}. Problem (\[rlxstp1\]) is clearly separable and the solution can be easily computed coordinatewise. $\Box$ Monte Carlo experiments ======================= In this section, using Monte Carlo experiments, we compare our Alternating $l_1$ approach to two recent methods proposed in the litterature: the Reweighted $l_1$ of Candès, Wakin and Boyd [@Candes:JFAA08] and the Iteratively Reweighted Least-Squares as proposed in [@Chartrand:ICASSP08]. The problem size was chosen to be the same as in Chartrand and Yin’s paper [@Chartrand:ICASSP08]: $n=256$, $m=100$. For each sparsity $k$ level a hundred different $k$-sparse vectors $x$ were drawn as follows: the support was chosen uniformly on all support with cardinal $k$ and the nonzero components were drawn from the Gaussian distribution $\mathcal N(0,4)$. The,n, the observation matrix was obtained in two steps: first draw a $m\times n$ matrix with i.i.d. Gaussian $\mathcal N(0,1)$ entries and then normalize each column to 2 as in [@Chartrand:ICASSP08]. The parameter $u$, namely the Lagrange multiplier for the complementarity constraint was tuned as follows: since on the one hand the natural breakdown point for $l_0/l_1$ equivalence, i.e. equivalence of using $l_0$ vs. $l_1$ minimization, lies around $k=\frac{m}4$ and on the other hand, the Alternating $l_1$ is nothing but a successive thresholding algorithm due to Lemma \[01\], we decided to chose the smallest possible $u$ so that the $\frac{m}4$ largest components $x_u^{(0)}$ the first step of the Alternating $l_1$ algorithm (which is nothing but the plain $l_1$ decoder whatever the value of $u$) be over $\frac1{u}$. Notice that this value of $u$ is surely not the solution of the dual problem but our choice is at least motivated by reasonable deduction based on pratical observations whereas the tuning parameter in the other two algorithms is not known to enjoy such an intuitive and meaningful selection rule. We chose $L=4$ in these experiments. The numerical results for the IRLS and the Reweighted $l_1$ were communicated to us by Rick Chartrand whom we greatly thank for his collaboration. Comparison between the success rates the three methods is shown in Figure 1. Our Alternating $l_1$ method outperformed both the Iteratively Reweighted Least Squares and the Reweighted $l_1$ methods for the given data size. As noted in [@Chartrand:ICASSP08], the IRLS and the Reweighted $l_1$ enjoy nearly the same exact recovery success rates. \[comp\] {width="8.5cm"} [**Remark**]{}. The Reweighted $l_1$ and the Reweighted LS both need a value of $\epsilon$ (or even a sequence of such values as in [@Chartrand:InvProb08]) which is hard to optimize ahead of time, whereas the value $u$ in the Alternating $l_1$ is a Lagrange multiplier, i.e. a dual variable. In the Monte Carlo experiments of the previous section, we decided to base our choice of $u$ on a simple an intuitive criterion suggested by the well known experimental behavior of the plain $l_1$ relaxation. On the other hand, it should be interesting to explore duality a bit further and perform experiments in the case where $u$ is approximately optimized (using any derivative free procedure for instance) based on our heuristic alternating $l_1$ approximation of the dual function $\theta$. [1]{} Candes, E., Romberg, J. and Tao T., Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information, IEEE Information Theory, 2006, 2, 52, pp. 489–509. Candes, E., Compressive sampling, 2006, 3, International Congress of Mathematics, pp. 1433–1452, EMS. Candes, E., Wakin, M. and Boyd S., Enhancing Sparsity by Reweighted $l_1$ Minimization, Journal of Fourier Analysis and Applications, 2008, 14, pp. 877–905. Cohen, A., Dahmen, W. and DeVore R., Compressed sensing and best $k$-term approximation, J. Amer. Math. Soc. 22 (2009), no. 1, 211–231. Chartrand, R. and Staneva, V., Restricted isometry properties and nonconvex compressive sensing, Inverse Problems, vol. 24, no. 035020, pp. 1–14, 2008. Chartrand, R. and Yin, W., Iterativement reweighted algorithms for Compressed Sensing, 33rd International Conference on Acoustics, Speech, and Signal Processing (ICASSP), 2008. A. d’Aspremont and L. El Ghaoui, Testing the Nullspace Property using Semidefinite Programming, http://arxiv.org/abs/0807.3520. Juditsky, A. and Nemirovsky, A., On Verifiable Sufficient Conditions for Sparse Signal Recovery via $\ell_1$ Minimization, http://arxiv.org/abs/0809.2650. Hiriart Urruty, J.-B. and Lemaréchal, C., Convex analysis and minimization algorithms II: Advanced theory and bundle methods, Springer- Verlag, 1993, 306, Grundlehren der Mathematischen Wissenschaften. [^1]: S. Chrétien is with the Laboratoire de Mathématiques, UMR CNRS 6623 and Université de Franche Comté, 16 route de Gray, 25030 Besan[ç]{}on Cedex, France. Email: stephane.chretien@univ-fcomte.fr [^2]: In the general case where $x$ is not the unique sparsest solution of (\[l0\]) using this approach for recovery is of course possibly not pertinent. Moreover, in such a case, this problem has several solutions with equal $l_0$-“norm” and one may rather define $\Delta_0(y)$ as an arbitrary element of the solution set. [^3]: Indeed, if $z_0^*=-1$, multiply by $-1$ the whole vector $[z_0^*,z^*,x^*]$

--- abstract: 'We establish the Berry-phase formulas for the angular momentum (AM) and the Hall viscosity (HV) to investigate chiral superconductors (SCs) in two and three dimensions. The AM is defined by the temporal integral of the antisymmetric momentum current induced by an adiabatic deformation, while the HV is defined by the symmetric momentum current induced by the symmetric torsional electric field. Without suffering from the system size or geometry, we obtain the macroscopic AM $L_z = \hbar m N_0/2$ at zero temperature in full-gap chiral SCs, where $m$ is the magnetic quantum number and $N_0$ is the total number of electrons. We also find that the HV is equal to half the AM at zero temperature not only in full-gap chiral SCs as is well known but also in nodal ones, but its behavior at finite temperature is different in the two cases.' author: - Atsuo Shitade - Taro Kimura title: Bulk angular momentum and Hall viscosity in chiral superconductors --- Introduction {#sec:intro} ============ Chiral superfluids and superconductors (SCs) are exotic states whose time-reversal symmetry is spontaneously broken and Cooper pairs carry nonzero angular momentum (AM). One well-known example is $^3$He-A whose pairing symmetry is $p_x + i p_y$ [@RevModPhys.47.331]. Among electron systems, there are a few candidates for chiral SCs such as Sr$_2$RuO$_4$ with $p_x + i p_y$ [@RevModPhys.75.657; @JPSJ.81.011009] and URu$_2$Si$_2$ with $d_{zx} + i d_{yz}$ [@PhysRevLett.99.116402; @PhysRevLett.100.017004; @1367-2630-11-5-055061; @JPSJ.81.023704]. Recently the $d_{x^2 - y^2} + i d_{xy}$ pairing symmetry was theoretically proposed in SrPtAs [@PhysRevB.86.100507; @PhysRevB.87.180503; @PhysRevB.89.020509]. There is a long-standing problem on the AM in chiral $\ell$-wave SCs, the so-called intrinsic AM paradox. This paradox is summarized as $L_z = \hbar m N_0/2 \times (\Delta_0/E_{\rm F})^{\gamma}$, where $|m| \leq \ell$, $N_0$, $\Delta_0$, and $E_{\rm F}$ are the magnetic quantum number, the total number of electrons, the gap strength, and the Fermi energy, respectively. $\gamma = 0$ [@Ishikawa01061977; @Ishikawa01041980; @PhysRevB.21.980; @volovik1995; @JPSJ.67.216; @Goryo1998549; @PhysRevB.69.184511; @PhysRevB.84.214509; @PhysRevB.85.100506] is the most natural if all electrons form Cooper pairs with the AM $\ell_z = \hbar m$. On the other hand, $\gamma = 1$ [@PhysRev.123.1911; @RevModPhys.47.331] is intuitively plausible if a few electrons near the Fermi surface form Cooper pairs. $\gamma = 2$ was also proposed by using the Ginzburg-Landau theory [@volovik1975; @cross1975]. Recent microscopic studies have settled this paradox, at least theoretically, to $\gamma = 0$ [@JPSJ.67.216; @PhysRevB.69.184511; @PhysRevB.84.214509; @PhysRevB.85.100506] but have been employed only in finite systems such as a cylinder [@JPSJ.67.216] and a disk [@PhysRevB.69.184511; @PhysRevB.84.214509; @PhysRevB.85.100506]. Generally, the physical quantities involving the position operator are ill defined in periodic systems. The most famous examples are the charge polarization and the orbital magnetization. The former is classically coupled to an electric field but is quantum-mechanically defined by the temporal integral of the charge current under an adiabatic deformation [@PhysRevB.47.1651; @PhysRevB.49.14202; @PhysRevB.88.155121; @JPSJ.83.033708]. On the other hand, the latter is coupled to a magnetic field and can be defined by the magnetic field derivative of the free energy [@PhysRevLett.99.197202; @PhysRevB.84.205137; @PhysRevB.86.214415]. Consequently, the charge polarization is associated with the Berry connection in the Bloch basis, while the orbital magnetization with the Berry curvature and the magnetic moment. These formalisms were extended to heat analogs such as the heat polarization [@JPSJ.83.033708] and magnetization [@1310.8043]. However, the AM has not been formulated yet. Another interesting clue to the AM is the Hall viscosity (HV), which has been intensively discussed in the context of the quantum Hall effect [@PhysRevLett.75.697; @avron1998]. Similar to the Hall conductivity, the HV is nonzero only when the time-reversal symmetry is broken. The important relation $\eta_{\rm H} = \hbar N_0 {\bar s}/2$ holds in general gapped systems at zero temperature [@PhysRevB.79.045308; @PhysRevB.84.085316; @PhysRevB.89.174507], in which the orbital spin ${\bar s}$ is equal to $\ell/2$ in chiral $\ell$-wave SCs. In addition, it was also related to the momentum-dependent Hall conductivity [@PhysRevLett.108.066805; @PhysRevB.86.245309]. The torsional Chern-Simons term was proposed to describe the quantum HV in the two-dimensional massive Dirac system [@PhysRevLett.107.075502; @Hidaka01012013; @PhysRevD.88.025040]. In this paper, we derive the Berry-phase formulas for the AM and the HV to apply to chiral SCs in two and three dimensions. The antisymmetric and symmetric components of a torsional electric field describe an angular velocity and a strain-rate tensor, respectively. Since the AM is conjugate to the former, it can be formulated in the same way as in the charge polarization, namely, by the temporal integral of the antisymmetric momentum current induced by an adiabatic deformation. Viscosity is defined by the symmetric momentum current, i.e., the stress tensor, induced by the latter. In contrast to the previous works regarding the intrinsic AM paradox, we obtain $L_z = \hbar m N_0/2$ without suffering from the finite-size effects. We also investigate the temperature dependence of the HV for two-dimensional gapped chiral SCs and three-dimensional nodal ones. Hereafter we assign the Latin ($a, b, \dots = {\hat 0}, {\hat 1}, \dots, {\hat d}$) and Greek ($\mu, \nu, \dots = 0, 1, \dots, d$) alphabets to locally flat and global coordinates, respectively. We follow the Einstein convention, which implies summation over the spacetime dimension $D = d + 1$ when an index appears twice in a single term. The Minkowski metric is taken as $\eta_{ab} = \operatorname{diag}(-1, +1, \dots, +1)$. The Planck constant and the charge are denoted by $\hbar$ and $q$, while the speed of light and the Boltzmann constant are put to $c = k_{\rm B} = 1$. The upper or lower signs in equations correspond to boson or fermion. Cartan Formalism {#sec:cartan} ================ To begin with, we examine an angular velocity from the gauge-theoretical viewpoint. Now that the system is rotated, we have to deal with a theory in a curved spacetime. Here we use the Cartan formalism, which consists of two gauge potentials such as a vielbein and a spin connection [@9789812791719]. A vielbein $h^a_{\phantom{a} \mu}$ is a gauge potential corresponding to local spacetime translations, while a spin connection $\omega^{ab}_{\phantom{ab} \mu}$ is that corresponding to local Lorentz transformations. In these gauge potentials as well as a vector potential $A_{\mu}$, the partial derivative is replaced by the covariant derivative $\partial_a \to D_a \equiv h_a^{\phantom{a} \mu} (\partial_{\mu} - i q A_{\mu}/\hbar - i \omega^{ab}_{\phantom{\mu}} S_{ab}/2 \hbar)$. Here $S_{ab}$ is the generator of local Lorentz transformations. The spatial component of a vielbein is related to a displacement vector [@PhysRevLett.107.075502; @Hidaka01012013; @PhysRevD.88.025040] as $h^{\hat k}_{\phantom{\hat k} i} = \delta^{\hat k}_{\phantom{\hat k} i} + \partial_i u^{\hat k}$. Moreover, $h^{\hat k}_{\phantom{\hat k} 0}$ describes rotation as illustrated below. For simplicity, let us consider a Dirac fermion in a curved spacetime. The Dirac Lagrangian density is given by $$h {\cal L} = h {\bar \psi} (\hbar \gamma^a D_a - m) \psi, \label{eq:dirac1}$$ in which $h \equiv \det h^a_{\phantom{a} \mu}$ is the determinant of a vielbein, ${\bar \psi} \equiv i^{-1} \psi^{\dag} \gamma^{\hat 0}$ is the Dirac conjugate, and the $\gamma$ matrices satisfy the Clifford algebra $\{\gamma^a, \gamma^b\} = 2 \eta^{ab}$. When we introduce the nonzero off-diagonal component $h^{\hat k}_{\phantom{\hat k} 0} = \phi_{\rm r}^{\hat k}$ in addition to the identity, the square line element is given by $$d s^2 = -d t^2 + (d {\vec x} + {\vec \phi}_{\rm r} d t)^2, \label{eq:line}$$ and the Dirac Lagrangian is reduced to $$h {\cal L} = \psi^{\dag} [i \hbar D_0 + i \hbar {\vec \alpha} \cdot {\vec D} - m \beta - i \hbar {\vec \phi}_{\rm r} \cdot {\vec D}] \psi. \label{eq:dirac2}$$ Here ${\vec \alpha}$ and $\beta$ are the Dirac matrices. If we put ${\vec \phi}_{\rm r} = {\vec \Omega} \times {\vec x}$, ${\vec \Omega}$ is assigned to an angular velocity, and the fourth term represents the coupling between an angular velocity and the AM. Such a discussion relies on the translational symmetry and gauge principle of gravity and hence is not restricted to relativistic systems. Since a vielbein is a gauge potential, it induces a field strength called torsion. Here a spin connection is neglected, and hence torsion is written by $$\begin{aligned} T^{\hat l}_{\phantom{\hat l} j0} = & \partial_j h^{\hat l}_{\phantom{\hat l}0} - \partial_0 h^{\hat l}_{\phantom{\hat l} j}, \label{eq:torsion1a} \\ T^{\hat l}_{\phantom{\hat l} ij} = & \partial_i h^{\hat l}_{\phantom{\hat l} j} - \partial_j h^{\hat l}_{\phantom{\hat l} i}. \label{eq:torsion1b}\end{aligned}$$ \[eq:torsion1\] The former is “electric.” The first term describes an angular velocity if ${\hat l}$ and $j$ are antisymmetric, while the second term describes a strain-rate tensor if symmetric. On the other hand, the latter is “magnetic” and characterizes edge and screw dislocations in crystals. Especially its flux is identified as the Burgers vector. It is a natural question why rotation is described by a vielbein, i.e., a gauge potential corresponding to space translations. In fact, global space translations do not include rotations. However, a gauge potential is associated with a local symmetry, and local space translations do include rotations. Note that an angular velocity is coupled not only to the AM but to the spin, which is implemented by a spin connection. Such spin responses are out of scope here. Angular Momentum {#sec:am} ================ Since an angular velocity is “electric,” the AM is similar to the charge polarization rather than to the orbital magnetization. Below we define the momentum polarization by the temporal integral of the nonsymmetric momentum current induced by an adiabatic deformation. The momentum current is conjugate to a “vector potential” $h^{\hat k}_{\phantom{\hat k} i}$ and is given by the product of the momentum and the velocity. In the Wigner representation of the Keldysh formalism, it is written by $$T_{\hat k}^{\phantom{\hat k} {\hat \imath}} = \pm \frac{i \hbar}{2} \int \frac{d^D \pi}{(2 \pi \hbar)^D} \operatorname{tr}[(-\partial_{\pi_{\hat \imath}} {{\hat {\bm G}}}^{-1}) \star {{\hat {\bm G}}} \star \pi_{\hat k}]^< + {{\rm c.c.}}\label{eq:momcurr}$$ Here ${{\hat {\bm G}}}$ is the Keldysh Green’s function, $(-\partial_{\pi_{\hat \imath}} {{\hat {\bm G}}}^{-1})$ is the renormalized velocity to satisfy the local conservation law, and $\pi_{\hat k}$ is the Wigner representation of the covariant derivative corresponding to the momentum. Note that this momentum current is Hermitian but not symmetric over ${\hat k}$ and ${\hat \imath}$. An adiabatic deformation is implemented by the gradient expansion up to the first order [@JPSJ.83.033708]. Since both the Keldysh Green’s function ${{\hat {\bm G}}}$ and the renormalized velocity $(-\partial_{\pi_{\hat \imath}} {{\hat {\bm G}}}^{-1})$ are perturbed, the gradient expansion of the momentum current Eq.  is given by $$\begin{aligned} T_{\hat k}^{\phantom{\hat k} {\hat \imath}} = & \pm \frac{i \hbar^2}{4} \int \frac{d^D \pi}{(2 \pi \hbar)^D} \pi_{\hat k} \operatorname{tr}[\partial_{\pi_{\hat \imath}} {\hat \Sigma}_1 {\hat G}_0 - \partial_{\pi_{\hat \imath}} {\hat G}_0^{-1} {\hat G}_1]^< \notag \\ & + {{\rm c.c.}}, \label{eq:mp1}\end{aligned}$$ in which the gradient expansion of the Keldysh Green’s function ${\hat G}_1$ is given by $${\hat G}_1 = {\hat G}_0 {\hat \Sigma}_1 {\hat G}_0 + i [{\hat G}_0 \partial_{X^0} {\hat G}_0^{-1} {\hat G}_0 \partial_{\pi_{\hat 0}} {\hat G}_0^{-1} {\hat G}_0 - (X^0 \leftrightarrow \pi_{\hat 0})], \label{eq:gd}$$ and the corresponding self-energy ${\hat \Sigma}_1$ is determined self-consistently. Now the momentum current is transformed into $$\begin{aligned} T_{\hat k}^{\phantom{\hat k} {\hat \imath}} = & \pm \frac{i \hbar^2}{4} \int \frac{d^D \pi}{(2 \pi \hbar)^D} \pi_{\hat k} \operatorname{tr}[\partial_{\pi_{\hat \imath}} {\hat \Sigma}_1 {\hat G}_0 - \partial_{\pi_{\hat \imath}} {\hat G}_0^{-1} {\hat G}_0 {\hat \Sigma}_1 {\hat G}_0]^< + {{\rm c.c.}}\notag \\ & \pm \frac{\hbar^2}{4} \int \frac{d^D \pi}{(2 \pi \hbar)^D} \pi_{\hat k} \operatorname{tr}[\partial_{\pi_{\hat \imath}} {\hat G}_0^{-1} {\hat G}_0 \partial_{X^0} {\hat G}_0^{-1} {\hat G}_0 \partial_{\pi_{\hat 0}} {\hat G}_0^{-1} {\hat G}_0 - (X^0 \leftrightarrow \pi_{\hat 0})]^< + {{\rm c.c.}}\label{eq:mp2}\end{aligned}$$ By extracting the lesser component and employing the temporal integral, we obtain the change in the momentum polarization, but not the momentum polarization itself, $$\begin{aligned} \Delta P_{\hat k}^{\phantom{\hat k} {\hat \imath}} \equiv & \int d X^0 T_{\hat k}^{\phantom{\hat k} {\hat \imath}} \notag \\ = & \frac{\hbar^2}{6} \epsilon_{ABC} \int \frac{d^D \pi}{(2 \pi \hbar)^D} \int d X^0 f(-\pi_{\hat 0}) \pi_{\hat k} \operatorname{tr}[G_0^{\rm R} \partial_A G_0^{{\rm R} -1} G_0^{\rm R} \partial_B G_0^{{\rm R} -1} G_0^{\rm R} \partial_C G_0^{{\rm R} -1}] + {{\rm c.c.}}\label{eq:mp3a} \\ & + \frac{\hbar^2}{4} \int \frac{d^D \pi}{(2 \pi \hbar)^D} \int d X^0 f^{\prime}(-\pi_{\hat 0}) \pi_{\hat k} \operatorname{tr}[(G_0^{\rm R} - G_0^{\rm A}) \partial_{\pi_{\hat \imath}} (G_0^{{\rm R} -1} + G_0^{{\rm A} -1}) G_0^{\rm R} \partial_{X^0} G_0^{{\rm R} -1}] + {{\rm c.c.}}\label{eq:mp3b} \\ & \mp \frac{i \hbar^2}{4} \delta_{\hat k}^{\phantom{\hat k} {\hat \imath}} \int \frac{d^D \pi}{(2 \pi \hbar)^D} \int d X^0 \operatorname{tr}[{\hat \Sigma}_1 {\hat G}_0]^< + {{\rm c.c.}}\label{eq:mp3c} \\ & \pm \frac{i \hbar^2}{4} \int \frac{d^D \pi}{(2 \pi \hbar)^D} \int d X^0 f^{\prime}(-\pi_{\hat 0}) \pi_{\hat k} \operatorname{tr}[\Sigma_1^{< (1)} (G_0^{\rm R} - G_0^{\rm A}) \partial_{\pi_{\hat \imath}} G_0^{{\rm R} -1} G_0^{\rm R}] + {{\rm c.c.}}, \label{eq:mp3d}\end{aligned}$$ \[eq:mp3\] where $\epsilon_{ABC}$ in Eq.  is the antisymmetric tensor with $\epsilon_{\pi_{\hat 0} \pi_{\hat \imath} X^0} = 1$. $f(\omega) = (e^{\beta \omega} \mp 1)^{-1}$ is the distribution function for boson or fermion. Below we focus on the clean and noninteracting limit ${{\hat {\bm \Sigma}}} = 0$ to derive the Berry-phase formula in the Bloch basis. The retarded Green’s function is given by $G_0^{\rm R} = [(-\pi_{\hat 0}) - {\cal H}(X^0) + \mu + i \eta]^{-1}$ with $\eta \to +0$, leading to $\partial_{\pi_{\hat 0}} G_0^{{\rm R} -1} = -1$, $\partial_{\pi_{\hat \imath}} G_0^{{\rm R} -1} = -v^{\hat \imath}$, and $\partial_{X^0} G_0^{{\rm R} -1} = - {{\dot {\cal H}}}$. In an adiabatic deformation, the trace can be expanded with respect to the eigenstates satisfying ${\cal H}(X^0) | u_{n {\vec \pi} X^0} \rangle = \epsilon_{n {\vec \pi} X^0} | u_{n {\vec \pi} X^0} \rangle$. We evaluate the integral over $(-\pi_{\hat 0})$ by the residue theorem to obtain $$\Delta P_{\hat k}^{\phantom{\hat k} {\hat \imath}} = -\sum_n \int \frac{d^d \pi}{(2 \pi \hbar)^d} \int d X^0 \pi_{\hat k} \Omega^{\hat \imath}_{n {\vec \pi} X^0} f_{n {\vec \pi} X^0}, \label{eq:mp4}$$ with $f_{n {\vec \pi} X^0} \equiv f(\epsilon_{n {\vec \pi} X^0} - \mu)$ and the Berry curvature in the $(\pi_{\hat \imath}/\hbar, X^0)$-space being defined by $$\Omega_{n {\vec \pi} X^0}^{\hat \imath} \equiv i \hbar \langle \partial_{\pi_{\hat \imath}} u_{n {\vec \pi} X^0} | \partial_{X^0} u_{n {\vec \pi} X^0} \rangle - (X^0 \leftrightarrow \pi_{\hat \imath}). \label{eq:berrycurv-t}$$ Similar to the charge polarization, Eq.  depends on the choice of an adiabatic deformation at finite temperature and hence is not well defined. At zero temperature in a gapped fermion system, its integrand becomes the total derivative with respect to $X^0$ when ${\hat k} \not= {\hat \imath}$, and the momentum polarization itself is given by $$P_{\hat k}^{\phantom{\hat k} {\hat \imath}} = \sum_n^{\rm occ} \int \frac{d^d \pi}{(2 \pi \hbar)^d} \pi_{\hat k} A_{n {\vec \pi}}^{\hat \imath}, \label{eq:mp5}$$ where we introduce the Berry connection, $$A_{n {\vec \pi}}^{\hat \imath} \equiv i \hbar \langle u_{n {\vec \pi}} | \partial_{\pi_{\hat \imath}} u_{n {\vec \pi}} \rangle. \label{eq:berrycon}$$ Indeed this expression is quite similar to that for the charge polarization, which is given by the integral of the Berry connection itself. Note that at the initial time, the reference system has time-reversal symmetry, and the momentum polarization should be zero. After all, the AM is obtained by the antisymmetric part of the momentum polarization, $$L_{\hat k} \equiv \epsilon_{{\hat \imath} {\hat \jmath} {\hat k}} P^{{\hat \jmath} {\hat \imath}} = \sum_n^{\rm occ} \int \frac{d^d \pi}{(2 \pi \hbar)^d} \epsilon_{{\hat \imath} {\hat \jmath} {\hat k}} A^{\hat \imath}_{n {\vec \pi}} \pi^{\hat \jmath}. \label{eq:am}$$ Since the Berry connection is regarded as the expectation value of the position operator in the Wannier basis, this Berry-phase formula really indicates ${\vec \ell} = {\vec x} \times {\vec p}$ in the momentum space. In the above derivation, it is not obvious why the system should be gapped. Wave functions have the phase degree of freedom; namely, physical quantities should be invariant under a unitary transformation $| u^{\prime}_{n {\vec \pi}} \rangle = e^{-i \theta_{n {\vec \pi}}} | u_{n {\vec \pi}} \rangle$. Correspondingly, the Berry connection Eq.  is transformed as ${\vec A}_{n {\vec \pi}}^{\prime} = {\vec A}_{n {\vec \pi}} + \hbar {\vec \partial}_{\pi} \theta_{n {\vec \pi}}$, and the integrand in Eq.  is transformed as ${\vec A}_{n {\vec \pi}}^{\prime} \times {\vec \pi} = {\vec A}_{n {\vec \pi}} \times {\vec \pi} + \hbar {\vec \partial}_{\pi} \times (\theta_{n {\vec \pi}} {\vec \pi})$. In a gapless system where the momentum space is restricted, the AM is not invariant. On the other hand, in a gapped system, the AM is found to be invariant by using the Stokes theorem and the single-valued property of wave functions. Hall Viscosity {#sec:eta} ============== Here we begin with a brief introduction of elasticity and viscosity. These are the mechanical properties of a system and are characterized by $$T^{(ki)} = \lambda^{(ki)(lj)} u_{(lj)} + \eta^{(ki)(lj)} {\dot u}_{(lj)}. \label{eq:visco}$$ Here $T^{(ki)}$, $u_{(lj)}$, and ${\dot u}_{(lj)}$ are stress, strain, and strain-rate tensors, respectively, and round brackets indicate the symmetry over their indexes. The linear coefficients $\lambda^{(ki)(lj)}$ and $\eta^{(ki)(lj)}$ are dubbed the elastic modulus and the viscosity, although their sign convention is not fixed. We can decompose the viscosity into the symmetric and antisymmetric parts: $\eta^{(ki)(lj)} = \eta^{(ki)(lj)}_{\rm S} + \eta^{(ki)(lj)}_{\rm A}$, where $\eta^{(ki)(lj)}_{\rm S} = \eta^{(lj)(ki)}_{\rm S}$ and $\eta^{(ki)(lj)}_{\rm A} = -\eta^{(lj)(ki)}_{\rm A}$. The latter is dubbed the HV and is formulated below. As discussed above, a strain-rate tensor is the symmetric torsional electric field, and the stress tensor is the symmetric part of the momentum current. Therefore the viscosity can be formulated by the perturbation theory of the momentum current Eq.  with respect to torsion [@1310.8043]. Here we define the nonsymmetric viscosity by $$\begin{aligned} \eta_{{\hat k} \phantom{\hat \imath} {\hat l}}^{\phantom{\hat k} {\hat \imath} \phantom{\hat l} {\hat \jmath}} \equiv & \frac{\partial T_{\hat k}^{\phantom{\hat k} {\hat \imath}}}{\partial (-T^{\hat l}_{\phantom{\hat l} {\hat \jmath} {\hat 0}})} \notag \\ = & \mp \frac{i \hbar^2}{2} \int \frac{d^D \pi}{(2 \pi \hbar)^D} \pi_{\hat k} \operatorname{tr}[\partial_{\pi_{\hat \imath}} {\hat \Sigma}_{T^{\hat l}_{\phantom{\hat l} {\hat \jmath} {\hat 0}}} {\hat G}_0 - \partial_{\pi_{\hat \imath}} {\hat G}_0^{-1} {\hat G}_{T^{\hat l}_{\phantom{\hat l} {\hat \jmath} {\hat 0}}}]^< \notag \\ & + {{\rm c.c.}}\label{eq:eta1}\end{aligned}$$ Note that the negative sign in the definition is necessary because $\partial_0 h^{\hat l}_{\phantom{\hat l} j}$ in Eq.  gives rise to a strain-rate tensor. The Keldysh Green’s function in the presence of torsion is given by $$\begin{aligned} {\hat G}_{T^a_{\phantom{a} cd}} = & {\hat G}_0 {\hat \Sigma}_{T^a_{\phantom{a} cd}} {\hat G}_0 \notag \\ & - \pi_a [{\hat G}_0 \partial_{\pi_c} {\hat G}_0^{-1} {\hat G}_0 \partial_{\pi_d} {\hat G}_0^{-1} {\hat G}_0 - (c \leftrightarrow d)]/2 i, \label{eq:gtem}\end{aligned}$$ and the corresponding self-energy $\Sigma_{T^a_{\phantom{a} cd}}$ is determined self-consistently. Equation  is rewritten by $$\begin{aligned} \eta_{{\hat k} \phantom{\hat \imath} {\hat l}}^{\phantom{\hat k} {\hat \imath} \phantom{\hat l} {\hat \jmath}} = & \mp \frac{i \hbar^2}{2} \int \frac{d^D \pi}{(2 \pi \hbar)^D} \pi_{\hat k} \operatorname{tr}[\partial_{\pi_{\hat \imath}} {\hat \Sigma}_{T^{\hat l}_{\phantom{\hat l} {\hat \jmath} {\hat 0}}} {\hat G}_0 - \partial_{\pi_{\hat \imath}} {\hat G}_0^{-1} {\hat G}_0 {\hat \Sigma}_{T^{\hat l}_{\phantom{\hat l} {\hat \jmath} {\hat 0}}} {\hat G}_0]^< + {{\rm c.c.}}\notag \\ & \mp \frac{\hbar^2}{4} \int \frac{d^D \pi}{(2 \pi \hbar)^D} \pi_{\hat k} \pi_{\hat l} \operatorname{tr}[\partial_{\pi_{\hat \imath}} {\hat G}_0^{-1} {\hat G}_0 \partial_{\pi_{\hat \jmath}} {\hat G}_0^{-1} {\hat G}_0 \partial_{\pi_{\hat 0}} {\hat G}_0^{-1} {\hat G}_0 - (j \leftrightarrow 0)]^< + {{\rm c.c.}}, \label{eq:eta2}\end{aligned}$$ and then $$\begin{aligned} \eta_{{\hat k} \phantom{\hat \imath} {\hat l}}^{\phantom{\hat k} {\hat \imath} \phantom{\hat l} {\hat \jmath}} = & -\frac{\hbar^2}{6} \epsilon^{{\hat \imath} {\hat \jmath} {\hat k}} \epsilon_{abc {\hat k}} \int \frac{d^D \pi}{(2 \pi \hbar)^D} f(-\pi_{\hat 0}) \pi_{\hat k} \pi_{\hat l} \operatorname{tr}[G_0^{\rm R} \partial_{\pi_a} G_0^{{\rm R} -1} G_0^{\rm R} \partial_{\pi_b} G_0^{{\rm R} -1} G_0^{\rm R} \partial_{\pi_c} G_0^{{\rm R} -1}] + {{\rm c.c.}}\label{eq:eta3a} \\ & - \frac{\hbar^2}{4} \int \frac{d^D \pi}{(2 \pi \hbar)^D} f^{\prime}(-\pi_{\hat 0}) \pi_{\hat k} \pi_{\hat l} \operatorname{tr}[(G_0^{\rm R} - G_0^{\rm A}) \partial_{\pi_{\hat \imath}} (G_0^{{\rm R} -1} + G_0^{{\rm A} -1}) G_0^{\rm R} \partial_{\pi_{\hat \jmath}} G_0^{{\rm R} -1}] + {{\rm c.c.}}\label{eq:eta3b} \\ & \pm \frac{i \hbar^2}{2} \delta_{\hat k}^{\phantom{\hat k} {\hat \imath}} \int \frac{d^D \pi}{(2 \pi \hbar)^D} \operatorname{tr}[{\hat \Sigma}_{T^{\hat l}_{\phantom{\hat l} {\hat \jmath} {\hat 0}}} {\hat G}_0]^< + {{\rm c.c.}}\label{eq:eta3c} \\ & \mp \frac{i \hbar^2}{2} \int \frac{d^D \pi}{(2 \pi \hbar)^D} f^{\prime}(-\pi_{\hat 0}) \pi_{\hat k} \operatorname{tr}[\Sigma_{T^{\hat l}_{\phantom{\hat l} {\hat \jmath} {\hat 0}}}^{< (1)} (G_0^{\rm R} - G_0^{\rm A}) \partial_{\pi_{\hat \imath}} G_0^{{\rm R} -1} G_0^{\rm R}] + {{\rm c.c.}}\label{eq:eta3d}\end{aligned}$$ \[eq:eta3\] Again we focus on the clean and noninteracting limit ${{\hat {\bm \Sigma}}} = 0$. The retarded Green’s function is given by $G_0^{\rm R} = [(-\pi_{\hat 0}) - {\cal H} + \mu + i \eta]^{-1}$. We expand the trace with respect to the Bloch basis and evaluate the integral over $(-\pi_{\hat 0})$ by the residue theorem. As a result, we obtain the Berry-phase formula for the nonsymmetric HV, $$\eta_{{\hat k} \phantom{\hat \imath} {\hat l}}^{\phantom{\hat k} {\hat \imath} \phantom{\hat l} {\hat \jmath}} = \frac{1}{\hbar} \epsilon^{{\hat \imath} {\hat \jmath} {\hat m}} \sum_n \int \frac{d^d \pi}{(2 \pi \hbar)^d} \pi_{\hat k} \pi_{\hat l} \Omega_{n {\vec \pi} {\hat m}} f_{n {\vec \pi}}, \label{eq:eta4}$$ with $f_{n {\vec \pi}} \equiv f(\epsilon_{n {\vec \pi}} - \mu)$ and the Berry curvature being defined by $$\Omega_{n {\vec \pi} {\hat k}} = i \hbar^2 \epsilon_{{\hat \imath} {\hat \jmath} {\hat k}} \langle \partial_{\pi_{\hat \imath}} u_{n {\vec \pi}} | \partial_{\pi_{\hat \jmath}} u_{n {\vec \pi}} \rangle. \label{eq:berrycurv}$$ The proper HV should be symmetric as shown in Eq. . In the conventional approach involving a metric, the stress tensor is defined by $T^{\mu \nu} \equiv 2 \delta S/\delta g_{\mu \nu}$. This is manifestly symmetric because a metric is symmetric. By using $g_{\mu \nu} = \eta_{ab} h^a_{\phantom{a} \mu} h^b_{\phantom{b} \nu}$, it can be related to the momentum current $T_a^{\phantom{a} \mu} \equiv \delta S/\delta h^a_{\phantom{a} \mu}$ by $T^{\mu \nu} = (h^{a \mu} T_a^{\phantom{a} \nu} + h^{a \nu} T_a^{\phantom{a} \mu})/2$. Therefore the symmetric HV is given by $$\eta^{({\hat k} {\hat \imath}) ({\hat l} {\hat \jmath})} \equiv (\eta^{{\hat k} {\hat \imath} {\hat l} {\hat \jmath}} + \eta^{{\hat \imath} {\hat k} {\hat l} {\hat \jmath}} + \eta^{{\hat k} {\hat \imath} {\hat \jmath} {\hat l}} + \eta^{{\hat \imath} {\hat k} {\hat \jmath} {\hat l}})/4. \label{eq:eta5}$$ Although a strain-rate tensor is described by torsion, Eq.  is different from the torsional HV discussed in Refs. . In two dimensions, only three components may be nonzero, $$\begin{aligned} \eta^{(xx)(xy)} = & \frac{1}{2 \hbar} \sum_n \int \frac{d^2 \pi}{(2 \pi \hbar)^2} \pi^{x 2} \Omega_{n {\vec \pi} z} f_{n {\vec \pi}}, \label{eq:eta2da} \\ \eta^{(xx)(yy)} = & \frac{1}{\hbar} \sum_n \int \frac{d^2 \pi}{(2 \pi \hbar)^2} \pi^x \pi^y \Omega_{n {\vec \pi} z} f_{n {\vec \pi}}, \label{eq:eta2db} \\ \eta^{(xy)(yy)} = & \frac{1}{2 \hbar} \sum_n \int \frac{d^2 \pi}{(2 \pi \hbar)^2} \pi^{y 2} \Omega_{n {\vec \pi} z} f_{n {\vec \pi}}. \label{eq:eta2dc}\end{aligned}$$ \[eq:eta2d\] Furthermore, if a system is rotationally invariant, there is only one nonzero component $\eta_{\rm H} = \eta^{(xx)(xy)} = \eta^{(xy)(yy)}$, $$\eta_{\rm H} = \frac{1}{4 \hbar} \sum_n \int \frac{d^2 \pi}{(2 \pi \hbar)^2} {\vec \pi}^2 \Omega_{n {\vec \pi} z} f_{n {\vec \pi}}. \label{eq:etah}$$ In three dimensions, all the components of the HV should vanish if a system is rotationally invariant. On the other hand, if a system is axially invariant along the $z$ axis, two components are independent, $$\begin{aligned} \eta^{(xx)(xy)} = & \frac{1}{2 \hbar} \sum_n \int \frac{d^3 \pi}{(2 \pi \hbar)^3} \pi^{x 2} \Omega_{n {\vec \pi} z} f_{n {\vec \pi}}, \label{eq:eta3da} \\ \eta^{(zx)(yz)} = & \frac{1}{4 \hbar} \sum_n \int \frac{d^3 \pi}{(2 \pi \hbar)^3} \notag \\ & \times (\pi^{z 2} \Omega_{n {\vec \pi} z} - \pi^z \pi^x \Omega_{n {\vec \pi} x} - \pi^z \pi^y \Omega_{n {\vec \pi} y}) f_{n {\vec \pi}}. \label{eq:eta3db}\end{aligned}$$ \[eq:eta3d\] As in the case of the momentum polarization Eq. , these expressions are quite analogous to that for the Hall conductivity, corresponding to the charge transport. The integrand just differs in the factor of ${\vec \pi}$. Finally, let us comment on the interacting cases. In the conventional metric approach, the effects of interactions are fully taken into account by using the many-body ground-state wave function. On the other hand, in our approach, such effects are compiled into the self-energy and can be taken into account by using the Feynman diagrams or the dynamical mean-field theory. Therefore these two approaches are complementary and coincide in the noninteracting limit, which reminds us that the charge polarization can be calculated by averaging over boundary conditions [@PhysRevB.49.14202] or by using the Green’s function formula [@PhysRevB.88.155121; @JPSJ.83.033708]. Applications to Chiral Superconductors {#sec:chiral} ====================================== Now we apply our results to chiral SCs. For simplicity, we restrict ourselves to the single-band model with a singlet or unitary triplet pairing described by $$H - \mu N = \sum_{\vec k} \begin{bmatrix} c^{\dag}_{{\vec k} \uparrow} & c_{-{\vec k} \downarrow} \end{bmatrix} \begin{bmatrix} \xi_{\vec k} & \Delta_{\vec k} \\ \Delta^{\ast}_{\vec k} & -\xi_{\vec k} \end{bmatrix} \begin{bmatrix} c_{{\vec k} \uparrow} \\ c_{-{\vec k} \downarrow} \end{bmatrix}. \label{eq:bcs}$$ in which $\xi_{\vec k} \equiv \epsilon_{\vec k} - \mu$ is even, and the gap $\Delta_{\vec k}$ is even or odd for singlet or triplet with ${\vec d}_{\vec k} \parallel {\vec z}$, respectively. This Hamiltonian has the positive and negative dispersions $\pm E_{\vec k} = \pm \sqrt{\xi_{\vec k}^2 + |\Delta_{\vec k}|^2}$, whose wave functions are given by $$\begin{bmatrix} | u_{{\vec k} +} \rangle & | u_{{\vec k} -} \rangle \end{bmatrix} = \begin{bmatrix} u_{\vec k} & -v_{\vec k} \\ v_{\vec k}^{\ast} & u_{\vec k} \end{bmatrix}. \label{eq:bdg}$$ Here $u_{\vec k} = \sqrt{(1 + \xi_{\vec k}/E_{\vec k})/2}$ and $v_{\vec k} = \sqrt{(1 - \xi_{\vec k}/E_{\vec k})/2} \Delta_{\vec k}/|\Delta_{\vec k}|$ satisfy the normalization condition $u_{\vec k}^2 + |v_{\vec k}|^2 = 1$. The temperature dependence of the gap strength is obtained by solving the gap equation. Now that we are not interested in the competition between several phases but in the AM and the HV for chiral SCs, we assume the attractive interaction favoring a chiral pairing symmetry $\Delta_{\vec k} = \Delta w_{\vec k}$ with $w_{\vec k}$ being normalized by $\int d \Omega_{\vec k} |w_{\vec k}|^2/4 \pi = 1$ in three dimensions. By using the standard approximations, the gap equation is given by $$\ln |\Delta_0/\Delta| = \int_0^{\infty} d \xi_{\vec k} \int \frac{d \Omega_{\vec k}}{4 \pi} 2 |w_{\vec k}|^2 f(E_{\vec k})/E_{\vec k}. \label{eq:gap}$$ Next, we calculate the Berry connection and curvature. The Berry connections for the positive- and negative-energy states have opposite signs; namely, $A_{+ {\vec k}}^{\hat \imath} = -A_{- {\vec k}}^{\hat \imath} \equiv A_{\vec k}^{\hat \imath}$, $$A_{\vec k}^{\hat \imath} = i (v_{\vec k} \partial_{k_{\hat \imath}} v_{\vec k}^{\ast} - \partial_{k_{\hat \imath}} v_{\vec k} v_{\vec k}^{\ast})/2. \label{eq:sccon1}$$ Correspondingly, the Berry curvatures change their signs; i.e, $\Omega_{+ {\vec k} {\hat m}} = -\Omega_{- {\vec k} {\hat m}} \equiv \Omega_{{\vec k} {\hat m}}$, $$\Omega_{{\vec k} {\hat m}} = i \epsilon_{{\hat \imath} {\hat \jmath} {\hat m}} \partial_{k_{\hat \imath}} v_{\vec k} \partial_{k_{\hat \jmath}} v_{\vec k}^{\ast}. \label{eq:sccurv1}$$ For chiral SCs with $w_{\vec k} \propto e^{i m \phi}$ in the polar coordinate ${\vec k} = (k \sin \theta \cos \phi, k \sin \theta \sin \phi, k \cos \theta)$, the Berry connection and curvature are given by $$\begin{aligned} A_{\vec k}^x = & -m |v_{\vec k}|^2 \sin \phi/k \sin \theta, \label{eq:sccon2x} \\ A_{\vec k}^y = & m |v_{\vec k}|^2 \cos \phi/k \sin \theta, \label{eq:sccon2y} \\ A_{\vec k}^z = & 0, \label{eq:sccon2z}\end{aligned}$$ \[eq:sccon2\] and $$\begin{aligned} \Omega_{{\vec k} z} = & m (k \partial_k + \cot \theta \partial_{\theta}) |v_{\vec k}|^2/k^2, \label{eq:sccurv2z} \\ \Omega_{{\vec k} x} = & m \cos \phi (-k \cot \theta \partial_k + \partial_{\theta}) |v_{\vec k}|^2/k^2, \label{eq:sccurv2x} \\ \Omega_{{\vec k} y} = & m \sin \phi (-k \cot \theta \partial_k + \partial_{\theta}) |v_{\vec k}|^2/k^2, \label{eq:sccurv2y}\end{aligned}$$ \[eq:sccurv2\] respectively. Consequently, the AM for gapped chiral SCs at zero temperature is given by $$L_z = -\hbar \sum_{\vec k} ({\vec A}_{\vec k} \times {\vec k})_z = \hbar m \sum_{\vec k} |v_{\vec k}|^2 = \hbar m N_0/2, \label{eq:scam}$$ where we introduce the number of electrons at zero temperature $N_0 = \sum_{\vec k} 2 |v_{\vec k}|^2$. This result is consistent with the recent microscopic studies suggesting $\gamma = 0$ [@Ishikawa01061977; @Ishikawa01041980; @PhysRevB.21.980; @volovik1995; @JPSJ.67.216; @Goryo1998549; @PhysRevB.69.184511; @PhysRevB.84.214509; @PhysRevB.85.100506]. However, the bulk AM is well defined only in a gapped system at zero temperature. Therefore, we cannot apply this result to three-dimensional chiral SCs with point or line nodes. Below we calculate the HV, which is well defined in a gapless system or at finite temperature. Hall Viscosity in Two Dimensions {#sub:2d} -------------------------------- First, we consider $w_{\vec k} = e^{i \ell \phi}$ in two dimensions. For odd $\ell$, the system is triplet and is classified into a class-D topological SC in terms of the topological periodic table [@PhysRevB.78.195125; @kitaev2009]. On the other hand, for even $\ell$, it is singlet and is classified into a class-C topological SC. Anyway it is gapped and the AM Eq.  is well defined at zero temperature. Since there is the rotational symmetry, only Eq.  is nonzero, $$\begin{aligned} 2 \eta_{\rm H} = & -\frac{\hbar}{2} \sum_{\vec k} k^2 \Omega_{{\vec k} z} [1 - 2 f(E_{\vec k})] \notag \\ = & -\frac{\hbar \ell}{2} \sum_{\vec k} k \partial_k |v_{\vec k}|^2 [1 - 2 f(E_{\vec k})]. \label{eq:sceta2d1}\end{aligned}$$ At zero temperature, we employ the partial integral to obtain $2 \eta_{\rm H} = \hbar \ell N_0/2$, which is consistent with the previous result for gapped chiral SCs [@PhysRevB.79.045308; @PhysRevB.84.085316; @PhysRevB.86.245309]. Let us turn to finite temperature. By solving the gap equation Eq.  and using the temperature dependence of the gap strength, we evaluate the HV normalized by that at zero temperature, i.e., half the AM, $$\frac{2 \eta_{\rm H}}{\hbar \ell N_0/2} = \int_0^{\infty} d \xi \partial_{\xi} (\xi/E) [1 - 2 f(E)], \label{eq:sceta2d2}$$ As shown in Fig. \[fig:2d\], both the gap strength and the HV exponentially converge since the system is gapped. ![ (Color online) The gap strength (red cross) and the HV (green filled square) for two-dimensional chiral SCs as functions of (a) temperature $T/|\Delta_0|$ and (b) inverse temperature $|\Delta_0|/T$. The green broken line indicates $e^{-|\Delta_0|/T}$. []{data-label="fig:2d"}](2d){width="48.00000%"} Hall Viscosity in Three Dimensions {#sub:3d} ---------------------------------- Next, we consider $w_{\vec k} = \sqrt{4 \pi} Y_{\ell m}(\theta, \phi)$ in three dimensions, where $Y_{\ell m}$ is a spherical harmonic function. As emphasized above, three-dimensional chiral SCs have nodes generally, and the bulk AM is not well defined. For example, $p_x \pm i p_y$ with $(\ell, m) = (1, \pm 1)$ and $d_{x^2 - y^2} \pm i d_{xy}$ with $(\ell, m) = (2, \pm 2)$ have point nodes at poles, while $d_{zx} \pm i d_{yz}$ with $(\ell, m) = (2, \pm 1)$ has both point and line nodes. Since there is the axial symmetry along the $z$ axis, two components Eq.  are nonzero, $$\begin{aligned} 2 \eta^{(xx)(xy)} = & -\hbar \sum_{\vec k} k^{x 2} \Omega_{{\vec k} z} [1 - 2 f(E_{\vec k})] \notag \\ = & -\hbar m \sum_{\vec k} \sin^2 \theta \cos^2 \phi (k \partial_k + \cot \theta \partial_{\theta}) |v_{\vec k}|^2 \notag \\ & \times [1 - 2 f(E_{\vec k})], \label{eq:sceta1a} \\ 2 \eta^{(zx)(yz)} = & -\frac{\hbar}{2} \sum_{\vec k} (k^{z 2} \Omega_{{\vec k} z} - k^z k^x \Omega_{{\vec k} x} - k^z k^y \Omega_{{\vec k} y}) \notag \\ & \times [1 - 2 f(E_{\vec k})] \notag \\ = & -\frac{\hbar m}{2} \sum_{\vec k} (2 \cos^2 \theta k \partial_k + \cos 2 \theta \cot \theta \partial_{\theta}) |v_{\vec k}|^2 \notag \\ & \times [1 - 2 f(E_{\vec k})], \label{eq:sceta1b}\end{aligned}$$ \[eq:sceta1\] both of which are reduced to $\hbar m N_0/2$ at zero temperature. We numerically calculate the normalized HV at finite temperature, $$\begin{aligned} \frac{2 \eta^{(xx)(xy)}}{\hbar m N_0/2} = & \frac{3}{4} \int_0^{\infty} d \xi \int_0^{\pi} \sin \theta d \theta \sin^2 \theta \partial_{\xi} (\xi/E) [1 - 2 f(E)], \label{eq:sceta2a} \\ \frac{2 \eta^{(zx)(yz)}}{\hbar m N_0/2} = & \frac{3}{2} \int_0^{\infty} d \xi \int_0^{\pi} \sin \theta d \theta \cos^2 \theta \partial_{\xi} (\xi/E) [1 - 2 f(E)], \label{eq:sceta2b}\end{aligned}$$ \[eq:sceta2\] instead of Eq. . In Figs. \[fig:3d11\], \[fig:3d21\], and \[fig:3d22\], we show the temperature dependences of the gap strength and the HV for $p_x + i p_y$, $d_{zx} + i d_{yz}$, and $d_{x^2-y^2} + i d_{xy}$, respectively. All of them converge by the power laws since the systems have nodes. These powers can be analytically obtained by expanding the integrands in Eqs.  and around nodes as in the appendix. Especially for $d_{x^2 - y^2} + i d_{xy}$, we find that the temperature dependence of $|\Delta/\Delta_0|$ and $2 \eta^{(xx)(xy)}/(\hbar m N_0/2)$ is almost determined by the line node, while that of $2 \eta^{(zx)(yz)}/(\hbar m N_0/2)$ is by the point nodes. We also summarize their powers in Table  \[tab:pow\]. ![ (Color online) Temperature dependences of $|\Delta/\Delta_0|$ (red cross), $2 \eta^{(xx)(xy)}/(\hbar m N_0/2)$ (green filled square), and $2 \eta^{(zx)(yz)}/(\hbar m N_0/2)$ (blue open circle) for a three-dimensional $p_x + i p_y$ SC with $(\ell, m) = (1, 1)$. The green broken and blue dotted lines indicate $(T/|\Delta_0|)^4$ and $(T/|\Delta_0|)^2$, respectively. []{data-label="fig:3d11"}](3d11){width="48.00000%"} ![ (Color online) Temperature dependences of $|\Delta/\Delta_0|$ (red cross), $2 \eta^{(xx)(xy)}/(\hbar m N_0/2)$ (green filled square), and $2 \eta^{(zx)(yz)}/(\hbar m N_0/2)$ (blue open circle) for a three-dimensional $d_{zx} + i d_{yz}$ SC with $(\ell, m) = (2, 1)$. The red solid, green broken, and blue dotted lines indicate $(T/|\Delta_0|)^3$, $T/|\Delta_0|$, and $(T/|\Delta_0|)^2$, respectively. []{data-label="fig:3d21"}](3d21){width="48.00000%"} ![ (Color online) Temperature dependences of $|\Delta/\Delta_0|$ (red cross), $2 \eta^{(xx)(xy)}/(\hbar m N_0/2)$ (green filled square), and $2 \eta^{(zx)(yz)}/(\hbar m N_0/2)$ (blue open circle) for a three-dimensional $d_{x^2-y^2} + i d_{xy}$ SC with $(\ell, m) = (2, 2)$. The red solid, green broken, and blue dotted lines indicate $(T/|\Delta_0|)^3$, $(T/|\Delta_0|)^2$, and $T/|\Delta_0|$, respectively. []{data-label="fig:3d22"}](3d22){width="48.00000%"} symmetry $(\ell, m)$ nodes $|\Delta|$ $2 \eta^{(xx)(xy)}$ $2 \eta^{(zx)(yz)}$ -------------------------- ------------- ---------------- ------------ --------------------- --------------------- $p_x + i p_y$ $(1, 1)$ point $T^4$ $T^4$ $T^2$ $d_{zx} + i d_{yz}$ $(2, 1)$ point and line $T^3$ $T$ $T^2$ $d_{x^2-y^2} + i d_{xy}$ $(2, 2)$ point $T^3$ $T^2$ $T$ : Power-law behaviors of the gap strength and the HV at low temperature for several pairing symmetries in three dimensions. See also Figs. \[fig:3d11\], \[fig:3d21\], and \[fig:3d22\]. []{data-label="tab:pow"} Discussion and Summary {#sec:summary} ====================== The temperature dependence of the AM for $p_x + i p_y$ was calculated in finite systems such as a mesoscopic cylinder [@JPSJ.67.216] and a macroscopic disk [@PhysRevB.84.214509; @PhysRevB.85.100506] compared to the coherence length. In three dimensions, the AM decreases from $\hbar N_0/2$ by $T^2$ owing to the presence of the point nodes [@JPSJ.67.216]. In two dimensions, the AM also decreases from $\hbar N_0/2$ by $T^2$, which is attributed to the presence of the Majorana edge modes [@PhysRevB.84.214509; @PhysRevB.85.100506]. Although the HV is equal to half the AM at zero temperature not only in two dimensions but also in three dimensions, the temperature dependence of the AM and the HV is generally different. This discrepancy does not conflict with the relation between the AM and the HV [@PhysRevB.79.045308; @PhysRevB.84.085316], because it is available only for gapped systems at zero temperature. Indeed such a difference was recently found in two-dimensional gapless systems by using the holographic approach [@1403.6047], too. As pointed out in this paper, the AM is given by the polarization of the momentum. In this sense, it is seen as a thermodynamic quantity, while the HV is just a transport coefficient but not a bulk quantity. Therefore it is natural to think that physical origins of those quantities and also their temperature dependences are different in general, even if they have a special relation at zero temperature. It may be interesting to study other models, e.g., quantum Hall systems and chiral SCs with higher $\ell$, at finite temperature. Let us note that such a relation between a thermodynamic quantity and a transport coefficient was also proposed for the entropy density and the shear viscosity [@PhysRevLett.94.111601]. Nevertheless it is expected that they would have a universal relation only in the extremely strong coupling situation. To summarize, we derive the Berry-phase formulas for the AM and the HV by using the Keldysh formalism in a curved spacetime. First, we examine the physical quantity conjugate to the AM, namely, an angular velocity of rotation, from the gauge-theoretical viewpoint of gravity. Since an angular velocity is the antisymmetric torsional electric field, we define the AM by the temporal integral of the antisymmetric momentum current induced by an adiabatic deformation, which is implemented by the gradient expansion. Viscosity is the response of the stress tensor, i.e., the symmetric momentum current, to a strain-rate tensor. It can be derived by the perturbation theory with respect to torsion because a strain-rate tensor is described by the symmetric torsional electric field. We also apply these results to chiral SCs in two and three dimensions. In two dimensions at zero temperature, we reproduce $L_z = 2 \eta_{\rm H} = \hbar \ell N_0/2$ without any finite-size effects. In three dimensions at zero temperature, where the AM is not well defined owing to the presence of nodes, we find that the HV is equal to half the AM calculated in finite systems previously. Although it is not related to the AM at finite temperature, it is useful to determine the gap structure for chiral SCs. We thank H. Sumiyoshi and Y. Tada for fruitful discussions. This work was supported by Grants-in-Aid for the Japan Society for the Promotion of Science, Fellows No. $24$-$600$ and No. $25$-$4302$. Temperature Dependence of Hall Viscosity {#sec:temp} ======================================== In this appendix, we approximately but analytically calculate the gap strength and the HV to explain their power-law behaviors at low temperature. For $p_x + i p_y$, where $|w_{\vec k}|^2 = 3 \sin^2 \theta/2$, we expand Eqs.  and around the point node $\theta = 0$ to obtain $$\begin{aligned} \ln |\Delta_0/\Delta| \simeq & 2 \int_0^{\infty}d \xi \int_0^{\pi/2} \theta d \theta \frac{3 \theta^2/2}{\sqrt{\xi^2 + 3 |\Delta|^2 \theta^2/2}} (e^{\beta \sqrt{\xi^2 + 3 |\Delta|^2 \theta^2/2}} + 1)^{-1} \notag \\ = & \frac{4}{3} (T/|\Delta|)^4 \int_0^{\infty} d x \int_0^{\sqrt{3/2} \beta |\Delta| \pi/2} d y \frac{y^3}{r} (e^r + 1)^{-1} \to \frac{7 \pi^4}{135} (T/|\Delta_0|)^4, \label{eq:gap3d11} \\ 1 - \frac{2 \eta^{(xx)(xy)}}{\hbar m N_0/2} \simeq & 3 \int_0^{\infty} d \xi \int_0^{\pi/2} \theta d \theta \theta^2 \frac{3 |\Delta|^2 \theta^2/2}{(\xi^2 + 3 |\Delta|^2 \theta^2/2)^{3/2}} (e^{\beta \sqrt{\xi^2 + 3 |\Delta|^2 \theta^2/2}} + 1)^{-1} \notag \\ = & \frac{4}{3} (T/|\Delta|)^4 \int_0^{\infty} d x \int_0^{\sqrt{3/2} \beta |\Delta| \pi/2} d y \frac{y^5}{r^3} (e^r + 1)^{-1} \to \frac{28 \pi^4}{675} (T/|\Delta_0|)^4, \label{eq:sceta3d11a} \\ 1 - \frac{2 \eta^{(zx)(yz)}}{\hbar m N_0/2} \simeq & 6 \int_0^{\infty} d \xi \int_0^{\pi/2} \theta d \theta \frac{3 |\Delta|^2 \theta^2/2}{(\xi^2 + 3 |\Delta|^2 \theta^2/2)^{3/2}} (e^{\beta \sqrt{\xi^2 + 3 |\Delta|^2 \theta^2/2}} + 1)^{-1} \notag \\ = & 4 (T/|\Delta|)^2 \int_0^{\infty} d x \int_0^{\sqrt{3/2} \beta |\Delta| \pi/2} d y \frac{y^3}{r^3} (e^r + 1)^{-1} \to \frac{2 \pi^2}{9} (T/|\Delta_0|)^2. \label{eq:sceta3d11b} \end{aligned}$$ \[eq:3d11\] In the second lines, we change the variables by $x = \beta \xi$ and $y = \sqrt{3/2} \beta |\Delta| \theta$, and in the third lines, we take the low-temperature limit $\sqrt{3/2} \beta |\Delta| \pi/2 \to \infty$. These integrals can be analytically estimated in the polar coordinate. For $d_{zx} + i d_{yz}$, where $|w_{\vec k}|^2 = 15 \sin^2 \theta \cos^2 \theta/2$, there are both point and line nodes. First, we expand Eqs.  and around the point node $\theta = 0$ and change the variables by $x = \beta \xi$ and $y = \sqrt{15/2} \beta |\Delta| \theta$, which results in $$\begin{aligned} \ln |\Delta_0/\Delta| \simeq & 2 \int_0^{\infty}d \xi \int_0^{\pi/2} \theta d \theta \frac{15 \theta^2/2}{\sqrt{\xi^2 + 15 |\Delta|^2 \theta^2/2}} (e^{\beta \sqrt{\xi^2 + 15 |\Delta|^2 \theta^2/2}} + 1)^{-1} \notag \\ = & \frac{4}{15} (T/|\Delta|)^4 \int_0^{\infty} d x \int_0^{\sqrt{15/2} \beta |\Delta| \pi/2} d y \frac{y^3}{r} (e^r + 1)^{-1} \to \frac{7 \pi^4}{675} (T/|\Delta_0|)^4, \label{eq:gap3d21-p} \\ 1 - \frac{2 \eta^{(xx)(xy)}}{\hbar m N_0/2} \simeq & 3 \int_0^{\infty} d \xi \int_0^{\pi/2} \theta d \theta \theta^2 \frac{15 |\Delta|^2 \theta^2/2}{(\xi^2 + 15 |\Delta|^2 \theta^2/2)^{3/2}} (e^{\beta \sqrt{\xi^2 + 15 |\Delta|^2 \theta^2/2}} + 1)^{-1} \notag \\ = & \frac{4}{75} (T/|\Delta|)^4 \int_0^{\infty} d x \int_0^{\sqrt{15/2} \beta |\Delta| \pi/2} d y \frac{y^5}{r^3} (e^r + 1)^{-1} \to \frac{28 \pi^4}{16875} (T/|\Delta_0|)^4, \label{eq:sceta3d21a-p} \\ 1 - \frac{2 \eta^{(zx)(yz)}}{\hbar m N_0/2} \simeq & 6 \int_0^{\infty} d \xi \int_0^{\pi/2} \theta d \theta \frac{15 |\Delta|^2 \theta^2/2}{(\xi^2 + 15 |\Delta|^2 \theta^2/2)^{3/2}} (e^{\beta \sqrt{\xi^2 + 15 |\Delta|^2 \theta^2/2}} + 1)^{-1} \notag \\ = & \frac{4}{5} (T/|\Delta|)^2 \int_0^{\infty} d x \int_0^{\sqrt{15/2} \beta |\Delta| \pi/2} d y \frac{y^3}{r^3} (e^r + 1)^{-1} \to \frac{2 \pi^2}{45} (T/|\Delta_0|)^2. \label{eq:sceta3d21b-p} \end{aligned}$$ \[eq:3d21-p\] On the other hand, for the line node, we redefine $\theta \to \pi/2 - \theta$ and expand Eqs.  and around $\theta = 0$, leading to $$\begin{aligned} \ln |\Delta_0/\Delta| \simeq & 2 \int_0^{\infty}d \xi \int_0^{\pi/2} d \theta \frac{15 \theta^2/2}{\sqrt{\xi^2 + 15 |\Delta|^2 \theta^2/2}} (e^{\beta \sqrt{\xi^2 + 15 |\Delta|^2 \theta^2/2}} + 1)^{-1} \notag \\ = & 2 \sqrt{\frac{2}{15}} (T/|\Delta|)^3 \int_0^{\infty} d x \int_0^{\sqrt{15/2} \beta |\Delta| \pi/2} d y \frac{y^2}{r} (e^r + 1)^{-1} \to \frac{\sqrt{3} \pi \zeta(3)}{2 \sqrt{10}} (T/|\Delta_0|)^3, \label{eq:gap3d21-l} \\ 1 - \frac{2 \eta^{(xx)(xy)}}{\hbar m N_0/2} \simeq & 3 \int_0^{\infty} d \xi \int_0^{\pi/2} d \theta \frac{15 |\Delta|^2 \theta^2/2}{(\xi^2 + 15 |\Delta|^2 \theta^2/2)^{3/2}} (e^{\beta \sqrt{\xi^2 + 15 |\Delta|^2 \theta^2/2}} + 1)^{-1} \notag \\ = & 3 \sqrt{\frac{2}{15}} (T/|\Delta|) \int_0^{\infty} d x \int_0^{\sqrt{15/2} \beta |\Delta| \pi/2} d y \frac{y^2}{r^3} (e^r + 1)^{-1} \to \frac{\sqrt{3} \pi \ln 2}{2 \sqrt{10}} (T/|\Delta_0|), \label{eq:sceta3d21a-l} \\ 1 - \frac{2 \eta^{(zx)(yz)}}{\hbar m N_0/2} \simeq & 6 \int_0^{\infty} d \xi \int_0^{\pi/2} d \theta \theta^2 \frac{15 |\Delta|^2 \theta^2/2}{(\xi^2 + 15 |\Delta|^2 \theta^2/2)^{3/2}} (e^{\beta \sqrt{\xi^2 + 15 |\Delta|^2 \theta^2/2}} + 1)^{-1} \notag \\ = & \frac{4 \sqrt{2}}{5 \sqrt{15}} (T/|\Delta|)^3 \int_0^{\infty} d x \int_0^{\sqrt{15/2} \beta |\Delta| \pi/2} d y \frac{y^4}{r^3} (e^r + 1)^{-1} \to \frac{3 \sqrt{3} \pi \zeta(3)}{20 \sqrt{10}} (T/|\Delta_0|)^3. \label{eq:sceta3d21b-l} \end{aligned}$$ \[eq:3d21-l\] By comparing each power, the gap strength $|\Delta/\Delta_0|$ and one component of the HV $2 \eta^{(xx)(xy)}/(\hbar m N_0/2)$ are mainly contributed from the line node, while the other component $2 \eta^{(zx)(yz)}/(\hbar m N_0/2)$ is from the point nodes. For $d_{x^2 - y^2} + i d_{xy}$, where $|w_{\vec k}|^2 = 15 \sin^4 \theta/8$, we expand Eqs.  and around the point node $\theta = 0$ and introduce $y = \sqrt{15/8} \beta |\Delta| \theta^2$. Then we obtain $$\begin{aligned} \ln |\Delta_0/\Delta| \simeq & 2 \int_0^{\infty}d \xi \int_0^{\pi/2} \theta d \theta \frac{15 \theta^4/8}{\sqrt{\xi^2 + 15 |\Delta|^2 \theta^4/8}} (e^{\beta \sqrt{\xi^2 + 15 |\Delta|^2 \theta^4/8}} + 1)^{-1} \notag \\ = & \sqrt{\frac{8}{15}} (T/|\Delta|)^3 \int_0^{\infty} d x \int_0^{\sqrt{15/8} \beta |\Delta| (\pi/2)^2} d y \frac{y^2}{r} (e^r + 1)^{-1} \to \frac{\sqrt{3} \pi \zeta(3)}{2 \sqrt{10}} (T/|\Delta_0|)^3, \label{eq:gap3d22} \\ 1 - \frac{2 \eta^{(xx)(xy)}}{\hbar m N_0/2} \simeq & 3 \int_0^{\infty} d \xi \int_0^{\pi/2} \theta d \theta \theta^2 \frac{15 |\Delta|^2 \theta^4/8}{(\xi^2 + 15 |\Delta|^2 \theta^4/8)^{3/2}} (e^{\beta \sqrt{\xi^2 + 15 |\Delta|^2 \theta^4/8}} + 1)^{-1} \notag \\ = & \frac{4}{5} (T/|\Delta|)^2 \int_0^{\infty} d x \int_0^{\sqrt{15/8} \beta |\Delta| (\pi/2)^2} d y \frac{y^3}{r^3} (e^r + 1)^{-1} \to \frac{2 \pi^2}{45} (T/|\Delta_0|)^2, \label{eq:sceta3d22a} \\ 1 - \frac{2 \eta^{(zx)(yz)}}{\hbar m N_0/2} \simeq & 6 \int_0^{\infty} d \xi \int_0^{\pi/2} \theta d \theta \frac{15 |\Delta|^2 \theta^4/8}{(\xi^2 + 15 |\Delta|^2 \theta^4/8)^{3/2}} (e^{\beta \sqrt{\xi^2 + 15 |\Delta|^2 \theta^4/8}} + 1)^{-1} \notag \\ = & 3 \sqrt{\frac{8}{15}} (T/|\Delta|) \int_0^{\infty} d x \int_0^{\sqrt{15/8} \beta |\Delta| (\pi/2)^2} d y \frac{y^2}{r^3} (e^r + 1)^{-1} \to \frac{\sqrt{3} \pi \ln 2}{\sqrt{10}} (T/|\Delta_0|). \label{eq:sceta3d22b}\end{aligned}$$ \[eq:3d22\] Thus, all the power-law behaviors can be explained by nodal excitations. [45]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [****,  ()](\doibase 10.1103/RevModPhys.47.331) [****, ()](\doibase 10.1103/RevModPhys.75.657) [****,  ()](\doibase 10.1143/JPSJ.81.011009) [****,  ()](\doibase 10.1103/PhysRevLett.99.116402) [****,  ()](\doibase 10.1103/PhysRevLett.100.017004) [****, ()](http://ads.nao.ac.jp/abs/2009NJPh...11e5061K) [****,  ()](\doibase 10.1143/JPSJ.81.023704) [****,  ()](\doibase 10.1103/PhysRevB.86.100507) [****,  ()](\doibase 10.1103/PhysRevB.87.180503) [****,  ()](\doibase 10.1103/PhysRevB.89.020509) [****,  ()](\doibase 10.1143/PTP.57.1836) [****,  ()](\doibase 10.1143/PTP.63.1083) [****, ()](\doibase 10.1103/PhysRevB.21.980) [****,  ()](\doibase 10.1143/JPSJ.67.216) [****,  ()](\doibase 10.1016/S0375-9601(98)00438-1) [****,  ()](\doibase 10.1103/PhysRevB.69.184511) [****, ()](\doibase 10.1103/PhysRevB.84.214509) [****,  ()](\doibase 10.1103/PhysRevB.85.100506) [****, ()](\doibase 10.1103/PhysRev.123.1911) [****,  ()](\doibase 10.1007/BF01141607) [****,  ()](\doibase 10.1103/PhysRevB.47.1651) [****, ()](\doibase 10.1103/PhysRevB.49.14202) [****,  ()](\doibase 10.1103/PhysRevB.88.155121) [****,  ()](\doibase 10.7566/JPSJ.83.033708) [****,  ()](\doibase 10.1103/PhysRevLett.99.197202) [****,  ()](\doibase 10.1103/PhysRevB.84.205137) [****,  ()](\doibase 10.1103/PhysRevB.86.214415) [****,  ()](\doibase 10.1103/PhysRevLett.75.697) [****, ()](\doibase 10.1023/A:1023084404080) [****, ()](\doibase 10.1103/PhysRevB.79.045308) [****,  ()](\doibase 10.1103/PhysRevB.84.085316) [****,  ()](\doibase 10.1103/PhysRevB.89.174507) [****,  ()](\doibase 10.1103/PhysRevLett.108.066805) [****,  ()](\doibase 10.1103/PhysRevB.86.245309) [****,  ()](\doibase 10.1103/PhysRevLett.107.075502) [****,  ()](\doibase 10.1093/ptep/pts063) [****,  ()](\doibase 10.1103/PhysRevD.88.025040) @noop [**]{} (, , ) [****,  ()](\doibase 10.1103/PhysRevB.78.195125) [****, ()](\doibase 10.1063/1.3149495) [****, ()](\doibase 10.1103/PhysRevLett.94.111601)

--- abstract: 'The statistical properties of a classical electromagnetic field in interaction with matter are numerically investigated on a one–dimensional model of a radiant cavity, conservative and with finite total energy. Our results suggest a trend towards equipartition of energy, with the relaxation times of the normal modes of the cavity increasing with the mode frequency according to a law, the form of which depends on the shape of the charge distribution.' address: - '$^{(a)}$International Centre for the Study of Dynamical Systems,' - ' Università di Milano, sede di Como, via Lucini 3, 22100 Como, Italy' - '$^{(b)}$Istituto Nazionale di Fisica della Materia, Unità di Milano, Via Celoria 16, 20133 Milano, Italy' - '$^{(c)}$ INFN, Sezione di Milano, Via Celoria 16, 20133 Milano, Italy' - '$^{(d)}$ INFN, sezione di pavia, Via Bassi 6, 27100 Pavia, Italy' author: - 'Giuliano Benenti$^{(a,b,c,*)}$, Giulio Casati$^{(a,b,c)}$ and Italo Guarneri$^{(a,b,d)}$' date: 'October 5, 1998' title: '[**Chaotic dynamics of a classical radiant cavity**]{} ' --- u [2]{} The study of quantum deviations from classical ergodicity has occupied much of Quantum Chaology since its origins. Remarkably enough, the historical development of quantum mechanics started with the Blackbody problem, which displays a deviation as blatant as possible from classical ergodicity. When a classical radiation field interacts with matter inside an enclosure with perfectly reflecting walls, approach to statistical equilibrium – if at all possible – appears to entail unending escape of energy towards higher and higher frequencies, in sharp contrast to the Planck distribution law[@JEANS]. Thus the problem of blackbody radiation was at once the first problem in quantum mechanics, the first problem in quantum field theory, and the first problem in quantum chaos. Considerable progress has been meanwhile attained in understanding the complex behaviour of nonlinear classical dynamical systems and its quantum counterparts, so a re–examination of the blackbody problem in the light of such developments appears necessary. One would like, first, to build a Hamiltonian model of a radiant cavity, which does indeed exhibit the sort of tendency to equipartition expected in Jeans’s time; second, to understand how does Planck’s law emerge from the quantal dynamics of that very model. Neither of these issues seems to have been satisfactorily dealt with as yet. In this Letter we accomplish the first part of the task by presenting a model whose classical dynamics leads to equipartition in the sense of Jeans. Our model is a variant of one which was introduced years ago [@BCL72] to this purpose, and which was later investigated in several papers [@BLVG74]. None of those investigations was able to detect a tendency towards energy equipartition among the normal modes of the cavity. Analogous results were obtained on different models [@GAFGG89; @P83], such as a one–dimensional linear string interacting with nonlinear oscillators etc. The general picture which emerges from all these works, in which the Newton–Maxwell equations were numerically solved, is that there is no tendency to energy equipartition among the field normal modes. The reason lies with two fully general aspects of the classical field–matter interaction. First, the total energy is finite, whereas the number of freedoms is infinite; second, field modes can only exchange energy via interaction with finitely many mechanical freedoms. Thus mechanical nonlinearities become less and less effective as energy flows from the matter to the field: this fact prevents the appearance of an altogether chaotic dynamics, thus causing high frequency modes to be nonergodically “frozen”. One therefore needs a model, giving rise to chaotic behavior of the mechanical freedoms, no matter how small their energy is. Though extremely simplified, our model displays this property. Let us first consider [@BCL72] an electromagnetic field confined inbetween two parallel, perfectly reflecting plane mirrors, a distance $2l$ apart. We take Cartesian coordinates $XYZ$ with the $X$ axis normal to the mirrors, and restrict to excitations only dependent on $X$, thus getting a 1–dimensional radiant cavity, the normal modes of which have angular frequencies $\omega_n=(\pi c/2l)n$, $n=1,2,...$. Then we introduce a uniformly charged, infinite plate of thickness $2\delta$, situated midway between the mirrors and parallel to them, bound to move along the $Z$–direction only. We denote $z$ its displacement in that direction, $\sigma$ and $m$ the charge and mass densities per unit surface of the plate, $f(x)$ the normalized (transverse) distribution of charge in the plate. Finally, the plate is subject to a mechanical restoring force per unit surface, $F(z)=-m\omega_0^2z$. Using the Coulomb gauge, plus zero boundary conditions on the mirrors for the $Z$ component of the vector potential, we obtain the following Hamiltonian for the full system plate plus field: $$\begin{aligned} \begin{array}{c} \displaystyle{ H_0=\frac{1}{2m}\left(p_z-2\left(\frac{\pi}{l}\right)^{1/2} \!\sigma\sum_{n=1}^{\infty}{}^{'}a_n q_n\right)^2+} \cr\cr \displaystyle{ \frac{1}{2}m\omega_0^2 z^2+ \frac{1}{2}\sum_{n=1}^{\infty}{}^{'}(p_{n}^2+ \omega_n^2 q_n^2)}, \end{array}\end{aligned}$$ where $(z,p_z)$ and $(q_n,p_n)$ are canonical conjugated variables for the plate and the $n-$th mode of the field respectively. In particular, $q_n(t)$ is the amplitude of the $Z$ component of the vector potential on the $n$–th normal mode of the free field, and the coefficients $a_n$ are given by $a_n=\int_{-\delta}^{\delta}dx f(x)\cos(\omega_n x/c)$. $\sum$’ means the sum over odd $n$’s only, because, with the chosen boundary conditions, even modes do not interact with the plate. Finite–energy states of our hamiltonian system correspond to vectors in the Hilbert space ${\cal H}_0$ of square–summable, $\infty$–dimensional vectors $$\q=\left\{m^{1/2}\omega_0 z,m^{1/2}\dot{z}, ...,\omega_{2n-1}q_{2n-1}, p_{2n-1},...\right\}$$ whose squared norm is just twice the energy. Since Hamilton’s equations are linear, the evolution of states in ${\cal H}_0$ is given by a unitary group $\exp(iH_0t)$, with a complete set of orthonormal eigenvectors , $${\bf u}_{k}=C_k\left\{\frac{\omega_0}{\Omega_k},1,..., \frac{\epsilon\,a_{2n-1}\,\omega_{2n-1}} {\Omega_k^2+\omega_{2n-1}^2}\,, \frac{\epsilon\,a_{2n-1}\,\Omega_k} {\Omega_k^2+\omega_{2n-1}^2},...\right\},$$ where $\epsilon=2\sigma(\pi/ml)^{1/2}$ and $C_k$ is a normalization constant. These eigenvectors define normal modes of the total system, with eigenfrequencies $\Omega_k$ given by the imaginary roots of the secular equation: $$\Omega_k^2\,\left(1+\epsilon^2\sum_{n=1}^{\infty}\,\frac{a_{2n-1}^2} {\Omega_k^2+\omega_{2n-1}^2}\right)+\omega_0^2=0.$$ We shall now introduce a nonlinear mechanism, which will couple these normal modes, giving rise to energy exchanges between them. To this end we introduce a second plate, parallel to the first, bound to move in the same direction, and with the same mass density. We assume for simplicity this plate to carry no charge, so that its motion is not influenced by the field. The interaction between plates is purely mechanical, and simulated by elastic bounces when $|z(t)-z_1(t)|=R$, with $z_1$ the $Z$–coordinate of the neutral plate. Between collisions, the system is integrable: the motion of the charged plate and the field is given by the action of the group $\exp(iH_0t)$, while the second plate is moving at a uniform speed. This is most easily described by adding to the vector ${\bf q}$ one more component $m^{1/2}w$, with $w$ the velocity of the uncharged plate. The new Hilbert space ${\cal H}$ of such ${\bf q}$–vectors is the phase space of our model. The evolution between collisions is again unitary, given by $\exp(iHt)$, with a generator $H$, and a complete set of normal modes, which are trivially related to the above described ones. At collisions the two plates exchange their velocities, thus mixing all the amplitudes in the expansion of the state vector over normal modes. The evolution from immediately after one collision to immediately after the next is given by a map, which, in Hilbert space notations, has the following simple form: $${\cal S}({\bf q}) =(Id-2P)e^{iHt({\bf q})}{\bf q}, \label{map}$$ The 1st (operator) factor describes a collision: $Id$ is the identity operator, $P$ is a one–dimensional projection: $P{\bf q}=<{\bf e}\vert{\bf q}>{\bf e}$, where ${\bf e}$ is the unit vector such that the scalar product $<{\bf e}\vert{\bf q}>$ yields the relative velocity of the two plates in the state ${\bf q}$. The 2nd factor describes evolution over the free-flight time $t({\bf q})$, which is the smallest positive root of the equations $$\vert z(e^{iHt}{\bf q})-z({\bf q})-w({\bf q})t\vert=0,2R.$$ The map allows for efficient numerical simulation: e.g., in the case of $200$ oscillators, we were able to follow a trajectory up to $5\times 10^6$ bounces with a relative error in energy conservation less than $10^{-10}$. In numerical simulations, one has of course to consider a finite number $N$ of field oscillators. In our computations we have varied $N$ and all other parameters, except $l=\pi,m=1,c=1$; moreover, since the dynamics depends on energy only via the scaled parameter $R/\sqrt{E}$, we have always taken $E=1$ and varied the “free path” $R$ instead. The choice of the charge density $f(x)$ is important, because the coupling of individual modes to the charged plate is scaled by the coefficients $a_n$ of the Fourier expansion of $f(x)$. Choosing a singular density $f(x)$, as in earlier studies, results, at all times, in a power law decay of the distribution of energy over the field modes, so that truncation effects are already significant at small integration times. We have therefore chosen $f(x)=k\exp(-\delta^2/(\delta^2-x^2))$ (the standard compactly supported $C^{\infty}$ function), with the constant $k$ fixed from normalization; this ensures a faster than algebraic, albeit nonexponential, decay of the distributions. Even though we do not have rigorous results, the dynamics of this billiard–type model appears to be completely chaotic independently of the total energy. Moreover, the finite–dimensional reduced dynamics has positive maximal Lyapunov exponents $\lambda,\lambda_c$ (the former being defined with respect to real time, the latter to the number of collisions). These were numerically computed by multiplying matrices obtained from linearization of the map (\[map\]) along a trajectory. The exponent $\lambda$ decreases with the number of normal modes taken into account, because bounces become less frequent, the two plates going to rest, in time average, for $N\to\infty$ (see below). On the contrary $\lambda_c$ was observed to increase with $N$; indeed, as collisions become more distant in time, phases change more drastically in between them, and their randomization is faster. If instead $N$ is increased keeping the energy per mode $E/N$ fixed, both $\lambda$ and $\lambda_n$ appear to saturate, suggesting that Lyapunov exponents converge to a finite non zero value in the thermodynamic limit. The maximal Lyapunov exponents remain positive on reducing $\sigma$, with no stochasticity threshold displayed. However, the time required to reach a converged value becomes larger, because trajectories need more time to fill the phase space. =3.8in =3.in Most of our numerical experiments were meant to understand how the energy is distributed among all the degrees of freedom (in time average). In Fig.\[fig1\] the time–average kinetic energy $\overline{T}=\lim_{t\to\infty} \,\overline{T}(t)$ of the charged plate is shown as a function of the number $N$ of field modes considered. In the above definition, $\overline{T}(t)$ is the time average up to time $t$ of $\frac{1}{2}m{\dot{z}^2}(t)$, while the limit means that the motion has been followed until stabilization of the time–average. Results are in accordance with the equipartition theorem: the total energy is equally shared between the $2N+3$ relevant canonical variables. For $N\to\infty$ we can extrapolate $\overline{T}=0$, that is, the electromagnetic field acts as a friction force on the plate. The approach to equilibrium is not uniform, because the relaxation time associated with the $n$–th overall normal mode increases with $n$. To analyze this increase we have used the equipartition indicator $$n_{eff}(t)=\exp\left\{-\sum_{n=0}^{N+1}\overline{E}_n(t) \ln\overline{E}_n(t)\right\}, \label{neff}$$ where $\overline{E}_n(t)$ indicates the normalized time average energy (up to the time $t$) of the $n$–th normal mode $(\overline{E}_{N+1}$ refers to the energy of the neutral plate). The parameter $n_{eff}$ is a measure of the number of modes significantly excited at time $t$; if only a finite number of modes is considered, it also measures the degree of equipartition, because $n_{eff}=1$ if only one normal mode is excited, whereas the maximal value $n_{eff}\sim N+3/2$ is only attained in the presence of complete equipartition. As far as the numerical simulation is truly representative of the infinite–dimensional system, $n_{eff}$ appears to increase with $t$ slower than any power, but faster than logarithmically (Fig.\[fig2\]). 0.cm =3.8in =3.in A quantitative description of the relaxation process and analytical estimates of expression (\[neff\]) can be obtained from eqn.(\[map\]) by a random–phase approximation. Let us consider an ensemble of trajectories which start at time zero from a single normal mode, with randomly distributed phases. Denoting $E_n(\tau)$ the ensemble–averaged (normalized) energy on the $n-$th normal mode after the $\tau-$th collision, and assuming complete randomness of phases at the collision time, from (\[map\]) we get: $$\label{rf} E_k(\tau)= (1-4\vert e_k\vert^2)E_k(\tau-1)+4W(\tau-1)\vert e_k\vert^2,$$ where $$\label{rf1} e_k=<u_k\vert {\bf e}>;\quad W(\tau)=\sum_k\vert e_k\vert^2 E_k(\tau).$$ Recalling the meaning of ${\bf e}$, one easily realizes that $W(\tau)$ is proportional to the average kinetic energy of the relative motion of the plates. Eqns. (\[rf\],\[rf1\]) can be solved numerically, to find how $n_{eff}$ increases with the number of collisions. The result (shown by the dashed line in the insert of Fig.(\[fig2\])) matches quite well with the numerical solution of the exact equations of motion, confirming the validity of the random phase approximation, hence the chaotic nature of dynamics. One can also solve (\[rf\],\[rf1\]) analytically, by implementing a continuous time approximation, plus standard Laplace transform techniques. Omitting details, one finds that the large–$\tau$ asymptotics of the solution is determined by the large–$k$ asymptotics of the coefficients $a_k$. In case of algebraic decay $a_k\sim \vert k\vert^{-\alpha}$, dispensing with prefactors which depend on $\epsilon, l, c$ and estimating the average time delay between the $\tau+1$–th and the $\tau$-th collision as $t\sim R/{\sqrt W(\tau)}$, one finds $n_{eff}(t)\sim t^{\frac{2}{4\alpha+5}}$ [@inprep]. With the charge distribution $f(x)$ used in our numerical simulations, we cannot give likewise explicit formulas, due to the complicated decay of coefficients $a_k$. However, it is possible to prove that $n_{eff}$ increases with $t$ faster than logarithmically, but slower than any power of $t$, as found in Fig.\[fig2\]. Thus the way the relaxation time of modes increases with their frequencies is determined by the choice of the charge density. To verify that the truncated system numerically investigated here really represents (up to a certain time) the real, infinite–dimensional, model, we plotted in Fig.\[fig3\] $n_{eff}$ as a function of the number $N$ of field oscillators taken into account. We found that, at any fixed time, $n_{eff}$ converges on increasing $N$, its limit value giving the number of overall normal modes significantly excited in the infinite–dimensional system. As this value increases with time, an equilibrium state is never reached. In summary, our numerical experiments have exposed a chaotic dynamics in the unusual case of an [*infinite*]{} dimensional, conservative system with a [*finite*]{} total energy. The dependence of time-averages on initial conditions gets lost, and, for any finite–dimensional reduction, the system reaches an equilibrium state, with equipartition of energy among the degrees of freedom of the field and of the matter. From our results we infer that, in the real infinite–dimensional problem, there is a trend towards equipartition, the finite energy of matter being removed to higher and higher frequencies of the field. However, this process takes place at a nonuniform rate, as relaxation times of normal modes increase with their frequency. Therefore, as in an old hypothesis of Jeans[@JEANS], a real equilibrium state is never reached. Artificial though it may appear, our model is actually [*the simplest, one-dim. model*]{} with charged particles undergoing elastic collisions inside a reflecting enclosure. We believe that more realistic models displaying the same basic features will display a similar behaviour. We also submit that quantization of this, or similar, field models may open a challenging new direction in the field of Quantum Chaos, starting from a very old problem. 0.cm =3.8in =3.in Present address: CEA, Service de Physique de l’Etat Condensé, Centre d’Etudes de Saclay, F–91191 Gif–sur–Yvette, France J.H.Jeans, [*The dynamical theory of gases*]{}, 2nd ed., Cambridge 1916. P. Bocchieri, A. Crotti and A. Loinger, Lett. Nuovo Cimento [**4**]{} (1972) 741. G. Casati, I. Guarneri and F. Valz–Gris, Phys. Rev. A [**16**]{} (1977) 1237; G. Benettin and L. Galgani, J. Stat. Phys. [**27**]{} (1982) 153; G. Casati, I. Guarneri and F. Valz Gris, J. Stat. Phys. [**30**]{} (1983) 195; R. Livi, M. Pettini, S. Ruffo and A. Vulpiani, J. Phys. A [**20**]{} (1987) 577; C. Alabiso, M. Casartelli and S. Sello, J. Stat. Phys. [**54**]{} (1989) 361. L. Galgani, C. Angaroni, L. Forti, A. Giorgilli and F. Guerra, Phys. Lett. A [**139**]{} (1989) 221; C. Alabiso, M. Casartelli and A. Scotti, Phys. Lett. A [**147**]{} (1990) 292. A. Patrascioiu, Phys. Rev. Lett. [**50**]{} (1983) 1879; K.R.S. Devi and A. Patrascioiu, Physica D [**11**]{} (1984) 359; A. Patrascioiu, E. Seiler and I.O. Stamatescu, Phys. Rev. A [**31**]{} (1985) 1906. G. Benenti, G. Casati and I. Guarneri, in preparation.

--- abstract: 'We develop and extend the theory of Mackey functors as an application of enriched category theory. We define Mackey functors on a lextensive category ${\ensuremath{\mathscr{E}}}$ and investigate the properties of the category of Mackey functors on ${\ensuremath{\mathscr{E}}}$. We show that it is a monoidal category and the monoids are Green functors. Mackey functors are seen as providing a setting in which mere numerical equations occurring in the theory of groups can be given a structural foundation. We obtain an explicit description of the objects of the Cauchy completion of a monoidal functor and apply this to examine Morita equivalence of Green functors.' address: 'Centre of Australian Category Theory, Macquarie University, New South Wales, 2109, Australia' author: - Elango Panchadcharam - Ross Street title: Mackey functors on compact closed categories --- [^1] Introduction ============ Groups are used to mathematically understand symmetry in nature and in mathematics itself. Classically, groups were studied either directly or via their representations. In the last 40 years, arising from the latter, groups have been studied using Mackey functors. Let $k$ be a field. Let ${\ensuremath{\mathbf{Rep}}}(G) = {\ensuremath{\mathbf{Rep}}}_k(G)$ be the category of $k$-linear representations of the finite group $G$. We will study the structure of a monoidal category ${\ensuremath{\mathbf{Mky}}}(G)$ where the objects are called Mackey functors. This provides an extension of ordinary representation theory. For example, ${\ensuremath{\mathbf{Rep}}}(G)$ can be regarded as a full reflective sub-category of ${\ensuremath{\mathbf{Mky}}}(G)$; the reflection is strong monoidal (= tensor preserving). Representations of $G$ are equally representations of the group algebra $kG$; Mackey functors can be regarded as representations of the “Mackey algebra” constructed from $G$. While ${\ensuremath{\mathbf{Rep}}}(G)$ is compact closed (= autonomous monoidal), we are only able to show that ${\ensuremath{\mathbf{Mky}}}(G)$ is star-autonomous in the sense of [@Ba]. Mackey functors and Green functors (which are monoids in ${\ensuremath{\mathbf{Mky}}}(G)$) have been studied fairly extensively. They provide a setting in which mere numerical equations occurring in group theory can be given a structural foundation. One application has been to provide relations between $\lambda$- and $\mu$-invariants in Iwasawa theory and between Mordell-Weil groups, Shafarevich-Tate groups, Selmer groups and zeta functions of elliptic curves (see [@BB]). Our purpose is to give the theory of Mackey functors a categorical simplification and generalization. We speak of Mackey functors on a compact (= rigid = autonomous) closed category ${\ensuremath{\mathscr{T}}}$. However, when ${\ensuremath{\mathscr{T}}}$ is the category ${\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})$ of spans in a lextensive category ${\ensuremath{\mathscr{E}}}$, we speak of Mackey functors on ${\ensuremath{\mathscr{E}}}$. Further, when ${\ensuremath{\mathscr{E}}}$ is the category (topos) of finite $G$-sets, we speak of Mackey functors on $G$. Sections \[Se2\]-\[Se4\] set the stage for Lindner’s result [@Li1] that Mackey functors, a concept going back at least as far as [@Gr], [@Dr1] and [@Di] in group representation theory, can be regarded as functors out of the category of spans in a suitable category ${\ensuremath{\mathscr{E}}}$. The important property of the category of spans is that it is compact closed. So, in Section \[Se5\], we look at the category ${\ensuremath{\mathbf{Mky}}}$ of additive functors from a general compact closed category ${\ensuremath{\mathscr{T}}}$ (with direct sums) to the category of $k$-modules. The convolution monoidal structure on ${\ensuremath{\mathbf{Mky}}}$ is described; this general construction (due to Day [@Da]) agrees with the usual tensor product of Mackey functors appearing, for example, in [@Bo1]. In fact, again for general reasons, ${\ensuremath{\mathbf{Mky}}}$ is a closed category; the internal hom is described in Section \[Se6\]. Various convolution structures have been studied by Lewis [@Le] in the context of Mackey functors for compact Lie groups mainly to provide counter examples to familiar behaviour. Green functors are introduced in Section \[Se7\] as the monoids in ${\ensuremath{\mathbf{Mky}}}$. An easy construction, due to Dress [@Dr1], which creates new Mackey functors from a given one, is described in Section \[Se8\]. We use the (lax) centre construction for monoidal categories to explain results of [@Bo2] and [@Bo3] about when the Dress construction yields a Green functor. In Section \[Se9\] we apply the work of [@Da4] to show that finite-dimensional Mackey functors form a $*$-autonomous [@Ba] full sub-monoidal category ${\ensuremath{\mathbf{Mky}_\textit{fin}}}$ of ${\ensuremath{\mathbf{Mky}}}$. Section \[Se11\] is rather speculative about what the correct notion of Mackey functor should be for quantum groups. Our approach to Morita theory for Green functors involves even more serious use of enriched category theory: especially the theory of (two-sided) modules. So Section \[Se12\] reviews this theory of modules and Section \[Se13\] adapts it to our context. Two Green functors are Morita equivalent when their ${\ensuremath{\mathbf{Mky}}}$-enriched categories of modules are equivalent, and this happens, by the general theory, when the ${\ensuremath{\mathbf{Mky}}}$-enriched categories of Cauchy modules are equivalent. Section \[Se14\] provides a characterization of Cauchy modules. The compact closed category ${\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})$ {#Se2} ====================================================================================== Let ${\ensuremath{\mathscr{E}}}$ be a finitely complete category. Then the category ${\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})$ can be defined as follows. The objects are the objects of the category ${\ensuremath{\mathscr{E}}}$ and morphisms $U {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}V$ are the isomorphism classes of *spans* from $U$ to $V$ in the bicategory of spans in ${\ensuremath{\mathscr{E}}}$ in the sense of [@Be]. (Some properties of this bicategory can be found in [@CKS].) A *span* from $U$ to $V$, in the sense of [@Be], is a diagram of two morphisms with a common domain $S$, as in $$(s_1, S, s_2): \quad \vcenter{\xygraph{ S="s" (:[d(1)r(0.9)] {V~.}^-{s_2}, :[d(1)l(0.9)] {U}_-{s_1} )}}$$ An isomorphism of two spans $(s_1, S, s_2): U {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}V$ and $(s'_1, S', s'_2): U {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}V$ is an invertible arrow $h: S {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}S'$ such that $s_1=s'_1 {\ensuremath{\circ}}h$ and $s_2=s'_2 {\ensuremath{\circ}}h$. $$\xygraph{ S="s" (:[d(1)r(1.3)] {V}="b"^-{s_2}, :[d(1)l(1.3)] {U}="a"_-{s_1}, :[d(2.1)] {S'}="c"^-{h}_{\cong} "c" : "a"^-{s'_1} "c" : "b"_-{s'_2} )}$$ The composite of two spans $(s_1, S, s_2): U {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}V$ and $(t_1, T, t_2): V {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}W$ is defined to be $(s_1{\ensuremath{\circ}}{\text{proj}}_1, T{\ensuremath{\circ}}S, t_2 {\ensuremath{\circ}}{\text{proj}}_2): U {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}W$ using the pull-back diagram as in $$\xygraph{ {S \times_V T = T {\ensuremath{\circ}}S}="a" (:[d(1)r(0.8)] {T}="t"^-{{\text{proj}}_2} :[d(1)r(0.8)] {W~.}^-{t_2}, :[d(1)l(0.8)] {S}="s"_-{{\text{proj}}_1} (:[d(1)l(0.8)] {U}_-{s_1}, :[d(1)r(0.8)] {V}="v"^-{s_2}) "t" : "v"_{t_1} "a" :@{}[d(1.7)] |{\textstyle \txt{pb}} )}$$ This is well defined since the pull-back is unique up to isomorphism. The identity span $(1, U, 1): U {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}U$ is defined by $$\xygraph{ U="s" (:[d(1)r(0.9)] {U}^-{1}, :[d(1)l(0.9)] {U}_-{1} )}$$ since the composite of it with a span $(s_1, S, s_2): U {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}V$ is given by the following diagram and is equal to the span $(s_1, S, s_2): U {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}V$ $$\xygraph{ {S}="a" (:[d(1)r(0.8)] {S}="t"^-{1} :[d(1)r(0.8)] {V~.}^-{s_2}, :[d(1)l(0.8)] {U}="s"_-{s_1} (:[d(1)l(0.8)] {U}_-{1}, :[d(1)r(0.8)] {U}="v"^-{1}) "t" : "v"_{s_1} "a" :@{}[d(1.7)] |{\textstyle \txt{pb}} )}$$ This defines the category ${\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})$. We can write ${\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})(U,V) \cong [{\ensuremath{\mathscr{E}}}/ (U \times V)]$ where square brackets denote the isomorphism classes of morphisms. ${\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})$ becomes a monoidal category under the tensor product $$\xymatrix@1{ {\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}}) \times {\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}}) \ar[r]^-{\times} & {\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})}$$ defined by $$\xymatrix@1{(U,V) \ar@{|->}[r] & U \times V}$$ $$[ \xymatrix@1{U \ar[r]^S & U'}, \xymatrix@1{V \ar[r]^T & V'}] \xymatrix@1{\ar@{|->}[r] &} [ \xymatrix@1{U \times V \ar[r]^-{S \times T} & U' \times V'}].$$ This uses the cartesian product in ${\ensuremath{\mathscr{E}}}$ yet is not the cartesian product in ${\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})$. It is also compact closed; in fact, we have the following isomorphisms: ${\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})(U,V) \cong {\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})(V, U)$ and ${\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})(U \times V, W) \cong {\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})(U, V \times W)$. The second isomorphism can be shown by the following diagram $$\vcenter{\xygraph{ S="s" (:[d(1)r(0.9)] {W}, :[d(1)l(0.9)] {U \times V} )}} \quad {\ensuremath{\xymatrix@1{\ar@{<->}[r] |-{\object@{|}} & }}}\quad \vcenter{\xygraph{ S="s" (:[d(1)r(0.9)] {W}, :[d(1)l(0.9)] {U}, :[d(1)] {V}, )}} \quad {\ensuremath{\xymatrix@1{\ar@{<->}[r] |-{\object@{|}} & }}}\quad \vcenter{\xygraph{ S="s" (:[d(1)r(0.9)] {V \times W~.}, :[d(1)l(0.9)] {U} )}}$$ Direct sums in ${\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})$ {#Se3} ======================================================================== Now we assume ${\ensuremath{\mathscr{E}}}$ is lextensive. References for this notion are [@Sc], [@CLW], and [@CL]. A category ${\ensuremath{\mathscr{E}}}$ is called lextensive when it has finite limits and finite coproducts such that the functor $${\ensuremath{\mathscr{E}}}/ A \times {\ensuremath{\mathscr{E}}}/ B {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathscr{E}}}/ A+B\ ; \quad \vcenter{\xymatrix{X \ar[d]^f \\ A}} \quad , \quad \vcenter{\xymatrix{Y \ar[d]^g \\ B}} \xymatrix{\ar@{|->}[r] &} \vcenter{\xymatrix{X+Y \ar[d]^{f+g} \\ A+B}}$$ is an equivalance of categories for all objects $A$ and $B$. In a lextensive category, coproducts are disjoint and universal and $0$ is strictly initial. Also we have that the canonical morphism $$(A \times B) + (A \times C) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}A \times (B+C)$$ is invertible. It follows that $ A \times 0 \cong 0.$ In ${\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})$ the object $U+V$ is the direct sum of $U$ and $V.$ This can be shown as follows (where we use lextensivity): $$\begin{aligned} {\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}}) (U+V, W) & \cong [{\ensuremath{\mathscr{E}}}/ ((U+V) \times W)] \\ & \cong [{\ensuremath{\mathscr{E}}}/ ((U \times W) +( V \times W)) ] \\ & \simeq [{\ensuremath{\mathscr{E}}}/ (U \times W)] \times [{\ensuremath{\mathscr{E}}}/ (V \times W)] \\ &\cong {\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})(U,W) \times {\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})(V,W);\end{aligned}$$ and so $ {\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})(W, U+V) \cong {\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})(W,U) \times {\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})(W,V)$. Also in the category ${\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})$, $0$ is the zero object (both initial and terminal): $${\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})(0, X) \cong [ {\ensuremath{\mathscr{E}}}/ (0 \times X)] \cong [{\ensuremath{\mathscr{E}}}/ 0] \cong 1$$ and so ${\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})(X,0) \cong 1$. It follows that ${\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})$ is a category with homs enriched in commutative monoids. The addition of two spans $(s_1, S, s_2) : U {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}V$ and $(t_1, T, t_2): U {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}V$ is given by $(\nabla {\ensuremath{\circ}}(s_1 + t_1), S+T, \nabla {\ensuremath{\circ}}(s_2 + t_2)): U {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}V$ as in $$\vcenter{\xygraph{ S="s" (:[d(1)r(0.9)] {V}^-{s_2}, :[d(1)l(0.9)] {U}_-{s_1} )}} \quad + \quad \vcenter{\xygraph{ T="s" (:[d(1)r(0.9)] {V}^-{t_2}, :[d(1)l(0.9)] {U}_-{t_1} )}} \quad = \vcenter{\xygraph{ {S + T}="a" (:[d(1)r(0.7)] {V+V}="t" ^-{s_2 + t_2} :[d(1)r(0.7)] {V~.}="v" ^-{\nabla}, :[d(1)l(0.7)] {U+U}="s" _-{s_1 + t_1} :[d(1)l(0.7)] {U}="u" _-{\nabla}) "a" : @/_6ex/ _-{[s_1,t_1]} "u" "a" : @/_-6ex/ ^-{[s_2,t_2]} "v" }}$$ Summarizing, ${\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})$ is a monoidal commutative-monoid-enriched category. There are functors $(-)_*: {\ensuremath{\mathscr{E}}}{\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})$ and $(-)^*: {\ensuremath{\mathscr{E}}}^{\text{op}} {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})$ which are the identity on objects and take $f: U {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}V$ to $f_*=(1_U, U, f)$ and $f^*=(f, U, 1_U)$, respectively. For any two arrows $\xymatrix@1{U \ar[r]^f &V \ar[r]^g & W}$ in ${\ensuremath{\mathscr{E}}}$, we have $(g {\ensuremath{\circ}}f)_* \cong g_* {\ensuremath{\circ}}f_*$ as we see from the following diagram $$\xygraph{ {U}="a" (:[d(0.8)r(0.7)] {V}="t"^-{f} :[d(0.8)r(0.7)] {W~.}^-{g}, :[d(0.8)l(0.7)] {U}="s"_-{1} (:[d(0.8)l(0.7)] {U}_-{1}, :[d(0.8)r(0.7)] {V}="v"^-{f}) "t" : "v"_{1} "a" :@{}[d(1.5)] |{\textstyle \txt{pb}} )}$$ Similarly $(g {\ensuremath{\circ}}f)^* \cong f^* {\ensuremath{\circ}}g^*$. Mackey functors on ${\ensuremath{\mathscr{E}}}$ {#Se4} =============================================== A *Mackey functor* $M$ from ${\ensuremath{\mathscr{E}}}$ to the category ${\ensuremath{\mathbf{Mod}_k}}$ of $k$-modules consists of two functors $$M_*: {\ensuremath{\mathscr{E}}}{\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathbf{Mod}_k}}, \quad M^*: {\ensuremath{\mathscr{E}}}^{\text{op}} {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathbf{Mod}_k}}$$ such that: 1. $M_*(U) = M^*(U) \quad (=M(U))$ for all $U$ in ${\ensuremath{\mathscr{E}}}$ 2. for all pullbacks $$\xygraph{ P="a" (:[r(1.8)] {V}^-{q} :[d(1.4)] {W}="t"^-{s}, :[d(1.4)] {U}_-{p} :"t"_-{r} )}$$ in ${\ensuremath{\mathscr{E}}}$, the square (which we call a *Mackey square*) $$\xygraph{ {M(U)}="a" (:[r(1.8)] {M(W)}_-{M_*(r)} :[u(1.4)] {M(V)}="t"_-{M^*(s)}, :[u(1.4)] {M(P)}^-{M^*(p)} :"t"^-{M_*(q)} )}$$ commutes, and 3. for all coproduct diagrams $$\xymatrix@1{ U \ar[r]^-{i} & {U+V} & V \ar[l]_-{j} }$$ in ${\ensuremath{\mathscr{E}}}$, the diagram $$\xymatrix@C=7ex{ {M(U)} \ar@<-0.8ex>[r]_-{M_*(i)} & {M(U+V)} \ar@<-1.1ex>[l]_-{M^*(i)} \ar@<1.1ex>[r]^-{M^*(j)} & {M(V)} \ar@<1.0ex>[l]^-{M_*(j)} }$$ is a direct sum situation in [$\mathbf{Mod}_k$]{}. (This implies $M(U+V) \cong M(U)\oplus M(V)$.) A morphism $\theta : M {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}N$ of Mackey functors is a family of morphisms $\theta_U : M(U) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}$ $ N(U)$ for $U$ in ${\ensuremath{\mathscr{E}}}$ which defines natural transformations $\theta_* : M_* {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}N_*$ and $\theta^* : M^* {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}N^*$. (Lindner [@Li1]) The category ${\ensuremath{\mathbf{Mky}}}({\ensuremath{\mathscr{E}}},{\ensuremath{\mathbf{Mod}_k}})$ of Mackey functors, from a lextensive category ${\ensuremath{\mathscr{E}}}$ to the category ${\ensuremath{\mathbf{Mod}_k}}$ of $k$-modules, is equivalent to $[{\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}}), {\ensuremath{\mathbf{Mod}_k}}]_+$, the category of coproduct-preserving functors. Let $M$ be a Mackey functor from ${\ensuremath{\mathscr{E}}}$ to ${\ensuremath{\mathbf{Mod}_k}}$. Then we have a pair $(M_*, M^*)$ such that $M_*: {\ensuremath{\mathscr{E}}}{\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathbf{Mod}_k}}$, $M^*: {\ensuremath{\mathscr{E}}}^{\text{op}} {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathbf{Mod}_k}}$ and $M(U) = M_*(U) = M^* (U)$. Now define a functor $M: {\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}}) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathbf{Mod}_k}}$ by $M(U) = M_*(U) = M^*(U)$ and $$M \left ( \vcenter{\xygraph{ S="s" (:[d(1)r(0.8)] {V}^-{s_2}, :[d(1)l(0.8)] {U}_-{s_1} ) }} \right ) \quad = \quad \big ( \xygraph{ {M(U)} (:[r(1.8)]{M(S)}^-{M^*(s_1)} :[r(1.8)]{M(V)}^-{M_*(s_2)} )} \big ).$$ We need to see that $M$ is well-defined. If $h: S {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}S'$ is an isomorphism, then the following diagram $$\xygraph{ {S'}="a" (:[r(2)] {S'}^-{1} :[d(1.2)] {S'}="t"^-{1}, :[d(1.2)] {S}_-{h^{-1}} :"t"_-{h} )}$$ is a pull back diagram. Therefore $M^*(h^{-1}) = M_*(h)$ and $M_*(h^{-1}) = M^*(h)$. This gives, $M_*(h)^{-1} = M^*(h)$. So if $h:(s_1,S,s_2) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}(s'_1,S',s'_2)$ is an isomorphism of spans, we have the following commutative diagram. $$\xymatrix @=9ex@R=11ex@C=11ex@ur{ {M(U)} \ar[r]^-{M^*(s_1)} \ar[d]_-{M^*(s'_1)} & {M(S)} \ar[d]^-{M_*(s_2)} \ar@<1ex>[dl]^-{M_*(h)} \\ {M(S')} \ar[r]_-{M_*(s'_2)} \ar@<1ex>[ur]^-{M^*(h)} & {M(V)} }$$ Therefore we get $$M_*(s_2) M^*(s_1) = M_*(s'_2)M^*(s'_1).$$ From this definition $M$ becomes a functor, since $$\begin{split} \hspace{-5pt} M \left ( \vcenter{\xygraph{ {P}="a" (:[d(1)r(0.9)] {T}="t"^-{p_2} :[d(1)r(0.9)] {W}^-{t_2}, :[d(1)l(0.9)] {S}="s"_-{p_1} (:[d(1)l(0.9)] {U}_-{s_1}, :[d(1)r(0.9)] {V}="v"^-{s_2}) "t" : "v"_{t_1} "a" :@{}[d(2)] |{\textstyle \txt{pb}} )}} \right ) & = \vcenter{\xygraph{ {M(U)}="u" (:[r(1.7)]{M(P)}="p"^-{M^*(p_1s_1)} :[r(1.7)]{M(W)}="w"^-{M_*(t_2p_2)}, :[d(1.1)r(0.9)]{M(S)}="s"_-{M^*(s_1)} :[d(1.1)r(0.9)] {M(V)}="v"_-{M_*(s_2)} :[u(1.1)r(0.9)] {M(T)}="t"_-{M^*(t_1)} :"w"_-{M_*(t_2)} "s":"p"^-{M^*(p_1)} "p":"t"^{M_*(p_2)} "p" :@{}[d(2.3)r(0.3)] |{\textstyle \txt{Mackey}} )}} \\ & = ( \xygraph{ {M(U)} (:[r(1.7)]{M(V)}^-{M(s_1, S, s_2)} :[r(1.7)]{M(W)}^-{M(t_1, T, t_2)} )} ), \end{split}$$ where $P=S \times_V T$ and $p_1$ and $p_2$ are the projections 1 and 2 respectively, so that $$M((t_1, T, t_2) {\ensuremath{\circ}}(s_1, S, s_2)) = M(t_1, T, t_2) {\ensuremath{\circ}}M(s_1, S, s_2).$$ The value of $M$ at the identity span $(1, U, 1): U {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}U$ is given by $$\begin{split} M \left ( \vcenter{\xygraph{ {U} (:[d(1)r(0.9)] {U}^-{1}, :[d(1)l(0.9)] {U}_-{1} )}} \right ) & \quad = \quad ( \xygraph{ {M(U)} (:[r(1.5)]{M(U)}^-{1} :[r(1.5)]{M(U)}^-{1} )} ) \\ & \quad = \quad ( 1 : \xygraph{ {M(U)} (:[r(1.5)]{M(U)} )} ). \end{split}$$ Condition (3) on the Mackey functor clearly is equivalent to the requirement that $M: {\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}}) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathbf{Mod}_k}}$ should preserve coproducts. Conversely, let $M: {\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}}) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathbf{Mod}_k}}$ be a functor. Then we can define two functors $M_*$ and $ M^*$, referring to the diagram $$\xygraph{ {{\ensuremath{\mathscr{E}}}}="a" (:[r(1.7)]{{\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})}="b"^-{(-)_*} :[r(2.2)]{{\ensuremath{\mathbf{Mod}_k}}~,}^-{M}, [d(0.8)]{{\ensuremath{\mathscr{E}}}^{\text{op}}}="c" "c":"b"_-{(-)^*} )}$$ by putting $M_* = M {\ensuremath{\circ}}(-)_*$ and $M^* = M {\ensuremath{\circ}}(-)^*$. The Mackey square is obtained by using the functoriality of $M$ on the composite span $$s^* {\ensuremath{\circ}}r_* = (p, P, q) = q_* {\ensuremath{\circ}}p^* .$$ The remaining details are routine. Tensor product of Mackey functors {#Se5} ================================= We now work with a general compact closed category ${\ensuremath{\mathscr{T}}}$ with finite products. It follows (see [@Ho]) that ${\ensuremath{\mathscr{T}}}$ has direct sums and therefore that ${\ensuremath{\mathscr{T}}}$ is enriched in the monoidal category ${\ensuremath{\mathscr{V}}}$ of commutative monoids. We write ${\ensuremath{\otimes}}$ for the tensor product in ${\ensuremath{\mathscr{T}}}$, write $I$ for the unit, and write $(-)^*$ for the dual. The main example we have in mind is ${\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})$ as in the last section where ${\ensuremath{\otimes}}= \times, I=1,$ and $V^*=V$. A Mackey functor on ${\ensuremath{\mathscr{T}}}$ is an additive functor $M: {\ensuremath{\mathscr{T}}}{\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathbf{Mod}_k}}$. Let us review the monoidal structure on the category ${\ensuremath{\mathscr{V}}}$ of commutative monoids; the binary operation of the monoids will be written additively. It is monoidal closed. For $A, B \in {\ensuremath{\mathscr{V}}},$ the commutative monoid $$[A,B] = \lbrace f: A {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}B ~ | ~ f ~\text{is a commutative monoid morphism} \rbrace ,$$ with pointwise addition, provides the internal hom and there is a tensor product $A {\ensuremath{\otimes}}B$ satisfying $${\ensuremath{\mathscr{V}}}(A{\ensuremath{\otimes}}B, C) \cong {\ensuremath{\mathscr{V}}}(A, [B,C]).$$ The construction of the tensor product is as follows. The free commutative monoid $FS$ on a set $S$ is $$FS = \lbrace u: S {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}\mathbb{N} ~ | ~ u(s)=0 ~ \text{for all but a finite number of} ~ s \in S \rbrace \subseteq \mathbb{N}^{S}.$$ For any $A,B \in {\ensuremath{\mathscr{V}}}$, $$A {\ensuremath{\otimes}}B = \left ( \begin{aligned} F(A \times B) / & (a+a', b) \sim (a,b) + (a',b) \\ & (a, b+b') \sim (a,b) + (a,b') \end{aligned} \right ).$$ We regard ${\ensuremath{\mathscr{T}}}$ and ${\ensuremath{\mathbf{Mod}_k}}$ as ${\ensuremath{\mathscr{V}}}$-categories. Every ${\ensuremath{\mathscr{V}}}$-functor ${\ensuremath{\mathscr{T}}}{\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathbf{Mod}_k}}$ preserves finite direct sums. So $[{\ensuremath{\mathscr{T}}}, {\ensuremath{\mathbf{Mod}_k}}]_+$ is the ${\ensuremath{\mathscr{V}}}$-category of ${\ensuremath{\mathscr{V}}}$-functors. For each $V \in {\ensuremath{\mathscr{V}}}$ and $X$ an object of a ${\ensuremath{\mathscr{V}}}$-category ${\ensuremath{\mathscr{X}}}$, we write $V {\ensuremath{\otimes}}X$ for the object (when it exists) satisfying $${\ensuremath{\mathscr{X}}}(V {\ensuremath{\otimes}}X, Y) \cong [V, {\ensuremath{\mathscr{X}}}(X,Y)]$$ ${\ensuremath{\mathscr{V}}}$-naturally in $Y$. Also the coend we use is in the ${\ensuremath{\mathscr{V}}}$-enriched sense: for the functor $T: {\ensuremath{\mathscr{C}}}^{\text{op}} {\ensuremath{\otimes}}{\ensuremath{\mathscr{C}}}{\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathscr{X}}}$, we have a coequalizer $$\xymatrix@C=5ex{ {\displaystyle \sum_{V, W} {\ensuremath{\mathscr{C}}}(V,W) {\ensuremath{\otimes}}T(W,V)} \ar@<0ex>[r] \ar@<1.7ex>[r] & {\displaystyle \sum_V T(V, V)} \ar@<0.5ex>[r] & {\displaystyle \int^V T(V,V)}}$$ when the coproducts and tensors exist. The tensor product of Mackey functors can be defined by convolution (in the sense of [@Da]) in $[{\ensuremath{\mathscr{T}}}, {\ensuremath{\mathbf{Mod}_k}}]_+$ since ${\ensuremath{\mathscr{T}}}$ is a monoidal category. For Mackey functors $M$ and $N$, the tensor product $M*N$ can be written as follows: $$\begin{split} (M * N)(Z) & = \int^{X,Y} {\ensuremath{\mathscr{T}}}(X {\ensuremath{\otimes}}Y, Z) {\ensuremath{\otimes}}M(X) {\ensuremath{\otimes}}_k N(Y) \\ & \cong \int^{X,Y} {\ensuremath{\mathscr{T}}}(Y, X^* {\ensuremath{\otimes}}Z) {\ensuremath{\otimes}}M(X) {\ensuremath{\otimes}}_k N(Y) \\ & \cong \int^{X} M(X) {\ensuremath{\otimes}}_k N(X^* {\ensuremath{\otimes}}Z) \\ & \cong \int^{Y} M(Z{\ensuremath{\otimes}}Y^*) {\ensuremath{\otimes}}_k N(Y) . \end{split}$$ the last two isomorphisms are given by the Yoneda lemma. The *Burnside functor* $J$ is defined to be the Mackey functor on ${\ensuremath{\mathscr{T}}}$ taking an object $U$ of ${\ensuremath{\mathscr{T}}}$ to the free $k$-module on ${\ensuremath{\mathscr{T}}}(I,U)$. The Burnside functor is the unit for the tensor product of the category ${\ensuremath{\mathbf{Mky}}}$. This convolution satisfies the necessary commutative and associative conditions for a symmetric monoidal category (see [@Da]). $[{\ensuremath{\mathscr{T}}}, {\ensuremath{\mathbf{Mod}_k}}]_+$ is also an abelian category (see [@Fr]). When ${\ensuremath{\mathscr{T}}}$ and $k$ are understood, we simply write ${\ensuremath{\mathbf{Mky}}}$ for this category $[{\ensuremath{\mathscr{T}}}, {\ensuremath{\mathbf{Mod}_k}}]_+$. The Hom functor {#Se6} =============== We now make explicit the closed structure on ${\ensuremath{\mathbf{Mky}}}$. The Hom Mackey functor is defined by taking its value at the Mackey functors $M$ and $N$ to be $${\ensuremath{\mathrm{Hom}}}(M, N)(V) = {\ensuremath{\mathbf{Mky}}}(M(V^* {\ensuremath{\otimes}}-), N),$$ functorially in $V$. To see that this hom has the usual universal property with respect to tensor, notice that we have the natural bijections below (represented by horizontal lines). $$\infer{L(V) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathbf{Mky}}}(M(V^* {\ensuremath{\otimes}}-), N) ~ ~ \text{natural in $V$}}{ \infer{L(V) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\displaystyle\int_U \text{Hom}_k (M(V^* {\ensuremath{\otimes}}U), N(U)) ~ ~ \text{natural in $V$}}}{ \infer{L(V) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathrm{Hom}}}_k (M(V^* {\ensuremath{\otimes}}U), N(U))~ ~ \text{dinatural in $U$ and natural in $V$}}{ \infer{L(V) {\ensuremath{\otimes}}_k M(V^* {\ensuremath{\otimes}}U) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}N(U) ~ ~ \text{natural in $U$ and dinatural in $V$}} {(L*M)(U) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}N(U) ~ ~ \text{natural in~} U }}}}$$ We can obtain another expression for the hom using the isomorphism $${\ensuremath{\mathscr{T}}}(V {\ensuremath{\otimes}}U, W) \cong {\ensuremath{\mathscr{T}}}(U, V^* {\ensuremath{\otimes}}W)$$ which shows that we have adjoint functors $$\xymatrix@C=8ex{{\ensuremath{\mathscr{T}}}\ar@{}[r]|{\bot} \ar@<0.3ex>@/^/ [r] ^{V {\ensuremath{\otimes}}-} & {\ensuremath{\mathscr{T}}}~. \ar@<0.3ex>@/^/ [l] ^{V^* {\ensuremath{\otimes}}-}}$$ Since $M$ and $N$ are Mackey functors on ${\ensuremath{\mathscr{T}}},$ we obtain a diagram $$\xymatrix@C=2ex{{\ensuremath{\mathscr{T}}}\ar@{}[rr]|{\bot} \ar@<0.3ex>@/^/ [rr] ^{V {\ensuremath{\otimes}}-} \ar[dr]_N && {\ensuremath{\mathscr{T}}}\ar@<0.3ex>@/^/ [ll] ^{V^* {\ensuremath{\otimes}}-} \ar[dl]^M \\ & {\ensuremath{\mathbf{Mod}_k}}}$$ and an equivalence of natural transformations $$\infer{M(V^* {\ensuremath{\otimes}}-) \Longrightarrow N}{M \Longrightarrow N(V {\ensuremath{\otimes}}-)}.$$ Therefore, the Hom Mackey functor is also given by $${\ensuremath{\mathrm{Hom}}}(M, N)(V) = {\ensuremath{\mathbf{Mky}}}(M, N(V {\ensuremath{\otimes}}-)).$$ Green functors {#Se7} ============== A *Green functor* $A$ on ${\ensuremath{\mathscr{T}}}$ is a Mackey functor (that is, a coproduct preserving functor $A: {\ensuremath{\mathscr{T}}}{\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathbf{Mod}_k}}$) equipped with a monoidal structure made up of a natural transformation $$\mu: A(U) {\ensuremath{\otimes}}_k A(V) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}A(U {\ensuremath{\otimes}}V),$$ for which we use the notation $\mu(a {\ensuremath{\otimes}}b)= a.b$ for $a\in A(U)$, $b \in A(V)$, and a morphism $$\eta: k {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}A(1),$$ whose value at $1 \in k$ we denote by 1. Green functors are the monoids in ${\ensuremath{\mathbf{Mky}}}$. If $A, B: {\ensuremath{\mathscr{T}}}{\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathbf{Mod}_k}}$ are Green functors then we have an isomorphism $${\ensuremath{\mathbf{Mky}}}(A*B,C) \cong \text{Nat}_{U,V} (A(U) {\ensuremath{\otimes}}_k B(V), C(U {\ensuremath{\otimes}}V) ).$$ Referring to the square $$\xygraph{ {{\ensuremath{\mathscr{T}}}{\ensuremath{\otimes}}{\ensuremath{\mathscr{T}}}}="a" (:[r(3.2)] {{\ensuremath{\mathbf{Mod}_k}}{\ensuremath{\otimes}}{\ensuremath{\mathbf{Mod}_k}}}^-{A {\ensuremath{\otimes}}B} :[d(1.2)] {{\ensuremath{\mathbf{Mod}_k}}~~~,}="t"^-{{\ensuremath{\otimes}}_k}, :[d(1.2)] {{\ensuremath{\mathscr{T}}}}_-{{\ensuremath{\otimes}}} :"t"_-{C} )}$$ we write this more precisely as $${\ensuremath{\mathbf{Mky}}}(A*B,C) \cong [{\ensuremath{\mathscr{T}}}{\ensuremath{\otimes}}{\ensuremath{\mathscr{T}}}, {\ensuremath{\mathbf{Mod}_k}}]({\ensuremath{\otimes}}_k {\ensuremath{\circ}}(A {\ensuremath{\otimes}}B), C {\ensuremath{\circ}}{\ensuremath{\otimes}}).$$ The Burnside functor $J$ and $\text{Hom}(A, A)$ (for any Mackey functor $A$) are monoids in ${\ensuremath{\mathbf{Mky}}}$ and so are Green functors. We denote by ${\ensuremath{\mathbf{Grn}}}({\ensuremath{\mathscr{T}}}, {\ensuremath{\mathbf{Mod}_k}})$ the category of Green functors on ${\ensuremath{\mathscr{T}}}$. When ${\ensuremath{\mathscr{T}}}$ and $k$ are understood, we simply write this as ${\ensuremath{\mathbf{Grn}}}(=\mathbf{Mon}({\ensuremath{\mathbf{Mky}}}))$ consisting of the monoids in ${\ensuremath{\mathbf{Mky}}}$. Dress construction {#Se8} ================== The Dress construction ([@Bo2], [@Bo3]) provides a family of endofunctors $D(Y,-)$ of the category ${\ensuremath{\mathbf{Mky}}}$, indexed by the objects $Y$ of ${\ensuremath{\mathscr{T}}}$. The Mackey functor defined as the composite $$\xymatrix@1{ {\ensuremath{\mathscr{T}}}\ar[r]^{-{\ensuremath{\otimes}}Y} & {\ensuremath{\mathscr{T}}}\ar[r]^M & {\ensuremath{\mathbf{Mod}}}_k}$$ is denoted by $M_Y$ for $M\in {\ensuremath{\mathbf{Mky}}}$; so $M_Y(U)=M(U {\ensuremath{\otimes}}Y)$. We then define the *Dress construction* $$D: {\ensuremath{\mathscr{T}}}{\ensuremath{\otimes}}{\ensuremath{\mathbf{Mky}}}{\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathbf{Mky}}}$$ by $D(Y,M)=M_Y$. The ${\ensuremath{\mathscr{V}}}$-category ${\ensuremath{\mathscr{T}}}{\ensuremath{\otimes}}{\ensuremath{\mathbf{Mky}}}$ is monoidal via the pointwise structure: $$(X,M) {\ensuremath{\otimes}}(Y,N) = (X{\ensuremath{\otimes}}Y, M*N).$$ \[Pro8.1\] The Dress construction $$D: {\ensuremath{\mathscr{T}}}{\ensuremath{\otimes}}{\ensuremath{\mathbf{Mky}}}{\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathbf{Mky}}}$$ is a strong monoidal ${\ensuremath{\mathscr{V}}}$-functor. We need to show that $D((X,M) {\ensuremath{\otimes}}(Y,N)) \cong D(X,M)*D(Y,N)$; that is, $M_X*M_Y \cong (M*N)_{X {\ensuremath{\otimes}}Y}$. For this we have the calculation $$\begin{split} (M_X*N_Y)(Z) & \cong \int^U M_X(U) {\ensuremath{\otimes}}_k N_Y(U^* {\ensuremath{\otimes}}Z) \\ & = \int^U M(U {\ensuremath{\otimes}}X) {\ensuremath{\otimes}}_k N(U^* {\ensuremath{\otimes}}Z {\ensuremath{\otimes}}Y) \\ & \cong \int^{U,V} {\ensuremath{\mathscr{T}}}(V,U {\ensuremath{\otimes}}X) {\ensuremath{\otimes}}M(V) {\ensuremath{\otimes}}_k N(U^* {\ensuremath{\otimes}}Z {\ensuremath{\otimes}}Y) \\ & \cong \int^{U,V} {\ensuremath{\mathscr{T}}}(V {\ensuremath{\otimes}}X^*,U) {\ensuremath{\otimes}}M(V) {\ensuremath{\otimes}}_k N(U^* {\ensuremath{\otimes}}Z {\ensuremath{\otimes}}Y) \\ & \cong \int^V M(V) {\ensuremath{\otimes}}_k N(V^* {\ensuremath{\otimes}}X {\ensuremath{\otimes}}Z {\ensuremath{\otimes}}Y) \\ & \cong (M*N)(Z {\ensuremath{\otimes}}X {\ensuremath{\otimes}}Y) \\ & \cong (M*N)_{X {\ensuremath{\otimes}}Y}(Z). \end{split}$$ Clearly we have $D(I,J) \cong J$. The coherence conditions are readily checked. We shall analyse this situation more fully in Remark \[Re8.5\] below. We are interested, after [@Bo2], in when the Dress construction induces a family of endofunctors on the category ${\ensuremath{\mathbf{Grn}}}$ of Green functors. That is to say, when is there a natural structure of Green functor on $A_Y=D(Y,A)$ if $A$ is a Green functor? Since $A_Y$ is the composite $$\xymatrix@1{ {\ensuremath{\mathscr{T}}}\ar[r]^{-{\ensuremath{\otimes}}Y} & {\ensuremath{\mathscr{T}}}\ar[r]^A & {\ensuremath{\mathbf{Mod}}}_k}$$ with $A$ monoidal, what we require is that $-{\ensuremath{\otimes}}Y$ should be monoidal (since monoidal functors compose). For this we use Theorem 3.7 of [@DPS]: *if $Y$ is a monoid in the lax centre ${\ensuremath{\mathcal{Z}}}_l({\ensuremath{\mathscr{T}}})$ of ${\ensuremath{\mathscr{T}}}$ then $-{\ensuremath{\otimes}}Y: {\ensuremath{\mathscr{T}}}{\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathscr{T}}}$ is canonically monoidal.* Let ${\ensuremath{\mathscr{C}}}$ be a monoidal category. The lax centre ${\ensuremath{\mathcal{Z}}}_l({\ensuremath{\mathscr{C}}})$ of ${\ensuremath{\mathscr{C}}}$ is defined to have objects the pairs $(A, u)$ where $A$ is an object of ${\ensuremath{\mathscr{C}}}$ and $u$ is a natural family of morphisms $u_B: A {\ensuremath{\otimes}}B {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}B {\ensuremath{\otimes}}A$ such that the following two diagrams commute $$\vcenter{\xygraph{ {A {\ensuremath{\otimes}}B {\ensuremath{\otimes}}C}="s" (:[r(3.2)] {B {\ensuremath{\otimes}}C {\ensuremath{\otimes}}A}="t"^-{u_{B {\ensuremath{\otimes}}C}}, :[d(1.2)r(1.6)] {B {\ensuremath{\otimes}}A {\ensuremath{\otimes}}C}_-{u_B {\ensuremath{\otimes}}1_C} :"t" _-{1_B {\ensuremath{\otimes}}u_C} )}} \qquad \vcenter{\xygraph{ {A {\ensuremath{\otimes}}I}="s" (:[r(3.2)] {I {\ensuremath{\otimes}}A}^-{u_I} :[d(1.2)l(1.6)] {A ~.}="t"^-{\cong}, :"t" _-{\cong} )}}$$ Morphisms of ${\ensuremath{\mathcal{Z}}}_l({\ensuremath{\mathscr{C}}})$ are morphisms in ${\ensuremath{\mathscr{C}}}$ compatible with the $u$. The tensor product is defined by $$(A,u) {\ensuremath{\otimes}}(B,v) = (A {\ensuremath{\otimes}}B, w)$$ where $w_C=(u_C {\ensuremath{\otimes}}1_B) {\ensuremath{\circ}}(1_A {\ensuremath{\otimes}}v_C)$. The centre ${\ensuremath{\mathcal{Z}}}({\ensuremath{\mathscr{C}}})$ of ${\ensuremath{\mathscr{C}}}$ consists of the objects $(A,u)$ of ${\ensuremath{\mathcal{Z}}}_l({\ensuremath{\mathscr{C}}})$ with each $u_B$ invertible. It is pointed out in [@DPS] that, when ${\ensuremath{\mathscr{C}}}$ is cartesian monoidal, an object of ${\ensuremath{\mathcal{Z}}}_l({\ensuremath{\mathscr{C}}})$ is merely an object $A$ of ${\ensuremath{\mathscr{C}}}$ together with a natural family $A \times X {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}X$. Then we have the natural bijections below (represented by horizontal lines) for ${\ensuremath{\mathscr{C}}}$ cartesian closed: $$\infer{A {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}\displaystyle{\int_X [X, X] } ~ ~ \text{in} ~ {\ensuremath{\mathscr{C}}}.}{ \infer{A {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}[X, X] ~ ~ \text{dinatural in}~ X} {A \times X {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}X ~ ~ \text{natural in}~ X}}$$ Therefore we obtain an equivalence ${\ensuremath{\mathcal{Z}}}_l({\ensuremath{\mathscr{C}}}) \simeq {\ensuremath{\mathscr{C}}}/ \int_X [X, X] .$ The internal hom in ${\ensuremath{\mathscr{E}}}$, the category of finite $G$-sets for the finite group $G$, is $[X,Y]$ which is the set of functions $r: X {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}Y$ with $(g.r)(x)=gr(g^{-1}x)$. The $G$-set $\int_X [X, X]$ is defined by $$\int_X [X, X] = \biggl \lbrace r=(r_X: X \longrightarrow X) \quad \Big \rvert~ f {\ensuremath{\circ}}r_X = r_Y {\ensuremath{\circ}}f~ \text{for all}~ G \text{-maps}~ f: X \longrightarrow Y \biggr \rbrace$$ with $(g.r)_X(x)=gr_X(g^{-1}x)$. \[5.1\] The $G$-set $\displaystyle \int_X [X, X]$ is isomorphic to $G_c$, which is the set $G$ made a $G$-set by conjugation action. Take $r \in \int_X [X,X]$. Then we have the commutative square $$\xygraph{ {G}="a" (:[r(2.2)] {G}^-{r_G} :[d(1.2)] {X}="t"^-{\hat{x}}, :[d(1.2)] {X}_-{\hat{x}} :"t"_-{r_X} )}$$ where $\hat{x}(g)=gx$ for $x \in X$. So we see that $r_X$ is determined by $r_G(1)$ and $$\begin{split} (g.r)_G(1) & = gr_G(g^{-1}1) \\ & = gr_G(g^{-1})\\ & = gr_G(1)g^{-1}. \end{split}$$ As a consequence of this Lemma, we have ${\ensuremath{\mathcal{Z}}}_l({\ensuremath{\mathscr{E}}}) \simeq {\ensuremath{\mathscr{E}}}/ G_c$ where ${\ensuremath{\mathscr{E}}}/ G_c$ is the category of crossed $G$-sets of Freyd-Yetter ([@FY1], [@FY2]) who showed that ${\ensuremath{\mathscr{E}}}/ G_c$ is a braided monoidal category. Objects are pairs $(X, |~|)$ where $X$ is a $G$-set and $|~|: X {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}G_c$ is a $G$-set morphism (“equivariant function”) meaning $|gx|=g|x|g^{-1}$ for $g \in G$, $x \in X$. The morphisms $f: (X, |~|) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}(Y, |~|)$ are functions $f$ such that the following diagram commutes. $$\xygraph{ {X}="s" (:[r(2.8)] {Y}^-{f} :[d(1.2)l(1.4)] {G_c}="t"^-{|~|}, :"t" _-{|~|} )}$$ That is, $f(gx)=gf(x)$. Tensor product is defined by $$(X,|~|) {\ensuremath{\otimes}}(Y,|~|) = (X \times Y, \Vert ~ \Vert),$$ where $\Vert (x,y) \Vert = |x| |y|$. \[5.2\] [@DPS Theorem 4.5] The centre ${\ensuremath{\mathcal{Z}}}({\ensuremath{\mathscr{E}}})$ of the category ${\ensuremath{\mathscr{E}}}$ is equivalent to the category ${\ensuremath{\mathscr{E}}}/ G_c$ of crossed $G$-sets. We have a fully faithful functor ${\ensuremath{\mathcal{Z}}}({\ensuremath{\mathscr{E}}}) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathcal{Z}}}_l({\ensuremath{\mathscr{E}}})$ and so ${\ensuremath{\mathcal{Z}}}({\ensuremath{\mathscr{E}}}) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathscr{E}}}/ G_c$. On the other hand, let $|~|: A {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}G_c$ be an object of ${\ensuremath{\mathscr{E}}}/ G_c$; so $|ga|g=g|a|$. Then the corresponding object of ${\ensuremath{\mathcal{Z}}}_l({\ensuremath{\mathscr{E}}})$ is $(A,u)$ where $$u_X: A \times X {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}X \times A$$ with $$u_X(a,x) = (|a|x, a).$$ However this $u$ is invertible since we see that $${u_X}^{-1}(x, a) = (a, |a|^{-1}x).$$ This proves the proposition. If $Y$ is a monoid in ${\ensuremath{\mathscr{E}}}/ G_c$ and $A$ is a Green functor for ${\ensuremath{\mathscr{E}}}$ over $k$ then $A_{Y}$ is a Green functor for ${\ensuremath{\mathscr{E}}}$ over $k$, where $A_{Y}(X)=A(X \times Y)$. We have ${\ensuremath{\mathcal{Z}}}({\ensuremath{\mathscr{E}}}) \simeq {\ensuremath{\mathscr{E}}}/ G_c $, so $Y$ is a monoid in ${\ensuremath{\mathcal{Z}}}({\ensuremath{\mathscr{E}}})$. This implies $- \times Y : {\ensuremath{\mathscr{E}}}{\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathscr{E}}}$ is a monoidal functor (see Theorem 3.7 of [@DPS]). It also preserves pullbacks. So $- \times Y : {\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}}) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})$ is a monoidal functor . If $A$ is a Green functor for ${\ensuremath{\mathscr{E}}}$ over $k$ then $A: {\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}}) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathbf{Mod}_k}}$ is monoidal. Then we get $A_{Y} = A {\ensuremath{\circ}}(- \times Y): {\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}}) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathbf{Mod}_k}}$ is monoidal and $A_{Y}$ is indeed a Green functor for ${\ensuremath{\mathscr{E}}}$ over $k$. \[Re8.5\] The reader may have noted that Proposition \[Pro8.1\] implies that $D$ takes monoids to monoids. A monoid in ${\ensuremath{\mathscr{T}}}{\ensuremath{\otimes}}{\ensuremath{\mathbf{Mky}}}$ is a pair $(Y,A)$ where $Y$ is a monoid in ${\ensuremath{\mathscr{T}}}$ and $A$ is a Green functor; so in this case, we have that $A_Y$ is a Green functor. A monoid $Y$ in ${\ensuremath{\mathscr{E}}}$ is certainly a monoid in ${\ensuremath{\mathscr{T}}}$. Since ${\ensuremath{\mathscr{E}}}$ is cartesian monoidal (and so symmetric), each monoid in ${\ensuremath{\mathscr{E}}}$ gives one in the centre. However, not every monoid in the centre arises in this way. The full result behind Proposition \[Pro8.1\] and the centre situation is: the Dress construction $$D : {\ensuremath{\mathcal{Z}}}({\ensuremath{\mathscr{T}}}) {\ensuremath{\otimes}}{\ensuremath{\mathbf{Mky}}}{\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathbf{Mky}}}$$ is a strong monoidal ${\ensuremath{\mathscr{V}}}$ -functor; it is merely monoidal when the centre is replaced by the lax centre. It follows that $A_Y$ is a Green functor whenever $A$ is a Green functor and $Y$ is a monoid in the lax centre of ${\ensuremath{\mathscr{T}}}$. Finite dimensional Mackey functors {#Se9} ================================== We make the following further assumptions on the symmetric compact closed category ${\ensuremath{\mathscr{T}}}$ with finite direct sums: - there is a finite set ${\ensuremath{\mathscr{C}}}$ of objects of ${\ensuremath{\mathscr{T}}}$ such that every object $X$ of ${\ensuremath{\mathscr{T}}}$ can be written as a direct sum $$X \cong \bigoplus_{i=1}^{n} C_i$$ with $C_i$ in ${\ensuremath{\mathscr{C}}}$; and - each hom ${\ensuremath{\mathscr{T}}}(X,Y)$ is a finitely generated commutative monoid. Notice that these assumptions hold when ${\ensuremath{\mathscr{T}}}={\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})$ where ${\ensuremath{\mathscr{E}}}$ is the category of finite $G$-sets for a finite group $G$. In this case we can take ${\ensuremath{\mathscr{C}}}$ to consist of a representative set of connected (transitive) $G$-sets. Moreover, the spans $S: X {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}Y$ with $S \in {\ensuremath{\mathscr{C}}}$ generate the monoid ${\ensuremath{\mathscr{T}}}(X,Y)$. We also fix $k$ to be a field and write ${\ensuremath{\mathbf{Vect}}}$ in place of ${\ensuremath{\mathbf{Mod}}}_k$. A Mackey functor $M: {\ensuremath{\mathscr{T}}}{\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathbf{Vect}}}$ is called *finite dimensional* when each $M(X)$ is a finite-dimensional vector space. Write ${\ensuremath{\mathbf{Mky}_\textit{fin}}}$ for the full subcategory of ${\ensuremath{\mathbf{Mky}}}$ consisting of these. We regard ${\ensuremath{\mathscr{C}}}$ as a full subcategory of ${\ensuremath{\mathscr{T}}}$. The inclusion functor ${\ensuremath{\mathscr{C}}}{\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathscr{T}}}$ is dense and the density colimit presentation is preserved by all additive $M: {\ensuremath{\mathscr{T}}}{\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathbf{Vect}}}$. This is shown as follows: $$\begin{aligned} \int^C {\ensuremath{\mathscr{T}}}(C,X) {\ensuremath{\otimes}}M(C) & \cong \int^C {\ensuremath{\mathscr{T}}}(C,\bigoplus_{i=1}^{n} C_i) {\ensuremath{\otimes}}M(C) \\ & \cong \bigoplus_{i=1}^{n} \int^C {\ensuremath{\mathscr{T}}}(C,C_i) {\ensuremath{\otimes}}M(C) \\ & \cong \bigoplus_{i=1}^{n} \int^C {\ensuremath{\mathscr{C}}}(C,C_i) {\ensuremath{\otimes}}M(C) \\ & \cong \bigoplus_{i=1}^{n} M(C_i) \\ & \cong M(\bigoplus_{i=1}^{n} C_i) \\ &\cong M(X).\end{aligned}$$ That is, $$M \cong \int^C {\ensuremath{\mathscr{T}}}(C,-) {\ensuremath{\otimes}}M(C).$$ The tensor product of finite-dimensional Mackey functors is finite dimensional. Using the last isomorphism, we have $$\begin{split} (M * N)(Z) & = \int^{X,Y} {\ensuremath{\mathscr{T}}}(X {\ensuremath{\otimes}}Y, Z) {\ensuremath{\otimes}}M(X) {\ensuremath{\otimes}}_k N(Y) \\ & \cong \int^{X,Y,C,D} {\ensuremath{\mathscr{T}}}(X {\ensuremath{\otimes}}Y,Z) {\ensuremath{\otimes}}{\ensuremath{\mathscr{T}}}(C,X) {\ensuremath{\otimes}}{\ensuremath{\mathscr{T}}}(D,Y) {\ensuremath{\otimes}}M(C) {\ensuremath{\otimes}}_k N(D) \\ & \cong \int^{C,D} {\ensuremath{\mathscr{T}}}(C {\ensuremath{\otimes}}D,Z) {\ensuremath{\otimes}}M(C) {\ensuremath{\otimes}}_k N(D) . \end{split}$$ If $M$ and $N$ are finite dimensional then so is the vector space ${\ensuremath{\mathscr{T}}}(C{\ensuremath{\otimes}}D,Z){\ensuremath{\otimes}}M(C) {\ensuremath{\otimes}}_k N(D)$ (since ${\ensuremath{\mathscr{T}}}(C {\ensuremath{\otimes}}D,Z)$ is finitely generated). Also the coend is a quotient of a finite direct sum. So $M*N$ is finite dimensional. It follows that ${\ensuremath{\mathbf{Mky}_\textit{fin}}}$ is a monoidal subcategory of ${\ensuremath{\mathbf{Mky}}}$ (since the Burnside functor $J$ is certainly finite dimensional under our assumptions on ${\ensuremath{\mathscr{T}}}$). The promonoidal structure on ${\ensuremath{\mathbf{Mky}_\textit{fin}}}$ represented by this monoidal structure can be expressed in many ways: $$\begin{split} P(M,N;L) & = {\ensuremath{\mathbf{Mky}_\textit{fin}}}(M*N, L) \\ & \cong \text{Nat}_{X,Y,Z} ({\ensuremath{\mathscr{T}}}(X {\ensuremath{\otimes}}Y, Z) {\ensuremath{\otimes}}M(X) {\ensuremath{\otimes}}_k N(Y),L(Z)) \\ & \cong \text{Nat}_{X,Y} (M(X) {\ensuremath{\otimes}}_k N(Y),L(X {\ensuremath{\otimes}}Y)) \\ & \cong \text{Nat}_{X,Z} (M(X) {\ensuremath{\otimes}}_k N(X^* {\ensuremath{\otimes}}Z),L(Z)) \\ & \cong \text{Nat}_{Y,Z} (M(Z {\ensuremath{\otimes}}Y^*) {\ensuremath{\otimes}}_k N(Y),L(Z)). \end{split}$$ Following the terminology of [@DS], we say that a promonoidal category ${\ensuremath{\mathscr{Ms}}}$ is *$*$-autonomous* when it is equipped with an equivalence $S: {\ensuremath{\mathscr{Ms}}}^{\text{op}} {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathscr{Ms}}}$ and a natural isomorphism $$P(M, N; S(L)) \cong P(N, L; S^{-1}(M)).$$ A monoidal category is $*$-autonomous when the associated promonoidal category is. As an application of the work of Day [@Da4] we obtain that ${\ensuremath{\mathbf{Mky}_\textit{fin}}}$ is $*$-autonomous. We shall give the details. For $M \in {\ensuremath{\mathbf{Mky}_\textit{fin}}}$, define $S(M)(X)=M(X^*)^*$ so that $S: {\ensuremath{\mathbf{Mky}_\textit{fin}}}^{\text{op}} {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathbf{Mky}_\textit{fin}}}$ is its own inverse equivalence. The monoidal category ${\ensuremath{\mathbf{Mky}_\textit{fin}}}$ of finite-dimensional Mackey functors on ${\ensuremath{\mathscr{T}}}$ is $*$-autonomous. With $S$ defined as above, we have the calculation: $$\begin{split} P(M,N;S(L)) & \cong \text{Nat}_{X,Y} (M(X) {\ensuremath{\otimes}}_k N(Y),L(X^* {\ensuremath{\otimes}}Y^*)^*) \\ & \cong \text{Nat}_{X,Y} (N(Y) {\ensuremath{\otimes}}_k L(X^* {\ensuremath{\otimes}}Y^*), M(X)^*) \\ & \cong \text{Nat}_{Z,Y} (N(Y) {\ensuremath{\otimes}}_k L(Z {\ensuremath{\otimes}}Y^*), M(Z^*)^*) \\ & \cong \text{Nat}_{Z,Y} (N(Y) {\ensuremath{\otimes}}_k L(Z {\ensuremath{\otimes}}Y^*), S(M)(Z)) \\ &\cong P(N,L;S(M)). \end{split}$$ Cohomological Mackey functors {#Se10} ============================= Let $k$ be a field and $G$ be a finite group. We are interested in the relationship between ordinary $k$-linear representations of $G$ and Mackey functors on $G$. Write ${\ensuremath{\mathscr{E}}}$ for the category of finite $G$-sets as usual. Write ${\ensuremath{\mathscr{R}}}$ for the category ${\ensuremath{\mathbf{Rep}}}_k(G)$ of finite -dimensional $k$-linear representations of $G$. Each $G$-set $X$ determines a $k$-linear representation $kX$ of $G$ by extending the action of $G$ linearly on $X$. This gives a functor $$k: {\ensuremath{\mathscr{E}}}{\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathscr{R}}}.$$ We extend this to a functor $$k_* : {\ensuremath{\mathscr{T}}}^{\text{op}} {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathscr{R}}},$$ where ${\ensuremath{\mathscr{T}}}= {\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})$, as follows. On objects $X \in {\ensuremath{\mathscr{T}}}$, define $$k_*X=kX.$$ For a span $(u,S,v): X {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}Y$ in ${\ensuremath{\mathscr{E}}}$, the linear function $k_*(S):kY {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}kX$ is defined by $$k_*(S)(y) = \sum_{v(s)=y} u(s) ~;$$ this preserves the $G$-actions since $$k_*(S)(gy) = \sum_{v(s)=gy} u(s) = \sum_{v(g^{-1}s)=y} gu(g^{-1}s) ~ = ~ gk_*(S)(y).$$ Clearly $k_*$ preserves coproducts. By the usual argument (going back to Kan, and the geometric realization and singular functor adjunction), we obtain a functor $$\widetilde{k_*}: {\ensuremath{\mathscr{R}}}{\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathbf{Mky}}}(G)_{\textit{fin}}$$ defined by $$\widetilde{k_*}(R)= {\ensuremath{\mathscr{R}}}(k_*-,R)$$ which we shall write as $R^{-}: {\ensuremath{\mathscr{T}}}{\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathbf{Vect}}}_k$. So $$R^X= {\ensuremath{\mathscr{R}}}(k_*X,R) \cong {\ensuremath{G\text{-}\mathbf{Set}}}(X,R)$$ with the effect on the span $(u,S,v): X {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}Y$ transporting to the linear function $${\ensuremath{G\text{-}\mathbf{Set}}}(X,R) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{G\text{-}\mathbf{Set}}}(Y,R)$$ which takes $\tau: X {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}R$ to $\tau_S: Y {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}R$ where $$\tau_S(y) = \sum_{v(s)=y} \tau(u(s)).$$ The functor $\widetilde{k_*}$ has a left adjoint $$\textit{colim}(-,k_*): {\ensuremath{\mathbf{Mky}}}(G)_{\textit{fin}} {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathscr{R}}}$$ defined by $$\textit{colim}(M,k_*) = \int^C M(C) {\ensuremath{\otimes}}_k k_*C$$ where $C$ runs over a full subcategory ${\ensuremath{\mathscr{C}}}$ of ${\ensuremath{\mathscr{T}}}$ consisting of a representative set of connected $G$-sets. The functor $\widetilde{k_*}: {\ensuremath{\mathbf{Rep}}}_k(G) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathbf{Mky}}}(G)$ is fully faithful. For $R_1, R_2 \in {\ensuremath{\mathscr{R}}}$, a morphism $\theta: R_1^- {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}R_2^-$ in ${\ensuremath{\mathbf{Mky}}}(G)$ is a family of linear functions $\theta_X$ such that the following square commutes for all spans $(u,S,v): X {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}Y$ in ${\ensuremath{\mathscr{E}}}$. $$\xygraph{ {{\ensuremath{G\text{-}\mathbf{Set}}}(X,R_1)} (:[r(3)] {{\ensuremath{G\text{-}\mathbf{Set}}}(X,R_2)}^-{\theta_X} :[d(1.4)] {{\ensuremath{G\text{-}\mathbf{Set}}}(Y,R_2)}="t"^-{(~-~)_S}, :[d(1.4)] {{\ensuremath{G\text{-}\mathbf{Set}}}(Y,R_1)}_-{(~-~)_S} :"t"_-{\theta_Y} )}$$ Since $G$ (with multiplication action) forms a full dense subcategory of ${\ensuremath{G\text{-}\mathbf{Set}}}$, it follows that we obtain a unique morphism $f: R_1 {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}R_2$ in ${\ensuremath{G\text{-}\mathbf{Set}}}$ such that $$f(r)= \theta_G(\hat{r})(1)$$ (where $\hat{r}: G {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}R$ is defined by $\hat{r}(g)=gr$ for $r \in R$); this is a special case of Yoneda’s Lemma. Clearly $f$ is linear since $\theta_G$ is. By taking $Y=G, S=G$ and $v=1_G: G {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}G$, commutativity of the above square yields $$\theta_X(\tau)(x) = f(\tau(x));$$ that is, $\theta_X = \widetilde{k_*}(f)_X$. An important property of Mackey functors in the image of $\widetilde{k_*}$ is that they are *cohomological* in the sense of [@We], [@Bo4] and [@TW]. First we recall some classical terminology associated with a Mackey functor $M$ on a group $G$. For subgroups $K \leq H$ of $G$, we have the canonical $G$-set morphism $\sigma^H_K: G/K {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}G/H$ defined on the connected $G$-sets of left cosets by $\sigma^H_K(gK)=gH$. The linear functions $$\begin{gathered} r^H_K=M_*(\sigma^H_K): M(G/H) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}M(G/K) \quad \text{and} \\ t^H_K=M^*(\sigma^H_K): M(G/K) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}M(G/H) \qquad \quad\end{gathered}$$ are called *restriction* and *transfer* (or *trace* or *induction*). A Mackey functor $M$ on $G$ is called *cohomological* when each composite $t^H_K r^H_K: M(G/H)$ $ {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}M(G/H)$ is equal to multiplication by the index $[H:K]$ of $K$ in $H$. We supply a proof of the following known example. For each $k$-linear representation $R$ of $G$, the Mackey functor $\widetilde{k_*}(R)=R^-$ is cohomological. With $M=R^-$ and $\sigma=\sigma^H_K$, notice that the function $$t^H_Kr^H_K = M^*(\sigma)M_*(\sigma) = M(\sigma, G/K,1)M(1,G/K, \sigma) =M(\sigma, G/K, \sigma)$$ takes $\tau \in {\ensuremath{\mathscr{E}}}(G/H,R)$ to $\tau_{G/K} \in {\ensuremath{\mathscr{E}}}(G/H,R)$ where $$\tau_{G/K}(H) = \sum_{\sigma(s)=H} \tau(\sigma(s)) = \sum_{\sigma(s)=H} \tau(H) = \quad (\sum_{\sigma(s)=H} 1) \tau(H)$$ and $s$ runs over the distinct $gK$ with $\sigma(s)=gH=H$; the number of distinct $gK$ with $g \in H$ is of course $[H:K]$. So $\tau_{G/K}(xH)=[H:K] \tau(xH)$. \[10.3\] The functor $k_*: {\ensuremath{\mathscr{T}}}^{\text{op}} {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathscr{R}}}$ is strong monoidal. Clearly the canonical isomorphisms $$k(X_1 \times X_2) \cong kX_1 {\ensuremath{\otimes}}kX_2, \quad k1\cong k$$ show that $k:{\ensuremath{\mathscr{E}}}{\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathscr{R}}}$ is strong monoidal. All that remains to be seen is that these isomorphisms are natural with respect to spans $(u_1,S_1,v_1):X_1 {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}Y_1, (u_2,S_2,v_2):X_2 {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}Y_2$. This comes down to the bilinearity of tensor product: $$\underset{v_2(s_2)=y_2}{\underset{v_1(s_1)=y_1}{\sum}} u_1(s_1){\ensuremath{\otimes}}u_2(s_2) = \sum_{v_1(s_1)=y_1} u_1(y_1) {\ensuremath{\otimes}}\sum_{v_2(s_2)=y_2} u_2(y_2).$$ We can now see that the adjunction $$\xymatrix@C=13pt{\text{\emph{colim}}(-,k_*) & \ar@{|-}[l] \widetilde{k_*}}$$ fits the situation of Day’s Reflection Theorem [@Da2] and [@Da3 pages 24 and 25]. For this, recall that a fully faithful functor $\Phi: {\ensuremath{\mathscr{A}}}{\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathscr{X}}}$ into a closed category ${\ensuremath{\mathscr{X}}}$ is said to be *closed under exponentiation* when, for all $A$ in ${\ensuremath{\mathscr{A}}}$ and $X$ in ${\ensuremath{\mathscr{X}}}$, the internal hom $[X,\Phi A]$ is isomorphic to an object of the form $\Phi B$ for some $B$ in ${\ensuremath{\mathscr{A}}}$. The functor $\textit{colim}(-,k_*): {\ensuremath{\mathbf{Mky}}}(G)_{\text{fin}} {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathscr{R}}}$ is strong monoidal. Consequently, $\widetilde{k_*}: {\ensuremath{\mathscr{R}}}{\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathbf{Mky}}}(G)_{\text{fin}}$ is monoidal and closed under exponentiation. The first sentence follows quite formally from Lemma \[10.3\] and the theory of Day convolution; the main calculation is: $$\begin{aligned} \text{\emph{colim}}(M*N,k_*)(Z) & = \int^C (M*N)(C) {\ensuremath{\otimes}}_k k_* C \\ & = \int^{C,X,Y} {\ensuremath{\mathscr{T}}}(X \times Y,C) {\ensuremath{\otimes}}M(X) {\ensuremath{\otimes}}_k N(Y) {\ensuremath{\otimes}}_k k_*C \\ & \cong \int^{X,Y} M(X) {\ensuremath{\otimes}}_k N(Y) {\ensuremath{\otimes}}_k k_*(X \times Y) \\ & \cong \int^{X,Y} M(X) {\ensuremath{\otimes}}_k N(Y) {\ensuremath{\otimes}}_k k_*X {\ensuremath{\otimes}}k_*Y \\ & \cong \text{\emph{colim}}(M,k_*) {\ensuremath{\otimes}}\text{\emph{colim}}(N,k_*).\end{aligned}$$ The second sentence then follows from [@Da2 Reflection Theorem]. In fancier words, the adjunction $$\xymatrix@C=13pt{\text{\emph{colim}}(-,k_*) & \ar@{|-}[l] \widetilde{k_*}}$$ lives in the $2$-category of monoidal categories, monoidal functors and monoidal natural transformations (all enriched over ${\ensuremath{\mathscr{V}}}$). Mackey functors for Hopf algebras {#Se11} ================================= In this section we provide another example of a compact closed category ${\ensuremath{\mathscr{T}}}$ constructed from a Hopf algebara $H$ (or *quantum group*). We speculate that Mackey functors on this ${\ensuremath{\mathscr{T}}}$ will prove as useful for Hopf algebras as usual Mackey functors have for groups. Let $H$ be a braided (semisimple) Hopf algebra (over $k$). Let ${\ensuremath{\mathscr{R}}}$ denote the category of left $H$-modules which are finite dimensional as vector spaces (over $k$). This is a compact closed braided monoidal category. We write ${\ensuremath{\mathbf{Comod}}}({\ensuremath{\mathscr{R}}})$ for the category obtained from the bicategory of that name in [@DMS] by taking isomorphisms classes of morphisms. Explicitly, the objects are comonoids $C$ in ${\ensuremath{\mathscr{R}}}$. The morphisms are isomorphism classes of comodules $S : \xymatrix@1@C=15pt{C \ar[r] |-{\object@{|}} & D}$ from $C$ to $D$; such an $S$ is equipped with a coaction $\delta: S {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}C {\ensuremath{\otimes}}S {\ensuremath{\otimes}}D$ satisfying the coassociativity and counity conditions; we can break the two-sided coaction $\delta$ into a left coaction $\delta_l:S {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}C {\ensuremath{\otimes}}S$ and a right coaction $\delta_r:S {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}S {\ensuremath{\otimes}}D$ connected by the bicomodule condition. Composition of comodules $S : \xymatrix@1@C=15pt{C \ar[r] |-{\object@{|}} & D}$ and $T : \xymatrix@1@C=15pt{D \ar[r] |-{\object@{|}} & E}$ is defined by the (coreflexive) equalizer $$\xymatrix@C=7ex{ {S{\ensuremath{\otimes}}_D T} \ar@<0.1ex>[r] & {S {\ensuremath{\otimes}}T} \ar@<0.7ex>[r]^-{1 {\ensuremath{\otimes}}\delta_l} \ar@<-0.7ex>[r]_-{\delta_r {\ensuremath{\otimes}}1} & {S {\ensuremath{\otimes}}D {\ensuremath{\otimes}}T~.} }$$ The identity comodule of $C$ is $C : \xymatrix@1@C=15pt{C \ar[r] |-{\object@{|}} & C}$. The category ${\ensuremath{\mathbf{Comod}}}({\ensuremath{\mathscr{R}}})$ is compact closed: the tensor product is just that for vector spaces equipped with the extra structure. Direct sums in ${\ensuremath{\mathbf{Comod}}}({\ensuremath{\mathscr{R}}})$ are given by direct sum as vector spaces. Consequently, ${\ensuremath{\mathbf{Comod}}}({\ensuremath{\mathscr{R}}})$ is enriched in the monoidal category ${\ensuremath{\mathscr{V}}}$ of commutative monoids: to add comodules $S_1 : \xymatrix@1@C=15pt{C \ar[r] |-{\object@{|}} & D}$ and $S_2 : \xymatrix@1@C=15pt{C \ar[r] |-{\object@{|}} & D}$, we take the direct sum $S_1 \oplus S_2$ with coaction defined as the composite $$\xymatrix@C=7.5ex{ {S_1 \oplus S_2} \ar@<0.1ex>[r]^-{\delta_1\oplus \delta_2} & {C {\ensuremath{\otimes}}S_1 {\ensuremath{\otimes}}D} \oplus {C {\ensuremath{\otimes}}S_2 {\ensuremath{\otimes}}D \cong C {\ensuremath{\otimes}}(S_1 \oplus S_2) {\ensuremath{\otimes}}D}. }$$ We can now apply our earlier theory to the example ${\ensuremath{\mathscr{T}}}={\ensuremath{\mathbf{Comod}}}({\ensuremath{\mathscr{R}}})$. In particular, we call a ${\ensuremath{\mathscr{V}}}$-enriched functor $M: {\ensuremath{\mathbf{Comod}}}({\ensuremath{\mathscr{R}}}) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathbf{Vect}}}_k$ a *Mackey functor on* $H$. In the case where $H$ is the group algebra $kG$ (made Hopf by means of the diagonal $kG {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}k(G \times G) \cong kG {\ensuremath{\otimes}}_k kG)$, a Mackey functor on $H$ is not the same as a Mackey functor on $G$. However, there is a strong relationship that we shall now explain. As usual, let ${\ensuremath{\mathscr{E}}}$ denote the cartesian monoidal category of finite $G$-sets. The functor $k: {\ensuremath{\mathscr{E}}}{\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathscr{R}}}$ is strong monoidal and preserves coreflexive equalizers. There is a monoidal equivalence $${\ensuremath{\mathbf{Comod}}}({\ensuremath{\mathscr{E}}}) \simeq {\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}}),$$ so $k: {\ensuremath{\mathscr{E}}}{\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathscr{R}}}$ induces a strong monoidal ${\ensuremath{\mathscr{V}}}$-functor $$\hat{k}: {\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}}) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathbf{Comod}}}({\ensuremath{\mathscr{R}}}).$$ With ${\ensuremath{\mathbf{Mky}}}(G) =[{\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}}), {\ensuremath{\mathbf{Vect}}}]_+$ as usual and with ${\ensuremath{\mathbf{Mky}}}(kG)= [{\ensuremath{\mathbf{Comod}}}({\ensuremath{\mathscr{R}}}), {\ensuremath{\mathbf{Vect}}}]_+$, we obtain a functor $$[\hat{k},1]: {\ensuremath{\mathbf{Mky}}}(kG) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathbf{Mky}}}(G)$$ defined by pre-composition with $\hat{k}$. Proposition 1 of [@DS2] applies to yield: The functor $[\hat{k},1]$ has a strong monoidal left adjoint $$\exists_{\hat{k}}: {\ensuremath{\mathbf{Mky}}}(G) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathbf{Mky}}}(kG).$$ The adjunction is monoidal. The formula for $\exists_{\hat{k}}$ is $$\exists_{\hat{k}}(M)(R)= \int^{X \in {\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})} {\ensuremath{\mathbf{Comod}}}({\ensuremath{\mathscr{R}}})(\hat{k}X,R) {\ensuremath{\otimes}}M(X).$$ On the other hand, we already have the compact closed category ${\ensuremath{\mathscr{R}}}$ of finite-dimensional representations of $G$ and the strong monoidal functor $$k_*: {\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})^{\text{op}} {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathscr{R}}}.$$ Perhaps ${\ensuremath{\mathscr{R}}}^{\text{op}} (\simeq {\ensuremath{\mathscr{R}}})$ should be our candidate for ${\ensuremath{\mathscr{T}}}$ rather than the more complicated ${\ensuremath{\mathbf{Comod}}}({\ensuremath{\mathscr{R}}})$. The result of [@DS2] applies also to $k_*$ to yield a monoidal adjunction $$\xymatrix{ [{\ensuremath{\mathscr{R}}}^{\text{op}}, {\ensuremath{\mathbf{Vect}}}] \ar@<-1.2ex>[r]^-{\perp}_-{[k_*,1]} & \ar@<-1.2ex>[l]_-{\exists_{k^*}} {\ensuremath{\mathbf{Mky}}}(G).}$$ Perhaps then, additive functors ${\ensuremath{\mathscr{R}}}^{\text{op}} {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathbf{Vect}}}$ would provide a suitable generalization of Mackey functors in the case of a Hopf algebra $H$. These matters require investigation at a later time. Review of some enriched category theory {#Se12} ======================================= The basic references are [@Ke], [@La] and [@St]. Let ${\ensuremath{\mathbf{\small{COCT}}}}_{\ensuremath{\mathscr{V}}}$ denote the $2$-category whose objects are cocomplete ${\ensuremath{\mathscr{V}}}$-categories and whose morphisms are (weighted-) colimit-preserving ${\ensuremath{\mathscr{V}}}$-functors; the $2$-cells are ${\ensuremath{\mathscr{V}}}$-natural transformations. Every small ${\ensuremath{\mathscr{V}}}$-category ${\ensuremath{\mathscr{C}}}$ determines an object $[{\ensuremath{\mathscr{C}}},{\ensuremath{\mathscr{V}}}]$ of ${\ensuremath{\mathbf{\small{COCT}}}}_{\ensuremath{\mathscr{V}}}$. Let $$Y: {\ensuremath{\mathscr{C}}}^{\text{op}} {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}[{\ensuremath{\mathscr{C}}},{\ensuremath{\mathscr{V}}}]$$ denote the Yoneda embedding: $YU = {\ensuremath{\mathscr{C}}}(U,-)$. For any object ${\ensuremath{\mathscr{X}}}$ of ${\ensuremath{\mathbf{\small{COCT}}}}_{\ensuremath{\mathscr{V}}}$, we have an equivalence of categories $${\ensuremath{\mathbf{\small{COCT}}}}_{\ensuremath{\mathscr{V}}}([{\ensuremath{\mathscr{C}}},{\ensuremath{\mathscr{V}}}],{\ensuremath{\mathscr{X}}}) \simeq [{\ensuremath{\mathscr{C}}}^{\text{op}},{\ensuremath{\mathscr{X}}}]$$ defined by restriction along $Y$. This is expressing the fact that $[{\ensuremath{\mathscr{C}}},{\ensuremath{\mathscr{V}}}]$ is the free cocompletion of ${\ensuremath{\mathscr{C}}}^{\text{op}}$. It follows that, for small ${\ensuremath{\mathscr{V}}}$-categories ${\ensuremath{\mathscr{C}}}$ and ${\ensuremath{\mathscr{D}}}$, we have $$\begin{split} {\ensuremath{\mathbf{\small{COCT}}}}_{\ensuremath{\mathscr{V}}}([{\ensuremath{\mathscr{C}}},{\ensuremath{\mathscr{V}}}],[{\ensuremath{\mathscr{D}}},{\ensuremath{\mathscr{V}}}]) & \simeq [{\ensuremath{\mathscr{C}}}^{\text{op}},[{\ensuremath{\mathscr{D}}},{\ensuremath{\mathscr{V}}}]] \\ & \simeq [{\ensuremath{\mathscr{C}}}^{\text{op}}{\ensuremath{\otimes}}{\ensuremath{\mathscr{D}}},{\ensuremath{\mathscr{V}}}]. \end{split}$$ The way this works is as follows. Suppose $F: {\ensuremath{\mathscr{C}}}^{\text{op}} {\ensuremath{\otimes}}{\ensuremath{\mathscr{D}}}{\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathscr{V}}}$ is a (${\ensuremath{\mathscr{V}}}$-) functor. We obtain a colimit-preserving functor $$\widehat{F}: [{\ensuremath{\mathscr{C}}},{\ensuremath{\mathscr{V}}}] {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}[{\ensuremath{\mathscr{D}}},{\ensuremath{\mathscr{V}}}]$$ by the formula $$\widehat{F}(M)V = \int^{U\in {\ensuremath{\mathscr{C}}}} F(U,V) {\ensuremath{\otimes}}MU$$ where $M \in [{\ensuremath{\mathscr{C}}},{\ensuremath{\mathscr{V}}}]$ and $V\in {\ensuremath{\mathscr{D}}}$. Conversely, given $G: [{\ensuremath{\mathscr{C}}},{\ensuremath{\mathscr{V}}}] {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}[{\ensuremath{\mathscr{D}}},{\ensuremath{\mathscr{V}}}]$, define $${\ensuremath{\overset{\vee}}}{G} : {\ensuremath{\mathscr{C}}}^{\text{op}} {\ensuremath{\otimes}}{\ensuremath{\mathscr{D}}}{\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathscr{V}}}$$ by $${\ensuremath{\overset{\vee}}}{G}(U,V) = G({\ensuremath{\mathscr{C}}}(U,-))V.$$ The main calculations proving the equivalence are as follows: $$\begin{split} {\ensuremath{\overset{\vee}}}{\widehat{F}}(U,V) & = \widehat{F}({\ensuremath{\mathscr{C}}}(U,-))V \\ & \cong \int^{U'}F(U',V) {\ensuremath{\otimes}}{\ensuremath{\mathscr{C}}}(U,U') \\ & \cong F(U,V) \quad \quad \text{by Yoneda; } \end{split}$$ and, $$\begin{split} \widehat{{\ensuremath{\overset{\vee}}}{G}}(M)V & = \int^{U} {\ensuremath{\overset{\vee}}}{G}(U,V){\ensuremath{\otimes}}MU \\ & \cong (\int^{U}G({\ensuremath{\mathscr{C}}}(U,-)) {\ensuremath{\otimes}}MU)V \\ & \cong G(\int^{U}{\ensuremath{\mathscr{C}}}(U,-){\ensuremath{\otimes}}MU)V \quad \text{since $G$ preserves weighted colimits} \\ & \cong G(M)V \quad \quad \text{by Yoneda again.} \end{split}$$ Next we look how composition of $G$s is transported to the $F$s. Take $$F_1: {\ensuremath{\mathscr{C}}}^{\text{op}} {\ensuremath{\otimes}}{\ensuremath{\mathscr{D}}}{\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathscr{V}}}, \quad F_2: {\ensuremath{\mathscr{D}}}^{\text{op}} {\ensuremath{\otimes}}{\ensuremath{\mathscr{E}}}{\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathscr{V}}}$$ so that $\widehat{F_1}$ and $\widehat{F_2}$ are composable: $$\xygraph{ {[{\ensuremath{\mathscr{C}}},{\ensuremath{\mathscr{V}}}]}="a" (:[u(1.2)r(2.2)] {[{\ensuremath{\mathscr{D}}},{\ensuremath{\mathscr{V}}}]}^-{\widehat{F_1}} :[d(1.2)r(2.2)] {[{\ensuremath{\mathscr{E}}},{\ensuremath{\mathscr{V}}}].}="t"^-{\widehat{F_2}}, "a" : @/_4ex/ _-{\widehat{F_2} {\ensuremath{\circ}}\widehat{F_1}} "t" )}$$ Notice that $$\begin{split} (\widehat{F_2} {\ensuremath{\circ}}\widehat{F_1})(M) & =\widehat{F_2}(\widehat{F_1}(M)) \\ & = \int^{V\in {\ensuremath{\mathscr{D}}}} F_2(V,-) {\ensuremath{\otimes}}\widehat{F_1}(M)V \\ & \cong \int^{U,V} F_2(V,-) {\ensuremath{\otimes}}F_1(U,V) {\ensuremath{\otimes}}MU \\ & \cong \int^{U} (\int^{V} F_2(V,-) {\ensuremath{\otimes}}F_1(U,V)) {\ensuremath{\otimes}}MU. \end{split}$$ So we define $F_2 {\ensuremath{\circ}}F_1 : {\ensuremath{\mathscr{C}}}^{\text{op}} {\ensuremath{\otimes}}{\ensuremath{\mathscr{E}}}{\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathscr{V}}}$ by $$\label{1} (F_2 {\ensuremath{\circ}}F_1)(U,W) = \int^{V} F_2(V,W) {\ensuremath{\otimes}}F_1(U,V);$$ the last calculation then yields $$\widehat{F_2} {\ensuremath{\circ}}\widehat{F_1} \cong \widehat{F_2 {\ensuremath{\circ}}F_1}.$$ The identity functor $1_{[{\ensuremath{\mathscr{C}}},{\ensuremath{\mathscr{V}}}]} : [{\ensuremath{\mathscr{C}}},{\ensuremath{\mathscr{V}}}] {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}[{\ensuremath{\mathscr{C}}},{\ensuremath{\mathscr{V}}}]$ corresponds to the hom functor of ${\ensuremath{\mathscr{C}}}$; that is, $${\ensuremath{\overset{\vee}}}{1}_{[{\ensuremath{\mathscr{C}}},{\ensuremath{\mathscr{V}}}]}(U,V) = {\ensuremath{\mathscr{C}}}(U,V).$$ This gives us the bicategory ${\ensuremath{\mathscr{V}\text{-}\mathbf{Mod}}}$. The objects are (small) ${\ensuremath{\mathscr{V}}}$-categories ${\ensuremath{\mathscr{C}}}$. A morphism $F: \xymatrix@1{{\ensuremath{\mathscr{C}}}\ar[r] |-{\object@{/}} & {\ensuremath{\mathscr{D}}}}$ is a ${\ensuremath{\mathscr{V}}}$-functor $F: {\ensuremath{\mathscr{C}}}^{\text{op}} {\ensuremath{\otimes}}{\ensuremath{\mathscr{D}}}{\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathscr{V}}}$; we call this a *module from* ${\ensuremath{\mathscr{C}}}$ *to* ${\ensuremath{\mathscr{D}}}$ (others call it a *left* ${\ensuremath{\mathscr{D}}}$-, *right* ${\ensuremath{\mathscr{C}}}$-*bimodule*). Composition of modules is defined by (\[1\]) above. We can sum up now by saying that $$\widehat{(~)} : {\ensuremath{\mathscr{V}\text{-}\mathbf{Mod}}}{\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathbf{\small{COCT}}}}_{{\ensuremath{\mathscr{V}}}}$$ is a pseudofunctor (= homomorphism of bicategories) taking ${\ensuremath{\mathscr{C}}}$ to $[{\ensuremath{\mathscr{C}}},{\ensuremath{\mathscr{V}}}]$, taking $F: \xymatrix@1{{\ensuremath{\mathscr{C}}}\ar[r] |-{\object@{/}} & {\ensuremath{\mathscr{D}}}}$ to $\widehat{F}$, and defined on $2$-cells in the obious way; moreover, this pseudofunctor is a local equivalence (that is, it is an equivalence on hom-categories): $$\widehat{(~)}: {\ensuremath{\mathscr{V}\text{-}\mathbf{Mod}}}({\ensuremath{\mathscr{C}}},{\ensuremath{\mathscr{D}}}) \simeq {\ensuremath{\mathbf{\small{COCT}}}}_{{\ensuremath{\mathscr{V}}}}([{\ensuremath{\mathscr{C}}},{\ensuremath{\mathscr{V}}}],[{\ensuremath{\mathscr{D}}},{\ensuremath{\mathscr{V}}}]).$$ A monad $T$ on an object ${\ensuremath{\mathscr{C}}}$ of ${\ensuremath{\mathscr{V}\text{-}\mathbf{Mod}}}$ is called a *promonad on* ${\ensuremath{\mathscr{C}}}$. It is the same as giving a colimit-preserving monad $\widehat{T}$ on the ${\ensuremath{\mathscr{V}}}$-category $[{\ensuremath{\mathscr{C}}},{\ensuremath{\mathscr{V}}}]$. One way that promonads arise is from monoids $A$ for some convolution monoidal structure on $[{\ensuremath{\mathscr{C}}},{\ensuremath{\mathscr{V}}}]$; then $$\widehat{T}(M) = A \ast M.$$ That is, ${\ensuremath{\mathscr{C}}}$ is a promonoidal ${\ensuremath{\mathscr{V}}}$-category [@Da]: $$P: {\ensuremath{\mathscr{C}}}^{\text{op}} {\ensuremath{\otimes}}{\ensuremath{\mathscr{C}}}^{\text{op}}{\ensuremath{\otimes}}{\ensuremath{\mathscr{C}}}{\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathscr{V}}}$$ $$J: {\ensuremath{\mathscr{C}}}{\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathscr{V}}}$$ so that $$\widehat{T}(M) = A \ast M = \int^{U,V} P(U,V;-) {\ensuremath{\otimes}}AU {\ensuremath{\otimes}}MV.$$ This means that the module $T: \xymatrix@1{{\ensuremath{\mathscr{C}}}\ar[r] |-{\object@{/}} & {\ensuremath{\mathscr{C}}}}$ is defined by $$\begin{split} T(U,V) & = \widehat{T}({\ensuremath{\mathscr{C}}}(U,-))V \\ & = \int^{U',V'} P(U',V';V) {\ensuremath{\otimes}}AU' {\ensuremath{\otimes}}{\ensuremath{\mathscr{C}}}(U,V') \\ & \cong \int^{U'} P(U',U;V) {\ensuremath{\otimes}}AU'. \end{split}$$ A promonad $T$ on ${\ensuremath{\mathscr{C}}}$ has a unit $\eta: {\ensuremath{\overset{\vee}}}{1} {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}T$ with components $$\eta_{U,V} : {\ensuremath{\mathscr{C}}}(U,V) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}T(U,V)$$ and so is determined by $$\eta_{U,V}(1_U) : I {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}T(U,U),$$ and has a multiplication $\mu: T {\ensuremath{\circ}}T {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}T$ with components $$\mu_{U,W} : \int^{V} T(V,W) {\ensuremath{\otimes}}T(U,V) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}T(U,W)$$ and so is determined by a natural family $$\mu'_{U,V,W}: T(V,W) {\ensuremath{\otimes}}T(U,V) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}T(U,W).$$ The *Kleisli category* ${\ensuremath{\mathscr{C}}}_T$ for the promonad $T$ on ${\ensuremath{\mathscr{C}}}$ has the same objects as ${\ensuremath{\mathscr{C}}}$ and has homs defined by $${\ensuremath{\mathscr{C}}}_T(U,V) = T(U,V);$$ the identites are the $\eta_{U,V}(1_U)$ and the composition is the $\mu'_{U,V,W}$. \[13.1\] $[{\ensuremath{\mathscr{C}}}_T,{\ensuremath{\mathscr{V}}}] \simeq [{\ensuremath{\mathscr{C}}},{\ensuremath{\mathscr{V}}}]^{\widehat{T}}.$ That is, the functor category $[{\ensuremath{\mathscr{C}}}_T,{\ensuremath{\mathscr{V}}}]$ is equivalent to the category of Eilenberg-Moore algebras for the monad $\widehat{T}$ on $[{\ensuremath{\mathscr{C}}},{\ensuremath{\mathscr{V}}}]$. (sketch) To give a $\widehat{T}$-algebra structure on $M\in [{\ensuremath{\mathscr{C}}},{\ensuremath{\mathscr{V}}}]$ is to give a morphism $\alpha: \widehat{T}(M) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}M$ satisfying the two axioms for an action. This is to give a natural family of morphisms $$T(U,V) {\ensuremath{\otimes}}MU {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}MV;$$ but that is to give $$T(U,V) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}[MU,MV];$$ but that is to give $$\label{2} {\ensuremath{\mathscr{C}}}_T(U,V) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathscr{V}}}(MU,MV).$$ Thus we can define a ${\ensuremath{\mathscr{V}}}$-functor $$\overline{M}:{\ensuremath{\mathscr{C}}}_T {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathscr{V}}}$$ which agrees with $M$ on objects and is defined by (\[2\]) on homs; the action axioms are just what is needed for $\overline{M}$ to be a functor. This process can be reversed. Modules over a Green functor {#Se13} ============================ In this section, we present work inspired by Chapters $2, 3$ and $4$ of [@Bo1], casting it in a more categorical framework. Let ${\ensuremath{\mathscr{E}}}$ denote a lextensive category and ${\ensuremath{\mathbf{CMon}}}$ denote the category of commutative monoids; this latter is what we called ${\ensuremath{\mathscr{V}}}$ in earlier sections. The functor $U:{\ensuremath{\mathbf{Mod}_k}}{\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}$ ${\ensuremath{\mathbf{CMon}}}$ (which forgets the action of $k$ on the $k$-module and retains only the additive monoid structure) has a left adjoint $K: {\ensuremath{\mathbf{CMon}}}{\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathbf{Mod}_k}}$ which is strong monoidal for the obvious tensor products on ${\ensuremath{\mathbf{CMon}}}$ and ${\ensuremath{\mathbf{Mod}_k}}$. So each category ${\ensuremath{\mathscr{A}}}$ enriched in ${\ensuremath{\mathbf{CMon}}}$ determines a category $K_*{\ensuremath{\mathscr{A}}}$ enriched in ${\ensuremath{\mathbf{Mod}_k}}$: the objects of $K_*{\ensuremath{\mathscr{A}}}$ are those of ${\ensuremath{\mathscr{A}}}$ and the homs are defined by $$(K_*{\ensuremath{\mathscr{A}}})(A,B) = K{\ensuremath{\mathscr{A}}}(A,B)$$ since ${\ensuremath{\mathscr{A}}}(A,B)$ is a commutative monoid. The point is that a ${\ensuremath{\mathbf{Mod}_k}}$-functor $K_*{\ensuremath{\mathscr{A}}}$ $ {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathscr{B}}}$ is the same as a ${\ensuremath{\mathbf{CMon}}}$-functor ${\ensuremath{\mathscr{A}}}{\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}U_*{\ensuremath{\mathscr{B}}}$. We know that ${\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})$ is a ${\ensuremath{\mathbf{CMon}}}$-category; so we obtain a monoidal ${\ensuremath{\mathbf{Mod}_k}}$-category $${\ensuremath{\mathscr{C}}}= K_*{\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}}).$$ The ${\ensuremath{\mathbf{Mod}_k}}$-category of *Mackey functors* on ${\ensuremath{\mathscr{E}}}$ is $\mathbf{Mky}_k({\ensuremath{\mathscr{E}}}) = [{\ensuremath{\mathscr{C}}},{\ensuremath{\mathbf{Mod}_k}}]$; it becomes monoidal using convolution with the monoidal structure on ${\ensuremath{\mathscr{C}}}$ (see Section \[Se5\]). The ${\ensuremath{\mathbf{Mod}_k}}$-category of *Green functors* on ${\ensuremath{\mathscr{E}}}$ is $\mathbf{Grn}_k({\ensuremath{\mathscr{E}}}) =\mathbf{Mon}[{\ensuremath{\mathscr{C}}},{\ensuremath{\mathbf{Mod}_k}}]$ consisting of the monoids in $[{\ensuremath{\mathscr{C}}},{\ensuremath{\mathbf{Mod}_k}}]$ for the convolution. Let $A$ be a Green functor. A *module* $M$ over the Green functor $A$, or *$A$-module* means $A$ acts on $M$ via the convolution $\ast $. The monoidal action $\alpha^M: A*M {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}M$ is defined by a family of morphisms $$\bar{\alpha}^M_{U,V} : A(U) {\ensuremath{\otimes}}_k M(V) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}M(U \times V) ,$$ where we put $\bar{\alpha}^M_{U,V}(a {\ensuremath{\otimes}}m)=a.m$ for $a \in A(U)$, $m \in M(V)$, satisfing the following commutative diagrams for morphisms $f: U {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}U'$ and $g: V {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}V'$ in ${\ensuremath{\mathscr{E}}}$. $$\vcenter{\xygraph{ {A(U) {\ensuremath{\otimes}}_k M(V)}="a" (:[r(2.5)] {M(U \times V)}^-{\bar{\alpha}^M_{U,V}} :[d(1.2)] {M(U' \times V')}="t"^-{M_*(f \times g)}, :[d(1.2)] {A(U') {\ensuremath{\otimes}}_k M(V')}_-{A_*(f) {\ensuremath{\otimes}}_k M_*(g)} :"t"_-{\bar{\alpha}^M_{U',V'}} )}} \qquad \vcenter{\xygraph{ {M(U)}="s" (:[r(2.3)] {A(1) {\ensuremath{\otimes}}_k M(U)}^-{\eta {\ensuremath{\otimes}}1} :[d(1.2)] {M(1 \times U)}="t"^-{\bar{\alpha}^M}, :"t" _-{\cong} )}}$$ $$\xygraph{ {A(U) {\ensuremath{\otimes}}_k A(V) {\ensuremath{\otimes}}_k M(W)}="a" (:[r(4.0)] {A(U) {\ensuremath{\otimes}}_k M(V \times W)}^-{1 {\ensuremath{\otimes}}\bar{\alpha}^M} :[d(1.2)] {M(U \times V \times W)~.} ="t"^-{\bar{\alpha}^M}, :[d(1.2)] {A(U \times V) {\ensuremath{\otimes}}_k M(W)}_-{\mu {\ensuremath{\otimes}}1} :"t"_-{\bar{\alpha}^M} )}$$ If $M$ is an $A$-module, then $M$ is in particular a Mackey functor. \[10.1\] Let $A$ be a Green functor and $M$ be an $A$-module. Then $M_U$ is an $A$-module for each $U$ of ${\ensuremath{\mathscr{E}}}$, where $M_U(X)=M(X \times U)$. Simply define $\bar{\alpha}^{M_U}_{V,W }=\bar{\alpha}^M_{V,W \times U}$. Let ${\ensuremath{\mathbf{Mod}(A)}}$ denote the category of left $A$-modules for a Green functor $A$. The objects are $A$-modules and morphisms are $A$-module morphisms $\theta: M {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}N$ (that is, morphisms of Mackey functors) satisfying the following commutative diagram. $$\xygraph{ {A(U) {\ensuremath{\otimes}}_k M(V)}="a" (:[r(2.5)] {M(U \times V)}^-{\bar{\alpha}^M_{U,V}} :[d(1.2)] {N(U \times V)}="t"^-{\theta(U \times V)}, :[d(1.2)] {A(U) {\ensuremath{\otimes}}_k N(V)}_-{1 {\ensuremath{\otimes}}_k \theta(U)} :"t"_-{\bar{\alpha}^N_{U,V}} )}$$ The category ${\ensuremath{\mathbf{Mod}(A)}}$ is enriched in ${\ensuremath{\mathbf{Mky}}}$. The homs are given by the equalizer $$\xygraph{ {{\ensuremath{\mathbf{Mod}}}(A)(M,N)}="a" (:[r(2.5)] {{\ensuremath{\mathrm{Hom}}}(M,N)} (:[r(3.4)] {{\ensuremath{\mathrm{Hom}}}(A*M,N)}="t"^-{{\ensuremath{\mathrm{Hom}}}(\alpha^M,1)}, :[d(1.2)r(1.8)] {{\ensuremath{\mathrm{Hom}}}(A*M,A*N)~.}_-{(A*-)} :"t"_-{{\ensuremath{\mathrm{Hom}}}(1, \alpha^N)} ))}$$ Then we see that ${\ensuremath{\mathbf{Mod}(A)}}(M,N)$ is the sub-Mackey functor of ${\ensuremath{\mathrm{Hom}}}(M,N)$ defined by $$\begin{split} {\ensuremath{\mathbf{Mod}(A)}}(M,N)(U)= & \{ \theta \in {\ensuremath{\mathbf{Mky}}}(M(- \times U),N-) \quad \mid~ \theta_{V \times W}(a.m)= a.\theta_W(m) \\ & \quad \text{for all} ~ V,W,~ \text{and} ~ a \in A(V), m \in M(W \times U) \}. \end{split}$$ In particular, if $A=J$ (Burnside functor) then ${\ensuremath{\mathbf{Mod}(A)}}$ is the category of Mackey functors and ${\ensuremath{\mathbf{Mod}(A)}}(M,N)={\ensuremath{\mathrm{Hom}}}(M,N)$. The Green functor $A$ is itself an $A$-module. Then by the Lemma \[10.1\], we see that $A_U$ is an $A$-module for each $U$ in ${\ensuremath{\mathscr{E}}}$. Define a category ${\ensuremath{\mathscr{C}}}_A$ consisting of the objects of the form $A_U$ for each $U$ in ${\ensuremath{\mathscr{C}}}$. This is a full subcategory of ${\ensuremath{\mathbf{Mod}(A)}}$ and we have the following equivalences $${\ensuremath{\mathscr{C}}}_A(U,V) \simeq {\ensuremath{\mathbf{Mod}(A)}}(A_U,A_V) \simeq A(U \times V).$$ In other words, the category ${\ensuremath{\mathbf{Mod}(A)}}$ of left $A$-modules is the category of Eilenberg-Moore algebras for the monad $T=A \ast -$ on $[{\ensuremath{\mathscr{C}}},{\ensuremath{\mathbf{Mod}_k}}]$; it preserves colimits since it has a right adjoint (as usual with convolution tensor products). By the above, the ${\ensuremath{\mathbf{Mod}_k}}$-category ${\ensuremath{\mathscr{C}}}_A$ (technically it is the Kleisli category ${\ensuremath{\mathscr{C}}}_{{\ensuremath{\overset{\vee}}}{T}}$ for the promonad ${\ensuremath{\overset{\vee}}}{T}$ on ${\ensuremath{\mathscr{C}}}$; see Proposition \[13.1\]) satisfies an equivalence $$[{\ensuremath{\mathscr{C}}}_A,{\ensuremath{\mathbf{Mod}_k}}] \simeq {\ensuremath{\mathbf{Mod}(A)}}.$$ Let ${\ensuremath{\mathscr{C}}}$ be a ${\ensuremath{\mathbf{Mod}_k}}$-category with finite direct sums and $\Omega$ be a finite set of objects of ${\ensuremath{\mathscr{C}}}$ such that every object of ${\ensuremath{\mathscr{C}}}$ is a direct sum of objects from $\Omega$. Let $W$ be the algebra of $\Omega \times \Omega$ - matrices whose $(X,Y)$ - entry is a morphism $X {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}Y$ in ${\ensuremath{\mathscr{C}}}$. Then $$W=\left \{ (f_{XY})_{X,Y \in \Omega} ~\mid ~ f_{XY} \in {\ensuremath{\mathscr{C}}}(X,Y) \right \}$$ is a vector space over $k$, and the product is defined by $$(g_{XY})_{X,Y \in \Omega} (f_{XY})_{X,Y \in \Omega} = \Big(\sum_{Y \in \Omega} g_{YZ} {\ensuremath{\circ}}f_{XY}\Big)_{X,Z \in \Omega}.$$ $[{\ensuremath{\mathscr{C}}},{\ensuremath{\mathbf{Mod}_k}}] \simeq {\ensuremath{\mathbf{Mod}_k}}^{W}$ (= the category of left $W$-modules). Put $$P=\bigoplus_{X \in \Omega} {\ensuremath{\mathscr{C}}}(X,-).$$ This is a small projective generator so Exercise F (page 106) of [@Fr] applies and $W$ is identified as End(P). In particular; this applies to the category ${\ensuremath{\mathscr{C}}}_A$ to obtain the *Green algebra* $W_A$ of a Green functor $A$: the point being that $A$ and $W_A$ have the same modules. Morita equivalence of Green functors {#Se14} ==================================== In this section, we look at the Morita theory of Green functors making use of adjoint two-sided modules rather than *Morita contexts* as in [@Bo1]. As for any symmetric cocomplete closed monoidal category ${\ensuremath{\mathscr{W}}},$ we have the monoidal bicategory ${\ensuremath{\mathbf{Mod}}}({\ensuremath{\mathscr{W}}})$ defined as follows, where we take ${\ensuremath{\mathscr{W}}}={\ensuremath{\mathbf{Mky}}}$. Objects are monoids $A$ in ${\ensuremath{\mathscr{W}}}$ (that is, $A: {\ensuremath{\mathscr{E}}}{\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathbf{Mod}_k}}$ are Green functors) and morphisms are modules $M: \xymatrix@1@C=15pt{A \ar[r] |-{\object@{|}} & B}$ (that is, algebras for the monad $A*-*B$ on ${\ensuremath{\mathbf{Mky}}}$) with a two-sided action $$\infer{\bar{\alpha}^M_{U,V,W} : A(U) {\ensuremath{\otimes}}_k M(V) {\ensuremath{\otimes}}_k B(W) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}M(U \times V \times W)} {\alpha^M : A * M * B {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}M}.$$ Composition of morphisms $M: \xymatrix@1@C=15pt{A \ar[r] |-{\object@{|}} & B}$ and $N: \xymatrix@1@C=15pt{B \ar[r] |-{\object@{|}} & C}$ is $M*_B N$ and it is defined via the coequalizer $$\xymatrix@C=7.5ex{ {M*B*N} \ar@<0.7ex>[r]^-{\alpha^M *1_N} \ar@<-0.7ex>[r]_-{1_M *\alpha^N} & {M*N} \ar@<0.1ex>[r] & {M*_B N} = N {\ensuremath{\circ}}M}$$ that is, $$(M*_B N)(U) = \sum_{X,Y} {\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})(X \times Y, U) {\ensuremath{\otimes}}M(X) {\ensuremath{\otimes}}_k N(Y) / \sim_B~.$$ The identity morphism is given by $A: \xymatrix@1@C=15pt{A \ar[r] |-{\object@{|}} & A.}$ The 2-cells are natural transformations $\theta :M {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}M'$ which respect the actions $$\xygraph{ {A(U) {\ensuremath{\otimes}}_k M(V) {\ensuremath{\otimes}}_k B(W)}="a" (:[r(4.5)] {M(U \times V \times W)}^-{\bar{\alpha}^M_{U,V,W}} :[d(1.2)] {M'(U \times V \times W)~.}="t"^-{\theta_{U \times V \times W}}, :[d(1.2)] {A(U) {\ensuremath{\otimes}}_k M'(V) {\ensuremath{\otimes}}_k B(W)}_-{1 {\ensuremath{\otimes}}_k \theta_V {\ensuremath{\otimes}}_k 1} :"t"_-{\bar{\alpha}^{M'}_{U,V,W}} )}$$ The tensor product on ${\ensuremath{\mathbf{Mod}}}({\ensuremath{\mathscr{W}}})$ is the convolution $\ast$. The tensor product of the modules $M: \xymatrix@1@C=15pt{A \ar[r] |-{\object@{|}} & B}$ and $N: \xymatrix@1@C=15pt{C \ar[r] |-{\object@{|}} & D}$ is $M*N: \xymatrix@1@C=15pt{A*C \ar[r] |-{\object@{|}} & B*D}$. Define Green functors $A$ and $B$ to be *Morita equivalent* when they are equivalent in ${\ensuremath{\mathbf{Mod}}}({\ensuremath{\mathscr{W}}})$. If $A$ and $B$ are equivalent in ${\ensuremath{\mathbf{Mod}}}({\ensuremath{\mathscr{W}}})$ then ${\ensuremath{\mathbf{Mod}(A)}}\simeq {\ensuremath{\mathbf{Mod}(B)}}$ as categories. ${\ensuremath{\mathbf{Mod}}}({\ensuremath{\mathscr{W}}})(-,J): {\ensuremath{\mathbf{Mod}}}({\ensuremath{\mathscr{W}}})^{\text{op}} {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}{\ensuremath{\mathbf{CAT}}}$ is a pseudofunctor and so takes equivalences to equivalences. Now we will look at the Cauchy completion of a monoid $A$ in a monoidal category ${\ensuremath{\mathscr{W}}}$ with the unit $J$. The ${\ensuremath{\mathscr{W}}}$-category ${\ensuremath{\mathcal{P} A}}$ has underlying category ${\ensuremath{\mathbf{Mod}}}({\ensuremath{\mathscr{W}}})(J,A)= {\ensuremath{\mathbf{Mod}}}(A^{\text{op}})$ where $A^{\text{op}}$ is the monoid $A$ with commuted multiplication. The objects are modules $M: \xymatrix@1@C=15pt{J \ar[r] |-{\object@{|}} & A}$; that is, right $A$-modules. The homs of ${\ensuremath{\mathcal{P} A}}$ are defined by $({\ensuremath{\mathcal{P} A}})(M,N)= {\ensuremath{\mathbf{Mod}}}(A^{\text{op}})(M,N)$ (see the equalizer of Section \[Se13\]). The Cauchy completion ${\ensuremath{\mathcal{Q} A}}$ of $A$ is the full sub-${\ensuremath{\mathscr{W}}}$-category of ${\ensuremath{\mathcal{P} A}}$ consisting of the modules $M: \xymatrix@1@C=15pt{J \ar[r] |-{\object@{|}} & A}$ with right adjoints $N: \xymatrix@1@C=15pt{A \ar[r] |-{\object@{|}} & J}$. We will examine what the objects of ${\ensuremath{\mathcal{Q} A}}$ are in more explicit terms. For motivation and preparation we will look at the monoidal category ${\ensuremath{\mathscr{W}}}= [{\ensuremath{\mathscr{C}}}, {\ensuremath{\mathscr{S}}}]$ where $({\ensuremath{\mathscr{C}}}, {\ensuremath{\otimes}}, I)$ is a monoidal category and ${\ensuremath{\mathscr{S}}}$ is the cartesian monoidal category of sets. Then $[{\ensuremath{\mathscr{C}}}, {\ensuremath{\mathscr{S}}}]$ becomes a monoidal category by convolution. The tensor product $\ast$ and the unit $J$ are defined by $$\begin{split} (M * N)(U) & = \int^{X,Y} {\ensuremath{\mathscr{C}}}(X {\ensuremath{\otimes}}Y, U) \times M(X) \times N(Y) \\ J(U)& = {\ensuremath{\mathscr{C}}}(I,U). \end{split}$$ Write ${\ensuremath{\mathbf{Mod}}}[{\ensuremath{\mathscr{C}}},{\ensuremath{\mathscr{S}}}]$ for the bicategory whose objects are monoids $A$ in $[{\ensuremath{\mathscr{C}}},{\ensuremath{\mathscr{S}}}]$ and whose morphisms are modules $M: \xymatrix@1@C=15pt{A \ar[r] |-{\object@{|}} & B}$. These modules have two-sided action $$\infer{\bar{\alpha}^M_{X,Y,Z} : A(X) \times M(Y) \times B(Z) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}M(X {\ensuremath{\otimes}}Y {\ensuremath{\otimes}}Z)~.} {\alpha^M : A * M * B {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}M}$$ Composition of morphisms $M: \xymatrix@1@C=15pt{A \ar[r] |-{\object@{|}} & B}$ and $N: \xymatrix@1@C=15pt{B \ar[r] |-{\object@{|}} & C}$ is given by the coequalizer $$\xymatrix@C=7.5ex{ {M*B*N} \ar@<0.7ex>[r]^-{\alpha^M *1_N} \ar@<-0.7ex>[r]_-{1_M *\alpha^N} & {M*N} \ar@<0.1ex>[r] & {M*_B N} }$$ that is, $$(M*_B N)(U) = \sum_{X,Z} {\ensuremath{\mathscr{C}}}(X {\ensuremath{\otimes}}Z, U) \times M(X) \times N(Z) / \sim_B$$ where $$\begin{aligned} (u, m {\ensuremath{\circ}}b, n) & \sim_B (u, m, b {\ensuremath{\circ}}n) \\ (t {\ensuremath{\circ}}(r {\ensuremath{\otimes}}s), m, n) & \sim_B (t, (Mr)m, (Ns)n)\end{aligned}$$ for   $ u: X {\ensuremath{\otimes}}Y {\ensuremath{\otimes}}Z {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}U,~ m \in M(X),~ b \in B(Y),~ n \in N(Z),~ t: X' {\ensuremath{\otimes}}Z' {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}U, ~r: X {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}X',$ $s: Z {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}Z'$. For each $K \in {\ensuremath{\mathscr{C}}}$, we obtain a module $A(K {\ensuremath{\otimes}}-): \xymatrix@1@C=15pt{J \ar[r] |-{\object@{|}} & A}$. The action $$A(K {\ensuremath{\otimes}}U) {\ensuremath{\otimes}}A(V) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}A(K {\ensuremath{\otimes}}U {\ensuremath{\otimes}}V)$$ is defined by the monoid structure on $A$. Every object of the Cauchy completion ${\ensuremath{\mathcal{Q} A}}$ of the monoid $A$ in $[{\ensuremath{\mathscr{C}}}, {\ensuremath{\mathscr{S}}}]$ is a retract of a module of the form $A(K {\ensuremath{\otimes}}-)$ for some $K \in {\ensuremath{\mathscr{C}}}$. Take a module $M: \xymatrix@1@C=15pt{J \ar[r] |-{\object@{|}} & A}$ in ${\ensuremath{\mathbf{Mod}}}[{\ensuremath{\mathscr{C}}}, {\ensuremath{\mathscr{S}}}]$. Suppose that $M$ has a right adjoint $N: \xymatrix@1@C=15pt{A \ar[r] |-{\object@{|}} & J}$. Then we have the following actions: $ A(V) \times A(W) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}A(V {\ensuremath{\otimes}}W)$, $M(V) \times A(W) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}M(V {\ensuremath{\otimes}}W), A(V) \times N(W) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}N(V {\ensuremath{\otimes}}W)$ since $A$ is a monoid, $M$ is a right $A$-module, and $N$ is a left $A$-module respectively. We have a unit $\eta: J {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}M*_A N$ and a counit $\epsilon: N*M {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}A$ for the adjunction. The component $\eta_U: {\ensuremath{\mathscr{C}}}(I,U) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}(M*_A N)(U)$ of the unit $\eta$ is determined by $$\eta' = \eta_U(1_I) \in \sum_{X,Z} {\ensuremath{\mathscr{C}}}(X {\ensuremath{\otimes}}Z, I) \times M(X) \times N(Z) / \sim_A;$$ so there exist $u: H {\ensuremath{\otimes}}K {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}I, \quad p \in M(H), \quad q \in N(K)$ such that $\eta'=[u,p,q]_A$. Then $$\eta_u(f: I {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}U) =[fu: H {\ensuremath{\otimes}}K {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}U, p, q]_A.$$ We also have $\bar{\epsilon}_{Y,Z}: NY \times MZ {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}A(Y {\ensuremath{\otimes}}Z)$ coming from $\epsilon$. The commutative diagram $$\xygraph{ {M(U)}="s" (:[r(5.3)] {\displaystyle {\sum_{X,Y,Z} {\ensuremath{\mathscr{C}}}(X {\ensuremath{\otimes}}Y {\ensuremath{\otimes}}Z, U) \times M(X) \times N(Y) \times M(Z) / \sim}}^-{\eta_U * 1} :[d(1.8)] {M(U)}="t"^-{1* \epsilon_U}, :"t" _-{1} )}$$ yields the equations $$\label{3} \begin{split} m & = (1*\epsilon_U)(\eta_U *1)(m) \\ & = (1*\epsilon_U)[u {\ensuremath{\otimes}}1_U, p,q,m]_A \\ & = M(u {\ensuremath{\otimes}}1_U)(p ~ \bar{\epsilon}_{K,U} (q,m)) \end{split}$$ for all $m \in M(U)$. Define $$\xymatrix@C=10ex{M(U) \ar@/_/ [r] _{i_U} & A(K {\ensuremath{\otimes}}U) \ar@<-0.2ex>@/_/ [l] _{r_U}}$$ by $i_U(m)=\bar{\epsilon}_{K,U} (q,m), ~ r_U(a)=M(u {\ensuremath{\otimes}}1_U)(p.a)$. These are easily seen to be natural in $U$. Equation (\[3\]) says that $r {\ensuremath{\circ}}i = 1_M$. So $M$ is a retract of $A(K {\ensuremath{\otimes}}-)$. Now we will look at what are the objects of ${\ensuremath{\mathcal{Q} A}}$ when ${\ensuremath{\mathscr{W}}}$= ${\ensuremath{\mathbf{Mky}}}$ which is a symmetric monoidal closed, complete and cocomplete category. The Cauchy completion ${\ensuremath{\mathcal{Q} A}}$ of the monoid $A$ in ${\ensuremath{\mathbf{Mky}}}$ consists of all the retracts of modules of the form $$\bigoplus^k_{i=1} A(Y_i \times -)$$ for some $Y_i \in {\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})$, $i=1,\ldots, k$. Take a module $M: \xymatrix@1@C=15pt{J \ar[r] |-{\object@{|}} & A}$ in ${\ensuremath{\mathbf{Mod}}}({\ensuremath{\mathscr{W}}})$ and suppose that $M$ has a right adjoint $N: \xymatrix@1@C=15pt{A \ar[r] |-{\object@{|}} & J}$. For the adjunction, we have a unit $\eta: J {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}M*_A N$ and a counit $\epsilon: N*M {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}A$. We write $\eta_U: {\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})(1,U) {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}(M*_A N)(U)$ is the component of the unit $\eta$ and it is determined by $$\eta' = \eta_1(1_1) \in \sum_{i=1}^k {\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})(X \times Y, 1) {\ensuremath{\otimes}}M(X) {\ensuremath{\otimes}}N(Y) / \sim_A.$$ Put $$\eta' = \eta_1(1_1) = \sum_{i=1}^k [(S_i: X_i \times Y_i {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}1) {\ensuremath{\otimes}}m_i {\ensuremath{\otimes}}n_i ]_A$$ where $m_i \in M(X_i)$ and $n_i \in N(Y_i)$. Then $$\eta_U(T: 1 {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}U) = \sum_{i=1}^k [(S_i \times T) {\ensuremath{\otimes}}m_i {\ensuremath{\otimes}}n_i ]_A.$$ We also have $\bar{\epsilon}_{Y,Z}:NY {\ensuremath{\otimes}}MZ {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}A(Y \times Z)$ coming from $\epsilon$. The commutative diagram $$\xygraph{ {M(U)}="s" (:[r(5.3)] {\displaystyle {\sum_{i=1}^k {\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})(X_i \times Y_i \times U, U) {\ensuremath{\otimes}}M(X_i) {\ensuremath{\otimes}}N(Y_i) {\ensuremath{\otimes}}M(U) / \sim_A}}^-{\eta_U * 1} :[d(1.8)] {M(U)}="t"^-{1* \epsilon_U}, :"t" _-{1} )}$$ yields $$m = \sum_{i=1}^k [M(P_i \times U) {\ensuremath{\otimes}}m_i {\ensuremath{\otimes}}\epsilon(n_i {\ensuremath{\otimes}}m) ]$$ where $m \in M(U)$ and $P_i: X_i \times Y_i {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}U$. Define a natural retraction $$\xymatrix@C=10ex{M(U) \ar@<-0.2ex>@/_/ [r] _-{i_U} & {\displaystyle \bigoplus_{i=1}^k } A(Y_i \times U) \ar@/_/ [l] _-{r_U}}$$ by $$r_U(a_i) = M(P_{i_k} \times U)(m_i.a_i), \quad i_U(m) = \sum_{i=1}^k \bar{\epsilon}_{Y_i ,U} (n_i {\ensuremath{\otimes}}m).$$ So $M$ is a retract of $\displaystyle{\bigoplus_{i=1}^k} A(Y_i \times -)$. It remains to check that each module $A(Y \times -)$ has a right adjoint since retracts and direct sums of modules with right adjoints have right adjoints. In ${\ensuremath{\mathscr{C}}}= {\ensuremath{\mathbf{Spn}}}({\ensuremath{\mathscr{E}}})$ each object $Y$ has a dual (in fact it is its own dual). This implies that the module ${\ensuremath{\mathscr{C}}}(Y,-): \xymatrix@1@C=15pt{J \ar[r] |-{\object@{|}} & J}$ has a right dual (in fact it is ${\ensuremath{\mathscr{C}}}(Y,-)$ itself) since the Yoneda embedding ${\ensuremath{\mathscr{C}}}^{\text{op}} {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}[{\ensuremath{\mathscr{C}}}, {\ensuremath{\mathbf{Mod}_k}}]$ is a strong monoidal functor. Moreover, the unit $\eta: J {\ensuremath{\xymatrix@1@C=15pt{\ar[r]&}}}A$ induces a module $\eta_*=A : \xymatrix@1@C=15pt{J \ar[r] |-{\object@{|}} & A}$ with a right adjoint $\eta^* : \xymatrix@1@C=15pt{A \ar[r] |-{\object@{|}} & J}$. Therefore, the composite $$\xymatrix@C=7ex{J \ar[r] |-{\object@{|}}^{{\ensuremath{\mathscr{C}}}(Y,-)} & J \ar[r] |-{\object@{|}}^{\eta_*} & A},$$ which is $A(Y \times -)$, has a right adjoint. Green functors $A$ and $B$ are Morita equivalent if and only if ${\ensuremath{\mathcal{Q} A}}\simeq \mathcal{Q} B$ as ${\ensuremath{\mathscr{W}}}$-categories. See [@Li2] and [@St]. 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--- abstract: 'Given a graph $G = (V,E)$, a vertex subset $S \subseteq V$ is called [*$t$-stable*]{} (or [*$t$-dependent*]{}) if the subgraph $G[S]$ induced on $S$ has maximum degree at most $t$. The [*$t$-stability number*]{} $\alpha_t(G)$ of $G$ is the maximum order of a $t$-stable set in $G$. The theme of this paper is the typical values that this parameter takes on a random graph on $n$ vertices and edge probability equal to $p$. For any fixed $0 < p < 1$ and fixed non-negative integer $t$, we show that, with probability tending to $1$ as $n\to \infty$, the $t$-stability number takes on at most two values which we identify as functions of $t$, $p$ and $n$. The main tool we use is an asymptotic expression for the expected number of $t$-stable sets of order $k$. We derive this expression by performing a precise count of the number of graphs on $k$ vertices that have maximum degree at most $k$.' author: - | Nikolaos Fountoulakis\ Max-Planck-Institut für Informatik\ Campus E1 4\ Saarbrücken 66123\ Germany\ - | Ross J. Kang[^1]\ School of Engineering and Computing Sciences\ Durham University\ South Road, Durham DH1 3LE\ United Kingdom\ - | Colin McDiarmid\ Department of Statistics\ University of Oxford\ 1 South Parks Road\ Oxford OX1 3TG\ United Kingdom bibliography: - 'tstable.bib' date: | 26 October 2010\ Mathematics Subject Classification: 05C80, 05A16 title: 'The $t$-stability number of a random graph' --- Introduction ============ Given a graph $G = (V,E)$, a vertex subset $S \subseteq V$ is called [*$t$-stable*]{} (or [*$t$-dependent*]{}) if the subgraph $G[S]$ induced on $S$ has maximum degree at most $t$. The [*$t$-stability number*]{} $\alpha_t(G)$ of $G$ is the maximum order of a $t$-stable set in $G$. The main topic of this paper is to give a precise formula for the $t$-stability number of a dense random graph. The notion of a $t$-stable set is a generalisation of the notion of a stable set. Recall that a set of vertices $S$ of a graph $G$ is *stable* if no two of its vertices are adjacent. In other words, the maximum degree of $G[S]$ is 0, and therefore a stable set is a 0-stable set. The study of the order of the largest $t$-stable set is motivated by the study of the *$t$-improper chromatic number* of a graph. A $t$-*improper colouring* of a graph $G$ is a vertex colouring with the property that every colour class is a $t$-stable set, and the $t$-*improper chromatic number* $\chi_t(G)$ of $G$ is the least number of colours necessary for a $t$-improper colouring of $G$. Obviously, a 0-improper colouring is a proper colouring of a graph, and the 0-improper chromatic number is the chromatic number of a graph. The $t$-improper chromatic number is a parameter that was introduced and studied independently by Andrews and Jacobson [@AnJa85], Harary and Fraughnaugh (née Jones) [@Har85; @HaJo85], and by Cowen *et al.* [@CCW86]. The importance of the $t$-stability number in relation to the $t$-improper chromatic number comes from the following obvious inequality: if $G$ is a graph that has $n$ vertices, then $$\begin{aligned} \chi_t(G) \ge \frac{n}{\alpha_t(G)}.\end{aligned}$$ The $t$-improper chromatic number also arises in a specific type of radio-frequency assignment problem. Let us assume that the vertices of a given graph represent transmitters and an edge between two vertices indicates that the corresponding transmitters interfere. Each interference creates some amount of noise which we denote by $N$. Overall, a transmitter can tolerate up to a specific amount of noise which we denote by $T$. The problem now is to assign frequencies to the transmitters and, more specifically, to assign as few frequencies as possible, so that we minimise the use of the electromagnetic spectrum. Therefore, any given transmitter cannot be assigned the same frequency as more than $T/N$ nearby transmitters — that is, neighbours in the transmitter graph — as otherwise the excessive interference would distort the transmission of the signal. In other words, the vertices/transmitters that are assigned a certain frequency must form a $T/N$-stable set, and the minimum number of frequencies we can assign is the $T/N$-improper chromatic number. Given a graph $G=(V,E)$, we let $S_{t}=S_{t}(G)$ be the collection of all subsets of $V$ that are $t$-stable. We shall determine the order of the largest member of $S_t$ in a random graph $G_{n,p}$. Recall that $G_{n,p}$ is a random graph on a set of $n$ vertices, which we assume to be $V_n:=\{1,\ldots ,n\}$, and each pair of distinct vertices is present as an edge with probability $p$ independently of every other pair of vertices. Our interest is in dense random graphs, which means that we take $0<p<1$ to be a fixed constant. We say that an event occurs *asymptotically almost surely (a.a.s.)* if it occurs with probability that tends to 1 as $n\to \infty$. Related background ------------------ The $t$-stability number of $G_{n,p}$ for the case $t = 0$ has been studied thoroughly for both fixed $p$ and $p(n)=o(1)$. Matula [@Mat70; @Mat72; @Mat76] and, independently, Grimmett and McDiarmid [@GrMc75] were the first to notice and then prove asymptotic concentration of the stability number using the first and second moment methods. For $0 < p < 1$, define $b := 1/(1-p)$ and $${\alpha_{0,p}(n)} := 2\log_b n -2\log_b\log_b n +2\log_b(e/2)+1.$$ For fixed $0<p<1$, it was shown that for any ${\varepsilon}> 0$ a.a.s. $$\begin{aligned} \lfloor {\alpha_{0,p}(n)} - {\varepsilon}\rfloor \le \alpha_0 (G_{n,p}) \le \lfloor {\alpha_{0,p}(n)} + {\varepsilon}\rfloor, \label{alpha0}\end{aligned}$$ showing in particular that $\chi(G_{n,p}) \ge (1 - {\varepsilon}) n/{\alpha_{0,p}(n)}$. Assume now that $p=p(n)$ is bounded away from 1. Bollobás and Erdős [@BoEr76] extended  to hold with $p(n) > n^{-\delta}$ for any $\delta >0$. Much later, with the use of martingale techniques, Frieze [@Fri90] showed that for any ${\varepsilon}> 0$ there exists some constant $C_{\varepsilon}$ such that if $p(n) \ge C_{\varepsilon}/n$ then  holds a.a.s. Efforts to determine the chromatic number of $G_{n,p}$ took place in parallel with the study of the stability number. For fixed $p$, Grimmett and McDiarmid conjectured that $\chi(G_{n,p}) \sim n/{\alpha_{0,p}(n)}$ a.a.s. This conjecture was a major open problem in random graph theory for over a decade, until Bollobás [@Bol88] and Matula and Kučera [@MaKu90] used martingales to establish the conjecture. It was crucial for this work to obtain strong upper bounds on the probability of nonexistence in $G_{n,p}$ of a stable set with just slightly fewer than ${\alpha_{0,p}(n)}$ vertices. Ł[u]{}czak [@Luc91a] fully extended the result to hold for sparse random graphs; that is, for the case $p(n) = o(1)$ and $p(n) \ge C/n$ for some large enough constant $C$. Consult Bollobás [@Bol01] or Janson, [Ł]{}uczak and Ruciński [@JLR00] for a detailed survey of these as well as related results. For the case $t \ge 1$, the first results on the $t$-stability number were developed indirectly as a consequence of broader work on hereditary properties of random graphs. A graph property — that is, an infinite class of graphs closed under isomorphism — is said to be *hereditary* if every induced subgraph of every member of the class is also in the class. For any given $t$, the class of graphs that are $t$-stable is an hereditary property. As a result of study in this more general context, it was shown by Scheinerman [@Sch92] that, for fixed $p$, there exist constants $c_{p,1}$ and $c_{p,2}$ such that $c_{p,1} \ln n \le {\alpha_t(G_{n,p})}\le c_{p,2} \ln n$ a.a.s. This was further improved by Bollobás and Thomason [@BoTh00] who characterised, for any fixed $p$, an explicit constant $c_p$ such that $(1-{\varepsilon})c_p \ln n \le {\alpha_t(G_{n,p})}\le (1+{\varepsilon})c_p \ln n$ a.a.s. For any fixed hereditary property, not just $t$-stability, the constant $c_p$ depends upon the property but essentially the same result holds. Recently, Kang and McDiarmid [@KaMc07; @KaMc10] considered $t$-stability separately, but also treated the situation in which $t = t(n)$ varies (i.e. grows) in the order of the random graph. They showed that, if $t = o(\ln n)$, then a.a.s. $$\begin{aligned} (1-{\varepsilon})2 \log_b n \le {\alpha_t(G_{n,p})}\le (1+{\varepsilon})2 \log_b n \label{alphatbasic}\end{aligned}$$ (where $b = 1/(1-p)$, as above). In particular, observe that the estimation  for ${\alpha_t(G_{n,p})}$ and the estimation  for $\alpha_0(G_{n,p})$ agree in their first-order terms. This implies that as long as $t = o(\ln n)$ the $t$-improper and the ordinary chromatic numbers of $G_{n,p}$ have roughly the same asymptotic value a.a.s. The results of the present work ------------------------------- In this paper, we restrict our attention to the case in which the edge probability $p$ and the non-negative integer parameter $t$ are fixed constants. Restricted to this setting, our main theorem is an extension of  and a strengthening of . \[1stability\] Fix $0 < p < 1$ and $t \ge 0$. Set $b:=1/(1-p)$ and $${\alpha_{t,p}(n)} := 2\log_b n + (t-2)\log_b\log_b n + \log_b (t^t/t!^2) + t\log_b(2 b p/e) + 2\log_b (e/2)+1.$$ Then for every ${\varepsilon}> 0$ a.a.s. $$\lfloor {\alpha_{t,p}(n)} - {\varepsilon}\rfloor \le {\alpha_t(G_{n,p})}\le \lfloor {\alpha_{t,p}(n)} + {\varepsilon}\rfloor.$$ We shall see that this theorem in fact holds if ${\varepsilon}={\varepsilon}(n)$ as long as ${\varepsilon}\gg \ln \ln n / \sqrt{\ln n}$. We derive the upper bound with a first moment argument, which is presented in Section \[1stMom\]. To apply the first moment method, we estimate the expected number of $t$-stable sets that have order $k$. In particular, we show the following. \[Expectation\] Fix $0 < p < 1$ and $t \ge 0$. Let $\alpha_t^{(k)}(G)$ denote the number of $t$-stable sets of order $k$ that are contained in a graph $G$. If $k=O(\ln n)$ and $k\to \infty$ as $n\to \infty$, then $${\,\mathbb{E}}({\alpha_t^{(k)}(G_{n,p})}) = \left( e^2 n^2 b^{-k+1} k^{t-2} \left(\frac{t b p}{e}\right)^t \frac{1}{t!^2} \right)^{k/2} (1 + o(1))^k.$$ (Note that by  the condition on $k$ is not very restrictive.) Using this formula, we will see in Section \[1stMom\] that the expected number of $t$-stable sets with $\lfloor {\alpha_{t,p}(n)} + {\varepsilon}\rfloor+1$ vertices tends to zero as $n\to \infty$. The key to the calculation of this expected value is a precise formula for the number of degree sequences on $k$ vertices with a given number of edges and maximum degree at most $t$. In Section \[DegSeq\], we obtain this formula by the inversion formula of generating functions — applied in our case to the generating function of degree sequences on $k$ vertices and maximum degree at most $t$. This formula is an integral of a complex function that is approximated with the use of an analytic technique called saddle-point approximation. Our proof is inspired by the application of this method by Chvátal [@Chv91] to a similar generating function. For further examples of the use of the saddle-point method, consult Chapter VIII of Flajolet and Sedgewick [@FlSe09]. The lower bound in Theorem \[1stability\] is derived with a second moment argument in Section \[2ndMom\]. We remark that Theorems \[1stability\] and \[Expectation\] are both stated to hold for the case $t = 0$ (if we assume that $0^0 = 1$) in order to stress that these results generalise the previous results of Matula [@Mat70; @Mat72; @Mat76] and Grimmett and McDiarmid [@GrMc75]. Our methods apply for this special case, however in our proofs our main concern will be to establish the results for $t \ge 1$. In Section \[chrom\] we give a quite precise formula for the $t$-improper chromatic number of $G_{n,p}$. For $t = 0$, that is, for the chromatic number, McDiarmid [@McD90] gave a fairly tight estimate on $\chi(G_{n,p})(=\chi_0(G_{n,p}))$ proving that for any fixed $0< p < 1$ a.a.s. $$\begin{aligned} \frac{n}{{\alpha_{0,p}(n)} - 1 -o(1)} \le \chi_0(G_{n,p}) \le \frac{n}{{\alpha_{0,p}(n)} -1 -\frac{1}{2} - \frac{1}{1-(1-p)^{1/2}}+o(1)}.\end{aligned}$$ Panagiotou and Steger [@PaSt09] recently improved the lower bound showing that a.a.s. $$\begin{aligned} \chi_0(G_{n,p}) \ge \frac{n}{{\alpha_{0,p}(n)} - \frac{2}{\ln b} -1 +o(1)},\end{aligned}$$ and asked if better upper or lower bounds could be developed. In Section \[chrom\], we improve upon McDiarmid’s upper bound and we generalise (for $t \ge 1$) both this new bound and the lower bound of Panagiotou and Steger. \[tchrom\] Fix $0 < p < 1$ and $t\ge 0$. Then a.a.s. $$\begin{aligned} \frac{n}{{\alpha_{t,p}(n)} -\frac{2}{\ln b} -1 + o(1)} \le \chi_t(G_{n,p}) \le \frac{n}{{\alpha_{t,p}(n)} -\frac{2}{\ln b} - 2 - o(1)}.\end{aligned}$$ Given a graph $G$, let the [*colouring rate*]{} $\overline{\alpha_0}(G)$ of $G$ be $|V(G)| /\chi_0(G)$, which is the maximum average size of a colour class in a proper colouring of $G$. Then the case $t=0$ of Theorem \[tchrom\] implies for any fixed $0 < p < 1$ that a.a.s. $$\begin{aligned} {\alpha_{0,p}(n)} -\frac{2}{\ln b} - 2 - o(1) \le \overline{\alpha_0} (G_{n,p}) \le {\alpha_{0,p}(n)} -\frac{2}{\ln b} - 1 + o(1).\end{aligned}$$ Thus the colouring rate of $G_{n,p}$ is a.a.s. contained in an explicit interval of length $1+o(1)$. We remark that Shamir and Spencer [@ShSp87] showed a.a.s. $\tilde{O}(\sqrt{n})$-concentration of $\chi_0(G_{n,p})$ — see also a recent improvement by Scott [@Sco08+]. (The $\tilde{O}$ notation ignores logarithmic factors.) It therefore follows that $\overline{\alpha_0}(G_{n,p})$ is a.a.s. $\tilde{O}( n^{-1/2})$-concentrated. The above discussion extends easily to $t$-improper colourings. Counting degree sequences of maximum degree $t$ {#DegSeq} =============================================== Given non-negative integers $k, t$ with $t < k$, we let $$C_{2 m}(t,k):=\sum_{(d_1,\ldots,d_k),\sum_i d_i =2 m,d_i\le t}\frac{1}{\prod_i d_i!}.$$ (Here, the $d_i$ are non-negative integers.) Given a fixed degree sequence $(d_1,\ldots, d_k)$ with $\sum_i d_i = 2 m$, the number of graphs on $k$ vertices $(v_1,\ldots , v_k)$ where $v_i$ has degree $d_i$ is at most $$\begin{aligned} \frac{1}{\prod_i d_i!} \frac{(2 m)!}{m!2^m}.\end{aligned}$$ See for example [@Bol01] in the proof of Theorem 2.16 or Section 9.1 in [@JLR00] for the definition of the configuration model, from which the above claim follows easily. Therefore, $C_{2 m}(t,k){(2 m)!/(m!2^m)}$ is an upper bound on the number of graphs with $k$ vertices and $m$edges such that each vertex has degree at most $t$. Note also that $(2 m)! C_{2 m}(t,k)$ is the number of allocations of $2m$ balls into $k$ bins with the property that no bin contains more than $t$ balls. In the proof of Theorem \[Expectation\], we need good estimates for $C_{2m}(t,k)$, when $2m$ is close to $tk$. In particular, as we will see in the next section (Lemma \[mstar\]) we will need a tight estimate for $C_{2m} (t,k)$ when $t - \ln k/\sqrt{k} < 2m/k < t - 1/(\sqrt{k} \ln k)$, since in this range the expected number of $t$-stable sets having $m$ edges is maximised. We require a careful and specific treatment of this estimation due to the fact that $2m/k$ is not bounded below $t$. For $t\ge 1$, note that $C_{2 m}(t,k)$ is the coefficient of $z^{2 m}$ in the following generating function: $$G(z)=R_t(z)^{k}= \left(\sum_{i=0}^t \frac{z^i}{i!} \right)^{k}.$$ Cauchy’s integral formula gives $$C_{2 m}(t,k)= \frac{1}{2\pi i} \int_{C} \frac{R_t(z)^k}{z^{2 m+1}} dz,$$ where the integration is taken over a closed contour containing the origin. Before we state the main theorem of this section, we need the following lemma, which follows from Note IV.46 in [@FlSe09]. \[lemma:r\_0,s\] Fix $t\ge 1$. The function $r R_t'(r)/R_t(r)$ is strictly increasing in $r$ for $r > 0$. For each $y \in (0, t)$, there exists a unique positive solution $r_0 = r_0(y)$ to the equation $r R_t'(r)/R_t(r)=y$ and furthermore the function $r_0(y)$ is a continuous bijection between $(0,t)$ and $(0,\infty)$. Thus, if we set $$s(y)=r_0(y) \frac{d}{d x} \left. \frac{x R_t'(x)}{R_t(x)}\right|_{x=r_0(y)},$$ then $s(y) > 0$. We will prove a “large powers” theorem to obtain a very tight estimate on $C_{2m}(t,k)$ when $2m/k$ is quite close to $t$. A version of this theorem holds if we instead assume that $2m/k$ is bounded away from $t$; indeed, this immediately follows from Theorem VIII.8 of [@FlSe09]. However, our version, where $2m/k$ approaches $t$, is necessary in light of Lemma \[mstar\] below. \[coupons\] Assume that $t\ge 1$ is fixed and $k \to \infty$. Suppose that $m$ and $k$ are such that $t - \ln k/\sqrt{k} \le 2 m/k \le t - 1/(\sqrt{k}\ln k)$ for any ${\varepsilon}> 0$, and $r_0$ and $s$ are defined as in Lemma \[lemma:r\_0,s\]. Then uniformly $$\begin{aligned} C_{2 m}(t,k)= \frac{1}{\sqrt{2\pi k s(2 m/k)}} \frac{R_t(r_0(2 m/k))^k}{r_0(2 m/k)^{2 m}}(1+o(1)).\end{aligned}$$ In the proof of the theorem (as well as in later sections), we make frequent use of the following lemma, whose proof is postponed until the end of the section. \[ytot\] If $y=y(k) \to t$ as $k \to \infty$ (and $y<t$) and $r_0$ and $s$ are defined as in Lemma \[lemma:r\_0,s\], then $$\begin{aligned} &r_0 = \frac{t}{t-y} + O(1),\label{r_asympt} &\\ &\frac{d r_0}{d y} = \frac{{r_0}^2}{t}\left(1+O\left(\frac{1}{r_0}\right)\right), \text{ and} \label{derivative} &\\ &s = \frac{t}{r_0}\left(1+O\left(\frac{1}{r_0}\right)\right). \label{lemma:s}&\end{aligned}$$ [**of Theorem \[coupons\]**]{} The proof is inspired by [@Chv91]. Throughout, we for convenience drop the subscript and write $R(z)$ in the place of $R_t(z)$. Recall that $r_0 = r_0(2m/k)$ is the unique positive solution of the equation $rR'(r)/R(r) = 2m/k$, where $t- \ln k/\sqrt{k} \le 2m/k \leq t-1/(\sqrt{k} \ln k)$, and let $C$ be the circle of radius $r_0$ centred at the origin. Using polar coordinates, we obtain $$\begin{aligned} C_{2 m}(t,k)&=\frac{1}{2\pi i} \int_{C} \frac{R(r_0 e^{i\varphi})^k} {{r_0}^{2 m+1}e^{i 2 m\varphi}e^{i\varphi}} d(r_0 e^{i\varphi}) = \frac{1}{2\pi {r_0}^{2 m}} \int_{-\pi}^{\pi} \frac{R(r_0 e^{i\varphi})^k} {e^{i 2 m\varphi}} d \varphi.\end{aligned}$$ We let $\delta = \delta(k) := \ln k \sqrt{r_0/k}$ and write $$\begin{aligned} \label{eqn:C} C_{2 m}(t,k)= \frac{1}{2\pi {r_0}^{2 m}} \left( \int_{\delta}^{2\pi-\delta} \frac{R(r_0 e^{i\varphi})^k}{e^{i 2 m\varphi}} d \varphi + \int_{-\delta}^{\delta} \frac{R(r_0 e^{i\varphi})^k}{e^{i 2 m\varphi}} d \varphi \right) .\end{aligned}$$ Note that, since $2 m/k < t - 1/(\ln k \sqrt{k})$, it follows from  that $\delta \to 0$ as $k \to \infty$. We shall analyse the two integrals of  separately. To begin, we consider the first integral of  and we wish to show that it makes a negligible contribution to the value of $C_{2 m}(t,k)$. Note that $$\begin{aligned} \left|R(r_0 e^{i \varphi})\right|^2 &= \left( \sum_{j=0}^{t} \frac{{r_0}^{j}}{j!} \cos(j \varphi) \right)^2 + \left( \sum_{j=0}^{t} \frac{{r_0}^{j}}{j!} \sin(j \varphi) \right)^2 \nonumber\\ &= \sum_{0 \le j_1,j_2 \le t} \frac{{r_0}^{j_1+j_2}}{j_1!j_2! } \left(\cos (j_1 \varphi) \cos (j_2 \varphi)+ \sin (j_1 \varphi) \sin (j_2 \varphi) \right) \nonumber\\ &= \sum_{0 \le j_1,j_2 \le t} \frac{{r_0}^{j_1+j_2}}{j_1!j_2!} \cos \left( (j_1-j_2) \varphi \right) \nonumber\\ &= R(r_0)^2 - \sum_{0 \le j_1 < j_2 \le t} \frac{2 {r_0}^{j_1+j_2}}{j_1!j_2!} \left(1-\cos \left( (j_1-j_2) \varphi \right) \right). \label{eqn:R}\end{aligned}$$ Note that $r_0\to \infty$ as $k \to \infty$. Hence, from , $$\begin{aligned} \left|R(r_0 e^{i \varphi})\right|^2 &\le R(r_0)^2 \left( 1 - \frac{\frac{2 {r_0}^{2 t-1}}{t!(t-1)!} (1 - \cos \varphi)}{\frac{{r_0}^{2 t}}{t!^2} + \Theta({r_0}^{2 t-1})} \right) = R(r_0)^2 \left( 1-(1+o(1))\frac{2 t}{r_0}( 1-\cos \varphi) \right).\end{aligned}$$ It follows that for $k$ large enough $$\begin{aligned} \label{caseBfirst} \left| \int_{\delta}^{2\pi-\delta} \frac{R(r_0 e^{i\varphi})^k}{e^{i 2 m\varphi}} d\varphi \right| & \le 2\pi R(r_0)^k\left( 1-(1+o(1))\frac{2 t}{r_0}( 1-\cos \delta) \right)^{k/2} \nonumber\\ & \le 2\pi R(r_0)^k\exp \left( -\frac{t k}{2r_0}( 1-\cos \delta) \right) \nonumber\\ & = 2\pi R(r_0)^k\exp \left(-\frac{t}{ 2} \cdot \frac{k\delta^2}{r_0 \ln k} \cdot \frac{1-\cos \delta}{\delta^2} \cdot \ln k \right).\end{aligned}$$ Since $\delta \to 0$, we have that $(1-\cos \delta)/\delta^2 \to 1/2$. By the choice of $\delta$, we also have that $k\delta^2/(r_0 \ln k) \to \infty$ as $k \to \infty$, and it follows from Inequality  that $$\begin{aligned} \label{first.approx} \left|\int_{\delta}^{2\pi-\delta} \frac{R(r_0 e^{i\varphi})^k}{e^{i 2 m\varphi}} d\varphi \right| < R(r_0)^k/k,\end{aligned}$$ for large enough $k$. In order to precisely estimate the second integral of , we consider the function $f: \mathbb{R} \to \mathbb{C}$ given by $$\begin{aligned} f(\varphi) &:= R(r_0 e^{i\varphi})\exp \left(-i\frac{2 m}{k} \varphi \right) = \exp \left(-i\frac{2 m}{k} \varphi \right) \left( \sum_{j=0}^{t} \frac{{r_0}^j}{j!} (\cos(j \varphi) + i \sin(j \varphi) )\right).\end{aligned}$$ The importance of the function $f$ is that $$\begin{aligned} \int_{-\delta}^{\delta} \frac{R(r_0e^{i\varphi})^k}{e^{i 2 m\varphi}} d \varphi = \int_{-\delta}^{\delta} f(\varphi)^k d \varphi.\end{aligned}$$ We will show that the real part of $f(\varphi)^k$ is well approximated by $R(r_0)^k\exp(-s k \varphi^2/2)$ when $|\varphi|$ is small — see  below. The imaginary part can be ignored as the integral approximates a real quantity. To this end we will apply Taylor’s Theorem, and in order to do this we shall need the first, second and third derivatives of $f$ with respect to $\varphi$. First, $$\begin{aligned} f'(\varphi) = \exp \left(-i\frac{2 m}{k} \varphi \right) \left( \sum_{j=0}^{t} \frac{{r_0}^j}{j!} \left(\frac{2 m}{k}-j \right)(\sin (j \varphi) - i \cos (j \varphi ) )\right). \end{aligned}$$ Note that $$\begin{aligned} f'(0) &= -i \left( \frac{2 m}{k} \sum_{j=0}^{t} \frac{{r_0}^{j}}{j!} - \sum_{j=0}^{t} \frac{{r_0}^{j}}{j!}j \right) =-i \left( \frac{2 m}{k}R(r_0) - r_0R'(r_0) \right)= 0\end{aligned}$$ by the choice of $r_0$. Next, $$\begin{aligned} f''(\varphi) =& -i \frac{2 m}{k} f'(\varphi) +\exp \left(-i \frac{2 m}{k} \varphi \right) \left( \sum_{j=0}^{t} \frac{{r_0}^j}{j!} \left(\frac{2 m}{k}-j \right)j(\cos (j \varphi )+i\sin (j \varphi) )\right).\end{aligned}$$ Therefore, $$\begin{aligned} f''(0)&= -i\frac{2 m}{k}f'(0) +\sum_{j=0}^{t} \frac{{r_0}^j}{j!} \left(\frac{2 m}{k}-j \right)j \nonumber \\ &= \frac{2 m}{k} \sum_{j=1}^{t} \frac{{r_0}^j}{j!}j - \sum_{j=1}^{t} \frac{{r_0}^j}{j!}j(j-1) - \sum_{j=1}^{t} \frac{{r_0}^j}{j!}j \nonumber \\ &= \left(\frac{r_0 R'(r_0)}{R(r_0)}\right) r_0 R'(r_0) - {r_0}^2 R''(r_0) - r_0 R'(r_0) \nonumber \\ &= -r_0 \left(\frac{-r_0 R'(r_0)^2}{R(r_0)} + r_0 R''(r_0) + R'(r_0) \right) \nonumber \\ &= -R(r_0) r_0 \left(\frac{(r_0 R''(r_0) + R'(r_0))R(r_0)-r_0 R'(r_0)^2}{R(r_0)^2}\right) \nonumber \\ &= -R(r_0)r_0 \frac{d}{d x} \left.\frac{xR'(x)}{R(x)}\right|_{x=r_0} = -R(r_0)s(2 m/k). \label{sec.par}\end{aligned}$$ Thus, $f''(0) < 0$ by Lemma \[lemma:r\_0,s\]. Last, we have $$\begin{aligned} f'''(\varphi) =& - i \frac{2 m}{k} f''(\varphi) - i \frac{2 m}{k} \exp \left(-i \frac{2 m}{k} \varphi \right) \left( \sum_{j=0}^{t} \frac{{r_0}^j}{j!} \left(\frac{2 m}{k}-j \right)j(\cos (j \varphi )+i\sin (j \varphi) )\right)\\ & + \exp \left(-i \frac{2 m}{k} \varphi \right) \left( \sum_{j=0}^{t} \frac{{r_0}^j}{j!} \left(\frac{2 m}{k}-j \right)j^2(-\sin (j \varphi )+i\cos (j \varphi) )\right).\end{aligned}$$ Since $r_0\rightarrow \infty$ as $k \rightarrow \infty$, there is a positive constant $a$ such that $a \le r_0$, for $k$ sufficiently large. Clearly, $f(0) = R(r_0) > a^t/t! > 0$. The continuity of $f$ on the compact set $-\pi \le \varphi \le \pi$ implies that there is a positive constant $\delta_0$ such that whenever $|\varphi| \le \delta_0$ we have $Re (f(\varphi)) > 0$. Since the first two derivatives of $Im (f(\varphi))$ with respect to $\varphi$ vanish when $\varphi =0$, and also $Im (f(0))=0$, Taylor’s Theorem implies that $$\begin{aligned} |Im (f(\varphi))| \le \sup_{|\varphi| \le \delta_0} |Im(f'''(\varphi))|\frac{\varphi^3}{6}\end{aligned}$$ if $|\varphi| \le \delta_0$. Now, note that $Re (f (\varphi))$ and $Im(f'''(\varphi))$ can be considered as polynomials of degree $t$ with respect to $r_0$. The leading term of $Re (f (\varphi))$ is $$\begin{aligned} Re\left( \exp \left(-i\frac{2 m}{k} \varphi \right) (\cos (t \varphi) + i \sin (t \varphi)) \right) \frac{{r_0}^t}{t!};\end{aligned}$$ thus, $Re (f (\varphi)) = \Omega({r_0}^t)$. On the other hand, using the derivative computations above and simplifying, it follows that the leading term of $Im(f'''(\varphi))$ is $$\begin{aligned} &Im\left( \exp \left(-i\frac{2 m}{k} \varphi \right) (\sin (t \varphi) + i \cos (t \varphi)) \right) \left(t - \frac{2 m}{k} \right)^3 \frac{{r_0}^t}{t!}.\end{aligned}$$ By , $t - 2 m/k = (1 + o(1)) t / r_0$ and thus $Im(f'''(\varphi)) = O({r_0}^{t-1})$. So, there exists $c_1 > 0$ such that for every $\varphi$ with $|\varphi|\le \delta_0$ $$\frac{\sup_{|\varphi |\le \delta_0} |Im(f'''(\varphi))|}{|Re(f(\varphi))|} < \frac{c_1}{r_0},$$ and therefore $$\left| \frac{Im (f(\varphi))}{Re (f(\varphi))}\right| \le \frac{c_1 \varphi^3}{6 r_0},$$ for any $\varphi$ with $|\varphi| \le \delta_0$. On the other hand, we have (see pages 15–16 of [@Chv91] for the details) $$\left| \frac{Re(z^k)}{Re(z)^k}-1\right| \le \epsilon \left(k, \left|\frac{Im (z)}{Re (z)} \right| \right),$$ with $$\epsilon(k,x)=(1+x)^k-1-x k \le e^{x k} - 1$$ (for $x \ge 0$). Since $\epsilon(k,x)$ increases in $x$ for $x\ge 0$, we have $$\begin{aligned} \label{eqn:f^k} 1- \epsilon \left(k,\frac{c_1 \delta^3}{6 r_0}\right) \le \frac{Re ( f(\varphi)^k)}{Re ( f(\varphi))^k} \le 1+ \epsilon \left(k,\frac{c_1 \delta^3}{6 r_0} \right),\end{aligned}$$ whenever $|\varphi| \le \delta \le \delta_0$. Next, we approximate the function $\ln Re (f(\varphi))$. First, $$\begin{aligned} \left. \frac{d}{d \varphi} (\ln Re (f(\varphi))) \right|_{\varphi=0} = \left. \frac{Re (f'(\varphi))}{Re (f(\varphi))} \right|_{\varphi=0} = 0.\end{aligned}$$ Second, we have $$\begin{aligned} \frac{d^2}{d \varphi^2} (\ln Re (f(\varphi))) & = \frac{d}{d \varphi} \left( \frac{Re (f'(\varphi))}{Re (f(\varphi))} \right) = \frac{Re(f''(\varphi))Re(f(\varphi))-Re(f'(\varphi))^2}{Re(f(\varphi))^2};\end{aligned}$$ therefore, by Equation , $$\begin{aligned} \left. \frac{d^2}{d \varphi^2} (\ln Re (f(\varphi))) \right|_{\varphi=0} & = \frac{Re(f''(0))Re(f(0))-Re(f'(0))^2}{Re(f(0))^2} = \frac{-R(r_0) s}{R(r_0)} = -s\end{aligned}$$ Now, the numerator of the third derivative with respect to $\varphi$ is $$\begin{aligned} & (Re(f''(\varphi))Re(f(\varphi))-Re(f'(\varphi))^2)'Re(f(\varphi))^2 \\ &- 2 Re(f(\varphi))(Re(f''(\varphi))Re(f(\varphi))-Re(f'(\varphi))^2) \\ &= Re (f(\varphi))\Big((Re(f''(\varphi))Re(f(\varphi))-Re(f'(\varphi))^2)'Re(f(\varphi)) \\ & - 2 (Re(f''(\varphi))Re(f(\varphi))-Re(f'(\varphi))^2) \Big). \end{aligned}$$ Thus an elementary calculation gives that (for $|\varphi| \le \delta_0$) $$\begin{aligned} &\frac{d^3}{d \varphi^3} (\ln Re(f(\varphi))) \\ &= \frac{Re(f'''(\varphi))Re(f(\varphi))^2-3 Re(f''(\varphi))Re(f'(\varphi))Re(f(\varphi))+2 Re(f'(\varphi))^3}{Re(f(\varphi))^3}.\end{aligned}$$ If, as we did earlier for $Re(f(\varphi))$ and $Im(f'''(\varphi))$, we consider $Re(f'(\varphi))$, $Re(f''(\varphi))$ and $Re(f'''(\varphi))$ as polynomials with respect to $r_0$, we can show that $Re(f'(\varphi)) = O({r_0}^{t-1})$, $Re(f''(\varphi)) = O({r_0}^{t-1})$ and $Re(f'''(\varphi)) = O({r_0}^{t-1})$. It then follows that there exists $c_2 > 0$ such that for every $\varphi$ with $|\varphi| \le \delta_0$ $$\begin{aligned} \left|\frac{d^3}{d \varphi^3} (\ln Re (f(\varphi)))\right| \le \frac{c_2}{r_0}.\end{aligned}$$ Therefore, Taylor’s Theorem implies that for every $\varphi$ with $|\varphi| \le \delta_0$ we have $$\left| \ln Re(f(\varphi))- \left(\ln R(r_0) - \frac{s\varphi^2}{2} \right)\right| \le \frac{c_2 \varphi^3}{6 r_0}.$$ It follows that $$\exp \left(- \frac{c_2 k \delta^3}{6 r_0}\right) \le \frac{Re (f(\varphi))^k}{R(r_0)^k \exp (-s k\varphi^2/2)} \le \exp \left( \frac{c_2 k \delta^3}{6 r_0}\right).$$ The condition that $2 m/k < t - 1/(\ln k\sqrt{k})$ and  together imply that $r_0 < t \ln k\sqrt{k} + O(1)$. Therefore, $k\delta^3/r_0 = \sqrt{r_0/k}\ln^3 k \to 0$ as $k \to \infty$, and we have $$\exp \left( \frac{c_2 k \delta^3}{6 r_0}\right)=1+o(1) \ \text{ and } \ \epsilon \left(k,\frac{c_1 \delta^3}{6 r_0} \right)\le \exp\left(\frac{c_1 k\delta^3}{6 r_0} \right)-1= o(1),$$ proving that $$\begin{aligned} Re (f(\varphi)^k)= R(r_0)^k \exp (-s k\varphi^2/2) (1+o(1)) \label{???}\end{aligned}$$ uniformly for $|\varphi| \le \delta$. From ,  and , we obtain $$\begin{aligned} \label{eqn:approx} 2\pi{r_0}^{2 m} C_{2 m}(t,k) &= R(r_0)^k\left(\int_{-\delta}^{\delta} \exp (-s k\varphi^2/2) d \varphi+o(1)\right).\end{aligned}$$ Using a change of variables $\psi =\sqrt{s k}\varphi$, observe that $$\int_{-\delta}^{\delta} \exp \left(-\frac{s k\varphi^2}{2} \right) d \varphi = \frac{1}{\sqrt{s k}} \int_{-\delta\sqrt{s k}}^{\delta\sqrt{s k}} \exp \left(-\frac{\psi^2}{2} \right) d \psi = \sqrt{\frac{2\pi}{s k}}(1 + o(1)),$$ as $k \to \infty$ since $\delta \sqrt{s k} \sim \sqrt{t}\ln k \to \infty$. Thus, Equation  becomes $$2\pi{r_0}^{2 m} C_{2 m}(t,k) =\sqrt{ \frac{2\pi}{ks}} R(r_0)^k(1+o(1))$$ and the result follows. Proof of Lemma \[ytot\] ----------------------- [**of Equation **]{} First, note that $r_0 = r_0(y)\to \infty$ as $k\to \infty$ by Lemma \[lemma:r\_0,s\]. So $$\begin{aligned} r_0 R'(r_0) & = \frac{{r_0}^t}{(t-1)!}\left(1+ \frac{t-1}{r_0} + O\left(\frac{1}{{r_0}^2}\right) \right), \\ R(r_0) & = \frac{{r_0}^t}{t!}\left(1+\frac{t}{r_0} + O\left(\frac{1}{{r_0}^2} \right) \right).\end{aligned}$$ Thus, $$\begin{aligned} \label{eq:expand} \frac{r_0 R'(r_0)}{R(r_0)} &= t\frac{1+\frac{t-1}{r_0} + O\left(\frac{1}{{r_0}^2} \right)}{1+\frac{t}{r_0} + O\left(\frac{1}{{r_0}^2} \right)} = t\left(1+\frac{t-1}{r_0} + O\left(\frac{1}{{r_0}^2} \right) \right)\left( 1-\frac{t}{r_0} + O\left(\frac{1}{{r_0}^2} \right) \right) \nonumber \\ &= t\left(1-\frac{t}{r_0}+\frac{t-1}{r_0} + O\left(\frac{1}{{r_0}^2} \right)\right) = t\left(1-\frac{1}{r_0}+ O\left(\frac{1}{{r_0}^2} \right) \right).\end{aligned}$$ Since $r_0 R'(r_0)/R(r_0) = y = t(1-(t-y)/t)$, we obtain $$\begin{aligned} \label{eq:Reltoy} 1 - \frac{t-y}{t} = 1 - \frac{1}{r_0} + O\left(\frac{1}{{r_0}^2} \right)\end{aligned}$$ which can be rewritten as $$r_0 = \frac{t}{t-y}\left(1 + O\left(\frac{1}{r_0} \right)\right),$$ and this implies the desired expression. [**of Equation **]{} A more careful treatment of the computations for the proof of  shows that the $O(1/{r_0}^2)$ error term in  may instead be written $\eta(1/r_0)/{r_0}^2$ where $\eta$ is a power series with positive radius of convergence. In particular, as $r_0R'(r_0)$ and $R(r_0)$ are polynomial functions of $r_0$, yields, for some power series $\eta_1$, $\eta_2$ and $\hat{\eta}_2$ with positive radius of convergence, that $$\begin{aligned} \frac{y}{t} &= \frac{r_0 R'(r_0)}{t R(r_0)} = \frac{1+\frac{t-1}{r_0} + \frac{\eta_1(1/r_0)}{{r_0}^2}}{1+\frac{t}{r_0} + \frac{\eta_2(1/r_0)}{{r_0}^2}} = \left( 1+\frac{t-1}{r_0} + \frac{\eta_1(1/r_0)}{{r_0}^2}\right) \left(1-\frac{t}{r_0} + \frac{\hat{\eta}_2(1/r_0)}{{r_0}^2} \right) \\ & = 1 - \frac{1}{r_0} + \eta\left(\frac{1}{r_0} \right)\frac{1}{{r_0}^2}.\end{aligned}$$ Then, by differentiating both sides of this expression with respect to $y$, we obtain $$\begin{aligned} \frac{1}{ t} = \frac{d }{ d r_0} \left(1 - \frac{1}{r_0} + \eta\left(\frac{1}{r_0} \right)\frac{1}{{r_0}^2} \right) \frac{d r_0}{d y}.\end{aligned}$$ We have that $$\begin{aligned} \frac{d }{ d r_0} \left(1 - \frac{1}{r_0} + \eta\left(\frac{1}{r_0} \right)\frac{1}{{r_0}^2} \right) & = \frac{1}{ {r_0}^2} - \eta\left(\frac{1}{r_0} \right)\frac{2}{{r_0}^3} - \eta'\left(\frac{1}{r_0} \right)\frac{1}{{r_0}^4} = \frac{1}{ {r_0}^2} + O\left( \frac{1}{ {r_0}^3} \right)\end{aligned}$$ and  immediately follows. [**of Equation **]{} By the definition of $r_0$, it follows from the chain rule that $$1=\frac{d}{d y}\frac{r_0R'(r_0)}{R(r_0)} = \frac{d}{d r_0} \frac{r_0R'(r_0)}{R(r_0)} \frac{d r_0}{d y}.$$ Thus, $$\begin{aligned} \left.\frac{d}{d x}\frac{x R'(x)}{R(x)}\right|_{x=r_0(y)} = \left(\left.\frac{d r_0(y')}{d y'}\right|_{y' = y} \right)^{-1},\end{aligned}$$ implying that $$\begin{aligned} s(y) &= r_0(y)\left(\left.\frac{d r_0(y')}{d y'}\right|_{y'=y} \right)^{-1} \stackrel{\mathtt{\eqref{derivative}}}{=} \frac{t}{r_0(y)} \left(1+O\left(\frac{1}{r_0(y)}\right)\right)\end{aligned}$$ as required. The expected number of $t$-stable sets of order $k$ - proof of Theorem \[Expectation\] {#1stMom} ====================================================================================== In this section, we give an asymptotic expression for the expected number of $t$-stable subsets of $V_n$ of order $k$ in $G_{n,p}$, proving Theorem \[Expectation\]. In light of , we will consider $k$ such that $k=k(n)=O(\ln n)$ and $k\to \infty$ as $n\to \infty$. Towards the end of the section, we will specify $k$ and derive the upper bound of Theorem \[1stability\] by a first moment argument. Let $A$ be a subset of $V_n$ that has order $k$. If ${\alpha_t^{(k)}(G_{n,p})}$ denotes the number of subsets of $V_n$ of order $k$ that are $t$-stable, then $$\begin{aligned} {\,\mathbb{E}}({\alpha_t^{(k)}(G_{n,p})}) = \binom{n}{k} {\,\mathbb{P}}(A \in S_t). \end{aligned}$$ Partitioning according to the number of edges that $A$ induces, we have $$\begin{aligned} \label{expansion} {\,\mathbb{P}}(A \in S_t) = \sum_{m=0}^{\lfloor t k/2 \rfloor} {\,\mathbb{P}}(A \in S_t, \; e(A)=m).\end{aligned}$$ By the definition of $C_{2 m}(t,k)$ (given at the beginning of Section \[DegSeq\]), it follows that $$\begin{aligned} \label{EdgesExpansion} {\,\mathbb{P}}(A \in S_t, \; e(A)=m) \le p^m (1-p)^{\binom{k}{2} -m} C_{2 m}(t,k)\frac{(2 m)!}{m! 2^m}=:f(m).\end{aligned}$$ First, we find the value of $m$ for which the expression $f(m)$ on the right-hand side of  is maximised. If $m^*$ is such that $f(m^*) = \max \{f(m): 0 \le 2 m \le t k \}$, it turns out that the following holds. \[mstar\] $ 2 m^* = t k -\sqrt{t k/b p}+o(\sqrt{k}). $ Let ${\lambda}_m = {\lambda}_m (t,k) =f(m+1)/f(m)$. Thus, $$\begin{aligned} {\lambda}_m &= \frac{p}{1-p}\frac{C_{2 m+2}(t,k)}{C_{2 m}(t,k)}\frac{1}{2} \frac{(2 m+2)(2 m+1)}{m+1}=\frac{p}{1-p}\frac{C_{2 m+2}(t,k)}{C_{2 m}(t,k)}(2 m+1).\end{aligned}$$ We will estimate ${\lambda}_m$ for all $m$ with $0 \le 2 m \le t k$ and treat three separate cases: 1. \[1stcase\] $2 m < t k - \sqrt{k} \ln k$; 2. \[2ndcase\] $2 m > t k - \sqrt{k} / \ln k$; and 3. \[3rdcase\] $t k - \sqrt{k} \ln k \le 2 m \le t k - \sqrt{k} / \ln k$. We will use Theorem \[coupons\] in Case \[3rdcase\], as we will determine those values $m$ for which ${\lambda}_m \approx 1$ within that range. In the other cases we will use a cruder argument, which is nonetheless sufficient for our purposes. ### Case \[1stcase\] {#case1stcase .unnumbered} We will show that ${\lambda}_m > 1$ for any such $m$. We set $S_{2 m}(t,k)=(2 m)!C_{2 m}(t,k)$. Note that this is equal to the number of ways of allocating $2 m$ labelled balls into $k$ bins so that each bin does not receive more than $t$ balls — we also denote the set of such allocations by ${\mathcal S}_{2 m}(t,k)$. We have $$\begin{aligned} \label{eq:CoeffRatio} \frac{C_{2(m+1)}(t,k)}{C_{2 m} (t,k)} = \frac{S_{2(m+1)}(t,k)}{S_{2 m} (t,k)}\frac{1}{(2 m+2)(2 m+1)}. \end{aligned}$$ We will obtain a lower bound on the left-hand side, by first obtaining a lower bound on the ratio $S_{2(m+1)}(t,k)/S_{2 m} (t,k)$. Let us consider $2 m+2$ distinct balls which we label $1,\ldots, 2 m+1, 2 m+2$. We construct an auxiliary bipartite graph whose parts are ${\mathcal S}_{2 m}(t,k)$ and ${\mathcal S}_{2 m+2}(t,k)$. If $c\in {\mathcal S}_{2 m}(t,k)$ and $c' \in {\mathcal S}_{2 m+2}(t,k)$, then $(c,c')$ forms an edge in the auxiliary graph if $c'$ restricted to balls $1,\ldots, 2 m$ is $c$. So any $c' \in {\mathcal S}_{2 m+2}(t,k)$ is adjacent to exactly one configuration $c \in {\mathcal S}_{2 m}(t,k)$, that is, its degree in the auxiliary graph is equal to 1. Also, if $e(c)$ is the number of non-full bins in a configuration $c \in {\mathcal S}_{2 m}(t,k)$, then $c$ has at least $e(c)(e(c) - 1)$ neighbours in ${\mathcal S}_{2 m+2} (t,k)$. This is the case since there are at least $e(c)(e(c) - 1)$ ways of allocating balls $2 m+1$ and $2 m+2$ into the non-full bins of $c$, therefore giving a lower bound on the number of configurations in ${\mathcal S}_{2 m+2} (t,k)$ whose restriction on the first $2 m$ balls is $c$. But $2 m < t k - \sqrt{k} \ln k$ and therefore $e(c)\ge \sqrt{k} (\ln k) / t$. These observations imply that for $k$ large enough $$S_{2 m+2} (t,k) \ge \frac{k \ln^2 k}{2 t^2} S_{2 m}(t,k),$$ and therefore $$\begin{aligned} \frac{C_{2(m+1)}(t,k)}{C_{2 m} (t,k)} &= \frac{S_{2(m+1)}(t,k)}{S_{2 m} (t,k)}\frac{1}{(2 m+2)(2 m+1)}\ge \frac{k \ln^2 k}{2(2 m+2)(2 m+1)} = \Omega\left(\frac{\ln^2 k}{ m} \right). \end{aligned}$$ So ${\lambda}_m = \Omega (\ln^2 k) > 1$ in Case \[1stcase\]. ### Case \[2ndcase\] {#case2ndcase .unnumbered} We treat this case similarly. We consider an auxiliary bipartite graph as above. Let $c \in {\mathcal S}_{2 m}(t,k)$ be a configuration of balls $1,\ldots, 2 m$. Since there are at most $\sqrt{k}/\ln k$ places available in the non-full bins, there are at most $k/\ln^2 k$ ways of allocating balls $2 m+1$ and $2 m+2$ into the non-full bins of $c$. In other words, the degree of any vertex in ${\mathcal S}_{2 m}(t,k)$ is at most $k/\ln^2 k$. Also, as above, the degree of any vertex/configuration $c' \in {\mathcal S}_{2 m+2}(t,k)$ is equal to one. Therefore, $$\frac{S_{2 m+2}(t,k) }{ S_{2 m}(t,k)} \le \frac{k}{ \ln^2 k}.$$ Substituting this into , we obtain $$\frac{C_{2(m+1)}(t,k) }{ C_{2 m} (t,k)} \le \frac{k }{ \ln^2 k}\frac{1}{ (2 m+2)(2 m+1)}.$$ Therefore, in Case \[2ndcase\] we have $$\begin{aligned} {\lambda}_m = O\left(\frac{k }{ m \ln^2 k}\right) = O\left(\frac{1}{ \ln^2 k}\right) =o(1).\end{aligned}$$ ### Case \[3rdcase\] {#case3rdcase .unnumbered} In this range, we need more accurate estimates, as we will identify those $m$ for which ${\lambda}_m$ is approximately equal to 1. We appeal to Theorem \[coupons\] for asymptotic estimates of $C_{2 m}(t,k)$ and $C_{2 m+2}(t,k)$ and write ${\lambda}_m = (1 + o(1)) {\tilde {\lambda}}_m$ where $$\begin{aligned} \label{ratio} {\tilde {\lambda}}_m &= \frac{p}{1-p}\left(\frac{s(2 m/k)}{s(2(m+1)/k)}\right)^{1/2} \left(\frac{R(r_0(2(m+1)/k))}{R(r_0(2 m/k))} \right)^k \frac{r_0(2 m/k)^{2 m}}{r_0(2(m+1)/k)^{2 m+2}}(2 m+1). \nonumber \\ & \end{aligned}$$ Writing $2 m=t k - x k$, we have $x = o(1)$. So, by  and , uniformly for every $z \in [t-x, t-x+2/k]$, we have $$\left. \frac{d r_0}{d y}\right|_{y=z} = \frac{t}{x^2}(1+o(1));$$ thus, the Mean Value Theorem yields $$\begin{aligned} r_0(2(m+1)/k) & = r_0(2 m/k) + \frac{2t}{x^2 k}(1+o(1)) \stackrel{\text{\eqref{r_asympt}}}{=} r_0(2 m/k)\left(1 + \frac{2}{x k}(1+o(1))\right). \label{perturb1}\end{aligned}$$ So, since $x k \to \infty$ as $k \to \infty$, Equation  and  yield $$\begin{aligned} \label{1st_rat} \left(\frac{s(2 m/k)}{s(2(m+1)/k)}\right)^{1/2} = 1+o(1).\end{aligned}$$ To estimate the third ratio of , we write $r_0(2(m+1)/k) = r_0(2 m/k)(1 + \eta)$ where $\eta = (2/x k)(1+o(1))$ by . We also write $$\begin{aligned} R(r_0(2(m+1)/k) & = \frac{{r_0}^t(2(m+1)/k)}{t!} \sum_{t=0}^t \frac{t!}{(t-\ell)!} \frac{1}{{r_0}^{\ell}(2(m+1)/k)}.\end{aligned}$$ Note that $$\begin{aligned} \sum_{t=0}^t &\frac{t!}{(t-\ell)!} \frac{1}{{r_0}^{\ell}(2(m+1)/k)} = \sum_{t=0}^t \frac{t!}{(t-\ell)!} \frac{(1 + \eta)^{-\ell}}{{r_0}^{\ell}(2 m/k)} = \sum_{t=0}^t \frac{t!}{(t-\ell)!} \frac{1 - \ell \eta(1 + O(\eta))}{{r_0}^{\ell}(2 m/k)} \\ & = 1 + \frac{t}{r_0(2 m/k)} (1 - \eta) + \frac{t(t-1)}{{r_0}^2(2 m/k)} + O\left( \frac{\eta^2}{r_0(2 m/k)} + \frac{\eta}{{r_0}^2(2 m/k)} + \frac{1}{{r_0}^3(2 m/k)}\right).\end{aligned}$$ Since this last big-O term is $o(1/k)$, it follows that $$\begin{aligned} \frac{R(r_0(2(m+1)/k)}{r_0(2(m+1)/k)^t} & = \frac{1}{t!} \left(1 + \frac{t}{r_0(2 m/k)} (1 - \eta) + \frac{t(t-1)}{{r_0}^2(2 m/k)} + o(1/k)\right)\end{aligned}$$ and similar calculations show that $$\begin{aligned} \frac{R(r_0(2 m/k)}{r_0(2 m/k)^t} & = \frac{1}{t!} \left(1 + \frac{t}{r_0(2 m/k)} + \frac{t(t-1)}{{r_0}^2(2 m/k)} + o(1/k)\right).\end{aligned}$$ So the third ratio in  becomes $$\begin{aligned} \left(\frac{R(r_0(2(m+1)/k))}{R(r_0(2 m/k))} \right)^k & = \left(\frac{r_0(2(m+1)/k)}{r_0(2 m/k)} \right)^{t k} \left( 1 - \frac{t \eta}{r_0(2 m/k)} + o(1/k) \right)^k \nonumber\\ & = \left(\frac{r_0(2(m+1)/k)}{r_0(2 m/k)} \right)^{t k} e^{-2} (1 + o(1)) \label{rat2}\end{aligned}$$ where the last equality holds by the fact that $$\begin{aligned} \frac{t \eta k}{r_0(2 m/k)} = \frac{t (2/x k) k}{t/x} (1 + o(1)) = 2(1 + o(1)).\end{aligned}$$ Since $x k\to \infty$, we have by  and  that $r_0(2(m+1)/k)=r_0(2 m/k)(1+o(1)) = (1+o(1))t/x$. So using  and  we can write the product of the third and the fourth terms in  as follows: $$\begin{aligned} \left(\frac{R(r_0(2(m+1)/k))}{R(r_0(2 m/k))} \right)^k &\frac{{r_0}^{2 m}(2 m/k)}{{r_0}^{2 m+2}(2(m+1)/k)} \\ & = e^{-2} \left(\frac{r_0(2(m+1)/k)}{r_0(2 m/k)} \right)^{t k - 2 m} \frac{1 + o(1)}{{r_0}^2(2(m+1)/k)} \\ & = e^{-2} \left(1 + \frac{2}{x k}(1+o(1))\right)^{x k} \frac{x^2}{t^2}(1+o(1)) \stackrel{x k \to \infty}{=} \frac{x^2}{t^2}(1+o(1)). \end{aligned}$$ If $x\ge \omega(k)/\sqrt{k}$, where $\omega(k)\to \infty$, then substituting this last equation and  into  and recalling that ${\lambda}_m = (1 + o(1)){\tilde {\lambda}}_m$, we obtain $$\begin{aligned} \lambda_m = \Omega(1)\frac{x^2}{t^2} (2 m+1) = \Omega\left(\frac{\omega(k)^2 m}{k} \right) = \Omega(\omega(k)^2) \to \infty.\end{aligned}$$ If $x\le 1/(\omega(k) \sqrt{k})$, then these substitutions yield $$\lambda_m = O(1) \frac{x^2}{t^2} (2 m+1) = O\left( \frac{m}{\omega(k)^2 k} \right)= O\left(\frac{1}{\omega(k)^2}\right) = o(1).$$ Assume now that $x=\alpha/\sqrt{k}$, for some $\alpha = \Theta(1)$. In this case, $$\lambda_m = \frac{p}{1-p}\frac{\alpha^2}{t^2 k}(t k -x k + 1) (1+o(1)) = \frac{p}{1-p}\frac{\alpha^2}{t}(1+o(1))\stackrel{b=1/(1-p)}{=} \frac{b p \alpha^2}{t}(1+o(1)).$$ Thus for any fixed $\alpha' < \sqrt{t / b p}<\alpha'' $ and for $k$ large enough we have $tk - \alpha'' \sqrt{k} \leq 2 m^* \leq t k - \alpha'\sqrt{k}$. Putting all these different cases together, we deduce that, if $m^*$ is such that $f(m^*)$ is maximised over the set $0 \le 2 m \le t k$, then $ 2 m^* = t k - \sqrt{t k/b p} +o(\sqrt{k}).$ This concludes the proof of Lemma \[mstar\]. Before we proceed to the proof of Theorem \[Expectation\], let us use Lemma \[mstar\] to compute a precise asymptotic expression for $f(m^*)$. Recall that $b=1/(1-p)$ and observe that $$\begin{aligned} p^{m^*} (1-p)^{\binom{k}{2} - m^*} & = b^{-\binom{k}{2}} (b p)^{t k/2 - \sqrt{t k/b p} + o(\sqrt{k})} = b^{-\binom{k}{2}} (b p)^{t k/2} \left(1 + O\left(\frac{1}{\sqrt{k}}\right)\right)^k. \label{pmstar}\end{aligned}$$ For the second part of the expression for $f(m^*)$, note that, by Theorem \[coupons\], $$\begin{aligned} \label{C_expr} C_{2 m^*}(t,k) = \frac{1}{\sqrt{2\pi s(2 m^*/k)}}\frac{R(r_0(2 m^*/k))^k}{r_0(2 m^*/k)^{2 m^*}} (1+o(1)).\end{aligned}$$ By , we have $$\begin{aligned} r_0(2 m^*/k) = \sqrt{t b p k} + o(\sqrt{k}). \end{aligned}$$ Thus, by , $s(2 m^*/k)=\Theta (1/\sqrt{k})$. Now, it follows that $$\begin{aligned} R(r_0(2 m^*/k)) & = \frac{{r_0}^t(2 m^*/k)}{t!} \sum_{\ell=0}^{t} \frac{t!}{(t-\ell)!} \frac{1}{{r_0}^\ell(2 m^*/k)} \\ & = \frac{{r_0}^t(2 m^*/k)}{t!} \left( 1 + \frac{t}{r_0(2 m^*/k)} + O\left(\frac{1}{{r_0}^2(2 m^*/k)}\right) \right) \\ & = \frac{{r_0}^t(2 m^*/k)}{t!} \left( 1 + \sqrt{\frac{t}{b p k}} + o\left(\frac{1}{\sqrt{k}}\right) \right);\end{aligned}$$ therefore, $$\begin{aligned} R(r_0(2 m^*/k))^k & = \frac{(r_0(2 m/k))^{t k}}{t!^k} e^{\sqrt{t k/b p}+o(\sqrt{k})}.\end{aligned}$$ Substituting this into , we obtain $$\begin{aligned} C_{2 m^*}(t,k) &= \Theta(k^{1/4}) \frac{(r_0(2 m^*/k))^{t k - 2 m^*}}{t!^k} e^{\sqrt{t k/b p}+o(\sqrt{k})} \nonumber \\ & = \Theta(k^{1/4}) \left(\sqrt{t b p k} + o(\sqrt{k}) \right)^{\sqrt{t k/b p} + o(\sqrt{k})} e^{\sqrt{t k/b p}+ o(\sqrt{k})} \frac{1}{t!^k} \nonumber\\ & = \frac{1}{t!^k} \left(1 + O\left(\frac{\ln k}{\sqrt{k}}\right)\right)^k. \label{Coef}\end{aligned}$$ For the last part of the expression for $f(m^*)$, we apply Stirling’s formula to obtain $$\begin{aligned} \frac{(2 m^*)!}{m^*! 2^{m^*}} & = \frac{(2 m^*/e)^{2 m^*} \sqrt{2 \pi (2 m^*)} e^{o(1)}}{(m^*/e)^{m^*} \sqrt{2 \pi m^*} e^{o(1)}} \frac{1}{2^{m^*}} = \Theta(1) \left(\frac{2 m^*}{e}\right)^{m^*} \nonumber \\ & = \Theta(1) \left(\frac{t k-\sqrt{t k/b p}+o(\sqrt{k})}{e} \right)^{t k/2 - \sqrt{t k/b p}/2 +o(\sqrt{k})} \nonumber\\ & = \left(\frac{t k}{e}\right)^{t k/2} \left(1 + O\left(\frac{\ln k}{\sqrt{k}}\right)\right)^k. \label{Match}\end{aligned}$$ Now, substituting ,  and  into the expression for $f$ (given in ), we obtain the following: $$\begin{aligned} f(m^*) & = b^{-\binom{k}{2}} (b p)^{t k/2} \frac{1}{t!^k} \left(\frac{t k}{e}\right)^{t k/2} \left(1 + O\left(\frac{\ln k}{\sqrt{k}}\right)\right)^k \nonumber\\ & = \left( b^{-k+1} \left(\frac{t b p k}{e}\right)^t \frac{1}{t!^2} \right)^{k/2} \left(1 + O\left(\frac{\ln k}{\sqrt{k}}\right)\right)^k. \label{fmstar}\end{aligned}$$ Upper bound on ${\,\mathbb{E}}\left(\alpha_t^{(k)} (G_{n,p}) \right)$ {#upper-bound-on-mathbbeleftalpha_tk-g_np-right .unnumbered} --------------------------------------------------------------------- By  and , we deduce that $$\begin{aligned} {\,\mathbb{P}}(A\in S_t) \le \left(\frac{t k}{2} + 1\right) f(m^*) = \left( b^{-k+1} \left(\frac{t b p k}{e}\right)^t \frac{1}{t!^2} \right)^{k/2} \left(1 + O\left(\frac{\ln k}{\sqrt{k}}\right)\right)^k. \label{ProbtStable}\end{aligned}$$ Thus, we obtain, $$\begin{aligned} {\,\mathbb{E}}({\alpha_t^{(k)}(G_{n,p})}) & \le \binom{n}{k} \left( b^{-k+1} \left(\frac{t b p k}{e}\right)^t \frac{1}{t!^2} \right)^{k/2} \left(1 + O\left(\frac{\ln k}{\sqrt{k}}\right)\right)^k \nonumber \\ & = \left( e^2 n^2 b^{-k+1} k^{t-2} \left(\frac{t b p}{e}\right)^t \frac{1}{t!^2} \right)^{k/2} \left(1 + O\left(\frac{\ln k}{\sqrt{k}}\right)\right)^k. \label{FinalCalc}\end{aligned}$$ Now, if we set $ k=\lceil{\alpha_{t,p}(n)}+{\varepsilon}(n) \rceil $ for some function ${\varepsilon}(n) \gg \ln \ln n/\sqrt{\ln n}$, then, substituting this into , we obtain $$\begin{aligned} {\,\mathbb{E}}({\alpha_t^{(k)}(G_{n,p})}) \le \left( \left(1+O\left(\frac{\ln \ln n}{\ln n}\right)\right) b^{-{\varepsilon}} \right)^{k/2} \left(1 + O\left(\frac{\ln k}{\sqrt{k}}\right)\right)^k = o(1), \end{aligned}$$ thus proving the right-hand side inequality in Theorem \[1stability\]. Lower bound on ${\,\mathbb{E}}(\alpha_t^{(k)} (G_{n,p}) )$ {#lower-bound-on-mathbbealpha_tk-g_np .unnumbered} ---------------------------------------------------------- To derive the lower bound on ${\,\mathbb{E}}({\alpha_t^{(k)}(G_{n,p})})$, we observe $${\,\mathbb{E}}({\alpha_t^{(k)}(G_{n,p})}) \ge \binom{n}{k}{\,\mathbb{P}}(A \in S_t, \; e(A)=m^*).$$ Let $(d_1,\ldots ,d_k)$ be a degree sequence such that, for every $1\le i\le k$, $d_i\le t$ and $\sum_i d_i =2 m^*$. By Theorem 2.16 in [@Bol01], with ${\lambda}:=\frac{1}{m^*}\sum_i \binom{d_i}{2}$, the number of graphs with this degree sequence is $$(1+o(1)) e^{-{\lambda}/2-{\lambda}^2/4} \frac{(2 m^*)!}{m^*! 2^{m^*}}.$$ But, since $d_i\le t$ for every $i$, then using the estimate from Lemma \[mstar\] we obtain ${\lambda}\le t^2 k/2 m^* \le 2 t$ for $k$ large enough. So the total number of graphs on $k$ vertices, $m^*$ edges and with maximum degree at most $t$ is at least $$\frac{e^{-t - t^2}}{2} C_{2 m^*}(t,k)\frac{(2 m^*)!}{m^*!2^{m^*}}.$$ Since $k=O(\ln n)$, we have $\binom{n}{k}=\Omega(\sqrt{1/k})(n e/k)^k$. Hence, using , we obtain $$\begin{aligned} {\,\mathbb{E}}({\alpha_t^{(k)}(G_{n,p})}) & \ge \binom{n}{k} {\,\mathbb{P}}(A \in S_t, \; e(A)=m^*) \ge \binom{n}{k} \frac{e^{-t - t^2}}{2} f(m^*) \nonumber\\ & = \left( e^2 n^2 b^{-k+1} k^{t-2} \left(\frac{t b p}{e}\right)^t \frac{1}{t!^2} \right)^{k/2} \left(1 + O\left(\frac{\ln k}{\sqrt{k}}\right)\right)^k. \label{FinalCalcLow}\end{aligned}$$ If $k = \lfloor {\alpha_{t,p}(n)}-{\varepsilon}(n) \rfloor$ ($> {\alpha_{t,p}(n)}-{\varepsilon}(n)-1$) where ${\varepsilon}(n)$ is some function satisfying $\ln \ln n/\sqrt{\ln n} \ll {\varepsilon}(n) \ll \ln n$, then by  we obtain $$\begin{aligned} {\,\mathbb{E}}({\alpha_t^{(k)}(G_{n,p})}) & \ge \left( \left(1+O\left(\frac{\ln \ln n}{\ln n}\right)\right) b^{ {\varepsilon}(n)}\right)^{k/2} \left(1 + O\left(\frac{\ln k}{\sqrt{k}}\right)\right)^k = n^{{\varepsilon}(n)(1 + o(1)) } \rightarrow \infty. \label{explow}\end{aligned}$$ In the next section, we use a sharp concentration inequality to show moreover that the following holds. \[ExpConc\] If ${\varepsilon}(n) \gg \ln \ln n/\sqrt{\ln n}$ is a function that satisfies $\limsup_{n \to \infty} {\varepsilon}(n) \le 2$, then $${\,\mathbb{P}}\left( \alpha_t (G_{n,p}) < \lfloor {\alpha_{t,p}(n)}-{\varepsilon}(n) \rfloor \right) \le \exp \left( - n^{{\varepsilon}(n)(1+o(1))} \right).$$ Since the right-hand side is $o(1)$, we obtain the lower bound of Theorem \[1stability\]. This lemma will also be a key step in the proof of the upper bound of Theorem \[tchrom\], when we need the fact that the right-hand side tends to $0$ quickly. A second moment calculation - Proof of Lemma \[ExpConc\] {#2ndMom} ======================================================== Let $(x_n)$ be a bounded sequence of real numbers such that for $$k = 2\log_b n+(t-2)\log_b\log_b n+x_n \in \mathbb{N}$$ we have ${\,\mathbb{E}}({\alpha_t^{(k)}(G_{n,p})})\rightarrow \infty$ as $n \rightarrow \infty$. In this section, we prove that a.a.s. there is a $k$-subset of $V_n$ which is $t$-stable, using a second moment argument. For this, we use Janson’s Inequality ([@Jan90], [@JLR90] or Theorems 2.14, 2.18 in [@JLR00]): $$\begin{aligned} \label{Janson} {\,\mathbb{P}}({\alpha_t^{(k)}(G_{n,p})}= 0) \le \exp \left( - \frac{{\,\mathbb{E}}^2({\alpha_t^{(k)}(G_{n,p})})}{{\,\mathbb{E}}({\alpha_t^{(k)}(G_{n,p})}) + \Delta }\right),\end{aligned}$$ where $$\Delta = \sum_{A, B \subseteq V_n, k-1\ge |A \cap B|\ge 2} {\,\mathbb{P}}(A, B \in S_t).$$ Let $p(k,\ell)$ be the probability that two $k$-subsets of $V_n$ that overlap on exactly $\ell$ vertices are both in $S_t$. We write $$\begin{aligned} \Delta = & \sum_{\ell=2}^{k-\lfloor (t+3) \log_b \log_b n\rfloor} \binom{n}{k} \binom{k}{\ell} \binom{n - k}{k-\ell} p(k,\ell) \nonumber \\ & + \sum_{\ell=k-\lfloor (t+3) \log_b \log_b n \rfloor+1}^{k-1} \binom{n}{k} \binom{k}{\ell} \binom{n - k}{k-\ell} p(k,\ell) =: \Delta_1 + \Delta_2.\end{aligned}$$ We conclude the proof of Lemma \[ExpConc\] by showing that $$\Delta_1 = O\left(\frac{\ln^5 n}{n^2}\right) {\,\mathbb{E}}^2 ({\alpha_t^{(k)}(G_{n,p})}) \ \mbox{and} \ \Delta_2 = o({\,\mathbb{E}}({\alpha_t^{(k)}(G_{n,p})})).$$ By , ${\,\mathbb{E}}({\alpha_t^{(k)}(G_{n,p})}) \ge n^{{\varepsilon}(n)(1 + o(1))}$. If ${\,\mathbb{E}}({\alpha_t^{(k)}(G_{n,p})}) < \Delta$, then by the above estimates for $\Delta_1$ and $\Delta_2$, we have ${\,\mathbb{E}}({\alpha_t^{(k)}(G_{n,p})})+\Delta = O(\ln^5 n/n^2) {\,\mathbb{E}}^2({\alpha_t^{(k)}(G_{n,p})})$ and, by , $${\,\mathbb{P}}({\alpha_t^{(k)}(G_{n,p})}= 0) \le \exp \left( -\Omega \left(\frac{n^2}{\ln^5 n}\right)\right) = \exp\left(-n^{2 + o(1)}\right) \le \exp\left(-n^{{\varepsilon}(n)(1 + o(1))}\right)$$ (where the last inequality uses $\limsup_{n\to\infty}{\varepsilon}(n)\le 2$). Otherwise, we have ${\,\mathbb{E}}({\alpha_t^{(k)}(G_{n,p})})+\Delta\le2{\,\mathbb{E}}({\alpha_t^{(k)}(G_{n,p})})$ and $${\,\mathbb{P}}({\alpha_t^{(k)}(G_{n,p})}= 0) \le \exp \left( -\frac12 {\,\mathbb{E}}({\alpha_t^{(k)}(G_{n,p})})\right) \le \exp\left(-n^{{\varepsilon}(n)(1 + o(1))}\right).$$ ### [**Bounding**]{} $\Delta_1$ {#bounding-delta_1 .unnumbered} Let us begin by bounding $\Delta_1$, first estimating $p(k,\ell)$. Let $A$ and $B$ be two $k$-subsets of $V_n$ that overlap on exactly $\ell$ vertices, i.e. $|A \cap B|=\ell$. Then $p(k,\ell)={\,\mathbb{P}}(A,B \in S_t) = {\,\mathbb{P}}(A \in S_t \; | \; B \in S_t) {\,\mathbb{P}}(B \in S_t)$. The property of having maximum degree at most $t$ is monotone decreasing; so if we condition on the set $E$ of edges induced by $A\cap B$, then the conditional probability that $A \in \mathcal{S}_{t}$ is maximized when $E = \emptyset$. Thus, $${\,\mathbb{P}}\left( A \in S_{t} \ | \ B \in S_{t} \right) \le {\,\mathbb{P}}\left( A \in S_{t} \ | \ E = \emptyset \right) \le \frac{{\,\mathbb{P}}(A \in S_{t})}{{\,\mathbb{P}}(E = \emptyset)} = b^{\binom{\ell}{2}} {\,\mathbb{P}}(A \in S_{t}).$$ Therefore, $$\begin{aligned} p(k,\ell) = {\,\mathbb{P}}(A \in S_t \; | \; B \in S_t){\,\mathbb{P}}(B \in S_t) \le b^{\binom{\ell}{2}} \left({\,\mathbb{P}}(A \in S_t)\right)^2. \label{Bound}\end{aligned}$$ On the other hand, for every $\ell\le k$, $$\binom{k}{\ell} \binom{n-k}{k-\ell} \le k^\ell \frac{k^\ell}{(n-k)^{\ell}}\binom{n}{k}.$$ Using the estimate of  along with the above inequality, we have $$\begin{aligned} \label{Delta1} \Delta_1 & \le \left(\binom{n}{k}{\,\mathbb{P}}(A \in S_t) \right)^2 \ \sum_{\ell=2}^{k-\lfloor (t+3) \log_b \log_b n \rfloor} \left(\frac{k^2}{n-k}\right)^\ell b^{\binom{\ell}{2}}\nonumber \\ & \le {\,\mathbb{E}}^2 (\alpha_t^{(k)} (G_{n,p})) \sum_{\ell=2}^{k-\lfloor (t+3) \log_b \log_b n \rfloor} \left(\frac{k^2}{n-k}\right)^\ell b^{\binom{\ell}{2}}.\end{aligned}$$ If we set $s_{\ell} = (k^2/(n-k))^\ell b^{\binom{\ell}{2}}$, then $ s_{\ell + 1}/s_{\ell} = b^{\ell} k^2/(n-k). $ So the sequence $\{s_{\ell}\}$ is strictly decreasing for $\ell < \log_b (n-k) - 2 \log_b k$ and is strictly increasing for $\ell > \log_b (n-k) - 2 \log_b k$. So $$\max \{ s_{\ell} : 2 \le \ell \le k- \lfloor(t+3) \log_b \log_b n \rfloor \} \le \max \left\{ s_2 , s_{\lceil 2\log_b n-4.5\log_b \log_b n \rceil} \right\}.$$ We have that $s_2 = b k^4/(n-k)^2$, but $$\begin{aligned} s_{\lceil 2\log_b n-4.5\log_b \log_b n \rceil} &\le \left( \frac{k^2}{n-k} b^{\log_b n-2.25\log_b \log_b n}\right)^{2\log_b n-4.5\log_b \log_b n} \\ &\le \left( \frac{4 \log_b^2 n}{\log_b^{2.25} n} \right)^{2\log_b n-4.5\log_b \log_b n} \le \left( \frac{4}{\log_b^{0.25} n} \right)^{\log_b n} = o(s_2).\end{aligned}$$ Thus, Inequality  now becomes for $n$ large enough $$\begin{aligned} \Delta_1 \le \frac{b k^5}{(n-k)^2} {\,\mathbb{E}}^2 (\alpha_t^{(k)}(G_{n,p})) = O\left(\frac{\ln^5 n}{n^2}\right){\,\mathbb{E}}^2 (\alpha_t^{(k)}(G_{n,p})). \end{aligned}$$ ### [**Bounding** ]{} $\Delta_2$ {#bounding-delta_2 .unnumbered} Now, we will show that $\Delta_2 = o({\,\mathbb{E}}(\alpha_t^{(k)} (G_{n,p})))$. First, we have $$\begin{aligned} \binom{k}{\ell} \binom{n-k}{k-\ell} \le (k n)^{k-\ell}.\end{aligned}$$ We now give a rough estimate on $p(k,\ell)$. If $A, \ B$ are two $k$-sets of vertices that overlap on $\ell$ vertices (and if $\deg_S(v)$ denotes the number of neighbours of $v$ in $S$), then $$\begin{aligned} {\,\mathbb{P}}(B \in S_t \; | \; A \in S_t) & \le {\,\mathbb{P}}(\forall v \in B\setminus A, \ \deg_{A\cap B}(v) \le t) \le \left( \binom{\ell}{\ell-t}(1-p)^{\ell-t} \right)^{k-\ell} \\ & \le \left( k^{t} b^{t-\ell}\right)^{k-\ell} \le b^{\left(t\log_b k+t-k+\lfloor (t+3) \log_b \log_b n \rfloor \right)(k-\ell)}\\ & = b^{\left(-2 \log_b n + (t + 5) \log_b \log_b n + \Theta(1)\right)(k-\ell)} \le \left(\frac{\log_b^{t+6} n}{n^2}\right)^{k-\ell}.\end{aligned}$$ Substituting these estimates into the expression for $\Delta_2$, we obtain $$\begin{aligned} \Delta_2 & \le \binom{n}{k} {\,\mathbb{P}}(A\in S_t)\sum_{\ell=k-\lfloor (t+3) \log_b \log_b n \rfloor+1}^{k-1} \left(k n\frac{\log_b^{t+6} n}{n^2 } \right)^{k-\ell} \\ & \le {\,\mathbb{E}}(\alpha_t^{(k)} (G_{n,p})) k \left(\frac{k \log_b^{t+6} n}{n} \right) = o({\,\mathbb{E}}(\alpha_t^{(k)} (G_{n,p}))).\end{aligned}$$ The $t$-improper chromatic number {#chrom} ================================= The upper bound --------------- Our general approach follows Bollobás [@Bol88] — see also [@McD90]. We revisit the analysis in order to obtain an improved upper bound to match the lower bound of Panagiotou and Steger [@PaSt09]. For a fixed $0 < {\varepsilon}< 1$, we set $\hat{\alpha}_{t,p} (n) = \lfloor {\alpha_{t,p}(n)} -1 - {\varepsilon}\rfloor $. First, we will show the following. A.a.s. for all $V' \subseteq V_n$ with $|V'| \ge n/\ln^3 n$, we have $\alpha_t (G_{n,p}[V']) \ge \hat{\alpha}_{t,p}(|V'|)$. Note that  implies that for any $V'\subseteq V_n$ with $|V'| \ge n/\ln^3 n$, we have $${\,\mathbb{E}}\left( \alpha_{t,p}^{(\hat{\alpha}_{t,p}(|V'|))} (G_{n,p}[V']) \right) \ge |V'|^{1+{\varepsilon}+o(1)}.$$ So, applying Lemma \[ExpConc\], we deduce that $${\,\mathbb{P}}\left( \alpha_t (G_{n,p}[V']) < \hat{\alpha}_{t,p}(|V'|) \right) = \exp \left( - |V'|^{1+{\varepsilon}+o(1)}\right) \le \exp \left(-\left(\frac{n}{\ln^3 n}\right)^{1+{\varepsilon}+o(1)} \right).$$ Since there are at most $2^n$ choices for $V'$, the probability that there exists a set $V' \subseteq V_n$ with $|V'| \ge n/\ln^3 n$ and $\alpha_t (G_{n,p}[V']) < \hat{\alpha}_{t,p}(|V'|)$ is at most $2^n \exp \left( -(n/\ln^3 n)^{1+{\varepsilon}+o(1)}\right) = o(1)$. We consider the following algorithm for $t$-improperly colouring $G_{n,p}$. Let $V'=V_n$. While $|V'|\ge n/\ln^3 n$, we choose and remove a $t$-stable set from $G_{n,p}[V']$ of size $\hat{\alpha}_t (|V'|)$. At the end, we obtain a collection of $t$-stable sets and each of them will form a colour class. The above lemma implies that a.a.s. we will be able to perform this algorithm, and end up with a set of at most $n/\ln^3 n$ vertices. We give a different a colour to each of these vertices. Thus, if the above algorithm “runs" for $f(n)$ steps, then $\chi_t(G_{n,p}) \le f(n) + n/\ln^3 n$. Since $\alpha_{t,p}(s)- 1 - {\varepsilon}$ is strictly increasing for all $s$ that are sufficiently large, for these $s$ the function $\hat{\alpha}_{t,p} (s)$ is non-decreasing. It is easy to see that $$\begin{aligned} \hat{\alpha}_{t,p}\left(\left\lceil\frac{n}{\ln^3 n}\right\rceil\right) & = 2\log_b n \left( 1+O\left(\frac{\ln \ln n}{\ln n} \right)\right) = \hat{\alpha}_{t,p}(n) \left( 1+O\left(\frac{\ln \ln n}{\ln n} \right)\right).\end{aligned}$$ Since $\hat{\alpha}_{t,p} (\lceil n/\ln^3 n \rceil)\le \hat{\alpha}_{t,p}(s) \le \hat{\alpha}_{t,p}(n)$ for all integers $n/\ln^3 n \le s \le n$, $$\begin{aligned} \label{alphaest} \hat{\alpha}_{t,p} (s) = \hat{\alpha}_{t,p}(n) \left( 1+O\left(\frac{\ln \ln n}{\ln n} \right)\right),\end{aligned}$$ and therefore $$\begin{aligned} \label{fnrough} f(n)=\frac{n}{\hat{\alpha}_{t,p}(n)}\left(1+O\left(\frac{\ln \ln n}{\ln n}\right)\right).\end{aligned}$$ Assume that there are $n_i$ vertices available when we have removed $i$ $t$-stable sets from $V_n$. Thus, the $t$-stable set that will be picked during the $(i+1)$th iteration will have size $\hat{\alpha}_{t,p} (n_i)$. Since the colouring algorithm stops as soon as there are less than $n/\ln^3 n$ vertices available, the following inequality holds: $$\begin{aligned} \label{fnbound} \sum_{i=0}^{f(n)-2} \hat{\alpha}_{t,p}(n_i) \le n\left(1 - \frac{1}{\ln^3 n}\right) \le n. \end{aligned}$$ Note that for all $i\ge 0$, $n_i = n -\sum_{j=0}^{i-1} \hat{\alpha}_{t,p} (n_j)$. Therefore, $$\begin{aligned} \log_b n_i & = \log_b \left(n- \sum_{j=0}^{i-1} \hat{\alpha}_{t,p} (n_j)\right) = \log_b n + \log_b \left(1 - \frac{\sum_{j=0}^{i-1} \hat{\alpha}_{t,p} (n_j)}{n} \right).\end{aligned}$$ We have $$\begin{aligned} \sum_{i=0}^{f(n)-2} \log_b \left(1 - \frac{\sum_{j=0}^{i-1}\hat{\alpha}_{t,p} (n_j)}{n} \right) & =\frac{1}{\ln b}\sum_{i=0}^{f(n)-2} \frac{n}{\hat{\alpha}_{t,p}(n_i)} \ln \left(1 - \frac{\sum_{j=0}^{i-1} \hat{\alpha}_{t,p} (n_j)}{n} \right) \frac{\hat{\alpha}_{t,p}(n_i)}{n} \\ & \stackrel{\eqref{alphaest}}{=} \frac{n (1+o(1))}{\hat{\alpha}_{t,p}(n) \ln b} \sum_{i=0}^{f(n)-2} \ln \left(1 - \frac{\sum_{j=0}^{i-1} \hat{\alpha}_{t,p} (n_j)}{n} \right) \frac{\hat{\alpha}_{t,p}(n_i)}{n} \\ & =\frac{n (1+o(1))}{\hat{\alpha}_{t,p}(n) \ln b} \int_{0}^{1} \ln (1-x)d x = -\frac{n (1+o(1))}{\hat{\alpha}_{t,p}(n) \ln b}. \end{aligned}$$ So $$\begin{aligned} \label{logs} \sum_{i=0}^{f(n)-2} 2\log_b n_i = (f(n)-1)2\log_b n - \frac{2n (1+o(1))}{\hat{\alpha}_{t,p}(n) \ln b}.\end{aligned}$$ Also, $$\begin{aligned} \log_b \log_b n_i & \ge \log_b \log_b \left(\frac{n}{\ln^3 n}\right) = \log_b \log_b n + \log_b \left(1 - \frac{3\log_b \ln n}{\log_b n} \right) \\ & = \log_b \log_b n - O\left(\frac{\ln \ln n}{\ln n} \right).\end{aligned}$$ Moreover, $\log_b \log_b n_i \le \log_b \log_b n$ so, for every $t\ge 0$, $$\begin{aligned} \label{loglogn} (t-2) \sum_{i=0}^{f(n)-2} \log_b \log_b n_i \ge (f(n)-1)(t-2)\log_b \log_b n - O\left(\frac{f(n) \ln \ln n}{\ln n} \right).\end{aligned}$$ Now, Equality  and Inequality  imply that for every $t \ge 0$ we have $$\begin{aligned} \sum_{i=0}^{f(n)-2} \hat{\alpha}_{t,p} (n_i) & \ge (f(n)-1)\left( {\alpha_{t,p}(n)} - {\varepsilon}- 2 \right) -\frac{2n (1+o(1))}{\hat{\alpha}_{t,p}(n) \ln b} - O\left(\frac{f(n) \ln \ln n}{\ln n} \right) \\ & \ge (f(n)-1)\left( {\alpha_{t,p}(n)} -{\varepsilon}- 2 - \frac{2n (1+o(1))}{f(n) \hat{\alpha}_{t,p}(n) \ln b} - O\left(\frac{\ln \ln n}{\ln n} \right) \right) \\ & \stackrel{\eqref{fnrough}}{=} (f(n)-1) \left( {\alpha_{t,p}(n)} -{\varepsilon}- 2 - \frac{2}{\ln b}- o(1)\right). \end{aligned}$$ So by  we obtain $$\begin{aligned} f(n)-1\le \frac{n}{{\alpha_{t,p}(n)} - 2/\ln b -2 - {\varepsilon}- o(1)}. \end{aligned}$$ The lower bound --------------- This proof is the generalisation of a proof of the lower bound on the chromatic number of a dense random graph given recently by Panagiotou and Steger [@PaSt09]. We let $\alpha_C (n) = 2\log_b n + (t-2)\log_b \log_b n - C$, where $C =C_n > 2\log_b n + (t-2)\log_b \log_b n - {\alpha_{t,p}(n)}$ is some function which is $\Theta (1)$, such that $\alpha_C(n)$ is integral. We specify $C$ at a later stage. Let $r= r_C := \lfloor n/\alpha_C (n) \rfloor$. By Theorem \[1stability\], a.a.s. there are no $t$-stable sets in $G_{n,p}$ of size more than ${\alpha_{t,p}(n)}+1$. (In fact, according to Theorem \[1stability\], we could have used the bound ${\alpha_{t,p}(n)}+{\varepsilon}$, but this would not give any improvement.) We will estimate the expected number of $t$-improper colourings of $G_{n,p}$ with $r$ colours such that each colour set has size at most ${\alpha_{t,p}(n)}+1$. In particular, we show that, if $C < 2\log_b n + (t-2)\log_b \log_b n - {\alpha_{t,p}(n)} + 1 + 2/\ln b - {\varepsilon}$, then this expectation converges to zero, proving that $\chi_t (G_{n,p}) > r_C$ a.a.s. Let ${{\mathcal D}}$ denote the set of $r$-tuples of positive integers $(k_1,\ldots , k_r)$ such that $\sum_{i=1}^r k_i =n$ and $k_i \le {\alpha_{t,p}(n)} + 1$ for all $i$. For some $(k_1,\ldots , k_r) \in {{\mathcal D}}$, let ${{\mathcal P}}= (P_1,\ldots, P_r)$ denote a partition of $V_n$ into $r$ non-empty parts $P_1,\ldots, P_r$ such that $|P_i|=k_i$. From , we obtain $$\begin{aligned} {\,\mathbb{P}}(P_i\in S_t) \le \left( b^{-k_i+1} \left(\frac{t b p k_i}{e}\right)^t \frac{1}{t!^2} \right)^{k_i/2} \left(1 + O\left(\frac{\ln k_i}{\sqrt{k_i}}\right)\right)^{k_i}.\end{aligned}$$ $$\begin{aligned} {\,\mathbb{P}}(P_i \in S_t, \ \forall i) & = \prod_{i=1}^r {\,\mathbb{P}}(P_i \in S_t) \le \prod_{i=1}^r \left( b^{-k_i+1} \left(\frac{t b p k_i}{e}\right)^t \frac{1}{t!^2} \right)^{k_i/2} \left(1 + O\left(\frac{\ln k_i}{\sqrt{k_i}}\right)\right)^{k_i} \nonumber \\ &= b^{-\left(\sum_{i=1}^r {k_i}^2/2\right) + n/2} \left(\frac{t b p}{e}\right)^{t n/2}\left(\prod_{i=1}^r {k_i}^{t k_i/2} \right) \frac{1}{t!^{n}} (1 + o(1))^n \nonumber \\ &= \left(\frac{t b^{1 + 1/t}p}{e t!^{2/t}}\right)^{t n/2} b^{-\sum_{i=1}^r {k_i}^2/2} \left(\prod_{i=1}^r {k_i}^{t k_i/2}\right) (1 + o(1))^n,\end{aligned}$$ uniformly over all $(k_1,\ldots, k_r) \in {{\mathcal D}}$. So, if $X_{t,r} = X_{t,r}(G_{n,p})$ denotes the number of $t$-improper colourings with $r$ colours and with each colour class of size at most ${\alpha_{t,p}(n)} +1$, then $$\begin{aligned} \label{ExpectCols} {\,\mathbb{E}}(X_{t,r}) = \frac{1}{r!} \left(\frac{t b^{1 + 1/t}p}{e t!^{2/t}}\right)^{t n/2} \sum_{(k_1,\ldots, k_r) \in {{\mathcal D}}} \binom{n}{k_1 \cdots k_r} b^{-\sum_{i=1}^r\left(\frac{k_i^2}{2} - \frac{t}{2} k_i \log_b k_i\right)} (1 + o(1))^n.\end{aligned}$$ We call a partition where all parts differ by at most one pairwise *balanced*. In the next subsection, we give a routine proof of the following property of balanced partitions. \[MaxBal\] For large enough $n$, the function $$h(P) :=-\sum_{i=1}^r \left(\frac{k_i^2}{2} - \frac{t}{2} k_i \log_b k_i\right),$$ where $P=\{P_1,\ldots, P_r \}$ is a partition of $V_n$ with $|P_i|=k_i$, is maximised over ${{\mathcal D}}$ when $P$ is a balanced partition. Let $B$ be a balanced partition. Then all parts have sizes either equal to $\alpha_C (n)$ or to $\alpha_C(n)+1$ and there are less than $\alpha_C(n)$ parts that take the latter quantity. Then $$\begin{aligned} \label{Balanced} h(B) &= -\frac{n}{\alpha_C (n)} \left( \frac{\alpha_C^2(n)}{2} - \frac{t}{2} \alpha_C(n) \log_b \alpha_C(n) \right) +o(n) \nonumber \\ &= -\frac{1}{2} n \alpha_C(n) + \frac{t}{2}n \log_b \alpha_C(n) +o(n) \nonumber \\ &= -n \log_b n - \frac{t-2}{2}n \log_b \log_b n +\frac{Cn}{2} + \frac{t}{2}n \log_b 2 + \frac{t}{2}n \log_b \log_b n +o(n) \nonumber \\ &= -n \log_b n + n \log_b \log_b n + \frac{Cn}{2} + \frac{t}{2}n \log_b 2 + o(n).\end{aligned}$$ Also, for any $(k_1,\ldots, k_r) \in {{\mathcal D}}$, we have (for $n$ large enough) $$\begin{aligned} \binom{n}{k_1 \cdots k_r} &\le \frac{n!}{\left(\alpha_C(n)!\right)^{r}} = O(n^{1/2}) \frac{n^n}{(\alpha_C(n))^{n} \left(\sqrt{2\pi \alpha_C(n)}\right)^{r}} \le \frac{n^n}{(\alpha_C(n))^{n+r/2} } \nonumber \\ &= b^{n \log_b n - n \log_b \alpha_C(n) - \frac{r}{2} \log_b \alpha_C(n)} = b^{n \log_b n - n \log_b 2 - n \log_b \log_b n +o(n)} \label{Multi}\end{aligned}$$ since $r \log_b \alpha_C (n) \le (n/\alpha_C(n)) \log_b \alpha_C(n) =o(n)$. Finally, $r!\ge r^re^{-r}$ and therefore $$\begin{aligned} \label{factorial} \frac{1}{r!} \le b^{-r\log_b r +r\log_b e} = b^{-\frac{n}{\alpha_C(n)} \log_b \left(\frac{n}{\alpha_C(n)} \right) +o(n)} = b^{-n \frac{\log_b n}{\alpha_C(n)} +o(n)} = b^{-\frac{n}{2} + o(n)}.\end{aligned}$$ As there are at most $\binom{n}{r}\le (e n/r)^{r} \le \left({2 e \alpha_C(n)} \right)^{r} \le b^{r\log_b \alpha_C(n) + O(r)} = b^{o(n)}$ summands in , we obtain from ,  and  that $$\begin{aligned} {\,\mathbb{E}}(X_{t,r})&\le \left(\frac{t b^{1 + 1/t}p}{e t!^{2/t}}\right)^{t n/2} b^{\frac{Cn}{2} + \frac{t}{2}n \log_b 2- n \log_b 2 -\frac{n}{2}} (1 + o(1))^n\\ &= b^{ \frac{n}{2}\left(t\log_b (2 t p/e) + t -2 \log_b t!+ C - 2 \log_b 2 \right) } (1 + o(1))^n.\end{aligned}$$ Therefore, if $C = C_n < -\log_b (t^t/t!^2) - t\log_b (2 b p/e) - \log_b (1/4) - {\varepsilon}$, i.e. if $\alpha_C(n) > {\alpha_{t,p}(n)} - 2/\ln b -1 + {\varepsilon}$ for an arbitrary ${\varepsilon}>0$, then ${\,\mathbb{E}}(X_{t,r}) =o(1)$. Thus, a.a.s. $$\begin{aligned} \chi_t (G_{n,p}) & \ge \frac{n}{{\alpha_{t,p}(n)} -\frac{2}{\ln b} -1 + {\varepsilon}}. \end{aligned}$$ Proof of Lemma \[MaxBal\] ------------------------- Suppose $h(P)$, $P$, $k_i$ are defined as in Lemma \[MaxBal\] and furthermore assume that the parts of $P$ are ordered by increasing size, i.e., $k_1 \le \cdots \le k_r$. Let $\tilde{P}=\{\tilde{P}_1,\ldots, \tilde{P}_r \}$ be a partition of $V_n$ where for some $v \in P_r$ we have $\tilde{P}_1=P_1 \cup \{ v\}$ and $\tilde{P}_r=P_r\setminus \{ v\}$, whereas $\tilde{P}_i=P_i$ for all $1 < i < r$. In other words, we obtain $\tilde{P}$ by moving a vertex from $P_r$ to $P_1$. Lemma \[MaxBal\] easily follows from the repeated application of the following. \[Swap\] For large enough $n$, it holds that, if $k_1 < k_r - 1$, then $h(\tilde{P}) > h(P)$. First, $k_1 \le \alpha_C(n)$ and $k_r \ge \alpha_C(n) + 1$, since the number of parts is $r = \lfloor n /\alpha_C(n) \rfloor$. $$\begin{aligned} \label{Difference} 2(h(\tilde{P})-h(P)) = &-(k_1+1)^2 + t(k_1+1) \log_b (k_1 + 1) -(k_r-1)^2 + t(k_r-1) \log_b (k_r - 1) \nonumber \\ & + {k_1}^2 -t k_1 \log_b k_1 + {k_r}^2 - t k_r\log_b k_r \nonumber \\ = & 2(k_r-k_1-1) + t ((k_1+1) \log_b (k_1 + 1) - k_1 \log_b k_1) \nonumber \\ & + t((k_r-1) \log_b (k_r - 1) - k_r\log_b k_r). \end{aligned}$$ Note that $$\begin{aligned} (k_1+1) \log_b (k_1 + 1) & = (k_1+1) \log_b k_1 + (k_1+1) \log_b \left( 1+1/k_1 \right) \\ & \ge (k_1+1) \log_b k_1 + (k_1+1)\left( 1/k_1 - 1/(2 {k_1}^2) \right) \\ & = k_1 \log_b k_1 + \log_b k_1 + 1 + O\left(1/k_1 \right),\end{aligned}$$ and similarly, since $k_r\ge \alpha_C (n) + 1 \to \infty$ as $n \to \infty$, $$\begin{aligned} (k_r-1) \log_b (k_r - 1) & = k_r \log_b k_r - \log_b k_r + 1 - o(1).\end{aligned}$$ Substituting these estimates into , we obtain $$\begin{aligned} \label{Diff1} 2(h(\tilde{P})-h(P)) \ge 2(k_r-k_1-1) - t(\log_b k_r - \log_b k_1) + O\left( 1/k_1 \right).\end{aligned}$$ Assume first that $k_r - k_1 \le \ln \ln n$. Then $\log_b (k_r/k_1) \le \log_b (k_r/(k_r -\ln \ln n)) = \log_b ( 1 + \ln \ln n/(k_r -\ln \ln n) ) =o(1)$. But $k_r -k_1-1\ge 1$ and $k_1 \ge \alpha_C(n) + 1 - \ln \ln n$ and therefore the right-hand side of  is positive for $n$ large enough. If, on the other hand $k_r - k_1 > \ln \ln n$, we write $\log_b k_r = \log_b (k_r - k_1 + k_1) = \log_b (k_r-k_1) + \log_b \left(1+k_1/(k_r- k_1) \right)$. So $$\begin{aligned} \log_b k_r - \log_b k_1 & = \log_b (k_r-k_1) + \log_b \left( 1+k_1/(k_r- k_1) \right) - \log_b k_1 \\ & = \log_b (k_r-k_1) + \log_b \left( 1/k_1+1/(k_r- k_1) \right) \\ & = \log_b (k_r-k_1) + \log_b \left( 1/k_1+o(1) \right) \le \log_b (k_r-k_1) + 1.\end{aligned}$$ So $$2(k_r-k_1-1) - t (\log_b k_r - \log_b k_1) + O\left( 1/k_1 \right)\ge 2(k_r -k_1 -1) - t (\log_b (k_r-k_1) - 1) \rightarrow \infty$$ as $n \to \infty$ and, by , $h(\tilde{P})-h(P) >0$ for $n$ large enough. Acknowledgement {#acknowledgement .unnumbered} --------------- We would like to thank an anonymous referee for a number of helpful comments. [^1]: Part of this work was completed while this author was a doctoral student at the University of Oxford; part while he was a postdoctoral fellow at McGill University. He was supported by NSERC (Canada) and the Commonwealth Scholarship Commission (UK).

--- abstract: 'We study certain $q$-deformed analogues of the maximal abelian subalgebras of the group von Neumann algebras of free groups. The radial subalgebra is defined for Hecke deformed von Neumann algebras of the Coxeter group $(\mathbb{Z}/{2{\mathbb{Z}}})^{\star k}$ and shown to be a maximal abelian subalgebra which is singular and with Pukánszky invariant $\{\infty\}$. Further all non-equal generator masas in the $q$-deformed Gaussian von Neumann algebras are shown to be mutually non-unitarily conjugate.' address: - 'Mathematisch Instituut, Universiteit Utrecht, Budapestlaan 6, 3584 CD Utrecht, The Netherlands' - 'Institute of Mathematics of the Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warsaw, Poland' - 'Institute of Mathematics of the Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warsaw, Poland' author: - Martijn Caspers - Adam Skalski - Mateusz Wasilewski title: 'On MASAs in $q$-deformed von Neumann algebras' --- Introduction ============ Our aim is to investigate maximal abelian subalgebras in certain ${\rm II}_1$-factors that can be viewed as deformations of ${\textup{VN}}(\mathbb{F}_n)$. Our particular interest lies in the analysis of counterparts of the radial masa $A_r$ in ${\textup{VN}}(\mathbb{F}_n)$, studied for example in [@BocaRad] and in [@Stuart] (see also [@Trenholme]). The main open problem concerning the radial masa in ${\textup{VN}}(\mathbb{F}_n)$ is the question whether it is isomorphic to the generator masa(s); so far they share all the known properties, such as maximal injectivity, same Pukánszky invariant, etc. They are also known not to be unitarily conjugate (see Proposition 3.1 of [@Stuart]). More generally, radial masas have been studied for von Neumann algebras of groups of the type $({\mathbb{Z}}/_{n{\mathbb{Z}}})^{\star k}$ in [@Trenholme] and [@BocaRad]. Here we want to analyse the behaviour of counterparts of the radial/generator masa in some deformed versions of ${\textup{VN}}(\mathbb{F}_n)$ or ${\textup{VN}}(({\mathbb{Z}}/_{n{\mathbb{Z}}})^{\star k})$; more specifically in Hecke deformed von Neumann algebras of right-angled Coxeter groups ${{\textup{VN}}_q(W)}$ of Dymara ([@Dymara], see also [@Garncarek] and [@Caspers]) and in $q$-deformed Gaussian von Neumann algebras $\Gamma_q({\mathcal{H}_{\mathbb{R}}})$ of Bożejko, Kümmerer and Speicher ([@BozejkoSpeicher]). In the former case we can naturally define the radial subalgebra (and not the generator one), and in the latter the object that intuitively corresponds to the radial subalgebra is in fact obviously isomorphic to the generator one (as studied by Ricard in [@ricard05qfactor] and further by Wen in [@Wen] and Parekh, Shimada and Wen in [@qMaxInjective]). We show in Section \[SectionqGaussian\] however that the different generator masas inside the $\Gamma_q({\mathcal{H}_{\mathbb{R}}})$ are not unitarily conjugate. Note that another example of a counterpart of the radial subalgebra in ${\textup{VN}}(\mathbb{F}_n)$ was studied and shown to be maximal abelian and singular in [@AmauryRoland]. It was a von Neumann subalgebra of the algebra $L^{\infty}(O_N^+)$, which shares many properties with ${\textup{VN}}(\mathbb{F}_n)$, although very recently was shown to be non-isomorphic to the latter in [@MikeRoland]. The plan of the paper is as follows: after finishing this section with introducing certain notations, in Section 2 we define the radial subalgebra of the Hecke deformed von Neumann algebra ${{\textup{VN}}_q(W)}$ and show it to be maximal abelian. In Section 3 we compute its Pukánszky invariant and deduce its singularity. Finally Section 4 discusses the non-unitary-conjugacy of a (continuous family of) different generator masas in the $q$-deformed Gaussian von Neumann algebras. [**Notation.**]{} Throughout this paper by a [**masa**]{} we mean a maximal abelian von Neumann subalgebra of a given von Neumann algebra ${\mathsf{M}}$. Let $U({\mathsf{M}})$ be the group of unitaries in ${\mathsf{M}}$. For a (unital) subalgebra ${\mathsf{A}}\subseteq {\mathsf{M}}$ we define the [**normalizer**]{} of ${\mathsf{A}}$ in ${\mathsf{M}}$ as $$N_{{\mathsf{M}}}({\mathsf{A}}) = \{ u \in U({\mathsf{M}}) \mid u {\mathsf{A}}u^\ast \subseteq {\mathsf{A}}\}.$$ A subalgebra ${\mathsf{A}}\subseteq {\mathsf{M}}$ is called [**singular**]{} if $N_{{\mathsf{M}}}({\mathsf{A}}) \subseteq {\mathsf{A}}$. ${\mathbb{N}}_0$ denotes the natural numbers including 0. The radial Hecke masa {#Sect=Masa} ===================== In this section we show that right angled Hecke von Neumann algebras admit a radial algebra and prove that it is in fact a masa. Let $W$ denote a [**right-angled Coxeter group**]{}. Recall that this is the universal group generated by a finite set $S$ of elements of order 2, with the relations forcing some of the distinct elements of $S$ to commute, and some other to be free. This is formally encoded by a function $m:S \times S \setminus \{(s,s):s \in S\} \to \{2, \infty\}$ such that for all $s,t \in S, s \neq t$ we have $$(st)^{m(s,t)} = e$$ (and $(st)^{\infty}=e$ means that $s$ and $t$ are free; necessarily $m(s,t) = m(t,s)$). We will always associate to $W$ the length function $|\cdot|:W \to {\mathbb{N}}_0$ given by the generating set $S$. All the information about $W$ is encoded by a graph $\Gamma$ with a vertex set $V\Gamma= S$ and the edge set $E\Gamma=\{(s,t) \in S \times S: m(s,t)=2\}$. Let $q\in(0, 1]$ and put $ p = \frac{q-1}{q^\frac{1}{2}}$ (note that our convention on $q$ means that $p\leq 0$). The algebra ${{\mathbb{C}}_q[W]}$ is a \*-algebra with a linear basis $\{T_w: w \in W\}$ satisfying the conditions ($s \in S, w \in W$) $$T_s T_w = \begin{cases} T_{sw} & \textup{ if } |sw| > |w|, \\ T_{sw} + p T_w & \textup{ if } |sw| < |w|. \end{cases}$$ The algebra ${{\mathbb{C}}_q[W]}$ acts in a natural way (via bounded operators) on the space $\ell^2(W)$ and its von Neumann algebraic closure in $B(\ell^2(W))$ will be denoted by ${{\textup{VN}}_q(W)}$. The vector $\delta_e \in \ell^2(W)$ will sometimes be denoted by $\Omega$; the corresponding vector state $\tau:=\omega_\Omega$ on ${{\textup{VN}}_q(W)}$ is a faithful trace. More generally to any element $T \in {{\textup{VN}}_q(W)}$ we can associate its [**symbol**]{} $T\Omega$, and as $\Omega$ is a separating vector for ${{\textup{VN}}_q(W)}$ this correspondence is injective. Finally note that using the right action of the Hecke algebra on itself we can define another von Neumann algebra acting on $\ell^2(W)$, say ${{\textup{VN}}_q(W)}^r$. It is obviously contained in the commutant of ${{\textup{VN}}_q(W)}$; in fact Proposition 19.2.1 of [@Davisbook] identifies it with ${{\textup{VN}}_q(W)}'$. We will write in what follows $L$ to denote the cardinality of $S$. Hecke von Neumann algebras were first considered in [@Dymara] and [@DymaraEtAl] in order to study weighted $L^2$-cohomology of Coxeter groups. In [@DymaraEtAl] the authors raised a natural question: how large is the centre of ${{\textup{VN}}_q(W)}$? A precise answer for the right-angled case was found in [@Garncarek], where Garncarek showed the following result. \[Thm=Factoriality\] Let $\vert S \vert \geq 3$ and assume that $\Gamma$ is irreducible. Then for $q \in [ \rho, 1]$ the right-angled Hecke von Neumann algebra ${{\mathbb{C}}_q[W]}$ is a ${\rm II}_1$ factor and for $(0, \rho)$ we have that ${{\mathbb{C}}_q[W]}$ is a direct sum of a ${\rm II}_1$-factor and $\mathbb{C}$. Here $\rho$ is the radius of convergence of the fundamental power series $\sum_{k =0}^\infty \vert \{ w \in W \mid \vert w \vert = k \} \vert z^k$. In particular ${{\textup{VN}}_q(W)}$ is diffuse if and only if $q \in [\rho, 1]$. Further structural results were obtained in [@Caspers], [@CaspersConnes], [@CaspersFima] where for example non-injectivity, approximation properties, absence of Cartan subalgebras, the Connes embedding property and the existence of graph product decompositions were established for ${{\textup{VN}}_q(W)}$. In this paper we consider the special case $W=({\mathbb{Z}}_2)^{*L}$, i.e. the case where $m$ is constantly equal infinity. We assume also that $L\geq 3$. Here the main result of [@Garncarek], c.f. Theorem \[Thm=Factoriality\], says that ${{\textup{VN}}_q(W)}$ is a factor if and only if $q \in [\frac{1}{L-1},1]$, and results of [@Dykema] together with a calculation in Section 5 of [@Garncarek] show that for that range of $q$ we have ${{\textup{VN}}_q(W)}\approx {\textup{VN}}(\mathbb{F}_{\frac{2Lq}{(1+q)^2}})$, where ${\textup{VN}}(\mathbb{F}_s)$ for $s\geq 1$ denote the interpolated free group factors of Dykema and Radulescu. An element $T \in {{\textup{VN}}_q(W)}$ is said to be radial if for its symbol decomposition $T\Omega= \sum_{w \in W} c_w \delta_w$, where $c_w \in {\mathbb{C}}$ we have $c_w = c_v$ for every $v,w \in W$ with $l(v) = l(w)$. We say that $T$ has radius (at most) $n$ if the frequency support (i.e. the set of those $w \in W$ for which $c_w\neq 0$) of $T_w$ is contained in the ball $\{ w \in W: |w| \leq n \}$. Define $h \in {{\mathbb{C}}_q[W]}\subset {{\textup{VN}}_q(W)}$ by the formula $h = \sum_{s \in S} T_s$ and put ${\mathsf{B}}:= \{h\}''$. \[Prop=Radial\] The von Neumann algebra ${\mathsf{B}}$ coincides with the collection of all radial operators in ${{\textup{VN}}_q(W)}$. In particular the set of all radial operators forms an algebra. For each $n \in {\mathbb{N}}$ consider the radial operator $h_n:=\sum_{w \in W, \vert w \vert =n} T_w \in {{\mathbb{C}}_q[W]}$ and put $h_0:=I$. For each $n \in {\mathbb{N}}$, $n\geq 2$, we have $$\label{recurrence} \begin{split} h h_n = & \sum_{s \in S} \sum_{\vert w \vert =n, \vert sw \vert > \vert w \vert} T_s T_w + \sum_{s \in S} \sum_{\vert w \vert =n, \vert sw \vert < \vert w \vert} T_s T_w \\ =& \sum_{s \in S} \sum_{\vert w \vert =n , \vert sw \vert > \vert w \vert} T_{sw} + \sum_{s \in S} \sum_{\vert w \vert =n , \vert sw \vert < \vert w \vert} T_{sw} + \sum_{s \in S} \sum_{\vert w \vert =n , \vert sw \vert < \vert w \vert} p T_{w} \\ =& h_{n+1} + (L-1) h_{n-1} + p h_{n}. \end{split}$$ We also have $h^2 = h_2 + p h + L h_0$. This shows in particular that the algebra generated by $h$ consists of radial operators. Moreover viewing the above as a recurrence formula we see that each $h_n$ can be expressed as a polynomial in $h$ and $I$, so that the subspace $A$ generated by $\{h_n: n \in {\mathbb{N}}\}$ coincides with the unital $^*$-algebra generated by $h$. Further define the radial subspace $\ell^2(W)_r:=\{(c_w)_{w \in W} \in \ell^2(W): \forall_{w,v \in W, |w|=|v|}\, c_v = c_w\}$ and denote the orthogonal projection from $\ell^2(W)$ onto $\ell^2(W)_r$ by $P_r$. It is easy to see that $A\Omega$ is norm dense in $\ell^2(W)_r$. Thus the unique trace-preserving conditional expectation $\mathbb{E}$ onto $A''\subset {{\textup{VN}}_q(W)}$ is given by the formula $$\mathbb{E}(T) \Omega = P_r T\Omega, \;\;\; T \in {{\textup{VN}}_q(W)}.$$ This shows that the set of radial operators in ${{\textup{VN}}_q(W)}$ coincides with $A''$ and passing now to ultraweak closures we see that $h$ generates the von Neumann algebra of all radial operators. Note that the above fact is not true (even for $p=0$) for a general right-angled Coxeter group. Also note that formulae as (and the subsequent line in the proof) play a very relevant role in our proof of singularity in Section \[Sect=Singular\]. The first main theorem of this paper is based on the idea of Pytlik for the radial algebra in ${\textup{VN}}(\mathbb{F}_n)$ ([@Pytlik]; see also [@SinclairSmith]). By $R_h \in {{\textup{VN}}_q(W)}^r$ we understand the operator on $\ell^2(W)$ given by the *right* action of $\sum_{s\in S} T_s$. \[Lem=Eta\] For every $v,w \in W$ with $\vert v \vert = \vert w \vert$ and for every $\epsilon > 0$ there exists a vector $\eta \in \ell^2(W)$ such that $$\Vert e_v - e_w - (h \eta - R_h\eta ) \Vert_2 < \varepsilon.$$ We first assume that $w = az$ and $v = zb$ for some word $z \in W$ with $\vert z \vert = \vert v \vert -1$ and some letters $a, b \in S$. In the proof $x$ and $y$ will always be words in $W$ and summations are always over $x$ and $y$. Put for $k \in \mathbb{N}$ $$\psi_k = \sum_{ \vert x \vert = \vert y \vert = k, \vert x a \vert = \vert b y \vert = k+1} e_{x azb y} \in \ell^2(W),$$ and define also $\psi_0=e_{azb}$. Let $\delta > 0$. As for each $k \in {\mathbb{N}}$ there are $L(L-1)^{k-1}$ reduced words in $W$ of length $k$, $$\label{Eqn=PsiKEstimate} \Vert \left( \frac{1-\delta}{L-1} \right)^k \psi_k \Vert_2^2 \leq \left( \frac{1-\delta }{L-1 }\right)^{2k} (L-1)^{2k-2}L^2 \leq 4 (1- \delta)^{2k}.$$ This means that we can define $$\eta_{\delta} = \sum_{k = 0}^{\infty} \left( \frac{1-\delta}{L-1} \right)^k \psi_k \in \ell^2(W).$$ We claim that the vector $\eta_{\delta}$, for $\delta$ small enough (dependent on $\epsilon$) satisfies the condition of the lemma. To show that we need to analyse the actions of $h$ and $R_h$ on $\psi_k$. For $k \geq 1$ we have (the bracket term included; the brackets are there in order to define further vectors in the remainder of the proof) $$\label{Eqn=LeftAction} \begin{split} h \psi_k = & \sum_{s\in S} \sum_{\vert x \vert = \vert y \vert = k, \vert x a \vert = \vert b y \vert = k+1, \vert s x \vert = k+1} e_{ s x azb y}\\ & + \sum_{s\in S} \sum_{\vert x \vert = \vert y \vert = k, \vert x a \vert = \vert b y \vert = k+1, \vert s x \vert = k-1} e_{ s x azb y} \left( + p e_{ x azb y} \right). \end{split}$$ and similarly, for $k \geq 1$, $$\label{Eqn=RightAction} \begin{split} R_h \psi_k = & \sum_{s\in S} \sum_{\vert x \vert = \vert y \vert = k, \vert x a \vert = \vert b y \vert = k+1, \vert y s \vert = k+1} e_{ x azb y s}\\ & + \sum_{s\in S} \sum_{\vert x \vert = \vert y \vert = k, \vert x a \vert = \vert b y \vert = k+1, \vert ys \vert = k-1} e_{ x azb y s} \left( + p e_{ x azb y} \right). \end{split}$$ Finally $$\label{Eqn=HonZeroAction} h \psi_0 = e_{zb} + pe_{azb} + \sum_{s \in S\setminus\{a\}} e_{sazb}, \qquad R_h \psi_0 = e_{az} + p e_{azb} + \sum_{s \in S\setminus\{b\}} e_{azbs}.$$ We now analyze the ‘commutators’ $h \psi_k - R_h\psi_k $ and their sum. Note first that for each $k\in {\mathbb{N}}_0$ the summand in $h \psi_k$ given by $p e_{ x azb y}$ also occurs in $R_h \psi_k $. We define (compare to ), $$\phi_{1,0} = \sum_{s \in S\setminus\{a\}} e_{sazb}, \:\: \phi_{2,0} = e_{zb}, \:\: \chi_{1,0} = \sum_{s \in S\setminus\{b\}} e_{azbs}, \chi_{2,0} = e_{az}.$$ For $k \geq 1$ we set the following notation: let $\phi_{1,k}$ and $\phi_{2,k}$ be the two large sums on respectively the first and second line of , without the vectors between brackets. Similarly we define $\chi_{1,k}$ and $\chi_{2,k}$ to be the two large sums on respectively the first and second line of , without the vectors between brackets. Then we have for all $k \in {\mathbb{N}}_0$ $$\phi_{1,k} = \frac{1}{L-1} \chi_{2,k+1}, \qquad \chi_{1,k} = \frac{1 }{L-1} \phi_{2,k+1},$$ so that $$\phi_{1,k} - \frac{1-\delta}{L-1} \chi_{2,k+1} = \delta \phi_{1,k}, \qquad \chi_{1,k} - \frac{1-\delta}{L-1} \phi_{2,k+1} = \delta \chi_{1,k}.$$ Thus a version of the telescopic argument yields the equality $$\begin{split} h \eta_\delta - R_h \eta_\delta = & \sum_{k=0}^\infty \left( \frac{1-\delta}{L-1} \right)^k \left( \phi_{1,k} + \phi_{2, k} - \chi_{1,k} - \chi_{2,k} \right) \\ = & e_{zb} - e_{az} + \delta \left( \sum_{k=1}^\infty \left( \frac{1-\delta}{L-1} \right)^k \left( \phi_{1,k} - \chi_{1,k} \right) \right). \end{split}$$ As $\delta \searrow 0$ this can be shown via a similar $\ell^2$-counting estimate as above to converge in norm to $e_{zb} - e_{az} $. From this we conclude the claim. For general $v = v_1 \ldots v_n $ and $w = w_1 \ldots w_n$ with $v_n \not = w_1$ the proposition follows from a triangle inequality and an application of the proof above to each pair $w_k \ldots w_n v_1 \ldots v_{k-1}$ and $w_{k+1} \ldots w_n v_1 \ldots v_{k}$. In case $v_n = w_1$ one can apply the above to the pairs $v_k \ldots v_n b w_1 \ldots w_{k-2}$ and $v_{k+1} \ldots v_n b w_1 \ldots w_{k-1}$ for some letter $b \not = v_n$. We are ready to formulate the first main result in this section. \[Thm=MASA\] The radial algebra ${\mathsf{B}}$ is a masa in ${{\textup{VN}}_q(W)}$. Suppose that $T \in {\mathsf{B}}' \cap {{\textup{VN}}_q(W)}$ and write $T \Omega= \sum_{u \in W} c_u e_u$. Let $v,w \in W$ with $\vert v \vert = \vert w \vert$, let $\varepsilon > 0$ and let $\eta$ be as in Lemma \[Lem=Eta\]. Note that as $T$ commutes with $h$ we have $\langle T \Omega, h\eta - R_h\eta \rangle = \langle (hT - R_h T ) \Omega, \eta \rangle = \langle T(h-R_h) \Omega, \eta \rangle= 0$. Then we get $$\vert \langle T \Omega, e_v - e_w \rangle \vert \leq \vert \langle T \Omega, e_v - e_w + h \eta - R_h \eta \rangle \vert \leq \varepsilon.$$ As $\varepsilon > 0$ is arbitrary, we see that $c_w = c_v$. Thus $T$ is radial, which is equivalent to the fact that $T \in {\mathsf{B}}$ by Proposition \[Prop=Radial\]. The recurrence formula allows us to compute explicitly the distribution of $h$ with respect to the canonical trace. As the formula is valid only from $n=2$ we first define ‘new’ $h_0$ as $\frac{L}{{\widetilde}{L}}$, where ${\widetilde}{L}:=L-1$, so that with respect to the new variables it holds for all $n\in \mathbb{N}$. For simplicity assume that $q\in [\frac{1}{{\widetilde}{L}}, 1]$, so that ${{\textup{VN}}_q(W)}$ is a (finite) factor. Then the distribution of $h$ is continuous (as ${\mathsf{B}}$ is diffuse) and the main result of [@CohenTrenholme] implies that the corresponding density is given (up to a normalising factor) by $$\frac{{\widetilde}{L} \sqrt{4{\widetilde}{L} - (x-p)^2}}{\pi \left[-(x-p)^2 - p (2-L)(x-p) +p^2(L-1)+ L^2\right]} dx.$$ Note that for $p=0$ we obtain, as expected, the distribution of the radial element in the group $({\mathbb{Z}}_2)^{*L}$ as computed in Theorem 4 of [@CohenTrenholme]. The Pukánszky invariant and singularity of the Hecke MASA {#Sect=Singular} ========================================================= The Pukánszky invariant $\mathcal{P}({\mathsf{A}})$ of a masa ${\mathsf{A}}\subseteq {\mathsf{M}}$ is determined by the von Neumann algebra generated by all ${\mathsf{A}}$-${\mathsf{A}}$ bimodule homomorphisms of $L^2({\mathsf{M}})$. We refer to [@SinclairSmith] for further discussion of $\mathcal{P}({\mathsf{A}})$. In [@PopaScan] Popa showed that the Pukánszky invariant can be used to prove singularity of certain masas (and indeed this was successfully applied by Radulescu [@Radulescu] in order to obtain singularity of the radial masa in ${\textup{VN}}(\mathbb{F}_n)$). We will use this strategy in this section, following very closely the proof of [@Radulescu], to show that the Hecke radial masa discussed in Section \[Sect=Masa\] is singular. In particular we determine its Pukánszky invariant. We need some terminology. Let again $L\geq 3$, $W=({\mathbb{Z}}_2)^{*L}$, $q \in [\frac{1}{L-1},1]$ and let ${\mathsf{B}}$ be the radial subalgebra of the factor ${{\textup{VN}}_q(W)}$ (shown to be a masa in Theorem \[Thm=MASA\]). The Pukánszky invariant of ${\mathsf{B}}\subseteq {{\textup{VN}}_q(W)}$ is defined as the type of the von Neumann algebra $\langle h, R_h \rangle' \subseteq {\mathsf{B}}(\ell^2(W))$, where $h$ and $R_h$ were defined in Section \[Sect=Masa\]. Next we introduce the necessary notation in order to determine the Pukánszky invariant of ${\mathsf{B}}\subseteq {{\textup{VN}}_q(W)}$. We need to construct certain bases, which are inspired by Radulescu’s bases in free group factors (see [@Radulescu]). For $l \in \mathbb{N}_0$ let $q_l: {{\mathbb{C}}_q[W]}\rightarrow {{\mathbb{C}}_q[W]}$ be the natural projection onto the span of $\{T_w, \vert w \vert = l\}$. Write ${{\mathbb{C}}_q^l[W]}= q_l({{\mathbb{C}}_q[W]})$. As before set $h_l = \sum_{\vert w \vert = l} T_w$. We have for $m \geq 1$ (see and its subsequent line) $$\label{Eqn=ChiProduct} h_1 h_m = h_m h_1 = h_{m+1} + p h_m + (L_m-1) h_{m-1},$$ where $L_m = L$ if $m \geq 2$ and $L_m = L+1$ if $m = 1$. Let $$S_l = {\rm span} \{ q_l( h_1 x), q_l( x h_1) \mid x \in q_{l-1}({{\mathbb{C}}_q[W]}) \};$$ in particular $S_1 = {\mathbb{C}}h_1$. Further for $ l \in \mathbb{N}, \gamma \in {{\mathbb{C}}_q^l[W]}$, set $$\gamma_{m,n} = q_{m+n+l}(h_m \gamma h_n), \qquad m,n \in \mathbb{N}_0.$$ We also set $\gamma_{m,n} = 0$ in case $m <0$ or $n <0$. Finally for $l \in \mathbb{N}$ and $\gamma \in {{\mathbb{C}}_q^l[W]}\ominus S_l$ set $$X_\gamma = \overline{{\textrm{span}}}^{\Vert \: \Vert_2} \{ \gamma_{m,n} \mid m,n \in \mathbb{N}_0 \}\subset \ell^2(W).$$ The following Lemma \[Lem=Blackbox\] collects all computational results we need further. As all the (rather easy) arguments are basically contained in [@Radulescu Lemma 1] we merely sketch the proof; all other proofs we give in this section will then be self-contained. \[Lem=Blackbox\] We have the following: 1. \[Item=BB1\] For $\gamma \in {{\mathbb{C}}_q^l[W]}, l \geq 1, m \geq 1, n \geq 0$, we have $$h_1 \gamma_{m,n} = \gamma_{m+1,n} + p \gamma_{m,n} + (L-1) \gamma_{m-1, n}.$$ 2. \[Item=BB2\] For $\gamma \in {{\mathbb{C}}_q^l[W]}\ominus S_l, l \geq 2, m \geq 0, n \geq 0$, we have $$h_1 \gamma_{m,n} = \gamma_{m+1,n} + p \gamma_{m,n} + (L-1) \gamma_{m-1, n}.$$ (Note that only the case $m=0$ was not already covered by ). 3. \[Item=BB3\] For $\beta \in {{\mathbb{C}}_q^1[W]}\ominus S_1, n \geq 0$, we have $$h_1 \beta_{0,n} = \beta_{1,n} + p \beta_{0,n} - \beta_{0, n-1}.$$ 4. \[Item=BB4\] \[Item=BlackboxOne\] For $\gamma \in {{\mathbb{C}}_q^l[W]}, l \geq 1$ we have, $$\begin{split} & q_{l+m+n+1}( h_1 h_m \gamma h_n) = q_{l+n+m+1}( h_1 q_{l+m+n}(h_m \gamma h_n)), \qquad \:m,n \in \mathbb{N}, \\ & q_{l-m-n-1}( h_1 h_m \gamma h_n) = q_{l-m-n-1}( h_1 q_{l-m-n}(h_m \gamma h_n)), \qquad \: m,n \textrm{ such that } 0\leq m + n \leq l. \end{split}$$ 5. \[Item=BB5\] For $\gamma \in {{\mathbb{C}}_q^l[W]}, l \geq 1$ we have $q_l(h_1 q_{l+1}(h_1 \gamma)) = (L-1) \gamma$. 6. \[Item=BB6\] For $\beta \in {{\mathbb{C}}_q^1[W]}\ominus S_1$ we have $q_n(h_1 q_{n+1}(\beta h_n ) ) = - q_{n}(\beta h_{n-1} )$. 7. \[Item=BB7\] For all $\gamma \in {{\mathbb{C}}_q^l[W]}\ominus S_l, l \geq 2, n\in \mathbb{N}, m \geq 1$ we have $q_{l}( q_{m+n+l}(h_m \gamma h_n) h_{m+n} ) = 0$. The proofs of – are easy consequences of , see also [@Radulescu Lemma 1 (a) and (b)]. The proof of is essentially the same as [@Radulescu Lemma 1.(c)]. is a direct consequence of . and follow from and respectively. follows from and . The following theorem gives the cornerstone in our computation of the Pukánszky invariant. The idea is based on first showing that for suitable $\beta$ and $\gamma$ the mapping $T: X_\beta \rightarrow X_\gamma$ defined by the formula is bounded and invertible. Then one uses a basis transition to the respective basis $\{ h_m \beta h_n \}_{m,n \in \mathbb{N}} and \{h_m \gamma h_n\}_{ m,n \in \mathbb{N}}$ to show that $T$ is actually a ${\mathsf{B}}-{\mathsf{B}}$-bimodule map. \[Thm=PukanszkyCore\] Let $l \in {\mathbb{N}}$, $l \geq2$, let $\beta \in {{\mathbb{C}}_q^1[W]}\ominus S_1$ and let $\gamma \in {{\mathbb{C}}_q^l[W]}(W) \ominus S_l$. Then the following hold: 1. \[Item=PukanszkyOne\] There exists a bounded invertible linear map $T: X_\beta \rightarrow X_\gamma$ determined by $$\label{Eqn=TmnBasis} T: \beta_{m,n} \mapsto \gamma_{m,n} + \gamma_{m-1, n-1}, \qquad m,n \in \mathbb{N}_0.$$ 2. \[Item=PukanszkyTwo\] We have $X_\beta = \overline{{\mathsf{B}}\beta {\mathsf{B}}}^{\Vert \: \Vert_2}$ and $X_\gamma = \overline{{\mathsf{B}}\gamma {\mathsf{B}}}^{\Vert \: \Vert_2}$. Moreover the map $T$ defined by agrees with the linear map $$\label{Eqn=TchiBasis} T: h_m \beta h_n \mapsto h_m \gamma h_n, \qquad m,n \in \mathbb{N}_0.$$ The proof of Theorem \[Thm=PukanszkyCore\] proceeds through a couple of lemmas, which we prove in two separate subsections. Proof of Theorem \[Thm=PukanszkyCore\] part -------------------------------------------- The first statement of Theorem \[Thm=PukanszkyCore\] is essentially a consequence of the following orthogonality property. \[Lem=Orthogonality\] Let $l \in {\mathbb{N}}$, $l \geq2$ and let $\beta, \beta' \in {{\mathbb{C}}_q^1[W]}\ominus S_1$, $\gamma \in {{\mathbb{C}}_q^l[W]}\ominus S_l$, $\gamma' \in {{\mathbb{C}}_q^l[W]}, l \geq 2$. We have then for each $m,n,m',n' \in {\mathbb{N}}_0$ $$\label{Eqn=BetaOrthogonality} \langle \beta_{m,n}, \beta'_{m', n'} \rangle = \delta_{m +n, n'+m'} (L-1)^{m+n - \vert n - n' \vert} {(-1)^{\vert n - n' \vert} } \langle \beta, \beta' \rangle;$$ similarly, $$\label{Eqn=GammaOrthogonality} \langle \gamma_{m,n}, \gamma'_{m', n'} \rangle = \delta_{m, m'} \delta_{n, n'} (L-1)^{m+n} \langle \gamma, \gamma' \rangle.$$ Let us first prove . Firstly, as $\gamma_{m,n}$ (resp. $\gamma'_{m', n'})$ is in the range of $q_{m+n+l}$ (resp. $q_{m'+n'+l}$) we must have $m+n = m'+n'$ or else both sides of are non-zero. We claim that $$\label{Eqn=Ind} q_l(h_{m'} q_{m+n+l}(h_m \gamma h_n) h_{n'}) = \delta_{m, m'} \delta_{n, n'} (L-1)^{m+n} \gamma.$$ For $k := m+n = 0$ this is obvious. We proceed by induction on $k$ and assume the assertion for $k-1$. For $k \geq 1$ one of $m$ and $n$ is non-zero and we may assume without loss of generality that $m \not = 0$ (the proof for $n$ can be done in the same way, or one considers the adjoint of which interchanges the roles of $m$ and $n$). If the left hand side of is non-zero, then we must have that $m'$ is non-zero, because otherwise this expression reads $q_l( q_{m+n+l}(h_m \gamma h_n) h_{n+m})$ which is zero by Lemma \[Lem=Blackbox\] . Using together with the fact that $q_l(h_r q_{m+n+l}(x) h_n)=0$ for every $r < m$ and $x \in {{\mathbb{C}}_q[W]}$ and $q_{m+n+l}(h_s \gamma h_n) = 0$ for $s < m$, we get $$q_l(h_{m'} q_{m+n+l}(h_m \gamma h_n) h_{n'}) = q_l(h_{m'-1} h_1 q_{m+n+l}( h_1 h_{m-1} \gamma h_n) h_{n'}).$$ Using then Lemma \[Lem=Blackbox\] and for the first two of the following equalities and then the induction hypothesis yields $$\label{Eqn=InductiveStep} \begin{split} & q_l(h_{m'} q_{m+n+l}(h_m \gamma h_n) h_{n'}) = q_l(h_{m'-1} q_{m+n+l-1} (h_1 q_{m+n+l}( h_1 q_{m+n+l-1} (h_{m-1} \gamma h_n)) h_{n'}) \\ = & (L-1) q_l(h_{m'-1} q_{m+n+l-1}( h_{m-1} \gamma h_n) h_{n'}) = (L-1) (L-1)^{m+n-1} \delta_{m, m'} \delta_{n, n'} \gamma. \end{split}$$ This completes the proof of . Then using the fact that $h_{m'}$ and $h_{n'}$ are self-adjoint we get $$\label{Eqn=EndConclusion} \begin{split} & \langle \gamma_{m,n}, \gamma'_{m', n'} \rangle = \langle q_{m+n+l}(h_m \gamma h_n), q_{m'+n'+l}( h_{m'} \gamma' h_{n'} ) \rangle = \langle h_{m'} q_{m+n+l}(h_m \gamma h_n) h_{n'} , \gamma' \rangle \\ = & \langle q_l(h_{m'} q_{m+n+l}(h_m \gamma h_n) h_{n'}) , \gamma' \rangle = (L-1)^{m+n} \delta_{m, m'} \delta_{n, n'} \langle \gamma, \gamma' \rangle. \end{split}$$ Next we sketch the proof of ; it is largely the same as . The claim gets replaced by the equality $$\label{Eqn=Ind2} q_l(h_{m'} q_{m+n+l}(h_m \beta h_n) h_{n'}) = (L-1)^{\vert m +n \vert - \vert n -n'\vert} { (-1)^{\vert n - n' \vert} } \delta_{m+n, m'+n'} \beta.$$ Again the proof proceeds by induction with respect to $k := m+n = m' +n'$. The case $k=0$ is obvious so assume $k \geq 1$. First assume that both $m, m' \geq 1$. Similar to and using the same results from Lemma \[Lem=Blackbox\] we find that $$\label{Eqn=mnbiggerone} \begin{split} & q_l( h_{m'} q_{m+n+l}(h_m \beta h_n) h_{n'}) = q_l( h_{m'-1} h_1 q_{m+n+l}( h_1 h_{m-1} \beta h_n) h_{n'}) \\ = & (L-1) q_l( h_{m'-1} q_{m+n+l-1}( h_{m-1} \beta h_n) h_{n' -1} ) = (L-1)^{m+n - \vert n - n'\vert} { (-1)^{\vert n - n' \vert} } \delta_{m+n, m'+n'} \langle \beta, \beta' \rangle. \end{split}$$ The proof of the equality (disregarding the intermediate steps) for the case $n, n' \geq 1$ proceeds in the same manner (or follows by taking adjoints of which swaps the roles of $m,m'$ and $n, n'$). The only case that remains is then $m = 0$ and $n' = 0$ (again the case $m' = 0$ and $n = 0$ follows by taking adjoints, or by symmetry). Then $n \geq 1, m'\geq 1$ and using Lemma \[Lem=Blackbox\] for the second equality and then applying the induction hypothesis we obtain $$\begin{split} & q_1( h_{m'} q_{n+1}(\beta h_n) ) = q_1( h_{m'-1} q_{n}( h_1 q_{n+1}( \beta h_{n-1} h_1) ) ) \\ = & { -} q_1( h_{m'-1} q_{n}( \beta h_{n-1} ) ) = (L-1)^{m+n - \vert n - n'\vert} \delta_{m+n, m'+n'} { (-1)^{\vert n - n' \vert} } \langle \beta, \beta' \rangle. \end{split}$$ Then the lemma follows by replacing $\gamma$ by $\beta$ in . Recall the elementary fact (see [@Radulescu Lemma 5] for a proof) that for a real number $a, \vert a \vert <1$ there exist constants $B_a>0$ and $C_a>0$ such that for any $ k \in \mathbb{N}, \lambda_1, \ldots, \lambda_k \in \mathbb{C}$ we have $$\label{Eqn=SquareEquivalence} B_a \sum_{i=1}^k \vert \lambda_i \vert^2 \leq \sum_{i=1}^k \lambda_i \overline{\lambda}_j a^{\vert i - j \vert} \leq C_a \sum_{i=1}^k \vert \lambda_i \vert^2.$$ [*Proof of Theorem \[Thm=PukanszkyCore\] (\[Item=PukanszkyOne\]).*]{} By Lemma \[Lem=Orthogonality\] and we see that the assignment $\beta_{m,n} \mapsto \gamma_{m,n}$ extends to a bounded invertible linear mapping $T_0: X_\beta \rightarrow X_\gamma$. By Lemma \[Lem=Orthogonality\] we see that $S: X_\gamma \mapsto X_\gamma: \gamma_{m,n} \mapsto \gamma_{m-1,n-1}$ is bounded with norm $\Vert S \Vert \leq (L-1)^{-2}$. Therefore ${\textrm{Id}}_{X_\gamma} + S$ is bounded and invertible. As the composition $(I + S) \circ T_0$ is bounded and invertible and agrees with we are done. Proof of Theorem \[Thm=PukanszkyCore\] part -------------------------------------------- The following Lemma \[Lem=abcomputations\] is the crucial part of the proof of Theorem \[Thm=PukanszkyCore\] . \[Lem=abcomputations\] Let $l \geq 2$, $\beta \in {{\mathbb{C}}_q^1[W]}\ominus S_1$ and let $\gamma \in {{\mathbb{C}}_q^l[W]}\ominus S_l$. For every $m,n \in \mathbb{N}_0$ there exist certain constants $b_{k,j}^{m,n}, c_{k,j}^{m,n} \in \mathbb{R}$, $k=0,\ldots,m$, $j=0,\ldots n$ such that we have the expansions $$\label{Eqn=Decomposition} h_m \beta h_n = \sum_{k \leq m, j \leq n} b_{k,j}^{m,n} \beta_{k,j}, \qquad h_m \gamma h_n = \sum_{k \leq m, j \leq n} c_{k,j}^{m,n} \gamma_{k,j}.$$ Moreover, these constants satisfy the following equalities: $$\label{Eqn=CoeffDependence} c_{k,j}^{m,n} = b_{k,j}^{m,n} + b_{k+1,j+1}^{m,n}, \qquad m,n \in \mathbb{N}, k=0,\ldots, m, j= 0, \ldots, n,$$ where $b^{m,n}_{m+1, n+1} = 0$. If $m = 0$ and $n \in \mathbb{N}$ arbitrary, then the existence of decompositions is a consequence of Lemma \[Lem=Blackbox\]. The relation for $m = 0$ becomes $c_{k,j}^{0,n} = b_{k,j}^{0,n}$ which is a rather direct consequence of Lemma \[Lem=Blackbox\] as well. The proof proceeds by induction on $m$. Let $L_k = L$ if $k >1$ and let $L_1 = L+1$. We have by and then Lemma \[Lem=Blackbox\] and , $$\label{Eqn=BigComputation} \begin{split} & h_m \beta h_n = (h_1 - p) h_{m-1} \beta h_n - (L_{m-1}-1) h_{m-2} \beta h_n \\ = & (h_1 - p) \sum_{k = 0}^{m-1} \sum_{j = 0}^{n} b_{k,j}^{m-1, n} \beta_{k, j} - (L_{m-1}-1) \sum_{k = 0}^{m-2} \sum_{j = 0}^{n} b_{k,j}^{m-2, n} \beta_{k,j} \\ = & \sum_{k = 0}^{m-1} \sum_{j = 0}^{n} b_{k,j}^{m-1, n} (\beta_{k+1, j} + (L -1) \beta_{k-1, j}) - \sum_{j = 0}^n b_{0,j}^{m-1, n} \beta_{0, j-1} - (L_{m-1}-1) \sum_{k = 0}^{m-2} \sum_{j = 0}^{n} b_{k,j}^{m-2, n} \beta_{k,j} \\ = & \sum_{k = 0}^{m} \sum_{j = 0}^{n} ( b_{k-1,j}^{m-1, n} + (L -1) b_{k+1,j}^{m-1, n}) \beta_{k, j} - \sum_{j = 0}^{n-1} b_{0,j+1}^{m-1, n} \beta_{0, j} - (L_{m-1}-1) \sum_{k = 0}^{m-2} \sum_{j = 0}^{n} b_{k,j}^{m-2, n} \beta_{k,j} \\ \end{split}$$ This shows that for all $0 \leq k \leq m, 0 \leq j \leq n$ we obtain $$b_{k,j}^{m,n} = b_{k-1, j}^{m-1, n} + (L -1 ) b_{k+1, j}^{m-1, n} - (L_{m-1} -1) b_{k,j}^{m-2, n} - \delta_{k, 0} b_{0, j+1}^{m-1, n}.$$ Let $\delta_{k \geq 1}$ be 1 if $k \geq 1$ and 0 otherwise. We get then $$b_{k,j}^{m,n} + b_{k+1,j+1}^{m,n} = \delta_{k \geq 1} (b_{k-1, j}^{m-1, n} + b_{k, j+1}^{m-1, n}) + (L -1 ) (b_{k+1, j}^{m-1, n} + b_{k+2, j+1}^{m, n+1} ) - (L_{m-1} -1) (b_{k,j}^{m-2, n} + b_{k+1,j+1}^{m-2, n} ).$$ So that by induction $$\label{Eqn=BTransition} \begin{split} b_{k,j}^{m,n} + b_{k+1,j+1}^{m,n} = &\delta_{k \geq 1} c_{k-1, j}^{m-1, n} + (L -1 ) c_{k+1, j}^{m-1, n} - (L_{m-1} -1) c_{k,j}^{m-2, n} \\ = & c_{k-1, j}^{m-1, n} + (L -1 ) c_{k+1, j}^{m-1, n} - (L_{m-1} -1) c_{k,j}^{m-2, n}. \end{split}$$ Exactly as we computed (with the difference that Lemma \[Lem=Blackbox\] is replaced by Lemma \[Lem=Blackbox\] ) we get the equalities $$h_m \gamma h_n = \sum_{k = 0}^{m+1} \sum_{j = 0}^{n} ( c_{k-1,j}^{m-1, n} + (L -1) c_{k+1,j}^{m-1, n}) \gamma_{k, j} - (L_{m-1}-1) \sum_{k = 0}^{m-2} \sum_{j = 0}^{n} c_{k,j}^{m-2, n} \gamma_{k,j}.$$ Thus $$c^{m,n}_{k,j} = c_{k-1, j}^{m-1, n} + (L -1 ) c_{k+1, j}^{m-1, n} - (L_m -1) c_{k,j}^{m-2, n}.$$ Combining the above with gives $c^{m,n}_{k,j} = b_{k,j}^{m,n} + b_{k+1,j+1}^{m,n}$ for all $0 \leq k \leq m, 0 \leq j \leq n$. [*Proof of Theorem \[Thm=PukanszkyCore\] .*]{} Lemma \[Lem=abcomputations\] shows that ${\mathsf{B}}\gamma {\mathsf{B}}\subseteq X_{\gamma}$ and ${\mathsf{B}}\beta {\mathsf{B}}\subseteq X_\beta$ and hence the inclusions hold also for the $\Vert \: \Vert_2$-closures. For the converse inclusion proceed by induction: take $h_n \gamma h_m \in {\mathsf{B}}\gamma {\mathsf{B}}$ and assume that all vectors $h_r \beta h_s$ with $r < n, s \leq m$ are contained in $X_\gamma$ (if $n = 0$ then assume that $r \leq n, s < m$ and consider adjoints, or use a similar induction argument on $m$). By we have $$h_n \gamma h_m = (h_1 - p) h_{n-1} \gamma h_m - (L_n-1) h_{n-2} \gamma h_m \in h_1 X_{\gamma} + X_\gamma.$$ Here again $L_n = L$ if $n \geq 2$ and $L_1 = L+1$. So it suffices to show that $h_1 X_\gamma \subseteq X_\gamma$, but this is a consequence of Lemma \[Lem=Blackbox\] . The proof for $\beta$ instead of $\gamma$ is the same but uses Lemma \[Lem=Blackbox\] and for the latter argument. The fact that agrees with is now a direct consequence of Lemma \[Lem=abcomputations\]. Indeed, $$\begin{split} & T( h_m \beta h_n) = T\left( \sum_{k \leq m, j \leq n} b_{k,j}^{m,n} \beta_{k,j} \right) = \sum_{k \leq m, j \leq n} b_{k,j}^{m,n} (\gamma_{k,j} + \gamma_{k-1, j-1}) \\ = & \sum_{k \leq m, j \leq n} (b_{k,j}^{m,n} + b_{k+1,j+1}^{m,n} ) \gamma_{k,j} = \sum_{k \leq m, j \leq n} c_{k,j}^{m,n} \gamma_{k,j} = h_m \gamma h_n. \end{split}$$ Consequences of Theorem \[Thm=PukanszkyCore\] --------------------------------------------- Let ${\mathsf{B}}_r = \langle R_h \rangle''$ (note that as ${{\textup{VN}}_q(W)}$ is in the standard form on $\ell^2(W)$, it is also equal to $J {\mathsf{B}}J$, where $J$ is the anti-linear Tomita-Takesaki modular conjugation $\delta_x \mapsto \delta_{x^{-1}}$). For a vector $\gamma \in \bigcup_{l \in {\mathbb{N}}_0} {{\mathbb{C}}_q^l[W]}$ we let $p_\gamma$ be the central support in $({\mathsf{B}}\cup {\mathsf{B}}_r)''$ of the vector state $\omega_{\gamma, \gamma}$. The operator $p_\gamma$ is then given by the projection onto the closure of ${\mathsf{B}}\gamma {\mathsf{B}}$. \[Lem=OrthSupport\] If vectors $\xi, \xi' \in \cup_{l \geq 1} {{\mathbb{C}}_q^l[W]}\ominus S_l$ are orthogonal then $p_{\xi}$ and $p_{\xi'}$ are orthogonal projections. Let $\xi \in {{\mathbb{C}}_q^l[W]}\ominus S_l$ and let $\xi' \in {{\mathbb{C}}_q^{l'}[W]}\ominus S_{l'}$ with $l, l' \geq 1$. If $l = l'$ then the lemma follows directly from Lemma \[Lem=Orthogonality\]. So assume that $l \not = l'$ and say that $l' \leq l$. It suffices to show that $$\label{Eqn=XiOrth} \xi'_{r,s} \perp \xi_{m,n} \qquad \textrm{ for every } r,s, m,n \in \mathbb{N}_0.$$ If $m+n+l \not = r+s+l'$ this is obvious as then the images of $q_{m+n+l}$ and $q_{r+s+l'}$ are mutually orthogonal. We may then assume $m+n+l = r+s+l'$, so that $r+s \geq m+n$. If $m+n = 0$ then is obvious, as $\xi \perp S_l$ whereas $\xi'_{r,s} \in S_l$. But then note that $\xi'_{r,s} = (\xi'_{a, b} )_{r-a, s-b}$ for any $a =0,\ldots,r,$ $b=0, \ldots s$ such that $l'+a+b = l$. As $\xi'_{a,b} \in S_l$ we see from Lemma \[Lem=Orthogonality\] that $(\xi'_{a, b} )_{r-a, s-b} \perp \xi_{m,n}$. We can now state and prove the main result of this section. \[Thm=Pukanszky\] The von Neumann algebra $({\mathsf{B}}\cup {\mathsf{B}}_r)' (1 - p_\Omega)$ is homogeneous of type I$_\infty$. Because $({\mathsf{B}}\cup {\mathsf{B}}_r)''$ is abelian the commutant $({\mathsf{B}}\cup {\mathsf{B}}_r)'$ is necessarily of type I; moreover $({\mathsf{B}}\cup {\mathsf{B}}_r)'$ is a direct integral of type I factors (see [@DixmierBook] for direct integration). Let $(\xi_i)_{i \in {\mathbb{N}}}$ be an orthonormal basis in $\cup_{l \geq 1} {{\mathbb{C}}_q^l[W]}\ominus S_l$. By Lemma \[Lem=OrthSupport\] the projections $(p_{\xi_i})_{i\in {\mathbb{N}}}$ are mutually orthogonal and by Theorem \[Thm=PukanszkyCore\] they have the same central support in $({\mathsf{B}}\cup {\mathsf{B}}_r)'$. As by Lemma \[Lem=OrthSupport\] we have $\sum_{i\in {\mathbb{N}}} p_{\xi_i} = 1 - p_\Omega$ and $1- p_\Omega$ is central in $({\mathsf{B}}\cup {\mathsf{B}}_r)'$ (c.f. [@PopaScan Lemma 3.1]) we see that the central support of each $p_{\xi_i}$ in $({\mathsf{B}}\cup {\mathsf{B}}_r)'$ is $1 - p_\Omega$. This shows that $(1- p_\Omega) ({\mathsf{B}}\cup {\mathsf{B}}_r)'$ is a direct integral of I$_\infty$-factors (which by definition means that it is homogeneous of type I$_\infty$). Theorem \[Thm=Pukanszky\] is phrased in the literature as follows: the Pukánszky invariant of ${\mathsf{B}}$ is $\{ \infty \}$. This is because in the ${\mathsf{B}}$-${\mathsf{B}}$-bimodule $(1-p_\Omega) L^2({\mathsf{M}})$, the only factors occuring in the direct integral decomposition of the commutant of ${\mathsf{B}}\cup {\mathsf{B}}_r$ are infinite (and necessarily of type I). The radial subalgebra ${\mathsf{B}}$ is a singular MASA of ${{\textup{VN}}_q(W)}$. This follows from Theorem \[Thm=Pukanszky\] by [@PopaScan Remark 3.4]. Generator MASAS in $q$-deformed Gaussian von Neumann algebras {#SectionqGaussian} ============================================================= In this section we consider masas in a different deformation of the free group factors, i.e. so-called q-Gaussian algebras. The starting point of the construction of $q$-Gaussian algebras is a real Hilbert space ${\mathcal{H}_{\mathbb{R}}}$. We complexify it, obtaining a complex Hilbert space ${\mathcal{H}}$, and form an algebraic direct sum $\bigoplus_{n\geqslant 0} {\mathcal{H}}^{\otimes n}$, where ${\mathcal{H}}^{\otimes 0}={\mathbb{C}}$. Following [@BozejkoSpeicher] (see that paper for all facts stated below without proofs), we will define an inner product on this space using the parameter $q \in (-1,1)$. For each $n\in {\mathbb{N}}$ we define an operator $P_q^n: {\mathcal{H}}^{\otimes n} \to {\mathcal{H}}^{\otimes n}$ by the formula $P_q^n(e_1 \otimes\dots \otimes e_n) = \sum_{\pi \in S_n} q^{i(\pi)} e_{\pi(1)} \otimes \dots \otimes e_{\pi(n)}$, where $e_1,\dots, e_n \in {\mathcal{H}}$, $S_n$ is the permutation group on $n$ letters and $i(\pi)$ denotes the number of inversions in the permutation $\pi$. These operators are strictly positive, so they define an inner product on $\bigoplus_{n\geqslant 0} {\mathcal{H}}^{\otimes n}$ – the Hilbert space that we get after completion is called the $q$-Fock space and is denoted by ${\mathcal{F}_q}({\mathcal{H}})$. The direct sum decomposition of the $q$-Fock space allows us to define shift-like operators. Let $\xi \in {\mathcal{H}}$. We define the **creation operator** $a_{q}^{\ast}(\xi): {\mathcal{F}_q}({\mathcal{H}}) \to {\mathcal{F}_q}({\mathcal{H}})$ by $a_q^{\ast}(\xi) (e_1\otimes\dots\otimes e_n) = \xi \otimes e_1 \otimes\dots \otimes \dots e_n$. The **annihilation operator** $a_q(\xi):{\mathcal{F}_q}({\mathcal{H}}) \to {\mathcal{F}_q}({\mathcal{H}})$ is defined as the adjoint of $a_q^{\ast}(\xi)$. Using the definition of the $q$-deformed inner product we can find the formula for $a_q(\xi)$: $a_q(\xi)(e_1\otimes\dots\otimes e_n) = \sum_{i=1}^{n} q^{i-1} \langle\xi, e_i\rangle e_1 \otimes\dots \widehat{e_{i}} \dots\otimes e_n$, where $\widehat{e_i}$ means that the factor $e_i$ is omitted. All the above operators extend to bounded operators on ${\mathcal{F}_q}({\mathcal{H}})$. Creation and annihilation operators will allow us to define $q$-Gaussian algebras. Let ${\mathcal{H}_{\mathbb{R}}}$ be a real Hilbert space and let ${\mathcal{H}}$ be its complexification. The von Neumann subalgebra of $\textrm{B}({\mathcal{F}_q}({\mathcal{H}}))$ generated by the set $\{a_q^{\ast}(\xi) + a_q(\xi): \xi \in {\mathcal{H}_{\mathbb{R}}}\}$ is called the **$q$-Gaussian algebra** associated with ${\mathcal{H}_{\mathbb{R}}}$ and is denoted by $\Gamma_q({\mathcal{H}_{\mathbb{R}}})$. The vector $\Omega=1 \in {\mathbb{C}}\subset {\mathcal{H}}^{\otimes 0} \subset {\mathcal{F}_q}({\mathcal{H}})$ is called the **vacuum vector**. It is a cyclic and separating vector for $\Gamma_q({\mathcal{H}_{\mathbb{R}}})$ and the associated vector state $\omega(x):= \langle \Omega, x\Omega \rangle$ is a normal faithful trace on $\Gamma_q({\mathcal{H}_{\mathbb{R}}})$. For $q=0$ the assignment ${\mathcal{H}_{\mathbb{R}}}\mapsto \Gamma_q({\mathcal{H}_{\mathbb{R}}})$ is precisely Voiculescu’s free Gaussian functor. In particular $\Gamma_{0}({\mathcal{H}_{\mathbb{R}}}) \simeq \textrm{L}(\mathbb{F}_{\textrm{dim}({\mathcal{H}_{\mathbb{R}}})})$. We will study problems pertaining to conjugacy of masas in the $q$-Gaussian algebras. It is a nice feature of these objects that the orthogonal operators on ${\mathcal{H}_{\mathbb{R}}}$ give rise to automorphisms of $\Gamma_q({\mathcal{H}_{\mathbb{R}}})$. To introduce these automorphisms, we need to present the **first quantisation**. Let $T: {\mathcal{H}}\to {\mathcal{H}}$ be a contraction. The assignment $ \bigoplus_{k\geqslant 0} {\mathcal{H}}^{\otimes k} \ni e_1 \otimes\dots\otimes e_n \mapsto Te_1 \otimes \dots \otimes Te_n \in \bigoplus_{k\geqslant 0} {\mathcal{H}}^{\otimes k}$ extends to a contraction ${\mathcal{F}_q}(T): {\mathcal{F}_q}({\mathcal{H}}) \to {\mathcal{F}_q}({\mathcal{H}})$ and is called the first quantisation of $T$. If $U: {\mathcal{H}}\to {\mathcal{H}}$ is a unitary then ${\mathcal{F}_q}(U)$ is also a unitary. To work with $\Gamma_q({\mathcal{H}_{\mathbb{R}}})$ we need a convenient notation for its generators. For any $\xi \in {\mathcal{H}_{\mathbb{R}}}$ we put $W(\xi):=a_q^{\ast}(\xi) + a_q(\xi)$. If $\eta=\xi_1 + i \xi_2 \in {\mathcal{H}}$ then we denote $W(\eta)=W(\xi_1)+i W(\xi_2)$, therefore $W(\eta)$ is complex-linear in $\eta$. Recall that the vacuum vector $\Omega$ is cyclic and separating. One can check that for any vectors $\eta_1,\dots,\eta_n \in {\mathcal{H}}$ we have $\eta_1\otimes\dots\otimes \eta_n \in \Gamma_q({\mathcal{H}_{\mathbb{R}}})\Omega$; the unique operator $W(\eta_1\otimes\dots\otimes \eta_n) \in \Gamma_q({\mathcal{H}_{\mathbb{R}}})$ such that $W(\eta_1\otimes\dots\otimes \eta_n)\Omega = \eta_1\otimes\dots\otimes \eta_n$ is called a **Wick word**. The span of all such operators associated with finite simple tensors forms a strongly dense $\ast$-subalgebra of $\Gamma_q({\mathcal{H}_{\mathbb{R}}})$, which we call the **algebra of Wick words**. Finally note that similarly to Section 2 we can also consider the ‘right’ version of $\Gamma_q({\mathcal{H}_{\mathbb{R}}})$, generated by the combinations of right creation and annihilation operators, in particular containing the right Wick words, to be denoted $W_r(\xi)$. We are ready to introduce the **second quantisation**. Let ${\mathcal{H}_{\mathbb{R}}}$ be a real Hilbert space and let ${\mathcal{H}}$ be its complexification. Suppose that $T:{\mathcal{H}}\to {\mathcal{H}}$ is a contraction such that $T({\mathcal{H}_{\mathbb{R}}}) \subset {\mathcal{H}_{\mathbb{R}}}$. Then the assignment $\Gamma_q({\mathcal{H}_{\mathbb{R}}}) \ni W(\eta_1\otimes\dots\otimes \eta_n) \mapsto W(T\eta_1\otimes\dots\otimes T\eta_n) \in \Gamma_q({\mathcal{H}_{\mathbb{R}}})$, where $\eta_1,\dots,\eta_n \in {\mathcal{H}}$, may be extended to a normal, unital, completely positive map on $\Gamma_q({\mathcal{H}_{\mathbb{R}}})$, denoted by $\Gamma_q(T)$. Note that the condition $T({\mathcal{H}_{\mathbb{R}}}) \subset {\mathcal{H}_{\mathbb{R}}}$ is essential, otherwise $\Gamma_q(T)$ would not even preserve the adjoint, let alone be completely positive. We will only deal with automorphisms and, in this construction, they come from orthogonal operators on ${\mathcal{H}_{\mathbb{R}}}$. If $U: {\mathcal{H}_{\mathbb{R}}}\to {\mathcal{H}_{\mathbb{R}}}$ is orthogonal then $\Gamma_q(U)(x) = {\mathcal{F}_q}(U) x {\mathcal{F}_q}(U)^{\ast}$, where we still denote by $U$ its canonical unitary extension to ${\mathcal{H}}$. It is easy to check that $\Gamma_q(U) W(\xi) = W(U\xi)$. One can verify that none of these automorphisms is inner, besides the identity. To find candidates for masas, we draw inspiration from the case $q=0$, in which the most basic masas are the so-called generator masas. In our picture they correspond to subalgebras generated by a single element $W(\xi)$, where $\xi \in {\mathcal{H}_{\mathbb{R}}}$. In [@ricard05qfactor] Ricard proved they are also masas in the case of $q$-Gaussian algebras. As an application, he established factoriality of all $q$-Gaussian algebras $\Gamma_q({\mathcal{H}_{\mathbb{R}}})$ with $\textrm{dim}({\mathcal{H}_{\mathbb{R}}}) \geqslant 2$. Recently these generator masas were also shown to be singular ([@Wen]) and maximally injective [@qMaxInjective]. Using the automorphisms produced by the second quantisation procedure, we can easily show that all these masas are conjugate by an outer automorphism. Indeed, consider masas generated by $W(\xi)$ and $W(\eta)$, where $\xi,\eta \in {\mathcal{H}_{\mathbb{R}}}$. By rescaling, we may assume that $\|\xi\|=\|\eta\|=1$. Therefore one can find an orthogonal operator $U$ such that $U\xi=\eta$; then $\Gamma_q(U) ((W(\xi))'') = (W(\eta))''$. Our aim now is to show that they are never conjugate by a unitary. Case of orthogonal vectors -------------------------- We first want to deal with the case when $\mathsf{A}:= (W(e_{1}))''$ and $\mathsf{B}:=(W(e_{2}))''$ are masas in $\mathsf{M}:=\Gamma_{q}({\mathcal{H}_{\mathbb{R}}})$ coming from two orthogonal vectors. In the case $q=0$ these masas correspond to two different generator masas of the free group factor. One can prove that these are not unitarily conjugate using Popa’s notion of orthogonal pairs of subalgebras (cf. [@Popa Corollary 4.3]). We will use another technique due to Popa giving a criterion for embedding $\mathsf{A}$ into $\mathsf{B}$ inside $\mathsf{M}$ (in a certain technical sense). We will actually only state the part of the theorem that is useful for us; for the full statement consult [@PopaIntertwining Theorem 2.1 and Corollary 2.3]. Let $\mathsf{A}$ and $\mathsf{B}$ be von Neumann subalgebras of a finite von Neumann algebra $(\mathsf{M},\tau)$. Suppose that there exists a sequence of unitaries $(u_{k})_{k\in {\mathbb{N}}} \subset \mathcal{U}(\mathsf{A})$ such that for any $x,y \in \mathsf{M}$ we have $\lim_{k\to\infty} \|\mathbb{E}_{\mathsf{B}} (xu_{k} y)\|_{2}=0$, where $\mathbb{E}_{\mathsf{B}}$ is the unique $\tau$-preserving conditional expectation from $\mathsf{M}$ onto $\mathsf{B}$. Then there does not exist a unitary $u \in \mathsf{M}$ such that $u\mathsf{A} u^{\ast} = \mathsf{B}$. Note that it suffices to check that $\lim_{k\to\infty} \|\mathbb{E}_{\mathsf{B}} (xu_{k} y)\|_{2}=0$ only for $x,y \in \widetilde{\mathsf{M}}$, where $\widetilde{\mathsf{M}}$ is a strongly dense $\ast$-subalgebra. It follows from Kaplansky’s density theorem, because we can approximate in the strong operator topology (in particular in $L^2$) and control the norm of the approximants at the same time. \[Prop:orthogonal\] Let $e_1, e_2 \in {\mathcal{H}_{\mathbb{R}}}$, $\|e_1\|=\|e_2\|=1$, $e_1 \perp e_2$. Set $\mathsf{A}=(W(e_1))''$, $\mathsf{B}=(W(e_2))''$, and $\mathsf{M}=\Gamma_q({\mathcal{H}_{\mathbb{R}}})$. There exists a sequence of unitaries $(u_{k})_{k \in {\mathbb{N}}} \subset \mathcal{U}(\mathsf{A})$ such that we have $\lim_{k\to\infty} \|\mathbb{E}_{\mathsf{B}}(xu_k y)\|_{2}=0$ for all $x,y \in \widetilde{\mathsf{M}}$, where $\widetilde{\mathsf{M}}$ is the algebra of Wick words. Let $(e_n)_{n\in{\mathbb{N}}}$ be an orthonormal basis of ${\mathcal{H}_{\mathbb{R}}}$. Assume that $x=W(e_{i_{1}}\otimes\dots\otimes e_{i_{n}})$ and $y=W(e_{j_{1}}\otimes\dots\otimes e_{j_m})$; it clearly suffices because the span of such elements is equal to $\widetilde{\mathsf{M}}$. By definition of the trace on $\Gamma_q({\mathcal{H}_{\mathbb{R}}})$ we have $\|\mathbb{E}_{\mathsf{B}} (xu_k y)\|_{2} = \|(\mathbb{E}_{\mathsf{B}}(xu_k y))\Omega\|$. Since the conditional expectation on the level of the Fock space is just the orthogonal projection (denoted $P$) onto the closed linear span of the set $\{e_{2}^{\otimes n}:n\in {\mathbb{N}}\}$, we get $\|(\mathbb{E}_{\mathsf{B}}(xu_k y))\Omega\|= \|P(x u_k y\Omega)\|$. Note now that as the left and right actions of $y$ on $\Omega$ produce the same result, $e_{j_{1}}\otimes\dots\otimes e_{j_m}$, we can change $y$ to its right version, $W_{r}(e_{j_{1}}\otimes\dots\otimes e_{j_m})$, denoted now by $\widetilde{y}$. Since $\widetilde{y} \in \mathsf{M}'$, we get $\|P(x u_k y\Omega)\|=\|P(x \widetilde{y} u_{k}\Omega)\|$. We now choose the sequence $(u_k)_{k \in {\mathbb{N}}}$ – it is an arbitrary sequence of unitaries in $\mathsf{A}$ such that the corresponding vectors $\eta_k:=u_{k}\Omega$ converge weakly to zero (such a sequence exists, because $\mathsf{A}$ is diffuse). Let $Q_{l}$ be the orthogonal projection from $\mathcal{F}_{q}({\mathbb{C}}e_{1})$ onto $\textrm{span}\{e_{1}^{\otimes j}: j \leqslant l\}$. Then for any $l$ the sequence $(Q_{l} \eta_{k})_{k \in {\mathbb{N}}}$ converges to zero in norm. Therefore to check that $\lim_{k\to\infty} \|P(x\widetilde{y} \eta_k)\|=0$, it suffices to do it for $\eta_k$ replaced by $(\mathds{1}-Q_l)\eta_k$. We now choose $l=n+m$. Therefore any $\eta_k$ consists solely of tensors $e_{1}^{\otimes d}$, where $d\geqslant n+m+1$. Since $x$ can be written as a sum of products of $n$ (in total) creation and annihilation operators and $y$ can be decomposed similarly into products of $m$ creation and annihilation operators, any simple tensor appearing in $x\widetilde{y}(\mathds{1} - Q_{n+m})\eta_{k}$ will contain at least one $e_{1}$. But all such simple tensors are orthogonal to ${\mathcal{F}_q}({\mathbb{C}}e_{2})$, so they are killed by $P$. There does not exist a unitary $u \in \Gamma_q({\mathcal{H}_{\mathbb{R}}})$ such that $u(W(e_1))''u^{\ast}=(W(e_{2}))''$, where the vectors $e_{1}$ and $e_{2}$ are orthogonal. General case ------------ Let us check now if the method used for a pair of orthogonal vectors can be applied in a more general setting. Assume now that $e_{1}$ and $v$ are two unit vectors and write $v=\alpha e_{1} + \beta e_{2}$, where $e_{2} \perp e_{1}$, $\alpha^2+\beta^2=1$, and $\beta \neq 0$. We fix now an orthonormal basis $(e_{n})_{n \in {\mathbb{N}}}$ of ${\mathcal{H}_{\mathbb{R}}}$ (if ${\mathcal{H}_{\mathbb{R}}}$ is finite-dimensional then this should be a finite sequence). The masas $\mathsf{A}:= W(v)''$ and $\mathsf{B}:=(W(e_1))''$ are not unitarily conjugate. We proceed exactly as in the proof of Proposition \[Prop:orthogonal\] and also use the same notation; note however that this time $P$ will be the orthogonal projection onto $\overline{\text{span}}\{e_1^{\otimes n}: n\geqslant 0\}$. The only problem is that now we do not have orthogonality. Write $\eta_{k} = \sum_{j\in{\mathbb{N}}} a_{j}^{(k)} v^{\otimes j}$. We have $\|v^{\otimes j}\|\simeq \left(\frac{1}{\sqrt{1-q}}\right)^{j}$ (cf. [@ricard05qfactor Third displayed formula on page 660]). Let us compute $v^{\otimes j}$: $$v^{\otimes j} = \sum_{k=0}^{j} \alpha^{j-k} \beta^{k} R_{j,k}(e_{1}^{\otimes (j-k)}\otimes e_{2}^{\otimes k}),$$ where $R_{j,k}(e_{1}^{\otimes (j-k)} \otimes e_2^{\otimes k})$ is equal to the sum of all simple tensors such that $j-k$ factors are equal to $e_{1}$ and $k$ factors are equal to $e_{2}$; there are $\binom{j}{k}$ such simple tensors. Note now that if $k\geqslant n+m+1$ then after applying $x\widetilde{y}$ at least one $e_2$ remains as a factor, so the orthogonal projection $P$ kills it. We conclude that it suffices to perform the summation in the displayed formula above only up to $j\wedge (n+m)$; we call the resulting tensors $\widetilde{v}^{\otimes j}$ and the corresponding $\eta_k$ is dubbed $\widetilde{\eta}_k$. Since $k$ is bounded, the number $\binom{j}{k}$ is polynomial in $j$, so if we get exponential decay of the norm of the individual factors in the sum, the factor $\binom{j}{k}$ does not affect the overall convergence. After neglecting the terms with $k > n+m$, we use the trivial estimate $\|P(x\widetilde{y} \widetilde{\eta}_k)\|\leqslant C \|\widetilde{\eta}_k\|$. The proof will be completed if we show that $\Vert \widetilde{\eta}_k\Vert$ converges to $0$. Note now that the square of the norm of $\widetilde{\eta}_k$ is equal to $\sum_{j\in {\mathbb{N}}} |a_{j}^{(k)}|^2 \cdot \|\widetilde{v}^{\otimes j}\|^2$. Recall that $\|\eta_k\|\leqslant 1$ and $\|v^{\otimes j}\| \simeq \left(\frac{1}{\sqrt{1-q}}\right)^{j}$, so the coefficients $a_{j}^{(k)}$ satisfy $\sum_{j\in {\mathbb{N}}} |a_{j}^{(k)}|^2 \left(\frac{1}{1-q}\right)^{j} \lesssim 1$. It therefore suffices to show that $\lim_{j\to \infty} (1-q)^{j} \|\widetilde{v}^{\otimes j}\|^2=0$, remembering that the vectors $\eta_k$ converge weakly to $0$, so we only care about large $j$. We estimate the norm of $\widetilde{v}^{\otimes j}$ by the triangle inequality: $$\|\widetilde{v}^{\otimes j}\| \leqslant \sum_{k=0}^{j \wedge (n+m)} |\alpha|^{j-k} |\beta|^{k} \binom{j}{k} \|e_{1}^{\otimes k}\otimes e_{2}^{j-k}\|.$$ Since $k$ is bounded, one can easily get an estimate of the form $\|e_{1}^{\otimes k} \otimes e_{2}^{\otimes (j-k)}\| \leqslant C \left(\frac{1}{\sqrt{1-q}}\right)^{j}$ (cf. [@ricard05qfactor Remark 2]). This yields $\|\widetilde{v}^{\otimes j}\| \leqslant C\left(\frac{1}{\sqrt{1-q}}\right)^{j} |\alpha|^{j} \cdot j^{k}$. It is the inequality that we wanted, i.e. we find out that $(1-q)^{j} \|\widetilde{v}^{\otimes j}\|^2$ is bounded by $C j^{k} |\alpha|^{j}$, which converges to zero very fast, as we assumed that $|\alpha|<1$. This finishes the proof of the proposition. The result above exhibits in particular explicitly a continuuum of non-mutually conjugate singular masas in $\Gamma_q({\mathcal{H}_{\mathbb{R}}})$ (in fact the proofs show that they do not even embed into each other inside $\Gamma_q({\mathcal{H}_{\mathbb{R}}})$, see [@PopaIntertwining Theorem 2.1 and Corollary 2.3]). Very recently Popa showed in [@Popanew] the existence of such uncountable families in a broad class of von Neumann algebras. Generator masas can be also studied for the so-called mixed q-Gaussians (cf. [@Speicher]). They are known to be masas by [@AdamSimeng], and in fact an application of methods of that paper and general results of [@bikrammukherjee16qawfactor] show that they are singular, as noted by Simeng Wang. There seems to be however nothing known about the ‘radial’ subalgebra in this more general context. Is it a masa? Is it isomorphic to a generator one? Acknowledgements {#acknowledgements .unnumbered} ---------------- For the first author: {width="10mm"} This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 702139. ------------------------------------ ---------------------------------------------------------------------------------------------------------------------------------------------- The second author was partially supported by the National Science Centre (NCN) grant no. 2014/14/E/ST1/00525. The third author was partially supported by the National Science Centre (NCN) grant no. 2016/21/N/ST1/02499. The work on the paper was started during the visit of the first author to IMPAN in January 2017, partially funded by the Warsaw Center of Mathematics and Computer Science. 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--- abstract: 'We introduce a general class of continuous univariate distributions with positive support obtained by transforming the class of two-piece distributions. We show that this class of distributions is very flexible, easy to implement, and contains members that can capture different tail behaviours and shapes, producing also a variety of hazard functions. The proposed distributions represent a flexible alternative to the classical choices such as the log-normal, Gamma, and Weibull distributions. We investigate empirically the inferential properties of the proposed models through an extensive simulation study. We present some applications using real data in the contexts of time-to-event and accelerated failure time models. In the second kind of applications, we explore the use of these models in the estimation of the distribution of the individual remaining life.' author: - '[Francisco J. Rubio[^1]]{} and Yili Hong[^2]' title: Survival and lifetime data analysis with a flexible class of distributions --- [*Key Words: AFT model; composite models; hazard function; logarithmic transformation; remaining life..*]{} Introduction {#sec:intro} ============ In many areas, including medical applications, the quantities of interest take positive values. For instance, in survival analysis, the interest typically focuses on modelling the survival times of a group of patients in terms of a set of covariates (see *e.g.* [@L03]). Other areas where positive observations appear naturally are finance (*e.g.* in modelling the size of reinsurance claims), network traffic modelling [@M01], reliability theory [@ME98], environmental science [@MG10], among many others. Parametric distributions provide a parsimonious way of describing the distribution of those quantities. Some of the most popular choices for modelling positive observations are the lognormal, log-logistic, Gamma, and Weibull distributions. We refer the readers to [@M12] for an extensive overview of these sorts of distributions as well as a study of their inferential properties in the presence of censored observations. However, these distributions do not always provide a good fit of the data. For example, when the data present heavier tails and/or a different shape around the mode than those captured by these distributions. In recent years, there has been an increasing interest in the development of flexible distributions with positive support in order to cover departures from the classical choices. Two popular strategies for generating new flexible distributions with positive support consist of: *Adding a shape parameter to an existing distribution with positive support*. For instance, in the context of reliability and survival analysis, [@MO97] proposed a transformation of a distribution $F(y;\theta)$, $y>0$, that introduces a new parameter $\gamma>0$. This transformation is defined through the cumulative distribution function (CDF) $$\begin{aligned} \label{MOdistribution} G(y;\theta,\gamma)=\dfrac{ F(y;\theta)}{F(y;\theta)+\gamma(1-F(y;\theta))}.\end{aligned}$$ The interpretation of the parameter $\gamma$ is given in [@MO97] in terms of the behavior of the ratio of hazard rates of $F$ and $G$. This ratio is increasing in $y$ for $\gamma\geq 1$ and decreasing in $y$ for $0<\gamma\leq 1$. This transformation is then proposed for the exponential and Weibull distributions in [@MO97] in order to generate more flexible models for lifetime data. Clearly, for $\gamma=1$, $G$ and $F$ coincide. Many choices of $F(;\theta)$ have already been studied in the literature. We refer the reader to [@FS06] for a general mechanism for adding parameters to a distribution. *Using transformations from ${\mathbb R}$ to ${\mathbb R}_+$*. The most common choice for this transformation is the exponential function. The idea is to define a positive variable $Y$ by transforming a real variable $X$ through $Y=\exp(X)$. This method is used to produce the class of log-symmetric distributions. This is, the family of positive random variables such that their logarithm is symmetrically distributed. Some members of this class are the lognormal, log-logistic, and log-Student-$t$ distributions, which are obtained by transforming the normal, logistic, and Student-$t$ distributions, respectively. More recently, other families of distributions have been proposed by using this idea, such as the log Birnbaum-Saunders distribution [@B08], log skew-elliptical distributions [@MG10], log-generalised extreme value distributions [@RD15], and log-scale mixtures of normals [@VS15]. In this paper, we propose a new class of flexible distributions with support on ${\mathbb R}_+$ by applying the second method to the family of two-piece distributions [@FS98; @A05; @RS14]. In Section \[LTPDistributions\], we introduce the proposed class of distributions and show that it contains very flexible members that can capture a wide variety of shapes and tail behaviours. We show that these models can be seen as a subclass of composite models, which are of great interest in finance. The associated hazard functions are non-monotone with either increasing or decreasing right tails. These distributions are easy to implement using the R packages [@R13] ‘twopiece’ and ‘TPSAS’, which are available upon request. In Section \[MaximumLikelihoodEstimation\], we discuss the properties of the maximum likelihood estimators (MLEs) associated to these models. Although a formal study of the asymptotic properties of the proposed models is beyond the scope of this paper, we present a simulation study which reveals that adding a shape parameter, via two-piece transformations, has little effect on the performance of the maximum likelihood estimators. In Section \[NumericalExamples\], we present two kinds of applications with real data. In the first example, we illustrate the use of the proposed distributions in the context of data fitting. The main application is presented in the second and third examples, where we employ the proposed distributions for modelling the errors in an accelerated failure time model (AFT) with applications to medical data. In the third example, we discuss the use of a certain class of prediction intervals of the remaining life, which are informative for individual prognosis. In all of these examples, we discuss model selection between some appropriate competitors and the selection of the baseline distribution in the proposed family of distributions. Log Two-piece Distributions {#LTPDistributions} =========================== For the sake of completeness, let us first recall the definition of two-piece distributions. Let $s(\cdot;\delta)$ be a symmetric unimodal density, with mode at $0$, with support on $\mathbb R$, and let $\delta\in\Delta\subset{\mathbb R}$ be a shape parameter (location and scale parameters can be added in the usual way). The corresponding CDF will be denoted as $S(\cdot;\delta)$. The shape parameter $\delta$ typically controls the tails of the density. For example, in the cases where $s(\cdot;\delta)$ is either a Student-$t$ density with $\delta>0$ degrees of freedom or an exponential power density with power parameter $\delta>0$ (see Appendix A). A real random variable $X$ is said to be distributed according to a two-piece distribution if its probability density function (PDF) is given by (see *e.g.* [@RS14]): $$\begin{aligned} \label{TP} s_{tp}(x;\mu,\sigma_1,\sigma_2,\delta) = \dfrac{2}{\sigma_1+\sigma_2}\left[s\left(\dfrac{x-\mu}{\sigma_1};\delta\right) I(x<\mu) + s\left(\dfrac{x-\mu}{\sigma_2};\delta\right) I(x\geq \mu) \right].\end{aligned}$$ This is, a two-piece density is obtained by continuously joining two half-$s$ densities with different scale parameters on either side of the location $\mu$. The density (\[TP\]) is unimodal, with mode at $\mu$, it is asymmetric for $\sigma_1\neq \sigma_2$, and coincides with the original density $s$ for $\sigma_1=\sigma_2$. Moreover, the tail behaviour of the PDF in (\[TP\]) is the same in each direction, by construction. A popular reparameterisation is obtained by redefining $\sigma_1=\sigma a(\gamma)$ and $\sigma_2 = \sigma b(\gamma)$, where $a(\cdot)$ and $b(\cdot)$ are positive functions of the parameter $\gamma$ [@A05]. Two common choices for $a(\cdot)$ and $b(\cdot)$ are the inverse scale factors $\{a(\gamma),b(\gamma)\}=\{\gamma,1/\gamma\}$, $\gamma \in {\mathbb R}_+$ [@FS98], and the epsilon-skew parameterisation $\{a(\gamma),b(\gamma)\}=\{1-\gamma,1+\gamma\}$, $\gamma \in (-1,1)$ [@MH00]. Other parameterisations are explored in [@RS14]. The PDF associated to this reparameterisation is given by $$\begin{aligned} \label{TP2} s_{tp}(x;\mu,\sigma,\gamma,\delta) = \dfrac{2}{\sigma[a(\gamma)+b(\gamma)]}\left[s\left(\dfrac{x-\mu}{\sigma b(\gamma)};\delta\right) I(x<\mu) + s\left(\dfrac{x-\mu}{\sigma a(\gamma)};\delta\right) I(x\geq \mu) \right]\end{aligned}$$ This transformation preserves the existence of moments and the ease of use of the original distribution $s$. The corresponding cumulative distribution function and quantile function can be easily obtained from this expression (see [@A05]). This class of distributions has been shown to have good inferential properties for regular choices of the baseline density $s$ [@A05; @JA10]. By applying method (ii), described in Section \[sec:intro\], to the family of two-piece distributions, we can produce distributions with support on ${\mathbb R}_+$ as follows. A positive random variable $Y$ is said to be distributed according to a log two-piece (LTP) distribution if its PDF is given by: $$\begin{aligned} \label{LTPPDF} s_l(y; \mu,\sigma,\gamma,\delta) = \dfrac{2}{y\sigma[a(\gamma)+b(\gamma)]} &\left[ s\left(\dfrac{\log(y)-\mu}{\sigma b(\gamma)};\delta\right)I(y<e^{\mu})\right.\nonumber\\ &\quad\quad+\left.s\left(\dfrac{\log(y)-\mu}{\sigma a(\gamma)};\delta\right)I(y\geq e^{\mu}) \right].\end{aligned}$$ Given that the class of two-piece distributions contains all the symmetric unimodal distributions with support on the real line, it follows that the class of LTP distributions contains the class of log-symmetric distributions as well as models such that the distribution of $\log Y$ is asymmetric. The LTP Laplace distribution, which is obtained by using a Laplace baseline density $s$ in (\[LTPPDF\]), has been studied in [@K01]. However, other types of log two–piece distributions have not been studied to the best of our knowledge. The corresponding CDF is given by $$\begin{aligned} \label{LTPCDF} S_l(y; \mu,\sigma,\gamma,\delta) &=& \dfrac{2b(\gamma)}{a(\gamma)+b(\gamma)} S\left(\dfrac{\log(y)-\mu}{\sigma b(\gamma)};\delta\right)I(y<e^{\mu}) \notag\\ &+&\left[\dfrac{b(\gamma)-a(\gamma)}{a(\gamma)+b(\gamma)} + \dfrac{2a(\gamma)}{a(\gamma)+b(\gamma)}S\left(\dfrac{\log(y)-\mu}{\sigma a(\gamma)};\delta\right)\right]I(y\geq e^{\mu}) .\end{aligned}$$ We can observe that the ratio of the mass cumulated on either side of the value $y=e^{\mu}$ is given by $$\begin{aligned} R(\gamma) &=& \dfrac{S_l(e^{\mu}; \mu,\sigma,\gamma,\delta)}{1-S_l(e^{\mu}; \mu,\sigma,\gamma,\delta)} = \dfrac{b(\gamma)}{a(\gamma)}.\end{aligned}$$ This helps us to identify the different roles of the parameters $\gamma$ and $\delta$. The parameter $\gamma$ controls the cumulation of mass on either side of $y=e^{\mu}$, while the parameter $\delta$ controls the tails of the density. In Figure \[fig:LTPN\]a we present some examples of a two-piece normal PDF with different values of the parameter $\gamma$. In these cases, the parameter $\gamma$ only affects the asymmetry of the density. Figure \[fig:LTPN\]b shows the corresponding LTP normal PDFs. We can observe that in these cases the parameter $\gamma$ affects the shapes of the density. That is, it controls the mass cumulated above and below the value $y=1$ as well as the spread and mode of the density. The corresponding hazard function can be easily constructed from (\[LTPPDF\]) and (\[LTPCDF\]). Figure \[fig:DH\] shows the variety of shapes of the density and hazard functions obtained for a log two-piece sinh-arcsinh distribution (LTP SAS, which is obtained by using a symmetric sinh-arcsinh baseline density function in (\[TP2\]), see also [@R15]. The corresponding expression is provided in Appendix A). The implementation of LTP distributions is straightforward in R by using the packages ‘twopiece’ and ‘TPSAS’, which are freely available upon request. Moreover, the $p$th moment of a LTP distribution exists, whenever the $p$th moment of the underlying (log-symmetric) log-$s$ distribution exists. In particular, all moments of the LTP normal distribution exist. ----- ----- (a) (b) ----- ----- ----- ----- (a) (b) (c) (d) (e) (f) ----- ----- An alternative construction {#an-alternative-construction .unnumbered} --------------------------- The family of two-piece distributions (\[TP\])-(\[TP2\]) can be seen as a special kind of finite mixtures of truncated PDFs, as shown in [@RS14]. In a similar fashion, the family of log two-piece distributions can be obtained as a particular class of finite mixtures of truncated distributions with positive support. In the context of survival and size distributions these sorts of mixtures are known as composite models (see [@NB13] for a literature review). Recall first that the PDF of a composite model can be written as: $$\begin{aligned} \label{CompPDF} s_c(y) = \omega s_1^*(y) I(y\leq \theta) + (1-\omega) s_2^*(y) I(y>\theta),\,\,\, y>0,\end{aligned}$$ where $\omega = \dfrac{s_2(\theta)S_1(\theta)}{s_2(\theta)S_1(\theta) + s_1(\theta)[1-S_2(\theta)]}$, $\theta>0$ is a threshold parameter, $s_1^*(y)=\dfrac{s_1(y)}{S_1(\theta)}$, $s_2^*(y)=\dfrac{s_2(y)}{1-S_2(\theta)}$, $s_1$ and $s_2$ are continuous PDFs with support on ${\mathbb R}_+$, and $S_1$ and $S_2$ are the corresponding CDFs. If we fix $s_1(y)=\dfrac{1}{\sigma_1 y}s\left(\dfrac{\log(y)-\mu}{\sigma_1}\right)$, $s_2(y)=\dfrac{1}{\sigma_1 y}s\left(\dfrac{\log(y)-\mu}{\sigma_2}\right)$, for some symmetric density $s$ with support on ${\mathbb R}$, and $\theta=e^{\mu}$, then it follows that (\[CompPDF\]) coincides with (\[LTPPDF\]), up to a reparameterisation. From this alternative construction, we conclude that the family of LTP distributions represents a subclass of composite models with the appealing properties and interpretability of parameters discussed above. This also allows us to motivate the use of LTP distributions as survival and size distributions. Models and Maximum Likelihood Estimation {#MaximumLikelihoodEstimation} ======================================== In this section, we present the parameter estimation procedure for time-to-event and accelerated failure time (AFT) models. Time-to-event model ------------------- Let ${\bf T}=(T_1,\dots,T_n)$ be an independent sample of survival times distributed as in (\[LTPPDF\]). The likelihood function of the parameters $(\mu,\sigma,\gamma,\delta)$ is defined as: $$\begin{aligned} L(\mu,\sigma,\gamma,\delta) = \prod_{j=1}^n s_l(T_j;\mu,\sigma,\gamma,\delta).\end{aligned}$$ The MLE is defined as the parameter values that maximise the likelihood function. By noting that $$\begin{aligned} L(\mu,\sigma,\gamma,\delta) \propto \prod_{j=1}^n s_{tp}(\log(T_j);\mu,\sigma,\gamma,\delta),\end{aligned}$$ it follows that the MLEs of the parameters of LTP distributions are the same as the MLEs of the parameters of the underlying two-piece distribution for the sample $\log({\bf T}) = [\log(T_1),\dots,$ $\log(T_n)]$. Inferential aspects of 3- and 4-parameter two-piece distributions have been largely discussed. For example, [@A05] show that, under certain regularity conditions on the baseline density $s$ in (\[TP2\]), the maximum likelihood estimators of the parameters of these distributions are consistent and asymptotically normal under the epsilon-skew parameterisation. [@JA10] and [@RS14] study some parameterisations that induce parameter orthogonality between the parameters $\mu$ and $\sigma$, showing that the epsilon-skew parameterisation induces this property. Parameter orthogonality, in turn implies a good asymptotic behaviour of the MLE [@JA10]. In most cases, the MLE is not available in closed-form, and it has to be obtained numerically. Samples containing censored observations are common in the context of survival analysis. The most common types of censoring in this context correspond to: (i) Left-censoring: when the phenomenon of interest has already occurred before the start of the study. A left-censored observation is an interval of the type $[0,T_j)$, where $T_j$ represents the start of the study for subject $j$. (ii) Interval censoring: when the phenomenon of interest occurs within a finite period of time $[T_j^L,T_j^R]$. (iii) Right-censoring: when the phenomenon of interest is not observed during the duration of the study. A right-censored observation is an interval of the type $(T_j,\infty]$, where $T_j$ represents the duration of the study for subject $j$. Ignoring censoring induces bias in the estimation of the parameters. Different types of censoring imply different contributions of the observations to the likelihood function. The contribution of a left-censored observation to the likelihood is $S_l(T_j;\mu,\sigma,\gamma,\delta)$; while the contribution of an interval-censored observation to the likelihood is $S_l(T_j^R;\mu,\sigma,\gamma,\delta)-S_l(T_j^L;\mu,\sigma,\gamma,\delta)$; and the contribution of a right-censored observation to the likelihood is $1-S_l(T_j;\mu,\sigma,\gamma,\delta)$. If we define the sets $\text{Left}=\{j: T_j \text{ is left-censored}\}$, $\text{Int}=\{j: T_j \text{ is interval-censored}\}$, $\text{Right}=\{j: T_j \text{ is right-censored}\}$, and $\text{Obs}=\{j: T_j \text{ in uncensored}\}$, then we can write the likelihood function as follows: $$\begin{aligned} L(\mu,\sigma,\gamma,\delta) = &\prod_{j\in\text{Obs}} s_l(T_j;\mu,\sigma,\gamma,\delta) \times \prod_{j\in\text{Left}} S_l(T_j;\mu,\sigma,\gamma,\delta) \\&\times \prod_{j\in\text{Right}}\left[1-S_l(T_j;\mu,\sigma,\gamma,\delta)\right] \times \prod_{j\in\text{Int}} \left[ S_l(T_j^R;\mu,\sigma,\gamma,\delta)-S_l(T_j^L;\mu,\sigma,\gamma,\delta)\right].\end{aligned}$$ The latter expression emphasises the practical importance of using distributions with a tractable distribution function. Accelerated failure time models ------------------------------- AFT models are a useful tool for modelling the set of survival times ${\bf T}=(T_1,\dots,T_n)$ in terms of a set of covariates $\bm{\beta} = (\beta_1,\dots,\beta_p)$ through the model equation: $$\begin{aligned} \label{AFTModel} h(T_j) = x_j^{\top}\bm{\beta} + \varepsilon_j, \,\,\, j=1,\dots,n,\end{aligned}$$ where ${\bf X}= (x_1,\dots,x_n)^{\top}$ is an $n\times p$ known design matrix and $\varepsilon_j \stackrel{ind.}{\sim} F(\cdot;\bm{\theta})$, $F$ is a continuous distribution with support on ${\mathbb R}$ and parameters $\bm{\theta}\in \Theta \subset {\mathbb R}^d$, and $h:{\mathbb R}_+\rightarrow{\mathbb R}$ is a continuous increasing function. The most common choice for $h$ is the logarithmic function, while the distribution of the errors $\varepsilon_j$ is typically assumed to be normal. Given that the assumption of normality of the errors can be restrictive in practice, other distributional assumptions have been recently studied such as the log Birnbaum-Saunders [@B08], finite mixtures of normal distributions [@KL08], the symmetric family of scale mixtures of normals [@VS15], and the log-generalised extreme value distribution [@RD15]. AFT models are extremely relevant in medicine, given that survival data naturally arise in many medical studies, which typically involve the follow-up of other covariates. The presence of different types of censored observations is common in this context [@B08; @KL08; @VS15]. If we assume that the errors $\varepsilon_j$ are distributed according to a LTP distribution with location $0$ and $\bm{\theta}=(\sigma,\gamma,\delta)$, then we can write the likelihood function as follows, $$\begin{aligned} L(\bm{\beta},\sigma,\gamma,\delta) =& \prod_{j\in\text{Obs}} s_l(T_j;x_j^{\top}\bm{\beta},\sigma,\gamma,\delta) \times \prod_{j\in\text{Left}} S_l(T_j;x_j^{\top}\bm{\beta},\sigma,\gamma,\delta)\\ &\times \prod_{j\in\text{Right}}\left[1-S_l(T_j;x_j^{\top}\bm{\beta},\sigma,\gamma,\delta)\right]\\ &\times \prod_{j\in\text{Int}} \left[ S_l(T_j^R;x_j^{\top}\bm{\beta},\sigma,\gamma,\delta)-S_l(T_j^L;x_j^{\top}\bm{\beta},\sigma,\gamma,\delta)\right],\end{aligned}$$ with the notation discussed previously. It is important to notice that by using asymmetric errors, we obtain a curve that does not represent the mean response. However, as discussed in [@AG08], this lack of centring can be calibrated after estimating the parameters by adding a suitable quantity $M_{\varepsilon}$ which reflects the lack of centring of the errors. For instance, in order to obtain the mean response, we can use $M_{\varepsilon}=-{\mathbb E}[\varepsilon_j]$, computed at the MLE of the parameters of the error distribution. This strategy will only affect the intercept parameter. When using baseline models with infinite variance (such as a log-Cauchy distribution), one might opt for centring around the median (or another quantile), instead of the mean. A formal study of the asymptotic properties of the MLEs under different types of censoring is beyond the scope of this paper. However, in Section \[SimulationStudy\], we illustrate the performance of the MLEs in a linear regression model with censored observations through a simulation study. Simulation Study {#SimulationStudy} ================ In this section, we present a simulation study in order to illustrate the performance of the MLEs of the parameters of some LTP distributions. Throughout, we employ the epsilon-skew parameterisation discussed previously. In our first simulation scenario, we simulate $N=10,000$ samples of sizes $n=30,50,100,250,500,1000$ from a LTP normal (log two-piece normal) with different combinations of the parameter values: $\mu=0$, $\sigma=1$, and $\gamma=0,0.25,0.5,0.75$. Negative values of $\gamma$ would induce similar results, since they produce the corresponding reflected density about $e^{\mu}$, and are therefore omitted. For each of these samples, we calculate the corresponding MLEs, using the R command ‘optim’, and calculate the bias, variance, and root-mean-square error (RMSE) of these. In our second simulation scenario, we simulate $N=10,000$ samples from a LTP $t$ (log two-piece Student-$t$) with parameters: $\mu=0$, $\sigma=1$, $\gamma=0,0.25,0.5,0.75$, and $\delta=1$. The third simulation scenario is analogous to the second scenario, with $\delta=2$. In the fourth scenario, we simulate $N=10,000$ samples from a LTP SAS with parameters : $\mu=0$, $\sigma=1$, $\gamma=0,0.25,0.5,0.75$, and $\delta=0.75$. Tables \[table:LTPN\]–\[table:LTPSAS2\] present the results of these simulations. In second class of simulations, we investigate the performance of the use of log two-piece errors in AFT models (\[AFTModel\]). For this purpose, we simulate from the linear regression model: $$\begin{aligned} \log(y_j) = {\bf x}_j^{\top}\bm{\beta} + \varepsilon_j, \,\,\, j=1,\dots,n,\end{aligned}$$ with $n=100,250,500$, $\beta = (1,2,3)^{\top}$, and ${\bf x}_j = (1,x_{j1},x_{j2})^{\top}$. The second and third entries of the covariates ${\bf x}_j$ are simulated from a right-half-normal with scale parameter $1/3$. For the distribution of the errors $\varepsilon_j$ we consider the following cases: (i) a two-piece normal distribution with parameters $\mu=0$, $\sigma=0.25$, and $\gamma=0,0.25,0.5$; and (ii) a TP SAS distribution [@R15] with parameters $\mu=0$, $\sigma=0.25$, $\gamma=0,0.25,0.5$, and $\delta=0.75$. We truncate the observations $y_j$ that are greater than $17.5$. This censoring mechanism produces samples with 15%–35% censored observations. Tables \[table:LTPNR\]–\[table:LTPSASR\] present the results of these simulations. The overall conclusions of this extensive simulation study are that the value of the shape parameter $\gamma$ does not seem to greatly affect the performance of the MLEs, while the use of models with a tail parameter $\delta$ have a clear effect on the performance of the MLEs. The performance of the MLEs of $\delta$ in LTP $t$ and LTP SAS models for small samples is different: the bias is smaller in the LTP SAS model. However, the estimation of $\sigma$ is more accurate in the LTP $t$ model. This is, perhaps, an unsurprising conclusion, given that it is well–known that it is difficult to learn about tail parameters with small samples and that tail parameters control the tail behaviour differently in different models. However, this analysis helps us to quantify the order of observations required for an accurate estimation. For LTP models with 4 parameters, such as the LTP $t$ and LTP SAS models, it is necessary to have at least $200$ observations in order to accurately estimate the tail parameters. In fact, the proposed flexible models are not recommended with small samples since, intuitively, these do not contain information about the features captured by the shape parameters $\gamma$ and $\delta$. Par. ------ ------ ------------------ -------------- -------------- --------------- ----------- ------------------ ------------------- --------------- $n$ $\hat\mu$ $\hat\sigma$ $\hat\gamma$ $\hat\delta$ $\hat\mu$ $\hat\sigma$ $\hat\gamma$ $\hat\delta$ 30 0.0052 -0.0347 0.0018 -2858.8 -0.0205 -0.0350 -0.0197 -2502.9 50 -0.0026 -0.0269 -0.0017 -623.9 -0.0168 -0.0247 -0.0114 -476.8 100 0.0005 -0.0096 0.0005 -5.8 -0.0047 -0.0090 -0.0037 27.0 250 0.0010 -0.0028 -0.0001 -0.0883 0.0008 -0.0035 -0.0007 -0.0890 500 -0.0001 -0.0003 0.0001 -0.0387 -0.0008 -0.0004 -0.0007 -0.0391 1000 4$\times10^{-5}$ -0.0001 0.0001 -0.0202 0.0002 5$\times10^{-5}$ -5$\times10^{-5}$ -0.0187 30 0.3303 0.0867 0.1034 7$\times10^8$ 0.3054 0.0874 0.0952 5$\times10^8$ 50 0.1465 0.0493 0.0468 1$\times10^8$ 0.1377 0.0490 0.0441 4$\times10^7$ 100 0.0590 0.0215 0.0191 7$\times10^4$ 0.0557 0.0214 0.0181 5$\times10^6$ 250 0.0203 0.0081 0.0069 0.1494 0.0195 0.0082 0.0065 0.1505 500 0.0102 0.0039 0.0034 0.0605 0.0096 0.0039 0.0032 0.0605 1000 0.0050 0.0019 0.0016 0.0283 0.0047 0.0019 0.0016 0.0277 30 0.5747 0.2965 0.3216 3$\times10^4$ 0.5529 0.2977 0.3092 2$\times10^4$ 50 0.3827 0.2237 0.2165 1$\times10^4$ 0.3714 0.2226 0.2103 6553.9 100 0.2428 0.1470 0.1382 277.2 0.2360 0.1465 0.1345 2271.3 250 0.1426 0.0902 0.0835 0.3965 0.1399 0.0906 0.0809 0.3979 500 0.1011 0.0629 0.0583 0.2491 0.0981 0.0629 0.0566 0.2490 1000 0.0711 0.0445 0.0410 0.1694 0.0690 0.0443 0.0398 0.1677 : Simulation results: LTP $t$, $\delta=2$.[]{data-label="table:LTPt21"} Applications {#NumericalExamples} ============ In this section, we present several medical applications with real data that illustrate the performance and usefulness of the proposed distributions. Throughout, we adopt the epsilon-skew parameterisation $\{a(\gamma),b(\gamma)\}=\{1-\gamma,1+\gamma\}$, $\gamma \in (-1,1)$, for the LTP distributions. Example 1: Nerve data --------------------- In our first example we analyse the data set reported in [@CL66] which contains $n=799$ observations rounded to the nearest half in units of 1/50 second, which correspond to the time between $800$ successive pulses along a nerve fibre. We consider two baseline distributions $s$ in (\[LTPPDF\]): a Student $t$ density with $\delta>0$ degrees of freedom (LTP $t$), and a symmetric sinh-arcsinh density [@JP09; @R15] (LTP SAS). The choice for these two baseline densities is motivated as follows. The Student-$t$ distribution is a parametric family of distributions with heavier tails than the normal ones; having the normal distribution as a limit case when $\delta \rightarrow\infty$. The behaviour of the tails of the Student-$t$ density is polynomial. On the other hand, the symmetric sinh-arcsinh density (reported in Appendix A) is a parametric density function which contains a parameter that controls the tail behaviour. This distribution can capture tails heavier or lighter than those of the normal density ($\delta\lessgtr 1$), being the normal distribution a particular case ($\delta=1$). The tails of the symmetric sinh-arcsinh density are lighter than any polynomial [@JP09]. Therefore, with these two choices of the baseline density we can cover a wide range of tail behaviours. Moreover, with the additional shape parameter $\gamma$ we also cover a wide range of shapes around the shoulders of the density. Table \[table:NerveMLE\] shows the MLEs and the Akaike information criterion (AIC) associated to these models as well as some natural competitors. We also report the estimators of the LTP Normal and the lognormal distributions, which are particular cases of the LTP SAS model. The AIC favours the LTP SAS model overall, closely followed by the LTP Normal. The MLE of the parameter $\delta$ in the LTP SAS model is larger than one, indicating that the data favour a model with lighter tails than those of the lognormal distribution. The 95% confidence intervals for the parameters $\gamma$ and $\delta$ (obtained as the 0.147-level profile likelihood intervals, see [@K85]) in the LTP SAS model are $(0.31,0.53)$ and $(1.02, 1.56)$, respectively. It is worth noticing that the confidence interval for $\delta$ only include values greater than one, which are associated to tails lighter than normal. Figures \[fig:NFM\]a–\[fig:NFM\]c show the probability plots and hazard functions corresponding to the LTP SAS, lognormal, and Weibull models, which visually illustrates the fit of these models. From Figure \[fig:NFM\]d We can observe that the fitted Gamma model produces an increasing hazard function, while the LTP SAS model produces a non monotonic hazard function with decreasing tail. This behaviour coincides with that of the fitted kernel estimation of the hazard function (which was obtained using lognormal kernels). Model $\hat\mu$ $\hat\sigma$ $\hat\gamma$ $\hat\delta$ AIC ------------ ----------- --------------- -------------- -------------- ----------------- LTP $t$ 2.59 1.04 0.40 111.90 5401.80 LTP SAS 2.63 1.39 0.42 1.22 [**5395.71**]{} LTP Normal 2.59 1.05 0.40 – 5398.45 Log-normal 1.91 1.08 – – 5443.70 Weibull – (scale) 11.27 (shape) 1.08 – 5415.40 Gamma – (scale) 9.31 (shape) 1.17 – 5411.11 : Nerve data: Maximum likelihood estimates, AIC (best value in bold).[]{data-label="table:NerveMLE"} ----- ----- (a) (b) (c) (d) ----- ----- Example 2: PBC data ------------------- In this section, we analyse the popular Mayo primary biliary cirrhosis (PBC) data, reported in Appendix D from [@FH91], in order to illustrate the performance of the proposed distributions in the context of AFT models. This data set contains information about the survival time and prognostic factors for 418 patients in a study conducted at Mayo Clinic between 1974 and 1984. The survival times are reported in days together with an indicator variable associated to the status of the patient at the end of the study (0/1/2 for censored, transplant, dead). [@J03] fitted, using a semiparametric method, an AFT model with five covariates: age (in years), logarithm of the serum albumin (in mg/dl), logarithm of the serum bilirubin (in mg/dl), edema, and logarithm of the prothrombin time (in seconds). Similarly, [@D10] reports the semiparametric estimators of the AFT model with an intercept parameter as follows: $(8.692,-0.025,1.498,-0.554,-0.904,-2.822)$. We consider a maximum likelihood estimation approach of the AFT model (\[AFTModel\]) containing an intercept and LTP $t$ errors with parameters $(0, \sigma_{\varepsilon}, \gamma_{\varepsilon}, \delta_{\varepsilon})$. The estimators and the AIC values are reported in Table \[table:PBCMLE\]. We can see that the estimators obtained for the model with LTP $t$ and Log-$t$ models are close to those reported by [@D10] using a semiparametric method. The AIC values favour the models with LTP-$t$ and Log Student-$t$ (Log-$t$) errors. However, these values do not provide strong evidence to distinguish between the two models, and therefore the model choice deserves further investigation. The MLE of the skewness parameter $\gamma_{\varepsilon}$ is relatively far from zero in the Log-$t$ model. However, the inclusion of this parameter produces little effect in the estimation of the degrees of freedom $\delta_{\varepsilon}$ and the regression parameters. The 95% confidence interval for $\gamma_{\varepsilon}$ in the LTP-$t$ model is $(-0.374,0.167)$, which does not rule out the value $\gamma_{\varepsilon}=0$ as a likely value of the parameter. Then, a parsimony argument favours the model with log-Student $t$ errors (Log-$t$) in this case. Moreover, we can observe that the MLEs of $\gamma_{\varepsilon}$ in the LTP $t$ and LTP Normal model have different signs. The 95% confidence interval for $\gamma_{\varepsilon}$ in the LTP Normal model is $(-0.072,0.493)$ (which indicates that $\gamma_{\varepsilon}=0$ is an unlikely value of the parameter). The reason for this difference is that the data seem to favour a model with heavier tails than normal. The lack of flexibility in the tails and the presence of extreme observations affect the estimation of the shape parameter $\gamma_{\varepsilon}$ in the LTP Normal model by pulling out this estimator in the opposite direction. This emphasises the importance of assessing the type of flexibility required for properly modelling the data. Model LTP $t$ LTP Normal Log-$t$ Log-normal ------------------------ ----------------- ------------ ----------------- ------------ Intercept 7.704 7.518 7.539 7.731 Age -0.026 -0.026 -0.027 -0.025 log(Albumin) 1.552 1.529 1.554 1.472 log(Bilirubin) -0.587 -0.620 -0.595 -0.606 Edema -0.762 -0.710 -0.706 -0.840 log(Protime) -2.464 -2.189 -2.313 -2.371 $\sigma_{\varepsilon}$ 0.773 0.908 0.770 0.973 $\gamma_{\varepsilon}$ -0.133 0.190 – – $\delta_{\varepsilon}$ 4.446 – 5.602 – AIC [**635.019**]{} 642.318 [**633.702**]{} 642.222 : PBC data: Maximum likelihood estimates, AIC (best values in bold).[]{data-label="table:PBCMLE"} Example 3: NCCTG Lung Cancer Data {#Lung} --------------------------------- In this section, we revisit the popular NCCTG Lung Cancer Data. This data set contains the survival times of $n=227$ patients (the total number of patients is 228 but we have removed one patient with a missing covariate, for the sake of simplicity) with advanced lung cancer from the North Central Cancer Treatment Group. The goal of this study was to compare the descriptive information from a questionnaire applied to a group of patients against the information obtained by the patient’s physician, in terms of prognostic power [@L94]. We fit an AFT model with three covariates “age” (in years),“sex” (Male=1 Female=2), “ph.ecog” (ECOG performance score, 0=good–5=dead) as well as an intercept, with different choices for the distribution of the errors in (\[AFTModel\]). Table \[table:LungMLE\] shows the MLEs associated to each of these models together with the AIC values. The AIC favours the model with LTP logistic errors, closely followed by the model with LTP SAS errors. One explanation for this is that the estimators of the LTP SAS model indicate tails heavier than normal ($\hat\delta=0.6674$), which is a tail behaviour naturally captured by the LTP logistic distribution without additional shape parameters that control the tail. The 95% confidence intervals for $\gamma$ and $\delta$ in the LTP SAS model are $(-0.05,0.60)$ and $(0.46,0.86)$, respectively, while the corresponding confidence interval for the parameter $\gamma$ in the LTP logistic model is $(0.16, 0.62)$. Model LTP SAS LTP Normal Log-normal LTP logistic Log-logistic ----------- ---------- ------------ ------------ ------------------ -------------- Intercept 6.2077 6.9505 6.4940 6.5538 5.9500 Age -0.0068 -0.0149 -0.0191 -0.0100 -0.0082 Sex 0.4614 0.4259 0.5219 0.4243 0.4857 ph.ecog -0.3824 -0.3121 -0.3557 -0.3541 -0.4042 $\sigma$ 0.4639 0.8835 1.0286 0.4847 0.5360 $\gamma$ 0.3095 0.5051 – 0.4083 – $\delta$ 0.6674 – – – – AIC 538.2100 545.9405 563.8323 [**536.0556**]{} 545.0486 : NCCTG Lung Cancer data: Maximum likelihood estimates, AIC (best value in bold).[]{data-label="table:LungMLE"} It is sometimes of interest to obtain information about the remaining life of individual cancer patients. This information is used for future planning of health care, which is of financial and medical importance. Specifically, the probability that patient $i$ survives until time $t$, given that he/she was alive at time $t_i$ is given by, $$\begin{aligned} \label{RemainingLife} G(t\vert t_i;\bm{\theta}) = {\mathbb P}(T\leq t\vert T> t_i) = \dfrac{G(t;\bm{\theta})-G(t_i;\bm{\theta})}{1-G(t_i;\bm{\theta})},\,\,\, t\geq t_i,\end{aligned}$$ where $G$ is the distribution under the model of interest. For an AFT model, the parameter $\bm{\theta}$ contains both the regression parameters as well as the parameters of the distribution of the errors. The simplest way to obtain an estimator of this probability consists of plugging in the MLE of $\bm{\theta}$ in (\[RemainingLife\]). The 100(1-$\alpha$)% prediction interval [@H09] for a patient that survived until time $t_i$ is $[T_i^L,T_i^R]$, which satisfies $G(T_i^L\vert t_i;\bm{\theta})=\alpha_1$ and $G(T_i^R\vert t_i;\bm{\theta})=1-\alpha_2$, with $\alpha_1 + \alpha_2=\alpha$. In our application we choose $\alpha_1=\alpha_2=0.05$, and we centre the prediction intervals at the mean of the regression model with LTP logistic errors. Figure \[fig:PredInt\] shows the 90% prediction interval for the remaining life for 10 censored patients. We can observe from Table \[table:LungMLE\] that the estimators of the regression parameters are very similar for the different choices of the distribution of the errors. At first glance, one might think that the choice of the distribution of the residual errors has little impact on the inference. However, if the interest in on predicting the remaining life of censored patients, we may obtain different intervals for different models. For instance, Figure \[fig:Patient3\] shows how different the survival functions of the remaining life for a particular censored patient, associated to the models with LTP logistic and logistic errors, can be. This emphasises the importance of the correct specification of the distribution of the residual errors. [c]{} [c]{} Discussion ========== We have proposed a flexible class of parametric distributions (LTP) with positive support that can be used for the modelling of survival data. We have shown that some members of this class of distributions represent a flexible extension of the classical choices such as the lognormal, log-logistic, and log Student-$t$ distributions. The genesis of LTP distributions allows the user to play with different baseline log-symmetric distributions in order to properly model the tail behaviour of the data. These distributions can be used to produce models that are robust to departures from the assumption of log-symmetry. Moreover, LTP distributions preserve the ease of use of the baseline log-symmetric distribution. For instance, in models that assume lognormality, a LTP-normal can be implemented with virtually the same parsimony level. In practice, we recommend to conduct a model selection between 4-parameter LTP models and the corresponding 3- and 2-parameter submodels. Given that the parameters of LTP distributions are easily-interpreted, this model selection provides information about the features favoured by the data, such as asymmetry and tail behaviour, providing in turn more insights on the phenomenon of interest. Model selection between these nested models can be conducted either using AIC or the likelihood ratio test. The good behaviour of the MLE in this family can be established by appealing to the literature on the study of inferential properties of the family of two-piece distributions, which are linked to the proposed models via a logarithmic transformation. Confidence intervals for the model parameters can be obtained by using the profile likelihood. This approach avoids relying on asymptotic results, such as normal confidence intervals (standard errors), that may not be accurate for small or moderate sample sizes. We conclude by pointing out possible extensions of our work. Multivariate extensions of the family of LTP distributions can be produced by using copulas. This approach has the advantage of separating the role of the parameters that control the shape of the distribution of the marginals and the dependencies between the marginals. As discussed in [@H09], the plug-in estimators considered in Section \[Lung\] may produce prediction intervals of the remaining life with a smaller coverage probability. The calibration of these intervals to improve their coverage in the context of LTP models represents an interesting research line. Acknowledgements {#acknowledgements .unnumbered} ================ We thank the Editor and two referees for their very helpful comments. FJR gratefully acknowledges research support from EPSRC grant EP/K007521/1. Appendix A: Some density functions {#appendix-a-some-density-functions .unnumbered} ================================== Throughout we use the notation $t=\dfrac{x-\mu}{\sigma}$. - The symmetric sinh-arcsinh distribution [@JP09]: $$\begin{aligned} f(x;\mu,\sigma,\delta) = \dfrac{\delta}{\sigma}\phi\left[\sinh\left(\delta \operatorname{arcsinh}\left(t\right)\right)\right] \dfrac{\cosh\left(\delta \operatorname{arcsinh}\left(t\right)\right)}{\sqrt{1+t^2}},\end{aligned}$$ where $\delta>0$ controls the tails of the density, and $\phi$ is the standard normal density function. Note that for $\delta=1$, this density corresponds to the normal density. - The exponential power distribution: $$\begin{aligned} f(x;\mu,\sigma,\delta) = \dfrac{\delta}{2\sigma\Gamma(1/\delta)}\exp\left(-\vert t \vert^\delta\right).\end{aligned}$$ where $\Gamma(\cdot)$ is the gamma function. This family contains the Laplace distribution for $\delta=1$ and the normal distribution (with variance $\sigma^2/2$) for $\delta=2$. - The Student-$t$ distribution: $$\begin{aligned} f(x;\mu,\sigma,\delta) = \dfrac{\Gamma\left(\dfrac{\delta+1}{2}\right)}{\sigma\sqrt{\pi\delta } \Gamma\left(\dfrac{\delta }{2}\right)} \left(1+\dfrac{t^2}{\delta}\right)^{-\dfrac{\delta +1}{2}},\end{aligned}$$ where $\Gamma(\cdot)$ is the gamma function. - The logistic distribution: $$\begin{aligned} f(x;\mu,\sigma) = \dfrac{1}{\sigma} \dfrac{\exp(-t)}{[1+\exp(-t)]^2}.\end{aligned}$$ [9]{} Azzalini, A. and Genton, M. G. (2008). Robust likelihood methods based on the skew-$t$ and related distributions. [*International Statistical Review*]{} 76: 106–129. Arellano-Valle, R. B., G[ó]{}mez, H. W. and Quintana, F. A. (2005). Statistical inference for a general class of asymmetric distributions. [*Journal of Statistical Planning and Inference*]{} 128: 427–443. Barros, M., Paula, G. A. and Leiva, V. (2008). 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--- abstract: 'The ground and excited states of $^{8}$He were investigated with a method of antisymmetrized molecular dynamics(AMD). We adopted effective nuclear interactions which systematically reproduce the binding energies of $^4$He, $^6$He and $^8$He. The ground state of $^8$He has both the $j$-$j$ coupling feature($p_{3/2}$ closure) and the $L$-$S$ coupling feature($^4$He$+2n+2n$) with a slight tail of dineutron at the long distance region. The theoretical results give an indication of the $0^+_2$ state with dineutron gas-like structure. The dineutron structure, $^4$He+$2n$+$2n$, of this state is similar to the $3\alpha$-cluster structure of the $^{12}$C($0^+_2$) state which has been interpreted as an $\alpha$ condensate state. Since the $^8$He($0^+_2$) state has a significant overlap with the dineutron condensate wave function where two dineutrons are moving in $S$ wave around the $\alpha$ core with a dilute density, we suggest that this theoretically predicted $0^+_2$ state is a candidate of the dineutron condensate state.' address: 'Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, Japan' author: - 'Yoshiko Kanada-En’yo' title: 'Dineutron structure in $^{8}$He' --- Introduction ============ In the recent progress of unstable nuclear physics, various kinds of exotic structure have been discovered. Many of these phenomena in light nuclear region are often related to cluster physics. From the viewpoints of nuclear cluster, there are many theoretical works on halo structure in neutron-rich nuclei and molecular structure in Be isotopes. Recently, Tohsaki [*et al.*]{} proposed a new type of cluster structure in the second $0^+$ state $^{12}$C, where 3 $\alpha$ clusters are weakly interacting[@Tohsaki01]. This is a dilute gas state of $\alpha$ particles which behave as bosonic particles in the dilute density. This phenomena is associated with Bose-Einstein Condensation(BEC) and is called “alpha condensation”. The alpha condensation was originally suggested in dilute nuclear matter by Röpke et al.[@Ropke98]. The $0^+_2$ of $^{12}$C is regarded as an example, where the alpha condensation is realized in a finite nuclear system. Then, it is challenging to search for such cluster-gas states in other nuclei. In analogy to the alpha condensation, dineutron condensation in neutron matter is a recent key issue in physics of unstable nuclei. Matsuo suggested that the dineutron correlation can be enhanced in dilute neutron matter[@Matsuo06]. In real systems, one should focus on dineutron correlation in finite nuclei such as halo nuclei and extremely neutron-rich nuclei, or that in neutron skin at a surface region of neutron-rich nuclei. In fact, the dineutron correlation in two-neutron halo nuclei like $^6$He and $^{11}$Li attracts great interests in these days. In case of $^6$He, where the $^4$He is the good core, the dineutron correlation of the valence neutrons has been demonstrated in three-body model calculations (for example, [@Bertsch91; @Zhukov93; @Aoyama01; @Arai01] and references therein). Now, let us consider structure of $^8$He from a point of view associated with the dineutron condensation. Firstly, more than one dineutrons are required to construct a dineutron condensate state. In $^8$He, two pairs of neutrons are possible from four valence neutrons around the $^4$He core. In second, $^8$He system may have some correspondence with the $^{12}$C system, because both of them have the same neutron number, $N=6$. In analogy to $^{12}$C, the ground state of $^8$He may have a feature of the neutron $p_{3/2}$ closure or the SU(3)-limit $p$-shell configuration. Instead of the ground state, one can speculate the dineutron gas-like state with developed $^4$He+$2n$+$2n$ structure in excited states. There are many theoretical works on He isotopes. Application of [*ab initio*]{} calculations such as GFMC and NCSM with realistic nuclear forces have now reached to the mass $A\sim 10$ region including $^6$He and $^8$He[@Pieper04; @Pieper05; @Caurier06]. Systematic studies of He isotopes have been performed also by model calculations with effective interactions such as cluster models as well as GSM[@Michel03; @Volya05; @Hagen05] and mean field approaches[@Sugahara96]. Three-body model with an assumption of the $^4$He core has been often adopted to study $^6$He [@Bertsch91; @Zhukov93; @Aoyama01; @Arai01; @Csoto93; @Baye94] and it has been applied to heavier He isotopes [@Aoyama02]. $^8$He and $^{10}$He have been also studied by such models as $^4$He+$Xn$ models [@Suzuki90; @Varga94; @Itagaki00; @Masui07] and by extended models [@Dote00; @Aoyama06] which have less assumption of the core. With Fermionic molecular dynamics, the study of He isotopes has been performed based on a realistic nuclear force [@Neff05]. However, many of these studies are concentrated on the ground states except for three-body models, GSM and GFMC. After the experimental indication of neutron skin structure in $^8$He[@Tanihata92], many experimental works on $^8$He have been recently performed to reveal the detailed properties of the ground state. The core excitation $^6$He$(2^+)$ in the ground state, which has been experimentally suggested[@Korsheninnikov03], indicates that $^8$He is different from a simple three-body state of $^6$He$(0^+)$+$2n$. Recent experiments using $^8$He beams suggested the significant component of the $(p_{3/2})^2(p_{1/2})^2$ configuration [@Chulkov05; @Keeley07]. They may support dineutron correlation in the $^8$He ground state rather than the pure $(p_{3/2})$ closure of neutrons. On the other hand, a measurement of spectroscopic factor of $^7$He($3/2^-$)[@Skaza06] in $^8$He suggested the pure sub-shell closed structure contradictory to the other experimental results. Thus, the neutron structure of the $^8$He ground state is controversial. Concerning excited states, although some levels are known to exist in the energy $E_x=3\sim 8$ MeV region, the experimental information is very poor for these states except for the $2^+_1$ state [@Korsheninnikov93]. In this paper, we investigated structure of $^8$He. In particular, we focused on $0^+$ states and discuss their dineutron component, because one of our major aims is to search for the dineutron gas-like state. We applied a method of antisymmetrized molecular dynamics(AMD)[@ENYObc; @ENYOsup; @AMDrev], which has been already proved to be useful in describing cluster structure in light nuclei. AMD has been applied to various light unstable nuclei such as He, Li, Be isotopes as well as stable nuclei. It has been applied also for study of cluster gas-like states in $^{12}$C and $^{11}$C($^{11}$B)[@Enyo-c12v2; @Enyo-c11]. In the present work, we adopted a AMD+generator coordinate method(GCM). Namely, we superposed a number of AMD wave functions, which were obtained by energy variation with constraints, to take various configurations into account. We comment that the theoretical method AMD+GCM of the present calculation is similar to those of the AMD+GCM and AMD+SSS works on He isotopes by Itagaki and his collaborators [@Itagaki00; @Aoyama06] in a sense that multi configurations of AMD wave functions are superposed. In [@Itagaki00; @Aoyama06], $^4$He+$Xn$ and $t+t+Xn$ configurations were [*a priori*]{} assumed. Another claim is that they used an effective interaction which makes a bound $^2n$. In the present work, we have no assumption of the cluster core and chose effective interactions by taking care of subsystem energies such as $\alpha$-$n$ and $^6$He as well as nucleon-nucleon scattering. We used some sets of interaction parameters and showed the calculated results of the ground and excited states of He isotopes. By assuming $(0s)^2$ configuration as the interior structure of a dineutron, we analyzed dineutron structure of $^8$He and compared it with the $\alpha$-cluster structure of $^{12}$C. The paper is organized as follows. In the next section, we briefly explain the theoretical method of the present work. Results are given in \[sec:results\], and dineutron structure is discussed in \[sec:discussions\]. Finally, we give a summary in \[sec:summary\]. Formulation {#sec:formulation} =========== In this section, we briefly explain the formulation of AMD+GCM in the present calculation. The detailed formulation of the AMD method for nuclear structure study is described in [@ENYOsup; @AMDrev]. There are various versions of practical methods of the AMD framework. In the present work, we performed superposition of a number of AMD wave functions obtained by energy variation with constraints based on the concept of GCM. The procedure of the variation, spin and parity projection and superposition is similar to those of AMD+GCM calculations in [@Itagaki00; @Kimura04; @Enyo04], though the details of model wave functions and effective interactions are different from each other. An AMD wave function is a Slater determinant of Gaussian wave packets; $$\Phi_{\rm AMD}({\bf Z}) = \frac{1}{\sqrt{A!}} {\cal{A}} \{ \varphi_1,\varphi_2,...,\varphi_A \},$$ where the $i$th single-particle wave function is written by a product of spatial($\phi$), intrinsic spin($\chi$) and isospin($\tau$) wave functions as, $$\begin{aligned} \varphi_i&=& \phi_{{\bf X}_i}\chi_i\tau_i,\\ \phi_{{\bf X}_i}({\bf r}_j) &=& \left(\frac{2\nu}{\pi}\right )^{\frac{3}{4}} \exp\bigl\{-\nu({\bf r}_j-\frac{{\bf X}_i}{\sqrt{\nu}})^2\bigr\}, \label{eq:spatial}\\ \chi_i &=& (\frac{1}{2}+\xi_i)\chi_{\uparrow} + (\frac{1}{2}-\xi_i)\chi_{\downarrow}.\end{aligned}$$ $\phi_{{\bf X}_i}$ and $\chi_i$ are spatial and spin functions, and $\tau_i$ is isospin function which is fixed to be up(proton) or down(neutron). The width parameter $\nu$ is chosen to be the optimum value for each system. Accordingly, an AMD wave function is expressed by a set of variational parameters, ${\bf Z}\equiv \{{\bf X}_1,{\bf X}_2,\cdots, {\bf X}_A,\xi_1,\xi_2,\cdots,\xi_A \}$. The energy variation was performed for the parity-projected AMD wave function $\Phi^\pm_{\rm AMD}({\bf Z})$ under constraints. In order to obtain basis wave functions, we adopted the total oscillator quanta and deformation as the constraints. Hereafter, we note the expectation value of an operator $\hat O$ with respect to a normalized parity-projected AMD wave function as $\langle \hat O \rangle$. Expectation values $\langle \hat N^{\rm ho} \rangle$ of the total oscillator quanta is given by the creation and annihilation operators of harmonic oscillator in the same way as [@Enyo04]. In the AMD+GCM calculations with the $\beta$-constraint (for example [@Kimura04]), the deformation is usually constrained by using the rotational invariant value $D\equiv Tr(QQ)/Tr^2(Q)$, where the matrix $Q$ is calculated by quadrupole operators as $Q_{\sigma\rho}=\langle \sum_i \hat\sigma_i \hat\rho_i \rangle$ ($\hat\sigma=\hat{x},\hat{y},\hat{z}$ and $\hat\rho=\hat{x},\hat{y},\hat{z}$) [@Dote97]. Here $D$ is approximately related to the quadrupole deformation parameter $\beta$ as $D(\beta)= (5\beta^2/2\pi+1)/3$. In the present work, we used the modified quadrupole matrix $Q'_{\sigma\rho}\equiv Q_{\sigma\rho}-A\delta_{\sigma\rho}$ ($A$ is the mass number) instead of the original $Q_{\sigma\rho}$ and imposed the constraint on the $D'\equiv Tr(Q'Q')/Tr^2(Q')$. This is useful for He isotopes to obtain basis wave functions with various configurations on mesh points of the two-dimensional parameters, $\beta$ and $\langle \hat N^{\rm ho} \rangle$. The energy variation with the constraint values $N_{\rm const}$ and $\beta_{\rm const}$ was performed with respect to the parity-projected AMD wave function by minimizing the energy defined as, $$E\equiv \langle \hat{H}\rangle+ V^{N}(N_{\rm const}-\langle \hat N^{\rm ho} \rangle)^2 +V^{\beta}(D(\beta_{\rm const})-D')^2.\label{eq:const}$$ Here the artificial potentials are introduced to satisfy the condition of the constraints. With a given set of constraint values $(N_{\rm const},\beta_{\rm const})$ the optimum wave function $\Phi^\pm_{\rm AMD}(N_{\rm const},\beta_{\rm const})$ was obtained. Finally, we superposed the spin-parity eigen states projected from the obtained wave functions, $$|^8{\rm He}(J^\pm_n)\rangle = \sum_{N_{\rm const},\beta_{\rm const}}c^{J\pm}_n(N_{\rm const},\beta_{\rm const}) |P^{J}_{MK}\Phi^\pm_{\rm AMD}(N_{\rm const},\beta_{\rm const})\rangle,$$ where the coefficients $c^{J\pm}_(N_{\rm const},\beta_{\rm const})$ were determined by diagonalizing the Hamiltonian and Norm matrices. In the present calculations, we took only $M=K=0$ states. Results {#sec:results} ======= Calculations ------------ $^6$He, $^8$He and $^{10}$He were calculated by the AMD+GCM method. The strengths, $V^{N}$ and $V^{\beta}$, for the constraint potentials in eq.\[eq:const\] are chosen to be 30 MeV and 2000 MeV, respectively. We chose the width parameter $\nu$ to optimize the energy for the $P^{J=0}_{(MK)=(00)}\Phi^+_{\rm AMD}(N_{\rm const} =N_{\rm min}+2)$, which gives the minimum energy among the states $P^{J=0}_{(MK)=(00)}\Phi^+_{\rm AMD}(N_{\rm const})$ in most cases. Here, $N_{\rm min}$ is the minimum value of the harmonic-oscillator quanta, $N_{\rm min}=$2, 4, and 6 for $^6$He, $^8$He, and $^{10}$He, respectively. A common $\nu$ value for each He isotope are used in the calculation with each interaction. The adopted $\nu$ values are listed in table \[tab:int\]. We adopted the constraint values of the mesh points $(i,j)$ on the $N_{\rm const}$-$\beta_{\rm const}$ plane as $N^{(i)}_{\rm const}=N_{\rm min}+\Delta^{(i)}$($\Delta^{(i)}$=0,1,2,3,4,6,8,10 for positive parity states and $\Delta^{(i)}$=1,2,3,4,6,8,10 for negative parity states) and $\beta^{(j)}_{\rm const}$=0, 0.2, 0.4, 0.6, $\cdots$, 1.6. Then, the total number of the basis wave functions are 72(63) for positive(negative)-parity states. On the $N_{\rm const}$-$\beta_{\rm cont}$ plane, we first obtained the wave function $\Phi^\pm_{\rm AMD}(N_{\rm const},\beta_{\rm const})$ at $N_{\rm const}=N_{\rm min}+2$ and $\beta_{\rm const}$=0, 0.2, 0.4, 0.6, $\cdots$, 1.6. Then we searched for $\Phi^\pm_{\rm AMD}(N_{\rm const}+1,\beta_{\rm const})$ (or $\Phi^\pm_{\rm AMD}(N_{\rm const}-1,\beta_{\rm const})$) starting from the $\Phi^\pm_{\rm AMD}(N_{\rm const},\beta_{\rm const})$ by increasing(or decreasing) $N_{\rm const}$ one by one. Some of the basis wave functions with the constraints have the breaking of the $^4$He-core. Such the basis wave functions with the $^4$He-core breaking have high energies in general, and therefore, they practically give only small contribution to the low-lying states of $^6$He, $^8$He and $^{10}$He isotopes. It means that the $^4$He cluster is a rather good core in $^6$He, $^8$He and $^{10}$He isotopes, while the motion of valence neutrons is relatively important. Parameter set v58 v56 m62 m56 ----------------------------------- ------------------- ------------- ------------- ------------- ------------- Central force Volkov No.2 Volkov No.2 MV1 case(3) MV1 case(3) Wigner $w$ 0.42 0.44 0.38 0.44 Bartlett $b$ 0 0.15 0 0.15 Heisenberg $h$ 0 0.15 0 0.15 Majorana $m$ 0.58 0.56 0.62 0.56 $\nu(^4$He) (fm$^{-2}$) 0.265 0.265 0.210 0.210 $\nu(^6$He) (fm$^{-2}$) 0.245 0.245 0.210 0.210 $\nu(^8$He) (fm$^{-2}$) 0.240 0.240 0.185 0.185 $\nu(^{10}$He) (fm$^{-2}$) 0.185 0.175 0.165 0.165 exp. v58 v56 m62 m56 $a_t$ (fm) 5.42 ($p$-$n$) 9.7 5.4 6.4 4.2 $a_s$ (fm) $-$16.5 ($n$-$n$) 9.7 $-$23.9 6.4 $>$100 $S_{n}(^5$He) (MeV) $-$0.9 $-$0.7 $-$0.7 $-$1.0 $-$0.4 $2E(^4$He)$-E(^4$He-$^4$He) (MeV) $-$0.1 0.6 1.4 $-$1.3 $-$0.6 $S_{2n}(^6$He) (MeV) 1.0 1.3 $-$0.2 2.1 1.1 $S_{2n}(^8$He) (MeV) 2.1 3.0 3.2 1.2 2.0 : \[tab:int\] Parameter sets of the effective interaction and the values of width parameter $\nu$ adopted in the present work. The theoretical values of scattering length $a_s$($a_t$) for singlet(triplet) even channel, neutron separation energy of $^5$He ($S_{n}(^5$He)$\equiv E(^4$He)$-E(^4$He-$n$)), $2\alpha$ threshold energy of $^8$Be, two-neutron separation energies of $^6$He and $^8$He ($S_{2n}(^6$He)$\equiv E(^4$He)$-E(^6$He) and $S_{2n}(^8$He)$\equiv E(^6$He)$-E(^8$He)) are also listed. Interactions ------------ We used effective nuclear interaction consisting of the central force, the spin-orbit force and Coulomb force. As for the central force, we adopted the Volkov force[@Volkov] used in the work on He isotopes with AMD+GCM($^4$He+$Xn$)[@Itagaki00], and also the MV1 force[@MV1] used in the AMD calculations of $^{12}$C [@Enyo-c12; @Enyo-c12v2]. We used the spin-orbit force of the G3RS force[@LS] as done in [@Itagaki00; @Enyo-c12]. We fixed the strengths of the spin-orbit term as $u_{ls}$ = 2000 MeV, which is the same value as in [@Itagaki00]. By taking care of energies of subsystems, we tuned the interaction parameters, $w$, $b$, $h$, $m$, for Wigner, Bartlett, Heisenberg and Majorana exchange terms in the the central force(Volkov or MV1), respectively. $^6$He, $^8$He and $^{10}$He were calculated with AMD+GCM by using totally 4 cases of central force. The parametrization for the central force is summarized in table \[tab:int\]. In order to demonstrate characteristics of the effective interactions, we also show the relative energies of subsystems and the nucleon-nucleon scattering lengths with these 4 types of interaction. We estimate the energies of the $^4$He, $^4$He-$n$ state with $J^\pi=3/2^-$, and $^4$He-$^4$He state with $J^\pi=0^+$, by assuming the $(0s)^4$ state of $^4$He and performing cluster-GCM calculations within the $\alpha$-$n$ and $\alpha$-$\alpha$ cluster models for simplicity. The first case of interaction is Volkov No.2 force[@Volkov] with interaction parameters $m=0.58$, $b=h=0$. This is the same effective interaction as that used in the AMD+GCM($^4$He+$Xn$) by Itagaki et al. [@Itagaki00], which succeeded to systematically reproduce the binding energies of He isotopes. We note this interaction ’v58’ in this paper. In spite of good agreement of the binding energies of He isotopes, the v58 force has a fault that 2 neutrons are bound in a free space. It is well known that the Volkov force with $b=h=0$ has too strong neutron-neutron attraction, because such the parametrization with no Bartlett term nor Heisenberg term gives the same interaction in the singlet-even channel as that in the triplet-even channel. In reality, the singlet-even channel has weaker attraction, and two neutrons are unbound. In order to describe dineutron correlation in neutron-rich nuclei it might be crucial to reproduce such the feature of two-nucleon system, though it does not matter in case of spin-isospin saturated systems like $Z=N$ nuclei. In the second case of interaction, we used Volkov No.2 force with modified interaction parameters as $m=0.56$, $b=h=0.15$. This interaction (noted as ’v56’) describes well the experimental $S$-wave scattering lengths of the $n$-$n$ and $p$-$n$ channels, and the unbound feature of 2-neutron system. The Majorana parameter $m=0.56$ was determined by adjusting the binding energy of $^8$He to the experimental data. However, this interaction fails to reproduce $2n$ separation energies of $^6$He and $^8$He, and it also gives too strong attraction in $^4$He-$^4$He system. The third interaction(’m62’) and the forth one(’m56’) listed in table \[tab:int\] are based on the MV1 force[@MV1]. The parametrization of the m62 interaction is $m=0.62$ and $b=h=0$, which is the same as used in the AMD calculations of $^{12}$C [@Enyo-c12v2; @Enyo-c12]. In case of the m62 interaction, two neutrons are bound in a free space as well as the Volkov force with $b=h=0$ like the v58 interaction. In the ’m56’ interaction, we used the modified Bartlett and Heisenberg terms, $b=h=0.15$, and the Majorana term $m=0.56$ which was adjusted to reproduce the binding energy of $^8$He. With the $m=0.56$ interaction, two neutrons are almost unbound in a free space, and other energies of subsystems are reasonably reproduced. Ground states of He isotopes ---------------------------- We show the calculated results of the ground states of He isotopes. The energies of He isotopes are shown in Fig. \[fig:he-be\]. The v58 and m56 interactions systematically reproduce the energies of $^4$He, $^6$He and $^8$He, though they overestimate the $^{10}$He energy. On the other hand, the v56 and m62 interactions are poor in reproduction of the $^6$He energy, and therefore, they fail to reproduce two-neutron separation energies of $^6$He and $^8$He as shown in table \[tab:int\]. Hereafter, we discuss the results obtained with the v58 and m56 interactions. We stress again that the v58 interaction well describes the energies of subsystems except for the fault of the too strong neutron-neutron interaction, while the m56 interaction reasonably reproduces the global features of the subsystem energies. =7 cm The calculated root-mean-square radii of proton, neutron and matter density are given in table \[tab:rmsr\] with the experimental data. The theoretical results of other calculations are also listed. Experimentally, extremely large radii of $^6$He and $^8$He have been reported by the reaction cross sections [@Tanihata85; @Tanihata88; @Tanihata92]. It has been suggested that the large radii originate in the remarkable enhancement of neutron radii due to the neutron-halo and neutron-skin structures in $^6$He and $^8$He, respectively. The empirical neutron radii are well described by the present calculations with the m56 interaction. On the other hand, the neutron radii calculated with the v58 interaction are slightly smaller than the empirical ones as well as the former AMD+GCM($^4$He+$Xn$) calculations with the same v58 interaction[@Itagaki00]. The proton radii calculated with the m56 interaction are consistent with the observed data except for that of $^4$He. Figure \[fig:rdens\] shows the proton density and neutron density. In $^6$He, the neutron density has a long tail at a large distance region. This is the neutron halo structure and is similar to the neutron density obtained by other calculations such as SVM[@Varga94]. In $^8$He, the neutron and proton density shows the neutron skin structure at the surface, which well corresponds to the discussion in [@Tanihata92; @Varga94]. Thus, the present calculations with the m56 interaction systematically describe the ground-state properties of $^6$He and $^8$He such as energies and radii. Let us discuss the effect of the spin-orbit force, which may induce the $j$-$j$ coupling feature of neutrons. The expectation values of the spin-orbit force $\langle V_{ls} \rangle$ and those of the squared total intrinsic spin of neutrons $\langle S_n^2 \rangle$ are listed in table \[tab:he8ex\]. From the values of $\langle S_n^2 \rangle$, the $S=1$ component in the $^6$He$(0^+_1)$ state is estimated to be 0.13 and 0.07 in the m56 and v58 results, respectively. It means that the $(p_{3/2})^2$ configuration is contained due to the spin-orbit force. However, the $S=0$ component is still significant because of $L$-$S$ coupling feature of spin-zero $2n$ correlation. We note that the fraction 0.87 in the m56 results for the $S=0$ component in $^6$He is in good agreement with three-body model calculations [@Arai01; @Csoto93; @Baye94; @Hagino05]. Compared with the results of $^6$He, where the $L$-$S$ coupling configuration is significant as well as the $j$-$j$ coupling configuration, the $j$-$j$ coupling feature increases in the $^8$He$(0^+_1)$ state because of the $(p_{3/2})^4$ closure. As a result, the spin-orbit force gives much larger attraction in $^8$He by factor $3\sim 4$ than in $^6$He. It is interesting that the the value $\langle S_n^2 \rangle=0.86$(0.72) of the $^8$He$(0^+_1)$ in the m56(v58) results is different from the value $\langle S_n^2 \rangle=1.33$ for the pure $(p_{3/2})^4$ closed state. This deviation is because the $L$-$S$ coupling configuration is still contained in $^8$He due to the spin-zero $2n$ correlation of neutron pairs. The detailed dineutron structure of $^6$He and $^8$He will be discussed later. exp. AMD-v58 AMD-m56 SVM[@Varga94] RMF[@Sugahara96] AMD($^4$He+$Xn$)[@Itagaki00] NCSM[@Caurier06] ----------- ------- --------------------- --------- --------- --------------- ------------------ ------------------------------ ------------------ $^4$He $r_p$ 1.455(1) 1.46 1.64 1.45 $r_n$ 1.46 1.64 1.45 $r_m$ 1.46 1.64 1.76 $^6$He $r_p$ 1.912(18)$^{(a)}$ 1.83 1.90 1.80 1.89 $r_n$ $2.59-2.61$$^{(b)}$ 2.40 2.49 2.67 2.67 $r_m$ $2.33-2.48$$^{(b)}$ 2.23 2.31 2.46 2.43 2.32 $^8$He $r_p$ $1.76-2.15$$^{(b)}$ 1.76 1.96 1.71 1.88 $r_n$ $2.64-2.69$$^{(b)}$ 2.37 2.63 2.53 2.8 $r_m$ $2.49-2.52$$^{(b)}$ 2.24 2.48 2.40 2.55 2.31 $^{10}$He $r_p$ 2.04 2.13 $r_n$ 2.88 2.97 $r_m$ 2.73 2.82 3.17 : \[tab:rmsr\] Root-mean-square radii (fm) of point-proton, point-neutron and point-matter density of the ground states of He isotopes. The experimental value(a) is deduced from the charge radius[@Wang04], and empirical values(b) are taken from [@Tanihata92; @Tanihata88]. Theoretical values of other calculations, NCSM[@Caurier06], SVM[@Varga94] AMD+GCM($^4$He+$Xn$)[@Itagaki00], RMF[@Sugahara96] are also given. =5. cm Excited states of $^8$He ------------------------ The calculated energy levels of $^8$He are illustlated in Fig. \[fig:he8spe\], and the properties of the excited states are shown in table \[tab:he8ex\]. In both of the m56 and v58 results, the $2^+_1$ state is the lowest excited state and the $0^+_2$ state appears just above the $2^+_1$ state. The $1^-_1$ and $3^-_1$ states are obtained in a higher energy region. In addition, in the present calculations with the m56 interaction, the $1^+_1$, $0^-_1$ and $2^-_1$ states are obtained in almost the same energy region as the $1^-_1$ and $3^-_1$ states. The present AMD framework is regarded as a kind of bound state approximation because of the restricted model space, and therefore, coupling with continuum states is not taken into account. In such a case, only resonance states remain in low-energy region while continuum states rise to a high excitation energy region in principle. However, in order to check the stability of the resonances against neutron decays, their properties should be carefully examined. In the present m56 results, the negative-parity states contain large component of $^6$He+$n+n$-like configurations with the valence neutron far from the core. Since they have extremely large neutron radii and show somehow escaping behavior of neutrons, further investigation is required for these negative-parity states. In particular, the $1^-_1$, $2^-_1$ and $0^-_1$ states can couple with $(0s)^2(0p)^3(1s)^1$ neutron configuration which has a valence $1s_{1/2}$ neutron with no centrifugal barrier. =8. cm ----------- ----------- ------- --------- ------- ------- ------- ------------------------- -------------------------- --------- ------- ------- ------- ------------------------- -------------------------- exp. AMD-v58 AMD-m56 nucleus $J^\pi_n$ $E_x$ $E_x$ $r_p$ $r_n$ $r_m$ $\langle S^2_n \rangle$ $\langle V_{ls} \rangle$ $E_x$ $r_p$ $r_n$ $r_m$ $\langle S^2_n \rangle$ $\langle V_{ls} \rangle$ (MeV) (MeV) (fm) (fm) (fm) (MeV) (MeV) (fm) (fm) (fm) (MeV) $^6$He $2^+_1$ 1.797 3.2 1.82 2.42 2.23 0.19 $-$2.3 2.6 1.87 2.46 2.28 0.27 $-$2.3 $^6$He $0^+_1$ 0 0.0 1.83 2.40 2.23 0.16 $-$2.6 0.0 1.90 2.49 2.31 0.26 $-$2.3 $^8$He $0^-_1$ 10.8 2.13 3.63 3.32 2.05 $-$5.9 $^8$He $2^-_1$ 10.8 2.07 3.41 3.13 2.00 $-$6.2 $^8$He $1^+_1$ 9.0 1.94 2.81 2.62 2.03 $-$2.5 $^8$He $3^-_1$ 7.16 13.5 1.90 2.89 2.68 0.64 $-$6.7 11.5 2.09 3.31 3.05 1.02 $-$5.3 $^8$He $1^-_1$ 4.36 12.1 1.95 3.05 2.82 0.81 $-$7.9 9.8 2.13 3.52 3.23 1.24 $-$5.8 $^8$He $0^+_2$ 10.3 1.97 2.94 2.73 0.67 $-$4.7 8.5 2.11 3.12 2.90 0.99 $-$1.0 $^8$He $2^+_1$ 3.1 9.3 1.76 2.48 2.32 0.39 $-$4.8 6.5 1.93 2.65 2.49 0.40 $-$2.8 $^8$He $0^+_1$ 0 0.0 1.76 2.37 2.24 0.72 $-$11.4 0.0 1.96 2.63 2.48 0.86 $-$7.3 $^{10}$He $0^+_1$ 0 0.0 2.04 2.88 2.73 0.13 $-$2.6 0.0 2.13 2.97 2.82 0.11 $-$1.7 ----------- ----------- ------- --------- ------- ------- ------- ------------------------- -------------------------- --------- ------- ------- ------- ------------------------- -------------------------- : \[tab:he8ex\] Excitation energies, Root-mean-square radii of point-proton, point-neutron and point-matter density, the expectation values of squared total intrinsic spin of neutrons $\langle S_n^2 \rangle$, and those of the spin-orbit force $\langle V_{ls} \rangle$. Compared with the experimental data, the theoretical values of the $2^+_1$ excitation energy are higher than the experimental one. However, it is important that the level structure for the excited states, $2^+_1$, $0^+_2$, $1^-_1$ and $3^-_1$, is not sensitive to the adopted interaction though the relative position to the ground energy depends on the interaction. The $0^+_2$ state is theoretically suggested to appear just above the $2^+_1$ state. What is striking is that the $0^+_2$ state has a remarkably large neutron radius compared with the ground state because of developed $^4{\rm He}+2n+2n$ structure. In the obtained wave function of the $0^+_2$ state, which is given by a superposition of the basis AMD wave functions, the amplitude is found to be widely distributed into the basis wave functions with various spatial configuration of $^4{\rm He}+2n+2n$. This indicates a gas-like feature that the dineutrons are rather freely moving around the $^4$He core. Therefore, we consider that the $0^+_2$ state is the candidate of the cluster gas-like state with two dineutrons around the $\alpha$ core. The detailed discussion of the dineutron-like structure is given later. In the experimental energy spectra, some excited states were observed above the $2^+_1$ state. Spins and parities of these states are not definitely assigned yet. In the present calculations, the predicted $0^+_2$ state has the strong monopole neutron transition from the ground states as the matrix element $M_n(0^+_1\rightarrow 0^+_2)=13.5(13.9)$ fm$^2$ in the m56(v58) results. This neutron matrix element is much larger than the observed proton matrix element $M_n(0^+_1\rightarrow 0^+_2)=5.4$ fm$^2$ of $^{12}$C by more than factor 2. Therefore, we consider that the $^8$He($0^+_2$) might be excited by inelastic scattering on nuclear target. The excited states of $^8$He have been theoretically predicted by a few other calculations such as CSM and GFMC. The CSM gives better agreement of the $2^+_1$ excitation energy with the experimental data[@Volya05]. We also comment that the GFMC calculation with AV18/IL2, which is an [*ab initio*]{} calculation with the realistic 2-body force and the empirical 3-body force, gives similar level structure to the present m56 results. Namely, the GFMC with AV18/IL2 gives the $2^+$ state at $E_x=4.72$ MeV and the $1^+_1$, $0^+_2$ and $2^+_2$ states in the $E_x >5 $ MeV region. Dineutron structure {#sec:discussions} =================== What is dineutron($^2n$) cluster ? ---------------------------------- There is no bound state in an isolate $nn$ system. However, it has been emphasized in many theoretical works that the spatial neutron-neutron correlation plays an important in the binding mechanism of the Borromean systems with two-neutron halo such as $^6$He and $^{11}$Li (for example, [@Bertsch91; @Zhukov93; @Aoyama01; @Aoyama94] and references therein). The neutron-neutron correlation is characterized by a spin-zero $nn$ pair with spatial correlation in $S$ wave. In the correlation density of two-neutron halo nuclei, a peak of the probability appears at the region with a small $n$-$n$ distance($R(nn)$) and a large $n$-core distance in general. This corresponds to the dineutron correlation. In an extended meaning, it is regarded as a “dineutron cluster” which can virtually exist in loosely bound neutron-rich nuclei. As mentioned above, the characteristics of the dineutron are the zero spin and the spatial correlation. In the correlation density for $^6$He, $^{11}$Li and $^{14}$Be given by three-body calculations [@Zhukov93; @Arai01; @Descouvemont-he6; @Descouvemont-be14], the peak for the dineutron correlation are seen typically around the $R(nn)\sim =2$ fm with a ridge in the $R(nn)=2\sim 3$ fm region. It is important that this $n$-$n$ distance at the peak nearly depends on the system size among these three systems, $^6$He, $^{11}$Li and $^{14}$Be. From this most probable $n$-$n$ distance, the typical size of the spatial correlation of the $nn$ pair can be estimated to be about 2 fm. Then, we here approximately describe the dineutron cluster, $^2n$, by a spin-zero neutron pair written by the simple harmonic-oscillator $(0s)^2$ state with the size parameter $b$ in order to investigate dineutron structure in $^8$He. Then, the $^2n$-cluster wave function $\phi^{^2n}({\bf S})$ which is localized at the position ${\bf S}$ is expressed as, $$\begin{aligned} &\phi^{^2n}({\bf S})={\cal{A}}\left\{\phi^{0s}_{{\bf S}} ({\bf r}_1) \chi_\uparrow \phi^{0s}_{{\bf S}}({\bf r}_2)\chi_\downarrow \right\},\\ & \phi^{0s}_{\bf S}({\bf r}_i) = \frac{1}{(b^2\pi)^{\frac{3}{4}}} \exp\bigl\{-\frac{1}{2b^2}({\bf r}_i-{\bf S})^2\bigr\}.\end{aligned}$$ In this definition, the relative motion between two neutrons in the $^2n$ cluster is given by a Gaussian, $$\phi^r({\bf r}_1-{\bf r}_2) = \frac{1}{(b_r^2\pi)^{\frac{3}{4}}} \exp\bigl\{-\frac{1}{2b_r^2}({\bf r}_1-{\bf r}_2)^2\bigr\},$$ with the size $b_r=\sqrt{2}b$, which should be the typical $nn$ distance $b_r=2\sim 3$ fm. With this approximation of the $^2n$ cluster, major component of the dineutron correlation might be taken into account, though the tail part at the large correlation length is omitted. For simplicity, we chose the size parameter $b$ for the $(0s)^2$ dineutron cluster as $b=1/\sqrt{2\nu}$, where $\nu$ is the width parameter $\nu(^6$He) and $\nu(^8$He) optimized for the $^6$He and $^8$He, respectively, in the AMD calculations. The values $\nu$, which are listed in table \[tab:int\], correspond to $b_r=2.0-2.3$ fm and satisfy the typical $nn$ distance of the dineutron correlation. dineutron-cluster motion ------------------------ =7. cm In order to investigate features of dineutron cluster structure in the $0^+$ states of $^8$He, we extracted the $^2n$-cluster motion from the obtained $^8$He($0^+$) wave functions. We assume a simple core $(^4{\rm He}+^2n)_{0^+}$ which is equivalent to the SU(3)-limit $^6$He($0^+$), and form the $^6$He$^{SU(3)}(0^+)$-$^2n$ cluster wave function with the $L=0$ relative motion between the core $^6$He$^{SU(3)}(0^+)$ and the $^2n$ cluster. In the same way as [@Enyo-be12; @Enyo-c12v2] for $\alpha$-cluster motion, we calculated the reduced width amplitudes $ry(r)$ for the $^2n$-cluster motion and the cluster probability $S^{\rm fac}$ by taking the overlap of the $^6$He$^{SU(3)}(0^+)$-$^2n$ cluster wave functions with the $^8$He wave functions. In Fig. \[fig:yl\], we show the reduced width amplitudes in the $^8$He$(0^+_1)$ and the $^8$He$(0^+_2)$ wave functions obtained by the v58 and m56 interactions. These indicate the $^6$He$^{SU(3)}(0^+)$-$^2n$ relative motion. We also show the reduced width amplitudes for the $^8$Be$^{SU(3)}(0^+)-\alpha$ relative motion in the $^{12}$C($0^+_1)$ and $^{12}$C($0^+_2)$ given in [@Enyo-c12]. Surprisingly, the $^2n$-cluster motion in the $^8$He is quite similar to the $\alpha$-cluster motion in the $^{12}$C. First we discuss the features of the dineutron clustering in the $0^+_2$ state. The most striking thing is that the $^8$He$(0^+_2)$ state has the large amplitude of the dineutron cluster in the long distance region around $r=4-6$ fm, which well corresponds to the peak position of the $\alpha$-cluster motion in the $^{12}$C($0^+_2$). The enhancement of the $^2n$-cluster component at the long distance is more remarkable in the v58 results than the m56 results. The cluster probability of the $^8$He($0^+_2$), which is defined by the integrated overlap with the $^6$He$^{SU(3)}(0^+)$-$^2n$ cluster wave functions, is $S^{\rm fac}=0.50$ and $S^{\rm fac}=0.43$ in the v58 and the m56 results. The larger development of the $^2n$ clustering in the v58 results is considered to be because of the stronger $n$-$n$ interaction in the v58 than the m56 interaction. It is very important that, even with the weaker $n$-$n$ interactions of the m56, the $^2n$-cluster structure survives with the significant component in the $^8$He($0^+_2$). Considering that the other $^2n$ cluster exists inside the $^6$He$^{SU(3)}(0^+)$ core, it is regarded that the $^8$He($0^+_2$) has the component of the developed $^4$He+$^2n$+$^2n$ clustering, where two dineutrons are moving in $L=0$ orbits. Furthermore, from the analogy of the $^2n$-cluster structure in the $^8$He($0^+_2$) with the $\alpha$-cluster structure in the $^{12}$C, the $^8$He($0^+_2$) is considered to contain the dineutron gas-like structure. Next, we discuss dineutron structure in the ground state of $^8$He. In the $^8$He($0^+_1$), the reduced width amplitude has a peak at the distance less than 3 fm. It means that the spatial development of the $^2n$ cluster is not so remarkable as that of the $^8$He$(0^+_2)$. After discussing dineutron structure in the $^6$He($0^+_1$), we shall compare it with the dineutron structure in the $^8$He($0^+_1$). In Fig. \[fig:yl\], we show the reduced width amplitudes of the $^4$He-$^2n$ cluster motion in the $^6$He($0^+_1$) obtained by the present calculations, and that in the $^6$He$^{SU(3)}(0^+)$ given by the SU(3)-limit $^4$He-$^2n$ state. Compared with the SU(3)-limit, the calculated $^6$He($0^+_1$) wave function has a long tail of dineutron structure at the surface. The $^2n$-cluster probability in the $^6$He($0^+_1$) state is $S^{\rm fac}=0.91$ and 0.84 in the v58 and the m56 calculations. This is consistent with the fraction, 0.92 and 0.87, of the $S=0$ component, which are estimated from $\langle S^2_n \rangle $. The $^2n$-cluster probability is reduced by the $S=1$ component because of the mixing of the $(p_{3/2})^2$ state. The dineutron wave function in the inner region is similar to that of the SU(3)-limit $^4$He-$^2n$ state. In this region, we have better to call it the spin-zero $2n$ correlation(dineutron correlation) rather than the $^2n$ cluster, because the antisymmetrization effect is important there. Comparing the result of $^8$He$(0^+_1)$ with that of $^6$He($0^+_1$), we found that the reduced width amplitude for the dineutron component is suppressed in the $^8$He$(0^+_1)$. This is because of the $p_{3/2}$ sub-shell closure effect. As mentioned in the previous section, the $j$-$j$ coupling feature is more remarkable in the $^8$He$(0^+_1)$ than the $^6$He($0^+_1$). However, the cluster probability of the $^8$He$(0^+_1)$ is still significant as $S^{\rm fac}=0.57$ and 0.52 in the v58 and the m56 results, respectively. This probability dominantly originates in the SU(3)-limit $^4$He+$^2n$+$^2n$ configuration, which is equivalent to the $L$-$S$ coupling $p$-shell configuration. It means that the dineutron correlation is still important in the $^8$He($0^+_1$). This situation is quite similar to that of the $^{12}$C$(0^+_1)$ which is the admixture of the $p_{3/2}$ closure and the SU(3)-limit $3\alpha$ state. As a result of the $L$-$S$ coupling feature due to the dineutron correlation, the $^8$He$(0^+_1)$ state should contain the significant $(p_{3/2})^2(p_{1/2})^2$ contamination. This result is consistent with the experimental indication of the $p_{1/2}$ component in the $^8$He ground state reported by the recent observations[@Chulkov05; @Keeley07]. As seen in Fig. \[fig:yl\], it is also interesting that the $^8$He($0^+_1$) state has a tail of the $^2n$-cluster motion at the surface, though the tail is slight compared with the long tail in the $^6$He($0^+_1$). In conclusion, the $^8$He($0^+_1$) is the admixture of the $p_{3/2}$ closure and the $L$-$S$ coupling $p$-shell configuration of neutrons with a small tail of the dineutron clustering. $^2n$ condensate wave function ------------------------------ In the previous subsection, we discuss the $^2n$-cluster wave function by assuming the core $(^4{\rm He}+^2n)_{0^+}$ which is equivalent to the SU(3)-limit $^6$He($0^+$). In this description, one of the $^2n$ clusters is confined in the the core $(^4{\rm He}+^2n)_{0^+}$, and its relative wave function to the $^4$He is given by the $1s$ orbit of the harmonic oscillator potential with the oscillator frequency $\omega=8\nu/3$. As shown in Fig. \[fig:yl\], in this SU(3)-limit, the radial wave function of the $^2n$-cluster around the $^4$He remains in the inner region. In such the case, although the $^2n$-cluster is moving in the $S$ wave, the $^2n$-cluster receives much effect of antisymmetrization from the $^4$He core and it does not necessarily indicate a gas-like state. In order to see more directly the $^2n$-cluster gas-like nature, where two $^2n$’s are moving in $S$ wave far from the the $^4$He core, we assumed the $^2n$ condensate wave function in the $^4{\rm He}+^2n+^2n$ system and calculated the overlap with the obtained $^8$He($0^+$) wave functions. We define the $^2n$ condensate wave function by naturally extending the $\alpha$ condensate wave function proposed by Tohsaki et al.[@Tohsaki01] as follows, $$\Psi_{\rm cond}(B)\equiv n_0 \int \prod^k_{i=1} \left\{ d^3{\bf S}_i \exp\left( -\frac{({\bf S}_i-{\bf S}_C)^2}{B^2}\right ) \right\} \Phi_{\rm Brink}({\bf S}_C,{\bf S}_1,{\bf S}_2,\cdots {\bf S}_k),$$ where $n_0$ is the normalization factor and $\Phi_{\rm Brink}({\bf S}_C,{\bf S}_1,{\bf S}_2,\cdots {\bf S}_k)$ is the Brink wave function for the $C+k(^2n)$-cluster system consisting the core($C$) and $k$ dineutrons($^2n$) as, $$\Phi_{\rm Brink}({\bf S}_C,{\bf S}_1,{\bf S}_2,\cdots {\bf S}_k)\equiv {\cal{A}}\left\{\phi^{\rm C}({\bf S}_C)\phi^{^2n}({\bf S}_1) \phi^{^2n}({\bf S}_2)\cdots \phi^{^2n}({\bf S}_k) \right \}.$$ Here, the wave function of the $i$th $^2n$, $\phi^{^2n}({\bf S}_i)$, is given by the $(0s)^2$ state localized around ${\bf S}_i$. ${\bf S}_C$ is the mean position of the center of mass motion of the core, and is chosen to be ${\bf S}_C=-\frac{2}{A}({\bf S}_1+{\bf S}_2+\cdots+{\bf S}_k)$. In heavy limit of the core mass $A$, this wave function is equivalent to the dineutron condensate wave function proposed by Horiuchi[@Horiuchi06]. In the present calculation for $^4{\rm He}+^2n+^2n$, the core $C$ is $^4$He, and the number of $^2n$ clusters is $k=2$. We assumed the $(0s)^4$ state of the core wave function, $\phi^{^4{\rm He}}$, and adopted the common size parameter $b=1/\sqrt(2 \nu(^8{\rm He})$ for the $^4{\rm He}$ and $^2n$ clusters. In the practical calculations, the 6-dimensional integrals for the coordinates, ${\bf S}_1$ and ${\bf S}_2$, are performed by taking mesh points on $(\theta_{12},|{\bf S}_1|,|{\bf S}_2|)$ and the total-angular-momentum projection ($\theta_{12}$ is the angle between ${\bf S}_1$ and ${\bf S}_2$). =6. cm In Fig. \[fig:bec\], we show the squared overlap, $|\langle ^8{\rm He}|\Psi_{\rm cond}(B)\rangle|^2$, between the $^2n$ condensate wave function and the $^8$He wave functions obtained by AMD+GCM. The calculated values are plotted as a function $B$ which indicates the size of the spatial distribution of $^2n$ clusters in the condensate wave function. The $^8$He$(0^+_1)$ has the overlap, about 0.5, at $B < 2$ fm. The condensate wave function $\Psi_{\rm cond}(B)$ with such a small size $B$ is almost equivalent to the SU(3)-limit $^4$He+$2n$+$2n$ state. On the other hand, the $^8$He$(0^+_2)$ has the maximum overlap, about 0.5, at remarkably large size $B=4-5$ fm. This is an strong indication of the dineutron gas-like component in the calculated $^8$He$(0^+_2)$. The dineutron gas-like feature is further enhanced in case of the v58 interaction than the m56 interaction. These results are consistent with the discussion of the $^2n$-cluster wave function in the previous subsection. Summary {#sec:summary} ======= We studied the structure of $^8$He with a method of AMD+GCM. We chose the effective nuclear interactions by taking care of energies of subsystems, and reproduced the properties of ground states of $^4$He, $^6$He and $^8$He. In the ground state of $^8$He, the component of the $p_{3/2}$ sub-shell closure is dominant. However, the $L$-$S$ coupling feature is also significantly contained because of the spin-zero dineutron correlation. This is consistent with the experimental report on the significant mixing of $(p_{3/2})^2(p_{1/2})^2$ component in the $^8$He($0^+_1)$. It is concluded that the $^8$He($0^+_1)$ is the admixture of $p_{3/2}$ sub-shell closure and $L$-$S$ coupling $p$-shell configurations with a slight dineutron tail at the surface. This result is also consistent with the experimentally suggested large spectroscopic factor of the $^6$He($2^+$) in the $^8$He($0^+_1)$. The present results suggest that the $0^+_2$ state may appear a few MeV above the $2^+_1$ state. By analyzing dineutron structure, it was found that this state has a significant component of the developed $^4$He+$^2n$+$^2n$ structure where two dineutrons are moving around the $^4$He core in $S$ wave with a dilute density. The $^2n$-cluster wave function of the $^8$He($0^+_2$) state is similar to the $\alpha$-cluster wave function of the $^{12}$C($0^+_2$) state. Therefore, we consider that the predicted $0^+_2$ state is the candidate of the dineutron gas-like state, which is analogy to the $\alpha$ condensate state suggested in the $^{12}$C($0^+_2$). In the experimental energy spectra of $^8$He, some excited states were observed above the $2^+_1$ state. Spins and parities of these states have not been definitely assigned yet. Since the present calculations predicted the remarkable neutron matrix element for the monopole transitions $^8$He($0^+_1)\rightarrow ^8$He($0^+_2)$, we expect that the $^8$He($0^+_2$) might be excited in inelastic scattering on nuclear target. Since the AMD framework is regarded as a kind of bound state approximation because of the restricted model space, coupling with continuum states is not taken into account. In future study, widths of the excited states should be carefully investigated by taking into account the continuum coupling in order to confirm the stability of the resonances against particle decays. In the present work, the calculations were performed within the AMD model space by using effective interactions. We chose the interaction parameters by taking care of subsystem energies such as $\alpha$-$n$, $^6$He as well as nucleon-nucleon systems. Although it is difficult to completely reproduce all of the subsystem energies with a unique effective interaction, we found the interaction which can reasonably reproduce the global feature of the subsystem energies. We here stress that the level structure of the excited states is not sensitive to the adopted nuclear forces within the reasonable choice of effective interaction, though the excitation energy relative to the ground state depends on the interaction. It is also important that the dineutron structure of the $^8$He$(0^+$) states is qualitatively similar among four sets of interaction adopted in the present calculations. For further investigations of He isotopes, more extended calculations based on the realistic forces should be important as well as [*ab initio*]{} calculations. Acknowledgments {#acknowledgments .unnumbered} =============== The authors would like to thank Prof. Horiuchi, Prof. Tohsaki and their collaborators for valuable discussions. They are also thankful to members of Yukawa Institute for Theoretical Physics(YITP) and Department of Physics in Kyoto University, especially Dr. Takashina for fruitful discussions. 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--- abstract: 'National parks often serve as hotspots for environmental crime such as illegal deforestation and animal poaching. Previous attempts to model environmental crime were either discrete and network-based or required very restrictive assumptions on the geometry of the protected region and made heavy use of radial symmetry. We formulate a level set method to track criminals inside a protected region which uses real elevation data to determine speed of travel, does not require any assumptions of symmetry, and can be applied to regions of arbitrary shape. In doing so, we design a Hamilton-Jacobi equation to describe movement of criminals while also incorporating the effects of patrollers who attempt to deter the crime. We discuss the numerical schemes that we use to solve this Hamilton-Jacobi equation. Finally, we apply our method to Yosemite National Park and Kangaroo Island, Australia and design practical patrol strategies with the goal of minimizing the area that is affected by criminal activity.' author: - 'D. J. Arnold[^1]' - 'D. Fernandez' - 'R. Jia' - 'C. Parkinson' - 'D. Tonne[^2]' - 'Y. Yaniv[^3]' - 'A. L. Bertozzi' - 'S. J. Osher' bibliography: - 'bibliography.bib' title: Modeling Environmental Crime in Protected Areas Using the Level Set Method --- Introduction ============ Environmental crime in protected national parks is a concern to authorities around the world. National parks often serve as hotspots for illegal logging and animal poaching, henceforth referred to as illegal extraction. In recent years, scientists have aided law enforcement agencies in the prevention of such crimes. One way in which they do this is by building models to describe deforestation, track animal movement, and predict adversarial behavior of criminals. @leader1993policies examined a case study of poaching in the Luangwa Valley, Zambia and concluded that increasing detection rates was a more effective deterrent to environmental crime than increasing severity of punishment. This means that models that can help improve detection rates of environmental criminals are a useful tool for patrollers of protected areas. Authorities employ different strategies to patrol protected areas and detect illegal extractors, and research into the effectiveness of these strategies is of major interest. Researchers are challenged to find efficient and practical patrol strategies to police vast areas with limited resources. An important model of illegal extraction was given by @Albers. They describe illegal extraction as a continuous spatial game between patrol units and extractors, constructing a model where extractors maximize their expected profit by trading off costs of penetrating further into a protected region (increasing effort required and risk of capture) against increased benefits of extracting further from the boundary of the protected region. Albers’ model was formulated as a Stackelberg game, an adversarial game with a defender (patrollers) and many attackers (extractors) with perfect information about the defender’s strategy. Albers gave some qualitative results in a very simplified situation where the protected region was circular, and all quantities were radially symmetric. @johnson2012patrol proved an additional result regarding optimal patrol strategies for this simple case. Subsequently, models for illegal extraction in protected regions using discrete rather than continuous methods have been developed by Fang et al. [@fang2013optimal; @fang2017paws] and Kar et al. [@kar2015game; @Intercept2017]. These models incorporate different methods for modeling human behavior and different ways to treat the protected region. used repeated Stackelberg games to attempt to understand the evolution of attackers strategies, but did not consider realistic spatial domains. developed a model including detailed terrain and spatial information by describing the protected region as a series of nodes connected by edges corresponding to natural pathways through the region (for example along rivers or walking trails). The PAWS algorithm developed by @fang2017paws has been deployed in Queen Elizabeth National Park (QENP) in Uganda, and in Malaysian forests. Taking another approach, @Intercept2017 used machine learning techniques to construct a model for predicting poaching attacks, again deploying the model for a field test in QENP, Uganda. Our goal is to generalize the model of @Albers so that it is applicable to realistic protected regions, can include the effects of terrain and geometry, and can be executed for real protected regions. The most important mathematical technique that enables this extension is the level set method of @osherSethian1988, which provides a simple way to track moving fronts through the protected region. Indeed, the level set method and Hamilton-Jacobi equations are now used extensively in modeling travel in a variety of contexts. Sethian and Vladimirsky [@SethVlad2; @SethVlad1] discuss a level set method for optimal travel on manifolds. In application, @Dubins, and later @Takei, modeled movement of simple cars through terrain with obstacles using methods from control theory which include Hamilton-Jacobi-Bellman equations. Tomlin has used similar methods to determine reachable sets for aircraft autolanders [@TomlinAircraft], and to model human movement in adversarial reach-avoid games [@TomlinReachAvoid]. Thus there is a precedent for using Hamilton-Jacobi type equations to model movement throughout domains. However, as discussed above, attempts to model environmental crime have either required overly restrictive assumptions regarding radially symmetry and geometry of the protected region, or have been discrete in nature. This work serves to fill a gap in the literature. We suggest a continuum model for environmental crime which is still able to account for realistic terrain information, and bridges the divide between environmental crime modeling and Hamilton-Jacobi formulations for modeling human movement. This paper is structured as follows. In section \[sec:albersmodel\], we describe the model of @Albers. In section \[sec:levelset\], we introduce the level set method, describe the numerical methods used to solve it, and show how it can be applied to model movement through regions with terrain. In section \[sec:model\], we describe our model for illegal extraction from protected regions, and provide an algorithm to calculate the regions through which extractors will pass. In section \[sec:results\], we present results and discuss their implications towards both patrol strategies and our modeling approach. Finally, in section \[sec:conclusion\], we make conclusions and identify potential directions for further research. The Illegal Logging Model of @Albers {#sec:albersmodel} ==================================== presented a model for illegal extraction from protected regions that includes spatial effects. They use a Stackelberg game in a circular region between defenders and extractors. In their model, an extractor gains benefit and incurs cost based on the depth $d$ into the region where they choose to extract. The benefit is associated with the extraction point, and is modeled as a concave increasing function $B(d)$. Cost is based on the trip distance, and is modeled as a quasiconvex increasing function $C(d)$, but no specific functions are given. Then the extractor’s profit associated with extracting at depth $d$ is $$\label{eq:AlbersProfitNoPatrol} P(d) = B(d) - C(d).$$ If the region is patrolled with density $\psi(d)$, the probability that the extractor is detected is the cumulative probability $\Psi(d)=\int^d_0\psi(r)dr$ over the trip out of the protected region. If the illegal extractor is caught, any resources they’ve extracted are confiscated and they must leave empty-handed. The extractor’s expected profit can therefore be written as $$\label{eq:AlbersProfit} P(d) = (1 - \Psi(d))B(d) - C(d).$$ Given the extractors have perfect information of the patrol strategy, they know the function $\Psi(d)$, and hence their task is to find the optimal distance $d^*$ to penetrate into the protected region. Simultaneously, the patrollers task is to pick the patrol strategy $\psi(d)$ that minimizes $d^*$. Figure \[fig:albersmodel\] illustrates the model, showing the pristine region where extractors are never present and the outer region through which they pass while travelling into or out of the protected region. (0,0) circle (1.5); (0,0) circle (3); (0,1.5)–(0,3); (0,-1.5)–(0,-3); (1.5,0)–(3,0); (-1.5,0)–(-3,0); (1.0607,1.0607) – (2.1213,2.1213); (-1.0607,-1.0607) – (-2.1213,-2.1213); (-1.0607,1.0607) – (-2.1213,2.1213); (1.0607,-1.0607) – (2.1213,-2.1213); at (-2.2,0.3) [$d$]{}; at (0,0) [Pristine area]{}; The Level-Set Method {#sec:levelset} ==================== The level-set method is a powerful technique that we use to model human movement through complex domains. Originally developed by @osherSethian1988, the level set method models fronts which propagate with a prescribed velocity. The front is embedded as the zero level set of an auxiliary function, which is then evolved according to a partial differential equation. We describe a simple version of the method in ${\mathbb R}^2$, sufficient for our needs. Suppose we are given a simple, closed curve (or a collection of non-intersecting, simple, closed curves) denoted by $\Gamma$. To evolve $\Gamma$ via level set flow, we begin by finding a Lipschitz continuous function $\phi_0 : {\mathbb R}^2 \to {\mathbb R}$ such that $\phi_0$ is positive inside $\Gamma$ and negative outside $\Gamma$, and thus $\Gamma = \{ x \in {\mathbb R}^2 \, : \, \phi_0(x) = 0\}$. Next, we evolve $\phi:{\mathbb R}^2\times[0,\infty)\to{\mathbb R}$ using the PDE $$\label{eq:genLevelSet} \begin{aligned} \phi_t + v(x) {\left \lvert \nabla \phi \right \rvert} = 0,& \\ \phi(x,0) = \phi_0(x),& \end{aligned}$$ for some non-negative velocity function $v$. Define $\Gamma(t) = \{x \in \mathbb R^2 \, : \, \phi(x,t) = 0\}$. This represents an evolution of $\Gamma = \Gamma(0)$. Indeed, we can re-write equation \[eq:genLevelSet\] as $\phi_t + v(x) \left( \frac{\nabla \phi}{{\left \lvert \nabla \phi \right \rvert}} \right) \cdot \nabla \phi = 0.$ Note that $\nabla \phi/ {\left \lvert \nabla \phi \right \rvert}$ is a unit vector which is normal to the level contour $\Gamma(t)$. Thus, locally, this equation models advection in the direction normal to $\Gamma(t)$ with velocity $v(x)$. This causes $\Gamma(t)$ to deform with normal velocity $v(x)$. With this in mind, one can view $\Gamma(t)$ as the set of points which could be reached after walking inward from the boundary for a time $t$ with normal velocity $v(x)$. For example, if $v(x) \equiv 1$, then for $t > 0$, the curve $\Gamma(t)$ is the set of points a distance $t$ from the original curve. This procedure is shown in figure \[fig:levelSetExample\]. Figure \[fig:distEx\] shows the boundary of a region defined by $(x(\theta),y(\theta)) = (\cos(\theta) ,\sin(\theta)+\sin(3\theta)/2)$ and a series of contours of points accessible after traveling a certain time from the boundary. Figure \[fig:initFun\] shows the graph of the level set function and with the same level sets overlaid on the surface. This example also displays one of the strengths of the level set method. Depending on the original curve, the the level contours may break apart into disconnected pieces or merge into into a single piece. More primitive methods for curve evolution (many of which involve tracking points along the curve) have trouble handling these changes in topology but the level set method accounts for them with no special considerations. For more information regarding level set methods, see @osher2003level. [0.35]{} {width="\textwidth"}   [0.45]{} {width="\textwidth"} Illegal Deforestation {#sec:model} ===================== Our model for the actions of illegal extractors committing crimes in protected regions is based on @Albers, as described in section \[sec:albersmodel\]. They derived a game-theoretic model for illegal deforestation in an idealized setting, assuming the protected region was circular to significantly simplify the problem, and neglecting the effects of terrain. We wish to relax these restrictions which leads to several significant difficulties. We are interested in real national parks and hence the protected region will not be symmetric and the benefit and cost functions could be arbitrary. In @Albers, the profit is maximized on a ring with some radius and extractors will enter from any point on the boundary of the region to travel to the maximum profit ring. Without radial symmetry however, the profit will, in general, be maximized at an isolated point rather than along a closed curve, and correspondingly there will be a single optimal route from this point to the boundary of the protected region. Hence illegal extractors only ever occupy a set of points of measure zero. In our model, we resolve this issue by assuming that extractors will tolerate extracting anywhere where the profit is close to the maximum possible profit. The cost function of @Albers does not generalize easily to irregular geometries. Instead of using a predefined formula based on distance, we constructively determine the cost by considering the effects that would detract from the final profit with the help of the level set method. Given terrain and elevation information for a real protected park, the paths taken by extractors will not be simple straight lines. Therefore it is also necessary to find the paths extractors will follow to exit of the protected region. In the next section we describe an algorithm to find the pristine and non-pristine regions in a protected area with arbitrary shape, terrain, benefit distribution and patrol strategy. Problem Description ------------------- Let $\Omega$ represent the area that needs to be protected and let $\psi(x,y)$ be the patrol density function measuring the likelihood of an extractor being caught at position $(x,y)$. The patrol budget $E$ is defined as $E=\int_{\Omega} \psi(x,y) \,\mathrm dx\,\mathrm dy$. The budget measures how many patrolling resources can be allocated to protect the domain $\Omega$. Effectively, the budget scales $\psi$. Extracting at a position $(x,y)$ gives the extractor an amount of benefit $B(x,y)$. For example, the benefit could depend on the quantity, value, or species of trees. We make no assumptions about $B$ except that it is known. $C(x,y)$ is the expected cost associated with extracting at position $(x,y)$. The cost is based on two factors, the effort involved in traveling from the extraction point to the boundary of the protected region, and the risk of being caught by patrols whilst inside the domain. The further into the protected the region the extractors penetrate, the higher the cost as they must expend more time/energy traveling, and are more likely to be captured by patrollers. This highlights a difference between our model and @Albers: we include the effect of the patrol directly in the cost function, whereas they include it as a modification to the benefit. In our model, the profit an extractor expects from extracting at a point $(x,y)$ in the protected region is simply $P(x,y)=B(x,y)-C(x,y)$. We assume the extractors accept a position to extract if the profit is within some tolerance of the maximum obtainable profit. Finally, we determine the paths that extractors will take when leaving the protected area (where they can be captured by the patrol). We assume the extractors will always take an optimal path from the extraction point to the boundary, and can only be caught after they have finished extracting and started heading back to the boundary (since no crime has been committed before they have extracted). Calculating the Profit Function ------------------------------- Since the benefit function is known in advance, it remains to calculate the expected cost function. As explained in the previous section, the cost function depends on many factors, including the optimal route to a given point. The level set method avoids most of the difficulties one might expect when calculating optimal paths from all points inside the protected region to the boundary. We use the level set method to find contours of equal cost in the protected region. The level set method implicitly uses the optimal path between the boundary and an interior point, and can be efficiently implemented. The expected cost associated with extracting at a given point $ (x_0,y_0) $ is calculated with the following level set equation, using the boundary of the domain $\Omega$ as the zero-level set of the function $\phi(x,y,0)$: $$\label{eq:costLevelSet} {\frac{\partial \phi}{\partial t}} = -\frac{1}{1/v(x,y)+\alpha\psi(x,y)B(x_0,y_0)} {\left \lvert \nabla \phi \right \rvert}.$$ The cost $C(x_0,y_0)$ is implicitly defined by $\phi(x_0,y_0,C(x_0,y_0)) = 0$. Intuitively, the point $(x_0,y_0)$ lies somewhere inside the curve $\Gamma = \partial \Omega$. When we evolve the boundary using \[eq:costLevelSet\], the zero level-sets $\Gamma(t) = \{\phi(x,y,t) = 0 \}$ represent the set of points which require equal cost to reach from the boundary. For some $C > 0$, we will have $(x_0,y_0) \in \Gamma(C)$. This $C$ is the cost of extracting at $(x_0,y_0)$. This is demonstrated in figure \[fig:demonstrateCost\]. [r]{}[0.37]{} The important term from a modeling perspective is the coefficient of ${\left \lvert \nabla\phi \right \rvert}$. The denominator of this term has two summands, $1/v(x,y)$ and $\alpha\psi(x,y)B(x_0,y_0)$. For the first summand, $v(x,y)$ is the walking speed at $(x,y)$ which is determined by the terrain at $(x,y)$. It is calculated by using the steepness at $(x,y)$ and a walking speed function adapted from [@IC2017] (discussed in section \[sec:walk\]). Increasing $v$ makes the level sets move faster, and hence decreases the cost to reach a given point. The second summand, $\alpha\psi(x,y)B(x_0,y_0)$, is a term that penalizes extractors for spending time in highly-patrolled areas. The patrol density function $\psi$ measures the likelihood of an extractor getting caught at position $(x,y)$, and $B(x_0,y_0)$ is the benefit acquired at $ (x_0,y_0) $. If the extractor gets caught, they lose all of the accrued benefit, so the impact of being caught should scale with the benefit gained. When $\psi(x,y)B(x_0,y_0)$ is larger, the level sets move more slowly, and the cost associated with extracting at a point increases. To compare the summands we include a parameter $\alpha$, which can be thought of as a risk aversion factor. This parameter weighs the willingness of an extractor to assume risk of being captured versus the willingess to expend time or energy. A large $\alpha$ signifies that the extractor will accept less risk of being captured, and will be willing to exert more effort to reach their chosen extraction point. Alternatively, a small $\alpha$ means the extractor is willing to accept a higher risk of being captured to decrease the physical effort required to travel in and out of the protected region. Note that this summand includes information about the extraction point, through the benefit $B(x_0,y_0)$. This means that the cost function must be calculated separately for each extraction point. ### The walking speed equation {#sec:walk} ![Our velocity function \[eq:ourIC\] (solid blue line), and the @IC2017 function \[eq:IC\] (dashed red line). Our velocity function decays to zero for high slopes while the function suggest by @IC2017 does not.[]{data-label="fig:velocityFunctions"}](VelFuncBoth.pdf){width="75.00000%"} gave a formula for walking speed based on slope. Recently, @IC2017 gave an improved formula that they claimed more accurately predicted travel times for humans walking on roads, $$\label{eq:IC} v = 0.11+\exp\left(-\frac{\left(100s+2\right)^2}{1800}\right),$$ where $100s$ is the grade in percent, and $v$ is the corresponding speed in $\textrm{m}/\textrm{s}$. This formula agrees well with experimental results given in [@IC2017], but has a drawback for our model, that the speed does not vanish as the slope goes to infinity. We hence modify the given equation so that the speed drops to zero as the slope becomes more extreme and use $$\label{eq:ourIC} v = 1.11\exp\left(-\frac{\left(100s+2\right)^2}{2345}\right),$$ which matches the maximum speed, gives similar results for grades less than $60 \%$ and decays to zero for more extreme slopes as can be seen in figure \[fig:velocityFunctions\]. We note that the specific walking velocity function is not crucial in our model, and other functions could be used. We also are able to incorporate areas where the walking speed is faster or slower which could model, for example, overgrown scrub or walking trails. It is also possible to set $v$ to zero in places to model impassable obstacles. The Expected Cost Algorithm {#sec:costAlgorithm} --------------------------- The expected profit function is calculated using the following algorithm. 1. Since the cost level set equation \[eq:costLevelSet\] depends on the particular benefit level where extraction takes place, we must solve it for all possible values of $B$. We pick $N$ benefit values in $[\min_\Omega B,\max_\Omega B]$ and evaluate the cost function using each benefit value $B_i$, by performing steps 2-4 for each $B_i$. 2. For a given $B_i$, first find the contour(s) of points where $B(x,y)=B_i$. In figure \[fig:profit1\], the black contour is the boundary of the domain, and the blue contour is an equal benefit contour with the same benefit $B_i$. 3. Evolve the expected cost level set function with the benefit $B_i$ using equation \[eq:costLevelSet\]. In figure \[fig:profit2\], the red contours are the equal-cost level set contours. Each of the contours represents a different cost value. 4. The cost level sets only apply to the points whose benefit has been used to calculate the level sets. That is, we have calculated the cost assuming the benefit gained is $B_i$, so our results are only valid at the points $(x,y)$ such that $B(x,y)=B_i$. Therefore we find the intersections between the cost level contours $\phi(x,y,C)=0$ and the benefit contour $B(x,y)=B_i$. At these intersections (marked in green), we can calculate $C(x,y)$. 5. Repeat steps 2 to 4 for each benefit level $B_i$, until values of $C(x,y)$ are known throughout the region. [0.31]{}   [0.31]{}   [0.31]{} Once the expected cost has been found it can be interpolated, and can be used to find the expected profit $P(x,y)=B(x,y)-C(x,y)$. This expression for the profit includes the effect of the defenders. Finding the Pristine Region --------------------------- Once the expected profit function is found, the next step is to find the pristine region, the region that extractors will never occupy. In @Albers, the pristine region was simply the part of the circular forest between its center and the ring of maximum profit. In general, however, the pristine region must be defined differently. We assume that extractors will extract anywhere where the profit is close to the maximum possible profit. Denote $P_{max}$ as the maximum expected profit, and $(1-\varepsilon)P_{max}$ as the lowest expected profit that the extractor will accept for some tolerance $\varepsilon$. We also assume that the extractors will take the optimal path away from their extraction point to the boundary of the region. The pristine region is determined using the following algorithm. 1. Find the equal profit contour corresponding to $(1 - \varepsilon)P_{max}$. The extractors will only extract within this high-profit region. 2. Pick points inside the high-profit region uniformly at random, and calculate the optimal path from that point to the boundary of the region. The optimal path can be found easily by solving the gradient descent equation $${\frac{\mathrm d\vec x}{\mathrm dt}}=-\nabla C_{B(\vec x_0)}(\vec x),$$ where $\vec x(0)=\vec x_0$ and $C_{B(\vec x_0)}(\vec x)$ is the cost function associated with the benefit value at the extraction point $\vec x_0$. 3. The non-pristine region includes the high-profit region, and all points within some (small) distance of one of the optimal paths found in the previous step. The rest of the region is pristine, and is not traversed by extractors. Metrics for Measuring Patrol Effectiveness ------------------------------------------ To measure the effectiveness of a given patrol strategy $\psi$, we use two metrics, although others could be used. A simple measure, as used in @Albers, is to calculate the proportion of the protected area that is pristine, using, $$\textrm{Pristine Proportion} = \frac{\int_{\Omega} \chi(x,y)\,\mathrm d x\,\mathrm d y}{\int_\Omega\,\mathrm dx\,\mathrm dy},$$ where $\chi(x,y)$ is an indicator function which is 1 in the pristine region and 0 in the non-pristine region. Some parts of the protected region may be more valuable to protect than other regions, and we could instead choose to measure the proportion of the net value protected using the following integral, $$\textrm{Proportion of Value Protected} = \frac{\int_{\Omega} B(x,y)\chi(x,y) \,\mathrm d x\,\mathrm d y}{\int_\Omega B(x,y)\,\mathrm dx\,\mathrm dy},$$ where $\chi$ is defined as previously. In place of the benefit function, other functions could be used that measure the relative importance of protecting certain areas. Equivalence of this model and [@Albers] {#sec:equiv} --------------------------------------- Although our model is more sophisticated than that given in [@Albers], it does give the same profit function in some simple symmetric cases. Consider a circular forest of radius 1 with walking speed 1 and a uniform patrol strategy with patrol budget 1 (so $\psi=1/\pi$). Further assume that the benefit associated with extracting at depth $d$ is $B(d)=2d$ and that the cost is $C(d)=d$ (equivalent to the travel time with walking speed 1 in our model). The capture probability associated with extracting at depth $d$ is $\Psi(d)= \int^d_0 \psi(r) dr = d/\pi$, and so the profit function based on equation \[eq:AlbersProfit\] is $$\label{eq:AlbersProfitCircle}P(d) = \left(1-\frac{d}{\pi}\right)2d-d=d-\frac{2d^2}{\pi}.$$ We can arrive at the same result using our model. We assume that the protected region is the unit circle centered at the origin so $\phi_0(x) = 1 - {\left \lvert x \right \rvert}.$ For $x$ in the unit circle, set $B(x) = 2(1-{\left \lvert x \right \rvert})$; this is analogous to setting $B(d) = 2d$ in the Albers model. Further, consider constant walking velocity $v = 1$, constant patrol density $\psi = 1/\pi$ and risk factor $\alpha = 1$. Fix $x_0$ in the unit circle. Note that ${\left \lvert \nabla \phi \right \rvert} = 1$ almost everywhere and the normal velocity $\left(1+2(1-{\left \lvert x_0 \right \rvert})/\pi\right)^{-1}$ in equation \[eq:costLevelSet\] is constant with respect to $x$. Thus the solution to equation \[eq:costLevelSet\] is given by $$\phi(x,t) = 1 - {\left \lvert x \right \rvert} - \frac{t}{1 + \frac{2(1-{\left \lvert x_0 \right \rvert})}{\pi}}.$$ Recall that the cost $C(x_0)$ associated with extracting at the point $x_0$ is given implicitly by $\phi(x_0,C(x_0)) = 0$. Solving this equation, we have $$C(x_0) = (1-{\left \lvert x_0 \right \rvert})\left(1 + \frac{2(1-{\left \lvert x_0 \right \rvert})}{\pi} \right)$$ and the profit $P(x_0) = B(x_0) - C(x_0)$ is given by $$P(x_0) = (1- {\left \lvert x_0 \right \rvert}) \left(1 - \frac{2(1-{\left \lvert x_0 \right \rvert})}{\pi}\right).$$ Note that this profit function is radial and that for any $x_0$ at depth $d$, we have $P(x_0) = d\left(1 - \frac{2d}{\pi}\right)$ exactly as in equation \[eq:AlbersProfitCircle\]. In general, with arbitrary benefit function $B(d)$, linear cost $C(d)=cd$, and homogenous patrol density $\psi(d)=p$, Albers gives $$P(d)=(1-pd)B(d)-cd,$$ while our proposed model gives (with speed set to $v=1/c$ and $\alpha=1$) $$C(d) = d\left(c+pB(d)\right),$$ and so $$P(d) = B(d)-cd-pdB(d)=(1-pd)B(d)-cd.$$ The two models agree even if the patrol density is piecewise constant and radially symmetric. If the cost function and/or patrol density are more complicated, then $\phi_t$ will not be constant, and the PDE solution will be more complicated. Our approach to calculating the cost and benefit functions is therefore equivalent to the approach given by @Albers in simple scenarios but is capable of modeling much more general cases. Numerical Implementation ------------------------ We briefly discuss the numerical implementation of our model. Numerical solutions to Hamilton-Jacobi equations have been well-studied for several years. Recent approaches overcome the curse of dimensionality by using Hopf-Lax formulations and optimization [@YTChow], but since we are concerned with two spatial dimensions, grid-based finite difference methods are sufficient. Recall, a Hamilton-Jacobi equation is an equation of the form $\phi_t + H(x,\nabla \phi) = 0$. Due to the nonlinear dependence on $\nabla \phi$, we cannot use basic differencing methods since they may lead to instabilities or fail to revolve the viscosity solution to the equation [@CrandallLions]. Rather, in order to minimize oscillation and track the direction in which information is traveling, we replace the Hamiltonian $H(x,y,\phi_x,\phi_y)$ with a numerical Hamiltonian $\hat H(x,y,\phi_x^+, \phi_x^-, \phi_y^+, \phi_y^-)$ where $\phi_x^+,\phi_x^-$ are the forward and backward difference approximations to $\phi_x$ respectively, and similarly for $\phi_y^+, \phi_y^-$. There are many different acceptable numerical Hamiltonians [@OsherShu1991]. We use the Godunov Hamiltonian given by $$\label{eq:godunovHamil}\hat H(x,y, {\phi_{x}^{+}},{\phi_{x}^{-}},{\phi_{y}^{+}},{\phi_{y}^{-}}) = {\underset{u \in I({\phi_{x}^{-}},{\phi_{x}^{+}})}{\text{ext}}}\,\,\, {\underset{v \in I({\phi_{y}^{-}},{\phi_{y}^{+}})}{\text{ext}}} H(x,y,u,v)$$ where $$I(a,b) = [\min(a,b), \max(a,b)]$$ and $${\underset{x \in I(a,b)}{\text{ext}}} = \left \{\begin{matrix} \min_{a \le x \le b} & \text{if } a \le b, \\ \max_{b \le x \le a} & \text{if } a > b. \end{matrix} \right.$$ Performing the minimization or maximization can be difficult if the Hamiltonian is complicated, but in simple cases, they can be resolved explicitly. In our case, the Hamiltonian is $H(x,y,\phi_x,\phi_y) = \tilde v(x,y) {\left \lvert \nabla \phi \right \rvert}$ where the velocity function $\tilde v$ is positive. When the intervals $I({\phi_{x}^{-}},{\phi_{x}^{+}})$ and $I({\phi_{y}^{-}},{\phi_{y}^{+}})$ do not include $0$, this Hamiltonian is monotone in each of $\phi_x$ and $\phi_y$ on the rectangle $I({\phi_{x}^{-}},{\phi_{x}^{+}}) \times I({\phi_{y}^{-}},{\phi_{y}^{+}})$ and the extrema in \[eq:godunovHamil\] must occur at the corners of the rectangle. If these intervals do contain $0$, some adjustment is necessary, but one can still explicitly resolve the extrema and find that $$\label{eq:GodunovHamiltonianExplicit}\hat H(x,y,\phi^+_x, \phi_x^-,\phi^+_y,\phi_y^-) = \tilde v(x,y) \sqrt{\max\{(\phi_x^-)_+^2,(\phi_x^+)_-^2\} + \max\{(\phi_y^-)_+^2,(\phi_y^+)_-^2\} }$$ where $(A)_+ = \max(A,0)$ and $(A)_- = \min(A,0)$. The Godunov Hamiltonian $\hat H$ gives a first-order approximation to the Hamiltonian $H$. Following @OsherShu1991, we approximate the derivatives $\phi_x,\phi_y$ at second order and we use second-order total variation diminshing Runge-Kutta time stepping to evolve the solution. In doing so, we have constructed a second order accurate essentially non-oscillatory scheme for \[eq:costLevelSet\]. There is an implementation issue which is worth mentioning. Note that the initial function $\phi(x,0)$ can be taken to be precisely the signed distance from $x$ to the initial contour: $$\phi(x,0) =\text{dist}(x,\Gamma) = \left\{\begin{matrix} \inf_{y \in \Gamma} {\left \lvert x-y \right \rvert}, & x \text{ inside } \Gamma, \\ -\inf_{y \in\Gamma} {\left \lvert x-y \right \rvert}, & x \text{ outside } \Gamma. \end{matrix} \right.$$ This is desirable since the distance functions has gradient $1$ almost everywhere and thus if we can preserve this property (that is, if we can ensure that $\phi(x,t)$ is the signed distance to $\Gamma(t)$ for all $t > 0$) we will observe the exact level set motion that we want. However as the level sets evolve, there is some distortion so that numerically for $t > 0$, we no longer have $\phi(x,t) = \text{dist}(x,\Gamma(t))$. If ${\left \lvert \nabla \phi \right \rvert}$ becomes too small, it is difficult to resolve the level sets accurately and if ${\left \lvert \nabla \phi \right \rvert}$ becomes to large, we are required to choose exceedingly fine time discretization or else the numerical solution may develop instabilities. We can prevent these from happening by occasionally replacing $\phi$ with the signed distance function to $\Gamma(t)$. That is, every so often, we halt the time integration, find the current zero level contour $\Gamma(t)$, reset $\phi(x,t) = \text{dist}(x,\Gamma(t))$ and continue. This process is known as *re-distancing* and is discussed by @redistancing. Results {#sec:results} ======= We present results for our algorithm applied to two real world locations: Yosemite National Park in California, and Kangaroo Island in South Australia. Yosemite National Park is a mountainous area with steep mountains and long valleys. Kangaroo Island has an interesting shape, with a narrow neck separating the main part of the island from a smaller part at the eastern end. For both locations we use real elevation data, sourced from the United States Geographical Survey (Yosemite National Park data) and the Foundation Spatial Data Framework (Kangaroo Island data). The elevation profiles for Yosemite National Park and Kangaroo Island are displayed in figure \[fig:elevationPlots\]. The data was processed using QGIS [@QGIS] and imported to MATLAB using TopoToolbox [@topotoolbox2; @topotoolbox1]. We apply several patrol strategies identified by @Albers and @johnson2012patrol before suggesting some simple and more effective patrols that account for the geometry of the regions. [0.49]{} [![Elevation profiles for Yosemite National Park and Kangaroo Island. Yellow corresponds to higher elevation, blue to lower elevation (scales different in each figure).[]{data-label="fig:elevationPlots"}](YosemiteElevationPlot2.png "fig:"){width="\textwidth"}]{}   [0.49]{} ![Elevation profiles for Yosemite National Park and Kangaroo Island. Yellow corresponds to higher elevation, blue to lower elevation (scales different in each figure).[]{data-label="fig:elevationPlots"}](KangarooIslandElevationPlot2.png "fig:"){width="\textwidth"} Yosemite National Park without patrols -------------------------------------- We first consider the case without any patrols, so the cost function depends only on the effort required to travel from the extraction point to the boundary of the park. In figure \[:nopatrolling\], we present results for two cases with different benefit functions. Both benefit functions have the same form, a quadratic increase from 0 at the boundary to a maximum value at the point furthest from the boundary, but figure \[:nopatrol2\] has maximum benefit double that as in figure \[:nopatrol\]. The lower benefit case has a larger pristine area and smaller high-profit region. In the high-benefit case, there is enough incentive for extractors to expend more effort and extract from more locations within the protected region, obtaining much higher profits (although not doubled). A selection of the optimal paths from extraction points to the boundary of the protected region are also shown. In the sections that follow, we will use $k=8$, the high-benefit case. [0.45]{} ![Two cases for Yosemite National Park with no patrols, benefit based on distance from boundary $d$ as $B(d)=kd(2d_m-d)/d_m$ (where $d_m=\max_\Omega d$) and $k=4$ in (a) and 8 in (b). The dark gray denotes the high-profit region where extraction occurs, and light gray shows the envelope of paths the extractors follow when exiting the protected region. Some of the individual paths are shown for illustrative purposes.[]{data-label="fig:nopatrolling"}](YosemiteNoPatrolBenefit8.png "fig:"){width="\textwidth"}   [0.45]{} ![Two cases for Yosemite National Park with no patrols, benefit based on distance from boundary $d$ as $B(d)=kd(2d_m-d)/d_m$ (where $d_m=\max_\Omega d$) and $k=4$ in (a) and 8 in (b). The dark gray denotes the high-profit region where extraction occurs, and light gray shows the envelope of paths the extractors follow when exiting the protected region. Some of the individual paths are shown for illustrative purposes.[]{data-label="fig:nopatrolling"}](YosemiteNoPatrolBenefit4.png "fig:"){width="\textwidth"} Homogeneous patrols ------------------- In figure \[:hompatrolling\] we consider the simplest nonzero patrol strategy, a homogeneous patrol in which the entire protected area is patrolled with equal intensity. Two cases are shown, both with the high-benefit case $k=8$ from the previous section, but with different patrol budgets $E$. The patrol strategy is simply $\psi=E/A$ where $E$ is the budget and $A$ is the area of the protected region. We specify $\alpha=1$ as the risk-aversion parameter, as this choice gives agreement between our model and that of @Albers in simple symmetric case (discussed in section \[sec:equiv\]). As expected, when the patrol budget increases the pristine proportion increases and the maximum profit obtained by the extractor decreases. [0.45]{} ![Two cases for Yosemite National Park with homogeneous patrols, benefit based on distance from boundary $d$ as $B(d)=kd(2d_m-d)/d_m$ (where $d_m=\max_\Omega d$ and $k=8$) and different budgets $E=3\times 10^4$ in (a) and $E=6\times10^4$ in (b).[]{data-label="fig:hompatrolling"}](HomogeneousPatrol_Budget30000_dtisdx8_BenefitLevels40 "fig:"){width="\textwidth"}   [0.45]{} ![Two cases for Yosemite National Park with homogeneous patrols, benefit based on distance from boundary $d$ as $B(d)=kd(2d_m-d)/d_m$ (where $d_m=\max_\Omega d$ and $k=8$) and different budgets $E=3\times 10^4$ in (a) and $E=6\times10^4$ in (b).[]{data-label="fig:hompatrolling"}](HomogeneousPatrol_Budget60000_dtisdx8_BenefitLevels40 "fig:"){width="\textwidth"} In figure \[:hompatrolling\], doubling the patrol budget decreases the maximum profit obtained by extractors by roughly 22%, increases the pristine proportion by roughly 4%, and increases the proportion of benefit protected by roughly 8.6%. These are clear improvements, but one perhaps would expect more change upon doubling the patrol density. claimed that the homogeneous patrol was not optimal, and this agrees with our intuition that some areas of the protected region will never be penetrated by extractors, so there is no need to patrol there. Band Patrols ------------ identified the band patrol as the optimal strategy in symmetric circular protected regions. In the circular case, the band patrol is a band between two distances from the boundary, $0<d_o<d_i<d_m$, where $d_m$ is the maximum distance from the boundary, with highest patrol density at the outer extent $d_o$, and decreasing density moving towards the inner-most extent of the band $d_i$. In the general case, we set up a band patrol by patrolling between $0.3d_m$ and $0.7d_m$ from the boundary. figure \[:bandresults\] shows the patrol strategy and the results of our algorithm. We did not implement the algorithm presented by @johnson2012patrol to find the optimal band patrol, instead testing a number of band patrols and choosing the one that gave the best results. [0.45]{} ![A band patrol in Yosemite National Park. The patrol is based on distance from boundary $d$ and decreases linearly from $d=0.3$ to $d=0.7$, and is zero elsewhere. The benefit is $B(d)=kd(2d_m-d)/d_m$ (where $d_m=\max_\Omega d$ and $k=8$) and the budget $E=3\times 10^4$.[]{data-label="fig:bandresults"}](BandPatrolStratp3top7.png "fig:"){width="\textwidth"}   [0.45]{} ![A band patrol in Yosemite National Park. The patrol is based on distance from boundary $d$ and decreases linearly from $d=0.3$ to $d=0.7$, and is zero elsewhere. The benefit is $B(d)=kd(2d_m-d)/d_m$ (where $d_m=\max_\Omega d$ and $k=8$) and the budget $E=3\times 10^4$.[]{data-label="fig:bandresults"}](BandPatrol_0p3to0p7Budget30000_dtisdx8_BenefitLevels40 "fig:"){width=".95\textwidth"} The patrol budget in figure \[:bandpatrol\] is $3\times10^4$, the same as for the homogeneous patrol in figure \[:hompatrol\], but the outcome is significantly better for the patrollers. The homogeneous patrol had a pristine proportion 0.809 and was able to protect 76.9% of the total benefit, whereas the band patrol has a pristine proportion of 0.901 and protects 86% of the total benefit. The optimal band patrol of @johnson2012patrol has the property that extractors do not enter the patrolled region, which ours does not. Despite this, it is still superior to the the homogenous patrol (even the homogeneous patrol with twice the budget!) and the other band patrols tested. Asymmetric Patrols ------------------ The patrol strategies employed above were identified by @Albers and are radially symmetric with respect to the geometry of Yosemite National Park. That is, these strategies depended only on the depth $d$ of the point $(x,y)$: $\psi(x,y) = \psi^* (d)$. Our method does not require patrols to be radially symmetric, and we here present an asymmetric patrol that outperforms the symmetric patrols discussed previously. Observing the results of the other patrols, it seems that extractors prefer to enter and leave the park at certain portions of the boundary: the portions which are most concave. This is intuitive, as entering at those areas will ensure less travel distance to reach the center. In response, we can design strategies to preferentially patrol those regions through which extractors are more likely to enter. figure \[:starresults\] shows just such a patrol strategy. [0.45]{} ![A custom patrol in Yosemite National Park which is designed to patrol the concavities in the boundary of the park. The benefit is $B(d)=kd(2d_m-d)/d_m$ (where $d_m=\max_\Omega d$ and $k=8$) and the budget $E=3\times 10^4$.[]{data-label="fig:starresults"}](star_patrol.png "fig:"){width="\textwidth"}   [0.45]{} ![A custom patrol in Yosemite National Park which is designed to patrol the concavities in the boundary of the park. The benefit is $B(d)=kd(2d_m-d)/d_m$ (where $d_m=\max_\Omega d$ and $k=8$) and the budget $E=3\times 10^4$.[]{data-label="fig:starresults"}](StarPatrol2_Budget30000_dtisdx8_BenefitLevels40 "fig:"){width=".95\textwidth"} Exploiting the geometry of the park, rather than patrolling in a depth-dependent way, the pristine proportion is increased to 0.92 and 90.7% of the benefit is protected. These results are better than the band or homogeneous patrols with the same budget, and could likely be improved upon even further by finding even more specialized strategies. This result shows the critical importance of the geometry of the protected region. To a very rough approximation Yosemite is fairly circular, but the relatively small-scale concavities are very important in understanding the behaviour of extractors. Kangaroo Island --------------- We now apply our method to Kangaroo Island, a small island off the southern coast of Australia. We present two patrol strategies (a homogeneous patrol and a custom designed patrol) which once again emphasize the importance of accounting for the geometry of the protected area. Kangaroo Island is fairly long and narrow with a portion at the eastern end connected to the rest of the island by a small neck. [0.45]{} ![Two cases for Kangaroo Island with homogeneous patrols, benefit based on distance from boundary $d$ as $B(d)=kd(2d_m-d)/d_m$ (where $d_m=\max_\Omega d$ and $k=8$) and different budgets $E=3\times 10^4$ in (a) and $E=6\times10^4$ in (b).[]{data-label="fig:kangahompatrolling"}](KangarooHomogeneousBudget30000_dtisdx8_BenefitLevels40 "fig:"){width="\textwidth"}   [0.45]{} ![Two cases for Kangaroo Island with homogeneous patrols, benefit based on distance from boundary $d$ as $B(d)=kd(2d_m-d)/d_m$ (where $d_m=\max_\Omega d$ and $k=8$) and different budgets $E=3\times 10^4$ in (a) and $E=6\times10^4$ in (b).[]{data-label="fig:kangahompatrolling"}](KangarooHomogeneousBudget60000_dtisdx8_BenefitLevels40 "fig:"){width="\textwidth"} Figure \[fig:kangahompatrolling\] shows results of homogeneous patrols applied to Kangaroo Island. In this case, we observe that, somewhat counter-intuitively, the increased patrol budget results in a smaller pristine area proportion. The reason for this is that increasing the patrol budget decreases the profit function but causes the high profit area to spread out. That is, with increased patrol the surface profile of the profit function will look more like a plateau which has a smaller maximum than in the low budget case, but has a larger near-maximal area. Although the pristine region is smaller, the maximum profit obtained by the extractors is much lower, decreasing from $1.34\times10^5$ to $1.15\times10^5$, so the patrol does have a significant effect on the extractors. We also observe that a large portion of the non-pristine area comprises the paths leaving the region as opposed to the high-profit region itself. This should inform our decision regarding how to more effectively patrol the region. In any radially symmetric patrol, the high-profit region will also be radially symmetric meaning that it will likely occupy the middle of the island and the paths will cover a very large area as they enter and exit from the north and south side of the island. If we can force the high profit region out of the middle of the island, then the paths will instead travel throughout the east and west portions of the island, hopefully occupying a smaller area. Another key geographic feature of the island is the peninsula at the eastern tip of the island. The peninsula is connected to the island by an isthmus thin enough that, for our purposes, the peninsula can almost be considered an independent region. Extractors will likely be uninterested by the peninsular region since it has much less depth and thus offers much less benefit than the main body of the island. Hence any patrol in that region is likely wasted effort. With the homogeneous results in mind, figure \[:kangaresults\] shows the results of a more effective patrol, where the middle of the island is patrolled uniformly and the eastern and western ends are not patrolled. This patrol was able to increase the pristine area proportion to 0.875 compared to the homogeneous patrol which gave 0.349 with the same budget. Again, designing this patrol required some simple observations regarding the geometry of the region, and once again shows the importance of explicitly incorporating geographical information into the model. [0.45]{} ![A custom patrol in Kangaroo Island which is designed to push the high profit region away from the center of the island. The benefit is $B(d)=kd(2d_m-d)/d_m$ (where $d_m=\max_\Omega d$ and $k=8$) and the budget $E=3\times 10^4$.[]{data-label="fig:kangaresults"}](kangaroo_vertBarPatrol "fig:"){width="\textwidth"}   [0.45]{} ![A custom patrol in Kangaroo Island which is designed to push the high profit region away from the center of the island. The benefit is $B(d)=kd(2d_m-d)/d_m$ (where $d_m=\max_\Omega d$ and $k=8$) and the budget $E=3\times 10^4$.[]{data-label="fig:kangaresults"}](KangarooVerticaBarBudget30000_dtisdx8_BenefitLevels40 "fig:"){width=".95\textwidth"} Conclusion {#sec:conclusion} ========== Modeling of environment crime in protected regions has been the subject of previous research, most of which has been focused on discrete network-based methods. Attempts at modelling environmental crime in continuous settings have required restrictive assumptions of symmetry and are only applicable to regions with very simple geometries. In this paper we have formulated a generalized version of the model of @Albers which can account for regions of arbitrary geometry, does not require any assumptions of symmetry and can incorporate topographical features. In doing so, we have unified environmental crime modeling and continuum, PDE models for tracking human movement. The level set method of @osherSethian1988 allows us to perform calculations in regions with complicated geometry. Using the level set method, we track the movement of environmental criminals (extractors) by treating them as a front propagating from the boundary of the protected region toward its center while accounting for changes in travel velocity due to elevation. The level set method is very powerful, and is suitable for many more applications involving movement through regions with complicated geometry. Working with arbitrary geometries increases the complexity of the model. Accordingly, we have proposed a novel method for incorporating patrol strategy into the calculation of cost to the extractors and suggested an efficient algorithm for resolving the cost function while performing relatively few level set computations. Further, we have re-defined the pristine area by considering an extraction area (an area of near-maximal profit to extractors) as well as paths that extractors will traverse when leaving the region. This is a necessary deviation from the @Albers model that arises due to the different topology of the point(s) that maximize the extractors’ profit. We have applied our model to two different regions: Yosemite National Park in California and Kangaroo Island in South Australia. In doing so, we tested several different patrol strategies, including some proposed by @Albers and @johnson2012patrol. Then, making some basic observations regarding the geometry of each region, we designed asymmetric patrols which protected the regions more effectively that any of those previously suggested. The success of these asymmetric patrols shows the importance of explicitly treating the geometry. We expect that even more effective patrol strategies exist, and determining them is a potential avenue for further research. The model presented here could be extended and modified in many ways. For example, the velocity function could be made to depend on ground cover (i.e. higher speeds on open plains and lower speeds in heavy scrub) or other effects such as lakes or paths. Likewise, the benefit and cost functions could be calculated differently depending on the situation. In the proposed model, the cost of travelling to the extraction point is not considered, only the trip out of the region after the crime has been committed. This follows @Albers, who neglected the cost of the inbound trip. Adding the extra cost of the inbound trip would be a minor modification of the algorithm as presented. We have only considered benefit functions that depend on distance from the boundary, which is likely not realistic. The proposed model can be evaluated with any benefit function (continuity is not necessary), however defining a realistic and accurate benefit function for a given national park would require input from agencies involved in managing the park. We have defined patrol strategies as density functions, but these do not correspond to patrol routes that would be followed. Determining specific patrol routes that give the desired patrol density is a task requiring significant work. used a discrete model to find explicit patrol routes but in our continuous setting the problem is very complicated. Another opportunity for further research is studying the effects of time-dependent benefit functions. We have used a constant benefit function but in some situations this may not be appropriate. For example, considering animal poaching, one might wish to account for the movement of animals over time. This would require significant modification to the model we have presented. Acknowledgements ================ The authors acknowledge the financial support of the NSF (grant DMS-1737770) and the Department of Defense (approved for public release, \#18-666). [^1]: Department of Mathematics, University of California Los Angeles. [^2]: Department of Mathematics, California State University Long Beach. [^3]: Department of Mathematics, University of Maryland.

--- author: - Mike Blake title: Universal Diffusion in Incoherent Black Holes --- Introduction ============ #### Many strongly correlated metals display a robust linear resistivity. It has long been suggested that such behaviour could be understood if transport properties were controlled by a universal dissipative timescale $\tau \sim {\hbar}/(k_B T)$ [@subirbook; @sachdev; @planckian]. In particular recent experiments have directly observed this same ‘Planckian’ time-scale in the scattering rates of a wide range of materials exhibiting a linear resistivity [@mackenzie] . #### Nevertheless, it remains unclear how such universality could occur across a range of materials whose microscopic physics and scattering mechanisms can differ massively. Inspired by an analogy with the viscosity bound of Kovtun, Son and Starinets [@kss2], [@incoherent] proposed that universal transport could arise from the saturation of a ‘Planckian’ bound on the charge and energy diffusion constants D \~ \[diff\] where $v$ is a characteristic velocity of the system. In a metal, one expects the characteristic velocity to be set by the Fermi energy. Since the diffusion constants are related to the conductivities via the Einstein relations, then the appeal of [(\[diff\])]{} is that the saturation of such a bound would lead to a universal linear resistivity[^1]. #### A natural environment in which to expect a universal relationship such as [(\[diff\])]{} to hold is in the charge diffusion constant of a particle-hole symmetric theory [@kovtun]. In this case the electrical conductivity decouples from momentum and hence is finite even in a translationally invariant theory. In [@butterflymike] we studied the charge diffusion constant of general holographic scaling theories and found a universal regime in which the diffusion constant was given by D\_[c]{} = C \[chargediff\] where $v_B$ is the velocity of the ‘butterfly effect’ [@butterflyeffect; @localisedshocks; @stringyeffects; @chaos; @multiple; @kitaev; @roberts; @cft; @channels; @eric] and $C$ is a constant that depended on the universality class of theory. #### In contrast the behaviour of energy diffusion, or diffusion in a metal where the electrical current couples to momentum, is more complicated. In a translationally invariant theory $D_{e}$ will diverge and hence it is extremely sensitive to the momentum relaxation rate $\Gamma$. Since this will depend on the precise way in which the translational symmetry is broken, one cannot expect to see universal behaviour in general. However the proposal of [@incoherent] is that [(\[diff\])]{} might apply to incoherent metals, where strong momentum relaxation is an intrinsic property of the theory. #### Whilst there are very few theoretical approaches to incoherent metals, explicit examples of holographic models with strong momentum relaxation have been constructed [@vegh; @blaise2; @donos1; @donos2; @andrade]. The purpose of this paper is to study the diffusion constants of these theories. In order that charge and energy transport decouple, we will focus on incoherent theories with a particle-hole symmetry[^2]. Since the electrical current decouples from momentum we find, just as in [@butterflymike], that these models have a universal regime in which the charge diffusion constant is given by [(\[chargediff\])]{}. #### As expected the behaviour of the energy diffusion constant is more complicated. When the translational symmetry is weakly broken, the energy diffusion constant is non-universal and depends on the momentum relaxation rate $\Gamma$. However when momentum relaxation is very strong the details of how we break the translational symmetry become unimportant. Rather, in this incoherent regime we find that the energy diffusion constant is always related to the butterfly effect as[^3] D\_[e]{} = E \[resultenergy\] where $E$ is another universal constant (which is different from $C$). Our results therefore support the suggestion that diffusion in incoherent metals saturates [(\[diff\])]{}. Indeed we find it striking how, now that we have identified a characteristic velocity $v_B$, these models precisely illustrate the proposal of [@incoherent]. #### In Section \[axionsection\] we begin by studying the diffusion constants of a neutral black hole in which momentum relaxation is incorporated through linear sources for massless scalars [@andrade]. This provides a simple toy model in which we can illustrate how universality can emerge in the incoherent limit. In Section \[qlattices\] we consider a more general family of solutions known as ‘Q-lattices’ [@donos1; @donos2; @blaise2] and demonstrate that this universality continues to hold. Finally, we close with a discussion of our results in the context of [@mackenzie; @incoherent]. Linear Axions {#axionsection} ============= #### In order explain the basic ideas of this paper, we will start with the simplest holographic model of transport. This just consists of the Einstein-Maxwell action coupled to massless scalars[^4]. In particular, if we work in four bulk dimensions we can consider the action S = \^[4]{}[x]{} \[axionaction\] where ${\cal A} = 1, 2$ runs over the two spatial coordinates of the boundary quantum field theory. #### As we mentioned in the introduction, we will restrict our attention to the case where there is no net charge density. The electrical current therefore decouples from momentum and hence the conductivity will be insensitive to the momentum relaxation rate. In contrast, to get a finite thermal conductivity we need to break the translational symmetry. #### In order to do this we will introduce linear sources for the axion fields $\chi_{\cal A} = k x_{\cal A}$ that implement momentum relaxation in the boundary theory. Whilst the use of linear sources may appear unphysical, we will see in Section \[qlattices\] that these axion fields can more generally be viewed as the phase of a scalar ‘Q-lattice’. In this context such sources are therefore related to a periodic deformation of the boundary theory. #### The advantage of using these linear sources is then that, even though they break translational symmetry, the background metric remains homogeneous. Indeed a black-hole solution can be written down analytically as [@andrade] $$\begin{aligned} ds^2 = -\frac{r^2 U(r)}{L^2} dt^2 &+& \frac{L^2}{r^2 U(r)} dr^2 + \frac{r^2}{L^2}(dx^2 + dy^2) \nonumber \\ A_t = 0 \;\;\;\;\;\;\;\ \chi_1 &=& k x \;\;\;\;\;\;\;\; \chi_2 = k y \label{axionmetric}\end{aligned}$$ where the emblackening factor $U(r)$ is given by U(r) = 1 - - ([1 - ]{}) This black hole has a horizon at radius $r_0$ that determines the temperature, $T$ according to 4 T L\^2= (3 r\_0 - ) \[temperature\] and the entropy density is given by the Bekenstein-Hawking formula s = \[entropy\] The physics of the boundary theory can be described by a single dimensionless parameter $k/T$. When $k/T \ll 1$ then the translational symmetry is only weakly broken and the theory is described as ‘coherent’. In this limit the background metric is well-approximated by the Schwarzchild solution and the only effect of the axions is to cause momentum to slowly relax [@davison] at a rate $\Gamma \sim k^2/T$. #### Conversely once $k/T \gtrsim 1$ we have an ‘incoherent’ metal in which momentum relaxation is a strong effect and cannot be treated perturbatively. In particular in this regime it is the axions themselves that are now responsible for sourcing the background geometry. From [(\[temperature\])]{} we can see that at low temperatures the horizon radius approaches a constant $r_0^2 = k^2 L^4/6$ and so the near-horizon geometry will now correspond to an $AdS_2 \times R^2$ metric ds\^2 = - dt\^2 + d[r]{}\^2 + ( dx\^2 + dy\^2 ) \[nearhorizon\] Charge Diffusion {#charge-diffusion .unnumbered} ---------------- #### The reason for starting with this simple model is that it is straightforward to write down analytic expressions for the transport coefficients and hence the diffusion constants. In particular, because we are dealing with the particle-hole symmetric theory, the electrical conductivity is just a constant = Whilst this is a simple formula, it is important to stress that the electrical conductivity itself is not a universal quantity - it explicitly depends on the normalisation of the current. To construct something independent of this normalisation, we can divide through by the charge susceptibility to obtain the diffusion constant D\_[c]{} = = ( )\_[T]{} \[einsteincharge\] For our axion model we simply have $\chi^{-1} = e^2 {L^2}/{r_0}$ and hence the diffusion constant is given by D\_[c]{} = \[diffaxion\] #### In order to write this diffusion constant in the form [(\[diff\])]{} we first need to identify a characteristic velocity of our theory. As we argued in [@butterflymike], one natural way to define such a velocity in a holographic theory is provided by the butterfly effect [@butterflyeffect; @localisedshocks; @roberts; @multiple; @kitaev; @stringyeffects; @chaos; @cft; @eric; @channels]. In particular by studying this effect one can identify the characteristic speed, $v_B$, at which quantum information propagates in a strongly coupled theory. #### Whilst the discussion in [@butterflymike] centred around translationally invariant theories, this velocity is only sensitive to the background geometry. It is therefore straightforward to apply the the shock-wave techniques of [@localisedshocks; @stringyeffects; @butterflymike; @roberts] to calculate this velocity for our homogeneous metric. In particular for any metric of the form [(\[axionmetric\])]{} this velocity is given by (see Appendix \[appendixa\]) v\_B\^2 = where the effects of momentum relaxation are implicitly contained in the dependence of the horizon radius on the ratio $k/T$[^5]. We therefore see that the diffusion constant of our axion models can be written as D\_[c]{} = \[diffcharge\] which holds independently of any of the parameters $e, L, k, T$ in our bulk theory. #### It is worth emphasising that whilst the conductivity is just a constant, $D_{c}$ itself actually depends on the strength of momentum relaxation $k/T$ through $r_0$. In fact, the dimensionless diffusion constant $D_{c} T$ vanishes in the incoherent limit, and hence it was suggested in [@andrea] that it is not possible to formulate a bound on the diffusion constants for models such as [(\[axionaction\])]{}. #### However, we can now see that this dependence simply reflects the fact that turning on sources for the axion fields will change the characteristic velocity, $v_B$, of the theory. Having identified this velocity we see that the relationship [(\[diffcharge\])]{} always holds and as such the timescale we would extract from the diffusion constant as $D_{c} \sim v_B^2 \tau$ is consistent with a Planckian bound [(\[diff\])]{} \~ \[tau\] Energy Diffusion {#energy-diffusion .unnumbered} ---------------- #### In contrast to the electrical conductivity, energy transport in our theory is sensitive to the details of momentum relaxation. Indeed for this axionic model the thermal conductivity, ${\kappa}$, is given by [@donos3] = \[thermal\] and hence diverges in the translationally invariant theory. In order to calculate the diffusion constant, we can once again make use of an Einstein relation to extract $D_{e}$ from the thermal conductivity and the specific heat $c_{\rho}$ D\_[e]{} =  c\_ = T ( ) \[einsteinthermo\] which yields [@davison] D\_[e]{} = ( 3 r\_0+ ) We can now explicitly see that, as discussed in the introduction, it is not in general possible to write the energy diffusion constant in the same form as $D_{c}$ [(\[chargediff\])]{}. In particular, when momentum relaxation is weak the diffusion constant exhibits the expected divergence D\_[e]{} \~T/k\^2 k/T 1 and hence its value is set by the momentum relaxation rate $\Gamma \sim k^2/T$. Since this depends on how strongly the translational symmetry is broken, it is highly non-universal[^6]. #### However the proposal of [@incoherent] is that universality could emerge in the incoherent limit. Indeed in this limit we see that the diffusion constant just approaches D\_[e]{} = k/T 1 which, up to an order one number, is precisely the same value that the charge diffusion constant took. Therefore whilst the energy diffusion constant still depends on $k/T$, this is again solely due to the fact that the characteristic velocity $v_B$ is changing. In other words, in the incoherent regime the energy diffusion constant of our axion model can be written as D\_[e]{} = and hence is governed by the same universal timescale \~ that controlled charge transport. \[neutralratio\] #### In order to illustrate this behaviour we have plotted the diffusion constants in Figure 1. The left hand figure shows the diffusion constants themselves which are non-universal and vanish in the limit $k/T \gg 1$. On the right hand side we have divided through by the butterfly velocity to construct the ratios $2 \pi D T/v_B^2$. In our simple geometry $D_{c}$ is just a constant in these units. In contrast the energy diffusion constant is more complicated, and can be seen to diverge in the translationally invariant limit. #### However, as we increase the strength of momentum relaxation then we find that $2 \pi D_e T/v_B^2$ cannot be made arbitrarily small but instead saturates at an ${\cal O}(1)$ value. As soon as we reach the incoherent regime $k \sim 4 \pi T$ then we will be close to this saturating value and hence the diffusion constants will be universally given by D\_[c]{} = D\_[e]{} \[diffscaling\] Precisely this phenomenology was proposed in [@incoherent] in the context of a fundamental bound on the diffusion constant. We find it remarkable that the simplest holographic model of an incoherent metal realises this behaviour. One might worry, however, that these results are an artefact of our choice of action [(\[axionaction\])]{}. In the next section we will therefore consider a much wider class of incoherent holographic geometries. We will find that the universality we see in the diffusion constants of this axion model continues to hold more generally. Q-Lattice Models {#qlattices} ================ #### We have just seen that in the incoherent limit $k \gg T$ both the charge and energy diffusion constants of the axion model were universal. That is we had $D T \sim v_B^2$ independently of the strength of momentum relaxation. In this section we want to show that such behaviour occurs more generally in holographic models of incoherent metals. #### In particular we wish to consider the so-called holographic ‘Q-lattice models’ that were introduced in [@donos1]. These models consist of coupling the axion model of the last section to a dilaton field $$\begin{aligned} S = \int \mathrm{d}^4x \sqrt{-g} \bigg [ R - \frac{c}{2} ( (\partial \phi)^2 + Y(\phi) ((\partial \chi_1)^2 + (\partial \chi_2)^2) - V(\phi) - \frac{1}{4} Z(\phi) F^{\mu \nu} F_{\mu \nu} \bigg] \nonumber \\ \label{qlattice}\end{aligned}$$ Roughly speaking, one can think of these models as arising from decomposing a complex scalar field into its magnitude and phase as $\Psi \sim \phi e^{i \chi }$. Once again the solutions we are interested correspond to breaking translational symmetry by turning on a linear source for the axions $\chi_{\cal A} = k x_{A}$. The name ‘Q-lattice’ then reflects the fact that, in terms of the original complex scalar $\Psi$ this appears to be a periodic deformation of the boundary theory. #### By choosing different actions for $\Psi$ one can engineer different potentials for the dilaton field $\phi$. Here, we will assume that when the dilaton becomes large the leading form of these potentials is an exponential Y() = e\^[2 ]{}  V() = - V\_0 e\^ Z() = Z\_0\^2 e\^ Note that the parameter $c$ cannot be set to one without changing these exponents, and so it is important in determining the form of the solutions. Since the scalar potential is unbounded, then this action can be used to construct solutions where the size of the lattice diverges $\phi \rightarrow \infty$ in the infra-red [@donos2; @blaise2]. These solutions therefore describe incoherent metals, in which the effects of the lattice are becoming extremely strong at low temperatures. Incoherent Scaling Geometries {#incoherent-scaling-geometries .unnumbered} ----------------------------- #### Just like in the axion model, the resulting metrics are homogeneous. They correspond to the well-studied class of metrics known as hyperscaling violating geometries [@dong; @kachru; @strangemetals; @huijse; @elias1; @elias2]. In particular there exist a family of neutral solutions of the form[^7] $$\begin{aligned} ds^2 = - U(r) dt^2 &+& \frac{dr^2}{U(r)} + V(r)(dx^2 + dy^2) \nonumber \\ A_t = 0 \;\;\;\;\;\;\;\ \chi_1 &=& k x \;\;\;\;\;\;\;\; \chi_2 = k y \label{metricgeneral}\end{aligned}$$ where the metric functions are given by U(r)= L\_t\^[-2]{} r\^[u\_1]{} V(r) &=& L\_x\^[-2]{} r\^[2 v\_1]{} e\^[(r)]{} = e\^[ \_0]{} r\^[ \_1]{} \[ansatz\] The powers in our geometry are then related to a dynamical critical exponent, $z<0$, and a hyperscaling violation exponent $\theta>2$[^8] as u\_1=  2 v\_1 = 2 \_1 = and the exponents are determined by the choice of parameters $c, \alpha$ z = = Additionally, the Einstein equations tell us how the scales of the metric $L_t, L_x$ are generated by the Q-lattice $$\begin{aligned} (z - \theta)^2 V_0 e^{\alpha \phi_0} &=& L_t^{-2} (2 z - \theta) (2 + z - \theta) \nonumber \\ (2 z -\theta) c k^2 e^{2 \phi_0} &=& L_x^{-2} V_0 e^{\alpha \phi_0}(2 z- 2) \label{constraint}\end{aligned}$$ #### Whilst these metrics appear quite complicated, the main point is that they correspond to generalisations of the near-horizon geometry we saw in the Section \[axionsection\]. In particular, for the case our axion model $z = \infty, V_0 = 6/L^2, \phi =0, c =1$ these constraints reduce to u\_1 = 2 v\_1 = 0 L\_t\^[-2]{} = L\_x\^[-2]{} = . and so we reproduce the metric [(\[nearhorizon\])]{} we studied in the last section. However rather than just the simple $AdS_2\times R^2$ geometry, the Q-lattices can now support far more general scaling geometries parameterised by $(z, \theta)$. #### Finally, in order to calculate the diffusion constants, we need to heat these solutions up to a finite temperature. We can do this by turning on an emblackening factor in our ansatz [(\[ansatz\])]{} U(r)= L\_t\^[-2]{} r\^[u\_1]{}(1 - ) where we have $\delta = u_1 + 2v_1-1$. It is simple to check that this deformation can be turned on without changing the rest of our bulk solution and corresponds to a temperature $4 \pi T = U'(r_0)$ for the boundary field theory. Charge Diffusion {#charge-diffusion-1 .unnumbered} ---------------- #### Before focusing on energy diffusion, let us begin by studying the diffusion of charge in these theories. In [@butterflymike] we discussed charge diffusion for general holographic scaling geometries with a particle-hole symmetry. Since the charge diffusion constant is only sensitive to the background metric, our analysis can also be applied to these Q-lattice models as well. #### To extract the diffusion constant we will again use the Einstein relation [(\[einsteincharge\])]{}. The conductivity of a dilaton model just corresponds to the effective Maxwell coupling at the horizon. This is now no longer a constant, but rather has a non-trivial temperature dependence = Z()|\_[r\_0]{} \~T\^[[(2 - ]{})/z]{} where $\Phi$ is an anomalous scaling dimension for the charge density that arises due to the coupling, $\gamma$, between the gauge field and the dilaton [@blaise1; @blaise2; @andreas1; @andreas2] \_1 = #### Although the conductivity of these dilaton models is now more complicated, the diffusion constant can still take a simple form. This is because the charge susceptibility is also sensitive to the profile of the dilaton [@kss1; @iqballiu] \^[-1]{} = \_\^[r\_0]{} \[susc\] and hence the effects of the running Maxwell coupling can effectively cancel. #### In particular, the behaviour of diffusion constant will depend on which region of the geometry dominates the integral in [(\[susc\])]{}. In order to characterise this, it is useful to introduce the scaling dimension, $\Delta_{\chi} = 2 - \theta + 2 \Phi - z $ of the susceptibility. Now since the contribution to $\chi^{-1}$ from near the horizon scales like $ T^{-\Delta_{\chi}/z}$ then for low temperatures the infra-red region of the geometry will dominate the integral whenever $\Delta_{\chi}/z > 0$. #### Since diffusion is then controlled by the near-horizon physics, it is natural to expect a connection with the butterfly effect. Upon evaluating the integral [(\[susc\])]{} we have that the diffusion constant is related to the horizon radius by D\_[c]{} = L\_x\^2 r\_0\^[1 - 2 v\_1]{} To compare with [(\[diff\])]{} we need the characteristic velocity of our theory. As was shown in [@butterflymike; @roberts], and we review in Appendix \[appendixa\], the butterfly velocity for a general metric of the form [(\[metricgeneral\])]{} is v\_B\^2 = = L\_x\^2 r\_0\^[1 - 2 v\_1]{} \[butterflylattice1\] and so we see that the diffusion constant is universally given by D\_[c]{} = \[diffcharge2\] #### It is worth stressing that, just as in the the axion model, both the diffusion constant and the butterfly velocity depend through $L_x$ and $r_0$ on the details of the Q-lattice solution (and in particular on the strength of momentum relaxation). However provided we are in this universal regime[^9] $\Delta_{\chi}/z > 0$ then we see that the relationship between them is always given by [(\[diffcharge2\])]{}. As such in all these different theories we will have that charge diffusion is universally controlled by the same Planckian timescale $\tau \sim 1/T$ that we saw in the axion model. Energy Diffusion {#energy-diffusion-1 .unnumbered} ---------------- #### We can now turn to the question of energy diffusion in these Q-lattice models. Recall that in the axion model we saw that the effects of the axion fields on the geometry were precisely such that the diffusion constant became universally related to $v_B$ in the incoherent limit. Our goal is to understand whether the same thing happens for our more general metrics [(\[ansatz\])]{}. For the final time, we therefore invoke the Einstein relation [(\[einsteinthermo\])]{} to compute $D_{e}$ from knowledge of the thermal conductivity $\kappa$ and the specific heat $c_{\rho}$. #### For these holographic Q-lattice models, it is now well known that one can obtain an analytic expression for the thermal conductivity in terms of properties of the black hole horizon [@donos3]. In particular this formula relates the thermal conductivity to the size of the lattice at the horizon according to[^10] = |\_[r\_0]{} \~T\^[(z-)/z]{} If we extract the specific heat from the scaling of the entropy density s \~T\^[(2 - )/z]{} then we can deduce that the diffusion constant must be given by D\_[e]{} = |\_[r\_0]{} \[difflattice\] #### Upon comparing this expression with the butterfly velocity [(\[butterflylattice1\])]{} then it is clear that for an arbitrary bulk metric, there will not be a simple relationship between the energy diffusion constant and $v_B$. This just reflects the fact that, as we explicitly saw in the axion model, we should not always expect to see universal behaviour in the diffusion constant. #### However, in our incoherent theories, the metric is not some arbitrary function. Remember the key point is that the lattice itself is now responsible for creating the geometry. As a result, the Einstein equations relate the profile of the scalar field to the metric and hence can be used to relate the diffusion constant [(\[difflattice\])]{} to the butterfly effect. #### In particular we can note that second equation in [(\[constraint\])]{} tells us the length-scale $L_x$ in our scaling solution is not arbitrary, but rather is fixed in terms of the lattice profile $k^2 e^{2 \phi_0}$. In the axion model, this condition is equivalent to our observation that the axions source a ground state entropy. In these more general scaling theories we can again use this equation to relate the lattice profile to the metric function $V(r)$ and hence the butterfly effect. #### Indeed, after using the equations [(\[constraint\])]{} we find that the diffusion constant of our scaling theories can be rewritten as[^11] D\_[e]{} = L\_x\^2 r\_0\^[1 - 2 v\_1]{} which is exactly the same combination of parameters $L_x, r_0$ that appeared in the charge diffusion constant. Comparing with [(\[butterflylattice1\])]{} we therefore see that the energy diffusion constant of these models is universally related to the butterfly effect by D\_[e]{} = \[latticeenergy\] and hence we extract the same timescale $\tau \sim 1/T$ independently of our choice of Q-lattice model. #### In order to emphasise why this result is so surprising, it is instructive to again recall what happened when momentum relaxation was a weak effect. As we explicitly saw in the axion model, in such a case the energy diffusion constant is not related to $v_B$ in any simple manner, but rather depends on the details of momentum relaxation. However what we are seeing is that when momentum relaxation is a strong effect this dependence goes away. The key point in that in these incoherent theories it is now the Q-lattice itself that is responsible for supporting the scaling geometry. The Einstein equations then imply that whatever type of Q-lattice we turn on the resulting geometry is always such that the diffusion constant and the butterfly effect of our models are related by [(\[latticeenergy\])]{}. Discussion ========== #### In this paper we have studied diffusion in simple holographic theories with broken translational symmetry. In particular we found that both the charge and energy diffusion constants of these models could be universally related to the butterfly velocity D\_[c]{} = C D\_[e]{} = E \[bound\] where the constants of proportionality depended only on the scaling exponents of our theories. For the charge diffusion constant, the relationship [(\[bound\])]{} held regardless of the strength of momentum relaxation. Conversely, for the energy diffusion constant it was necessary to be in the incoherent regime where the lattice itself supports the background geometry. #### Since we have been considering theories with a particle-hole symmetry, the universality of the charge diffusion constant could have been anticipated from the results of [@butterflymike] . In contrast, one would expect that energy diffusion should be highly sensitive to the way in which translational symmetry is broken. However we saw that in the incoherent limit this dependence went away, and the energy diffusion constant also became universally tied to the butterfly velocity. Heuristically, it seems that once we reach the Planckian rate we can no longer increase the rate of dissipation by breaking the translational symmetry any further[^12]. The microscopic details of momentum relaxation are therefore unimportant and the result is universality. #### Throughout this paper we focused for technical reasons on theories with a particle-hole symmetry. Now that we understand that in an incoherent theory both charge and energy diffusion can be universally governed by [(\[bound\])]{} then it should be possible to extend our considerations to finite density. In this situation, the energy and charge currents overlap and the result is a pair of coupled diffusion equations with eigenvalues $D_{\pm}$. So long as we are in the incoherent regime, i.e. that the dominant degrees of freedom sourcing the geometry are the Q-lattice fields, then we expect that these diffusion constants will continue to take similar values as to[^13] [(\[bound\])]{}. #### These results therefore lend support to the proposal of [@incoherent] that diffusion in incoherent metals can be universal. To reiterate, our central observation is that even when momentum relaxation is strong the diffusion constants of our models could not be made parametrically smaller than the butterfly velocity. That is in the incoherent regime they were given by D \~ \[mikeproposal\] independently of the details of the theory or the strength of momentum relaxation. These holographic models therefore provide concrete examples of how universal transport properties governed by $\tau \sim {\hbar}/{k_ B T}$ could emerge in an incoherent metal [@incoherent; @mackenzie]. #### An important question for future work is to understand to what extent [(\[mikeproposal\])]{} holds outside the simple class of theories studied here and in [@butterflymike]. For instance, one can consider more general holographic models in which the metric is not homogeneous. Recently [@steinberg] studied the connection between the charge diffusion constant and the butterfly velocity of such theories by considering a hydrodynamic treatment of striped inhomogeneities. Interestingly they found that the diffusion constant continues to obey the scaling $D_{c} \sim v_B^2/T$ expected from our proposal [(\[mikeproposal\])]{}, albeit with a coefficient of proportionality that is no longer universal[^14]. It would be worthwhile to perform a more detailed analysis of this inhomogeneous setting, and in particular to address the question of energy diffusion in the presence of strong disorder. #### Finally, we note that it would be of great interest to investigate whether [(\[mikeproposal\])]{} also holds in non-holographic theories. In particular [@stanford] has recently proposed a generalised Sachdev-Ye-Kitaev model that provides a soluble quantum mechanical model of an incoherent metal. Remarkably the energy diffusion constant and butterfly velocity of this model were found to obey a simple relationship $D_{e} = \hbar v_B^2/(2 \pi k_B T)$ in consistency with our proposal[^15]. The results of [@stanford] therefore suggest a more general validity of our proposal, and provide an exciting new context in which to explore it further. The Butterfly Effect {#appendixa} ==================== #### A detailed discussion of the butterfly effect and the connection to quantum chaos can be found in [@localisedshocks; @stringyeffects; @multiple; @butterflyeffect; @chaos]. In the interests of making this paper self-contained, we will simply review how to extract the butterfly velocity $v_B$ for general metrics of the form [(\[metricgeneral\])]{} using the shock-wave techniques of [@localisedshocks; @stringyeffects]. #### The butterfly effect is associated with the exponential growth of a small perturbation to a quantum system. In holography this effect has a beautiful realisation in terms of a gravitational shock-wave at the horizon of a black-hole [@butterflyeffect]. In particular, if one considers releasing a particle from the boundary at a time $t_w$ in the past, then for late times $t_w > \beta$ the energy density of this particle is localised near the horizon. In Kruskal coordinates $(u,v)$ the resulting stress tensor of this particle is then given by T\_[u u]{} \~E e\^[ t\_w]{} (u) () \[stresstensor0\] where $E$ is the asymptotic energy of the particle. #### Due to the exponential boosting of the energy density the back reaction of this stress tensor eventually becomes significant and results in the formation of a shock-wave geometry [@butterflymike; @localisedshocks; @stringyeffects; @butterflyeffect; @chaos; @multiple; @roberts]. To construct these solutions we first need to write our metrics in Kruskal coordinates as ds\^2 &=& A( u v) du dv + V( u v) ( dx\^2 + dy\^2) Then the shock-wave corresponds to a solution where there is a shift in the $v$ coordinate $v \rightarrow v + h(x)$ as one crosses the $u=0$ horizon. Such a metric can be described by an ansatz ds\^2 = A( u v) du dv + V( u v ) d\^2 - A(u v) (u) h(x) du\^2 where for quite general theories of Einstein gravity coupled to matter one finds a solution to the Einstein equations provided the shift obeys a Poisson equation [@sfetsos] (\_[i]{}\_[i]{} - m\^2 ) h(x) \~ E e\^[ t\_w]{} () \[poisson\] with a screening length m\^2 = |\_[u=0]{} \[musquare\] In particular, as is discussed in detail in [@sfetsos; @roberts], one finds that after using the background equations of motion [(\[poisson\])]{} and [(\[musquare\])]{} continue to hold even when there are non-trivial matter fields supporting the background geometry[^16]. As such the only way the matter content of the theory effects the shock wave is through the determination of the background metric $A( u v)$, $V(u v)$. The net result is that, even though we have introduced scalar fields to break the translational symmetry, we can still apply these equations to construct the shock wave solutions for our homogeneous metrics [(\[metricgeneral\])]{}. #### It is then straightforward to solve [(\[poisson\])]{}. At large distances one finds that the shift is given by h(x) \~ where $t_{*} \sim \beta \mathrm{log} N^2$ is the scrambling time [@scrambling]. From the form of this solution we can then read off that the effects of the particle grow with a Lyapunov exponent $\lambda_L = 2 \pi/\beta$ and propagate at the butterfly velocity $v_B = 2 \pi/ (\beta m)$. 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Y. Gu, X. L. Qi and D. Stanford, “Local criticality, diffusion and chaos in generalized Sachdev-Ye-Kitaev models,” arXiv:1609.07832 \[hep-th\]. K. Sfetsos, “On gravitational shock waves in curved space-times,” Nucl. Phys. B [**436**]{} (1995) 721 \[hep-th/9408169\]. Y. Sekino and L. Susskind, “Fast Scramblers,” JHEP [**0810**]{} (2008) 065 \[arXiv:0808.2096 \[hep-th\]\]. [^1]: Here we are assuming that the susceptibilities are temperature independent. [^2]: We will briefly discuss how our ideas generalise to finite density in Section \[discussion\]. [^3]: Note that the diffusion constants in [(\[chargediff\])]{} and [(\[resultenergy\])]{} do themselves depend on the way translational symmetry is broken. However these effects solely reflect how the characteristic velocity, $v_B$, is changed due to the presence of momentum relaxation. [^4]: The diffusion constant of a similar massive gravity model were studied in the context of [@incoherent] in [@andrea]. However, without a definition of the characteristic velocity $v$ they were unable to see the universality we observe. [^5]: Note that since $v_B$ is sensitive only to the background geometry information can still spread ballistically even in theories with momentum relaxation. Intuitively information can be carried by degrees of freedom, such as ‘particle-hole pairs’, that do not carry any net momentum. [^6]: At high temperatures the characteristic velocity $v_B$ is just a constant. [^7]: Note our radial coordinate differs from the more familiar one in [@dong] by $r = {\tilde r}^{\theta-z}$. [^8]: These restrictions on the exponents are necessary to order to have a consistent geometry in which the scalar field diverges in the infra-red. In terms of our exponents this corresponds to the regime $2 \leq \alpha < \sqrt{4 + c}$. [^9]: On the other hand when $\Delta_{\chi}/z < 0$ it will be the UV region of the geometry which dominates the integral. We therefore cannot calculate the diffusion constant just from knowledge of our infra-red scaling theory. The diffusion constant is then no longer related to the butterfly effect in a universal manner, but rather it will be parametrically larger than $v_B^2/T$ by powers of the UV cutoff [@butterflymike]. [^10]: It is worth noting that the thermal conductivity bound of [@saso2] does not apply to models with an unbounded potential, hence why $\kappa/T$ can vanish at low temperatures. Nevertheless the diffusion constant will still be universal and hence it would provide a more natural object than the conductivity on which to formulate rigorous ‘bounds’. [^11]: Note that the condition that the axion fields remain present in the equations of motion imposes the constraint $u_1 + 2 v_1 - 2 \phi_1 = 2$ which we have used in simplifying [(\[difflattice\])]{}. [^12]: Since the coefficients in [(\[bound\])]{} depend on the universality class, and can be arbitrary small, it will not be possible to formulate a strict bound using $v_B$. Nevertheless the spirit that the diffusion constants are generically set by $v_B^2/T$ remains. [^13]: Note however that when the charge density becomes sufficiently strong to dominate the geometry we would expect to move away from the universal regime. This suggests that incoherent metals saturating [(\[bound\])]{} have an approximate particle-hole symmetry in the infrared. [^14]: In particular the constant of proportionality depends on the profile of the inhomogeneities. It would be interesting to understand if there is another velocity, which differs only by a numerical factor from $v_B$, in terms of which this relationship can be made more universal. [^15]: Intriguingly, this is precisely the same as the result [(\[diffscaling\])]{} for the axion theory we studied in Section \[axionsection\]. [^16]: Note that if there is a non-zero background stress tensor then in these coordinates there is a shift in the components of the stress tensor in addition to the change in the metric [@sfetsos; @roberts].

--- abstract: 'We show that any polyhomogeneous asymptotically hyperbolic constant-mean-curvature solution to the vacuum Einstein constraint equations can be approximated, arbitrarily closely in Hölder norms determined by the physical metric, by shear-free smoothly conformally compact vacuum initial data.' address: 'Department of Mathematical Sciences, Lewis & Clark College' author: - 'Paul T. Allen and Iva Stavrov Allen' bibliography: - 'ShearFreeDensity.bib' title: 'Smoothly compactifiable shear-free hyperboloidal data is dense in the physical topology' --- Introduction {#introduction .unnumbered} ============ In the study of asymptotically flat (or asymptotically simple) spacetimes, initial data corresponding to spacelike slices extending towards null infinity has asymptotically hyperbolic geometry. Lars Andersson and Piotr Chruściel, building on their work with Helmut Friedrich [@AnderssonChruscielFriedrich], construct in [@AnderssonChrusciel-Dissertationes] a large number of constant-mean-curvature (CMC) vacuum initial data sets with asymptotically hyperbolic geometry using the conformal method of Yvonne Choquet-Bruhat, André Lichnerowicz, and James York. In the work [@AnderssonChrusciel-Dissertationes], particular attention is paid to the regularity of solutions at the conformal boundary. Data constructed in [@AnderssonChrusciel-Dissertationes] typically admits a $C^2$, but not $C^3$ conformal compactification. In particular, they showed that data which is smooth in the interior “physical” manifold is typically polyhomogeneous, rather than smooth, at the conformal boundary. In their detailed analysis [@AnderssonChrusciel-Obstructions], Andersson and Chruściel show that initial data must satisfy the shear-free condition along the conformal boundary (see §\[SF:Define\]) in order for any resulting spacetime geometry to admit a $C^2$ conformal compactification. This suggests that one might require the shear-free condition hold in order for a solution to the Einstein constraint equations to be “admissible” in the asymptotically hyperbolic setting. Thus we refer to initial data satisfying the shear-free condition as [[***hyperboloidal***]{}]{}, distinguished among those solutions to the constraint equations having asymptotically hyperbolic geometry. Our recent work [@AHEM-Preliminary], joint with James Isenberg and John M. Lee, contains a systematic study of CMC hyperboloidal initial data, including a parametrization of all such data in the “weakly asymptotically hyperbolic” setting (see also [@WAH-Preliminary]). This is not, however, the end of the story. Even if one restricts attention to shear-free data, the initial data constructed in [@AnderssonChrusciel-Dissertationes] and [@AHEM-Preliminary] may not be sufficiently regular at the conformal boundary to obtain a spacetime development admitting conformal compactification. For example, the existing evolution theorems of Helmut Friedrich [@Friedrich-ConformalFieldEquations], [@Friedrich-StaticRadiative], etc., all require more regularity of the conformal compactification. (The regularity issue is not unrelated to the shear-free condition: Andersson, Chruściel, and Friedrich show in [@AnderssonChruscielFriedrich] that initial data, constructed from smooth “free data” using the conformal method (see §\[ConformalMethod\]), with pure-trace extrinsic curvature is shear-free if and only if it is smoothly conformally compact.) In addition to issues of regularity, one may be concerned about whether the collection of hyperboloidal data is sufficiently general for modeling a wide variety of physical situations. Here we address these issues by showing that any polyhomogeneous asymptotically hyperbolic CMC solution to vacuum constraint equations can be approximated, arbitrarily closely in Hölder norms determined by the physical (non-compactified) spatial metric, by hyperboloidal (i.e. shear-free) vacuum initial data that is smoothly conformally compact. In the case that the conformal boundary is a $2$-sphere, the work of [@Friedrich-ConformalFieldEquations] implies that the approximating data has a spacetime development admitting a smooth conformal infinity. There are a number of ways in which one might interpret our result. From the perspective of modeling isolated gravitational systems, it is an indication that some version of Bondi-Sachs-Penrose approach to using conformal compactness for studying asymptotically flat spacetimes is feasible for studying a large class of physical systems. However, it is also an indication that the Hölder topology determined by the physical metric is insufficiently strong for studying the conformal boundary of asymptotically hyperbolic initial data sets. (For example, it is observed in [@AnderssonChrusciel-Obstructions] that among the initial data constructed in [@AnderssonChrusciel-Dissertationes] from smooth “free data” by means of the conformal method, the shear-free condition does not hold generically with respect to the $C^\infty$ topology determined by the conformally compactified metric.) Indeed, it was the approximation result here that motivated several of the results in [@AHEM-Preliminary], where continuity of the conformal method for construction of solutions to the constraint equations is established in a topology strong enough to detect the shear-free condition. Discussion of main result ========================= Here we present a discussion of the details needed in order to make precise our approximation result. As we make use of several results from [@Lee-FredholmOperators], [@WAH-Preliminary], and [@AHEM-Preliminary], we maintain conventions similar to the conventions in those works. Asymptotically hyperbolic initial data -------------------------------------- Let $M$ be the interior of a smooth three-dimensional compact manifold ${\overline}M$ having boundary $\partial M$. We say that a smooth function $\rho\colon {\overline}M \to [0,\infty)$ is a [[***defining function***]{}]{} if $\rho^{-1}(0) = \partial M$ and if ${d}\rho\neq 0$ on $\partial M$. A metric $g$ on $M$ is said to be [[***$C^k$ conformally compact***]{}]{} if ${\overline}g:= \rho^2 g$ extends to a metric of class $C^k$ on ${\overline}M$ for one, and hence all, smooth defining functions $\rho$. A $C^2$ conformally compact metric $g$ is [[***asymptotically hyperbolic***]{}]{} if $|{d}\rho|_{{\overline}g} =1$ along $\partial M$ for one, and hence all, smooth defining functions $\rho$. The sectional curvatures of such metrics approach $-1$ as $\rho \to 0$; see [@WAH-Preliminary] for generalizations of this definition. A vacuum initial data set $(g,K)$ consists of a Riemannian metric $g$ and symmetric covariant $2$-tensor $K$, both defined on $M$ and satisfying the vacuum Einstein constraint equations \[Constraints\] $$\label{HamiltonianConstraint} R[g] -|K|^2_g + (\operatorname{tr}_g K)^2 =0,$$ $$\label{MomentumConstraint} \operatorname{div}_g K-{d}(\operatorname{tr}_g K) =0.$$ It is convenient to introduce the notation $\tau = \operatorname{tr}_gK$ for the trace of $K$ and $\Sigma = K - \frac13\tau g$ for the traceless part of $K$. We say that [[***$(g,K)$ is an asymptotically hyperbolic initial data set***]{}]{} if $g$ is asymptotically hyperbolic and if the tensor ${\overline}\Sigma = \rho\Sigma$ extends to a $C^1$ tensor field on ${\overline}M$. Such a data set is said to be [[***smoothly conformally compact***]{}]{} if for any defining function $\rho$ the tensor fields ${\overline}g=\rho^2 g$ and ${\overline}\Sigma=\rho \Sigma$ extend smoothly to ${\overline}M$. We note that there exist “weakly asymptotically hyperbolic” solutions to , satisfying less stringent regularity conditions; see [@AHEM-Preliminary]. Asymptotically hyperbolic data sets may be viewed as intersecting future null infinity in the asymptotically flat spacetime containing a future development of the data set; we refer the reader to [@AHEM-Preliminary], and the references therein, for a more detailed discussion of asymptotically hyperbolic initial data sets and asymptotic flatness. The formula for the change of scalar curvature under conformal deformation, together with , implies $$\label{LichRho} 4\rho \Delta_{{\overline}g}\rho + (R[{\overline}g] - |{\overline}\Sigma|^2_{{\overline}g}) \rho^2 + 6\left( \frac{\tau^2}{9} - |{d}\rho|^2_{{\overline}g}\right) =0,$$ where our sign convention on the scalar Laplace operator is $\Delta_{{\overline}g} = \operatorname{tr}_{{\overline}g}\operatorname{Hess}_{{\overline}g}$. Evaluating at $\rho =0$ we find that $\tau^2 = 9$ along $\partial M$. Thus in the constant-mean-curvature (CMC) setting we have $\tau = \pm 3$, with the sign indicating whether the initial data intersects future or past null infinity (relative to the notion of “future” determined by $K$). Henceforth we restrict attention to the CMC case and set $\tau = -3$, which (due to our sign convention for $K$) corresponds to future null infinity; see the discussion in [@AHEM-Preliminary]. Note that when $\tau =-3$ the constraint equations reduce to $$\label{CMC-constraints} R[g]-|\Sigma|^2_g + 6 =0 \quad\text{ and }\quad \operatorname{div}_g\Sigma =0.$$ The shear-free condition {#SF:Define} ------------------------ While any sufficiently regular solution to the Einstein constraint equations gives rise to some spacetime development thereof (see [@ChoquetBruhat-GRBook] and the references therein), it was shown in [@AnderssonChrusciel-Obstructions] that the development of an asymptotically hyperbolic initial data set admits a conformal compactification along future null infinity only if the [[***shear-free condition***]{}]{} $$\label{FirstShearFree} \left[\operatorname{Hess}_{{\overline}g}\rho -\frac13(\Delta_{{\overline}g}\rho){\overline}g -{\overline}\Sigma\right]_{\partial M} =0$$ holds. We say that an asymptotically hyperbolic initial data set is a [[***hyperboloidal initial data set***]{}]{} if holds. The conformal method {#ConformalMethod} -------------------- The existence of asymptotically hyperbolic initial data sets is addressed in [@AnderssonChruscielFriedrich] and [@AnderssonChrusciel-Dissertationes]. The existence of hyperboloidal data is discussed in [@AHEM-Preliminary]. All these works make use of the conformal method, which we now describe. We first introduce the [[***conformal Killing operator***]{}]{} $\mathcal D_g$, which maps vector fields to trace-free symmetric covariant $2$-tensors by $$\label{DefineCK} \mathcal D_g W = \frac12 \mathcal L_W g - \frac13( \operatorname{div}_g W) g.$$ The formal $L^2$ adjoint $\mathcal D_g^*$ is given by $\mathcal D_g^* T = -(\operatorname{div}_g T)^\sharp$, and can be used to construct the self-adoint, elliptic operator $L_g := \mathcal D_g^* \mathcal D_g$, which is called the [[***vector Laplacian***]{}]{}. In the CMC setting, with $\tau = -3$, the conformal method seeks a solution $(g,K)$ to of the form \[Conformal\] $$\begin{aligned} \label{Conformal-g} g &= \phi^4 \lambda \\ \label{Conformal-K} K&= \phi^{-2}\left( \mu + \mathcal D_\lambda W\right) - \phi^4 \lambda,\end{aligned}$$ for some Riemannian metric $\lambda$, symmetric covariant $2$-tensor field $\mu$, vector field $W$, and positive function $\phi$. Replacing $g$ and $K$ in by the expressions in , we find that the constraints are satisfied if $W$ and $\phi$ satisfy the elliptic system \[ConformalConstraints\] $$\begin{aligned} \label{ConformalMomentumConstraint} L_\lambda W &= -\operatorname{div}_\lambda \mu \\ \label{Lich} \Delta_\lambda\phi &= \frac18 \operatorname{R}[\lambda] \phi - \frac18 |\mu + \mathcal D_\lambda W|^2_\lambda \phi^{-7} + \frac34 \phi^5.\end{aligned}$$ Thus if $\lambda$ and $\mu$ are specified, it remains only to solve in order to obtain a solution to . We make use of the nomenclature of [@AHEM-Preliminary] and refer to $(\lambda,\mu)$ as a [[***free data set***]{}]{}. If $\lambda$ is an asymptotically hyperbolic metric on $M$, then $g = \phi^4 \lambda$ is an asymptotically hyperbolic metric provided $\phi \in C^2({\overline}M)$ and $\phi =1$ along $\partial M$. If $\phi$ satisfies these conditions, then metric $g$ and tensor $K$ given by form an asymptotically hyperbolic data set provided $\rho\mu$ and $\rho \mathcal D_\lambda W$ extend to $C^1$ tensor fields on ${\overline}M$. We remark that the conformal method, as described above, does not necessarily yield hyperboidal data (i.e., data satisfying the shear-free condition). However, with appropriately constructed free data, one can ensure that the resulting initial data does in fact satisfy the shear-free condition; see [@AHEM-Preliminary]. Polyhomogeneous data {#polyhomog:sec} -------------------- The two works [@AnderssonChruscielFriedrich] and [@AnderssonChrusciel-Dissertationes], where large classes of asymptotically hyperbolic initial data are constructed, contain detailed analyses of the regularity of solutions at $\partial M$ and show the following: Even if the free data $\lambda$ and $\mu$ are smoothly conformally compact, the solutions $W$ and $\phi$ to need not give rise to smoothly conformally compactifiable fields $g$ and $K$. Rather, the resulting metric $g$ and tensor field $K$ admit formal expansions at $\partial M$ given, in terms of an arbitrary smooth defining function $\rho$, by \[PhysicalAsymptotics\] $$\begin{aligned} \label{PhysicalAymptotics-g} {\overline}g &\sim {\overline}g_0 + \sum_{i=0}^{\infty} \sum_{p=0}^{p_i} \rho^{s_i} (\log\rho)^{p} {\overline}g_{ip}, \\ \label{PhysicalAymptotics-Sigma} {\overline}\Sigma &\sim {\overline}\Sigma_0 + \sum_{i=0}^{\infty}\sum_{q=0}^{q_i} \rho^{t_i} (\log\rho)^{q} {\overline}\Sigma_{iq},\end{aligned}$$ where the barred terms are smooth tensor fields. Tensor fields which admit such expansions are called “polyhomogeneous.” We remark that a number of closely-related definitions of polyhomogeneous tensor fields exist in the literature; see §\[section:boundary\] below for a precise definition of the notion of polyhomogenity used here. The asymptotic expansions of the polyhomogeneous data constructed in [@AnderssonChrusciel-Dissertationes] take the form with ${\mathrm{Re}}{(s_0)} >2$ and ${\mathrm{Re}}{(q_0)} >1$. Thus, letting $C^k_{\textup{\normalfont phg}}({\overline}M)$ denote the collection of polyhomogeneous tensor fields on $M$ which extend to fields of class $C^k$ on ${\overline}M$, we have ${\overline}g\in C^2_{\textup{\normalfont phg}}({\overline}M)$ and ${\overline}\Sigma\in C^1_{\textup{\normalfont phg}}({\overline}M)$. The polyhomogeneous hyperboloidal data sets constructed in [@AHEM-Preliminary] also have this regularity. The approximation theorem ------------------------- We now give a careful statement of our main result. \[Density\] Suppose that $(g,K)$ is a polyhomogeneous asymptotically hyperbolic vacuum initial data set on $3$-manifold $M$. Then there exists a family $(g_{\varepsilon}, K_{\varepsilon})$ of solutions to the vacuum Einstein constraint equations , defined for sufficiently small ${\varepsilon}>0$, such that 1. each initial data set is hyperboloidal, meaning that each $(M,g_{\varepsilon})$ is asymptotically hyperbolic, that $(g_{\varepsilon}, K_{\varepsilon})$ each satisfy the constraint equations , and that $(g_{\varepsilon}, K_{\varepsilon})$ each satisfy the shear-free condition ; 2. each initial data set in the family is smoothly conformally compact, in the sense that ${\overline}g_{\varepsilon}= \rho^2 g_{\varepsilon}\in C^\infty({\overline}M)$ and ${\overline}\Sigma_{\varepsilon}= \rho(K_{\varepsilon}+ g_{\varepsilon}) \in C^\infty({\overline}M)$; and 3. we have $(g_{\varepsilon}, K_{\varepsilon}) \to (g,K)$ as ${\varepsilon}\to 0$ in the $C^{k,\alpha}(M) \times C^{k,\alpha}(M)$ topology, for any $k$ and $\alpha$. We now describe the proof of Theorem \[Density\]; the details are contained in §\[FreeData\]–§\[SolveConstraint\] below. First we construct a family of free data $(\lambda_{\varepsilon}, \mu_{\varepsilon})$ for small ${\varepsilon}>0$. Our construction is such that the metrics $\lambda_{\varepsilon}$ agree with $g$ away from a neighborhood of $\partial M$, but are smoothly conformally compact. We also arrange that the fields $\mu_{\varepsilon}$ agree with $\Sigma$ away from a neighborhood of $\partial M$, but are deformed near the boundary in order that the shear-free condition holds upon deformation to a solution of the constraint equations. The free data $(\lambda_{\varepsilon}, \mu_{\varepsilon})$ is furthermore carefully constructed so that application of the conformal method yields smoothly conformally compact initial data sets. (The construction is motivated by the analysis in [@AnderssonChrusciel-Obstructions].) The proof proceeds by applying the conformal method to the free data $(\lambda_{\varepsilon}, \mu_{\varepsilon})$. In order to show that the resulting solutions to the constraint equations approach $(g,K)$ as ${\varepsilon}\to 0$, it is necessary to obtain uniform estimates for family of solutions $W_{\varepsilon}$, $\phi_{\varepsilon}$ to . Technical preliminaries ======================= We present several technical results needed for the proof of Theorem \[Density\]. Function spaces --------------- We fix a smooth defining function $\rho$ on ${\overline}M$, and we make use of weighted Hölder spaces $C^{k,\alpha}_\delta(M)$ of tensor fields on $M$ as defined in [@WAH-Preliminary] (see also [@Lee-FredholmOperators]). These spaces are defined independently of any Riemannian structure, but have equivalent norms determined by any sufficiently regular asymptotically hyperbolic metric. We emphasize that the convention regarding the weight $\delta$ is such that tensor field $u\in C^{0}_\delta(M)$ when $|u|_g \leq C \rho^\delta$ for any asymptotically hyperbolic metric $g = \rho^{-2}{\overline}g$. Recall that the [[***weight of a tensor bundle***]{}]{} is the covariant rank less the contravariant rank. (Thus the weight of a vector field is $-1$, while the weight of a metric tensor is $2$.) The weight of a tensor field is important to keep in mind: If $u$ is a tensor field of weight $r$, then $|u|_g = \rho^{-r}|u|_{{\overline}g}$. In particular, for tensors of weight $r$ we have the following inclusion: $$C^{k,\alpha}({\overline}M)\subseteq C^{k,\alpha}_r(M);$$ compare with Lemma 3.7 of [@Lee-FredholmOperators]. It is convenient to distinguish the following class of metrics: We say that an asymptotically hyperbolic metric $h$ is a [[***preferred background metric***]{}]{} if ${\overline}h =\rho^2 h$ extends smoothly to ${\overline}M$ and if in a neighborhood of $\partial M$ we have that ${\overline}h$ is a product metric of the form ${d}\rho \otimes {d}\rho +{\overline}b$ for some metric ${\overline}b$ on $\partial M$. We denote by $\nabla$ and ${\overline}\nabla$ the Levi-Civita connections associated to $h$ and ${\overline}h$ respectively, and note that the difference tensor ${\overline}\nabla-\nabla$ is an element of $C^{k,\alpha}(M)$ for all $k$ and $\alpha$. Throughout this section and the next, $h$ represents any preferred background metric, and ${\overline}h = \rho^2 h$. In §\[FreeData\] we fix a preferred background metric, adapted to the metric $g$ appearing in Theorem \[Density\], which we retain throughout the proof of that theorem. The following is an immediate consequence of Lemma 2.2(d) of [@WAH-Preliminary]. \[BoundaryLemma\] Suppose $u\in C^{k,\alpha}_r(M)$ is a tensor field of weight $r$ such that ${\overline}\nabla u\in C^{k-1,\alpha}_{r+1}(M)$ and such that $|u|_{{\overline}h}\to 0$ as $\rho \to 0$. Then $u\in C^{k,\alpha}_{r+1}(M)$ and $$\|u\|_{C^{k,\alpha}_{r+1}(M)} \leq C \left( \|u\|_{C^{k,\alpha}_r(M)} + \|{\overline}\nabla u\|_{C^{k-1,\alpha}_{r+1}(M)} \right)$$ Differential operators {#DiffOpSec} ---------------------- We now record several results concerning differential operators arising in the conformal method. A differential operator $\mathcal P = \mathcal P[g]$ of order $l$ arising from a metric $g$ is said to be [[***geometric***]{}]{} (in the sense of [@Lee-FredholmOperators]) if in any coordinate frame the components of $\mathcal P u$ are linear functions of $u$ and its derivatives, whose coefficients are universal polynomials in the components of $g$, their partial derivatives, and $\sqrt{\det g_{ij}}$, such that the coefficient of the $j$th derivative of $u$ involves no more than $l-j$ derivatives of the metric. Such operators are [*uniformly degenerate*]{}; the mapping properties of such operators have been studied in [@Mazzeo-Edge], [@Lee-FredholmOperators], [@AnderssonChrusciel-Dissertationes]. Recently, in work [@WAH-Preliminary] with James Isenberg and John M. Lee we have extended some of these results to the [*weakly asympotically hyperbolic*]{} setting. We recall here several results needed for the proof of Theorem \[Density\]; the aforementioned works apply in much more general settings. The following proposition allows us to compare corresponding operators arising from different metrics. \[prop:GeometricOperatorsContinuous\] Let $k\geq 0$, $\alpha\in [0,1)$, and $\delta\in \mathbb R$. Suppose $g\in C^{k,\alpha}(M)$ is an asymptotically hyperbolic metric, and that $\mathcal P$ is a geometric operator of order $l\leq k$. Then there exists ${\varepsilon}_*>0$ and $C>0$ such that for any asymptotically hyperbolic metric $g^\prime\in C^{k,\alpha}(M)$ with $\|g - g^\prime\|_{C^{k,\alpha}(M)}\leq {\varepsilon}_*$ we have $$\|\mathcal P[g]u - \mathcal P[g^\prime]u \|_{C^{k-l,\alpha}_\delta(M)} \leq C \| g-g^\prime\|_{C^{k,\alpha}(M)} \|u\|_{C^{k,\alpha}_\delta(M)}$$ for all $u\in C^{k,\alpha}_\delta(M)$. We now turn attention to elliptic geometric operators. The operators arising in the results here satisfy the following. \[Assume-P\] Suppose $(M,g)$ is an asymptotically hyperbolic manifold. We assume $\mathcal P = \mathcal P[g]$ is a second-order linear elliptic operator acting on sections of a tensor bundle $E$. Furthermore we assume that $\mathcal P$ is [geometric]{} in the sense defined above, and that $\mathcal P$ is formally self-adjoint. The mapping properties of operators satisfying Assumption \[Assume-P\] can be understood by studying the [[***indicial map***]{}]{} $I_s(\mathcal P)$, defined for $s\in \mathbb C$ to be the bundle map $$E\otimes \mathbb C\big|_{\partial M} \to E\otimes \mathbb C\big|_{\partial M}$$ given by $I_s(\mathcal P) {\overline}u = \rho^{-s} \mathcal P (\rho^s {\overline}u)\big|_{\rho =0}$; see [@Mazzeo-Edge], [@Lee-FredholmOperators]. The [[***characteristic exponents***]{}]{} of $\mathcal P$, which we denote by $\mathcal E$, are defined to be those values of $s$ for which $I_s(\mathcal P)$ has a non-trivial kernel at some point on $\partial M$. In [@Lee-FredholmOperators] it is shown that, under Assumption \[Assume-P\], these exponents and their multiplicities are constant on $\partial M$, and agree with those associated to the corresponding operator in the half-space model of hyperbolic space. Furthermore, due to the self-adjointness of $\mathcal P$, the characteristic exponents are symmetric about the line ${\mathrm{Re}}{(s)} = 1-r$, where $r$ is the weight of $E$. The [[***indicial radius***]{}]{} $R$ of $\mathcal P$ is defined to be the smallest number $R\geq 0$ such that ${\mathrm{Re}}{(s)} \leq 1-r+R$ for all $s\in \mathcal E$. The importance of the indicial radius is the following result from [@Lee-FredholmOperators]: If $(M,g)$ is asymptotically hyperbolic of class $C^{k,\alpha}$ with $\alpha \in (0,1)$, if Assumption \[Assume-P\] is satisfied, and if there is a compact set $K\subset M$ and a constant $C>0$ such that $$\|u\|_{L^2(M)}\leq C\|\mathcal P u\|_{L^2(M)} \quad\text{ for all }\quad u\in C^\infty_c(M{\!\smallsetminus\!}K),$$ then $\mathcal P\colon C^{k+2,\alpha}_\delta(M) \to C^{k,\alpha}_\delta(M)$ is Fredholm if and only if $|1-\delta|<R$. The following proposition is a consequence of [@AHEM-Preliminary Proposition 6.3], [@WAH-Preliminary Proposition 6.1], and [@WAH-Preliminary Lemma 5.6] (see also [@Lee-FredholmOperators Lemma 6.4]); we emphasize that the results cited apply in much more general situations. \[MappingProperties\] Suppose $(M,g)$ is an asymptotically hyperbolic $3$-manifold, and suppose that $g$ is smoothly conformally compact. 1. \[Prop4a\] For each $k\geq 0$, $\alpha \in (0,1)$, and $\delta\in (-1,3)$ the vector Laplacian is an isomorphism $$L_g\colon C^{k+2,\alpha}_\delta(M) \to C^{k,\alpha}_\delta(M).$$ In particular, there exists a constant $C>0$ such that $$\|X \|_{C^{k+2,\alpha}_\delta(M)} \leq C \|L_gX\|_{C^{k,\alpha}_\delta(M)}$$ for all vector fields $X\in C^{k+2,\alpha}_\delta(M)$. 2. \[Prop4b\] Let $k\geq 0$ and $\alpha\in (0,1)$. Suppose $\kappa \in C^{k,\alpha}_\sigma(M)$ for some $\sigma >0$ and that $c$ is a constant satisfying $c>-1$ and $c+\kappa\geq 0$. Then so long as $$|\delta - 1|\leq \sqrt{1+c}$$ the map $$\Delta_g - (c+\kappa)\colon C^{k+2,\alpha}_\delta(M) \to C^{k,\alpha}_\delta(M)$$ is an isomorphism. In particular, there exists a constant $C>0$ such that $$\|u \|_{C^{k+2,\alpha}_\delta(M)} \leq C \|\Delta_gu - (c+\kappa)u\|_{C^{k,\alpha}_\delta(M)}$$ for all functions $u\in C^{k+2,\alpha}_\delta(M)$. Furthermore, if $w\in C^{0}_\delta(M)$ is such that $\Delta_g w - (c+\kappa)w \in C^{k,\alpha}_{\delta^\prime}(M)$ then $w\in C^{k,\alpha}_{\delta^\prime}(M)$ whenever $|\delta^\prime - 1|\leq \sqrt{1+c}$. The tensor ${\mathcal{H}}_{{\overline}g}(\rho)$ ----------------------------------------------- Together with James Isenberg and John M. Lee, we introduced in [@WAH-Preliminary] a conformally invariant version of the trace-free Hessian that is used in [@AHEM-Preliminary] to characterize the shear-free condition; we now recall its definition and basic properties. Let $$A_{{\overline}g}(\rho) = \frac{1}{2} |{d}\rho|_{{\overline}g} \operatorname{div}_{{\overline}g}\left[ |{d}\rho|_{{\overline}g}\operatorname{grad}_{{\overline}g}\rho\right].$$ We define the tensor field ${\mathcal{H}}_{{\overline}g}(\rho)$ by $$\label{DefineB} {\mathcal{H}}_{{\overline}g}(\rho) :=|{d}\rho|_{{\overline}g}^6\,\mathcal D_{{\overline}g}(|{d}\rho|^{-2}_{{\overline}g} \operatorname{grad}_{{\overline}g}\rho) + A_{{\overline}g}(\rho) \left( {d}\rho \otimes {d}\rho - \frac{1}{3}|{d}\rho|^2_{{\overline}g}\, {\overline}g \right),$$ where $\mathcal D_{{\overline}g}$ is the conformal Killing operator defined in . We have the following basic properties of ${\mathcal{H}}_{{\overline}g}(\rho)$. \[B-BasicProperties\] 1. ${\mathcal{H}}_{{\overline}g}(\rho)$ is symmetric and trace-free. 2. \[B-TransverseProperty\] ${\mathcal{H}}_{{\overline}g}(\rho)(\operatorname{grad}_{{\overline}g}\omega, \cdot)=0$. 3. \[B-Scaling\] ${\mathcal{H}}_{{\overline}g}(c \rho)=c^5{\mathcal{H}}_{{\overline}g}(\rho)$ for all constants $c$. 4. \[B-ConformalScaling\] If $\theta$ is a strictly positive function then ${\mathcal{H}}_{\theta^4{\overline}g}(\rho)=\theta^{-8}{\mathcal{H}}_{{\overline}g}(\rho)$ and $A_{\theta^4{\overline}g}(\rho) = \theta^{-8}A_{{\overline}g}(\rho)$. Suppose $(g,K) = (g, \Sigma - g)$ is a polyhomogeneous asymptotically-hyperbolic CMC initial data set. Then the shear-free condition is satisfied if and only if $$\left.{\overline}\Sigma\right|_{\partial M} = \left.{\mathcal{H}}_{{\overline}g}(\rho)\right|_{\partial M},$$ where ${\overline}g = \rho^2 g$ and ${\overline}\Sigma = \rho\Sigma$. Analysis on ${\overline}M$ {#section:boundary} ========================== The solution to a geometric elliptic equation of the form $\mathcal P u = f$ on an asymptotically hyperbolic manifold $(M,g)$ may be smooth on $M$, but may not extend smoothly to ${\overline}M$, even if ${\overline}g\in C^\infty({\overline}M)$; see [@Mazzeo-Edge], [@AnderssonChrusciel-Dissertationes], [@ChruscielDelayLeeSkinner], et. al. Rather, many solutions to elliptic equations have asymptotic expansions at $\partial M$ containing powers of $\rho$ and powers of $\log\rho$. The logarithmic terms arise in situations where there is a resonance (see §\[bdry:sec\] or [@WAH-Preliminary Remark A.12]) and are thus features of the algebraic structure of $\mathcal P$. Tensor fields with expansions involving powers of $\rho$ and $\log\rho$ are called [*polyhomogeneous*]{}. We now present a more careful definition, and subsequently discuss conditions under which the solution itself is in fact smooth on ${\overline}M$. We note that a number of related definitions of polyhomogeneity appear in the literature; see [@Mazzeo-Edge], [@AnderssonChrusciel-Dissertationes], [@WAH-Preliminary], [@IsenbergLeeStavrov-AHP], [@ChruscielDelayLeeSkinner], et. al. For convenience, we work with a fixed preferred background metric $h = \rho^{-2}{\overline}h$, denoting by ${\overline}\nabla$ the Levi-Civita connection associated to ${\overline}h$. (The following, however, is independent of the choice of $h$.) We subsequently make frequent and implicit use of the following construction: If $E$ is a tensor bundle over ${\overline}M$ and ${\overline}u$ is a smooth section of $E\big|_{\partial M}$, we may extend ${\overline}u$ to the neighborhood of $\partial M$ by parallel transport along $\operatorname{grad}_{{\overline}h}\rho$; using a smooth cutoff function, the resulting tensor may be extended further to all of ${\overline}M$. Furthermore, when working in the neighborhood of $\partial M$ where ${\overline}h = {d}\rho\otimes {d}\rho + {\overline}b$, we abuse notation by writing $\rho\partial_\rho$ for $\rho{\overline}\nabla_{\operatorname{grad}_{{\overline}h}\rho}$. Polyhomogeneity --------------- In order to carefully define polyhomogeneity for tensor fields, we first introduce for each $\delta \in \mathbb R$ the class $\mathcal B_\delta(M)$ of tensor fields, defined by $$\mathcal B_\delta(M) = \bigcap_{\substack{0\leq k\\ t<\delta}} C^k_t(M).$$ (The reader may wish to compare these spaces to the conormality spaces appearing in [@WAH-Preliminary] and the references therein.) The importance of this definition is that if $s\in \mathbb C$ then $\rho^s \log\rho$ is contained in $\mathcal B_\delta(M)$ with $\delta = {\mathrm{Re}}{(s)}$, but is not of class $C^0_\delta(M)$. Furthermore, $u\in \mathcal B_\delta(M)$ if and only if $(\log\rho) u\in \mathcal B_\delta(M)$. If tensor $u$ of weight $r$ satisfies $u\in \mathcal B_{\delta+r}(M)$ for some $\delta>0$, then $(\rho\partial_\rho)^ku$ vanishes at $\partial M$ for all $k\geq 0$. The same holds for certain other fields, such as the functions $\rho^s(\log\rho)^{-n}$ with ${\mathrm{Re}}{(s)}=0$ and $n$ a positive integer. Consequently, we obtain the following. \[LinInd\] Suppose $E$ is a tensor bundle of weight $r$, and that $t_i\in \mathbb{C}$, $q_i\in \mathbb{N}_0$, and sections ${\overline}u_i$ of $E\big{|}_{\partial M}$ are such that $$u=\sum_{i=1}^N \rho^{t_i}(\log \rho)^{q_i}{\overline}u_i \in \mathcal B_{\delta+r}(M)$$ for some $\delta>\max_{1\le i\le N}{\mathrm{Re}}(t_i)$. Then ${\overline}u_i=0$ for all $1\le i\le N$. It suffices to fix a point of $\partial M$ and consider the case when $u$ is a function. Under such restrictions our claim is a consequence of the fact that a finite $\mathbb{R}$-linear combination of single-variable functions of the form $\rho^t(\log \rho)^q$ with ${\mathrm{Re}}(t)=0$ $$\sum_{j=1}^{J} a_j \rho^{ib_j}(\log \rho)^{q_j}$$ vanishes at $\rho=0$ together with all of its $\rho\partial_\rho$ derivatives if and only if all of the coefficients $a_j$ vanish. A smooth section $u$ of tensor bundle $E$ over $M$ having weight $r$ is defined to be [[***polyhomogeneous***]{}]{} if 1. there exist sequences $s_i\in \mathbb C$ and $p_i\in \mathbb N_0$ with ${\mathrm{Re}}{(s_i)}$ non-decreasing and diverging to $+\infty$ as $i\to\infty$, 2. \[part:phg-bdy-sections\] for $i,p\in \mathbb N_0$ with $0\leq p \leq p_i$ there exists smooth section ${\overline}u_{ip}$ of $E\big|_{\partial M}$, and 3. \[part:phg-expansion\] for each $k\in \mathbb N_0$ there exists $N_k\in \mathbb N_0$ such that $$\label{GenericTensorExpansion} u - \sum_{i=0}^{N_k} \sum_{p=0}^{p_i} \rho^{s_i-r} (\log\rho)^p {\overline}u_{ip} \in \mathcal B_k(M).$$ We assume that those exponents $s_i$ having the same real part are ordered such that their imaginary parts are increasing. (The factor of $\rho^{-r}$ in is motived by the fact that $|u|_g = \rho^r |u|_{{\overline}g}$; thus the leading order behavior of $|u|_{ g}$ will be as $\rho^{{\mathrm{Re}}{(s_0)}}(\log\rho)^{p_0}$.) If $u$ satisfies the definition above, we write $$u\sim \sum_{i=0}^{\infty} \sum_{p=0}^{p_i} \rho^{s_i-r} (\log\rho)^p {\overline}u_{ip}.$$ Let $\mathcal B_{\textup{\normalfont phg}}(M)$ be the collection of all tensor fields on $M$ which are polyhomogeneous as defined above. We furthermore denote by $\mathcal B^{\textup{\normalfont phg}}_\delta(M)$ those polyhomogeneous tensor fields that are of class $\mathcal B_\delta(M)$, and by $C^{k}_{\textup{\normalfont phg}}({\overline}M)$ those polyhomogeneous tensor fields extending to tensor fields of class $C^k$ on ${\overline}M$. \[PhgSplitting\] 1. It follows from Lemma \[LinInd\] that if $u\in \mathcal B^{\textup{\normalfont phg}}_\delta(M)$ then we have ${\mathrm{Re}}(s_0)\geq \delta$. 2. Polyhomogeneous expansions are unique in the sense that if $$v\sim \sum_{i=0}^\infty \sum_{p=0}^{p_i} \rho^{s_i} (\log \rho)^p {\overline}v_{ip} \text{\ \ and \ \ } v\sim \sum_{i=0}^\infty \sum_{q=0}^{q_i} \rho^{t_i} (\log \rho)^p {\overline}w_{ip},$$ then $s_i=t_i$, $p_i=q_i$, and ${\overline}v_{ip}={\overline}w_{jp}$. 3. Tensor fields $u$ which are smooth on ${\overline}M$ are polyhomogeneous with a Taylor-series like expansion $$u\sim \sum_{n=0}^\infty \frac{\rho^n}{n!} {\overline}u_n.$$ The fields ${\overline}u_n$ are the restrictions of ${\overline}\nabla {}^n u(\operatorname{grad}_{{\overline}h} \rho,\dots, \operatorname{grad}_{{\overline}h}\rho, \cdot, \dots, \cdot)$ to the boundary. We emphasize that this holds regardless of whether $u$ is analytic or not. 4. A tensor field $u\in \mathcal B^{\textup{\normalfont phg}}_\delta(M)$ of weight $r$ is in $C^l_{\textup{\normalfont phg}}({\overline}M)$ if $\delta> l+r$; see Lemma 3.7 in [@Lee-FredholmOperators]. Thus $u\in C^\infty({\overline}M)$ if and only if $u\in C^\infty({\overline}M)+\mathcal B^{\textup{\normalfont phg}}_k(M)$ for all $k\in \mathbb{N}$. Furthermore, a polyhomogeneous tensor field $$u\sim \sum_{i=0}^\infty \sum_{p=0}^{p_i} \rho^{s_i} (\log \rho)^p {\overline}u_{ip}.$$ is smooth on ${\overline}M$ if and only if $s_i\in \mathbb{N}_0$ and $p_i=0$ for all $i$. PDE results ----------- The relationship between the uniformly degenerate elliptic operators and polyhomogeneity has been extensively studied in [@Mazzeo-Edge]; see also [@AnderssonChrusciel-Dissertationes], [@WAH-Preliminary], [@AHEM-Preliminary] for studies focusing on operators arising in the study of the Einstein constraint equations. It this paper we make use of the following result, which is a consequence of Proposition 6.3 of [@AHEM-Preliminary] and Proposition 6.4 of [@WAH-Preliminary]. \[SolveGeneric\] Suppose that $(M,g)$ is a smoothly conformally compact asymptotically hyperbolic $3$-manifold. 1. \[SolveGenericVL\] If $Y$ is a vectorfield on $M$ which extends smoothly to ${\overline}M$, then the solution $W$ to $$L_g W = Y$$ satisfies $W\in\rho^3 C^0_{\textup{\normalfont phg}}({\overline}M)$ and $\mathcal D_gW \in C^0_{\textup{\normalfont phg}}({\overline}M)$. 2. \[SolveGenericLich\] For any function $A\in \rho^2 C^\infty({\overline}M)$, there exists a unique positive solution $\phi\in C^2_{\textup{\normalfont phg}}({\overline}M)$ to $$\Delta_g\phi = \frac18 R[g] \phi - A \phi^{-7} + \frac{3}{4}\phi^5, \qquad \left.\phi\right|_{\partial M} =1.$$ Furthermore, if $R[g]+6 = \mathcal O(\rho^2)$ then $\phi -1= \mathcal O(\rho^2)$. Boundary regularity {#bdry:sec} ------------------- Even if $g$ is smoothly conformally compact and $f$ extends smoothly to ${\overline}M$, solutions to $\mathcal P[g]u = f$ may not extend smoothly to ${\overline}M$. To understand why this is the case, and to understand those circumstances where $u$ [*does*]{} extend smoothly to ${\overline}M$, we examine more closely the relationship between $\mathcal{P}$ and its indicial map $I_s(\mathcal{P})$. For a more general treatment of the subject the reader is referred to [@Mazzeo-Edge]; see also [@AnderssonChrusciel-Dissertationes], In background coordinates $(\rho,\theta^1,\theta^2)=(\theta^0,\theta^1,\theta^2)$ near $\partial M$ (see [@WAH-Preliminary], [@Lee-FredholmOperators]), we have $$\mathcal P = a^{ij}(\rho\partial_i)(\rho\partial_j) + b^i(\rho\partial_i) + c,$$ where the matrix-valued functions $a^{ij}$, $b^i$, and $c$ extend smoothly to $\rho=0$. Computing in these coordinates one sees that $$I_s(\mathcal P) {\overline}u = \rho^{-s} \mathcal P (\rho^s {\overline}u)\big|_{\rho =0}=({\overline}a^{\rho\rho} s^2 + {\overline}b^\rho s+ {\overline}c) {\overline}u,$$ where ${\overline}a^{\rho\rho} = a^{\rho\rho}\big|_{\rho =0}$, ${\overline}b^\rho= b^\rho\big|_{\rho =0}$, and ${\overline}c = c\big|_{\rho =0}$ are smooth (matrix-valued) functions of $(\theta^1,\theta^2)$. As in [@Mazzeo-Edge], we define the [[***indicial operator***]{}]{} $I(\mathcal P)$ to be the unique dilation-invariant operator on $\partial M\times (0,\infty)$ such that $$I(\mathcal P) (\rho^s {\overline}u) = \rho^{s}I_s(\mathcal P){\overline}u$$ for all smooth sections ${\overline}u$ of $E\big|_{\partial M}$. Thus $$\label{IndicialFull} I(\mathcal P)(\rho^s (\log\rho)^p{\overline}u) = \sum_{k=0}^p \binom{p}{k}\rho^s (\log\rho)^{p-k} I_s^{(k)}(\mathcal P){\overline}u,$$ where $I_s^{(k)}(\mathcal P) = \frac{d^k}{ds^k}I_s(\mathcal P)$. In coordinates we have $$I(\mathcal P) = {\overline}a^{\rho\rho}(\rho\partial_\rho)^2 + {\overline}b^\rho(\rho\partial_\rho) + {\overline}c,$$ with ${\overline}a^{\rho\rho}$, ${\overline}b^{\rho}$ and ${\overline}c$ as above. It should be noted that $I(\mathcal P)$ can be extended to a differential operator $\mathcal I(\mathcal P) u = I(\mathcal P)(\varphi u)$ on $M$ by means of a cut-off function $\varphi$ supported in a collar neighborhood of $\partial M$. We furthermore set $\mathcal R = \mathcal P - \mathcal I(\mathcal P)$. Careful examinations of coordinate expressions for $\mathcal{P}$, $I(\mathcal P)$ and $\mathcal R$ yield the following: \[lemma:I-and-R\] Suppose $(M,g)$ is a smoothly conformally compact asymptotically hyperbolic manifold and that $\mathcal P$ satisfies Assumption \[Assume-P\]. Then for any $\delta\in \mathbb R$ we have 1. $\mathcal I(\mathcal P)\colon \mathcal B^{\textup{\normalfont phg}}_\delta(M) \to \mathcal B^{\textup{\normalfont phg}}_\delta(M)$ and 2. $\mathcal R\colon \mathcal B^{\textup{\normalfont phg}}_\delta(M) \to \mathcal B^{\textup{\normalfont phg}}_{\delta+1}(M)$. This lemma can be interpreted as saying that $\mathcal I(\mathcal P)$ is an approximation of $\mathcal{P}$ near $\partial M$. It is crucial to notice that $I(\mathcal P)$ is an operator of Cauchy-Euler type. The method advertised in entry-level courses for solving a constant coefficient Cauchy-Euler ODE such as $$\label{Math235} {\overline}a(\rho\partial_\rho)^2 u + {\overline}b(\rho\partial_\rho)u + {\overline}c u=f$$ involves studying the roots $s_1$ and $s_2$ of the associated characteristic polynomial equation $${\overline}a s^2 + {\overline}b s + {\overline}c =0.$$ In the PDE setting, this corresponds to a study of characteristic exponents as defined in §\[DiffOpSec\]. Typical solutions to the ODE have expansions in terms of powers of $\rho$, where the exponents present are the same as the exponents in the expansion of $f$, as well as the roots $s_i$. However, when the expansion of $f$ includes $\rho^{s_i}$, we have a resonance that leads to the presence of terms of the form $\rho^{s_i} \log\rho$ in the expansion of the solution $u$. Further resonances arise when $s_1=s_2$, in which case the two homogeneous solutions are $\rho^{s_1}$ and $\rho^{s_1}\log\rho$. The situation in the case of a (self-adjoint, geometric, elliptic) PDE in asymptotically hyperbolic setting is extremely similar to the ODE case. We now present conditions which ensure that no resonances, and thus no log terms, occur. The proofs presented below are inspired by computations done in [@AnderssonChrusciel-Obstructions]. \[prop:ForcingSmoothness\] Let $(M,g)$ be an asymptotically hyperbolic manifold that is smoothly conformally compact. Suppose $\mathcal P = \mathcal P[g]$ acts on tensors of weight $r$ and satisfies Assumption \[Assume-P\], and let $\mu$ denote the maximum real part of the characteristic exponents of $\mathcal{P}$. If $u\in \mathcal B_{\textup{\normalfont phg}}(M)$ is such that 1. $\mathcal P u$ extends to a tensor field in $C^\infty({\overline}M)$, and 2. there exists $\delta >\mu$ such that $u\in C^\infty({\overline}M)+\mathcal B_{\delta+r}^{\textup{\normalfont phg}}(M)$, then $u$ extends to a tensor field in $C^\infty({\overline}M)$. Without loss of generality we may assume $u\in \mathcal B_{\delta+r}^{\textup{\normalfont phg}}(M)$. By Remark \[PhgSplitting\] we may then assume that $u$ has polyhomogeneous expansion with ${\mathrm{Re}}{(s_i - r)} >\mu$ for all $i$. Let $\{\delta_j\}_{j=0}^\infty$ be the strictly increasing sequence listing the elements of ${\mathrm{Re}}\{s_i\}$. It suffices to show that for each $j\in \mathbb{N}_0$ there exists ${\overline}u_j \in C^\infty({\overline}M)$ and $u_j \in \mathcal B^{\textup{\normalfont phg}}_{\delta_j}(M)$ such that $u = {\overline}u_j + u_j$; we do so inductively. When $j=0$ there is nothing to prove as we may set ${\overline}u_0 =0$. Thus we assume for some $j\geq 0$ that $u = {\overline}u_j + u_j$ as above. Let $$w_j = \sum_{{\mathrm{Re}}{(s_i)}= \delta_j} \sum_{p=0}^{p_i} \rho^{s_i-r} (\log\rho)^p {\overline}u_{ip}$$ and define $u_{j+1} = u_j - w_j$; note that $u_{j+1} \in \mathcal B^{\textup{\normalfont phg}}_{\delta_{j+1}}(M)$ as desired. From Lemma \[lemma:I-and-R\] we have that $$\label{IndicialAux} \mathcal I(\mathcal P) w_j \in C^\infty({\overline}M) + \mathcal B^{\textup{\normalfont phg}}_{\delta^\prime}(M), \quad \delta^\prime >\delta_j.$$ On the other hand, a direct computation in a collar neighborhood of the boundary $\partial M$ shows that $$\mathcal I(\mathcal P) w_j = \sum_{{\mathrm{Re}}{(s_i)}= \delta_j} \sum_{p=0}^{p_i} \rho^{s_i-r} (\log\rho)^p{\overline}w_{ip},$$ where by we have $$\label{IndicialNotSoFull} {\overline}w_{ip_i}=I_{s_i-r}(\mathcal P) {\overline}u_{ip_i},\quad {\overline}w_{i(p_i-1)}=I_{s_i-r}(\mathcal P) {\overline}u_{i(p_i-1)}+p_i I_{s_i-r}^{(1)}(\mathcal P) {\overline}u_{ip_i},$$ etc. In view of Remark \[PhgSplitting\] it follows that each exponent $s_i-r$ in the expansion of $w_j$ is a non-negative integer and that $I_{s_i-r}(\mathcal P) {\overline}u_{ip_i}={\overline}w_{ip_i}= 0$ whenever $p_i\neq 0$. However, since ${\mathrm{Re}}{(s_i - r)} >\mu$ we can only have $I_{s_i-r}(\mathcal P) {\overline}u_{ip_i}= 0$ if ${\overline}u_{ip_i}=0$. Thus $p_i=0$, and the proof of our induction step is complete. For simplicity, we now restrict attention to a special class of operators, which includes those arising in the conformal method. \[Assume-L\] Suppose $(M,g)$ is an asymptotically hyperbolic manifold, and that $\mathcal P=\mathcal P[g]$ is a geometric operator acting on sections of tensor bundle $E$ and satisfying Assumption \[Assume-P\]. We furthermore assume that the indicial operator $I_s(\mathcal P)$ is a product of a polynomial $p(s)$ and an isomorphism of $E\big|_{\partial M}$, where $p(s)$ has simple integer roots. \[prop:Linear-L-Smooth\] Let $(M,g)$ be an asymptotically hyperbolic manifold that is smoothly conformally compact. Suppose $\mathcal P = \mathcal P[g]$ acts on tensor field of weight $r$ and satisfies Assumption \[Assume-L\]. Let $\mu$ denote the highest characteristic exponent of $\mathcal{P}$. If $u\in \mathcal B^{\textup{\normalfont phg}}_{\mu+r}(M)$ satisfies $\mathcal P u \in C^\infty({\overline}M)$ and $\mathcal Pu \in \mathcal B_{\delta+r}(M)$ for some $\delta>\mu\ge 0$, then $u$ extends to a smooth tensor field on ${\overline}M$. Since $u\in \mathcal B^{\textup{\normalfont phg}}_{\mu+r}(M)$, it admits an expansion with ${\mathrm{Re}}{(s_i-r)}\geq \mu$. Let $$\label{rndwexpansion} w = \sum_{{\mathrm{Re}}{(s_i-r)}= \mu} \sum_{p=0}^{p_i} \rho^{s_i-r} (\log\rho)^p {\overline}u_{ip}.$$ From Lemma \[lemma:I-and-R\] we have $$\mathcal I(\mathcal P) w \in \mathcal B^{\textup{\normalfont phg}}_{\delta^\prime+r}(M) \text{\ \ for some\ \ }\delta^\prime >\mu.$$ The computation and Remark \[PhgSplitting\] now imply that $I_{s_i-r}(\mathcal P) {\overline}u_{ip_i}=0$ and, if $p_i\ge 1$, that $$\label{moreaux} I_{s_i-r}(\mathcal P) {\overline}u_{i(p_i-1)}+p_i I_{s_i-r}^{(1)}(\mathcal P) {\overline}u_{ip_i}=0.$$ Therefore, the only non-vanishing term in the expansion has to correspond to $s_i-r=\mu$ which, by our assumptions, is a nonnegative integer. Furthermore, we must have $p_i=0$ because otherwise contradicts Assumption \[Assume-L\]. Thus $w$ extends smoothly to ${\overline}M$ and our result is now immediate from Proposition \[prop:ForcingSmoothness\]. We conclude this section with a regularity result for semilinear scalar equations of the form $\mathcal P u = f(u)$, where $f$ satisfies the following. \[Assume-F\] We assume that $f$ is a smooth real function on $M\times I$ where $0\in I$ is an open interval. Furthermore, we assume that on a neighborhood of zero $f$ is given by an absolutely and uniformly convergent power series $$f(x,u) = \sum_{l=0}^\infty a_l(x) u^l$$ with functions $a_0, a_1 \in \rho C^\infty({\overline}M)$ and $a_l\in C^\infty({\overline}M)$ for $l\geq 2$. In what follows we simply write $f(u)$ for $f(\cdot, u(\cdot))$. \[fcondition\] If $u\in \rho C^\infty({\overline}M)+\mathcal{B}^{\textup{\normalfont phg}}_\delta(M)$ with $\delta>1$, and if $f$ satisfies Assumption \[Assume-F\] then $f(u)\in \rho C^\infty({\overline}M)+\mathcal{B}^{\textup{\normalfont phg}}_{\delta+1}(M)$. \[prop:SemilinearSmoothness\] Let $(M,g)$ be an asymptotically hyperbolic manifold that is smoothly conformally compact and suppose $\mathcal P = \mathcal P[g]$ is an elliptic operator acting on functions and satisfying Assumption \[Assume-L\]. Let $\mu$ denote the largest characteristic exponent of $\mathcal P$. Furthermore, let $f$ be a function satisfying Assumption \[Assume-F\]. Suppose that $\mathcal P u = f(u)$, where $u \in \mathcal B^{\textup{\normalfont phg}}_\mu(M)$ and $f(u) \in \mathcal B_\delta(M)$ for some $\delta>\mu$. Then $u$ extends to a function in $C^\infty({\overline}M)$. Since $u\in \mathcal B^{\textup{\normalfont phg}}_{\mu}(M)$, it admits an expansion with ${\mathrm{Re}}{(s_i)}\geq \mu$. Note that $\mu \geq 1$, as a consequence of the fact that the set of the characteristic exponents of $\mathcal{P}$ is symmetric about ${\mathrm{Re}}{(s)}=1$ (cf. Corollary 4.5 in [@Lee-FredholmOperators]). Furthermore, by Assumption \[Assume-L\] we have that $\mu$ is an integer. As in the proof of Proposition \[prop:Linear-L-Smooth\] we consider the function $$w = \sum_{{\mathrm{Re}}{(s_i)}= \mu} \sum_{p=0}^{p_i} \rho^{s_i} (\log\rho)^p {\overline}u_{ip}.$$ From Lemma \[lemma:I-and-R\] and the assumption that $f(u) \in \mathcal B_\delta(M)$ for some $\delta>\mu$ we have $$I(\mathcal P)w\in \mathcal B^{\textup{\normalfont phg}}_{\delta'}(M) \quad\text{ for some }\quad \delta' >\mu.$$ Arguing as in the proof of Proposition \[prop:Linear-L-Smooth\] we obtain $s_i=\mu$ and $p_i=0$ for all $i$ in the above expression for $w$. Thus $$u = \rho^\mu {\overline}u_{\mu 0} + v\in \rho C^\infty({\overline}M)+\mathcal B^{\textup{\normalfont phg}}_{\delta''}(M) \quad \text{ for some }\quad \delta'' >\mu.$$ It remains to establish smoothness of the function $v$. We do so by using the inductive argument from the proof of Proposition \[prop:ForcingSmoothness\] to show that for each $j$ there exist ${\overline}v_j \in \rho C^\infty({\overline}M)$ and $v_j \in \mathcal B_{\delta_j}^{\textup{\normalfont phg}}(M)$ such that $v = {\overline}v_j + v_j$. The inductive step relies on $$\mathcal I(\mathcal P) v_j = f({\overline}u + {\overline}v_j + v_j) - \mathcal P{\overline}u - \mathcal P {\overline}v_j - \mathcal R v_j\in \rho C^\infty({\overline}M)+\mathcal B^{\textup{\normalfont phg}}_{\delta_j+1}(M),$$ which in turn is a consequence of Remark \[fcondition\]. The free data {#FreeData} ============= We now commence the proof of Theorem \[Density\], and assume that $(g,K)$ is a polyhomogeneous constant-mean-curvature asymptotically hyperbolic initial data set. In this section we construct a family of free data $(\lambda_{\varepsilon}, \mu_{\varepsilon})$, and subsequently establish several estimates for geometric quantities and differential operators associated to the family of metrics $\lambda_{\varepsilon}$. It is important that these estimates are uniform in ${\varepsilon}>0$, in order that they lead to the convergence portion of Theorem \[Density\]. It is our convention that, unless otherwise stated, all constants are independent of ${\varepsilon}$, provided ${\varepsilon}$ is sufficiently small. Construction of the free data ----------------------------- In order to construct a family of free data, we define a family of smooth cutoff functions. Let $\chi\colon \mathbb R \to [0,1]$ be a smooth, decreasing function such that $$\chi(x) = 1\text{ if }x\geq 2 \quad \text{ and }\quad \chi(x) = 0 \text{ if } x \leq 1.$$ For ${\varepsilon}\in (0,1)$ define $\chi_{\varepsilon}\colon M \to [0,1]$ by $ \chi_{\varepsilon}= \chi( { \rho}/{{\varepsilon}}). $ We note that $\operatorname{supp}\chi_{\varepsilon}\subset \{\rho>{\varepsilon}\}$ and that $\chi_{\varepsilon}=1$ if $\rho\geq2{\varepsilon}$. Furthermore, ${d}\chi_{\varepsilon}= \chi^\prime(\rho/{\varepsilon}) {\varepsilon}^{-1}{d}\rho$ is supported in $\{{\varepsilon}\leq \rho \leq 2{\varepsilon}\}$. Thus, since ${d}\rho\in C^{k,\alpha}_1(M)$ for all $k\geq 1$ and $\alpha\in (0,1)$, we see that $\chi_{\varepsilon}\in C^{k,\alpha}(M)$, with bound independent of ${\varepsilon}$: $$\label{chistuff} \|\chi_{\varepsilon}\|_{C^{k,\alpha}(M)}\leq C.$$ Let ${\overline}b$ be the smooth metric induced on $\partial M$ by ${\overline}g = \rho^2 g$. We define a preferred background metric $h$ by choosing ${\overline}h$ to be a smooth metric on ${\overline}M$ such that in a neighborhood of $\partial M$ we have $$\label{DefineBoundaryMetric} {\overline}h = {d}\rho \otimes {d}\rho + {\overline}b$$ and setting $h = \rho^{-2}{\overline}h$. Let ${\overline}\nabla$ be the Levi-Civita connection associated to ${\overline}h$, and note that in the neighborhood of $\partial M$ where holds we have $\Delta_{{\overline}h}\rho =0$. We define, for sufficiently small ${\varepsilon}>0$, the smooth metrics ${\overline}\lambda_{\varepsilon}$ on ${\overline}M$ by $$\label{DefineLambdaBar} {\overline}\lambda_{\varepsilon}:= \chi_{\varepsilon}\, {\overline}g + (1-\chi_{\varepsilon}) {\overline}h.$$ Setting $\lambda_{\varepsilon}= \rho^{-2}{\overline}\lambda_{\varepsilon}$, we define the family of free data $(\lambda_{\varepsilon}, \mu_{\varepsilon})$ by $$\lambda_{\varepsilon}:= \rho^{-2}{\overline}\lambda_{\varepsilon}\quad\text{ and }\quad \mu_{\varepsilon}:= \chi_{\varepsilon}\Sigma = \chi_{\varepsilon}\rho^{-1} {\overline}\Sigma.$$ We emphasize that $\lambda_{\varepsilon}$ are each a smoothly conformally compact asymptotically hyperbolic metric on $M$. Estimates for $\lambda_{\varepsilon}$ ------------------------------------- We note the following properties of the metrics $\lambda_{\varepsilon}$. \[lemma:IntrinsicMetricEstimate\] Let $k\geq 0$ and $\alpha\in(0,1)$. We have $$\| g - \lambda_{\varepsilon}\|_{C^{k,\alpha}_1(M)}\leq C \quad \text{ and } \quad \| g - \lambda_{\varepsilon}\|_{C^{k,\alpha}(M)}\leq C{\varepsilon}.$$ This furthermore implies that for sufficiently small ${\varepsilon}>0$ we have $\| g^{-1} - \lambda_{\varepsilon}^{-1}\|_{C^{k,\alpha}(M)}\leq C{\varepsilon}$. Since ${\overline}h$ agrees with ${\overline}g$ at $\rho=0$, we may apply Lemma \[BoundaryLemma\] to conclude that ${\overline}h - {\overline}g \in C^{k,\alpha}_3(M)$ with bound independent of ${\varepsilon}$. Also recall , which shows that the functions $1-\chi_{\varepsilon}$ are uniformly bounded in $C^{k,\alpha}(M)$. Our first claim now follows from the identity $\lambda_{\varepsilon}- g = \rho^{-2}(1-\chi_{\varepsilon})({\overline}h - {\overline}g)$. Since the support of $\lambda_{\varepsilon}- g$ is in $\{\rho \leq 2{\varepsilon}\}$, the first estimate implies the second. Finally, the estimate for the inverses comes from the second estimate applied to the series expansion $\lambda_{\varepsilon}^{-1} - g^{-1}$, centered at $g$. The following is immediate from the fact that ${\overline}\lambda_{\varepsilon}={\overline}h={d}\rho\otimes {d}\rho+{\overline}b$ in a collar neighborhood of the boundary. \[BoundaryH\] We have $ {\mathcal{H}}_{{\overline}\lambda_{\varepsilon}}(\rho) =0 $ and $ \lvert{d}\rho\rvert^2_{{\overline}\lambda_{\varepsilon}} =1 $ along $\partial M$. We now obtain estimates on the scalar curvature of the metrics $\lambda_{\varepsilon}$. We first note that $$\label{FirstScalarLambda} R[\lambda_{\varepsilon}] +6 = -6(|{d}\rho|^2_{{\overline}\lambda_{\varepsilon}} -1) +4\rho \Delta_{{\overline}\lambda_{\varepsilon}}\rho +\rho^2 R[{\overline}\lambda_{\varepsilon}].$$ In a neighborhood of $\partial M$, where $\lambda_{\varepsilon}= \rho^{-2}{\overline}h$, we have $$\label{CompactifyR} R[\lambda_{\varepsilon}] + 6 = \rho^2 R[{\overline}h] \in C^{k,\alpha}_2(M),$$ due to the fact that $|{d}\rho|^2_{{\overline}h} \equiv 1$ and $\Delta_{{\overline}h}\rho \equiv 0$ near $\partial M$. However, we do not have a uniform estimate on $R[\lambda_{\varepsilon}] + 6 $ in $C^{k,\alpha}_2(M)$. Rather, we obtain the following. \[prop:ScalarCurvature\] Let $k\geq 0$ and $\alpha\in(0,1)$. For sufficiently small ${\varepsilon}>0$ we have $$\| R[\lambda_{\varepsilon}] - R[g]\|_{C^{k,\alpha}_1(M)}\leq C \quad\text{ and }\quad \| R[\lambda_{\varepsilon}] - R[g]\|_{C^{k,\alpha}(M)}\leq C{\varepsilon}.$$ We make use of the formula , analyzing each term on the right side. The scalar curvature $R[{\overline}\lambda_{\varepsilon}]$ is the sum of contractions of terms of the form $$({\overline}\lambda_{\varepsilon})^{-1}\otimes ({\overline}\lambda_{\varepsilon})^{-1}\otimes ({\overline}\lambda_{\varepsilon})^{-1}\otimes {\overline}\nabla\,{\overline}\lambda_{\varepsilon}\otimes {\overline}\nabla\,{\overline}\lambda_{\varepsilon}\quad\text{ and }\quad ({\overline}\lambda_{\varepsilon})^{-1}\otimes ({\overline}\lambda_{\varepsilon})^{-1}\otimes {\overline}\nabla{}^2{\overline}\lambda_{\varepsilon};$$ The scalar curvature of ${\overline}g$ is comprised of analogous terms. From Lemma \[lemma:IntrinsicMetricEstimate\] we have $$\|({\overline}\lambda_{\varepsilon})^{-1} - ({\overline}g)^{-1}\|_{C^{k,\alpha}_{-2}(M)} \leq \|\lambda_{\varepsilon}^{-1} - g^{-1}\|_{C^{k,\alpha}(M)}\leq C.$$ Likewise, both $\|{\overline}\nabla({\overline}\lambda_{\varepsilon}- {\overline}g)\|_{C^{k,\alpha}_3(M)}$ and $\|{\overline}\nabla^2({\overline}\lambda_{\varepsilon}- {\overline}g)\|_{C^{k,\alpha}_3(M)}$ can be bounded by $$\|{\overline}\lambda_{\varepsilon}- {\overline}g\|_{C^{k+2,\alpha}_3(M)} \leq \|\lambda_{\varepsilon}- g\|_{C^{k+2,\alpha}_1(M)}\leq C.$$ We now conclude that $$\|\rho^2(R[{\overline}\lambda_{\varepsilon}]- R[{\overline}g])\|_{C^{k,\alpha}_{1}(M)} \leq C.$$ Similar reasoning, using that ${d}\rho \in C^{k,\alpha}_1(M)$ and ${\overline}\nabla{d}\rho\in C^{k,\alpha}_2(M)$, yields $$\|\rho(\Delta_{{\overline}\lambda_{\varepsilon}}\rho- \Delta_{{\overline}g}\rho)\|_{C^{k,\alpha}_1(M)} \leq C.$$ Finally, we estimate the function $$\eta = (|{d}\rho|^2_{{\overline}\lambda_{\varepsilon}}-1) - (|{d}\rho|^2_{{\overline}g}-1) = (({\overline}\lambda_{\varepsilon})^{-1} - ({\overline}g)^{-1})({d}\rho, {d}\rho).$$ Lemma \[lemma:IntrinsicMetricEstimate\] implies that $\|\eta\|_{C^{2,\alpha}(M)}$ and $\|{\overline}\nabla \eta\|_{C^{1,\alpha}_1(M)}$ are uniformly bounded in ${\varepsilon}$. Since $\eta$ vanishes at $\rho =0$ we may apply Lemma \[BoundaryLemma\] to conclude that $\|\eta\|_{C^{2,\alpha}_1(M)}$ is uniformly bounded in ${\varepsilon}$. This establishes the first estimate in the lemma. The second estimate follows from the first due to the fact that $\lambda_{\varepsilon}$ agrees with $g$ for $\rho\geq 2{\varepsilon}$. Estimates for geometric operators defined by $\lambda_{\varepsilon}$ -------------------------------------------------------------------- Here we record several consequences of Proposition \[prop:GeometricOperatorsContinuous\] and Lemma \[lemma:IntrinsicMetricEstimate\]. \[ell-stuff\] For any $k\geq 0$ and $\alpha\in [0,1)$, and for any $\delta\in \mathbb R$, there is a constant $C>0$, independent of sufficiently small ${\varepsilon}$, such that the following hold: 1. \[DivergenceEstimate\] For any tensor field $u\in C^{k+1,\alpha}_\delta(M)$ we have $$\| \operatorname{div}_{\lambda_{\varepsilon}} u \|_{C^{k,\alpha}_\delta(M)} \leq C \| u\|_{C^{k+1,\alpha}_\delta(M)}.$$ 2. \[Depsilon:est\] For any vector field $X\in C^{k+1,\alpha}_\delta(M)$ we have $$\|\mathcal{D}_{\lambda_{\varepsilon}}X\|_{C^{k,\alpha}_\delta(M)} \leq C \|X\|_{C^{k+1,\alpha}_\delta(M)}.$$ For the first claim, we note that the estimate holds with $\lambda_{\varepsilon}$ replaced by $g$. Since $$\| \operatorname{div}_{\lambda_{\varepsilon}}u\|_{C^{k,\alpha}_\delta(M)} \leq \| \operatorname{div}_{\lambda_{\varepsilon}}u- \operatorname{div}_{g}u\|_{C^{k,\alpha}_\delta(M)} +\| \operatorname{div}_{g}u\|_{C^{k,\alpha}_\delta(M)}$$ we may invoke Proposition \[prop:GeometricOperatorsContinuous\] and Lemma \[lemma:IntrinsicMetricEstimate\] to obtain the desired estimate. The proof of the second claim follows from analogous reasoning. Due to Proposition \[MappingProperties\] the vector Laplacian $L_{\lambda_{\varepsilon}}:C^{k+2,\alpha}_\delta(M) \to C^{k,\alpha}_\delta(M)$ is invertible for each ${\varepsilon}>0$, $k\geq 0$, $\alpha\in (0,1)$ and $\delta \in (-1,3)$. In particular, there exist constants $C_{{\varepsilon}}$, depending on ${\varepsilon}$, such that $\|X\|_{C^{k+2,\alpha}_\delta(M)} \leq C_{{\varepsilon}} \| L_{\lambda_{\varepsilon}}X\|_{C^{k,\alpha}_\delta(M)}$. The linearized Licherowicz operator that appears in §\[SolveConstraint\] is similarly invertible for each $\lambda_{\varepsilon}$. We now show that the invertibility estimates can be made uniform in ${\varepsilon}$. \[UniformInvertVL\] Let $k\geq 0$, $\alpha\in (0,1)$, and $\delta\in (-1,3)$. Furthermore, let the functions $\kappa, \kappa_{\varepsilon}\in C^{k,\alpha}_1(M)$ be such that $\|\kappa_{\varepsilon}-\kappa\|_{C^{k,\alpha}(M)}\leq C{\varepsilon}$ and $3+\kappa\geq 0$. Then there exists a constant $C>0$ such that: 1. for all vector fields $X\in C^{k+2,\alpha}_\delta(M)$ and for all sufficiently small ${\varepsilon}>0$ we have $$\|X\|_{C^{k+2,\alpha}_\delta(M)} \leq C\| L_{\lambda_{\varepsilon}}X\|_{C^{k,\alpha}_\delta(M)},$$ and 2. for all functions $u\in C^{k+2,\alpha}_\delta(M)$ and for all sufficiently small ${\varepsilon}>0$ we have $$\|u\|_{C^{k+2,\alpha}_\delta(M)} \leq C\| \Delta_{\lambda_{\varepsilon}}u-(3+\kappa_{\varepsilon})u\|_{C^{k,\alpha}_\delta(M)}.$$ From Proposition \[MappingProperties\](\[Prop4a\]) we have $$\label{VL-Intermediate} \begin{aligned} \|X\|_{C^{k+2,\alpha}_\delta(M)} &\leq C \|L_gX\|_{C^{k,\alpha}_\delta(M)} \\ &\leq C\left( \|L_gX-L_{\lambda_{\varepsilon}}X\|_{C^{k,\alpha}_\delta(M)} +\|L_{\lambda_{\varepsilon}}X\|_{C^{k,\alpha}_\delta(M)} \right). \end{aligned}$$ From Proposition \[prop:GeometricOperatorsContinuous\] we have $$\|L_gX-L_{\lambda_{\varepsilon}}X\|_{C^{k,\alpha}_\delta(M)} \leq C \|g-\lambda_{\varepsilon}\|_{C^{k+2,\alpha}(M)} \|X\|_{C^{k+2,\alpha}_\delta(M)}.$$ Making use of Lemma \[lemma:IntrinsicMetricEstimate\], we see that this term may be absorbed in to the left side of when ${\varepsilon}>0$ is small; this proves the first invertibility estimate. The second estimate follows from a similar argument applied to Proposition \[MappingProperties\](\[Prop4b\]); the details are left to the reader. Estimates for $\mu_{\varepsilon}$ --------------------------------- \[EstimateMuEpsilon\] Let $k\geq 0$ and $\alpha\in (0,1)$. There exists a constant $C>0$ such that $$\begin{gathered} \| \mu_{\varepsilon}- \Sigma \|_{C^{k,\alpha}_1(M)} \leq C, \qquad \| \mu_{\varepsilon}- \Sigma \|_{C^{k,\alpha}(M)} \leq C{\varepsilon}, \\ \|\operatorname{div}_{\lambda_{\varepsilon}} \mu_{\varepsilon}\|_{C^{k,\alpha}_1(M)} \leq C, \qquad \| \operatorname{div}_{\lambda_{\varepsilon}}\mu_{\varepsilon}\|_{C^{k,\alpha}(M)} \leq C{\varepsilon}.\end{gathered}$$ Furthermore, $\operatorname{div}_{\lambda_{\varepsilon}}\mu_{\varepsilon}\in C^{k,\alpha}_\delta(M)$ for all $\delta\in \mathbb R$. First recall and note that ${\overline}\Sigma \in C^{1,\alpha}_2(M)$; thus $\mu_{\varepsilon}= \rho^{-1}\chi_{\varepsilon}{\overline}\Sigma$ is uniformly bounded in $C^{1,\alpha}_1(M)$, which implies the first estimate. The second estimate follows from this and the fact that the support of $\mu_{\varepsilon}- \Sigma$ is contained in the region where $\rho \leq 2{\varepsilon}$. The uniform bound on $\mu_{\varepsilon}$ in $C^{1,\alpha}_1(M)$, together with Proposition \[ell-stuff\], implies that $\operatorname{div}_{\lambda_{\varepsilon}}\mu_{\varepsilon}$ is uniformly bounded in $C^{0,\alpha}_1(M)$. Since $\lambda_{\varepsilon}$ agrees with $g$ and $\mu_{\varepsilon}$ agrees with $\rho^{-1}{\overline}\Sigma$ for $\rho \geq 2{\varepsilon}$, we see from that $\operatorname{div}_{\lambda_{\varepsilon}}\mu_{\varepsilon}$ is supported in the region ${\varepsilon}\leq \rho \leq 2{\varepsilon}$. This, together with the third estimate, yields the fourth estimate. Finally, the fact that $\mu_{\varepsilon}$ is compactly supported implies that $\operatorname{div}_{\lambda_{\varepsilon}}\mu_{\varepsilon}\in C^{0,\alpha}_\delta(M)$ for all $\delta$. Construction of approximating initial data {#SolveConstraint} ========================================== Analysis of the conformal momentum constraint --------------------------------------------- For each free data set $(\lambda_{\varepsilon},\mu_{\varepsilon})$, Propositions \[MappingProperties\] and \[SolveGeneric\] guarantee that there exists a unique $W_{\varepsilon}\in \rho^3 C^0_{\textup{\normalfont phg}}({\overline}M)$ such that $$\label{VLepsilon} L_{\lambda_{\varepsilon}}W_{\varepsilon}= -\operatorname{div}_{\lambda_{\varepsilon}}\mu_{\varepsilon}$$ and $\mathcal D_{\lambda_{\varepsilon}} W_{\varepsilon}\in C^0_{\textup{\normalfont phg}}({\overline}M)$. By Proposition \[UniformInvertVL\] there is a constant $C$ such that for all sufficiently small ${\varepsilon}>0$ we have $$\|W_{\varepsilon}\|_{C^{k+2,\alpha}_\delta(M)} \leq C \| \operatorname{div}_{\lambda_{\varepsilon}}\mu_{\varepsilon}\|_{C^{k,\alpha}_\delta(M)}$$ for $\delta=0,1$. The estimates for $\mu_{\varepsilon}$ in Lemma \[EstimateMuEpsilon\] now imply the following estimates for the solutions $W_{\varepsilon}$ to . \[EstimateWepsilon\] Let $k\geq 0$ and $\alpha\in(0,1)$. There exists a constant $C>0$ such that $$\begin{gathered} \| W_{\varepsilon}\|_{C^{k,\alpha}_1(M)} \leq C, \qquad \| W_{\varepsilon}\|_{C^{k,\alpha}(M)} \leq C{\varepsilon}, \\ \| \mathcal D_{\lambda_{\varepsilon}}W_{\varepsilon}\|_{C^{k,\alpha}_1(M)} \leq C, \qquad \| \mathcal D_{\lambda_{\varepsilon}} W_{\varepsilon}\|_{C^{k,\alpha}(M)} \leq C{\varepsilon}.\end{gathered}$$ We define the tensors $\sigma_{\varepsilon}$ by $$\sigma_{\varepsilon}:= \mu_{\varepsilon}+ \mathcal D_{\lambda_{\varepsilon}} W_{\varepsilon}$$ and record the following consequence of Lemmas \[lemma:IntrinsicMetricEstimate\], \[EstimateMuEpsilon\], and \[EstimateWepsilon\]. \[EstimateSigmaNorms\] Let $k\geq 0$, $\alpha\in (0,1)$ and let ${\varepsilon}>0$ be sufficiently small. The function $|\sigma_{\varepsilon}|^2_{\lambda_{\varepsilon}}$ is in $C^{k,\alpha}_2(M)$, and satisfies $$\| |\sigma_{\varepsilon}|^2_{\lambda_{\varepsilon}} - |\Sigma|^2_{\lambda_{\varepsilon}} \|_{C^{k,\alpha}_2(M)} \leq C \quad\text{ and }\quad \| |\sigma_{\varepsilon}|^2_{\lambda_{\varepsilon}} - |\Sigma|^2_{\lambda_{\varepsilon}} \|_{C^{k,\alpha}_1(M)} \leq C{\varepsilon}.$$ Finally, we address smoothness of the tensor fields $\sigma_{\varepsilon}$. The strategy is to employ Proposition \[prop:Linear-L-Smooth\] of §\[section:boundary\]. \[VL-NoLogs\] The solution $W_{\varepsilon}$ of and the tensor field $\sigma_{\varepsilon}$ extend smoothly to ${\overline}M$. A direct computation shows that the indicial map $I_s(L_{\lambda_{\varepsilon}})$ is $$I_s(L_{\lambda_{\varepsilon}}) Y = -\frac12\left( Y + \frac13 Y(\rho) \operatorname{grad}_{{\overline}\lambda_{\varepsilon}}\rho \right)(s^2 - 4s).$$ Thus $L_{\lambda_{\varepsilon}}$ satisfies Assumption \[Assume-L\] with the highest characteristic exponent of $\mu=4$. By Proposition \[SolveGeneric\] we have that $W_{\varepsilon}$ is polyhomogeneous, while Proposition \[MappingProperties\] implies $W_{\varepsilon}\in C^k_\delta(M)$ for all $k\geq 0$ and $\delta<3$. Since $L_{\lambda_{\varepsilon}}W_{\varepsilon}=-\operatorname{div}_{\lambda_{\varepsilon}}\mu_{\varepsilon}$ extends smoothly to ${\overline}M$ and since by Lemma \[EstimateMuEpsilon\] $\operatorname{div}_{\lambda_{\varepsilon}}\mu_{\varepsilon}\in C^k_\delta(M)$ for all $k\geq 0$ and all $\delta\in \mathbb{R}$, we are in position to apply Proposition \[prop:Linear-L-Smooth\]. Consequently, $W_{\varepsilon}$ extends smoothly to ${\overline}M$, and thus $\sigma_{\varepsilon}$ does as well. Analysis of the Lichnerowicz equation {#AnalyzeLich} ------------------------------------- From Proposition \[SolveGeneric\] there exists, for each sufficiently small ${\varepsilon}>0$, a [unique]{} positive polyhomogeneous function $\phi_{\varepsilon}\in C^2_{\textup{\normalfont phg}}({\overline}M)$ such that $$\label{LichEpsilon} \begin{gathered} 0=\mathcal N_{\varepsilon}(\phi_{\varepsilon}):= \Delta_{\lambda_{\varepsilon}}\phi_{\varepsilon}-\frac18 R[\lambda_{\varepsilon}] \phi_{\varepsilon}+ \frac18|\sigma_{\varepsilon}|^2_{\lambda_{\varepsilon}} \phi_{\varepsilon}^{-7} - \frac{3}{4}\phi_{\varepsilon}^5, \\ \qquad \left.\phi_{\varepsilon}\right|_{\partial M} =1. \end{gathered}$$ In order to obtain estimates on $\phi_{\varepsilon}-1$ we first show that the constant function $\phi=1$ is an approximate solution of . \[N1-Estimate\] Let $k\geq 0$ and $\alpha\in (0,1)$. For each sufficiently small ${\varepsilon}>0$ we have $\mathcal N_{\varepsilon}(1)\in C^{0,\alpha}_1(M)$ with $$\label{N1Estimate} \|\mathcal N_{\varepsilon}(1)\|_{C^{k,\alpha}_1(M)} \leq C \quad\text{ and }\quad \|\mathcal N_{\varepsilon}(1)\|_{C^{k,\alpha}(M)} \leq C{\varepsilon}.$$ Using we have $$\mathcal N_{\varepsilon}(1) = -\frac18\left(R[\lambda_{\varepsilon}]-R[g]\right) + \frac18\left(|\sigma_{\varepsilon}|^2_{\lambda_{\varepsilon}}- |\Sigma|^2_{\lambda_{\varepsilon}} \right) +\frac18 \left(|\Sigma|^2_{\lambda_{\varepsilon}}-|\Sigma|^2_{g}\right).$$ Estimates are now immediate from Proposition \[prop:ScalarCurvature\], Lemma \[EstimateSigmaNorms\], and Lemma \[lemma:IntrinsicMetricEstimate\]. The linearization of $\mathcal N_{\varepsilon}$ at $\phi =1$ is the operator $$\label{LinearizedN} \mathcal L_{\varepsilon}:= \Delta_{\lambda_{\varepsilon}} - (3+\kappa_{\varepsilon}),$$ where $$\kappa_{\varepsilon}= \frac18 \left( R[\lambda_{\varepsilon}] + 6\right) + \frac78 |\sigma_{\varepsilon}|^2_{\lambda_{\varepsilon}}.$$ We now prove the properties of $\kappa_{\varepsilon}$ needed in order to apply Proposition \[UniformInvertVL\]. To that end we set $\kappa=|\Sigma|^2_g$ and note that $\kappa\in C^{k,\alpha}_2(M)$ for all $k\geq 0$ and $\alpha\in (0,1)$. \[kappa-BasicProperties\] For all $k\geq 0$ and $\alpha\in (0,1)$ and sufficiently small ${\varepsilon}>0$ we have: 1. $\kappa_{\varepsilon}\in C^{k,\alpha}_1(M)$. 2. $\|\kappa_{\varepsilon}-\kappa\|_{C^{k,\alpha}(M)} \leq C{\varepsilon}$. The fact that $\kappa_{\varepsilon}\in C^{k,\alpha}_1(M)$ is immediate from Proposition \[prop:ScalarCurvature\] and Lemma \[EstimateSigmaNorms\]. Using we can express $\kappa_{\varepsilon}-\kappa$ as $$\kappa_{\varepsilon}-\kappa=\frac18 \left( R[\lambda_{\varepsilon}] -R[g]\right) + \frac78 \left(|\sigma_{\varepsilon}|^2_{\lambda_{\varepsilon}}-|\Sigma|^2_g\right).$$ The $C^{k,\alpha}(M)$ estimate on the difference of scalar curvatures follows from Proposition \[prop:ScalarCurvature\], while the remaining estimate follows from Lemmas \[lemma:IntrinsicMetricEstimate\] and \[EstimateSigmaNorms\]. We now see from Proposition \[UniformInvertVL\] that for all $\delta \in (-1,3)$ and sufficiently small ${\varepsilon}>0$ the mapping $$\mathcal L_{\varepsilon}\colon C^{k+2,\alpha}_\delta(M) \to C^{k,\alpha}_\delta(M)$$ defined in is an isomorphism with inverse bounded uniformly in ${\varepsilon}$. We now proceed to obtain estimates for $\phi_{\varepsilon}$ by viewing $\phi_{\varepsilon}-1$ as a fixed point of the map $$\mathcal G_{\varepsilon}\colon u \to -\mathcal L_{\varepsilon}^{-1}\left( \mathcal N_{\varepsilon}(1) + \mathcal Q_{\varepsilon}(u)\right),$$ where $$\begin{aligned} \mathcal Q_{\varepsilon}(u) &:= \mathcal N_{{\varepsilon}}(1+u) - \mathcal N_{\varepsilon}(1) -\mathcal L_{\varepsilon}(u) \\ &= \frac18|\sigma_{\varepsilon}|^2_{\lambda_{\varepsilon}} \left( (1+u)^{-7} -1 + 7u\right) -\frac34 \left( (1+u)^5 -1 -5u\right). \end{aligned}$$ In preparation, we define for each $r_0, r_1>0$, which are not necessarily independent of ${\varepsilon}$ and a fixed integer $k\in \mathbb{N}_0$, the collection of functions $$X(r_0, r_1) = \{ u \in C^{k+2,\alpha}_1(M) \mid \| u\|_{C^{k+2,\alpha}_1(M)} \leq r_1 \text{ and } \| u\|_{C^{k+2,\alpha}(M)} \leq r_0\}.$$ Note that for all $r_0, r_1>0$, the set $X(r_0, r_1)$ is a complete metric space with respect to the norm ${{\left\|\kern-0.24ex\left|{ u } \right|\kern-0.24ex\right\|}}_X := \|u\|_{C^{k+2,\alpha}_1(M)} + \|u\|_{C^{k+2,\alpha}(M)}$. We require the following mapping properties of $\mathcal Q_{\varepsilon}$. \[Q-Mapping\] Let $k\geq 0$ and $\alpha\in (0,1)$. There exists $r_*>0$ and continuous function $F\colon [0,r_*]\times [0,r_*] \to [0,\infty)$, both independent of ${\varepsilon}>0$, such that $F(0,0) =0$ and such that for each $\delta\in [0,1]$ we have $$\| \mathcal Q_{\varepsilon}(u) - \mathcal Q_{\varepsilon}(v)\|_{C^{k,\alpha}_\delta(M)} \leq F\left(\|u\|_{C^{k,\alpha}(M)} , \|v\|_{C^{k,\alpha}(M)}\right)\, \|u-v\|_{C^{k,\alpha}_\delta(M)}$$ for all $u,v\in X(r_0, r_1)$ with $r_0\in [0,r_*]$ and $r_1>0$. In particular $$\| \mathcal Q_{\varepsilon}(u) \|_{C^{k,\alpha}_\delta(M)} \leq F(r_0,0 ) \,\|u\|_{C^{k,\alpha}_\delta(M)}.$$ Set $$H_l(u,v) = u^{l-1}v + u^{l-2}v^2 + \dots + u v^{l-1}.$$ With $$Q_1(u):= u^5 \quad\text{ and }\quad Q_2(u):=u^{-7}$$ we have $$Q_1(u) - Q_1(v) = (u-v) \sum_{l=2}^5 \binom{5}{l} H_l(u,v)$$ and $$Q_2(u) - Q_2(v) = (u-v) \sum_{l=2}^\infty(-1)^l \binom{l+6}{l} H_l(u,v)$$ provided $|u|$ and $|v|$ are less than $1$. The uniform bound on $|\sigma_{\varepsilon}|^2_{\lambda_{\varepsilon}}$ provided by Lemma \[EstimateSigmaNorms\] implies that there are constants $C_*$ and $C_l$, $l\in \mathbb N$, such that $$F(u,v) = C_* \sum_{l=2}^\infty C_l H_l(u,v)$$ converges uniformly and has the desired properties. We now obtain the desired contraction property of $\mathcal G_{\varepsilon}$. There exists constants ${\varepsilon}_*>0$ and $C_*>0$ such that for each ${\varepsilon}\in (0,{\varepsilon}_*]$ the map $\mathcal G_{\varepsilon}$ is a contraction mapping $X(C_*{\varepsilon}, C_*) \to X(C_*{\varepsilon}, C_*)$. Let $u\in X(r_0, r_1)$ and $\delta\in \{0,1\}$. By the uniform invertibility of $\mathcal L_{\varepsilon}^{-1}$ (cf. Proposition \[UniformInvertVL\] and Lemma \[kappa-BasicProperties\]) we have $$\begin{aligned} \| \mathcal G_{\varepsilon}(u)\|_{C^{k+2,\alpha}_\delta(M)} &= \| \mathcal L_{\varepsilon}^{-1} \left( \mathcal N_{\varepsilon}(1) + \mathcal Q_{\varepsilon}(u)\right)\|_{C^{k+2,\alpha}_\delta(M)} \\ &\leq C \|\mathcal N_{\varepsilon}(1)\|_{C^{k,\alpha}_\delta(M)} + C\|\mathcal Q_{\varepsilon}(u)\|_{C^{k,\alpha}_\delta(M)}. \end{aligned}$$ Using Lemma \[N1-Estimate\] and Lemma \[Q-Mapping\] we obtain $$\begin{gathered} \| \mathcal G_{\varepsilon}(u)\|_{C^{k+2,\alpha}(M)} \leq C^\prime {\varepsilon}+ C^{\prime\prime} F(r_0, 0)\|u\|_{C^{k+2,\alpha}(M)} \\ \| \mathcal G_{\varepsilon}(u)\|_{C^{k+2,\alpha}_1(M)} \leq C^\prime + C^{\prime\prime} F(r_0, 0)\|u\|_{C^{k+2,\alpha}_1(M)}\end{gathered}$$ for some constants $C^\prime,C^{\prime\prime}>0$ independent of ${\varepsilon}$. Choosing $C_*>2 C^\prime$ and, using the fact that $F(0,0)=0$, choosing ${\varepsilon}_*$ small enough that $C^{\prime\prime}F(C_*{\varepsilon}_*, 0)<1/2$ ensures that $$\mathcal G_{\varepsilon}\colon X(C_* {\varepsilon},C_*) \to X(C_* {\varepsilon},C_*)$$ whenever ${\varepsilon}\in (0,{\varepsilon}_*]$. We furthermore have $$\| \mathcal G_{\varepsilon}(u)- \mathcal G_{\varepsilon}(v)\|_{C^{k+2,\alpha}_\delta(M)} \leq C \| \mathcal Q_{\varepsilon}(u) - \mathcal Q_{\varepsilon}(v)\|_{C^{k,\alpha}_\delta(M)}.$$ Since $F(0,0)=0$, Lemma \[Q-Mapping\] implies that we can choose ${\varepsilon}_*>0$ such that $\mathcal G_{\varepsilon}$ is a contraction for ${\varepsilon}\in (0,{\varepsilon}_*]$. The contraction property of $\mathcal G_{\varepsilon}$, together with the Banach fixed point theorem, immediately leads to following. \[PhiConverges\] Let $k\geq 0$ and $\alpha\in (0,1)$. There exists ${\varepsilon}_*>0$ and constant $C>0$ such that whenever ${\varepsilon}\in (0,{\varepsilon}_*)$ we have $\phi_{\varepsilon}-1\in C^{k,\alpha}_1(M)$ and $$\| \phi_{\varepsilon}-1\|_{C^{k,\alpha}(M)} \leq C {\varepsilon}.$$ We now analyze the regularity on ${\overline}M$ of solutions $\phi_{\varepsilon}$ of . We do so by writing in terms of the auxiliary variable $$u=\phi_{\varepsilon}- u_0, \quad\text{ where }\quad u_0=1-\frac{1}{24}\rho^2 \operatorname{R}[{\overline}h].$$ This particular change of variable is motivated by the fact that, while Lemma \[N1-Estimate\] shows that the function $1$ is an approximate solution to , the function $u_0$ constitutes a better approximate solution. We make this precise in the following lemma. \[N2-Estimate\] Let $k\geq 0$ and $\alpha\in (0,1)$. For each sufficiently small ${\varepsilon}>0$ we have $\mathcal N_{\varepsilon}(u_0)\in \rho^4C^{\infty}({\overline}M)$. It suffices to perform the computation in the collar neighborhood of the boundary where $\lambda_{\varepsilon}= h$. There we have $\operatorname{R}[h]=-6+\rho^2\operatorname{R}[{\overline}h]$ and $\langle {d}\rho, {d}\operatorname{R}[{\overline}h]\rangle_{{\overline}h}=0$; the latter can be seen as a consequence of the fact that $\operatorname{grad}_{{\overline}h}\rho$ is a Killing vector field in the collar neighborhood of $\partial M$. A direct computation now shows that $$\Delta_h u_0=\rho^2\Delta_{{\overline}h}u_0-\rho\langle {d}\rho, {d}u_0\rangle_{{\overline}h}=\rho^4\Delta_{{\overline}h}\operatorname{R}[{\overline}h]\in \rho^4 C^\infty({\overline}M).$$ On the other hand, Propositions \[SolveGeneric\] and \[VL-NoLogs\] imply $|\sigma_{\varepsilon}|^2_{\lambda_{\varepsilon}} \in \rho^4 C^\infty({\overline}M)$. Using this fact we obtain $$\begin{gathered} \frac{1}{8}\operatorname{R}[h]u_0-\frac{1}{8}|\sigma_{\varepsilon}|^2_{h}u_0^{-7}+\frac{3}{4}u_0^5 \\ =\frac{1}{8}\left(-6+\frac{5}{4}\rho^2\operatorname{R}[{\overline}h]\right)+\frac{3}{4}\left(1-\frac{5}{24}\rho^2\operatorname{R}[{\overline}h]\right)+\rho^4 C^\infty({\overline}M)\in \rho^4 C^\infty({\overline}M).\end{gathered}$$ This completes the proof. \[LichNoLogs\] The solution $\phi_{\varepsilon}$ of extends to a smooth function on ${\overline}M$. We see from Lemma \[N2-Estimate\] that $$\Delta_hu=\frac{1}{8}\operatorname{R}[h]u-\frac{1}{8}|\sigma_{\varepsilon}|^2_h\left((u_0+u)^{-7}-u_0^{-7}\right)+\frac{3}{4}\left((u_0+u)^5-u_0^5\right)+\rho^4C^\infty({\overline}M).$$ Since $\frac{1}{8}\operatorname{R}[h]=-\frac{3}{4}+\rho^2C^\infty({\overline}M)$, $|\sigma_{\varepsilon}|^2_{\lambda_{\varepsilon}} \in \rho^4 C^\infty({\overline}M)$, and $\frac{15}{4}u_0^4=\frac{15}{4}+\rho^2C^\infty({\overline}M)$, we have $$\label{NewLichU} \Delta_{\lambda_{\varepsilon}}u - 3 u = f(u),$$ where near $\rho=0$ the function $f(u)$ has the uniformly and absolutely convergent power series $$f(u) = \sum_{l=0}^\infty a_l u^l$$ with $a_0 \in \rho^4 C^\infty({\overline}M)$, $a_1\in \rho^2 C^\infty({\overline}M)$, and $a_l \in C^\infty({\overline}M)$ for $l\geq 2$. In particular, $f$ satisfies Assumption \[Assume-F\]. Also note that, by Proposition \[SolveGeneric\], $u\in \rho^2C^0_{\textup{\normalfont phg}}({\overline}M)$; consequently $f(u)\in C^{k,\alpha}_4$ for all $k\geq 0$ and $\alpha\in(0,1)$. Applying Proposition \[MappingProperties\] now yields $u\in C^k_\delta(M)$ for all $k\geq 0$ and $\delta<3$. Finally, we observe that the indicial map of $\Delta_{\lambda_{\varepsilon}} -3$ is $$I_s(\Delta_{\lambda_{\varepsilon}} -3) = (s-3)(s+1).$$ In particular, $\Delta_{\lambda_{\varepsilon}}-3$ satisfies Assumption \[Assume-L\] with the highest characteristic exponent of $\mu=3$. Invoking Proposition \[prop:SemilinearSmoothness\] we conclude that $u$ and $\phi_{\varepsilon}$ extend to functions in $C^\infty({\overline}M)$. The proof of Theorem \[Density\] -------------------------------- We now construct the approximating initial data and show that they satisfy the shear-free condition, are smoothly conformally compact, and have the desired convergence property. The solutions $W_{\varepsilon}$ to and $\phi_{\varepsilon}$ to give rise to initial data sets $(g_{\varepsilon}, K_{\varepsilon})$ determined by $$\label{BuildApproximates} \begin{aligned} g_{\varepsilon}&= \phi_{\varepsilon}^4 \lambda_{\varepsilon}\\ K_{\varepsilon}&= \Sigma_{\varepsilon}- g_{\varepsilon}= \phi_{\varepsilon}^{-2} \sigma_{\varepsilon}- \phi_{\varepsilon}^4\lambda_{\varepsilon}. \end{aligned}$$ By Propositions \[VL-NoLogs\] and \[LichNoLogs\] we see that ${\overline}g_{\varepsilon}=\rho^2g_{\varepsilon}$ and ${\overline}\Sigma_{\varepsilon}=\rho(K_{\varepsilon}+g_{\varepsilon})$ extend smoothly to ${\overline}M$. To see that $(g_{\varepsilon}, K_{\varepsilon})$ is shear-free note that Lemma \[BoundaryH\], Proposition \[B-BasicProperties\], and the fact that $\phi_{\varepsilon}=1$ along $\partial M$ imply $$\left.{\mathcal{H}}_{{\overline}g_{\varepsilon}}(\rho)\right|_{\partial M} =0.$$ In addition, we have $$\left.{\overline}\Sigma_{{\varepsilon}}\right|_{\partial M} = \left.\rho\sigma_{\varepsilon}\right|_{\partial M} =\left.\rho \left(\mu_{\varepsilon}+ \mathcal D_{\lambda_{\varepsilon}}W_{\varepsilon}\right)\right|_{\partial M}.$$ By definition, $\mu_{\varepsilon}$ vanishes along $\partial M$. Furthermore, Proposition \[SolveGeneric\] implies that $\mathcal D_{\lambda_{\varepsilon}}W_{\varepsilon}\in C^0_{\textup{\normalfont phg}}({\overline}M)$, and thus we see that ${\overline}\Sigma_{{\varepsilon}}$ vanishes along $\partial M$. Consequently, the approximating family of initial data $(g_{\varepsilon}, K_{\varepsilon})$ satisfies the shear-free condition. Finally, we prove the following convergence property. Let $k\geq 0$ and $\alpha\in (0,1)$. Then $$\|g_{\varepsilon}-g\|_{C^{k,\alpha}(M)}\leq C{\varepsilon}, \ \ \|K_{\varepsilon}-K\|_{C^{k,\alpha}(M)}\leq C{\varepsilon}$$ for some constant $C$ independent of ${\varepsilon}$. We have $$g_{\varepsilon}- g = \phi_{\varepsilon}^4 (\lambda_{\varepsilon}- g) + (\phi_{\varepsilon}^4 -1) g.$$ From Lemma \[lemma:IntrinsicMetricEstimate\] we see that the $C^{k,\alpha}$ norm of the first term is $\mathcal O({\varepsilon})$, while the second term can be estimated using Proposition \[PhiConverges\]. Note that $K - K_{\varepsilon}= \Sigma - \Sigma_{\varepsilon}-(g-g_{\varepsilon})$. Thus it suffices to estimate $$\Sigma - \Sigma_{\varepsilon}= \Sigma - \phi_{\varepsilon}^{-2}(\mu_{\varepsilon}+ \mathcal D_{\lambda_{\varepsilon}}W_{\varepsilon}).$$ Due to Lemma \[EstimateWepsilon\] and Proposition \[PhiConverges\], it suffices to estimate $\Sigma-\mu_{\varepsilon}$. This, however, is accomplished in Lemma \[EstimateMuEpsilon\].

--- abstract: 'We propose a three-mode optomechanical system to realize optical nonreciprocal transmission with unidirectional amplification, where the system consists of two coupled cavities and one mechanical resonator which interacts with only one of the cavities. Additionally, the optical gain is introduced into the optomechanical cavity. It is found that for a strong optical input, the optical transmission coefficient can be greatly amplified in a particular direction and suppressed in the opposite direction. The expressions of the optimal transmission coefficient and the corresponding isolation ratio are given analytically. Our results pave a way to design high-quality nonreciprocal devices based on optomechanical systems.' author: - 'L. N. Song$^{1}$' - 'Qiang Zheng$^{2,1}$' - 'Xun-Wei Xu$^{3}$' - 'Cheng Jiang$^{1,4}$' - 'Yong Li$^{1,5}$' title: | Optimal undirectional amplification induced by optical gain\ in optomechanical systems --- Introduction ============ The study of optomechanical systems [@aspe] based on the parametric coupling between the photonic and phononic fields, excites a wide range of interests. Many interesting properties of the optomechanical systems, such as optomechanically induced transparency (OMIT) [@agarwal; @painter; @weiss], quantum entanglement [@lt; @ydwang], Bell-nonlocality [@bell], and imaging structure of tumors [@tumors], have been reported. These properties indicate that the optomechanical system is a key quantum coherent device for precise measurement and quantum information processing. In a network based on electrical or optical elements, one of the key coherent devices is the nonreciprocal one, such as isolator or circulator, where the signals have significantly different transmission behaviors in two opposite directions due to the breaking of time-reversal symmetry. Traditionally, the approach to break the time-reversal symmetry is utilizing the magneto-optical effect [@MO; @Effect], which usually makes the system bulky and unrobust to the external magnetic field. Recently, several magnetic-free mechanisms have been proposed to implement nonreciprocal devices, such as spatio-temporal asymmetry of refractive-index [@ri1; @ri2], angular momentum biasing in photonic or acoustic systems [@DLsoun13; @amb1; @amb2; @amb3]. As an all-optical and magnetic-free platform, the optomechanical system has also been suggested to implement the optical nonreciprocal devices. Up to now, there exist at least two kinds of optical nonreciprocity based on optomechanical systems. For the first kind, the transmitted signal is the weak light field, and its transmission behavior is assisted by another strong control field which enhances significantly the effective optomechanical coupling. This kind of nonreciprocity has been achieved in physical systems displayed OMIT  [@hafezi; @dchomit; @jh], frequency conversion between optical and microwave fields [@tl; @ok1], and quantum-limited amplification [@clerk; @mercier; @nunnenkamp; @malz; @fang; @zxz; @jc; @ok; @dch; @zl]. And the second kind of optical nonreciprocity is based on the nonlinear interaction in the system, suggested in Ref. [@manipat]. Here the input field (that is, the transmitted signal) is usually very strong, and it is not necessary to introduce the additional strong control field. A variety of nonlinear interactions, induced by coupling the cavity fields to a qubit [@zheng], atomic ensemble [@song; @xia], mechanical resonators [@Ruesink; @Rodriguez; @xu], or nonlinear optical medium [@xm], have been used to investigate this kind of optical nonreciprocity. We would like to note that a nonreciprocal device of optical diode based on the nonlinear interaction has recently been proposed [@xu] in a three-mode system, which is composed by a standard optomechanical system plus another cavity coupled with the optomechanical cavity (shown in Fig. \[Fig-1\]). In this work, we will further investigate the optical nonreciprocal phenomenon in the similar three-mode optomechanical system with introducing an additional optical gain for the optomechanical cavity. For the case without optical gain [@xu], the value of the transmission coefficient is usually smaller than $1$ and the optical diode was achieved. With the aid of the optical gain in the three-mode optomechanical system, we find in this work that the value of the transmission coefficient in one direction can be much larger than $1$, while in the opposite direction it can be much smaller than $1$. Thus, the optical unidirectional amplification can be achieved with good isolation rate due to the presence of the additional optical gain. And the analytical expression of the optimal transmission coefficient in the amplifying direction is obtained, which is only determined by the product of two factors, with the first (second) term representing the proportion of the external decay rate into the effective (total) decay of the cavity. ![(Color online) Schematic diagram of the three-mode optomechanical system with optical gain. The whispering-gallery cavity $1$ is coupled to the mechanical mode induced by radial radiation-pressure onto the cavity boundary [@kippenberg], and the additional optical gain $% \mathcal{G}$ is introduced for cavity ${1}$ [@Peng; @xm]. The second whispering-gallery cavity ${2}$ is coupled to the cavity ${1}$ via optical hopping interaction. The input field is injected either from the cavity ${1}$ or the cavity ${2}$.[]{data-label="Fig-1"}](scheme){width="8.5cm"} Model and steady-state solution =============================== For concreteness, the optomechanical system under consideration is schematically shown in Fig. \[Fig-1\], which consists of two coupled whispering-gallery cavities and one mechanical resonator induced by radial radiation-pressure onto the cavity boundary [@kippenberg] of one of the cavitis (cavity 1). In addition, the optical gain is introduced for cavity 1, which can be achieved by doped Er$^{3+}$ ions in silica with pumping the Er$^{3+}$ ions by a laser [@Peng; @xm]. The Hamiltonian of such an optomechanical system can be written as ($\hbar =1 $) $$\begin{aligned} H &=&\omega _{1}a_{1}^{\dagger }a_{1}+\omega _{2}a_{2}^{\dagger }a_{2}+\frac{% 1}{2}\omega _{m}\left( q^{2}+p^{2}\right) +J(a_{1}^{\dagger }a_{2}+a_{2}^{\dagger }a_{1}) \notag \\ &&+ga_{1}^{\dagger }a_{1}q+i\sqrt{\kappa _{1,e}}\left( \alpha _{1,\mathrm{in}% }a_{1}^{\dagger }e^{-i\omega _{d}t}-\alpha _{1,\mathrm{in}}^{\ast }a_{1}e^{i\omega _{d}t}\right) \notag \\ &&+i\sqrt{\kappa _{2,e}}\left( \alpha _{2,\mathrm{in}}a_{2}^{\dagger }e^{-i\omega _{d}t}-\alpha _{2,\mathrm{in}}^{\ast }a_{2}e^{i\omega _{d}t}\right), \label{H}\end{aligned}$$where $a_{1}$ and $a_{2}$ are the annihilation operators of the optical fields in two cavities (with the frequencies of $\omega _{1}$ and $\omega _{2}$); $p$ and $q$ are the momentum and displacement operators of the mechanical resonator (with the resonance frequency of $\omega _{m}$), respectively. $\kappa _{j,e} $ ($j=1,2$) is the external decay rate of cavity $j$. In Eq. (\[H\]), the fourth term denotes the coupling between two cavities with strength $J$, and the fifth term represents the radiation-pressure optomechanical coupling with the single-photon optomechanical coupling $g$. The last two terms stand for the coupling between the classical input fields (with the amplitude $\alpha _{j,\mathrm{in% }}$ and the frequency of $\omega _{d}$) and the cavity fields. According to Hamiltonian (\[H\]), the quantum Langevin equations (QLEs) are obtained in the rotating frame of the driving frequency $\omega _{d}$ as $$\begin{aligned} \dot{a}_{1}=&-\left( i\Delta _{1}+\frac{\kappa _{\mathrm{eff}}}{2}\right) a_{1}-igqa_{1}-iJa_{2}+\sqrt{\kappa _{1,e}}\alpha _{1,\mathrm{in}} && \notag \\ & +\sqrt{\kappa _{1,o}}a_{1,\mathrm{vac}}+\sqrt{\mathcal{G}}a_{1,\mathrm{gain% }}\text{,} & & \label{l1} \\ \dot{a}_{2}=& -\left( i\Delta _{2}+\frac{\kappa _{2}}{2}\right) a_{2}-iJa_{1}+\sqrt{\kappa _{2,e}}\alpha _{2,\mathrm{in}}+\sqrt{\kappa _{2,o}% }a_{2,\mathrm{vac}}\text{,} & & \notag \\ & & \label{l2} \\ \dot{q}=& \,\,\omega _{m}p\text{,} & & \label{l3} \\ \dot{p}=& -\omega _{m}q-ga_{1}^{\dagger }a_{1}-\gamma _{m}p+\zeta \text{,} & & \label{l4}\end{aligned}$$ where $\Delta _{j}=\omega _{j}-\omega _{d}$ ($j=1,2$) is the detuning of cavity $j$ from the input field, respectively. $\kappa _{j}=\kappa _{j,o}+\kappa _{j,e}$ is the total decay rate of cavity $j$, where $\kappa _{j,o}$ is the intrinsic decay rate. $\kappa _{\mathrm{eff}}=\kappa _{1}-% \mathcal{G}$ is the effective decay rate of cavity $1$, where $\mathcal{G% }$ is the gain rate induced by the doped $\mathrm{Er}^{3+}$ ions with optical pumping. $\gamma _{m}$ is the decay rate of the mechanical resonator, $a_{j,\mathrm{vac}}$, $a_{1,\mathrm{% gain}}$, and $\zeta $ are the noise operators with zero mean values. Assuming the input signal field(s) to be strong enough, the operators can be replaced by their average values with the mean-field approximation $\alpha _{j}=\left\langle a_{j}\right\rangle $, $\bar{p}=\left\langle p\right\rangle $, and $\bar{q}% =\left\langle q\right\rangle $. From Eqs. (\[l1\]-\[l4\]), one can obtain the following steady-state equations $$\begin{aligned} 0=& -\left( i\Delta _{1}+\frac{\kappa _{\mathrm{eff}}}{2}\right) \alpha _{1}-ig\bar{q}\alpha _{1}-iJ\alpha _{2}+\sqrt{\kappa _{1,e}}\alpha _{1,% \mathrm{in}}\text{,} & & \notag \label{a1} \\ & & & \\ 0=& -\left( i\Delta _{2}+\frac{\kappa _{2}}{2}\right) \alpha _{2}-iJ\alpha _{1}+\sqrt{\kappa _{2,e}}\alpha _{2,\mathrm{in}}\text{,} & & \label{a2} \\ \bar{p}=& \,\,0\text{,} & & \label{a3} \\ \bar{q}=& \,\,\frac{g\left\vert \alpha _{1}\right\vert ^{2}}{\omega _{m}}% \text{.} & & \label{a4}\end{aligned}$$ To study the optical nonreciprocal transmission, we will focus on two cases. In the first case, the input field is only injected into cavity ${1}$ with amplitudes $|\alpha _{1,\mathrm{in}}|=\sqrt{p_{\mathrm{in}}/(\hbar \omega _{d})}$ and $\alpha _{2,\mathrm{in}}=0$, where $p_{\mathrm{in}}$ is the power of the input field. With the input-output relation [@Gardiner] $$\alpha _{j,\mathrm{out}}-\alpha _{j,\mathrm{in}}=\sqrt{\kappa _{j,e}}\alpha _{j}\text{,} \label{io}$$ the equation of the output field $\alpha _{2,\mathrm{out}}$ can be given as $$0=-\left( \frac{\kappa }{2}+i\Delta \right) \alpha _{2,\mathrm{out}% }+iU\left\vert \alpha _{2,\mathrm{out}}\right\vert ^{2}\alpha _{2,\mathrm{out% }}+\varepsilon \alpha_{1,\mathrm{in}}\text{,} \label{a1in}$$ where $$\begin{aligned} \kappa \equiv &\,\,\kappa _{\mathrm{eff}}+\frac{4J^{2}\kappa _{2}}{\kappa _{2}^{2}+4\Delta _{2}^{2}}\text{,} \\ \Delta \equiv &\,\,\Delta _{1}-\frac{4J^{2}\Delta _{2}}{\kappa _{2}^{2}+4\Delta _{2}^{2}}\text{,} \\ U \equiv &\,\,\frac{g^{2}\left( \kappa _{2}^{2}+4\Delta _{2}^{2}\right) }{% 4\omega _{m}J^{2}\kappa _{2,e}}\text{,} \\ \varepsilon \equiv &-\frac{2iJ\sqrt{\kappa _{1,e}\kappa _{2,e}}}{\kappa _{2}+2i\Delta _{2}}\text{.}\end{aligned}$$ In the second case, the input field is only injected into cavity ${2}$ with the amplitude $|\tilde{\alpha}_{2,\mathrm{in}}|=\sqrt{\tilde{p}_{% \mathrm{in}}/(\hbar \omega _{d})}$ and $\tilde{\alpha}_{1,\mathrm{in}}=0$, where $\tilde{p}_{\mathrm{in}}$ is the power of input field. Here we have added tildes $\tilde{\text{ }}$for ${\alpha }_{j,\mathrm{in}}$, ${\alpha }_{j,\mathrm{out}}$, and ${p}_{% \mathrm{in}}$ in order to distinguish them from that in the first case. Similarly, the equation of the output field $\tilde{\alpha}_{1,\mathrm{out}}$ is obtained as $$0=-\left( \frac{\kappa }{2}+i\Delta \right) \tilde{\alpha}_{1,\mathrm{out}% }+i \tilde{U}\left\vert \tilde{\alpha}_{1,\mathrm{out}}\right\vert ^{2}% \tilde{\alpha}_{1,\mathrm{out}}+{\varepsilon} \tilde{\alpha}_{2,\mathrm{in}} \text{,} \label{a2in}$$ where $$\tilde{U}\equiv \frac{g^{2}}{\omega _{m}\kappa_{1,e}}.$$ To describe the transmission properties quantitatively, we define the following transmission coefficients $$T\equiv \left\vert \frac{\alpha _{2,\mathrm{out}}}{\alpha _{1,\mathrm{in}}}% \right\vert ^{2}\text{,}\ \ \tilde{T}\equiv \left\vert \frac{\tilde{\alpha}% _{1,\mathrm{out}}}{\tilde{\alpha}_{2,\mathrm{in}}}\right\vert ^{2},$$respectively, for the two cases with opposite transmission directions. By making use of Eqs. (\[a1in\]) and (\[a2in\]), the transmission coefficients are determined by $$\begin{aligned} 0 =&\,\,4U^{2}T^{3}s_{\mathrm{in}}^{2}+8\Delta UT^{2}s_{\mathrm{in}}+T\left( \kappa ^{2}+4\Delta ^{2}\right) -\lambda \text{,} & \notag \\ & & \label{t21}\\ 0 =&\,\,4\tilde{U}^{2}\tilde{T}^{3}\tilde{s}_{\mathrm{in}}^{2}+8\Delta \tilde{U}\tilde{T}^{2}\tilde{s}_{\mathrm{in}}+% \tilde{T}\left( \kappa ^{2}+4\Delta ^{2}\right) -\lambda & \notag \\ & & \label{t12}\end{aligned}$$ with $s_{\mathrm{in}}=|\alpha_{1,\mathrm{in}}|^{2}$, $\tilde{s}_{\mathrm{in}}=|\alpha_{2,\mathrm{in}}|^{2}$, and $\lambda =16J^{2}\kappa _{1,e}\kappa _{2,e}/\left( \kappa_{2}^{2}+4\Delta _{2}^{2}\right) $. The optical nonreciprocity requires $T\neq \tilde{T}$ when the input fields have the same powers in the two cases, i.e. $p_{\mathrm{in}}=\tilde{p}_{% \mathrm{in}}$ and $s_{\mathrm{in}}=\tilde{s}_{\mathrm{in}}$. Thus it is clear from Eqs. (\[t21\]) and (\[t12\]) that the necessary condition to observe the optical nonreciprocity is $U\neq \tilde{U}$, which can be explicitly written as $$\kappa _{1,e}\left( \kappa _{2}^{2}+4\Delta _{2}^{2}\right) \neq 4\kappa _{2,e}J^{2}\text{.} \label{non}$$ We would like to note that the similar condition of optical nonreciprocity has also been reported in Ref. [@xu] in a similar three-mode opotomechanical system without the optical gain. Unidirectional amplification ============================ In this section we will study the transmission behavior in the three-mode optomechanical system under consideration. It is found that the optical signal field can be unidirectionally amplified with the additional optical gain. And the expressions of the optimal transmission coefficient and the isolation ratio are given analytically. Stability condition ------------------- Since both the optical gain and the nonlinear interaction are introduced in our system, the first step is to ensure the stability of the system in steady state. By splitting each operator into its mean value and fluctuation: $a_{j}=\alpha _{j}+\delta a_{j}$, $q=\bar{q}+\delta q$, $p=\bar{% p}+\delta p$, the linearized QLEs corresponding to Eqs. (\[l1\])-(\[l4\]) can be written in a matrix form as $$\dot{\mu}=-M\mu +\mu _{\mathrm{in}}\text{,} \label{muuu}$$where $\mu =(\delta a_{1},\delta a_{1}^{\dagger },a_{2},\delta a_{2}^{\dagger },\delta p,\delta q)^{T}$, [$\mu _{\mathrm{in}}=(\sqrt{\kappa _{1,o}}a_{1,\mathrm{vac}}+\sqrt{\mathcal{G}}a_{1,\mathrm{gain}},\sqrt{\kappa _{1,o}}a_{1,\mathrm{vac}}^{\dagger }+\sqrt{\mathcal{G}}a_{1,\mathrm{gain}% }^{\dagger },\newline \sqrt{\kappa _{2,o}}a _{2,\mathrm{in}},\sqrt{\kappa _{2,o}}a _{2,% \mathrm{in}}^{\dagger }{,0},\zeta )^{T}$]{}, and the coefficient matrix $$M=\left( \begin{array}{cccccc} \frac{\kappa _{\mathrm{eff}}}{2}+i\left( \Delta _{1}+g\bar{q} \right) & 0 & iJ & 0 & ig\alpha _{1} & 0 \\ 0 & \frac{\kappa _{\mathrm{eff}}}{2}-i\left( \Delta _{1}+g\bar{q}\right) & 0 & -iJ & -ig\alpha _{1}^{\ast } & 0 \\ iJ & 0 & \frac{\kappa _{2}}{2}+i\Delta _{2} & 0 & 0 & 0 \\ 0 & -iJ & 0 & \frac{\kappa _{2}}{2}-i\Delta _{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -\omega _{m} \\ g\alpha _{1}^{\ast } & g\alpha _{1} & 0 & 0 & \omega _{m} & \gamma _{m}% \end{array}% \right) \text{.}$$ The stability condition can be derived by using the Routh-Hurwitz criterion [@dejusus], which requires all the real parts of eigenvalues of the matrix $M$ to be positive. The explicit forms of such a criterion in the current model are cumbersome and not given here. However, in the following discussions all the stability conditions have been checked numerically. Optical amplification induced by optical gain --------------------------------------------- For the nonreciprocal device based on the nonlinearity in the three-mode optomechanical system [@xu], the optical diode is achieved and the value of the maximum transmission coefficients is usually smaller than one. This subsection will show the optical unidirectional amplification assisted by the optical gain. That is, the transmission coefficient along one of the two directions is larger than one, and the one in the opposite direction is much smaller than one. In Fig. \[Fig-2\], the transmission coefficients $T$ and $\tilde{T}$ are plotted as a function of the input power $p_{\mathrm{in}}$. It is apparent that in Fig. \[Fig-2\] the optical unidirectional amplification appears in two regions where $T>1>\tilde{T}$ (i.e. $p_{\mathrm{% in}}\in \lbrack p_{\mathrm{in},l},p_{\mathrm{in},u}]$) and $\tilde{T}>1>T$, respectively. However, the isolation ratio in the first region is better than that in the second region. Then in what follows, we just focus on the first region with $p_{\mathrm{in}}\in \lbrack p_{\mathrm{in},l},p_{\mathrm{in% },u}]$, where we only consider the upper branch of $T$. As shown in Fig. \[Fig-2\](a), when the system works in the upper branch of $T$ with $p_{% \mathrm{in}}\in \lbrack p_{\mathrm{in},l},~p_{\mathrm{in},u}]$, it has obvious optical nonreciprocity with the tremendous difference between the values of (upper-branch) $T$ and $\tilde{T}$. Here, $p_{\mathrm{in},l}=2.03\,\mathrm{{\mu}W} $ and $p_{\mathrm{in},u}=0.68\,\mathrm{mW}$ corresponding to $T=T_{\mathrm{% max,num}}$ and $T=1$, respectively, are the lower and upper bounds of input field power. To quantify optical nonreciprocity, the isolation ratio is introduced as $% E\left( \mathrm{dB}\right) =10\times \log _{10}({T}/\tilde{T})$. Accordingly, with $p_{\mathrm{in}}\in \lbrack p_{\mathrm{in},l},p_{\mathrm{in% },u}]$ in Fig. \[Fig-2\](a), it is found numerically that $|E\left( \mathrm{dB}\right) |\in \lbrack 26.99,52.04]$. Moreover, in Fig. \[Fig-2\](a) the value of $T$ is larger than $1$ while that of $\tilde{T}$ is much smaller than $1$ in the working region. It clearly displays that the signal is amplified when the input field is injected from cavity $1$. With the aid of the optical gain, Fig. \[Fig-2\](b) and Fig. \[Fig-2\](c) also show the similar unidirectional amplification as that in Fig. \[Fig-2\](a). Note that in Fig. \[Fig-2\], the parameters satisfy the nonreciprocity condition Eq. (\[non\]). The effect of the external decay rate $\kappa _{1,e}$ on the transmission behavior is also investigated in Fig. \[Fig-5\]. This figure shows that with the increase of $\kappa _{1,e}$, all the values of the transmission coefficients are collectively lifted upward. This means with the increase of the external decay in cavity ${1}$, both the transmission coefficients in the two directions can be increased with the unidirectional amplification remained. ![(Color online) The black solid line represents the equation $T_{% \mathrm{max}, \mathrm{num}}=T_{\mathrm{max}, \mathrm{theor}}$, and each blue circle denotes the point with the coordinates ($T_{\mathrm{max},\mathrm{num}% } $, $T_{\mathrm{max}, \mathrm{theor}}$) which are respectively obtained numerically and analytically for the same parameters. Actually these values of $T_{\mathrm{max},\mathrm{num}} $ are taken from the data marked as black circles in Fig. \[Fig-2\] and Fig. \[Fig-5\], and that of $T_{\mathrm{max}, \mathrm{theor}}$ are calculated based on Eq. (\[tup\]) with the corresponding parameters. One can find all the blue circles collapse into the line.[]{data-label="Fig-3"}](t){width="8cm"} ![(Color online) Transmission coefficients $T$ as a function of the input power $p_{\mathrm{in}}$ for different values of the detuning. The dotted lines represent unstable values and solid lines represent stable values. The cavity coupling strength is kept to be the optimal one $J=J_{% \text{opt}}\equiv\protect\sqrt{\protect\kappa _{\mathrm{eff}}(\protect\kappa % _{2}^{2}+\Delta _{2}^{2})/(4\protect\kappa _{2})}$. Here the other parameters are chosen based on a recent optomechanical experiment with whispering gallery [@par]: $\protect\kappa _{1}/2\protect\pi =50\,\mathrm{MHz}$, $\protect\kappa _{2}/2\protect\pi =100\,\mathrm{MHz}$, $\protect\kappa _{\mathrm{eff}}/2\protect\pi =200\,% \mathrm{kHz}$, $\protect\kappa _{1,e}/2\protect\pi =50\,\mathrm{MHz}$, $% \protect\kappa _{2,e}/2\protect\pi =100\,\mathrm{MHz}$, $\protect\omega % _{d}/2\protect\pi =200\,\mathrm{THz}$, $\protect\omega _{m}/2\protect\pi % =200\,\mathrm{MHz}$, $\protect\gamma _{m}/2\protect\pi =50\,\mathrm{kHz}$, $% g/2\protect\pi =0.8\,\mathrm{kHz}$, $\Delta _{0}/2\protect\pi =1\,\mathrm{MHz% }$.[]{data-label="Fig-4"}](delta1delta2){width="8cm"} Optimal transmission coefficient and the corresponding isolation ratio ---------------------------------------------------------------------- In Sec. III B, it is found that with $p_{\mathrm{in}}\in \lbrack p_{\mathrm{% in},l},~p_{\mathrm{in},u}]$, our optomechanical system displays the optical nonreciprocal transmission of unidirectional amplification. This inspires us to ask the following question: What are the optimal maximum transmission coefficient and the corresponding isolation ratio in our system? We will study such a question in details in this section. Eq. (\[t21\]) is a cubic equation for the transmission coefficient $T$. However, the analytical solution of $T$ has somewhat complex dependence on the system parameters and makes it less informative. This difficulty can be circumvented by solving $s_{\mathrm{in}}$ in Eq. (\[t21\]). The solution to $s_{\mathrm{in}}$ in Eq. (\[t21\]) is formally given as $$s_{\mathrm{in}}=\frac{2T\Delta \pm \sqrt{T\lambda -T^{2}\kappa ^{2}}}{% 2T^{2}U}\text{.} \label{s1in}$$Because $s_{\mathrm{in}}$ must be positive, under the condition $\Delta >0$, the valid region of $T$ with $T\lambda -T^{2}\kappa ^{2}\geq 0$ should be $$0<T\leq T_{\mathrm{max}\mathrm{,theor}}\text{,} \label{tr21}$$where the possible maximum transmission coefficient $$T_{\mathrm{max}\mathrm{,theor}}=\frac{\lambda }{\kappa ^{2}}=\frac{% 16J^{2}\kappa _{1,e}\kappa _{2,e}\left( \kappa _{2}^{2}+4\Delta _{2}^{2}\right) }{\left[ 4J^{2}\kappa _{2}+\kappa _{\mathrm{eff}}\left( \kappa _{2}^{2}+4\Delta _{2}^{2}\right) \right] ^{2}}\text{.} \label{tup}$$With the optical amplification requirement $T_{\mathrm{max}\mathrm{,theor}% }>1 $, the condition for $\kappa _{\mathrm{eff}}$ is determined as $$0<\kappa _{\mathrm{eff}}<\sqrt{16J^{2}\kappa _{1,e}\kappa _{2,e}\left( \kappa _{2}^{2}+4\Delta _{2}^{2}\right) }-\frac{4J^{2}\kappa _{2}}{\kappa _{2}+4\Delta _{2}^{2}}\text{.} \label{keff}$$ The numerical counterpart $T_{\mathrm{max},\mathrm{num}}$ of the maximum transmission coefficient $T_{\mathrm{max}}$ can be easily obtained by the numerical solutions to Eq. (\[t21\]), such as that in Fig. \[Fig-2\]. For the parameters considered in Figs. \[Fig-2\]-\[Fig-3\], it is checked that the relation $T_{% \mathrm{max},\mathrm{num}}=T_{\mathrm{max}\mathrm{,theor}}$ is always valid in the working region. As an example, in Fig. \[Fig-3\] all the blue circles representing the point ($T_{\mathrm{max}\mathrm{,num}}$, $T_{\mathrm{% max}\mathrm{,theor}}$) collapse into the line with equation $T_{\mathrm{max}% \mathrm{,num}}=T_{\mathrm{max}\mathrm{,theor}}$. This suggests that the expression given in Eq. (\[tup\]) is a good approximate result for $T_{% \mathrm{max}}$ for the parameters considered in Figs. \[Fig-2\]-\[Fig-3\]. From now on, for simplicity we set $T_{\mathrm{max}}=T_{\mathrm{max}\mathrm{,theor}}$. Then, $T_{\mathrm{max}}$ can be further optimized with respect to the coupling strength $J$ between the two cavities. Solving $\partial T_{\mathrm{max}}/\partial J=0$ under the condition $\kappa _{\mathrm{eff}}>0$, the optimal coupling strength is given as $$J=J_{\mathrm{opt}}:=\sqrt{\frac{\kappa _{\mathrm{eff}}\left( \kappa _{2}^{2}+4\Delta _{2}^{2}\right) }{4\kappa _{2}}}\text{.} \label{jopt}$$Substituting Eq. (\[jopt\]) into Eq. (\[tup\]), the optimized value of $% T_{\mathrm{max}}$ is obtained as $$T_{\mathrm{max}}^{\mathrm{opt}}=\frac{\kappa _{1,e}}{\kappa _{\mathrm{eff}}}% \cdot \frac{\kappa _{2,e}}{\kappa _{2}}\text{.} \label{tupopt}$$ There are two terms in Eq. (\[tupopt\]), in which the first (second) term represents the proportion of the external decay rate into the effective (total) decay of the cavity. This indicates that $T_{\mathrm{max}}^{\mathrm{% opt}}$ is determined only by the intrinsic parameters of the system. As a result, $T_{\mathrm{max}}^{\mathrm{opt}}$ should remain as a constant, when the other parameters (e.g., the detunings) are changed. This invariance of $% T_{\mathrm{max}}^{\mathrm{opt}}$ is displayed in Fig. \[Fig-4\]: although the detuning $\Delta _{1}$ and $\Delta _{2}$ change, $T_{\mathrm{max}}^{\mathrm{opt}}$ is unaltered. Finally, the isolation ratio $E_{0}$ corresponding to $T_{\mathrm{max}}^{\mathrm{opt}}$ is derived. According to Eqs. (\[t21\],\[t12\],\[jopt\]), the absolute value of isolation ratio is given as $$|E_{0}|\simeq 10\times \log _{10}\left( 1+\frac{% \left( \kappa _{2}\Delta _{1}-\kappa _{\mathrm{eff}}\Delta _{2}\right) ^{2}}{% \kappa _{2}^{2}\kappa _{\mathrm{eff}}^{2}}\right), \label{E}$$ where we have used the fact that $T_{\mathrm{max}}^{\mathrm{opt}} \gg 1$ and the corresponding value of $\tilde T$ at $p_{\rm{in}}=p_{{\rm{in}},l}$ is much less than 1. For the special case $\kappa _{2}\gg \kappa _{\mathrm{eff}}$ and $\Delta _{1}\sim \Delta _{{2}}\gg \kappa _{\mathrm{eff}}>0$, Eq. (\[E\]) is simplified as $$|E_{0}|\simeq 10\times \log _{10} \frac{% \Delta^2 _{1}}{\kappa^2 _{\mathrm{eff}}} . \label{E2}$$That means one can obtain good isolation ratio by modifying the optical gain so that the effective decay rate $\kappa_{\rm{eff}}$ of cavity 1 is very small compared with $\kappa_2$ and $\Delta_{1,2}$. Conclusions =========== In summary, it is found that assisted by the optical gain, the nonreciprocal transmission with unidirectional amplification can be realized for a strong optical input signal in our three-mode optomechanical system. The origin of the optical amplification comes from the optical gain. An interesting property of our system is that it simultaneously has high isolation ratio and high transmission coefficient in a particular direction. 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--- abstract: 'We used optical microscopy to investigate flows inside water rivulets that were inkjet-printed onto different surfaces and under different ambient conditions. The acquired fluid dynamics videos were submitted to the 2013 Gallery of Fluid Motion.' author: - | Vadim Bromberg, Timothy J. Singler\ \ Mechanical Engineering Department,\ State University of New York at Binghamton, Binghamton, NY 13902, USA title: 'Flows in inkjet-printed aqueous rivulets' --- Introduction ============ Understanding the flow inside sessile liquid structures of different shapes is important in a variety of solution-based material deposition and patterning processes. Solvent evaporation inherent in these processes is already known to lead to a rich variety of flows [@Deegan1997]. The small length scale and general lack of shape symmetry implies the potential for capillary pressure gradients and corresponding flow phenomena. Finally, the non-instantaneous nature of the formation of these liquid structures adds another element to the flow complexity. In the linked videos, we investigated the internal flow of inkjet-printed water rivulets of finite length using optical microscopy. Six millimeter-long rivulets were formed by printing a pre-determined number of drops ($D_{0}=52\mu$m) at controlled frequency ($f=20$Hz) and spatial overlap ($\Delta x=20\mu$m). Microscope cover slips made of borosilicate glass were surface coated and used as substrates. Two surface coatings were investigated - S1805 photoresist with and without KOH etching. Rivulets were printed inside a controlled humidity chamber at two relative humidity levels - $45\%$ and $25\%$. The ambient, ink, and substrate were kept at room temperature ($T=25$C). For fluorescent microscopy videos, the water was seeded with Nile Red polystyrene spheres ($1.1\mu$m) at $0.1\%$ volume fraction. We report the experimental details and results of a wider range of both printing parameters, ink properties, and substrate surfaces elsewhere [@BrombergDFD2013]. The two surface coatings resulted in different values of the static advancing and receding contact angles ($\theta_{A}$ and $\theta_{R}$) for water and in the rivulet formation process during printing. Water droplets on the photoresist-coated surface exhibited $\theta_{A}=80^{\circ}$ and $\theta_{R}=35^{\circ}$while those on the etched surface $\theta_{A}=45^{\circ}$ and $\theta_{R}=0$. Complete rivulet formation was inhibited on the first surface type due to a non-zero $\theta_{R}$, which allowed capillary-driven contact line recession and rivulet breakup into individual droplets [@Schiaffino1997]. The zero $\theta_{R}$ on the second surface type prevented rivulet breakup [@Schiaffino1997] but resulted in the formation of a distinctive bulge at the starting end of the water rivulet. The bulge grows immediately after the coalescence of the first few drops during rivulet formation. We investigated the flow that causes bulge growth by using optical fluorescent microscopy. During rivulet formation, a distinctly pulsatile axial flow drives fluid away from the terminal end of the growing rivulet where printed drops are landing. The frequency of the flow is the same as the drop frequency. It has been hypothesized that the large local mean curvature in the region where drops land causes a capillary pressure gradient along the length of the rivulet [@Duineveld2003]. Using $\mu$PIV, the height- and width-averaged axial velocity was measured and showed that the amplitude of the pulsatile flow decreases with reduced relative humidity in the ambient, all other conditions being fixed. The decreased flow magnitude results in the inhibition of bulge formation. Videos ====== The video entry to the 2013 Gallery of Fluid motion is shown in - [Video 1 - Low resolution](URL of video) - [Video 2 - High resolution](URL of video) In the video, the following subjects are shown: - A schematic of the flow observation approach - Visualization of rivulet break up during printing onto a substrate with non-zero receding contact angle (surface type 1) - Visualization of successful rivulet formation on surface type 2 and concequent bulge growth - Fluorescence microscopy video showing pulsatile axial flow and corresponding mean axial speed - The effect of liquid evaporation on axial flow and bulge growth inhibition [1]{} Deegan, R. D., Bakajin, O., Dupont, T. F., Huber, G., Nagel, S. R., Witten, T. A. Capillary flow as the cause of ring stains from dried liquid drops. , 389, 827–829, 1997. Bromberg, V. and Singler, T. J. Experimentally observed flows inside inkjet-printed aqueous rivulets. American Physical Society, 66th Annual Meeting of the APS Division of Fluid Dynamics, 2013. Schiaffino, S., and Sonin, A. Formation and stability of liquid and molten beads on a solid surface. Journal of Fluid Mechanics, 343, 95–110, 1997. Duineveld, P. C. The stability of ink-jet printed lines of liquid with zero receding contact angle on a homogeneous substrate. Journal of Fluid Mechanics, 477, 175–200, 2003.

--- abstract: 'In past few years the flavor physics made important transition from the work on confirmation the standard model of particle physics to the phase of search for effects of a new physics beyond standard model. In this paper we review current state of the physics of $b$-hadrons with emphasis on results with a sensitivity to new physics.' author: - | [*Michal Kreps*]{}\ KIT, Wolfgang-Gaede-Stra[ß]{}e 1, 77131 Karlsruhe, Germany title: 'B Physics (Experiment)' --- Introduction ============ The start of the $b$-physics dates back to 1964 when decay of the long lived kaon to two pions and thus the $CP$ violation was observed [@Christenson:1964fg]. It didn’t took very long until a proposal for theoretical explanation of $CP$ violation was made. In their famous work, Kobayashi and Maskawa showed that with 4 quarks there is no reasonable way to include the $CP$ violation [@Kobayashi:1973fv]. Together with it they also proposed several models to explain the $CP$ violation in kaon system, amongst which the 6 quark model got favored over time. The explanation of the $CP$ violation in the six quark model of Kobayashi and Maskawa builds on the idea of quark mixing introduced by Cabibbo. The quark mixing introduces difference between eigenstates of the strong and weak interaction. The $CP$ violation requires a complex phase in order to provide a difference between process and its charge conjugate. In the four quark model, the quark mixing is described by $2\times2$ unitarity matrix. With only four quarks, states can be always rotated in order to keep the mixing matrix real and thus quark mixing cannot accommodate the $CP$ violation. Other arguments, which we are not going to discuss here, prevent also suitable inclusion of the $CP$ violation in other parts of the theory. With extension to six quarks, the mixing matrix becomes $3\times3$ unitarity matrix, called Cabibbo-Kobayashi-Maskawa matrix, $V_{CKM}$. In this case there is no possibility to rotate away all phases and one complex phase always remains in the matrix. This complex phase of $V_{CKM}$ provides the $CP$ violation in the standard model. The idea has two important implications. First, in addition to three quarks known in early 1970’s and predicted charm quark it postulates existence of other two quarks, called bottom and top. Second, despite the tiny $CP$ violation in the kaon system, proposed mechanism predicts large $CP$ violation in the $B$ system. It took almost three decades, but both predictions were experimentally confirmed, first by discovering the bottom quark in 1977 [@Herb:1977ek] followed by the top quark discovery in 1995 [@Abachi:1994td; @Abe:1995hr] and finally by the measurement of large $CP$ violation in the $B^0$ system in 2001 [@Aubert:2001sp; @Abe:2001xe]. In order to test the Kobayashi-Maskawa mechanism of the $CP$ violation many measurements are performed. In those main aim is to determine the $V_{CKM}$ with a highest possible precision. Tests are often presented in a form of the so called unitarity triangle. It follows from the unitarity requirement of the $V_{CKM}$. The product of the two columns of the matrix has to be zero in the standard model. As elements of the matrix are complex numbers, this requirement graphically represents triangle in the complex plane. In the last decade the flavor physics moved towards search for inconsistencies which would indicate presence of a new physics. We omit the charm mixing and $CP$ violation and prospects of starting experiments which are discussed elsewhere in these proceedings. Here we concentrate on the big picture with some emphasis on tensions in various measurements performed by , Belle, CDF, CLEO-c and DØ experiments. Sides of the unitarity triangle =============================== Looking to the unitarity triangle there are two sets of quantities one can determine, namely angles and sides. In this section we will discuss the status of sides determinations. The sides itself are determined by the $V_{td}$, $V_{ub}$ and $V_{cb}$ elements of the $V_{CKM}$. To determine those quantities, two principal measurements are used. First type is the measurement of the $B^0$ oscillation frequency which determines the $V_{td}$. Second type is the measurement of branching fraction of semileptonic $B$ decays, which can be translated to the $V_{ub}$ or $V_{cb}$. As there are no recent results on the $B$ mixing, we concentrate on semileptonic decays and determination of the $V_{ub}$ and $V_{cb}$. The determination of the $V_{ub}$ and $V_{cb}$ is based on the $b\rightarrow ul\nu$ and $b\rightarrow cl\nu$ transitions. Advantage of semileptonic transitions is in confinement of the all soft QCD effects into single form factor. In general two complementary approaches exists. The first one is inclusive measurements, where one tries to measure the inclusive rate of the $B\rightarrow X_{(c,u)}l\nu$ rate with $X_{(c,u)}$ denoting any possible hadron containing charm or up quark . The second approach uses exclusive measurements where one picks up a well defined hadron like $D^{*}$ in the case of $V_{cb}$ measurement. The two approaches are complementary with inclusive being theoretically clean in a first order, while exclusive being much cleaner for experiment, but more difficult for theory. In addition, part of the good properties of the inclusive approach on the theory side is destroyed by a necessity of kinematic requirements on the experimental side. As one needs good control over background in those measurements, it is practically domain of B-factories running with the $e^+e^-$ at the $\Upsilon(4S)$ resonance. Coming to the current status, determinations of the $V_{cb}$ as well as the $V_{ub}$ has some issues and inconsistencies [@Barberio:2008fa]. In the inclusive determination of the $V_{cb}$ the fit to all information has consistently too small $\chi^2$. On the other hand in the exclusive determination using $B\rightarrow D^*l\nu$ decays, different measurements are not fully consistent with $\chi^2/ndf=56.9/21$. This inconsistency is due to the differences between Belle and results rather than inconsistence between old and new measurements. The world average determined from the inclusive measurement is $V_{cb}=(41.5\pm0.44 \pm0.58)\cdot 10^{-3}$, from the $B\rightarrow Dl\nu$ we obtain $V_{cb}=(39.4\pm1.4\pm0.9)\cdot 10^{-3}$ and from the $B\rightarrow D^*l\nu$ $V_{cb}=(38.6\pm0.5\pm1.0)\cdot 10^{-3}$. As can be seen, despite the tension in the experimental information from $B\rightarrow D^*l\nu$ decays, the two exclusive determinations agree with each other, but the inclusive approach yields value which is about $2.3\sigma$ higher than the one from exclusive determination. While the determination of the $V_{ub}$ is in principle same as the determination of the $V_{cb}$, in practice the $V_{ub}$ is much more difficult due to the smallness of the $b\rightarrow ul\nu$ branching fraction compared to the $b\rightarrow cl\nu$. The $b\rightarrow cl\nu$ in this case is a significant background. Kinematic selection to reduce this background destroys possibilities of theory for the precise and reliable calculations. On the inclusive determination side, there are several groups which perform fits to the experimental data of inclusive decays. ![Distribution of the remaining energy in the $B\rightarrow\tau\nu$ searches using semileptonic tag at Belle (left) and fully hadronic tag at (right).[]{data-label="fig:Btaunu"}](kreps_michal.fig1.eps){width="65.00000%"} [![Distribution of the remaining energy in the $B\rightarrow\tau\nu$ searches using semileptonic tag at Belle (left) and fully hadronic tag at (right).[]{data-label="fig:Btaunu"}](kreps_michal.fig2a.eps "fig:"){width="37.00000%"} ![Distribution of the remaining energy in the $B\rightarrow\tau\nu$ searches using semileptonic tag at Belle (left) and fully hadronic tag at (right).[]{data-label="fig:Btaunu"}](kreps_michal.fig2b.eps "fig:"){width="60.00000%"}]{} On the exclusive side, experiment provides new result on the $B\rightarrow\pi l\nu$ and $B\rightarrow\rho l\nu$. Using their partial branching fraction in different momentum transfer regions together with lattice QCD calculations they derive $|V_{ub}|=(2.95 \pm 0.31) \times 10^{-3}$ [@Aubert:2010uj], which is about $2\sigma$ below the inclusive determinations. If this stands, than we have another discrepancy in the sides of unitarity triangle. Another way of accessing $V_{ub}$ is to use $B^+\rightarrow\tau\nu$ leptonic decays which proceed through weak annihilation. In the standard model its rate is given by expression $$BF=\frac{G_F^2m_B}{8\pi}m_l^2\left(1-\frac{m_l^2}{m_B^2}\right)^2f_B^2|V_{ub}|^2\tau_B,$$ where all quantities except of $f_B^2$ and $V_{ub}$ are well known. Typically one takes input on the $f_B^2$ and $V_{ub}$ from other measurements and puts constraints on a new physics. Alternatively one can take measured branching fraction together with the prediction for $f_B^2$ and extract $V_{ub}$. B-factories provided recently evidence for this decay. Both, Belle and reconstruct one $B$ in a semileptonic or a fully hadronic decay, called tagged, together with identified charged products of the $\tau$ decay. In such events, all what should be remaining are neutrinos and therefore one expects zero additional energy in the event. In Fig. \[fig:Btaunu\] we show examples of the distribution of additional energy. The Belle experiment sees evidence on the level of $3.5\sigma$ in both tags [@Ikado:2006un; @Adachi:2008ch] while experiment obtains excess of about $2.2\sigma$ [@Aubert:2007xj; @Aubert:2008gx]. The world average of the branching fraction of $(1.73\pm0.35)\cdot 10^{-4}$ is little higher than the SM prediction of $(1.20\pm0.25)\cdot 10^{-4}$ and yields $V_{ub}$ which is in some tension with other determinations. The result of the $B^+\rightarrow\tau\nu$ branching fraction brings up the question whether theory prediction from the lattice QCD for the $f_B^2$ is correct. One way to test predictions is to turn to charm sector where we expect smaller contributions from a new physics. Decay $D_s^+\rightarrow \tau^+\nu$ is a usual testing ground for calculations. The branching fraction is given by same formula as for $B^+\rightarrow\tau\nu$ with replacing $f_B^2$ and $V_{ub}$ by their appropriate counterparts. The branching fraction for $D_s^+\rightarrow \tau^+\nu$ was measured by CLEO, and Belle experiments and there used to be some discrepancy between the prediction for $f_{D_s}$ and its value extracted from the $D_s^+\rightarrow \tau^+\nu$ data. Summary of the evolution of this discrepancy is shown in Fig. \[fig:fds\] [@Kronfeld:2009cf]. Current situation is not too critical anymore as the discrepancy went down from $4\sigma$ to $2\sigma$. ![Confidence regions in the plane of strong phase $\delta$ and the CKM angle $\gamma/\phi_3$ from Belle experiment (left) and 1-CL for the CKM angle $\gamma$ from experiment (right). On the left plot, contours correspond to 1, 2 and 3 standard deviations. On the right plot, separate contours for decays $B^+\rightarrow D^0K^+$, $B^+\rightarrow D^{*0}K^+$, $B^+\rightarrow D^0K^{*+}$ and combination of all is shown.[]{data-label="fig:gamma"}](kreps_michal.fig3.eps){width="80.00000%"} ![Confidence regions in the plane of strong phase $\delta$ and the CKM angle $\gamma/\phi_3$ from Belle experiment (left) and 1-CL for the CKM angle $\gamma$ from experiment (right). On the left plot, contours correspond to 1, 2 and 3 standard deviations. On the right plot, separate contours for decays $B^+\rightarrow D^0K^+$, $B^+\rightarrow D^{*0}K^+$, $B^+\rightarrow D^0K^{*+}$ and combination of all is shown.[]{data-label="fig:gamma"}](kreps_michal.fig4a.eps "fig:"){width="49.00000%"} ![Confidence regions in the plane of strong phase $\delta$ and the CKM angle $\gamma/\phi_3$ from Belle experiment (left) and 1-CL for the CKM angle $\gamma$ from experiment (right). On the left plot, contours correspond to 1, 2 and 3 standard deviations. On the right plot, separate contours for decays $B^+\rightarrow D^0K^+$, $B^+\rightarrow D^{*0}K^+$, $B^+\rightarrow D^0K^{*+}$ and combination of all is shown.[]{data-label="fig:gamma"}](kreps_michal.fig4b.eps "fig:"){width="49.00000%"} With this we conclude discussion of sides of the unitarity triangle, where despite lot of the experimental work and large progress several tensions remains. Angles of the unitarity triangle ================================ The angles of the unitarity triangle are defined as $$\begin{aligned} \alpha&=&\arg\left(-{V_{td}V^*_{tb}}/{V_{ud}V^*_{ub}}\right),\\ \beta&=&\arg\left(-{V_{cd}V^*_{cb}}/{V_{td}V^*_{tb}}\right),\\ \gamma&=&\arg\left(-{V_{ud}V^*_{ub}}/{V_{cd}V^*_{cb}}\right).\end{aligned}$$ As they are give by the phases of complex numbers, their determination is possible only through $CP$ violation measurements. Here we omit determination of the angle $\alpha$, briefly mention status of the angle $\beta$ and concentrate on the angle $\gamma$ which received most of the new experimental information. The angle $\beta$ is practically give by the $V_{td}$ phase. One of the process where this CKM matrix element enters is the $B^0$ mixing. Its best determination comes from the measurement of $CP$ violation due to the interference of decays with and without mixing to a common final state. Using decays to $c\overline c$ resonance with neutral kaon extracts using final dataset $\sin 2\beta=0.687\pm0.028\pm0.012$ [@Aubert:2009yr]. The latest measurement from Belle experiment gives $\sin2\beta=0.642\pm0.031\pm0.017$ [@Chen:2006nk]. It is worth to note that both experiments are still statistically limited. Determination of the angle $\gamma$ provides important information for tests of a physics beyond standard model. It is determined from the interference of tree level $b\rightarrow c$ and $b\rightarrow u$ transitions and thus having small sensitivity to a new physics. While several different decays are suggested for the determination, all current experimental information comes from the $B^+\rightarrow D^0K^+$. In those decays, the $b\rightarrow c$ transition provides $B^+\rightarrow D^0K^+$ decay while the $b\rightarrow u$ transitions yields $B^+\rightarrow \overline{D}^0K^+$ final state. Thus measurement of the $CP$ violation in the final states which are common to $D^0$ and $\overline{D}^0$ is needed. Three different final states are currently used. The first one uses q Cabibbo favored $\overline{D}^0\rightarrow K^-\pi^-$ with q doubly Cabibbo suppressed $D^0\rightarrow K^-\pi^+$ [@Atwood:1996ci; @Atwood:2000ck]. The second method uses a Cabibbo suppressed $D^0$ decays like $\pi^+\pi^-$, $K^+K^-$ [@Gronau:1990ra]. The third approach uses a Dalitz plot analysis of a $D^0\rightarrow K_s\pi^+\pi^-$ [@Giri:2003ty]. The main limitation is that rates are small and up to now there was no significant measurement of the $CP$ violation in those decays. Recently Belle and experimental announced $\approx 3.5\sigma$ evidence for the $CP$ violation in the $B^+\rightarrow D^0K^+$ decays with $D^0\rightarrow K_s\pi^+\pi^-$ [@delAmoSanchez:2010rq; @Poluektov:2010wz]. The extracted confidence regions on the angle $\gamma$ are shown in Fig. \[fig:gamma\]. Belle experiment extracts $\gamma=(78^{+11}_{-12}\pm4\pm9)^\circ$ and obtains $\gamma=(68\pm14\pm4\pm3)^\circ$. $B_s$ sector ============ The $CP$ violation in the $B_s$ meson sector is currently the most exciting place and widely discussed in relation to a new physics. Two results, which are in many models of a new physics related are the measurement of the $CP$ violation in the $B_s\rightarrow J/\psi\phi$ decays and the flavor specific asymmetry in a semileptonic $B_s$ decays. The origin of the first one is in the interference of the decays with and without $B_s$ mixing. The standard model predicts only tiny $CP$ violation which comes from the fact that all CKM matrix elements entering are almost real. The previous results from Tevatron experiments showed about $1.5$-$1.8\sigma$ deviation from the standard model [@cdf:betas; @d0:betas] with combination being $2.2\sigma$ away. Recently CDF collaboration updated its result with more data and few improvements, which yield the better constraints on the $CP$ violation in $B_s\rightarrow J/\psi\phi$. Resulting 2-dimensional $\Delta\Gamma_s$-$\beta_s$ contour is shown in Fig. \[fig:bs\]. Overall CDF experiment now observes better agreement between the data and standard model with difference of about $0.8\sigma$. More details on this update can be find in Ref. [@proc:Elisa]. ![The $\Delta\Gamma_s$-$\beta_s$ confidence regions in $B_s\rightarrow J/\psi\phi$ decays from CDF experiment using $5.2$ fb$^{-1}$ of data (right). Latest results on the flavor specific asymmetry in semileptonic $B_{(s)}$ decays from DØ experiment (right).[]{data-label="fig:bs"}](kreps_michal.fig5a.eps "fig:"){width="30.00000%"} ![The $\Delta\Gamma_s$-$\beta_s$ confidence regions in $B_s\rightarrow J/\psi\phi$ decays from CDF experiment using $5.2$ fb$^{-1}$ of data (right). Latest results on the flavor specific asymmetry in semileptonic $B_{(s)}$ decays from DØ experiment (right).[]{data-label="fig:bs"}](kreps_michal.fig5b.eps "fig:"){width="30.00000%"} The second measurement we present here is measurement of the flavor specific asymmetry in semileptonic $b$-hadron decays. In the standard model as well as in a large class of new physics models this quantity is predicted to be small. It can be generated or by a direct $CP$ violation or by an asymmetry in the mixing rate between $b$- and $\overline{b}$-mesons. Typically direct $CP$ violation is zero as we talk about the most allowed decay amplitude $b\rightarrow cl\nu$ which would need a second contribution to interfere with. As it is not easy to construct a model where a second amplitude with reasonable size would exists typically the direct $CP$ violation is predicted to be zero. The effect of different mixing rates is small for the $B^0$ due to the small decay width difference and small in the standard model for the $B_s$ due to the small phase involved. DØ experiment announced a new measurement this year, with a highly improved treatment of systematic uncertainties. They measure $A_{fs}^b=(-96\pm25\pm15)\times 10^{-4}$ which is significantly different from the standard model expectation of $A_{fs}^b=(-2.3^{+0.5}_{-0.6})\times 10^{-4}$ [@Lenz:2006hd]. If this result is confirmed, it is clear sign of the physics beyond the standard model. For more details see Ref. [@proc:bertram]. Rare decays =========== Rare FCNC transitions are best known outside the flavor physics community for searches of a physics beyond standard model. Prime example is rare $B_s\rightarrow\mu^+\mu^-$ decay, where previous results could put strong constraints on some new physics model, even with limits, which are far from the standard model expectations. The standard model prediction for the branching fraction of $B_s\rightarrow\mu^+\mu^-$ is $(3.6\pm0.3)\times 10^{-9}$ [@Buras:2009us]. The main difficulty is in suppressing and controlling background. The search for this decays is dominated by the Tevatron experiments. Recently DØ experiment updated their result using 6.1 fb$^{-1}$ of data which yields upper limit on the branching fraction of $5.2\cdot 10^{-8}$ at 95% C.L. [@Abazov:2010fs]. The best limit at this moment is one from the CDF experiment using 3.7 fb$^{-1}$ of data and the upper limit of $4.3\cdot 10^{-8}$ at 95% C.L. [@cdf:bsmumu]. Those are about an order of magnitude above the standard model prediction. Another example of an FCNC rare process which generates lot of excitement these days is a class of the decays governed by the $b\rightarrow sl^+l^-$ quark level transition with $l$ being a charged lepton. Decays $B^{0,\pm}\rightarrow K^{0,\pm}\mu^+\mu^-$ and $B^{0,\pm}\rightarrow K^{*0,\pm}\mu^+\mu^-$ were already observed. Recently CDF experiment observed also decay $B_s\rightarrow\phi\mu^+\mu^-$ with $\approx 6.3\sigma$ significance using 4.4 fb$^{-1}$ of data [@cdf:bkmumu]. The measured branching fraction is $(1.44\pm0.33\pm0.46)\cdot10^{-6}$. As those decays proceed even in the standard model through more than one amplitude, there is a rich phenomenology of interferences. From the interference effects, the forward-backward asymmetry of the muons as a function of dimuon invariant mass is the one which is responsible for the excitement. It is measurement in Belle [@Wei:2009zv], [@Aubert:2008ju] and CDF [@cdf:bkmumu] experiments and we show results in Fig. \[fig:afb\]. ![The forward-backward asymmetry of the muon in $B\rightarrow K^*\mu+\mu^-$ decays as a function of dimuon invariant mass from CDF (left), Belle (middle) and (right). The points represent measurement, the red line in CDF and Belle case and the blue line in result show the standard model prediction and the other curves represent a different beyond standard model scenarios. The areas without data points correspond to the charmonium regions which are excluded from the analysis.[]{data-label="fig:afb"}](kreps_michal.fig6a.eps "fig:"){width="25.00000%"} ![The forward-backward asymmetry of the muon in $B\rightarrow K^*\mu+\mu^-$ decays as a function of dimuon invariant mass from CDF (left), Belle (middle) and (right). The points represent measurement, the red line in CDF and Belle case and the blue line in result show the standard model prediction and the other curves represent a different beyond standard model scenarios. The areas without data points correspond to the charmonium regions which are excluded from the analysis.[]{data-label="fig:afb"}](kreps_michal.fig6b.eps "fig:"){width="30.00000%"} ![The forward-backward asymmetry of the muon in $B\rightarrow K^*\mu+\mu^-$ decays as a function of dimuon invariant mass from CDF (left), Belle (middle) and (right). The points represent measurement, the red line in CDF and Belle case and the blue line in result show the standard model prediction and the other curves represent a different beyond standard model scenarios. The areas without data points correspond to the charmonium regions which are excluded from the analysis.[]{data-label="fig:afb"}](kreps_michal.fig6c.eps "fig:"){width="35.00000%"} While not statistically significant, all three experiments show some departure in the same direction from the standard model. It is going to be interesting to follow future measurements of this quantity. Conclusions =========== Globally, except of the flavor specific asymmetry in semileptonic $b$-decays, there is not a significant discrepancy in the global picture of $CP$ violation. On the other hand, there are few discrepancies which are worth to follow in the future. In Fig. \[fig:ckmfit\] we show the global status of the CKM fit [@Charles:2004jd]. Other determination [@Ciuchini:2000de; @Lunghi:2009ke] provide similar picture. ![The global status of the unitarity triangle fit from CKMFitter group (left), the graphical representation of the $B\rightarrow \tau\nu$ versus $\sin2\beta$ disagreement (middle) and the situation with indirect and direct determinations of the parameter proportional to $\epsilon_K$ from UTFit group (right). In the middle plot, colored confidence regions show expectation for the $B\rightarrow \tau\nu$ branching fraction from the fit where two quantities are excluded while the point shows experimental results. In the right plot, the colored areas show confidence region of the $B_K$ from the fit without constraint from the $CP$ violation in the kaon system and the cross represents experimental measurement of the quantity.[]{data-label="fig:ckmfit"}](kreps_michal.fig7a.eps "fig:"){width="30.00000%"} ![The global status of the unitarity triangle fit from CKMFitter group (left), the graphical representation of the $B\rightarrow \tau\nu$ versus $\sin2\beta$ disagreement (middle) and the situation with indirect and direct determinations of the parameter proportional to $\epsilon_K$ from UTFit group (right). In the middle plot, colored confidence regions show expectation for the $B\rightarrow \tau\nu$ branching fraction from the fit where two quantities are excluded while the point shows experimental results. In the right plot, the colored areas show confidence region of the $B_K$ from the fit without constraint from the $CP$ violation in the kaon system and the cross represents experimental measurement of the quantity.[]{data-label="fig:ckmfit"}](kreps_michal.fig7b.eps "fig:"){width="30.00000%"} ![The global status of the unitarity triangle fit from CKMFitter group (left), the graphical representation of the $B\rightarrow \tau\nu$ versus $\sin2\beta$ disagreement (middle) and the situation with indirect and direct determinations of the parameter proportional to $\epsilon_K$ from UTFit group (right). In the middle plot, colored confidence regions show expectation for the $B\rightarrow \tau\nu$ branching fraction from the fit where two quantities are excluded while the point shows experimental results. In the right plot, the colored areas show confidence region of the $B_K$ from the fit without constraint from the $CP$ violation in the kaon system and the cross represents experimental measurement of the quantity.[]{data-label="fig:ckmfit"}](kreps_michal.fig7c.eps "fig:"){width="30.00000%"} All groups see $\approx 2.5\sigma$ improvement of the fit if either constraint from the $B\rightarrow \tau\nu$ or $\sin2\beta$ is removed from the fit. Other main small discrepancies are in the $V_{ub}$ and the $CP$ violation parameter $\epsilon_K$ in the kaon system. It is worth to note that the discrepancy between measured $\sin(2\beta)$ and its prediction from the fit without $\sin(2\beta)$ was pointed out already in 2007 [@Lunghi:2007ak]. On the limited space we could not discuss the charm quark sector, which has strong potential. Its status and prospects at the time of conference can be find in Ref. [@proc:Mannel]. The prospects of the LHC in the bottom quark sector were discussed in several contribution, with most relevant one with respect to this work being Ref. [@proc:Conti]. With large expectations whole community is positive about future interesting results and the importance of the flavor physics for discovering and/or understanding a physics beyond standard model. Acknowledgments {#acknowledgments .unnumbered} =============== The author would like to thank organizers for the kind invitation to the conference and would like to acknowledge support from the Bundesministerium für Bildung und Forschung, Germany. [99]{} J. H. Christenson, J. W. Cronin, V. L. Fitch and R. Turlay, Phys. Rev. Lett.  [**13**]{} (1964) 138. M. Kobayashi and T. Maskawa, Prog. Theor. Phys.  [**49**]{} (1973) 652. S. W. Herb [*et al.*]{}, Phys. Rev. Lett.  [**39**]{} (1977) 252. S. Abachi [*et al.*]{} \[D0 Collaboration\], Phys. Rev. Lett.  [**74**]{} (1995) 2422 \[arXiv:hep-ex/9411001\]. 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--- abstract: 'The low-energy structures of mixed Ar–Xe and Kr–Xe Lennard-Jones clusters are investigated using a newly developed parallel Monte Carlo minimization algorithm with specific exchange moves between particles or trajectories. Tests on the 13- and 19- atom clusters show a significant improvement over the conventional basin-hopping method, the average search length being reduced by more than one order of magnitude. The method is applied to the more difficult case of the 38-atom cluster, for which the homogeneous clusters have a truncated octahedral shape. It is found that alloys of dissimilar elements (Ar–Xe) favor polytetrahedral geometries over octahedra due to the reduced strain penalty. Conversely, octahedra are even more stable in Kr–Xe alloys than in Kr$_{38}$ or Xe$_{38}$, and they show a core-surface phase separation behavior. These trends are indeed also observed and further analysed on the 55-atom cluster. Finally, we correlate the relative stability of cubic structures in these clusters to the glassforming character of the bulk mixtures.' author: - 'F. Calvo' - 'E. Yurtsever' title: 'Composition-induced structural transitions in mixed rare-gas clusters' --- Introduction ============ Clusters of heterogeneous materials show a much richer behavior than their homogeneous counterparts. In many bulk compounds, doping can significantly affect some global property, and alloying is a common way to tailor a completely new kind of material. At the mesoscale level, size is another complicating factor, giving rise to further changes with respect to the macroscopic object. To a large extent, most expectations of nanotechnology have been put into the electronic and catalytic properties of small atomic clusters. Therefore, it should not be surprising that numerous theoretical studies of mixed clusters were devoted to bimetallic clusters. In particular, there has been a significant amount of work at the level of sophisticated electronic structure calculations,[@chacko; @bromley; @rao] but these were often limited to small sizes due to the numerical effort involved. On a different scale of chemical complexity, many studies have been carried out using explicit, empirical force fields in order to investigate the segregation properties of these clusters. There are several driving forces toward mixing or segregation in binary systems: - the difference in atomic sizes; - the difference in surface energies; - minimization of the overall strain; - the number of interactions between unlike atoms. These factors can often compete with each other. For instance, minimizing surface energies does usually not increase with the number of interactions between different atoms. Also, even though this is not our prime interest here, it should be noted that kinetic factors can be crucial in this problem.[@baletto] In particular, Vach and coworkers have found from experiments and simulations of mixed rare-gas clusters that some anomalous enrichment effects could be observed due to the growth by pick-up of these systems.[@vach] Very recently, radial segregation and layering have been observed in large Ar/Xe clusters formed in an adiabatic expansion by Tchaplyguine [*et al.*]{}[@tchaplyguine] using photoelectron spectroscopy measurements. These data have also been theoretically interpreted by Amar and Smaby.[@amar] Fortunately, mixed rare-gas systems can be quite safely described using simple pairwise potentials such as the Lennard-Jones (LJ) potential. More accurate potentials are of course also available, even though we will have no need for them in the present, mostly methodological work. Hence they are much more convenient to study in a broad size range, not only for their structure but also their dynamics or thermodynamics. It is known from previous studies that the topography of the potential energy surfaces of homogeneous LJ clusters can be very peculiar, as for the sizes 38 or 75.[@miller] The multiple-funnel structure of these energy landscapes makes it especially hard to locate the most stable structures (global minima) or to simulate the finite-temperature behavior of these clusters in an ergodic way. The effects of mixing different rare-gas atoms on cluster structure and thermodynamics have been studied for the specific size 13 by Frantz on the examples of Ar–Kr mixtures[@frantzar] as well as Ne–Ar mixtures.[@frantzne] Fanourgakis [*et al.*]{} have also investigated these latter compounds.[@fanourgakis] A systematic work of Ar–Xe mixed clusters of 13 and 19 atoms has been carried out by Munro and coworkers,[@jordan] including some global optimization and Monte Carlo simulations. Mixed clusters involving lighter species such as H$_2$ and D$_2$ have been investigated using path-integral Monte Carlo simulations (PIMC) by Chakravarty.[@chakravarty] More recently, Sabo, Doll and Freeman reported a rather complete study of the energy landscapes[@sabo1; @sabo2]and melting phase change[@sabo3] in mixed Ar–Ne clusters. In this work quantum delocalization and the effects of impurities on cluster properties were also accounted for using PIMC techniques. The main conclusion of these studies is that atomic heterogeneity can be responsible for a drastic increase in complexity of the energy landscapes of rare-gas clusters. This complexity is manifested by numerous new low-lying minima in competitive funnels, characterized by the same overall geometrical arrangement but different permutations of unlike atoms. Following Jellinek and Krissinel,[@jellinek] we will refer to such isomers as “homotops.” The presence of several homotops on a given energy landscape often induces solid-solid transitions, which can be detected by some feature in the heat capacity,[@frantzar; @frantzne; @fanourgakis; @jordan; @parneix] even though they can be washed out by quantum effects.[@sabo3] As shown by Munro [*et al.*]{},[@jordan] the various funnels corresponding to different homotops of a same geometry are separated by significant energy barriers. This explains the difficulty or even failure of simulation methods to achieve ergodic sampling of these systems, albeit small.[@jordan] A similar situation is found in Lennard-Jones polymers,[@polymer] where a large number of isomers are based on the same geometrical arrangement, differing only in the path linking the monomers. Beyond the actual rare-gases, binary Lennard-Jones compounds have been investigated in both the cluster and bulk regimes. Clarke and coworkers looked at phase separation of small particles with equal compositions.[@clarke] Based on Monte Carlo simulations, they sketched a phase diagram in the general structure of liquid clusters. Bulk binary Lennard-Jones systems have been seen to provide relatively simple numerical models for glass formation.[@jonsson; @kob; @coluzzi; @utz; @broderix; @yamamoto] Most often, the LJ interactions in such studies have been tuned in a non-additive way in order to hinder crystallisation. In another related work, Lee and coworkers[@goddard] have investigated the role of atomic size ratio in binary and ternary metallic alloys. Interestingly, severaly links between the physics and chemistry of clusters and those of supercooled liquids and glasses have been established since the pioneering work by Frank.[@frank] The initial suggestion that the local order in simple liquids is not crystalline but icosahedral[@frank] (more generally polytetrahedral) has since been verified experimentally[@schenk] and theoretically.[@jonsson; @tarjus] From the clusters viewpoint, the favored finite-size structures of good model glassformers have been shown by Doye and coworkers to be polytetrahedral.[@doyeglass] The 38-atom homogeneous Lennard-Jones cluster is known to show some glassy properties, especially slow relaxation to the ground state,[@doye38] due to the competition between two stable funnels on the energy landscape, corresponding to truncated octahedral and icosahedral shapes, respectively. Due to entropic effects,[@doye38; @ptmc38] a solid-solid transition occurs between the two funnels, at temperatures lower than the melting point. The crystal-like configuration of this cluster makes it a good candidate to further investigate the relationship between cluster structure and criteria for glassification. Because homogeneous LJ$_{38}$ constitutes a relatively difficult task for global optimization algorithms, binary clusters of the same size can be expected to be much worse. In this paper, we propose a simple but efficient way to deal with the multiple new minima introduced by unlike atoms within a general Monte Carlo global minimization scheme. This algorithm will then be applied to the 38- and 55-atom cases, in mixtures of Xe with either Ar or Kr atoms. In the next section, we present the method and test it on the simple cases of the 13- and 19-atom clusters. In Sec. \[sec:res\] we give our results obtained at sizes 38 and 55 and we correlate them to the different glassforming abilities of the bulk mixtures. We finally conclude in Sec. \[sec:ccl\]. Methods {#sec:meth} ======= Global optimization of cluster structure[@walescheraga] is currently best achieved using either genetic algorithms[@hartke] or the Monte Carlo+minimization method,[@mcmin] also known as basin-hopping (BH).[@bh] The case of homogeneous Lennard-Jones clusters is among the most documented of cluster physics, and an up-to-date table of putative global minima can be found in Ref. . Even though it can never be guaranteed that global minimization has been successful, it is likely that all important structural forms of LJ clusters have been found up to more than 100 atoms. These include icosahedral, truncated octahedral, decahedral as well as tetrahedral arrangements. Compared to homogeneous clusters, the available data on heterogeneous systems is rather scarce. Besides the specific works by Frantz on the 13-atom Ne–Ar and Kr–Ar clusters,[@frantzar; @frantzne] Munro [*et al.*]{} used a parallel version of the BH scheme, similar to the replica-exchange Monte Carlo method,[@geyer] where several trajectories are run simultaneously at various temperatures.[@jordan] Although these authors looked at moderately large clusters, they reported significant difficulties to locate global minima at specific compositions, as in Xe$_{10}$Ar$_3$ or Xe$_{13}$Ar$_6$, for instance.[@jordan] Optimization algorithm ---------------------- A natural problem occuring using the basin-hopping method is that many of the low-lying minima are expected to be related to each other via particle exchange. Such a process only occurs via large deformations of the remaining cluster, hence it is quite unprobable. As in condensed matter physics,[@gazzillo; @karaaslan; @grigera] allowing exchange moves between particles as a possible Monte Carlo step may result in notably faster convergence, provided that the interactions are not too dissimilar. Actually, optimization of mixed clusters on the lattice formed by the homogeneous system has already been studied by Robertson and coworkers.[@lattice] Here we do not wish to restrict to such situations. In the framework of global optimization methods, the local minimization stage removes the possible energetic penalty associated to replacing a small atom by a bigger one. We can thus expect some increased efficiency of the algorithm in case of multiple homotops. Now we convert the extra numerical cost of running parallel trajectories at various temperatures into running them at various compositions, at the same fixed temperature $T$ for all compositions. For a X$_p$Y$_{n-p}$ compound, each of the $n$ trajectories is then labelled with the number $p$ of X atoms, running from 0 to $n$. Exchange moves between adjacent trajectories (from $p$ to $p+1$) thus need to incorporate the transmutation of two atoms (one for each configuration) into the other atom type to preserve composition. As in most Monte Carlo processes, the probability of attempting such moves must be set in advance as a parameter. The global optimization algorithm can thus be summarized into its main steps. Keeping the above notations for atom types, and denoting ${\bf R}_i^{(p)}$ the configuration at step $i$ of trajectory $p$, we start the optimization process using fully random configurations, but locally optimized. 1. With probability $P_{\rm ex}$, it is decided whether an exchange between adjacent trajectories will be attempted or not. If so, then the two trajectories involved in the exchange are determined randomly. 2. For each composition $p$ not concerned by any exchange, a new configuration ${\bf R}_{i+1}^{(p)}$ is generated from ${\bf R}_i^{(p)}$ using either several particle exchanges or large atomic moves. The probability to select particle exchanges is denoted $P_{\rm swap}$, and the number of simultaneous exchanges is allowed to fluctuate randomly between 1 and $N_{\rm swap}^{\rm max}$. If atomic moves are selected, then each atom is displaced randomly around its previous location in the three directions by a random amount of maximum magnitude $h^{(p)}$. In both cases, ${\bf R}_{i+1}^{(p)}$ is obtained after local minimization. 3. In case of an exchange between adjacent trajectories, the two configurations ${\bf R}_i^{(p)}$ and ${\bf R}_i^{(p+1)}$ corresponding to these trajectories are then swapped, one X atom of ${\bf R}_i^{(p)}$ being transmuted into Y, and one Y atom of ${\bf R}_i^{(p+1)}$ being transmuted into X. Again, the configurations ${\bf R}_{i+1}^{(p)}$ and ${\bf R}_{i+1}^{(p+1)}$ are obtained after local minimization. 4. Each new configuration is accepted with the usual Metropolis acceptance probability at temperature $T$. The algorithm has two main parameters, namely $P_{\rm ex}$ and $P_{\rm swap}$. The maximum number of particle exchange moves, $N_{\rm swap}^{\rm max}$, was set to 4 in this study. We expect that better results could be obtained by adjusting this parameter appropriately, probably taking higher values for larger clusters or for compostitions close to 50%. The amplitude of atomic displacement, $h^{(p)}$, is set to half the equilibrium distance in the X$_2$ dimer for $p=0$, half the equilibrium distance in the Y$_2$ dimer for $p=n$, and is interpolated linearly between these two values for $0<p<n$. In the present work, the exchange probabilities were taken as $P_{\rm ex}=0.5$ and $P_{\rm swap}=0.9$, hence allowing a rather large probability of sampling among homotops of a same structure. Benchmark calculations ---------------------- Low-energy structures for mixtures of xenon with either argon or krypton atoms have been first investigated for the sizes $n=13$ and $n=19$, as there are quantitative global optimization data available for Ar–Xe clusters from the Jordan group.[@jordan] We have adjusted the LJ values used by Leitner [*et al.*]{}[@leitner] to reproduce the clusters energies found by Munro and coworkers.[@jordan]. With respect to argon, the present data for $\sigma$ and $\varepsilon$ are thus $\sigma_{\rm KrKr}=1.12403$, $\sigma_{\rm XeXe}=1.206$, $\sigma_{\rm KrXe}=1.16397$, $\sigma_{\rm ArXe}=1.074$, $\varepsilon_{\rm KrKr}=1.373534$, $\varepsilon_{\rm XeXe}=1.852$, $\varepsilon_{\rm KrXe}=1.59914$, and $\varepsilon_{\rm ArXe}=1.48$. Global optimization of Ar–Xe and Kr–Xe clusters was performed using the parallel algorithm previously described, simultaneously for all compositions, for a maximum number of 10000 minimization steps per trajectory, and at $T=0$. Ten independent runs were carried out to estimate an average search length for each composition. All global minima reported by Munro [*et al.*]{} were always found within the number of MC steps allowed. [l|rr|l|rr]{} Ar$_n$Xe$_{13-n}$ & Global minimum & Average & Kr$_n$Xe$_{13-n}$ & Global minimum & Average\ cluster & energy & search length & cluster & energy & search length\ Xe$_{13}$ & -82.093 & 3.2 & Xe$_{13}$ & -82.093 & 3.0\ ArXe$_{12}$ & -78.698 & 7.9 & KrXe$_{12}$ & -81.014 & 9.6\ Ar$_2$Xe$_{11}$&-76.274 & 9.6 & Kr$_2$Xe$_{11}$& -79.263 & 4.3\ Ar$_3$Xe$_{10}$&-74.015 & 5.8 & Kr$_3$Xe$_{10}$& -77.550 & 5.7\ Ar$_4$Xe$_9$& -71.597 & 8.6 & Kr$_4$Xe$_9$ & -75.869 & 26.1\ Ar$_5$Xe$_8$& -69.017 & 14.0 & Kr$_5$Xe$_8$ & -74.186 & 25.8\ Ar$_6$Xe$_7$& -66.584 & 37.2 & Kr$_6$Xe$_7$ & -72.498 & 26.4\ Ar$_7$Xe$_6$& -63.791 & 19.4 & Kr$_7$Xe$_6$ & -70.844 & 45.2\ Ar$_8$Xe$_5$& -60.733 & 13.9 & Kr$_8$Xe$_5$ & -69.141 & 18.3\ Ar$_9$Xe$_4$& -57.851 & 22.7 & Kr$_9$Xe$_4$ & -67.473 & 4.7\ Ar$_{10}$Xe$_3$&-54.594 & 12.0 & Kr$_{10}$Xe$_3$ & -65.802 & 11.5\ Ar$_{11}$Xe$_2$&-51.122 & 7.5 & Kr$_{11}$Xe$_2$ & -64.128 & 4.1\ Ar$_{12}$Xe& -47.698 & 4.1 & Kr$_{12}$Xe & -62.490 & 2.4\ Ar$_{13}$& -44.327 & 2.7 & Kr$_{13}$ & -60.884 & 2.3\ The results for Ar$_n$Xe$_{13-n}$ and Kr$_n$Xe$_{13-n}$ clusters are given in Table \[tab:arxe13\]. The average search length is generally higher for compositions close to 50%, for which the number of homotops is maximum for a given isomer, regardless of symmetry. The statistics presently obtained for Ar–Xe clusters show that the average search is between 10 and 1000 times faster than using conventional parallel basin-hopping.[@jordan] Kr–Xe clusters roughly exhibit the same level of difficulty, but we do not see any strong evidence for particularly severe cases: Ar$_3$Xe$_{10}$ even seems to be one of the easiest. Similarly, the results obtained for Ar$_n$Xe$_{19-n}$ clusters show a significant improvement over fixed-composition basin-hopping.[@jordan] They are given in Table \[tab:arxe19\] along with the corresponding data for Kr$_n$Xe$_{19-n}$ clusters. This time, the algorithm is about 1–100 times faster depending on $n$, the average search length being still longer for equal compositions. For both the 13- and 19-atom clusters, all global minima are homotops of either the single or double icosahedron. This situation is particularly suited for our algorithm, especially the exchange moves. [l|rr|l|rr]{} Ar$_n$Xe$_{19-n}$ & Global minimum & Average & Kr$_n$Xe$_{19-n}$ & Global minimum & Average\ cluster & energy & search length & cluster & energy & search length\ Xe$_{19}$ &-134.566 & 72.4 & Xe$_{19}$ & -134.566& 70.7\ ArXe$_{18}$ &-131.819 & 64.3 & KrXe$_{18}$ & -133.651& 94.0\ Ar$_2$Xe$_{17}$&-129.116 & 80.3 & Kr$_2$Xe$_{17}$& -132.701& 109.8\ Ar$_3$Xe$_{16}$&-126.547 & 85.2 & Kr$_3$Xe$_{16}$& -130.088& 84.3\ Ar$_4$Xe$_{15}$&-123.764 & 238.2& Kr$_4$Xe$_{15}$& -129.067& 167.2\ Ar$_5$Xe$_{14}$&-120.786 & 196.6& Kr$_5$Xe$_{14}$& -127.284& 175.4\ Ar$_6$Xe$_{13}$&-118.284 & 221.2& Kr$_6$Xe$_{13}$& -125.498& 265.9\ Ar$_7$Xe$_{12}$&-115.681 & 391.8& Kr$_7$Xe$_{12}$& -123.709& 334.6\ Ar$_8$Xe$_{11}$&-113.075 & 387.9& Kr$_8$Xe$_{11}$& -121.951& 319.5\ Ar$_9$Xe$_{10}$&-110.242 & 264.2& Kr$_9$Xe$_{10}$& -120.115& 287.1\ Ar$_{10}$Xe$_9$&-107.531 & 295.8& Kr$_{10}$Xe$_9$& -118.304& 243.6\ Ar$_{11}$Xe$_8$&-104.576 & 193.8& Kr$_{11}$Xe$_8$& -116.521& 187.4\ Ar$_{12}$Xe$_7$&-101.811 & 235.5& Kr$_{12}$Xe$_7$& -114.736& 214.3\ Ar$_{13}$Xe$_6$&-98.110 & 158.5& Kr$_{13}$Xe$_6$& -112.947& 201.3\ Ar$_{14}$Xe$_5$&-94.396 & 247.3& Kr$_{14}$Xe$_5$& -111.189& 188.8\ Ar$_{15}$Xe$_4$&-90.438 & 121.3& Kr$_{15}$Xe$_4$& -108.863& 176.5\ Ar$_{16}$Xe$_3$&-86.328 & 131.2& Kr$_{16}$Xe$_3$& -106.609& 115.1\ Ar$_{17}$Xe$_2$&-81.907 & 97.3 & Kr$_{17}$Xe$_2$& -104.332& 10.2\ Ar$_{18}$Xe &-77.298 & 86.8 & Kr$_{18}$Xe & -102.036& 98.1\ Ar$_{19}$ &-72.660 & 62.2 & Kr$_{19}$ & -99.801 & 69.8\ Initially, the configurations at all compositions are random. The chances to locate the proper structure (without any consideration of the homotops) increase linearly with the number of trajectories. As soon as the right structure is found, the algorithm naturally optimizes atom types to find the most stable homotop, hence the global minimum. But it can also communicate the structure to the adjacent trajectories, until all compositions only need to sample among the permutational homotops. When the interactions are not too dissimilar (as in Kr–Xe clusters), it is likely that the mixed clusters share the same isomer as the global minimum of the homogeneous cluster, which justifies the lattice approach of Robertson [*et al.*]{}[@lattice] The problem is then reduced to locating the most stable homotops. By setting $P_{\rm swap}$ to one and starting all trajectories from this minimum, the algorithm can be even more successful, and we estimated the average search length to be further reduced by a factor about 3 with respect to the values given in Table \[tab:arxe19\]. However, when the interactions differ significantly among atoms types, or when the energy landscape of the homogeneous cluster does not display a single steep funnel, it becomes much harder to make a guess about structure in these binary clusters. Fig. \[fig:bht\] shows the mean first passage time needed to locate the global minima of Ar$_{19-n}$Xe$_n$ using the algorithm under different conditions. Disabling swap moves between atom types or exchange moves between adjacent trajectories usually attenuates the efficiency. Employing a rather high temperature is even worse, because the cluster may easily leave its optimal lattice. This contrasts with optimizing homogeneous clusters, where using a nonzero temperature helps the system to escape from a funnel.[@thbh] However, if the energy gap between homotops of the same lattice increases and gets close to the gap between different lattices, we expect the zero temperature method not to be the best. But in such cases, even the notion of a lattice should be questionned. Structural transitions {#sec:res} ====================== In this section we focus on two larger sizes, for which no global optimization result is available. The LJ$_{38}$ cluster is characterized by its archetypal two-funnel energy landscape.[@doye38] The high free-energy barrier separating these two funnels and the higher entropy of the less stable minima of the icosahedral funnel make it particularly hard to locate the truncated octahedral lowest-energy minimum using unbiased global optimization algorithms. Hence it is not surprising that this peculiar structure was first found by construction.[@lj38; @morse] Composition-induced transitions in the 38-atom clusters ------------------------------------------------------- We have attempted to locate global minima for binary Ar–Xe and Kr–Xe clusters of size 38, using the parallel basin-hopping algorithm previously described. Because of the huge number of homotops at this size, and most importantly because of the structural competition between icosahedra and truncated octahedra, we cannot be fully confident that the global optimization was successful. Therefore, the energies reported in Table \[tab:38\] for Ar–Xe clusters should be taken with caution, as they could probably be bettered. The same data for 38-atom Kr–Xe clusters is also reported in Table \[tab:krxe38\]. [l|lrc|l|lrc]{} $n$ & Mixing ratio & Energy & Point group & $n$ & Mixing ratio & Energy & Point group\ 0 & 0 & -173.928 & $O_h$ & 20 & 0.58 & -268.683 & $C_1$\ 1 & 0.03 & -179.232 & $C_s$ & 21 & 0.59 & -272.465 & $C_1$\ 2 & 0.06 & -186.333 & $C_s$ & 22 & 0.56 & -276.924 & $C_s$\ 3 & 0.15 & -191.890 & $C_1$ & 23 & 0.61 & -280.169 & $C_s$\ 4 & 0.19 & -197.767 & $C_1$ & 24 & 0.60 & -283.955 & $C_{2v}$\ 5 & 0.22 & -203.421 & $C_s$ & 25 & 0.60 & -287.679 & $C_s$\ 6 & 0.25 & -208.709 & $C_s$ & 26 & 0.59 & -290.973 & $C_1$\ 7 & 0.28 & -213.815 & $C_1$ & 27 & 0.58 & -294.157 & $C_s$\ 8 & 0.32 & -218.631 & $C_s$ & 28 & 0.58 & -297.320 & $C_{2v}$\ 9 & 0.34 & -223.491 & $C_1$ & 29 & 0.56 & -300.202 & $C_{2v}$\ 10& 0.36 & -228.209 & $C_1$ & 30 & 0.50 & -303.484 & $C_1$\ 11& 0.39 & -232.771 & $C_1$ & 31 & 0.51 & -305.987 & $C_1$\ 12& 0.40 & -237.337 & $C_1$ & 32 & 0.46 & -308.404 & $C_1$\ 13& 0.43 & -241.887 & $C_s$ & 33 & 0.43 & -310.521 & $C_1$\ 14& 0.44 & -246.117 & $C_1$ & 34 & 0.38 & -311.708 & $C_1$\ 15& 0.45 & -249.677 & $C_s$ & 35 & 0.31 & -313.772 & $C_1$\ 16& 0.46 & -253.593 & $C_s$ & 36 & 0.25 & -315.988 & $C_1$\ 17& 0.46 & -257.184 & $C_s$ & 37 & 0.14 & -318.860 & $C_1$\ 18& 0.46 & -261.079 & $C_s$ & 38 & 0 & -322.115 & $O_h$\ 19& 0.56 & -264.927 & $C_1$ & & & &\ Specifically to this cluster size, all minima found during the optimization process were categorized as either icosahedral or cubic-like, depending on the energy of the corresponding homogeneous isomer found by quenching. In cases where the cubic isomer was not found among the isomers, we performed additional optimizations starting from this structure, setting $P_{\rm swap}$ to one. This mainly occured for Ar–Xe clusters. Eventually, two series of minima were obtained for each of the icosahedral and octahedral funnels. We did not find any decahedral isomer that could compete with these structural types, even though some evidence for stabilizing decahedra by doping was reported in Ref. . We have represented in Fig. \[fig:dener\] the relative energy differences $\Delta E = E_{\rm fcc}-E_{\rm ico}$ between the most stable cubic isomers and the most stable icosahedral isomers, as they were obtained from our optimization scheme, for both the Ar–Xe and Kr–Xe mixtures. Besides some strong variations sometimes seen from one composition to the next, and which can be attributed to usual finite-size effects, general trends can be clearly observed. First, Kr$_{38-n}$Xe$_n$ clusters are always most stable in the cubic shape. Actually, changing the composition most often further stabilizes truncated octahedra, and only rarely enhances the stability of icosahedra, which occurs for $n >29$ and $n=21$. Conversely Ar$_{38-n}$Xe$_n$ clusters are preferentially found icosahedral, exceptions being $n>35$ and $n=0$. This is an example of a composition-induced structural transition between the two funnels of the energy landscape. From a computational point of view, it should be noted that the optimization algorithm was able to locate the truncated octahedral minima for Kr–Xe clusters by itself, starting from disordered minima, and that the extra runs starting from this structure only produced slightly more stable homotops. This is another illustration of the efficiency of the present parallel optimization method. The general degree of disorder is higher in icosahedral structures than in the cubic-like isomer. Hence it is more difficult to put up the latter geometry with very unlike interactions, as in Ar–Xe clusters. Cubic homotops of argon with xenon are rather distorted, but the strain is much lower with krypton instead of argon. Examples of global minima obtained at compositions $n=9$, 19, and 29 are represented in Fig. \[fig:structex\]. In Kr–Xe compounds, a progressive core-surface phase separation is seen with Kr atoms outside, in agreement with energetic arguments: atoms with the larger $\varepsilon$ prefer to occupy interior sites. In icosahedral clusters, the strain increases at such sites, especially in polytetrahedral systems. Icosahedral Kr–Xe clusters also prefer to have Xe atoms at the center, but the increased strain is too high a penalty, which explains that cubic structures are favored over icosahedra. In general, no complete phase separation is found in Ar–Xe clusters, even though Ar atoms seem to fit best at the centre of the cluster. In both cases, surface energies thus play an important role. Mixing in these clusters can be estimated using radial distribution functions.[@radial] Here we use the same index as Jellinek and Krissinel,[@jellinek] namely the overall mixing ratio $\gamma$ defined as[@jellinek] $$\begin{aligned} &\gamma(X_pY_{n-p})=\nonumber \\ &\displaystyle\frac{ E_{X_pY_{n-p}} - E_{X_p}(X_pX_{n-p})-E_{Y_{n-p}}(Y_pY_{n-p})} {E_{X_pY_{n-p}}}, \label{eq:mixing}\end{aligned}$$ where $E_{X_pY_{n-p}}$ is the binding energy of cluster $X_pY_{n-p}$, $E_{X_p}(X_pX_{n-p})$ the binding energy of subcluster $X_p$ in the homogeneous cluster $X_pX_{n-p}$ at the same atomic configuration as $X_pY_{n-p}$, and a similar definition for the last term of Eq. (\[eq:mixing\]). As seen from Table \[tab:38\], the mixing ratio increases notably in Ar–Xe clusters, up to more than 60% for some compositions. Kr–Xe clusters, despite exhibiting some core-surface segregation, show similar variations of the mixing ratio with composition, with only slightly smaller values of $\gamma$. Therefore the mixing ratio, as defined in Eq. (\[eq:mixing\]), is a rather ambiguous parameter for quantifying the extent of mixing in this small cluster. The optimal structure of an homogeneous cluster described with a pairwise potential results from a competition between maximizing the number of nearest neighbors and minimizing the strain energy, or penalty induced by distorting these bonds.[@morse] Binary Lennard-Jones systems exhibit several extra complications due to the various ways of rearranging atom types in a given structure. In these systems, the strain varies notably among the homotops, especially in clusters made of very unlike atoms. However, our results indicate that the same general rules hold for homogeneous and heterogeneous systems. In Ar$_{19}$Xe$_{19}$, the strain is rather high, but the number of contacts is also high. In Kr$_{19}$Xe$_{19}$, both the strain and the number of nearest neighbors are much smaller. To investigate the role of heterogeneity on the strain, we have computed the various contributions to the reduced strain energies in Ar$_{38-n}$Xe$_n$ clusters. The strain energies are defined for Ar–Ar, Xe–Xe, and Ar–Xe interactions as follows:[@morse] $$\begin{aligned} E^{\rm strain}_{\rm Ar-Ar} &=& V^{\rm LJ}_{\rm Ar-Ar} + N^{nn}_{\rm Ar-Ar} \varepsilon_{\rm Ar-Ar},\\ E^{\rm strain}_{\rm Ar-Xe} &=& V^{\rm LJ}_{\rm Ar-Xe} + N^{nn}_{\rm Ar-Xe} \varepsilon_{\rm ArXe},\\ E^{\rm strain}_{\rm Xe-Xe} &=& V^{\rm LJ}_{\rm Xe-Xe} + N^{nn}_{\rm Xe-Xe} \varepsilon_{\rm Xe-Xe}.\\\end{aligned}$$ In these equations, $V^{\rm LJ}_{\rm X-Y}$ is the (negative) total binding energy between atoms $X$ and $Y$, $N^{nn}_{\rm X-Y}$ is the number of X–Y nearest neighbors, and $\varepsilon_{\rm X-Y}$ is the Lennard-Jones well depth corresponding to the interaction between $X$ and $Y$ atoms. Reduced strain energies are then defined as $e^{\rm strain}= E^{\rm strain}/N^{nn}\varepsilon$, in order to account for the different magnitudes of the interactions among atom types. According to these definitions, the strain energies are always positive quantities. The strain energies in Ar$_{38-n}$Xe$_n$ clusters are represented versus composition in Fig. \[fig:strain\]. They give us some insight about the possible ways of reducing strain. The pattern exhibited by the reduced strain versus composition shows different behaviors for clusters having mostly argon or xenon atoms. For $n<20$, most strain is carried by interactions between alike atoms. This case is illustrated by Ar$_9$Xe$_{29}$ in Fig. \[fig:structex\], where a kind of core/surface phase separation occurs. Here surface energies are also important, but the situation is rather different from mixed cubic Kr–Xe clusters. Because having the xenon atoms at the inner sites of the icosahedral structure would maximise the strain of these atoms, it is much more favorable to have the smaller atoms inside and the xenon atoms outside. The cubic Kr–Xe structures, on the other hand, are not especially strained, and having the smaller atoms inside would lead to an energetic penalty. When the number of Ar atoms increases above about 19 in the 38-atom cluster, interactions between unlike atoms are significantly more strained. The case of Ar$_{29}$Xe$_9$ depicted in Fig. \[fig:structex\] is perticularly informative: Xe atoms are located scarcely among the icosahedral cluster, and relieve the structure from too much strain at the expense of only few Xe–Xe interactions. In this case, cluster structure is driven by the number of unlike interactions. It is also worth noting that a few compositions are especially weakly strained; this occurs when the global minimum is octahedral, but also in the range $19<n<24$. For these latter clusters, the core/surface segregation and the number of unlike interactions are both optimal. Polytetrahedral transitions in the 55-atom Ar–Xe clusters --------------------------------------------------------- The cubic to icosahedral transition seen above actually favors polyicosahedral (or anti-Mackay) structures. The strain reduction produced by size disparity in 38-atom Ar–Xe clusters helps in stabilizing these kinds of structures, which are otherwise replaced by multilayer (or Mackay) geometries in the homogeneous clusters. Since most LJ clusters under the size of 38 atoms are most stable as polytetrahedra,[@northby] we do not expect that changing composition will affect them to a large extent. As a notable exception, the 6-atom homogeneous LJ cluster is more stable in its octahedral isomer. The lowest energy geometries of mixed Ar–Xe clusters containing 6 atoms, represented in Fig. \[fig:arxe6\], show polytetrahedral transitions for two compositions, namely Ar$_4$Xe$_2$ and Ar$_3$Xe$_3$. This behavior mimics somewhat what was observed for the larger 38-atom cluster, only at a smaller scale. In particular, and as in Fig. \[fig:dener\], polytetrahedral arrangements are seen to be more convenient for Xenon compositions under 50%. Possible polytetrahedral structures of mixed Ar–Xe clusters have been investigated for the size 55, whose most stable isomer is well known as a perfect double layer (Mackay) icosahedron for the homogeneous system. The global optimization results are summarized in Table \[tab:55\] for all compositions. For this we also conducted complementary calculations on the 2-layer lattice. Each putative global minimum was labelled either as Mackay icosahedron or, when the lattice structure does not exactly match the multilayer icosahedron, as polytetrahedral. Most compositions become increasingly polytetrahedral as the ratio of Xenon atoms increases, even though the polytetrahedral character may often be only local. [l|lrcc|l|lrcc]{} $n$ & Mixing ratio & Energy & Point group & Type & $n$ & Mixing ratio & Energy & Point group & Type\ 0 & 0 & -279.248 & $I_h$ & MI & 28 & 0.53 & -417.785 & $C_1$ & PT\ 1 & 0.05 & -284.276 & $C_{2v}$ & MI & 29 & 0.53 & -421.225 & $C_s$ & PT\ 2 & 0.10 & -289.313 & $C_s$ & MI & 30 & 0.51 & -424.566 & $C_1$ & PT\ 3 & 0.15 & -294.360 & $C_s$ & MI & 31 & 0.49 & -427.888 & $C_1$ & PT\ 4 & 0.28 & -302.344 & $C_s$ & PT & 32 & 0.47 & -431.565 & $C_1$ & PT\ 5 & 0.34 & -310.780 & $C_s$ & PT & 33 & 0.46 & -435.680 & $C_1$ & PT\ 6 & 0.38 & -316.331 & $C_s$ & PT & 34 & 0.45 & -439.565 & $C_1$ & PT\ 7 & 0.41 & -321.770 & $C_1$ & PT & 35 & 0.42 & -443.335 & $C_1$ & PT\ 8 & 0.43 & -327.241 & $C_s$ & PT & 36 & 0.41 & -446.892 & $C_1$ & PT\ 9 & 0.44 & -332.356 & $C_1$ & PT & 37 & 0.41 & -449.965 & $C_1$ & PT\ 10& 0.47 & -337.497 & $C_s$ & PT & 38 & 0.36 & -454.356 & $C_s$ & MI\ 11& 0.50 & -342.655 & $C_s$ & PT & 39 & 0.35 & -458.203 & $C_s$ & MI\ 12& 0.55 & -347.608 & $C_1$ & PT & 40 & 0.34 & -462.564 & $C_{2v}$ & MI\ 13& 0.57 & -352.520 & $C_1$ & PT & 41 & 0.30 & -466.709 & $C_{2v}$ & MI\ 14& 0.56 & -357.586 & $C_s$ & PT & 42 & 0.28 & -472.191 & $I_h$ & MI\ 15& 0.58 & -362.483 & $C_{2v}$ & PT & 43 & 0.26 & -475.967 & $C_{5v}$ & MI\ 16& 0.58 & -367.231 & $C_1$ & PT & 44 & 0.24 & -479.739 & $D_{5d}$ & MI\ 17& 0.59 & -372.016 & $C_s$ & PT & 45 & 0.22 & -483.495 & $C_{3v}$ & MI\ 18& 0.59 & -376.740 & $C_1$ & PT & 46 & 0.20 & -487.219 & $C_{2v}$ & MI\ 19& 0.59 & -381.501 & $C_{2v}$ & PT & 47 & 0.18 & -490.910 & $C_s$ & MI\ 20& 0.60 & -386.007 & $C_s$ & PT & 48 & 0.16 & -494.585 & $C_2$ & MI\ 21& 0.60 & -390.135 & $C_1$ & PT & 49 & 0.14 & -498.231 & $C_s$ & MI\ 22& 0.59 & -395.065 & $C_1$ & PT & 50 & 0.12 & -501.860 & $C_{2v}$ & MI\ 23& 0.59 & -399.213 & $C_1$ & PT & 51 & 0.10 & -505.467 & $C_{3v}$ & MI\ 24& 0.60 & -403.205 & $C_1$ & PT & 52 & 0.08 & -509.052 & $D_{5d}$ & MI\ 25& 0.58 & -406.983 & $C_1$ & PT & 53 & 0.06 & -512.616 & $C_{5v}$ & MI\ 26& 0.56 & -410.686 & $C_1$ & PT & 54 & 0.04 & -516.170 & $I_h$ & MI\ 27& 0.56 & -414.427 & $C_s$ & PT & 55 & 0 & -517.168 & $I_h$ & MI\ Two examples of lowest energy structures are represented in Fig. \[fig:arxe55\], corresponding to $n=15$ and $n=40$. The obvious deviations of the geometry of the former from the Mackay icosahedron and the various occupation sites of the heavy atoms for both structures illustrate again that there is no simple rule that determine the most stable minima when the atomic sizes do significantly differ from each other. Temperature-induced transitions ------------------------------- We now go back to the 38-atom clusters of Kr and Xe atoms, for which the global minimum was always found to be a truncated octahedron. The extremely large number of isomers (including homotops) in the energy landscape of binary Lennard-Jones clusters, added to the expected presence of significant energy barriers between icosahedral and cubic isomers,[@doye38] prevent finite-temperature simulations from being conducted in a reliably ergodic way with the presently available tools. For example, the particle exchange moves used to accelerate convergence of sampling among homotops will likely have very low acceptance probabilities in MC simulations at low temperatures, especially for Ar–Xe clusters. Therefore, even with powerful methods such as parallel tempering or multicanonical Monte Carlo, reaching convergence in 38-atom LJ clusters does not seem currently feasible to us. As an alternative, we have chosen to investigate solid-solid transitions by means of the superposition approach.[@stillinger; @hsa] For a given cluster, databases of minima in each of the icosahedral (ICO) and truncated octahedral (FCC) funnels were constructed using the optimization algorithm. For each composition and each of the two funnels, no more than 2000 distinct minima were considered. The classical partition function of the Y$_{38-p}$Xe$_p$ cluster (Y=Ar or Kr) restricted to funnel A=FCC or ICO is approximated by a harmonic superposition over all minima of the databases, which belong to this funnel:[@hsa] $$Q_A(\beta) = \sum_{i\in A} n_i \frac{\exp(-\beta E_i)}{(\beta h\bar \nu_i)^{3n-6}}, \label{eq:qx}$$ where $\beta=1/k_BT$ is the inverse temperature, $\bar \nu_i$ the geometric mean vibrational frequency, $n_i = 2p!(n-p)!/h_i$ with $h_i$ the order of the point group of minimum $i$ and $n=38$. We do not consider quantum effects here, although they may be important at low temperatures,[@quantum] since delocalization or zero-point effects are not expected to be significant for rare gases as heavy as krypton or xenon. Within the harmonic superposition approximation, a solid-solid transition occurs when $Q_{\rm FCC}=Q_{\rm ICO}$.[@doyecalvo] This latter equation is solved numerically in $\beta$ or $T$, its solution is denoted $T_{\rm ss}$. In cases where icosahedra are energetically more stable than octahedra, a solid-solid transition can occur if some cubic structures are entropically favored, which requires lower vibrational frequencies and/or lower symmetries. We did not find such situations in our samples of Ar–Xe clusters, therefore we restrict to Kr–Xe clusters in the following. Similar to transitions between funnels, transitions between homotops will happen if their partition functions are equal. The huge number of homotops gives rise to as many values for the corresponding temperatures, and we define the homotop transition temperature $T_h$ such that $$T_h = \min_j \{ T_h^{(j)} \, | \, T_h^{(j)}>0 \}, \label{eq:th}$$ where $T_h^{(j)}$ is the transition temperature between the global minimum (homotop 0) and its homotop $j$. Equating the harmonic partition functions for these two isomers leads to the expression of $T_h^{(j)}$:[@doyecalvo] $$k_B T_h^{(j)} = \frac{E_j - E_0}{(3n-6)\ln \bar\nu_0/\bar\nu_j + \ln n_j/n_0}. \label{eq:thj}$$ Since all homotops are characterized by different vibrational and symmetry properties, the transition temperatures $T_h^{(j)}$ are not ordered exactly as the energy differences $E_j-E_0$. This reflects that solid-solid transitions involve crossover in free energy rather than binding energy. The above equation also shows that $T_h^{(j)}$ can take negative values if homotop $j$ has a higher symmetry and/or a higher vibrational frequency than the ground state. In this case the global minimum is always the free energy minimum, and no solid-solid transition occurs, hence the form of Eq. (\[eq:th\]). Finally, a third temperature has a strong consequence on cluster structure, namely the melting temperature. Its estimation from either simulations or superpositions approximations is already quite difficult for the homogeneous LJ$_{38}$ cluster,[@doye38; @ptmc38] and we did not attempt to compute it for binary clusters. However, the previous study by Frantz[@frantzar] has shown that the melting point in mixed, 13-atom Ar–Kr clusters varies quite regularly (approximately quadratically) with composition. As a simple rule, we will assume that the melting point of Kr$_{38-n}$Xe$_n$, $T_{\rm melt}(n)$, lies inside some range between the approximate melting points of Kr$_{38}$ and Xe$_{38}$, respectively. From the results obtained by Doye and Wales[@doye38] and the Monte Carlo data of Ref.  for the LJ$_{38}$ cluster, we get $T_{\rm melt}(0)\simeq 0.234$ and $T_{\rm melt}(38)\simeq 0.315$ in reduced LJ units of argon. This provides rough limits to the actual melting points of Kr–Xe clusters, for the price of neglecting finite-size effects. The transition temperatures are represented in Fig. \[fig:tss\] for all compositions in Kr$_{38-n}$Xe$_n$ clusters. We notice first that the structural transition temperature $T_{\rm ss}$ varies quite regularly with composition in both the ranges $n<19$ and $n>21$, and that it shows strong size effects between these limits. Several situations are predicted to occur depending on the relative values of $T_{\rm ss}$, $T_h$ and $T_{\rm melt}$. In most cases, $T_{\rm melt} < T_{\rm ss}$. That melting takes place at temperatures lower than the cubic/icosahedral transition simply nullifies the transition between structural types. However, this extra stability of the octahedral funnel may have a consequence on the melting point itself, which is likely to increase. Still, this situation implies that simulations will more easily reach convergence. However, there are notable exceptions for this behavior, at $n<4$, $n=21$ and $n>34$. In these clusters, heterogeneity is not sufficient for the thermodynamical behavior of the cluster to deviate too much from those of the homogeneous system. The transition between homotops usually occurs prior to melting. Thermal equilibrium within the cubic funnels thus involves several homotops (and “restricted” solid-solid transitions), and the corresponding thermodynamical state could be probably simulated using specifically designed exchange moves between outer particles within a Monte Carlo scheme. A few clusters melt before exhibiting any transition between homotops. This occurs for instance at $n=11$, 13 or 15. For these sizes the structural transition also occurs at temperatures higher than the estimated melting point. These cases should pose less problems to conventional simulations than the homogeneous cluster. Glassy behavior --------------- The previous results have shown that finite-size Ar–Xe compounds show a preferential polytetrahedral order, even for very low doping rates, over octahedral order. On the other hand, Kr–Xe clusters at the same sizes further favor cubic order. Since polytetrahedral order is known to be present in liquids and, more generally, in disordered structural glasses, it seems natural to correlate the behavior observed in these clusters to the dynamics of the corresponding bulk materials.[@jonsson] We have simulated the cooling of 108-atom binary rare-gas liquids, using a simple Metropolis Monte Carlo scheme under constant volume and temperature. Initially the atoms are placed randomly into a cubic box of side $L$, and periodic boundaries are treated in the minimum image convention. The LJ interactions were not truncated, and the simulations consisted of 100 stages of 10$^5$ MC cycles each, linearly spaced in temperature. Three compositions have been selected, following our knowledge of the cluster structure. For each composition, different length sizes $L$ and different temperature ranges $[T_{\rm min},T_{\rm max}]$ were chosen in order to cover both sides of the melting point. In the first mixture, 24 xenon atoms and 84 argon atoms are simulated with $L=4.8815$ LJ units of argon, with $0.1\leq T\leq 1$. In the second mixture, 24 xenon atoms are added to 84 krypton atoms at $L=5.487$ and $0.15\leq T\leq 1.5$. The third mixture consists of 9 argon atoms and 99 xenon atoms at $L=5.887$ and $0.2\leq T\leq 2$. Even though we did not attempt to locate the most stable crystalline forms for these mixtures, our searches close to the face-centred cubic morphology showed that the most stable configurations for these mixtures always had some cubic order. It is likely that the actual ground states for such systems are indeed crystalline.[@middleton] The average root mean square fluctuation of the bond distances, also known as the Lindemann index $\delta$, universally characterizes the thermodynamical state of the condensed system as either solid or liquid, depending on its value being lower or higher than about 0.15. To quantify the extent of crystalline order, we have used the bond order parameter $Q_4$ introduced by Steinhardt and coworkers.[@q4] The two parameters $\delta$ and $Q_4$ allow us to follow in Monte Carlo time the cooling processes for all materials in a simultaneous way, independently of thermodynamical characteristics such as the melting temperature. The correlation between $\delta$ and $Q_4$ for ten cooling simulations of each of the three bulk binary compounds is represented in Fig. \[fig:glass\]. In all cases, the Lindemann parameter covers the whole range $0.01<\delta<0.18$, indicating that the melting point was indeed crossed. However, the three compounds display very contrasted cooling behaviors. In the (Ar$_{84}$Xe$_{24}$) system, $\delta$ regularly decreases but $Q_4$ always remain below 0.05. Therefore crystallization never takes place, and the final state obtained by quenching is significantly higher in energy than some crystalline forms; this is typical of glass formation. In (Kr$_{84}$Xe$_{24}$), all simulations show some rather sharp transition from a (high $\delta$, low $Q_4$) state to a (low $\delta$, high $Q_4$) state as $\delta$ crosses about 0.1. The temperatures where crystallisation occurs may vary somewhat among the cooling runs, in the same way as they are expected to depend on the cooling rate. Lastly, the case of (Ar$_9$Xe$_{99}$) is intermediate: while most simulations end up in a nearly fully crystalline phase ($Q_4\sim 0.15$), a few of them show a limited degree of cubic ordering in the solid phase, $Q_4$ having values close to 0.07. These results very closely reflect our previous data on binary, 38-atom clusters of the same materials. In terms of composition, the first mixture corresponds to Ar$_{30}$Xe$_8$, which clearly favors icosahedral shapes over truncated octahedra. The second mixture reminds of Kr$_{30}$Xe$_8$, for which the cubic structure is even more stable than in the homogeneous cluster. The third mixture should be compared to Ar$_3$Xe$_{35}$, which favors icosahedra only moderately. This correlation found here between cluster structure and the glassforming ability of the bulk material confirms previous analyses on the icosahedral local order in liquids and glasses,[@jonsson; @tarjus] as well as the recent conclusions obtained by Doye [*et al.*]{}[@doyeglass] that clusters of good glassformers indeed show a polytetrahedral order. Conclusion {#sec:ccl} ========== As far as structural and dynamical properties are concerned, binary compounds show a significantly richer complexity with respect to homogeneous clusters. The work reported in the present paper was intended to achieve several goals. First, a parallel global optimization algorithm was designed to locate the most stable structures of mixed rare-gas clusters, beyond the lattice approximations of Robertson and co-workers.[@lattice] Based on the basin-hopping or Monte Carlo+minimization algorithm,[@mcmin; @bh] this algorithm includes exchange moves between particles at fixed composition as well as exchange moves between configurations at different compositions. Tests on simple Ar$_n$Xe$_{13-n}$ and Ar$_n$Xe$_{19-n}$ clusters show that the method is quite efficient, in addition to being easy to implement. For these systems, we have found that the choice of a very low temperature works best as it allows some significant time to be spent for optimizing the search for homotops on a same common lattice. Putative global minima for Ar$_{38-n}$Xe$_n$ and Kr$_{38-n}$Xe$_n$ clusters have been investigated for all compositions. The structure of Ar–Xe compounds is mainly polytetrahedral, except at very low doping rates. Kr–Xe clusters not only remain as truncated octahedra, but mixing the two rare gases even favors these cubic structures over icosahedra. We see some significant trend toward core/surface phase separation in Ar–Xe clusters with $n>20$ and in all Kr–Xe clusters. However, these demixing behaviors are not due to the same factors, as Xe atoms favor outer sites to reduce strain in Ar–Xe icosahedra, while they occupy interior sites to maximize the number of bonds in Kr–Xe truncated octahedra. Conversely, Ar$_{38-n}$Xe$_n$ clusters with $n<20$ exhibit a higher degree of mixing. Analysing the strain in these stable structures confirms the presence of a structural transition near $n=20$ in these systems. Polytetrahedral morphologies were also found as the most stable structures of many mixed Ar–Xe clusters with 55 atoms, as soon as the relative number of Xe atoms was large enough. The general conclusion thus seems that the extra strain introduced by mixing these different elements penalizes the highly ordered (cubic or 2-layer icosahedron) structures. Within the harmonic superposition approximation, we have estimated the temperatures required by the 38-atom Kr–Xe clusters to undergo a structural transition toward the icosahedral funnel, or toward other octahedral homotops. For compositions with a doping rate higher than 3/38, the structural transition temperature was seen to occur at temperatures higher than the extrapolated melting point. This mainly reflects the special stability of the octahedral structures, and has the probable consequence that actual melting points increase somewhat. These predictions could probably be checked with numerical simulations. For most compositions, the transitions between different homotops of the truncated octahedron are seen to be potentially induced by relatively small temperatures. Therefore particle exchange moves will be necessary in order that simulations remain close to ergodic. Following previous results by other researchers,[@jonsson; @tarjus; @doyeglass] we have found some further evidence that criteria for glass formation in bulk materials may also lie in the parameters, which are responsible for stable cluster structures. Since the atomistic simulation of the dynamical vitrification process can generally be much harder than obtaining stable configurations of atomic clusters, we expect the approach followed in the present theoretical effort to be also useful in the community of glasses and supercooled liquids. The method is obviously not limited to rare-gases, and its application to other compounds, especially metallic nanoalloys, should be straightforward, except maybe for fine tuning its intrinsic parameters. From a methodological point of view, it could also be applied to materials with more than two components. 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[l|lrc|l|lrc]{} $n$ & Mixing ratio & Energy & Point group & $n$ & Mixing ratio & Energy & Point group\ 0 & 0 & -238.897 & $O_h$ & 20 & 0.45 & -286.209 & $C_s$\ 1 & 0.09 & -241.200 & $C_s$ & 21 & 0.44 & -288.240 & $C_s$\ 2 & 0.18 & -243.604 & $C_{2v}$ & 22 & 0.42 & -290.323 & $C_{2v}$\ 3 & 0.25 & -245.962 & $C_s$ & 23 & 0.42 & -292.310 & $C_s$\ 4 & 0.27 & -248.453 & $C_s$ & 24 & 0.41 & -294.347 & $O_h$\ 5 & 0.33 & -250.927 & $C_s$ & 25 & 0.37 & -296.352 & $C_{3v}$\ 6 & 0.35 & -253.489 & $C_{2v}$ & 26 & 0.35 & -298.371 & $C_{2v}$\ 7 & 0.39 & -256.005 & $C_s$ & 27 & 0.33 & -300.382 & $C_s$\ 8 & 0.41 & -258.570 & $D_{4h}$ & 28 & 0.32 & -302.414 & $C_{4v}$\ 9 & 0.43 & -261.156 & $C_s$ & 29 & 0.29 & -304.411 & $C_{4v}$\ 10& 0.45 & -263.740 & $C_{2v}$ & 30 & 0.25 & -306.457 & $C_s$\ 11& 0.47 & -266.294 & $C_s$ & 31 & 0.22 & -308.390 & $C_s$\ 12& 0.48 & -268.867 & $C_s$ & 32 & 0.20 & -310.370 & $C_s$\ 13& 0.49 & -271.420 & $C_s$ & 33 & 0.16 & -312.315 & $C_s$\ 14& 0.50 & -273.996 & $C_{2v}$ & 34 & 0.13 & -314.287 & $C_{3v}$\ 15& 0.50 & -275.996 & $C_s$ & 35 & 0.10 & -316.230 & $C_s$\ 16& 0.50 & -278.046 & $D_{4h}$ & 36 & 0.07 & -318.200 & $D_{4h}$\ 17& 0.48 & -280.082 & $C_s$ & 37 & 0.04 & -320.133 & $C_{4v}$\ 18& 0.47 & -282.129 & $C_{2v}$ & 38 & 0 & -322.115 & $O_h$\ 19& 0.46 & -284.164 & $C_s$ & & & &\

--- abstract: | The semileptonic $B_c$-meson decays into the heavy quarkonia $J/\psi (\eta_c)$ and a pair of leptons are investigated in the framework of three-point sum rules of QCD and NRQCD. Calculations of analytical expressions for the spectral densities of QCD and NRQCD correlators with account for the Coulomb-like $\alpha_s/v$ terms are presented. The generalized relations due to the spin symmetry of NRQCD for the form factors of $B_c\to J/\psi (\eta_c)l\nu_l$ transitions with $l$ denoting one of the leptons $e, \mu $ or $\tau $, are derived at the recoil momentum close to zero. This allows one to express all NRQCD form factors through a single universal quantity, an analogue of Isgur-Wise function at the maximal invariant mass of lepton pair. The gluon condensate corrections to three-point functions are calculated both in full QCD in the Borel transform scheme and in NRQCD in the moment scheme. This enlarges the parameteric stability region of sum rule method, that makes the results of the approach to be more reliable. Numerical estimates of widths for the transitions of $B_c\to J/\psi (\eta_c)l\nu_l$ are presented. --- =-20mm =-32mm [**Semileptonic $B_c$-meson decays in sum rules of QCD and NRQCD.**]{}\ V.V. Kiselev$^{a)}$, A.K. Likhoded$^{a)}$, A.I. Onishchenko$^{b)}$\ $^{a)}$[*State Research Center of Russia “Institute for High Energy Physics”,\ Protvino, Moscow region, 142284 Russia*]{}\ $^{b)}$[*Institute for Theoretical and Experimental Physics,\ Moscow, 117218 Russia*]{}. Introduction ============ Recently, the CDF collaboration reported on the first experimental observation of the $B_c$ meson, the heavy quarkonium with the mixed heavy flavour [@cdf]. This meson stands among the families of charmonium $\bar c c$ and bottominum $\bar b b$ in what concerns its spectroscopic properties: two heavy quarks move nonrelativistically, since the confinement scale, determining the presence of light degrees of freedom (sea of gluons and quarks), is suppressed with respect to the heavy quark masses $m_Q$ as well as the Coulomb-like exchanges result in transfers about $\alpha_s m_Q^2$, which is again much less than the heavy quark mass. So, the nonrelativistic picture of binding the quarks leads to the well-known arrangement of system levels, which is very similar for the families mentioned above. The calculations of $\bar b c$-mass spectrum were reviewed in [@eichten; @prd1]. So, we expect $$M_{B_c} = 6.25\pm 0.03\; {\rm GeV,}$$ when the measurement gave $$M^{exp}_{B_c} = 6.40\pm 0.39\pm 0.13\; {\rm GeV.}$$ There is an essential difference in the production mechanisms of heavy quarkonia $\bar c c$, $\bar b b$ and $B_c$. To bind the $\bar b$ and $c$ quarks, one has to produce four heavy quarks in the flavour conserving interactions[^1], which allows one to use the perturbative QCD, since the virtualities are determined by the scale of heavy quarks mass. Thus, we see that the production of $B_c$ is relatively suppressed $\sigma(B_c)/\sigma(\bar b b) \sim 10^{-3}$ because of the additional heavy quark pair in final states. The basic peculiarities of production mechanisms appear due to the higher orders of QCD even in the leading approximation: the fragmentation regime at high transverse momenta much greater than the quark masses and a strong role of nonfragmentational contributions at $p_T\sim m_Q$, which can be exactly calculated perturbatively; a negligible contribution of octet mechanism [@octet] because there is no enhancement due to a lower order in $\alpha_s$. The predictions for the cross sections and distributions of $B_c$ in various interactions are discussed in [@prod], where we see a good agreement with the meausrements of CDF \[1\]. In contrast to $\bar c c$ and $\bar b b$ states decaying due to the annihilation into the light quarks and gluons, the $B_c$ meson is a long lived particle, since it decays due to the weak interaction. The lifetime and various modes of decays were analyzed in the framework of a) potential models [@pm], b) technique of Operator Product Expansion in the effective theory of NRQCD [@nrqcd], considering the series in both a small relative velocity $v$ of heavy quarks inside the meson and the inverse heavy quark mass [@beneke], c) QCD sum rules [@Shif; @Rein] applied to the three-point correlators [@Col; @bagan; @Kis2]. The results of potential models and NRQCD are in agreement with each other. So, we expect that the total lifetime is equal to $$\tau_{B_c} = 0.55\pm 0.15\; {\rm ps,}$$ which agrees with the experimental value given by CDF \[1\]: $$\tau^{exp}_{B_c} = 0.46^{+0.18}_{-0.16}\pm 0.03\; {\rm ps,}$$ within the accuracy available. Further, the consideration of exclusive $B_c$ decays in the framework of QCD sum rules indicated that the role of Coulomb corrections to the bare quark loop results could be very important to reach the agreement with the other approaches mentioned [@Kis2]. This requires working out the $\alpha_s/v$ corrections in NRQCD, which possesses a spin symmetry providing some relations between the exclusive form factors. For the semileptonic decays, such a relation was derived in [@Jenkins]. Note, that the CDF Collaboration observed 20 events of $B_c^+\to \psi e^+(\mu^+) \nu$, so that the consistent calculation of semileptonic decay modes is of interest. For theoretical reviews on the $B_c$ meson physics, see [@review]. In this paper we perform a detailed analysis of semileptonic $B_c$ decays in the framework of sum rules in QCD and NRQCD. We recalculate the double spectral densities available previously in [@Col] in full QCD for the massless leptons and add the analytical expressions for the form factors necessary in evaluation of decays to massive leptons and P-wave levels of quarkonium with different quark masses. We analyze the NRQCD sum rules for the three-point correlators for the first time. We derive generalized relations between the NRQCD form factors, which extends the consideration in [@Jenkins], because we explore a soft limit of recoil momentum close to zero, wherein the velocities of initial and final heavy quarkonia $v_{1,2}$ are not equal to each other, when their product tends to zero, in contrast to the hard limit $v_1=v_2$. The spin symmetry relations between the form factors are conserved after the taking into account the Coulomb $\alpha_s/v$ corrections, which can be written down in covariant form. We investigate numerical estimates in the sum rules schemes of spectral density moments and Borel transform and show an important role of Coulomb corrections. Next, we perform the calculation of gluon condensate contribution to the three-point sum rules of both full QCD and NRQCD for the case of three massive quarks, for the first time. The paper is organized as follows. The QCD sum rules of three-point correlators are considered in Section 2, where the spectral densities are calculated in the bare quark-loop approximation and with account of the Coulomb corrections, and the gluon condensate term in the Borel transform scheme is presented. Numerical estimates of semileptonic decay modes are also given here. Section 3 is devoted to the NRQCD sum rules for recoil close to zero. The spin symmetry relations are derived and the gluon condensate is taken into account in the scheme of moments. The results are summarized in Section 4. Appendices A and B contain technical details in evaluation of decay widths for the massive leptons and gluon condensate in full QCD, respectively. Three-point QCD sum rules ========================= In this paper we will use the approach of three-point QCD sum rules [@Shif; @Rein] in the study of form factors and decay rates for the transitions $B_c^{+}\to \psi (\eta_c)l^+\nu_l$, where $l$ denotes one of the leptons $e, \mu$ or $\tau$. This procedure is similar to that of two-point sum rules and the information from the latter on the coupling of mesons to their currents is required in order to extract the values for the form factors. Thus, in our work we will use the meson couplings, defined by the following equations: $$\langle 0|\bar q_1 i\gamma_5 q_2|P(p)\rangle = \frac{f_{P}M_{P}^2}{m_1 + m_2},$$ and $$\langle 0|\bar q_1\gamma_{\mu} q_2|V(p,\epsilon)\rangle = i\epsilon_{\mu}M_Vf_V,$$ where $P$ and $V$ represent the scalar and vector mesons with desired flavour quantum numbers, respectively, and $m_1, m_2$ are the quark masses. Now we would like to describe the method used. Description of the method ------------------------- As we have already said, for the calculation of hadronic matrix elements relevant to the semileptonic $B_c$-decays into the pseudoscalar and vector mesons in the framework of QCD, we explore the QCD sum rule method. The hadronic matrix elements for the transition $B_c^{+}\to \psi (\eta_c)l^+\nu_l$ can be written down as follows: $$\begin{aligned} \langle\eta_c(p_2)|V_{\mu}|B_c(p_1)\rangle &=& f_{+}(p_1 + p_2)_{\mu} + f_{-}q_{\mu},\\ \frac{1}{i}\langle J/\psi (p_2)|V_{\mu}|B_c(p_1)\rangle &=& i F_V\epsilon_{\mu\nu\alpha\beta}\epsilon^{*\nu}(p_1 + p_2)^{\alpha}q^{\beta},\\ \frac{1}{i}\langle J/\psi (p_2)|A_{\mu}|B_c(p_1)\rangle &=& F_0^A\epsilon_{\mu}^{*} + F_{+}^{A}(\epsilon^{*}\cdot p_1)(p_1 + p_2)_{\mu} + F_{-}^{A}(\epsilon^{*}\cdot p_1)q_{\mu}, \end{aligned}$$ where $q_{\mu} = (p_1 - p_2)_{\mu}$ and $\epsilon^{\mu} = \epsilon^{\mu}(p_2)$ is the polarization vector of $J/\psi$-meson. $V_{\mu}$ and $A_{\mu}$ are the flavour changing vector and axial electroweak currents. The form factors $f_{\pm}, F_V, F_0^A$ and $F_{\pm}^A$ are functions of $q^2$ only. It should be noted, that by virtue of transversality of the lepton current $l_{\mu} = l\gamma_{\mu}(1 + \gamma_5)\nu_l$ in the limit $m_l\to 0$, the probabilities of semileptonic decays into $e^{+}\nu_e$ and $\mu^{+}\nu_{\mu}$ are independent of $f_{-}$ and $F_{-}^A$. Thus, in calculation of these particular decay modes of $B_c$-meson these form factors can be consistently neglected [@Col; @pm; @Kis2]. However, since the calculation of both the semileptonic decay modes, including $e,\mu$ or $\tau$, and some hadronic decays, stands among the goals of this paper, we will present the results for the complete set of form factors given in Eqs. (3)-(5). Following the standard procedure for the evaluation of form factors in the framework of QCD sum rules, we consider the three-point functions: $$\begin{aligned} \Pi_{\mu}(p_1, p_2, q^2) &=& i^2 \int dxdye^{i(p_2\cdot x - p_1\cdot y)} \cdot \nonumber\\ &&\langle 0|T\{\bar q_1(x)\gamma_5 q_2(x), V_{\mu}(0), \bar b(y)\gamma_5 c(y)\}|0 \rangle,\\ \Pi_{\mu\nu}^{V, A}(p_1, p_2, q^2) &=& i^2 \int dxdye^{i(p_2\cdot x - p_1\cdot y)} \cdot \nonumber\\ && \langle 0|T\{\bar q_1(x)\gamma_{\mu} q_2(x), J_{\mu}^{V, A}(0), \bar b(y)\gamma_5 c(y)\}|0\rangle,\end{aligned}$$ where $\bar q_1(x)\gamma_5 q_2(x)$ and $\bar q_1(x)\gamma_{\nu}q_2(x)$ are interpolating currents for states with the quantum numbers of $\eta_c$ and $J/\psi$, correspondingly. $J_{\mu}^{V, A}$ are the currents $V_{\mu}$ and $A_{\mu}$ of relevance to the various cases. The Lorentz structures in the correlators can be written down as: $$\begin{aligned} \Pi_{\mu} &=& \Pi_{+}(p_1 + p_2)_{\mu} + \Pi_{-}q_{\mu},\label{R1}\\ \Pi_{\mu\nu}^V &=& i\Pi_V\epsilon_{\mu\nu\alpha\beta}p_2^{\alpha}p_1^{\beta},\\ \Pi_{\mu\nu}^A &=& \Pi_{0}^{A}g_{\mu\nu} + \Pi_{1}^{A}p_{2, \mu}p_{1, \nu} + \Pi_{2}^{A}p_{1, \mu}p_{1, \nu} + \Pi_{3}^{A}p_{2, \mu}p_{2, \nu} + \Pi_{4}^{A}p_{1, \mu}p_{2, \nu}.\end{aligned}$$ The form factors $f_{\pm}$, $f_V$, $F_{0}^{A}$ and $F_{\pm}^{A}$ will be determined, respectively, from the amplitudes $\Pi_{\pm}$, $\Pi_V$, $\Pi_{0}^{A}$ and $\Pi_{\pm}^{A} = \frac{1}{2}(\Pi_{1}^{A}\pm \Pi_{2}^{A})$. In (8)-(10) the scalar amplitudes $\Pi_i$ are the functions of kinematical invariants, i.e. $\Pi_i = \Pi_i(p_1^2, p_2^2, q^2)$. To calculate the QCD expression for the three-point correlators we employ the Operator Product Expansion (OPE) for the $T$-ordered product of currents in (6)-(7). The vacuum correlations of heavy quarks are related to their contribution to the gluon operators. For example, for the $\langle\bar QQ\rangle$ and $\langle\bar QGQ\rangle$ condensates the heavy quark expansion gives $$\begin{aligned} \langle\bar QQ\rangle &=& -\frac{1}{12m_Q}\frac{\alpha_s}{\pi}\langle G^2\rangle - \frac{1}{360m_Q^3}\frac{\alpha_s}{\pi}\langle G^3\rangle + ...\nonumber\\ \langle\bar QGQ\rangle &=& \frac{m_Q}{2}\log (m_Q^2)\frac{\alpha_s}{\pi}\langle G^2\rangle - \frac{1}{12m_Q}\frac{\alpha_s}{\pi}\langle G^3\rangle + ... \nonumber\end{aligned}$$ Then, in the lowest order for the energy dimension of operators the only nonperturbative correction comes from the gluon condensate: $$\Pi_i(p_1^2, p_2^2, q^2) = \Pi_i^{pert}(p_1^2, p_2^2, q^2) + \Pi_i^{G^2}(p_1^2, p_2^2, q^2)\langle\frac{\alpha_s}{\pi}G^2\rangle . \label{OPE}$$ The leading QCD term is a triangle quark loop diagram, for which we can write down the double dispersion representation at $q^2\leq 0$: $$\Pi_i^{pert}(p_1^2, p_2^2, q^2) = -\frac{1}{(2\pi)^2}\int \frac{\rho_i^{pert}(s_1, s_2, Q^2)}{(s_1 - p_1^2)(s_2 - p_2^2)}ds_1ds_2 + \mbox{subtractions}, \label{pertdisp}$$ where $Q^2 = -q^2 \geq 0$. The integration region in (\[pertdisp\]) is determined by the condition $$-1 < \frac{2s_1s_2 + (s_1 + s_2 - q^2)(m_b^2 - m_c^2 - s_1)} {\lambda^{1/2}(s_1, s_2, q^2)\lambda^{1/2}(m_c^2, s_1, m_b^2)} < 1,$$ and $$\lambda(x_1, x_2, x_3) = (x_1 + x_2 - x_3)^2 - 4x_1x_2.$$ The calculation of spectral densities $\rho_i^{pert}(s_1, s_2, Q^2)$ and gluon condensate contribution to (\[OPE\]) will be considered in underlying sections. Now let us proceed with the physical part of three-point sum rules. The connection to hadrons in the framework of QCD sum rules is obtained by matching the resulting QCD expressions of current correlators with the spectral representation, derived from a double dispersion relation at $q^2\leq 0$. $$\Pi_i(p_1^2, p_2^2, q^2) = -\frac{1}{(2\pi)^2}\int \frac{\rho_i^{phys}(s_1, s_2, Q^2)}{(s_1 - p_1^2)(s_2 - p_2^2)}ds_1ds_2 + \mbox{subtractions}. \label{physdisp}$$ Assuming that the dispersion relation (\[physdisp\]) is well convergent, the physical spectral functions are generally saturated by the lowest lying hadronic states plus a continuum starting at some effective thresholds $s_1^{th}$ and $s_2^{th}$: $$\begin{aligned} \rho_i^{phys}(s_1, s_2, Q^2) &=& \rho_i^{res}(s_1, s_2, Q^2) + \\ && \theta (s_1-s_1^{th})\cdot\theta (s_2-s_2^{th})\cdot \rho_i^{cont}(s_1, s_2, Q^2), \nonumber\end{aligned}$$ where $$\begin{aligned} \rho_i^{res}(s_1, s_2, Q^2) &=& {\langle 0|\bar c\gamma_{\mu}(\gamma_5) c|J/\psi (\eta_c) \rangle\langle J/\psi (\eta_c)|F_i(Q^2)|B_c\rangle\langle B_c|\bar b\gamma_5 c|0)\rangle}\cdot \nonumber\\ && {(2\pi)^2 \delta(s_1-M_1^2) \delta(s_2-M_2^2)} + \mbox{higher~state~contributions},\end{aligned}$$ where $M_{1,2}$ denote the masses of quarkonia in the initial and final states. The continuum of higher states is modelled by the perturbative absorptive part of $\Pi_i$, i.e. by $\rho_i$. Then, the expressions for the form factors $F_i$ can be derived by equating the representations for the three-point functions $\Pi_i$ in (\[OPE\]) and (\[physdisp\]), which means the formulation of sum rules. Calculating the spectral densities ---------------------------------- In this section we present the analytical expressions to one loop approximation for the perturbative spectral functions. We have recalculated their values, already available in the literature [@Col]. Among new results there are the expressions for $\rho_{-}, \rho_{-}^{A}$ and $\rho_{\pm}^{'A}$, where $\rho_{\pm}^{'A}$ are spectral functions, which come from the double dispersion representation of $\Pi_{\pm}^{'A} = \frac{1}{2}(\Pi_{3}^{A}\pm \Pi_{4}^{A})$. These spectral densities are not required for the purposes of this paper, but they will be useful for calculation of form factors for the transition of $B_c$-meson into a scalar meson[^2]. The procedure of evaluating the spectral functions involves the standard use of Cutkosky rules [@Cutk]. There is, however, one subtle point in using these rules. At $Q^2 < 0$ there is no problem in applying the Cutkosky rules in order to determine $\rho_i(s_1, s_2, Q^2)$ and the limits of integration over $s_1, s_2$. At $Q^2 >0$, which is the physical region, non-Landau-type singularities appear [@Ball1; @Ball2], what makes the determination of spectral functions to be quite complicated. In our case we restrict the region of integration in $s_1$ and $s_2$ by $s_1^{th}$ and $s_2^{th}$, so that at moderate values of $Q^2$ the non-Landau singularities do not contribute to the values of spectral functions. For spectral densities $\rho_i(s_1, s_2, Q^2)$ we have the following expressions: $$\begin{aligned} \rho_{+}(s_1, s_2, Q^2) &=& \frac{3}{2k^{3/2}}\{\frac{k}{2}(\Delta_1 + \Delta_2) - k[m_3(m_3 - m_1) + m_3(m_3 - m_2)] - \nonumber\\ && [2(s_2\Delta_1 + s_1\Delta_2) - u(\Delta_1 + \Delta_2)]\\ && \cdot [m_3^2 - \frac{u}{2} + m_1m_2 - m_2m_3 - m_1m_3]\}, \nonumber \\ \rho_{-}(s_1, s_2, Q^2) &=& - \frac{3}{2k^{3/2}}\{\frac{k}{2}(\Delta_1 - \Delta_2) - k[m_3(m_3 - m_1) - m_3(m_3 - m_2)] + \nonumber\\ && [2(s_2\Delta_1 - s_1\Delta_2) + u(\Delta_1 - \Delta_2)]\\ && \cdot [m_3^2 - \frac{u}{2} + m_1m_2 - m_2m_3 - m_1m_3]\}, \nonumber \\ \rho_{V}(s_1, s_2, Q^2) &=& \frac{3}{k^{3/2}}\{(2s_1\Delta_2 - u\Delta_1)(m_3 - m_2) \nonumber \\ && + (2s_2\Delta_1 - u\Delta_2)(m_3 - m_1) + m_3k\}, \\ \rho_{0}^A(s_1, s_2, Q^2) &=& \frac{3}{k^{1/2}}\{ (m_1 - m_3)[m_3^2 + \frac{1}{k}(s_1\Delta_2^2 + s_2\Delta_1^2 - u\Delta_1\Delta_2)] \nonumber \\ && - m_2(m_3^2 - \frac{\Delta_1}{2}) - m_1(m_3^2 - \frac{\Delta_2}{2}) \\ && + m_3[ m_3^2 - \frac{1}{2}(\Delta_1 + \Delta_2 - u) + m_1m_2]\}, \nonumber\\ \rho_{+}^A(s_1, s_2, Q^2) &=& \frac{3}{2k^{3/2}}\{m_1[2s_2\Delta_1 - u\Delta_1 + 4\Delta_1\Delta_2 + 2\Delta_2^2]\nonumber \\ && m_1m_3^2[4s_2 - 2u] + m_2[2s_1\Delta_2 - u\Delta_1] - m_3[2(3s_2\Delta_1 + s_1\Delta_2) \nonumber \\ && - u(3\Delta_2 + \Delta_1) + k + 4\Delta_1\Delta_2 + 2\Delta_2^2 + m_3^2(4s_2 - 2u)] \\ && + \frac{6}{k}(m_1 - m_3)[4s_1s_2\Delta_1\Delta_2 - u(2s_2\Delta_1\Delta_2 + s_1\Delta_2^2 + s_2\Delta_1^2)\nonumber \\ && + 2s_2(s_1\Delta_2^2 + s_2\Delta_1^2)]\}, \nonumber \\ \rho_{-}^A(s_1, s_2, Q^2) &=& -\frac{3}{2k^{5/2}}\{ kum_3(2m_1m_3 - 2m_3^2 + u) + 12(m_1 - m_3)s_2^2\Delta_1^2 + \nonumber \\ && k\Delta_2[(m_1 + m_3)u - 2s_1(m_2 - m_3)] + 2\Delta_2^2(k + 3us_1)(m_1-m_3) \nonumber \\ && + \Delta_1[ku(m_2 - m_3) + 2\Delta_2(k - 3u^2)(m_1 - m_3)] + \\ && 2s_2(m_1 - m_3)[2km_3^2 - k\Delta_1 + 3u\Delta_1^2 - 6u\Delta_1\Delta_2] - \nonumber\\ && 2s_1s_2(km_3 - 3\Delta_2^2(m_1 - m_3))],\nonumber \\ \rho_{+}^{'A}(s_1, s_2, Q^2) &=& -\frac{3}{2k^{5/2}}\{ -2(m_1 - m_3) [(k - 3us_2)\Delta_1^2 + 6s_1^2\Delta_2^2] + \nonumber \\ && ku(m_1 - m_3)(2m_3^2 + \Delta_2) + ku^2m_3 + \Delta_1[ku(2m_1 - m_2 - 3m_3)\nonumber \\ && - 2(m_1 - m_3)(ks_2 - k\Delta_2 + 3u^2\Delta_2)] - \\ && 2s_1[(m_1-m_3)(2km_3^2 - 6u\Delta_1\Delta_2 - 3u\Delta_2^2) + \nonumber \\ && 2s_2(km_3 + 3m_1\Delta_1^2 - 3m_3\Delta_1^2) + k\Delta_2(2m_1 - m_2 - 3m_3)]\},\nonumber \\ \rho_{-}^{'A}(s_1, s_2, Q^2) &=& \frac{3}{2k^{5/2}}\{ 2(m_1 - m_3) [(k + 3us_2)\Delta_1^2 + 6s_1^2\Delta_2^2] + \nonumber \\ && ku(m_1 - m_3)(2m_3^2 + \Delta_2) + ku^2m_3 + \Delta_1[ku(- 2m_1 - m_2 + m_3)\nonumber \\ && - 2(m_1 - m_3)(ks_2 - k\Delta_2 + 3u^2\Delta_2)] + \\ && 2s_1[(m_1-m_3)(2km_3^2 - 6u\Delta_1\Delta_2 + 3u\Delta_2^2) - \nonumber \\ && 2s_2(km_3 - 3m_1\Delta_1^2 + 3m_3\Delta_1^2) + k\Delta_2(2m_1 + m_2 - m_3)]\}.\nonumber\end{aligned}$$ Here $k = (s_1 + s_2 + Q^2)^2 - 4s_1s_2$, $u = s_1 + s_2 + Q^2$, $\Delta_1 = s_1 - m_1^2 + m_3^2$ and $\Delta_2 = s_2 - m_2^2 + m_3^2$. $m_1, m_2$ and $m_3$ are the masses of quark flavours relevant to the various decays, see prescriptions in Fig. 1. (80,80) (0,0)[ ]{} (0,20)[$p_1$]{} (60,20)[$p_2$]{} (30,15)[$k$]{} (30,56)[$q$]{} (30,25)[$m_3$]{} (6,38)[$p_1+k$]{} (48,38)[$p_2+k$]{} (21,38)[$m_1$]{} (36,38)[$m_2$]{} We neglect hard $O(\frac{\alpha_s}{\pi})$ corrections to the triangle diagrams, as they are not available yet. Nevertheless, we expect that their contributions are quite small $\sim 10\%$ and so, taking into account the accuracy of QCD sum rules, the correction will not change drastically our results. In expressions (\[pertdisp\]) the integration over $s_1$ and $s_2$ is performed in the near-threshold region, where instead of $\alpha_s$, the expansion should be done in the parameters $(\alpha_s/v_{13(23)})$, with $v_{13(23)}$ meaning the relative velocities of quarks in $(b\bar c)$ and $(c\bar c)$ systems. For the heavy quarkonia, where the quark velocities are small, these corrections take an essential role (as it is the case for two-point sum rules [@Novikov; @Schw]). The $\alpha_s/v$ corrections, caused by the Coulomb-like interaction of quarks, are related with the ladder diagrams, shown in Fig. 2. It is well known, that an account of these corrections in two-point sum rules numerically leads to a double-triple multiplication of Born value of spectral density [@fbc; @Grigor]. (80,80) (0,0)[ ]{} (0,0)[$E_1$]{} (60,0)[$E_2$]{} (30,-3)[$p$]{} (30,36)[$q$]{} (9,-2)[$p_1\cdots p_n$]{} (44,-2)[$q_1\cdots q_k$]{} (16,7)[$\cdots$]{} (42,7)[$\cdots$]{} (10.5,4.5)[$k_1$]{} (24,6)[$k_n$]{} (35,6)[$l_1$]{} (50,4.5)[$l_k$]{} Now, let us comment the effect of these corrections in the case of three-point sum rules [@Kis2]. Consider, for example, the three-point function $\Pi_{\mu}(p_1, p_2, q)$ at $q^2 = q_{max}^2$, where $q_{max}^2$ is the maximum invariant mass of the lepton pair in the decay $B_c\to\eta_c l\nu_l$. Introduce the notations $p_1\equiv (m_b + m_c +E_1,\vec 0)$ and $p_2\equiv (2m_c + E_2,\vec 0)$. At $s_1 = M_1^2$ and $s_2 = M_2^2$ we have $E_1\ll (m_b + m_c)$ and $E_2\ll 2m_c$. In this kinematics, the quark velocities are small, and, thus, the diagram in Fig. 2 may be considered in the nonrelativistic approximation. We will use the Coulomb gauge, in which the ladder diagrams with the Coulomb-like gluon exchange are dominant. Then the gluon propagator has the form $$D^{\mu\nu} = i\delta^{\mu 0}\delta^{\nu 0}/{\bf k}^2.$$ In this approximation, the nonrelativistic potential of heavy quark interaction in the momentum representation is given by $$\begin{aligned} \tilde V({\bf k}) &=& -\frac{4}{3}\alpha_s({\bf k}^2)\frac{4\pi}{{\bf k}^2}, \quad \alpha_s({\bf k}^2) = \frac{4\pi}{b_0 ln({\bf k}^2/\Lambda^2)}, \nonumber\\ b_0 &=& 11 -\frac{2}{3}n_f, \quad \Lambda = \Lambda_{\overline{MS}}\exp\left [ \frac{1}{b_0}\left (\frac{31}{6} - \frac{5}{9}n_f\right )\right ],\nonumber\end{aligned}$$ with $n_f$ being the number of flavours, while the fermionic propagators, corresponding to either a particle or antiparticle, have the following forms: $$\begin{aligned} S_F(k + p_i) &=& \frac{i(1 + \gamma_0)/2}{E_i + k^0 - \frac{|{\bf k}|^2}{2m} + i0},\nonumber\\ S_F(p) &=& \frac{-i(1-\gamma_0)/2}{-k^0 - \frac{|{\bf k}|^2}{2m} + i0}.\nonumber\end{aligned}$$ The notations, concerning Fig. 2, are given by $$k_i = p_{i+1} - p_i,\quad l_i = q_i - q_{i-1},\quad p_{n+1}\equiv q_0\equiv p.$$ Integration over $p_i^0, p^0$ and $q_i^0$ by means of residues yields the following expression $$\begin{aligned} \Pi_{\mu}(E_1, E_2, q) &=& 2 N_c g_{\mu 0}\sum_{n=1}^{\infty} \prod_{i=1}^n \frac{d {\bf p}_i}{(2\pi)^3(\frac{|{\bf p}_i|^2}{2\mu_1} - E_1 -i0)} \tilde V (({\bf p}_{i+1} - {\bf p}_i)^2)\cdot \nonumber\\ && \frac{1}{\frac{|{\bf p}|^2}{2\mu_1} - E_1 - i0}\cdot \frac{1}{\frac{|{\bf p}|^2}{2\mu_2} - E_2 - i0}\cdot\label{C1}\\ && \sum_{k=1}^{\infty}\prod_{j=1}^{k}\frac{d {\bf q}_j} {(2\pi )^3(\frac{|{\bf q}_j|^2}{2\mu_2} - E_2 - i0)}\tilde V (({\bf q}_j - {\bf q}_{j-1})^2)\frac{d {\bf p}}{(2\pi)^3}\nonumber ,\end{aligned}$$ $$\tilde V(({\bf p}_{n+1} - {\bf p}_n)^2)\equiv \tilde V(({\bf p} - {\bf p}_n)^2), \quad \tilde V(({\bf q}_{1} - {\bf q}_0)^2) = \tilde V(({\bf q}_{1} - {\bf p})^2),$$ where $N_c$ denotes the number of colors, $\mu_1$ and $\mu_2$ are the reduced masses of the $(b\bar c)$ and $(c\bar c)$ systems, correspondingly. This three-point function may be expressed in terms of the Green’s functions for the relative motion of heavy quarks in the $(b\bar c)$ and $(c\bar c)$ systems in the Coulomb field, $G_E^{(i)}(\bf x, y)$: $$\begin{aligned} G_E^{(i)}({\bf x, y}) &=& \sum_{n=1}^{\infty}\left (\prod_{k=1}^n\int \frac{d {\bf p}_k}{(2\pi )^3(\frac{|{\bf p}_k|^2}{2\mu_i} - E_i - i0)}\right )\nonumber\\ && \times\prod_{k=1}^{n-1}\tilde V (({\bf p}_k - {\bf p}_{k+1})^2) e^{i{\bf p}_1{\bf x} - i{\bf p}_n{\bf y}}. \label{C2} \end{aligned}$$ Comparing the expressions (\[C1\]) and (\[C2\]), we find $$\begin{aligned} \Pi_{\mu} (E_1, E_2, q_{max}^2) &=& 2 N_c g_{\mu 0}\int {G_{E_1}^{(1)}({\bf x} = 0, {\bf p})G_{E_2}^{(2)}({\bf p}, {\bf y} = 0)} %%{(\frac{|{\bf p}|^2}{2\mu_1} - E_1 - i0)\; %%(\frac{|{\bf p}|^2}{2\mu_2} - E_2 - %%i0)}\; \frac{d {\bf p}} {(2\pi )^3}. \nonumber \\ &=& 2 N_c g_{\mu 0}\int G_{E_1}^{(1)}(0, z)G_{E_2}^{(2)}(z, 0) d^3 z.\end{aligned}$$ For the Green’s function we use the representation $$G_E ({\bf x, y}) = \sum_{l, m}\left (\sum_{n = l + 1}^{\infty} \frac{\Psi_{nlm}({\bf x})\Psi_{nlm}^{*}({\bf y})}{E_{nl} - E - i0} + \int %%_0^{\infty} \frac{d{ k}}{(2\pi)}\frac{\Psi_{klm}({\bf x})\Psi_{klm}^{*}({\bf y})} {{k} - E - i0}\right ).$$ Provided ${\bf x} = 0$, only the terms with $l = 0$ are retained in the sum. Then for the spectral density one has $$\begin{aligned} \rho_{\mu}(E_1, E_2, q_{max}^2)=-2 N_c g_{\mu 0}\Psi_1^C (0)\Psi_2^C (0) \int \tilde \Psi_{1E_1}^C ({\bf p}) \tilde \Psi_{2E_2}^C ({\bf p}) \cdot %\nonumber \\ %%(2\pi)^2 %%\delta (\frac{|{\bf p}|^2}{2\mu_1} - E_1 - i0)\; %%\delta (\frac{|{\bf p}|^2}{2\mu_2} - E_2 - i0)\; \frac{d {\bf p}} {(2\pi )^3},\end{aligned}$$ where $\Psi_i^C$ are the Coulomb wave functions for the $(b\bar c)$ or $(c\bar c)$ systems. An analogous expression can also be derived in the Born approximation: $$\begin{aligned} \rho_{\mu}^B(E_1, E_2, q_{max}^2)=- 2 N_c g_{\mu 0}\Psi_1^f (0)\Psi_2^f (0) \int \tilde \Psi_{1E_1}^f ({\bf p}) \tilde \Psi_{2E_2}^f ({\bf p}) \cdot %\nonumber \\ %%(2\pi)^2 %%\delta (\frac{|{\bf p}|^2}{2\mu_1} - E_1 - i0)\; %%\delta (\frac{|{\bf p}|^2}{2\mu_2} - E_2 - i0)\; \frac{d {\bf p}} {(2\pi )^3}.\end{aligned}$$ Here $\Psi_i^f$ stands for the function of free quark motion. Since the continuous spectrum Coulomb functions have the same normalization as the free states, we obtain the approximation $$\rho_{\mu}(E_1, E_2, q_{max}^2) \approx \rho_{\mu}^B(E_1, E_2, q_{max}^2) \frac{\Psi_1^C (0)\Psi_2^C (0)}{\Psi_1^f (0)\Psi_2^f (0)} \equiv\rho_{\mu}^B (E_1, E_2, q_{max}^2){\bf C}, \label{r1}$$ $${\bf C} = \left \{\frac{4\pi\alpha_s}{3v_{13}}\left [1 - \exp\left ( -\frac{4\pi\alpha_s}{3v_{13}}\right )^{-1}\right ] \frac{4\pi\alpha_s}{3v_{23}}\left [1 - \exp\left ( -\frac{4\pi\alpha_s}{3v_{23}}\right )^{-1}\right ]\right \}^{\frac{1}{2}},\label{C}$$ where $v_{13}$, $v_{23}$ are relative velocities in the $(b\bar c)$ and $(c\bar c)$ systems, respectively. For them we have the following expressions: $$\begin{aligned} v_{13} &=& \sqrt{1 - \frac{4m_1m_3}{p_1^2 - (m_1 - m_3)^2}},\\ v_{23} &=& \sqrt{1 - \frac{4m_2m_3}{p_2^2 - (m_2 - m_3)^2}}.\end{aligned}$$ Eq.(\[r1\]) is exact for the identical quarkonia in the initial and final states of transition under consideration. However, if the reduced masses are different, then the overlapping of Coulomb functions can deviate from unity, which breaks the exact validity of (\[r1\]). From a pessimistic viewpoint, this relation can serve as the estimate of upper bound on the form factor at zero recoil. In reality, this boundary is practically saturated, which means that in sum rules at low momenta inside the quarkonia, i.e. in the region of physical resonances, the most essential effect comes from the normalization factor $\bf C$, determined by the Coulomb function at the origin. The latter renormalizes the coupling constant in the quark-mesonic vertex from the bare value to the “dressed" one. After that, the motion of heavy quarks in the triangle loop is very close to that of free quarks. In accordance with (\[R1\]) for the Lorentz decomposition of $\rho_{\mu} (p_1, p_2, q)$ we have $$\rho_{\mu}(p_1, p_2, q^2) = (p_1 + p_2)_{\mu}\rho_{+}(q^2) + q_{\mu}\rho_{-}(q^2).$$ As we have seen, the nonrelativistic expression of $\rho_\mu (E_1, E_2, q_{max}^2)$ is proportional to the vector $(g_{\mu 0})$, which allows us to isolate the evident combination of form factors $f_{\pm}$. The relations between the form factors appearing in NRQCD at the recoil momentum close to zero will be considered below. Here we stress only that we have $$\rho_{+}(q_{max}^2) = \rho_{+}^B(q_{max}^2){\bf C}, \label{AR}$$ where the factor ${\bf C}$ has been specified in (\[C\]). In the case of $B_c\to J/\psi l\nu_l$ transition, one can easily obtain an analogous result for $\rho_0^A(q^2)$ (note that the form factor $F_0^A\sim \rho_0^A$ gives the dominant contribution to the width of this decay [@Jenkins]). In the nonrelativistic approximation we have $$\rho_0^A(q_{max}^2) = \rho_0^{A,B} (q_{max}^2){\bf C}.\label{AC}$$ To conclude this section, note that the derivation of formulas (\[AR\]) and (\[AC\]) is purely formal, since the spectral densities are not specified at $q_{max}^2$ (one can easily show $\rho_i^B (q^2)$ to be singular in this point). Therefore, the resultant relations are valid only for $q^2$ approaching $q_{max}^2$. Unfortunately, this derivation does not give the $q^2$ dependence of the factor ${\bf C}$. But we suppose that ${\bf C}$ does not crucially effect the pole behaviour of the form factors. Therefore, the resultant widths of transitions can be treated as the saturated upper bounds in the QCD sum rules. Gluon condensate contribution ----------------------------- In this subsection we will discuss the calculation of Borel transformed Wilson coefficient of the gluon condensate operator for the three-point sum rules with arbitrary masses. The technique used is the same as in [@Ball2] with some modifications to simplify the resulting expression. As was noted in [@Ball2], this method does not allow for the subtraction of continuum contributions, which, however, only a little change our results as the total contribution of the gluon condensate to the three point sum rule is small by itself ($\leq 10\%$), and, thus, its continuum portion is small, too. The form of the obtained expression does not permit us to use the same argument as in [@Ball2] to argue on the absence of those contributions at all. Their argument was based on an expectation, that the typical continuum contribution can show up as incomplete $\Gamma$ functions in the resulting expression and the absence of them in the final answer leads authors of [@Ball2] to the conclusion, that such contributions are actually absent in the processes, they considered. The gluon condensate contribution to three-point sum rules is given by diagrams, depicted in Fig. \[cond\]. For calculations we have used the Fock-Schwinger fixed point gauge [@Fock; @Schw]: $$x^{\mu}A_{\mu}^a (x) = 0,$$ where $A_{\mu}^a$, $a = \{1, 2, ...,8\}$ is the gluon field. (150,180) (0,120)[ ]{} (75,120)[ ]{} (0,60)[ ]{} (75,60)[ ]{} (0,0)[ ]{} (75,0)[ ]{} (0,140)[$p_1$]{} (60,140)[$p_2$]{} (30,135)[$k$]{} (30,176)[$0$]{} (62,160)[$k_1$]{} (52,172)[$k_2$]{} (6,158)[$p_1+k$]{} (54,148)[$p_2+k$]{} (0,80)[$p_1$]{} (60,80)[$p_2$]{} (30,75)[$k$]{} (30,116)[$0$]{} (16,62)[$k_1$]{} (43,62)[$k_2$]{} (-3,98)[$p_1+k+k_1$]{} (46,98)[$p_2+k-k_2$]{} (0,20)[$p_1$]{} (60,20)[$p_2$]{} (30,15)[$k$]{} (30,56)[$0$]{} (16,2)[$k_1$]{} (-2,42)[$k_2$]{} (46,38)[$p_2+k$]{} (75,20)[$p_1$]{} (135,20)[$p_2$]{} (105,15)[$k$]{} (105,56)[$0$]{} (82,52)[$k_1$]{} (126,52)[$k_2$]{} (78,38)[$p_1+k$]{} (121,38)[$p_2+k$]{} (75,80)[$p_1$]{} (135,80)[$p_2$]{} (105,75)[$k$]{} (105,116)[$0$]{} (82,112)[$k_2$]{} (73,102)[$k_1$]{} (121,98)[$p_2+k$]{} (75,140)[$p_1$]{} (135,140)[$p_2$]{} (105,135)[$k$]{} (105,176)[$0$]{} (137,160)[$k_2$]{} (118,122)[$k_1$]{} (81,158)[$p_1+k$]{} In the evaluation of diagrams in Fig. \[cond\] we encounter integrals of the type[^3] $$\begin{aligned} I_{\mu_1\mu_2...\mu_n}(a,b,c) = \int\frac{d^4 k}{(2\pi)^4} \frac{k_{\mu_1}k_{\mu_2}...k_{\mu_n}}{[k^2 - m_3^2]^a[(p_1 + k)^2 - m_1^2]^b [(p_2 + k)^2 - m_2^2]^c}.\end{aligned}$$ Continuing to Euclidean space-time and employing the Schwinger representation for propagators, $$\frac{1}{[p^2 + m^2]^a} = \frac{1}{\Gamma (a)}\int_0^{\infty} d\alpha\alpha^{a-1}e^{-\alpha (p^2 + m^2)},$$ we find the following expression for the scalar integral ($n = 0$): $$\begin{aligned} I_0 (a, b, c) &=& \frac{(-1)^{a + b + c}i}{\Gamma (a)\Gamma (b)\Gamma (c)} \int_0^{\infty}\int_0^{\infty}\int_0^{\infty} d\alpha d\beta d\gamma \alpha^{a-1}\beta^{b-1}\gamma^{c-1}\nonumber\\ && \int\frac{d^4 k}{(2\pi )^4} e^{-\alpha (k^2 + m_3^2) - \beta (s_1 + k^2 + 2p_1\cdot k + m_1^2) - \gamma (s_2 + k^2 + 2p_2\cdot k + m_2^2)} .\end{aligned}$$ This representation proves to be very convenient for applying the Borel transformation with $$\hat B_{p^2}(M^2)e^{-\alpha p^2} = \delta (1 - \alpha M^2).$$ Then, we have $$\begin{aligned} \hat I_0 (a, b, c) &=& \frac{(-1)^{a + b + c}i}{\Gamma (a)\Gamma (b)\Gamma (c) 16\pi^2} (M_{cc}^2)^{2 - a -c}(M_{bc}^2)^{2 - a - b}\cdot\nonumber\\ && U_0 (a + b + c - 4,1 - c -b),\\ \hat I_{\mu} (a, b, c) &=& \frac{(-1)^{a + b + c + 1}i}{\Gamma (a)\Gamma (b) \Gamma (c)16\pi^2}\left (\frac{p_{1\mu}}{M_{bc}^2} + \frac{p_{2\mu}}{M_{cc}^2} \right )(M_{cc}^2)^{3 - a - c}(M_{bc}^2)^{3 - a - b}\cdot\nonumber\\ && U_0 (a + b + c - 5, 1 - c - b),\\ \hat I_{\mu\nu} (a, b, c) &=& \frac{(-1)^{a + b + c}i}{\Gamma (a)\Gamma (b) \Gamma (c)16\pi^2}\left (\frac{p_{1\mu}}{M_{bc}^2} + \frac{p_{2\mu}}{M_{cc}^2} \right )\cdot\nonumber \\ && \left (\frac{p_{1\nu}}{M_{bc}^2} + \frac{p_{2\nu}}{M_{cc}^2} \right )(M_{cc}^2)^{4 - a - c}(M_{bc}^2)^{4 - a - b}U_0 (a + b + c - 6, 1 - c - b) \nonumber \\ && + \frac{g_{\mu\nu}}{2}\frac{(-1)^{a + b + c +1}i}{\Gamma (a)\Gamma (b) \Gamma (c)16\pi^2}(M_{cc}^2)^{3 - a - c}(M_{bc}^2)^{3 - a - b}\cdot\\ && U_0 (a + b + c - 6, 2 - c -b),\nonumber\end{aligned}$$ where $M_{bc}^2$ and $M_{cc}^2$ are the Borel parameters in the $s_1$ and $s_2$ channels, respectively. Here we have introduced the $U_0(a,b)$ function, which is given by the following expression: $$U_0 (a, b) = \int_0^{\infty} dy (y + M_{bc}^2 + M_{cc}^2)^ay^b \exp \left [-\frac{B_{-1}}{y} - B_0 - B_1y\right ],$$ where $$\begin{aligned} B_{-1} &=& \frac{1}{M_{cc}^2M_{bc}^2}(m_2^2M_{bc}^4 + m_1^2M_{cc}^4 + M_{cc}^2M_{bc}^2(m_1^2 + m_2^2 - Q^2)), \nonumber\\ B_{0} &=& \frac{1}{M_{bc}^2M_{cc}^2}(M_{cc}^2(m_1^2 + m_3^2) + M_{bc}^2(m_2^2 + m_3^2))\label{CD},\\ B_{1} &=& \frac{m_3^2}{M_{bc}^2M_{cc}^2}. \nonumber\end{aligned}$$ Then, one can express the results of calculation for any diagram in Fig. \[cond\] through $\hat I_0(a,b,c)$, $\hat I_{\mu}(a,b,c)$, $\hat I_{\mu\nu}(a,b,c)$ and their derivatives over the Borel parameters, using the partial fractioning of the integrand expression together with the following relation: $$\begin{aligned} && \hat B_{p_1^2}(M_{bc}^2)\hat B_{p_2^2}(M_{cc}^2)[p_1^2]^{m}[p_2^2]^{n} I_{\mu_1\mu_2 ...\mu_n} (a,b,c) =\nonumber\\ && [M_{bc}^2]^{m}[M_{cc}^2]^{n}\frac{d^m}{d(M_{bc}^2)^{m}} \frac{d^n}{d(M_{cc}^2)^{n}}[M_{bc}^2]^{m}[M_{cc}^2]^{n}\hat I_{\mu_1\mu_2 ...\mu_n} (a,b,c).\end{aligned}$$ The obtained expression can be further written down in terms of three quantities $\hat I_0(1,1,1)$, $\hat I_{\mu}(1,1,1)$, $\hat I_{\mu\nu}(1,1,1)$ and their derivatives over the Borel parameters and quark masses by means of $$\hat I_{\mu_1\mu_2 ...\mu_n}(a, b, c) = \frac{1}{\Gamma (a) \Gamma (b)\Gamma (c)}\frac{d^{a - 1}}{d (m_3^2)^{a - 1}} \frac{d^{b - 1}}{d (m_1^2)^{b-1}}\frac{d^{c - 1}}{d (m_2^2)^{c-1}} \hat I_{\mu_1\mu_2 ...\mu_n}(1, 1, 1).$$ However, contrary to the case, discussed in [@Ball2], in such calculations the values of parameters $a, b$ will arise, for which the $U_0(a, b)$-function has no analytical expression (it is connected to nonzero $m_3$ mass in our case). The analytical approximations for $U_0(a, b)$ at these values of parameters lead to very cumbersome expressions. The search for the most compact form of the final answer leads to the conclusion, that the best decision in this case is to express the result in terms of $U_0(a, b)$-function at different values of its parameters. For this purpose we have used the following transformation properties of $U_0(a, b)$: $$\begin{aligned} \frac{d U_0(a, b)}{d M_{bc}^2} &=& aU_0(a - 1, b) - \left (\frac{m_2^2}{M_{cc}^2} - \frac{m_1^2M_{cc}^2}{M_{bc}^4}\right ) U_0(a, b - 1) + \nonumber\\ && \frac{m_1^2 + m_3^2}{M_{bc}^4}U_0(a, b) + \frac{m_3^2}{M_{bc}^4M_{cc}^2}U_0(a, b + 1),\\ \frac{d U_0(a, b)}{d M_{cc}^2} &=& aU_0(a - 1, b) - \left (\frac{m_1^2}{M_{bc}^2} - \frac{m_2^2M_{bc}^2}{M_{cc}^4}\right ) U_0(a, b - 1) + \nonumber\\ && \frac{m_2^2 + m_3^2}{M_{cc}^4}U_0(a, b) + \frac{m_3^2}{M_{bc}^2M_{cc}^4}U_0(a, b + 1).\end{aligned}$$ In Appendix B we have presented an analytical expression, obtained in this way, for the Wilson coefficient of gluon condensate operator, contributing to the $\Pi_1 = \Pi_{+} + \Pi_{-}$ amplitude. One can see that, even in this form the obtained results are very cumbersome. So, we have realized the gluon condensate corrections as C++ codes, where the functions $U_0(a, b)$ are evaluated numerically. Analytical approximations, which can be made for the $U_0(a, b)$ functions are discussed in Appendix B. In Fig. \[qcdf+\] we have shown the effect of gluon condensate on the $f_{1}(0)$ form factor in the Borel transformed three-point sum rules. (140,100) (0,10)[ ]{} (100,10)[$M^2_{cc}$, GeV$^2$]{} (10,75)[$f_1(0)$]{} We can draw the conclusion that the calculation of gluon condensate term in full QCD sum rules allows one to enlarge the stability region in the parameter space for the form factors, which indicates the reliability of sum rules technique. Numerical results on the form factors ------------------------------------- First, we evaluate the form factors in the scheme of spectral density moments. This scheme is not strongly sensitive to the values of threshold energies, determining the region of resonance contribution. In the calculations we put $$\begin{aligned} k_{th}(\bar b c) & = & 1.5\; {\rm GeV},\nonumber \\ k_{th}(\bar c c) & = & 1.2\; {\rm GeV}, \\ m_b & = & 4.6\; {\rm GeV},\nonumber \\ m_c & = & 1.4\; {\rm GeV},\nonumber \end{aligned}$$ where $k_{th}$ is the momentum of quark motion in the rest frame of quarkonium. The chosen values of threshold momenta correspond to the minimal energy of heavy meson pairs in specified channels. The typical behaviour of form factors in the moment scheme of QCD sum rules is presented in Fig. \[f2mom\]. The evaluation of Coulomb corrections strongly depends on the appropriate set of $\alpha_s$ for the quarkonia under consideration. The corresponding scale of gluon virtuality is determined by quite a low value close to the average momentum transfer in the system. So, the expected $\alpha_s$ is about 0.5. To decrease the uncertainty we consider the contribution of Coulomb rescattering in the two-point sum rules giving the leptonic constants of heavy quarkonia. These sum rules are quite sensitive to the value of strong coupling constant, as the perturbative contribution depends on it linearly. The observed value for the charmonium, $f_\psi\approx 410$ MeV, can be obtained in this technique at $\alpha_s^{coul}(\bar c c) = 0.6$. The value $f_{B_c}=385$ MeV, as it is predicted in QCD sum rules [@fbc], gives $\alpha_s^{coul}(\bar b c) = 0.45$. We present the results of the Coulomb enhancement for the form factors in Table \[form\]. (140,100) (0,-120)[ ]{} (0,60)[$F_{0}^A(0)$, GeV]{} (30,10)[$n_{bc}$]{} (80,10)[$m_{cc}$]{} (140,100) (0,-120)[ ]{} (0,60)[$F_{0}^A(0)$, GeV]{} (30,10)[$n_{bc}$]{} (80,10)[$m_{cc}$]{} The result after the introduction of the Coulomb correction is shown in Fig. \[f3mom\]. Such large corrections to the form factors should not lead to a confusion, as they are resulted from the fare account of the Coulomb corrections both for bare quark loop diagram and meson coupling constant. In the scheme of the Borel transformation we find a strong dependence on the thresholds of continuum contribution. We think that this dependence reflects the influence of contributions coming from the excited states. So, the choice of $k_{th}$ values in the same region as in the scheme of spectral density moments results in the form factors, which are approximately 50% greater than the predictions in the moments scheme, where the higher excitations numerically are not essential. In this case we can explore the ideology of finite energy sum rules [@fesr], wherein the choice of interval for the quark-hadron duality, expressed by means of sum rules, allows one to isolate the contribution of basic states only. So, if we put $$\begin{aligned} k_{th}(\bar b c) & = & 1.2\; {\rm GeV},\nonumber \\ k_{th}(\bar c c) & = & 0.9\; {\rm GeV}, \end{aligned}$$ then the region of the lowest bound states is taken into account in both channels of initial and final states, and the Borel transform scheme leads to the results, which are very close to those of moment scheme. The dependence of calculated values on the Borel parameters is presented in Figs. \[f4bor\] and \[f5bor\], in the bare and Coulomb approximations, respectively. approx. $f_+$ $f_-$ $F_V$, GeV$^{-1}$ $F_0^A$, GeV $F_+^A$, GeV$^{-1}$ $F_-^A$, GeV$^{-1}$ --------- ------- -------- ------------------- -------------- --------------------- --------------------- bare 0.10 -0.057 0.016 0.90 -0.011 0.018 coul. 0.66 -0.36 0.11 5.9 -0.074 0.12 : The form factors of $B_c^+$ decay modes into the heavy quarkonia at $q^2=0$ in the bare quark-loop approximation taking into account for the Coulomb correction.[]{data-label="form"} (140,100) (4,-160)[ ]{} (0,60)[$F_{0}^A(0)$, GeV]{} (30,10)[$M^2_{bc}$, GeV$^2$]{} (90,10)[$M^2_{cc}$, GeV$^2$]{} As for the dependence of form factors on $q^2$, the consideration of bare quark loop term shows that, say, for $F_0^A(q^2)$ it can be approximated by the pole function: $$F_0^A(q^2) = \frac{F_0^A(0)}{1-\frac{q^2}{M^2_{pole}}},$$ with $M_{pole} \approx 4.5$ GeV. The latter is in a good agreement with the value given in [@bagan]. However, we believe that the pole mass can change after the inclusion of $\alpha_s$ corrections[^4]. From the naive meson dominance model we expect that $M_{pole} \approx 6.3 - 6.5$ GeV. (140,100) (7,-160)[ ]{} (0,60)[$F_{0}^A(0)$, GeV]{} (30,10)[$M^2_{bc}$, GeV$^2$]{} (90,10)[$M^2_{cc}$, GeV$^2$]{} We have calculated the total widths of semileptonic decays in the region of $M_{pole}= 4.5 - 6.5$ GeV, which result in the 30% variation of predictions for the modes with the massless leptons and more sizable dependence for the modes with the $\tau$ lepton (see Table \[tot\]). To compare with other estimates we calculate the width of $\bar b\to \bar c e^+\nu$ transition as the sum of decays into the pseudoscalar and vector states and find[^5] BR$(B_c^+\to c\bar c e^+\nu)\approx 3.4\pm 0.6$ %, which is in a good agreement with the value obtained in potential models [@pm] and in OPE calculations [@beneke], where the following estimate was obtained BR$(B_c^+\to c\bar c e^+\nu)\approx 3.8$ %. In the presented results we have supposed the quark mixing matrix element $|V_{bc}|=0.040$. mode $\Gamma$, $10^{-15}$ GeV BR, % -------------------------- -------------------------- ---------------- $\eta_c e^+ \nu_e$ $11\pm 1$ $0.9\pm 0.1$ $\eta_c \tau^+ \nu_\tau$ $3.3\pm 0.9$ $0.27\pm 0.07$ $\psi e^+ \nu_e$ $28\pm 5$ $2.5\pm 0.5$ $\psi \tau^+ \nu_\tau$ $7\pm 2$ $0.60\pm 0.15$ : The width of $B_c^+$ decay modes into the heavy quarkonia and leptonic pair and the branching fractions, calculated at $\tau_{B_c}=0.55$ ps.[]{data-label="tot"} As for the hadronic decays, in the approach of factorization [@blokshif] we assume that the width of transition $B_c^+\to J/\psi (\eta_c)+$ [*light hadrons*]{} can be calculated with the same form factors after the introduction of QCD corrections, which can be easily written down as the factor $H= N_c a_1^2$. The factor $a_1$ represent the hard $\alpha_s$ corrections to the four fermion weak interaction. Numerically we put $a_1=1.2$, which yields $H \approx 4.3$. So, we find $$\begin{aligned} {\rm BR}[B_c^+\to J/\psi+\mbox{\it light hadrons}] & = & 11\pm 2 \%,\\ {\rm BR}[B_c^+\to \eta_c+\mbox{\it light hadrons}] & = & 4.0\pm 0.5 \%.\end{aligned}$$ Neglecting the decays of $\bar b\to \bar c c \bar s$, which are suppressed by both the small phase space and the negative Pauli interference of decay product with the charmed quark in the initial state [@beneke], we evaluate the branching fraction of beauty decays in the total width of $B_c$ as $${\rm BR}[B_c^+\to \bar c c+\mbox{\it X}] = 23\pm 5 \% ,$$ which is in agreement with the estimates in other approaches [@pm; @beneke], where this value is equal to 25 %. Three-point NRQCD sum rules =========================== The formulation of sum rules in NRQCD follows the same lines as in QCD, the only difference is the lagrangian, describing strong interactions of heavy quarks. Symmetry of form factors in NRQCD and one-loop approximation ------------------------------------------------------------ At the recoil momentum close to zero, the heavy quarks in both the initial and final states have small relative velocities, so that the dynamics of heavy quarks is essentially nonrelativistic. This allows us to use the NRQCD approximation in the study of mesonic form factors. As in the case of heavy quark effective theory (HQET), the expansion in the small relative velocities to the leading order leads to various relations between the different form factors. Solving these relations results in the introduction of an universal form factor (an analogue of the Isgur-Wise function) at $q^2\to q^2_{max}$. In this subsection we consider the limit $$\begin{aligned} v_1^{\mu} &\neq & v_2^{\mu}, \label{cs}\\ w &=& v_1\cdot v_2\to 1,\nonumber\end{aligned}$$ where $v_{1,2}^{\mu} = p_{1,2}^{\mu}/\sqrt{p_{1,2}^2}$ are the four-velocities of heavy quarkonia in the initial and final states. The study of region (\[cs\]) is reasonable enough, because in the rest frame of $B_c$-meson ($p_1^{\mu} = (\sqrt{p_1^2}, \vec 0)$), the four-velocities differ only by a small value $|\vec p_2|$ $(p_2^{\mu} = (\sqrt{p_2^2}, \vec p_2)$, whereas their scalar product $w$ deviates from unity only due to a term, proportional to the square of $|\vec p_2|$: $w = \sqrt{1 + \frac{|\vec p_2|^2}{p_2^2}}\sim 1 + \frac{1}{2}\frac{|\vec p_2|^2}{p_2^2}$. Thus, in the linear approximation at $|\vec p_2|\to 0$, relations (\[cs\]) are valid and take place. Here we would like to note, that (\[cs\]) generalizes the investigation of [@Jenkins], where the case of $v_1 = v_2$ was considered. This condition severely restricts the relations of spin symmetry for the form factors and, as a consequence, it provides a single connection between the form factors. As can be seen in Fig. 1, since the antiquark line with the mass $m_3$ is common to the heavy quarkonia, the four-velocity of antiquark can be written down as a linear combination of four-velocities $v_1$ and $v_2$: $$\tilde v_3^{\mu} = av_1^{\mu} + bv_2^{\mu}. \label{v3}$$ In the leading order of NRQCD for the kinematical invariants, determining the spin structure of quark propagators in the limit $w\to 1$, we have the following expressions: $$\begin{aligned} p_1^2 &\to & (m_1 + m_3)^2 = {\cal M}_{1}^2,\nonumber \\ p_2^2 &\to & (m_2 + m_3)^2 = {\cal M}_{2}^2,\nonumber \\ \Delta_1 &\to & 2m_3{\cal M}_{1},\nonumber \\ \Delta_2 &\to & 2m_3{\cal M}_{2}, \\%%\nonumber \\ u &\to & 2{\cal M}_{1}{\cal M}_{2}\cdot w,\nonumber \\ \lambda (p_1^2, p_2^2, q^2) &\to & 4{\cal M}_{1}^2{\cal M}_{2}^2 (w^2 - 1).\nonumber\end{aligned}$$ In this kinematics it is an easy task to show, that in (\[v3\]) $a = b = -\frac{1}{2}$, i.e. $$\tilde v_3^{\mu} = -\frac{1}{2}(v_1 + v_2)^{\mu}.$$ Applying the momentum conservation in the vertices on Fig. 1 we derive the following formulae for the four-velocities of quarks with the masses $m_1$ and $m_2$: $$\begin{aligned} \tilde v_1^{\mu} &=& v_1^{\mu} + \frac{m_3}{2m_1}(v_1 - v_2)^{\mu}, \\ \tilde v_2^{\mu} &=& v_2^{\mu} + \frac{m_3}{2m_2}(v_2 - v_1)^{\mu},\end{aligned}$$ and in the limit $w\to 1$, we have $\tilde v_1^2 = \tilde v_2^2 = 1$, as it should be. After these definitions have been done, it is straightforward to write down the transition form factor for the current $J_{\mu} = \bar Q_1\Gamma_{\mu} Q_2$ with the spin structure $\Gamma_{\mu} = \{\gamma_{\mu}, \gamma_5\gamma_{\mu}\}$ $$\begin{aligned} \langle H_{Q_1\bar Q_3}|J_{\mu}|H_{Q_2\bar Q_3}\rangle &=& tr[\Gamma_{\mu}\frac{1}{2}(1 + \tilde v_1^{\mu}\gamma_{\mu})\Gamma_1\frac{1}{2} (1 + \tilde v_3^{\nu}\gamma_{\nu})\cdot \nonumber \\ && \Gamma_2\frac{1}{2}(1 + \tilde v_2^{\lambda}\gamma_{\lambda})]\cdot h(m_1, m_2, m_3),\end{aligned}$$ where $\Gamma_{1}$ determines the spin state in the heavy quarkonium $Q_1\bar Q_3$ (in our case it is pseudoscalar, so that $\Gamma_{1} = \gamma_5$), $\Gamma_2$ determines the spin wave function of quarkonium in the final state: $\Gamma_2 = \{\gamma_5, \epsilon^{\mu}\gamma_{\mu}\}$ for the pseudoscalar and vector states, respectively ($H = {P,V}$). The quantity $h$ is an universal function at $w\to 1$, independent of the quarkonium spin state. So, for the form factors, discussed in our paper, we have $$\begin{aligned} \langle P_{Q_1\bar Q_3}|\bar Q_1\gamma^{\mu} Q_3|P_{Q_2\bar Q_3}\rangle &=& (c_1^{P}\cdot v_1^{\mu} + c_2^{P}\cdot v_2^{\mu})\cdot h, \\ \langle P_{Q_1\bar Q_3}|\bar Q_1\gamma^{\mu} Q_3|V_{Q_2\bar Q_3}\rangle &=& i c_V\cdot\epsilon^{\mu\nu\alpha\beta}\epsilon_{\nu}v_{1\alpha}v_{2\beta}\cdot h, \\ \langle P_{Q_1\bar Q_3}|\bar Q_1\gamma_5\gamma^{\mu} Q_3|V_{Q_2\bar Q_3}\rangle &=& (c_{\epsilon}\cdot\epsilon^{\mu} + c_1\cdot v_1^{\mu}(\epsilon\cdot v_1) + c_2\cdot v_2^{\mu}(\epsilon\cdot v_1))\cdot h,\end{aligned}$$ where $$\begin{aligned} c_{\epsilon} &=& -2,\nonumber\\ c_1 &=& -\frac{m_3(3m_1 + m_3)}{4m_1m_2},\nonumber\\ c_2 &=& \frac{1}{4m_1m_2}(4m_1m_2 + m_1m_3 + 2m_2m_3 + m_3^2),\nonumber\\ c_V &=& -\frac{1}{2m_1m_2}(2m_1m_2 + m_1m_3 + m_2m_3),\\%%\nonumber\\ c_1^{P} &=& 1 + \frac{m_3}{2m_1} - \frac{m_3}{2m_2},\nonumber\\ c_2^{P} &=& 1 - \frac{m_3}{2m_1} + \frac{m_3}{2m_2}.\nonumber \end{aligned}$$ Then for the form factors in NRQCD we have the following symmetry relations: $$\begin{aligned} f_{+}(c_1^{P}\cdot{\cal M}_2 - c_2^{P}{\cal M}_1) - f_{-}(c_1^{P}\cdot{\cal M}_2 + c_2^{P}\cdot{\cal M}_1) &=& 0,\nonumber\\ F_{0}^{A}\cdot c_V - c_{\epsilon}\cdot F_V{\cal M}_1{\cal M}_2 &=& 0, \label{Fsym}\\ F_{0}^{A}(c_1 + c_2) - c_{\epsilon}{\cal M}_1 (F_{+}^{A}({\cal M}_1 + {\cal M}_2) + F_{-}^{A}({\cal M}_1 - {\cal M}_2)) &=& 0,\nonumber \\ F_{0}^{A}c_1^{P} + c_{\epsilon}\cdot{\cal M}_1(f_{+} + f_{-}) &=& 0. \nonumber\end{aligned}$$ Thus, we can claim, that in the approximation of NRQCD, the form factors of weak currents responsible for the transitions between two heavy quarkonium states are given in terms of the single form factor, say, $F_{0}^{A}$. The exception is observed for the form factors $F_{+}^{A}$ and $F_{-}^{A}$, since the definite value is only taken by their linear combination $F_{+}^{A}({\cal M}_1 + {\cal M}_2) + F_{-}^{A}({\cal M}_1 - {\cal M}_2)$. This fact has a simple physical explanation. Indeed, the polarization of vector quarkonium $\epsilon^{\mu}$ has two components: the longitudinal term $\epsilon_L^{\mu}$ and transverse one $\epsilon_T^{\mu}$ (i.e. $(\epsilon_T\cdot v_1) = 0$). $\epsilon_L^{\mu}$ can be decomposed in terms of $v_1^{\mu}$ and $v_2^{\mu}$: $$\epsilon^{\mu} = \alpha\epsilon_L^{\mu} + \beta\epsilon_T^{\mu},\quad \alpha^2 + \beta^2 = 1,$$ where $$\begin{aligned} \epsilon_L^{\mu} &=& \frac{1}{\sqrt{s_2k}}(-2s_2p_1^{\mu} + up_2^{\mu})\to \frac{1}{\sqrt{w^2 - 1}}(-v_1^{\mu} + wv_2^{\mu}),\nonumber \\ \alpha &=& -\frac{2\sqrt{s_2}}{\sqrt{k}}(\epsilon_L\cdot p_1)\to -\frac{1}{\sqrt{w^2 - 1}}(\epsilon\cdot v_1). \label{xe}\end{aligned}$$ From (\[xe\]) one can see that the decomposition of polarization vector $\epsilon$ into the longitudinal and transverse parts in NRQCD is singular in the limit $w\to 1$: $$\epsilon^{\mu} = -\frac{1}{w^2 - 1}(-v_1^{\mu} + wv_2^{\mu})(\epsilon\cdot v_1) + \beta\cdot\epsilon_T^{\mu}.$$ It means, that the introduction to the form factor $F_{0}^{A}$ of an additional term $\Delta F_0^{A} = (w^2 - 1)\cdot\delta h$, which vanishes at $w\to 1$ and, thus, is not under control in NRQCD, leads to a finite correction for the form factors $F_{+}^{A}$ and $F_{-}^{A}$. This correction is cancelled in the special linear combination of form factors, presented in (\[Fsym\]). In the case of $v_1 = v_2$ we reproduce the single relation between form factors $F_{0}^{A}$ and $f_{\pm}$, as it was obtained early in [@Jenkins]. Thus, we have obtained the generalized relations due to the spin symmetry of NRQCD lagrangian for the case $v_1\neq v_2$ in the limit, where the invariant mass of lepton pair takes its maximum value, i.e at the recoil momentum close to zero. In the one-loop approximation for the three-point NRQCD sum rules, i.e. in the calculation of bare quark loop, the symmetry relations (\[Fsym\]) take place already for the double spectral densities $\rho_j^{NR}$ in the limit $|\vec p_2|\to 0$. We have checked, that the spectral densities of the full QCD in the NRQCD limit $w\to 1$ satisfy the symmetry relations (\[Fsym\]). It is easily seen that in this approximation, $$\rho_{0}^{A, NR} = -\frac{6m_1m_2m_3}{|\vec p_2|(m_1 + m_3)}.$$ When integrating over the resonance region, we must take into account that $$\frac{|\omega_2 - \omega_1\frac{m_{13}}{m_{23}}|m_2}{|\vec p_2|\sqrt{2\omega_1m_{13}}} \leq 1,$$ and so we see, that in the limit $|\vec p_2|\to 0$ the integration region tends to a single point. Here $p_1 = (m_1 + m_3 + \omega_1, \vec 0)$, $p_2 = (m_2 + m_3 + \omega_2, \vec p_2)$ and $m_{ij} = \frac{m_im_j}{m_i + m_j}$ is the reduced mass of system $(Q_i\bar Q_j)$. After the substitutions of variables $\omega_1 = \frac{k^2}{2m_{13}}$ and $x = (\omega_2 - \frac{k^2}{2m_{23}})\cdot\frac{m_2}{|\vec p_2|k}$, in the limit $|\vec p_2|\to 0$ for the correlator $\Pi_{0}^{A, NR}$ we have the following expression: $$\begin{aligned} \Pi_{0}^{A, NR} &=& -\frac{1}{(2\pi)^2}\int\frac{d\tilde \omega_1 d\tilde \omega_2} {(\tilde \omega_1 - \omega_1)(\tilde \omega_2 - \omega_2)}\rho_{0}^{A, NR} = \nonumber \\ &=& \frac{3}{\pi^2}\int_0^{k_{th}}\frac{k^2dk}{(\omega_1 - \frac{k^2}{2m_{13}}) (\omega_2 - \frac{k^2}{2m_{23}})},\end{aligned}$$ where $k_{th}$ denotes the resonance region boundary. In the method of moments in NRQCD sum rules we put $\omega_1 = -(m_1 + m_3) + q_1$ and $\omega_2 = -(m_2 + m_3) + q_2$, so that in the limit $q_{1,2}\to 0$ we have $$\begin{aligned} \frac{1}{n!}\frac{1}{m!}\frac{d^{n + m}}{dq_1^ndq_2^m}\Pi_{0}^{A, NR} &=& \Pi_{0}^{A, NR}[n, m] \nonumber \\ &=& \frac{3}{\pi^2}\int_0^{k_{th}}\frac{k^2dk}{({\cal M}_{1} + \frac{k^2}{2m_{13}})^{n+1} ({\cal M}_{2} + \frac{k^2}{2m_{23}})^{m+1}}.\end{aligned}$$ In the hadronic part of NRQCD sum rules in the limit $|\vec p_2|\to 0$ we model the resonance contribution by the following presentation: $$\Pi_{0}^{A, res} = \sum_{i,j}\frac{f_i^{Q_1\bar Q_3}M_{1, i}^2}{(m_1 + m_3)M_{1, i}^2} \frac{f_j^{Q_2\bar Q_3}M_{2, j}}{M_{2, j}^2}F_{0,ij}^{A}\sum_{l,m}\left (\frac{q_1^2}{M_{1,i}^2}\right )^l\left (\frac{q_2^2}{M_{2,j}^2}\right )^m , \label{nrhad}$$ Saturating the $p_1^2$ and $p_2^2$ channels by ground states of mesons under consideration we have $$F_{0, 1S\to 1S}^{A} = \frac{\Pi_{0}^{A, NR}[n, m](m_1 + m_3)}{f_{1S}^{Q_1\bar Q_3} M_{1,1S}^2 f_{1S}^{Q_2\bar Q_3}M_{2,1S}^2}M_{1, 1S}^nM_{2, 1S}^m. \label{nrf0}$$ The value of $F_{0, 1S\to 1S}^{A}$ at fixed $n=4$ is presented in Fig. \[nrfig1\] at $$\begin{aligned} k_{th} &=& 1.3\; {\rm GeV},\nonumber \\ m_b &=& 4.6\; {\rm GeV},\nonumber \\ m_c &=& 1.4\; {\rm GeV}.\nonumber\end{aligned}$$ Further, in (\[nrhad\]) we can take into account the dominant subleading term, which is the contribution by the transition of $2S\to 2S$. In this case one could expect that the form factor is not suppressed in comparison with the contribution by the $1S\to 2S$ transition, since in the potential picture the latter decay has to be neglected because the overlapping between the wave functions at zero recoil is close to zero for the states with the different quantum numbers [^6]. So, we can easily modify the relation (\[nrf0\]) due to the second transition and justify the value of $F_{0, 2S\to 2S}^{A}$ to reach the stability of $F_{0, 1S\to 1S}^{A}$ at low values of moment numbers. We find $F_{0, 2S\to 2S}^{A}/F_{0, 1S\to 1S}^{A} \approx 3.7$ and present the behaviour of the form factor $F_0^A$ at zero recoil in Fig. \[nrfig2\]. (140,100) (20,0)[ ]{} (100,-3)[$m$]{} (10,67)[$F_0^A(0)$, GeV]{} (140,100) (20,0)[ ]{} (100,-3)[$m$]{} (10,67)[$F_0^A(0)$, GeV]{} Contribution of the gluon condensate ------------------------------------ Following a general formalism for the calculation of gluon condensate contribution in the Fock-Schwinger gauge [@Fock; @Schw] we have considered diagrams, depicted in Fig. \[cond\]. In our calculations, we have used the NRQCD approximation and analysed the limit, where the invariant mass of the lepton pair takes its maximum value. Here we would like to note that the spin structure, in the leading order of relative velocity of heavy quarks, does not change, in comparison with the bare loop result for the nonrelativistic quarks. Thus, we can conclude, that in this approximation, the symmetries of NRQCD lagrangian lead to a universal Wilson coefficient for the gluon condensate operator. As a consequence, relations (\[Fsym\]) remain valid. Below we perform calculations for the form factor $F_{0}^{A}(p_1^2, p_2^2, Q^2)$ in the limit $q^2\to q_{max}^2$. The contribution of gluon condensate to the corresponding correlator is given by the following expression: $$\Delta F_{0}^{G^2} = \langle \frac{\alpha_s}{\pi}G_{\mu\nu}^2\rangle \cdot\frac{\pi}{48}[3R_{0} - R_{2}], \label{x1}$$ where $$\begin{aligned} R_{0} &=& -\frac{1}{\pi^2i}\int\frac{k^2dkdk_0}{ P_3(k, k_0)P_1(k, k_0, \omega_1)P_2(k, k_0, \omega_2)}R_g ,\nonumber\\ R_{2} &=& -\frac{1}{\pi^2i}\int\frac{k^4dkdk_0}{ P_3(k, k_0)P_1(k, k_0, \omega_1)P_2(k, k_0, \omega_2)}R_k , \label{x2}\end{aligned}$$ In these expressions the inverse propagators are $$\begin{aligned} P_1(k, k_0, \omega_1) &=& \omega_1 + k_0 - \frac{k^2}{2m_1}, \nonumber \\ P_2(k, k_0, \omega_2) &=& \omega_2 + k_0 - \frac{k^2}{2m_2}, \\ P_3(k, k_0) &=& - k_0 - \frac{k^2}{2m_3}, \nonumber \end{aligned}$$ The functions $R_g$ and $R_k$ are symmetric under the permutation of indices 1 and 2 and they have the following forms: $$\begin{aligned} R_g &=& -\frac{1}{m_1}\left [\frac{1}{P_3} + \frac{1}{P_1}\right ]\frac{1}{P_1^2} - \frac{1}{m_2}\left [\frac{1}{P_3} + \frac{1}{P_1}\right ]\frac{1}{P_2^2}, \nonumber\\ R_k &=& \frac{3}{m_1^2}\frac{1}{P_1^4} + \frac{3}{m_2^2}\frac{1}{P_2^4} + \frac{3}{m_1^2}\frac{1}{P_1^3P_3} + \frac{3}{m_2^2}\frac{1}{P_2^3P_3} + \nonumber \\ && \frac{1}{m_1m_3}\frac{1}{P_1^2P_3^2} + \frac{1}{m_2m_3}\frac{1}{P_2^2P_3^2} - \frac{1}{m_3^2}\frac{1}{P_3^4} - \label{x3} \\ && \frac{1}{m_1m_3}\frac{1}{P_1P_3^3} - \frac{1}{m_2m_3}\frac{1}{P_2P_3^3} - \frac{1}{m_1m_2}\frac{1}{P_1P_2P_3^2} - \frac{1}{m_1m_2}\frac{1}{P_1^2P_2^2}. \nonumber\end{aligned}$$ Let us note, that in the calculations of diagrams in Fig. \[cond\] we have used the following vertex for the interaction of heavy quark with the gluon $$L_{int}^{v} = -g_s\bar h_v v_{\mu}A^{\mu}h_v,$$ where $A_{\mu} = A_{\mu}^{a}\cdot\frac{\lambda^{a}}{2}$ and its Fourier-transform in Fock-Schwinger gauge has the form $$A_{\mu}^{a}(k_g) = -\frac{i}{2}G_{\mu\nu}^{a}(0)\frac{\partial}{\partial k_g} (2\pi)^4 \delta (k_g).$$ After two differentiations of the nonrelativistic propagators $$\frac{1}{P(p, m_Q)} = \frac{1}{pv + \frac{p^2}{2m_Q}},$$ two types of contributions to the gluon condensate correction appear. The first is equal to $$v_{\mu}v_{\nu}\alpha_s G_{\mu\alpha}^aG_{\nu\beta}^a\cdot g^{\alpha\beta}\to \langle \alpha_s G_{\mu\nu}^2\rangle \cdot\frac{3}{12},$$ and it leads to the term with $R_{0}$. The second expression $$v_{\mu}v_{\nu}\alpha_s G_{\mu\alpha}^aG_{\nu\beta}^ak^{\alpha}k^{\beta}\to \langle \alpha_sG_{\mu\nu}^2\rangle \frac{k^2 - (v\cdot k)^2}{12} = - \langle \alpha_sG_{\mu\nu}^2\rangle \frac{|\vec k|^2}{12}$$ leads to the term with $R_{2}$, which is of the same order in the relative velocity of heavy quarks as $R_{0}$, but it is suppressed numerically in the region of moderate numbers for the momenta of spectral densities, where the non-perturbative contributions (condensates) of higher dimension operators are not essential. It is easy to show, that the contributions of $R_{0}$ and $R_{2}$ can be obtained by the differentiation of two basic integrals: $$\begin{aligned} E_{0} &=& -\frac{1}{\pi^2i}\int_{0}^{\infty}\frac{k^2dkdk_0}{P_1P_2P_3} = \frac{2m_{13}2m_{23}}{k_{13} + k_{23}}, \label{x4} \\ E_{2} &=& -\frac{1}{\pi^2i}\int_{0}^{\infty}\frac{k^4dkdk_0}{P_1P_2P_3} = - \frac{2m_{13}2m_{23}}{k_{13} + k_{23}}(k_{13}^2 + k_{13}k_{23} + k_{23}^2),\nonumber\end{aligned}$$ where $k_{13} = \sqrt{-2m_{13}\omega_1}$, $k_{23} = \sqrt{-2m_{23}\omega_2}$. Then for $R_{0}$ and $R_{2}$ we have: $$\begin{aligned} R_{0} &=& \hat R_{g}\cdot E_{0},\nonumber \\ R_{2} &=& \hat R_{k}\cdot E_{2},\nonumber \end{aligned}$$ where operators $\hat R_g$ and $\hat R_k$ can be obtained from $R_g$ and $R_k$ in (\[x3\]) after the substitutions: $$\begin{aligned} \frac{1}{P_1^n}&\to& \frac{(-1)^n}{n!} \frac{\partial^n}{\partial\omega_1^n}, \nonumber \\ \frac{1}{P_2^m}&\to& \frac{(-1)^m}{m!} \frac{\partial^m}{\partial\omega_2^m}, \nonumber \\ \frac{1}{P_3^l}&\to& \frac{(-1)^l}{l!} \left [\frac{\partial}{\partial\omega_1} + \frac{\partial}{\partial\omega_2}\right ]^l, \nonumber\end{aligned}$$ As a result we have: $$\begin{aligned} R_0 &=& \{\frac{1}{6m_1}\frac{\partial^3}{\partial\omega_1^3} + \frac{1}{6m_2}\frac{\partial^3}{\partial\omega_2^3} + \frac{1}{2m_1}\left ( \frac{\partial^3}{\partial\omega_1^3} + \frac{\partial^3}{\partial\omega_1^2\partial\omega_2}\right ) + \nonumber\\ && \frac{1}{2m_2}\left (\frac{\partial^3}{\partial\omega_2^3} + \frac{\partial^3}{\partial\omega_2^2\partial\omega_1}\right )\}\cdot \frac{2m_{13}2m_{23}}{k_{13} + k_{23}}, \label{x5}\\ R_2 &=& \{\frac{3}{4!m_1^2}\frac{\partial^4}{\partial\omega_1^4} + \frac{3}{4!m_2^2}\frac{\partial^4}{\partial\omega_2^4} + \frac{3}{3!m_1^2}\left ( \frac{\partial^4}{\partial\omega_1^4} + \frac{\partial^4}{\partial\omega_1^3\partial\omega_2} \right ) + \nonumber \\ && \frac{1}{3!m_2^2}\left (\frac{\partial^4}{\partial\omega_2^4} + \frac{\partial^4}{\partial\omega_2^3\partial\omega_1}\right ) + \frac{1}{4m_1m_3} \left (\frac{\partial^4}{\partial\omega_1^4} + 2\frac{\partial^4}{\partial\omega_1^3 \partial\omega_2} + \frac{\partial^4}{\partial\omega_1^2\partial\omega_2^2}\right ) + \nonumber \\ && \frac{1}{4m_2m_3}\left (\frac{\partial^4}{\partial\omega_2^4} + 2\frac{\partial^4}{\partial\omega_2^3 \partial\omega_1} + \frac{\partial^4}{\partial\omega_1^2\partial\omega_2^2}\right ) - \nonumber \\ && \frac{1}{4!m_3^2}\left (\frac{\partial^4}{\partial\omega_1^4} + 4\frac{\partial^4}{\partial\omega_1^3\partial\omega_2} + 6\frac{\partial^4}{\partial\omega_1^2\partial\omega_2^2} + 4\frac{\partial^4}{\partial\omega_1\partial\omega_2^3} + \frac{\partial^4}{\partial\omega_2^4} \right ) - \label{x6}\\ && \frac{1}{3!m_1m_3}\left (\frac{\partial^4}{\partial\omega_1^4} + 3\frac{\partial^4}{\partial\omega_1^3\partial\omega_2} + 3\frac{\partial^4}{\partial\omega_1^2\partial\omega_2^2} + \frac{\partial^4}{\partial\omega_1\partial\omega_2^3}\right ) - \nonumber\\ && \frac{1}{3!m_2m_3}\left (\frac{\partial^4}{\partial\omega_2^4} + 3\frac{\partial^4}{\partial\omega_2^3\partial\omega_1} + 3\frac{\partial^4}{\partial\omega_1^2\partial\omega_2^2} + \frac{\partial^4}{\partial\omega_2\partial\omega_1^3}\right ) - \nonumber\\ && \frac{1}{2m_1m_2}\left ( \frac{\partial^4}{\partial\omega_1^3\partial\omega_2} + 2\frac{\partial^4}{\partial\omega_1^2\partial\omega_2^2} + \frac{\partial^4}{\partial\omega_1\partial\omega_2^3}\right ) - \frac{1}{4m_1m_2}\left ( \frac{\partial^4}{\partial\omega_1^2\partial\omega_2^2}\right )\}\cdot \nonumber\\ && \left (-\frac{2m_{13}2m_{23}}{m_{13} + m_{23}}\right )(k_{13}^2 + k_{13}k_{23} + k_{23}^2), \nonumber\end{aligned}$$ Equations (\[x1\]), (\[x5\]) and (\[x6\]) represent the most compact analytical expression for the contribution of gluon condensate to the form factor $F_0^A$, whereas performing the differentiations leads to very cumbersome expressions. In the moment scheme of sum rules we suppose $$\begin{aligned} \omega_1 &=& -(m_1 + m_3) + q_1, \nonumber\\ \omega_2 &=& -(m_2 + m_3) + q_2, \nonumber\end{aligned}$$ and expand $\Delta F_{0}^{G^2}$ in a series over $\{q_1, q_2\}$ at the point $\{ 0, 0\}$, which allows us to determine $\Delta F_{0}^{G^2}[n, m] = \frac{1}{n!} \frac{1}{m!}\frac{d^{n+m}}{dq_1^ndq_2^m}\Delta F_{0}^{G^2}$. Further analysis has been performed numerically with the help of MATHEMATICA. The evaluation of the $\Delta F_0^{G^2}[n,m]$ dependence in a broad range of $[n,m]$ takes too much calculation time, so we restrict ourselves by showing the results in Figs. \[nrfig1\] and \[nrfig2\] for the fixed $n=4$ with the following set of parameters $$\begin{aligned} m_b &=& 4.6\; \mbox{GeV},\nonumber\\ m_c &=& 1.4\; \mbox{GeV},\nonumber\\ \langle\frac{\alpha_s}{\pi}G_{\mu\nu}^2\rangle &=& 1.7\cdot 10^{-2}\; \mbox{GeV}^4.\nonumber \end{aligned}$$ by taking into account (Fig. \[nrfig1\]) and without accounting (Fig. \[nrfig2\]) the contribution of transition $2S\to 2S$, where $F_{0}(2S\to 2S)/F_{0}(1S\to 1S)\approx 3.7$. As can be seen in these figures, the gluon contribution, while varying the moment number in the region of bound $\bar cc$-states, plays an important role, because it allows us to extend the stability region for the form factor $F_{0}^A$ up to three times (from $m < 5$ till $m < 15$), and, thus, the reliability of sum rule predictions. Let us also note, that in the scheme of saturation for the hadronic part of sum rules by the ground states in both variables $s_1$ and $s_2$, the account for the gluon condensate leads to the $20\%$ reduction for the value of form factor $F_{0}^{A}$. The analysis of the dependence on the moment number $m$ in the region of bound states $\bar b c$ at fixed $m$ shows that the contribution of gluon condensate in this case does not affect the character of this dependence. This may be explained by the fact that the Coulomb corrections to the Wilson coefficient of gluon operator $G_{\mu\nu}^2$ play an essential role[^7]. The summation of $\alpha_s/v$-terms for $\bar bc$ may give a sizeable effect, contrary to the situation with the $\bar cc$ system, where the relative velocity of heavy quarks is not too small. To conclude, we have calculated the contribution of gluon condensate to the three-point sum rules for heavy quarkonia in the leading order of relative velocity of heavy quarks and in the first order of $\alpha_s$. Due to the symmetry of NRQCD in the limit, when the invariant mass of the lepton pair takes its maximum value, i.e. at the recoil momentum close to zero, the Wilson coefficient for the form factor $F_{0}^A$ is universal in the sense that it determines the contributions of gluon condensate to other form factors, in accordance with relations (\[Fsym\]). Conclusions =========== We have calculated the semileptonic decays of $B_c$ meson in the framework of sum rules in QCD and NRQCD. We have extended the previous evaluations in QCD to the case of massive leptons: the complete set of double spectral densities in the bare quark-loop approximation have been presented. The analysis in the sum rule schemes of density moments and Borel transform has been performed and consistent results have been obtained. We have taken into account the gluon condensate contribution for the form factors of semipeltonic transitions between the heavy quarkonia in the Borel transform sum rules of QCD, wherein the analitycal expressions for the case of three nonzero masses of quarks have been presented. We have considered the soft limit on the form factors in NRQCD at the recoil momentum close to zero, which has allowed one to derive the generalized relations due to the spin symmetry of effective lagrangian. The relations have shown a good agreement with the numerical estimates in full QCD, which means the corrections in both the relative velocities of heavy quarks inside the quarkonia and the inverse heavy quark masses to be small within the accuracy of the method. Next, we have presented the analytical results on the gluon condensate term in the NRQCD sum rules within the moments scheme. In both the QCD and NRQCD sum rules, the account for the gluon condensate has allowed one to enforce the reliability of predictions, since the region of physical stability for the form factors evaluated has significantly expanded in comparison with the leading order calculations of bare quark-loop contribution. Next, we have investigated the role played by the Coulomb $\alpha_s/v$ corrections for the semileptonic transitions between the heavy quarkonia. We have shown that as in the case of two-point sum rules, the three-point spectral densities are enhanced due to the Coulomb renormalization of quark-meson vertices. The complete analysis shows that the numerical estimates of various branching fractions: $$\begin{aligned} {\rm BR}[B_c^+\to J/\psi l^+ \nu] &=& 2.5\pm 0.5 \% ,\nonumber\\ {\rm BR}[B_c^+\to \bar c c+\mbox{\it X}] &=& 23\pm 5 \% , \nonumber\end{aligned}$$ agree with the results obtained in the framework of potential models and Operator Product Expansion in NRQCD. More detailed results are presented in tables. Thus, we draw the conclusion that at present the theoretical predictions on the semileptonic decays of $B_c$ meson give consistent and reliable results. The authors would like to express the gratitude to Prof. A.Wagner and members of DESY Theory Group for their kind hospitality during the visit to DESY, where this paper was written, as well as to Prof. S.S.Gershtein for discussions. The authors also thank Prof. A. Ali for reading this manuscript and valuable remarks. V.V.K. and A.K.L. express the gratitude to A.L.Kataev for fruitful discussions and notes. This work is in part supported by the Russian Foundation for Basic Research, grants 99-02-16558 and 96-15-96575. The work of A.I.Onishchenko was supported, in part, by International Center of Fundamental Physics in Moscow, Intenational Science Foundation, and INTAS-RFBR-95I1300 grants. Appendix A ========== In this Appendix we present the derivation of exclusive semileptonic widths for the $B_c$-meson decays into the $J/\psi$, $\eta_c$ mesons with account of lepton masses. The exclusive semileptonic width $\Gamma_{SL}$ for the decay $B_c\to J/\psi (\eta_c) l\bar\nu_l$, where $l = e,\mu$ or $\tau$, can be written down in the form [@Bigi] $$\Gamma_{SL} = \frac{1}{(2\pi)^3}\frac{G_F^2|V_{cb}|^2}{M_{B_c}}\int d^4q \int d\tau_l L^{\alpha\beta}W_{\alpha\beta}, \label{SL1}$$ where $d^4q = 2\pi |\vec q|dq^2dq_0$, $d\tau_l = |\vec p_l|d\Omega_l /(16\pi^2\sqrt{q^2})$ is the leptonic pair phase space, $d\Omega_l$ is the solid angle of charged lepton $l$, $|\vec p_l| = \sqrt{q^2}\Phi_l/2$ is its momentum in the dilepton center of mass system and $\Phi_l\equiv\sqrt{1 - 2\lambda_{+} + \lambda_{-}^2}$, with $\lambda_{\pm}\equiv (m_l^2\pm m_{\nu_l}^2)/q^2$. The tensors $L^{\alpha\beta}$ and $W_{\alpha\beta}$ in Eq. (\[SL1\]) are given by $$L^{\alpha\beta} = \frac{1}{4}\sum_{spins} (\bar lO^{\alpha}\nu_l) (\bar\nu_l O^{\beta}l) = 2[p_l^{\alpha}p_{\nu_l}^{\beta} + p_l^{\beta} p_{\nu_l}^{\alpha} - g^{\alpha\beta}(p_l\cdot p_{\nu_l}) + i\epsilon^{\alpha\beta\gamma\delta}p_{l\gamma}p_{\nu_l\delta}], \label{SL2}$$ and $$\begin{aligned} W_{\alpha\beta} &=& \int\frac{d^3\vec p_2}{2E_2}\delta^4(p_1 - p_2 - q)\tilde W_{\alpha\beta}\nonumber \\ &=& \theta (E_2)\delta (M_1^2 - 2M_1\cdot q_0 + q^2 - M_2^2) \tilde W_{\alpha\beta}|_{p_2 = p_1 - q},\label{SL3}\end{aligned}$$ where $$\tilde W_{\alpha\beta} = (f_{+}(t)(p_1 + p_2)_{\alpha} + f_{-}(t)q_{\alpha}) (f_{+}(t)(p_1 + p_2)_{\beta} + f_{-}(t)q_{\beta})$$ for the pseudoscalar particle in the final state, and $$\begin{aligned} \tilde W_{\alpha\beta}& =& -(iF_{0}^{A}g_{\alpha e} + iF_{+}^{A}p_{1e}(p_1 + p_2)_{\alpha} + iF_{-}^{A}p_{1e}q_{\alpha} - F_V\epsilon_{\alpha eij}(p_1 + p_2)^iq^j)\cdot \nonumber\\ && (iF_{0}^{A} + iF_{+}^{A}p_{1k}(p_1 + p_2)_{\beta} + iF_{-}^{A}p_{1k}q_{\beta} + F_V\epsilon_{\beta kmn}(p_1 + p_2)^mq^n )\cdot \nonumber\\ && (\sum_{polarizations}\epsilon^{e}\epsilon^{*k}) \end{aligned}$$ for the vector meson in the final state with the polarization $\epsilon_{\mu}$, where $$\sum_{polarizations}\epsilon^{e}\epsilon^{*k} = \frac{p_{2}^{e}p_{2}^{k}} {M_2^2} - g^{ek}$$ $M_2$ is the mass of final state meson ($J/\psi$ or $\eta_c$). The integral over the leptonic phase space in Eq.(\[SL1\]) is given by $$\int d\tau_l L^{\alpha\beta} = \frac{1}{4\pi}\frac{|\vec p_l|}{\sqrt{q^2}} \langle L^{\alpha\beta}\rangle,$$ with $$\langle L^{\alpha\beta}\rangle = \frac{1}{4\pi}\int d\Omega_l L^{\alpha\beta} = \frac{2}{3}\{ (1 + \lambda_1)(q^{\alpha}q^{\beta} - g^{\alpha\beta}q^2) + \frac{3}{2}\lambda_2 g^{\alpha\beta}q^2\},$$ where $\lambda_1\equiv \lambda_{+} - 2\lambda_{-}^2$ and $\lambda_2\equiv \lambda_{+} - \lambda_{-}^2$. Introducing the dimensionless kinematical variable $t\equiv q^2/m_b^2$ and integrating over $q_0$ the semileptonic width takes the following form: $$\Gamma_{SL} = \frac{1}{64\pi^3}\frac{G_F^2|V_{bc}|^2m_b^2}{M_1^2} \int_{t_{min}}^{t_{max}} dt\Phi_l(t)|\vec q|\langle L^{\alpha\beta}\rangle \tilde W_{\alpha\beta}, \label{SL4}$$ where $$|\vec q| = \frac{1}{2M_1}\sqrt{(M_1^2 + m_b^2t - M_2^2)^2 - 4m_b^2M_1^2t}.$$ In Eq. (\[SL4\]) the limits of integration are given by $t_{min} = \frac{m_l^2}{m_b^2}$ and $t_{max} = \frac{1}{m_b^2}(M_1 - M_2)^2$. Calculation of $\langle L^{\alpha\beta}\rangle\tilde W_{\alpha\beta}$ yields the following expressions $$\begin{aligned} \langle L^{\alpha\beta}\rangle\tilde W_{\alpha\beta} &=& \frac{1}{3}( 3t^2f_{-}^2(t)\lambda_2 m_b^4 + 6tf_{-}(t)f_{+}(t)(M_1^2-M_2^2) \lambda_2 m_b^2 +\nonumber\\ && f_{+}^2(t)(t^2(2\lambda_1 - 3\lambda_2 + 2)m_b^4 - 2t(M_{1}^2 + M_{2}^2)(2\lambda_1 - 3\lambda_2 + 2)m_b^2 +\nonumber \\ && 2(M_{1}^2 - M_{2}^2)^2(\lambda_1 + 1))) \nonumber\end{aligned}$$ for the pseudoscalar meson in the final state, and $$\begin{aligned} \langle L^{\alpha\beta}\rangle\tilde W_{\alpha\beta} &=& \frac{1}{12M_{2}} (2(t^2(\lambda_1+1)m_b^4-2t((1+\lambda_1)M_{1}^2 - 5(1+\lambda_1)M_{2}^2 + 9M_{2}^2\lambda_2)m_b^2+\nonumber\\ && (M_{1}^2-M_{2}^2)^2(1+\lambda_1))(F_{0}^{A})^2 - 2(t^2m_b^4 - 2t(M_{1}^2+M_{2}^2)m_b^2 + \nonumber \\ && (M_{1}^2-M_{2}^2)^2) (F_{+}^{A}(2(1+\lambda_1)tm_b^2-3t\lambda_2m_b^2-2(1+\lambda_1)M_{1}^2+ \nonumber\\ && 2(1+\lambda_1)M_{2}^2) - 3tF_{-}^{A}m_b^2\lambda_2)F_{0}^{A} + (t^2m_b^4-2t(M_{1}^2+M_{2}^2)m_b^2+\nonumber \\ && (M_{1}^2-M_{2}^2)^2)((2t^2m_b^4+ 2t^2\lambda_1m_b^4 - 3t^2\lambda_2m_b^4 - 4(1+\lambda_1)tM_{1}^2m_b^2 - \nonumber\\ && 4(1+\lambda_1)tM_{2}^2 + 6tM_{1}^2\lambda_2m_b^2 + 6tM_{2}^2\lambda_2m_b^2 + 2(1+\lambda_1)M_{1}^4 +\nonumber\\ && 2(1+\lambda_1)M_{2}^4 - 4(1+\lambda_1)M_{1}^2M_{2}^2)(F_{+}^{A})^2 + \nonumber\\ && 6tF_{-}^{A}F_{+}^{A}\lambda_2m_b^2(M_{1}^2-M_{2}^2) + tm_b^2(3t(F_{-}^{A})^2\lambda_2m_b^2 + \nonumber\\ && 16(1+\lambda_1)(F_{V})^2M_{2}^2 - 24\lambda_2M_{2}^2(F_{V})^2))) \nonumber\end{aligned}$$ for the vector meson in the final state. Appendix B ========== In this Appendix we illustrate the kind of expressions, which arise for the gluon condensate contribution to the form factors $F^{i}(t)$ in the framework of Borel transformed three point sum rules, in the case of $f_1(t) = f_{+}(t) + f_{-}(t)$ form factor. Following the algorithm, described in section [**2.3**]{}, for $\Pi_1^{\langle G^2\rangle}$ we have the following expression $$\begin{aligned} \Pi_1^{\langle G^2\rangle} &=& C_1^{(-1,-2)}U_0(-1,-2) + C_1^{(-1,-1)}U_0(-1,-1) + \sum_{i=-4}^{0} C_1^{(0,i)}U_0(0,i) +\nonumber\\ && \sum_{i=-5}^{-1} C_1^{(1,i)}U_0(1,i) + \sum_{i=-6}^{0} C_1^{(2,i)}U_0(2,i) + \sum_{i=-6}^{0} C_1^{(3,i)}U_0(3,i),\end{aligned}$$ where $$\begin{aligned} C_1^{(-1,-2)} &=& -\frac{M_{bc}^2+M_{cc}^2}{12M_{bc}^2}, \\ C_1^{(-1,-1)} &=& -\frac{1}{12M_{bc}^2},\\ C_1^{(0,-4)} &=& \frac{1}{48M_{bc}^6M_{cc}^4}(M_{bc}^{10}(6m_2^2+m_3^2) + 8m_2^2M_{bc}^8M_{cc}^2 + (-5m_1^2+5m_2^2+\nonumber\\ && m_3^2)M_{bc}^6M_{cc}^4 + (-7m_1^2+3m_2^2+2m_1(m_2-m_3))M_{bc}^4M_{cc}^6 - 4m_1^2M_{bc}^2M_{cc}^8 \quad\quad\quad\\ && - 2m_1^2M_{cc}^{10}),\nonumber\\ C_1^{(0,-3)} &=& \frac{1}{48M_{bc}^6M_{cc}^4}((m_1^2-2m_1m_3-2m_3^2)M_{cc}^8 + (6m_2^2+4M_{cc}^2)M_{bc}^8 - \nonumber\\ && (2m_1^2+2m_2^2+6m_2m_3+2m_3^2+2m_1(-5m_2+m_3)- 11M_{cc}^2\nonumber\\ && -2Q^2)M_{bc}^6M_{cc}^2 + (m_1^2-m_2^2+2m_1(m_2-5m_3)-2m_2m_3+2m_3^2 \\ && + 5M_{cc}^2+Q^2)M_{bc}^2M_{cc}^6 + (-7m_1^2+4m_1m_2-2m_2^2-10m_1m_3- \nonumber\\ && 8m_2m_3+2m_3^2+12M_{cc}^2+2Q^2)M_{bc}^4M_{cc}^4),\nonumber\\ C_1^{(0,-2)} &=& \frac{1}{48M_{bc}^6M_{cc}^4}((3m_1^2-2m_1m_3-2m_3^2)M_{cc}^6 + (m_2^2-5m_3^2+9M_{cc}^2)M_{bc}^6 +\nonumber\\ && (-4m_1^2-7m_2^2+2m_1(6m_2-5m_3)-8m_2m_3- 13m_3^2+16M_{cc}^2+\nonumber\\ && 4Q^2)M_{bc}^4M_{cc}^2 - (5m_1^2-4m_1(m_2-4m_3)+2(m_2^2+ 2m_2m_3+m_3^2-\\ && M_{cc}^2-Q^2)M_{bc}^2M_{cc}^4),\nonumber\\ C_1^{(0,-1)} &=& \frac{1}{48M_{bc}^6M_{cc}^4}(-2m_3^2M_{cc}^4 + (-6m_3^2+5M_{cc}^2)M_{bc}^4 + \\ && (-m_1^2+2m_1m_2-m_2^2+4m_2m_3-16m_3^2+M_{cc}^2+Q^2)M_{bc}^2M_{cc}^2,\nonumber\\ C_1^{(0,0)} &=& \frac{m_3^2(M_{bc}^2+M_{cc}^2)}{48M_{bc}^6M_{cc}^4},\\ C_1^{(1,-5)} &=& -\frac{1}{48M_{bc}^8M_{cc}^6}(m_2^2M_{bc}^4-m_1^2M_{cc}^4) ((5m_2^2+m_3^2)M_{bc}^8 - m_3^2M_{bc}^6M_{cc}^2 - \\ && (4m_1^2+3m_2^2)M_{bc}^4M_{cc}^4 + 2m_1(m_2-m_3)M_{bc}^2M_{cc}^6 + 2m_1^2M_{cc}^4) ,\nonumber\\ C_1^{(1,-4)} &=& \frac{1}{48M_{bc}^8M_{cc}^6}(m_1^2(3m_1^2+2m_1m_3+6m_3^2) M_{cc}^{10} + \nonumber\\ && (-2m_2^4-7m_2^2M_{cc}^2+m_3^2M_{cc}^2)M_{bc}^{10} + (8m_2^4+4m_2^3m_3+3m_2^2m_3^2+\nonumber\\ && 2m_2m_3^3+m_1^2(4m_2^2+m_3^2)-2m_1(4m_2^3-3m_2^2m_3+ m_2m_3^2-m_3^3- \\ && 11m_2^2M_{cc}^2-3m_3^2M_{cc}^2-4m_2^2Q^2-m_3^2Q^2)M_{bc}^8M_{cc}^2 - (3m_1^4+6m_2^2m_3^2+\nonumber\\ && m_1^3(-6m_2+4m_3)+2m_1(m_2^2m_3+2m_2M_{cc}^2-2m_3M_{cc}^2) +\nonumber\\ && m_1^2(10m_2^2+2m_2m_3+2m_3^2-8M_{cc}^2-3Q^2))M_{bc}^4M_{cc}^6+ (4m_1m_2^2m_3+\nonumber\\ && m_1^2(9m_2^2+16M_{cc}^2)+m_2^2(2m_2^2+4m_2M-3-4m_3^2-M_{cc}^2-\nonumber\\ && 2Q^2))M_{bc}^6M_{cc}^4-m_1(6m_1^3+4m_1^2m_3+2m_3^2(-m_2+m_3)+ m_1(m_2^2+\nonumber\\ && 2m_2m_3-4m_3^2+5M_{cc}^2-Q^2))M_{bc}^2M_{cc}^8),\nonumber\\ C_1^{(1,-3)} &=& \frac{1}{48M_{bc}^{8}M_{cc}^6}(2m_3(m_1^3+5m_1^2m_3+2m_3^3) M_{cc}^8 + (m_3^4+m_2^2(9m_3^2-4M_{cc}^2+\nonumber\\ && 8m_2m_3M_{cc}^2-3m_3^2M_{cc}^2-11M_{cc}^4) M_{bc}^8 + (2m_2^4+10m_2^2m_3^2+m_3^4+\nonumber\\ && 7m_2^2M_{cc}^2+18m_2m_3M_{cc}^2+ 2m_3^2M_{cc}^2-20M_{cc}^2-4m_1(m_2-m_3)(m_2^2\\ && +2M_{cc}^2)+m_1^2(2m_2^2+5M_{cc}^2)- 2m_2^2Q^2-5M_{cc}^2Q^2)M_{bc}^6M_{cc}^2 + (-7m_2^2m_3^2\nonumber\\ && -4m_3^4+4m_2^2M_{cc}^2+10m_2m_3M_{cc}^2-7M_{cc}^4+2m_1^2(2m_2m_3- m_3^2+5M_{cc}^2)+\nonumber\\ &&m_1(6m_2-6m_3- 2m_2^2m_3+4m_2m_3^2+4m_2M_{cc}^2+10m_3M_{cc}^2)+2m_3^2Q^2\nonumber\\ && -4M_{cc}^2Q^2)M_{bc}^4 M_{cc}^6 + (m_1^4-2m_1^3(m_2-4m_3)+10m_3^2M_{cc}^2+2m_1m_3(m_2^2+\nonumber\\ && m_2m_3+m_3^2-M_{cc}^2-Q^2)+m_1^2(m_2^2+2m_2m_3+7M_{cc}^2-Q^2)),\nonumber\\ C_1^{(1,-2)} &=& \frac{1}{48M_{bc}^8M_{cc}^6}(m_3^2(3m_1^2-2m_1m_3+6m_3^2) M_{cc}^6 + (14m_2m_3M_{cc}^2+(16m_3^2-\nonumber\\ && 13M_{cc}^2)M_{cc}^2+m_2^2(4m_3^2+M_{cc}^2)) M_{bc}^6 + (-5m_2^2m_3^2-2m_2m_3^3-2m_3^4+\nonumber\\ && 4m_2^2M_{cc}^2+8m_2m_3M_{cc}^2+ 32m_3^2M_{cc}^2-8M_{cc}^4+m_1^2(-3m_3^2+4M_{cc}^2)+\\ && 2m_1(3m_2m_3^2-2m_3^3-2m_2M_{cc}^2 +6m_3M_{cc}^2)+3m_3^2Q^2-4M_{cc}^2Q^2)M_{bc}^4M_{cc}^2\nonumber\\ &&-(2m_1^3m_3+m_1^2(-4m_2m_3+4m_3^2+M_{cc}^2)+m_3^2(m_2^2+2m_2m_3+ 4m_3^2\nonumber\\ && -7M_{cc}^2-Q^2)+2m_1m_3 (m_2^2-m_2m_3+3m_3^2-2M_{cc}^2-Q^2))M_{bc}^2M_{cc}^6),\nonumber\\ C_1^{(1,-1)} &=& -\frac{1}{48M_{bc}^8M_{cc}^6}(2m_1m_3^3M_{cc}^4 + (4m_3^4-17m_3^2M_{cc}^2+M_{cc}^4)M_{bc}^4 +\nonumber\\ && 2m_3^2(m_1^2-2m_1m_2+m_2^2+2m_1m_3+4m_3^2-6M_{cc}^2-Q^2)),\\ C_1^{(2,-6)} &=& \frac{(M_{bc}^2-M_{cc}^2)(m_2^2M_{bc}^4-m_1^2M_{cc}^4)^3} {96M_{bc}^{10}M_{cc}^8},\\ C_1^{(2,-5)} &=& -\frac{1}{96M_{bc}^8M_{cc}^6}(m_2^2M_{bc}^4-m_1^2M_{cc}^4) ((m_2^2+2m_3^2)M_{bc}^6 + 7m_1^2M_{bc}^2M_{cc}^4 +\nonumber\\ && m_2^2(m_1^2-2m_1m_2+m_2^2+ 2m_1m_3+2m_2m_3+3M_{cc}^2-Q^2)M_{bc}^4 - \\ && m_1^2(m_1^2-2m_1m_2+m_2^2+2m_1m_3+2m_2m_3+ 7M_{cc}^2-Q^2)M_{cc}^4),\nonumber\\ C_1^{(2,-4)} &=& \frac{1}{96M_{bc}^{10}M_{cc}^8}(-3m_1^4m_3^2M_{cc}^{10}+ m_1^3(6m_2M_{bc}^2-6m_3M_{bc}^2+m_1m_3^2-\nonumber\\ && 2m_1M_{cc}^2-4m_3M_{cc}^2)M_{bc}^2M_{cc}^8 - (12m_2^3m_3M_{cc}^2+4m_2m_3^3M_{cc}^2-12m_2^2M_{cc}^4\nonumber\\ && -4m_3^2M_{cc}^4+m_2^4(3m_3^2+4M_{cc}^2))M_{bc}^{10} + 2m_1(-3m_2^3M_{bc}^2+4m_1m_2m_3M_{cc}^2\nonumber\\ &&-8m_1M_{cc}^4+m_2^2(3m_3M_{bc}^2-m_1m_3^2+3m_1M_{cc}^2+2m_3M_{cc}^2))M_{bc}^6 M_{cc}^4 + \nonumber\\ && 2m_1^2(-4m_2(m_1-2m_3)M_{cc}^2+m_2^2(m_3^2+3M_{cc}^2)+(3m_1^2+6m_1m_3+ \\ && 2M_{cc}^2-3Q^2)M_{cc}^2)M_{bc}^4M_{cc}^6 + m_2^2(m_2^2(m_3^2-4M_{cc}^2)+ 4m_2(m_1-4m_3)M_{cc}^2\nonumber\\ && +2(-2m_1^2-4m_1m_3+3M_{cc}^2+2Q^2)M_{cc}^2)M_{bc}^8M_{cc}^2,\nonumber\\ C_1^{(2,-3)} &=& \frac{1}{48M_{bc}^8M_{cc}^6}(m_1(9m_3M_{bc}^2-m_1m_3^2+ m_2(-9M_{bc}^2-4m_1m_3+4m_3^2)+\nonumber\\ && 4m_1M_{cc}^2)M_{bc}^2M_{cc}^4 - (2m_2^3m_3+ 9m_2^2m_3^2+m_3^4+10m_2m_3M_{cc}^2-\\ && 10M_{cc}^4)M_{bc}^6 + (m_2^4m_3^2+2m_2^3m_3^3- 3m_2^2m_3^2M_{cc}^2-3m_2^2M_{cc}^4-14m_2m_3M_{cc}^4\nonumber\\ && +9M_{cc}^6+m_1^2(m_2^2m_3^2+ 2m_3^2M_{cc}^2-3M_{cc}^4)+2m_1(-m_2^3m_3^2+m_2^2m_3^3+\nonumber\\ && m_2^2m_3M_{cc}^2- 2m_2m_3^2M_{cc}^2+m_2M_{cc}^4-3m_3M_{cc}^4)-m_2^2m_3^2Q^2-2m_3^2M_{cc}^2Q^2+ \nonumber\\ && 3M_{cc}^2Q^2)M_{bc}^4+m_1(m_1^3m_3^2+3(m_2-m_3)m_3^2M_{bc}^2+ m_1^2(3m_2M_{bc}^2-3m_3M_{bc}^2\nonumber\\ && -2m_2m_3^2+2m_3^3-2m_3M_{cc}^2)+ m_1m_3^2(m_2^2+2m_2m_3-3M_{cc}^2-Q^2))M_{cc}^4),\nonumber\\ C_1^{(2,-2)} &=& \frac{1}{96M_{bc}^{10}M_{cc}^8}(3m_1^2m_3^4M_{cc}^6+ m_1m_3^2M_{bc}^2M_{cc}^4(6m_2M_{bc}^2-6m_3M_{bc}^2+m_1m_3^2-\nonumber\\ && 2m_1M_{cc}^2+ 4m_3M_{cc}^2)+M_{bc}^6(3m_2^2m_3^4+8m_2m_3(m_3^2-M_{cc}^2)M_{cc}^2+\\ && 4M_{cc}^4 (-11m_3^2+2M_{cc}^2)) + M_{bc}^4m_3M_{cc}^2(-8m_2(m_1-2m_3)m_3M_{cc}^2+\nonumber\\ && m_2^2(m_3^3+6m_3M_{cc}^2)-2M_{cc}^2(-3m_1^2m_3-6m_1m_3^2+2m_1M_{cc}^2+\nonumber \\ && 20m_3M_{cc}^2+3m_3Q^2))),\nonumber\\ C_1^{(2,-1)} &=& \frac{1}{96M_{bc}^8M_{cc}^6}m_3^2(M_{bc}^2(4m_2m_3+15m_3^2- 16M_{cc}^2)+m_3(-m_1^2m_3+\\ && 2m_1(m_2m_3-m_3^2+2M_{cc}^2)+m_3(-m_2^2-2m_2m_3+13M_{cc}^2 +Q^2))),\nonumber\\ C_1^{(2,0)} &=& -\frac{m_3^4(M_{bc}^2(m_3^2-4M_{cc}^2)+m_3^2M_{cc}^2} {96M_{bc}^{10}M_{cc}^8},\\ C_1^{(3,-6)} &=& \frac{(M_{bc}^4m_2^2-m_1^2M_{cc}^2)^3} {96M_{bc}^{10}M_{cc}^8},\\ C_1^{(3,-5)} &=& \frac{1}{96M_{bc}^8M_{cc}^8}(M_{bc}^4m_2^2-m_1^2M_{cc}^4) (M_{bc}^4m_2^2(2m_2m_3+M_{cc}^2) -\nonumber\\ && m_1^2M_{cc}^4(2m_2m_3+7M_{cc}^2)),\\ C_1^{(3,-4)} &=& -\frac{1}{96M_{bc}^{10}M_{cc}^8}(-3m_1^4m_3^2M_{cc}^8 + M_{bc}^8m_2^2(m_2^2m_3^2-8m_2m_3M_{cc}+2M_{cc}^4)+\nonumber\\ && 2M_{bc}^4m_1^2M_{cc}^2(m_2^2m_3^2+6m_2m_3M_{cc}^2+3M_{cc}^4)),\\ C_1^{(3,-3)} &=& \frac{1}{48M_{bc}^8M_{cc}^8}(m_1^2m_3^2M_{cc}^4 (-2m_2m_3+M_{cc}^2) + M_{bc}^4(-2m_2^3m_3^3+2m_2^2m_3^2M_{cc}^2+\nonumber\\ && 6m_2m_3M_{cc}^4-3M_{cc}^6)),\\ C_1^{(3,-2)} &=& -\frac{m_3^2(3m_1^2m_3^2M_{cc}^4 + M_{bc}^4(m_2^2m_3^2+ 12m_2m_3M_{cc}^2-18M_{cc}^4))}{96M_{bc}^{10}M_{cc}^8},\\ C_1^{(3,-1)} &=& \frac{m_3^4(2m_2M-3-9M_{cc}^2)}{96M_{bc}^8M_{cc}^8},\\ C_1^{(3,0)} &=& \frac{m_3^6}{96M_{bc}^{10}M_{cc}^8}.\end{aligned}$$ For functions $U_0(a, b)$ we have the following expressions $$%\begin{equation} U_0(a,b) = \sum_{n = 1+b}^{1+a+b}2C_{n-b-1}^{a}\exp[-B_0] (M_{bc}^2 + M_{cc}^2)^{a+b+1-n}\left (\frac{B_{-1}}{B_{1}}\right )^{\frac{n} {2}}K_{-n} [2\sqrt{B_{-1}B_{1}}], \nonumber$$for $a\geq 0$. Here $K_{n}[z]$ is the modified Bessel function of the second order. In the case of $U_0(-1,-2)$ and $U_0(-1,-1)$ we have failed to obtain exact analytical expressions, so we present their analytical approximations: $$\begin{aligned} U_0(-1,-2) &=& \frac{\exp [-B_0]}{2v^3B_{-1}B_{1}}\{ 2 \exp[-\sqrt{\frac{B_{-1}}{B_{1}}}]B_{1}v^2 + 2\exp [-B_{-1}^{1/2}B_{1}^{3/2}] \sqrt{B_{-1}B_{1}}v^2\nonumber\\ && + \exp [-B_{-1}^{1/2}B_{1}^{3/2}]B_{-1}^{3/2}B_{1}^{1/2}(2+vB_{1})v - \nonumber\\ && 2v^2B_{-1}B_{1}^2\left (\exp [\frac{B_{-1}}{v}]\Gamma \left (0, \frac{B_{-1}}{v} + \sqrt{\frac{B_{-1}}{B_{1}}} \right ) + \Gamma (0, B_{-1}^{1/2}B_{1}^{3/2})\right )\nonumber\\ && -2vB_{-1}B_{1}(\exp [\frac{B_{-1}}{v}]\Gamma\left (0, \frac{B_{-1}}{v} + \sqrt{\frac{B_{-1}}{B_{1}}} \right) + \Gamma (0, B_{-1}^{1/2}B_{1}^{3/2}) \nonumber\\ && - \exp[vB_{1}]\Gamma (0, B_{-1}^{1/2}B_{1}^{3/2} + vB_{1})) - v^2B_{-1}\exp [B_{-1}^{1/2}B_{1}^{3/2}]\\ && - B_{-1}^2B_{1}(v^2B_{1}^2\Gamma (0, B_{-1}^{1/2}B_{1}^{3/2}) + 2vB_{1}\Gamma (0, B_{-1}^{1/2}B_{1}^{3/2}) \nonumber\\ && + 2\Gamma (0, B_{-1}^{1/2} B_{1}^{3/2} - 2\exp [vB_{1}]\Gamma (0, B_{-1}^{1/2}B_{1}^{3/2} + vB_{1}))\}, \nonumber\\ U_0 (-1, -1) &=& \frac{-\exp[B_0]}{v^2}\{ (\exp\frac{B_{-1}}{v} \Gamma \left (0, \frac{B_{-1}}{v}+\sqrt{\frac{B_{-1}}{B_{1}}}\right ) + \nonumber\\ && \Gamma (0, B_{-1}^{1/2}B_{1}^{3/2}) - \exp vB_{1}\Gamma (0, B_{-1}^{1/2}B_{1}^{3/2} + vB_{1}) +\nonumber\\ && v(\exp\frac{B_{-1}}{v}\Gamma\left (0, \frac{B_{-1}}{v} + \sqrt{\frac{B_{-1}}{B_{1}}}\right ) - \Gamma\left (0,\sqrt{\frac{B_{-1}} {B_{1}}}\right ))B_{1} )\\ && - \sqrt{\frac{B_{-1}}{B_{1}}}v\exp [-B_{-1}^{1/2}B_{1}^{3/2}] + B_{-1}(vB_{1}\Gamma (0, B_{-1}^{1/2}B_{1}^{3/2}) + \nonumber\\ && \Gamma (0, B_{-1}^{1/2}B_{1}^{3/2}) - \exp [vB_1]\Gamma (0, B_{-1}^{1/2}B_{1}^{3/2}+vB_{1}))\}, \nonumber\end{aligned}$$ where $v = M_{bc}^2 + M_{cc}^2$ and $\Gamma (a, z)$ is the incomplete gamma function, which is given by the integral $\Gamma (a, z) =\int_z^{\infty} t^{a-1}e^{-t}dt$. 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[^1]: We do not consider for the production in weak interactions. [^2]: The meson at the $P$-wave level, for which $\langle 0|\bar q_1\gamma_{\mu}q_2|P(p)\rangle = i f_Pp_{\mu},$ where $P(p)$ denotes the scalar $P$-wave meson under consideration, and $m_1\neq m_2$. [^3]: Since the diagrams under consideration do not have UV divergencies, there is no need for a dimensional regularization. [^4]: In HQET, the slope of Isgur-Wise function acquires a valuable correction due to the $\alpha_s$-term. [^5]: For normalization of the calculated branching ratios we used the total $B_c$ width obtained in the framework of the OPE approach \[8\]. [^6]: The corresponding estimates were performed in [@cheng], where the $1S\to 2S$ transition is suppressed with respect to $1S\to 1S$ as 1/25. [^7]: At present, there is an analytical calculation of initial six moments for the Wilson coefficient of gluon condensate in the second order of $\alpha_s$ [@iliyn]. In this region, the influence of gluon condensate to the sum rule results is negligibly small for the heavy quarkonia, containing the $b$-quark, which does not allow one to draw definite conclusions on the role of such $\alpha_s$-corrections.

--- abstract: 'We study the Higgs sector of supersymmetric models containing two Higgs doublets with a light MSSM-like CP odd Higgs, $m_A \lesssim 10$ GeV, and $\tan \beta \lesssim 2.5$. In this scenario all of the Higgses resulting from two Higgs doublets: light and heavy CP even Higgses, $h$ and $H$, the CP odd Higgs, $A$, and the charged Higgs, $H^\pm$, could have been produced at LEP or the Tevatron, but would have escaped detection because they decay in modes that have not been searched for or the experiments are not sensitive to. Especially $H \to ZA$ and $H^\pm \to W^{\pm \star} A$ with $A \to c \bar c, \tau^+ \tau^-$ present an opportunity to discover some of the Higgses at LEP, the Tevatron and also at B factories. Typical $\tau$- and $c$-rich decay products of all Higgses require modified strategies for their discovery at the LHC.' author: - 'Radovan Derm'' išek' date: 'June 5, 2008' title: 'Light CP-odd Higgs and Small $\tan \beta$ Scenario in the MSSM and Beyond' --- [*Introduction:*]{} In our current understanding of particle physics all known elementary particles acquire mass in the process of electroweak symmetry breaking. In the standard model (SM) this is achieved by the Higgs mechanism: a complex scalar electroweak doublet gets a vacuum expectation value, spontaneously breaks the electroweak symmetry, gives masses to $W$ and $Z$ bosons and leaves a scalar Higgs boson in the spectrum. The Higgs boson is the last missing piece of the standard model. In theories beyond the SM the Higgs sector is typically more complicated, e.g. in the minimal supersymmetric standard model (MSSM) there are two Higgs doublets which lead to five Higgs bosons in the spectrum: light and heavy CP even Higgses, $h$ and $H$, the CP odd Higgs, $A$, and a pair of charged Higgs bosons, $H^\pm$; and there are many simple models with even more complicated Higgs sector. Discovery of Higgs bosons and exploration of their properties is the key to understanding the electroweak symmetry breaking and a major step in uncovering the ultimate theory of particle physics. Since the searches for Higgs bosons rely on detection of their decay products, it is crucial to understand the way Higgs bosons decay. Although it is usually the case that there is one Higgs boson with properties (couplings to $W$ and $Z$ bosons) of the SM Higgs it is not necessarily true that such a Higgs decays in the way the SM Higgs does [@Chang:2008cw]. A significant model dependence of decay modes applies to other Higgses as well. In this letter we would like to bring attention to the class of models with Higgs sectors that resemble the Higgs sector of the MSSM in the region with a light CP odd Higgs boson, $m_A \lesssim 10$ GeV, and $\tan \beta \lesssim 2.5$. Although this region is ruled out in the MSSM, after careful review of experimental limits we argue that it is easy to make this region phenomenologically viable in simple extensions of the MSSM. We focus on features that involve the two Higgs doublet part of a possible extension. Perhaps the most interesting observation is that all the Higgses resulting from two Higgs doublets: $h$, $H$, $A$ and $H^\pm$ could have been produced already at LEP or the Tevatron, but would have escaped detection because they decay in modes that have not been searched for or the experiments are not sensitive to. We discuss several search modes that present an opportunity to discover some of the Higgses at both LEP and the Tevatron. These decay modes also require modified strategies for Higgs discovery at the LHC. In addition, the light CP odd Higgs might be within the reach of current B factories. [*MSSM at $m_A \ll m_Z$ and $\tan \beta \lesssim 2.5$:*]{} Let us start with the discussion of the Higgs sector of the MSSM [@higgs_review]. In the MSSM, the CP-even Higgs mass-squared matrix at the tree level can be written in terms of CP odd Higgs boson mass, $m_A$, the mass of the Z boson, $m_Z$, and the ratio of the vacuum expectation values of the two Higgs doublets, $\tan \beta = v_u/v_d$. Radiative corrections can be approximated by the contribution to $H_u-H_u$ element of the Higgs mass-squared matrix. This contribution, $\Delta$, is dominated by stop loops and thus depends on stop masses and mixing in the stop sector (it contains a $1/\sin^2 \beta$ factor). The Higgs mass eigenstates are obtained by an orthogonal transformation parameterized by $\alpha$. The coupling squared of the lighter CP-even Higgs boson to $ZZ$ divided by the SM value is given by: $$C_{ZZh} = \frac{g^2_{ZZh}}{g^2_{ZZh_{SM}}} = \sin^2 (\beta - \alpha).$$ The two CP even Higgs bosons share the SM Higgs coupling to Z and thus $C_{ZZH} = \cos^2 (\beta - \alpha)$. For Higgs searches also the couplings of $h$ and $H$ to $ZA$ states are important, and their values squared (appropriately rescaled) are given as: $C_{ZAh} = \cos^2 (\beta - \alpha)$ and $C_{ZAH} = \sin^2 (\beta - \alpha)$. Note that the couplings are complementary which is important for Higgs searches. The Higgs spectrum is particularly simple in two distinct regimes. In the decoupling regime, $m_A \gg m_Z$, masses of light and heavy CP even Higgses are given by $$\begin{aligned} m_h^2 &\simeq & m_Z^2 \cos^2 2 \beta + \Delta \sin^2 \beta , \\ m_H^2 &\simeq & m_A^2 + m_Z^2 \sin^2 2 \beta + \Delta \cos^2 \beta \simeq m_A^2 . \label{eq:mH2decoupled}\end{aligned}$$ The light CP even Higgs boson is SM-like in its couplings to $ZZ$ and $WW$ for any $\tan \beta \geq 1$: $$C_{ZZh} \simeq 1 - \frac{m_Z^4}{4 m_A^4} \sin^2 4\beta \simeq 1.$$ In the opposite limit, characterized by $m_A \ll m_Z$, neglecting radiative corrections we have: $$\begin{aligned} m_h^2 &\simeq& m_A^2 \cos^2 2 \beta , \label{eq:mh2_mAlessmZ} \\ m_H^2 &\simeq& m_Z^2(1+ \frac{m_A^2}{m_Z^2}\sin^2 2 \beta),\end{aligned}$$ and the $ZZH$ coupling is given by: $$C_{ZZH} \simeq \cos^2 2\beta \left( 1 - \frac{m_A^2}{m_Z^2} \sin^2 2\beta \right). \label{eq:ZZH_mAlessmZ}$$ Formulas including radiative corrections are somewhat long and not particularly revealing in this limit. Note however, that for $\tan \beta > few$ the radiative corrections contribute mostly to the Heavy CP even Higgs which plays the role of the SM Higgs, $C_{ZZH} \simeq 1$. This regime of the parameter space does not receive much attention because it is beyond any doubt ruled out by Higgs searches at LEP. The mass of the light CP even Higgs cannot be raised by radiative corrections and pairs of $h A$ would be copiously produced since $g_{ZAh} \simeq 1$. In addition, light $h$ and $A$ would significantly contribute to the Z width and thus are ruled out by the Z-pole measurements [@:2004qh]. As $\tan \beta $ approaches 1 for $m_A \ll m_Z$ the situation described above dramatically changes.[^1] The light CP even Higgs boson becomes SM-like, $C_{ZZh} \simeq 1$, since $C_{ZZH} \simeq 0$, see Eq. (\[eq:ZZH\_mAlessmZ\]), and although it is massless at the tree level (\[eq:mh2\_mAlessmZ\]), it will receive a contribution from superpartners and the tree level relation between the light CP even and CP odd Higgses, $m_h < m_A$ is typically not valid. Even for modest superpartner masses the light CP even Higgs boson will be heavier than $2m_A$ and thus $h \to AA$ decay mode is open and generically dominant. For small $\tan \beta$ the width of $A$ is shared between $\tau^+ \tau^-$ and $c \bar c$ for $m_A < 2m_b$ and thus the width of $h$ is spread over several different final states, $4\tau$, $4c$, $2\tau 2c$ and highly suppressed $b \bar b$ and thus the LEP limits in each channel separately are highly weakened. Since $h$ is SM-like, $e+ e^- \to hA$ is highly suppressed and the limits from the Z width measurements can be easily satisfied even for $m_h + m_A < m_Z$. In addition, we will see that decay modes of the heavy CP even Higgs (that also turns out to be within the reach of LEP) are modified in this region and even the charged Higgs boson can be below LEP or Tevatron limits due to decay modes that have not been searched for. For $m_A \ll m_Z$ and $\tan \beta \simeq 1$ including radiative corrections we find: $$\begin{aligned} m_h^2 &\simeq& \Delta/2, \\ m_H^2 &\simeq& m_Z^2 + m_A^2 + \Delta/2. \end{aligned}$$ As already mentioned, $h$ is SM-like and thus $C_{ZZh} = C_{ZAH} \simeq 1$ and $C_{ZZH} = C_{ZAh} \simeq 0$. For $\tan \beta \simeq 2.5$ both Higgses equally share the coupling to $Z$, $C_{ZZh} \simeq C_{ZZH} \simeq 0.5$. The mass of the charged Higgs is given as, $$\begin{aligned} m_{H^\pm}^2 &=& m_W^2 + m_A^2 - \Delta^\prime \; \simeq \; m_W^2, \end{aligned}$$ where $\Delta^\prime$ represents radiative correction which is typically not significant (it is positive and has a tendency to decrease the mass of the charged Higgs). Using [*FeynHiggs2.6.3*]{} [@Heinemeyer:1998yj] (with $m_t = 172.6$ GeV) for $\tan \beta = 1.01$, $\mu = 100$ GeV, $m_A = 8$ GeV and varying soft susy breaking scalar and gaugino masses between 300 GeV and 1 TeV and mixing in the stop sector, $X_t/m_{\tilde t}$, between 0 and -2, we typically find: $m_h \simeq 38 - 56$ GeV with $g_{ZZh}/g_{ZZh_{SM}} \simeq 0.84 - 0.97$, $m_H \simeq 108 - 150$ GeV and $m_{H^\pm} \simeq 78 - 80$ GeV. The dominant branching ratios of the light CP even Higgs are typically: $$B (h \to A A, \; b \bar b) \; \simeq \; 90 \%, \; 10 \% \label{eq:Bh}$$ with $$B (A \to \tau^+ \tau^-, \; c \bar c, \; gg ) \; \simeq \; 50 \%, \; 40 \%, \; 10\% , \label{eq:BA}$$ for $2m_\tau \lesssim m_A \lesssim 10$ GeV. Branching ratios of the Heavy CP even Higgs vary with SUSY spectrum. For 1 TeV SUSY and $X_t/m_{\tilde t} = 0$ we find: $$B (H \to ZA, \; A A, \; hh, \; b \bar b) \; \simeq \; 37 \%, \; 34 \%, \; 28 \% , \; 0.4 \% \label{eq:BH}$$ (similar branching ratios apply for 300 GeV SUSY with $X_t/m_{\tilde t} = -2$). Finally, the dominant branching ratios of the charged Higgs are: $$B (H^+ \to W^{+ \star } A, \; \tau^+ \nu, \; c \bar s) \; \simeq \; 70 \%, \; 20 \%, \; 10 \% . \label{eq:BHpm}$$ For discussion of experimental constraints let us also include branching ratios of the top quark: $$B(t \to H^+ b) \simeq 40 \%, \quad B(t \to W^+ b) \simeq 60 \% . \label{eq:Bt}$$ These results (except (\[eq:BH\])) are not very sensitive to superpartner masses nor the mass of the CP odd Higgs as far as $m_A < 2m_b$. Increasing $\tan \beta$ to $2.5$ only the following branching ratios significantly change: $B (A \to \tau^+ \tau^-, gg ) \simeq 90 \%, 10\%$, $B (H^+ \to W^{+ \star } A, \tau^+ \nu) \simeq 35 \%, 65 \%$ and $B(t \to H^+ b, W^+ b) \simeq 10 \%, 90 \%$. In spite of all Higgses being within the reach of LEP we will see that only the light CP even Higgs is significantly constrained and actually in the MSSM ruled out by LEP data. For limits on various search modes and complete list of references see Ref. [@Schael:2006cr]. [*Light CP even Higgs:*]{} One of the strongest limits on the CP even Higgs boson comes from the search for $h \to b \bar b$. The limit for $m_h \lesssim 60$ GeV is $C_{ZZh} B(h \to b \bar b) \lesssim 0.04$. In order to satisfy this limit we need $B (h \to A A) \gtrsim 96 \%$. This is not impossible, but it is not generically (in large regions of SUSY parameter space) satisfied (\[eq:Bh\]). With generic $B (h \to b \bar b) \simeq 10 \%$ we need $m_h \gtrsim 85$ GeV or somewhat reduced $C_{ZZh}$. Note also that combined $h \to b \bar b$ and $h \to AA \to 4b$ limit requires $m_h > 110$ GeV for $C_{ZZh} \simeq 1$. Thus $m_A < 10$ GeV is favored. For $m_A < 10$ GeV the limits on the dominant decay mode $h \to AA$ are not so strong since it is spread over several final states: $h \to AA \to 2\tau 2c, 4\tau, 4c, \dots$ with about 36%, 23%, 14%, $\dots$ branching ratios. These limits can be satisfied for any $m_h$ provided $m_A\gtrsim 9$ GeV with limits weakening for heavier $h$ and completely expiring for $m_h \gtrsim 86$ GeV [@Jack_TeV4LHC]. The limits from $e^+ e^- \to hA$ are typically comfortably satisfied for masses and branching ratios of interest as a consequence of a very small $C_{ZAh}$. The strongest limit comes from $hA \to AAA \to 6 \tau$ which for $m_A \simeq 10$ GeV and $m_h = 40 - 60$ GeV requires $C_{ZAh} \lesssim 0.07$ and this limit runs out for $m_h \simeq 65$ GeV. A comparable constraint on $C_{ZAh}$ comes from the $Z$-width measurement for $m_h = 40$ GeV and it becomes weaker for larger $m_h$. We see that this scenario can avoid all decay-mode specific limits in the region of SUSY parameter space that leads to $B (h \to A A) > 96 \%$ and $C_{ZZh} \gtrsim 0.93$ and we will se later that there are no other relevant constraints from searches for $H$ and $H^\pm$. However, the decay-mode independent search from OPAL [@Abbiendi:2002qp] sets the limit on the Higgs mass by looking only for reconstructed Z boson decaying leptonically and excludes $m_h < 82$ GeV for $C_{ZZh} = 1$. Taking the decay-mode independent limit at face value[^2] our scenario is ruled out in the MSSM, since $m_h$ cannot be pushed above 82 GeV with radiative corrections. There are however various ways to increase the mass of the SM-like Higgs boson in extensions of the MSSM. A simple possibility is to consider singlet extensions of the MSSM containing $\lambda S H_u H_d$ term in the superpotential. It is known that this term itself contributes $\lambda^2 v^2 sin^2 2 \beta$, where v = 174 GeV, to the mass squared of the CP even Higgs [@Ellis:1988er] and thus can easily push the Higgs mass above $82$ GeV. Note this contribution is maximized for $\tan \beta \simeq 1$. Singlet extensions can also alter the couplings of the Higgses to $Z$ and $W$ through mixing [@mixed] or provide new Higgs decay modes [@4tau], [@Chang:2008cw]. Considering this possibility would lead to very model dependent predictions and thus in this paper we assume that a possible extension does not significantly alter the two Higgs doublet part of the Higgs sector besides increasing the Higgs mass above the decay-mode independent limit. [*Heavy CP even Higgs:*]{} The heavy CP even Higgs is too heavy to be produced in association with $Z$ at LEP. In addition, $C_{ZZH}$ is small. However, since $C_{ZAH} \simeq 1$ and A is light, pairs of $AH$ would be produced at LEP. The strongest limit comes from $HA \to b \bar b \tau^+ \tau^-$ which is however comfortably satisfied due to small $B(H\to b \bar b)$. The limits on $AH \to Ahh$ mode are not constraining since $h$ decays dominantly into $AA$. This mode might not be open in extensions of the MSSM which increase the mass of $h$ in which case $ZA$, $AA$ and $b\bar b$ will share the width of $H$. The searches in various final states of $AH \to AAA$ are either not sensitive to or were not done in the range of masses typical in our scenario. [*The dominant $H \to ZA$ mode has never been searched for!*]{} Interestingly, $e^+ e^- \to H A \to (ZA) A$ events could be mistaken for $e^+ e^- \to ZH \to Z (AA)$ or simply $Z +jets$ for which searches have been done. Of course the interpretation (the reconstructed Higgs mass) would be wrong and this might be responsible for various local excesses of events and small excesses over large range of reconstructed Higgs masses in flavor independent searches, see e.g. Refs. [@Achard:2003ty; @Abdallah:2004bb]. [*Charged Higgs:*]{} Searches for pair produced charged Higgs bosons were performed by LEP collaborations [@:2001xy; @Achard:2003gt; @Heister:2002ev; @Abdallah:2003wd]. Three different final states, $\tau^+ \nu \tau^- \bar \nu$, $c \bar s \bar c s$ and $c \bar s \tau^- \bar \nu$ were considered and lower limits were set on the mass $m_{H^\pm}$ as a function of the branching ratio $B(H^+ \to \tau^+ \nu)$, assuming $B(H^+ \to \tau^+ \nu) + B(H^+ \to c \bar s)=1$. In addition, DELPHI considered a possibility $H^+ \to W^{+\star} A$ which is important if the CP odd Higgs boson is not too heavy [@Akeroyd:2002hh] and limits were obtained under the assumption that $A$ is heavy enough to decay into $b \bar b$ [@Abdallah:2003wd]. The strongest limits are set by ALEPH [@Heister:2002ev]. Assuming $B(H^+ \to \tau^+ \nu) + B(H^+ \to c \bar s)=1$, charged Higgs bosons with mass below 79.3 GeV are excluded at 95% C.L., independent of $B(H^+ \to \tau^+ \nu)$. Somewhat lower limits have been obtained by DELPHI [@Abdallah:2003wd] and L3 [@Achard:2003gt] collaborations due to local excesses of events. In the scenario discussed above the charged Higgs decays dominantly into $W^\star A$ with $A \to c \bar c$ or $\tau^+ \tau^-$ (\[eq:BHpm\]). LEP limits thus apply to the remaining branching ratios and can be comfortably satisfied for $m_{H^\pm} \gtrsim 75$ GeV. At the Tevatron the charged Higgs is searched for in the decay of the top quark. If kinematically allowed, the top quark can decay to $H^+ b$, competing with the standard model decay $W^+ b$. The strongest limits come from CDF [@Abulencia:2005jd] from data samples corresponding to an integrated luminosity of 193 ${\rm pb}^{-1}$. It is assumed that $H^+$ can decay only to $\tau^+ \nu$, $c\bar s$, $t^\star \bar b$ or $W^+ A$ with $A \to b \bar b$. If charged Higgs decays exclusively to $\tau^+ \nu$, the $B(t \to H^+ b)$ is constrained to be less than 0.4 at 95 % C.L. For MSSM benchmark scenarios, assuming $H^+ \to \tau^+ \nu$ or $H^+ \to c \bar s$ only, stronger limits than at LEP are set for $\tan \beta \lesssim 1.3$ on the mass of the charged Higgs. For $\tan \beta \lesssim 1$ the limit is $m_{H^\pm} \gtrsim 100$ GeV. If no assumption is made on the charged Higgs decay (but still allowing only those that were searched for) the $B(t \to H^+ b)$ is constrained to be less than $\sim 0.8$ for $m_{H^\pm} \simeq 80$ GeV. For $H^\pm \to W^{\pm \star} A$ with $A \to c \bar c$ or $\tau^+ \tau^-$, [*the decay modes that were not search for*]{}, and in addition modes that can easily mimic $W$ decay modes, especially the dominant hadronic mode, it is reasonable to expect that the limits would be even weaker. In our scenario $B(t \to H^+ b) \lesssim 40 \%$, see Eq. (\[eq:Bt\]), and thus the Tevatron does not place stronger limits than LEP. [*CP odd Higgs:*]{} The light CP odd Higgs might be within the reach of current B factories where it can be produced in Upsilon decays, $\Upsilon \to A \gamma$. This was recently suggested in the framework of the next-to-minimal supersymmetric model (NMSSM) with a light CP odd Higgs boson being mostly the singlet [@Dermisek:2006py] (in our scenario $A$ is doublet-like) and it overlaps with searches for lepton non-universality in $\Upsilon$ decays [@Sanchis-Lozano]. It is advantageous to look for a light CP odd Higgs in $\Upsilon (1S)$, $(2S)$ and $(3S)$ since these states cannot decay to B mesons and thus the $A \gamma$ branching ratio is enhanced. Predictions for the branching ratio $B(\Upsilon \to A \gamma)$ for $\tan \beta = 1$ can be readily (although only approximately) obtained from the results of Ref. [@Dermisek:2006py] taking $\tan \beta \cos \theta_A \simeq 1$ ($\cos \theta_A$ is the doublet component which is 1 for MSSM-like CP odd Higgs). We find, that the $B(\Upsilon (1 S) \to A \gamma) $ ranges between $5 \times 10^{-5}$ for $m_A \simeq 2 m_\tau$ and $10^{-7}$ for $m_A \simeq 9.2$ GeV. CLEO recently reported results from $\Upsilon (2S) \to \pi^+ \pi^- \Upsilon (1S) \to \pi^+ \pi^- A \gamma$ [@Kreinick:2007gh] and also preliminary results from $\Upsilon (1S)$ data [@CLEOprelim]. The limits, assuming $B(A \to \tau^+ \tau^-)= 1$, range between $7 \times 10^{-5}$ and $8 \times 10^{-6}$ depending on the mass of the CP odd Higgs. For large $\tan \beta$ these limits rule out a significant part of the parameter space. For small $\tan \beta$ the $B(A \to \tau^+ \tau^-)$ is reduced and thus these limits are less constraining. The BaBar results from recently taken $\Upsilon (3S)$ and $\Upsilon (2S)$ data assuming $B(A \to \tau^+ \tau^-) = 1$ should be available soon [@BaBar]. [*Discussion:*]{} Supersymmetric models with Higgs sectors that resemble the Higgs sector of the MSSM in the region with light CP odd Higgs boson, $m_A \lesssim 10$ GeV, and $\tan \beta \lesssim 2.5$ can have SM-like CP even Higgs boson as light as 82 GeV (the decay-mode independent limit) dominantly decaying into $h \to AA \to 2\tau 2c, 4\tau, 4c$. This feature is similar to the scenario with a light singlet-like CP-odd Higgs in the NMSSM [@4tau]. In addition, this scenario predicts that the CP odd and the heavy CP even Higgses were also produced at LEP in $e^+ e^- \to HA$, and could be discovered searching for the dominat decay mode $H \to Z A$. Furthermore, up to $\sim 40 \%$ of top quarks produced at the Tevatron could have decayed into charged Higgs and the $b$ quark with $H^\pm \to W^{\pm \star} A$ and $A \to c \bar c$ or $\tau^+ \tau^-$. These decay modes of $H$ and $H^\pm$ have never been searched for and thus provide an opportunity to discover some of the Higgses in LEP data or at the Tevatron. The search for $H^\pm$ including these modes is especially desirable at the Tevatron with currently available large data sample. The need for these searches is further amplified by the fact that the charged Higgs with properties emerging in this scenario could explain the $2.8 \sigma$ deviation from lepton universality in $W$ decays measured at LEP [@Dermisek:2008dq]. The CP odd Higgs could also be discovered at B factories. It would be desirable to include $A \to c \bar c$ and combine searches assuming $B(A \to \tau^+ \tau^-) + B(A \to c \bar c) = 1$ or, optimally, allowing a small $B(A\to gg)$. Since dominant decay modes of $h$, $H$, $A$ and $H^\pm$ lead to $\tau$- and $c$-rich final states (this is very different from usual scenarios where Higgs decays are dominated by $b$ final states) modified strategies for their discovery at the LHC are needed. Although this scenario is ruled out in the MSSM we have argued that it can be easily viable in simple extensions. For example, in the NMSSM the scenario with a light doublet-like CP odd Higgs and small $\tan \beta$ is viable and has all the features (Higgs mass ranges and decay modes) of the MSSM in this limit [@NMSSM_small_tb]. It should be stressed however that this scenario is not limited to singlet extensions of the MSSM and it would be viable in many models beyond the MSSM that increase the mass of the SM-like Higgs boson. I would like to thank A. Akeroyd, J. Gunion, P. Langacker, B. McElrath, V. Rychkov and D. Shih for discussions. RD is supported by the U.S. Department of Energy, grant DE-FG02-90ER40542. 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G. Abbiendi [*et al.*]{} \[OPAL Collaboration\], Eur. Phys. J.  C [**27**]{}, 311 (2003) \[arXiv:hep-ex/0206022\]. J. R. Ellis, J. F. Gunion, H. E. Haber, L. Roszkowski and F. Zwirner, Phys. Rev.  D [**39**]{}, 844 (1989). See, e.g. V. Barger, P. Langacker, H. S. Lee and G. Shaughnessy, Phys. Rev.  D [**73**]{}, 115010 (2006); R. Dermisek and J. F. Gunion, Phys. Rev.  D [**77**]{}, 015013 (2008). R. Dermisek and J. F. Gunion, Phys. Rev. Lett.  [**95**]{}, 041801 (2005); Phys. Rev.  D [**73**]{}, 111701 (2006); Phys. Rev.  D [**75**]{}, 075019 (2007); Phys. Rev.  D [**76**]{}, 095006 (2007). P. Achard [*et al.*]{} \[L3 Collaboration\], Phys. Lett.  B [**583**]{}, 14 (2004) \[arXiv:hep-ex/0402003\]. J. Abdallah [*et al.*]{} \[DELPHI Collaboration\], Eur. Phys. J.  C [**44**]{}, 147 (2005) \[arXiv:hep-ex/0510022\]. LEP Collaborations, arXiv:hep-ex/0107031. P. Achard [*et al.*]{} \[L3 Collaboration\], Phys. Lett.  B [**575**]{}, 208 (2003) \[arXiv:hep-ex/0309056\]. A. Heister [*et al.*]{} \[ALEPH Collaboration\], Phys. Lett.  B [**543**]{}, 1 (2002) \[arXiv:hep-ex/0207054\]. J. Abdallah [*et al.*]{} \[DELPHI Collaboration\], Eur. Phys. J.  C [**34**]{}, 399 (2004) \[arXiv:hep-ex/0404012\]. A. G. Akeroyd, S. Baek, G. C. Cho and K. Hagiwara, Phys. Rev.  D [**66**]{}, 037702 (2002) \[arXiv:hep-ph/0205094\]. A. Abulencia [*et al.*]{} \[CDF Collaboration\], Phys. Rev. Lett.  [**96**]{}, 042003 (2006) \[arXiv:hep-ex/0510065\]. R. Dermisek, J. F. Gunion and B. McElrath, Phys. Rev.  D [**76**]{}, 051105 (2007) \[arXiv:hep-ph/0612031\]. M. A. Sanchis-Lozano, Mod. Phys. Lett.  A [**17**]{}, 2265 (2002); Int. J. Mod. Phys.  A [**19**]{}, 2183 (2004); E. Fullana and M. A. Sanchis-Lozano, Phys. Lett.  B [**653**]{}, 67 (2007). D. Kreinick, arXiv:0710.5929 \[hep-ex\]. S. Stone, at FPCP 08, Taipei, Taiwan, May 5-9, 2008. Y. Kolomensky, private communication. R. Dermisek, arXiv:0807.2135 \[hep-ph\]. R. Dermisek and J. F. Gunion, in preparation. [^1]: This region of the parameter space has an interesting physical meaning. A light CP odd Higgs is a signal of an approximate global $U(1)_R$ symmetry of the Higgs potential which allows the $\mu$-term in the superpotential, $W \supset \mu H_u H_d$, but forbids the corresponding $B_\mu$-term in the soft SUSY breaking lagrangian. This symmetry is broken by gaugino masses and thus a small $B_\mu$-term is generated by radiative corrections that lift the mass of the CP odd Higgs, $m_A = 2 B_\mu \sin 2 \beta$. The small $\tan \beta$ on the other hand signals that the top Yukawa couplings becomes non-perturbative close to the grand unification (GUT) scale. The exact value of $\tan \beta$ consistent with perturbativity all the way to the GUT scale depends on superpartner masses through threshold corrections to the top Yukawa coupling, and it is about $\tan \beta \gtrsim 1.2$. However, adding extra vector-like complete SU(5) matter multiplets at the TeV scale, e. g. parts of the sector that mediates SUSY breaking (messengers) or present for no particular reason, does not affect the unification of gauge couplings while it slows down the running of the top Yukawa coupling [@Masip:1998jc; @Barbieri:2007tu] and even $\tan \beta \simeq 1$ can be consistent with perturbative unification of gauge couplings. [^2]: We would like to make two comments however. First of all, this search was not expected to be sensitive to $m_h \gtrsim 55$ GeV and much stronger limits than expected were set due to a significant deficit ($\sim 2\sigma$) of background events for a large range of $m_h$ between 60 an 80 GeV (deficit of background events is also responsible for much stronger limits on $C_{ZZh} B(h \to b \bar b)$ in the same region). Second of all, the efficiencies for signal events to pass the selection cuts are estimated assuming the Higgs decays into two body final states. Clearly, the efficiencies for Higgs decaying into four body final states $4\tau, 4c, 2\tau 2c, \dots$ (that further decay into states typically containing $e^\pm$, $\mu^\pm$ either from $\tau$ or semi-leptonic $c$ decays which might make the Z reconstruction less efficient) are smaller and the exclusion limits would be weaker.

--- abstract: 'We show that by combining measurements of the temperature and polarization anisotropies of the Cosmic Microwave Background (CMB), future experiments will tightly constrain the expansion rate of the universe during recombination. A change in the expansion rate modifies the way in which the recombination of hydrogen proceeds, altering the shape of the acoustic peaks and the level of CMB polarization. The proposed test is similar in spirit to the examination of abundances of light elements produced during Big Bang Nucleosynthesis and it constitutes a way to study possible departures from standard recombination. For simplicity we parametrize the change in the Friedmann equation by changing the gravitational constant $G$. The main effect on the temperature power spectrum is a change in the degree of damping of the acoustic peaks on small angular scales. The effect can be compensated by a change in the shape of the primordial power spectrum. We show that this degeneracy between the expansion rate and the primordial spectrum can be broken by measuring CMB polarization. In particular we show that the MAP satellite could obtain a constraint for the expansion rate $H$ during recombination of $\delta H/H \simeq 0.09$ or $\delta G/G \simeq 0.18$ after observing for four years, whereas Planck could obtain $\delta H/H \leq 0.014$ or $\delta G/G \leq 0.028$ within two years, even after allowing for further freedom in the shape of the power spectrum of primordial fluctuations.' address: - '$^1$Fakultät für Physik, Ludwig-Maximilians-Universität, Geschwister-Scholl-Platz 1, 80799 München' - '$^2$Max-Planck-Institut für Astrophysik, P.O. Box 1317, 85741 Garching' - '$^3$Dept. of Physics, New York University, 4 Washington Pl., New York, NY 10003' - | $^4$Institute for Advanced Study, Einstein Drive, Princeton NJ 08540\ [(Received 16 December 2002; revised manuscript received 19 February 2003; published 18 March 2003)]{} author: - 'Oliver Zahn$^{1,2,3}$[^1] and Matias Zaldarriaga$^{3,4}$[^2]' title: Probing the Friedmann equation during recombination with future CMB experiments --- Introduction\[introduction\] ============================ The parameters of our cosmological model will be determined with great accuracy by upcoming data from Cosmic Microwave Background experiments, galaxy surveys, weak lensing surveys, Lyman alpha forest studies, and other observations. With the new data it will be possible to perform a number of consistency checks that will strengthen our confidence in the underlying model. Some of these consistency checks have already been performed with existing data. Recent analysis of the CMB data have resulted in constraints on the baryon density $\Omega_b h^2$ that are in excellent agreement with its determination based on the study of the primordial abundances of light elements (e.g. [@Wang; @Efsthathiou]). The combination of CMB data with local measures of the Hubble constant [@keyproject] and measures of the local strength of galaxy clustering result in a determination of the cosmological constant that is in good agreement with results from the study of the luminosity of distant supernovae (e.g. [@Riess; @revpar]). Recently also joint BBN and CMB constraints on different dark energy models have been discussed in [@Kneller:2002zh]. One of the assumptions of the cosmological model that has been hard to test is the explicit validity of the Friedmann equation – the relation between the expansion rate of the universe and its matter content. The difficulty lies in finding an epoch in the evolution of the universe during which both the energy density and the expansion rate can be determined independently. The two obvious candidates are the present time and Big Bang Nucleosynthesis (BBN). Precise measures of both the expansion rate and the matter density in the local universe are difficult to accomplish and suffer from various systematic problems. At present the expansion rate has been measured with errors on the order of $10\%$ [@keyproject]. Direct determinations of the matter density however are more uncertain. It is fair to say that the general conclusion from these studies is that the Friedmann equation only holds true if either a cosmological constant or a curvature term is added. This is because direct determinations of the present matter density almost always point to $\Omega_m < 1$ (e.g. see [@revpar]). Neither the cosmological constant nor the curvature scale can be constrained independently, that is without going through the Friedmann equation, so at best we can say that the Friedmann equation has not been tested accurately at the present epoch. A more radical interpretation would be that the Friedmann equation has been tested but that the test has failed. Although we do not support this interpretation, at present some infrared modification of gravity cannot be ruled out observationally. Such modifications are being explored for example as ways to solve the cosmological constant problem (see for example [@Arkani-Hamed]) or to explain the accelerated expansion inferred from the luminosity distance to high redshift supernovae without resorting in a cosmological constant or a quintessence field (e.g. see [@Deffayet:2001pu; @Deffayet2]). During the epoch of Big Bang Nucleosynthesis (BBN) the situation is more fortunate. The energy density is dominated by radiation which we think we can estimate accurately. On the other hand the expansion rate affects the freeze-out abundances of light elements, so that a precision test of the Friedmann equation can be performed. The standard procedure is to constrain the number of relativistic degrees of freedom, $g_*$, which can be translated into a limit on the number of neutrino species. A lot of progress has been made in determining the primordial abundances. Recently the deuterium abundance in hydrogen clouds at high redshift was accurately determined [@O'Meara:2000dh; @Burles:1997ez; @Burles:1997fa]. Building on a prior of $N_\nu \geq 3$ these data have been exploited to enforce an upper limit of 3.2 at $2\sigma$ for the number of neutrino species, based purely on BBN considerations [@Burles:1999zt]. Progress has also been made in appreciating the systematical uncertainties that impair the determination of primordial $^4He$ (see e.g. [@Olive:2000qm; @Peimbert:2002ks]). Built on the safely established abundance ranges for Deuterium, Helium and Lithium, it can be shown that the uncertainty in the number of relativistic degrees of freedom during BBN is around 9 % (68% C.L.)[@Olive:1998vj]. This constraint can equally well be phrased as a constraint on the validity of the Friedmann equation: during Nucleosynthesis the ratio of the squared expansion rate to the energy density can depart by only $9 \%$ from what is predicted by the Friedmann equation. Constraints on the expansion history during BBN have also been established in [@Carroll:2001bv]. In this paper we propose using the anisotropies in the CMB to perform a test similar to the one that has been done using BBN. We will show that such a test can ultimately constrain the validity of the Friedmann equation during recombination more accurately than what has so far been reached within Nucleosynthesis, albeit in a more model dependent way. During recombination, the energy density is dominated by the density of non-relativistic matter which we cannot estimate directly. However the dark matter energy density enters in two different ways and one can exploit this to simultaneously determine the dark matter density and the expansion rate during recombination. The ratio of matter to radiation energy density sets the redshift of matter radiation equality. At that time the expansion rate changes from a scaling as $t^{1/2}$ to $t^{2/3}$. Perturbation modes of the photon-baryon fluid that entered the horizon during the radiation dominated era behave differently than those that entered during the matter dominated era. Modes that entered during radiation domination provided the dominant contribution to the total density perturbation that generated the gravitational potential. On the contrary modes that entered the horizon during matter domination, were sub-dominant in their contribution to the total density perturbation which was dominated by the dark matter fluctuations. The gravitational potential acts as a source for perturbations in the photon baryon fluid. As a result, small scale modes that entered the horizon in the radiation era go through a sort of feedback loop that increases their amplitude as they cross the horizon (for a review of CMB physics see for example [@DodlesonHu]). The anisotropy power spectrum is very sensitive to the redshift of matter radiation equality and thus the CMB should very accurately determine the ratio of dark matter to radiation energy density, i.e. the parameter $\Omega_m h^2$. The dark matter density dominates over other energy components in the Friedmann equation during recombination. In the standard scenario, it sets both the redshift at which matter and radiation become equal and the rate of expansion during recombination through the Friedmann equation. In this paper we break the link between energy density and expansion rate by introducing a free parameter to modify the Friedmann equation. We investigate how well such a parameter can be constrained. From a pragmatic perspective our study can be regarded as the investigation of a particular departure from standard recombination. Different variations have been studied in the literature. For instance the possibility that energetic sources of Ly-$\alpha$-photons could be present during recombination to delay it was considered in [@Peebles:200l]. Also, the possiblility of a time variation of the fine structure constant was investigated [@Kaplinghat:1998ry; @Landau:2000dd; @Hannestad:2000ys]. In [@Uzan:2002vq] the effects of a time dependence of the gravitational constant have been outlined. The conclusion of these investigations and ours is that with future CMB data departures from standard recombination will be severely constrained and perhaps modifications that point to interesting new physics could be discovered. In section \[lambda\], we will introduce our model and in section \[analysis\] we will investigate the constraints that can be set with currently available data and forecast what future CMB experiments might be able to achieve. We will conclude in section \[discussion\] with discussion. The Model: Variation of the gravitational constant\[lambda\] ============================================================ In this section we introduce the model we will use to investigate how well one can test the Friedmann equation using the CMB. The problem is somewhat more subtle than in the case of BBN because we are dealing with the dynamics of perturbations that could be affected by the “new physics” in ways other than through a change in the expansion rate. Thus we need to find a self-consistent way of modifying both the dynamics of the universe and that of the perturbations. One possibility is to add another component that contributes to the energy density during recombination, increasing the rate of expansion at that time. The additional component could be a quintessence field with a potential and initial conditions tuned so that it has some effect during recombination and is unimportant or only marginally important at other times (except perhaps today when it could start to dominate). This approach has the virtue of only modifying the expansion rate but it introduces too much freedom because results depend on when exactly this extra component is important. In such a model we also expect the ratio of the sound horizon at recombination to the angular diameter distance to the last scattering surface to change and thus that the acoustic peaks be slightly shifted. At late times the evolution of the gravitational potentials will also induce an integrated Sachs-Wolfe (ISW) effect. As a result, constraints on any specific model of this kind will come from these three effects [@UzanRiaz]. In our study we want to isolate the information encoded in the change of the expansion rate at recombination so we will use a simpler prescription and assume that the gravitational constant $G$ is somewhat different from its locally measured value. We introduce a single parameter $\lambda$ such that, $$G \rightarrow \lambda^2 G.$$ The expansion rate is proportional to $\lambda$. With this prescription not only the Friedmann equation gets modified but also the dynamics of the perturbations changes because it depends on the strength of gravity. We will show in the following that our prescription has the nice feature that it only changes the CMB power spectrum through the change in recombination, allowing us to isolate the observable effects of this change. The basic reason is that gravity does not have a preferred scale and that we only measure angles when studying the CMB. If $G$ were slightly different all that would happen is that the universe would be expanding a bit faster or slower by a factor $\lambda$ so that the “expansion clock" would be running at a different rate. Such a change cancels in the ratios of distances that we measure with the CMB. The only way we can find out that such an alteration had occurred is by having an independent clock that measures the expansion rate. In our case this independent clock will be the physics of hydrogen recombination. In this sense our simple test is very similar to what has been done in the context of BBN. Effect of $\lambda$ on the CMB anisotropies ------------------------------------------- The dependence of the Hubble parameter and the dynamics of perturbations on the gravitational constant will lead to modifications of the CMB anisotropies as we vary the parameter $\lambda$. We will discuss the physics in this section. We start by considering the modification to the Friedmann equation, $$H^2=\left(\frac{\dot{a}}{a} \right)^2=\frac{8 \pi}{3} G \rho \rightarrow \frac{8 \pi}{3} \lambda^2 G \rho$$ where $\rho$ is the total energy density. As a function of the expansion factor $a$ and $\lambda$, the expansion rate $H$ satisfies: $$H(a,\lambda)=\lambda f(a) \ ,$$ where the function $f(a)$ is independent of $\lambda$. Thus with this simple prescription, the shape of the function $H$ of $a$ is not changed by $\lambda$, only the amplitude changes. For example the redshift at which matter and radiation contribute equally to the energy density does not change. The change introduced is a simple rescaling of the “expansion rate clock". In order to understand how the anisotropies get modified, we start by writing down the integral solution for the temperature anisotropies produced by a mode of wavevector $\k$ observed towards direction $\n$ [@Seljak:1996is]. The temperature can be written as an integral along the line of sight over sources, $$\Delta T(\n,\k) = \int^{\tau_0}_0 d \tau \; S(k,\tau) e^{i \k\cdot \n D(\tau)} g(\tau) \label{lineofsight}$$ In this equation $S(k,\tau)$ is the source term, $g(\tau)$ is the visibility function, and $D(\tau)$ is the distance from the observer to a point along the line of sight corresponding to the conformal time $\tau$ ($ad\tau=dt$). Dots indicate differentiation with respect to $\tau$. The visibility function $g(\tau)$ can be written in terms of the opacity for Thomson scattering $\kappa$ as $$g(\tau)=\dot{\kappa} \exp(-\kappa)=-d/d\tau \exp(-\kappa)$$ with $$\kappa = \sigma_T \int_\tau^{\tau_0} a n_e(\tau) d\tau, \label{opacity}$$ where $\sigma_T$ is the Thomson scattering cross section and $n_e(\tau)$ is the number density of free electrons. We have also defined $\dot{\kappa}=\sigma_T a n_e $. Finally, the source term in the integral equation is given by $$S=\phi + \frac{\delta_\gamma}{4} + \hat{n} \cdot \mathbf{v}_b$$ where $\phi$ is the gravitational potential, $\delta_\gamma$ is the fractional perturbation in the photon energy density and $\mathbf{v}_b$ is the baryon velocity. The acoustic oscillations in the photon-baryon plasma satisfy (see e.g. [@Hu:1994jd]) $$\begin{aligned} \ddot{\delta}_\gamma & +& \frac{\dot{R}}{(1+R)}\dot{\delta}_\gamma + k^2 c_S^2\delta_\gamma = \nonumber\\ &=& 4\left[\ddot{\phi} + \frac{\dot{R}}{1+R}\dot{\phi} -\frac{1}{3} k^2 \phi \right] \; \label{photbarglg}\end{aligned}$$ with the sound speed $c_S^2=1/3(1+R)$, and the baryon-photon momentum density ratio $R = (p_b+\rho_b)/(p_\gamma + \rho_\gamma)\simeq 3\rho_b/4 \rho_\gamma$. The velocity satisfies, $$\dot{\delta}_\gamma+kv_\gamma+\dot{\phi}=0$$ Finally the gravitational potential satisfies the Poisson equation $$-k^2 \phi = 4 \pi \lambda^2 G \rho \delta^{total} , \label{po}$$ where $\rho \delta^{total}$ gives the combined perturbation due to all the fluids. We are now ready to study the dependence of $\Delta T$ on $\lambda$. For this purpose it is best to consider the expansion factor as a time variable rather than $\tau$. We note that, $$\frac{d}{d \tau}=\frac{da}{d \tau}\frac{d}{da} = a^2 \cdot H\cdot \frac{d}{da}=\lambda f(a) a^2 \frac{d}{da}. \label{tautoa}$$ As a result, when we change time variables, every time derivative introduces a factor of $\lambda$. By inspection of equations (\[photbarglg\]) and (\[po\]) it is clear that the dynamics of a mode with wavenumber $k$ in a universe with $\lambda \neq 1$ is equivalent to the dynamics of a mode with $k'=k/\lambda$ in a universe with $\lambda=1$. That is, $$S(k,a,\lambda)=S(k/\lambda ,a, \lambda=1).$$ We have explicitly included the $\lambda$ dependence of the source to make our argument clearer. To obtain the CMB power spectrum, we first need to expand equation (\[lineofsight\]) in Legendre polynomials. The amplitude of the $l$ expansion coefficient is $$\Delta T_l(k,\lambda) = \int^{1}_0 da \; \tilde S(k,a,\lambda) j_l(k D(a,\lambda)) \tilde g(a,\lambda). \label{lineofsight2}$$ We have introduced $\tilde g(a,\lambda) = -d/da \exp(-\kappa)$. The conformal distance $D$ is given by $$D(a,\lambda) =\int_{a}^1 \frac{da}{H(a) a^2}=\lambda^{-1} D(a,\lambda=1) .$$ Thus if the visibility function where to be independent of $\lambda$ we would have, $$\Delta T_l(k,\lambda) = \Delta T_l(k/\lambda,\lambda=1). \label{lineofsight3}$$ The power spectrum is calculated from $\Delta T_l(k,\lambda)$ using, $$\begin{aligned} C_l(\lambda) &=& \int \frac{dk}{k} P(k) |\Delta_{Tl}(k,\lambda)|^2 \nonumber \\ &=& \int \frac{dk'}{k'} P(k' \lambda) |\Delta_{Tl}(k',\lambda=1)|^2,\end{aligned}$$ where $P(k)$ is the power spectrum of primordial fluctuations which is usually taken to be a power law $P(k)\propto k^{n-1}$. Thus we see that provided we adjust the amplitude of the primordial power spectrum appropriately $C_l(\lambda)=C_l(\lambda=1)$. Our result is qualitatively very easy to understand: gravity introduces no preferred scale, so the dynamics of the perturbations remains the same when scales are measured in units of the expansion time. As a result, the angular power spectrum does not change as we change $\lambda$. Of course this conclusion only holds true if the visibility function is not affected by $\lambda$. However the physics of recombination does introduce a preferred timescale, so the power spectra of the anisotropies will actually change. In other words, in our simple minded prescription the only source of change is the difference in the way recombination proceeds as we change the expansion rate of the universe at recombination. This is the sense in which our model resembles the studies done in the context of Big Bang Nucleosynthesis. Let us now turn to study how the visibility function changes with $\lambda$. It depends on the ionization fraction $x_e = n_e/n_H$, where $n_e$ again is the free electron density and $n_H$ is the number density of hydrogen atoms. The evolution of the ionization fraction is modified when $G$ is changed. It evolves according to (e.g. [@Ma:1994ub]): $$\frac{dx_e}{d \tau} = a C_r \left[\beta(T_b)(1-x_e)-n_H\alpha^{(2)}(T_b)x_e^2\right] \label{recombdgl}$$ where $a(t)$ is just the scale factor, $\beta(T_b)$ is the collisional ionization rate from the ground state and $\alpha^{(2)}(T_b)$ is the recombination rate to excited states. The baryon temperature is $T_b$ and the Peebles correction coefficient (which also depends on the expansion rate) is denoted $C_r$ [@Peebles:1968]. The transformation from $\tau$ to $a(\tau)$ as a time variable using equation (\[tautoa\]), makes clear, that contrary to what happens to the perturbation equations, $x_e(a)$ depends on $\lambda$. We plotted $x_e$ for different values of $\lambda$ in Figure \[freeeminus\]. The behavior is easy to understand; the faster the universe is expanding at a given redshift (i.e. the larger the $\lambda$), the more difficult it is for hydrogen to recombine and hence the larger is $x_e$. The change in $x_e$ leads to a change in the visibility function which we show in Figure \[visibility\]. As $\lambda$ is increased, the visibility function becomes broader. This broadening leads to a larger damping of the anisotropies on small (angular) scales, as shown in Figure \[cmbGplot\]. We note however that even for a factor of four change in $\lambda$ the changes in the visibility function are rather small. What happens is that if we increase $\lambda$, $x_e$ at a given redshift after the start of recombination increases. However, when calculating optical depths this change almost exactly cancels with the decrease in the time intervals between different redshifts due to the increased expansion rate. As a result changes in both the location and shape of the visibility function are small even for large changes in $\lambda$. Figure \[cmbGplot\] shows that the effect of $\lambda$ is to change the relative amplitudes of the acoustic peaks on different scales. This effect can be compensated by changing the relative amplitude of modes of different scales in the primordial power spectrum. We are now going to study what happens to CMB polarization and to show that it can lift the degeneracy between $\lambda$ and the shape of the primordial power spectrum. To understand the effect of $\lambda$ on the polarization we will employ the simple analytic expression for the amplitude of the $Q$ Stokes parameter produced by a single Fourier mode $k$ [@Zaldarriaga:1995gi]: $$Q \propto c_s k \delta\tau_D \sin (k c_S \tau_D) e^{-k^2/k_D^2}$$ where $\tau_D$ is the conformal time corresponding to the peak of the visibility function, $\delta \tau_D$ is its width and $k_D$ describes the damping of the small scale modes. As we discussed above, the damping increases with the width of the visibility function, so the exponential factor will lead to the same effect we described for the temperature. The difference in the case of the polarization is the extra $\delta \tau_D$ in the amplitude of the polarization. This extra factor comes from the fact that if the visibility function is wider the photons will travel on average longer between their last two scatterings which will enhance the quadrupole anisotropy and will thus lead to a higher level of polarization [@Zaldarriaga:1995gi]. For that reason, there exists a characteristic wavemode value $k^*$, for scales larger than which the polarization power spectrum will increase with $\lambda$, while it will decrease for scales smaller than $k^*$. The polarization power spectrum is plotted in Figure \[ppower\]. We see that for $l$s larger than around $800$, the polarization behaves just as the temperature does, decreasing for increasing $\lambda$. On larger scales the effect is opposite, the amplitude of polarization relative to temperature roughly increases by 10% when $\lambda$ increases by 20%. This response of the anisotropies on $\lambda$ is what will help to break the degeneracy between $\lambda$ and the primordial power spectrum when information from polarization is included. For completeness we show the temperature polarization cross correlation power spectrum in Figure \[cross\]. The cross correlation will be easier to detect in experiments such as the MAP satellite where the accuracy of the polarization measurement is limited by detector noise. The cross correlation power spectrum behaves similarly to the polarization power spectrum; when $\lambda$ is increased the power on small scales is suppressed while on large scales it is amplified. Constraints on $\lambda$\[analysis\] ==================================== In our likelihood analysis of currently available and simulated future data we let vary $\Omega_\Lambda$, $\omega_\text{dark matter}=\Omega_\text{dark matter} h^2$, $\omega_\text{baryon}=\Omega_\text{baryon} h^2$, the optical depth due to reionization $\tau_{ri}$, $\lambda$ and the amplitude and shape of the primordial power spectrum. We explicitly assume that the universe is flat. As we mention above, we expect there to be a degeneracy between the shape of the primordial power spectrum and the parameter $\lambda$. To study this effect in the case of future satellite missions which will measure polarization and could break this degeneracy, we introduce additional freedom in the shape of the spectrum. Rather than just assuming that $P(k)$ is of power law form $$P(k)=k^{n-1} \Leftrightarrow \frac{\ln P(k)}{\ln (k)}=n-1,$$ we also allowed spectra with curvature by adding another term in the expansion of $\ln P$ as a function of $\ln k$. $$\ln P(k)/P(k_0) = (n-1) \ln(k/k_0) + \alpha [\ln (k/k_0)]^2+ ...$$ where $k_0$ is the pivot point. With this prescription the effective slope of the power spectrum changes slightly with scale, $$\frac{\partial \ln P(k)}{\partial \ln k}= n-1 + \alpha \ln (k/k_0) .$$ We employed two kinds of likelihood analysis. For an evaluation of what currently available data can tell us about $\lambda$ we used an importance sampling Markov-chain method to generate a large number of cosmological models distributed according to the likelihood distribution $\mathcal{L}(\text{model}|\text{data})$. This is more efficient than an exploration of the entire parameter space, because the sampling is weighted and statistics can be established over the target distribution itself. Moreover the algorithm is very easy to implement. The weighted sampling is achieved through the Metropolis-Hastings algorithm. The application of the Markov method to the extraction of cosmological parameters from CMB information has been suggested in [@Christensen:2001gj]. To speed up the power spectrum computations for the Markov chain we made use of the k-splitting technique discussed in [@Tegmark:2000qy]. For each model we marginalized analytically over the amplitude of the scalar fluctuations. In the end we constructed histograms and computed expectation values and variances for each parameter directly from the Markov chain. In order to estimate what the satellite missions MAP and Planck will be able to tell us about the relation between the energy density and the expansion rate of the universe during recombination, we investigated the shape of the likelihood function $\mathcal{L}(C_l|\theta_i)$ (for a power spectrum $C_l$ given a model consisting of the cosmological parameters $\theta_i$) in the vicinity of its maximum directly by a Fisher matrix evaluation. This method has been widely used to make predictions for the errors that are to be expected in the extraction of cosmological parameters from planned CMB-experiments. The Fisher matrix is given by the expectation value of the second derivative of the logarithm of the likelihood function $\mathcal{L}(C_l|\theta_i)$. Assuming Gaussianity of the likelihood it is of the form $$F_{ij}=\sum_l \sum_{A,B} \frac{\partial C_{Al}}{\partial \theta_i}\mathbb{C}^{-1}(\hat{C}_{Al},\hat{C}_{Bl})\frac{\partial C_{Bl}}{\partial \theta_j} \; ,$$ where $A$ and $B$ run over the three observables: temperature, E-type polarization, T-E cross correlation and $i,j$ run over the cosmological parameters. The covariance matrix between parameters is given by the inverse of the Fisher matrix. Overall we verified a good agreement between Fisher matrix and Markov chain results which confirms that the likelihood function $\mathcal{L}(C_l|\theta_i)$ resembles relatively well a Gaussian in the vicinity of its maximum value. Constraints from currently available temperature data {#tempanalysis} ----------------------------------------------------- In order to find the constraints which can be imposed on the parameter $\lambda$ using available temperature data, we employed a compilation of 30 experiments which functions as a complete account of pre-MAP CMB temperature anisotropy information [^3]. The data, which have been compressed to 29 bins, are plotted in Figure \[temp-data\]. In this context, we actually assumed a simple power law behaviour of the primordial scalar perturbations. The results we obtained from the Markov-chain analysis are shown in Table \[curtres\] and Figure \[curthist\]. As expected from the rather weak dependence of the anisotropies on $\lambda$ we find that current data cannot put severe constraints on the expansion rate during recombination even though other cosmological parameters are well determined. Param. mean $\sigma_\text{Markov}$ ----------- ------- ------------------------ $\lambda$ 1.749 0.471 $n_S$ 1.038 0.0553 : Mean value and standard deviation $\sigma$ for the currently available temperature data.\[curtres\] Having emphasized the importance of measuring the linearly polarized component of the CMB, we should also note that its recent first detection by DASI [@Kovac:2002fg], unfortunately has too large errorbars to deliver much information about $\lambda$. Adding the DASI data in fact only changes the standard error in the determination of $\lambda$ by 3%. Future satellite missions and the relevance of measuring polarization {#polanalysis} --------------------------------------------------------------------- An up to date estimate of the expected angular resolutions and sensitivities of the MAP and Planck satellites has been obtained from the experimental groups websites. For both satellites we combined the three frequency channels with the highest angular resolution and took into account the number of polarized instruments. In the case of MAP the sensitivity estimate has been provided for 2 years of observation and the angular resolutions $\theta_{\text{fwhm}}$ are 13.2, 21.0 and 31.8 arcminutes for the three channels observing at 90, 60 and 40 $GHz$. This leads to a raw sensitivity of $$\begin{aligned} w_T^{-1} &=& (0.081 \mu K)^2 \\ w_P^{-1} &=& (0.114 \mu K)^2 \end{aligned}$$ If MAP observes for four years the raw sensitivities ($w^{-1}$) will be halved. For the Planck satellite mission (here the sensitivity estimates are given for a one year observation period), the three channels (217, 143 and 100 GHz) at $\theta_{\text{fwhm}}=$ 5.0,7.1 and 9.2 arcminutes give together $$\begin{aligned} w_T^{-1} &=& (0.0084 \mu K)^2 \\ w_P^{-1} &=& (0.0200 \mu K)^2 \, .\end{aligned}$$ Again the raw sensitivities for a two year observation will be half of these values. For both satellites, a sky coverage of $f_{Sky}=0.8$ was assumed. From these experimental characterisics the full estimator covariance matrices for each multipole $l$ can be constructed (e.g. [@Zaldarriaga:1997ch]). The diagonal terms of the covariance matrices for temperature, polarization and cross correlation are [ $$\begin{aligned} {\rm Cov}(\hat{C}_{T l}^2)&=&\frac{2}{(2l+1) f_{Sky}} (C_{T l}+w_{T}^{-1} B_l^{-2})^2 \\ {\rm Cov}(\hat{C}_{E l}^2)&=&\frac{2}{(2l+1) f_{Sky}} (C_{E l}+w_{E}^{-1} B_l^{-2})^2 \\ {\rm Cov}(\hat{C}_{C l}^2)&=&\frac{1}{(2l+1) f_{Sky}} [C_{Cl}^2+(C_{Tl}+w_{T}^{-1}B_l^{-2})\\ &\times& (C_{El}+w_{P}^{-1}B_l^{-2})] \label{Clerror}\end{aligned}$$ ]{}where the beam window function $\mathcal{B}_l$ is to be constructed from the relevant frequency channels with their individual sensitivities $w_c$ as $$\begin{aligned} \mathcal{B}^2_l &=& \sum_c B_{l,c}^2 \frac{w_c}{w} \\ B_{l,c}^2 &=& e^{-l (l+1) \theta_b^2,c}\end{aligned}$$ Here, the standard width of the beam $\theta_b$ is obtained from the full width half maximum resolution by $$\theta_b,c=\frac{\theta_{\text{fwhm,c}}}{\sqrt{8 \ln 2}}$$ As a fiducial model we have adopted the parameter values $$\begin{array}{|c|c|c|c|c|c|c|c|} \hline \tau & \Omega_K & \Omega_\Lambda & \omega_{\text{dm}} & \omega_{\text{ba}} & n_S & \lambda \\ \hline 0.05 & 0 & 0.7 & 0.14 & 0.02 & 1 & 1 \\ \hline \end{array}$$ They imply a hubble constant of $h=0.73$. ### Expected constraints from MAP The analysis of expected constraints from the *MAP* satellite mission shows the constraints it can put on $\lambda$ are rather weak. Better sensitivity is needed to accurately determine the polarization and higher angular resolution to map the damping tail. On the other hand our results illustrate how the inclusion of a curvature term for the primordial spectrum significantly weakens the constraints that one obtains from the temperature data. A precision test of the Friedmann equation will have to wait until both polarization and the damping tail are measured accurately. In the near future experiments such as Boomerang, BICEP, Polatron and others are expected to significantly improve polarization measurements while CBI, ACBAR and others will map the damping tail. In a few more years, the Planck satellite will measure both temperature and polarization accurately enough to severely constrain any change in $\lambda$. We will present an analysis of Planck’s sensitivity in the next section. In Section \[lambda\] we discussed the different response to a change of $\lambda$ of the temperature and the polarization anisotropy. We found that on large angular scales polarization power is increased if we increase the expansion rate during recombination, while the temperature anisotropy is almost not affected on these scales. When only temperature is being measured, changes in the expansion rate of the universe during recombination are strongly degenerate with the slope of the spectrum of the initial scalar perturbations, which can be clearly seen in Figures \[maphist\] and \[mapcontours\]. Because polarization responds differently on different scales to a change of $\lambda$, it breaks this degeneracy. We show this in Figure \[mapcontours\]. The result of the likelihood analysis which includes polarization information is included in Table \[maplitem\]. We noticed that a major part of the information on $\lambda$ from MAP’s polarization measurements will come from the cross correlation between temperature and polarization. When adding just the cross correlation information we found that we gain almost 90 % of the information that is gained in the case in which all three estimators are included. T $(F_{ii}^{-1})^{1/2}$ $(F_{ii}^{-1})^{1/2}$, $\alpha$ T+P $(F_{ii}^{-1})^{1/2}$ $(F_{ii}^{-1})^{1/2}$, $\alpha$ ----------- ----------------------- --------------------------------- ----------- ----------------------- --------------------------------- $\lambda$ 0.1928 (0.1279) 0.2616 (0.1992) $\lambda$ 0.1373 (0.0903) 0.1658 (0.1165) $n$ 0.0294 (0.0248) 0.0646 (0.0595) $n$ 0.0170 (0.0144) 0.0489 (0.0418) $\alpha$ $\times$ 0.0237 (0.0222) $\alpha$ $\times$ 0.0151 (0.0137) ### Expected constraints from Planck T $(F_{ii}^{-1})^{1/2}$ $(F_{ii}^{-1})^{1/2}$, $\alpha$ T+P $(F_{ii}^{-1})^{1/2}$ $(F_{ii} ^{-1})^{1/2}$, $\alpha$ ----------- ----------------------- --------------------------------- ----------- ----------------------- ---------------------------------- $\lambda$ 0.0170 (0.0152) 0.0325 (0.0278) $\lambda$ 0.0115 (0.0093) 0.0174 (0.0141) $n$ 0.0118 (0.0106) 0.0182 (0.0160) $n$ 0.0072 (0.0060) 0.0098 (0.0080) $\alpha$ $\times$ 0.0072 (0.0620) $\alpha$ $\times$ 0.0039 (0.0033) The Planck satellite explores the very small structures in the primeaval plasma in a multipole range up to nearly $l=3000$. Even if only Planck’s temperature data are used, the standard error in $\lambda$ will after a one year observation be as small as $3.2\%$, even if one marginalizes over the parameters that describe the shape of the primordial power spectrum (Table \[plancklitem\]). This corresponds to a constraint on the gravitational constant of $\frac{\delta G}{G}=0.064$. Finally, the improvement gained from Planck’s polarization data ($\frac{\delta \lambda}{\lambda} \simeq 0.017 \Leftrightarrow \frac{\delta G}{G} \simeq 0.034$) is shown on the right hand side of that same Table. If one assumes that there is no curvature in the primordial power spectrum the constraint on G would be as small as $\frac{\delta G}{G} = 1.8 \%$ after an observation of two years. Figure \[planckhist\] and \[planckcont\] show again, analogously to the case of MAP, how polarization helps break the degeneracy between $\lambda$ and the parameters that describe the shape of primodial perturbations. We have extended our investigation beyond Planck to the case of an experiment which has essentially no noise and where the errors in the CMB power spectra are on all scales dominated by the cosmic variance term. For example experiments currently being considered for measuring the B-modes of CMB polarization would be cosmic variance limited for E polarization over a wide range of $ls$ (e.g. [@bmodes]). Such an optimal experiment, exploring structures into a multipole range of $l=4000$, represents the limit of how much information on $\lambda$ one could extract from the CMB in principle. We found an expected error for $\lambda$ of order $0.3\%$ which translates into a constraint of the value of the gravitational constant during recombination of $\frac{\delta G}{G} \simeq 0.6 \%$. Discussion\[discussion\] ======================== We have made the expansion rate of the universe a free parameter in a likelihood analysis within an eight-dimensional cosmological parameter space. For simplicity we assumed that the gravitational constant is changed by a factor $\lambda$. We showed that an increase of $\lambda$ leads to a wider visibility function which in turn increases the damping of anisotropies on small scales and increases the level of large scale polarization. We calculated the constraints that current CMB data can impose on the expansion rate. The constraints that can be imposed on the parameter $\lambda$ using the information from the damping tail are severely weakened by our lack of knowledge about the shape of the primordial power spectrum. We showed that measuring polarization helps to break this degeneracy. Current data can only constrain $\lambda$ to about 47% at 1 $\sigma$. We showed that MAP could obtain 9% error bars for $\lambda$ while for Planck errorbars go down to 0.9%. We also explored the ultimate limit that could be achieved by a cosmic variance limited experiment measuring anisotropies up to $l=4000$ and found errors of under a percent in that case. Thus next generation experiments should be able to deliver very accurate constraints on the expansion rate of the universe during recombination. We acknowledge that if the variation of the gravitational constant during recombination is taken seriously a model needs to be built where $G$ changes after recombination and converges towards the stable value observed in laboratory experiments today and where its current rate of change is less than the experimental bound $\frac{\dot{G}}{G} \simeq 10^{-12} \text{yr}^{-1}$[@Uzan:2002vq; @Guenther:1998]. If we introduce a scalar field to control the value of $G$ we would also have to require that this field does not lead to an unacceptably large fifth force and does not violate solar system constraints such as shifting the orbit of the moon through the Nordvedt effect [@Will; @Nordtvedt]. The shift of $G$ after recombination will induce a change in the angular diameter distance to recombination, shifting the CMB power spectrum in $l$. We have shown that future experiments will be able to constrain the change of $G$ to a few percent due to its effect at recombination. As a result the induced shift in the peak positions would be small and could be interpreted as slightly different values of $\Omega_m$ and/or $\Omega_\Lambda$, the two parameters that control the distance to the last scattering surface in conventional models. In the same way, any induced integrated Sachs-Wolfe (ISW) effect would be difficult to observe because of the cosmic variance limitation. 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D, [**169**]{}, 1014 (1968) [^1]: Email address: zahn@mpa-garching.mpg.de [^2]: Email address: mz31@nyu.edu [^3]: The results of the Archeops [@Benoit:2002mk] and ACBAR experiments [@Kuo:2002ua] as well as the new Boomerang data [@Ruhl:2002cz] were added to the compilation described in [@Tegmark:2002cy].

--- author: - | [Hauke Petersen]{}, [Martine Lenders]{}, [Matthias Wählisch]{}\ Freie Universität Berlin, Germany - | [Oliver Hahm]{}, [Emmanuel Baccelli]{}\ INRIA, France bibliography: - 'references.bib' - 'rfcs.bib' title: '**Old Wine in New Skins? Revisiting the Software Architecture for IP Network Stacks on Constrained IoT Devices**' ---

--- abstract: 'In distributed storage systems that employ erasure coding, the issue of minimizing the total [*repair bandwidth*]{} required to exactly regenerate a storage node after a failure arises. This repair bandwidth depends on the structure of the storage code and the repair strategies used to restore the lost data. Minimizing it requires that undesired data during a repair align in the smallest possible spaces, using the concept of interference alignment (IA). Here, a points-on-a-lattice representation of the symbol extension IA of Cadambe [*et al.*]{} provides cues to perfect IA instances which we combine with fundamental properties of Hadamard matrices to construct a new storage code with favorable repair properties. Specifically, we build an explicit $(k+2,k)$ storage code over $\mathbb{GF}(3)$, whose single systematic node failures can be repaired with bandwidth that matches exactly the theoretical minimum. Moreover, the repair of single parity node failures generates at most the same repair bandwidth as any systematic node failure. Our code can tolerate any single node failure and any pair of failures that involves at most one systematic failure.' author: - | Dimitris S. Papailiopoulos and Alexandros G. Dimakis\ Department of Electrical Engineering\ University of Southern California\ Los Angeles, CA 90089\ Email:`{papailio, dimakis}@usc.edu` title: Distributed Storage Codes through Hadamard Designs --- Introduction ============ The demand for large scale data storage has increased significantly in recent years with applications demanding seamless storage, access, and security for massive amounts of data. When the deployed nodes of a storage network are individually unreliable, as is the case in modern data centers, or peer-to-peer networks, redundancy through erasure coding can be introduced to offer reliability against node failures. However, increased reliability does not come for free: the encoded representation needs to be maintained posterior to node erasures. To maintain the same redundancy when a storage node leaves the system, a new node has to join the array, access some existing nodes, and regenerate the contents of the departed node. This problem is known as the [*Code Repair Problem*]{} [@DimakisGWWR:08], [@storagewiki]. The interest in the code repair problem, and specifically in designing repair optimal $(n,k)$ erasure codes, stems from the fact that there exists a fundamental minimum repair bandwidth needed to regenerate a lost node that is substantially less than the size of the encoded data object. MDS erasure storage codes have generated particular interest since they offer maximum reliability for a given storage capacity; such an example is the EvenOdd construction [@evenodd]. However, most practical solutions for storage use existing off-the-shelf erasure codes that are repair inefficient: a single node repair generates network traffic equal to the size of the [*entire*]{} stored information. Designing repair optimal MDS codes, i.e., ones achieving the minimum repair bandwidth bound that was derived in [@DimakisGWWR:08], seems to be challenging especially for high rates $\frac{k}{n}\ge\frac{1}{2}$. Recent works by Cadambe [*et al.*]{} [@CadambeWinc] and Suh [*et al.*]{} [@SuhCodes] used the symbol extension IA technique of Cadambe [*et al.*]{} [@CadambeJ:08] to establish the existence, for all $n$, $k$, of asymptotically optimal MDS storage codes, that come arbitrarily close to the theoretic minimum repair bandwidth. However, these asymptotic schemes are impractical due to the arbitrarily large file size and field size that they require. Explicit and practical designs for optimal MDS storage codes are constructed roughly for rates $\frac{k}{n}\le\frac{1}{2}$ [@WuD:09]-[@Wu:09c], [@RashmiProduct], and most of them are based upon the concept of interference alignment. Interestingly, as of now no explicit MDS storage code constructions exist with optimal repair properties for the high data rate regime.[^1] [**Our Contribution**]{}: In this work we introduce a new high-rate, explicit, $(k+2,k)$ storage code over $\mathbb{GF}(3)$. Our storage code exploits fundamental properties of Hadamard designs and perfect IA instances pronounced by the use of a lattice representation for the symbol extension IA of Cadambe [*et al.*]{} [@CadambeJ:08]. This representation gives hints for coding structures that allow [*exact*]{} instead of asymptotic alignment. Our code exploits these structures and achieves perfect IA without requiring the file size or field size to scale to infinity. Any single systematic node failure can be repaired with bandwidth matching the theoretic minimum and any single parity node failure generates (at most) the same repair bandwidth as any systematic node repair. Our code has two parities but cannot tolerate any two failures: the form presented here can tolerate any single failure and any pair of failures that involves at most one systematic node failure[^2]. Here, in contrast to MDS codes, slightly more than $k$, that is, $k\left(1+\frac{1}{2k}\right)$, encoded pieces are required to reconstruct the file object. Distributed Storage Codes with $2$ Parity Nodes =============================================== In this section, we consider the code repair problem for storage codes with $2$ parity nodes. Let a file of size $M=kN$ denoted by the vector ${\bf f}\in\mathbb{F}^{kN}$ be partitioned in $k$ parts ${\bf f}=\left[{\bf f}^T_1\ldots{\bf f}^T_k\right]^T$, each of size $N$.[^3] We wish to store this file with rate $\frac{k}{k+2}$ across $k$ systematic and $2$ parity storage units each having storage capacity $\frac{M}{k}=N$. To achieve this level of redundancy, the file is encoded using a $(k+2,k)$ distributed storage code. The structure of the storage array is given in Fig. 1, where ${\bf A}_i$ and ${\bf B}_i$ are $N\times N$ matrices of coding coefficients used by the parity nodes $a$ and $b$, respectively, to “mix” the contents of the $i$th file piece ${\bf f}_{i}$. Observe that the code is in systematic form: $k$ nodes store the $k$ parts of the file and each of the $2$ parity nodes stores a linear combination of the $k$ file pieces. $$\begin{aligned} &\begin{array}{|c|c|} \hline \text{systematic node} & \text{systematic data}\\ \hline 1&{\bf f}_1\\ \hline \vdots&\vdots\\ \hline k&{\bf f}_k\\ \hline \text{parity node} & \text{parity data}\\ \hline a&{\bf A}_1^T{\bf f}_1+\ldots+{\bf A}_k^T{\bf f}_k\\ \hline b&{\bf B}_1^T{\bf f}_1+\ldots+{\bf B}_k^T{\bf f}_k\\ \hline \end{array}\nonumber\end{aligned}$$ To maintain the same level of redundancy when a node fails or leaves the system, the code repair process has to take place to exactly restore the lost data in a [*newcomer*]{} storage component. Let for example a systematic node $i\in\{1,\ldots,k\}$ fail. Then, a newcomer joins the storage network, connects to the remaining $k+1$ nodes, and has to download sufficient data to reconstruct ${\bf f}_i$. Observe that the missing piece ${\bf f}_i$ exists as a term of a linear combination [*only*]{} at each parity node, as seen in Fig. 1. To regenerate it, the newcomer has to download from the parity nodes at least the size of what was lost, i.e., $N$ linearly independent data elements. The downloaded contents from the parity nodes can be represented as a stack of $N$ equations [$$\begin{aligned} \hspace{-0.1cm}\left[ \begin{array}{c} {\bf p}_i^{(a)}\\ {\bf p}_i^{(b)} \end{array} \right]\hspace{-0.1cm}&{\stackrel{\triangle}{=}}\hspace{-0.1cm}\underbrace{\left[ \begin{array}{@{}c@{}} \left({\bf A}_{i}{\bf V}^{(a)}_i\right)^T\\ \left({\bf B}_i{\bf V}^{(b)}_i\right)^T \end{array} \right]\hspace{-0.1cm}{\bf f}_i}_{\text{useful data}}\hspace{-0.1cm}+\hspace{-0.35cm}\sum_{j=1,j\ne i}^k\hspace{-0.1cm} \underbrace{\left[ \begin{array}{@{}c@{}} \left({\bf A}_{j}{\bf V}^{(a)}_i\right)^T\\ \left({\bf B}_j{\bf V}^{(b)}_i\right)^T\label{Yrep} \end{array} \right]\hspace{-0.1cm}{\bf f}_j}_{\text{interference by ${\bf f}_j$}}\end{aligned}$$ ]{}where ${\bf p}_i^{(a)},{\bf p}_i^{(b)}\in\mathbb{F}^{\frac{N}{2}}$ are the equations downloaded from parity nodes $a$ and $b$ respectively. Here, ${\bf V}_i^{(a)},{\bf V}_i^{(b)}\in\mathbb{F}^{N\times \frac{N}{2}}$ denote the [*repair matrices*]{} used to mix the parity contents.[^4] Retrieving ${\bf f}_i$ from (\[Yrep\]) is equivalent to solving an underdetermined set of $N$ equations in the $kN$ unknowns of ${\bf f}$, with respect to only the $N$ desired unknowns of ${\bf f}_i$. However, this is not possible due to the additive [*interference*]{} components that corrupt the desired information in the received equations. These terms are generated by the undesired unknowns ${\bf f}_j$, $j\ne i$, as noted in (\[Yrep\]). Additional data need to be downloaded from the systematic nodes, which will “replicate” the interference terms and will be subtracted from the downloaded equations. To erase a single interference term, a download of a basis of equations that generates the corresponding interference term, say $\left[ \begin{smallmatrix} \left({\bf A}_{s}{\bf V}^{(a)}_i\right)^T\\ \left({\bf B}_s{\bf V}^{(b)}_i\right)^T\label{Yrep} \end{smallmatrix} \right]\hspace{-0.1cm}{\bf f}_j$, suffices. Eventually, when all undesired terms are subtracted, a full rank system of $N$ equations in $N$ unknowns $\left[ \begin{smallmatrix} \left({\bf A}_{i}{\bf V}^{(a)}_i\right)^T\\ \left({\bf B}_i{\bf V}^{(b)}_i\right)^T \end{smallmatrix} \right]{\bf f}_i$ has to be formed. Thus, it can be proven that the [*repair bandwidth*]{} to exactly regenerate systematic node $i$ is given by [$$\begin{aligned} \gamma_i=N+\sum_{j=1,j\ne i}^{k}{\text{rank}}\left(\left[{\bf A}_j{\bf V}^{(a)}_i \;{\bf B}_j{\bf V}^{(b)}_i\right]\right),\nonumber\end{aligned}$$ ]{}where the sum rank term is the aggregate of interference dimensions. Interference alignment plays a key role since the lower the interference dimensions are, the less repair data need to be downloaded. We would like to note that the theoretical minimum repair bandwidth of any node for optimal $(k+2,k)$ MDS codes is exactly $(k+1)\frac{N}{2}$, i.e. half of the remaining contents; this corresponds to each interference spaces having rank $\frac{N}{2}$. This is also true for the systematic parts of non-MDS codes, as long as they have the same problem parameters that were discussed in the beginning of this section, and all the coding matrices have full rank $N$. An abstract example of a code repair instance for a $(4,2)$ storage code is given in Fig. 2, where interference terms are marked in red. {width="1\columnwidth"} To minimize the repair bandwidth $\gamma_i$, we need to carefully design both the storage code and the repair matrices. In the following, we provide a $2$-parity code that achieves optimal systematic and near optimal parity repair. $$\begin{aligned} {\bf X}_1 = {\text{diag}}\left(\left[ \begin{smallmatrix} 1\\ 1\\ 1\\ 1\\ -1\\ -1\\ -1\\ -1 \end{smallmatrix} \right] \right), \;\; {\bf X}_2 = {\text{diag}}\left( \left[ \begin{smallmatrix} 1\\ 1\\ -1\\ -1\\ 1\\ 1\\ -1\\ -1 \end{smallmatrix} \right] \right), \;\; {\bf X}_3 = {\text{diag}}\left( \left[ \begin{smallmatrix} 1\\ -1\\ 1\\ -1\\ 1\\ -1\\ 1\\ -1 \end{smallmatrix} \right] \right)\nonumber $$ A New Storage Code ================== We introduce a $(k+2,k)$ storage storage code over $\mathbb{GF}(3)$, for file sizes $M = k2^k$, with coding matrices $$\begin{aligned} {\bf A}_i &= {\bf I}_{N},\;\; {\bf B}_i = {\bf X}_{i},\label{code}\end{aligned}$$ where $N=2^k$, ${\bf X}_i = {\bf I}_{2^{i-1}}\otimes \text{blkdiag}\left({\bf I}_{\frac{N}{2^{i}}},-{\bf I}_{\frac{N}{2^{i}}}\right)$, and $i\in\{1,\ldots,k\}$. In Fig. 3, we give the coding matrices of the $(5,3)$ version of the code. The code in (\[code\]) has optimally repairable systematic nodes and its parity nodes can be repaired by generating as much repair bandwidth as a systematic repair does. It can tolerate any single node failure, and any pair of failures that contains at most one systematic failure. Moreover, to reconstruct the file at most $k+\frac{1}{2}$ coded blocks are required. In the following, we present the tools that we use in our derivations. Then, in Sections V and VI we prove Theorem 1. Dots-on-a-Lattice and Hadamard Designs ====================================== Optimality during a systematic repair, requires interference spaces collapsing down to the minimum of $\frac{N}{2}$, out of the total $N$, dimensions. At the same time, useful data equations have to span $N$ dimensions. For the constructions presented here, we consider that the same repair matrix is used by both parities, i.e., ${\bf V}^{(1)}_i={\bf V}^{(2)}_i={\bf V}_i$. Hence, for the repair of systematic node $i\in\{1,\ldots,k\}$ we optimally require $${\text{rank}}\left(\left[{\bf V}_i\; {\bf X}_{j}{\bf V}_i\right]\right)=\frac{N}{2},$$ for all $j\in\{1,\ldots,k\}\backslash i$, and at the same time $${\text{rank}}\left(\left[{\bf V}_i\;\;{\bf X}_i{\bf V}_i\right]\right)=N.$$ The key ingredient of our approach that eventually provides the above is Hadamard matrices. To motivate our construction, we start by briefly discussing the repair properties of the asymptotic coding schemes of [@CadambeWinc], [@SuhCodes]. Consider a $2$-parity MDS storage code that requires file sizes $M = k2\Delta^{k-1}$, i.e., $N =2\Delta^{k-1}$. Its $N\times N$ diagonal coding matrices $\{{\bf X}_s\}_{s=1}^{k}$ have i.i.d. elements drawn uniformly at random from some arbitrarily large finite field $\mathbb{F}$. During the repair of a systematic node $i\in\{1,\ldots,k\}$, the repair matrix ${\bf V}_i$ that is used by both parity nodes to mix their contents, has as columns the $\frac{N}{2}=\Delta^{k-1}$ elements of the set [$$\mathcal{V}_i=\left\{\prod_{s=1,s\ne i}^k{\bf X}_s^{x_s}{\bf w}: x_s\in\{0,\ldots,\Delta-1\}\right\}.$$ ]{}Then, we define a map $\mathcal{L}$ from vectors in the set $\left\{\prod_{s=1}^k{\bf X}_s^{x_s}{\bf w}:x_s\in\mathbb{Z}\right\}$ to points on the integer lattice $\mathbb{Z}^{k}$: $\prod_{s=1}^{k}{\bf X}_s^{x_s}{\bf w} \overset{\mathcal{L}}{\rightarrow} \sum_{s=1}^{k}x_s{\bf e}_s$, where ${\bf e}_s$ is the $s$-th column of ${\bf I}_{k+1}$. Now, consider the induced lattice representation of ${\bf V}_i$ [$$\mathcal{L}({\bf V}_i){\stackrel{\triangle}{=}}\left\{\sum_{s=1,s\ne i}^kx_s{\bf e}_s;\; x_s\in\{0,\ldots,\Delta-1\}\right\}.$$ ]{}Observe that the $i$-th dimension of the lattice where $\mathcal{L}({\bf V}_i)$ lies on, indicates all possible exponents $x_i$ of ${\bf X}_i$. {width="0.7\columnwidth"} Then, the products ${\bf X}_j{\bf V}_i$, $j\ne i$, and ${\bf X}_i{\bf V}_i$ map to [$$\begin{aligned} \mathcal{L}({\bf X}_j{\bf V}_i)&=\Biggl\{\hspace{-0.1cm}(x_j+1){\bf e}_j\hspace{-0.1cm}+\hspace{-0.3cm}\sum_{s=1,s\ne j}^k\hspace{-0.3cm}x_s{\bf e}_s;\; x_s\in\{0,\ldots,\Delta-1\}\Biggr\}\nonumber\\ \text{and }\mathcal{L}({\bf X}_i{\bf V}_i)&=\Biggl\{e_i+\sum_{i=1,s\ne i}^kx_i{\bf e}_i;\; x_s\in\{0,\ldots,\Delta-1\}\Biggr\},\nonumber\end{aligned}$$ ]{}respectively. In Fig. 2, we give an illustrative example for $k=3$, and $\Delta=2$. Observe how matrix multiplication of ${\bf X}_i$ and elements of $\mathcal{V}_i$ manifests itself through the dots-on-a-lattice representation: the product of ${\bf X}_i$ with the elements of $\mathcal{V}_i$ shifts the corresponding arrangement of dots along the $x_i$-axis, i.e., the $x_i$-coordinate of the initial points gets increased by one. Asymptotically optimal repair of node $i$ is possible due to the fact that interference spaces asymptotically align [$$\begin{aligned} \frac{{\text{rank}}\left(\left[{\bf V}_i\;\;{\bf X}_j{\bf V}_i\right]\right)}{\frac{N}{2}}&=\frac{\left|\mathcal{L}({\bf V}_i)\cup\mathcal{L}({\bf X}_j{\bf V}_i)\right|}{{\Delta^{k-1}}}\nonumber\\ & = \frac{\left|\mathcal{L}({\bf V}_i)\right|+o(\Delta^{k-1})}{{\Delta^{k-1}}} \overset{\Delta\rightarrow\infty}{\longrightarrow} 1,\end{aligned}$$ ]{}and useful spaces span $N$ dimensions, that is, ${\text{rank}}\left(\left[{\bf V}_i\;\;{\bf X}_i{\bf V}_i\right]\right) = \left|\mathcal{L}({\bf V}_i)\cup\mathcal{L}({\bf X}_i{\bf V}_i)\right| = 2\Delta^{k-1}$, with arbitrarily high probability for sufficiently large field sizes. The question that we answer here is the following: How can we design the coding and the repair matrices such that [*i)*]{} [*exact*]{} interference alignment is possible and [*ii)*]{} the full rank property is satisfied, for fixed in $k$ file size and field size? We first address the first part. We want to design the code such that the space of the repair matrix is invariant to any transformation by matrices generating its columns, i.e., $\mathcal{L}({\bf X}_j{\bf V}_i)=\mathcal{L}({\bf V}_i)$. This is possible when [$$\begin{aligned} \mathcal{L}({\bf X}_j{\bf V}_i)&=\Biggl\{\hspace{-0.1cm}(x_j+1){\bf e}_j\hspace{-0.1cm}+\hspace{-0.3cm}\sum_{s=1,s\ne j}^k\hspace{-0.3cm}x_s{\bf e}_s;\; x_s\in\{0,\ldots,\Delta-1\}\Biggr\}\nonumber\\ &=\Biggl\{\hspace{-0.1cm}x_j{\bf e}_j\hspace{-0.1cm}+\hspace{-0.3cm}\sum_{s=1,s\ne j}^k\hspace{-0.3cm}x_s{\bf e}_s;\; x_s\in\{0,\ldots,\Delta-1\}\Biggr\}=\mathcal{L}({\bf V}_i),\nonumber\end{aligned}$$ ]{}that is, when the matrix powers “wrap around” upon reaching their modulus $\Delta$. This wrap-around property is obtained when the diagonal coding matrices have elements that are roots of unity. For diagonal matrices, ${\bf X}_1,\ldots,{\bf X}_k$, whose elements are $\Delta$-th roots of unity, i.e., ${\bf X}_s^{\Delta} = {\bf X}_s^0$, for all $s\in\{1,\ldots,k\}$, we have that $\mathcal{L}({\bf X}_j{\bf V}_i)=\mathcal{L}({\bf V}_i)$, for all $i\in\{1,\ldots,k\}\backslash j$. However, arbitrary diagonal matrices whose elements are roots of unity are not sufficient to ensure the full rank property of the useful data repair space $\left[{\bf V}_i\;\;{\bf X}_i{\bf V}_i\right]$. In the following we prove that the full rank property along with perfect IA is guaranteed when we set $N=2^k$, ${\bf X}_i = {\bf I}_{2^{i-1}}\otimes \text{blkdiag}\left({\bf I}_{\frac{N}{2^{i}}},-{\bf I}_{\frac{N}{2^{i}}}\right)$, and consider the set $$\mathcal{H}_{N} = \left\{\prod_{i = 1}^{k}{\bf X}_{i}^{x_i}{\bf w}: x_{i}\in \{0,1\}\right\}. \label{Hprod}$$ Interestingly, there is a one-to-one correspondence between the elements of $\mathcal{H}_{N}$ and the columns of a Hadamard matrix. Let an $N\times N$ Hadamard matrix of the Sylvester’s construction [$${\bf H}_{N} {\stackrel{\triangle}{=}}\left[ \begin{array}{rr} {\bf H}_{\frac{N}{2}} &{\bf H}_{\frac{N}{2}}\\ {\bf H}_{\frac{N}{2}} & -{\bf H}_{\frac{N}{2}} \end{array} \right],$$ ]{}with ${\bf H}_{1} = 1$. Then, ${\bf H}_{N}$ is full-rank with mutually orthogonal columns, that are the $N$ elements of $\mathcal{H}_{N}$. Moreover, any two columns of ${\bf H}_{N}$ differ in $\frac{N}{2}$ positions. \[HadamardLem\] The proof is omitted due to lack of space. To illustrate the connection between $\mathcal{H}_{N}$ and ${\bf H}_{N}$ we “decompose” the Hadamard matrix of order $4$ [$$\begin{aligned} {\bf H}_4 &= \left[ \begin{smallmatrix} 1 & 1 & 1 & 1\\ 1 & -1 & 1 & -1\\ 1 & 1 & -1 & -1\\ 1 & -1 & -1 & 1 \end{smallmatrix} \right] = \left[{\bf w}\;\;{\bf X}_2{\bf w}\;\;{\bf X}_1{\bf w}\;\;{\bf X}_2{\bf X}_1{\bf w}\right],\end{aligned}$$ ]{}where ${\bf X}_1 = \text{diag}\left( \begin{smallmatrix} 1\\ 1\\ -1\\ -1 \end{smallmatrix} \right) \text{ and } {\bf X}_2 = {\text{diag}}\left( \begin{smallmatrix} 1 \\ -1\\ 1\\ -1 \end{smallmatrix} \right)$. Due to the commutativity of ${\bf X}_1$ and ${\bf X}_2$, the columns of ${\bf H}_{4}$ are also the elements of $\mathcal{H}_{4}=\left\{{\bf w},{\bf X}_1{\bf w},{\bf X}_2{\bf w},{\bf X}_1{\bf X}_2{\bf w}\right\}$. By using $\mathcal{H}_N$ as our “base” set, we are able to obtain perfect alignment condition due to the wrap around property of it elements; the full rank condition will be also satisfied due to the mutual orthogonality of these elements. {width="0.7\columnwidth"} Repairing Single Node Failures ============================== Systematic Repairs ------------------ Let systematic node $i\in\{1,\ldots,k\}$ fail. Then, we pick the columns of the repair matrix as a set of $\frac{N}{2}$ vectors whose lattice representation is invariant to all ${\bf X}_j$s but to one key matrix ${\bf X}_i$. We specifically construct the $N\times\frac{N}{2}$ repair matrix ${\bf V}_i$ whose columns have a one-to-one correspondence with the elements of the set $$\mathcal{V}_i = \left\{\prod_{s=1,s\ne i}^{k} {\bf X}^{x_s}_s{\bf w}:x_s\in\{0,1\}\right\}. \label{Vi}$$ First, observe that ${\bf V}_i$ is full column rank since it is a collection of $\frac{N}{2}$ distinct columns from $\mathcal{H}_{N}$. Then, we have the following lemma. For any $i,j\in\{1,2,\ldots,k\}$, we have that $$\begin{aligned} {\text{rank}}(\left[{\bf V}_i \;{\bf X}_j{\bf V}_i\right])&=\left|\mathcal{L}({\bf V}_i)\cup\mathcal{L}\left({\bf X}_j{\bf V}_i\right)\right|\nonumber\\ &=\left\{ \begin{array}{lc} N, & i=j\\ \frac{N}{2}, & i \ne j \end{array} \right..\end{aligned}$$ The above holds due to each element of $\mathcal{H}_N$ being associated with a unique power tuple. Then, the columns of $\left[{\bf V}_i \;{\bf X}_i{\bf V}_i\right]$ are exactly the elements of $\mathcal{H}_N$, since [$$\begin{split} \mathcal{L}\left({\bf V}_i\right)\cup\mathcal{L}\left({\bf X}_{i}{\bf V}_i\right) &=\left\{\sum_{s=1,s\ne i}^{k}x_i{\bf e}_i;\; x_i\in\{0,1\}\right\}\\ &\bigcup \left\{e_i+\sum_{s=1,s\ne i}^{k}x_i{\bf e}_i;\; x_i\in\{0,1\}\right\}\\ &=\mathcal{L}\left({\bf H}_{N}\right). \end{split}$$ ]{}Moreover, the set of columns in ${\bf V}_i$ are identical to the set of columns of ${\bf X}_j{\bf V}_i$, i.e., $\mathcal{L}({\bf V}_i)=\mathcal{L}({\bf X}_j{\bf V}_i)$, for $j\ne i$, due to Lemmata 1 and 2. Therefore, the interference spaces span $\frac{N}{2}$ dimensions, which is the theoretic minimum, and the desired data space during any systematic node repair is full-rank, since it has as columns all columns of ${\bf H}_N$. Hence, we conclude that a single systematic node of the code can be repaired with bandwidth $(k+1)\frac{N}{2}=\frac{k+1}{2k}M$. In Fig. 4, we depict a $(6,4)$ code of our construction, along with the illustration of the repair spaces. Parity repairs -------------- Here, we prove that a single parity node repair generates at most the repair bandwidth of a single systematic repair. Let parity node $a$ fail. Then, observe that if the newcomer uses the $N\times N$ repair matrix ${\bf V}_a^{(b)}={\bf X}_1$ to multiply the contents of parity node $b$, then it downloads ${\bf X}_1\left(\sum_{i=1}^k{\bf X}_1{\bf f}_i\right)={\bf f}_1+\sum_{i=2}^k{\bf X}_1{\bf X}_i{\bf f}_i$. Observe, that the component corresponding to systematic part ${\bf f}_1$ appears the same in the linear combination stored at the lost parity. By Lemma 2, each of the remaining blocks, ${\bf X}_1{\bf X}_i{\bf f}_i$ share exactly $\frac{N}{2}$ indices with equal elements to the same $\frac{N}{2}$ indices of ${\bf X}_i{\bf f}_i$ which was lost, for any $i\in\{2,\ldots,k\}$. This is due to the fact that the diagonal elements of matrices ${\bf X}_1{\bf X}_i$ and ${\bf X}_i$ are the elements of some two columns of ${\bf H}_{N}$. Therefore, the newcomer has to download from systematic node $j\in\{2,\ldots,k\}$, the $\frac{N}{2}$ entries that parity $a$’s component ${\bf X}_j{\bf f}_j$ differs from the term ${\bf X}_1{\bf X}_j{\bf f}_j$ of the downloaded linear combination. Hence, the first parity can be repaired with bandwidth at most $N+(k-1)\frac{N}{2}=(k+1)\frac{N}{2}$.[^5] The repair of parity node $b$ can be performed in the same manner. Erasure Resiliency ================== Our code can tolerate any single node failure and any two failures with at most one of them being a systematic one. A double systematic and parity node failure can be treated by first reconstructing the lost systematic node from the remaining parity, and then reconstructing the lost parity from all the systematic nodes. However, two simultaneous systematic node failures cannot be tolerated. Consider for example the corresponding matrix when we connect to nodes $\{1,\ldots,k-2\}$ and both parities: [$$\left[ \begin{array}{ccc|cc} {\bf I}_{N} & \ldots & {\bf 0}_{N\times N}&{\bf 0}_{N\times N}&{\bf 0}_{N \times N}\\ \vdots&&\vdots&\vdots\\ {\bf 0}_{N\times N} &\ldots & {\bf I}_{N}&{\bf 0}_{N \times N}&{\bf 0}_{N \times N }\\ \hline {\bf I}_{N} &\ldots&{\bf I}_{N}&{\bf I}_{N}&{\bf I}_{N}\\ {\bf X}_1 &\ldots&{\bf X}_{k-2}&{\bf X}_{k-1}&{\bf X}_k \end{array} \right]{\bf f}. \label{DC_2}$$ ]{}The rank of this $kN\times kN$ matrix is $(k-1)N+\frac{N}{2}$ due to the submatrix $\left[ \begin{smallmatrix} {\bf I}_{N}&{\bf I}_{N}\\ {\bf X}_{k-1}&{\bf X}_k \end{smallmatrix} \right]$ having rank $\frac{3N}{2}$. For these cases, an extra download of $\frac{N}{2}$ equations is required to decode the file, i.e., an aggregate download of $kN+\frac{N}{2}$ equations, or $k+\frac{1}{2}$ encoded pieces. 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Ho, “Searching for minimum storage regenerating codes," In [*Allerton Conf. on Control, Comp., and Comm.*]{}, Urbana-Champaign, IL, September 2009. K.V. Rashmi, N. B. Shah, P. V. Kumar, and K. Ramchandran “Exact regenerating codes for distributed storage," In [*Allerton Conf. on Control, Comp., and Comm.*]{}, Urbana-Champaign, IL, September 2009. N. B. Shah, K. V. Rashmi, P. V. Kumar, and K. Ramchandran, “Explicit codes minimizing repair bandwidth for distributed storage," in *Proc. IEEE ITW*, Jan. 2010. C. Suh and K. Ramchandran, “Exact regeneration codes for distributed storage repair using interference alignment," in *Proc. 2010 IEEE Int. Symp. on Inform. Theory (ISIT)*, Seoul, Korea, Jun. 2010. Y. Wu. “A construction of systematic MDS codes with minimum repair bandwidth," Submitted to *IEEE Transactions on Information Theory*, Aug. 2009. Preprint available at http://arxiv.org/abs/0910.2486. V. Cadambe, S. Jafar, and H. Maleki, “Distributed data storage with minimum storage regenerating codes - exact and functional repair are asymptotically equally efficient,” in [*2010 IEEE Intern. Workshop on Wireless Network Coding (WiNC)*]{}, Apr 2010. C. Suh and K. Ramchandran, “On the existence of optimal exact-repair MDS codes for distributed storage,” Apr. 2010. Preprint available online at http://arxiv.org/abs/1004.4663 K. Rashmi, N. B. Shah, and P. V. Kumar, “Optimal exact-regenerating codes for distributed storage at the MSR and MBR points via a product-matrix construction,” submitted to IEEE Transactions on Information Theory, Preprint available online at http://arxiv.org/pdf/1005.4178. I. Tamo, Z. Wang, and J. Bruck “MDS Array Codes with Optimal Rebuilding,” [*to appear at ISIT 2011*]{}, preprint available at http://arxiv.org/abs/1103.3737 V. R. Cadambe, C. Huang, and J. Li, “Permutation codes: optimal exact-repair of a single failed node in MDS code based distributed storage systems,” [*to appear at ISIT 2011*]{}, preprint available at http://newport.eecs.uci.edu/$\sim$vcadambe/permutations.pdf D. S. Papailiopoulos, A. G. Dimakis, and V. R. Cadambe, “Repair optimal erasure codes through hadamard designs,” preprint available at http://www-scf.usc.edu/$\sim$papailio/ [^1]: During the submission of this manuscript, two independent works appeared that constructed MDS codes of arbitrary rate that can optimally repair their systematic nodes, see [@Tamo], [@PermCodes]. [^2]: Our latest work expands Hadamard designs to construct $2$-parity MDS codes that can optimally repair any systematic or parity node failure and $m$-parity MDS codes that can optimally repair any systematic node failure [@PVD]. [^3]: $\mathbb{F}$ denotes the finite field over which all operations are performed. [^4]: Here, we consider that the newcomer downloads the same amount of information from both parities. In general this does not need to be the case. [^5]: By “at most” we mean that this result is proved using an achievable scheme, however, we do not prove that it is optimal.

--- abstract: 'Most stars in galactic disks are believed to be born as a member of star clusters or associations. Star clusters formed in disks are disrupted due to the tidal stripping and the evolution of star clusters themselves, and as a results new stars are supplied to the galactic disks. We performed $N$-body simulations of star clusters in galactic disks, in which both star clusters and galactic disks are modeled as $N$-body (“live”) systems, and as a consequence the disks form transient and recurrent spiral arms. In such non-steady spiral arms, star clusters migrate radially due to the interaction with spiral arms. We found that the migration timescale is a few hundreds Myr and that the angular momentum changes of star clusters are at most $\sim 50$% in 1 Gyr. Radial migration of star clusters to the inner region of galaxies results in a fast disruption of the star clusters because of a stronger tidal field in the inner region of the galaxy. This effect is not negligible for the disruption timescale of star clusters in galactic disks. Stars stripped from clusters form tidal tails which spread over 1–2 kpc. While the spatial distribution of tidal tails change in a complicated way due to the non-steady spiral arms, the velocity distribution conserve well even if the tidal tails are located at a few kpc from their parent clusters. Tidal tails of clusters in galactic disks might be detected using velocity plots.' author: - | M. S. Fujii$^{1}$ [^1] and J. Baba$^{2}$\ $^{1}$Leiden Observatory, Leiden University, NL-2300RA Leiden, The Netherlands\ $^{2}$Interactive Research Center of Science, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro, Tokyo 152-8551, Japan date: 'Accepted 1988 December 15. Received 1988 December 14; in original form 1988 October 11' title: Destruction of star clusters due to the radial migration in spiral galaxies --- \[firstpage\] galaxies: star clusters — galaxies:spiral — galaxies: kinematics and dynamics — methods: N-body simulations Introduction ============ Star clusters are one of the fundamental building blocks of galactic disks because most stars are formed in star clusters . In disk galaxies, new clusters born in galactic disks travel in their host disks experiencing disruptions and supply new stars to the disks. The disruption of clusters is caused by the tidal force from their host galaxy and also the internal evolution of star clusters themselves such as dynamical evolution, mass loss due to the stellar evolution, and gas expulsion . Non-axisymmetric structures in galactic disks, such as spiral arms [@2007MNRAS.376..809G] and bars [@2012MNRAS.419.3244B], have also been expected to play important roles for the dynamical disruption of clusters. For example, @2007MNRAS.376..809G investigated the effect of spiral-arm passages on the evolution of star clusters assumed to rotate in a fixed pattern speed, as in the stationary density wave theory [@1964ApJ...140..646L; @1996ssgd.book.....B]. However, self-consistent simulations of galactic disks have shown that self-excited spiral arms are not stationary regardless of the existence of gas (or some kind of dissipation) [@1984ApJ...282...61S; @2002MNRAS.336..785S; @2003MNRAS.344..358B; @2009ApJ...706..471B; @2011ApJ...730..109F; @2011MNRAS.410.1637S; @2011ApJ...735....1W; @2012MNRAS.421.1529G; @Baba+2012]. Such non-steady spiral arms do not have a single pattern speed but roughly follow the galactic rotation [@2011ApJ...735....1W; @2012MNRAS.421.1529G]. Therefore, these arms scatter stars everywhere in the disk by the co-rotation resonance [@2002MNRAS.336..785S; @2012MNRAS.421.1529G; @Baba+2012]. Non-steady spiral arms change the gravitational fields around star clusters chaotically rather than periodically as in the stationary density waves. In order to know the dynamical evolution of star clusters in galactic disks with non-steady spiral arms, we need to model both star clusters and galactic disks as $N$-body (“live”) systems. Such self-consistent $N$-body simulations are technically more difficult than those with rigid potential disks because the dynamical timescale of star clusters is much shorter than that of galactic disks and a large number of particles are required for the modeling of the disks [@2011ApJ...730..109F]. We solve this problem using a direct-tree hybrid code, Bridge [@2007PASJ...59.1095F]. In this letter, we perform self-consistent $N$-body simulations of star clusters in live disks using Bridge and demonstrate that the angular-momentum exchange between star clusters and spiral arms causes the radial migration of star clusters of a few kpc from their initial galacto-centric radii. The migration timescale is shorter than the galactic rotation timescale, i.e., a few hundred Myr. The radial migration causes the tidal disruption of star clusters bringing them to closer to the galactic center. Star clusters lose their mass in their perigalacticon passage, and their tidal tails spread over a few kpc. We also find that tidal-tail stars stay close to their parent clusters in their velocity space even if they are already a few kpc from the parent cluster. The tidal tail of young clusters might be detectable using their velocities. $N$-body simulations ==================== We performed a series of $N$-body simulations of star clusters embedded in a live galactic disk with spiral arms. We modeled both the disk and the clusters as $N$-body systems, but the halo is modeled as a potential. We set up the disks following models used in @2011ApJ...730..109F. We adopted an exponential disk model with a total disk mass of $3.2 \times 10^{10} M_{\odot}$ with $3\times 10^6$ (3M) particles and as a consequence the mass of a disk-particle is $\sim 10^4 M_{\odot}$. The scale radius and scale height of the disk are 3.4 kpc and 0.34 kpc, respectively. For the dark matter halo, we adopted the NFW model [@1997ApJ...490..493N] with the concentration parameter of the halo, $c=10$. The virial radius and the mass of the halo are 122 kpc and $6.4\times 10^{11}M_{\odot}$. We modeled star clusters as a King model with the dimensionless central potential $W_0=3$ [@1966AJ.....71...64K]. We adopted a total cluster mass of $10^5 M_{\odot}$ and a half-mass radius of 8 pc, and therefore the tidal radius is 30 pc. Our model is similar to young massive clusters in M51 and M82 (see Figure 9 in ) rather than those in in the Milky Way disk, which are more compact and therefore would be tidally disrupted less than our model. Included the stellar evolution, however, the Milky-Way clusters might expand a factor of 5–10 in the first 10 Myr as seen in observations (see Figure 8 in ). We used 8192 (8k) equal-mass particles for the cluster. We first integrated only the disk up to 5 Gyr, in which self-excited spiral arms fully developed from the initial Poisson noise due to the swing amplification. Then, we detected dense regions in the disk using the procedure below and put star clusters in the dense regions assuming that they are born there. We detected disk particles whose eighth-nearest-neighbor position is closer than 70% of the Jacobi radius of a cluster with $10^5 M_{\odot}$. We chose the densest position from the detected locations and rejected other candidates within five Jacobi radii from the selected one in order to avoid that star clusters initially collide with each other. Repeating this procedure, we chose 97 positions in a spiral arm in 4–10 kpc from the galactic center. The initial positions of the star clusters are shown in the top left panel of Figure \[fig:snap\]. We adopted the center-of-mass velocity of the eight neighbors as the cluster velocity. The initial radial, azimuthal, and vertical velocity dispersion among the star clusters are 10.2, 7.7, and 7.5 $\rm km~s^{-1}$, respectively. Although these values are smaller than the mean of the disk stars (19, 14, and 14 $\rm km~s^{-1}$ at 8 kpc respectively), star clusters would have rather smaller velocity dispersion than those of old stars if we assume that star clusters form from giant molecular clouds as is observed in the Milky Way . The simulations are performed using a direct-tree hybrid code, Bridge [@2007PASJ...59.1095F]. In Bridge, only the inner motion of star clusters are integrated using a direct $N$-body code and the other interactions are integrated using a tree code [@1986Natur.324..446B]. We adopted a sixth-order Hermite scheme for the direct method [@2008NewA...13..498N] without any softening and with an accuracy parameter of 0.9. For the disk particles, the gravitational potential is softened using Plummer softening with a length of 10 pc. We adopted an opening angle of 0.4 with the center-of-mass approximation and a time step of 0.29 Myr for the tree code. ![Snapshots of the initial condition and at 0.25, 0.5, and 1 Gyr. Gray scale shows the column density of the disk, and color points indicate cluster particles. Since these are only twelve colors but 97 clusters, we plot several clusters with the same color. The color points are as large as the cluster size to make the points visible. More than 80% of cluster-particles are still bound at 1 Gyr. \[fig:snap\]](snap0000new.eps "fig:"){width="40mm"} ![Snapshots of the initial condition and at 0.25, 0.5, and 1 Gyr. Gray scale shows the column density of the disk, and color points indicate cluster particles. Since these are only twelve colors but 97 clusters, we plot several clusters with the same color. The color points are as large as the cluster size to make the points visible. More than 80% of cluster-particles are still bound at 1 Gyr. \[fig:snap\]](snap0052new.eps "fig:"){width="40mm"} ![Snapshots of the initial condition and at 0.25, 0.5, and 1 Gyr. Gray scale shows the column density of the disk, and color points indicate cluster particles. Since these are only twelve colors but 97 clusters, we plot several clusters with the same color. The color points are as large as the cluster size to make the points visible. More than 80% of cluster-particles are still bound at 1 Gyr. \[fig:snap\]](snap0106new.eps "fig:"){width="40mm"} ![Snapshots of the initial condition and at 0.25, 0.5, and 1 Gyr. Gray scale shows the column density of the disk, and color points indicate cluster particles. Since these are only twelve colors but 97 clusters, we plot several clusters with the same color. The color points are as large as the cluster size to make the points visible. More than 80% of cluster-particles are still bound at 1 Gyr. \[fig:snap\]](snap0212new.eps "fig:"){width="40mm"} Results ======= Radial migration of star clusters --------------------------------- Figure \[fig:snap\] shows the time evolution of the star clusters and disk. Although each spiral arm looks like a single coherent structure in each snapshot, it is transient and recurrent. They are wound up due to the differential rotation of the galactic disk and break up into multiple segments with a few kpc-scale, but the segments reconnect and form new long coherent arms (see online material movie). The orbits of star clusters in such transient spiral arms are complicated rather than a simple epicyclic motion. The motion of star clusters in spiral arms is similar to that of gas and stars in the disk as shown in @2011ApJ...735....1W [@2012MNRAS.421.1529G]. Clusters tend to stay in or close to spiral arms and migrate along the arms. One of the orbits (the distance from the galactic center) obtained from our simulation is shown in the top left panel of Figure \[fig:cluster76\] (see also online material movie). The cluster is initially located at around 9 kpc and migrates inward down to around 6.5 kpc losing its angular momentum, but it migrates outward up again to 8.5 kpc (see the top left panel in Figure \[fig:cluster76\]). Such radial migration is caused by the dynamical interaction between star clusters and spiral arms. The bottom left panel of Figure \[fig:cluster76\] shows time evolution of the azimuthal force from the disk on the star cluster. The cluster loses its angular momentum, when it is moving ahead of a spiral arm (see top right panel of Figure \[fig:cluster76\], in which spiral arms are moving from right to left). On the other hand, the star cluster gains angular momentum, when it is moving behind a spiral arm (see middle right panel of Figure \[fig:cluster76\]). The angular momentum of clusters does not change when clusters are located at just the middle of two arms (see bottom right panel of figure \[fig:cluster76\]). Thus, star clusters in non-steady disks lose or gain angular momentum and as a result migrate a few kpc from their initial positions. Such angular-momentum changes occur for all star clusters. The clusters in our simulation lost or gained at most $\sim 50$% of their initial angular momenta within 1 Gyr, which corresponds to radial migration of a few kpc. The angular-momentum change in the clusters is similar to that of stars investigated in simulations of stellar disks [@2002MNRAS.336..785S; @2008ApJ...675L..65R; @2012MNRAS.421.1529G; @Baba+2012]. Since star clusters in transient spiral arms migrate a few kpc in their galactic rotation timescale, i.e., a few hundred Myr, open clusters in the Galactic disk older than $\sim 100$ Myr are expected to have already migrated from their initial galactcentric radii. This rapid migration of star clusters may also change our understanding of the evolution of the Galactic disk. Indeed open clusters in the Milky Way disk have been used as tracers for the dynamical and chemical evolution of the Galactic disk . In these studies, the spiral arms are considered to be stationary density waves, but these results might change in non-steady arms. {width="130mm"} Another important effect of the radial migration is the tidal disruption of star clusters due to the smaller Jacobi radii at smaller distance from the galactic center. In our models it changes from 64 pc at 8 kpc to 45 pc at 4 kpc. In Figure \[fig:mass\] we see a clear correlation between the bound mass at the end of our simulation (1.5 Gyr) and the minimum perigalacticon distance of the clusters. In our simulation, clusters lose masses mainly during their perigalacticon passages, and therefore the mass loss becomes larger when they migrates inward. However, internal heating due to the passage of spiral arms [@2007MNRAS.376..809G] does not work efficiently in the case of transient arms, because both star clusters and spiral arms are always corotating. ![ The relation between the bound mass at 1.5 Gyr normalized by the initial mass and the minimum perigalacticon distance. \[fig:mass\]](mass_peri.eps){width="80mm"} Tidal tails ----------- Stars tidally stripped from their parent clusters form tidal tails spreading over a few kpc (see Figure \[fig:snap\]). Their shapes are more complicated than those in simple spherical external potentials and change in time. When star clusters migrate from apogalacticon to perigalacticon, they move along a spiral arm, and their tidal tails also elongate along the spiral arm. When clusters are in the other phases (e.g., perigalacticon or apogalacticon), the tails come closer to the parent cluster again even if they are no longer bound to the cluster (see top right panel of Figure \[fig:cluster76\]). This behavior is similar to that in the case in axisymmetric external potentials [@2005AJ....129.1906C]. Figure \[fig:xyv\] shows spacial (left) and velocity (right) distributions of two clusters. The cluster shown in the top panels is at its perigalacticon passage, and the one in the bottom panels is moving from its perigalacticon to apogalacticon. The cluster shown in the top panels is the same as that shown in Figure \[fig:cluster76\]. In spacial distribution plots (left panels in Figure \[fig:xyv\]), cluster particles (crosses) show tidal tails spreading over a few kpc which are close to the orbits (black doted curves) or the circular orbits at the positions of the clusters (black dashed curve). In spite of such a large spatial distribution, we find that the tidal-tail stars still remain close to their parent clusters in velocity space. In the right panels, we plot the position of cluster stars in $v_{R}$-$v_{\phi}$ space as crosses. Colors in both panels indicate the velocity deviation from the parent clusters; blue, red, green, and black indicate $<$2, 2–5, 5–7, and $>$7 $\rm km~s^{-1}$, respectively. We also plot disk particles within 1 kpc from the cluster center (cyan points). In contrast to the cluster particles, the disk particles have a larger distribution in the velocity spaces. In spatial plots we plot disk particle which are within 1 kpc from the cluster center and have velocity deviation of $<$5 $\rm km~s^{-1}$ in the velocity space (cyan points). If we detect star cluster tails using only their velocities, these stars would be detected as contamination. They are located randomly in spatial plots, while star cluster particles distribute in tidal tails. ![Distribution of cluster particles and disk particles around clusters in $x$-$y$ plane (left) and $v_{R}$-$v_{\phi}$ phase (right). The cluster shown in the top panels is the same one shown in Figure \[fig:cluster76\] at a time of 1.75 Gyr, when the cluster is located at its perigalacticon. The bottom panels show a cluster, which is moving from its perigalacticon to apogalacticon at a time of 1 Gyr. Crosses and points indicate cluster and disk particles, respectively. In the left panels, colors indicate the velocity shown in right panels. Black dotted and dashed curves show orbits obtained from the simulation and circular orbits at the distance of the cluster from the galactic center, respectively. Yellow circles show the 1 and 2 kpc radius from the center-of-mass position of the cluster. In the right panels, blue, red, green, and black crosses indicate cluster particles with $<$2, 2–5, 5–7, and $>$7 $\rm km~s^{-1}$ from the mean velocity of the cluster particles. Cyan points indicate disk particles which are located within 1 kpc from the cluster center. Disk particles which are located within 1 kpc and have velocities $<$2 and 2–5 $\rm km~s^{-1}$ from the mean velocity of the cluster are shown in the left panels as blue and red points, respectively.\[fig:xyv\]](xyv_cl76_247.eps "fig:"){width="90mm"} ![Distribution of cluster particles and disk particles around clusters in $x$-$y$ plane (left) and $v_{R}$-$v_{\phi}$ phase (right). The cluster shown in the top panels is the same one shown in Figure \[fig:cluster76\] at a time of 1.75 Gyr, when the cluster is located at its perigalacticon. The bottom panels show a cluster, which is moving from its perigalacticon to apogalacticon at a time of 1 Gyr. Crosses and points indicate cluster and disk particles, respectively. In the left panels, colors indicate the velocity shown in right panels. Black dotted and dashed curves show orbits obtained from the simulation and circular orbits at the distance of the cluster from the galactic center, respectively. Yellow circles show the 1 and 2 kpc radius from the center-of-mass position of the cluster. In the right panels, blue, red, green, and black crosses indicate cluster particles with $<$2, 2–5, 5–7, and $>$7 $\rm km~s^{-1}$ from the mean velocity of the cluster particles. Cyan points indicate disk particles which are located within 1 kpc from the cluster center. Disk particles which are located within 1 kpc and have velocities $<$2 and 2–5 $\rm km~s^{-1}$ from the mean velocity of the cluster are shown in the left panels as blue and red points, respectively.\[fig:xyv\]](xyv_cl46.eps "fig:"){width="90mm"} Summary ======= We performed a series of $N$-body simulations of star clusters in live stellar disks with multiple spiral arms. In these simulations, both galactic disks and star clusters are modeled as $N$-body systems and integrated self-consistently. Our results show that star clusters migrate radially a few kpc in the time scale of their orbital period in the disk (a few 100 Myr) because of the angular-momentum exchange with transient spiral arms. The angular-momentum change of the clusters is at most $\sim 50$% of the initial angular momentum within 1 Gyr. In the case of non-steady transient spiral arms, the radial migration strongly affects the tidal disruption of star clusters because star clusters lose more mass when they approach the galactic center due to the smaller Jacobi radii. This kind of disruption mechanism does not appear in stationary density waves. The heating due to the non-steady (corotating) spiral-arm passage would not be as strong as that by the density-wave spiral arms, because the adiabatic change of the energy due to the slow passages of spiral arms suppresses the heating per passage, and the number of spiral passages is quite few (see the corotation case in Figure 8 in @2007MNRAS.376..809G). Furthermore, the radial migration of star clusters can carry stars far from their original orbital radii and finally the distribution of stars would be much wider than those in the case of the density waves [@2010ApJ...713..166B]. With transient spiral arms, star clusters and their tidal tails tend to stay in or close to spiral arms. The shape of tidal tails of clusters change in a complicated way in time compared to those in a smooth tidal field like a halo potential. When a star cluster approach the galactic center, star clusters move along a spiral arm as is the case of stars and gas moving in spiral arms. In this phase, the tidal tails also spread along the spiral arm. During the apogalacticon passage, on the other hand, the tidal tails are compressed and distribute around 1 kpc from the cluster even though they are unbound. The tidal tails of clusters might be detectable even if they spread over a few kpc, because the tidal-tail stars still remain very close to the cluster in velocity space after they become unbound. If we know the velocity of stars, we might be able to detect the tidal tails of star clusters in the Galactic disk using a future astrometry such as [*Gaia*]{} and [*JASMINE*]{}. Acknowledgments {#acknowledgments .unnumbered} =============== The authors thank Dan Caputo and Jeroen Bédorf for careful reading of the manuscript. This work was supported by Postdoctoral Fellowship for Research Abroad of the Japan Society for the Promotion of Science (JSPS) and HPCI Strategic Program Field 5 “The Origin of Matter and the Universe.” Numerical computations were carried out on GRAPE-DR at the Center for Computational Astrophysics (CfCA) of the National Astronomical Observatory of Japan and the Little Green Machine at Leiden University. 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--- abstract: 'In this correspondence, we study the secure multi-antenna transmission with artificial noise (AN) under imperfect channel state information in the presence of spatially randomly distributed eavesdroppers. We derive the optimal solutions of the power allocation between the information signal and the AN for minimizing the secrecy outage probability (SOP) under a target secrecy rate and for maximizing the secrecy rate under a SOP constraint, respectively. Moreover, we provide an interesting insight that channel estimation error affects the optimal power allocation strategy in opposite ways for the above two objectives. When the estimation error increases, more power should be allocated to the information signal if we aim to decrease the rate-constrained SOP, whereas more power should be allocated to the AN if we aim to increase the SOP-constrained secrecy rate.' author: - 'Tong-Xing Zheng, , and Hui-Ming Wang,  [^1] [^2] [^3]' title: Optimal Power Allocation for Artificial Noise under Imperfect CSI against Spatially Random Eavesdroppers --- Physical layer security, artificial noise, multi-antenna, secrecy outage, power allocation, imperfect CSI. Introduction ============ Physical layer security (PLS), which achieves secure transmissions by exploiting the randomness of wireless channels, has drawn considerable attention recently [@Yang2015Safeguarding], [@WangMaga15]. It has been shown that we are able to greatly improve PLS using multi-antenna techniques with global channel state information (CSI). However, to acquire the CSI of an eavesdropper is very difficult in real wiretap scenarios, since the eavesdropper is usually passive. Without the eavesdropper’s CSI, Goel *et al.* [@Goel2008Guaranteeing] proposed a so-called artificial noise (AN) aided multi-antenna transmission strategy, in which the transmitter masked the information-bearing signal by injecting isotropic AN into the null space of the main channel (from the transmitter to a legitimate receiver), thus creating non-decodable interference to potential eavesdroppers while without impairing the legitimate receiver. This seminal work has unleashed a wave of innovation [@Zhang2013Design]-[@WangTSP], and the AN scheme has become a promising approach to safeguarding wireless communications. In practice, the CSI of the main channel is acquired by training, channel estimation and feedback, which inevitably result in CSI imperfection. Some endeavors have studied the AN scheme allowing for imperfect CSI. For example, robust beamforming schemes have been proposed in [@Mukherjee2011Robust] for MIMO systems and in [@WangTVT] for cooperative relay systems. The effects of channel quantized feedback to the AN scheme are discussed in [@LinTWC] and [@Zhang2015Artificial], while in [@WangTSP], training and feedback have been jointly investigated and optimized. However, all the aforementioned works ignored the uncertainty of eavesdroppers’ spatial positions. Generally, eavesdroppers are geographically distributed randomly, especially in large-scale wireless networks. Analyzing secrecy performance in such random wiretap scenarios is fundamentally different from that with deterministic eavesdroppers’s locations. Recently, stochastic geometry theory has provided a powerful tool to analyze network performance by modeling nodes’ positions according to some spatial distributions such as a Poisson point process (PPP) [@Haenggi2009Stochastic]; it facilitates the study of the AN scheme against random eavesdroppers [@Ghogho2011Physical]-[@Zheng2015Multi]. However, the impact of imperfect CSI on designing the AN is still an open problem. Particularly, it is yet unknown what the optimal power allocation strategy is, and how a channel estimation error influences power allocation and secrecy performance. Due to the complicated/implicit forms of the objective functions caursed by location randomness and CSI imperfection, previous works can only obtain the optimal power allocation either by exhaustive search or by numerical calculation instead of providing a tractable expression. This makes it challenging to reveal an explicit analytical relationship between the optimal power allocation and the channel estimation error. Our research are motivated by the above observations and challenges. In this correspondence, we study an AN-aided multi-input single-output (MISO) secure transmission against randomly located eavesdroppers under imperfect channel estimation. We investigate two important performance metrics, namely, secrecy outage probability (SOP) and secrecy rate, respectively. The SOP reflects the quality difference between the main and wiretap channels; the secrecy rate measures the rate efficiency of secure transmission. We provide the optimal power allocation strategies for the following optimization problems: 1. Minimizing the SOP subject to a secrecy rate constraint; 2. Maximizing the secrecy rate subject to a SOP constraint. Furthermore, we draw an interesting conclusion that channel estimation error influences the optimal power allocation in *opposite ways* for the above two objectives. *When the estimation error increases, more power should be allocated to the information signal if we aim to decrease the rate-constrained SOP, whereas more power should be given to the AN if we aim to increase the SOP-constrained secrecy rate.* To the best of our knowledge, we are the first to reveal an explicit analytical relationship between the optimal power allocation and channel estimation error through strict mathematical proofs. Although existing works have also shown that AN should be exploited to increase the secrecy rate under imperfect CSI in point-to-point transmissions, their conclusions are just extracted from simulations under specific parameter settings, which may not apply to more general cases. *Notations*: $(\cdot)^{\dag}$, $(\cdot)^{\mathrm{T}}$, $|\cdot|$, $\|\cdot\|$ denote conjugate, transpose, absolute value, and Euclidean norm, respectively. $\mathcal{CN}$ denotes the circularly symmetric complex Gaussian distribution with zero mean and unit variance. $\mathbb{C}^{m\times n}$ denotes the $m\times n$ complex number domain. System Model and Problem Description ==================================== Consider a secure transmission from a transmitter (Alice) to a legitimate receiver (Bob) overheard by randomly located eavesdroppers (Eves)[^4]. Alice has $N$ antennas, Bob and Eves each has a single antenna. Without loss of generality, we place Alice at the origin and Bob at a deterministic position with a distance $r_B$ from Alice. The locations of Eves are modeled as a homogeneous PPP $\Phi_E$ of density $\lambda_E$ on a 2-D plane with the $k$-th Eve a distance $r_k$ from Alice. All wireless channels are assume to undergo flat Rayleigh fading together with a large-scale path loss governed by the exponent $\alpha>2$. The channel vector of a node with a distance $r$ from Alice is characterized as $\bm{h}r^{-\frac{\alpha}{2}}$, where $\bm{h}\in\mathbb{C}^{N\times 1}$ denotes the small-scale fading vector, with independent and identically distributed (i.i.d.) entries $h_{i}\thicksim \mathcal{CN}$. We focus on a *frequency-division duplex* (FDD) system in which the channel reciprocity no longer holds. We assume Bob estimates the main channel with estimation errors, and sends the estimated channel to Alice via an ideal feedback link (e.g., a high-quality link with negligible quantization error). In this case, the exact main channel $\bm{h}_b$ can be modeled as $$\label{channel_model} \bm{h}_b = \sqrt{1-\tau^2}\hat{\bm{h}}_{b} + \tau\tilde{\bm{h}}_{b},$$ where $\hat{\bm{h}}_{b}$ and $\tilde{\bm{h}}_{b}$ denote the estimated channel and estimation error with i.i.d. entries $\hat h_{b,i}, \tilde h_{b,i}\sim\mathcal{CN}(0,1)$. This assumption arises from employing the minimum mean square error (MMSE) estimation[^5] [@Mukherjee2011Robust], [@Geraci2014Physical]. Here, $\tau\in[0,1]$ denotes the error coefficient; $\tau=0$ corresponds to a perfect channel estimation, and $\tau=1$ means no CSI is acquired at all. For each eavesdropper, although its CSI is unknown, we assume its channel statistics information is available, which is a general assumption when dealing with PLS [@Zhang2013Design]-[@Zheng2015Multi]. Recalling the AN scheme in [@Goel2008Guaranteeing], the transmitted signal vector $\bm{x}$ at Alice is designed in the form of $$\label{x} \bm{x}=\sqrt{\xi P}\bm{w}s+\sqrt{(1-\xi) P/(N-1)}\bm{Gv},$$ where $s$ is the information signal with $\mathbb{E}[|s|^2]=1$, $\bm{v}\in\mathbb{C}^{(N-1)\times 1}$ is an AN vector with i.i.d. entries $v_i\sim\mathcal{CN}$, and $\xi$ is the power allocation ratio (PAR) of the desired signal power to the total power $P$. $\bm{w}\triangleq{\hat{\bm{h}}_{b}^{\dag}}/{\|\hat{\bm{h}}_{b}\|}$ is the beamforming vector for the information signal, $\bm{G}\in\mathbb{C}^{N\times(N-1)}$ is a weighting matrix for the AN. The columns of $\bm{W}\triangleq[\bm{w} ~\bm{G}]$ constitute an orthogonal basis. Let $\bm{s}\triangleq\left[s~ \bm{v}^{\mathrm{T}}\right]$, and the received signals at Bob and the $k$-th Eve are given from $$\begin{aligned} \label{signal_Bob} y_B &=\sqrt{1-\tau^2}\|\hat{\bm{h}}_{b}\|^2r_B^{-\frac{\alpha}{2}}s +\underbrace{\tau\tilde{\bm{h}}_{b}\bm{W} \bm{s}^{\mathrm{T}} r_B^{-\frac{\alpha}{2}} + n_B}_{n^{o}_B},\\ \label{signal_Evek} y_k&=\bm{h}_{e,k}\bm{w}r_k^{-\frac{\alpha}{2}}s + \bm{h}_{e,k}\bm{Gv}r_k^{-\frac{\alpha}{2}} + n_k,\ \forall k\in\Phi_E,\end{aligned}$$ where $\bm{h}_{e,k}$ denotes the channel from Alice to the $k$-th Eve, and $n_B^{o}$ combines the residual channel estimation error and thermal noise. Without loss of generality, we assume $n_B, n_{k\in\Phi_E}\thicksim \mathcal{CN}$. The exact capacity expression of the main channel under imperfect receiver CSI is still unavailable. A commonly used approach is to examine a capacity lower bound by treating $n_B^{o}$ as the worst-case Gaussian noise[^6]. By doing so, the SINRs of Bob and the $k$-th Eve are respectively given by $$\begin{aligned} \label{sinr_Bob} \gamma_B &= \xi\kappa(\tau),\\ \label{sinr_Evek} \gamma_k &=\frac{\xi P|\bm{h}_{e,k}^{\mathrm{T}}\bm{w}|^2r_k^{-\alpha} }{(1-\xi)P\|\bm{h}_{e,k}^{\mathrm{T}} \bm{G}\|^2 r_k^{-\alpha}/(N-1)+1},\end{aligned}$$ where $\kappa(\tau)= \frac{(1-\tau^2)P\gamma} {\tau^2 P+r_B^{\alpha}}$ with $\gamma\triangleq \|\hat{\bm{h}}_{b}\|^2$. Eq. holds for the pessimistic assumption that the $k$-th Eve has perfect knowledge of both $\hat{\bm{h}}_{b}$ and $\tilde{\bm{h}}_{b}$. Given that $\tau\in[0,1]$, $\kappa(\tau)$ is a monotonically decreasing function of $\tau$; it reflects the accuracy of channel estimation. Specifically, a small value of $\kappa(\tau)$ corresponds to a low estimation accuracy and vice versa. Hereafter, we omit $\tau$ from $\kappa(\tau)$ for notational brevity. We consider the wiretap scenario in which each Eve individually decodes a secret message. This corresponds to a *compound* wiretap channel model [@Liang2009Compound], and the capacities of the main channel and the equivalent wiretap channel are $C_B = \log_2(1+\gamma_B)$ and $C_E = \log_2(1+\gamma_E)$ with $\gamma_E \triangleq \max_{k\in\Phi_E}\gamma_k$. Note that the capacity of Eves is determined by the maximum capacity among all links connecting Alice with Eves. As done in [@Zhang2013Design] and [@Zheng2015Multi], after encoding secret information, Alice transmits the codewords and embedded secret messages at rates $C_B$ and $R_S$, respectively. If *at least one Eve decodes the secret messages*, i.e., $C_E$ exceeds the *rate $ C_B-R_S$ of redundant information* (to protect from eavesdropping), perfect secrecy is compromised and a secrecy outage occurs; the corresponding SOP is defined as $$\label{sop_def} \mathcal{O}\triangleq \mathbb{P}\{C_E>C_B-R_S\}, ~\forall ~C_B> R_S.$$ In the following, we will optimize the PAR to minimize the SOP under a target secrecy rate, and to maximize the secrecy rate under a SOP constraint $\mathcal{O}\le\epsilon\in(0,1)$, respectively. We emphasize that different from existing research with deterministic Eves’ positions, the analysis and design here is much more complicated due to the extra spatial randomness. Secrecy Outage Probability Minimization ======================================= In this section, we optimize the PAR that minimizes the SOP under a target secrecy rate. Recalling , Alice transmits only when $C_B=\log_2(1+\xi\kappa)> R_S$, i.e., $\xi>\frac{2^{R_S}-1}{\kappa}$ should hold to guarantee a reliable connection between Alice and Bob. For ease of notation, throughout the paper we define $T\triangleq 2^{R_S}$, $\omega\triangleq \frac{T-1}{\kappa}$, $\delta\triangleq \frac{2}{\alpha}$, $\beta \triangleq\pi \Gamma\left(1+\delta\right)$, $\theta\triangleq\frac{T-1}{T}$, and $\varphi\triangleq\frac{\xi^{-1}-1}{N-1}$. The problem of minimizing $\mathcal{O}$ in is formulated as $$\label{min_sop} \min_{\xi}~\mathcal{O},\qquad \mathrm{s.t.} \quad \omega<\xi\leq 1.$$ Before proceeding to this optimization problem, we provide a closed-form expression of $\mathcal{O}$ over the PPP network. \[lemma\_sop\] *If $\xi>\omega$, the SOP defined in is given by $$\label{pso} \mathcal{O}=1-\exp\left(-\beta\lambda_E \left({P\theta^{-1}}\right)^{\delta} \mathcal{J}(\xi)\right),$$ where $\mathcal{J}(\xi)= \left({\omega}^{-1}-{\xi}^{-1}\right)^{-\delta} \left(1+\left({\xi}{\omega}^{-1}-1\right)\theta\varphi\right)^{1-N}$.* Substitute $C_B$ and $C_E$ along with and into , and after some algebraic manipulations, we obtain $\mathcal{O}=1-\mathcal{F}_{\gamma_E} \left(x\right)$ with $x\triangleq\frac{1+\kappa\xi}{T}-1$, where $\mathcal{F}_{\gamma_E}(x)$ is the cumulative distribution function (CDF) of $\gamma_{E}$, which is $$\begin{aligned} \label{cdf_eta_e_app} \mathcal{F}_{\gamma_E}(x) &=\mathbb{P} \left\{\max_{k\in\Phi_E}\gamma_k<x\right\} =\mathbb{E}_{\Phi_E}\left[\prod_{k\in\Phi_E} \mathbb{P}\{\gamma_k<x\}\right]\nonumber\\ & \stackrel{\mathrm{(a)}} =\mathbb{E}_{\Phi_E} \left[\prod_{k\in\Phi_E}\left( 1-e^{-\frac{r_k^{\alpha}x}{P\xi}}(1+\varphi x)^{1-N} \right) \right]\nonumber\\ &\stackrel{\mathrm{(b)}}= \exp\left(-2\pi\lambda_E (1+\varphi x)^{1-N} \int_0^{\infty} e^{-\frac{r^{\alpha}x}{P\xi}} rdr\right)\nonumber\\ & =\exp\left(-\beta\lambda_E(P\xi)^{\delta} x^{-\delta}(1+\varphi x)^{1-N}\right),\end{aligned}$$ where (a) holds for the CDF of $\gamma_k$ [@Zhang2013Design], and (b) holds for the probability generating functional (PGFL) over a PPP [@Stoyan1996Stochastic]. Substituting into $\mathcal{O}$ completes the proof. The theoretical values of $\mathcal{O}$ are well verified by Monte-Carlo simulations, as shown in Fig. \[SOP\_PHI\]. We see that adding transmit antennas is beneficial for decreasing the SOP. We also observe that as $\xi$ increases, $\mathcal{O}$ first decreases and then increases; there exists a unique $\xi$ that minimizes $\mathcal{O}$. In the following we are going to calculate the value of this unique $\xi$. From , it is apparent that minimizing $\mathcal{O}$ is equivalent to minimizing $\mathcal{J}(\xi)$. The first-order derivative of $\mathcal{J}(\xi)$ on $\xi$ is given by $$\label{dJ1} \frac{d\mathcal{J}(\xi)}{d\xi} = \frac{\theta\mathcal{J}(\xi)(\xi^3+a\xi^2+b\xi+c)} {\xi^2\left(\xi-\omega\right)(\omega+(\xi-\omega)\theta\varphi)},$$ where $a\triangleq-l_1\omega$, $b\triangleq-\frac{\delta}{\theta}\omega^2-l_0\omega^2-l_2\omega$, and $c\triangleq l_2\omega^2$, with $l_0\triangleq\frac{\delta}{N-1}$, $l_1\triangleq 1-l_0$, and $l_2\triangleq 1+l_0$. Let $\mathcal{K}(\xi)=\xi^3+a\xi^2+b\xi+c$. Since $\xi>\omega$, the sign of $\frac{d\mathcal{J}(\xi)}{d\xi}$ follows that of $\mathcal{K}(\xi)$. In other words, to investigate the monotonicity of $\mathcal{J}(\xi)$ on $\xi$, we need to just examine the sign of $\mathcal{K}(\xi)$. In the following theorem, we provide the solution to problem . \[opt\_par\_sop\_theorem\] *The optimal PAR that minimizes $\mathcal{O}$ in is $$\begin{aligned} \label{opt_par_sop} \xi^* = \begin{cases} ~\varnothing, &0<\kappa\le T-1\\ ~ 1, &T-1<\kappa\le (T-1)\left(1+\sqrt{{\delta}/{\theta}}\right)\\ ~\xi_o,&\text{otherwise} \end{cases}\end{aligned}$$ where $\xi_o= \sqrt[3]{q+p}+ \sqrt[3]{q-p} -\frac{a}{3}$ with $p\triangleq \sqrt{\left(\frac{b}{3}-\frac{a^2}{9}\right)^3 +q^2}$ and $q\triangleq \frac{ab}{6}-\frac{c}{2}-\frac{2a^3}{54}$, and $a$, $b$, $c$ have been defined in . $\xi^* =\varnothing$ means that transmission is suspended.* We know that Alice transmits only when $\xi>\omega$. 1) If $\omega\ge 1$, i.e., $\kappa\le T-1$, no feasible $\xi\in[0,1]$ satisfies $\xi>\omega$, and transmission is suspended. 2) If $\omega<1$, Alice transmits in the range of $\xi\in(\omega,1]$. Next, we derive the optimal value of $\xi$ that minimizes $\mathcal{J}(\xi)$. We first prove the convexity of $\mathcal{K}(\xi)$ on $\xi\in(\omega,1]$. From the expression of $\mathcal{K}(\xi)$, we have $\frac{d^2\mathcal{K}(\xi)}{d\xi^2}=6\xi-2l_1\omega>0$, i.e., $\mathcal{K}(\xi)$ is a *convex* function of $\xi$. Then we determine the sign of $\mathcal{K}(\xi)$. The values of $\mathcal{K}(\xi)$ at boundaries $\xi=\omega$ and $\xi=1$ are $\mathcal{K}(\omega)=-\frac{\delta}{\theta}\omega^3$ and $\mathcal{K}(1)=(1-\omega)^2 -\frac{\delta}{\theta}\omega^2$, respectively. Obviously, $\mathcal{K}(\omega)<0$ always holds. Next, we discuss the optimal value of $\xi$ for the following two cases. *Case 1*: $\mathcal{K}(1)\le0$. Since $\mathcal{K}(\xi)$ is convex on $\xi\in(\omega, 1]$, $\mathcal{K}(\xi)$ or $\frac{d\mathcal{J}(\xi)}{d\xi}$ is always negative. Hence $\mathcal{J}(\xi)$ monotonically decreases with $\xi$, and the minimum $\mathcal{J}(\xi)$ is achieved at $\xi=1$, with the corresponding condition obtained from $\mathcal{K}(1)\le 0$, which is $\frac{1}{1+\sqrt{\delta/\theta}}\le\omega<1$. *Case 2*: $\mathcal{K}(1)>0$. It means $\mathcal{K}(\xi)$ or $\frac{d\mathcal{J}(\xi)}{d\xi}$ becomes first negative and then positive as $\xi$ increases from $\omega$ to 1, i.e., $\mathcal{J}(\xi)$ first decreases and then increases with $\xi$, and the optimal value of $\xi$ is the unique root of the cubic equation $\mathcal{K}(\xi)=0$. Solving this equation using Cardano’s formula yields $\xi_o$. Combining *Case 1* and *Case 2* completes the proof. Theorem \[opt\_par\_sop\_theorem\] indicates that when the value of $\kappa$ is small which corresponds to a poor link quality or a large channel estimation error, Alice either suspends the transmission or transmits with full power. When the value of $\kappa$ becomes large enough, it is wise to create AN to decrease the SOP. The resulting minimum SOP, denoted as $\mathcal{O}^*$, is obtained by substituting $\xi^*$ into . Next, we investigate the influence of channel estimation error on the optimal PAR. Although we obtain a closed-form expression of $\xi_o$ in , it is complicated to reveal the explicit connection between $\xi_o$ and $\kappa$. Nevertheless, by leveraging the equation $\mathcal{K}(\xi_o)=0$, we develop some insights into the behavior of $\xi_o$ with respect to $\kappa$ in the following proposition. \[opt\_par\_tau\_proposition\] *$\xi_o$ monotonically decreases with $\kappa$.* Since $\omega=\frac{T-1}{\kappa}$, to complete the proof, we need to just prove the monotonicity of $\xi_o$ on $\omega$. Utilizing the derivative rule for implicit functions [@Jittorntrum1978Implicit] with $\mathcal{K}(\xi_o) = 0$, we obtain $$\label{dxi_domega} \frac{d\xi_o}{d\omega}= \frac{-\partial\mathcal{K} /\partial\omega}{\partial \mathcal{K}/\partial\xi_o}= \frac{l_1\xi_o^2+{2\frac{\delta}{\theta}\omega}\xi_o +2l_0\omega\xi_o+l_2\xi_o-2l_2\omega} {3\xi_o^2-2l_1\omega\xi_o-\frac{\delta}{\theta}\omega^2 -l_2\omega-l_0\omega^2}.$$ Substituting $a$, $b$ and $c$ defined in into $\mathcal{K}(\xi_o) = 0$ yields $$\label{theta_xi} \frac{\delta}{\theta} =\frac{(\xi_o^2+l_0\omega\xi_o-l_2\omega)(\xi_o-\omega)} {\omega^2\xi_o}.$$ Since $\xi_o>\omega$ and $\frac{\delta}{\theta}>0$, the term $\xi_o^2+l_0\omega\xi_o-l_2\omega$ in satisfies the following inequality $$\label{theta_xi_inequ} 0<\xi_o^2+l_0\omega\xi_o-l_2\omega <\xi_o^2+l_0\xi_o^2-l_2\omega <l_2(\xi_0^2-\omega)\Rightarrow \xi_o>\sqrt{\omega}.\nonumber \vspace{-0.0cm}$$ Substituting $\frac{\delta}{\theta}$ in into yields the numerator $-\frac{\partial\mathcal{K}}{\partial\omega}= \frac{\xi_o}{\omega}[l_1\xi_o(\xi_o-\omega) +l_2(\xi_o^2-\omega)]>0$ and denominator $\frac{\partial \mathcal{K}}{\partial\xi_o}=l_1\xi_o(\xi_o-\omega) +\frac{l_2}{\xi_o}(\xi_o^3-\omega^2)>0$, hence we have $\frac{d\xi_o}{d\omega}>0$. Combined with $\omega=\frac{T-1}{\kappa}$, we directly obtain $\frac{d\xi_o}{d\kappa}= \frac{d\xi_o}{d\omega}\frac{d\omega}{d\kappa}<0$, which completes the proof. Proposition \[opt\_par\_tau\_proposition\] shows that, *when the channel estimation error gets larger, if we aim to decrease the SOP under a target secrecy rate, we should increase the information signal power*, which is validated in Fig. \[PAR\_SOP\_RS\]. It is because that, in order to minimize the SOP, we should first guarantee the link quality of the main channel to support the target secrecy rate. Hence, we should increase the information signal power to balance the deterioration caused by the channel estimation error. When $\tau$ exceeds a certain value, transmission is suspended, which is just as analyzed previously. We also find from Fig. \[PAR\_SOP\_RS\] that the value of $\xi^*$ increases as $R_S$ increases, which can be easily confirmed by the fact $\frac{d\xi_o}{dT}= \frac{d\xi_o}{d\omega}\frac{d\omega}{dT}>0$. Fig. \[SOP\_TAU\_P\_RS\] shows that the minimum SOP $\mathcal{O}^*$ increases with $\tau$. For a given $P$, $\mathcal{O}^*$ increases with $R_S$. For a given $R_S$, the two curves with different values of $P$ cross as $\tau$ increases (see the intersection $\mathcal{P}$). Specifically, before $\tau$ exceeds $\mathcal{P}$, increasing $P$ decreases $\mathcal{O}^*$, and after that the opposite happens. This transition occurs because for too large an estimation error, increasing transmit power does not significantly improve Bob’s capacity, whereas it is of great benefit to Eves. This result implies that using full power is not always advantageous, particularly when the estimation error is large. Secrecy Rate Maximization ========================= In this section, we optimize the PAR that maximizes the secrecy rate subject to a SOP constraint. We first transform the SOP constraint $\mathcal{O}\le\epsilon$ into the following equivalent form $$\begin{aligned} \label{SOP_constraint} 1-\mathcal{F}_{\gamma_E} \left(\frac{1+\kappa\xi}{2^{R_S}}-1\right) &\stackrel{\mathrm{(c)}}\le\epsilon \Rightarrow \frac{1+\kappa\xi}{2^{R_S}}-1 \geq \mathcal{F}_{\gamma_E}^{-1}(1-\epsilon)\nonumber\\ & \Rightarrow R_S\le\log_2\frac{1+\kappa\xi}{1+\varrho(\xi)\xi},\end{aligned}$$ where (c) holds due to the monotonically increasing feature of the CDF $\mathcal{F}_{\gamma_E}(x)$ on $x$. $\varrho(\xi)\triangleq \frac{\mathcal{F}_{\gamma_E}^{-1}(1-\epsilon)}{\xi}$ with $\mathcal{F}_{\gamma_E}^{-1}(\cdot)$ the inverse function of $\mathcal{F}_{\gamma_E}(\cdot)$. Clearly, a positive value of $R_S$ that satisfies the SOP constraint exits only when $\varrho(\xi)<\kappa$. The problem of maximizing $R_S$ can be formulated as $$\label{rs_max} \max_{\xi} R_S=\log_2\frac{1+\kappa\xi} {1+\varrho(\xi)\xi}\quad \mathrm{s.t.}~ \varrho(\xi)<\kappa,~0\leq \xi\leq 1.$$ An illustration on the relationship between the secrecy rate and the PAR is shown in Fig. \[RS\_PHI\]. It is intuitive that increasing the number of antennas helps to improve the secrecy rate. We observe that, $R_S$ first increases with $\xi$, then decreases with it, and even reduces to zero for too large a $\xi$. This implies we should carefully choose the PAR to achieve a high secrecy rate. From , we see that the value of $R_S$ is bottlenecked by $\varrho(\xi)$, which implicitly reflects the influence of the density of PPP Eves $\lambda_E$ and the SOP threshold $\epsilon$. For example, a larger $\lambda_E$ or a smaller $\epsilon$ increases $\varrho(\xi)$ (see and the definition of $\varrho(\xi)$), and then decreases $R_S$ (see ). Therefore, $\varrho(\xi)$ plays a critical role in maximizing $R_S$. Although it is intractable to obtain an analytical expression of $\varrho(\xi)$ due to the transcendental equation $1-\mathcal{F}_{\gamma_E}(\xi\varrho(\xi))=\epsilon$ (see ), we provide an explicit connection between $\varrho(\xi)$ and $\xi$ in the following lemma, which is very critical for the subsequent optimization. \[varrho\_lemma\] *$\varrho(\xi)$ is a monotonically increasing and convex function of $\xi\in[0,1]$.* For notational brevity, we omit $\xi$ from $\varrho(\xi)$. Plugging $x=\xi\varrho$ into $1-\mathcal{F}_{\gamma_E}(x)=\epsilon$ yields $$\label{Z} \mathcal{Z}(\xi,\varrho) - L = 0,$$ where $\mathcal{Z}(\xi,\varrho)= \varrho^{\delta} \left(1+\varrho\frac{1-\xi}{N-1}\right)^{N-1}$, and $L\triangleq \frac{\beta\lambda_EP^{\delta}}{-\ln(1-\epsilon)}$. Using the derivative rule for implicit functions with , the first- and second-order derivatives of $\varrho$ on $\xi$ are given by $$\begin{aligned} \label{dx1} \frac{d\varrho}{d\xi} &=-\frac{{\partial \mathcal{Z}}/{\partial\xi}} {{\partial \mathcal{Z}}/{\partial \varrho}} =\frac{\varrho^2}{\delta+l_2(1-\xi)\varrho},\\ \label{dx2} \frac{d^2\varrho}{d\xi^2} &=\frac{2}{\varrho}\left(\frac{d\varrho}{d\xi}\right)^2 +\frac{l_2\varrho^2\left(\varrho-(1-\xi)\frac{d\varrho} {d\xi}\right)} {\left(\delta+l_2(1-\xi)\varrho\right)^2}.\end{aligned}$$ Clearly, $\frac{d\varrho}{d\xi}>0$ always holds. With , we have $\varrho-(1-\xi)\frac{d\varrho}{d\xi}= \frac{\delta\varrho+l_0(1-\xi)\varrho^2} {\delta+l_2(1-\xi)\varrho}>0$ in . Removing the second term from the right-hand side of yields $\frac{d^2\varrho}{d\xi^2} >\frac{2}{\varrho}\left(\frac{d\varrho}{d\xi}\right)^2>0$. With $\frac{d\varrho}{d\xi}>0$ and $\frac{d^2\varrho}{d\xi^2}>0$, we complete the proof. Lemma \[varrho\_lemma\] indicates the maximum value of $\varrho(\xi)$ is achieved at $\xi=1$, which is $\varrho_{max}=\varrho(1)=L^{{1}/{\delta}}$ from . Besides, it is clearly that $\mathcal{Z}(\xi,\varrho) - L$ monotonically increases with $\varrho$ for a given $\xi$. Generally, we can calculate the unique value of $\varrho(\xi)$ that satisfies using the bisection method in the range $[0,\varrho_{max}]$. For the special case of large antennas, i.e., $N\rightarrow\infty$, we provide an approximate value of $\varrho(\xi)$, denoted as $\varrho^{o}(\xi)$. Simulation results show that, when $N\ge 20$, the maximum value of $R_S$ calculated based on $\varrho^{o}(\xi)$ is quite close to that based on the exact $\varrho(\xi)$, i.e., $\varrho^{o}(\xi)$ can be a computationally convenient alternative to $\varrho(\xi)$ when $N$ is large. *As $N\rightarrow\infty$, $\varrho(\xi)$ in approximates to $$\begin{aligned} \label{varrho} \varrho^{o}(\xi) = \begin{cases} \qquad L^{{1}/{\delta}}, & \xi = 1\\ \frac{\delta}{1-\xi}\ln\left( \frac{{\delta}^{-1}(1-\xi)L ^{{1}/{\delta}}} {\mathcal{W}\left({\delta}^{-1}(1-\xi)L ^{{1}/{\delta}}\right)}\right), & \text{otherwise} \end{cases}\end{aligned}$$ where $\mathcal{W}(\cdot)$ is the Lambert-W function.* Since $\lim_{N\rightarrow\infty}\left(1+\frac{x}{N}\right)^{-N} = e^{-x}$, we have $\mathcal{Z}(\xi,\varrho^o)= (\varrho^o)^{\delta}e^{(1-\xi)\varrho^o}$ where $\varrho^o\triangleq \lim_{N\rightarrow\infty}\varrho$, and transforms to $L= (\varrho^o)^{\delta}e^{(1-\xi)\varrho^o}$. 1) When $\xi=1$, we easily obtain $\varrho^o= L^{\frac{1}{\delta}}$. 2) When $\xi\ne 1$, we find that $\frac{1-\xi}{\delta}L^{\frac{1}{\delta}}= \frac{(1-\xi)\varrho^o}{\delta} e^{\frac{(1-\xi)\varrho^o}{\delta}}$. Let $\mu\triangleq \frac{1-\xi}{\delta}\varrho^o$, and we obtain $\frac{1-\xi}{\delta}L^{\frac{1}{\delta}}=\mu e^\mu\Rightarrow e^{\frac{1-\xi}{\delta}L^{\frac{1}{\delta}}}=e^{\mu e^\mu}$. We further let $\nu\triangleq e^\mu$ and $t\triangleq e^{\frac{1-\xi}{\delta}L^{\frac{1}{\delta}}}$, such that $\nu^\nu = t\Rightarrow\nu=\frac{\ln t}{\mathcal{W}(\ln t)}$. The solution $\varrho^o$ can be given by $\varrho^o = \frac{1-\xi}{\delta}\mu= \frac{1-\xi}{\delta}\ln \nu$, with yields the final expression in by substituting in $\nu$ along with $t$. Due to the implicit function of $\varrho(\xi)$ on $\xi$, we can hardly derive an explicit expression of $R_S$. Nevertheless, we still reveal the concavity of $R_S$ on $\xi$, and provide the solution to problem in the following theorem. \[opt\_par\_rs\_theorem\] *$R_S$ in is a concave function of $\xi$. The optimal $\xi$ that maximizes $R_S$ is given by $$\begin{aligned} \label{opt_par_rs} \xi^*=\begin{cases} ~\varnothing, & \kappa\le\varrho_{min}\\ ~1,& \kappa>\frac{\delta L^{{\alpha}/{2}}+L^{\alpha}} {\delta-L^{\alpha}} ~\textrm{and}~L<\sqrt[\alpha]{\delta}\\ ~\xi_r,&\textrm{otherwise} \end{cases}\end{aligned}$$ where $\varrho_{min}\triangleq \varrho(0)$ denotes the minimum value of $\varrho(\xi)$. $\xi_r$ is the unique root of $\frac{dR_S}{d\xi}=0$, where $$\label{drs} \frac{dR_S}{d\xi}=\frac{1}{\ln2}\left[\frac{\kappa}{1+\kappa\xi} -\frac{\varrho(\xi)+\frac{\xi d\varrho(\xi)}{d\xi}} {1+\xi\varrho(\xi)} \right].$$* Alice transmits only when $\varrho<\kappa$. Obviously, if $\kappa\le \varrho_{min}$, then $\varrho<\kappa$ never holds for an arbitrary $\varrho$ since $\varrho_{min}\le\varrho$, such that transmission is suspended. If $\kappa>\varrho_{min}$, Alice transmits for a $\xi$ that satisfies $\varrho<\kappa$. To maximize $R_S$, we first give the second-order derivative of $R_S$ on $\xi$ from $$\label{drs2} \frac{d^2R_S}{d\xi^2}=\frac{1}{\ln2} \left[\frac{-\kappa^2}{(1+\kappa\xi)^2} -\frac{2\frac{d\varrho}{d\xi} +\xi\frac{d^2\varrho}{d\xi^2}}{(1+\varrho\xi)} +\frac{\left(\varrho+\xi\frac{d\varrho}{d\xi}\right)^2}{(1+\varrho\xi)^2} \right],\nonumber$$ with $\frac{d\varrho}{d\xi}$ and $\frac{d^2\varrho}{d\xi^2}$ given in Lemma \[varrho\_lemma\]. Substituting $\frac{d^2\varrho}{d\xi^2} >\frac{2}{\varrho}\left(\frac{d\varrho}{d\xi}\right)^2>0$ (see Lemma \[varrho\_lemma\]) into the above equation yields $$\label{drs22} \frac{d^2R_S}{d\xi^2}<-\frac{1}{\ln2} \left(\frac{\kappa^2}{(1+\kappa\xi)^2} -\frac{\varrho^2}{(1+\varrho\xi)^2}\right).$$ Since $\varrho<\kappa$, we have $\frac{\kappa^2}{(1+\kappa\xi)^2} -\frac{\varrho^2}{(1+\varrho\xi)^2}>0 \Rightarrow\frac{d^2R_S}{d\xi^2}<0$, i.e., $R_S$ is a *concave* function of $\xi$. Due to the concavity of $R_S$ on $\xi$, the maximum value of $R_S$ is achieved either at boundaries or at stationary points. From , the boundary values are $\frac{dR_S}{d\xi}|_{\xi=0} =\frac{\kappa-\varrho_{min}}{\ln2}$ and $\frac{dR_S}{d\xi}|_{\xi=1}=\frac{1}{\ln2} \left(\frac{\kappa}{1+\kappa} -\frac{L^{{\alpha}/{2}}+\frac{\alpha}{2} L^{\alpha}} {1+L^{{\alpha}/{2}}}\right)$. Obviously, $\frac{dR_S}{d\xi}|_{\xi=0}>0$. 1) If $\frac{dR_S}{d\xi}|_{\xi=1}>0$, $R_S$ monotonically increases with $\xi$, and the optimal value of $\xi$ is 1, with the corresponding condition directly obtained from $\frac{dR_S}{d\xi}|_{\xi=1}>0$. 2) If $\frac{dR_S}{d\xi}|_{\xi=1}\le0$, $R_S$ first increases and then decreases with $\xi$, and the optimal value of $\xi$ is the unique root of $\frac{dRs}{d\xi}=0$. Theorem \[opt\_par\_rs\_theorem\] shows only for a large $\kappa$ (small estimation error) and a small $L$ (a sparse-eavesdropper scenario or a moderate SOP constraint), allocating full power to the information signal provides a higher secrecy rate than the AN scheme does, otherwise generating AN is advantageous. Since $R_S$ is a concave function of $\xi$, we can efficiently calculate the unique root $\xi_r$ of $\frac{dR_S}{d\xi_r}=0$ in using the bisection method. Substituting $\xi^*$ and $\varrho(\xi^*)$ into yields $R_S^*$. Although $\xi_r$ can only be calculated numerically, we show how $\xi_r$ is affected by $\kappa$ in the following. \[opt\_par\_tau\_proposition2\] *$\xi_r$ in monotonically increases with $\kappa$.* From , $\frac{dR_S}{d\xi_r}=0$ transforms to $\mathcal{A}(\xi_r) = 0$, and $$\label{A} \mathcal{A}(\xi_r)= (\kappa\xi_r^2-l_0\xi_r+l_2)\varrho_r^2 +\left(l_2\kappa\xi_r-l_2\kappa+\delta\right)\varrho_r -\delta\kappa,$$ with $\varrho_r\triangleq \varrho(\xi_r)$. Using the derivative rule for implicit functions with the equation $\mathcal{A}(\xi_r) = 0$ yields $$\label{dpar_dkappa} \frac{d\xi_r}{d\kappa}=-\frac{{\partial \mathcal{A}}/{\partial\kappa}}{{\partial \mathcal{A}}/{\partial\xi_r}} =-\frac{\varrho_r^2\xi_r^2-\left(\delta +l_2(1-\xi_r)\varrho_r\right)} {\psi_1(\xi_r)+\psi_2(\xi_r)\frac{d\varrho_r}{d\xi_r}},$$ where $\psi_1(\xi_r)=(1+2\kappa\xi_r)\varrho_r^2 +l_2(\kappa-\varrho_r)\varrho_r$ and $\psi_2(\xi_r) =2\left(\kappa\xi_r^2+l_2\right)\varrho_r+ \left(l_2\kappa\xi_r+\delta\right) -2l_0\xi_r\varrho_r-l_2\kappa$. Obviously, $\kappa>\varrho_r\Rightarrow\psi_1(\xi_r)>0$ and $\frac{d\varrho_r}{d\xi_r}>0$ (see Lemma \[varrho\_lemma\]). $\mathcal{A}(\xi_r) = 0$ can be further reformed as $\left(\kappa\xi_r^2+l_2)\varrho_r +(l_2\kappa\xi_r+\delta\right) =l_0\xi_r\varrho_r+l_2\kappa+\frac{\delta\kappa}{\varrho_r}$, substituting which into $\psi_2(\xi_r)$ directly yields $\psi_2(\xi_r)>0$. Hence we have $\frac{\partial \mathcal{A}}{\partial\xi_r}>0$. Leveraging , $\frac{dR_S}{d\xi_r}=0$ can be reformed by $\xi_r^2\frac{d\varrho_r}{d\xi_r}=1-\frac{1+\varrho_r\xi_r} {1+\kappa\xi_r}<1$. Substituting $\frac{d\varrho_r}{d\xi_r}$ in into this inequality yields $\varrho_r^2\xi_r^2<\left(\delta+l_2(1-\xi_r)\varrho_r\right)$, i.e., $\frac{\partial \mathcal{A}}{\partial\kappa}<0$. With $\frac{\partial \mathcal{A}}{\partial\xi_r}>0$ and $\frac{\partial \mathcal{A}}{\partial\kappa}<0$, we see from that $\frac{d\xi_r}{d\kappa}>0$, which completes the proof. Proposition \[opt\_par\_tau\_proposition2\] indicates that, *when channel estimation error becomes larger, if we aim to increase the secrecy rate under a SOP constraint, we should increase the AN power*, just as shown in Fig. \[PAR\_RS\_EP\_LE\]. The reason is: channel estimation error heavily degrades the main channel while has no effect on the wiretap channels. For a large estimation error, although increasing the information signal power improves Bob’s capacity, the improvement is not significant. On the contrary, increasing AN power always greatly deteriorates the wiretap channels regardless of CSI imperfection. Therefore, when estimation error becomes larger, increasing AN power is more beneficial to the secrecy rate than increasing signal power. Nevertheless, transmission is suspended if $\tau$ exceeds a certain value, which corresponds to the case $\kappa\le\varrho_{min}$ as indicated in Theorem \[opt\_par\_rs\_theorem\]. We can also prove $\frac{d\xi_r}{d\lambda_E}<0$ and $\frac{d\xi_r}{d\epsilon}>0$ in a similar way as the proof of Proposition \[opt\_par\_tau\_proposition2\]. Due to space limit, we omit the relevant proofs, and the results are verified in Fig. \[PAR\_RS\_EP\_LE\]. We see that the optimal PAR $\xi^*$ decreases for a larger $\lambda_E$ or a smaller $\epsilon$. It means that, when transmission is more vulnerable to wiretapping, we should increase AN power. Fig. \[RS\_TAU\_P\] depicts the maximum secrecy rate $R_S^*$ versus $\tau$. The approximated value of $R_S^*$ is quite close to the exact one. We observe that $R_S^*$ monotonically decreases with $\tau$. Interestingly, $R_S^*$ increases with $P$ at the small $\tau$ region, whereas decreases with it at the large $\tau$ region. The underlying reason is just similar to the explanation for the intersection in Fig. \[SOP\_TAU\_P\_RS\]. Conclusions =========== In this correspondence, we investigate the AN-aided multi-antenna transmission under imperfect CSI against PPP Eves. We provide explicit solutions of the optimal PARs with channel estimation errors for minimizing the SOP under a secrecy rate constraint and for maximizing the secrecy rate subject to a SOP constraint, respectively. 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Geraci, H. S. Dhillon, J. G. Amdrews, J. Yuan, and I. B. Collings, “Physical layer security in downlink multi-antenna cellular networks," *IEEE Trans. Commun.*, vol. 62, no. 6, pp. 2006-2021, June 2014. B. Hassibi and B. M. Hochwald, “How much training is needed in multiple-antenna wireless links?” *IEEE Trans. Inform. Theory*, vol. 49, no. 4, pp. 951-963, Apr. 2003. X. Zhou, P. Sadeghi, T. A. Lamahewa, and S. Durrani, “Design guidelines for training-based MIMO systems with feedback," *IEEE Trans. Signal Process.*, vol. 57, no. 10, pp. 4014-4026, Oct. 2009. Y. Liang, G. Kramer, H. V. Poor, and S. Shamai, “Compound wiretap channels," *EURASIP J. Wireless Commun. Network.*, 2009. K. Jittorntrum, “An implicit function theorem," *J. of Optimization Theory and Applications*, vol. 25, no. 4, pp. 575-577, 1978. D. Stoyan, W. Kendall, and J. Mecke, *Stochastic Geometry and its Applications, 2nd ed*. John Wiley and Sons, 1996. [^1]: ©2015 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to pubs-permissions@ieee.org. [^2]: This work was partially supported by the Foundation for the Author of National Excellent Doctoral Dissertation of China under Grant 201340, the National High-Tech Research and Development Program of China under Grant No. 2015AA011306, the New Century Excellent Talents Support Fund of China under Grant NCET-13-0458, the Fok Ying Tong Education Foundation under Grant 141063, the Fundamental Research Funds for the Central University under Grant No. 2013jdgz11, and the Young Talent Support Fund of Science and Technology of Shaanxi Province under Grant 2015KJXX-01. The review of this paper was coordinated by Prof. G. Mao. [^3]: The authors are with the School of Electronics and Information Engineering, and also with the MOE Key Lab for Intelligent Networks and Network Security, Xi¡¯an Jiaotong University, Xi’an, 710049, Shaanxi, China (e-mail: txzheng@stu.xjtu.edu.cn; xjbswhm@gmail.com). H.-M. Wang is the corresponding author. [^4]: This may correspond to such a scenario that a multi-antenna transmitter Alice provides specific service to a specified subscriber Bob, while the service should be kept secret to eavesdroppers (also named unauthorized users). [^5]: The Gaussian error model is a stochastic uncertainty model. Another widely used model is the deterministic bounded error model, which is more convenient for analyzing the quantized CSI [@Zhang2015Artificial]. [^6]: The tightness of this capacity lower bound was verified for Gaussian inputs with MMSE channel estimation in [@Hassibi2003How; @Zhou2009Design].

--- abstract: 'We generalize the standard quantum adiabatic approximation to the case of open quantum systems. We define the adiabatic limit of an open quantum system as the regime in which its dynamical superoperator can be decomposed in terms of independently evolving Jordan blocks. We then establish validity and invalidity conditions for this approximation and discuss their applicability to superoperators changing slowly in time. As an example, the adiabatic evolution of a two-level open system is analyzed.' author: - 'M.S. Sarandy' - 'D.A. Lidar' title: Adiabatic approximation in open quantum systems --- Introduction ============ The adiabatic theorem [@Born:28; @Kato:50; @Messiah:book] is one of the oldest and most useful general tools in quantum mechanics. The theorem posits, roughly, that if a state is an instantaneous eigenstate of a sufficiently slowly varying Hamiltonian $H$ at one time, then it will remain an eigenstate at later times, while its eigenenergy evolves continuously. Its role in the study of slowly varying quantum mechanical systems spans a vast array of fields and applications, such as energy-level crossings in molecules [@Landau:32; @Zener:32], quantum field theory [@Gellmann:51], and geometric phases [Berry:84,Wilczek:84]{}. In recent years, geometric phases have been proposed to perform quantum information processing [@ZanardiRasseti:99; @ZanardiRasseti:2000; @Ekert-Nature], with adiabaticity assumed in a number of schemes for geometric quantum computation (e.g., [@Pachos:00; @Duan-Science:01; @Pachos:02; @Fazio:03]). Moreover, additional interest in adiabatic processes has arisen in connection with the concept of adiabatic quantum computing, in which slowly varying Hamiltonians appear as a promising mechanism for the design of new quantum algorithms and even as an alternative to the conventional quantum circuit model of quantum computation [@Farhi:00; @Farhi:01]. Remarkably, the notion of adiabaticity does not appear to have been extended in a systematic manner to the arena of *open* quantum systems, i.e., quantum systems coupled to an external environment [@Breuer:book]. Such systems are of fundamental interest, as the notion of a closed system is always an idealization and approximation. This issue is particularly important in the context of quantum information processing, where environment-induced decoherence is viewed as a fundamental obstacle on the path to the construction of quantum information processors (e.g., [@LidarWhaley:03]). The aim of this work is to systematically generalize the concept of adiabatic evolution to the realm of open quantum systems. Formally, an open quantum system is described as follows. Consider a quantum system $S$ coupled to an environment, or bath $B$ (with respective Hilbert spaces $\mathcal{H}_{S},\mathcal{H}_{B}$), evolving unitarily under the total system-bath Hamiltonian $H_{SB}$. The exact system dynamics is given by tracing over the bath degrees of freedom [@Breuer:book] $$\rho (t)=\mathrm{Tr}_{B}[U(t)\rho _{SB}(0)U^{\dag }(t)], \label{system}$$where $\rho (t)$ is the system state, $\rho _{SB}(0)=\rho (0)\otimes \rho _{B}(0)$ is the initially uncorrelated system-bath state, and $U(t)=\mathcal{T}\mathsf{\exp }[-i\int_{0}^{t}H_{SB}(t^{\prime })dt^{\prime }]$ ($\mathcal{T}$ denotes time-ordering; we set $\hbar =1$). Such an evolution is completely positive and trace preserving [@Breuer:book; @Kraus:71; @Alicki:87]. Under certain approximations, it is possible to convert Eq. (\[system\]) into the convolutionless form $$\begin{aligned} {\dot{\rho}}(t) &=& \mathcal{L}(t) \rho (t). \label{eq:t-Lind}\end{aligned}$$ An important example is $$\begin{aligned} {\dot{\rho}}(t) &=& -i\left[ H(t),\rho (t) \right] +\frac{1}{2} \sum_{i=1}^{N}\left([\Gamma _{i}(t),\rho (t) \Gamma^{\dagger }_{i}(t)] \right. \nonumber \\ &&\left.+[\Gamma_{i}(t)\rho (t), \Gamma^{\dagger }_{i}(t)]\right). \label{eq:t-Lind2}\end{aligned}$$Here $H(t)$ is the time-dependent effective Hamiltonian of the open system and $\Gamma _{i}(t)$ are time-dependent operators describing the system-bath interaction. In the literature, Eq. (\[eq:t-Lind2\]) with time-*in*dependent operators $\Gamma _{i}$ is usually referred to as the Markovian dynamical semigroup, or Lindblad equation [@Breuer:book; @Alicki:87; @Gorini:76; @Lindblad:76] \[see also Ref. [@Lidar:CP01] for a simple derivation of Eq. (\[eq:t-Lind2\]) from Eq. (\[system\])\]. However, the case with time-dependent coefficients is also permissible under certain restrictions [@Lendi:86]. The Lindblad equation requires the assumption of a Markovian bath with vanishing correlation time. Equation (\[eq:t-Lind\]) can be more general; for example, it applies to the case of non-Markovian convolutionless master equations studied in Ref. [@Breuer:04]. In this work we will consider the class of convolutionless master equations (\[eq:t-Lind\]). In a slight abuse of nomenclature, we will henceforth refer to the time-dependent generator $\mathcal{L}(t)$ as the Lindblad superoperator, and the $\Gamma _{i}(t)$ as Lindblad operators. Returning to the problem of adiabatic evolution, conceptually, the difficulty in the transition from closed to open systems is that the notion of Hamiltonian eigenstates is lost, since the Lindblad superoperator – the generalization of the Hamiltonian – cannot in general be diagonalized. It is then not *a priori* clear what should take the place of the adiabatic eigenstates. Our key insight in resolving this difficulty is that this role is played by *adiabatic Jordan blocks of the Lindblad superoperator*. The Jordan canonical form [@Horn:book], with its associated left and right eigenvectors, is in this context the natural generalization of the diagonalization of the Hamiltonian. Specifically, we show that, for slowly varying Lindblad superoperators, the time evolution of the density matrix, written in a suitable basis in the state space of linear operators, occurs separately in sets of Jordan blocks related to each Lindblad eigenvalue. This treatment for adiabatic processes in open systems is potentially rather attractive as it can simplify the description of the dynamical problem by breaking down the Lindblad superoperator into a set of decoupled blocks. In order to clearly exemplify this behavior, we analyze a simple two-level open system for which the exact solution of the master equation (\[eq:t-Lind\]) can be analytically determined. The paper is organized as follows. We begin, in Sec. \[closed\], with a review of the standard adiabatic approximation for closed quantum systems. In Sec. \[open\] we describe the general dynamics of open quantum systems, review the superoperator formalism, and introduce a strategy to find suitable bases in the state space of linear operators. Section \[adiabatic\] is devoted to deriving our adiabatic approximation, including the conditions for its validity. In Sec. \[example\], we provide a concrete example which illustrates the consequences of the adiabatic behavior for systems in the presence of decoherence. Finally, we present our conclusions in Sec. \[conclusions\]. The adiabatic approximation in closed quantum systems {#closed} ===================================================== Condition on the Hamiltonian ---------------------------- To facilitate comparison with our later derivation of the adiabatic approximation for open systems, let us begin by reviewing the adiabatic approximation in closed quantum systems, subject to unitary evolution. In this case, the evolution is governed by the time-dependent Schrödinger equation $$H(t)\,|\psi (t)\rangle =i\,|{\dot{\psi}}(t)\rangle , \label{se}$$where $H(t)$ denotes the Hamiltonian and $|\psi (t)\rangle $ is a quantum state in a $D$-dimensional Hilbert space. For simplicity, we assume that the spectrum of $H(t)$ is entirely discrete and nondegenerate. Thus we can define an instantaneous basis of eigenenergies by $$H(t)\,|n(t)\rangle =E_{n}(t)\,|n(t)\rangle , \label{ebh}$$with the set of eigenvectors [$|n(t)\rangle $]{} chosen to be orthonormal. In this simplest case, where to each energy level there corresponds a unique eigenstate, *adiabaticity is then defined as the regime associated to an independent evolution of the instantaneous eigenvectors of* $H(t)$. This means that instantaneous eigenstates at one time evolve continuously to the corresponding eigenstates at later times, and that their corresponding eigenenergies do not cross. In particular, if the system begins its evolution in a particular eigenstate $|n(0)\rangle$, then it will evolve to the instantaneous eigenstate $|n(t)\rangle $ at a later time $t$, without any transition to other energy levels. In order to obtain a general validity condition for adiabatic behavior, let us expand $|\psi (t)\rangle $ in terms of the basis of instantaneous eigenvectors of $H(t)$, $$|\psi (t)\rangle =\sum_{n=1}^{D}a_{n}(t)\,e^{-i\int_{0}^{t}dt^{\prime }E_{n}(t^{\prime })}\,|n(t)\rangle , \label{ep}$$with $a_{n}(t)$ being complex functions of time. Substitution of Eq. ([ep]{}) into Eq. (\[se\]) yields $$\sum_{n}\left( {\dot{a}}_{n}|n\rangle +a_{n}|{\dot{n}}\rangle \right) \,e^{-i\int_{0}^{t}dt^{\prime }E_{n}(t^{\prime })}=0, \label{an1}$$where use has been made of Eq. (\[ebh\]). Multiplying Eq. (\[an1\]) by $\langle k(t)|$, we have $${\dot{a}}_{k}=-\sum_{n}a_{n}\langle k|{\dot{n}}\rangle \,e^{-i\int_{0}^{t}dt^{\prime }g_{nk}(t^{\prime })}, \label{an2}$$where $$g_{nk}(t)\equiv E_{n}(t)-E_{k}(t). \label{eq:g}$$A useful expression for $\langle k|{\dot{n}}\rangle $, for $k\neq n$, can be found by taking the time derivative of Eq. (\[ebh\]) and multiplying the resulting expression by $\langle k|$, which reads $$\langle k|{\dot{n}}\rangle =\frac{\langle k|{\dot{H}}|n\rangle }{g_{nk}}\quad (n\neq k). \label{knee}$$Therefore, Eq. (\[an2\]) can be written as $${\dot{a}}_{k}=-a_{k}\langle k|{\dot{k}}\rangle -\sum_{n\neq k}a_{n}\frac{\langle k|{\dot{H}}|n\rangle }{g_{nk}}\,e^{-i\int_{0}^{t}dt^{\prime }g_{nk}(t^{\prime })}. \label{anf}$$Adiabatic evolution is ensured if the coefficients $a_{k}(t)$ evolve independently from each other, i.e., if their dynamical equations do not couple. As is apparent from Eq. (\[anf\]), this requirement is fulfilled by imposing the conditions $$\max_{0\le t\le T} \left\vert \frac{\langle k|{\dot{H}}|n\rangle }{g_{nk}}\right\vert \,\ll\, \min_{0\le t\le T} \left\vert{g_{nk}}\right\vert, \label{vcc}$$ which serves as an estimate of the validity of the adiabatic approximation, where $T$ is the total evolution time. Note that the left-hand side of Eq. (\[vcc\]) has dimensions of frequency and hence must be compared to the relevant physical frequency scale, given by the gap $g_{nk}$ [@Messiah:book; @Mostafazadeh:book]. For a discussion of the adiabatic regime when there is no gap in the energy spectrum see Refs. [@Avron:98; @Avron:99]. In the case of a degenerate spectrum of $H(t)$, Eq. (\[knee\]) holds only for eigenstates $|k\rangle $ and $|n\rangle $ for which $E_{n}\neq E_{k}$. Taking into account this modification in Eq. (\[anf\]), it is not difficult to see that the adiabatic approximation generalizes to the statement that each degenerate eigenspace of $H(t)$, instead of individual eigenvectors, has independent evolution, whose validity conditions given by Eq. (\[vcc\]) are to be considered over eigenvectors with distinct energies. Thus, in general one can define adiabatic dynamics of closed quantum systems as follows: \[defc\] A closed quantum system is said to undergo adiabatic dynamics if its Hilbert space can be decomposed into decoupled Schrödinger eigenspaces with distinct, time-continuous, and noncrossing instantaneous eigenvalues of $H(t)$. It is conceptually useful to point out that the relationship between slowly varying Hamiltonians and adiabatic behavior, which explicitly appears from Eq. (\[vcc\]), can also be demonstrated directly from a simple manipulation of the Schrödinger equation: recall that $H(t)$ can be diagonalized by a unitary similarity tranformation $$H_{d}(t)=U^{-1}(t)\,H(t)\,U(t), \label{hdc}$$where $H_{d}(t)$ denotes the diagonalized Hamiltonian and $U(t)$ is a unitary transformation. Multiplying Eq. (\[se\]) by $U^{-1}(t)$ and using Eq. (\[hdc\]), we obtain $$H_{d}\,|\psi \rangle _{d}=i\,|{\dot{\psi}}\rangle _{d}-i\,{\dot{U}}^{-1}|\psi \rangle , \label{sed}$$where $|\psi \rangle _{d}\equiv U^{-1}|\psi \rangle $ is the state of the system in the basis of eigenvectors of $H(t)$. Upon considering that $H(t)$ changes slowly in time, i.e., $dH(t)/dt\approx 0$, we may also assume that the unitary transformation $U(t)$ and its inverse $U^{-1}(t)$ are slowly varying operators, yielding $$H_{d}(t)\,|\psi (t)\rangle _{d}=i\,|{\dot{\psi}}(t)\rangle _{d}. \label{eq:Had}$$Thus, since $H_{d}(t)$ is diagonal, the system evolves separately in each energy sector, ensuring the validity of the adiabatic approximation. In our derivation of the condition of adiabatic behavior for open systems below, we will make use of this semi-intuitive picture in order to motivate the decomposition of the dynamics into Lindblad-Jordan blocks. Condition on the total evolution time ------------------------------------- The adiabaticity condition can also be given in terms of the total evolution time $T$. We shall consider for simplicity a nondegenerate $H(t)$; the generalization to the degenerate case is possible. Let us then rewrite Eq. (\[anf\]) as follows [@Gottfried:book]: $$e^{i\gamma _{k}(t)}\,\frac{\partial }{\partial t}[ a_{k}(t)\,e^{-i\gamma _{k}(t)}] = -\sum_{n\neq k}a_{n}\frac{\langle k|{\dot{H}}|n\rangle }{g_{nk}}\,e^{-i\int_{0}^{t}dt^{\prime }g_{nk}(t^{\prime })}, \label{adtti}$$ where $\gamma _{k}(t)$ denotes the Berry’s phase [@Berry:84] associated to the state $|k\rangle $, $$\gamma _{k}(t)=i\int_{0}^{t}dt^{\prime }\langle k(t^{\prime })|{\dot{k}}(t^{\prime })\rangle .$$Now let us define a normalized time $s$ through the variable transformation $$t=sT,\,\,\,\,\,0\leq s\leq 1. \label{nt}$$Then, by performing the change $t\rightarrow s$ in Eq. (\[adtti\]) and integrating, we obtain $$\begin{aligned} &&a_{k}(s)\,e^{-i\gamma _{k}(s)}= \nonumber \\ &&a_{k}(0)-\sum_{n\neq k}\int_{0}^{s}ds^{\prime }\frac{F_{nk}(s^{\prime })}{g_{nk}(s^{\prime })}e^{-iT\int_{0}^{s^{\prime }}ds^{\prime \prime }g_{nk}(s^{\prime \prime })}, \label{akint}\end{aligned}$$where $$\begin{aligned} F_{nk}(s)=a_{n}(s)\,\langle k(s)|\frac{dH(s)}{ds}|n(s)\rangle \,e^{-i\gamma _{k}(s)}. \end{aligned}$$ However, for an adiabatic evolution as defined above, the coefficients $a_{n}(s)$ evolve without any mixing, which means that $a_{n}(s)\approx a_{n}(0)\,e^{i\gamma _{n}(s)}$. Therefore, $$\begin{aligned} F_{nk}(s)=a_{n}(0)\,\langle k(s)|\frac{dH(s)}{ds}|n(s)\rangle \,e^{-i[\gamma _{k}(s)-\gamma _{n}(s)]}. \end{aligned}$$ In order to arrive at a condition on $T$, it is useful to separate out the fast oscillatory part from Eq. (\[akint\]). Thus, the integrand in Eq. (\[akint\]) can be rewritten as $$\begin{aligned} &&\frac{F_{nk}(s^{\prime })}{g_{nk}(s^{\prime })}e^{-iT\int_{0}^{s^{\prime }}ds^{\prime \prime }g_{nk}(s^{\prime \prime })}= \nonumber \\ &&\frac{i}{T}\left[ \frac{d}{ds^{\prime }}\left( \frac{F_{nk}(s^{\prime })}{g_{nk}^{2}(s^{\prime })}e^{-iT\int_{0}^{s^{\prime }}ds^{\prime \prime }g_{nk}(s^{\prime \prime })}\right) \right. \nonumber \\ &&\left. -\,e^{-iT\int_{0}^{s^{\prime }}ds^{\prime \prime }g_{nk}(s^{\prime \prime })}\frac{d}{ds^{\prime }}\left( \frac{F_{nk}(s^{\prime })}{g_{nk}^{2}(s^{\prime })}\right) \right] . \label{ricc}\end{aligned}$$Substitution of Eq. (\[ricc\]) into Eq. (\[akint\]) results in $$\begin{aligned} &&a_{k}(s)\,e^{-i\gamma _{k}(s)}= \nonumber \\ &&a_{k}(0)+\frac{i}{T}\sum_{n\neq k}\left( \frac{F_{nk}(0)}{g_{nk}^{2}(0)}-\frac{F_{nk}(s)}{g_{nk}^{2}(s)}e^{-iT\int_{0}^{s}ds^{\prime }g_{nk}(s^{\prime })}\right. \nonumber \\ &&\left. +\,\int_{0}^{s}ds^{\prime }\,e^{-iT\int_{0}^{s^{\prime }}ds^{\prime \prime }g_{nk}(s^{\prime \prime })}\frac{d}{ds^{\prime }}\frac{F_{nk}(s^{\prime })}{g_{nk}^{2}(s^{\prime })}\right). \label{akfinal}\end{aligned}$$A condition for the adiabatic regime can be obtained from Eq. (\[akfinal\]) if the integral in the last line vanishes for large $T$. Let us assume that, as $T\rightarrow \infty $, the energy difference remains nonvanishing. We further assume that $d\{F_{nk}(s^{\prime })/g_{nk}^{2}(s^{\prime })\}/ds^{\prime }$ is integrable on the interval $\left[ 0,s\right] $. Then it follows from the Riemann-Lebesgue lemma [@Churchill:book] that the integral in the last line of Eq. (\[akfinal\]) vanishes in the limit $T\rightarrow \infty $ (due to the fast oscillation of the integrand) [@RiemannLebesgue]. What is left are therefore only the first two terms in the sum over $n\neq k$ of Eq. (\[akfinal\]). Thus, a general estimate of the time rate at which the adiabatic regime is approached can be expressed by $$\begin{aligned} T\gg \frac{F}{g^{2}}, \label{timead}\end{aligned}$$ where $$\begin{aligned} &&F=\max_{0\leq s\leq 1}|a_{n}(0)\,\langle k(s)|\frac{dH(s)}{ds}|n(s)\rangle |, \nonumber \\ &&g=\min_{0\leq s\leq 1}|g_{nk}(s)|\, ,\end{aligned}$$with $\max $ and $\min $ taken over all $k$ and $n$. A simplification is obtained if the system starts its evolution in a particular eigenstate of $H(t)$. Taking the initial state as the eigenvector $|m(0)\rangle $, with $a_{m}(0)=1$, adiabatic evolution occurs if $$\begin{aligned} T\gg \frac{\mathcal{F}}{\mathcal{G}^{2}}, \label{timead2}\end{aligned}$$ where $$\begin{aligned} &&\mathcal{F}=\max_{0\leq s\leq 1}|\langle k(s)|\frac{dH(s)}{ds}|m(s)\rangle |\,, \nonumber \\ &&\mathcal{G}=\min_{0\leq s\leq 1}|g_{mk}(s)|\,.\end{aligned}$$Equation (\[timead2\]) gives an important validity condition for the adiabatic approximation, which has been used, e.g., to determine the running time required by adiabatic quantum algorithms [@Farhi:00; @Farhi:01]. The dynamics of open quantum systems {#open} ==================================== In this section, we prepare the mathematical framework required to derive an adiabatic approximation for open quantum systems. Our starting point is the convolutionless master equation (\[eq:t-Lind\]). It proves convenient to transform to the superoperator formalism, wherein the density matrix is represented by a $D^{2}$-dimensional coherence vector  $$|\rho \rangle \rangle =\left( \begin{array}{cccc} \rho _{1} & \rho _{2} & \cdots & \rho _{D^{2}}\end{array}\right) ^{t}, \label{vcv}$$and the Lindblad superoperator $\mathcal{L}$ becomes a $(D^{2}\times D^{2})$-dimensional supermatrix [@Alicki:87]. We use the double bracket notation to indicate that we are not working in the standard Hilbert space of state vectors. Such a representation can be generated, e.g., by introducing a basis of Hermitian, trace-orthogonal, and traceless operators \[e.g., su($D$)\], whence the $\rho _{i}$ are the expansion coefficients of $\rho $ in this basis [@Alicki:87], with $\rho _{1}$ the coefficient of $I$ (the identity matrix). In this case, the condition $\mathrm{Tr}\rho ^{2}\leq 1$ corresponds to $\left\Vert |\rho \rangle \rangle \right\Vert \leq 1$, $\rho =\rho ^{\dag }$ to $\rho _{i}=\rho _{i}^{\ast }$, and positive semidefiniteness of $\rho $ is expressed in terms of inequalities satisfied by certain Casimir invariants \[e.g., of $su(D)$\] [@byrd:062322; @Kimura:2003-1; @Kimura:2003-2]. A simple and well-known example of this procedure is the representation of the density operator of a two-level system (qubit) on the Bloch sphere, via $\rho =(I_{2}+\overrightarrow{v}\cdot \overrightarrow{\sigma })/2$, where $\overrightarrow{\sigma }=(\sigma _{x},\sigma _{y},\sigma _{z})$ is the vector of Pauli matrices, $I_{2}$ is the $2\times 2$ identity matrix, and $\overrightarrow{v}\in \mathbb{R}^{3}$ is a three-dimensional coherence vector of norm$\leq 1$. More generally, coherence vectors live in Hilbert-Schmidt space: a state space of linear operators endowed with an inner product that can be defined, for general vectors $u$ and $v$, as $$(u,v)\equiv \langle \langle u|v\rangle \rangle \equiv \frac{1}{{\mathcal{N}}}{\text{Tr}}\left( u^{\dagger }v\right) , \label{ip}$$where ${\mathcal{N}}$ is a normalization factor. Adjoint elements $\langle \langle v|$ in the dual state space are given by row vectors defined as the transpose conjugate of $|v\rangle \rangle $: $\langle \langle v|=(v_{1}^{\ast },v_{2}^{\ast },...,v_{D^{2}}^{\ast })$. A density matrix can then be expressed as a discrete superposition of states over a complete basis in this vector space, with appropriate constraints on the coefficients so that the requirements of Hermiticity, positive semidefiniteness, and unit trace of $\rho $ are observed. Thus, representing the density operator in general as a coherence vector, we can rewrite Eq. (\[eq:t-Lind\]) in a superoperator language as $$\mathcal{L}(t)\,|\rho (t)\rangle \rangle =|{\dot{\rho}}(t)\rangle \rangle , \label{le}$$where $\mathcal{L}$ is now a supermatrix. This master equation generates nonunitary evolution, since $\mathcal{L}(t)$ is non-Hermitian and hence generally nondiagonalizable. However, it is always possible to obtain an elegant decomposition in terms of a block structure, the Jordan canonical form [@Horn:book]. This can be achieved by the similarity transformation $$\mathcal{L}_{J}(t)=S^{-1}(t)\,\mathcal{L}(t)\,S(t), \label{jd}$$where $\mathcal{L}_{J}(t)=\mathrm{diag}(J_{1},...,J_{m})$ denotes the Jordan form of $\mathcal{L}(t)$, with $J_{\alpha }$ representing a Jordan block related to an eigenvector whose corresponding eigenvalue is $\lambda _{\alpha }$, $$J_{\alpha }=\left( \begin{array}{ccccc} \lambda _{\alpha } & 1 & 0 & \cdots & 0 \\ 0 & \lambda _{\alpha } & 1 & \cdots & 0 \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ 0 & \cdots & 0 & \lambda _{\alpha } & 1 \\ 0 & \cdots & \cdots & 0 & \lambda _{\alpha }\end{array}\right) . \label{ljmatg}$$The number $m$ of Jordan blocks is given by the number of linearly independent eigenstates of $\mathcal{L}(t)$, with each eigenstate associated to a different block $J_{\alpha }$. Since $\mathcal{L}(t)$ is in general non-Hermitian, we generally do not have a basis of eigenstates, whence some care is required in order to find a basis for describing the density operator. A systematic procedure for finding a convenient discrete vector basis is to start from the instantaneous right and left eigenstates of $\mathcal{L}(t)$, which are defined by $$\begin{aligned} \mathcal{L}(t)\,|\mathcal{P}_{\alpha }(t)\rangle \rangle &=&\lambda _{\alpha }(t)\,|\mathcal{P}_{\alpha }(t)\rangle \rangle , \label{rleb0} \\ \langle \langle Q_{\alpha }(t)|\,\mathcal{L}(t) &=&\langle \langle Q_{\alpha }(t)|\,\lambda _{\alpha }(t), \label{rleb}\end{aligned}$$where, in our notation, possible degeneracies correspond to $\lambda _{\alpha }=\lambda _{\beta }$, with $\alpha \neq \beta $. In other words, we reserve a different index $\alpha $ for each independent eigenvector since each eigenvector is in a distinct Jordan block. It can immediately be shown from Eqs. (\[rleb0\]) and (\[rleb\]) that, for $\lambda _{\alpha }\neq \lambda _{\beta }$, we have $\langle \langle Q_{\alpha }(t)|\mathcal{P}_{\beta }(t)\rangle \rangle =0$. The left and right eigenstates can be easily identified when the Lindblad superoperator is in the Jordan form $\mathcal{L}_{J}(t)$. Denoting $|\mathcal{P}_{\alpha }(t)\rangle \rangle _{J}=S^{-1}(t)\,|\mathcal{P}_{\alpha }(t)\rangle \rangle $, i.e., the right eigenstate of $\mathcal{L}_{J}(t)$ associated to a Jordan block $J_{\alpha }$, then Eq. (\[rleb0\]) implies that $|\mathcal{P}_{\alpha }(t)\rangle \rangle _{J}$ is time-independent and, after normalization, is given by $$\left. |\mathcal{P}_{\alpha }\rangle \rangle _{J}\frac{{}}{{}}\right\vert _{J_{\alpha }}=\left( \begin{array}{c} 1 \\ 0 \\ \vdots \\ 0 \\ \end{array}\right) , \label{pj}$$where only the vector components associated to the Jordan block $J_{\alpha }$ are shown, with all the others vanishing. In order to have a complete basis we shall define new states, which will be chosen so that they preserve the block structure of $\mathcal{L}_{J}(t)$. A suitable set of additional vectors is $$\left. |\mathcal{D}_{\alpha }^{(1)}\rangle \rangle _{J}\frac{{}}{{}}\right\vert _{J_{\alpha }}=\left( \begin{array}{c} 0 \\ 1 \\ 0 \\ \vdots \\ 0 \\ \end{array}\right) ,\,...\,,\,\left. |\mathcal{D}_{\alpha }^{(n_{\alpha }-1)}\rangle \rangle _{J}\frac{{}}{{}}\right\vert _{J_{\alpha }}=\left( \begin{array}{c} 0 \\ 0 \\ 0 \\ \vdots \\ 1 \\ \end{array}\right) , \label{dj}$$where $n_{\alpha }$ is the dimension of the Jordan block $J_{\alpha }$ and again all the components outside $J_{\alpha }$ are zero. This simple vector structure allows for the derivation of the expression $$\mathcal{L}_{J}(t)\,|\mathcal{D}_{\alpha }^{(j)}\rangle \rangle _{J}=|\mathcal{D}_{\alpha }^{(j-1)}\rangle \rangle _{J}+\lambda _{\alpha }(t)\,|\mathcal{D}_{\alpha }^{(j)}\rangle \rangle _{J}, \label{ldj}$$with $|\mathcal{D}_{\alpha }^{(0)}\rangle \rangle _{J}\equiv |\mathcal{P}_{\alpha }\rangle \rangle _{J}$ and  $|\mathcal{D}_{\alpha }^{(-1)}\rangle \rangle _{J}\equiv 0$. The set $\left\{ |\mathcal{D}_{\alpha }^{(j)}\rangle \rangle _{J},\text{\thinspace with}\,j=0,...,(n_{\alpha }-1)\right\} $ can immediately be related to a right vector basis for the original $\mathcal{L}(t)$ by means of the transformation $|\mathcal{D}_{\alpha }^{(j)}(t)\rangle \rangle =S(t)\,|\mathcal{D}_{\alpha }^{(j)}\rangle \rangle _{J}$ which, applied to Eq. (\[ldj\]), yields $$\mathcal{L}(t)\,|\mathcal{D}_{\alpha }^{(j)}(t)\rangle \rangle =|\mathcal{D}_{\alpha }^{(j-1)}(t)\rangle \rangle +\lambda _{\alpha }(t)\,|\mathcal{D}_{\alpha }^{(j)}(t)\rangle \rangle . \label{ldo}$$Equation (\[ldo\]) exhibits an important feature of the set $\left\{ |\mathcal{D}_{\beta }^{(j)}(t)\rangle \rangle \right\} $, namely, it implies that Jordan blocks are invariant under the action of the Lindblad superoperator. An analogous procedure can be employed to define the left eigenbasis. Denoting by $_{J}\langle \langle \mathcal{Q}_{\alpha }(t)|=\langle \langle \mathcal{Q}_{\alpha }(t)|S(t)$ the left eigenstate of $\mathcal{L}_{J}(t)$ associated to a Jordan block $J_{\alpha }$, Eq. (\[rleb\]) leads to the normalized left vector $$\left. _{J}\langle \langle \mathcal{Q}_{\alpha }|\frac{{}}{{}}\right\vert _{J_{\alpha }}=\left( \frac{{}}{{}}0,\,.\,.\,.\,,0,1\frac{{}}{{}}\right) . \label{qj}$$The additional left vectors are defined as $$\begin{aligned} \left. _{J}\langle \langle \mathcal{E}_{\alpha }^{(0)}|\frac{{}}{{}}\right\vert _{J_{\alpha }} &=&\left( \frac{{}}{{}}1,0,0,\,.\,.\,.\,,0\frac{{}}{{}}\right) , \nonumber \\ \vspace{0.1cm} &.\,\,.\,\,.& \nonumber \\ \vspace{0.1cm}\left. _{J}\langle \langle \mathcal{E}_{\alpha }^{(n_{\alpha }-2)}|\frac{{}}{{}}\right\vert _{J_{\alpha }} &=&\left( \frac{{}}{{}}0,\,.\,.\,.\,,0,1,0\frac{{}}{{}}\right) , \label{rj}\end{aligned}$$which imply the following expression for the left basis vector $\langle \langle \mathcal{E}_{\alpha }^{(i)}(t)|=\,_{J}\langle \langle \mathcal{E}_{\alpha }^{(i)}|\,S^{-1}(t)$ for $\mathcal{L}(t)$: $$\langle \langle \mathcal{E}_{\alpha }^{(i)}(t)|\,\mathcal{L}(t)=\langle \langle \mathcal{E}_{\alpha }^{(i+1)}(t)|+\langle \langle \mathcal{E}_{\alpha }^{(i)}(t)|\,\lambda _{\alpha }(t). \label{lro}$$Here we have used the notation $_{J}\langle \langle \mathcal{E}_{\alpha }^{(n_{\alpha }-1)}|\equiv \,_{J}\langle \langle \mathcal{Q}_{\alpha }|$ and $_{J}\langle \langle \mathcal{E}_{\alpha }^{(n_{\alpha })}|\equiv 0$. A further property following from the definition of the right and left vector bases introduced here is $$\langle \langle \mathcal{E}_{\alpha }^{(i)}(t)|\mathcal{D}_{\beta }^{(j)}(t)\rangle \rangle =\,_{J}\langle \langle \mathcal{E}_{\alpha }^{(i)}|\mathcal{D}_{\beta }^{(j)}\rangle \rangle _{J}=\delta _{\alpha \beta }\delta ^{ij}. \label{lrr}$$This orthonormality relationship between corresponding left and right states will be very useful in our derivation below of the conditions for the validity of the adiabatic approximation. The adiabatic approximation in open quantum systems {#adiabatic} =================================================== We are now ready to derive our main result: an adiabatic approximation for open quantum systems. We do this by observing that the Jordan decomposition of $\mathcal{L}(t)$ \[Eq. (\[jd\])\] allows for a nice generalization of the standard quantum adiabatic approximation. We begin by defining the adiabatic dynamics of an open system as a generalization of the definition given above for closed quantum systems: \[def:open-ad\] An open quantum system is said to undergo adiabatic dynamics if its Hilbert-Schmidt space can be decomposed into decoupled Lindblad–Jordan eigenspaces with distinct, time-continuous, and noncrossing instantaneous eigenvalues of $\mathcal{L}(t)$. This definition is a natural extension for open systems of the idea of adiabatic behavior. Indeed, in this case the master equation (\[eq:t-Lind\]) can be decomposed into sectors with different and separately evolving Lindblad-Jordan eigenvalues, and we show below that the condition for this to occur is appropriate slowness of the Lindblad superoperator. The splitting into Jordan blocks of the Lindblad superoperator is achieved through the choice of a basis which preserves the Jordan block structure as, for example, the sets of right $\left\{ |\mathcal{D}_{\beta }^{(j)}(t)\rangle \rangle \right\} $ and left $\left\{ \langle \langle \mathcal{E}_{\alpha }^{(i)}(t)|\right\} $ vectors introduced in Sec. \[open\]. Such a basis generalizes the notion of Schrödinger eigenvectors. Intuitive derivation -------------------- Let us first show how the adiabatic Lindblad-Jordan blocks arise from a simple argument, analogous to the one presented for the closed case \[Eqs. (\[hdc\])-(\[eq:Had\])\]. Multiplying Eq. (\[le\]) by the similarity transformation matrix $S^{-1}(t)$, we obtain $$\begin{aligned} \mathcal{L}_{J}\,|\rho \rangle \rangle _{J}=|{\dot{\rho}}\rangle \rangle _{J}-{\dot{S}}^{-1}\,|\rho \rangle \rangle , \label{ljinter}\end{aligned}$$ where we have used Eq. (\[jd\]) and defined $|\rho \rangle \rangle _{J}\equiv S^{-1}|\rho \rangle \rangle $. Now suppose that $\mathcal{L}(t)$, and consequently $S(t)$ and its inverse $S^{-1}(t)$, changes slowly in time so that ${\dot{S}}^{-1}(t)\approx 0$. Then, from Eq. (\[ljinter\]), the adiabatic dynamics of the system reads $$\mathcal{L}_{J}(t)\,|\rho (t)\rangle \rangle _{J}=|{\dot{\rho}}(t)\rangle \rangle _{J}. \label{ljrj}$$Equation (\[ljrj\]) ensures that, choosing an instantaneous basis for the density operator $\rho (t)$ which preserves the Jordan block structure, the evolution of $\rho (t)$ occurs separately in adiabatic blocks associated with distinct eigenvalues of $\mathcal{L}(t)$. Of course, the conditions under which the approximation ${\dot{S}}^{-1}(t)\approx 0$ holds must be carefully clarified. This is the subject of the next two subsections. Condition on the Lindblad superoperator --------------------------------------- Let us now derive the validity conditions for open-system adiabatic dynamics by analyzing the general time evolution of a density operator under the master equation (\[le\]). To this end, we expand the density matrix for an arbitrary time $t$ in the instantaneous right eigenbasis $\left\{ |{\mathcal{D}_{\beta }^{(j)}(t)\rangle \rangle }\right\} $ as $$|\rho (t)\rangle \rangle =\frac{1}{2}\sum_{\beta =1}^{m}\sum_{j=0}^{n_{\beta }-1}r_{\beta }^{(j)}(t)\,|\mathcal{D}_{\beta }^{(j)}(t)\rangle \rangle , \label{rtime}$$where $m$ is the number of Jordan blocks and $n_{\beta }$ is the dimension of the block $J_{\beta }$. We emphasize that we are assuming that there are no eigenvalue crossings in the spectrum of the Lindblad superoperator during the evolution. Requiring then that the density operator Eq. (\[rtime\]) evolves under the master equation (\[le\]) and making use of Eq. (\[ldo\]), we obtain $$\begin{aligned} \sum_{\beta =1}^{m}\sum_{j=1}^{n_{\beta }-1}r_{\beta }^{(j)}\,\left( |\mathcal{D}_{\beta }^{(j-1)}\rangle \rangle +\lambda _{\beta }\,|\mathcal{D}_{\beta }^{(j)}\rangle \rangle \right) = \nonumber \\ \sum_{\beta =1}^{m}\sum_{j=0}^{n_{\beta }-1}\left( {\dot{r}}_{\beta }^{(j)}\,|\mathcal{D}_{\beta }^{(j)}\rangle \rangle +r_{\beta }^{(j)}\,|{\dot{\mathcal{D}}}_{\beta }^{(j)}\rangle \rangle \right) . \label{lindg1}\end{aligned}$$Equation (\[lindg1\]) multiplied by the left eigenstate $\langle \langle \mathcal{E}_{\alpha }^{(i)}|$ results in $${\dot{r}}_{\alpha }^{(i)}=\lambda _{\alpha }\,r_{\alpha }^{(i)}+r_{\alpha }^{(i+1)}-\sum_{\beta =1}^{m}\sum_{j=0}^{n_{\beta }-1}r_{\beta }^{(j)}\langle \langle \mathcal{E}_{\alpha }^{(i)}|{\dot{\mathcal{D}}}_{\beta }^{(j)}\rangle \rangle , \label{rdot}$$with $r_{\alpha }^{(n_{\alpha })}(t)\equiv 0$. Note that the sum over $\beta $ mixes different Jordan blocks. An analogous situation occurred in the closed system case, in Eq. (\[anf\]). Similarly to what was done there, in order to derive an adiabaticity condition we must separate this sum into terms related to the eigenvalue $\lambda _{\alpha }$ of $\mathcal{L}(t)$ and terms involving mixing with eigenvalues $\lambda _{\beta }\neq \lambda _{\alpha }$. In this latter case, an expression can be found for $\langle \langle \mathcal{E}_{\alpha }^{(i)}|{\dot{\mathcal{D}}}_{\beta }^{(j)}\rangle \rangle $ as follows: taking the time derivative of Eq. ([ldo]{}) and multiplying by $\langle \langle \mathcal{E}_{\alpha }^{(i)}|$ we obtain, after using Eqs. (\[lro\]) and (\[lrr\]), $$\begin{aligned} \langle \langle \mathcal{E}_{\alpha }^{(i)}|{\dot{\mathcal{D}}}_{\beta }^{(j)}\rangle \rangle =\frac{1}{\omega _{\beta \alpha }}\,\left( \,\langle \langle \mathcal{E}_{\alpha }^{(i)}|\,{\dot{\mathcal{L}}}\,|\mathcal{D}_{\beta }^{(j)}\rangle \rangle \,\right. \nonumber \\ \left. +\,\langle \langle \mathcal{E}_{\alpha }^{(i+1)}|{\dot{\mathcal{D}}}_{\beta }^{(j)}\rangle \rangle -\langle \langle \mathcal{E}_{\alpha }^{(i)}|{\dot{\mathcal{D}}}_{\beta }^{(j-1)}\rangle \rangle \,\right) ,\,\,\,\, \label{rddi}\end{aligned}$$where we have defined $$\omega _{\beta \alpha }(t)\equiv \lambda _{\beta }(t)-\lambda _{\alpha }(t) \label{eq:omab}$$and assumed $\lambda _{\alpha }\neq \lambda _{\beta }$. Note that, while $\omega _{\beta \alpha }$ plays a role analogous to that of the energy difference $g_{nk}$ in the closed case \[Eq. (\[eq:g\])\], $\omega _{\beta \alpha }$ may be complex. A similar procedure can generate expressions for all the terms $\langle \langle \mathcal{E}_{\alpha }^{(i)}|{\dot{\mathcal{D}}}_{\beta }^{(j-k)}\rangle \rangle $, with $k=0,...,j$. Thus, an iteration of Eq. (\[rddi\]) yields $$\begin{aligned} \langle \langle \mathcal{E}_{\alpha }^{(i)}|{\dot{\mathcal{D}}}_{\beta }^{(j)}\rangle \rangle &=&\sum_{k=0}^{j}\frac{(-1)^{k}}{\omega _{\beta \alpha }^{k+1}}\left( \langle \langle \mathcal{E}_{\alpha }^{(i)}|\,{\dot{\mathcal{L}}}\,|\mathcal{D}_{\beta }^{(j-k)}\rangle \rangle \,\right. \nonumber \\ &&\left. +\,\langle \langle \mathcal{E}_{\alpha }^{(i+1)}|{\dot{\mathcal{D}}}_{\beta }^{(j-k)}\rangle \rangle \right) . \label{rddi3}\end{aligned}$$ From a second recursive iteration, now for the term $\langle \langle \mathcal{E}_{\alpha }^{(i+1)}|{\dot{\mathcal{D}}}_{\beta }^{(j-k)}\rangle \rangle $ in Eq. (\[rddi3\]), we obtain $$\begin{aligned} && \langle \langle \mathcal{E}_{\alpha }^{(i)}|{\dot{\mathcal{D}}}_{\beta }^{(j)}\rangle \rangle = \nonumber \\ && \sum_{p=1}^{(n_{\alpha }-i)}\left( \prod_{q=1}^{p}\sum_{k_{q}=0}^{(j-S_{q-1})}\right) \frac{\langle \langle \mathcal{E}_{\alpha }^{(i+p-1)}|{\dot{\mathcal{L}}}|\mathcal{D}_{\beta }^{(j-S_{p})}\rangle \rangle }{(-1)^{S_{p}}\,\omega _{\beta \alpha }^{p+S_{p}}}, \label{rdd}\end{aligned}$$where $$S_{q}=\sum_{s=1}^{q}k_{s}\,\,\,\,,\,\,\,\,\left( \prod_{q=1}^{p}\sum_{k_{q}=0}^{(j-S_{q-1})}\right) =\sum_{k_{1}=0}^{j-S_{0}}\cdots \sum_{k_{p}=0}^{j-S_{p-1}},$$with $S_{0}=0$. We can now split Eq. (\[rdot\]) into diagonal and off-diagonal terms $$\begin{aligned} &&{\dot{r}}_{\alpha }^{(i)}=\lambda _{\alpha }r_{\alpha }^{(i)}+r_{\alpha }^{(i+1)}-\sum_{\beta \,|\,\lambda _{\beta }=\lambda _{\alpha }}\sum_{j=0}^{n_{\beta }-1}r_{\beta }^{(j)}\langle \langle \mathcal{E}_{\alpha }^{(i)}|{\dot{\mathcal{D}}}_{\beta }^{(j)}\rangle \rangle \hspace{0.3cm} \nonumber \\ &&-\sum_{\beta \,|\,\lambda _{\beta }\neq \lambda _{\alpha }}\sum_{j=0}^{n_{\beta }-1}r_{\beta }^{(j)}\langle \langle \mathcal{E}_{\alpha }^{(i)}|{\dot{\mathcal{D}}}_{\beta }^{(j)}\rangle \rangle , \label{rfinal}\end{aligned}$$where the terms $\langle \langle \mathcal{E}_{\alpha }^{(i)}|{\dot{\mathcal{D}}}_{\beta }^{(j)}\rangle \rangle $, for $\lambda _{\beta }\neq \lambda _{\alpha }$, are given by Eq. (\[rdd\]). In accordance with our definition of adiabaticity above, the adiabatic regime is obtained when the sum in the second line is negligible. Summarizing, by introducing the normalized time $s$ defined by Eq. (\[nt\]), we thus find the following from Eqs. (\[rdd\]) and (\[rfinal\]). \[t1\] A sufficient condition for open quantum system adiabatic dynamics as given in Definition \[def:open-ad\] is: $$\begin{aligned} &&\max_{0\le s\le 1}\,\left\vert \sum_{p=1}^{(n_{\alpha }-i)}\left( \prod_{q=1}^{p}\sum_{k_{q}=0}^{(j-S_{q-1})}\right) \frac{\langle \langle \mathcal{E}_{\alpha }^{(i+p-1)}|\frac{d\mathcal{L}}{ds}|\mathcal{D}_{\beta }^{(j-S_{p})}\rangle \rangle }{(-1)^{S_{p}}\,\omega _{\beta \alpha }^{p+S_{p}}}\right\vert \nonumber \\ &&\ll 1, \label{vc}\end{aligned}$$with $\lambda _{\beta }\neq \lambda _{\alpha }$ and for arbitrary indices $i$ and $j$ associated to the Jordan blocks $\alpha $ and $\beta $, respectively. The condition (\[vc\]) ensures the absence of mixing of coefficients $r_{\alpha }^{(i)}$ related to distinct eigenvalues $\lambda _{\alpha }$ in Eq. ([rfinal]{}), which in turn guarantees that sets of Jordan blocks belonging to different eigenvalues of $\mathcal{L}(t)$ have independent evolution. Thus the accuracy of the adiabatic approximation can be estimated by the computation of the time derivative of the Lindblad superoperator acting on right and left vectors. Equation (\[vc\]) can be simplified by considering the term with maximum absolute value, which results in: \[c1os\] A sufficient condition for open quantum system adiabatic dynamics is $$\begin{aligned} &&{\cal N}_{ij}^{n_\alpha n_\beta} \max_{0\le s\le 1}\,\left\vert \frac{\langle \langle \mathcal{E}_{\alpha }^{(i+p-1)}|\frac{d\mathcal{L}}{ds}|\mathcal{D}_{\beta }^{(j-S_{p})}\rangle \rangle }{\omega _{\beta \alpha }^{p+S_{p}}}\right\vert \ll 1, \end{aligned}$$where the $\max$ is taken for any $\alpha \ne \beta$, and over all possible values of $i\in\{0,...,n_{\alpha}-1\}$, $j\in\{0,...,n_{\beta}-1\}$, and $p$, with $$\begin{aligned} &&{\cal N}_{ij}^{n_\alpha n_\beta} = \sum_{p=1}^{(n_{\alpha }-i)}\left( \prod_{q=1}^{p}\sum_{k_{q}=0}^{(j-S_{q-1})}\right) 1 \label{numberterms} \\ &&= \left( \begin{array} [c]{c} n_\alpha-i+1+j\\ 1+j \end{array} \right)-1 = \frac{(n_\alpha-i+1+j)!}{(1+j)!(n_\alpha-i)!}-1. \nonumber\end{aligned}$$ Observe that the factor ${\cal N}_{ij}^{n_\alpha n_\beta}$ defined in Eq. (\[numberterms\]) is just the number of terms of the sums in Eq. (\[vc\]). We have included a superscript $n_\beta$, even though there is no explicit dependence on $n_\beta$, since $j\in\{0,...,n_{\beta}-1\}$. Furthermore, an adiabatic condition for a slowly varying Lindblad super-operator can directly be obtained from Eq. (\[vc\]), yielding the following. A simple sufficient condition for open quantum system adiabatic dynamics is ${\dot{\mathcal{L}}}\approx 0$. Note that this condition is in a sense too strong, since it need not be the case that $\dot{\mathcal{L}}$ is small in general (i.e., for all its matrix elements). Indeed, in Sec. \[example\] we show via an example that adiabaticity may occur due to the *exact* vanishing of relevant matrix elements of ${\dot{\mathcal{L}}}$. The general condition for this to occur is the presence of a *dynamical symmetry* [@Bohm:88]. Let us end this subsection by mentioning that we can also write Eq. (\[vc\]) in terms of the time variable $t$ instead of the normalized time $s$. In this case, the natural generalization of Eq. (\[vc\]) is $$\begin{aligned} &&\max_{0\le t\le T}\,\left\vert \sum_{p=1}^{(n_{\alpha }-i)}\left( \prod_{q=1}^{p}\sum_{k_{q}=0}^{(j-S_{q-1})}\right) \frac{\langle \langle \mathcal{E}_{\alpha }^{(i+p-1)}|{\dot{\mathcal{L}}}|\mathcal{D}_{\beta }^{(j-S_{p})}\rangle \rangle }{(-1)^{S_{p}}\,\omega _{\beta \alpha }^{p+S_{p}}}\right\vert \nonumber \\ &&\ll \,\min_{0\le t\le T} \, |\omega_{\beta \alpha}|. \label{vch}\end{aligned}$$ Note that, as in the analogous condition (\[vcc\]) in the closed case, the left-hand side has dimensions of frequency, and hence must be compared to the natural frequency scale $\omega_{\beta \alpha}$. However, unlike the closed systems case, where Eq. (\[vcc\]) can immediately be derived from the time condition (\[timead\]), we cannot prove here that $\omega_{\beta\alpha}$ is indeed the relevant physical scale. Therefore, Eq. (\[vch\]) should be regarded as a heuristic criterion. Condition on the total evolution time {#sec:tot-t} ------------------------------------- As mentioned in Sec. \[closed\], for closed systems the rate at which the adiabatic regime is approached can be estimated in terms of the total time of evolution, as shown by Eqs. (\[timead\]) and (\[timead2\]). We now provide a generalization of this estimate for adiabaticity in open systems. ### One-dimensional Jordan blocks Let us begin by considering the particular case where $\mathcal{L}(t)$ has only one-dimensional Jordan blocks and each eigenvalue corresponds to a single independent eigenvector, i.e., $\lambda _{\alpha }=\lambda _{\beta }\Rightarrow \alpha =\beta $. Bearing these assumptions in mind, Eq. ([rfinal]{}) can be rewritten as $${\dot{r}}_{\alpha }=\lambda _{\alpha }r_{\alpha }-r_{\alpha }\langle \langle \mathcal{E}_{\alpha }|{\dot{\mathcal{D}}}_{\alpha }\rangle \rangle -\sum_{\beta \neq \alpha }r_{\beta }\langle \langle \mathcal{E}_{\alpha }|{\dot{\mathcal{D}}}_{\beta }\rangle \rangle , \label{rfsc}$$where the upper indices $i,j$ have been removed since we are considering only one-dimensional blocks. Moreover, for this special case, we have from Eq. (\[rdd\]) $$\langle \langle \mathcal{E}_{\alpha }|{\dot{\mathcal{D}}}_{\beta }\rangle \rangle =\frac{\langle \langle \mathcal{E}_{\alpha }|\,{\dot{\mathcal{L}}}\,|\mathcal{D}_{\beta }\rangle \rangle }{\omega _{\beta \alpha }}. \label{edsc}$$In order to eliminate the term $\lambda _{\alpha }r_{\alpha }$ from Eq. ([rfsc]{}), we redefine the variable $r_{\alpha }(t)$ as $$r_{\alpha }(t)=p_{\alpha }(t)\,e^{\int_{0}^{t}\lambda _{\alpha }(t^{\prime })dt^{\prime }},$$which, applied to Eq. (\[rfsc\]), yields $${\dot{p}}_{\alpha }=-p_{\alpha }\,\langle \langle \mathcal{E}_{\alpha }|{\dot{\mathcal{D}}}_{\alpha }\rangle \rangle -\sum_{\beta \neq \alpha }p_{\beta }\,\langle \langle \mathcal{E}_{\alpha }|{\dot{\mathcal{D}}}_{\beta }\rangle \rangle \,e^{\Omega _{\beta \alpha }}, \label{eq:rfsc2}$$with $$\Omega _{\beta \alpha }(t)=\int_{0}^{t}dt^{\prime }\,\omega _{\beta \alpha }(t^{\prime }). \label{omos}$$Equation (\[eq:rfsc2\]) is very similar to Eq. (\[anf\]) for closed systems, but the fact that $\Omega _{\beta \alpha }$ is in general complex-valued leads to some important differences, discussed below. We next introduce the scaled time $s=t/T$ and integrate the resulting expression. Using Eq. ([edsc]{}), we then obtain $$\begin{aligned} &&p_{\alpha }(s)\,=\,p_{\alpha }(0)-\int_{0}^{s}ds^{\prime }p_{\alpha }(s^{\prime })\,\Phi _{\alpha }(s^{\prime }) \nonumber \\ &&-\sum_{\beta \neq \alpha }\int_{0}^{s}ds^{\prime }\frac{V_{\beta \alpha }(s^{\prime })}{\omega _{\beta \alpha }(s^{\prime })}\,e^{T\,\Omega _{\beta \alpha }(s^{\prime })}, \label{scint}\end{aligned}$$where $\Phi _{\alpha }(s)$ is defined by $$\Phi _{\alpha }(s)=\langle \langle \mathcal{E_{\alpha }}(s)|\frac{d}{ds}|\mathcal{D_{\alpha }}(s)\rangle \rangle$$and $V_{\beta \alpha }(s)$ by $$V_{\beta \alpha }(s)=p_{\beta }(s)\,\langle \langle \mathcal{E_{\alpha }}(s)|\frac{d\mathcal{L}(s)}{ds}|\mathcal{D_{\beta }}(s)\rangle \rangle . \label{vba}$$The integrand in the last line of Eq. (\[scint\]) can be rearranged in a similar way to Eq. (\[ricc\]) for the closed case, yielding $$\begin{aligned} &&\frac{V_{\beta \alpha }(s)}{\omega _{\beta \alpha }(s)}\,e^{T\,\Omega _{\beta \alpha }(s)} \nonumber \\ &=&\frac{1}{T}\left[ \frac{d}{ds}\left( \frac{V_{\beta \alpha }}{\omega _{\beta \alpha }^{2}}\,e^{T\,\Omega _{\beta \alpha }(s)}\right) -e^{T\,\Omega _{\beta \alpha }(s)}\frac{d}{ds}\frac{V_{\beta \alpha }}{\omega _{\beta \alpha }^{2}}\right] . \label{ninte}\end{aligned}$$Therefore, from Eq. (\[scint\]) we have $$\begin{aligned} &&p_{\alpha }(s)\,=\,p_{\alpha }(0)-\int_{0}^{s}ds^{\prime }p_{\alpha }(s^{\prime })\,\Phi _{\alpha }(s^{\prime }) \nonumber \\ &&+\frac{1}{T}\sum_{\beta \neq \alpha }\left( \frac{V_{\beta \alpha }(0)}{\omega _{\beta \alpha }^{2}(0)}-\frac{V_{\beta \alpha }(s)}{\omega _{\beta \alpha }^{2}(s)}\,e^{T\,\Omega _{\beta \alpha }(s)}\right. \nonumber \\ &&\left. +\int_{0}^{s}ds^{\prime }\,e^{T\,\Omega _{\beta \alpha }(s^{\prime })}\frac{d}{ds^{\prime }}\frac{V_{\beta \alpha }(s^{\prime })}{\omega _{\beta \alpha }^{2}(s^{\prime })}\right) . \label{afsc}\end{aligned}$$ Thus a condition for adiabaticity in terms of the total time of evolution can be given by comparing $T$ to the terms involving indices $\beta \neq \alpha $. This can be formalized as follows. \[t2\] Consider an open quantum system whose Lindblad superoperator $\mathcal{L}(s)$ has the following properties: $(a)$ The Jordan decomposition of $\mathcal{L}(s)$ is given by one-dimensional blocks. $(b)$ Each eigenvalue of $\mathcal{L}(s)$ is associated to a unique Jordan block. Then the adiabatic dynamics in the interval $0\leq s\leq 1$ occurs if and only if the following time conditions, obtained for each Jordan block $\alpha$ of $\mathcal{L}(s)$, are satisfied: $$\begin{aligned} T &\gg& \max_{0\leq s\leq 1} \left\vert \,\sum_{\beta \neq \alpha }\left( \frac{V_{\beta \alpha }(0)}{\omega _{\beta \alpha }^{2}(0)}-\frac{V_{\beta \alpha }(s)}{\omega _{\beta \alpha }^{2}(s)}\,e^{T\,\Omega _{\beta \alpha }(s)}\right. \right. \nonumber \\ &&\left. \left. +\int_{0}^{s}ds^{\prime }\,e^{T\,\Omega _{\beta \alpha }(s^{\prime })}\frac{d}{ds^{\prime }}\frac{V_{\beta \alpha }(s^{\prime })}{\omega _{\beta \alpha }^{2}(s^{\prime })}\right) \right\vert, \label{eq:Tad}\end{aligned}$$ Equation (\[eq:Tad\]) simplifies in a number of situations. - Adiabaticity is guaranteed whenever $V_{\beta \alpha }$ vanishes for all $\alpha \neq \beta $. An example of this case will be provided in Sec. \[example\]. - Adiabaticity is similarly guaranteed whenever $V_{\beta \alpha }(s)$, which can depend on $T$ through $p_{\beta }$, vanishes for all $\alpha ,\beta $ such that $\mathrm{Re}(\Omega _{\beta \alpha })>0$ and does not grow faster, as a function of $T$, than $\exp (T|\,{\mathrm{Re}}\Omega _{\beta \alpha }|)$ for all $\alpha ,\beta $ such that $\mathrm{Re}(\Omega _{\beta \alpha })<0$. - When $\mathrm{Re}(\Omega _{\beta \alpha })=0$ and $\mathrm{Im}(\Omega _{\beta \alpha })\neq 0$ the integral in inequality (\[eq:Tad\]) vanishes in the infinite time limit due to the Riemann-Lebesgue lemma [Churchill:book]{}, as in the closed case discussed before. In this case, again, adiabaticity is guaranteed provided $p_{\beta }(s)$ \[and hence $V_{\beta \alpha }(s)$\] does not diverge as a function of $T$ in the limit $T \rightarrow \infty$. - When $\mathrm{Re}(\Omega _{\beta \alpha })>0$, the adiabatic regime can still be reached for large $T$ provided that $p_{\beta }(s)$ contains a decaying exponential which compensates for the growing exponential due to $\mathrm{Re}(\Omega _{\beta \alpha })$. - Even if there is an overall growing exponential in inequality ([eq:Tad]{}), adiabaticity could take place over a finite time interval $[0,T_{\ast }]$ and, afterwards, disappear. In this case, which would be an exclusive feature of open systems, the crossover time $T_{\ast }$ would be determined by an inequality of the type $T\gg a+b\exp (cT)$, with $c>0$. The coefficients $a,b$ and $c$ are functions of the system-bath interaction. Whether the latter inequality can be solved clearly depends on the values of $a,b,c$, so that a conclusion about adiabaticity in this case is model dependent. ### General Jordan blocks We show now that the hypotheses $(a)$ and $(b)$ can be relaxed, providing a generalization of Proposition \[t2\] for the case of multidimensional Jordan blocks and Lindblad eigenvalues associated to more than one independent eigenvector. Let us redefine our general coefficient $r_{\alpha }^{(i)}(t)$ as $$r_{\alpha }^{(i)}(t)=p_{\alpha }^{(i)}(t)\,e^{\int_{0}^{t}\lambda _{\alpha }(t^{\prime })dt^{\prime }},$$which, applied to Eq. (\[rfinal\]), yields $$\begin{aligned} &&{\dot{p}}_{\alpha }^{(i)}\,=\,p_{\alpha }^{(i+1)} \nonumber \\ &&-\sum_{\beta \,|\,\lambda _{\beta }=\lambda _{\alpha }}\sum_{j=0}^{n_{\beta }-1}p_{\beta }^{(j)}\langle \langle \mathcal{E}_{\alpha }^{(i)}|{\dot{\mathcal{D}}}_{\beta }^{(j)}\rangle \rangle \,e^{\Omega _{\beta \alpha }} \nonumber \\ &&-\sum_{\beta \,|\,\lambda _{\beta }\neq \lambda _{\alpha }}\sum_{j=0}^{n_{\beta }-1}p_{\beta }^{(j)}\langle \langle \mathcal{E}_{\alpha }^{(i)}|{\dot{\mathcal{D}}}_{\beta }^{(j)}\rangle \rangle \,e^{\Omega _{\beta \alpha }}. \label{rfscg}\end{aligned}$$The above equation can be rewritten in terms of the scaled time $s=t/T$. The integration of the resulting expression then reads $$\begin{aligned} &&p_{\alpha }^{(i)}(s)\,=\,p_{\alpha }^{(i)}(0)+T\int_{0}^{s}ds^{\prime }p_{\alpha }^{(i+1)}(s^{\prime }) \nonumber \\ &&\hspace{-0.9cm}-\sum_{\beta \,|\,\lambda _{\beta }=\lambda _{\alpha }}\sum_{j}\int_{0}^{s}ds^{\prime }p_{\beta }^{(j)}(s^{\prime })\,\Phi _{\beta \alpha }^{(ij)}(s^{\prime })\,e^{T\,\Omega _{\beta \alpha }(s^{\prime })} \nonumber \\ &&\hspace{-0.9cm}-\sum_{\beta \,|\,\lambda _{\beta }\neq \lambda _{\alpha }}\sum_{j,p}\int_{0}^{s}ds^{\prime }\frac{(-1)^{S_{p}}\,V_{\beta \alpha }^{(ijp)}(s^{\prime })}{\omega _{\beta \alpha }^{p+S_{p}}(s^{\prime })}\,e^{T\,\Omega _{\beta \alpha }(s^{\prime })}, \label{scintg}\end{aligned}$$where use has been made of Eq. (\[rdd\]), with the sum over $j$ and $p$ in the last line denoting $$\sum_{j,p}\equiv \sum_{j=0}^{n_{\beta }-1}\sum_{p=1}^{(n_{\alpha }-i)}\left( \prod_{q=1}^{p}\sum_{k_{q}=0}^{(j-S_{q-1})}\right) .$$The function $\Phi _{\beta \alpha }^{(ij)}(s)$ is defined by $$\Phi _{\beta \alpha }^{(ij)}(s)=\langle \langle \mathcal{E}_{\alpha }^{(i)}(s)|\frac{d}{ds}|{\mathcal{D}}_{\beta }^{(j)}(s)\rangle \rangle , \label{phabij}$$and $V_{\beta \alpha }^{(ijp)}(s)$ by $$V_{\beta \alpha }^{(ijp)}(s)=p_{\beta }^{(j)}(s)\langle \langle \mathcal{E}_{\alpha }^{(i+p-1)}(s)|\frac{d\mathcal{L}(s)}{ds}|\mathcal{D}_{\beta }^{(j-S_{p})}(s)\rangle \rangle .\, \label{vbapj}$$The term $T\int_{0}^{s}ds^{\prime }p_{\alpha }^{(i+1)}(s^{\prime })$ in the first line of Eq. (\[scintg\]), which was absent in the case of one-dimensional Jordan blocks analyzed above, has no effect on adiabaticity, since it does not cause any mixing of Jordan blocks. Therefore, the analysis can proceed very similarly to the case of one-dimensional blocks. Rewriting the integral in the last line of Eq. (\[scintg\]), as we have done in Eqs. (\[akfinal\]) and (\[afsc\]), and imposing the absence of mixing of the eigenvalues $\lambda _{\beta }\neq \lambda _{\alpha }$, i.e., the negligibility of the last line of Eq. (\[scintg\]), we find the following general theorem ensuring the adiabatic behavior of an open system. \[t3\] Consider an open quantum system governed by a Lindblad superoperator $\mathcal{L}(s)$. Then the adiabatic dynamics in the interval $0\leq s\leq 1$ occurs if and only if the following time conditions, obtained for each coefficient $p_\alpha^{(i)}(s)$, are satisfied: $$\begin{aligned} T &\gg& \max_{0\leq s\leq 1} \left\vert \,\sum_{\beta \,|\,\lambda _{\beta }\neq \lambda _{\alpha }}\sum_{j,p}\,(-1)^{S_{p}}\right. \nonumber \\ &&\times \left[ \frac{V_{\beta \alpha }^{(ijp)}(0)}{\omega _{\beta \alpha }^{p+S_{p}+1}(0)}-\frac{V_{\beta \alpha }^{(ijp)}(s)\,e^{T\,\Omega _{\beta \alpha }(s)}}{\omega _{\beta \alpha }^{p+S_{p}+1}(s)}\right. \nonumber \\ &&\left. \left. +\int_{0}^{s}ds^{\prime }\,e^{T\,\Omega _{\beta \alpha }(s^{\prime })}\frac{d}{ds^{\prime }}\frac{V_{\beta \alpha }^{(ijp)}(s^{\prime })}{\omega _{\beta \alpha }^{p+S_{p}+1}(s^{\prime })}\right] \right\vert . \label{eq:tadscg}\end{aligned}$$ Theorem \[t3\] provides a very general condition for adiabaticity in open quantum systems. The comments made about simplifying circumstances, in the case of one-dimensional blocks above, hold here as well. Moreover, a simpler sufficient condition can be derived from Eq. (\[eq:tadscg\]) by considering the term with maximum absolute value in the sum. This procedure leads to the following corollary: \[ct3\] A sufficient time condition for the adiabatic regime of an open quantum system governed by a Lindblad superoperator $\mathcal{L}(t)$ is $$\begin{aligned} T &\gg& \mathcal{M}_{ij}^{n_\alpha n_\beta} \, \max_{0\le s\le 1} \left\vert \, \frac{V_{\beta \alpha }^{(ijp)}(0)}{\omega _{\beta \alpha}^{p+S_{p}+1}(0)} -\frac{V_{\beta \alpha }^{(ijp)}(s)\,e^{T\,\Omega_{\beta \alpha }(s)}}{\omega _{\beta \alpha}^{p+S_{p}+1}(s)} \right. \nonumber \\ &&\left.+\int_{0}^{s}ds^{\prime }\,e^{T\,\Omega _{\beta \alpha }(s^{\prime })}\frac{d}{ds^{\prime }} \frac{V_{\beta \alpha }^{(ijp)}(s^{\prime })}{\omega _{\beta\alpha }^{p+S_{p}+1}(s^{\prime })} \right\vert, \label{eq:tadcol}\end{aligned}$$ where $\max $ is taken over all possible values of the indices $\lambda_\alpha \neq \lambda_\beta $, $i$, $j$, and $p$, with $$\begin{aligned} &&\mathcal{M}_{ij}^{n_\alpha n_\beta} = \sum_{\beta \,|\,\lambda _{\beta }\neq \lambda _{\alpha}} \sum_{j=0}^{(n_{\beta}-1)}\sum_{p=1}^{(n_{\alpha }-i)}\left( \prod_{q=1}^{p}\sum_{k_{q}=0}^{(j-S_{q-1})}\right) 1 \nonumber \\ &&= \Lambda_{\beta\alpha} \left[ \frac{(n_\alpha+n_\beta-i+1)!}{(n_\alpha-i+1)!n_\beta!}-n_\beta-1 \right], \label{Nlt}\end{aligned}$$ where $\Lambda_{\beta\alpha}$ denotes the number of Jordan blocks such that $\lambda_\alpha \neq \lambda_\beta$. Physical interpretation of the adiabaticity condition ----------------------------------------------------- There are various equivalent ways in which to interpret the adiabatic theorem for *closed* quantum systems [@Messiah:book]. A particularly useful interpretation follows from Eq. (\[timead2\]): the evolution time must be much longer than the ratio of the norm of the time derivative of the Hamiltonian to the square of the spectral gap. In other words, either the Hamiltonian changes slowly, or the spectral gap is large, or both. It is tempting to interpret our results in a similar fashion, which we now do. The quantity $V_{\beta \alpha }^{(ijp)}$, by Eq. (\[vbapj\]), plays the role of the time derivative of the Lindblad superoperator. However, the appearance of $\exp [T\,\mathrm{Re}\,\Omega _{\beta \alpha }(s)]$ in Eq. (\[eq:tadscg\]) has no analog in the closed-systems case, because the eigenvalues of the Hamiltonian are real, while in the open-systems case the eigenvalues of the Lindblad superoperator may have imaginary parts. This implies that adiabaticity is a phenomenon which is not guaranteed to happen in open systems even for very slowly varying interactions. Indeed, from Theorems \[t2\] and \[t3\], possible pictures of such system evolutions include the decoupling of Jordan blocks only over a finite time interval (disappearing afterwards), or even the case of complete absence of decoupling for any time $T$, which implies no adiabatic evolution whatsoever. The quantity $\omega _{\beta \alpha}$, by Eq. (\[eq:omab\]), clearly plays the role of the spectral gap in the open-system case. There are two noteworthy differences compared to the closed-system case. First, the $\omega _{\beta \alpha }$ can be complex. This implies that the differences in decay rates, and not just in energies, play a role in determining the relevant gap for open systems. Second, for multidimensional Jordan blocks, the terms $\omega_{\beta\alpha}$ depend on distinct powers for distinct pairs $\beta,\alpha$. Thus certain $\omega _{\beta \alpha }$ (those with the higher exponents) will play a more dominant role than others. The conditions for adiabaticity are best illustrated further via examples, one of which we provide next. Example: The adiabatic evolution of an open quantum two-level system {#example} ==================================================================== In order to illustrate the consequences of open quantum system adiabatic dynamics, let us consider a concrete example that is analytically solvable. Suppose a quantum two-level system, with internal Hamiltonian $H=(\omega /2)\,\sigma _{z}$, and described by the master equation (\[eq:t-Lind\]), is subjected to two sources of decoherence: spontaneous emission $\Gamma _{1}(t)=\epsilon (t)\,\sigma _{-}$ and bit flips $\Gamma _{2}(t)=\gamma (t)\,\sigma _{x}$, where $\sigma _{-}=\sigma _{x}-i\sigma _{y}$ is the lowering operator. Writing the density operator in the basis $\left\{ I_{2},\sigma _{x},\sigma _{y},\sigma _{z}\right\} $, i.e., as $\rho =(I_{2}+\overrightarrow{v}\cdot \overrightarrow{\sigma })/2$, Eq. (\[le\]) results in $$|{\dot{\rho}}(t)\rangle \rangle =\frac{1}{2}\left( \begin{array}{c} 0 \\ -\omega v_{y}-2\epsilon ^{2}v_{x} \\ \omega v_{x}-2(\gamma ^{2}+\epsilon ^{2})v_{y} \\ -4\epsilon ^{2}-2(\gamma ^{2}+2\epsilon ^{2})v_{z} \\ \end{array}\right) =\frac{1}{2}\left( \begin{array}{c} 0 \\ {\dot{v}}_{x} \\ {\dot{v}}_{y} \\ {\dot{v}}_{z} \\ \end{array}\right) , \label{rhoex}$$where $v_{x}(t)$, $v_{y}(t)$, and $v_{z}(t)$ are real functions providing the coordinates of the quantum state $|\rho (t)\rangle \rangle $ on the Bloch sphere. The Lindblad superoperator is then given by $$\mathcal{L}(t)=\left( \begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & -2\,\epsilon ^{2} & -\omega & 0 \\ 0 & \omega & -2\epsilon ^{2}-2\gamma ^{2} & 0 \\ -4\,\epsilon ^{2} & 0 & 0 & -4\epsilon ^{2}-2\gamma ^{2}\end{array}\right) . \label{lmat}$$In order to exhibit an example that has a nontrivial Jordan block structure, we now assume $\gamma ^{2}=\omega $ (which can in practice be obtained by measuring the relaxation rate $\gamma $ and correspondingly adjusting the system frequency $\omega $). We then have three different eigenvalues for $\mathcal{L}(t)$, $$\begin{aligned} \lambda _{1} &=&0, \\ \lambda _{2} &=&-2\epsilon ^{2}-\gamma ^{2}\text{ }\mathrm{(twofold\,\, degenerate)} \\ \lambda _{3} &=&-4\epsilon ^{2}-2\gamma ^{2},\end{aligned}$$which are associated with the following three independent (unnormalized) right eigenvectors: $$|\mathcal{D}_{1}^{(0)}\rangle \rangle =\left( \begin{array}{c} f(\gamma ,\epsilon ) \\ 0 \\ 0 \\ 1 \\ \end{array}\right) ,|\mathcal{D}_{2}^{(0)}\rangle \rangle =\left( \begin{array}{c} 0 \\ 1 \\ 1 \\ 0 \\ \end{array}\right) ,|\mathcal{D}_{3}^{(0)}\rangle \rangle =\left( \begin{array}{c} 0 \\ 0 \\ 0 \\ 1 \\ \end{array}\right) , \label{dex}$$with $f(\gamma ,\epsilon )=-1-(\gamma ^{2}/2\epsilon ^{2})$. Similarly, for the left eigenvectors, we find $$\begin{aligned} \langle \langle \mathcal{E}_{1}^{(0)}| &=&\left( \frac{{}}{{}}1/f(\gamma ,\epsilon ),0,0,0\frac{{}}{{}}\right) , \nonumber \\ \vspace{0.1cm}\langle \langle \mathcal{E}_{2}^{(1)}| &=&\left( \frac{{}}{{}}0,\gamma ^{2},-\gamma ^{2},0\frac{{}}{{}}\right) , \nonumber \\ \vspace{0.1cm}\langle \langle \mathcal{E}_{3}^{(0)}| &=&\left( \frac{{}}{{}}-1/f(\gamma ,\epsilon ),0,0,1\frac{{}}{{}}\right) . \label{rex}\end{aligned}$$The Jordan form of $\mathcal{L}(t)$ can then be written as $$\mathcal{L}_{J}(t)=\left( \begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & -2\epsilon ^{2}-\gamma ^{2} & 1 & 0 \\ 0 & 0 & -2\epsilon ^{2}-\gamma ^{2} & 0 \\ 0 & 0 & 0 & -4\epsilon ^{2}-2\gamma ^{2}\end{array}\right) , \label{ljmat}$$(observe the two-dimensional middle Jordan block), with the transformation matrix leading to the Jordan form being $$S(t)=\left( \begin{array}{cccc} f(\gamma ,\epsilon ) & 0 & 0 & 0 \\ 0 & 1 & \gamma ^{-2} & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1\end{array}\right) . \label{smat}$$Note that, in our example, each eigenvalue of $\mathcal{L}(t)$ is associated to a unique Jordan block, since we do not have more than one independent eigenvector for each $\lambda _{\alpha }$. We then expect that the adiabatic regime will be characterized by an evolution which can be decomposed by single Jordan blocks. In order to show that this is indeed the case, let us construct a right and left basis preserving the block structure. To this end, we need to introduce a right and a left vector for the Jordan block related to the eigenvalue $\lambda _{2}$. As in Eqs. (\[dj\]) and (\[rj\]), we define the additional states as $$|\mathcal{D}_{2}^{(1)}\rangle \rangle _{J}=\left( \begin{array}{c} 0 \\ 0 \\ 1 \\ 0 \\ \end{array}\right) ,\,\,\,_{J}\langle \langle \mathcal{E}_{2}^{(0)}|=\left( \frac{{}}{{}}0,1,0,0\frac{{}}{{}}\right) . \label{drjex}$$We then obtain, after applying the transformations $|\mathcal{D}_{2}^{(1)}(t)\rangle \rangle =S(t)\,|\mathcal{D}_{2}^{(1)}\rangle \rangle _{J}$ and $\langle \langle \mathcal{E}_{2}^{(0)}(t)|=\,_{J}\langle \langle \mathcal{E}_{2}^{(0)}|\,S^{-1}(t)$, the right and left vectors $$|\mathcal{D}_{2}^{(1)}\rangle \rangle =\left( \begin{array}{c} 0 \\ \gamma ^{-2} \\ 0 \\ 0 \\ \end{array}\right) ,\,\,\,\langle \langle \mathcal{E}_{2}^{(0)}|=\left( \frac{{}}{{}}0,0,1,0\frac{{}}{{}}\right) . \label{drex}$$Expanding the coherence vector in the basis $\left\{ |\mathcal{D}_{\alpha }^{(j)}(t)\rangle \rangle \right\} $, as in Eq. (\[rtime\]), the master equation (\[le\]) yields $$\begin{aligned} &&\hspace{-0.3cm}f(\gamma ,\epsilon )\,{\dot{r}}_{1}^{(0)}+{\dot{f}}(\gamma ,\epsilon )\,r_{1}^{(0)}=0, \nonumber \\ &&\hspace{-0.3cm}{\dot{r}}_{2}^{(0)}-2\frac{{\dot{\gamma}}}{\gamma ^{3}}r_{2}^{(1)}+\frac{{\dot{r}}_{2}^{(1)}}{\gamma ^{2}}=-\left( 2\epsilon ^{2}+\gamma ^{2}\right) r_{2}^{(0)}-2\frac{\epsilon ^{2}}{\gamma ^{2}}r_{2}^{(1)}, \nonumber \\ &&\hspace{-0.3cm}{\dot{r}}_{2}^{(0)}=r_{2}^{(1)}-\left( 2\epsilon ^{2}+\gamma ^{2}\right) r_{2}^{(0)}, \nonumber \\ &&\hspace{-0.3cm}{\dot{r}}_{1}^{(0)}+{\dot{r}}_{3}^{(0)}=\left( -4\epsilon ^{2}-2\gamma ^{2}\right) \,r_{3}^{(0)}, \label{glee}\end{aligned}$$It is immediately apparent from Eq. (\[glee\]) that the block related to the eigenvalue $\lambda _{2}$ is already decoupled from the rest. On the other hand, by virtue of the last equation, the blocks associated to $\lambda _{1}$ and $\lambda _{3}$ are coupled, implying a mixing in the evolution of the coefficients $r_{1}^{(0)}(t)$ and $r_{3}^{(0)}(t)$. The role of the adiabaticity will then be the suppression of this coupling. We note that in this simple example, the coupling between $r_{1}^{(0)}(t)$ and $r_{3}^{(0)}(t)$ would in fact also be eliminated by imposing the probability conservation condition $\mathrm{Tr}\rho =1$. However, in order to discuss the effects of the adiabatic regime, let us permit a general time evolution of all coefficients (i.e., probability leakage) and analyze the adiabatic constraints. The validity condition for adiabatic dynamics, given by Eq. (\[vch\]), yields $$\left\vert \frac{\langle \langle \mathcal{E}_{3}^{(0)}|\,{\dot{\mathcal{L}}}\,|\mathcal{D}_{1}^{(0)}\rangle \rangle }{\lambda _{1}-\lambda _{3}}\right\vert =\left\vert \frac{2\gamma ^{2}{\dot{\epsilon}}/\epsilon -2\gamma {\dot{\gamma}}}{\gamma ^{2}+2\epsilon ^{2}}\right\vert \ll \left\vert \lambda_{1}-\lambda_{3} \right\vert. \label{exvc}$$ We first note that we have here the possibility of an adiabatic evolution even without ${\dot{\mathcal{L}}}(t)\approx 0$ in general (i.e., for all its matrix elements). Indeed, solving $\gamma ^{2}{\dot{\epsilon}}/\epsilon =\gamma {\dot{\gamma}}$, Eq. (\[exvc\]) implies that independent evolution in Jordan blocks will occur for $\epsilon (t)\propto \gamma (t)$.  Since $f(\gamma ,\epsilon )=-1-(\gamma ^{2}/2\epsilon ^{2})$ is then constant in time, it follows, from Eq. (\[glee\]), that $r_{1}^{(0)}(t)$ is constant in time, which in turn ensures the decoupling of $r_{1}^{(0)}(t)$ and $r_{3}^{(0)}(t)$. In this case, it is a *dynamical symmetry* (constancy of the ratio of magnitudes of the spontaneous emission and bit-flip processes), rather than the general slowness of ${\dot{\mathcal{L}}}(t)$, that is responsible for the adiabatic behavior. The same conclusion is also obtained from the adiabatic condition (\[vc\]). Of course, Eq. (\[exvc\]) is automatically satisfied if $\mathcal{L}(t)$ is slowly varying in time, which means ${\dot{\gamma}}(t)\approx 0$ and ${\dot{\epsilon}}(t)\approx 0$. Assuming this last case, the following solution is found: $$\begin{aligned} &&r_{1}^{(0)}(t)=r_{1}^{(0)}(0), \nonumber \\ &&r_{2}^{(0)}(t)=\left[ r_{2}^{(1)}(0)\,t+r_{2}^{(0)}(0)\right] \,{e}^{(-2\epsilon ^{2}-\gamma ^{2})\,t}, \nonumber \\ &&r_{2}^{(1)}(t)=r_{2}^{(1)}(0)\,{e}^{(-2\epsilon ^{2}-\gamma ^{2})\,t}, \nonumber \\ &&r_{3}^{(0)}(t)=r_{3}^{(0)}(0)\,{e}^{(-4\epsilon ^{2}-2\gamma ^{2})\,t}. \label{solex}\end{aligned}$$It is clear that the evolution is independent in the three distinct Jordan blocks, with functions $r_{\alpha }^{(i)}(t)$ belonging to different sectors evolving separately. The only mixing is between $r_{2}^{(0)}(t)$ and $r_{2}^{(1)}(t)$, which are components of the the same block. The decoupling of the coefficients $r_{1}^{(0)}(t)$ and $r_{3}^{(0)}(t)$ in the adiabatic limit is exhibited in Fig. \[f1\]. Observe that the adiabatic behavior is recovered as the dependence of $\epsilon (t)$ and $\gamma (t)$ on $t$ becomes negligible. The original coefficients $v_{x}$, $v_{y}$, and $v_{z}$ in the Bloch sphere basis $\left\{ I_{2},\sigma _{x},\sigma _{y},\sigma _{z}\right\} $ can be written as combinations of the functions $r_{\alpha }^{(i)}$. Equation ([solex]{}) yields $$\begin{aligned} v_{x}(t) &=&\left( v_{x}(0)+(v_{x}(0)-v_{y}(0))\gamma ^{2}\,t\right) \,e^{(-2\epsilon ^{2}-\gamma ^{2})\,t}, \nonumber \\ v_{y}(t) &=&\left( v_{y}(0)+(v_{x}(0)-v_{y}(0))\gamma ^{2}\,t\right) \,e^{(-2\epsilon ^{2}-\gamma ^{2})\,t}, \nonumber \\ v_{z}(t) &=&\left( v_{z}(0)-\frac{1}{f(\gamma ,\epsilon )}\right) e^{(-4\epsilon ^{2}-2\gamma ^{2})\,t}+\frac{1}{f(\gamma ,\epsilon )} \label{evs}\end{aligned}$$with the initial conditions $$\begin{aligned} &&v_{x}(0)=r_{2}^{(0)}(0)+\gamma ^{-2}r_{2}^{(1)}(0), \nonumber \\ &&v_{y}(0)=r_{2}^{(0)}(0), \nonumber \\ &&v_{z}(0)=\frac{1}{f(\gamma ,\epsilon )}+r_{3}^{(0)}(0),\end{aligned}$$where now $r_{1}^{(0)}(0)=1/f(\gamma ,\epsilon )$ has been imposed in order to satisfy the $\mathrm{Tr}\rho =1$ normalization condition. The Bloch sphere is then characterized by an asymptotic decay of the Bloch coordinates $v_{x}$ and $v_{y}$, with $v_{z}$ approaching the constant value $1/f(\gamma ,\epsilon )$. Finally, let us comment on the analysis of adiabaticity in terms of the conditions derived in Sec. \[sec:tot-t\] for the total time of evolution. Looking at the matrix elements of ${\dot {\cal L}} (t)$, it can be shown that, for $\beta\ne \alpha$, the only term $V_{\beta \alpha }^{(ijp)}$ defined by Eq. (\[vbapj\]) which can be *a priori* nonvanishing is $V_{13}$. Therefore, we have to consider the energy difference $\omega _{13}=4\epsilon ^{2}+2\gamma ^{2}$. Assuming that the decoherence parameters $\epsilon $ and $\gamma $ are nonvanishing, we have $\omega _{13}>0$ and hence $\Omega _{13}>0$. This signals the breakdown of adiabaticity, unless $V_{13}=0$. However, as we saw above, $V_{13}\propto \langle \langle \mathcal{E}_{3}^{(0)}|\,{\dot{\mathcal{L}}}\,|\mathcal{D}_{1}^{(0)}\rangle \rangle =2\gamma ^{2}{\dot{\epsilon}}/\epsilon -2\gamma {\dot{\gamma}}$ and thus $V_{13}=0$ indeed implies the adiabaticity condition $\epsilon (t)\propto \gamma (t)$, in agreement with the results obtained from Theorem \[t1\]. In this (dynamical symmetry) case adiabaticity holds exactly, while if $\epsilon (t)$ is *not* proportional to $\gamma (t)$, then there can be no adiabatic evolution. Thus, the present example, despite nicely illustrating our concept of adiabaticity in open systems, does not present us with the opportunity to derive a nontrivial condition on $T$; such more general examples will be discussed in a future publication. Conclusions and outlook {#conclusions} ======================= The concept of adiabatic dynamics is one of the pillars of the theory of closed quantum systems. Here we have introduced its generalization to open quantum systems. We have shown that under appropriate slowness conditions the time-dependent Lindblad superoperator decomposes into dynamically decoupled Jordan blocks, which are preserved under the adiabatic dynamics. Our key results are summarized in Theorems \[t1\] and \[t3\], which state sufficient (and necessary in the case of Theorem \[t3\]) conditions for adiabaticity in open quantum systems. In particular, Theorem \[t3\] also provides the condition for breakdown of the adiabatic evolution. This feature has no analog in the more restricted case of closed quantum systems. It follows here from the fact that the Jordan eigenvalues of the dynamical superoperator – the generalization of the real eigenvalues of a Hamiltonian – can have an imaginary part, which can lead to unavoidable transitions between Jordan blocks. It is worth mentioning that all of our results have been derived considering systems exhibiting gaps in the Lindblad eigenvalue spectrum. It would be interesting to understand the notion of adiabaticity when no gaps are available, as similarly done for the closed case in Refs. [@Avron:98; @Avron:99]. Moreover, two particularly intriguing applications of the theory presented here are to the study of geometric phases in open systems and to quantum adiabatic algorithms, both of which have received considerable recent attention [@Farhi:00; @Farhi:01; @Thomaz:03; @Carollo:04; @Sanders:04]. We leave these as open problems for future research. M.S.S. gratefully acknowledges the Brazilian agency CNPq for financial support. D.A.L. gratefully acknowledges financial support from NSERC and the Sloan Foundation. 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--- abstract: 'We develop a quantum circuit model describing unitary interactions between quantum fields and a uniformly accelerated object, and apply it to a semi-transparent mirror which uniformly accelerates in the Minkowski vacuum. The reflection coefficient $R_{\omega}$ of the mirror varies between 0 and 1, representing a generalization of the perfect mirror ($R_{\omega}=1$) discussed extensively in the literature. Our method is non-perturbative, not requiring $R_{\omega} \sim 0$. We use the circuit model to calculate the radiation from an eternally accelerated mirror and obtain a finite particle flux along the past horizon provided an appropriate low frequency regularization is introduced. More importantly, it is straightforward to see from our formalism that the radiation is squeezed. The squeezing is closely related to cutting the correlation across the horizon, which therefore may have important implications to the formation of a black hole firewall.' author: - 'Daiqin Su$^{1}$' - 'C. T. Marco Ho$^{1}$' - 'Robert B. Mann$^{1,2,3}$' - 'Timothy C. Ralph$^{1}$' title: Quantum Circuit Model for a Uniformly Accelerated Mirror --- Introduction ============ It has been well known since the 1970s that a moving mirror can radiate particles [@Moore70; @Fulling76]. A perfect moving mirror acts as a moving boundary and thus changes the states, especially the vacuum, of the quantum fields. For an appropriately chosen accelerated trajectory the radiation flux is thermal, and an analogy [@Davies77; @Walker85; @Carlitz87] can be drawn with Hawking radiation from a collapsing star [@Hawking75] that eventually forms a black hole. Since the thermal fluxes are correlated with the final vacuum fluctuations, some authors [@Wilczek93; @Hotta15] have proposed that the emission of the large amounts of information left in the black hole need not be accompanied by the eventual emission of a large amount of energy, providing a new perspective to the solution of the black hole information paradox [@Hawking76]. The trajectory of a uniformly accelerated mirror is of particular interest. When the mirror is uniformly accelerating, its trajectory is a hyperbola in spacetime, and both the energy flux and particle flux are zero [@Fulling76; @Davies77; @Birrell82; @Grove86]. Particles and energy are only radiated when the acceleration of the mirror changes. In the case that the mirror eternally accelerates, the energy flux along the horizon is divergent [@Frolov79; @Frolov80; @Frolov99]. This divergence is evidently related to the ideal assumption that the mirror accelerates for infinitely long time. One way to get rid of the divergence is to turn on and off the mirror so that effectively it interacts with the fields for a finite time [@Obadia01; @Obadia03; @Parentani03]. In this paper, we develop a quantum circuit model to describe unitary interactions between quantum fields and a uniformly accelerated object (such as a mirror, cavity, squeezer [*etc.*]{}). Our circuit model can be considered a further development of the matrix formalism first proposed by Obadia and Parentani [@Obadia01] to describe a mirror following general trajectories. We concentrate on a uniformly accelerated object because the transformations between Minkowski modes, Rindler modes and Unruh modes are well known [@Unruh76; @Takagi86; @Crispino08] and can be represented by some simple quantum optical elements, like two-mode squeezers, beamsplitters [*etc.*]{} As an application of our circuit model, we revisit the uniformly accelerated mirror problem in $(1+1)$-dimensional Minkowski spacetime. Unlike the self-interaction model proposed by Obadia and Parentani [@Obadia01], which requires a perturbative expansion and is valid only for low reflection coefficients, our circuit model is non-pertubative insofar as it is valid for any value of the reflection coefficient. For the eternally accelerated mirror, the radiation flux in a localized wave packet mode is divergent. We can regularize this infrared divergence by introducing a low-frequency cutoff for the mirror, which means the mirror is transparent for the low-frequency field modes (to some extent, this is physically equivalent to having the mirror interact with the field for a finite period of time). After infrared regularization the particle number in a localized wave packet mode is finite. We further study the properties of the radiation flux and find that the radiation field is squeezed. This squeezing effect has gone unnoticed up to now, but in our circuit model it is a very straightforward result. We show that the generation of squeezing is closely related to cutting the correlations across the horizon. This mechanism of transferring correlations to squeezing may have important implications for black hole firewalls [@AMPS; @Braunstein13], as we shall subsequently discuss. Our paper is organized as follows. In Sec. \[RindlerUnruh\], we briefly review the relations between Rindler modes and Unruh modes. Motivated from these transformations, we introduce our circuit model in Sec. \[circuit\] and calculate the radiation flux from an eternally accelerated mirror in Sec. \[radiation\]. In Sec. \[squeezing\], we show that the radiation field from the accelerated mirror is squeezed and the squeezing is related to the correlations across the horizon. In Sec. \[firewall\], we propose that a Rindler firewall can be generated by a uniformly accelerated mirror and we conjecture that a black hole firewall could be squeezed. We conclude in Sec. \[conclusion\]. In this paper, we take the unit $\hbar = c =1$. Rindler modes and Unruh modes {#RindlerUnruh} ============================= In this section we describe the relations between Rindler modes and Unruh modes, which act as the foundation of our quantum circuit model. We begin with a brief review of the three ways of quantizing a massless scalar field $\hat{\Phi}$ in $(1+1)$-dimensional Minkowski spacetime (for comprehensive reviews, see [@Takagi86; @Crispino08]). A massless scalar field $\hat{\Phi}$ satisfies the Klein-Gordon equation $\Box \hat{\Phi} = 0$, where the d’ Alembertian $\Box \equiv (\sqrt{-g})^{-1} \partial_{\mu} [\sqrt{-g}g^{\mu\nu} \partial_{\nu}]$ and $g_{\mu\nu}$ is the metric of the spacetime [@Birrell82]. In the inertial frame, Minkowski coordinates $(t, x)$ are used and the metric $g_{\mu\nu} = \eta_{\mu\nu} = \text{diag}\{-1, +1\}$. The scalar field $\hat{\Phi}$ can be quantized in the standard way, $$\label{MinkowskiMode} \hat{\Phi} = \int d k \big(\hat{a}_{1k} u_{1k} + \hat{a}_{2k} u_{2k} + \text{h.c.} \big),$$ where h.c. represents Hermitian conjugate, $u_{1k}~(u_{2k})$ are single-frequency left-moving (right-moving) mode functions $$u_{1k}(V) = (4 \pi k)^{-1/2} e^{-i k V}, \,\,\,\,\,\, u_{2k}(U) =(4 \pi k)^{-1/2} e^{-i k U},$$ with $V = t+x, U = t-x$. $\hat{a}_{1k}(\hat{a}_{2k})$, $\hat{a}_{1k}^{\dag} (\hat{a}_{2k}^{\dag} )$ are the corresponding annihilation and creation operators satisfying the bosonic commutation relations $$\begin{aligned} [\hat{a}_{mk}, \hat{a}_{nk'}^{\dag}] = \delta_{mn} \delta({k-k'}), \,\,\, [\hat{a}_{mk}, \hat{a}_{nk'}] = [\hat{a}_{mk}^{\dag}, \hat{a}_{nk'}^{\dag}] = 0, \nonumber\end{aligned}$$ with $m,n = 1,2$. The Minkowski vacuum state $| 0_M \rangle$ is defined as $\hat{a}_{mk} | 0_M \rangle = 0$. ![Four wedges of $(1+1)$-dimensional Minkowski spacetime: $R, L, F$ and $P$. The right Rindler wedge ($R$) is causally disconnected to the left Rindler wedge ($L$). The Rindler coordinates $(\tau, \xi)$ only cover the $R$ wedge and $(\bar{\tau}, \bar{\xi})$ only cover the $L$ wedge. []{data-label="RindlerFrame"}](f1.png){width="8.0cm"} As shown in Fig. \[RindlerFrame\], Minkowski spacetime can be divided into four wedges: $R, L, F$ and $P$. We introduce Rindler coordinates $(\tau, \xi)$ in the $R$ wedge and $(\bar{\tau}, \bar{\xi})$ in the $L$ wedge, $$\begin{aligned} t &=& a^{-1} e^{a \xi} \sinh(a \tau), \,\,\, x = a^{-1} e^{a \xi} \cosh(a \tau), \nonumber \\ t &=& -a^{-1} e^{a \bar{\xi}} \sinh(a \bar{\tau}), \,\,\, x = - a^{-1} e^{a \bar{\xi}} \cosh(a \bar{\tau}),\end{aligned}$$ where $\tau$ is the proper time of the uniformly accelerated observer with proper acceleration $a$ in the $R$ wedge. The metric is $g_{\mu\nu} = e^{2 a \xi} \text{diag}\{- 1, +1 \}$ in $R$ and is $g_{\mu\nu} = e^{2 a \bar{\xi}} \text{diag}\{- 1, +1 \}$ in $L$. It is obvious that the vector field $\partial_{\tau}~(\text{or}~ \partial_{\bar{\tau}})$ is the timelike Killing vector field of the spacetime [@Birrell82]. In the Rindler frame, the scalar field $\hat{\Phi}$ can be quantized as [@Fulling73; @Unruh76] $$\begin{aligned} \hat{\Phi} = \int d \omega (\hat{b}_{1\omega}^R g_{1\omega}^R + \hat{b}_{1\omega}^L g_{1\omega}^L + \hat{b}_{2\omega}^R g_{2\omega}^R + \hat{b}_{2\omega}^L g_{2\omega}^L + \text{h.c.} ) $$ where the superscripts $``R"$ and $``L"$ represent modes and operators in the $R$ and $L$ wedge, respectively. The modes $g_{m\omega}^R~(g_{m\omega}^L)$ only have support in the $R$ ($L$) wedge, $$g^R_{1\omega}(v) = (4 \pi \omega)^{-1/2} e^{-i \omega v}, \,\,\, g^R_{2\omega}(u) = (4 \pi \omega)^{-1/2} e^{-i \omega u},$$ where $v = \tau + \xi$, $u = \tau - \xi$, and by replacing $v, u$ by $\bar{v} = - \bar{\tau} - \bar{\xi}$ and $\bar{u} = - \bar{\tau} + \bar{\xi}$ we obtain modes in the $L$ wedge. Note that we have used the prescription that $\partial_{\bar{\tau}}$ is past-directed. The commutation relations of the operators are $$\begin{aligned} [\hat{b}_{m\omega}^R, \hat{b}_{n\omega'}^{R\dag} ] = \delta_{mn} \delta({\omega-\omega'}), \,\,\,\, [\hat{b}_{m\omega}^L, \hat{b}_{n\omega'}^{L\dag} ] = \delta_{mn} \delta({\omega-\omega'}), \end{aligned}$$ with all others vanishing. The Rindler vacuum state $| 0_R \rangle$ is defined as $\hat{b}_{m\omega}^R | 0_R \rangle = \hat{b}_{m\omega}^L | 0_R \rangle =0$. It proves useful to introduce Unruh modes (instead of Minkowski modes) that cover the whole Minkowski spacetime for two reasons: 1) the Unruh and Minkowski modes share the same vacuum; 2) the transformation between Rindler modes and Unruh modes is a two-mode squeezing transformation. The Unruh modes are defined as $$\begin{aligned} \label{UnruhRindler} \hat{c}_{m\omega} &=& \text{cosh}(r_{\omega}) \hat{b}_{m\omega}^R - \text{sinh}(r_{\omega}) \hat{b}_{m\omega}^{L\dag}, \nonumber \\ \hat{d}_{m\omega} &=& \text{cosh}(r_{\omega}) \hat{b}_{m\omega}^L - \text{sinh}(r_{\omega}) \hat{b}_{m\omega}^{R\dag},\end{aligned}$$ where $r_{\omega}$ satisfies $\text{tanh}(r_{\omega}) = e^{-\pi \omega/a}$. It is easy to find the inverse transformation, $$\begin{aligned} \label{UnruhRindler:inverse} \hat{b}_{m\omega}^R &=& \text{cosh}(r_{\omega}) \hat{c}_{m\omega} + \text{sinh}(r_{\omega}) \hat{d}_{m\omega}^{\dag}, \nonumber\\ \hat{b}_{m\omega}^L &=& \text{cosh}(r_{\omega}) \hat{d}_{m\omega} + \text{sinh}(r_{\omega}) \hat{c}_{m\omega}^{\dag}. \end{aligned}$$ We can see that the Rindler modes $(\hat{b}_{m\omega}^R, \hat{b}_{m\omega}^L)$ and Unruh modes $(\hat{c}_{m\omega}, \hat{d}_{m\omega} )$ are related by a two-mode squeezing operator with a frequency dependent squeezing parameter $r_{\omega}$. In terms of Unruh modes, the scalar field $\hat{\Phi}$ can be expressed as $$\begin{aligned} \label{UnruhMode} \hat{\Phi} &=& \int d \omega (\hat{c}_{1\omega} G_{1\omega} + \hat{d}_{1\omega} \bar{G}_{1\omega} \nonumber \\&&\qquad\qquad + \hat{c}_{2\omega} G_{2\omega} + \hat{d}_{2\omega} \bar{G}_{2\omega} + \text{h.c.} )\end{aligned}$$ where $$\begin{aligned} \label{Unruhmodes} G_{1\omega}(V) &=& F(\omega, a) (aV)^{-i \omega/a}, \nonumber\\ \bar{G}_{1\omega}(V) &=& F(\omega, a) (-aV)^{i \omega/a}, \nonumber\\ G_{2\omega}(U) &=& F(\omega, a) (-aU)^{i \omega/a}, \nonumber\\ \bar{G}_{2\omega}(U) &=& F(\omega, a) (aU)^{-i \omega/a},\end{aligned}$$ with $F(\omega, a) \equiv \frac{e^{\pi \omega/2a}}{\sqrt{4\pi \omega} \sqrt{2\sinh(\pi \omega/a)}}$. $G_{1\omega}(V)$ and $\bar{G}_{2\omega}(U)$ are analytic in the lower-half complex plane while $\bar{G}_{1\omega}(V)$ and $G_{2\omega}(U)$ are analytic in the upper-half complex plane. The Unruh modes annihilate the Minkowski vacuum state $$\begin{aligned} \hat{c}_{m\omega} | 0_M \rangle = \hat{d}_{m\omega} | 0_M \rangle = 0\end{aligned}$$ as noted above. Circuit model {#circuit} ============= General formalism ----------------- How are the states of a quantum field affected by an object (such as a beamsplitter) that is uniformly accelerated in the $R$ wedge? This is the question of central interest in this paper. A straightforward way to study this problem is to work in the accelerated frame in which the object is static. It is obvious that the object only interacts with Rindler modes in the $R$ wedge and the Rindler modes in the $L$ wedge remain unaffected. The interaction between the object and the Rindler modes is unitary and it transforms the Rindler modes as $$\label{generalcoupling} \hat{b}_{mk}^{\prime R} = \int d \omega \bigg( \alpha^{m1}_{k \omega} \hat{b}_{1\omega}^R + \beta^{m1}_{k \omega} \hat{b}_{1\omega}^{R \dagger} + \alpha^{m2}_{k \omega} \hat{b}_{2 \omega}^R + \beta^{m2}_{k \omega} \hat{b}_{2\omega}^{R \dagger} \bigg). \\$$ This is the most general interaction which not only couples the left-moving and right-moving Rindler modes but also Rindler modes with different frequencies. Together with Eqs. (\[UnruhRindler\]) and (\[UnruhRindler:inverse\]), we can construct a quantum circuit model (or input-output formalism) for the uniformly accelerated object. We start from the inertial frame in which Unruh modes are used instead of Minkowski modes. This makes the model simpler although we still have to transform the Minkowski modes to the Unruh modes and vice versa. First, based on Eq. (\[UnruhRindler:inverse\]), the Unruh modes pass through a collection of two-mode squeezers each of which couples a pair of Unruh modes $(\hat{c}_{m\omega}, \hat{d}_{m\omega} )$ with frequency dependent squeezing parameter $r_{\omega}$. Second, the output right Rindler modes $\hat{b}_{m\omega}^R $ interact with the object and are transformed to $\hat{b}_{mk}^{\prime R}$ according to Eq. (\[generalcoupling\]) while the left Rindler modes $\hat{b}_{m\omega}^L$ remain unchanged. Finally, based on Eq. (\[UnruhRindler\]), the Rindler modes pass through a collection of two-mode antisqueezers and are transformed to output Unruh modes $(\hat{c}^{\prime}_{m\omega}, \hat{d}^{\prime}_{m\omega} )$. If we use an inertial detector to detect the radiation field from the accelerated object, we have to transform the Unruh modes $(\hat{c}^{\prime}_{m\omega}, \hat{d}^{\prime}_{m\omega} )$ to Minkowski modes to model the coupling with the detector. In the special case that the interaction does not couple Rindler modes with different frequencies, the input-output formalism is substantially simplified. The coefficients $\alpha^{mn}_{k \omega}$ and $\beta^{mn}_{k \omega}$ are now proportional to $\delta(k - \omega)$ so Eq. (\[generalcoupling\]) becomes $$\label{nomixingfrequency} \hat{b}_{m\omega}^{\prime R} = \alpha^{m1}_{\omega \omega} \hat{b}_{1\omega}^R + \beta^{m1}_{\omega \omega} \hat{b}_{1\omega}^{R \dagger} + \alpha^{m2}_{\omega \omega} \hat{b}_{2 \omega}^R + \beta^{m2}_{\omega \omega} \hat{b}_{2\omega}^{R \dagger}.$$ Since modes with different frequencies are independent, we can propose a quantum circuit model for each single frequency. The quantum circuit is shown in Fig. \[circuit:Fig\]. A pair of left-moving Unruh modes $(\hat{c}_{1\omega}, \hat{d}_{1\omega})$ and a pair of right-moving Unruh modes $(\hat{c}_{2\omega}, \hat{d}_{2\omega})$ pass through the two-mode squeezers $S_{\omega}$, from which emerge left-moving Rindler modes $(\hat{b}_{1\omega}^R, \hat{b}_{1\omega}^L)$ and right-moving Rindler modes $(\hat{b}_{2\omega}^R, \hat{b}_{2\omega}^L)$, respectively. $\hat{b}_{1\omega}^R$ and $\hat{b}_{2\omega}^R$ interact with each other when passing through the object (symbolized by the black dot in Fig. \[circuit:Fig\]) and emerge as $\hat{b}_{1\omega}^{\prime R}$ and $\hat{b}_{2\omega}^{\prime R}$, which can be described by a unitary transformation $U_{\omega}$ according to Eq. (\[nomixingfrequency\]). After that, the Rindler modes are combined by two-mode antisqueezers $S^{-1}_{\omega}$, ending up with Unruh modes again. ![(color online). Unruh modes pass through the squeezers and then become Rindler modes. The Rindler modes in the right Rindler wedge interact with the object ($U_{\omega}$) and then combine with the Rindler modes from the left Rindler wedge in the antisqueezers, going back to Unruh modes again. []{data-label="circuit:Fig"}](f2.png){width="8.0cm"} For computational purposes, we introduce operator vectors $\hat{\bf c}_{\omega}$, $\hat{\bf d}_{\omega}$, $\hat{\bf b}_{\omega}^R$ and $\hat{\bf b}_{\omega}^L$, which are defined as $$\begin{aligned} \hat{\bf c}_{\omega} = {\hat{c}_{\omega} \choose \hat{c}_{\omega}^{\dag}}, \,\,\,\, \hat{\bf d}_{\omega} = {\hat{d}_{\omega} \choose \hat{d}_{\omega}^{\dag}}, \,\,\,\, \hat{\bf b}_{\omega}^R = {\hat{b}_{\omega}^R \choose \hat{b}_{\omega}^{R\dag}},\,\,\,\, \hat{\bf b}_{\omega}^L = {\hat{b}_{\omega}^L \choose \hat{b}_{\omega}^{L\dag}}.\end{aligned}$$ Then Eqs. (\[UnruhRindler\]) and (\[UnruhRindler:inverse\]) can be rewritten as $$\begin{aligned} {\hat{\bf c}_{m\omega} \choose \hat{\bf d}_{m\omega}} = S^{-1}_{\omega} {\hat{\bf b}_{m\omega}^R \choose \hat{\bf b}_{m\omega}^L}, \,\,\,\,\,\, {\hat{\bf b}_{m\omega}^R \choose \hat{\bf b}_{m\omega}^L} =S_{\omega}{\hat{\bf c}_{m\omega} \choose \hat{\bf d}_{m\omega}},\end{aligned}$$ with $$\begin{aligned} S_{\omega} \equiv \left({I \text{cosh}(r_{\omega}) \atop \sigma_x \text{sinh}(r_{\omega})}{ \sigma_x \text{sinh}(r_{\omega}) \atop I \text{cosh}(r_{\omega})}\right)\end{aligned}$$ where $I = \left({1 \atop 0}{0 \atop 1} \right)$ is the identity matrix and $\sigma_x = \left({0 \atop 1}{1 \atop 0} \right)$ is one of the Pauli matrices. The transformation between the input Unruh modes $(\hat{\bf c}_{1\omega}, \hat{\bf d}_{1\omega}, \hat{\bf c}_{2\omega}, \hat{\bf d}_{2\omega})^T$ and the output Unruh modes $(\hat{\bf c}^{\prime}_{1\omega} \hat{\bf d}^{\prime}_{1\omega}, \hat{\bf c}^{\prime}_{2\omega}, \hat{\bf d}^{\prime}_{2\omega})^T$ can be represented as $$\begin{aligned} \label{input-output} \begin{pmatrix} \hat{\bf c}^{\prime}_{1\omega} \\ \hat{\bf d}^{\prime}_{1\omega} \\ \hat{\bf c}^{\prime}_{2\omega} \\ \hat{\bf d}^{\prime}_{2\omega} \end{pmatrix} =\mathcal{S}^{-1}_{\omega} \mathcal{U}_{\omega} \mathcal{S}_{\omega} \begin{pmatrix} \hat{\bf c}_{1\omega} \\ \hat{\bf d}_{1\omega} \\ \hat{\bf c}_{2\omega} \\ \hat{\bf d}_{2\omega} \end{pmatrix}.\end{aligned}$$ $\mathcal{S}_{\omega}$ characterizes the transformation from Unruh modes to Rindler modes $$\begin{aligned} \mathcal{S}_{\omega} = \left({S_{\omega} \atop 0}{ 0 \atop S_{\omega}}\right)\end{aligned}$$ and $\mathcal{U}_{\omega}$ characterizes the action of the object $$\begin{aligned} \mathcal{U}_{\omega} = \begin{pmatrix} U^{11}_{\omega} & 0 & U^{12}_{\omega} & 0 \\ 0 & I & 0 & 0 \\ U^{21}_{\omega} & 0 & U^{22}_{\omega} & 0\\ 0 & 0 & 0 & I \end{pmatrix}\end{aligned}$$ where $$\begin{aligned} U^{mn}_{\omega} = \left({\alpha^{mn}_{\omega \omega} \atop \beta^{mn*}_{\omega \omega}}{ \beta^{mn}_{\omega \omega} \atop \alpha^{mn*}_{\omega \omega}}\right).\end{aligned}$$ We emphasize that the general formalism developed here is valid for a wide class of quantum optical devices (objects), such as beamsplitters, single-mode squeezers, two-mode squeezers, cavities, and even for devices with time-dependent parameters, for example, beamsplitters with time-dependent transmission coefficients. In this paper, we mainly apply the formalism to the simplest case, a beamsplitter. Circuit model for a uniformly accelerated mirror ------------------------------------------------ The perfect moving mirror problem has been extensively studied for several decades. A perfect moving mirror provides a clear boundary for a quantum field, which vanishes along the mirror’s trajectory. The standard method for calculating the radiation from a perfect moving mirror is to find the Bogoliubov transformation between the input and output modes by taking into account the Dirichlet boundary condition. However a realistic mirror is not perfect but usually partially transparent, for which the Dirichlet boundary condition is not satisfied. In this paper, we are interested in a uniformly accelerated imperfect mirror whose motion looks nontrivial for an inertial observer. Rather than use the standard method (which is still valid if appropriate boundary conditions are considered), we shall employ the circuit model developed in the previous section, leading to a much simpler way to attack this problem. The idea is to work in the accelerated frame, in which the mirror is static and can be considered as a beamsplitter. Without loss of generality, we assume that the mirror uniformly accelerates in the $R$ wedge. The beamsplitter transforms the right Rindler modes as $$\begin{aligned} \label{beamsplitter} \hat{b}^{\prime R}_{1\omega} &=&\text{cos}\,\theta_{\omega} \hat{b}^R_{1\omega}- i e^{i \phi_{\omega}} \text{sin}\,\theta_{\omega} \hat{b}^R_{2\omega}, \nonumber\\ \hat{b}^{\prime R}_{2\omega} &=&\text{cos} \,\theta_{\omega} \hat{b}^R_{2\omega}- i e^{-i \phi_{\omega}} \text{sin}\,\theta_{\omega} \hat{b}^R_{1\omega},\end{aligned}$$ where $\theta_{\omega}$ an $\phi_{\omega}$ are frequency dependent. The relative phase shift $i e^{\pm i \phi_{\omega}} $ ensures that the transformation is unitary. The intensity reflection and transmission coefficients of the beamsplitter are $$\begin{aligned} R_{\omega} = \text{sin}^2\,\theta_{\omega}, \,\,\,\,\,\,\,\, T_{\omega} = \text{cos}^2\,\theta_{\omega}. \end{aligned}$$ By comparing Eqs. (\[beamsplitter\]) and (\[nomixingfrequency\]) we have $$\begin{aligned} \alpha^{11}_{\omega \omega} &=& \alpha^{22}_{\omega \omega} = \cos \theta_{\omega}, \nonumber \\ \alpha^{12}_{\omega \omega} &=& - \alpha^{21*}_{\omega \omega} = -i e^{i \phi_{\omega}} \sin \theta_{\omega},\end{aligned}$$ and all $\beta^{mn}_{\omega \omega}$ are zero. We can therefore express the action of the beamsplitter as $$\begin{aligned} \label{mirrortransformation} \mathcal{U}_{\omega} = \begin{pmatrix} I \text{cos}\,\theta_{\omega}& 0 & Z \text{sin}\,\theta_{\omega} & 0 \\ 0 & I & 0 & 0 \\ -Z^* \text{sin}\,\theta_{\omega} & 0 & I \text{cos}\,\theta_{\omega} & 0\\ 0 & 0 & 0 & I \end{pmatrix},\end{aligned}$$ where $I$ is the $2\times 2$ identity matrix and $$Z = \left({-i e^{i \phi_{\omega}} \atop 0}{0 \atop i e^{-i \phi_{\omega}}} \right). $$ The explicit expressions for the transformation Eq. (\[input-output\]) can be calculated straightforwardly and are summarized in Appendix \[appendixA\]. With these transformations, it is easy to calculate the expectation value of the particle number of the output $\hat{c}^{\prime}_{1\omega}$, $$\begin{aligned} \label{ParticleNumber:1} &&\langle 0_M | \hat{c}_{1\omega}^{\prime \dag} \hat{c}_{1\omega^{\prime}}^{\prime} | 0_M \rangle \nonumber \\ &=& 2(1-\text{cos} \,\theta_{\omega})\text{cosh}^2(r_{\omega}) \text{sinh}^2(r_{\omega}) \delta(\omega-\omega^{\prime}) \nonumber \\ &=& 2(1-\text{cos} \,\theta_{\omega})\frac{e^{2\pi \omega/a}}{(e^{2\pi \omega/a}-1)^2} \delta(\omega-\omega^{\prime}) \nonumber \\ &\equiv& n(\omega) \delta(\omega-\omega^{\prime}).\end{aligned}$$ The corresponding expectation values for the other three outputs is the same as Eq. (\[ParticleNumber:1\]). Hence the number of Unruh particles in every output is generally not zero. The particle-number distribution is $$\label{ParticleNumber:2} n(\omega) = 2 (1-\text{cos}\,\theta_{\omega})\frac{e^{2\pi \omega/a}}{(e^{2\pi \omega/a}-1)^2},$$ depending on the transmission coefficient of the uniformly accelerated mirror. Note that $n(\omega) = 0$ only when $\theta_{\omega} = 0$; in other words when the mirror is completely transparent to the field mode with frequency $\omega$. We also note that the distribution of the output Unruh particles is not thermal. Radiation from an eternally accelerated mirror {#radiation} ============================================== ![(color online). A uniformly accelerated mirror on the right Rindler wedge. An inertial detector is placed at an appropriate position to detect left-moving particles coming from the uniformly accelerated mirror. []{data-label="Fig:Mirror"}](f3.png){width="8.0cm"} As an application of the quantum circuit model, we calculate the radiation flux from an eternally accelerated mirror. As shown in Fig. \[Fig:Mirror\], an inertial detector is placed at an appropriate position to detect the left-moving particles radiated by the accelerated mirror. In the previous section, we have shown that the accelerated mirror radiates Unruh particles. However, the inertial detector responds only to Minkowski particles. In order to calculate the response of the inertial detector we need to find the transformation between Unruh modes and Minkowski modes. This can be done by comparing Eqs. (\[MinkowskiMode\]) and (\[UnruhMode\]), and then using the Klein-Gordon inner product [@Birrell82], $$\begin{aligned} \label{MinkowskiUnruh} \hat{a}_k &=& \int d \omega \bigg( \langle u_k, G_{\omega} \rangle\hat{c}^{\prime}_{\omega} + \langle u_k, \bar{G}_{\omega} \rangle\hat{d}^{\prime}_{\omega} \bigg) \nonumber \\ &\equiv& \int d \omega ( A_{k \omega} \hat{c}^{\prime}_{\omega} + B_{k \omega} \hat{d}^{\prime}_{\omega} ),\end{aligned}$$ where $A_{k\omega} = \langle u_k, G_{\omega} \rangle$ and $B_{k\omega} = \langle u_k, \bar{G}_{\omega} \rangle$ are the Bogoliubov transformation coefficients. Since we only consider left-moving modes here, without introducing any confusion, we have omitted the subscript $``1"$. Using the relation between Unruh modes and Rindler modes Eq. (\[UnruhRindler\]) and the relation between Rindler modes and Minkowski modes [@Crispino08], We can find the transformation between Unruh modes and Minkowski modes. A more straightforward way is to directly calculate the Klein-Gordon inner product using the explicit expressions of Unruh modes Eq. (\[Unruhmodes\]). The result is $$\begin{aligned} A_{k\omega} = B^*_{k\omega} = \frac{i \sqrt{2 \sinh(\pi \omega/a)}}{2\pi \sqrt{\omega k}} \Gamma(1-i\omega/a) \bigg(\frac{k}{a}\bigg)^{i\omega/a}, \nonumber \\ $$ where $\Gamma(z)$ is the Gamma function. In realistic quantum optics experiments a detector normally detects localized wave packet modes. In order to take this into account we consider Gaussian wave packet modes defined as $$\label{GaussianMode} \hat{a}(f) = \int_0^{\infty} d k f(k; k_0, \sigma, V_0) \hat{a}_k, $$ where $$f(k; k_0, \sigma, V_0) = \bigg(\frac{1}{2\pi \sigma^2} \bigg)^{1/4} \exp\bigg\{-\frac{(k-k_0)^2}{4\sigma^2} - i k V_0\bigg\}$$ with $k_0$, $\sigma$ and $V_0$ the central frequency, bandwidth and central position, respectively. In the narrow bandwidth limit ($k_0 \gg \sigma$), the integration over $k$ can be approximately calculated to a very good accuracy. When $k_0 \gg \sigma$, the Gaussian wave packet $f(k; k_0, \sigma, V_0)$ is significantly nonzero only for positive $k$, so the range of integration of $k$ can be extended to $(-\infty, \infty)$ without introducing large errors. Secondly, since $f(k; k_0, \sigma, V_0)$ is well localized around $k_0$, those values of $A_{k\omega}$ and $B_{k\omega}$ only near $k_0$ are relevant. Writing [@Downes13] $$\frac{1}{\sqrt{k}}\bigg( \frac{k}{a}\bigg)^{i \omega/a} \approx \frac{1}{\sqrt{k_0}}e^{i\frac{\omega}{k_0}\frac{k}{a}}e^{i\frac{\omega}{a}[\ln(\frac{k_0}{a})-1]}$$ and then expanding $A_{k\omega}$ and $B_{k\omega}$ around $k_0$ yields $$\begin{aligned} \label{A} A_{\omega} \equiv \int_0^{\infty} d k f(k) A_{k \omega} &\approx& i \sqrt{\frac{\sigma}{\pi \omega k_0}} \bigg(\frac{1}{2\pi} \bigg)^{1/4} \sqrt{2\sinh(\pi \omega/a)} \Gamma(1-i\omega/a) e^{i\frac{\omega}{a}\ln(\frac{k_0}{a})} e^{-i k_0 V_0}\exp\bigg\{-\frac{\sigma^2(\omega/a-k_0 V_0)^2}{k_0^2}\bigg\}, \nonumber \\\end{aligned}$$ $$\begin{aligned} \label{B} B_{\omega} \equiv \int_0^{\infty} d k f(k) B_{k \omega} &\approx& -i \sqrt{\frac{\sigma}{\pi \omega k_0}} \bigg(\frac{1}{2\pi} \bigg)^{1/4} \sqrt{2\sinh(\pi \omega/a)} \Gamma(1+i\omega/a) e^{-i\frac{\omega}{a}\ln(\frac{k_0}{a})} e^{-i k_0 V_0}\exp\bigg\{-\frac{\sigma^2(\omega/a+k_0 V_0)^2}{k_0^2}\bigg\} \nonumber \\\end{aligned}$$ up to first order in $k-k_0$. Using Eq. (\[ParticleNumber:1\]) and $$|\Gamma(1-i\omega/a)|^2 = |\Gamma(1+i\omega/a)|^2 = \frac{\pi \omega/a}{\sinh(\pi \omega/a)}$$ the expectation value $N(f) = \langle 0_M | \hat{a}^{\dagger}(f)\hat{a}(f)|0_M \rangle$ of the Gaussian mode particle number is $$\begin{aligned} \label{ParticleNumber} N(f) &=& \int d \omega \int d \omega^{\prime} \langle 0_M |(A^*_{\omega} \hat{c}^{\prime \dagger}_{\omega} +B^*_{\omega}\hat{d}^{\prime \dagger}_{\omega}) (A_{\omega^{\prime}} \hat{c}^{\prime}_{\omega^{\prime}} +B_{\omega'}\hat{d}^{\prime}_{\omega^{\prime}}) |0_M \rangle \nonumber \\ \nonumber \\ &=& 2 \int d \omega (|A_{\omega}|^2+|B_{\omega}|^2)(1-\cos \theta_{\omega})\frac{e^{2\pi \omega/a}}{(e^{2\pi \omega/a}-1)^2}, \nonumber \\ \nonumber \\ &=& \sqrt{\frac{8}{\pi}}\frac{\sigma}{k_0} \int_0^{\infty} d \Omega \bigg\{\exp\bigg[-\frac{2\sigma^2(\Omega-k_0 V_0)^2}{k_0^2}\bigg]+\exp\bigg[-\frac{2\sigma^2(\Omega+k_0 V_0)^2}{k_0^2}\bigg] \bigg\} (1-\cos \theta_{\Omega})\frac{e^{2\pi \Omega}}{(e^{2\pi \Omega}-1)^2}\end{aligned}$$ where $\Omega = \omega/a$ is the dimensionless Rindler frequency. Two special cases are of particular interest. Consider first that the mirror is completely transparent for all modes, that is $\cos^2 \, \theta_{\omega} = 1$. From Eq. (\[ParticleNumber\]), the particle number vanishes, $N(f)=0$. This is not surprising because a completely transparent mirror does nothing to the Minkowski vacuum. The second case is that the mirror is perfect for all modes, that is, $\cos^2 \, \theta_{\omega} = 0$. When $\Omega \rightarrow 0$, $(e^{2\pi \Omega}-1)^{-2} \sim \Omega^{-2} $ and all other factors in the integrand of Eq. (\[ParticleNumber\]) are finite. Therefore, the particle number $N(f)$ is divergent. This infrared divergence occurs because we naively assume that the mirror accelerates for an infinitely long time, which seems physically unreasonable. In the framework of the self-interaction model, the mirror is switched on and off so that one obtains finite particle flux [@Obadia01]. In our circuit model, we could also switch on and off the mirror. However instead we shall use a simpler method of regularization. The idea is to directly introduce a low frequency cutoff for the mirror, that is, the mirror is completely transparent for low-frequency field modes. The mechanism for a physical mirror to reflect electromagnetic waves is that the atoms consisting of the mirror absorb electromagnetic waves and then reemit them again. If the wavelength of the electromagnetic wave is very long, the response time of the mirror is very long. Hence if the mirror accelerates for a finite time, it cannot respond to Rindler modes with characteristic oscillation period longer than the accelerating time. In this sense, introducing a low-frequency cutoff is equivalent to switching on and off the mirror. In higher dimensional spacetime, e.g., $(1+3)$-dimensional spacetime, there is another reason justifying a low-frequency cutoff. A physical mirror with finite size cannot reflect field modes whose wavelengths are much larger than its size. This infrared divergence is not due to the pathological character of a massless scalar field in $(1+1)$-dimensional spacetime [@Coleman73]; it also appears in higher dimensional spacetime [@Frolov99] if the mirror is accelerated for an infinitely long time. ![(color online). Particle number versus central position of the Gaussian wave packet: $k_0/a = 20, a g = 10$. For larger bandwidth (narrower wave packet in time domain), the particle number distribution is narrower, showing that particles are localized around the past event horizon. []{data-label="Fig:pn"}](f4.png){width="8.0cm"} ![(color online). Energy of the wave packets versus the central frequency: $\sigma/a = 1.0, a V_0 =0, a g = 10$. The energy is almost constant in the high central frequency limit. []{data-label="Fig:energy"}](f5.png){width="8.0cm"} If we assume that the reflectivity $R_{\omega}$ of the mirror is a power law of $\omega$ as $\omega \rightarrow 0$ ($R_{\omega} \sim \omega^{\gamma}$) then in order to obtain finite particle number we must have $\gamma > 1$. As a concrete example, we choose $$R_{\omega} = \sin^2 \theta_{\omega} = \frac{g^2\omega^2}{1+g^2\omega^2},$$ where $g$ is a parameter characterizing the low-frequency cutoff. Fig. \[Fig:pn\] shows the particle number $N(f)$ versus the central position of the Gaussian wave packet. We can see that the particle-number distribution is symmetric with respect to $V_0 = 0$. In addition, for larger bandwidth (narrower wave packet in time domain), the distribution is more localized around $V_0 = 0$. These two facts indicate that the particle flux radiated by the uniformly accelerated mirror is well localized around the past horizon $V_0 = 0$. Since the mirror starts to accelerate in the distant past, that means the mirror only radiates particles when it starts accelerating. It radiates no particles when it is uniformly accelerating. Although in Eq. (\[ParticleNumber\]) the integrand explicitly depends on the central frequency $k_0$ of the Gaussian wave packet, in the large $k_0$ limit the integration turns out to be almost independent of $k_0$. That means the particle number $N(f) \sim \frac{1}{k_0}$ in the large central frequency limit (see Appendix \[appendixC\]), yielding the relationship $E(f) \approx k_0 N(f) \sim \mathcal{O}(1)$, for the energy of the wave packet, as shown in Fig. \[Fig:energy\]. Adding up the energy of all wave packets yields a divergent result. This ultraviolet divergence arises as a consequence of the physically unrealistic assumption that the mirror is accelerated eternally, so that it appears to any inertial observers when they cross the past horizon. This ultraviolet divergence can be removed by smoothly switching on the mirror [@Obadia01], or by considering an accelerated mirror whose acceleration was slowly increased from zero. For a switch-on timescale of $\Delta T$, the particle number is suppressed for wave packets with central frequency $k_0 > \frac{1}{\Delta T}$ while it remains the same for wave packets with central frequency $k_0 < \frac{1}{\Delta T}$. Therefore Eq. (\[ParticleNumber\]) is not applicable to wave packets with very high central frequency because it does not take into account physical initial conditions. Squeezing from accelerated mirrors {#squeezing} ================================== A well known mechanism for generating particles from the vacuum is the two-mode squeezing process. Examples include non-degenerate parametric down conversion [@BachorRalph] and the Unruh effect [@Unruh76]. The two output modes are entangled with each other so that the composite state is a pure state. Another important mechanism is the single-mode squeezing process, for example degenerate parametric down conversion [@BachorRalph]. It is possible that a particle generation process is the combination of the two, which we now show is the case for the uniformly accelerated mirror. Using the quantum circuit model for the uniformly accelerated mirror, it is very easy to show that the wavepacket mode is squeezed at some quadrature phase depending on the central frequency and central position of the wave packet. The correlations between various output Unruh modes are summarized in Appendix \[appendixB\]. If we consider left-moving and narrow bandwidth Gaussian wave packet modes, using Eqs. (\[MinkowskiUnruh\]), (\[GaussianMode\]), (\[A\]), (\[B\]) and (\[B1\]), we have $$\begin{aligned} \label{CroCorrelation} &&\langle 0_M |\hat{a}(f) \hat{a}(f) |0_M \rangle = \int dk \int dk^{\prime} f(k) f(k^{\prime}) \int d\omega \int d\omega^{\prime} \big[A_{k\omega} B_{k^{\prime} \omega^{\prime}}\langle 0_M |\hat{c}^{\prime}_{\omega} \hat{d}^{\prime}_{\omega^{\prime}} |0_M \rangle + B_{k\omega} A_{k^{\prime} \omega^{\prime}}\langle 0_M |\hat{d}^{\prime}_{\omega} \hat{c}^{\prime}_{\omega^{\prime}} |0_M \rangle\big] \nonumber \\ \nonumber \\ &=& - \sqrt{\frac{8}{\pi}}\frac{\sigma}{k_0} e^{-2ik_0V_0} \int_0^{\infty} d \Omega \exp\bigg[-\frac{\sigma^2(\Omega-k_0 V_0)^2}{k_0^2}\bigg] \exp\bigg[-\frac{\sigma^2(\Omega+k_0 V_0)^2}{k_0^2}\bigg] (1-\cos \theta_{\Omega})e^{\pi \Omega}\frac{e^{2\pi \Omega}+1}{(e^{2\pi \Omega}-1)^2}. \nonumber \\\end{aligned}$$ The quadrature observable of the localized wave packet mode $\hat{a}(f)$ is defined as $$\hat{X}(\phi) \equiv \hat{a}(f)e^{-i\phi} + \hat{a}^{\dagger}(f)e^{i \phi},$$ where $\phi$ is the quadrature phase. From Eqs. (\[ParticleNumber\]) and (\[CroCorrelation\]), we find that for a narrow bandwidth Gaussian wave packet the variance is $$\begin{aligned} \label{VarianceGaussian} &&\big(\Delta X({\phi})\big)^2 = \langle 0_M | \hat{X}^2({\phi}) |0_M \rangle - \langle 0_M | \hat{X}({\phi}) |0_M \rangle^2 = 1+ 2 \langle 0_M | \hat{a}^{\dagger}(f)\hat{a}(f) |0_M \rangle + 2~ \text{Re} \bigg[\langle 0_M | \hat{a}(f)\hat{a}(f) |0_M \rangle e^{-2i\phi} \bigg] \nonumber \\ &=& 1+ 4\sqrt{\frac{2}{\pi}}\frac{\sigma}{k_0} \int_0^{\infty} d \Omega \bigg\{\exp\bigg[-\frac{2\sigma^2(\Omega-k_0 V_0)^2}{k_0^2}\bigg]+\exp\bigg[-\frac{2\sigma^2(\Omega+k_0 V_0)^2}{k_0^2}\bigg] \bigg\} (1-\cos \theta_{\Omega})\frac{e^{2\pi \Omega}}{(e^{2\pi \Omega}-1)^2}\nonumber \\ &&- 4\sqrt{\frac{2}{\pi}}\frac{\sigma}{k_0} \cos(2\phi+2k_0V_0)\int_0^{\infty} d \Omega \exp\bigg[-\frac{\sigma^2(\Omega-k_0 V_0)^2}{k_0^2}\bigg] \exp\bigg[-\frac{\sigma^2(\Omega+k_0 V_0)^2}{k_0^2}\bigg] (1-\cos \theta_{\Omega})e^{\pi \Omega}\frac{e^{2\pi \Omega}+1}{(e^{2\pi \Omega}-1)^2}, \nonumber \\ \end{aligned}$$ where we have used the fact that in the Minkowski vacuum state, $\langle 0_M | \hat{X}({\phi}) |0_M \rangle = 0$. The variance of the wave packet mode could be smaller than one if the third term of Eq. (\[VarianceGaussian\]) is larger than the second term. In order to show that single-mode squeezing is possible, we consider a Gaussian wave packet centered at $V_0 = 0$. Eq. (\[VarianceGaussian\]) considerably simplifies, yielding $$\begin{aligned} \big(\Delta X^{min}\big)^2 = 1 - 4\sqrt{\frac{2}{\pi}}\frac{\sigma}{k_0} \int_0^{\infty} d \Omega \exp \bigg(-\frac{2\sigma^2\Omega^2}{k_0^2}\bigg) \times (1-\cos \theta_{\Omega})\frac{e^{\pi \Omega}}{(e^{\pi \Omega}+1)^2} <1\end{aligned}$$ for the minimum of $\big(\Delta X({\phi})\big)^2$, which is at $\phi = 0$. The variance of the quadrature beats the quantum shot noise, showing that the Gaussian wave packet mode is squeezed. When the center of the Gaussian wave packet is away from the past horizon $V_0 = 0$, the mode is squeezed at a different quadrature phase angle. According to Eq. (\[VarianceGaussian\]), the minimum of the variance is reached when $\phi_s + k_0 V_0 =0$ is satisfied, that is $$\label{phase} \phi_s = - k_0 V_0.$$ The squeezing phase angle $\phi_s$ depends on both the central frequency and central position of the Gaussian wave packet. Other than the rotation of the squeezing phase angle, the squeezing amplitude decreases when the center of the wave packet is away from the past horizon. Fig. \[Fig:variance\] shows the minimum variance of various wave packet modes (different central position and bandwidth), where the condition (\[phase\]) has been satisfied. From Fig. \[Fig:variance\] we see that the squeezing is stronger for a larger bandwidth Gaussian wave packet, which implies that different single-frequency Minkowski modes are also correlated. This can be verified if we replace $f(k)$ in Eq. (\[CroCorrelation\]) by a Dirac delta function $\delta(k - k_0)$. For a very large bandwidth wave packet mode (such as a broad bandwidth tophat mode), we find that the minimum variance approaches but never exceeds 0.5. We also note that when $\cos(2\phi + 2k_0 V_0) = -1$, the variance is maximal and larger than unity. ![(color online). Minimum variance versus central position of the Gaussian wave packet: $k_0/a = 20, a g = 10$. Maximum squeezing is achieved when the wave packet centers on the past horizon $V_0 = 0$. The squeezing is stronger for larger bandwidth wave packets. []{data-label="Fig:variance"}](f6.png){width="8.0cm"} According to the quantum circuit model, it is easy to understand the origin of the single-mode squeezing. In Fig. \[circuit:Fig\], after passing through the mirror the left-moving Rindler mode $\hat{b}_{\omega}^{\prime R}$ in the $R$ wedge is in thermal state, as well as the left-moving Rindler mode $\hat{b}_{\omega}^{L}$ in the $L$ wedge. The entanglement between $\hat{b}_{\omega}^{\prime R}$ and $\hat{b}_{\omega}^{L}$ depends on the transmission coefficient of the mirror. If the mirror is completely transparent, they are perfectly entangled; while if the mirror is perfect, the entanglement is completely severed. The Rindler modes $\hat{b}_{\omega}^{\prime R}$ and $\hat{b}_{\omega}^{L}$ further pass through a two-mode antisqueezer $S^{-1}_{\omega}$, ending up with two Unruh modes $\hat{c}^{\prime}_{ \omega}$ and $\hat{d}^{\prime}_{ \omega}$, which are also entangled. The amount of entanglement between $\hat{c}^{\prime}_{ \omega}$ and $\hat{d}^{\prime}_{ \omega}$ depends on the amount of entanglement between $\hat{b}_{\omega}^{\prime R}$ and $\hat{b}_{\omega}^{L}$. If $\hat{b}_{\omega}^{\prime R}$ and $\hat{b}_{\omega}^{L}$ are perfectly entangled, there is no entanglement between $\hat{c}^{\prime}_{ \omega}$ and $\hat{d}^{\prime}_{ \omega}$; otherwise, $\hat{c}^{\prime}_{ \omega}$ and $\hat{d}^{\prime}_{ \omega}$ are partially entangled. From Eq. (\[MinkowskiUnruh\]), the Minkowski mode $\hat{a}_k$ is a linear combination of the Unruh modes $\hat{c}^{\prime}_{ \omega}$ and $\hat{d}^{\prime}_{ \omega}$. It is a general result in quantum optics that a linear combination of entangled modes would produce single-mode squeezing, e.g., a $50:50$ beamsplitter transforms a two-mode squeezed state into single-mode squeezed sate in each output mode. Therefore, the Minkowski mode $\hat{a}_k$ is squeezed. It is clear that the single-mode squeezing is closely related to the correlations across the horizon. If the mirror is transparent ($\cos \theta_{\Omega} = 1$), the correlations across the horizon are preserved and there is no single-mode squeezing. When one uses a partially transmitting mirror ($\cos \theta_{\Omega} < 1$) to sever the correlations across the horizon, single-mode squeezing is inevitably produced according to Eq. (\[VarianceGaussian\]). Squeezed Firewall ? {#firewall} =================== Recently three assertions about black hole evaporation were shown to be mutually inconsistent[@AMPS]: (i) Hawking radiation is a unitary process, (ii) low energy effective field theory is valid near the event horizon, and (iii) an infalling observer encounters nothing unusual at the horizon. One of the proposed solutions to this paradox is that the infalling observer burns up at the horizon. A black hole firewall forms at the horizon for an old black hole and the correlations across the horizon are severed. Recently this firewall state was modeled for a Rindler horizon in Minkowski spacetime by severing correlations across the horizon. The response of an Unruh-DeWitt detector was seen to be finite [@Louko14]. The correlations across the horizon are severed by requiring the Wightman function to be zero, disregarding the underlying dynamics. Furthermore, a low-frequency cutoff in the Wightman function was introduced, implying that correlations between high-frequency modes are cut whilst correlations between low-frequency modes are preserved. This is a warm firewall. We propose that a uniformly accelerated mirror is a possible mechanism for generating a Rindler firewall. From the quantum circuit model we can see that the accelerated mirror acts as a pair of scissors cutting the correlations across the past horizon. If the mirror is perfect, the correlations across the horizon are completely severed and the particle flux along the horizon is divergent. This is a hot firewall, destroying everything that crosses it. However, if the mirror is not perfect but transparent for low-frequency modes, the high-frequency correlations are cut while low-frequency correlations are preserved, and the particle flux in a localized wave packet mode along the horizon is finite, similar to the warm firewall proposed by Louko [@Louko14]. In Sec. \[squeezing\], we showed that the radiation field from the accelerated mirror is squeezed, which implies that the Rindler firewall is squeezed. It seems that squeezing is a general property of a Rindler firewall because in order to form a firewall one has to cut the correlations across the horizon, which inevitably generates single-mode squeezing. Is a black hole firewall squeezed? Black hole firewalls are introduced in order to preserve the unitarity of black hole evolution [@AMPS; @Braunstein13]. For an old black hole, the late time Hawking radiation should be correlated with early time Hawking radiation but not with the degrees of freedom inside the event horizon. The correlations across the horizon are severed during the evaporation. According to the arguments for the Rindler firewall, it is reasonable to conjecture that the black hole firewalls are also squeezed. In addition, if the single-mode squeezing is strong enough, black hole firewalls do not have to be entangled with other unknown systems. Conclusions {#conclusion} =========== We have developed a quantum circuit formalism to describe unitary interactions between a uniformly accelerated object and the quantum fields. The key point is to work in the accelerated frame where the object is stationary and couples only to Rindler modes in one of the Rindler wedges. If the initial state of the quantum fields is given in the inertial frame and the response of inertial detectors is considered, we have to transform modes from the inertial frame to the accelerated frame, which turns out to be a two-mode squeezing operation if we consider Unruh modes instead of Minkowski modes in the inertial frame. We thus can construct a quantum circuit using two-mode squeezers and devices depending on the interaction of the object with the Rindler modes. As an example, we studied a uniformly accelerated mirror. In the accelerated frame, the mirror is stationary and is simply a beamsplitter with frequency dependent reflection coefficient. The input-output relation of a beamsplitter is well known and is widely used in quantum optics [@BachorRalph]. The quantum circuit for the uniformly accelerated mirror is shown in Fig. \[circuit:Fig\]. As an application, we calculated the radiation flux from an eternally accelerating mirror in the Minkowksi vacuum. We found that the particles are localized around the horizon and the particle number in a localized wave packet mode is divergent if no low frequency regularization is introduced. Our results are consistent with earlier results obtained using different methods [@Frolov99; @Obadia01]. The infrared divergence occurs due to the ideal assumption that the mirror accelerates for an infinitely long time. We emphasize that the infrared divergence is not due to the particular pathological character of a massless scalar field in $(1+1)$-dimensional spacetime [@Coleman73] because it also appears in higher dimensional spacetime [@Frolov99]. We regularize the radiation flux by introducing a low-frequency cutoff for the mirror, that is, the mirror is completely transparent for low frequency field modes. Physically, this is equivalent to having the mirror interact with the field for a finite time. After regularizing the infrared divergence, the particle number of a localized wave packet mode is finite. However the energy of the wave packet mode does not decay as the central frequency increases, in turn implying that the total energy of the radiation flux is infinite. This ultraviolet divergence arises because of the naive assumption that the mirror is accelerated eternally so that it appears to inertial observers when they cross the past horizon. If the mirror slowly increased its acceleration or was switched on smoothly, the number of high frequency particles would be suppressed, removing this ultraviolet divergence. Using perturbation theory it is straightforward to show that the energy flux is finite if the mirror is smoothly turned on and off [@Obadia01]. A further application of our circuit model would be in the study a uniformly accelerated cavity. Previous work on this topic [@Alsing03; @Downes11; @Bruschi12] studied how the quantum states stored inside a perfect cavity are affected by uniform acceleration. While Unruh-Davies radiation [@Unruh76; @Davies75] cannot affect the field modes inside a perfect cavity, it can affect field modes inside an imperfect one. Because the circuit model is designed to study an imperfect uniformly accelerated mirror, we believe that by generalizing the model from one mirror to two mirrors, one can study the interaction between Unruh-Davies radiation and the field modes inside an imperfect cavity. One limitation of our circuit model is that it is only suitable for studying hyperbolic trajectories in Minkowski spacetime; more general trajectories are not straightforwardly incorporated. One might expect this to severely limit the utility of the circuit model because physically it is not possible to accelerate a mirror for an infinitely long time. However our use of the transparency term shows that we can turn on and off the mirror so that it is transparent in the distant past and distant future. This could be used to model a mirror that initially undergoes inertial motion, accelerates for a finite period of time, and then returns to inertial motion. We will leave this topic for future work. We find that the radiation flux from the uniformly accelerated mirror is squeezed. To the best of our knowledge, the contribution of single-mode squeezing to the generation of particles by a moving mirror has not been discussed previously. The squeezing angle depends on the central frequency and position of the localized detector mode function. Maximum squeezing occurs when the detector mode function centers on the horizon. It is clear from the circuit model that the squeezing is related to the correlations across the horizon. When the mirror is completely transparent, the correlations across the horizon are preserved and there is no squeezing. When the mirror completely reflects a Rindler mode with a particular frequency, it destroys the correlation across the horizon and generates some squeezing in the Minkowski mode. It therefore provides a mechanism for transferring the correlations across the horizon to the squeezing of the radiation flux on the horizon. Recently, Louko [@Louko14] proposed a Rindler firewall state by severing the correlations across the horizon by hand and claimed that the response of a particle detector is finite. It was subsequently shown that entanglement survives this Rindler firewall [@Martinez15]. Our calculation suggests that one way of generating a Rindler firewall is to uniformly accelerate a mirror. We conjecture that if the firewall is formed in an old black hole, the radiation flux at the horizon could be squeezed as the price of severing the entanglement across the event horizon. In addition, the black hole firewall may not need to be highly entangled with other systems [@Susskind14] because the squeezing may be enough to account for the particle flux on the horizon. ACKNOWLEGEMENTS {#acknowlegements .unnumbered} =============== We would like to thank Antony Lee, Shih-Yuin Lin and Yiqiu Ma for useful discussions. This research was supported in part by Australian Research Council Centre of Excellence of Quantum Computation and Communication Technology (Project No. CE110001027), and in part by the Natural Sciences and Engineering Research Council of Canada. Input-output relations\[appendixA\] =================================== We summarize the input-output relations of the quantum circuit Fig. \[circuit:Fig\] with the object a beamsplitter. The action of the beamsplitter is represented by Eq. (\[mirrortransformation\]). Substituting it into Eq. (\[input-output\]), we have $$\begin{aligned} \label{UnruhLeftC} \hat{\bf c}^{\prime}_{1\omega} &=& \hat{\bf c}_{1\omega}[\text{cosh}^2(r_{\omega})\text{cos} \,\theta_{\omega}-\text{sinh}^2(r_{\omega})] - \sigma_x \hat{\bf d}_{1\omega} \text{cosh}(r_{\omega})\text{sinh}(r_{\omega})(1-\text{cos} \,\theta_{\omega}) +Z \hat{\bf c}_{2\omega} \text{cosh}^2(r_{\omega})\text{sin} \,\theta_{\omega} \nonumber \\ &&+Z \sigma_x \hat{\bf d}_{2\omega} \text{cosh}(r_{\omega})\text{sinh}(r_{\omega}) \text{sin} \,\theta_{\omega}. \nonumber \\ &=& [\text{cosh}^2(r_{\omega})\text{cos} \,\theta_{\omega}-\text{sinh}^2(r_{\omega})] {\hat{c}_{1\omega} \choose \hat{c}_{1\omega}^{\dag}} - \text{cosh}(r_{\omega})\text{sinh}(r_{\omega})(1-\text{cos} \,\theta_{\omega}){\hat{d}_{1\omega}^{\dag} \choose \hat{d}_{1\omega}} + \text{cosh}^2(r_{\omega})\text{sin} \,\theta_{\omega} {-i e^{i \phi_{\omega}}\hat{c}_{2\omega} \choose i e^{- i \phi_{\omega}}\hat{c}_{2\omega}^{\dag}} \nonumber \\ &&+ \text{cosh}(r_{\omega})\text{sinh}(r_{\omega}) \text{sin} \,\theta_{\omega} {-i e^{i \phi_{\omega}}\hat{d}_{2\omega}^{\dag} \choose i e^{- i \phi_{\omega}}\hat{d}_{2\omega}}, \end{aligned}$$ $$\begin{aligned} \label{UnruhLeftD} \hat{\bf d}^{\prime}_{1\omega} &=& \sigma_x \hat{\bf c}_{1\omega} \text{cosh}(r_{\omega})\text{sinh}(r_{\omega})(1-\text{cos}\,\theta_{\omega}) + \hat{\bf d}_{1\omega}[\text{cosh}^2(r_{\omega})-\text{sinh}^2(r_{\omega})\text{cos}\,\theta_{\omega}] - \sigma_x Z \hat{\bf c}_{2\omega} \text{cosh}(r_{\omega})\text{sinh}(r_{\omega}) \text{sin}\,\theta_{\omega} \nonumber \\ && - \sigma_x Z \sigma_x \hat{\bf d}_{2\omega} \text{sinh}^2(r_{\omega})\text{sin}\,\theta_{\omega} \nonumber \\ &=& \text{cosh}(r_{\omega})\text{sinh}(r_{\omega})(1-\text{cos}\,\theta_{\omega}){\hat{c}_{1\omega}^{\dag} \choose \hat{c}_{1\omega}} +[\text{cosh}^2(r_{\omega})-\text{sinh}^2(r_{\omega})\text{cos}\,\theta_{\omega}]{\hat{d}_{1\omega} \choose \hat{d}_{1\omega}^{\dag}} \nonumber \\ &&- \text{cosh}(r_{\omega})\text{sinh}(r_{\omega}) \text{sin}\,\theta_{\omega}{i e^{- i \phi_{\omega}}\hat{c}_{2\omega}^{\dag} \choose -i e^{i \phi_{\omega}}\hat{c}_{2\omega}} - \text{sinh}^2(r_{\omega})\text{sin}\,\theta_{\omega}{ i e^{- i \phi_{\omega}}\hat{d}_{2\omega} \choose -i e^{i \phi_{\omega}}\hat{d}_{2\omega}^{\dag} }, \end{aligned}$$ $$\begin{aligned} \label{UnruhRightC} \hat{\bf c}^{\prime}_{2\omega} &=& -Z^* \hat{\bf c}_{1\omega} \text{cosh}^2(r_{\omega})\text{sin}\,\theta_{\omega} -Z^* \sigma_x \hat{\bf d}_{1\omega} \text{cosh}(r_{\omega})\text{sinh}(r_{\omega}) \text{sin}\,\theta_{\omega} + \hat{\bf c}_{2\omega}[\text{cosh}^2(r_{\omega})\text{cos}\,\theta_{\omega}-\text{sinh}^2(r_{\omega})] \nonumber \\ &&-\sigma_x \hat{\bf d}_{2\omega} \text{cosh}(r_{\omega})\text{sinh}(r_{\omega})(1-\text{cos}\,\theta_{\omega}) \nonumber \\ &=&\text{cosh}^2(r_{\omega})\text{sin}\,\theta_{\omega} {-ie^{-i \phi_{\omega}}\hat{c}_{1\omega} \choose ie^{i \phi_{\omega}}\hat{c}_{1\omega}^{\dag}} + \text{cosh}(r_{\omega})\text{sinh}(r_{\omega}) \text{sin}\,\theta_{\omega} {-i e^{-i \phi_{\omega}}\hat{d}_{1\omega}^{\dag} \choose i e^{i \phi_{\omega}}\hat{d}_{1\omega}} +[\text{cosh}^2(r_{\omega})\text{cos}\,\theta_{\omega}-\text{sinh}^2(r_{\omega})] {\hat{c}_{2\omega} \choose \hat{c}_{2\omega}^{\dag}}\nonumber \\ &&- \text{cosh}(r_{\omega})\text{sinh}(r_{\omega})(1-\text{cos}\,\theta_{\omega}){\hat{d}_{2\omega}^{\dag} \choose \hat{d}_{2\omega}}, \end{aligned}$$ $$\begin{aligned} \label{UnruhRightD} \hat{\bf d}^{\prime}_{2\omega} &=& -\sigma_x Z^* \hat{\bf c}_{1\omega} \text{cosh}(r_{\omega})\text{sinh}(r_{\omega}) \text{sin}\,\theta_{\omega} + \sigma_x Z^* \sigma_x \hat{\bf d}_{1\omega} \text{sinh}^2(r_{\omega})\text{sin}\,\theta_{\omega} + \sigma_x \hat{\bf c}_{2\omega} \text{cosh}(r_{\omega})\text{sinh}(r_{\omega})(1-\text{cos}\,\theta_{\omega}) \nonumber \\ &&+ \hat{\bf d}_{2\omega}[\text{cosh}^2(r_{\omega})-\text{sinh}^2(r_{\omega})\text{cos}\,\theta_{\omega}] \nonumber \\ &=&-\text{cosh}(r_{\omega})\text{sinh}(r_{\omega}) \text{sin}\,\theta_{\omega}{i e^{ i \phi_{\omega}}\hat{c}_{1\omega}^{\dag} \choose -i e^{-i \phi_{\omega}}\hat{c}_{1\omega}} - \text{sinh}^2(r_{\omega})\text{sin}\,\theta_{\omega}{ i e^{ i \phi_{\omega}}\hat{d}_{1\omega} \choose -i e^{-i \phi_{\omega}}\hat{d}_{1\omega}^{\dag} } \nonumber \\ &&+\text{cosh}(r_{\omega})\text{sinh}(r_{\omega})(1-\text{cos}\,\theta_{\omega}) {\hat{c}_{2\omega}^{\dag} \choose \hat{c}_{2\omega}} +[\text{cosh}^2(r_{\omega})-\text{sinh}^2(r_{\omega})\text{cos}\,\theta_{\omega}]{\hat{d}_{2\omega} \choose \hat{d}_{2\omega}^{\dag}} \label{UnruhRightD}.\end{aligned}$$ Correlations between output Unruh modes\[appendixB\] ==================================================== Using Eqs. (\[UnruhLeftC\])-(\[UnruhRightD\]), it is straightforward to calculate the correlations between various output Unruh modes in the Minkowski vacuum state. $$\begin{aligned} \label{B1} \langle 0_M | \hat{c}'_{m \omega} \hat{d}'_{m \omega'} |0_M \rangle &=& \langle 0_M | \hat{d}'_{m \omega} \hat{c}'_{m \omega'} |0_M \rangle = \langle 0_M | \hat{c}^{\prime \dag}_{m \omega} \hat{d}^{\prime \dag}_{m \omega'} |0_M \rangle = \langle 0_M | \hat{d}^{\prime \dag}_{m \omega} \hat{c}^{\prime \dag}_{m \omega'} |0_M \rangle \nonumber \\ \nonumber \\ &=& - (1 - \cos \theta_{\omega}) \cosh(r_{\omega}) \sinh(r_{\omega}) \bigg [\sinh^2(r_{\omega}) + \cosh^2(r_{\omega}) \bigg] \delta(\omega - \omega'),\end{aligned}$$ $$\begin{aligned} \label{B2} \langle 0_M | \hat{c}'_{1 \omega} \hat{d}'_{2 \omega'} |0_M \rangle &=& \langle 0_M | \hat{d}'_{2 \omega} \hat{c}'_{1 \omega'} |0_M \rangle = \langle 0_M | \hat{c}^{\prime \dag}_{1 \omega} \hat{d}^{\prime \dag}_{2 \omega'} |0_M \rangle ^* = \langle 0_M | \hat{d}^{\prime \dag}_{2 \omega} \hat{c}^{\prime \dag}_{1 \omega'} |0_M \rangle^* \nonumber \\ \nonumber \\ &=& i e^{i \varphi_{\omega}}\sin \theta_{\omega} \cosh(r_{\omega}) \sinh(r_{\omega}) \delta(\omega - \omega'),\end{aligned}$$ $$\begin{aligned} \label{B3} \langle 0_M | \hat{c}'_{2 \omega} \hat{d}'_{1 \omega'} |0_M \rangle &=& \langle 0_M | \hat{d}'_{1 \omega} \hat{c}'_{2 \omega'} |0_M \rangle = \langle 0_M | \hat{c}^{\prime \dag}_{2 \omega} \hat{d}^{\prime \dag}_{1 \omega'} |0_M \rangle ^* = \langle 0_M | \hat{d}^{\prime \dag}_{1 \omega} \hat{c}^{\prime \dag}_{2 \omega'} |0_M \rangle^* \nonumber \\ \nonumber \\ &=& - i e^{- i \varphi_{\omega}}\sin \theta_{\omega} \cosh(r_{\omega}) \sinh(r_{\omega}) \delta(\omega - \omega'),\end{aligned}$$ with others zero and here $m = 1,2$. High central frequency limit {#appendixC} ============================ We derive an analytically approximate expression for the particle number $N(f)$ in the high central frequency limit. From Eq. (\[ParticleNumber\]), one expects that the term in the braces has two peaks at $k_0 V_0$ and $-k_0 V_0$. If $k_0$ is large then the peaks are far away from the origin. However, the factor $\frac{e^{2\pi \Omega}}{(e^{2\pi \Omega}-1)^2}$ exponentially decays for large $\Omega$ so that it strongly suppresses one of the Gaussian peaks. Therefore, the main contribution to the integration is from the low frequency. We Taylor expand the term in the braces to second order, $$\begin{aligned} \exp\bigg[-\frac{2\sigma^2(\Omega-k_0 V_0)^2}{k_0^2}\bigg]+\exp\bigg[-\frac{2\sigma^2(\Omega+k_0 V_0)^2}{k_0^2}\bigg] \approx 2 e^{-2 \sigma^2 V_0^2} + \frac{4 \sigma^2 \Omega^2}{k_0^2} (4 \sigma^2 V_0^2 -1) e^{-2 \sigma^2 V_0^2}. \end{aligned}$$ In order to get an analytic expression, we introduce sharp low frequency cutoff, $R_{\omega} = 1$ for $\Omega \ge \epsilon$ and zero for $0 < \Omega < \epsilon$. Therefore we have $1-\cos \theta_{\Omega} = 1$ for $\Omega \ge \epsilon$ and zero for $0 < \Omega < \epsilon$. The particle number $N(f)$ can be approximated as $$\begin{aligned} \label{ParticleNumber:2} N(f) &\approx& 4 \sqrt{\frac{2}{\pi}}\frac{\sigma}{k_0} e^{-2 \sigma^2 V_0^2} \bigg[ \int_{\epsilon}^{\infty} d \Omega \frac{e^{2\pi \Omega}}{(e^{2\pi \Omega}-1)^2} + \frac{2 \sigma^2}{k_0^2} (4 \sigma^2 V_0^2 -1) \int_{\epsilon}^{\infty} d \Omega \frac{\Omega^2 e^{2\pi \Omega}}{(e^{2\pi \Omega}-1)^2} \bigg] \nonumber \\ \nonumber \\ &\approx& \bigg(\frac{2}{\pi} \bigg)^{3/2} \bigg(\frac{\sigma}{k_0} \bigg) e^{-2 \sigma^2 V_0^2} \bigg[\frac{1}{e^{2 \pi \epsilon} - 1} + \frac{2 \sigma^2}{k_0^2} (4 \sigma^2 V_0^2 -1)\bigg(\frac{1}{12} - \frac{\epsilon^2}{2 \pi} \bigg) \bigg]. \end{aligned}$$ Comparison with direct numerical calculation shows that Eq. (\[ParticleNumber:2\]) is a very good approximation when $\epsilon$ is small. We can see that the particle number is dependent on the low frequency cutoff $\epsilon$. The first term of Eq. (\[ParticleNumber:2\]) is proportional to $\frac{1}{e^{2 \pi \epsilon} - 1}$ which is divergent when $\epsilon \rightarrow 0$. Furthermore, in the high central frequency limit $k_0 \rightarrow \infty$, the leading order of $N(f)$ is proportional to $\frac{1}{k_0}$. [99]{} G. T. Moore, J. Math. Phys. [**11**]{}, 2679(1970). S. A. Fulling and P. C. W. Davies, Proc. R. Soc. Lond. A. [**348**]{}, 393(1976). P. C. W. Davies and S. A. Fulling, Proc. R. Soc. Lond. A. [**356**]{}, 237(1977). W. R. Walker, Phys. Rev. D [**31**]{}, 767(1985). R. D. Carlitz and R. S. Willey, Phys. Rev. D [**36**]{}, 2327(1987). S. Hawking, Commun. Math. Phys. [**43**]{}, 199(1975). F. Wilczek, arXiv:hep-th/9302096. M. Hotta, R. Schützhold and W. G. Unruh, Phys. Rev. D [**91**]{}, 124060(2015). S. Hawking, Phys. Rev. D [**14**]{}, 2460(1976). N. D. Birrell and P. C. W. Davies, [*Quantum Fields in Curved Space*]{}(Cambridge University Press, Cambridge, England, 1982). P. G. Grove, Class. Quantum Grav. 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--- author: - 'Hua-Xing <span style="font-variant:small-caps;">Chen</span>$^{1,2,}$[^1], Atsushi <span style="font-variant:small-caps;">Hosaka</span>$^{1,}$[^2] and Shi-Lin <span style="font-variant:small-caps;">Zhu</span>$^{2,}$[^3]' title: Light Scalar Mesons in the QCD Sum Rule --- Introduction ============ The nature of light scalar mesons of $up$, $down$ and $strange$ quarks is not fully understood [@scalar; @Yao:2006px]. The expected members are $\sigma(600)$, $\kappa(800)$, $f_0(980)$ and $a_0(980)$ forming a nonet of flavor SU(3). Because they have the same spin and parity as the vacuum, $J^P = 0^+$, they reflect the bulk properties of the non-perturbative QCD vacuum. So far, several different pictures for the scalar mesons have been proposed. In the conventional quark model, they have a $\bar q q$ configuration of $^3P_0$ whose masses are expected to be larger than 1 GeV due to the $p$-wave orbital excitation. Furthermore, the mass ordering in a naive quark mass counting of $m_u \sim m_d < m_s$ implies $m_\sigma \sim m_{a_0} < m_\kappa < m_{f_0}$. In chiral models, they are regarded as chiral partners of the Nambu-Goldstone bosons ($\pi, K, \eta, \eta^\prime)$ [@Hatsuda:1994pi]. Due to the collective nature, their masses are expected to be lower than those of the quark model. Yet another interesting picture is that they are tetraquark states [@Jaffe:1976ig; @Lee:2006vk; @Brito:2004tv; @Zhang:2006xp]. In contrast with the $\bar q q$ states, their masses are expected to be around 0.6 – 1 GeV with the ordering of $m_\sigma < m_\kappa < m_{f_0, a_0}$, consistent with the recent experimental observation  [@scalar; @Yao:2006px; @experiment]. If such tetraquarks survive, they may be added to members of exotic multiquark states. In this contribution, we would like to report the results of a systematic study of the masses of the tetraquark scalar mesons in the QCD sum rule. We find that the QCD sum rule analysis with tetraquark currents implies the masses of scalar mesons in the region of 600 – 1000 MeV with the ordering, $m_\sigma < m_\kappa < m_{f_0, a_0}$, while the conventional $\bar q q$ currents imply masses around 1.5 GeV. Independent Currents ==================== Let us start with currents for the scalar tetraquark, which we consider only local currents. Using the antisymmetric combination for diquark flavor structure, we arrive at the following five independent currents [@Chen:2006hy] $$\begin{aligned} \nonumber\label{define_udud_current} S^\sigma_3 &=& (u_a^T C \gamma_5 d_b)(\bar{u}_a \gamma_5 C \bar{d}_b^T - \bar{u}_b \gamma_5 C \bar{d}_a^T)\, , \\ \nonumber V^\sigma_3 &=& (u_a^T C \gamma_{\mu} \gamma_5 d_b)(\bar{u}_a \gamma^{\mu}\gamma_5 C \bar{d}_b^T - \bar{u}_b \gamma^{\mu}\gamma_5 C \bar{d}_a^T)\, , \\ T^\sigma_6 &=& (u_a^T C \sigma_{\mu\nu} d_b)(\bar{u}_a \sigma^{\mu\nu} C \bar{d}_b^T + \bar{u}_b \sigma^{\mu\nu} C \bar{d}_a^T)\, , \\ \nonumber A^\sigma_6 &=& (u_a^T C \gamma_{\mu} d_b)(\bar{u}_a \gamma^{\mu} C \bar{d}_b^T + \bar{u}_b \gamma^{\mu} C \bar{d}_a^T)\, , \\ \nonumber P^\sigma_3 &=& (u_a^T C d_b)(\bar{u}_a C \bar{d}_b^T - \bar{u}_b C \bar{d}_a^T)\, ,\end{aligned}$$ where the sum over repeated indices ($\mu$, $\nu, \cdots$ for Dirac, and $a, b, \cdots$ for color indices) is taken. Either plus or minus sign in the second parentheses ensures that the diquarks form the antisymmetric combination in the flavor space. The currents $S$, $V$, $T$, $A$ and $P$ are constructed by scalar, vector, tensor, axial-vector, pseudoscalar diquark and antidiquark fields, respectively. The subscripts $3$ and $6$ show that the diquarks (antidiquark) are combined into the color representation $\mathbf{\bar 3_c}$ and $\mathbf{6_c}$ ($\mathbf{3_c}$ or $\mathbf{\bar 6_c}$), respectively. The currents for other members are formed similarly. We can also use a symmetric combination for diquark flavor structure. However, they are related to the antisymmetric ones by the axial U(1) transformation [@Umekawa:2004js]. QCD Sum Rule Analysis ===================== For the past decades QCD sum rule has proven to be a very powerful and successful non-perturbative method [@Shifman:1978bx; @Reinders:1984sr]. In sum rule analyses, we consider two-point correlation functions: $$\Pi(q^2)\,\equiv\,i\int d^4x e^{iqx} \langle0|T\eta(x){\eta^\dagger}(0)|0\rangle \, , \label{eq_pidefine}$$ where $\eta$ is an interpolating current for the tetraquark. We compute $\Pi(q^2)$ in the operator product expansion (OPE) of QCD up to certain order in the expansion, which is then matched with a hadronic parametrization to extract information of hadron properties. At the hadron level, we express the correlation function in the form of the dispersion relation with a spectral function: $$\Pi(p)=\int^\infty_0\frac{\rho(s)}{s-p^2-i\varepsilon}ds \, , \label{eq_disper}$$ where $$\begin{aligned} \rho(s) & \equiv & \sum_n\delta(s-M^2_n)\langle 0|\eta|n\rangle\langle n|{\eta^\dagger}|0\rangle \ \nonumber\\ &=& f^2_X\delta(s-M^2_X)+ \rm{higher\,\,states}\, . \label{eq_rho}\end{aligned}$$ For the second equation, as usual, we adopt a parametrization of one pole dominance for the ground state $X$ and a continuum contribution. The mass of the state $X$ can be obtained $$M^2_X=\frac{\int^{s_0}_0 e^{-s/M_B^2}s\rho(s)ds}{\int^{s_0}_0 e^{-s/M_B^2}\rho(s)ds}\, . \label{eq_LSR}$$ We performed the sum rule analysis using all currents and their various linear combinations, and found a good sum rule by a linear combination of $A_6^\sigma$ and $V_3^\sigma$ $$\begin{aligned} \eta^\sigma_1 = \cos\theta A^\sigma_6 + \sin\theta V^\sigma_3\, ,\end{aligned}$$ where the best choice of the mixing angle turns out to be $\cot\theta = 1 / \sqrt{2}$. For $\kappa$, $f_0$ and $a_0$, we have also found that similar linear combinations give better sum rules. The results of OPE can be found in Ref. [@Chen:2006zh] Numerical Analysis ================== For numerical calculations, we use the following values of condensates [@Yang:1993bp; @Ioffe:2002be; @Gimenez:2005nt]: $\langle\bar qq \rangle=-(0.240 \mbox{ GeV})^3$, $\langle\bar ss\rangle=-(0.8\pm 0.1)\times(0.240 \mbox{ GeV})^3$,$\langle g_s^2GG\rangle =(0.48\pm 0.14) \mbox{ GeV}^4$, $ m_u = 5.3 \mbox{ MeV}$, $m_d = 9.4 \mbox{ MeV}$, $m_s(1\mbox{ GeV})=125 \pm 20 \mbox{ MeV}$, $\langle g_s\bar q\sigma G q\rangle=-M_0^2\times\langle\bar qq\rangle$, $M_0^2=(0.8\pm0.2)\mbox{ GeV}^2$. The sum rules are written as power series of the Borel mass $M_B$. Since the Borel transformation suppresses the contributions from $s > M_B$, smaller values are preferred to suppress the continuum contributions also. However, for smaller $M_B$ convergence of the OPE becomes worse. Therefore, we should find an optimal $M_B$ preferably in a small value region. We have found that the minima of such a region are 0.4 GeV for $\sigma$, 0.5 GeV for $\kappa$ and 0.8 GeV for $f_0$ and $a_0$, where the pole contributions reach around 50 % for all cases [@Chen:2006zh]. As $M_B$ is increased, the pole contributions decrease, but the resulting tetraquark masses are stable as shown in Fig. \[pic\_tetra\]. After careful test of the sum rule for a wide range of parameter values of $M_B$ and $s_0$, we have found reliable sum rules, with which we find the masses $ m_\sigma = (0.6 \pm 0.1) \; {\rm GeV}$, $ m_\kappa = (0.8 \pm 0.1) \; {\rm GeV}$, $m_{f_0,a_0} = (1 \pm 0.1) \; {\rm GeV}\; ,$ which are consistent with the experimental results [@Yao:2006px]. For comparison, we have also performed the QCD sum rule analysis using the $\bar q q$ current within the present framework. The stable (weak $M_B$) behavior is obtained with the masses of all four mesons around 1.5 GeV. Here again we have tested various values of $M_B$ and $s_0$, and confirmed that the result shown is optimal. Conclusions =========== We have performed the QCD sum rule analysis with tetraquark currents, which implies the masses of scalar mesons in the region of 600 – 1000 MeV with the ordering, $m_\sigma < m_\kappa < m_{f_0, a_0}$. We have also performed the QCD sum rule analysis with the conventional $\bar q q$ currents, which implies masses around 1.5 GeV. We have tested all possible independent tetraquark currents as well as their linear combinations. Our observation supports a tetraquark structure for low-lying scalar mesons. To test the validity of the tetraquark structure, it is also important to study decay properties, which is often sensitive to the structure of wave functions. Such a tetraquark structure will open an alternative path toward the understanding exotic multiquark dynamics which one does not experience in the conventional hadrons. Acknowledgements {#acknowledgements .unnumbered} ================ H. X. C and A. H. thank the Yukawa Institute for Theoretical Physics at Kyoto University for hospitality during the YKIS2006 on “New Frontiers on QCD”. H.X.C. is grateful to the Monkasho fellowship for supporting his stay at RCNP, Osaka University. A.H. is supported in part by the Grant for Scientific Research ((C) No.16540252) from the Ministry of Education, Culture, Science and Technology, Japan. S.L.Z. was supported by the National Natural Science Foundation of China under Grants 10375003 and 10421503, Ministry of Education of China, FANEDD, Key Grant Project of Chinese Ministry of Education (NO 305001) and SRF for ROCS, SEM. [10]{} E. M. Aitala et al., Phys. Rev. Lett. 86, 770 (2001); M. Ablikim et al., Phys. Lett. B 598, 149 (2004). W. M. Yao [*et al.*]{} \[Particle Data Group\], J. Phys. G [**33**]{}, 1 (2006). T. Hatsuda and T. Kunihiro, Phys. Rept.  [**247**]{}, 221 (1994). 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H. X. Chen, A. Hosaka and S. L. Zhu, arXiv:hep-ph/0609163. [^1]: e-mail address: hxchen@rcnp.osaka-i.ac.jp [^2]: e-mail address: hosaka@rcnp.osaka-u.ac.jp [^3]: e-mail address: zhusl@th.phy.pku.edu.cn

--- abstract: | We report on the discovery of a very narrow-line star forming object beyond redshift of 5. Using the prime-focus camera, Suprime-Cam, on the 8.2 m Subaru telescope together with a narrow-passband filter centered at $\lambda_{\rm c}$ = 8150 Å with passband of $\Delta\lambda$ = 120 Å, we have obtained a very deep image of the field surrounding the quasar SDSSp J104433.04$-$012502.2 at a redshift of 5.74. Comparing this image with optical broad-band images, we have found an object with a very strong emission line. Our follow-up optical spectroscopy has revealed that this source is at a redshift of $z=5.655\pm0.002$, forming stars at a rate $\sim 13 ~ h_{0.7}^{-2} ~ M_\odot$ yr$^{-1}$. Remarkably, the velocity dispersion of Ly$\alpha$-emitting gas is only 22 km s$^{-1}$. Since a blue half of the Ly$\alpha$ emission could be absorbed by neutral hydrogen gas, perhaps in the system, a modest estimate of the velocity dispersion may be $\gtrsim$ 44 km s$^{-1}$. Together with a linear size of 7.7 $h_{0.7}^{-1}$ kpc, we estimate a lower limit of the dynamical mass of this object to be $\sim 2 \times 10^9 M_\odot$. It is thus suggested that LAE J1044$-$0123 is a star-forming dwarf galaxy (i.e., a subgalactic object or a building block) beyond redshift 5 although we cannot exclude a possibility that most Ly$\alpha$ emission is absorbed by the red damping wing of neutral intergalactic matter. author: - 'Yoshiaki Taniguchi , Masaru Ajiki , Takashi Murayama , Tohru Nagao , Sylvain Veilleux , David B. Sanders , Yutaka Komiyama , Yasuhiro Shioya , Shinobu S. Fujita , Yuko Kakazu , Sadanori Okamura , Hiroyasu Ando , Tetsuo Nishimura , Masahiko Hayashi , Ryusuke Ogasawara , & Shin-ichi Ichikawa' title: 'THE DISCOVERY OF A VERY NARROW-LINE STAR FORMING OBJECT AT A REDSHIFT OF 5.66' --- INTRODUCTION ============ Searches for Ly$\alpha$ emitters (hereafter LAEs) at high redshift are very useful in investigating the early star formation history of galaxies (Partridge & Peebles 1967). Recent progress in the observational capability of 10-m class optical telescopes has enabled us to discover more than a dozen of Ly$\alpha$ emitting galaxies beyond redshift 5 (Dey et al. 1998; Weymann et al. 1998; Hu et al. 1999, 2001, 2002; Dawson et al. 2001, 2002; Ellis et al. 2001; Ajiki et al. 2002; see also Spinrad et al. 1998; Rhoads & Malhotra 2001). The most distant object known to date is HCM-6A at $z=6.56$ (Hu et al. 2002). These high-$z$ Ly$\alpha$ emitters can be also utilized to investigate physical properties of the intergalactic medium (IGM) because the epoch of cosmic reionization ($z_{\rm r}$) is considered to be close to the redshifts of high-$z$ Ly$\alpha$ emitters; i.e., $z_{\rm r} \sim 6$ – 7 (Djorgovski et al. 2001; Becker et al. 2001; Fan et al. 2002). In other words, emission-line fluxes of the Ly$\alpha$ emission from such high-$z$ galaxies could be absorbed by neutral hydrogen in the IGM if neutral gas clouds are located between the source and us (Gunn & Peterson 1965; Miralda-Escudé 1998; Miralda-Escudé & Rees 1998; Haiman 2002 and references therein). Therefore, careful investigations of Ly$\alpha$ emission-line properties of galaxies with $z > 5$ provides very important clues simultaneously both on the early star formation history of galaxies and on the physical status of IGM at high redshift. The first step is to look for a large number of Ly$\alpha$ emitter candidates at high redshift through direct imaging surveys. The detectability of high-redshift objects is significantly increased if they are hosts to recent bursts of star formation which ionize the surrounding gas and result in strong emission lines like the hydrogen recombination line Ly$\alpha$. These emission-line objects can be found in principle through deep optical imaging with narrow-passband filters customized to the appropriate redshift. Indeed, recent attempts with the Keck 10 m telescope have revealed the presence of Ly$\alpha$ emitters in blank fields at high redshift (e.g., Cowie & Hu 1998; Hu et al. 2002). These recent successes have shown the great potential of narrow-band imaging surveys with 8-10 m telescopes in the search for high-$z$ Ly$\alpha$ emitters. It is also worthwhile noting that subgalactic populations at high redshift have been recently found thanks to the gravitational lensing (Ellis et al. 2001; Hu et al. 2002). In an attempt to find star-forming objects at $z \approx 5.7$, we have carried out a very deep optical imaging survey in the field surrounding the quasar SDSSp J104433.04$-$012502.2 at redshift of 5.74[^1] (Fan et al. 2000; Djorgovski et al. 2001; Goodrich et al. 2001), using Suprime-Cam (Miyazaki et al. 1998), the wide-field ($34^\prime \times 27^\prime$ with a 0.2 arcsec/pixel resolution) prime-focus camera on the 8.2 m Subaru telescope (Kaifu 1998). In this Letter, we report on our discovery of a very narrow-line star-forming system at $z \approx 5.7$. OBSERVATIONS ============ Optical Imaging --------------- In this survey, we used the narrow-passband filter, NB816, centered on 8150 Å with a passband of $\Delta\lambda$(FWHM) = 120 Å; the central wavelength corresponds to a redshift of 5.70 for Ly$\alpha$ emission. We also used broad-passband filters, $B$, $R_{\rm C}$, $I_{\rm C}$, and $z^\prime$. A summary of the imaging observations is given in Table 1. All of the observations were done under photometric condition and the seeing size was between 0.7 arcsec and 1.3 arcsec during the run. Note that we analyzed only two CCD chips, in which quasar SDSSp J104433.04$-$012502.2 is present, to avoid delays for follow-up spectroscopy. The CCD data were reduced and combined using $IRAF$ and the mosaic-CCD data reduction software developed by Yagi et al. (2002). Photometric and spectrophotometric standard stars used in the flux calibration are SA101 for the $B$, $R_{\rm C}$, and $I_{\rm C}$ data, and GD 108, GD 58 (Oke 1990), and PG 1034+001 (Massey et al. 1988) for the NB816 data. The $z^\prime$ data were calibrated by using the magnitude of SDSSp J104433.04$-$012502.2 (Fan et al. 2000). The total size of the field is 1167 by 1167, corresponding to a solid angle of $\approx$ 136 arcmin$^{2}$. The volume probed by the NB816 imaging has (co–moving) transverse dimensions of 27.56 $h_{0.7}^{-1}\times 27.56 h_{0.7}^{-1}$ Mpc$^2$, and the half–power points of the filter correspond to a co–moving depth along the line of sight of 44.34 $h_{0.7}^{-1}$ Mpc ($z_{\rm min} \approx 5.653$ and $z_{\rm max} \approx 5.752$; note that the transmission curve of our NB816 filter has a Gaussian-like shape). Therefore, a total volume of $3.4 \times 10^4 h_{0.7}^{-3}$ Mpc$^{3}$ is probed in our NB816 image. Here, we adopt a flat universe with $\Omega_{\rm matter} = 0.3$, $\Omega_{\Lambda} = 0.7$, and $h=0.7$ where $h = H_0/($100 km s$^{-1}$ Mpc$^{-1}$). Source detection and photometry were performed using SExtractor version 2.2.1 (Bertin, & Arnouts 1996). Our detection limit (a 3$\sigma$ detection within a $2^{\prime\prime}$.8 diameter aperture) for each band is listed in Table 1. As for the source detection in the NB816 image, we used a criterion that a source must be a 13-pixel connection above 5$\sigma$ noise level. Adopting the criterion for the NB816 excess, $I_{\rm C} - NB816 > 1.0$ mag, we have found two strong emission-line sources. Our follow-up optical spectroscopy of these sources reveals that one source found at $\alpha$(J2000)=10$^{\rm h}$ 44$^{\rm m}$ 27$^{\rm s}$ and $\delta$(J2000)=$-01^\circ$ 23$^\prime$ 45$^{\prime\prime}$ (hereafter LAE J1044$-$0123) is a good candidate to be a subgalactic object at high redshift[^2]. Its AB magnitude in the $NB816$ band is 24.73. The optical thumb-nail images of LAE J1044$-$0123 are given in Fig. 1. As shown in this figure, LAE J1044$-$0123 is seen clearly only in the NB816 image. Although it is seen in the $I_{\rm C}$ image, its flux is below the $3\sigma$ noise level. The observed equivalent width is $EW_{\rm obs} > 238$ Å. The NB816 image reveals that LAE J1044$-$0123 is spatially extended; its angular diameter is 1.6 arcsec (above the 2$\sigma$ noise level). The size of the point spread function in the NB816 image is 0.90 arcsec. Correcting for this spread, we obtain an angular diameter of 1.3 arcsec. Optical Spectroscopy -------------------- Our optical spectroscopy was made by using the Keck II Echelle Spectrograph and Imager (ESI: Sheinis et al. 2000) on 2002 March 15 (UT). We used the Echelle mode with the slit width of 1 arcsec, resulting in a spectral resolution $R \simeq 3400$ at 8000 Å. The integration time was 1800 seconds. The spectrum of LAE J1044$-$0123, shown in Fig. 2, presents a narrow emission line at $\lambda = 8090$ Å. This is the only emission line that was detected within the ESI wavelength range (from 4000 Å to 9500 Å). This line may be either Ly$\alpha$ or \[O [ii]{}\]$\lambda$3727. The emission-line profile appears to show a sharper cutoff at wavelengths shortward of the line peak, providing some evidence that this line is Ly$\alpha$. A stronger argument in favor of this line identification comes from the lack of structure in the profile. If this line were \[O [ii]{}\] emission, the redshift would be $z \approx 1.17$. Since the \[O [ii\]]{} feature is a doublet line of \[O [ii]{}\]$\lambda$3726.0 and \[O [ii]{}\]$\lambda$3728.8, the line separation would be larger than 6.1 Å and the lines would be resolved in the ESI observations. Further, if the line were H$\beta$, \[O [iii]{}\]$\lambda$4959, \[O [iii]{}\]$\lambda$5007, or H$\alpha$ line, we would detect some other emission lines in our spectrum. Therefore, we conclude that the emission line at 8090 Å is Ly$\alpha$, giving a redshift of 5.655$\pm$0.002. RESULTS AND DISCUSSION ====================== Star Formation Activity in LAE J1044$-$0123 ------------------------------------------- The rest-frame equivalent width of Ly$\alpha$ emission becomes $EW_0 > 36$ Å. Our Keck/ESI spectrum gives the observed Ly$\alpha$ flux of $f$(Ly$\alpha$) = $(1.3 \pm 0.1) \times 10^{-17}$ ergs cm$^{-2}$ s$^{-1}$ and the rest-frame equivalent width of Ly$\alpha$ emission $EW_0 > 36$ Å. On the other hand, our NB816 magnitude of LAE J1044$-$0123 gives $f$(Ly$\alpha$) $\simeq 4.1 \times 10^{-17}$ ergs cm$^{-2}$ s$^{-1}$, being higher than by a factor of 3 than the Keck/ESI flux. Since the Keck/ESI spectrum was calibrated by a single measurement of a spectroscopic standard star, HZ 44, the photometric accuracy may not be good. Further, our slit width may not cover the entire Ly$\alpha$ nebula of LAE J1044$-$0123. Therefore, we use the NB816-based flux to estimate the star formation rate. The NB816 flux gives the Ly$\alpha$ luminosity $L$(Ly$\alpha$) $\simeq 1.4 \times 10^{43} ~ h_{0.7}^{-2}$ ergs s$^{-1}$. Using the relation $SFR = 9.1 \times 10^{-43} L({\rm Ly}\alpha) ~ M_\odot {\rm yr}^{-1}$ (Kennicutt 1998; Brocklehurst 1971), we obtain $\sim 13 ~h_{0.7}^{-2} ~ M_\odot$ yr$^{-1}$. This is a lower limit because no correction was made for possible internal extinction by dust grains in the system. The lack of UV continuum from this object prevents us from determining the importance of this effect. The most intriguing property of LAE J1044$-$0123 is that the observed emission-line width (full width at half maximum; FWHM) of redshifted Ly$\alpha$ is only 2.2 $\pm$ 0.3 Å. Since the instrumental spectral resolution is 1.7 $\pm$ 0.1 Å, the intrinsic width is only 1.4 $\pm$ 0.5 Å; note that this gives a upper limit because the line is barely resolved. It corresponds to $FWHM_{\rm obs} \simeq$ 52 $\pm$ 19 km s$^{-1}$ or a velocity dispersion $\sigma_{\rm obs} = FWHM_{\rm obs}/(2 \sqrt{2 {\rm ln} 2}) \simeq$ 22 km s$^{-1}$. This value is comparable to those of luminous globular clusters (Djorgovski 1995). It is interesting to compare the observational properties of LAE J1044$-$0123 with similar LAEs at $z \gtrsim 5$. For this comparison, we choose Abell 2218 a (Ellis et al. 2001), LAE J1044$-$0130 (Ajiki et al. 2002), and J123649.2+621539 (Dawson et al. 2002) because these objects were also observed using KecK/ESI. A summary is given in Table 2. This comparison shows that LAE J1044$-$0123 has the narrowest line width that may be roughly comparable to that of Abell 2218 a although the mass of Abell 2218, $\sim 10^6 M_\odot$, is much smaller than that of LAE J1044$-$0123 (see next subsection). Another important point appears that the line profile of LAE J1044$-$0123 does not show intense red wing emission which is evidently seen in those of the other three LAEs. The diversity of the observational properties of these LAEs suggest that the H [i]{} absorptions affect significantly the visibility of the Ly$\alpha$ emission line. Further, the contribution of superwinds may be different from LAE to LAE. What is LAE J1044$-$0123 ? -------------------------- Now a question arises as; \`\`What is LAE J1044$-$0123 ?". There are two alternative ideas: (1) LAE J1044$-$0123 is a part of a giant system and we observe only the bright star-forming clump, or (2) LAE J1044$-$0123 is a single star-forming system. Solely from our observations, we cannot judge which is the case. If this is the first case, LAE J1044$-$0123 may be similar to Abell 2218 a found by Ellis et al. (2001). One problem in this interpretation seems that the spatial extension, $\sim$ 7.7 kpc, of LAE J1044$-$0123 is fairly large for such a less-massive system. Therefore, adopting the second case, it seems important to investigate possible dynamical status of LAE J1044$-$0123 for future consideration. If a source is surrounded by neutral hydrogen, Ly$\alpha$ photons emitted from the source are heavily scattered. Furthermore, the red damping wing of the Gunn-Peterson trough could also suppress the Ly$\alpha$ emission line (Gunn & Peterson 1965; Miralda-Escudé 1998; Miralda-Escudé & Rees 1998; Haiman 2002 and references therein). If this is the case for LAE J1044$-$0123, we may see only a part of the Ly$\alpha$ emission. Haiman (2002) estimated that only 8% of the Ly$\alpha$ emission is detected in the case of HCM-6A at $z=6.56$ found by Hu et al. (2002). However, the observed Ly$\alpha$ emission-line profile of LAE J1044$-$0123 shows the sharp cutoff at wavelengths shortward of the line peak. This property suggests that the H [i]{} absorption is dominated by H [i]{} gas in the system rather than that in the IGM Therefore, it seems reasonable to adopt that blue half of the Ly$\alpha$ emission could be absorbed in the case of LAE J1044$-$0123. Then we estimate a modest estimate of the velocity dispersion, $\sigma_0 \sim 2 \sigma_{\rm obs} \sim$ 44 km s$^{-1}$. Given the diameter of this object probed by the Ly$\alpha$ emission, $D \simeq 7.7 h_{0.7}^{-1}$ kpc, we obtain the dynamical timescale of $\tau_{\rm dyn} \sim D/\sigma_0 \sim 1.7 \times 10^8$ yr. This would give a upper limit of the star formation timescale in the system; i.e., $\tau_{\rm SF} \lesssim \tau_{\rm dyn}$. However, if the observed diameter is determined by the so-called Strömgren sphere photoionized by a central star cluster, it is not necessary to adopt $\tau_{\rm SF} \sim \tau_{\rm dyn}$. It seems more appropriate to adopt a shorter timescale for such a high-$z$ star-forming galaxies, e.g., $\tau_{\rm SF} \sim 10^7$ yr, as adopted for HCM-6A at $z \approx 6.56$ (Hu et al. 2002) by Haiman (2002). One may also derive a dynamical mass $M_{\rm dyn} = (D/2) \sigma_0 ^2 G^{-1} \sim 2 \times 10^9 M_\odot$ (neglecting possible inclination effects). At the source redshift, $z=5.655$, the mass of a dark matter halo which could collapse is estimated as $M_{\rm vir} \sim 9 \times 10^6 r_{\rm vir, 1}^3 h_{0.7}^{-1} ~ M_\odot$ where $r_{\rm vir, 1}$ is the Virial radius in units of 1 kpc \[see equation (24) in Barkana & Loeb (2001)\]. If we adopt $r_{\rm vir} = D/2$ = 3.85 kpc, we would obtain $M_{\rm vir} \sim 5 \times 10^8 ~ M_\odot$. However, the radius of dark matter halo could be ten times as long as $D/2$. If this is the case, we obtain $M_{\rm vir} \sim 5 \times 10^{10} ~ M_\odot$ and $\sigma_0 \sim 75$ km s$^{-1}$. Comparing this velocity dispersion with the observed one, we estimate that the majority of Ly$\alpha$ emission would be absorbed by neutral hydrogen. The most important issue related to LAE J1044$-$0123 seems how massive this source is; i.e., $\sim 10^9 M_\odot$ or more massive than $10^{10} M_\odot$. If the star formation timescale is as long as the dynamical one, the stellar mass assembled in LAE J1044$-$0123 at $z=5.655$ exceeds $10^9 ~ M_\odot$, being comparable to the nominal dynamical mass, $M_{\rm dyn} \sim 2 \times 10^9 ~ M_\odot$. Since it is quite unlikely that most mass is assembled to form stars in the system, the dark matter halo around LAE J1044$-$0123 would be more massive by one order of magnitude at least than the above stellar mass. If this is the case, we could miss the majority of the Ly$\alpha$ emission and the absorption cloud be attributed to the red damping wing of neutral hydrogen in the IGM. Since the redshift of LAE J1044$-$0123 ($z=5.655$) is close to that of SDSSp J104433.04$-$012502.2 ($z=5.74$), it is possible that these two objects are located at nearly the same cosmological distance. The angular separation between LAE J1044$-$0123 and SDSSp J104433.04$-$012502.2, 113 arcsec, corresponds to the linear separation of 4.45 $h_{0.7}^{-1}$ Mpc. The Strömgren radius of SDSSp J104433.04$-$012502.2 can be estimated to be $r_{\rm S} \sim 6.3 (t_{\rm Q}/2\times 10^7 ~ {\rm yr})^{1/3}$ Mpc using equation (1) in Haiman & Cen (2002) where $t_{\rm Q}$ is the lifetime of the quasar (see also Cen & Haiman 2000). Even if this quasar is amplified by a factor of 2 by the gravitational lensing (Shioya et al. 2002), we obtain $r_{\rm S} \sim 4.9$ Mpc. Therefore, it seems likely that the IGM around LAE J1044$-$0123 may be ionized completely. If this is the case, we cannot expect that the Ly$\alpha$ emission of LAE J1044$-$0123 is severely absorbed by the red damping wing emission. In order to examine which is the case, $L$-band spectroscopy is strongly recommended because the redshifted \[O [iii]{}\]$\lambda$5007 emission will be detected at 3.33 $\mu$m. However, we need the James Webb Space Telescope to complete it. [lcccc]{} $B$ & 2002 February 17 & 1680 & 26.6 & 1.2\ $R_{\rm C}$ & 2002 February 15, 16 & 4800 & 26.2 & 1.4\ $I_{\rm C}$ & 2002 February 15, 16 & 3360 & 25.9 & 1.2\ $NB816$ & 2002 February 15 - 17 & 36000 & 26.0 & 0.9\ $z'$ & 2002 February 15, 16 & 5160 & 25.3 & 1.2\ [lcccc]{} LAE J1044$-$0123 & 5.655 & 4.1 & 52 & This paper\ Abell 2218 a & 5.576 & 6.2 & $\sim$70 & 1\ LAE J1044$-$0130 & 5.687 & 1.5 & 340 & 2\ J123649.2+621539 & 5.190 & 3.0 & 280 & 3\ We would like to thank T. Hayashino and the staff at both the Subaru and Keck Telescopes for their invaluable help. We would like to thank Paul Shapiro and Renyue Cen for their encouraging discussion on high-$z$ Ly$\alpha$ emitters. We would also like to thank Zoltan Haiman and an anonymous referee for their useful comments. [^1]: The discovery redshift was $z=5.8$ (Fan et al. 2000). Since, however, the subsequent optical spectroscopic observations suggested a bit lower redshift; $z=5.73$ (Djorgovski et al. 2001) and $z=5.745$ (Goodrich et al. 2001), we adopt $z=5.74$ in this Letter. [^2]: Another source has been identified as a Ly$\alpha$ emitter at $z=5.687$ (Ajiki et al. 2002)

--- abstract: 'The Pearcey kernel is a classical and universal kernel arising from random matrix theory, which describes the local statistics of eigenvalues when the limiting mean eigenvalue density exhibits a cusp-like singularity. It appears in a variety of statistical physics models beyond matrix models as well. We consider the Fredholm determinant of a trace class operator acting on $L^2\left(-s, s\right)$ with the Pearcey kernel. Based on a steepest descent analysis for a $3\times 3$ matrix-valued Riemann-Hilbert problem, we obtain asymptotics of the Fredholm determinant as $s\to +\infty$, which is also interpreted as large gap asymptotics in the context of random matrix theory.' author: - 'Dan Dai,  Shuai-Xia Xu and Lun Zhang' title: Asymptotics of Fredholm determinant associated with the Pearcey kernel --- Introduction and statement of the results ========================================= In a series of papers [@BH]–[@BH96a], Brézin and Hikami initiated the studies of deformed complex Gaussian unitary ensemble (GUE) of the form $$\frac{1}{Z_n}e^ {-n \textrm{Tr}\left(\frac{M^2}{2}-AM \right)}{\,\mathrm{d}}M,$$ defined on the space of $n \times n$ Hermitian matrices, where $Z_n$ is a normalization constant and $A$ is a deterministic matrix also known as the external source. An interesting feature of this matrix ensemble is that it provides a simple model to create a phase transition for the eigenvalues of $M$ in the large $n$ limit. Indeed, by assuming the matrix $A$ is diagonal with two eigenvalues $a$ and $-a$ of equal multiplicity, it follows from the work of Pastur [@Pastur] that if $a>1$, the eigenvalues are distributed on two disjoint intervals, while for $0<a<1$, the eigenvalues are distributed on a single interval. In the critical case when $a\to 1$ as $n\to \infty$, the gap closes at the origin and the limiting mean eigenvalue density exhibits a cusp-like singularity, i.e., the density vanishes like $|x|^{1/3}$ as $x\to 0$. Upon letting $n\to\infty$ and after proper scaling, a new local eigenvalue process characterized by the so-called Pearcey kernel emerges near the origin. The Pearcey kernel $K^{\mathrm{Pe}}$ is defined as (see [@BH; @BH1]) $$\begin{aligned} \label{eq: pearcey kernel} K^{\mathrm{Pe}}(x,y;\rho)&=\int_0^{\infty}p(x+z)q(y+z){\,\mathrm{d}}z \nonumber \\ &=\frac{p(x)q''(y)-p'(x)q'(y)+p''(x)q(y)-\rho p(x)q(y)}{x-y},\end{aligned}$$ where $\rho\in\mathbb{R}$, $$\label{eq:pearcey integral} p(x)=\frac{1}{2\pi}\int_{-\infty}^\infty e^{-\frac14s^4-\frac{\rho}{2}s^2+isx} {\,\mathrm{d}}s \qquad \text{and} \qquad q(y)=\frac{1}{2\pi} \int_\Sigma e^{\frac14 t^4+\frac{\rho}{2}t^2+ity} {\,\mathrm{d}}t.$$ The contour $\Sigma$ in the definition of $q$ consists of the four rays $\arg t=\pi/4,3\pi/4,5\pi/4,7\pi/4$, where the first and the third rays are oriented from infinity to zero while the second and the last rays are oriented outwards; see Figure \[fig: sigma\] for an illustration. The functions $p$ and $q$ in are solutions of the third order differential equations $$\begin{aligned} p'''(x)&=xp(x)+\rho p'(x), \label{eq:Pearcey1} \\ q'''(y)&=-yq(y)+\rho q'(y),\end{aligned}$$ respectively. Since $p$ and $q$ were first introduced by Pearcey in the context of electromagnetic fields [@Pear], the kernel $K^{\mathrm{Pe}}$ bears the name Pearcey kernel. To see how $K^{\mathrm{Pe}}$ describe the aforementioned phase transition, note that the eigenvalues of $M$ form a determinantal point process with a correlation kernel $K_n(x,y;a)$ depending on $a$ (see [@BH; @BH1; @Zinn]), it was established in [@BH; @BH1] (for $\rho=0$) and in [@BK3; @TW] (for general $\rho \in \mathbb{R}$) that $$\lim_{n\to\infty}\frac{1}{n^{3/4}}K_n\left(\frac{x}{n^{3/4}},\frac{y}{n^{3/4}}; 1+\frac{\rho}{2\sqrt{n}}\right)=K^{\mathrm{Pe}}(x,y;\rho),$$ i.e., the correlation kernel $K_n$ converges to the Pearcey kernel near the origin as $n\to \infty$ in a double scaling regime. (100,70)(-5,2) (20,40)(60,40) (40,40)[(1,1)[15]{}]{} (40,40)[(-1,-1)[15]{}]{} (40,40)[(-1,1)[15]{}]{} (40,40)[(1,-1)[15]{}]{} (40,40) (39.3,36)[$0$]{} (43,41)[$\pi/4$]{} (50,50)[(-1,-1)[.0001]{}]{} (30,50)[(-1,1)[.0001]{}]{} (50,30)[(1,-1)[.0001]{}]{} (30,30)[(1,1)[.0001]{}]{} (56,52)[$\Sigma$]{} Like the classical kernels (sine kernel and Airy kernel) arising from random matrix theory [@Forrester; @metha], the Pearcey kernel is a universal object as evidenced by its appearance in a variety of stochastic models. On one hand, the Pearcey statistics have been established in specific matrix models including large complex correlated Wishart matrices [@HHNa; @HHNb], a two-matrix model with special quartic potential [@GZ], and quite recently for general complex Hermitian Wigner-type matrices at the cusps [@EKS], where the requirement on the identical distribution in Wigner matrices is dropped. It is worthwhile to mention that, for Wigner-type matrices, the density of states exhibits only square root or cubic root cusp singularities; see the classification theorem in [@AjEK2017; @AlEK2018]. On the other hand, one also encounters the Pearcey kernel beyond matrix models, as can be seen from its connection with non-intersecting Brownian motions at cusps [@AOV; @AM; @BK3] and a combinatorial model on random partitions [@OR]. Let $K^{\mathrm{Pe}}_{s,\rho}$ be the trace class operator acting on $L^2\left(-s, s\right)$ with the Pearcey kernel , it is well-known that the associated Fredholm determinant $\det\left(I-K^{\mathrm{Pe}}_{s,\rho}\right)$ gives us the probability of finding no particles (also known as the gap probability) on the interval $(-s,s)$ in a determinantal point process on the real line characterized by the Pearcey kernel. Moreover, it is shown in [@AM; @BC1; @BH; @TW] that the gap probability satisfies some nonlinear differential equations under more general settings. Since one cannot evaluate the Fredholm determinant explicitly for any fixed $s$, a natural and fundamental question is then to ask for its large $s$ asymptotics, which will be the aim of the present work. Denote by $$\label{def:Fnotation} F(s;\rho):=\ln \det\left(I-K^{\mathrm{Pe}}_{s,\rho}\right)$$ the logarithm of Fredholm determinant associated with the Pearcey kernel, our main result is the following theorem. \[main-thm\] With $F(s;\rho)$ defined in , we have, as $s\to +\infty$, $$\label{main-F-asy} F(s;\rho)= -\frac{9 s^{\frac83}}{2^{\frac{17}3}} + \frac{\rho s^2}{4} - \frac{\rho^2 s^{{\frac43}}}{2^{{\frac{10}3}}} - \frac{2}{9} \ln s +\frac{\rho^4}{216} + C + {\mathcal{O}}(s^{-\frac{2}{3}}),$$ uniformly for $\rho$ in any compact subset of $\mathbb{R}$, where $C$ is an undetermined constant independent of $\rho$ and $s$. In the literature, the large $s$ asymptotics of $F(s;\rho)$ was formally derived in [@BH] for $\rho = 0$, based on the coupled nonlinear differential equations satisfied by $F(s;0)$. Moreover, the asymptotics therein contains the leading term alone, without providing any information about the error estimate or the sub-leading terms. Our asymptotic expansion includes more terms and improves the result in [@BH]. After a change of variable $s\mapsto s/2$, the leading term of the asymptotic formula , i.e., $-9 s^{\frac83}/2^{\frac{17}3}$, agrees with that obtained in [@BH (3.36)]. Furthermore, we wish to emphasize that our derivation is rigorous, which makes use of integrable structure of the Pearcey kernel in the sense of Its-Izergin-Korepin-Slavnov [@IIKS90] and involves a steepest descent analysis of the revelent Riemann-Hilbert (RH) problem. As also observed in [@BH], the leading term of the asymptotic formula confirms the so-called Forrester-Chen-Eriksen-Tracy conjecture [@CET95; @Forrester93]. This conjecture asserts that if the density of state behaves as $|x-x^*|^\beta$ near a point $x^*$, then the probability $E(s)$ of emptiness of the interval $(x^* -s, x^* +s)$ behaves like $$E(s) \sim \exp \biggl(-C s^{2\beta +2} \biggr), \qquad \textrm{as } s \to +\infty.$$ In the present Pearcey case, we have $\beta = \frac{1}{3}$ and $2\beta +2 = \frac83$. Evaluation of the constant $C$ in is a challenging problem in the studies of large gap asymptotics [@Kra1]. For the classical sine, Airy and Bessel kernels encountered in random matrix theory, one could resolve this problem either by investigating the relevant Hankel or Toeplitz determinants which approximate the Fredholm determinants [@DIK2008; @DIKZ; @dkv; @Kra2], or by studying the total integrals of the Painlevé transcendents on account of Tracy-Widom type formulas for the gap probability [@BRD08]; see also [@BE; @E10; @E06] for the approach of operator theory. It seems unlikely that these methods are applicable in the present case. One rough idea to tackle this problem is based on the observation that, as the parameter $\rho$ tends to $-\infty$, the cusp singularity at the origin disappears and the origin becomes a regular point inside the bulk. Thus, we expect that $F(s;\rho)$ might be related to the determinant of (generalized) sine kernel under certain scaling limits when $\rho \to -\infty$, from which the constant term can be derived. We will leave this issue to a future publication. Finally, we note that the asymptotics of Fredholm determinant associated with the Pearcey kernel is also investigated from the viewpoint of phase transition in [@ACV; @BC1], i.e., to show how the Pearcey process becomes an Airy process by sending both $s$ and the parameter $\rho$ to positive infinity. We emphasize the asymptotic results therein are essentially different from ours. The rest of this paper is devoted to the proof of Theorem \[main-thm\]. We mainly follow the general strategy established in [@Bor:Dei2002; @DIZ97]. In Section \[sec:DiffIdentity\], we relate the partial derivatives of $F(s;\rho)$ to a $3 \times 3$ RH problem with constant jumps, which is essential in the proof. After introducing some auxiliary functions defined on a Riemann surface with a specified sheet structure in Section \[sec:auxiliary function\], we then perform a Deift-Zhou steepest descent analysis [@DZ93] on this RH problem for large positive $s$ in Section \[sec:asymanalyX\]. This asymptotic outcome, together with the differential identities for $F(s;\rho)$, will finally lead to the proof of Theorem \[main-thm\], as presented in Section \[sec:proof\]. Differential identities for the Fredholm determinant {#sec:DiffIdentity} ==================================================== A Riemann-Hilbert characterization of the Pearcey kernel -------------------------------------------------------- The starting point toward the proof of Theorem \[main-thm\] is an alternative representation of the Pearcey kernel $K^{\mathrm{Pe}}$ via a $3 \times 3$ RH problem, as shown in [@BK3] and stated next. \[rhp: Pearcey\] We look for a $3 \times 3$ matrix-valued function $\Psi(z)=\Psi(z;\rho)$ satisfying - $\Psi(z)$ is defined and analytic in ${\mathbb{C}}\setminus \{\cup_{j=0}^5\Sigma_j \cup \{ 0 \} \}$, where $$\label{def:sigmai} \begin{aligned} &\Sigma_0=(0,+\infty), ~~ \Sigma_1=e^{\frac{\pi i}{4}}(0,+\infty), ~~\Sigma_2=e^{\frac{ 3 \pi i}{4}}(0,+\infty), \\ &\Sigma_3=(-\infty, 0), ~~ \Sigma_4=e^{-\frac{3\pi i}{4}}(0,+\infty),~~ \Sigma_5=e^{-\frac{\pi i}{4}}(0,+\infty), \end{aligned}$$ with the orientations as shown in Figure \[fig:Pearcey\]. (100,70)(-5,2) (40,40)[(-1,-1)[20]{}]{} (40,40)[(-1,1)[20]{}]{} (40,40)[(-1,0)[30]{}]{} (40,40)[(1,0)[30]{}]{} (40,40)[(1,1)[20]{}]{} (40,40)[(1,-1)[20]{}]{} (30,50)[(1,-1)[1]{}]{} (30,40)[(1,0)[1]{}]{} (30,30)[(1,1)[1]{}]{} (50,50)[(1,1)[1]{}]{} (50,40)[(1,0)[1]{}]{} (50,30)[(1,-1)[1]{}]{} (39,36.3)[$0$]{} (20,15)[$\Sigma_4$]{} (20,63)[$\Sigma_2$]{} (3,40)[$\Sigma_3$]{} (60,15)[$\Sigma_5$]{} (60,63)[$\Sigma_1$]{} (75,40)[$\Sigma_0$]{} (22,44)[$\Theta_2$]{} (22,34)[$\Theta_3$]{} (55,44)[$\Theta_0$]{} (55,34)[$\Theta_5$]{} (38,53)[$\Theta_1$]{} (38,22)[$\Theta_4$]{} (40,40) - For $z\in \Sigma_k$, $k=0,1,\ldots,5$, the limiting values $$\Psi_+(z) = \lim_{\substack{\zeta \to z \\\zeta\textrm{ on $+$-side of }\Sigma_k}}\Psi(\zeta), \qquad \Psi_-(z) = \lim_{\substack{\zeta \to z \\\zeta\textrm{ on $-$-side of }\Sigma_k}}\Psi(\zeta),$$ exist, where the $+$-side and $-$-side of $\Sigma_k$ are the sides which lie on the left and right of $\Sigma_k$, respectively, when traversing $\Sigma_k$ according to its orientation. These limiting values satisfy the jump relation $$\label{jumps:M} \Psi_{+}(z) = \Psi_{-}(z)J_{\Psi}(z),\qquad z\in \cup_{j=0}^5\Sigma_j,$$ where $$\label{def:JPsi} J_{\Psi}(z):= \left\{ \begin{array}{ll} \begin{pmatrix} 0&1&0 \\ -1&0&0 \\ 0&0&1 \end{pmatrix}, & \qquad \hbox{$z\in \Sigma_0$,} \\ \begin{pmatrix} 1&0&0 \\ 1&1&1 \\ 0&0&1 \end{pmatrix}, & \qquad \hbox{$z\in \Sigma_1$,} \\ \begin{pmatrix} 1&0&0 \\ 0&1&0 \\ 1&1&1 \end{pmatrix}, & \qquad \hbox{$z\in \Sigma_2$,} \\ \begin{pmatrix} 0&0&1 \\ 0&1&0 \\ -1&0&0 \end{pmatrix}, & \qquad \hbox{$z\in \Sigma_3$,} \\ \begin{pmatrix} 1&0&0 \\ 0&1&0 \\ 1&-1&1 \end{pmatrix}, & \qquad \hbox{$z\in \Sigma_4$,} \\ \begin{pmatrix} 1&0&0 \\ 1&1&-1 \\ 0&0&1 \end{pmatrix}, & \qquad \hbox{$z\in \Sigma_5$.} \end{array} \right.$$ - As $z \to \infty$ and $\pm{\mathrm{Im}\,}z>0$, we have $$\label{eq:asyPsi} \Psi(z)= \sqrt{\frac{2\pi}{3}}i e^{\frac{\rho^2}{6}} \Psi_0 \left(I+ \frac{\Psi_1}{z} +\mathcal O(z^{-2}) \right)\operatorname{diag}\left(z^{-\frac13},1,z^{\frac13} \right)L_{\pm} e^{\Theta(z)},$$ where $$\label{asyPsi:coeff} \Psi_0 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ \kappa_3(\rho)+\frac{2\rho}{3} & 0 & 1 \end{pmatrix}, \qquad \Psi_1 = \begin{pmatrix} 0 & \kappa_3(\rho) & 0 \\ \widetilde\kappa_6(\rho) & 0 & \kappa_3(\rho) + \frac{\rho}{3} \\ 0 & \widehat \kappa_6(\rho) & 0 \end{pmatrix},$$ with $$\label{poly:kappa3} \kappa_3(\rho) = \frac{\rho^3}{54} - \frac{\rho}{6},$$ $\widetilde\kappa_6(\rho) = \kappa_6(\rho) + \frac{\rho}{3} \kappa_3(\rho) - \frac{1}{3}$, $\widehat\kappa_6(\rho) = \kappa_6(\rho) -\kappa_3^2(\rho) + \frac{\rho^2}{9} - \frac{1}{3},$ and $$\label{poly:kappa6} \kappa_6(\rho) = \frac{\rho^6}{5832} - \frac{\rho^4}{162} - \frac{\rho^2}{72} + \frac{7}{36}.$$ Moreover, $L_{\pm}$ are constant matrices $$\begin{aligned} \label{def:Lpm} L_{+}= \begin{pmatrix} -\omega & \omega^2 & 1 \\ -1&1&1 \\ -\omega^2 & \omega & 1 \end{pmatrix}, \qquad L_{-}= \begin{pmatrix} \omega^2 & \omega & 1 \\ 1&1&1 \\ \omega & \omega^2 & 1 \end{pmatrix},\end{aligned}$$ with $\omega=e^{2\pi i/3}$, and $\Theta(z)$ is given by $$\begin{aligned} \label{def:Theta} \Theta(z)=\Theta(z;\rho)&= \begin{cases} \operatorname{diag}(\theta_1(z;\rho),\theta_2(z;\rho),\theta_3(z;\rho)), & \text{${\mathrm{Im}\,}z >0$,} \\ \operatorname{diag}(\theta_2(z;\rho),\theta_1(z;\rho),\theta_3(z;\rho)), & \text{${\mathrm{Im}\,}z <0$,} \\ \end{cases}\end{aligned}$$ with $$\label{eq: theta-k-def} \theta_k(z;\rho)=\frac34 \omega^{2k}z^{\frac43}+\frac{\rho}{2}\omega^kz^{\frac23}, \qquad k=1,2,3.$$ - $\Psi(z)$ is bounded near the origin. It is shown in [@BK3 Section 8.1] that the above RH problem has a unique solution expressed in terms of solutions of the Pearcey differential equation . Indeed, note that admits the following solutions: $$\label{def:pj} p_j(z)=p_j(z;\rho)=\int_{\Gamma_j}e^{-\frac14 s^4-\frac{\rho}{2}s^2+is z}{\,\mathrm{d}}s, \qquad j=0,1,\ldots,5,$$ where $$\begin{aligned} \Gamma_0&=(-\infty,+\infty), \quad && \Gamma_1=(i\infty, 0]\cup[0,\infty), \\ \Gamma_2&=(i\infty,0]\cup[0,-\infty), \quad && \Gamma_3=(-i\infty, 0]\cup[0,-\infty), \\ \Gamma_4&=(-i\infty,0]\cup[0,\infty), \quad && \Gamma_5=(-i\infty, i\infty).\end{aligned}$$ We then have $$\label{def:soltopsi} \Psi(z)=\left\{ \begin{array}{ll} \begin{pmatrix} -p_2(z) & p_1(z) & p_5(z)\\ -p_2'(z) & p_1'(z) & p_5'(z) \\-p_2''(z) & p_1''(z) & p_5''(z) \end{pmatrix}, & \quad \hbox{$z\in \Theta_0$,} \\ \begin{pmatrix} p_0(z) & p_1(z) & p_4(z)\\ p_0'(z) & p_1'(z) & p_4'(z) \\ p_0''(z) & p_1''(z) & p_4''(z) \end{pmatrix}, & \quad \hbox{$z\in \Theta_1$,} \\ \begin{pmatrix} -p_3(z) & -p_5(z) & p_4(z) \\ -p_3'(z) & -p_5'(z) & p_4'(z) \\-p_3''(z) & -p_5''(z) & p_4''(z) \end{pmatrix}, & \quad \hbox{$z\in \Theta_2$,} \\ \begin{pmatrix} p_4(z) & -p_5(z) & p_3(z)\\ p_4'(z) & -p_5'(z) & p_3'(z) \\ p_4''(z) & -p_5''(z) & p_3''(z)\end{pmatrix}, & \quad \hbox{$z\in \Theta_3$,} \\ \begin{pmatrix} p_0(z) & p_2(z) & p_3(z)\\ p_0'(z) & p_2'(z) & p_3'(z) \\ p_0''(z) & p_2''(z) & p_3''(z) \end{pmatrix}, & \quad \hbox{$z\in \Theta_4$,} \\ \begin{pmatrix} p_1(z) & p_2(z) & p_5(z)\\ p_1'(z) & p_2'(z) & p_5'(z) \\ p_1''(z) & p_2''(z) & p_5''(z) \end{pmatrix}, & \quad \hbox{$z\in \Theta_5$,} \end{array} \right.$$ where $\Theta_k$, $k=0,1,\ldots,5$, is the region bounded by the rays $\Sigma_k$ and $\Sigma_{k+1}$ (with $\Sigma_6:=\Sigma_0$); see Figure \[fig: sigma\] for an illustration. For our purpose, we present a refined asymptotics of $\Psi$ at infinity in , which can be verified directly from . To see this, let us focus on the case $z\in\Theta_1$, since the asymptotics in other regions can be derived in a similar way. Carrying out a steepest descent analysis to the integrals defined (cf. [@BK3; @Miyamoto]), we have, as $z\to \infty$, $$\label{eq:asyp0} p_0(z) = \begin{cases} -\sqrt{\frac{2\pi}{3}}i e^{\frac{\rho^2}{6}} \omega z^{-\frac{1}{3}} e^{\theta_1(z;\rho)} \biggl( 1 + \frac{\kappa_3(\rho)}{\omega} z^{-\frac{2}{3}} + \frac{\kappa_6(\rho)}{\omega^2} z^{-\frac{4}{3}} + \mathcal O(z^{-2} ) \biggr), & {\mathrm{Im}\,}z >0, \\ \sqrt{\frac{2\pi}{3}}i e^{\frac{\rho^2}{6}} \omega^2 z^{-\frac{1}{3}} e^{\theta_2(z;\rho)} \biggl( 1 + \frac{\kappa_3(\rho)}{\omega^2} z^{-\frac{2}{3}} + \frac{\kappa_6(\rho)}{\omega^4} z^{-\frac{4}{3}} + \mathcal O(z^{-2} ) \biggr), & {\mathrm{Im}\,}z <0, \end{cases}$$ $$p_1(z) = \sqrt{\frac{2\pi}{3}}i e^{\frac{\rho^2}{6}} \omega^2 z^{-\frac{1}{3}} e^{\theta_2(z;\rho)} \biggl( 1 + \frac{\kappa_3(\rho)}{\omega^2} z^{-\frac{2}{3}} + \frac{\kappa_6(\rho)}{\omega^4} z^{-\frac{4}{3}} + \mathcal O(z^{-2} ) \biggr)$$ for $-\frac{3\pi}{4} < \arg z < \frac{5\pi}{4},$ and $$\label{eq:asyp4} p_4(z) = \sqrt{\frac{2\pi}{3}}i e^{\frac{\rho^2}{6}} z^{-\frac{1}{3}} e^{\theta_3(z;\rho)} \biggl( 1 + \kappa_3(\rho) z^{-\frac{2}{3}} + \kappa_6(\rho) z^{-\frac{4}{3}} + \mathcal O(z^{-2} ) \biggr)$$ for $-\frac{\pi}{4} < \arg z < \frac{7\pi}{4},$ where $\theta_k(z;\rho)$, $k=1,2,3$, are given in , $\kappa_3(\rho)$ and $\kappa_6(\rho)$ are polynomials in $\rho$ given in and . The asymptotic expansions of $p_j'(z)$ and $p_j''(z)$, $j=0,1,4$, can be derived in a similar fashion or simply by taking derivatives on the right hand sides of –. A combination of all these asymptotic results and then gives us after a straightforward calculation. Now, define $$\label{eq: tilde-psi} \widetilde \Psi (z)=\widetilde \Psi (z;\rho)= \begin{pmatrix} p_0(z) & p_1(z) & p_4(z) \\ p_0'(z) & p_1'(z) & p_4'(z) \\ p_0''(z) & p_1''(z) & p_4''(z) \end{pmatrix}, \qquad z\in \mathbb{C},$$ that is, $\widetilde \Psi$ is the analytic extension of the restriction of $\Psi$ on the region $\Theta_1$ to the whole complex plane. The Pearcey kernel then admits the following equivalent representation in terms of $\widetilde \Psi$ (see [@BK3 Equation (10.19)]): $$\label{eq:PearceyRH} K^{\mathrm{Pe}}(x,y;\rho)=\frac{1}{2 \pi i(x-y)} \begin{pmatrix} 0 & 1 & 1 \end{pmatrix} \widetilde \Psi^{-1}(y;\rho)\widetilde \Psi(x;\rho) \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \qquad x,y \in \mathbb{R}.$$ Differential equations for $\Psi$ --------------------------------- For later use, we need the following linear differential equations for $\Psi$ with respect to $z$ and $\rho$. \[prop:ODE\] Let $\Psi=\Psi(z;\rho)$ be the unique solution of the RH problem \[rhp: Pearcey\]. We then have $$\label{eq: ODE1} \frac{\partial \Psi}{\partial z}= \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ z & \rho & 0 \end{pmatrix}\Psi,$$ and $$\label{eq:ODE2} \frac{\partial \Psi}{\partial \rho}=\frac12 \begin{pmatrix} 0 & 0 & 1 \\ z & \rho & 0 \\ 1 & z & \rho \end{pmatrix}\Psi.$$ The differential equation follows directly from , and . To show , we obtain from , and direct calculations that, for $j=1,\ldots,5$, $$\frac{\partial p_j}{\partial \rho}=-\frac12 \int_{\Gamma_j}s^2e^{-\frac14 s^4-\frac{\rho}{2}s^2+is z}{\,\mathrm{d}}s=\frac12 p_j'',$$ $$\frac{\partial p_j'}{\partial \rho}=-\frac i 2 \int_{\Gamma_j}s^3 e^{-\frac14 s^4-\frac{\rho}{2}s^2+is z}{\,\mathrm{d}}s= \frac12 p_j'''=\frac12 (zp_j+\rho p_j'),$$ and $$\frac{\partial p_j''}{\partial \rho}= \frac 1 2 \int_{\Gamma_j}s^4 e^{-\frac14 s^4-\frac{\rho}{2}s^2+is z}{\,\mathrm{d}}s = \frac12 p_j'''' =\frac12 (zp_j+\rho p_j')'=\frac12 (p_j+zp_j'+\rho p_j''),$$ which leads to . This completes the proof of Proposition \[prop:ODE\]. Differential identities for $F$ ------------------------------- By , it is readily seen that $$\label{eq:tildeKdef} K^{\mathrm{Pe}}(x,y;\rho) = \frac{{\mathbf}{f}^t(x){\mathbf}{h}(y)}{x-y},$$ where $$\label{def:fh} {\mathbf}{f}(x)=\begin{pmatrix} f_1 \\ f_2 \\ f_3 \end{pmatrix}:=\widetilde \Psi(x) \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \qquad {\mathbf}{h}(y)=\begin{pmatrix} h_1 \\ h_2 \\ h_3 \end{pmatrix} := \frac{1}{2 \pi i} \widetilde \Psi^{-t}(y) \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}.$$ With the function $F$ defined in , we have $$\label{eq:derivatives} \frac{\partial}{\partial s}F(s;\rho)=\frac{{\,\mathrm{d}}}{{\,\mathrm{d}}s} \ln \det(I-K_{s,\rho}^{\mathrm{Pe}}) =-\textrm{tr}\left((I-K_{s,\rho}^{\mathrm{Pe}})^{-1} \frac{{\,\mathrm{d}}}{{\,\mathrm{d}}s}K_{s,\rho}^{\mathrm{Pe}}\right)=-R(s,s)-R(-s,-s),$$ where $R(u,v)$ stands for the kernel of the resolvent operator, that is, $$R=\left(I-K_{s,\rho}^{\mathrm{Pe}}\right)^{-1}-I=K_{s,\rho}^{\mathrm{Pe}}\left(I-K_{s,\rho}^{\mathrm{Pe}}\right)^{-1}=\left(I-K_{s,\rho}^{\mathrm{Pe}}\right)^{-1}K_{s,\rho}^{\mathrm{Pe}}.$$ Since the kernel of the operator $K_{s,\rho}^{\mathrm{Pe}}$ is integrable in the sense of [@IIKS90], its resolvent kernel is integrable as well; cf. [@DIZ97; @IIKS90]. Indeed, by setting $$\label{def:FH} {\mathbf}{F}(u)= \begin{pmatrix} F_1 \\ F_2 \\ F_3 \end{pmatrix}:=\left(I-K_{s,\rho}^{\mathrm{Pe}}\right)^{-1}{\mathbf}{f}, \qquad {\mathbf}{H}(v)=\begin{pmatrix} H_1 \\ H_2 \\ H_3 \end{pmatrix} :=\left(I-K_{s,\rho}^{\mathrm{Pe}}\right)^{-1}{\mathbf}{h},$$ we have $$\label{eq:resolventexpli} R(u,v)=\frac{{\mathbf}{F}^t(u){\mathbf}{H}(v)}{u-v}.$$ We could also represent $\frac{\partial}{\partial \rho}F(s;\rho)$ in terms of ${\mathbf}{F}$ and ${\mathbf}{H}$ defined in . To proceed, we note from and that $$\label{def:fderivative} \frac{\partial {\mathbf}{f}}{\partial \rho}(x) = \frac12 \begin{pmatrix} 0 & 0 & 1 \\ x & \rho & 0 \\ 1 & x & \rho \end{pmatrix} {\mathbf}{f}(x).$$ Moreover, by taking derivative with respect to $\rho$ on both sides of $\Psi \cdot \Psi^{-1}=I$, it is readily seen from that $$\frac{\partial \Psi^{-1}}{\partial \rho} = -\frac{\Psi^{-1}}{2} \begin{pmatrix} 0 & 0 & 1 \\ z & \rho & 0 \\ 1 & z & \rho \end{pmatrix},$$ which gives us $$\frac{\partial {\mathbf}{h}}{\partial \rho}(y) = -\frac12 \begin{pmatrix} 0 & y & 1 \\ 0 & \rho & y \\ 1 & 0 & \rho \end{pmatrix} {\mathbf}{h}(y).$$ This, together with and , implies $$\begin{gathered} \frac{{\,\mathrm{d}}}{{\,\mathrm{d}}\rho}K^{{\mathrm{Pe}}}=\frac{\frac{\partial {\mathbf}{f}^t}{\partial \rho}(x){\mathbf}{h}(y)+{\mathbf}{f}^t(x)\frac{\partial {\mathbf}{h}}{\partial \rho}(y)}{x-y} \\ ={\mathbf}{f}^t(x) \begin{pmatrix} 0 & 1/2 & 0 \\ 0 & 0 & 1/2 \\ 0 & 0 & 0 \end{pmatrix}{\mathbf}{h}(y)=\frac 12 (f_1(x)h_2(y)+f_2(x)h_3(y)).\end{gathered}$$ Hence, we obtain $$\begin{gathered} \label{eq:lamdaderivative} \frac{\partial}{\partial \rho} F(s;\rho)= \frac{{\,\mathrm{d}}}{{\,\mathrm{d}}\rho} \ln \det\left(I-K_{s,\rho}^{\mathrm{Pe}}\right) =-\textrm{tr}\left(\left(I-K_{s,\rho}^{\mathrm{Pe}}\right)^{-1} \frac{{\,\mathrm{d}}}{{\,\mathrm{d}}\rho}K_{s,\rho}^{\mathrm{Pe}}\right) \\ =-\frac 12 \int_{-s}^s (F_1(v)h_2(v)+F_2(v)h_3(v)){\,\mathrm{d}}v.\end{gathered}$$ We next establish the connection between the functions $\frac{\partial}{\partial s} F(s;\rho)$, $\frac{\partial}{\partial \rho} F(s;\rho)$ and an RH problem with constant jumps, which is based on the fact that the resolvent kernel $R(u,v)$ is related to the following RH problem. \[rhp:Y\] We look for a $3 \times 3$ matrix-valued function $Y(z)$ satisfying the following properties: 1. $Y(z)$ is defined and analytic in $\mathbb{C}\setminus [-s,s]$, where the orientation is taken from the left to the right. 2. For $x\in(-s,s)$, we have $$\label{eq:Y-jump} Y_+(x)=Y_-(x)(I-2\pi i {\mathbf}{f}(x){\mathbf}{h}^t(x)),$$ where the functions ${\mathbf}{f}$ and ${\mathbf}{h}$ are defined in . 3. As $z \to \infty$, $$\label{eq:Y-infty} Y(z)=I+\frac{\mathsf{Y}_1}{z}+\mathcal O(z^{-2}).$$ 4. As $z \to \pm s$, we have $Y(z) = \mathcal O(\ln(z \mp s))$. By [@DIZ97], it follows that $$\label{eq:Yexpli} Y(z)=I-\int_{-s}^s\frac{{\mathbf}{F}(w){\mathbf}{h}^t(w)}{w-z}{\,\mathrm{d}}w$$ and $$\label{def:FH2} {\mathbf}{F}(z)=Y(z){\mathbf}{f}(z), \qquad {\mathbf}{H}(z)=(Y^t(z))^{-1}{\mathbf}{h}(z).$$ Recall the RH problem \[rhp: Pearcey\] for $\Psi$, we make the following undressing transformation to arrive at an RH problem with constant jumps. To proceed, the four rays $\Sigma_k$, $k=1,2,4,5$, emanating from the origin are replaced by their parallel lines emanating from some special points on the real line. More precisely, we replace $\Sigma_1$ and $\Sigma_5$ by their parallel rays $\Sigma_1^{(s)}$ and $\Sigma_5^{(s)}$ emanating from the point $s$, replace $\Sigma_2$ and $\Sigma_4$ by their parallel rays $\Sigma_2^{(s)}$ and $\Sigma_4^{(s)}$ emanating from the point $-s$. Furthermore, these rays, together with the real axis, divide the complex plane into six regions $\texttt{I-VI}$, as illustrated in Figure \[fig:X\]. (100,70)(-5,2) (25,40)[(-1,0)[30]{}]{} (55,40)[(1,0)[30]{}]{} (25,40)(55,40) (25,40)[(-1,-1)[25]{}]{} (25,40)[(-1,1)[25]{}]{} (55,40)[(1,1)[25]{}]{} (55,40)[(1,-1)[25]{}]{} (15,40)[(1,0)[1]{}]{} (65,40)[(1,0)[1]{}]{} (10,55)[(1,-1)[1]{}]{} (10,25)[(1,1)[1]{}]{} (70,25)[(1,-1)[1]{}]{} (70,55)[(1,1)[1]{}]{} (39,36.3)[$0$]{} (-2,11)[$\Sigma_4^{(s)}$]{} (-2,67)[$\Sigma_2^{(s)}$]{} (3,42)[$\Sigma_3^{(s)}$]{} (80,11)[$\Sigma_5^{(s)}$]{} (80,67)[$\Sigma_1^{(s)}$]{} (73,42)[$\Sigma_0^{(s)}$]{} (10,46)[$\texttt{III}$]{} (10,34)[$\texttt{IV}$]{} (68,46)[$\texttt{I}$]{} (68,34)[$\texttt{VI}$]{} (38,55)[$\texttt{II}$]{} (38,20)[$\texttt{V}$]{} (40,40) (25,40) (55,40) (24,36.3)[$-s$]{} (54,36.3)[$s$]{} We now define $$\begin{aligned} \label{eq:YtoX} X(z) = \left\{ \begin{array}{ll} Y(z)\Psi(z), & \hbox{for $z$ in the region $\texttt{I}\cup \texttt{III}\cup \texttt{IV} \cup \texttt{VI}$,} \\ Y(z)\widetilde \Psi(z), & \hbox{for $z$ in the region $\texttt{II}$,} \\ Y(z)\widetilde \Psi(z) \begin{pmatrix} 1 & -1 & -1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}, & \hbox{for $z$ in the region $\texttt{V}$,} \end{array} \right.\end{aligned}$$ where $\widetilde \Psi$ is defined in . Then, $X$ satisfies the following RH problem. \[rhp:X\] The function $X$ defined in has the following properties: 1. $X(z)$ is defined and analytic in $\mathbb{C}\setminus \{\cup^5_{j=0}\Sigma_j^{(s)}\cup \{-s\} \cup\{s\}\}$, where $$\label{def:sigmais} \begin{aligned} &\Sigma_0^{(s)}=(s,+\infty), ~~ &&\Sigma_1^{(s)}=s+e^{\frac{\pi i}{4}}(0,+\infty), ~~ &&&\Sigma_2^{(s)}=-s+e^{\frac{ 3 \pi i}{4}}(0,+\infty), \\ &\Sigma_3^{(s)}=(-\infty, -s), ~~ &&\Sigma_4^{(s)}=-s+e^{-\frac{3\pi i}{4}}(0,+\infty), ~~ &&&\Sigma_5^{(s)}=s+e^{-\frac{\pi i}{4}}(0,+\infty), \end{aligned}$$ with the orientations from the left to the right; see the solid lines in Figure \[fig:X\]. 2. $X$ satisfies the jump condition $$\label{eq:X-jump} X_+(z)=X_-(z)J_X(z), \qquad z\in \cup^5_{j=0}\Sigma_j^{(s)},$$ where $$\label{def:JX} J_X(z):=\left\{ \begin{array}{ll} \begin{pmatrix} 0&1&0 \\ -1&0&0 \\ 0&0&1 \end{pmatrix}, & \qquad \hbox{$z\in \Sigma_0^{(s)}$,} \\ \begin{pmatrix} 1&0&0 \\ 1&1&1 \\ 0&0&1 \end{pmatrix}, & \qquad \hbox{$z\in \Sigma_1^{(s)}$,} \\ \begin{pmatrix} 1&0&0 \\ 0&1&0 \\ 1&1&1 \end{pmatrix}, & \qquad \hbox{$z\in \Sigma_2^{(s)}$,} \\ \begin{pmatrix} 0&0&1 \\ 0&1&0 \\ -1&0&0 \end{pmatrix}, & \qquad \hbox{$z\in \Sigma_3^{(s)}$,} \\ \begin{pmatrix} 1&0&0 \\ 0&1&0 \\ 1&-1&1 \end{pmatrix}, & \qquad \hbox{$z\in \Sigma_4^{(s)}$,} \\ \begin{pmatrix} 1&0&0 \\ 1&1&-1 \\ 0&0&1 \end{pmatrix}, & \qquad \hbox{$z\in \Sigma_5^{(s)}$.} \end{array} \right.$$ 3. As $z \to \infty$ and $\pm {\mathrm{Im}\,}z>0$, we have $$\label{eq:asyX} X(z)= \sqrt{\frac{2\pi}{3}}i e^{\frac{\rho^2}{6}} \Psi_0 \left(I+ \frac{\mathsf{X}_1}{z} +\mathcal O(z^{-2}) \right)\operatorname{diag}\left(z^{-\frac13},1,z^{\frac13} \right)L_{\pm} e^{\Theta(z)},$$ where $\Psi_0$, $L_{\pm}$ and $\Theta(z)$ are given in , and , respectively, and $$\label{X1-formula} \mathsf{X}_1 = \Psi_1 + \Psi_0^{-1} \mathsf{Y}_1 \Psi_0$$ with $\Psi_1$ and $\mathsf{Y}_1$ given in and . 4. As $z \to \pm s$, we have $X(z) = \mathcal O(\ln(z \mp s))$. We only need to show that $X$ does not have a jump over $(-s,s)$, while the other claims follow directly from and the RH problem \[rhp: Pearcey\] for $\Psi$. By , we have, for $-s<x<s$, $$I-2\pi i {\mathbf}{f}(x){\mathbf}{h}^t(x)=\widetilde\Psi(x) \begin{pmatrix} 1 & -1 & -1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\widetilde\Psi(x)^{-1}.$$ This, together with and , implies that for $x\in(-s,s)$, $$\begin{gathered} X_+(x)=Y_+(x)\widetilde \Psi(x)=Y_-(x)(I-2\pi i {\mathbf}{f}(x){\mathbf}{h}^t(x))\widetilde \Psi(x) \\ =Y_-(x)\widetilde \Psi(x)\begin{pmatrix} 1 & -1 & -1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}=X_-(x),\end{gathered}$$ as desired. This completes the proof of Proposition \[rhp:X\]. The connections between the above RH problem and the partial derivatives of $F(s;\rho)$ are revealed in the following proposition. \[prop:derivativeandX\] With $F$ defined in , we have $$\begin{aligned} \frac{\partial}{\partial \rho} F(s;\rho) &= \frac{{\,\mathrm{d}}}{{\,\mathrm{d}}\rho} \ln \det\left(I-K_{s,\rho}^{\mathrm{Pe}}\right) = -\frac{1}{2} \left[ (\mathsf{X}_1)_{12} + (\mathsf{X}_1)_{23} \right] + \frac{\rho^3}{54}, \label{eq:derivativein-rho-X}\end{aligned}$$ where $\mathsf{X}_1$ is given in and $(M)_{ij}$ stands for the $(i,j)$th entry of a matrix $M$, and $$\begin{aligned} \frac{\partial}{\partial s} F(s;\rho)&=\frac{{\,\mathrm{d}}}{{\,\mathrm{d}}s} \ln \det\left(I-K_{s,\rho}^{{\mathrm{Pe}}}\right) \nonumber \\ &=-\frac{1}{ \pi i} \left[\lim_{z \to -s} \left(\left(X^{-1}(z)X'(z)\right)_{21}+\left(X^{-1}(z)X'(z)\right)_{31}\right) \right], \label{eq:derivativeinsX-2}\end{aligned}$$ where the above limit is taken from the region $\texttt{II} \, \cup \texttt{V}$. We begin with the proof of . From and , it follows that $$\mathsf{Y}_1 = \int_{-s}^s {\mathbf}{F}(v){\mathbf}{h}^t(v) {\,\mathrm{d}}v = \int_{-s}^s \begin{pmatrix} F_1(v) \\ F_2(v) \\ F_3(v) \end{pmatrix} \begin{pmatrix} h_1(v) & h_2(v) & h_3(v) \end{pmatrix} {\,\mathrm{d}}v .$$ This, together with , implies that $$\label{eq:Frho1} \frac{\partial}{\partial \rho} F(s;\rho) = - \frac{1}{2} \left[ (\mathsf{Y}_1)_{12} + (\mathsf{Y}_1)_{23} \right].$$ By , it is readily seen that $\mathsf{Y}_1 = \Psi_0 (\mathsf{X}_1 - \Psi_1 ) \Psi_0^{-1}$. With the aid of the explicit expressions of $\Psi_0$ and $\Psi_1$ in , we then have $(\mathsf{Y}_1)_{12} + (\mathsf{Y}_1)_{23}=(\mathsf{X}_1)_{12} + (\mathsf{X}_1)_{23}-\frac{\rho^3}{27}$, which gives us in view of . We next consider $\frac{\partial}{\partial s} F(s;\rho)$. For $z\in \texttt{II}$, we see from , and that $$\label{eq:FHinX} {\mathbf}{F}(z)=Y(z){\mathbf}{f}(z)=Y(z)\widetilde \Psi(z)\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}=X(z) \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$$ and $${\mathbf}{H}(z)=Y^{-t}(z){\mathbf}{h}(z)=X^{-t}(z)\widetilde \Psi^{t}(z)\cdot \frac{\widetilde \Psi^{-t}(z)}{2 \pi i}\begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}=\frac{X^{-t}(z)}{2 \pi i} \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}.$$ Combining the above formulas, , and L’Hôspital’s rule then gives us $$\begin{aligned} \frac{\partial}{\partial s} F(s;\rho) =&-\frac{1}{2 \pi i} \bigg[\lim_{z \to -s} \left(\left(X^{-1}(z)X'(z)\right)_{21}+\left(X^{-1}(z)X'(z)\right)_{31}\right) \nonumber \\ & + \lim_{z \to s} \left(\left(X^{-1}(z)X'(z)\right)_{21}+\left(X^{-1}(z)X'(z)\right)_{31}\right)\bigg], \label{eq:derivativeinsX} $$ where the above limits are taken from the region $\texttt{II}$. Similarly, one can show that also holds provided the limits are taken from the region $\texttt{V}$. The expression can be further simplified via the following symmetric relation of $X(z)$: $$\label{eq: Xz+-} X(z) = \widetilde {C} X(-z) {\Lambda},$$ where $$\label{def:tildeC} \widetilde {C}= \Psi_0 \begin{pmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{pmatrix} \Psi_0^{-1}$$ is a nonsingular matrix with $\Psi_0$ given in , and $$\label{eq: MatLam} {\Lambda}= \begin{pmatrix} -1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}.$$ To see , let us consider the function $$\label{def:tildeX} \widetilde X(z):=X(-z) {\Lambda}.$$ It is straightforward to check that $\widetilde X(z)$ satisfies the same jump condition as $X(z)$ shown in and . This means $X(z)\widetilde X^{-1}(z)$ is analytic in the complex plane with possible isolated singular points located at $z = \pm s$. Since $ X(z) = \mathcal O(\ln(z \mp s)) $ as $z \to \pm s$, we conclude that the possible singular points $\pm s$ are removable. In view of the asymptotics of $X(z)$ given in , we further obtain from and a bit more cumbersome calculation that $$X(z)\widetilde X^{-1}(z) = \widetilde{C} + {\mathcal{O}}(z^{-1}), \qquad \textrm{as } z \to \infty,$$ with $\widetilde{C}$ given in . An appeal to Liouville’s theorem then leads us to . As a consequence of , we have $$X'(z) = -\widetilde {C} X'(-z) {\Lambda}.$$ This, together with and the fact that ${\Lambda}^{-1}={\Lambda}$, implies $$\begin{aligned} \lim_{z \to s} \left( X^{-1}(z)X'(z) \right) = - \lim_{z \to s} \left( {\Lambda}X^{-1}(-z) X'(-z) {\Lambda}\right) = - \lim_{z \to -s} \left( {\Lambda}X^{-1}(z) X'(z) {\Lambda}\right) .\end{aligned}$$ To this end, we observe that for an arbitrary $3\times 3$ matrix $ M = (m_{ij})_{i,j=1}^3$, $$\left( {\Lambda}M {\Lambda}\right)_{21} = - m_{31} \quad \textrm{and} \quad \left( {\Lambda}M {\Lambda}\right)_{31} = - m_{21}.$$ A combination of the above two formulas shows that $$\lim_{z \to -s} \left(X^{-1}(z)X'(z)\right)_{21}+\left(X^{-1}(z)X'(z)\right)_{31} = \lim_{z \to s} \left(X^{-1}(z)X'(z)\right)_{21}+\left(X^{-1}(z)X'(z)\right)_{31}.$$ Inserting this formula into gives us . This completes the proof of Proposition \[prop:derivativeandX\]. Auxiliary functions {#sec:auxifuncs} =================== \[sec:auxiliary function\] In this section, we introduce some auxiliary functions and study their properties. The aim is to construct the so-called $\lambda$-functions, of which the analytic continuation defines a meromorphic function on a Riemann surface with a specified sheet structure. The $\lambda$-functions have desired behavior around each branch point, and will be crucial in our further asymptotic analysis of the RH problem \[rhp:X\] for $X$. Throughout this section, unless specified differently, we shall take the principal branch for all fractional powers. A three-sheeted Riemann surface and the $w$-functions {#sec:w function} ----------------------------------------------------- We introduce a three-sheeted Riemann surface $\mathcal R$ with sheets $$\begin{aligned} \mathcal R_1 &= {\mathbb{C}}\setminus \{(-\infty,-1]\cup[1,+\infty) \}, & \mathcal R_2 &= {\mathbb{C}}\setminus [1,+\infty), \\ \mathcal R_3 &={\mathbb{C}}\setminus (-\infty,-1] .\end{aligned}$$ We connect the sheets $\mathcal R_j$, $j=1,2,3$, to each other in the usual crosswise manner along the cuts $(-\infty,-1]$ and $[1,+\infty)$. More precisely, $\mathcal R_1$ is connected to $\mathcal R_2$ along the cut $[1,+\infty)$ and $\mathcal R_1$ is connected to $\mathcal R_3$ along the cut $(-\infty,-1]$. We then compactify the resulting surface by adding a common point at $\infty$ to the sheets $\mathcal R_1$ and $\mathcal R_2$, and a common point at $\infty$ to the sheets $\mathcal R_1$ and $\mathcal R_3$. We denote this compact Riemann surface by $\mathcal R$, which has genus zero and is shown in Figure \[fig: Riemann surface\]. (0,0)–(6,0)–(8,1)–(2,1)–cycle (0,-2)–(6,-2)–(8,-1)–(2,-1)–cycle (0,-4)–(6,-4)–(8,-3)–(2,-3)–cycle; (1,0.5)–(3,0.5) (5,0.5)–(7,0.5) (1,-3.5)–(3,-3.5) (5,-1.5)–(7,-1.5); (8,0.5) node[$\mathcal R_1$]{} (8,-1.5) node[$\mathcal R_2$]{} (8,-3.5) node[$\mathcal R_3$]{}; (3,0.5)–(3,-3.5); (5,0.5)–(5,-1.5); (3,0.5) circle (2pt) node \[above\] (q1) [$-1$]{}; (5,0.5) circle (2pt) node \[above\] (q1) [$1$]{}; (5,-1.5) circle (2pt) node \[above\] (q1) ; (3,-3.5) circle (2pt) node \[above\] (q1) ; We intend to find functions $\lambda_j$, $j=1,2,3$, on these sheets, such that each $\lambda_j$ is analytic on $\mathcal R_j$ and admits an analytic continuation across the cuts. For this purpose, we start with an elementary function $w(z)$ that is meromorphic on $\mathcal R$, which satisfies the following algebraic equation $$\label{eq: alg eq w} w(z)^3-3w(z)+2z=0.$$ We choose three solutions $w_j(z)$, $j = 1,2,3$, to such that they are defined and analytic on $\mathcal R_j$, respectively. Each function $w_j(z)$ maps $\mathcal R_j$ to certain domain in the extended complex $w$-plane $\overline {\mathbb{C}}$. The correspondences between some points $z\in \mathcal R$ and the points $w \in \overline {\mathbb{C}}$ are given in Table \[Table:z-w-corres\], where $z^{(j)}$ denotes the point $z$ on the closure of the $j$-th sheet $\mathcal R_j$. ----------------------------------------- ------------ ---------------- ----------- ----------- ------------ ------------- \[-1em\] $z \in \mathcal R$ $-1^{(1)}$ $\infty^{(1)}$ $1^{(1)}$ $0^{(1)}$ $0^{(2)}$ $0^{(3)}$ \[1ex\] \[-1em\] $w \in \overline {\mathbb{C}}$ $-1$ $\infty$ 1 0 $\sqrt{3}$ $-\sqrt{3}$ \[1ex\] ----------------------------------------- ------------ ---------------- ----------- ----------- ------------ ------------- : Images of some points $z\in \mathcal R$ under the mappings $w_j$, $j=1,2,3$.[]{data-label="Table:z-w-corres"} Due to the sheet structure shown in Figure \[fig: Riemann surface\], it follows that $\infty^{(1)} = \infty^{(2)} = \infty^{(3)}$, $-1^{(1)} = -1^{(3)}$ and $1^{(1)} = 1^{(2)}$. Moreover, we actually have the following explicit expressions for $w_j(z)$. \[prop:w-expression\] The three solutions $w_j(z)$, $j=1,2,3$, to the algebraic equation with the mapping properties shown in Table \[Table:z-w-corres\] are given by $$\label{eq: w-eta} w_j(z) = \omega^{j-2}\eta (z)^{\frac{1}{3}} + \omega^{2-j}\eta (z)^{-\frac{1}{3}},$$ where $$\label{eq: eta-def} \eta(z) = (z^2 -1 )^{\frac{1}{2}} - z, \qquad z \in \mathbb{C} \setminus \{ (-\infty, -1] \cup [1,+\infty)\},$$ with $$\label{eq: eta-arg} \arg \eta(z) \in (0, \pi).$$ Clearly, $\eta(z)\neq 0$ and satisfies the quadratic equation $$\label{eq:etaeq} \eta(z)^2 + 2z \eta(z) + 1 = 0.$$ Hence, with $w_j(z)$, $j=1,2,3$, defined in , we have $$\begin{aligned} w_j(z)^3 -3w_j(z) + 2z & = \left( \omega^{j-2}\eta (z)^{\frac{1}{3}} + \omega^{2-j}\eta (z)^{-\frac{1}{3}} \right)^3 - 3 \left( \omega^{j-2}\eta (z)^{\frac{1}{3}} + \omega^{2-j}\eta (z)^{-\frac{1}{3}} \right) + 2z \\ & = \eta(z) + \frac{1}{\eta(z)} + 2z = 0, \end{aligned}$$ as expected, where in the last step we have made use of . Furthermore, an elementary analysis shows that ${\mathrm{Im}\,}\eta(z) >0$ for $z \in \mathbb{C} \setminus \{ (-\infty, -1] \cup [1,+\infty)\}$. Thus, $\arg \eta(z) \in (0, \pi)$ and $\eta(0) = e^{\frac{\pi i}{2}}=i $. This, together with , implies that $w_1(0) = 0$, $ w_2(0) = \sqrt{3}$, and $w_3(0) = -\sqrt{3}$. The other relations in Table \[Table:z-w-corres\] can then be verified similarly, and we omit the details here. This completes the proof of Proposition \[prop:w-expression\]. As a consequence of Proposition \[prop:w-expression\], we have the following properties of $w_j(z)$. \[prop:w\] The functions $w_j(z)$, $j=1,2,3$, given in satisfy the following properties. - $w_j(z)$ is analytic on $\mathcal R_j$, $j=1,2,3$, and $$\begin{aligned} w_{1,\pm}(x) &= w_{3,\mp}(x), \qquad x \in (-\infty,-1), \label{eq: w1pm-w3pm} \\ w_{1,\pm}(x) &= w_{2,\mp}(x), \qquad x \in (1,\infty). \label{eq: w1pm-w2pm}\end{aligned}$$ Here, we orient $(-\infty,-1)$ and $(1,\infty)$ from the left to the right. Hence, the function $\cup_{j=1}^3 \mathcal R_j \to {\mathbb{C}}: \mathcal R_j \ni z \mapsto w_j(z)$ has an analytic continuation to a meromorphic function $w: \mathcal R \to \overline {\mathbb{C}}$. This function is a bijection. - $w$ satisfies the symmetry properties $$\begin{aligned} w_j(\overline z) &= \overline{w_j(z)}, \qquad~~\,~~\, z\in\mathcal R_j, \quad j=1,2,3, \label{eq: wz-wzbar} \\ w_1(-z) & = - w_1(z) , \qquad~~\, z\in {\mathbb{C}}\setminus \{ (-\infty,-1] \cup [1,+\infty) \} , \label{eq: w1z+-} \\ w_2(-z)&=-w_3(z), \qquad~~\, z \in {\mathbb{C}}\setminus (-\infty,-1]. \label{eq: w2-w3}\end{aligned}$$ - As $z \to \infty$ and $-\pi<\arg z <\pi$, we have $$\label{eq: w1-large-z} w_1(z) =\left\{ \begin{array}{ll} -2^{\frac13}\omega^2z^{\frac13}-\frac{\omega}{2^{\frac13}} z^{-\frac13}+\frac{\omega^2}{6\cdot 2^{\frac23}}z^{-\frac53}-\frac{\omega}{12\cdot 2^{\frac13}}z^{-\frac73}+\frac{\omega^2}{18\cdot 4^{\frac13}}z^{-\frac{11}{3}}+{\mathcal{O}}(z^{-\frac{13}{3}}), & \hbox{${\mathrm{Im}\,}z>0$,} \\ [2mm] -2^{\frac13}\omega z^{\frac13}-\frac{\omega^2}{2^{\frac13}} z^{-\frac13}+\frac{\omega}{6\cdot 2^{\frac23}}z^{-\frac53}-\frac{\omega^2}{12\cdot 2^{\frac13}}z^{-\frac73}+\frac{\omega}{18\cdot 4^{\frac13}}z^{-\frac{11}{3}}+{\mathcal{O}}(z^{-\frac{13}{3}}), & \hbox{${\mathrm{Im}\,}z<0$,} \end{array} \right.$$ and $$\begin{gathered} \label{eq: w3-large-z} w_3(z) = -2^{\frac13} z^{\frac13}-\frac{1}{2^{\frac13}} z^{-\frac13}+\frac{1}{6\cdot 2^{\frac23}}z^{-\frac53}-\frac{1}{12\cdot 2^{\frac13}}z^{-\frac73} \\ +\frac{1}{18\cdot 4^{\frac13}}z^{-\frac{11}{3}}+{\mathcal{O}}(z^{-\frac{13}{3}}), \quad z\in{\mathbb{C}}\setminus (-\infty,-1].\end{gathered}$$ - As $z \to -1$ and $-\pi<\arg(z+1) <\pi$, we have $$\label{eq: w1-n1-asy} w_1(z)=-1+\sqrt{\frac23}(z+1)^{\frac12}+\frac{z+1}{9}+\frac{5}{54\cdot \sqrt{6}}(z+1)^{\frac32}+{\mathcal{O}}(z+1)^2,$$ and $$\label{eq: w3-n1-asy} w_3(z)=-1-\sqrt{\frac23}(z+1)^{\frac12}+\frac{z+1}{9}-\frac{5}{54\cdot \sqrt{6}}(z+1)^{\frac32}+{\mathcal{O}}(z+1)^2.$$ - As $z \to 1$ and $-\pi<\arg(z-1) <\pi$, we have $$\label{eq: w1-p1-asy} w_{1}(z)= \left\{ \begin{array}{ll} 1+i\sqrt{\frac23}(z-1)^{\frac12}+\frac{z-1}{9}-i\frac{5}{54\cdot \sqrt{6}}(z-1)^{\frac32}+{\mathcal{O}}(z-1)^2, & \hbox{${\mathrm{Im}\,}z>0$,} \\ [2mm] 1-i\sqrt{\frac23}(z-1)^{\frac12}+\frac{z-1}{9}+i\frac{5}{54\cdot \sqrt{6}}(z-1)^{\frac32}+{\mathcal{O}}(z-1)^2, & \hbox{${\mathrm{Im}\,}z<0$.} \end{array} \right.$$ and $$\label{eq: w2-p1-asy} w_{2}(z)= \left\{ \begin{array}{ll} 1-i\sqrt{\frac23}(z-1)^{\frac12}+\frac{z-1}{9}+i\frac{5}{54\cdot \sqrt{6}}(z-1)^{\frac32}+{\mathcal{O}}(z-1)^2, & \hbox{${\mathrm{Im}\,}z>0$,} \\ [2mm] 1+i\sqrt{\frac23}(z-1)^{\frac12}+\frac{z-1}{9}-i\frac{5}{54\cdot \sqrt{6}}(z-1)^{\frac32}+{\mathcal{O}}(z-1)^2, & \hbox{${\mathrm{Im}\,}z<0$.} \end{array} \right.$$ To show , we see from that, if $x<-1$, $$\begin{aligned} w_{1,+}(x)&=\omega^{-1}\eta_{+}(x)^{\frac13}+\omega\eta_{+}(x)^{-\frac13}=\omega^{-1}(-x-\sqrt{x^2-1})^{\frac13}+\omega(-x-\sqrt{x^2-1})^{-\frac13} \nonumber \\ &=\omega^{-1}\eta_{-}(x)^{-\frac13}+\omega\eta_{-}(x)^{\frac13}=w_{3,-}(x).\end{aligned}$$ Similarly, it is easy to check $w_{1,-}(x)=w_{3,+}(x)$ for $x<-1$ and . While follows directly from , the proof of relies on the fact that $$\eta(z) \eta(-z) = -1.$$ Hence, by , it follows that $$\begin{aligned} w_1(-z) &= \omega^{-1} \eta(-z)^{\frac{1}{3}} + \omega \eta(-z)^{-\frac{1}{3}} = \omega^{-1} e^{\frac{1}{3} \pi i} \eta(z)^{-\frac{1}{3}} + \omega e^{-\frac{1}{3} \pi i} \eta(z)^{\frac{1}{3}} \\ & = - \omega^{-1} \eta(z) ^{\frac{1}{3}} -\omega \eta(z) ^{-\frac{1}{3}} = -w_1(z),\end{aligned}$$ which is . The relation follows in a similar manner. To obtain the asymptotics of $w_j(z)$, $j=1,3$, as $z \to \infty$, we observe from the definition of $\eta(z)$ in that $$\begin{aligned} \eta(z) = \begin{cases} {\displaystyle}-\frac{1}{2z} - \frac{1}{8z^3} - \frac{1}{16z^5} + {\mathcal{O}}(z^{-7}), & {\mathrm{Im}\,}z >0, \vspace{1mm} \\ {\displaystyle}-2z +\frac{1}{2z} + \frac{1}{8z^3} + \frac{1}{16z^5} + {\mathcal{O}}(z^{-7}), & {\mathrm{Im}\,}z <0. \end{cases}\end{aligned}$$ Inserting the above formula into , it is readily seen that, if ${\mathrm{Im}\,}z>0$ and $z \to \infty$, $$\begin{aligned} w_1(z) & = \omega^{-1} \eta(z) ^{\frac{1}{3}} + \omega \eta(z)^{-\frac{1}{3}} \\ & = \frac{e^{-\frac{\pi i}{3}}}{\sqrt[3]{2} z^{\frac{1}{3}}} \left(1 + \frac{1}{4z^2} + \frac{1}{8z^4} + {\mathcal{O}}(z^{-6}) \right)^{\frac{1}{3}} + e^{\frac{\pi i}{3}} \sqrt[3]{2} z^{\frac{1}{3}} \left(1 + \frac{1}{4z^2} + \frac{1}{8z^4} + {\mathcal{O}}(z^{-6}) \right)^{-\frac{1}{3}} \\ &= -2^{\frac13}\omega^2z^{\frac13}-\frac{\omega}{2^{\frac13}} z^{-\frac13}+\frac{\omega^2}{6\cdot 2^{\frac23}}z^{-\frac53}-\frac{\omega}{12\cdot 2^{\frac13}}z^{-\frac73}+\frac{\omega^2}{18\cdot 4^{\frac13}}z^{-\frac{11}{3}}+{\mathcal{O}}(z^{-\frac{13}{3}}),\end{aligned}$$ which is the first formula in . This, together with the fact that $w_1(-z)=-w_1(z)$, implies the second formula of . The asymptotics of $w_3(z)$ in can be derived through similar computations, we omit the details here. We next come to the asymptotics of $w_j(z)$, $j=1,3$, as $z \to -1$. Since $$(z-1)^{\frac12}=\sqrt{2}i-\frac{i}{2\sqrt{2}}(z+1)+{\mathcal{O}}(z+1)^2, \qquad z\in\mathbb{C}\setminus [1,+\infty),\qquad z\to -1,$$ it follows from that $$\begin{aligned} \eta(z) = (z^2-1)^{\frac12}-z = 1 + i \sqrt{2} (z+1)^{\frac{1}{2}} - (z+1) - i \frac{ (z + 1)^{\frac{3}{2}}}{ 2 \sqrt2} + {\mathcal{O}}(z+1)^{\frac52}, \qquad z\to -1,\end{aligned}$$ with $-\pi <\arg(z+1)< \pi$. A combination of the above formula and then gives us $$\begin{aligned} w_1(z) & = \omega^{-1} \eta(z)^{\frac{1}{3}} + \omega \eta(z)^{-\frac{1}{3}} \\ & = \omega^{-1} \left(1 + i \sqrt{2} (z+1)^{\frac{1}{2}} - (z+1) - i \frac{ (z + 1)^{\frac{3}{2}}}{ 2 \sqrt2} + {\mathcal{O}}(z+1)^{\frac52} \right)^{\frac{1}{3}} \\ & \quad + \omega \left(1 + i \sqrt{2} (z+1)^{\frac{1}{2}} - (z+1) - i \frac{ (z + 1)^{\frac{3}{2}}}{ 2 \sqrt2} + {\mathcal{O}}(z+1)^{\frac52} \right)^{-\frac{1}{3}} \\ &=-1+\sqrt{\frac23}(z+1)^{\frac12}+\frac{z+1}{9}+\frac{5}{54\cdot \sqrt{6}}(z+1)^{\frac32}+{\mathcal{O}}(z+1)^2,\end{aligned}$$ as shown in . The asymptotics of $w_3$ in can be proved in a similar manner, we omit the details here. Finally, we note that, as $z \to 1$, $$\begin{aligned} \eta(z) = \left\{ \begin{array}{ll} -1 + \sqrt{2}(z-1)^{\frac{1}{2}} - (z-1) + \frac{(z - 1)^{\frac{3}{2}}}{ 2 \sqrt2} + {\mathcal{O}}(z-1)^{\frac52}, & \hbox{${\mathrm{Im}\,}z>0$,} \\ -1 - \sqrt{2}(z-1)^{\frac{1}{2}} - (z-1) - \frac{(z - 1)^{\frac{3}{2}}}{ 2 \sqrt2} + {\mathcal{O}}(z-1)^{\frac52}, & \hbox{${\mathrm{Im}\,}z<0$,} \end{array} \right.\end{aligned}$$ with $-\pi <\arg(z-1)< \pi$. Inserting the above formula into then gives us and after straightforward calculations. This completes the proof of Proposition \[prop:w\]. The image of the map $w:\mathcal R\mapsto \overline{\mathbb{C}}$ is illustrated in Figure \[fig: image of w\]. (-7.5,0)–(7.5,0) ; (-4,0).. controls (-4,2) and (-5,4)..(-7,5) (4,0).. controls (4,2) and (5,4)..(7,5); (-4,0).. controls (-4,-2) and (-5,-4)..(-7,-5) (4,0).. controls (4,-2) and (5,-4)..(7,-5); (-7,5.5)–(7,-5.5) (-7,-5.5)–(7,5.5); (5mm,0mm) arc(0:45:5mm); (1.3,-0.2) node\[above\][$\pi/3$]{}; (0,0) circle (2pt) (-4,0) circle (2pt) (4,0) circle (2pt); (0,0) node\[below right\][$0$]{} (-4,0) node\[below right\][$-1$]{} (4,0) node\[below right\][$1$]{}; (-6.1,-3)node\[right\][$\gamma_1^-$]{} (-6.1, 3)node\[right\][$\gamma_1^+$]{} (5,-3)node\[right\][$\gamma_2^-$]{} (5,3)node\[right\][$\gamma_2^+$]{} (-0.5,4.5)node\[right\][$ \widehat{\mathcal R}_1$]{} (-6,1)node\[right\][$ \widehat{\mathcal R}_3$]{} (6,1)node\[right\][$ \widehat{\mathcal R}_2$]{}; The $\lambda$-functions {#sec:lamda function} ----------------------- With the functions $w_j(z)$ given in Proposition \[prop:w-expression\], we define the $\lambda$-functions as $$\label{def: lambda function} \lambda_j(z)=\frac{3}{4^{\frac53}}w_j(z)^4+\left(\frac{\rho}{2^{\frac53}s^{\frac23}}-\frac{3}{2^{\frac43}}\right)w_j(z)^2, \qquad j=1,2,3,$$ which depend on the parameters $s>0$ and $\rho \in \mathbb{R}$. The properties of the $\lambda$-functions are listed in the following proposition. \[prop: prop of lambda\] The functions $\lambda_j(z)$, $j=1,2,3$, defined by have the following properties. - $\lambda_j(z)$ is analytic on $\mathcal R_j$, $j=1,2,3$, and $$\begin{aligned} \lambda_{1,\pm}(x) &= \lambda_{3,\mp}(x), \qquad x \in (-\infty,-1), \label{eq:lamda 1 lamda 3} \\ \lambda_{1,\pm}(x) &= \lambda_{2,\mp}(x), \qquad x \in (1,\infty),\end{aligned}$$ Hence the function $\cup_{j=1}^3 \mathcal R_j \to {\mathbb{C}}: \mathcal R_j \ni z \mapsto \lambda_j(z)$ has an analytic continuation to a meromorphic function on the Riemann surface $\mathcal R $. - We have the following symmetry properties $$\begin{aligned} \lambda_j(\overline z) & = \overline{\lambda_j(z)}, \qquad z \in \mathcal R_j, \quad j=1,2,3, \label{eq:symlambdaj} \\ \lambda_1(-z) & = \lambda_1(z) , \qquad z\in {\mathbb{C}}\setminus \{ (-\infty,-1] \cup [1,+\infty) \} , \label{eq: lambda1z+-} \\ \lambda_2(-z)&=\lambda_3(z), \qquad z \in {\mathbb{C}}\setminus (-\infty,-1]. \label{eq: lambda23z}\end{aligned}$$ - As $z \to \infty$ and $-\pi<\arg z <\pi$, we have $$\lambda_1(z)=\left\{ \begin{array}{ll} \frac34 \omega^2 z^{\frac43}+\frac{\rho \omega}{2s^{\frac23}}z^{\frac23}-D_0+D_1 \omega^2 z^{-\frac23}+{\mathcal{O}}(z^{-\frac43}), & \hbox{${\mathrm{Im}\,}z>0$,} \\ [1ex] \frac34 \omega z^{\frac43}+\frac{\rho \omega^2}{2s^{\frac23}}z^{\frac23}-D_0+D_1 \omega z^{-\frac23} +{\mathcal{O}}(z^{-\frac43}), & \hbox{${\mathrm{Im}\,}z<0$,} \end{array} \right.$$ and $$\lambda_3(z)= \frac34 z^{\frac43}+\frac{\rho}{2s^{\frac23}}z^{\frac23}-D_0+D_1 z^{-\frac23} + {\mathcal{O}}(z^{-\frac43}), \quad z\in{\mathbb{C}}\setminus (-\infty,-1],$$ where $$\label{D0-def} D_0:=3\cdot 2^{-\frac73}-(2s)^{-\frac23}\rho, \qquad D_1:=\frac{1}{8}\left(-2+\left(\frac2s\right)^{\frac23}\rho\right).$$ - As $z\to -1$ and $-\pi<\arg (z+1) <\pi$, we have $$\label{eq:lambda-1 at -1} \lambda_1(z)=C_0+C_1(z+1)^{\frac12}+C_2(z+1)+C_3(z+1)^{\frac32}+{\mathcal{O}}(z+1)^2,$$ and $$\label{eq:lambda-3 at -1} \lambda_3(z)=C_0-C_1(z+1)^{\frac12}+C_2(z+1)-C_3(z+1)^{\frac32}+{\mathcal{O}}(z+1)^2,$$ where $$\label{def:Ci} \begin{aligned} C_0&:=\frac{\rho}{2^{\frac 53}s^{\frac 23}}-\frac{9}{2^{\frac {10}{3}}}, \quad &&C_1:=\frac{\sqrt{3}}{2^{\frac56}}-\frac{\rho}{\sqrt{3}\cdot2^{\frac16}s^{\frac23}}, \\ C_2&:=\frac{2^{\frac23}}{3}+\frac{2^{\frac13}\rho}{9s^{\frac23}}, \quad &&C_3:=\frac{7\rho}{108\sqrt{3}\cdot 2^{\frac16}s^{\frac23}}-\frac{165}{108\sqrt{3}\cdot2^{\frac56}}. \end{aligned}$$ - As $z\to 1$ and $-\pi<\arg (z-1) <\pi$, we have $$\lambda_1(z)=\left\{ \begin{array}{ll} C_0-iC_1(z-1)^{\frac12}-C_2(z-1)+iC_3(z-1)^{\frac32}+{\mathcal{O}}(z-1)^2, & \hbox{${\mathrm{Im}\,}z>0$,} \\ C_0+iC_1(z-1)^{\frac12}-C_2(z-1)-iC_3(z-1)^{\frac32}+{\mathcal{O}}(z-1)^2, & \hbox{${\mathrm{Im}\,}z<0$,} \end{array} \right.$$ and $$\lambda_2(z)=\left\{ \begin{array}{ll} C_0+iC_1(z-1)^{\frac12}-C_2(z-1)-iC_3(z-1)^{\frac32}+{\mathcal{O}}(z-1)^2, & \hbox{${\mathrm{Im}\,}z>0$,} \\ C_0-iC_1(z-1)^{\frac12}-C_2(z-1)+iC_3(z-1)^{\frac32}+{\mathcal{O}}(z-1)^2, & \hbox{${\mathrm{Im}\,}z<0$,} \end{array} \right.$$ where the constants $C_i$, $i=0,1,2,3$, are given in . - For $z\in {\mathbb{C}}\setminus \{ (-\infty,-1] \cup [1,+\infty) \}$, we have $$\label{eq:sumlambda} \lambda_1(z) + \lambda_2(z) + \lambda_3(z) = - \frac{9}{2^{\frac{7}{3}}} + \frac{3 \rho}{2^{\frac{2}{3}} s^{\frac{2}{3}}}.$$ The proofs of items (a)–(e) follow directly from the definition of $\lambda_j(z)$ in and Proposition \[prop:w\]. It then remains to prove . Since $w_j$, $j=1,2,3$, are three solutions of the algebraic equation , it follows from Vieta’s rule that, for $z\in {\mathbb{C}}\setminus \{ (-\infty,-1] \cup [1,+\infty) \}$, $$\begin{aligned} w_1(z)+w_2(z)+w_3(z)&=0, \\ w_1(z)w_2(z)+w_1(z)w_3(z)+w_2(z)w_3(z)&=-3, \\ w_1(z)w_2(z)w_3(z)&=-2z.\end{aligned}$$ Hence, $$\begin{aligned} \label{eq:sumwsquare} &w_1(z)^2+w_2(z)^2+w_3(z)^2 \nonumber \\ &=(w_1(z)+w_2(z)+w_3(z))^2-2(w_1(z)w_2(z)+w_1(z)w_3(z)+w_2(z)w_3(z))=6,\end{aligned}$$ $$\begin{aligned} &w_1(z)^2w_2(z)^2+w_1(z)^2w_3(z)^2+w_2(z)^2w_3(z)^2 \\ &=(w_1(z)w_2(z)+w_1(z)w_3(z)+w_2(z)w_3(z))^2-2w_1(z)w_2(z)w_3(z)(w_1(z)+w_2(z)+w_3(z)) \\ &=9,\end{aligned}$$ and $$\begin{aligned} \label{eq:sumwquartic} &w_1(z)^4+w_2(z)^4+w_3(z)^4 \nonumber \\ &=(w_1(z)^2+w_2(z)^2+w_3(z)^2)^2-2(w_1(z)^2w_2(z)^2+w_1(z)^2w_3(z)^2+w_2(z)^2w_3(z)^2) \nonumber \\ &=36-18=18.\end{aligned}$$ A combination of , and yields . This completes the proof of Proposition \[prop: prop of lambda\]. Asymptotic analysis of the Riemann-Hilbert problem for $X$ {#sec:asymanalyX} ========================================================== In this section, we shall perform a Deift-Zhou steepest descent analysis [@DZ93] to the RH problem for $X$ as $s \to +\infty$. It consists of a series of explicit and invertible transformations which leads to an RH problem tending to the identity matrix as $s \to +\infty$. First transformation: $X \to T$ ------------------------------- This transformation is a rescaling of the RH problem for $X$, which is defined by $$\label{def:XtoT} T(z)=X(sz).$$ It is then straightforward to check that $T$ satisfies the following RH problem. The function $T$ defined in has the following properties: 1. $T(z)$ is defined and analytic in $\mathbb{C}\setminus \{\cup^5_{j=0}\Sigma_j^{(1)}\cup \{-1\} \cup\{1\}\}$, where the contours $\Sigma_j^{(1)}$ are defined in with $s=1$. 2. $T$ satisfies the jump condition $$\label{eq:T-jump} T_+(z)=T_-(z)J_T(z), \qquad z\in\cup^5_{j=0}\Sigma_j^{(1)},$$ where $$\label{def:JT} J_T(z):=\left\{ \begin{array}{ll} \begin{pmatrix} 0&1&0 \\ -1&0&0 \\ 0&0&1 \end{pmatrix}, &\qquad \hbox{$z\in \Sigma_0^{(1)}$,} \\ \begin{pmatrix} 1&0&0 \\ 1&1&1 \\ 0&0&1 \end{pmatrix}, &\qquad \hbox{$z\in \Sigma_1^{(1)}$,} \\ \begin{pmatrix} 1&0&0 \\ 0&1&0 \\ 1&1&1 \end{pmatrix}, &\qquad \hbox{$z\in \Sigma_2^{(1)}$,} \\ \begin{pmatrix} 0&0&1 \\ 0&1&0 \\ -1&0&0 \end{pmatrix}, &\qquad \hbox{$z\in \Sigma_3^{(1)}$,} \\ \begin{pmatrix} 1&0&0 \\ 0&1&0 \\ 1&-1&1 \end{pmatrix}, &\qquad \hbox{$z\in \Sigma_4^{(1)}$,} \\ \begin{pmatrix} 1&0&0 \\ 1&1&-1 \\ 0&0&1 \end{pmatrix}, &\qquad \hbox{$z\in \Sigma_5^{(1)}$.} \end{array} \right.$$ 3. As $z \to \infty$ and $\pm{\mathrm{Im}\,}z>0$, we have $$\label{eq:asyT} T(z)= \sqrt{\frac{2\pi}{3}}i e^{\frac{\rho^2}{6}} \Psi_0 \left(I+\frac{\mathsf{X}_1}{sz} + \mathcal O(z^{-2}) \right)\operatorname{diag}\left((sz)^{-\frac13},1,(sz)^{\frac13} \right)L_{\pm}e^{\Theta(sz)},$$ where we recall that $$\begin{aligned} \Theta(sz)&= \begin{cases} \operatorname{diag}(\theta_1(sz;\rho),\theta_2(sz;\rho),\theta_3(sz;\rho)), & \text{${\mathrm{Im}\,}z >0$,} \\ \operatorname{diag}(\theta_2(sz;\rho),\theta_1(sz;\rho),\theta_3(sz;\rho)), & \text{${\mathrm{Im}\,}z <0$,} \\ \end{cases}\end{aligned}$$ with $$\label{eq:thetasz} \theta_k(sz;\rho)=s^{\frac43}\left(\frac34 \omega^{2k}z^{\frac43}+\frac{\rho\omega^k}{2s^{\frac23}}z^{\frac23}\right), \qquad k=1,2,3.$$ 4. As $z \to \pm 1$, we have $T(z) = \mathcal O(\ln(z \mp 1))$. Second transformation: $T \to S$ -------------------------------- In this transformation we normalize the large $z$ behavior of $T$ using the $\lambda$-functions introduced in Section \[sec:lamda function\]. We define $$\label{def:TtoS} S(z)= S_0 \frac{\operatorname{diag}\left(s^{\frac13},1,s^{-\frac13} \right)}{e^{D_0s^{\frac43}} \sqrt{\frac{2\pi}{3}}i e^{\frac{\rho^2}{6}}} \Psi_0^{-1} T(z)\operatorname{diag}\left(e^{-s^{\frac43}\lambda_1(z)},e^{-s^{\frac43}\lambda_2(z)},e^{-s^{\frac43}\lambda_3(z)}\right),$$ where $$S_0 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ D_1 s^{\frac{4}{3}} & 0 & 1 \end{pmatrix}$$ with $D_0$ and $D_1$ being the constants given in , and the functions $\lambda_i$, $i=1,2,3$, are defined in . Then, $S$ satisfies the following RH problem. \[prop:S\] The function $S$ defined in has the following properties: 1. $S(z)$ is defined and analytic in $\mathbb{C}\setminus \{\cup^5_{j=0}\Sigma_j^{(1)}\cup \{-1\} \cup\{1\}\}$. 2. $S$ satisfies the jump condition $$\label{eq:S-jump} S_+(z)=S_-(z)J_S(z), \qquad z\in \cup^5_{j=0}\Sigma_j^{(1)},$$ where $$\label{def:JS} J_S(z):=\left\{ \begin{array}{ll} \begin{pmatrix} 0&1&0 \\ -1&0&0 \\ 0&0&1 \end{pmatrix}, &\quad \hbox{$z\in \Sigma_0^{(1)}$,} \\ \begin{pmatrix} 1&0&0 \\ e^{s^{\frac43}(\lambda_2(z)-\lambda_1(z))}&1&e^{s^{\frac43}(\lambda_2(z)-\lambda_3(z))} \\ 0&0&1 \end{pmatrix}, &\quad \hbox{$z\in \Sigma_1^{(1)}$,} \\ \begin{pmatrix} 1&0&0 \\ 0&1&0 \\ e^{s^{\frac43}(\lambda_3(z)-\lambda_1(z))} & e^{s^{\frac43}(\lambda_3(z)-\lambda_2(z))} &1 \end{pmatrix}, &\quad \hbox{$z\in \Sigma_2^{(1)}$,} \\ \begin{pmatrix} 0&0&1 \\ 0&1&0 \\ -1&0&0 \end{pmatrix}, &\quad \hbox{$z\in \Sigma_3^{(1)}$,} \\ \begin{pmatrix} 1&0&0 \\ 0&1&0 \\ e^{s^{\frac43}(\lambda_3(z)-\lambda_1(z))} &-e^{s^{\frac43}(\lambda_3(z)-\lambda_2(z))}&1 \end{pmatrix}, &\quad \hbox{$z\in \Sigma_4^{(1)}$,} \\ \begin{pmatrix} 1&0&0 \\ e^{s^{\frac43}(\lambda_2(z)-\lambda_1(z))}&1&-e^{s^{\frac43}(\lambda_2(z)-\lambda_3(z))} \\ 0&0&1 \end{pmatrix}, &\quad \hbox{$z\in \Sigma_5^{(1)}$.} \end{array} \right.$$ 3. As $z \to \infty$ and $\pm{\mathrm{Im}\,}z>0$, we have $$\label{eq:asyS} S(z)= \left(I + \frac{\mathsf{S}_1}{z} + \mathcal O(z^{-2}) \right) \operatorname{diag}\left(z^{-\frac13},1,z^{\frac13} \right)L_{\pm},$$ where $$\label{asyS:coeff} \mathsf{S}_1 = \begin{pmatrix} * & \frac{(\mathsf{X}_1)_{12}}{ s^{\frac{2}{3}}} - D_1 s^{\frac{4}{3}} & * \\ * & * & \frac{(\mathsf{X}_1)_{23}}{ s^{\frac{2}{3}}} - D_1 s^{\frac{4}{3}} \\ * & * & * \end{pmatrix}$$ with the constant $D_1$ given in , and $*$ stands for some unimportant entry. 4. As $z \to \pm 1$, we have $S(z) = \mathcal O(\ln(z \mp 1))$. By , it is clear that $S(z)$ is defined and analytic in $\mathbb{C}\setminus \{\cup^5_{j=0}\Sigma_j^{(1)}\cup \{-1\} \cup\{1\}\}$ and $$\begin{aligned} J_S(z)&=S_-^{-1}(z)S_+(z) \nonumber \\ &=\operatorname{diag}\left(e^{s^{\frac43}\lambda_{1,-}(z)},e^{s^{\frac43}\lambda_{2,-}(z)},e^{s^{\frac43}\lambda_{3,-}(z)}\right)J_T(z) \nonumber \\ & \quad \times \operatorname{diag}\left(e^{-s^{\frac43}\lambda_{1,+}(z)},e^{-s^{\frac43}\lambda_{2,+}(z)},e^{-s^{\frac43}\lambda_{3,+}(z)}\right),\end{aligned}$$ for $z\in \cup^5_{j=0}\Sigma_j^{(1)}$, where $J_T$ is given in . This, together with item (a) in Proposition \[prop: prop of lambda\], gives us . Checking the asymptotics is a bit more cumbersome. To that end, we observe from Proposition \[prop: prop of lambda\] and that the following formula is useful $$\begin{gathered} \operatorname{diag}\left(e^{-s^{\frac43}\lambda_1(z)+\theta_1(sz)},e^{-s^{\frac43}\lambda_2(z)+\theta_2(sz)},e^{-s^{\frac43}\lambda_3(z)+\theta_3(sz)}\right) \\ =e^{D_0s^{\frac43}}\left(I- \frac{D_1s^{\frac43}}{z^{\frac23}}\operatorname{diag}\left(\omega^2,\omega,1\right)+ {\mathcal{O}}(z^{-\frac43})\right), \qquad z\to \infty.\end{gathered}$$ We omit the details here. This completes the proof of Proposition \[prop:S\]. Estimate of $J_S$ for large $s$ ------------------------------- The jump matrix $J_S(z)$ defined in is constant on $\Sigma_0^{(1)}$ and on $\Sigma_3^{(1)}$. On the other jump contours, it comes out that the nonzero off-diagonal entries of $J_S$ are all exponentially small for large $s$, which can be seen from the following proposition. In what follows, we denote by $U(z_0,\delta)$ the fixed open disk centered at $z_0$ with small radius $\delta>0$. \[prop:lambda\] 1. There exist positive constants $c_1,c_2>0$ such that $$\begin{aligned} {\mathrm{Re}\,}(\lambda_2(z)-\lambda_1(z) ) &\leq - c_1 |z|^{\frac43},&& z\in (\Sigma_1^{(1)} \cup \Sigma_5^{(1)} )\setminus U(1,\delta), \label{eq:lambda21}\\ {\mathrm{Re}\,}(\lambda_3(z)-\lambda_1(z) ) &\leq - c_2 |z|^{\frac43},&& z\in (\Sigma_2^{(1)} \cup \Sigma_4^{(1)} )\setminus U(-1,\delta),\end{aligned}$$ for $s$ large enough. 2. There exists a constant $c_3>0$ such that $$\begin{aligned} {\mathrm{Re}\,}\left(\lambda_2(z)-\lambda_3(z) \right) &\leq - c_3 |z|^{\frac43},&& z\in \Sigma_1^{(1)} \cup \Sigma_5^{(1)}, \label{eq:lambda23ineq} \\ {\mathrm{Re}\,}\left(\lambda_3(z)-\lambda_2(z) \right) &\leq - c_3 |z|^{\frac43},&& z\in \Sigma_2^{(1)} \cup \Sigma_4^{(1)}, \label{eq:lambda32ineq} \end{aligned}$$ for $s$ large enough. We will only present the proof of , since the proofs of other estimates are analogous. In view of the symmetry properties , it suffices to show for $z\in \Sigma_1^{(1)}\setminus U(1,\delta)$. To this end, let us define $$\lambda_j^*(z)=\frac{3}{4^{\frac53}}w_j(z)^4-\frac{3}{2^{\frac43}}w_j(z)^2, \qquad j=1,2,3.$$ From the behavior of the $\lambda$-functions at infinity given in Proposition \[prop: prop of lambda\], it follows from an elementary analysis that $$|\lambda_j(z)-\lambda_j^*(z)|\leq \varrho s^{-\frac23}\max\{1,|z|^\frac43\}, \qquad z\in \Sigma_1^{(1)}\setminus U(1,\delta),$$ for some $\varrho>0$, if $s$ is large enough. This, together with the triangle inequality, implies that it is sufficient to show for $\lambda_j^*(z)$, $j=1,2$. For large value of $z\in\Sigma_1^{(1)}$, the estimate follows from the asymptotics of $\lambda_j^*$ at infinity, which can be obtained by taking $s\to +\infty$ in item (c) of Proposition \[prop: prop of lambda\]. For bounded $z$, the claim is supported by Figure \[fig: estimates 1\]. The sign of ${\mathrm{Re}\,}\left( \lambda_2^*- \lambda_1^*\right)$ remains unchanged in the region bounded by the solid lines in Figure \[fig: estimates 1\]. By using the asymptotics of $\lambda_1^*$ and $\lambda_2^*$ at infinity, it is readily seen that ${\mathrm{Re}\,}\left( \lambda_2^*(z)- \lambda_1^*(z)\right)<0$ for all $z$ on the right side of the solid curve excluding $[1, \infty)$, which particularly holds for the finite part of $\Sigma_1^{(1)}\setminus U(1,\delta)$. This completes the proof of Proposition \[prop:lambda\]. ![Curves where ${\mathrm{Re}\,}\lambda_1^*={\mathrm{Re}\,}\lambda_2^*$ (solid lines), ${\mathrm{Re}\,}\lambda_1^*={\mathrm{Re}\,}\lambda_3^*$ (dashed lines), and ${\mathrm{Re}\,}\lambda_2^*={\mathrm{Re}\,}\lambda_3^*$ (dashed-dotted lines).[]{data-label="fig: estimates 1"}](Lambda-value.pdf){width="340pt"} As a consequence of the above proposition, the following corollary about the estimate of $J_S$ is immediate. \[coro:JS\] There is a constant $c > 0$ such that $$J_{S}(z) = I + {\mathcal{O}}\left(e^{-c s^{\frac43} |z|^{\frac43}}\right), \qquad \text{ as } s \to +\infty,$$ uniformly for $z \in \bigcup\limits_{i=1,2,4,5}\Sigma_i^{(1)}\setminus\{U(1,\delta) \cup U(-1,\delta)\}$. Global parametrix ----------------- By Corollary \[coro:JS\], if we suppress all entries of the jump matrices for $S$ that decay exponentially as $s \to +\infty$, we are led to the following RH problem for the global parametrix $N$. \[rhp:global para\] We look for a $3 \times 3$ matrix-valued function $N$ satisfying - $N(z)$ is defined and analytic in $\mathbb{C}\setminus \{(-\infty,-1]\cup[1,+\infty)\}$. - $N$ satisfies the jump condition $$\label{eq:N-jump} N_+(x)=N_-(x)\left\{ \begin{array}{ll} \begin{pmatrix} 0&1&0 \\ -1&0&0 \\ 0&0&1 \end{pmatrix}, & \qquad \hbox{$x>1$,} \\ \begin{pmatrix} 0&0&1 \\ 0&1&0 \\ -1&0&0 \end{pmatrix}, & \qquad \hbox{$x<-1$.} \end{array} \right.$$ - As $z \to \infty$, we have $$\label{eq:asyN1} N(z)= \left( I+ {\mathcal{O}}(z^{-1}) \right) \operatorname{diag}\left( z^{-\frac13},1,z^{\frac13} \right) L_{\pm},$$ where the constants $L_{\pm}$ are given in . The above RH problem can be solved explicitly. Let the scalar functions $N_j(\cdot)$, $j = 1,2,3,$ be defined as $$\begin{aligned} N_1(w)=\frac{1-\frac13 w^2}{(1-w^2)^{\frac12}}, \quad N_2(w)=-\frac{i}{\sqrt{6}}\frac{w-\frac{\sqrt{3}}{3}w^2}{(1-w^2)^{\frac12}}, \quad N_3(w)=-\frac{i}{\sqrt{6}}\frac{w+\frac{\sqrt{3}}{3}w^2}{(1-w^2)^{\frac12}},\end{aligned}$$ where the branch cut for the square root is taken along $\gamma_1^- \cup \gamma_2^-$, i.e., the curve defined by $w_{1,-}\left((-\infty,1] \cup [1,+\infty)\right)$; see Figure \[fig: image of w\] for an illustration. It is shown in [@Des Section 6.1.5] that the solution to RH problem \[rhp:global para\] is given by $$\label{Nz-def} N(z)= \mathcal{N}_0 \begin{pmatrix} N_1(w_1(z)) & N_1(w_3(z)) & N_1(w_2(z)) \\ N_2(w_1(z)) & N_2(w_3(z)) & N_2(w_2(z)) \\ N_3(w_1(z)) & N_3(w_3(z)) & N_3(w_2(z)) \end{pmatrix} {\Lambda},$$ where ${\Lambda}$ is defined in and $$\label{eq: MatN0} \mathcal{N}_0= \frac14 \operatorname{diag}\left(\frac{4}{2^{\frac16}},\sqrt{6}, 3\cdot 2^{\frac16} \right) \begin{pmatrix} \sqrt{2}i & 1 & -1 \\ 0 & 2 & 2 \\ -\sqrt{2}i & 1 & -1 \end{pmatrix}$$ is an invertible constant matrix. It is worthwhile to mention that $\mathcal{N}_0 ^{-1}N(z)$ satisfies the following symmetric relation (see [@Des Equations (2.2.30) and (6.1.38)]) $$\label{eq: Nz+-1} \mathcal{N}_0 ^{-1}N(z) = {\Lambda}\mathcal{N}_0 ^{-1}N(-z) {\Lambda},$$ or equivalently, $$\label{eq: Nz+-} N(z) = \Upsilon N(-z) {\Lambda},$$ where $$\label{eq: Upsilon-def} \Upsilon := \mathcal{N}_0 {\Lambda}\mathcal{N}_0 ^{-1}.$$ Finally, from the asymptotic behaviors of the $w$-functions given in Proposition \[prop:w\], it is readily seen that the following proposition regarding the refined asymptotic behaviors of the global parametrix $N$ near $-1$ and $\infty$. With $N$ defined in , we have $$\begin{aligned} N(z) = \, & \frac{3^{\frac14}}{2^{\frac34} (z + 1)^{\frac14}} \begin{pmatrix} -2^{\frac13} i & 0 & 2^{\frac13} \\ - i & 0 & 1 \\ 2^{-\frac43} i & 0 & -2^{-\frac43} \end{pmatrix} + \frac{1}{2^{\frac43} \cdot 3^{\frac12}} \begin{pmatrix} 0 & -2^{\frac53} & 0 \\ 0 & 2^{\frac73} & 0 \\ 0 & -5 & 0 \end{pmatrix} \nonumber \\ & + \frac{(z + 1)^{\frac14}}{6^{\frac54}} \begin{pmatrix} -2^{\frac13} i & 0 & -2^{\frac13} \\ 5 i & 0 & 5 \\ \frac{25}{2^{\frac43}} i & 0 & \frac{25}{2^{\frac43}} \end{pmatrix} + \frac{(z + 1)^{\frac34}}{72 \cdot 6^{\frac34}} \begin{pmatrix} -13 \cdot 2^{\frac13} i & 0 & 13 \cdot 2^{\frac13} \\ 35 i & 0 & -35 \\ -\frac{83}{2^{\frac43} } i & 0 & \frac{83}{2^{\frac43} } \end{pmatrix} \nonumber \\ & + \frac{ 2^{\frac23} (z + 1)}{27 \sqrt{3}} \begin{pmatrix} 0 & -2^{\frac53} & 0 \\ 0 & 2^{\frac13} & 0 \\ 0 & 7 & 0 \end{pmatrix} + {\mathcal{O}}(z+1)^{\frac54},\qquad z \to -1, \label{Nz-exp-z=-1}\end{aligned}$$ and $$\label{eq:asyN} N(z)= \left( I + \frac{ \mathsf{N}_1}{z} + \mathcal {\mathcal{O}}(z^{-2}) \right) \operatorname{diag}\left(z^{-\frac13},1,z^{\frac13} \right)L_{\pm},\qquad z \to \infty,$$ where $$\label{N1-coeff} \mathsf{N}_1 = \begin{pmatrix} 0 & -2^{-\frac{5}{3}} & 0 \\ - \frac{5}{16 \cdot 2^{\frac{1}{3}}} & 0 & 2^{ -\frac{5}{3}} \\ 0 & \frac{5}{16 \cdot 2^{\frac{1}{3}}} & 0 \end{pmatrix},$$ and the constants $L_{\pm}$ are given in . Local parametrix near $-1$ -------------------------- Due to the fact that the convergence of the jump matrices to the identity matrices on $\Sigma_2^{(1)}$ and $\Sigma_4^{(1)}$ is not uniform near $-1$, we intend to find a function $P^{(-1)}(z)$ satisfying an RH problem as follows. \[rhp:localpara1\] We look for a $3 \times 3$ matrix-valued function $P^{(-1)}(z)$ satisfying - $P^{(-1)}(z)$ is defined and analytic in $\overline{U(-1,\delta)} \setminus \{\Sigma_2^{(1)} \cup \Sigma_3^{(1)} \cup \Sigma_4^{(1)} \}$. - $P^{(-1)}(z)$ satisfies the jump condition $$\begin{aligned} \label{eq:P1-jump} P^{(-1)}_{+}(z)=P^{(-1)}_{-}(z)J_S(z), \qquad z\in U(-1,\delta)\cap \{\cup\Sigma_{i=2,3,4}^{(1)}\},\end{aligned}$$ where $J_S(z)$ is defined in . - As $s \to +\infty$, $P^{(-1)}(z)$ matches $N(z)$ on the boundary $\partial U(-1,\delta)$ of $U(-1,\delta)$, i.e., $$\label{eq:mathching1} P^{(-1)}(z)=\left(I+\mathcal {\mathcal{O}}(s^{-\frac43} ) \right)N(z), \qquad z\in \partial U(-1,\delta).$$ The RH problem \[rhp:localpara1\] for $P^{(-1)}(z)$ can be solved explicitly with the aid of the Bessel parametrix $\Phi^{({\mathrm{Bes}})}_{\alpha}$ described in Appendix \[Append: BP\]. To this aim, we introduce the local conformal mapping $$\label{def: cm f} f(z)=\frac{1}{4}\left(\lambda_1(z)-\lambda_3(z) \right)^2, \qquad z\in U(-1,\delta).$$ By and , we have that $f(z)$ is analytic in $U(-1,\delta)$ and, as $z\to -1$, $$\label{def: expf-1} f(z)=C_1^2(z+1)+2C_1 C_3(z+1)^2 + {\mathcal{O}}(z+1)^3$$ with the constants $C_1$ and $C_3$ given in . Let $\Phi^{({\mathrm{Bes}})}_{0}$ be the Bessel parametrix in with $\alpha=0$, we set, for $z\in \overline{U(-1,\delta)} \setminus \{\Sigma_2^{(1)} \cup \Sigma_3^{(1)} \cup \Sigma_4^{(1)} \}$, $$\begin{aligned} \label{eq: local par} P^{(-1)}(z)=\, & E(z)\begin{pmatrix} \left( \Phi^{({\mathrm{Bes}})}_{0} \right)_{11}(s^{\frac{8}{3}}f(z))& 0 & \left( \Phi^{({\mathrm{Bes}})}_{0}\right)_{12}(s^{\frac{8}{3}}f(z)) \\ 0 & 1 & 0 \\ \left( \Phi^{({\mathrm{Bes}})}_{0}\right)_{21}(s^{\frac{8}{3}}f(z)) & 0 & \left( \Phi^{({\mathrm{Bes}})}_{0}\right)_{22}(s^{\frac{8}{3}}f(z)) \end{pmatrix} \nonumber \\ &\times \begin{pmatrix} e^{\frac{s^{\frac43}}{2}(\lambda_3(z)-\lambda_1(z))} & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0& e^{\frac{s^{\frac43}}{2}(\lambda_1(z)-\lambda_3(z))} \end{pmatrix} \nonumber \\ & \times \left\{ \begin{array}{ll} \begin{pmatrix} 1&0&0 \\ 0&1&0 \\ 0& e^{s^{\frac43}(\lambda_3(z)-\lambda_2(z))} &1 \end{pmatrix}, &\quad \hbox{$|\arg f(z)|< \frac{3\pi}{4} $,} \\ I, & \quad \hbox{$ \frac{3\pi}{4} <|\arg f(z)|<\pi$,} \end{array} \right.\end{aligned}$$ where $f(z)$ is defined in and $$\label{def:E} E(z)=\frac{1}{ \sqrt{2} }N(z)\begin{pmatrix} 1 & 0 &-i \\ 0 & \sqrt{2} & 0 \\ -i & 0& 1 \end{pmatrix}\begin{pmatrix} \pi^{\frac{1}{2}} s^{\frac{2}{3}}f(z) ^{\frac{1}{4}}& 0 &0 \\ 0 & 1 & 0 \\ 0 & 0& \pi^{-\frac{1}{2}} s^{-\frac{2}{3}}f(z) ^{-\frac{1}{4}} \end{pmatrix}$$ with $N(z)$ given in . \[prop:p-1\] The local parametrix $P^{(-1)}(z)$ defined in solves the RH problem \[rhp:localpara1\]. From , it is straightforward to check that $P^{(-1)}(z)$ satisfies the jump condition in provided the prefactor $E(z)$ is analytic in $U(-1,\delta)$. To see this, we note that the only possible jump for $E(z)$ is on the interval $(-1 - \delta , -1)$. For $x \in (-1 - \delta , -1)$, it is readily seen from that $f_+(x)^{\frac{1}{4}} = i\, f_-(x)^{\frac{1}{4}}$. Hence, we obtain from the jump of $N(z)$ in that $$\begin{aligned} E_{-}^{-1} (x) E_+(x) = \, & \begin{pmatrix} \pi^{-\frac{1}{2}} s^{-\frac{2}{3}} f_-(x) ^{-\frac{1}{4}}& 0 &0 \\ 0 & 1 & 0 \\ 0 & 0& \pi^{\frac{1}{2}} s^{\frac{2}{3}} f_-(x) ^{-\frac{1}{4}} \end{pmatrix} \begin{pmatrix} \frac{1}{\sqrt{2}} & 0 & \frac{i}{\sqrt{2}} \\ 0 & 1 & 0 \\ \frac{i}{\sqrt{2}}& 0& \frac{1}{\sqrt{2}} \end{pmatrix} \begin{pmatrix} 0&0&1 \\ 0&1&0 \\ -1&0&0 \end{pmatrix}\\ & \times \begin{pmatrix} \frac{1}{\sqrt{2}} & 0 & -\frac{i}{\sqrt{2}} \\ 0 & 1 & 0 \\ -\frac{i}{\sqrt{2}}& 0& \frac{1}{\sqrt{2}} \end{pmatrix} \begin{pmatrix} \pi^{\frac{1}{2}} s^{\frac{2}{3}}f_+(x) ^{\frac{1}{4}}& 0 &0 \\ 0 & 1 & 0 \\ 0 & 0& \pi^{-\frac{1}{2}} s^{-\frac{2}{3}} f_+(x) ^{-\frac{1}{4}} \end{pmatrix}=I.\end{aligned}$$ This shows that $E(z)$ is indeed analytic in $U(-1,\delta)$. It remains to verify the matching condition . From the local behaviors of $\lambda_1$ and $\lambda_3$ near $-1$ given in item (d) of Proposition \[prop: prop of lambda\], the function $e^{\frac{s^{\frac43}}{2}(\lambda_1(z)-\lambda_3(z))}$ appearing in the last term of is exponentially small as $s\to +\infty$ for $z \in \partial U(-1,\delta)$. Thus, it follows from the asymptotic behavior of the Bessel parametrix $\Phi^{({\mathrm{Bes}})}_{0}$ at infinity in that, for $z \in \partial U(-1,\delta)$, $$\begin{aligned} \label{eq:mathching1-exp} P^{(-1)}(z) N^{-1}(z)= I+ \frac{J_{1}^{(-1)} (z) }{ s^{\frac{4}{3}} } + \mathcal {\mathcal{O}}(s^{-\frac83} ) , \qquad s \to +\infty,\end{aligned}$$ where $$\label{j1-1-formula} J_{1}^{(-1)} (z) = \frac{1}{8 f(z)^{\frac{1}{2}}} N(z) \begin{pmatrix} -1 & 0 & -2i \\ 0 & 0 & 0 \\ -2i & 0 & 1 \end{pmatrix} N^{-1}(z),$$ which gives us . This completes the proof of Proposition \[prop:p-1\]. We conclude this section by evaluating $E(-1)$ and $E'(-1)$ for later use. The calculations are straightforward and cumbersome by combining and the asymptotics of $N(z)$ and $f(z)$ given in and . We omit the details but present the results below. $$\label{eq: E(-1)} E(-1) = \begin{pmatrix} -i 2^{\frac{1}{12}} 3^{\frac14} C_1^{\frac12} \pi^{\frac12} s^{\frac23} & - \frac{2^{\frac13}}{3^{\frac12}} & - \frac{1}{2^{\frac5{12}} 3^{\frac54} C_1^{\frac12} \pi^{\frac12} s^{\frac23} } \\ - i 2^{-\frac14} 3^{\frac14} C_1^{\frac12} \pi^{\frac12} s^{\frac23} & \frac{2}{3^{\frac12}} & \frac{5}{ 2^{\frac34} 3^{\frac54} C_1^{\frac12} \pi^{\frac12} s^{\frac23}} \\ i\frac{3^{\frac14} C_1^{\frac12} \pi^{\frac12} s^{\frac23} }{ 2^{\frac{19}{12}}} & -\frac{5}{2^{\frac43} 3^{\frac12}} & \frac{25}{ 2^{\frac{25}{12}} 3^{\frac54} C_1^{\frac12} \pi^{\frac12} s^{\frac23}} \end{pmatrix},$$ and $$\label{eq: Ep(-1)} E'(-1) = \begin{pmatrix} -i\frac{ (13 C_1 + 108 C_3) \pi^{\frac12} s^{\frac23}}{36 \cdot 2^{\frac{11}{12}} 3^{\frac34} C_1^{\frac12}} & * & * \\ i \frac{(35 C_1 - 108 C_3) \pi^{\frac12} s^{\frac23}}{72 \cdot 2^{\frac14} 3^{\frac34} C_1^{\frac12}} & * & * \\ -i \frac{(83 C_1 - 108 C_3) \pi^{\frac12} s^{\frac23}}{144 \cdot 2^{\frac{7}{12}} 3^{\frac34} C_1^{\frac12}} & * & * \end{pmatrix},$$ where the constants $C_1$ and $C_3$ are given in , and $*$ stands for some unimportant entry. Local parametrix near $1$ ------------------------- Similar to the situation encountered near $z=-1$, we intend to find a function $P^{(1)}(z)$ satisfying the following RH problem near $z=1$. \[rhp:localpara2\] We look for a $3 \times 3$ matrix-valued function $P^{(1)}(z)$ satisfying - $P^{(1)}(z)$ is defined and analytic in $\overline{U(1,\delta)} \setminus \{\Sigma_0^{(1)} \cup \Sigma_1^{(1)} \cup \Sigma_5^{(1)}\}$. - $P^{(1)}(z)$ satisfies the jump condition $$\label{eq:P2-jump} P^{(1)}_{+}(z)=P^{(1)}_{-}(z)J_S(z), \qquad z\in U(-1,\delta)\cap \{\cup\Sigma_{i=0,1,5}^{(1)}\},$$ where $J_S(z)$ is defined in . - As $s \to +\infty$, we have $$\label{eq:mathching2} P^{(1)}(z)=\left(I+\mathcal {\mathcal{O}}(s^{-\frac43} ) \right)N(z),\qquad z\in \partial U(1,\delta).$$ Again, the RH problem \[rhp:localpara2\] can be solved with the help of the Bessel parametrix $\Phi^{({\mathrm{Bes}})}_{0}$, following the same spirit in the construction of $P^{(-1)}(z)$. The conformal mapping now reads $$\label{def: cm f-2} \widetilde{f}(z)=\frac{1}{4} ( \lambda_1(z)-\lambda_2(z) )^2,\qquad z\in U(1,\delta).$$ From and , one can see that $$\label{eq: fz+-} \widetilde{f}(z) = f(-z),$$ where $f(z)$ is defined in . In view of , this also implies that, as $z\to 1$, $$\label{def: expf-2} \widetilde f(z)= - C_1^2(z-1)+2C_1 C_3(z-1)^2+{\mathcal{O}}(z-1)^3,$$ with the constants $C_1$ and $C_3$ given in . For $z\in \overline{U(1,\delta)} \setminus \{\cup\Sigma_{i=0,1,5}^{(1)}\}$, we then set $$\begin{aligned} \label{eq: local par-2} P^{(1)}(z)=&\, \widetilde E(z)\begin{pmatrix} \left( \Phi^{({\mathrm{Bes}})}_{0} \right)_{11}(s^{\frac{8}{3}} \widetilde f(z))& -\left( \Phi^{({\mathrm{Bes}})}_{0} \right)_{12}(s^{\frac{8}{3}}\widetilde f(z)) & 0 \\ -\left( \Phi^{({\mathrm{Bes}})}_{0} \right)_{21}(s^{\frac{8}{3}}\widetilde f(z)) & \left( \Phi^{({\mathrm{Bes}})}_{0} \right)_{22}(s^{\frac{8}{3}}\widetilde f(z)) & 0 \\ 0 & 0 & 1 \end{pmatrix} \nonumber \\ & \times \begin{pmatrix} e^{\frac{s^{\frac43}}{2}(\lambda_2(z)-\lambda_1(z))} & 0 & 0 \\ 0 & e^{\frac{s^{\frac43}}{2}(\lambda_1(z)-\lambda_2(z))} & 0 \\ 0 & 0 & 1 \end{pmatrix} \nonumber \\ & \times \left\{ \begin{array}{ll} \begin{pmatrix} 1&0&0 \\ 0&1& e^{s^{\frac43}(\lambda_2(z)-\lambda_3(z))} \\ 0& 0 &1 \end{pmatrix}, &\quad \hbox{$|\arg \widetilde f(z)|< \frac{3\pi}{4} $,} \\ I, & \quad \hbox{$ \frac{3\pi}{4} <|\arg \widetilde f(z)|<\pi$,} \end{array} \right.\end{aligned}$$ where $\widetilde f(z)$ is defined in and $$\label{def:E-2} \widetilde E(z)=\frac{1}{ \sqrt{2} }N(z)\begin{pmatrix} 1 & i & 0 \\ i & 1 & 0 \\ 0 & 0 & \sqrt{2} \end{pmatrix}\begin{pmatrix} \pi^{\frac{1}{2}} s^{\frac{2}{3}} \widetilde f(z) ^{\frac{1}{4}}& 0 & 0 \\ 0 & \pi^{-\frac{1}{2}} s^{-\frac{2}{3}} \widetilde f(z) ^{-\frac{1}{4}} & 0 \\ 0 & 0 & 1 \end{pmatrix}.$$ It comes out that $P^{(1)}(z)$ is closely related to $P^{(-1)}(z)$. To see the relation, we observe from and that $$\widetilde E(z) = \Upsilon E(-z) {\Lambda},$$ where the constant matrices ${\Lambda}$ and $\Upsilon$ are given in and , respectively. A further appeal to the symmetric relations and then implies $$P^{(1)}(z) = \Upsilon P^{(-1)}(-z) {\Lambda}.$$ This in turn shows that the $ P^{(1)}(z)$ fulfills the jump condition and the matching condition . In particular, we have, as $s \to +\infty$, $$\begin{aligned} \label{eq:mathching2-exp} P^{(1)}(z) N^{-1}(z)= I+ \frac{J_{1}^{(1)} (z) }{ s^{\frac{4}{3}} } + \mathcal {\mathcal{O}}(s^{-\frac83} ) , \qquad z \in \partial U(1,\delta),\end{aligned}$$ where $$\label{j1-2-formula} J_{1}^{(1)} (z) = \frac{1}{8 \widetilde f(z)^{\frac{1}{2}}} N(z) \begin{pmatrix} -1 & 2i & 0 \\ 2i & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix} N^{-1}(z).$$ Again, using and , it is readily seen from that $$\begin{aligned} \label{j1-symmetric} J_{1}^{(1)} (z) &=\frac{1}{8 f(-z)^{\frac{1}{2}}} \Upsilon N(-z) \Lambda \begin{pmatrix} -1 & 2i & 0 \\ 2i & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix} \Lambda^{-1} N^{-1}(-z) \Upsilon^{-1} \nonumber \\ &=\Upsilon J_1^{(-1)}(-z) \Upsilon,\end{aligned}$$ where we have made use of the fact that $\Upsilon^{-1}=\Upsilon$; see . In summary, we have proved the following proposition. \[prop:p-2\] The local parametrix $P^{(1)}(z)$ defined in solves the RH problem \[rhp:localpara2\]. Final transformation -------------------- Our final transformation is defined by $$\label{def:StoR} R(z)=\left\{ \begin{array}{ll} S(z)N^{-1}(z), &\quad \hbox{$z\in \mathbb{C}\setminus \{U(-1,\delta) \cup U(1,\delta)\cup \Sigma_S\}$,} \\ S(z)(P^{(-1)}(z))^{-1}, &\quad \hbox{$z \in U(-1,\delta)$,} \\ S(z)(P^{(1)}(z))^{-1}, &\quad \hbox{$z \in U(1,\delta)$.} \end{array} \right.$$ It is then easily seen that $R(z)$ satisfies the following RH problem. \[rhp:R\] The $3 \times 3$ matrix-valued function $R(z)$ defined in has the following properties: - $R(z)$ is analytic in $\mathbb{C}\setminus \Sigma_R$, where the contour $\Sigma_R$ is shown in Figure \[fig:ContourR\]. - $R(z)$ satisfies the jump condition $$\label{eq:R-jump} R_{+}(z)=R_{-}(z)J_R(z),\qquad z\in\Sigma_R,$$ where $$J_R(z)= \left\{ \begin{array}{ll} P^{(-1)}(z)N^{-1}(z), &\quad \hbox{$z\in \partial U(-1,\delta)$,} \\ P^{(1)}(z)N^{-1}(z), &\quad \hbox{$z\in \partial U(1,\delta)$,}\\ N(z)J_S(z)N^{-1}(z), &\quad \hbox{$z \in \cup\Sigma_{i=1,2,4,5}^{(1)}\setminus \{U(-1,\delta)\cup U(1,\delta)\}$.} \end{array} \right.$$ - As $z \to \infty$, we have $$\label{eq:asyR1} R(z)=I+{\mathcal{O}}(z^{-1}).$$ (100,70)(-5,2) (37.5,41)[(-2,1)[28]{}]{} (37.5,39)[(-2,-1)[28]{}]{} (62.5,41)[(2,1)[28]{}]{} (62.5,39)[(2,-1)[28]{}]{} (40,40) (40,40) (60,40) (60,40) (41,42.3)[(1,0)[.0001]{}]{} (61,42.3)[(1,0)[.0001]{}]{} (38,34)[$-1$]{} (59,34)[$1$]{} (20,50)[(2,-1)[.0001]{}]{} (20,30.5)[(2,1)[.0001]{}]{} (80,30.5)[(2,-1)[.0001]{}]{} (80,50)[(2,1)[.0001]{}]{} In view of Corollary \[coro:JS\], it follows that the jumps of $R(z)$ tend to the identity matrix exponentially fast as $s\to +\infty$, except for those on $\partial U(-1,\delta)\cup \partial U(1,\delta)$; see also and for the expansions of $J_R(z)$ on $\partial U(1,\delta)$ and $\partial U(-1,\delta)$. Then, by a standard argument (cf. [@DeiftBook]), we conclude that, as $s\to +\infty$, $$\label{eq:estR} R(z)=I+ \frac{R_1(z)}{s^{\frac{4}{3}}}+ {\mathcal{O}}(s^{-\frac{8}{3}}) \quad \textrm{and} \quad \frac{{\,\mathrm{d}}}{{\,\mathrm{d}}z}R(z)=\frac{R_1'(z)}{s^{\frac{4}{3}}}+ {\mathcal{O}}(s^{-\frac{8}{3}}),$$ uniformly for $z \in \mathbb{C} \setminus \Sigma_R$. Moreover, a combination of and RH problem \[rhp:R\] shows that $R_1(z)$ solves the following RH problem. \[rhp:R1 z\] The $3 \times 3$ matrix-valued function $R_1(z)$ appearing in satisfies the following properties: - $R_1(z)$ is defined and analytic in $\mathbb{C}\setminus \{\partial U(-1,\delta)\cup \partial U(1,\delta)\}$. - $R_1$ satisfies the jump condition $$\label{eq:R1-jump} R_{1,+}(z)-R_{1,-}(z) = \begin{cases} J_{1}^{(-1)}(z), & z \in \partial U(-1,\delta), \\ J_{1}^{(1)}(z), & z \in \partial U(1,\delta), \end{cases}$$ where $J_{1}^{(-1)}(z)$ and $J_{1}^{(1)}(z)$ are given in and , respectively. - As $z \to \infty$, we have $R_1(z) = {\mathcal{O}}(z^{-1}).$ By Cauchy’s residue theorem, the solution to the above RH problem is given by $$\begin{aligned} & R_1(z) \nonumber \\ &= \frac{1}{2 \pi i} \oint_{\partial U(-1,\delta)} \frac{J_{1}^{(-1)}(\zeta)}{z-\zeta } {\,\mathrm{d}}\zeta + \frac{1}{2 \pi i} \oint_{\partial U(1,\delta)} \frac{J_{1}^{(1)}(\zeta)}{z- \zeta} {\,\mathrm{d}}\zeta \nonumber \\ &= \begin{cases} {\displaystyle}\frac{\mathop{\textrm{Res}}\limits_{\zeta = -1}J_{1}^{(-1)}(\zeta)}{z+1} + \frac{\mathop{\operatorname*{Res}}\limits_{\zeta = 1}J_{1}^{(1)}(\zeta)}{z-1} , & z \in \mathbb{C}\setminus \{U(-1,\delta)\cup U(1,\delta)\}, \medskip \\ {\displaystyle}\frac{\mathop{\textrm{Res}}\limits_{\zeta = -1} J_{1}^{(-1)}(\zeta)}{z+1} + \frac{\mathop{\operatorname*{Res}}\limits_{\zeta = 1}J_{1}^{(1)}(\zeta)}{z-1} - J_{1}^{(-1)}(z) , & z \in U(-1,\delta), \medskip \\ {\displaystyle}\frac{\mathop{\operatorname*{Res}}\limits_{\zeta = -1}J_{1}^{(-1)}(\zeta)}{z+1} + \frac{\mathop{\operatorname*{Res}}\limits_{\zeta = 1}J_{1}^{(1)}(\zeta)}{z-1} - J_{1}^{(1)}(z), & z \in U(1,\delta). \end{cases} \label{R1-expression}\end{aligned}$$ For our purpose, we need to know the exact value of $R_1'(-1)$. From the local behaviors of $N(z)$ and $f(z)$ near $z=-1$ given in and , we obtain from that $$\begin{aligned} \label{J1-1-expan} J_{1}^{(-1)}(z) &= \frac{1}{8 f(z)^{\frac{1}{2}}} N(z) \begin{pmatrix} -1 & 0 & -2i \\ 0 & 0 & 0 \\ -2i & 0 & 1 \end{pmatrix} N^{-1}(z) \nonumber \\ &=\frac{\mathcal{J}_{-1}}{z+1} + \mathcal{J}_0 + \mathcal{J}_1 \cdot (z+1) + {\mathcal{O}}(z+1)^2, \qquad z \to -1,\end{aligned}$$ where $$\label{j1-1-residue} \mathcal{J}_{-1}= \operatorname*{Res}_{\zeta = -1}J_{1}^{(-1)}(\zeta) = \frac{1}{16 \sqrt{6} \, C_1} \begin{pmatrix} 1 & - 2^{\frac43} & - 2^{\frac53} \\ 2^{-\frac13} & -2 & -2^{\frac43} \\ -2^{-\frac53} & 2^{-\frac13} & 1 \end{pmatrix}$$ with $C_1$ given in . Although the explicit formulas of $\mathcal{J}_0$ and $\mathcal{J}_1$ in are also available and $\mathcal{J}_{1}$ is indeed involved in our later calculation, we decide not to include the exact formulas here due to their complicated forms. Also note that $J_1^{(1)}(z)=\Upsilon J_1^{(-1)}(-z) \Upsilon$ (see ), it is then readily seen that $$\begin{aligned} \operatorname*{Res}_{\zeta = 1}J_{1}^{(1)}(\zeta) &= - \Upsilon \, \operatorname*{Res}_{\zeta = -1}J_{1}^{(-1)}(\zeta) \, \Upsilon = -\Upsilon \, \mathcal{J}_{-1} \, \Upsilon \nonumber \\ & = \frac{1}{16 \sqrt{6} C_1} \begin{pmatrix} -1 & -2^{\frac43} & 2^{\frac53} \\ 2^{-\frac13} & 2 & - 2^{\frac43} \\ 2^{-\frac53} & 2^{-\frac13} & -1 \end{pmatrix}. \label{j1-2-residue}\end{aligned}$$ Hence, in view of , we arrive at $$\label{Rp(-1)-expression} R_1'(-1) = - \mathcal{J}_{1} - \frac{1}{4} \operatorname*{Res}_{\zeta = 1}J_{1}^{(1)}(\zeta)= -\mathcal{J}_{1}-\frac{1}{64 \sqrt{6} C_1} \begin{pmatrix} -1 & -2^{\frac43} & 2^{\frac53} \\ 2^{-\frac13} & 2 & - 2^{\frac43} \\ 2^{-\frac53} & 2^{-\frac13} & -1 \end{pmatrix},$$ where $\mathcal{J}_{1}$ is given in . Finally, we point out that, $$\label{eq:asyR} R(z)=I + \frac{\mathsf{R}_1}{z} +{\mathcal{O}}(z^{-2}), \qquad z\to \infty,$$ where $$\mathsf{R}_1=\frac{i}{2 \pi}\int_{\Sigma_R} R_-(w)\left(J_{ R}(w)-I\right){\,\mathrm{d}}w.$$ Comparing , with , we obtain $$\begin{aligned} \label{R1-coeff} \mathsf{R}_1 &=\frac{1}{s^{\frac{4}{3}}} \left(\operatorname*{Res}_{\zeta = -1}J_{1}^{(-1)}(\zeta)+\operatorname*{Res}_{\zeta = 1}J_{1}^{(1)}(\zeta)\right) + {\mathcal{O}}(s^{-\frac{8}{3}}) \nonumber \\ &= \frac{1}{s^{\frac{4}{3}}} \left(\mathcal{J}_{-1} - \Upsilon \mathcal{J}_{-1} \Upsilon \right) + {\mathcal{O}}(s^{-\frac{8}{3}}),\end{aligned}$$ on account of and . We are now ready to prove Theorem \[main-thm\]. Proof of Theorem \[main-thm\] {#sec:proof} ============================= Our strategy is to find the large $s$ asymptotics of $\frac{\partial}{\partial s} F(s;\rho)$ and $\frac{\partial}{\partial \rho} F(s;\rho)$ by making use of the differential identities established in Proposition \[prop:derivativeandX\], which will in turn give us the asymptotics of $F(s;\rho)$. We start with deriving the asymptotics of $\frac{\partial}{\partial s} F(s;\rho)$. Substituting into the differential identity gives us $$\begin{aligned} \frac{\partial}{\partial s} F(s;\rho)&=-\frac{1}{ \pi i s} \lim_{z \to -1} \left[\left(T^{-1}(z)T'(z)\right)_{21}+\left(T^{-1}(z) T'(z)\right)_{31}\right], \label{eq:derivativeinsT}\end{aligned}$$ where $'$ denotes the derivative with respect to $z$. Tracing back the invertible transformations $T\mapsto S$ and $S\mapsto R$ in and , it follows that $$\begin{gathered} \label{eq:TtoR} T(z)=e^{D_0s^{\frac43}} \sqrt{\frac{2\pi}{3}}i e^{\frac{\rho^2}{6}} \Psi_0 \operatorname{diag}\left(s^{-\frac13},1,s^{\frac13} \right) S_0^{-1} \\ \times R(z)P^{(-1)}(z)\operatorname{diag}\left(e^{s^{\frac43}\lambda_1(z)},e^{s^{\frac43}\lambda_2(z)},e^{s^{\frac43}\lambda_3(z)}\right), \qquad z \in U(-1,\delta).\end{gathered}$$ In addition, from the explicit expression of $P^{(-1)}(z)$ in , we further obtain $$\begin{aligned} \label{Tformula-A-B} T(z)&=e^{D_0s^{\frac43}} \sqrt{\frac{2\pi}{3}}i e^{\frac{\rho^2}{6}} \Psi_0 \operatorname{diag}\left(s^{-\frac13},1,s^{\frac13} \right) S_0^{-1} R(z)E(z) \mathscr{B}(s^{\frac{8}{3}}f(z)) \mathscr{A}(z)\end{aligned}$$ for $z \in U(-1,\delta)$ and $|\arg f(z)|<\frac34\pi$, where $$\label{eq: A(z) def} \mathscr{A}(z) := \begin{pmatrix} e^{\frac{s^{\frac43}}{2}(\lambda_1(z)+\lambda_3(z))} & 0 & 0\\ 0 & e^{s^{\frac43}\lambda_2(z)} & 0 \\ 0 & e^{\frac{s^{\frac43}}{2}(\lambda_1(z)+\lambda_3(z))} & e^{\frac{s^{\frac43}}{2}(\lambda_1(z)+\lambda_3(z))} \end{pmatrix}$$ and $$\label{def:Bz} \mathscr{B}(z):=\begin{pmatrix} \left( \Phi^{({\mathrm{Bes}})}_{0}\right)_{11}(z)& 0 & \left( \Phi^{({\mathrm{Bes}})}_{0}\right)_{12}(z) \\ 0 & 1 & 0 \\ \left( \Phi^{({\mathrm{Bes}})}_{0}\right)_{21}(z) & 0 & \left( \Phi^{({\mathrm{Bes}})}_{0} \right)_{22}(z) \end{pmatrix}.$$ Since the prefactor $e^{D_0s^{\frac43}} \sqrt{\frac{2\pi}{3}}i e^{\frac{\rho^2}{6}} \Psi_0 \operatorname{diag}\left(s^{-\frac13},1,s^{\frac13} \right) S_0^{-1}$ in is independent of $z$, we have $$\begin{aligned} T^{-1}(z)T'(z) = \, & \mathscr{A}^{-1}(z) \mathscr{A}'(z) + s^{\frac{8}{3}}f'(z) \mathscr{A}^{-1}(z) \mathscr{B}^{-1}(s^{\frac{8}{3}}f(z)) \mathscr{B}'(s^{\frac{8}{3}}f(z)) \mathscr{A}(z) \nonumber \\ &+ \mathscr{A}^{-1}(z) \mathscr{B}^{-1}(s^{\frac{8}{3}}f(z)) E^{-1}(z) E'(z) \mathscr{B}(s^{\frac{8}{3}}f(z)) \mathscr{A}(z)\nonumber \\ & + \mathscr{A}^{-1}(z) \mathscr{B}^{-1}(s^{\frac{8}{3}}f(z)) E^{-1}(z) R^{-1}(z) R'(z) E(z) \mathscr{B}(s^{\frac{8}{3}}f(z)) \mathscr{A}(z). \label{T'formula-A-B}\end{aligned}$$ We next evaluate the four terms on the right hand side of the above formula one by one. For the first term in , we see from that $$\begin{aligned} \label{Term1-final} \mathscr{A}^{-1}(z) \mathscr{A}'(z) = \displaystyle \frac{s^{\frac{4}{3}}}{2} \begin{pmatrix} \lambda_1'(z) + \lambda_3'(z) & 0 & 0 \\ 0 & 2\lambda_2'(z) & 0 \\ 0 & \lambda_1'(z) +\lambda_3'(z) - 2\lambda_2'(z) & \lambda_1'(z)+ \lambda_3'(z) \end{pmatrix}. $$ Thus, $$\label{eq:term1} \left(\mathscr{A}^{-1}(z) \mathscr{A}'(z)\right)_{21}=\left(\mathscr{A}^{-1}(z) \mathscr{A}'(z)\right)_{31}=0.$$ For the second term in , we recall the following properties of the modified Bessel functions (cf. [@DLMF Chapter 10]): $$\begin{aligned} I_0(z)&= \sum_{k=0}^{\infty}\frac{(z/2)^{2k}}{(k!)^2}, \\ K_0(z)&= -\left(\ln(z/2)+\gamma\right) I_0(z)+{\mathcal{O}}(z^2), \qquad z \to 0,\end{aligned}$$ where $\gamma$ is the Euler’s constant. This, together with and , implies that, as $z \to 0$, $$\label{eq:Bzero} \mathscr{B}(z)= \begin{pmatrix} 1+{\mathcal{O}}(z)& 0 & {\mathcal{O}}(\ln z) \\ 0 & 1 & 0 \\ \frac{\pi i}{2}z+{\mathcal{O}}(z^2) & 0 &1+{\mathcal{O}}(z \ln z) \end{pmatrix}$$ and $$\begin{gathered} \label{eq:B-1zero} \mathscr{B}^{-1}(z)=\begin{pmatrix} \left( \Phi^{({\mathrm{Bes}})}_{0}\right)_{22}(z)& 0 & -\left( \Phi^{({\mathrm{Bes}})}_{0} \right)_{12}(z) \\ 0 & 1 & 0 \\ -\left( \Phi^{({\mathrm{Bes}})}_{0}\right)_{21}(z) & 0 & \left( \Phi^{({\mathrm{Bes}})}_{0} \right)_{11}(z) \end{pmatrix} \\ = \begin{pmatrix} 1+{\mathcal{O}}(z\ln z)& 0 & {\mathcal{O}}(\ln z) \\ 0 & 1 & 0 \\ -\frac{\pi i}{2}z +{\mathcal{O}}(z^2) & 0 &1+{\mathcal{O}}(z) \end{pmatrix}.\end{gathered}$$ A combination of these two formulas shows $$\label{eq:B-1B} \lim_{z\to 0}\left( \mathscr{B}^{-1}(z)\mathscr{B}'(z) \right)_{21}=0, \qquad \lim_{z\to 0}\left( \mathscr{B}^{-1}(z)\mathscr{B}'(z) \right)_{31} =\frac{\pi i}{2}.$$ To this end, we note that for an arbitrary $3\times 3$ matrix $ M = (m_{ij})_{i,j=1}^3$, it is readily seen from that $$\label{eq: A-1MA} \begin{split} &\lim_{z \to -1} \left(\mathscr{A}^{-1}(z) M \mathscr{A}(z) \right)_{21} = m_{21} e^{\frac{s^{\frac{4}{3}}}{2} [\lambda_1(-1) + \lambda_3(-1) -2 \lambda_2(-1)]}, \\ &\lim_{z \to -1} \left(\mathscr{A}^{-1}(z) M \mathscr{A}(z) \right)_{31} = m_{31} - m_{21} e^{\frac{s^{\frac{4}{3}}}{2} [\lambda_1(-1) + \lambda_3(-1) -2 \lambda_2(-1)]}. \end{split}$$ We then obtain from , and the facts $f(-1) = 0, f'(-1) = C_1^2$ (see ) that $$\begin{aligned} \label{Term2-final} &\lim_{z \to -1} \left(s^{\frac{8}{3}}f'(z) \mathscr{A}^{-1}(z) \mathscr{B}^{-1}(s^{\frac{8}{3}}f(z)) \mathscr{B}'(s^{\frac{8}{3}}f(z)) \mathscr{A}(z) \right)_{21} = 0, \\ \label{Term2-final2} &\lim_{z \to -1} \left(s^{\frac{8}{3}}f'(z) \mathscr{A}^{-1}(z) \mathscr{B}^{-1}(s^{\frac{8}{3}}f(z)) \mathscr{B}'(s^{\frac{8}{3}}f(z)) \mathscr{A}(z) \right)_{31} = \frac{C_1^2 \pi i}{2} s^{\frac{8}{3}},\end{aligned}$$ where $C_1$ is given in . Regrading the last two terms in , we first observe from items (d) and (f) of Proposition \[prop: prop of lambda\] that $$\begin{aligned} \lambda_1(-1) + \lambda_3(-1) - 2 \lambda_2(-1) &= \lambda_1(-1) + \lambda_3(-1) -2\left( - \frac{9}{2^{\frac{7}{3}}} + \frac{3 \rho}{2^{\frac{2}{3}} s^{\frac{2}{3}}} - \lambda_1(-1) - \lambda_3(-1)\right) \nonumber \\ &=6C_0+\frac{9}{2^{\frac43}}-\frac{3\cdot 2^{\frac13} \rho}{s^{\frac23}}= -\frac{9}{2^{\frac73}} - \frac{3 \rho}{2^{\frac23} s^{\frac23}}. \label{lambda-123-value}\end{aligned}$$ This means the exponential terms on the right hand side of are exponentially small as $s \to + \infty$ for $\rho$ in any compact subset of $\mathbb{R}$. Next, it is readily seen from , and that, for an arbitrary $3\times 3$ matrix $ M = (m_{ij})_{i,j=1}^3$, $$\label{eq: B-1MB} \begin{split} &\lim_{z \to -1} \left(\mathscr{B}^{-1}(s^{\frac{8}{3}}f(z)) M \mathscr{B}(s^{\frac{8}{3}}f(z)) \right)_{21} = m_{21}, \\ &\lim_{z \to -1} \left(\mathscr{B}^{-1}(s^{\frac{8}{3}}f(z)) M \mathscr{B}(s^{\frac{8}{3}}f(z)) \right)_{31} = m_{31}. \end{split}$$ This, together with and , implies that $$\begin{aligned} &\lim_{z\to -1}\left(\mathscr{A}^{-1}(z) \mathscr{B}^{-1}(s^{\frac{8}{3}}f(z)) E^{-1}(z) E'(z) \mathscr{B}(s^{\frac{8}{3}}f(z)) \mathscr{A}(z)\right)_{21} = {\mathcal{O}}(e^{-( 9 \cdot 2^{-\frac{10}{3}} - \delta ) s^{\frac43}}), \nonumber \\ & \lim_{z\to -1}\left( \mathscr{A}^{-1}(z) \mathscr{B}^{-1}(s^{\frac{8}{3}}f(z)) E^{-1}(z) R^{-1}(z) R'(z) E(z) \mathscr{B}(s^{\frac{8}{3}}f(z)) \mathscr{A}(z)\right)_{21} \\ & \hspace{10cm} ={\mathcal{O}}(e^{-( 9 \cdot 2^{-\frac{10}{3}} - \delta ) s^{\frac43}}) \nonumber\end{aligned}$$ for an arbitrary small constant $\delta>0$, i.e., the above two terms are exponentially small as $s \to +\infty$. Similarly, we also have $$\begin{gathered} \label{eq:3term1} \lim_{z\to -1}\left(\mathscr{A}^{-1}(z) \mathscr{B}^{-1}(s^{\frac{8}{3}}f(z)) E^{-1}(z) E'(z) \mathscr{B}(s^{\frac{8}{3}}f(z)) \mathscr{A}(z)\right)_{31} \\ =\lim_{z \to -1}\left( E^{-1}(z) E'(z) \right)_{31} + {\mathcal{O}}(e^{-( 9 \cdot 2^{-\frac{10}{3}} - \delta ) s^{\frac43}}),\end{gathered}$$ $$\begin{gathered} \label{eq:4term1} \lim_{z\to -1}\left( \mathscr{A}^{-1}(z) \mathscr{B}^{-1}(s^{\frac{8}{3}}f(z)) E^{-1}(z) R^{-1}(z) R'(z) E(z) \mathscr{B}(s^{\frac{8}{3}}f(z)) \mathscr{A}(z) \right)_{31} \\ =\lim_{z \to -1}\left( E^{-1}(z) R^{-1}(z) R'(z) E(z) \right)_{31}+ {\mathcal{O}}(e^{-( 9 \cdot 2^{-\frac{10}{3}} - \delta ) s^{\frac43}}).\end{gathered}$$ For the limit of the right hand side on of , it follows from the explicit expressions of $E(-1)$ and $E'(-1)$ given in and that $$\label{Term3-final} \lim_{z \to -1}\left(E^{-1}(z) E'(z) \right)_{31} = 0.$$ For the limit of the right hand side on of , we note from that, as $s\to +\infty$, $$E^{-1}(z) R^{-1}(z) R'(z) E(z) = E^{-1}(z) \left( \frac{R_1'(z)}{s^{\frac43}} + {\mathcal{O}}(s^{-\frac{8}{3}}) \right) E(z).$$ In view of the explicit expressions of $E(-1)$ and $R_1'(-1)$ in and , it is then readily seen that $$\label{Term4-final} \lim_{z \to -1}\left( E^{-1}(z) R^{-1}(z) R'(z) E(z) \right)_{31} = \pi i \frac{C_1 - 12 C_3 }{32 C_1} + {\mathcal{O}}(s^{-\frac{4}{3}}), \quad s\to +\infty,$$ where the constants $C_i$, $i=1,3$, are given in . Finally, by , , , , , , and , we obtain $$\begin{aligned} \frac{\partial}{\partial s} F(s;\rho)&=-\frac{1}{ \pi i s} \lim_{z \to -1} \left[\left(T^{-1}(z)T'(z)\right)_{21}+\left(T^{-1}(z) T'(z)\right)_{31}\right] \nonumber \\ & = -\frac{1}{s} \left( \frac{1}{2}s^{\frac{8}{3}} C_1^2 + \frac{C_1 - 12 C_3 }{32 C_1} + {\mathcal{O}}(s^{-\frac{4}{3}}) \right) \nonumber \\ &=-\frac{3 s^{\frac53}}{2^{\frac83}} + \frac{\rho s}{2} - \frac{\rho^2 s^{\frac13}}{3 \cdot 2^{\frac43}} - \frac{2}{9 s} + {\mathcal{O}}(s^{-\frac{5}{3}}).\end{aligned}$$ as $s \to +\infty$. Integrating the above formula gives us $$\label{main-F-asy-2} F(s;\rho)= -\frac{9 s^{\frac83}}{2^{\frac{17}3}} + \frac{\rho s^2}{4} - \frac{\rho^2 s^{{\frac43}}}{2^{{\frac{10}3}}} - \frac{2}{9} \ln s + \kappa(\rho) + {\mathcal{O}}(s^{-\frac{2}{3}}),$$ uniformly for $\rho$ in any compact subset of $\mathbb{R}$, where $\kappa(\rho)$ is the constant of integration that might be dependent on $\rho$. To find more information about $\kappa(\rho)$, we come to $\frac{\partial}{\partial \rho} F(s;\rho)$. From and , we have $$\label{eq:derivativeinRhoS} \frac{\partial}{\partial \rho} F(s;\rho) = -\frac{s^{\frac{2}{3}}}{2} \left( ( \mathsf{S}_1 )_{12} + ( \mathsf{S}_1 )_{23} \right) - D_1 s^2 + \frac{\rho^3}{54},$$ where $\mathsf{S}_1$ and $D_1$ are given in and , respectively. Recall that (see ) $$S(z) = R(z) N(z), \qquad z\in \mathbb{C}\setminus \{U(-1,\delta) \cup U(1,\delta) \cup \Sigma_S\},$$ it follows from the large $z$ behaviors of $S(z)$, $N(z)$ and $R(z)$ in , and that $$\mathsf{S}_1 = \mathsf{R}_1 + \mathsf{N}_1,$$ where $\mathsf{R}_1$ and $\mathsf{N}_1$ are the coefficients of $1/z$ for $R(z)$ and $N(z)$ at infinity. A combination of the above formula and the expressions of $\mathsf{R}_1$ and $\mathsf{N}_1$ in and gives us $$( \mathsf{S}_1 )_{12} + ( \mathsf{S}_1 )_{23} = {\mathcal{O}}(s^{-\frac{4}{3}}), \qquad s \to +\infty.$$ This, together with and $D_1$ given in , further implies $$\frac{\partial}{\partial \rho} F(s;\rho) = \frac{s^2}{4} - \frac{\rho s^{\frac{4}{3}}}{2^{\frac{7}{3}}} + \frac{\rho^3}{54} + {\mathcal{O}}(s^{-\frac{2}{3}}).$$ Comparing this approximation with the asymptotics of $F(s;\rho)$ given in , it is easily seen that $$\label{eq:kapparho} \kappa'(\rho) = \frac{\rho^3}{54} \Longrightarrow \kappa(\rho) = \frac{\rho^4}{216} + C,$$ where $C$ is an undetermined constant independent of $s$ and $\rho$. Inserting into leads to our final asymptotic result . This completes the proof of Theorem \[main-thm\]. Bessel parametrix {#Append: BP} ================= Define $$\label{Phi-B-solution} \Phi^{({\mathrm{Bes}})}_{\alpha}(z)=\left\{ \begin{array}{ll} \begin{pmatrix} I_{\alpha}(z^{1/2}) & \frac{i}{\pi}K_{\alpha}(z^{1/2}) \\ \pi iz^{1/2}I'_{\alpha}(z^{1/2}) & -z^{1/2}K_{\alpha}'(z^{1/2}) \end{pmatrix}, & z\in \texttt{I},\\ [4mm] \begin{pmatrix} I_{\alpha}(z^{1/2}) & \frac{i}{\pi}K_{\alpha}(z^{1/2}) \\ \pi iz^{1/2}I'_{\alpha}(z^{1/2}) & -z^{1/2}K_{\alpha}'(z^{1/2}) \end{pmatrix}\begin{pmatrix} 1 & 0\\ -e^{\alpha \pi i} & 1 \end{pmatrix}, & z\in \texttt{II}, \\ [4mm] \begin{pmatrix} I_{\alpha}(z^{1/2}) & \frac{i}{\pi}K_{\alpha}(z^{1/2}) \\ \pi iz^{1/2}I'_{\alpha}(z^{1/2}) & -z^{1/2}K_{\alpha}'(z^{1/2}) \end{pmatrix}\begin{pmatrix} 1 & 0 \\ e^{-\alpha\pi i} & 1 \end{pmatrix}, & z\in \texttt{III}, \end{array} \right.$$ where $I_\alpha(z)$ and $K_\alpha(z)$ denote the modified Bessel functions (cf. [@DLMF Chapter 10]), the principle branch is taken for $z^{1/2}$ and the regions $\texttt{I-III}$ are illustrated in Fig. \[fig:jumps-Phi-B\]. By [@KMVV], we have that $\Phi^{({\mathrm{Bes}})}_{\alpha}(z)$ satisfies the RH problem below. (80,70)(-5,2) (40,40)[(-2,-3)[16]{}]{} (40,40)[(-2,3)[16]{}]{} (40,40)[(-1,0)[30]{}]{} (30,55)[(2,-3)[1]{}]{} (30,40)[(1,0)[1]{}]{} (30,25)[(2,3)[1]{}]{} (39,36.3)[$0$]{} (20,11)[$\widehat \Gamma_3$]{} (20,68)[$\widehat \Gamma_1$]{} (3,40)[$\widehat \Gamma_2$]{} (52,39)[$\texttt{I}$]{} (25,44)[$\texttt{II}$]{} (25,34)[$\texttt{III}$]{} (40,40) RH problem for $\Phi^{({\mathrm{Bes}})}_\alpha$ {#rh-problem-for-phimathrmbes_alpha .unnumbered} ----------------------------------------------- \(a) $\Phi^{({\mathrm{Bes}})}_{\alpha}(z)$ is defined and analytic in $\mathbb{C}\setminus \{\cup^3_{j=1}\widehat \Gamma_j\cup\{0\}\}$, where the contours $\widehat \Gamma_j$, $j=1,2,3$, are indicated in Figure \[fig:jumps-Phi-B\]. \(b) $\Phi^{({\mathrm{Bes}})}_{\alpha}(z)$ satisfies the jump condition $$\label{Bessel-jump} \Phi^{({\mathrm{Bes}})}_{\alpha,+}(z)=\Phi^{({\mathrm{Bes}})}_{\alpha,-}(z) \left\{ \begin{array}{ll} \begin{pmatrix} 1 & 0\\ e^{\alpha \pi i} & 1 \end{pmatrix}, & \qquad z \in \widehat \Gamma_1, \\ \begin{pmatrix} 0 & 1\\ -1 & 0 \end{pmatrix}, & \qquad z \in \widehat \Gamma_2, \\ \begin{pmatrix} 1 & 0 \\ e^{-\alpha\pi i} & 1 \\ \end{pmatrix}, & \qquad z \in \widehat \Gamma_3. \end{array} \right .$$ \(c) $\Phi^{({\mathrm{Bes}})}_{\alpha}(z)$ satisfies the following asymptotic behavior at infinity: $$\begin{gathered} \label{eq:Besl-infty} \Phi^{({\mathrm{Bes}})}_{\alpha}(z)= \frac{( \pi^2 z )^{-\frac{1}{4} \sigma_3}}{\sqrt{2}} \begin{pmatrix} 1 & i \\ i & 1 \end{pmatrix} \\ \times \left( I + \frac{1}{8 z^{1/2}} \begin{pmatrix} -1-4\alpha^2 & -2i \\ -2i & 1+4\alpha^2 \end{pmatrix} + {\mathcal{O}}\left(\frac{1}{z}\right) \right)e^{z^{1/2}\sigma_3},\quad z\to \infty. \end{gathered}$$ \(d) $\Phi^{({\mathrm{Bes}})}_{\alpha}(z)$ satisfies the following asymptotic behaviors near the origin: If $\alpha<0$, $$\Phi^{({\mathrm{Bes}})}_{\alpha}(z)= {\mathcal{O}}\begin{pmatrix} |z|^{\alpha/2} & |z|^{\alpha/2} \\ |z|^{\alpha/2} & |z|^{\alpha/2} \end{pmatrix}, \qquad \textrm{as $z \to 0$}.$$ If $\alpha=0$, $$\Phi^{({\mathrm{Bes}})}_{\alpha}(z)= {\mathcal{O}}\begin{pmatrix} \ln|z| & \ln|z| \\ \ln|z| & \ln|z| \end{pmatrix}, \qquad \textrm{as $z \to 0$}.$$ If $\alpha>0$, $$\Phi^{({\mathrm{Bes}})}_{\alpha}(z)= \left\{ \begin{array}{ll} {\mathcal{O}}\begin{pmatrix} |z|^{\alpha/2} & |z|^{-\alpha/2} \\ |z|^{\alpha/2} & |z|^{-\alpha/2} \end{pmatrix}, & \hbox{as $z \to 0$ and $z\in \texttt{I}$,} \\ {\mathcal{O}}\begin{pmatrix} |z|^{-\alpha/2} & |z|^{-\alpha/2} \\ |z|^{-\alpha/2} & |z|^{-\alpha/2} \end{pmatrix}, & \hbox{as $z \to 0$ and $z\in \texttt{II}\cup \texttt{III} $.} \end{array} \right.$$ Acknowledgements {#acknowledgements .unnumbered} ================ Dan Dai was partially supported by grants from the City University of Hong Kong (Project No. 7005032, 7005252), and grants from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 11303016). 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--- author: - 'J. Gosset[^1], A. Baldisseri, H. Borel, F. Staley, Y. Terrien' date: | Received: 7 June 1999 / Revised version: 13 September 1999 /\ Published online: 3 February 2000 – © Springer-Verlag 2000 title: | Another look at anomalous $J/\Psi$ suppression\ in ${\mathrm {Pb + Pb}}$ collisions at $P/A = 158\,{\mathrm {GeV}}/c$ --- Introduction {#sec:intro} ============ Very interesting results have been obtained recently by the NA50 experiment at CERN concerning $J/\Psi$ production in ${\mathrm {Pb + Pb}}$ collisions at $P/A= 158$GeV/$c$ [@ABR97; @ABR97a; @RAM98; @ROM98; @ABR99]. In most central collisions, the $J/\Psi$ events are significantly suppressed with respect to what is expected from normal nuclear absorption as measured in lighter systems [@ABR97a; @RAM98; @ROM98; @ABR99]. According to theoretical predictions made more than ten years ago by Matsui and Satz [@MAT86], this anomalous $J/\Psi$ suppression could be a sign of the awaited formation of a quark–gluon plasma in nucleus–nucleus collisions at very high energy. The deficiencies of customary data presentations, using the ratio between $J/\Psi$ and Drell–Yan events or the differential cross section for $J/\Psi$ production with respect to some measured centrality variable, will be stressed first. A new data presentation [@GOS99] will then be proposed, which removes the previous deficiencies with the help of one key additional ingredient, the inclusive differential cross section with respect to the centrality variable. For any process “p”, the yield of “p” events per nucleus–nucleus collision is plotted as a function of an estimated squared impact parameter. This new presentation will be applied to NA50 results from their 1995 data taking, available in a thesis [@BEL97]. Implications of this new presentation will also be discussed. Finally, conclusions will be drawn in the perspective of RHIC and LHC experiments on nucleus–nucleus collisions at very high energy. Usual data presentation {#sec:usual} ======================= In the usual presentation of NA50 results concerning $J/\Psi$ production in nucleus–nucleus collisions [@ABR97; @ABR97a; @RAM98; @ROM98], the ratio between $J/\Psi$ and Drell–Yan events is plotted as a function of the transverse energy $E_{\mathrm {T}}$ measured in an electromagnetic calorimeter. This centrality variable is aimed primarily at sorting out all events according to the impact parameter of the collision. The Drell–Yan process, supposed to be insensitive to nuclear matter effects, is indeed a good reference for normalizing $J/\Psi$ events from the physics point of view. Moreover, some systematic effects cancel in such a ratio. However, when one sees an interesting feature in a ratio, it is not always obvious to know whether it is due to the numerator or the denominator. More importantly, since the Drell–Yan continuum is much less populated than the $J/\Psi$ peak in the dimuon mass spectrum, the statistical uncertainty on the ratio comes essentially from the denominator. It is between 5 and 10 times larger – 5 for central collisions, 10 for peripheral ones – than the contribution from the number of events in the $J/\Psi$ peak. In other words, one would need between 25 and 100 times less running time, all other conditions staying equal, to get a given relative statistical uncertainty on $J/\Psi$ production from the number of events in the $J/\Psi$ peak than from the ratio between $J/\Psi$ and Drell–Yan events. This is a first deficiency of the usual presentation. One would like to use another quantity for normalizing $J/\Psi$ events without losing so much in statistical accuracy. The ratio between $J/\Psi$ and Drell–Yan events is obtained from the differential $E_{\mathrm {T}}$ distributions of cross section for both classes of events, which are the basic experimental data one has to start with. These raw experimental data, $d \sigma_{\mathrm {p}} / d E_{\mathrm {T}}$ and $E_{\mathrm {T}}$, where “p” stands for either $J/\Psi$ or Drell–Yan process, do not have a very direct physical meaning. From the increase or the decrease of $\mathrm {d} \sigma_{\mathrm {p}}/\mathrm {d}_{\mathrm {T}} $ as a function of $E_{\mathrm {T}}$, one cannot even infer whether the production of “p” events increases or decreases with the centrality of the collision. In particular, the decrease of $\mathrm {d} \sigma_{\mathrm {p}} / \mathrm {d} E_{\mathrm {T}}$ at high $E_{\mathrm {T}}$ for any “p” process simply reflects the fact that, for most central collisions, there is a maximum value of $E_{\mathrm {T}}$ beyond which there is no more cross section. $E_{\mathrm {T}}$ is surely increasing with centrality, but at which rate? The answer to this question is needed if one wants to go from $E_{\mathrm {T}}$ to a more direct centrality variable like the impact parameter. For all these reasons, ${\mathrm {d}} \sigma_{\mathrm {p}} / {\mathrm {d}} E_{\mathrm {T}}$ and $E_{\mathrm {T}}$ are rather difficult to understand and compare directly with simple models. This is the second deficiency of the usual presentation. One would like to find other experimental quantities, not too far from $\mathrm {d} \sigma_{\mathrm {p}} / \mathrm {d} E_{\mathrm {T}}$ and $E_{\mathrm {T}}$, which would have a more direct physical meaning, from which one could directly say something on the variation of the production of “p” events with centrality, and which could be compared directly with simple models. New data presentation {#sec:new} ===================== One key quantity that could be used to remove the above-mentioned deficiencies is the inclusive distribution of the cross section for the centrality variable, denoted as $C$ hereafter for more generality. This inclusive distribution will be needed in its differential form $\mathrm {d} \sigma_{\mathrm {inc}} / \mathrm {d} C$ for normalizing $\mathrm {d} \sigma_{\mathrm {p}} / \mathrm {d} C$, and in its integral form $\sigma_{\mathrm {inc}}(C)$ for getting an estimated squared impact parameter $(b^2)_{\mathrm {e}} $. When we use $\mathrm {d} \sigma_{\mathrm {p}} / \mathrm {d} C$ we mix the probability to get a given centrality with the probability to get the “p” process at this centrality. This is precisely why the increase or the decrease of $\mathrm {d} \sigma_{\mathrm {p}} / \mathrm {d} C$ as a function of centrality has no straightforward meaning. This distribution $\mathrm {d} \sigma_{\mathrm {p}} / \mathrm {d} C$ is in fact the product of two quantities which themselves have a more direct physical meaning than their product. It can be written as $Y_{\mathrm {p}}\cdot {\mathrm {d}} \sigma_{\mathrm {inc}} / \mathrm {d} C$, where the inclusive distribution ${\mathrm {d}} \sigma_{\mathrm {inc}} / {\mathrm {d}} C$ carries the probability that a nucleus–nucleus collision occurs at a given value $C$ of the centrality variable, and $Y_{\mathrm {p}}$ is the yield of “p” events per nucleus–nucleus collision at this given centrality. This yield $Y_{\mathrm {p}}$, which, being equal to ($\mathrm {d} \sigma_{\mathrm {p}} / \mathrm {d} C)/\-({\mathrm {d}} \sigma_{\mathrm {inc}} / {\mathrm {d}} C$), is a well-defined physical quantity. For copiously produced particles it is simply their average multiplicity per nucleus–nucleus collision at a given centrality. As it is a ratio, it should be insensitive to some systematic uncertainties. Its variation as a function of $C$ should accurately reflect whether the production of “p” events increases or decreases with the centrality of the collision. It should not be subject to any artificial decrease for most central collisions. For both $J/\Psi$ and Drell–Yan processes, one expects that this yield steadily increases towards more central collisions, like the number of nucleon–nucleon collisions they originate from, unless the $J/\Psi$ is very strongly suppressed. The first step in the new data presentation is thus to use the yield $Y_{\mathrm {p}}$ of “p” events per nucleus–nucleus collision instead of ${\mathrm {d}} \sigma_{\mathrm {p}} / {\mathrm {d}} C$. The idea behind tagging a process with a centrality variable in nucleus–nucleus collisions is always to sort out events according to the impact parameter. If a centrality variable $C$ is assumed to vary monotonically as a function of the impact parameter $b$ – and centrality variables are purposely chosen for that reason –, it is very easy to go from $C$ to an estimate of $b$, or more precisely $b^2$. One only has to use the integral inclusive cross section $\sigma_{\mathrm {inc}} (C)$, from most central collisions to any given value of $C$. From the geometrical dependence of the inclusive cross section, ${\mathrm {d}}\sigma_{\mathrm {inc}}=2{\pi}{\cdot}b\cdot {\mathrm {d}} b=\pi\cdot {\mathrm {d}} (b^2)$, one simply gets $\sigma_{\mathrm {inc}}(C)=\pi(b^2)_{\mathrm {e}}$ where $(b^2)_{\mathrm {e}}$ is an estimate of the squared impact parameter corresponding to the value of $C$. There is a one-to-one correspondence between $C$ and $(b^2)_{\mathrm {e}} $. $C$ slices are transformed into $(b^2)_{\mathrm {e}}$ slices with a width proportional to the number of counts in the $C$ slices. For this reason, which is also related to the fact that $\mathrm {d} \sigma_{\mathrm {inc}} / \mathrm {d} b$ goes to zero at zero impact parameter, $(b^2)_{\mathrm {e}}$ seems a better variable than the estimated impact parameter $b_{\mathrm {e}}$. Instead of dividing $\sigma_{\mathrm {inc}}(C)$ by $\pi$, one could divide it by the geometrical cross section $\sigma_{\mathrm {geo}} $ and get a quantity proportional to $(b^2)_{\mathrm {e}}$, but with such a normalization that it varies between 0 and 1 from most central to most peripheral collisions. Another quantity which could be interesting to use for plotting results from different systems would be $(b_{\mathrm {max}}^2-(b^2)_{\mathrm {e}})$, where $b_{\mathrm {max}}^2=\sigma_{\mathrm {geo}}/\pi$. It has the advantage of being correlated, and not anticorrelated, with the centrality, and extends to larger and larger values for larger and larger systems, with the zero value corresponding always to most peripheral collisions. Such a transformation from $\sigma_{\mathrm {inc}}(C)$ to $(b^2)_{\mathrm {e}} $ has been used in more or less details by several experiments in the field of nucleus–nucleus collisions at various incident energies. Its reliability has been discussed thoroughly, and has been checked to be excellent within the framework of the intranuclear cascade model at energies per nucleon around 1GeV [@CAV90]. Model calculations are obviously needed to evaluate the method and to compare the quality factors of various centrality variables [@CUG83], which combine the fluctuations of $C$ at any given $b$ and the variation rate of $C$ with respect to $b$ or $b^2$. For some AGS or SPS experiments, even though the data are plotted as a function of a centrality variable, a scale for the impact parameter estimated along the preceding lines is indicated in parallel [@AGG98; @BAR99]. Finally, the second step in the new data presentation consists in replacing the measured centrality variable $C$ with an estimate of the squared impact parameter, $(b^2)_{\mathrm {e}} = \sigma_{\mathrm {inc}}(C)/\pi$. After applying both steps one gets the yield $Y_{\mathrm {p}}$ of “p” events per nucleus–nucleus collision as a function of the estimated squared impact parameter. Both quantities are raw experimental data and have a clear physical meaning. As compared with the usual ratio between $J/\Psi$ and Drell–Yan events, there is a huge gain in statistical accuracy for $J/\Psi$. With this new data presentation one can consider any process independently of all others, with the best statistics available for each of them. In the same way as the integral of $\mathrm {d} \sigma_{\mathrm {p}} / \mathrm {d} C$ as a function of $C$ is the total cross section for the “p” process, the integral of $Y_{\mathrm {p}}$ as a function of $(b^2)_{\mathrm {e}}$ is equal to the total cross section for the “p” process divided by $\pi$. There is no loss of information in going from the centrality variable $C$ to the estimated squared impact parameter $(b^2)_{\mathrm {e}}$. The limits of the slices used for looking at the variation with centrality have simply to be specified for both $C$ and $(b^2)_{\mathrm {e}}$. The main requirement for this new presentation is a good inclusive centrality distribution, with high enough statistical accuracy and proper corrections for efficiency and empty-target contribution. The yield $Y_{\mathrm {p}}$ being the ratio of cross sections, part of systematic uncertainties are removed if inclusive measurements are taken simultaneously with the measurements of the “p” process. The normalization uncertainty of the inclusive cross section has an effect on the abscissa rather than on the ordinate, which may be unusual but does not bring about any practical problem. Such a presentation provides an excellent starting point for comparisons between experiment and theory, and also between experimental results themselves. Since there is no explicit appearance of $C$ in the new presentation, results obtained with various centrality variables should be identical as long as these variables sample the impact parameter the same way. Anyway, such a comparison could help to check systematic uncertainties. For comparisons between experiment and theory, it is straightforward to compare the measured yields as a function of the estimated $b^2$ with the calculated ones as a function of the real $b^2$. This is particularly interesting for a quick comparison with simple models. However, in order to take into account the fluctuations of any centrality variable as a function of the impact parameter, a comparison with better quality would result from using the same procedure of $b^2$ estimation for both experiment and theory, even if inclusive cross sections do not agree within a high degree of accuracy [@CAV90], or from unfolding the experimental results from these fluctuations. It is also clear that this whole presentation could be applied advantageously to other processes than $J/\Psi$ and Drell–Yan production. Application {#sec:appl} =========== In a thesis by Bellaiche [@BEL97] from the NA50 collaboration, all necessary pieces of information are available from the 1995 data taking for applying this new data presentation. They have not been published as such. The results presented below thus have to be considered with care. They are only indicative, and they are to be used simply as an illustration of the advantages inherent to the new data presentation. Basic experimental data are the differential $E_{\mathrm {T}}$ distributions of the cross section for the $J/\Psi$ and Drell–Yan events (Fig. \[fig:basic\_data\_PsiDY\]). The key additional ingredient is the differential $E_{\mathrm {T}}$ distribution of the inclusive cross section, by the integration of which one can estimate the squared impact parameter $(b^2)_{\mathrm {e}}$ for each value of $E_{\mathrm {T}}$ (Fig. \[fig:basic\_data\_inc\]). Uncertainties on $\mathrm {d} \sigma_{\mathrm {inc}} / \mathrm {d} E_{\mathrm {T}}$ have been neglected in the following. The inclusive cross section is only available with arbitrary units in the thesis. A normalization factor had to be introduced to get the $(b^2)_{\mathrm {e}} $ values in units of fm$^2$. It has been adjusted in such a way that the dependence of $(b^2)_{\mathrm {e}} $ upon $E_{\mathrm {T}}$ agrees with the correlation between the average values of $E_{\mathrm {T}}$ and $b$ listed in NA50 publications [@ABR97; @ABR99] for successive $E_{\mathrm {T}}$ slices, as fitted on the basis of a Glauber model calculation. These average values are also shown in the bottom part of Fig. \[fig:basic\_data\_inc\]. $E_{\mathrm {T}}$ values from [@ABR99] have been divided by 0.74 to take into account the different $E_{\mathrm {T}}$ scales used in the NA50 publications. After the first step, i.e. the normalization of the $J/\Psi$ and Drell–Yan cross sections to the inclusive one, one gets (Fig. \[fig:first\_step\]) the yields of $J/\Psi$ and Drell–Yan events per Pb–Pb collision as a function of $E_{\mathrm {T}}$. Both yields increase with $E_{\mathrm {T}}$, without any artificial decrease at large $E_{\mathrm {T}}$. Whereas this increase is rather steady for Drell–Yan events, there is a change of behaviour for $J/\Psi$ at about 40GeV, an $E_{\mathrm {T}}$ value beyond which the increase is definitely slower for $J/\Psi$ than for Drell–Yan events. This is an indication for $J/\Psi$ suppression in central collisions, relative to Drell–Yan events. After the second step, i.e. replacing $E_{\mathrm {T}}$ by $\sigma_{\mathrm {inc}}(E_{\mathrm {T}})/\pi$, one gets (Fig. \[fig:second\_step\]) the yields of $J/\Psi$ and Drell–Yan events per Pb–Pb collision as a function of $(b^2)_{\mathrm {e}}$, the squared impact parameter estimated from the $E_{\mathrm {T}}$ inclusive cross section. Slices with almost constant width in $(b^2)_{\mathrm {e}}$ have been used. The same remarks could be made as from Fig. \[fig:first\_step\] after the first step. The $E_{\mathrm {T}}$ value of 40GeV for the change of behaviour of the $J/\Psi$ yield is changed into a $(b^2)_{\mathrm {e}}$ value of 80fm$^2$. Points with large error bars at large $E_{\mathrm {T}}$ in Fig. \[fig:first\_step\] are all contained in the point at the smallest value of $(b^2)_{\mathrm {e}} $ in Fig. \[fig:second\_step\]. The limits of the yields for most central collisions are more easily readable from Fig. \[fig:second\_step\] than from Fig. \[fig:first\_step\]. They could be directly compared with the $J/\Psi$ and Drell–Yan yields in p–p collisions – $J/\Psi$ and Drell–Yan cross sections divided by the total inelastic p–p cross section – times the number of nucleon–nucleon collisions in most central Pb–Pb collisions from a Glauber model calculation. From integration of the yields in Fig. \[fig:second\_step\] one can get the total cross sections for $J/\Psi$ and Drell–Yan production divided by $\pi$. A comparison is also made in Fig. \[fig:second\_step\] with a model calculation à la Blaizot and Ollitrault [@BLA96]. The yield of Drell–Yan events per Pb–Pb collision as a function of $(b^2)_{\mathrm {e}}$ is compared, within a scale factor, to the number of nucleon–nucleon collisions calculated in the Glauber model as a function of the real $b^2$, without taking into account the fluctuations between the estimated and actual $b^2$. The agreement is reasonable. For $J/\Psi$ events this number of nucleon–nucleon collisions is multiplied by two correction factors for absorption. The first one corresponds to normal $J/\Psi$ absorption in nuclear matter with some cross section $\sigma_{\mathrm {abs}}$. The second one is intended for simulating complete $J/\Psi$ suppression due to quark–gluon plasma formation. It goes down from one to zero when nucleon–nucleon collisions producing $J/\Psi$ occur in a tube of nuclear matter with nucleon density per unit area larger than a critical value $\rho_{\mathrm {crit}}$. With $\sigma_{\mathrm {abs}}=6.0$mb and $\rho_{\mathrm {crit}}=2.9$fm$^{-2}$, the model reasonably accounts for the experiment, in particular for the clear change of behaviour at an impact parameter of about 9fm. Finally, very accurate results are obtained for $J/\Psi$ production which can be compared easily with simple models. The onset of the anomalous $J/\Psi$ suppression can be looked at with much better accuracy than on the basis of the usual $J/\Psi$ over Drell–Yan ratio. However, we want to recall the word of caution from the beginning of this section. Definite conclusions about $J/\Psi$ anomalous suppression need to be drawn from official data. This is also why there was no attempt to calculate error bars for the values of $\sigma_{\mathrm {abs}}$ and $\rho_{\mathrm {crit}}$. Moreover the Drell–Yan production remains an essential result. One has to check its normal behaviour within its inherently limited accuracy. Discussion {#sec:disc} ========== One idea from this new data presentation, the normalization of $J/\Psi$ events to the inclusive, or minimum bias, $E_{\mathrm {T}}$ distribution, has been used recently by the NA50 collaboration [@ABR99], making the most of the whole statistics available for $J/\Psi$ production in their 1996 data taking. However, this new presentation is not used as such, except the first step for Drell–Yan production only. For easy comparison with previously published results, the $J/\Psi$ yield is transformed into a “minimum bias” $J/\Psi$ over Drell–Yan ratio, through a division by a model calculation for the Drell–Yan yield. In order to stick more closely to the raw experimental data, it would be very interesting if the new presentation were to be applied as a whole to these most recent and also to future NA50 data. One would not have to worry anymore because of the different $E_{\mathrm {T}}$ scales used in successive presentations. More importantly, it would be particularly helpful to compare between one another the results obtained with the three centrality variables available in this experiment, namely the transverse energy $E_{\mathrm {T}} $ measured in an electromagnetic calorimeter, the zero-degree energy measured in a hadronic calorimeter, and the multiplicity measured in a silicon detector. Perhaps it would also be possible to get more accurate information on $J/\Psi$ production from older data takings, for example in ${\mathrm {S+U}}$ collisions. Finally, it is interesting to try and quantify the gain brought about by the normalization to the inclusive centrality distribution in the assessment of the anomalous $J/\Psi$ suppression in ${\mathrm {Pb + Pb}}$ collisions. It can be done for example on NA50 results as shown in Fig. 9 from [@ABR99]. In this figure, the “minimum bias” as well as the measured $J/\Psi$ over Drell–Yan ratios are divided by the normal absorption factor and plotted as a function of the mean nuclear path length $L$ (Fig. \[fig:absorption\_band\]). The normal absorption appears as a horizontal line at a constant value of 1, without any information on its uncertainty. In Fig. \[fig:absorption\_band\], an uncertainty band, necessary for a quantitative comparison to ${\mathrm {Pb + Pb}}$ results, has been added around the straight reference line. It has been calculated from the same p nucleus and ${\mathrm {S+U}}$ data as used in [@ABR99], with correct error bars as compared to previous NA50 publications (see note added in proof to [@RAM98]), and taking into account the correlation between the normalization and the slope of the exponential fit. An uncertainty band had already been shown in [@KHA97] but it had been calculated with the old error bars for ${\mathrm {S+U}}$ data and without taking into account the correlation between the normalization and the slope. By chance this uncertainty band was not too much wrong since both effects were roughly compensating for each other. One way to quantify the discrepancy of ${\mathrm {Pb + Pb}}$ data from normal nuclear absorption consists in fitting the points corresponding to most central collisions, i.e. beyond $L=8$fm, with an exponential function of $L$ (Fig. \[fig:absorption\_band\]). The result is an effective additional absorption cross section of 9mb, with uncertainties of 0.6 and 2.6mb depending on whether one uses the “minimum bias” or the measured $J/\Psi$ over Drell–Yan ratio. With respect to the reference absorption cross section of $ 5.8\pm 0.7$mb, the significance of this additional absorption amounts to 9.8 or 3.3 standard deviations, respectively, with a clear advantage to the “minimum bias” ratio, because it uses the normalization to the inclusive centrality distribution. Conclusion and perspectives {#sec:conc} =========================== A new data presentation has been proposed for results from nucleus–nucleus collisions. Its domain of application is not limited to $J/\Psi$ and Drell–Yan production processes which have been chosen for illustration. For any “p” process one ends with its yield per nucleus–nucleus collision as a function of the estimated squared impact parameter. Since the normalization of the yield refers to the most probable, i.e. inclusive, process, one keeps the best statistical accuracy for each process. It seems to be a good way for going as far as possible with raw experimental data, sticking as closely as possible to them while trying to show physical quantities of interest. Could it be the best way to present experimental results concerning nucleus–nucleus collisions before comparison to any model? From the theoretical side, one would like to compare experimental results with results from model calculations on plots using the most relevant variable from the model, for instance the number of participants, the number of nucleon–nucleon collisions, the mean path length in nuclear matter, etc. The proposed data presentation could serve as the common basis before going to any of these plots. For future nucleus–nucleus experiments that will take place at RHIC and LHC, the present work shows that it is possible to study any process independently of all others. The experimental results to be presented for each process have a direct physical meaning and are easily compared with model calculations. The only requirement is the measurement of the inclusive differential cross section with respect to at least one centrality variable used to sort out events according to impact parameter. It is essential that such inclusive measurements be available in experiments to be performed at RHIC and LHC. The authors want to express their thanks to F. Bellaiche for providing them with the values of $J/\Psi$ and Drell–Yan cross sections from his thesis. Discussions with J.-P. Blaizot, J. Hüfner, J.-Y. Ollitrault and H. Satz are gratefully acknowledged. [88.]{} M.C. Abreu et al. (NA50 Collaboration), Phys. Lett. B [**410**]{}, 327 (1997) M.C. Abreu et al. (NA50 Collaboration), Phys. Lett. B [**410**]{}, 337 (1997) L. Ramello for the NA50 Collaboration, Nucl. Phys. A [**638**]{}, 261c (1998) Talk presented by A. 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--- author: - 'T. Corbard' - 'F. Morand' - 'F. Laclare' - 'R. Ikhlef' - 'M. Meftah' date: 'March 31, 2013' title: On the importance of astronomical refraction for modern Solar astrometric measurements --- [Several efforts are currently made from space missions in order to get accurate solar astrometric measurements i.e. to probe the long term variations of solar radius or shape, their link with solar irradiance variations and their influence on earth climate. These space missions use full disk solar imagery. In order to test our ability to perform such measurements from ground on the long term, we need to use similar techniques and instruments simultaneously from ground and space. This should help us to model and understand how the atmosphere affect ground based metrologic measurements. However, using full imagery from ground instead of the traditional astrolabe technique immediatly raise the question of the effect of refraction and how well we can correct from it.]{} [The goal is to study in details the influence of pure astronomical refraction on solar metrologic measurements made from ground-based full disk imagery and to provide the tools for correcting the measurements and estimating the associated uncertainties.]{} [We use both analytical and numerical methods in order to confront commonly or historically used approximations and exact solutions.]{} [We provide the exact formulae for correcting solar radius measurements at any heliographic angle and for any zenith distance. We show that these corrections can be applyed up to $80\degr$ of zenith distance provided that full numerical integration of the refraction integral is used. We also provide estimates of the absolute uncertainties associated with the differential refraction corrections and shows that approximate formulae can be used up to $80\degr$ of zenith distance for computing these uncertainties. For a given instrumental setup and the knowledge of the uncertainties associated with local weather records, this can be used to fix the maximum zenith distance one can observe depending on the required astrometric accuracy. ]{} Introduction ============ Ground based solar astrometric measurements have up to now been based on the so-called equal altitudes method (Débarbat & Guinot [@Debarbat]). They have historically been made from transit instruments or astrolabes. Several instruments, derived from Danjon astrolabe, have been dedicated to solar diameter measurements, as DORaySol experiment (Morand et al. [@Morand]). Observations consist in determining the transit times, through the same equal zenith distance circle, of the two solar limbs which are the extremities of a vertical solar diameter. Accuracy of these measurements is then mainly determined by datation accuracy and not by the optical resolution of the instrument. As the two limbs are observed at equal zenith distances, influence of astronomical refraction is inherently reduced (e.g. Laclare et al. [@Laclare]). Only the small climatic conditions variations (temperature, pressure, relative humidity) between the two crossings, distant from a few minutes of time, can still play a role. The main drawback of this method is that only vertical diameters can be determined. Recent work in the field of solar metrology involve measurements from space using full disk solar images (Dame et al. [@Dame]; Kuhn et al. [@Kuhn]). PICARD-SOL (Meftah et al. [@Meftah]) is a ground based project that was set up at Calern observatory in order to use the same technique simultaneously from ground and space and in order to inter-calibrate the different measurements. Using full disk imagery from ground raise however the question of the influence of astronomical refraction and how well we can correct for it. The effect of atmospheric refraction is to change the true topocentric zenith angle $z^t$ of a celestial object to a lower observed one $z$. The refraction function $R(z)$ is defined by: $$\label{eq:gen} z=z^t-R(z)$$ Alternatively, we may take the true angles as argument and define the associated refraction function $\bar{R}$ by: $$\label{eq:gen_alt} z=z^t-\bar{R}(z^t)$$ If the refraction function $R(z)$ is known, the associated function $\bar{R}(z^t)$ can easily be evaluated for any true zenith distance $z^t$ by solving the non linear equation $x-R(z^t-x) = 0$. fundamental equations for astronomical refraction ================================================= From Snell’s law of refraction applied to a spherical atmosphere, the curvature of a light path is linked to the local refractive index $n$ through the so-called refractive invariant : $$\label{eq:invarient} n \ r \sin(\xi)=\mathrm{constant}$$ where $\xi$ is the local zenith distance i.e. the angle between the light ray and the radius vector $r$ from Earth center. From this, the differential refraction along the light ray is obtained by: $$\label{eq:diff} dR=- \tan \xi \ {dn \over n}$$ In order to find the total amount of refraction at observer position, we can integrate along the full ray path from $n=n_{\mbox{obs}}$ and $\xi=z$ at observer position up to $n=1$ outside the atmosphere. $$\label{eq:int} R= \int_1^{n_{\mathrm{obs}}} \tan \xi \ {dn \over n}$$ This can be done either by direct numerical integration of Eq. (\[eq:int\]) after an appropriate change of variable (Auer & Standish [@Auer]) or by using a full ray-tracing procedure solving the system of coupled differential equations provided by Eq. (\[eq:diff\]) and the differentiation of Eq. (\[eq:invarient\]) (van der Werf [@Werf2003], [@Werf2008]). This, in principle, requires a model of the full atmosphere i.e. temperature, pressure, density etc..at any point through the light path. In the next section we recall why this is in fact not needed if we avoid areas close to the horizon and give some usual approximations of the refraction integral. Approximation to the refraction integral {#sec:approx} ======================================== For zenith distance up to $70\degr$, the refraction integral can be evaluated with good accuracy without any hypothesis about the structure of the atmosphere: it depends only on temperature and pressure at the observer (Oriani’s theorem, see also: Ball [@Ball]; Young [@Young]). This justifies that, over time, a large number of nearly equivalent approximate formulae have been derived that do not require the full knowledge of the structure of the real atmosphere. A development of the refraction integral into semi-convergent series of odd power of $\tan(z)$ is what is commonly found in textbooks (e.g. Ball [@Ball]; Smart [@Smart]; Woolard & Clemence [@Woolard]; Danjon [@Danjon]). An example of this will be given in Sect. \[sec:calern\]. In fact the first two terms of such expansion (up to $\tan^3$) corresponds to what is known as Laplace formulae of which Fletcher ([@Fletcher]) said that *no reasonable theory differs by more than a few thousandths, hundredths, tenths of a second at $z= 60\degr$, $70\degr$, $75\degr$ respectively*. For large zenith distance, $\tan(z)$ power series will diverge at the horizon and are not appropriate. Closed formula valid at low zenith distance and that are finite at the horizon can however still be found (see e.g. Wittmann [@Wittmann]). Assuming an exponential law for the variation of air density with height, it’s possible for instance to derive a formula involving the error function (Fletcher [@Fletcher]; Danjon [@Danjon]). Another example is Cassini’s exact formula for an homogeneous atmosphere model. While physically un-realistic, the model of Cassini, thanks to Oriani’s theorem, gives also excellent results up to at least $70\degr$ of zenith distance while remaining finite down to the horizon (Young [@Young]). For large zenith distances however, Young ([@Young]) have shown that the lowest layers of the atmosphere and especially the lapse rate at observer becomes progressively dominant as one observe closer to the horizon. This therefore should be included in atmospheric models and we can not avoid anymore the full numerical evaluation of the refraction integral. In the following sub-sections we present first in details the refraction model as it was used for reducing solar astrolabe data at Calern observatory, then we give the full error function model from which the Calern model was actually derived and finally we recall Cassini’s formula. In Sect. \[sec:res\], these three approximations will then be compared to full numerical integration of the refraction integral using a standard atmosphere model. Refraction model used at Calern observatory for Solar metrology {#sec:calern} --------------------------------------------------------------- The refraction model that was used for the reduction of astrolabe measurements at Calern observatory is a truncation of the expansion in odd power of $\tan(z)$ (Danjon [@Danjon]). For an observer at geodetic latitude $\varphi$ and altitude $h$ above the reference ellipsoid, the refraction $R$ is obtained as a function of the observed zenith angle, the wavelength ($\lambda$) and local atmospheric conditions i.e. pressure ($P$), absolute temperature ($T$), and relative humidity ($f_h\in[0,1]$) by: $$\begin{aligned} \label{eq:Danjon} R(z,\lambda,P,T,f_h,h,\varphi)&=\alpha(1-\beta) \tan(z)-\alpha(\beta-{\alpha \over 2}) \tan^3(z) \nonumber\\ & +3\alpha\left(\beta-{\alpha \over 2}\right)^2 \tan^5(z) \end{aligned}$$ where $$\alpha(T,P,f_h,\lambda)=n_{\mathrm{obs}}-1$$ is the air refractivity for local atmospheric conditions and the given wavelength, and $$\label{eq:beta} \beta(T,h,\varphi)=\ell(T)/r_c(\varphi,h)$$ is the ratio between the height $\ell$ of the homogeneous atmosphere and the earth radius of curvature $r_c$ at observer position. The homogeneous atmosphere has by definition a constant air density $\rho$ equal to the one at observer position and its height is such that it would give the same pressure as the one recorded at observer position. Note that we do not assume here that the atmosphere is homogeneous, we just use the reduced height that can be obtained for any real atmosphere just from the pressure and density at observer. Assuming furthermore ideal gas law for dry air we have: $$\label{eq:ell} \ell(T)={P \over {\rho \ g}}={P_0 \over {\rho_0 \ g_0}}{T \over T_0},$$ where ${\rho_0=1.293\ \mbox{kg}\,\mbox{m}^{-3}}$ for ${T_0=273.15\,\mbox{K}}$, ${P_0=101325\,\mbox{Pa}}$ and normal gravity ${g_0=9.80665\,\mbox{m}\,\mbox{s}^{-2}}$. The radius of curvature for Calern observatory (${\varphi=43\degr45\arcmin7\arcsec}$, ${h=1323\,\mbox{m}}$) was approximated by the minimum reference ellipsoid curvature at latitude $45\degr$ and sea level (Chollet [@Chollet], see Appendix \[App:curvature\]): $$r_c(45\degr,0)=6367.512\ \mathrm{km}$$ Ambient air refractivity was deduced from the refractive index $n_0(\lambda)$ under standard conditions and the partial pressure of water vapor $p$ by applying the formula recommended by the first resolution of the 13th General Assembly of the International Union of Geodesy and Geophysics (IUGG [@IUGG63]; Baldini [@Baldini]). After conversion to Pa (Pascal) as the pressure unit, the equation becomes: $$\label{eq:alpha} \alpha(T,P,f_h,\lambda)={T_0 \over T} \left\{(n_0(\lambda)-1) {P \over P_0} - 4.13\, 10^{-10}\ p(f_h,T)\right\}$$ Refractivity under standard condition (sea level, ${T=T_0}$, ${P=P_0}$, $0\%$ humidity, $0.03\%$ of carbon dioxide) was taken from the work of Barrel and Sears ([@Barrel]): $$\label{eq:Barrel} n_0(\lambda)-1=\left\{2876.04+{16.288 \over (10^6\lambda)^2}+{0.136 \over (10^6\lambda)^4}\right\}\ 10^{-7}.$$ Partial pressure of water vapor for the current temperature and relative humidity was deduced from a fit of water vapor pressure data published by the Bureau Des Longitudes ([@BDL]) for temperatures between $-15\ {\degr}\mbox{C}$ and $+25\ {\degr}\mbox{C}$. The resulting equation, converted to Pa, is (Chollet [@Chollet]): $$\label{eq:water} p(f_h,T)=f_h\ 6.1075\, 10^2\ e^{7.292\, 10^{-2}(T-T_0)-2.84\, 10^{-4}(T-T_0)^2}$$ Finally, we note that local atmospheric pressure $P$ was measured from the height $H$ (in mm) of a mercurial barometer and its temperature $\theta$ (in $\degr\mbox{C}$). Taking into account corrections for local gravity (latitude and altitude) and for temperature (through the volume thermal expansion of mercury and the coefficient of linear thermal expansion of the tube), $P$ was obtained by[^1] (see Appendix \[App:baro\]): $$\label{eq:baro} P=H\left\{1-2.64\, 10^{-3}\cos(2\varphi)-1.96\, 10^{-7} h - 1.63\, 10^{-4}\ \theta\right\}$$ Error function formula ---------------------- In fact, in Eq. (\[eq:Danjon\]), only the first two terms which correspond to Laplace formula can be found without any hypothesis on the real atmosphere (only the reduced height $\ell$ and the refractivity at observer are needed). The term in $\tan^5$ comes from an additional assumption, namely the fact that air density follows an exponential decrease with height (actually with a well chosen variable which vary almost linearly with height, see Danjon ([@Danjon]), Fletcher ([@Fletcher])). This leads to the following equation: $$\label{eq:erfc} R=\alpha \left({{2-\alpha}\over {\sqrt{2\beta-\alpha}}}\right)\sin(z)\ \Psi\left({{\cos(z)}\over{\sqrt{2\beta-\alpha}}}\right)$$ with : $$\Psi(x)=e^{x^2}\int_x^{\infty}e^{-t^2}dt={{\sqrt{\pi}}\over{2}}e^{x^2}\left(1-\mathrm{erf}(x)\right)$$ from which Eq. (\[eq:Danjon\]) was derived by keeping only the three first terms of its asymptotic expansion. Cassini ------- By comparing the results with a full integration method, Young ([@Young]) shows the superiority of Cassini’s formula over the series-expansion approach and advocates its use by astronomers. Cassini assumed an homogeneous atmosphere for which he obtained the exact formula: $$\label{eq:Cassini} R=\mathrm{asin}\left({{n_{obs}\ r_c\sin(z)}\over{r_c+\ell}}\right)-\mathrm{asin}\left({{r_c\sin(z)}\over{r_c+\ell}}\right)$$ Again, it can be shown (Ball [@Ball]) that expanding this formula also leads to the to first two terms of Eq. (\[eq:Danjon\]) i.e. to Laplace formula. On the observed shape of the Sun due to pure astronomical refraction ==================================================================== ![Geometry for the solar shape due to astronomical refraction.The dashed circle represents the true solar disk of centre C$^t$ and radius $R_{\sun}$ while the elliptical shape (full line) represents the observed Sun of centre C. The point at the top represents observer’s zenith. []{data-label="Fig:Dessin"}](dessin.eps){width=".3\textwidth"} In this section, we assume that the Sun is a perfect sphere of angular radius $R_{\sun}$ at 1 AU and that there is no other effect affecting its observed shape than astronomical refraction defined by Eq. (\[eq:gen\_alt\]). In the horizontal coordinate system (zenith distance-azimuth), we note ($z_{\sun}^t$, $A_{\sun}$) the true position of the Sun centre ($C^t$) observed at zenith angle $z_{\sun}$; ($z^{t}$, $A$) the true position of a point ($L^t$) of the solar limb observed at zenith angle $z$; ${\delta z=z-z_{\sun}}$ and ${\delta A=A-A_{\sun}}$. Figure \[Fig:Dessin\] shows all the angles involved. Each true limb point position can be defined by the angle $\psi^t\in[-\pi,\pi[$ between the direction $\overline{C^tL^t}$ and the vertical circle. Similarly, each observed limb point can be located by the angle $\psi\in[-\pi,\pi[$ between the observed direction $\overline{CL}$ and the vertical circle. However, because the figure is symmetric with respect to the vertical circle, we consider only the interval $[0,\pi]$ for $\psi$ and $\psi^t$ in the following. For observation with an Alt-Az mount this would correspond directly to the angle with one of the CCD axis. For an equatorial mount, one CCD axis is aligned with the hour circle passing through the celestial poles and the Sun and therefore the vertical circle can be materialized on the solar image by computing first the parallactic angle between these two circles. If ${d(\psi)=\bar{d}(\psi^t)}$ is the angular distance between the observed position of the Sun centre and the observed limb points, we define by: $$\label{eq:ddef} <d>=\left[{1 \over {\pi}}\int_0^{\pi}d(\psi)^2d\psi\right]^{1/2}=\left[{1 \over {\pi}}\int_0^{\pi}\bar{d}(\psi^t)^2d\psi^t\right]^{1/2}$$ the geometric mean radius of the observed Sun. The horizontal and vertical angular extent of the observed Sun are noted $D_{h}$ and $D_{v}$ respectively and, the flattening is given by: $$\label{eq:flatdef} f={{D_{h} - D_{v}} \over D_{h}}$$ Following Mignard ([@Mignard]), we define the magnification $\Gamma$ as the ratio between the vertical size of the image ($\delta z$) of a small object to its true size ($\delta z^t$). From Eqs. (\[eq:gen\]) and (\[eq:gen\_alt\]), we have: $$\label{eq:gamma} \Gamma={{dz}\over{dz^t}}=1-{{d\bar{R}}\over{dz^t}}=\left({{dz^t}\over{dz}}\right)^{-1}=\left({1+{{dR}\over{dz}}}\right)^{-1}$$ The distorsion $\Delta$ is then defined as the rate of change of the magnification: $$\label{eq:delta} \Delta={{d\Gamma}\over{dz}}=-\Gamma^2{{d^2R}\over{dz^2}}=-{1\over\Gamma}{{d^2\bar{R}}\over{d{z^t}^2}}$$ Approximate formulae for all zenith angles ------------------------------------------ Any limb point true position can be located by its projections on the vertical circle passing through the true Sun centre, and on the great circle perpendicular to this vertical circle passing through the limb point (see Fig. \[Fig:Dessin\]). Because all the angles involved are small, we can write: $$\begin{aligned} x^t&=&R_{\sun}\cos(\psi^t)\\ y^t&=&R_{\sun}\sin(\psi^t)\end{aligned}$$ and: $$\label{eq:circle} {x^t}^2+{y^t}^2=R_{\sun}^2$$ By looking at the expression of the observed values $x$ and $y$ of these projections, one can obtain an approximate formula for the observed shape of the Sun. The projection $x^t$ on the vertical circle can be approximated by keeping the two first terms of a Taylor expansion of the refraction: $$\label{eq:xt} x^t\simeq z^t-z_{\sun}^t=\delta z+R(z)-R(z_{\sun})\simeq\delta z\left(1+ {{dR}\over{dz}}\right)+ {{(\delta z)^2}\over{2}}{{d^2R}\over{dz^2}}$$ The observed projection $y$ is linked to $z$ and $\delta A$ both by the cosine and sine rules: $$\begin{aligned} \cos(y^t)&\simeq&\cos^2(z^t)+\sin^2(z^t)\cos(\delta A) \label{eq:cos}\\ \sin(y^t)&=&\sin(\delta A) \sin(z^t) \label{eq:sin}\end{aligned}$$ Differentiating Eq. (\[eq:cos\]) and using Eq. (\[eq:sin\]) with ${\sin(y^t)\simeq y^t}$, ${\sin(\delta A)\simeq \delta A}$ and ${dz^t=-\bar{R}(z^t)}$ leads to: $$dy^t={{-y^t\ \bar{R}(z^t)}\over{\tan(z^t)}}$$ The observed distance $y$ is then obtained by: $$\label{eq:yt} y\simeq \delta A \sin(z)=y^t+dy^t=y^t\left({ 1-{{\bar{R}(z^t)}\over{\tan(z^t)}}}\right)$$ Finally, by reporting Eqs. (\[eq:xt\]) and (\[eq:yt\]) in Eq. (\[eq:circle\]) and using Eqs. (\[eq:gamma\]) and (\[eq:delta\]), we obtain: $$\label{eq:shape} \left[{{\delta z}\over{\Gamma}}-{\Delta\over 2}\left({{\delta z} \over \Gamma}\right)^2\right]^2+\left[{{\delta A \sin z}\over{1-{{\displaystyle\bar{R}(z^t)}\over{\displaystyle\tan(z^t)}}}}\right]^2=R_{\sun}^2$$ where the magnification and distortion are taken at $z_{\sun}$. From this we can deduce the position of the two vertical limb points and the observed vertical extent of the image. For ${\Delta \ll R_{\sun}}$ and ${\delta A=0}$, we find: $$\begin{aligned} d(\pi)&\simeq&\Gamma\, R_{\sun} \left(1+{{\Delta R_{\sun}}\over 2}\right) \label{eq:dpi}\\ d(0)&\simeq&\Gamma\, R_{\sun} \left(1-{{\Delta R_{\sun}}\over 2}\right)\label{eq:d0}\end{aligned}$$ and thus: $$\label{eq:Dv} D_{v}=d(0)+d(\pi)\simeq 2\,\Gamma\, R_{\sun}$$ In the horizontal direction we obtain from Eq. (\[eq:shape\]) with ${\delta z =0}$: $$\label{eq:approxdmax} D_{h}=2d(\pi/2)\simeq 2R_{\sun} \left(1-{{\bar{R}\left(z_{\sun}^t\right)}\over{\tan\left(z_{\sun}^t\right)}}\right)$$ Approximate formulae for small zenith angles - elliptic shape ------------------------------------------------------------- Keeping only the first term in Eq. (\[eq:Danjon\]) is equivalent to neglecting Earth curvature. We obtain the following approximation valid close to the zenith only (${z<45\degr}$): $$R(z)=k\tan(z) \ \ \mathrm{with:} \ \ k=\alpha(1\!-\!\beta)$$ For this flat-Earth approximation we can also write: $$\label{eq:kkp} \bar{R}(z^t)\simeq k'\tan(z^t) \ \ \mathrm{with:} \ \ k'= k\big(1-k\ \mathrm{sec}^2(z^t)\big)$$ In that case and if we neglect the distortion, Eq. (\[eq:shape\]) is reduced to the equation of a simple ellipse (see also e.g. Ball [@Ball]): $${{x^2}\over{\left(1-k'\ \mathrm{sec}^2\left(z_{\sun}^t\right)\right)^2}}+{{y^2}\over{(1-k')^2}}=R_{\sun}^2$$ where ${x=\delta z}$ and ${y=\sin(z)\,\delta A}$ can be assimilated to Cartesian coordinates on two perpendicular axes on the image. The major axis of the observed ellipse is thus given by: $$\label{eq:dmax} {D_{h}\over 2} =R_{\sun}(1-k')$$ while the observed minor axis is: $$\label{eq:dmin} {D_{v}\over 2} =R_{\sun}\left(1-k'\ \mathrm{sec}^2\left(z_{\sun}^t\right)\right)$$ We note from these equations that the Sun is shrunken in all directions. The observed horizontal diameter is smaller than the true diameter but remains the same for all zenith angles (c.f. Fig.\[Fig:contract\]) while the observed vertical diameter decreases with increasing zenith distance. The combination of these two effects leads to the apparent flattening of the setting Sun (but keeping in mind that this approximate formula is not valid close to the horizon). From Eqs. (\[eq:flatdef\]), (\[eq:dmax\]) and (\[eq:dmin\]), the flattening for small zenith angles is: $$f\simeq k \tan^2(z^t_{\sun}).$$ while, near the horizon, Eq. (\[eq:shape\]) implies that the flattening is simply given by the vertical magnification taken at the the Sun’s centre. For small zenith angles, the observed elliptic shape can be written as: $$d(\psi)={{D_{v}} \over { 2\sqrt{1-(2f-f^2)\sin^2(\psi)}}}$$ which can be approximated by: $$\label{eq:dist_app} d(\psi)\simeq R_{\sun}\left(1-k'\left(1+\cos^2(\psi)\tan^2(z^t_{\sun})\right)\right),$$ and the mean radius is obtained by: $$\label{eq:meanell} <d>={{\sqrt{D_{v}D_{h}}}\over{2}}\simeq R_{\sun} \left(1-k'-{k'\over 2}\tan^2(z^t_{\sun})\right)$$ Exact formulae for all zenith angles ------------------------------------ The classical approximate formulae above are useful for understanding the shape of the observed Sun in terms of magnification and distortion induced by refraction. Equation (\[eq:shape\]) shows that the general shape is a distorted ellipse with more flattening in the lower part than in the upper’s. However, the shape of the observed Sun can also easily be obtained, in the general case, without any approximation. In the following, we obtain first the solution of the forward problem: for given true Sun radius $R_{\sun}$ and true zenith distance $z_{\sun}^t$, we obtain the shape of the observed Sun for any given refraction model. Then, we give the solution of the inverse problem: from the observed solar shape, the knowledge of $z_{\sun}^t$ (from ephemeris) and assuming a refraction model, we deduce the true angular solar radius. Forward problem --------------- Here we assume that the true zenith distance of the Sun centre $z_{\sun}^t$ and its true angular radius $R_{\sun}$ are known. For any refraction model $\bar{R}(z^t)$, and true angle $\psi^t$, we deduce the observed angle $\psi$ and angular distance $d(\psi)$. Applying the cosine and sine formulae respectively, we have : $$\label{eq:limb} \left\{ \begin{array}{r c l} z^t&=&\mathrm{acos}\left[\cos\left(z_ {\sun}^t\right)\cos\left(R_{\sun}\right)+\sin\left(z_{\sun}^t\right)\sin\left(R_{\sun}\right)\cos(\psi^t)\right]\\ \\ \delta A&=&\mathrm{asin}\left({\displaystyle\sin(R_{\sun})\sin(\psi^t)}\over{\displaystyle\sin(z^t)}\right) %\mathrm{atan}\left[{\sin(\psi^t)}\over{\sin\left(z_{\sun}^t\right)\cot\left(R_{\sun}\right)-\cos\left(z_{\sun}^t\right)\cos(\psi^t)}\right] \end{array} \right.$$ From Eq. (\[eq:gen\_alt\]), we can get the observed zenith distances: $$\label{eq:limb2} z=z^t-\bar{R}\left(z^t\right) \ \ \ \mathrm{and} \ \ \ z_{\sun}=z_{\sun}^t-\bar{R}\left(z_{\sun}^t\right)$$ and finally angular distances $\bar{d}(\psi^t)$ between the observed Sun centre and the observed positions of each limb point are obtained by application of the cosine rule: $$\label{eq:dist} \bar{d}(\psi^t)\!=\!d(\psi)\!=\!\mathrm{acos}\big(\cos(z)\cos(z_{\sun})+\sin(z)\sin(z_{\sun})\cos(\delta A)\big)$$ where the observed angle $\psi$ can be deduced from the true one by applying the sine rule: $$\psi=\mathrm{asin}\left({{\sin(\delta A)\sin(z)}\over{\sin(\bar{d}(\psi^t))}}\right)=\mathrm{asin}\left({{\sin(z)}\over{\sin(z^t)}}{{\sin(R_{\sun})}\over{\sin(\bar{d}(\psi^t))}}\sin(\psi^t)\right)$$ The smallest observed diameter of the Sun is obtained on the vertical direction: $$D_{v}={d}(0)+{d}(\pi)=2R_{\sun}-\left(\bar{R}\left(z_{\sun}^t+R_{\sun}\right)-\bar{R}\left(z_{\sun}^t-R_{\sun}\right) \right)$$ and the largest angular extent, observed in the direction parallel to the astronomical horizon is obtained by: $$D_{h}=2{d}(\pi/2)$$ We note that Eqs. (\[eq:limb\]) and (\[eq:limb2\]) lead back to the approximation Eq. (\[eq:approxdmax\]) for the largest observed angular extent. This is however more easily obtained using the sine rule rather than Eq. (\[eq:dist\]). With ${\sin(d(\pi/2))\simeq d(\pi/2)}$, ${\sin(R_{\sun})\simeq R_{\sun}}$ and ${\cos(R_{\sun})\simeq 1}$, we obtain: $$\label{eq:sinerule} {d}(\pi/2)\simeq\sin(z)\, \sin(\delta A)=\sin\left(z_{\sun}^t-\bar{R}\left(z_{\sun}^t\right)\right){{R_{\sun}}\over{\sin(z_{\sun}^t)}}.$$ which, with a first order expansion of the sine function around $z_{\sun}^t$, leads to Eq. (\[eq:approxdmax\]). Inverse problem {#Sec:inverse} --------------- Here we give the solution of the inverse problem: given a refraction model ($R(z)$, $\bar{R}(z^t)$), knowing $z_{\sun}^t$ from ephemeris and the observed angular distance $d(\psi)$ between the observed Sun centre and a limb point at an observed angle $\psi$ with the vertical circle, we deduce the true angular radius $R_{\sun}$. One can compute successively: $$\label{eq:inverse} \left\{ \begin{array}{r c l} z_{\sun}&=&z_{\sun}^t-\bar{R}(z_{\sun}^t)\\ \\ \delta A&=&\mathrm{atan}\left[{{\displaystyle\sin(\psi)}\over{\displaystyle\sin(z_{\sun})\mathrm{cot}(d(\psi))+\cos(z_{\sun})\cos(\psi)}}\right]\\ \\ z&=&\mathrm{asin}\left[{{\displaystyle\sin(\psi)\sin(d(\psi))}\over{\displaystyle\sin(\delta A)}}\right]\\ \\ z^t&=&z+R(z)\\ \\ \psi^t&=&\mathrm{atan}\left[{{\displaystyle\sin(\delta A)}\over{\displaystyle \cos(z_{\sun}^t)\cos(\delta A)-\sin(z_{\sun}^t)\mathrm{cot}(z^t)}}\right]\\ \\ R_{\sun}&=&\mathrm{asin}\left[{{\displaystyle\sin(\delta A)\sin(z^t)}\over{\displaystyle\sin(\psi^t)}}\right]\\ \end{array} \right.$$ Results {#sec:res} ======= On the absolute value of refraction ----------------------------------- We first look at the absolute value of refraction and compare the various approximate formulae of Sect. \[sec:approx\] to the full numerical integration of the refraction integral using a standard atmosphere (Sinclair [@Sinclair]). This atmosphere is assumed to be spherically symmetric, in hydrostatic equilibrium and made of a mixture of dry air and water vapor that follows the perfect gas law. It is made of two layers: the troposphere with a constant temperature gradient which extends from the ground up to the tropopause at 11 km, and an upper isothermal stratosphere. Like in the US Standard Atmosphere ([@US76]), the temperature and pressure at the surface are 288.15 K and 101325 Pa and the constant tropospheric lapse rate is 6.5 K km$^{-1}$. In the troposphere, the relative humidity $f_h$ is assumed constant and equal to its value at the observer. The partial pressure of water vapor in a tropospheric layer at temperature $T$ is then obtained by: $$\label{eq:vaporbis} p(f_h,T)=f_h\left({T}\over{247.1}\right)^\delta 10^2$$ which, with ${\delta=18.36}$, never depart by more than $0.5$ hPa from Eq. (\[eq:water\]) for temperature lower than $30\degr$. Dry air is assumed in the stratosphere. Finally, Eq. (\[eq:alpha\]) and its derivatives with respect to $T$ and $P$ are used to find air refractivity along the integral path. The numerical integration was performed by using the method of Auer & Sandish ([@Auer]) also recommended by the Astronomical Almanac (Seidelmann [@Seidelmann]). The program used is based on the one published by Hohenkerk & Sinclair ([@Hohenkerk]) but adapted in order to use a dispersion equation based on the work of Peck & Reeder ([@Peck]) in replacement of the less accurate equation of Barrel & Sears ([@Barrel]) (Eq. (\[eq:Barrel\])). For ${T=15\ \degr\mbox{C}}$, ${P=P_0}$, $0\%$ humidity and $0.045\%$ of carbon dioxide, we take: $$\label{eq:Peck} n_0(\lambda)-1=\left\{ { {0.05792105}\over{238.0185-\left(10^6\lambda\right)^{-2}} } + { {0.00167917}\over{57.362-\left(10^6\lambda\right)^{-2}} } \right\}$$ This dispersion equation was also used by Ciddor ([@Ciddor]) who derived a new set of equations for calculating the refractive index of air which was subsequently adopted by the International Association of Geodesy (IAG [@IAG99]) as a new standard. In the following, all computations have been made using ${\lambda=535.7\ \mbox{nm}}$ which is one of the wavelengths used by the PICARD-SOL project. ![Absolute differences (in mas) between a reference model and the different approximate refraction formulae as a function of the true zenith distance. The reference model is obtained by full numerical integration of the US Standard Atmosphere ([@US76]) and Ciddor ([@Ciddor]) equation for air refractivity. From top to bottom: $\tan^5$ expansion Eqs. (\[eq:Danjon\])-(\[eq:water\]), full error function Eq. (\[eq:erfc\]), $\tan^5$ expansion Eq. (\[eq:Danjon\]), Cassini’s formula Eq. (\[eq:Cassini\]). All approximate formulae but the top one use Ciddor ([@Ciddor]) air refractivity[]{data-label="Fig:compabs"}](compare_refrac_p_1.ps){width=".3\textwidth"} Figure \[Fig:compabs\] shows the absolute differences in milliarcseconds (mas) between the approximate formulae and the exact integral evaluation for zenith distances up to $80\degr$. We immediately see that for zenith distance lower than $75\degr$, all the approximate formulae lead to less than 50 mas of absolute error. The full line corresponds to the $\tan^5$ formula Eq. (\[eq:Danjon\]) described in Sect. \[sec:calern\] while the dashed line corresponds to the same formula but using the new Ciddor ([@Ciddor]) equations instead of Eqs. (\[eq:alpha\])-(\[eq:water\]) for computing air refractivity. For zenith distances lower than $80^\circ$, the impact of using the old formula for refractivity never exceed 80 mas. The superiority of Ciddor equations to better fit observations and this for a wider range in wavelengths is however clearly established. The two other lines correspond to the error function (dot-dash) and Cassini (triple dots-dash) formulae both using the Ciddor ([@Ciddor]) equation for refractivity. These two last formulae were selected mainly because, unlike the series expansions in $\tan(z)$, they are finite at the horizon. The full integration with standard atmosphere conditions leads to a refraction of about $1980\arcsec$ at the horizon. The error function and Casini formulae lead respectively to $2088\arcsec$ and $1180\arcsec$ corresponding to relative errors of $5\%$ and $40\%$ respectively. This tends to favour the use of the error function formula over Cassini’s one very close to the horizon. The hypothesis made to derive the error function formula are indeed more realistic than Cassini’s hypothesis of an homogeneous atmosphere. It has however been shown that refraction below $5\degr$ of the horizon is variable and strongly depend on the local lapse rate and properties of the boundary layer above or below the observer’s eye (e.g. Young [@Young]). Within few degrees from the horizon, refraction may be influenced by thermal inversion boundary layers, ducting or other phenomena leading to extreme refraction. In this range, the local lapse rate must be known and it is not expected that any formula using just the temperature and pressure at observer could give an accurate absolute refraction. It is however probably more interesting to look in the range between $60\degr$ and $85\degr$ of zenith distance, which is more important to astronomers willing to push in that range the limits of their astrometric measurements using only temperature and pressure recorded at observer position. We first note from Fig. \[Fig:compabs\] that, between $60\degr$ and $77\degr$, the $\tan^5$ expansion formula is actually giving slightly better absolute refraction values than the error function formula. If we now assume that temperature and pressure at observer position are perfectly known, the only remaining important unknown in the atmospheric model is the tropospheric lapse rate. We can however fix limits for a realistic lapse rate: it must lie between an isothermal model and a lapse rate of ${10\ \mbox{K\,km}^{-1}}$ which would correspond to an adiabatic atmosphere (Young [@Young]). Figure \[Fig:comprel\] shows the absolute value of the relative error for such models with lapse rate ranging from 0 to ${10\ \mbox{K\,km}^{-1}}$ when they are compared to the standard model with a lapse rate of ${6.8\ \mbox{K\,km}^{-1}}$. From this we can deduce that, no matter what is the real atmosphere, if the conditions at observer are known, the relative error on refraction is lower than $0.01\%$ for zenith angles below $77\degr$ and lower than $0.4\%$ for zenith angles between $77\degr$ and $85\degr$. ![Relative error on refraction as a function of zenith distance for different tropospheric lapse rate. The reference model use US Standard Atmosphere ([@US76]) with a lapse rate of ${6.8\ \mbox{K\,km}^{-1}}$. The top curve correspond to an isothermal model and other atmosphere models have lapse rate of $2.5$, $5$, $10$ and ${7.5\ \mbox{K\,km}^{-1}}$ (from top to bottom at low zenith distance). All models are computed using full numerical integration.[]{data-label="Fig:comprel"}](compare_refrac_laps_2.ps){width=".3\textwidth"} On the mean solar radius correction {#Sec:mean} ----------------------------------- ![Difference between the true solar radius and the observed one as a function of the true zenith distance. The full line corresponds to average weather conditions at Calern (${T=15\ {\degr}\mbox{C}}$, ${P=875\ \mbox{hPa}}$). The dashed and dot-dashed lines correspond respectively to ${T=-10\ {\degr}\mbox{C}}$, ${P=900\ \mbox{hPa}}$ and ${T=30\ {\degr}\mbox{C}}$, ${P=850\ \mbox{hPa}}$. All calculations are made using the exact formulae Eqs. (\[eq:ddef\]) and (\[eq:dist\]) for Calern station assuming $50\%$ humidity.[]{data-label="Fig:refrac2diam"}](refrac2diam_p_1.ps){width=".3\textwidth"} ![Difference between the correction due to refraction on the mean solar radius as calculated from integrating the exact formula Eq. (\[eq:dist\]) or using the approximate formula Eq. (\[eq:meanell\]). The dashed line is obained by replacing $k'$ by $k$ in Eq. (\[eq:meanell\])[]{data-label="Fig:refrac2diam2"}](refrac2diam_p_2.ps){width=".3\textwidth"} Figure \[Fig:refrac2diam\] shows the difference between the true radius of the Sun and the mean radius of the observed Sun as defined by Eq. (\[eq:ddef\]) as a function of the true zenith distance of the centre of the Sun. The exact formula Eq. (\[eq:dist\]) was used and we took standard conditions for Calern observatory (${T=15\ {\degr}\mbox{C}}$, ${P=875\ \mbox{hPa}}$). The dashed and dot-dashed lines are for ${T=-10\ {\degr}\mbox{C}}$, ${P=900\ \mbox{hPa}}$ and ${T=30\ {\degr}\mbox{C}}$, ${P=850\ \mbox{hPa}}$ respectively in order to illustrate the maximum amplitude of the effect at Calern station. The difference in the mean radius correction between the two extreme weather conditions range from 50 mas at the zenith up to 1850 mas at ${z^t=85\degr}$. It reaches 100 mas around ${z^t=55\degr}$ and 200 mas around ${z^t=70\degr}$. This represents always less than $0.2\%$ of the correction. Figure \[Fig:refrac2diam2\] shows the difference between the exact formula obtained by integrating Eq. (\[eq:dist\]) and the approximate formula Eq. (\[eq:meanell\]) corresponding to an elliptical shape. The dashed line illustrates the result if $k'$ is approximated by $k$ (see Eq. (\[eq:kkp\])). In both cases the difference remains less than 20 mas for zenith distances lower than $70\degr$. For larger zenith distances however, errors increase rapidly and the refraction function should be evaluated using full numerical integration. On the angular dependence of solar radius correction ---------------------------------------------------- ![Difference between the true solar radius and the angular distances between the observed Sun centre and the observed positions of each limb points between the vertical (north for ${\psi=0\degr}$ and south for ${\psi=180\degr}$) and the horizon (${\psi=90\degr}$). The full lines are for ${z_{\sun}^t=70\degr}$, $50\degr$, $30\degr$ and $10\degr$ respectively from top to bottom and are for average weather conditions at Calern. The dashed and dot-dashed lines are for ${z_{\sun}^t=70\degr}$ and the same extreme weather conditions as in Fig. \[Fig:refrac2diam\].[]{data-label="Fig:refrac2diam3"}](refrac2diam_p_3.ps){width=".3\textwidth"} For precise metrologic measurements of the Sun and in order to correct for other effects (optical aberrations, turbulence, etc..) that are dependent on the position on the image, one may want to correct not the mean radius but each individual radius measured at all angles $\psi$. This can be done by following the procedure given in Sect. \[Sec:inverse\]. Figure \[Fig:refrac2diam3\] is obtained from Eq. (\[eq:dist\]) and illustrates the amplitude of the correction as a function of $\psi$ for different values of $z_{\sun}^t$, the true zenith distance of the Sun centre. We see that the horizontal diameter (${\psi=90\degr}$) is affected by refraction (by about ${2\times 0.23\arcsec=0.46\arcsec}$ for the chosen weather conditions) in agreement with Eq. (\[eq:approxdmax\]). The north and south vertical corrections (${\psi=0\degr}$ and $180\degr$ respectively) are also slightly different in agreement with Eqs. (\[eq:dpi\])-(\[eq:d0\]). Figure \[Fig:contract\] shows that the contraction of the horizontal radius lies between 210 and 260 mas depending on the actual weather conditions and remains constant for all zenith distances below $80\degr$. It then decreases rapidly towards zero at the horizon. ![Contraction of the horizontal radius (${R_{\sun}-d(\pi/2)}$) as a function of the true zenith distance $z^t_{\sun}$. The full line is for average weather conditions at Calern. The dashed and dot-dashed lines are for the same extreme weather conditions as in Fig. \[Fig:refrac2diam\].[]{data-label="Fig:contract"}](contract.ps){width=".3\textwidth"} On uncertainties associated to radius corrections ------------------------------------------------- ![Partial derivatives of the vertical diameter correction ($\delta_v$) as a function of the true zenith distance. Partial derivatives in temperature, pressure, zenith distance and relative humidity are given in $\mbox{mas\,K}^{-1}$, $\mbox{mas\,hPa}^{-1}$, $\mbox{mas\,arcmin}^{-1}$ and $\mbox{mas/\%}$ from top to bottom (at $40\degr$) respectively. The full, dashed and dot-dashed lines are for the same weather conditions as on Fig. \[Fig:refrac2diam\].[]{data-label="Fig:refrac2diam5"}](refrac2diam_p_5.ps){width=".3\textwidth"} ![Uncertainties on the vertical diameter correction assuming ${\Delta T=0.5\ \mbox{K}}$ (dotted line), ${\Delta P=1\ \mbox{hPa}}$ (dashed line), ${\Delta f_h=5\%}$ and ${\Delta z_{\sun}^t=5.4\arcmin}$, $1.0\arcmin$ or $0.2\arcmin$ (full lines from top to bottom). The total error is obtained by summing the four contributions.[]{data-label="Fig:refrac2diam6"}](refrac2diam_p_6.ps){width=".3\textwidth"} We have shown that, apart from weather conditions at observer’s position, differences in atmospheric models and especially different tropospheric lapses rate will not play any significant role at least up to $85\degr$ of zenith distance. The four main contributions are therefore uncertainties in temperature, pressure, humidity and, for large zenith distance, uncertainties on the true zenith distance itself. $$\label{eq:deriv0} \Delta d(\psi)=\sum_{i=1}^4\left|{{\partial d(\psi)}\over{\partial X_i}}\right|\Delta X_i \ \ \ \ \ \ \ X=\left\{T,P,f_h,z_{\sun}^t\right\} %\Delta\delta_v=\left|{{\partial\delta_v}\over{\partial T}}\right|\Delta T+\left|{{\partial\delta_v}\over{\partial P}}\right|\Delta P+\left|{{\partial\delta_v}\over{\partial f_h}}\right|\Delta f_h+\left|{{\partial\delta_v}\over{\partial z_{\sun}^t}}\right|\Delta z_{\sun}^t$$ It should be noted that we assume here observations made using filters with a narrow bandwidth around $\lambda$. For broadband filters, an additional term ${{\partial d(\psi)}/{\partial \lambda}}$ should be added by differentiating Eq. (\[eq:Peck\]). The largest uncertainty will be obtained for the vertical diameter (${D_v=d(0)+d(\pi)}$) which is the most affected by refraction. Figure \[Fig:refrac2diam5\] shows the four partial derivatives contributing to $\Delta D_v$ between the two extreme weather conditions chosen above for Calern (see Sect. \[Sec:mean\]). The partial derivatives shown have been obtained by numerically differentiating Eq. (\[eq:dist\]) but we have also checked that the analytical expressions that can be derived from the approximate elliptical shape Eq. (\[eq:dist\_app\]) are actually valid up to $80\degr$ of zenith distance. Closer to the horizon the partial derivative over the zenith distance becomes significantly overestimated (c.f. Fig. \[Fig:refrac2diamfinal\]). From Eq. (\[eq:dist\_app\]) and taking $k'\simeq k$, we obtain: $$\label{eq:deriv1} \left\{ \begin{array}{r c l} \left|{{\displaystyle\partial d(\psi)}\over{\displaystyle\partial X_i}}\right|&=&\left|{{\displaystyle\partial k}\over{\displaystyle\partial X_i}}\right|\left(1+\cos^2(\psi)\tan^2\left(z^t_{\sun}\right)\right)R_{\sun} \ \ \ \ \ i=1..3\\ \\ \left|{{\displaystyle\partial d(\psi)}\over{\displaystyle\partial z^t_{\sun}}}\right|&=&2 k \cos^2(\psi){\rm sec}^2\left(z^t_{\sun}\right)\tan\left(z^t_{\sun}\right)R_{\sun} \end{array} \right.$$ and from Eqs. (\[eq:beta\]), (\[eq:ell\]), (\[eq:alpha\]), (\[eq:kkp\]), (\[eq:vaporbis\]), we obtain: $$\label{eq:deriv2} \left\{ \begin{array}{r c l} {{\displaystyle\partial k}\over{\displaystyle\partial T}}&=&-C_1{{P}\over{T^2}}(n_0(\lambda)-1)+C_3 T^{\delta-1}f_h\left(C_2-{{\delta-1}\over{T}}\right)\\ \\ {{\displaystyle\partial k}\over{\displaystyle\partial P}}&=&C_1(T^{-1}-C_2)(n_0(\lambda)-1) \\ \\ {{\displaystyle\partial k}\over{\displaystyle\partial f_h}}&=&-C_3(T^{-1}-C_2)T^\delta\\ \end{array} \right.$$ where: $$\label{eq:deriv3} C_1=T_0/P_0,\ \ C_2^{-1}=C_1r_c\rho_0g_0, \ \ C_3=4.13\,10^{-8}T_0(247.1)^{-\delta}$$ For temperature, pressure and humidity, we assume uncertainties of ${\Delta T = 0.5\ \mbox{K}}$, ${\Delta P = 1\ \mbox{hPa}}$ and ${\Delta f_h = 5\%}$ which are typical for a standard weather station. The precision on the true zenith distance relies on ephemeris calculations and a correct timing. At any given time ephemeris can give not only $z_{\sun}^t$ but also the instantaneous rate ${dz_{\sun}^t}/{dt}$ and, from the knowledge of the image exposure time $\Delta t$, one can deduce an uncertainty on $z_{\sun}^t$ by: $$\label{eq:deriv4} \Delta z_{\sun}^t=\left|{{dz_{\sun}^t}\over{dt}}\right| \Delta t$$ The maximum rate is about $650\arcsec\,\mbox{min}^{-1}$ at summer solstice. Image exposures of 1 s, 5.5 s or 30 s would then correspond to a maximum uncertainty $\Delta z_{\sun}^t$ of $0.18\arcmin$, $1\arcmin$ or $5.4\arcmin$ respectively. Figure \[Fig:refrac2diam6\] shows the contribution of these uncertainties to the total uncertainty on vertical diameter correction for large zenith distances. We can see for instance that for $1\arcmin$ precision on the zenith distance (or 5.5 s exposure), the uncertainty coming from zenith distance can become, above $70\degr$, of the same importance as the combined uncertainties coming from temperature and pressure records. The total relative error on the vertical diameter correction (${\Delta D_v/(2R_{\sun}-D_v)}$) remains however below $1\%$ up to ${z^t_{\sun} = 85\degr}$. Conclusions =========== We have obtained in Sec. \[Sec:inverse\], the exact formulae that can be used to correct solar radius measurements at any heliographic angle and any zenith distance from the effect of astronomical refraction. Using full integration of the refraction integral, we have shown that this correction can be applied for any true zenith distance up to $85\degr$ with a relative uncertainty on the correction that remains less than $1\%$. Absolute uncertainties on these corrections are also derived that allows us to fix the maximum zenith distance one should observe depending on the needed metrologic accuracy. Figure \[Fig:refrac2diamfinal\] shows the maximum total absolute uncertainty obtained on the solar radius assuming that the vertical radii have been observed at different zenith distances. We use ${\lambda = 535.7\ \mbox{nm}}$ and two exposure times used by the PICARD-SOL project at this wavelength. Because we took the maximum value for $dz_{\sun}^t/dt$, this curves represent only upper limits, the actual value of $dz_{\sun}^t/dt$ should be use for each measurement. From this, one can deduce that observing below $70\degr$, $75\degr$ or $80\degr$ of zenith distances will keep the absolute uncertainties on refraction corrections below 10, 20 and 50 mas respectively. The comparison between numerical derivatives (full lines) and the use of approximate formulae Eqs. (\[eq:deriv0\])-(\[eq:deriv4\]) (dashed lines) shows that, even if the approximate formulae should not be used above $70\degr$ for correcting the measurements (c.f. Fig. \[Fig:refrac2diam2\]), they can be used at least up to ${z_{\sun}^t=80\degr}$ for estimating the uncertainties. In summary, the process that we suggest to correct ground based radii measurements from refraction for true zenith distances up to $80\degr$ is as follow. Inputs are: the measurements $d(\phi)$ and eventually their associated errors $\delta d(\phi)$ where $\phi$ is an arbitrary angle defined on the solar image; the time of image record and the exposure time $\Delta t$; weather records ($P$, $T$, $f_h$) and their associated uncertainties ($\Delta T$, $\Delta P$ and $\Delta f_h$); the wavelength ($\lambda$) and observer’s geodetic coordinates ($\varphi$, $h$). One can then successively: - find the direction of the zenith on the image and associate each angle $\phi$ to its corresponding angle $\psi$ (c.f. Fig. \[Fig:Dessin\]). Depending on the instrumental setup, this may require the computation of the parallactic angle from ephemeris, - determine $z_{\sun}^t$ and $dz_{\sun}^t/dt$ from ephemeris at the time of image record, - calculate $R_{\sun}$ using Eqs. (\[eq:inverse\]) and full numerical integration for the refraction function $R(z,\lambda,P,T,f_h,h,\varphi)$, - estimate $\Delta d(\psi)$ from Eqs. (\[eq:deriv0\])-(\[eq:deriv4\]) and the knowledge of $\Delta T$, $\Delta P$, $\Delta f_h$, $\Delta t$ and $dz_{\sun}^t/dt$, - estimate $\Delta R_{\sun}$ from: $$\Delta R_{\sun}=R_{\sun} { {\Delta d(\psi)+\delta d(\psi)} \over {d(\psi)} }.$$ For zenith distances lower than $70\degr$ full numerical integration can be replaced by Eq. (\[eq:Danjon\]) in order to evaluate the refraction function (c.f. Fig. \[Fig:compabs\]). In both cases Ciddor ([@Ciddor]) equations should be used for computing air refractivity at observer position. The corresponding codes are available from the authors upon request. It is important to keep in mind that, at all zenith distances, other phenomena such as extinction or optical turbulence must be taken into account for ground based solar metrology. We know that they will dominate refraction effects at low zenith distances. Close to the horizon extinction is proportional to refraction (Laplace’s extinction theorem) and effects of optical turbulence (e.g. Ikhlef et al. [@Ikhlef] and reference therein) will become increasingly important knowing that the Fried parameter varies as sec(z)$^{-0.6}$. It is interesting however to know that for any zenith distance up to $80\degr$ refraction can be reliably corrected and uncertainties on this correction estimated. After these correction are applied, all other phenomena impacting metrologic measurements can therefore be investigated without fearing contamination by astronomical refraction even at high zenith distances. The mean radius correction presented here (c.f. Fig. \[Fig:refrac2diam\]) as well as mean turbulence corrections have been applied to correct the first PICARD-SOL measurements (Meftah et al. [@Meftah]). The corrections that can be applied individually for each heliographic angles should be used in future work in order to disentangle the different effects. Finally we note that we have considered only the radial symmetric-component of refraction also called pure or normal refraction. There also exists an asymmetric component known as anomalous refraction (e.g. Teleki [@Teleki]) resulting from the tilted atmospheric layers. Anomalous refraction may depend not only on zenith distance but also on azimuth and it can lead to seasonal or high frequency effects (see e.g. Hirt ([@Hirt]) and references therein). The amplitude of such effect has however been found to be lower than $0.2\arcsec$ for local effects and one order of magnitude less for regional effects that may originate higher in the atmosphere (e.g. Hu [@Hu]). Moreover it has been shown that anomalous refraction is spatially coherent at scales of at least $2\degr$ (Pier et al. [@Pier]) and it has been established from dedicated observations that its main source is confined in the layer immediately above ground level (less than 60 m, see Taylor et al. ([@Taylor])). It is therefore difficult to believe that differential effects of anomalous refraction and especially the one that may be triggered in the Upper Troposphere - Lower Stratosphere (UTLS) interface (c.f. Badache-Damiani et al. [@Badache]) could lead to significant bias on solar astrometric measurements relying on direct solar disk imaging. ![Maximum absolute uncertainties on radius estimates using measurements in the vertical direction corrected from refraction. This curves are obtained for ${\lambda=535.7\ \mbox{nm}}$, ${T=(15 \pm 0.5)\ {\degr}\mbox{C}}$, ${P=(875 \pm 1)\ \mbox{hPa}}$, ${f_h=50\% \pm 5\%}$ and ${dz_{\sun}^t/dt = 650\arcsec\,\mbox{min}^{-1}}$. The top and bottom lines are for 8.9 s and 1.3 s of exposure time respectively. Full lines give the results from full numerical derivatives calculations while dashed lines are obtained using approximate formulae Eqs. (\[eq:deriv0\])-(\[eq:deriv4\]).[]{data-label="Fig:refrac2diamfinal"}](refrac2diam_p_final.ps){width=".3\textwidth"} Tha authors acknowledge financial support from CNES in the framework of PICARD-SOL project. We thank P. Exertier and J. Paris for useful discussions on the precise geodetic coordinates of Calern station, S. 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T. 2004, AJ, 127, 3622 Note on the radius of curvature at Calern observatory {#App:curvature} ===================================================== According to the WGS84 reference ellipsoid, the Earth’s equatorial and polar radii are given respectively by ${a=6378.137\ \mbox{km}}$ and ${b = 6356.752\ \mbox{km}}$. The curvature in the (north-south) meridian and at the geodetic latitude of Calern solar astrometric instruments ${\varphi= 43\degr 45\arcmin 7\arcsec}$ is then given by: $$r_c^0={(ab)^2 \over {\left(a^2\cos^2(\varphi)+b^2\sin^2(\varphi) \right)^{3/2}}}=6365.985 \ \mathrm{km}$$ One could also consider the mean radius of curvature calculated for Calern. From the curvature in the prime vertical (normal to the meridian): $$r_c^{90}={a^2 \over {\sqrt{a^2\cos^2(\varphi)+b^2\sin^2(\varphi)} }}=6388.371 \ \mathrm{km}$$ we can deduce the radius of curvature for any azimuth angle $A$ by: $$r_c^A={1 \over {{\cos^2(A)\over r_c^0}+{\sin^2(A)\over r_c^{90}} } }$$ from which we can deduce the mean radius of curvature averaging over all directions, by: $$<\!\!r_c\!\!>=\sqrt{r_c^0 r_c^{90}}= {a^2b \over {a^2\cos^2(\varphi)+b^2\sin^2(\varphi) }}=6377.168 \ \mathrm{km}$$ If, instead of the radius of curvature, one considers the distance from geocenter, we have: $$R=\sqrt{ {{a^4\cos^2(\varphi)+b^4\sin^2(\varphi)} }\over {{a^2\cos^2(\varphi)+b^2\sin^2(\varphi)} }}=6367.955 \ \mathrm{km}$$ One should add to these values the elevation of the observer above the reference ellipsoid (${h=1.323\ \mbox{km}}$ for Calern observatory). If we consider that, on average, we observe the sun closer to the north-south direction than east-west direction we can take: $$r_c=r_c^0+h=6367.308 \ \mathrm{km}$$ which is very close to the value used by Chollet ([@Chollet]). Finally we note that, for ephemeris calculations, the geodedic latitude should be corrected for the local gravimetric deflection. For Calern solar astrometric instruments this lead to an astronomic latitude ${\varphi_{\mathrm{ast}}= 43\degr 44\arcmin 53\arcsec}$ which is also compatible within $1\arcsec$ with the direct measurements made using a full entry pupil astrolabe on the same site. Similarly, we note that taking into account the local undulation with respect to the reference ellipsoid leads to a height above sea level of ${h_{\mathrm{sl}}=1.271\ \mbox{km}}$ for Calern solar astrometric station. Note on the corrections applied to mercurial barometer reading {#App:baro} ============================================================== The two corrections (for gravity and barometer temperature) can be written as multiplicative factors (e.g. Princo [@Princo]): $$\label{eq:manuel} P=H\left({1+L\ \theta}\over{1+M\ \theta}\right){g \over g_0}$$ where $P$ is the corrected atmospheric pressure, $H$ is the barometer reading, ${M=1.818\, 10^{-4}\ \mbox{K}^{-1}}$ is the coefficient of volume thermal expansion of mercury, and ${L=1.84\,10^{-5}\ \mbox{K}^{-1}}$ is the coefficient of linear thermal expansion of brass. According to the 1967 reference system formula (Helmert’s equation), we have: $$\label{eq:helmert} g=g_{45}\left(1-a\cos(2\varphi)-b\cos^2(2\varphi)\right)$$ where ${g_{45}=9.8061999\ \mbox{ms}^{-2}}$ is the gravity acceleration at mid latitude, ${a = 2.64\,10^{-3}}$ and ${b = 1.96\,10^{-6}}$. This can be corrected from the so-called Free Air Correction (FAC) which accounts for the fact that gravity decreases with height above sea level (${C_{\mathrm{FAC}} = -3.086\, 10^{-6}\ \mbox{s}^{-2}}$), itself corrected in order to take into account the increasing gravity due to the extra mass assumed for a flat terrain (Bouger correction, ${C_{\mathrm{B}} = 4.2\, 10^{-10}\ \mbox{m}^3\,\mbox{s}^{-2}\,\mbox{kg}^{-1}}$). For a mean rock density of ${\rho_r = 2.67\, 10^3\ \mbox{kg}\,\mbox{m}^{-3}}$ this leads to: $$C_g=(C_{\mathrm{FAC}}+\rho_r C_{\mathrm{B}})=-1.96\,10^{-6}\ \mbox{s}^{-2}$$ Close to $45\degr$ of latitude, the second term of Eq. (\[eq:helmert\]) can be neglected and, if we note ${\epsilon = 1-g_{45}/g_0 = 4.6\,10^{-5}}$, Eq. (\[eq:manuel\]) can be approximated by: $$P=H\left(1-\epsilon\right)\ \big(1-(M\!\!-\!\!L)\ \theta\big) \ \left\{1-a\cos(2\varphi) +{C_g\over g_{45}}\ h\right\}$$ Neglecting second order terms leads to Eq. (\[eq:baro\]). We note that absolute gravity measurements have now been made at Calern geodetic observatory leading to ${g=(980215549.2 \pm 12.6)\,10^{-8}\ \mbox{m}\,\mbox{s}^{-2}}$ (Nicolas et al. [@Nicolas]). This shows that the relative error on the correction $g/g_0$ discussed above and previously used for refraction calculations was less than $5\,10^{-5}$. One could however now directly use Eq. (\[eq:manuel\]) with the measured value of local gravity. [^1]: Chollet ([@Chollet]) used erroneously $2.64\, 10^{-4}$ in this equation.

--- abstract: 'Dimensional restrictions in a theorem of Christ, Li, Tao, and Thiele on multilinear oscillatory integral forms can be relaxed.' address: | Michael Christ\ Department of Mathematics\ University of California\ Berkeley, CA 94720-3840, USA author: - Michael Christ date: 'May 27, 2010.' title: | Multilinear Oscillatory Integrals\ Via Reduction of Dimension --- \[theorem\][Proposition]{} \[theorem\][Conjecture]{} \[theorem\][Corollary]{} \[theorem\][Lemma]{} \[theorem\][Sublemma]{} \[theorem\][Observation]{} \[definition\][Notation]{} \[definition\][Remark]{} \[definition\][Question]{} \[definition\][Questions]{} \[definition\][Example]{} \[definition\][Problem]{} \[definition\][Exercise]{} [^1] Introduction ============ By a multilinear oscillatory integral we mean a complex scalar-valued multilinear form $(f_1,\cdots,f_{n})\mapsto I(P;f_1,\cdots,f_{n})$ defined by an integral expression $$I(P;f_1,\cdots,f_{n}) = \int_{\reals^m} e^{iP(x)}\prod_{j=1}^{n} f_j(\pi_j(x))\,dx.$$ This expression involves parameters $m,n,(\kappa_1,\cdots,\kappa_{n}),(\pi_1,\cdots,\pi_{n})$ where $m$ is the ambient dimension, $\pi_j:\reals^m\to\reals^{\kappa_j}$ are surjective linear transformations, and $1\le \kappa_j\le m-1$. Each function $f_j$ is assumed to belong to $L^\infty(\reals^{\kappa_j})$, and to have support in a specified compact set $B_j\subset\reals^{\kappa_j}$. Here $n\ge 2$, $m\ge 2$. The phase function $P$ will always be assumed to be a real-valued polynomial. In this note we continue the study, initiated in [@cltt], of inequalities of the form $$\label{lambdadecay} |I(\lambda P;f_1,\cdots,f_{n})|\le C(1+|\lambda|)^{-\rho}\prod_j{ \| f_j \|}_{L^\infty},$$ where $\lambda\in\reals$ is arbitrary, while $C,\rho\in\reals^+$ are constants which are permitted to depend on $P$ and on the supports of $\{f_j\}$. In the “linear” case $n=2$, there is an extensive literature concerning such inequalities, typically phrased in terms of $\prod_j{ \| f_j \|}_{L^{p_j}}$ for more general exponents $p_j$. See for instance [@stein] for an introduction. Much less is known concerning the multilinear case $n\ge 3$. A central notion, investigated in [@cltt] and [@sublevel], is that of nondegeneracy of the phase. A polynomial $P$ is said to be degenerate relative to $\{\pi_j\}$ if $P$ can be decomposed as $\sum_j Q_j\circ\pi_j$, for some polynomials $Q_j$. Various forms of this condition are equivalent; in particular, if $P$ has degree $D$, then $P$ admits a decomposition $P=\sum_j \pi_j^*(h_j)$ where $h_j$ are distributions on $\reals^{\kappa_j}$ and $\pi_j^*$ is the natural pull back operation, if and only if $P$ admits a decomposition $P=\sum_j Q_j\circ\pi_j$ where each $Q_j$ is a polynomial of degree $\le D$. $P$ is said to be nondegenerate, relative to $\{\pi_j\}$, if it is not degenerate. Whenever $\pi_j,\tilde\pi_j$ are surjective mappings with identical nullspaces and with ranges of equal dimensions, $\tilde\pi_j=L\circ\pi_j$ for some linear transformation $L$. Therefore nondegeneracy is a property only of the collection of subspaces $\scriptv_j=\kernel(\pi_j)$, rather than of the mappings $\pi_j$, so we may equivalently speak of nondegeneracy relative to a collection of subspaces $\{\scriptv_j\}$. Let $D\ge 1$ be a positive integer, and fix $\{\scriptv_j=\kernel(\pi_j)\}$. The vector space of all degenerate polynomials $P:\reals^m\to\reals$ of degree $\le D$ is a subspace $\scriptp_{\text{degen}}$ of the vector space $\scriptp(D)$ of all polynomials $P:\reals^m\to\reals$ of degree $\le D$. Denote the quotient space by $\scriptp(D)/\scriptp_{\text{degen}}$, by $[P]$ the equivalence class of $P$ in $\scriptp(D)/\scriptp_{\text{degen}}$, and by ${ \| \cdot \|}_{\text{ND}}$ some fixed choice of norm for the quotient space. A family of subspaces $\scriptv_j\subset\reals^m$ of codimensions $\kappa_j$ is said to have the [*uniform power decay*]{} property if for each degree $D$ there exists an exponent $\gamma>0$ such that for any linear mappings $\pi_j$ with nullspaces equal to $\scriptv_j$, and for any collection of bounded subsets $B_j\subset\reals^{\kappa_j}$, there exists $C<\infty$ such that whenever each $f_j$ is supported in $B_j$, $$\label{decaydef1} |I(P;f_1,\cdots,f_{n})|\le C{ \| P \|}_{\text{ND}}^{-\gamma} \prod_{j=1}^{n}{ \| f_j \|}_{L^\infty}.$$ Certain variations on this definition are also natural. One can consider only one-parameter families of polynomials $\{\lambda P_0: \lambda\in\reals\}$, where $P_0$ remains fixed. One might allow the exponent $\gamma$ to depend on the supports $B_j$; this would be a more natural hypothesis in an extension to nonpolynomial $C^\infty$ phases $P$. The case of polynomial phases $P$, with bounds which depend only on ${ \| P \|}_{\text{ND}}$, is fundamental, so we restrict to this case in this paper. For polynomial phases, the methods of [@cltt] and of this paper show that $\gamma$ can be taken to be independent of $\{B_j\}$. The uniform decay property is defined in the same way, with ${ \| P \|}_{\text{ND}}^{-\gamma}$ replaced by $\Theta({ \| P \|}_{\text{ND}})$ for some function satisfying $\Theta(R)\to 0$ as $R\to\infty$. Nondegeneracy is a necessary condition even for a yet weaker form of the decay property [@cltt]. No other necessary conditions are known to this author. In the nonsingular case in which the mapping $\reals^m\owns x\mapsto(\pi_j(x))_{j=1}^{n} \in\times_{j=1}^{n}\reals^{\kappa_j}$ is bijective, it has been shown by Phong and Stein that $P$ is nondegenerate relative to $\{\kernel(\pi_j)\}$ if and only if holds; in that case, nondegeneracy admits a simple characterization in terms of nonvanishing of some mixed partial derivative of $P$. The singular case, where this embedding is not bijective, is the object of our investigation. As is explained in [@sublevel], the singular situation only genuinely arises for $n\ge 3$. It was shown in [@cltt] that the uniform power decay property holds in two primary cases: firstly, when $\kappa_j=m-1$ for all $j$, and secondly, when $\kappa_j=1$ for all $j$ and $n<2m$, provided in this second case that $\{\kernel(\pi_j)\}$ is in general position. It was subsequently proved in [@sublevel] that certain uniform upper bounds for measures of sublevel sets, bounds which would be implied by the uniform decay property, are valid for all $\{\pi_j\}$, subject only to the hypothesis that it is possible to choose coordinates in $\reals^m$ and in all $\reals^{\kappa_j}$ in which all $\pi_j$ are represented by matrices with rational entries. In that result the rate of decay proved to hold was not of the form of a negative power of ${ \| P \|}_{\text{ND}}$, but merely some slowly decaying function; the proof relied on a strong form of Szemerédi’s theorem. This note extends the second result of [@cltt] to more general codimensions. \[thm:reduction\] If a finite family of subspaces $\{\scriptv_\alpha\}$ of $\reals^m$ of codimensions $\kappa_\alpha\in [1,m-1]$ is in general position and satisfies $$\label{newhypothesis} 2\max_\beta\kappa_\beta + \sum_{\alpha}\kappa_\alpha\le 2m,$$ then $\{\scriptv_\alpha\}$ has the uniform power decay property. The coefficient of $2$ in is unnatural, and the proof still applies in many cases with $2\max_\beta\kappa_\beta$ replaced by $\max_\beta\kappa_\beta$, or even a smaller quantity, but it seems difficult to formulate a simple general result. When all $\kappa_j=1$, the hypothesis reduces to $n\le 2m-2$, whereas the hypothesis $n\le 2m-1$ actually suffices by [@cltt]. It remains to define the notion of general position in this theorem. The following notation will be useful in that regard. Let $\whole$ be a real vector space of some dimension $m\ge 2$. For any index set $A$ and any $A$-tuple $(\kappa_\alpha: \alpha\in A)\in [1,m-1]^A$, $G(\whole,A,(\kappa_\alpha: \alpha\in A))$ denotes the manifold consisting of all $|A|$-tuples of linear subspaces of $\whole$ of codimensions $\kappa_\alpha$. An element of $G(\whole,A,(\kappa_\alpha: \alpha\in A))$ will be called a snarl. We will sometimes set $A=\{1,2,\cdots,n\}$ and identify $\whole$ with $\reals^m$, and write $G(m,A,(\kappa_\alpha: \alpha\in A))$, or instead $G(m,\kappa_j: 1\le j\le n)$, to simplify notation. $G(m,A,(\kappa_\alpha: \alpha\in A))$ is a product of standard Grassmann manifolds $G(m,\kappa_\alpha)$, and thus carries a natural real analytic structure. A precise statement of Theorem \[thm:reduction\] is that whenever $m,(\kappa_\alpha)$ satisfy , there exists an analytic subvariety $X\subset G(m,A,(\kappa_\alpha: \alpha\in A))$ of positive codimension, such that every snarl in the complement of $X$ has the uniform decay property. We will not describe $X$ explicitly, for to do so would be prohibitively complicated, but it is constructed in principle through a recursive procedure defined in the proof of the theorem. However, in the special case where every subspace $\scriptv_j$ has codimension one, an explicit definition of general position is given in Definition \[defn:specialgeneral\]. Whenever we speak of general position with all $\kappa_j=1$, it is understood that we refer to that explicit definition. The proof of Theorem \[thm:reduction\] proceeds by induction on the codimensions $\kappa_j$, which reduces the general case to the case where all codimensions equal one, already treated in [@cltt]. In a companion paper [@christdosilva], a limited class of special cases of Theorem \[thm:reduction\] is treated by a rather different method, which we believe to be of interest despite its currently more restricted scope. The symbols $C,c$ will denote constants in $(0,\infty)$, whose values are permitted to change from one occurrence to the next. They typically depend only on $m,n$, $\{\pi_j\}$, an upper bound for the degree of the polynomial phase $P$, and the supports $B_j$ of $f_j$. $\langle x\rangle$ is shorthand for $(1+|x|^2)^{1/2}$. The author thanks Diogo Oliveira e Silva for useful corrections and comments on the exposition. An Example ========== Heavy notation in the general discussion below obscures a straightforward idea, so we discuss here a simple example, in the hope of illuminating the proof. Consider $$\iint_{\reals^4} e^{iP(x_1,x_2,y_1,y_2)}f_0(x_1,y_1)f_1(x_2,y_2)f_2(x_1+x_2,y_1+y_2) \,dx_1\,dx_2\,dy_1\,dy_2.$$ Rewrite this as $$\iint\Big(\iint e^{iP(s,u,t,-t+v)}f_0(s,t)f_1(u,-t+v)f_2(s+u,v)\,ds\,dt\Big) \,du\,dv.$$ The inner integral can be rewritten as $$\iint f_0(s,t)\cdot e^{iQ_{u,v}(s,t)}F_{1,u,v}(t)F_{2,u,v}(s+t)\,ds\,dt = \Big\langle e^{iQ_{u,v}}(F_{1,u,v}\circ L_1)(F_{2,u,v}\circ L_2),\, \overline{f_0} \Big\rangle$$ where $F_{1,u,v}(t)=f_1(u,-t+v)$, $F_{2,u,v}$ has a similar expression in terms of $f_2$, $Q_{u,v}$ is a certain polynomial in $(s,t)$, $L_1(s,t)=t$, and $L_2(s,t)=s+t$. If the $4$-fold integral is not suitably small, then there exists $(u_0,v_0)$ for which this inner product is not suitably small. Therefore $f_0=f_0(s,t)$ has a nonnegligible inner product with a function of a special form, namely, a product of a function of $L_1(s,t)$, a function of $L_2(s,t)$, and a polynomial $q(s,t)$ whose degree does not exceed that of $P$. By an argument used in [@cltt] (see the derivation of below), it suffices to analyze the case where $f_0$ is [*equal*]{} to such a product. Substitute this product back into the original integral over $\reals^4$. Then $P$ is replaced by $\tilde P = P(x_1,x_2,y_1,y_2)+q(x_1,y_1)$. As a function of $(x_1,x_2,y_1,y_2)$, $q(x_1,y_1)$ is degenerate. Therefore $\tilde P$ belongs to the same equivalence class as $P$. The effect is a reduction to the case where $f_0(x_1,y_1)$ is replaced by a product of two factors, each of which depends only on the image of $(x_1,x_2,y_1,y_2)$ under a mapping $L_j$. The same reasoning can be applied to similarly reduce $f_1,f_2$. There results a multilinear form involving $6$ functions $g_\alpha(L_\alpha(x_1,x_2,y_1,y_2))$, where each $L_\alpha$ is a linear mapping from $\reals^4$ to $\reals^1$, rather than to the original $\reals^2$. The case of one-dimensional target spaces was treated in [@cltt]. In §\[section:resolution\] we will formalize the concept of a resolution of a snarl, a sequence of moves which, in the example just presented, transforms the given collection of three subspaces of codimension two into a collection of six subspaces of codimension 1. In §\[section:linearalgebra\] we will prove that any snarl in general position admits a resolution by a sequence of such moves. Finally, in §\[section:slicing\], we will carry out the analytic argument outlined in the preceding paragraphs to demonstrate that each move preserves the uniform power decay property. Resolution {#section:resolution} ========== A splitting of a snarl $(\whole,A,\{\scriptv_\alpha: \alpha\in A\})$ is a snarl $(\whole,B,\{\scriptw_\beta: \beta\in B\})$ with index set $B$ satisfying $|B|=|A|+1$, $|A\cap B|=|A|-1$, if $\alpha\in A\cap B$ then $\scriptw_\alpha=\scriptv_\alpha$, and if indices $\alpha_0,\beta',\beta''$ are specified so that $B\setminus A = \{\beta',\beta''\}$ and $A\setminus B=\{\alpha_0\}$, then $$\begin{gathered} \scriptw_{\beta'}\cap\scriptw_{\beta''}=\scriptv_{\alpha_0} \\ \codim(\scriptw_{\beta'}) + \codim(\scriptw_{\beta''}) = \codim(\scriptv_{\alpha_0}).\end{gathered}$$ Direct consequences of the definition are $$\begin{aligned} &\sum_{\alpha\in A}\codim(\scriptv_\alpha) = \sum_{\beta\in B}\codim(\scriptw_\beta). \\ &\max_\alpha\codim(\scriptv_\alpha)\ge \max_\beta\codim(\scriptw_\beta).\end{aligned}$$ Therefore if a snarl satisfies our main hypothesis , any splitting continues to satisfy that hypothesis. If $(\whole,A,\{\scriptv_\alpha: \alpha\in A\})$ is a snarl with index set $A$, then for any nonempty subset $A'\subset A$, $\scriptv_{A'}$ is defined to be $\cap_{\alpha\in A'} \scriptv_\alpha$. Let $(\whole,B,\{\scriptw_\beta: \beta\in B\})$ be a splitting of a snarl $(\whole,A,\{\scriptv_\alpha: \alpha\in A\})$. Let $\beta',\beta'',\alpha_0$ be the three distinguished indices which appear in the preceding definition. A splitting $(\whole,B,\{\scriptw_\beta: \beta\in B\})$ of a snarl $(\whole,A,\{\scriptv_\alpha: \alpha\in A\})$ is transverse if $A\setminus \{\alpha_0\}$ can be partitioned as the disjoint union of two nonempty sets $A',A''$ such that $$\begin{gathered} \dimension(\scriptw_{\beta'}\cap\scriptv_{A'})>0, \\ \dimension(\scriptw_{\beta''}\cap\scriptv_{A''})>0, \\ \whole = \scriptw_{\beta'} + \scriptw_{\beta''}, \\ \scriptw_{\beta'} + \scriptv_{\alpha_0} \text{ and } \scriptw_{\beta''} + \scriptv_{\alpha_0} \text{ are proper subspaces of } \whole. $$ In §\[section:slicing\] we will establish: \[prop:induction\] Suppose that the snarl $\snarl^\sharp$ is a transverse splitting of a snarl $\snarl$. If $\snarl^\sharp$ has the uniform power decay property, then so does $\snarl$. A chain of transverse splittings of a snarl $\snarl$ is a finite sequence of snarls $(\snarl_k)_{k=0}^N$ such that $\snarl_0=\snarl$, and $\snarl_{k+1}$ is a transverse splitting of $\snarl_k$ for each $k\in\{0,1,2,\cdots,N-1\}$. A snarl $(\whole,A,\{\scriptv_\alpha: \alpha\in A\})$ one-dimensional if for every $\alpha\in A$, $\scriptv_\alpha$ has codimension one. A resolution $(\snarl_k)_{k=0}^N$ of a snarl $\snarl$ is a chain of transverse splittings of $\snarl$ such that $\snarl_N$ is one-dimensional. $\snarl_N$ is called the terminal element of this resolution. \[defn:specialgeneral\] A one-dimensional snarl $(\whole,A,\{\scriptv_\alpha: \alpha\in A\})$ is said to be in general position if for any index set $A'\subset A$, $\{\scriptv_\alpha: \alpha\in A'\}$ spans a subspace of dimension $\min(|A'|, m)$. It was shown in Theorem 2.1 of [@cltt] that any one-dimensional snarl in $\reals^m$ with index set $A$ satisfying $|A|<2m$ has the uniform power decay property, provided that it is in general position in this sense. Combining that theorem with Proposition \[prop:induction\] gives: \[prop:resolutionsuffices\] Let $\snarl=(\whole,A,\{\scriptv_\alpha: \alpha\in A\})$ be a snarl satisfying $$\max_{\alpha\in A}\codim(\scriptv_\alpha) + \sum_{\alpha\in A}\codim(\scriptv_\alpha)\le 2\dimension(\whole).$$ Suppose that $\snarl$ admits a resolution with terminal element in general position. Then $\snarl$ has the uniform power decay property. There remains the question of the existence and abundance of snarls admitting resolutions with the desired properties. \[prop:abundance\] Fix $m>1$, a finite index set $A$, and $\{\kappa_\alpha: \alpha\in A\}$ satisfying . There exists an analytic variety $X$ of positive codimension in $G(m,(\kappa_\alpha: \alpha\in A))$, such that any snarl $Q\notin X$ admits a resolution with terminal element in general position. Propositions \[prop:resolutionsuffices\] and \[prop:abundance\] together establish our main theorem. By a straightforward induction, Proposition \[prop:abundance\] is a consequence of the following result. \[prop:atlast\] Let $m,n$ and an index set $A$ of cardinality $n$ be given. Let $\{\kappa_\alpha: \alpha\in A\}$ satisfy . There exist an index set $B=(A\setminus\{\alpha_0\})\cup\{\beta',\beta''\}$ of cardinality $n+1$ and parameters $\{\kappa_{\beta'},\kappa_{\beta''}\}$ such that $\{\kappa_\beta: \beta\in B\}$ continues to satisfy , and such that for any analytic variety $Y\subset G(m,(\kappa_\alpha: \alpha\in B))$ of positive codimension, there exists an analytic variety $X\subset G(m,(\kappa_\alpha: \alpha\in A))$ of positive codimension such that any snarl in $G(m,(\kappa_\alpha: \alpha\in A))\setminus X$ admits a transverse splitting belonging to $G(m,(\kappa_\alpha: \alpha\in B))\setminus Y$. A defect of our theory is that the variety $X$ in Proposition \[prop:abundance\] has been defined not explicitly, but only by a rather complicated recursive procedure. However, Proposition \[prop:resolutionsuffices\] can be applied directly to any snarl for which a resolution can be found. Proof of Proposition \[prop:atlast\] {#section:linearalgebra} ==================================== Identify $A$ with $\{0,1,\cdots,n-1\}$ in such a way that $\kappa_{0}=\max_j\kappa_j$. Partition the set of indices $\{1,2,\cdots,n-1\}$ into $2$ nonempty disjoint subsets, $S',S''$. Consider $$\begin{aligned} {2} &\scriptv_{S'}=\cap_{j\in S'}\scriptv_j \qquad\qquad && \scriptv_{S''}=\cap_{j\in S''}\scriptv_j \\ &\kappa_{S'}=\sum_{j\in S'}\kappa_j && \kappa_{S''}=\sum_{j\in S''}\kappa_j.\end{aligned}$$ Choose this partition so that $|\kappa_{S'}-\kappa_{S''}|\le\kappa_0$, which is possible because $\kappa_0\ge\kappa_j$ for all $j$. \[lemma:constructW\] Suppose that $\sum_{j=0}^{n-1}\kappa_j<2m$, and that $\max_j\kappa_j>1$. There exist integers $\kappa',\kappa''\in \{1,2,\cdots,\kappa_0-1\}$, depending only on $m$ and on $\{\kappa_j: 0\le j<n\}$ and satisfying $\kappa'+\kappa''=\kappa_0$, together with an analytic variety $X_0\subset G(m,\kappa_j: 0\le j<n)$ of positive codimension, such that whenever $(\scriptv_j:0\le j<n)\notin X_0$, there exist subspaces $W'\subset\scriptv_{S'}$ and $W''\subset\scriptv_{S''}$ of dimensions $\kappa',\kappa''$ respectively, which satisfy $$\begin{aligned} &W'\cap W''=\{0\} \\ &(W'+W'')\cap\scriptv_0=\{0\}.\end{aligned}$$ If $W',W''$ satisfy these conclusions, define $\scriptv_n=\scriptv_0+W''$ and $\scriptv_{n+1}=\scriptv_0+W'$. Then $(\scriptv_i: 1\le i\le n+1)$ is a transverse splitting of $(\scriptv_j: 0\le j<n)$. Suppose without loss of generality that $\kappa_{S'}\ge \kappa_{S''}$. Since $\kappa_{S'}\le \kappa_{S''}+\kappa_0$ and $\kappa_{S'}+\kappa_{S''}<2m-\kappa_0$, $2\kappa_{S'}\le \kappa_{S'}+(\kappa_{S''}+\kappa_0)<2m$. Therefore $\max(\kappa_{S'},\kappa_{S''})<m$. If $\{\scriptv_j\}$ is in general position, $$\begin{aligned} &\dimension(\scriptv_{S'}) =\max(0, m-\kappa_{S'}) =m-\kappa_{S'}\ge 1 \\ &\dimension(\scriptv_{S''})=\max(0,m-\kappa_{S''}) =m-\kappa_{S''}\ge 1\end{aligned}$$ and since $2m-\kappa_{S'}-\kappa_{S''}-\kappa_0\ge 0$ by , if $(\scriptv_j)$ is in general position then $$\label{tritransverse} \dimension\big(\scriptv_{S'} +\scriptv_{S''} +\scriptv_0\big) =\min\big(m,m-\kappa_{S'}+m-\kappa_{S''}+m-\kappa_0\big) =m.$$ Furthermore, since $\scriptv_0$ has positive codimension and $\scriptv_{S'},\scriptv_{S''}$ have positive dimensions, if $(\scriptv_i: 0\le i<n)$ is in general position, then neither of $\scriptv_{S'},\scriptv_{S''}$ is contained in $\scriptv_0$. Moreover, $\scriptv_0$ has codimension $\kappa_0\ge 2$. These facts, together with , ensure that there exist $\kappa',\kappa''\in[1,\kappa_0]$ satisfying $\kappa'+\kappa''=\kappa_0$, and subspaces $W'\subset\scriptv_{S'}$ and $W''\subset\scriptv_{S''}$ of dimensions $\kappa',\kappa''$ respectively, such that $W'\cap W''=\{0\}$ and $$\label{tridirectsum} W'+W''+\scriptv_0=\reals^m.$$ Since $$\dimension(W')+ \dimension(W'')+\dimension(\scriptv_0) = \kappa'+\kappa'' + (m-\kappa_0)=m,$$ is a direct sum decomposition. Fix such $\kappa',\kappa''$. Choose subspaces $U',U''\subset\reals^m$ of codimensions $m-\kappa_{S'}-\kappa'$ and $m-\kappa_{S''}-\kappa''$ respectively, which are transverse to one another. To an arbitrary $\snarl=(\scriptv_j: 0\le j<n)\in G(m,\kappa_j: 0\le j<n)$ associate $W'(\snarl)=U'\cap\scriptv_{S'}$ and $W''(\snarl) = U''\cap\scriptv_{S''}$. The set of all $\snarl$ for which they fail to do so, is an analytic variety $X_0$ of positive codimension. The hypothesis $\sum_{j=0}^{n-1}\kappa_j<2m$ is not sufficient to ensure that the splitting $(\scriptv_j: 1\le j\le n+1)$ lies in general position. Indeed, the sum of the dimensions of $\scriptv_n\cap\scriptv_{S'}$ and $\scriptv_{n+1}\cap\scriptv_{S''}$ is required by the above construction to be $\ge\kappa_0$. For $(\scriptv_i: 1\le i\le n+1)$ in general position, these two intersections will have dimensions equal to $m-\kappa_{S'}-\kappa_n, m-\kappa_{S''}-\kappa_{n+1} $, respectively. Thus the construction requires $2m-\kappa_{S'}-\kappa_{S''}-\kappa_n-\kappa_{n+1}\ge \kappa_0$. Since $\kappa_n+\kappa_{n+1}=\kappa_0$, this is equivalent to $2m-\sum_{j=0}^{n-1}\kappa_j\ge \kappa_0$, that is, to $\max_i\kappa_i+\sum_j\kappa_j\le 2m$. Let $(\kappa_j: 0\le j<n)$ satisfy $2\max_i\kappa_i+ \sum_{j}\kappa_j\le 2m$ and $\max_i\kappa_i>1$. There exist $\kappa_n,\kappa_{n+1}\in[1,\kappa_0-1]$ satisfying $\kappa_n+\kappa_{n+1}=\kappa_0$ with the following property. For any analytic subvariety $Y\subset G(m,\kappa_i: 1\le i\le n+1)$ of positive codimension, there exists an analytic subvariety $X\subset G(m,\kappa_j: 0\le j<n)$ of positive codimension, such that if $(\scriptv_j: 0\le j<n)\notin X$, then in the above construction, $W',W''$ can be chosen so that $(\scriptv_1,\cdots,\scriptv_{n-1},\scriptv_0+W',\scriptv_0+W'')\notin Y$. Choose $S',S''$ as above, so that $|\kappa_{S'}-\kappa_{S''}|\le \kappa_0$. Then $(m-\kappa_{S'})+(m-\kappa_{S''})\ge 3\kappa_0$ by and the choice $\kappa_0=\max_{j}\kappa_j$. Therefore $m-\kappa_{S'}$ and $m-\kappa_{S''}$ are both $\ge\kappa_0$; it is here that the full strength of is used. Consequently if $\kappa_n,\kappa_{n+1}\in[1,\kappa_0-1]$ are chosen to satisfy $\kappa_n+\kappa_{n+1}=\kappa_0$, then $$\begin{gathered} \label{eq:allof2} m-\kappa_{S'}-\kappa_n\ge\kappa_{n+1} \\ \label{eq:allof3} m-\kappa_{S''}-\kappa_{n+1}\ge\kappa_{n}.\end{gathered}$$ Consider any $\snarl^\sharp=(\scriptv_j: 1\le j\le n+1)\in G(m,n+1,\kappa_1,\cdots,\kappa_{n+1})$ in general position, where the precise meaning of general position remains to be specified. Define $\scriptv_0=\scriptv_n\cap\scriptv_{n+1}$. Then $\dimension(\scriptv_n\cap\scriptv_{n+1})=\max(0,m-\kappa_n-\kappa_{n+1})=m-\kappa_0$, so $\scriptv_0=\scriptv_n\cap\scriptv_{n+1}$ has codimension $\kappa_0$. Moreover, general position ensures that $$\begin{aligned} \dimension(\scriptv_{S'}\cap\scriptv_n)&=m-\kappa_{S'}-\kappa_n \\ \dimension(\scriptv_{S''}\cap\scriptv_{n+1})&=m-\kappa_{S''}-\kappa_{n+1}.\end{aligned}$$ Therefore if $\snarl^\sharp$ is in general position, $$\dimension(\scriptv_n\cap\scriptv_{S'}) +\dimension(\scriptv_{n+1}\cap\scriptv_{S''}) +\dimension(\scriptv_n\cap\scriptv_{n+1}) \ge \kappa_n+\kappa_{n+1}+(m-\kappa_0)=m.$$ Since the index sets $S',S'',\{n,n+1\}$ are pairwise disjoint, general position then implies that $ (\scriptv_n\cap\scriptv_{S'}) +(\scriptv_{n+1}\cap\scriptv_{S''}) +(\scriptv_n\cap\scriptv_{n+1})=\reals^m$. The two subspaces $\scriptv_n\cap\scriptv_{S'}$ and $\scriptv_0=\scriptv_n\cap\scriptv_{n+1}$ are contained in $\scriptv_n$ and have dimensions $m-\kappa_{S'}-\kappa_n$ and $m-\kappa_0$, respectively. If $\{\scriptv_j: j\in S'\}$ and $\scriptv_{n+1}$ are jointly in general position relative to $\scriptv_n$, these two subspaces will be transverse; their sum will have dimension equal to $$\begin{gathered} \max(m-\kappa_n, m-\kappa_{S'}-\kappa_n + m-\kappa_0) = 2m-\kappa_{S'}-\kappa_n-\kappa_0 \\ \ge m-\kappa_0+\kappa_{n+1} =\dimension(\scriptv_0)+\kappa_{n+1},\end{gathered}$$ using . Therefore there exists a subspace $W'\subset\scriptv_n\cap\scriptv_{S'}$ of dimension exactly $\kappa_{n+1}$, satisfying $\dimension(W'+\scriptv_0) = \dimension(W')+\dimension(\scriptv_0)$. Since $W',\scriptv_0$ are both contained in $\scriptv_n$ and the sum of their dimensions equals the dimension $\kappa_{n+1}+m-\kappa_0 =m-\kappa_n$ of $\scriptv_n$, their span equals $\scriptv_n$. For the same reasons, there exists a subspace $W''\subset\scriptv_{n+1}\cap\scriptv_{S''}$ of dimension $\kappa_{n}$ which is transverse to $\scriptv_0$, such that $W'',\scriptv_0$ together span $\scriptv_{n+1}$. Since the three index sets $S',S'',\{0\}$ are disjoint, general position implies that $W',W''$ can be chosen so that $W''$ is transverse to $W'+\scriptv_0$. Thus $(\scriptv_j: 1\le n\le n+1)$ is a transverse splitting of $(\scriptv_j: 0\le j<n)$. Let $(\kappa_j: 0\le j<n)$ satisfy , and choose $\kappa_n,\kappa_{n+1}$ as above. We have proved that there exists an analytic subvariety $Y_0\subset G(m,\kappa_j: 1\le j\le n+1)$ of positive codimension, such that for any $\snarl^\sharp\in G(m,\kappa_j: 1\le j\le n+1)\subset Y_0$, there exists at least one $\snarl\in G(m,\kappa_j: 0\le j<n)$ which admits at least one transverse splitting equal to $\snarl^\sharp$. Indeed, each mention of “general position” in the above discussion can be expressed as the condition that $\snarl^\sharp$ satisfies none of a finite set of analytic equations. The union of the varieties defined by each of these equations defines $Y_0$, which has positive codimension. Given any analytic subvariety $Y\subset G(m,\kappa_j: 1\le j\le n+1)$ of positive codimension, set $\tilde Y=Y\cup Y_0$ and let $X$ be the set of all $\snarl\in G(m,\kappa_j: 0\le j<n)$ for which the subspaces $W'(\snarl),W''(\snarl)$ defined in the proof of Lemma \[lemma:constructW\] either fail to define a transverse splitting of $\snarl$, or define a splitting which belongs to $\tilde Y$. Then $X$ is an analytic subvariety, for all restrictions encountered can be expressed as analytic equations for $(\scriptv_j: 0\le j<n)$ together with the subspaces $U',U''$ used to define the functionals $W'(\cdot),W''(\cdot)$ We have shown that there exists at least one $\snarl_0\in G(m,\kappa_j: 0\le j<n)$ which admits some transverse splitting $\snarl_0^\sharp\in G(m,\kappa_j: 1\le j\le n+1)\setminus\tilde Y$. The subspaces $U',U''$ may be chosen so that $W'(\snarl_0),W''(\snarl_0)$ define this splitting $\snarl_0^\sharp$. Then $X$ is nonempty, so $X$ has positive codimension. The inductive step {#section:slicing} ================== We now prove Proposition \[prop:induction\]. Let $W',W''$ and $\scriptv_n=\scriptv_0+W''$, $\kappa_n=\kappa''$, $\scriptv_{n+1}=\scriptv_0+W'$, and $\kappa_{n+1}=\kappa'$ be as in Lemma \[lemma:constructW\]. Set $W=W'+W''$, and $W^\star =\scriptv_0=\kernel(\pi_0)$. $W,W^\star$ are a pair of supplementary subspaces, so $\reals^m=W+W^\star$ may be identified with $W\times W^\star$. Thus an arbitrary element of $\reals^m$ can be expressed in a unique way as $x+y$ with $x\in W$ and $y\in W^\star$; $x+y$ will henceforth be identified with $(x,y)$. Define linear transformations $\tilde\pi_j:W\mapsto\reals^{\kappa_j}$ by $$\tilde\pi_j(x)=\pi_j(x,0).$$ For any $(x,y)$, $\pi_j(x,y) = \pi_j(x,0)+\pi_j(0,y)$, so $$f_j(\pi_j(x,y))=f_{j,y}(\tilde\pi_j(x))$$ where $$f_{j,y}(t) = f_j(t+\pi_j(0,y)).$$ We will use the equivalence, with the mappings $\pi_j$, sets $B_j$, and phase function $P$ fixed, between an [*a priori*]{} inequality of the form $$|I(P;f_0,\cdots,f_{n-1})|\le \scriptc\prod_{j=0}^{n-1}{ \| f_j \|}_\infty,$$ and the formally stronger inequality $$\label{L2inequality} |I(P;f_0,\cdots,f_{n-1})|\le \tilde\scriptc{ \| f_0 \|}_2\prod_{j=1}^{n-1}{ \| f_j \|}_\infty.$$ If the latter holds, then the former holds with $\scriptc\le C\tilde\scriptc$. If the former holds, then the latter follows with $\tilde\scriptc\le C\scriptc^{1/2}$, by interpolation with the trivial inequality $$|I(P;f_0,\cdots,f_{n-1})|\le \tilde C'{ \| f_0 \|}_1\prod_{j=1}^{n-1}{ \| f_j \|}_\infty.$$ Our argument is not phrased exclusively in terms of one inequality or the other, but uses their equivalence at each step of an induction. Our oscillatory integral may be written as $$I(P;f_0,\cdots,f_{n-1}) =\int_{E} I_y(P_y;f_{0,y},\cdots,f_{n-1,y})\,dy$$ for some bounded subset $E\subset W^\star$, where $$I_y(P_y;g_{0},\cdots,g_{n-1}) = \int e^{iP(x,y)}\prod_{j=0}^{n-1} g_j(\tilde\pi_j(x))\,dx.$$ Note that $$f_{0,y}(\tilde\pi_0(x)) = f_0(\pi_0(x,0)+\pi_0(0,y)) \equiv f_0(\pi_0(x,0)).$$ $x\mapsto\pi_0(x,0)$ is a linear isomorphism of $W$ with $\reals^{\kappa_0}$. Therefore by a linear change of variables in $\reals^{\kappa_0}$, we may arrange that $$\pi_0(x,0)\equiv x.$$ With this simplification, $$I_y(P_y;f_{0,y},\cdots,f_{n-1,y}) = \big\langle e^{iP(x,y)}\prod_{j=1}^{n-1} f_{j,y}(\tilde\pi_j(x)), \, \overline{f_{0}} \big\rangle,$$ where the inner product is taken with respect to $x$ for fixed $y$. Fix bounded sets $B_j\subset\reals^{\kappa_j}$, and consider only functions $f_j$ supported in $B_j$. Define $\Lambda=\Lambda(P,\{\pi_j\})$ to be the optimal constant in the inequality . Let $\{f_j: 1\le j\le n-1\}$ and $f_0$ be functions satisfying ${ \| f_j \|}_\infty\le 1$ for $j\ge 1$, and ${ \| f_0 \|}_2=1$, such that $$|I(P;f_0,\cdots,f_{n-1})|\ge \tfrac12 \Lambda{ \| f_0 \|}_2.$$ There exists $z$ such that $\big|\big \langle e^{iP(x,z)}\prod_{j=1}^{n-1} f_{j,z}(\tilde\pi_j(x)), \, \overline{f_0} \big\rangle\big|\ge c\Lambda$. Decompose $$f_0(x) = a e^{-iP(x,z)}\prod_{j=1}^{n-1} h_j(\tilde\pi_j(x)) +g_0(x)$$ where $$\begin{aligned} |a|&\le C{ \| f_0 \|}_2 \\ { \| g_0 \|}_{\lt(\reals^{\kappa_0})}^2&\le { \| f_0 \|}_2^2 - c\Lambda^2{ \| f_0 \|}_2^2\end{aligned}$$ and $h_j=\overline{f_{j,z}}$. Then $$\begin{gathered} I(P;f_0,\cdots,f_{n-1}) = I(P;f_1,\cdots,f_{n-1},g_0) \\ + a\iint e^{iP(x,y)} \prod_{j=1}^{n-1}f_j(\pi_j(x,y)) \cdot e^{-iP(x,z)} \prod_{k=1}^{n-1} h_j(\tilde\pi_j(x)) \,dx\,dy.\end{gathered}$$ The second term may be written as $$a \iint e^{iQ(x,y)} \prod_{j=1}^{n-1}f_j(\pi_j(x,y)) \cdot \prod_{k=1}^{n-1} h_k(\pi_k^\sharp(x,y)) \,dx\,dy$$ where $\pi_k^\sharp:\reals^m\to\reals^{\kappa_k}$ is defined by $$\pi_k^\sharp(x,y) = \tilde\pi_k(x) =\pi_k(x,0) = \pi_k(\pi_0(x,y),0) = \pi_k(\pi_0(x,0),0)$$ and $$Q(x,y) = P(x,y)-P(x,z).$$ Since $x=\pi_0(x,y)$, $(x,y)\mapsto P(x,z)$ is a polynomial function of $\pi_0(x,y)$. Therefore $[Q]=[P]$, where $[\cdot]$ denotes the equivalence class in the space of polynomials modulo those polynomials which are degenerate relative to $\{\pi_j: 0\le j\le n-1\}$. Now $$\begin{gathered} \iint e^{iQ(x,y)} \prod_{j=1}^{n-1}f_j(\pi_j(x,y)) \cdot \prod_{k=1}^{n-1} h_k(\pi_k^\sharp(x,y)) \,dx\,dy \\ = I\Big(Q;f_1,\cdots,f_{n-1},h_1,\cdots,h_{n-1}, \{\pi_j\}_{j=1}^{n-1},\{\pi^\sharp_k\}_{k=1}^{n-1}\Big).\end{gathered}$$ This is not what we are aiming for; for instance, this expression is $2n-2$–multilinear, while we are aiming for an $n+1$–multilinear form. Elements $(x,0)\in W$ may be decomposed as $(x,0)=(x',x'',0)$ where $(x',0,0)\in W'$ and $(0,x'',0)\in W''$. Thus $\pi_k^\sharp(x',x'',y)=\pi_k(x',x'',0)= \pi_k(x',0,0)+\pi_k(0,x'',0)$ depends only on $x''$ for $k\in S'$, and depends only on $x'$ for $k\in S''$; the nullspace of $\pi_k^\sharp$ contains $W''+\scriptv_0$ for each $k\in S'$. Therefore we may write $$\prod_{k\in S'}h_k(\pi_k^\sharp)(x',x'',y) = f_n\big(\pi_n(x',x'',y)\big)$$ where $\pi_n$ is a surjective linear mapping from $\reals^m$ to a Euclidean space of dimension $\kappa_n=\dimension(W'')$, the nullspace of $\pi_n$ equals $\scriptv_0+W''=\scriptv_n$, $$\pi_n(x',x'',y) = \pi_k^\sharp(x',x'',y) = \pi_k^\sharp(x',x'',0) = \pi_k^\sharp(0,x'',0),$$ and ${ \| f_n \|}_\infty \le\prod_{k\in S'}{ \| h_k \|}_\infty\le 1$; this can be done, albeit in an artificial way, even if the intersection of the nullspaces of all such $\pi_k^\sharp$ has dimension strictly greater than $m-\kappa_n$, by defining $f_n$ to be independent of one or more coordinates in $\pi_n(\reals^m)$ in a sufficiently large bounded set. Likewise $$\prod_{k\in S''}h_k(\pi_k^\sharp)(x',x'',y) = f_{n+1}\big(\pi_{n+1}(x',x'',y)\big)$$ where $\pi_{n+1}$ is a surjective linear mapping with nullspace $\scriptv_{n+1}$ from $\reals^m$ to a Euclidean space of dimension $\kappa_{n+1}=\dimension(W')$, and ${ \| f_{n+1} \|}_\infty\le 1$. With these definitions, $$\begin{gathered} \iint e^{iQ(x,y)} \prod_{j=1}^{n-1}f_j(\pi_j(x,y)) \cdot \prod_{k=1}^{n-1} h_k(\pi_k^\sharp(x)) \,dx\,dy = I\big(Q;f_1,\cdots,f_{n+1}, \{\pi_j\}_{j=1}^{n+1}\big).\end{gathered}$$ ${ \| f_j \|}_\infty\le 1$ for all $j\in\{1,2,\cdots,n+1\}$, and $f_i$ is supported in a bounded subset of $\reals^{\kappa_i}$ which depends only on $\{B_j: 0\le j\le n-1\}$, on $\{\pi_j: 0\le j\le n-1\}$, on the choices of $S',S''$, and on the choice of $W$. Now $Q$ is nondegenerate[^2] relative to $\{\pi_j\}_{j=1}^{n+1}$, because $Q$ is nondegenerate relative to $\{\pi_j\}_{j=0}^{n-1}$ and the projections $\pi_n,\pi_{n+1}$ both factor through $\pi_0$. The norm of $Q$ in the quotient space of polynomials modulo sums of polynomials $q\circ\pi_j$ with $1\le j\le n+1$ is at least as large as the norm of $P$ in the quotient space of polynomials modulo $q\circ\pi_j$ with $0\le j\le n-1$, up to a constant factor which depends only on choices of norms for these spaces. We are reasoning under the induction hypothesis that for any collection of bounded subsets $B_j\subset \reals^{\kappa_j}$, there exist $C<\infty$ and an exponent $\gamma>0$ such that for all continuous functions $f_j$ supported in $B_j$ respectively, $$\big| I\big(Q;f_1,\cdots,f_{n+1}, \{\pi_j\}_{j=1}^{n+1}\big)\big| \le C\langle { \| Q \|}_{\text{ND}} \rangle^{-\gamma} \prod_{j=1}^{n+1}{ \| f_j \|}_\infty \le C\langle { \| P \|}_{\text{ND}} \rangle^{-\gamma}.$$ $C,\gamma$ depend on $\{\pi_j: 1\le j\le n+1\}$ and on $\{B_j\}$, which in turn depend on $\{\pi_j: 0\le j\le n-1\}$ and on the designation of bounded subsets on which the functions $f_j$ are supported for all $j\in\{0,1,\cdots,n-1\}$. Therefore whenever $\{f_j\}$ are continuous functions supported in $B_j\subset\reals^{\kappa_j}$, satisfying ${ \| f_0 \|}_2=1$ and ${ \| f_j \|}_\infty\le 1$ for all $j\in\{1,2,\cdots,n-1\}$, $$\begin{aligned} |I(f_0,\cdots,f_{n-1},&\{\pi_j\}_{j=0}^{n-1})| \\ &\le |I(g_0,f_1,\cdots,f_{n-1},\{\pi_j\}_{j=0}^{n-1})| +C \big| I\big(Q;f_1,\cdots,f_{n+1}, \{\pi_j\}_{j=1}^{n+1}\big)\big| \\ &\le \Lambda{ \| g_0 \|}_2 + C\langle { \| P \|}_{\text{ND}} \rangle^{-\gamma} \\ &\le \Lambda { \| f_0 \|}_2(1-c\Lambda^2) + C\langle { \| P \|}_{\text{ND}} \rangle^{-\gamma} \\ &\le \Lambda (1-c\Lambda^2) + C\langle { \| P \|}_{\text{ND}} \rangle^{-\gamma}.\end{aligned}$$ By taking the supremum over all $f_0,\cdots,f_{n-1}$ which are supported in the sets $B_j$ and satisfy ${ \| f_0 \|}_2\le 1$ and ${ \| f_j \|}_\infty=1$ for all $j\ge 1$, we conclude that $$\Lambda\le \Lambda (1-c\Lambda^2) + C\langle { \| P \|}_{\text{ND}} \rangle^{-\gamma}.$$ Subtracting $\Lambda$ from both sides and rearranging yields $$\label{swallowingresult} \Lambda^3 \le C\langle { \| P \|}_{\text{ND}} \rangle^{-\gamma}.$$ This completes the proof of Proposition \[prop:induction\], hence of Theorem \[thm:reduction\]. [20]{} M. Christ, [*Bounds for multilinear sublevel sets via Szemerédi’s theorem*]{}, preprint. M. Christ, X. Li, C. Thiele, and T. Tao, [*On multilinear oscillatory integrals, nonsingular and singular*]{}, Duke Math. J. 130 (2005), no. 2, 321–351. M. Christ and D. Oliveira e Silva, [*On trilinear oscillatory integrals*]{}, preprint. E. M. Stein, [*Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. With the assistance of Timothy S. Murphy*]{}. Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, 1993. [^1]: The author was supported in part by NSF grant DMS-0901569. [^2]: In fact, $Q$ is nondegenerate relative to $\{\pi_j\}_{j=1}^{n+1}$, if and only if $P$ is nondegenerate relative to $\{\pi_j\}_{j=0}^{n-1}$.

--- author: - 'V. D’Elia$^{1,2}$, E. Pian$^{3,4}$, A. Melandri$^{5}$, P. D’Avanzo$^{5}$, M. Della Valle$^{6,7}$, P.A. Mazzali$^{8,9,10}$, S. Piranomonte$^{1}$, G. Tagliaferri$^{5}$, L.A. Antonelli$^{1,2}$, F. Bufano$^{11,12}$, S. Covino$^{5}$, D. Fugazza$^{5}$, D. Malesani$^{13}$, P. Møller$^{14}$, E. Palazzi$^{3}$' title: 'SN 2013dx associated with GRB130702A: a detailed photometric and spectroscopic monitoring and a study of the environment [^1]' --- [Long-duration gamma-ray bursts (GRBs) and broad-line, type Ic supernovae (SNe) are strongly connected. We aim at characterizing SN 2013dx, which is associated with GRB130702A, through sensitive and extensive ground-based observational campaigns in the optical-IR band.]{} [We monitored the field of the Swift GRB 130702A (redshift z = 0.145) using the 8.2 m VLT, the 3.6 m TNG and the 0.6 m REM telescopes during the time interval between 4 and 40 days after the burst. Photometric and spectroscopic observations revealed the associated type Ic SN 2013dx. Our multiband photometry allowed constructing a bolometric light curve.]{} The bolometric light curve of SN 2013dx resembles that of 2003dh (associated with GRB030329), but is $\sim$10% faster and $\sim$25% dimmer. From this we infer a synthesized $^{56}$Ni mass of $\sim$0.2 M${_\odot}$. The multi-epoch optical spectroscopy shows that the SN 2013dx behavior is best matched by SN 1998bw, among the other well-known low-redshift SNe associated with GRBs and XRFs, and by SN 2010ah, an energetic type Ic SN not associated with any GRB. The photospheric velocity of the ejected material declines from $\sim 2.7\times 10^{4}$ km s$^{-1}$ at $8$ rest frame days from the explosion, to $\sim 3.5\times 10^{3}$ km s$^{-1}$ at $40$ days. These values are extremely close to those of SN1998bw and 2010ah. We deduce for SN 2013dx a kinetic energy of $\sim 35 \times 10^{51}$ erg and an ejected mass of $\sim 7$ M${_\odot}$. This suggests that the progenitor of SN2013dx had a mass of $\sim$25-30 M$_\odot$, which is 15-20% less massive than that of SN 1998bw. Finally, we studied the SN 2013dx environment through spectroscopy of the closeby galaxies: $9$ out of the $14$ inspected galaxies lie within $0.03$ in redshift from $z=0.145$, indicating that the host of GRB130702A/SN 2013dx belongs to a group of galaxies, an unprecedented finding for a GRB-associated SN and, to our knowledge, for long GRBs in general. Introduction ============ The connection between gamma-ray bursts and supernovae has been firmly established on the basis of a handful of nearby events ($z < 0.3$) for which a decent spectroscopic monitoring was possible (Galama et al. 1998; Patat et al. 2001; Hjorth et al. 2003; Stanek et al. 2003; Malesani et al. 2004; Pian et al. 2006; Bufano et al. 2012), which enabled deriving the physical properties of the SNe (Mazzali et al. 2006a,b; Woosley & Bloom 2006; Hjorth & Bloom 2012). In these few cases, observations have revealed that SNe accompanying GRBs are explosions of bare stellar cores, that is, their progenitors (whose estimated mass is higher than $\sim$20 M$_\odot$) have lost all their hydrogen and helium envelopes before collapse (a.k.a. supernovae of type Ic). A stripped star of Wolf-Rayet type (Crowther 2007) appears indeed more suited to favor the propagation of a relativistic jet. However, many unknowns still surround the GRB-SN connection: the properties of their progenitors (e.g., their multiplicity and metallicity, Podsiadlowski et al. 2010; Levesque et al. 2014), the nature of their remnants (e.g., Woosley & Heger 2012; Mazzali et al. 2014), and the formation and propagation of the jets through the stellar envelope (Zhang et al. 2003; Uzdensky & MacFadyen 2006; Fryer et al. 2009; Lyutikov 2011; Bromberg et al. 2014). The detailed analysis of the physics of the supernovae can elucidate these problems, and accurate optical spectroscopy is a fundamental tool to this aim. It is well known, in fact, that light curve models alone do not have the ability to uniquely determine all supernova parameters. In particular, while the mass of $^{56}$Ni can be reasonably well established from the light-curve peak if the time of explosion is known to within a good accuracy, or from the late, exponential decline phase of the light curve if this is observed, the two parameters that determine the shape of the light curve, the ejecta mass $M_{ej}$ and the kinetic energy $E_K$, are degenerate (Eq. 2; Arnett 1982). Only the simultaneous use of light curve and spectra can break this degeneracy (Mazzali et al. 2013, and references therein). Time-resolved optical spectra of GRB-SNe can be acquired only at $z \ls 0.3, $ however. At higher redshift, the subtraction of the host galaxy and afterglow components from the spectrum of the GRB optical counterpart lowers the signal-to-noise ratio dramatically, so that the resulting spectral residuals are noisy (e.g., Melandri et al. 2012; 2014). In fact, in the redshift range $0.3 \ls z \ls 1$ the current generation of telescopes can test the SN-GRB connection only on the basis of a single spectrum acquired at the epoch of maximum light (Della Valle et al. 2003; Soderberg et al. 2005; Berger et al. 2011; Sparre et al. 2011; Jin et al. 2013, Cano et al. 2014) or through the detection of rebrightenings in the GRB afterglow light-curves, due to emerging SNe (e.g. Bloom et al. 1999; Lazzati et al. 2001; Greiner et al. 2003; Garnavich et al. 2003; Masetti et al. 2003; Zeh et al. 2004; Gorosabel et al. 2005; Bersier et al. 2006; Della Valle et al. 2006; Soderberg et al. 2006, 2007; Cobb et al. 2010; Tanvir et al. 2010; Cano et al. 2011a,b; 2014). Finally, at redshift higher than one, even the detection of a rebrightening is in general not possible. SN 2013dx, associated with GRB130702A at $z=0.145$, represents a remarkable new entry into the group of the most consistently investigated SN-GRB events. Here, we present the results of our extensive photometric and spectroscopic campaign, carried out with the VLT, TNG, and REM telescopes, covering an interval of about $40$ days after GRB detection. In particular, we obtained 16 spectra spaced apart by 2-3 days. The paper is organized as follows: Section 2 summarizes the properties of GRB130702A and its associated SN 2013dx as reported in the literature; Sect. 3 introduces our dataset and illustrates the data reduction process; Sect. 4 presents our results; in Sect. 5 we draw our conclusions. We assume a cosmology with $H_0=70$ km s$^{-1}$ Mpc$^{-1}$, $\Omega_{\rm m} = 0.3$, $\Omega_\Lambda = 0.7$. GRB130702A/SN 2013 dx ===================== GRB130702A was detected by [*Fermi*]{}-LAT and -GBM instruments on July 02, 2013 at 00:05 UT (Fermi trigger 394416326, Cheung et al. 2013; Collazzi et al. 2013). In the following, this time represents our $t0$. The intermediate Palomar Transient Factory reported a possible counterpart after following up the burst in the optical band at the position RA(J2000) = $14h 29m 14.78s$ DEC(J2000) = +$15d 46' 26.4"$. The transient was $0.6"$ away from an SDSS faint source ($r = 23.01$) classified as a star, which was later identified as its host galaxy (Singer et al. 2013; Kelly et al. 2013). A [*Swift*]{} target-of-opportunity observation was then activated, and analysis of the XRT and UVOT data revealed a coincident, new X-ray source in the sky (D’Avanzo et al. 2013a). The fading behavior was confirmed both in the X-ray by subsequent XRT observations (D’Avanzo et al. 2013c) and in the optical bands by several ground-based telescopes (see, e.g., Guidorzi et al. 2013; Xu et al. 2013; D’Avanzo et al. 2013b). GRB130702A was also observed by Konus-Wind in the $20-1200$ keV band (Golenetskii et al. 2013) and by INTEGRAL (Hurley et al. 2013) in the mm (Perley and Kasliwal 2013) and in the radio band (van der Horst 2013). The redshift of the transient was reported to be $z=0.145$ (Mulchaey et al. 2013a; D’Avanzo et al. 2013d; Mulchaey et al. 2013b). The emerging of a new supernova associated with GRB130702A was detected by several spectroscopic measurements (Schulze et al. 2013; Cenko et al. 2013; D’Elia et al. 2013), and was named SN 2013dx (Schulze et al. 2013b, D’Elia et al. 2013b). Observations and data reduction =============================== We observed the field of SN 2013dx in imaging and spectroscopic modes using VLT, TNG, and REM. Details on the data acquisition and reduction process are given below. Figure 1 shows the SN 2013dx field, with two nearby galaxies clearly visible. Details on the analysis of this field can be found in Sect. 4.6. {width="9cm"} Photometry ---------- ### TNG photometry TNG photometry was conducted using the DOLORES camera in imaging mode, with the SDSS g,r,i,z and the Johnson U filters. Image reduction was carried out by following the standard procedures: subtraction of an averaged bias frame, division by a normalized flat frame. Astrometry was performed using the USNOB1.0[^2] catalogs. Aperture photometry was made with the Starlink[^3] PHOTOM package. The aperture was set to $10$ pixels, which is equivalent to $2.5"$. The calibration was made against the SDSS catalog and Landolt standard field stars (for the U filter). To minimize any systematic effect, we performed differential photometry with respect to local isolated and non-saturated standard stars selected from these two catalogs. The log of the TNG photometric observations can be found in Table 1. Date$^a$ t-t0 (d)$^b$ texp(s)$^c$ filt.$^d$ mag$^e$ flux ($\mu$Jy)$^f$ ---------- -------------- ---------------- ----------- ---------------- -------------------- 0705 $ 3.89692$ $ 2\times300$ U $19.95\pm0.04$ $23.4\pm0.9$ 0720 $ 18.90617$ $ 5\times120$ U $21.27\pm0.26$ $7.0\pm1.9$ 0723 $ 21.90367$ $ 10\times120$ U $21.94\pm0.27$ $3.7\pm1.0$ 0804 $ 33.89719$ $ 5\times240$ U $22.66\pm0.28$ $1.9\pm0.6$ 0703 $ 1.96469$ $ 2\times300$ g $19.29\pm0.06$ $80.4\pm4.3$ 0705 $ 3.88074$ $ 1\times300$ g $20.12\pm0.08$ $37.4\pm2.8$ 0714 $ 12.93708$ $ 1\times300$ g $20.33\pm0.12$ $30.9\pm3.5$ 0716 $ 14.92913$ $ 1\times300$ g $20.40\pm0.10$ $28.9\pm2.9$ 0717 $ 15.90906$ $ 1\times300$ g $20.38\pm0.04$ $29.6\pm1.0$ 0720 $ 18.89108$ $ 1\times300$ g $20.76\pm0.07$ $20.7\pm1.4$ 0723 $ 21.88002$ $ 1\times300$ g $21.05\pm0.10$ $16.0\pm1.6$ 0703 $ 1.95541$ $ 2\times300$ r $19.10\pm0.01$ $91.9\pm1.2$ 0705 $ 3.89015$ $ 1\times300$ r $19.84\pm0.02$ $46.6\pm0.8$ 0709 $ 7.91513$ $ 1\times300$ r $19.91\pm0.05$ $43.5\pm2.1$ 0714 $ 12.94571$ $ 1\times300$ r $19.77\pm0.03$ $49.8\pm1.2$ 0716 $ 14.92508$ $ 1\times300$ r $19.76\pm0.03$ $50.2\pm1.2$ 0717 $ 15.92203$ $ 1\times300$ r $19.71\pm0.06$ $52.6\pm2.7$ 0720 $ 18.89967$ $ 1\times300$ r $19.85\pm0.04$ $46.1\pm1.6$ 0723 $ 21.89224$ $ 1\times300$ r $20.06\pm0.04$ $38.1\pm1.4$ 0804 $ 33.88716$ $ 1\times300$ r $20.92\pm0.06$ $17.3\pm1.0$ 0703 $ 1.97366$ $ 2\times300$ i $18.97\pm0.05$ $101\pm4.5$ 0705 $ 3.88548$ $ 1\times300$ i $19.89\pm0.04$ $43.5\pm1.7$ 0714 $ 12.94145$ $ 1\times300$ i $19.95\pm0.05$ $41.2\pm2.0$ 0716 $ 14.92082$ $ 1\times300$ i $19.95\pm0.06$ $41.2\pm2.4$ 0717 $ 15.92766$ $ 1\times300$ i $19.89\pm0.07$ $43.3\pm2.7$ 0720 $ 18.89542$ $ 1\times300$ i $19.82\pm0.06$ $46.5\pm2.8$ 0723 $ 21.88747$ $ 1\times300$ i $20.15\pm0.06$ $34.1\pm1.9$ 0705 $ 3.90667$ $ 5\times120$ z $19.71\pm0.03$ $50.1\pm1.3$ : **Log of the TNG photometric observations** $^a$: Observation date (month-day) $^b$: Time after the GRB (observer frame) $^c$: Exposure time $^d$: Adopted filter $^e$: Magnitudes in AB system, except U-band magnitudes, which are Vega (not corrected for the Galactic extinction) $^f$: Fluxes (corrected for the Galactic extinction). ### VLT photometry VLT photometry was carried out using the FORS2 camera in imaging mode with the Johnson U,B,R,I filters. Image reduction was performed using the same standard techniques as were adopted for the TNG data. Images were calibrated against Landolt standard field stars. Again, differential photometry was performed to minimize any systematic effects. The aperture was set to $10$ pixels, which is equivalent to $1.3"$. The log of the VLT photometric observations can be found in Table 2. Date$^a$ t-t0 (d)$^b$ texp(s)$^c$ filt.$^d$ mag$^e$ flux ($\mu$Jy)$^f$ ---------- -------------- ------------- ----------- ------------------ -------------------- 0710 8.032 $2\times60$ u $ 21.36\pm0.04$ $12.7\pm0.5$ 0711 9.974 $2\times60$ u $ 21.42\pm0.04$ $12.0\pm0.5$ 0713 11.966 $2\times60$ u $ 21.51\pm0.07$ $11.1\pm0.7$ 0716 14.974 $2\times60$ u $ 21.70\pm0.14$ $9.3 \pm1.3$ 0720 18.971 $2\times60$ u $ 22.53\pm0.60$ $4.3 \pm3.2$ 0722 20.972 $2\times60$ u $ 22.54\pm0.51$ $4.3 \pm2.6$ 0727 25.975 $6\times60$ u $ 23.06\pm0.12$ $2.7 \pm0.3$ 0730 28.980 $8\times60$ u $ 23.21\pm0.11$ $2.3 \pm0.2$ 0804 33.002 $8\times60$ u $ 23.20\pm0.16$ $2.3 \pm0.4$ 0806 35.966 $8\times60$ u $ 23.24\pm0.15$ $2.2 \pm0.3$ 0810 39.984 $8\times60$ u $ 23.46\pm0.21$ $1.8 \pm0.4$ 0710 8.034 $1\times60$ g $ 20.20\pm0.01$ $37.2\pm0.3$ 0711 9.976 $1\times60$ g $ 20.12\pm0.01$ $39.8\pm0.4$ 0713 11.968 $1\times60$ g $ 20.30\pm0.01$ $33.9\pm0.4$ 0716 14.976 $1\times60$ g $ 20.40\pm0.03$ $30.8\pm0.8$ 0720 18.973 $1\times60$ g $ 20.85\pm0.07$ $20.3\pm1.4$ 0722 20.974 $1\times60$ g $ 20.87\pm0.09$ $20.0\pm1.8$ 0727 25.979 $2\times60$ g $ 21.21\pm0.03$ $14.6\pm0.4$ 0730 28.987 $2\times60$ g $ 21.67\pm0.03$ $9.5 \pm0.2$ 0804 33.009 $2\times60$ g $ 21.85\pm0.04$ $8.1 \pm0.3$ 0806 35.973 $2\times60$ g $ 22.12\pm0.04$ $6.3 \pm0.2$ 0810 39.991 $2\times60$ g $ 22.17\pm0.06$ $6.0 \pm0.3$ 0710 8.035 $1\times60$ r $ 19.99\pm0.01$ $44.9\pm0.3$ 0710 8.040 $1\times60$ r $ 19.97\pm0.01$ $45.7\pm0.3$ 0711 9.977 $1\times60$ r $ 19.91\pm0.01$ $48.4\pm0.3$ 0711 9.982 $1\times60$ r $ 19.90\pm0.01$ $48.9\pm0.3$ 0713 11.969 $1\times60$ r $ 19.84\pm0.01$ $51.6\pm0.3$ 0716 14.977 $1\times60$ r $ 19.82\pm0.01$ $52.7\pm0.5$ 0720 18.974 $1\times60$ r $ 19.93\pm0.01$ $47.4\pm0.6$ 0722 20.968 $1\times60$ r $ 19.94\pm0.02$ $47.1\pm0.9$ 0727 25.981 $1\times60$ r $ 20.25\pm0.01$ $35.4\pm0.3$ 0730 28.989 $1\times60$ r $ 20.35\pm0.01$ $32.2\pm0.3$ 0804 33.011 $1\times60$ r $ 20.68\pm0.02$ $23.7\pm0.3$ 0806 35.975 $1\times60$ r $ 20.95\pm0.02$ $18.6\pm0.3$ 0810 39.994 $1\times60$ r $ 21.20\pm0.03$ $14.8\pm0.4$ 0710 8.037 $1\times60$ i $ 20.22\pm0.02$ $36.3\pm0.5$ 0711 9.979 $1\times60$ i $ 20.14\pm0.01$ $39.3\pm0.4$ 0713 11.971 $1\times60$ i $ 20.07\pm0.01$ $41.8\pm0.5$ 0716 14.978 $1\times60$ i $ 19.93\pm0.02$ $47.3\pm0.7$ 0720 18.976 $1\times60$ i $ 19.76\pm0.08$ $55.3\pm4.0$ 0722 20.977 $1\times60$ i $ 19.99\pm0.02$ $44.8\pm1.0$ 0727 25.982 $1\times60$ i $ 20.23\pm0.02$ $35.9\pm0.6$ 0730 28.990 $1\times60$ i $ 20.28\pm0.02$ $34.3\pm0.6$ 0804 33.013 $1\times60$ i $ 20.49\pm0.05$ $28.4\pm1.3$ 0806 35.977 $1\times60$ i $ 20.60\pm0.03$ $25.5\pm0.6$ 0810 39.995 $1\times60$ i $ 21.02\pm0.06$ $17.4\pm0.9$ : **Log of the VLT photometric observations** $^a$: Observation date (month-day) $^b$: Time after the GRB (observer frame) $^c$: Exposure time $^d$: Adopted filter $^e$: Magnitudes in AB system (not corrected for the Galactic extinction) $^f$: Fluxes (corrected for the Galactic extinction). ### REM photometry Optical and NIR observations were performed with the REM telescope (Zerbi et al. 2001; Chincarini et al. 2003; Covino et al. 2004) equipped with the ROSS2 optical imager and the REMIR NIR camera. The ROSS2 instrument is able to observe simultaneously in the g, r, i, and z SDSS filters. Observations of GRB130702A/SN2013dx were carried out over fifteen epochs between 2013 July 10 and Aug 13. Image reduction was carried out by following the same standard procedures as for the TNG and VLT photometry, including differential photometry. The aperture was set to $10$ pixels, which is equivalent to $5"$. Astrometry was performed using the USNOB1.0 and the 2MASS[^4] catalogs for the optical and NIR frames, respectively. The calibration was made against the SDSS catalog for the optical filters and the 2MASS catalog for NIR filters. The log of the REM photometric observations can be found in Table 3, where we only report the g and r photometry. In the i band the S/N was too low, and the SN 2013dx too much contaminated by the host galaxy to obtain firm detections. In the z and H bands we did not detect the SN down to typical upper limits of $z\sim 19.7$ (AB, $3\sigma$) and $H\sim 18$ (Vega, $3\sigma)$. The columns are organized in the same way as in Table 1. Date$^a$ t-t0 (d)$^b$ texp(s)$^c$ filt.$^d$ mag$^e$ flux ($\mu$Jy)$^f$ ---------- -------------- -------------- ----------- ---------------- -------------------- 0711 $9.04484 $ $9\times300$ g $20.54\pm0.19$ $25.4\pm9.2$ 0713 $11.05959$ $9\times300$ g $20.56\pm0.08$ $25.1\pm3.6$ 0715 $13.10455$ $9\times300$ g $20.40\pm0.11$ $28.9\pm5.9$ 0716 $14.10692$ $9\times300$ g $20.45\pm0.13$ $27.6\pm6.9$ 0718 $16.06110$ $9\times300$ g $20.32\pm0.22$ $31.3\pm13$ 0719 $17.09980$ $9\times300$ g $21.56\pm0.41$ $10.0\pm8.7$ 0725 $22.99765$ $9\times300$ g $21.36\pm0.13$ $12.0\pm3.0$ 0711 $ 9.04484$ $9\times300$ r $19.96\pm0.09$ $43.5\pm4.9$ 0713 $11.05959$ $9\times300$ r $19.91\pm0.07$ $45.3\pm4.1$ 0715 $13.10455$ $9\times300$ r $19.75\pm0.10$ $52.6\pm6.7$ 0716 $14.10692$ $9\times300$ r $19.62\pm0.14$ $59.5\pm10$ 0718 $16.06110$ $9\times300$ r $19.81\pm0.16$ $49.8\pm10$ 0719 $17.09980$ $9\times300$ r $19.88\pm0.16$ $46.6\pm9.5$ 0725 $22.99765$ $9\times300$ r $20.01\pm0.15$ $41.6\pm7.4$ : **Log of the REM photometric observations** $^a$: Observation date (month-day) $^b$: Time from the GRB (observed frame) $^c$: Exposure time $^d$: Adopted filter $^e$: Magnitudes in AB system (not corrected for the Galactic extinction) $^f$: Fluxes (corrected for the Galactic extinction). {width="9cm"} Spectroscopy ------------ ### TNG spectroscopy TNG spectroscopy was carried out using the DOLORES camera in slit mode, with the LR-B grism. This configuration covers the spectral range $3000-8430$Å with a resolution of $\lambda/\Delta\lambda = 585$ for a slit width of $1$” at the central wavelength $5850$Å. Spectra were acquired at five epochs. A slit width of $1$” was used in all but the first observation, for which a $1.5$” slit was adopted, owing to a seeing angle higher than $1$”. The slit position angle was set to the parallactic angle in all our observations. Table 4 contains a summary of the TNG spectroscopic observations. The spectra were extracted using standard procedures (bias and background subtraction, flat fielding, wavelength and flux calibration) under the packages ESO-MIDAS[^5] and IRAF[^6]. Ne-Hg or helium lamps and spectrophotometric stellar spectra acquired in the same nights as the target were used for wavelength and flux calibration. To account for slit losses, we matched the flux-calibrated spectra with our multiband TNG photometry acquired in the same nights. ### VLT spectroscopy VLT spectroscopy was carried out using the FORS2 camera in slit mode, with the 300V grism. This configuration covers the spectral range $3300-9500$ Å with a resolution of $\lambda/\Delta\lambda = 440$ for a slit width of $1$” at the central wavelength $5900$ Å. Eleven spectroscopic epochs were acquired. A slit width of $1$” was used in all observations. The slit position angle was set to different values in each observation to place different field galaxies in the slit to study the SN 2013dx surroundings (see Sect. 4.2 for details). Table 5 gives a summary of the VLT spectroscopic observations. As for the TNG data, the spectra were extracted using the packages ESO-MIDAS and IRAF. A He+Ag/Cd+Ar lamp and spectrophotometric stars acquired the same night of the target were used for wavelength and flux calibration. For a few epochs, some spectrophotometric stars were not observed in the same night as the target. In these cases, the flux calibration was performed using archival data. In none of these epochs, however, was the difference beween the observation time of the scientific files and the adopted standard longer than three days. To account for slit losses, we checked the flux-calibrated spectra against our simultaneous VLT multiband photometry. Results ======= Optical light curves -------------------- Our optical photometric data, only corrected for Galactic extinction, are shown in Fig. 2. The SN component starts to emerge and dominate its host galaxy and GRB optical afterglow at about $\text{five}$ days after the GRB explosion. We fit empirical curves to the data past day five to determine the peak times of SN2013dx in the different bands. The results in the rest frame are $T_{peak,i} = 16 \pm 2$ days, $T_{peak,r} = 15 \pm 1$ days, $T_{peak,g} = 10 \pm 2$ days, and $T_{peak,U} = 8 \pm 1$ days. SN 2013dx is one of the few cases in which a SN is observed in the U band. The U peak is observed very early, at about one week in the rest-frame. As commonly observed in other SNe, the peak occurs at later times while moving to redder bands. As an example, the I-band peak is observed about a week later than that in the U-band. {width="18cm"} Optical spectra --------------- First, all the VLT and TNG optical spectra of SN 2013dx were dereddened using the Galactic value $E(B-V)=0.04$ mag toward its line of sight (Schlegel, Finkbeiner, & Davis 1998). The intrinsic extinction appears to be negligible both from X-ray data and from analysis of the X-ray-to-optical spectral energy distribution. Then, we estimated the host galaxy contribution. Since SN 2013dx was still bright at the time of our last observation (10 Aug.) and later on its field became not observable anymore because of solar constraints, we could not acquire a reliable spectrum to subtract the host galaxy contribution from our data. Therefore, we considered the SDSS magnitudes of the host galaxy, $u=24.42\pm 0.96$, $g= 23.81\pm0.85$, $r = 23.01\pm0.24$ $i=23.22\pm0.39,$ and $z=23.04\pm0.59$, and noted that after they were reduced to rest-frame, they were consistent with the normalized template of a star-forming galaxy with moderate intrinsic absorption ($E(B-V) <0.10$, Kinney et al. 1996). Then we subtracted this rescaled template from our dereddened fluxes. The host galaxy contributes about 15% of the total flux, and possibly less, because not all its light is captured in the slit when acquiring the SN spectra. Finally, we subtracted the afterglow contribution. The afterglow light curve can be modeled with a broken power law, as shown in Singer et al. (2013). We thus adopted their first decay index ($\alpha_1=0.57$) and their temporal break $t_{b}=1.17$ days. However, the second decay index $\alpha_2$ does not take into account the emerging SN contribution. $\alpha_2=1.85$ is the lowest index for which the afterglow is not oversubtracted in our early time photometry, while $\alpha_2=2.5$ is the highest index allowed by the closure relations linking spectral and temporal indices (Zhang & Mészáros 2004). Thus, we chose an average $\alpha_2 = 2.2$. The adopted spectral index is $\beta_{\nu}=0.7$ (Singer et al. 2013). Even assuming the two extreme decay indices, the difference in the afterglow contribution can only be appreciated in the first three spectra, and it is lower than 20% in the worst case. Figure 3 shows our sixteen dereddened, host- and afterglow-subtracted TNG and VLT spectra. The spectra are shown starting from $4000$Å in the rest-frame, because the flux calibration becomes unreliable at shorter wavelengths. In Fig. 4 we compare in the rest-frame 11 of our spectra with those of SN 1998bw at comparable phases after explosion (Patat et al. 2001). Each SN 1998bw spectrum is scaled in flux by an arbitrary constant to find the best match with our spectral dataset. In most pairs of spectra the same continuum shape and broad absorption features can be seen, even if some diversities are present (at t$<20$ d the $4400$Å pseudo-emission peak is completely absent in 1998bw; the pseudo-peak around $6300$Å in 1998bw is absent in 2013dx; the strong absorption around $7000$Å in 1998bw is much less pronounced in 2013dx at t$<15$ d). This fact leads us to classify SN 2013dx as a broad-line type Ic SN, that is, one with a highly stripped progenitor (no hydrogen or helium left before explosion), similar to those of previously studied GRB- and XRF-SNe (Mazzali et al. 2006a,b). Other GRB-associated SNe, such as 2003dh, 2006aj, and 2010bh, do not compare spectrally as convincingly with SN 2013dx. {width="9cm"} Of the broad-lined SNe that are not accompanied by GRBs/XRFs, the type Ic SN 2010ah shows a remarkable spectral similarity. In Fig. 5 we show the comparison of the models that best describe the two available spectra of SN2010ah (Mazzali et al. 2013) with the spectra of SN2013dx (corrected as detailed above) taken at comparable phases. The agreement between the two SNe is quite good at $4500 \ls \lambda \ls 6000$ Å, where the most relevant absorption features are located. At wavelengths shorter than $4500$ Å some residual contamination from the afterglow or a shock breakout component (e.g., Campana et al. 2006; Ferrero et al. 2006) might still be present in the SN2013dx spectrum, while at wavelengths higher than $6000$ Å the model shown for SN2010ah clearly does not match SN 2013dx accurately. We note that both SNe show a SiII absorption line ($\sim6000$ Å) and O I absorption line ($\sim7300$ Å), but they are weaker in SN2013dx than in SN2010ah. This may suggest a reduced abundance of silicon and oxygen in the former SN. {width="9cm"} Bolometric light curve ---------------------- We computed a bolometric light curve in the range $3000--10000$ Åfrom our multicolor light curves. After correcting the photometric data in an analogous way as done for the spectra (Galactic extinction, host galaxy, and afterglow contribution, see Sect. 4.2) and after applying k-corrections using our VLT spectra (that cover a wider wavelength range than the TNG spectra), we splined the residual monochromatic light curves, which should represent the supernova component, and the broad-band flux at each photometric observation epoch was integrated. The flux was linearly extrapolated blueward of the U-band flux down to 3000 Å and redward of the I-band flux to $10000$ Å. The result is reported in Fig. 6. The errors associated with the photometry, host galaxy template, and afterglow were propagated and summed in quadrature. The figure also shows the bolometric light curves of SN 1998bw (Patat et al. 2001), SN 2006aj (Pian et al. 2006), SN 2010bh (Bufano et al. 2012), SN 2003dh (Hjorth et al. 2003), SN 2003lw (Mazzali et al. 2006a), SN 1994I (Richmond et al. 1996) and SN 2012bz (Melandri et al. 2012). {width="9.2cm"} Photospheric velocities ----------------------- To estimate the velocity of the SN ejected material, we started out from the general similarity between our spectra and those of SN1998bw. Following Patat et al. (2001, see their Fig. 5), we identified the [$\lambda$6355]{} feature and followed its redward shift along our spectra. In the spectra acquired between 17 and 22 July, the cosmological redshift and the blueshift due to the photospheric expansion combine to cause this absorption doublet to overlap with the telluric feature at 6870 Å. For these spectra, the position of the [$\lambda$6355]{} cannot be measured. ![Temporal evolution of the photosphere expansion velocity of SN 2013dx (large circles) measured directly on the spectra from the position of the minimum of the [$\lambda$6355]{} absorption line. For comparison we report the photospheric velocities of other core-collapse Ic SNe (1994I, small circles; 2002ap, stars; 2003dh, triangles; 2006aj, small filled squares), as derived from spectral models (Sauer et al. 2006; Mazzali et al. 2002; 2003; 2006b). For SN1998bw we report both the velocities derived from direct measurements (crosses, Patat et al. 2001) and those derived from models (large open squares, Nakamura et al. 2001).](Photo4.pdf){width="9cm"} The time evolution of the photosphere expansion velocity as deduced from the minimum of the absorption trough of the $\lambda$6355 line is shown in Fig. 7 (large circles). The velocity decline extends from $\sim 2.7\times 10^{4}$ km s$^{-1}$ at $8$ rest frame days from the explosion to $\sim 3.5\times 10^{3}$ km s$^{-1}$ at $40$ days. Figure 7 also displays the photospheric velocities of other SNe, in particular that of SN 1998bw, computed by Patat et al. (2001) using the same procedure as that adopted by us. The photospheric velocities of SN 2013dx and SN 1998bw agree very well, although at very late time the latter retains a slightly higher velocity expansion (see also Fig. 4 of Patat et al. 2001). Comparing the expansion velocity of SN 2013dx with that of other SNe, we find that the former is higher than that of SN 1994D (Patat et al. 1996), SN 1994I (Millard et al. 1999), SN 1997ef (Patat et al. 2001) and SN 2006aj (Mazzali et al. 2006), but not as extreme as that of SN 2003dh (Hjorth et al. 2003). Physical parameters of SN 2013dx -------------------------------- The bolometric light curve of SN 2013dx shows that the peak luminosity of this SN is intermediate between those of the nearby GRB-SNe 1998bw and 2003dh on one hand and those of the two XRF-SNe 2006aj and 2010bh (Fig. 6) on the other. The spectrum of SN 2013dx is much more similar to that of the most energetic Ic SNe, and especially to SN 2010ah, which, among the broad-lined Ic SNe not accompanied by a GRB/XRF, is the most spectroscopically similar to SN 1998bw (Mazzali et al. 2013), although because of its lower ejecta mass, it has a significantly lower kinetic energy ($1.2 \times 10^{52}$ erg vs $5 \times 10^{52}$ erg inferred by Nakamura et al. 2001 for SN 1998bw). The similarity of the light-curve shape of SN 2013dx to the shapes of those of SN 1998bw and SN 2003dh and the spectral resemblance to SN 1998bw and SN 2010ah led us to adopt these three previously known and well-studied SNe as “templates” for estimating the physical parameters of SN 2013dx through the relationships that link the SN ejecta mass $M_{ej}$ and kinetic energy $E_K$ to the observables, that is, to the width of the bolometric light curve and the photospheric velocities (Arnett 1982, see Sect. 4.1 in Mazzali et al. 2013). This method, described in detail by Mazzali et al. (2013; see also Valenti et al. 2008; Walker et al. 2014), should be applied with caution when only few SNe are available that provide a good match and the data coverage of both target and templates is not excellent. However, when the observational information is adequate, the outcome satisfactorily agrees with the results of modeling based on radiative transport: for SN 2010ah, which has a well-sampled light curve, but only two spectra taken around maximum light, the physical parameters obtained with the two approaches differ by no more than 25% (Mazzali et al. 2013). The dataset presented here for SN 2013dx is detailed and rich, and we can compare it with as many as three template SNe with good data for which models were developed (Nakamura et al. 2001; Deng et al. 2005; Corsi et al. 2011; Mazzali et al. 2003; 2006a; 2013). This allows us to use the rescaling method to describe the physics of SN 2013dx with good accuracy. Developing a dedicated model is beyond the scope of this paper, but will be presented in the future. Based on the similarity of the light-curve shape of SN2013dx and SN2003dh, we adapted the model curve of 2003dh (Mazzali et al. 2003; Deng et al. 2005) by compressing it by 10% in time and dimming it by 25% in flux. We thus obtained the synthetic curve reported in Fig. 6 (solid line). Since SN 2003dh produced an estimated $^{56}Ni$ mass of $\sim$ 0.35 M$_\odot$ (Mazzali et al. 2006a), the 25% flux dimming corresponds to a $^{56}Ni$ mass of 0.26 M$_\odot$. However, this is further reduced by the effect of the 10% faster temporal evolution of SN 2013dx with respect to SN 2013dx. By taking this into account and assuming $^{56}Ni$ decay dominates at early times (i.e., neglecting the contribution of $^{56}Co$ decay), we obtain a further reduction of 20%, meaning that the final estimated $^{56}Ni$ mass synthesized by SN 2013dx should be $\sim$0.2 M$_\odot$. From a spectroscopic point of view, SN2013dx is a close analog of the GRB-SN prototype SN1998bw and a very close analog of the broad-lined type Ic SN2010ah, although in the latter case the comparison is limited to only two spectra. This suggests that the photospheric velocities are similar and that the kinetic energy of SN2013dx may then just simply scale like the ejecta mass. Using the observed light curve width $\tau$ and measured photospheric velocities of SNe 1998bw, 2003dh, 2013dx (Figs. 6 and 7) and SN 2010ah (Mazzali et al. 2013), we applied the scaling relationships to SN2013dx and to each of the adopted template SN. Then we averaged the results and found for SN2013dx $M_{ej} = 7 \pm 2$ M$_\odot$ and $E_K = (35 \pm 10) \times 10^{51}$ erg. Our errors on these physical quantities reflect conservatively the empirical nature of the method and its uncertainties (for instance, the fact that the phases at which the measured photospheric velocities are compared are never identical for the target and the template SN). The progenitor of SN 2013dx is thus probably $25-30 M_{\odot}$, which is $15- 20\%$ less massive than that estimated for SN 1998bw ($30-35 M_{\odot}$, Maeda et al. 2006) Field of SN2013dx ----------------- The field of SN 2013dx shows a bright galaxy southward of the transient, at $\sim 8"$ from the SN position (see Fig. 8). This could have been the SN host, since the faint source ($R=23.01$) at $0.6"$ from SN 2013dx was initially misclassified as a star in the SDSS (Singer et al. 2013). For this reason, Leloudas et al. (2013) obtained a spectrum of this galaxy, reporting a $z=0.145$, which is the same as that of the transient (Mulchaey et al. 2013a,b; D’Avanzo et al. 2013d). Kelly et al. (2013) performed an extensive study of this galaxy and its outskirt, concluding that the SN 2013dx host galaxy is a dwarf satellite of the bright one. Many more galaxies are present in the SN 2013dx field. Kelly et al. 2013 noted that many of them have an SDSS photometric redshift compatible with that of the host and the bright galaxy. In addition, they also pointed out that the SDSS spctroscopic galaxy survey targeted five galaxies brighter than $17.7$ within 15’ from SN 2013dx, their redshift being close to $z=0.145$. We decided to use different slit position angles during our VLT spectroscopic campaign. This allowed us to study both the spectroscopic evolution of the SN and to determine the redshift of more nearby galaxies. The different position angles adopted are described in Table 5, and the corresponding slit orientations are displayed in Fig. 8. This figure also marks with numbers the $14$ galaxies that we placed in the slit in our observations (“S” marks a star). We determine the redshift of all these galaxies, although for three of them the determination is not completely certain because it relies on faint absorption or emission features. Table 6 illustrates the redshift of the galaxies, which are numbered as in Fig. 8. The uncertain redshifts are marked with “?”, while the last two columns of the table list the emission and absorption features on which the redshift determination is based. The spectra of our $14$ field galaxies are shown in appendix A1. It is interesting to note that $9$ out of $14$ galaxies lie within $0.03$ from $z=0.145$, that is, the redshift of SN 2013dx, and one more is within $0.2$. These $\text{ten}$ galaxies are marked in red in Fig. 8, while the remaining $\text{\\ four}$ are marked in blue. In particular, we confirm the photometric redshifts reported for two of the field galaxies in Fig. 2 of Kelly et al (2013), their sources S3 and S5. This is clear evidence that SN 2013dx occurred in a group or a small cluster of galaxies. This conclusion is strengthened by the fact that many more galaxies in the SDSS, falling just outside the field of view of our images, have a spectroscopic or photometric redshift consistent with $z=0.145$. In detail, since the angular separation among the two farthest galaxies with the same redshift in the field is $\sim 3'$, the physical extent of the group at $z=0.145$ is $\sim 600$ kpc. This is the first SN associated with a long GRB detected in such an environment (and, to our knowledge, the first long GRB in general). For the SN 1998bw environment, a candidate cluster was first claimed (Duus & Newell 1977), but then not confirmed by spectroscopic analysis (Foley et al. 2006). For SN 2012bz, associated with GRB120422A, two objects have been reported at the redshift of the GRB (Schulze et al 2014). However, this looks like an interacting system consisting of two or possibly three galaxies, and not like a group. Intuitively, one may argue that the probability of hosting a GRB should be higher in the largest galaxy of the cluster, where the maximum star-formation occurs. However, the dwarf galaxies have comparatively high or higher specific star-formation rates, which might bias the probability of hosting a GRB in their favor. Indeed, studies on complete samples of GRBs (see, e.g., Perley et al. 2014; Vergani et al. 2014) show that at $z<1$ long GRBs strongly prefer low-mass galaxies (confirming previous studies on incomplete samples). This is probably because GRB progenitors are more easily developed in low-metallicity environments, which possess a high specific SFR. {width="9cm"} Conclusions =========== We have presented an extensive and sensitive ground-based observational campaign on the SN associated with GRB130702A at $z =0.145$, one of the nearest GRB-SNe detected so far. Its relative proximity guaranteed the construction of a nice dataset with a good S/N. The properties of SN 2013dx are similar to those of previous GRB- and XRF-SNe: the peak luminosity is intermediate between those of GRB-SNe and XRF-SNe, and the photospheric velocities are more similar to those of GRB-SNe. Accordingly, the physical parameters of SN 2013dx, derived with the empirical method based on the rescaling of the quantities known for other SNe, are similar to those determined for the previous GRB-SNe, but somewhat lower than those of SN 1998bw: we estimate a synthesized $^{56}Ni$ mass of $\sim$ 0.2 M$_\odot$, an ejecta mass of $M_{ej} \sim 7$ M$_\odot$, and a kinetic energy of $E_K \sim 35 \times 10^{51}$ erg. Furthermore, we performed a study of the SN 2013dx environment through spectroscopy of the field galaxies close to the host of GRB130702A. We find that $65\%$ of the observed targets have the same redshift as SN 2013dx, indicating that this is a group of galaxies. This represents the first report of a GRB-SN association taking place in a galaxy group or cluster. We thank an anonymous referee for several helpful comments. We are grateful to the ESO Director for awarding Discretionary Time to this project. We thank S. Valenti for several helpful discussions and F. Patat for providing the photospheric velocities for SN 1998bw. We thank the TNG staff, in particular G. Andreuzzi, L. Di Fabrizio, and M. Pedani, for their valuable support with TNG observations, and the Paranal Science Operations Team, in particular H. Boffin, S. Brillant, C. Cid, O. Gonzales, V. D. Ivanov, D. Jones, J. Pritchard, M. Rodrigues, L. Schmidtobreick, F. J. Selman, J. Smoker and S. Vega. The Dark Cosmology Centre is funded by the Danish National Research Foundation. VDE acknowledges partial support from PRIN MIUR 2009. This research was partially supported by INAF PRIN 2011, PRIN MIUR 2010/2011, and ASI-INAF grants I/088/06/0 and I/004/11/1. 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[^2]: http://www.nofs.navy.mil/data/fchpix/ [^3]: http://starlink.jach.hawaii.edu/starlink [^4]: http://www.ipac.caltech.edu/2mass/ [^5]: http://www.eso.org/projects/esomidas/ [^6]: http://iraf.noao.edu/

--- abstract: | The main focus of this work is to understand the dynamics of non regulated markets. The present model can describe the dynamics of any market where the pricing is based on supply and demand. It will be applied here, as an example, for the German stock market presented by the Deutscher Aktienindex (DAX), which is a measure for the market status. The duality of the present model consists of the superposition of the two components - the long and the short term behaviour of the market. The long term behaviour is characterised by a stable development which is following a trend for time periods of years or even decades. This long term growth (or decline) is based on on the development of fundamental market figures. The short term behaviour is described as a dynamical evaluation (trading) of the market by the participants. The trading process is described as an exchange between supply and demand. In the framework of this model there the trading is modelled by a system of nonlinear differential equations. The model also allows to explain the chaotic behaviour of the market as well as periods of growth or crashes.\ PCAS numbers: 01.75.+m, 05.40.+j, 02.50.Le\ Contribution to the technical seminar 22/12/98, DESY-IfH Zeuthen author: - | A. Schaale\ \ title: A dynamical model of non regulated markets --- The traditional approaches of pricing models (indices, stocks, currencies, gold, etc.) are related to combinations of economic figures like profit or cash-flow and their expected development. Indeed, these fundamental figures are related to the approximate price. However, it is well known that similar objects (companies, goods, ...) can be priced on the same market quiet different. One can observe quick changes in the pricing, which can’t be explained by any change of the underlying basic figures [@davidson]. The present model consists of two basic components: - (Long term trend) Scaling of the price (index) based on the long term development of basic figures [@mandelbrot1] - (Short term trend) Pricing by the exchange between buyers (optimists), sellers (pessimists) and neutral market members Studying, for example, the DAX $I$ for a time period of one decade one will recognise, that the basic trend $I_0(t)$ shows an exponential behaviour with deviations (fig. 1.). This trend can be presented as: $$I_0(t)=\hat{I}_0\,e^{\lambda t}, \;\; \mbox{with} \;\; \hat{I}_0, \lambda=const. \label{defi0}$$ The parameter $\hat{I}_0$ is the starting value: $\hat{I}_0(t)\equiv I_0(t=0)$. The growth rate $\lambda$ can variate on different markets. This parameter summarises all basic influences on the market, such as economic freedom, taxes, social-economic parameters, infrastructure and others. Comparing different markets one will find, that certain economics are growing (US, Europe) while others are declining over years (Japan [^1]). The value for the parameter $\hat{I}_0$ and $\lambda$ can be fitted from the historical market data using the least square method. The development $I_0(t)$ symbolises the average growth of the economy which is measured in various economic figures. The growth in (\[defi0\]) fulfils the Euler equation, describing the “natural” growth of unlimited systems: $$I_0'(t)-\lambda I_0(t)=0 \label{euler}$$ where $y'(t)\equiv \frac{d}{dt}y(t)$. The function (\[defi0\]) describes a real growth process. As far as there is no universal pricing model, the individual evaluations by the market participants differ and the price deviates from the fundamental average. These different evaluations which are changing in time lead to some kind of spontaneous oscillations. As far as each market has another scale it is useful to normalise the market index (price) to make different markets better comparable: $$I(t) \rightarrow i(t)=\frac{I(t)}{I_0(t)} \label{itrans}$$ The function (\[itrans\]) performs a normalisation which will project all indices of real markets to a unitarian index $i$ with a constant basic trend: $i_0(t) \equiv 1$ and $\lambda=0$. This way the development of markets can be compared in a single scheme. For further discussions it is necessary to define the market structure. A market is the totality of all market members participating in the trading process [@caldarelli]. The total amount of market members on normalised markets (\[itrans\]) is constant. The normalised DAX can be found in (fig. 2.). As already mentioned above, the subjective evaluations of the market status differ from each other [@davidson]. The market participants can be separated into three groups: optimists, pessimists and neutral market participants. Each group has a certain concentration which evolves in time $c_k(t)$. Based on the normalisation there is: $$c_o(t)+c_p(t)+c_n(t)=1, \label{marktnorm}$$ with $c_o(t)$, $c_p(t)$ and $c_n(t)$ as the corresponding concentrations [^2] The dynamics of the market is a result of the development of the $c_k(t)$ and the index $i(t)$. Each market group has certain features and react on market changes in a different way: - [**Optimists**]{} consider the market to be priced low. They want to buy. - [**Pessimists**]{} consider the market to be priced high. They want to sell. - [**Neutral market members**]{} consider the market to be priced fair. They are passive. The groups have different sizes. Comparing a typical daily trading volume with the total market capitalisation one will find that it is orders of magnitude smaller ($<1\%$). This leads to the following relation between the concentrations: $$c_o(t), c_p(t) << c_n(t). \label{relationen}$$ Using (\[marktnorm\]) the dimension of the problem reduces from 3 to 2 independent functions $c_p(t)$ and $c_p(t)$. The system dynamics can be written in the form of a system of differential equations: $$c_k'(t)= L_k\big(c_o(t),c_p(t),t\big),\;\; k=o,p \label{ct-system2}$$ Now it is necessary to describe in $L$ the structure of the market drivers, which determine the dynamics of trading: On non regulated markets there the price is determined by supply and demand. The ratio of the concentration of optimists and pessimists defines the price level [@farmer]. In general the functional relation between the concentrations of different market members and the index $i$ can be expressed in the following form: $$i(t)=f\Bigg(\frac{c_o(t)}{c_p(t)}\Bigg). \label{marktdruck}$$ At present it is not possible to derive the explicit form of $f$ from economic principles. The function $f$ expresses the [ *subjective*]{} evaluations of market participants. Here and in the following there will be made extensive use of Taylors theorem. Unknown functions will be expanded in Taylor series in order to parametrise them. As far as the higher order terms of each expansions will be neglected, it is possible to [*define*]{} the function $f$ in the following form: $$i(t)=f\Bigg(\frac{c_o(t)}{c_p(t)}\Bigg)\equiv \frac{c_o(t)}{c_p(t)} \label{ip}$$ In the equilibrium state there the equation (\[ip\]) gives sensible results: $$c_p(t)=c_o(t) \leftrightarrow i(t)=i_0(t)\equiv 1 \label{gleichgewicht}$$ After defining the basic conceptions there will be studied now the development of the concentrations $c_o(t)$ and $c_p(t)$. Their changes in time can be expressed by the following system of equations: $$\begin{aligned} c_o'(t) &=& F_o(\Delta i(t))+\xi_o U(t), \nonumber \\ c_p'(t) &=& F_p(\Delta i(t))-\xi_p U(t), \nonumber \\ \Delta i(t) &=& i(t)-1 \label{defcpprime} \end{aligned}$$ The system describes the exchange of concentrations as functions of the current index and as external influences. The functions $F_k(\Delta i(t)),\; k=o,p$ describe the subjective evaluations of the market members as a function of supply (pessimists) and demand (optimists). The function $U(t)$ represents an “external field”. It models effects that influence the market, but which are [*not*]{} related to the present value of the index $i(t)$. Typical external influences could be related to interest rates, taxes, political events or persons. The constants $\xi_k$ describe the difference in [*perception*]{} of external influences by the different market groups. The external influences lead to periods of continues optimism or depression, as they are observed on real markets. In general the functions $F_k$ in (\[defcpprime\]) are unknown. They will be expanded in Taylor series around the equilibrium state $i_0$: $$F_k(\Delta i(t))= \sum_{n=0}^{\infty} \alpha_{k,n} \cdot \big[\Delta i(t)\big]^n \label{taylor}$$ In the following the will be used the following approach: $$F_k(\Delta i(t))= \alpha_{k,1} \Delta i(t) + O([\Delta i(t)]^2). \label{fp2}$$ In case without external influences $U(t)=0$ it makes sense to assume that the system is symmetric concerning optimism and pessimism. Otherwise the system would follow a systematic trend, which has been already taken into account in (\[marktnorm\]). This leads to the relation $$F_p(\Delta i(t))=-F_o(\Delta i(t)) \equiv F(\Delta i(t))\; . \label{fop2}$$ On ideal markets the perception of external influences would be symmetric too. Real markets show deviations from this symmetry $\xi_o\neq \xi_p$. Performing a redefinition of $U(t) \rightarrow \xi_o U(t)$ one can substitute the the $\xi$ such as $\xi_o=1$ and $\frac{\xi_p}{\xi_o}=1+\varepsilon$, where $\varepsilon$ is an empirical parameter defining the asymmetry of perception of optimists and pessimists. Based on several reasonable assumptions, it has become possible to construct a nonlinear system of differential equations that reflects the market dynamics: $$\begin{aligned} && c'_o(t)-\alpha \Big[c_o(t) c_p^{-1}(t)-1\Big]-U(t)=0,\nonumber \\ && c'_p(t)+\alpha \Big[c_o(t) c_p^{-1}(t)-1\Big]+(1+\varepsilon)\, U(t)=0, \label{finalsystem} \end{aligned}$$ with the starting conditions: $$c_o(0)=c_{o0},\; c_p(0)=c_{p0}. \label{finalsystemstart}$$ and $\alpha \equiv \alpha_1$. The equations of system (\[finalsystem\]) describe the principal relation between the concentrations and the market index, where the exchange between the concentration levels can be performed in [*infinite small steps*]{} (continuum limit). That means that the ideal market would react on infinite small deviations from the equilibrium with infinite small trading reactions (exchange of fractions of stocks). This is not possible on real markets, which react with the exchange of [*finite sized trading units*]{}. This causes discontinuous changes of the index. Each new trading process is related to the former trading process which itself has caused a change of the index. One can realize this discontinuous trading behaviour by transforming the system of differential equations (\[finalsystem\]) into a system of logistic equations, where the trading process becomes described as a [*sequence of finite exchange transactions*]{} [@caldarelli; @busshaus]: $$\begin{aligned} c_o^{(n+1)} &=& c_o^{(n)}+ \Delta c_o^{(n)}, \nonumber \\ c_p^{(n+1)} &=& c_p^{(n)}+ \Delta c_p^{(n)}, \nonumber \\ \Delta c_o^{(n)} &=& \alpha \Big[c_o^{(n)} \Big(c_p^{(n)}\Big)^{-1}-1\Big] + U^{(n)},\nonumber \\ \Delta c_p^{(n)}&=& -\alpha \Big[c_o^{(n)} \Big(c_p^{(n)}\Big)^{-1}-1\Big]-(1+\varepsilon)\, U^{(n)}, \nonumber \\ U^{(n)}&\equiv& U(t_n), \nonumber \\ n&=&0,1,... \label{finalsystemlog} \end{aligned}$$ with the starting conditions $$c_o^{(0)}=c_o(0),\;\; c_p^{(0)}=c_p(0). \label{finalsystemstartlog}$$ Now there will be shown the results of the application of the model to real markets. At first there will be studied growth periods and crashes, which are observed regularly on all financial markets. Using historical data of the DAX one can find, that the growth periods are caused by an exponential growing external optimism $$U(t)=U_0\,\Big(e^{\beta(t-t_0)}-1\Big) \rightarrow U^{(n)}=U_0\Big(e^{\beta(t_n-t_0)}-1\Big)\; . \label{uexp}$$ As one can see in (fig. 3.) that an exponential growing external optimism leads to an exponential growing index $i(t)$. Starting from a certain deviation the system starts to generate oscillations and becomes instable. This fact may cause pessimism (or even panic) in a self reinforcing process [@bouchaud]. After some time this leads to a collapse of the market [@hogg]. Therefore crashes are not only the result of changes in the external influences $U$, but they are caused by the internal instability when the system is far from the equilibrium [@caldarelli; @johansen; @illinski]. Even if the external optimism would continue growing, the system would start to collapse starting from a critical deviation (DAX: critical deviation at $\pm 35\%$). An external potential of the type (\[uexp\]) is mathematically equivalent to a redefinition of the long term trend $I_0(t)$: $$I_0(t)=\hat{I_0} e^{\beta (t-t_0)} \rightarrow I^*_0(t)=\hat{I^*_0} e^{\beta^* (t-t_0)}, \;\;\; \beta^*>\beta \label{istar}$$ This “excited” state exists usually only a certain time period, until the system reaches the critical deviation. After the begin of the collapse the external optimism vanishes and the system returns to the equilibrium state. This behaviour can be found in the historical data of the DAX and other markets. Phases of continuous growth over several month are followed by phases of decline. All of these periods show an exponential behaviour. The market system is very sensitive concerning changes in the neutral component of the market $c_n$. Relatively small external influences on the neutral component become enhanced by a leverage effect on the index. This effect is caused by the different orders of magnitude of the concentrations (\[relationen\]): $$\Delta i(t) \sim \frac{c_n(t)}{c_{p,o}(t)} \, \Delta U(t), \;\;\; \frac{c_n(t)}{c_{p,o}(t)}\sim 100...1000 \label{faktor1}$$ Another essential feature of the dynamics of markets is the chaotic behaviour, for example in the daily changes of the index. The reason for the appearance of chaos is the feedback of the market to itself. The strength of response on deviations of the equilibrium is described by the model parameter $\alpha$. In (fig. 4) there are shown examples of the development of the market system (\[finalsystemlog\]) in dependence of $\alpha$. In (fig. 4a) there the response of the market is relatively small, so that the market compensates after several transactions. If $\alpha$ reaches a critical value (fig. 4b) the reaction on a deviation $\Delta i$ is that strong, that it creates a new deviation with the same size but opposite sign. As the result of this the system starts to oscillate. A further increase of $\alpha$ causes a permanent overcompensation of the market deviations. The system becomes chaotic (fig. 4c). The parameter $\alpha$ is proportional to the volatility of markets. It is worth to remark, that the market shows a typical feature of non linear problems - fractal patterns. The basic trend over years or decades has an exponential behaviour. The different fragments (medium term trends) have an exponential behaviour as well (but a different growth rate). In this work the model was applied on financial markets, but it can be generalised to all markets which are based on supply and demand. The model describes the long and the short term dynamics of markets within a single theoretical framework, using a few empirical parameters. The model can describe crashes as phase transitions, caused by it’s internal instability. Important features of real markets like chaotic behaviour and a fractal structure are described by a system of non linear differential equations. Using this model it is possible to determine basic parameters, which can describe the status of the market in both, the short and the long term trend. I would like to thank Gerhardt Bohm and Klaus Behrndt for their helpful support and discussion. {width="150mm"} {width="150mm"} DAX with medium term trends Jan. 2 1998 - Feb. 4 1999 {width="150mm"} {width="150mm"} [99]{} C.Davidson, “The New Science”, 1st International Conference on High Frequency Data in Finance, March 29-31, 1995, Zurich G.Caldarelli, M.Marsili and Y.-C. Zhang, “A Prototype Model of Stock Exchange”, cond-math/9709118, SISSA Ref 22/97/CM, 1997 B.Mandelbrot, Jour. of Business of the Chicago University, 39, p. 242, (1966); dt. 40, p. 393, (1967) A.Johansen and D.Sornette, “Critical Crashes”, Risk, v.12, No. 1, p.91, (1999) K.Illinski, “Critical Crashes?”, Preprint IPHYS-99-5, (1999) J.Agyris, G.Faust, M.Haase, “Die Erforschung des Chaos”, Vieweg, (1995) B.Mandelbrot, “Fractals and Scaling in Finance”, Springer, (1997) J.D.Farmer, “Market force, ecology and evolution”, Preprint adap-org/9812005, (1998) J.-P.Bouchaud and R.Cont, “A Langevin Approach to Stock Market Fluctuations and Crashes”, Preprint cond-mat/98012379, (1998) C.Busshaus and H.Rieger, “A Prognosis Oriented Microscopic Stock Market Model”, Preprint cond-mat/9903079, (1999) T.Hogg, B.A.Huberman and M.Youssefmir, “The Instability of Markets”, Preprint adap-org/9507002, (1995)\ T.Hogg, B.A.Huberman and M.Youssefmir, “Bubbles and Market Crashes”, Preprint adap-org/9409001, (1994) [^1]: A long term decline of national economies is often caused by massive regulations, reducing the economic freedom. [^2]: The concentration is the weighted average of individual market members with a similiar market view, but a different capitalization $$c_i(t)=\frac{1}{M(t)}\, \sum_{k=1}^{N_i}m_k(t)$$ where $i=o,p,n$ represents the corresponding market views of optimists, pessimists and neutral market members. $m$ is their individual capital and $M$ the summary market capitalization. $N_i$ is the total number of individual market members with the same market view.

--- author: - - bibliography: - 'references.bib' title: | An Operational Semantic Basis\ for OpenMP Race Analysis --- OpenMP; operational semantics; concurrency; formal definition; data race; data race detection tool; structured parallelism

--- abstract: 'In the context of cosmological perturbation theory, we derive the second order Boltzmann equation describing the evolution of the distribution function of radiation without a specific gauge choice. The essential steps in deriving the Boltzmann equation are revisited and extended given this more general framework: i) the polarisation of light is incorporated in this formalism by using a tensor-valued distribution function; ii) the importance of a choice of the tetrad field to define the local inertial frame in the description of the distribution function is emphasized; iii) we perform a separation between temperature and spectral distortion, both for the intensity and for polarisation for the first time; iv) the gauge dependence of all perturbed quantities that enter the Boltzmann equation is derived, and this enables us to check the correctness of the perturbed Boltzmann equation by explicitly showing its gauge-invariance for both intensity and polarization. We finally discuss several implications of the gauge dependence for the observed temperature.' author: - 'Atsushi Naruko$^{1}$, Cyril Pitrou$^{2,3}$, Kazuya Koyama$^{4}$, Misao Sasaki$^{5}$' date: 'May 8, 2013' title: 'Second order Boltzmann equation : gauge dependence and gauge invariance' --- Introduction ============ The non-Gaussianity in the Cosmic Microwave Background (CMB) has been one of the hottest topics in cosmology because it could open a new window for probing the primordial universe. Recent CMB observations, especially WMAP [@Komatsu:2010fb] and Planck [@Ade:2013ydc], have confirmed to a very high accuracy that the primordial curvature perturbations have a nearly scale invariant initial power spectrum and the associated statistics is nearly Gaussian. These observations are consistent with the predictions of an early inflationary era driven by a single slow-rolling scalar field. The possibility of non-Gaussianity in the primordial curvature perturbations was discussed for the first time quantitatively by Komatsu and Spergel [@Komatsu:2001rj]. They parameterized the level of non-Gaussianity in the potential $\Phi(x)$ (the curvature potential in the Newton or Poisson gauge) by $$\Phi (x) = \Phi_{{\mathrm{L}}} (x) + f_{{\mathrm{NL}}}^{{\mathrm{local}}} \Bigl[ \Phi^2_{{\mathrm{L}}} (x) - \langle \Phi_{{\mathrm{L}}}^2 (x) \rangle \Bigr] \,.$$ Here, $\Phi_{{\mathrm{L}}} (x)$ denotes the Gaussian part of the perturbation, or in perturbation theory its linear part, and $\langle \cdots \rangle$ designates the statistical average. This type of non-Gaussianity leads to a non-vanishing three point correlation function, or equivalently in reciprocal space to a non-vanishing bispectrum. The prediction for the possible values of this parameter $f^{{\mathrm{local}}}_{{{\mathrm{NL}}}}$ from a phase of single-field slow-roll inflation was first performed by Maldacena [@Maldacena:2002vr], and it was shown that it is of order of the slow-roll parameters and thus highly suppressed. Hence, if a $f^{{\mathrm{local}}}_{{{\mathrm{NL}}}}$ of order unity or greater is detected, this simplest model of inflation will be ruled out. Unfortunately, the interpretation of the measured non-Gaussianity is not so straightforward because we do not observe directly the primordial non-Gaussianity in the curvature perturbation but its effect on the CMB fluctuations. Therefore, to relate the primordial curvature perturbation to CMB, we first have to compute the evolution of perturbations after inflation. These effects can be split unambiguously in two parts: i) a linear transfer that cannot create a non-Gaussian signal if the initial conditions are purely Gaussian, and ii) a non-linear transfer that generates a non-Gaussian signal in the observables even if the initial conditions are purely Gaussian. The resulting non-Gaussian signal from i) is often called [*primordial non-Gaussianity*]{} and all the possible sources of non-linear evolutions which enter the category ii) are called [*secondary non-Gaussianity*]{}. Recently the Planck collaboration provided a constraint $f^{{\mathrm{local}}}_{{{\mathrm{NL}}}} = 2.7 \pm 5.8$ [@Ade:2013ydc]. This result was obtained by subtracting one of the secondary non-Gaussianities that arises from the correlation between the lensing and integrated Sachs-Wolfe effect. This clearly demonstrates the importance of subtracting all the secondary non-Gaussianity consistently in order to obtain an accurate constraint on the primordial non-Gaussianity. The evolution of the perturbations on super-horizon scales is well understood, even fully non-linearly using either a covariant approach [@Langlois:2005ii; @Langlois:2005qp; @Enqvist:2006fs; @Pitrou:2007xy], or a separate universe approach with the so-called $\delta N$ formalism [@Comer:1994np; @Kolb:2004jg; @Lyth:2004gb], since it leads to a conservation law for the curvature perturbations in the case of adiabatic perturbations. On the other hand, the evolution for modes below the horizon scale is not so simple analytically, especially at the non-linear order, and the use of a kinetic description cannot be avoided on small scales since radiation starts to develop an anisotropic stress. In order to obtain numerical results for the non-linear evolution, we need to derive and solve without approximations the coupled system of non-linear a) Einstein equation for the metric, b) conservation and Euler equations for fluids and c) Boltzmann equation for radiation (photons and neutrinos). Note that the conservation and Euler equations can always be deduced from the lowest moments of the Boltzmann equation, and the full set of equations is often only referred to as [*Einstein-Boltzmann system*]{} of equations. In order to follow this roadmap, the second order Boltzmann equation was written down in the Poisson gauge in Refs. [@Bartolo:2006cu; @Bartolo:2006fj; @Pitrou:2008hy; @Pitrou:2008ut; @Pitrou:2010sn; @Beneke:2010eg]. The gauge dependence of the distribution function was obtained at linear order [@Durrer:1993db] and then at second order [@Pitrou:2007jy] but leaving aside the problem of polarisation. It was then extended to include polarised light in Ref. [@Pitrou:2008hy]. The system of equations was then solved numerically in Fourier space in Poisson gauge in [@Pitrou:2010sn], and it was reported that the secondary effects around the last-scattering surface could mimic a primordial signal of $f_{{\mathrm{N L}}}^{{\mathrm{local}}} \sim 4$. Recently there have been a huge progress in improving the numerical calculations and clarifying the amplitude of various secondary non-Gaussianities at recombination [@Huang:2012ub; @Su:2012gt; @Pettinari:2013he], and a consensus emerged that when including all the non-linear effects around recombination *and* the integrated early effects after recombination, it could mimic a primordial signal of $f_{{\mathrm{N L}}}^{{\mathrm{local}}} \sim 0.8$, as expected from analytic approximations [@Creminelli:2011sq]. The description of the spectral dependence of the distribution function is also crucial at second order. Indeed, at first order there are no spectral distortions and the perturbation of the photon distribution function can be understood as a single, spectrum-independent temperature fluctuation. However, at second order, there appears a deviation from the Planck distribution, resulting in a continuum of spectral distortions, which in principle must superimpose the thermal Sunyaev-Zeldovich effect [@Ade:2013qta]. To describe this distortion, we use a direction and position dependent Compton $y$ parameter [@Stebbins:2007ve; @Pitrou:2009bc], and also introduce a similar tensor-valued variable to describe the distortion in the polarisation, thus extending the formalism introduced in Ref. [@Pitrou:2009bc]. In this paper, we derive the second order Boltzmann equation without restricting to a specific gauge, and including polarisation. Reflecting on the above, our motivation is two-fold. First, since the structure of the second order Einstein and Boltzmann equations depends very much on a choice of the gauge, we have to find a gauge in which we can numerically solve this system accurately and quickly. Therefore, it is preferable not to specify the gauge from the beginning but to formulate the equations without specifying it. We can impose different gauge restrictions in their final form to explore the stability and efficiency of the numerical integration. Second, we would like to check the equations derived in Refs. [@Pitrou:2008hy; @Pitrou:2010sn; @Beneke:2010eg]. As a direct check, we recover them in the specific case of the Poisson gauge. Then, as an indirect check, we revisit the transformation properties of the distribution function and the metric perturbations and confirm that the perturbed Boltzmann equation is gauge-invariant up to second order in perturbations, thus increasing our confidence in the rather lengthy derivation. In Ref. [@Pettinari:2013he], it was found that the inclusion or omission of certain line of sight terms can make a large impact on the estimation of the bias to the primordial non-Gaussianity due to the secondary non-Gaussianity. In Refs [@Huang:2012ub; @Pettinari:2013he] all physical effects were included except for lensing and time-delay. These time-integrated effects require a separate analysis because at later times small-scale multipoles get excited and numerically it is very difficult to evolve the equations. In this paper, we point out that the separation of these effects depends on a gauge. Given that the lensing-ISW cross correlation gives the largest bias to the primordial local type non-Gaussianity, one should bear this gauge dependence in mind when separating these time integrated effects in the calculations. The choice of the gauge and the associated choice of the tetrad field for the distribution function is also crucial in the interpretation of the quantities as observables. These subtle details do not affect our interpretation of observables in the linear theory since it is only relevant for the monopole and the dipole. However it is no longer the case at second order in perturbations. We must understand the transformation properties of the distribution function under a gauge transformation or a change of the inertial frame and determine what is a gauge and a choice of inertial frame that is related to CMB experiments. The structure of this paper is as follows. In section \[sec:def\], we give the definitions of the variables that we use for the metric, momentum and distribution function. Especially, to express the perturbation of the metric, we use a geometrical $(3+1)$ decomposition, or the ADM [@Arnowitt:1962hi] parametrisation of the metric. At first order, there is no particular advantage in using this formalism, but various expressions are simplified at second order for the choice of the inertial frame that we make. In section \[sec:Boltz\], we derive the second order Boltzmann equation with polarisation without restricting to a specific gauge. In section \[sec:gauge\], we discuss the gauge dependence of the variables. We carefully investigate the gauge transformation of the metric, momentum and the distribution function. We then check explicitly the gauge invariance of the perturbed Boltzmann equation up to second order as a consistency test. Finally, in Section \[sec:conc\], we summarize our results and we comment briefly on the relevance of our formalism for the observed CMB anisotropy. Useful technical details are gathered in the appendices. Definitions {#sec:def} =========== In this section we build all the tools which are used for the description of polarized radiation in cosmology. We first review briefly the parametrization of cosmological perturbations, and explain how a photon momentum can be uniquely described by its energy and direction once a suitable tetrad choice has been made. We then introduce the tensor-valued distribution function which is used to treat statistically a gas of polarized photons, and which is the key object in the Boltzmann equation, and we finally present how it can be decomposed into its main spectral components. Spacetime coordinates and local inertial frame {#ssec:deftet} ---------------------------------------------- We shall use the ADM formalism to write down the expression of the perturbed metric where the metric can be decomposed as $$\begin{aligned} \label{MetricADM} {{\rm d}}s^2 &= a^2(\eta) \Bigl[ - N^2 {{\rm d}}\eta^2 + \gamma_{ij} ({{\rm d}}x^i + \beta^i {{\rm d}}\eta) ({{\rm d}}x^j + \beta^j {{\rm d}}\eta) \Bigr] \nonumber\\ &= a^2(\eta) \Bigl[ - (N^2 - \gamma_{i j} \beta^i \beta^j) {{\rm d}}\eta^2 + 2 \gamma_{ij} \beta^j {{\rm d}}x^i {{\rm d}}\eta + \gamma_{ij} {{\rm d}}x^i {{\rm d}}x^j \Bigr] \,,\end{aligned}$$ where $N$ is the lapse function, $\beta^i$ is the shift vector, $\gamma_{i j}$ is the spatial metric, and indices of the Latin type ($i,j,k\cdots$) run from $1$ to $3$. To describe the perturbations around the flat Friedmann-Lemaître-Robertson-Walker (FLRW) space-time, perturbation variables, $\alpha$ and $h_{i j}$, are introduced as $$\label{DefPertNNi} N \equiv 1 + \alpha \,, \qquad \gamma_{i j} \equiv \delta_{i j} + 2 h_{i j} \,.$$ For simplicity, we use the following definitions $$\beta_i\equiv \delta_{ij} \beta^j \,, \quad h^i{}_j \equiv \delta^{ik} h_{kj} \,, \quad h^{ij} \equiv \delta^{ik} \delta^{jl}h_{kl} \,,$$ where the spatial indices are raised and lowered with $\delta_{ij}$ and $\delta^{ij}$, rather than with $\gamma_{ij}$ and $\gamma^{ij}$. As is clearly seen below, the ADM form of the metric perturbation will simplify the expressions of the perturbed Boltzmann equation. Any perturbation $X$ will be expanded into its first and second order parts as $$X= X^{(1)} + \frac{1}{2} X^{(2)}\,.$$ The relations between the ADM variables and the usual definitions of cosmological perturbations are provided in Appendix \[app:ADM\]. The Boltzmann equation is better formulated by explicitly using a local inertial frame at every point of the space-time and this can be achieved by using a tetrad field. It is a set of four vector fields which satisfy $$\eta_{(a) (b)} = g_{\mu \nu} e_{(a)}{}^\mu e_{(b)}{}^\nu \,, \qquad g_{\mu \nu} = \eta_{(a) (b)} e^{(a)}{}_\mu e^{(b)}{}_\nu \,.$$ These conditions determine the choice of tetrad only up to rotations and boosts. Here the following particular tetrads are chosen up to second order accuracy $$\begin{aligned} e^{(0)}{}_\mu &= a (- N, 0, 0, 0) \,,\qquad e^{(i)}{}_\mu = a \left( \beta^i + h^i{}_j \beta^j, \delta^i{}_j + {h^i}_j - \frac{1}{2} h^{i k} h_{k j} \right) \,, \label{tetrads}\end{aligned}$$ and the inverse tetrads are given by $$\begin{aligned} e_{(0)}{}^\mu = - e^{(0) \mu} &= - \frac{1}{a} \left( \frac{1}{N}, - \frac{\beta^i}{N} \right) \,, \qquad e_{(i)}{}^\mu = \frac{1}{a} \left[ 0, \delta_{(i) (k)} \left( \delta^{j k} - h^{j k} + \frac{3}{2} h^{j l} h_l{}^k \right) \right] \,.\end{aligned}$$ The time-like tetrad is chosen to be orthogonal to the constant time hypersurfaces since ${\bm e}^{(0)} \propto {{\rm d}}\eta$. As for the spatial tetrads, this choice corresponds to asking that there is no rotation between the background and the perturbed tetrads [@Pitrou:2007jy]. Momentum -------- To facilitate the separation between the magnitude of the momentum and its direction in a covariant manner, let us consider the projection of the momentum of photon $p^\mu$ onto the set of tetrads, $$\begin{aligned} p^{(a)} = e^{(a)}{}_\mu p^\mu \,. \end{aligned}$$ We introduce the conformal momentum of photon rather than the physical momentum $p^{(a)}$ $$\label{defpP} q^{(a)} \equiv a p^{(a)} \,.$$ Since the momentum of photon satisfies the null condition $p^\mu p_\mu = 0$, or equivalently $q^{(a)} q_{(a)} = 0$, only three components among four are independent, that is $$\begin{aligned} q^{(a)} q_{(a)} = 0 \,, \quad \Leftrightarrow \quad (q^{(0)})^2 = \delta_{(i) (j)} q^{(i)} q^{(j)} \,. \end{aligned}$$ Thus the three spatial components $q^{(i)}$ can be regarded as such independent variables. Furthermore, $q^{(i)}$ can be decomposed into its magnitude $q$ and direction $n^{(i)}$ as $$\begin{aligned} q \equiv \sqrt{\delta_{(i) (j)} q^{(i)} q^{(j)}} = |q^{(0)}| \,, \qquad n^{(i)} \equiv \frac{q^{(i)}}{q} \,. \end{aligned}$$ Physically the above $q$ can be understood as the conformal (re-scaled) energy, $q = a E_{{\mathrm{phys}}}$, seen by an observer orthogonal to time constant hypersurfaces. From Eq. (\[defpP\]), the components of momentum $p^{\mu}$ are expressed as functions of $(q, n^{(i)})$ up to the second order as $$\begin{aligned} \label{p0pi} p^0 &= \frac{q}{a^2} (1 - \alpha + \alpha^2) \,, \\ p^i &= \frac{q}{a^2} \left( n^{(i)} - \beta^i - {h^i}_j n^{(j)} + \alpha \beta^i + \frac{3}{2} h^{i k} h_{k j} n^{(j)} \right) \,. \label{momentum}\end{aligned}$$ Conversely, ($q, n^{(i)}$) are given by the components of momentum as $$\begin{aligned} q &= a^2 (1 + \alpha) p^0 \,, \label{def:q} \\ n^{(i)} &= \left[ (1 - \alpha + \alpha^2) \delta^i{}_j + (1 - \alpha) h^i{}_j - \frac{1}{2} h^{ik} h_{k j} \right] \frac{p^j}{p^0} + (1 - \alpha) \beta^i + \beta^j {h^i}_j \,. \label{def:n^i}\end{aligned}$$ One can introduce a projection operator in terms of $e^{(0)}{}_\mu$ and $n_\mu$. The projection operator, often called the screen projector, is defined as $$\begin{aligned} S_{\mu \nu} &\equiv g_{\mu \nu} + e^{(0)}{}_\mu e^{(0)}{}_\nu - n_\mu n_\nu \,,\end{aligned}$$ where $n^\mu$, the direction vector of photon, is defined by $$\begin{aligned} n^\mu \equiv e_{(i)}{}^\mu n^{(i)} \,. \end{aligned}$$ Clearly $S_{\mu \nu}$ is a projection of the tangent space onto a two dimensional plane orthogonal to both $e^{(0)}{}_\mu$ and $n_\mu$ since $S^{\mu \nu} e^{(0)}{}_\mu$ and $S^{\mu \nu} n_\mu$ vanish. Its expression in tetrad components reduces necessarily to the identity of the two-dimensional subspace which is left invariant by the projector, that is $$\begin{aligned} S_{(i) (j)} = \delta_{(i) (j)} - n_{(i)} n_{(j)} \,,\qquad S_{(0) (0)}=S_{(0) (i)}=0 \,.\end{aligned}$$ Distribution function for photons and Stokes parameters {#ssec:f} ------------------------------------------------------- In order to describe the polarisation of radiation, we introduce a tensor-valued distribution function $f_{\mu \nu}$, which is complex valued and Hermitian. The construction of this distribution function is discussed in Appendix \[app:Dist\]. It is independent of the choice of the electromagnetic gauge and contains only four physical degrees of freedom since it satisfies the conditions $$f_{\mu\nu} e_{(0)}{}^\mu = f_{\mu\nu} e_{(0)}{}^\nu = f_{\mu\nu} n^\mu = f_{\mu\nu} n^\nu = 0 \,.$$ Note that the distribution function depends on the observer’s velocity, $u^\mu \equiv e_{(0)}{}^{\mu}$, used in its definition. As long as no confusion arises from such dependence, we omit to specify it. In the case where this is needed, mainly when studying the transformation properties of such a quantity, we shall use the notation $f_{\mu\nu}^{\bm{e}_{(0)}}$ to stress that the tensor-valued distribution function is dependent on the observer’s velocity and thus on the choice of the tetrad field. The four degrees of freedom can be extracted by decomposing $f_{\mu \nu}$ into a trace part, a symmetric traceless part and an antisymmetric part as $$f_{\mu \nu} \equiv \frac{1}{2} I S_{\mu \nu} + P_{\mu \nu}+ \frac{{{\rm i}}}{2} \epsilon_{\rho \mu \nu \sigma} e_{(0)}{}^\rho n^\sigma V \,, \label{Pol}$$ where the antisymmetric tensor is defined by $$\begin{gathered} \epsilon_{\alpha \beta \gamma \delta} = \epsilon_{[ \alpha \beta \gamma \delta] } \,, \quad \epsilon_{0 1 2 3} = \sqrt{- g} \,, \qquad {\rm or} \qquad \epsilon_{(0)(1)(2)(3)}=-\epsilon^{(0)(1)(2)(3)} = 1 \,.\end{gathered}$$ [*I*]{} is the intensity and $V$ is the degree of circular polarisation. $P_{\mu \nu}$ encodes the two degrees of linear polarisation (so called $Q$ and $U$ Stokes parameters). All these functions, together with the original tensor-valued distribution function, are functions of the position on space-time $x^\mu=(\eta,x^i)$ and on the point in tangent space. This point in the tangent space can be chosen to be parametrized either by the components $p^\mu$ in the basis canonically associated with the coordinates system, or alternatively by their Cartesian counterparts $p^{(a)}$. In fact we will choose to parametrize the tangent space by the components of the conformal momentum in tetrad space, $q^{(i)} = a p^{(i)}$, expressed in their spherical coordinates $q$ and $n^{(i)}$, as this leads to the most simple form for the Boltzmann equation as we shall see further. Spectral distortion ------------------- On the background space-time, the distribution function, which is characterized only by the intensity $I$, is given by a Planck distribution whose temperature $\bar T$ depends only on $\eta$ due to the symmetries of the FLRW universe. As we will check later, the background temperature scales as $\propto 1/a$. We thus have $$\bar I(\eta,q) = {{I_{\rm BB}}}\left[\frac{q}{a(\eta)\bar T(\eta)}\right] \,, \quad \text{with} \quad {{I_{\rm BB}}}(x) \equiv \frac{2}{( e^x - 1 )} \,.$$ At first order in perturbation, the fluctuation of intensity can be described as a fluctuation of temperature $\delta T$ which is independent of $q$. There are two reasons for this. First, as we shall discuss further, gravitational interactions do not induce spectral distortions in the sense that they shift all wavelengths by the same ratio. Second, the collisions at linear order in perturbation do not induce spectral distortions and the redistribution of the photon directions resulting from it can be described by a direction dependent temperature. A similar procedure can be followed for the description of polarisation at first order. However at second order the situation becomes more complicated since the Compton scattering at this order of perturbation induces spectral distortions which cannot be reabsorbed in a simple direction dependent temperature. As a result, the photon distribution is not described by a Planck distribution function, but fortunately it is sufficient to use two direction dependent quantities. The first remains the temperature and the second describes the type of spectral distortion generated at second order. Actually in general, at the $n$-th order, $n$ directional dependent functions would be needed [@Stebbins:2007ve; @Pitrou:2009bc] to characterize fully the spectrum. In order to parametrize this distortion, we introduce on top of the temperature $T$, the so-called Compton $y$ parameter. In this section we will omit the dependence of all quantities on the coordinates $x^\mu$ and we will focus on the dependence on the tangent space coordinates $(q,n^{(i)})$. The distribution function can be expanded around a Planck distribution in the so-called Fokker-Planck expansion as  [@Stebbins:2007ve] $$\begin{aligned} \label{defy} I \Bigl( q,n^{(i)} \Bigr) &\simeq {{I_{\rm BB}}}\left( \frac{q}{a T} \right) + y \bigl( n^{(i)} \bigr) q^{-3} {{\frac{\partial}{\partial \ln q}}}\left[ q^3 {{\frac{\partial}{\partial \ln q}}}{{I_{\rm BB}}}\left( \frac{q}{a T} \right) \right] \notag\\ &= {{I_{\rm BB}}}\left( \frac{q}{a T} \right) + y \bigl( n^{(i)} \bigr) {{\cal D}}_q^2{{I_{\rm BB}}}\left( \frac{q}{aT} \right) \,,\end{aligned}$$ where $${{\cal D}}_q^2 \equiv q^{-3} {{\frac{\partial}{\partial \ln q}}}\left( q^3 {{\frac{\partial}{\partial \ln q}}}\right) = {{\frac{\partial^2}{\partial \ln q^2}}}+3 {{\frac{\partial}{\partial \ln q}}}\,.$$ Because the number density of photon is given by $n \propto a^{-3}\int I q^2 {{\rm d}}q$, the $y$ term does not contribute to the photon number density and the temperature $T$ is the temperature of the black-body that would have the same number density (see Ref. [@Pitrou:2010sn] for a discussion on other possible definitions for the temperature) and we call it here [*number density temperature*]{}. It can be expanded around the background temperature as $$T \bigl( n^{(i)} \bigr) \equiv \bar T(\eta) \left[ 1 + \Theta \bigl( n^{(i)} \bigr) \right] \,.$$ Note that the expansion (\[defy\]) is not the same as Eq. ($11$) nor Eq. ($15$) of Ref. [@Stebbins:2007ve]. Indeed, the temperature of the Planck spectrum around which we expand is neither the physically motivated logarithmic averaged temperature of Ref. [@Stebbins:2007ve] nor a fiducial temperature, but another physically motivated temperature (the number density temperature) that suits better to describe the spectral distortion of the type that appears in CMB. However, when performing perturbations in cosmology, we need to refer to the background space-time temperature $\bar T$, not to the local number density temperature. Thus it is convenient to expand the distribution function around a Planck distribution at $\bar T$ rather than $T$. Expanding Eq. (\[defy\]) in $\Theta$ up to the second order, we obtain the expansion as $$\begin{aligned} I &= {{I_{\rm BB} \left( \frac{q}{a \bar T} \right)}}- \bigl( \Theta+\Theta^2 \bigr) {{\frac{\partial}{\partial \ln q}}}{{I_{\rm BB} \left( \frac{q}{a \bar T} \right)}}+\left( y + \frac{1}{2} \Theta^2 \right) {{\cal D}}_q^2 {{I_{\rm BB} \left( \frac{q}{a \bar T} \right)}}\,, \label{intensity}\end{aligned}$$ where we used the fact that $y$ is at least a second order quantity. Here, in order to simplify the notation, it is implied that $\Theta$ and $y$ depend on $x^\mu$ and $n^{(i)}$. For a given $I$, the spectral components $\Theta$ and $y$ can be extracted by performing different types of integrals on $q$ (see appendix \[AppExtraction\] for details). This expansion is similar to Eq. ($11$) of Ref. [@Stebbins:2007ve] when only second derivatives of the Planck distribution are kept. Now we want to obtain a similar decomposition for polarisation. Indeed, when dealing with polarisation we also need to expand its spectral dependence in a way similar to what has been performed for the intensity in Eqs. (\[defy\]) and (\[intensity\]), that is we want to separate the polarisation tensor into a spectral distortion $Y_{\mu\nu}$ and non-distorted component ${{\cal P}}_{\mu \nu}$. However, this separation is slightly different given that there is no polarisation on the background and hence there is no term corresponding to the first term in Eq. (\[intensity\]). In the appendix of Ref. [@Stebbins:2007ve], it has been shown that the expansion should be $$\label{defyPbase} P_{\mu \nu} \Bigl( q, n^{(i)} \Bigr) \simeq - {{\cal P}}_{\mu\nu} \bigl( n^{(i)} \bigr) {{\frac{\partial}{\partial \ln q}}}{{I_{\rm BB}}}\left( \frac{q}{a T} \right) + Y_{\mu \nu} \bigl( n^{(i)} \bigr) {{\cal D}}_q^2 {{I_{\rm BB}}}\left( \frac{q}{a T} \right) \,,$$ which is just a consequence of the fact that there is no background polarisation. We will check that $Y_{\mu\nu}$ vanishes at first order as it is not generated by collisions at this order. Similarly to the expansion of the intensity part, we want to expand the distribution function around a Planck spectrum at the background temperature $\bar T$ rather than the local number density temperature $T$. Thus we expand Eq. (\[defyPbase\]) in $\Theta$ up to first order to get $$\begin{aligned} \label{defyP} P_{\mu \nu} &= - (1 + 3 \Theta) {{\cal P}}_{\mu\nu} {{\frac{\partial}{\partial \ln q}}}{{I_{\rm BB} \left( \frac{q}{a \bar T} \right)}}+ (Y_{\mu \nu} + \Theta {{\cal P}}_{\mu\nu}) {{\cal D}}_q^2 {{I_{\rm BB} \left( \frac{q}{a \bar T} \right)}}\,.\end{aligned}$$ Again here, in order to simplify the notation, it is implied that $\Theta$, ${{\cal P}}_{\mu \nu}$ and $Y_{\mu\nu}$ depend on $x^\mu$ and $n^{(i)}$. Boltzmann equation {#sec:Boltz} ================== Now that we have all the tools at hand, we are ready to formulate the Boltzmann equation for polarized radiation in the cosmological context, and extract its spectral components. This section is entirely dedicated to this task. Given that the complete and detailed derivation can be rather lengthy, all details which are not necessary in a first reading are gathered in Appendix \[app:Boltz\_pol\]. We first present the general expression of the Boltzmann equation for a tensor-valued distribution function. Since the Boltzmann equation is the description of how this distribution function evolves along a photon geodesic, it is necessary to perturb the geodesic equation up to second order. We then show how the Boltzmann equation can be split into its main spectral components, that is into a temperature and a distortion. Finally we write the explicit forms of the free-streaming part and the collision part of the Boltzmann equation. Boltzmann equation {#boltzmann-equation} ------------------ The evolution of the tensor-valued distribution function is dictated by the Boltzmann equation [@Tsagas:2007yx] $$\label{Evolfmunu} S_\mu{}^\rho S_\nu{}^\sigma \frac{{{\cal D}}f_{\rho \sigma}}{{{\cal D}}\lambda} = C_{\mu \nu} \,,$$ where ${{\cal D}}/ {{\cal D}}\lambda$ is the covariant derivative along a photon trajectory $x^{\mu}(\lambda)$ and the momentum and $C_{\mu \nu}$ is the associated collision term. The explicit form of ${{\cal D}}f_{\mu \nu}/ {{\cal D}}\lambda$ is $$\frac{{{\cal D}}f_{\mu \nu}}{{{\cal D}}\lambda} \equiv \nabla_\rho f_{\mu \nu} \frac{{{\rm d}}x^\rho}{{{\rm d}}\lambda} + \frac{{{\partial}}f_{\mu \nu}}{{{\partial}}q^{(i)}} \frac{{{\rm d}}q^{(i)}}{{{\rm d}}\lambda} \,,$$ where $\nabla_\mu$ indicates a covariant derivative associated with $g_{\mu \nu}$. Using spherical coordinates $(q,n^{(i)})$ instead of $q^{(i)}$ for the momentum space, the Liouville operator, that is the l.h.s of Eq. (\[Evolfmunu\]), reads $$\label{DDLoperatortensor} S_\mu{}^\rho S_\nu{}^\sigma \frac{{{\cal D}}f_{\rho \sigma}}{{{\cal D}}\lambda} = S_\mu{}^\rho S_\nu{}^\sigma \nabla_\tau f_{\rho \sigma} \frac{{{\rm d}}x^\tau}{{{\rm d}}\lambda} + \frac{{{\partial}}f_{\mu \nu}}{{{\partial}}\ln q} \frac{{{\rm d}}\ln q}{{{\rm d}}\lambda} + D_{(i)} f_{\mu \nu} \frac{{{\rm d}}n^{(i)}}{{{\rm d}}\lambda} \,,$$ where $D_{(i)}$ is a covariant derivative for momentum. The detail of the construction of such derivative is discussed in Appendix \[sapp:covp\]. Thanks to the operation of the projection $S_\mu{}^\rho S_\nu{}^\sigma$ onto $ {{\cal D}}f_{\rho \sigma} / {{\cal D}}\lambda$, the left hand side of the Boltzmann equation can be decomposed into the $I$, $V$ and $P_{\mu \nu}$ parts similarly to Eq. (\[Pol\]); $$\begin{gathered} S_\mu{}^\rho S_\nu{}^\sigma \frac{{{\cal D}}f_{\rho \sigma}}{{{\cal D}}\lambda} = \frac{1}{2} L[I] S_{\mu \nu} + L[{\bf P}\,]_{\mu \nu} + \frac{{{\rm i}}}{2} L[V] \epsilon_{\rho \mu \nu \sigma} e_{(0)}{}^\rho n^\sigma \,,\end{gathered}$$ where the corresponding Liouville operators are defined as $$\label{defofLs} L[I] \equiv \frac{{{\cal D}}I}{{{\cal D}}\lambda} \,, \qquad L[{\bf P} \, ]_{\mu \nu} \equiv S_\mu{}^\rho S_\nu{}^\sigma \frac{{{\cal D}}P_{\rho \sigma}}{{{\cal D}}\lambda} \,, \qquad L[V]\equiv \frac{{{\cal D}}V}{{{\cal D}}\lambda} \,.$$ Here, the operator ${{\cal D}}/ {{\cal D}}\lambda$ on a scalar distribution function $f$, takes the simpler form $$\frac{{{\cal D}}f}{{{\cal D}}\lambda} \Bigl( x^\mu, q, n^{(i)} \Bigr) \equiv {{\partial}}_\mu f \frac{{{\rm d}}x^\mu}{{{\rm d}}\lambda} + \frac{{{\partial}}f}{{{\partial}}\ln q} \frac{{{\rm d}}\ln q}{{{\rm d}}\lambda} + D_{(i)} f \frac{{{\rm d}}n^{(i)}}{{{\rm d}}\lambda}\,.$$ In a similar manner, one can also decompose the collision term, that is the r.h.s of Eq. (\[Evolfmunu\]) as $$\begin{gathered} C_{\mu \nu} \equiv \frac{1}{2} C^I S_{\mu \nu} + C^P_{\mu \nu} + \frac{{{\rm i}}}{2} C^V \epsilon_{\rho \mu \nu \sigma} e_{(0)}{}^\rho n^\sigma \,, \label{PolC}\end{gathered}$$ so that after extracting the trace, symmetric traceless and antisymmetric parts we get obviously $L[I]=C^I$, $L[{\bf P}]_{\mu\nu} = C^P_{\mu\nu}$ and $L[V]=C^V$. Beware that this does not mean that the intensity, linear polarization and circular polarization evolve independently, since for instance $C^I$ is the “intensity part” of the collision term but it may involve in general all components $I$, $P_{\mu\nu}$ and $V$ of the tensor-valued distribution function. As a matter of fact, Compton collision does indeed intermix intensity and linear polarization, whereas circular polarization evolves independently. Geodesic equation and momentum evolution ---------------------------------------- From Eq (\[p0pi\]) and the definition of momentum, ${{\rm d}}x^\mu/{{\rm d}}\lambda=p^\mu$ we obtain $$\begin{aligned} \frac{{{\rm d}}\eta}{ {{\rm d}}\lambda} &= \frac{q}{a^2} (1 - \alpha) \,, \label{eta-evo} \\ \frac{{{\rm d}}x^i}{ {{\rm d}}\lambda} &= \frac{q}{a^2} \left( n^{(i)} - \beta^i - {h^i}_j n^{(j)} \right) \,, \label{xi-evo}\end{aligned}$$ where only the first order terms are kept. In terms of $(q, n^{(i)})$, the geodesic equation leads to the evolution equation for the conformal energy $$\begin{aligned} \frac{{{\rm d}}\ln q}{{{\rm d}}\lambda} &= \frac{q}{a^2} \Bigl[ - \alpha_{,i} n^{(i)} + \beta_{i, j} n^{(i)} n^{(j)} - h_{ij}{}' n^{(i)} n^{(j)} + \alpha \Bigl( \alpha_{,i} n^{(i)} - \beta_{i, j} n^{(i)} n^{(j)} + h_{ij}{}' n^{(i)} n^{(j)} \Bigr) \notag\\ & \qquad \qquad + \alpha_{, j} h^j{}_i n^{(i)} + \beta^k h_{i j, k} n^{(i)} n^{(j)} + (\beta_{k, i} - \beta_{i ,k} + 2 h_{i k}{}') h^k{}_j n^{(i)} n^{(j)} \Bigr] \,. \label{q-evo}\end{aligned}$$ As for the direction evolution, up to first order in perturbations, we obtain $$\frac{{{\rm d}}n^{(i)}}{{{\rm d}}\lambda} = -\frac{q}{a^2} S^{(i) (j)} \Bigl[ \alpha_{,j} - (\beta_{k ,j} - h_{j k}{}') n^{(k)} + (h_{j l,k} - h_{k l,j}) n^{(k)}n^{(l)} \Bigr] \,. \label{n-evo}$$ It is obvious that $n_{(i)}{{\rm d}}n^{(i)}/{{\rm d}}\lambda=1$ as it ought to be since $n^{(i)}$ is a unit vector. We need these expressions only at first order, except for the evolution of $q$ because the background distribution function is constant in space-time (see below). Before closing this subsection, we mention that we are free to choose another affine parameter than $\lambda$, to label a point on a geodesic. A convenient choice is to take the conformal time $\eta$ at each point of space-time crossed by the geodesic. The advantage of such choice, is that for photons having the same direction (and which thus follow the same path), but not the same energy, the same conformal time $\eta$ would correspond to the same point of the geodesic. We can trade $\lambda$ for $\eta$ using Eq (\[eta-evo\]), that is with $$\begin{aligned} \frac{{{\rm d}}\eta}{ {{\rm d}}\lambda} = \frac{q}{a^2} (1 - \alpha) \quad & \Longrightarrow \quad \frac{{{\rm d}}\lambda}{ {{\rm d}}\eta} = \frac{a^2}{q} (1 + \alpha ) \,.\end{aligned}$$ The evolution of position, conformal energy, and direction, take then the form $$\begin{aligned} \frac{{{\rm d}}x^i}{{{\rm d}}\eta} = \frac{{{\rm d}}x^i}{ {{\rm d}}\lambda} \frac{{{\rm d}}\lambda}{{{\rm d}}\eta} &= n^{(i)} + \alpha n^{(i)} - \beta^i - {h^i}_j n^{(j)} \,, \\ \frac{{{\rm d}}\ln q}{{{\rm d}}\eta} = \frac{{{\rm d}}\ln q}{ {{\rm d}}\lambda} \frac{{{\rm d}}\lambda}{{{\rm d}}\eta} &= - \alpha_{,i} n^{(i)} + \beta_{i, j} n^{(i)} n^{(j)} - h_{ij}{}' n^{(i)} n^{(j)} \notag\\ & \qquad + \alpha_{, j} h^j{}_i n^{(i)} + \beta^k h_{i j, k} n^{(i)} n^{(j)} + (\beta_{k, i} - \beta_{i ,k} + 2 h_{i k}{}') h^k{}_j n^{(i)} n^{(j)} \,, \\ \frac{{{\rm d}}n^{(i)}}{{{\rm d}}\eta} = \frac{{{\rm d}}n^{(i)}}{ {{\rm d}}\lambda} \frac{{{\rm d}}\lambda}{{{\rm d}}\eta} &= - S^{(i) (j)} \left[ \alpha_{,j} - (\beta_{k ,j} - h_{j k}{}') n^{(k)} + (h_{j l,k} - h_{k l,j}) n^{(k)}n^{(l)} \right] \,.\end{aligned}$$ Spectral decomposition of the Boltzmann equation ------------------------------------------------ We are now in position of writing down explicitly the Boltzmann equation, expanding the orders of perturbations, and separating the spectral components. Let us first look at the formal structure of the Boltzmann equation, especially focusing on the spectral decomposition. At the background level, the Boltzmann equation yields $$\label{EqbackgroundI} \frac{{{\cal D}}}{{{\cal D}}\lambda} {{I_{\rm BB} \left( \frac{q}{a \bar T} \right)}}= \frac{q}{a^2} \left. \frac{{{\partial}}{{I_{\rm BB}}}(x)}{{{\partial}}\eta} \right|_{q} = - \frac{{{\rm d}}\ln (a \bar T)}{{{\rm d}}\lambda} \left.\frac{{{\rm d}}{{I_{\rm BB}}}(x)}{{{\rm d}}\ln x} \right|_{x=q/(a\bar T)} = 0 \,.$$ This implies that $\bar I$ has no time dependence and $\bar T$ scales as $1/a$. One can conclude that the Planck distribution does not change in time if the initial distribution is given by the Planck one. This does not mean that the radiation is not losing energy as the universe expands. Indeed, since the physical energy of a photon is not the conformal energy $q$ but $q/a$, then $\bar \rho \propto \int \bar I (q/a)^3 {{\rm d}}q/a \propto a^{-4}$ as expected. This background result for the scaling of $\bar T$ is useful as it implies that only the partial derivative with respect to $q$ on ${{I_{\rm BB}}}[q/(a \bar T)]$ are relevant, and this motivates our use of the conformal energy. Now the action of the Liouville operator on the intensity, Eq (\[intensity\]), is expanded up to the second order as $$\begin{aligned} L[I] &= \frac{{{\rm d}}\ln q}{{{\rm d}}\lambda} {{\frac{{{\partial}}I_{\rm BB}}{{{\partial}}\ln q}}}- L \Bigl[ \Theta + \Theta^2 \Bigr] {{\frac{{{\partial}}I_{\rm BB}}{{{\partial}}\ln q}}}- \Theta \frac{{{\rm d}}\ln q}{{{\rm d}}\lambda} \frac{{{\partial}}^2 {{I_{\rm BB}}}}{{{\partial}}\ln q^2} + L \left[ y + \frac{1}{2} \Theta^2 \right] {{\cal D}}_q^2 {{I_{\rm BB}}}\notag\\ &= - \left[ L [\Theta] - \frac{{{\rm d}}\ln q}{{{\rm d}}\lambda} - \Theta \frac{{{\rm d}}\ln q}{{{\rm d}}\lambda} + 2 \Theta \left( L [\Theta] - \frac{{{\rm d}}\ln q}{{{\rm d}}\lambda} \right) \right] {{\frac{{{\partial}}I_{\rm BB}}{{{\partial}}\ln q}}}+ \left[ L [y] + \Theta \left( L [\Theta] - \frac{{{\rm d}}\ln q}{{{\rm d}}\lambda} \right) \right] {{\cal D}}^2_q {{I_{\rm BB}}}\,, \label{original}\end{aligned}$$ where we used ${{\partial}}{{I_{\rm BB}}}/{{\partial}}\eta = 0$. When we compare this formulation of the Liouville operator with the spectral decomposition (\[intensity\]), we are tempted to say that the expression inside the first square brackets contributes to the evolution of the temperature due to free-streaming, and that the expression inside the second square brackets is very closely related to the evolution of the distortion. It order to give a clear meaning to this assertion we decompose Eq. (\[original\]) according to $$\label{spectraldecI} L[I] \equiv \frac{q}{a^2} \left[ - \Bigl( {{\cal L}}^\Theta + 2 \Theta {{\cal L}}^\Theta \Bigr) {{\frac{{{\partial}}I_{\rm BB}}{{{\partial}}\ln q}}}+ \Bigl( {{\cal L}}^Y + \Theta {{\cal L}}^\Theta \Bigr) {{\cal D}}^2_q {{I_{\rm BB}}}\right] \,.$$ This spectral decomposition is motivated by the fact that i) in the case where the conformal energy is not affected by free-streaming, ${{\rm d}}\ln q / {{\rm d}}\lambda = 0$, then $(q/a^2) {{\cal L}}^\Theta$ simply reduces to $L[\Theta]$; and ii) the prefactor $q/a^2$ is introduced because the Liouville term is expected to be proportional to $q/a^2$, as it can be inferred from the explicit form (\[eta-evo\]) of ${{\rm d}}\eta/{{\rm d}}\lambda$. From a comparison of Eq (\[original\]) with this decomposition, we have $$\begin{aligned} \label{SpectraldecLI} \frac{q}{a^2}{\cal L}^\Theta&=L[\Theta] -(1+ \Theta )\frac{{{\rm d}}\ln q}{{{\rm d}}\lambda}\,,\\ \frac{q}{a^2}{\cal L}^Y&=L[y]\,.\end{aligned}$$ We must bear in mind that in these expressions, even though the operator $L[.]$ has been defined in Eq. (\[defofLs\]) for functions of $(\eta,x^i,q,n^{(i)})$, it is applied on the spectral components $\Theta$ and $y$ which do not depend on $q$. Since the Liouville operator is equated to the collision term in the Boltzmann equation, it is convenient to decompose the collision term in the same manner as the Liouville term. That is, it is decomposed as $$C^I \equiv \frac{q}{a^2} \left[ - \Bigl( {{\cal C}}^\Theta + 2 \Theta {{\cal C}}^\Theta \Bigr) {{\frac{{{\partial}}I_{\rm BB}}{{{\partial}}\ln q}}}+ \Bigl( {{\cal C}}^Y + \Theta {{\cal C}}^\Theta \Bigr) {{\cal D}}^2_q {{I_{\rm BB}}}\right] \,,$$ such that the spectral components of the Boltzmann equation can be formally very simple and are given by $$\begin{aligned} {{\cal L}}^\Theta= {{\cal C}}^\Theta \,,\qquad {{\cal L}}^Y={{\cal C}}^Y. \end{aligned}$$ This decomposition means that once the spectral decomposition of the collision term is known (${\cal C}^\Theta$ and ${\cal C}^Y$), then we only need to obtain the spectral decomposition of the Liouville term from Eqs (\[SpectraldecLI\]). We follow the same logic for polarization. First, the corresponding Liouville operator reads $$\begin{aligned} L [ {\bf P} ]_{\mu \nu} &= - L \Bigl[ (1 + 3 \Theta) {{\cal P}}_{\mu\nu} \Bigr] {{\frac{{{\partial}}I_{\rm BB}}{{{\partial}}\ln q}}}- {{\cal P}}_{\mu \nu} \frac{{{\rm d}}\ln q}{{{\rm d}}\lambda} \frac{{{\partial}}^2 {{I_{\rm BB}}}}{{{\partial}}\ln q^2} + L \Bigl[ Y_{\mu \nu} + \Theta {{\cal P}}_{\mu \nu} \Bigr] {{\cal D}}_q^2 {{I_{\rm BB}}}\notag\\ &= - \left[ (1 + 3 \Theta) L [ {\bm {{\cal P}}} ]_{\mu \nu} + 3 \left(L[\Theta]- \frac{{{\rm d}}\ln q}{{{\rm d}}\lambda}\right) {{\cal P}}_{\mu \nu} \right] {{\frac{{{\partial}}I_{\rm BB}}{{{\partial}}\ln q}}}+ \left[ L [ {\bf Y} ]_{\mu \nu} + \left(L[\Theta]- \frac{{{\rm d}}\ln q}{{{\rm d}}\lambda}\right){{\cal P}}_{\mu \nu} + \Theta L [{\bm {{\cal P}}}]_{\mu \nu} \right] {{\cal D}}_q^2 {{I_{\rm BB}}}\,. \end{aligned}$$ For the same reasons as in the case of intensity, it appears natural to decompose this Liouville operator into spectral components according to $$\label{spdec:calP_lhs} L [ {\bf P} ]_{\mu \nu}\equiv\frac{q}{a^2} \left\{ - \Bigl[ (1 + 3 \Theta) {{\cal L}}_{\mu \nu}^P + 3 {{\cal L}}^\Theta {{\cal P}}_{\mu \nu} \Bigr] {{\frac{{{\partial}}I_{\rm BB}}{{{\partial}}\ln q}}}+ \Bigl( {{\cal L}}^Y_{\mu \nu} + {{\cal L}}^\Theta {{\cal P}}_{\mu \nu} + \Theta {{\cal L}}_{\mu \nu}^P \Bigr) {{\cal D}}_q^2 {{I_{\rm BB}}}\right\} \,,$$ which implies that the spectral components are given by $$\label{SpectraldecLP} \frac{q}{a^2} {{\cal L}}_{\mu \nu}^P \equiv L [ {\bm {{\cal P}}} ]_{\mu \nu},\qquad \frac{q}{a^2} {{\cal L}}_{\mu \nu}^Y \equiv L [ {\bf Y} ]_{\mu \nu} \,.$$ The collision term must then follow the same type of decomposition, that is $$\begin{aligned} C^P_{\mu \nu} &= \frac{q}{a^2} \left\{ - \Bigl[ (1 + 3 \Theta) {{\cal C}}_{\mu \nu}^P + 3 {{\cal C}}^\Theta {{\cal P}}_{\mu \nu} \Bigr] {{\frac{{{\partial}}I_{\rm BB}}{{{\partial}}\ln q}}}+ \Bigl( {{\cal C}}^Y_{\mu \nu} + {{\cal C}}^\Theta {{\cal P}}_{\mu \nu} + \Theta {{\cal C}}_{\mu \nu}^P \Bigr) {{\cal D}}_q^2 {{I_{\rm BB}}}\right\} \,, \label{spdec:calP_rhs}\end{aligned}$$ so that, again, the spectral components of the polarized part of the Boltzmann equation take the formally simple form $$\label{eq:Boltz_pol_dis} {{\cal L}}^P_{\mu \nu}={{\cal C}}^P_{\mu \nu} \,,\qquad {{\cal L}}^Y_{\mu \nu} ={{\cal C}}^Y_{\mu \nu} \,.$$ Again, this decomposition means that once the spectral decomposition of the collision term is known (${\cal C}^P_{\mu\nu}$ and ${\cal C}^Y_{\mu\nu}$), then we only need to obtain the spectral decomposition of the Liouville term from Eqs (\[SpectraldecLP\]) bearing in mind that the Liouville operator $L[.]$ applies on functions which do not depend on $q$, but only on $(\eta,x^i,n^{(i)})$. Temperature and spectral distortion of Liouville operators ---------------------------------------------------------- Now that the spectral separation of the Boltzmann equation is performed, it is time to expand the equations obtained in orders of perturbations. In the next two sections, we present such expansion for the temperature and spectral distortion parts of the Boltzmann equation. The case of polarization is reported in Appendix \[sapp:Boltzeq\_pol\]. At first order in perturbation, with Eqs. (\[eta-evo\]) and (\[q-evo\]), the Boltzmann equation leads to $$\begin{aligned} {\cal L}^\Theta&= \Theta' + \Theta_{, i} n^{(i)} + \alpha_{, i} n^{(i)} - \beta_{i, j} n^{(i)} n^{(j)} + h_{i j}{}' n^{(i)} n^{(j)} \,.\end{aligned}$$ Note that there is absolutely no q-dependence, nor scale factor $a$ in this expression, meaning that our spectral decomposition performed in Eq. (\[spectraldecI\]) is adequate. Concerning the spectral distortion part, the Liouville part at first order is ${\cal L}^Y= y'+y_{,i} n^{(i)}$, but since ${\cal C}^Y=0$ at first order (see section \[ssec:Collision\]), one can conclude that only the temperature part evolves at first order and no spectral distortion is induced. Up to the second order, the Boltzmann equation for the temperature is given by $$\begin{aligned} \label{LCThetanotOpened} {\cal L}^\Theta &= \frac{a^2}{q}\left[L[\Theta] - (1 + \Theta) \frac{{{\rm d}}\ln q}{{{\rm d}}\lambda}\right] \notag\\ &= \Theta' + \Theta_{, i} n^{(i)} + \alpha_{, i} n^{(i)} - \beta_{i, j} n^{(i)} n^{(j)} + h_{i j}{}' n^{(i)} n^{(j)} \notag\\ & \qquad +\frac{a^2}{q} \left[ \left. \frac{{{\rm d}}\eta}{{{\rm d}}\lambda} \right|^{(1)} \Theta' + \left. \frac{{{\rm d}}x^i}{{{\rm d}}\lambda} \right|^{(1)} \Theta_{, i} + \left. \frac{{{\rm d}}n^{(i)}}{{{\rm d}}\lambda} \right|^{(1)} D_i \Theta - \left. \frac{{{\rm d}}\ln q}{{{\rm d}}\lambda} \right|^{(1) \times (1)} - \Theta \left. \frac{{{\rm d}}\ln q}{{{\rm d}}\lambda} \right|^{(1)} \right] = {{\cal C}}^\Theta \,.\end{aligned}$$ As for the spectral distortion, the Boltzmann equation is given, even at second order by $$\begin{aligned} {\cal L}^Y &= y' + y_{, i} n^{(i)} = {{\cal C}}^Y \,.\end{aligned}$$ This means simply that gravitational effects do not induce spectral distortions, and this result holds actually non-perturbatively. Before ending this subsection, for the sake of completeness, we shall write down the most general form of the second order Boltzmann equation for the intensity. Using Eqs. (\[eta-evo\]), (\[q-evo\]), (\[xi-evo\]) and (\[n-evo\]), the detailed form of the evolution equation for temperature obtained in Eq. (\[LCThetanotOpened\]) is $$\begin{aligned} {\cal L}^\Theta=& \Theta' + \Theta_{, i} n^{(i)} + \alpha_{,i} n^{(i)} - \beta_{i ,j} n^{(i)} n^{(j)} + h_{i j}{}' n^{(i)} n^{(j)} \notag\\ & \quad - \alpha \Theta' - (\beta^i + h^i{}_j n^{(j)}) \Theta_{, i} - \Bigl[ \alpha_{,i} - (\beta_{j ,i} - h_{i j}{}') n^{(j)} + (h_{k i, j} - h_{j k, i}) n^{(j)} n^{(k)} \Bigr] D^i \Theta \notag\\ & \quad - \Bigl( \alpha \alpha_{,i} + \alpha_{, j} h^j{}_i \Bigr) n^{(i)} - \Bigl[ \alpha (- \beta_{i ,j} + h_{i j}{}') + \beta^k h_{i j, k} + (\beta_{k, i} - \beta_{i, k} + 2 h_{i k}{}') h^k{}_j \Bigr] n^{(i)} n^{(j)} \notag\\ & \quad - \Theta n^{(i)} \Bigl( - \alpha_{,i} + \beta_{i ,j} n^{(j)} - h_{i j}{}' n^{(j)} \Bigr) = {{\cal C}}^\Theta \,.\end{aligned}$$ Temperature and spectral distortion of collision terms {#ssec:Collision} ------------------------------------------------------ The expression of the collision term has been derived by taking into account only the intensity in [@Dodelson1993; @Bartolo:2006cu] and then it was extended to include the effect of polarisation in [@Pitrou:2008hy; @Pitrou:2008ut; @Beneke:2010eg]. Here we summarize the result obtained by Beneke et al. [@Beneke:2010eg] applying the decomposition of the distribution function into intensity and linear polarisation. The complete expression of the collision term for intensity is given by $$\begin{aligned} {{\cal C}}^\Theta &= a \, \bar{n}_e \sigma_T \left( - \Theta + \langle \Theta \rangle - \frac{3}{4} S^{(i) (j)} \Bigl[ \langle \Theta m_{(i) (j)} \rangle - 2 \langle {{\cal P}}_{(i) (j)} \rangle \Bigr] + v^{(i)} n_{(i)} + {{\cal S}}^T + S^{(i) (j)} {{\cal Q}}^T_{(i) (j)} + \delta_e {{\cal C}}^\Theta \right) \,, \\ {{\cal C}}^Y &= a \, \bar{n}_e \sigma_T \left( - y + \langle y \rangle - \frac{3}{4} S^{(i) (j)} \Bigl[ \langle y m_{(i) (j)} \rangle - 2 \langle Y_{(i) (j)} \rangle \Bigr] + {{\cal S}}^Y + S^{(i) (j)} {{\cal Q}}^Y_{(i) (j)} \right) \,,\end{aligned}$$ where ${{\cal S}}^T$, ${{\cal S}}^Y$, ${{\cal Q}}^T_{(i) (j)}$ and ${{\cal Q}}^Y_{(i) (j)}$ are quadratic contributions defined by $$\begin{aligned} {{\cal S}}^T &= \Theta^2 + \langle \Theta^2 \rangle - 2 \Theta \langle \Theta \rangle - \Theta v^{(i)} n_{(i)} + 2 \langle \Theta \rangle v^{(i)} n_{(i)} - 2 \langle \Theta n_{(i)} \rangle v^{(i)} - \frac{1}{5} v^{(i)} v_{(i)} + (v^{(i)} n_{(i)})^2 \,, \\ {{\cal S}}^Y &= \frac{1}{2} \Theta^2 + \frac{1}{2} \langle \Theta^2 \rangle - \Theta \langle \Theta \rangle - \Theta v^{(i)} n_{(i)} + \langle \Theta \rangle v^{(i)} n_{(i)} - \langle \Theta n_{(i)} \rangle v^{(i)} + \frac{1}{5} v^{(i)} v_{(i)} + \frac{1}{2} (v^{(i)} n_{(i)})^2 \,, \end{aligned}$$ $$\begin{aligned} {{\cal Q}}^T_{(i) (j)} &= - \frac{3}{4} \langle \Theta^2 m_{(i) (j)} \rangle + \frac{3}{2} \Theta \langle \Theta m_{(i) (j)} \rangle + \frac{3}{4} \Bigl[ \langle \Theta n_{(j)} \rangle v_{(i)} + \langle \Theta n_{(i)} \rangle v_{(j)} \Bigr] \notag\\ & \qquad - \frac{3}{4} n^{(k)} \Bigl[ v_{(j)} \langle \Theta m_{(i) (k)} \rangle + v_{(i)} \langle \Theta m_{(j) (k)} \rangle \Bigr] - \frac{3}{4} v^{(k)} \Bigl[ 2 n_{(k)} \langle \Theta m_{(i) (j)} \rangle + \langle \Theta m_{(i) (j)} n_{(k)} \rangle \Bigr] - \frac{1}{5} v_{(i)} v_{(j)} \notag\\ & \qquad + \frac{9}{2} \langle \Theta {{\cal P}}_{(i) (j)} \rangle - 3 \Theta \langle {{\cal P}}_{(i) (j)} \rangle + \frac{3}{2} n^{(k)} \Bigl[ v_{(j)} \langle {{\cal P}}_{(i) (k)} \rangle + v_{(i)} \langle {{\cal P}}_{(j) (k)} \rangle \Bigr] \notag\\ & \qquad + \frac{3}{2} v^{(k)} \Bigl[ \langle {{\cal P}}_{(i) (k)} n_{(j)} \rangle + \langle {{\cal P}}_{(j) (k)} n_{(i)} \rangle \Bigr] - \frac{3}{2} v^{(k)} \Bigl[ \langle {{\cal P}}_{(i) (j)} n_{(k)} \rangle - 2 n_{(k)} \langle {{\cal P}}_{(i) (j)} \rangle \Bigr] \,, \\ {{\cal Q}}^Y_{(i) (j)} &= - \frac{3}{8} \langle \Theta^2 m_{(i) (j)} \rangle + \frac{3}{4} \Theta \langle \Theta m_{(i) (j)} \rangle - \frac{3}{4} v^{(k)} \Bigl[ n_{(k)} \langle \Theta m_{(i) (j)} \rangle - \langle \Theta m_{(i) (j)} n_{(k)} \rangle \Bigr] - \frac{1}{20} v_{(i)} v_{(j)} \notag\\ & \qquad + \frac{3}{2} \langle \Theta {{\cal P}}_{(i) (j)} \rangle - \frac{3}{2} \Theta \langle {{\cal P}}_{(i) (j)} \rangle + \frac{3}{2} v^{(k)} \Bigl[ n_{(k)} \langle {{\cal P}}_{(i) (j)} \rangle - \langle {{\cal P}}_{(i) (j)} n_{(k)} \rangle \Bigr] \,,\end{aligned}$$ and $m_{(i) (j)} \equiv n_{(i)} n_{(j)} - \delta_{(i) (j)}/3$. Note that we have introduced the notation $\langle Q \rangle = \int_{\Omega} Q = \int {{\rm d}}^2 n^{(i)} Q$, which corresponds to a multipole extraction that we do not perform explicitly here. Note also that $\delta_e=\delta n_e/\bar n_e$ is the fractional perturbation of the baryons number density, and $v^{(i)}$ are the tetrad components of the baryons spatial velocity. Again, we also defer the expression of the collision term for polarisation to Appendix \[sapp:Boltzeq\_pol\]. Gauge dependence of the distribution function {#sec:gauge} ============================================= Now that we have established the Boltzmann equation, up to second order, with its spectral components separated, we investigate the gauge dependence of its constituents. Eventually the Boltzmann equation itself should be gauge-invariant, so if we are able to check explicitly that the Boltzmann equation is gauge-invariant, this means that it is very likely that i) the perturbative expansion of the equation is correct; and ii) the gauge transformation rules for all its constituents (metric and distribution function perturbations) are correctly understood. We thus consider this verification as a *consistency test*. This section is dedicated entirely to this task. We first review the gauge dependence for tensors, and deduce how it can be extended to a scalar distribution function. The case of a tensor-valued distribution function, even though it is the less trivial part, is treated in appendix \[AppGTtensor\]. We then infer what should be the transformation rule of the Liouville and collision operators, and in order to complete the consistency test, we check that the perturbed expressions of the Liouville and collision operators do indeed transform following these rules. Coordinates on the manifold --------------------------- We need to specify how the functional dependence of the quantities appearing in the Boltzmann equation (and in the Einstein equation) is obtained. If we consider a scalar function $f:{\cal M}\mapsto {\mathbb{R}}$ on the space-time manifold ${\cal M}$, then the choice of a coordinates system [^1] $c:{\mathbb{R}}^4 \mapsto {\cal M}$ does not affect the geometrical meaning of this function, but it affects its functional form $f \circ c:{\mathbb{R}}^4 \mapsto {\mathbb{R}}$, where $\circ$ designates the composition rule, in the sense that for another coordinates system $\tilde c:{\mathbb{R}}^4 \mapsto {\cal M}$, then $f\circ \tilde c \neq f \circ c$. The confusion only arises from the fact that we often refer to $f \circ c$ as $f$ only. A distribution function $f$ (that we take as a scalar-valued for simplicity here) is a function on the tangent bundle $T {\cal M}$ of the manifold and can be regarded as a function on manifold ${\cal M}$ describing the space-time and on the tangent space ${\mathbb{R}}^4$ (more precisely a restriction to the mass shell ${\mathbb{R}}^3$) of each point. It is thus a function $$f: (T{\cal M})\mapsto {\mathbb{R}}\,.$$ Again the particular choice of coordinates on the tangent space does not affect the geometrical meaning of the function but its functional form. Once a choice $c$ of coordinates on the manifold ${\cal M}$ has been made, there is a natural basis, called canonical basis, that is made of the partial derivatives with respect to the coordinates. This leads to a natural coordinates system $Tc$ for the tangent space at each point. Thus $(c,Tc) : {\mathbb{R}}^4 \times {\mathbb{R}}^3 \mapsto T{\cal M}$, is a coordinates system for the tangent bundle. In order to simplify the notation we will note $(c,Tc)$ as simply as $c$. Furthermore, in this paper we use coordinates in the tangent space described by the tetrad components. More specifically we use the components of the conformal momentum in spherical coordinates in this tetrad basis, $(q,n^{(i)})$. There is a problem with such a choice since the tetrad basis is not unique. However, once a choice $c$ of coordinates on the manifold ${\cal M}$ has been made, the tetrad basis might be completely fixed from the metric through a prescription described in § \[ssec:deftet\]. Once a coordinates system $c$ has been chosen, and the tetrad is fixed thanks to this choice, we obtain the functional form of $f$ in the form $f \circ c:{\mathbb{R}}^4\times {\mathbb{R}}^3\mapsto {\mathbb{R}}$. Geometrical interpretation of the gauge --------------------------------------- When performing perturbations around a background FLRW space-time, we need to have a one-to-one correspondence between the background space-time $\overline{\cal M}$ and the physical (and perturbed) space-time ${\cal M}$. This can be completely defined geometrically [@Bruni:1996im] but we take a shorter approach. If we have two sets of coordinates [^2] $\bar c:{\mathbb{R}}^4 \mapsto \overline{\cal M}$ and $c:{\mathbb{R}}^4 \mapsto {\cal M}$ on the background and the perturbed space-time, then we identify points with the same coordinates, that is, we identify points with $c \circ \bar c^{-1}:\overline{\cal M}\mapsto {\cal M}$. Since we could have chosen different sets of coordinates, there is some freedom in this choice, which is known as the gauge freedom. On the background, the symmetries can justify that we can find a preferred choice of coordinates. For instance for a flat FLRW space-time within a given background cosmology, it is enough to choose that the time coordinate is the proper time of observers with 4-velocity orthogonal to the homogeneous surfaces, that is the proper time of comoving observers. On a given homogeneous surface, there are preferred choices of Cartesian coordinates, since it is conformally related to ${\mathbb{R}}^3$, and all these Cartesian systems on the spatial homogeneous surfaces are related by global translation and rotation in ${\mathbb{R}}^3$ which are irrelevant given the homogeneity. And once a coordinate system has been chosen on a homogeneous surface it can be Lie dragged by the comoving observers to any homogeneous surface. So essentially there is a unique mapping $\bar c$ from ${\mathbb{R}}^4$ to the background manifold $\overline{\cal M}$. This point is illustrated in the left part of Fig. \[fig1\]. However on the physical space-time we could consider another coordinates system $\tilde c:{\mathbb{R}}^4 \mapsto {\cal M}$, and this leads to a different identification through $\tilde c \circ \bar c^{-1}$. The fact that we fix the system of coordinates on the background space-time but there is still some freedom on the physical space-time for the choice of coordinates leads to a freedom in the identification between points of these two space-times. With $c$, a given point $P \in {\cal M}$ would be labelled by the coordinates $x^\mu$, that is $c( {\bm x}) = P$, and with $\tilde c$ it would be labeled by the coordinates $\tilde x^\mu$, that is $\tilde c( \tilde{\bm x}) = P$ (see the upper part of Fig. \[fig1\]). For every point, there exist four numbers ${\xi}^\mu= ({ T},{ L}^i)$ such that $$\begin{gathered} \label{Trulexmu} \tilde{x}^\mu( {\bm x}) = x^\mu + {\xi}^\mu( {\bm x}) \,.\end{gathered}$$ We should note that, although there is an index in the notation, ${\xi}^\mu$ is not a vector field on ${\cal M}$ nor on $\overline{\cal M}$, but it can be seen as a vector field on ${\mathbb{R}}^4$. In the literature, a vector field $\zeta^\nu$ is often used to generate a coordinate transformation  [@Bruni:1996im] $$\tilde x^\mu = \exp\left({\cal L}_{{{\boldsymbol{\zeta}}}}\right)x^\mu =x^\mu+\zeta^\mu+\frac{1}{2}\zeta^\mu{}_{,\nu} \zeta^\nu+\cdots\,.$$ In this paper we adopt the former definition Eq. (\[Trulexmu\]) and care must be taken when comparing our transformation rules with those in the literature. Note that in the rest of this section, we will use extensively the notation $$\tilde{\bm x} \equiv \tilde x^\mu\,,\qquad {\bm x}\equiv x^\mu\,,$$ even though $x^\mu$ and $\tilde x^\mu$ are not vectors but just coordinates. ![\[fig1\] In this figure, we represent all four dimensional spacetimes with only two dimensions. First, we have noted that there is essentially a unique way to map ${\mathbb{R}}^4$ to the background manifold, that is to relate the top-left to the bottom left. However, there are several ways to relate ${\mathbb{R}}^4$ to the perturbed manifold, and consequently to relate the background manifold to the perturbed manifold. We represented a coordinates system $c$ and another coordinates system $\tilde c$ which relate ${\mathbb{R}}^4$ (top-left) to the physical manifold (top-right). For each point $P$ of the physical manifold, there is a set of four numbers $x^\mu$ and another set of four numbers $\tilde x^\mu$ such that $c(x^\mu) = \tilde c(\tilde x^\mu)$. Furthermore, each coordinates system has different surfaces of constant time, and thus different set of tetrads, given that the null tetrad is always chosen to be orthogonal to the constant-time surfaces. When using the tetrad field to extract the components of a given momentum (that is a point in the tangent bundle), this will lead to different components, depending on the coordinates system chosen, but since this is the same momentum at the same point and the tetrad are normalized, we can relate the components by a Lorentz transformation. For a given point of the tangent bundle, that is, for a given point of the manifold and a given momentum, we always have $c(x^\mu,p^{(a)})=\tilde c(\tilde x^\mu, p^{\tilde{(a)}})$.](PlotGauge.pdf){width="\textwidth"} Metric transformation --------------------- In general the coordinate transformation rule of any tensorial quantity is given by $$\label{GTTensor} {\bm T} \circ \tilde c{(\tilde {\bm x})} = {\bm T}\circ c({\bm x})\,.$$ This means that it is invariant under a coordinates transformation since it is geometrically defined and independent of the coordinates used to parametrise the manifold. From this we can deduce the gauge transformation of its components, which is defined as the transformation when the tensors are compared at the same coordinate in two different coordinates systems. Let us consider in particular the metric. Since $g_{\mu \nu} \equiv {\bm g}({{\partial}}/{{\partial}}x^\mu,{{\partial}}/{{\partial}}x^\nu)$ and $g_{\tilde \mu \tilde \nu} \equiv {\bm g}({{\partial}}/{{\partial}}\tilde x^\mu,{{\partial}}/{{\partial}}\tilde x^\nu)$, we then deduce the usual coordinate transformation rule of a 2-form $$\label{Transfog1} g_{\tilde \mu \tilde \nu} \circ \tilde c{(\tilde {\bm x})} = \frac{{{\partial}}x^\alpha}{{{\partial}}\tilde x^\mu} \frac{{{\partial}}x^\beta}{{{\partial}}\tilde x^\nu} g_{\alpha \beta}\circ c({\bm x})\,.$$ Expanding the left hand side which is evaluated at $\tilde {\bm x}$ around ${\bm x}$ leads the gauge transformation rule up to the second order $$\begin{gathered} \label{EqTmetric} g_{\tilde \alpha \tilde \beta}\circ \tilde c{\left({\bm x}\right)} = g_{\alpha \beta}\circ c({\bm x}) - {{\cal L}}_{{{{\boldsymbol{\xi}}}}} g_{\alpha \beta}\circ c({\bm x}) + \frac{1}{2} {{\cal L}}^2_{{{{\boldsymbol{\xi}}}}} g_{\alpha \beta}\circ c({\bm x}) + \frac{1}{2} {{\cal L}}_{{{{\boldsymbol{\xi}}}} {{{\boldsymbol{\xi}}}}} g_{\alpha \beta}\circ c({\bm x}) \,,\end{gathered}$$ where ${{{\boldsymbol{\xi}}}} {{{\boldsymbol{\xi}}}}$ designates the quantity ${\xi}^\mu{}_{,\nu}{\xi}^\nu$. We emphasize that the coordinate transformation  is the transformation of components at the same point of space-time, whereas the gauge transformation  is the transformation of the components at different points which have the same coordinates in two different coordinates systems. Since the notation can become rather cumbersome if we specify which coordinates system $c$ or $\tilde c$ is to be used, we shall indicate it only when it is the new coordinates system $\tilde c$. Throughout this paper, we use the symmetrization and anti-symmetrization definitions $$X_{(\mu\nu)} \equiv \frac{1}{2}\left(X_{\mu\nu}+ X_{\nu\mu}\right) \,, \qquad X_{[\mu\nu]} \equiv \frac{1}{2}\left(X_{\mu\nu}-X_{\nu\mu}\right)\,.$$ Up to second order, from the transformation (\[EqTmetric\]) and the decomposition (\[MetricADM\]), we obtain the gauge transformation for the ADM variables as \[TrulemetricADM\] $$\begin{aligned} \tilde{\alpha} &= \alpha - {{\cal H}}T - T' + \frac{1}{2} ({{\cal H}}^2 + {{\cal H}}') T^2 + {{\cal H}}(2 T' T + T_{,i} L^i) - {{\cal H}}\alpha T \notag\\ & \qquad - \alpha T' - \alpha' T - \alpha_{,i} L^i + \beta^i T_{,i} + T'' T + T'{}^2 + T'_{,i} L^i + \frac{1}{2} T^{, i} T_{, i} \,, \\ \tilde{\beta}^i &= \beta^i + T^{,i} - L^i{}' + 2 \alpha T^{, i} - \beta^i T' - \beta^i{}' T + \beta^j L^i{}_{,j} - \beta^i{}_{,j} L^j - 2 h^{i j} T_{,j} \notag\\ & \qquad - (2 T' T^{, i} + T T'{}^{, i}) + T L^i{}'' + T' L^i{}' + T_{,j} L^{i ,j} - T^{, i}{}_{,j} L^j + L^i{}_{,j}{}' L^j \,, \\ 2 \tilde{h}_{i j} &= 2 h_{i j} - 2 {{\cal H}}T \delta_{i j} - 2 L_{(i ,j)} - 4 {{\cal H}}T h_{i j} + (2 {{\cal H}}^2 + {{\cal H}}') T^2 \delta_{i j} + 2 {{\cal H}}( T T' + T_{,k} L^k) \delta_{i j} \notag\\ & \qquad + 4 {{\cal H}}T L_{(i ,j)} - 2 \beta_{(i} T_{,j)} - 2 h_{i j}{}' T - 2 h_{i j, k} L^k - 4 h_{(i| k} L^k{}_{,|j)} \notag\\ & \qquad - T_{,i} T_{,j} + 2 T L_{(i ,j)}{}' + 2 T_{(,i} L_{j)}{}' + 2 L_{(i ,j) k} L^k + 2 L_{(i|, k} L^k{}_{,|j)} + L_{k, i} L^k{}_{, j} \,.\end{aligned}$$ These relations must be understood as follows. $a^2 \alpha$ at first order is the $00$ component of the first order metric in the $c$ coordinates system, and taken at the point of coordinates ${\bm x}$, and is thus equal to $-\frac{1}{2}g^{(1)}_{00}\circ c({\bm x})$. Instead, $a^2 \tilde \alpha$ which means $a^2 \tilde \alpha({\bm x})$, when considered at first order is the $\tilde 0 \tilde 0$ component of the first order metric in the $\tilde c$ coordinates system, but also taken at the point of coordinates ${\bm x}$, that is $-\frac{1}{2}g^{(1)}_{\tilde 0 \tilde 0}\circ \tilde c{({\bm x})}$. Tangent space basis and tetrads ------------------------------- As discussed in Section \[ssec:f\], the transformation of the basis on the tangent space is entirely linked to the coordinates change in the base manifold. Usually, the canonical basis ${{\partial}}/{{\partial}}x^\mu$ and the corresponding forms ${{\rm d}}x^\mu$ are used as a basis of the tangent space and they transform according to $$\label{TruleCanonicalBasis} \frac{{{\partial}}}{{{\partial}}\tilde x^\mu}\circ \tilde c{(\tilde{\bm x})} = \frac{{{\partial}}x^\nu}{{{\partial}}\tilde x^\mu} \frac{{{\partial}}}{{{\partial}}x^\nu}{({\bm x})} \,, \qquad {{\rm d}}\tilde x^\mu \circ \tilde c {(\tilde{\bm x})} = \frac{{{\partial}}\tilde x^\mu}{{{\partial}}x^\nu} {{\rm d}}x^\nu {c({\bm x})} \,.$$ When we consider the components of the metric, these components refer to this canonical basis. However we will use the tetrad field as a basis for the tangent space. We thus need to relate $\tilde{\bm e}_{(a)}\circ \tilde c{(\tilde{\bm x})}$ with ${\bm e}_{(a)}\circ c{({\bm x})}$ in a similar fashion. Since this is a relation between two orthonormal basis at the same point of space-time, there exists a Lorentz transformation $\Lambda^{(a)}_{~~ (b)}$ such that we can relate the two tetrad fields associated with the coordinates systems $c$ and $\tilde c$ (see the bottom right part of Fig. \[fig1\] for an illustration of this) as $$\tilde{\bm e}_{(a)}\circ \tilde c{(\tilde{\bm x})} = \Lambda_{(a)}^{~~ (b)} ({\bm x}) {\bm e}_{(b)} ({\bm x}) \,,\qquad \tilde{\bm e}^{(a)} \circ \tilde c{(\tilde{\bm x})} = \Lambda^{(a)}_{~~ (b)} ({\bm x}) {\bm e}^{(b)} ({\bm x})\,,\qquad \eta_{(c) (d)} \Lambda^{(c)}_{~~ (a)} \Lambda^{(d)}_{~~ (b)} = \eta_{(a) (b)} \,.$$ The components of this Lorentz transformation are given, up to first order, by $$\label{Reltetradssamephysicalpoint} \Lambda^{(0)}_{~~ (0)} = 1 \,,\qquad \Lambda^{(0)}_{~~ (i)} = \Lambda^{(i)}_{~~ (0)} = {{\partial}}_i T \,,\qquad \Lambda^{(i)}_{~~ (j)} = \delta^{(i)}_{~~ (j)} +L^{[i}_{~~ , j]}\,.$$ At second order, it proves more useful to relate the components of these quantities, that is, to relate $\widetilde{e}_{(a)}{}^{\widetilde{\mu}}\circ \tilde c{(\tilde{\bm x})} \equiv {{\rm d}}\tilde x^\mu\circ \tilde c{(\tilde{\bm x})} [\widetilde{\bm e}_{(a)}]$ to $ e_{(a)}{}^\mu({\bm x})\equiv {{\rm d}}x^\mu({\bm x})[{\bm e}_{(a)}]$. For the former components, we must use  with $a(\tilde{\eta})$, $\tilde \alpha(\tilde{\bm x})$, $\tilde \beta_i(\tilde{\bm x})$ and $\tilde h_{ij}(\tilde{\bm x})$ \[that is $g_{\tilde \mu \tilde \nu}\circ \tilde c{(\tilde{\bm x})}$\] which can be deduced from the rule  just by shifting the argument of the left hand side of the rules  from ${\bm x}$ to $\tilde{\bm x}$. For the latter we must use  with $a(\eta)$, $\alpha({\bm x})$, $\beta_i({\bm x})$ and $h_{ij}({\bm x})$ \[that is $g_{\mu \nu}{({\bm x})}$\]. We do not report the corresponding expression since we will work instead directly with the perturbation components of the metric in the next section. Momentum, energy $q$ and direction $n^{(i)}$ -------------------------------------------- The components of the momentum of a particle in the canonical basis transform as $$\label{Transfopmu} p^{\widetilde{\mu}}\circ \tilde c(\tilde{\bm x}) = \frac{{{\partial}}\tilde{x}^\mu}{{{\partial}}x^\nu} p^\nu({\bm x})\,, \qquad \text{with} \qquad p^{\widetilde{\mu}} \equiv {{\rm d}}\tilde x^\mu({\bm p})\,,\quad p^{\mu} \equiv {{\rm d}}x^\mu({\bm p})\,.$$ However, we are going to use the tetrad basis in the tangent space rather than the canonical basis. We thus want to express $p^{\widetilde{(a)}}\circ \tilde c{(\tilde{\bm x})} \equiv \widetilde{\bm e}^{{(a)}}\circ \tilde c{(\tilde{\bm x})}[{\bm p}]$, \[or $\tilde q\circ \tilde c{(\tilde{ \bm x})}$ [^3] and $n^{\widetilde{(\imath)}}\circ \tilde c{(\tilde{ \bm x})}$\] as a function of $p^{(a)}({\bm x}) \equiv {\bm e}^{{(a)}} ({\bm x})[{\bm p}]$ \[or $q({\bm x})$ and $n^{(i)}({\bm x})$\]. Eventually, we will prefer to use $q$ and $n^{(i)}$ rather than $p^{(i)}$. From the definition of $q$, Eq (\[def:q\]) we then obtain the desired transformation relation $$\begin{aligned} \tilde{q}\circ \tilde c{(\tilde{\bm x})} &\equiv q ({\bm x})+\delta q ({\bm x}) \equiv q ({\bm x})[1+\delta \ln q ({\bm x})] \,, \end{aligned}$$ as $$\begin{aligned} \label{EqTp} \tilde{q}\circ \tilde c{(\tilde{\bm x})} &\equiv a^2(\tilde \eta) \tilde{N} p^{\tilde{0}} \circ \tilde c{(\tilde{\bm x})} \notag\\ &= q ({\bm x}) \left[ 1 + {{\cal H}}T + T_{,i}n^{(i)} + \frac{1}{2} ({{\cal H}}' + {{\cal H}}^2) T^2 + {{\cal H}}T T_{,i}n^{(i)} - T' T_{,i}n^{(i)} + \frac{1}{2} T^{,i} T_{,i} + T_{,i} \left( \alpha n^{(i)} - h^i{}_j n^{(j)} \right)\right] \,,\end{aligned}$$ As for $n^{(i)}$ from Eq (\[def:n\^i\]), its transformation $$\begin{aligned} n^{\widetilde{(i)}}\circ \tilde c{(\tilde{\bm x})} &\equiv n^{(i)}({\bm x}) + \delta n^{(i)}({\bm x}) \,,\end{aligned}$$ is given by $$\begin{aligned} n^{\widetilde{(i)}}\circ \tilde c{(\tilde{\bm x})} &\equiv \left[ \tilde{\beta}^i + \left( \frac{1}{\tilde{N}} \delta^i{}_j + \tilde{h}^i{}_j \right) \frac{p^{\tilde{j}}}{p^{\tilde{0}}} \right] \circ \tilde c{(\tilde{\bm x})} \notag\\ &= n^{(i)}({\bm x}) + S^{(i)(j)} T_{,j} + L^{[i,j]} n_{(j)} \,. \end{aligned}$$ Here when the argument is not specified, it is $({\bm x})$. Note that for future use, we have defined in these expressions the differences $\delta q$, $\delta \ln q$, and $\delta n^{(i)}$. The first order and second order perturbation of these can be read directly from the expressions above. We also define $\delta q^{(i)}\equiv q\Bigl[(\delta \ln q) n^{(i)} + \delta n^{(i)}\Bigr]$. It is worth stressing that these differences are measured at the same point. For instance $\delta n^{(i)}({\bm x})= n^{\widetilde{(i)}} \circ \tilde c{(\tilde{\bm x})} - n^{(i)}({\bm x}) $, and they do not vanish because in one case we use the tetrads ${\bm e}^{(i)}$ associated with the coordinate system $c$ to obtain the components, and in another case we use the tetrads $\tilde{\bm e}^{(i)}$ associated with the coordinate system $\tilde c$. It is the basis at a given point of space-time that changes when we change the coordinate system, not the momentum itself. This point is illustrated in the bottom right part of Fig. (\[fig1\]) Scalar distribution function ---------------------------- If we were using the natural basis associated with a coordinate system (the canonical basis) for the tangent space, then any scalar function on the tangent bundle $T{\cal M}$, that is a function of the space-time position and of the tangent space at each point, would transform as $$I \circ \tilde c{(\tilde {\bm x},p^{\widetilde{\mu}})} = I({\bm x},p^\mu) \,,$$ where $\tilde {\bm x}$ and ${\bm x}$ are related by  and $p^{\widetilde{\mu}}$ and $p^{\mu}$ are related by . Again this rule is a statement that the function is invariant under a change of coordinates because it is defined purely geometrically. However, as mentioned earlier, we use the basis of the tetrad field, ${\bm e}_{(i)}$ and ${\bm e}^{(i)}$, to obtain the components of momentum not the canonical basis. The tetrads are also completely determined by the choice of coordinates due to our prescription . The tetrad field, though being of tensorial nature, is not invariant as in Eq. . Furthermore, as mentioned earlier, we also work with the conformal momentum ${\bm q}$ rather than the momentum itself ${\bm p}$. Given this choice for the basis of the tangent space, the scalar function transforms as $$\label{Trulef} I\circ \tilde c{(\tilde {\bm x},q^{\widetilde{(\imath)}})} = I ({\bm x},q^{(\imath)})\,,$$ that is, it is unchanged when it is evaluated at the same point of the tangent bundle. On the other hand, the gauge transformation is a transformation rule at the same coordinate point and it is the relation between $I\circ \tilde c{({\bm x},q^{{(i)}})}$ and $I{( {\bm x},q^{{(i)}})}$. We then need the expressions of $q^{\widetilde{(\imath)}}\circ \tilde c{(\tilde{\bm x})}$ in terms of $q^{{(\imath)}}({\bm x})$ in spherical coordinates, which are derived in the previous section in . At first order, using the fact that the background distribution function cannot depend on the direction $n^{(i)}$, we obtain $$I \circ \tilde c{({\bm x},q,n^{(i)})} + \left(\xi^\mu{{\partial}}_\mu +\delta q \frac{{{\partial}}}{{{\partial}}q}\right) I \circ \tilde c{({\bm x},q,n^{(i)})}= I ({\bm x},q,n^{(i)})\,.$$ Given that the background distribution function also depends neither on time nor on space but only on $q$, we obtain $$I^{(1)}\circ \tilde c{({\bm x},q,n^{(i)})} = I^{(1)} ({\bm x},q,n^{(i)})-\delta \ln q \,{{\frac{\partial \bar I{(q)}}{\partial \ln q}}}\,.$$ At second order, we obtain $$I \circ \tilde c{({\bm x},q,n^{(i)})} + \left(\xi^\mu{{\partial}}_\mu +\delta q \frac{{{\partial}}}{{{\partial}}q} +\delta n^{(i)} D_{(i)} +\frac{1}{2}\delta q^2\frac{{{\partial}}^2 }{{{\partial}}q^2}\right) I \circ \tilde c{({\bm x},q,n^{(i)})}= I ({\bm x},q,n^{(i)}) \,.$$ Using the first order expressions, the gauge transformation rule reads $$\begin{aligned} \label{GaugeTRuleI2} \frac{1}{2}I^{(2)}\circ \tilde c{({\bm x},q,n^{(i)})} &=& \frac{1}{2} I^{(2)} ({\bm x},q,n^{(i)}) - \left(\xi^\mu{{\partial}}_\mu +\delta \ln q {{\frac{\partial}{\partial \ln q}}}+\delta n^{(i)} D_{(i)}\right) I^{(1)}{({\bm x},q,n^{(i)})}\nonumber\\* &&+ \left(\xi^\mu{{\partial}}_\mu +\delta n^{(i)}\frac{{{\partial}}}{{{\partial}}n^{(i)}}\right) (\delta \ln q) {{\frac{\partial}{\partial \ln q}}}\bar I{(q)} +\frac{1}{2}(\delta \ln q)^2 \left({{\cal D}}_q^2-2 {{\frac{\partial}{\partial \ln q}}}\right)\bar I(q)\,,\end{aligned}$$ where we used ${{\partial}}\delta \ln q / {{\partial}}\ln q = 0$. This transformation rule can be applied to $I$ or $V$ since these are scalar valued distribution functions. However we are interested in the transformation rule for the spectral components of $I$. Using the decomposition Eq (\[intensity\]) for $I$ we obtain that the temperature is transforming under a gauge transformation as (noting for simplicity $\tilde \Theta \equiv \Theta \circ \tilde c$) $$\begin{aligned} \tilde{\Theta}^{(1)} &= \Theta^{(1)} + (\delta \ln q)^{(1)} \,,\\ \frac{1}{2} \tilde{\Theta}^{(2)} &= \frac{1}{2} \Theta^{(2)} + \frac{1}{2} (\delta \ln q)^{(2)} +\Theta^{(1)} (\delta \ln q)^{(1)} - \left(\xi^\mu \frac{{{\partial}}}{{{\partial}}x^\mu}+\delta n^{(i)} D_{(i)} \right) \Bigl[ \Theta^{(1)}+(\delta \ln q)^{(1)} \Bigr]\,,\end{aligned}$$ where it is implied that all quantities are evaluated either at ${\bm x}$ or at $({\bm x},q,n^{(i)})$. The detailed form of the transformation rule can then be obtained just by considering the perturbations of $q$ and $n^i$, $\delta \ln q$ and $\delta n^{(i)}$, which have been obtained in Eqs. (\[EqTp\]). For completeness we report it here $$\begin{aligned} \tilde{\Theta} &= \Theta + {{\cal H}}T + T_{,i}n^{(i)} + \frac{1}{2} ({{\cal H}}^2 - {{\cal H}}') T^2 + {{\cal H}}T (T_{, i} n^{(i)} - T') - (T' T_{,i} + T T_{, i}') n^{(i)} \notag\\ &\qquad + T_{,i} \Bigl( \alpha n^{(i)} - h^i{}_j n^{(j)} \Bigr) - L^i ({{\cal H}}T_{, i} + T_{, i j} n^{(j)}) + \left( n^{(i)}n^{(j)} - \frac{1}{2} \delta^{i j} \right) T_{,j} T_{, i} - L^{[i}{}_{,j]} T_{, i} n^{(j)} \notag\\ &\qquad + ({{\cal H}}T + T_{, i} n^{(i)}) \Theta - \xi^\mu \Theta_{, \mu} - \Bigl( S^{(i) (j)} T_{,j} + L^{[i}{}_{,j]} n^{(j)} \Bigr) D_{(i)} \Theta \,.\end{aligned}$$ Finally, the gauge transformation of $y$ is trivial. Since $y$ vanishes at first order, $y$ is gauge invariant at second order, and it can also be checked directly by extracting $y$ out of the transformation rule of $I$ at second order . Baryons fluid description ------------------------- In order to obtain the complete gauge transformation of the collision term, we need the gauge transformation rule of the baryons fluid velocity in the tetrad frame up to second order, since it appears in the collision term. We obtain $$\begin{aligned} v^{\widetilde{(\imath)}}\circ \tilde c{({\bm x})} &= v^{(i)}({\bm x}) + T^{, i} + \alpha T^{, i} - h^i{}_j T^{, j} - T' T^{, i} + L^{[i}{}_{, j]} (v^{(j)} + T^{, j}) - T (v^{(i)}{}' + T^{, i}{}') - L^j (v^{(i)} + T^{, i})_{, j} \,.\end{aligned}$$ We also need the gauge transformation rule up to first order of the electrons density and it is easily obtained to be $$\delta_e\circ \tilde c{({\bm x})} = \delta_e({\bm x}) + 3 {\cal H} T n_e \,.$$ Gauge dependence of the Boltzmann equation and Gauge invariant form {#ssec:lc} ------------------------------------------------------------------- Having derived all the necessary gauge transformation rules, it is now possible to check the gauge dependence of the derived second order Boltzmann equation and collision term explicitly. More precisely we shall check that the Liouville and the collision terms of the Boltzmann equation transform as they should do. The Liouville term and the Collision term are distribution functions (scalar or tensor valued depending whether or not we are considering the intensity or polarisation). From the transformation rule of a scalar distribution function  and the spectral decomposition (\[spectraldecI\]) we deduce that ${\cal L}^Y$ and ${{\cal C}}^Y$ are gauge invariant and ${{\cal L}}^\Theta$ and ${{\cal C}}^\Theta$ should transforms up to the second order as (noting $\widetilde{{{\cal L}}^\Theta} \equiv {{\cal L}}^\Theta \circ \tilde c$ and $\widetilde{{{\cal C}}^\Theta} \equiv {{\cal C}}^\Theta \circ \tilde c$ ) $$\begin{aligned} \widetilde{{{\cal L}}}^\Theta &= {{\cal L}}^\Theta - \Bigl( \xi^\mu {{\partial}}_\mu + \delta n^{(i)} D_{(i)} \Bigr) {{\cal L}}^{\Theta\,(1)} + 2 {{\cal H}}T {{\cal L}}^{\Theta\,(1)} \,, \\ \widetilde{{{\cal C}}^\Theta} &= {{\cal C}}^\Theta - \Bigl( \xi^\mu {{\partial}}_\mu + \delta n^{(i)} D_{(i)} \Bigr) {{\cal C}}^{\Theta (1)} + 2 {{\cal H}}T {{\cal C}}^{\Theta (1)} \,. \end{aligned}$$ After very long and tedious but straightforward calculations using all the transformation rules derived so far for the distribution function, the metric components and the baryons velocity and energy density, we have checked that the Liouville operator ${{\cal L}}^\Theta$ and the collision term ${{\cal C}}^\Theta$ actually transform as in the above equations. This completes the consistency test of the Boltzmann equation that we have derived as well as all the gauge transformation rules obtained for its constituents. The same property is found of course for the circular polarisation since in that case the collision term vanishes. As for polarisation, we have also checked that the Liouville operator ${{\cal L}}^P_{(i)(j)}$ and the collision term ${{\cal C}}^P_{(i)(j)}$ transform like tensor valued quantities (see the details in Appendix \[AppGTLC\]). More importantly, we have checked that the Liouville and collision terms for the spectral distortions (${{\cal L}}^Y_{(i)(j)}$ and ${{\cal C}}^Y_{(i)(j)}$) are gauge invariant as it should be since they vanish on the background and first-order spacetimes. The gauge invariance of the Boltzmann equation, as in the case of Einstein equation and in general for covariant equations, enables us to write it down in terms of gauge invariant variables. In practice, it is equivalent to completely fix the gauge and write down the equations in term of the perturbation in this gauge. Summary and discussion {#sec:conc} ====================== In this paper we derived the second order Boltzmann equation in the most general manner incorporating polarisation and without fixing a gauge. In order to describe the polarisation of photon, we used a formalism based on a tensor-valued distribution function. We performed the separation between temperature and spectral distortion for the intensity and we also extended this separation to polarisation. We then derived the gauge transformation rules for the metric, the momentum and the distribution function to see how those quantities are mixed under the gauge transformation. As an application, we checked the gauge dependence of the derived Boltzmann equation under a gauge transformation and obtained consistent transformation rules. This is a non-trivial check of the correctness of the derived equations as well as the gauge transformation rules. We now discuss two issues related to the gauge dependence in the Boltzmann equation. Gauge dependence of lensing term -------------------------------- It is well known that the lensing term $$\begin{aligned} {{\cal L}}\supset \frac{{{\rm d}}n^{(i)}}{{{\rm d}}\lambda} D_i \Theta ~ : ~ ({{\mathrm{lensing ~ term}}}) \,,\end{aligned}$$ which is written in terms of the conventional lensing potential in the Newtonian gauge significantly affects the bispectrum of CMB [@Hanson:2009kg; @Lewis:2011fk; @Ade:2013ydc]. Indeed, the correlation between the lensing and ISW effect is the dominant contribution to the bias for the local-type non-Gaussianity in Planck. However it is very hard to include this contribution in the line of sight integration and evaluate it until today. Usually, the lensing effect is added separately to the final result obtained in the Poisson gauge. However the effect from this lensing term depends on the gauge choice. Actually, as we have seen above, the lensing term is mixed with other terms under the gauge transformation. This means that some of lensing effects in a specific gauge are absorbed into other effects in another gauge. In principle, there exists a gauge where we can avoid the difficult computation of this lensing term to some extent by evaluating other more tractable terms. Since we have derived the Boltzmann equation without choosing any specific gauge, it should be possible to investigate this possibility further. Observed temperature anisotropies --------------------------------- Here we make a comment on the observed temperature anisotropies. In the main part of this paper, we have shown that the second order Boltzmann equation is gauge invariant and thus it can be written in terms of gauge invariant quantities. However, there is a subtlety in the meaning of “gauge invariance”. This originates from our choice of the local inertial frame. As is clear from Eq (\[tetrads\]), we always choose the local inertial frame so that the three-velocity vanishes, $\hat{v}^i=0$ and there is no rotation of the spatial axis relative to the background spatial coordinate axis, let us call $\theta_i=0$. In order to achieve this, the local inertial frame has to be changed when we perform a gauge transformation. If we were to identify this local inertial frame as the one of an observer, we would be lead to consider different observers in different gauges. This is clear from the gauge transformation at the first order: $$\Theta \to \Theta + {\cal H} T + T_{,i} n^i\,. \label{1sttempg}$$ The last term comes from a change of local inertial frame. By fixing the gauge we can promote $\Theta$ to the gauge invariant temperature fluctuations but these are temperature fluctuations observed by an observer with $\hat{v}^i=0$ and $\theta_i=0$ in this gauge. In order to evaluate temperature fluctuations observed by a different observer, we need to change the local inertial frame. Alternatively, we can perform a gauge transformation keeping the conditions $\hat{v}^i=0$ and $\theta_i=0$ for the local inertial frame so that this frame coincides with the one of the observer. In all the literature, the second order temperature anisotropies are calculated in the Poisson gauge so far, with a specific choice of the local inertial frame. Strictly speaking this is not the temperature anisotropies that we observe as there is no reason for us to be comoving with the local inertial frame associated with such a gauge. One thus needs to change the local inertial frame or change a gauge. At first order, this was not an issue. As is clear from (\[1sttempg\]), the change of gauge and the local inertial frame only affect the monopole $\ell =0$ and dipole $\ell =1$ if we expand the temperature anisotropies into multipole components. Thus, the $\ell \geq 2$ modes are not affected by the change of observers. However this is no longer the case at the second order. In the second order gauge transformation, there are terms that are convolutions of the first order temperature anisotropies and the gauge transformation; $$\begin{aligned} \Theta \to \Theta - \xi^\mu \Theta_{, \mu} - \delta n^{(i)} D_{(i)} \Theta + \cdots \,. \end{aligned}$$ These terms affect the observed temperatures even for the $\ell \geq 2$ modes. In order to define the "observed temperature anisotropies”, we should keep the conditions $\hat{v}^i=0$ and $\theta_i=0$ for the local inertial frame and specify the gauge so that this local inertial frame coincides with our local inertial frame where we perform experiments. Thus special care must be taken when we compare theoretical predictions to observations. Our formula for the gauge transformation will be useful to investigate this issue further. A.N is grateful to Shuichiro Yokoyama and Ryo Saito for their continuous encouragement and fruitful discussion. This work was supported in part by Monbukagaku-sho Grant-in-Aid for the Global COE programs, "The Next Generation of Physics, Spun from Universality and Emergence” at Kyoto University, by JSPS Grant-in-Aid for Scientific Research (A) No. 21244033, and by the Long-term Workshop at Yukawa Institute on Gravity and Cosmology 2012, YITP-T-12-03. AN is partly supported by Grant-in-Aid for JSPS Fellows No. 21-1899 and JSPS Postdoctoral Fellowships for Research Abroad. CP was supported by the STFC (UK) grant ST/H002774/1 during the first part of this research, and was then supported by French state funds managed by the ANR within the Investissements d’Avenir programme under reference ANR-11-IDEX-0004-02. K.K. is supported by STFC grant ST/H002774/1, ST/K0090X/1, the European Research Council and the Leverhulme trust. ADM variables and usual perturbation variables {#app:ADM} ============================================== From the form of the metric in the ADM parametrisation (\[MetricADM\]), and the perturbation of the lapse function and the spatial metric given in the equations (\[DefPertNNi\]), the perturbed metric is expressed up to second order as $${{\rm d}}s^2 = a^2 (\eta) \Bigl[ - (1 + 2 \alpha + \alpha^2 - \beta_i \beta^i) {{\rm d}}\eta^2 + 2 (\delta_{i j} + 2 h_{i j}) \beta^j {{\rm d}}x^i {{\rm d}}\eta + (\delta_{i j} + 2 h_{i j}) {{\rm d}}x^i {{\rm d}}x^j \Bigr] \,.$$ This has to be compared with the usual parametrisation of the perturbations of the metric which is in the form $$\begin{gathered} {{\rm d}}s^2 = a^2 (\eta) \Bigl[ - (1 + 2 A) {{\rm d}}\eta^2 + 2 B_i {{\rm d}}x^i {{\rm d}}\eta + (\delta_{i j} + 2 C_{i j}) {{\rm d}}x^i {{\rm d}}x^j \Bigr] \,, \end{gathered}$$ where $C_{i j}$ can be further split into 2 scalar, 2 vector and 2 tensor degrees of freedom. By a direct comparison of these two parametrisations, the relation between the two metric parametrisations is $$\begin{aligned} g_{0 0} &: 1 + 2 \alpha + \alpha^2 - \beta_i \beta^i = 1 + 2 A\,,\\ g_{0 i} &: \beta_i + 2 h_{i j} \beta^j = B_i\,, \\ g_{i j} &: \delta_{i j} + 2 h_{i j} = \delta_{i j} + 2 C_{i j}\,.\end{aligned}$$ At first order we obtain $$\begin{gathered} A^{(1)} = \alpha^{(1)} \,, \qquad B_i^{(1)} = \beta^{(1)}_i \,, \qquad C_{i j}^{(1)} = h_{i j}^{(1)} \,,\end{gathered}$$ and the two parametrisations are the same. However, at second order we get the relations $$\begin{gathered} A^{(2)}= \alpha^{(2)} + \alpha^{(1) 2} - \beta^{(1)}_i \beta^{(1) i} \,, \qquad B_i^{(2)} = \beta^{(2)}_i + 4 h_{i j}^{(1)} \beta^{(1) j} \,,\qquad C_{i j}^{(2)} = h_{i j}^{(2)} \,.\end{gathered}$$ Construction of the distribution function for polarised light {#app:Dist} ============================================================= We consider a two-dimensional polarisation plane defined by two unit complex vectors $\hat {{\boldsymbol{\epsilon}}}_{({{\mathrm{I}}})}$ and $\hat {{\boldsymbol{\epsilon}}}_{({{\mathrm{II}}})}$, which are mutually orthogonal, $\hat \epsilon_{({{\mathrm{A}}})}{}^{\star\mu} \hat \epsilon_{({{\mathrm{B}}})}{}^\nu g_{\mu\nu} =\delta_{({{\mathrm{A}}})({{\mathrm{B}}})}$. Any polarisation $\boldsymbol{\epsilon}$ can be represented by a superposition of $\hat {{\boldsymbol{\epsilon}}}_{(A)}$ in the form $${{\boldsymbol{\epsilon}}}= \sum_{A={{\mathrm{I}}},{{\mathrm{II}}}} \epsilon^A \hat {{\boldsymbol{\epsilon}}}_{(A)} \label{pol}\,.$$ The orthogonal vectors $\hat {{\boldsymbol{\epsilon}}}_{(A)}$ define a polarisation plane and we choose them to be orthogonal to the direction of the photon $n^{(i)}$ and to the observer velocity ${\bm e}_{(0)}$, $$\label{EqProjepsilons} \hat \epsilon_{(A)}{}^\mu n^{\nu} g_{\mu \nu} = \hat \epsilon_{(A)}{}^\mu {e}_{(0)}{}^\nu g_{\mu\nu}=0\,.$$ We can also associate canonically polarisation forms through $\hat \epsilon^{(A)}{}_\mu \equiv g_{\mu\nu} \hat \epsilon_{(A)}{}^\nu$, and they will be also complex unit forms and mutually orthogonal. The polarisation density matrix $f_{A B}$ is defined so that the expected number of photon in a phase-space element with a polarisation state ${{\boldsymbol{\epsilon}}}$ is given by $$\begin{gathered} f( {\bf x}, {\bf p}, {{\boldsymbol{\epsilon}}}) \equiv f_{A B}({\bf x}, {\bf p}) \epsilon^{\star A} \epsilon^B \,. \end{gathered}$$ With such a parametrisation, all the electromagnetic gauge degrees of freedom have been fixed and we parametrise the physical degrees of freedom of this density matrix by the usual Stokes parameters as $$\begin{gathered} f_{A B} = \frac{1}{2} \begin{pmatrix} I + Q & U - i V \\ U + i V & I - Q \end{pmatrix} \,,\end{gathered}$$ where it is implied that $f_{AB}$ and the Stokes parameters depend on the position $x^\mu$ and on the momentum $p^\mu$ \[or $(q,n^{(i)})$ in spherical coordinates\]. $f_{AB}$ is a Hermitian matrix since $f_{AB}=f_{BA}^\star$. From the four-dimensional point of view, the polarisation density matrix is a tensor-valued distribution function. It is a 2-form defined by $$\begin{gathered} f_{\mu \nu} \equiv f_{A B} \epsilon^{\star(A)}{}_\mu \epsilon^{(B)}{}_\nu \,,\end{gathered}$$ and the expected number of photon in a phase-space element for a polarisation state ${{\boldsymbol{\epsilon}}}$ is given by $$\label{Eqdefffromfmunu} f ({\bf x}, {\bf p}, {{\boldsymbol{\epsilon}}}) \equiv f_{\mu\nu} ({\bf x}, {\bf p}) \epsilon^{\star\mu} \epsilon^\nu \,.$$ This can be viewed as a multipolar expansion in the polarisation state. From (\[EqProjepsilons\]), the tensor-valued distribution function is a projected quantity such that $$f_{\mu \nu} =S_\mu{}^\alpha S_{\mu}{}^\beta f_{\alpha \beta}\,.$$ It is then straightforward to realize that it can be decomposed according to (\[Pol\]). Boltzmann equation for the tensor-valued distribution function {#app:Boltz_pol} ============================================================== From a scalar valued to a tensor-valued distribution function ------------------------------------------------------------- In this section, we explain in detail how the Boltzmann equation for the tensor-valued distribution function can be obtained from the Boltzmann equation of a scalar distribution function. Since this scalar distribution function $f$ depends on $x^\mu,p^\mu$ but also on $\epsilon^\mu$, the action of the Liouville operator is given by $$\label{CrudeBoltzmann} \frac{{{\cal D}}}{{{\cal D}}\lambda} f (x^\mu, p^\mu, \epsilon^\mu) = \frac{{{\rm d}}x^\alpha}{{{\rm d}}\lambda} \frac{{{\partial}}f}{{{\partial}}x^\alpha} + \frac{{{\rm d}}p^\alpha}{{{\rm d}}\lambda} \frac{{{\partial}}f}{{{\partial}}p^\alpha} + \frac{{{\rm d}}\epsilon^\alpha}{{{\rm d}}\lambda} \frac{{{\partial}}f}{{{\partial}}\epsilon^\alpha} = C [f]\,.$$ In the geometric optics approximation, $p^\mu$ and $\epsilon^\mu$ are parallel transported and we obtain $$\begin{aligned} 0 =& \frac{{{\cal D}}p^\alpha}{{{\cal D}}\lambda} = \frac{{{\rm d}}p^\alpha}{{{\rm d}}\lambda} + \Gamma^\alpha_{\beta \gamma} p^\beta p^\gamma \,, \\ 0 =& \frac{{{\cal D}}\epsilon^\alpha}{{{\cal D}}\lambda} = \frac{{{\rm d}}\epsilon^\alpha}{{{\rm d}}\lambda} + \Gamma^\alpha_{\beta \gamma} \epsilon^\beta p^\gamma \,.\end{aligned}$$ Using (\[Eqdefffromfmunu\]), the term involving the evolution of polarisation is obtained as $$\begin{gathered} \frac{{{\rm d}}\epsilon^\alpha}{{{\rm d}}\lambda} \frac{{{\partial}}f}{{{\partial}}\epsilon^\alpha} = f_{\alpha \beta} \frac{{{\rm d}}\epsilon^{\alpha}}{{{\rm d}}\lambda} \epsilon^{* \beta} + f_{\alpha \beta} \frac{{{\rm d}}\epsilon^{\star \beta}}{{{\rm d}}\lambda} \epsilon^{\alpha} = - \Gamma^{\alpha}_{\gamma \delta} \epsilon^\gamma p^\delta f_{\alpha \beta} \epsilon^{* \beta} - \Gamma^{\beta}_{\gamma \delta} \epsilon^{\star\gamma} p^\delta f_{\alpha \beta} \epsilon^{\alpha}\,.\end{gathered}$$ Combining this result with the space-time derivative term of the Liouville operator, we get $$\begin{gathered} p^\alpha \frac{{{\partial}}f}{{{\partial}}x^\alpha} + \frac{{{\rm d}}\epsilon^\alpha}{{{\rm d}}\lambda} \frac{{{\partial}}f}{{{\partial}}\epsilon^\alpha} = p^\gamma \epsilon^\alpha (\nabla_\gamma f_{\alpha \beta}) \epsilon^{* \beta} \,,\end{gathered}$$ and thus the Boltzmann equation (\[CrudeBoltzmann\]) can be rewritten as $$\label{EqBoltzmannfee} \frac{{{\cal D}}f}{{{\cal D}}\lambda} = \epsilon^\mu \left( p^\alpha \nabla_\alpha f_{\mu \nu} + \frac{{{\rm d}}p^\alpha}{{{\rm d}}\lambda} \frac{{{\partial}}f_{\mu \nu}}{{{\partial}}p^\alpha} \right) \epsilon^{* \nu} = C [f] \equiv \epsilon^\mu C_{\mu \nu} \epsilon^{* \nu} \,.$$ The last equality defines the tensor-valued collision term. If we do fix the electromagnetic gauge condition for the collision term in the same manner as what we did for $f_{\mu \nu}$, that is, if $C_{\mu \nu} =S_\mu{}^\alpha S_{\nu}{}^\beta C_{\alpha \beta}$, then the Boltzmann equation for $f_{\mu\nu}$ is given by $$\label{EqBaseBoltzmann} S_\mu{}^\alpha S_{\mu}{}^\beta \frac{{{\cal D}}f_{\alpha \beta}}{{{\cal D}}\lambda} \equiv S_\mu{}^\alpha S_{\mu}{}^\beta \left(p^\sigma \nabla_\sigma f_{\alpha \beta} + \frac{{{\rm d}}p^\sigma}{{{\rm d}}\lambda} \frac{{{\partial}}f_{\alpha \beta}}{{{\partial}}p^\sigma}\right) = C_{\mu\nu}\,.$$ Note that the use of the projectors is required because the components of the equation which are not in the polarisation plane are not fixed by (\[EqBoltzmannfee\]). From the canonical basis to the tetrad basis {#sapp:covp} -------------------------------------------- In the equation (\[EqBaseBoltzmann\]), the Greek indices refer to a given coordinate system and its canonical basis for the tangent space, and the distribution function is a function of $(x^\mu,p^\mu)$. If we want to use instead an orthonormal basis for the tangent space, that is, to use the components $p^{(i)}=e^{(i)}{}_\mu p^\mu$ or the conformal momentum components $q^{(i)} = a p^{(i)}$ in the tetrad basis, then the Boltzmann equation can be modified accordingly. In order to do so, we need to be explicit about the partial derivatives to emphasize which variables are to be kept constant when the partial derivatives are evaluated. The Boltzmann equation reads indeed $$\label{Boltzbaseopen} S_\mu{}^\alpha S_{\nu}{}^\beta \frac{{{\cal D}}f_{\alpha \beta}}{{{\cal D}}\lambda} = S_\mu{}^\alpha S_{\nu}{}^\beta \left( p^\gamma \left.\frac{{{\partial}}f_{\alpha \beta}}{{{\partial}}x^\gamma}\right|_{p^\mu} - p^\gamma \Gamma^\delta_{\gamma \alpha} f_{\delta \beta} - p^\gamma \Gamma^\delta_{\gamma \beta} f_{\alpha \delta} + \left.\frac{{{\rm d}}p^\gamma}{{{\rm d}}\lambda} \frac{{{\partial}}f_{\alpha \beta}}{{{\partial}}p^\gamma}\right|_{x^\mu}\right) \,.$$ Using the properties $$\left.\frac{{{\partial}}f_{\alpha \beta}}{{{\partial}}x^\mu} \right|_{p^\mu} = \left.\frac{{{\partial}}f_{\alpha \beta}}{{{\partial}}x^\mu}\right|_{q^{(i)}} + \frac{{{\partial}}f_{\alpha \beta}}{{{\partial}}q^{(i)}} \frac{{{\partial}}(a e^{(i)}{}_\nu)}{{{\partial}}x^\mu}p^\nu \,, \qquad \left.\frac{{{\partial}}f_{\alpha \beta}}{{{\partial}}p^\mu}\right|_{x^\mu} = \left.\frac{{{\partial}}f_{\alpha \beta}}{{{\partial}}q^{(i)}}\right|_{x^\mu} ae^{(i)}{}_\mu \,, \qquad \frac{{{\rm d}}q^{(i)}}{{{\rm d}}\lambda} = a e^{(i)}{}_\mu \frac{{{\rm d}}p^\mu}{{{\rm d}}\lambda} + p^\mu \frac{{{\rm d}}(a e^{(i)}{}_\mu)}{{{\rm d}}\lambda} \,,$$ we then deduce that $$\begin{aligned} \label{Boltzbasetetrad} S_\mu{}^\alpha S_{\nu}{}^\beta \frac{{{\cal D}}f_{\alpha \beta}}{{{\cal D}}\lambda} &=S_\mu{}^\alpha S_\nu{}^\beta \left(p^\gamma \left.\frac{{{\partial}}f_{\alpha \beta}}{{{\partial}}x^\gamma} \right|_{q^{(i)}} - p^\gamma \Gamma^\delta_{\gamma \alpha} f_{\delta \beta} - p^\gamma \Gamma^\delta_{\gamma \beta} f_{\alpha\delta} + \left.\frac{{{\rm d}}q^{(i)}}{{{\rm d}}\lambda} \frac{{{\partial}}f_{\alpha \beta}}{{{\partial}}q^{(i)}} \right|_{x^\mu}\right) \notag\\ &= S_\mu{}^\alpha S_{\mu}{}^\beta \left(p^\gamma \nabla_\gamma f_{\alpha \beta} + \frac{{{\rm d}}q^{(i)}}{{{\rm d}}\lambda} \frac{{{\partial}}f_{\alpha \beta}}{{{\partial}}q^{(i)}}\right)\,.\end{aligned}$$ Comparing Eqs. (\[Boltzbaseopen\]) and (\[Boltzbasetetrad\]), we notice that the notation $p^\gamma \nabla_\gamma f_{\alpha \beta}$ could be ambiguous. Indeed if we use the canonical coordinate system for the tangent space, then ${{\partial}}/{{\partial}}x^\mu$ is to be taken at $p^\mu$ fixed, but if we take the tetrad basis (or another coordinate system for the tangent space), then ${{\partial}}/{{\partial}}x^\mu$ is to be taken with $q^{(i)}$ fixed. Eq. (\[Boltzbasetetrad\]) is not exactly the desired form of the Boltzmann equation when the distribution function depends on $(x^\mu,q^{(i)})$. In fact, the use of the tetrad basis makes it natural to work with spherical coordinates in the tangent space. In order to introduce them, we first relate the Cartesian derivative in the tangent space, that is, the derivative with respect to $q^{(i)}$, to the covariant derivative on the unit sphere which is described by the possible directions $n^{(i)}$ of the momentum. We must stress that at any point of space-time, $x^\mu$, a distribution function (tensor-valued like $f_{\mu\nu}$ or scalar valued like its trace $I$) which depends on $(x^\mu,q^{(i)})$ can be considered as a field in the tangent space because the tangent space at a given point can be considered as a flat three-dimensional manifold whose points are labelled by $q^{(i)}$ and the natural covariant derivative in this manifold is ${{\partial}}/{{\partial}}q^{(i)}$. Using that $q^{(i)} = q n^{(i)}$, and the property $$\label{ExtrinsicK} \frac{{{\partial}}n^{(i)}}{{{\partial}}q^{(j)}} = \frac{1}{q} S^{(i)}{}_{(j)} \,,$$ it is possible to show the following relations; $$\begin{aligned} \label{Expandderpi} q \frac{{{\partial}}T_{\mu\nu}}{{{\partial}}q^{(i)}} &= \frac{{{\partial}}T_{\mu\nu}}{{{\partial}}\ln q} n_{(i)} + D_{(i)} T_{\mu\nu} - e_{(i)}{}^\rho (T_{\mu \rho} n_{\nu} + T_{\nu \rho}n_{\mu}) \,, &\quad\text{with}\quad & D_{(i)} T_{\mu\nu} \equiv q S_{(i)}{}^{(j)} S_\mu{}^\alpha S_\nu{}^\beta \frac{{{\partial}}T_{\alpha\beta}}{{{\partial}}q^{(j)}} \,, \\* q \frac{{{\partial}}S}{{{\partial}}q^{(i)}} &= \frac{{{\partial}}S}{{{\partial}}\ln q} n^{(i)} + D_{(i)} S \,, &\quad\text{with}\quad & D_{(i)} S \equiv q S_{(i)}{}^{(j)} \frac{{{\partial}}S}{{{\partial}}q^{(j)}} \,,\end{aligned}$$ where $T_{\mu\nu}(q^{(i)})$ is a projected tensor field in the tangent space such that $T_{\mu\nu}(q^{(i)}) n^{\nu}=T_{\mu\nu}(q^{(i)}) n^{\nu}=0$, and $S(q^{(i)})$ is a scalar field in the tangent space. Here $D_{(i)}$ is the covariant derivative on the two-sphere associated with the unit direction vector $n^{(i)}$, and it appears naturally as an induced derivative on the sphere, given that this is the surface orthogonal to $n^{(i)}$ (see Ref. [@Gourgoulhon:2007ue] for more details on induced derivatives). We can then deduce the useful property $$\label{KeyRuleForCovDSpherical} q S_\mu{}^\alpha S_\nu{}^\beta \frac{{{\partial}}f_{\alpha \beta}}{{{\partial}}q^{(i)}} = \frac{{{\partial}}f_{\mu\nu}}{{{\partial}}\ln q} n_{(i)} + D_{(i)} f_{\mu\nu} \,.$$ Given that $$\frac{{{\rm d}}q^{(i)}}{{{\rm d}}\lambda} = q \left(\frac{{{\rm d}}\ln q }{{{\rm d}}\lambda} n^{(i)} + \frac{{{\rm d}}n^{(i)}}{{{\rm d}}\lambda} \right)\,,$$ from Eqs. (\[Expandderpi\]) and Eq. (\[Boltzbasetetrad\]), we then find that the Boltzmann equation takes the form $$S_\mu{}^\alpha S_\nu{}^\beta \frac{{{\cal D}}f_{\alpha \beta}}{{{\cal D}}\lambda} = S_\mu{}^\alpha S_\nu{}^\beta \nabla_\gamma f_{\alpha \beta} \frac{{{\rm d}}x^\gamma}{{{\rm d}}\lambda} + \frac{{{\partial}}f_{\mu \nu}}{{{\partial}}\ln q} \frac{{{\rm d}}\ln q}{{{\rm d}}\lambda} + D_{(i)} f_{\mu \nu} \frac{{{\rm d}}n^{(i)}}{{{\rm d}}\lambda} = C_{\mu \nu} \,.$$ Decomposition of the Boltzmann equation --------------------------------------- In order to obtain equations for the components of $f_{\mu\nu}$, $I, P_{\mu \nu}$ and $V$, we want to apply the same type of decomposition on the equation itself. Applying ${{\cal D}}/{{\cal D}}\lambda$ on the decomposition  of $f_{\mu\nu}$ leads to $$\begin{gathered} \frac{{{\cal D}}f_{\mu \nu}}{{{\cal D}}\lambda} = \frac{1}{2} \left( \frac{{{\cal D}}I}{{{\cal D}}\lambda} S_{\mu \nu} + I \frac{{{\cal D}}S_{\mu \nu}}{{{\cal D}}\lambda} \right) + \frac{{{\cal D}}P_{\mu \nu}}{{{\cal D}}\lambda} + \frac{{{\rm i}}}{2} \epsilon_{\alpha \mu \nu \beta} \left( \frac{{{\cal D}}V}{{{\cal D}}\lambda} e_{(0)}{}^\alpha n^\beta + V \frac{{{\cal D}}(e_{(0)}{}^\alpha n^\beta)}{{{\cal D}}\lambda} \right) \,.\end{gathered}$$ We then need to screen-project this equation in order to obtain the tensor-valued Boltzmann equation. Then the last two terms vanish. Indeed, first $$S_\mu{}^\alpha S_\nu{}^\beta \frac{{{\cal D}}S_{\alpha \beta}}{{{\cal D}}\lambda} = S_\mu{}^\alpha S_\nu{}^\beta \left[ \frac{{{\rm d}}p}{{{\rm d}}\lambda} \left( - \frac{p_\alpha e^{(0)}{}_\beta + e^{(0)}{}_\alpha p_\beta}{p^2} + 2 \frac{ p_\alpha p_\beta}{p^3} \right) + \frac{1}{p} \left( p_\alpha \frac{{{\cal D}}e^{(0)}{}_\beta}{{{\cal D}}\lambda} + \frac{{{\cal D}}e^{(0)}{}_\alpha}{{{\cal D}}\lambda} p_\beta \right) \right] = 0\,.$$ Second, from the normalization condition of $e_{(0)}{}^\mu$ and $n^\mu$, we can show that $$e^{(0)}{}_\mu \frac{{{\cal D}}e_{(0)}{}^\mu}{{{\cal D}}\lambda} = 0 \,, \qquad n_\mu \frac{{{\cal D}}n^\mu}{{{\cal D}}\lambda} = 0 \,.$$ This means that the derivative of $e_{(0)}{}^\mu$ is orthogonal to $e^{(0)}{}_\mu$ and the derivative of $n^\mu$ is orthogonal to $n_\mu$. We thus find that $$S_\mu{}^\alpha S_\nu{}^\beta \epsilon_{\gamma \alpha \beta \delta} \frac{{{\cal D}}(e_{(0)}{}^\gamma n^\delta)}{{{\cal D}}\lambda} = S_\mu{}^\alpha S_\nu{}^\beta \epsilon_{\gamma \alpha \beta \delta} \left( \frac{{{\cal D}}e_{(0)}{}^\gamma}{{{\cal D}}\lambda} n^\delta + e_{(0)}{}^{\gamma} \frac{{{\cal D}}n^\delta}{{{\cal D}}\lambda} \right) = 0 \,.$$ Finally, we obtain that the Boltzmann equation for the tensor-valued distribution functions can be split into the desired form as $$\begin{gathered} S_\mu{}^\alpha S_\nu{}^\beta \frac{{{\cal D}}f_{\alpha \beta}}{{{\cal D}}\lambda} = \frac{1}{2} \frac{{{\cal D}}I}{{{\cal D}}\lambda} S_{\mu \nu} + S_\mu{}^\alpha S_\nu{}^\beta \frac{{{\cal D}}P_{\alpha \beta}}{{{\cal D}}\lambda} + \frac{{{\rm i}}}{2} \frac{{{\cal D}}V}{{{\cal D}}\lambda} \epsilon_{\alpha \mu \nu \beta} e_{(0)}{}^\alpha n^\beta = C_{\mu \nu} \,.\end{gathered}$$ Expression of the Boltzmann equation for polarisation {#sapp:Boltzeq_pol} ----------------------------------------------------- First of all, the Boltzmann equation for the circular polarisation is the same as that for the intensity, but with $\bar V=0$ and with a vanishing collision term since it is not generated by the Compton scattering. We shall not study further the equation dictating the evolution of $V$ since it should remain null at all time unless generated by other types of collisions. As for the linear polarisation, let us write down the basic equations for the tetrad components as commonly done at first order. By moving to the tetrad components, one can rewrite the covariant derivative in the Liouville operator as $$\begin{aligned} e_{(a)}{}^\mu e_{(b)}{}^\nu L [{\bf P}\,]_{\mu \nu} &= S_{(a)}{}^{(c)} S_{(b)}{}^{(d)} \left( P_{(c) (d) | (e)} e^{(e)}{}_\mu \frac{{{\rm d}}x^\mu}{{{\rm d}}\lambda} + \frac{{{\partial}}P_{(c) (d)}}{{{\partial}}\ln q} \frac{{{\rm d}}\ln q}{{{\rm d}}\lambda} + D_{(i)} P_{(c) (d)} \frac{{{\rm d}}n^{(i)}}{{{\rm d}}\lambda} \right) \,,\end{aligned}$$ where $$\begin{aligned} f_{(a) (b) | (c)} &\equiv e_{(c)}{}^\mu {{\partial}}_\mu f_{(a) (b)} - w^{(d)}{}_{(a) (c)} f_{(d) (b)} - w^{(d)}{}_{(b) (c)} f_{(a) (d)} \,,\end{aligned}$$ and $w^{(a)}{}_{(b) (c)}$ is the Ricci rotation coefficient defined by $w^{(a)}{}_{(b) (c)} \equiv e^{(a)}{}_\mu \nabla_{(c)} e_{(b)}{}^\mu$. Here we also notice that only the spatial component has non-vanishing term because the projection of $S_\mu{}^\nu$ onto $e^\mu{}_{(0)}$ vanishes by construction. After the decomposition of the Boltzmann equation for polarisation, one obtains the following equation for the temperature part as in Eq. (\[eq:Boltz\_pol\_dis\]) up to the second order $$\begin{aligned} {{\cal L}}^P_{(i) (j)} &= {{\cal P}}_{(i) (j)}{}' + {{\cal P}}_{(i) (j), k} n^{(k)} + \frac{a^2}{q}\left[\left. \frac{{{\rm d}}\eta}{{{\rm d}}\lambda} \right|^{(1)} {{\cal P}}_{(i) (j)}{}' + \left. \frac{{{\rm d}}x^k}{{{\rm d}}\lambda} \right|^{(1)} {{\cal P}}_{(i) (j), k} + \left. \frac{{{\rm d}}n^{(k)}}{{{\rm d}}\lambda} \right|^{(1)} D_{(k)} {{\cal P}}_{(i) (j)}\right] \notag\\ & \qquad - a\Bigl( w^{(k)}{}_{(l) (0)} + w^{(k)}{}_{(l) (m)} n^{(m)} \Bigr)^{(1)} \Bigl( S_{(i)}{}^{(l)} {{\cal P}}_{(k) (j)} + S_{(j)}{}^{(l)} {{\cal P}}_{(k) (i)} \Bigr)\,.\end{aligned}$$ As for the spectral distortion, the equation is given by $$\begin{aligned} {{\cal L}}^Y_{(i) (j)} &= Y_{(i) (j)}{}' + Y_{(i) (j), k} n^{(k)} \,.\end{aligned}$$ The complete expression of the collision term for the linear polarisation is given by $$\begin{aligned} {{\cal C}}_{(i) (j)}^P &= a \, \bar{n}_e \sigma_T \left( - {{\cal P}}_{(i) (j)} - \frac{3}{4} {{\cal T}}_{(i) (j)}{}^{(k) (l)} \Bigl[ \langle \Theta m_{(k) (l)} \rangle - 2 \langle {{\cal P}}_{(k) (l)} \rangle \Bigr] + v^{(k)} n_{(k)} {{\cal P}}_{(i) (j)} \right) \notag\\ &\qquad + a \, \bar{n}_e \sigma_T \biggl( {{\cal T}}_{(i) (j)}{}^{(k) (l)} \Bigl[ {{\cal Q}}_{(k) (l)}^T - \Theta {{\cal Q}}_{(k) (l)}^{T ~~~ (1)} \Bigr] - 3 {{\cal C}}^\Theta {{\cal P}}_{(i) (j)} + \delta_e {{\cal C}}^P_{(i) (j)} \biggr) \,, \\ C^Y_{(i) (j)} &= a \, \bar{n}_e \sigma_T \left( - Y_{(i) (j)} - \frac{3}{4} {{\cal T}}_{(i) (j)}{}^{(k) (l)} \Bigl[ \langle y m_{(k) (l)} \rangle - 2 \langle Y_{(k) (l)} \rangle \Bigr] + {{\cal T}}_{(i) (j)}{}^{(k) (l)} {{\cal Q}}_{(k) (l)}^Y - {{\cal C}}^\Theta {{\cal P}}_{(i) (j)} \right) \,,\end{aligned}$$ where ${{\cal T}}_{(i) (j)}{}^{(k) (l)}$ is a traceless projection operator with respect to $S_{(i) (j)}$ $$\begin{aligned} {{\cal T}}_{(i) (j)}{}^{(k) (l)} \equiv S_{(i)}{}^{(k)} S_{(j)}{}^{(l)} - \frac{1}{2} S^{(k) (l)} S_{(i) (j)} \,.\end{aligned}$$ Finally before closing this section let us explicitly write down the equation for the temperature part for the sake of completeness. It is of the form $$\begin{aligned} {{\cal L}}^P_{(i) (j)} = {{\cal C}}_{(i) (j)}^P \,, \end{aligned}$$ and using that at first order $$w^{(k)}{}_{(l) (0)} = \frac{1}{a}\beta^{[k}{}_{, l]} \,,\qquad w^{(k)}{}_{(l) (m)} =\frac{2}{a} h_m{}^{[k}{}_{, l]} \,,$$ the explicit form of ${{\cal L}}^P_{(i) (j)}$ is $$\begin{aligned} {{\cal L}}_{(i) (j)}^P \equiv &{{\cal P}}_{(i) (j)}{}' + {{\cal P}}_{(i) (j), k} n^{(k)} \,, - \alpha {{\cal P}}_{(i) (j)}{}' - (\beta^k + h^k{}_l n^{(l)}) {{\cal P}}_{(i) (j), k} \notag\\ & \qquad - \Bigl( \alpha_{, k} - \beta_{l, k} n^{(l)} + h_{k l}{}' n^{(l)} + 2 h_{m [k, l]} n^{(l)} n^{(m)} \Bigr) D^{(k)} {{\cal P}}_{(i) (j)} \notag\\ & \qquad \qquad - \Bigl( \beta^{[k}{}_{, l]} + 2 h_m{}^{[k}{}_{, l]} n^{(m)} \Bigr) \Bigl( S_{(i)}{}^{(l)} {{\cal P}}_{(k) (j)} + S_{(j)}{}^{(l)} {{\cal P}}_{(k) (i)} \Bigr) \,.\end{aligned}$$ Technical details about linear polarisation =========================================== Gauge transformation of a tensor-valued distribution function {#AppGTtensor} ------------------------------------------------------------- In this section, we investigate the transformation property of the distribution matrix up to the second order. The transformation properties of $I$ and $V$, which are scalar distribution functions, have already been investigated in the main text. In this section we focus on the case of the linear polarisation and set $I=V=0$ such that $f_{\mu\nu}=P_{\mu\nu}$. Since the polarisation matrix is at least first order, this means that we need only to keep terms which are first order in the coordinates transformation $(T,L^i)$. First we must understand the transformation properties of the screen projector. As for the tetrads, it is not a purely geometric quantity and it is not invariant under a change of coordinates. Indeed, it is defined with respect to the time-like tetrad which depends on the choice of coordinates. We thus have the two related screen projectors $${\bm S}({\bm p})={\bm g}+{\bm e}^{(0)} \otimes {\bm e}^{(0)}-{\bm n}\otimes{\bm n}\,,\quad \tilde{\bm S}({\bm p})={\bm g}+\tilde {\bm e}^{(0)} \otimes \tilde {\bm e}^{(0)}-\tilde{\bm n}\otimes\tilde{\bm n}\,,\quad\text{with}\quad {\bm n}\equiv \frac{{\bm p}}{{p}^\mu{e}^{(0)}_\mu}-{\bm e}^{(0)}\,,\quad \tilde{\bm n}\equiv \frac{{\bm p}}{{p}^\mu \tilde {e}^{(0)}_\mu}-\tilde{\bm e}^{(0)}\,.$$ If we use the transformation rule at the same point  we find that at first order in the transformation the relation between these projectors is given by  [@Tsagas:2007yx] $$\label{RelSStilde} \tilde S_{\mu \nu}\circ \tilde c{(\tilde{\bm x},q^{\widetilde{(a)}})}= S_{\mu\nu}({\bm x},q^{(a)})+ 2 \left(e^{(0)}_{(\mu}+n_{(\mu}\right) S_{\nu) \alpha}({\bm x},q^{(a)}) V^\alpha\,,\quad \text{with}\quad {\bm V} \equiv -\Lambda^{(i)}_{\,\,(0)} {\bm e}_{(i)}=-{{\partial}}^i T {\bm e}_i\,.$$ Note that if we use tetrad coordinates for the argument of the screen projectors, then $${\bm S}(q,n^{(i)})={\bm g}+{\bm e}^{(0)} \otimes {\bm e}^{(0)}-n_{(i)} n_{(j)}{\bm e}^{(i)}\otimes {\bm e}^{(j)}\,,\qquad \tilde{\bm S}\circ \tilde c{(q,n^{(i)})}={\bm g}+\tilde{\bm e}^{(0)} \otimes \tilde{\bm e}^{(0)}-n_{(i)} n_{(j)}\tilde {\bm e}^{(i)} \otimes \tilde {\bm e}^{(j)}\,.$$ This means that when the coordinates of the projectors are expressed in the tetrad basis associated with the corresponding coordinates system, we obtain $$\label{SijisSij} S_{(i)(j)}(q,n^{(k)})=\delta_{ij}-n_{(i)}n_{(j)} = \tilde S_{\widetilde{(i)}\widetilde{(j)}}\circ \tilde c{(q,n^{(k)})}\,.$$ Thus in any gauge, the expression of the related screen projector in the tetrad basis is the same by construction, and $S_{(i)(j)}$ depends actually only on $n^{(k)}$. However it must remain clear that these two tensors are geometrically different since they are associated with different coordinates systems and indeed their relation is given at first order by . The distribution tensor is also not geometrically invariant since for every observer used to define the screen projector, we must consider a different distribution tensor. However all the possible distribution matrices are related through projections and we find that the distribution tensors defined by the tetrad $\tilde{\bm e}_{(0)}$ and ${\bm e}_{(0)}$ are related at the same point of the tangent bundle by [@Tsagas:2007yx] $$f^{\tilde{\bm e}_{(0)}}_{\mu\nu}\circ \tilde c{(\tilde {\bm x},q^{\widetilde{(\imath)}})} = \tilde S_\mu^\alpha \circ \tilde c{(\tilde {\bm x},q^{\widetilde{(\imath)}})} \tilde S_\nu^\beta \circ \tilde c{(\tilde {\bm x},q^{\widetilde{(\imath)}})} f^{{\bm e}_{(0)}}_{\alpha \beta}({\bm x},q^{(\imath)})\,.$$ This means that the only requirement is to project the distribution function so that it is projected with respect to the new observer and the new direction. Combining this transformation rule with  we obtain at first order the transformation rule as $$f^{\tilde{\bm e}_{(0)}}_{\mu\nu}\circ \tilde c{(\tilde {\bm x},q^{\widetilde{(\imath)}})} = f^{{\bm e}_{(0)}}_{\mu \nu}({\bm x},q^{(\imath)}) + 2 \left(e^{(0)}_{(\mu}+n_{(\mu}\right) f^{{\bm e}_{(0)}}_{\nu)\alpha}({\bm x},q^{(\imath)})V^\alpha\,.$$ If we project this expression onto the tetrad components, and noting $\tilde f_{(i)(j)}\equiv f^{\tilde{\bm e}_{(0)}}_{(i)(j)}$ and $f_{(i)(j)}\equiv f^{{\bm e}_{(0)}}_{(i)(j)}$, we obtain $$\label{Rulefijstrange} \tilde f_{\widetilde{(i)}\widetilde{(j)}}\circ \tilde c{(\tilde {\bm x},q^{\widetilde{(\imath)}})} = f_{(i)(j)}({\bm x},q^{(\imath)}) - f_{(i)(k)}({\bm x},q^{(\imath)})L_{\,\,\,,j]}^{[k} -f_{(k)(j)}({\bm x},q^{(\imath)})L_{\,\,\,,i]}^{[k} - 2 n_{((i)} f_{(j))(k)}({\bm x},q^{(\imath)}){{\partial}}^k T\,,$$ find the transformation rule under a gauge transformation, we need to expand the left hand side around $({\bm x},q^{(\imath)})$. At first order we get $$\begin{aligned} \tilde f_{\widetilde{(i)}\widetilde{(j)}}\circ \tilde c{(\tilde {\bm x},q^{\widetilde{(\imath)}})} &\simeq\left(1+\xi^\mu \frac{{{\partial}}}{{{\partial}}x^\mu} +\delta q^{(i)}\frac{{{\partial}}}{{{\partial}}q^{(i)}} \right) \tilde f_{\widetilde{(i)}\widetilde{(j)}}\circ \tilde c{({\bm x},q^{{(\imath)}})}+\cdots \nonumber\\ &\simeq \tilde f_{\widetilde{(i)}\widetilde{(j)}}\circ \tilde c{({\bm x},q^{{(\imath)}})} + \left(\xi^\mu \frac{{{\partial}}}{{{\partial}}x^\mu}+\delta q^{(i)}\frac{{{\partial}}}{{{\partial}}q^{(i)}} \right) f_{{(i)}{(j)}}+\cdots \,.\end{aligned}$$ Using the expansion (\[Expandderpi\]) we then obtain the gauge transformation rule for the tensor valued distribution function in tetrad coordinates. Noting $\widetilde{f_{(i)(j)}} \equiv \tilde f_{\widetilde{(i)}\widetilde{(j)}}\circ \tilde c$ for simplicity, this reads $$\label{Gaugeruletensor} \widetilde{f_{{(i)}{(j)}}} = f_{(i)(j)}-\left(\xi^\mu \frac{{{\partial}}}{{{\partial}}x^\mu}+\delta \ln q {{\frac{\partial}{\partial \ln q}}}+\delta n^{(i)}D_{(i)} \right)f_{(i)(j)}-f_{(i)(k)} L_{\,\,\,,l]}^{[k}S^{(l)}_{(j)} -f_{(k)(j)} L_{\,\,\,,l]}^{[k} S^{(l)}_{(i)}\,,$$ where it is implied that all quantities are evaluated either at ${\bm x}$ or at $({\bm x},q,n^{(i)})$. For completeness we report the explicit form of the gauge transformation for ${{\cal P}}_{(i) (j)}$ which is obtained from the above transformation rule and the spectral decomposition (\[defyP\]) $$\begin{aligned} \label{GT:calP} \tilde{{{\cal P}}}_{(i) (j)} &= {{\cal P}}_{(i) (j)} - L^{[k}{}_{, l]} \Bigl( S^{(l)}{}_{(i)} {{\cal P}}_{(k) (j)} + S^{(l)}{}_{(j)} {{\cal P}}_{(k) (i)} \Bigr) - \xi^\mu {{\partial}}_\mu {{\cal P}}_{(i) (j)} - \delta n^{(k)} D_{(k)} {{\cal P}}_{(i) (j)} \,. \end{aligned}$$ It is also found, as expected, that the spectral distortion $Y_{(i)(j)}$ part is gauge invariant since it vanishes on the background and at first order. Gauge transformation for Liouville and Collision terms {#AppGTLC} ------------------------------------------------------ We deduce from the transformation rule (\[GT:calP\]) and the spectral decomposition (\[spdec:calP\_lhs\]) and (\[spdec:calP\_rhs\]) that the spectral distortion part, ${{\cal L}}^Y_{(i)(j)} $ and ${{\cal C}}^Y_{(i)(j)}$, must be gauge invariant. Concerning the temperature part, they should transform as (noting $\widetilde{{{\cal L}}^P_{(i)(j)}} \equiv {{\cal L}}^P_{\tilde{(i)}\tilde{(j)}} \circ \tilde c$ and $\widetilde{{{\cal C}}^P_{(i)(j)}} \equiv {{\cal C}}^P_{\tilde{(i)}\tilde{(j)}} \circ \tilde c$ ) $$\begin{aligned} \widetilde{{{\cal L}}^P_{(i)(j)}} &= {{\cal L}}^P_{(i)(j)} - L^{[k}{}_{, l]} \Bigl( S^{(l)}{}_{(i)} {{\cal L}}^P_{(k) (j)} + S^{(l)}{}_{(j)} {{\cal L}}^P_{(k) (i)} \Bigr) - \xi^\mu {{\partial}}_\mu {{\cal L}}^P_{(i)(j)} - \delta n^{(k)} D_{(k)} {{\cal L}}^P_{(i) (j)} \,, \\ \widetilde{{{\cal C}}^P_{(i)(j)}} &= {{\cal C}}^P_{(i)(j)} - L^{[k}{}_{, l]} \Bigl( S^{(l)}{}_{(i)} {{\cal C}}^P_{(k) (j)} + S^{(l)}{}_{(j)} {{\cal C}}^P_{(k) (i)} \Bigr) - \xi^\mu {{\partial}}_\mu {{\cal C}}^P_{(i)(j)} - \delta n^{(k)} D_{(k)} {{\cal C}}^P_{(i) (j)} \,. \end{aligned}$$ Using the transformation rules derived in this paper, we checked that this is indeed the case when using the detailed form of the Liouville and collision operators. Extraction of temperature and spectral distortion {#AppExtraction} ================================================= The functions $y$ and $\Theta$ can be extracted thanks to the integrals of the type $${{\cal M}}_n[f]\equiv \frac{\int f q^{2+n} {{\rm d}}q}{(3+n) \int \bar I(q) q^{2+n} {{\rm d}}q} \,,$$ just by applying them order by order to $I(q)$, using that ${{\cal M}}_0[{{\cal D}}_q^2 \bar I]=0$. We then obtain $$\begin{aligned} \Theta^{(1)} &= {{\cal M}}_1 [I^{(1)}] = {{\cal M}}_0[I^{(1)}] \,, \label{ExtractTheta1} \\ \frac{1}{2}\Theta^{(2)} &= \frac{1}{2} {{\cal M}}_0[I^{(2)}]-\Theta^{(1)2} \,, \\ \frac{1}{2}y^{(2)} &= \frac{1}{2} \left( {{\cal M}}_1[I^{(2)}] - {{\cal M}}_0 [I^{(2)}] \right) - \frac{1}{2}\Theta^{(1)}{}^2 \,.\end{aligned}$$ Similarly to what can be done for the intensity part, the spectral components of polarisation can be extracted thanks to $$\begin{aligned} {{\cal P}}^{(1)}_{\mu\nu} &= {{\cal M}}_1 [{\bf P}_{\mu\nu}^{(1)}] = {{\cal M}}_0 [P_{\mu\nu}^{(1)}] \,, \\ \frac{1}{2} {{\cal P}}^{(2)}_{\mu\nu} &= \frac{1}{2} {{\cal M}}_0 [P_{\mu\nu}^{(2)}] - 3 \Theta^{(1)} {{\cal P}}^{(1)}_{\mu\nu} \,,\\ \frac{1}{2}Y^{(2)}_{\mu\nu} &= \frac{1}{2} \Bigl( {{\cal M}}_1[P_{\mu\nu}^{(2)}]-{{\cal M}}_0[P_{\mu\nu}^{(2)}] \Bigr) - \Theta {{\cal P}}^{(1)}_{\mu\nu} \,.\end{aligned}$$ [10]{} WMAP, E. Komatsu [*et al.*]{}, Astrophys. J. Suppl. 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--- abstract: 'We derive an analytical solution to the computation of the output of a Lyot coronagraph for a given complex amplitude on the pupil plane. This solution, which does not require any simplifying assumption, relies on an expansion of the entrance complex amplitude on a Zernike base. According to this framework, the main contribution of the paper is the expression of the response of the coronagraph to a single base function. This result is illustrated by a computer simulation which describes the classical effect of propagation of a tip-tilt error in a coronagraph.' author: - André Ferrari title: Analytical analysis of Lyot coronographs response --- Introduction ============ The discovery of extrasolar planets is at the origin of a renewed interest in stellar coronagraphy. Considering the ambition of the targeted objectives, many authors have pointed out the necessity for a very accurate analysis of the system in order to study various undesired effects. For example, the specific properties of the light intensity measured by a system based on an extreme adaptive optics system and a coronagraph are the result that neither the residuals of the turbulence, nor the ideal coronagraphed point-spread function can be neglected with respect to the faint object (planet). @aimeithd04 analyzed the fact that the wavefront amplitudes associated to these two contributions will interfere leading to the so-called “pinned” speckles. Another example is given by @lloyd05 which pointed out that a small misalignment of the star with the center of the stop can result in a fake source. A related problem is also present in [@soummer2005APJ] wich derives the optimal apodization for an arbitrary shaped aperture using an algorithm proposed independently in [@guyon2000] which relies on iterated simulations of the coronagraph response. More generally, an intense activity aims to optimize the different coronagraph parameters (mask size, apodization shape,...) for a number of projects dedicated to devise high-dynamic range imaging on the VLT (Sphere), Gemini (GPI) or the Subaru telescope (HiCIAO), see for example [@IAUC200]. The input/output relation of a coronagraph is in this case simulated by numerical computations based on discrete Fourier transforms. However, such a numerical technique suffers from the well-known problems related to the choice of the extent of the sampled surface and the sampling frequency which both define the sampling in the transformed domain. Note that this compromise is coupled with the difficulty to evaluate numerically the simulation errors. This work focuses on the analytical characterization of the response of a Lyot coronagraph. The objective is obviously also to gain deeper insight in the behaviour of the system. This problem has already been studied in the literature and analytical results were obtained under various assumptions. In the one-dimensional case, @lloyd05 assume that the Lyot stop is band limited and the phase on the telescope aperture is small. This last hypothesis is removed in [@sivara05] where the computation is carried for a rectangular pupil assuming again that the Lyot stop is band-limited. The development presented herein for a circular pupil differs from these approaches substituting these simplifying assumptions by an expansion of the complex amplitude on an orthogonal basis. Section 1 recalls the general formalism of Lyot coronagraphy and justifies the choice of an expansion of the complex amplitude on a Zernike base. Section 2 contains the main results of the paper; the response of the coronagraph to a Zernike polynomial is computed. The result involving an infinite sum, a bound on the truncation error is then derived. Section 3 presents two simulations. First the response of the coronagraph to the 6 first Zernike functions is computed. Then the formalism derived in this paper is used to illustrate the effect of a tip-tilt error in a coronagraph. A short appendix containing the material required for the mathematical derivations of section 2 is included at the end of the paper. Notations and hypothesis {#secNotaandHyp} ======================== Coronagraph formalism --------------------- We follow the notations of @art_aa01b and @aim03. The successive planes of the coronagraph are denoted by $A$, $B$, $C$ and $D$. $A$ is the entrance aperture, $B$ denotes the focal plane with the mask (without loss of generality we assume that the amplitude of the mask is $1-\epsilon$ where $\epsilon=1$ corresponds to the classical Lyot coronagraph and $\epsilon=2$ to the Roddier coronagraph), $C$ is the image of the aperture with the Lyot stop and $D$ is the image in the focal plane after the coronagraph. The aperture transmission function is $p(x,y)$ and the wavefront complex amplitude in $A$ is $\Psi(x,y)$. In the case of an apodized pupil, we assume that the apodization function is included in $\Psi(x,y)$. In order to simplify the notations, the mask function in $B$ is defined with coordinates proportional to $1/\lambda f$ and decomposed as: $$1 -\epsilon m\left(\frac{x}{\lambda f},\frac{y}{\lambda f}\right)$$ where function $m(.)$ equals to 1 inside the coronagraphic mask and 0 outside. We will make in the sequel the usual approximations of paraxial optics. Moreover we neglect the quadratic phase terms associated with the propagation of the waves or assume that the optical layout is properly designed to cancel it [@aimeithd03]. The expression in cartesian coordinates of the complex amplitude in the successive planes are: $$\begin{aligned} \Psi_A(x,y)&=&\Psi(x,y) p(x,y) \\ \Psi_B(x,y)&=&\frac{1}{\jmath \lambda f} \widehat{\Psi_A}\left(\frac{ x}{\lambda f},\frac{ y}{\lambda f}\right)\left( 1 -\epsilon m\left(\frac{x}{\lambda f},\frac{y}{\lambda f}\right)\right) \label{planCcart}\\ \Psi_C(x,y)&=&\frac{1}{\jmath \lambda f} \widehat{\Psi_B}\left(\frac{ x}{\lambda f},\frac{ y}{\lambda f}\right) p(-x,-y) \\ &=&- \left( \Psi_A(-x,-y) - \epsilon \left[\Psi_A(-u,-v)\ast \widehat{m}\left( u,v\right)\right] \left(x,y\right) \right)p(-x,-y) \label{geneplanC}\\ \Psi_D(x,y) &=&\frac{-1}{\jmath \lambda f} \widehat{\Psi_A}\left(\frac{- x}{\lambda f},\frac{- y}{\lambda f}\right) \nonumber\\ && \qquad +\epsilon \frac{1}{\jmath \lambda f} \left( \widehat{\Psi_A}(-x,-y)m(-x,-y) \ast \widehat{p}(-x,-y) \right) \left(\frac{ x}{\lambda f},\frac{ y}{\lambda f}\right) \label{geneplanD} \end{aligned}$$ where $\hat{f}$ is the Fourier transform of $f$ and $\ast$ denotes convolution. Eqs. (\[geneplanC\],\[geneplanD\]) assume that the Lyot stop is the same as the pupil. However for classical “unapodized” Lyot coronagraph the residual intensity in plane $C$ is concentrated at the edges of the pupil and a reduction of the Lyot stop size is needed in order to improve the rejection. The case of a reduced Lyot stop, which consists in convolving Eq. (\[geneplanD\]) by the appropriate function, has not been considered in Eqs. (\[geneplanC\],\[geneplanD\]) to alleviate the notations but will be discussed in section 3. It is important to note that the reduction of the Lyot stop can be avoided using a prolate apodized entrance pupil which will optimally concentrate the residual amplitude in $C$, see for example [@art_aa01b]. The coronagraph response being derived herein for a circular pupil, the use of polar coordinates will be preferred. Transcription of previous equations to polar coordinates is straightforward. Moreover, as long as the aperture transmission function and the stop have a circular symmetry, their Fourier transform will verify the same symmetry, as proved by Eq. (\[hankelgeneral\]) with $m=0$, i.e. the Hankel transform. This leads to the following expression of the complex amplitude in $D$: $$\Psi_D(r\lambda f,\theta)=\frac{-1}{\jmath \lambda f} % \widehat{\Psi_A}( r, \theta+\pi) % +\frac{\epsilon }{j \lambda f} \left( \left(\widehat{\Psi_A}( r,\theta+\pi)m(r)\right) \ast \widehat{p}(r) \right) (r,\theta) \label{ampcompD}$$ where the convolution of the two functions is still computed with respect to to the cartesian coordinates $(x,y)$. Choice of a base ---------------- As mentioned in the introduction, the analytical computation of the coronagraph response proposed herein relies on the expansion of the complex amplitude in $A$ on an orthogonal basis. Eq. (\[ampcompD\]) shows that the coronagraph *acts linearly* on the complex amplitude, consequently the problems simplifies to the computation of the response of each basis function. The retained solution consists in the expansion of the complex amplitude in $A$ on Zernike polynomials. Basic properties of the Zernike polynomials required in the paper are recalled in appendix \[appendA\]. Adopting the usual ordering of the Zernike circle polynomial [@maha94] we can write: $$\begin{aligned} \Psi_A(r,\theta) &=& \sum_{(m,n)} a_{(m,n)} U_n^m(r/R,\theta)\label{expanA0}\\ &=& \sum_k a_k Z_k(r/R,\theta),\;a_k\equiv a_{(m,n)}\in \mathbb{C} \label{expanA}\end{aligned}$$ where $R$ is the radius of the aperture. This expansion is rather unusual, the Zernike polynomials being generally used for the expansion of the wavefront. However it is worthy to note that, as Eq. (\[geneplanC\]) shows, a coronagraphic system will always introduce amplitude aberration. Hence, even in the case of a perfect wave with no aberration in $A$, an expansion of only the phase in $C$ will not be appropriate. Finally, Eq. (\[expanA\]) can also be justified by the fact that it coincides (up to a linear transform) with the classical approximation of the complex amplitude in the case of sufficiently small phase errors assuming a first order development of the exponential function. We will illustrate the expansion (\[expanA\]) in the case of tip-tilt error with an apodized pupil: $$\Psi_A(rR,\theta) = a(r)\Pi(r)e^{\jmath \beta r \cos(\theta) }$$ where $a(r)$ denotes the pupil apodization and $\Pi(r) = 1$ for $r\in [0,1)$ and $0$ if $r\geq 1$. Computation of the projection of $\Psi_A(r,\theta)$ on $U_n^m(r/R,\theta)$ is straightforward using the definition of the Bessel functions of integer order [@abramb]: $$\begin{aligned} \int_0^{2\pi} \int_0^R \Psi_A(r,\theta) U_n^m(r/R,\theta) r dr d\theta &=&R^2\int_0^1\int_0^{2\pi} R_n^m(r) \cos(m\theta) a(r) e^{\jmath \beta r \cos(\theta) } r dr d\theta \\ &=&2\pi R^2 \jmath^m \int_0^1 a(r)R_n^m(r)J_m(\beta r)rdr \label{apotilt}\end{aligned}$$ The projection of $\Psi_A(r,\theta)$ on $U_n^{-m}(r/R,\theta)$ equals 0. - In the unapodized case, $a(r)=1$, integral in Eq. (\[apotilt\]) can be computed using Eq. (\[intbesselradial\]): $$2\pi R^2 \jmath^m \int_0^1 a(r)R_n^m(r)J_m(\beta r)rdr=2\pi R^2 \jmath^m (-1)^\frac{n-m}{2}\frac{J_{n+1}(\beta)}{\beta}$$ The coefficient $a_{k}$ is then obtained dividing this quantity by the $L^2$ norm of the Zernike polynomials [@bornb], leading to: $$a_k = \jmath^m (-1)^\frac{n-m}{2} \frac{4(n+1)}{1+\delta(m)}\frac{J_{n+1}(\beta)}{\beta} \label{amntilt}$$ - A particularly important case is that where $a(r)$ is proportional to the circular prolate function $\varphi_{0,0}(c,rR)$, [@aim03]. In this case the integral in Eq. (\[apotilt\]) can be computed using the expansion of $\varphi_{0,0}(c,r)$ derived in [@slep64]: $$\varphi_{0,0}(c,r) = \sum_{k=0}^\infty d_k^{0,0}(c) \sqrt{r} F(k+1,-k;1;r^2)$$ The function $F(k+1,-k;1;r^2)$ defined in Eq. (\[hyperg\]) reduces to a polynomial of order $2k$ which, as mentioned in [@slep64] “is closely related to the Zernike polynomials”. Indeed using Eq. (\[zernger\]) and the results below it can be easily checked that: $F(k+1,-k;1;r^2) = (-1)^k R_{2k}^0(r)$. Inserting this expansion in Eq. (\[apotilt\]) and integrating terms by terms leads to integrals which generalize Eq. (\[intbesselradial\]). These integrals can be computed for example using of integrals of the type $\int_0^1 r^{\nu} J_m(\beta r)dr$ [@gradb]. This derivation will not be presented herein for sake of brevity. Finally, for more complicated complex amplitudes, the $a_k$ can be of course computed numerically. This problem as been addressed in [@pawl02] using a piecewise approximation of $\Psi_A(x,y)$ over a lattice of squares with size $\Delta \times \Delta$ and centered on point $(x_i,y_j)$. In this case the estimation of $a_{k} $ is given by: $$\hat{a}_{k} = \sum_{(x_i,y_j) \in \mathcal{D}} \Psi_A(x_i,y_j)w_n^m( x_i,y_j)^\ast \label{eqpawl}$$ where $w_n^m(x_i,y_j)$ is the integral of the Zernike polynomial $U_n^m(\rho/R,\phi)$ over the square centered on $(x_i,y_j)$. [@pawl02] gives bound for the mean integrated squared error on the reconstruction of $\Psi_A(x,y)$ when the coefficients are given by Eq. (\[eqpawl\]). This analysis is particularly important in our case because it quantifies the dependence of the error on the smoothness of $\Psi_A(x,y)$, the sampling rate $\Delta$ and the geometrical error due to the circular geometry of the pupil. Coronagraph response ==================== Response of the coronagraph to a Zernike polynomial --------------------------------------------------- The purpose of this section is to compute the complex amplitude in $D$ when the complex amplitude in $A$ is the Zernike polynomial with radial degree $n$ and azimuthal frequency $m$. In this case the complex amplitude $\Psi_D(r,\theta)$ will be denoted as $\mathcal{D}_n^m(r,\theta)$. According to Eq. (\[ampcompD\]), the difficulty in the computation of $\mathcal{D}_n^m(r,\theta)$ lies in the evaluation of the convolution: $$\Xi(r,\theta)= \left( \left(\widehat{\Psi_A}( r,\theta+\pi)m(r)\right) \ast \widehat{p}(r) \right) (r,\theta) \label{convol1}$$ In this expression $m(r)$ is an “annular” mask of radius $d$ which, with the definition adopted in Eq. (\[planCcart\]) is defined as: $$m(r) = \Pi\left( r \frac{\lambda f}{d} \right) \label{theMask}$$ The computation of the convolution in $\Xi(r,\theta)$ is sketched in Fig. \[compconv\]. Using Eq. (\[theMask\]), $\Xi(r,\theta)$ simplifies to: $$\Xi(r,\theta)= \int_0^{d/\lambda f }\int_0^{2\pi} \widehat{\Psi_A}(\rho,\phi+\pi) \widehat{p}\left(\sqrt{r^2+\rho^2-2r\rho \cos(\theta-\phi)} \right) \rho d\rho d\phi$$ The next step consists in substituting in this equation: - $\widehat{p}(r)$ by the Fourier transform of $p(r) = \Pi\left(r/R\right)$: $$\hat{p}(r)=\frac{RJ_1(2\pi R r)}{r} \label{Airy}$$ - $\Psi_A(\rho,\phi)$ by $U_n^m(\rho/R,\phi)$ and consequently $\widehat{\Psi_A}(\rho,\phi)$ by $R^2\widehat{U_n^m}(rR,\phi)$ where $\widehat{U_n^m}(r,\phi)$ is given in Eq. (\[fourier1zern\]). In order to simplify the notations we define the new “standardized” integral $\tilde{\Xi}(r,\theta,\xi)$ by: $$\tilde{\Xi}(r,\theta,\xi)= \int_0^\xi \int_0^{2\pi} \cos(m\phi)J_{n+1}(\rho) \frac{J_1\left( \sqrt{r^2+\rho^2-2r\rho \cos(\theta-\phi)} \right)} {\sqrt{r^2+\rho^2-2r\rho \cos(\theta-\phi)}} d\rho d\phi \label{convol2}$$ It can be easily checked in this case that: $$\Xi(r,\theta)= R^2 \jmath^{m} (-1)^\frac{n-m}{2} \tilde{\Xi}\left(2\pi R r,\theta,\frac{2\pi R d}{\lambda f} \right) \label{xixitild}$$ Analytical computation of $\tilde{\Xi}(r,\theta,\xi)$ relies on the properties of the Gegenbauer polynomials defined in appendix \[appendA\]. Substituting Eq. (\[eqGegen\]) for $\nu=1$ in Eq. (\[convol2\]) allows indeed to separate the integrations with respect to $\rho$ and $\phi$: $$\tilde{\Xi}(r,\theta,\xi)= 2\sum_{k=0}^\infty (k+1) \frac{J_{k+1}(r)}{r} % \int_0^\xi \frac{J_{k+1}(\rho)J_{n+1}(\rho)}{\rho}d\rho % \int_0^{2\pi} \cos(m\phi) C_k^{(1)}(\cos(\theta-\phi)) d\phi \label{convol3}$$ - Computation of the integral on $\phi$ is straightforward using (\[cheby\]): $$\int_0^{2\pi} \cos(m\phi) C_k^{(1)}(\cos(\theta-\phi))d\phi = \pi \cos(m\theta) \sum_{q=0}^k \delta(m-k+2q)$$ - Computation of the integral on $\rho$ relies on recursion formulas on indefinite integrals of products of Bessel functions, [@abramb]: $$\begin{aligned} && k\not= n,\;\int_0^\xi \frac{J_{n}(\rho)J_{k}(\rho)}{\rho} d\rho= \frac{\xi J_{k-1}(\xi)J_{n}(\xi)-\xi J_{k}(\xi)J_{n-1}(\xi) ) +(n-k) J_{n}(\xi)J_{k}(\xi) }{k^2-n^2} \label{int1} \\ && \int_0^\xi \frac{J_{n}(\rho)^2}{\rho} d\rho= \frac{1}{2n}(1-J_0(\xi)^2 -2\sum_{q=1}^{n-1}J_q(\xi)^2 -J_n(\xi)^2) \label{int2} \end{aligned}$$ After computation of the integral of Eq. (\[convol3\]), substitution of Eq. (\[xixitild\]) in Eq. (\[ampcompD\]) gives the complex amplitude in $D$ for a single basis function $\Psi_A( r,\theta)= U_n^m(r/R,\theta)$: $$\mathcal{D}_n^m( r,\theta)= \jmath^{m-1} (-1)^\frac{n-m}{2}R \cos(m\theta) \left( -\frac{J_{n+1}(2\pi \mu r)}{r} + \epsilon \sum_{k=0}^\infty \eta_{m,n,k}(2\pi \mu d) \frac{J_{k+1}(2\pi \mu r )}{r} \right) \label{leresultat}$$ with $\mu = R/{\lambda f}$ and: $$\eta_{m,n,k}(\xi) = (k+1) \big(\sum_{q=0}^k \delta(m-k+2q) \big)\int_0^\xi \frac{J_{n+1}(\rho)J_{k+1}(\rho)}{\rho} d\rho \label{defeta}$$ The corresponding complex amplitude in $C$ for $r<R$ can be directly computed from Eq. (\[leresultat\]) using the inverse Fourier transform of $\cos(m\theta)J_{k+1}(2\pi r)/r$ obtained in Eq. (\[zernger\]): $$\mathcal{C}_n^m(r,\theta)= (-1)^\frac{n-m}{2} \cos(m\theta) \left( -\mathcal{R}_n^m\left( \frac{r}{R}\right) + \epsilon \sum_{k=0}^\infty \eta_{m,n,k}(2\pi \mu d) \mathcal{R}_k^m\left( \frac{r}{R}\right) \right) \label{leresultat2}$$ Eqs. (\[leresultat\],\[leresultat2\]) give an analytical expression of the complex amplitude in $C$ for $r<R$ and in $D$ when a single basis function is applied in $A$ and when the size of the Lyot stop equals the size of the entrance pupil. In the general where the amplitude in $A$ is given by Eqs. (\[expanA0\],\[expanA\]), the complex amplitudes in $C$ and $D$ become: $$\Psi_C(r,\theta) = \sum_{(m,n)} a_{(m,n)} \mathcal{C}_n^m(r,\theta),\; \Psi_D(r,\theta) = \sum_{(m,n)} a_{(m,n)} \mathcal{D}_n^m(r,\theta) \label{repcomplete}$$ As mentioned in section 2.1, if the entrance pupil is not apodized a reduction of the Lyot stop must be considered. This is achieved replacing $p(r)$ by $p(\alpha^{-1}r)$ with $\alpha<1$. The expression of the complex amplitude in $C$ is of course straightforward and for example Eq. (\[leresultat2\]) becomes $\mathcal{C}_n^m(r,\theta) p(\alpha^{-1}r)$. This result allows numerical computation of the complex amplitude in $D$ using a single Fourier transform. Unfortunately it is much more complicated to obtain an analytical expression of the complex amplitude in $D$. The derivation presented above can be of course redeveloped replacing $\hat{p}(r)$ by $\alpha^2 \hat{p}(\alpha r)$ and straightforward computation shows that: 1. Similarly to Eq. (\[leresultat\]), the convolution (\[convol1\]) will expand in an infinite sum of functions $\cos(m\theta)J_{k+1}(2\pi \alpha \mu r )/r$. However, the “radial contribution” to the coefficients weighting these functions, see Eq. (\[defeta\]), becomes: $$\int_0^\xi \frac{J_{k+1}(\rho)J_{n+1}(\alpha^{-1}\rho)}{\rho}d\rho$$ which cannot be computed straightforwardly as in Eqs. (\[int1\],\[int2\]). 2. The first term in Eq. (\[ampcompD\]) is now replaced by the Fourier transform of $U_n^m(\rho/R,\phi) \Pi(r/\alpha R)$ which cannot be anymore calculated using Eq. (\[intbesselradial\]). Bound for the truncation error of $\mathcal{D}_n^m(r,\theta)$ ------------------------------------------------------------- As we are interested in the computation of $\mathcal{C}_n^m(r,\theta)$ or $\mathcal{D}_n^m(r,\theta)$ from the implementation of formula (\[leresultat\]), the errors produced when the infinite sum is truncated must be studied. In order to reduce mathematical developments we only present herein the results for $\mathcal{D}_n^m(r,\theta)$ when the size of the Lyot stop equals the size of the pupil. We define the truncation error on $\mathcal{D}_n^m(r,\theta)$: $$\mathcal{E}_N(r,\theta;m,n,\mu,d) = \epsilon R \left|\cos(m\theta) \sum_{k=N+1}^\infty \eta_{m,n,k}(2\pi \mu d) \frac{J_{k+1}(2\pi \mu r)}{r} \right|$$ Computation of a bound on the truncation error relies on the classical upper bound for the Bessel functions of integer order [@abramb]: $$\left| J_{k+1}(r)\right| \leq \frac{(r/2)^{k+1}}{k!},\; r\geq 0 \label{majorbess}$$ Substitution of this result in Eq. (\[defeta\]) gives: $$\begin{aligned} \eta_{m,n,k}(\xi) &\leq& (k+1) \big(\sum_{q=0}^k \delta(m-k+2q) \big) \frac{1}{k+n+2}\frac{1}{k!n!} \left(\frac{\xi}{2} \right)^{k+n+2}\\ & \leq & \frac{k+1}{k!n!} \left(\frac{\xi}{2} \right)^{k+n+2}\end{aligned}$$ which leads to the following bound for the truncation error: $$\mathcal{E}_N(r,\theta;m,n,\mu,d) \leq \frac{\epsilon R(\pi \mu)^{3+n}d^{2+n}}{n!} \sum_{k=N+1}^\infty \frac{k+1}{(k!)^2}\left( (\pi \mu)^2 rd \right)^k \label{laborne}$$ The above serie is absolutely convergent for $r>0$. As a consequence the expansion in Eq. (\[leresultat\]) converges uniformly for $(r,\theta)\in [0,\infty)\times [0,2\pi)$. Finally, it is worthy to note that the computation of the infinite sum in the upper bound (\[laborne\]) can be avoided using the equality: $$\sum_{k=0}^\infty \frac{k+1}{(k!)^2}x^k = I_0(2\sqrt{x})+\sqrt{x}I_1(2\sqrt{x})$$ where $I_\nu(x)$ is the modified Bessel function. Simulation results ================== Response of the coronagraph to the first Zernike function --------------------------------------------------------- Figures \[figresu1\] and \[figresu2\] give the intensity in the $D$ plane of the coronagraph when the complex amplitude in the $A$ plane is one of the first six Zernike polynomials. The complex amplitudes have been computed using Eq. (\[leresultat\]). Each raw contains $U_n^m(r,\theta)$ and $\mathcal{D}_n^m(r,\theta)$ for a given couple $(n,m)$. These plots have been obtained truncating the infinite summation of Eq. (\[leresultat\]) to the first 40 terms. The relevance of the truncation error bound is verified in Fig. \[plotbound\]. This plot shows the error bound (\[laborne\]) as a function of $r$ for the parameters used in Figs. \[figresu1\] and \[figresu2\]. The increase of the bound with $r$ is simply due to the fact that the majoration of $|J_{k+1}(r)|$ given by Eq. (\[majorbess\]) is only relevant for small values of $r$ as long as $|J_{k+1}(r)|$ is bounded on $[0,\infty)$. It is important to note that this plot justifies, at least for this configuration, the validity of a truncation to $N=40$ for the computation of $\mathcal{D}_n^m(r,\theta)$. In this case the truncation error is in fact always less than $10^{-10}$. Application to tip-tilt error analysis -------------------------------------- The effects of a tip-tilt error in Lyot coronagraphs has been extensively studied by @lloyd05 and @sivara05. The scope of the simulation presented here is only to validate the results derived in section 2 simulating the particular case where there is a misalignment of the star with the center of the stop. According to the previous notations the complex amplitude in $D$ decomposes as Eq. (\[repcomplete\]). In the case of a tip-tilt error in $A$, the values of the coefficients $a_{(m,n)}$ are given by Eq. (\[amntilt\]). Fig. \[plottilt\] shows $|\Psi_D(r,\theta)|$ for different values of $\beta>0$ (the case $\beta=0$ is given in the first row of Fig. \[figresu1\]). The truncation in the summation (\[repcomplete\]) has been chosen taking into account that Eq. (\[amntilt\]) implies: $$|a_{(m,n)}| \sim \frac{ 4}{\sqrt{2\pi}\beta (1+\delta(m))} \sqrt{n} \left( \frac{e\beta}{2n}\right)^n,\; \mbox{when}\; n\rightarrow \infty$$ Note that according to the notations of Eq. (\[expanA\]), $\Psi_B(r,\theta)$ equals Eq. (\[Airy\]) shifted of $-\beta\lambda f/(2\pi R)$ on axis $x$. Consequently, the star is behind the focal stop in the first two images and outside in the last one. Conclusion ========== In this paper we have presented a theoretical formalism for the analytical study of the Lyot coronagraph response. The main purposes of this work are of course to assist coronagraph design but also to improve data processing performances for the detection and characterization of extrasolar planets. - The first application is the computation of the response of the coronagraph to a planet at a given position. This is achieved for example in the case of a classical Lyot coronagraph using Eqs. (\[repcomplete\],\[amntilt\]). This point is essential for the derivation of an optimal decision scheme to test the presence of a planet at a given location. - This formalism can also be applied to fully characterize the statistical properties of the complex amplitude in the $D$ plane. For a given spatial covariance in $A$ which is fixed through the covariance of coefficients $a_k$, the spatial covariance in $D$ becomes: $${\mathsf{cov}}[\Psi_D(r,\theta) \Psi_D(r',\theta') ] = \sum_{k,l} \mathsf{cov}[a_k,a_l] \mathcal{D}_k(r,\theta) \mathcal{D}_l(r',\theta')$$ Although detection algorithms based solely on the marginal distribution of the complex amplitude can be developed as in [@IAUC05a], the use of an accurate model for the spatial correlation of the complex amplitude is essential in order to derive detection algorithms with optimal performances, as demonstrated in [@icassp06]. The author thanks the anonymous referee who helped improve the paper. The author is also grateful to Claude Aime and Rémi Soummer for helpful discussions and insightful comments. Appendix {#appendA} ======== This section presents some facts about Fourier transform in polar coordinates, Zernike and Gegenbauer polynomials. Among the various available possibilities to define an orthogonal set of functions on the unit radius disk a central position is hold by the Zernike polynomials, see for example [@maha94] and included references. They are defined for $n\geq m $ by: $$U_n^m(r,\theta) = R_n^m(r) \cos(m\theta)\Pi(r),\; U_n^{-m}(r,\theta) = R_n^m(r) \sin(m\theta)\Pi(r) \label{leszern}$$ when $n$ et $m$ share the same parity. The $R_n^m(r)$ are the radial polynomials. Different normalizations exist for $R_n^m(r)$, we retain herein the definition of [@bornb]: $R_n^m(1)=1$. Among many properties verified by these polynomials, we focus on: $$\int_{0}^{1} r R_n^m(r) J_m(v r) dr = (-1)^\frac{n-m}{2}\frac{J_{n+1}(v)}{v} \label{intbesselradial}$$ see [@bornb appendix VII] for the proof. This equality allows straightforward computation of the Fourier transform of the Zernike polynomials. In fact recall first that when $f(r,\theta)=g(r)\cos(m\theta)$, $m\in \mathbb{Z}$, a simple change of variables in the Fourier transform integral leads to: $$\hat{f}(\rho,\phi)= 2\pi (-\jmath) ^m \cos(m\phi) \int_{0}^{\infty} r g(r) J_m(2\pi r \rho) dr \label{hankelgeneral}$$ An analog result for the inverse Fourier transform of $\hat{f}(\rho,\phi)=h(\rho)\cos(m\phi)$ is: $$f(r,\theta)= 2\pi \jmath ^m \cos(m\theta) \int_{0}^{\infty} \rho h(\rho) J_m(2\pi r \rho) d\rho \label{hankelinvgeneral}$$ Applying the result of Eq. (\[hankelgeneral\]) with Eq. (\[intbesselradial\]) immediately gives: $$\begin{aligned} &&\widehat{U_n^m}(\rho,\phi) = \jmath^m (-1)^\frac{n+m}{2} \cos(m\phi)\frac{J_{n+1}(2\pi \rho)}{\rho} \label{fourier1zern}\\ &&\widehat{U_n^{-m}}(\rho,\phi) = \jmath^m (-1)^\frac{n+m}{2} \sin(m\phi) \frac{J_{n+1}(2\pi \rho)}{\rho}\end{aligned}$$ The previous equation gives the inverse Fourier transform of $\cos(m\phi)\frac{J_{n+1}(2\pi \rho)}{\rho}$ when $n\geq m\geq 0$ and $n$ et $m$ share the same parity. In the general case where $n\geq 0$ and $m\geq 0$ this inverse Fourier transform, denoted as $f(r,\theta)$ must be computed independently. If we subsitute $h(r)$ by $J_{n+1}(2\pi \rho)/\rho$ in Eq. (\[hankelinvgeneral\]) the resulting integral is a Weber-Schafheitlin type integral [@abramb]. This results in $f(r,\theta) =\jmath^m\cos(m\theta) \mathcal{R}_n^m(r)$ where: $$\mbox{if}\;r<1, \; \mathcal{R}_n^m(r)= r^m \frac{\Gamma\left( \frac{n+m}{2} +1 \right)} {\Gamma(m+1)\Gamma\left( \frac{n-m}{2} +1 \right)} F\left( \frac{n+m}{2} +1,\frac{m-n}{2} ;m+1,r^2\right) \label{zernger}$$ $F(a,b;c;z)$ is the Gauss hypergeometric function, see [@gradb]: $$F(a,b;c;z) = 1 + \frac{ab}{1!c}z+ \frac{a(a+1)b(b+1)}{2!c(c+1)}z^2+\cdots \label{hyperg}$$ It is interesting to note from Eqs. (\[zernger\]) and (\[hyperg\]) that if $b=(m-n)/2 \in \mathbb{Z}^-$ the sum in Eq. ($\ref{hyperg}$) reduces to a polynom in $z$ of order $-(m-n)/2$. Consequently $\mathcal{R}_n^m(r)$ reduces to a polynom with degree $n$ which of course coincides up to $(-1) ^{(m-n)/2}$ with $R_n^m(r)$ for $r\leq 1$. For this reason $\mathcal{R}_n^m(r)$ can be considered as a natural generalization of the Zernike polynomials. Note that, contrarily to the generalization proposed in [@myri66] or [@wuns05], this generalization is not a polynomial. We now briefly give the principal results related to the Gegenbauer polynomials. See for example [@SpecialFunctions] or [@abramb] for detailed properties. The Gegenbauer (or ultraspherical) polynomials, noted as $t \mapsto C_k^{(\nu)}(t)$ are defined as the coefficients of the power series expansion of $r \mapsto (1-2rt+r^2)^{-\nu}$: $$\frac{1}{(1-2rt+r^2)^\nu} = \sum_{k=0}^\infty C_k^{(\nu)}(t)r^k$$ For example $ C_k^{(1)}(t)$ gives the Chebyshev polynomial of the second kind $U_k(t)$: $$C_k^{(1)}(\cos(\psi))=\sum_{q=0}^k \cos((k-2q)\psi) \label{cheby}$$ Among the numerous beautiful properties of the Gegenbauer polynomials, we focus on the expansion: $$\frac{J_\nu(w)}{w}= 2^\nu \Gamma(\nu) \sum_{k=0}^\infty (k+\nu ) \frac{J_{k+\nu}(r)}{r^\nu} \frac{J_{k+\nu}(\rho)}{\rho^\nu} C_k^{(\nu)} (\cos(\gamma)) \label{eqGegen}$$ where $w=\sqrt{r^2+\rho^2-2r\rho \cos(\gamma)}$. [99]{} Abramowitz, M. and Stegun, I.A.: 1972, [*Handbook of Mathematical Functions*]{}, Dover Aime, C.: 2003, in C. Aime and R. Soummer (eds.), [*Astronomy with High Contrast Imaging*]{}, pp 65–87, E.A.S Publications Series Aime, C. and Soummer, R.: 2004, in C. Aime and R. Soummer (eds.), [*Astronomy with High Contrast Imaging II*]{}, pp 89–101, E.A.S Publications Series Aime, C., Soummer, R., and Ferrari, A.: 2002, [*Astronomy and Astrophysics*]{}, Vol. 389, pp 334–344 Aime, C. and Vakili, F. (eds.): 2006, [*Direct Imaging of Exoplanets: Science & Techniques*]{}, International Astronomical Union Colloquium 200, Cambridge University Press Andrews, G.-E., Askey, R., Roy, R., and Rota, G.-C.: 1999, [*Special Functions*]{}, Encyclopedia of Mathematics and its Applications, Cambridge University Press Born, M. and Wolf, E.: 1991, [*Principle of Optics*]{}, Pergamon Press Chatelain, F., Ferrari, A., and Tourneret, J.-Y.: 2006, IEEE ICASSP Ferrari, A., Carbillet, M., Aime, C., Serradel, E. and Soummer, R.: 2005 International Astronomical Union Colloquium 200, Cambridge University Press. Gradshteyn, I. S., Ryzhik, I. M., Jeffrey, A., and Zwillinger, D.: 2000, [*Table of Integrals, Series, and Products, Sixth Edition*]{}, Academic Press Guyon, O. and Roddier, F.: 2000, in [*SPIE Interferometry in Optical Astronomy*]{}, Vol. 4006, pp 377–387 Lloyd, J. and Sivaramakrishnan, A.: 2005, [*The Astrophysical Journal*]{}, Vol. 14, No. 3, pp 476–489 Mahajan, V.: 1994, [*Applied Optics*]{}, Vol. 33, pp 8121–8124 Myrick, D.: 1966, [*J. SIAM Appl. Math.*]{}, Vol. 14, pp 476–489 Pawlak, M. and Liao, X.L.: 2002, [*IEEE trans. on Information theory*]{}, Vol. 48, pp 2736–2753 Sivaramakrishnan, A., Soummer, R., Lloyd, A. S. J., Oppenheimer, B., and Makidon, R.: 2005, [*The Astrophysical Journal*]{}, Vol. 634, pp 1416–1422 Slepian, D.: 1964, [*The Bell System Technical Journal*]{}, Vol. 43, pp 3009–3058 Soummer, R.: 2005, [*The Astrophysical Journal*]{}, Vol. 618, pp 161–164 Soummer, R., Aime, C., and Falloon, P.: 2003, [*Astronomy and Astrophysics*]{}, Vol. 397, pp 1161–1172 Wünsche, A.: 2005, [*J. Comput. Appl. Math.*]{}, Vol. 174, pp 135–163 ![Computation of the convolution between $\widehat{\Psi_A}( r,\theta+\pi)m(r)$ and $\widehat{p}(r)$. \[compconv\]](f1.eps){width=".6\textwidth"} ![Complex amplitude in $A$ and squared root of the amplitude in $D$, i.e. $|\mathcal{D}_n^m(r,\theta)|$. The parameters used in the simulation are: $\lambda f =1$, $R=1$, $d=3$, $\epsilon = 1$ (Lyot coronagraph). \[figresu1\]](f2a.eps "fig:"){width=".9\textwidth"}\ ![Complex amplitude in $A$ and squared root of the amplitude in $D$, i.e. $|\mathcal{D}_n^m(r,\theta)|$. The parameters used in the simulation are: $\lambda f =1$, $R=1$, $d=3$, $\epsilon = 1$ (Lyot coronagraph). \[figresu1\]](f2b.eps "fig:"){width=".9\textwidth"}\ ![Complex amplitude in $A$ and squared root of the amplitude in $D$, i.e. $|\mathcal{D}_n^m(r,\theta)|$. The parameters used in the simulation are: $\lambda f =1$, $R=1$, $d=3$, $\epsilon = 1$ (Lyot coronagraph). \[figresu1\]](f2c.eps "fig:"){width=".9\textwidth"}\ ![Complex amplitude in $A$ and squared root of the amplitude in $D$, i.e. $|\mathcal{D}_n^m(r,\theta)|$. The parameters used in the simulation are: $\lambda f =1$, $R=1$, $d=3$, $\epsilon = 1$ (Lyot coronagraph). \[figresu2\]](f3a.eps "fig:"){width=".9\textwidth"}\ ![Complex amplitude in $A$ and squared root of the amplitude in $D$, i.e. $|\mathcal{D}_n^m(r,\theta)|$. The parameters used in the simulation are: $\lambda f =1$, $R=1$, $d=3$, $\epsilon = 1$ (Lyot coronagraph). \[figresu2\]](f3b.eps "fig:"){width=".9\textwidth"}\ ![Complex amplitude in $A$ and squared root of the amplitude in $D$, i.e. $|\mathcal{D}_n^m(r,\theta)|$. The parameters used in the simulation are: $\lambda f =1$, $R=1$, $d=3$, $\epsilon = 1$ (Lyot coronagraph). \[figresu2\]](f3c.eps "fig:"){width=".9\textwidth"}\ ![Bounds on the truncation error as a function of $r$. The parameters are the same as the parameters used for Figs. \[figresu1\] and \[figresu2\]. For each value of $N$, the bound is plot for the first 6 Zernike polynomials. \[plotbound\]](f4.eps){width=".8\textwidth"} ![$|\Psi_D(r,\theta)|$ for different values of $\beta$. The parameters are the same as the parameters used for Figs. \[figresu1\] and \[figresu2\]. \[plottilt\]](f5a.eps "fig:"){width=".45\textwidth"}![$|\Psi_D(r,\theta)|$ for different values of $\beta$. The parameters are the same as the parameters used for Figs. \[figresu1\] and \[figresu2\]. \[plottilt\]](f5b.eps "fig:"){width=".45\textwidth"}\ ![$|\Psi_D(r,\theta)|$ for different values of $\beta$. The parameters are the same as the parameters used for Figs. \[figresu1\] and \[figresu2\]. \[plottilt\]](f5c.eps "fig:"){width=".45\textwidth"}![$|\Psi_D(r,\theta)|$ for different values of $\beta$. The parameters are the same as the parameters used for Figs. \[figresu1\] and \[figresu2\]. \[plottilt\]](f5d.eps "fig:"){width=".45\textwidth"}

--- abstract: - 'Dans [@CT] la biquantification des paires symétriques a été étudié à l’aide du formalisme graphique de M. Kontsevich. Dans ce papier on corrige, compte tenu des résultats montrés récemment dans [@CFFR], une erreur mineure dans [@CT], qui en tout cas n’invalide pas les résultats plus importants dans [@CT]: cette erreur consiste au fait que les auteurs avaient oublié une contribution provenante d’une certaine composante du bord dans le propagateur à quatre couleurs. La correction qu’on apporte ici a l’avantage de remettre finalement en jeu la “translation quantique” des charactères qui apparaît dans la méthode des orbites, et qui était mystèrieusement absente dans [@CT]. En plus, on présente une comparaison détaillée des deux façons différentes de construire la biquantification, [*i.e.*]{} en utilisant ou le démi-plan de Poincaré ou le premier quadrant, ainsi qu’un approche plus conceptuel à la biquantification selon [@CFFR] et toutes les corrections dues des résultats dans [@CT] qu’il faut corriger à cause de la présence de la translation quantique. Finalement on reconsidère la construction de la triquantification dévéloppée dans la partie finale de [@CT] pour l’étude des charactères compte tenu du même problème du bord dans la biquantification.' - 'The biquantization of symmetric pairs was studied in [@CT] in terms of Kontsevich-like graphs. This paper, also in view of recent results in [@CFFR], amends a minor mistake that did not spoil the main results of the paper. The mistake consisted in ignoring a regular term in the boundary contribution of some propagators. On the other hand, its correction brings back the quantum shift, present in the approaches by the orbit method, that was otherwise puzzlingly missing. In addition a detailed comparison of the two, equivalent, ways of defining biquantization working on the upper half plane or on one quadrant is presented, as well as a more conceptual approach to biquantization and the due corrections of some results of [@CT] in view of the aforementioned correction by the quantum shift. Finally, we review the triquantization construction developed for the treatment of characters by taking into accounts the same boundary problem as for the biquantization.' address: - 'Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190 CH-8057 Zürich (Switzerland)' - 'MPIM Bonn, Vivatsgasse 7, 53111 Bonn (Germany)' - 'Université Denis-Diderot-Paris 7, UFR de mathmatiques, Site Chevaleret, Case 7012 , 75205 Paris cedex 13 (France)' author: - 'A. S. Cattaneo' - 'C. A. Rossi' - 'C. Torossian' title: Biquantization of symmetric pairs and the quantum shift --- [^1] Introduction {#s-0} ============ In [@CT] the biquantization of symmetric pairs was studied in terms of Kontsevich-like graphs. A puzzling result was the absence of the quantum shift, otherwise present in the treatments using the orbit method, by the natural character of the adjoint representation of $\mathfrak k$ on $\mathfrak g/\mathfrak k=\mathfrak p$, where $\mathfrak g=\mathfrak k\oplus\mathfrak p$ is the symmetric pair under consideration. It turns out that due to a (fortunately minor) mistake in [@CT] the quantum shift is actually there. Apart from this, the mistake does not spoil the other results of the paper. The whole construction of [@CT Section 1.6] relies on the 4-colored propagators introduced in [@CFb] for the Poisson sigma model with two branes. It was recently observed by G. Felder and the second author in the preparation of [@CFFR], that, unlike in Kontsevich [@K], the boundary contributions of the $4$-colored propagators on the first quadrant for the collapse of the two endpoints may have a regular term in addition to the usual singular one. The regular term turns out simply to be the differential of the angle of the position where the two points collapsed, measured with respect to the origin, up to a sign, which depends on the boundary conditions themselves (roughly speaking, if we consider the same boundary conditions on the two half-lines bounding the first quadrant, then the sign is positive, while, for different boundary conditions on the two half-lines, we have a negative sign). Recall that these propagators are constructed from the Euclidean propagator (the differential of the angle of the line joining the two points) by reflecting the second argument with respect to the two boundaries of the first quadrant (producing four distinct closed $1$-forms on the compactified configuration space of two points in the interior of the first quadrant) and then summing them up with signs; concretely, $$\omega^{\varepsilon_1,\varepsilon_2}(z_1,z_2)=\frac{1}{2\pi}\left[\mathrm d\ \mathrm{arg}(z_2-z_1)+\varepsilon_1\mathrm d\ \mathrm{arg}(z_2-\overline z_1)+\varepsilon_2 \mathrm d\ \mathrm{arg}(z_2+\overline z_1)+\varepsilon_1\varepsilon_2 \mathrm d\ \mathrm{arg}(z_2+z_1)\right],$$ where $\varepsilon_i=\pm$, $i=1,2$. The contribution where the second argument is reflected w.r.t. the origin (corresponding to the situation where the second argument is reflected w.r.t. both boundaries of the first quadrant) is responsible for the regular term in all four situations, see Figure 1. \ \ The presence of this regular term was mistakenly neglected in [@CT]. Its net effect is that more boundary contributions have to be taken into account and extra terms are needed for cancellation. It turns out [@CFFR] that it is enough to allow for the presence of short loops and to assign each of them the regular term. This also has the pleasant effect of restoring the quantum shift. Some by-products, in particular in [@CT Sections 4 and 5], have to be modified accordingly. Regular terms also appear in the 8-colored propagators introduced in [@CT Section 6] for the three brane case that is needed to show the independence from the choice of polarization. Also in this case the introduction of short loops, consistent with what was observed above, saves the game: we will review these aspects, as well as their relationship with the Harish-Chandra homomorphism. Notation and conventions {#s-1} ======================== We work over a ground field $\mathbb K$, which may be $\mathbb R$ or $\mathbb C$. We consider a finite-dimensional symmetric pair $\mathfrak g$ over $\mathbb K$, [*i.e.*]{} a Lie algebra $\mathfrak g$, endowed with a Lie algebra automorphism $\sigma$, which is additionally an involution. In particular, $\mathfrak g=\mathfrak k\oplus\mathfrak p$, where $\mathfrak k$, resp. $\mathfrak p$, is the eigenspace w.r.t. the eigenvalue $+1$, resp. $-1$, of $\sigma$. For a Lie subalgebra $\mathfrak h$ of a Lie algebra $\mathfrak g$, we denote by $\mathfrak h^\perp$ its annihilator. As $\mathfrak k$ is a Lie subalgebra of $\mathfrak g$ and $\mathfrak g/\mathfrak k=\mathfrak p$ is a $\mathfrak k$-module, we introduce the short-hand notation $$\delta(\bullet)=\frac{1}2\mathrm{tr}_{\mathfrak p}(\mathrm{ad}_\mathfrak k(\bullet)),$$ see [*e.g.*]{} also [@T1; @T2]. Biquantization in the framework of the $2$-brane formality {#s-2} ========================================================== We consider a finite-dimensional Lie algebra $\mathfrak g$ over $\mathbb K$; further, we consider two Lie subalgebras $\mathfrak h_i$, $i=1,2$. To these data, we associate a Poisson manifold $X$ and two coisotropic submanifolds $U_i$, $i=1,2$, as follows: we set $X=\mathfrak g^*$, endowed with the linear Kirillov–Kostant–Souriau Poisson bivector $\pi$, and $U_i=\mathfrak h_i^\perp$. We want to regard biquantization as analyzed in [@CT] in the more general framework of the $2$-brane Formality Theorem [@CFFR Theorem 7.2]: thus, before entering into the details of biquantization, we need to review in some detail the main result of [@CFFR] and draw a bridge between it and the computations in [@CT]. On compactified configuration spaces {#ss-2-1} ------------------------------------ For the upcoming discussion of the $4$-colored propagators, we need to fix certain issues regarding compactified configuration spaces: in particular, we discuss two types of compactified configuration spaces, which arise in the context of biquantization, and we prove that they are in fact diffeomorphic. We observe that the following discussion may be viewed as an extension of certain computations in [@CFFR]. ### The compactified configuration space of points in $\mathbb H^+\sqcup \mathbb R$ {#sss-2-1-1} For two non-negative integers $m$, $n$, we consider the (open) configuration space $C_{n,m}^+$ of $n$ distinct points in the complex upper half-plane $\mathbb H^+$ and $m$ ordered points on the real axis $\mathbb R$. Its precise definition is $$C_{n,m}^+=\left\{(z_1,\dots,z_n,x_1,\dots,x_m)\in (\mathbb H^+)^n\times \mathbb R^m:\ z_i\neq z_j,\ i\neq j,\ x_1<\cdots<x_m\right\}/G_2,$$ where $G_2$ is the two-dimensional real Lie group $\mathbb R^+\ltimes \mathbb R$, acting on $\mathbb H^+\sqcup \mathbb R$ by rescalings and real translations. Provided $2n+m-2\geq 0$, $C_{n,m}^+$ is a smooth manifold of dimension $2n+m-2$; it is obviously oriented. We further consider the open configuration space $$C_n=\left\{(z_1,\dots,z_n)\in \mathbb C^n:\ z_i\neq z_j,\ i\neq j\right\}/G_3,$$ where $G_3$ is the $3$-dimensional real Lie group $\mathbb R^+\ltimes \mathbb C$, acting on $\mathbb C$ by rescalings and complex translations. It is obvious that, provided $2n-3\geq 0$, $C_n$ is a smooth manifold of dimension $2n-3$, which admits an obvious orientation from $\mathbb C^n$ and from the obvious orientability of $G_3$. Kontsevich [@K Subsection 5.1] provides compactifications $\mathcal C_{n,m}^+$ and $\mathcal C_n$ of $C_{n,m}^+$ and $C_n$ respectively in the sense of Fulton–MacPherson [@FMcP] (to be more precise, the smooth version of the algebraic compactification of [@FMcP], exploited in detail by Axelrod–Singer [@AS]): both compactified configuration spaces admit structures of smooth manifolds with corners ([*i.e.*]{} locally modeled on $(\mathbb R^+)^k\times\mathbb R^l$), and as such they admit boundary stratifications. We observe that the permutation group $\mathfrak S_n$ acts naturally on $C_n$, and it can be proved that its action extends to $\mathcal C_n$: in particular, we may consider more general compactified configuration spaces $\mathcal C_A$, for a finite subset of $\mathbb N$. Because of similar reasons, we may consider compactified configuration spaces $\mathcal C_{A,B}^+$, for a finite subset $A$ and a finite, ordered subset $B$ of $\mathbb N$. The stratifications of $\mathcal C_{n,m}^+$ and $\mathcal C_n$ admit a beautiful description in terms of trees [@K Subsection 5.1]. We first consider the boundary stratification of $\mathcal C_n$: for simplicity, we illustrate the boundary strata of codimension $1$, namely such boundary strata are labeled by subsets $A$ of $[n]=\{1,\dots,n\}$ of cardinality $2\leq |A|\leq n$, $$\partial_A \mathcal C_n\cong \mathcal C_A\times\mathcal C_{([n]\smallsetminus A)\sqcup \{\bullet\}},$$ where the first, resp. second, factor on the right-hand side of the previous identification represents the configuration of distinct points in $\mathbb C$ labeled by $A$ which collapse together in $\mathbb C$ to a single point $\bullet$, resp. the final configuration of points after the collapse. The boundary strata of codimension $1$ of $\mathcal C_{n,m}^+$ are of two types, namely, - there exists a subset $A$ of $[n]$, of cardinality $2\leq |A|\leq n$, such that $$\partial_A \mathcal C_{n,m}^+\cong \mathcal C_A\times \mathcal C_{([n]\smallsetminus A)\sqcup \{\bullet\},m}^+,$$ where the first, resp. second, factor on the right-hand side of the previous identification describes the collapse of the points in $\mathbb H^+$ labeled by $A$ to a single point $\bullet$ in $\mathbb H^+$, resp. the final configuration of points after the collapse; - there exist a subset $A$ of $[n]$ and an ordered subset of $[m]$ consisting of consecutive non-negative integers, such that $0\leq |A|\leq n$, $0\leq |B|\leq m$, $1\leq |A|+|B|\leq n+m-1$, for which we have $$\partial_{A,B}\mathcal C_{n,m}^+\cong \mathcal C_{A,B}^+\times \mathcal C_{[n]\smallsetminus A,([m]\smallsetminus B)\sqcup \{\bullet\}}^+,$$ where the first, resp. second, factor on the right-hand side of the previous identification describes the collapse of the points in $\mathbb H^+$ labeled by $A$ and the consecutive, ordered points on $\mathbb R$ labeled by $B$ to a single point $\bullet$ in $\mathbb R$, resp. the final configuration of points after the collapse. ### The compactified configuration space of points in $Q^{+,+}\sqcup i\mathbb R^+\sqcup \mathbb R^+$ {#sss-2-1-2} For three non-negative integers $l$, $m$ and $n$, we consider the (open) configuration space $C_{n,k,l}^+$ of $n$ distinct points in the first quadrant $Q^{+,+}$, $k$ ordered points on the positive imaginary axis $i\mathbb R^+$ and $l$ ordered points on the positive real axis $\mathbb R$. We observe that the order of the points on the positive imaginary axis is the opposite of the intuitive one, [*i.e.*]{} $i x\leq i y$ if and only if $y\leq x$, for $x$, $y$ in $\mathbb R$. The precise definition of $C_{n,k,l}^+$ is $$\begin{aligned} C_{n,k,l}^+=&\left\{(z_1,\dots,z_n,ix_1,\dots,ix_k,y_1,\dots,y_l)\in (Q^{+,+})^n\times (i\mathbb R^+)^k\times (\mathbb R^+)^l:\ z_i\neq z_j,\ i\neq j,\right.\\ &\left.\ x_k<\cdots<x_1,\ y_1<\cdots< y_l\right\}/G_1, \end{aligned}$$ where $G_1=\mathbb R^+$ acts on $Q^{+,+}\sqcup i\mathbb R^+\sqcup \mathbb R^+$ by rescalings. Provided $2n+k+l-1\geq 0$, $C_{n,k,l}^+$ is a smooth manifold of dimension $2n+k+l-1$. It inherits an obvious orientation from the natural orientation of $(Q^{+,+})^n\times (i\mathbb R^+)^k\times(\mathbb R^+)^l$ and the one of $\mathbb R^+$. We may provide a compactification $\mathcal C_{n,k,l}^+$ in the sense of Fulton–MacPherson [@FMcP] of $C_{n,k,l}^+$ in a way similar to Kontsevich: the compactified configuration space $\mathcal C_{n,k,l}^+$ admits a structure of smooth manifold with corners. We now consider the boundary strata of $\mathcal C_{n,k,l}^+$ of codimension $1$, which are of the three following types: - there exists a subset $A$ of $[n]$, of cardinality $2\leq |A|\leq n$, such that $$\partial_A \mathcal C_{n,k,l}^+\cong \mathcal C_A\times \mathcal C_{([n]\smallsetminus A)\sqcup \{\bullet\},k,l}^+,$$ where the first, resp. second, factor on the right-hand side of the previous identification describes the collapse of the points in $Q^{+,+}$ labeled by $A$ to a single point $\bullet$ in $Q^{+,+}$, resp. the final configuration of points after the collapse; - there exist a subset $A$ of $[n]$ and an ordered subset of $[k]$, resp. $[l]$, consisting of consecutive non-negative integers, such that $0\leq |A|\leq n$, $0\leq |B|\leq k$, resp. $0\leq |B|\leq l$, for which we have $$\partial_{A,B}\mathcal C_{n,k,l}^+\cong \mathcal C_{A,B}^+\times \mathcal C_{[n]\smallsetminus A,([k]\smallsetminus B)\sqcup \{\bullet\},l}^+,\ \text{resp.}\ \partial_{A,B}\mathcal C_{n,k,l}^+\cong \mathcal C_{A,B}^+\times \mathcal C_{[n]\smallsetminus A,k,([l]\smallsetminus B)\sqcup \{\bullet\}}^+$$ where the first, resp. second, factor on the right-hand side of the previous identification describes the collapse of the points in $Q^{+,+}$ labeled by $A$ and the consecutive, ordered points on $i\mathbb R^+$ or $\mathbb R^+$ labeled by $B$ to a single point $\bullet$ in $i\mathbb R^+$ or $\mathbb R^+$, resp. the final configuration of points after the collapse. - there exist a subset $A$ of $[n]$ and an ordered subset $B=B_1\sqcup B_2$ of $[k]\sqcup [l]$, for which $B_1$ and $B_2$ are ordered subsets of consecutive points in $[k]$ and $[l]$ respectively, such that $k\in B_1$ if $B_1\neq \emptyset$, $1\in B_2$ if $B_2\neq \emptyset$, $0\leq |A|\leq n$, $0\leq |B|\leq k+l$, resp. $1\leq |A|+|B|\leq n+k+l-1$, for which we have $$\partial_{A,B_1,B_2}\mathcal C_{n,k,l}^+\cong \mathcal C_{A,B_1,B_2}^+\times \mathcal C_{[n]\smallsetminus A,[k]\smallsetminus (B\cap[k]),[l]\smallsetminus (B\cap [l])}^+,$$ where the first, resp. second, factor on the right-hand side of the previous identification describes the collapse of the points in $Q^{+,+}$ labeled by $A$ and the consecutive, ordered points on $i\mathbb R^+$ and $\mathbb R^+$ labeled by $B=B_1\sqcup B_2$ to the origin of the axes, resp. the final configuration after the collapse. ### The relationship between $\mathcal C_{n,m}^+$ and $\mathcal C_{n,k,l}^+$ {#sss-2-1-3} First of all, we consider the open configuration spaces $C_{n,m}^+$ and $C_{n,k,l}^+$, where $m=k+l+1$. We observe that the holomorphic function $z\mapsto z^2$ on $\mathbb C$, when restricted to $Q^{+,+}\sqcup i\mathbb R^+\sqcup \mathbb R^+$, gives rise to a biholomorphism to $\mathbb H^+\sqcup (\mathbb R\smallsetminus \{0\})$, whose inverse we denote by $z\mapsto \sqrt z$: in fact, we have to choose a well-suited branch-cut for the complex square root, [*e.g.*]{} we cut out from the complex plane the negative imaginary axis plus the origin. For $m$, $k$ and $l$ as before, we choose the $k+1$-st point on $\mathbb R$. Then, there is an obvious map from $C_{n,m}^+$ to $C_{n,k,l}^+$, which is defined by the following explicit formula: $$\label{eq-square} \begin{aligned} &C_{n,m}^+\ni [(z_1,\dots,z_n,x_1,\dots,x_{k+1},\dots,x_m)]\mapsto\\ &\mapsto\left[\left(\sqrt{z_1-x_{k+1}},\dots,\sqrt{z_n-x_{k+1}},\sqrt{x_1-x_{k+1}},\dots,\sqrt{x_k-x_{k+1}},\sqrt{x_{k+1}-x_{k+1}},\dots,\sqrt{x_m-x_{k+1}}\right)\right]\in C_{n,k,l}^+. \end{aligned}$$ First of all, we observe that, because of the order on the points on the real axis, the difference $x_i-x_{k+1}$ is strictly negative, resp. positive, if $1\leq i\leq k$, resp. $k+2\leq i\leq m$: thus, because of the said choice of a complex square root, $\sqrt{x_i-x_{k+1}}=i\sqrt{x_{k+1}-x_i}$, if $1\leq i\leq k$, or $\sqrt{x_i-x_{k+1}}=\sqrt{x_i-x_{k+1}}$, if $k+2\leq i\leq m$, where now both square roots on the right-hand side of both equalities are real, positive numbers. Again, the order on the points $x_i$, $1\leq i\leq k$ implies that $\sqrt{x_{k+1}-x_i}>\sqrt{x_{k+1}-x_{i+1}}$, therefore, the natural order on $x_i$, $1\leq i\leq k$, is mapped to the natural order on $i\sqrt{x_{k+1}-x_i}$ discussed in Subsubsection \[sss-2-1-2\]. We may depict the morphism  graphically [*via*]{} \ \ An easy computation proves that the above morphism is well-defined, [*i.e.*]{} it does not depend on the choice of representatives; furthermore, the morphism is obviously smooth, and is in fact a diffeomorphism, whose inverse is $$C_{n,k,l}^+\ni [(z_1,\dots,z_n,ix_1,\dots,i x_k,y_1,\dots,y_l)]\mapsto\left[\left(z_1^2,\dots,z_n^2,-x_1^2,\dots,-x_k^2,0,y_1^2,\dots,y_l^2\right)\right]\in C_{n,m}^+.$$ The important point is that the complex square function and the chosen inverse (the above complex square root) extend to smooth functions between the compactified configuration spaces $\mathcal C_{n,m}^+$ and $\mathcal C_{n,k,l}^+$. \[p-square\] For non-negative integers $n$, $m$, $k$, $l$, such that $m=k+l+1$, the smooth manifolds with corners $\mathcal C_{n,m}^+$ and $\mathcal C_{n,k,l}^+$ are diffeomorphic [*via*]{} the choice of a complex square root with branch cut $i\mathbb R^-\sqcup \{0\}$. We prove that the diffeomorphism  extends to a diffeomorphism on the compactified configuration spaces by computing its expression w.r.t. local coordinates for the relevant boundary strata of codimension $1$. In fact, as sketched in [@K Subsection 5.2], the boundary strata of higher codimension correspond to products with more than two factors of compactified configuration spaces of the same kind, representing configuration of points collapsing together, be it in the complex upper half-plane or on the real axis, resp. in the first quadrant or on the positive complex or real axis or on the origin. It suffices therefore to prove the claim on the interior of the boundary strata of codimension $1$ of $\mathcal C_{n,m}^+$ and $\mathcal C_{n,k,l}^+$: these have been characterized explicitly in Subsubsections \[sss-2-1-1\] and \[sss-2-1-2\]. Furthermore, without loss of generality, we may assume $A=[i]$ and $B=[j]$. We have to prove that the map  maps diffeomorphically the interior of boundary strata of codimension $1$ of $\mathcal C_{n,m}^+$ to the interior of boundary strata of codimension $1$ of $\mathcal C_{n,k,l}^+$. We consider first the boundary stratum of type $i)$ of $\mathcal C_{n,m}^+$ labeled by $A=[i]$, for $2\leq i\leq n$. Local coordinates of the interior $C_i\times \mathcal C_{n-i+1,m}^+$ are provided by $$C_i\times C_{n-i+1,m}^+\ni \left(\left(e^{i\varphi},z_1,\dots,z_{i-2}\right),\left(e^{it},w_1,\dots,w_{n-i},x_1,\dots,x_k,0,x_{k+2},\dots,x_m\right)\right),$$ where $\varphi$ is in $[0,2\pi)$, $t$ in $(0,\pi)$, $z_i$ in $\mathbb C$, $w_i$ in $\mathbb H^+$, and all points in $\mathbb C$ and $\mathbb H^+$ are distinct, while the points on the real axis are lexicographically (strictly) ordered. On the other hand, the interior of the boundary stratum $\mathcal C_i\times\mathcal C_{n-i+1,k,l}^+$ is described by the following local coordinates: $$C_i\times C_{n-i+1,k,l}^+\ni \left(\left(e^{i\varphi},z_1,\dots,z_{i-2}\right),\left(e^{it},w_1,\dots,w_{n-i},i x_1,\dots,i x_k,y_1,\dots,y_l\right)\right),$$ where $\varphi$ is in $[0,2\pi)$, $t$ in $(0,\frac{\pi}2)$, $z_i$ in $\mathbb C$, $w_i$ in $Q^{+,+}$, and all points in $\mathbb C$ and $Q^{+,+}$ are distinct, and $x_1>\cdots x_k>0$, $0<y_1<\cdots<y_l$. For $\varepsilon>0$ sufficiently small, local coordinates for $\mathcal C_{n,m}^+$, resp. $\mathcal C_{n,k,l}^+$, near the interior of the boundary stratum $\mathcal C_i\times\mathcal C_{n-i,m}^+$, resp. $\mathcal C_i\times\mathcal C_{n-i+1,k,l}^+$, are given by $$\begin{aligned} &\left[\left(e^{it},e^{it}+\varepsilon e^{i\varphi},e^{it}+\varepsilon z_1,\dots,e^{it}+\varepsilon z_{i-2},w_1,\dots,w_n,x_1,\dots,x_k,0,x_{k+2},\dots,x_m\right)\right],\ \text{resp.}\\ &\left[\left(e^{it},e^{it}+\varepsilon e^{i\varphi},e^{it}+\varepsilon z_1,\dots,e^{it}+\varepsilon z_{i-2},w_1,\dots,w_n,ix_1,\dots,i x_k,y_1,\dots,y_l\right)\right]. \end{aligned}$$ We apply the morphism  to the first of the previous expressions, getting $$\label{eq-sq-1} \left[\left(\sqrt{e^{it}},\sqrt{e^{it}+\varepsilon e^{i\varphi}},\sqrt{e^{it}+\varepsilon z_1},\dots,\sqrt{e^{it}+\varepsilon z_{i-2}},\sqrt{w_1},\dots,\sqrt{w_n},i\sqrt{-x_1},\dots,i\sqrt{-x_k},\sqrt{x_{k+2}},\dots,\sqrt{x_m}\right)\right].$$ We rewrite the terms in the previous expression containing the infinitesimal parameter $\varepsilon$ using the fact that the chosen complex square root is holomorphic on $\mathbb H^+$, thus getting $$\sqrt{e^{it}+\varepsilon z}=e^{i\frac{t}2}+\frac{\varepsilon z}{2 e^{i\frac{t}2}}+\mathcal O(\varepsilon^2)=e^{i\frac{t}2}+\frac{\varepsilon}2 e^{-i\frac{t}2}z+\mathcal O(\varepsilon^2),\ t\in (0,\pi),\ z\in \mathbb C.$$ To compare expressions, we may neglect terms in the expansion of order strictly higher than $2$: rescaling by $\frac{1}2$ the infinitesimal parameter $\varepsilon$, Expression  can be rewritten as $$\left[\left(e^{i\frac{t}2},e^{i\frac{t}2}+\varepsilon e^{i\left(\varphi-\frac{t}2\right)},e^{i\frac{t}2}+\varepsilon e^{-i\frac{t}2}z_1,\dots,e^{i\frac{t}2}+\varepsilon e^{-i\frac{t}2}z_{i-2},\sqrt{w_1},\dots,\sqrt{w_n},i\sqrt{-x_1},\dots,i\sqrt{-x_k},\sqrt{x_{k+2}},\dots,\sqrt{x_m}\right)\right],$$ whence it follows immediately that the morphism  maps the interior of $\mathcal C_i\times\mathcal C_{n-i+1,m}^+$ diffeomorphically to the interior of $\mathcal C_i\times \mathcal C_{n-i+1,k,l}^+$, where the diffeomorphism is explicitly the product of the morphism  from $C_{n-i+1,m}^+$ to $C_{n-i+1,k,l}^+$ with the obvious diffeomorphism of $C_i$ given by $$C_i\ni \left[\left(z_1,\dots,z_i\right)\right]\mapsto \left[\left(\frac{z_1}{\sqrt{w_1-x_{k+1}}},\dots,\frac{z_i}{\sqrt{w_1-x_{k+1}}}\right)\right]\in C_i,$$ where $w_1$ and $x_{k+1}$ are taken from $C_{n-i+1,m}^+$. We now consider the interior of the boundary stratum $\mathcal C_{i,B}^+\times \mathcal C_{n-i,([m]\smallsetminus B)\sqcup \{\bullet\}}^+$, where $B$ is an ordered subset of $[m]$ consisting of consecutive elements, and we assume that $1\leq i+|B|\leq n+m-1$. We have to further distinguish between two situations: $|B|=0$ (and consequently $1\leq i\leq n$), and $|B|\neq 0$. We consider the situation $|B|=0$, and we further distinguish between the case, where the new point $\bullet$ on the real axis (corresponding to the cluster of points labeled by $[i]$ in $\mathbb H^+$ approach $\mathbb R$) lies on the left or on the right of the distinguished point $x_{k+1}$. We do the explicit computations only in the case, where $\bullet$ is on the left of $x_{k+1}$, leaving the other case to the reader. If $\bullet$ lies on the left of $x_{k+1}$, we may safely assume that $\bullet=x$ lies on the left of $x_1$: then, local coordinates for the interior of $\mathcal C_{i,0}^+\times \mathcal C_{n-i,m+1}^+$ are given by $$C_{i,0}^+\times C_{n-i,m+1}^+\ni \left(\left(i,z_1,\dots,z_{i-1}\right),\left(e^{it},w_1,\dots,w_{n-i-1},x,x_1,\dots,x_k,0,x_{k+2},\dots,x_m\right)\right),$$ where $t$ in $(0,\pi)$, $z_i$ and $w_j$ are in $\mathbb H^+$, and all points in $\mathbb H^+$ are distinct, while the points on the real axis are lexicographically (strictly) ordered. Similarly, local coordinates for the interior of $\mathcal C_{i,0}^+\times \mathcal C_{n-i,m+1}^+$ are given by $$C_{i,0}^+\times C_{n-i,k+1,l}^+\ni \left(\left(i,z_1,\dots,z_{i-1}\right),\left(e^{it},w_1,\dots,w_{n-i-1},ix,ix_1,\dots,ix_k,y_1,\dots,y_l\right)\right),$$ where $t$ in $(0,\frac{\pi}2)$, $z_i$ in $\mathbb H^+$ and $w_i$ in $Q^{+,+}$, all points in $\mathbb H^+$ and $Q^{+,+}$ are distinct, $x>x_1>\cdots>x_k>0$ and $0<y_1<\cdots<y_l$. For $\varepsilon>0$ sufficiently small, local coordinates for $\mathcal C_{n,m}^+$, resp. $\mathcal C_{n,k,l}^+$, near the interior of the boundary stratum $\mathcal C_{i,0}^+\times\mathcal C_{n-i,m+1}^+$, resp. $\mathcal C_{i,0}^+\times\mathcal C_{n-i,k+1,l}^+$, are given by $$\begin{aligned} &\left[\left(x+\varepsilon i,x+\varepsilon z_1,\dots,x+\varepsilon z_{i-1},e^{it},w_1,\dots,w_{n-i-1},x_1,\dots,x_k,0,x_{k+2},\dots,x_m\right)\right],\ \text{resp.}\\ &\left[\left(ix+\varepsilon,ix-i\varepsilon z_1,\dots,ix-i\varepsilon z_{i-1},e^{it},w_1,\dots,w_{n-i-1},ix_1,\dots,ix_k,y_1,\dots,y_l\right)\right] \end{aligned}$$ The image of the first expression w.r.t. the morphism  is simply $$\label{eq-sq-2} \left[\left(\sqrt{x+\varepsilon i},\sqrt{x+\varepsilon z_1},\dots,\sqrt{x+\varepsilon z_{i-1}},\sqrt{e^{it}},\sqrt{w_1},\dots,\sqrt{w_{n-i-1}},i\sqrt{-x_1},\dots,i\sqrt{-x_k},\sqrt{x_{k+2}},\dots,\sqrt{x_m}\right)\right].$$ We consider the first $i$ entries in the previous expression: once again, using the holomorphy of the chosen complex square root, and recalling that $x<x_1<0$ and that $\varepsilon$ is chosen sufficiently small, we find $$\begin{aligned} \sqrt{x+\varepsilon i}&=\sqrt{x}+\frac{\varepsilon i}{2\sqrt x}+\mathcal O(\varepsilon^2)=i\sqrt{-x}+\frac{\varepsilon}{2\sqrt{-x}}+\mathcal O(\varepsilon^2),\\ \sqrt{x+\varepsilon z_j}&=i\sqrt{-x}-\frac{i\varepsilon}2 \frac{z_j}{\sqrt{-x}}+\mathcal O(\varepsilon^2),\ 1\leq j\leq i-1. \end{aligned}$$ Once again, rescaling by $\frac{1}2$ the infinitesimal parameter $\varepsilon$, and neglecting terms of order higher than $1$ w.r.t. $\varepsilon$ in the above expressions, we may rewrite Expression  as $$\left[\left(i\sqrt{-x}+\varepsilon,i\sqrt{-x}+\varepsilon \frac{z_1}{\sqrt{-x}},\dots,i\sqrt{-x}+\varepsilon\frac{z_{i-1}}{\sqrt{-x}},e^{i\frac{t}2},\sqrt{w_1},\dots,\sqrt{w_{n-i-1}},i\sqrt{-x_1},\dots,i\sqrt{-x_k},\sqrt{x_{k+2}},\dots,\sqrt{x_m}\right)\right],$$ and it is easy to see that the morphism  maps $C_{i,0}^+\times C_{n-i,m+1}^+$ diffeomorphically to $C_{i,0}^+\times C_{n-i,k+1,l}^+$, and the induced morphism is precisely given by the product of the morphism  from $C_{n-i,m+1}^+$ to $C_{n-i,k+1,l}^+$ with the obvious diffeomorphism of $C_{i,0}^+$ given by $$C_{i,0}^+\ni \left[\left(z_1,\dots,z_i\right)\right]\mapsto \left[\left(\frac{z_1}{\sqrt{x_{k+1}-x}},\dots,\frac{z_i}{\sqrt{x_{k+1}-x}}\right)\right]\in C_{i,0}^+,$$ where $x$ denotes the first point on the real axis in lexicographical order, and $x_{k+1}$ is the special point on real axis. For the situation $|B|\neq 0$, we need to distinguish between two cases, namely $i)$ $B$ contains $k+1$ or $ii)$ $b$ does not contain $k+1$ (in which case, either the minimum of $B$ is greater or equal than $k+2$ or the maximum of $B$ is less or equal than $k$). We first consider the case, where $B$ contains $k+1$, and we assume $A=[i]$ and $B=[p,q]=\{p,\dots,q\}$, where $1\leq p\leq k+1\leq q\leq m$; we further write $B=\{k\}\sqcup B_1\sqcup B_2$, where $B_1=[p,k]$ and $B_2=[k+2,q]$ (of course, $B_1$ and/or $B_2$ may be empty). The interior of the corresponding boundary stratum of $C_{n,m}^+$, resp. $C_{n,k,l}^+$, is $C_{i,B}^+\times C_{n-i,([m]\smallsetminus B)\sqcup \{\bullet\}}^+$, resp. $C_{i,B_1,B_2}^+\times C_{n-i,[k]\smallsetminus B_1,[l]\smallsetminus B_2}^+$, and corresponding local coordinates are given by $$\begin{aligned} C_{i,B}^+\times C_{n-i,([m]\smallsetminus B)\sqcup \{\bullet\}}^+\ni &\left(\left(e^{it_1},z_1,\dots,z_{i-1},x_1,\dots,x_{k-p+1},0,x_{k-p+3},\dots,x_{q-p+1}\right),\right.\\ &\left.\left(e^{it_2},w_1,\dots,w_{n-i-1},x_1',\dots,x_{p-1}',0,x_{p+1}',\dots,x_{m-2k-q}'\right)\right),\\ C_{i,B_1,B_2}^+\times C_{n-i,[k]\smallsetminus B_1,[l]\smallsetminus B_2}^+\ni &\left(\left(e^{it_1},z_1,\dots,z_{i-1},i x_1,\dots,i x_{k-p+1},y_1,\dots,y_{q-k-1}\right),\right.\\ &\left.\left(e^{it_2},w_1,\dots,w_{n-i-1},i x_1',\dots,i x_{p-1}',y_1',\dots,y_{l-q+k+1}'\right)\right), \end{aligned}$$ where $t_i$, $i=1,2$, is in $(0,\pi)$, resp. $(0,\frac{\pi}2)$, all points in $\mathbb H^+$, resp. $Q^{+,+}$, are distinct in the first, resp. second, expression. In the first, resp. second, expression, the $x_i$ and $x_i'$ are lexicographically ordered, resp. $x_1>\cdots x_{k-p+1}>0$, $x_1'>\cdots x_{p-1}'>0$, $0<y_1<\cdots y_{q-k-1}$ and $0<y_1'<\cdots< y'_{l-q+k+1}$. Choosing a positive number $\varepsilon$ sufficiently small as before, we may write local coordinates of $\mathcal C_{n,m}^+$, resp. $\mathcal C_{n,k,l}^+$, near the interior of the boundary stratum $\mathcal C_{i,B}^+\times\mathcal C_{n-i,([m]\smallsetminus B)\sqcup \{\bullet\}}^+$, resp. $\mathcal C_{i,B_1,B_2}^+\times \mathcal C_{n-i,[k]\smallsetminus B_1,[l]\smallsetminus B_2}^+$, namely $$\begin{aligned} &\left[\left(\varepsilon e^{i t_1},\varepsilon z_1,\dots,\varepsilon z_{i-1},e^{it_2},w_1,\dots,w_{n-i-1},\right.\right.\\ &\left.\left.x_1',\dots,x_{p-1}',\varepsilon x_1,\dots,\varepsilon x_{k-p+1},0,\varepsilon x_{k-p+3},\dots,\varepsilon x_{q-p+1},x_{p+1}',\dots,x_{m-2k-q}'\right)\right],\ \text{resp.}\\ &\left[\left(\varepsilon e^{i t_1},\varepsilon z_1,\dots,\varepsilon z_{i-1},e^{i t_2},w_1,\dots,w_{n-i-1},\right.\right.\\ &\left.\left.ix_1',\dots,ix_{p-1}',\varepsilon ix_1,\dots,\varepsilon ix_{k-p+1},0,\varepsilon y_1,\dots,\varepsilon y_{q-k+1},y_1',\dots,y'_{l-q+k+1}\right)\right]. \end{aligned}$$ We now apply the morphism  to the first of the two previous expressions, getting $$\begin{aligned} &\left[\left(\sqrt{\varepsilon e^{i t_1}},\sqrt{\varepsilon z_1},\dots,\sqrt{\varepsilon z_{i-1}},\sqrt{e^{it_2}},\sqrt{w_1},\dots,\sqrt{w_{n-i-1}},\right.\right.\\ &\left.\left.i\sqrt{-x_1'},\dots,i\sqrt{-x_{p-1}'},i\sqrt{-\varepsilon x_1},\dots,i\sqrt{-\varepsilon x_{k-p+1}},\sqrt{\varepsilon x_{k-p+3}},\dots,\sqrt{\varepsilon x_{q-p+1}},\sqrt{x_{p+1}'},\dots,\sqrt{x_{m-2k-q}'}\right)\right]=\\ &\left[\left(\sqrt{\varepsilon} e^{i \frac{t_1}2},\sqrt{\varepsilon}\sqrt{z_1},\dots,\sqrt{\varepsilon}\sqrt{z_{i-1}},e^{i\frac{t_2}2},\sqrt{w_1},\dots,\sqrt{w_{n-i-1}},\right.\right.\\ &\left.\left.i\sqrt{-x_1'},\dots,i\sqrt{-x_{p-1}'},\sqrt{\varepsilon}i\sqrt{-x_1},\dots,\sqrt{\varepsilon}i\sqrt{-x_{k-p+1}},\sqrt{\varepsilon}\sqrt{x_{k-p+3}},\dots,\sqrt{\varepsilon}\sqrt{x_{q-p+1}},\sqrt{x_{p+1}'},\dots,\sqrt{x_{m-2k-q}'}\right)\right], \end{aligned}$$ from which we read immediately that the morphism  maps diffeomorphically $C_{i,B}^+\times C_{n-i,([m]\smallsetminus B)\sqcup\{\bullet\}}^+$ to $C_{i,B_1,B_2}^+\times C_{n-i,[k]\smallsetminus B_1,[l]\smallsetminus B_2}^+$. Finally, we consider the case $|B|\neq 0$, such that the maximum of $B=[j]$ with $j\leq k$. The interior of the corresponding boundary stratum of $C_{n,m}^+$, resp. $C_{n,k,l}^+$, is $C_{i,j}^+\times C_{n-i,m-j+1}^+$, resp. $C_{i,j}^+\times C_{n-i,k-j+1,l}^+$, and corresponding local coordinates are given by $$\begin{aligned} C_{i,j}^+\times C_{n-i,m-j+1}^+\ni &\left(\left(i,z_1,\dots,z_{i-1},x_1,\dots,x_j\right),\left(e^{it},w_1,\dots,w_{n-i-1},x_1',\dots,x_{k-j+1}',0,x_{k-j+2}',\dots,x_{m-j+1}'\right)\right),\ \text{resp.}\\ C_{i,j}^+\times C_{n-i,k-j+1,l}^+\ni &\left(\left(i,z_1,\dots,z_{i-1},x_1,\dots,x_j\right),\left(e^{it},w_1,\dots,w_{n-i-1},i x_1',\dots,i x_{k-j+1}',y_1',\dots,y_l'\right)\right), \end{aligned}$$ where $t$ is in $(0,\pi)$, resp. $(0,\frac{\pi}2)$, all points in $\mathbb H^+$ and $Q^{+,+}$ are distinct in both expressions. In the first, resp. second, expression, the $x_i$ and $x_i'$ are lexicographically ordered, resp. $x_1'>\cdots x_{k-j+1}'>0$ and $0<y_1'<\cdots< y'_l$. Choosing a positive number $\varepsilon$ sufficiently small, we now write local coordinates of $\mathcal C_{n,m}^+$, resp. $\mathcal C_{n,k,l}^+$, near the interior of the boundary stratum $\mathcal C_{i,j}^+\times\mathcal C_{n-i,m-j+1}^+$, resp. $\mathcal C_{i,j}^+\times \mathcal C_{n-i,k-j+1,l}^+$, namely $$\begin{aligned} &\left[\left(x_1'+\varepsilon i,x_1'+\varepsilon z_1,\dots,x_1'+\varepsilon z_{i-1},e^{it},w_1,\dots,w_{n-i-1},x_1',x_1'+\varepsilon x_1,\dots,x_1'+\varepsilon x_j,x_2',\dots,x_{k-j+1}',0,\right.\right.\\ &\left.\left. x_{k-j+2}',\dots,x_{m-j+1}'\right)\right],\ \text{resp.}\\ &\left[\left(ix_1'+\varepsilon,ix_1'-i\varepsilon z_1,\dots,ix_1'-i\varepsilon z_{i-1},e^{it},w_1,\dots,w_{n-i-1},ix_1',ix_1-i'\varepsilon x_1,\dots,ix_1'-i\varepsilon x_j,ix_2',\dots,ix_{k-j+1}',y_1',\dots,y_l'\right)\right]. \end{aligned}$$ If we apply the morphism  to the first of the two previous expressions, we get $$\begin{aligned} &\left[\left(\sqrt{x_1'+\varepsilon i},\sqrt{x_1'+\varepsilon z_1},\dots,\sqrt{x_1'+\varepsilon z_{i-1}},\sqrt{e^{it}},\sqrt{w_1},\dots,\sqrt{w_{n-i-1}},\right.\right.\\ &\left.\left.\sqrt{x_1'},\sqrt{x_1'+\varepsilon x_1},\dots,\sqrt{x_1'+\varepsilon x_j},\sqrt{x_2'},\dots,\sqrt{x_{k-j+1}'},\sqrt{x_{k-j+2}'},\dots,\sqrt{x_{m-j+1}'}\right)\right]. \end{aligned}$$ Once again, we find $$\begin{aligned} \sqrt{x_1'+\varepsilon i}&=i\sqrt{-x_1'}+\frac{\varepsilon}2\frac{1}{\sqrt{-x_1'}}+\mathcal O(\varepsilon^2),\ &\ \sqrt{x_1'+\varepsilon iz_e}&=i\sqrt{-x_1'}-i\frac{\varepsilon}2\frac{z_e}{\sqrt{-x_1'}}+\mathcal O(\varepsilon^2),\\ \sqrt{x_1'+\varepsilon x_e}&=i\sqrt{-x_1'}-i\frac{\varepsilon}2\frac{x_e}{\sqrt{-x_1'}}+\mathcal O(\varepsilon^2), \end{aligned}$$ and using the same arguments as in the previous computations, we see that the morphism  maps $C_{i,j}^+\times C_{n-i,m-j+1}^+$ diffeomorphically to $C_{i,j}^+\times C_{n-i,k-j+1,l}^+$. The choice of propagators {#ss-2-2} ------------------------- We now discuss the propagators needed for the computations in the framework of (bi)quantization. In particular, we discuss in detail the $4$-colored propagators: we will mainly work here with the $4$-colored propagators as introduced originally in [@CFb], and used extensively in [@CT]. The point is that we will view the biquantization techniques in [@CT] in the framework of the $2$-brane formality of [@CFFR]. In [@CFFR], the authors preferred to work with the $4$-colored propagators on $\mathcal C_{2,1}^+$, in order to use the (simpler) compactified configuration spaces $\mathcal C_{n,m}^+$ of Kontsevich’s type: in order to tie in with the computations in [@CT], we want to establish a more precise relationship than the one sketched in [@CFFR] about the $4$-colored propagators in [@CT] and in [@CFFR]. ### The Kontsevich propagator {#sss-2-2-1} We consider a pair $(z_1,z_2)$ of distinct points in $\mathbb H^+$, and we associate to it a closed $1$-form by the formula $$\label{eq-prop-K} \omega(z_1,z_2)=\frac{1}{2\pi}\left[\mathrm d\ \mathrm{arg}(z_1-z_2)-\mathrm d\ \mathrm{arg}(\overline z_1-z_2)\right]=\frac{1}{2\pi}\left[\mathrm d\ \mathrm{arg}(z_1-z_2)+\mathrm d\ \mathrm{arg}(z_1-\overline z_2)\right].$$ In Formula , the function $\mathrm{arg}(z)$ denotes the Euclidean angle of the complex number $z$: it can be made into a smooth function by restricting its domain of definition on $\mathbb C\smallsetminus i\mathbb R^-$. In particular, the restriction of $\mathrm{arg}(z)$ on $\mathbb H^+$ is a smooth function. However, we want to consider $\omega(z_1,z_2)$ as a closed $1$-form on $(\mathbb H^+\times\mathbb H^+)\smallsetminus \Delta$, $\Delta$ being the diagonal in $\mathbb H^+\times\mathbb H^+$: as such, $\omega(z_1,z_2)$ is the sum of a closed form on $(\mathbb H^+\times\mathbb H^+)\smallsetminus \Delta$ and of an exact $1$-form, where the corresponding function is $\mathrm{arg}(z_1-\overline z_2)/2\pi$. We observe, for the sake of later computations (see [@VdB] for a very nice application of this idea), that the closed $1$-form can be made into a truly exact $1$-form by restricting the domain of definition to $$\{(z_1,z_2)\in(\mathbb H^+\times\mathbb H^+)\smallsetminus \Delta:\ \mathrm{Re}(z_1)=\mathrm{Re}(z_2)\Rightarrow \mathrm{Im}(z_1)>\mathrm{Im}(z_2)\}.$$ It is not difficult to prove that the $1$-form  descends to $C_{2,0}^+$; a bit more involved is the proof that it extends to a smooth $1$-form $\omega$ on the compactified configuration space $\mathcal C_{2,0}^+$. The function $\eta(z_1,z_2)=\mathrm{arg}(z_1-\overline z_2)/2\pi$ also descends to $C_{2,0}^+$ and extends to a smooth function on $\mathcal C_{2,0}^+$. \[l-prop-K\] The closed $1$-form  determines a smooth, closed $1$-form $\omega$ on $\mathcal C_{2,0}^+$, which further enjoys the following properties: - $$\omega\vert_{\mathcal C_2\times\mathcal C_{1,0}^+}=\mathrm d\varphi,$$ where $\mathrm d\varphi$ denotes (improperly) the normalized volume form of $\mathcal C_2\cong S^1$; - $$\omega\vert_{\mathcal C_{1,0}^+\times\mathcal C_{1,1}^+}=0,$$ where $\mathcal C_{1,0}^+\times\mathcal C_{1,1}^+$ denotes the boundary stratum of $\mathcal C_{2,0}^+$ corresponding to the approach of the first argument $z_1$ to $\mathbb R$. The function $\eta(z_1,z_2)$ determines a smooth function $\eta$ on $\mathcal C_{2,0}^+$, which restricts on the boundary stratum $\mathcal C_2\times\mathcal C_{1,0}^+$ to the constant function $\pi/2$; observe that $\mathcal C_{1,0}^+\cong \{i\}$. The $1$-form $\omega$ is usually called Kontsevich’s angle form [@K Subsection 6.2]: it will be useful, for certain computations, to recall that Kontsevich’s angle function is the sum of a closed $1$-form and of an exact $1$-form, constructed by means of the function $\eta$. We finally observe that the natural involution $(z_1,z_2)\overset{\tau}\mapsto (z_2,z_1)$ of $(\mathbb H^+\times\mathbb H^+)\smallsetminus \Delta$ yields an involution $\tau$ of $\mathcal C_{2,0}^+$: we may then consider two Kontsevich’s angle forms $\omega^{\pm}$ defined through $$\omega^+=\omega,\ \omega^-=\tau^*(\omega).$$ The angle forms $\omega^\pm$ have been first introduced in [@CFb; @CF]: they have opposite boundary conditions when one of their arguments approaches $\mathbb R$, as can be easily deduced from Lemma \[l-prop-K\]. ### The $4$-colored propagators on $\mathcal C_{2,1}^+$ {#sss-2-2-2} We consider a triple $(z_1,z_2,x)$ in $(\mathbb H^+\times\mathbb H^+)\smallsetminus \Delta\times \mathbb R$. There is a natural smooth projection from $(\mathbb H^+\times\mathbb H^+)\smallsetminus \Delta\times \mathbb R$ to $(\mathbb H^+\times\mathbb H^+)\smallsetminus \Delta$, thus we may consider the pull-back $\omega^{+,+}$ of the closed $1$-form $\omega^+(z_1,z_2)$ to $(\mathbb H^+\times\mathbb H^+)\smallsetminus \Delta\times \mathbb R$. We set $\omega^{-,-}(z_1,z_2,x)$ to be the pull-back of $\omega^-$ w.r.t. the very same projection. We recall the complex square root discussed in Subsubsection \[sss-2-1-3\]: as already remarked, it is a biholomorphism from $\mathbb H^+$ to $Q^{+,+}$, and we associate to a triple $(z_1,z_2,x)$ in $(\mathbb H^+\times\mathbb H^+)\smallsetminus \Delta\times \mathbb R$ a pair $(\sqrt{z_1-x},\sqrt{z_2-x})$ in $(Q^{+,+}\times Q^{+,+})\smallsetminus \Delta$ (compare with the morphism of Proposition \[p-square\]). We then set $$\begin{aligned} \omega^{+,-}(z_1,z_2,x)&=\frac{1}{2\pi}\left[\mathrm d\ \mathrm{arg}\!\left(\sqrt{z_1-x}-\sqrt{z_2-x}\right)+\mathrm d\ \mathrm{arg}\!\left(\overline{\sqrt{z_1-x}}-\sqrt{z_2-x}\right)-\right.\\ &\phantom{=\frac{1}{2\pi}[}\left.-\mathrm d\ \mathrm{arg}\!\left(\overline{\sqrt{z_1-x}}+\sqrt{z_2-x}\right)-\mathrm d\ \mathrm{arg}\!\left(\sqrt{z_1-x}+\sqrt{z_2-x}\right)\right]\\ \omega^{-,+}(z_1,z_2,x)&=\frac{1}{2\pi}\left[\mathrm d\ \mathrm{arg}\!\left(\sqrt{z_1-x}-\sqrt{z_2-x}\right)-\mathrm d\ \mathrm{arg}\!\left(\overline{\sqrt{z_1-x}}-\sqrt{z_2-x}\right)+\right.\\ &\phantom{=\frac{1}{2\pi}[}\left.+\mathrm d\ \mathrm{arg}\!\left(\overline{\sqrt{z_1-x}}+\sqrt{z_2-x}\right)-\mathrm d\ \mathrm{arg}\!\left(\sqrt{z_1-x}+\sqrt{z_2-x}\right)\right]. \end{aligned}$$ The two $1$-forms $\omega^{+,-}$ and $\omega^{-,+}$ are smooth and obviously closed on $(\mathbb H^+\times\mathbb H^+)\smallsetminus \Delta\times \mathbb R$. We need to characterize more explicitly the compactified configuration space $\mathcal C_{2,1}^+$ (for whose more precise description we refer to [@CFFR Section 5]): here, we content ourselves to describe all boundary strata of codimension $1$, which we depict as follows \ \ We observe that the boundary stratum $\alpha$ corresponds to $\mathcal C_2\times\mathcal C_{1,1}^+$, the boundary strata $\beta$ and $\gamma$ to two copies of $\mathcal C_{2,0}^+\times\mathcal C_{0,2}^+$, the boundary strata $\delta$ and $\varepsilon$ to two copies of $\mathcal C_{1,1}^+\times\mathcal C_{1,1}^+$, and the boundary strata $\eta$, $\theta$, $\zeta$ and $\xi$ to four copies of $\mathcal C_{1,0}^+\times\mathcal C_{1,2}^+$. When it is clear from the context, we will omit to write the projections $\pi_i$, $i=1,2$, from the these spaces to the each of the factors. We finally recall, once again, that the function $\mathrm{arg}(z)$ is well-defined and smooth on $\mathbb H^+$: in particular, the function $\eta$ from Lemma \[l-prop-K\], Subsubsection \[sss-2-2-1\], yields a smooth function (denoted again by $\eta$) on $\mathcal C_{1,1}^+$, when the second argument approaches $\mathbb R$. In more down-to-earth terms, $\eta=\mathrm{arg}(z-x)/2\pi$. It is not difficult to prove that the $4$ $1$-forms $\omega^{+,+}$, $\omega^{+,-}$, $\omega^{-,+}$ and $\omega^{-,-}$ descend to smooth, closed $1$-forms on $C_{2,1}^+$. In fact, as the following Lemma shows (for whose proof we refer to [@CFFR Lemma 5.4]), these in turn extend to smooth, closed $1$-forms on the compactified configuration space $\mathcal C_{2,1}^+$. \[l-CF\] The $1$-forms $\omega^{+,+}$, $\omega^{+,-}$, $\omega^{-,+}$ and $\omega^{-,-}$ determine smooth, closed $1$-forms on the compactified configuration space $\mathcal C_{2,1}^+$, which enjoy the following properties: - $$\omega^{+,+}\vert_\alpha=\mathrm d\varphi,\ \omega^{+,-}\vert_\alpha=\mathrm d\varphi-\mathrm d\eta,\ \omega^{-,+}\vert_\alpha=\mathrm d\varphi-\mathrm d\eta,\ \omega^{-,-}\vert_\alpha=\mathrm d\varphi,$$ where $\mathrm d\varphi$ is the normalized volume form of $\mathcal C_2\cong S^1$. - $$\begin{aligned} \omega^{+,+}\vert_\beta&=\omega^+,\ & \omega^{+,-}\vert_\beta&=\omega^+,\ & \omega^{-,+}\vert_\beta&=\omega^-,\ & \omega^{-,-}\vert_\beta&=\omega^-\quad \text{and}\\ \omega^{+,+}\vert_\gamma&=\omega^+,\ & \omega^{+,-}\vert_\gamma&=\omega^-,\ & \omega^{-,+}\vert_\gamma&=\omega^+,\ & \omega^{-,-}\vert_\gamma&=\omega^-, \end{aligned}$$ where $\omega^\pm$ have to be understood on $\mathcal C_{2,0}^+$. - $$\begin{aligned} &\omega^{+,+}\vert_\delta=\omega^{+,-}\vert_\delta=\omega^{-,+}\vert_\delta=0,\\ &\omega^{+,-}\vert_\varepsilon=\omega^{-,+}\vert_\varepsilon=\omega^{-,-}\vert_\varepsilon=0. \end{aligned}$$ - $$\begin{aligned} &\omega^{+,-}\vert_\eta=\omega^{-,-}\vert_\eta=0,\ & &\omega^{+,+}\vert_\theta=\omega^{-,+}\vert_\theta=0,\\ &\omega^{-,+}\vert_\zeta=\omega^{-,-}\vert_\zeta=0,\ & &\omega^{+,+}\vert_\xi=\omega^{+,-}\vert_\xi=0. \end{aligned}$$ ### The $4$-colored propagators on $\mathcal C_{2,0,0}^+$ {#sss-2-2-3} We now define on $(Q^{+,+}\times Q^{+,+})\smallsetminus \Delta$ $4$ closed, smooth $1$-forms, which, by an (apparent) abuse of notation, are denoted by $\omega^{\pm,\pm}$: namely, we set $$\begin{aligned} \omega^{+,+}&=\frac{1}{2\pi}\left[\mathrm d\ \mathrm{arg}(z_1-z_2)-\mathrm d\ \mathrm{arg}(\overline z_1-z_2)-\mathrm d\ \mathrm{arg}(\overline z_1+z_2)+\mathrm d\ \mathrm{arg}(z_1+z_2)\right],\\ \omega^{+,-}&=\frac{1}{2\pi}\left[\mathrm d\ \mathrm{arg}(z_1-z_2)+\mathrm d\ \mathrm{arg}(\overline z_1-z_2)-\mathrm d\ \mathrm{arg}(\overline z_1+z_2)-\mathrm d\ \mathrm{arg}(z_1+z_2)\right],\\ \omega^{-,+}&=\frac{1}{2\pi}\left[\mathrm d\ \mathrm{arg}(z_1-z_2)-\mathrm d\ \mathrm{arg}(\overline z_1-z_2)-\mathrm d\ \mathrm{arg}(\overline z_1+z_2)+\mathrm d\ \mathrm{arg}(z_1+z_2)\right],\\ \omega^{+,+}&=\frac{1}{2\pi}\left[\mathrm d\ \mathrm{arg}(z_1-z_2)+\mathrm d\ \mathrm{arg}(\overline z_1-z_2)+\mathrm d\ \mathrm{arg}(\overline z_1+z_2)+\mathrm d\ \mathrm{arg}(z_1+z_2)\right], \end{aligned}$$ for an element $(z_1,z_2)$ of $(Q^{+,+}\times Q^{+,+})\smallsetminus \Delta$. We first observe that the last three summands in the previous $1$-forms are exact $1$-forms: namely, as has been previously remarked, the function $\mathrm{arg}(z)$ is smooth and well-defined on $\mathbb C\smallsetminus (i\mathbb R^-\sqcup\{0\})$, hence the three functions appearing in the last three summands of the previous formulæ are well-defined and smooth on $(Q^{+,+}\times Q^{+,+})\smallsetminus \Delta$. It is not difficult to prove that the closed $1$-forms $\omega^{\pm,\pm}$ descends to smooth, closed $1$-forms on the open configuration space $C_{2,0,0}^+$, and that these in turn determine smooth, closed $1$-forms $\omega^{\pm,\pm}$ on the compactified configuration space $\mathcal C_{2,0,0}^+$. Because of the results of Subsubsection \[sss-2-1-3\], we already know that there is a diffeomorphism between $\mathcal C_{2,0,0}^+$ and $\mathcal C_{2,1}^+$, which smoothly extends to the compactified configuration spaces the diffeomorphism $$C_{2,1}^+\ni [(z_1,z_2,x)]\mapsto [(\sqrt{z_1-x},\sqrt{z_2-x})]\in C_{2,0,0}^+.$$ We leave it to the reader to reinterpret on $\mathcal C_{2,0,0}^+$ the boundary strata of codimension $1$ of $\mathcal C_{2,1}^+$. It is not difficult to prove that the pull-backs w.r.t. the morphism  from $\mathcal C_{2,1}^+$ to $\mathcal C_{2,0,0}^+$ of the $1$-forms $\omega^{\pm,\pm}$ on $\mathcal C_{2,0,0}^+$ are exactly the $1$ forms $\omega^{\pm,\pm}$ introduced in Subsubsection \[sss-2-2-2\]: [*e.g.*]{} for $\omega^{+,+}$, we have the obvious identity $$\omega^{+, +}=\frac1{2\pi} \left[\mathrm{d}\ \mathrm{arg} (z_1^2-z_2^2) -\mathrm{d}\ \mathrm{arg}(\overline{z}_1^2-z_2^2)\right],$$ whence the claim follows. Similar arguments work for the other cases. According to the boundary stratification of $\mathcal C_{2,0,0}^+$, we have the following variant of Lemma \[l-CF\]. \[l-CF-q\] The $1$-forms $\omega^{+,+}$, $\omega^{+,-}$, $\omega^{-,+}$ and $\omega^{-,-}$ determine smooth, closed $1$-forms on the compactified configuration space $\mathcal C_{2,0,0}^+$, which enjoy the following properties: - $$\omega^{+,+}\vert_\alpha=\mathrm d\varphi+\mathrm d\eta,\ \omega^{+,-}\vert_\alpha=\mathrm d\varphi-\mathrm d\eta,\ \omega^{-,+}\vert_\alpha=\mathrm d\varphi-\mathrm d\eta,\ \omega^{-,-}\vert_\alpha=\mathrm d\varphi+\mathrm d \eta,$$ where $\mathrm d\varphi$ is the normalized volume form of $\mathcal C_2\cong S^1$, and $\eta=\mathrm{arg}(z)/2\pi$ is a well-defined, smooth function on $\mathcal C_{1,0,0}^+$. - $$\begin{aligned} \omega^{+,+}\vert_\beta&=\omega^+,\ & \omega^{+,-}\vert_\beta&=\omega^+,\ & \omega^{-,+}\vert_\beta&=\omega^-,\ & \omega^{-,-}\vert_\beta&=\omega^-\quad \text{and}\\ \omega^{+,+}\vert_\gamma&=\omega^+,\ & \omega^{+,-}\vert_\gamma&=\omega^-,\ & \omega^{-,+}\vert_\gamma&=\omega^+,\ & \omega^{-,-}\vert_\gamma&=\omega^-, \end{aligned}]$$ where $\omega^\pm$ have to be understood on $\mathcal C_{2,0}^+$. - $$\begin{aligned} &\omega^{+,+}\vert_\delta=\omega^{+,-}\vert_\delta=\omega^{-,+}\vert_\delta=0,\\ &\omega^{+,-}\vert_\varepsilon=\omega^{-,+}\vert_\varepsilon=\omega^{-,-}\vert_\varepsilon=0. \end{aligned}$$ - $$\begin{aligned} &\omega^{+,-}\vert_\eta=\omega^{-,-}\vert_\eta=0,\ & &\omega^{+,+}\vert_\theta=\omega^{-,+}\vert_\theta=0,\\ &\omega^{-,+}\vert_\zeta=\omega^{-,-}\vert_\zeta=0,\ & &\omega^{+,+}\vert_\xi=\omega^{+,-}\vert_\xi=0. \end{aligned}$$ We observe that the $1$-forms $\omega^{\pm,\pm}$, be they defined either on $\mathcal C_{2,1}^+$ or on $\mathcal C_{2,0,0}^+$, satisfy the same boundary conditions $ii)$, $iii)$ and $iv)$; on the other hand, the behavior of the $4$-colored propagators on the boundary strata $\mathcal C_2\times\mathcal C_{1,1}^+$ and $\mathcal C_2\times\mathcal C_{1,0,0}^+$ are quite different. This can be traced back to the proof of Proposition \[p-square\], when analyzing the shape of the morphism  on the boundary stratum $\mathcal C_2\times \mathcal C_{1,1}^+$. Still, we have to be careful about these (seemingly) different boundary conditions for the $4$-colored propagators $\omega^{\pm,\pm}$: namely, the fact that the $4$-colored propagators, quite opposite to Kontsevich’s angle form, can be written as a sum of a regular and of a singular term (the $1$-form living on $\mathcal C_{1,1}^+$ or $\mathcal C_{1,0,0}^+$ and on $\mathcal C_2$ respectively) produces a significant change in the application of Stokes’ Theorem, which is the fundamental tool for proving the $2$-brane Formality Theorem, from which biquantization follows. Formality Theorems {#ss-2-3} ------------------ In this Subsection, we recall the $2$-brane Formality Theorem of [@CFFR], from which we will derive the biquantization techniques we apply later on. Although the main computations of this Subsection are already contained in [@CFFR], we review them in some detail because of the following reasons: first, the $2$-brane Formality Theorem has been proved using superpropagators along the same patterns of [@CF], and superpropagators are better suited for keeping track of all different colors of propagators w.r.t. the treatment in [@CT], and second, because we deserve here a more careful treatment than in [@CFFR] of the $1$-loop correction arising because of the aforementioned regular term in the $4$-colored propagators. We thus profit of the space here to correct a slight mistake in [@CFFR Subsection 7.1] (in the sense that the computations therein are correct, but a subtle point has been missed regarding the multidifferential operator associated to the $1$-loop correction, which we illustrate here in detail) and, more importantly, to correct a more serious mistake in [@CT], where the regular part of the restriction to the boundary stratum $\mathcal C_2\times \mathcal C_{1,1}^+$ or $\mathcal C_2\times \mathcal C_{1,0,0}^+$ is missing completely. The correction term arising from the presence of the regular part is responsible for a quantum shift, which will be illustrated explicitly in Section \[s-3\], which is predicted by representation-theoretic arguments and was otherwise absent. We also prove a version of [@K Lemmata 7.3.1.1, 7.3.3.1] for the $4$-colored propagators: such vanishing lemmata are central in some computations in [@CT] regarding the Harish–Chandra homomorphism. The main idea of the proof is, once again, Stokes’ Theorem, but of course here we have to be a bit more careful and slightly change the final argument. ### Admissible graphs {#sss-2-3-1} Before entering into the technicalities of the $1$-brane and $2$-brane Formality Theorems, we need to spend some words on admissible graphs. For a pair of non-negative integers $(n,m)$, such that $2n+m-2\geq 0$, we consider the set $\mathcal G_{n,m}$ of admissible graphs of type $(n,m)$: the integer $n$, resp. $m$, refers to the number of vertices of the first, resp. second type, [*i.e.*]{} vertices in $\mathbb H^+$, resp. on $\mathbb R$. An admissible graph $\Gamma$ of type $(n,m)$ in the framework of the $1$-brane Formality Theorem [@CF; @CFFR] is an oriented graph, which may admit double edges, [*i.e.*]{} given any two vertices $(v_1,v_2)$, there can be more than one edge connecting $v_1$ to $v_2$, and edges departing from vertices of the second type; it does not possess short loops, [*i.e.*]{} there can no edge in $\Gamma$ with coincident initial and final point. The presence of multiple edges and edges departing from $\mathbb R$ is in opposition to the definition of admissible graphs of type $(n,m)$ as in [@K]. Further, for a triple of non-negative integers $(n,k,l)$, such that $2n+k+l-1\geq 0$, we consider the set $\mathcal G_{n,k,l}$ of admissible graphs of type $(n,k,l)$, where $n$ is the number of vertices of the first type ([*i.e.*]{} in $Q^{+,+}$), $k$, resp. $l$, is the number of vertices of the first type on $i\mathbb R^+$, resp. $\mathbb R^+$. A general element $\Gamma$ of $\mathcal G_{n,k,l}$ is an oriented graphs with $n$, resp. $k+l$, vertices of the first, resp. second type, which may admit multiple edges, edges departing from $i\mathbb R^+\sqcup\{0\}\sqcup\mathbb R^+$ and even short loops. We observe that we may also equivalently consider, for $m=k+l+1$, the set $\mathcal G_{n,m}$ of admissible graphs of type $(n,m)$, consisting of oriented graphs with $n$, resp. $m$, vertices of the first, resp. second type ([*i.e.*]{} lying in $\mathbb H^+$ and on $\mathbb R$ respectively), such that one vertex of the first type is marked and which admit multiple edges, edges departing from $\mathbb R$ and short loops: the notation is abused, but it will be clear from the context if we allow elements of $\mathcal G_{n,m}$ to possess or not short loops, which is the only additional feature that the admissible graphs for the $2$-brane Formality Theorem admit w.r.t. the ones in the $1$-brane Formality Theorem. The algebraic counterpart of the geometric results of Subsubsection \[sss-2-1-3\] is the fact that we may freely pass from $\mathcal G_{n,m}$ to $\mathcal G_{n,k,l}$, for $m=k+l+1$, by noting that the vertex labeled by $k+1$ on $\mathbb R$ corresponds to the origin $\{0\}$. ### Superpropagators {#sss-2-3-2} We now pick an admissible graph $\Gamma$ of type $(n,m)$ for the $1$-brane Formality Theorem of [@CF]. As $\Gamma$ is of type $(n,m)$, its vertices correspond to a point of $\mathcal C_{n,m}^+$, and an edge $e$ determines a natural projection $\pi_e:\mathcal C_{n,m}^+\to \mathcal C_{2,0}^+$. If we pick an admissible graph $\Gamma$ of $\mathcal G_{n,m}$ in the framework of the $2$-brane Formality Theorem, then the vertices of $\Gamma$ still define a configuration of points in $\mathcal G_{n,m}$. An edge $e$ defines, as in the previous situation, either a natural projection $\pi_e:\mathcal C_{n,m}^+\to\mathcal C_{2,1}^+$, if $e=(v_e^i,v_e^f)$, $v_e^i\neq v_e^f$, or $\pi_e:\mathcal C_{n,m}^+\to\mathcal C_{1,1}^+$, if $e$ is a short loop. The point in $\mathbb R$ in either $\mathcal C_{2,1}^+$ or $\mathcal C_{1,1}^+$ is the marked point of $\mathcal C_{n,m}^+$. If, equivalently, we consider the corresponding admissible graph $\Gamma$ of type $(n,k,l)$, then an edge $e$ of $\Gamma$ determines either a projection $\pi_e:\mathcal C_{n,k,l}^+\to\mathcal C_{2,0,0}^+$ or $\pi_e:\mathcal C_{n,k,l}^+\to\mathcal C_{1,0,0}^+$. We now consider the vector space $X=\mathbb K^d$ and two linear (or affine) subspaces $U_i$, $i=1,2$, for which we assume there is a direct sum decomposition $$\label{eq-orth-split} X=(U_1\cap U_2)\overset{\perp}\oplus (U_1^\perp\cap U_2)\overset{\perp}\oplus (U_1\cap U_2^\perp)\overset{\perp}\oplus (U_1+U_2)^\perp,$$ w.r.t. a chosen inner product over $X$. Clearly, we have $$U_1=(U_1\cap U_2)\overset{\perp}\oplus (U_1\cap U_2^\perp),\ U_2=(U_1\cap U_2)\overset{\perp}\oplus (U_1^\perp\cap U_2).$$ We choose linear coordinates $\{x_i\}$ on $X$ which are adapted to the orthogonal decomposition , [*i.e.*]{} there are two non-disjoint subsets $I_i$, $i=1,2$, of $[d]$, such that $$[d]=\left(I_1\cap I_2\right)\sqcup\left(I_1\cap I_2^c\right)\sqcup\left(I_1^c\cap I_2\right)\sqcup\left(I_1^c\cap I_2^c\right),$$ w.r.t. which $\{x_i\}$ is a set of linear coordinates on $U_1\cap U_2$, $U_1\cap U_2^\perp$, $U_1^\perp\cap U_2$ or $(U_1+U_2)^\perp$, if the index $i$ belongs to $I_1\cap I_2$, $I_1\cap I_2^c$, $I_1^c\cap I_2$ or $I_1^c\cap I_2^c$ respectively. Accordingly, for $I$ either one of the previous subsets of $[d]$, and $e$ an edge of admissible graph $\Gamma$ of type $(n,m)$, we set $$\tau_e^I=\sum_{i\in I}\iota_{\mathrm d x_i}^{(v_e^i)}\partial_{x_i}^{(v_e^f)}\in\mathrm{End}\!\left(T_\mathrm{poly}(X)^{\otimes(m+n)}\right),\ T_\mathrm{poly}(X)=\mathrm S(X^*)\otimes\wedge^\bullet X,$$ and $\partial_{x_i}^{(v)}$ denotes the action of the differential operator on the copy of $T_\mathrm{poly}(X)$ sitting at the $v$-th position, and similarly for $\iota_{\mathrm d x_i}^{(v)}$. We observe that $\tau_e^I$ is well-defined and has degree $-1$ w.r.t. the natural grading on $T_\mathrm{poly}(X)$. We now set $$\begin{aligned} A&=\mathrm S(U_1^*)\otimes \wedge (X/U_1)=\mathrm S(U_1^*)\otimes \wedge (U_1^\perp\cap U_2)\otimes \wedge (U_1+U_2)^\perp,\\ B&=\mathrm S(U_2^*)\otimes \wedge (X/U_2)=\mathrm S(U_2^*)\otimes \wedge (U_1\cap U_2^\perp)\otimes \wedge (U_1+U_2)^\perp,\\ K&=\mathrm S((U_1\cap U_2)^*)\otimes \wedge (U_1+U_2)^\perp.\end{aligned}$$ It is clear that $A$ and $B$ both admit a (trivial) structure of $A_\infty$-algebra, and $K$ is naturally an $A$-$B$-bimodule. With respect to the previously introduced notation, the relevant superpropagators are then given by $$\begin{aligned} \label{eq-A-form}\omega^A_e&=\pi_e^*(\omega^+)\otimes\left(\tau^{I_1\cap I_2}_e+\tau^{I_1\cap I_2^c}_e\right)+\pi_e^*(\omega^-)\otimes\left(\tau^{I_1^c\cap I_2}_e+\tau^{I_1^c\cap I_2^c}_e\right),\\ \label{eq-B-form}\omega^B_e&=\pi_e^*(\omega^+)\otimes\left(\tau^{I_1\cap I_2}_e+\tau^{I_1^c\cap I_2}_e\right)+\pi_e^*(\omega^-)\otimes\left(\tau^{I_1\cap I_2^c}_e+\tau^{I_1^c\cap I_2^c}_e\right),\\ \label{eq-K-form}\omega^K_e&=\pi_e^*(\omega^{+,+})\otimes \tau^{I_1\cap I_2}_e+\pi_e^*(\omega^{+,-})\otimes \tau^{I_1\cap I_2^c}_e+\pi_e^*(\omega^{-,+})\otimes \tau^{I_1^c\cap I_2}_e+\pi_e^*(\omega^{-,-})\otimes \tau^{I_1^c\cap I_2^c}_e,\end{aligned}$$ for an edge $e=(v_e^i,v_e^f)$, $v_e^i\neq v_e^f$, of an admissible graph $\Gamma$ of type $(n,m)$. We observe that the superpropagators ,  and are closed $1$-forms on $\mathcal C_{2,0}^+$ and $\mathcal C_{2,1}^+$ with values in $\mathrm{End}\!\left(T_\mathrm{poly}(X)^{\otimes(m+n)}\right)$ (of course, $A$, $B$ and $K$ may be viewed as subalgebras of $T_\mathrm{poly}(X)$). Equivalently, we may regard the superpropagator  as a closed $1$-form on $\mathcal C_{2,0,0}^+$ with values in $\mathrm{End}\!\left(T_\mathrm{poly}(X)^{\otimes(m+n)}\right)$. Lemma \[l-prop-K\], Subsubsection \[sss-2-2-1\], implies the following useful boundary conditions for the superpropagators  and : - their restrictions to the boundary stratum $\mathcal C_{2,0}^+\times\mathcal C_{1,0}^+$ equal $$\omega_e^A\vert_{\mathcal C_{2,0}^+\times\mathcal C_{1,0}^+}=\omega_e^B\vert_{\mathcal C_{2,0}^+\times\mathcal C_{1,0}^+}=\mathrm d\varphi\otimes\tau_e^{[d]};$$ - their restrictions to the boundary stratum $\mathcal C_{1,0}^+\times\mathcal C_{1,1}^+$ corresponding to the approach of the first, resp. second, argument to $\mathbb R$ equal $$\begin{aligned} \omega_e^A\vert_{\mathcal C_{1,0}^+\times\mathcal C_{1,1}^+}&=\pi_e^*(\omega^-)\otimes\left(\tau^{I_1^c\cap I_2}_e+\tau^{I_1^c\cap I_2^c}_e\right),\ &\text{resp.}\quad \omega_e^A\vert_{\mathcal C_{1,0}^+\times\mathcal C_{1,1}^+}&=\pi_e^*(\omega^+)\otimes\left(\tau^{I_1\cap I_2}_e+\tau^{I_1\cap I_2^c}_e\right),\\ \omega_e^B\vert_{\mathcal C_{1,0}^+\times\mathcal C_{1,1}^+}&=\pi_e^*(\omega^-)\otimes\left(\tau^{I_1\cap I_2^c}_e+\tau^{I_1^c\cap I_2^c}_e\right),\ &\text{resp.}\quad \omega_e^A\vert_{\mathcal C_{1,0}^+\times\mathcal C_{1,1}^+}&=\pi_e^*(\omega^+)\otimes\left(\tau^{I_1\cap I_2}_e+\tau^{I_1^c\cap I_2}_e\right). \end{aligned}$$ In particular, we see why admissible graphs appearing in the $1$-brane Formality Theorem may admit edges departing from $\mathbb R$, see for more details [@CFb; @CF]. We now concentrate on the boundary conditions for the superpropagator  on $\mathcal C_{2,1}^+$: Lemma \[l-CF\] yields - the restriction of the superpropagator  to the boundary stratum $\alpha$ of $\mathcal C_{2,1}^+$ equals $$\omega_e^K\vert_\alpha=\mathrm d\varphi\otimes \tau_e^{[d]}-\mathrm d\eta\otimes\left(\tau_e^{I_1\cap I_2^c}+\tau_e^{I_1^c\cap I_2}\right);$$ - the restriction of the superpropagator  to the boundary strata $\beta$ and $\gamma$ equals $$\omega_e^K\vert_\beta=\omega_e^A,\ \omega_e^K\vert_\gamma=\omega_e^B;$$ - the restriction of the superpropagator  to the boundary strata $\delta$ and $\varepsilon$ equals $$\omega_e^K\vert_\delta=\pi_e^*(\omega^{-,-})\otimes \tau_e^{I_1^c\cap I_2^c},\ \omega_e^K\vert_\varepsilon=\pi_e^*(\omega^{+,+})\otimes \tau_e^{I_1\cap I_2};$$ - the restriction of the superpropagator  to the boundary strata $\\eta$, $\theta$, $\zeta$ and $\xi$ equals $$\begin{aligned} \omega_e^K\vert_\eta&=\pi_e^*(\omega^{+,+})\otimes \tau_e^{I_1\cap I_2^c}+\pi_e^*(\omega^{-,+})\otimes \tau_e^{I_1^c\cap I_2},\ &\ \omega_e^K\vert_\theta=\pi_e^*(\omega^{+,-})\otimes \tau_e^{I_1\cap I_2^c}+\pi_e^*(\omega^{-,-})\otimes \tau_e^{I_1^c\cap I_2^c},\\ \omega_e^K\vert_\zeta&=\pi_e^*(\omega^{+,+})\otimes \tau_e^{I_1\cap I_2}+\pi_e^*(\omega^{+,-})\otimes \tau_e^{I_1\cap I_2^c},\ &\ \omega_e^K\vert_\xi=\pi_e^*(\omega^{-,+})\otimes \tau_e^{I_1^c\cap I_2}+\pi_e^*(\omega^{-,-})\otimes \tau_e^{I_1^c\cap I_2^c}. \end{aligned}$$ If we choose the superpropagator  on $\mathcal C_{2,0,0}^+$, it satisfies the same boundary conditions, with the exception of the first one, which takes the form $$\omega_e^K\vert_\alpha=\mathrm d\varphi\otimes \tau_e^{[d]}+\mathrm d\eta\otimes\left(\tau_e^{I_1\cap I_2}+\tau_e^{I_1^c\cap I_2^c}-\tau_e^{I_1\cap I_2^c}-\tau_e^{I_1^c\cap I_2}\right).$$ For the sake of simplicity, we write $\tau_e^+=\tau_e^{I_1\cap I_2}+\tau_e^{I_1^c\cap I_2^c}$ and $\tau_e^-=\tau_e^{I_1^c\cap I_2}+\tau_e^{I_1\cap I_2^c}$. We observe that the boundary conditions of type $iii)$ and $iv)$ explain why the admissible graphs appearing in the $2$-brane Formality Theorem admit edges departing from $\mathbb R$; when considering such admissible graphs in $\mathbb Q^{+,+}\sqcup i\mathbb R^+\sqcup\mathbb R^+\sqcup\{0\}$, we observe that the boundary conditions $iii)$ imply that such graphs admit edges departing from or arriving at the origin. We now deal with the so-called superloop propagator: its origin will be explained carefully in the proof of the $2$-brane Formality Theorem, which will come later on. For the time being, we content ourselves by noting that the superloop propagator appear only first in the $2$-brane Formality Theorem as a consequence of the boundary condition $i)$ satisfied by the superpropagator , more precisely it arises because of the “regular term” containing the form $\mathrm d\eta$. With the same notation as before, the superloop propagator associated to a short loop $e$ of an admissible graph $\Gamma$ of type $(n,m)$ is defined as the closed $1$-form on $\mathcal C_{2,1}^+$ with values in $\mathrm{End}\!\left(T_\mathrm{poly}(X)^{(m+n)}\right)$ $$\omega_e^K=\frac{1}2\pi_e^*(\mathrm d\eta)\otimes (\mathrm{div}_{(v)}^+-\mathrm{div}^-_{(v)}),\ e=(v,v),$$ where $$\mathrm{div}^+_{(v)}=\sum_{k\in (I_1\cap I_2)\sqcup (I_1^c\cap I_2^c)}\iota_{\mathrm d x_k}^{(v)}\partial_{x_k}^{(v)},\quad \mathrm{div}^-_{(v)}=\sum_{k\in (I_1^c\cap I_2)\sqcup (I_1\cap I_2^c)}\iota_{\mathrm d x_k}^{(v)}\partial_{x_k}^{(v)}.$$ We observe that the superloop propagator is exact: this fact will be used in all subsequent computations. Notice that the superloop propagator on $\mathcal C_{2,0,0}^+$ is defined by the same formula without the rescaling by $1/2$ (because of the morphism  from $\mathcal C_{2,1}^+$ to $\mathcal C_{2,0,0}^+$). ### The formality morphisms {#sss-2-3-3} We consider $X$, $U_1$ and $U_2$ as before, to which we associate the graded vector spaces $A$, $B$ and $K$. Using the superpropagators ,  and , and keeping in mind the notation in the previous Subsubsections, we set $$\begin{aligned} \label{eq-c-A}\mathcal O^A_\Gamma(\gamma_1|\cdots|\gamma_n|a_1|\cdots|a_m)&=\mu_{n+m}^B\left(\int_{\mathcal C_{n,m}^+}\prod_{e\in E(\Gamma)}\omega^A_e(\gamma_1|\cdots|\gamma_n|a_1|\cdots|a_m)\right),\\ \label{eq-c-B}\mathcal O^B_\Gamma(\gamma_1|\cdots|\gamma_n|a_1|\cdots|a_m)&=\mu_{n+m}^B\left(\int_{\mathcal C_{n,m}^+}\prod_{e\in E(\Gamma)}\omega^B_e(\gamma_1|\cdots|\gamma_n|b_1|\cdots|b_m)\right),\\ \label{eq-c-K}\mathcal O_\Gamma^K(\gamma_1|\cdots|\gamma_n|a_1|\cdots|a_k|k|b_1|\cdots|b_l)&=\mu_{m+n}^K\left(\int_{\mathcal C_{n,m}^+}\prod_{e\in E(\Gamma)}\omega^K_e(\gamma_1|\cdots|\gamma_n|a_1|\cdots|a_k|k|b_1|\cdots|b_l)\right),\end{aligned}$$ where $\gamma_i$, $i=1,\dots,n$, are elements of $T_\mathrm{poly}(X)$, $a_i$ and $b_i$ are elements of $A$ and $B$ respectively, $k$ is an element of $K$; $E(\Gamma)$ is the set of edges of an admissible graph $\Gamma$ of type $(n,m)$; $\mu^A$, $\mu^B$ and $\mu^K$ denotes the multiplication map on $T_\mathrm{poly}(X)$, followed by the projection onto $A$, $B$ and $K$ respectively. Since $\Gamma$ may have multiple edges, there is a combinatorial subtlety to be taken into account: in all previous formulæ, whenever there are multiple edges between two vertices $(v^i,v_f)$, for $v^i\neq v_f$, we must divide by the factorial of the number of such edges. We observe that short loops cannot be multiple edges, as the superpropagator for a short loop squares obviously to $0$. We also observe that the product on formulæ ,  and  are well-defined and do not depend on the order of the factors: namely, the total degree of any superpropagator appearing in these formulæ is $0$, as the $1$-form piece has (form) degree $1$, while the multidifferential operator piece has degree $-1$. Using the multidifferential operators defined in ,  and , we set $$\begin{aligned} \label{eq-A-mor}\mathcal U_A^n(\gamma_1|\cdots|\gamma_n)(a_1|\dots|a_m)&=(-1)^{\left(\sum_{i=1}^n|\gamma_i|-1\right)m}\sum_{\Gamma\in\mathcal{G}_{n,m}}\mathcal{O}_{\Gamma}^A(\gamma_1|\cdots|\gamma_n|a_1|\cdots|a_m),\\ \label{eq-B-mor}\mathcal U_B^n(\gamma_1|\cdots|\gamma_n)(a_1|\dots|a_m)&=(-1)^{\left(\sum_{i=1}^n|\gamma_i|-1\right)m}\sum_{\Gamma\in\mathcal{G}_{n,m}}\mathcal{O}_{\Gamma}^B(\gamma_1|\cdots|\gamma_n|b_1|\cdots|b_m),\\ \label{eq-K-mor}\mathcal U_K^n(\gamma_1|\cdots|\gamma_n)(a_1|\cdots|a_k|k|b_1|\cdots|b_l)&=(-1)^{\left(\sum_{i=1}^n|\gamma_i|-1\right)m}\sum_{\Gamma\in\mathcal{G}_{n,m}}\mathcal{O}_{\Gamma}^K(\gamma_1|\cdots|\gamma_n|a_1|\cdots|a_k|k|b_1|\cdots|b_l),\\ \label{eq-A_inf-bimod}\mathrm d_K^{k,l}(a_1|\cdots|a_k|k|b_1|\cdots|b_l)&=\sum_{\Gamma\in\mathcal G_{0,m}}\mathcal O_\Gamma^K(a_1|\cdots|a_k|k|b_1|\cdots|b_l),\end{aligned}$$ with the above notation. Some observations are necessary here. The morphisms  and  and  are multilinear maps from $T_\mathrm{poly}(X)$ to the multidifferential operators on $A$ and $B$ respectively; the morphisms  and  are multilinear maps from $T_\mathrm{poly}(X)$ to the multidifferential operators from $A^{\otimes k}\otimes K\otimes B^{\otimes l}$ to $K$. All multidifferential operators appearing in the previous formulæ are non-trivial only if the number of edges of the admissible graphs of type $(n,m)$ equals $2n+m-2$: since to each edge of an admissible graph is associated a contraction operator (which lowers degrees by $1$), it follows immediately that the morphisms \[eq-A-mor\], ,  have degree $2-n$, and that the morphism  has degree $2-m$. We refer to [@CFFR Section 3] for a short introduction to $A_\infty$-categories in the present framework (see [@Kel; @Lef-Has] for more details on $A_\infty$-categories and related issues), which is needed for the statement of the main theorem ($1+2$-brane Formality Theorem) of the present Section. We only recall that $T_\mathrm{poly}(X)$ has a structure of dg (short for differential graded) Lie algebra with trivial differential and Schouten–Nijenhuis bracket (extending the natural Lie bracket on polynomial vector fields on $X$); similarly, the Hochschild cochain complex of an $A_\infty$-category $\mathcal A$ (roughly, an abelian category, whose spaces spaces of morphisms admit the structure of $A_\infty$-algebras and $A_\infty$-bimodules) is also a dg Lie algebra with Hochschild differential (the $A_\infty$-structure itself) and Gerstenhaber bracket (which is well-defined an any sort of Hochschild cochain complex). \[t-form\] We may regard $A$, $B$ as $A_\infty$-algebras, whose only non-trivial Taylor component is given by the corresponding natural (graded) commutative products: then, the morphisms  fit into the Taylor components of a non-trivial $A_\infty$ $A$-$B$-bimodule structure over $K$, which restricts to the natural $A$ left- and $B$-right module structures on $K$. Furthermore, the morphisms ,  and  fit into the Taylor components of an $L_\infty$-morphism $\mathcal U$ from $T_\mathrm{poly}(X)$ to the (completed) Hochschild cochain complex of the $A_\infty$-category $\mathcal A$ with two objects $U_i$, $=1,2$, and spaces of morphisms given by $$\mathrm{Hom}_{\mathcal A}(U_1,U_1)=A,\ \mathrm{Hom}_{\mathcal A}(U_1,U_1)=B,\ \mathrm{Hom}_{\mathcal A}(U_1,U_2)=K,\ \mathrm{Hom}_{\mathcal A}(U_1,U_1)=\{0\},$$ with the respective $A_\infty$-structures. Finally, the $L_\infty$-morphism $\mathcal U$ extends to an $L_\infty$-quasi-isomorphism by suitably completing the graded vector spaces $A$, $B$, $K$. The first claim has been proved in detail in [@CFFR Proposition 6.5], to which we refer. The second claim splits into three claims, namely $\mathcal U$ consists of three morphisms $\mathcal U_A$, $\mathcal U_B$ and $\mathcal U_K$, where $\mathcal U_A$, resp. $\mathcal U_B$, is a pre-$L_\infty$-morphism from $T_\mathrm{poly}(A)$, resp. $T_\mathrm{poly}(B)$, to the (completed) Hochschild cochain complex of $A$, resp. $B$, and $\mathcal U_K$ is a collection of maps from $T_\mathrm{poly}(X)$ to the mixed component $\mathrm C^\bullet(A,B,K)$ of the (completed) Hochschild cochain complex of the above $A_\infty$-category $\mathcal A$. Here, we have used the (non-canonical) identification of dg Lie algebras $T_\mathrm{poly}(X)=T_\mathrm{poly}(A)=T_\mathrm{poly}(B)$. The fact that $\mathcal U_A$ and $\mathcal U_B$ are $L_\infty$-morphisms has been proved in detail in [@CF]; they extend to $L_\infty$-quasi-isomorphisms by suitably completing $A$ and $B$. The fact that the morphism $\mathcal U_K$ satisfies the required $L_\infty$-identities has been proved in detail in [@CFFR Theorem 7.2]: we profit nonetheless for discussing an incorrect issue in the proof regarding the superloop propagator. The superloop propagator, which has been defined above, is manifestly different from the one considered in [@CFFR Subsection 7.1]: the point is that the actual superloop propagator is the correct one. We may repeat the proof of [@CFFR Theorem 7.2] [*verbatim*]{} until the discussion of boundary strata of codimension $1$ of the form $\mathcal C_A\times C_{([n]\smallsetminus A)\sqcup\{\bullet\},m}^+$, where $|A|=2$: the following discussion on how the corresponding integral contribution looks like is precisely the same, [*i.e.*]{} the only situation that matter arise when there are at least one and at most two edges connecting the two vertices labeled by $A$, [*i.e.*]{} pictorially \ \ We are interested only in the contributions from the last three subgraphs (which we denote collectively by $\Gamma_A$). Taking into account the fact that the second graph $\Gamma_A$ has $2$ multiple edges (thus recalling the normalization factor $1/2$), its contribution equals $$\int_{\mathcal C_2}\omega_{\Gamma_A}^K=-\pi_e^*(\mathrm d\eta)\otimes \tau^{[d]}_e\tau^-_e=\frac{1}2\pi_e^*(\mathrm d\eta)\otimes \tau^{[d]}_e(\tau^+_e-\tau_e^-),$$ where $\pi_e$ is here the projection with respect to the “phantom” short loop arising from the contraction of the vertices of the subgraph $\Gamma_A$. The novelty with respect to the corresponding computations in the proof of [@CFFR Theorem 7.2] lies in the re-writing of the second term in the previous chain of equalities; of course, we have used the obvious fact that $(\tau_e^{[d]})^2=0$. The fourth graph in Figure 4 yields a similar contribution. The third graph, on the other hand, yields the contribution $$\int_{\mathcal C_2}\omega_{\Gamma_A}^K=-\pi_{e}^*(\mathrm d\eta)\otimes\tau^{[d]}_{e_1}\tau^-_{e_2}-\pi_{e}^*(\mathrm d\eta)\otimes\tau^{[d]}_{e_2}\tau^-_{e_1}=\frac{1}2\pi_{e}^*(\mathrm d\eta)\otimes\tau^{[d]}_{e_1}(\tau^+_{e_2}-\tau_{e_2}^-)+\frac{1}2\pi_{e}^*(\mathrm d\eta)\otimes\tau^{[d]}_{e_2}(\tau^+_{e_1}-\tau_{e_1}^-),$$ where $e_1=(i,j)$, $e_2=(j,i)$, and $e$ is (improperly) the “phantom” short loop arising from the contraction of the two vertices $i$, $j$. Here, we have used the obvious fact that $\tau_{e_1}^{[d]}\tau_{e_2}^{[d]}=-\tau_{e_2}^{[d]}\tau_{e_1}^{[d]}$. The factor $1/2$ before the function $\eta$ on $\mathcal C_{1,1}^+$ (which we have tacitly omitted) is compatible with the fact that the pull-back of $\eta$ from $\mathcal C_{1,0,0}^+$ to $\mathcal C_{1,1}^+$ is precisely the rescaled function $\eta$ on $\mathcal C_{1,1}^+$. Therefore, the same arguments as in the corresponding part of the proof of [@CFFR Theorem 7.2] show that the right compensation for the contributions coming from the last three graphs in Figure 4 is given precisely by the superloop propagator $\omega^K_e$, which differ from the superloop propagator chosen in the proof of [@CFFR Theorem 7.2] in its multidifferential operator part: the trick is to prove that we may rewrite the multidifferential operator parts of the contributions coming from the last three graphs in Figure 4 using the difference $\tau_e^+-\tau_e^-$, which is exactly the term appearing if we do the computations using the compactified configuration spaces $\mathcal C_{n,k,l}^+$ instead of $\mathcal C_{n,m}^+$. ### Biquantization as a consequence of Theorem \[t-form\] {#sss-2-3-4} We consider now the particular situation $X=\mathfrak g^*$, for $\mathfrak g$ a finite-dimensional Lie algebra over $\mathbb K$, and for a given Lie subalgebra $\mathfrak h$ thereof, we set $U_1=X$ and $U_2=\mathfrak h^\perp$, the annihilator of $\mathfrak h$ in $\mathfrak g$. We observe that, later on, we will consider $U_2$ to be the affine space $\lambda+\mathfrak h^\perp$, where $\lambda$ is a character of $\mathfrak h$: the results of the previous Subsubsection still hold true in this situation. For the sake of explicit computations, we choose a complementary subspace of $\mathfrak h$ in $\mathfrak g$, [*i.e.*]{} we choose a subspace $\mathfrak p$ of $\mathfrak g$, such that $\mathfrak g=\mathfrak h\oplus\mathfrak p$. We observe that, in general, $\mathfrak p$ is not $\mathfrak h$-invariant with respect to the restriction of the adjoint representation. Still, in the case of symmetric pairs $(\mathfrak k,\mathfrak p)$, $\mathfrak p$ is a $\mathfrak k$-module. We thus apply [@CFFR Theorem 7.2] to this situation (we only observe that, in this framework, we do not consider completed algebras, as in [@CFFR]: still, Theorem 7.2 holds true, the only difference is that we have to drop the property of the $L_\infty$-morphism to be an $L_\infty$-quasi-isomorphism): we may view the Poisson structure on $X$ as a Maurer–Cartan (shortly, form now on, MC) element of $T_\mathrm{poly}(X)$, and its image with respect to the $L_\infty$-morphism from [@CFFR Theorem 7.2] is a MC element in the Hochschild cochain complex (with mixed component completed) of the $A_\infty$-category $\mathrm{Cat}_\infty(A,B,K)$, with objects $U_i$, $i=1,2$. Using the previous prescriptions, we have $$A=\mathrm S(\mathfrak g),\ B=\mathrm S(\mathfrak p)\otimes\wedge \mathfrak h^*,\ K=\mathrm S(\mathfrak p);$$ a bit improperly, we sometimes write $\mathfrak p=\mathfrak g/\mathfrak h$ (as it is an identification only of vector spaces, obviously not of $\mathfrak h$-modules). Since $\mathfrak g$ is a Lie algebra, $X=\mathfrak g^*$ is a Poisson manifold with linear Poisson bivector $\pi$, and $U_1$ and $U_2$ are coisotropic submanifolds thereof. The linear Poisson structure on $X$ determines a Maurer–Cartan element of $T_\mathrm{poly}(X)$: for a choice of a formal parameter $\hbar$, the image of $\hbar\pi$ with respect to the $L_\infty$-morphism $\mathfrak U$ from Theorem \[t-form\] is a Maurer–Cartan element $\mathcal U(\hbar\pi)$ in the (completed) Hochschild cochain complex of the $A_\infty$-category $\mathcal A$, which is a concept needing some unraveling. The Hochschild cochain complex of $\mathcal A$ splits into three terms, namely the Hochschild cochain complex of $A$, the one of $B$ and a graded vector space which contains $A$, $B$ and $K$: general elements of the mixed term $\mathrm C^\bullet(A,B,K)$ are multilinear maps from $A^{\otimes k}\otimes K\otimes B^{\otimes l}$ to $K$. From the general theory of Hochschild cochain complexes it is known that Maurer–Cartan elements of Hochschild cochain complexes correspond to $A_\infty$-structures on the underlying graded vector spaces: in our situation, a Maurer–Cartan element is precisely a structure of $A_\infty$-algebra on both $A$ and $B$, and a corresponding structure of $A_\infty$-$A$-$B$-bimodule on $K$, or, equivalently, to an $A_\infty$-structure on the category $\mathcal A$. Now we consider $A_\hbar=A[\![\hbar]\!]$, and similarly for $B_\hbar$ and $K_\hbar$: it is clear that the structure of $A_\infty$-category on $\mathcal A$ extends to $\hbar$-linearly to $\mathcal A_\hbar$, whose objects are the same objects of $\mathcal A$, but whose morphism spaces are replaced by $A_\hbar$, $B_\hbar$ and $K_\hbar$ endowed with the $\hbar$-linearly extended $A_\infty$-structure $\mu$. We have a natural Hochschild differential $\mathrm d_\mathrm H$ on the Hochschild cochain complex of $\mathcal A_\hbar$, given by the adjoint representation of $\mu$ with respect to the Gerstenhaber bracket. The element $\mathcal U(\hbar)$ satisfies the Maurer–Cartan equation $$\mathrm d_\mathrm H\mathcal U(\hbar\pi)+\frac{1}2\left[\mathcal U(\hbar\pi),\mathcal U(\hbar\pi)\right]=\frac{1}2\left[\mu+\mathcal U(\hbar\pi),\mu+\mathcal U(\hbar\pi)\right]=0,$$ [*i.e.*]{} $\mu+\mathcal U(\hbar\pi)$ is a Maurer–Cartan element for $\mathcal A_\hbar$, which deforms (with respect to the formal parameter $\hbar$) the “classical” $A_\infty$-structure on $\mathcal A$. Since $A_\hbar$ is concentrated in degree $0$ by construction, and $\mu_A$ (the component of $\mu$ in the Hochschild cochain complex of $A$) is the obvious $\hbar$-linear commutative, associative product on $A_\hbar$, then $\mu_A+\mathcal U_A(\hbar\pi)$ is an associative product $\star_{A_\hbar}$ on $A_\hbar$, which deforms non-trivially $\mu_A$: $(A_\hbar,\star_{A_\hbar})$ is a deformation quantization of $(A,\mu_A)$ in the sense of [@K]. The image of $\pi$ in $T_\mathrm{poly}(A)$ with respect to the dg Lie algebra isomorphism $T_\mathrm{poly}(X)\cong T_\mathrm{poly}(A)$ (depending on a choice of $\mathfrak p$) is a Maurer–Cartan element in $T_\mathrm{poly}(A)$, which is a sum of a three polyvector fields, $\pi_0$, $\pi_1$ and $\pi_2$, where $\pi_i$ is an $i$-th polyvector field of polynomial degree $2-i$. We observe that $A=\mathrm S(\mathfrak g)$ and $B=\mathrm S(\mathfrak p\oplus\mathfrak h^*[-1])$, thus it makes sense to speak about polynomial degree for elements of $A$ and $B$; $[\bullet]$ is the degree-shifting functor (hence, the polynomial grading of $B$ does not coincide with the internal grading coming from the functor $[-1]$), [*e.g.*]{} the internal degree of $\pi_i$ is $1$, $i=0,1,2$. As a Maurer–Cartan element of $T_\mathrm{poly}(A)$, $\pi$ defines a $P_\infty$-structure on $B$, in other words, $\pi$ defines a Poisson algebra structure on $B$ up to homotopy: for example, $\pi_0$ is a homological vector field over $B$, whose cohomology identifies with the Chevalley–Eilenberg cohomology of the $\mathfrak h$-module $\mathrm S(\mathfrak g/\mathfrak h)$, which in turn inherits from $\pi_1$ (which is a bivector field of internal degree $1$) a structure of graded Poisson algebra. We notice that, in degree $0$, this corresponds to the well-known fact that Poisson reduction endows the commutative algebra $\mathrm S(\mathfrak g/\mathfrak h)^\mathfrak h$ with a Poisson structure coming from the natural one on $A=\mathrm S(\mathfrak g)$. The Maurer–Cartan element $\mu_B+\mathcal U_B(\hbar\pi)$ is an $A_\infty$-structure on $B_\hbar$, deforming the obvious $A_\infty$-structure on $B$: thus, a $P_\infty$-algebra structure on $B$ produces an $A_\infty$-structure [*via*]{} the graded version of deformation quantization, see also [@CF]. We observe that the $A_\infty$-structure $\mu_B+\mathcal U_B(\hbar\pi)$ is the sum of (possibly) infinitely many components of different internal degree: in particular, the component of internal degree $2$ is an element of $B_\hbar$ of degree $2$, the curvature of the $A_\infty$-structure. If it non-trivial, then we cannot talk about the cohomology of $A_\infty$-algebra $(B_\hbar,\mu_B+\mathcal U_B(\hbar\pi))$, and some problems may arise: luckily, in the present framework, the curvature vanishes, see [*e.g.*]{} [@CFb; @CT] and later on. We finally observe that the term of order $1$ with respect to $\hbar$ of the $A_\infty$-structure on $B_\hbar$ is precisely the ($\hbar$-shifted) $P_\infty$-structure on $B_\hbar$: thus, if we select its vector field piece, we get the $\hbar$-shifted Chevalley–Eilenberg differential on $B_\hbar$. Finally, the mixed component $\mu_K+\mathcal U_K(\hbar\pi)$ determines a deformation of the $A_\infty$-$A$-$B$-bimodule $\mu_K$ structure on $K$: we do not spend here much words, because we will deal with $\mu_K+\mathcal U_K(\hbar\pi)$ in the rest of the paper, at least in degree $0$. We only observe that, through $\mu_K$, we may re-prove classical Koszul duality between $A$ and $B$ (both are graded quadratic algebras), and its deformation quantization permits to extend the Koszul duality to the deformed case $(A_\hbar,\star_\hbar)$ and $(B_\hbar,\mu_B+\mathcal U_B(\hbar\pi))$. Biquantization as in [@CT] is the specialization to degree $0$ of the data presented above. In particular, $(A_\hbar,\star_{A_\hbar})$ is an $A_\infty$-algebra concentrated in degree $0$, hence its cohomology equals itself; the piece of $B_\hbar$ of degree $0$ equals $\mathrm S(\mathfrak p)\cong\mathrm S(\mathfrak g/\mathfrak h)$ endowed with a differential $\mathrm d_{B_\hbar}^0$ and with an associative product $\star_{B_\hbar}$ up to homotopy. Finally, $K_\hbar$ is also concentrated in degree $0$, hence its cohomology with respect to $\mathrm d_{K_\hbar}^{0,0}$ (the $(0,0)$-component of $\mu_K+\mathcal U_K(\hbar\pi)$) equals itself, hence $K_\hbar$ becomes with respect to $d_{K_\hbar}^{1,0}=\star_L$ a left $(A_\hbar,\star_{A_\hbar})$- and with respect to $\mathrm d_{K_\hbar}^{0,1}=\star_R$ a right $(\mathrm H^0(B_\hbar),\star_{B_\hbar})$-module (the latter also because of the vanishing of the curvature of the $A_\infty$-structure $\mu_B+\mathcal U(\hbar\pi)$). Later on, still in the framework of finite-dimensional Lie algebras and Lie subalgebras thereof, we will consider the more general framework, where both $A_\hbar$ and $B_\hbar$ are $A_\infty$-algebras with no curvature, and $K_\hbar$ is a graded $A_\infty$-$A_\hbar$-$B_\hbar$-bimodule, hence the $0$-th cohomologies $\mathrm H^0(A_\hbar)$, $\mathrm H^0(B_\hbar)$ become associative algebras and $\mathrm H^0(K_\hbar)$ is an $\mathrm H^0(A_\hbar)$-$\mathrm H^0(B_\hbar)$-bimodule. ### Symmetries of the $4$-colored propagators {#sss-2-3-5} For later purposes, we now exhibit certain symmetries of the $2$-colored and $4$-colored propagators, which we now discuss in some detail. The complex upper half-plane $\mathbb H^+$ has two obvious symmetries, namely the reflection with respect to the imaginary axis $i\mathbb R$, given by $z\overset{\sigma}\mapsto -\overline z$, and the inversion with respect to the unit half-circle, given by $z\overset{\tau}\mapsto 1/\overline z$: both maps extend to $\mathbb H^+\sqcup \mathbb R$, and they define two orientation-reversing involutions $\sigma$ and $\tau$ of it. Equivalently, $\mathcal Q^{+,+}\sqcup i\mathbb R^+\sqcup\{0\}\sqcup\mathbb R^+$ admits two orientation-reversing involutions $\sigma$ and $\tau$, where $z\overset{\sigma}\mapsto i\overline{z}$ and $z\overset{\tau}\mapsto \frac{1}{\overline z}$. It is not difficult to prove that $\sigma$ and $\tau$ descend both to involutions of $C_{n,m}^+$ and $C_{n,k,l}^+$, and that, using the same techniques as in the proof of Proposition \[p-square\], Subsubsection \[sss-2-1-3\], $\sigma$ and $\tau$ extend to involutions of the compactified configuration spaces $\mathcal C_{n,m}^+$ and $\mathcal C_{n,k,l}^+$. We observe that $\sigma$ and $\tau$ are orientation-preserving, resp. -reversing, if and only if $n+m-1$ is even, resp. odd. We then have the following technical Lemma about the behavior of the $2$-colored and $4$-colored propagators with respect to the action of $\sigma$ and $\tau$. \[l-symm-4\] The $2$-colored and $4$-colored propagators behave as follows with respect to the involutions $\sigma$ and $\tau$ on the respective compactified configuration spaces $\mathcal C_{2,0}^+$ and $\mathcal C_{2,1}^+$: $$\begin{aligned} \sigma^*(\omega^+)&=-\omega^+,\ & \sigma^*(\omega^-)&=-\omega^-,\ & \tau^*(\omega^+)&=-\omega^++2\pi_1^*(\mathrm d\eta),\ & \tau^*(\omega^-)&=-\omega^-+2\pi_2^*(\mathrm d\eta),\\ \sigma^*(\omega^{+,+})&=-\omega^{+,+}, & \sigma^*(\omega^{+,-})&=-\omega^{-,+}, & \sigma^*(\omega^{-,+})&=-\omega^{+,-}, & \sigma^*(\omega^{-,-})&=-\omega^{-,-},\\ \tau^*(\omega^{+,+})&=-\omega^{+,+}+2 \pi_1^*(\mathrm d\eta), & \tau^*(\omega^{+,-})&=-\omega^{+,-}, & \tau^*(\omega^{-,+})&=-\omega^{-,+}, & \tau^*(\omega^{-,-})&=-\omega^{-,-}+2 \pi_2^*(\mathrm d\eta), \end{aligned}$$ where now $\pi_i$, $i=1,2$, denotes the two natural projections from $\mathcal C_{2,0}^+$ onto $\mathcal C_{1,0}^+$ or from $\mathcal C_{2,1}^+$ to $\mathcal C_{1,1}^+$. Similar formulæ hold true for the $4$-colored propagators on $\mathcal C_{2,0,0}$, keeping in track a rescaling before the exact $1$-form $\eta$. ### Kontsevich’s Vanishing Lemmata {#sss-2-3-6} We now need a Vanishing Lemma for the $4$-colored propagators, reminiscent of the Vanishing Lemmata in [@K Subsubsubsection 7.3.3.1]. We observe that Kontsevich’s Vanishing Lemmata in [@K Subsubsubsection 7.3.3.1] are key ingredients in the proof of the globalization of its $L_\infty$-Formality-quasi-isomorphism: in this sense, the Vanishing Lemma we are going to state and prove here (the main application being for later computations regarding the generalized Harish-Chandra homomorphism) play also a central [*rôle*]{} in the globalization of the $2$-brane $L_\infty$-Formality-quasi-isomorphism of [@CFFR], but do not indulge here on this point, referring to upcoming work for more details. We consider the three natural projections $\pi_{ij}$, $i\leq i<\leq 3$, from $\mathcal C_{3,0}^+$ onto $\mathcal C_{2,0}^+$, which smoothly extend the projections $[(z_1,z_2,z_3)]\overset{\pi_{ij}}\to [(z_i,z_j)]$ to the corresponding compactified configuration spaces. The typical fiber of the projection $\pi_{ij}$ is $2$-dimensional, and it is not difficult to verify that it is a smooth manifold with corners (hence, it admits a natural stratification, whose description, at least in codimension $1$, will be made explicit later on). We improperly denote by the same symbol the natural projection $\pi_{ij}$, $i\leq i<\leq 3$, from $\mathcal C_{3,1}^+$ onto $\mathcal C_{2,1}^+$, which this times extends the projection $[(z_1,z_2,z_3,x)]\overset{\pi_{ij}}\mapsto[(z_i,z_j,x)]$: again, its fiber is a smooth $2$-dimensional manifold with corners. For any two smooth $1$-forms $\eta_i$, $i=1,2$, on $\mathcal C_{2,0}^+$ or $\mathcal C_{2,1}^+$, we define a smooth function on $\mathcal C_{2,0}^+$ or $\mathcal C_{2,1}^+$ [*via*]{} the integral $$\label{eq-int-van} \Omega(\eta_1,\eta_2)=\pi_{13,*}\!\left(\pi_{12}^*(\eta_1)\wedge\pi_{23}^*(\eta_2)\right)=\int_{z_2\in\mathcal Q^{+,+}\smallsetminus\{z_1,z_3\}}\eta_1(z_1,z_2)\wedge\eta_2(z_2,z_3),$$ where $\pi_{ij,*}$ denotes integration along the fiber of the projection $\pi_{ij}$. \[l-vanish-4\] The function $\Omega(\eta_1,\eta_2)$ vanishes, whenever $\eta_1=\eta_2$ is either one of the $2$-colored propagators or either one of the $4$-colored propagators. The claim for the $2$-colored propagators $\omega^+$ and $\omega^-$ is precisely the content of the vanishing lemmata in [@K Subsubsubsection 7.3.3.1], to which we refer for a proof. We observe that the proof below for the $4$-colored propagators applies with minor changes ([*e.g.*]{} the final argument involves the involution $\sigma$ and not $\tau$) applies to the statement for $2$-colored propagators. For symmetry reasons, it suffices to prove the claim for the $4$-colored propagators $\omega^{+,+}$ and $\omega^{+,-}$. We first prove the claim for $\eta_1=\eta_2=\omega^{+,+}$: the idea is to show first that $\Omega(\eta_1,\eta_2)$ is a constant function, and then to use Lemma \[l-symm-4\] to prove that, for a well-suited choice of arguments, $\Omega(\eta_1,\eta_2)$ equals minus itself. We therefore compute the exterior derivative of $\Omega(\eta_1,\eta_2)$: we make use of generalized Stokes’ Theorem, and the fact that $\eta_1=\eta_2$ is a closed $1$-form, yields $$\mathrm d\Omega(\eta_1,\eta_2)=\pi_{13,*}^\partial(\pi_{12}^*(\eta_1)\pi_{23}^*(\eta_2)),$$ where $\pi_{13,*}^\partial$ denotes integration along the codimension $1$-boundary strata of the fiber of $\pi_{13}$: there are five such boundary strata, which correspond to $i)$ the point labeled by $z_2$ approaching either $\mathbb R$ on the left or on the right of the marked point on $\mathbb R$ or the marked point itself, or to $ii)$ the point labeled by $z_2$ approaching either the point labeled by $z_1$ or $z_2$. In the three situations in $i)$, the corresponding contribution vanishes in view of Lemma \[l-CF\], $iii)$ and $iv)$. In both situations described in $ii)$, the boundary fibration is trivial, namely $\mathcal C_2\times \mathcal C_{2,1}^+$: an easy computation in local coordinates for $\mathcal C_{3,1}^+$ near the boundary strata in $i)$ shows that there are no orientation signs appearing, and we finally get, using Lemma \[l-CF\], $i)$, $$\mathrm d\Omega(\eta_1,\eta_2)=\int_{\mathcal C_2^+}\mathrm d\varphi\ \omega^{+,+}+\int_{\mathcal C_2^+}\omega^{+,+}\mathrm d\varphi=0.$$ Therefore, $\Omega(\eta_1,\eta_2)$ is a constant function on $\mathcal C_{2,1}^+$, whose value is completely determined by a choice of a point in $\mathcal C_{2,1}^+$, and a natural choice is $(i,2i,0)$. Since $i\mathbb R$ is the fixed point set of $\sigma$, $\sigma$ preserves $\Omega(\eta_1,\eta_2)$, whence $$\Omega(\eta_1,\eta_2)=\sigma^*(\Omega(\eta_1,\eta_2)=-\pi_{13,*}(\sigma^*(\pi_{12}^*(\eta_1))\wedge \sigma^*(\pi_{23}^*(\eta_1))=-\Omega(\eta_1,\eta_2),$$ where the minus sign in the second equality arises because $s$ is orientation-reversing on the first quadrant, while the third equality is a consequence of Lemma \[l-symm-4\]. The very same arguments can be applied in the situation $\eta_1=\eta_2=\omega^{-,-}$. We consider now the case $\eta_1=\eta_2=\omega^{+,-}$. The computation of the exterior derivative of $\Omega(\eta_1,\eta_2)$ in this case is similar to the previous one: we only observe that the boundary condition for boundary strata of type $i)$ let appear a regular term $\mathrm d\eta$, whose contribution to integration is trivial. The arguments for dealing with the other strata are, once again, a consequence of Lemma \[l-CF\], $iii)$ and $iv)$. Therefore, $\Omega(\eta_1,\eta_2)$ is uniquely determined by a given point in $\mathcal C_{2,1}^+$: quite differently from the previous case, we choose a pair of points lying on the unit circle. Recalling that the unit circle is the fixed point [*locus*]{} of the involution $\tau$, we get in this case $$\Omega(\eta_1,\eta_2)=\tau^*(\Omega(\eta_1,\eta_2)=-\pi_{13,*}(\tau^*(\pi_{12}^*(\eta_1))\wedge \tau^*(\pi_{23}^*(\eta_1))=-\Omega(\eta_1,\eta_2),$$ by the very same arguments as in the previous case, because $t$ is orientation-reversing and because of Lemma \[l-symm-4\]. As an application of Lemma \[l-vanish-4\], we briefly sketch the vanishing of the curvature of the $A_\infty$-algebra $B_\hbar$, with the previously introduced notations. As already mentioned, the curvature of $B_\hbar$ is the piece of degree $2$ in $B_\hbar$ of the Maurer–Cartan element $\mu_B+\mathcal U_B(\hbar\pi)$: more explicitly, the curvature is given by the formal power series $$\mathcal U(\hbar\pi)_0=\sum_{n\geq 1}\frac{1}{n!}\mathcal U_B^n(\underset{n}{\underbrace{\hbar\pi|\cdots|\hbar\pi}})=\sum_{n\geq 1}\frac{1}{n!}\sum_{\Gamma\in\mathcal{G}_{n,0}}\mathcal{O}_{\Gamma}^B(\underset{n}{\underbrace{\hbar\pi|\cdots|\hbar\pi}})=\sum_{n\geq 1}\frac{1}{n!}\sum_{\Gamma\in\mathcal{G}_{n,0}}\mu_{n}^B\left(\int_{\mathcal C_{n,0}^+}\prod_{e\in E(\Gamma)}\omega^B_e(\underset{n}{\underbrace{\hbar\pi|\cdots|\hbar\pi}})\right).$$ For an admissible graph $\Gamma$ in $\mathcal G_{n,0}$, the rightmost integral is non-trivial only if the degree of the integrand equals $2n-2$. To each vertex of the first type of $\Gamma$ is associated a copy of the linear Poisson bivector $\hbar\pi$, hence from each vertex depart exactly two arrows: each arrow, by definition, corresponds to a derivation and a contraction. In particular, the differential operator $\mathcal O_\Gamma^B$ has degree $2n-2$: since $\hbar\pi$ is linear, the polynomial degree of the object on the rightmost part of the previous chain of equalities is $n-(2n-2)=-n+2$, whence $1\leq n\leq 2$. When $n=1$, $\mu_1^B(\hbar\pi)$ vanishes because of the coisotropy of $U_2$. For $n=2$, there is only one possible admissible graph $\Gamma$ of type $(2,0)$, namely the loop graph connecting the two vertices of the first type: the corresponding operator $\mathcal O_\Gamma^B$ vanishes because of Lemma \[l-vanish-4\]. Quantum reduction algebras {#ss-2-4} -------------------------- The present Subsection presents the results of [@CT Section 2] using a slightly different perspective, coherent with the approach to biquantization we have introduced in the previous Subsection: however, we think it useful to review many results mainly because of the notation, which will be then used extensively in the rest of the paper. Thus, we will mostly concentrate on the dg vector space $B_\hbar$, where we denote by $\mathrm d_{B_\hbar}$ its differential ([*i.e.*]{} the piece of degree $1$ of its $A_\infty$-structure): we have already observed that $\mathrm d_{B_\hbar}=\hbar\mathrm d_\mathrm{CE}+\mathcal O(\hbar^2)$, where $\mathrm d_\mathrm{CE}=\pi_0$ is the Chevalley–Eilenberg differential on $B_\hbar$. The admissible graphs $\Gamma$ appearing in $\mathrm d_{B_\hbar}$ are of type $(n,1)$, possibly with multiple edges and admitting edges departing from $\mathbb R$: each edge $e$ of an element $\Gamma$ of $\mathcal C_{n,1}$ is the sum of two colored edges, namely $e^+$ and $e^-$ according to Formula , Subsubsection \[sss-2-3-2\]. The properties of the superpropagator  imply that an edge $e$ of $\Gamma$ arriving to the only vertex of the second type has color $+$, while an edge departing from it has color $-$. One of the main tools we will use throughout the present Subsection is Lemma \[l-vanish-4\], Subsubsection \[sss-2-3-6\]. Namely, we assume $v$ is a vertex of the first type of $\Gamma$ of type $(n,1)$ with two edges at it of the form $e_1=(\bullet_1,v)$ and $e_2=(v,\bullet_2)$, where $\bullet_i$, $i=1,2$, denotes some other vertex (notice that now we allow $\bullet_1=\bullet_2$): then the configuration at $v$ is either $(e_1^+,e_1^-)$ or $(e_1^-,e_2^+)$. Of course, since to each edge of the first type of $\Gamma$ is associated a copy of the linear Poisson bivector $\hbar\pi$, we assume that from $v$ as above departs a third edge, which does not join any other vertex of $\Gamma$: this edge has color $-$. This “phantom” edge is the “edge to $\infty$”, using the terminology of [@CT]: dimensional arguments imply that each admissible graph $\Gamma$ of type $(n,1)$ admit precisely one vertex of the first type with a phantom edge. ### Symmetric pairs and, more generally, Lie subalgebras of trivial extension class {#sss-2-4-1} We consider here the case of a symmetric pair $\mathfrak g=\mathfrak k\oplus\mathfrak p$, or, more generally, of a Lie subalgebra $\mathfrak h\subseteq\mathfrak g$ admitting an $\mathfrak h$-invariant complementary subspace $\mathfrak p$: we observe that this is equivalent to the triviality of the extension class $\alpha$ of the short exact sequence of $\mathfrak h$-modules $\mathfrak h\hookrightarrow\mathfrak g\twoheadrightarrow\mathfrak g/\mathfrak h$. By definition, a symmetric pair is a pair $(\mathfrak g,\sigma)$, where $\mathfrak g$ is a finite-dimensional Lie algebra over $\mathbb K$ and $\sigma$ is an involutive Lie algebra automorphism: thus, $\mathfrak g=\mathfrak k\oplus\mathfrak p$ is the direct sum of the $+1$-eigenspace $\mathfrak k$ and the $-1$-eigenspace $\mathfrak p$ of $\sigma$. In particular, we have the Cartan relations $$\label{eq-cartan} [\mathfrak k,\mathfrak k]\subseteq \mathfrak k,\ [\mathfrak k,\mathfrak p]\subseteq \mathfrak k,\ [\mathfrak p,\mathfrak p]\subseteq \mathfrak k.$$ In particular, $\mathfrak k$ is a Lie subalgebra of $\mathfrak g$ and $\mathfrak p$ is a $\mathfrak k$-module. The graded algebra $B$ in the case of a symmetric pair equals $B=\mathrm S(\mathfrak k)\otimes\wedge^\bullet(\mathfrak p)$. Of course, if the above extension class $\alpha$ vanishes, then $\mathfrak g$ admits simply a decomposition $\mathfrak g=\mathfrak h\oplus\mathfrak p$, with relations $[\mathfrak h,\mathfrak h]\subseteq\mathfrak h$ and $[\mathfrak h,\mathfrak p]\subseteq\mathfrak p$. The claim is that in the case of a symmetric pair $(\mathfrak g,\sigma)$ or of a pair $(\mathfrak g,\mathfrak h)$ with trivial extension class the quantized differential $\mathrm d_{B_\hbar}$ equals simply the ($\hbar$-shifted) Chevalley–Eilenberg differential $\hbar\mathrm d_\mathrm{CE}$ on $B_\hbar=\mathrm C^\bullet(\mathfrak h,\mathrm S(\mathfrak g/\mathfrak h))[\![\hbar]\!]$. Namely, we consider an admissible graph $\Gamma$ of type $(n,1)$, $n\geq 2$, appearing in Formula , Subsubsection \[sss-2-3-3\], and we know from the above considerations that $\Gamma$ possesses a vertex $v$ of the first type with a phantom edge $e_\mathrm{gh}$ and two edges $e_1=(\bullet_1,v)$ and $e_2=(v,\bullet_2)$. The configuration at the vertex is either $(e_1^+,e_2^-,e_\mathrm{gh}^-)$ or $(e_1^-,e_2^+,e_\mathrm{gh}^-)$: in Lie algebraic terms, these two configurations correspond to $[\mathfrak h,\mathfrak p]\subseteq\mathfrak h$ and $[\mathfrak h,\mathfrak h]\subseteq\mathfrak p$ respectively, which is a contradiction to the Cartan relations, for $\mathfrak g$ a symmetric pair, or to the fact that $\mathfrak k$ has a $\mathfrak k$-invariant complement. This implies that only the contributions of order $1$ with respect to $\hbar$ matter, whence the claim. ### Cohomology of degree $0$ {#sss-2-4-2} The content of the present Subsubsection presents some arguments for dealing with the classification of admissible graphs appearing in the computation of the differential $\mathrm d_{B_\hbar}$, which will also appear in other contexts related to biquantization. We consider here the case of a Lie subalgebra $\mathfrak h$ of $\mathfrak g$ with no assumptions on the extension class of $\mathfrak h\subseteq \mathfrak g$: thus, we only assume to have picked out some complementary subspace $\mathfrak p$ for explicit computations. \[p-quant-0\] The admissible graphs of type of $(n,1)$, $n\geq 1$, appearing in the restriction of the differential $\mathrm d_{B_\hbar}$ to $\mathrm S(\mathfrak p)$ are either of type Bernoulli ([*i.e.*]{} connected graphs with one root and one phantom edge), or of type wheel ([*i.e.*]{} connected graphs whose edges between the vertices of the first type form an oriented loop with one phantom edge), or of mixed type ([*i.e.*]{} a Bernoulli-type graphs attached to a wheel-type graph), see also Figure 5. We consider, for $n\geq 1$, an admissible graph of type $(n,1)$, and we denote by $p$ the number of edges of $\Gamma$ arriving at the vertex of the second type. Degree reasons imply that such a graph admits one phantom edge, hence the actual edges connecting vertices of $\Gamma$ are $2n-1$. Dimensional reasons imply also that $p\geq 1$: namely, if $p$ were $0$, since $n\geq 1$ by assumption, there would be a $1$-dimensional submanifold of $\mathcal C_{n,1}^+$ over which there is nothing to integrate (a subset of $\mathbb R$), hence integration would yield $0$. The admissible graph $\Gamma$ is connected in the sense that, if we remove from it the edges to the vertex of the second type, we obtain a connected graph in the strict sense of the world, as $p\geq 1$. The connectedness of $\Gamma$ is also a consequence of dimensional reasons: if $\Gamma=\Gamma_1\sqcup\Gamma_2$, then either $\Gamma_1$ or $\Gamma_2$ would have a phantom edge. W.l.o.g. we assume $\Gamma_1$ has a phantom edge, hence $\Gamma_2$ does not, which means that all its edges provide differential forms to be integrated: if $n_i$ is the number of vertices of the first type of $\Gamma_i$, $i=1,2$, then the integral over $\mathcal C_{n_2,1}^+$ of the differential form associated to $\Gamma_2$ vanishes, as its degree is $2 n_2$, while the dimension of $\mathcal C_{n_2,1}^+$ is $2n_2-1$. The boundary conditions for the superpropagator  imply immediately that edges arriving at the vertex of second type are all colored by $+$, whence $p\leq (n-1)+1=n$: in fact, from the vertex of the first type from which departs the phantom edge departs another edge, which may or may not arrive at the vertex of the second type. On the other hand, there are $2n-1-p$ edges from vertices of the second type arriving at (distinct) vertices of the first type: this implies that the polynomial degree of the differential operator on $B_\hbar$ associated to $\Gamma$ is $n-(2n-1-p)=-n+1+p\geq 0$, whence $p\geq n-1$: it follows then immediately $p=n-1$ or $p=n$. We first consider the case $p=n$: from every vertex of the first type of $\Gamma$ departs one edge to the vertex of the second type, $\Gamma$ is connected, has a phantom edge and there is a single vertex of the first type, which is the final point of none of its edges. Such an admissible graph is obviously of Bernoulli type. Then, assume $p=n-1$: the only vertex from which does not depart an edge to the vertex of the second type may or may not be the vertex from which departs the phantom edge. In the first case, the connectedness of $\Gamma$ implies that it is of wheel-type, while in the second case, it must be a wheel type, from which departs an edge hitting the root of a Bernoulli-type graph. Here, the root of a Bernoulli-type graph is the only vertex of the second type, which is the final point of none of its edges. Here is a pictorial representation of the three types of graphs appearing in $\mathrm d_{B_\hbar}$ according to the previous Proposition: \ \ Although for most of the computations in this framework we do not really need it, we want to understand the differential $\mathrm d_{B_\hbar}$ on the whole of $B_\hbar$: as the next proposition shows, the results of Proposition \[p-quant-0\] with a slight addition suffice. \[p-quant-d\] The admissible graphs of type $(n,1)$ appearing in $\mathrm d_{B_\hbar}$ are either the admissible graphs of Proposition \[p-quant-0\] or connected graphs which are brackets of two Bernoulli-type graphs ([*i.e.*]{} there is an edge departing from the vertex of the second type to a vertex of the first type, whose two outgoing edges arrive at the roots of two Bernoulli-type graphs, see Figure 6). We first observe that, since we are considering $\mathrm d_{B_\hbar}$ on the whole of $B_\hbar$, admissible graphs $\Gamma$ can have edges departing from the vertex of the second type. We first assume that that all edges departing from the vertex of the second type are phantom edges: in this case, we are reduced to the very same analysis as in the proof of Proposition \[p-quant-0\]. We now assume that the admissible graph $\Gamma$ of type $(n,1)$ possesses $k$ edges departing from the vertex of the second type and arriving to $k$ vertices of the first type (these vertices are distinct because of the linearity of the Poisson bivector $\hbar\pi$). Because of degree reasons, $\Gamma$ has $k+1$ phantom edges departing from vertices of the first type. Dimensional reasons imply further that no vertex of the first type may have two phantom edges. We denote by $p$ the number of edges departing from vertices of the first type and arriving to the vertex of the second type: the very same arguments as in the proof of Proposition \[p-quant-0\] imply that $p\geq n$. On the other hand, the polynomial degree of the differential operator associated to $\Gamma$ equals $(n-k)-(2n-(k+1)-p)=-n+1+p\geq 0$, whence either $p=n-1$ or $p=n$. We first consider the case $p=n$: in this situation, from every vertex of the first type departs exactly one edge to the vertex of the second type. Thus, $\Gamma$ is the disjoint union of exactly one Bernoulli-type graph and of either wheel-like graphs or Bernoulli-wheel-type graphs or Bernoulli-type graphs, whose root is the endpoint of an edge departing from the vertex of the second type (at least such a graph appears here, as $k\geq 1$ by assumption). Since $\Gamma$ is a disjoint union of subgraphs, the corresponding integral factors into integrals over compactified configuration spaces of the form $\mathcal C_{m,1}^+$, for $1\leq m\leq n$. For a subgraph of the last type as above, the corresponding integral vanishes, because the degree of the integrand is $2m$, while the dimension of the fiber is $2m-1$. We consider now the case $p=n-1$: the corresponding differential operator is translation-invariant. If $\Gamma$ is the disjoint union of connected subgraphs, at least one of which is a Bernoulli-type graph, whose root is the final point of an edge departing from the vertex of the second type, the last argument in the previous paragraph yields triviality of the corresponding differential operator. A subgraph like the one we have analyzed always appear, if there is a vertex of the first type of $\Gamma$, from which departs one edge to the vertex of the second type and which is the endpoint of an edge issued from the said vertex. As $p=n-1$, there can be exactly one edge departing from the vertex of the second type and arriving at the only vertex of the first type, from which no edge depart to $\mathbb R$: these two edges meet two distinct roots (again, because $p=n-1$) of Bernoulli-type graphs. Therefore, $\Gamma$ is the disjoint union of subgraphs as in Proposition \[p-quant-0\] and of a connected subgraph of the said type. Here is a pictorial representation of the new type of connected graphs appearing in $\mathrm d_{B_\hbar}$ in higher degrees according to the previous Proposition: \ \ We consider now a general admissible graph $\Gamma$ of type $(n,1)$ appearing in $\mathrm d_{B_\hbar}$: we may consider the involution $\sigma$ of $\mathcal C_{n,1}^+$ from Subsubsection \[sss-2-3-5\]. It preserves, resp. reverses, orientation if $n$ is even, resp. odd; the pull-back of integrand in Formula  with respect to $\sigma$ equals itself up to a global $-1$-sign, hence if $n$ is even, the contributions to $\mathrm d_{B_\hbar}$ from admissible graphs of type $(n,1)$ with $n$ even are trivial. We may therefore write $\mathrm d_{B_\hbar}$ as $$\mathrm d_{B_\hbar}=\hbar\mathrm d_\mathrm{CE}+\hbar^3 \mathrm d_3+\hbar^5 \mathrm d_5+\cdots,$$ where only odd powers of $\hbar$ appear. Therefore, if $f$ is a general element of $B_\hbar$, then $f$ is $\mathrm d_{B_\hbar}$-closed if and only if both its even and odd part with respect to $\hbar$ are $\mathrm d_{B_\hbar}$-closed. ### The generalized Iwasawa decomposition {#sss-2-4-3} In this Subsubsection we review in some detail the generalized Iwasawa decomposition of a symmetric pair $\mathfrak g=\mathfrak k\oplus\mathfrak p$: we want to stress that we do not present here any new results, but simply need to fix notation and conventions in view of later applications, namely the Harish-Chandra homomorphism in the framework of deformation quantization. A detailed discussion of the Iwasawa decomposition for semisimple symmetric pairs can be found in [@Dix Chapter 1, Section 13]; the main reference to the generalized Iwasawa decomposition for a general symmetric pair is [@T1 Subsections 1.1, 1.2 and 1.3]. We consider a general symmetric pair $(\mathfrak g,\sigma)$ with Cartan decomposition $\mathfrak g=\mathfrak k\oplus\mathfrak p$. For an element $\xi$ of $\mathfrak k^\perp$, we set $\mathfrak g(\xi)=\left\{x\in\mathfrak g:\ \mathrm{ad}^*(x)(\xi)=0\right\}$: it is a Lie subalgebra of $\mathfrak g$ and the Lie algebra involution $\sigma$ restricts to $\mathfrak g(\xi)$, whose Cartan decomposition we denote by $\mathfrak g(\xi)=\mathfrak k(\xi)\oplus\mathfrak p(\xi)$. An element $\xi$ of $\mathfrak k^\perp$ is said to be regular, if the dimension of $\mathfrak g(\xi)$ is minimal among the subalgebras $\mathfrak g(\eta)$, $\eta$ in $\mathfrak k^\perp$. The set of regular elements of $\mathfrak k^\perp$ is a Zarisky-open subset of $\mathfrak k^\perp=\mathfrak p^*$, and for every regular element $\xi$ of $\mathfrak k^\perp$, $[\mathfrak k(\xi),\mathfrak p(\xi)]=0$, see [@Dix Chapter 1.11] for more details on regular linear functionals on a Lie algebra and [@T1 Definition 1.1.2.1 and Lemma 1.2.2.1] for a similar discussion in the present situation. For $\xi$ in $\mathfrak k^\perp$ regular as above, we set $\mathfrak a(\xi)=\mathfrak p(\xi)\oplus[\mathfrak p(\xi),\mathfrak p(\xi)]$: the Cartan relations together with the previous remark imply that it is a nilpotent Lie subalgebra of $\mathfrak g$. We denote by $\mathfrak s_\xi$ a maximal torus of $\mathfrak a(\xi)$, which is additionally preserved by $\sigma$: according to [@T1 Subsubsection 1.2.3], there exists only one maximal torus $\mathfrak s_\xi$ of $\mathfrak a(\xi)$, which is exactly the set of semisimple elements of $\mathfrak p(\xi)$. It is moreover central in $\mathfrak g(\xi)$. The regular element $\xi$ is said to be generic (or, following the terminology of [@T1 Definition 1.2.3.1], very regular), if the dimension of $\mathfrak s_\xi$ is maximal among all $\mathfrak s_\eta$, for $\eta$ regular. According to [@T1 Lemma 1.3.1.1], for $\xi$ very regular in $\mathfrak k^\perp$, $\mathfrak g$ admits a root decomposition $\mathfrak g=\mathfrak g_0\oplus\bigoplus_{\alpha\in\Delta}\mathfrak g_\alpha$, where $\mathfrak g_0$ is the centralizer of $\mathfrak s_\xi$ in $\mathfrak g$, and we set $\mathfrak g_\alpha=\left\{x\in\mathfrak g:\ \mathrm{ad}(t)(x)=\alpha(t)x,\ t\in\mathfrak s_\xi\right\}$, for $\alpha$ in $\mathfrak s_\xi^*$. An element $\alpha$ of $\mathfrak s_\xi$ is said to be a root, if $\mathfrak g_\alpha\neq\{0\}$; the set of all roots is denoted by $\Delta$. Moreover, $\mathfrak g_0$ is $\sigma$-stable and inherits the structure of symmetric pair, whence $\mathfrak g_0=\mathfrak k_0\oplus\mathfrak p_0$ its Cartan decomposition. Finally, for a root $\alpha$ in $\Delta$, $\sigma(\mathfrak g_\alpha)=\mathfrak g_{-\alpha}$, whence $\mathfrak g$ admits a triangular decomposition $\mathfrak g=\mathfrak n_-\oplus\mathfrak g_0\oplus\mathfrak n_+$, where $\mathfrak n_\pm$ denotes the direct sum of all root spaces associated to positive/negative roots. From the triangular decomposition follows the generalized Iwasawa decomposition $\mathfrak g=\mathfrak k\oplus\mathfrak p_0\oplus \mathfrak n_+$: in fact, see also [@Dix Proposition 1.13.11], $\mathfrak g=\mathfrak k+\mathfrak p_0+\mathfrak n_+$ directly from the previous triangular decomposition. Namely, a general element $x$ of $\mathfrak g$ can be (uniquely) written as $x=x_-+k_0+p_0+x_+$, with $x_\pm$ in $\mathfrak n_\pm$, $k_0$ in $\mathfrak k_0$ and $p_0$ in $\mathfrak p_0$. Then, $x_-=(x_-+\sigma(x_-))-\sigma(x_-)$, and obviously $x_-+\sigma(x_-)$ belongs to $\mathfrak k$, while $\sigma(x_-)$ is in $\mathfrak n_+$, whence the first claim follows. Now, we assume $k+p_0+x_+=0$: then, $\sigma(k+p_0+x_+)=k-p_0+\sigma(x_+)=0$, whence $2p_0+x_+-\sigma(x_+)=0$. From the triangular decomposition of $\mathfrak g$ follows automatically $p_0=x_+=0$, and thus also $k=0$. The relations for the triangular decomposition of $\mathfrak g$ imply that $\mathfrak k_0\oplus \mathfrak n_+$ is a Lie subalgebra of $\mathfrak g$, and thus is $(\mathfrak k_0\oplus\mathfrak n_+)^\perp$ a coisotropic submanifold of $\mathfrak g^*$. \[p-iwasawa\] For a general symmetric pair $(\mathfrak g,\xi)$ and a very regular element $\xi$ of $\mathfrak k^\perp$, the reduction space $\mathrm H^0(B_\hbar)$ for the coisotropic submanifold $(\mathfrak k_0\oplus\mathfrak n_+)^\perp$ of $\mathfrak g^*$ equals $\mathrm S(\mathfrak p_0)^{\mathfrak k_0}[\![\hbar]\!]$. We perform a slight change in the proof, namely we consider the triangular decomposition $\mathfrak g=\mathfrak n_-\oplus\mathfrak k_0\oplus\mathfrak p_0\oplus\mathfrak n_+$: we observe that $\mathfrak n_-\oplus\mathfrak p_0$ is, in general, not a module for the Lie subalgebra $\mathfrak k_0\oplus\mathfrak n_+$. According to previous arguments, the differential $\mathrm d_{B_\hbar}$ can be written as a formal power series $\hbar\mathrm d_\mathrm{CE}+\hbar^3\mathrm d_3+\cdots$, where only odd powers of $\hbar$ appear, and a general element $f$ satisfying $\mathrm d_{B_\hbar}(f)=0$ can be written as $f=f_0+\hbar^2 f_2+\cdots$, where only even powers of $\hbar$ appear. In particular, $f_0$ is $\mathrm d_\mathrm{CE}$-closed: according to [@T1 ???], $f_0$ belongs then to $\mathrm S(\mathfrak p_0)^{\mathfrak k_0}$. Therefore, to prove the claim, it suffices to prove that the operators $\mathrm d_{2n+1}$, $n\geq 1$ act trivially on $\mathrm S(\mathfrak p_0)^{\mathfrak k_0}$. The differential operator $\mathrm d_{2n+1}$ is the sum of differential operators associated to admissible graphs in $\mathcal G_{2n+1,1}$, see Subsubsection \[sss-2-3-1\]. In view of Proposition \[p-quant-0\], an admissible graph $\Gamma$ yielding a (possibly) non-trivial contribution to $\mathrm d_{2n+1}$ is either of type Bernoulli or of type wheel or of mixed type. We first consider $\Gamma$ of type Bernoulli in $\mathcal G_{2n+1,1}$: by Proposition \[p-quant-0\], $\Gamma$ has a root ([*i.e.*]{} a vertex of the first type with no incoming edges), a phantom edge ([*i.e.*]{} an edge with no final point) and from all vertices of the first type there is exactly on outgoing edge to the unique vertex of the first type. To every edge $e$ of $\Gamma$ corresponds a superpropagator, which in turn yields a coloring $e=e^++e^-$, where the color $\pm$ corresponds to linear coordinates on $\mathfrak n_-\oplus\mathfrak p_0$ and $\mathfrak k_0\oplus\mathfrak n_+$ respectively. If the vertex of the second type is associated to $f_0$ in $\mathrm S(\mathfrak p_0)^{\mathfrak k_0}$, all edges pointing to the said vertex are colored by $+$, as they correspond to derivative with respect to $\mathfrak p_0$. The phantom edge is obviously colored by $-$, as it corresponds to an element of $\mathfrak k_0^*\oplus\mathfrak n^*_+$. We consider the vertex $v$ of the first type from which departs the phantom edge: the edge departing from $v$ to the only vertex of the second type is colored by $+$, thus in view of Lemma \[l-vanish-4\], the only incoming edge to $v$ with initial point a distinct vertex of the first type is colored by $-$. Therefore, all edges connecting two distinct vertices of the first type are colored by $-$: to be even more precise, they correspond to derivatives and contractions with respect to $\mathfrak n_+$. Finally, the root of $\Gamma$ carries an element of $\mathfrak n_+$, which vanishes upon restriction. We then consider an admissible graph $\Gamma$ of type wheel. Therefore, $\Gamma$ has a phantom edge with initial point $v$, a vertex of the first type, and no root, and from each vertex of the first type different from $v$ depart exactly one edge to the only vertex of the second type. The phantom edge is colored by $-$, and it corresponds to an element of $\mathfrak k_0^*\oplus\mathfrak n_+^*$; as the differential operator corresponding to $\Gamma$ acts on $\mathrm S(\mathfrak p_0)^{\mathfrak k_0}$, all edges pointing to the only vertex of the second type are colored by $+$. If we first assume that the phantom edge of $\Gamma$ corresponds to an element of $\mathfrak k_0^*$, the relations $[\mathfrak k_0,\mathfrak n_\pm]\subseteq \mathfrak n_\pm$ and $[\mathfrak k_0,\mathfrak p_0]\subseteq \mathfrak p_0$ imply that the vertex $v$ presents automatically a configuration of the form $(e_1,e_2,e_\mathrm{gh})$, $e_1=(\bullet,v)$, $e_2=(v,\bullet)$, with the same coloring, [*i.e.*]{} $(e_1^+,e_2^+,e_\mathrm{gh}^-)$ or $(e_1^-,e_2^-,e_\mathrm{gh}^-)$, thus the corresponding integral weight vanishes because of Lemma \[l-vanish-4\]. If the phantom edge of $\Gamma$ corresponds to an element of $\mathfrak n_+^*$, we first assume that the edge $e_2$ departing from $v$ corresponds to derivation and contraction with respect to $\mathfrak k_0$ or $\mathfrak p_0$ ([*i.e.*]{} the color of $e_2$ is $-$ and $+$ respectively): the relations $[\mathfrak k_0,\mathfrak n_+]\subseteq \mathfrak n_+$ and $[\mathfrak p_0,\mathfrak n_+]\subseteq \mathfrak n_+$, together with the Cartan relations for the small symmetric pair $\mathfrak g_0$ imply that the edge $e_1$ carries simultaneously derivation and contraction with respect to $\mathfrak n_+$ and $\mathfrak k_0$ or $\mathfrak p_0$, which is impossible. If $e_2$ corresponds to either $\mathfrak n_-$ or $\mathfrak n_-$, the relation $[\mathfrak n_\pm,\mathfrak p_0]\subseteq \mathfrak n_\pm$ implies that to $v$ corresponds a configuration of the form either $(e_1^+,e_2^+,e_\mathrm{gh}^-)$ or $(e_1^+,e_2^+,e_\mathrm{gh}^-)$, which implies triviality of the corresponding integral weight. We finally consider an admissible graph $\Gamma$ in $\mathcal G_{2n+1,1}$ of mixed type Bernoulli-wheel. In this case, $\Gamma$ has a unique vertex $v$ of the first type from which departs the phantom edge, and there is a vertex $w$, from which departs an edge from a wheel-like graph $\Gamma_1$ to the root of a Bernoulli-like graph $\Gamma_2$. The coloring of $\Gamma_2$ can be deduced [*via*]{} the same arguments used previously for a Bernoulli type graph: in particular, the color of the edge with the root of $\Gamma_2$ as final point is $-$: more precisely, it corresponds to derivation and contraction with respect to $\mathfrak n_+^*$. The Cartan relations for the small symmetric pair $\mathfrak g_0$ together with $[\mathfrak k_0,\mathfrak n_\pm]\subseteq \mathfrak n_\pm$ and $[\mathfrak p_0,\mathfrak n_\pm]\subseteq \mathfrak \mathfrak n_\pm$ imply that the internal edges of the wheel-like graph $\Gamma_1$ have the same color, either $+$ or $-$; the corresponding edges correspond to derivation and contraction with respect to either $\mathfrak n_-$ and $\mathfrak n_+$. The corresponding differential operators are of the form $$\text{either}\ \mathrm{tr}_{\mathfrak n_-}\!\left(\mathrm{ad}(X_1)\cdots\mathrm{ad}(X_p)\mathrm{ad}(Y)\right)\ \text{or}\ \mathrm{tr}_{\mathfrak n_+}\!\left(\mathrm{ad}(X_1)\cdots\mathrm{ad}(X_p)\mathrm{ad}(Y)\right),$$ where $X_i$ are general elements of $\mathfrak p_0$ and $Y$ is an element either of $\mathfrak n_-$ or $\mathfrak n_+$. Such operators are clearly trivial due to the nilpotence of both $\mathfrak n_\pm$. ### Polarizations {#sss-2-4-4} This short Subsubsection also serves the purpose of fixing notation and conventions for certain issues, which will be dealt later on by means of deformation quantization. For more details on polarizations of Lie algebras, we refer to [@Dix Chapter 1.12]. A general element $\xi$ in $\mathfrak g^*$ defines a skew-symmetric bilinear form on a finite-dimensional Lie algebra $\mathfrak g$ [*via*]{} the assignment $B_\xi(x_1,x_2)=\langle\xi,[x_1,x_2]\rangle$. It is clear that $B_\xi$ restricts to a non-degenerate skew-symmetric bilinear form on $\mathfrak g/\mathfrak g(\xi)$, whence $\dim \mathfrak g+\dim\mathfrak g(\xi)$ is even. A polarization $\mathfrak b$ of $\xi$ as above is a Lie subalgebra of $\mathfrak g$, which is isotropic with respect to $B_\xi$ ([*i.e.*]{} $\xi$ defines a character for $\mathfrak b$) and of maximal dimension ([*i.e.*]{} the dimension of $\mathfrak b$ equals $(\dim\mathfrak g+\dim\mathfrak g(\xi))/2$). The isotropy condition on $\mathfrak b$ makes it automatically an algebraic subalgebra of $\mathfrak g$. A polarization $\mathfrak b$ of $\xi$ satisfies Pukanszky’s condition, if $\mathfrak b=\mathfrak g(\xi)+\mathfrak b_u$, where $\mathfrak b_u$ denotes the unipotent radical of $\mathfrak b$, see also [@T1 Subsubsection 1.5.2] for equivalent characterizations of Pukanszky’s condition. \[p-polar\] For a finite-dimensional Lie algebra $\mathfrak g$ and a general element $\xi$ of $\mathfrak g^*$, the reduction space $\mathrm H^0(B_\hbar)$ associated to the coisotropic submanifold $\xi+\mathfrak b^\perp$ equals $\mathbb K[\![\hbar]\!]$. The proof makes use of [@T1 Lemma 1.5.2.2] concerning equivalent characterizations of Pukanszky’s condition: in fact, we use the equivalence between the above characterization of Pukanszky’s condition and the one stating that $\mathrm{Ad}^*(B)(\xi)=\xi+\mathfrak b^\perp$, for $B$ an algebraic, connected subgroup of $G$ (an algebraic group with Lie algebra $\mathfrak g$) with Lie algebra $\mathfrak b$. According to [@T1 Subsubsection 1.5.2], $\mathrm{Ad}^*(B)(\xi)$ is a Zarisky open subset of $\xi+\mathfrak b^\perp$. We consider an element $f=f_0+\mathcal O(\hbar)$ of $B_\hbar^0$, $f_i$ in $\mathbb K[\xi+\mathfrak b^\perp]$. If $f$ is $\mathrm d_{B_\hbar}$-closed, then $f_0$ is $B$-invariant, which, by the above, means that $f_0$ is constant. By recurrence, the higher order identities reduce to $f_i$ $B$-invariant, $i\geq 1$, whence the claim follows. Products on quantum reduction algebras {#ss-2-5} -------------------------------------- After having discussed in some detail the relevant quantum reduction algebras which we will encounter in the sequel, we are now interested in a detailed discussion of the existence of associative products on quantum reduction algebras. We consider the dg vector space $B_\hbar$ in its full generality for a finite-dimensional Lie algebra $\mathfrak g$ and a Lie subalgebra $\mathfrak h$ thereof, to which we associate a dg algebra $B$, whose deformation quantization $B_\hbar$ is a flat $A_\infty$-algebra: in particular, this means that the graded vector space $\mathrm H^\bullet(B_\hbar)$ is endowed with an associate product. More precisely, the $A_\infty$-structure $\mu_B+\mathcal U_B(\hbar\pi)$, where $\pi$ denotes here the $P_\infty$-structure on $B$ Fourier-dual to the Poisson bivector on $X=\mathfrak g^*$, consists of infinitely many Taylor components $$\mathcal U_B(\hbar\pi)^n:B_\hbar^{\otimes n}\to B_\hbar[2-n],\ n\geq 1,$$ which satisfy an infinite series of quadratic identities between them. For our purposes, we need only know that $\mathcal U_B(\hbar\pi)^1=\mathrm d_{B_\hbar}$, and that $\mu_B+\mathcal U_B(\hbar\pi)^2$ defines a $\mathbb K$-bilinear pairing of degree $0$ on $B_\hbar$, which is compatible with $\mathrm d_{B_\hbar}$ and which is associative up to a the homotopy $\mathcal U_B(\hbar\pi)^3$ with respect to $\mathrm d_{B_\hbar}$. Therefore, $\mu_B+\mathcal U_B(\hbar\pi)^2$ descends to the quantum reduction space $\mathrm H^\bullet(B_\hbar)$ to an associative product, which we denote for simplicity by $\star_{B_\hbar}$: its restriction to $\mathrm H^0(B_\hbar)$ defines an obvious deformation of the commutative product on $B^0$. We refer to [@CT Section 3] for a careful description of the deformed product $\star_{B_\hbar}$ on $\mathrm H^0(B_\hbar)$ in the case of a symmetric pair $(\mathfrak g,\sigma)$ (with Cartan decomposition $\mathfrak g=\mathfrak k\oplus\mathfrak p$) and of a character $\chi$ of the Lie subalgebra $\mathfrak k$. We only observe that in [@CT Section 3], the authors use the notation $\star_{\mathrm{CF},\lambda}=\star_{B_\hbar}$. Applications of biquantization in Lie theory for symmetric pairs {#s-3} ================================================================ In the present Section, we discuss the first relevant applications of biquantization, as discussed in detail in the previous Section, to concrete problems in Lie theory in the framework of a symmetric space $(\mathfrak g,\sigma)$ with standard Cartan decomposition $\mathfrak g=\mathfrak k\oplus\mathfrak p$. In fact, the results presented here are the revisited versions of the results of [@CT Section 4] taking into account the more precise and correct approach to biquantization presented in the previous Section. A comparison between the quantum deformed product and Rouvière’s product {#ss-3-1} ------------------------------------------------------------------------ We consider the quantum reduction algebra $(\mathrm H^0(B_\hbar),\star_{B_\hbar})$ for a symmetric pair $(\mathfrak g,\sigma)$ with the standard Cartan decomposition. On the other hand, we consider the quantum deformed algebra $(A_\hbar,\star_{A_\hbar})$ and the $A_\infty$-$A_\hbar$-$B_\hbar$-bimodule $K_\hbar$: unraveling the $A_\infty$-identities for $K_\hbar$, we see that $K_\hbar$, which is concentrated in degree $0$, becomes actually an $(A_\hbar,\star_{A_\hbar})$-$(\mathrm H^0(B_\hbar),\star_{B_\hbar})$-bimodule, and we denote by $\star_L$ and $\star_R$ the respective left $(A_\hbar,\star_{A_\hbar})$- and right $(\mathrm H^0(B_\hbar),\star_{B_\hbar})$-action on $K_\hbar$. More explicitly, in the present framework, we have $A_\hbar=\mathrm S(\mathfrak g)[\![\hbar]\!]$, $B_\hbar^0=K_\hbar=\mathrm S(\mathfrak p)[\![\hbar]\!]$. We observe that the linearity of the Poisson bivector $\pi$ on $X=\mathfrak g^*$ has an interesting by-product, as already remarked in [@K Subsubsection 8.3.1]: although there are infinitely many bidifferential operators appearing in the deformed product $\star_{A_\hbar}$, the action of the infinite series $\star_{A_\hbar}$, for $\hbar=1$, on $\mathrm S(\mathfrak g)\otimes\mathrm S(\mathfrak g)$ is well-defined, as can be proved by inspecting the integral weights and counting degrees. Similar arguments hold true also for the components of the $A_\infty$-structure $\mu_B+\mathcal U_B(\hbar\pi)$ on $B_\hbar$ and for the $A_\infty$-$A_\hbar$-$B_\hbar$-bimodule structure on $K_\hbar$. As a consequence, we may consider the $A_\infty$-algebras $A_\hbar$ and $B_\hbar$ and the $A_\infty$-$A_\hbar$-$B_\hbar$-bimodule $K_\hbar$ as polynomial deformations with respect to $\hbar$, and in particular we may safely consider the value of the parameter $\hbar=1$. Therefore, we consider the associative algebras $(A,\star_A)$, the $A_\infty$-algebra $(B,\mu_B+\mathcal U_B(\pi))$, with corresponding quantum reduction algebra $(\mathrm H^0(B),\star_B)$, and the $(A,\star_A)$-$(\mathrm H^0(B),\star_B)$-bimodule $(K,\star_L,\star_R)$. The deformed differential $\mathrm d_B$ on $B^0$ is now a differential operator from $\mathrm S(\mathfrak p)$ to $\mathrm S(\mathfrak p)\otimes \mathfrak k^*$ of infinite order, whose action is well-defined. In a similar way, the pairing $\mu_B+\mathcal U_B(\pi)^2$ is a well-defined bidifferential operator on $\mathrm S(\mathfrak p)$ of infinite order, and so $\star_L$ and $\star_R$ (we should more precise on $\star_R$, as we should speak of the infinite-order bidifferential operator on $\mathrm S(\mathfrak p)$ coming from the $A_\infty$-$A$-$B$-bimodule structure on $K$). The deformed product $\star_A$ on $A=\mathrm S(\mathfrak g)$ has been explicitly characterized in [@K; @BCKT]. More precisely, we consider the following function on $\mathfrak g$, $$\label{eq-duf} q(x)=\underset{\mathfrak g}\det \!\left(\frac{\sinh\!\left(\frac{\mathrm{ad}(x)}2\right)}{\frac{\mathrm{ad}(x)}2}\right),$$ which is analytic in a neighborhood of $0$. It can be expanded in a power series of the polynomials $c_n(x)=\mathrm{tr}_\mathfrak g(\mathrm{ad}(x)^n)$, $n\geq 1$. Alternatively, it may be viewed as an element of the completed symmetric algebra $\widehat{\mathrm S}(\mathfrak g^*)$ of the dual of $\mathfrak g$, and as such, as an invertible, $\mathfrak g$-invariant, infinite-order differential operator with constant coefficients acting on $A$. Similar arguments hold true also if we consider its square root $\sqrt{q}$: we denote by $\partial_{\sqrt{q}}$ the corresponding invertible, $\mathfrak g$-invariant, infinite-order differential operator on $A$. We further denote by $\beta$ the Poincaré–Birkhoff–Witt (shortly, PBW) isomorphism from $\mathrm S(\mathfrak g)$ to $\mathrm U(\mathfrak g)$, [*i.e.*]{} $\beta$ is the symmetrization morphism $$\mathrm S(\mathfrak g)\ni x_1\cdots x_n\mapsto \frac{1}{n!}\sum_{\sigma\in\mathfrak S_n}x_{\sigma(1)}\cdots x_{\sigma(n)}\in\mathrm U(\mathfrak g),\ x_j\in \mathfrak g,\ j=1,\dots,n.$$ Then, the deformed product $\star_A$ on $\mathrm S(\mathfrak g)$ is related with the product in $\mathrm U(\mathfrak g)$ [*via*]{} $$\label{eq-DK} \beta\!\left(\partial_{\sqrt{q}}(f_1)\star_A\partial_{\sqrt{q}}(f_2)\right)=\beta(\partial_{\sqrt{q}}(f_1))\cdot\beta(\partial_{\sqrt{q}}(f_2)),\ f_i\in A,\ i=1,2,$$ and $\cdot$ denotes here the product in $\mathrm U(\mathfrak g)$. The way the operator $\partial_{\sqrt{q}}$ arises in the framework of deformation quantization has been elucidated in detail in [@K Subsubsections 8.3.1, 8.3.2 and 8.3.3], combining the results therein with [@Sh]. We also refer to [@BCKT Part II] for a complete overview of the applications of deformation quantization as in [@K] in Lie theory. The motivation for the following computations lies in the comparison in Identity  between the UEA $\mathrm U(\mathfrak g)$ and the quantum deformed algebra $(A,\star_A)$, which are related precisely by the “strange” automorphism $\partial_{\sqrt{q}}$ of the symmetric algebra $\mathrm S(\mathfrak g)$, which appears also in Duflo’s Theorem: namely, the composition $\beta\circ \partial_{\sqrt{q}}$ defines an algebra isomorphism between $\mathrm S(\mathfrak g)^\mathfrak g$ and the center of $\mathrm U(\mathfrak g)$. The main point is that Kontsevich’s deformed product $\star_A$ contains bidifferential operators, which are represented in terms of wheel-like graphs: such graphs are precisely responsible for the appearance of the “strange” automorphism $\partial_{\sqrt{q}}$. Quite similarly, in the case of a symmetric pair $(\mathfrak g,\sigma)$, we may consider the associative algebra $(\mathrm H^0(B),\star_B)$, where $\mathrm H^0(B)=\mathrm S(\mathfrak p)^\mathfrak k$. It is worth observing that Poisson reduction methods yield a Poisson structure on $\mathrm S(\mathfrak p)^\mathfrak k$ simply by restriction, and the product $\star_B$ defines a deformation quantization of $\mathrm S(\mathfrak p)^\mathfrak k$ in the sense of Kontsevich. On the other hand, for any choice of a character $\chi$ of $\mathfrak k$ ([*i.e.*]{} a $1$-dimensional $\mathfrak k$-representation on $\mathbb K$), the PBW isomorphism $\beta$ induces a direct sum decomposition $\mathrm U(\mathfrak g)=\beta(\mathrm S(\mathfrak p))\oplus \mathrm U(\mathfrak g)\cdot \mathfrak k^{-\chi}$, where $\mathfrak k^{-\chi}$ denotes the affine subspace of $\mathrm U(\mathfrak g)$ spanned by elements of the form $x-\chi(x)$, $x$ in $\mathfrak k$. Then, Rouvière defines also a “deformation quantization” of the Poisson algebra $\mathrm S(\mathfrak p)^\mathfrak k$ [*via*]{} the formula $$\label{eq-rouv} \beta(f_1\# f_2)=\beta(f_1)\cdot\beta(f_2)\ \text{modulo $\mathrm U(\mathfrak g)\cdot \mathfrak k^{-\chi}$,}\ f_i\in\mathrm S(\mathfrak p)^\mathfrak k,\ i=1,2.$$ The PBW isomorphism (of vector spaces) is obviously $\mathfrak g$-invariant, hence it is automatically $\mathfrak k$-invariant: therefore, $\beta$ restricts to a $\mathfrak k$-invariant isomorphism of vector spaces from $\mathrm S(\mathfrak p)$ to $\beta(\mathrm S(\mathfrak p))\subseteq \mathrm U(\mathfrak g)$. In particular, from the above decomposition $\mathrm U(\mathfrak g)=\beta(\mathrm S(\mathfrak p))\oplus \mathrm U(\mathfrak g)\cdot \mathfrak k^{-\chi}$, it follows immediately that the right-hand side of Identity  defines a unique bilinear pairing $\#$ on $\mathrm S(\mathfrak p)$, which restricts to an associative product on $\mathfrak k$-invariant elements. We are now going to compare the products $\star_B$ and $\#$ [*via*]{} biquantization techniques. As one could naturally guess from Identity , the two products on $\mathrm S(\mathfrak p)^\mathfrak k$ do not coincide, but are related to each other in a similar fashion, [*i.e.*]{} through a “relative” counterpart of Duflo’s “strange” automorphism. The novelty of the approach through biquantization is the fact that we use it to compare $\star_B$ on $\mathrm H^0(B)$ with $\star_A$ on $A$; the rest of the proof, [*i.e.*]{} the comparison of different automorphisms of $\mathrm S(\mathfrak p)$ similar in shape to Duflo’s “strange” automorphism, is really similar to the proof presented in [@K Subsection 3.1] with due modifications. Of course, the upcoming discussion can be generalized to the framework of some Lie subalgebra $\mathfrak h$ of any finite-dimensional Lie algebra $\mathfrak g$ over $\mathbb K$: we will discuss generalizations of the results presented here elsewhere, in particular in relationship with equivalences of categories of representations of Lie algebras and corresponding subalgebras and with the relative Duflo conjecture. ### A version of Duflo’s “strange” automorphism for symmetric pairs {#sss-3-1-1} Using the previous notation and conventions, we define the following operator $$\label{eq-A-wheel} \mathcal A(f)=f\star_L 1,\ f\in A=\mathrm S(\mathfrak g)$$ from $A$ to $K=\mathrm S(\mathfrak p)$. Of course, we could have first defined the operator $\mathcal A_\hbar$ from $A_\hbar$ to $K_\hbar$, for $\hbar$ a formal parameter. By its very construction, $\mathcal A_\hbar$ is a deformation of the surjective projection from $A$ to $K$, which we denote by $\pi$. By its very construction, $\mathcal A_\hbar=\pi\circ \mathbb A_\hbar$, where $\pi$ is extended $\hbar$-linearly to $A_\hbar$, while $\mathbb A_\hbar$ is a formal series of differential operators on $A$, where $\mathbb A_0=\mathrm{id}$, and $\mathbb A_n$ (the coefficient of degree $n$ with respect to $\hbar$) is a differential operator of order $n$. In particular, it is clear that $\mathbb A_\hbar$ is invertible. By the same arguments as before, we may safely set $\hbar=1$, and thus we get an invertible differential operator $\mathbb A$ on $A$ of infinite order. We consider $(A,\star_A)$ as a left $(A,\star_A)$-module: then, $\mathcal A$ is a surjective morphism of $(A,\star_A)$-modules from $(A,\star_A)$ to $(K,\star_L)$, whence $K\cong A/I$, where $I=\mathrm{Ker}(\mathcal A)$. By the very definition of $\mathcal A$, $I=A\star_A \mathbb A^{-1}(\mathfrak k)$: in fact, $A\star_A \mathbb A^{-1}(\mathfrak k)\subseteq I$ by its very construction. To prove the opposite inclusion, we re-introduce momentarily the formal parameter $\hbar$. For $\hbar=0$, the ideal $I=\langle\mathfrak k\rangle$ of $A$ is finitely-generated. In fact, as $I$ is the two-sided ideal of $A$, viewed here as a commutative algebra, generated by the ideal $\mathfrak k$, viewed here as an ideal of linear functions on $X=\mathfrak g^*$. In particular, there is a surjective morphism $A^{\oplus \dim\mathfrak k}\to I\to 0$ of $A$-modules: by the previous argument, there is a morphism of left $A_\hbar$-modules $A\star_{A_\hbar} \mathbb A_\hbar^{-1}(\mathfrak k)\to I_\hbar$, which is a formal $\hbar$-deformation of the surjective morphism $A^{\oplus \dim\mathfrak h}\to I$, whence the surjectivity of the deformed morphism follows. Therefore, $I_\hbar=A_\hbar\star_{A_\hbar} \mathbb A_\hbar^{-1}(\mathfrak k)$, whence the claim follows by setting safely $\hbar=1$. It remains to compute $\mathbb A^{-1}(\mathfrak k)$. Writing $\mathbb A=\mathrm{id}+\sum_{n\geq 1}\mathbb A_n$, $\mathbb A_n$, $n\geq 1$, has order $n$ by construction; furthermore, $\mathbb A_n$, $n\geq 1$, has no constant term. Namely, $\mathbb A_n$ is specified by differential operators associated to admissible graphs in $\mathcal G_{n,1}$: recalling from the previous Section the construction of the differential operator associated to $\Gamma$ admissible of type $(n,1)$, if $\Gamma$ has no edge pointing to the only vertex of the second type, the corresponding integral weight vanishes by a dimensional argument. The operator $\mathbb A^{-1}$ is completely determined by the power series expansion of $\mathbb A$: once again, it is of the form $\mathrm{id}+\sum_{n\geq 1} \widetilde{\mathbb A}_n$, where $\widetilde{\mathbb A}_n$ has no constant term, for $n\geq 1$, as follows by an easy computation. We thus compute $\mathbb A(k)$, for $k$ a general element of $\mathfrak k$. Because of degree reasons, see also [@K Subsubsection 8.3.1], $\mathbb A(k)=k+\mathbb A_1(k)$. Further, it is readily checked that $\mathbb A_1$ is the sum of two differential operators, associated to the following admissible graphs of type $(1,2)$ \ The contribution of the first graph is trivial, because $k$ is viewed as a linear function on $X$. We consider the second graph: we want to observe that such a graph did not appear in the computations performed in [@CT]. First of all, the corresponding integral weight is $$\label{eq-loop-weight} \int_{\mathcal C_{1,2}^+}\rho\omega^{+,-},$$ omitting wedge products. In fact, in the superpropagator $\omega_e$, only two of the $4$-colored propagators are non-trivial, namely $\omega^{+,+}$ and $\omega^{+,-}$ by construction; since it acts as a derivation on an element of $\mathfrak k$, by its very definition, the part with $\omega^{+,+}$ vanishes. \[l-loop-weight\] The integral $\int_{\mathcal C_{1,2}^+}\mathrm d\eta\wedge\omega^{+,-}$ equals $\frac{1}4$. The integral weight associated to the previous admissible graph is $\int_{\mathcal C_{1,2}^+}\mathrm d\eta\omega^{+,-}$, where we have suppressed wedge products between the forms in the integrand. The $1$-form $\rho$ is exact, whence $$\int_{\mathcal C_{1,2}^+}\mathrm d\eta\omega^{+,-}=\int_{\partial C_{1,2}^+}\eta\omega^{+,-},$$ where we have used notation from Subsection \[ss-2-2\]. Hence, it suffices to compute all boundary contributions to evaluate the integral. The boundary strata of $\mathcal C_{1,2}^+$ are of the type $\mathcal C_{A,B}^+\times \mathcal C_{1\smallsetminus A,[2]\smallsetminus B\sqcup \{\bullet\}}^+$, for $A$ a subset of $[1]$ and $B$ an ordered subset of $[2]$, such that $0\leq |A|\leq 1$, $0\leq |B|\leq 2$ and $|A|+|B|\leq 2$. Dimensional arguments imply that there are only two types of such boundary strata, $\mathcal C_{0,2}^+\times \mathcal C_{1,1}^+$ and $\mathcal C_{0,3}^+\times \mathcal C_{1,0}^+$, which correspond to five different situations. We consider the boundary stratum $\mathcal C_{0,2}^+\times\mathcal C_{1,1}^+$, which corresponds to the situation where $i_1)$ the point on the positive real axis approaches the origin, $ii_1)$ the point in the interior of the first quadrant collapses to the point on the positive real axis or $iii_1)$ the point $1$ approaches the origin. The boundary conditions for $\omega^{-,+}$ yield triviality of the contributions $i_1)$ and $iii_1)$; the second one yields $\frac{1}4\int_{\mathcal C_{1,1}^+}\omega^+=\frac{1}4$, and we have already included orientation signs. The boundary stratum $\mathcal C_{0,3}^+\times\mathcal C_{1,0}^+$ corresponds to the point $1$ approaching either the positive imaginary axis or the positive real axis: in the first case, the corresponding contribution vanishes by means of the boundary conditions for $\omega^{+,-}$, while in the second case, the function $\eta$ vanishes when its argument approaches the real axis. Recalling now the construction of the superpropagators in biquantization from Subsubsection \[sss-2-3-2\], the differential operator corresponding to the second graph in Figure 7 is $$\frac{1}4\left[\mathrm{tr}_{\mathfrak p}(\mathrm{ad}_\mathfrak{k}(\bullet))-\mathrm{tr}_{\mathfrak k}(\mathrm{ad}_\mathfrak{k}(\bullet)\right)]=\delta(\bullet)-\frac{1}4\mathrm{tr}_\mathfrak g(\mathrm{ad}(\bullet)),$$ whence $\mathbb A(k)=k+\delta(k)-\frac{1}4\mathrm{tr}_\mathfrak g(\mathrm{ad}(k))$. Therefore, we have $$\mathbb A^{-1}(\mathbb A(k))=k=\mathbb A^{-1}(k)+\delta(k)-\frac{1}4\mathrm{tr}_\mathfrak g(\mathrm{ad}(k)),\ \text{whence}\ \mathbb A^{-1}(k)=k-\delta(k)+\frac{1}4\mathrm{tr}_\mathfrak g(\mathrm{ad}(k)),\ k\in\mathfrak k.$$ We observe that we have used the fact that the terms of $\mathbb A_\hbar^{-1}$ of degree higher or equal than $1$ are differential operators without constant term. Putting all previous arguments together, we have the identification of right $(A,\star_A)$-modules $$I=A\star_A \mathfrak k^{-\delta+\frac{1}4\mathrm{tr}_\mathfrak g\circ\mathrm{ad}}.$$ Further, we consider the restriction of $\mathcal A$ to $K$, viewed here as a subalgebra of $A$. First of all, any admissible graph $\Gamma$ of type $(n,2)$ yielding a possibly non-trivial contribution to $\mathcal A$ has exactly $2n$ edges because of dimensional reasons. To any edge $e$ of $\Gamma$ corresponds a superpropagator $\omega_e$, whose components are only of type $(+,+)$ or $(+,-)$: it follows immediately from the boundary conditions for both of them that $\Gamma$ has no edge departing from the only vertex on the positive real axis, and that any edge pointing to this vertex has a corresponding superpropagator with color $(+,+)$. This excludes immediately double edges pointing to the only vertex of $\Gamma$ on the positive real axis. In particular, this means that any admissible graph $\Gamma$ of type $(n,2)$ yielding a non-trivial contribution to $\mathcal A$ in the present situation must satisfy the following rule: from any vertex of the first of $\Gamma$ departs at most one edge pointing to the only vertex on the positive real axis (of course, because of the presence of short loops, the initial and final point of such an edge may coincide). We assume therefore that $p\leq n$ edges have the only vertex of the second type on the positive real axis as endpoint. If $p<n$, there are then $2n-p$ edges, whose endpoints are both vertices of the first type (double edges and short loops are allowed). Since every edge is associated to a derivation, the polynomial degree associated to the vertices of the first type of $\Gamma$ is $-n+p$ (counting $n$ because of the linearity of the Poisson structure and $p-2n$ derivations), which is strictly negative, leading to a contradiction. Therefore, form any vertex of the first type of $\Gamma$ depart exactly one edge to the only vertex of the positive real axis and one to a vertex of the first type; the polynomial degree of the corresponding differential operator on $K$ is immediately $0$. Because of the linearity of the Poisson structure, exactly one edge has a vertex of the first type as final point, whence admissible graphs of type $(n,2)$ contributing non-trivially to $\mathcal A$ are disjoint unions of wheel-type graphs; we observe that the $1$-wheel may in principle appear. Further, only wheel-like graphs with $n$ even contribute (possibly) non-trivially to $\mathcal A$. Namely, by the previous argument, from every vertex of the first type departs exactly one edge to the only vertex of the second type on the positive real axis, whose color is $(+,+)$ and whose operator-valued part corresponds to derivation and contraction with respect to $\mathfrak p$. The Cartan relation $[\mathfrak k,\mathfrak p]\subseteq \mathfrak p$ implies that at each vertex of the first type in a wheel-like graph $\Gamma$, the edge arriving at such a vertex must have color either $(+,+)$ or $(+,-)$, while the edge departing from it on th wheel must have opposite color. In other words, the edges of the cycle in a wheel-like graph $\Gamma$ must have alternating colours $(+,+)$ and $(+,-)$: this, in turn, excludes immediately $n$-wheels with $n$ odd. Summarizing all previous arguments, the restriction of the operator $\mathcal A$ to $K$, which we denote (improperly) by the same symbol, defines an invertible, translation-invariant differential operator on $K$. Its symbol, regarded as an element of the completed symmetric algebra $\widehat{\mathrm S}(\mathfrak p)$ and defined through $j_\mathcal A(x)=e^{-x}\mathcal A(e^x)$, has the explicit form $$j_\mathcal A(x)=\exp\left(\sum_{n\geq 1}W_{2n}^\mathcal A\mathrm{tr}_\mathfrak p(\mathrm{ad}^{2n}(x))\right),\ x\in\mathfrak p,$$ where $W_{2n}^\mathcal A$, $n\geq 1$, denotes the integral weight of following wheel-like graph: \ We observe that $j_\mathcal A$ is analytic in a neighborhood of $0$ in $\mathfrak p$. We observe that the Cartan relations for the symmetric pair $(\mathfrak g,\sigma)$ imply immediately that, for a general element $x$ of $\mathfrak p$, $\mathrm{ad}(x)^2$ is a well-defined endomorphism of $\mathfrak p$, thus all even powers of the adjoint representation restricted to $\mathfrak p$: hence the above expression is well-defined. Finally, all previous computations imply also the direct sum decomposition of $(A,\star_A)$: $$A=K\oplus (A\star_A\mathfrak k^{-\delta+\frac{1}4\mathrm{tr}_\mathfrak g\circ\mathrm{ad}}).$$ On the other hand, $K=B^0=\mathrm S(\mathfrak p)$ by definition. We may therefore consider the endomorphism $\mathcal B$ of $K$ defined through $$\label{eq-B-wheel} \mathcal B(f)=1\star_R f,\ f\in K=B^0=\mathrm S(\mathfrak p).$$ We may repeat almost [*verbatim*]{} the previous arguments to evaluate explicitly the operator $\mathcal B$: it is an invertible, translation-invariant differential operator on $K$ of infinite order, with symbol in $\widehat{\mathrm S}(\mathfrak p)$ given by $$j_\mathcal B(x)=\exp\left(\sum_{n\geq 1}W_{2n}^\mathcal B\mathrm{tr}_\mathfrak p(\mathrm{ad}^{2n}(x))\right),\ x\in\mathfrak p,$$ where $W_{2n}^\mathcal B$, $n\geq 1$, denotes the integral weight of following wheel-like graph: \ Once again, notice that $j_\mathcal B$ is analytic in a neighborhood of $0$ in $\mathfrak p$. At this point one could wonder whether or not the integral weights $W_{2n}^\mathcal A$ and $W_{2n}^\mathcal B$ coincide (which would imply that $j_\mathcal A=j_\mathcal B$). First of all, we observe that in the Formulæ for $j_\mathcal A$ and $j_\mathcal B$, only weights of even wheels appear because of the vanishing of the the differential operators corresponding to odd wheel-like graphs. In fact, [*e.g.*]{} the $1$-wheels $\mathcal W_1^\mathcal A$ and $\mathcal W_1^\mathcal B$ are both computable and yield distinct results. This can be seen either by computing separately both integral weights or by computing [*e.g.*]{} $W_1^\mathcal A$ and finding then a relationship between $W_1^\mathcal A$ and $W_1^\mathcal B$. First of all, the integral weight $W_1^\mathcal A$ is explicitly $$W_1^\mathcal A=\int_{\mathcal C_{1,2}^+}\rho\omega^{+,+}.$$ It can be computed by the same technique used in Lemma \[l-loop-weight\]. Of course, there are certain differences to be taken into account, namely, the different boundary conditions satisfied by the $4$-colored propagator $\omega^{+,+}$. Here, the only boundary strata of $\mathcal C_{1,2}^+\cong\mathcal C_{1,1,0}^+$ which yield non-trivial contributions are $i)$ the stratum corresponding to the approach of the point in $Q^{+,+}$ to the only point on $i\mathbb R^+$ and $ii)$ the approach of the only point on $i\mathbb R^+$ to the origin. Both integrals are readily computed, as well as their orientation signs, which then yield the desired result. We now consider the following admissible graph of type $(2,1)$: \ It represents a $2$-form on the $3$-dimensional smooth manifold with corners $\mathcal C_{2,1}^+$, therefore, in virtue of Stokes’ Theorem, $$\int_{\mathcal C_{2,1}^+}\mathrm d(\mathrm d\eta\omega^{+,+})=0=\int_{\partial C_{2,1}^+}\mathrm d\eta\omega^{+,+}.$$ The boundary strata of codimension $1$ of $\mathcal C_{2,1}^+$ have been illustrated explicitly in Subsubsection \[sss-2-2-2\]: either because of the boundary conditions for $\omega^{+,+}$ and $\rho$ or because of dimensional reasons, it is not difficult to verify that only three boundary strata yield non-trivial contributions, namely the strata $\alpha$, $\theta$ and $\zeta$. The boundary stratum $\theta$, resp. $\zeta$, yields precisely $W_1^\mathcal B$, resp. $W_1^\mathcal A$; on the other hand, it is not difficult to verify that the boundary stratum $\alpha$ yields the non-trivial contribution $1/4$, as can be verified by a direct computation. Thus, in general, we cannot expect the weights $W_n^\mathcal A$ and $W_n^\mathcal B$ to coincide. ### Explicit comparison of the products of Rouvière and Cattaneo–Felder {#sss-3-1-2} The operator $\mathcal A$ is surjective, and its restriction to $K=\mathrm S(\mathfrak p)\subseteq A$ is an automorphism, while $\mathcal B$ is an automorphism of $K$, whence $$1\star_R f=\mathcal B(f)=\mathcal A(\mathcal A^{-1}(\mathcal B(f)))=\mathcal A^{-1}(\mathcal B(f))\star_L 1,\ f\in K=\mathrm S(\mathfrak p).$$ We now recall from Subsubsection \[sss-2-4-1\] that the quantum reduction algebra (at $\hbar=1$) $\mathrm H^0(B)=\mathrm S(\mathfrak p)^\mathfrak k$. We thus consider two elements $f_i$, $i=1,2$, of $\mathrm H^0(B)$, endowed with the deformed associative product $\star_B$; then $(K,\star_R)$ becomes a right $(\mathrm H^0(B),\star_B)$-module, and the latter module structure is compatible with the left $(A,\star_A)$-module structure on $(K,\star_L)$, whence $$\begin{aligned} &1\star_R (f_1\star_B f_2)=\left(\mathcal A^{-1}(\mathcal B(f_1\star_B f_2))\right)\star_L 1=\\ =& (1\star_R f_1)\star_R f_2=\left(\mathcal A^{-1}(\mathcal B(f_1))\star_L 1\right)\star_R f_2=\mathcal A^{-1}(\mathcal B(f_1))\star_L (1\star_R f_2)=\mathcal A^{-1}(\mathcal B(f_1))\star_L \left(\mathcal A^{-1}(\mathcal B(f_2))\star_L 1\right)=\\ =& \left(\mathcal A^{-1}(\mathcal B(f_1))\star_A \mathcal A^{-1}(\mathcal B(f_2))\right)\star_L 1, \end{aligned}$$ whence from the previous computations follows $$\label{eq-rouviere} (\mathcal A^{-1}\circ \mathcal B)(f_1\star_B f_2)=(\mathcal A^{-1}\circ \mathcal B)(f_1)\star_A (\mathcal A^{-1}\circ \mathcal B)(f_2)\ \text{modulo $A\star_A \mathfrak k^{-\delta+\frac{1}4\mathrm{tr}_\mathfrak g\circ\mathrm{ad}}$}.$$ We apply the operator $\beta\circ\partial_{\sqrt{q}}$ on both sides of Identity  and because of Identity , we get $$(\beta\circ\partial_{\sqrt{q}}\circ\mathcal A^{-1}\circ\mathcal B)(f_1\star_B f_2)=(\beta\circ\partial_{\sqrt{q}}\circ\mathcal A^{-1}\circ\mathcal B)(f_1)\cdot (\beta\circ\partial_{\sqrt{q}}\circ\mathcal A^{-1}\circ\mathcal B)(f_2)\ \text{modulo $\mathrm U(\mathfrak g)\cdot \mathfrak k^{-\delta+\frac{1}4\mathrm{tr}_\mathfrak g\circ\mathrm{ad}}$},$$ for $f_i$ in $\mathrm S(\mathfrak p)^\mathfrak k$, $i=1,2$. We introduce the function $J$ on $\mathfrak p$ defined [*via*]{} $$j(x)=\underset{\mathfrak p}\det\!\left(\frac{\sinh(\mathrm{ad}(x))}{\mathrm{ad}(x)}\right),\ x\in\mathfrak p.$$ This function is the determinant of the exponential map for the symmetric pair $(\mathfrak g,\sigma)$: it can be written as a formal power series of traces in $\mathfrak p$ of even powers of the restriction to $\mathfrak p$ of the adjoint representation of $\mathfrak g$. Similarly to what has been done before, we define $\partial_{\sqrt{j}}$ as the invertible, translation-invariant, $\mathfrak k$-invariant differential operator of infinite order on $\mathrm S(\mathfrak p)$, whose symbol is exactly the square root of $j$. We define a modified version of the previously introduced Rouvière’s product [*via*]{} $$\beta\!\left(\partial_{\sqrt{j}}(f_1\# f_2)\right)=\beta(\partial_{\sqrt{j}}(f_1))\cdot\beta(\partial_{\sqrt{j}}(f_2))\ \text{modulo $\mathrm U(\mathfrak g)\cdot\mathfrak k^{-\delta+\frac{1}4\mathrm{tr}_\mathfrak g\circ\mathrm{ad}}$.}$$ We observe that the left-hand side of the previous Identity is well-defined, because of the $\mathfrak k$-invariance and invertibility of $\partial_{\sqrt{j}}$. The key point is now the following identity, which puts into relationship the functions $q$, $J$, $j_\mathcal A$ and $j_\mathcal B$: $$\label{eq-DR-symb} j_\mathcal A(x)\sqrt{j(x)}=j_\mathcal B(x)\sqrt{q(x)},\ \forall x\in\mathfrak p.$$ The previous identity is the relative version, in the case of a symmetric pair $(\mathfrak g,\sigma)$, of the results of [@K Subsubsection 8.3.4]. Its proof is a consequence of [@CT Lemma 14 and Proposition 12]: it is result which is left unaltered by the changes to biquantization which have been previously discussed. \[t-Rou-CF\] For a general symmetric pair $(\mathfrak g,\sigma)$, Rouvière’s product $\#$ on $\mathrm S(\mathfrak p)^\mathfrak k$ coincides with the product $\star_B$ on $\mathrm H^0(B)=\mathrm S(\mathfrak p)^\mathfrak k$, [*i.e.*]{} $$\beta\!\left(\partial_{\sqrt{j}}(f_1\star_B f_2)\right)=\beta\!\left(\partial_{\sqrt{j}}(f_1)\right)\cdot \beta\!\left(\partial_{\sqrt{j}}(f_2)\right)\ \text{modulo $\mathrm U(\mathfrak g)\cdot\mathfrak k^{-\delta+\frac{1}4\mathrm{tr}_\mathfrak g\circ\mathrm{ad}}$},$$ for $f_i$, $i=1,2$, a general element of $\mathrm H^0(B)=\mathrm S(\mathfrak p)^\mathfrak g$. We observe that the characters $\delta$ and $\delta-1/4\ \mathrm{tr}_\mathfrak g\circ \mathrm{ad}$ differ precisely by a character of the Lie algebra $\mathfrak g$ itself (the trace of its adjoint representation) restricted to a character of the Lie subalgebra $\mathfrak h$; as a consequence, $1/4\ \mathrm{tr}_\mathfrak g\circ\mathrm{ad}$ yields an $\mathfrak h$-equivariant map from $\mathfrak g$ to the base field $\mathbb K$ ([*i.e.*]{} a linear functional on $\mathfrak g$ which vanishes on the subspace $[\mathfrak h,\mathfrak g]$), thanks to which we may actual consider only the character $\delta$, instead of the sum $\delta-1/4\mathrm{tr}_\mathfrak g\circ\mathrm{ad}$. More precisely, to the symmetric pair $\mathfrak g=\mathfrak k\oplus \mathfrak p$ we may associate a sub-symmetric pair $\mathfrak g_1=\mathfrak k_1\oplus \mathfrak p$, where $\mathfrak k_1=[\mathfrak p,\mathfrak p]$; it is clear that $\mathfrak g_1$ is an ideal of $\mathfrak g$. The reason is that, to pass from the expression on the right-hand side of the Identity in Theorem \[t-Rou-CF\] to the one on the left-hand side, we produce terms which are actually in $\mathrm U(\mathfrak g)\cdot \mathfrak k_1^{-\delta+\frac{1}4\ \mathrm{tr}_\mathfrak g\circ\mathrm{ad}}$, because we reverse two elements in $\mathfrak p$, actually producing an element of $\mathfrak k_1$ (whose commutator with elements of $\mathfrak p$ remains in $\mathfrak p$), and it is obvious that the second summand in the character vanishes on $\mathfrak k_1$, being a character of $\mathfrak g$. If we now consider a non-trivial character $\chi$ of $\mathfrak k$, we may associate $X=U_1=\mathfrak g^*$, $U_2=\chi+\mathfrak k^\perp$. Obviously, $A=\mathrm S(\mathfrak g)$, $B^0=K=\mathrm S(\mathfrak g)/\langle \mathfrak k^{-\chi}\rangle\cong \mathrm S(\mathfrak p)$, the last isomorphism being induced by an affine morphism. The arguments of Subsubsection \[sss-3-1-1\] can be repeated almost [*verbatim*]{}. The only relevant difference is that the kernel of the surjective module homomorphism $\mathcal A$ from $(A,\star_A)$ to $(K,\star_L)$ is identified with $A\star_A \mathfrak k^{-\chi-\delta+\frac{1}4\mathrm{tr}_\mathfrak g\circ \mathrm{ad}}$. Further, the restriction of $\mathcal A$ to $B^0$ and the operator $\mathcal B$ have the same shape as previously. Still, there is an associative product $\star_B$ on $\mathrm H^0(B)\cong\mathrm S(\mathfrak p)^\mathfrak k$: of course, now the product $\star_B$ depends explicitly on the character $\chi$. Identity  is consequently modified as $$(\mathcal A^{-1}\circ \mathcal B)(f_1\star_B f_2)=(\mathcal A^{-1}\circ \mathcal B)(f_1)\star_A (\mathcal A^{-1}\circ \mathcal B)(f_2)\ \text{modulo $A\star_A \mathfrak k^{-\chi-\delta+\frac{1}4\mathrm{tr}_\mathfrak g\circ\mathrm{ad}}$},$$ from which we deduce $$\beta\!\left(\partial_{\sqrt{j}}(f_1\star_B f_2)\right)=\beta\!\left(\partial_{\sqrt{j}}(f_1)\right)\cdot \beta\!\left(\partial_{\sqrt{j}}(f_2)\right)\ \text{modulo $\mathrm U(\mathfrak g)\cdot\mathfrak k^{-\chi-\delta+\frac{1}4\mathrm{tr}_\mathfrak g\circ\mathrm{ad}}$},$$ for any two $\mathfrak k$-invariant elements of $\mathrm S(\mathfrak p)$. Of course, once again, we may safely remove the character $1/4\ \mathrm{tr}_\mathfrak g\circ\mathrm{ad}$ in the previous identity. \[r-comm-corr\] The product $\star_B$ is commutative on $\mathrm S(\mathfrak p)^\mathfrak k$ because of the symmetry of the function $E(X, Y)$, see [@CT Lemma 11], whence the algebra $(\mathrm U(\mathfrak g)/\mathrm U(\mathfrak g)\cdot \mathfrak k^{-\delta})^\mathfrak k$ is also commutative. Now the arguments of [@CT Subsubsection 4.2.1] are no longer correct because of the contribution of short loop terms that make the symmetry argument inefficient. Therefore, Theorem 5 in [@CT], which states the commutativity of $(\mathrm U(\mathfrak g)/\mathrm U(\mathfrak g)\cdot \mathfrak k^{-z\delta})^\mathfrak k$, for any real number $z$, is also no longer correct. Differential operators expressed [*via*]{} exponential coordinates in a symmetric pair {#ss-3-3} -------------------------------------------------------------------------------------- We consider the triple $X=U_1=\mathfrak g^*$ and $U_2=\mathfrak k^\perp$, for a symmetric pair $(\mathfrak g,\sigma)$. Therefore, we have two associative algebras $(A,\star_A)$, $(\mathrm H^0(B),\star_B)$, and a $(A,\star_A)$=$(\mathrm H^0(B),\star_B)$-bimodule $K$, where $A=\mathrm S(\mathfrak g)$, $\mathrm H^0(B)=\mathrm S(\mathfrak p)^\mathfrak k$ and $K=\mathrm S(\mathfrak p)$. Through the function $q$ on $\mathfrak g$, we define the Kashiwara–Vergne density function $D(x,y)$, for $(x,y)$ a general element of $\mathfrak g\times\mathfrak g$, [*via*]{} $$D(x,y)=\frac{\sqrt{q(x)}\sqrt{q(y)}}{\sqrt{q(\mathrm{BCH}(x,y))}},$$ where the Baker–Campbell–Hausdorff formula $\mathrm{BCH}(x,y)$ is defined by $\exp(x)\exp(y)=\exp(\mathrm{BCH}(x,y))$, for $(x,y)$ in a neighborhood of $(0,0)$. The Kashiwara–Vergne density function has been introduced in to formulate the famous Kashiwara–Vergne conjecture, a general statement for finite-dimensional Lie algebras about deformations of the Baker–Campbell–Hausdorff formula for the product of exponentials in a Lie group: the Kashiwara–Vergne conjecture leads to a proof of Duflo’s Theorem about the center of $\mathrm U(\mathfrak g)$ in terms of the $\mathfrak g$-invariant symmetric algebra $\mathrm S(\mathfrak g)^\mathfrak g$. The Kashiwara–Vergne conjecture has been proved in the general case in using deformation quantization techniques to find a suitable deformation of the BCH formula; recently, a different approach to the Kashiwara–Vergne conjecture using Drinfel’d associators has been found, and a relationship between the latter approach and the former [*via*]{} deformation quantization has been elucidated. We will also discuss a relative Kashiwara–Vergne conjecture in the framework of symmetric pairs later on. For $x$, $y$ general elements of $\mathfrak p$, we denote by $e^x$ and $e^y$ the exponential of $x$ and $y$, viewed as linear functions on $X=\mathfrak g^*$. Then, we $$e^x\star_A e^y=\frac{D(x,y)}{D(P(x,y),K(x,y))}e^{P(x,y)}\star_A e^{K(x,y)},\quad P=P(x,y)\in\mathfrak p,\quad K=K(x,y)\in\mathfrak k,$$ and the power series $P$ and $K$ are defined [*via*]{} the exponential map for symmetric spaces. More precisely, it has been proved that symmetric spaces admit an exponential map, which is a diffeomorphism from a neighborhood of $0$ in $\mathfrak p$ into its image in $G/K$, where $G$ is a symmetric pair (in the sense of Lie groups) and $K$ is the fixed point set of an involution $\sigma$ of $G$ (which is a Lie group automorphism): it is simply the restriction of the exponential map of $\mathfrak g$ to right $K$-cosets in $G$. For $x$, $y$ in a sufficiently small neighborhood of $0$ in $\mathfrak p$, such that $\exp(x)$ and $\exp(y)$ both exist, we have $\exp(x)\exp(x)=\exp(P(x,y))\exp(K(x,y))$. An important observation at this point is that both $P$ and $K$, as previously defined, are power series in the free Lie algebra generated by $x$, $y$: in particular, the Cartan relations for $(\mathfrak g,\sigma)$ imply that $K$ is an element of $\mathfrak k_1=[\mathfrak p,\mathfrak p]$. For $x$, $y$ in a sufficiently small neighborhood of $0$ of $\mathfrak p$ as before, we consider the expression $e^{K(x,y)}\star_L 1=\mathcal A(e^{K(x,y)})$. First of all, $\mathcal A(e^K)$ is a constant element of $K=\mathrm S(\mathfrak p)$. By its very definition, $K=K(x,y)$ is an element of $\mathfrak k_1$: in the computation of a summand $K^n\star_L 1$, only the part of the differential operator acting on $K^n$ of degree $n$ survives, because either of degree reasons or of the fact that the restriction of $K$ as a linear function to $\mathfrak k^\perp$ vanishes. Using the involution $s$ of the preceding Subsection together with the computations of Subsection \[ss-3-1\], it is easy to prove that $K\star_L 1=\delta(K)-1/4\ \mathrm{tr}_\mathfrak g(\mathrm{ad}(K))=\delta(K)$, because, as observed before, $K$ belongs to $\mathfrak k_1$. We consider, more generally, graphs appearing in the computation of $K^n\star_2 1$, $n\geq 2$. We may actually repeat almost [*verbatim*]{} the arguments about the shape of the graphs appearing in $\mathcal A$, $\mathcal B$: the same arguments imply that short loops may appear only at vertices of the first type, which are linked to the only vertex of the second type on the positive real axis corresponding to $K^n$ through a single edge, while more complicated graphs are wheels. The Cartan relations for the symmetric pair $(\mathfrak g,\sigma)$ imply that the rays of such wheels are colored by $(+,-)$, while the wheel itself has all edges either colored by $(+,+)$ or $(+,-)$. In particular, it follows that $\mathcal A(e^K)$ consists of an infinite series of wheels, where the short loop graph is considered as the $1$-wheel: in this case, the $1$-wheel contribution appears explicitly. \[l-sym-K\] For $x$ a general element of $\mathfrak k$, the function $\mathcal A(e^x)=e^x\star_L 1$ satisfies $$\mathcal A(e^x)=\sqrt{q(x)}e^{\delta(x)-\frac{1}4\mathrm{tr}_\mathfrak g(\mathrm{ad}(x))};$$ in particular, if $x$ belongs to the Lie subalgebra $\mathfrak k_1$, the exponent on the right-hand side of the previous equality simplifies to $\delta(x)$. To compute $\mathcal A(e^x)$, for $x$ in $\mathfrak k$, it suffices to replace $K$ in the previous computations by $x$. The rest of the proof follows along the same lines of the proof of [@CT Lemma 15], but we have to observe now that $$\mathcal A(e^x)=e^x\star_L 1=\exp\!\left(\delta(x)-\frac{1}4\mathrm{tr}_\mathfrak g(\mathrm{ad}(x))+\sum_{n\geq 2} w_n^\mathfrak k \mathrm{tr}_\mathfrak k(\mathrm{ad}(x)^n)+\sum_{n\geq 2} w_n^\mathfrak p \mathrm{tr}_\mathfrak p(\mathrm{ad}(x)^n)\right),$$ for certain integral weights $w_n^\mathfrak k$ and $w_n^\mathfrak p$. More precisely, such weights are associated to the colored wheel-like graphs \ Further, also $\sqrt{q(x)}$ can be written in a similar form, with the only difference that it does not contain terms proportional to the trace of the adjoint representations of $\mathfrak k$ on itself or on $\mathfrak p$. Therefore, the very same computations of [@CT Lemma 15] imply the above identity. We observe that we may get rid of the factor $e^{\delta-\frac{1}4\mathrm{tr}_\mathfrak g\circ\mathrm{ad}}$, viewed as an element of the completion $\widehat{\mathrm S}(\mathfrak k)$ simply by applying the biquantization techniques to the modified triple $X=U_1=\mathfrak g$, $U_2=-\delta+1/4\ \mathrm{tr}_\mathfrak g\circ\mathrm{ad}+\mathfrak k^\perp$. To the previous triple are associated two associative algebra $(A,\star_A)$ and $(\mathrm H^0(B),\star_B)$ and a bimodule $K$, where $A=\mathrm S(\mathfrak g)$, $B^0=K=\mathrm S(\mathfrak p)$, and $H^0(B)=\mathrm S(\mathfrak p)^\mathfrak k$. Notice that the product $\star_B$ on $\mathrm H^0(B)$, for $B$ defined by the modified triple, does not coincide with $\star_B$ for the initial triple; similarly, the operator $\mathcal A$ on $A$ for the modified triple also does not coincide with the operator $\mathcal A$ for the initial triple, as their kernels do not obviously coincide. Still, both their restrictions to $B^0$ coincide, thus also their symbols, and similarly for $\mathcal B$. On the other hand, as already observed, for the above modified triple we have the identity $$\mathcal A(e^x)=e^x\star_L 1=\sqrt{q(x)},\ x\in\mathfrak k.$$ We consider the following expression, using the notation from above, $$\begin{aligned} (e^x\star_A e^y)\star_L 1&=\frac{D(x,y)}{D(P,K)} \left(e^P\star_A e^K\right)\star_L 1=\frac{\sqrt{q(x)}\sqrt{q(y)}}{\sqrt{q(P)}\sqrt{q(K)}} e^P\star_L(e^K\star_L 1)=\frac{\sqrt{q(x)}\sqrt{q(y)}}{\sqrt{q(P)}}j_\mathcal A(P)e^P=\\ &=\frac{\sqrt{q(x)}\sqrt{q(y)}}{\sqrt{j(P)}}j_\mathcal B(P)e^P=\\ &=e^x\star_L(e^y\star_L1)=j_\mathcal A(y) e^x\star_L e^y=\frac{j_\mathcal A(y)}{j_\mathcal B(y)} e^x\star_L (1\star_R e^y),\ x,y\in\mathfrak p. \end{aligned}$$ Comparing the expression on the second line with the rightmost expression on the third line, we find $$e^x\star_L (1\star_R e^y)=\frac{\sqrt{q(x)}\sqrt{j(y)}j_\mathcal B(P)}{\sqrt{j(P)}} e^P$$ We may view the expressions on both sides of the previous identity as analytic functions on a sufficiently small neighborhood of $(0,0)$ in $\mathfrak p\times \mathfrak p$. We consider further an element $T$ of $\mathrm H^0(B)=\mathrm S(\mathfrak p)^\mathfrak k$, for which we obtain $$\begin{aligned} e^x\star_L (1\star_R T)&=(e^x\star_L 1)\star_R T=j_\mathcal A(x) e^x\star_R T=\\ &=T_y\!\left(e^x\star_L (1\star_R e^y)\right)|_{y=0}=T_y\!\left(\frac{\sqrt{q(x)}\sqrt{j(y)}j_\mathcal B(P)}{\sqrt{j(P)}} e^P\right)\bigg\vert_{y=0}, \end{aligned}$$ where we regard in the second line $T$ as a differential operator on $\mathfrak p^*$ with respect to the variable $y$. Therefore, using Identity , we get the following expression, $$e^x\star_R T=T_y\!\left(\frac{\sqrt{j(x)}\sqrt{j(y)}j_\mathcal B(P)}{\sqrt{j(P)}j_\mathcal B(P)} e^P\right)\bigg\vert_{y=0},$$ whose right-hand side is, according to the arguments of [@Rouv Section 6]. precisely the differential operator $\beta(\partial_{sqrt{j}}(T))$ expressed [*via*]{} exponential coordinates on the symmetric space $G/K$, for $G$, $K$ connected, simply connected Lie groups with Lie algebras $\mathfrak g$, $\mathfrak k$ respectively, up to a modification by the analytic function $\sqrt{j}/j_\mathcal B$ on $\mathfrak p$. A deep consequence of the fact that the product $\star_B$ coincides with Rouvière’s product, together with a result about the existence of polarizations compatible with the structure of symmetric pair, for which we refer to [@T1; @T2], implies that the symbol $j_\mathcal B$ is constant and thus equal to $1$. Therefore, the expression $e^x\star_L T$, for $T$ in $\mathrm S(\mathfrak p)^\mathfrak k$, viewed here as an element of $K$, truly expresses in terms of biquantization the differential operator $T$ through exponential coordinates on $G/K$. Deformation of the Baker–Campbell–Hausdorff formula for symmetric pairs {#ss-3-4} ----------------------------------------------------------------------- As already briefly remarked in the previous Subsection, the problem of deforming the BCH formula and the BCH density function $D$ (see above) for a finite-dimensional Lie algebra $\mathfrak g$ is related to Duflo’s Theorem [*via*]{} the Kashiwara–Vergne conjecture. As the KV conjecture relies on the exponential map on Lie algebras, it seems natural to formulate a similar conjecture for a general symmetric pair, because also symmetric pairs admit an exponential map. We refer to [@Rouv] for more details on the KV conjecture for symmetric pairs. The exponential map for a symmetric pair $(\mathfrak g,\sigma)$ with Cartan decomposition $\mathfrak g=\mathfrak k\oplus\mathfrak p$ is a well-defined diffeomorphism $\exp_{G/K}$ from a sufficiently small neighborhood of $0$ in $\mathfrak p$ to its image in $G/K$, for $G$, $K$ as in the previous Subsection: it is induced from the exponential map $\exp_G$ of $\mathfrak g$. Standard manipulations, see [@Rouv Section 2] for more details, imply that the exponential map for the symmetric pair $(\mathfrak g,\sigma)$ yields a BCH formula $\mathrm{BCH}_\mathfrak p$ [*via*]{} $$\exp_G(2\mathrm{BCH}_\mathfrak p(x,y))=\exp_G(x)\exp_G(2y)\exp_G(x),$$ for $(x,y)$ in a sufficiently small neighborhood of $(0,0)$ in $\mathfrak p\times\mathfrak p$. The element $\mathrm{BCH}_\mathfrak p(x,y)$ belongs to $\mathfrak p$, and is expressible in terms of even iterated brackets of $x$ and $y$. The KV conjecture for symmetric pairs expresses a deformation of the function $\mathrm{BCH}_\mathfrak p$ in terms of $\mathfrak k$-adjoint vector fields on $\mathfrak p\times\mathfrak p$; we refer once again to [@Rouv] for a complete introduction to this issue and for a discussion of some consequences. There is also another claim, which expresses the corresponding variation in terms of the said $\mathfrak k$-adjoint vector fields of the density function for the symmetric pair, which is defined as $$D_\mathfrak p(x,y)=\frac{\sqrt{j(x)}\sqrt{j(y)}}{\sqrt{j(\mathrm{BCH}_\mathfrak p(x,y))}},$$ again for $(x,y)$ in a sufficiently small neighborhood of $(0,0)$, where all functions make sense. The deformation of $D_\mathfrak p$ contains traces of the adjoint $\mathfrak k$-action on $\mathfrak p$. We are now going to illustrate how biquantization techniques yield two different deformations of $\mathrm{BCH}_\mathfrak p$ and $D_\mathfrak p$ in the sense elucidated above. These two deformations can be characterized in terms of the $A_\infty$-structures on $A$, $B$ and $K$ through the quadratic relations between the corresponding Taylor coefficients: this is not immediately recognizable from the approach we take, which is in turn essentially motivated by the results of [@T3; @AM]. We consider the triple $X=U_1=\mathfrak g^*$ and $U_2=\delta-1/4\ \mathrm{tr}_\mathfrak g\circ\mathrm{ad}+\mathfrak k^\perp$ and the corresponding algebras $A$, $\mathrm H^0(B)$ and $A$-$\mathrm H^0(B)$-bimodule $K$. ### First deformation {#sss-3-4-1} First of all, we may consider, for $(x,y)$ in a sufficiently small neighborhood of $(0,0)$ in $\mathfrak p\times\mathfrak p$, where $\mathrm{BCH}_\mathfrak p$ and $D_\mathfrak p$, as well as $\sqrt{q}$, $\sqrt{j}$, $j_\mathcal A$ and $j_\mathcal B$ are well-defined, the function $$\begin{aligned} (e^x\star_A e^y)\star_L 1&=\frac{\sqrt{q(x)}\sqrt{q(y)}}{\sqrt{j(\mathrm{BCH}_\mathfrak p(x,y))}} e^{\mathrm{BCH}_\mathfrak p(x,y)}=\\ &=e^x\star_L(e^y\star_L 1)=j_\mathcal A(y) e^x\star_L e^y, \end{aligned}$$ where we have used results of the previous Subsection, setting $P(x,y)=\mathrm{BCH}_\mathfrak p(x,y)$. On the other hand, we may also consider $$e^x\star_L(1\star_R e^y)=j_\mathcal B(y) e^x\star_L e^y=e^x\star_L e^y,$$ where we have used once again the aforementioned fact that $j_\mathcal B\equiv 1$, whence, recalling Identity , $$e^x\star_L e^y=\frac{\sqrt{q(x)}\sqrt{j(y)}}{\sqrt{j(\mathrm{BCH}_\mathfrak p(x,y))}} e^{\mathrm{BCH}_\mathfrak p(x,y)}$$ Finally, we also have $$(e^x\star_L 1)\star_R e^y=j_\mathcal A(x) e^x\star_R e^y.$$ The problem is therefore, how to relate $(e^x\star_L 1)\star_R e^y$ with $e^x\star_L (1\star_R e^y)$ in order to draw a bridge between the previous two formulæ: the two expressions do not coincide, as both actions are compatible to each other only in cohomology. But the $A_\infty$-nature of $B$ and of the bimodule $K$ permit to control explicitly the failure for the compatibility between $\star_L$ and $\star_R$: in facts, we find $$\label{eq-deform-1} (e^x\star_L 1)\star_R e^y-e^x\star_L(1\star_R e^y)=\mathrm d_K^{1,1}(e^x,1,\mathrm d_B(e^y))=\mathrm d_K^{1,1}(e^x,1,e^y\mathrm d_B(y))$$ where $\mathrm d_B$ is the Chevalley–Eilenberg differential on the complex $B=\mathrm S(\mathfrak p)\otimes \wedge(\mathfrak k^*)$, and $\mathrm d^{1,1}_K$ denotes the $(1,1)$-Taylor component of the $A_\infty$-bimodule structure on $K$. More precisely, $$\mathrm d_K^{1,1}(a_1|k|b_1)=\sum_{n\geq 0}\frac{1}{n!}\sum_{\Gamma\in\mathcal G_{n,3}}\mu_{n+3}^K\left(\int_{\mathcal C_{n,3}^+}\prod_{e\in E(\Gamma)}\omega^K_e(\underset{n}{\underbrace{\pi|\cdots|\pi}}|a_1|k|b_1)\right),$$ and dimensional arguments imply that $b_1$ must be an element of $B^1$, otherwise the previous expression is trivial. Identity  may be derived in a slightly different way, which makes apparent the fact that it embodies the KV deformation problem sketched at the beginning of the present Subsection. Namely, for $n\geq 1$, we consider the forgetful projection $\pi_{n,1,1}$ from $\mathcal C_{n,1,1}^+$ to $\mathcal C_{0,1,1}^+$: in this situation, we prefer to consider the compactified configuration spaces $\mathcal C_{n,1,1}^+$ to highlight the fact that we consider the functions $e^x$, $1$ and $e^y$ to be put on the positive imaginary axis, at the origin and on the positive real axis respectively. We observe that the fiber of $\pi_{n,1,1}$ at a generic point of $\mathcal C_{0,1,1}^+$ is an orientable compact smooth manifold with corners of dimension $2n$, hence we may consider the push-forward (or integration along the fiber) $\pi_{n,1,1,*}$ with respect to $\pi_{n,1,1}$. In particular, we may consider the expression $$\sum_{n\geq 0}\frac{1}{n!}\sum_{\Gamma\in\mathcal G_{n,3}}\mu_{n+3}^K\left(\pi_{n,1,1,*}\!\left(\prod_{e\in E(\Gamma)}\omega^K_e\right)(\underset{n}{\underbrace{\pi|\cdots|\pi}}|e^x|1|e^y)\right),$$ for $(x,y)$ as above. First of all, for $(x,y)$ as above, the previous expression is a smooth function on $\mathcal C_{0,1,1}^+$: it is a consequence of the fact that, for $n\geq 1$, the push-forward $\pi_{n,1,1,*}$ selects the piece of the integrand of (form) degree bigger or equal than $2n$. To the three vertices of the second type are associated functions, while to each vertex of the first type of an admissible graph $\Gamma$ of type $(n,3)$ is associated a copy of the linear Poisson bivector $\pi$: as a consequence, the form degree of each integrand must be precisely $2n$, whence the first claim. We may further divide the previous expression by $j_\mathcal A(x)$: this “normalization” is due to previous computations, and it does not affect the following computations. Because of the fact that $\pi$ is a linear Poisson bivector, we may use the arguments in [@BCKT Chapter 2] or [@Kath] to prove that the “normalized” function on $\mathcal C_{0,1,1}^+$ given by the previous expression can be re-written in the form $$\label{eq-1-def} D_\mathfrak p^1(x,y)e^{\mathrm{BCH}_\mathfrak p^1(x,y)},$$ where the exponent $\mathrm{BCH}_\mathfrak p^1(x,y)$ is a smooth $\mathfrak p$-valued function on $\mathcal C_{0,1,1}^+$, corresponding to the connected graphs of Lie type, whose unique root is in $\mathfrak p$, while $D_\mathfrak p^1(x,y)$ is a smooth $\mathbb K$-valued function on $\mathcal C_{0,1,1}^+$, corresponding to graphs of Lie type with roots in $\mathfrak k$ and wheel-like graphs (possibly with graphs of type Lie attached to their spokes). Both $\mathrm{BCH}_\mathfrak p^1(x,y)$ and $D_\mathfrak p^1(x,y)$ are weighted sums over the graphs highlighted right above, where now the corresponding integral weights are smooth functions on $\mathcal C_{0,1,1}^+$. The deformation formulæ we are interested into can be therefore computed by taking the exterior derivative of  as a function on $\mathcal C_{0,1,1}^+$: we may therefore apply the generalized Stokes Theorem for integration along the fiber of $\pi_{n,1,1}$ in the first integral formula for . For any admissible graph $\Gamma$ of type $(n,3)$ as above, the corresponding integrand is a closed form, whence it suffices to consider the corresponding integral along the boundary strata of codimension $1$ of the generic fiber. For $n\geq 1$, a general boundary stratum of codimension $1$ in the generic fiber corresponds either $i)$ to the collapse of points in $Q^{+,+}$ labeled by a subset $A$ of $[n]$ of cardinality $2\leq |A|\leq n$ to a single point in $Q^{+,+}$, or $ii)$ to the approach of points in $Q^{+,+}$ labeled by a subset $A$ of $[n]$ of cardinality $0\leq |A|\leq n$ to $i\mathbb R^+$, to the origin or to $\mathbb R^+$. Notice that no boundary stratum appears, where either the point on $i\mathbb R^+$ or $\mathbb R^+$ approaches the origin (this is because we are considering the boundary strata of codimension $1$ of the generic fiber). Standard dimensional arguments, Kontsevich’s Vanishing Lemma [@K Lemma] and the boundary conditions for the $4$-colored propagators $\omega^{+,+}$ and $\omega^{+,-}$ imply that the only boundary strata yielding non-trivial contributions correspond to boundary strata of type $ii)$, where points in $Q^{+,+}$ labeled by $A\subseteq [n]$ approach the point on $\mathbb R^+$. We refer to [@T3] or [@BCKT Chapter 2] for similar computations. The sum over all such contributions yields a smooth $\widehat{\mathrm S}(\mathfrak p)$-valued $1$-form on $\mathcal C_{0,1,1}^+$, depending on $(x,y)$ as above, whose integral over $\mathcal C_{0,1,1}^+$ identifies with the right-hand side of Identity . The generalized Stokes Theorem, see [*e.g.*]{} the computations in [@T3], implies that the previous $1$-form specifies a smooth $\mathfrak k$-valued $1$-form $\omega_1(x,y)$ satisfying the identities $$\begin{aligned} \mathrm d\mathrm{BCH}_\mathfrak p^1(x,y)&=\langle \left[y,\omega_1(x,y)\right],\partial_y\mathrm{BCH}_\mathfrak p^1(x,y)\rangle,\\ \mathrm d D_\mathfrak p^1(x,y)&=\langle \left[y,\omega_1(x,y)\right],\partial_y D_\mathfrak p^1(x,y)\rangle+\mathrm{tr}_\mathfrak p\!\left(\mathrm{ad}(y)\partial_y\omega_1(x,y)\right)D_\mathfrak p^1(x,y), \end{aligned}$$ where we have used the notation $$\langle [y,\omega_1(x,y)],\partial_y\mathrm{BCH}_\mathfrak p^1(x,y)\rangle=\frac{\mathrm d}{\mathrm d t}\mathrm{BCH}_\mathfrak p^1(x,y+t[y,\omega_1(x,y)])\big\vert_{t=0},$$ and analogously for other similar expressions in the previous identities. Finally, we consider the function  on $\mathcal C_{0,1,1}^+$, which is a smooth, compact $1$-dimensional manifold with corners: its two boundary strata of codimension $1$ correspond to the approach of the point either on $i\mathbb R^+$ or on $\mathbb R^+$ to the origin. If we choose the smooth section of $\mathcal C_{0,1,1}^+$ which corresponds to fixing the point on $i\mathbb R^+$ to $i$, then $\mathcal C_{0,1,1}^+\cong [0,\infty]$: the boundary point $\{0\}$, resp. $\{\infty\}$, corresponds to the approach of the point on $\mathbb R^+$, resp. on $i\mathbb R^+$, to the origin. Taking into account the “normalization” with respect to $j_\mathcal A(x)$ in the function  and the computations at the beginning of the Subsubsection, the value of the function  at $0$ yields the values of both $\mathrm{BCH}_\mathfrak p^1$ and $D_\mathfrak p^1(x,y)$ at $0$, which are precisely $\mathrm{BCH}_\mathfrak p(x,y)$ and $D_\mathfrak p(x,y)$, whence the function  produces a genuine deformation of the BCH formula and corresponding density for the symmetric pair $(\mathfrak g,\sigma)$. The value of the “normalized” function  at $\infty$ is also readily computed, namely it is simply $e^x\star_R e^y$. ### Second deformation {#sss-3-4-2} We begin by considering, for $(x,y)$ as in the previous Subsubsection, the two expressions $(1\star_R e^x)\star_R e^y$ and $1\star_R(e^x\star_B e^y)$. We observe that neither the pairing $\star_B$ is associative, nor it is compatible with the pairing $\star_R$ in both expressions. Since, as observe before, $j_\mathcal B\equiv 1$, the first expression equals $e^x\star_R e^y$, while the second equals $e^x\star_B e^y$. The fact that $\star_R$ and $\star_B$ are the $(0,1)$- and $2$-Taylor components of the $A_\infty$-structures on $K$ and on $B$, we get $$\label{eq-deform-2} (1\star_R e^x)\star_R e^y-1\star_R(e^x\star_B e^y)=e^x\star_R e^y-e^x\star_B e^y=\mathrm d_K^{1,1}(1|\mathrm d_B(e^x)|e^y)+\mathrm d_K^{1,1}(1|e^x|\mathrm d_B(e^y)).$$ We observe that, due to the choice of the triple $X=U_1=\mathfrak g^*$ and $U_2=\delta-1/4\ \mathrm{tr}_\mathfrak g\circ\mathrm{ad}+\mathfrak k^\perp$, the pairing $\star_B$ depends on the (natural) character $\delta-1/4\ \mathrm{tr}_\mathfrak g\circ\mathrm{ad}$. We now define, for $(x,y)$ as above, a smooth function on $\mathcal C_{0,0,2}^+$ [*via*]{} $$\sum_{n\geq 0}\frac{1}{n!}\sum_{\Gamma\in\mathcal G_{n,3}}\mu_{n+3}^K\left(\pi_{n,0,2,*}\!\left(\prod_{e\in E(\Gamma)}\omega^K_e\right)(\underset{n}{\underbrace{\pi|\cdots|\pi}}|1|e^x|e^y)\right),$$ where, for $n\geq 1$, $\pi_{n,0,2}$ is the forgetful projection from $\mathcal C_{n,0,2}^+$ onto $\mathcal C_{0,0,2}^+$, and $\pi_{n,0,2,*}$ denotes the push-forward with respect to $\pi_{n,0,2}$. Again, because of the fact that $\pi$ is a linear Poisson bivector, the arguments of [@BCKT Chapter 2] or [@Kath] yield an explicit expression for the previous function on $\mathcal C_{0,0,2}^+$ in the shape $$\label{eq-2-def} D_\mathfrak p^2(x,y)e^{\mathrm{BCH}_\mathfrak p^2(x,y)}.$$ The $\mathfrak p$-valued function $\mathrm{BCH}_\mathfrak p^2(x,y)$ and the $\mathbb K$-valued function $D_\mathfrak p^2(x,y)$ are as in Formula , with due modifications in the integral weights. As in the previous Subsubsection, the exterior derivative of the function  may be computed by means of the generalized Stokes Theorem in a way similar to the sketch of the computation of the exterior derivative of the function . The relevant contributions come from the boundary strata of codimension $1$ of the generic fiber of $\pi_{n,0,2}$, for $n\geq 1$: such strata correspond to either $i)$ the collapse of points in $Q^{+,+}$ labeled by a subset $A$ of $[n]$ of cardinality $2\leq |A|\leq n$ to a single point in $Q^{+,+}$, or $ii)$ the approach of points in $Q^{+,+}$ labeled by $A\subseteq [n]$, $0\leq |A|\leq n$, to $i\mathbb R^+$, the origin, or $\mathbb R^+$. As we are considering a generic fiber, we do not consider strata, where the two points on $\mathbb R^+$ approach to each other or where the first point on $\mathbb R^+$ approaches the origin. Standard arguments imply that the only non-trivial contributions come from boundary strata of type $ii)$, corresponding to the approach of points in $Q^{+,+}$ either to the first or to the second point on $\mathbb R^+$. More precisely, the sum over all such contributions yields a smooth $\widehat{\mathrm S}(\mathfrak p)$-valued $1$-form on $\mathcal C_{0,0,2}^+$, whose integral over $\mathcal C_{0,0,2}^+$ is precisely the rightmost expression in the chain of identities . The previous $1$-form yields in turn a $\mathfrak k\times\mathfrak k$-valued $1$-form on $\mathcal C_{0,0,2}^+$ $\omega_2(x,y)=(\omega_2^1(x,y),\omega_2^2(x,y))$, which obeys the identities $$\begin{aligned} \mathrm d\mathrm{BCH}_\mathfrak p^2(x,y)&=\langle \left[x,\omega_2^1(x,y)\right],\partial_x\mathrm{BCH}_\mathfrak p^2(x,y)\rangle+\langle \left[y,\omega_2^2(x,y)\right],\partial_y\mathrm{BCH}_\mathfrak p^2(x,y)\rangle,\\ \mathrm d D_\mathfrak p^1(x,y)&=\langle \left[x,\omega_2^1(x,y)\right],\partial_x D_\mathfrak p^2(x,y)\rangle+\langle \left[y,\omega_2^2(x,y)\right],\partial_y D_\mathfrak p^2(x,y)\rangle+\\ &\phantom{=}+\mathrm{tr}_\mathfrak p\!\left(\mathrm{ad}(x)\partial_x\omega_2^1(x,y)+\mathrm{ad}(y)\partial_y\omega_2^2(x,y)\right)D_\mathfrak p^2(x,y). \end{aligned}$$ The previous results imply due modifications of the results of [@CT Subsubsection 4.4.3]: we observe that changes are caused by the fact that we have chosen at the beginning the modified triple $X=U_1=\mathfrak g^*$ and $U_2=\delta-1/4\ \mathrm{tr}_\mathfrak g\circ\mathrm{ad}+\mathfrak k^\perp$, thus introducing in many formulæ the (natural) character $\delta-1/4\ \mathrm{tr}_\mathfrak g\circ\mathrm{ad}$ of $\mathfrak k$. Harish-Chandra homomorphism in diagrammatical terms re-visited {#s-5} ============================================================== Once again, we borrow notation and conventions from [@CT Section 5], in particular for what concerns the generalized Iwasawa decomposition. Geometrically, to the decomposition $\mathfrak g=\mathfrak k\oplus\mathfrak p_0\oplus\mathfrak n_+$, and $\mathfrak k=\mathfrak k_0\oplus \mathfrak k/\mathfrak k_0$, we associate $U_1=\mathfrak k^\perp$ and $U_2=(\mathfrak k_0\oplus\mathfrak n_+)^\perp$, whence $$A=\mathrm S(\mathfrak p_0)\otimes\mathrm S(\mathfrak n_+)\otimes\wedge(\mathfrak k_0^*)\otimes\wedge(\mathfrak r^*),\ B=\mathrm S(\mathfrak p_0)\otimes\mathrm S(\mathfrak r)\otimes\wedge(\mathfrak k_0^*)\otimes\wedge(\mathfrak n_+^*),\ K=\mathrm S(\mathfrak p_0)\otimes\wedge(\mathfrak k_0^*),$$ where $\mathrm r=\mathfrak k/\mathfrak k_0$. We observe that, in this situation, we have all $4$-colored propagators coming into play; thus, in admissible graphs, multiple edges may appear. Harish-Chandra graphs and reduction spaces re-visited {#ss-5-1} ----------------------------------------------------- The $A_\infty$-$A_\hbar$-$B_\hbar$-bimodule $K_\hbar$ identifies now, as a vector space, with $\mathrm S(\mathfrak p_0)\otimes\wedge(\mathfrak k_0^*)[\![\hbar]\!]$, and has a differential $\mathrm d_{K_\hbar}^{0,0}$, due to the flatness of both $A_\hbar$, $B_\hbar$. The next proposition is new, and we need it to identify correctly the $0$-th cohomology of $\mathrm d_{K_\hbar}^{0,0}$: this result is implicitly used in [@CT], but we have realized that its proof needs a more involved argument which requires Lemma \[l-vanish-4\]. \[p-red-bimod\] The reduction space $\mathrm H^0(K_\hbar)$ identifies with $\mathrm S(\mathfrak p_0)^{\mathfrak k_0}[\![\hbar]\!]$. By its very construction, $\mathrm d_{K_\hbar}^{0,0}$ on $\mathrm S(\mathfrak p_0)$ is determined by admissible graphs in $\mathcal G_{n,1}$; we observe that the only vertex of the second type is the origin. We consider an admissible graph $\Gamma$ in $\mathcal G_{n,1}$, for $n\geq 2$: there is a vertex of the first type, from which departs one edge to $\infty$, and such an edge is colored by $(-,-)$. We concentrate on the remaining $n-1$ vertices: every edge hitting the origin is colored by $(+,+)$, therefore, from each one of the $n-1$ vertices of the first type can depart at most one edge to the origin. We denote by $p$ the number of edges departing from the $n$ vertices of the first type of $\Gamma$ and hitting the origin: by the previous arguments, we know that $p\leq (n-1)+1=n$, because the vertex with an edge to $\infty$ has an additional edge, which may or may not hit the origin. The polynomial degree of the differential operator associated to $\Gamma$ equals $n-(2n-1-p)=-n+1+p$, which must be greater or equal than $0$: this forces immediately $p\geq n-1$. This fact, combined with the previous condition on $p$, yields that either $p=n$ or $p=n-1$. In the case $p=n$, the corresponding differential operator has polynomial degree $1$, and from each vertex of the first type departs exactly one edge to the origin: this implies that the graph $\Gamma$ must have a vertex of the form \ \ Of course, the vertex from which departs the arrow to $\infty$ may be also isolated, [*i.e.*]{} no edge has it as the endpoint: in this case, as $n\geq 2$, standard dimensional arguments imply that the corresponding weight is trivial. If it is not isolated, the property $[\mathfrak k_0,\mathfrak p_0]\subset \mathfrak p_0$ implies that the two consecutive edges correspond to propagators of type $(+,+)$, and Lemma \[l-vanish-4\] yields triviality of the corresponding weight. We now consider the case $p=n-1$: the corresponding differential operator has polynomial degree $0$, [*i.e.*]{} it is a translation-invariant differential operator. We first assume that the vertex from which departs an edge to $\infty$ has the other edge hitting the origin: the fact that the differential operator has constant coefficients implies that this vertex cannot be isolated, otherwise Lemma \[l-vanish-4\] would imply triviality of its weight. Therefore, $\Gamma$ must be a disjoint union of wheel-like graphs, one of which must look like as follows: \ \ Lemma \[l-vanish-4\] implies that the two edges meeting at the vertex with the edge to $\infty$ must have distinct colors, otherwise the corresponding weight vanishes. Therefore, the relations $[\mathfrak k_0,\mathfrak k_0]\subseteq \mathfrak k_0$, $[\mathfrak k_0,\mathfrak p_0]\subseteq \mathfrak p_0$, $[\mathfrak k_0,\mathfrak n_+]\subseteq \mathfrak n_+$ imply immediately that the outgoing edge from this special vertex admits only the color $\mathfrak r$. But then again, the previous relations imply that all edges in the cycle of the wheel-like graph are colored by $\mathfrak r$, hence we may apply once again Lemma \[l-vanish-4\]. We have thus proved that the only non-trivial graph corresponds exactly to the adjoint action of $\mathfrak k_0$ on $\mathrm S(\mathfrak p_0)$, whence the claim follows. Therefore, the three reduction algebras associated to $A$, $B$ and $K$, are exactly as in [@CT Subsection 5.1]. The discussion in [@CT Subsection 5.2] is not modified by the previously discussed changes: in fact, it deals only the reduction algebra placed on the positive imaginary axis, which is left untouched by the changes occurring in biquantization. Construction of characters re-visited {#s-6} ===================================== As before, we consider a symmetric pair $(\mathfrak g,\sigma)$, with a Cartan decomposition $\mathfrak g=\mathfrak k\oplus\mathfrak p$. The techniques of biquantization are applied, in this framework, to the case of two coisotropic subvarieties $f+\mathfrak k^\perp$ and $\mathfrak k^\perp$, where $f$ is an element of $\mathfrak k^\perp$, and $\mathfrak b$ is a polarization for $f$ ([*i.e.*]{} $f$ determines a skew-symmetric form on $\mathfrak g$, and a polarization $\mathfrak b$ for $f$ is a subalgebra of $\mathfrak g$, which is isotropic for the said skew-symmetric form and of maximal dimension among the isotropic subalgebras). Thus, we observe that in this situation, too, all $4$-colored propagators are involved. The aim of [@CT Subsections 2.5 and 6.1] is the construction of characters for the reduction algebra $(\mathrm H^0_{\hbar,\mathfrak b}(\mathfrak k^\perp),\star_\mathrm{CF})$, where $\mathrm H^0_{\hbar,\mathfrak b}(\mathfrak k^\perp)$ denotes the $0$-th cohomology of $A_\hbar$, where $A=\mathcal O_{\mathrm N^\vee_{\mathfrak k^\perp,\mathfrak g^*}[-1]}$. In order to perform explicit computations using biquantization, a decomposition of $\mathfrak g$ into direct summands, two of them being complements of $\mathfrak k$ and $\mathfrak b$, is needed, because in this situation, all $4$-colored propagators are involved: hence, the suffix denotes an explicit dependence (in the explicit computations) of the polarization $\mathfrak b$. As has been proved in detail in [@CT Subsection 1.5], there is an explicit (non canonical) isomorphism $\mathrm H^0(A_\hbar)\cong \mathrm H^0_{\hbar,\mathfrak b}(\mathfrak k^\perp)$: thus, [*via*]{} biquantization, it is possible to construct characters for $\mathrm H^0(A_\hbar)$, which still depend from a choice of a polarization $\mathfrak b$ of some element $f$ in $\mathfrak k^\perp$. Therefore, to really deal with a canonical construction of characters for $H^0(A_\hbar)$ [*via*]{} biquantization, one needs to prove independence of the choice of polarizations of the characters constructed on $H^0_{\hbar,\mathfrak b}(\mathfrak k^\perp)$. The main technical tool in the proof of independence of polarizations is a Stokes’ argument, reminiscent of the Stokes’ argument leading to biquantization, in presence of three coisotropic submanifolds, [*i.e.*]{} for $f$ in $\mathfrak k^\perp$ as before, $f+\mathfrak b_i^\perp$, $i=1,2$, and $\mathfrak k^\perp$, where now $\mathfrak b_i$ is a polarization for $f$, $i=1,2$. Of course, in this new situation, we need the $8$-colored propagators $\theta_{j_1,j_2,j_3}$, whose explicit construction and relative discussion of the main properties is the content of [@CT Subsubsection 6.2.1]. Roughly speaking, the $8$-colored propagators interpolate the $4$-colored ones (in a sense that will be made precise later on): therefore, as already pointed out in the Introduction, the appearance of “regular terms” in the $4$-colored propagators is likely to cause the appearance of similar “regular terms” in the $8$-colored propagators. This is the main novelty in the present discussion. Construction of the $8$-colored propagators re-visited {#ss-6-1} ------------------------------------------------------ We will write down in more detail the construction outlined in [@CT Subsubsection 6.2.1], from which we borrow notation and conventions. The $8$-colored propagators $\theta_{j_1,j_2,j_3}$, $j_k\in\{1,2\}$, $k=1,2,3,$, are $8$ distinct smooth, closed $1$-forms on the compactified configuration space $\mathcal C^+_{2,0,0,0}(\sqsubset)$, where $\sqsubset=\left\{z=x+\mathrm i y\in \mathbb C:\ x\geq 0,\ -\frac{\pi}2\leq y\leq \frac{\pi}2\right\}$. We observe that the half-strip $\sqsubset$ is a smooth manifold with corners of dimension $2$ with three boundary components $\sqsubseteq_i$, $i=1,2,3$ of codimension $1$, where $\sqsubseteq_1$ is the lower horizontal half-line, $\sqsubseteq_2$ is the vertical segment and $\sqsubseteq_3$ is the upper horizontal half-line. The space $\mathcal C^+_{2,0,0,0}(\sqsubset)$, which is a smooth manifold with corners of dimension $4$, is the correct generalization in the framework of $3$ branes of Kontsevich’s eye $\mathcal C_{2,0}^+$ (the compactified configuration space of $2$ points in the complex upper half-plane), needed to define the propagators in the case of no branes and one brane, see [@K; @CF], and of the I-cube $\mathcal C_{2,1}^+$, needed to define the $4$-colored propagators, see [@CFFR]. Without going here into further details, the manifold $\mathcal C_{2,0,0,0}^+(\sqsubset)$ is diffeomorphic, as a smooth manifold with corners, to $\mathcal C_{2,2}^+$: the latter space is a quotient with respect to rescalings and translations, which can be used to fix the two ordered vertices on the real axis to $\{0,1\}$, and then we may in a conformal way map the complex upper half-plane to the half-strip $\sqsubset$, the point $1$ to $-\mathrm i\frac{\pi}2$, $0$ to $\frac{\pi}2$, the half-line right to $1$ to $\sqsubseteq_1$, the segment $[0,1]$ to $\sqsubseteq_2$ and the negative real axis to $\sqsubseteq_3$. This has been done explicitly in [@F], where propagators for the Poisson $\sigma$-model in presence of many branes have been discussed, and was sketched in the seminal paper [@CFb]. The construction of the $8$-colored propagators has been split in two pieces, namely, the case $j_1\neq j_3$ and $j_1=j_3$. ### The case $j_1\neq j_3$ {#sss-6-1-1} There are $4$ propagators which fall into this class, namely $\theta_{112}$, $\theta_{122}$, $\theta_{211}$ and $\theta_{221}$, which are constructed starting from the normalized, closed $1$-form on $\mathcal C_2\cong S^1$, the compactified configuration space of $2$ points in $\mathbb C$ by using the reflections with respect to $\sqsubseteq_i$, $i=1,2,3$. We consider the boundary stratum of codimension $1$ of $\mathcal C_{2,0,0,0}^+(\sqsubset)$ corresponding to the collapse of the two points in the interior of the half-strip $\sqsubset$: more precisely, such a stratum is $\mathcal C_2\times \mathcal C_{1,0,0,0}^+(\sqsubset)\cong \mathcal C_2\times \mathcal C_{1,2}^+$, and local coordinates near such a stratum are given by $$\mathcal C_2\times \mathcal C_{1,0,0,0}^+(\sqsubset)\cong S^1\times C_{1,0,0,0}^+(\sqsubset)\ni (\varphi, z)\mapsto (z,z+\varepsilon \mathrm e^{\mathrm i\varphi})\in \mathcal C_{2,0,0,0}^+(\sqsubset),$$ where the stratum is recovered as $\varepsilon$ tends to $0$. An easy computation unraveling the formula for $\theta_{j_1,j_2,j_3}$, $j_1\neq j_3$, in [@CT Subsubsection 6.2.1] in the same spirit of the proof of [@CFFR Lemma 5.4], yields the following behavior of $\theta_{112}$, $\theta_{122}$, $\theta_{211}$ and $\theta_{221}$, when restricted to the boundary stratum $\mathcal C_2\times \mathcal C_{1,0,0,0}^+(\sqsubset)$ of $\mathcal C^+_{2,0,0,0}(\sqsubset)$: $$\begin{aligned} \theta_{112}\vert_{\mathcal C_2\times \mathcal C_{1,0,0,0}^+(\sqsubset)}&=\pi_1^*(\omega)+\pi_2^*(\widehat\rho), & \theta_{122}\vert_{\mathcal C_2\times \mathcal C_{1,0,0,0}^+(\sqsubset)}&=\pi_1^*(\omega)-\pi_2^*(\widehat\rho),\\ \theta_{221}\vert_{\mathcal C_2\times \mathcal C_{1,0,0,0}^+(\sqsubset)}&=\pi_1^*(\omega)+\pi_2^*(\widehat\rho), & \theta_{211}\vert_{\mathcal C_2\times \mathcal C_{1,0,0,0}^+(\sqsubset)}&=\pi_1^*(\omega)-\pi_2^*(\widehat\rho),\\ \end{aligned}$$ where $\omega$ is the normalized volume form of $\mathcal C_2\cong S^1$ (the “singular part” of the propagator), and $\widehat\rho$ is the smooth, closed $1$-form on $\mathcal C_{1,0,0,0}^+(\sqsubset)$ given by the formula $$\widehat\rho(z)=\frac{1}{2\pi}\left[\mathrm d\ \mathrm{arg}\!\left(\mathrm i\frac{\pi}2+z\right)-\mathrm d\ \mathrm{arg}\!\left(-\mathrm i\frac{\pi}2+z\right)-\mathrm d\ \mathrm{arg}\!\left(\mathrm i\pi+\mathrm{Re}(z)\right)\right],$$ the “regular part”. Finally, $\pi_i$, $i=1,2$, is the projection from $\mathcal C_2\times \mathcal C_{1,0,0,0}^+(\sqsubset)$ onto the $i$-th factor. The configuration space $\mathcal C_{1,0,0,0}^+(\sqsubset)$ is a smooth manifold with corners of dimension $2$: it has six boundary strata of codimension $1$, which correspond to the collapse of the point in the interior of the half-strip $\sqsubset$ to the boundary components $\sqsubseteq_i$ or to the two angle points $\pm\mathrm i\frac{\pi}2$, and the boundary stratum “at infinity”, where the point in $\sqsubset$ tends to $\infty$ in $\sqsubset$ ([*i.e.*]{} along a horizontal line in $\sqsubset$). In the first three cases, the boundary strata are simply $\sqsubseteq_i$, while on the remaining two, they are $\mathcal C_{1,1}^+\times \left\{\pm\mathrm i\frac{\pi}2\right\}$: again, using local coordinates near such boundary strata, we may prove that $\widehat\rho$ vanishes on the first three boundary strata, while on the remaining two we have $$\widehat\rho\vert_{\mathcal C_{1,1}^+\times \left\{\mathrm i\frac{\pi}2\right\}}=-\rho,\quad \widehat\rho\vert_{\mathcal C_{1,1}^+\times \left\{-\mathrm i\frac{\pi}2\right\}}=\rho,$$ where $\rho$ is the (normalized) angle form on $\mathcal C_{1,1}^+$ (or, in previous terminology, the short loop contribution). Finally, $\widehat\rho$ vanishes on the boundary stratum “at infinity”. ### The case $j_1=j_3$ {#sss-6-1-2} The construction of the propagators $\theta_{111}$, $\theta_{121}$, $\theta_{212}$ and $\theta_{222}$, on the other hand, relies on a trickier argument, which we now review in more details. We consider the strip $S=\left\{z\in\mathbb C:\ -\frac{\pi}2\leq \mathrm{Im}(z)\leq \frac{\pi}2\right\}$; on its interior, we consider the metric $g=\frac{\mathrm d x^2+\mathrm d y^2}{\cos^2 y}$, where $z=x+\mathrm iy$. It is not difficult to prove, by a direct computation, that $g$ tends to the standard Poincaré hyperbolic metric, when we approach the two boundary lines of $S$: the basic propagator in deformation quantization (Kontsevich’s angle form) is constructed [*via*]{} hyperbolic geometry, and the main idea behind the construction of $\theta_{j_1,j_2,j_1}$ is to use the geometry of $S$ determined by the metric $g$. A slight variation of the computations leading to the general form of geodesics in the complex upper half-plane $\mathbb H^+$, endowed with the hyperbolic Poincaré metric, leads to the following general form of geodesics in the interior of $S$, endowed with the metric $g$: $$\label{eq-geod} \sin(y)=A\mathrm e^x+B\mathrm e^{-x},$$ where $A$, $B$ are real constants; we additionally have geodesic vertical segments in the interior of $S$, see also [@CT Figure 6] for a pictorial description of the geodesics in the interior of $S$ with respect to metric $g$. It follows immediately from  that, for any two points $z_1$, $z_2$ in $S$, there is a unique geodesic passing through them. Thus, it makes sense to define the geodetic angle function $\widetilde\vartheta$ on the open configuration space $C^+_{2,0,0}(S)$ of two points in the interior of $S$ and no point on the two boundary lines in a way similar to Kontsevich’s angle function, [*i.e.*]{} $\widetilde\vartheta(z_1,z_2)$, for $z_1\neq z_2$ in the interior of $S$, is the angle between the geodesic vertical segment going through $z_1$ and the unique geodesic joining $z_1$ and $z_2$ (obviously, $\widetilde\vartheta$ is well-defined up to the addition of integer multiples of $2\pi$). More explicitly, the geodetic angle function $\vartheta(z_1,z_2)$ is defined through $$\tan\!\left(\widetilde\vartheta(z_1,z_2)+\frac{\pi}2\right)=\frac{\sin(y_1)\cosh(x_1-x_2)-\sin(y_2)}{\cos(y_1)\sinh(x_1-x_2)},\quad z_i=x_i+\mathrm i y_i,\ i=1,2.$$ It is easily verified that $\widetilde\vartheta(z_1,z_2)$ extends to $S$. The compactified configuration space $\mathcal C^+_{2,0,0}(S)$ has a boundary stratification: we are particularly interested into the boundary strata of codimension $1$, which correspond to the collapse of exactly one point in the interior of $S$ to one of the two boundary lines of $S$, of both points together in the interior of $S$ and of both points together to a point on one of the two boundary lines of $S$. The stratum corresponding to the collapse of exactly one point in the interior of $S$ to one of the two boundary axis is represented either by $\mathcal C_{1,0}^+\times\mathcal C_{1,1,0}^+(S)$ or $\mathcal C_{1,0}^+\times\mathcal C_{1,0,1}^+(S)$; the stratum corresponding to the collapse of the two points in the interior of $S$ by $\mathcal C_2\times\mathcal C_{1,0,0}^+(S)$, and the remaining two boundary strata are represented either $\mathcal C_{2,0}^+\times\mathcal C_{0,1,0}^+(S)$ or $\mathcal C_{2,0}^+\times\mathcal C_{0,0,1}^+(S)$. Using local coordinates near the boundary strata of codimension $1$ previously analyzed, we can prove that the exterior derivative of $\widetilde\vartheta$, which we denote as $\vartheta(z_1,z_2)$ as in [@CT Subsubsection 6.2.1], is a well-defined closed $1$-form on the compactified configuration space $\mathcal C_{2,0,0,0}^+(S)$, which satisfies the following properties: 1. the restriction of $\vartheta$ to the boundary stratum $\mathcal C_2\times\mathcal C_{1,0,0}^+(S)$ equals $\pi_1^*(\omega)$, $\pi_i$ being the natural projection onto the $i$-th factor, and $\omega$ the normalized volume form on $\mathcal C_2\cong S^1$. 2. The restriction of $\vartheta$ to either one of the boundary strata $\mathcal C_{1,0}^+\times \mathcal C_{1,1,0}^+(S)$ or $\mathcal C_{1,0}^+\times \mathcal C_{1,0,1}^+(S)$ corresponding to the collapse of the first argument to either one of the two boundary lines of $S$ vanishes. 3. The restriction of $\vartheta$ to either one of the boundary strata $\mathcal C_{2,0}^+\times\mathcal C_{0,1,0}^+(S)$ or $\mathcal C_{2,0}^+\times\mathcal C_{0,0,1}^+(S)$ equals $\pi_1^*(\omega^\pm)$, where $\pi_i$ is, once again, the projection onto the $i$-th factor, and $\omega^\pm$ denotes the $2$-colored propagators (Kontsevich’s angle function and its image with respect to the involution exchanging the two arguments). We now borrow from [@CT Subsubsection 6.2.1] the definition of the propagators $\theta_{1,j_2,1}$ and $\theta_{2,j_2,2}$, $j_2=1,2$, which are smooth, closed $1$-forms on the compactified configuration space $\mathcal C_{2,0,0}^+(\sqsubset)$: they are constructed using the angle form $\vartheta$ on $\mathcal C_{2,0,0}^+(S)$ and the natural involution $\sigma$ of $S$ associated to the reflection with respect to the imaginary axis, see [@CT Subsubsection 6.2.1] for the explicit formulæ. We observe that the angle form $\vartheta$ is equivariant with respect to the $\mathbb Z_2$-action induced on $\mathcal C_{2,0,0}^+(S)$ by the diagonal action of $\sigma$ and the natural sign action. Once again, we are interested to the boundary stratum $\mathcal C_2\times \mathcal C_{1,0,0}^+(\sqsubset)$ of $\mathcal C_{2,0,0}^+(\sqsubset)$, where the points in $\sqsubset$ collapse together in $\sqsubset$. As an example we consider the case $j_1=j_3=1$: then, we have the explicit formulæ$$\theta_{1,1,1}(z_1,z_2)=\frac{1}{2\pi}\left[\vartheta(z_1,z_2)-\vartheta(\sigma(z_1),z_2)\right],\quad \theta_{1,2,1}(z_1,z_2)=\frac{1}{2\pi}\left[\vartheta(z_1,z_2)-\vartheta(z_1,\sigma(z_2))\right],\quad (z_1,z_2)\in\mathcal C_{2,0,0}^+(\sqsubset).$$ We may use the local coordinates of Subsubsection \[sss-6-1-1\] near the said boundary stratum to perform explicit computations: using the properties of the angle form $\vartheta$, we see that the restriction of $\theta_{1,1,1}$ and $\vartheta_{1,2,1}$ to $\mathcal C_2\times \mathcal C_{1,0,0}^+(\sqsubset)$ splits into a singular part and a regular part; more precisely $$\theta_{1,1,1}\vert_{\mathcal C_2\times \mathcal C_{1,0,0}^+(\sqsubset)}=\pi_1^*(\omega)+\pi_2^*(\widetilde\rho),\quad \theta_{1,1,1}\vert_{\mathcal C_2\times \mathcal C_{1,0,0}^+(\sqsubset)}=\pi_1^*(\omega)-\pi_2^*(\widetilde\rho),$$ where $\pi_i$, $i=1,2$, and $\omega$ are as before. On the other hand, the regular term $\widetilde\rho$ is a smooth, closed $1$-form on $\mathcal C_{1,0,0}^+(\sqsubset)$ defined [*via*]{} $$\widetilde\rho(z)=\frac{1}{2\pi}\vartheta(z,\sigma(z))=\frac{1}{2\pi}\mathrm d\arctan\!\left(\tanh(x)\tan(y)\right),\quad z=x+\mathrm i y\in\mathcal C_{1,0,0}^+(\sqsubset).$$ Direct computations show that $\widetilde\rho$ has a behavior similar to the regular term of Subsubsection \[sss-6-1-1\] on the boundary strata of $\mathcal C_{1,0,0}^+(\sqsubset)$, with the only difference that $\widetilde\rho$ is non-trivial, when restricted to the boundary stratum “at infinity” (which is equivalent to a closed interval; in more familiar terms, it is $\mathcal C_{1,1}^+$): in fact, it equals $\rho$, the short loop contribution. Final considerations on polarizations {#ss-6-2} ------------------------------------- Having constructed the $8$-colored propagators in detail, we now want to discuss, in light of the appearance of “regular terms” also in the $8$-colored propagators, how such changes affect the arguments and the computations in the final stages of [@CT]. The main direct consequence of the construction of the $8$-colored propagators is explained in [@CT Proposition 21, Subsubsection 6.2.2]: therein, for the case of a symmetric pair $\mathfrak g=\mathfrak k\oplus \mathfrak p$, an element $f$ of $\mathfrak k^\perp$, and two polarizations $\mathfrak b_i$, $i=1,2$ (we observe that $\mathfrak k$ and $\mathfrak b_i$, $i=1,2$, are assumed to be in a position of normal intersection, [*i.e.*]{} $\mathfrak k\cap(\mathfrak b_1+\mathfrak b_2)=\mathfrak k\cap\mathfrak b_1+\mathfrak k\cap\mathfrak b_2$), it has been proved that the corresponding characters, depending upon the choice of either $\mathfrak b_1$ or $\mathfrak b_2$, are equal. The key argument of the proof relies, as the principle of biquantization, on Stokes’ Theorem applied to the situation, where we consider sums over admissible graphs in $\mathcal G_{n,3}$ and corresponding integral weights: here we view an admissible graph $\Gamma$ in $\mathcal G_{n,3}$ as an embedded graph in $\sqsubset$, where the first, resp. the third, vertex of the second type is $\mathrm i\frac{\pi}2$, resp. $-\mathrm i\frac{\pi}2$ (consequently, the second vertex lies on the vertical boundary segment $\sqsubseteq_2$ of $\sqsubset$), graphically \ \ In the previous picture, $P$ is a general element of $\mathrm H^0_{\hbar,\mathfrak b_1,\mathfrak b_2}(\mathfrak k^\perp)$, using notations from [@CT Subsubsection 6.2.2]. As has been already observed about biquantization, we need to consider admissible graphs with short loop contributions on vertices of the first type, if we want Stokes’ Theorem to do the job, because of the “regular term”: similarly, the results of Subsubsections \[sss-6-1-1\] and \[sss-6-1-2\] imply that what may be called “triquantization” can be performed using the techniques of Deformation Quantization with some changes, which should keep into account the “regular terms” in the $8$-colored propagators (which, by the way, are completely consequent with the “regular term” in the $4$-colored propagators). The first obvious observation about admissible graphs $\Gamma$ in $\mathcal G_{n,3}$ is that between any two vertices there can be [*at most*]{} eight edges, because to any edge we assign one of the $8$-colored propagators; therefore, we consider admissible graphs with multiple edges. From the point of view of combinatorics, we will have to take into account in the integral weight the (possible) multiplicity of edges of an admissible graph $\Gamma$ of $\mathcal G_{n,3}$. To $\Gamma$, we may associate an integral weight by standard prescriptions as in [@K; @CF; @CFFR], using the $8$-colored propagators: in this particular situation, as we want to apply Stokes’ Theorem, the integral weight $w_\Gamma$ of a general admissible graph $\Gamma$ in $\mathcal G_{n,3}$, for $n\geq 0$, is defined as an integral of the exterior derivative of a product $\omega_\Gamma$ of $8$-colored propagators (specified by the shape of $\Gamma$) over the compactified configuration space $\mathcal C_{n,0,1,0}^+(\sqsubset)$, which is orientable and of dimension $2n+1$. Such integrals are, on the one hand, obviously trivial (because the $8$-colored propagators are closed); on the other hand, if the degree of the integrand is $2n+1$ ([*i.e.*]{} if the degree of $\omega_\Gamma$ is $2n$), such a trivial contribution equals the sum over all boundary strata of codimension $1$ of $\mathcal C_{n,0,1,0}^+(\sqsubset)$: of course, this argument is similar to the proof [*e.g.*]{} of associativity of the $\star$-product in [@K]. More precisely, we have $$0=\int_{\mathcal C_{n,0,1,0}^+(\sqsubset)}\mathrm d\omega_\Gamma=\sum_{i}\pm\int_{\partial_i\mathcal C_{n,0,1,0}^+(\sqsubset)}\omega_\Gamma,$$ where the sum is over all boundary strata of codimension $1$ of $\mathcal C_{n,0,1,0}^+(\sqsubset)$, and the signs are dictated by their orientations. \[r-triquant\] We are purposefully sketchy here, but we plan to return to these issues somewhere else, as the $8$-colored propagators are the central tool in the construction of a “Formality Theorem in presence of $3$ branes” generalizing the main result of [@CFFR]; triquantizazion should be then related to the evaluation of the corresponding formality quasi-isomorphism at a Poisson structure on some linear space $X$. Let us consider only the boundary strata of codimension $1$ of $\mathcal C_{n,0,1,0}^+(\sqsubset)$ corresponding to the collapse of a subset $A$ of $\{1,\dots,n\}$ of cardinality $2\leq |A|\leq n$ in the interior of $\sqsubset$: such boundary strata are simply $\mathcal C_A\times \mathcal C_{n-|A|+1,0,1,0}^+(\sqsubset)$, where $\mathcal C_A$ is the compactified configuration space of $|A|$ points in $\mathbb C$ (modulo rescalings and complex translations, see [@K]), which is an orientable manifold with corners of dimension $2|A|-3$. We denote by $\Gamma_A$, resp. $\Gamma^A$, the subgraph of $\Gamma$, whose vertices are labeled by $A$ and whose edges are all edges connecting vertices labeled by $A$, resp. the graph obtained by contracting $\Gamma_A$ to a single vertex (necessarily of the first type). The restriction of $\Gamma_A$ to $\mathcal C_A\times \mathcal C_{n-|A|+1,0,1,0}^+(\sqsubset)$ is a product of forms splitting into the sum of a “singular term” (living on $\mathcal C_A$) and of “regular terms” $\widehat\rho$ or $\widetilde\rho$ (living, on the other hand, on $\mathcal C_{n-|A|+1,0,1,0}^+(\sqsubset)$). Recalling [@K Lemma 6.6] and the fact that only the “singular part” of the integrand ([*i.e.*]{} the product of all “singular terms” in $\omega_\Gamma$) is to be integrated over $\mathcal C_A$, we reduce to the case $|A|=2$. Further, the “singular term” and both “regular terms” are all $1$-forms: in particular, the shape of $\omega_\Gamma$ forces that possible non-trivial factors in the restriction of $\omega_\Gamma$ on such a boundary stratum are associated to two vertices of the first type, labeled by $A$, which are joined by [*at most*]{} three edges (either multiple edges or not). \ \ If there are only two edges between $v_A^1$ and $v_A^2$, then the weight contribution after integrating over $\mathcal C_2\cong S^1$ can be a sum of $\widehat\rho$ and $\widetilde\rho$ of the form $\pm a\widehat\rho\pm b\widetilde\rho$, $a$, $b$ in $\{0,1\}$. On the other hand, if there are three edges between $v_A^1$ and $v_A^2$, the weight contribution after integration is the product of $\widehat\rho$ and $\widetilde\rho$. Further, to $\Gamma$ in $\mathcal G_{n,3}$ we associate a polydifferential operator, which acts on the ($\hbar$-shifted) Poisson bivector $\pi_\hbar$ (a copy of which is placed at an vertex of the first type) and to the triple $(1|P|1)$, where $1$ is regarded as a constant function and $P$ is as above. The rule associates to each oriented edge of $\Gamma$ a derivation operator (in the graded sense, as we deal here with graded vector spaces). Finally, we multiply the end result as an element of $\mathfrak g^*$, and take its restriction to $f+(\mathfrak k^\perp+\mathfrak b_1+\mathfrak b_2)$. We now recall that $X=\mathfrak g^*$ is endowed with a ($\hbar$-shifted) linear Poisson structure, whence no multiple edges are possible between vertices of the first type. As a consequence, every vertex of the first type admits [*at most*]{} one incoming edge, [*i.e.*]{} $\Gamma_A$ can be only of the second type in Figure 15; any of the vertices of $\Gamma_A$ may further have [*at most*]{} one outgoing edge. We briefly discuss the coloring of an admissible graph. We choose a system of coordinates on $\mathfrak g$ which is adapted to the coisotropic submanifolds $\mathfrak k^\perp$, $f+\mathfrak b_1$ and $f+\mathfrak b_2$: in other words, we assume there is a partition of $\left\{1,\dots,d\right\}$, where $d$ is the dimension of $\mathfrak g$ of the form $$\begin{aligned} \{1,\dots,d\}=&(I_1\cap I_2\cap I_3)\sqcup(I_1^c\cap I_2\cap I_3)\sqcup(I_1\cap I_2^c\cap I_3)\sqcup(I_1\cap I_2\cap I_3^c)\sqcup\\ &(I_1^c\cap I_2^c\cap I_3)\sqcup(I_1^c\cap I_2\cap I_3^c)\sqcup(I_1\cap I_2^c\cap I_3^c)\sqcup(I_1^c\cap I_2^c\cap I_3^c), \end{aligned}$$ labeling a basis of $\mathfrak g$. This means that [*e.g.*]{} the elements of the basis indexed by $i$ in $I_1\cap I_2\cap I_3$ constitute a coordinate system for the intersection of $\mathfrak k^\perp$, $f+\mathfrak b_1$ and $f+\mathfrak b_2$, the elements indexed by $i$ in $I_1^c\cap I_2\cap I_3$ a coordinate system for the intersection of $\mathfrak k^*$ with $f+\mathfrak b_1$ and $f+\mathfrak b_2$, [*et similiter*]{}. The choice of labeling the coordinates on $\mathfrak g^*$ with respect to the above partition yields an obvious coloring of an admissible graph $\Gamma$ in $\mathcal G_{n,3}$: in fact, to any edge we may associate a triple $(j_1,j_2,j_3)$, $j_k$ in $\{1,2\}$, by the rule that the “parity” of $I_k$, resp. $I_k^c$, is $1$, resp. $2$, $k=1,2,3$. This rule simultaneously determines, for any colored edge of $\Gamma$, the coordinate set with respect to which the edge takes derivation, and the labeling of the edge by one of the $8$-colored propagators. Therefore, adjusting the arguments of the discussion at the beginning of [@CFFR Subsection 7.1], and taking into account the polydifferential operator associated to the only possibly non-trivial subgraph $\Gamma_A$ as in Figure 15, we see that these problems may be corrected by allowing for the presence of admissible graphs with short loops: the main difference between the results of [@CFFR] and the triquantization discussed here is the fact that triquantization requires the presence of two distinct short loops, namely, one taking care of the “regular term” $\widehat\rho$ and one of the “regular term” $\widetilde\rho$. Of course, there is a geometric counterpart to regular terms (which we discuss here very briefly, hoping to return to a more precise statement in the context of a formality theorem in presence of $3$ branes): as in [@CFFR], the geometric counterpart is played by a partial divergence operator, whose main property is Leibniz’ rule with respect to the Schouten–Nijenhuis bracket, which is essential in the computations. Since both “regular terms” are closed $1$-forms on $\mathcal C_{1,0,0,0}^+(\sqsubset)$, an admissible graph $\Gamma$ may admit [*at most*]{} two short loops of different type at each vertex of the first type: in fact, a vertex of the first type admits in this situation [*at most*]{} one short loop contribution, because of the linearity of the Poisson structure and because a divergence operator is of order $1$. In particular, a vertex of the first type with a short loop (either $\widehat\rho$ or $\widetilde\rho$) must have an outgoing edge, otherwise dimensional argument imply triviality of the corresponding integral weight. Previous computations imply that both short loop contributions vanish, when restricted to boundary strata $\mathcal C_{A,1}^+\times\mathcal C_{n-|A|,0,1,0}^+(\sqsubset)$, for subset $A$ of $\{1,\dots,n\}$ of cardinality $1\leq |A|\leq n$ of $\mathcal C_{n,0,1,0}^+(\sqsubset)$. Furthermore, when we consider restrictions to boundary strata of the form $\mathcal C_{A,2}^+\times \mathcal C_{n-|A|,0,0,0}^+(\sqsubset)$, the short loop contributions $\widehat\rho$ and $\widetilde\rho$ restrict to the short loop contribution in biquantization. Pictorially we now have to consider admissible graphs of the following shape: \ \ Summarizing the discussion so far, Proposition 21 in [@CT Subsubsection 6.2.2] remains valid, provided we enlarge the set $\mathcal G_{n,3}$ of admissible graphs, for $n\geq 0$, so as to contain graphs with multiple edges and short loop contributions of two distinct types at vertices of the first type. Finally, the results of [@CT Subsubsections 6.3.2 and 6.3.3] are easily proved to be still valid with the previous modification of admissible graphs. [^1]: The first author acknowledges partial support by SNF Grant $200020\_131813/1$

--- abstract: 'We have found that long-wavelength neutrino oscillations induced by a tiny breakdown of the weak equivalence principle of general relativity can provide a viable solution to the solar neutrino problem.' address: - 'Instituto de Física, Universidade de São Paulo, C. P. 66.318, 05315-970 São Paulo, Brazil' - 'Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas, 13083-970 Campinas, Brazil' author: - 'A. M. Gago[^1] H. Nunokawa and R. Zukanovich Funchal$^{\mbox{\scriptsize a}}$' title: 'The Solar Neutrino Problem in the Light of a Violation of the Equivalence Principle[^2]' --- INTRODUCTION ============ Neutrinos have had, since their childhood in the early 30’s, profound consequences on our understanding of the forces of nature. In the past they led to the discovery of neutral currents and provided the first indication in favour of the standard model of electroweak interaction. They may be today at the very hart of yet another breakthrough in our perceptions of the physical world. Today the results coming from solar neutrino experiments [@homestake; @sage; @gallex; @sk99] as well as from atmospheric neutrino experiments [@atmospheric] are difficult to be understood without admitting neutrino flavour conversion. Nevertheless the dynamics underlying such conversion is yet to be established and in particular does not have to be a priori related to the electroweak force. The interesting idea that gravitational forces may induce neutrino mixing and flavour oscillations, if the weak equivalence principle of general relativity is violated, was proposed about a decade ago [@gasper; @hl], and thereafter, many works have been performed on this subject [@muitas]. Many authors have investigated the possibility of solving the solar neutrino problem (SNP) by such gravitationally induced neutrino oscillations [@pantaleone; @bkl; @kuo], generally finding it necessary, in this context, to invoke the MSW like resonance [@hl] since they conclude that it is impossible that this type of long-wavelength vacuum oscillation could explain the specific energy dependence of the data [@pantaleone; @bkl]. Nevertheless we demonstrate that all the recent solar neutrino data coming from gallium, chlorine and water Cherenkov detectors can be well accounted for by long-wavelength neutrino oscillations induced by a violation of the equivalence principle (VEP). THE VEP FORMALISM ================= We assume that neutrinos of different types will suffer different time delay due to the weak, static gravitational field in the space on their way from the Sun to the Earth. Their motion in this gravitational field can be appropriately described by the parameterized post-Newtonian formalism with a different parameter for each neutrino type. Neutrinos that are weak interaction eigenstates and neutrinos that are gravity eigenstates will be related by a unitary transformation that can be parameterized, assuming only two neutrino flavours, by a single parameter, the mixing angle $\theta_G$ which can lead to flavour oscillation [@gasper]. In this work we assume oscillations only between two species of neutrinos, which are degenerate in mass, either between active and active ($\nu_e \leftrightarrow \nu_\mu,\nu_\tau$) or active and sterile ($\nu_e \leftrightarrow \nu_s$, $\nu_s$ being an electroweak singlet) neutrinos. The evolution equation for neutrino flavours $\alpha$ and $\beta$ propagating through the gravitational potential $\phi(r)$ in the absence of matter can be found in Ref. [@us]. In the case we take $\phi$ to be a constant, this can be analytically solved to give the survival probability of $ \nu_e$ produced in the Sun after travelling the distance $L$ to the Earth $$P( \nu_e \rightarrow \nu_e) = 1 - \sin^2 2\theta_G \sin^2 \frac{\pi L}{\lambda}, \label{prob}$$ where the oscillation wavelength $\lambda$ for a neutrino with energy $E$ is given by $$\lambda = \left[\frac{\pi {\mbox{ km}}}{5.07}\right] \left[\frac{10^{-15}} {|\phi \Delta \gamma|}\right] \left[\frac{ {\mbox{MeV}}}{E}\right], \label{wavelength}$$ which in contrast to the wavelength for mass induced neutrino oscillations in vacuum, is inversely proportional to the neutrino energy. Here $\Delta \gamma$ is the quantity which measures the magnitude of VEP. ANALYSIS ======== We will discuss here our analysis and results for active to active conversion. The same analysis for the $\nu_e \to \nu_s$ channel can be found in Ref. [@us], given similar results. In order to examine the observed solar neutrino rates in the VEP framework we have calculated the theoretical predictions for gallium, chlorine and Super-Kamiokande (SK) water Cherenkov solar neutrino experiments, as a function of the two VEP parameters ($\sin^2 2 \theta_G$ and $ | \phi \Delta \gamma |$), using the solar neutrino fluxes predicted by the Standard Solar Model by Bahcall and Pinsonneault (BP98) [@BP98] taking into account the eccentricity of the Earth orbit around the Sun. We do a $\chi^2$ analysis to fit these parameters and an extra normalization factor $f_B$ for the $^8$B neutrino flux, to the most recent experimental results coming from Homestake [@homestake] $R_{\mbox{Cl}}= 2.56 \pm 0.21$ SNU, GALLEX[@gallex] and SAGE[@sage] combined $R_{\mbox{Ga}}= 72.5 \pm 5.5$ SNU and SK [@sk99] $R_{\mbox{SK}}= 0.475 \pm 0.015$ normalized to BP98. We use the same definition of the $\chi^2$ function to be minimized as in Ref. [@chi2], except that our theoretical estimations were computed by convoluting the survival probability given in Eq. (\[prob\]) with the absorption cross sections [@bhp], the neutrino-electron elastic scattering cross section with radiative corrections [@xsec] and the solar neutrino flux corresponding to each reaction, $pp$, $pep$, $^7$Be, $^8$B, $^{13}$N and $^{15}$O; other minor sources were neglected. -1.cm -0.35cm We present in Fig. 1 (a) the allowed region determined only by the rates with free $f_B$, for fixed $^8$B flux ($f_B=1$) the allowed region is similar. In Ref. [@us] one can find a table which gives more details on best fitted parameters as well as the $\chi^2_{\mbox{\scriptsize min}}$ values for fixed and free $f_B$. We found for $f_B=1$ that $\chi^2_{\mbox{\scriptsize min}} = 1.49$ for 3-2=1 degree of freedom and for $f_B=0.81$ that $\chi^2_{\mbox{\scriptsize min}} = 0.32$ for 3-3=0 degree of freedom. We then perform a spectral shape analysis fitting the $^8$B spectrum measured by SK  [@sk99] using the following $\chi^2$ definition: $$\chi^2 = \sum_i \left[\frac{S^{\mbox{\scriptsize obs}}(E_i)-f_B S^{\mbox{\scriptsize theo}}(E_i)}{\sigma_i}\right]^2,$$ where the sum is performed over all the 18 experimental points $S^{\mbox{\scriptsize obs}}(E_i)$ normalized by BP98 prediction for the recoil-electron energy $E_i$, $\sigma_i$ is the total experimental error and $S^{\mbox{\scriptsize theo}}$ is our theoretical prediction that was calculated using the BP98 $^8$B differential flux, the $\nu-e$ scattering cross section [@xsec], the survival probability as given by Eq. (\[prob\]) taking into account the eccentricity as we did for the rates, the experimental energy resolution as in Ref. [@res] and the detection efficiency as a step function with threshold $E_{\mbox{\scriptsize th}}$ = 5.5 MeV. After the $\chi^2$ minimization with $f_B=0.80$ we have obtained $\chi^2_{\mbox{\scriptsize min}}=15.8$ for 18-3 =15 degree of freedom. The best fitted parameters that also can be found in Ref. [@us] permit us to compute the allowed region displayed in Fig. 1 (b). Finally, we perform a combined fit of the rates and the spectrum obtaining the allowed region presented in Fig. 1 (c). The combined allowed region is essentially the same as the one obtained by the rates alone. In all cases presented in Figs. 1 (a)-(c) we have two isolated islands of 90% C.L. allowed regions. See Ref. [@us] for a table with the best fitted parameters for this global fit as well as for the fitted values corresponding to the local minimum in these islands. Note that only the upper corner of the Fig. 1 (c), for $|\phi \Delta \gamma | > 2 \times 10^{-23}$ and maximal mixing in the $\nu_e \to \nu_\mu$ channel can be excluded by CCFR [@pkm]. Moreover, there are no restrictions in the range of parameters we have considered in the case of $\nu_e \to \nu_\tau, \nu_s$ oscillations. DISCUSSIONS AND CONCLUSIONS =========================== Let’s finally remark that, in contrast to the usual vacuum oscillation solution to the SNP, in this VEP scenario no strong seasonal effect is expected in any of the present or future experiments, even the ones that will be sensitive to $^7$Be neutrinos [@borexino; @hellaz]. Contrary to the usual vacuum oscillation case, the oscillation length for the low energy $pp$ and $^7$Be neutrinos are very large, comparable to or only a few times smaller than the Sun-Earth distance, so that the effect of the eccentricity in the oscillation probability is small. On the other hand, for higher energy neutrinos relevant for SK, the effect of the eccentricity in the probability could be large, but it is averaged out after the integration over a certain neutrino energy range. These observations are confirmed by Fig. 4 of Ref. [@us]. We have found a new solution to the SNP which is comparable in quality of the fit to the other suggested ones and can, in principle, be discriminated from them in the near future. In fact a very-long-baseline neutrino experiment in a $\mu$-collider [@geer] could directly probe the entire parameter region where this solution was found. ACKNOWLEDGMENTS {#acknowledgments .unnumbered} =============== We thank P. Krastev, E. Lisi, G. Matsas, H. Minakata, M. Smy, P. de Holanda and GEFAN for valuable discussions and comments. H.N. thanks W. Haxton and B. Balantekin and the Institute for Nuclear Theory at the University of Washington for their hospitality and the Department of Energy for partial support during the final stage of this work. This work was supported by the Brazilian funding agencies FAPESP and CNPq. [99]{} K. Lande [*et al.*]{}, Astrophys. J. [**496**]{} (1998) 505. J. N. Abdurashitov [*et al.*]{},astro-ph/9907113. W. Hampel [*et al.*]{}, Phys.Lett.  B [**447**]{} (1999) 127. See M. Nakahata in these Proceedings. See F. Ronga and M. Nakahata in these Proceedings. M. Gasperini, Phys. Rev. D [**38**]{} (1988) 2635; [*ibid.*]{} [**39**]{} (1980) 3606. A. Halprin and C. N. Leung, Phys. Rev. Lett. [**67**]{} (1991) 1833; Nucl. Phys. B (Proc. Suppl.) [**28A**]{} (1992) 139. See Ref. [@us] and references therein. J. Pantaleone, A. Halprin, and C. N. Leung, Phys. Rev. D [**47**]{} (1993) R4199. J. N. Bahcall, P. I. Krastev, and C. N. Leung, Phys. Rev. D [**52**]{} (1995) 1770. S. W. Mansour and T. K. Kuo, hep-ph/9810510. A. M. Gago, H. Nunokawa and, R. Zukanovich Funchal, hep-ph/9909250. J. N. Bahcall, S. Basu, and M. H. Pinsonneault, Phys. Lett.  B [**433**]{} (1998) 1. M. M. Guzzo and H. Nunokawa, Astropart.  Phys. [**12**]{} (1999) 87; G. L. Fogli and E. Lisi, Astropart. Phys.  [**3**]{} (1995) 185. See http://www.sns.ias.edu/$\sim$jnb/. J. N. Bahcall, M. Kamionkowski, and A. Sirlin, Phys. Rev. D [**51**]{} (1995) 6146. B. Faïd, [*et al.*]{}, Astropart. Phys. [**10**]{} (1999) 93. J. Pantaleone, T. K. Kuo, and S. W. Mansour, hep-ph/9907478. R. S. Raghavan, Science [**267**]{} (1995) 45. G. Bonvicini, Nucl. Phys. [**B35**]{} (1994) 438. S. Geer, Phys. Rev. D [**57**]{} (1998) 1. [^1]: On leave of absence from : Sección Física, Departamento de Ciencias, Pontificia Universidad Católica del Perú, Apartado 1761, Lima, Perú. [^2]: Talk given by R. Zukanovich Funchal.

--- abstract: 'We study experimentally and theoretically the effect of Eu doping and partial oxygen isotope substitution on the transport and magnetic characteristics and spin-state transitions in (Pr$_{1-y}$Eu$_y$)$_{0.7}$Ca$_{0.3}$CoO$_3$ cobaltites. The Eu doping level $y$ is chosen in the range of the phase diagram near the crossover between the ferromagnetic and spin state transitions ($0.10 < y < 0.20$). We prepared a series of samples with different degrees of enrichment by the heavy oxygen isotope $^{18}$O, namely, containing 90%, 67%, 43 %, 17 %, and 0 % of $^{18}$O. Based on the measurements of ac magnetic susceptibility $\chi(T)$ and electrical resistivity $\rho(T)$, we analyze the evolution the sample properties with the change of Eu and $^{18}$O content. It is demonstrated that the effect of increasing $^{18}$O content on the system is similar to that of increasing Eu content. The band structure calculations of the energy gap between $t_{2g}$ and $e_g$ bands including the renormalization of this gap due to the electron-phonon interaction reveal the physical mechanisms underlying such similarity.' author: - 'N. A. Babushkina' - 'A. N. Taldenkov' - 'S. V. Streltsov' - 'T. G. Kuzmova' - 'A. A. Kamenev' - 'A. R. Kaul' - 'D. I. Khomskii' - 'K. I. Kugel' title: 'Effect of Eu doping and partial oxygen isotope substitution on magnetic phase transitions in (Pr$_{1-y}$Eu$_y$)$_{0.7}$Ca$_{0.3}$CoO$_3$ cobaltites' --- Introduction \[intro\] ====================== Most magnetic oxides are characterized by a strong interplay of electron, lattice, and spin degrees of freedom giving rise to multiple phase transitions and different types of ordering. The phase transitions are often accompanied by the formation of different inhomogeneous states. In such situation, the oxygen isotope substitution provides a unique tool for investigating inhomogeneous states in magnetic oxides, which allows studying the evolution of their properties in a wide range of the phase diagram. Sometimes, particularly if a system is close to the crossover between different states (usually leading to a phase separation), the isotope substitution can lead to significant changes in the ground state of the system. [@Babushkina1998]. A good example of such phenomena is provided by cobaltites. These perovskite cobalt oxides have attracted a special interest owing to the possibility of the spin-state transitions (SST) for the Co ions induced by temperature or doping [@goodenough.jap65; @asai.jpsj98; @saitoh.prb97; @tokura.prb98; @korotin.prb96; @berggold.prb08; @ovchinnikov.UFN10] and the related phase separation phenomena. [@maignan.jpcm02; @leighton.prb03; @louca.prl06; @podlesnyak.prl08; @sboychakov.prb09; @louca.prb09; @leighton.prb10; @podlesnyak.prb11] The effect of $^{16}$O $\rightarrow ^{18}$O isotope substitution on the properties of (Pr$_{1-y}$Eu$_y$)$_{0.7}$Ca$_{0.3}$CoO$_3$ cobaltites ($0.12 <y < 0.26$) was studied earlier in our paper Ref. . It was found that with increasing Eu content, the ground state of the compound changes from a “nearly-metallic" ferromagnet (ferromagnetic metallic clusters embedded into an insulating host) to a “weakly-magnetic insulator" at $y < y_{cr} \approx 0.18$, regardless the isotope content. A pronounced SST was observed in the insulating phase (in the samples with $y > y_{cr}$), whereas in the metallic phase (at $y < y_{cr}$), the magnetic properties were quite different, without any indications of a temperature-induced SST. Using the magnetic, electrical, and thermal data, we constructed the phase diagram for this material. The characteristic feature of this phase diagram is a broad crossover range near $y_{cr}$ corresponding to a competition of the phases mentioned above. The $^{16}$O $\to ^{18}$O substitution gives rise to an increase in temperature $T_{SS}$ of the SST and to a slight decrease in the ferromagnetic (FM) transition temperature $T_{FM}$. However, a number of problems important for understanding the physics of the systems with spin-state transitions have not been touched upon in the study reported in Ref. . The most important question is the relation between the changes caused by varying the composition (increase of concentration $y$ of the smaller rare-earth ions Eu) and that due to isotope substitution, and the physical mechanism underlying these changes. We see, [*cf.*]{} phase diagram in Ref.  and in Fig. \[PhDiag\] below, that there exists some correlation between these changes, but the situation is not so simple: thus, in the right part of the phase diagram the spin state transition increases both with the increase of Eu content $y$ and with the increase of isotope mass (going from $^{16}$O to $^{18}$O). At the same time, in the left part of this phase diagram the effect of the increase of Eu content and of increasing oxygen mass on the phase transition (in this case transition to a nearly ferromagnetic state) is just the opposite: an increase in Eu content leads to the decrease of $T_{FM}$, but the increase of oxygen mass – to the increase of it. Another important open question concerns the behavior of separate phases in the regime of phase separation. There are many different correlated systems, in which phase separation was detected in some range of compositions, temperatures, external fields, etc. Typically, the measured transition temperatures in this case changes e.g. with doping. However, it often remains unclear whether this change is the effect occurring in separate regions of different phases, or is just the result of averaging out over the inhomogeneous system. To answer these questions, we now carried out the detailed study of the behavior of the (PrEu)CoO$_3$, using as a tool the possibility of fine tuning the properties of the system by partial isotope substitution. This partial substitution plays in effect the same role as doping, external pressure, etc. The obtained results established the possibility of “rescaling" the changes in the system with doping and with isotope substitution and allowed us to clarify the questions formulated above. As to the second question formulated above, just the possibility of fine tuning the properties of the system inside the region of phase separation, provided by partial isotope substitution, allowed us to study separately the behavior of different phases within this phase separated regime – the possibility, which would be very difficult to get by other means. Our results obtained in this way demonstrate that not only the average critical temperatures change with doping and with isotope substitution, but also “individual" transition temperatures (temperature of ferromagnetic transition in more metallic regions and the temperature of the spin-state transition in more insulating parts of the sample) do change with chemical and isotope composition. As to the main, the first question formulated above, about the mechanisms governing the change of properties of the system with chemical and isotope composition, the experimental findings reported in the present paper provided us an opportunity to formulate a realistic theoretical model clarifying the mechanisms underlying the pronounced isotope effects in cobaltites exhibiting spin-state transitions. The theoretical analysis demonstrates that the main factor is the change of the effective bandwidth with the change both of chemical and isotope composition. The opposite trends in two parts of phase diagram mentioned above find natural explanation in this picture. To analyze the effects of the partial oxygen isotope substitution for the doped cobaltites in the crossover region of the phase diagram, we have prepared a series of oxide materials with nearly continuous tuning of their characteristics. This allows us to trace the evolution of relative content of different phases as a function of $^{18}$O/$^{16}$O ratio. Note that there were only few investigations of this kind, one of which we undertook earlier for (La$_{1-y}$Pr$_y$)$_{0.7}$Ca$_{0.3}$MnO$_3$ manganites. [@Babushkina2000] Here, the pronounced isotope effect manifesting itself in (Pr$_{1-y}$Eu$_y$)$_{0.7}$Ca$_{0.3}$CoO$_3$ cobaltites provides indeed a unique possibility to address the problems discussed above through the use of the partial oxygen isotope substitution. Samples ======= Polycrystalline (Pr$_{1-y}$Eu$_y$)$_{0.7}$Ca$_{0.3}$CoO$_3$ samples were prepared by the chemical homogenization (“paper synthesis") method [@Balagurov99] through the use of the following operations. At first, non-concentrated water solutions of metal nitrates Pr(NO$_3$)$_3$, Eu(NO$_3$)$_3$, Ca(NO$_3$)$_2$, and Co(NO$_3$)$_2$ of 99.95% purity were prepared. The exact concentration of dissolved chemicals was established by gravimetric titration and, in the case of Co-based solution, by means of potentiometric titration. The weighted amounts of metal nitrate solutions were mixed in stoichiometric ratio and the calculated mixture of nitrates were dropped onto the ash-free paper filters. The filters were dried out at about 80$^{\circ}$C and the procedure of the solutions dropping was performed repeatedly. Then, the filters were burned out and the remaining ash was thoroughly ground. It was annealed at 800$^{\circ}$C for 2 hours to remove the carbon. The powder obtained was pressed into the pellets and sintered at 1000$^{\circ}$C in the oxygen atmosphere for 100 hours. Finally, the samples were slowly cooled down to room temperature by switching off the furnace. Samples were analyzed at room temperature by the powder X-ray diffraction using Cu K$_{\alpha}$ radiation. All detectable peaks were indexed by the space group, Pnma. According to the X-ray diffraction patterns, all (Pr$_{1-y}$Eu$_y$)$_{0.7}$Ca$_{0.3}$CoO$_3$ samples were obtained as single-phase crystalline materials. We prepared a series of ceramic cobaltite samples with the degrees of enrichment by $^{18}$O equal to 90%, 67%, 43%, 17%, and 0%. These values were determined by the changes in the sample mass in the course of the isotope exchange and by the mass spectrometry of the residual gas in the oxygen exchange contour. The samples were annealed in the appropriate $^{16}$O–$^{18}$O gas mixture at 950 $^\circ$C during 48 h at total pressure of 1 bar. The Eu doping of the samples was chosen to be near and on the both sides of the crossover doping level $y_{cr}$: a composition with the high Eu content exhibiting the SST, a low-Eu composition corresponding to the nearly ferromagnetic state, and the sample at the phase crossover, where both $T_{SS}$ and $T_{FM}$, were observed. For all samples, we measured the temperature dependence of the real $\chi'(T)$ and imaginary $\chi''(T)$ parts of the ac magnetic susceptibility and electrical resistivity $\rho(T)$. The resistivity of the samples was measured by the conventional four-probe technique in the temperature range from 4.2 to 300 K . The measurements of ac magnetic susceptibility $\chi(T)$ were performed in ac magnetic field with a frequency 667 Hz and an amplitude of about 0.1 Oe. Based on these measurements, we were able to analyze the evolution the sample properties with the change Eu and $^{18}$O content. Experimental results ==================== 1\. For the samples with $y > y_{cr}$ (Eu content $y=0.20$), the material correspond to a “weakly-magnetic insulator", in notation of Ref. . The $\chi'(T)$ curves give clear indications of a spin-state transition (SST) at $T_{SS}$ manifesting itself as a peak in the $\chi'(T)$, see Fig. \[Fig\_chi\_02\]. The value of $T_{SS}$ increases with the $^{18}$O content. The high-temperature phase is a paramagnet and a relatively good conductor, see Fig. \[Fig\_rho\_02\]. The low-spin (LS) insulating phase is dominant below the crossover to a LS state occurring at about 100 K. The increase of $\chi'$ at low temperatures is most probable caused by an incomplete transition, after which there may remain small magnetic (and presumably more conducting) clusters immersed into the LS state bulk insulator. As a result of this crossover to a LS state, the electrical resistance $R$ increases by 10–12 orders of magnitude, which can be treated as a metal–insulator transition (MI), see Fig. \[Fig\_rho\_02\]). In these compounds, the metal–insulator transition is accompanied (or caused) by the SST. The $^{18}$O content does not produce a significant effect on $R(T)$ at low $T$, although $R$ is larger in the samples with heavy oxygen. The value of the metal–insulator transition temperature $T_{MI}$ can be determined from the logarithmic derivative of $R(T)$, see inset of Fig. \[Fig\_rho\_02\]. The mass dependence of $T_{MI}$ correlates well with the isotopic shift of $T_{SS}$, see Fig. \[Fig\_Tss\_y02\]. According to the calculation of isotopic constant $T \sim M^{-\alpha}$; $\alpha = -d \ln T/d\ln M = -(\Delta T/\Delta M)(M/T)$, we have the value $\alpha_{SS}$ and $\alpha_{MI} = -(1.5 \pm 0.07)$. The increasing in the oxygen mass promotes the development of the LS state. ![\[Fig\_chi\_02\] (Color online) Temperature dependence of magnetic susceptibility for (Pr$_{1-y}$Eu$_y$)$_{0.7}$Ca$_{0.3}$CoO$_3$ with $y=0.2$.](Fig_chi_02.eps){width="0.9\columnwidth"} ![\[Fig\_rho\_02\] (Color online) Temperature dependence of electrical resistivity for (Pr$_{1-y}$Eu$_y$)$_{0.7}$Ca$_{0.3}$CoO$_3$ with $y=0.2$.](Fig_rho_02.eps){width="0.9\columnwidth"} ![\[Fig\_Tss\_y02\] (Color online) Isotopic shift of the characteristic temperatures of the spin-state $T_{SS}$ and metal-insulator $T_{MI}$ transitions at the $^{16}$O$\to$$^{18}$O substitution for the sample with $y=0.2$.](Fig_Tss_y02.eps){width="0.9\columnwidth"} 2\. The most important results are obtained for the samples with Eu content $y \sim y_{cr}$ ($y=0.14$ and 0.16). They correspond to a wide concentration range of the phase separation. For the sample with $y=0.14$, we observe in $\chi'(T)$ curves a feature corresponding to a steep increase of the magnetization on cooling at 60–70 K, see Fig. \[Fig\_chi\_014\]. This behavior is caused by the FM phase arising in these samples. In addition, in the temperature dependence of resistivity, we see a steep increase of resistivity characteristic of the metal–insulator transition similar to that observed for the samples with $y = 0.2$, see Fig. \[Fig\_rho\_014\]. The transition temperature corresponds to the minimum in the logarithmic derivative of the resistivity, see inset of Fig. \[Fig\_rho\_014\]. This means that here we deal also with the change in the relative content of metallic and insulating phases suggesting the existence of the regions corresponding to the low-spin insulating state and the correlation between the metal–insulator and SS transitions. At the same time we do not observe any clear indications of the SST in the temperature dependence of the magnetic susceptibility. Thus, the samples with $y =0.14$ turn out to be at the boundary of the phase separation range. The increasing in the oxygen mass favors the LS state as well as suppression of the FM phase and of the metallization. The values of the isotopic constant calculated for the SS and FM transitions in this sample are $\alpha_{SS,MI} = -(1.7 \pm 0.06)$ and $\alpha_{FM} = (0.5 \pm 0.05)$, respectively. For this sample, the isotope shifts of the characteristic transition temperatures are illustrated in Fig. \[Fig\_Tss\_y014\]. ![\[Fig\_chi\_014\] (Color online)Temperature dependence of magnetic susceptibility for (Pr$_{1-y}$Eu$_y$)$_{0.7}$Ca$_{0.3}$CoO$_3$ with $y=0.14$.](Fig_chi_014.eps){width="0.9\columnwidth"} ![\[Fig\_rho\_014\] (Color online) Temperature dependence of electrical resistivity for (Pr$_{1-y}$Eu$_y$)$_{0.7}$Ca$_{0.3}$CoO$_3$ with $y=0.14$ (solid line) and $y=0.1$ with $^{16}$O (dashed line) and $^{18}$O (dash-dot line). ](Fig_rho_014.eps){width="0.9\columnwidth"} ![\[Fig\_Tss\_y014\] (Color online) Isotopic shift of the characteristic temperatures of the spin-state $T_{SS}$, metal-insulator $T_{MI}$ and ferromagnetic $T_{FM}$ transitions for the sample with $y=0.14$.](Fig_Tss_y014.eps){width="0.9\columnwidth"} For the samples with $y =0.16$, the effect of the partial oxygen isotope substitution by $^{18}$O manifests itself even more clearly. Here, we observed the features characteristic both of FM and SS transitions. On the one hand, the temperatures dependence of $\chi'$ exhibits a steep growth at 60–70 K similar to that in the sample 0.14 indicating the existence of the FM transition. On the other hand, in of $\chi'(T)$ , we observed a clearly pronounced peak, which can be attributed to SST at $T_{SS}$ (Fig. \[Fig\_chi\_016\]). $\chi'(T)$ curves also demonstrate that the transition to the LS state gradually disappears with the decrease of the oxygen mass. In samples with small $^{18}$O content ($<17$%), this transition is hardly seen due to the gradual transformation from the LS to FM state when the oxygen mass decreases. Note also that for $y = 0.16$ (as well as in the samples with $y =0.14$) the temperature $T_{FM}$ decreases with the growth of the average mass of oxygen. ![\[Fig\_chi\_016\] (Color online) Temperature dependence of magnetic susceptibility for (Pr$_{1-y}$Eu$_y$)$_{0.7}$Ca$_{0.3}$CoO$_3$ with $y=0.16$.](Fig_chi_016.eps){width="0.9\columnwidth"} ![\[Fig\_revchi\_016\] (Color online) Temperature dependence of inverse magnetic susceptibility for (Pr$_{1-y}$Eu$_y$)$_{0.7}$Ca$_{0.3}$CoO$_3$ with $y=0.16$.](Fig_revchi_016.eps){width="0.9\columnwidth"} In addition, the Curie–Weiss temperature $\theta$ in the formulas for the inverse magnetic susceptibility $\chi^{-1}(T)$ considerably decreases with the increase of the average mass of oxygen isotopes (see Fig. \[Fig\_revchi\_016\]). For the samples with the largest oxygen mass, we have $\theta=$ 184 K, whereas for the samples with the $^{16}$O, $\theta=$ 45 K. This phenomenon may be related to the transition from the antiferromagnetic interaction to ferromagnetic one. For sample with $y = 0.16$ as well as in the samples with $y = 0.14$ and 0.2, we have found the growth of resistivity at decreasing temperatures (by 5-7 orders of magnitude) and the metal-insulator transition in the vicinity of 70 K (Fig. \[Fig\_rho\_016\]). Both the resistivity and magnetic susceptibility data give clear indication that this sample is in the phase-separation range. The temperatures of MI and SS transitions increase with the average mass oxygen (see inset of Fig. \[Fig\_rho\_016\]). In the sample with $y = 0.16$, we see the same general tendency, namely the growth of $T_{SS}$ and the decrease of $T_{FM}$ with the growth of the average oxygen isotope mass. Here, the values of the isotope constant are $\alpha_{SS,MI} = -(1.7 \pm 0.06)$ and $\alpha_{FM} = (0.4 \pm 0.1)$, respectively; the isotope shifts of the characteristic transition temperatures are illustrated in Fig. \[Fig\_Tss\_y016\]. ![\[Fig\_rho\_016\] (Color online) Temperature dependence of electrical resistivity for (Pr$_{1-y}$Eu$_y$)$_{0.7}$Ca$_{0.3}$CoO$_3$ with $y=0.16$.](Fig_rho_016.eps){width="0.9\columnwidth"} ![\[Fig\_Tss\_y016\] (Color online) Isotopic shift of the characteristic temperatures of the spin-state $T_{SS}$, metal-insulator $T_{MI}$ and ferromagnetic $T_{FM}$ transitions for the sample with $y=0.16$.](Fig_Tss_y016.eps){width="0.9\columnwidth"} 3\. Finally, the samples with $y < y_{cr}$ (Eu content $y=0.10$) fall into the range of “nearly-metallic" ferromagnet. In Ref. , it was shown that at $T < T_{FM}$ the compositions with a low Eu content correspond to the domains of the metallic ferromagnetic phase embedded in a weakly magnetic nonconducting matrix. According to the temperature dependence $\chi'(T)$ plotted in Fig. \[Fig\_chi\_01\], the magnetization steeply increases on cooling at about $60-70$ K (i.e. the ferromagnetic phase arises). With the growth of the average oxygen mass, the value of $\chi'(T)$ decreases at low temperatures. Here, we have $T_{FM}$($^{18}$O)$ < T_{FM}$($^{16}$O) and the maximum isotope shift of $T_{FM}$ does not exceed 2–3 K. ![\[Fig\_chi\_01\] (Color online) Temperature dependence of magnetic susceptibility for (Pr$_{1-y}$Eu$_y$)$_{0.7}$Ca$_{0.3}$CoO$_3$ with $y=0.1$.](Fig_chi_01.eps){width="0.9\columnwidth"} The electrical resistivity for this samples ($y=0.10$) increases at low temperatures, even when the ferromagnetic phase arises. This behavior is quite similar to that observed in the samples with $y=0.14$ and $y=0.16$. Note that the electrical resistance in the samples with $^{18}$O is higher than in those with $^{16}$O. However, with the decrease of Eu content down to $y =0.10$, the MI peculiarity in the resistance is suppressed. Such behavior $R(T)$ for samples with $y =0.10$ can be compared with that for $y =0.14$ samples, see Fig. \[Fig\_rho\_014\]) and inset of this figure. The resistivity for samples $y=0.10$ corresponds to a more smooth curve than for samples with $y=0.14$, although the regular course of the temperature dependence remains nearly unchanged. Thus, the samples with $y=0.10$ do not become really metallic but their behavior differs from the behavior of the samples with $y=0.14$ (they do not exhibit indications of SST). Therefore, we argue that in the phase diagram, the composition with $y=0.10$ lies outside the crossover region, on the left-hand side of it. Discussion of experimental results ================================== The obtained data for the temperatures of the phase transitions can be presented in the form of the phase diagram (Fig. \[PhDiag\]). It illustrates that with the growth of Eu content the system transforms from the nearly ferromagnetic metal to the LS insulating state undergoing LS $\to$ IS spin-state transitions . Between these states, we have a broad crossover region corresponding to the phase separation. Indeed, the simultaneous observation both of $T_{FM}$ and $T_{SS}$ in the samples with Eu content $y =0.14$ and 0.16 is a clear evidence of the phase separation in the system. These samples also provide a spectacular illustration of the effect related to the variable content of $^{18}$O. In particular, for the samples with $y =0.16$, the temperature dependence of magnetic susceptibility (Fig. \[Fig\_chi\_014\]) exhibits at high values of $^{18}$O content a pronounced feature corresponding to the SST. With the lowering of $^{18}$O content, this feature becomes weaker and disappears below 17% of $^{18}$O. Hence, we see that the change on the average oxygen mass can drastically affect the phase composition of the cobaltite samples. The oxygen isotope substitution $^{16}$O $\to ^{18}$O shifts the phase equilibrium toward the insulating state. For the heavier isotope, the spin-state transition temperature $T_{SS}$ grows, while the ferromagnetic transition temperature $T_{FM}$ decreases. The change of average oxygen mass is a unique tool for investigating special properties of phase separation in cobaltites near the crossover between the FM and LS phases. We also see that the effect of increasing $^{18}$O content on the system is similar to that of increasing Eu content. ![\[PhDiag\] (Color online) Phase diagram of (Pr$_{1-y}$Eu$_y$)$_{0.7}$Ca$_{0.3}$CoO$_3$ compound with $^{16}$O and 90% $^{18}$O. PM, FM, IS, and LS stand for the paramagnetic, ferromagnetic, intermediate-spin, and low-spin state, respectively. The hatched area corresponds to the phase-separated state (PS).](PhDiag.eps){width="0.95\columnwidth"} The analysis of these results for different chemical and isotope composition demonstrates that the effect of the increase in Eu content $y$ and of average oxygen mass are qualitatively similar. One can rescale the dependence of the transition temperature on both parameters using the combined variable $y + 0.01x$, where $x$ is the relative content of $^{18}$O, as shown in Fig. \[Tss(x,y)\]. Both the temperatures of SS and FM transitions depend almost linearly on this combined variable. We see that the change of Eu content by 1% is equivalent to the change of the isotope content by 100 % . This actually the most important result of the present study. The theoretical analysis of these results is given in the following sections. Calculation details =================== To explain the composition and isotope dependence of the properties of our system (see Fig. \[PhDiag\]), especially the similar dependence of the SST temperature and the temperature of the FM transition illustrated in Fig. \[Tss(x,y)\], we propose a realistic model (Section \[SecIsotope\]) based predominantly on the change of the electron bandwidth with chemical and isotope composition. Some input, as well as the estimates of relevant parameters are taken from the [*ab initio*]{} band structure calculations for the limiting “pure" compositions corresponding to $y =0$ (PrCoO$_3$) and $y =1$ (EuCoO$_3$). The crystal structure of PrCoO$_3$ obtained in Ref.  for $T=300$ K was utilized in those calculations. For EuCoO$_3$, the lattice parameters were taken from Ref. . The exact atomic positions for EuCoO$_3$ are unknown, therefore we used the same positions as for PrCoO$_3$ (with the correct unit cell volume for EuCoO$_3$). The splitting between different one-electron energy levels $\Delta_{CF}$ was calculated within the local density approximation (LDA) in the framework of the linear muffin-tin orbitals method (LMTO). [@Andersen1984] Partially filled, but physically unimportant $4f$ states of the Eu and Pr were treated as frozen. [@Nekrasov2003] The Brillouin-zone (BZ) integration in the course of self-consistency iterations was performed over a mesh of 144 [**k**]{}-points in the irreducible part of the BZ. Doping dependence: LDA results ============================== There are different ways to estimate the temperature of the spin-state transition with the use of the band structure calculation. The most direct way is to calculate the total energies of different spin states. [@Korotin96; @Streltsov11]. However, in the case of the doped system this would require very large supercells. Moreover, currently,we have no single commonly accepted model, which can explain all experimental facts. Various combinations of a static or dynamic order of the different spin states are discussed in the literature. [@Nekrasov2003; @Knizek2009-2; @Haverkort-06; @Ren-11; @Lamonova-11] This is the reason why we chose an alternative approach. The energy of any of the spin states depends on two important parameters: the single-electron energy difference $\Delta_{CFS}$ between the highest $t_{2g}$ and the lowest $e_g$ levels, and the intra-atomic Hund’s rule exchange coupling $J_H$. The Hund’s rule energy $J_H$ is an atomic characteristic and does not change appreciably either with the Eu doping or with the isotope substitution. So,to investigate the dependence of the spin-state transition temperature on the doping or isotope substitution in the first approximation one can focus on the study of the single parameter, $\Delta_{CFS}$. If $\Delta_{CFS}$ is large enough ($\Delta_{CFS} > 2J_H$), it will be energetically favorable to localize all electrons in the low-lying $t_{2g}$ subshell of Co$^{3+}$, i.e. the system will be in the low-spin state. With decrease of the crystal-field splitting, some electrons can be transferred to $e_g$ subshell, which allows the system to gain the exchange energy, since there will be more electrons with the same spin. As we will show below both the isotope substitution and the doping can be related to the crystal-field splitting. In order to estimate $\Delta_{CFS}$, we used the Wannier function projection procedure proposed in Ref. , which allows us to project the full-orbital band Hamiltonian onto the subspace of few states (five $d$ states of Co). With the Fourier transformation one obtains the Hamiltonian in the real space, from which the splitting between highest in energy $t_{2g}$ and lowest $e_g$ can be easily calculated. For PrCoO$_3$, we obtain $\Delta_{CFS} = 2.07$ eV. The total and partial densities of states (DOS) obtained for PrCoO$_3$ in the LDA calculations are presented in Fig. \[PrCoO3\]. In the octahedral symmetry, the 3$d$ states of Co are split into $t_{2g}$ and $e_g$ subbands. In the LDA, the valence band is mostly formed by the Co$-t_{2g}$ states, while the conduction band is determined by the Co$-e_{g}$ states. The O$-2p$ band is located in the energy range from -7 to -1.5 eV. The DOS for EuCoO$_3$ is qualitatively very similar and corresponding calculations of $\Delta_{CFS}$ result in the value of 2.14 eV. The increase of the $t_{2g}-e_g$ excitation energy on going from PrCoO$_3$ to EuCoO$_3$ is caused by two factors. The first is the lanthanide contraction: the substitution of the large Pr$^{3+}$ by smaller Eu$^{3+}$ ions leads to some decrease of the Co–O distance and to the corresponding increase in the $p-d$ hybridization, which leads to an increase of the difference between the centers of the $t_{2g}$ and $e_g$ bands. The second effect is related to the decrease of the effective widths of $t_{2g}$ and $e_g$ energy bands with the corresponding increase of the energy gap between them. This narrowing of energy bands on going from PrCoO$_3$ to EuCoO$_3$ is also related, in effect, with the lanthanide contraction, because of which the tilting of CoO$_6$ octahedra increases and the Co–O–Co angle and the corresponding bandwidth decrease on going from PrCoO$_3$ to EuCoO$_3$. Both these effects finally lead to the increase of $\Delta_{CFS}$ with Eu content, which leads to the enhanced stabilization of of the LS state of Co$^{3+}$ (see the more detailed discussion of these effects in Section \[SecIsotope\]). This change of the crystal-field splitting (CFS) results in the modification of the spin-state transition temperature, since this transition is due to the competition of the Hund’s rule exchange coupling $J_H$ and the CFS. [@Nekrasov2003] It was found in Refs.  and (see also Fig. \[PhDiag\]) that the change of the Eu content $y$ in (Pr$_{1-y}$Eu$_y$)$_{0.7}$Ca$_{0.3}$CoO$_3$ by 0.02 leads to the change of the spin-state transition temperature by about 14 K. In the first approximation, it is possible to neglect the presence of Ca and interpolate the change of the CFS for the complex system like (Pr$_{1-y}$Eu$_y$)$_{0.7}$Ca$_{0.3}$CoO$_3$ using the values of the CFS for $y=1$ (PrCoO$_3$) and $y=0$ (EuCoO$_3$). Such a linear interpolation predicts the change of the spin-state transition temperature by 16 K, if $y$ changes by 0.02, which is in an excellent agreement with the experiment. [@Babushkina2010; @Kalinov2010]  The total and partial densities of states (DOS) for PrCoO$_3$. The Fermi energy corresponds to zero.](PrCoO3_DOS.eps){width="0.95\columnwidth"} Isotope substitution: Model results \[SecIsotope\] ================================================== The isotope substitution does not change the chemical properties of the ions such as the oxidation numbers or bonding energies. However, it affects the crystal lattice through the modification of the phonon spectra. Below, following the approach of Ref. , we demonstrate the effect of this modification on the electronic and magnetic properties and hence on the spin-state transition. In the tight-binding model, the band spectra of a solid is determined by the on-site ionic energy levels $\varepsilon_i^{nlm}$ and the hopping matrix elements between different sites $t_{ij}^{ll',mm'}$. The ionic energies $\varepsilon_i^{nlm}$ obviously do not depend on the mass of the ions, being determined by the quantum numbers and the intra-atomic Coulomb and exchange interactions. The hopping parameters depend on the type of the orbitals ($s,p,d,f$), bonding type ($\pi,\sigma,\delta$), and the distance between ions, $u$. According to the famous Harrison parametrization [@Harrison1999; @Andersen1977] in the absence lattice vibrations the hopping integrals between e.g. $p$ orbitals of the oxygen and transition metal $d$-orbitals equal to $$\label{hopdist} t_{pd} = \frac{C_{pdm}}{u^{4}},$$ where coefficients $C_{pdm}$ depend on the type of the bonding and can be different for different metals and ligands. [@Harrison1999; @Slatter-Koster] The static version of Eq.  can be generalized taking into account the presence of lattice vibrations, i.e. phonons, which depend on the ion masses. The mean $pd$ hopping matrix element can be calculated as $$\begin{aligned} \langle t_{pd} \rangle &=& \frac 1{2v} \int_{u_0-v}^{u_0+v} \frac{C_{pdm}}{u^4} du =\nonumber \\ &=& \frac{C_{pdm}} {6v} \Big(\frac 1{(u_0-v)^3} - \frac 1{(u_0+v)^3} \Big), \end{aligned}$$ where $v=\sqrt {\langle \delta u^2 \rangle}$ is the mean square displacement from the equilibrium position $u_0$ due to phonons. Since $v/u_0 \ll 1$ one may simplify last equation expanding it to series to the 4th order $$\begin{aligned} \label{hopdist-phonon} \langle t_{pd} \rangle = \frac{C_{pdm}}{u_0^4} \Big( 1 + \frac{10}3 \Big(\frac v {u_0}\Big)^2\Big) + O(\Big(\frac v{u_0}\Big)^4). \end{aligned}$$ In the static limit $v \to 0$, the last formula coincides with Eq. . In the Debye model at zero temperature, the mean square displacement is written as [@Reissland1973] $$\label{v} v = \sqrt {\langle \delta u^2 \rangle} = \frac{9\hbar^2}{4k_B \theta_D} \frac1{m}.$$ Here $m$ is the mass of vibrating ions and $\theta_D$ is the Debye temperature. Due to different mass the mean square displacement in the compounds enriched by $^{16}$O or $^{18}$O will be different. The Debye temperature for the very similar system, LaCoO$_3$ was found to be $\sim$600 K.[@Stolen1997]. One may use this value to estimate $v$ in (Pr$_{1-y}$Eu$_y$)$_{0.7}$Ca$_{0.3}$CoO$_3$. Then the mean square displacement for $^{18}$O is $v_{18} = 0.100$ $\AA$ and $v_{16} = 0.107$ $\AA$ for $^{16}$O. According to Eq. , this leads to the decrease of the effective bandwidth in going from $^{16}$O to $^{18}$O. Qualitative explanation of this effect is presented in Fig. \[Fig\_model\]a. For strong electron-phonon coupling the same effect could be attributed to polaron band narrowing depending on the isotope mass. The temperature of the spin-state transition depends on the energy difference between $t_{2g}$ and $e_g$ subbands, which is defined by the widths of corresponding bands and the positions of their centers. We start with the study of the bandwidths dependence on the ligand ion mass, see schematic illustration in Fig. \[Fig\_model\]b. In order to calculate the change in the bandwidth caused by the $^{16}$O $\to$ $^{18}$O substitution, one needs to know the hopping integrals, which depend on two unknown parameters $C_{pdm}$ and $u_0$. The $C_{pdm}$ coefficients can be in principle evaluated as it is prescribed for instance in Ref. . However, for the better precision we calculated $C_{pdm}$ parameters from the LDA $t_{2g}$ and $e_g$ bandwidths in pure PrCoO$_3$. The equilibrium Co–O distance $u_0$ in its turn can be evaluated from the actual crystal structure data for EuCoO$_3$ and PrCoO$_3$. Finally, performing all these calculations, one gets that the $t_{2g}$ bandwidth decreases on 22 K when one substitutes $^{16}$O by $^{18}$O. The decrease of the $e_g$ bandwidth is two times larger and equals 44 K. In effect, the minimum energy of the $t_{2g} \to e_g$ transition increases by 33 K. Hence the spin-state transition temperature due to the change of the bandwidth must increase by the same value going from $^{16}$O to $^{18}$O in qualitative accordance with the experiment. [@Kalinov2010] Let us consider the second mechanism, which affects the spin-state transition and which is related to the dependence of changes in the centers of gravity of corresponding bands (i.e. crystal-field splitting) with the isotope substitution. It turns out the this effect counteracts the first one(change of the effective $t_{2g}$ and $e_g$ bandwidths), but numerically this second effect is much smaller, see below. Generally speaking, there are two main contributions to the crystal-field splitting, $\Delta_{CFS}$. One comes from the Coulomb interaction of the $3d$ electrons with the negatively charged ligands, another is due to the hybridization between $d$ orbitals of metal ions metal and $p$ orbitals of the ligands. [@Ballhausen1962; @Sugano1970] For most oxides of $3d$ transition metals both terms are acting in “the same direction”, resulting to the same sequence of levels. [@Ushakov2011] That is why so crude approaches as the atomic sphere approximation (ASA) [@Skiver1984] often used in the [*ab initio*]{} calculations provide quite precise band structure in most cases. The effect of the Coulomb term can be omitted or effectively incorporated into the kinetic energy contribution. Below we will follow the same strategy by considering the kinetic energy only, keeping in the mind that the Coulomb contribution can be taken into account via the parameters renormalization. In the second order of the perturbation theory, the crystal-field splitting between $t_{2g}$ and $e_g$ subbands is written as $$\label{deltaCF} \Delta_{CFS} = \frac{t^2_{pd\sigma} - t^2_{pd\pi}}{\Delta_{CT}},$$ where $\Delta_{CT}$ is the charge-transfer energy (which corresponds to the $d^n p^6 \to d^{n+1} p^5$ transition), $t_{pd\sigma}$ and $t_{pd\pi}$ are the hopping matrix elements for different types of the bonds. Since the average hopping $\langle t_{pd} \rangle$ decreases on going from $^{16}$O to $^{18}$O according to Eqs.  and , the crystal-field splitting should also decrease as follows from Eq. . As a result, this contribution should lead to the opposite tendency: decrease of the spin-state transition temperature going from $^{16}$O to $^{18}$O. However, this effect does not exceed few kelvins, at the realistic values of the charge-transfer energy in cobaltites. [@Chainani1992] Note also that we estimated here the changes in the distance between the [*edges*]{} of $t_{2g}$ and $e_g$ subbands. However, at finite temperatures, we should have not only the transitions between the band edge but just from one subband to another. Such a temperature-induced smearing could diminish somehow our estimates of the isotope effect in SST. In Fig. \[Tss(x,y)\]b, we see that the isotope effect for $T_{FM}$, being much weaker (than for $T_{SS}$), is of the opposite sign. Nevertheless, there is the same similarity between the effects of the Eu content and the oxygen isotope substitution. This is in agreement with our expectations, because the ferromagnetism of the low-Eu doped samples with metallic clusters should be stabilized by the double-exchange mechanism, according to which $T_{FM}$ is proportional to the effective bandwidth of the itinerant electrons. This bandwidth decreases for the heavier isotope, that is why $T_{FM}$ goes down at the $^{16}$O $\to ^{18}$O substitution as well as at increasing Eu content, see Fig. \[Tss(x,y)\]b. A schematic illustration of the mechanism underlying the oxygen isotope effect discussed above is given in Fig. \[Fig\_model\]. Conclusions =========== Experimental studies carried out for (Pr$_{1-y}$Eu$_y$)$_{0.7}$Ca$_{0.3}$CoO$_3$ cobaltites with varying isotope substitution of $^{16}$O by $^{18}$O demonstrated that there exists a strong similarity in the changes caused by the chemical composition (increasing Eu content) and those arising from the oxygen isotope substitution. The chemical composition $y \sim 0.1-0.2$ was chosen because in this range a crossover occurs between the ferromagnetic near-metallic state with magnetic Co ions to the nonmagnetic insulator with the low-spin Co$^{3+}$ ($t_{2g}^6e_g^0$, $S = 0$), see Fig. \[PhDiag\]. The main experimental conclusion presented in Fig. \[Tss(x,y)\] is that one can rescale the behavior of this system. The dependence of the spin-state transition temperature and of the ferromagnetic transition temperature on content $y$ of Eu and on the content $x$ of the heavier isotope $^{18}$O can be represented by the same almost linear plot as function of the combined variable $y + 0.01x$. This means, for example, that the change of Eu content $y$ by 0.005 is equivalent to the substitution of 50% of $^{16}$O by $^{18}$O. In addition, it clearly demonstrates that not only the average transition temperatures change with doping and with isotope substitution, but also the transition temperatures for each separate phase vary with chemical and isotope composition. Based on this similarity between the role of chemical and isotope composition for the spin-state transition and for the transition to the ferromagnetic state at a smaller Eu content, we propose a theoretical explanation of the isotope effect in these transitions. We investigate the corresponding changes and estimate the relevant parameters using the [*ab initio*]{} band structure calculations. These results together with the analytical model allow us to explain the observed behavior. In particular, the isotope effect both in the spin-state and ferromagnetic transitions is interpreted in terms of the change in the corresponding widths of the $d$ bands occurring due to the electron-phonon renormalization, which depends on the atomic masses of the respective isotopes. All this demonstrates once again that the oxygen isotope substitution is a powerful tool for revealing salient features in the behavior of strongly correlated magnetic oxides. Summarizing, we can say that using this approach, we established, first, that in the aforementioned crossover range, we can clearly distinguish two coexisting phases, nearly insulating exhibiting a spin-state transition and nearly metallic “ferromagnetic", with the different behavior of the transition temperatures. Second, we have found that these transition temperatures depend almost linearly on the content of the heavy oxygen isotope, which is a non-trivial observation clearly demonstrating that the electronic structure could be effectively controlled by isotopes. Third, based on these observations and using the parameters deduced from our band-structure calculations, we put forward a simplified model capturing the main physics of the isotope effect in the systems with spin-state transitions and quantitatively describing the experimental data. Acknowledgments {#acknowledgments .unnumbered} =============== This work is supported by the Russian Foundation for Basic Research (projects 10-02-00140-a, 10-02-00598-a, 10-02-96011-a, 11-02-00708-a, and 11-02-91335-NNIO-a), by the Ural Branch of Russian Academy of Sciences through the young-scientist program, by the German projects SFB 608, DFG GR 1484/2-1, FOR 1346, and by the European network SOPRANO. [99]{} N. A. Babushkina, L. M. Belova, O. Yu. 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--- abstract: | We design an FPRAS to count the number of bases of any matroid given by an independent set oracle, and to estimate the partition function of the random cluster model of any matroid in the regime where $0<q<1$. Consequently, we can sample random spanning forests in a graph and (approximately) compute the reliability polynomial of any matroid. We also prove the thirty year old conjecture of Mihail and Vazirani that the bases exchange graph of any matroid has expansion at least 1. Our algorithm and the proof build on the recent results of Dinur, Kaufman, Mass and Oppenheim [@KM17; @DK17; @KO18] who show that high dimensional walks on simplicial complexes mix rapidly if the corresponding localized random walks on 1-skeleton of links of all complexes are strong spectral expanders. One of our key observations is a close connection between pure simplicial complexes and multiaffine homogeneous polynomials. Specifically, if $X$ is a pure simplicial complex with positive weights on its maximal faces, we can associate with $X$ a multiaffine homogeneous polynomial $p_{X}$ such that the eigenvalues of the localized random walks on $X$ correspond to the eigenvalues of the Hessian of derivatives of $p_{X}$. author: - Nima Anari - Kuikui Liu - Shayan Oveis Gharan - Cynthia Vinzant bibliography: - 'refs.bib' title: 'Log-Concave Polynomials II: High-Dimensional Walks and an FPRAS for Counting Bases of a Matroid' --- Introduction ============ Let $\mu:2^{[n]}\to {{\mathbb{R}}}_{\geqslant 0}$ be a probability distribution on the subsets of the set $[n]=\set{1,2,\dots,n}$. We assign a multiaffine polynomial with variables $x_1,\dots,x_n$ to $\mu$, $$g_\mu(x) = \sum_{S\subseteq [n]} \mu(S)\cdot \prod_{i\in S} x_i.$$ The polynomial $g_\mu$ is also known as the *generating polynomial* of $\mu$. A polynomial $p\in{{\mathbb{R}}}[x_1,\dots,x_n]$ is $d$-homogeneous if every monomial of $p$ has degree $d$. We say $\mu$ is *$d$-homogeneous* if the polynomial $g_\mu$ is $d$-homogeneous, meaning that $\card{S}=d$ for any $S$ with $\mu(S)>0$. A polynomial $p \in {{\mathbb{R}}}[x_1,\dots,x_n]$ with nonnegative coefficients is *log-concave* on a subset $K\subseteq {{\mathbb{R}}}_{\geqslant 0}^n$ if $\log p$ is a concave function at any point in $K$, or equivalently, its Hessian $\nabla^{2}\log p$ is negative semidefinite on $K$. We say a polynomial $p$ is *strongly log-concave* on $K$ if for any $k\geqslant 0$, and any sequence of integers $1\leqslant i_1,\dots,i_k\leqslant n$, $$(\partial_{i_1} \cdots \partial_{i_k} p)(x_1,\dots,x_n)$$ is log-concave on $K$. In this paper, for convenience and clarity, we only work with (strong) log-concavity with respect to the all-ones vector, ${{\mathds{1}}}$. So, unless otherwise specified, $K=\set{{{\mathds{1}}}}$ in the above definition. We say the distribution $\mu$ is *strongly log-concave* at ${{\mathds{1}}}$ if $g_\mu$ is strongly log-concave at ${{\mathds{1}}}$. The notion of strong log-concavity was first introduced by @Gur09 [@Gur10] to study approximation algorithms for mixed volume and multivariate generalizations of Newton’s inequalities. In this paper, we show that the “natural” Monte Carlo Markov Chain (MCMC) method on the support of a $d$-homogeneous strongly log-concave distribution $\mu:2^{[n]}\to {{\mathbb{R}}}_{\geqslant 0}$ mixes rapidly. This chain can be used to generate random samples from a distribution arbitrarily close to $\mu$. The chain ${{\mathcal{M}}}_{\mu}$ is defined as follows. We take the state space of ${{\mathcal{M}}}_\mu$ to be the support of $\mu$, namely $\operatorname{supp}(\mu) = \set{S\subseteq [n]{}\mu(S)\neq 0}$. For $\tau\in \operatorname{supp}(\mu)$, first we drop an element $i \in \tau$, chosen uniformly at random from $\tau$. Then, among all sets $\sigma \supset \tau \setminus \set{i}$ in the support of $\mu$ we choose one with probability proportional to $\mu(\sigma)$. It is easy to see that ${{\mathcal{M}}}_{\mu}$ is reversible with stationary distribution $\mu$. Furthermore, assuming $g_\mu$ is strongly log-concave, we will see that ${{\mathcal{M}}}_\mu$ is irreducible. We prove that this chain mixes rapidly. More formally, for a state $\tau$ of the Markov chain ${{\mathcal{M}}}$, and $\epsilon > 0$, the *total variation mixing time* of ${{\mathcal{M}}}$ started at $\tau$ with transition probability matrix $P$ and stationary distribution $\pi$ is defined as follows: $$t_{\tau}(\epsilon)=\min\set{t\in {{\mathbb{Z}}}_{\geqslant 0}{}\norm{P^t(\tau,\cdot)-\pi}_1 \leqslant \epsilon},$$ where $P^t(\tau,\cdot)$ is the distribution of the chain started at $\tau$ at time $t$. The following theorem is the main result of this paper. \[thm:SLCmixing\] Let $\mu:2^{[n]} \rightarrow {{\mathbb{R}}}_{\geqslant0}$ be a $d$-homogeneous strongly log-concave probability distribution. If $P_{\mu}$ denotes the transition probability matrix of ${{\mathcal{M}}}_{\mu}$ and $X(k)$ denotes the collection of size-$k$ subsets of $[n]$ which are contained in some element of $\operatorname{supp}(\mu)$, then for every $0 \leqslant k \leqslant d-1$, $P_{\mu}$ has at most $\card{X(k)} \leqslant \binom{n}{k}$ eigenvalues of value $>1 - \frac{k+1}{d}$. In particular, ${{\mathcal{M}}}_{\mu}$ has spectral gap at least $1/d$, and if $\tau$ is in the support of $\mu$ and $0<\epsilon<1$, the total variation mixing time of the Markov chain ${{\mathcal{M}}}_{\mu}$ started at $\tau$ is at most $$t_\tau({{\epsilon}}) \ \leqslant \ d\log\parens*{\frac1{\epsilon\mu(\tau)}}.$$ To state the key corollaries of this theorem, we will need the following definition. Given a domain set $\Omega$, which is compactly represented by, say, a membership oracle, and a nonnegative weight function $w:\Omega \rightarrow {{\mathbb{R}}}_{\geqslant0}$, a *fully polynomial-time randomized approximation scheme (FPRAS)* for computing the partition function $Z = \sum_{x \in \Omega} w(x)$ is a randomized algorithm that, given an error parameter $0 < \epsilon < 1$ and error probability $0 < \delta < 1$, returns a number $\tilde{Z}$ such that ${\rm Prob}[(1 - \epsilon)Z \leqslant \tilde{Z} \leqslant (1 + \epsilon)Z] \geqslant 1 - \delta$. The algorithm is required to run in time polynomial in the problem input size, $1/\epsilon$, and $\log(1/\delta)$. Equipped with this definition, we can now concisely state the main applications of \[thm:SLCmixing\]. \[thm:SLCmixing\] gives us an algorithm to efficiently sample from a distribution which approximates $\mu$ closely in total variation distance. By the equivalence between approximate counting and approximate sampling for self-reducible problems [@JVV86], this gives an FPRAS for each of the following: 1. counting the bases of a matroid, and 2. estimating the partition function of the random cluster model for a new range of parameter values For real linear matroids, we also give an algorithm for estimating the partition function of a generalized version of a $k$-determinantal point process. Note that these problems are all instantiations of the following: estimate the partition function of some efficiently computable nonnegative weights on bases of a matroid. Furthermore, as the restriction and contraction of a matroid by a subset of the ground set are both (smaller) matroids, problems of this form are indeed self-reducible. In the following sections we discuss these applications in greater depth. Counting Problems on Matroids ----------------------------- Let $M=([n],{{\mathcal{I}}})$ be an arbitrary matroid on $n$ elements (see \[subsec:matroids\]) of rank $r$. Let $\mu$ be the uniform distribution on the bases of the matroid $M$. It follows that $\mu$ is $r$-homogeneous. Using the Hodge-Riemann relation proved by @AHK18, a subset of the authors proved [@AOV18] that for any matroid $M$, $\mu$ is strongly log-concave.[^1] This implies that the chain ${{\mathcal{M}}}_{\mu}$ converges rapidly to stationary distribution. This gives the first polynomial time algorithm to generate a uniformly random base of a matroid. Note that to run ${{\mathcal{M}}}_{\mu}$ we only need an oracle to test whether a given set $S\subseteq [n]$ is an independent set of $M$. Therefore, with only polynomially many queries (in $n,r,\log(1/{{\epsilon}})$) we can generate a random base of $M$. For any matroid $M=([n],{{\mathcal{I}}})$ of rank $r$, any basis $B$ of $M$ and $0<{{\epsilon}}<1$, the mixing time of the Markov chain ${{\mathcal{M}}}_{\mu}$ starting at $B$ is at most $$t_B({{\epsilon}}) \ \leqslant \ r\log(n^{r}/\epsilon) \leqslant \ r^2 \log(n/{{\epsilon}}).$$ To prove this we simply used the fact that a matroid of rank $r$ on $n$ elements has at most $\binom{n}{r}\leqslant n^r$ bases. There are several immediate consequences of the above corollary. Firstly, by equivalence of approximate counting and approximate sampling for self-reducible problems [@JVV86] we can count the number of bases of any matroid given by an independent set oracle up to a $1+{{\epsilon}}$ multiplicative error in polynomial time. There is a randomized algorithm that for any matroid $M$ on $n$ elements with rank $r$ given by an independent set oracle, and any $0<{{\epsilon}}<1$, counts the number of bases of $M$ up to a multiplicative factor of $1\pm{{\epsilon}}$ with probability at least $1-\delta$ in time polynomial in $n,r,1/{{\epsilon}},\log(1/\delta)$. As an immediate corollary for any $1\leqslant k\leqslant r$ we can count the number of independent sets of $M$ of size $k$. This is because if we truncate $M$ to independent sets of size at most $k$ it remains a matroid. As a consequence we can generate uniformly random forests in a given graph, and compute the reliability polynomial $$\begin{aligned} C_{M}(p) = \sum_{S \subseteq [n] : \operatorname{rank}(S) = r} (1 - p)^{\card{S}} p^{n - \card{S}} \end{aligned}$$ for any matroid and $0 \leqslant p \leqslant 1$, all in polynomial time. Note this latter fact follows from the ability to count the number of independent sets of a fixed size, as the complements of rank-$r$ subsets $S \subseteq [n]$ are precisely the independent sets of the dual of $M$. Prior to our work, we could only compute the reliability polynomial for graphic matroids due to a recent work of Guo and Jerrum [@GJ18a]. One can associate a graph $G_M$ to any matroid $M$, called *the bases exchange* graph. This graph has a vertex for every basis of $M$ and two bases $B,B'$ are connected by an edge if $\card{B\Delta B'}=2$. It follows by the bases exchange property of matroids that this graph is connected. For an unweighted graph $G=(V,E)$, the expansion of a set $S\subset V$ and the graph $G$ are defined as $$\operatorname{h}(S)=\frac{\card{E(S,\overline{S})}}{\card{S}} \ \ \ \ \text{ and } \ \ \ \ \operatorname{h}(G)=\min_{S:\card{S}\leqslant \card{\overline{S}}} \operatorname{h}(S).$$ @MV89 conjectured that the bases exchange graph has expansion at least one, i.e., that $\operatorname{h}(G_M)\geqslant 1$, for any matroid $M$. It turns out that the bases exchange graph is closely related to the Markov chain ${{\mathcal{M}}}_{\mu}$. The following theorem is an immediate consequence of the above corollary. \[thm:basesexchange\] For any matroid $M$, the expansion of the bases exchange graph is at least $1$, $\operatorname{h}(G_M)\geqslant 1$. The Random Cluster Model {#subsec:RandomCluster} ------------------------ Another application of this theory is estimating the partition function of the random cluster model. For a matroid $M = ([n],{{\mathcal{I}}})$ of rank $r$ and parameters $p, q$, the partition function of the random cluster model from statistical mechanics due to Fortuin and Kasteleyn [@For71; @FK72I; @For72II; @For72III] is the following polynomial function associated to $M$, $$\begin{aligned} Z_M(p,q)=\sum_{S\subseteq [n]} q^{r+1-\operatorname{rank}(S)} p^{\card{S}} \end{aligned}$$ where $\operatorname{rank}(S)$ is the size of the largest independent set contained in $S$. We note that typically one scales each term by $(1-p)^{n-\card{S}}$ but up to a normalization factor (and change of variables) the two polynomials are equivalent. We refer interested readers to a recent book of @Grim09 for further information. Typically, one considers the special case where $M$ is a graphic matroid, in which case the exponent of $q$ is simply the number of connected components of $S$. To the best of our knowledge, prior to this work, one could only compute $Z_M$ when $q=2$ because of the close connection to the Ising model [@JS93; @GJ17]. Our next result is a polynomial time algorithm that estimates $Z_{M}(p,q)$ for any $0< q\leqslant 1$ and $p\geqslant 0$. \[thm:randomclusterSLC\] For a matroid $M$ with rank function $\operatorname{rank}:2^{[n]}\rightarrow {{\mathbb{Z}}}_{\geqslant 0}$, parameter $0 < q \leqslant 1$ and choice of “external field” $\bm{\lambda} = (\lambda_{1},\dots,\lambda_{n}) \in {{\mathbb{R}}}_{>0}^{n}$, the polynomial $$\begin{aligned} f_{M,k,q}(x_{1},\dots,x_{n}) = \sum_{S \in \binom{[n]}{k}} q^{-\operatorname{rank}(S)} \prod_{i\in S}\lambda_{i}x_i \end{aligned}$$ is strongly log-concave. Together with \[thm:SLCmixing\], this gives an FPRAS for estimating $f_{M,k,q}({{\mathds{1}}})$ given an independence oracle for the matroid $M$. Estimating $Z_{M}(p, q)$ then follows as $$\begin{aligned} Z_{M}(p,q) = q^{r+1}\sum_{k=0}^{n} p^{k}f_{M,k,q}({{\mathds{1}}}) \end{aligned}$$ and each term is nonnegative. In fact, the polynomial $Z_{M}$ is closely related to the Tutte polynomial $$\begin{aligned} T_{M}(x,y) = \sum_{S \subseteq [n]} (x - 1)^{r - \operatorname{rank}(S)}(y - 1)^{\card{S} - \operatorname{rank}(S)}. \end{aligned}$$ Indeed, we can write $$\begin{aligned} T_{M}(x,y) = \frac{1}{(x-1)(y-1)^{r+1}}Z_{M}\parens*{y-1,(x-1)(y-1)} \end{aligned}$$ Hence, an FPRAS for estimating $Z_{M}(p,q)$ for $p \geqslant 0$ and $0 \leqslant q \leqslant 1$ gives an FPRAS for estimating $T_{M}(x, y)$ in the region described by the inequalities $y \geqslant 1$ and $0 \leqslant (x - 1)(y - 1) \leqslant 1$. Determinantal Distributions on Real Linear Matroids --------------------------------------------------- Finally, we show that the class of homogeneous multiaffine strongly log-concave polynomials is closed under raising all coefficients to a fixed exponent less than $1$. \[thm:cpow\] Let $f= \sum_{S \subseteq [n]} c_{S}\prod_{i\in S}x_i$ be a homogeneous degree-$k$ multiaffine strongly log-concave polynomial. Then $f_{\alpha} = \sum_{S \subseteq [n]} c_{S}^{\alpha} \prod_{i\in S}x_i$ is strongly log-concave for every $0 \leqslant \alpha \leqslant 1$. We use the above theorem to design a sampling algorithm for determinantal point processes. A *determinantal point process (DPP)* on a set of elements $[n]$ is a probability distribution $\mu:2^{[n]}\rightarrow {{\mathbb{R}}}_{\geqslant 0}$ identified by a positive semidefinite matrix $L \in {{\mathbb{R}}}^{n\times n}$ where for any $S\subseteq [n]$ we have $$\mu(S) \ \propto \ \det(L_S),$$ where $L_S$ is the principal sub-matrix of $L$ indexed by the elements of $S$. Determinantal point processes are fundamental to the study of a variety of tasks in machine learning, including text summarization, image search, news threading, and diverse feature selection [see, e.g., @KT12]. A *$k$-determinantal point process ($k$-DPP)* is a determinantal point process conditioned on the sets $S$ having size $k$. Given a positive semidefinite matrix $L$, let $\mu$ be the corresponding $k$-DPP. We have $$g_\mu(x)\propto \sum_{S\in \binom{[n]}{k}} \det(L_S)\cdot \prod_{i\in S}x_i.$$ It turns out that the above polynomial is real stable and so it is strongly log-concave over ${{\mathbb{R}}}^n_{\geqslant 0}$ [see, e.g., @AOR16]. @AOR16 show that a natural Markov chain with the Metropolis rule mixes rapidly and generates a random sample of $\mu$. The above theorem immediately implies the following log-concavity result. \[cor:dpp\] For every positive semidefinite matrix $L \succcurlyeq 0$ and exponent $0 \leqslant \alpha \leqslant 1$, the polynomial $$\begin{aligned} \sum_{S\in \binom{[n]}{k}} \det(L_S)^{\alpha} \prod_{i\in S}x_i \end{aligned}$$ is strongly log-concave. It follows from \[thm:SLCmixing\] that for any $0\leqslant\alpha\leqslant 1$ we can generate samples from a “smoothed” $k$-DPP distribution, where for any set $S$, $\Pr{S}\propto \det(L_S)^\alpha$, in polynomial time. The weights $\det(L_{S})^{\alpha}$ may be thought of as a way to interpolate between two extremes for selecting diverse data points. We also note that for $\alpha = 1/2$, it is known that \[cor:dpp\] follows from the Brunn-Minkowski theorem applied to appropriately defined zonotopes. For $\alpha = 0$ when the $k$-DPP has full support, and for $\alpha = 1$ as mentioned earlier, the above polynomial is actually real stable, and hence strongly log-concave. gives a unified proof that all of these polynomials are strongly log-concave. Related Works ------------- There is a long line of work on designing approximation algorithms to count the bases of a matroid. Most of these works focus on expansion properties of bases exchange graph. @FM92 showed that for a special class of matroids known as *balanced matroids* [@MS91; @FM92], the bases exchange graph has expansion at least 1. A matroid $M$ is balanced if for any minor of $M$ (including $M$ itself), the uniform distribution over its bases satisfies the pairwise negative correlation property. Many of the extensive results in this area [@Gam99; @JS02; @JSTV04; @Jur06; @Clo10; @CTY15; @AOR16] only study approximation algorithms for this limited class of matroids, and not much is known beyond the class of balanced matroids. Unfortunately, many interesting matroids are not balanced. An important example is the matroid of all acyclic subsets of edges of a graph $G=(V,E)$ of size at most $k$ (for some $k<\card{V}-1$) [@FM92]. There has been other approaches for counting bases. @GJ18b used the popping method to count bases of bicircular matroids. @BS07 designed a *randomized* algorithm that gives, roughly, a $\log(n)^r$ approximation factor to the number of bases of a given matroid with $n$ elements and rank $r$. In [@AOV18], a subset of the authors gave a deterministic $e^r$ approximation to the number of bases using the fact that $g_\mu(M)$ is log-concave over ${{\mathbb{R}}}^n_{\geqslant 0}$. There is an extensive literature on hardness of exact computation and inapproximability of the Tutte polynomial and the partition function of the random cluster model. It is known that exact computation of the Tutte polynomial for a graph is -hard at all points $(x,y)$ except at $(1,1), (-1,-1), (0,-1), (-1,0)$, along the hyperbola $(x-1)(y-1) = 1$, and for planar graphs, along the hyperbola $(x-1)(y-1) = 2$ [@JVW90], [@Ver91], [@Wel94]. In the realm of inapproximability, it is known that even for planar graphs, there is no FPRAS to approximate the Tutte polynomial for $x > 1, y < -1$ or $y > 1, x < -1$ assuming $\neq$ [@GJ08; @GJ12II]. Furthermore, there is no FPRAS for estimating the partition function $Z_{M}$ of the random cluster model on general graphic matroids when $q > 2$, nor is there an FPRAS for $Z_{M}$ at $q = 2$ for general binary matroids, unless there is an FPRAS for counting independent sets in a bipartite graph [@GJ12I; @GJ13; @GJ14]. Independent Work ---------------- In a closely related upcoming work, Brändén and Huh, in a slightly different language, independently prove the strong log-concavity of several of the polynomials that appear in this paper. In upcoming papers, both groups of authors use these techniques to prove the strongest form of Mason’s conjecture and further study closure properties of (strongly) log-concave polynomials. Techniques ---------- One of our key observations is a close connection between pure simplicial complexes and multiaffine homogeneous polynomials. Specifically, if $X$ is a pure simplicial complex with positive weights on its maximal faces, we can associate with $X$ a multiaffine homogeneous polynomial $p_{X}$ such that the eigenvalues of the localized random walks on $X$ correspond to the eigenvalues of the Hessian of derivatives of $p_{X}$. Weighted Simplicial Complex $X$ Multiaffine Polynomial $p_{X}$ --------------------------------- -------------------------------- Dimension-$d$ Degree-$d$ Weight of $\emptyset$ Evaluation at ${{\mathds{1}}}$ Connectivity of Links Indecomposability Link Differentiation Local Random Walk (Normalized) Hessian \[tab:my\_label\] Using this correspondence, one can study multiaffine homogeneous polynomials using techniques from simplicial complexes, and vice versa. To study the walk $\mathcal{M}_{\mu}$ corresponding to a polynomial $g_{\mu}$, we analyze the simplicial complex corresponding to $g_{\mu}$. To do this, we leverage recent developments in the area of high-dimensional expanders, which we discuss below. Given a simplicial complex $X$ (see \[sec:SimplicialComplex\]) and an ordering of its vertices, one can associate a high dimensional Laplacian matrix to the $k$-dimensional faces of $X$. These matrices generalize the classical graph Laplacian and there has been extensive research to study their eigenvalues [see @Lub17 and the references therein]. A method known as Garland’s method [@Gar73] relates the eigenvalues of graph Laplacians of 1-skeletons of links of $X$ to eigenvalues of high dimensional Laplacians of $X$ [see @BS97; @Opp18]. Recently, @KM17 studied a high dimensional walk on a simplicial complex, which is closely related to the walk ${{\mathcal{M}}}_\mu$ that we defined above (see \[sec:highdimwalk\]). Their goal is to argue that, similar to classical expander graphs, high dimensional walks mix rapidly on a high dimensional expander. Their bounds were improved in a work of @DK17, who showed that if all nontrivial eigenvalues of the simple random walk matrix on all 1-skeletons of links of $X$ have absolute value at most $\lambda$, then the high dimensional walk on $k$-faces of $X$ has spectral gap at least $\frac1{k+2}-O((k+1)\lambda)$. This was further improved in a recent work of @KO18: They showed that if all non-trivial eigenvalues of the simple random walk matrix on all 1-skeleton of links of $X$ are at most $\lambda$, then the spectral gap of the high dimensional walk is at least $\frac1{k+2}-(k+1)\lambda$. In other words, negative eigenvalues of the random walk matrix do not matter. One only needs positive eigenvalues to be small. Note that in order to make the spectral gap bounds meaningful one needs $\lambda \ll \frac1{k^2}$. In other words, one needs that, except the trivial eigenvalue of 1, all other eigenvalues are either negative or very close to $0$. Here is the place where the connection to (strong) log-concavity comes into the picture. A polynomial $p$ is log-concave at ${{\mathds{1}}}$ if $\nabla^2 p({{\mathds{1}}})$ has at most one positive eigenvalue. A polynomial $p$ is strongly log-concave if the same holds for all partial derivatives of $p$. Our main observation is that this property is equivalent to taking $\lambda=0$ in the corresponding simplicial complex. Namely, we obtain the best possible spectral gap of $\frac{1}{k+2}$ when the simplicial complex comes from a strongly log-concave polynomial. Our approach has a close connection to the original plan of @FM92 who used the negative correlation property of balanced matroids to show that the bases exchange walk mixes rapidly. Unfortunately, most interesting matroids do not satisfy negative correlation. But it was observed [@AHK18; @HW17; @AOV18] that all matroids satisfy a *spectral* negative dependence property. Namely, consider the uniform distribution $\mu$ over the bases of a matroid $M$, and consider the Hessian $\nabla^2 \log(g_{\mu})$ of the $\log$ of the generating polynomial $g_{\mu}$ at the point $x ={{\mathds{1}}}$. Then $M$ is negatively correlated if and only if all off-diagonal entries of this matrix are non-positive, whereas $M$ being spectrally negatively correlated means that this matrix is negative semidefinite. Spectral negative correlation is precisely what one needs to bound the mixing time of the high dimensional walk on the corresponding simplicial complex. #### Structure of the paper. In \[sec:prelim\] we discuss necessary background on linear algebra, matroids, simplicial complexes and strongly log-concave polynomials. We also provide a useful characterization of strong log-concavity. In \[sec:highdimwalk\] we discuss and reprove a version of the main theorem of @KO18 on mixing time of high dimensional walks, \[thm:localexpander\]. In \[sec:SLCtoLSE\] we use this to prove \[thm:SLCmixing\] and the Mihail-Vazirani conjecture, \[thm:basesexchange\]. Finally, in \[sec:applications\] we first prove our new characterization of strong log-concavity and discuss its applications. Specifically, we give a self-contained proof that the uniform distribution over the bases of a matroid is strongly log-concave and we prove \[thm:randomclusterSLC,thm:cpow\]. #### Acknowledgements. Part of this work was started while the first and last authors were visiting the Simons Institute for the Theory of Computing. It was partially supported by the DIMACS/Simons Collaboration on Bridging Continuous and Discrete Optimization through NSF grant CCF-1740425. Shayan Oveis Gharan and Kuikui Liu are supported by the NSF grant CCF-1552097 and ONR-YIP grant N00014-17-1-2429. Cynthia Vinzant was partially supported by the National Science Foundation grant DMS-1620014. We thank Lap Chi Lau, Mark Jerrum and Alan Frieze for helpful comments on an earlier version of this manuscript. Preliminaries {#sec:prelim} ============= First, let us establish some notational conventions. Unless otherwise specified, all logarithms are in base $e$. All vectors are assumed to be column vectors. For two vectors $\phi, \psi\in {{\mathbb{R}}}^n$, we use $\dotprod{\phi, \psi}$ to denote the standard Euclidean inner product between $\phi$ and $\psi$. We use ${{\mathbb{R}}}_{>0}$ and ${{\mathbb{R}}}_{\geqslant 0}$ to denote the set of positive and nonnegative real numbers, respectively, and $[n]$ to denote $\set{1,\dots,n}$. For a vector $x\in {{\mathbb{R}}}^n$ and a set $S\subseteq [n]$, we let $x^S$ denote $\prod_{i\in S} x_i$. We use $\partial_{x_i}$ or $\partial_i$ to denote the partial differential operator $\partial/\partial x_i$. We denote the gradient of a function or polynomial $p$ by $\nabla p$ and the Hessian of $p$ by ${\nabla^2 p}$. Linear Algebra -------------- We say a matrix $A\in{{\mathbb{R}}}^{n\times n}$ is *stochastic* if all entries of $A$ are nonnegative and every row adds up to exactly 1. It is well-known that the largest eigenvalue in magnitude of any stochastic matrix is $1$ and its corresponding eigenvector is the all-ones vector, ${{\mathds{1}}}$. If the eigenvalues of a matrix $A\in{{\mathbb{R}}}^{n\times n}$ are all real, then we order them as $$\lambda_n(A) \leqslant \hdots \leqslant \lambda_1(A).$$ A symmetric matrix $A\in{{\mathbb{R}}}^{n\times n}$ is positive semidefinite (PSD), denoted $A\succcurlyeq 0$, if all its eigenvalues $\lambda_k(A)$ are nonnegative, or equivalently if for all $v\in{{\mathbb{R}}}^n$, $$v^\intercal Av \geqslant 0.$$ Similarly, $A$ is negative semidefinite (NSD), denoted $A\preccurlyeq 0$, if $v^\intercal Av \leqslant 0$ for all $v\in{{\mathbb{R}}}^n$. Equivalently, a real symmetric matrix is PSD (NSD) if its eigenvalues are nonnegative (nonpositive), respectively. \[thm:schur\] If $A,B \in {{\mathbb{R}}}^{n \times n}$ are positive semidefinite, then their Hadamard product $A \circ B$, whose entries are $(A \circ B)_{i,j} = A_{i,j}B_{i,j}$, is positive semidefinite. \[thm:perron\] Let $A \in {{\mathbb{R}}}^{n \times n}$ be symmetric and have strictly positive entries. Then $A$ has an eigenvalue $\lambda$ which is strictly positive. Furthermore, it has multiplicity one and its corresponding eigenvector $v$ has strictly positive entries. \[thm:CauchyInterlacing\] For a symmetric matrix $A\in{{\mathbb{R}}}^{n\times n}$ and vector $v\in {{\mathbb{R}}}^n$, the eigenvalues of $A$ interlace the eigenvalues of $A+vv^\intercal$. That is, for $B = A+vv^\intercal,$ $$\lambda_n(A)\leqslant \lambda_n(B)\leqslant\lambda_{n-1}(A) \leqslant \dots \leqslant \lambda_{2}(B) \leqslant \lambda_{1}(A) \leqslant \lambda_{1}(B).$$ The following is an immediate consequence: \[lem:Cauchy1eigenvalue\] Let $A\in{{\mathbb{R}}}^{n\times n}$ be a symmetric matrix and let $P\in{{\mathbb{R}}}^{m\times n}$. If $A$ has at most one positive eigenvalue, then $PAP^\intercal$ has at most one positive eigenvalue. Since $A$ has at most one positive eigenvalue, we can write $A=B+vv^\intercal$ for some vector $v\in {{\mathbb{R}}}^n$ and some negative semidefinite matrix $B$. Then $PAP^\intercal = PBP^\intercal + Pvv^\intercal P^\intercal$. First, observe that $PBP^\intercal\preccurlyeq 0$, since for $x\in{{\mathbb{R}}}^m$, $x^\intercal PB P^\intercal x = (P^\intercal x)^\intercal B (P^\intercal x)\leqslant 0$. Second, let $w = Pv\in {{\mathbb{R}}}^m$. Then $Pvv^\intercal P^\intercal=ww^\intercal$ and by \[thm:CauchyInterlacing\], the eigenvalues of $PBP^\intercal$ interlace the eigenvalues of $PBP^\intercal+(Pv)(Pv)^\intercal$. Since all eigenvalues of $PBP^\intercal$ are nonpositive, $PAP^\intercal=PBP^\intercal+ww^\intercal$ has at most one positive eigenvalue. The following fact is well-known. \[fact:eigenvaluesinvert\] Let $A\in {{\mathbb{R}}}^{n\times k}$ and $B\in {{\mathbb{R}}}^{k\times n}$ be arbitrary matrices. Then, non-zero eigenvalues of $AB$ are equal to non-zero eigenvalues of $BA$ with the same multiplicity. \[lem:stochastic\] Let $A\in {{\mathbb{R}}}^{n\times n}$ be a symmetric matrix with at most one positive eigenvalue. Then, for any PSD matrix $B\in{{\mathbb{R}}}^{n\times n}$, $BA$ has at most one positive eigenvalue. Since $B\succcurlyeq 0$, we can write $B=C^\intercal C$ for some $C\in {{\mathbb{R}}}^{n\times n}$. By \[fact:eigenvaluesinvert\], $BA=C^\intercal C A$ has the same nonzero eigenvalues as the matrix $C A C^\intercal$. Since $A$ has at most one positive eigenvalue, by \[lem:Cauchy1eigenvalue\], $C A C^\intercal$ has at most one positive eigenvalue and so does $BA$. \[lem:adjacency1poseig\] Let $A\in{{\mathbb{R}}}^{n\times n}$ be a symmetric matrix with nonnegative entries and at most one positive eigenvalue, and let $w(i)=\sum_{j=1}^n A_{i,j}$. Then, $$A\preccurlyeq \frac{ww^\intercal}{\sum_i w(i)}.$$ Let $W=\operatorname{diag}(w)$. Then, by \[lem:Cauchy1eigenvalue\], $\mathcal{A}=W^{-1/2} A W^{-1/2}$ has at most one positive eigenvalue. Observe that the top eigenvector of $\mathcal{A}$ is the $\sqrt{w}$ vector, where $\sqrt{w}(i) = \sqrt{w(i)}$, for all $i$. In particular, $\mathcal{A}\sqrt{w}=\sqrt{w}$. So, $\sqrt{w}$ is the only eigenvector of $\mathcal{A}$ with positive eigenvalue and we have $${\cal A} \preccurlyeq \frac{\sqrt{w}\sqrt{w}^\intercal}{\norm{\sqrt{w}}^2} = \frac{\sqrt{w}\sqrt{w}^\intercal}{\sum_i w(i)}.$$ Multiplying both sides of the inequality on the left and right by $W^{1/2}$ proves the lemma. In this paper, we will often switch between different inner products. As such, we highlight the following variational characterization of eigenvalues of a linear operator that is self-adjoint with respect to an arbitrary inner product. In particular, the matrix of the linear operator need not be symmetric. \[thm:Courant-Fischer\] Let $T:{{\mathbb{R}}}^{n} \rightarrow {{\mathbb{R}}}^{n}$ be a linear operator that is self-adjoint with respect to some inner product $\dotprod{\cdot,\cdot}$ (not necessarily Euclidean). If $\lambda_{n} \leqslant \dots \leqslant \lambda_{1}$ are the eigenvalues of $T$, then $$\begin{aligned} \lambda_{k} &= \min_{U } \ \max_{v } \ \dotprod{v, Tv}, \end{aligned}$$ where the minimum is taken over all $(n-k)$-dimensional subspaces $U\subseteq {{\mathbb{R}}}^n$ and the maximum is taken over all vectors $v\in U$ with $\dotprod{v,v}= 1$. When the inner product $\dotprod{\cdot,\cdot}$ is clear, we call a matrix $A$ self-adjoint when $\dotprod{u, Av}=\dotprod{Au, v}$, for all $u, v$. Similarly we call a self-adjoint $A$ positive semidefinite when for all $v$ $$\dotprod{v, Av}\geqslant 0.$$ By \[thm:Courant-Fischer\], this is equivalent to $A$ having nonnegative eigenvalues. Markov Chains and Random Walks {#subsec:randomwalks} ------------------------------ For this paper, we consider a Markov chain as a triple $(\Omega,P,\pi)$ where $\Omega$ denotes the (finite) state space, $P\in {{\mathbb{R}}}_{\geqslant 0}^{\Omega\times \Omega}$ denotes the transition probability matrix and $\pi \in {{\mathbb{R}}}_{\geqslant 0}^{\Omega}$ denotes a stationary distribution of the chain (which will be unique for all chains we consider). For $\tau,\sigma \in \Omega$, we use $P(\tau,\sigma)$ to denote the corresponding entry of $P$, which is the probability of moving from $\tau$ to $\sigma$. We say a Markov chain is $\epsilon$-lazy if for any state $\tau \in \Omega$, $P(\tau,\tau) \geqslant \epsilon$. A chain $(\Omega, P, \pi)$ is *reversible* if there is a nonzero nonnegative function $f:\Omega\to{{\mathbb{R}}}_{\geqslant0}$ such that for any pair of states $\tau, \sigma \in \Omega$, $$f(\tau) P(\tau,\sigma) = f(\sigma)P(\sigma,\tau).$$ If this condition is satisfied, then $f$ is proportional to a stationary distribution of the chain. In this paper we only work with reversible Markov chains. Note that being reversible means that the transition matrix $P$ is self-adjoint w.r.t. the following $\dotprod{\cdot, \cdot}$ defined for $\phi,\psi\in {{\mathbb{R}}}^\Omega$: $$\dotprod{\phi,\psi}=\sum_{x\in\Omega} f(x)\phi(x)\psi(x).$$ Reversible Markov chains can be realized as random walks on weighted graphs. Given a weighted graph $G=(V,E,w)$ where every edge $e\in E$ has weight $w(e)$, the non-lazy *simple* random walk on $G$ is the Markov chain that from any vertex $u\in V$ chooses an edge $e=\set{u,v}$ with probability proportional to $w(e)$ and jumps to $v$. We can make this walk $\epsilon$-*lazy* by staying at every vertex with probability $\epsilon$. It turns out that if $G$ is connected, then the walk has a unique stationary distribution where $\pi(u)\propto w(u)$, where $w(u)=\sum_{v\sim u} w(\set{u,v})$ is the weighted degree of $u$. For any reversible Markov chain $(\Omega, P, \pi)$, the largest eigenvalue of $P$ is $1$. We let $\lambda^*(P)$ denote the second largest eigenvalue of $P$ in absolute value. That is, if $-1\leqslant \lambda_n\leqslant \dots\leqslant \lambda_1=1$ are the eigenvalues of $P$, then $\lambda^*(P)$ equals $\max\set{\abs{\lambda_2},\abs{\lambda_n}}$. \[thm:mixingtime\] For any reversible irreducible Markov chain $(\Omega, P, \pi)$, ${{\epsilon}}>0$, and any starting state $\tau\in \Omega$, $$t_\tau({{\epsilon}}) \ \leqslant \ \frac1{1-\lambda^*(P)}\cdot \log\parens*{\frac{1}{{{\epsilon}}\cdot \pi(\tau)}}.$$ For our results, it will be enough to look at the second largest eigenvalue $\lambda_2(P)$, which we can bound using the conductance of a weighted graph. Consider a weighted graph $G=(V,E,w)$ and a subset $S\subseteq V$ of vertices. We let $\overline{S}$ denote the complement $V\setminus S$. Then the *conductance* of $S$, denoted by $\operatorname{cond}(S)$, is defined as $$\operatorname{cond}(S) \ = \ \frac{w(E(S,\overline{S}))}{\operatorname{vol}(S)} \ = \ \frac{\sum_{e\in E(S,\overline{S})} w(e)}{\sum_{v\in S} w(v)},$$ where $E(S,\overline{S})=\set{ \set{u,v}\in E{}u\in S,v\notin S}$ is the set of edges between $S$ and $\overline{S}$, $w(E(S,\overline{S}))$ is the sum of weights of these edges, and the volume $\operatorname{vol}(S)$ is the sum of the weighted degrees of the vertices in $S$. The conductance of $G$ is then $$\operatorname{cond}(G)\ = \ \min_{S} \ \operatorname{cond}(S),$$ where the minimum is taken over subsets $\emptyset \subsetneq S\subsetneq V$ for which $w(S)\leqslant w(\overline{S})$. We say $G$ is $d$-regular if $w(v)=d$ for all $v\in V$. \[thm:Cheeger\] For any $d$-regular weighted graph $G=(V,E,w)$, $$\frac{d-\lambda_2(A_G)}{2} \ \leqslant \ \operatorname{cond}(G) \ \leqslant \ \sqrt{2(d-\lambda_2(A_{G}))},$$ where $A_G$ is the weighted adjacency matrix of $G$ given by $(A_G)_{ij} = w(\set{i,j})$. A direct consequence of the above theorem is that if the (weighted) graph $G$ is connected, i.e., for all proper nonempty subsets $S$ of vertices, $w(E(S,\overline{S}))>0$, then $\lambda_2(A_G)<d$. If the matrix $A_G$ is stochastic, then the graph is $1$-regular, which gives the following. \[cor:cheegerconnected\] If $A\in {{\mathbb{R}}}^{n\times n}$ is a stochastic matrix corresponding to a reversible Markov chain with the property that $\sum_{\card{\set{i,j}\cap S}=1} A_{i,j}>0$ for all subsets $\emptyset \subsetneq S\subsetneq [n]$, then $\lambda_2(A)<1$. Matroids {#subsec:matroids} -------- A *matroid* $M=([n],{{\mathcal{I}}})$ is a combinatorial structure consisting of a ground set $[n]$ of elements and a nonempty collection ${{\mathcal{I}}}$ of *independent* subsets of $[n]$ satisfying: i) If $S\subseteq T$ and $T\in {{\mathcal{I}}}$, then $S\in {{\mathcal{I}}}$ (hereditary property). ii) If $S,T\in {{\mathcal{I}}}$ and $\card{T}>\card{S}$, then there exists an element $i\in T \setminus S$ such that $S\cup \set{i}\in {{\mathcal{I}}}$ (exchange axiom). The *rank*, denoted by $\operatorname{rank}(S)$, of a subset $S \subset [n]$ is the size of any maximal independent set of $M$ contained in $S$. Thus, the independent sets of $M$ are precisely those subsets $S \subset [n]$ for which $\operatorname{rank}(S) = \card{S}$. We call $\operatorname{rank}([n])$ the rank of $M$, and if $M$ has rank $r$, any set $S\in {{\mathcal{I}}}$ of size $r$ is called a *basis* of $M$. An element $i \in [n]$ is a *loop* if $\set{i} \notin {{\mathcal{I}}}$, that is, $\set{i}$ is dependent. Two non-loops $i,j \in [n]$ are *parallel* if $\set{i,j} \notin {{\mathcal{I}}}$, that is, $\set{i,j}$ is dependent. Let $M = ([n],{{\mathcal{I}}}$) be a matroid and $S \in {{\mathcal{I}}}$. Then the *contraction $M/S$* is the matroid with ground set $[n] \setminus S$ and independent sets $\set{T \subseteq [n] \setminus S {}T \cup S \in {{\mathcal{I}}}}$. We will use a key property of matroids called the *matroid partition property*. For any matroid $M=([n],{{\mathcal{I}}})$, the non-loops of $M$ can be partitioned into sets $S_1,S_2,\dots,S_k$ for some $1 \leqslant k\leqslant n$ with the property that non-loops $j,k \in [n]$ are parallel if and only if they belong to the same set $S_i$. Indeed, one can check from the axioms for a matroid that being parallel defines an equivalence relation on the non-loop elements of $[n]$ and $S_{1},\dots,S_{k}$ are then the corresponding equivalence classes. Simplicial Complexes {#sec:SimplicialComplex} -------------------- A *simplicial complex* $X$ on the ground set $[n]$ is a nonempty collection of subsets of $[n]$ that is downward closed, namely if $\tau \subset \sigma$ and $\sigma \in X$, then $\tau \in X$. The elements of $X$ are called *faces/simplices*, and the *dimension* of a face $\tau \in X$ is defined as $\dim(\tau) = \card{\tau}$. Note that for convenience and clarity of notation, our definition deviates from the standard definition of $\dim(\tau) = \card{\tau}-1$ used by topologists. The empty set $\emptyset$ has dimension-$0$. A face of dimension 1 is a *vertex* of $X$ and a face of dimension 2 is called an *edge*. More generally, we write $$X(k) \ = \ \set{\tau\in X{}\dim(\tau) =k}$$ for the collection of dimension-$k$ faces, or *$k$-faces/$k$-simplices*, of $X$. The dimension of $X$ is the largest $k$ for which $X(k)$ is nonempty, and we say that $X$ is *pure* of dimension $d$ if all maximal faces of $X$ have the dimension $d$. In this paper we will only consider pure simplicial complexes. The *link* of a face $\tau\in X$ denoted by $X_\tau$ is the simplicial complex on $[n]\setminus \tau$ obtained by taking all faces in $X$ that contain $\tau$ and removing $\tau$ from them, $$X_\tau \ = \ \set{\sigma\setminus\tau{}\sigma\in X, \sigma\supset\tau}.$$ Note that if $X$ is pure of dimension $d$ and $\tau\in X(k)$, then $X_\tau$ is pure and has dimension $(d-k)$. For any matroid $M = ([n],{{\mathcal{I}}})$ of rank $r$, the independent sets ${{\mathcal{I}}}$ form a pure $r$-dimensional simplicial complex on $[n]$ called its *independence (or matroid) complex*. Furthermore, for any $S \in {{\mathcal{I}}}$, the link ${{\mathcal{I}}}_{S}$ of the independence complex consists precisely of the independent sets of the contraction $M/S$. There are many other beautiful simplicial complexes associated to matroids, but here we will be mainly interested in the independence complex. We can equip a simplicial complex with a weight function: $w:X\to{{\mathbb{R}}}_{>0}$ which assigns a positive weight to each face of $X$. We say that $w$ is *balanced* if for every non-maximal face $\tau\in X$ of dimension $k$, $$\begin{aligned} w(\tau) \ = \ \sum_{\sigma \in X(k+1) : \sigma \supset \tau} w(\sigma)\label{eq:balancedweight} \end{aligned}$$ For a pure simplicial complex $X$ we can define a balanced weight function by assigning arbitrary positive weights to maximal faces and defining the weight of each lower dimensional face recursively. Indeed, if $X$ is a pure simplicial complex of dimension $d$ and $w$ is a balanced weight function, then, for any $\tau\in X(k)$, $$w(\tau)=(d-k)! \sum_{\sigma\in X(d): \sigma\supset \tau} w(\sigma).$$ One natural choice is the function which assigns a weight of one to each maximal face, but there are many other interesting choices. Any balanced weight function on $X$ induces a weighted graph on the vertices of $X$ as follows. The *1-skeleton* of $X$ is the graph on vertices $X(1)$ with edges $X(2)$. Then, restricting $w$ to $X(1)$ and $X(2)$ determines a weighted graph, where $w(v)$ gives the weighted degree of each $v \in X(1)$. The weighted graphs coming from both $X$ and its links $X_\tau$ will be useful in later sections. Log-Concave Polynomials ----------------------- We say a polynomial $p\in{{\mathbb{R}}}[x_1,\dots,x_n]$ is $d$-homogeneous if every monomial of $p$ has degree $d$; equivalently, $p$ is $d$-homogeneous if $p(\lambda x_1,\dots,\lambda x_n)=\lambda^d p(x_1,\dots,x_n)$ for every $\lambda \in {{\mathbb{R}}}$. For a $d$-homogeneous polynomial $p$, the following identity, known as Euler’s identity, holds: $$\begin{aligned} d\cdot p(x) &= \sum_{k=1}^{n} x_{k}\partial_k p(x). \label{eq:eulerppartial} \end{aligned}$$ Note that if $p$ is homogeneous then all directional derivatives of $p$ are also homogeneous, so one can apply this to $\partial_i p(x) $ and $\partial_i\partial_j p(x)$ to find that $$\begin{aligned} \label{eq:Hesssum} (d-1) \cdot (\nabla p) = \sum_{k=1}^{n} x_{k} \cdot (\nabla (\partial_k p)) = (\nabla^{2}p) \cdot x \quad \text{ and } \quad (d-2) \cdot (\nabla^{2}p) = \sum_{k=1}^{n} x_{k} \cdot (\nabla^{2}(\partial_k p)). \end{aligned}$$ A polynomial $p \in {{\mathbb{R}}}[x_1,\dots,x_n]$ with nonnegative coefficients is log-concave if $\log p$ is a concave function over ${{\mathbb{R}}}_{>0}^{n}$. For simplicity we also consider the zero polynomial to be log-concave. Equivalently, $p$ is log-concave if the Hessian of $\log p$ $$\nabla^2\log p=\frac{p\cdot (\nabla^2 p) - (\nabla p) (\nabla p)^\intercal}{p^{2}}$$ is negative semi-definite at any point $x \in {{\mathbb{R}}}_{>0}^{n}$, where $\nabla p$ is the gradient of $p$. Since $(\nabla p)(\nabla p)^\intercal$ is a rank-1 matrix, by Cauchy’s interlacing theorem, $p\cdot (\nabla^2 p)$ has at most one positive eigenvalue at any $x\in{{\mathbb{R}}}_{>0}^{n}$. Since $p$ has nonnegative coefficients and $x$ has strictly positive entries, $p(x)>0$ so $\nabla^{2} \log p$ being negative semidefinite is equivalent to $\nabla^{2}p \preccurlyeq \frac{(\nabla p)(\nabla p)^{\intercal}}{p}$, where the right-hand side is a rank-1 positive semidefinite matrix. In particular, $\nabla^{2}p$ has at most one positive eigenvalue at $x$. In [@AOV18] it is shown that for homogeneous polynomials $p$, the converse of this is also true, i.e., if $\nabla^2 p$ has at most one positive eigenvalue at all $x>0$ then $p$ is log-concave. \[prop:oneposeigenvalue\] A degree-$d$ homogeneous polynomial $p\in{{\mathbb{R}}}[x_1,\dots,x_n]$ with nonnegative coefficients is log-concave over ${{\mathbb{R}}}_{>0}^{n}$ iff $(\nabla^2 p)(x)$ has at most one positive eigenvalue at all $x\in {{\mathbb{R}}}_{>0}^{n}$. Whenever $p$ has degree at least 2 and nonnegative coefficients, $(\nabla^{2}p)(x)$ has at least one positive entry for any $x \in {{\mathbb{R}}}_{>0}^{n}$. Hence, one can see (via, for example, the variational characterization of eigenvalues \[thm:Courant-Fischer\] and using a test vector with positive entries) that $(\nabla^{2}p)(x)$ must have at least one strictly positive eigenvalue. Thus, whenever we write “at most one positive eigenvalue”, we also mean it has “exactly one positive eigenvalue”. We say a polynomial $p\in {{\mathbb{R}}}[x_1, \dots, x_n]$ is *decomposable* if it can be written as a sum of polynomials in disjoint subset of the variables, that is, if there exists a nonempty subset $I \subsetneq [n]$ and nonzero polynomials $g\in {{\mathbb{R}}}[x_i : i\in I]$, $h \in {{\mathbb{R}}}[x_i : i\not\in I]$ for which $f = g+h$. We call $f$ *indecomposable* otherwise. \[prop:LCtoDecomp\] If $p\in{{\mathbb{R}}}[x_1,\dots,x_n]$ has nonnegative coefficients, is homogeneous of degree at least 2, and log-concave at ${{\mathds{1}}}$, then $p$ is indecomposable. Suppose that $p$ has nonnegative coefficients, is homogeneous of degree $\geqslant 2$, and is decomposable, with decomposition $p = g+h$ where $g\in {{\mathbb{R}}}[x_i : i\in I]$ and $h \in {{\mathbb{R}}}[x_i : i\not\in I]$. Both $g$ and $h$ are restrictions of $p$ obtained by setting some variables equal to zero, therefore both $g$ and $h$ are log-concave. Then, at ${{\mathds{1}}}$, the Hessians of $g$ and $h$ each have precisely one positive eigenvalue. However, the Hessian of $p$ at this point is a block diagonal matrix with these two blocks, $\nabla^2g,\nabla^2h$, $$\nabla^2 p= \begin{bmatrix} \nabla^2 g & 0 \\ 0& \nabla^2h \end{bmatrix}.$$ So, $p$ has exactly two positive eigenvalues, meaning that $p$ is not log-concave, a contradiction. In order to prove several distributions of interest are strongly log-concave, we will prove an equivalent characterization of strongly log-concave polynomials. \[thm:SLCdeg2\] Let $p\in {{\mathbb{R}}}[x_1,\dots,x_n]$ be a $d$-homogeneous polynomial such that: 1. for any $0\leqslant k\leqslant d-2$ and any $(i_{1},\dots,i_{k}) \in [n]^k$, $\partial_{i_{1}}\dotsb \partial_{i_{k}} p$ is indecomposable, and 2. for any $(i_1,\dots,i_{d-2})\in [n]^{d-2}$, the quadratic $\partial_{i_1}\dots \partial_{i_{d-2}} p$ is either identically zero, or log-concave at ${{\mathds{1}}}$. Then $p$ is strongly log-concave at ${{\mathds{1}}}$. In \[prop:LCtoDecomp\] we show that the condition that all partial derivatives are indecomposable is necessary for a polynomial to be (strongly) log-concave. Walks on Simplicial Complexes {#sec:highdimwalk} ============================= Consider a pure $d$-dimensional complex $X$ with a balanced weight function $w:X\to{{\mathbb{R}}}_{>0}$. We will call $(X,w)$ a *weighted complex*. For $1\leqslant k< d$, we define a random walk on $X(k)$ known as the *upper $k$-walk* based on movement from an face in $X(k)$ to a higher-dimensional face and then returning to $X(k)$. Similarly, for $1 \leqslant k < d$, we define a random walk on $X(k+1)$ known as the *lower $k$-walk* based on movement from a face in $X(k+1)$ to a lower-dimensional face and then returning to $X(k+1)$. To define these walks we construct a bipartite graph $G_k$ with one side corresponding to $X(k)$ and the other side corresponding to $X(k+1)$. We connect $\tau\in X(k)$ to $\sigma\in X(k+1)$ with an edge of weight $w(\sigma)$ iff $\tau\subset \sigma$. Now, consider the simple (weighted) random walk on $G_k$. Given a vertex we choose a neighbor proportional to the weight of the edge connecting the two vertices. This is a walk on a bipartite graph and is naturally periodic. We can consider the odd steps and even steps, in order to obtain two random walks; one on $X(k)$ called $P_{k}^{\wedge}$, and the other on $X(k+1)$ called $P_{k+1}^{\vee}$, where given $\tau\in X(k)$ you take two steps of the walk in $G_k$ to transition to the next $k$-face with respect to the $P_{k}^{\wedge}$ matrix, and similarly, you take two steps in $G_k$ from $\sigma\in X(k+1)$ to transition with respect to $P_{k+1}^{\vee}$. Now, let us formally write down the entries of $P_{k}^{\wedge}$ and $P_{k+1}^{\vee}$. Given a simplex $\tau\in X(k)$, first among all $k+1$ dimensional simplices $\sigma\in X(k+1)$ that contain $\tau$ we choose one proportional to $w(\sigma)$. Then, we delete one of the $\card{\sigma}=k+1$ elements of $\sigma$ uniformly at random to obtain a new state $\tau'$. It follows that the probability of transition to $\tau'$ is equal to the probability of choosing $\sigma=\tau\cup\tau'$ in the first step, which is equal to $\frac{w(\tau\cup\tau')}{w(\tau)}$ since $w$ is balanced, times the probability of choosing $\tau'$ conditioned on $\sigma=\tau\cup\tau'$, which is $\frac1{k+1}$. In summary, for $1 \leqslant k < d$, $$\begin{aligned} \label{eq:P+def} P_{k}^{\wedge}(\tau,\tau') &= \begin{cases} \frac{1}{k+1}, &\quad \text{if } \tau = \tau' \\ \frac{w(\tau \cup \tau')}{(k+1)w(\tau)}, &\quad \text{if } \tau \cup \tau' \in X(k+1) \\ 0, &\quad \text{otherwise} \end{cases} \end{aligned}$$ Note that upper walk is not defined for $k=d$, because there is no $(d+1)$-dimensional simplex in $X$. Analogously, given $\sigma\in X(k+1)$, first we remove a uniformly random element of $\sigma$ to obtain $\tau$. Then, among all all $k+1$ simplices $\sigma'\in X(k+1)$ that contain $\tau$ we choose one proportional to $w(\sigma')$. It follows that for $1 \leqslant k < d$, $$\begin{aligned} \label{eq:P-def} P_{k+1}^{\vee}(\sigma,\sigma') &= \begin{cases} \sum_{\tau \in X(k) : \tau \subset \sigma} \frac{w(\sigma)}{(k+1)w(\tau)}, &\quad \text{if } \sigma = \sigma' \\ \frac{w(\sigma')}{(k+1)w(\sigma \cap \sigma')}, &\quad \text{if } \sigma \cap \sigma' \in X(k) \\ 0, &\quad \text{otherwise} \end{cases} \end{aligned}$$ Observe the corresponding random walks are reversible with respect to the weight function $w$, i.e., for all $\tau,\tau'\in X(k)$, we have $$w(\tau) P^{\wedge}_k(\tau,\tau')=w(\tau')P^{\wedge}_k(\tau',\tau) \quad\quad w(\tau) P^{\vee}_k(\tau,\tau')=w(\tau')P^{\vee}_k(\tau',\tau).$$ This implies that both chains have the same stationarity distribution where the probability of $\tau\in X(k)$ is proportional to $w(\tau)$. \[lem:upperlower\] For any $1\leqslant k<d$, $P_{k}^{\wedge}$ and $P_{k+1}^{\vee}$ are stochastic, self-adjoint w.r.t. the $w$-induced inner product, PSD, and have the same (with multiplicity) non-zero eigenvalues. Let $P_k$ be the transition probability matrix of the simple random walk on $G_k$. Since $G_k$ is bipartite and we can write $$P_k=\begin{bmatrix} 0 & P_k^{\downarrow} \\ P_k^{\uparrow} & 0\end{bmatrix}$$ where $P_k^{\downarrow}\in {{\mathbb{R}}}^{X(k+1)\times X(k)}$ and $P_k^{\uparrow}\in {{\mathbb{R}}}^{X(k)\times X(k+1)}$ are stochastic matrices. Note that $P_k$ is self-adjoint w.r.t. the weight-induced inner product given by weights of the stationary distribution. It follows that $P_k$ is self-adjoint w.r.t. the inner product $$\begin{aligned} \langle \phi, \psi \rangle = \sum_{\tau \in X(k)} w(\tau)\phi(\tau)\psi(\tau) + (k+1)\sum_{\sigma \in X(k+1)} w(\sigma)\phi(\sigma)\psi(\sigma) \end{aligned}$$ Observe that $$P_k^2 = \begin{bmatrix} P_k^{\downarrow}P_k^{\uparrow} & 0 \\ 0 & P_k^{\uparrow} P_k^{\downarrow} \end{bmatrix}$$ So, in particular, $P_k^2$ is PSD and stochastic. Since $P_{k}^{\wedge}$ and $P_{k+1}^{\vee}$ correspond to two step walks on $G_k$, indeed we can write $$\begin{aligned} P_{k}^{\wedge} &=& P_k^{\uparrow}P_k^{\downarrow}\\ P_{k+1}^{\vee} &=& P_k^{\downarrow}P_k^{\uparrow} \end{aligned}$$ It follows that both matrices are self-adjoint w.r.t. the $w$-induced inner product, are PSD, and stochastic, and by \[fact:eigenvaluesinvert\] they have the same eigenvalues. Let us specifically study $P_{1}^{\wedge}$. Observe that $P_{1}^{\wedge}$ is the transition probability matrix of the simple $(1/2)$-lazy random walk on the weighted 1-skeleton of $X$ where the weight of each edge $e \in X(2)$ is $w(e)$. We also need to consider the non-lazy variant of this random walk, given by the transition matrix $${{\tilde{P}}}_{1}^{\wedge} \ = \ 2(P_{1}^{\wedge}-I/2).$$ Similarly, for any face $\tau\in X(k)$, we define the upper random walk on the faces of the link $X_{\tau}$. Specifically, let $P_{\tau,1}^{\wedge}$ denote the transition matrix of the upper walk, as above, on the 1-dimensional faces of $X_{\tau}$, and $${{\tilde{P}}}^{\wedge}_{\tau, 1} \ = \ 2(P_{\tau,1}^{\wedge}-I/2)$$ be the transition matrix for the non-lazy version. For $\lambda\geqslant 0$, a pure $d$-dimensional weighted complex $(X,w)$ is a $\lambda$-local-spectral-expander if for every $0 \leqslant k < d - 1$, and for every $\tau \in X(k)$, we have $\lambda_2({{\tilde{P}}}^{\wedge}_{\tau, 1})\leqslant \lambda$. In other words, $X$ is $\lambda$-local spectral expander if the spectral gap of the natural random walk on the 1-skeleton of the link of all simplices of $X$ has a spectral gap of at least $1-\lambda$. In this section we give a somewhat simpler proof of the following special case of the main theorem of [@KO18]. \[thm:localexpander\] Let $(X,w)$ be a pure $d$-dimensional weighted $0$-local spectral expander and let $0\leqslant k<d$. Then, for all $-1 \leqslant i\leqslant k$, $P_{k}^{\wedge}$ has at most $\card{X(i)} \leqslant \binom{n}{i}$ eigenvalues of value $> 1-\frac{i+1}{k+1}$, where for convenience, we set $X(-1) = \emptyset$ and $\binom{n}{-1} = 0$. In particular, the second largest eigenvalue of $P_{k}^{\wedge}$ is at most $\frac{k}{k+1}$. In other words, $P_{k}^{\wedge}$ has very few “big” eigenvalues. For example, $P_{k}^{\wedge}$ has exactly one eigenvalue strictly larger than $\frac{k}{k+1}$ corresponding to the maximum eigenvalue (which has value 1) and at most $n = \card{X(1)}$ eigenvalues strictly larger than $\frac{k-1}{k+1}$. Hence, $P_{k}^{\wedge}$ has at most $n -1$ eigenvalues between $\frac{k-1}{k+1}$ and $\frac{k}{k+1}$. Note that the significance of this theorem is that we are able to establish an estimate on *all* eigenvalues of $P_{k}^{\wedge}$. For the proof, we will need the following lemma. We remark that the inner product on the space ${{\mathbb{R}}}^{X(k)}$ is given by $\dotprod{\phi,\psi}=\sum_{\tau\in X(k)} w(\tau)\phi(\tau)\psi(\tau)$, and that being self-adjoint, PSD, and the Loewner order are defined w.r.t. this inner product. \[lem:lowerupper\] $P_{k}^{\wedge} \preccurlyeq \frac{k}{k+1} P_{k}^{\vee} + \frac{1}{k+1}I$ for all $0 \leqslant k < d$. For convenience, let $M = P_{k}^{\wedge} - \parens*{\frac{k}{k+1} P_{k}^{\vee} + \frac{1}{k+1}I}$. Fix $\eta \in X(k-1)$. We will first consider submatrices $M_{\eta}$ whose entries are given by the following: $$\begin{aligned} M_{\eta}(\tau,\sigma) = \begin{cases} M(\tau,\sigma), & \quad\text{if } \tau \neq \sigma, \eta = \tau \cap \sigma \\ -\frac{1}{k+1} \cdot \frac{w(\tau)}{w(\eta)}, &\quad\text{if } \tau = \sigma, \tau \supset \eta \\ 0, &\quad\text{otherwise} \end{cases}\end{aligned}$$ Note that $M = \sum_{\eta \in X(k-1)} M_{\eta}$ and hence, it suffices to prove that $M_{\eta} \preccurlyeq 0$ for every $\eta \in X(k-1)$.\  \ Fix $\eta \in X(k-1)$. Let $\tau,\sigma \in X(k)$ with $\tau \neq \sigma$ and $\tau \cap \sigma = \eta$. Then $$\begin{aligned} M_{\tau \cap \sigma}(\tau,\sigma) = M(\tau, \sigma) = \frac{1}{k+1}\parens*{\frac{w(\tau \cup \sigma)}{w(\tau)} - \frac{w(\sigma)}{w(\tau \cap \sigma)}} = \frac{1}{k+1} \parens*{\frac{w(\tau\cup\sigma)w(\tau \cap \sigma) - w(\tau)w(\sigma)}{w(\tau)w(\tau \cap \sigma)}}\end{aligned}$$ Furthermore, by definition, if $\tau \in X(k)$ with $\tau \supset \eta$, then $M_{\eta}(\tau,\tau) = -\frac{1}{k+1} \cdot \frac{w(\tau)}{w(\eta)}$. A matrix calculation reveals that $$\begin{aligned} M_{\eta} = \frac{1}{(k+1)w(\eta)}\operatorname{diag}(w_{\eta})^{-1} \cdot (w(\eta) \cdot A_{\eta} - w_{\eta}w_{\eta}^{\intercal})\end{aligned}$$ where $w_{\eta}$ is the $\card{X(k)}$-dimensional vector whose non-zero entries are $w(\tau)$ for $\tau \supset \eta$, and $A_{\eta}$ is the $\card{X(k)} \times \card{X(k)}$ matrix whose non-zero entries are $w(\tau \cup \sigma)$ for $\tau,\sigma \in X(k)$ satisfying $\tau \cup \sigma \in X(k+1)$ and $\tau \cap \sigma = \eta$. Note that $M_\eta$ is NSD w.r.t. the inner product defined by $w$, if and only if $\operatorname{diag}(w_\eta)M_\eta$ is NSD in the usual sense, because for any $v$ $$\dotprod{v, M_\eta v}=v^\intercal \operatorname{diag}(w_k) M_\eta v=v^\intercal \operatorname{diag}(w_\eta)M_\eta v,$$ where $w_k$ is the vector of $w$ values on $X(k)$ and for the last equality we used that $w_k$ is the same as $w_\eta$ on all $\tau\supset \eta$. Thus, it suffices to prove that $A_{\eta} \preccurlyeq \frac{w_{\eta}w_{\eta}^{\intercal}}{w(\eta)}$. We view $A_{\eta}$ as the weighted adjacency matrix of the 1-skeleton (which we recall is a graph) of the link $X_{\eta}$. Then ${{\tilde{P}}}_{\eta,1}^{\wedge} = \frac{1}{k+1}\operatorname{diag}(w_{\eta})^{-1}A_{\eta}$ gives its non-lazy simple random walk matrix. As $(X,w)$ is a 0-local spectral expander, ${{\tilde{P}}}_{\eta,1}^{\wedge}$ has at most one positive eigenvalue, whence $A_{\eta} = (k+1)\operatorname{diag}(w_{\eta}) \cdot {{\tilde{P}}}_{\eta,1}^{\wedge}$ has at most one positive eigenvalue by \[lem:stochastic\]. Finally, observe that the weights being balanced enforces that $w(\tau) = \sum_{\sigma \in X(k) : \tau \cup \sigma \in X(k+1)} w(\tau \cup \sigma)$ and $w(\eta) = \sum_{\tau \in X(k) : \tau \supset \eta} w(\tau)$. That $A_{\eta} \preccurlyeq \frac{w_{\eta}w_{\eta}^{\intercal}}{w(\eta)}$ then follows immediately by \[lem:adjacency1poseig\]. We go by induction on $k$. The case $k=0$ is trivial, as $P_{0}^{\wedge}$ is $1 \times 1$. When $k = 1$, we have $P_{1}^{\wedge} = \frac{1}{2}\parens*{{{\tilde{P}}}_{1}^{\wedge} + I}$. As $(X,w)$ is a 0-local spectral expander, ${{\tilde{P}}}_{1}^{\wedge}$ has exactly one positive eigenvalue, with value 1. Hence, $P_{1}^{\wedge}$ has eigenvalue 1 with multiplicity 1. All other eigenvalues of $P_{1}^{\wedge}$ are less than or equal to $1/2$, of which, there are $\card{X(1)} - 1$ many. Thus, the base case holds. Assume the claim holds for some $d - 1 > k \geqslant 0$. Recall by \[lem:upperlower\], $P_{k+1}^{\vee}$ has the same non-zero eigenvalues as $P_{k}^{\wedge}$. By \[lem:lowerupper\], $$\begin{aligned} P_{k+1}^{\wedge} \preccurlyeq \frac{k+1}{k+2} P_{k+1}^{\vee} + \frac{1}{k+2}I\end{aligned}$$ For $-1 \leqslant i \leqslant k$, $P_{k}^{\vee}$ as at most $\card{X(i)}$ eigenvalues $> 1 - \frac{i+1}{k+1}$. Hence, $P_{k+1}^{\wedge}$ has at most $\card{X(i)}$ eigenvalues $> \frac{k+1}{k+2} \cdot \parens*{1 - \frac{i+1}{k+1}} + \frac{1}{k+2} = 1 - \frac{i+1}{k+2}$. For $i = k+1$, we trivially have that $P_{k+1}^{\wedge}$ has at most $\card{X(k+1)}$ eigenvalues $> 0$, as $P_{k+1}^{\wedge}$ is $\card{X(k+1)} \times \card{X(k+1)}$. From Strongly Log-Concave Polynomials to Local Spectral Expanders {#sec:SLCtoLSE} ================================================================= In this section, we prove \[thm:SLCmixing\]. Let $p = \sum_Sc_Sx^S \in {{\mathbb{R}}}[x_1,\dots,x_n]$ be a $d$-homogeneous multiaffine strongly log-concave polynomial with nonnegative coefficients. We can construct a pure $d$-dimensional complex $X^p$ from $p$ as follows: For every term $c_S x^S$ of $p$ we include the $d$-dimensional simplex $S$ with weight $$\label{eq:wX^p} w(S)=c_S.$$ Note that since $p$ has nonnegative coefficients, the above weight function is nonnegative. We turn $X^p$ into a simplicial complex by including all subsets of the $d$-dimensional simplices and weighting each lower dimensional simplex inductively according to \[eq:balancedweight\]. \[prop:SLCtoHDE\] Let $p\in {{\mathbb{R}}}[x_1,\dots,x_n]$ be a multiaffine homogeneous polynomial with nonnegative coefficients. If $p$ is strongly log-concave then $(X^p,w)$ is a $0$-local-spectral-expander, where $w(S) = c_{S}$ for every maximal face $S \in X^{p}$. The converse of the above statement also holds true and we will discuss it in the next section. We build up to the proof of \[prop:SLCtoHDE\] and thereby the proof of \[thm:SLCmixing\]. We now fix a multiaffine $d$-homogeneous strongly log-concave polynomial $p$ with nonnegative coefficients. Fix a simplex $\tau\in X^p(k)$ and let $p_\tau=\left(\prod_{i\in\tau} \partial_i\right) p$. Note that $p_\tau$ is $(d-k)$-homogeneous. \[lem:wTau\_pTau\] For any $0\leqslant k\leqslant d$, and any simplex $\tau\in X^p(k)$, $ w(\tau)=(d-k)!\cdot p_{\tau}({{\mathds{1}}})$. We prove this by induction on $d-k$. If $\dim(\tau)=d$ then $p_{\tau}=c_{\tau}$ and the statement follows immediately from \[eq:wX\^p\]. So, suppose the statement holds for all simplices $\sigma \in X^p(k+1)$ and fix a simplex $\tau\in X^p(k)$. Then by definition, $$w(\tau) \ = \ \sum_{\sigma\in X^p(k+1): \sigma\supset \tau} w(\sigma) \ = \ \sum_{i\in X^p_\tau(1)} w(\tau\cup i).$$ Using the inductive hypothesis and the fact that $\partial_i p_\tau=0$ for $i\notin X^p_\tau(1)$, we then find that $$w(\tau) \ = \ (d-k-1)!\sum_{i\in X^p_\tau(1)} p_{\tau \cup \set{i}}({{\mathds{1}}}) \ = \ (d-k-1)!\sum_{i=1}^n \partial_i p_\tau({{\mathds{1}}}) \ = \ (d-k)!\cdot p_{\tau}({{\mathds{1}}}),$$ where the last equality follows from Euler’s identity. Recall that ${{\tilde{P}}}^{\wedge}_{\tau,1}$ is the transition probability matrix of the non-lazy random walk on the 1-skeleton of the link $X^p_\tau$. To prove the \[prop:SLCtoHDE\] it is enough to show that $\lambda_2({{\tilde{P}}}^{\wedge}_{\tau,1})\leqslant 0$, i.e. that ${{\tilde{P}}}^{\wedge}_{\tau,1}$ has at most one positive eigenvalue. Since $p$ is strongly log-concave, ${\nabla^2 p_\tau}({{\mathds{1}}})$ has at most one positive eigenvalue. Let $$\label{eq:tHessian} \tilde{\nabla}^{2} p_\tau = \frac{1}{d-k-1}\operatorname{diag}(\nabla p_\tau({{\mathds{1}}}))^{-1} {\nabla^2 p_\tau({{\mathds{1}}})}.$$ We claim that $$\tilde{\nabla}^2 p_\tau={{\tilde{P}}}^{\wedge}_{\tau,1}. \label{eq:tnablatP}$$ To see this, note that by \[eq:P+def\], for $i,j\in X_{\tau}^p$, $${{\tilde{P}}}^{\wedge}_{\tau,1}(i,j) =\frac{w_{\tau}(\set{i,j})}{w_\tau(\set{i})}=\frac{w(\tau\cup \set{i,j})}{w(\tau\cup \set{i})}.$$ On the other hand, by \[eq:tHessian\], $$(\tilde{\nabla}^2 p_\tau) (i,j) = \frac{(\partial_i\partial_j p_\tau)({{\mathds{1}}})}{(d-k-1)\cdot (\partial_i p_\tau)({{\mathds{1}}})}.$$ The above two are equal by \[lem:wTau\_pTau\], which proves \[eq:tnablatP\]. Since $p$ has nonnegative coefficients, the vector $\nabla p_\tau({{\mathds{1}}})$ has nonnegative entries, which implies $\operatorname{diag}(\nabla p_\tau({{\mathds{1}}}))\succcurlyeq 0$. Since $\nabla^2 p_\tau({{\mathds{1}}})$ has at most one positive eigenvalue, $\tilde{\nabla}^2p_\tau({{\mathds{1}}})$ has at most one positive eigenvalue by \[lem:stochastic\] as desired. Therefore by \[eq:tnablatP\], $\tilde{\nabla}^2p_\tau={{\tilde{P}}}^{\wedge}_{\tau,1}$ has at most one positive eigenvalue and $\lambda_2({{\tilde{P}}}^{\wedge}_{\tau,1})\leqslant 0$. With this we are ready to prove our main theorem. Let $\mu$ be a $d$-homogeneous strongly log-concave distribution and let $P_{\mu}$ be the transition probability matrix of the chain ${{\mathcal{M}}}_\mu$. By \[thm:mixingtime\] it is enough to show that $\lambda^*(P_{\mu})\leqslant 1-1/d$. Observe that the chain ${{\mathcal{M}}}_\mu$ is exactly the same as the chain $P^{\vee}_{d}$ for the simplicial complex $X^{g_\mu}$ defined above. Therefore, $\lambda^*(P_{\mu})=\lambda^*(P^{\vee}_{d}) = \lambda^{*}(P_{d-1}^{\wedge})$, where the last equality follows by \[lem:upperlower\]. Since $g_\mu$ is strongly log-concave, by \[prop:SLCtoHDE\], $X^{g_\mu}$ is $0$-local-spectra-expander. Therefore, by \[thm:localexpander\], $$\lambda^{*}(P_{d-1}^{\wedge}) \leqslant 1-\frac{1}{(d-1)+1}=1-\frac1d,$$ as desired. Bases Exchange Walk ------------------- In this part we prove \[thm:basesexchange\]. Fix a rank $r$ matroid $M=([n],{{\mathcal{I}}})$, let $\mu$ denote the uniform distribution on the bases of $M$, and consider the simplicial complex $X^{g_{\mu}}$. As before, recall that $P_{r}^{\vee}$ is the transition matrix for the Markov chain ${{\mathcal{M}}}_{\mu}$ defined in the introduction. As discussed in \[subsec:randomwalks\], each reversible Markov chain is equivalent to a random walk in a (weighted) undirected graph. Let $H_M$ be the graph corresponding to $P_{r}^{\vee}$. Then the vertices of $H_M$ correspond to bases of $M$, and the weight of an edge between two bases $\tau,\tau'$ is, $$P_{r}^{\vee}(\tau,\tau')=P_{r}^{\vee}(\tau',\tau).$$ The symmetry of $P_{r}^{\vee}$ follows by the fact that $P_{r}^{\vee}$ is reversible and $w(\tau)=w(\tau')=1$. Observe that $P_{r}^{\vee}$ is the adjacency matrix of $H_M$. Furthermore, note that by \[thm:gMclc\] (proof in \[sec:slc\]), $g_{\mu}$ is a strongly log-concave polynomial and hence, $X^{g_{\mu}}$ is a 0-local spectral expander by \[prop:SLCtoHDE\]. By \[thm:localexpander\], $\lambda_2(P_{r}^{\vee}) \leqslant 1-\frac1{r}$. Note that the weighted degree of each vertex of $H_M$ is 1 since $P_{r}^{\vee}$ is a stochastic matrix. Therefore, by Cheeger’s inequality, \[thm:Cheeger\], $$\label{eq:condHM}\operatorname{cond}(H_M) \ \geqslant \ \frac{1-\lambda_2(P_{r}^{\vee})}{2} \ \geqslant \ \frac{1-(1-1/r)}{2}=\frac1{2r}.$$ Let $G_M=({{\mathcal{B}}},E)$ be the (unweighted) bases exchange graph associated to $M$ as defined in the introduction. It follows that $G_M$ is the unweighted base graph of $H_M$. Fix a nonempty set $S\subset {{\mathcal{B}}}$ of bases such that $\card{S}\leqslant \card{{{\mathcal{B}}}}/2$. We need to show that the expansion $\operatorname{h}(S)$ is $\geqslant 1$. Note that since the weighted degree of each vertex of $H_M$ is 1, $\operatorname{vol}_H(S)=\card{S}$. It follows that $$\operatorname{cond}(H_M)\leqslant \operatorname{cond}(S) = \frac{\sum_{\tau,\tau':\tau \in S,\tau'\notin S} P_{r}^{\vee}(\tau,\tau')}{\card{S}} \leqslant \frac{\sum_{\tau,\tau':\tau \in S,\tau'\notin S} \frac1{2r}}{\card{S}} = \frac{\frac1{2r} \card{E(S,\overline{S})}}{\card{S}}=\frac{\operatorname{h}(S)}{2r}$$ The second inequality follows by the following fact. Putting the above together with \[eq:condHM\], implies $\operatorname{h}(S)\geqslant 1$ as desired. This completes the proof of \[thm:basesexchange\]. For any pair of bases, $\tau,\tau'\in X^{g_{\mu}}(r)$, $$P^{\vee}_{r}(\tau,\tau')\leqslant \frac1{2r}.$$ Suppose $P_{r}^{\vee}(\tau,\tau')>0$. This means that $\card{\tau\cap\tau'}=r-1$. Therefore, $$w(\tau\cap \tau')\geqslant w(\tau)+w(\tau') =2.$$ So, $P_{r}^{\vee}(\tau,\tau')=\frac{w(\tau')}{rw(\tau\cap\tau')}\leqslant \frac1{2r}.$ Proof of Strong Log-Concavity and Applications {#sec:slc} ============================================== \[sec:applications\] In this part we prove \[thm:SLCdeg2\] using connections with high dimensional expanders. \[thm:SLCdeg2\] can be seen as the converse of \[prop:SLCtoHDE\]. In fact, the following statement was proved by @Opp18. \[thm:opp\] Let $(X,w)$ be a pure $d$-dimensional weighted simplicial complex such that: 1. for all $\tau\in X(k)$, for $0\leqslant k\leqslant d-2$, the 1-skeleton of $X_\tau$ is a connected (weighted) graph, and 2. for any $\tau\in X(d-2)$, $\lambda_2({{\tilde{P}}}^{\wedge}_{\tau,1})\leqslant 0$. Then $(X,w)$ is a weighted $0$-local-spectral-expander. Our proof of \[thm:SLCdeg2\] can be seen as a translation of Oppenheim’s result into the language of polynomials.[^2] We actually prove a slightly stronger statement, in the sense that a direct translation of \[thm:opp\] corresponds to testing strong log-concavity of a *multiaffine* homogeneous polynomial. In this case, each derivative $\partial_{\tau}p$ corresponds to taking the link of $\tau$ in $X^{p}$. Indecomposability of each derivative then corresponds to connectivity of the 1-skeleton of the corresponding link, and log-concavity of quadratics corresponds to $\lambda_{2}({{\tilde{P}}}_{\tau,1}^{\wedge})\leqslant 0$. We proceed by induction on the degree of $p$. If the degree of $p$ is at most 2, the claim obviously holds. So, suppose $d\geqslant 3$. For any $1\leqslant i\leqslant n$, let $p_i$ denote $\partial_i p$. By induction, we can assume that for all $i$, $p_i$ is strongly log-concave (at ${{\mathds{1}}}$). First, by \[eq:Hesssum\], ${\nabla^2 p}({{\mathds{1}}})=\frac1{d-2}\sum_{i=1}^n {\nabla^2 p}_i ({{\mathds{1}}})$. By induction and \[prop:oneposeigenvalue\], each matrix ${\nabla^2 p}_i({{\mathds{1}}})$ has at most one positive eigenvalue. Instead, we work with the normalized Hessian matrix, $\tilde{\nabla}^2 p=\frac{1}{d-1} \operatorname{diag}(\nabla p({{\mathds{1}}}))^{-1} {\nabla^2 p}({{\mathds{1}}})$ as defined in \[eq:tHessian\]. Since the normalized Hessian matrix is stochastic, its top eigenvector is the all-ones vector. When working with the normalized Hessian we need to use the correct inner product operators. For a $d$-homogeneous polynomial $p$ with nonnegative coefficients and $d > 1$, and vectors $\phi,\psi\in{{\mathbb{R}}}^n$, define $$\begin{aligned} \dotprod{\phi,\psi}_{p} = (d-1) \sum_{j=1}^{n} \phi(j)\psi(j) (\partial_j p({{\mathds{1}}})),\end{aligned}$$ which gives the norm $\norm{\phi}_p^2=\dotprod{\phi,\phi}_p$. The following identity is immediate: $$\begin{aligned} \dotprod{\phi, (\tilde{\nabla}^{2}p)\psi}_{p} = \dotprod{\phi, {\nabla^2 p}({{\mathds{1}}}) \psi} = \dotprod{(\tilde{\nabla}^{2}p)\phi, \psi}_{p} \end{aligned}$$ In particular, $\tilde{\nabla}^2 p$ is self-adjoint with respect to $\dotprod{\cdot ,\cdot}_{p}$. Furthermore, by \[eq:Hesssum\], $$\label{eq:tHessumi}\dotprod{\phi,(\tilde{\nabla}^2 p)\psi}_{p} = \dotprod{\phi, {\nabla^2 p}({{\mathds{1}}})\psi}= \frac1{d-2}\sum_{k=1}^n \dotprod{\phi, \nabla^2 p_k({{\mathds{1}}}) \psi}=\frac1{d-2}\sum_{k=1}^n \dotprod{\phi,\tilde{\nabla}^2 p_k \psi}_{p_k}.$$ We highlight that the Hessian $\nabla^{2}p({{\mathds{1}}})$ may be viewed as the weighted adjacency matrix of a graph with edge weights $\partial_{i}\partial_{j}p({{\mathds{1}}})$. Thus, our normalized Hessian may be viewed as the associated random walk matrix, and the inner product $\langle\cdot,\cdot\rangle_{p}$ may be viewed as a change of basis, which converts the random walk matrix into the normalized adjacency matrix. Let $\mu$ be an eigenvalue of $\tilde{\nabla}^{2}p$ with eigenvector $\phi$. We prove that $\mu\leqslant \mu^{2}$. We claim that this is enough for the induction step: First, since $\tilde{\nabla}^{2}p$ is stochastic $\mu\leqslant 1$. Therefore, we either $\mu=1$ or $\mu\leqslant 0$. So, to prove that $\tilde{\nabla}^2 p$ has (exactly) one positive eigenvalue, it is enough to show that $\lambda_2(\tilde{\nabla}^{2}p)<1$. But, since $p$ is indecomposable, the underlying (weighted) graph of $\tilde{\nabla}^2 p$ is connected, so by \[cor:cheegerconnected\], $\lambda_2(\tilde{\nabla}^2 p)<1$. It remains to prove that $\mu\leqslant \mu^2$. From \[eq:tHessumi\], $$\begin{aligned} \label{eq:muphitnabla} \mu \norm{\phi}_{p}^{2} &= \dotprod{ \phi, (\tilde{\nabla}^{2} p)\phi }_{p} = \frac{1}{d-2} \sum_{k=1}^{n} \dotprod{ \phi, (\tilde{\nabla}^{2} p_{k})\phi }_{p_k}\end{aligned}$$ Decomposing $\phi$ orthogonally along ${{\mathds{1}}}$ write $$\phi=\phi_{k}^{\perp{{\mathds{1}}}} + \phi_{k}^{{{\mathds{1}}}}$$ where $\phi_k^{{{\mathds{1}}}}=\frac{\dotprod{ \phi,{{\mathds{1}}}}_{p_k}}{\norm{{{\mathds{1}}}}_{p_k}^2}{{\mathds{1}}}$ and $\phi_k^{\perp {{\mathds{1}}}}$ is orthogonal to ${{\mathds{1}}}$, i.e., $\dotprod{ \phi_k^{\perp{{\mathds{1}}}},{{\mathds{1}}}}_{p_k}=0$. It follows that $$\dotprod{ \phi_k^{\perp{{\mathds{1}}}}, (\tilde{\nabla}^2 p_k) \phi_k^{\perp{{\mathds{1}}}}}_{p_k}\leqslant 0.$$ This is because $\tilde{\nabla}^{2}p_{k}$ has exactly one positive eigenvalue with corresponding eigenvector of ${{\mathds{1}}}$. Therefore, $$\begin{aligned} \mu\norm{\phi}^2_p\leqslant \frac{1}{d-2} \sum_{k=1}^{n} \dotprod{ \phi_{k}^{{{\mathds{1}}}}, (\tilde{\nabla}^{2}p_{k})\phi_{k}^{{{\mathds{1}}}} }_{p_k} = \frac{1}{d-2} \sum_{k=1}^{n} \dotprod{ \phi_{k}^{{{\mathds{1}}}}, \phi_{k}^{{{\mathds{1}}}} }_{p_k} = \frac{1}{d-2} \sum_{k=1}^{n} \frac{\dotprod{ \phi, {{\mathds{1}}}}_{p_k}^{2}}{\dotprod{ {{\mathds{1}}},{{\mathds{1}}}}_{p_k}}\label{eq:muphitnabla2}\end{aligned}$$ Next, we rewrite the numerator and denominator of each ratio in the righthand side. We have $$\begin{aligned} \dotprod{ {{\mathds{1}}},{{\mathds{1}}}}_{p_k} = (d-2)\sum_{i=1}^{n} (\partial_{i}p_{k}({{\mathds{1}}})) = (d-2)(d-1) \cdot p_{k}({{\mathds{1}}})\end{aligned}$$ where we used \[eq:eulerppartial\] for polynomial $p_k$. Furthermore, $$\begin{aligned} \label{eq:phionepk} \dotprod{ \phi, {{\mathds{1}}}}_{p_k} = (d-2)\sum_{i=1}^{n} \phi(i) (\partial_{i}p_{k}({{\mathds{1}}})) = (d-2) \cdot ((\nabla^{2}p({{\mathds{1}}}))\phi)(k)\end{aligned}$$ So, putting the above identities together, we obtain $$\begin{aligned} \frac{\dotprod{ \phi, {{\mathds{1}}}}_{p_k}}{\dotprod{ {{\mathds{1}}},{{\mathds{1}}}}_{p_k}} = \frac{1}{(d-1) \cdot p_{k}} ((\nabla^{2}p({{\mathds{1}}})) \phi)(k) = ((\tilde{\nabla}^{2}p({{\mathds{1}}}))\phi)(k) = \mu \cdot \phi(k)\end{aligned}$$ where the last identity we crucially used $\phi$ is the eigenvector of $\tilde{\nabla}^2p$ corresponding to $\mu$. Plugging into \[eq:muphitnabla2\] we get $$\begin{aligned} \mu\norm{\phi}_p^2&\leqslant \frac{1}{d-2} \sum_{k=1}^{n} \frac{\dotprod{ \phi, {{\mathds{1}}}}_{p_k}^{2}}{\dotprod{ {{\mathds{1}}},{{\mathds{1}}}}_{p_k}} =\frac{\mu}{d-2}\sum_{k=1}^n \phi(k) \dotprod{ \phi,{{\mathds{1}}}}_{p_k}\\ &=\mu\sum_{k=1}^n \phi(k) (({\nabla^2 p}({{\mathds{1}}}))\phi)(k) = \mu \dotprod{ \phi, ({\nabla^2 p}({{\mathds{1}}}))\phi}=\mu\dotprod{ \phi,(\tilde{\nabla}^2 p)\phi}_p = \mu^2 \norm{\phi}_p^2. $$ The second equality uses \[eq:phionepk\] and the last equality uses definition of $\phi$. So, $\mu\leqslant \mu^2$ as desired. An Alternative Proof of Strong Log-concavity of Bases Generating Polynomial --------------------------------------------------------------------------- \[thm:gMclc\] Let $M = ([n],{{\mathcal{I}}})$ be a matroid of rank $r$. Then, for every choice of “external field” $\bm{\lambda} = (\lambda_{1},\dots,\lambda_{n}) \in {{\mathbb{R}}}_{>0}^{n}$, its (weighted) bases generating polynomial $$\begin{aligned} g_{M}(x_{1},\dots,x_{n}) = \sum_{B \text{ basis}} \bm{\lambda}^{B}x^{B}\end{aligned}$$ is strongly log-concave (at ${{\mathds{1}}}$). We verify the indecomposability and log-concavity conditions of \[thm:SLCdeg2\]. If $i_{1},\dots,i_{k}$ contains duplicate elements, then multiaffine-ness of $g_M$ forces $\partial_{i_{1}}\dotsb \partial_{i_{k}}g_M$ to be identically zero. For any subset $S = \set{i_{1},\dots,i_{k}}$, we use $\partial_{S}$ as a shorthand notation for $\partial_{i_{1}}\dotsb \partial_{i_{k}}$. If $S$ is not independent, then no basis contains $S$, and again, $\partial_{S}g_M = 0$ identically. Hence, we assume $S \in {{\mathcal{I}}}$. We first argue that $\partial_{S}g_{M}$ is indecomposable. Observe that $\partial_{S}g_{M}$ equals the weighted basis generating polynomial $ g_{M/S}$ of the contraction $M/S$. As $M/S$ is a matroid of rank $\geqslant 2$, applying the exchange property immediately tells us that $g_{M/S}$ is indecomposable. Now, we verify log-concavity of all quadratics. Assume $S \in {{\mathcal{I}}}$ and $\card{S} = r-2$. As $S$ has rank-$(r-2)$, $M/S$ has rank two. In particular, $\partial_{S}g_{M} = g_{M/S}$ is quadratic and $\nabla^{2}g_{M/S}$ has entries $$\begin{aligned} (\nabla^{2}g_{M/S})_{ij} = \begin{cases} \lambda_{i}\lambda_{j}, &\quad \text{if } \set{i,j} \text{ is independent in } M/S \\ 0, &\quad \text{otherwise}. \end{cases}\end{aligned}$$ We need to prove that the above matrix has at most one positive eigenvalue. For any set $T\subseteq [n]$, let $\bm{\lambda}_T$ denote the vector with $i$th entry $\lambda_i$ for $i\in T$ and $0$ for $i\not\in T$. The matroid partition property tells us that we can partition the non-loops of $M/S$ into blocks $B = B_{1} \cup \dots \cup B_{k}$. Then $$\begin{aligned} \nabla^{2}g_{M/S} \ = \ \bm{\lambda}_{B}\bm{\lambda}_{B}^{\intercal} - \sum_{i=1}^{k} \bm{\lambda}_{B_{i}} \bm{\lambda}_{B_{i}}^{\intercal} \preccurlyeq \bm{\lambda}_{B}\bm{\lambda}_{B}^{\intercal} .\end{aligned}$$ This proves that $\nabla^{2} \partial_{S}g_{M}$ has at most one positive eigenvalue and thus $\partial_{S}g_{M}$ is log-concave. Let $M = ([n],{{\mathcal{I}}})$ be a matroid of rank $r$. Then the polynomial $$\begin{aligned} \sum_{S \in {{\mathcal{I}}}: \card{S} = k} \bm{\lambda}^{S}x^{S}\end{aligned}$$ is strongly log-concave (at ${{\mathds{1}}}$), for every choice of “external field” $\bm{\lambda} = (\lambda_{1},\dots,\lambda_{n}) \in {{\mathbb{R}}}_{>0}^{n}$. This is the weighted basis generating polynomial of the rank-$k$ truncation of $M$, which is still a matroid. Thus, the claim follows from \[thm:gMclc\]. The Random Cluster Model on Matroids ------------------------------------ Using the the simplified characterization of strong log-concavity, we can also prove \[thm:randomclusterSLC\]. We verify the indecomposability and log-concavity conditions of \[thm:SLCdeg2\]. Let $f$ denote $f_{M,k,q}$. As before, for the derivatives $\partial_{i_1}\hdots \partial_{i_k}f$, by multiaffine-ness of $f$, we may assume $i_{1},\dots,i_{k}$ does not contain any duplicate elements. Consider $S = \set{i_{1},\dots,i_{k}}$. Note that $\partial_{S}f$ has a monomial $x^{T}$ with nonzero coefficient for every $T \subset [n] \setminus S$ with $\card{T} \leqslant k - \card{S}$. Hence, indecomposability of $\partial_{S}f$ for every $S \subset [n]$ with $\card{S} \leqslant k$ is immediate. Now, we verify log-concavity of quadratics. Let $S \subset [n]$ with $\card{S} = k-2$. First, we calculate that $$(\partial_{S}f)(x_{1},\dots,x_{n}) = \sum_{T \in \binom{[n]}{k} : T \supset S} q^{-\operatorname{rank}(T)}\bm{\lambda}^{T \setminus S}x^{T \setminus S} = \sum_{\set{i,j} \in \binom{[n] \setminus S}{2}} q^{-\operatorname{rank}(S \cup \set{i,j})} \lambda_{i}\lambda_{j}x_{i}x_{j}.$$ Then for elements $i\neq j$ of $[n]\setminus S$, the $(i,j)$th entry of $\nabla^{2}\partial_{S}f$ is $$\begin{aligned} (\nabla^{2}\partial_{S}f)_{ij} = q^{-\operatorname{rank}(S \cup \set{i,j})}\lambda_{i}\lambda_{j} = q^{-\operatorname{rank}(S)}q^{-\operatorname{rank}_{M/S}(\set{i,j})}\lambda_{i}\lambda_{j}\end{aligned}$$ We will show that the matrix $A = q^{\operatorname{rank}(S)}\nabla^{2}\partial_{S}f$ at most one positive eigenvalue. Note that for $i\neq j$ in $[n]\setminus S$, $A_{ij} = q^{-\operatorname{rank}_{M/S}(\set{i,j})}\lambda_{i}\lambda_{j}$. From here, consider the vector $v\in {{\mathbb{R}}}^n$ with $v_i = 0$ for $i\in S$, $v_i = \lambda_{i}$ for loops of $M/S$ and $v_i = q^{-1}\lambda_{i}$ for non-loops of $M/S$. Now consider the matrix $vv^{\intercal} - A$, we can check that for $i,j\in [n]\setminus S$, $$\begin{aligned} (vv^{\intercal} - A)_{ij} = \begin{cases} (q^{-2} - q^{-1})\lambda_{i}\lambda_{j}&\quad\text{if } i,j \text{ are parallel non-loops in $M/S$, and} \\ 0&\quad\text{otherwise} \end{cases}\end{aligned}$$ In particular, by the matroid partition property, if $B_{1},\dots,B_{k}$ denote the equivalence classes of non-loops of $M/S$ which are parallel to each other, then $$\begin{aligned} vv^{\intercal} - A = (q^{-2} - q^{-1})\sum_{j=1}^{k} \bm{\lambda}_{B_{j}}\bm{\lambda}_{B_{j}}^{\intercal}\end{aligned}$$ where $\bm{\lambda}_{B_{j}}$ is the vector with entries $\bm{\lambda}_{B_{j}}(i) = \lambda_{i}$ if $i \in B_{j}$ and $\bm{\lambda}_{B_{j}}(i) = 0$ otherwise. As $0 < q \leqslant 1$, $q^{-2} - q^{-1} \geqslant 0$, in which case the right-hand side is positive semidefinite and $A \preccurlyeq vv^{\intercal}$. We conclude $A$ has at most one positive eigenvalue as desired. Observe that $$\begin{aligned} q^{r}f_{M,r,q}(x_{1},\dots,x_{n})\end{aligned}$$ converges to the bases generating polynomial coefficient-wise as $q \rightarrow 0$. Hence, one can view this as a stronger result than strong log-concavity of the bases generating polynomial. Geometric Scaling of Coefficients --------------------------------- In this section we prove \[thm:cpow\]. If $k = 0,1$, the claim is obvious so assume $k \geqslant 2$. The claim is obvious when $\alpha = 1$, and the case $\alpha = 0$ follows by taking coefficient-wise limits as $\alpha \rightarrow 0$. Hence, we will also assume $0 < \alpha < 1$. Finally, we will assume that all coefficients $c_{S}$ are strictly positive. The result for general strongly log-concave polynomials then follows by taking coefficient-wise limits.\  \ Let $T \in \binom{[n]}{k-2}$. We must prove that $\nabla^{2} \partial_{T}f_{\alpha}$ has at most one positive eigenvalue. Observe that we may concisely write $$\begin{aligned} \nabla^{2} \partial_{T}f &= \brackets*{c_{T \cup \set{i,j}}}_{ij} \\ \nabla^{2} \partial_{T}f_{\alpha} &= \brackets*{c_{T \cup \set{i,j}}^{\alpha}}_{ij}\end{aligned}$$ As $\nabla^{2}\partial_{T}f$ has at most one positive eigenvalue, and all entries are nonnegative, we may write $$\begin{aligned} \nabla^{2} \partial_{T}f = vv^{\intercal} - A\end{aligned}$$ for some vector $v \in {{\mathbb{R}}}^{n}$ and a positive semidefinite matrix $A$. Note that since $\nabla^{2}\partial_{T}f$ has strictly positive entries, the Perron-Frobenius Theorem (see \[thm:perron\]) tells us that the entries of $v$ are strictly positive. In particular, $c_{T \cup \set{i,j}} = v_{i}v_{j} - A_{ij} > 0$ where $v_{i}v_{j} > 0$. Our goal is to write $$\begin{aligned} c_{T\cup\set{i,j}}^{\alpha} = (v_{i}v_{j} - A_{ij})^{\alpha} = v_{i}^{\alpha}v_{j}^{\alpha} \parens*{1 - \frac{A_{ij}}{v_{i}v_{j}}}^{\alpha}\end{aligned}$$ and then Taylor expand $\parens*{1 - \frac{A_{ij}}{v_{i}v_{j}}}^{\alpha}$. Consider the function $\varphi_{\alpha}(x) = (1 - x)^{\alpha}$, whose Taylor expansion about zero we recall is $$\begin{aligned} \sum_{k=0}^{\infty} \frac{\prod_{j=0}^{k-1} (\alpha - j)}{k!} \cdot (-1)^{k}x^{k} = \sum_{k=0}^{\infty} \parens*{\prod_{j=0}^{k-1} \frac{\alpha - j}{1 + j}} \cdot (-1)^{k}x^{k} = 1 - \sum_{k=1}^{\infty} \parens*{\prod_{j=0}^{k-1} \abs*{\frac{\alpha - j}{1 + j}}} \cdot x^{k}\end{aligned}$$ where for the last equality, we crucially use the fact that $0 < \alpha < 1$. The interval of convergence of this power series contains $(-1,1)$, as if $a_{k} = (-1)^{k}\prod_{j=0}^{k-1} \frac{\alpha - j}{1 + j}$, then $$\begin{aligned} \abs*{\frac{a_{k+1}x^{k+1}}{a_{k}x^{k}}} = \abs{x} \cdot \frac{\alpha - k}{1 + k} \rightarrow \abs{x} \quad \text{as} \quad k \rightarrow \infty\end{aligned}$$ gives a radius of convergence of 1 by the Ratio Test. Hence, to apply this power series representation to our values of $x$, we verify that $x = \frac{A_{ij}}{v_{i}v_{j}} \in (-1,1)$, i.e. $\abs{A_{ij}}< v_{i}v_{j}$, for every $i,j$.\  \ For $i = j$, we have $A_{ii} \geqslant 0$ so $v_{i}^{2} - A_{ii} > 0$ is gives the desired inequality. For $i \neq j$, observe that $A$ being positive semidefinite means that its principal minors are nonnegative. In particular, for $S = \set{i,j}$, we have $$\begin{aligned} \det(A_{S,S}) = A_{ii}A_{jj} - A_{ij}^{2} \geqslant 0 \implies \abs{A_{ij}} \leqslant \sqrt{A_{ii}A_{jj}} < v_{i}v_{j}\end{aligned}$$ Having verified that the power series is valid for every entry of our matrix, we have $$\begin{aligned} \nabla^{2}\partial_{T}f_{\alpha} &= \brackets*{v_{i}^{\alpha}v_{j}^{\alpha}}_{ij} \circ \parens*{{{\mathds{1}}}{{\mathds{1}}}^{\intercal} - \sum_{k=1}^{\infty} \parens*{\prod_{j=0}^{k-1} \abs*{\frac{\alpha - j}{1 + j}}} \cdot \brackets*{\frac{A_{ij}}{v_{i}v_{j}}}^{\circ k}} \\ &= \underset{(1)}{\underbrace{\brackets*{v_{i}^{\alpha}v_{j}^{\alpha}}_{ij}}} - \underset{(2)}{\underbrace{\sum_{k=1}^{\infty} \parens*{\prod_{j=0}^{k-1} \abs*{\frac{\alpha - j}{1 + j}}} \cdot \parens*{\brackets*{v_{i}^{\alpha}v_{j}^{\alpha}}_{ij} \circ \brackets*{\frac{A_{ij}}{v_{i}v_{j}}}^{\circ k}}}}\end{aligned}$$ Here, we recall that $A \circ B$ denotes the Hadamard product of $A,B$, where $(A \circ B)_{ij} = A_{ij}B_{ij}$. Similarly, $A^{\circ k}$ denotes the $k$-iterated Hadamard product of $A$ with itself.\  \ All we must do is prove that (1) and (2) are both positive semidefinite, and that (1) is rank-1. Observe that $$\begin{aligned} [v_{i}^{\alpha}v_{j}^{\alpha}]_{ij} = [v_{i}^{\alpha}]_{i} \cdot [v_{i}^{\alpha}]_{i}^{\intercal} \quad\quad \brackets*{\frac{1}{v_{i}v_{j}}}_{ij} = \brackets*{\frac{1}{v_{i}}}_{i} \cdot \brackets*{\frac{1}{v_{i}}}_{i}^{\intercal}\end{aligned}$$ This tells us (1) is positive semidefinite and rank-1. For the second, observe that $$\begin{aligned} \brackets*{\frac{A_{ij}}{v_{i}v_{j}}}_{ij} = A \circ \brackets*{\frac{1}{v_{i}v_{j}}}_{ij}\end{aligned}$$ As $A \succcurlyeq 0$ by assumption, this matrix is positive semidefinite by the Schur Product Theorem (see \[thm:schur\]). Again, inductively applying the Schur Product Theorem, we have $\brackets*{v_{i}^{\alpha}v_{j}^{\alpha}}_{ij} \circ \brackets*{\frac{A_{ij}}{v_{i}v_{j}}}^{\circ k} \succcurlyeq 0$ for every $k$. As (2) is a nonnegative linear combination of positive semidefinite matrices, it is positive semidefinite. Not that this operation does not preserve complete log-concavity when $f$ is not assumed to be multiaffine. For example, consider the degree-2 bivariate polynomial $f(x,y) = ax^{2} + bxy + cy^{2}$, where $a,b,c > 0$. Here, $$\begin{aligned} \nabla^{2}f = \begin{bmatrix} 2a & b \\ b & 2c \end{bmatrix}\end{aligned}$$ so log-concavity amounts to $\det(\nabla^{2} f) = 4ac - b^{2} \leqslant 0$, i.e. $b^{2} \geqslant 4ac$. Now, raise each coefficient to the power $\alpha$. Then, $$\begin{aligned} \nabla^{2} f_{\alpha} = \begin{bmatrix} 2a^{\alpha} & b^{\alpha} \\ b^{\alpha} & 2c^{\alpha} \end{bmatrix}\end{aligned}$$ so log-concavity amounts to $\det(\nabla^{2} f_{\alpha}) = 4a^{\alpha}c^{\alpha} - b^{2\alpha} \leqslant 0$, i.e. $b^{2} \geqslant 4^{1/\alpha}ac$. Clearly, as one decreases $\alpha$ to 0, this inequality gets stronger, which certainly isn’t implied by log-concavity of $f$.\  \ The problem lies in the fact that when you differentiate a monomial that contains variables with multiplicities, you will obtain “factorial coefficients” which are not raised to the power $\alpha$. The operation must be modified appropriately to take this into account. We defer the discussion of the appropriate generalization of this operation for non-multiaffine polynomials to a future article. [^1]: Indeed, in [@AOV18], it is shown that $g_{\mu}$ satisfies a seemingly stronger property known as “complete log-concavity”, namely that $\partial_{v_{1}} \dotsb \partial_{v_{k}} g_M$ is log-concave (at ${{\mathds{1}}}$) for any sequence of *directional* derivatives $\partial_{v_{1}}\dotsb \partial_{v_{k}}$ with nonnegative directions $v_{1},\dots,v_{k} \in {{\mathbb{R}}}_{\geqslant0}^{n}$. We will prove in a future companion paper that complete log-concavity is equivalent to strong log-concavity. [^2]: In a companion paper, we will give an alternative proof based purely on elementary calculus and linear algebra.

--- abstract: 'This paper examines the problem of locating outlier columns in a large, otherwise low-rank, matrix. We propose a simple two-step adaptive sensing and inference approach and establish theoretical guarantees for its performance; our results show that accurate outlier identification is achievable using very few linear summaries of the original data matrix – as few as the squared rank of the low-rank component plus the number of outliers, times constant and logarithmic factors. We demonstrate the performance of our approach experimentally in two stylized applications, one motivated by robust collaborative filtering tasks, and the other by saliency map estimation tasks arising in computer vision and automated surveillance, and also investigate extensions to settings where the data are noisy, or possibly incomplete.' author: - 'Xingguo Li and Jarvis Haupt[^1]' bibliography: - 'salbib\_jdh.bib' title: | Identifying Outliers in Large Matrices via\ Randomized Adaptive Compressive Sampling --- Adaptive sensing, compressed sensing, robust PCA, sparse inference Introduction ============ In this paper we address a matrix *outlier identification* problem. Suppose $\Mb\in\RR^{n_1\times n_2}$ is a data matrix that admits a decomposition of the form $$\Mb = \Lb + \Cb,$$ where $\Lb$ is a low-rank matrix, and $\Cb$ is a matrix of outliers that is nonzero in only a fraction of its columns. We are ultimately interested in identifying the locations of the nonzero columns of $\Cb$, with a particular focus on settings where $\Mb$ may be very large. The question we address here is, can we accurately (and efficiently) identify the locations of the outliers from a small number of linear measurements of $\Mb$? Our investigation is motivated in part by robust collaborative filtering applications, in which the goal may be to identify the locations (or even quantify the number) of corrupted data points or outliers in a large data array. Such tasks may arise in a number of contemporary applications, for example, when identifying malicious responses in survey data or anomalous patterns in network traffic, to name a few. Depending on the nature of the outliers, conventional low-rank approximation approaches based on principal component analysis (PCA) [@Pearson:01; @Jolliffe:05] may be viable options for these tasks, but such approaches become increasingly computationally demanding as the data become very high-dimensional. Here, our aim is to leverage dimensionality reduction ideas along the lines of those utilized in randomized numerical linear algebra, (see, e.g., [@Halko:11; @Mahoney:11] and the references therein) and compressed sensing (see, e.g., [@Candes:06:Freq; @Donoho:06:CS; @Candes:06:UES]), in order to reduce the size of the data on which our approach operates. In so doing, we also reduce the computational burden of the inference approach relative to comparable methods that operate on “full data.” We are also motivated by an image processing task that arises in many computer vision and surveillance applications – that of identifying the “saliency map” [@Koch:87] of a given image, which (ideally) indicates the regions of the image that tend to attract the attention of a human viewer. Saliency map estimation is a well-studied area, and numerous methods have been proposed for obtaining saliency maps for a given image – see, for example, [@Tsotsos:95:Visual; @Itti:98; @Harel:06; @Liu:07; @Rao:10]. In contrast to these (and other) methods designed to identify saliency map of an image as a “post processing” step, our aim here is to estimate the saliency map *directly from compressive samples* – i.e., without first performing full image reconstruction as an intermediate step. We address this problem here using a linear subspace-based model of saliency, wherein we interpret an image as a collection of distinct (non-overlapping) patches, so that images may be (equivalently) represented as matrices whose columns are *vectorized* versions of the patches. Previous efforts have demonstrated that such local patches extracted from natural images may be well *approximated* as vectors in a union of low-dimensional linear subspaces (see, e.g., [@Yu:11:Models]). Here, our approach to the saliency map estimation problem is based on an assumption that salient regions in an image may be modeled as outliers from a single common low-dimensional subspace; the efficacy of similar saliency models for visual saliency has been established recently in [@Shen:12]. Our approach here may find utility in rapid threat detection in security and surveillance applications in high-dimensional imaging tasks where the goal is not to image the entire scene, but rather to merely identify regions in the image space corresponding to anomalous behavior. Successful identification of salient regions could comprise a first step in an active vision task, where subsequent imaging is restricted to the identified regions. Innovations and Our Approach ---------------------------- We propose a framework that employs dimensionality reduction techniques within the context of a two-step adaptive sampling and inference procedure, and our approach is based on a few key insights. First, we exploit the fact that the enabling geometry of our problem (to be formalized in the following section) is approximately preserved if we operate not on $\Mb$ directly, but instead on a “compressed” version $\bPhi\Mb$ that has potentially many fewer rows. Next, we use the fact that we can learn the (ostensibly, low-dimensional) linear subspace spanned by the columns of the low rank component of $\bPhi\Mb$ using a small, randomly selected subset of the columns of $\bPhi\Mb$. Our algorithmic approach for this step utilizes a recently proposed method called *Outlier Pursuit* (OP) [@Xu:12] that aims to separate a matrix $\Yb$ into its low-rank and column-sparse components using the convex optimization $$\label{eqn:OP} \argmin_{\bL,\bC} \ \ \|\bL\|_{*} + \lambda \|\bC\|_{1,2} \ \ \mbox{s.t.} \ \Yb = \bL + \bC$$ where $\|\bL\|_*$ denotes the nuclear norm of $\bL$ (the sum of its singular values), $\|\bC\|_{1,2}$ is the sum of the $\ell_2$ norms of the columns of $\bC$, and $\lambda>0$ is a regularization parameter. Finally, we leverage the fact that correct identification of the subspace spanned by the low-rank component of $\bPhi \Mb$ facilitates (simple) inference of the column outliers. We analyze two variants of this overall approach. The first (depicted as Algorithm \[alg:main\]) is based on the notion that, contingent on correct identification of the subspace spanned by the low-rank component of $\bPhi\Mb$, we may effectively transform the overall outlier identification problem into a compressed sensing problem, using a carefully-designed linear measurement operator whose net effect is to *(i)* reduce the overall $n_1\times n_2$ matrix to a $1\times n_2$ vector whose elements are (nominally) nonzero only at the locations of the outlier columns, and *(ii)* compressively sample the resulting vector. This reduction enables us to employ well-known theoretical results (e.g., [@Candes:08:RIP]) to facilitate our overall analysis. We call this approach Adaptive Compressive Outlier Sensing (ACOS). **Assume:** $\Mb\in \mathbb{R}^{n_1\times n_2}$ Column sampling Bernoulli parameter $\gamma\in[0,1]$, regularization parameter $\lambda>0$, Measurement matrices   $\bPhi\in\mathbb{R}^{m\times n_1}$, $\Ab\in\mathbb{R}^{p\times n_2}$, measurement vector $\bphi \in\mathbb{R}^{1\times m}$ **Initalize:** Column sampling matrix $\Sbb = \Ib_{:,\cS}$, where\ $\cS = \{i:S_i = 1\}$ with $\{S_i\}_{i\in[n_2]}$ i.i.d. Bernoulli$(\gamma)$ **** Collect Measurements: $\Yb_{(1)} = \bPhi \Mb \Sbb$\ Solve: $\{\widehat{\Lb}_{(1)},\widehat{\Cb}_{(1)}\} = \argmin_{\bL,\bC} \|\bL\|_{*} + \lambda \|\bC\|_{1,2}$ $\mbox{ s.t. } \ \Yb_{(1)} = \bL + \bC$\ Let: $\widehat{\cL}_{(1)}$ be the linear subspace spanned by col’s of $\widehat{\Lb}_{(1)}$ **** Compute: $\Pb_{\widehat{\mathcal{L}}_{(1)}}$, the orthogonal projector onto $\widehat{\cL}_{(1)}$ Set: $\Pb_{\widehat{\cL}_{(1)}^{\perp}} \triangleq \Ib - \Pb_{\widehat{\mathcal{L}}_{(1)}}$ Collect Measurements: $\yb_{(2)} =\bphi \ \Pb_{\widehat{\cL}_{(1)}^{\perp}} \bPhi \Mb \Ab^{T} $ Solve: $\widehat{\cbb} = \argmin_{\cbb} \ \ \|\cbb\|_{1} \ \ \mbox{s.t.} \ \yb_{(2)} = \cbb\Ab^{T}$ $\widehat{\cI}_{\Cb} = \{i: \widehat{\rm c}_{i} \neq 0\}$ The second approach, which we call Simplified ACOS (SACOS) and summarize as Algorithm \[alg:simple\], foregoes the additional dimensionality reduction in the second step and identifies as outliers those columns of $\bPhi \Mb$ having a nonzero component orthogonal to the subspace spanned by the low-rank component of $\bPhi\Mb$. The simplified approach has a (perhaps significantly) higher sample complexity than ACOS, but (as we will see in Section \[sec:exp\]) benefits from an ability to identify a larger number of outlier columns relative to the ACOS method. In effect, this provides a trade-off between detection performance and sample complexity for the two methods. **Assume:** $\Mb\in \mathbb{R}^{n_1\times n_2}$ Column sampling Bernoulli parameter $\gamma\in[0,1]$, regularization parameter $\lambda>0$, Measurement matrices   $\bPhi\in\mathbb{R}^{m\times n_1}$, $\Ab\in\mathbb{R}^{p\times n_2}$, measurement vector $\bphi \in\mathbb{R}^{1\times m}$ **Initalize:** Column sampling matrix $\Sbb = \Ib_{:,\cS}$, where\ $\cS = \{i:S_i = 1\}$ with $\{S_i\}_{i\in[n_2]}$ i.i.d. Bernoulli$(\gamma)$ **** Collect Measurements: $\Yb = \bPhi \Mb$ Form: $\Yb_{(1)} = \Yb \Sbb$ Solve: $\{\widehat{\Lb}_{(1)},\widehat{\Cb}_{(1)}\} = \argmin_{\bL,\bC} \|\bL\|_{*} + \lambda \|\bC\|_{1,2}$ $\mbox{ s.t. } \ \Yb_{(1)} = \bL + \bC$\ Let: $\widehat{\cL}_{(1)}$ be the linear subspace spanned by col’s of $\widehat{\Lb}_{(1)}$\ **** Compute: $\Pb_{\widehat{\mathcal{L}}_{(1)}}$, the orthogonal projector onto $\widehat{\cL}_{(1)}$ Set: $\Pb_{\widehat{\cL}_{(1)}^{\perp}} \triangleq \Ib - \Pb_{\widehat{\mathcal{L}}_{(1)}}$ Form: $\Yb_{(2)} =\Pb_{\widehat{\cL}_{(1)}^{\perp}} \Yb$ Form: $\widehat{\cbb}$ with $\widehat{c}_i = \|(\Yb_{(2)})_{:,i}\|_2$ for all $i\in[n_2]$ $\widehat{\cI}_{\Cb} = \{i: \widehat{\rm c}_{i} \neq 0\}$ Related Work ------------ Our effort here leverages results from Compressive Sensing (CS), where parsimony in the object or signal being acquired, in the form of *sparsity*, is exploited to devise efficient procedures for acquiring and reconstructing high-dimensional objects [@Candes:06:Freq; @Donoho:06:CS; @Candes:06:UES; @Candes:08:RIP]. The sequential and adaptive nature of our proposed approach is inspired by numerous recent works in the burgeoning area of adaptive sensing and adaptive CS (see, for example, [@Ji:08:BCS; @Bashan:08; @Haupt:09:CBS; @Haupt:11:DS; @Bashan:11; @Indyk:11; @Iwen:12; @Malloy:12; @Singh:12; @Castro:12; @Price:12; @Malloy:12:NOACS; @Davenport:12:CBS; @Singh:13; @Arias-Castro:13; @Wei:13; @Krishnamurthy:13; @Soni:14] as well as the summary article [@Haupt:11:Chapter] and the references therein). The column subsampling inherent in the first step of our approaches is also reminiscent of the data partitioning strategy of the *divide-and-conquer* parallelization approach of [@Mackey:11] (though our approach only utilizes one small partition of the data for the first inference step). Our efforts here utilize a generalization of the notion of sparsity, formalized in terms of a low-rank plus outlier matrix model. In this sense, our efforts here are related to earlier work in Robust PCA [@Chandrasekaran:11:Rank; @Candes:11:PCA] that seek to identify low-rank matrices in the presence of sparse impulsive outliers, and their extensions to settings where the outliers present as entire columns of an otherwise low-rank matrix [@Xu:12; @Chen:11; @McCoy:11; @Hardt:13; @Lerman:14]. In fact, the computational approach and theoretical analysis of the first step of our approach make direct utilization of the results of [@Xu:12]. We also note a related work [@Wright:13], which seeks to decompose matrices exhibiting some simple structure (e.g., low-rank plus sparse, etc.) into their constituent components from compressive observations. Our work differs from that approach in both the measurement model and scope. Namely, our linear measurements take the form of row and column operations on the matrix and our overall approach is adaptive in nature, in contrast to the non-adaptive “global” compressive measurements acquired in [@Wright:13], each of which is essentially a linear combination of all of the matrix entries. Further, the goal of [@Wright:13] was to exactly recover the constituent components, while our aim is only to identify the locations of the outliers. We discuss some further connections with [@Wright:13] in Section \[sec:disc\]. A component of our numerical evaluation here entails assessing the performance of our approach in a stylized image processing task of saliency map estimation. We note that several recent works have utilized techniques from the sparse representation literature in salient region identification, and in compressive imaging scenarios. A seminal effort in this direction was [@Olshausen:97:Sparse], which proposed a model for feature identification via the human visual cortex based on parsimonious (sparse) representations. More recently, [@Yan:10] applied techniques from *dictionary learning* [@Olshausen:97:Sparse; @Aharon:06:KSVD] and low-rank-plus-sparse matrix decomposition [@Chandrasekaran:11:Rank; @Candes:11:PCA] in a procedure to identify salient regions of an image from (uncompressed) measurements. Similar sparse representation techniques for salient feature identification were also examined in [@Li:09]. An adaptive compressive imaging procedure driven by a saliency “map” obtained via low-resolution discrete cosine transform (DCT) measurements was demonstrated in [@Yu:10]. Here, unlike in [@Li:09; @Yan:10], we consider salient feature identification based on compressive samples, and while our approach is similar in spirit to the problem examined in [@Yu:10], here we provide theoretical guarantees for the performance of our approach. Finally, we note several recent works [@Aksoylar:13; @Haupt:13] that propose methods for identifying salient elements in a data set using compressive samples. Outline ------- The remainder of the paper is organized as follows. In Section \[sec:main\] we formalize our problem, state relevant assumptions, and state our main theoretical results that establish performance guarantees for the Adaptive Compressive Outlier Sensing (ACOS) approach of Algorithm \[alg:main\] and the Simplified ACOS approach of Algorithm \[alg:simple\]. In Section \[sec:proof\] we outline the proofs of our main results. Section \[sec:exp\] contains the results of a comprehensive experimental evaluation of our approach on synthetic data, as well as in a stylized image processing application of saliency map estimation. In section \[extension\] we empirically investigate several extensions of our methods to noisy and “missing data” scenarios. In Section \[sec:disc\] we provide a brief discussion of the computational complexity of our approach, and discuss a few potential future directions. We relegate proofs and other auxiliary material to the appendix. A Note on Notation ------------------ We use bold-face upper-case letters ($\Mb, \Lb, \Cb, \bPhi, \bL, \bC, \bI$ etc.) to denote matrices, and use the [MATLAB]{}-inspired notation $\Ib_{:,\cS}$ to denote the sub matrix formed by extracting columns of $\Ib$ indexed by $i\in\cS$. We typically use bold-face lower-case letters ($\xb, \vb, \bc, \bphi$, etc.) to denote vectors, with an exception along the lines of the indexing notation above – i.e., that $\Cb_{:,i}$ denotes the $i$-th column of $\Cb$. Note that we employ both “block” and “math” type notation (e.g., $ \Lb, \bL$), where the latter are used to denote variables in the optimization tasks that arise throughout our exposition. Non-bold letters are used to denote scalar parameters or constants; the usage will be made explicit, or will be clear from context. The $\ell_1$ norm of a vector $\xb = [{\rm x}_1 \ {\rm x}_2 \ \dots \ {\rm x}_n]$ is $\|\xb\|_1 = \sum_{i} |{\rm x}_i|$ and the $\ell_2$ norm is $\|\xb\|_2 = \left(\sum_{i} |{\rm x}_i|^2\right)^{1/2}$. We denote the nuclear norm (the sum of singular values) of a matrix $\bL$ by $\|\bL\|_*$ and the $1,2$ norm (the sum of column $\ell_2$ norms) of a matrix $\bC$ by $\|\bC\|_{1,2}$. We denote the operator norm (the largest singular value) of a matrix $\bL$ by $\|\bL\|$. Superscript asterisks denote complex conjugate transpose. For positive integers $n$, we let $[n]$ denote the set of positive integers no greater than $n$; that is, $[n]=\{1,2,\dots,n\}$. Main Results {#sec:main} ============ Problem Statement ----------------- Our specific problem of interest here may be formalized as follows. We suppose $\Mb\in\RR^{n_1\times n_2}$ admits a decomposition of the form $\Mb = \Lb + \Cb$, where $\Lb$ is a low-rank matrix having rank at most $r$, and $\Cb$ is a matrix having some $k\leq n_2$ nonzero columns that we will interpret as “outliers” from $\Lb$, in the sense that they do not lie (entirely) within the span of the columns of $\Lb$. Formally, let $\cL$ denote the linear subspace of $\RR^{n_1}$ spanned by the columns of $\Lb$ (and having dimension at most $r$), denote its orthogonal complement in $\RR^{n_1}$ by $\cL^{\perp}$, and let $\Pb_{\cL}$ and $\Pb_{\cL^{\perp}}$ denote the orthogonal projection operators onto $\cL$ and $\cL^{\perp}$, respectively. We assume that the nonzero columns of $\Cb$ are indexed by a set $\cI_{\Cb}$ of cardinality $k$, and that $i \in \cI_{\Cb}$ if and only if $\|\Pb_{\cL^{\perp}}\Cb_{:,i}\|_2 > 0$. Aside from this assumption, the elements of the nonzero columns of $\Cb$ may be *arbitrary*. Notice that *without loss of generality*, we may assume that the columns of $\Lb$ are zero at the locations corresponding to the nonzero columns of $\Cb$ (since those columns of $\Lb$ can essentially be aggregated into the nonzero columns of $\Cb$, and the resulting column will still be an outlier according to our criteria above). We adopt that model here, and assume $\Lb$ has a total of $n_{\Lb}$ nonzero columns[^2], including all $k$ of the indices in $\cI_{\Cb}$ where $\Cb$ has a nonzero column, but also potentially others, to allow for the case where some $n_{\Lb} - k$ columns of $\Mb$ itself to be zero. Clearly $n_{\Lb} \leq n_2 - k$. Given this setup, our problem of interest here may be stated concisely – our aim is to identify the set $\cI_{\Cb}$ containing the locations of the outlier columns. Assumptions ----------- It is well-known in the matrix completion and robust PCA literature that separation of low-rank and sparse matrices from observations of their sum may not be a well-posed task – for example, matrices having only a single nonzero element are simultaneously low rank, sparse, column-sparse, row-sparse, etc. To overcome these types of identifiability issues, it is common to assume that the linear subspace spanned by the rows and/or columns of the low-rank matrix be “incoherent” with the canonical basis (see, e.g., [@Candes:09; @Chandrasekaran:11:Rank; @Candes:11:PCA; @Xu:12; @Chen:11]). In a similar vein, since our aim is to identify column outliers from an otherwise low-rank matrix we seek conditions that make the factors distinguishable so that any of the directions of the column space of $\Lb$ that we seek to identify are not defined by a single vector (stated another way, we would like the vectors whose columns comprise $\Lb$ to be “spread out” in the subspace spanned by columns of $\Lb$). To this end, we assume an incoherence condition on the row space of the low-rank component $\Lb$. We formalize this notion via the following definition from [@Xu:12]. Let $\Lb\in\RR^{n_1\times n_2}$ be a rank $r$ matrix with at most $n_{\Lb}\leq n_2$ nonzero columns, and compact singular value decomposition (SVD) $\Lb=\Ub\bSigma\Vb^*$, where $\Ub$ is $n_1\times r$, $\bSigma$ is $r\times r$, and $\Vb$ is $n_2\times r$. The matrix $\Lb$ is said to satisfy the *column incoherence property* with parameter $\mu_{\Lb}$ if $$\max_i \|\Vb^*\eb_i\|_2^2 \leq \mu_{\Lb} \frac{r}{n_{\Lb}},$$ where $\{\eb_i\}$ are basis vectors of the canonical basis for $\RR^{n_2}$. Note that $\mu_{\Lb} \in[1,n_{\Lb}/r]$; the lower limit is achieved when all elements of $\Vb^*$ have the same amplitude, and the upper limit when any one element of $\Vb^*$ is equal to $1$ (i.e., when the row space of $\Lb$ is aligned with the canonical basis). For our purposes, an undesirable case occurs when $\Vb^*$ is such that $\max_i \|\Vb^*\eb_i\|_2^2 =1$, since this implies that (at least) one of the directions in the span of the columns of $\Lb$ is described by only a single vector (and thus distinguishing that vector from a column outlier becomes ambiguous). With this, we may state our assumptions concisely, as follows: we assume that the components $\Lb$ and $\Cb$ of the matrix $\Mb=\Lb + \Cb$ satisfy the following *structural conditions*: - ${{\rm rank}}(\Lb) = r$, - $\Lb$ has $n_{\Lb}$ nonzero columns, - $\Lb$ satisfies the *column incoherence property* with parameter $\mu_{\Lb}$, and - $|\cI_{\Cb}|=k$, where $\cI_{\Cb} = \{i: \|\Pb_{\cL^{\perp}}\Cb_{:,i}\|_2 > 0, \Lb_{:,i} = \mathbf{0}\}$. Recovery Guarantees and Implications ------------------------------------ Our main results identify conditions under which the procedures outlined in Algorithm \[alg:main\] and Algorithm \[alg:simple\] succeed. Our particular focus is on the case where the measurement matrices are random, and satisfy the following property. An $m\times n$ matrix $\bPhi$ is said to satisfy the *distributional JL property* if for any fixed $\vb\in\RR^{n}$ and any $\epsilon\in(0,1)$, $$\label{eqn:distJL} {{\rm Pr}}\left(\ \left| \ \|\bPhi\vb\|_2^2 - \|\vb\|_2^2 \ \right| \geq \epsilon \|\vb\|_2^2 \ \right) \leq 2e^{-mf(\epsilon)},$$ where $f(\epsilon)>0$ is a constant depending only on $\epsilon$ that is specific to the distribution of $\bPhi$. Random matrices satisfying the distributional JL property are those that preserve the length of any fixed vector to within a multiplicative factor of $(1\pm \epsilon)$ with probability at least $1-2e^{-mf(\epsilon)}$. By a simple union bounding argument, such matrices can be shown to approximately preserve the lengths of a finite collection of vectors, all vectors in a linear subspace, all vectors in a union of subspaces, etc., provided the number of rows is sufficiently large. As noted in [@Gilbert:12], for many randomly constructed and appropriately normalized $\bPhi$, (e.g., such that entries of $\bPhi$ are i.i.d. zero-mean Gaussian, or are drawn as an ensemble from any subgaussian distribution), $f(\epsilon)$ is quadratic[^3] in $\epsilon$ as $\epsilon\rightarrow 0$. This general framework also allows us to directly utilize other specially constructed *fast* or *sparse* JL transforms [@Ailon:06; @Dasgupta:10]. With this, we are in position to formulate our first main result. We state it here as a theorem; its proof appears in Section \[sec:proof\]. \[thm:main\] Suppose $\Mb = \Lb + \Cb$, where the components $\Lb$ and $\Cb$ satisfy the structural conditions (**c1**)-(**c4**) with $$\label{eqn:kmin} k \leq \frac{1}{40(1+121\ r\mu_{\Lb})} \ n_2.$$ For any $\delta\in(0,1)$, if the column subsampling parameter $\gamma$ satisfies $$\label{eqn:gammamin} \gamma \geq \max\left\{\frac{1}{20}, \ \frac{200 \log(\frac{5}{\delta})}{n_{\Lb}}, \ \frac{24\log(\frac{10}{\delta})}{n_2},\ \frac{10 r \mu_{\Lb} \log(\frac{5r}{\delta})}{n_{\Lb}} \right\},$$ the measurement matrices are each drawn from any distribution satisfying with $$\label{eqn:mmin} m \geq \frac{5(r+1) + \log(k) + \log(2/\delta)}{f(1/4)}$$ and $$\label{eqn:pmin} p \geq \frac{11k + 2k\log(n_2/k) + \log(2/\delta)}{f(1/4)},$$ the elements of $\bphi$ are i.i.d. realizations of any continuous random variable, and for any upper bound $k_{\rm ub}$ of $k$ the regularization parameter is set to $\lambda = \frac{3}{7\sqrt{ k_{\rm ub} }}$, then the following hold simultaneously with probability at least $1-3\delta$: - the ACOS procedure in Algorithm \[alg:main\] correctly identifies the salient columns of $\bC$ (i.e., $\widehat{\cal I}_{\bC} = {\cal I}_{\bC}$), and - the total number of measurements collected is no greater than $\left(\frac{3}{2}\right)\gamma m n_2 + p$. It is interesting to compare this result with that of [@Xu:12], which established that the Outlier Pursuit procedure succeeds in recovering the true low-rank subspace and locations of the outlier columns provided $\Mb$ satisfy conditions analogous to (**c1**)-(**c4**) with $k \leq n_2/(1+(121/9)\ r\mu_{\Lb})$. The sufficient condition on the number of recoverable outliers that we identify for the ACOS procedure differs from the condition identified in that work by only constant factors. Further, the number of identifiable outliers could be as large as a fixed fraction of $n_2$ when both the rank $r$ and coherence parameter $\mu_{\Lb}$ are small. It is also interesting to note the sample complexity improvements that are achievable using the ACOS procedure. Namely, it follows directly from our analysis that for appropriate choice of the parameters $\gamma,m,$ and $p$ the ACOS algorithm correctly identifies the salient columns of $\Cb$ with high probability from relatively few observations, comprising only a fraction of the measurements required by other comparable (non-compressive) procedures [@Xu:12] that produce the same correct salient support estimate but operate directly on the full ($n_1 \times n_2$) matrix $\bM$. Specifically, our analysis shows that the ACOS approach succeeds with high probability with an effective sampling rate of $\frac{\# {\rm obs}}{n_1 n_2} = {\cal O}\left( \ \max\left\{\frac{(r+ \log k) (n_2/n_{\Lb})\mu_{\Lb} r\log r}{n_1n_2}, \frac{(r+ \log k)}{n_1} \right\} + \frac{k\log(n_2/k)}{n_1n_2}\ \right)$, which may be small when $r$ and $k$ are each small relative to the problem dimensions (and $n_{\Lb} \sim n_2$, so that $\Lb$ does not have a large number of zero columns outside of $\cI_{\Cb}$). Another point of comparison for our result comes from the related work [@Chen:11], which addresses a different (and in a sense, more difficult) task of identifying both the column space and the set of outlier columns of a matrix $\Mb = \Lb + \Cb$ from observations that take the form of samples of the elements of $\Mb$. There, to deal with the fact that observations take the form of point samples of the matrix (rather than more general linear measurements as here), the authors of [@Chen:11] assume that $\Lb$ also satisfy a row incoherence property in addition to a column incoherence property, and show that in this setting that the column space of $\Lb$ and set of nonzero columns of $\Cb$ may be recovered from only $\cO\left(n_2 r^2 \mu^2 \log(n_2)\right)$ observations via a convex optimization, where $\mu\in[1,n_1/r]$ is the row incoherence parameter. Normalizing this sample complexity by $n_1n_2$ facilitates comparison with our result above; we see that the sufficient conditions for the sample complexity of our approach are smaller than for the approach of [@Chen:11] by a factor of at least $1/r$, and, our approach does not require the row incoherence assumption. We provide some additional, experimental, comparisons between our ACOS method and the RMC method in Section \[sec:exp\]. We may also obtain performance guarantees for Algorithm \[alg:simple\] (in effect, using a simplified version of the analysis used to establish Theorem \[thm:main\]). This yields the following corollary. \[cor:main\] Suppose $\Mb = \Lb + \Cb$, where the components $\Lb$ and $\Cb$ satisfy the structural conditions (**c1**)-(**c4**) with $k$ as in . Let the measurement matrix $\bPhi$ be drawn from a distribution satisfying , and assume and hold. If for any upper bound $k_{\rm ub}$ of $k$ the regularization parameter is set to $\lambda = \frac{3}{7\sqrt{ k_{\rm ub} }}$, then the following hold simultaneously with probability at least $1-2\delta$: - the ACOS procedure in Algorithm \[alg:simple\] correctly identifies the salient columns of $\bC$ (i.e., $\widehat{\cal I}_{\bC} = {\cal I}_{\bC}$), and - the total number of measurements collected is no greater than $m n_2 $. We leave the proof (which is straightforward, using the lemmata in the following section) to the interested reader. Proof of Theorem \[thm:main\] {#sec:proof} ============================= First, we note that in both of the steps of Algorithm \[alg:main\] the prescribed observations are functions of $\Mb$ only through $\bPhi\Mb$; stated another way, $\Mb$ never appears in the algorithm in isolation from the measurement matrix $\bPhi$. Motivated by this, we introduce $$\label{eqn:Mtilde} \widetilde{\Mb} \triangleq \bPhi \Mb = \bPhi \Lb + \bPhi\Cb = \widetilde{\Lb} + \widetilde{\Cb},$$ to effectively subsume the action of $\bPhi$ into $\widetilde{\Mb}$. Now, our proof is a straightforward consequence of assembling three intermediate probabilistic results via a union bounding argument. The first intermediate result establishes that for $\Mb = \Lb + \Cb$ with components $\Lb$ and $\Cb$ satisfying the structural conditions **(${\bc}$1**)-**(${\bc}$4**), the components $\widetilde{\Lb}$ and $\widetilde{\Cb}$ of $\widetilde{\Mb}$ as defined in satisfy analogous structural conditions provided that $m$, the number of rows of $\bPhi$, be sufficiently large. We state this result here as a lemma; its proof appears in Appendix \[a:lem1\]. \[lem:Mtilde\] Suppose $\Mb = \Lb + \Cb$, where $\Lb$ and $\Cb$ satisfy the structural conditions (**c1**)-(**c4**). Fix any $\delta\in(0,1)$, suppose $\bPhi$ is an $m\times n_1$ matrix drawn from a distribution satisfying the distributional JL property with $m$ satisfying and let $\widetilde{\Mb}=\widetilde{\Lb} + \widetilde{\Cb}$ be as defined in . Then, the components $\widetilde{\Lb}$ and $\widetilde{\Cb}$ satisfy the following conditions simultaneously with probability at least $1-\delta$: - ${{\rm rank}}(\widetilde{\Lb}) = r$, - $\widetilde{\Lb}$ has $n_{\Lb}$ nonzero columns, - $\widetilde{\Lb}$ satisfies the column incoherence property with parameter $\mu_{\Lb}$, and - ${\cI}_{\widetilde{\Cb}} \triangleq \{i: \|\Pb_{{\widetilde{\cL}}^{\perp}} \widetilde{\Cb}_{:,i}\|_2 > 0, \widetilde{\Lb}_{:,i} = \mathbf{0}\} = {\cI}_{\Cb}$, where $\widetilde{\cL}$ is the linear subspace of $\RR^{m}$ spanned by the columns of $\widetilde{\Lb}$, and $\Pb_{{\widetilde{\cL}}^{\perp}}$ denotes the orthogonal projection onto the orthogonal complement of $\widetilde{\cL}$ in $\RR^{m}$. The second intermediate result guarantees two outcomes – first, that Step 1 of Algorithm \[alg:main\] succeeds in identifying the correct column space of $\widetilde{\cL}$ (i.e., that $\widehat{\cL}_{(1)} = \widetilde{\cL}$) with high probability provided the components $\widetilde{\Lb}$ and $\widetilde{\Cb}$ of $\widetilde{\Mb}$ as specified in satisfy the structural conditions **($\widetilde{\bc}$1**)-**($\widetilde{\bc}$4**) and the column sampling probability parameter $\gamma$ be sufficiently large, and second, that the number of columns of the randomly generated sampling matrix $\Sbb$ be close to $\gamma n_2$. We also provide this result as a lemma; its proof appears in Appendix \[a:lem2\]. \[lem:Step1\] Let $\widetilde{\Mb}=\widetilde{\Lb} + \widetilde{\Cb}$ be an $m\times n_2$ matrix, where the components $\widetilde{\Lb}$ and $\widetilde{\Cb}$ satisfy the conditions **($\widetilde{\bc}$1**)-**($\widetilde{\bc}$4**) with $k$ satisfying . Fix $\delta\in(0,1)$ and suppose the column sampling parameter $\gamma$ satisfies . When $\lambda = \frac{3}{7\sqrt{ k_{\rm ub} }}$ for any $k_{\rm ub}\geq |\cI_{\widetilde{\Cb}}|$, the following hold simultaneously with probability at least $1-\delta$: $\Sbb$ has $|\cS|\leq (3/2) \gamma n_2$ columns, and the subspace $\widehat{\cL}_{(1)}$ resulting from Step 1 of Algorithm \[alg:main\] satisfies $\widehat{\cL}_{(1)} = \widetilde{\cL}$. Our third intermediate result shows that the support set of the vector $\widehat{\cbb}$ produced in Step 2 of Algorithm \[alg:main\] is the same as the set of salient columns of $\widetilde{\Cb}$, provided that $\widehat{\cL}_{(1)} = \widetilde{\cL}$ and that $p$, the number of rows of $\Ab$, is sufficiently large. We state this result here as a lemma; its proof appears in Appendix \[a:lem3\] \[lem:Step2\] $\widetilde{\Mb}=\widetilde{\Lb} + \widetilde{\Cb}$ be an $m\times n_2$ matrix, where the components $\widetilde{\Lb}$ and $\widetilde{\Cb}$ satisfy the conditions **($\widetilde{\bc}$1**)-**($\widetilde{\bc}$4**) for any $k\leq n_2$, and suppose $\widehat{\cL}_{(1)} = \widetilde{\cL}$, the subspace spanned by the columns of $\widetilde{\Lb}$. Let $\bPhi\Mb = \widetilde{\Mb}$ in Step 2 of Algorithm \[alg:main\]. Fix $\delta\in(0,1)$, suppose $\Ab$ is a $p\times n_2$ matrix drawn from a distribution satisfying the distributional JL property with $p$ satisfying , and suppose the elements of $\bphi$ are i.i.d. realizations of any continuous random variable. Then with probability at least $1-\delta$ the support $\cI_{\widehat{\cbb}}\triangleq \{i: \widehat{\rm c}_i \neq 0\}$ of the vector $\widehat{\cbb}$ produced by Step 2 of Algorithm \[alg:main\] satisfies $\cI_{\widehat{\cbb}} = \cI_{\widetilde{\Cb}}$. Our overall result follows from assembling these intermediate results via union bound. In the event that the conclusion of Lemma \[lem:Mtilde\] holds, then so do the requisite conditions of Lemma \[lem:Step1\]. Thus, with probability at least $1-2\delta$ the conclusions of Lemmata \[lem:Mtilde\] and \[lem:Step1\] both hold. This implies that the requisite conditions of Lemma \[lem:Step2\] hold also with probability at least $1-2\delta$, and so it follows that the conclusions of all three Lemmata hold with probability at least $1-3\delta$. Experimental Evaluation {#sec:exp} ======================= In this section we provide a comprehensive experimental evaluation of the performance of our approaches for both synthetically generated and real data, the latter motivated by a stylized application of saliency map estimation in an image processing task. We compare our methods with the Outlier Pursuit (OP) approach of [@Xu:12] and the Robust Matrix Completion (RMC) approach of [@Chen:11], each of which employs a convex optimization to identify both the subspace in which the columns of the low rank matrix lie, and the locations of the nonzero columns in the outlier matrix. We implement the RMC method using an accelerated approximate alternating direction method of multipliers (ADMM) method inspired by [@Goldstein:12] (as well as [@Xu:12; @Beck:09]). We implement the OP methods (as well as the intermediate execution of the OP-like optimization in Step 1 of our approach) using the procedure in [@Chen:11]. We implement the $\ell_1$-regularized estimation in Step 2 of our procedure by casting it as a LASSO problem and using an accelerated proximal gradient method [@Beck:09]. \ \ \ Synthetic Data {#sec:syn} -------------- We experiment on synthetically generated $n_1 \times n_2$ matrices $\Mb$, with $n_1 = 100$ and $n_2=1000$, formed as follows. For a specified rank $r$ and number of outliers $k$, we let the number of nonzero columns of $\Lb$ be $n_{\Lb} = n_2-k$, generate two random matrices $\Ub \in \RR^{n_1 \times r}$ and $\Vb \in \RR^{n_{\Lb} \times r}$ with i.i.d. $\cN \left( 0,1 \right)$ entries, and we take $\Lb = [\Ub\Vb^{T} \ {\mathbf 0}_{n_1\times k}]$. We generate the outlier matrix $\Cb$ as $\Cb=[{\mathbf 0}_{n_1\times n_{\Lb}} \ \Wb]$ where $\Wb\in\RR^{n_1\times k}$ has i.i.d. $\cN(0,r)$ entries (which are also independent of entries of $\Ub$ and $\Vb$). Then, we set $\Mb = \Lb + \Cb$. Notice that the outlier vector elements have been scaled, so that all columns of $\Mb$ have the same squared $\ell_2$ norm, in expectation. In all experiments we generate $\bphi$, $\bPhi$, and $\Ab$ with i.i.d. zero-mean Gaussian entries. Our first experiment investigates the “phase transition” behavior of our ACOS approach; our experimental setting is as follows. First, we set the average sampling rate by fixing the column downsampling fraction $\gamma=0.2$, and choosing a row sampling parameter $m\in\{0.1n_1, 0.2n_1, 0.3n_1\}$ and column sampling parameter $p\in\{0.1n_2, 0.2n_2, 0.3n_2\}$. Then, for each $(r,k)$ pair with $r\in\{1, 2, 3, \dots,40\}$ and $k\in\{2, 4, 6, \dots,100\}$ we generate a synthetic matrix $\Mb$ as above, and for each of $3$ different values of the regularization parameter $\lambda \in \{0.3, 0.4, 0.5\}$ we perform $100$ trials of Algorithm \[alg:main\] recording in each whether the recovery approach succeeded[^4] in identifying the locations of the true outliers for that value of $\lambda$, and associate to each $(r,k)$ pair the (empirical) average success rate. Then, at each $(r,k)$ point examined we identify the point-wise maximum of the average success rates for the $3$ different values of $\lambda$; in this way, we assess whether recovery for that $(r,k)$ is achievable by our method for the specified sampling regime for *some* choice of regularization parameters. The results in Figure \[fig:sim1\] depict the outcome of this experiment for the $9$ different sampling regimes examined. For easy comparison, we provide the average sampling rate as fraction of observations obtained (relative to the full matrix dimension) in the caption in each figure. The results of this experiment provide an interesting, and somewhat intuitive, illustration of the efficacy of our approach. Namely, we see that increasing the parameter $m$ of the matrix $\bPhi$ in Step 1 of our algorithm while keeping the other sampling parameters fixed (i.e., moving from top to bottom in any one column) facilitates accurate recovery for increasing ranks $r$ of the matrix $\Lb$. Similarly, increasing the parameter $p$ of the matrix $\Ab$ in Step 2 of our algorithm while keeping the other sampling parameters fixed (i.e., moving from left to right in any one row) facilitates accurate recovery for an increasing number $k$ of outlier columns. Overall, our approach can successfully recover the locations of the outliers for non-trivial regimes of $r$ and $k$ using very few measurements – see, for instance, panel $(i)$, where $\sim30$ outlier columns can be accurately identified in the presence of a rank $\sim30$ background using an effective sampling rate of only $\sim6.3\%$. We adopt a similar methodology to evaluate the Simplified ACOS approach, except that we set $k\in \{20,40,60,\ldots,980\}$ (and the parameter $p$ is no longer applicable, since there is no additional compression in Step 2 for this method). The results are shown in Figure \[fig:sim\_simple\]. As noted above the SACOS approach has a higher average sampling rate than ACOS for the same $m$, but the results show this facilitates recovery of much larger numbers $k$ of outlier columns (notice the difference in the vertical scales in Figures \[fig:sim1\] and \[fig:sim\_simple\]). Overall, we may view ACOS and SACOS as complementary; when the number $k$ of outlier columns is relatively small and low sampling ratio $\frac{\# {\rm obs}}{n_1 n_2}$ is a primary focus, ACOS may be preferred, while if the number $k$ of outlier columns is relatively large, SACOS is more favorable (at the cost of increased sample complexity). We also compute phase transition curves for RMC using a similar methodology to that described above. The results are provided in Figure \[fig:rmc\] . We observe[^5] that RMC approach is viable for identifying the outliers from subsampled data provided the sampling rate exceeds about $10\%$, but even then only for small values of the rank $r$. As alluded in the discussion in previous sections, the relative difference in performance is likely due in large part to the difference in the observation models between the two approaches – the RMC approach is inherently operating in the presence of “missing data” (a difficult scenario!) while our approach permits us to observe linear combinations of any row or column of the entire matrix (i.e., we are allowed to “see” each entry of the matrix, albeit not necessarily individually, throughout our approach). ---------- ------ ------ ----- ----- ------- ------- ------- ------ ------ ------ Method GBVS OP RMC RMC SACOS SACOS SACOS ACOS ACOS ACOS Sampling 100% 100% 20% 5% 20% 5% 3% 4.5% 2.5% 1.5% ---------- ------ ------ ----- ----- ------- ------- ------- ------ ------ ------ ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- Method GBVS OP RMC RMC SACOS SACOS SACOS ACOS ACOS ACOS Sampling 100% 100% 20% 5% 20% 5% 3% 4.5% 2.5% 1.5% Step 1 0.9926 2.9441 2.6324 2.7254 0.0538 0.0107 0.0074 0.0533 0.0214 0.0105 (0.2742) (0.3854) (0.3237) (0.3660) (0.0121) (0.0034) (0.0017) (0.0118) (0.0056) (0.0025) Step 2 – – – – 0.0015 0.0011 0.0009 0.2010 0.2014 0.2065 – – – – (0.0003) (0.0003) (0.0003) (0.0674) (0.0692) (0.0689) ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- \[table:timing\] Real Data --------- We also evaluate the performance of our proposed methods on real data in the context of a stylized image processing task that arises in many computer vision and automated surveillance – that of identifying the “saliency map” of an image. For this, we use images from the MSRA Salient Object Database [@Liu:07], available online at <http://research.microsoft.com/en-us/um/people/jiansun/SalientObject/salient_object.htm>. As discussed above, our approach here is based on representing each test image as a collection of (vectorized) non-overlapping image patches. We transform each (color) test image to gray scale, decompose it into non-overlapping $10 \times 10$-pixel patches, vectorize each patch into a $100 \times 1$ column vector, and assemble the column vectors into a matrix. Most of the images in the database are of the size $300 \times 400$ (or $400 \times 300$), which here yields matrices of size $100\times 1200$, corresponding to $1200$ patches. Notice that we only used gray scale values of image as the input feature rather than any high-level images feature – this facilitates the use of our approach, which is based on collecting linear measurements of the data (e.g., using a spatial light modulator, or an architecture like the *single pixel camera* [@Duarte:08:SPC]). Here, our experimental approach is (somewhat necessarily) a bit more heuristic than for the synthetic data experiments above, due in large part to the fact that the data here may not adhere exactly to the low-rank plus outlier model. To compensate for this, we augment Step 1 of Algorithm \[alg:main\] and Algorithm \[alg:simple\] with an additional “rank reduction” step, where we further reduce the dimension of the subspace spanned by the columns of the learned $\widehat{\Lb}_{(1)}$ by truncating its SVDs to retain the smallest number of leading singular values whose sum is at least $0.95 \times \|\widehat{\Lb}_{(1)}\|_*$. Further, we generalize Step 2 of each procedure by declaring an image patch to be salient when its (residual) column norm is sufficiently large, rather than strictly nonzero. We used visual heuristics to determine the “best” outputs for Step 2 of each method, selecting LASSO parameters (for ACOS) or thresholds (for SACOS) in order to qualitatively trade off false positives with misses. We implement our ACOS and SACOS methods using three different sampling regimes for each, with the fixed column downsampling parameter $\gamma=0.2$ throughout. For ACOS, we examine settings where $m=0.2n_1$, $0.1n_1$ and $0.05n_1$ with $p=0.5n_2$, which result in average sampling rates of 4.5%, 2.5% and 1.5%, respectively. For SACOS, we examine settings where $m=0.2n_1$, $0.05n_1$ and $0.03n_1$, resulting in average sampling rates of 20%, 5% and 3%, respectively. As before, we generate the $\bPhi$ and $\Ab$ matrices to have i.i.d. zero-mean Gaussian entries. We compare our approaches with two “benchmarks” – the Graph-based visual saliency (GBVS) method from the computer vision literature [@Harel:06] and the OP approach (both of which use the full data) – as well as with the RMC approach at sampling rates of $20\%$ and $5\%$. The results of this experiment are provided in Figure \[fig:real\]. We note first that the OP approach performs fairly well at identifying the visually salient regions in the image, essentially identifying the same salient regions as the GBVS procedure and providing evidence to validate the use of the low-rank plus outlier model for visual saliency (see also [@Shen:12]). Next, comparing the results of the individual procedures, we see that the OP approach appears to uniformly give the best detection results, which is reasonable since it is using the full data as input. The RMC approach performs well at the 20% sampling rate, but its performance appears to degrade at the 5% sampling rate. The SACOS approach, on the other hand, still produces reasonably accurate results using only 3% sampling. Moreover, ACOS provides acceptable results even with 1-2% sampling rate. We also compare implementation times of the algorithms on this saliency map estimation task. Table \[table:timing\] provides the average execution times (and standard deviations) for each approach, evaluated over 1000 images in the MSRA database[^6]. Here, we only execute each procedure for one choice of regularization parameter, and we also include the additional “rank reduction” step discussed above for the ACOS and SACOS methods. Overall, we see the ACOS approach is up to $4\times$ faster than the GBVS method and $15\times$ faster than the OP and RMC methods, while the SACOS approach could result overall in relative speedups of $100\times$ over GBVS and $300\times$ over the OP and RMC methods. Overall, our results suggest a significant improvement obtained via ACOS and SACOS for both detection consistency and timing, which may have a promising impact in a variety of salient signal detection tasks. \ \ \ (a) 2.1% (b) 4.2% (c) 6.3% Extensions {#extension} ========== Noisy Observations ------------------ We demonstrate the outlier detection performance of our approaches under the scenario when $\Mb$ is contaminated by unknown random noise or modeling error. Formally, we consider the setting where $\Lb$ and $\Cb$ are as above, but $$\begin{aligned} \Mb =\Lb + \Cb + \Nb,\end{aligned}$$ where $\Nb$ has i.i.d. $\cN(0,\sigma^2)$ entries. \ \ \ (a) 10% (b) 20% (c) 30% We first investigate the performance of the ACOS method, following a similar experimental methodology as in Section \[sec:exp\] to generate $\Lb$ and $\Cb$, except that now we renormalize each column of $(\Lb + \Cb)$ to have unit Euclidean norm (essentially to standardize the noise levels). We consider three different noise levels ($\sigma = 0.001$, 0.0005 and 0.0001), three pairs of the row sampling parameter $m$ and the column sampling parameter $p$ ($m=0.1n_1,~p=0.1n_2$;  $m=0.2n_1,~p=0.2n_2$; and  $m=0.3n_1,~p=0.3n_2$) and for each we fix the column downsampling fraction to be $\gamma=0.2$; the corresponding sampling ratios are 2.1%, 4.2% and 6.3%, respectively. We again perform 100 trials of Algorithm \[alg:main\] and record the success frequency for each. The results are given in Figure \[fig:noisy\]. It can be observed from the results that increasing $m$ and $p$ promote accurate estimation of outlier column indices for increasing rank $r$ and numbers $k$ of outlier columns, which is exactly what we have seen in Figure \[fig:sim1\] for the noiseless case. However, the presence of noise degrades the estimation performance, albeit gracefully. This is reasonable, since in Step 2 of Algorithm \[alg:main\], the measurements $\yb_2$ might be perturbed more seriously as the energy of noise increases, which results in more difficult recovery of true supports of $\cbb$. Under this scenario, we will require larger $p$ to enable better recovery of the underlying true supports. We also evaluate the SACOS procedure in noisy settings for three choices of $m$ ($m=0.1n_1$, $0.2n_1$ and $0.3n_2$) and fixed column downsampling fraction $\gamma=0.2$. Here, we again normalize columns of $(\Lb + \Cb)$, but consider three higher noise levels, corresponding to $\sigma=0.03$, $0.02$ and $0.01$. The results are presented in Figure \[fig:noise\_sacos\]. Here, we again observe a graceful performance degradation with noise. Notice, however, that higher level of variances of noise can be tolerated for SACOS compared with ACOS, which is an artifact of the difference between the second (inference) steps of the two procedures. Missing Data ------------ We also describe and demonstrate an extension of our SACOS method that is amenable to scenarios characterized by missing data. Suppose that there exists some underlying matrix $\Mb$ that admits a decomposition of the form $\Mb = \Lb + \Cb$ with $\Lb$ and $\Cb$ as above, but we are only able to observe $\Mb$ at a subset of its locations. Formally, we denote by $\bOmega \subseteq [n_1]\times[n_2]$ the set of indices corresponding to the available elements of $\Mb$, and let $\Pb_{\bOmega}(\cdot)$ be the operator that masks its argument at locations not in $\bOmega$. Thus, rather than operate on $\Mb$ itself, we consider procedures that operate on the sampled data $\Pb_{\bOmega}(\Mb)$. In this setting, we can modify our SACOS approach so that the observations obtained in Step 1 are of the form $\Yb_{(1)} = \bPhi \Pb_{\bOmega} (\Mb)\Sbb$, where (as before) $\Sbb$ is a column selection matrix but $\bPhi$ is now a *row* subsampling matrix (i.e., it is comprised of a subset of rows of the $n_1\times n_1$ identity matrix) containing some $m$ rows. The key insight here is that the composite operation of sampling elements of $\Mb$ followed by row subsampling can be expressed in terms of a related operation of subsampling elements of a row-subsampled version of $\Mb$. Specifically, we have that $\bPhi \Pb_{\bOmega} (\Mb) = \Pb_{\bOmega_{\bPhi}}(\bPhi \Mb)$, where $\Pb_{\bOmega_{\bPhi}}(\cdot)$ masks the same elements as $\Pb_{\bOmega}(\cdot)$ in the rows selected by $\bPhi$. Now, given $\Yb_{(1)}$, we solve a variant of RMC [@Chen:11] $$\begin{aligned} \{\widehat{\Lb}_{(1)},\widehat{\Cb}_{(1)}\} = \argmin_{\bL,\bC} &\|\bL\|_{*} + \lambda \|\bC\|_{1,2}\\ \mbox{ s.t. } & \Yb_{(1)} = \Pb_{\bOmega_{\bPhi}}( \bL + \bC)\end{aligned}$$ in an initial step, identifying (as before) an estimate $\widehat{\Lb}_{(1)}$ whose column span is an estimate of the subspace spanned by the low-rank component of $\bPhi\Mb$. Then (in a second step) we perform the “missing data” analog of the orthogonal projection operation on every column $j \in [n_2]$ of $\bPhi \Pb_{\bOmega} (\Mb)$, as follows. For each $j\in[n_2]$, we let $\cI_j\in[m]$ denote the locations at which observations of column $j$ of $\bPhi \Pb_{\bOmega} (\Mb)$ are available, and let $(\bPhi\Pb_{\bOmega} (\Mb))_{\cI_j,j}$ be the sub vector of $(\bPhi \Pb_{\bOmega} (\Mb))_{:,j}$ containing only the elements indexed by $\cI_j$. Similarly, let $(\widehat{\Lb}_{(1)})_{\cI_j,:}$ be the row submatrix of $\widehat{\Lb}_{(1)}$ formed by retaining rows indexed by $\cI_j$. Now, let $\Pb_{\widehat{\cL}_{(1)_{j}}}$ denote the orthogonal projection onto the subspace spanned by columns of $(\widehat{\Lb}_{(1)})_{\cI_j,:}$ and compute the residual energy of the $j$-th column as $\|(\Ib-\Pb_{\widehat{\cL}_{(1)_{j}}}) (\bPhi \Pb_{\bOmega} (\Mb))_{\cI_j,j}\|_2$. Overall, the orthogonal projection for the $j$-th column of $\bPhi \Pb_{\bOmega} (\Mb)$ is only computed over the nonzero entries of that column, an approach motivated by a recent effort examining subsampling methods in the context of matrix completion [@Krishnamurthy:14]. \ \ \ We evaluate this approach empirically using the same data generation methods as above, and using an independent *Bernoulli* model to describe the subsampling operation $\Pb_{\bOmega}(\cdot)$ (so that each $(i,j)\in\bOmega$ independently with probability $p_{\bOmega}$). We consider noise-free settings, fix the column subsampling parameter $\gamma = 0.2$, and examine three different row-sampling scenarios ($m=0.1n_1$, $0.2n_1$ and $0.3n_1$) in each choosing subsets of $m$ rows uniformly at random from the collection of all $n_1 \choose m$ sets of cardinality $m$. The results are in Figure \[fig:missing\]. Again, increasing $m$ and $p$ permits accurate estimation of outlier column indices for increasing rank $r$ and numbers $k$ of outlier columns. Further, we do observe the performance degradation as the number of missing entries of $\Mb$ increases. Discussion and Future Directions {#sec:disc} ================================ It is illustrative here to note a key difference between our approach and more conventional compressive sensing (CS) tasks. Namely, the goal of the original CS works [@Candes:06:Freq; @Donoho:06:CS; @Candes:06:UES] and numerous follow-on efforts was to exactly recover or reconstruct a signal from compressive measurements, whereas the nature of our task here is somewhat simpler, amounting to a kind of multidimensional “support recovery” task (albeit in the presence of a low-rank “background”). Exactly recovering the low-rank and column-sparse components would be sufficient for the outlier identification task we consider here, but as our analysis shows it is not strictly necessary. This is the insight that we exploit when operating on the “compressed” data $\bPhi\Mb$ instead of the original data matrix $\Mb$. Ultimately, this allows us to successfully identify the locations of the outliers *without first estimating the original (full size) low-rank matrix or the outliers themselves*. For some regimes of $\mu_{\Lb}$, $r$ and $k$, we accomplish the outlier identification task using as few as $\cO\left(( r + \log k)(\mu_{\Lb} r\log r) + k\log(n_2/k) \right)$ observations. Along related lines, it is reasonable to conjecture that any procedure would require at least $r^2 + k$ measurements in order to identify $k$ outliers from an $r$-dimensional linear subspace. Indeed, a necessary condition for the existence of outliers of a rank-$r$ subspace, as we have defined them, is that the number of rows of $\Mb$ be at least $r+1$. Absent any additional structural conditions on the outliers and the subspace spanned by columns of the low-rank matrix, one would need to identify a collection of $r$ vectors that span the $r$-dimensional subspace containing the column vectors of the low-rank component (requiring specification of some $\cO(r^2)$ parameters) as well as the locations of the $k$ outliers (which would entail specifying another $k$ parameters). In this sense, our approach may be operating near the sample complexity limit for this problem, at least for some regimes of $\mu_{\Lb}$, $r$ and $k$. It would be interesting to see whether the dimensionality reduction insight that we exploit in our approach could be leveraged in the context of the Compressive Principal Component Pursuit (Compressive PCP) of [@Wright:13] in order to yield a procedure with comparable performance as ours, but which acquires only *non-adaptive* linear measurements of $\Mb$. Direct implementation of that approach in our experimental setting was somewhat computationally prohibitive (e.g., simulations at a $10\%$ sampling rate would require generation and storage of random matrices having $10^9$ elements). Alternatively, it is interesting to consider implementing the Compressive PCP method not on the full data $\Mb$, but on the a priori compressed data $\bPhi\Mb$. Our Lemma \[lem:Mtilde\] establishes that the row compression step preserves rank and column incoherence properties, so it is plausible that the Compressive PCP approach may succeed in recovering the components of the compressed matrix, which would suffice for the outlier identification task. We defer this investigation along these lines to a future effort. Method Complexity -------- ---------------------------------------------------------------------------------------------------------------------- OP $\cO \left( {\rm IT} \cdot \left[ n_1 n_2\cdot \min\{n_1,n_2\} \right] \right)$ RMC $\cO \left( {\rm IT} \cdot \left[ n_1 n_2\cdot \min\{n_1,n_2\} \right] \right)$ ACOS $\cO \left( {\rm IT}_1 \left[ m(\gamma n_2) \min\{m,\gamma n_2 \}\right] + {\rm IT}_2\left[ p n_2 \right] \right)$ SACOS $\cO \left( {\rm IT}_1 \left[ m(\gamma n_2) \min\{m,\gamma n_2 \}\right] + m^2 n_2\right)$ : Computational complexities of outlier identification methods. The stated results assume use of an accelerated first order method for all solvers (see text for additional details). \[table:complex\] We also comment briefly on the computational complexities of the methods we examined. We consider first the OP and RMC approaches, and assume that the solvers for each utilize an iterative accelerated first-order method (like those mentioned in the first part of Section \[sec:exp\]). In this case, the computational complexity will be dominated by SVD steps in each iteration. Now, for an $n_1\times n_2$ matrix the computational complexity of the SVD is $\cO(n_1 n_2 \cdot \min\{n_1,n_2\})$; with this, and assuming some ${\rm IT}$ iterations are used, we have that the complexities of both OP and RMC scale as $\cO \left( {\rm IT} \cdot \left[ n_1 n_2\cdot \min\{n_1,n_2\} \right] \right)$. By a similar analysis, we can conclude that the complexity of Step 1 of the ACOS and SACOS methods scales like $\cO \left( {\rm IT}_1 \cdot \left[ m (\gamma n_2)\cdot \min\{m,\gamma n_2\} \right] \right)$, where ${\rm IT}_1$ denotes the number of iterations for the solver in Step 1. If we further assume an iterative accelerated first-order method for the LASSO in Step 2 of the ACOS approach, and that ${\rm IT}_2$ iterations are used, then the second step of the ACOS approach would have overall computational complexity $\cO \left( {\rm IT}_2 \cdot \left[ p n_2] \right) \right)$. Along similar lines, Step 2 of SACOS would entail ${\cal O}(m^2 n_2 + mn_2) = {\cal O}(m^2 n_2)$ operations to compute the orthogonal projections and their $\ell_2$ norms. We summarize the overall complexity results in Table \[table:complex\]. Since we will typically have $\gamma$ small, $m \ll n_1$, and $p\ll n_2$ in our approaches, the computational complexity of our approaches can be much less than methods that operate on the full data or require intermediate SVD’s of matrices of the same size as $\Mb$. Note that we have not included here the complexity of acquiring or forming the observations in any of the methods. For the ACOS method, this would comprise up to an additional $\cO \left(mn_1(\gamma n_2)\right)$ operations for Step 1 and $\cO(m^2 + mn_1 + n_1n_2 + n_2p) = \cO(mn_1 + n_1n_2 + n_2p)$ operations for Step 2, where the complexity for the second step is achieved by iteratively multiplying together the left-most two factors in the overall product, and using the fact that $m\leq n_1$. Similarly, observations obtained via the SACOS approach could require up $\cO(m n_1 n_2)$ operations. On the other hand, depending on the implementation platform, forming the observations themselves could also have a negligible computational effect e.g., in our imaging example when linear observations are formed “implicitly” using a spatial light modulator or single pixel camera [@Duarte:08:SPC]. Finally, we note that further reductions in the overall complexity of our approach may be achieved using fast or sparse JL embeddings along the lines of [@Ailon:06; @Dasgupta:10]. Finally, it is worth noting[^7] that the performance in our our visual saliency application could likely be improved using an additional assumption that the salient regions be spatially clustered. This could be implemented here using *group sparse* regularization (e.g. [@Yuan:06]) in Step 2 of ACOS, or (more simply) by directly identifying groups of nonzero elements in Step 2 of SACOS. We defer investigations along these lines to a future effort. Proof of Lemma \[lem:Mtilde\] {#a:lem1} ----------------------------- We proceed using the formalism of *stable embeddings* that has emerged from the dimensionality reduction and compressive sensing literature (see, e.g., [@Davenport:10]). For $\epsilon\in[0,1]$ and $\cU,\cV\subseteq \RR^n$, we say $\bPhi$ is an $\epsilon$-*stable embedding* of $(\cU,\cV)$ if $$(1-\epsilon) \|\ub-\vb\|_2^2 \leq \|\bPhi\ub-\bPhi\vb\|_2^2 \leq (1+\epsilon) \|\ub-\vb\|_2^2$$ for all $\ub\in\cU$ and $\vb\in\cV$. Our proof approach is comprised of two parts. First, we show that each of the four claims in the lemma follow when $\bPhi$ is an $\epsilon$-stable embedding of $$\label{eqn:epsdes} \left(\cL, \cup_{i\in\cI_{\Cb}} \{\bC_{:,i}\} \cup \{{\mathbf 0}\}\right)$$for any choice of $\epsilon<1/2$. Second, we show that for any $\delta\in(0,1)$, generating $\bPhi$ as a random matrix as specified in the lemma ensures it will be a $\sqrt{2}/4$-stable embedding of with probability at least $1-\delta$. The choice of $\sqrt{2}/4$ in the last step is somewhat arbitrary – we choose this fixed value for concreteness here, but note that the structural conclusions of the lemma follow using any choice of $\epsilon<1/2$ (albeit with slightly different conditions on $m$). ### Part 1 Throughout this portion of the proof we assume that $\bPhi$ is an $\epsilon$-stable embedding of for some $\epsilon<1/2$, and establish each of the four claims in turn. First, to establish that ${{\rm rank}}(\bPhi \Lb)=r={{\rm rank}}(\Lb)$, we utilize an intermediate result of [@Gilbert:12], stated here as a lemma (without proof) and formulated in the language of stable embeddings. \[lem:Sketched\] Let $\Lb$ be an $n_1\times n_2$ matrix of rank $r$, and let $\cL$ denote the column space of $\Lb$, which is an $r$-dimensional linear subspace of $\RR^{n_1}$. If for some $\epsilon\in(0,1)$, $\bPhi$ is an $\epsilon$-stable embedding of $(\cL, \{{\mathbf 0}\})$ then ${{\rm rank}}(\bPhi \Lb)=r={{\rm rank}}(\Lb)$. Here, since $\bPhi$ being an $\epsilon$-stable embedding of implies it is also an $\epsilon$-stable embedding of $(\cL,\{{\mathbf 0}\})$, the first claim (of Lemma \[lem:Mtilde\]) follows from Lemma \[lem:Sketched\]. Next we show that $\bPhi\Lb$ has $n_{\Lb}$ nonzero columns. Since $\bPhi$ is a stable embedding of $(\cL,\{{\mathbf 0}\})$, it follows that for each of the $n_{\Lb}$ nonzero columns $\Lb_{:,i}$ of $\Lb$ we have $\|\bPhi\Lb_{:,i}\|_2^2 > (1-\epsilon) \|\Lb_{:,i}\|_2^2 > 0$, while for each of the remaining $n_2-n_{\Lb}$ columns $\Lb_{:,j}$ of $\Lb$ that are identically zero we have $\|\bPhi\Lb_{:,j}\|_2^2 = 0$ so that $\bPhi\Lb_{:,j}=0$. Continuing, we show next that $\bPhi \Lb$ satisfies the column incoherence property with parameter $\mu_{\Lb}$. Recall from above that we write the compact SVD of $\Lb$ as $\Lb = \Ub \bSigma \Vb^*$, where $\Ub$ is $n_1\times r$, $\Vb$ is $n_2\times r$, and $\bSigma$ is an $r\times r$ nonnegative diagonal matrix of singular values (all of which are strictly positive). The incoherence condition on $\Lb$ is stated in terms of column norms of the matrix $\Vb^*$ whose rows form an orthonormal basis for the row space of $\Lb$. Now, when the rank of $\bPhi\Lb$ is the same as that of $\Lb$, which is true here on account of Lemma \[lem:Sketched\], the row space of $\bPhi\Lb$ is *identical* to that of $\Lb$, since each are $r$-dimensional subspaces of $\RR^{n_2}$ spanned by linear combinations of the columns of the $\Vb^*$. Thus since the rank and number of nonzero columns of $\bPhi\Lb$ are the same as for $\Lb$, the coherence parameter of $\bPhi\Lb$ is just $\mu_{\Lb}$, and the third claim is established. Finally, we establish the last claim, that the set of salient columns of $\bPhi\Cb$ is the same as for $\Cb$. Recall that the condition that a column $\Cb_{:,i}$ be salient was equivalent to the condition that $\|\Pb_{\cL^{\perp}}\Cb_{:,i}\|_2 > 0$, where $\Pb_{\cL^{\perp}}$ is the orthogonal projection operator onto the orthogonal complement of $\cL$ in $\RR^{n_1}$. Here, our aim is to show that an analogous result holds in the “projected” space – that for all $i\in\cI_{\Cb}$ we have $\|\Pb_{{(\bPhi\cL)}^{\perp}} \bPhi\Cb_{:,i}\|_2 > 0$, where $\bPhi\cL$ is the linear subspace spanned by the columns of $\bPhi\Lb$. For this we utilize an intermediate result of [@Davenport:10] formulated there in terms of a “compressive interference cancellation” method. We state an adapted version of that result here as a lemma (without proof). \[lem:IntCanc\] Let $\cV_1$ be an $r$-dimensional linear subspace of $\RR^{n}$ with $r<n$, let $\cV_2$ be any subset of $\RR^{n}$, and let $\check{\cV}_2 = \ \{\Pb_{\cV_1^{\perp}} \vb: \vb\in\cV_2\}$, where $\Pb_{\cV_1^{\perp}}$ is the orthogonal projection operator onto the orthogonal complement of $\cV_1$ in $\RR^{n}$. If $\bPhi$ is an $\epsilon$-stable embedding of $(\cV_1, \check{\cV}_2 \cup \{{\mathbf 0}\})$, then for all $\check{\vb} \in \check{\cV}_2$ $$\|\Pb_{(\bPhi\cV_1)^{\perp}} (\bPhi \check{\vb}) \|_2^2 \geq \left(1-\frac{\epsilon}{1-\epsilon}\right) \|\check{\vb} \|_2^2,$$ where $\Pb_{(\bPhi\cV_1)^{\perp}}$ is the orthogonal projection operator onto the orthogonal complement of the subspace of $\RR^{n}$ spanned by the elements of $\bPhi \cV_1 = \{\bPhi \vb: \vb\in\cV_1\} $. Before applying this result we first note a useful fact, that $\bPhi$ being an $\epsilon$-stable embedding of $(\cV_1, \check{\cV}_2 \cup \{{\mathbf 0}\})$ is equivalent to $\bPhi$ being an $\epsilon$-stable embedding of $(\cV_1, \cV_2 \cup \{{\mathbf 0}\})$, which follows directly from the definition of stable embeddings and the (easy to verify) fact that $\left\{\vb_1 - \check{\vb}_2: \vb_1\in\cV_1, \check{\vb}_2\in\check{\cV}_2\cup \{{\mathbf 0}\} \right\}=\left\{\vb_1 - \vb_2: \vb_1\in\cV_1, \vb_2\in\cV_2 \cup \{{\mathbf 0}\}\right\}$. Now, to apply Lemma \[lem:IntCanc\] here, we let $\cV_1 = \cL$, $\cV_2 = \cup_{i\in\cI_{\Cb}} \{\Cb_{:,i}\}$, and $\check{\cV}_2 = \cup_{i\in\cI_{\Cb}} \{\Pb_{\cL^{\perp}}\Cb_{:,i}\}$. Since $\bPhi$ is an $\epsilon$-stable embedding of , we have that for all $i\in\cI_{\Cb_{:,i}}$, $\|\Pb_{(\bPhi\cL)^{\perp}} (\bPhi \Cb)_{:,i} \|_2^2 \geq \left(1-\frac{\epsilon}{1-\epsilon}\right) \|\Pb_{\cL^{\perp}}\Cb_{:,i}\|_2^2$. Since $\epsilon<1/2$, the above result implies $\|\Pb_{(\bPhi\cL)^{\perp}} \bPhi \Cb_{:,i} \|_2>0$ for all $i\in\cI_{\Cb}$, while for all $j\notin\cI_{\Cb}$ we have $\Cb_{:,j}={\mathbf 0}$, implying that $\bPhi\Cb_{:,j}={\mathbf 0}$ and hence $\|\Pb_{(\bPhi\cL)^{\perp}} \bPhi \Cb_{:,j} \|_2=0$. Using this, and the fact that the nonzero columns of $\bPhi \Lb$ coincide with the nonzero columns of $\Lb$, we conclude that $\cI_{\bPhi\Cb} = \{i:\|\Pb_{(\bPhi\cL)^{\perp}} \bPhi \Cb_{:,j} \|_2 > 0, (\bPhi\Lb)_{:,i} = \mathbf{0}\}$ is the same as $\cI_{\Cb}$. ### Part 2 Given the structural result established in the previous step, the last part of the proof entails establishing that a random matrix $\bPhi$ generated as specified in the statement of Lemma \[lem:Mtilde\] is an $\sqrt{2}/4$-stable embedding of . Our approach here begins with a brief geometric discussion, and a bit of “stable embedding algebra.” Appealing to the definition of stable embeddings, we see that $\bPhi$ being an $\epsilon$-stable embedding of is equivalent to $\bPhi$ being such that $$\label{eqn:JLcond} (1-\epsilon) \|\vb\|_2^2 \leq \|\bPhi \vb\|_2^2 \leq (1+\epsilon) \|\vb\|_2^2$$ holds for all $\vb \in \cL \cup \bigcup_{i\in\cI_{\Cb}} \cL-\Cb_{:,i}$, where $\cL-\Cb_{:,i}$ denotes the $r$-dimensional *affine* subspace of $\RR^{n_1}$ comprised of all elements taking the form of a sum between a vector in $\cL$ and the fixed vector $\Cb_{:,i}$. Thus, in words, establishing our claim here entails showing that a random $\bPhi$ (generated as specified in the lemma, with appropriate dimensions) approximately preserves the lengths of all vectors in a *union of subspaces* comprised of one $r$-dimensional linear subspace and some $|\cI_{\Cb}|=k$, $r$-dimensional affine subspaces. Stable embeddings of linear subspaces using random matrices is, by now, well-studied (see, e.g., [@Baraniuk:08; @Davenport:10; @Gilbert:12], as well as a slightly weaker result [@Sarlos:06 Lemma $10$]), though stable embeddings of *affine* subspaces has received less attention in the literature. Fortunately, using a straightforward argument we may leverage results for the former in order to establish the latter. Recall the discussion above, and suppose that rather than establishing that holds for all $\vb \in \cL \cup \bigcup_{i\in\cI_{\Cb}} \cL-\Cb_{:,i}$ we instead establish a slightly stronger result, that holds for all $\vb \in \cL \cup \bigcup_{i\in\cI_{\Cb}} \cL^i$, where for each $i\in\cI_{\Cb}$, $\cL^i$ denotes the $(r+1)$-dimensional *linear* subspace of $\RR^{n_1}$ spanned by the columns of the matrix $[\Lb \ \Cb_{:,i}]$. (That the dimension of each $\cL^i$ be $r+1$ follows from the assumption that columns $\Cb_{:,i}$ for $i\in\cI_{\Cb}$ be outliers.) Clearly, if for some $i\in\cI_{\Cb}$ the condition holds for all $\vb\in\cL^i$, then it holds for all vectors formed as linear combinations of $[\Lb \ \Cb_{:,i}]$, so it holds in particular for all vectors in the $r$ dimensional affine subspace denoted by $\cL-\Cb_{:,i}$. Further, that holds for any $i\in\cI_{\Cb}$ implies it holds for linear combinations that use a weight of zero on the component $\Cb_{:,i}$, so in this case holds also for all $\vb\in\cL$. Based on the above discussion, we see that a sufficient condition to establish that $\bPhi$ be an $\epsilon$-stable embedding of is that hold for all $\vb \in \bigcup_{i\in\cI_{\Cb}} \cL^i$; in other words, that $\bPhi$ preserve (up to multiplicative $(1\pm \epsilon)$ factors) the squared lengths of all vectors in a union of (up to) $k$ unique $(r+1)$-dimensional linear subspaces of $\RR^{n_1}$. To this end we make use of another result adapted from [@Gilbert:12], and based on the union of subspaces embedding approach utilized in [@Baraniuk:08]. \[lem:JLSubsp\] Let $\bigcup_{i=1}^k \cV^i $ denote a union of $k$ linear subspaces of $\RR^{n}$, each of dimension at most $d$. For fixed $\epsilon\in(0,1)$ and $\delta\in(0,1)$, suppose $\bPhi$ is an $m\times n$ matrix satisfying the distributional JL property with $$m \geq \frac{d \log(42/\epsilon) + \log(k) + \log(2/\delta)}{f(\epsilon/\sqrt{2})}$$ Then $(1-\epsilon) \|\vb\|_2^2 \leq \|\bPhi \vb\|_2^2 \leq (1+\epsilon) \|\vb\|_2^2$ holds simultaneously for all $\vb\in \bigcup_{i=1}^k \cV^i$ with probability at least $1-\delta$. Applying this lemma here with $d=r+1$ and $\epsilon=\sqrt{2}/4$, and using the fact that $\log(84\sqrt{2})< 5$ yields the final result. Proof of Lemma \[lem:Step1\] {#a:lem2} ---------------------------- Our approach is comprised of two parts. In the first, we show that the two claims of Lemma \[lem:Step1\] follow directly when the following five conditions are satisfied - $\Sbb$ has $(1/2) \gamma n_2 \leq |\cS| \leq (3/2) \gamma n_2$ columns, - $\widetilde{\Lb}\Sbb$ has at most $(3/2) \gamma n_{\Lb}$ nonzero columns, - $\widetilde{\Cb}\Sbb$ has at most $k$ nonzero columns, - $\sigma^2_1(\widetilde{\Vb}^*\Sbb)\leq (3/2) \gamma$, and - $\sigma^2_r(\widetilde{\Vb}^*\Sbb)\geq (1/2) \gamma$, where the matrix $\widetilde{\Vb}^*$ that arises in (**a4**)-(**a5**) is the matrix of right singular vectors from the compact SVD $\widetilde{\Lb}=\widetilde{\Ub}\widetilde{\bSigma}\widetilde{\Vb}^*$ of $\widetilde{\Lb}$, and $\sigma_i(\widetilde{\Vb}^*\Sbb)$ denotes the $i$-th largest singular value of $\widetilde{\Vb}^*\Sbb$. Then, in the second part of the proof we show that (**a1**)-(**a5**) hold with high probability when $\Sbb$ is a random subsampling matrix generated with parameter $\gamma$ in the specified range. We briefly note that parameters $(1/2)$ and $(3/2)$ arising in the conditions (**a1**)-(**a5**) are somewhat arbitrary, and are fixed to these values here for ease of exposition. Analogous results to that of Lemma \[lem:Step1\] could be established by replacing $(1/2)$ with any constant in $(0,1)$ and $(3/2)$ with any constant larger than $1$, albeit with slightly different conditions on $\gamma$. ### Part 1 Throughout this portion of the proof, we assume that conditions (**a1**)-(**a5**) hold. Central to our analysis is a main result of [@Xu:12], which we state as a lemma (without proof). \[lem:OP\] Let $\check{\Mb} = \check{\Lb} + \check{\Cb}$ be an $\check{n}_1\times \check{n}_2$ matrix whose components $\check{\Lb}$ and $\check{\Cb}$ satisfy the structural conditions - ${{\rm rank}}(\check{\Lb}) = \check{r}$, - $\check{\Lb}$ has $n_{\check{\Lb}}$ nonzero columns, - $\check{\Lb}$ satisfies the *column incoherence property* with parameter $\mu_{\check{\Lb}}$, and - $|\cI_{\check{\Cb}}|=\{i: \|\Pb_{\check{\cL}^{\perp}} \check{\Cb}_{:,i}\|_2 > 0, \check{\Lb}_{:,i} = \mathbf{0}\} = \check{k}$, where $\check{\cL}$ denotes the linear subspace spanned by columns of $\check{\Lb}$ and $\Pb_{\check{\cL}^{\perp}}$ is the orthogonal projection operator onto the orthogonal complement of $\check{\cL}$ in $\RR^{\check{n}_1}$, with $$\check{k} \leq \left(\frac{1}{1+(121/9)\ \check{r}\mu_{\check{\Lb}}}\right)\check{n}_2.$$ For any upper bound $\check{k}_{\rm ub} \geq \check{k}$ and $\lambda = \frac{3}{7 \sqrt{\check{k}_{\rm ub}}}$ any solutions of the *outlier pursuit* procedure $$\label{eqn:OP2} \{\widehat{\check{\Lb}},\widehat{\check{\Cb}}\} = \argmin_{\bL,\bC} \|\bL\|_* + \lambda \|\bC\|_{1,2} \ \mbox{ s.t. } \ \check{\Mb} = \bL + \bC,$$ are such that the columns of $\widehat{\check{\Lb}}$ span the same linear subspace as the columns of $\check{\Lb}$, and the set of nonzero columns of $\widehat{\check{\Cb}}$ is the same as the set of locations of the nonzero columns of $\check{\Cb}$. Introducing the shorthand notation $\check{\Lb}=\widetilde{\Lb}\Sbb$, $\check{\Cb}=\widetilde{\Cb}\Sbb$, and $\check{n}_2=|\cS|$, our approach will be to show that conditions (**a1**)-(**a5**) along with the assumptions on $\widetilde{\Mb}$ ensure that ($\check{\cbb}$**1**)-($\check{\cbb}$**4**) in Lemma \[lem:OP\] are satisfied for some appropriate parameters $\check{r}$, $n_{\check{\Lb}}$, $\mu_{\check{\Lb}}$, and $\check{k}$ that depend on analogous parameters of $\widetilde{\Mb}$. First, note that (**a5**) implies that the matrix $\widetilde{\Vb}^*\Sbb$ has rank $r$, which in turn implies that $\check{\Lb}$ has rank $r$. Thus, ($\check{\cbb}$**1**) is satisfied with $\check{r}=r$. The condition ($\check{\cbb}$**2**) is also satisfied here for $n_{\check{\Lb}}$ no larger than $(3/2)\gamma n_{\Lb}$; this is a restatement of (**a2**). We next establish ($\check{\cbb}$**3**). To this end, note that since $\check{\Lb}$ has rank $r$, it follows that the $r$-dimensional linear subspace spanned by the rows of $\check{\Lb}=\widetilde{\Ub}\widetilde{\bSigma}\widetilde{\Vb}^*\Sbb$ is the same as that spanned by the rows of $\widetilde{\Vb}^*\Sbb$. Now, let $\Sbb^{T}\widetilde{\cV}$ denote the $r$-dimensional linear subspace of $\RR^{\check{n}_2}$ spanned by the columns of $\Sbb^{T}\widetilde{\Vb}$ and let $\Pb_{\Sbb^{T}\widetilde{\cV}}$ denote the orthogonal projection operator onto $\Sbb^{T}\widetilde{\cV}$. Then, bounding the column incoherence parameter of $\check{\Lb}$ entails establishing an upper bound on $\max_{i\in[\check{n}_2]} \|\Pb_{\Sbb^{T}\widetilde{\cV}}\eb_i\|_2^2$, where $\eb_i$ is the $i$-th canonical basis vector of $\RR^{\check{n}_2}$. Directly constructing the orthogonal projection operator (and using that $\widetilde{\Vb}^*\Sbb$ is a rank $r$ matrix) we have that $$\begin{aligned} \nonumber \lefteqn{\max_{i\in[\check{n}_2]} \|\Pb_{\Sbb^{T}\widetilde{\cV}}\eb_i\|_2^2 = \max_{i\in[\check{n}_2]} \left\|\Sbb^{T} \widetilde{\Vb}\left(\widetilde{\Vb}^*\Sbb \Sbb^{T} \widetilde{\Vb}\right)^{-1} \widetilde{\Vb}^*\Sbb \eb_i\right\|_2^2}\hspace{4em}&&\\ \nonumber &\stackrel{(a)}{\leq}&\max_{j\in[n_2]} \left\|\Sbb^{T} \widetilde{\Vb}\left(\widetilde{\Vb}^*\Sbb \Sbb^{T} \widetilde{\Vb}\right)^{-1} \widetilde{\Vb}^*\eb_j\right\|_2^2\\ \nonumber &\stackrel{(b)}{\leq}& \left(\frac{\sigma_1(\widetilde{\Vb}^*\Sbb)}{\sigma^2_r(\widetilde{\Vb}^*\Sbb)}\right)^2 \mu_{\Lb}\frac{r}{n_{\Lb}}\\ &\stackrel{(c)}{\leq}& \left(\frac{6}{\gamma}\right) \mu_{\Lb} \frac{r}{n_{\Lb}},\end{aligned}$$ where $(a)$ follows from the fact that for any $i\in[\check{n}_2]$ the vector $\Sbb \eb_j$ is either the zero vector or one of the canonical basis vectors for $\RR^{n_2}$, $(b)$ follows from straightforward linear algebraic bounding ideas and the column incoherence assumption on $\widetilde{\Lb}$, and $(c)$ follows from (**a4**)-(**a5**). Now, we let $n_{\check{\Lb}}$ denote the number of nonzero columns of $\check{\Lb}$, and write $$\max_{i\in[\check{n}_2]} \|\Pb_{\Sbb^{T}\widetilde{\cV}}\eb_i\|_2^2 \leq \left(\frac{6}{\gamma}\right) \mu_{\Lb} \frac{r}{n_{\Lb}} \left(\frac{n_{\check{\Lb}}}{n_{\check{\Lb}}}\right) \leq 9\mu_{\Lb} \frac{r}{n_{\check{\Lb}}},$$ where the last inequality uses (**a2**). Thus ($\check{\cbb}$**3**) holds with $$\label{eqn:mucheck} \mu_{\check{\Lb}}= 9\mu_{\Lb}.$$ Next, we establish ($\check{\cbb}$**4**). Recall from above that $\check{\Lb}$ has rank $r$, and is comprised of columns of $\widetilde{\Lb}$; it follows that the subspace $\check{\cL}$ spanned by columns of $\check{\Lb}$ is the same as the subspace $\widetilde{\cL}$ spanned by columns of $\widetilde{\Lb}$. Thus, $\|\Pb_{\check{\cL}^{\perp}} \check{\Cb}_{:,i}\|_2 = \|\Pb_{\widetilde{\cL}^{\perp}} \check{\Cb}_{:,i}\|_2$, so to obtain an upper bound on $\check{k}$ we can simply count the number $\check{k}$ of nonzero columns of $\check{\Cb}=\widetilde{\Cb}\Sbb$. By (**a3**) and , $$\begin{aligned} \nonumber \check{k} &\leq& \left(\frac{1}{20(1+ 121 r \mu_{\Lb})}\right)\ \left(\frac{1}{2}\right)n_2\\ \nonumber &\stackrel{(a)}{\leq}& \left(\frac{1}{1+ 121 r \mu_{\Lb}}\right)\ \left(\frac{1}{2}\right) \gamma n_2\\ &\stackrel{(b)}{\leq}& \left(\frac{1}{1+ (121/9) \check{r} \mu_{\check{\Lb}}}\right)\ \check{n}_2,\end{aligned}$$ where $(a)$ follows from the assumption that $\gamma\geq 1/20$, and $(b)$ follows from (**a1**) and as well as the fact that $\check{r}=r$. Finally, we show that the two claims of Lemma \[lem:Step1\] hold. The first follows directly from (**a1**). For the second, note that for any $k_{\rm ub}\geq k$ we have that $\check{k}_{\rm ub} \triangleq k_{\rm ub} \geq \check{k}$. Thus, since $\lambda =\frac{3}{7\sqrt{k_{\rm ub}}} = \frac{3}{7\sqrt{\check{k}_{\rm ub}}}$ and ($\check{\cbb}$**1**)-($\check{\cbb}$**4**) hold, it follows from Lemma \[lem:OP\] that the optimization produces an estimate $\widehat{\check{\Lb}}$ whose columns span the same linear subspace as that of $\check{\Lb}$. But, since $\check{\Lb}$ has rank $r$ and its columns are just a subset of columns of the rank-$r$ matrix $\widetilde{\Lb}$, the subspace spanned by the columns of $\check{\Lb}$ is the same as that spanned by columns of $\widetilde{\Lb}$. ### Part 2 The last part of our proof entails showing (**a1**)-(**a5**) hold with high probability when $\Sbb$ is randomly generated as specified. Let $\cE_1,\dots,\cE_5$ denote the events that conditions (**a1**)-(**a5**), respectively, hold. Then ${{\rm Pr}}\left( \ \left\{\bigcap_{i=1}^5 \cE_i\right\}^c \ \right)\leq \sum_{i=1}^5 {{\rm Pr}}(\cE_i^c)$, and we consider each term in the sum in turn. First, since $|\cS|$ is a Binomial($n_2,\gamma$) random variable, we may bound its tails using [@McDiarmid:98 Theorem 2.3 (b-c)]. This gives that ${{\rm Pr}}\left(|\cS| > 3\gamma n_2/2\right) \leq \exp\left(-3\gamma n_2/28\right)$ and ${{\rm Pr}}\left(|\cS| < \gamma n_2/2\right) \leq\exp\left(-\gamma n_2/8\right).$ By union bound, we obtain that ${{\rm Pr}}(\cE_1^c) \leq \exp\left(-3\gamma n_2/28\right) + \exp\left(-\gamma n_2/8\right).$ Next, observe that conditionally on $|\cS|=s$, the number of nonzero columns present in the matrix $\widetilde{\Lb} \Sbb$ is a hypergeometric random variable parameterized by a population of size $n_2$ with $n_{\Lb}$ positive elements and $s$ draws. Denoting this hypergeometric distribution here by ${\rm hyp}(n_2,n_{\Lb},s)$ and letting $H_{|\cS|}\sim {\rm hyp}(n_2,n_{\Lb},|\cS|)$, we have that ${{\rm Pr}}(\cE_2^c)={{\rm Pr}}\left(H_{|\cS|} > \left(\frac{3}{2}\right) \gamma n_{\Lb} \right)$. Using a simple conditioning argument, ${{\rm Pr}}(\cE_2^c) \leq \sum_{s=\lceil (2/3)\gamma n_2\rceil}^{\lfloor (4/3)\gamma n_2\rfloor} {{\rm Pr}}\left(H_{s} > \left(\frac{3}{2}\right) \gamma n_{\Lb} \right){{\rm Pr}}(|\cS| = s) + {{\rm Pr}}\left(\left| |\cS| -\gamma n_2 \right| > \left(\frac{1}{3}\right)\gamma n_2\right)$, and our next step is to simplify the terms in the sum. Note that for any $s$ in the range of summation, we have ${{\rm Pr}}\left(H_{s} > \left(\frac{3}{2}\right) \gamma n_{\Lb} \right) = {{\rm Pr}}\left(H_{s} > \left(\frac{3}{2}\right) \gamma n_{\Lb}\left(\frac{s n_2}{s n_2}\right) \right)$, and thus $$\begin{aligned} \nonumber {{\rm Pr}}\left(H_{s} > \left(\frac{3}{2}\right) \gamma n_{\Lb} \right) &\stackrel{(a)}{\leq}& {{\rm Pr}}\left(H_{s} > \left(\frac{9}{8}\right) s\left(\frac{n_{\Lb}}{n_2}\right)\right)\\ \nonumber &\stackrel{(b)}{\leq}& \exp\left(-\frac{3s(n_{\Lb}/n_2)}{400}\right)\\ &\stackrel{(c)}{\leq}& \exp\left(-\frac{\gamma n_{\Lb}}{200}\right), \end{aligned}$$ where $(a)$ utilizes the largest value of $s$ to bound the term $\gamma n_2/s$, $(b)$ follows from an application of Lemma \[lem:hyptail\] in Appendix \[a:hyp\], and $(c)$ results from using the smallest value of $s$ (within the range of summation) to bound the error term. Assembling these results, we have that ${{\rm Pr}}(\cE_2^c) \leq \exp\left(-\gamma n_{\Lb}/200\right) + \exp\left(-\gamma n_2/24\right) + \exp\left(-\gamma n_2/18\right)$, where we use the fact that the probability mass function of $|\cS|$ sums to one, and another application of [@McDiarmid:98 Theorem 2.3(b,c)]. Bounding ${{\rm Pr}}(\cE^c_3)$ is trivial. Since $\widetilde{\Cb}$ itself has $k$ nonzero columns, the subsampled matrix $\widetilde{\Cb}\Sbb$ can have at most $k$ nonzero columns too. Thus, ${{\rm Pr}}(\cE^c_3) = 0$. Finally, we can obtain bounds on the largest and smallest singular values of $\widetilde{\Vb}^*\Sbb$ using the Matrix Chernoff inequalities of [@Tropp:12]. Namely, letting $\Zb=\widetilde{\Vb}^*\Sbb$ we note that the matrix $\Zb\Zb^*$ may be expressed as a sum of independent positive semidefinite rank-one $r\times r$ Hermitian matrices, as $\Zb\Zb^* = \widetilde{\Vb}^*\Sbb\Sbb^{T}\widetilde{\Vb} = \sum_{i=1}^{n_2} S_i (\widetilde{\Vb}^*_{:,i})(\widetilde{\Vb}^*_{:,i})^*$, where the $\{S_i\}_{i=1}^{n_2}$ are i.i.d. Bernoulli($\gamma$) random variables as in the statement of Algorithm \[alg:main\] (and, $S_i^2=S_i$). To instantiate the result of [@Tropp:12], we note that $\lambda_{\rm max}(S_i (\widetilde{\Vb}^*_{:,i})(\widetilde{\Vb}^*_{:,i})^*) \leq \|\widetilde{\Vb}^*_{:,i}\|_2^2 \leq \mu_{\Lb} r/n_{\Lb}\triangleq R$ almost surely for all $i$, where the last inequality follows from the incoherence assumption ($\widetilde{\cbb}$**3**) (as well as ($\widetilde{\cbb}$**1**)-($\widetilde{\cbb}$**2**)). Further, direct calculation yields $\mu_{\rm min} \triangleq \lambda_{\rm min}\left(\EE\left[\Zb\Zb^*\right]\right) = \lambda_{\rm min}(\gamma \Ib) = \gamma$ and $\mu_{\rm max} \triangleq \lambda_{\rm max}\left(\EE\left[\Zb\Zb^*\right]\right) = \lambda_{\rm max}(\gamma \Ib) = \gamma$, where the identity matrices in each case are of size $r\times r$. Thus, applying [@Tropp:12 Corollary 5.2] (with $\delta=1/2$ in that formulation) we obtain that ${{\rm Pr}}(\cE_4^c) = {{\rm Pr}}\left(\sigma^2_{1}\left(\widetilde{\Vb}^*\Sbb\right) \geq 3\gamma/2 \right) \leq r \cdot\left(9/10\right)^{\frac{\gamma n_{\Lb}}{r \mu_{\Lb}}}$, and ${{\rm Pr}}(\cE_5^c) = {{\rm Pr}}\left(\sigma^2_{r}\left(\widetilde{\Vb}^*\Sbb\right) \leq \gamma/2 \right)\leq r \cdot\left(9/10\right)^{\frac{\gamma n_{\Lb}}{r \mu_{\Lb}}}$. Putting the results together, and using a further bound on ${{\rm Pr}}(\cE^c_1)$, we have ${{\rm Pr}}\left( \ \left\{\bigcap_{i=1}^5 \cE_i\right\}^c \ \right) \leq \exp\left(-\frac{\gamma n_{\Lb}}{200}\right) + 2\exp\left(-\frac{\gamma n_2}{24}\right) + 2\exp\left(-\frac{\gamma n_2}{18}\right) + r \cdot\left(\frac{9}{10}\right)^{\frac{\gamma n_{\Lb}}{r \mu_{\Lb}}} + r \cdot\left(\frac{9}{10}\right)^{\frac{\gamma n_{\Lb}}{r \mu_{\Lb}}}$, which is no larger than $\delta$ given that $\gamma$ satisfies (in particular, this ensures each term in the sum is no larger than $\delta/5$). Proof of Lemma \[lem:Step2\] {#a:lem3} ---------------------------- First, note that since $\widehat{\cL}_{(1)} = \widetilde{\cL}$, we have that $\|\Pb_{\widehat{\cL}_{(1)}^{\perp}} \widetilde{\Mb}_{:,i}\|_2 > 0$ for all $i\in\cI_{\widetilde{\Cb}}$, and $\|\Pb_{\widehat{\cL}_{(1)}^{\perp}} \widetilde{\Mb}_{:,i}\|_2 = 0$ otherwise. This, along with the fact that the entries of $\bphi$ be i.i.d. realizations of a *continuous* random variable, imply that with probability one the $1\times n_2$ vector $\xb^{T} \triangleq \bphi \Pb_{\widehat{\cL}_{(1)}^{\perp}} \widetilde{\Mb}$ is nonzero at every $i\in\cI_{\widetilde{\Cb}}$ and zero otherwise. Indeed, since for each $i\in\cI_{\widetilde{\Cb}}$ the distribution of ${\rm x}_i = \bphi \Pb_{\widehat{\cL}_{(1)}^{\perp}} \widetilde{\Mb}_{:,i}$ is a continuous random variable with nonzero variance, it takes the value zero with probability zero. On the other hand, for $j\notin \cI_{\widetilde{\Cb}}$, ${\rm x}_j = \bphi \Pb_{\widehat{\cL}_{(1)}^{\perp}} \widetilde{\Mb}_{:,j}=0$ with probability one. With this, we see that exact identification of $\cI_{\widetilde{\Cb}}$ can be accomplished if we can identify the support of $\xb$ from linear measurements of the form $\yb = (\yb_{(2)})^{T} = \Ab\xb $. To proceed, we appeal to (now, well-known) results from the compressive sensing literature. We recall one representative result of [@Candes:08:RIP] that is germane to our effort below. Here, we cast the result in the context of the stable embedding formalism introduced above, and state it as a lemma without proof. \[lem:RIP\] Let $\xb\in\RR^n$ and $\zb = \Ab \xb$. If $\Ab$ is an $\epsilon$-stable embedding of $(\cU_{\binom{n}{2k}},\{{\mathbf 0}\})$ for some $\epsilon < \sqrt{2}-1$ where $\cU_{\binom{n}{2k}}$ denotes the union of all $\binom{n}{2k}$ unique $2k$-dimensional linear subspaces of $\RR^n$ spanned by canonical basis vectors, and $\xb$ has at most $k$ nonzero elements, then the solution $\widehat{\xb}$ of $$\argmin_{\bx} \|\bx\|_1 \ \mbox{ s.t. } \zb = \Ab \bx.$$ is equal to $\xb$. Now, a straightforward application of Lemma \[lem:JLSubsp\] above provides that for any $\delta\in(0,1)$, if $$p\geq \frac{2k\log(42/\epsilon) + \log\binom{n}{2k} + \log(2/\delta)}{f(\epsilon/\sqrt{2})}$$ then the randomly generated $p\times n_2$ matrix $\bA$ will be an $\epsilon$-stable embedding of $(\cU_{\binom{n}{2k}},\{{\mathbf 0}\})$ with probability at least $1-\delta$. This, along with the well-known bound $\binom{n}{2k}\leq \left(\frac{en}{2k}\right)^{2k}$ and some straightforward simplifications, imply that the condition that $p$ satisfy is sufficient to ensure that with probability at least $1-\delta$, $\bA$ is a $(\sqrt{2}/4)$-stable embedding of $(\cU_{\binom{n}{2k}},\{{\mathbf 0}\})$. Since $\sqrt{2}/4 < \sqrt{2}-1$, the result follows. An Upper Tail Bound for the Hypergeometric Distribution {#a:hyp} ------------------------------------------------------- Let ${\rm hyp}(N,M,n)$ denote the hypergeometric distribution parameterized by a population of size $N$ with $M$ positive elements and $n$ draws, so $H\sim{\rm hyp}(N,M,n)$ is a random variable whose value corresponds to the number of positive elements acquired from $n$ draws (without replacement). The probability mass function of $H\sim{\rm hyp}(N,M,n)$ is ${{\rm Pr}}(H=k) = \binom{M}{k}\binom{N-M}{n-k}/\binom{N}{n}$ for $k\in\{\max\{0,n+M-N\},\dots,\min\{M,n\}\}$, and its mean value is $\EE[H]=nM/N$. It is well-known that the tails of the hypergeometric distribution are similar to those of the binomial distribution for $n$ trials and success probability $p=M/N$. For example, [@Chvatal:79] established that for all $t\geq 0$, ${{\rm Pr}}(H -np \geq nt) \leq e^{-2t^2 n}$, a result that follows directly from Hoeffding’s work [@Hoeffding:63], and exhibits the same tail behavior as predicted by the Hoeffding Inequality for a Binomial($n,p$) random variable (see, e.g., [@McDiarmid:98]). Below we provide a lemma that yields tighter bounds on the upper tail of $H$ when the fraction of positive elements in the population is near $0$ or $1$. Our result is somewhat analogous to [@McDiarmid:98 Theorem 2.3(b)] for the Binomial case. \[lem:hyptail\] Let $H\sim{\rm hyp}(N,M,n)$, and set $p=M/N$. For any $\epsilon\geq 0$, $${{\rm Pr}}(H \geq (1+\epsilon)np) \leq e^{-\frac{\epsilon^2 np}{2(1+\epsilon/3)}}.$$ We begin with an intermediate result of [@Chvatal:79], that for any $t\geq 0$ and $h\geq 1$, $${{\rm Pr}}(H-pn \geq tn) \leq \left(h^{-(p+t)}(1-p+hp)\right)^n.$$ Now, for the specific choices $t=\epsilon p$ and $h=1+\epsilon$ we have $$\begin{aligned} \nonumber {{\rm Pr}}(H-np \geq \epsilon np) &\leq& \left((1+\epsilon)^{-(1+\epsilon)p}(1+\epsilon p)\right)^n\\ \nonumber &\stackrel{(a)}\leq& \left((1+\epsilon)^{-(1+\epsilon)}e^{\epsilon}\right)^{np}\\ &\stackrel{(b)}\leq& e^{-\frac{\epsilon^2 np}{2(1+\epsilon/3)}},\end{aligned}$$ where $(a)$ follows from the inequality $1+x\leq e^x$ (with $x=\epsilon p$), and $(b)$ follows directly from [@McDiarmid:98 Lemma 2.4]. [Xingguo Li]{} received the B.E. degree in 2010 in Communications Engineering from Beijing University of Posts and Telecommunications, and M.S. degree in 2013 with honor in Applied and Computational Mathematics from University of Minnesota Duluth. In 2010, he held a visiting research appointment in the Robotics Institute of School of Computer Science, Carnegie Mellon University. He is currently a Ph.D. student in the Department of Electrical and Computer Engineering, University of Minnesota, under the supervision of Professor Jarvis Haupt. His current research interest focuses on statistical signal processing, high-dimensional sparse regression and optimization with applications in image processing, computer vision and machine learning. [Jarvis Haupt]{} (S’05–M’09) received the B.S., M.S. and Ph.D. degrees in electrical engineering from the University of Wisconsin-Madison in 2002, 2003, and 2009, respectively. From 2009-2010 he was a Postdoctoral Research Associate in the Dept. of Electrical and Computer Engineering at Rice University in Houston, Texas. He is currently an Assistant Professor in the Dept. of Electrical and Computer Engineering at the University of Minnesota. Professor Haupt is the recipient of several academic awards, including the Wisconsin Academic Excellence Scholarship, the Ford Motor Company Scholarship, the Consolidated Papers and Mead Witter Foundation Tuition Scholarships, the Frank D. Cady Mathematics Scholarship, and the Claude and Dora Richardson Distinguished Fellowship. He received the DARPA Young Faculty Award in 2014. His research interests generally include high-dimensional statistical inference, adaptive sampling techniques, and statistical signal processing and learning theory, with applications in the biological sciences, communications, imaging, and networks. [^1]: Submitted June 30, 2014; revised October 12, 2014. The authors are with the Department of Electrical and Computer Engineering at the University of Minnesota – Twin Cities. Tel/fax: (612) 625-3300  / (612) 625-4583. Emails: [{lixx1661, jdhaupt}@umn.edu]{}. (Corresponding author: J. Haupt.) A version of this work was submitted to ICASSP 2015. The authors graciously acknowledge support from the NSF under Award No. CCF-1217751. [^2]: As we will see, the conditions under which our column subsampling in Step 1 succeeds will depend on the number of *nonzero* columns in the low-rank component, since any all-zero columns are essentially non-informative for learning the low-rank subspace. Thus, we make the distinction between $n_2$ and $n_{\Lb}$ explicit throughout. [^3]: It was shown in [@Achlioptas:01], for example, that $f(\epsilon) = \epsilon^2/4 - \epsilon^3/6$ for matrices whose elements are appropriately normalized Gaussian or symmetric Bernoulli random variables. [^4]: We solve the optimization associated with Step 2 of our approach as a LASSO problem, with $10$ different choices of regularization parameter $\mu\in(0,1)$. We deem any trial a success if for at least one value of $\mu$, there exists a threshold $\tau>0$ such that $\min_{i\in\cI_{\Cb}} |\widehat{{\rm c}}_i(\mu)| > \tau > \max_{j\notin \cI_{\Cb}} |\widehat{{\rm c}}_j(\mu)|$ for the estimate $\widehat{\cbb}({\mu})$ produced in Step 2. An analogous threshold-based methodology was employed to assess the outlier detection performance of the Outlier Pursuit approach in [@Xu:12]. [^5]: Our evaluation of RMC here agrees qualitatively with results in [@Chen:11], where sampling rates around $10\%$ yielded successful recovery for small $r$. [^6]: Timing comparisons were done with [MATLAB]{} R2013a on an iMac with a 3.4 GHz Intel Core i7 processor, 32 GB memory, and running OS X 10.8.5. [^7]: Thanks to David B. Dunson and Alfred O. Hero for these suggestions.

=25.5cm =-3.5cm TIFR/TH/93-48 Like Sign Dilepton Signature for Gluino Production at LHC with or without R Conservation\ H. Dreiner$^{a^\dagger}$, Manoranjan Guchait$^b$, D.P. Roy$^{c^{\dagger\dagger}}$\ $^a$Physics Department, University of Oxford, Oxford OX1 3RH, UK\ $^b$Physics Department, Jadavpur University, Calcutta - 700 032, India\ $^c$Theory Group, Tata Institute of Fundamental Research, Bombay - 400 005, India\ \ The isolated like sign dilepton signature for gluino production is investigated at the LHC energy for the $R$ conserving as well as the $L$ and $B$ violating SUSY models over a wide range of the parameter space. One gets viable signals for gluino masses of 300 and 600 GeV for both $R$ conserving and $L$ violating models, while it is less promising for the $B$ violating case. For a 1000 GeV gluino, the $L$ violating signal should still be viable; but the $R$ conserving signal becomes too small at least for the low luminosity option of LHC. ------------------------------------------------------------------------ width 5cm $^\dagger$Present Address: Institute of Theoretical Physics, ETH, Zurich, Switzerland. $^{\dagger\dagger}$E-Mail: DPROY@TIFRVAX.BITNET I.\ The hadron colliders offer by far the best discovery limit for superparticles because of their higher energy reach. The superparticles having the largest production cross section at hadron colliders are the strongly interacting ones – the squark $\tilde q$ and gluino $\tilde g$. Therefore there has been a good deal of discussion on the search of these superparticles at the present and proposed hadron colliders \[1,2\]. So far the search programme has been largely based on the missing $p_T$ signature assuming $R$ conservation \[3\]. The latter implies pair production of superparticles followed by their decay into the lightest superparticle (LSP) which has to be stable. It is also required to be colourless and neutral for cosmological reasons \[4\]. The LSP escapes detection due to its feeble interaction with matter resulting in the missing $p_T$ signature for superparticle production. There is a growing realisation in recent years however that the multilepton signature, and in particular the like sign dilepton (LSD) signature, may play an equally important role in superparticle search for the following reasons. 1) The squark and gluino searches are expected to be carried over the mass range of several hundred GeV at LHC/SSC. The dominant decay mode for a squark or gluino in this mass range is not its direct decay into the LSP, which is generally assumed to be the lightest neutralino, but a cascade decay via the heavier neutralino and chargino states. The cascade decay proceeds through the emission of $W$ or $Z$ which have significant leptonic branching ratios. Thus one expects two (or more) leptons resulting from the cascade decay of the squark or gluino pair. More over, in the latter case the two leptons are expected have like sign half the time due to the Majorana nature of gluino \[5-7\]. 2) In the $R$ violating $(R\!\!\!/)$ SUSY models there is no missing $p_T$ signature, since the LSP is no longer stable. Instead it decays into a leptonic or baryonic channel depending on whether the $L$ or $B$ violating Yukawa coupling is the dominant one \[8-10\]. In the former case one expects two (or more) leptons resulting from the decay of the LSP pair. Again the two leptons are expected to have like sign half the time due to the Majorana nature of the LSP. In the latter case there is no viable signature from the baryonic decay of LSP; the like sign dileptons coming from the cascade decay process provides by far the best signature for this case \[9\]. Thus there are two contributions to the like sign dilepton signature for gluino production – 1) from the cascade decay of gluino into LSP, which holds for $R$ conserving as well as the $L\!\!\!/$ and $B\!\!\!/$ SUSY models, and 2) from the leptonic decay of LSP in the $L\!\!\!/$ moel. In the latter case the size of the like sign dilepton signal is expected to be large. In fact the Tevatron dilepton data is already known to give a gluino (squark) mass limit in the $L\!\!\!/$ SUSY model \[10\], which is as large as that obtained from the corresponing missing-$p_T$ data for the $R$ conserving case \[11\]. On the other hand the LSD signal arising from the cascade decay process is suppressed by the leptonic branching fractions of the two vector bosons, and hence expected to be relatively small in size. Nonetheless it is expected to provide a useful alternative signature for gluino production in the $R$ conserving SUSY model, since it has a smaller background compared to the missing-$p_T$ channel. Moreover it provides the only signature for gluino production in the $B\!\!\!/$ SUSY model as mentioned above. Thus it is important to make a systematic study of the LSD signal, arising from both these sources, for the gluino mass range of interest to LHC/SSC along with the corresponding background. The present work is devoted to this excercise. To be specific we shall concentrate on the LHC energy of = 16  [TeV]{}, and assume a typical luminosity of 10 events/fb corresponding to the low luminosity option of LHC. Of course any viable signal here shall be even more viable at SSC or the high luminosity option of LHC. We shall study gluino production and decay under the assumption m\_[g]{} &lt; m\_[q]{} in which case they provide the most important signal for superparticle search at hadron colliders. This inequality seems to be favoured by a large class of SUSY models. However, we shall briefly discuss how the results would change if the squarks are lighter than the gluino, in which case the dominant superparticle signal would come from the production and decay of squarks. The paper is organised as follows. The standard model (SM) background for the like sign dilepton channel is briefly discussed in section II. Section III gives the formalism of gluino cascade decay into the LSP via the heavier neutralino and chargino states. It tabulates the masses and the compositions of the neutralino and chargino states, resulting from the diagonalisation of their mass matrices, for a wide range of gluino mass and the other SUSY parameters. It also gives the branching fractions of gluino decay into these states. Section IV describes the LSP decay in the $R\!\!\!/$ SUSY models. Section V compares the resulting like sign dilepton signals with the SM background. The main conclusions are summarised in section VI. II.\ The SM background to the like sign dilepton channel at LHC has been studied in detail in \[12\]. We shall only summarise the essential points here. The two main sources of LSD background are from gg b|b  (b|bg) via $B\bar B$ mixing, and gg t|t  (t|tg) follwed by the sequencial decay of one of the $t$ quarks into $b$. In the first case both the leptons originate from $B$ particle decay, $B \rightarrow \ell \nu D ~(D^\star)$, while in the second case one of the leptons (the softer one) originates from $B$. Consequently both the contributions can be suppressed by imposing isolation cut on the leptons \[13\]. Moreover the isolation cut for the $B$ decay lepton is known to become more powerful with the lepton $p_T$, resulting in a $p_T$ cutoff for the isolated lepton \[14\] p\^\_T E\^[AC]{}\_T ([m\^2\_B - m\^2\_[D(D\^)]{} m\^2\_[D(D\^)]{}]{}). Substituting for the bottom and charm particle masses one ses that a typical isolation cut of E\^[AC]{}\_T &lt; 10  [GeV]{} at LHC \[15\] implies a lepton $p_T$ cutoff p\^\_T 60 [GeV]{}. Loss of visible $E^{AC}_T$ due to semileptonic $D$ decay and energy resolution leads to a small spill over of the isolated lepton background beyond this kinematic cutoff. On the other hand the contributions to $E^{AC}_T$ from the fragmentation of $b$ quark into $B$ particle as well as the underlying event tend to strengthen the bound. All these effects are taken into account in the ISAJET \[16\] Monte Carlo calculation of this background in \[12\], which shows that the background becomes negligibly small beyond the lepton $p_T$ of 60 GeV. This is evidently a powerful result, which can be exploited in the search of the gluino signal in the LSD channel. We shall use this result in our analysis. We shall also include the fake LSD background arising from the misidentification of one of the lepton charges in $t\bar t \rightarrow \ell^+\ell^- X$. III.\ We shall work within the framework of the minimal supersymmetric standard model (MSSM) so as to have the minimum number of parameters \[1,2\]. The gluino cascade decay into the LSP proceeds via the heavier neutralino and chargino states. There are 4 neutralino states, which are mixtures of the 4 basic interaction states, i.e. \^0\_i = N\_[i1]{} B + N\_[i2]{} W\^3 + N\_[i3]{} H\^0\_1 + N\_[i4]{} H\^0\_2. The masses and compositions of the neutralinos are obtained by diagonalising the mass matrix \[1,2,5,17\] M\_N = (). where $M_1$ and $M_2$ are the soft masses of the bino $\tilde B$ and wino $\tilde W$ respectively, $\mu$ is the supersymmetric higsino mass parameter and $\tan\beta$ is the ratio of the two higgs vacuum expectation values. The two soft gaugino masses are related to that of the gluino in the MSSM, i.e. M\_2 = [\^2\_W\_s]{} m\_[g]{} 0.3  m\_[g]{} M\_1 = [5 3]{} \^2\_W    M\_2 0.5  M\_2. Thus there are 3 independent parameters, $m_{\tilde g}$, $\mu$ and $\tan\beta$, defining the mass matrix. The Majorana nature of the neutralinos ensures that the mass matrix is in general complex symmetric and hence can be diagonalised by only one unitary matrix $N$, i.e. N\^M\_N N\^[-1]{} = M\^D\_N. We have followed the analytical prescription of diagonalising this mass matrix recently suggested in \[18\]. But we have also cross-checked our results extensively with the numerical diagonalisation program EISCH1.FOR of CERN library as well as the published results of \[2,5,19\]. The two chargino mass states are mixtures of the charged wino $\tilde W^\pm$ and Higgsino $\tilde H^\pm$. Their masses and compositions are obtained by diagonalising the corresponding chargino mass matrix M\_C = (). This is done via the biunitary transformation U  M\_C V\^[-1]{} = M\^D\_C where $U$ and $V$ are $2 \times 2$ unitary matrices, which diagonalise the hermitian (real symmetric) matrices $M_C M^\dagger_C$ and $M^\dagger_C M_C$ respectively. Explicit expressions for $U$ and $V$ may be found in \[1,5\] along with those of the mass eigenvalues. The corresponding chargino eigenstates are \^\_[iL]{} &=& V\_[i1]{} W\^\_L + V\_[i2]{} H\^\_L\ \^\_[iR]{} &=& U\_[i1]{} W\^\_R + U\_[i2]{} H\^\_[R]{} where $L$ and $R$ refer to the left and right handed helicity states. We shall use the real orthogonal representation for the unitary matrices $U,V$ and $N$. Moreover the chargino and neutralino eigenstates shall be labelled in increasing order of mass, with \^0\_1 representing the LSP. Table I (a,b,c) show the masses and compositions of the neutralino and chargino states for three representative values of the gluino mass which are of interest to LHC/SSC; i.e. m\_[g]{} = 300,600  [and]{}  1000 [GeV]{}. The results are not very sensitive to the variation of $\tan\beta$ over the range allowed by MSSM, i.e. $1 < \tan\beta < m_t/m_b ~(\simeq 30)$. We have chosen 2 representative values = 2  [and]{}  10; the current lower mass bounds of top quark and neutral higgs boson do not seem to favour $\tan\beta = 1$ \[2\]. On the other hand, the results are quite sensitive to the variation of $\mu$ over the range $-M_2 < \mu < M_2$. Therefore we have chosen 5 representative values of this variable, i.e. = 0.1 m\_W,  m\_W,  4 m\_W as in ref. \[5\]. This also helps us cross-check some of our results with theirs \[20\]. The SM parameters used are m\_W = 80  [GeV]{},  m\_Z = 91  [GeV]{},  \^2\_W = 0.233,  = 1/128,  \_s = 0.115. The masses and compositions of the neutralinos are shown in the 2nd and 3rd columns of Table I, while those of the charginos are shown in the 5th and 6th columns. The upper and lower entries of 6th column refer to the compositions of the left and right handed components of the chargino respectively. The sign of a mass affects the phases of the corresponding couplings \[5\]; these are irrelevant however in the approximation we shall be working in. One should note the following systematics in the masses and compositions of the neutralino and chargino states. 1. For $\mu = \pm 4~m_W$, the higgsino mass parameter is generally larger than $M_1$ and $M_2$. Consequently the LSP $(\chi^0_1)$ is dominated by the bino $\tilde B$ component, while the second lightest neutralino $\chi^0_2$ and the lightest chargino $\chi^\pm_1$ are dominated by the wino $\tilde W$. These are clearly reflected in their masses and compositions. Only at $m_{\tilde g} = 1000$ GeV, the $\chi^0_2$ and $\chi^\pm_1$ acquire significant higgsino components as $M_2 \simeq |\mu|$. Evidently the gaugino dominance of $\chi^0_1$, $\chi^0_2$ and $\chi^\pm_1$ is expected to hold even better at $|\mu| > 4 ~M_W$. 2. For $\mu = -m_W$, the higgsino mass parameter is comparable to $M_1,M_2$ at $m_{\tilde g} = 300$ GeV and smaller at $m_{\tilde g} = 600$ and 1000 GeV. Consequently the $\chi^0_1, \chi^0_2$ and $\chi^\pm_1$ contain significant admixtures of the gaugino and higgsino components at $m_{\tilde g} = 300$ GeV, while they are dominated by the higgsinos at $m_{\tilde g} = 600$ and 1000 GeV. 3. For $\mu = m_W$ and $0.1~m_W$, the $\chi^0_1, ~\chi^0_2$ and $\chi^\pm_1$ have large higgsino components as expected. However, this part of the parameter space is disallowed by the LEP data, which gives a lower mass limit of $\sim m_Z/2$ for the lightest chargino as well as higgs dominated neutralino \[3,21\]. Only at $m_{\tilde g} = 1000$ GeV does the $\mu = m_W$ value falls marginally within the allowed region. Thus we see that the values of $\mu = \pm 4 ~m_W$ and $-m_W$ are generally representative of the two extreme cases where the lighter neutralino and chargino states are dominated by the gaugino and higgsino components respectively, while it is the opposite for the heavier ones. Note that for $m_{\tilde g} = 300$ GeV the lighter states have substantial gaugino components even at $\mu = -m_W$. Nonetheless the latter represents the extreme composition as it lies close to the boundary of the experimentally allowed region \[3,19\]. We shall estimate the branching fractions of gluino decay into the above chargino and neutralino states by neglecting the contributions of top quark $(\tilde g \rightarrow t\bar t \chi^0_i, t\bar b \chi^-_i)$ as well as the loop induced processes $(\tilde g \rightarrow g \chi^0_i)$. This seems to be a reasonable approximation for the bulk of the parameter space under investigation \[2\]. Thus the decay processes of interest are g [ ]{} q |q \^0\_i g [ ]{} q’ |q \^\_i where $q$ and $q'$ are understood to represent the light quarks of a given generation. The relevant interaction terms for these processes are \_[qqg]{} &=& [ig\_s ]{} q\_L\^|[g]{}\_A \_A q\_L + [ig\_s ]{} q\^\_R |[g]{}\_A \_A q\_R + h.c.,\ [L]{}\_[qq\^0\_i]{} &=& iA\^q\_[\^0\_i]{} q\^\_L |\^0\_i q\_L + iB\^q\_[\^0\_i]{} q\^\_R |\^0\_i q\_R + h.c.,\ [L]{}\_[q’q\^\_i]{} &=& iA\^d\_[\^\_i]{} u\^\_L |\^-\_i d\_L + iA\^u\_[\^\_i]{} d\^\_L |\^+\_i u\_L + h.c., where $g_s$ is the QCD coupling and $\lambda_A$ are the generators of the colour $SU(3)$ group. In the absence of top quark one can safely neglect the small Yukawa couplings associated with the higgs sector, so that $A$ and OB$B$ are simply the $SU(2) \times U(1)$ gauge couplings of left handed (doublet) and right handed (singlet) quarks \[5\] respectively. Moreover we shall ignore the phase factors associated with these couplings, since the interference terms between left and right handed squark exchanges are negligible for final states involving only light quarks \[22\]. Thus we have A\^\_[\^0\_i]{} &=& [g’ 3]{} N\_[i1]{} + [g ]{} N\_[i2]{}\ A\^d\_[\^0\_i]{} &=& [g’ 3]{} N\_[i1]{} - [g ]{} N\_[i2]{}\ B\^u\_[\^0\_i]{} &=& [4 3]{} [g’ ]{} N\_[i1]{}\ B\^d\_[\^0\_i]{} &=& -[2 3]{} [g’ ]{} N\_[i1]{}\ A\^d\_[\^\_i]{} && A\_[\^\_[iR]{}]{} = g U\_[i1]{}\ A\^u\_[\^\_i]{} && A\_[\^\_[iL]{}]{} = g V\_[i1]{} where $g$ and $g'$ are the standard $SU(2)$ and $U(1)$ couplings and $u,d$ stand for up and down member of any quark generation. In terms of these couplings we have the following spin-averaged squared matrix elements for (21) and (22). |M\^2\_[g q|q \^0\_i]{} = (A\^[q\^2]{}\_[\^0\_i]{} + B\^[q\^2]{}\_[\^0\_i]{}) |M\^2\_[g q’ |q \^\_i]{} = (A\^[u\^2]{}\_[\^\_i]{} + A\^[d\^2]{}\_[\^\_i]{}) where particle indices have been used for their 4-momenta and we have ignored a common multiplicative constant involving the QCD coupling and colour factor, since it is not relevant for our calculation \[22\]. Note that in the limit $m_{\tilde g} \gg m_{\chi^0_i}, m_{\chi^\pm_i}$ the various partial widths are propertional to the respective factors infront of the square bracket. Thus one gets the following branching fractions as a simple first approximation. B\_[g \^0\_i]{} B\_[g \^\_i]{} . They show that the $SU(2)$ gauge interaction dominates over the $U(1)$ in gluino decay. Consequently the charginos account for a little over 50% of gluino decay and the neutralinos a little under 50%, of which only 12% goes into the $\tilde B$ dominated neutralino and the remainder into the $\tilde W^3$ dominated one. The gluino branching fractions into the different neutralino and chargino states, resulting from (25) and (26), are shown in Table I for the allowed range of the parameter space. At $m_{\tilde g} = 1000$ GeV they are shown only for $\tan\beta = 2$, since in this case the loop induced decay processes are expected to become significant for $\tan\beta = 10$ \[2\]. The branching fractions have been obtained with a common squark mass m\_[q]{} = m\_[g]{} + 200  [GeV]{}, but are insensitive to the choice of this parameter. In fact one can easily check that they are reasonably close to those obtained from the approximate formulae (27) and (28). One should note the following systematic features which will be useful for our subsequent analysis. 1. The branching fraction for direct gluino decay into the LSP has its maximum value for $\tilde B$ dominated LSP, i.e. B\_[g (B)]{} = .15 - .20. It holds for most of the parameter space at $m_{\tilde g} = 300$ GeV and for $\mu \simeq \pm 4 ~m_W$ at the higher values of gluino mass. 2. The largest branching fraction for gluino decay is into the $\tilde W$ dominated chargino state, i.e. B\_[g \^\_i (W)]{} 0.5. It generally corresponds to the lighter (heavier) chargino state for $\mu \simeq \pm 4 ~m_W ~(-m_W)$; but it has a substantial admixture of the heavier (lighter) one at $m_{\tilde g} = 1000~(300)$ GeV as discussed earlier. 3. The second largest branching fraction is into the corresponding $\tilde W$ dominated neutralino, i.e. B\_[g \^0\_i (W)]{} 0.3. It generally corresponds to the second lightest (heaviest) neutralino state for $\mu \simeq \pm 4 ~m_W ~(-m_W)$, but again with the same caveat as above. 4. Thus the $\tilde W$ dominated chargino and neutralino states together account for $\sim 80\%$ of gluino decay. Note that these two states have nearly degenerate mass m\_[\^\_i (W)]{} m\_[\^0\_i (W)]{} throughout the parameter space; and this common mass is also roughly equal to 1/3rd of the gluino mass, as expected from (10). The first equality implies very similar kinematics for the two major decay processes (31) and (32); while the second implies that the kinematics is mainly determined by the gluino mass and not by $\mu$ or $\tan\beta$. As a result one gets a fairly simple and robust signature as we shall see later. 5. The major decay modes of the $\tilde W$ dominated chargino and neutralino states are \^\_i (W) [W ]{} \^0\_[1,2,3]{} q’ |q (), \^0\_i (W) [Z ]{} \^0\_[1,2,3]{} q|q (|). Simple phase space considerations ensure that the decay into the LSP $\chi$ $(\equiv \chi^0_1)$ dominates over most of the parameter space for $m_{\tilde g} = 300$ GeV and at $\mu \simeq \pm 4~m_W$ for heavier gluinos. Note that the leptonic branching fractions of (34) and (35) are about 20% and 6% respectively, where $\ell$ includes both $e$ and $\mu$. These are the primary sources of isolated leptons from cascade decay \[23\]. 6. Finally since all the branching fractions and kinematics of gluino decay discussed above are very similar for $\mu = \pm 4~m_W$, we shall present the resulting signals for only one of these two values. IV.\ We shall concentrate on explicit $R$ parity violation \[8-10\], where the LSP decay arises from one of the following $R\!\!\!/$ Yukawa interaction terms in the Lagrangian. \_[R/]{} = \_[ijk]{} \_i \_j |e\_k + ’\_[ijk]{} \_i q\_j |d\_k + \^\_[ijk]{} |d\_i \_j |u\_k plus analogous terms from the permutation of the supertwiddle. Here $\ell$ and $\bar e$ $(q$ and $\bar u,\bar d)$ denote the left handed lepton doublet and antilepton singlet (quark doublet and antiquark singlet) and $i,j,k$ are the generation indices. The first two terms correspond to $L\!\!\!/$ and the third one to $B\!\!\!/$ Yukawa interaction. While proton stability prohibit simulataneous presence of both these interactions at any level of phenomenological significance, either one of them could be present at a significant level. Thus one has two types of models, corresponding to $L\!\!\!/$ and $B\!\!\!/$. In the $B\!\!\!/$ model d\_i d\_j u\_k, so that the only leptons in the signal are those coming from the cascade decay. Moreover the decay quarks from (37) lead to a stronger isolation cut for these leptons, so that one expects a weaker signal in this case compared to the $R$ conserving SUSY model. On the other hand the $L\!\!\!/$ model implies additional leptons from the LSP decay, resulting in a much stronger signal as we see below. In analogy with the standard Yukawa coupling of quarks and leptons to the higgs boson one expects a hierarchical structure for these Yukawa couplings as well. Thus the dominant LSP decay process is \_i q\_j |d\_k (e\_i u\_j |d\_k + \_i d\_j |d\_k) or \_i \_j |e\_k (e\_i \_j |e\_k + \_i e\_j |e\_k) depending on whether the dominant $L\!\!\!/$ Yukawa coupling is one of the $\lambda'_{ijk}$ or $\lambda_{ijk}$ couplings (in the latter case particle identity requires $i \not= j$). The spin averaged and squared matrix element for the decay process (39) is |M\^2\_[e\_i \_j |e\_k]{} &=& |M\^2\_[\_i e\_j |e\_k]{} = [A\^[e\^2]{}\_(e) (|e) D\^2\_[e]{}]{} + [A\^[\^2]{}\_() (e |e) D\^2\_]{}\ & & + [B\^[e\^2]{}\_(|e) (e ) D\^2\_]{} - [A\^e\_A\^\_G(,e,|e,) D\_[e]{} D\_]{} + [A\^e\_B\^e\_G(,e,,|e) D\_[e]{} D\_]{}\ & &                           + [A\^\_B\^e\_G(,,e,|e) D\_ D\_]{}, D\_[e]{} = m\^2\_[e]{} - (- e)\^2,  G(,e,,|e) = (e) (|e) - () (e |e) + (|e) (e), and A\^e\_= [-g’ ]{} N\_[11]{} - [g ]{} N\_[12]{},  A\^\_= [-g’ ]{} N\_[11]{} + [g ]{} N\_[12]{},  B\^e\_= [-2g’ ]{} N\_[11]{}, where we have dropped a common multiplicative factor involving the $\lambda$ coupling, since it is not relevant for our calculation. Moreover, one can factor out a common denominator assuming a common slepton mass m\_[e]{} = m\_ m\_. We shall be working in this limit. For simplicity we shall present the like sign dilepton signal for the $L\!\!\!/$ SUSY model assuming the leading Yukawa coupling to be $\lambda_{123}$. This corresponds to a lepton (i.e. $e$ and $\mu$) multiplicity of 1 for each LSP decay. The corresponding lepton multiplicities for the choices of different $\lambda$’s as the leading $L\!\!\!/$ Yukawa coupling is listed in Table II. The corresponding LSD signals can be simply obtained by scaling the present signal by the squares of the lepton multiplicities; for the lepton spectrum is insensitive to the detailed structure of the squared matrix element. If the leading $L\!\!\!/$ Yukawa coupling is a $\lambda'$ coupling, then the relevant LSP decay is (38). The corresponding squared matrix elements are easily obtained from (40) by obvious substitutions (see eq. 34 of \[24\]). One should note however that in this case the squared matrix elements for $\chi \rightarrow e u \bar d$ and $\nu d \bar d$ are not identical. Consequently the lepton multiplicity for this LSP decay is 1/2 only if the LSP is a pure $\tilde B$ or $\tilde W$, but not for a general composition of LSP. In the latter case it can be calculated from the relative rates of the two decay processes, i.e. = [A\^[e\^2]{}\_+ A\^[u\^2]{}\_+ B\^[d\^2]{}\_- A\^e\_A\^u\_+ A\^e\_B\^d\_+ A\^u\_B\^d\_A\^[\^2]{}\_+ A\^[d\^2]{}\_+ B\^[d\^2]{}\_- A\^\_A\^d\_+ A\^\_B\^d\_+ A\^d\_B\^d\_]{} assuming $m_{\tilde e} \simeq m_{\tilde q} \gg m_\chi$. The resulting lepton multiplicities for the parameter values of our interest are shown in Table III. The corresponding LSD signals can again be obtained by scaling the ones presented here by the squares of these multiplicities. Although the lepton isolation cut is somewhat stronger in this case it would not degrade the signal substantially. One should note that there would be no $e$ or $\mu$ in LSP decay if the leading $L\!\!\!/$ coupling is a $\lambda'_{3jk}$ or $\lambda'_{i3k}$ \[10\]. The first decay proceeds through $\tau$ or $\nu_\tau$ emission and the second through $\nu_i$ only due to the large top quark mass. Finally it should be noted that we have conservatively assumed the leading $R\!\!\!/$ Yukawa coupling to be $\ll 1$, so that the pair production of superparticles and their decays into LSP are not affected \[10\]. V.\ The gluino pair production cross-section has been calculated for the leading order QCD process \[25\] gg gg using the gluon structure functions of \[26\] with a QCD scale $Q = 2m_{\tilde g}$. Each of the gluinos is assumed to decay into the LSP via the cascade decay processes discussed above. The LSP escapes undetected in the $R$ conserving SUSY model, while it decays in to a baryonic (leptonic) channel in the $B\!\!\!/ ~(L\!\!\!/)$ violating models \[8-10\]. The resulting like sign dilepton signals have been calculated for the LHC energy using a parton level Monte Carlo program. The isolated LSD signals are shown in Figs. 1-5 along with the SM background against the $p_T$ of the 2nd (softer) lepton, with the isolation cut of eq. (6) and a rapidity cut of $|\eta| < 3$ on both the leptons. The SM background, arising from the $b\bar b$ production (crosses) and $t\bar t$ production (histogram), were calculated in \[12\] assuming $m_t = 150$ GeV \[27\]. The former dominates in the small $p_T$ region, while the latter dominates in the large $p_T$ region of our interest. But both are seen to become negligible for $p_{T2} \geq 60$ GeV. The dominant background in this region is expected to arise from the misidentification of one of the lepton charges in t|t \^+\^- X. We have calculated the resulting fake LSD background, assuming it to be about 1% of the above cross-section, i.e. a misidentification of one of the lepton charges at the 1/2% level. To avoid overcrowding, this background has been shown only in Figs. 4 and 5. Fig. 1 shows the isolated LSD signals for a 300 GeV gluino at $\mu = 4~m_W$ and both values of $\tan\beta$. A brief discussion of the signal curves is in order. 1. $R$ conserving Model: The main source of the signal in this case is the squence g [0.5 ]{} \^\_i (W) [0.2 ]{} \^, which has a leptonic branching fraction of 0.10. The corresponding branching fraction for the second largest source g [0.3 ]{} \^0\_i (W) [0.06 ]{} \^+ \^- is effectively 0.036. Moreover the degeneracy relation (33) along with $m_W \simeq m_Z$ imply very similar kinematic distributions for the two final states. Thus one can simply take account of the second source by increasing the branching fraction of the first by 36%. We have followed this prescription in obtaining the LSD signal. Following this procedure we have calculated the dilepton cross-section assuming the decay chain (46) for both the gluinos, with $\chi^\pm_i (\tilde W) = \chi^\pm_1$. The resulting dilepton branching fraction is $\simeq 2\%$. Deviding it by a factor of 2 gives the final LSD signal, shown as the dot-dashed lines. 2. $B\!\!\!/$ Model: The source of the dileptons in this case is the same as above. However, the quarks coming from the LSP decay (37) are included in the isolation cut of the leptons. This results in a substantial depletion of the LSD signal as shown by the short dashed lines. 3. $L\!\!\!/$ Model: The main source of the LSD signal in this case are the leptons from the LSP decay. The hardest component corresponds to the direct decay of each gluino into the LSP, i.e. g [.17-.20 ]{} (B) . Thus the dilepton branching fraction is $\simeq 3-4\%$, i.e. roughly similar to the $R$ conserving case. The resulting LSD signal, shown by the dotted lines, is similar to the later in both shape and size. However, the largest component comes from the decay sequence g [0.8 ]{} \^\_i (W),  \^0\_i (W) for each gluino. This can be combined with the cross-term between (48) and (49), since the 2nd (softer) lepton in either case comes from (49). Hence the combined dilepton branching fraction is $\simeq 1$. The resulting LSD signal is shown by the solid lines. Finally the cross-term between the combined decay sequence (48) and (49) for one gluino and (46) for the other has a dilepton branching fraction of $\simeq 0.2$, which is shown by the long dashed line \[28\]. There is negligible double counting in adding the above two components, since the probability of both the decay leptons coming from the same gluino to populate the large $p_T$ region of interest is negligible. As mentioned earlier, the $L\!\!\!/$ signals have been calculated for the LSP decay mode $\chi \rightarrow \ell \nu \tau$ having a leptonic multiplicity of 1. The corresponding signals for the other $L\!\!\!/$ decays of (38) and (39) can be obtained by multiplying them with the respective leptonic multiplicities of each LSP decay, shown in Tables II and III. As one sees from Fig. 1, the $L\!\!\!/$ LSD signal is clearly large compared to the SM background at large $p_T$. It is also larger than the fake LSD background shown in Fig. 5. The $R$ conserving signal is larger than the first but comparable to the second. Nonetheless it can be easily recognised by the large missing-$p_T$ carried by the $\chi$ and $\nu$ of (46). This is shown in Fig. 6. The $B\!\!\!/$ signal is somewhat larger than the SM background but smaller than that coming from the fake LSD by a factor of $\sim 5$. Thus identifying this signal would require identification of lepton charge to a 0.1% accuracy. Fig. 2 shows the corresponding signals for $\mu = -m_W$. In this case the gluino has significant branching fractions into both $\chi^\pm_1$ and $\chi^\pm_2$; but the former is still the larger one. Moreover if either of the gluinos decays via $\chi^\pm_1$, the softer lepton $p_T$ distribution would correspond to this decay mode. Therefore it is reasonable to approximate $\chi^\pm_i (\tilde W)$ by $\chi^\pm_1$ as in the previous case. Thus the decay sequences and branching fractions are identical to (46-49), except for a marginal reduction of the dilepton branching fraction from (48) from 3 to 2% at $\tan\beta = 10$. A comparison of the signal curves with those of Fig. 1 shows that all of them are qualitatively similar. Thus the 300 GeV gluino signals are seen to be fairly insensitive to the choice of $\mu$ as well as $\tan\beta$. Fig. 3 shows the LSD signals for a 600 GeV gluino at $\mu = 4~m_W$. Again in this case the LSP is dominated by $\tilde B$; and the second lightest neutralino and the lighter chargino are dominated by $\tilde W$. Therefore we can use the gluino decays of (46-49) with the same branching fractions; the dilepton branching fraction from (48) is 3% for both values of $\tan\beta$. The resulting LSD signals are of course smaller and harder than the previous case. Nonetheless the $L\!\!\!/$ and $R$ conserving signals are confortably above the SM background for $p_{T2} \gsim 60$ GeV, while the $B\!\!\!/$ signal is comparable to this background. Moreover the $L\!\!\!/$ violating signal is larger than the fake LSD background as well. Although the $R$ conserving signal is smaller than this background, it can again be distinguished by the large missing-$p_T$ accompanying this signal (Fig. 6). But it would be difficult to identify the $B\!\!\!/$ signal unless the fake LSD background can be further suppressed by an order of magnitude. Fig. 4 presents the corresponding LSD signals for $\mu = -m_W$. Here the gaugino dominated states are the heavier chargino and neutralinos. Consequently the direct decay of gluino into LSP (48) is negligible, while the cascade decays (46) and (47) proceed via $\chi^\pm_2$ and $\chi^0_4$. The signals of Fig. 4 have been obtained with this substitution. One noticable change is the enhancement of the $B\!\!\!/$ signal; the higher mass of $\chi^\pm_2 ~(\chi^0_4)$ ensures the isolation of the decay lepton even in the presence of (37). It should be added here that the $\chi^\pm_2 ~(\chi^0_4)$ decay has significant branching fraction into $\chi^0_2$ \[5\], which has not been taken into account in these curves. However, we have checked its effect on the $R$ conserving LSD signal by replacing $\chi$ by $\chi^0_2$ in (46). It has negligible effect on the lepton momentum spectrum and hence the resulting LSD signal, since $m_{\chi^0_2} - m_\chi \ll m_{\chi^\pm_2} - m_{\chi^0_2}$. It may be noted here that this mass difference is also small compared to $m_{\chi^0_2}$, particularly for $\tan\beta = 2$. Consequently $\chi$ should carry a large part of the $\chi^0_2$ momentum in the cascade decay $\chi^\pm_2 \rightarrow \chi^0_2 \rightarrow \chi$; and hence the resulting $L\!\!\!/$ LSD signal should not be substantially degraded. However, we have not checked this quantitatively since this signal is any way quite large. Finally, a comparison of the signal curves of Figs. 3 and 4 shows that they are quite similar for the $L\!\!\!/$ as well as the $R$ conserving case. The reason of course is that the masses of the respective $\chi^\pm_i (\tilde W)$ states are qualitatively similar, as remarked before. Since the LSD signals for a 1000 GeV gluino are less promising, we have presented them in Fig. 5 for only one set of parameters, i.e. $\mu = 4~m_W$ and $\tan\beta = 2$. It also shows the SM as well as the fake LSD background. The $L\!\!\!/$ signal is larger than the first at large $p_T$ but somewhat below the second. The latter can be reduced below the signal if one can identify lepton charge to within 0.2% accuracy. It should also be possible to separate the two via the accompanying ${p\!\!\!/}_T$ distribution. The size of the $R$ conserving LSD signal is much too low, while the $B\!\!\!/$ signal lies below the scale of this figure. One hopes the size of the $R$ conserving LSD signal to become viable at the SSC energy or the high luminosity option of LHC. Although the signal to background ratio is not expected to improve, the two can be separated via the accompanying ${p\!\!\!/}_T$ distribution (Fig. 6). Of course the canonical missing-$p_T$ signal of a 1000 GeV gluino is expected to be observable even at the low luminosity option of LHC \[2\]. Let us conclude this section by looking at the fate of the LSD signal if squarks are lighter than gluinos. In this case the superparticle production would be dominated by pair production of squarks, in which singlet and doublet pairs occur with equal probability. Since the singlet squarks do not couple to $\tilde W$, the major decay mode is through $\tilde B$ dominated neutralino. Consequently the direct decay into $\chi$ is expected to dominate over a large part of the parameter space \[5\]. The doublet squark decays are very similar to the gluino decays, except that the decay of the chargino pair would always lead to opposite sign dileptons. The largest source of LSD is the decay of one squark via $\chi^\pm_i (\tilde W)$ (46) and the other via $\chi^0_i (\tilde W)$ (47). The end result is a degradation of the $R$ conserving LSD signal by a factor of $\sim 5$, while the $L\!\!\!/$ signal is enhanced compared to the gluino case. VI.\ We have investigated the isolated LSD signal for gluino production at the LHC energy in the $R$ conserving as well as $L$ and $B$ violating SUSY models. The signals are investigated for representative values of gluino mass, $\mu$ and $\tan\beta$ assuming MSSM, against the standard model background as well as the fake LSD background coming from the misidentification of lepton charge. The main results are listed below. 1. The signals are fairly insensitive to the choice of $\tan\beta$ as well as the $\mu$ parameter. Thus for a given gluino mass one has fairly robust signals, which should hold for at least the bulk of the $\tan\beta$ and $\mu$ parameter space. 2. For the $L\!\!\!/$ SUSY model, the LSD signal is larger than both the SM and the fake LSD background for gluino masses of 300 and 600 GeV. Although the $R$ conserving signal is somewhat lower than the latter background, it can be identified via the large missing-$p_T$ accompanying the LSD. The signal is less promising, however, for the $B\!\!\!/$ case. 3. For a gluino mass of 1000 GeV, the $L\!\!\!/$ LSD signal should still be viable. However the size of the $R$ conserving LSD signal is too small in this case, at least for the low luminosity option of LHC. 4. In going from pair production of gluinos to that of squarks one expects a substantial degradition of the LSD signal for the $R$ conserving model while it is expected to be enhanced for the $L\!\!\!/$ model. : We gratefully acknowledge discussions with Amitava Datta, Manual Drees, Rohini Godbole, Graham Ross, Probir Roy and Xerxes Tata. One of us (MG) acknowledges the hospitality of the Theory Group of TIFR during the course of this work. \ 1. For a review see e.g. H. Haber and G. Kane, Phys. Rep. 117, 75 (1985). 2. Report of the Supersymmetry Working Group (C. Albajar et al.), Proc. of ECFA-LHC Workshop, Vol. II, p. 606-683, CERN 90-10 (1990). 3. The missing-$p_T$ signature is used for superparticle search at the LEP $e^+e^-$ collider as well; but it plays a less crucial role there, since most of their constraints on superparticle masses simply follow from the measurement of $Z$ total width. See e.g. ALEPH collaboration, D. Decamp et al., Phys. Rep. 216, 253 (1992). 4. J. Ellis, J. Hagelin, D.V. Nanopoulos, K. Olive and M. Srednicki, Nucl. Phys. B238, 453 (1984). 5. H. Baer, V. Barger, D. Karatas and X. Tata, Phys. Rev., D36, 96 (1987). 6. M. Barnett, J. Gunion and H. Haber, Proc. of 1988 Summer Study on High Energy Physics, Snowmass, Colorado (World Scientific, 1989) p. 230; Proc. of 1990 Summer Study on High Energy Physics, Snowmass, Colorado (World Scientific, 1992) p. 201. 7. H. Baer, X. Tata and J. Woodside, Phys. Rev. D41, 906 (1990). 8. S. Dawson, Nucl. Phys. B261, 297 (1985); S. Dimopoulos and L.J. Hall, Phys. Lett. B207, 210 (1987); S. Dimiopoulos et al., Phys. Rev. D41, 2099 (1990). 9. H. Dreiner and G.G. Ross, Nucl. Phys. B365, 597 (1991). 10. D.P. Roy, Phys. Lett. B283, 270 (1992). 11. CDF collaboration: F. Abe et al., Phys. Rev. Lett. 69, 3439 (1992). 12. N.K. Mondal and D.P. Roy, TIFR TH/93/23, submitted to the Phys. Rev. D. 13. R.M. Godbole, S. Pakvasa and D.P. Roy, Phys. Rev. Lett. 50, 1539 (1983); see also V. Barger, A.D. Martin and R.J.N. Phillips, Phys. Rev. D28, 145 (1990). 14. D.P. Roy, Phys. Lett. B196, 395 (1987). 15. F. Cavanna, D. Denegri and T. Rodrigo, Proc. of ECFA-LHC Workshop, Vol. II, p. 329, CERN 90-10 (1990). 16. F.E. Paige and S.D. Protopopescu, ISAJET Program, BNL-38034 (1986). 17. J.F. Gunion and H. Haber, Nucl. Phys. B272, 1 (1986). 18. M. Guchait, Z. Phys. C57, 157 (1993); see also M.M. Elkheishen, A.A. Shafik and A.A. Aboshousha, Phys. Rev. D45, 4345 (1992). 19. L. Roszkowski, Phys. Lett. B262, 59 (1991). 20. The $\epsilon$ parameter of \[5\] corresponds to $-\mu$. We thank Xerxes Tata for a clarification of this point. The main difference between our Table I and the corresponding Table of \[5\] lies in the range of $\tan\beta$ covered. Besides, the more up to date values of the SM parameters used here make a nonnegligible difference to the result. 21. Only a gauge dominated neutralino is not seriously constrained by the LEP data \[3\] since it does not couple to $Z$. 22. There are also interference terms between the left handed squark and right handed antisquark exchanges (and vice versa) which vanish in the limit $m_{\chi^0_i} ~(m_{\chi^\pm_i}) \ll m_{\tilde g}$. We shall neglect them since the gluino decays mainly into the gauge dominated chargino/neutralinos, for which this mass inequality is reasonably satisfied by eqs. (10) and (11). However we have checked that the gluino branching fractions evaluated by retaining these interference terms \[5\] agree with those shown in Table I to within 10%. 23. There is one exception to this however, which occurs for $m_{\tilde \ell}$ $< m_{\chi^\pm_i (\tilde W)}$ $< m_\chi + m_W$. In this case the chargino decays dominantly into a lepton via the slepton, so that the resulting LSD signal is comparable in size to the canonical missing-$p_T$ signal. This case has been recently investigated by R.M. Barnett, J.F. Gunion and H.E. Haber, LBL-34106 (1993); and H. Baer, C. Kao and X. Tata, FSU-HEP-930527 (1993). 24. J. Butlerworth and H. Dreiner, OUNP-92-15, Nucl. Phys. B (to be published). 25. P.R. Harrison and C.H. Llewellyn Smith, Nucl. Phys. B213, 223 (1983) and B223, 542 (E) (1983); E. Reya and D.P. Roy, Phys. Rev. D32, 645 (1985). 26. M. Gluck, E. Hoffman and E. Reya, Z. Phys. C13, 119 (1982). 27. These LSD background are 8 time larger than the ones shown in \[12\], since they include both $e$ and $\mu$ and both the charge combinations $++$ and $--$. 28. One could effectively include the decay sequence (47) along with (46) by increasing the dilepton branching fraction and hence the normalisation of the long dashed line by 36%, though we have not done this. =25.5cm =-3.5cm =25.5cm =-3.5cm [**Figure Captions**]{}\ 1. The LSD signals for 300 GeV gluino production at LHC at $\mu = 4~m_W$ and $\tan\beta = 2,10$ shown against the $p_T$ of the 2nd (softer) lepton. The dot-dashed and short dashed lines represent the signals in the $R$ conserving and $B$ violating SUSY models, while the solid, long dashed and dotted lines are three components of the signal in the $L$ violating model. The SM background from $t\bar t$ via sequencial decay and from $b\bar b$ via mixing are shown by the histogram and the crosses respectively. 2. The LSD signals for 300 GeV gluino production at $\mu = -m_W$ shown along with the SM background. The conventions are the same as in Fig. 1. 3. The LSD signal for 600 GeV gluino production at $\mu = 4~m_W$ shown along with the SM background. The conventions are the same as in Fig. 1. 4. The LSD signal for 600 GeV gluino production at $\mu = -m_W$ shown along with the SM background, with the same convention as in Fig. 1. The circles denote the fake LSD background from misidentification of one of the lepton charges. 5. The LSD signal for 1000 GeV gluino production at $\mu = 4~m_W$ and $\tan\beta = 2$ shown along with the SM background, with the same convention as in Fig. 1. The circles denote the fake LSD background. 6. The missing-$p_T$ $({p\!\!\!/}_T)$ distribution of the LSD signals for the $R$ conserving model are shown for gluino masses of 300, 600 and 1000 GeV with $p^\ell_T > 20$ GeV (solid lines) along with the fake LSD background (dashed line). The ${p\!\!\!/}_T$ distribution of the SM background from $t\bar t$ is marginally softer than the latter, while that from $b\bar b$ is extremely soft.

--- abstract: 'An isolated and fast decayed active region (NOAA 9729) was observed when passing through solar disk. There is only one CME related with it that give us a good opportunity to investigate the whole process of the CME. Filament in this active region rises up rapidly and then hesitates and disintegrates into flare loops. The rising filament from EIT images separates into two parts just before eruption. A new filament reforms several hours later after CME and the axis of this new one rotates clockwise about $22^{\circ}$ comparing with that of the former one. We also observed a bright transient J-shaped X-ray sigmoid immediately appears after filament eruption. It quickly develops into a soft X-ray cusp and rises up firstly then drops down. Two magnetic cancelation regions have been observed clearly just before filament eruption. Moreover, the magnetic flux rope erupted as the magnetic helicity approach the maximum and the normalized helicity is -0.036 when the magnetic flux rope erupted, which is close to the prediction value of @Zhang08 based on the theoretical non-linear force-free model.' address: - 'Key Laboratory of Solar Activity, National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China' - 'Jilin Normal University, 136000 Siping, Jilin Province, China' - 'Shijiazhuang University, 050035 Shijiazhuang, Hebei Province, China' author: - Shangbin Yang - Wenbin Xie - Jihong Liu title: Eruption of the magnetic flux rope in a fast decayed active region --- Sun; Magnetic helicity; CMEs; filament eruption =0.5 cm Introduction {#sec:Introduction} ============ Coronal mass ejections (CMEs) have been one of the outstanding problems of solar physics. The nature of driver and initiation mechanism for the sudden explosive release of the stored free magnetic energy are still unclear. It is proposed that CMEs are that metastable magnetic configurations are followed by some finite perturbation or some additional energy build-up and make a eventual catastrophic transition to a lower energy state or to non-equilibrium state [Ref. @PF02; @Lin03]. There are already several mechanisms to cause the catastrophic transition have been proposed such as slow reduction of the overlying flux [@FP95; @PF90], photospheric converging and shear motions [@Fob94; @Ant94], flux emergence [@Fey95], etc. As a candidate for the metastable configuration, @Stu01 considered a long twisted flux tube, anchored at both ends in the photosphere with overlying magnetic arcade. They argue from a simple order-of-magnitude calculation that part of the flux tube will open up to infinity if there are 1-2 full winds for each field line in the flux tube even this configuration is stable according to linear MHD stability theory. @Fan05 carried out simulations in a spherical geometry of the evolution of coronal magnetic field as an a twisted magnetic flux rope emerges slowly into a preexisting coronal potential arcade field. She find that the flux tube becomes kinked and ruptures through the arcade field and cause a eruption when the twist in the emerged tube reaches a critical amount. @TK05 use the flux rope model of @TD99 as the initial condition to get a simulation which have a good agreement with the development of helical shape and the rise profile of a failed filament eruption described by @Ji03. Kinking movement in the eruption of filament are also usually observed [@Liu07; @Liu08]. In addition, soft X-ray images of solar active regions frequently show S- or inverse-S (sigmoidal) morphology. @Can99 found that active regions containing X-ray sigmoids are more likely to erupt and many eruption also associated with sigmoid structure. While the question of whether there exit highly twisted flux ropes with more than one wind between anchored ends susceptible to the kink instability as precursors for eruptive flux tube remains a topic of debate [e.g. @RK96; @Lea03; @Lek05; @RL05] when these authors investigate the relation between the shape of X-ray sigmoid and eruption. @GF06 demonstrated the partial expulsion of a three dimensional magnetic flux rope erupts when enough twist has emerged to induce a loss of equilibrium. After multiple reconnections at current sheets that form during the eruption, the rope breaks in two, so that only a part of it escapes. The “degree of emergence” of a pre-eruption flux rope, whether it possess bald-patch (BP) or whether it is high enough in the coronal to possess an X-line determines whether the rope is expelled totally or partially. Their simulation result is well consistent with the partial eruption model described by @Gil01. But should a twisted or kinked or magnetic flux tube need kink instability or twist over such threshold to eruption? Is it kink instability or kink-induced instability in a filament eruption [@Gil07]? For example, the kink instability may be occurring in conjunction with a breakout scenario [@Wil05]. @Low01 also argued that a force-free magnetic field in the unbounded space outside a sphere cannot be in equilibrium if the amount of detached flux is too large compared to the amount of anchored flux. Eruption of magnetic flux tube can also be caused by another instabilities such as torus instability [@KT06] and Ballooning instability [e.g. @Fon01]. So it is essential to investigate the whole eruptive process of a magnetic flux tube since emergence and evolution of structure in different wavelength especially the soft X-ray sigmoid for study how the metastable state is established and how to loss equilibrium and erupt for this magnetic flux tube. On the other hand, @Zhang06 pointed out that the accumulation of magnetic helicity in the corona plays a significant role in storing magnetic energy. They propose a conjecture that there is an upper bound on the total magnetic helicity that a force-free field can contain. The accumulation of magnetic helicity in excess of this upper bound would initiate a non-equilibrium situation, resulting in a CME expulsion as a natural product of coronal evolution. @Ber84 proposed the concept of relative magnetic helicity and the formula to get the accumulated magnetic helicity across a surface into a volume from the movement in that surface. @Cha01 firstly calculated the accumulated magnetic helicity by applying LCT (Local Correlation Tracking) to MDI data. However, the relation between accumulated magnetic helicity and eruption of magnetic flux tube has not been checked in the past work. What is the accumulated helicity for a emerging flux tube when it erupts? We need to investigate the process of magnetic helicity accumulation in an eruption. NOAA 9729 is an active region emerging on Dec. 05, 2001. Its initial tilt angle is almost perpendicular to the equator of sun and then develop to a bipolar active region. It rapidly dispersed in the following days and disappeared on Dec. 08, 2001 from white light (WL) image. The evolution of such photospheric concentrations are usually been explained in terms of the rising of very distorted flux tubes. @wea70 [@wea72] also noticed the almost random distribution of the starting tilt of emerging bipoles, which subsequently became more parallel to the equator, and proposed that this was caused by the emergence of twisted flux tubes. Especially, a kinked flux tube arches upward and evolves into a buckled loop with a local change of tube orientation at the loop apex that exceeds 90 degrees from the original direction of the tube @Fan99. The characteristic of this emerging active region implies that there are strong twist in it. Moreover, there is only one CME related with it since emergence. This give us a good opportunity to investigate the whole process of a CME for a strong twisted flux tube. In this paper we use multiple wavelengths and instruments to investigate the whole CME process. We also calculate the accumulated magnetic helicity since this active region emerged. We give a quick survey of the instrument and data sets used in this study (Sec. \[sec:data\]). We describe the evolution of white light and emerging speed in Sec.  \[sec:wl\]. we describe the evolution of H-alpha, EIT, X-ray, and corresponding CMEs from Sec. \[sec:halpha\] to \[sec:CME\]. We calculate accumulated magnetic helicity in Sec. \[sec:helicity\]. The summary and discussion is represented in Sec. \[sec:summary\]. OBSERVATIONS AND DATA ANALYSIS {#sec:observation} ============================== Instrumentation and Data {#sec:data} ------------------------ We use data of SOHO/MDI to investigate the evolution of White light image and line-of-sight magnetograms [@Sche95]. We use data of high-resolution global $H\alpha$ network to investigate the evolution of Chromosphere. We use data of Extreme Ultraviolet Imaging Telescope to investigate the evolution of low Corona [EIT; @Del95]. We use soft X-ray images obtained with the Soft X-ray Telescope (SXT) on board the Yohkoh satellite to investigate the evolution of high Corona [@Tsu91]. We use Large Angle Spectrometric Coronagraph [LASCO; @Bru95] aboard SOHO to investigate the CME associated with this active region. Estimation of emerging speed {#sec:wl} ---------------------------- NOAA 09729 emerged from the convective zone at about 03:26UT on December 05, 2001 seen from WL image. It developed to a bipolar active region rapidly and disappeared on Dec. 08, 2001. The evolution of tilt angle and the distance between two polarities considering the barycenters of the leading and following polarities are shown in Fig. \[fig:DT\]. Its initial title angle is almost perpendicular to the equator of sun and still disobeyed Joy’s law when this active region disappeared despite the connecting line between two polarities became more parallel to the equator. The maximum distance ($dmax$) between two polarities is at 07:06UT on December 7. If we suppose the coronal part of one active region is represented by a single semicircular loop. The initial distance ($dmin$) will be the chord of the semicircle and the maximum distance ($dmax$) will be the diameter of this semicircle. The emerging hight in this time interval is approximately $\sqrt{(dmax ^2-dmin ^2)/4}=30.6Mm$ under this assumption and the corresponding average emerging speed is about 0.164km/s. The mean magnetic field in this active region is about 100 Gauss and the corresponding ${\textrm{Alfv}\acute{\textrm{e}}\textrm{n}}$ speed at photosphere is about 8.9Km/s . The emerging speed will be about 0.018 of local $V_A$. Fig. \[fig:mdi\_WL\] shows the time sequence of WL images. ![Evolution of tilt angle (dashed line) and distance (solid line) between two polarities.[]{data-label="fig:DT"}](WL_DT.eps) $\textrm{H}{\alpha}$ Evolution {#sec:halpha} ------------------------------ Fig. \[fig:halpha\] shows the evolution of NOAA 9729 in $\textrm{H}{\alpha}$. Top two rows (Fig. \[fig:halpha\]a-f) show the evolution of $\textrm{H}{\alpha}$ from December 05 to 09, 2001. The filament in this active region didn’t appeared before 22:30 UT December 4. At 16:59UT on December 5, a inverse-S shaped filament has appeared in the $\textrm{H}{\alpha}$ locating above polarity inversion line (PIL) which is shown in Fig. \[fig:halpha\]b. The shape of this filament didn’t changed a lot after this moment while the middle channel of this filament labeled by red solid line in Fig. \[fig:halpha\]c separated into several parts and is not clear than the initial one on December 5. The filament still existed after eruption at about 02:34UT on December 7 and there is an obvious clockwise rotation of the middle channel of the filament. The clockwise rotation angle of the filament middle channel is approximately $22^{\circ}$ denoted in the Fig. \[fig:halpha\]. The filament almost kept the same shape in the next two days after this active region disappeared from WL, which can be seen in the Fig. \[fig:halpha\] (e-f). Bottom two rows (Fig. \[fig:halpha\]g-n) show the evolution detail in $\textrm{H}{\alpha}$ when eruption occurred on December 07, 2001. There are two stages for the filament eruption. Firstly, the left part in the middle filament channel became thin and disappeared gradually from Fig. \[fig:halpha\]g-j. The left part of the middle filament channel has disappeared entirely at 01:45UT. The right part of the filament disappeared totally before 02:12UT in Fig. \[fig:halpha\]. One interesting phenomena is that the right part of filament became thicker before eruption which is different with the left one. At 03:41UT after eruption, two bright foot-points regions have appeared and the brightness decreased with time. The shape of the two bright regions is consistent with cancelation region of magnetic field region in Fig. \[fig:mdi\]f. The filament above the polarity inversion line (PIL) also gradually appeared again after the eruption, which is shown from Fig. \[fig:halpha\]l-n. EIT evolution {#sec:EIT} ------------- Fig. 4 shows the detail evolution of NOAA 9729 EIT images in the 171 [Å]{} filter when the eruption occurred on December 7. The filament in EUV wavelength can be seen clearly from the Fig. \[fig:EIT\]. There are similar stages for the filament eruption from EUV comparing with that in $\textrm{H}{\alpha}$. Two square boxes in Fig.  \[fig:EIT\]a denotes two EUV bright points. The left part risen up labeled by red arrow in Fig. \[fig:EIT\]b and it did not erupt immediately. In the same time, The upper EUV bright point disappeared . The right part of the filament risen up subsequently, which is labeled by blue arrows in Fig. \[fig:EIT\]b-c. In Fig. \[fig:EIT\]d, the risen filament separated in to two parts that labeled by two red arrows. At 02:34UT on December 07, the filament eruption occurred. Fig. \[fig:EIT\]e shows the rising filament and a kinked-like structure can also been found. After eruption, two bright EUV regions formed on the two foot-points of this flux tube which is labeled by red boxes in Fig. \[fig:EIT\]f. At 03:10UT, a bright region denoted by red boxes above the inversion line also has formed subsequently and it expanded along the inversion line to the opposite directions and post-flare loops formed in the same time, which is shown from Fig. \[fig:EIT\]h-j. In the following several hours, the formed post-flare loop continued rising up and it became cooling gradually and disappeared at last from Fig. \[fig:EIT\]k-l. Yohkoh X-ray Evolution {#sec:xray} ---------------------- Fig. \[fig:xrt\] shows the evolution of NOAA 9729 in soft xray. An Inverse S-shaped sigmoid structure formed clearly since December 05. Such inverse S-shaped sigmoid is an indication of the presence of negative twist in the magnetic field (Rust & Kumar 1996; Pevtsov et al. 2001). The negative chirality is the same as that deduced from the shape of filament in Fig. \[fig:halpha\]. The shape of this sigmoid didn’t change a lot in the following days before eruption and it rotated clockwise which is consistent with the rotation sense of connecting line of the two foot-points from WL image as presented in Fig. \[fig:mdi\_WL\]. An X-ray bright point existed long time before eruption as labeled by the white square box in Fig. \[fig:xrt\]f-i. At about 00:51UT on December 7, the left part of the sigmoid risen up as denoted by the green arrow in Fig. 4m. The time is consistent with the eruption of left part of the filament in the $\textrm{H}{\alpha}$ images Fig. \[fig:halpha\]g-j and the EUV images Fig. \[fig:EIT\]b. At 02:29UT, the sigmoid has erupted and a transient J-shaped sigmoid formed at the same time. This transient J-shaped sigmoid existed earlier than the two bright EUV foot-points denoted by the two red square boxes in Fig. \[fig:EIT\]f and it became broad subsequently which can be seen from Fig. \[fig:EIT\]o. At 03:53UT on December 07 in Fig. \[fig:EIT\]p, a X-ray cusp has formed and it risen up firstly as labeled by white arrow in Fig. \[fig:EIT\]q. This cusp was diffused at last, which can be seen in Fig. \[fig:EIT\]r. CME evolution {#sec:CME} ------------- Eruption of this active region brought one partial halo CME which is recorded in CME catalog maintained at the CDAW Data Center. The first appearance in the LASCO/C2 field of view (FOV) is at 03:06UT on December 7 and the central position angle (CPA) is $343^ \circ$. According to the description in this CME catalog list, the linear speed obtained by fitting a straight line to the height-time measurements is 803.2km/s. The acceleration of such CME is -51.01$m/s^2$. This CME slows down within the LASCO FOV while it has escaped out of ten solar radius. The eruptive part of the filament escaped successfully. The speed value and deceleration satisfy the character of the fast CMEs. Fast CMEs originate from an active region and their initial speeds are well above the CME median speed, 400 Km/s. They show no significant acceleration, but may show some deceleration [@Cyr00]. Evolution of magnetic field and helicity {#sec:helicity} ---------------------------------------- Fig. \[fig:mdi\] shows the evolution of line-of-sight magnetic field of this active region. Two magnetic cancelation regions along the PIL about four hours before flux tube eruption are labeled by red circles in fig.3e. After the eruption the two cancelation regions disappeared. The accumulated magnetic helicity was calculated using full-disk line-of-sight magnetograms taken by SOHO/MDI. From the photospheric magnetic field observations the helicity flux across the photosphere **S** can be calculated by $${\frac{dH_{R}}{dt}=-2\int(\vec{A}_{p}\cdot\vec{U}){B_n}\vec{dS},} \label{eq:Hrate}$$ where $\vec{U}$ denotes the horizontal velocity field. The vector potential $\vec{A}_{p}$ is obtained by applying Local Correlation Tracking (LCT) and Fast Fourier Transforms (FFT) to the normal components of the photospheric magnetic field $B_n$ [@Cha01]. After applying nonlinear mapping and flux density interpolation the geometrical foreshortening was corrected [@Liu06; @Yang09b]. To reduce the noise, we set the horizontal velocity to zero in regions where the magnetic field is small (${< 10 G}$). In order to better track the emerging regions and to exclude the effect of relative quiet regions outside the emergence sites, we set the horizontal velocity to zero in regions of a weak cross-correlation (${<0.9}$) of two magnetograms. We calculate the accumulated magnetic helicity as $${H(t)=\int_{0}^{t}\frac{dH_{R}(t)}{dt}dt}$$ where the starting moment of time t=0 corresponds to the beginning emergence of the active regions. Further, the following definition of the normalized magnetic helicity $H_{norm}$ is used: $$H_{norm}(t) = \frac{{|H_m(t)|}}{{\Phi _m ^2 }}, \label{eq:Hnorm}$$ where $\Phi_m$ is the maximum absolute value of the magnetic flux through the photosphere of the studied newly emerging AR. Fig. \[fig:hm\_norm\] depicts the evolution of parameter that the total relative magnetic helicity normalized by one-half of the maximum sum of the unsigned positive and negative magnetic fluxes $\phi$. This emerging flux tube take negative magnetic helicity to the corona. The flux tube in AR 9729 erupted at 02:34 UT on December 07. Just at this moment, this normalized parameter is -0.036. Recently, new methods have been used to calculate the horizontal motion of magnetic structures on the photosphere, in which the evolution of vertical magnetic fields satisfy the ideal induction equation [e.g. @Wel07]. The rotation motions of the magnetic features, is introduced to get a more precise measurement of the magnetic helicity [@Par05; @Sch11; @Rom11]. The difference of magnetic helicity fluxes obtained from different methods is usually within 15% [@Rom11]. The estimate the final normalized helicity is $-0.036\pm0.0054$ when the magnetic flux tube erupted in AR 9729. SUMMARY AND DISCUSSION {#sec:summary} ====================== In this paper, we have presented a multiple wavelengths study of emergence of a fast decayed active region NOAA 9729 and associated eruption. Our main observation results are summarized as follows. 1\. NOAA 9729 emerged from convective zone with initial tilt angle is almost perpendicular to the equator of sun. The connecting line between leading polarity and following polarity rotated clockwise. This active region still disobeyed Joy’s law when it disappeared from WL. 2\. The filament in $\textrm{H}\alpha$, EUV structure and sigmoid in X-ray all show the same inverse S-shape. This is a indication of negative helicity in the flux tube. The eruption stages in the three wavelength are also similar. The left part of the structure risen up and the right part risen subsequently. Then, the eruption occurred as the pre-eruption state evolved. 3\. A filament in $\textrm{H}\alpha$ reformed again after erutpion. The middle channel of filament in $\textrm{H}\alpha$ rotated clockwise about $22^\circ$ than before. 4\. EUV structure separated into two parts before eruption. The sudden rising flux tube in the eruption showed a kinked structure. The EUV bright regions disappeared after the flux tube risen up. The risen flux tube dropped down and flare loops formed. The intimal bright point formed above the PIL and expanded along the two opposite directions of this PIL. 5\. Inverse S-shaped sigmoid structure became thinner before eruption. A transient J-shaped sigmoid structure formed subsequently after eruption just before EUV bright region formed above the PIL formed. 6\. The associated CME recorded is a partial halo CME. The linear speed obtained by fitting a straight line to the height-time measurements is 803.2km/s. The acceleration of such CME is -51.01$m/s^2$. Associated CME in the eruption belongs to the fast CME. 7\. From line-of-sight magnetic field, two magnetic cancelation regions just underneath filament can be found before eruption and these regions disappeared after eruption. Negative helicity was taken by the emergence of magnetic flux and differential rotation. The parameter that the total relative magnetic helicity normalized by one-half of the maximum sum of the unsigned positive and negative magnetic fluxes is -0.036 when eruption taken place. What’s the metastable structure before eruption? Firstly, we can deduced it as a negative twist flux tube. As described in the sum. 2, all structures in the emerging flux tube showed a inverse S-shaped structure. These type of structures are associated with negative twist magnetic flux tube. The negative accumulated magnetic helicity also implies existence of negative twist in the flux tube. Secondly, we can deduce this flux tube is a kinked flux tube. As the description in the sum. 1, this active region disobeyed Joy’s law and the tilt angle rotated clockwise. If we consider this flux tube as a simple flux tube described in @Lop03, such type of flux tube should have a negative writhe (right hand) flux tube. In the framework of a kink instability model as simulated by @Fan99 , also noted in @Lek96 and @Lin98, the sign of the writhe of kinked flux tubes would be the same as that of the twist within the tubes due to conservation of helicity. When a horizontal flux tube is emerging twisted right-handed (left-handed) of negative (positive) magnetic helicity, the writhe of the tube axis resulting from kink instability is also right-handed (left-handed). This would leads to a clockwise (counter-clockwise) rotation of the apex portion of the rising tube as viewed from the top. Such scenario well explains the evolution in sum. 1. [@Yang09b] also pointed out that the emerging flux tube tends to be caused by kink instability if the accumulated helicity and writhe have the same sign. Hence, we conclude that the metastable structure before eruption is a kinked flux tube. What’s the trigger mechanism for this kinked flux tube? A kinked flux tube can erupts because of kink instability. There are already some evidences from observational results such as @RL05 and @Liu07 [@Liu08] or from simulation results such as @Fan05 and @TK05. Another possible trigger mechanism is the torus instability. When the confining poloidal field decreases with distance fast enough, radially outward perturbations of the flux rope could trigger the torus instability [@KT06] before the helical kink instability set in, and the toroidal flux rope would no longer be confined. @Fan07 performed 3D simulations to investigate two distinct mechanisms that led to the eruption of the flux tube. One case (case K) is the emerging flux rope is kinked and a kink instability develops in it, leading to an eruption at last. The other one (case T) is the overlying field declines more rapidly with height, and the emerging flux rope is found to lose equilibrium and erupt via the torus instability. The corresponding normal relative magnetic helicity $H_m/\Phi ^2$ reaches approximately -0.16 for case K and approximately -0.18 for case T. However, in our observation result the normalized magnetic helicity is about -0.036 when the flux tube in AR 9729 erupted, which is one order smaller than the simulation results. @Gil01 explored the various magnetic configurations for failed and partial eruptions of filaments and cavities. They suggested that reconnection at different positions of the flux rope that threads into the filament creates different topologies with implications of full, partial, or failed filament eruptions. Reconnection occurs within the prominence can cause partial filament to erupt and the rest part survive after eruption. @Moo01 also proposed a 3D Tether-Cutting model to explain the eruption process of a sigmoid structure. They conjecture that the magnetic explosion was unleashed by runaway tether-cutting via implosive/explosive reconnection in the middle of the sigmoid, as in the standard model. This internal reconnection apparently begins at the very start of the sigmoid eruption and grows in step with the explosion,caused the explosion to be ejective. In our observation, the bright point in the middle of sigmoid structure could be found clearly. Two magnetic filed cancelation regions also existed before eruption and disappeared after eruption reflect the existence of reconnection there. Therefore we propose that field lines rooted to the photosphere near the inversion line where for the formation of a magnetic tangential discontinuity are locally reconnection and cause an instability. Field lines above the surface are detached from the photosphere to form this CME and partially open the field which make the filament loses equilibrium to rise quickly and then be drawn back by the tension force of magnetic field after eruption to form a new filament in our observation. The eruption happened when the accumulated magnetic approach near the maximum. It probably related to the possible existence of upper bounds of total relative magnetic helicity for force-free magnetic field in unbounded space, as conjectured in @Zhang06 on the study of axis-symmetric force-free field solutions. 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--- abstract: 'Solving a delegation graph for transitive votes is already a non-trivial task for many programmers. When extending the current main paradigm, where each voter can only appoint a single transitive delegation, to a system where each vote can be separated over multiple delegations, solving the delegation graph becomes even harder. This article presents a solution of an example graph, and a non-formal proof of why this algorithm works.' author: - Jonas Degrave bibliography: - 'must.bib' title: 'Resolving multi-proxy transitive vote delegation' --- Introduction {#introduction .unnumbered} ============ In the area of voting systems, there is a growing interest into the topic of liquid democracy, in which votes can be delegated to others in order to deal with the complexity of the topic at hand. In these voting systems, many currently use a single proxy system, in which you delegate your vote to one other user. This other user can in its turn further delegate his and your vote in a transitive chain. However, when the user who receives the delegated vote decides not to use it (possibly unintentionally), the vote goes to waste. However, there is no mathematical need for a user to only delegate to a single other user. In this paper, I propose a multi-proxy delegation system. This system fulfills the following requirements: - Every user has one vote, which can be delegated. - The delegation is transitive, meaning that it can be further delegated. - This vote is distributed equally between all the user’s direct delegates who actually vote, potentially through further delegations. This approach can be used in addition to vote counting systems, as long as each vote in the counting system can be weighted with the total number of votes each user received. \(A) [A]{}; (B) \[right of=A\] [B]{}; (C) \[below of=A\] [C]{}; (D) \[right of=C\] [D]{}; (E) \[below of=C\] [E]{}; (F) \[right of=E\] [F]{}; (G) \[below of=E\] [G]{}; (H) \[right of=G\] [H]{}; (I) \[above right of=H\] [I]{}; (J) \[right of=H\] [J]{}; (K) \[above of=I\] [K]{}; (L) \[above right of=I\] [L]{}; (M) \[right of=I\] [M]{}; (N) \[below right of=J\] [N]{}; \(O) \[below of=G\] [O]{}; (P) \[right of=O\] [P]{}; (Q) \[below right of=O\] [Q]{}; (R) \[right of=P\] [R]{}; (S) \[right of=Q\] [S]{}; \(T) \[below of=Q\] [T]{}; (U) \[right of=T\] [U]{}; (V) \[below right of=U\] [V]{}; (W) \[below left of=V\] [W]{}; (X) \[left of=W\] [X]{}; (Y) \[above left of=X\] [Y]{}; \(A) edge node \[left\] (B) (C) edge \[bend left\] node \[left\] (D) (D) edge \[bend left\] node \[right\] (C) (E) edge \[bend left\] node \[left\] (F) (F) edge \[bend left\] node \[right\] (E) (G) edge node \[right\] (H) (H) edge node \[right\] (I) edge node \[right\] (J) (I) edge node \[right\] (K) edge node \[right\] (L) edge node \[right\] (M) (J) edge node \[right\] (N) edge node \[right\] (M) (O) edge node \[right\] (P) (P) edge node \[right\] (Q) edge node \[right\] (R) (Q) edge node \[right\] (O) edge node \[right\] (S) \(T) edge \[bend left\] node (U) edge \[bend left\] node (V) edge \[bend left\] node (W) edge \[bend left\] node (X) edge \[bend left\] node (Y) (U) edge \[bend left\] node (V) edge \[bend left\] node (W) edge \[bend left\] node (X) edge \[bend left\] node (Y) edge \[bend left\] node (T) (V) edge \[bend left\] node (W) edge \[bend left\] node (X) edge \[bend left\] node (Y) edge \[bend left\] node (T) edge \[bend left\] node (U) (W) edge \[bend left\] node (X) edge \[bend left\] node (Y) edge \[bend left\] node (T) edge \[bend left\] node (U) edge \[bend left\] node (V) (X) edge \[bend left\] node (Y) edge \[bend left\] node (T) edge \[bend left\] node (U) edge \[bend left\] node (V) edge \[bend left\] node (W) (Y) edge \[bend left\] node (T) edge \[bend left\] node (U) edge \[bend left\] node (V) edge \[bend left\] node (W) edge \[bend left\] node (X) ; Preparing the example {#preparing-the-example .unnumbered} ===================== Suppose we have the following delegation graph for 25 people, as presented in Figure \[fig:graph:original\]. These people either vote themselves, and are shown as green nodes, or they did not vote, and are shown as blue nodes. We chose this example to show all relevant difficulties encountered in solving the delegation graph. In order to tackle the problem using a flow graph, there are still multiple problems which need fixing. Therefore, we need the following initial steps: - Remove all edges starting from people who vote themselves. - Descend the tree starting from the people who vote themselves, and collect all nodes encountered this way. Remove all nodes not encountered from this descend, as their vote cannot be cast in a meaningful way. After this process, we end up with the graph depicted in Figure \[fig:graph:simplified\]. \(A) [$A$]{}; (C) \[below of=A\] [$C$]{}; (D) \[right of=C\] [$D$]{}; (E) \[below of=C\] [$E$]{}; (F) \[right of=E\] [$F$]{}; (G) \[below of=E\] [$G$]{}; (H) \[right of=G\] [$H$]{}; (I) \[above right of=H\] [$I$]{}; (J) \[right of=H\] [$J$]{}; (K) \[above of=I\] [$K$]{}; (L) \[above right of=I\] [$L$]{}; (M) \[right of=I\] [$M$]{}; (N) \[below right of=J\] [$N$]{}; \(O) \[below of=G\] [$O$]{}; (P) \[right of=O\] [$P$]{}; (Q) \[below right of=O\] [$Q$]{}; (R) \[right of=P\] [$R$]{}; (S) \[right of=Q\] [$S$]{}; \(T) \[below of=Q\] [$T$]{}; (U) \[right of=T\] [$U$]{}; (V) \[below right of=U\] [$V$]{}; (W) \[below left of=V\] [$W$]{}; (X) \[left of=W\] [$X$]{}; (Y) \[above left of=X\] [$Y$]{}; \(C) edge \[bend left\] node \[left\] (D) (G) edge node \[right\] (H) (H) edge node \[right\] (I) edge node \[right\] (J) (I) edge node \[right\] (K) edge node \[right\] (L) edge node \[right\] (M) (J) edge node \[right\] (N) edge node \[right\] (M) (O) edge node \[right\] (P) (P) edge node \[right\] (Q) edge node \[right\] (R) (Q) edge node \[right\] (O) edge node \[right\] (S) \(T) edge \[bend left\] node (U) edge \[bend left\] node (V) edge \[bend left\] node (W) edge \[bend left\] node (X) edge \[bend left\] node (Y) (U) edge \[bend left\] node (V) edge \[bend left\] node (W) edge \[bend left\] node (X) edge \[bend left\] node (Y) edge \[bend left\] node (T) (V) edge \[bend left\] node (W) edge \[bend left\] node (X) edge \[bend left\] node (Y) edge \[bend left\] node (T) edge \[bend left\] node (U) (W) edge \[bend left\] node (X) edge \[bend left\] node (Y) edge \[bend left\] node (T) edge \[bend left\] node (U) edge \[bend left\] node (V) ; Solving the delegation graph: the first method {#solving-the-delegation-graph-the-first-method .unnumbered} ============================================== In this section, I will introduce a first approach to solving this delegation graph, based on solving a system of linear equations. As you can see, $D$ receives one vote from $C$ in addition to the vote he already had. Therefore: $$D = 1 + C.$$ M started with a vote, and receives one third of a vote from I, and half of a vote from J: $$M = 1 + \frac{1}{3} I + \frac{1}{2} J$$ If we do this for all votes, we end up with the following set of linear equations. [$$\begin{aligned} A &= 1 &\\ C &= 1 &\\ D &= 1 + C &\\ E &= 1 &\\ F &= 1 &\\ G &= 1 &\\ H &= 1 + G &\\ I &= 1 + \frac{1}{2} H &\\ J &= 1 + \frac{1}{2} H &\\ K &= 1 + \frac{1}{3} I &\\ L &= 1 + \frac{1}{3} I &\\ M &= 1 + \frac{1}{3} I + \frac{1}{2} J &\\ N &= 1 + \frac{1}{2} J &\\ O &= 1 + \frac{1}{2} Q &\\ P &= 1 + O &\\ Q &= 1 + \frac{1}{2} P &\\ R &= 1 + \frac{1}{2} P &\\ S &= 1 + \frac{1}{2} Q &\\ T &= 1 + \frac{1}{5} U + \frac{1}{5} V + \frac{1}{5} W &\\ U &= 1 + \frac{1}{5} T + \frac{1}{5} V + \frac{1}{5} W &\\ V &= 1 + \frac{1}{5} T + \frac{1}{5} U + \frac{1}{5} W &\\ W &= 1 + \frac{1}{5} U + \frac{1}{5} V + \frac{1}{5} W &\\ X &= 1 + \frac{1}{5} T + \frac{1}{5} U + \frac{1}{5} V + \frac{1}{5} W &\\ Y &= 1 + \frac{1}{5} T + \frac{1}{5} U + \frac{1}{5} V + \frac{1}{5} W &\\\end{aligned}$$ ]{} We can solve system this linear system exactly for all variable $A$ through $Y$, by converting it to a a matrix system, as shown in equation \[eq:problem\]. This system can easily be solved by inverting the matrix ${\mathbf{B}}$, if it is not singular, and ${\mathbf{B}}$ is never singular, since it represents a bijective transformation. $${\mathbf{S}} = {\mathbf{B}}^{-1}\cdot{\mathbf{J}},\label{eq:path1}$$ This solution is shown in equation \[eq:solution\]. Solving the delegation graph: the second method {#solving-the-delegation-graph-the-second-method .unnumbered} =============================================== A second way to look at this, is by constructing the adjacency matrix $D$ of our directed graph. Now, we know that after zero steps in the delegation chain ${\mathbf{S}} = {\mathbf{J}}$. After one step in the delegation chain ${\mathbf{S}} = {\mathbf{A}}.{\mathbf{J}}$, after two steps ${\mathbf{S}} = {\mathbf{A}}^2.{\mathbf{J}}$, and so on. Therefore the total number of votes everybody has after an infinite number of delegation steps is $${\mathbf{S}} = \lim_{n \to +\infty}\sum\limits_{i=0}^n {\mathbf{A}}^n\cdot{\mathbf{J}},$$ and since $\displaystyle\lim_{n \to +\infty}{\mathbf{A}}^n={\mathbf{0}}$, $${\mathbf{S}} = \Bigg(\lim_{n \to +\infty}\sum\limits_{i=0}^n {\mathbf{A}}^n\Bigg)\cdot{\mathbf{J}}.$$ Because this is a Neumann series $${\mathbf{S}} = ({\mathbf{I}}-{\mathbf{A}})^{-1}\cdot{\mathbf{J}}. \label{eq:path2}$$ Note that ${\mathbf{B}}={\mathbf{I}}-{\mathbf{A}}$, and hence, the solutions in \[eq:path1\] and \[eq:path2\] are identical. How to interprete the results {#how-to-interprete-the-results .unnumbered} ============================= We could just look at the matrix ${\mathbf{S}}$ to find out the total amount of votes the people who actually voted received from the delegation graph. It might be tempting to interpret the elements of ${\mathbf{S}}$ of nodes not voting, as the number of votes they would have had, if they would have voted. This is not correct. Take for example the amount of votes of node $O$. The matrix ${\mathbf{S}}$ indicates that $O$ would have had 2.333 votes. This is not true, you can easily verify that $O$ actually only would have had 1.75 votes. The reason is that the edge $O \rightarrow P$ is removed if $O$ votes, changing the flow of votes altogether. But there is more information: the matrix ${\mathbf{B}}^{-1}$ contains more useful information about the origin of each vote. If we take for instance the solution to our specific problem shown in equation \[eq:solution\], we can see that each element of ${\mathbf{B}}_{ij}$ indicates the contribution of node $j$ to the amount of votes of $i$, if and only if $i$ actually votes. This allows for detailed feedback to each user as to why he voted exactly this, or to why he voted with a certain amount of votes. It is however not possible to trace the exact route of each vote, since it might be – and often is – infinitely long. How to implement the algorithm {#how-to-implement-the-algorithm .unnumbered} ============================== While inverting a matrix is a very common operation, this is non-trivial for large networks. A matrix inversion is of the order $O(N^3)$ for speed and $O(N^2)$ with $N$ the rank of the matrix, here the number of nodes. Since our matrix is however most likely sparse, since the expected number of delegations per user $E[D] \ll N$. This allows for more specialized approaches developed especially for sparse matrices, such that the the memory use scales with $O(ND)$ and the speed scales approximately with $O(N^2)$ as well [@li2009fast]. Since usually $E[D]$ is really small, we have used the algorithm provided in the scipy package without notable problems, even though it is slower. It does have the benefit of only needing a limited amount of memory. Further extensions {#further-extensions .unnumbered} ================== This approach can easily be extended to allow for other ideas in liquid voting systems, such as decaying votes when the trust chain grows (by making the sum of outgoing edges smaller than 1), or where the user can weigh the distribution of his vote for his proxies autonomously rather than presuming the vote is distributed equally. **Jonas Degrave** received a M.Sc. degree in electronics engineering at Ghent University in 2012, where he is currently pursuing a Ph.D. in computer science. His research interests are focused on machine learning and how it can be applied on robotics in order to generate more efficient locomotion.

--- abstract: 'Dimensional transmutation in classically conformal invariant theories may explain the electro-weak scale and the fact that so far nothing but the Standard Model (SM) particles have been observed. We discuss in this paper implications of this type of symmetry breaking for neutrino mass generation.' author: - Manfred Lindner - Steffen Schmidt - Juri Smirnov bibliography: - 'references.bib' title: 'Neutrino Masses and Conformal Electro-Weak Symmetry Breaking' --- \[sec:Introduction\]Introduction ================================ A key feature of quantum field theory (QFT) is that it can not predict overall scales. Scale ratios are, however, calculable and this leads to the question how large ratios can be explained or made natural. Symmetries play here an important role. The fermions of the Standard Model (SM) are protected by chiral symmetry such that only logarithmic corrections occur, while the SM Higgs mass is unprotected which leads to the famous hierarchy problem. As a consequence one expects either new physics in the TeV-range or a new symmetry which also leads to new particles in the TeV-range. This is one of the main motivations for the LHC, but so far no new particles or interactions showed up. Even though there are good reasons that e.g. supersymmetric particles show up at a somewhat higher scale one may wonder if the fact that so far no new particle has been found points into some other direction. A potential role of conformal symmetry has therefore recently been discussed as a solution and we would like to discuss in this paper the implications for neutrino mass generation. It is interesting to note that the standard model of particle physics (SM) is nearly conformal invariant. Only the mass term of the scalar field which is responsible for the breaking of $SU(2)_L \times U(1)_Y$ symmetry violates conformal symmetry explicitly and all SM masses are directly proportional to this scale. Note that the introduction of an explicit Higgs mass term in the SM does not only break conformal invariance, but it also creates the hierarchy problem, namely the quadratic sensitivity of quantum corrections to high scales. It is therefore tempting to relate the breaking of conformal symmetry with the generation of the electro-weak (EW) scale by dimensional transmutation. Scale invariance is broken at the quantum level (i.e. it has an anomaly) even in perturbation theory [@Callan:1970yg], but it has been argued that the protective features of conformal symmetry may not be completely destroyed [@Bardeen:1995kv]. Specifically logarithmic sensitivities still would exist, while quadratic divergencies would be absent. Various attempts of this type exist in the literature [@Coleman:1973jx; @Fatelo:1994qf; @Hempfling:1996ht; @Hambye:1995fr; @Meissner:2006zh; @Foot:2007as; @Foot:2007ay; @Chang:2007ki; @Hambye:2007vf; @Meissner:2007xv; @Meissner:2009gs; @Iso:2009ss; @Holthausen:2009uc; @Iso:2009nw; @Foot:2010et; @Khoze:2013uia; @Kawamura:2013kua; @Gretsch:2013ooa; @Heikinheimo:2013fta; @Gabrielli:2013hma; @Carone:2013wla; @Khoze:2013oga; @Englert:2013gz; @Farzinnia:2013pga; @Abel:2013mya; @Foot:2013hna; @Hill:2014mqa; @Guo:2014bha; @Radovcic:2014rea; @Khoze:2014xha; @Smirnov:2014zga; @Kannike:2014mia; @Chankowski:2014fva] and applications for the breaking of the EW symmetry have recently received more attention. Realizing these ideas within the SM corresponds to the Coleman-Weinberg effective potential, where $m_t < 79$ GeV would be required and where the Higgs mass would have to be $m_H \simeq 9$ GeV. This is obviously ruled out. However, we know that the SM is incomplete, since neutrino masses must be included. Furthermore, there is no dark matter (DM) candidate in the SM. Phenomenologically successful models which employ conformal electro-weak symmetry breaking require therefore some extension and a number of them predict also interesting DM candidates [@Farzinnia:2014xia; @Guo:2014bha; @Khoze:2013uia; @Carone:2013wla; @Ishiwata:2011aa; @Cheng:2004sd; @Foot:2010av; @Hambye:2007vf; @Radovcic:2014rea]. In this paper we focus on neutrino masses and we argue that the dynamical generation of scales forbids any explicit Majorana or Dirac mass term which would otherwise be possible and expected for a given set of fermions. This implies that all mass terms must be dimensionless Yukawa couplings times one of the vacuum expectation values (VEVs) generated by the dynamical symmetry breaking. This clearly alters expectations for neutrino masses and we will discuss how this leads naturally to a generic TeV scale see-saw, inverse see-saw and pseudo-Dirac scenarios. It has been shown in [@Meissner:2006zh; @Foot:2007ay] that extending the SM by merely right-handed neutrinos and an additional scalar field can result in the correct low energy phenomenology. The basic idea is that introducing additional scalar degrees of freedom makes the running of the couplings such that spontaneous symmetry breaking is possible. The additional scalar singlet gets a VEV and by its admixture to the Higgs a mass term is generated which can again induce EW symmetry breaking. This cascading symmetry breaking mechanism results in the discussed model in the correct Higgs mass and VEV. Thus, the EW scale appears naturally given the particle content of the model. The simplest model compatible with data contains a complex scalar singlet [@Meissner:2008gj] and the symmetry breaking takes place entirely in the new scalar sector, then it is transmitted via the Higgs portal to the SM boson. Explicit Majorana masses are not allowed and Majorana mass terms arise via Majorana-Yukawa couplings to the new scalar, which exemplifies nicely how neutrino mass generation is affected. Note that this has immediate consequences for the expected Majorana mass terms. Usually, an explicit mass is expected to have the largest possible value allowed by the symmetries of the system, while it is now the product of the symmetry breaking hidden scalar with a TeV-scale VEV with a Yukawa coupling. Since the Yukawa couplings of the SM show numerically a huge range, we assume the same to be true more general for all Yukawa couplings and Majorana mass terms can consequently have now any value between zero and the symmetry breaking scale. Motivated by this simple example we would like to discuss in this paper the changes for neutrino mass terms in conformally invariant theories in a more general way. We give therefore in [Sec. ]{}\[sec:Rules\] an overview of the considered cases for the generation of neutrino masses within the framework of conformal theories. The consequences for the possible structure of VEVs are elaborated in the same paragraph. On the other hand we investigate if different conformally invariant neutrino mass models are possible at all with regard to the occurrence of radiative symmetry breaking and the correct Higgs mass. The different models are presented in [Sec. ]{}\[sec:Models\] and are divided into two parts. The first part is based on mere extensions of the particle content of the SM, whereas the second part consists of theories that extend the SM gauge group by a $U(1)$ symmetry which separates a Hidden Sector (HS) from the SM. Different models within these parts are organized by their effects on the neutrino mass matrix $\mathcal{M}$. For neutrino masses it is in this context crucial that conformal symmetry forbids explicit mass scales in the classical Lagrangian. Phenomenological viable conformal EW symmetry breaking employs Higgs portals which connect to another sector with TeV scale dynamical mass generation. This implies that all Dirac and Majorana fermion masses are governed by this TeV scale or by the EW scale times some Yukawa coupling. This severely affects expectations for neutrino masses. A parameter scan for an effective model reveals that there are basically four phenomenological classes of theories. This scan is performed in [Sec. ]{}\[sec:Phenomenology\]. We summarize our findings and conclude with a discussion in [Sec. ]{}\[sec:Conclusion\]. \[sec:Rules\] Model Building Rules ================================== In this section we present model building rules for neutrino masses in a theory with classically conformal Lagrangian. Specifically we consider the following cases for neutrino masses in extensions of the standard model. - The SM can be embedded in a larger gauge group, which breaks to the required gauge group to describe the observed particle spectrum as is the case in GUT models. - The SM gauge group can be left unchanged and additional fields postulated. - A Hidden Sector (HS) with an additional symmetry group can be postulated. Resulting in the total symmetry group being a direct product of the SM and the new sector $G(SM) \times G(HS)$. In the following we will assume that the latter two possibilities are relevant, since the embedding of the SM in a larger gauge sector requires an additional scale of symmetry breaking which itself poses a little hierarchy problem, as in [@Holthausen:2009uc] where additional parameter tuning is required. Furthermore, the additional symmetry is assumed to be global to avoid the need for anomaly cancellation at this point. General Conformal Building Rules {#chap:rules} -------------------------------- A fermion mass term is a chirality flip of the field. Therefore, we will have an incoming particle of one chirality, e.g. the left-handed neutrino $\nu_L$ and its antiparticle of opposite chirality as an outgoing particle, which is right-handed. This particle can either be its own antiparticle with a Majorana mass or a distinct particle with a Dirac mass. The operators in the Lagrangian have dimension three and thus have to be augmented by a dimension one scalar field in order to fulfil the conformal requirements. Thus we assume the fermions only to couple via Yukawa couplings of the form $$\label{eq:Mass} \overline{\psi_L}\psi_R\varphi \; \text{ and } \; \overline{\psi_R}\psi_L\varphi \, ,$$ where the $\psi$ are fermions and $\varphi$ represents a scalar. Explicit mass terms are forbidden in the Lagrangian, i.e. any diagram like = 1mm [mass]{} $$\begin{aligned} \parbox{25mm}{ \begin{fmfgraph*}(35,20) \fmfleft{L} \fmfright{R} \fmf{plain}{L,V} \fmf{plain}{V,R} \fmfv{decor.shape=cross,decor.size=7}{V} \end{fmfgraph*} } \end{aligned}$$ with an explicit fermion mass term (cross) is forbidden. Yukawa couplings and mass terms which are generated via Yukawa and VEVs couplings like $$\label{eq:Yukawa} y \,\overline{\psi_L}\psi_R v_\varphi \; \text{ and } \; y\,\overline{\psi_R}\psi_L v_\varphi \, .$$ are allowed: [insertion]{} $$\begin{aligned} \parbox{40mm}{ \begin{fmfgraph*}(40,40) \fmfleft{L} \fmfright{R} \fmftop{T} \fmf{plain}{L,V} \fmf{plain}{V,R} \fmffreeze \fmf{dashes}{V,T} \end{fmfgraph*} } \hspace{3mm} \parbox{40mm}{ \begin{fmfgraph*}(40,40) \fmfleft{L} \fmfright{R} \fmftop{T} \fmflabel{$\langle \varphi \rangle$}{T} \fmf{plain}{L,V} \fmf{plain}{V,R} \fmffreeze \fmf{dashes}{T,V} \end{fmfgraph*} } \end{aligned}$$ Each neutrino mass diagram needs an odd number of mass insertions. Note that we work within the flavour basis, i.e. we use fields that appear in the unbroken Lagrangian. For the scalars conformal invariance only allows couplings which connect 4 scalars, i.e. diagrams of the form [potential]{} $$\begin{aligned} \parbox{25mm}{ \begin{fmfgraph*}(25,20) \fmfleft{L1,L2} \fmfright{R1,R2} \fmf{dashes}{L1,V} \fmf{dashes}{L2,V} \fmf{dashes}{R1,V} \fmf{dashes}{R2,V} \end{fmfgraph*} } \end{aligned}$$ \ These rules will be used throughout this work and will serve to derive rules with regard to specific neutrino mass questions. The Weinberg Operator Case -------------------------- We will argue that all neutrino mass diagrams, leading to a Majorana mass contribution for the active neutrinos, involve at least one vacuum expectation value other than the Higgs VEV and show that this is a topological necessity of conformally invariant theories including upto $SU(2)$ triplet representations.\ To prove this we first note that any diagram has an even number of doublet scalar mass insertions. This is because all diagrams generating left-handed Majorana masses have the left-handed doublet as the incoming and the outgoing particle, i.e. we have to start and end up with a doublet. If we assume that the theory has only upto $SU(2)$ triplet scalars and fermions, the only possibilities to connect two fermionic doublets are Yukawa couplings with a scalar triplet or singlet. Connecting a doublet fermion to a singlet fermion involves a doublet scalar. Equivalently a doublet and a triplet fermion are connected via a scalar doublet. Furthermore, two fermion singlets connect to a singlet scalar, two fermion triplets to a singlet scalar as well and a triplet and singlet fermion to a triplet scalar (see Table \[tab:Yukawa\]). Thus scalar doublets occur if and only if we connect a fermionic doublet to a fermionic non-doublet. Therefore, in order to start and end up with a fermion doublet we necessarily have an even number of scalar doublet mass insertions.\ Secondly, note that in any theory including upto $SU(2)$ triplets there are only potential couplings possible that involve an even number of $SU(2)$ doublets. Thus, each doublet line will couple to an odd number of doublet lines. As the product of an even and an odd number is an even number, the number of doublet lines will remain even. Connecting some of these lines and producing a loop will not change this fact as this closing reduces the number of external doublet lines by an even number.\ On the other hand two fundamental building rules for conformally invariant neutrino mass generation require that firstly there is always an odd number of mass insertions and secondly potential couplings always connect four lines. Both together yield that there has to be left an odd number of scalar external lines. Consequently as there has to be an odd number of VEVs but an even number of doublet VEVs, there has to be a singlet or a triplet VEV. Note, however, that this proof is based on the assumption that there are no fermion or gauge boson loops involved. This finding can be summarized as follows: If there are no gauge boson or fermion loops possible, a conformally invariant theory with upto $SU(2)$ triplet scalars and fermions needs a singlet or triplet scalar vacuum expectation value to generate left-handed Majorana neutrino masses. Radiative Models {#sec:Radiative} ----------------- In this subsection we deal with the question if it is possible to choose the particle content and the VEV structure of a theory such that the lowest order contribution to the left-handed Majorana masses is fully radiative i.e. there is no scalar with a VEV coupled to the neutrino line.\ We assume that there are no fermion or gauge boson loops. In this case, if in the potential only terms appear which couple fields in singlet pairs neutrinos can not gain mass via loops. This is the case, as scalars connected to the fermion line can only be coupled in such a way that they either produce one scalar of the own kind and two of another or couple to a particle of the own kind coming from the fermion line and thus reducing the number of its species by an even number. So either the number of a species stays the same, reduces or increases by an even number. As there has to be an odd number of mass insertions to the fermion line it is thus impossible to combine all scalars connected to the fermion line in a loop without producing at least one external line that already couples to the fermion line.\ An other way to understand this, is that for a loop induced active neutrino mass there has to be a lepton number violating term in the potential. Since the potential contains only four scalar operators, there has to be at least one among them with non pairwise coupled scalars. We can summarize this result: In a conformally invariant theory without fermion or gauge boson loops it is impossible to generate left-handed Majorana neutrino masses in a fully radiative way if the potential contains only terms coupling scalars in singlet pairs. We present models, which have not only pairwise scalar combinations in the potential, and yield fully radiative left-handed neutrino masses in [Appendix ]{}\[app:Exceptions\]. We only discuss models, which can yield neutrino masses with non vanishing diagonal elements, as those are excluded experimentally, as argued in [@Law:2013dya]. Furthermore, two possibilities to circumvent the above argument are presented, one is a model containing fermion loops. The other is the Ma model [@Ma:2006km] with a $Z_2$ symmetry, which forbids the Dirac tree level coupling and violates lepton number with the sterile neutrino Yukawa term. However, we do not consider discrete symmetries in the main body of the articles and the only way to have a model with this topology is with a hidden sector symmetry. The requirement of electrically neutral VEVs makes this model only viable for generating loop induced masses for the sterile neutrinos. This possibility will be discussed later on. \[sec:Models\] Overview of viable models ======================================== In this section we will give a summary of models in which it is possible to have neutrino masses and radiative scale symmetry breaking (RSSB). The criteria for the generation of neutrino masses are presented in [Sec. ]{}\[sec:Rules\]. As we will see the RSSB works in the case that at least two additional bosonic degrees of freedom are present, of which at least one must be a scalar. This modifies the beta function of the mass parameter in such a way that a scalar component gets a VEV, which is then cascaded to the Higgs sector through the Higgs-scalar mixing as described in [Sec. ]{}\[sec:Introduction\]. The symmetry breaking is consistently studied in the Gildener-Weinber approach [@Gildener:1976ih], which relies on the existence of a flat direction in the classical potential. Then the one loop effictive potential is computed. The minimal requirement of two bosonic degrees of freedom in the additional sector is crucial, since the RSSB relies on the bosonic degrees of freedom dominating over the top quark contributions. The symmetry breaking has to be triggered by the hidden sector and the pseudo-glodstone boson (PGB) associated with the scale symmetry breaking has to reside mainly in the hidden sector, see for example [@Radovcic:2014rea]. In the case of one additional bosonic degree of freedom, the Higgs boson is mainly the PGB which phenomenologically requires larger values of quartic couplings and leads to low scale Landau poles, see for example discussion in [@Foot:2007ay], which corresponds to model **3A** with only one real scalar field. It was demonstrated that RSSB is possible, but in our opinion the low scale Landau pole is problematic and we will take the model with two real scalars as the simplest realistic model. We will demonstrate the RSSB in a case with two bosonic degrees of freedom in the HS. The scalar field content is given by the $SU(2)$ doublet $H$ and two real SM singlets $\Phi$ and $S$. The potential has the form $$\begin{aligned} V(H, \Phi, S) = \frac{\l_H}{2}\,(H^\dagger\,H)^2 + \frac{\l_S}{2}\,S^4 + \frac{\l_\Phi}{2} \,\Phi^4 + \\ \nonumber \l_{HS}\,H^\dagger\,H\,S^2 + \l_{H\Phi}\,H^\dagger\,H\,\Phi^2 +\l_{S\Phi}\,\Phi^2\,S^2\,.\end{aligned}$$ For simplicity we will use spherical coordinates in field space with the replacements $$\begin{aligned} \label{eqn:fieldDefs} H = r\, \sin \theta \sin \omega \, , \\ \nonumber S = r \, \sin \theta \cos \omega\, , \\ \nonumber \Phi = r \, \cos \theta\,.\end{aligned}$$ We find with [Eq. ]{}\[eqn:fieldDefs\] and the definitions $(\tan \theta)^2 =: \epsilon$ and $(\sin \omega)^2 =:\delta$ that $$\begin{aligned} & (r\,\cos \theta )^4 \, V(r, \theta\, \Phi) = \frac{1}{2} \, \left( \l_\Phi + \epsilon (2\,\delta \l_{H\,\Phi} + 2 (1- \delta) \l_{S\Phi} + \right. \\ \nonumber & \left. \epsilon (\delta^2 \l_H+ 2 (1- \delta )\,\delta \,\l_{HS} + (1-\delta)^2 \,\l_S ) ) \right)= R(\Lambda)\,.\end{aligned}$$ The vanishig of this quantity at the scale of symetry breaking $R(\Lambda_{RSSB})=0$ defines the classically flat direction in the potential, it is the renormalization condition. Assuming that the mixing anomg the scalars is not large i.e. $\epsilon, \, \delta < 1$ a hierarchical VEV structure appears $$\begin{aligned} {\ensuremath{\left\langle \Phi \right\rangle}} & = {\ensuremath{\left\langle r \right\rangle}} (1+\epsilon)^{-1/2} = : v \\ \nonumber {\ensuremath{\left\langle S \right\rangle}} & = {\ensuremath{\left\langle r \right\rangle}} (1+\epsilon)^{-1/2} \sqrt{\epsilon} = v \, \sqrt{\epsilon}\\ \nonumber {\ensuremath{\left\langle H \right\rangle}} & = {\ensuremath{\left\langle r \right\rangle}} \, (1+\epsilon)^{-1/2} \sqrt{\epsilon\, \delta} = v\,\sqrt{\epsilon\,\delta} \\ \nonumber & \Rightarrow {\ensuremath{\left\langle \Phi \right\rangle}} > {\ensuremath{\left\langle S \right\rangle}} > {\ensuremath{\left\langle H \right\rangle}} \,.\end{aligned}$$ After the symmetry breaking the right handed neutrinos get their Majorana mass through Yukawa interactions with the HS scalars $M_{N_{i}}= Y_{N_{i}}/2 \,v^2 (1 +\epsilon)$. The scalar spectrum contains two massive excitations and one which is mass-less on tree level and corresponds to the flat direction in the potential. The idea behind the Gildener Weinberg approach is that the quantum effects are taken into account in the one loop correction to the mass of this particle, making it a PGB of broken scale symmetry. This procedure ensures perturbativity as discussed in detail in, [@Gildener:1976ih]. The mass of the PGB is given by $$\begin{aligned} M_S^2 = \frac{1}{8 \pi^2 {\ensuremath{\left\langle r \right\rangle}}^2}\left( M_H^4 + 6 m_W^4 + 3 m_Z^4 \right.\\ \nonumber \left. +M_\Phi^4 -12 m_t^4 - 2 \sum_i M_{N_{i}}^4 \right) \,,\end{aligned}$$ while the tree level scalar masses are (with $\lambda_{\Phi S}<0$ and $\lambda_{HS}, \,\lambda_{\Phi H}>0$ for explicitness) $$\begin{aligned} & M_H^2 = v^2 \left[ (\delta-1 )(1 + 16 \delta \, \epsilon) \,\lambda_{\Phi\,S} + \right.\\ \nonumber & \left. \delta\,\epsilon (3 \delta \,\lambda_H - (\delta -1 )\lambda_{HS} ) \right]\,\delta^{-1}\,, \\ & M_{\Phi}^2 = -v^2 \left[ (16 (\delta-1 )\epsilon -1)\lambda_{\Phi S} \right.\\ \nonumber & \left.-\epsilon (\delta \lambda_{HS} -3 (\delta-1 )\lambda_S) \right] \,.\end{aligned}$$ As can be seen the PGB resides mainly in the HS and thus the mixing with the Higgs can be brought in agreement with the experimentally constrained Higgs-scalar mixing [@Farzinnia:2014xia], while the potential parameters are perturbative and no low energy Landau pole appears. We plot the phenomenologically allowed mass regions in [Fig. ]{}\[fig:ThreeMasses\]. ![The phenomenlogically allowed mass region in the simplest neutrino mass model with RSSB, a Higgs mass of $125$ GeV, a higgs portal mixing compatible with the bound $\sin \theta < 0.37$, perturbative potential parameters and no low scale Landau pole. Here $M_N$ is the mass of the heaviest right handed neutrino, $M_\Phi$ is the heavy scalar dominating the spectrum and $M_S$ is the mass of the PGB.[]{data-label="fig:ThreeMasses"}](ThreeFieldFinite){width="45.00000%"} Next we study neutrino mass models with RSSB, which will be organized in the following way. Firstly we distinguish models with the SM model gauge group and secondly those where an additional hidden sector symmetry comprises with the SM symmetry group a direct product group. In the first case models are distinguished which affect the left handed neutrino mass directly (**$\#$A**) and those with an additional singlet fermion state which contributes to the left handed neutrino masses, as known from the type I see-saw mechanism (**$\#$B**). In the second scenario in all models there are additional SM singlet fermion states. We distinguish models with effect on the masses of the total singlets under the full gauge group (**$\#$C**), denoted by $\nu_R$ and those where masses of fermions are affected, which carry a hidden sector charge (**$\#$D**) and are denoted by $\nu_x$. The Dirac type masses in our framework are always determined by Yukawa couplings $y_D$ and the Higgs VEV, and assumed to exist if allowed by the symmetry. We comment on loop effects in models where those can lead to suppression of mass matrix entries. Furthermore, some comments on phenomenological implications will be made, but the main phenomenological discussion is omitted at this point and postponed to [Sec. ]{}\[sec:Phenomenology\]. All models carry an identification number and are described in detail in [Appendix ]{}\[app:Models\]. At first we focus on the models where the SM field content is extended. Assuming that we have singlet, doublet and triplet fermionic and scalar $SU(2)$ representations we list all combinations systematically and check whether a conformal neutrino mass model can be constructed, see Table \[tab:ModelsSM\]. Assuming only the above mentioned representations the presented list is complete. The models share features with the non-conformal analogues, nevertheless the scalar sector is in all cases enlarged to make the graph construction topologically possible without explicit mass insertions. Furthermore, the mass scales are all around the TeV scale, since the general spirit of the radiativly broken scale invariance forbids large scale separation. We again present a full catalogue of models with a $U(1)_\text{hidden}$, given that we only involve up to the triplet representation of the $SU(2)_L$ group, see Table \[tab:ModelsHS\]. This model sector could be enlarged by regarding more complex Hidden groups, but due to our little knowledge of the dark sector we stick here to the minimality condition. As a result we find a variety of tree level and radiative models with possible textures in the neutrino mass matrix. As one of the most promising models we point out **1D** and **2D**, which lead to an inverse see-saw (ISS) mass matrix structure which implies seizable active sterile mixing, discussed in [@Deppisch:2004fa; @Abada:2014vea]. The active-sterile mixing and the light masses are given by $$\begin{aligned} \label{ISSrelations} \epsilon = \frac{1}{2} m^{\dagger}_{D} (M^{-1}_{Rx})^*(M^{-1}_{Rx})^T m_D \approx \frac{y_D^2}{y_M^2} \frac{v^2}{\text{TeV}^2}, \\ \nonumber m_\nu = m^T_D (M^{-1}_{Rx})^T \mu M_{Rx}^{-1} m_D \approx \mu \,\epsilon.\end{aligned}$$ The $M_{Rx}$ scale is of the order of one to few TeV and the $\mu$ scale is loop induced in 2D and suppressed by heavier scales in 1D, which brings it to the keV scale. The Yukawa couplings in this region can be close to one, which makes it an attractive alternative to the fine tuned solutions. The effects of the active-sterile mixing can lead to an improved $\chi^2$ for the Electro-weak precision observables, as shown in [@Akhmedov:2013hec] and we will comment on it in the next section. In general the requirement of no scalar scale hierarchy restricts the vacuum expectation values of the new scalars not to be higher than the TeV scale. This leads with Yukawa couplings in the perturbative region to a particle spectrum below the TeV scale. However, this is not a necessity in all models. For instance if several additional scalar VEVs induce a cascade where the heaviest field begins with the symmetry breaking and transfers the scale by a portal to the next which in turn cascades down to the third scale, the scale separation can become larger without large tuning of the couplings, this can lift up the spectrum to a few TeV, as can be the case in the conformal inverse see-saw. In several models, see Table \[tab:ModelsSM\] and \[tab:ModelsHS\], the Majorana contribution to the light neutrino mass is suppressed and therefore an other neutrino mass scenario appears, in that case the active neutrinos are almost mass degenerate with the sterile components comprising pseudo Dirac pairs. This possibility is experimentally extremely challenging, but might be accessible in long baseline and low energy oscillation experiments [@Beacom:2003eu]. In general scale separation does not appear naturally in models with RSSB, thus the neutrino mass scale can appear if the Yukawa couplings are arranged in a way leading to a see-saw suppression. The other possibility is that the lightness is connected to a small lepton number violation parameter. This smallness can be argued to be natural in t’Hoft sense, as the symmetry of the theory would increase if this parameter would be exactly zero. Furthermore, in radiative neutrino mass models the smallness of the lepton number violation is augmented by a mass suppression by the loop factors. The most interesting possibility is, however, if both of this mechanisms are at work. This is the case if the Majorana scale is induced by a loop involving a lepton number violating coupling, leading to the Pseudo Dirac and Inverse see-saw scenarios. Where in the last scenario the Yukawa couplings can be of order one. \ [|&gt;m[1cm]{}|&gt;m[3.5cm]{}|&gt;m[3cm]{}|&gt;m[1.5cm]{} |&gt;m[1.8cm]{}|&gt;m[6cm]{}|]{} \# & particle content & non-conformal motivation & neutrino masses & correct Higgs mass & phenomenological note\ \ \ [|&gt;m[1cm]{}|&gt;m[3.5cm]{}|&gt;m[3cm]{}|&gt;m[1.5cm]{} |&gt;m[1.8cm]{}|&gt;m[6cm]{}|]{} 1A & Conformal SM (CSM) & $\diagup$ & **No** & **No** & This theory does not yield neutrino masses.\ 2A & CSM + $\nu_R:(1,0)$ & See-saw type I & **Yes** & **No** & Neutrinos in this theory are of Dirac type.\ 3A & CSM + $\nu_R:(1,0)$ + $\varphi:(1,0)$ & See-saw type I & **Yes** & **Yes** & In dependence of the coupling constants this theory can yield Sub TeV or Pseudo-Dirac neutrinos.\ 4A & CSM + $\Delta:(3,-2)$ & See-saw type II & **Yes** & **No** & This theory yields pure left-handed Majorana neutrinos.\ 5A & CSM + $\Delta:(3,-2)$ + $\varphi:(1,0)$ & See-saw type II & **Yes** & **Yes** & This theory yields pure left-handed Majorana neutrinos as well.\ 6A & CSM + $\nu_R:(1,0)$ + $\varphi:(1,0)$ + $\Delta:(3,-2)$ & See-saw type I/II & **Yes** & **Yes** & Sub TeV and Pseudo-Dirac neutrinos are possible.\ 7A & CSM + $\delta_-:(1,-2)$ & $\diagup$ & **No** & **No** & Neutrinos remain massless.\ 8A & CSM + $\delta_-:(1,-2)$ + $\Delta:(3,-2)$ & $\diagup$ & **Yes** & **No** & The additional $\delta_-$ only contributes corrections to the masses.\ 9A & CSM + $\Sigma:(3,0)$ & See-saw type III & **No** & **No** & Neutrinos remain massless.\ 10A & CSM + $\Sigma:(3,0)$ + $\varphi:(1,0)$ & See-saw type III & **Yes** & **Yes** & This theory yields the same neutrino phenomenology like the conformal See-saw type I.\ 11A & CSM + $\delta_-:(1,-2)$ + $\epsilon_{++}:(1,4)$ + $\varphi:(1,0)$ & Zee-Babu & **Yes** & **Yes** & Pure left-handed Majorana neutrino masses suppressed by 2 loops.\ 12A & CSM + $H_2:(2,1)$ + $\eta_{+}:(1,2)$ + $\varphi:(1,0)$ & Zee Model & **Yes** & **Yes** & Pure left-handed Majorana neutrino masses suppressed by 1 loop.\ 13A & CSM + $\phi_1:(2,3)$ $ H_2:(2,1) $ $ \eta:(1,-4) ;\; \phi_2:(1,0)$ &Law-McDonald & **Yes** & **Yes** & Pure left-handed Majorana neutrino masses suppressed by 2 loops.\ \ [|&gt;m[1cm]{}|&gt;m[3.5cm]{}|&gt;m[3cm]{}|&gt;m[1.5cm]{} |&gt;m[1.8cm]{}|&gt;m[6cm]{}|]{} 1B & CSM + $\nu_R:(1,0)$ + $\Sigma:(3,0)$ + $\Delta:(3,0)$ + $\varphi:(1,0)$ & $\diagup$ & **Yes** & **Yes** & This theory can generate conditions for the Pseudo-Dirac and the Sub TeV see-saw.\ 2B & CSM + $\nu_R:(1,0)$ + $\nu_x:(1,0)$ + $\varphi:(1,0)$ & $\diagup$ & **Yes** & **Yes** & The extension by further sterile neutrinos is trivial if they cannot be distinguished from the original sterile neutrinos.\ \ [|&gt;m[1cm]{}|&gt;m[3.5cm]{}|&gt;m[1.5cm]{}| &gt;m[3.5cm]{}|&gt;m[5cm]{}|]{} \# & particle content & $U(1)_H$ & VEV structure & phenomenological note\ \ \ [|&gt;m[1cm]{}|&gt;m[3.5cm]{}|&gt;m[1.5cm]{}| &gt;m[3.5cm]{}|&gt;m[5cm]{}|]{} & $\nu_R:(1,0)$ & 0 & & The double see-saw mass structure is implied.\ 1C & $\nu_x:(1,0)$ & 1 & all scalars get a VEV & Pseudo-Dirac and\ & $\varphi_1:(1,0)$ & 1 & & sub TeV scenarios\ & $\varphi_2:(1,0)$ & 2 & & are possible .\ & $\nu_R:(1,0)$ & 0 & & The minimal extended see-saw structure is implied.\ 2C & $\nu_x:(1,0)$ & 2 & all scalars get a VEV & Light sterile neutrinos\ & $\varphi_1:(1,0)$ & 0 & & with large\ & $\varphi_2:(1,0)$ & -2 & & active-sterile mixing .\ 3C & theory 1C + & theory 1C & $\varphi_1$ gets no VEV & radiative model,\ & $\varphi_3:(1,0)$ & -4 & & implies Pseudo-Dirac scenario\ & $\nu_R:(1,0)$ & 0 & &\ & $\Sigma:(3,0)$ & 1 & & Pseudo-Dirac and sub TeV\ 4C & $\Delta:(3,0)$ & 1 & all scalars get a VEV & scenarios\ & $\varphi_1:(1,0)$ & 1 & & are possible.\ & $\varphi_2:(1,0)$ & 2 & &\ 5C & theory 3C + & theory 3C & $\varphi_1$ gets no VEV & radiative model,\ & $\varphi_3:(1,0)$ & -4 & & implies Pseudo-Dirac scenario\ \ [|&gt;m[1cm]{}|&gt;m[3.5cm]{}|&gt;m[1.5cm]{}| &gt;m[3.5cm]{}|&gt;m[5cm]{}|]{} & $\nu_R:(1,0)$ & 0 & &\ & $\nu_x:(1,0)$ & 1 & &\ 1D & $\Sigma:(3,0)$ & -2 & all scalars get a VEV & generates small $\nu_x$ mass,\ & $\Delta:(3,0)$ & -3 & & implies the inverse see-saw scenario\ & $\varphi_1:(1,0)$ & -3 & &\ & $\varphi_2:(1,0)$ & -4 & &\ & $\varphi_4:(1,0)$ & 1 & &\ 2D & theory 1D + & theory 1D & $\varphi_1$ gets no VEV & radiative model,\ & $\varphi_3:(1,0)$ & 10 & & implies the inverse see-saw scenario\ \[sec:Phenomenology\]Phenomenology ================================== In this section we will check which of the proposed models can indeed reproduce the correct neutrino mass phenomenology i.e. the mass square differences and the correct mixing angles and at the same time be consistent with rare decay experiments and electroweak precision observables (EWPO). In a plot we will demonstrate viable regions of the allowed parameter space and estimate expected signals for future lepton flavour and number violation experiments. In most of the discussed models the PMNS matrix becomes not exactly unitary. This happens if the active-sterile mixing is considerable and induces a number of effects on physical quantities as the Weinberg angle, the W-boson mass, the left and right handed couplings $g_L$, $g_R$, the leptonic and invisible Z-boson decay width and the neutrino oscillation probabilities, for more detailed discussion and limits see [@Antusch:2006vwa] and references therein. Thus studying the non unitarity allows to narrow down the parameter space of a given model. However, some effects are not captured by this treatment only. Those are processes where explicit particle propagation is responsible for the new physics signal. To get an order of magnitude estimate we integrate out heavier degrees of freedom to obtain an effective scenario with a $(3+n)\times(3+n)$ nearly unitary mixing matrix $\mathbf{U}$. $$\label{mixingmatrix} \mathbf{U}=\left( \begin{array}{cc} \mathscr{U} & \mathscr{R}\\ \mathscr{W} & \mathscr{V} \end{array} \right)\,.$$ This corresponds to a scenario with three active and $n$ sterile neutrinos. Here $\mathscr{R}$ can be considered as the active-sterile mixing. $\mathscr{U}$ is the PMNS matrix and is not unitary any more. A measure for non-unitarity of the PMNS matrix is given by $$\begin{aligned} \label{epsilon} \e_\alpha\equiv {\textstyle\sum_{i\geq 4}} |\mathbf{U}_{\alpha i}|^2\,. \end{aligned}$$ This matrix diagonalizes the following mass matrix $$\mathcal{M}=\begin{pmatrix} m_L & m_D\\ m_D^T & M_R\end{pmatrix}.$$ Thus, the active and sterile neutrinos have a Majorana mass and mix due to the Dirac mass terms. This set up covers all the effects on neutrino physics of a given model. The Majorana mass nature opens the possibility for lepton number violation and neutrino-less double beta decay. The propagating sterile states lift the GIM suppression in the lepton flavour violating processes for the charged leptons and different non-unitraity parameters $\epsilon_\alpha$ in [Eq. ]{}\[epsilon\] parametrise deviation from lepton universality. Furthermore, in this set up we can get estimates on the oblique corrections [@Peskin:1990zt]. As shown in [@Akhmedov:2013hec] they can contribute significantly to EWPOs especially given large non-unitarity and heavy sterile neutrinos. The mass terms in the effective theory after integrating out heavier degrees of freedom have the following form. $$\begin{aligned} \label{eq:MassTerms} -\mathscr{L}_m =\frac{1}{2} m^{*}_{\!L,ij} \bar{\nu}^{c}_{\!L\!,i}\nu^{}_{\!L\!,j} + m^{*}_{\!D,ij}\bar{\nu}^{}_{\!L,i}\nu^{}_{\!R,j} \\ \nonumber +\frac{1}{2}M^{*}_{\!R,ij}\bar{\nu}^{c}_{\!R\!,i}\nu^{}_{\!R\!,j}+h.c., \end{aligned}$$ where $m^{*}_{\!D,ij}=g^{}_{H\!,ij}\cdot v_H$ and $M^{*}_{\!R,ij}=g^{}_{\varphi\!,ij}\cdot v_\varphi$. While direct masses for the left handed neutrinos are generated due to a scalar or fermionic triplet. $m^{*}_{\!L,ij}=g^{a}_{L\!,ij}\cdot v_\Delta + (g^{}_{\Sigma\!,ij}+g^{b}_{\Delta\!,ij})\cdot v_H^2/v_\phi$. The scalar triplet contributes a dimension four operator, both triplets generate terms proportional to the squared Higgs VEV, the fermionic triplet a dimension fife operator and the scalar triplet a dimension six operator. The $g$ parameters are effective Yukawa couplings which can contain corrections from heavier particles integrated out of the theory. Thus depending on the theory in question the perturbativity condition is not to be taken as a strict bound. Scanning over the effective Yukawa coupling space provides us with sets of viable solutions according to the above criteria. To visualize the solutions we set up a two dimensional map with the horizontal axis for the averaged right handed mass scale and the vertical axis for the averaged Dirac couplings. The average represents the order of magnitude of the Yukawas in the case they are in the same ball park, if they are spread apart the average is dominated by the largest. The spread over several orders of magnitude, however, is not considered as it would require unnaturally large tuning. The results of our study are presented in [Fig. ]{}\[fig:Masterplot\]. We would like to discuss four phenomenological scenarios separately. Pure left handed Majorana mass ------------------------------ In the case that the Dirac coupling is very small, or there are no fermionic singlets under the SM gauge group included in the theory, the only possible neutrino mass term is the left handed Majorana mass. In this scenario the charged lepton flavour violation is strongly GIM suppressed and beyond experimental precision. The PMNS mixing matrix is unitary and therefore the most promising signals are expected in the $0\nu\beta\beta$ experiments. The total mass scale enters the effective electron neutrino mass, since it is entirely Majorana. For a detailed study if the $0\nu\beta\beta$ sensitivity depending of the hierarchy see [@Rodejohann:2012xd] and references therein. The current experimental bound on the electron neutrino effective mass, the parameter controlling the $0\nu\beta\beta$ decay, is ${\ensuremath{\left\langle m_{ee} \right\rangle}}< 0.4 \, \text{eV}$ [@Macolino:2013ifa]. The models leading to a pure left handed Majorana mass are in our set up **11A**, **12A** and **13A** here the neutrino masses are suppressed by one or two loops. Pseudo Dirac Scenario --------------------- The other distinct region with only light neutrinos is around the point in the parameter space where the active and sterile neutrinos form mass degenerate pairs of Dirac fermions. This point has no lepton number violation and an effective GIM suppression of the charged lepton flavour violating process. Now the pairs can acquire a small Majorana mass, either through contributions of light sterile neutrinos or a small $m_L$ mass. This leads to a mass splitting among the degenerate Majorana pairs forming the effective Dirac neutrino states. Phenomenologically this is consistent with observations as long as this splitting is smaller than the experimental accuracy of the mass square difference measurement. For detailed bounds consult [@deGouvea:2009fp]. It turns out that the strongest constraints apply to the splitting of the first and second mass state and are of the order of $10^{-9}$ eV, while in the third mass state, with the dominant tau flavour, the splitting can be up to $10^{-3}$ eV. Since the right-handed neutrinos are light in this scenario, the PMNS matrix is unitary and there are no phenomenological bounds from EWPOs, lepton universality or lepton flavour violation. The origin of the mass splitting is not important for oscillation experiments, when it comes to lepton flavour violation, however, there is an interesting subtlety. Consider the two cases in the one flavour scenario, where in the first case the Majorana mass appears on tree level for the active neutrino and in the second case for the sterile component $$\mathcal{M}_1=\begin{pmatrix} \mu & m_D\\ m_D & 0\end{pmatrix} \text{ and } \mathcal{M}_2=\begin{pmatrix} 0 & m_D\\ m_D & \mu\end{pmatrix}.$$ In the limit $\mu << m_D$ the mass eigenvalues are given in both cases as $m_\pm = \pm m_D +\mu/2 $ and the diagonalizing mixing matrices are $$U_{1/2} \approx \sqrt{\frac{1}{2}} \begin{pmatrix} 1 \pm \epsilon & -1 + \epsilon \\ 1 \mp \epsilon & 1 + \epsilon \end{pmatrix} \text{ with } \epsilon = \frac{\mu}{4 m_D}.$$ We consider now the expansion of the effective mass for the neutrnoless double beta decay in powers of the momentum transfer. The effective mass is approximately given by ${\ensuremath{\left\langle m_{ee} \right\rangle}} \approx |q^2 \textstyle{\sum_{i}} \mathbf{U}_{e i}^2 \, m_i/(q^2 - m_i^2)|$, the relation holds that $ m_i^2 \ll |q^2| \approx (0.1 \,\text{GeV})^2$ and we can expand $$\begin{aligned} {\ensuremath{\left\langle m_{ee} \right\rangle}} \approx |\sum_i U_{ei}^2 m_i + 1/q^2 \sum_i U_{ei}^2 m_i^3 + O(1/q^4)|.\end{aligned}$$ Inserting the parameters we find that in the case where the active neutrino has a direct Majorana mass the effect is of order [@Rodejohann:2012xd], $$\begin{aligned} \label{eqn:directCont0NuBetaBeta} {\ensuremath{\left\langle m_{ee\, 1} \right\rangle}} \approx \mu \approx (m_+^2 - m_-^2)/(2 \, m_D) \,.\end{aligned}$$ In the case where the Majorana mass appears via the sterile component, the first order contribution vanishes, as the electron neutrino entry of the neutrino mass matrix is zero and one has to leading order $$\begin{aligned} {\ensuremath{\left\langle m_{ee\,2} \right\rangle}} \approx m_D\, (m_+^2 - m_-^2)/(2 \, q^2) \approx \mu \, m_D^2/q^2 \,, \end{aligned}$$ since $m_D$ is of the order of the absolute neutrino mass scale the effective mass is suppressed by the factor $(m_D/q)^2$ with respect to [Eq. ]{}\[eqn:directCont0NuBetaBeta\] which is at least 14 orders of magnitude. Thus there is an interesting experimental possibility to distinguish these scenarios. Assume, neutrino oscillation experiments on cosmic scales detect a small mass splitting testing oscillations on cosmic length scales as described in [@Beacom:2003eu]. If this splitting is in the phenomenologically allowed region today, the contribution to the effective mass for $0\nu\beta\beta$ decay can be up to a few $10^{-5}$ eV, as displayed in [Fig. ]{}\[fig:Masterplot\] in the zoomed in region. This is only the case, if the mass splitting originates from a direct active neutrino Majorana mass, in the second case it would be of the order $10^{-17}$ eV and beyond experimental reach. Thus, measuring the $0\nu \beta \beta$ decay provides evidence of scenario one and placing a limit smaller than the predicted value shows that scenario two is realized. Even though the possibility is interesting from the theoretical perspective, it is extremely challenging experimentally, since the maximal expected decay rate is four orders of magnitude below the current sensitivity. Among the presented models the pseudo Dirac scenario can be realized in **3A**, **6A** with a direct active neutrino Majorana mass and in 1B and possibly in all the (**$\#$C**) models with a sterile neutrino Majorana mass, leading to no observable $0\nu\beta\beta$ decays. The Yukawa couplings in the Majorana and Dirac sector have to be tiny, in fact below $10^{-12}$, but there is no hard theoretical argument which could exclude this possibility a priori. Sub-TeV Yukawa See-saw ---------------------- In several models where the right handed Majorana couplings are not loop or scale suppressed the right handed Majorana mass is below one TeV. Viable solutions lie in a triangular shaped region for Majorana couplings smaller than one. In this parameter region the contributions to $m_L$ are subdominant and thus the averaged effective Yukawa couplings represent the sterile Majorana mass on the x-axis and the Dirac mass on the y-axis respectively. For the scan the Casas-Ibarra parametrisation [@Ibarra:2010xw] was used, which parametrizes the active sterile-mixing as $R = -i D_{\sqrt{m_\nu}}\mathcal{O}^*D_{\sqrt{M_R}}U_{\text{PMNS}}$ with $\mathcal{O}^T\,\mathcal{O}=1$. Thus the physical effects connected to non-unitarity are controlled by the norm of $\mathcal{O}$. The shape of the Sub-TeV see-saw region is explained as follows. The see-saw relation ($m_D^2/M_R \approx m_\nu = 0.1$ eV) and $M_R > m_D$ sets the lowest value for $m_D$ given a $M_R$, which is $m_D > \sqrt{m_\nu \, M_R}$. This sets the lower boundary of this region and is represented by a black dotted diagonal line of gradient two in the log plot in [Fig. ]{}\[fig:Masterplot\]. Deviations to higher Dirac couplings induce a larger active-sterile mixing and thus larger non-unitarity. The unitarity bounds constrain the region to the left and are represented by a brown line, for non-unitarity of one percent. Since deviation from unitarity is proportional to $m_D/M_R$ the line has gradient one in the log plot. However, those turn out to be not the strongest constraints. The most stringent bounds come from neutrino-less double beta decay, displayed as a red line. In the region of interest the dominant contribution to the electron neutrino effective mass for $0\nu\beta\beta$ is given by $ {\ensuremath{\left\langle m_{ee} \right\rangle}} \approx | \textstyle{\sum_{i\geq 4}} \mathbf{U}_{e i}^2 \,\text{GeV}^2/M_i|$. The predicted effective electron neutrino mass violates the observational bound if the right handed neutrinos become too light. The constraints from rare lepton flavour violating decays and lepton universality are somewhat weaker. The EWPOs in this parameter region are consistent with their measured values. To this end the $\chi^2$ function as in [@Akhmedov:2013hec] has been calculated and loop effects of the right handed neutrinos included, the resulting $\chi^2$ values do not differ significantly from the SM values. The main characteristics of this scenario is lepton number violation, since the Majorana mass of the sterile neutrino is not suppressed. Besides the rare decay processes, lepton number violation can lead to beyond SM processes at colliders in decays of the heavy Majorana neutrinos, see [Fig. ]{}\[fig:DecayLNV\]. The production cross section for this process, is proportional to $ | \textstyle{\sum_{i}} \mathbf{U}_{\alpha i}^2 \,M_i^{-1}|$ [@Keung:1983uu] and has basically zero SM background. ![The lepton number violating decay as a collider signature for the Sub-TeV and multi TeV see-saw with a heavy Majorana Neutrino decay.[]{data-label="fig:DecayLNV"}](HeavyMajoranaDecay){width="45.00000%"} It is interesting to discuss the values of the Yukawa couplings in this region. Since in our models of spontaneous broken scale invariance all the masses are a result of a VEV coupled with a Yukawa term, the see-saw relation is induced entirely by the Yukawa coupling structure. While for the Majorana coupling the region of the Sub-TeV see-saw implies in the presented models couplings between $10^{-3}$ and one, the Dirac Yukawa couplings vary between $10^{-7}$ and $10^{-4}$. This values might be considered small and fine-tuned, however we have to stress here that in that case the electron Yukawa coupling, which is of order $10^{-6}$ is suspicious as well. In the models discussed above this scenario is realized in **3A**, **6A**, **10A** , **1B**, **1C** and **4C**. Inverse Yukawa see-saw ---------------------- The most interesting scenario from the theoretical point of view in the context of RSSB is the inverse see-saw, introduced in [@Deppisch:2004fa; @Abada:2014vea]. It naturally occurs in models **D1** and **D2**, where the mass matrix has the following texture and the scale $\mu$ is loop or scale suppressed $$\label{eq:seesawInv} \mathcal{M} = \begin{pmatrix} 0 & m_D & 0 \\ m_D^T & 0 & M_{Rx} \\ 0 & M_{Rx}^T & \mu \end{pmatrix} \, .$$ The spectrum of this models contains Pseudo Dirac pairs of heavy neutrinos, with masses of order $M_{Rx}$ and their mass splitting $\mu$ determines the amount of lepton number violation present. At the same time it is the parameter, which controls the smallness of the active neutrino masses. As given by [Eq. ]{}\[ISSrelations\] the active sterile mixing is determined by the ratio $m_D^2/M_{Rx}^2$ and the general spirit of RSSB together with no tuning in the Yukawa couplings suggests seizable values. Seizable mixing is only compatible with small active masses if a cancellation mechanism is at work. It can be seen in the Casas-Ibarra parametrization, we choose the two flavour case with $U_\text{PMNS}=1$ for simplicity here. The orthogonal complex matrix $\mathcal{O}(\theta)$ is in this case a simple $2\times 2$ rotation matrix with the complex angle $\theta = a +ib$. In the limit $a \ll 1$ and $1 \ll b$ we have $$\label{eq:ISSIbarra} \mathcal{O}(\theta) \approx \begin{pmatrix} \cosh(b) & - i\sinh(b) \\ i \sinh(b) & \cosh(b) \end{pmatrix} \, .$$ As shown in [@Kersten:2007vk], this leads in the limit of $m_\nu \rightarrow 0$ and with $\sinh(b)\approx \cosh(b) \approx e^b$, $e^b\sqrt{m_1} \rightarrow \sqrt{\mu}, \,\,e^b\sqrt{m_2} \rightarrow \alpha \sqrt{\mu}$ to an $$\label{eq:ISSIbarra} m_D \approx \begin{pmatrix} \sqrt{\mu M_1} & - i\sqrt{\mu M_2} \\ i\sqrt{\mu M_1}\alpha & \sqrt{\mu M_2}\alpha \end{pmatrix} \, .$$ Which in the limiting case has rank 1 and thus induces massless active neutrinos. This shows that the orthogonal matrix with dominating imaginary arguments is a good effective description of the ISS. Using this fact we study experimental constraints on this scenario. At first we consider the $0\nu\beta\beta$ decay, which placed the most severe bounds on the Sub-TeV scenario. The general expression useful to consider in this case is [@Abada:2014vea] ${\ensuremath{\left\langle m_{ee} \right\rangle}} \approx |q^2 \textstyle{\sum_{i}} \mathbf{U}_{e i}^2 \, m_i/(q^2 - m_i^2) |$ . Which now can be studied in three cases, depending on the ratio of $q^2/M_{Rx}^2$, where the neutrino momentum is $|q| \approx 0.1 \,\text{GeV}$. If we have $M_{Rx} \gg 0.1 \text{GeV}$ and using the facts that for $i>3$, $\mathbf{U}_{e i}^2 \approx m_D^2/M_{Rx}^2$ and $ \mu\,m_D^2/M_{Rx}^2 \approx m_\nu $ the following approximation holds $$\begin{aligned} {\ensuremath{\left\langle m_{ee} \right\rangle}} \approx \left|\textstyle{\sum_{i=1}^3} \mathbf{U}_{e i}^2 \, m_i - \frac{q^2}{2} \textstyle{\sum_{i>3}} \mathbf{U}_{e i}^2 \frac{\mu}{m_i^2} \right| \\ \nonumber \approx \left|\textstyle{\sum_{i=1}^3} \mathbf{U}_{e i}^2 \, m_i - m_\nu \frac{q^2}{M_{Rx}^2} \right| \approx \left|\textstyle{\sum_{i=1}^3} \mathbf{U}_{e i}^2 \, m_i \right|.\end{aligned}$$ Which means that the rate is purely given by the light neutrino spectrum with well known phenomenology. The other limit is $M_{Rx} \ll 0.1 \,\text{GeV}$, leading to ${\ensuremath{\left\langle m_{ee} \right\rangle}} \approx | \textstyle{\sum_{i}} (\mathbf{U}_{e i}^2 \, m_i + 1/q^2\,\mathbf{U}_{e i}^2 \, m_i^3) | = \mathcal{M}_{ee}+O(\mu m_D^2/q^2)$. This situation is similar to the discussed Pseudo Dirac scenario with light neutrinos and the lowest order contribution is $\mu m_D^2/q^2 < \mu M_{Rx}^2/q^2$, which in this limit is negligible. The only case when the heavy Pseudo Dirac states can measurably contribute to the $0\nu\beta\beta$ decay is when $M_{Rx} \approx 0.1\, \text{GeV}$. Then we have $$\begin{aligned} {\ensuremath{\left\langle m_{ee} \right\rangle}} \approx \left|m_{ee}^\text{light} + \textstyle{\sum_{i>3}} \mathbf{U}_{e i}^2 \,\mu \left( 1 + \frac{ m_i^2}{|q^2|} \right)^{-1} \right| \\ \nonumber \approx \left| m_{ee}^\text{light} + \textstyle{\sum_{i>3}} m_\nu \, \left( 1 + \frac{ m_i^2}{|q^2|} \right)^{-1}\right|,\end{aligned}$$ which is of the order of the light neutrino contributions. Thus, we see that neutrinoless double beta decay does not provide strong bounds in the ISS scenario, since the lepton number violation is suppressed as the scale $\mu$. This is not the case for the Lepton flavour violating processes. The best constrained value is the branching ratio $\text{Br}(\mu \rightarrow e + \gamma )$, where the limit is placed by the MEG collaboration [@Adam:2013mnn] and is $5,7 \cdot 10^{-13}$. The neutral fermion contribution to this loop induced decay is $$\begin{aligned} \text{Br}(\mu \rightarrow e + \gamma ) = \frac{3 \alpha_{\text{em}}}{32 \pi} \left| \textstyle{\sum_{i}} \mathbf{U}_{\mu i}^* \mathbf{U}_{e i} \,G \left(\frac{m_i^2}{M_W^2}\right)\right|^2,\end{aligned}$$ where in the loop function $G(x)$ the masses appear squared and the cancellation leading to a vanishing $0\nu\beta\beta$ process can not work. We find that the MEG bound together with the non-unitarity constraints [@Antusch:2006vwa] lead to the most severe constraints on this models, as shown in [Fig. ]{}\[fig:Masterplot\]. As stated before the ISS opens the possibility in the RSSB framework to have states above the TeV scale. The region of right handed masses between one and a few ten TeV is divided in two subregions, which are distinguished by the value of the active-sterile mixing. If this value is sizeable, in fact above $10^{-6}$, the phenomenology is considerably affected. The most sensitive observables are the Z boson invisible decay width and the Muon decay constant, which is used to determine the Fermi constant. The observables dependence on the non unitarity parameters, see [Eq. ]{}\[epsilon\] is given by $$\begin{aligned} \frac{\Gamma^\text{inv}_Z}{[\Gamma^{\text{inv}}_Z]_\text{SM}} = \frac{1}{3}\sum_\alpha (1-\epsilon_\alpha)^2 \,,\\ G_\mu = G_F (1 - \epsilon_e)(1 - \epsilon_\mu)\,.\end{aligned}$$ This region of seizable active sterile mixing with heavy particles is of particular interest, since here the oblique corrections can become large. So, on the one hand the $\chi^2$ with the EWPOs provides us with phenomenological bounds in this region. On the other hand this is an example of a theory where contributions from heavy sterile neutrinos can improve the electroweak fit, as discussed in [@Akhmedov:2013hec]. In [Fig. ]{}\[fig:Masterplot\] the region with an improved $\chi^2$ is bound towards lower mass values by experimental constraints from the $\mu \rightarrow e + \gamma$ decay and towards higher masses the radiative corrections become incompatible with observations in case of large active-sterile mixing. Having discussed constraints on the right handed mass, it is interesting to study which Dirac mass scales are allowed. The mass scale of the light neutrinos is set by the following scale relation $m_\nu \approx \mu m_D^2/M_{Rx}^2$, furthermore it is required that $\mu/M_{Rx}=:\delta \ll 1$ and $m_D< M_{Rx}$. Those relations imply that $m_D > \sqrt{m_\nu M_{Rx}/\delta}$. Given a right handed mass scale, the minimal Dirac mass is larger than in the usual see-saw scenario, which implies that the active-sterile mixing has to be larger as well. The most promising signature to distinguish the heavy Pseudo Dirac neutrino from the ISS scenario from a heavy Majorana neutrino is a direct test at a collider, which is feasible as all the particles involved are around the TeV scale. The difference lies in the dominant decay channel of the right handed neutrinos. While in the Majorana see-saw the lepton number violation is unsuppressed generically, the dominant process is expected to be the lepton number violating decay in [Fig. ]{}\[fig:DecayLNV\]. In the case of a decay of a heavy Pseudo Dirac neutrino, lepton number violation is suppressed by the smallness of the right handed Majorana scale $\mu$ [@Kersten:2007vk], thus the dominant processes are lepton number conserving decays. As argued in [@Das:2012ze; @Das:2014jxa], the opposite sign dilepton decay has a very large SM background and thus the relevant channel becomes the trilepton decay with missing energy, see [Fig. ]{}\[fig:DecayDirac\]. As shown by Das et al. the inclusive cross section of the trilepton final state is controlled by the branching ratio of the heavy neutrino in the W boson and a lepton, it has the partial decay width $$\begin{aligned} \label{eq:decay} \Gamma(N \rightarrow \ell_\alpha W) = \frac{g^2 \,\epsilon_\alpha}{64 \pi} \frac{m_i^3}{M_W^2} \left( 1 - \frac{M_W^2}{m_i^2}\right)^2 \left(1 + 2\frac{M_W^2}{m_i^2} \right).\end{aligned}$$ ![The dominant collider signature for the ISS scenario with the trilepton plus missing energy signature.[]{data-label="fig:DecayDirac"}](HeavyDiracDecay){width="45.00000%"} As shown by [Eq. ]{}\[eq:decay\], the decay width crucially depends on the non-unitarity parameter $\epsilon_\alpha$. The interesting feature of the ISS in the RSSB framework is, that the requirement of no large scale separation results in naturally large active-sterile mixing, as $\epsilon \approx m_D^2/M_{Rx}^2$. Thus the most natural value for $\epsilon$, given an order of magnitude between the scales and Yukawa couplings of order one is about one percent, close to the sensitivity threshold of modern experiments. Note that the recently proposed production mechanism for heavy sterile neutrinos via t-channel processes can further increase the collider sensitivity, as argued in [@Dev:2013wba]. Decoupled Hidden Sector ----------------------- The discussion of RSSB led us to the finding that generically all scalar scales are close to the TeV scale if no finetuning in the potential is involved. The most natural mechanism to generate the neutrino mass scale, far below was the connection to lepton number violation and thus models seem favourable where this scale is suppressed. We found the most natural model to be the ISS, in this scenario the Majorana scale is generically at the order of keV. We would like to point out that the connection to the dark matter sector in this context seems very promising by considering two set-ups. Suppose a scenario in which the Hidden sector contains a SM singlet fermion $\nu_x$ with the dark $U(1)$ charge 1 and a SM scalar singlet $\phi_D$ with a dark charge 2 which gets a VEV and thus via the term $\phi_D \bar{\nu}_x \nu_x^c$ generates a mass between the EW and the TeV scale for the fermion. This particle is stable but also almost decoupled from the SM, since the Higgs portal coupling to $\phi_D$ is so far the only allowed interaction channel and it is constrained to be small by experiment. Therefore, it is a decoupled sterile neutrino. It is however possible to switch on a fermionic portal of the form $\bar{\nu}_x \eta \,\ell_R $, as discussed in [@Cao:2009yy; @Giacchino:2013bta; @Toma:2013bka; @Kopp:2014tsa] and in a scale invariant context in [@Antipin:2013exa]. Here $\ell_R$ is a right handed lepton, which is a phenomenologically allowed interaction. The $\eta$ is an electrically charged scalar mediator which has to be of a similar mass due to the requirement of no scalar mass hierarchies. This interaction can lead with the appropriate parameter choice to the production of $\nu_x$ in the early universe with the correct abundance to be a cold dark matter candidate via the lepton portal interaction, as discussed in the literature. This class of models has a rich phenomenology including gamma ray signals which can be peaked and serve as a good DM detection signature [@Kopp:2014tsa]. The detailed discussion, however, goes beyond the scope of this work. The intriguing insight is, that the requirement of no scalar mass hierarchy leads automatically to the region of typical WIMP masses. In the second scenario the ISS, as in **1D** and **2D** with an additional fermionic state $\nu_x$ in the hidden sector, with the charges $(1,0,1)$ in $(SU(2),U(1)_Y, U(1)_\text{Hidden})$ is considered. It is thus a 3 active and 3+3+1 sterile scenario. The mass matrix after eliminating unphysical phases has the structure $$\label{eq:seesawInvDM} \mathcal{M} = \begin{pmatrix} 0 & m_D^{3 \times 3} & 0 & 0 \\ m_D^T & 0 & M_{Rx}^{3 \times 3} & A^{3 \times 1} \\ 0 & M_{Rx}^T & \mu_1^{3 \times 3} & 0 \\ 0 & A^T & 0 & \mu_2 \\ \end{pmatrix} \, .$$ The spectrum would be given by three Pseudo Dirac neutrinos of the scale $M_{Rx}$. The light neutrino mass is given by [Eq. ]{}\[ISSrelations\] and with $A \approx M_{Rx}$, which is natural given order one Yukawas, the additional sterile state has a mass of $\mu$ and a small mixing with the active neutrinos of the order $\mu^2/M_{Rx}^2$. The remarkable feature is that the scale $\mu \approx \text{keV}$ required by the see-saw relation is also the correct scale for this state to be a Dark Matter candidate [@Dolgov:2000ew; @Bezrukov:2009th]. We find that incorporating the neutrino mass generation in the RSSB framework naturally provides us with two scales of DM candidates, those are the TeV scale suitable for a cold Dark Matter particle and the keV scale leading to warm Dark Matter. {width="100.00000%"} \[sec:Conclusion\]Conclusion ============================ We studied in this paper consequences of conformal electro-weak symmetry breaking models for neutrino masses. Many phenomenologically viable models contain extra scalars which undergo dimensional transmutation. The VEV of this scalar triggers then via the Higgs portal electro-weak symmetry breaking. This over-all picture has interesting consequences for neutrino masses. First, no explicit Dirac or Majorana mass terms are allowed, since they would violate conformal symmetry explicitly. All fermion masses must therefore arise as some Yukawa coupling times the VEV of some scalar. The second generic feature is that Coleman-Weinberg type symmetry breaking leads to a loop-generated symmetry breaking effective potential where the Higgs mass (curvature of the minimum) is loop suppressed compared to the VEV. This can be seen in the Standard Model, where Coleman-Weinberg symmetry breaking leads to a Higgs mass of about $9$ GeV, which is excluded. This explains also why the scale which is generated by the extra scalar should be in the TeV range in order to match the EW scale. This implies finally that all neutrino masses come from Dirac and Majorana Yukawa couplings which are multiplied either with the EW scale or with the TeV-ish symmetry breaking scale of the extra scalar. We studied such scenarios in this paper in a rather general context. For that we distinguished three basic strategies of accommodating neutrino masses. Embedding of the SM in a larger gauge group, enlarging the field content of the SM by additional fields or extending the SM by a Hidden sector. In [Sec. ]{}\[sec:Models\] we present a catalogue of viable conformal neutrino mass models and describe them in more detail in the appendix. Note that any neutrino mass between zero and the VEVs (or even somewhat bigger) can be obtained by selecting the corresponding Yukawa couplings. The wide spectrum of Yukawa couplings for other fermions of the SM implies that a wide spectrum of neutrino mass terms is expected in these scenarios. Note that very tiny neutrino masses are still quite natural, since the discussed models suppress them via a see-saw or via loops. We show that in the Yukawa see-saw model the adjustment of the couplings can be reduced to the same amount as present in the charged lepton sector. The amount of tuning the Yukawa couplings can be largely reduced if the neutrino mass generation is related to lepton number violation. If lepton number is taken to be an approximate symmetry, which is broken explicitly in the Lagrangian, the smallness of the braking parameter is natural in the sense, that its absence would increase the symmetry. The lepton number violation parameters can lead to light neutrino masses via loops, or to small Majorana mass contributions of Dirac particle pairs. We present several models where small and or loop suppressed lepton number violation is the driving principle behind neutrino mass generation. In particular when combined with the ISS mechanism the small Majorana mass fraction in the heavy Dirac neutrino pair leads to small active neutrino mass and no fine-tuning is needed. In addition we perform in [Sec. ]{}\[sec:Phenomenology\] a phenomenological analysis with the goal to check whether the models can indeed reproduce the neutrino oscillation data and at the same time be consistent with rare decay experiments, as the $0\nu\beta\beta$ searches. Our finding is, that there are four phenomenologically viable regions. Scenario A has only light neutrinos with Majorana masses, which are generated on tree or loop level by particles with lepton number violating couplings. This scenario can lead to detectable signals in the $0\nu\beta\beta$ decays and the additional states, such as the triplet scalar can be produced at colliders, since their mass must be about the TeV scale. Scenario B is a Pseudo-Dirac scenario where pairs of light mass eigenstates are almost degenerate with only small Majorana mass fractions. This scenario requires, however, very small Dirac Yukawa couplings and is in general experimentally very challenging. The most promising searches for light Pseudo Dirac neutrinos are oscillations on cosmic scales which could probe the small mass splitting. Scenario C is the Sub-TeV scenario with right handed Majorana states below the TeV scale. This region is severely constrained by limits in the $0\nu\beta\beta$ decay, since the lepton number violation is unsuppressed. The collider signature which one would expect are decays to same sign dileptons, a process practically without SM background. In scenario D the right handed mass can be up to few ten TeV. This can be achieved in ISS models where several scalars are in the game and have a hierarchical VEV structure. The ISS scenario is of particular interest, since it improves the Electro-Weak fit with respect to its SM value. In this parameter region the active-sterile mixing is enlarged and can provide testable signals. This conformal ISS is also theoretically attractive since it contains Yukawas of order one and the smallness of the hidden sector parameters is implied by loop suppression and thus completely avoids fine-tuning. In this region the heavy sterile neutrinos are almost mass degenerate Pseudo Dirac pairs with small Majorana mass fractions. This leads to a suppression of lepton flavour violation and the most relevant constraints in this case, come from searches of lepton flavour violating decays, as $\mu \rightarrow e + \gamma$. At colliders a decay of such a heavy neutrino would have a trilepton final state and missing energy without lepton number violation as the smoking gun signal. We briefly comment that the Hidden sector can contain almost decoupled Dark Matter candidates, which can be either coupled via the lepton portal to the SM or due to small active-sterile neutrino mixing. The masses are either at the EW or the keV scale. The observation here is, that taking the gauge hierarchy problem seriously can provide us with a hint for a Dark matter scale. Additional signals in collider experiments are expected to appear in all viable neutrino mass models, since all require new scalar or fermionic states around the TeV mass region. Therefore, we expect that all those models can be tested by the LHC. The collider signatures and neutrino experiments combined will provide very powerful tools for studying and distinguishing among the different scenarios. Phenomenological details of such models and further theoretical aspects will be discussed in future work. Acknowledgments {#acknowledgments .unnumbered} =============== We would like to thank Hiren Patel, Martin Holthausen, Branimir Radovcic, Julian Heeck and Kher-Sham Lim for helpful discussions. JS acknowledges support from the IMPRS for Precision Tests of Fundamental Symmetries. \[app:Models\]Conformal Neutrino Mass Models ============================================ Models within the SM Gauge Group {#models-within-the-sm-gauge-group .unnumbered} -------------------------------- We begin with the systematic description of viable conformal neutrino mass models. It will be very important in this section to point out which particles have been integrated out and which picture of neutrino mass generation we are considering. ### Models with Dominant Contributions to the Left-Handed Majorana Entry {#models-with-dominant-contributions-to-the-left-handed-majorana-entry .unnumbered} - \ : $L:(2,-1);\; H:(2,1);\; \nu^{}_{\!R}:(1,0);\; \varphi:(1,0)$,\ : $-\mathscr{L}_Y=g^{}_{H} \overline{L}\tilde{H} \nu^{}_{\!R}+g^{}_{\varphi}\varphi\overline{\nu^{c}_{\!R}} \nu^{}_{\!R}$ + h.c.\ : $V_{\textrm{I}}=\lambda_H(H^\dagger H)^2 +\lambda_\varphi(\varphi^\dagger\varphi)^2 +\lambda_{H\varphi}(\varphi^\dagger\varphi)(H^\dagger H)$\ With this we find the diagrams\ [conformal1]{} $$\begin{aligned} \hspace{-2cm} \phantom{+} \hspace{1cm} \parbox{45mm}{ \begin{fmfgraph*}(45,40) \fmftop{X1,S1,S2,S3,X2} \fmflabel{$\langle H\rangle$}{S1} \fmflabel{$\langle \varphi\rangle$}{S2} \fmflabel{$\langle H\rangle$}{S3} \fmfbottom{XP1,P1,P2,P3,XP2} \fmfleft{L1} \fmflabel{L}{L1} \fmfright{L2} \fmflabel{L}{L2} \fmf{fermion,tension=1}{L1,I1} \fmf{fermion,tension=1,label=$\nu^{}_{\!R}$}{I1,I2} \fmf{fermion,tension=1,label=$\nu^{}_{\!R}$}{I3,I2} \fmf{fermion,tension=1}{L2,I3} \fmf{scalar,tension=1}{S1,I1} \fmf{scalar,tension=1}{S2,I2} \fmf{scalar,tension=1}{S3,I3} \fmf{phantom,tension=1}{P1,I1} \fmf{phantom,tension=1}{P2,I2} \fmf{phantom,tension=1}{P3,I3} \end{fmfgraph*} } \hspace{15mm} + \hspace{15mm} \parbox{45mm}{ \begin{fmfgraph*}(45,40) \fmftop{X1,S1,S2,X2} \fmflabel{$\langle H\rangle$}{S1} \fmflabel{$\langle H\rangle$}{S2} \fmfbottom{P1,P2} \fmfbottom{S3} \fmflabel{$\langle\varphi\rangle$}{S3} \fmftop{P3} \fmfleft{L1} \fmflabel{L}{L1} \fmfright{L2} \fmflabel{L}{L2} \fmf{fermion,tension=2}{L1,I1} \fmf{fermion,tension=1,label=$\nu^{}_{\!R}$}{I1,I2} \fmf{fermion,tension=1,label=$\nu^{}_{\!R}$}{I3,I2} \fmf{fermion,tension=2}{L2,I3} \fmf{scalar,tension=1}{S1,V} \fmf{scalar,tension=1}{S2,V} \fmf{scalar,right=0.5,label=H}{V,I1} \fmf{scalar,left=0.5,label=H}{V,I3} \fmf{scalar}{S3,I2} \fmf{phantom,tension=1}{VP,P1} \fmf{phantom,tension=1}{VP,P2} \fmf{phantom,left=0.5}{I1,VP} \fmf{phantom,right=0.5}{I3,VP} \fmf{phantom}{I2,P3} \end{fmfgraph*} } \end{aligned}$$ $$\begin{aligned} \hspace{-2cm} + \hspace{1cm} \parbox{45mm}{ \begin{fmfgraph*}(45,40) \fmftop{X4,S1,S2,S3,X3} \fmflabel{$\langle H\rangle$}{S1} \fmflabel{$\langle\varphi\rangle$}{S2} \fmflabel{$\langle H\rangle$}{S3} \fmfbottom{XP4,P1,P2,P3,XP3} \fmfleft{L1} \fmflabel{L}{L1} \fmfright{L2} \fmflabel{L}{L2} \fmf{fermion,tension=3}{L1,I1} \fmf{fermion,tension=1,label=$\nu^{}_{\!R}$}{I1,I2} \fmf{fermion,tension=2.5,label=$\nu^{}_{\!R}$}{I3,I2} \fmf{fermion,tension=2.5}{L2,I3} \fmf{scalar,tension=1}{S1,V} \fmf{scalar,tension=1}{S2,V} \fmf{scalar,right=0.5,tension=1,label=H}{V,I1} \fmf{scalar,left=0.5,tension=1,label=$\varphi$}{V,I2} \fmf{scalar}{S3,I3} \fmf{phantom,tension=1}{VP,P1} \fmf{phantom,tension=1}{VP,P2} \fmf{phantom,left=0.5,tension=1}{I1,VP} \fmf{phantom,right=0.5,tension=1}{I2,VP} \fmf{phantom}{I3,P3} \end{fmfgraph*} } \hspace{15mm} + \hspace{15mm} \parbox{45mm}{ \begin{fmfgraph*}(45,40) \fmftop{X1,S1,S2,S3,X2} \fmflabel{$\langle H\rangle$}{S1} \fmflabel{$\langle\varphi\rangle$}{S2} \fmflabel{$\langle H\rangle$}{S3} \fmfbottom{XP1,P1,P2,P3,XP2} \fmfleft{L1} \fmflabel{L}{L1} \fmfright{L2} \fmflabel{L}{L2} \fmf{fermion,tension=2.5}{L1,I1} \fmf{fermion,tension=2.5,label=$\nu^{}_{\!R}$}{I1,I2} \fmf{fermion,tension=1,label=$\nu^{}_{\!R}$}{I3,I2} \fmf{fermion,tension=3}{L2,I3} \fmf{scalar,tension=1}{S2,V} \fmf{scalar,tension=1}{S3,V} \fmf{scalar,right=0.5,tension=1,label=$\varphi$}{V,I2} \fmf{scalar,left=0.5,tension=1,label=H}{V,I3} \fmf{scalar}{S1,I1} \fmf{phantom,tension=1}{VP,P2} \fmf{phantom,tension=1}{VP,P3} \fmf{phantom,left=0.5,tension=1}{I2,VP} \fmf{phantom,right=0.5,tension=1}{I3,VP} \fmf{phantom}{I1,P1} \end{fmfgraph*} } \end{aligned}$$ The first diagram is the tree level contribution while the other three are one-loop corrections to the first diagram and have thus a smaller contribution to the total neutrino mass. Further contributions have either at least two loops or 9 mass insertions and thus have even smaller impact on the masses. The mass matrix has the following structure: $$\mathcal{M} = \begin{pmatrix} M_L & m_D \\ m_D & M_R \end{pmatrix} \, .$$ The masses are given by $M_R = g_\phi {\ensuremath{\left\langle \varphi \right\rangle}}$, $m_D = g_h {\ensuremath{\left\langle H \right\rangle}}$ and the loop supressed left handed contributions $$\begin{aligned} M_L \approx \frac{g_H^2 g_\phi {\ensuremath{\left\langle H \right\rangle}}^2{\ensuremath{\left\langle \varphi \right\rangle}}}{ (4\pi)^2 \Lambda^2}+ \frac{g_H^2 g_\phi {\ensuremath{\left\langle H \right\rangle}}^2{\ensuremath{\left\langle \varphi \right\rangle}}}{M_R (4\pi)^2 \Lambda} \end{aligned}$$ with $\Lambda$ the dominant loop mass contribution. Integrating out the heavier right handed states leads to an effective mass for the light species of the order $$\begin{aligned} m_\nu \approx g_H^2 \frac{{\ensuremath{\left\langle H \right\rangle}}^2}{M_R} + \frac{g_H^2 g_\phi {\ensuremath{\left\langle H \right\rangle}}^2{\ensuremath{\left\langle \varphi \right\rangle}}}{ (4\pi)^2 \Lambda^2}+ \frac{g_H^2 g_\phi {\ensuremath{\left\langle H \right\rangle}}^2{\ensuremath{\left\langle \varphi \right\rangle}}}{M_R (4\pi)^2 \Lambda} \\ \nonumber = \frac{g_H^2}{g_\phi} \frac{v_H^2}{v_\varphi} \left( 1 + \frac{g_\phi v_\varphi}{(4\pi)^2\Lambda} + \frac{g_\phi^2 \,v_\varphi^2 }{ (4\pi)^2 \Lambda^2}\right). \end{aligned}$$ With $g_H$ being the Dirac and $g_\phi$ the Majorana type Yukawa coupling. The tree level contribution dominates in this scenario. We refer to this model type as the Yukawa see-saw. We consider now in addition a moled with tree level correction to the left-handed Majorana entry by introducing a scalar triplet. - \ : $L:(2,-1);\; H:(2,1);\; \Delta:(3,-2);\; \varphi:(1,0)$\ : $-\mathcal{L}_Y=g_\Delta\bar{L}\vec{\sigma}\Delta L^c+h.c. =g_\Delta(\bar{L}\vec{\sigma}\Delta L^c+\bar{L}^c\vec{\sigma}\Delta^*L)$\ : $$\begin{aligned} \nonumber V_{\textrm{II}}= \lambda_H (H^\dagger H)^2 + \lambda_{\Delta T}Tr(\Delta^\dagger \Delta)^2 + \lambda_{T \Delta}(Tr(\Delta^\dagger \Delta))^2 \\ \nonumber + \lambda_{H\Delta, 1}(H^\dagger H) Tr\Delta^\dagger \Delta + \lambda_{H\Delta,2} H^\dagger\Delta \Delta^\dagger H \\ \nonumber +\lambda_\varphi(\varphi^\dagger \varphi)^2 +\lambda_{H\varphi}(\varphi^\dagger \varphi)(H^\dagger H) \\ \nonumber + \lambda_{\varphi\boldsymbol{\Delta}}(\varphi^\dagger \varphi)\operatorname{Tr}{\boldsymbol{\Delta}^\dagger \boldsymbol{\Delta}}+\lambda_{\varphi\boldsymbol{\Delta} H}[\varphi H^Ti\sigma_2 \boldsymbol{\Delta} H+h.c.] . \end{aligned}$$ All 1-Particle-Irreducible (1PI) diagrams with upto 3 mass insertions and maximum one loop are given by\ [conformalJuri2]{} $$\begin{aligned} \hspace{1cm} \phantom{+} \parbox{45mm}{ \begin{fmfgraph*}(45,40) \fmfleft{L1} \fmflabel{L}{L1} \fmfright{L2} \fmflabel{L}{L2} \fmftop{S1} \fmflabel{$\langle \Delta \rangle$}{S1} \fmfbottom{P1} \fmf{fermion}{L1,I1} \fmf{fermion}{L2,I1} \fmf{scalar}{I1,S1} \fmf{phantom}{I1,P1} \end{fmfgraph*} } \hspace{5mm} + \hspace{5mm} \parbox{45mm} { \begin{fmfgraph*}(45,40) \fmfleft{L1} \fmflabel{L}{L1} \fmfright{L2} \fmflabel{L}{L2} \fmftop{X1,S1,S2,X2} \fmflabel{$\langle\Delta\rangle$}{S1} \fmflabel{$\langle\Delta\rangle$}{S2} \fmfbottom{S3} \fmflabel{$\langle\Delta\rangle$}{S3} \fmfbottom{XP1,P1,P2,XP2} \fmftop{P3} \fmf{fermion,tension=2.5}{L1,I1} \fmf{fermion,label=L,tension=1}{I2,I1} \fmf{fermion,label=L,tension=1}{I2,I3} \fmf{fermion,tension=2.5}{L2,I3} \fmf{scalar,left=0.5,label=$\Delta$}{I1,V} \fmf{phantom,left=0.5}{I1,VP} \fmf{scalar,right=0.5,label=$\Delta$}{I3,V} \fmf{phantom,right=0.5}{I3,VP} \fmf{scalar}{V,S1} \fmf{phantom}{VP,P1} \fmf{scalar}{V,S2} \fmf{phantom}{VP,P2} \fmf{scalar}{S3,I2} \fmf{phantom}{P3,I2} \end{fmfgraph*} } \hspace{5mm} + \hspace{5mm} \parbox{45mm} { \begin{fmfgraph*}(40,50) \fmfleft{L1} \fmflabel{L}{L1} \fmfright{L2} \fmflabel{L}{L2} \fmftop{X1,S1,S2,S3,X2} \fmflabel{$\langle H\rangle$}{S1} \fmflabel{$\langle\varphi\rangle$}{S2} \fmflabel{$\langle H\rangle$}{S3} \fmfbottom{PX1,P1,P2,P3,PX2} \fmf{fermion}{L1,I1} \fmf{fermion}{L2,I1} \fmf{scalar,label=$\Delta$,tension=2.5}{I1,V} \fmf{phantom,tension=2.5}{I1,VP} \fmf{scalar}{S1,V} \fmf{phantom}{P1,VP} \fmf{scalar}{S2,V} \fmf{phantom}{P2,VP} \fmf{scalar}{S3,V} \fmf{phantom}{P3,VP} \end{fmfgraph*} } \end{aligned}$$ The theory at hand is the conformal analogue of the type II see-saw mechanism. Based on measurments of EWPOs the VEV $\langle \Delta_0 \rangle$ has to be orders of magnitude below the EW scale and in our single scale scenario it seems more natural for it to be exactly zero at tree level. Therefore, the main contribution comes from the third diagram which yields the neutrino mass $$\begin{aligned} M_L = g_\Delta \frac{\lambda_{\varphi \Delta H}}{M_\Delta^2} \langle \varphi \rangle \langle H \rangle^2, \end{aligned}$$ where $M_\Delta$ is the physical mass of the scalar triplet. This is controlled by the lepton number violating coupling $\lambda_{\varphi \Delta H} $, furthermore the neutrino mass is suppressed by the mass of the triplet scalar. The mass of the double charged triplet component is experimentally constrained to be above 450 GeV [@CMS:2012ulp] and since there should be no large splitting among the components we assume the neutral component to be at least of the same order. This model can be enlarged by right handed neutrinos, which leads us to - \ : $L:(2,-1);\; H:(2,1);\; \Delta:(3,-2);\; \varphi:(1,0);\; \nu^{}_{\!R}:(1,0)$\ : $-\mathscr{L}_Y=g^{}_{H}\bar{L}\tilde{H}\nu^{}_{\!R} +g^{}_{\varphi}\varphi\bar{\nu}^{c}_{\!R}\nu^{}_{\!R} +g_\Delta\bar{L}\vec{\sigma}\Delta L^c+h.c.$\ : $V=V_{\textrm{II}}$\ The following diagram is additional to those of **3A** and **5A** [conformalJuri3]{} $$\begin{aligned} \hspace{1cm} \phantom{+} \parbox{45mm}{ \begin{fmfgraph*}(45,40) \fmftop{X4,S1,S2,S3,X3} \fmflabel{$\langle H\rangle$}{S1} \fmflabel{$\langle\Delta\rangle$}{S2} \fmflabel{$\langle H\rangle$}{S3} \fmfbottom{XP4,P1,P2,P3,XP3} \fmfleft{L1} \fmflabel{L}{L1} \fmfright{L2} \fmflabel{L}{L2} \fmf{fermion,tension=3}{L1,I1} \fmf{fermion,tension=1,label=$\nu^{}_{\!R}$}{I1,I2} \fmf{fermion,tension=2.5,label=$\nu^{}_{\!R}$}{I3,I2} \fmf{fermion,tension=2.5}{L2,I3} \fmf{scalar,tension=1}{V,S1} \fmf{scalar,tension=1}{V,S2} \fmf{scalar,right=0.5,tension=1,label=H}{V,I1} \fmf{scalar,left=0.5,tension=1,label=$\varphi$}{V,I2} \fmf{scalar}{S3,I3} \fmf{phantom,tension=1}{VP,P1} \fmf{phantom,tension=1}{VP,P2} \fmf{phantom,left=0.5,tension=1}{I1,VP} \fmf{phantom,right=0.5,tension=1}{I2,VP} \fmf{phantom}{I3,P3} \end{fmfgraph*} } \end{aligned}$$ The diagram contributes to the left handed mass an approximate term of the order ${\ensuremath{\left\langle H \right\rangle}}^2{\ensuremath{\left\langle \Delta \right\rangle}}/((4\pi)^2 \Lambda \,M_R)$ which is supressed by the smallness of the triplet VEV and therefore subdominant. In this model the $\varphi$ field can have a VEV, which brings us to the Yukawa see-saw scenario, or it can have no VEV and the right handed neutrino only adds a Dirac contribution to the neutrino mass. In this case the phenomenology would be of the Pseudo Dirac scenario. Like seen in the non-conformal case it is also possible to introduce a triplet fermion to couple to the left-handed doublet. Unlike in the non-conformal scenario we now have to introduce an uncharged singlet scalar to generate neutrino masses. - \ : $L:(2,-1);\; H:(2,1);\; \Sigma:(3,0);\; \varphi:(1,0)$,\ : $-\mathscr{L}_Y=g_\Sigma \tilde{H}^\dagger \overline{\Sigma} L + g_\varphi \varphi \operatorname{Tr}{\left[ \overline{\Sigma^c} \Sigma \right]} + h.c. $\ : $V = V_{\textrm{I}}$\ The main contribution to the neutrino mass is given by\ [conformal4]{} $$\begin{aligned} \parbox{55mm}{ \begin{fmfgraph*}(55,40) \fmfstraight \fmfleft{L} \fmflabel{L}{L} \fmfright{R} \fmflabel{L}{R} \fmftop{P1,H1,F,H3,P2} \fmflabel{$\langle H \rangle$}{H1} \fmflabel{$\langle \varphi \rangle$}{F} \fmflabel{$\langle H \rangle$}{H3} \fmf{fermion}{L,V1} \fmf{fermion,label=$\Sigma$}{V1,V2} \fmf{fermion,label=$\Sigma$}{V3,V2} \fmf{fermion}{R,V3} \fmffreeze \fmf{scalar}{H1,V1} \fmf{scalar}{F,V2} \fmf{scalar}{H3,V3} \end{fmfgraph*} } \end{aligned}$$ This diagram yields the mass $$M_L = g_\Sigma^2 \frac{\langle H \rangle^2}{g_\varphi \langle \varphi \rangle}.$$ ### Models with Dominant Contributions to the Right-Handed Majorana Entry {#models-with-dominant-contributions-to-the-right-handed-majorana-entry .unnumbered} Already in model **3A** right handed neutrinos with Majorana mass were considered. There are, however, further ways to influence the right-handed Majorana mass. The first possibility we want to study is to introduce a scalar and a fermion triplet and a scalar singlet. - \ : $L:(2,-1) ;\; \nu_R:(1,0);\; \Sigma:(3,0);\; H:(2,1);\; \Delta:(3,0);\; \varphi:(1,0)$\ : $-\mathscr{L}_Y =g^{}_{H}\bar{L}\tilde{H}\nu^{}_{\!R} + g_{\Delta} \operatorname{Tr}{[\overline{\Sigma} \boldsymbol{\Delta} \nu_R]} + g_{\varphi,1}\operatorname{Tr}{[\varphi \overline{\Sigma^ c}\Sigma]} + g_{\varphi,2} \varphi \overline{\nu_R} \nu_R^c + h.c.$\ The relevant lepton number violating term in the potential is displayed. : $V \supset \lambda \varphi H^Ti\sigma_2\boldsymbol{\Delta}^\dagger \tilde{H} + h.c.$\ Furthermore we forbid the VEV of $\Delta$. In addition to the diagram of **3A** we get the diagram\ [conformalJuri5]{} $$\begin{aligned} \parbox{55mm}{ \begin{fmfgraph*}(55,40) \fmfstraight \fmfleft{L} \fmflabel{$\nu_R$}{L} \fmfright{R} \fmflabel{$\nu_R$}{R} \fmftop{A,T1,T2,T3,T4,T5,T6,T7,B} \fmflabel{$\langle H \rangle$}{T1} \fmflabel{$\langle \varphi \rangle$}{T2} \fmflabel{$\langle H \rangle$}{T3} \fmflabel{$\langle H \rangle$}{T5} \fmflabel{$\langle \varphi \rangle$}{T6} \fmflabel{$\langle H \rangle$}{T7} \fmfbottom{xA,xT1,xT2,xT3,xT4,xT5,xT6,xT7,xB} \fmflabel{$\langle \varphi \rangle$}{xT4} \fmf{fermion}{L,V1} \fmf{fermion,label=$\Sigma$}{V1,V2} \fmf{fermion,label=$\Sigma$}{V3,V2} \fmf{fermion}{R,V3} \fmffreeze \fmf{scalar,label=$\Delta$,tension=2.5}{TV1,V1} \fmf{scalar,tension=2.5}{xT4,V2} \fmf{scalar,label=$\Delta$,tension=2.5}{TV3,V3} \fmf{scalar}{T1,TV1} \fmf{scalar}{T2,TV1} \fmf{scalar}{TV1,T3} \fmf{scalar}{T5,TV3} \fmf{scalar}{T6,TV3} \fmf{scalar}{TV3,T7} \end{fmfgraph*} } \end{aligned}$$ Note that the scalar triplet $\Delta$ cannot be used to generate left-handed Majorana masses as it has the wrong hypercharge. Adding contributions from both diagrams the right-handed mass is given by $$\begin{split} M_R & = g_{\varphi,2} \langle \varphi \rangle + \lambda^2 g_{\Delta}^2 \frac{\langle H \rangle^4 \langle \varphi \rangle^2}{M_\Sigma \cdot M_\Delta^4} \\ & \approx \left( g_{\varphi,2} + g_{\Delta}^2 \frac{\text{GeV}^2}{g_{\varphi,1} {\ensuremath{\left\langle \varphi \right\rangle}}^2} \right) {\ensuremath{\left\langle \varphi \right\rangle}}. \end{split}$$ Here the fact was used, that the combination $\lambda {\ensuremath{\left\langle H \right\rangle}}^2{\ensuremath{\left\langle \varphi \right\rangle}}/M_\Delta$ from the diagram induces an effective VEV of the triplet field , which is experimentally constrained by measurements of the $\rho$ parameter to be ${\ensuremath{\left\langle \Delta \right\rangle}} \lesssim 1 \text{GeV}$. Thus the second term is subdominant. Models with an Additional Hidden Sector Symmetry {#models-with-an-additional-hidden-sector-symmetry .unnumbered} ------------------------------------------------ The particle content is extended by additional SM singlet fermions. However, those would not be distinguishable from the sterile neutrinos $\nu_R$ if they had all quantum numbers in common. Now with the Hidden Sector symmetry, which will be denoted by $U(1)_H$, there are observable effects. The SM singlet fermions with a hidden charge are denoted by $\nu_x$ and this requires the mass matrix to be extended to $3 \times 3$ in the one flavour case $$\mathcal{M} = \begin{pmatrix} M_L & m_D & 0 \\ m_D & M_R & M_{Rx} \\ 0 & M_{Rx} & M_x \end{pmatrix} \, .$$ Note that the sterile neutrino $\nu_R$ must not carry a hidden charge, as otherwise coupling to the Higgs would be forbidden and the complete sector would decouple. ### Modifying the $\nu_R$ Majorana Mass {#modifying-the-nu_r-majorana-mass .unnumbered} We begin with a theory in which the direct term $$g \varphi \overline{\nu_R} \nu_R^c$$ is forbidden by the additional HS symmetry. - \ : $L:(2,-1, 0) ;\; H:(2, 1,0) ;\; \nu_R:(1,0,0);\; \nu_x:(1,0,1);\; \varphi_1:(1,0,1);\; \varphi_2:(1,0,2)$,\ where the third number in brackets denotes the HS charge. This particle content yields the additional terms\ : $-\mathscr{L}_{Y_1} = g_1 \varphi_1 \overline{\nu_R} \nu_x^c + g_2 \varphi_2 \overline{\nu_x} \nu_x^c +g^{}_{H}\bar{L}\tilde{H}\nu^{}_{\!R}$\ If $\varphi_1$ and $\varphi_2$ get a VEV this theory yields the mass matrix $$\mathcal{M} = \begin{pmatrix} 0 & m_D & 0 \\ m_D & 0 & M_{Rx} \\ 0 & M_{Rx} & M_x \end{pmatrix} \, .$$ This mass matrix represents the double see-saw mechanism [@Barr:2003nn]. In language of diagrams this model is represented by\ [double\_seesaw]{} $$\begin{aligned} \parbox{55mm}{ \begin{fmfgraph*}(55,45) \fmfstraight \fmfleft{L} \fmflabel{$\nu_R$}{L} \fmfright{R} \fmflabel{$\nu_R$}{R} \fmftop{A,T1,T2,T3,B} \fmflabel{$\langle \varphi_1 \rangle$}{T1} \fmflabel{$\langle \varphi_2 \rangle$}{T2} \fmflabel{$\langle \varphi_1 \rangle$}{T3} \fmf{fermion}{L,V1} \fmf{fermion,label=$\nu_x$}{V2,V1} \fmf{fermion,label=$\nu_x$}{V2,V3} \fmf{fermion}{R,V3} \fmffreeze \fmf{scalar}{V1,T1} \fmf{scalar}{T2,V2} \fmf{scalar}{V3,T3} \end{fmfgraph*} } \end{aligned}$$ \ Integrating out $\nu_x$ we obtain an effective mass $M_R$ and find the contracted mass matrix $$\mathcal{M} = \begin{pmatrix} 0 & m_D \\ m_D & M_R \end{pmatrix} \, ,$$ where $M_R$ can be calculated from the diagram. Two cases are relevant, either if $M_{Rx}<<M_x$ one has $$M_R \approx \frac{g_1^2}{g_2} \frac{\langle \varphi_1 \rangle^2}{\langle \varphi_2 \rangle} \, ,$$ or in the other limit $M_{Rx}>>M_x$ the mass is $$M_R \approx M_{Rx} = g_1 \langle \varphi_1 \rangle \, .$$ This indicates that it is possible to have either the double or the inverse see-saw scenario realized. So far there is no reason to assume that $M_x$ is small, thus the more natural scenario in this model is the double see-saw, leading to a Sub-TeV see-saw scenario. - \ : $ L:(2,-1, 0) ;\; H:(2,1,0) ;\; \nu_R:(1,0,0);\; \nu_x:(1,0,2);\; \varphi_1:(1,0,0) ;\; \varphi_2:(1,0,-2)$\ : $-\mathscr{L}_Y \supset -\mathscr{L}_{Y_1} $ We see that the Majorana mass term for the hidden sector fermion can not be constructed and hence the matrix structure is $$\mathcal{M} = \begin{pmatrix} 0 & m_D & 0 \\ m_D & M_R & M_{Rx} \\ 0 & M_{Rx} &0 \end{pmatrix} \, .$$ This is a structure of the minimal extended see-saw, discussed in [@Heeck:2012bz], but here it is at the TeV scale. The interesting feature is that with $M_R > m_D$ and $M_R > M_{Rx}$ this see-saw scenario generates light active and sterile neutrinos which can have large mixing with the active sector. The light sterile neutrino could for instance explain the missing upturn in the Super Kamiokande data, as discussed in [@Smirnov:2014zga], a detailed discussion of this scenario is beyond the scope of this work. A phenomenologically different scenario occurs if we forbid the VEV $\langle \varphi_1 \rangle$. Consider the following theory. - \ : $ L:(2,-1,0) ;\; H:(2,1,0) ;\;\nu_R:(1,0,0);\; \nu_x:(1,0,1);\; \varphi_1:(1,0,1);\; \varphi_2:(1,0,2);\; \varphi_3:(1,0,-4)$,\ Note that the newly introduced SM singlet scalar $\varphi_3$ does not change the Yukawa Lagrangian. There is, however, an additional potential term:\ : $V \supset \lambda \varphi_1^2 \varphi_2 \varphi_3 \, + h.c.$\ Thus if we forbid, as mentioned, the VEV of $\varphi_1$, the diagram with the main contribution to $M_R$ is given by\ [right\_radiative]{} $$\begin{aligned} \parbox{55mm}{ \begin{fmfgraph*}(55,45) \fmfleft{L} \fmflabel{$\nu_R$}{L} \fmfright{R} \fmflabel{$\nu_R$}{R} \fmftop{TA,T1,T2,TB} \fmflabel{$\langle \varphi_2 \rangle$}{T1} \fmflabel{$\langle \varphi_3 \rangle$}{T2} \fmfbottom{BA,BB,B,BC,BD} \fmflabel{$\langle \varphi_2 \rangle$}{B} \fmf{fermion}{L,V1} \fmf{fermion,label=$\nu_x$}{V2,V1} \fmf{fermion,label=$\nu_x$}{V2,V3} \fmf{fermion}{R,V3} \fmffreeze \fmf{scalar,left=.5,label=$\varphi_1$}{V1,TV} \fmf{scalar,right=.5,label=$\varphi_1$}{V3,TV} \fmf{scalar,tension=1.3}{T1,TV} \fmf{scalar,tension=1.3}{T2,TV} \fmf{scalar}{B,V2} \end{fmfgraph*} } \end{aligned}$$ \ The mass of the right handed neutrino is generated at one loop and the effective mass matrix reads $$\mathcal{M} = \begin{pmatrix} 0 & m_D \\ m_D & M_R \end{pmatrix} \, .$$ To approximate the scale of $M_R$ we use the fact that this loop has the same topology as in the Ma model. Therefore, the right handed mass scale is $$\begin{aligned} M_R \approx \frac{\lambda}{16\pi^2} \frac{ g_1^2}{g_2} {\ensuremath{\left\langle \varphi_3 \right\rangle}} I\left(\frac{{\ensuremath{\left\langle \varphi_2 \right\rangle}}^2}{M_{\varphi_1}^2} \right) \, , \\ \text{ with } I(x) = \frac{1}{1-x}\left(1+\frac{x \log{x}}{1-x}\right) \,. \end{aligned}$$ Thus the right handed mass is loop suppressed and controlled by the parameter $\lambda$, which if set to zero increases the Lagrangian symmetry. Therefore, this model leads to a scenario with Pseudo-Dirac active neutrinos. - \ : $L:(2,-1,0) ;\; H:(2,1,0) ;\; \nu_R:(1,0,0); \; \nu_x:(1,0,1); \; \Sigma:(3,0,1); \; \Delta:(3,0,1); \; H:(2,1,0);\;\varphi_1:(1,0,1);\; \varphi_2:(1,0,2)$\ : $-\mathscr{L}_Y =-\mathscr{L}_{Y_1} + g_{\Delta} \operatorname{Tr}{[\overline{\Sigma} \Delta \nu_R]} + g_\Sigma \operatorname{Tr}{[\varphi_2 \overline{\Sigma^ c}\Sigma]} + h.c.$\ : $V \supset \lambda \varphi_1 H^Ti\sigma_2\boldsymbol{\Delta}^\dagger \tilde{H}+ h.c.$\ Note that we only displayed terms in the Yukawa Lagrangian and the potential that are relevant for the lowest order diagram of right-handed neutrino mass generation. The diagram additional to **1C** is given by\ [right\_hidden]{} $$\begin{aligned} \parbox{65mm}{ \begin{fmfgraph*}(65,45) \fmfstraight \fmfleft{L} \fmflabel{$\nu_R$}{L} \fmfright{R} \fmflabel{$\nu_R$}{R} \fmftop{A,T1,T2,T3,T4,T5,T6,T7,B} \fmflabel{$\langle H \rangle$}{T1} \fmflabel{$\langle \varphi_1 \rangle$}{T2} \fmflabel{$\langle H \rangle$}{T3} \fmflabel{$\langle H \rangle$}{T5} \fmflabel{$\langle \varphi_1 \rangle$}{T6} \fmflabel{$\langle H \rangle$}{T7} \fmfbottom{xA,xT1,xT2,xT3,xT4,xT5,xT6,xT7,xB} \fmflabel{$\langle \varphi_2 \rangle$}{xT4} \fmf{fermion}{L,V1} \fmf{fermion,label=$\Sigma$}{V1,V2} \fmf{fermion,label=$\Sigma$}{V3,V2} \fmf{fermion}{R,V3} \fmffreeze \fmf{scalar,label=$\Delta$,tension=2.5}{TV1,V1} \fmf{scalar,tension=2.5}{xT4,V2} \fmf{scalar,label=$\Delta$,tension=2.5}{TV3,V3} \fmf{scalar}{T1,TV1} \fmf{scalar}{T2,TV1} \fmf{scalar}{TV1,T3} \fmf{scalar}{T5,TV3} \fmf{scalar}{T6,TV3} \fmf{scalar}{TV3,T7} \end{fmfgraph*} } \end{aligned}$$ \ If $\Delta$ does not get a VEV at tree level this is the leading tree-level diagram in the $3 \times 3$ space as the term $\varphi \overline{\nu_R} \nu_R^c$ is forbidden. The right-handed mass $M_R$ can therefore be estimated using the same argument as in **1B** $$M_R = \frac{\lambda^2 g_{\Delta}^2}{g_\Sigma} \left( \frac{\langle H \rangle}{M_\Delta} \right)^4 \frac{\langle \varphi_1 \rangle^2}{\langle \varphi_2 \rangle} \lesssim \frac{ g_{\Delta}^2}{{g_\Sigma}} \frac{\text{GeV}^2}{{\ensuremath{\left\langle \varphi_2 \right\rangle}}}.$$ This means that the mass matrix is given by $$\mathcal{M} = \begin{pmatrix} 0 & m_D & 0 \\ m_D & M_R & M_{Rx} \\ 0 & M_{Rx} & M_x \end{pmatrix} \, .$$ Which is similar to **1C** but with a non vanishing $M_R$ at tree level. Now we turn to a theory with different phenomenology by forbidding the VEV of $\varphi_1$. - \ : $L:(2,-1,0) ;\; \nu_R:(1,0,0);\; \Sigma:(3,0,1);\; \Delta:(3,0,1);\; H:(2,1,0);\; \varphi_1:(1,0,1);\; \varphi_2:(1,0,2);\; \varphi_3:(1,0,-4)$\ The Yukawa Lagrangian is the same as in the previous theory, while we get an additional potential term.\ : $V \supset \lambda \varphi_1 H^Ti\sigma_2 \boldsymbol{\Delta}^\dagger \tilde{H} + \lambda' \varphi_1^2 \varphi_2 \varphi_3 + h.c. + ...$\ With $\langle \varphi_1 \rangle=0$ the lowest order diagram contributing to the right-handed neutrino mass is given by\ [right\_hidden\_radiative]{} $$\begin{aligned} \parbox{65mm}{ \begin{fmfgraph*}(65,55) \fmfleft{L} \fmflabel{$\nu_R$}{L} \fmfright{R} \fmflabel{$\nu_R$}{R} \fmftop{A,T1,T2,B,T3,T4,C,T5,T6,D} \fmflabel{$\langle H \rangle$}{T1} \fmflabel{$\langle H \rangle$}{T2} \fmflabel{$\langle \varphi_2 \rangle$}{T3} \fmflabel{$\langle \varphi_3 \rangle$}{T4} \fmflabel{$\langle H \rangle$}{T5} \fmflabel{$\langle H \rangle$}{T6} \fmfbottom{B} \fmflabel{$\langle \varphi_2 \rangle$}{B} \fmf{fermion}{L,V1} \fmf{fermion,lab=$\Sigma$}{V1,V2} \fmf{fermion,lab=$\Sigma$}{V3,V2} \fmf{fermion}{R,V3} \fmffreeze \fmf{scalar,right=1/4,tension=1.5,lab=$\Delta$}{TV1,V1} \fmf{scalar,right=1/4,lab=$\varphi_1$}{TV2,TV1} \fmf{scalar,left=1/4,lab=$\varphi_1$}{TV2,TV3} \fmf{scalar,left=1/4,tension=1.5,lab=$\Delta$}{TV3,V3} \fmf{scalar,tension=.3}{T1,TV1} \fmf{scalar,tension=.3}{TV1,T2} \fmf{scalar,tension=.3}{TV2,T3} \fmf{scalar,tension=.3}{TV2,T4} \fmf{scalar,tension=.3}{T5,TV3} \fmf{scalar,tension=.3}{TV3,T6} \fmf{scalar}{V2,B} \end{fmfgraph*} } \end{aligned}$$ \ Using the fact that the loop has the same topology as in **3C** and just the external VEVs are different we get $$\begin{aligned} M_R \approx \frac{\lambda^2 \lambda' }{16\pi^2} \left(\frac{{\ensuremath{\left\langle H \right\rangle}}}{M_\Delta}\right)^4 \frac{ g_\Delta^2}{g_\Sigma} {\ensuremath{\left\langle \varphi_3 \right\rangle}} I\left(\frac{{\ensuremath{\left\langle \varphi_2 \right\rangle}}^2}{M_{\varphi_1}^2} \right) \, . \end{aligned}$$ This loop suppression combined with a mass suppression to the fourth power with the Triplet mass can generate the Pseudo-Dirac scenario for active neutrinos without large fine tuning in the Majorana mass sector. ### Modifying the $\nu_x$ Majorana Mass {#modifying-the-nu_x-majorana-mass .unnumbered} The general mass matrix structure for the following models will be of the form $$\label{eq:inverse} \mathcal{M} = \begin{pmatrix} 0 & m_D & 0 \\ m_D & 0 & M_{Rx} \\ 0 & M_{Rx} & M_x \end{pmatrix} \, .$$ - \ : $ L:(2,-1, 0) ;\; \nu_R:(1,0,0) ;\; \nu_x:(1,0,1);\; \Sigma:(3,0,-2);\; H:(2,1,0);\; \varphi_1:(1,0,-3);\;\varphi_2:(1,0,-4);\; \Delta:(3,0,-3); \; \varphi_4:(1,0,1)$\ : $-\mathscr{L}_Y \supset g_H \overline{L}\tilde{H}\nu_R + g_{Rx} \varphi_4 \overline{\nu_R} \nu_x^c + g_{\Delta} \operatorname{Tr}{[\overline{\Sigma} \Delta \nu_x]} + g_\Sigma \operatorname{Tr}{[\varphi_2 \overline{\Sigma^ c}\Sigma]} + h.c.$\ : $V \supset \lambda \varphi_1 H^Ti\sigma_2\boldsymbol{\Delta}^\dagger \tilde{H}+ h.c. + ...$\ The leading diagram is\ [x\_hidden]{} $$\begin{aligned} \parbox{65mm}{ \begin{fmfgraph*}(65,45) \fmfstraight \fmfleft{L} \fmflabel{$\nu_x$}{L} \fmfright{R} \fmflabel{$\nu_x$}{R} \fmftop{A,T1,T2,T3,T4,T5,T6,T7,B} \fmflabel{$\langle H \rangle$}{T1} \fmflabel{$\langle \varphi_1 \rangle$}{T2} \fmflabel{$\langle H \rangle$}{T3} \fmflabel{$\langle H \rangle$}{T5} \fmflabel{$\langle \varphi_1 \rangle$}{T6} \fmflabel{$\langle H \rangle$}{T7} \fmfbottom{xA,xT1,xT2,xT3,xT4,xT5,xT6,xT7,xB} \fmflabel{$\langle \varphi_2 \rangle$}{xT4} \fmf{fermion}{L,V1} \fmf{fermion,label=$\Sigma$}{V1,V2} \fmf{fermion,label=$\Sigma$}{V3,V2} \fmf{fermion}{R,V3} \fmffreeze \fmf{scalar,label=$\Delta$,tension=2.5}{TV1,V1} \fmf{scalar,tension=2.5}{xT4,V2} \fmf{scalar,label=$\Delta$,tension=2.5}{TV3,V3} \fmf{scalar}{T1,TV1} \fmf{scalar}{T2,TV1} \fmf{scalar}{TV1,T3} \fmf{scalar}{T5,TV3} \fmf{scalar}{T6,TV3} \fmf{scalar}{TV3,T7} \end{fmfgraph*} } \end{aligned}$$ \ The mass matrix is given by eq. (\[eq:inverse\]) and the Majorana mass of $\nu_x$ is $$M_x = \frac{\lambda^2 g_{\Delta}^2}{g_\Sigma} \left( \frac{\langle H \rangle}{M_\Delta} \right)^4 \frac{\langle \varphi_1 \rangle^2}{\langle \varphi_2 \rangle} \lesssim \frac{ g_{\Delta}^2}{{g_\Sigma}} \frac{\text{GeV}^2}{{\ensuremath{\left\langle \varphi_2 \right\rangle}}} ,$$ where the suppression of the small lepton number violating contribution by the heavy scalar VEV makes it an inverse see-saw scenario. Implying sterile neutrinos with at the TeV scale and slightly above. Those form pseudo Dirac pairs and can have seizable mixing with the active neutrinos. With the mass scale of ${\ensuremath{\left\langle \varphi_2 \right\rangle}}$ around a few TeV and the yuakawa couplings of $g_\Delta \approx 10^{-1}$ and $ g_\Sigma \approx 1$, the scale $M_x$ is naturally at the keV scale, which is required phenomenologically to have sub eV active neutrino masses. The active-sterile mixing is approximately given by $(m_D/M_{Rx})^2$ and can in principle range from 1$\%$ to undetectable values below $10^{-10}$. The interesting observation is that small active-sterile mixing requires unnaturaly small Dirac Yukawa couplings in this model. It is possible that the $\nu_x$ Majorana masses are generated radiatively. - \ : $ L:(2,-1, 0) ;\; \nu_R:(1,0,0) ;\; \nu_x:(1,0,1);\; \Sigma:(3,0,-2);\; H:(2,1,0);\; \varphi_1:(1,0,-3);\; \varphi_2:(1,0,-4);\; \varphi_3:(1,0,10)\Delta:(3,0,-3);\; \varphi_4:(1,0,1)$\ Here ${\ensuremath{\left\langle \varphi_1 \right\rangle}}=0$ and the Yukawa Lagrangian is the same as in the theory before. The potential, however, is extended.\ : $V \supset \lambda \varphi_1 H^Ti\sigma_2\boldsymbol{\Delta}^\dagger \tilde{H} + \lambda' \varphi_1^2 \varphi_2 \varphi_3 + h.c.$\ We obtain the following diagram\ [x\_hidden\_radiative]{} $$\begin{aligned} \parbox{65mm}{ \begin{fmfgraph*}(65,55) \fmfleft{L} \fmflabel{$\nu_x$}{L} \fmfright{R} \fmflabel{$\nu_x$}{R} \fmftop{A,T1,T2,B,T3,T4,C,T5,T6,D} \fmflabel{$\langle H \rangle$}{T1} \fmflabel{$\langle H \rangle$}{T2} \fmflabel{$\langle \varphi_2 \rangle$}{T3} \fmflabel{$\langle \varphi_3 \rangle$}{T4} \fmflabel{$\langle H \rangle$}{T5} \fmflabel{$\langle H \rangle$}{T6} \fmfbottom{B} \fmflabel{$\langle \varphi_2 \rangle$}{B} \fmf{fermion}{L,V1} \fmf{fermion,lab=$\Sigma$}{V1,V2} \fmf{fermion,lab=$\Sigma$}{V3,V2} \fmf{fermion}{R,V3} \fmffreeze \fmf{scalar,right=1/4,tension=1.5,lab=$\Delta$}{TV1,V1} \fmf{scalar,right=1/4,lab=$\varphi_1$}{TV2,TV1} \fmf{scalar,left=1/4,lab=$\varphi_1$}{TV2,TV3} \fmf{scalar,left=1/4,tension=1.5,lab=$\Delta$}{TV3,V3} \fmf{scalar,tension=.3}{T1,TV1} \fmf{scalar,tension=.3}{TV1,T2} \fmf{scalar,tension=.3}{TV2,T3} \fmf{scalar,tension=.3}{TV2,T4} \fmf{scalar,tension=.3}{T5,TV3} \fmf{scalar,tension=.3}{TV3,T6} \fmf{scalar}{V2,B} \end{fmfgraph*} } \end{aligned}$$ \ As before the Majorana mass of $\nu_x$ can be approximated by $$M_R \sim 10^{-2} \cdot \frac{g_\Delta^2 \lambda^2 \lambda'}{g_\Sigma} \left( \frac{{\ensuremath{\left\langle H \right\rangle}}}{M_\Delta} \right)^4\cdot EWS \, .$$ We see that in this setup the $\nu_x$ mass is at the keV scale when the Yukawa couplings are of order one, the potential terms between 0.1 and one and the Triplet around the TeV scale. This is the right scale for the inverse see-saw scenario. Note that as before we need another scalar $\varphi_4$ for the connection between SM sector and Hidden Sector. \[app:Exceptions\] Fully Radiative Generated Left-Handed Masses =============================================================== As was shown by \[sec:Radiative\] there is no way of generating left-handed neutrino masses radiatively by pairwise coupling scalars in the potential. We go through the five possibilities for non pairwise coupling of scalars and study whether radiative mass generation is possible. Furthermore, we present possibilities to circumvent \[sec:Radiative\]. - **Possibility 1**: We can introduce a potential coupling of four different $SU(2)$ singlet scalars such that their hypercharges add up to zero. In this case one $SU(2)$ singlet with vanishing hypercharge has to be included as we need an electrically neutral scalar to gain a VEV.\ With this kind of coupling it is indeed possible to construct a theory that generates neutrino masses fully radiatively. Consider as an example the theory **11A**.\ : $L:(2,-1);\; \ell_R:(1,-2);\; H:(2,1);\;\delta_-:(1,-2);\; \epsilon_{++}:(1,4)$\ : $-\mathscr{L}_Y \supset g_\delta\bar{L}L^c\delta_- + g_\epsilon \overline{l_R^c} \ell^{}_R \epsilon_{++} + \overline{L} H \ell_R + h.c.$\ : $V \supset \lambda\varphi \delta_- \delta_- \epsilon_{++} + h.c. + ... $\ For this theory we find the radiative generation of neutrino masses represented by the diagram:\ [Zee\_conf]{} $$\begin{aligned} \parbox{75mm}{ \begin{fmfgraph*}(75,55) \fmfstraight \fmfleft{L} \fmflabel{L}{L} \fmfright{R} \fmflabel{L}{R} \fmftop{P1,P2,P3} \fmflabel{$\langle \varphi \rangle$}{P2} \fmf{fermion,tension=1}{L,V1} \fmf{fermion,label=L}{V2,V1} \fmf{fermion,label=$l_R$}{V3,V2} \fmf{fermion,label=$l_R$}{V3,V4} \fmf{fermion,label=L}{V4,V5} \fmf{fermion,tension=1}{R,V5} \fmffreeze \fmf{scalar,left=.3,label=$\delta_-$}{V1,VT} \fmf{scalar,right=.3,label=$\delta_-$}{V5,VT} \fmf{phantom,tension=2.5}{P1,VT} \fmf{scalar,tension=.5}{P2,VT} \fmf{phantom,tension=2.5}{P3,VT} \fmf{scalar,tension=3,label=$\epsilon_{++}$}{V3,VT} \fmfv{decor.shape=cross,decor.size=8}{V2} \fmfv{decor.shape=cross,decor.size=8}{V4} \end{fmfgraph*} } \end{aligned}$$ \ The crosses denote the insertion of a Higgs VEV, i.e. they represent the mass of the charged lepton. This theory is the conformally invariant analogue to the Zee-Babu model. The corresponding left-handed neutrino mass is given by $$M_L = 8 \lambda \langle \varphi \rangle m_l^2 g_\delta^2 g_\epsilon I \, ,$$ where $I$ is given by eq. (\[eq:Integral\]). $$\label{eq:Integral} \begin{split} I = \int \! \frac{\mathrm{d}^4p}{(2\pi)^4} \; \int \! \frac{\mathrm{d}^4q}{(2\pi)^4} \; \frac{1}{p^2-m_l^2} \frac{1}{q^2-m_l^2} \\ \frac{1}{p^2-m_\delta^2} \frac{1}{q^2-m_\delta^2} \frac{1}{(p-q)^2-m_\epsilon^2} \, . \end{split}$$ - **Possibility 2**: We can introduce an additional $SU(2)$ doublet $H_2$, an additional charged scalar singlet $\eta_+$ and a total singlet $\varphi: (1,0)$. As stated by before there has to be a term in the potential with non pairwise coupled scalars. This $\lambda_L$ term violates lepton number and its size controls the neutrino masses. : $L:(2,-1);\; \ell_R:(1,-2);\; H_1:(2,1);\ H_2:(2,1);\;\eta_+:(1,+2);\; \varphi:(1,0)$\ With additional terms in the Yukawa Lagrangian and potential. : $-\mathscr{L}_Y \supset g_1\, \eta_+ \bar{L}\,i \sigma_2 L^c + g_2\,\bar{L} H_2 \ell_R + h.c.$\ : $V \supset \lambda_L \eta \tilde{H}_1^\dagger H_2 \, \varphi + h.c. + ... $\ The loop diagram gives neutrino masses\ [Zee\_radiative]{} $$\begin{aligned} \parbox{55mm}{ \begin{fmfgraph*}(55,45) \fmfleft{L} \fmflabel{$L$}{L} \fmfright{R} \fmflabel{$L$}{R} \fmftop{TA,T1,T2,TB} \fmflabel{$\langle H_{2/1} \rangle$}{T1} \fmflabel{$\langle \varphi \rangle$}{T2} \fmfbottom{BA,BB,B,BC,BD} \fmf{fermion}{L,V1} \fmf{fermion,label=$\ell_R$}{V2,V1} \fmf{fermion,label=$L$}{V2,V3} \fmf{fermion}{R,V3} \fmffreeze \fmf{scalar,left=.5,label=$H_{1/2}$}{V1,TV} \fmf{scalar,right=.5,label=$\eta_+$}{V3,TV} \fmf{scalar,tension=1.3}{T1,TV} \fmf{scalar,tension=1.3}{T2,TV} \fmfv{decor.shape=cross,decor.size=8}{V2} \end{fmfgraph*} } \end{aligned}$$ \ which have the mass pattern as the non conformal Zee model [@Zee:1985rj], with the difference that the dimensionful parameter controlling the neutrino masses is replaced by the product of the coupling with the scalar VEV $\lambda_L \cdot {\ensuremath{\left\langle \varphi \right\rangle}}$, see **12A**. - **Possibility 3**: We can introduce a potential coupling of 3 different $SU(2)$ doublets such that their hypercharges add up to zero in the following structure $$\left( \phi_1^\dagger \vec{\sigma} H_i \right) \left( \tilde{H}_j^\dagger \vec{\sigma} H_j \right) \, .$$ As proposed in [@Law:2013dya] an additional doubly charged singlet scalar can be used to gain neutrino masses at two loop level. In a conformal model, however, an additional scalar is required to have a lepton number violating term in the Lagrangian without an explicit mass scale. : $ L:(2,-1);\; \ell_R (1 ,-2) ;\; \phi_1:(2,3);\; H_1:(2,1);\; H_2:(2,1) ;\; \eta:(1,-4) ;\; \phi_2:(1,0)$\ : $-\mathscr{L}_Y \supset g\, \eta \, \bar{\ell}_R \ell_R^c + h.c. $\ : $ V \supset \lambda_i \,\phi_2 \eta \phi_1^\dagger \tilde{H}_i + \lambda_{ij} \left( \phi_1^\dagger \vec{\sigma} H_i \right) \left( \tilde{H}_j^\dagger \vec{\sigma} H_j \right) $ Here, both doublets $H_1$ and $H_2$ as well as the singlet scalar get a VEV and generate neutrino masses at two loop level. - **Possibility 4**: A potential term coupling 4 different $SU(2)$ triplets such that their hypercharges add up to zero in the following way $$\left( \Delta_1^\dagger \Delta_2 \right) \left( \Delta_3^\dagger \Delta_4 \right) \, .$$ This term generates neutrino masses at the two loop level with the same topology as in the conformal Zee-Babu model in example 1. - **Possibility 5**: A further term that can be introduced is given by the coupling $$\varphi H_1^Ti\sigma_2\boldsymbol{\Delta}^\dagger H_2 \, ,$$ where $\varphi$ is a $SU(2)$ singlet, $H_1$ and $H_2$ are doublets and $\Delta$ is a $SU(2)$ triplet with hypercharges such that they add up to zero in this term. That with the help of such a coupling the fully radiative generation of neutrino masses is possible can be seen in the following theory:\ : $L_1:(2,-1);\; L_2:(2,-3);\; L_3:(2,0)$\ $\Delta_1:(3,-4);\; \Delta_2:(3,-3);\; \Delta_3:(3,-1)$\ $H_1:(2,1);\; H_2:(2,-1);\; H_3:(2,-3);\; H_4:(2,0)$\ $\varphi:(1,0)$\ : $-\mathcal{L}_Y \supset g_a\bar{L}_1\vec{\sigma}\Delta_1 L^c_2 + g_b\bar{L}_2\vec{\sigma}\Delta_2 L^c_3 + g_c\bar{L}_3\vec{\sigma}\Delta_3 L^c_1 + h.c.$\ : $$\begin{aligned} &\nonumber V = \lambda_a\varphi H_2^Ti\sigma_2 \boldsymbol{\Delta}_1^\dagger H_3\\ \nonumber &+ \lambda_b\varphi H_3^Ti\sigma_2\boldsymbol{\Delta}_2^\dagger H_4 + \lambda_c\varphi H_4^Ti\sigma_2\boldsymbol{\Delta}_3^\dagger H_2 + h.c. \\ \nonumber &+ \lambda_{13}(H_3^\dagger H_3)(H_1^\dagger H_1) + \lambda_{14}(H_4^\dagger H_4) (H_1^\dagger H_1) \\ \nonumber &+ \text{pairwise couplings} \end{aligned}$$ If we forbid the VEVs $\langle \Delta_1 \rangle$, $\langle \Delta_2 \rangle$ and $\langle \Delta_3 \rangle$, then the following diagram describes the radiative generation of neutrino masses:\ [scalar]{} $$\begin{aligned} \parbox{65mm} { \begin{fmfgraph*}(65,55) \fmfleft{L1} \fmflabel{$L_1$}{L1} \fmfright{L2} \fmflabel{$L_1$}{L2} \fmftop{A,S1,G,S2,C,E,S3,D,F,S4,H,S5,B} \fmflabel{$\langle\varphi\rangle$}{S1} \fmflabel{$\langle H_2 \rangle$}{S2} \fmflabel{$\langle\varphi\rangle$}{S3} \fmflabel{$\langle H_2 \rangle$}{S4} \fmflabel{$\langle\varphi\rangle$}{S5} \fmf{fermion,tension=2}{L1,I1} \fmf{fermion,label=$L_2$}{I2,I1} \fmf{fermion,label=$L_3$}{I2,I3} \fmf{fermion,tension=2}{L2,I3} \fmffreeze \fmf{scalar,left=1/4,label=$\Delta_1$,tension=4.3}{I1,V1} \fmf{scalar,left=1/4,label=$H_3$,tension=4.3}{V1,V2} \fmf{scalar,right=1/4,label=$H_4$,tension=4.3}{V3,V2} \fmf{scalar,right=1/4,label=$\Delta_3$,tension=4.3}{I3,V3} \fmf{scalar,label=$\Delta_2$,tension=.1}{V2,I2} \fmf{scalar,tension=1.7}{V1,S1} \fmf{scalar,tension=1.7}{V1,S2} \fmf{scalar,tension=5}{S3,V2} \fmf{scalar,tension=1.7}{V3,S4} \fmf{scalar,tension=1.7}{S5,V3} \end{fmfgraph*} } \end{aligned}$$ \ Admittedly this theory is very baroque and can be phenomenologically problematic. Especially to ensure anomaly cancellation the new fermions have to be vector like. The particle content in the loop is intended to show that it is possible to generate neutrino masses fully radiatively from a topological point of view. - **Alternative 1**: So far in the radiative models no additional symmetries were considered. However, the argument of \[sec:Radiative\] can be avoided if a new symmetry is present, which forbids tree level couplings for fermion singlets in the SM Dirac term. If there is a discrete symmetry, for example $Z_2$ under which all SM particles are even and the spectrum given by : $ L(2,1, (+)) ;\; H_1:(2,1, (+));\; H_2:(2,1, (-));\; \nu_x:(1,0,(-));\;\varphi:(1,0,(+))$\ \ Additional in the Yuakawa Lagrangian there are the following terms. : $-\mathscr{L}_Y \supset g_{1}\bar{L} H_2 \nu_x + g_2 \varphi \bar{\nu_x} \nu_x^c + h.c.$ The relevant coupling in the potential is: : $V \supset \lambda\, (H_2^\dagger H_1)^2 + \text{pairwise couplings}$ with $H_1$ being the SM Higgs. This would be the conformal analogue of the Ma model and generates neutrino masses at one loop level. Note, however, that in general discrete symmetries are not so restrictive. Therefore, in our models continuous symmetries are used. For example a hidden sector $U(1)$ would have the same effect on the Yukawa Lagrangian, but the potential term would be forbidden. Thus a model of this type can only generate neutrino masses in the hidden sector, as shown in the model **3C**. - **Alternative 2**: To circumvent the argument of \[sec:Radiative\] we can allow fermion loops. The following theory shows that it is possible to construct left-handed Majorana masses such that the lowest order diagram has to be a full loop diagram.\ : $L_1:(2,-1);\; L_2:(2,0);\; L_3:(2,2);\; L_4:(2,-2)$\ $\Delta_1:(3,-2);\; \Delta_2:(3,0);\; \Delta_3:(3,2)$\ : $-\mathscr{L}_Y \supset g_{11}\bar{L}_1\vec{\sigma}\Delta_1 L_1^c + g_{24}\bar{L}_2\vec{\sigma}\Delta_1 L_4^c + g_{22}\bar{L}_2\vec{\sigma}\Delta_2 L_2^c \\ \phantom{\underline{Yukawa Lagrangian}: -\mathcal{L}_Y=} + g_{23}\bar{L}_2\vec{\sigma}\Delta_3 L_3^c + g_{34}\bar{L}_3\vec{\sigma}\Delta_2 L_4^c + h.c.$\ Furthermore we require the VEV of the neutral component of $\Delta_1$ to vanish. The following diagram is then the lowest order contribution to the left-handed neutrino masses:\ [fermion]{} $$\begin{aligned} \parbox{35mm} { \begin{fmfgraph*}(35,85) \fmfleft{L1} \fmflabel{$L_1$}{L1} \fmfright{L2} \fmflabel{$L_1$}{L2} \fmftop{S1,S2,S3} \fmflabel{$\langle\Delta_2\rangle$}{S1} \fmflabel{$\langle\Delta_3\rangle$}{S2} \fmflabel{$\langle\Delta_2\rangle$}{S3} \fmfbottom{P1,P2,P3} \fmf{fermion,tension=3}{L1,I1} \fmf{fermion,tension=3}{L2,I1} \fmffreeze \fmf{scalar,tension=3.5,label=$\Delta_1$}{I1,V1} \fmf{fermion,left=1/3,tension=3,label=$L_2$}{V1,V2} \fmf{fermion,right=1/6,label=$L_2$}{V3,V2} \fmf{fermion,left=1/6,label=$L_3$}{V3,V4} \fmf{fermion,right=1/3,tension=3,label=$L_4$}{V1,V4} \fmf{scalar,tension=3}{V2,S1} \fmf{scalar}{S2,V3} \fmf{scalar,tension=3}{V4,S3} \end{fmfgraph*} } \end{aligned}$$ \ Like before this theory can be phenomenologically problematic. And again it is only intended to show that the topological possibility of fully radiative mass generation in conformally invariant theories with pairwise scalar coupling exists when introducing fermion loops.

--- abstract: | In many applications such as data compression, imaging or genomic data analysis, it is important to approximate a given tensor by a tensor that is sparsely representable. For matrices, i.e. $2$-tensors, such a representation can be obtained via the singular value decomposition, which allows to compute best rank $k$-approximations. For very big matrices a low rank approximation using SVD is not computationally feasible. In this case different approximations are available. It seems that variants of the CUR-decomposition are most suitable. For $d$-mode tensors ${\mathcal{T}}\in \otimes_{i=1}^d {\mathbb{R}}^{n_i}$, with $d>2$, many generalizations of the singular value decomposition have been proposed to obtain low tensor rank decompositions. The most appropriate approximation seems to be best $(r_1,\ldots,r_d)$-approximation, which maximizes the $\ell_2$ norm of the projection of ${\mathcal{T}}$ on $\otimes_{i=1}^d {\mathbf{U}}_i$, where ${\mathbf{U}}_i$ is an $r_i$-dimensional subspace ${\mathbb{R}}^{n_i}$. One of the most common methods is the *alternating maximization method* (AMM). It is obtained by maximizing on one subspace ${\mathbf{U}}_i$, while keeping all other fixed, and alternating the procedure repeatedly for $i=1,\ldots,d$. Usually, AMM will converge to a local best approximation. This approximation is a fixed point of a corresponding map on Grassmannians. We suggest a Newton method for finding the corresponding fixed point. We also discuss variants of CUR-approximation method for tensors. The first part of the paper is a survey on low rank approximation of tensors. The second new part of this paper is a new Newton method for best $(r_1,\ldots,r_d)$-approximation. We compare numerically different approximation methods. author: - | Shmuel Friedlandand Venu Tammali\ Department of Mathematics, Statistics and Computer Science,\ University of Illinois at Chicago,\ Chicago, Illinois 60607-7045, USA,\ E-mail: `friedlan@uic.edu, vtamma2@uic.edu` date: 'December 10, 2014' title: 'Low-rank approximation of tensors' --- [**2000 Mathematics Subject Classification.**]{} 14M15, 15A18, 15A69, 65H10, 65K10. [**Key words**]{} Tensor, best rank one approximation, best $(r_1,\ldots,r_d)$-approximation, sampling, alternating maximization method, singular value decomposition, Grassmann manifold, fixed point, Newton method. Introduction {#S:intro} ============ Let ${\mathbb{R}}$ be the field of real numbers. Denote by ${\mathbb{R}}^{\mathbf{n}}={\mathbb{R}}^{n_1\times\ldots\times n_d}:=\otimes_{i=1}^d {\mathbb{R}}^{n_j}$, where ${\mathbf{n}}=(n_1,\ldots,n_d)$, the tensor products of ${\mathbb{R}}^{n_1},\ldots,{\mathbb{R}}^{n_d}$. ${\mathcal{T}}=[t_{i_1,\ldots, i_d}]\in {\mathbb{R}}^{\mathbf{n}}$ is called a *$d$-mode tensor*. Note that the number of coordinates of ${\mathcal{T}}$ is $N=n_1\ldots n_d$. A tensor ${\mathcal{T}}$ is called a *sparsely representable tensor* if it can represented with a number of coordinates that is much smaller than $N$. Apart from sparse matrices, the best known example of a sparsely representable $2$-tensor is a low rank approximation of a matrix $A\in {\mathbb{R}}^{n_1\times n_2}$. A rank $k$-approximation of $A$ is given by $A_{\textrm{appr}}:=\sum_{i=1}^k {\mathbf{u}}_i{\mathbf{v}}_i{^\top}$, which can be identified with $\sum_{i=1}^k {\mathbf{u}}_i\otimes {\mathbf{v}}_i$. To store $A_{\textrm{appr}}$ we need only the $2k$ vectors ${\mathbf{u}}_1,\ldots,{\mathbf{u}}_k\in{\mathbb{R}}^{n_1},\;{\mathbf{v}}_1,\ldots,{\mathbf{v}}_k\in{\mathbb{R}}^{n_2}$. A best rank $k$-approximation of $A\in {\mathbb{R}}^{n_1\times n_2}$ can be computed via the *singular value decomposition*, abbreviated here as SVD, [@GolV96]. Recall that if $A$ is a real symmetric matrix, then the best rank $k$-approximation must be symmetric, and is determined by the spectral decomposition of $A$. The computation of the SVD requires $\mathcal {O}(n_1n_2^2)$ operations and at least $\mathcal{O} (n_1 n_2)$ storage, assuming that $n_2\le n_1$. Thus, if the dimensions $n_1$ and $n_2$ are very large, then the computation of the SVD is often infeasible. In this case other type of low rank approximations are considered, see e.g. [@AM01; @DV06; @DKM06; @FriKNZ06; @FMMN11; @FKV04; @GTZ97]. For $d$-tensors with $d>2$ the situation is rather unsatisfactory. It is a major theoretical and computational problem to formulate good generalizations of low rank approximation for tensors and to give efficient algorithms to compute these approximations, see e.g. [@deLdV00; @LMV00; @ES09; @FMMN11; @FMPS13; @Kho; @KB09; @MMD06; @Ose11; @OsT09; @ZhaG01]. We now discuss briefly the main ideas of the approximation methods for tensors discussed in this paper. We need to introduce (mostly) standard notation for tensors. Let $[n]:=\{1,\ldots,n\}$ for $n\in{\mathbb{N}}$. For ${\mathbf{x}}_i:=(x_{1,i},\ldots,x_{n_i,i}){^\top}\in {\mathbb{R}}^{n_i},i\in[d]$, the tensor $\otimes_{i\in[d]}{\mathbf{x}}_i={\mathbf{x}}_1\otimes\cdots\otimes {\mathbf{x}}_d={\mathcal{X}}=[x_{j_1,\ldots,j_d}]\in{\mathbb{R}}^{\mathbf{n}}$ is called a decomposable tensor, or rank one tensor if ${\mathbf{x}}_i\ne{\mathbf{0}}$ for $i\in [d]$. That is, $x_{j_1,\ldots,j_d}=x_{j_1,1}\cdots x_{j_d,d}$ for $j_i\in[n_i], i\in[d]$. Let ${{\langle}{\mathbf{x}}_i,{\mathbf{y}}_i{\rangle}}_i:={\mathbf{y}}_i{^\top}{\mathbf{x}}_i$ be the standard inner product on ${\mathbb{R}}^{n_i}$ for $i\in[d]$. Assume that ${\mathcal{S}}=[s_{j_1,\ldots,j_d}]$ and ${\mathcal{T}}=[t_{j_1,\ldots,j_d}]$ are two given tensors in ${\mathbb{R}}^{\mathbf{n}}$. Then ${{\langle}{\mathcal{S}},{\mathcal{T}}{\rangle}}:=\sum_{j_i\in[n_i],i\in[d]} s_{j_1,\ldots,j_d}t_{j_1,\ldots,j_d}$ is the standard inner product on ${\mathbb{R}}^{\mathbf{n}}$. Note that $$\begin{aligned} &&{{\langle}\otimes_{i\in[d]}{\mathbf{x}}_i,\otimes_{i\in[d]}{\mathbf{y}}_i{\rangle}}=\prod_{i\in[d]}{{\langle}{\mathbf{x}}_i,{\mathbf{y}}_i{\rangle}}_i,\\ &&{{\langle}{\mathcal{T}},\otimes_{i\in[d]}{\mathbf{x}}_i{\rangle}}=\sum_{j_i\in[n_i],i\in[d]} t_{j_1,\ldots,j_d}x_{j_1,1}\cdots x_{j_d,d}. \end{aligned}$$ The norm $\|{\mathcal{T}}\|:=\sqrt{{{\langle}{\mathcal{T}},{\mathcal{T}}{\rangle}}}$ is called the Hilbert-Schmidt norm. (For matrices, i.e. $d=2$, it is called the Frobenius norm.) Let $I=\{1\le i_1<\cdots <i_l\le d\}\subset [d]$. Assume that ${\mathcal{X}}=[x_{j_{i_1},\cdots,j_{i_l}}]\in \otimes_{k\in[l]} {\mathbb{R}}^{n_{i_k}}$. Then the contraction ${\mathcal{T}}\times {\mathcal{X}}$ on the set of indices $I$ is given by: $${\mathcal{T}}\times {\mathcal{X}}=\sum_{j_{i_k}\in [n_{i_k}], k\in[l]} t_{j_1,\ldots,j_d} x_{j_{i_1},\ldots,j_{i_l}}\in\otimes_{p\in[d]\setminus I} {\mathbb{R}}^{n_p}.$$ Assume that ${\mathbf{U}}_i\subset {\mathbb{R}}^{n_i}$ is a subspace of dimension $r_i$ with an orthonormal basis ${\mathbf{u}}_{1,i},\ldots,{\mathbf{u}}_{r_i,i}$ for $i\in[d]$. Let ${\mathbf{U}}:=\otimes_{i=1}^d {\mathbf{U}}_i\subset {\mathbb{R}}^{\mathbf{n}}$. Then $\otimes_{i=1}^d {\mathbf{u}}_{j_i,i}$, where $j_i\in[n_i],i\in[d]$, is an orthonormal basis in ${\mathbf{U}}$. We are approximating ${\mathcal{T}}\in{\mathbb{R}}^{n_1\times \cdots\times n_d}$ by a tensor $$\label{Sapprox} {\mathcal{S}}=\sum_{j_i\in[r_i],i\in[d]} s_{j_1,\ldots,j_d} {\mathbf{u}}_{j_1,1}\otimes\cdots \otimes{\mathbf{u}}_{j_d,d} \in {\mathbb{R}}^{\mathbf{n}}$$ The tensor ${\mathcal{S}}'=[s_{j_1,\ldots,j_d}]\in {\mathbb{R}}^{r_1\times \cdots\times r_d}$ is the *core tensor* corresponding to ${\mathcal{S}}$ in the terminology of [@Tuc66]. There are two major problems: The first one is how to choose the subspaces ${\mathbf{U}}_1,\ldots,{\mathbf{U}}_d$. The second one is the choice of the core tensor ${\mathcal{S}}'$. Suppose we already made the choice of ${\mathbf{U}}_1,\ldots,{\mathbf{U}}_d$. Then ${\mathcal{S}}=P_{{\mathbf{U}}}({\mathcal{T}})$ is the orthogonal projection of ${\mathcal{T}}$ on ${\mathbf{U}}$: $$\label{PUfor} P _{\otimes_{i\in[d]}{\mathbf{U}}_i}({\mathcal{T}})=\sum_{j_i\in[r_i],i\in[d]} {{\langle}{\mathcal{T}},\otimes_{i\in[d]} {\mathbf{u}}_{j_i,i}{\rangle}} \otimes_{i\in[d]} {\mathbf{u}}_{j_i,i}.$$ If the dimensions of $n_1,\ldots,n_d$ are not too big, then this projection can be explicitly carried out. If the dimension $n_1,\ldots,n_d$ are too big to compute the above projection, then one needs to introduce other approximations. That is, one needs to compute the core tensor ${\mathcal{S}}'$ appearing in accordingly. The papers [@AM01; @DV06; @DKM06; @FriKNZ06; @FMMN11; @FKV04; @GTZ97; @Kho; @MMD06; @Ose11; @OsT09] essentially choose ${\mathcal{S}}'$ in a particular way. We now assume that the computation of $P_{{\mathbf{U}}}({\mathcal{T}})$ is feasible. Recall that $$\label{normprojfor} \|P_{\otimes_{i\in[d]}{\mathbf{U}}_i}({\mathcal{T}})\|^2=\sum_{j_i\in [r_i],i\in[d]}|{{\langle}{\mathcal{T}},\otimes_{i=1}^d {\mathbf{u}}_{j_i,i}{\rangle}}|^2.$$ The best ${\mathbf{r}}$-approximation of ${\mathcal{T}}$, where ${\mathbf{r}}= (r_1,\ldots,r_d)$, in Hilbert-Schmidt norm is the solution of the minimal problem: $$\label{bestapproxprb} \min_{{\mathbf{U}}_i,\dim {\mathbf{U}}_i=r_i, i\in[d]}\min_{{\mathcal{X}}\in \otimes_{i\in[d]}^d{\mathbf{U}}_i} \|{\mathcal{T}}-{\mathcal{X}}\|.$$ This problem is equivalent to the following maximum $$\label{maxnormprb} \max_{{\mathbf{U}}_i,\dim {\mathbf{U}}_i=r_i, i\in[d]} \|P_{\otimes_{i\in[d]}{\mathbf{U}}_i}({\mathcal{T}})\|^2.$$ The standard *alternating maximization method*, denoted by AMM, for solving is to solve the maximum problem, where all but the subspace ${\mathbf{U}}_i$ is fixed. Then this maximum problem is equivalent to finding an $r_i$-dimensional subspace of ${\mathbf{U}}_i$ containing the $r_i$ biggest eigenvalues of a corresponding nonnegative definite matrix $A_i({\mathbf{U}}_1,\ldots,{\mathbf{U}}_{i-1},{\mathbf{U}}_{i+1},\ldots,{\mathbf{U}}_d)\in {\mathrm{S}}_{n_i}$. Alternating between ${\mathbf{U}}_1,{\mathbf{U}}_2,\ldots,{\mathbf{U}}_d$ we obtain a nondecreasing sequence of norms of projections which converges to $v$. Usually, $v$ is a critical value of $\|P_{\otimes_{i\in[d]}{\mathbf{U}}_i}({\mathcal{T}})\|$. See [@LMV00] for details. Assume that $r_i=1$ for $i\in [d]$. Then $\dim{\mathbf{U}}_i=1$ for $i\in[d]$. In this case the minimal problem is called a best rank one approximation of ${\mathcal{T}}$. For $d=2$ a best rank one approximation of a matrix ${\mathcal{T}}=T\in{\mathbb{R}}^{n_1\times n_2}$ is accomplished by the first left and right singular vectors and the corresponding maximal singular value $\sigma_1(T)$. The complexity of this computation is ${\mathcal{O}}(n_1n_2)$ [@GolV96]. Recall that the maximum is equal to $\sigma_1(T)$, which is also called the spectral norm $\|T\|_2$. For $d>2$ the maximum is called the spectral norm of ${\mathcal{T}}$, and denoted by $\|{\mathcal{T}}\|_\sigma$. The fundamental result of Hillar-Lim [@HL13] states that the computation of $\|{\mathcal{T}}\|_\sigma$ is NP-hard in general. Hence the computation of best ${\mathbf{r}}$-approximation is NP-hard in general. Denote by ${\mathop{\mathrm{Gr}}\nolimits}(r,{\mathbb{R}}^n)$ the variety of all $r$-dimensional subspaces in ${\mathbb{R}}^n$, which is called Grassmannian or Grassmann manifold. Let $${\mathbf{1}}_d:=(\underbrace{1,\ldots,1}_d), \quad {\mathop{\mathrm{Gr}}\nolimits}({\mathbf{r}},{\mathbf{n}}):={\mathop{\mathrm{Gr}}\nolimits}(r_1,n_1)\times \cdots \times {\mathop{\mathrm{Gr}}\nolimits}(r_d,n_d).$$ Usually, the AMM for best ${\mathbf{r}}$-approximation of ${\mathcal{T}}$ will converge to a fixed point of a corresponding map ${\mathbf{F}}_{\mathcal{T}}:{\mathop{\mathrm{Gr}}\nolimits}({\mathbf{r}},{\mathbf{n}})\to {\mathop{\mathrm{Gr}}\nolimits}({\mathbf{r}},{\mathbf{n}})$. This observation enables us to give a new Newton method for finding a best ${\mathbf{r}}$-approximation to ${\mathcal{T}}$. For best rank one approximation the map ${\mathbf{F}}_{\mathcal{T}}$ and the corresponding Newton method was stated in [@FMPS11]. This paper consists of two parts. The first part surveys a number of common methods for low rank approximation methods of matrices and tensors. We did not cover all existing methods here. We were concerned mainly with the methods that the first author and his collaborators were studying, and closely related methods. The second part of this paper is a new contribution to Newton algorithms related to best ${\mathbf{r}}$-approximations. These algorithms are different from the ones given in [@ZhaG01; @ES09; @SL10]. Our Newton algorithms are based on finding the fixed points corresponding to the map induced by the AMM. In general its known that for big size problem, where each $n_i$ is big for $i\in[d]$ and $d\ge 3$, Newton methods are not efficient. The computation associate the matrix of derivatives (Jacobian) is too expensive in computation and time. In this case AMM or MAMM (modified AMM) are much more cost effective. This well known fact is demonstrated in our simulations. We now briefly summarize the contents of this paper. In §\[S:SVD\] we review the well known facts of singular value decomposition (SVD) and its use for best rank $k$-approximation of matrices. For large matrices approximation methods using SVD are computationally unfeasible. Section \[S:lowrankapmat\] discusses a number of approximation methods of matrices which do not use SVD. The common feature of these methods is sampling of rows, or columns, or both to find a low rank approximation. The basic observation in §\[Sub:rowsamp\] is that, with high probability, a best $k$-rank approximation of a given matrix based on a subspace spanned by the sampled row is with in a relative $\epsilon$ error to the best $k$-rank approximation given by SVD. We list a few methods that use this observation. However, the complexity of finding a particular $k$-rank approximation to an $m\times n$ matrix is still ${\mathcal{O}}(kmn)$, as the complexity truncated SVD algorithms using Arnoldi or Lanczos methods [@GolV96; @LSY98]. In §\[Sub:CURap\] we recall the CUR-approximation introduced in [@GTZ97]. The main idea of CUR-approximation is to choose $k$ columns and rows of $A$, viewed as matrices $C$ and $R$, and then to choose a square matrix $U$ of order $k$ in such a way that $CUR$ is an optimal approximation of $A$. The matrix $U$ is chosen to be the inverse of the corresponding $k\times k$ submatrix $A'$ of $A$. The quality of CUR-approximation can be determined by the ratio of $|\det A'|$ to the maximum possible value of the absolute value of all $k\times k$ minors of $A$. In practice one searches for this maximum using a number of random choices of such minors. A modification of this search algorithm is given in [@FMMN11]. The complexity of storage of $C,R,U$ is ${\mathcal{O}}(k\max(m,n))$. The complexity of finding the value of each entry of $CUR$ is ${\mathcal{O}}(k^2)$. The complexity of computation of $CUR$ is ${\mathcal{O}}(k^2mn)$. In §\[S:fastapten\] we survey CUR-approximation of tensors given in [@FMMN11]. In §\[S:prelbrap\] we discuss preliminary results on best ${\mathbf{r}}$-approximation of tensors. In §\[sub:maxprob\] we show that the minimum problem is equivalent to the maximum problem . In §\[sub:singvten\] we discuss the notion of singular tuples and singular valuesva tensor introduced in [@Lim05]. In §\[sub:apprprb\] we recall the well known solution of maximizing $\|P_{\otimes_{i\in[d]}{\mathbf{U}}_i}({\mathcal{T}})\|^2$ with respect to one subspace, while keeping other subspaces fixed. In §\[S:AMM\] we discuss AMM for best ${\mathbf{r}}$-approximation and its variations. (In [@LMV00; @FMPS13] AMM is called *alternating least squares*, abbreviated as ALS.) In §\[sub:gendp\] we discuss the AMM on a product space. We mention a *modified alternating maximization method* and and *2-alternating maximization method*, abbreviated as MAMM and 2AMM respectively, introduced in [@FMPS13]. The MAMM method consists of choosing the one variable which gives the steepest ascend of AMM. 2AMM consists of maximization with respect to a pair of variables, while keeping all other variables fixed. In §\[sub:AMMbrapr\] we discuss briefly AMM and MAMM for best ${\mathbf{r}}$-approximations for tensors. In §\[sub:companbrap\] we give the complexity analysis of AMM for $d=3, r_1\approx r_2\approx r_3$ and $n_1\approx n_2\approx n_3$. In §\[S:fixpt\] we state a working assumption of this paper that AMM converges to a fixed point of the induced map, which satisfies certain smoothness assumptions. Under these assumptions we can apply the Newton method, which can be stated in the standard form in ${\mathbb{R}}^L$. Thus, we first do a number of AMM iterations and then switch to the Newton method. In §\[sub:newtboneappr\] we give a simple application of these ideas to state a Newton method for best rank one approximation. This Newton method was suggested in [@FMPS11]. It is different from the Newton method in [@ZhaG01]. The new contribution of this paper is the Newton method which is discussed in §\[S:bestbrapprox\] and §\[S:forDF\]. The advantage of our Newton method is its simplicity, which avoids the notions and tools of Riemannian geometry as for example in [@ES09; @SL10]. In simulations that we ran, the Newton method in [@ES09] was $20\%$ faster than our Newton method for best ${\mathbf{r}}$-approximation of $3$-mode tensors. However, the number of iterations of our Newton method was $40\%$ less than in [@ES09]. In the last section we give numerical results of our methods for best ${\mathbf{r}}$-approximation of tensors. In §\[S:NumRes\] we give numerical simulations of our different methods applied to a real computer tomography (CT) data set (the so-called MELANIX data set of OsiriX). The summary of these results are given in §\[S:Conclusion\]. Singular Value Decomposition {#S:SVD} ============================ Let $A\in{\mathbb{R}}^{m\times n}\setminus\{0\})$. We now recall well known facts on the SVD of $A$ [@GolV96]. See [@Ste93] for the early history of the SVD. Assume that $r={\mathrm{rank\;}}A$. Then there exist $r$-orthonormal sets of vectors ${\mathbf{u}}_1,\ldots,{\mathbf{u}}_r\in {\mathbb{R}}^m, {\mathbf{v}}_1,\ldots,{\mathbf{v}}_r\in {\mathbb{R}}^n$ such that we have: $$\begin{aligned} \notag &&A{\mathbf{v}}_i=\sigma_i(A){\mathbf{u}}_i, \quad {\mathbf{u}}_i{^\top}A=\sigma_i(A){\mathbf{v}}_i{^\top}, \quad i\in [r], \quad \sigma_1(A)\ge \cdots\ge \sigma_r(A)>0,\\ &&A_k=\sum_{i\in[k]} \sigma_i(A){\mathbf{u}}_i{\mathbf{v}}_i{^\top}, \quad k\in [r], \; A=A_r. \label{SVDid}\end{aligned}$$ The quantities ${\mathbf{u}}_i$, ${\mathbf{v}}_i$ and $\sigma_i(A)$ are called the left, right $i$-th singular vectors and $i$-th singular value of $A$ respectively, for $i\in[r]$. Note that ${\mathbf{u}}_k$ and ${\mathbf{v}}_k$ are uniquely defined up to $\pm 1$ if and only if $\sigma_{k-1}(A)>\sigma_k(A)>\sigma_{k+1}(A)$. Furthermore for $k\in [r-1]$ the matrix $A_k$ is uniquely defined if and only if $\sigma_k(A)>\sigma_{k+1}(A)$. Denote by ${\mathcal{R}}(m,n,k)\subset {\mathbb{R}}^{m\times n}$ the variety of all matrices of rank at most $k$. Then $A_k$ is a best rank-$k$ approximation of $A$: $$\min_{B\in{\mathcal{R}}(m,n,k)} \|A-B\|=\|A-A_k\|.$$ Let ${\mathbf{U}}\in {\mathop{\mathrm{Gr}}\nolimits}(p,{\mathbb{R}}^m), {\mathbf{V}}\in{\mathop{\mathrm{Gr}}\nolimits}(q,{\mathbb{R}}^n)$. We identify ${\mathbf{U}}\otimes {\mathbf{V}}$ with $$\label{defUVT} {\mathbf{U}}{\mathbf{V}}{^\top}:={\mathrm{span}}\{{\mathbf{u}}{\mathbf{v}}{^\top},\quad {\mathbf{u}}\in{\mathbf{U}},{\mathbf{v}}\in {\mathbf{V}}\}\subset {\mathbb{R}}^{m\times n}.$$ Then $P_{{\mathbf{U}}\otimes {\mathbf{V}}}(A)$ is identified with the projection of $A$ on ${\mathbf{U}}{\mathbf{V}}{^\top}$ with respect to the standard inner product on ${\mathbb{R}}^{m\times n}$ given by ${\lanX,Y{\rangle}}={\mathop{\mathrm{tr}}\nolimits}XY{^\top}$. Observe that $$\textrm{Range }A={\mathbf{U}}_r^\star, \; {\mathbb{R}}^m={\mathbf{U}}_r^\star\oplus({\mathbf{U}}_r^\star)^\perp, \textrm{ Range }A{^\top}={\mathbf{V}}_r^\star,\; {\mathbb{R}}^n={\mathbf{V}}_r^\star\oplus({\mathbf{V}}_r^\star)^\perp.$$ Hence $$P_{{\mathbf{U}}\otimes {\mathbf{V}}}(A)=P_{({\mathbf{U}}\cap{\mathbf{U}}_r^\star)\otimes({\mathbf{V}}\cap{\mathbf{V}}_r^\star)}(A)\Rightarrow {\mathrm{rank\;}}P_{{\mathbf{U}}\otimes {\mathbf{V}}}(A)\le \min(\dim {\mathbf{U}},\dim {\mathbf{V}},r).$$ Thus $$\begin{aligned} \notag &&\max_{{\mathbf{U}}\in{\mathop{\mathrm{Gr}}\nolimits}(p,{\mathbb{R}}^m),{\mathbf{V}}\in{\mathop{\mathrm{Gr}}\nolimits}(q,{\mathbb{R}}^n)}\|P_{{\mathbf{U}}\otimes {\mathbf{V}}}(A)\|^2=\|P_{{\mathbf{U}}_l^\star\otimes{\mathbf{V}}_l^\star}(A)\|^2=\sum_{j\in[l]}\sigma_j(A)^2,\\ &&\min_{{\mathbf{U}}\in{\mathop{\mathrm{Gr}}\nolimits}(p,{\mathbb{R}}^m),{\mathbf{V}}\in{\mathop{\mathrm{Gr}}\nolimits}(q,{\mathbb{R}}^n)}\|A-P_{{\mathbf{U}}\otimes {\mathbf{V}}}(A)\|^2=\|A-P_{{\mathbf{U}}_l^\star\otimes{\mathbf{V}}_l^\star}(A)\|^2=\sum_{j\in[r]\setminus[l]}\sigma_j(A)^2, \notag\\ &&l=\min(p,q,r). \label{maxprojmat}\end{aligned}$$ To compute ${\mathbf{U}}_l^\star,{\mathbf{V}}_l^\star$ and $\sigma_1(A),\ldots,\sigma_l(A)$ of a large scale matrix $A$ one can use Arnoldi or Lanczos methods [@GolV96; @LSY98], which are implemented in the partial singular value decomposition. This requires a substantial number of matrix-vector multiplications with the matrix $A$ and thus a complexity of at least ${\mathcal{O}}(lmn)$. Sampling in low rank approximation of matrices {#S:lowrankapmat} ============================================== Let $A=[a_{i,j}]_{i=j=1}^{m,n}\in{\mathbb{R}}^{m\times n}$ be given. Assume that ${\mathbf{b}}_1,\ldots,{\mathbf{b}}_m\in{\mathbb{R}}^n, {\mathbf{c}}_1,\ldots,{\mathbf{c}}_n\in{\mathbb{R}}^m$ are the columns of $A{^\top}$ and $A$ respectively. (${\mathbf{b}}_1{^\top},\ldots{\mathbf{b}}_m{^\top}$ are the rows of $A$.) Most of the known fast rank $k$-approximation are using sampling of rows or columns of $A$, or both. Low rank approximations using sampling of rows {#Sub:rowsamp} ---------------------------------------------- Suppose that we sample a set $$\label{defsubI} I:=\{1\le i_1<\ldots<i_s\le m\}\subset [m], \quad |I|=s,$$ of rows ${\mathbf{b}}_{i_1}{^\top},\ldots,{\mathbf{b}}_{i_s}{^\top}$, where $s\ge k$. Let ${\mathbf{W}}(I):={\mathrm{span}}({\mathbf{b}}_{i_1},\ldots,{\mathbf{b}}_{i_s})$. Then with high probability the projection of the first $i$-th right singular vectors ${\mathbf{v}}_i$ on ${\mathbf{W}}(I)$ is very close to ${\mathbf{v}}_i$ for $i\in[k]$, provided that $s\gg k$. In particular, [@DV06 Theorem 2] claims: \[DesVemtheo\] (Deshpande-Vempala) Any $A\in{\mathbb{R}}^{m\times n}$ contains a subset $I$ of $s=\frac{4k}{\epsilon}+2k \log(k +1)$ rows such that there is a matrix $\tilde A_k$ of rank at most $k$ whose rows lie in ${\mathbf{W}}(I)$ and $$\|A-\tilde A_k\|^2\le (1+\epsilon)\|A-A_k\|^2.$$ To find a rank-$k$ approximation of $A$, one projects each row of $A$ on ${\mathbf{W}}(I)$ to obtain the matrix $P_{{\mathbb{R}}^m\otimes {\mathbf{W}}(I)}(A)$. Note that we can view $P_{{\mathbb{R}}^m\otimes {\mathbf{W}}(I)}(A)$ as an $m\times s'$ matrix, where $s'=\dim{\mathbf{W}}(I)\le s$. Then find a best rank $k$-approximation to $P_{{\mathbb{R}}^m\otimes {\mathbf{W}}(I)}(A)$, denoted as $P_{{\mathbb{R}}^m\otimes {\mathbf{W}}(I)}(A)_k$. Theorem \[DesVemtheo\] and the results of [@FKV04] yield that $$\|A-P_{{\mathbb{R}}^m\otimes {\mathbf{W}}(I)}(A)_k\|^2\le (1+\epsilon)\|A-A_k\|^2 + \eta \|A-P_{{\mathbb{R}}^m\otimes {\mathbf{W}}(I)}(A)\|^2.$$ Here $\eta$ is proportional to $\frac{k}{s}$, and can be decreased with more rounds of sampling. Note that the complexity of computing $P_{{\mathbb{R}}^m\otimes {\mathbf{W}}(I)}(A)_k$ is ${\mathcal{O}}(ks'm)$. The key weakness of this method is that to compute $P_{{\mathbb{R}}^m\otimes {\mathbf{W}}(I)}(A)$ one needs ${\mathcal{O}}(s'mn)$ operations. Indeed, after having computed an orthonormal basis of ${\mathbf{W}}(I)$, to compute the projection of each row of $A$ on ${\mathbf{W}}(I)$ one needs $s'n$ multiplications. An approach for finding low rank approximations of $A$ using random sampling of rows or columns is given in Friedland-Kave-Niknejad-Zare [@FriKNZ06]. Start with a random choice of $I$ rows of $A$, where $|I|\ge k$ and $\dim {\mathbf{W}}(I)\ge k$. Find $P_{{\mathbb{R}}^m\otimes {\mathbf{W}}(I)}(A)$ and $B_1:=P_{{\mathbb{R}}^m\otimes {\mathbf{W}}(I)}(A)_k$ as above. Let ${\mathbf{U}}_{1}\in{\mathop{\mathrm{Gr}}\nolimits}(k,{\mathbb{R}}^m)$ be the subspace spanned by the first $k$ left singular vectors of $B_1$. Find $B_2=P_{{\mathbf{U}}_1\otimes {\mathbb{R}}^n}(A)$. Let ${\mathbf{V}}_2\in{\mathop{\mathrm{Gr}}\nolimits}(k,{\mathbb{R}}^n)$ correspond to the first $k$ right singular vectors of $B_2$. Continuing in this manner we obtain a sequence of rank $k$-approximations $B_1,B_2,\ldots$. It is shown in [@FriKNZ06] that $\|A-B_1\|\ge \|A-B_2\|\ge \ldots$ and $\|B_1\|\le \|B_2\|\le \ldots$. One stops the iterations when the relative improvement of the approximation falls below the specified threshold. Assume that $\sigma_k(A)>\sigma_{k+1}(A)$. Since best rank-$k$ approximation is a unique local minimum for the function $\|A-B\|, B\in {\mathcal{R}}(m,n,k)$ [@GolV96], it follows that in general the sequence $B_j, j\in{\mathbb{N}}$ converges to $A_k$. It is straightforward to show that this algorithm is the AMM for low rank approximations given in §\[sub:AMMbrapr\]. Again, the complexity of this method is ${\mathcal{O}}(kmn)$. Other suggested methods as [@AM01; @DKM06; @RV07; @Sar06] seem to have the same complexity ${\mathcal{O}}(kmn)$, since they project each row of $A$ on some $k$-dimensional subspace of ${\mathbb{R}}^n$. CUR-approximations {#Sub:CURap} ------------------ Let $$\label{defsubJ} J:=\{1\le j_1<\ldots<j_t\le n\}\subset [n], \quad |J|=t,$$ and $I\subset [m]$ as in be given. Denote by $A[I,J]:=[a_{i_p,j_q}]_{p=q=1}^{s,t}\in {\mathbb{R}}^{s\times t}$. CUR-approximation is based on sampling simultaneously the set of $I$ rows and $J$ columns of $A$ and the approximation matrix to $A(I,J,U)$ given by $$\label{defCUR} A(I,J,U):=CUR, \quad C:=A[[m],J], \; R:=A[I,[n]], \; U\in {\mathbb{R}}^{t\times s}.$$ Once the sets $I$ and $J$ are chosen the approximation $A(I,J,U)$ depends on the choice of $U$. Clearly the row and the column spaces of $A(I,J,U)$ are contained in the row and column spaces of $A[I,[n]]$ and $A[[m],J]$ respectively. Note that to store the approximation $A(I,J,U)$ we need to store the matrices $C$, $R$ and $U$. The number of these entries is $tm+sn+st$. So if $n,m$ are of order $10^5$ and $s,t$ are of order $10^2$ the storages of $C,R,U$ can be done in Random Access Memory (RAM), while the entries of $A$ are stored in external memory. To compute an entry of $A(I,J,U)$, which is an approximation of the corresponding entry of $A$, we need $st$ flops. Let ${\mathbf{U}}$ and ${\mathbf{V}}$ be subspaces spanned by the columns of $A[[m],J]$ and $A[I,[n]]{^\top}$ respectively. Then $A(I,J,U)\in {\mathbf{U}}{\mathbf{V}}{^\top}$, see . Clearly, a best CUR approximation is chosen by the least squares principle: $$\begin{aligned} \notag &&A(I,J,U^\star)=A([m],J)U^\star A(I,[n]),\\ &&U^\star=\arg\min\{\|A-A([m],J)UA(I,[n])\|,\; U\in{\mathbb{R}}^{|J|\times |I|}\}.\label{bestCURap}\end{aligned}$$ The results in [@FriT07] show that the least squares solution of is given by: $$\label{Ustarfor} U^\star=A([m],J)^\dagger A A(I,[n])^\dagger.$$ Here $F^\dagger$ denotes the Moore-Penrose pseudoinverse of a matrix $F$. Note that $U^\star$ is unique if and only if $$\label{uniqUsol} {\mathrm{rank\;}}A[[m],J]=|J|, \quad {\mathrm{rank\;}}A[I,[n]]=|I|.$$ The complexity of computation of $A([m],J)^\dagger$ and $A(I,[n])^\dagger$ are ${\mathcal{O}}(t^2m)$ and ${\mathcal{O}}(s^2n)$ respectively. Because of the multiplication formula for $U^\star$, the complexity of computation of $U^\star$ is ${\mathcal{O}}(stmn)$. One can significantly improve the computation of $U$, if one tries to best fit the entires of the submatrix $A[I',J']$ for given subsets $I'\subset [m], J'\subset [n]$. That is, let $$\begin{aligned} \notag &&U^\star(I',J'):=\arg\min\{\|A[I',J']-A(I',J)UA(I,J')\|,\; U\in{\mathbb{R}}^{|J|\times |I|}\}=\\ &&A[I',J]^\dagger A[I',J'] A^\dagger [I,J']. \label{UstarforI'J'}\end{aligned}$$ (The last equality follows from .) The complexity of computation of $U^\star(I',J')$ is ${\mathcal{O}}(st|I'||J'|)$. Suppose finally, that $I'=I$ and $J'$. Then and the properties of the Moore-Penrose inverse yield that $$\label{UstarforIJ} U^\star(I,J)=A[I,J]^\dagger, \quad A(I,J,U^\star(I,J))= B(I,J):=A[[m],J]A[I,J]^\dagger A[I,[n]].$$ In particular $B(I,J)[I,J]=A[I,J]$. Hence $$\begin{aligned} \notag &&A[[m],J]=B(I,J)[[m],J],\; A[I,[n]]=B(I,J)[I,[n]] \textrm{ if } |I|=|J|=k \textrm{ and } \det A[I,J]\ne 0,\\ &&B(I,J)=A[[m],J]A[I,J]^{-1} A[I,[n]].\label{invertAIJ}\end{aligned}$$ The original $CUR$ approximation of rank $k$ has the form $B(I,J)$ given by [@GTZ97]. Assume that ${\mathrm{rank\;}}A\ge k$. We want to choose an approximation $B(I,J)$ of the form which gives a good approximation to $A$. It is possible to give an upper estimate for the maximum of the absolute values of the entries of $A-B(I,J)$ in terms of $\sigma_{k+1}(A)$, provided that $\det A[I,J]$ is relatively close to $$\label{4.13.defmuk} \mu_k:=\max_{I\subset [m],J\subset[n],|I|=|J|=} |\det A[I,J]|>0.$$ Let $$\label{4.13.defentwnrm} \|F\|_{\infty,e}:=\max_{i\in[m],j\in[n]} |f_{i,j}|, \quad F=[f_{i,j}]\in{\mathbb{R}}^{m\times n}.$$ The results of [@GTZ97; @GorT01] yield: $$\label{4.13.basin} \|A- B(I,J)\|_{\infty,e}\le \frac{(k+1)\mu_k}{\det A[I,J]}\sigma_{p+1}(A).$$ (See also [@Fri15 Chapter 4, §13].) To find $\mu_k$ is probably an NP-hard problem in general [@Fri13a]. A standard way to find $\mu_k$ is either a random search or greedy search [@GOSTZ10; @Fri13a]. In the special case when $A$ is a symmetric positive definite matrix one can give the exact conditions when the greedy search gives a relatively good result [@Fri13a]. In the paper by Friedland-Mehrmann-Miedlar-Nkengla [@FMMN11] a good approximation $B(I,J)$ of the form is obtained by a random search on the maximum value of the product of the significant singular values of $A[I,J]$. The approximations found in this way are experimentally better than the approximations found by searching for $\mu_k$. Fast approximation of tensors {#S:fastapten} ============================= The fast approximation of tensors can be based on several decompositions of tensors such as: Tucker decomposition [@Tuc66]; matricizations of tensors, as unfolding and applying SVD one time or several time recursively, (see below); higher order singular value decomposition (HOSVD) [@deLdV00], Tensor-Train decompositions [@Ose09; @Ose11]; hierarchical Tucker decomposition [@Gra10; @Hac11]. A very recent survey [@GKT13] gives an overview on this dynamic field. In this paper we will discuss only the CUR-approximation. CUR-approximations of tensors {#sub:CURap} ----------------------------- Let ${\mathcal{T}}\in{\mathbb{R}}^{n_1\times \ldots n_d}$. In this subsection we denote the entries of ${\mathcal{T}}$ as ${\mathcal{T}}(i_1,\ldots,i_d)$ for $i_j\in [n_j]$ and $j\in [d]$. CUR-approximation of tensors is based on matricizations of tensors. The unfolding of ${\mathcal{T}}$ in the mode $l\in [d]$ consists of rearranging the entries of ${\mathcal{T}}$ as a matrix $T_l({\mathcal{T}})\in {\mathbb{R}}^{n_l\times N_l}$, where $N_l=\frac{\prod_{i\in [d]}n_i}{n_l}$. More general, let $K\cup L=[d]$ be a partition of $[d]$ into two disjoint nonempty sets. Denote $N(K)=\prod_{i\in K} n_i, N(L)=\prod_{j\in L} n_j$. Then unfolding ${\mathcal{T}}$ into the two modes $K$ and $L$ consists of rearranging the entires of ${\mathcal{T}}$ as a matrix $T(K,L,{\mathcal{T}})\in {\mathbb{R}}^{N(K)\times N(L)}$. We now describe briefly the $CUR$-approximation of $3$ and $4$-tensors as described by Friedland-Mehrmann-Miedlar-Nkengla [@FMMN11]. (See [@MMD06] for another approach to CUR-approximations for tensors.) We start with the case $d=3$. Let $I_i$ be a nonempty subset of $[n_i]$ for $i\in [3]$. Assume that the following conditions hold: $$|I_1|=k^2, \quad |I_2|=|I_3|=k, \quad J:=I_2\times I_3\subset [n_2]\times [n_3].$$ We identify $ [n_2]\times [n_3]$ with $[n_2n_3]$ using a lexicographical order. We now take the CUR-approximation of $T_1({\mathcal{T}})$ as given in : $$B(I_1,J)=T_1({\mathcal{T}})[[n_1],J]T_1({\mathcal{T}})[I_1,J]^{-1}T_1({\mathcal{T}})[I_1,[n_2n_3]].$$ We view $T_1({\mathcal{T}})[[n_1],J]$ as an $n_1\times k^2$ matrix. For each $\alpha_1\in I_1$ we view $T_1({\mathcal{T}})[\{\alpha_1\}, [n_2n_3]]$ as an $n_2\times n_3$ matrix $Q(\alpha_1):=[{\mathcal{T}}(\alpha_1,i_2,i_3)]_{i_2\in[n_2], i_3\in[n_3]}$. Let $R(\alpha_1)$ be the $CUR$-approximation of $Q(\alpha_1)$ based on the sets $I_2,I_3$: $$R(\alpha_1):=Q(\alpha_1)[[n_2],I_3] Q(\alpha_1)[I_2,I_3]^{-1} Q(\alpha_1)[I_2,[n_3]].$$ Let $F:=T_1({\mathcal{T}})[I_1,J]^{-1}\in {\mathbb{R}}^{k^2\times k^2}$. We view the entries of this matrix indexed by the row $(\alpha_2,\alpha_3)\in I_2\times I_3$ and column $\alpha_1\in I_1$. We write these entries as ${\mathcal{F}}(\alpha_1,\alpha_2,\alpha_3), \alpha_j\in I_j, j\in[3]$, which represent a tensor ${\mathcal{F}}\in {\mathbb{R}}^{I_1\times I_2 \times I_3}$. The entries of $Q(\alpha_1)[I_2,I_3]^{-1}$ are indexed by the row $\alpha_3\in I_3$ and column $\alpha_2\in I_2$. We write these entries as ${\mathcal{G}}(\alpha_1,\alpha_2,\alpha_3), \alpha_2\in I_2,\alpha_3\in I_3$, which represent a tensor ${\mathcal{G}}\in {\mathbb{R}}^{I_1\times I_2 \times I_3}$. Then the approximation tensor ${\mathcal{B}}=[{\mathcal{B}}(j_1,j_2,j_3)]\in {\mathbb{R}}^{n_1\times n_2\times n_3}$ is given by: $$\begin{aligned} &&{\mathcal{B}}(i_1,i_2,i_3)=\sum_{\alpha_1\in I_1,\alpha_j,\beta_j\in I_j, j=2,3}\\ &&{\mathcal{T}}(i_1,\alpha_2,\alpha_3){\mathcal{F}}(\alpha_1,\alpha_2,\alpha_3) {\mathcal{T}}(\alpha_1,j_2,\beta_3){\mathcal{G}}(\alpha_1,\beta_2,\beta_3){\mathcal{T}}(\alpha_1,\beta_2,j_3).\end{aligned}$$ We now discuss a CUR-approximation for $4$-tensors, i.e. $d=4$. Let ${\mathcal{T}}\in {\mathbb{R}}^{n_1\times n_2\times n_3\times n_4}$ and $K=\{1,2\},L=\{3,4\}$. The rows and columns of $X:=T(K,L,{\mathcal{T}})\in {\mathbb{R}}^{(n_1n_2)\times (n_3n_4)}$ are indexed by pairs $(i_1,i_2)$ and $(i_3,i_4)$ respectively. Let $$I_j\subset [n_j], \; |I_j|=k,\; j\in [4], \quad J_1:=I_1\times I_2,\;J_2:=I_3\times I_4.$$ First consider the CUR-approximation $X[[n_1n_2],J_2] X[J_1,J_2]^{-1} X[J_1,[n_3n_4]]$ viewed as tensor ${\mathcal{C}}\in{\mathbb{R}}^{n_1\times n_2\times n_3\times n_4}$. Denote by ${\mathcal{H}}(\alpha_1,\alpha_2,\alpha_3,\alpha_4)$ the $((\alpha_3,\alpha_4), (\alpha_1,\alpha_2))$ entry of the matrix $X[J_1,J_2]^{-1}$. So ${\mathcal{H}}\in {\mathbb{R}}^{I_1\times I_2\times I_3\times I_4}$. Then $${\mathcal{C}}(i_1,i_2,i_3,i_4)=\sum_{\alpha_j\in I_j,j\in[4]} {\mathcal{T}}(i_1,i_2,\alpha_3,\alpha_4){\mathcal{H}}(\alpha_1,\alpha_2,\alpha_3,\alpha_4){\mathcal{T}}(\alpha_1,\alpha_2,i_3,i_4).$$ For $\alpha_j\in I_j,j\in[4]$ view vectors $X[[n_1n_2],(\alpha_3,\alpha_4)]$ and $X[(\alpha_1,\alpha_2),[n_3n_4]]$ as matrices $Y(\alpha_3,\alpha_4)\in {\mathbb{R}}^{n_1\times n_2}$ and $Z(\alpha_1,\alpha_2)\in {\mathbb{R}}^{n_3\times n_4}$ respectively. Next we find the CUR-approximations to these two matrices using the subsets $(I_1,I_2)$ and $(I_3,I_4)$ respectively: $$\begin{aligned} &&Y(\alpha_3,\alpha_4)[[n_1],I_2]Y(\alpha_3,\alpha_4)[I_1,I_2]^{-1}Y(\alpha_3,\alpha_4)[I_1,[n_2]],\\ &&Z(\alpha_1,\alpha_2)[[n_3],I_4]Z(\alpha_1,\alpha_2)[I_3,I_4]^{-1}Z(\alpha_1,\alpha_2)[I_3,[n_4]].\end{aligned}$$ We denote the entries of $Y(\alpha_3,\alpha_4)[I_1,I_2]^{-1}$ and $Z(\alpha_1,\alpha_2)[I_3,I_4]^{-1}$ by ${\mathcal{F}}(\alpha_1,\alpha_2,\alpha_3,\alpha_4), \alpha_1\in I_1,\alpha_2\in I_2$ and ${\mathcal{G}}(\alpha_1,\alpha_2,\alpha_3,\alpha_4), \alpha_3\in I_3,\alpha_4\in I_4$ respectively. Then the CUR-approximation tensor ${\mathcal{B}}$ of ${\mathcal{T}}$ is given by: $$\begin{aligned} &&{\mathcal{B}}(i_1,i_2,i_3,i_4)=\sum_{\alpha_j,\beta_j\in I_j, j\in [4]} {\mathcal{T}}(i_1,\beta_2,\alpha_3,\alpha_4){\mathcal{F}}(\beta_1,\beta_2,\alpha_3,\alpha_4) {\mathcal{T}}(\beta_1,i_2,\alpha_3,\alpha_4)\\ &&{\mathcal{H}}(\alpha_1,\alpha_2,\alpha_3,\alpha_4){\mathcal{T}}(\alpha_1,\alpha_2,i_3,\beta_4){\mathcal{G}}(\alpha_1,\alpha_2,\beta_3,\beta_4){\mathcal{T}}(\alpha_1,\alpha_2,\beta_3,i_4).\end{aligned}$$ We now discuss briefly the complexity of the storage and computing an entry of the CUR-approximation ${\mathcal{B}}$. Assume first that $d=3$. Then we need to store $k^2$ columns of the matrices $T_1({\mathcal{T}})$, $k^3$ columns of $T_2({\mathcal{T}})$ and $T_3({\mathcal{T}})$, and $k^4$ entries of the tensors ${\mathcal{F}}$ and ${\mathcal{G}}$. The total storage space is $k^2 n_1+k^3(n_2+n_3)+2k^4$. To compute each entry of ${\mathcal{B}}$ we need to perform $4k^6$ multiplications and $k^6$ additions. Assume now that $d=4$. Then we need to store $k^3$ columns of $T_l({\mathcal{T}}), l\in[4]$ and $k^4$ entries of ${\mathcal{F}},{\mathcal{G}},{\mathcal{H}}$. Total storage needed is $k^3(n_1+n_2+n_3+n_4+3k)$. To compute each entry of ${\mathcal{B}}$ we need to perform $6k^8$ multiplications and $k^8$ additions. Preliminary results on best ${\mathbf{r}}$-approximation {#S:prelbrap} ======================================================== The maximization problem {#sub:maxprob} ------------------------ We first show that the best approximation problem is equivalent to the maximum problem , see [@LMV00] and [@Hac12 §10.3]. The Pythagoras theorem yields that $$\|{\mathcal{T}}\|^2 = \|P_{\otimes_{i=1}^d {\mathbf{U}}_i}({\mathcal{T}})\|^2 + \|P_{(\otimes_{i=1}^d {\mathbf{U}}_i)^\perp}({\mathcal{T}})\|^2,\quad \|{\mathcal{T}}-P_{\otimes_{i=1}^d {\mathbf{U}}_i}({\mathcal{T}})\|^2= \|P_{(\otimes_{i=1}^d {\mathbf{U}}_i)^\perp}({\mathcal{T}})\|^2.$$ (Here $(\otimes_{i=1}^d {\mathbf{U}}_i)^\perp$ is the orthogonal complement of $\otimes_{i=1}^d {\mathbf{U}}_i$ in $\otimes_{i=1}^d {\mathbb{R}}^{n_i}$.) Hence $$\label{bestapproxid} \min_{{\mathbf{U}}_i\in{\mathop{\mathrm{Gr}}\nolimits}(r_i,{\mathbb{R}}^{n_i}), i\in[d]}\|{\mathcal{T}}-P_{\otimes_{i=1}^d {\mathbf{U}}_i}({\mathcal{T}})\|^2=\|{\mathcal{T}}\|^2- \max_{{\mathbf{U}}_i\in {\mathop{\mathrm{Gr}}\nolimits}(r_i,{\mathbb{R}}^{n_i}), i\in [d]} \|P_{\otimes_{i=1}{\mathbf{U}}_i}({\mathcal{T}})\|^2.$$ This shows the equivalence of and . Singular values and singular tuples of tensors {#sub:singvten} ---------------------------------------------- Let ${\mathrm{S}}(n)=\{{\mathbf{x}}\in{\mathbb{R}}^n, \;\|{\mathbf{x}}\|=1\}$. Note that one dimensional subspace ${\mathbf{U}}\in{\mathop{\mathrm{Gr}}\nolimits}(1,{\mathbb{R}}^n)$ is ${\mathrm{span}}({\mathbf{u}})$, where ${\mathbf{u}}\in{\mathrm{S}}(n)$. Let ${\mathrm{S}}({\mathbf{n}}):={\mathrm{S}}(n_1)\times\cdots\times{\mathrm{S}}(n_d)$. Then best rank one approximation problem for ${\mathcal{T}}\in{\mathbb{R}}^{{\mathbf{n}}}$ is equivalent to finding $$\label{axprobbr1ap} \|{\mathcal{T}}\|_{\sigma}:=\max_{({\mathbf{x}}_1,\ldots,{\mathbf{x}}_d)\in{\mathrm{S}}({\mathbf{n}})} {\mathcal{T}}\times (\otimes_{i\in[d]} {\mathbf{x}}_i).$$ Let $f_{\mathcal{T}}:{\mathbb{R}}^{{\mathbf{n}}}\to {\mathbb{R}}$ is given by $f_{{\mathcal{T}}}({\mathcal{X}})={{\langle}{\mathcal{X}},{\mathcal{T}}{\rangle}}$. Denote by ${\mathrm{S}}'({\mathbf{n}})\subset {\mathbb{R}}^{\mathbf{n}}$ all rank one tensors of the form $\otimes_{i\in[d]} {\mathbf{x}}_i$, where $({\mathbf{x}}_1,\ldots,{\mathbf{x}}_n)\in{\mathrm{S}}({\mathbf{n}})$. Let $f_{\mathcal{T}}({\mathbf{x}}_1,\ldots,{\mathbf{x}}_d):=f_{\mathcal{T}}(\otimes_{i\in[d]}{\mathbf{x}}_i)$. Then the critical points of $f_{\mathcal{T}}|{\mathrm{S}}'({\mathbf{n}})$ are given by the Lagrange multipliers formulas [@Lim05]: $$\label{defsingvalstup} {\mathcal{T}}\times (\otimes_{j\in[d]\setminus\{i\}} {\mathbf{u}}_j)=\lambda {\mathbf{u}}_i,\quad i\in [d],\quad ({\mathbf{u}}_1,\ldots,{\mathbf{u}}_d)\in{\mathrm{S}}({\mathbf{n}}).$$ One calls $\lambda$ and $({\mathbf{u}}_1,\ldots,{\mathbf{u}}_d)$ a singular value and singular tuple of ${\mathcal{T}}$. For $d=2$ these are the singular values and singular vectors of ${\mathcal{T}}$. The number of *complex* singular values of a generic ${\mathcal{T}}$ is given in [@FO14]. This number increases exponentially with $d$. For example for $n_1=\cdots=n_d=2$ the number of distinct singular values is $d!$. (The number of real singular values as given by is bounded by the numbers given in [@FO14].) Consider first the maximization problem of $f_{\mathcal{T}}({\mathbf{x}}_1,\ldots,{\mathbf{x}}_d)$ over ${\mathrm{S}}({\mathbf{n}})$ where we vary ${\mathbf{x}}_i\in{\mathrm{S}}(n_i)$ and keep the other variables fixed. This problem is equivalent to the maximization of the linear form ${\mathbf{x}}_i{^\top}({\mathcal{T}}\times (\otimes_{j\in [d]\setminus\{i\}}{\mathbf{x}}_j))$. Note that if ${\mathcal{T}}\times (\otimes_{j\in [d]\setminus\{i\}}{\mathbf{x}}_j)\ne {\mathbf{0}}$ then this maximum is achieved for ${\mathbf{x}}_i=\frac{1}{\|{\mathcal{T}}\times (\otimes_{j\in [d]\setminus\{i\}}{\mathbf{x}}_j)\|}{\mathcal{T}}\times (\otimes_{j\in [d]\setminus\{i\}}{\mathbf{x}}_j)$. Consider second the maximization problem of $f_{\mathcal{T}}({\mathbf{x}}_1,\ldots,{\mathbf{x}}_d)$ over ${\mathrm{S}}({\mathbf{n}})$ where we vary $({\mathbf{x}}_i, {\mathbf{x}}_j)\in{\mathrm{S}}(n_i)\times{\mathrm{S}}(n_j), 1\le i < j\le d$ and keep the other variables fixed. This problem is equivalent to finding the first singular value and the corresponding right and left singular vectors of the matrix ${\mathcal{T}}\times (\otimes_{k\in[d]\setminus\{i,j\}}{\mathbf{x}}_k)$. This can be done by using use Arnoldi or Lanczos methods [@GolV96; @LSY98]. The complexity of this method is ${\mathcal{O}}(n_in_j)$, given the matrix ${\mathcal{T}}\times (\otimes_{k\in[d]\setminus\{i,j\}}{\mathbf{x}}_k)$. A basic maximization problem for best ${\mathbf{r}}$-approximation {#sub:apprprb} ------------------------------------------------------------------ Denote by ${\mathrm{S}}_n\subset {\mathbb{R}}^{n\times n}$ the space of real symmetric matrices. For $A\in {\mathrm{S}}_n$ denote by $\lambda_1(A)\ge \ldots\ge\lambda_n(A)$ the eigenvalues of $A$ arranged in a decreasing order and repeated according to their multiplicities. Let ${\mathrm{O}}(n,k)\subset {\mathbb{R}}^{n\times k}$ be the set of all $n\times k$ matrices $X$ with $k$ orthonormal columns, i.e. $X{^\top}X=I_k$, where $I_k$ is $k\times k$ identity matrix. We view $X\in {\mathbb{R}}^{n\times k}$ as composed of $k$-columns $[{\mathbf{x}}_1\ldots{\mathbf{x}}_k]$. The column space of $X\in {\mathrm{O}}(n,k)$ corresponds to a $k$-dimensional subspace ${\mathbf{U}}\subset{\mathbb{R}}^n$. Note that ${\mathbf{U}}\in{\mathop{\mathrm{Gr}}\nolimits}(k,{\mathbb{R}}^n)$ is spanned by the orthonormal columns of a matrix $Y\in O(n,k)$ if and only if $Y=XO$, for some $O\in {\mathrm{O}}(k,k)$. For $A\in {\mathrm{S}}_n$ one has the Ky-Fan maximal characterization [@HJ88 Cor. 4.3.18] $$\label{KyFanchar} \max_{[{\mathbf{x}}_1\ldots{\mathbf{x}}_k]\in {\mathrm{O}}(n,k)}\sum_{i=1}^k {\mathbf{x}}_i{^\top}A{\mathbf{x}}_i=\sum_{i=1}^k \lambda_i(A).$$ Equality holds if and only if the column space of $X=[{\mathbf{x}}_1\ldots{\mathbf{x}}_k]$ is a subspace spanned by $k$ eigenvectors corresponding to $k$-largest eigenvalues of $A$. We now reformulate the maximum problem in terms of orthonormal bases of ${\mathbf{U}}_i, i\in [d]$. Let ${\mathbf{u}}_{1,i},\ldots,{\mathbf{u}}_{n_i,i}$ be an orthonormal basis of ${\mathbf{U}}_i$ for $i\in [d]$. Then $\otimes_{i=1}^d {\mathbf{u}}_{j_i,i}, j_i\in [n_i],i\in[d]$ is an orthonormal basis of $\otimes_{i=1}^d {\mathbf{U}}_i$. Hence $$\|P_{\otimes_{i=1}^d{\mathbf{U}}_i}({\mathcal{T}})\|^2=\sum_{j_i\in [n_i],i\in[d]}{{\langle}{\mathcal{T}},\otimes_{i=1}^d {\mathbf{u}}_{j_i,i}{\rangle}}^2.$$ Hence is equivalent to $$\begin{aligned} \label{maxnormprb1} &&\max_{[{\mathbf{u}}_{1,i}\ldots{\mathbf{u}}_{r_i,i}]\in{\mathrm{O}}(n_i,r_i),i\in[d]} \sum_{j_i\in [n_i], i\in[d]}{{\langle}{\mathcal{T}},\otimes_{i=1}^d {\mathbf{u}}_{j_i,i}{\rangle}}^2=\\ &&\max_{{\mathbf{U}}_i\in{\mathop{\mathrm{Gr}}\nolimits}(r_i,{\mathbb{R}}^{n_i}),i\in[d]} \|P_{\otimes_{i=1}^d {\mathbf{U}}_i}({\mathcal{T}})\|^2. \notag \end{aligned}$$ A simpler problem is to find $$\begin{aligned} \label{maxnormprbbas} &&\max_{[{\mathbf{u}}_{1,i}\ldots{\mathbf{u}}_{r_i,i}]\in{\mathrm{O}}(n_i,r_i)} \sum_{j_i\in [n_i], i\in[d]}{{\langle}{\mathcal{T}},\otimes_{i=1}^d {\mathbf{u}}_{j_i,i}{\rangle}}^2=\\ &&\max_{{\mathbf{U}}_i\in{\mathop{\mathrm{Gr}}\nolimits}(r_i,{\mathbb{R}}^{n_i})} \|P_{\otimes_{i=1}^d {\mathbf{U}}_i}({\mathcal{T}})\|^2, \notag \end{aligned}$$ for a fixed $i\in [d]$. Let $$\begin{aligned} \notag &&{\underline U}:=({\mathbf{U}}_1,\ldots,{\mathbf{U}}_d)\in{\mathop{\mathrm{Gr}}\nolimits}({\mathbf{r}},{\mathbf{n}}),\\ &&{\mathop{\mathrm{Gr}}\nolimits}_i({\mathbf{r}},{\mathbf{n}}):={\mathop{\mathrm{Gr}}\nolimits}(r_1,n_1)\times\ldots\times {\mathop{\mathrm{Gr}}\nolimits}(r_{i-1},n_{i-1})\times{\mathop{\mathrm{Gr}}\nolimits}(r_{i+1},n_{i+1})\times\ldots\times{\mathop{\mathrm{Gr}}\nolimits}(r_d,n_d),\notag\\ &&{\underline U}_i:=({\mathbf{U}}_1,\ldots,{\mathbf{U}}_{i-1},{\mathbf{U}}_{i+1},\ldots,{\mathbf{U}}_d)\in {\mathop{\mathrm{Gr}}\nolimits}_i({\mathbf{r}},{\mathbf{n}}),\notag\\ &&A_i({\underline U}_i):=\sum_{j_l\in [r_l],l\in [d]\setminus\{i\}} ({\mathcal{T}}\times \otimes_{k\in [d]\setminus \{i\}} {\mathbf{u}}_{j_k,k})({\mathcal{T}}\times \otimes_{k\in [d]\setminus \{i\}} {\mathbf{u}}_{j_k,k}){^\top}. \label{defAUi}\end{aligned}$$ The maximization problem reduces to the maximum problem with $A=A_i({\underline U}_i)$. Note that each $A_i({\underline U}_i)$ is a positive semi-definite matrix. Hence $\sigma_j(A_i({\underline U}_i))=\lambda_j(A_i({\underline U}_i))$ for $j\in [n_i]$. Thus the complexity to find the first $r_i$ eigenvectors of $A_i({\underline U}_i)$ is ${\mathcal{O}}(r_in_i^2)$. Denote by ${\mathbf{U}}_i({\underline U}_i)\in{\mathop{\mathrm{Gr}}\nolimits}(r_i,{\mathbb{R}}^{n_i})$ a subspace spanned by the first $r_i$ eigenvectors of $A_i({\underline U}_i)$. Note that this subspace is unique if and only if $$\label{uniqcondUiAi} \lambda_{r_i}(A_i({\underline U}_i))>\lambda_{r_i+1}(A_i({\underline U}_i)).$$ Finally, if ${\mathbf{r}}={\mathbf{1}}_d$ then each $A_i({\underline U}_i)$ is a rank one matrix. Hence ${\mathbf{U}}_i({\underline U}_i)={\mathrm{span}}({\mathcal{T}}\times \otimes_{k\in [d]\setminus \{i\}} {\mathbf{u}}_{1,k})$. For more details see [@FM08]. Alternating maximization methods for best ${\mathbf{r}}$-approximation {#S:AMM} ====================================================================== General definition and properties {#sub:gendp} --------------------------------- Let $\Psi_i$ be a compact smooth manifold for $i\in[d]$. Define $$\Psi:=\Psi_1\times \cdots \times \Psi_d, \; \hat \Psi_i=(\Psi_1\times \cdots \times\Psi_{i-1}\times \Psi_{i+1}\times \cdots\times\Psi_d) \textrm{ for }i\in [d].$$ We denote by $\psi_i$, $\psi=(\psi_1,\ldots,\psi_d)$ and $\hat\psi_i=(\psi_1,\ldots,\psi_{i-1},\psi_{i+1},\ldots,\psi_d)$ the points in $\Psi_i$, $\Psi$ and $\hat \Psi_i$ respectively. Identify $\psi$ with $(\psi_i,\hat\psi_i)$ for each $i\in[d]$. Assume that $f:\Psi\to{\mathbb{R}}$ is a continuous function with continuous first and second partial derivatives. (In our applications it may happen that $f$ has discontinuities in first and second partial derivatives.) We want to find the maximum value of $f$ and a corresponding maximum point $\psi^\star$: $$\label{maxprobfPsi} \max_{\psi\in\Psi} f(\psi)=f(\psi^\star).$$ Usually, this is a hard problem, where $f$ has many critical points and a number of these critical points are local maximum points. In some cases, as best ${\mathbf{r}}$ approximation to a given tensor ${\mathcal{T}}\in{\mathbb{R}}^{\mathbf{n}}$, we can solve the maximization problem with respect to one variable $\psi_i$ for any fixed $\hat\psi_i$: $$\label{maxprobPsii} \max_{\psi_i\in \Psi_i} f((\psi_i,\hat\psi_i))=f((\psi_i^\star(\hat\psi_i),\hat\psi_i)),$$ for each $i\in[d]$. Then the *alternating maximization method*, abbreviated as AMM, is as follows. Assume that we start with an initial point $\psi^{(0)}=(\psi^{(0)}_1,\ldots,\psi^{(0)}_d)=(\psi_1^{(0)},\hat \psi_1^{(0,1)})$. Then we consider the maximal problem for $i=1$ and $\hat\psi_1:=\hat \psi_1^{(0,1)}$. This maximum is achieved for $\psi_1^{(1)}:=\psi_1^\star(\hat\psi_1^{(0,1)})$. Assume that the coordinates $\psi_1^{(1)},\ldots,\psi_{j}^{(1)}$ are already defined for $j\in [d-1]$. Let $\hat \psi_{j+1}^{(0,j+1)}:= (\psi_1^{(1)},\ldots,\psi_j^{(1)},\psi_{j+2}^{(0)},\ldots,\psi_d^{(0)})$. Then we consider the maximum problem for $i=j+1$ and $\hat\psi_{j+1}:=\hat \psi_{j+1}j^{(0,j+1)}$. This maximum is achieved for $\psi_{j+1}^{(1)}:=\psi_{j+1}^\star(\hat\psi_{j+1}^{(0,j+1)})$. Executing these $d$ iterations we obtain $\psi^{(1)}:=(\psi^{(1)}_1,\ldots,\psi^{(1)}_d)$. Note that we have a sequence of inequalities: $$f(\psi^{(0)})\le f(\psi_1^{(1)},\hat\psi_1^{(0,1)})\le f(\psi_2^{(1)},\hat\psi_2^{(0,2)})\le\cdots \le f(\psi_d^{(1)},\hat\psi_d^{(0,d)})=f(\psi^{(1)}).$$ Replace $\psi^{(0)}$ with $\psi^{(1)}$ and continue these iterations to obtain a sequence $\psi^{(l)}=(\psi_1^{(l)},\ldots,\psi_d^{(l)})$ for $l=0,\ldots,N$. Clearly, $$\label{nondecpropamm} f(\psi^{(l-1)})\le f(\psi^{(l)}) \textrm{ for }l\in{\mathbb{N}}\Rightarrow \lim_{l\to\infty} f(\psi^{(l)})=M.$$ Usually, the sequence $\psi^{(l)}, l=0,\ldots,$ will converge to $1$-semi maximum point $\phi=(\phi_1,\ldots,\phi_d)\in\Psi$. That is, $f(\phi)=\max_{\psi_i\in\Psi} f((\psi_i,\hat\phi_i))$ for $i\in [d]$. Note that if $f$ is differentiable at $\phi$ then $\phi$ is a critical point of $f$. Assume that $f$ is twice differentiable at $\phi$. Then $\phi$ does not have to be a local maximum point [@FMPS13 Appendix]. The *modified alternating maximization method*, abbreviated as MAMM, is as follows. Assume that we start with an initial point $\psi^{(0)}=(\psi^{(0)}_1,\ldots,\psi^{(0)}_d)$. Let $\psi^{(0)}=(\psi_i^{(0)},\hat \psi_i^{(0)})$ for $i\in[d]$. Compute $f_{i,0}= \max_{\psi_i\in\Psi_i} f((\psi_i,\hat\psi_i^{(0)}))$ for $i\in [d]$. Let $j_1\in\arg\max_{i\in[d]} f_{i,0}$. Then $\psi^{(1)}=(\psi_j^\star(\hat\psi_{j_1}^{(0)}),\hat\psi_{j_1}^{(0)})$ and $f_1=f_{1,{j_1}}=f(\psi^{(1)})$. Note that it takes $d$ iterations to compute $\psi^{(1)}$. Now replace $\psi^{(0)}$ with $\psi^{(1)}$ and compute $f_{i,1}= \max_{\psi_i\in\Psi_i} f((\psi_i,\hat\psi_i^{(1)}))$ for $i\in[d]\setminus\{j_1\}$. Continue as above to find $\psi^{(l)}$ for $l=2,\ldots,N$. Note that for $l\ge 2$ it takes $d-1$ iterations to determine $\psi^{(l)}$. Clearly, holds. It is shown in [@FMPS13] that the limit $\phi$ of each convergent subsequence of the points $\psi^{(j)}$ is $1$-semi maximum point of $f$. In certain very special cases, as for best rank one approximation, we can solve the maximization problem with respect to any pair of variables $\psi_i,\psi_j$ for $1\le i<j\le d$, where $d\ge 3$ and all other variables are fixed. Let $$\begin{aligned} &&\hat \Psi_{i,j}:=\Psi_1\times \cdots \times\Psi_{i-1}\times \Psi_{i+1}\times\cdots\times \Psi_{j-1}\times \Psi_{j+1}\times\cdots\times \Psi_d,\\ &&\hat\psi_{i,j}=(\psi_1,\ldots,\psi_{i-1},\psi_{i+1},\ldots,\psi_{j-1},\psi_{j+1},\ldots,\psi_d)\in \hat \Psi_{i,j},\quad \psi_{i,j}=(\psi_i,\psi_j)\in \Psi_i\times \Psi_j.\end{aligned}$$ View $\psi=(\psi_1,\ldots,\psi_d)$ as $(\psi_{i,j},\hat\psi_{i,j})$ for each pair $1\le i<j\le d$. Then $$\label{maxprobPsiij} \max_{\psi_{i,j}\in \Psi_i\times \Psi_j} f((\psi_{i,j},\hat\psi_{i,j}))=f((\psi_{i,j}^\star(\hat\psi_{i,j}),\hat\psi_{i,j})).$$ A point $\psi$ is called *2-semi maximum point* if the above maximum equals to $f(\psi)$ for each pair $1\le i<j\le d$. The *$2$-alternating maximization method*, abbreviated here as 2AMM, is as follows. Assume that we start with an initial point $\psi^{(0)}=(\psi^{(0)}_1,\ldots,\psi^{(0)}_d)$. Suppose first that $d=3$. Then we consider the maximization problem for $i=2, j=3$ and $\hat\psi_{2,3}=\psi_1^{(0)}$. Let $(\psi_2^{(0,1)},\psi_3^{(0,1)})=\psi_{2,3}^\star(\psi_1^{(0)})$. Next let $i=1,j=3$ and $\psi_{1,3}=\psi_2^{(0,1)}$. Then $(\psi_1^{(0,2)},\psi_3^{(0,2)})=\psi_{1,3}^\star(\psi_2^{(0,1)})$. Next let $i=1,2$ and $\hat\psi_{1,2}=\psi_3^{(0,2)}$. Then $\psi^{(1)}=(\hat\psi_{1,2}^\star(\psi_3^{(0,2)}),\psi_3^{(0,2)})$. Continue these iterations to obtain $\psi^{(l)}$ for $l=2,\ldots$. Again, holds. Usually the sequence $\psi^{(l)},l\in{\mathbb{N}}$ will converge to a 2-semi maximum point $\phi$. For $d\ge 4$ the 2AMM can be defined appropriately see [@FMPS13]. A *modified $2$-alternating maximization method*, abbreviated here as M2AMM, is as follows. Start with an initial point $\psi^{(0)}=(\psi^{(0)}_1,\ldots,\psi^{(0)}_d)$ viewed as $(\psi_{i,j}^{(0)},\hat\psi_{i,j}^{(0)})$, for each pair $1\le i<j\le d$. Let $f_{i,j,0}:=\max_{\psi_{i,j}\in \Psi_i\times \Psi_j} f((\psi_{i,j},\hat\psi_{i,j}^{(0)}))$. Assume that $(i_1,j_1)\in\arg\max_{1\le i <j\le d} f_{i,j,0}$. Then $\psi^{(1)}=(\psi_{i_1,j_1}^\star(\hat\psi_{i_1,j_1}^{(0)}),\hat\psi_{i_1,j_1}^{(0)})$. Let $f_{i_1,j_1,1}:=f(\psi^{(1)})$. Note that it takes $d \choose 2$ iterations to compute $\psi^{(1)}$. Now replace $\psi^{(0)}$ with $\psi^{(1)}$ and compute $f_{i,j,1}= \max_{\psi_{i,j}\in\Psi_i\times \Psi_j} f((\psi_{i,j},\hat\psi_{i,j}^{(1)}))$ for all pairs $1\le i<j\le d$ except the pair $(i_1,j_1)$. Continue as above to find $\psi^{(l)}$ for $l=2,\ldots,N$. Note that for $l\ge 2$ it takes ${d \choose 2}-1$ iterations to determine $\psi^{(l)}$. Clearly, holds. It is shown in [@FMPS13] that the limit $\phi$ of each convergent subsequence of the points $\psi^{(j)}$ is $2$-semi maximum point of $f$. AMM for best ${\mathbf{r}}$-aproximations of tensors {#sub:AMMbrapr} ---------------------------------------------------- Let ${\mathcal{T}}\in {\mathbb{R}}^{\mathbf{n}}$. For best rank one approximation one searches for the maximum of the function $f_{\mathcal{T}}={\mathcal{T}}\times (\otimes_{i\in [d]}{\mathbf{x}}_i)$ on ${\mathrm{S}}({\mathbf{n}})$, as in . For best ${\mathbf{r}}$-approximation one searches for the maximum of the function $f_{\mathcal{T}}=\|P_{\otimes_{i\in[d]}{\mathbf{U}}_i}\|^2$ on ${\mathop{\mathrm{Gr}}\nolimits}({\mathbf{r}},{\mathbf{n}})$, as in . A solution to the basic maximization problem with respect to one subspace ${\mathbf{U}}_i$ is given in §\[sub:apprprb\]. The AMM for best ${\mathbf{r}}$-approximation were studied first by de Lathauwer-Moor-Vandewalle [@LMV00]. The AMM is called in [@LMV00] *alternating least squares*, abbreviated as ALS. A crucial problem is the starting point of AMM. A high order SVD, abbreviated as HOSVD, for ${\mathcal{T}}$, see [@deLdV00], gives a good starting point for AMM. That is, let $T_l({\mathcal{T}})\in{\mathbb{R}}^{n_l\times N_l}$ be the unfolded matrix of ${\mathcal{T}}$ in the mode $l$, as in §\[sub:CURap\]. Then ${\mathbf{U}}_l$ is the subspace spanned by the first $l$-left singular vectors of $T_l({\mathcal{T}})$. The complexity of computing ${\mathbf{U}}_l$ is ${\mathcal{O}}(r_lN)$, where $N=\prod_{i\in[d]} n_i$. Hence for large $N$ the complexity of computing partial HOSVD is high. Another approach is to choose the starting subspaces at random, and repeat the AMM for several choices of random starting points. MAMM for best rank one approximation of tensors was introduced by Friedland-Mehrmann-Pajarola-Suter in [@FMPS13] by the name *modified alternating least squares*, abbreviated as MALS. 2AMM for best rank one approximation was introduced in [@FMPS13] by the name *alternating SVD*, abbreviated as ASVD. It follows from the observation that $A:={\mathcal{T}}\times (\otimes_{l\in[d]\setminus\{i,j\}}{\mathbf{x}}_l)$ is an $n_i\times n_j$ matrix. Hence the maximum of the bilinear form ${\mathbf{x}}{^\top}A{\mathbf{y}}$ on ${\mathrm{S}}((n_i,n_j))$ is $\sigma_1(A)$. See §\[S:SVD\]. M2AMM was introduced in [@FMPS13] by the names MASVD. We now introduce the following variant of 2AMM for best ${\mathbf{r}}$-rank approximation, called *$2$-alternating maximization method variant* and abbreviated as 2AMMV. Consider the maximization problem for a pair of variables as in . Since for ${\mathbf{r}}\ne {\mathbf{1}}_d$ we do not have a closed solution to this problem, we apply the AMM for two variables $\psi_i$ and $\psi_j$, while keeping $\hat \psi_{i,j}$ fixed. We then continue as in 2AMM method. Complexity analysis of AMM for best ${\mathbf{r}}$-approximation {#sub:companbrap} ---------------------------------------------------------------- Let ${\underline U}=({\mathbf{U}}_1,\ldots,{\mathbf{U}}_d)$. Assume that $$\begin{aligned} \notag &&{\mathbf{U}}_{i}={\mathrm{span}}({\mathbf{u}}_{1,i},\ldots,{\mathbf{u}}_{r_i,n_i}), \quad {\mathbf{U}}_i^\perp={\mathrm{span}}({\mathbf{u}}_{r_i+1},\ldots, {\mathbf{u}}_{n_i,i}),\\ &&{\mathbf{u}}_{j,i}{^\top}{\mathbf{u}}_{k,i}=\delta_{j,k},\;j,k\in [n_i], \quad i\in [d].\label{basisUi}\end{aligned}$$ For each $i\in[d]$ compute the symmetric positive semi-definite matrix $A_i({\underline U}_i)$ given by . For simplicity of exposition we give the complexity analysis for $d=3$. To compute $A_1({\underline U}_1)$ we need first to compute the vectors ${\mathcal{T}}\times ({\mathbf{u}}_{j_2,2}\otimes {\mathbf{u}}_{j_3,3})$ for $j_2\in[r_2]$ and $j_3\in[r_3]$. Each computation of such a vector has complexity ${\mathcal{O}}(N)$, where $N=n_1n_2n_3$. The number of such vectors is $r_2r_3$. To form the matrix $A_i({\underline U}_i)$ we need ${\mathcal{O}}(r_2r_3n_1^2)$ flops. To find the first $r_1$ eigenvectors of $A_1({\underline U}_1)$ we need ${\mathcal{O}}(r_1n_1^2)$ flops. Assuming that $n_1,n_2,n_3\approx n$ and $r_1,r_2,r_3\approx r$ we deduce that we need $O(r^2 n^3)$ flops to find the first $r_1$ orthonormal eigenvectors of $A_1({\underline U}_1)$ which span ${\mathbf{U}}_1({\underline U}_1)$. Hence the complexity of finding orthonormal bases of ${\mathbf{U}}_1({\underline U}_1)$ is ${\mathcal{O}}(r^2n^3)$, which is the complexity of computing $A_1({\underline U}_1)$. Hence the complexity of each step of AMM for best ${\mathbf{r}}$-approximation, i.e. computing $\psi^{(l)}$, is ${\mathcal{O}}(r^2n^3)$. It is possible to reduce the complexity of AMM for best ${\mathbf{r}}$-approximation is to ${\mathcal{O}}(r n^3)$ if we compute and store the matrices ${\mathcal{T}}\times {\mathbf{u}}_{j_1,1},{\mathcal{T}}\times {\mathbf{u}}_{j_2,2},{\mathcal{T}}\times {\mathbf{u}}_{j_3,3}$. See §\[sub:compNewtrap\]. We now analyze the complexity of AMM for rank one approximation. In this case we need only to compute the vector of the form ${\mathbf{v}}_i:={\mathcal{T}}\times(\otimes_{j\in[d]\setminus\{i\}}{\mathbf{u}}_j)$ for each $i\in[d]$, where ${\mathbf{U}}_j={\mathrm{span}}({\mathbf{u}}_j)$ for $j\in[i]$. The computation of each ${\mathbf{v}}_i$ needs ${\mathcal{O}}((d-2)N)$ flops, where $N=\prod_{j\in[d]}$. Hence each step of AMM for best rank one approximation is ${\mathcal{O}}(d(d-2)N)$. So for $d=3$ and $n_1\approx n_2\approx n_3$ the complexity is ${\mathcal{O}}(n^3)$, which is the same complexity as above with $r=1$. Fixed points of AMM and Newton method {#S:fixpt} ===================================== Consider the AMM as described in §\[sub:gendp\]. Assume that the sequence $\psi^{(l)}, l\in{\mathbb{N}}$ converges to a point $\phi\in\Psi$. Then $\phi$ is a fixed point of the map: $$\label{defFPsi} \tilde{\mathbf{F}}:\Psi\to \Psi, \quad \tilde{\mathbf{F}}=(\tilde F_1,\ldots,\tilde F_d),\; \tilde F_i:\Psi\to \Psi_i,\; \tilde F_i(\psi)=\psi_i^\star(\hat\psi_i),\; \psi=(\psi_i,\hat\psi_i),\; i\in[d].$$ In general, the map $\tilde {\mathbf{F}}$ is a multivalued map, since the maximum given in may be achieved at a number of points denoted by $\psi_i^\star(\hat\psi_i)$. In what follows we assume: \[fixpass\] The AMM converges to a fixed point $\phi$ of $\tilde {\mathbf{F}}$ i.e. $\tilde{\mathbf{F}}(\phi)=\phi$, such that the following conditions hold: 1. There is a connected open neighborhood $O\subset \Psi$ such that $\tilde{\mathbf{F}}:O\to O$ is one valued map. 2. $\tilde{\mathbf{F}}$ is a contraction on $O$ with respect to some norm on $O$. 3. $\tilde{\mathbf{F}}\in {\mathrm{C}}^2(O)$, i.e. $\tilde{\mathbf{F}}$ has two continuous partial derivatives in $O$. 4. $O$ is diffeomorphic to an open subset in ${\mathbb{R}}^L$. That is, there exists a smooth one-to-one map $H:O\to {\mathbb{R}}^L$ such that the Jacobian $D(H)$ is invertible at each point $\psi\in O$. Assume that the conditions of Assumption \[fixpass\] hold. Then the map $\tilde {\mathbf{F}}: O\to O$ can be represented as $${\mathbf{F}}:O_1\to O_1, \quad {\mathbf{F}}=H\circ \tilde {\mathbf{F}}\circ H^{-1}, \quad O_1=H(O).$$ Hence to find a fixed point of $\tilde {\mathbf{F}}$ in $O$ it is enough to find a fixed point of ${\mathbf{F}}$ in $O_1$. A fixed point of ${\mathbf{F}}$ is a zero point of the system $$\label{zeroGx} {\mathbf{G}}({\mathbf{x}})={\mathbf{0}}, \quad G({\mathbf{x}}):={\mathbf{x}}-{\mathbf{F}}({\mathbf{x}}).$$ To find a zero of ${\mathbf{G}}$ we use the standard Newton method. In this paper we propose new Newton methods. We make a few iterations of AMM and switch to a Newton method assuming that the conditions of Assumption \[fixpass\] hold as explained above. A fixed point of the map $\tilde{\mathbf{F}}$ for best rank one approximation induces a fixed point of map ${\mathbf{F}}: {\mathbb{R}}^{\mathbf{n}}\to {\mathbb{R}}^{\mathbf{n}}$ [@FMPS13 Lemma 2]. Then the corresponding Newton method to find a zero of ${\mathbf{G}}$ is straightforward to state and implement, as explained in the next subsection. This Newton method was given in [@FMPS11 §5]. See also Zhang-Golub [@ZhaG01] for a different Newton method for best $(1,1,1)$ approximation. Let $\tilde{\mathbf{F}}:{\mathop{\mathrm{Gr}}\nolimits}({\mathbf{r}},{\mathbf{n}})\to {\mathop{\mathrm{Gr}}\nolimits}({\mathbf{r}},{\mathbf{n}})$ be the induced map AMM. Each ${\mathop{\mathrm{Gr}}\nolimits}(r,{\mathbb{R}}^n)$ can be decomposed as a compact manifold to a finite number of charts as explained in §\[S:bestbrapprox\]. These charts induce standard charts of ${\mathop{\mathrm{Gr}}\nolimits}({\mathbf{r}},{\mathbf{n}})$. After a few AMM iterations we assume that the neighborhood $O$ of the fixed point of $\tilde{\mathbf{F}}$ lies in one the charts of ${\mathop{\mathrm{Gr}}\nolimits}({\mathbf{r}},{\mathbf{n}})$. We then construct the corresponding map ${\mathbf{F}}$ in this chart. Next we apply the standard Newton method to ${\mathbf{G}}$. The papers by Eldén-Savas [@ES09] and Savas-Lim [@SL10] discuss Newton and quasi-Newton methods for $(r_1,r_2,r_3)$ approximation of $3$-tensors using the concepts of differential geometry. Newton method for best rank one approximation {#sub:newtboneappr} --------------------------------------------- Let ${\mathcal{T}}\in{\mathbb{R}}^{{\mathbf{n}}}\setminus \{0\}$. Define: $$\begin{aligned} \notag &&\Psi_i={\mathbb{R}}^{n_i},\; i\in [d],\quad \Psi={\mathbb{R}}^{n_1}\times \cdots \times {\mathbb{R}}^{n_d}, \quad \psi=({\mathbf{x}}_1,\ldots,{\mathbf{x}}_d)\in \Psi,\\ &&f_{\mathcal{T}}:\Psi\to {\mathbb{R}}, \quad f_{\mathcal{T}}(\psi)={\mathcal{T}}\times (\otimes_{j\in[d]}{\mathbf{x}}_j),\\ &&{\mathbf{F}}=(F_1,\ldots,F_d):\Psi\to \Psi, \quad F_i(\psi)={\mathcal{T}}\times (\otimes_{j\in[d]\setminus\{i\}}{\mathbf{x}}_j), \quad i\in [d].\label{defbFfcT}\end{aligned}$$ Recall the results of §\[sub:singvten\]: Any critical point of $f_{\mathcal{T}}|{\mathrm{S}}({\mathbf{n}})$ satisfies . Suppose we start the AMM with $\psi^{(0)}=({\mathbf{x}}_1^{(0)},\ldots,{\mathbf{x}}^{(0)}_d)\in{\mathrm{S}}({\mathbf{n}})$ such that $f_{\mathcal{T}}(\psi^{(0)})\ne 0$. Then it is straightforward to see that $f_{\mathcal{T}}(\psi^{(l)})>0$ for $l\in{\mathbb{N}}$. Assume that $\lim_{l\to\infty}\psi^{(l)}=\omega=({\mathbf{u}}_1,\ldots,{\mathbf{u}}_d) \in{\mathrm{S}}({\mathbf{n}})$. Then $\omega$ is the singular tuple of ${\mathcal{T}}$ satisfying . Clearly, $\lambda=f_{\mathcal{T}}(\omega)>0$. Let $$\label{defnorphi} \phi=({\mathbf{y}}_1,\ldots,{\mathbf{y}}_d):=\lambda^{-\frac{1}{d-2}}\omega=\lambda^{-\frac{1}{d-2}}({\mathbf{u}}_1,\ldots,{\mathbf{u}}_d).$$ Then $\phi$ is a fixed point of ${\mathbf{F}}$. Our Newton algorithm for finding the fixed point $\phi$ of ${\mathbf{F}}$ corresponding to a fixed point $\omega$ of AMM is as follows. We do a number of iterations of AMM to obtain $\psi^{(m)}$. Then we renormalize $\psi^{(m)}$ according to : $$\label{phi0for} \phi_0:=(f_{\mathcal{T}}(\psi^{(m)})^{-\frac{1}{d-2}}\psi^{(m)}.$$ Let $D{\mathbf{F}}(\psi)$ denote the Jacobian of ${\mathbf{F}}$ at $\psi$, i.e. the matrix of partial derivatives of ${\mathbf{F}}$ at $\psi$. Then we perform Newton iterations of the form: $$\label{Newtitomeg} \phi^{(l)}=\phi^{(l-1)} -(I-D{\mathbf{F}}(\phi^{(l-1)}))^{-1}(\phi^{(l-1)}-{\mathbf{F}}(\phi^{(l-1)})), \quad l\in{\mathbb{N}}.$$ After performing a number of Newton iterations we obtain $\phi^{(m')}=({\mathbf{z}}_1,\ldots,{\mathbf{z}}_d)$ which is an approximation of $\phi$. We then renormalize each ${\mathbf{z}}_i$ to obtain $\omega^{(m')}:=(\frac{1}{\|{\mathbf{z}}_1\|}{\mathbf{z}}_1,\ldots,\frac{1}{\|{\mathbf{z}}_d\|}{\mathbf{z}}_d)$ which is an approximation to the fixed point $\omega$. We call this Newton method *Newton-1*. We now give the explicit formulas for $3$-tensors, where $n_1=m,n_2=n,n_3=l$. First $$\label{defmapF} {\mathbf{F}}({\mathbf{u}},{\mathbf{v}},{\mathbf{w}}):=({\mathcal{T}}\times({\mathbf{v}}\otimes{\mathbf{w}}),{\mathcal{T}}\times({\mathbf{u}}\otimes{\mathbf{w}}),{\mathcal{T}}\times({\mathbf{u}}\otimes{\mathbf{v}})), \quad {\mathbf{G}}:=({\mathbf{u}},{\mathbf{v}},{\mathbf{w}})-{\mathbf{F}}({\mathbf{u}},{\mathbf{v}},{\mathbf{w}}).$$ Then $$\label{JacDF} D{\mathbf{G}}({\mathbf{u}},{\mathbf{v}},{\mathbf{w}})=\left[\begin{array}{ccc}I_m&-{\mathcal{T}}\times {\mathbf{w}}&-{\mathcal{T}}\times{\mathbf{v}}\\-({\mathcal{T}}\times{\mathbf{w}}){^\top}&I_n&-{\mathcal{T}}\times{\mathbf{u}}\\ -({\mathcal{T}}\times {\mathbf{v}}){^\top}&-({\mathcal{T}}\times {\mathbf{u}}){^\top}&I_l\end{array}\right].$$ Hence Newton-1 iteration is given by the formula $$({\mathbf{u}}_{i+1},{\mathbf{v}}_{i+1},{\mathbf{w}}_{i+1})= ({\mathbf{u}}_{i},{\mathbf{v}}_{i},{\mathbf{w}}_i) -(D{\mathbf{G}}({\mathbf{u}}_i,{\mathbf{v}}_i,{\mathbf{w}}_i))^{-1}{\mathbf{G}}({\mathbf{u}}_i,{\mathbf{v}}_i,{\mathbf{w}}_i),$$ for $i=0,1,\ldots,$. Here we abuse notation by viewing $({\mathbf{u}},{\mathbf{v}},{\mathbf{w}})$ as a column vector $({\mathbf{u}}{^\top},{\mathbf{v}}{^\top},{\mathbf{w}}{^\top}){^\top}\in {\mathbb{C}}^{m+n+l}$. Numerically, to find $(D{\mathbf{G}}({\mathbf{u}}_i,{\mathbf{v}}_i,{\mathbf{w}}_i))^{-1}{\mathbf{G}}({\mathbf{u}}_i,{\mathbf{v}}_i,{\mathbf{w}}_i)$ one solves the linear system $$(D{\mathbf{G}}({\mathbf{u}}_i,{\mathbf{v}}_i,{\mathbf{w}}_i))({\mathbf{x}},{\mathbf{y}},{\mathbf{z}})={\mathbf{G}}({\mathbf{u}}_i,{\mathbf{v}}_i,{\mathbf{w}}_i).$$ The final vector $({\mathbf{u}}_j,{\mathbf{v}}_j,{\mathbf{w}}_j)$ of Newton-1 iterations is followed by a scaling to vectors of unit length ${\mathbf{x}}_j=\frac{1}{\|{\mathbf{u}}_j\|}{\mathbf{u}}_j, {\mathbf{y}}_j=\frac{1}{\|{\mathbf{v}}_j\|}{\mathbf{v}}_j, {\mathbf{z}}_j=\frac{1}{\|{\mathbf{w}}_j\|}{\mathbf{w}}_j$. We now discuss the complexity of Newton-1 method for $d=3$. Assuming that $m\approx n\approx l$ we deduce that the computation of the matrix $D{\mathbf{G}}$ is ${\mathcal{O}}(n^3)$. As the dimension of $D{\mathbf{G}}$ is $m+n+l$ it follows that the complexity of each iteration of Newton-1 method is ${\mathcal{O}}(n^3)$. Newton method for best ${\mathbf{r}}$-approximation {#S:bestbrapprox} =================================================== Recall that an $r$-dimensional subspace ${\mathbf{U}}\in {\mathop{\mathrm{Gr}}\nolimits}(r,{\mathbb{R}}^n)$ is given by a matrix $U=[u_{ij}]_{i,j=1}^{n,r}\in{\mathbb{R}}^{n\times r}$ of rank $r$. In particular there is a subset $\alpha\subset [n]$ of cardinality $r$ so that $\det U[\alpha,[r]]\ne 0$. Here $\alpha=(\alpha_1,\ldots,\alpha_r), 1\le \alpha_1<\ldots <\alpha_r\le n$. So $U[\alpha,[r]]:=[u_{\alpha_i j}]_{i,j=1}^r\in {\mathbf{GL}}(r,{\mathbb{R}})$, (the group of invertible matrices). Clearly, $V:=U U[\alpha,[r]]^{-1}$ represents another basis in ${\mathbf{U}}$. Note that $V[\alpha,[r]]=I_r$. Hence the set of all $V\in {\mathbb{R}}^{n\times r}$ with the condition: $V[\alpha,[r]]=I_r$ represent an open cell in ${\mathop{\mathrm{Gr}}\nolimits}(r,n)$ of dimension $r(n-r)$ denoted by ${\mathop{\mathrm{Gr}}\nolimits}(r,{\mathbb{R}}^n)(\alpha)$. (The number of free parameters in all such $V$’s is $(n-r)r$.) Assume for simplicity of exposition that $\alpha=[r]$. Note that $V_0=\left[\begin{array}{c}I_r\\0\end{array}\right] \in {\mathop{\mathrm{Gr}}\nolimits}(r,{\mathbb{R}}^n)([r])$. Let ${\mathbf{e}}_i=(\delta_{1i},\ldots,\delta_{ni}){^\top}\in {\mathbb{R}}^n, i=1,\ldots,n$ be the standard basis in ${\mathbb{R}}^n$. So ${\mathbf{U}}_0={\mathrm{span}}({\mathbf{e}}_1,\ldots,{\mathbf{e}}_r)\in {\mathop{\mathrm{Gr}}\nolimits}(r,{\mathbb{R}}^n)([r])$, and $V_0$ is the unique representative of ${\mathbf{U}}_0$. Note that ${\mathbf{U}}_0^\perp$, the orthogonal complement of ${\mathbf{U}}_0$, is ${\mathrm{span}}({\mathbf{e}}_{r+1},\ldots,{\mathbf{e}}_n)$. It is straightforward to see that ${\mathbf{V}}\in {\mathop{\mathrm{Gr}}\nolimits}(r,{\mathbb{R}}^n)([r])$ if and only if ${\mathbf{V}}\cap {\mathbf{U}}_0^\perp=\{{\mathbf{0}}\}$. The following definition is a geometric generalization of ${\mathop{\mathrm{Gr}}\nolimits}(r,{\mathbb{R}}^n)(\alpha)$: $$\label{openUcell} {\mathop{\mathrm{Gr}}\nolimits}(r,{\mathbb{R}}^n)({\mathbf{U}}):=\{{\mathbf{V}}\in {\mathop{\mathrm{Gr}}\nolimits}(r,{\mathbb{R}}^n),\; {\mathbf{V}}\cap {\mathbf{U}}^\perp=\{{\mathbf{0}}\}\} \textrm{ for } {\mathbf{U}}\in{\mathop{\mathrm{Gr}}\nolimits}(r,{\mathbb{R}}^n).$$ A basis for ${\mathop{\mathrm{Gr}}\nolimits}(r,{\mathbb{R}}^n)({\mathbf{U}})$, which can be identified the tangent hyperplane $T_{\mathbf{U}}{\mathop{\mathrm{Gr}}\nolimits}(r,{\mathbb{R}}^n)$, can be represented as $\oplus^r {\mathbf{U}}^\perp$: Let ${\mathbf{u}}_1,\ldots,{\mathbf{u}}_r$ and ${\mathbf{u}}_{r+1},\ldots,{\mathbf{u}}_n$ be orthonormal bases of ${\mathbf{U}}$ and ${\mathbf{U}}^\perp$ respectively Then each subspace ${\mathbf{V}}\in {\mathop{\mathrm{Gr}}\nolimits}(r,{\mathbb{R}}^n)({\mathbf{U}})$ has a unique basis of the form ${\mathbf{u}}_1+{\mathbf{x}}_1,\ldots,{\mathbf{u}}_r+{\mathbf{x}}_r$ for unique ${\mathbf{x}}_1,\ldots,{\mathbf{x}}_r\in {\mathbf{U}}^\perp$. Equivalently, every matrix $X\in {\mathbb{R}}^{(n-r)\times r}$ induces a unique subspace ${\mathbf{V}}$ using the equality $$\label{defx1xr} [{\mathbf{x}}_1\;\ldots\;{\mathbf{x}}_r]=[{\mathbf{u}}_1\;\ldots\;{\mathbf{u}}_{n-r}] X \textrm{ for each } X\in{\mathbb{R}}^{(n-r)\times r}.$$ Recall the results of §\[sub:apprprb\]. Let $\underline{U}=(U_1,\ldots,U_d)\in{\mathop{\mathrm{Gr}}\nolimits}({\mathbf{r}},{\mathbf{n}})$. Then $$\label{deftilFGrn} \tilde{\mathbf{F}}=(\tilde F_1,\ldots, \tilde F_d):{\mathop{\mathrm{Gr}}\nolimits}({\mathbf{r}},{\mathbf{n}})\to{\mathop{\mathrm{Gr}}\nolimits}({\mathbf{r}},{\mathbf{n}}), \quad \tilde F_i({\underline U})={\mathbf{U}}_i({\underline U}_i), \; i\in[d],$$ where ${\mathbf{U}}_i({\underline U}_i)$ a subspace spanned by the first $r_i$ eigenvectors of $A_i({\underline U}_i)$. Assume that $\tilde{\mathbf{F}}$ is one valued at ${\underline U}$, i.e. holds. Then it is straightforward to show that $\tilde{\mathbf{F}}$ is smooth (real analytic) in neighborhood of $\underline{U}$. Assume next that there exists a neighborhood $O$ of $\underline{U}$ such that $$\label{regFcond} O\subset {\mathop{\mathrm{Gr}}\nolimits}({\mathbf{r}},{\mathbf{n}})({\underline U}):={\mathop{\mathrm{Gr}}\nolimits}(r_1,{\mathbb{R}}^{n_1})({\mathbf{U}}_1)\times \cdots \times{\mathop{\mathrm{Gr}}\nolimits}(r_d,{\mathbb{R}}^{n_d})({\mathbf{U}}_d), \quad {\underline U}=({\mathbf{U}}_1,\ldots,{\mathbf{U}}_d),$$ such that the conditions *1-3* of Assumption \[fixpass\] hold. Observe next that ${\mathop{\mathrm{Gr}}\nolimits}({\mathbf{r}},{\mathbf{n}})({\underline U})$ is diffeomorphic to $${\mathbb{R}}^L:= {\mathbb{R}}^{(n_1-r_1)\times r_1} \ldots \times {\mathbb{R}}^{(n_d-r_d)\times r_d}, \quad L=\sum_{i\in[d]} (n_i-r_i)r_i$$ We say that $\tilde{\mathbf{F}}$ is *regular* at $\underline{U}$ if in addition to the above condition the matrix $I-D\tilde {\mathbf{F}}({\underline U})$ is invertible. We can view $X=[X_1\;\ldots \;X_d]\in {\mathbb{R}}^{(n_1-r_1)\times r_1} \ldots \times {\mathbb{R}}^{(n_d-r_d)\times r_d}$. Then $\tilde {\mathbf{F}}$ on $O$ can be viewed as $$\label{tildebFre} {\mathbf{F}}:O_1\to O_1, \quad O_1\subset {\mathbb{R}}^L,\quad {\mathbf{F}}(X)=[F_1(X),\ldots, F_d(X)],\; X=[X_1\;\ldots \;X_d]\in{\mathbb{R}}^L.$$ Note that $F_i(X)$ does not depend on $X_i$ for each $i\in[d]$. In our numerical simulations we first do a small number of AMM and then switch to Newton method given by . Observe that ${\underline U}$ corresponds to $X({\underline U})=[X_1({\underline U}),\ldots,X_d({\underline U})]$. When referring to we identify $X=[X_1,\ldots,X_d]$ with $\phi=(\phi_1,\ldots,\phi_d)$ and no ambiguity will arise. Note that the case ${\mathbf{r}}={\mathbf{1}}_d$ corresponds to best rank one approximation. The above Newton method in this case is different from Newton method given in §\[sub:newtboneappr\]. A closed formula for $D{\mathbf{F}}(X({\underline U}))$ {#S:forDF} ======================================================= Recall the definitions and results of §\[sub:apprprb\]. Given ${\underline U}$ we compute $\tilde F_i({\underline U})={\mathbf{U}}_i({\underline U}_i)$, which is the subspace spanned by the first $r_i$ eigenvectors of $A_i({\underline U}_i)$, which is given by , for $i\in[d]$. Assume that holds. Let $$\begin{aligned} \notag &&{\mathbf{U}}_{i}({\underline U}_i)={\mathrm{span}}({\mathbf{v}}_{1,i},\ldots,{\mathbf{v}}_{r_i,n_i}), \quad {\mathbf{U}}_i({\underline U}_i)^\perp={\mathrm{span}}({\mathbf{v}}_{r_i+1,i},\ldots, {\mathbf{v}}_{n_i,i}),\\ &&{\mathbf{v}}_{j,i}{^\top}{\mathbf{v}}_{k,i}=\delta_{jk},\;j,k\in [n_i], \quad i\in [d].\label{basistFUi}\end{aligned}$$ With each $X=[X_1,\ldots,X_d]\in{\mathbb{R}}^L$ we associate the following point $({\mathbf{W}}_1,\ldots,{\mathbf{W}}_d)\in{\mathop{\mathrm{Gr}}\nolimits}({\mathbf{r}},{\mathbf{n}})({\underline U})$. Suppose that $X_i=[x_{pq,i}]\in{\mathbb{R}}^{(n_i-r_i)\times r_i}$. Then ${\mathbf{W}}_i$ has a basis of the form $${\mathbf{u}}_{j_i,i}+\sum_{k_i\in [n_i-r_i]}x_{k_ij_i,i}{\mathbf{u}}_{r_i+k_i,i}, \quad j_i\in [r_i].$$ One can use the following notation for a basis ${\mathbf{w}}_{1,i},\ldots,{\mathbf{w}}_{r_i,i}$, written as a vector with vector coordinates $[{\mathbf{w}}_{1,i}\cdots{\mathbf{w}}_{r_i,i}]$: $$\label{basisWi} [{\mathbf{w}}_{1,i}\cdots{\mathbf{w}}_{r_i,i}]=[{\mathbf{u}}_{1,i}\cdots{\mathbf{u}}_{r_i,i}] + [{\mathbf{u}}_{r_i+1,i}\cdots{\mathbf{u}}_{n_i,i}]X_i, \quad i\in[d].$$ Note that to the point ${\underline U}\in {\mathop{\mathrm{Gr}}\nolimits}({\mathbf{r}},{\mathbf{n}})({\underline U})$ corresponds the point $X=0$. Since ${\mathbf{u}}_{1,i},\ldots,{\mathbf{u}}_{n_i,i}$ is a basis in ${\mathbb{R}}^{n_i}$ it follows that $$\begin{aligned} \notag &&[{\mathbf{v}}_{1,i}\cdots {\mathbf{v}}_{r_i,i}]=[{\mathbf{u}}_{1,i}\cdots {\mathbf{u}}_{r_i,i}]Y_{i,0}+[{\mathbf{u}}_{r_i+1,i},\ldots,{\mathbf{u}}_{n_i,i}]X_{i,0}=[{\mathbf{u}}_{1,i}\cdots {\mathbf{u}}_{n_i,i}]Z_{i,0}, \\ && Y_{i,0}\in {\mathbb{R}}^{r_i\times r_i}, \;X_{i,0}\in {\mathbb{R}}^{(n_i-r_i)\times r_i},\;Z_{i,0}=\left[\begin{array}{c}Y_{i,0}\\X_{i,0}\end{array}\right]\in {\mathbb{R}}^{n_i\times r_i}, \textrm{ for }i\in [d].\label{vforuXY}\end{aligned}$$ View $[{\mathbf{v}}_{1,i}\cdots {\mathbf{v}}_{r_i,i}],[{\mathbf{u}}_{1,i}\cdots {\mathbf{u}}_{n_i,i}]$ as $n_i\times r_i$ and $n_i\times n_i$ matrices with orthonormal columns. Then $$\label{forZi0} Z_{i,0}=[{\mathbf{u}}_{1,i}\cdots {\mathbf{u}}_{n_i,i}]{^\top}[{\mathbf{v}}_{1,i}\cdots {\mathbf{v}}_{r_i,i}], \quad i\in [d].$$ The assumption that $\tilde {\mathbf{F}}:O\to O$ implies that $Y_{i,0}$ is an invertible matrix. Hence $[{\mathbf{v}}_{1,i}\cdots {\mathbf{v}}_{r_i,i}]Y_{i,0}^{-1}$ is also a basis in ${\mathbf{U}}_i({\underline U}_i)$. Clearly, $$[{\mathbf{v}}_{1,i}\cdots {\mathbf{v}}_{r_i,i}]Y_{i,0}^{-1}=[{\mathbf{u}}_{1,i}\cdots {\mathbf{u}}_{r_i,n_i}]+[{\mathbf{u}}_{r_i+1,i},\ldots,{\mathbf{u}}_{n_i,i}]X_{i,0}Y_{i,0}^{-1}, \quad i\in [d].$$ Hence $\tilde{\mathbf{F}}({\underline U})$ corresponds to ${\mathbf{F}}(0)$ where $$\label{exprebF0} F_i(0)=X_{i,0}Y_{i,0}^{-1}, \;i\in [d], \quad {\mathbf{F}}(0)=(F_1(0),\ldots,F_d(0)).$$ We now find the matrix of derivatives. So $D_iF_j\in{\mathbb{R}}^{((n_i-r_i)r_i\times ((n_j-r_j)r_j}$ is the partial derivative matrix of $(n_j-r_j)r_j$ coordinates of $F_j$ with respect to $(n_i-r_i)r_i$ the coordinates of ${\mathbf{U}}_i$ viewed as the matrix $\left[\begin{array}{c}I_{r_i}\\G_i\end{array}\right]$. So $G_i\in {\mathbb{R}}^{(n_i-r_i)\times r_i}$ are the variables representing the subspace ${\mathbf{U}}_i$. Observe first that $D_iF_i=0$ just as in Newton method for best rank one approximation in §\[sub:newtboneappr\]. Let us now find $D_i F_j(0)$. Recall that $D_i F_j(0)$ is a matrix of size $(n_i-r_i)r_i\times (n_j-r_j)r_j$. The entries of $D_i F_j(0)$ are indexed by $((p,q),(s,t))$ as follows: The entries of $G_i=[g_{pq,i}]\in {\mathbb{R}}^{(n_i-r_i)\times r_i}$ are viewed as $(n_i-r_i)r_i$ variables, and are indexed by $(p,q)$, where $p\in[n_i-r_i],q\in[r_i]$. $F_j$ is viewed as a matrix $G_j\in {\mathbb{R}}^{(n_j-r_j)\times r_j}$.The entries of $F_j$ are indexed by $(s,t)$, where $s\in[n_j-r_j]$ and $t\in[r_j]$. Since ${\underline U}\in{\mathop{\mathrm{Gr}}\nolimits}({\mathbf{r}},{\mathbf{n}})({\underline U})$ corresponds to $0\in{\mathbb{R}}^L$ we denote by $A_j(0)$ the matrix $A_j({\underline U}_j)$ for $j\in[d]$. We now give the formula for $\frac{\partial A_j(0)}{\partial g_{pq,i}}$. This is done by noting that we vary ${\mathbf{U}}_{i}$ by changing the orthonormal basis ${\mathbf{u}}_{1,i},\ldots,{\mathbf{u}}_{r_i,i}$ up to the first perturbation with respect to the real variable $\varepsilon$ to $$\hat{\mathbf{u}}_{1,i}={\mathbf{u}}_{1,i},\ldots,\hat {\mathbf{u}}_{q-1,i}= {\mathbf{u}}_{q-1,i},\hat{\mathbf{u}}_{q,i}={\mathbf{u}}_{q,i}+\varepsilon{\mathbf{u}}_{r_i+p,i}, \hat{\mathbf{u}}_{q+1,i}={\mathbf{u}}_{q+1,i},\ldots,\hat{\mathbf{u}}_{r_i,i}={\mathbf{u}}_{r_i,i}$$ We denote the subspace spanned by these vectors as ${\mathbf{U}}_{i}(\varepsilon,p,q)$. That is, we change only the $q$ orthonormal vector of the standard basis in ${\mathbf{U}}_{i}$, for $q=1,\ldots,r_i$. The new basis is an orthogonal basis, and up order $\varepsilon$, the vector $ {\mathbf{u}}_{q,i}+\varepsilon{\mathbf{u}}_{r_i+p,i}$ is also of length $1$. Let ${\underline U}(\varepsilon,i,p,q)=({\mathbf{U}}_1,\ldots,{\mathbf{U}}_{i-1},{\mathbf{U}}_i(\varepsilon,p,q),{\mathbf{U}}_{i+1},\ldots,{\mathbf{U}}_d)$. Then ${\underline U}(\varepsilon,i,p,q)_j$ is obtained by dropping the subspace ${\mathbf{U}}_j$ from ${\underline U}(\varepsilon,i,p,q)$. We will show that $$\label{defAjvareps} A_j({\underline U}(\varepsilon,i,p,q)_j)=A_j({\underline U}_j)+\varepsilon B_{j,i,p,q}+O(\varepsilon^2).$$ We now give a formula to compute $B_{j,i,p,q}$. Assume that $i,j\in [d]$ is a pair of different integers. Let $J$ be a set of $d-2$ pairs $\cup_{l\in [d]\setminus\{i,j\}}\{(k_l,l)\}$, where $k_l\in[r_l]$. Denote by ${\mathcal{J}}_{ij}$ the set of all such $J$’s. Note that ${\mathcal{J}}_{ij}={\mathcal{J}}_{ji}$. Furthermore, the number of elements in ${\mathcal{J}}_{ij}$ is $R_{ij}=\prod_{l\in[d]\setminus\{i,j\}} r_l$. We now introduce the following matrices $$\label{defCijk} C_{ij}(J):={\mathcal{T}}\times (\otimes_{(k,l)\in J}{\mathbf{u}}_{k,l})\in {\mathbb{R}}^{n_i\times n_j}, \quad J\in{\mathcal{J}}_{ij}.$$ Note that $C_{ij}(J)=C_{ji}(J){^\top}$. \[AjBjpqfor\] Let $i,j\in [d],i\ne j$. Assume that $p\in[n_i-r_i],q\in[r_i]$. Then holds. Furthermore $$\begin{aligned} \label{newforAj} &&A_j({\underline U}_j)=\sum_{k\in[r_i], J\in{\mathcal{J}}_{ji}} (C_{ji}(J){\mathbf{u}}_{k,i})(C_{ji}(J){\mathbf{u}}_{k,i}){^\top},\\ &&B_{j,i,p,q}=\sum_{J\in{\mathcal{J}}_{ji}} (C_{ji}(J){\mathbf{u}}_{k_i+p,i})(C_{ji}(J){\mathbf{u}}_{q,i}){^\top}+(C_{ji}(J){\mathbf{u}}_{q,i})(C_{ji}(J){\mathbf{u}}_{k_i+p,i}){^\top}\label{forBjipq}\end{aligned}$$ [**Proof. **]{}The identity of is just a restatement of . To compute $A_j({\underline U}(\varepsilon,i,p,q)_j)$ use by replacing $u_{k,i}$ with $\hat u_{k,i}$ for $k\in[n_i]$. Deduce first and then .[ $\Box$\ ]{} Recall that ${\mathbf{v}}_{1,j},\ldots,{\mathbf{v}}_{r_j,j}$ is an orthonormal basis of ${\mathbf{U}}_j({\underline U}_j)$, and these vectors are the eigenvectors $A_j({\underline U}_j)$ corresponding its first $r_j$ eigenvalues. Let ${\mathbf{v}}_{r_j+1,j},\ldots,{\mathbf{v}}_{n_j,i}$ be the last $n_j-r_j$ orthonormal eigenvectors of $A_j({\underline U}_j)$. We now find the first perturbation of the first $r_i$ eigenvectors for the matrix $A_j({\underline U}_j)+\varepsilon B_{j,i,p,q}$. Assume first, for simplicity of exposition, that each $\lambda_k(A_j({\underline U}_j))$ is simple for $k\in [r_j]$: Then it is known, e.g. [@Fri15 Chapter 4, §19, (4.19.2)]: $$\label{pertforxj2} {\mathbf{v}}_{k,j}(\varepsilon,i,p,q)={\mathbf{v}}_{k,j}+\varepsilon (\lambda_{k}(A_j({\underline U}_j)) I_{n_j}-A_j({\underline U}_j))^\dagger B_{j,i,p,q}{\mathbf{v}}_{k,j} +O(\varepsilon^2), k\in[r_j].$$ The assumption that $\lambda_{k}(A_j({\underline U}_j))$ is a simple eigenvalue for $k\in[r_j]$ yields $$(\lambda_{k}(A_j({\underline U}_i))I_{n_j}-A_j({\underline U}_j))^\dagger{\mathbf{y}}=\sum_{l\in[n_j] \setminus\{k\}}\frac{1}{\lambda_{k}(A_j({\underline U}_j))-\lambda_{l}(A_j({\underline U}_j))} ({\mathbf{v}}_{l,j}{^\top}{\mathbf{y}}){\mathbf{v}}_{l,j},$$ for ${\mathbf{y}}\in{\mathbb{R}}^{n_j}$. Since we are interested in a basis of ${\mathbf{U}}_j({\underline U}(\varepsilon,i,p,q)_j)$ up to the order of $\varepsilon$ we can assume that this basis is of the form $$\tilde {\mathbf{v}}_{k,j}(\varepsilon,i,p,q)={\mathbf{v}}_{k,j}+\varepsilon{\mathbf{w}}_{k,j}(i,p,q), \quad {\mathbf{w}}_{k,j}(i,p,q)\in {\mathrm{span}}({\mathbf{v}}_{r_j+1,j}\ldots,{\mathbf{v}}_{n_j,j}).$$ Hence $$\begin{aligned} \notag &&{\mathbf{w}}_{k,j}(i,p,q)=\sum_{l\in[n_j]\setminus [r_j]}\frac{1}{\lambda_{k}(A_j({\underline U}_j))-\lambda_{l}(A_j({\underline U}_j))} ({\mathbf{v}}_{l,j}{^\top}{\mathbf{c}}_{k,j,i,p,q}){\mathbf{v}}_{l,j},\\ &&{\mathbf{c}}_{k,j,i,p,q}:= B_{j,i,p,q}{\mathbf{v}}_{k,j}.\label{pertforbasU2}\end{aligned}$$ Note that the assumption yields that ${\mathbf{w}}_{k,j}$ is well defined for $k\in[r_j]$. Let $$\begin{aligned} &&W_j(i,p,q)=[{\mathbf{w}}_{1,j}(i,p,q)\cdots{\mathbf{w}}_{r_j,j}(i,p,q)]=\left[\begin{array}{c}V_j(i,p,q)\\U_j(i,p,q)\end{array}\right],\\ &&V_j(i,p,q)\in{\mathbb{R}}^{r_j\times r_j},\quad U_j(i,p,q)\in{\mathbb{R}}^{(n_j-r_j)\times r_j}.\end{aligned}$$ Up to the order of $\varepsilon$ we have that a basis of ${\mathbf{U}}_{j}({\underline U}(\varepsilon,i,p,q)_j)$ is given by columns of matrix $Z_{j,0}+\varepsilon W_j(i,p,q)=\left[\begin{array}{c}Y_{j,0}+\varepsilon V_j(i,p,q)\\X_{j,0} +\varepsilon U_j(i,p,q)\end{array}\right]$. Note $$(Z_{j,0}+\varepsilon W_j(i,p,q))(Y_{j,0}+\varepsilon V_j(i,p,q))^{-1}=\left[\begin{array}{c} I_{r_j}\\ (X_{j,0}+\varepsilon U_j(i,p,q))(Y_{j,0}+\varepsilon V_j(i,p,q))^{-1}\end{array}\right].$$ Observe next $$\begin{aligned} &&Y_{j,0}+\varepsilon V_j(i,p,q)=Y_{j,0}(I_{r_j}+\varepsilon Y_{j,0}^{-1} V_j(i,p,q)), \\ &&(Y_{j,0}+\varepsilon V_j(i,p,q))^{-1}=(I_{r_j}+\varepsilon Y_{j,0}^{-1} V_j(i,p,q))^{-1}Y_{j,0}^{-1}=\\ &&Y_{j,0}^{-1}-\varepsilon Y_{j,0}^{-1} V_j(i,p,q)Y_{j,0}^{-1}+O(\varepsilon^2),\\ &&(X_{j,0}+\varepsilon U_j(i,p,q))(Y_{j,0}+\varepsilon V_j(i,p,q))^{-1}=\\ &&X_{j,0}Y_{j,0}^{-1}+\varepsilon(U_j(i,p,q) Y_{j,0}^{-1}-X_{j,0}Y_{j,0}^{-1}V_{j}(i,p,q) Y_{j,0}^{-1})+O(\varepsilon^2).\end{aligned}$$ Hence $$\label{D1F2pqfor} \frac{\partial F_j}{\partial g_{pq,i}}(0)=U_j(i,p,q)Y_{j,0}^{-1}-X_{j,0}Y_{j,0}^{-1}V_{j}(i,p,q) Y_{j,0}^{-1}.$$ Thus $D{\mathbf{F}}(0)=[D_iF_J]_{i,j\in[d]}\in{\mathbb{R}}^{L\times L}$. We now make one iteration of Newton method given by for $l=1$, where $\phi^{(0)}=0$: $$\label{Newtmetstrt0} \phi^{(1)}=-(I-D{\mathbf{F}}(0))^{-1}F(0), \quad \phi^{(1)}=[X_{1,1},\ldots,X_{d,1}]\in{\mathbb{R}}^L.$$ Let ${\mathbf{U}}_{i,1}\in{\mathop{\mathrm{Gr}}\nolimits}(r_i,{\mathbb{R}}^{n_i})$ be the subspace represented by the matrix $X_{i,1}$: $$\label{defUi1} {\mathbf{U}}_{i,1}={\mathrm{span}}(\tilde{\mathbf{u}}_{1,i,1},\ldots,\tilde{\mathbf{u}}_{r_i,i,1}),\; [\tilde{\mathbf{u}}_{1,i,1},\ldots,\tilde{\mathbf{u}}_{n_i,i,1}]= [{\mathbf{u}}_{1,i},\ldots,{\mathbf{u}}_{n_i,i}]\left[\begin{array}{c} I_{r_i}\\X_{j,1}\end{array}\right]$$ for $i\in[d]$. Perform the Gram-Schmidt process on $\tilde{\mathbf{u}}_{1,i,1},\ldots,\tilde{\mathbf{u}}_{r_i,i,1}$ to obtain an orthonormal basis ${\mathbf{u}}_{1,i,1},\ldots,{\mathbf{u}}_{r_i,i,1}$ of ${\mathbf{U}}_{i,1}$. Let ${\underline U}:=({\mathbf{U}}_{1,1},\ldots,{\mathbf{U}}_{d,1})$ and repeat the algorithm which is described above. We call this Newton method *Newton-2*. Complexity of Newton-2 {#sub:compNewtrap} ====================== In this section we assume for simplicity that $d=3$, $r_1=r_2=r_3=r$, $n_i\approx n$ for $i\in [3]$. We assume that executed a number of times the AMM for a given ${\mathcal{T}}\in {\mathbb{R}}^{\mathbf{n}}$. So we are given ${\underline U}=({\mathbf{U}}_1,\ldots,{\mathbf{U}}_d)$, and an orthonormal basis ${\mathbf{u}}_{1,i},\ldots,{\mathbf{u}}_{r,i}$ of ${\mathbf{U}}_i$ for $i\in[d]$. First we complete each ${\mathbf{u}}_{1,i},\ldots,{\mathbf{u}}_{r,i}$ to an orthonormal basis ${\mathbf{u}}_{1,i},\ldots,{\mathbf{u}}_{n_i,i}$of ${\mathbb{R}}^{n_i}$, which needs ${\mathcal{O}}(n^3)$ flops. Since $d=3$ we still need only ${\mathcal{O}}(n^3)$ to carry out this completion for each $i\in [3]$. Next we compute the matrices $C_{ij}(J)$. Since $d=3$, we need $n$ flops to compute each entry of $C_{ij}(J)$. Since we have roughly $n^2$ entries, the complexity of computing $C_{ij}(J)$ is ${\mathcal{O}}(n^3)$. As the cardinality of ${\mathcal{J}}_{ij}$ is $r$ we need ${\mathcal{O}}(rn^3)$ flops to compute all $C_{ij}(J)$ for $J\in{\mathcal{J}}_{ij}$. As the number of pairs in $[3]$ is $3$ it follows that the complexity of computing all $C_{ij}(J)$ is ${\mathcal{O}}(rn^3)$. The identity yields that the complexity of computing $A_j({\underline U}_j)$ is ${\mathcal{O}}(r^2n^2)$. Recall next that $A_j({\underline U}_j)$ is $n_j\times n_j$ symmetric positive semi-definite matrix. The complexity of computations of the eigenvalues and the orthonormal eigenvectors of $A_j({\underline U}_j)$ is ${\mathcal{O}}(n^3)$. Hence the complexity of computing ${\underline U}$ is ${\mathcal{O}}(r n^3)$, as we pointed out at the end of §\[sub:companbrap\]. The complexity of computing $B_{j,i,p.q}$ using is ${\mathcal{O}}(r n^2)$. The complexity of computing ${\mathbf{w}}_{k,j}(i,p,q)$, given by is ${\mathcal{O}}(n^2)$. Hence the complexity of computing $W_j(i,p,q)$ is ${\mathcal{O}}(rn^2)$. Therefore the complexity of computing $D_i F_j$ is ${\mathcal{O}}(r^2n^3)$. Since $d=3$, the complexity of computing the matrix $D{\mathbf{F}}(0)$ is also ${\mathcal{O}}(r^2n^3)$. As $D{\mathbf{F}}(0)\in {\mathbb{R}}^{L\times L}$, where $L\approx 3r n$, the complexity of computing $(I-D{\mathbf{F}}(0))^{-1}$ is ${\mathcal{O}}(r^3n^3)$. In summary, the complexity of one step in Newton-2 is ${\mathcal{O}}(r^3 n^3)$. Numerical Results {#S:NumRes} ================= We have implemented a Matlab library tensor decomposition using Tensor Toolbox given by [@KB09]. The performance was measured via the actual CPU-time (seconds) needed to compute. All performance tests have been carried out on a 2.8 GHz Quad-Core Intel Xeon Macintosh computer with 16GB RAM. The performance results are discussed for real data sets of third-order tensors. We worked with a real computer tomography (CT) data set (the so-called MELANIX data set of OsiriX) [@FMPS13]. Our simulation results are averaged over 10 different runs of the each algorithm. In each run, we changed the initial guess, that is, we generated new random start vectors. We always initialized the algorithms by random start vectors, because this is cheaper than the initialization via HOSVD. We note here that for Newton methods our initial guess is the subspaces returned by one iteration of AMM method. All the alternating algorithms have the same stopping criterion where convergence is achieved if one of the two following conditions are met: $iterations > 10;fitchange <0.0001$ is met. All the Newton algorithms have the same stopping criterion where convergence is achieved if one of the two following conditions are met: $iterations > 10;change <\exp(-10)$. Our numerical simulations demonstrate the well known fact that for large size tensors Newton methods are not efficient. Though the Newton methods converge in fewer iterations than alternating methods, the computation associated with the matrix of derivatives (Jacobian) in each iteration is too expensive making alternating maximization methods much more cost effective. Our simulations also demonstrate that our Newton-1 for best rank one approximation is as fast as AMM methods. However our Newton-2 is much slower than alternating methods. We also give a comparison between our Newton-2 and the Newton method based on Grassmannian manifold by [@ES09], abbreviated as Newton-ES. We also observe that for large tensors and large rank approximation two alternating maximization methods, namely MAMM and 2AMMV, seem to outperform the other alternating maximization methods. We would recommend Newton-1 for rank one approximation in case of rank one approximation both for large and small sized tensors. For higher rank approximation we recommend 2AMMV in case of large size tensors and AMM or MAMM in case of small size tensors. Our Newton-2 performs a bit slower than Newton-ES, however we would like to point couple of advantages. Our method can be easily extendable to higher dimensions ( for $d>3$ case) both analytically and numerically compared to Newton-ES. Our method is also highly parallelizable which can bring down the computation time drastically. Computation of $D_{i}F_{j}$ matrices in each iteration contributes to about $50\%$ of the total time, which however can be parallelizable. Finally the number of iterations in Newton-2 is at least $30\%$ less than in Newton-ES. It is not only important to check how fast the different algorithms perform but also what quality they achieve. This was measured by checking the Hilbert-Schmidt norm, abbreviated as HS norm, of the resulting decompositions, which serves as a measure for the quality of the approximation. In general, we can say that the higher the HS norm, the more likely it is that we find a global maximum. Accordingly, we compared the HS norms to say whether the different algorithms converged to the same stationary point. In Figure 4, we show the average HS norms achieved by different algorithms and compared them with the input norm. We observe all the algorithms seem to attain the same local maximum. Best (2,2,2) and rank two approximations {#sub:b222apr} ---------------------------------------- Assume that ${\mathcal{T}}$ is a $3$-tensor of rank three at least and let ${\mathcal{S}}$ be a best $(2,2,2)$-approximation to ${\mathcal{T}}$given by . It is easy to show that ${\mathcal{S}}$ has at least rank $2$. Let ${\mathcal{S}}'=[s_{j_1,j_2,j_3}]\in{\mathbb{R}}^{2\times 2 \times 2}$ be the core tensor corresponding to ${\mathcal{S}}$. Clearly ${\mathrm{rank\;}}{\mathcal{S}}={\mathrm{rank\;}}{\mathcal{S}}'\ge 2$. Recall that a real nonzero $2\times 2 \times 2$ tensor has rank one, two or three [@BK99]. So ${\mathrm{rank\;}}{\mathcal{S}}\in\{2,3\}$. Observe next that if ${\mathrm{rank\;}}{\mathcal{S}}={\mathrm{rank\;}}{\mathcal{S}}'=2$ then ${\mathcal{S}}$ is also a best rank two approximation of ${\mathcal{T}}$. Recall that a best rank two approximation of ${\mathcal{T}}$ may not always exist. In particular where ${\mathrm{rank\;}}{\mathcal{T}}>2$ and the border rank of ${\mathcal{T}}$ is $2$ [@DeSL08]. In all our numerical simulations for best $(2,2,2)$-approximation we performed on random large tensors, the tensor ${\mathcal{S}}'$ had rank two. Note that the probability of $2\times 2\times 2$ tensors, with entries normally distributed with mean $0$ and variance $1$, to have rank $2$ is $\frac{\pi}{4}$ [@Ber13]. Conclusions {#S:Conclusion} =========== We have extended the alternating maximization method (AMM) and modified alternating maximization method (MAMM) given in [@FMPS13] for the computation of best rank one approximation to best ${\mathbf{r}}$-approximations. We have also presented new algorithms such as $2$-alternating maximization method variant (2AMMV) and Newton method for best ${\mathbf{r}}$-approximation (Newton-2). We have provided closed form solutions for computing the $DF$ matrix in Newton-2. We implemented Newton-1 for best rank one approximation [@FMPS11] and Newton-2. From the simulations, we have found out that for rank one approximation of both large and small sized tensors, Newton-1 performed the best. 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