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the transport properties of nonlinear non - equilibrium dynamical systems are far from well - understood@xcite .
consider in particular so - called ratchet systems which are asymmetric periodic potentials where an ensemble of particles experience directed transport@xcite .
the origins of the interest in this lie in considerations about extracting useful work from unbiased noisy fluctuations as seems to happen in biological systems@xcite .
recently attention has been focused on the behavior of deterministic chaotic ratchets@xcite as well as hamiltonian ratchets@xcite .
chaotic systems are defined as those which are sensitively dependent on initial conditions . whether chaotic or not , the behavior of nonlinear systems including the transition from regular to chaotic behavior is in general sensitively dependent on the parameters of the system .
that is , the phase - space structure is usually relatively complicated , consisting of stability islands embedded in chaotic seas , for examples , or of simultaneously co - existing attractors .
this can change significantly as parameters change .
for example , stability islands can merge into each other , or break apart , and the chaotic sea itself may get pinched off or otherwise changed , or attractors can change symmetry or bifurcate .
this means that the transport properties can change dramatically as well .
a few years ago , mateos@xcite considered a specific ratchet model with a periodically forced underdamped particle .
he looked at an ensemble of particles , specifically the velocity for the particles , averaged over time and the entire ensemble .
he showed that this quantity , which is an intuitively reasonable definition of ` the current ' , could be either positive or negative depending on the amplitude @xmath0 of the periodic forcing for the system . at the same time , there exist ranges in @xmath0 where the trajectory of an individual particle displays chaotic dynamics .
mateos conjectured a connection between these two phenomena , specifically that the reversal of current direction was correlated with a bifurcation from chaotic to periodic behavior in the trajectory dynamics .
even though it is unlikely that such a result would be universally valid across all chaotic deterministic ratchets , it would still be extremely useful to have general heuristic rules such as this .
these organizing principles would allow some handle on characterizing the many different kinds of behavior that are possible in such systems .
a later investigation@xcite of the mateos conjecture by barbi and salerno , however , argued that it was not a valid rule even in the specific system considered by mateos .
they presented results showing that it was possible to have current reversals in the absence of bifurcations from periodic to chaotic behavior .
they proposed an alternative origin for the current reversal , suggesting it was related to the different stability properties of the rotating periodic orbits of the system .
these latter results seem fundamentally sensible . however , this paper based its arguments about currents on the behavior of a _ single _ particle as opposed to an ensemble .
this implicitly assumes that the dynamics of the system are ergodic .
this is not true in general for chaotic systems of the type being considered . in particular , there can be extreme dependence of the result on the statistics of the ensemble being considered .
this has been pointed out in earlier studies @xcite which laid out a detailed methodology for understanding transport properties in such a mixed regular and chaotic system . depending on specific parameter value , the particular system under consideration has multiple coexisting periodic or chaotic attractors or a mixture of both .
it is hence appropriate to understand how a probability ensemble might behave in such a system .
the details of the dependence on the ensemble are particularly relevant to the issue of the possible experimental validation of these results , since experiments are always conducted , by virtue of finite - precision , over finite time and finite ensembles .
it is therefore interesting to probe the results of barbi and salerno with regard to the details of the ensemble used , and more formally , to see how ergodicity alters our considerations about the current , as we do in this paper .
we report here on studies on the properties of the current in a chaotic deterministic ratchet , specifically the same system as considered by mateos@xcite and barbi and salerno@xcite .
we consider the impact of different kinds of ensembles of particles on the current and show that the current depends significantly on the details of the initial ensemble .
we also show that it is important to discard transients in quantifying the current .
this is one of the central messages of this paper : broad heuristics are rare in chaotic systems , and hence it is critical to understand the ensemble - dependence in any study of the transport properties of chaotic ratchets .
having established this , we then proceed to discuss the connection between the bifurcation diagram for individual particles and the behavior of the current .
we find that while we disagree with many of the details of barbi and salerno s results , the broader conclusion still holds .
that is , it is indeed possible to have current reversals in the absence of bifurcations from chaos to periodic behavior as well as bifurcations without any accompanying current reversals .
the result of our investigation is therefore that the transport properties of a chaotic ratchet are not as simple as the initial conjecture .
however , we do find evidence for a generalized version of mateos s conjecture .
that is , in general , bifurcations for trajectory dynamics as a function of system parameter seem to be associated with abrupt changes in the current .
depending on the specific value of the current , these abrupt changes may lead the net current to reverse direction , but not necessarily so .
we start below with a preparatory discussion necessary to understand the details of the connection between bifurcations and current reversal , where we discuss the potential and phase - space for single trajectories for this system , where we also define a bifurcation diagram for this system . in the next section ,
we discuss the subtleties of establishing a connection between the behavior of individual trajectories and of ensembles .
after this , we are able to compare details of specific trajectory bifurcation curves with current curves , and thus justify our broader statements above , after which we conclude .
the goal of these studies is to understand the behavior of general chaotic ratchets .
the approach taken here is that to discover heuristic rules we must consider specific systems in great detail before generalizing .
we choose the same @xmath1-dimensional ratchet considered previously by mateos@xcite , as well as barbi and salerno@xcite .
we consider an ensemble of particles moving in an asymmetric periodic potential , driven by a periodic time - dependent external force , where the force has a zero time - average .
there is no noise in the system , so it is completely deterministic , although there is damping .
the equations of motion for an individual trajectory for such a system are given in dimensionless variables by @xmath2 where the periodic asymmetric potential can be written in the form @xmath3 + \frac{1}{4 } \sin [ 4\pi ( x -x_0 ) ] \bigg ] .\ ] ] in this equation @xmath4 have been introduced for convenience such that one potential minimum exists at the origin with @xmath5 and the term @xmath6 .
( a ) classical phase space for the unperturbed system . for @xmath7 , @xmath8 ,
two chaotic attractors emerge with @xmath9 ( b ) @xmath10 ( c ) and a period four attractor consisting of the four centers of the circles with @xmath11.,title="fig:",width=302 ] the phase - space of the undamped undriven ratchet the system corresponding to the unperturbed potential @xmath12 looks like a series of asymmetric pendula .
that is , individual trajectories have one of following possible time - asymptotic behaviors : ( i ) inside the potential wells , trajectories and all their properties oscillate , leading to zero net transport . outside the wells , the trajectories either ( ii ) librate to the right or ( iii ) to the left , with corresponding net transport depending upon initial conditions .
there are also ( iv ) trajectories on the separatrices between the oscillating and librating orbits , moving between unstable fixed points in infinite time , as well as the unstable and stable fixed points themselves , all of which constitute a set of negligible measure . when damping is introduced via the @xmath13-dependent term in eq .
[ eq : dyn ] , it makes the stable fixed points the only attractors for the system .
when the driving is turned on , the phase - space becomes chaotic with the usual phenomena of intertwining separatrices and resulting homoclinic tangles .
the dynamics of individual trajectories in such a system are now very complicated in general and depend sensitively on the choice of parameters and initial conditions .
we show snapshots of the development of this kind of chaos in the set of poincar sections fig .
( [ figure1]b , c ) together with a period - four orbit represented by the center of the circles . a broad characterization of the dynamics of the problem as a function of a parameter ( @xmath14 or @xmath15 ) emerges in a bifurcation diagram
. this can be constructed in several different and essentially equivalent ways .
the relatively standard form that we use proceeds as follows : first choose the bifurcation parameter ( let us say @xmath0 ) and correspondingly choose fixed values of @xmath16 , and start with a given value for @xmath17 .
now iterate an initial condition , recording the value of the particle s position @xmath18 at times @xmath19 from its integrated trajectory ( sometimes we record @xmath20 .
this is done stroboscopically at discrete times @xmath21 where @xmath22 and @xmath23 is an integer @xmath24 with @xmath25 the maximum number of observations made . of these , discard observations at times less than some cut - off time @xmath26 and plot the remaining points against @xmath27 .
it must be noted that discarding transient behavior is critical to get results which are independent of initial condition , and we shall emphasize this further below in the context of the net transport or current .
if the system has a fixed - point attractor then all of the data lie at one particular location @xmath28 . a periodic orbit with period @xmath29 ( that is , with period commensurate with the driving ) shows up with @xmath30 points occupying only @xmath31 different locations of @xmath32 for @xmath27 .
all other orbits , including periodic orbits of incommensurate period result in a simply - connected or multiply - connected dense set of points . for the next value @xmath33 , the last computed value of @xmath34 at @xmath35 are used as initial conditions , and previously , results are stored after cutoff and so on until @xmath36 .
that is , the bifurcation diagram is generated by sweeping the relevant parameter , in this case @xmath0 , from @xmath27 through some maximum value @xmath37 .
this procedure is intended to catch all coexisting attractors of the system with the specified parameter range .
note that several initial conditions are effectively used troughout the process , and a bifurcation diagram is not the behavior of a single trajectory .
we have made several plots , as a test , with different initial conditions and the diagrams obtained are identical .
we show several examples of this kind of bifurcation diagram below , where they are being compared with the corresponding behavior of the current .
having broadly understood the wide range of behavior for individual trajectories in this system , we now turn in the next section to a discussion of the non - equilibrium properties of a statistical ensemble of these trajectories , specifically the current for an ensemble .
the current @xmath38 for an ensemble in the system is defined in an intuitive manner by mateos@xcite as the time - average of the average velocity over an ensemble of initial conditions .
that is , an average over several initial conditions is performed at a given observation time @xmath39 to yield the average velocity over the particles @xmath40 this average velocity is then further time - averaged ; given the discrete time @xmath39 for observation this leads to a second sum @xmath41 where @xmath25 is the number of time - observations made . for this to be a relevant quantity to compare with bifurcation diagrams , @xmath38 should be independent of the quantities @xmath42 but still strongly dependent on @xmath43 .
a further parameter dependence that is being suppressed in the definition above is the shape and location of the ensemble being used .
that is , the transport properties of an ensemble in a chaotic system depend in general on the part of the phase - space being sampled .
it is therefore important to consider many different initial conditions to generate a current . the first straightforward result we show in fig .
( [ figure2 ] ) is that in the case of chaotic trajectories , a single trajectory easily displays behavior very different from that of many trajectories .
however , it turns out that in the regular regime , it is possible to use a single trajectory to get essentially the same result as obtained from many trajectories . further consider the bifurcation diagram in fig .
( [ figure3 ] ) where we superimpose the different curves resulting from varying the number of points in the initial ensemble .
first , the curve is significantly smoother as a function of @xmath0 for larger @xmath44 . even more relevant is the fact that the single trajectory data ( @xmath45 ) may show current reversals that do not exist in the large @xmath44 data .
current @xmath38 versus the number of trajectories @xmath44 for @xmath7 ; dashed lines correspond to a regular motion with @xmath46 while solid lines correspond to a chaotic motion with @xmath47 .
note that a single trajectory is sufficient for a regular motion while the convergence in the chaotic case is only obtained if the @xmath44 exceeds a certain threshold , @xmath48.,title="fig:",width=302 ] current @xmath38 versus @xmath0 for different set of trajectories @xmath44 ; @xmath45 ( circles ) , @xmath49 ( square ) and @xmath50 ( dashed lines ) . note that a single trajectory suffices in the regular regime where all the curves match . in the chaotic regime
, as @xmath44 increases , the curves converge towards the dashed one.,title="fig:",width=302 ] also , note that single - trajectory current values are typically significantly greater than ensemble averages .
this arises from the fact that an arbitrarily chosen ensemble has particles with idiosyncratic behaviors which often average out . as our result , with these ensembles we see typical @xmath51 for example , while barbi and salerno report currents about @xmath52 times greater .
however , it is not true that only a few trajectories dominate the dynamics completely , else there would not be a saturation of the current as a function of @xmath44 .
all this is clear in fig .
( [ figure3 ] ) .
we note that the * net * drift of an ensemble can be a lot closer to @xmath53 than the behavior of an individual trajectory
. it should also be clear that there is a dependence of the current on the location of the initial ensemble , this being particularly true for small @xmath44 , of course .
the location is defined by its centroid @xmath54 . for @xmath45 , it is trivially true that the initial location matters to the asymptotic value of the time - averaged velocity , given that this is a non - ergodic and chaotic system .
further , considering a gaussian ensemble , say , the width of the ensemble also affects the details of the current , and can show , for instance , illusory current reversal , as seen in figs .
( [ current - bifur1],[current - bifur2 ] ) for example .
notice also that in fig .
( [ current - bifur1 ] ) , at @xmath55 and @xmath56 , the deviations between the different ensembles is particularly pronounced .
these points are close to bifurcation points where some sort of symmetry breaking is clearly occuring , which underlines our emphasis on the relevance of specifying ensemble characteristics in the neighborhood of unstable behavior .
however , why these specific bifurcations should stand out among all the bifurcations in the parameter range shown is not entirely clear . to understand how to incorporate this knowledge into calculations of the current ,
therefore , consider the fact that if we look at the classical phase space for the hamiltonian or underdamped @xmath57 motion , we see the typical structure of stable islands embedded in a chaotic sea which have quite complicated behavior@xcite . in such a situation , the dynamics always depends on the location of the initial conditions .
however , we are not in the hamiltonian situation when the damping is turned on in this case , the phase - space consists in general of attractors .
that is , if transient behavior is discarded , the current is less likely to depend significantly on the location of the initial conditions or on the spread of the initial conditions . in particular , in the chaotic regime of a non - hamiltonian system , the initial ensemble needs to be chosen larger than a certain threshold to ensure convergence .
however , in the regular regime , it is not important to take a large ensemble and a single trajectory can suffice , as long as we take care to discard the transients .
that is to say , in the computation of currents , the definition of the current needs to be modified to : @xmath58 where @xmath59 is some empirically obtained cut - off such that we get a converged current ( for instance , in our calculations , we obtained converged results with @xmath60 ) . when this modified form is used , the convergence ( ensemble - independence ) is more rapid as a function of @xmath61 and the width of the intial conditions .
armed with this background , we are now finally in a position to compare bifurcation diagrams with the current , as we do in the next section .
our results are presented in the set of figures fig .
( [ figure5 ] ) fig .
( [ rev - nobifur ] ) , in each of which we plot both the ensemble current and the bifurcation diagram as a function of the parameter @xmath0 .
the main point of these numerical results can be distilled into a series of heuristic statements which we state below ; these are labelled with roman numerals . for @xmath7 and @xmath8 , we plot current ( upper ) with @xmath62 and bifurcation diagram ( lower ) versus @xmath0 .
note that there is a * single * current reversal while there are many bifurcations visible in the same parameter range.,title="fig:",width=302 ] consider fig .
( [ figure5 ] ) , which shows the parameter range @xmath63 chosen relatively arbitrarily . in this figure , we see several period - doubling bifurcations leading to order - chaos transitions , such as for example in the approximate ranges @xmath64 . however , there is only one instance of current - reversal , at @xmath65 .
note , however , that the current is not without structure it changes fairly dramatically as a function of parameter .
this point is made even more clearly in fig .
( [ figure6 ] ) where the current remains consistently below @xmath53 , and hence there are in fact , no current reversals at all .
note again , however , that the current has considerable structure , even while remaining negative
. for @xmath66 and @xmath8 , plotted are current ( upper ) and bifurcation diagram ( lower ) versus @xmath0 with @xmath62 .
notice the current stays consistently below @xmath53.,title="fig:",width=302 ] current and bifurcations versus @xmath0 . in ( a ) and ( b )
we show ensemble dependence , specifically in ( a ) the black curve is for an ensemble of trajectories starting centered at the stable fixed point @xmath67 with a root - mean - square gaussian width of @xmath68 , and the brown curve for trajectories starting from the unstable fixed point @xmath69 and of width @xmath68 . in ( b ) , all ensembles are centered at the stable fixed point , the black line for an ensemble of width @xmath68 , brown a width of @xmath70 and maroon with width @xmath71 .
( c ) is the comparison of the current @xmath38 without transients ( black ) and with transients ( brown ) along with the single - trajectory results in blue ( after barbi and salerno ) .
the initial conditions for the ensembles are centered at @xmath67 with a mean root square gaussian of width @xmath68 .
( d ) is the corresponding bifurcation diagram.,title="fig:",width=302 ] it is possible to find several examples of this at different parameters , leading to the negative conclusion , therefore , that * ( i ) not all bifurcations lead to current reversal*. however , we are searching for positive correlations , and at this point we have not precluded the more restricted statement that all current reversals are associated with bifurcations , which is in fact mateos conjecture .
we therefore now move onto comparing our results against the specific details of barbi and salerno s treatment of this conjecture . in particular , we look at their figs .
( 2,3a,3b ) , where they scan the parameter region @xmath72 .
the distinction between their results and ours is that we are using _
ensembles _ of particles , and are investigating the convergence of these results as a function of number of particles @xmath44 , the width of the ensemble in phase - space , as well as transience parameters @xmath73 . our data with larger @xmath44 yields different results in general , as we show in the recomputed versions of these figures , presented here in figs .
( [ current - bifur1],[current - bifur2 ] ) .
specifically , ( a ) the single - trajectory results are , not surprisingly , cleaner and can be more easily interpreted as part of transitions in the behavior of the stability properties of the periodic orbits .
the ensemble results on the other hand , even when converged , show statistical roughness .
( b ) the ensemble results are consistent with barbi and salerno in general , although disagreeing in several details .
for instance , ( c ) the bifurcation at @xmath74 has a much gentler impact on the ensemble current , which has been growing for a while , while the single - trajectory result changes abruptly . note , ( d ) the very interesting fact that the single - trajectory current completely misses the bifurcation - associated spike at @xmath75 .
further , ( e ) the barbi and salerno discussion of the behavior of the current in the range @xmath76 is seen to be flawed
our results are consistent with theirs , however , the current changes are seen to be consistent with bifurcations despite their statements to the contrary . on the other hand ( f ) , the ensemble current shows a case [ in fig .
( [ current - bifur2 ] ) , at @xmath77 of current reversal that does not seem to be associated with bifurcations .
in this spike , the current abruptly drops below @xmath53 and then rises above it again .
the single trajectory current completely ignores this particular effect , as can be seen .
the bifurcation diagram indicates that in this case the important transitions happen either before or after the spike .
all of this adds up to two statements : the first is a reiteration of the fact that there is significant information in the ensemble current that can not be obtained from the single - trajectory current .
the second is that the heuristic that arises from this is again a negative conclusion , that * ( ii ) not all current reversals are associated with bifurcations .
* where does this leave us in the search for ` positive ' results , that is , useful heuristics ?
one possible way of retaining the mateos conjecture is to weaken it , i.e. make it into the statement that * ( iii ) _ most _ current reversals are associated with bifurcations . * same as fig .
( [ current - bifur1 ] ) except for the range of @xmath0 considered.,title="fig:",width=302 ] for @xmath78 and @xmath8 , plotted are current ( upper ) and bifurcation diagram ( lower ) versus @xmath0 with @xmath62 .
note in particular in this figure that eyeball tests can be misleading .
we see reversals without bifurcations in ( a ) whereas the zoomed version ( c ) shows that there are windows of periodic and chaotic regimes .
this is further evidence that jumps in the current correspond in general to bifurcation.,title="fig:",width=302 ] for @xmath7 and @xmath79 , current ( upper ) and bifurcation diagram ( lower ) versus @xmath0.,title="fig:",width=302 ] however , a * different * rule of thumb , previously not proposed , emerges from our studies .
this generalizes mateos conjecture to say that * ( iv ) bifurcations correspond to sudden current changes ( spikes or jumps)*. note that this means these changes in current are not necessarily reversals of direction .
if this current jump or spike goes through zero , this coincides with a current reversal , making the mateos conjecture a special case .
the physical basis of this argument is the fact that ensembles of particles in chaotic systems _ can _ have net directed transport but the details of this behavior depends relatively sensitively on the system parameters .
this parameter dependence is greatly exaggerated at the bifurcation point , when the dynamics of the underlying single - particle system undergoes a transition a period - doubling transition , for example , or one from chaos to regular behavior .
scanning the relevant figures , we see that this is a very useful rule of thumb . for example
, it completely captures the behaviour of fig .
( [ figure6 ] ) which can not be understood as either an example of the mateos conjecture , or even a failure thereof . as such
, this rule significantly enhances our ability to characterize changes in the behavior of the current as a function of parameter .
a further example of where this modified conjecture helps us is in looking at a seeming negation of the mateos conjecture , that is , an example where we seem to see current - reversal without bifurcation , visible in fig .
( [ hidden - bifur ] ) .
the current - reversals in that scan of parameter space seem to happen inside the chaotic regime and seemingly independent of bifurcation . however , this turns out to be a ` hidden ' bifurcation when we zoom in on the chaotic regime , we see hidden periodic windows .
this is therefore consistent with our statement that sudden current changes are associated with bifurcations .
each of the transitions from periodic behavior to chaos and back provides opportunities for the current to spike .
however , in not all such cases can these hidden bifurcations be found .
we can see an example of this in fig .
( [ rev - nobifur ] ) .
the current is seen to move smoothly across @xmath80 with seemingly no corresponding bifurcations , even when we do a careful zoom on the data , as in fig .
( [ hidden - bifur ] ) .
however , arguably , although subjective , this change is close to the bifurcation point .
this result , that there are situations where the heuristics simply do not seem to apply , are part of the open questions associated with this problem , of course .
we note , however , that we have seen that these broad arguments hold when we vary other parameters as well ( figures not shown here ) . in conclusion ,
in this paper we have taken the approach that it is useful to find general rules of thumb ( even if not universally valid ) to understand the complicated behavior of non - equilibrium nonlinear statistical mechanical systems . in the case of chaotic deterministic ratchets
, we have shown that it is important to factor out issues of size , location , spread , and transience in computing the ` current ' due to an ensemble before we search for such rules , and that the dependence on ensemble characteristics is most critical near certain bifurcation points .
we have then argued that the following heuristic characteristics hold : bifurcations in single - trajectory behavior often corresponds to sudden spikes or jumps in the current for an ensemble in the same system .
current reversals are a special case of this
. however , not all spikes or jumps correspond to a bifurcation , nor vice versa .
the open question is clearly to figure out if the reason for when these rules are violated or are valid can be made more concrete .
a.k . gratefully acknowledges t. barsch and kamal p. singh for stimulating discussions , the reimar lst grant and financial support from the alexander von humboldt foundation in bonn . a.k.p .
is grateful to carleton college for the ` sit , wallin , and class of 1949 ' sabbatical fellowships , and to the mpipks for hosting him for a sabbatical visit , which led to this collaboration .
useful discussions with j .-
m . rost on preliminary results are also acknowledged .
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e * 66 * , 041104 ( 2002 ) | in 84 , 258 ( 2000 ) , mateos conjectured that current reversal in a classical deterministic ratchet is associated with bifurcations from chaotic to periodic regimes .
this is based on the comparison of the current and the bifurcation diagram as a function of a given parameter for a periodic asymmetric potential .
barbi and salerno , in 62 , 1988 ( 2000 ) , have further investigated this claim and argue that , contrary to mateos claim , current reversals can occur also in the absence of bifurcations .
barbi and salerno s studies are based on the dynamics of one particle rather than the statistical mechanics of an ensemble of particles moving in the chaotic system .
the behavior of ensembles can be quite different , depending upon their characteristics , which leaves their results open to question . in this paper we present results from studies showing how the current depends on the details of the ensemble
used to generate it , as well as conditions for convergent behavior ( that is , independent of the details of the ensemble ) .
we are then able to present the converged current as a function of parameters , in the same system as mateos as well as barbi and salerno .
we show evidence for current reversal without bifurcation , as well as bifurcation without current reversal .
we conjecture that it is appropriate to correlate abrupt changes in the current with bifurcation , rather than current reversals , and show numerical evidence for our claims . | arxiv |
one surprising result that has come out of the more than 200 extrasolar planet discoveries to date is the wide range of eccentricities observed . unlike our own solar system
, many of the extrasolar planets which are not tidally locked to their host stars have moderate eccentricities ( @xmath1 ) , and 15 planets have high eccentricities ( @xmath0 ) .
these observations have spawned several theories as to the origin of highly eccentric extrasolar planets .
one such method , planet - planet scattering , occurs when multiple jovian planets form several astronomical units ( au ) from the host star and then interact , leaving one in an eccentric orbit and often ejecting the other @xcite .
this method has been proposed to explain the architecture of the @xmath2 and planetary system @xcite , which contains a hot jupiter as well as two jovian planets in moderately eccentric orbits .
@xcite suggested a merger scenario in which inner protoplanets perturb each other and merge to form a single massive , eccentric planet with @xmath3 and @xmath4 au .
interactions with stellar companions are another possible way to boost a planet s eccentricity .
of the 15 stars hosting a planet with @xmath0 , six are also known to possess stellar - mass companions in wide binary orbits : hd 3651 @xcite , hd 20782 @xcite , hd 80606 , hd 89744 @xcite , 16 cyg b , and hd 222582 @xcite .
if the inclination angle between the planetary orbit and a stellar companion is large , the kozai mechanism @xcite can induce large - amplitude oscillations in the eccentricity of the planet ( e.g. malmberg et al .
these oscillations can be damped by general relativistic effects and by interaction with other planets , and hence are most effective in systems with a single planet in an orbit @xmath51 au from the host star @xcite .
the kozai mechanism has been suggested to explain the high eccentricity of 16 cyg bb @xcite and hd 80606b @xcite .
@xcite found the inclination of 16 cyg b orbiting the system barycenter to lie between 100 and 160 degrees , where 90 degrees is an edge - on orientation .
however , it is the difference in inclination between the orbital planes of the planetary and stellar companion that is critical in determining the importance of the kozai mechanism , and the inclination of the planet s orbit is generally not known for non - transiting systems . of the 192 known planetary systems , 23 ( 12% )
are multi - planet systems .
recent discoveries of additional objects in systems known to host at least one planet @xcite suggest that multiple - planet systems are common .
of particular interest are systems which host a jovian planet and a low - mass `` hot neptune , '' e.g. 55 cnc ( = hd 75732 ) , gj 876 , @xmath6 arae ( = hd 160691 ) , gl 777a ( = hd 190360 ) . motivated by the discoveries of hot neptunes in known planetary systems , we have undertaken an intensive survey of selected single - planet systems to search for additional low - mass companions .
three of the planetary systems discussed in this paper ( hd 3651 , hd 80606 , hd 89744 ) are part of this campaign .
the excellent radial - velocity precision of the high resolution spectrograph on the hobby - eberly telescope ( het ) , combined with queue - scheduling , allow us to time the observations in such a way as to minimize phase gaps in the orbit of the known planet , and also to act quickly on potential new planet candidates .
the use of the het in this manner is discussed further in @xcite with regard to the discovery of hd 37605b . in this work
, we aim to combine observational limits on additional planets in known planetary systems with dynamical constraints obtained by n - body simulations .
the observations address the question : what additional planets are ( or are not ) in these systems ?
the dynamical simulations can answer the question : where are additional planets possible ?
section 2 describes the observations and the test particle simulations for six highly eccentric planetary systems : hd 3651 , hd 37605 , hd 45350 , hd 80606 , hd 89744 , and 16
cyg b. we have chosen these systems based on two criteria : ( 1 ) each hosts a planet with @xmath0 , and ( 2 ) each has been observed by the planet search programs at mcdonald observatory . in 3 , we present and discuss the results of the updated orbital fits , dynamical simulations , and detection limit computations .
five of the six stars considered in this work have been observed with the mcdonald observatory 9.2 m hobby - eberly telescope ( het ) using its high resolution spectrograph ( hrs ) @xcite .
a full description of the het planet search program is given in @xcite .
for 16 cyg b , observations from mcdonald observatory were obtained only with the 2.7 m harlan j. smith ( hjs ) telescope ; the long - term planet search program on this telescope is described in @xcite .
all available published data on these systems were combined with our data from mcdonald observatory in the orbit fitting procedures . to place constraints on the architecture of planetary systems , we would like to know where additional objects can remain in stable orbits in the presence of the known planet(s ) .
we performed test particle simulations using swifthal / swift.html . ]
@xcite to investigate the dynamical possibility of additional low - mass planets in each of the six systems considered here .
low - mass planets can be treated as test particles since the exchange of angular momentum with jovian planets is small .
we chose the regularized mixed - variable symplectic integrator ( rmvs3 ) version of swift for its ability to handle close approaches between massless , non - interacting test particles and planets .
particles are removed if they are ( 1 ) closer than 1 hill radius to the planet , ( 2 ) closer than 0.05 au to the star , or ( 3 ) farther than 10 au from the star .
since the purpose of these simulations is to determine the regions in which additional planets could remain in stable orbits , we set this outer boundary because the current repository of radial - velocity data can not detect objects at such distances .
the test particle simulations were set up following the methods used in @xcite , with the exception that only initially circular orbits are considered in this work . for each planetary system ,
test particles were placed in initially circular orbits spaced every 0.002 au in the region between 0.05 - 2.0 au .
we have chosen to focus on this region because the duration of our high - precision het data is currently only 2 - 4 years for the objects in this study .
the test particles were coplanar with the existing planet , which had the effect of confining the simulation to two dimensions .
input physical parameters for the known planet in each system were obtained from our keplerian orbit fits described in 3.1 , and from recent literature for 16 cyg b @xcite and hd 45350 @xcite .
the planetary masses were taken to be their minimum values ( sin @xmath7 ) .
the systems were integrated for @xmath8 yr , following @xcite and allowing completion of the computations in a reasonable time .
we observed that nearly all of the test - particle removals occurred within the first @xmath9 yr ; after this time , the simulations had essentially stabilized to their final configurations .
we present updated keplerian orbital solutions for hd 3651b , hd 37605b , hd 80606b , and hd 89744b in table 1 .
a summary of the data used in our analysis is given in table 2 , and the het radial velocities are given in tables 3 - 6 .
the velocity uncertainties given for the het data represent internal errors only , and do not include any external sources of error such as stellar `` jitter . ''
the parameters for the remaining two planets , hd 45350b and 16 cyg bb , are taken from @xcite and @xcite , respectively .
radial velocity measurements from the het are given for hd 45350 in @xcite , and velocities for 16 cyg b from the hjs telescope are given in @xcite .
as in @xcite , all available published data were combined with those from mcdonald , and the known planet in each system was fit with a keplerian orbit using gaussfit @xcite , allowing the velocity offset between each data set to be a free parameter .
examination of the residuals to our keplerian orbit fits revealed no evidence for additional objects in any of the six systems in this study .
the saturn - mass ( m sin @xmath10 ) planet hd 3651b was discovered by @xcite using observations from lick and keck .
we fit these data , which were updated in @xcite , in combination with observations from the hjs and het at mcdonald observatory .
the het data , which consist of multiple exposures per visit , were binned using the inverse - variance weighted mean value of the velocities in each visit .
the standard error of the mean was added in quadrature to the weighted rms about the mean velocity to generate the error bar of each binned point ( n=29 ) .
the rms about the combined fit for each dataset is : lick & keck6.6 , het9.4 , hjs12.2 .
the fitted orbital parameters for hd 3651b are of comparable precision to those reported in @xcite , and agree within 2@xmath11 .
the recent discovery of a t dwarf companion to hd 3651 @xcite prompts an interesting exercise : can the radial - velocity trend due to this object be detected in the residuals after removing the planet ?
we detect a slope of @xmath12 yr@xmath13 , indicating that we are indeed able to discern a trend which is possibly due to the binary companion .
however , the reduced @xmath14 of the orbital solution is not significantly improved by the inclusion of a linear trend ( @xmath15=0.18 ) .
the parameters given in table 1 were obtained from the fit which did not include a trend .
we present 23 new het observations for hd 37605 obtained since its announcement by @xcite .
the data now span a total of 1065 days .
the best fit is obtained by including an acceleration of @xmath16 yr@xmath13 , indicating a distant orbiting body .
such a finding would lend support to the hypothesis that very eccentric single - planet systems originate by interactions within a wide binary system .
the shortest period that this outer companion could have and still remain consistent with the observed acceleration and its uncertainty over the timespan of the observations is about 40 yr , assuming a circular orbit .
this object would then have a minimum mass in the brown dwarf range
. the planet orbiting hd 80606 , first announced by @xcite , is the most eccentric extrasolar planet known , with @xmath17 ( table 1 ) .
we have fit the coralie data in combination with the keck data given in @xcite and 23 observations from het .
the extreme velocity variations induced by this planet greatly increase the sensitivity of orbit fits to the weighting of individual measurements . since the uncertainties of the het velocities given in tables 3 - 6 represent internal errors only , we experimented with adding 1 - 7 of radial - velocity `` jitter '' in quadrature before fitting the data for hd 80606 .
for all of these jitter values , the fitted parameters remained the same within their uncertainties .
table 1 gives the parameters derived from a fit which added 3.5 of jitter @xcite to the het data .
the rms about the combined fit is : coralie18.7 , het7.5 , keck5.6 .
@xcite noted that the eccentricity @xmath18 and the argument of periastron @xmath19 had to be held fixed in their fit to the keck data alone .
however , the large number of measurements included in this work allowed gaussfit to converge with all parameters free . for hd 89744b , we combine data from the het with 6 measurements from the hjs telescope and lick data from @xcite .
the het data were binned in the same manner as for hd 3651 , resulting in n=33 independent visits .
the rms about the combined fit for each dataset is : lick17.1 , het10.7 , hjs9.5 . as with hd 3651b ,
our derived parameters agree with those of @xcite within 2@xmath11 .
the scatter about our fit remains large , most likely due to the star s early spectral type ( f7v ) , which hinders precision radial - velocity measurements due to the smaller number of spectral lines .
for example , the f7v star hd 221287 was recently found to host a planet @xcite ; despite the superb instrumental precision of the harps spectrograph , that orbital solution has a residual rms of 8.5 .
the results of the dynamical simulations are shown in figures 1 - 3 .
the survival time of the test particles is plotted against their initial semimajor axis .
as shown in figure 1 , the short - period planets hd 3651 and hd 37605 sweep clean the region inside of about 0.5 au . in both of these systems , however , a small number of test particles remained in low - eccentricity orbits near the known planet s apastron distance , near the 1:2 mean - motion resonance ( mmr ) . in the hd 3651 system , particles remained stable beyond about 0.6 au , which is not surprising given the low mass of the planet . for hd 37605 ,
two distinct strips of stability are seen in fig . 1 , corresponding to the 1:2 and 1:3 mmrs .
the eccentricity of the test particles in the region of the 1:2 mmr oscillated between 0.00 and 0.06 .
particles in 1:3 mmr oscillated in eccentricity with a larger range , up to @xmath20 , which is expected due to secular forcing . as with hd 3651 , the region beyond about 0.8 au
was essentially unaffected by the planet .
figure 2 shows the results for the hd 45350 and hd 80606 systems .
the long period ( 963.6 days ) and relatively large mass ( m sin @xmath21=1.8 ) of hd 45350b restricted stable orbits to the innermost 0.2 au .
these test particles oscillated in eccentricity up to @xmath22 .
the 4 planet orbiting hd 80606 removed all test particles to a distance of about 1.5 au , and only beyond 1.75 au did test particles remain in stable orbits for the duration of the simulation ( @xmath8 yr ) .
a region of instability is evident at 1.9 au due to the 8:1 mmr .
figure 3 shows that hd 89744b eliminated all test particles except for a narrow region near the 8:3 resonance .
for the 16 cyg b system , only particles inside of about 0.3 au remained stable , leaving open the possibility of short - period planets .
the surviving particles oscillated in eccentricity up to @xmath23 , but these simulations treat the star as a point mass , and hence tidal damping of the eccentricity is not included
. our results are consistent with those of @xcite , who investigated dynamical stability in extrasolar planetary systems and found that no test particles survived in the habitable zones of the hd 80606 , hd 89744 , and 16 cyg b systems .
three of these systems ( hd 3651 , hd 80606 , hd 89744 ) were monitored intensely with the het as part of a larger effort to search for low - mass , short period planets .
no evidence was found for any such objects in these or any of the six systems in this work .
we then asked what limits can be set on additional planets using the high - precision het data we have obtained .
the procedure for determining companion limits was identical to the method described in @xcite , except that in this work , the best - fit keplerian orbit for the known planet ( see 3.1 ) was removed before performing the limits computations . in this way
, we determined the radial - velocity amplitude @xmath24 for which 99% of planets would have been detected in the residuals .
the eccentricity of the injected test signals was chosen to be the mean eccentricity of the surviving particles from the simulations described in 3.2 . only the regions in which test particles survived were considered in these limits computations .
the results of these computations were highly varied , reflecting the differing observing strategies employed for these six objects .
in particular , hd 3651 , hd 80606 , and hd 89744 were monitored intensely with the het as part of a search for short - period objects , whereas hd 37605 and hd 45350 were only observed sporadically after the known planet orbits were defined and published @xcite , and 16 cyg b has only been observed with the hjs telescope at a frequency of at most once per month .
the companion limits are shown in figures 4 - 6 ; planets with masses above the solid line can be ruled out by the data with 99% confidence .
not surprisingly , the tightest limits were obtained for hd 3651 ( figure 4 ) , which had a total of 195 measurements , including 29 independent het visits .
for periods less than about 1 year , we can exclude planets with m sin @xmath21 2 neptune masses .
similar results were obtained for 16 cyg b ( n=161 ) , where the limits approach a neptune mass ( figure 6 ) .
since the detection limits generally improve with the addition of more data and with higher - quality data , we can define a quantity to measure the goodness of the limits .
a simple choice would be @xmath25 , where @xmath26 is the total number of observations , and @xmath27 is the mean uncertainty of the radial - velocity measurements .
the values of @xmath26 and @xmath27 are given in table 2 . in the hd 45350 system , the results of the dynamical simulations complement those of the detection limit determinations .
very tight limits are obtained in close orbits ( @xmath280.2 au ) . in this region ,
test particles were stable ( fig . 2 ) and our observations can exclude planets with m sin _ i _ between about 1 and 4 neptune masses .
similar results were obtained for the 16 cyg b system , in which test particles remained stable inward of @xmath29 0.3 au . in that region ,
planets of 1 - 3 neptune masses can be excluded by our limits determinations ( fig .
6 ) . in most of the limits determinations
, there are multiple `` blind spots '' evident where the periodogram method failed to significantly recover the injected signals .
typically this occurs at certain trial periods for which the phase coverage of the observational data is poor , and often at the 1-month and 1-year windows . for none of hd 37605 ( fig . 4 ) , hd 80606 ( fig . 5 ) , or hd 89744 ( fig .
6 ) could additional companions be ruled out below about 0.7 , and for most orbital periods tested , the limits were substantially worse .
one possible explanation for this result is that the sampling of the observations was poorly distributed in phase for many of the injected test signals , making significant recovery by the periodogram method difficult .
this is evidenced by the `` jagged '' regions in the plots .
also , the intrinsic scatter for those three systems was too large to permit tight limits determination .
this is certainly reasonable for the f7 star hd 89744 .
the three systems with the best limits ( hd 3651 , hd 45350 , and 16 cyg b ) also had the lowest rms scatter about their orbital solutions ( mean=@xmath30 ; table 1 ) .
in contrast , the mean rms for the remaining three systems was @xmath31 .
additional factors such as a paucity of data ( hd 37605 ) and short time baselines ( hd 80606 , hd 89744 ) made the determination of useful companion limits challenging for some of the planetary systems in this study .
we have shown that for a sample of six highly eccentric extrasolar planetary systems , there is no evidence for additional planets .
test particle simulations show that there are regions detectable by current surveys ( i.e. for @xmath32 au ) where additional objects can exist . for hd 3651 and hd 37605
, we find that protected resonances are also present .
combining these simulations with detection limits computed using new high - precision het data combined with all available published data is particularly effective for the hd 3651 and hd 45350 systems .
additional short - period planets can be ruled out down to a few neptune masses in the dynamically stable regions in these systems .
this material is based upon work supported by the national aeronautics and space administration under grant nos .
nng04g141 g and nng05g107 g issued through the terrestrial planet finder foundation science program .
we are grateful to the het tac for their generous allocation of telescope time for this project .
we also would like to thank barbara mcarthur for her assistance with gaussfit software .
we thank the referee , greg laughlin , for his careful review of this manuscript .
this research has made use of nasa s astrophysics data system ( ads ) , and the simbad database , operated at cds , strasbourg , france .
the hobby - eberly telescope ( het ) is a joint project of the university of texas at austin , the pennsylvania state university , stanford university , ludwig - maximilians - universit " at mnchen , and georg - august - universit " at g " ottingen the het is named in honor of its principal benefactors , william p. hobby and robert e. eberly .
lllllllll hd 3651 b & 62.197@xmath330.012 & 53932.2@xmath330.4 & 0.630@xmath330.046 & 250.7@xmath336.3 & 15.6@xmath331.1 & 0.20@xmath330.01 & 0.280@xmath330.006 & 7.1 + hd 37605 b & 55.027@xmath330.009 & 52992.8@xmath330.1 & 0.677@xmath330.009 & 218.4@xmath331.7 & 201.5@xmath333.9 & 2.39@xmath330.12 & 0.263@xmath330.006 & 13.0 + hd 45350 b & 963.6@xmath333.4 & 51825.3@xmath337.1 & 0.778@xmath330.009 & 343.4@xmath332.3 & 58.0@xmath331.7 & 1.79@xmath330.14 & 1.92@xmath330.07 & 9.1 + hd 80606 b & 111.428@xmath330.002 & 53421.928@xmath330.004 & 0.933@xmath330.001 & 300.4@xmath330.3 & 470.2@xmath332.5 & 4.10@xmath330.12 & 0.460@xmath330.007 & 13.5 + hd 89744 b & 256.78@xmath330.05 & 53816.1@xmath330.3 & 0.689@xmath330.006 & 194.1@xmath330.6 & 263.2@xmath333.9 & 7.92@xmath330.23 & 0.91@xmath330.01 & 14.4 + 16 cyg b b & 799.5@xmath330.6 & 50539.3@xmath331.6 & 0.689@xmath330.011 & 83.4@xmath332.1 & 51.2@xmath331.1 & 1.68@xmath330.07 & 1.68@xmath330.03 & 10.6 + lllll hd 3651 & 163 & 3.4 & & @xcite + hd 3651 & 3 & 6.1 & & hjs + hd 3651 & 29 & 2.1 & & het + hd 3651 ( total ) & 195 & 3.2 & 7083 & + hd 37605 ( total ) & 43 & 2.9 & 1065 & het + hd 45350 & 38 & 2.8 & & @xcite + hd 45350 & 28 & 4.2 & & het + hd 45350 & 47 & 8.9 & & hjs + hd 45350 ( total ) & 113 & 5.7 & 2265 & + hd 80606 & 61 & 13.7 & & @xcite + hd 80606 & 46 & 5.1 & & @xcite + hd 80606 & 23 & 2.5 & & het + hd 80606 ( total ) & 130 & 8.7 & 2893 & + hd 89744 & 50 & 11.2 & & @xcite + hd 89744 & 33 & 3.2 & & het + hd 89744 & 6 & 9.4 & & hjs + hd 89744 ( total ) & 89 & 8.1 & 2687 & + 16 cyg b & 95 & 6.3 & & @xcite + 16 cyg b & 29 & 19.7 & & hjs phase ii + 16 cyg b & 37 & 7.4 & & hjs phase iii + 16 cyg b ( total ) & 161 & 9.0 & 6950 & + lrr [ tbl-3 ] 53581.87326 & -19.1 & 2.9 + 53581.87586 & -19.4 & 2.7 + 53581.87846 & -20.7 & 2.7 + 53600.79669 & -11.5 & 2.4 + 53600.79860 & -15.5 & 3.0 + 53600.80050 & -22.8 & 2.9 + 53604.79166 & -15.8 & 1.9 + 53604.79356 & -18.8 & 2.1 + 53604.79548 & -21.3 & 2.1 + 53606.78169 & -19.3 & 1.8 + 53606.78360 & -14.8 & 2.1 + 53606.78551 & -24.0 & 1.8 + 53608.77236 & -18.8 & 1.9 + 53608.77426 & -18.0 & 1.9 + 53608.77617 & -18.8 & 1.8 + 53615.96280 & -28.0 & 2.6 + 53615.96471 & -31.9 & 2.4 + 53615.96662 & -37.8 & 2.5 + 53628.74050 & -6.8 & 2.2 + 53628.74240 & -14.5 & 2.4 + 53628.74431 & -5.5 & 2.2 + 53669.61012 & -18.2 & 2.1 + 53669.61203 & -19.2 & 2.2 + 53669.61394 & -17.7 & 2.4 + 53678.78954 & -10.6 & 2.4 + 53678.79141 & -8.6 & 2.3 + 53678.79332 & -2.3 & 2.1 + 53682.78423 & -15.4 & 2.2 + 53682.78609 & -15.0 & 2.3 + 53682.78801 & -11.9 & 2.3 + 53687.77684 & 11.3 & 2.2 + 53687.77875 & 8.7 & 2.2 + 53687.78066 & 15.9 & 2.2 + 53691.75967 & 9.6 & 2.2 + 53691.76158 & 20.3 & 2.1 + 53691.76349 & 15.9 & 2.0 + 53696.75837 & 16.1 & 1.8 + 53696.76028 & 18.6 & 1.8 + 53696.76220 & 20.0 & 2.0 + 53694.75275 & 18.0 & 1.9 + 53694.75466 & 15.1 & 2.0 + 53694.75656 & 17.8 & 2.0 + 53955.83401 & -0.5 & 1.9 + 53955.83593 & -1.2 & 2.0 + 53955.83785 & 1.3 & 1.9 + 53956.82850 & 0.4 & 2.0 + 53956.83046 & -1.0 & 2.0 + 53956.83236 & -5.4 & 2.2 + 53957.82201 & -2.1 & 2.0 + 53957.82392 & -1.3 & 2.0 + 53957.82583 & -3.6 & 2.0 + 53973.80721 & 9.8 & 7.3 + 53973.81020 & 3.5 & 2.3 + 53973.81200 & -3.5 & 2.0 + 53976.78393 & -10.4 & 2.4 + 53976.78586 & -5.4 & 2.1 + 53976.78778 & -6.7 & 2.3 + 53978.97197 & -3.8 & 2.6 + 53985.95886 & -9.0 & 2.3 + 53985.96079 & 4.3 & 3.3 + 53987.95335 & -8.3 & 2.2 + 53987.95527 & -8.0 & 2.2 + 53987.95719 & -12.0 & 2.3 + 53989.73817 & -13.2 & 2.2 + 53989.74009 & -13.2 & 2.1 + 53989.74203 & -18.6 & 2.1 + 54003.70719 & 2.0 & 2.2 + 54003.70915 & 4.7 & 2.4 + 54005.68297 & 7.0 & 2.5 + 54005.68488 & 11.1 & 2.0 + 54005.68690 & 10.2 & 2.1 + 54056.77919 & -7.5 & 2.2 + 54056.78110 & -11.5 & 2.1 + 54056.78302 & -9.6 & 2.3 + 54062.55119 & 20.1 & 1.8 + 54062.55312 & 21.9 & 2.0 + 54062.55505 & 20.9 & 2.0 + 54064.54710 & 12.8 & 2.0 + 54064.54902 & 16.7 & 2.1 + 54064.55094 & 16.6 & 2.1 + 54130.55316 & 19.1 & 2.4 + 54130.55508 & 16.9 & 2.5 + 54130.55701 & 17.6 & 2.5 + lrr [ tbl-4 ] 53002.67151 & 487.6 & 3.8 + 53003.68525 & 495.5 & 3.0 + 53006.66205 & 496.2 & 3.0 + 53008.66407 & 501.3 & 2.9 + 53010.80477 & 499.8 & 2.9 + 53013.79399 & 482.1 & 2.6 + 53042.72797 & 269.7 & 2.8 + 53061.66756 & 489.0 & 2.6 + 53065.64684 & 479.0 & 2.8 + 53071.64383 & 463.8 & 2.6 + 53073.63819 & 460.4 & 2.6 + 53082.62372 & 422.8 & 2.5 + 53083.59536 & 422.2 & 2.8 + 53088.59378 & 418.6 & 4.0 + 53089.59576 & 379.1 & 2.2 + 53092.59799 & 343.7 & 2.5 + 53094.58658 & 323.2 & 2.4 + 53095.58642 & 302.1 & 2.4 + 53096.58744 & 302.1 & 3.2 + 53098.57625 & 193.8 & 2.7 + 53264.95137 & 164.9 & 3.0 + 53265.94744 & 112.9 & 3.0 + 53266.94598 & 113.2 & 3.7 + 53266.95948 & 74.6 & 3.6 + 53266.97396 & 119.2 & 8.0 + 53283.92241 & 471.6 & 2.7 + 53318.81927 & 213.3 & 3.0 + 53335.92181 & 496.9 & 2.6 + 53338.90602 & 493.9 & 2.6 + 53377.81941 & 109.1 & 2.7 + 53378.81189 & 214.6 & 2.7 + 53379.80225 & 338.3 & 2.6 + 53381.64429 & 436.1 & 2.7 + 53384.64654 & 482.9 & 2.8 + 53724.85584 & 468.2 & 2.6 + 53731.69723 & 435.4 & 2.7 + 53738.67472 & 404.3 & 2.6 + 53743.81020 & 400.5 & 2.6 + 53748.64724 & 348.4 & 2.7 + 54039.85015 & 272.5 & 3.1 + 54054.96457 & 437.4 & 2.7 + 54055.95279 & 422.0 & 2.9 + 54067.76282 & 376.4 & 2.6 + lrr [ tbl-5 ] 53346.88103 & -20.8 & 3.0 + 53358.02089 & -49.5 & 2.7 + 53359.82400 & -60.4 & 3.0 + 53361.02985 & -64.7 & 2.5 + 53365.03079 & -77.4 & 2.4 + 53373.98282 & -88.4 & 3.0 + 53377.80112 & -105.5 & 2.4 + 53379.75230 & -109.3 & 2.7 + 53389.74170 & -115.3 & 2.5 + 53391.74400 & -129.4 & 2.4 + 53395.72763 & -146.4 & 2.3 + 53399.72518 & -158.4 & 2.5 + 53401.72497 & -174.7 & 2.7 + 53414.67819 & -219.8 & 3.0 + 53421.85529 & 261.0 & 2.2 + 53423.86650 & 322.1 & 2.0 + 53424.85231 & 245.9 & 2.1 + 53432.87120 & 87.5 & 1.9 + 53433.60628 & 70.0 & 2.1 + 53446.79322 & 4.5 & 1.9 + 54161.85400 & -109.5 & 2.8 + 54166.83797 & -119.3 & 2.4 + 54186.76189 & -184.2 & 2.3 + lrr [ tbl-6 ] 53709.89685 & -184.5 & 2.3 + 53723.85188 & -238.6 & 2.2 + 53723.85367 & -238.2 & 2.5 + 53723.85546 & -227.7 & 2.3 + 53727.84394 & -238.9 & 2.5 + 53727.84573 & -244.9 & 2.4 + 53727.84752 & -242.9 & 2.6 + 53736.81887 & -257.6 & 2.5 + 53736.82100 & -248.2 & 2.9 + 53736.82315 & -253.4 & 2.4 + 53738.03261 & -246.7 & 2.8 + 53738.03441 & -243.3 & 2.4 + 53738.03620 & -236.0 & 2.5 + 53738.80860 & -240.5 & 2.6 + 53738.81040 & -258.9 & 2.4 + 53738.81219 & -249.3 & 2.5 + 53734.81795 & -242.8 & 2.6 + 53734.81973 & -243.9 & 2.8 + 53734.82152 & -248.5 & 2.4 + 53742.79119 & -252.0 & 2.8 + 53742.79299 & -257.2 & 2.8 + 53742.79479 & -239.7 & 2.8 + 53751.78199 & -257.4 & 2.9 + 53751.78378 & -263.1 & 2.5 + 53751.78558 & -268.0 & 2.3 + 53753.78155 & -273.1 & 2.5 + 53753.78381 & -278.7 & 2.5 + 53753.78607 & -266.4 & 2.4 + 53755.76038 & -286.6 & 2.3 + 53755.76218 & -266.5 & 2.6 + 53755.76397 & -274.9 & 2.7 + 53746.81506 & -257.1 & 1.9 + 53746.81778 & -250.9 & 2.1 + 53746.82051 & -245.2 & 2.3 + 53757.77002 & -277.6 & 2.4 + 53757.77181 & -280.3 & 2.4 + 53757.77360 & -288.7 & 2.2 + 53797.64609 & -439.8 & 3.1 + 53797.64834 & -462.6 & 2.8 + 53797.65059 & -452.5 & 2.9 + 53809.62428 & -658.6 & 2.4 + 53809.62700 & -658.8 & 2.5 + 53809.62972 & -659.2 & 2.3 + 53837.76359 & -304.3 & 3.0 + 53837.76670 & -324.0 & 2.9 + 53837.78731 & -308.6 & 2.7 + 53837.79077 & -285.2 & 2.6 + 53866.69987 & -215.9 & 1.7 + 53866.70329 & -228.3 & 1.7 + 53866.70670 & -220.4 & 1.8 + 53868.68349 & -251.6 & 3.8 + 53868.68562 & -208.6 & 2.9 + 53868.68777 & -247.4 & 9.7 + 53875.66956 & -215.7 & 1.6 + 53883.65565 & -213.8 & 1.8 + 53883.65837 & -209.2 & 1.7 + 53883.66109 & -200.4 & 1.7 + 53890.63776 & -203.4 & 1.7 + 53890.63954 & -202.6 & 1.9 + 53890.64134 & -203.2 & 1.9 + 53893.62959 & -193.8 & 2.0 + 53893.63139 & -189.3 & 1.9 + 53893.63318 & -189.7 & 1.8 + 54047.94811 & -375.2 & 4.8 + 54047.94991 & -353.2 & 4.5 + 54047.95172 & -362.6 & 4.4 + 54050.96248 & -415.0 & 2.6 + 54050.96453 & -423.0 & 2.5 + 54050.96657 & -420.1 & 2.4 + 54052.96488 & -426.8 & 2.3 + 54052.96762 & -437.1 & 2.5 + 54052.97035 & -447.6 & 2.5 + 54056.94606 & -468.0 & 3.0 + 54056.94786 & -466.4 & 2.6 + 54056.94964 & -479.4 & 2.8 + 54063.92981 & -599.1 & 2.1 + 54063.93166 & -594.8 & 2.3 + 54063.93348 & -592.3 & 2.4 + 54073.91213 & -685.8 & 2.8 + 54073.91476 & -688.7 & 2.9 + 54073.91739 & -704.4 & 2.7 + 54122.01039 & -220.8 & 2.5 + 54122.01243 & -219.1 & 2.6 + 54122.01447 & -218.4 & 2.8 + 54129.74214 & -215.7 & 2.6 + 54129.74491 & -224.4 & 3.0 + 54129.74768 & -223.7 & 3.1 + 54160.65850 & -189.5 & 3.2 + 54160.66031 & -181.8 & 2.7 + 54160.66212 & -204.8 & 3.2 + 54163.66458 & -213.9 & 3.1 + 54163.66643 & -200.8 & 2.9 + 54163.66828 & -208.0 & 3.2 + 54165.88148 & -208.5 & 2.7 + | long time coverage and high radial velocity precision have allowed for the discovery of additional objects in known planetary systems .
many of the extrasolar planets detected have highly eccentric orbits , which raises the question of how likely those systems are to host additional planets .
we investigate six systems which contain a very eccentric ( @xmath0 ) planet : hd 3651 , hd 37605 , hd 45350 , hd 80606 , hd 89744 , and 16
cyg b. we present updated radial - velocity observations and orbital solutions , search for additional planets , and perform test particle simulations to find regions of dynamical stability .
the dynamical simulations show that short - period planets could exist in the hd 45350 and 16 cyg b systems , and we use the observational data to set tight detection limits , which rule out additional planets down to a few neptune masses in the hd 3651 , hd 45350 , and 16 cyg b systems . | arxiv |
the lep experiments at the resonance of @xmath1-boson have tested the standard model ( sm ) at quantum level , measuring the @xmath1-decay into fermion pairs with an accuracy of one part in ten thousands .
the good agreement of the lep data with the sm predictions have severely constrained the behavior of new physics at the @xmath1-pole .
taking these achievements into account one can imagine that the physics of @xmath1-boson will again play the central role in the frontier of particle physics if the next generation @xmath1 factory comes true with the generated @xmath1 events several orders of magnitude higher than that of the lep .
this factory can be realized in the gigaz option of the international linear collider ( ilc)@xcite .
the ilc is a proposed electron - positron collider with tunable energy ranging from @xmath12 to @xmath13 and polarized beams in its first phase , and the gigaz option corresponds to its operation on top of the resonance of @xmath1 boson by adding a bypass to its main beam line .
given the high luminosity , @xmath14 , and the cross section at the resonance of @xmath1 boson , @xmath15 , about @xmath16 @xmath1 events can be generated in an operational year of @xmath17 of gigaz , which implies that the expected sensitivity to the branching ratio of @xmath1-decay can be improved from @xmath18 at the lep to @xmath19 at the gigaz@xcite . in light of this , the @xmath1-boson properties , especially its exotic or rare decays which are widely believed to be sensitive to new physics , should be investigated comprehensively to evaluate their potential in probing new physics . among the rare @xmath1-decays , the flavor changing ( fc ) processes were most extensively studied to explore the flavor texture in new physics @xcite , and it was found that , although these processes are severely suppressed in the sm , their branching ratios in new physics models can be greatly enhanced to @xmath19 for lepton flavor violation decays @xcite and @xmath20 for quark flavor violation decays @xcite . besides the fc processes
, the @xmath1-decay into light higgs boson(s ) is another type of rare process that was widely studied , e.g. the decay @xmath21 ( @xmath22 ) with the particle @xmath0 denoting a light higgs boson was studied in @xcite , the decay @xmath23 was studied in the two higgs doublet model ( 2hdm)@xcite and the minimal supersymmetric standard model ( mssm)@xcite , and the decay @xmath4 was studied in a model independent way @xcite , in 2hdm@xcite and also in mssm@xcite .
these studies indicate that , in contrast with the kinematic forbidden of these decays in the sm , the rates of these decays can be as large as @xmath18 in new physics models , which lie within the expected sensitivity of the gigaz . in this work ,
we extend the previous studies of these decays to some new models and investigate these decays altogether .
we are motivated by some recent studies on the singlet extension of the mssm , such as the next - to - minimal supersymmetric standard model ( nmssm ) @xcite and the nearly minimal supersymmetric standard model ( nmssm ) @xcite , where a light cp - odd higgs boson @xmath0 with singlet - dominant component may naturally arise from the spontaneous breaking of some approximate global symmetry like @xmath24 or peccei - quuin symmetry @xcite .
these non - minimal supersymmetric models can not only avoid the @xmath25-problem , but also alleviate the little hierarchy by having such a light higgs boson @xmath0 @xcite .
we are also motivated by that , with the latest experiments , the properties of the light higgs boson are more stringently constrained than before .
so it is worth updating the previous studies .
so far there is no model - independent lower bound on the lightest higgs boson mass . in the sm
, it must be heavier than @xmath26 gev , obtained from the null observation of the higgs boson at lep experiments .
however , due to the more complex structure of the higgs sector in the extensions of the sm , this lower bound can be significantly relaxed according to recent studies , e.g. , for the cp - odd higgs boson @xmath0 we have @xmath27 gev in the nmssm @xcite , @xmath28 gev in the nmssm @xcite , and @xmath29 gev in the lepton - specific 2hdm ( l2hdm ) @xcite . with such a light cp - odd higgs boson , the z - decay into one or more
@xmath0 is open up . noting that the decay @xmath30 is forbidden due to bose symmetry , we in this work study the rare @xmath1-decays @xmath6 ( @xmath22 ) , @xmath31 and @xmath4 in a comparative way for four models , namely the type - ii 2hdm@xcite , the l2hdm @xcite , the nmssm and the nmssm . in our study
, we examine carefully the constraints on the light @xmath0 from many latest experimental results .
this work is organized as follows . in sec .
ii we briefly describe the four new physics models . in sec .
iii we present the calculations of the rare @xmath1-decays . in sec .
iv we list the constraints on the four new physics models . in sec .
v we show the numerical results for the branching ratios of the rare @xmath1-decays in various models . finally , the conclusion is given in sec .
as the most economical way , the sm utilizes one higgs doublet to break the electroweak symmetry . as a result ,
the sm predicts only one physical higgs boson with its properties totally determined by two free parameters . in new physics models ,
the higgs sector is usually extended by adding higgs doublets and/or singlets , and consequently , more physical higgs bosons are predicted along with more free parameters involved in .
the general 2hdm contains two @xmath32 doublet higgs fields @xmath33 and @xmath34 , and with the assumption of cp - conserving , its scalar potential can be parameterized as@xcite : @xmath35,\end{aligned}\ ] ] where @xmath36 ( @xmath37 ) are free dimensionless parameters , and @xmath38 ( @xmath39 ) are the parameters with mass dimension . after the electroweak symmetry breaking , the spectrum of this higgs sector includes three massless goldstone modes , which become the longitudinal modes of @xmath40 and @xmath1 bosons , and five massive physical states : two cp - even higgs bosons @xmath41 and @xmath42 , one neutral cp - odd higgs particle @xmath0 and a pair of charged higgs bosons @xmath43 . noting the constraint @xmath44 with @xmath45 and @xmath46 denoting the vacuum expectation values ( vev ) of @xmath33 and @xmath34 respectively , we choose @xmath47 as the input parameters with @xmath48 , and @xmath49 being the mixing angle that diagonalizes the mass matrix of the cp - even higgs fields .
the difference between the type - ii 2hdm and the l2hdm comes from the yukawa coupling of the higgs bosons to quark / lepton . in the type - ii 2hdm
, one higgs doublet @xmath34 generates the masses of up - type quarks and the other doublet @xmath33 generates the masses of down - type quarks and charged leptons ; while in the l2hdm one higgs doublet @xmath33 couples only to leptons and the other doublet @xmath34 couples only to quarks .
so the yukawa interactions of @xmath0 to fermions in these two models are given by @xcite @xmath50 with @xmath51 denoting generation index .
obviously , in the type - ii 2hdm the @xmath52 coupling and the @xmath53 coupling can be simultaneously enhanced by @xmath54 , while in the l2hdm only the @xmath53 coupling is enhanced by @xmath55 .
the structures of the nmssm and the nmssm are described by their superpotentials and corresponding soft - breaking terms , which are given by @xcite @xmath56 where @xmath57 is the superpotential of the mssm without the @xmath25 term , @xmath58 and @xmath59 are higgs doublet and singlet superfields with @xmath60 and @xmath61 being their scalar component respectively , @xmath62 , @xmath63 , @xmath64 , @xmath65 , @xmath66 and @xmath67 are soft breaking parameters , and @xmath68 and @xmath69 are coefficients of the higgs self interactions . with the superpotentials and the soft - breaking terms
, one can get the higgs potentials of the nmssm and the nmssm respectively . like the 2hdm ,
the higgs bosons with same cp property will mix and the mass eigenstates are obtained by diagonalizing the corresponding mass matrices : @xmath70 where the fields on the right hands of the equations are component fields of @xmath71 , @xmath72 and @xmath61 defined by @xmath73 @xmath74 and @xmath75 are respectively the cp - even and cp - odd neutral higgs bosons , @xmath76 and @xmath77 are goldstone bosons eaten by @xmath1 and @xmath78 , and @xmath79 is the charged higgs boson .
so both the nmssm and nmssm predict three cp - even higgs bosons , two cp - odd higgs bosons and one pair of charged higgs bosons . in general , the lighter cp - odd higgs @xmath0 in these model is the mixture of the singlet field @xmath80 and the doublet field combination , @xmath81 , i.e. @xmath82 and its couplings to down - type quarks are then proportional to @xmath83 .
so for singlet dominated @xmath0 , @xmath84 is small and the couplings are suppressed . as a comparison
, the interactions of @xmath0 with the squarks are given by@xcite @xmath85 i.e. the interaction does not vanish when @xmath86 approaches zero . just like the 2hdm where we use the vevs of the higgs fields as fundamental parameters , we choose @xmath68 , @xmath69 , @xmath87 , @xmath88 , @xmath66 and @xmath89 as input parameters for the nmssm@xcite and @xmath68 , @xmath54 , @xmath88 , @xmath65 , @xmath90 and @xmath91 as input parameters for the nmssm@xcite .
about the nmssm and the nmssm , three points should be noted .
the first is for the two models , there is no explicit @xmath92term , and the effective @xmath25 parameter ( @xmath93 ) is generated when the scalar component of @xmath59 develops a vev .
the second is , the nmssm is actually same as the nmssm with @xmath94@xcite , because the tadpole terms @xmath95 and its soft breaking term @xmath96 in the nmssm do not induce any interactions , except for the tree - level higgs boson masses and the minimization conditions . and
the last is despite of the similarities , the nmssm has its own peculiarity , which comes from its neutralino sector .
in the basis @xmath97 , its neutralino mass matrix is given by @xcite @xmath98 where @xmath99 and @xmath100 are @xmath101 and @xmath102 gaugino masses respectively , @xmath103 , @xmath104 , @xmath105 and @xmath106 .
after diagonalizing this matrix one can get the mass eigenstate of the lightest neutralino @xmath107 with mass taking the following form @xcite @xmath108 this expression implies that @xmath107 must be lighter than about @xmath109 gev for @xmath110 ( from lower bound on chargnio mass ) and @xmath111 ( perturbativity bound ) . like the other supersymmetric models , @xmath107 as the lightest sparticle acts as the dark matter in the universe , but due to its singlino - dominated nature , it is difficult to annihilate sufficiently to get the correct density in the current universe .
so the relic density of @xmath107 plays a crucial way in selecting the model parameters .
for example , as shown in @xcite , for @xmath112 , there is no way to get the correct relic density , and for the other cases , @xmath107 mainly annihilates by exchanging @xmath1 boson for @xmath113 , or by exchanging a light cp - odd higgs boson @xmath0 with mass satisfying the relation @xmath114 for @xmath115 . for the annihilation , @xmath54 and @xmath25
are required to be less than 10 and @xmath116 respectively because through eq.([mass - exp ] ) a large @xmath87 or @xmath25 will suppress @xmath117 to make the annihilation more difficult .
the properties of the lightest cp - odd higgs boson @xmath0 , such as its mass and couplings , are also limited tightly since @xmath0 plays an important role in @xmath107 annihilation .
the phenomenology of the nmssm is also rather special , and this was discussed in detail in @xcite .
in the type - ii 2hdm , l2hdm , nmssm and nmssm , the rare @xmath1-decays @xmath118 ( @xmath22 ) , @xmath3 and @xmath4 may proceed by the feynman diagrams shown in fig.[fig1 ] , fig.[fig2 ] and fig.[fig3 ] respectively . for these diagrams , the intermediate state @xmath119 represents all possible cp - even higgs bosons in the corresponding model , i.e. @xmath41 and @xmath42 in type - ii 2hdm and l2hdm and @xmath41 , @xmath42 and @xmath120 in nmssm and nmssm . in order to take into account the possible resonance effects of @xmath119 in fig.[fig1](c ) for @xmath2 and fig.[fig3 ] ( a ) for @xmath11 , we have calculated all the decay modes of @xmath119 and properly included the width effect in its propagator . as to the decay @xmath121 , two points should be noted .
one is , unlike the decays @xmath6 and @xmath11 , this process proceeds only through loops mediated by quarks / leptons in the type - ii 2hdm and l2hdm , and additionally by sparticles in the nmssm and nmssm .
so in most cases its rate should be much smaller than the other two .
the other is due to cp - invariance , loops mediated by squarks / sleptons give no contribution to the decay@xcite . in actual calculation , this is reflected by the fact that the coupling coefficient of @xmath122 differs from that of @xmath123 by a minus sign ( see eq.([asqsq ] ) ) , and as a result , the squark - mediated contributions to @xmath121 are completely canceled out . with regard to the rare decay @xmath11 , we have more explanations . in the lowest order , this decay proceeds by the diagram shown in fig.[fig3 ] ( a ) , and hence one may think that , as a rough estimate , it is enough to only consider the contributions from fig.[fig3](a ) . however , we note that in some cases of the type - ii 2hdm and l2hdm , due to the cancelation of the contributions from different @xmath119 in fig.[fig3 ] ( a ) and also due to the potentially largeness of @xmath124 couplings ( i.e. larger than the electroweak scale @xmath125 ) , the radiative correction from the higgs - mediated loops may dominate over the tree level contribution even when the tree level prediction of the rate , @xmath126 , exceeds @xmath20 . on the other hand
, we find the contribution from quark / lepton - mediated loops can be safely neglected if @xmath127 in the type - ii 2hdm and the l2hdm . in the nmssm and the nmssm , besides the corrections from the higgs- and quark / lepton - mediated loops , loops involving sparticles such as squarks , charginos and neutralinos can also contribute to the decay .
we numerically checked that the contributions from squarks and charginos can be safely neglected if @xmath127 .
we also calculated part of potentially large neutralino correction ( note that there are totally about @xmath128 diagrams for such correction ! ) and found they can be neglected too . since considering all the radiative corrections
will make our numerical calculation rather slow , we only include the most important correction , namely that from higgs - mediated loops , in presenting our results for the four models .
one can intuitively understand the relative smallness of the sparticle contribution to @xmath11 as follows .
first consider the squark contribution which is induced by the @xmath129 interaction ( @xmath130 denotes the squark in chirality state ) and the @xmath131 interaction through box diagrams . because the @xmath132 interaction conserves the chirality of the squarks while the @xmath133 interaction violates the chirality , to get non - zero contribution to @xmath11 from the squark loops , at least four chiral flippings are needed , with three of them provided by @xmath131 interaction and the rest provided by the left - right squark mixing .
this means that , if one calculates the amplitude in the chirality basis with the mass insertion method , the amplitude is suppressed by the mixing factor @xmath134 with @xmath135 being the off diagonal element in squark mass matrix .
next consider the chargino / neutralino contributions . since for a light @xmath0 , its doublet component , parameterized by @xmath84 in eq.([mixing ] ) ,
is usually small , the couplings of @xmath0 with the sparticles will never be tremendously large@xcite .
so the chargino / neutralino contributions are not important too . in our calculation of the decays
, we work in the mass eigenstates of sparticles instead of in the chirality basis .
for the type - ii 2hdm and the l2hdm , we consider the following constraints @xcite : * theoretical constraints on @xmath136 from perturbativity , unitarity and requirements that the scalar potential is finit at large field values and contains no flat directions @xcite , which imply that @xmath137 * the constraints from the lep search for neutral higgs bosons . we compute the signals from the higgs - strahlung production @xmath138 ( @xmath139 ) with @xmath140 @xcite and from the associated production @xmath141 with @xmath142 @xcite , and compare them with the corresponding lep data which have been inputted into our code .
we also consider the constraints from @xmath138 by looking for a peak of @xmath143 recoil mass distribution of @xmath1-boson @xcite and the constraint of @xmath144 mev when @xmath145 @xcite .
+ these constraints limit the quantities such as @xmath146 \times br ( h_i \to \bar{b } b ) $ ] on the @xmath147 plane with the the subscript @xmath148 denoting the coupling coefficient of the @xmath149 interaction .
they also impose a model - dependent lower bound on @xmath150 , e.g. , @xmath151 for the type - ii 2hdm ( from our scan results ) , @xmath152 for the l2hdm@xcite , and @xmath153 for the nmssm @xcite .
these bounds are significantly lower than that of the sm , i.e. @xmath154 , partially because in new physics models , unconventional decay modes of @xmath155 such as @xmath156 are open up . as to the nmssm , another specific reason for allowing a significantly lighter cp -
even higgs boson is that the boson may be singlet - dominated in this model .
+ with regard to the lightest cp - odd higgs boson @xmath0 , we checked that there is no lower bound on its mass so long as the @xmath157 interaction is weak or @xmath155 is sufficiently heavy . * the constraints from the lep search for a light higgs boson via the yukawa process @xmath158 with @xmath22 and @xmath61 denoting a scalar @xcite .
these constraints can limit the @xmath159 coupling versus @xmath160 in new physics models . * the constraints from the cleo - iii limit on @xmath161 and the latest babar limits on @xmath162 .
these constraints will put very tight constraints on the @xmath163 coupling for @xmath164 . in our analysis , we use the results of fig.8 in the second paper of @xcite to excluded the unfavored points . * the constraints from @xmath165 couplings .
since the higgs sector can give sizable higher order corrections to @xmath165 couplings , we calculate them to one loop level and require the corrected @xmath165 couplings to lie within the @xmath166 range of their fitted value .
the sm predictions for the couplings at @xmath1-pole are given by @xmath167 and @xmath168 @xcite , and the fitted values are given by @xmath169 and @xmath170 , respectively@xcite .
we adopt the formula in @xcite to the 2hdm in our calculation . * the constraints from @xmath171 leptonic decay .
we require the new physics correction to the branching ratio @xmath172 to be in the range of @xmath173 @xcite .
we use the formula in @xcite in our calculation . + about the constraints ( 5 ) and ( 6 ) , two points should be noted .
one is all higgs bosons are involved in the constraints by entering the self energy of @xmath171 lepton , the @xmath174 vertex correction or the @xmath175 vertex correction , and also the box diagrams for @xmath176@xcite .
since the yukawa couplings of the higgs bosons to @xmath171 lepton get enhanced by @xmath54 and so do the corrections , @xmath54 must be upper bounded for given spectrum of the higgs sector . generally speaking , the lighter @xmath0 is , the more tightly @xmath54 is limited@xcite .
the other point is in the type - ii 2hdm , @xmath177 , b - physics observables as well as @xmath178 decays discussed above can constraint the model in a tighter way than the constraints ( 5 ) and ( 6 ) since the yukawa couplings of @xmath171 lepton and @xmath179 quark are simultaneously enhanced by @xmath54 .
but for the l2hdm , because only the yukawa couplings of @xmath171 lepton get enhanced ( see eq.[yukawa ] ) , the constraints ( 5 ) and ( 6 ) are more important in limiting @xmath54 .
* indirect constraints from the precision electroweak observables such as @xmath180 , @xmath181 and @xmath182 , or their combinations @xmath183 @xcite .
we require @xmath184 to be compatible with the lep / sld data at @xmath185 confidence level@xcite .
we also require new physics prediction of @xmath186 is within the @xmath187 range of its experimental value .
the latest results for @xmath188 are @xmath189 ( measured value ) and @xmath190 ( sm prediction ) for @xmath191 gev @xcite . in our code , we adopt the formula for these observables presented in @xcite to the type - ii 2hdm and the l2hdm respectively .
+ in calculating @xmath180 , @xmath181 and @xmath182 , we note that these observables get dominant contributions from the self energies of the gauge bosons @xmath1 , @xmath192 and @xmath193 .
since there is no @xmath194 coupling or @xmath195 coupling , @xmath0 must be associated with the other higgs bosons to contribute to the self energies .
so by the uv convergence of these quantities , one can infer that , for the case of a light @xmath0 and @xmath196 , these quantities depend on the spectrum of the higgs sector in a way like @xmath197 at leading order , which implies that a light @xmath0 can still survive the constraints from the precision electroweak observables given the splitting between @xmath150 and @xmath198 is moderate@xcite . * the constraints from b physics observables such as the branching ratios for @xmath199 , @xmath200 and @xmath201 , and the mass differences @xmath202 and @xmath203 .
we require their theoretical predications to agree with the corresponding experimental values at @xmath187 level . + in the type - ii 2hdm and the l2hdm , only the charged higgs boson contributes to these observables by loops , so one can expect that @xmath198 versus @xmath54 is to be limited .
combined analysis of the limits in the type - ii 2hdm has been done by the ckmfitter group , and the lower bound of @xmath204 as a function of @xmath87 was given in fig.11 of @xcite .
this analysis indicates that @xmath198 must be heavier than @xmath205 at @xmath185 c.l .
regardless the value of @xmath54 . in this work
, we use the results of fig.11 in @xcite to exclude the unfavored points . as for
the l2hdm , b physics actually can not put any constraints@xcite because in this model the couplings of the charged higgs boson to quarks are proportional to @xmath206 and in the case of large @xmath54 which we are interested in , they are suppressed . in our analysis of the l2hdm , we impose the lep bound on @xmath198 , i.e. @xmath207@xcite . * the constraints from the muon anomalous magnetic moment @xmath208 .
now both the theoretical prediction and the experimental measured value of @xmath208 have reached a remarkable precision , but a significant deviation still exists : @xmath209 @xcite . in the 2hdm , @xmath208 gets additional contributions from the one - loop diagrams induced by the higgs bosons and also from the two - loop barr - zee diagrams mediated by @xmath0 and @xmath155@xcite .
if the higgs bosons are much heavier than @xmath25 lepton mass , the contributions from the barr - zee diagrams are more important , and to efficiently alleviate the discrepancy of @xmath208 , one needs a light @xmath0 along with its enhanced couplings to @xmath25 lepton and also to heavy fermions such as bottom quark and @xmath171 lepton to push up the effects of the barr - zee diagram@xcite .
the cp - even higgs bosons are usually preferred to be heavy since their contributions to @xmath208 are negative .
+ in the type - ii 2hdm , because @xmath54 is tightly constrained by the process @xmath210 at the lep@xcite and the @xmath178 decay@xcite , the barr - zee diagram contribution is insufficient to enhance @xmath208 to @xmath187 range around its measured value@xcite .
so in our analysis , we require the type - ii 2hdm to explain @xmath208 at @xmath211 level .
while for the l2hdm , @xmath54 is less constrained compared with the type - ii 2hdm , and the barr - zee diagram involving the @xmath171-loop is capable to push up greatly the theoretical prediction of @xmath208@xcite .
therefore , we require the l2hdm to explain the discrepancy at @xmath187 level .
+ unlike the other constraints discussed above , the @xmath208 constraint will put a two - sided bound on @xmath54 since on the one hand , it needs a large @xmath54 to enhance the barr - zee contribution , but on the other hand , too large @xmath54 will result in an unacceptable large @xmath208 . *
since this paper concentrates on a light @xmath0 , the decay @xmath212 is open up with a possible large decay width .
we require the width of any higgs boson to be smaller than its mass to avoid a too fat higgs boson@xcite .
we checked that for the scenario characterized by @xmath213 , the coefficient of @xmath214 interaction is usually larger than the electroweak scale @xmath125 , and consequently a large decay width is resulted .
for the nmssm and nmssm , the above constraints become more complicated because in these models , not only more higgs bosons are involved in , but also sparticles enter the constraints .
so it is not easy to understand some of the constraints intuitively .
take the process @xmath199 as an example . in the supersymmetric models , besides the charged higgs contribution , chargino loops , gluino loops as well as neutralino loops also contribute to the process@xcite , and depending on the susy parameters , any of these contributions may become dominated over or be canceled by other contributions . as a result , although the charged higgs affects the process in the same way as that in the type - ii 2hdm , charged higgs as light as @xmath215 is still allowed even for @xmath216@xcite . since among the constraints
, @xmath208 is rather peculiar in that it needs new physics to explain the discrepancy between @xmath217 and @xmath218 , we discuss more about its dependence on susy parameters . in the nmssm and the nmssm
, @xmath208 receives contributions from higgs loops and neutralino / chargino loops . for the higgs contribution , it is quite similar to that of the type - ii 2hdm except that more higgs bosons are involved in@xcite . for the neutralino / chargino contribution , in the light bino limit ( i.e.
@xmath219 ) , it can be approximated by@xcite @xmath220 for @xmath221 with @xmath222 being smuon mass . so combining the two contributions together
, one can learn that a light @xmath0 along with large @xmath54 and/or light smuon with moderate @xmath87 are favored to dilute the discrepancy . because more parameters are involved in the constraints on the supersymmetric models , we consider following additional constraints to further limit their parameters : * direct bounds on sparticle masses from the lep1 , the lep2 and the tevatron experiments @xcite . * the lep1 bound on invisible z decay @xmath223 ; the lep2 bound on neutralino production @xmath224 and @xmath225@xcite .
* dark matter constraints from the wmap relic density 0.0975 @xmath226 0.1213 @xcite .
note that among the above constraints , the constraint ( 2 ) on higgs sector and the constraint ( c ) on neutralino sector are very important . this is because in the supersymmetric models , the sm - like higgs is upper bounded by about @xmath227 at tree level and by about @xmath228 at loop level , and that the relic density restricts the lsp annihilation cross section in a certain narrow range . in our analysis of the nmssm , we calculate the constraints ( 3 ) and ( 5 - 7 ) by ourselves and utilize the code nmssmtools @xcite to implement the rest constraints .
we also extend nmssmtools to the nmssm to implement the constraints .
for the extension , the most difficult thing we faced is how to adapt the code micromegas@xcite to the nmssm case .
we solve this problem by noting the following facts : * as we mentioned before , the nmssm is actually same as the nmssm with the trilinear singlet term setting to zero .
so we can utilize the model file of the nmssm as the input of the micromegas and set @xmath229 . * since in the nmssm , the lsp is too light to annihilate into higgs pairs , there is no need to reconstruct the effective higgs potential to calculate precisely the annihilation channel @xmath230 with @xmath61 denoting any of higgs bosons@xcite .
we thank the authors of the nmssmtools for helpful discussion on this issue when we finish such extension@xcite .
with the above constraints , we perform four independent random scans over the parameter space of the type - ii 2hdm , the l2hdm , the nmssm and the nmssm respectively .
we vary the parameters in following ranges : @xmath231 for the type - ii 2hdm , @xmath232 for the l2hdm , @xmath233 for the nmssm , and @xmath234 for the nmssm . in performing the scans , we note that for the nmssm and the nmssm , some constraints also rely on the gaugino masses and the soft breaking parameters in the squark sector and the slepton sector .
since these parameters affect little on the properties of @xmath0 , we fix them to reduce the number of free parameters in our scan . for the squark sector
, we adopt the @xmath235 scenario which assumes that the soft mass parameters for the third generation squarks are degenerate : @xmath236 800 gev , and that the trilinear couplings of the third generation squarks are also degenerate , @xmath237 with @xmath238 . for the slepton sector , we assume all the soft - breaking masses and trilinear parameters to be 100 gev .
this setting is necessary for the nmssm since this model is difficult to explain the muon anomalous moment at @xmath239 level for heavy sleptons@xcite .
finally , we assume the grand unification relation @xmath240 for the gaugino masses with @xmath241 being fine structure constants of the different gauge group . with large number of random points in the scans ,
we finally get about @xmath242 , @xmath243 , @xmath244 and @xmath242 samples for the type - ii 2hdm , the l2hdm , the nmssm and the nmssm respectively which survive the constraints and satisfy @xmath245 . analyzing the properties of the @xmath0 indicates that for most of the surviving points in the nmssm and the nmssm , its dominant component is the singlet field ( numerically speaking , @xmath246 ) so that its couplings to the sm fermions are suppressed@xcite .
our analysis also indicates that the main decay products of @xmath0 are @xmath247 for the l2hdm@xcite , @xmath248 ( dominant ) and @xmath247 ( subdominant ) for the type - ii 2hdm , the nmssm and the nmssm , and in some rare cases , neutralino pairs in the nmssm@xcite . in fig.[fig4 ]
, we project the surviving samples on the @xmath249 plane .
this figure shows that the allowed range of @xmath54 is from @xmath250 to @xmath251 in the type - ii 2hdm , and from @xmath252 to @xmath253 in the l2hdm .
just as we introduced before , the lower bounds of @xmath254 come from the fact that we require the models to explain the muon anomalous moment , while the upper bound is due to we have imposed the constraint from the lep process @xmath255 , which have limited the upper reach of the @xmath256 coupling for light @xmath61 @xcite(for the dependence of @xmath256 coupling on @xmath54 , see sec .
this figure also indicates that for the nmssm and the nmssm , @xmath54 is upper bounded by @xmath257 .
for the nmssm , this is because large @xmath87 can suppress the dark matter mass to make its annihilation difficult ( see @xcite and also sec .
ii ) , but for the nmssm , this is because we choose a light slepton mass so that large @xmath54 can enhance @xmath208 too significantly to be experimentally unacceptable .
we checked that for the slepton mass as heavy as @xmath258 , @xmath259 is still allowed for the nmssm . in fig.[fig5 ] and fig.[fig6 ] , we show the branching ratios of @xmath260 and @xmath261 respectively .
fig.[fig5 ] indicates , among the four models , the type - ii 2hdm predicts the largest ratio for @xmath260 with its value varying from @xmath262 to @xmath263 .
the underlying reason is in the type - ii 2hdm , the @xmath264 coupling is enhanced by @xmath54 ( see fig.[fig4 ] ) , while in the other three model , the coupling is suppressed either by @xmath265 or by the singlet component of the @xmath0 .
fig.[fig6 ] shows that the l2hdm predicts the largest rate for @xmath266 with its value reaching @xmath5 in optimum case , and for the other three models , the ratio of @xmath261 is at least about one order smaller than that of @xmath267 .
this feature can be easily understood from the @xmath268 coupling introduced in sect .
we emphasize that , if the nature prefers a light @xmath0 , @xmath260 and/or @xmath269 in the type - ii 2hdm and the l2hdm will be observable at the gigaz . then by the rates of the two decays
, one can determine whether the type - ii 2hdm or the l2hdm is the right theory .
on the other hand , if both decays are observed with small rates or fail to be observed , the singlet extensions of the mssm are favored . in fig.[fig7
] , we show the rate of @xmath3 as the function of @xmath270 .
this figure indicates that the branching ratio of @xmath121 can reach @xmath271 , @xmath272 , @xmath273 and @xmath274 for the optimal cases of the type - ii 2hdm , the l2hdm , the nmssm and the nmssm respectively , which implies that the decay @xmath121 will never be observable at the gigaz if the studied model is chosen by nature .
the reason for the smallness is , as we pointed out before , that the decay @xmath121 proceeds only at loop level . comparing the optimum cases of the type - ii 2hdm , the nmssm and the nmssm shown in fig.5 - 7
, one may find that the relation @xmath275 holds for any of the decays .
this is because the decays are all induced by the yukawa couplings with similar structure for the models . in the supersymmetric models ,
the large singlet component of the light @xmath0 is to suppress the yukawa couplings , and the @xmath0 in the nmssm has more singlet component than that in the nmssm .
next we consider the decay @xmath11 , which , unlike the above decays , depends on the higgs self interactions . in fig.[fig8 ] we plot its rate as a function of @xmath270 and this figure indicates that the @xmath276 may be the largest among the ratios of the exotic @xmath1 decays , reaching @xmath277 in the optimum cases of the type - ii 2hdm , the l2hdm and the nmssm .
the underlying reason is , in some cases , the intermediate state @xmath119 in fig.[fig3 ] ( a ) may be on - shell .
in fact , we find this is one of the main differences between the nmssm and the nmssm , that is , in the nmssm , @xmath119 in fig.[fig3 ] ( a ) may be on - shell ( corresponds to the points with large @xmath278 ) while in the nmssm , this seems impossible .
so we conclude that the decay @xmath11 may serve as an alternative channel to test new physics models , especially it may be used to distinguish the nmssm from the nmssm if the supersymmetry is found at the lhc and the @xmath11 is observed at the gigaz with large rate .
before we end our discussion , we note that in the nmssm , the higgs boson @xmath0 may be lighter than @xmath279 without conflicting with low energy data from @xmath178 decays and the other observables ( see fig.[fig4]-[fig8 ] ) . in this case , @xmath0 is axion - like as pointed out in @xcite .
we checked that , among the rare @xmath1 decays discussed in this paper , the largest branching ratio comes from @xmath280 which can reach @xmath281 . since in this case , the decay product of @xmath0 is highly collinear muon pair , detecting the decay @xmath280 may need some knowledge about detectors , which is beyond our discussion .
in this paper , we studied the rare @xmath1-decays @xmath2 ( @xmath7 ) , @xmath282 and @xmath4 in the type - ii 2hdm , lepton - specific 2hdm , nmssm and nmssm , which predict a light cp - odd higgs boson @xmath0 . in the parameter space allowed by current experiments , the branching ratio can be as large as @xmath5 for @xmath118 , @xmath8 for @xmath3 and @xmath9 for @xmath4 , which implies that the decays @xmath2 and @xmath283 may be accessible at the gigaz option .
since different models predict different size of branching ratios , these decays can be used to distinguish different model through the measurement of these rare decays .
this work was supported in part by hastit under grant no .
2009hastit004 , by the national natural science foundation of china ( nnsfc ) under grant nos . 10821504 , 10725526 , 10635030 , 10775039 , 11075045 and by the project of knowledge innovation program ( pkip ) of chinese academy of sciences under grant no .
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we investigate these rare decays in several new physics models , namely the type - ii two higgs doublet model ( type - ii 2hdm ) , the lepton - specific two higgs doublet model ( l2hdm ) , the nearly minimal supersymetric standard model ( nmssm ) and the next - to - minimal supersymmetric standard model ( nmssm ) .
we find that in the parameter space allowed by current experiments , the branching ratios can reach @xmath5 for @xmath6 ( @xmath7 ) , @xmath8 for @xmath3 and @xmath9 for @xmath4 , which implies that the decays @xmath10 and @xmath11 may be accessible at the gigaz option .
moreover , since different models predict different patterns of the branching ratios , the measurement of these rare decays at the gigaz may be utilized to distinguish the models . | arxiv |
supersymmetry ( susy ) is one of the most attractive extensions of the standard model .
this symmetry solves the naturalness problem and predicts gauge coupling unification at the gut scale @xmath1 .
it also predicts the existence of superpartner of the standard model ( sm ) particles . from the naturalness argument , their masses should be below tev range , hence these particles will be discovered at tevatron or large hadron collider ( lhc )
. mechanisms of susy breaking and its mediation to the minimal supersymmetric standard model ( mssm ) sector are one of the most important problems in the susy phenomenology . in many models ,
this dynamics is related to high energy physics far above the electroweak(ew ) scale , e.g. , gut scale or planck scale .
once the mechanism is specified , mass spectrum and flavor structure of susy particle at the ew scale can be determined by a small number of parameters .
hence it may be possible to confirm or exclude the mechanism by direct search or flavor - changing - neutral - current ( fcnc ) experiments in near future .
if susy breaking is mediated by gravity , the structure of susy breaking masses of scalars are determined by khler potential . in the present paper ,
we focus on the no - scale type khler potential , in which the hidden sector and the observable sector are separated as follows : @xmath2 where @xmath3 and @xmath4 are hidden sector fields and observable sector fields , respectively
. this type of khler potential is originally investigated in ref .
@xcite with @xmath5 and @xmath6 .
characteristic features of the khler potential eq.([eq : noscalekahler ] ) is that all scalar masses and trilinear scalar couplings ( a - terms ) vanish as the cosmological constant vanishes@xcite .
the only source of susy breaking is gaugino masses .
hence this scenario is highly predictive , various phenomenological consequences are obtained with a few parameters . the separation in eq.([eq : noscalekahler ] ) implies that couplings of the hidden sector and the observable sector is flavor blind , and contributions of susy particles to fcnc are suppressed .
therefore this khler potential is also interesting from the viewpoint of the susy flavor problem .
the no - scale structure of the khler potential is obtained in various models .
it has been shown that in some classes of string theory , for example weakly coupled @xmath7 heterotic string theory , khler potential becomes the no - scale type@xcite . if the hidden sector and the observable sector are separated in the superspace density in the supergravity lagrangian , the khler potential is indeed given as in eq .
( [ eq : noscalekahler ] ) . in the two cases ,
the gaugino masses can be induced if the hidden sector fields couple to the gauge multiplets via the gauge kinetic function .
recently it has been pointed out that the form eq.([eq : noscalekahler ] ) is realized naturally in a five - dimensional setting with two branes , namely , sequestered sector scenario@xcite . in this scenario ,
the hidden sector fields live on one brane and the visible sector fields live on the other .
it has been shown that the form of the khler potential of the effective theory obtained by dimensional reduction is indeed eq.([eq : noscalekahler])@xcite .
if the sm gauge fields dwell in the bulk , gaugino mediate the susy breaking on the hidden sector brane to the visible sector brane and the no - scale boundary condition is given at the compactification scale of the fifth dimension ( gaugino mediation @xcite ) . in the no - scale scenario , degrees of freedom of susy particle mass spectrum
is limited because only non - zero soft susy breaking masses are gaugino masses and higgs mixing mass @xmath8 at the energy scale where the boundary condition is given . hence phenomenological aspects of this scenario have been investigated in the literature , mainly focusing on the mass spectrum .
direct search bounds and the cosmological constraint ( i.e. , a charged particle can not be the lsp if the r - parity is conserved ) were considered and allowed region in the parameter space was identified . for the boundary condition ,
the following three cases were considered .
first , universal gaugino masses are given at the gut scale .
in this case , the cosmological constraint is severe and only the region @xmath9 and @xmath10 is allowed since stau tends to be light@xcite .
the second case is that universal gaugino masses are given above the gut scale .
and the third case is that non - universal gaugino masses are given at the gut scale . in this case wino ,
higgsino or sneutrino can be the lsp . in the latter two cases ,
it is shown that the cosmological constraint is not severer than the first case . in the present paper ,
current limits from the lightest higgs mass @xmath11 and the branching ratio for @xmath0 are also used to constrain the no - scale scenario . combining these constraints
, we will show that almost all the parameter region is excluded when universal gaugino masses are given at the gut scale .
however , when the boundary condition is given above the gut scale , relatively large parameter region is allowed .
we also consider the case that the non - universal gaugino masses are given at the gut scale .
we will show that these constraints are important when the higgsino - like neutralino is the lsp .
this paper is organized as follows . in section [ sec : noscalsebc ] , we review some phenomenological aspects of the no - scale models , especially indications of the direct search bounds and the cosmological bound . in section [ sec : higgsbsgamma ] , we further constrain these models from the higgs mass bound and @xmath12 result . indications of these bounds for the tevatron are also discussed .
our conclusions are given in section [ sec : conclusions ] .
in this section , we briefly review phenomenological aspects of susy models with no - scale boundary condition , mainly indications of the cosmological bound and direct search limit at lep 2 .
we consider the following three cases . *
universal gaugino masses are given at the gut scale .
hereafter we call this case the minimal scenario . *
universal gaugino masses are given above the gut scale @xmath13 . throughout this paper
, we take the minimal su(5 ) to be the theory above the gut scale as a typical example . *
non - universal gaugino masses are given at the gut scale .
once one of the above boundary conditions is given , mass spectrum of susy particles at the ew scale and their contributions to fcnc can be calculated . in this paper
we solve the one - loop level rges to obtain the soft susy breaking mass parameters at the ew scale .
the higgsino mass parameter @xmath14 is determined by potential minimum condition at the one - loop level .
first , we discuss the minimal scenario . in this case
, the following boundary condition is given at the gut scale , @xmath15 where @xmath16 is the common scalar mass and @xmath17 is universal trilinear scalar coupling . with this boundary condition , bino and right - handed
sleptons are lighter than other susy particles .
their masses are approximately , @xmath18 from eq.([eq : minimalmneumtau ] ) we see that the charged right - handed slepton is the lsp if the d - term @xmath19 is negligible , i.e. , @xmath20 .
hence this parameter region is excluded by the cosmological consideration . on the other hand ,
lep 2 experiments yields the upper bound on the cross section for smuon pair production , @xmath21 for @xmath22 and @xmath23@xcite , so the parameter region @xmath24 is excluded in fig .
[ fig : limitmimposmu ] and [ fig : limitmimnegmu ] , allowed region of the parameter space are shown in the @xmath25 plane .
the regions above the dash - dotted line and the left side of the dash - dot - dotted line are excluded by cosmological bound and lep 2 bound on smuon pair production , respectively .
therefore the minimal scenario is constrained severely .
next we see the case that the universal gaugino masses are given above the gut scale . in the minimal su(5 ) case , the right - handed slepton belongs to 10-plet , so the large group factor makes slepton masses heavier .
for example , when @xmath26 the bino mass and the right - handed slepton mass at the weak scale are approximately given , @xmath27 hence the cosmological constraint is not severe because the stau mass is large enough and neutralino is the lsp in the large parameter region @xcite . in the fig.[fig : limitmbc1e17posmu ] and [ fig : limitmbc1e17negmu ] , the same figures as in the fig.[fig : limitmimposmu ] and [ fig : limitmimnegmu ] are shown .
unlike in the minimal case , the stau search bound at lep @xcite is also plotted because mass difference between @xmath28 and @xmath29 is larger than in the minimal case and it can be stronger than the smuon search . from these figures we see that the @xmath29 lsp is avoided unless @xmath30 is larger than about 20 .
the charged stau lsp can also be avoided if gaugino masses at the gut scale are non - universal@xcite , i.e. , the following boundary condition is given , @xmath31 this boundary condition can be given naturally within the gut framework @xcite . in this case ,
not only bino - like neutralino , but also wino - like , higgsino - like neutralino or sneutrino can be the lsp . for @xmath32 and @xmath33 ,
the lsp is wino - like neutralino .
for example , when @xmath34 and @xmath35 , then wino mass and charged slepton mass are ( notice that in this case the left - handed sleptons are lighter than right - handed sleptons ) ; @xmath36 the higgsino is the lsp if @xmath37 . for example , when @xmath38 and @xmath39 , then the higgsino mass and the right - handed slepton mass are @xmath40 in the two cases given above , neutral wino or higgsino is the lsp .
in fact from fig.[fig : limit12r432r2posmu ] - [ fig : limit12r232r0.5negmu ] we find that neural particle is the lsp in large parameter region , thus it is cosmologically viable .
in the previous section we take into account only lep 2 bound and the cosmological constraint .
we find that the minimal scenario is severely constrained , but the other two scenarios are not . in this section
we also include the current higgs mass bound and @xmath41 constraint .
as we will see , combining the above four constraint , not only the minimal case but also the other two scenarios can be constrained more severely .
we also discuss the possibility whether this scenario can be seen at the tevatron run 2 or not .
before we show the numerical results , some remarks on our calculation of the higgs mass and @xmath12 are in order .
it is well known that radiative correction is important when the lightest higgs mass is evaluated @xcite . in the present paper ,
the lightest higgs mass is evaluated by means of the one - loop level effective potential@xcite .
this potential is evaluated at the renormalization point of the geometrical mean of the two stop mass eigenvalues @xmath42 .
we compared our result with a two - loop result by using _
feynhiggs_@xcite , and checked that the difference between these two results is smaller than 5 gev as long as @xmath30 is bigger than 5 .
when @xmath30 is close to 2 , the difference can be 7 gev .
however , as we will see later , higgs mass bound plays an important rule around @xmath43 . and
the two - loop effects always make the higgs mass lighter than that obtained at the one - loop level .
so our conclusion is conservative and is not significantly changed by the two - loop effect .
we exclude the parameter region where the lightest higgs mass is lighter than the current 95% c.l .
limit from lep 2 experiments , @xmath44 @xcite . in the present paper , @xmath12
is calculated including leading order ( lo ) qcd corrections@xcite , and compare it to the current cleo measurement . in the mssm
, chargino contribution can be comparable to the sm and charged higgs contributions .
they interfere constructively ( destructively ) each other when @xmath45 ( @xmath46 ) .
the difference between the lo and the next - to - leading order ( nlo ) result can be sizable only when cancellation among different contributions at the lo is spoiled by the nlo contributions .
as we will see , however , the @xmath41 constraint is severe when the interference is constructive . in the case of destructive interference where the deviation from the nlo result may be large ,
this constraint is not so important .
hence we expect that our conclusion is not changed significantly by the inclusion of the nlo corrections . for the experimental value
, we use 95% c.l .
limit from cleo , @xmath47 @xcite .
first we show the numerical results for the minimal case .
the case for @xmath48 is shown in fig.[fig : limitmimposmu ] . in this case , for small @xmath30 region , the stop mass is not so large that radiative correction factor @xmath49 which raises the higgs mass is small .
( for example , @xmath50 gev and @xmath51 gev for @xmath52 gev and @xmath53 ) .
hence the higgs mass limit constrains this scenario severely . in fig.[fig : limitmimposmu ] , the higgs mass bound and @xmath12 constraints in the @xmath54 plane are shown .
the regions below the solid line and above the dashed line are excluded by the higgs mass and @xmath12 bound , respectively .
the indication of @xmath55 reported by lep 2@xcite is also shown in this figure . from the figure we find that the higgs mass bound almost excludes the region where the stau lsp is avoided .
note that , as we discussed earlier , the bound we put on the higgs mass may be conservative , because the two loop correction may further reduce the higgs mass . the same figure but for @xmath56 is shown in fig.[fig : limitmimnegmu ] .
now @xmath12 also constrains parameter region strongly since chargino contribution to @xmath0 interferes with sm and charged higgs ones constructively .
the region above the dashed line is excluded by @xmath12 constraint .
we find that only one of the two constraints is enough to exclude all the region where cosmological bound and the smuon mass bound are avoided .
hence if r - parity is conserved , i.e. , the cosmological bound is relevant , this scenario with @xmath56 is excluded .
next we show the numerical results in the case that the cutoff scale is larger than the gut scale . as a typical example , we choose the minimal su(5 ) as the theory above the gut scale . in fig.[fig :
limitmbc1e17posmu ] and [ fig : limitmbc1e17negmu ] , results are shown for positive and negative @xmath14 , respectively . in both figures ,
we take @xmath58 gev . for @xmath48 case , large parameter region
is allowed and susy scale @xmath59 can be as small as about 180 gev , which indicates the lsp mass @xmath60 gev . for @xmath56 , as in the minimal case , @xmath12 constraint is severer , and @xmath59 must be larger than around 280 gev .
we also considered other values of the boundary scale @xmath61 from @xmath62 to @xmath63 , and checked that the behavior of the contour plot does not change so much .
according to ref.@xcite , tevatron run 2 experiment can explore up to @xmath64 gev for integrated luminosity @xmath65 .
hence if @xmath66 and @xmath67 , susy particles can be discovered at the experiment . in this range ,
trilepton from chargino - neutralino associated production @xmath68 , @xmath69 , @xmath70 is one of clean signals for susy search .
notice that now two body decay @xmath71 opens .
so same flavor , opposite sign dilepton from @xmath72 decay may be useful .
the two body decay allows us to observe the peak edge of invariant mass of two leptons at the @xmath73 .
it is expressed in terms of the neutralino masses and the slepton mass as , @xmath74 in table [ tab : mllmax ] , the dependence of @xmath73 on @xmath61 is shown . here
we fix @xmath75 .
notice that as @xmath61 changes , the right - handed mass changes sizably while the neutralino masses do not .
hence we can obtain the mass relation among them and also cutoff scale @xmath61 , which corresponds to the compactification scale in the sequestered sector scenario , by measuring @xmath73 . on the other hand , since only @xmath76 gev is allowed for @xmath45 , the tevatron run 2 can not survey this scenario , and we have to wait lhc experiment .
next , we turn to the case that gaugino masses are non - universal at the gut scale .
we explore the following three cases , wino - like neutralino lsp , higgsino - like neutralino lsp and the tau sneutrino lsp .
we will see that in the wino - like neutralino lsp and tau sneutrino lsp cases , constraint is not so severe even if we combine higgs mass bound and @xmath12 data , but in the higgsino - like lsp case where stops are as light as sleptons and charginos , the predicted higgs mass tends to be small , and thus the higgs mass bound becomes important .
first , we discuss the wino - lsp case .
the results for @xmath34 , @xmath35 are shown in fig.[fig : limit12r432r2posmu ] and fig.[fig : limit12r432r2negmu ] , for @xmath46 and @xmath45 , respectively . in this case , we obtain a relatively large higgs mass since @xmath77 is large and so are the masses of stops .
hence , for @xmath46 , @xmath78 can be as small as 100 gev at @xmath43 , where the mass of the lsp @xmath79 is about 90 gev . for @xmath45
, though @xmath12 constraint is slightly severer than in the @xmath46 case , @xmath80 is allowed , which corresponds to @xmath81 .
hence the wino - lsp with mass around 100 gev is allowed .
examples of the mass spectrum in this case are listed as point a ( @xmath46 ) and point b ( @xmath45 ) in table [ tab : spectrum ] . at the both points ,
@xmath59 is chosen to be near the smallest value such that all constraints are avoided .
in general , however , masses of @xmath28 and @xmath82 are highly degenerate when wino is the lsp .
in fact , from table [ tab : spectrum ] , we see that the mass difference is less than 1 gev .
therefore a lepton from @xmath70 is very soft and trilepton signal search is not useful because acceptance cut usually requires the smallest transverse momentum of the three leptons @xmath83 to be larger than 5 gev@xcite .
recently collider phenomenology in such cases are studied in ref .
it is shown that certain range of @xmath84 and @xmath85 , susy signals which are different from those in the minimal case can be detected .
the high degeneracy requires to include radiative corrections to calculate @xmath85 @xcite , which is beyond of this work .
it deserves detail study to estimate the mass difference in the scenario .
since the constraint for the sneutrino lsp case in the @xmath86 plane is similar to those in the wino - lsp case , we show the result for @xmath46 only in fig.[fig : limit12r2.532r1.5posmu ] . in the figure
, we take @xmath87 and @xmath88 .
notice that the decomposition of the lsp depends on @xmath30 and the sneutrino is the lsp for @xmath89 .
an example of the mass spectrum is listed as the point c in table [ tab : spectrum ] . in this case ,
trilepton signal comes from @xmath90 , @xmath91 , @xmath92 .
since @xmath93 , @xmath94 of a lepton from @xmath82 decay is small and this signal may be hard to be detected .
we may need unusual trigger to explore this scenario .
next , we turn to the higgsino lsp case .
higgsino lsp scenario is realized when @xmath77 is smaller than half of @xmath78 , which indicates that colored particles are lighter than in the universal gaugino mass case .
hence the one - loop correction to the higgs potential which enhances the higgs mass is small and the higgs mass constraint is important . the same figures as fig.[fig : limitmimposmu ] and [ fig : limitmimnegmu ] are shown in fig.[fig : limit12r232r0.5posmu ] ( @xmath46 ) and fig.[fig : limit12r232r0.5negmu ] ( @xmath45 ) for @xmath95 and @xmath39 . in order to satisfy the higgs mass bound , @xmath78 must be larger than around 300 gev .
combining the bound with @xmath12 , constraint becomes severer , especially for @xmath45 case where @xmath96 is required .
example of mass spectrum in this scenario is listed as point d and e in table [ tab : spectrum ] .
again we choose almost the smallest value of @xmath78 where all constraints are avoided .
we see that the lsp mass must be at least @xmath97 for @xmath46 and @xmath98 for @xmath45 .
hence this scenario can not be explored at the tevatron run 2 .
the no - scale type boundary conditions are obtained in various types of susy models .
this scenario is attractive because it is highly predictive and can be a solution to the susy flavor problem . in this paper
we investigated the indication of the current higgs mass and @xmath41 constraint on susy models with the boundary condition .
first we considered the minimal case where the universal gaugino mass are given at the gut scale .
this scenario has been already constrained by direct search at lep and the cosmological bound severely , under the assumption of the exact r - parity .
we showed that the higgs mass bound and @xmath41 constraint are also taken into account , then almost all the parameter region is excluded , leaving very narrow allowed region for @xmath48 .
next we considered the case that the boundary condition is given above the gut scale .
since the cosmological constraint is not severe , wide region of the parameter space is allowed . in the @xmath46 case ,
tevatron have a chance to observe susy signatures like trilepton events .
the scale @xmath61 may be explored by measuring the peak edge of invariant mass of two leptons at the @xmath73 .
however for the @xmath45 case , since @xmath99 is required , we have to wait lhc .
finally we considered the case where non - universal gaugino masses are given at the gut scale .
we see that the higgs mass bound is strong in the higgsino lsp case because stop masses are as light as sleptons and charginos .
the mass of the higgsino - like neutralino must be larger than about 245 gev and 370 gev for @xmath46 and @xmath45 , respectively . in the wino lsp and sneutrino lsp case
, the mass of the lsp can be as small as 150 gev .
however , the mass difference between the lsp and parent particles produced at the collider is much smaller than in the minimal case , unusual acceptance cut may be required .
the author would like to thank m. yamaguchi for suggesting the subject , fruitful discussion and careful reading of the manuscript .
he also thanks t. moroi and m. m. nojiri for useful discussion .
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plane for the minimal case with @xmath48 . in the region above the dash - dotted line , the stau is the lsp .
the left side of the dash - dot - dotted line is excluded by the upper bound on smuon pair production cross section at lep .
the current higgs mass bound excludes the region below the solid line . in the region above the dashed line , @xmath12 is smaller than the lower limit obtained by the cleo .
the shaded region is allowed .
the @xmath100 curve is also shown as the dotted line .
noewsb means that radiative breaking does not occur.,width=340,height=264 ] but @xmath56 .
the region above the dashed line is excluded since @xmath12 is larger than the upper limit obtained by the cleo .
the other lines are the same as in fig.[fig : limitmimposmu ] .
notice that all region is excluded.,width=340,height=264 ] plane for universal gaugino masses , @xmath26 and @xmath48 .
the left sides of the dash - dot - dotted and dash - dot - dot - dotted line are excluded by the upper bound on the smuon and stau pair production cross section , respectively .
the other lines are the same as in fig.[fig : limitmimposmu ] , width=340,height=264 ] plane for the higgsino lsp case , @xmath95 , @xmath39 , @xmath101 and @xmath48 . in the region between the two dashed line ,
@xmath12 is smaller than the lower limit of the cleo result .
, width=340,height=264 ] | no - scale structure of the khler potential is obtained in many types of supersymmetric models . in this paper ,
phenomenological aspects of these models are investigated with special attention to the current higgs mass bound at lep and @xmath0 result at the cleo .
when the boundary condition is given at the gut scale and gaugino masses are universal at this scale , very narrow parameter region is allowed only for positive higgsino mass region if r - parity is conserved .
the negative higgsino mass case is entirely excluded . on the other hand , relatively large parameter region
is allowed when the boundary condition is given above the gut scale , and tevatron can discover susy signals for the positive higgsino mass case .
the no - scale models with wino , higgsino or sneutrino lsp are also considered .
we show that the higgs mass constraint is important for the higgsino lsp case , which requires the lsp mass to be larger than about 245 gev .
0.0 mm 0.0 mm 159.2 mm -16.0 mm 240.0 mm | arxiv |
it has been known for a long time that owing to planar property and mutual focussing , colliding plane waves ( cpw ) result in spacelike singularities [ 1 ] .
these singularities are somewhat weakened when the waves are endowed with a relative cross polarization prior to the collision . a solution given by chandrasekhar and xanthopoulos ( cx )
[ 2 ] , however constitutes an example contrasting this category , namely , it possesses a cauchy horizon ( ch ) instead of a spacelike singularity .
naturally , this solution initiated a literature devoted entirely on the quest of stability of horizons formed hitherto .
ch formed in spacetimes of cpw was shown by yurtsever to be unstable against plane - symmetric perturbations [ 3 ]
. a linear perturbation analysis by cx reveals also an analogues result [ 4 ] .
any such perturbation applied to a cpw spacetime will turn the ch into an essential singularity .
a second factor that proved effective in weakening the strength of a singularity in cpw is the electromagnetic ( em ) field itself . in other words , the degree of divergence in
the curvature scalars of colliding pure gravitational waves turn out to be stronger than the case when em field is coupled to gravity .
in particular , collision of pure em waves must have a special significance as far as singularity formation is concerned .
such an interesting solution was given by bell and szekeres ( bs ) which describes the collision of two linearly polarized step em waves [ 5 ] .
the singularity ( in fact a ch ) formed in the interaction region of the bs solution was shown to be removable by a coordinate transformation .
on the null boundaries , however it possesses esential curvature singularities which can not be removed by any means .
since cross polarization and em field both play roles in the nature of resulting singularity it is worthwhile to purse these features together .
this invokes a cross polarized version of the bs ( cpbs ) solution which was found long time ago [ 6,7 ] .
this metric had the nice feature that the weyl scalars are all regular in the interaction region .
cross polarization , however , does not remove the singularities formed on the null boundaries . in this paper
we choose cpbs solution as a test ground , instead of bs , with various added test fields to justify the validity of a ch stability conjecture proposed previously by helliwell and konkowski ( hk ) [ 8,9 ] . unlike the tedious perturbation analysis of both cx and yurtsever the conjecture seems to be much economical in reaching a direct conclusion about the stability of a ch .
this is our main motivation for considering the problem anew , for the case of untested solutions in cpw . in this paper
we look at the spacetimes : a ) single plane wave with added colliding test fields and b ) colliding plane waves having non - singular interaction regions with test field added , fig.1 illustrates these cases
. the terminology of singularities should be follwed from the classification presented by ellis and schmidt [ 10 ] .
singularities in maximal four dimensional spacetimes can be divided into three basic types : quasiregular ( qr ) , scalar curvature ( sc ) and non - scalar curvature ( nsc ) . the ch stability conjecture due to hk is defined as follows . for all maximally extended spacetimes with ch
, the backreaction due to a field ( whose test stress - energy tensor is @xmath0 ) will affect the horizon in one of the following manners .
@xmath1 , @xmath2 and any null dust density @xmath3 are finite , and if the stress energy tensor @xmath4 in all parallel propagated orthonormal ( ppon ) frames is finite , then the ch remains non - singular .
b)if @xmath1 , @xmath2 and any null dust density @xmath3 are finite , but @xmath4 diverges in some ppon frames , then an nsc singularity will be formed at the ch .
c)if @xmath1 , @xmath2 and any null dust density @xmath3 diverges , then an sc singularity will be formed at the ch
. expressed otherwise , the conjecture suggests to put a test field into the background geometry and study the reaction it will experience . if certain scalars diverge then in an exact back - reaction solution the field will respond with an infinite strength to the geometry ( i.e action versus reaction ) .
such an infinite back - reaction will render a ch unstable and convert it into a scalar singularity . + the paper is organized as follows . in section
ii , we review the cpbs solution and the correct nature of the singularity structure is presented in appendix a. section iii , deals with geodesics and test em and scalar field analyses . in section iv , we present an exact back reaction calculation for the collision of cross polarized em field coupled with scalar field .
the derivation of weyl and maxwell scalars are given in appendix b. the insertion of test null dusts to the background cpbs spacetime and its exact back reaction solution is studied in section v. appendix c is devoted for the properties of this solution .
the paper is concluded with a discussion in section vi .
the metric that describes collision of em waves with the cross polarization was found to be [ 7 ] + @xmath5 in this representation of the metric our notations are + @xmath6 in which @xmath7 is a constant measuring the second polarization , @xmath8 are constant of energy and @xmath9 stand for the usual null coordinates .
it can be seen easily that for @xmath10 the metric reduces to bs . unlike the bs metric , however , this is conformally non - flat for @xmath11 , where the conformal curvature is generated by the cross polarization . as a matter of fact
this solution is a minimal extension of the bs metric .
a completely different generalization of the bs solution with second polarization was given by cx [ 11 ] .
their solution , however , employs an ehlers transformation and involves two essential parameters which is therefore different from ours . both solutions
form ch in the interaction region .
our result drown out in this paper , namely , that the horizon is unstable against added sources can also be shown to apply to the cx metric as well . as it was shown before the interaction region @xmath12 of the above
metric is of type - d without scalar curvature singularities .
we wish to check now the possible singularities of metric ( 1 ) .
the single component of the weyl scalar suffices to serve our purpose .
we find that the real part of the weyl scalar @xmath13 is given by @xmath14\end{aligned}\ ] ] where we have used the abbreviations @xmath15 as @xmath16 we obtain @xmath17 which reduces to the singularity form of the bs spacetime given by @xmath18 .
we see that the same singularity remains unaffected by the introduction of the cross polarization . a similar calculation for @xmath19
gives the symmetrical singular hypersurface sitting on @xmath20 .
now in order to explore the true nature of the singularity we concentrate our account on the incoming region ii @xmath21 .
the metric in this region is expressed in the form @xmath22\ ] ] where @xmath23 we observe that for @xmath24 , @xmath25 is a bounded positive definite function which suggests that no additional singularities arise except the one occuring already in the bs case , namely at @xmath26 . to justify this
we have calculated all riemann components in local and ppon frames ( see appendix a ) . it is found that all riemann tensor components vanish as @xmath27 . in the ppon frame , however , they are all finite and according to the classification scheme of ellis and schmidt such a singularity is called a quasiregular ( qr ) singularity . this is said to be the mildest type among all types of singularities . to check whether the qr is stable or not we consider generic test fields added to such a background geometry and study the effects .
this we will do in the follwing sections .
we are interested in the stability of qr singularities that are developed at @xmath28 in region ii and @xmath29 in region iii . to investigate their stability we will express geodesics and behaviour of test em and scalar fields by calculating stress - energy tensor in local and ppon frames . +
our discussion on geodesics will be restricted in region ii only .
we shall consider the geodesics that originate at the wave front and move toward the quasiregular singularity .
solution of geodesics equations in region ii can be obtained by geodesics lagrangian method and using @xmath30 as a parameter .
the results are @xmath31}{a}\tan(au ) + \frac{3p_{x_{0}}\left[5q^{2}+2 - 2\sqrt{1+q^{2}}\right]u}{4 } \nonumber \\ & & - \frac{p_{x_{0}}\left[\sqrt{1+q^{2}}-1\right]^{2}}{8a } \sin(2au ) - \frac{2p_{y_{0}}q}{a\cos(au ) } \nonumber \\ y - y_{0}&=&-\frac{2qp_{x_{0}}}{a\cos(au)}-\frac{2p_{y_{0}}\tan(au)}{a } \nonumber \\ v - v_{0}&=&\frac{\tan(au)}{a}\left[p^{2}_{x_0}(1 + 2q^2 ) + p^{2}_{y_{0}}\right ] + u\left[\frac{\epsilon}{4}(1 + 3\sqrt{1+q^{2}})\right . \nonumber \\ & & \left.-\frac{3p^2_{x_0}}{8}(5q^{2}+2 - 2\sqrt{1+q^{2}})\right ] + \frac{2p_{x_{0}}p_{y_{0}}q}{a\cos(au ) } \nonumber \\ & & + \frac { ( \sqrt{1+q^{2}}-1 ) \sin(2au)}{8a}\left[\frac{p^{2}_{x_{0}}}{2 } ( \sqrt{1+q^{2}}-1)-\epsilon\right]\end{aligned}\ ] ] where @xmath32 for null and @xmath33 for time like geodesics and @xmath34 and @xmath35 are constants of integration .
it can be checked easily that for @xmath10 our geodesics agree with those of the region ii of the bs metric [ 8 ] .
it is clear to see that if either @xmath36 or @xmath35 is nonzero then @xmath37 becomes positive for @xmath38 , and particles can pass from region ii to the region iv .
geodesics that remain in region ii are @xmath39 where @xmath40 .
the effect of cross polarization is that more geodesics remains in region ii relative to the parallel polarization case . on physical grounds
this result could be anticipated because cross polarization behaves like rotation which creates a pushing out effect in the non - inertial frames . to test the stability of quasiregular singularity ,
let us consider a test em field whose vector potential is choosen appropriately as in [ 9 ] to be @xmath41 with arbitrary functions @xmath42 and @xmath43 .
the only nonzero energy - momentum for this test em field is @xmath44\ ] ] in which a prime denotes derivative with respect to @xmath37 .
both of scalars @xmath45 and @xmath2 vanish , predicting that qr singularities are not transformed into a scs . in the ppon frame .
@xmath46 we find that @xmath47 are given in terms of @xmath48 by @xmath49 for@xmath50 and @xmath51 , otherwise .
the divergence of this quantity predicts the occurence of nscs and therefore qr singularity must be unstable .
+ the stability conjecture therefore correctly finds that these qr singularities are unstable .
however , the same stability conjecture does not find correctly the nature of the singularity . as we have discussed in section ii
, the interior of the interaction region has no scs .
the only scs is on the null boundaries .
clarke and hayward have analysed these singular points for a collinear bs spacetime and found that the singularity nature of surfaces @xmath52 and @xmath53 are qr . this observation can also be used in the cross polarized version of bs spacetime , because the order of diverging terms in @xmath54 and @xmath13 are the same .
the qr singularity structure formed in the incoming region of bs problem remains unchanged in the case of cross polarized version of the same problem .
let us now consider the stability of these qr singularities by imposing a test scalar field in region ii which is the one of the incoming region bounded by the qr singularity .
the massless scalar field equation is given by @xmath55 where we consider @xmath56 independent scalar waves so that a particular solution to this equation is obtained as in the ref ( ) @xmath57 where @xmath58 and @xmath59 are arbitrary functions .
the stress energy tensor is given by @xmath60 the corresponding non - zero stress - energy tensors for the test scalar wave is obtained by taking @xmath61 as , @xmath62f(v)f'(v ) } { 8\pi f^{2 } } \nonumber \\
t_{xy}&=&t_{yx}=\frac{aq\sin(au ) \tan(au ) f(v)f'(v)}{8 \pi f^{2}}\end{aligned}\ ] ] it is observed that each component diverges as the qr singularity @xmath63 is approached .
+ next we consider the stress energy tensor in a ppon frame .
such frame vectors are given in equation ( 11 ) .
the stress - energy tensor is @xmath64 the nonzero components are ; @xmath65 \nonumber \\
t_{01}&=&t_{10}=\left(\frac{\sec^{2}(au)}{16 \pi}\right)\left[\frac{a^{2 } \tan^{2}(au)f^2}{f^{2}}-f'^{2}(v)\right ] \nonumber \\
t_{22}&=&t_{33}=\left(\frac{a\sec^{2}(au)\tan(au)f(v)f'(v)}{8\pi f^{3}}\right)\left[f^{2 } + 2q^{2}\sin^{2}(au)\right ] \nonumber \\ t_{32}&=&t_{23}=\frac{aq\sec^{3}(au)\sin^{2}(au)f'(v)f(v)\sqrt{f^{2 } + q^{2}\sin^{2}(au)}}{4\pi f^{3}}\end{aligned}\ ] ] these components also diverge as the singularity @xmath66 is approached . by the conjecture
, this indicates that the qr singularity will be transformed into a curvature singularity .
finally we calculate the scalar @xmath67 .
@xmath68 \right\}\end{aligned}\ ] ] which also diverges as @xmath69 . from these analyses
we conclude that the curvature singularity formed will be an scs .
+ hence , the conjecture predicts that the qr singularities of cross polarized version of bs spacetime are unstable .
it is predicted that the qr singularities will be converted to scalar curvature singularities if generic waves are added .
the similar results have also been obtained by hk for the bs spacetime .
hk was unable to compare the validity of the conjecture by an exact back - reaction solution .
in the next section we present an explicit example that represents cross - polarized em field coupled with scalar field .
in the former sections , we applied hk stability conjecture to test the stability of qr singularities in the incoming region of cpbs spacetime . according to the conjecture
these mild singularities are unstable . in order to see the validity of the conjecture we introduce this new solution .
+ let the metric be ; @xmath70\ ] ] the new solution is obtained from the electrovacuum solution .
the ems solution is generated in the following manner .
the lagrangian density of the system is defined by @xmath71\end{aligned}\ ] ] which correctly generate all ems field equations from a variational principle . here
@xmath72 is the scalar field and we define the em potential one - form ( with coupling constant @xmath73 ) by @xmath74 where @xmath75 and @xmath76 are the components in the killing directions . variation with respect to @xmath77 and @xmath72 yields the following ems equations .
@xmath78 \\ 2b_{uv}&=&-v_{v}b_{u}-v_{u}b_{v}-tanhw\left(w_{v}b_{u}+w_{u}b_{v}\right ) \nonumber \\ & & -e^{v}\left[2b_{uv}tanhw + w_{u}b_{v}+w_{v}b_{u}\right]\end{aligned}\ ] ] where @xmath79 and @xmath80 are the newmann - penrose spinors for em plane waves given as follows @xmath81 \nonumber \\ \phi_{0}&= & \frac{e^{u/2}}{\sqrt{2}}\left[e^{-v/2}\left(isinh\frac{w}{2 } + cosh\frac{w}{2}\right)a_{v}\right . \nonumber \\ & & \left.+e^{v/2}\left(icosh\frac{w}{2}+sinh\frac{w}{2}\right)b_{v}\right ] \end{aligned}\ ] ] the remaining two equations which do not follow from the variations , namely @xmath82 are automatically satisfied by virtue of integrability equations .
+ the metric function @xmath83 can be shifted now in accordance with + @xmath84 where @xmath85 and @xmath86 satisfy the electrovacuum em equations .
integrability condition for the equation ( 31 ) imposes the constraint , @xmath87=0\ ] ] from which we can generate a large class of ems solution . as an example , for any @xmath88 satisfying the massless scalar field equation the corresponding @xmath89 is obtained from @xmath90 the only effect of coupling a scalar field to the cross polarized em wave is to alter the metric into the form , @xmath91 here @xmath92 and @xmath93 represents the solution of electrovacuum equations and @xmath89 is the function that derives from the presence of the scalar field .
+ it can be easily seen that for @xmath10 our solution represents pure em bs solution coupled with scalar field .
it constitutes therefore an exact back reaction solution to the test scalar field solution in the bs spacetime considered by hk ( ) .
it is clear to see that the weyl scalars are nonzero and scs is forming on the surface @xmath94 .
this is in aggrement with the requirement of stability conjecture introduced by hk . for @xmath24 the obtained solution forms the exact back reaction solution of the test scalar field solution in the cpbs spacetime . in appendix b , we present the weyl and maxwell scalars explicitly .
+ from the explicit solutions we note that , the coulomb component @xmath95 remains regular but @xmath13 and @xmath54 are singular when @xmath96 or @xmath97 .
this indicates that the singularity structure of the new solution is a typical scs .
this result is in complete agreement with the stability conjecture .
* a ) * let us assume first null test dusts moving in the cpbs background . for simplicity
we consider two different cases the @xmath98 and @xmath99 projections of the spacetime .
we have in the first case @xmath100 where we have used the coordinates @xmath101 according to @xmath102 the energy - momentum tensor for two oppositely moving null dusts can be chosen as @xmath103 where @xmath104 and @xmath105 are the finite energy densities of the dusts .
the null propagation directions @xmath106 and @xmath107 are @xmath108 with @xmath109 @xmath110 we observe from ( 1 ) that @xmath111 which is finite for @xmath112 .
the components of energy - momentum tensors in ppon frames are proportional to the expression ( 38 ) .
this proportionality makes all the components of energy - momentum tensors finite . in the limit as @xmath113 which reduces our line element to the bs
this expression diverges on the horizon @xmath114 .
trace of the energy - momentum is obviously zero therefore we must extract information from the scalar @xmath115 .
one obtains , @xmath116 the projection on @xmath99 , however is not as promising as the @xmath98 case .
consider the reduced metric @xmath117 a similar analysis with the null vectors @xmath118 with @xmath119 @xmath120 yields the scalar @xmath121 from the metric ( 1 ) we see that @xmath122 which diverges on the horizon @xmath123 .
the scalar @xmath124 constructed from the test dusts therefore diverges .
the ppon components of the energy momentum tensors are also calculated and it is found that all of the components are proportional to the expression ( 42 ) .
this indicates that the components of energy - momentum tensor diverges as the singularity is approached .
when we consider the hk stability conjecture an exact back reaction solution must give a singular solution .
we present now an exact back reaction solution of two colliding null shells in the interaction region of the cpbs spacetime . + * b ) * the metric given by @xmath125 where @xmath126 with @xmath127 positive constants was considered by wang [ 12 ] to represent collision of two null shells ( or impulsive dusts ) .
the interaction region is transformable to the de sitter cosmological spacetime . in other words
the tail of two crossing null shells is energetically equivalent to the de sitter space .
it can be shown also that the choice of the conformal factor @xmath128 , with @xmath129 positive constants becomes isomorphic to the anti - de sitter space .
+ the combined metric of cpbs and colliding shells can be represented by @xmath130 this amounts to the substitutions @xmath131 where @xmath132 correspond to the metric functions of the cpbs solution . under this substitutions
the scale invariant weyl scalars remain invariant ( or at most multiplied by a conformal factor ) because @xmath133 is the combination that arise in those scalars . the scalar curvature , however , which was zero in the case of cpbs now arises as nonzero and becomes divergent as we approach the horizon .
appendix c gives the scalar quantities @xmath134 and @xmath135 .
thus the exact back reaction solution is a singular one as predicted by the conjecture .
it is further seen that choosing @xmath136 , which removes one of the shells leaves us with a single shell propagating in the @xmath37- direction . from the scalars we see that even a single shell gives rise to a divergent back reaction by the spacetime .
the horizon , in effect , is unstable and transforms into a singularity in the presence of two colliding , or even a single propagating null shell .
let us note as an alternative interpretation that the metric ( 43 ) may be considered as a colliding em waves in a de sitter background .
collision of em waves creates an unstable horizon in the de sitter space which is otherwise regular for @xmath137 and @xmath138 .
in this paper we have analysed the stability of qr singularities in the cpbs spacetime .
three types of test fields are used to probe the stability .
first we have applied test em field to the background cpbs spacetime . from the analyses we observed that the qr singularity in the incoming region becomes unstable according to the conjecture , and
it is transformed into nsc singularity .
this is the prediction of the conjecture .
however , the exact back - reaction solution shoes that beside the true singularities on the null boundaries @xmath139 and @xmath140
. there are quasiregular singularities in the incoming regions .
the interior of interaction region is singularity free and the hypersurface @xmath141 is a killing - cauchy horizon .
as it was pointed out by hk in the case of colliding em step waves , conjecture fails to predict the correct nature of the singularity in the interaction region .
we have also discovered the same behaviour for the cross polarized version of colliding em step waves .
the addition of cross polarization does not alter the existing property . +
secondly we have applied test scalar field to the background cpbs spacetime .
the effect of scalar field on the qr singularity is stronger than the effect of em test field case .
we have obtained that the qr singularity is unstable and transformes into a scs . to check the validity of the conjecture
, we have constructed a new solution constituting an exact back reaction solution to the test scalar field in the cpbs spacetime .
the solution represents the collision of cross polarized em field coupled to a scalar field .
an examination of weyl and maxwell scalars shows that @xmath142 and @xmath143 diverge as the singularity is approached and unlike the test em field case the conjecture predicts the nature of singularity formed correctly .
+ finally , we have introduced test null dusts into the interaction region of cpbs spacetime . the conjecture predicts that the ch are unstable and transforms into scs .
this result is compared with the exact back - reaction solution and observed that the conjecture works .
to determine the type of singularity in the incoming region of cpbs spacetime , the riemann tensor in local and in ppon frame must be evaluated .
non - zero riemann tensors in local coordinates are found as follows .
@xmath144 \nonumber \\
-r_{uyuy}&= & e^{-u - v } \left [ \phi_{22 } coshw + ( i m \psi_4 ) sinhw - re \psi_4 \right ] \nonumber \\ r_{uxuy}&= & e^{-u}\left [ \phi_{22 } sinhw + ( im\psi_4)coshw\right]\end{aligned}\ ] ] where @xmath145 \nonumber \\ i m \psi_4&=&-\frac{i}{2}\left(w_{uu } -u_uw_u + m_uw_u -v^2_u coshw sinhw \right ) \nonumber \\ \phi_{22}&=&\frac{1}{4}\left(2u_{uu}-u^2_u - w^2_u - v^2_u cosh^2w \right)\end{aligned}\ ] ] note that in region ii the weyl scalar @xmath146 , therefore only @xmath143 exists .
it is clear that @xmath147 in the limit @xmath27 , so that all of the components vanish @xmath148 to find the riemann tensor in a ppon frame , we define the following ppon frame vectors ; @xmath149 in this frame the non - zero components of the riemann tensors are : @xmath150 in the limit of @xmath27 , we have the following results @xmath151 which are all finite .
this indicates that the apparent singularity in region ii ( one of the incoming region ) is a quasiregular singularity .
in order to calculate the weyl and maxwell scalars we make use of the cx line element @xmath152 where @xmath153 @xmath154 is given in equation ( 4 ) and we have chosen @xmath155 , such that the new coordinates @xmath156 are related to @xmath157 by @xmath158 the weyl and maxwell scalars are found as @xmath159 \\ & & \nonumber \\ \psi_0&=&-e^{\gamma - i\lambda } \left [ 3r + \frac{1}{4f\sigma \sin \theta
\sin \psi } \left(\sigma \sin(\psi -\theta)-\sigma_{\theta } \sin \psi \sin\theta \right .
\nonumber \\ & & \nonumber \\ & & \left.\left.+ i\sin \alpha \sin^2 \theta \sin \psi \right ) \left(\gamma_{\psi}+\gamma_{\theta}\right)\right ] \\ & & \nonumber \\ 2\phi_{00}&= & e^{\gamma}\left[\frac{\cos \alpha}{\sigma^2}-\frac { \sin(\psi+\theta)(\gamma_{\theta}+\gamma_{\psi})}{2f\sin
\psi \sin \theta } \right ] \\ & & \nonumber \\ 2\phi_{22}&= & e^{\gamma}\left[\frac{\cos \alpha}{\sigma^2}-\frac{\sin(\theta- \psi)(\gamma_{\theta}-\gamma_{\psi})}{2f\sin \psi
\sin \theta } \right ] \\ & & \nonumber \\
-2\phi_{02}&=&e^{\gamma + i\lambda}\frac{\cos \alpha}{\sigma^2}\end{aligned}\ ] ] where @xmath160 \\ & & \\
e^{i\lambda}&=&\frac{\sin \theta + \sigma \sin \psi + i \sin \psi \sin \theta \cos \psi } { \sin \theta + \sigma \sin \psi - i \sin \psi \sin \theta \cos \psi}\end{aligned}\ ] ]
the non - zero weyl and maxwell scalars for the collision of null shells in the background of cpbs spacetime are found as follows .
@xmath161 \theta(u ) \theta(v ) \\ & & \nonumber
\\ 4\phi e^{-m}\lambda&= & \left [ ( a\beta + \alpha b)\tan(au+bv)+(a \beta -\alpha b)\tan(au - bv)\right .
\nonumber \\ & & \nonumber \\ & & \left.+\frac{4\alpha \beta}{\phi } \right ] \theta(u ) \theta(v ) \\ & & \nonumber \\ \phi_{22}&=&(\phi_{22})_{cpbs } + \left(\frac{\alpha e^m } { \phi}\right)\left [ \delta(u ) \right.\nonumber \\ & & \nonumber \\ & & \left .
- \theta(u)\left(a \pi + \frac{u}{(1-u^2)(1-v^2)}\right ) \right]\\ & & \nonumber \\ \phi_{00}&=&(\phi_{00})_{cpbs } + \left(\frac{\beta e^m } { \phi}\right)\left [ \delta(v ) \right .
\nonumber \\ & & \nonumber \\ & & \left.+ \theta(v)\left(b \pi - \frac{v}{(1-u^2)(1-v^2)}\right)\right ] \\ & & \nonumber \\ \phi_{02}&= & ( \phi_{02})_{cpbs}+\left ( \frac{e^m}{4fy\phi}\right ) \left[\frac{1}{f}\left(\alpha q \theta(u ) + \beta p \theta(v)\right ) \right .
\nonumber \\ & & \nonumber \\ & & \left .
+ iq\left(\alpha l \theta(u ) + \beta k \theta(v)\right)\right]\end{aligned}\ ] ] where @xmath162 \\ & & \\ p&=&a\left[2q^2\sin(au+bv)\cos(au - bv)+f^2\left(\tan(au - bv)-\tan(au+bv)\right ) \right . \\ & & \\ & & \left .
+ 2f\cos(au - bv)\sin(au - bv)\left(\sqrt{1+q^2}-1\right)\right ] \\ & & \\ y&=&\left(1+\frac{q^2}{f^2}\tan(au+bv)\sin(au+bv)\cos(au - bv)\right)^{1/2}\\ & & \\ k&=&\frac{a}{\sqrt{\cos(au+bv)\cos(au - bv)}}\left[\frac{\cos(au - bv ) } { \cos(au+bv)}+\sin2au \right .
\\ & & \\ & & \left .
-\frac{2\left(\sqrt{1+q^2}-1\right)\sin(au+bv)\cos(au - bv ) \tan(au - bv)}{f}\right ] \\ & & \\ l&=&\frac{b}{\sqrt{\cos(au+bv)\cos(au - bv)}}\left[\frac{\cos(au - bv ) } { \cos(au+bv)}+\sin2bv \right . \\ & & \\ & & \left
. + \frac{2\left(\sqrt{1+q^2}-1\right)\sin(au+bv)\cos(au - bv ) \tan(au - bv)}{f}\right ] \\ & & \\ \pi&=&\frac{\left(\sqrt{1+q^2}-1\right)\sin(2au-2bv)}{\sqrt{1+q^2}+1 + \left(\sqrt{1+q^2}-1\right)\sin^2(au - bv)}\end{aligned}\ ] ] 99 j.b .
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fig.1(a ) : single plane waves with added colliding test fields indicated by arrows .
ch exists on the surface @xmath163 .
+ ( b ) : colliding plane wave spacetime with ch s in the incoming regions at @xmath164 and @xmath165 .
test fields are added to test the stability of ch existing in region iv . | the quasiregular singularities ( horizons ) that form in the collision of cross polarized electromagnetic waves are , as in the linear polarized case , unstable .
the validity of the helliwell - konkowski stability conjecture is tested for a number of exact backreaction cases . in the test electromagnetic case
the conjecture fails to predict the correct nature of the singularity while in the scalar field and in the null dust cases the aggrement is justified . | arxiv |
x - ray studies of fairly normal " galaxies , with high - energy emission not obviously dominated by a luminous active galactic nucleus ( agn ) , have recently been extended to cosmologically interesting distances in the deep field ( cdf ) surveys , which have now reached 1 ms of exposure ( cdf - n : hornschemeier et al .
2001 , hereafter paper ii ; brandt et al .
2001b , hereafter paper v ; cdf - s : tozzi et al . 2001 ; p.
rosati et al . , in prep . ) .
galaxies with @xmath8 are detected in appreciable numbers at 0.52 kev fluxes below @xmath9 erg @xmath6 s@xmath7 ( e.g. , paper ii ) ; the cdf - n survey goes almost two orders of magnitude fainter , detecting significant numbers of normal galaxies among the population of x - ray sources making the diffuse x - ray background ( xrb ; paper ii ; a.j .
barger et al . , in prep . ) .
these normal galaxies contribute as much as 510% of the xrb flux in the 0.52 kev band .
the bulk of the energy density of the xrb is certainly explained by agn , but the investigation of the typical " galaxy , whether its x - ray emission is dominated by a population of x - ray binaries , hot interstellar gas , or even a low - luminosity agn , is an equally important function of deep x - ray surveys .
normal galaxies are likely to be the most numerous extragalactic x - ray sources in the universe and are expected to dominate the number counts at 0.52 kev fluxes of @xmath10@xmath11 erg @xmath6 s@xmath7 ( ptak et al .
2001 ) .
the cdf - n has reached the depths necessary to detect individually many normal [ @xmath12 ; @xmath13 is from 0.52 kev ] @xmath14 galaxies to @xmath15 , corresponding to a look - back time of @xmath16 gyr ( @xmath17 km s@xmath7 mpc@xmath7 , @xmath18 , and @xmath19 are adopted throughout this paper ) .
reaching larger look - back times presents the exciting possibility of detecting the bulk x - ray response to the heightened star - formation rate at @xmath203 ( e.g. , madau et al . 1996 ) .
one thus expects the x - ray luminosity per unit @xmath2-band luminosity to be larger at @xmath211 in the past due to the increased energy output of x - ray binary populations at @xmath203 ; this x - ray emission represents a fossil record " of past epochs of star formation ( e.g. , ghosh & white 2001 ; ptak et al .
therefore , measurements of the x - ray luminosities of typical galaxies can constrain models of x - ray binary production in galaxies . while x - ray emission from individual galaxies is not easily detected at @xmath22 ,
it is possible to estimate the emission at their extremely faint flux levels using statistical methods such as stacking , a technique implemented successfully on the cdf - n survey data in several previous studies .
these include the detection of x - ray emission from the average @xmath21 bright ( @xmath23 ) galaxy in the hubble deep field - north ( ) described in brandt et al .
( 2001a , hereafter paper iv ) and a study of x - ray emission from @xmath244 lyman break galaxies identified in the ( brandt et al . 2001c , hereafter paper vii ) . encouraged by the success of these analyses ,
we extend here the study of normal galaxies to the entire plus flanking fields region , now concentrating on galaxies at @xmath25 to complement the study of @xmath26 galaxies performed in paper vii .
we focus on this redshift range due to the extensive spectroscopic redshift coverage ( cohen et al . 2000 and references therein ) and superb imaging which has allowed a comprehensive galaxy morphology study ( van den bergh , cohen , & crabbe 2001 ) .
the cdf - n data provide extremely deep x - ray coverage over this area ( see figure 7 of paper v for the exposure map of this region ) ; the point - source detection limits in this region of the cdf - n survey in the 0.52 kev and 28 kev bands are @xmath27 erg @xmath6 s@xmath7 and @xmath28 erg @xmath6 s@xmath7 , respectively .
in this study , we place observational constraints on the evolution of the ratio of x - ray luminosity to @xmath2-band luminosity of normal " spiral galaxies up to @xmath29 ; this ratio is an indicator of the current level of star formation in a galaxy ( e.g. , david , jones , & forman 1992 ; shapley et al .
we also place constraints on the fraction of the diffuse xrb explained by galaxies lingering just below the cdf - n detection threshold , and thus the contribution to the xrb by normal galaxies .
spectroscopic redshifts for the galaxies are drawn from the catalogs of cohen et al .
( 2000 ) , cohen ( 2001 ) , and dawson et al .
( 2001 ) in the range @xmath30 .
spectroscopic redshift determination is difficult in the range @xmath31 due to the absence of strong features in the observed - frame optical band and the lack of the lyman break feature useful to identify higher redshift objects .
we have therefore used the deep photometric redshift catalog of fernndez - soto , lanzetta , & yahil ( 1999 ) for the redshift interval @xmath32 , which allows some overlap in redshift space with the spectroscopic catalogs for cross - checking .
the spectroscopic catalogs cover the entire hdf - n plus a substantial fraction of the flanking fields region , whereas the photometric catalog only covers the hdf - n .
we shall refer to these two samples as the spectroscopic sample " and the photometric sample " throughout the rest of this letter . for the spectroscopic sample ,
the @xmath33-band magnitude was used to filter the sources by optical luminosity , as this is best matched to rest - frame @xmath2 over most of the redshift range under consideration here .
the @xmath33 magnitudes are those given in barger et al .
( 1999 ) for the hawaii flanking fields area . for the photometric sample , the f814w ( hereafter @xmath34 ) magnitudes of fernndez - soto et al .
( 1999 ) were used .
we chose galaxies which had no x - ray detection within 40 in the 0.58 kev ( full ) , 0.52 kev ( soft ) and 28 kev ( hard ) bands down to a wavdetect ( freeman et al .
2002 ) significance threshold of @xmath35 in the restricted acis grade set of paper iv .
this low detection threshold ensures that our study does not include sources with x - ray emission just below the formal detection limits of paper v. we have attempted to construct a sample of galaxies similar to spiral galaxies in the local universe . to accomplish this ,
we have used the morphological classes of van den bergh et al .
( 2001 ) for galaxies from @xmath36 in the hdf - n and the flanking fields . to simplify the morphological filtering
, we have cast objects in the van den bergh et al .
( 2001 ) catalog into the following four classes : ( 1 ) e / s0 " and e " , ( 2 ) merger " , ( 3 ) sa"sc " , including proto - spirals and spiral / irregulars , and ( 4 ) irr " , peculiar " and/or tadpole " .
we then filtered the catalog to keep only classes ( 2 ) and ( 3 ) . filtering the photometric sample is more difficult due to the faintness of many of the sources and problems due to morphological evolution with redshift .
we have used the spectral energy distribution ( sed ) classifications of fernndez - soto et al .
( 2001 ) to exclude all galaxies of type e " .
comparison of the source lists reveals that , within the area covered by both , @xmath37 70% of galaxies identified through the two methods are in common .
since the evolution of x - ray properties with redshift is of interest , we have made an effort to study objects with comparable optical luminosities at different redshifts .
this is particularly important due to the non - linear relationship between x - ray luminosity and @xmath2-band luminosity for some types of spiral galaxies ( @xmath38 ; e.g. fabbiano & shapley 2001 ) . using the value of @xmath39 in the @xmath40-band as determined by blanton et al .
( 2001 ) for a large sample of galaxies in the sloan digital sky survey , we determined the value of @xmath39 in the @xmath2-band .
the sloan filter @xmath40 is best matched to @xmath2 ; the resulting value of @xmath39 in the @xmath2 band is @xmath41 . to ensure that our results are not sensitively dependent upon the galaxy sed used to determine the optical properties , we have used both the sa and sc galaxy seds of poggianti et al .
( 1997 ) to calculate @xmath33 and @xmath34 vs. @xmath42 for an @xmath39 galaxy using the synthetic photometry package synphot in iraf .
these calculations are shown in figure [ sample_definition ] .
note the close similarity between the sa and sc tracks ; this is because the @xmath33 band corresponds to rest - frame @xmath2 in the middle of our redshift range .
also plotted in figure 1 are the 151 galaxies in the spectroscopic sample with spiral or merger morphology having @xmath43 and the 651 galaxies in the photometric sample with sed class other than e " having @xmath44 .
these galaxies were filtered by optical flux to lie within 1.5 mags of the @xmath39 galaxy tracks discussed above ; the galaxy samples constructed assuming sa and sc seds were identical ( or nearly so ) for all redshifts up to @xmath45 .
galaxies meeting the optical magnitude filter were then divided by redshift into several bins ; these bins were constructed to ensure that there were @xmath46 galaxies per bin .
the number of galaxies , median redshift , median look - back time and median optical magnitude for each bin are listed in table 1 . in figure 1
, we mark all the objects in the sc sed sample with colors indicating the different redshift bins .
table 1 also includes the number of galaxies rejected from each redshift bin due to the presence of an x - ray detection within 40 ; this exclusion radius ensures that our results will not be adversely affected by the wings of the psf of very bright x - ray sources .
these galaxies satisfied both the optical magnitude and morphology filtering constraints and were rejected only due to x - ray detection .
this exclusion criterion is very conservative , however , considering that our astrometry is accurate to @xmath476 in the area under consideration ( see paper v ) . to allow for the off - nuclear nature of some of the x - ray sources found in normal galaxies ( e.g. , paper iv ) , we consider galaxies to be highly confident x - ray detections if the x - ray source is within 15 of the galaxy s center
this matching radius is also well matched to the chandra psf .
off - axis for 0.52 kev and the 83% encircled - energy radius for 0.58 kev .
] we therefore also give the number of galaxies having an x - ray detection within 15 in table 1 .
the x - ray imaging data at each position were stacked in the same manner as in paper vii , keeping the 30 pixels whose centers fall within an aperture of radius 15 .
the detection significance in each band was assessed by performing 100,000 monte - carlo stacking simulations using local background regions as in paper vii .
a source is considered to be significantly detected if the number of counts over background exceeds that of 99.99% of the simulations .
no single source in the stacking sample appeared to dominate the distribution , demonstrating the effectiveness of our selection criteria .
stacking of the galaxies in the redshift bins described in table [ sample_table ] and figure [ sample_definition ] resulted in significant detections in the soft band for all of the redshift bins up to @xmath48 ( see table 2 ) .
we also stacked galaxies in the redshift bin @xmath49 , but there was not a significant detection .
the results for the two different spiral galaxy sed samples are nearly or exactly identical except for the detection in the highest redshift bin ( @xmath50 ) .
we adopt a @xmath51 power law for the calculation of x - ray fluxes and luminosities , assuming that these galaxies are similar to spiral galaxies in the local universe and have their x - ray emission dominated by x - ray binaries ( e.g. , kim , fabbiano , & trinchieri 1992 ) .
while there were several cases of significant detections in the full band , there were no highly significant detections in the hard band .
given the variation of effective area and background rate with energy , the signal - to - noise ratio for sources with the assumed spectrum is highest in the soft band and lowest in the hard band , so this behavior is expected . the flux level of the soft - band detections for the spectroscopic sample is ( 56)@xmath52 erg @xmath6 s@xmath7 . the corresponding rest - frame 0.52 kev luminosities for the average galaxy are @xmath53 erg s@xmath7 for the lowest redshift bin and @xmath54 erg s@xmath7 for the highest redshift bin .
for the photometric sample , the soft - band flux level of the detections is ( 35)@xmath52 erg @xmath6 s@xmath7 .
we also give fluxes for the less significant detections in the full band for those redshift bins having highly significant soft - band detections .
we have investigated the properties of the sources which were rejected from the stacking samples due to an x - ray detection at or near the position of the galaxy ( the numbers of such sources are given in the last column of table 1 ) .
there are only 15 distinct galaxies with x - ray detections within 15 . of these 15 sources ,
three are broad - line agn , which are clearly not normal galaxies .
one object has a photometric redshift which differs significantly from its spectroscopic redshift . since the optical properties of this object at its spectroscopic redshift place it outside our sample boundaries
, we have rejected it .
one object is very near another x - ray source which has been positively identified with a narrow - line agn .
thus , there are a total of 10 normal " galaxies positively identified with x - ray sources within this sample , consituting a small minority of the galaxies under study here .
the worst case is in the lowest redshift bin where 15% of the galaxies had x - ray detections .
figure [ sbhistogram ] shows a histogram of @xmath55 values calculated for both the stacking samples and the individually x - ray detected galaxies .
the individually x - ray detected galaxy set does possibly contain some lower - luminosity agn , including the agn candidate ( see papers ii and iv ) . with the exception of the objects in the redshift bin with median @xmath56 ,
figure [ sbhistogram ] shows that typically the x - ray luminosities of the individually detected objects are on average much higher than those of the stacked galaxies ; they are sufficiently more luminous as to appear atypical of the normal galaxy population . for the lowest redshift bin ,
it is plausible that our results are moderately biased by the exclusion of the x - ray detected sources . in 4 , we therefore also give results which include the individually x - ray detected objects in the sample average for this lowest redshift bin . for additional comparison
, we have considered the radio properties of the individually detected galaxies and the stacked galaxies using the catalogs of richards et al .
( 1998 ) and richards ( 2000 ) .
the percentage of radio detections among the individually x - ray detected galaxies is higher than that among the stacked galaxies ( @xmath57% vs. @xmath585% ) .
this possible difference between the two populations is significant at the 93% level as determined using the fisher exact probability test for two independent samples ( see siegel & castellan 1988 ) . due to the x - ray luminosity difference and possible difference in radio properties , and to the fact that these galaxies constitute a small minority of those under study
, we are confident we have not biased our determination of the properties of the typical galaxy by omitting these x - ray detected objects from further consideration .
in figure [ lxlb]a , we plot the x - ray - to - optical luminosity ratio @xmath59 for each stacked detection , where @xmath13 is calculated for the rest - frame 0.52 kev band .
we have also plotted our approximate sensitivity limit in figure [ lxlb]a , which is simply the corresponding @xmath60 x - ray luminosity detection limit achieved for a 30 ms stacking analysis divided by @xmath61 .
we do not expect to detect galaxies having less x - ray emission per unit @xmath2-band luminosity than this value .
for comparison with the local universe , we also plot the mean @xmath59 for spiral galaxies of comparable @xmath62 from the sample of shapley et al .
this sample includes 234 spiral galaxies observed with _
einstein _ and excludes agn where the x - ray emission is clearly dominated by the nucleus .
the galaxies in the shapley et al .
( 2001 ) sample all have @xmath63 ; median redshift is @xmath64 . in figure [ lxlb]b ,
we plot @xmath59 versus @xmath65 ; the values up to @xmath66 are consistent with what is observed in the shapley et al .
sample for objects with comparable @xmath65 , although they are toward the high end of what is observed .
this is consistent with figure [ lxlb]a , which shows the average @xmath59 derived from stacking being somewhat higher than that for the shapley et al .
galaxies of comparable optical luminosity .
there is a slight increase ( factor of 1.6 ) in the average @xmath67 from the local universe to @xmath68 . for the highest redshift bin ( @xmath50 ) ,
the results become somewhat sensitive to the galaxy sed one assumes for determining the optical properties .
we suspect that an sc galaxy sed is more appropriate at this epoch due to the higher prevalence of star formation .
adopting this sed , we find that @xmath69 has increased somewhat more substantially ( @xmath70 times ) at @xmath50 .
one may also constrain star - formation models using only the x - ray properties of these galaxies .
the average x - ray luminosity of the spiral galaxies in the shapley et al .
( 2001 ) sample having the same range of @xmath65 as used in this study is @xmath71 erg s@xmath7 ( converted to 0.52 kev ) .
the average galaxy in our stacking sample has a luminosity @xmath72 times higher at @xmath68 .
this increases to @xmath73 times higher at @xmath74 .
the @xmath68 value is most likely affected by some bias due to the exclusion of legitimate normal galaxies in the lowest redshift bin ( see 3 ) .
if we include these individually x - ray detected objects , then the average galaxy in our stacking sample has a luminosity @xmath75 times higher at @xmath68 , consistent with the predictions of ghosh & white ( 2001 ) that the x - ray luminosity of the typical sa - sbc spiral galaxy should be @xmath75 times higher at @xmath76 . including the x - ray detected galaxies does not significantly affect our results for the interval @xmath77 ( the difference is @xmath78% ) .
however , since the x - ray luminosities of the x - ray detected objects with @xmath74 are substantially higher ( by an order of magnitude ) than the x - ray stacking averages ( see figure 2 ) , it is not appropriate to include these objects in the calculation of the average x - ray luminosity .
we thus find a smaller increase in the average x - ray luminosity of galaxies at @xmath79 than the increase by a factor of @xmath80 predicted by ghosh & white ( 2001 ) .
we find that the average x - ray luminosities of galaxies have not evolved upward by more than a factor of @xmath73 by @xmath74 , regardless of exclusion or inclusion of x - ray detected objects .
the range of average 0.52 kev fluxes for the spiral galaxies studied here is ( @xmath46@xmath5 erg @xmath6 s@xmath7 .
these x - ray fluxes are consistent with independent predictions made by ptak et al .
( 2001 ) that galaxies of this type will be detected at 0.52 kev flux levels of @xmath81 erg @xmath6 s@xmath7 ( converted from their 210 kev prediction assuming a @xmath82 power law ) . assuming a 0.52 kev xrb flux density of @xmath83 erg @xmath6 s@xmath7 deg@xmath84 ( garmire et al .
2001 ) , we have identified @xmath85% of the soft xrb as arising from spiral galaxies not yet individually detected in deep surveys .
many of these objects should be sufficently bright to be detected with acis exposures of @xmath86 ms , which should be achievable in the next several years of the mission .
we thank alice shapley for useful discussions and sharing data .
we gratefully acknowledge the financial support of nasa grant nas 8 - 38252 ( gpg , pi ) , nasa gsrp grant ngt5 - 50247 ( aeh ) , nsf career award ast-9983783 ( wnb , dma , feb ) , and nsf grant ast99 - 00703 ( dps ) .
this work would not have been possible without the enormous efforts of the entire team .
ccccccc + 0.400.75 & sc & 0.635 & 6.34 & 21.73 & 29 & 12/6 + 0.750.90 & sc & 0.821 & 7.39 & 21.83 & 38 & 2/2 + 0.901.10 & sc & 0.960 & 8.05 & 22.29 & 30 & 6/4 + + + 0.501.00 & sc & 0.920 & 7.87 & 22.99 & 37 & 9/4 + 1.001.50 & sc & 1.200 & 8.97 & 24.28 & 64 & 6/2 + 1.001.50 & sa & 1.240 & 9.09 & 24.69 & 80 & 8/2 + rrrrrrrrrcrrc + 0.400.75 & sc & 49.6 & 31.7 & 99.99 & @xmath87 & 25.41 & 25.44 & 1.52 & 0.65 & 2.94 & 1.26 & 0.67 + 0.750.90 & sc & 58.0 & 36.4 & @xmath87 & @xmath87 & 33.05 & 33.09 & 1.36 & 0.57 & 4.92 & 2.07 & 1.33 + 0.901.10 & sc & 54.4 & 25.9 & @xmath87 & @xmath87 & 26.54 & 26.56 & 1.60 & 0.51 & 8.44 & 2.68 & 1.40 + + + 0.501.00 & sc & 42.0 & 33.5 & 99.77 & @xmath87 & 34.51 & 34.54 & 0.95 & 0.51 & 4.51 & 2.40 & 1.12 + 1.001.50 & sc & 44.8 & 38.5 & 98.94 & @xmath87 & 59.67 & 59.73 & 0.58 & 0.34 & 5.34 & 3.06 & 0.79 + 1.001.50 & sa & 45.6 & 34.2 & 98.29 & 99.95 & 74.61 & 74.68 & 0.48 & 0.24 & 4.70 & 2.36 & 0.81 + .
the blue curves give @xmath33 vs. @xmath42 for @xmath39 sa and sc galaxies ( the lower curve at higher redshift is for the sa galaxy ) . galaxies without a colored circle were not included in the stacking sample because they either were not within the range of optical luminosity specified or because an x - ray detection was found within 40 .
( b ) the black filled circles are the photometric redshift sample of fernndez - soto et al .
( 1999 ) , excluding the e "- type galaxies .
the blue curves give @xmath34 vs. @xmath42 for @xmath39 sa and sc galaxies ( the lower curve at higher redshift is for the sa galaxy ) .
[ sample_definition ] ] as a function of redshift for the stacking samples .
the redshift error bars indicate the full extent of the redshift bin ; the data points are at the median redshift value for that bin .
the solid line indicates the 2@xmath88 x - ray sensitivity limit normalized by @xmath89 .
the dashed lines above and below the solid line indicate the effect of decreasing and increasing the optical luminosity by one magnitude , respectively .
objects which have less x - ray luminosity per unit @xmath2-band luminosity than this are not expected to be detected in the current data .
the error bar on the shapley et al .
( 2001 ) data point indicates the dispersion of values in this sample .
( b ) @xmath59 vs. @xmath65 for both the shapley et al .
( 2001 ) sample ( open circles ) and the stacked detections presented here .
the error bars on @xmath59 in both figures were calculated following the numerical method described in 1.7.3 . of lyons ( 1991 ) .
the solid line in ( b ) indicates @xmath89 ; again the dashed lines correspond to one magnitude fainter and brighter than @xmath89 . | we present a statistical x - ray study of spiral galaxies in the hubble deep field - north and its flanking fields using the chandra deep field north 1 ms dataset .
we find that @xmath0 galaxies with @xmath1 have ratios of x - ray to @xmath2-band luminosity similar to those in the local universe , although the data indicate a likely increase in this ratio by a factor of @xmath33 .
we have also determined that typical spiral galaxies at @xmath1 should be detected in the 0.52 kev band in the flux range ( @xmath46@xmath5 erg @xmath6 s@xmath7 .
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