Neutralino dark matter and Higgs mediated lepton flavor violation in the minimal supersymmetric standard model
Abstract
We reexamine the prospects for the detection of Higgs mediated lepton flavor violation at LHC, at a photon collider and in decays such as , . We allow for the presence of a large, model independent, source of lepton flavor violation in the slepton mass matrix in the sector by the mass insertion approximation and constrain the parameter space using the LFV decays together with the mesons physics observables, the anomalous magnetic moment of the muon and the dark matter relic density. We further impose the exclusion limit on spinindependent neutralinonucleon scattering from CDMS and the CDF limits from direct search of the heavy neutral Higgs at the TEVATRON. We find rates probably too small to be observed at future experiments if models have to accommodate for the relic density measured by WMAP and explain the anomaly: better prospects are found if these two constraints are applied only as upper bounds. The spinindependent neutralinonucleon cross section in the studied constrained parameter space is just below the present CDMS limit and the running XENON100 experiment will cover the region of the parameter space where the lightest neutralino has large gauginohiggsino mixing.
pacs:
11.30.Fs, 11.30.Pb, 12.60.Jv, 14.80.Ly, 14.80.CpI Introduction
One of the appealing features of the minimal supersymmetric standard model (MSSM) with Rparity conservation is the presence of a neutral, stable particle, the lightest neutralino, which presents all the characteristics to be a possible candidate for accounting for the cold dark matter in the Universe kamionkowski . The amount of dark matter , where is the dark matter density normalized to the critical density of the Universe and is the reduced Hubble constant, recently has been precisely measured by the WMAP experiment wmap .
The Higgs sector of the MSSM djouadi , especially the heavy neutral Higgses and , play a prominent role in the physics of neutralino dark matter in two ways. In some region of the supersymmetric (SUSY) parameter space neutralinos yield the desired amount of relic density by annihilating into fermions through the channel resonant exchange of neutral Higgs bosons , , , the so called funnel region where . As dark matter is expected to be distributed in a halo surrounding our galaxy, neutralinos can scatter off nuclei in terrestrial detectors: coherent scattering is mediated by scalar interactions through the channel exchange of squarks and channel exchange of the CPeven neutral Higgs bosons and . These effects become sizeable when squarks are heavy and is large in reason of the enhanced Higgs bosons coupling to downtype fermions, especially for the quark which has the largest Yukawa coupling: moreover this couplings receive large radiative SUSYQCD corrections at large that can be relevant for their production in hadronhadron collisions at TEVATRON and LHC carena1 ; carena2 ; carena3 . In this scenario it is well known that mesons physics observables are very sensible to Higgs physics bkq ; isidori1 ; isidori2 ; buras ; isidori3 and put strong constraints on the parameter space. The branching ratios for lepton flavor violating decays are also enhanced near the experimental bounds bkl ; sher ; dedes ; brignole1 ; brignole2 ; arganda ; kanemura2 ; parry ; paradisi2 ; paradisi1 ; chuan ; arganda2 ; herrero .
Once a source of LFV is present in the slepton mass matrix, for example the MSSM with the celebrated see saw mechanism for generation of small neutrino masses, two different mechanisms of LFV arise: gaugemediated LFV effects through the exchange of gauginos and sleptons borzumati ; hisano1 ; hisano2 and Higgsmediated LFV effects through effective nonholomorphic Yukawa interactions for quarks and leptons bkq ; bkl . LFV Yukawa couplings of the type are induced at loop level and become particularly sizable at large . In this case the effective flavorviolating Yukawa interactions are described by the lagrangian:
(1)  
(2) 
where is the mixing angle between the CPeven Higgs bosons and , is the physical CPodd boson, are flavor indices that in the following are understood to be different (). The coefficients in Eq. (2) are induced at one loop level by the exchange of gauginos and sleptons, provided a source of slepton mixing is present. The expressions of in the mass insertion approximation are given by paradisi1 :
(3)  
(4) 
where and are the and couplings respectively, is the the Higgs mixing parameter, the gaugino mass parameters and stands for the leftleft (rightright) slepton mass matrix entry. is the derivative of the three point oneloop integral. The LFV mass insertions , , where are the offdiagonal flavor changing entries of the slepton mass matrix, are free parameters which allow for a model independent study of LFV signals.
The connection between gauginomediated LFV signals and neutralino dark matter in the seesaw mechanism implemented in mSUGRA constrained MSSM has been recently studied in Refs. barger1 ; barger2 : here it is shown that large neutrino Yukawa coupling affects the renormalization group evolution equations of SUSY parameters from the grand unification (GUT) scale to the electroweak scale (in a SO(10) GUT scenario) enhancing some LFV rates by orders of magnitude and changing also the neutralino relic density and direct and indirect detection rates.
In this paper, on the other hand, we follow a different phenomenological approach: the study is done in the framework of MSSM with real parameters assigning the value of the parameters at the weak scale without any assumption on the mechanism of SUSY breaking or the high energy theory, nor on the origin of LFV entries in the slepton mass matrix and limitate our attention to Higgs mediated flavor violating effects. We study the interplay between the assumptions of the lightest neutralino as dark matter candidate and Higgs mediated flavor violation both to constrain the MSSM parameter space and to give prediction for the neutralinonucleus scattering and LFV signals at colliders and in decays. For related studies see carena3 ; isidori4 ; carena4 .
In Section II we discuss the scan of the parameter space in the real MSSM and the imposed constraints. Than we study their effects on the spinindependent neutralinonucleon cross section and the arising correlations between supersymmetric parameters in Sec. III. In Section IV we analyse LFV signals , , at LHC and at a future photon collider. Conclusions are given in Sec. V.
Ii Constrained parameter space
We introduce LFV in the model through the mass insertions . This value ensures the largest rates in LFV processes and allow us to study the more optimistic scenarios; higher values contradict the mass insertion approximation as an expansion of propagators in these small parameters. Higgs mediated effects become eneteresting at large and and low ; further, if SUSYQCD particles are heavy Higgs effects are dominant also for neutralino dark matter. We thus scan the following real MSSM parameter space:

100 GeV 1 TeV;

2060;

500 GeV5 TeV;
The sign of is taken positive, as preferred by the SUSY explanation of the anomaly. 
150 GeV1.5 TeV;
We do not impose any relation but let them vary independently. To have large masses for gluinos we choose:
1 TeV 5 TeV. 
1 TeV5 TeV;
for the third generation of squarks: these are are varied freely without imposing any relation. For the first and the second generation the soft masses are set to be equal, , where and is another free parameter which varies in the same range. 
300 GeV 2.5 TeV;
for sleptons of the third generation which are independent parameters. For the first and the second generation the soft masses are set to be equal, , where and is another free parameter which varies in the same range. Sleptons, diffrently from squarks, can be light in order to explain the anomaly. 
2, , ;
for first and second generation the trilinear scalar couplings are set to zero.
The outlined (16dimensional) large parameter space is restricted imposing the following experimental limits:

Light Higgs and SUSY masses.
LEP, TEVATRON bounds on sparticle masses and the LEP bound on light Higgs: 114.4 GeV pdg . 
physics observables.
For the physics observables we use the theoretical and experimental numbers of Table 1 in Ref. fit . Thus we require , the upper bound on rare decays branching ratio is set to and . Finally we require . ^{1}^{1}1There is at present a discrepancy between the SM value and the experimental value of the purely leptonic decays branching ratio due to a recent analysis utfit . Given the unclear situation both on the theoretical and experimental side we do not consider it here. 
Direct dark matter detection.
The most stringent limit up to date in the neutralino mass range GeV for the neutralinonucleus spin independent cross section comes from the CDMS experiment. The upper limits from the 2008 analysis cdms and the recent final combined results cdmsfinal are reported in Fig. 1 (left panel). 
Nonstandard Higgs search at TEVATRON.
Recently CDF collaboration has published the most stringent exclusion limits in the (, ) plane in the light of negative results in the search for heavy neutral Higgs bosons in the inclusive production and the successive decay into pairs cdf . The excluded region is limited in the low , high region and is depicted in Fig. 1 (right panel).
For numerical computations we use the code DarkSusy darksusy for accelerator bounds, the neutralino relic density and direct dark matter detection in our general MSSM. DarkSusy uses the code FeynHiggs feynhiggs for SUSY and Higgs mass spectrum and Higgs widths and branching ratios. For we used the routines in DarkSusy while for MSSM contribution to the muon anomalous magnetic moments those of FeynHiggs which include also the leading and subleading twoloop contributions. For the others physics observales we used the formulas of Refs. buras ; isidori3 . We generate random models, selecting the ones which evade the listed constraints. All of them are applied at the same time with the exception of the relic density and anomaly for which we also relax the lower bounds: thus requiring only and around survive, the light grey (turquoise) points in the Figures. Requiring also around are left, the plusshaped points, finally if only 52 remain, (the square points).
Iii Neutralino dark matter
The spinindependent neutralinonucleon cross section in the limit of heavy squarks and large can be approximated as carena1 ; carena2 ; carena3 ; carena4
(5) 
where and are the lightest neutralino unitary mixing matrix elements, the nucleon mass (neglecting the mass difference between the neutron and the proton) and a factor which depends on nucleon form factors. It scales like and it is able to constrain the low high region even if to a lesser extent than flavor physics observables that scale like .
The right panel of Fig. 1 presents the allowed region in the (, ) plane: the region delimited by the line is excluded by CDF search in the channel . The left panel of Fig. 1 shows the scatter plot for the spinindependent neutralinonucleon cross section as a function of and the region excluded by CDMS cdms ; cdmsfinal . We emphasize that CDF and CDMS limits are very mild constraints as can be seen in Fig. 1. The region excluded by CDF is practically excluded by the other constraints while the CDMS limit exclude only one plusshaped point (not reported in Fig. 1) leaving untouched the regions preferred by WMAP and the anomaly. Further, the final CDMS upper limits curve exclude around 300 lightgray (magneta) points between the solid and the dotted line in Fig. 1 and it is not still constraining the more interesting region. For clarity, in all the other plots only the final CDMS limits are applied. Actually, it is more meaningful to compare the CDMS results with the plusshaped and square points: in fact the limit on scattering with nuclei are extracted from rates which depend on the local density of dark matter in our galaxy halo which is assumed to be furnished by the weakly interacting massive particle, in this case the lightest neutralino. From this point of view the negative results of these experiments are natural in the present scenario and the two events found in the signal region by CDMS collaboration cannot be explained by our scenario.
The XENON100 experiment xenon should reach the sensitivity corresponding to the dashed gray (red) line in the Figure 1 (left panel). Such sensitivity is able to cover the region with the highest cross section, GeV, where there is large higgsino component, as we will discuss below. On the other hand the region preferred by anomaly cannot be covered. We also report the prospected sensitivity goal of the XENON experiment with 1 ton detector mass xenon , dotdashed grey (red) line, which is pb for neutralino mass in the range GeV: practically all of the parameter space will be probed.
As no relation has been imposed between the neutralino mass and and between gaugino mass parameters, it is interesting to explore which correlations may emerge by the imposition of all the applied constraints.
Fig. 2, left panel, presents the scatter plot of the ratio versus where is rightright mass parameter for the stau. Points with the correct relic density abundance accumulate along the line where neutralino pair annihilation into fermions through resonant channel exchange of neutral Higgs bosons , is the dominant mechanism in large portion of the parameter space. Stau coannihilation is at work for models where and coannihilation with the second neutralino and the lightest chargino are important for larger values of the ratios.
In the right panel of Fig. 2 the gaugino fraction is plotted against the neutralino mass. For neutralino masses below 400 GeV, the preferred region by the , is pure gaugino while for masses greater than 400 GeV higgsino component is present. This effect can be seen in Fig. 1 in the spinindependent neutralino cross section which depends on (): the models with the highest cross section are the ones with GeV where the coupling is enhanced in reason of a larger higgsino component.
The left and right panels of Fig. 3 present the scatter plot in the (, ) and (, ) planes respectively. We see that models with the correct relic abundance have . The prefer models with . The models with strong gauginohiggsino mixing, , , can be probed by XENON100. We further note that most of the point in WMAP and ranges are charactherized by having high degeneracy in the gaugino masses . Such conditions give a “welltempered bino/wino” neutralino which can be realized in model with split supersymmetry as shown for example in Ref. arkani
Iv Prospects for LFV signals
The LFV decay which is more sensible to Higgs mediated effects is sher ; brignole2 ; kanemura2 ; paradisi2 ; chuan ; arganda2 ; herrero and the branching ratio reads brignole2 ; paradisi2 :
(6) 
Here MeV, and a factor which depends on the hadronisation of quarks bilinears matrix elements: in the treatment of Ref. brignole2 it is such that .^{2}^{2}2The approach using chiral perturbation theory arganda ; herrero gives results different at most by a factor two. These uncertainties will not change our conclusions. Lepton flavor violation enters through the factor
(7) 
We also consider the radiative decay which receives also important contributions by gauginoslepton loop diagrams: the factors and enter separately in the branching ratio and not through the combination in Eq. (7). For the computation we used the formulas in Ref. paradisi1 including both gaugino mediated and Higgs mediated effects.
The CPodd Higgs boson decay width and branching ratio are cannoni
(8)  
(9)  
(10) 
where is defined in Eq. (7) and we used the fact that . For the CPeven Higgs boson , the right hand sides of Eq. (10), are multiplied by a factor which is order one in our scenario where , thus the previous formulas hold for both bosons.
In Fig. 4, left panel, we present the scatter plot for the effective vertex given by Eq. (7), while the right panel shows the scatter plot of the branching ratio given by Eq. (10) as a function of . We see that they reach and respectively for models preferred by WMAP measurements, two orders of magnitude lower than what found without imposing it on the parameter space brignole1 ; moretti ; cannoni . The branching ratios of LFV decays and versus are given in Fig. 5. For models with the correct neutralino relic density abundance and preferred by the anomaly, both are generally under while relaxing the constraints lower bound they can reach the level. We remind that at a SuperB factory the present limits () can be lowered to () for the final state because the branching ratio scales linearly with the luminosity due to practically negligible background. In the case for the presence of large background the branching ratio scale as the square root of the luminosity and the sensitivity reach is one order of magnitude lower belle .
We further revisit the prospects for detection of Higgs LFV signals in collisions at LHC brignole1 ; moretti and in collisions at the photon collider option of the future International Linear Collider cannoni .
At high the dominant production mechanisms for at LHC is fusion due to the enhanced couplings. We calculate the cross section with FeynHiggs which uses the approximation
(11) 
where is the total SM cross section for production of Higgs boson with mass via fusion: to obtain the value in the MSSM it is rescaled with the ratio of the decay width of the inverse process in the MSSM over the SM decay width feynhiggs ; hein1 ; hein2 . We calculate for each random model the product the . As masses and couplings of and are practically identical as discussed above, we have .
The scatter plot is shown in Fig. 6, left panel. We see that with the nominal integrated luminosity of fb per year models which satisfy both the relic density abundance and can give up to 10 events per year (squared points), up to 40 if we relax the condition on the lower limit of (plusshaped points) and up to 200300 relaxing both the lower limits (magenta (grey) points).
In collisions the main production mechanism for is fusion choi while the is suppressed by a factor which cannot be compensated by corrections to the Yukawa coupling. In Ref. cannoni we studied in detail the fusion process where the Higgs boson is produced in the channel via a virtual pair and can be detected from its decay mode .
A good analytical approximation for the cross section is obtained using the equivalent particle approximation wherein the colliding real photons split respectively into and pairs with the subsequent fusion into the Higgs boson. The splitting functions of the photon at leading order read choi :
(12) 
thus the cross section is given by:
(13)  
(14) 
with , , is the fraction of the energy of the photon carried by the virtual lepton. In cannoni we have shown that the effect of photons spectra can be neglected, we thus consider monochromatic photons with GeV, and photonphoton luminosity 500 fb yr based on the parameters of TESLA(800) tesla .
The scatter plot of the signal cross section versus is shown in Fig. 6, right panel. Here the models which satisfy both the relic density abundance and (squared points) have maximal cross section fb, which is too small. Relaxing the lower limits cross section values up to fb are possible, giving 10 events/year.
V Summary and conclusions
In the framework of the MSSM with heavy SUSYQCD particles and large we have studied lepton flavor violation mediated by the heavy neutral Higgs , in sector both in lepton decays, and , and at high energy collider through the production and decay at LHC, , and the fusion at a photon collider, . The approach to LFV has been model independent by the use of the mass insertion approximation. We used large mass insertions to estimate the number of events in the most favourable scenario that can be obtained in future experiments. With such a large source of LFV the branching ratios of rare decays can exceed the present experimental upper bounds from BABAR and BELLE which therefore provide constraints on the MSSM parameter space in presence of LFV. Other constraints that have been imposed are limits from direct search of sparticles and of the light Higgs, physics observables, the anomaly, and recent limits from TEVATRON search of non standard Higgs bosons in the channel. In the Rparity conserving MSSM the heavy neutral Higgs play an important role both in neutralino annihilation cross sections to satisfy the relic density of dark matter measured by WMAP and in the spinindependent neutralino nucleon cross section in direct dark matter search experiment. We thus have imposed on the MSSM parameter space the present CDMS exclusion limit and the WMAP limits on .
We have found that in models with and : the branching ratios of , are both under the thus probably undetectable even at superB factory; at LHC the cross section for , can reach fb in the range GeV giving up to 10 events with 100 fb; the cross section of reach fb, thus too small even for the large value of the expected luminosity of 500 fb. Prospects are somewhat more encouraging if we relax the lower limits, imposing only and . In this case branching ratios of LFV decays can reach , the cross sections at LHC about 2 fb for low masses and around fb in collisions.
We derive two main indications from this analysis. On one hand, even with large sources of lepton flavor violation in the slepton mass matrix, the process where Higgs mediated transitions should manifest could be beyond the sensitivity reach of future experiments. On the other hand, to observe such effects, in any case, the full luminosity of the machine is needed.
We emphasize that our not optimistic conclusions are specific to Higgs mediated effects. As shown in barger1 , in typical SUSY parameter space where gauginomediated effects are dominant over Higgs mediated ones and in the context of SUSY seesaw mechanism, LFV rates are detectable by future experiments.
We have also studied the spinindependent neutralino nucleus cross section: we have shown that in models that satisfy and , the cross section lies just below the sensitivity of XENON100 which should report results soon. The full XENON 1 ton is needed to cover all the parameter space. However, if the lower limit on is not considered XENON100 is sensitive to the neutralino mass range 3001000 GeV in models where the higgsino component is large.
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