Implications of the 125 GeV Higgs boson for scalar dark matter and for the CMSSM phenomenology
Abstract
We study phenomenological implications of the ATLAS and CMS hint of a GeV Higgs boson for the singlet, and singlet plus doublet nonsupersymmetric dark matter models, and for the phenomenology of the CMSSM. We show that in scalar dark matter models the vacuum stability bound on Higgs boson mass is lower than in the standard model and the 125 GeV Higgs boson is consistent with the models being valid up the GUT or Planck scale. We perform a detailed study of the full CMSSM parameter space keeping the Higgs boson mass fixed to GeV, and study in detail the freezeout processes that imply the observed amount of dark matter. After imposing all phenomenological constraints except for the muon we show that the CMSSM parameter space is divided into well separated regions with distinctive but in general heavy sparticle mass spectra. Imposing the constraint introduces severe tension between the high SUSY scale and the experimental measurements – only the slepton coannihilation region survives with potentially testable sparticle masses at the LHC. In the latter case the spinindependent DMnucleon scattering cross section is predicted to be below detectable limit at the XENON100 but might be of measurable magnitude in the general case of light dark matter with large binohiggsino mixing and unobservably large scalar masses.
1112.3647
1 Introduction
In the standard model (SM) of particle interactions the only unknown quantity is the Higgs boson mass Higgs:1964ia (); Guralnik:1964eu (); Englert:1964et (); Higgs:1964pj (). Any assumption that fixes the Higgs boson quartic selfcoupling at any scale implies a prediction for the Higgs boson mass. Many models of that sort have been proposed in the past based on different arguments of new physics beyond the SM. In general, the properties of the SM Higgs potential are among the best studied quantities in particle physics (Xing:2011aa (); for a review and references see EliasMiro:2011aa ()).
Based on data collected in 2011, both the ATLAS and CMS experiments have published their results for searches for the SMlike Higgs boson Chatrchyan:2012tx (); ATLAS:2012si () confirming and improving their earlier claims ATLASCONF2011163 (); CMSPASHIG11032 () for the inconclusive evidence of a signal of a GeV (CMS) or GeV (ATLAS) Higgs boson; we will assume that the mass is in this GeV range. The corresponding local significances of the excess in ATLAS and CMS are and , respectively, while the global significances after taking into account the lookelsewhereeffect are and . Although definitive confirmation of the observed evidence requires more data, the LHC result motivates studies of fundamental scalars in particle physics and in cosmology.
If the present inconclusive evidence for GeV Higgs boson will be confirmed, this result will have a profound impact on building models beyond the SM and on their phenomenology. In the context of the SM, the Higgs boson mass 125 GeV is below the vacuum stability bound GeV coming from the requirement of the SM validity up to the scale of gauge coupling unification Vanishing SM Higgs boson selfcoupling below the GUT scale, implies that the fundamental scale of new physics related to electroweak symmetry breaking and, perhaps, to flavour generation, might be lower than the GUT scale. On the other hand, the Higgs boson mass GeV may imply that there is new physics beyond the SM not too far from the electroweak scale that modifies the Higgs boson mass prediction. The most popular such a framework is low energy supersymmetry (SUSY) that prefers a light Higgs boson. For SUSY scenarios the lightest Higgs boson mass GeV is unusually high, close to the upper bound in popular models, and implies a higher SUSY breaking scale than one expects from naturalness arguments. Clearly those arguments mean that the present hint for the Higgs boson mass requires reassessment of several “standard” concepts both in SUSY and in nonSUSY models.
The aim of this work is twofold. First, assuming that the Higgs boson mass is in the range GeV, we study the implications of this assumption on the vacuum stability in scalar dark matter (DM) models. In those models the DM and Higgs sectors are related via the Higgs portal and the scalar potentials are in general rather complicated. Due to many new selfinteractions in the scalar sector, the SM Higgs quartic coupling renormalization is modified and one might expect that the triviality may be achieved for higher values of We show that this is indeed the case and the SM vacuum stability results will be changed in the nonSUSY scalar DM models compared to the SM prediction. As a new result we show that in those scenarios the 125 GeV Higgs boson is consistent with the vacuum stability up to and, therefore, the scalar DM models do not require new fundamental scales between TeV and the GUT scales.
Second, a technically much more involved question is what is the implication of the GeV LHC result for SUSY predictions of generating DM relic abundance, DM direct detection and for the LHC phenomenology. Generically such a heavy Higgs boson requires rather heavy stops, i.e., a large SUSY breaking scale^{1}^{1}1In the context of the 125 GeV Higgs boson this point has already been noted in Carena:2011aa (); Moroi:2011aa (); Moroi:2011ab (); Draper:2011aa (); Arbey:2011ab (); Heinemeyer:2011aa (); Li:2011ab (); Baer:2011ab (); Hall:2011aa (); Arbey:2011aa ().. This, in general, implies a large fine tuning to obtain the correct electroweak scale, very fine tuned DM annihilation channels and poor prospects for discovering SUSY at the LHC. We analyze those issues in detail in the constrained minimal supersymmetric standard model (CMSSM) and show that the requirements of GeV and correct DM relic abundance together select out parameter regions with well defined sparticle spectra. We work out CMSSM predictions for DM direct detection cross sections in those parameter regions. The most important new result of this paper is to predict sharp linear relationship between the gluino, lightest stop and slepton masses in the stop and slepton coannihilation regions that are the only ones accessible to the LHC experiments.
If, in addition, also the muon anomalous magnetic moment constraint is imposed on the CMSSM, only a tiny parameter region is singled out that induces DM via the slepton coannihilation channel. In this parameter space the LHC has a good chance to observe gluinos and the lightest stop but the DM direct detection experiments like XENON100 are predicted to obtain null result. In the other DM freezeout channels that also predict the correct amount of DM the situation might be an opposite – only TeV scale DM is observable in DM direct detection experiments while the heavy gluinos and scalars decouple from the spectrum. We classify all those possibilities and discuss their phenomenology.
2 Scalar dark matter and vanishing Higgs selfcoupling
Triviality of the SM Higgs boson selfcoupling, at some scale is an interesting possibility. From theoretical point of view this may indicate a scale where some new fundamental theory beyond the SM generates electroweak symmetry breaking and Higgs boson Yukawa couplings, i.e., flavour physics. From the phenomenological point of view this scale uniquely predicts the Higgs boson mass due to the evolution of the Higgs selfcoupling via renormalization group equations. Examples of this running at two loop level in the SM are presented in Fig. 1 for different values of the SM Higgs boson masses as indicated in the figure. Our results agree with the recent works Xing:2011aa (); EliasMiro:2011aa (). This result shows that the LHC indications for the Higgs boson imply the triviality scale to be about GeV rather than the GUT scale GeV. Such a low scale can be associated with the seesaw scale GellMann:1979kx (); Yanagida:1979uq (); Mohapatra:1979ia (); Glashow:1979nm (); Minkowski:1977sc () where neutrino masses are generated rather than with the GUT scale.
The natural question to ask is that what happens to the vacuum stability in models with extended scalar sector? Particularly interesting among those models are the scalar DM models that have been already addressed in the 125 GeV Higgs boson scenario Djouadi:2011aa ().^{2}^{2}2Singlet fermion DM has also been studied Baek:2011aa ().
2.1 Scalar singlet model
The simplest DM model is obtained by extending the SM scalar potential with a real McDonald:1993ex (); Burgess:2000yq (); Barger:2007im (); Gonderinger:2009jp () or complex Barger:2008jx () singlet scalar field. In view of embedding this scenario into a GUT framework Kadastik:2009dj (), we study the complex singlet scalar , but the phenomenology in the real singlet case is similar. The vacuum stability of the real singlet model has previously been studied in Gonderinger:2009jp ().
Denoting the SM Higgs boson with the most general Lagrangian invariant under the transformations , is given by
(1) 
The vacuum stability conditions for the complex singlet model with a global are given in Barger:2008jx (). However, those conditions are not applicable here because this model is far too simple compared to the general case (1). For the general model the full vacuum stability conditions are rather complicated and have been addressed previously in Ref. Kadastik:2009cu (). However, the conditions of Kadastik:2009cu () turn out to be too restrictive because they are derived by requiring the matrix of quartic couplings to be positive. This is required only if the coefficients of biquadratic terms are negative and, in general, cut out some allowed parameter space.
The conditions arising from pure quartic terms of the potential (1) are
(2) 
For simplicity we consider in addition only the case when the coefficents of the terms biquadratic in real fields (e.g. the coefficient of ) are all nonnegative, giving
(3) 
Doing this, we exclude a part of the points that would be allowed by the full vacuum stability conditions. However, this is sufficient for our purposes because our aim is to show that regions of the parameter space exist that lower the SM Higgs boson mass vacuum stability bound.
The oneloop RGEs can be obtained from those in Kadastik:2009cu () by setting all couplings of the inert doublet to zero. The RGEs show that nonzero or give a positive contribution to the function of , pushing the scale where higher. For qualitative understanding of the model, we let . Fig. 2 shows one loop level running for the GeV Higgs quartic coupling for (the SM case) and for . In the latter case, the minimum bound on Higgs boson mass from the vacuum stability argument is lowered and the vacuum can be stable up to the GUT or Planck scale.
2.2 Inert doublet model
In the inert doublet model LopezHonorez:2006gr (); Barbieri:2006dq (); Ma:2006km (); Deshpande:1977rw () there is, besides the SM Higgs , an additional scalar doublet that is odd under a new symmetry and thus does not have Yukawa couplings. The neutral component of the inert doublet is a DM candidate. The most general Lagrangian invariant under the transformations , is
(4) 
The requirement of vacuum stability imposes
(5) 
We will not perform a detailed study of the inert doublet model alone here, because it is a limiting case of the singlet plus doublet model studied below.
2.3 Singlet plus doublet model
This model has been previously studied in the context of GUT Kadastik:2009dj (); Kadastik:2009cu (); Kadastik:2009ca (); Kadastik:2009gx (); Huitu:2010uc (). Here, however, we present a general scan of parameters without imposing any GUT boundary conditions.
The Lagrangian with even and odd and is
(6) 
Just as for the complex singlet model, we consider here only the case of positive biquadratic terms for real fields (with the exception of the purely inert doublet conditions that are completely general). The simplified vacuum stability conditions for this model are given by (2), (3) and (5) together with an additional constraint^{3}^{3}3Again, similarly to the singlet model the constraints in Ref. Kadastik:2009cu () that were used in the previous version of the current paper are too restrictive.
(7) 
The RGEs for couplings and mass parameters are given in Kadastik:2009cu (). We have performed a scan of the parameters for the values of couplings randomly generated in the ranges
(8)  
with the rest of the parameters set to zero. In the case of every generated point we check that it satisfies the requirements of vacuum stability and perturbativity in the whole range from to , positivity of masses at and lie within the range of the WMAP cosmic abundance. The points that satisfy all the constraints are shown in Fig. 3.
In the left panel of Fig. 3, the region excluded by the CMS is shown in red; the GeV Higgs mass range is shown in green. Because the points were calculated using oneloop RGEs for the doublet plus singlet model, we show the GUT scale vacuum stability bound for the SM at oneloop level with the blue line (the twoloop bound is lower by about GeV). The points excluded by the XENON100 experiment Aprile:2011hi () are shown in gray while the black points satisfy the present direct detection constraints. The shortage of points in the range from about GeV to about GeV is due to the DM being mostly singletlike: in the low mass range it annihilates via the Higgs resonance, in the mass range above GeV the quartic scalar interactions can be large enough to allow for efficient annihilation via contact terms, but in between annihilation is not efficient, resulting in overaboundance of DM and exclusion by CMB bounds.
The right panel of Fig. 3 shows the XENON100 direct detection constraints in detail. The points in the Higgs boson mass range GeV are green. The low mass region below GeV is excluded. Between GeV and GeV there is a region that accommodates GeV, having vacuum stability up to the GUT scale with a low mass Higgs. Thus we conclude that the scalar DM models are perfectly consistent with the 125 GeV higgs mass and do not require the existence of new fundamental scale below the GUT or Planck scale.
The scan is no exhaustive, but for 124126 GeV Higgs mass range, the noticeable differences with the rest of the parameter space are in the soft coupling and couplings between dark sector and the SM Higgs that tend to be smaller than with a freely varying Higgs mass.
3 CMSSM dark matter and LHC phenomenology for the 125 GeV Higgs boson
The CMSSM is the most thoroughly studied SUSY model. Naturally, if the Higgs boson is discovered with the mass GeV, one would like to know what is the implication of this discovery for the phenomenology of this model. Here we show that if all the phenomenological constraints are taken into account, the CMSSM parameter space shrinks into well defined small regions according to the dominant DM freezeout process. We study whether the CMSSM can be tested at the LHC and in DM direct detection experiments such as XENON100 and conclude that, despite of heavy Higgs boson, discovery of CMSSM gluinos and/or stops is not excluded at the LHC. In addition, if the sparticle spectrum is too heavy for the LHC discovery, DM direct detection experiments may still discover the CMSSM DM.
It is well known that such a heavy Higgs boson imposes challenges on SUSY models in which the Higgs boson mass is predicted to be
(9) 
where is the stop dominated loop contribution. For GeV the loop contribution must be as large as the tree level one which requires very heavy stops unless there is an extremely large trilinear scalar coupling that makes the lightest stop light due to large mixing. A heavy SUSY scale, in turn, makes the lightness of electroweak symmetry breaking scale unnatural. In addition, a heavy sparticle spectrum imposes fine tunings on the processes that contribute to the DM freezeout in SUSY models. Taking those facts into account, the phenomenological constraints that are commonly addressed in the context of SUSY models, summarized in Table 1, the constraints from SUSY searches at the LHC and the constraints from DM direct detection, the CMSSM parameter space is known to be rather fine tuned Farina:2011bh (); Buchmueller:2011ki (); Buchmueller:2011sw (); Bertone:2011nj (); Fowlie:2011mb ().
At the GUT scale the parameter space of the CMSSM is described by five parameters,
(10) 
the common scalar mass, the common gaugino mass, the common trilinear coupling, ratio of two Higgs vevs and the sign of the higgsino mass parameter. To scan over the CMSSM parameter space we randomly generate the parameters in the following ranges:
(11)  
We use the MicrOMEGAs package Belanger:2010gh (); Belanger:2006is () to compute the electroweak scale sparticle mass spectrum, the Higgs boson masses, the DM relic abundance , the spinindependent DMnucleon direct detection cross section and the other observables in Table 1. In addition, we require GeV. We do not attempt to find the best fit regions of the parameter space because there is no Higgs mass measurement yet. In addition, there is a few GeV theoretical uncertainty in the computation of SUSY Higgs masses in the available codes. Therefore, to select the phenomenologically acceptable parameter space we impose hard cuts for the observables in Table 1.^{4}^{4}4The new constraints on from the LHCb and CMS Aaij:2012ac (); CMSbstomumu () have an impact on points with low stop mass at high . Qualitatively, however, the regions and channels remain the same. Our approach should be regarded as an example study of the CMSSM parameter space for heavy Higgs boson; qualitatively similar results should hold if the real Higgs boson mass deviates from 125 GeV by a few GeV.
Quantity  Experiment  Standard Model 

Bethke:2009jm ()  parameter  
Lancaster:2011wr ()  parameter  
PDG ()  parameter  
Larson:2010gs ()  0  
Davier:2010nc ()  0  
BR Misiak:2006zs ()  
BR Bsmumu ()  at 95%C.L.  
BR/SM Buchmueller:2009fn ()  1 
Our results are presented in Figs. 47. Because there is a tension between the observables that push the SUSY scale to high values and the measurement of Farina:2011bh (), we disregard the constraint for the moment. The reason is that the CMSSM parameter fit is largely dominated by two observables, the DM relic abundance and the , the latter constraining mostly the scale. We would first like to study the parameter space that induces correct and . Therefore we discuss the implications of the constraint later.
In Fig. 4 we present our results in scatter plots without the constraint. In the upper left panel the results are presented in plane, in the upper right panel in plane, in the lower left panel in plane, and in the lower right panel in plane. The first 100 days XENON100 constraint Aprile:2011hi () is also shown.
We identify five distinctive parameter regions according the dominant DM annihilation processes.

The light blue points with small and represent the slepton coannihilation region. They are featured by very large values of Those points represent the best fit value of the CMSSM Farina:2011bh () and have low enough sparticle masses that allow potential SUSY discovery at the LHC. However, their spinindependent direct detection cross section is predicted to be below cm and remains unobservable at the XENON100. The present XENON100 experimental bound is plotted in the upper right panel with solid red line. This is the only parameter region that survives at level after the constraint is imposed.

The green dots represent the stop coannihilation region. Consequently those points have the lowest possible stop mass and, due to the mass degeneracy with DM, stops can be long lived and seen as stable very slow particles (hadrons) at the LHC. The feature of those points is an enormous trilinear coupling and very large stop mixing. In addition, the gluino mass can be reachable at the LHC. For stop coannihilation region the spinindependent DM direct detection cross section is, unfortunately, unobservable.

The dots represented by continuous colour code from red to orange represent the so called welltempered neutralino ArkaniHamed:2006mb (), i.e., neutralinos with large binohiggsino mixing. The colour varies according to the higgsino component from red (predominantly bino) to yellow (pure higgsino). Therefore those points can simultaneously have small DM mass and large DMnucleon scattering cross section that can be well tested at the XENON100. However, apart from the DM, all other sparticle masses are predicted to be too heavy for direct production at the LHC.

The yellow dots around TeV represent the pure higgsino DM that is almost degenerate in mass with chargino. The sparticle mass spectrum is predicted to be even heavier than in the previous case because the DM scale is fixed to be high. These points represent the most general and most abundant bulk of the GeV Higgs scenario – apart from the light DM and heavy Higgs boson there are no other observable consequences because stops can completely decouple. In our case the 10 TeV bound on stops is imposed only because we did not generate larger values of

The dark blue points represent heavy Higgs resonances. Those points are featured by very large values of and give the heaviest mass spectrum. In essence those points are just smeared out higgsino points due to additional Higgsmediated processes.
In order to study the testability of those parameter regions at the LHC we plot in Fig. 5 the physical gluino mass against the lightest stop mass and the lightest slepton mass against the lightest stop mass. Clearly, the only two regions of interest for the LHC are the slepton and stop coannihilation regions. Therefore we plot in lower panels the low mass scale zoom of the upper panels. According to Ref. Baer:2011aa () both regions have a chance to be discovered already in the 7 TeV LHC. Interestingly, due to the stop mass degeneracy with DM the stops can be longlived. In this case one must search for hadrons at the LHC experiments.
To study the and heavy Higgs mass dependence of the generated parameter space we plot in Fig. 6 scatter plots in and plains. The slepton coannihilation points have a preferably large that implies large contributions to the observables like and the Those allow for indirect testing of this parameter region. Unfortunately the heavy Higgses are predicted to be too heavy to detect at the LHC.
We remind that so far we have disregarded the constraint. If we impose a hard cut on the generated parameter space, only the slepton coannihilation region survives. The result is plotted in Fig. 7 where we repeat the content of Fig. 4 but with the additional constraint. As expected, the observed deviation in the from the SM prediction is hard to explain in SUSY models with heavy spectrum. Therefore the two measurements, and GeV, are in conflict in the CMSSM Baer:2011ab (). The conflict is mildest in the slepton coannihilation case because of large and the lightest sparticle spectrum. Therefore, for the GeV Higgs boson, we predict definite sparticle masses and correlations between them, shown in Fig. 7, for the LHC. If the CMSSM is realized in Nature and if it contributes significantly to the , the sparticle spectrum is essentially fixed and potentially observable at the LHC.
4 Conclusions
The recent LHC searches for the SMlike Higgs boson motivate studies of the fundamental scalars in particle physics models and in cosmology. In this paper we analyzed the implications of the GeV Higgs boson for the vacuum stability in scalar DM models and for the phenomenology of CMSSM. This value of the Higgs boson mass is interesting in both cases because it does not fit to the standard expectation neither in the SM nor in minimal supersymmetric models with SUSY breaking scale below 1 TeV.
We have shown that in the case of nonSUSY scalar DM models the vacuum can be stable up to the GUT scale even for Higgs boson masses much below the corresponding SM bound. Therefore, unlike the SM, the scalar DM models can be valid up to the GUT or Planck scales even for the Higgs boson mass as low as GeV.
In minimal SUSY models, to the contrary, the GeV Higgs boson is heavier than expected in scenarios that address naturalness of the electroweak scale. In order to generate such a large Higgs boson mass at loop level, the SUSY breaking scale must be rather high and could be unobservable at the LHC. This problem can be overcome with extremely large stop term so that the lightest stop is light due to large mixing. At the same time the DM neutralino can also be light, either because of dominant slepton coannihilation processes or because of large binohiggsino mixing. In the latter case the DMnucleon scattering cross section can be observable in direct detection experiments like the XENON100.
To quantify those results we studied the CMSSM by scanning over its parameter space allowing the sparticle mass parameters to be very large. We first considered the case without attempting to explain the in the context of CMSSM. We confirmed that for very large terms there exists a stop coannihilation region where all DM, stop and gluino are preferably light. Due to the mass degeneracy between stop and DM the stops can also be long lived resulting in nontrivial LHC phenomenology. The second parameter region that is potentially reachable at the LHC is the slepton coannihilation region. The most important result of this work is to make sharp predictions of gluino, stop and slepton masses, shown in Fig. 5, for the CMSSM parameter regions that remain testable at the LHC.
For other channels of generating the correct DM relic abundance the GeV Higgs boson implies very heavy sparticle masses. The exception is, of course, the DM that can be light due to binohiggsino mixing even if other sparticles are as heavy as 10 TeV. In this case the CMSSM cannot be tested at the LHC but the DM spinindependent scattering cross section off nuclei may be large due to the large higgsino component. The latter scenario may be discoverable already in the running XENON100 experiment.
If, in addition, one attempts to explain also the in this framework, there is immediate tension between the high SUSY scale and the large value of the needed contribution. We found that after imposing the constraint on the CMSSM, only the slepton coannihilation region survived at level, see Fig. 7. This implies that the CMSSM has a definite prediction for the sparticle masses and spectrum to be tested at the LHC experiments.
Acknowledgements
We thank A. Strumia for several discussions.
This work was supported by the ESF grants 8090, 8499, 8943, MTT59, MTT60, MJD140, JD164, by the recurrent financing SF0690030s09 project
and by the European Union through the European Regional Development Fund.
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