The Upper Bound of the Second Higgs Boson Mass in Minimal Gauge Mediation with the Gravitino Warm Dark Matter
TThe Upperbound of the Second Higgs Boson Mass in Minimal Gauge Mediation withthe Gravitino Warm Dark Matter
Gongjun Choi, , ∗ Tsutomu T. Yanagida, , , † Norimi Yokozaki, , ‡ Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai 200240, China Kavli IPMU (WPI), UTIAS, The University of Tokyo,5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8583, Japan Zhejiang Institute of Modern Physics and Department of Physics,Zhejiang University, Hangzhou, Zhejiang 310027, China Theory Center, IPNS, KEK, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan (Dated: December 8, 2020)A keV-scale gravitino arsing from a minimal supersymmetric (SUSY) Standard Model (MSSM) isan interesting possibility since the small scale problems that ΛCDM model encounters in the moderncosmology could be alleviated with the keV-scale gravitino serving as the warm dark matter (WDM).Such a light gravitino asks for a low scale supersymmetry (SUSY) breaking for which the gaugemediation (GM) is required as a consistent SUSY-breaking mediation mechanism. In this paper,we show upperbounds of the masses of the second CP-even Higgs boson H and the CP-odd Higgsboson A , assuming the keV-scale gravitino to be responsible for the current DM relic abundance:the upperbound on the mass of H/A is found to be ∼ O (10 - 100)keV. Interestingly, the mass of H/A can be as small as 2-3 TeV and the predicted tan β is as largeas 55-60 for the gravitino mass of O (10) keV. This will be tested in the near future Large HadronCollider (LHC) experiments. I. INTRODUCTION
The physical Higgs boson mass 125GeV in the Stan-dard Model (SM) has been regarded as one of theoreti-cally most challenging problems to understand since it issubject to radiative corrections as large as heavy particlemasses in a potential extension of the SM. Thereby thisunnatural separation between the electroweak scale anda UV-cutoff such as GUT or Planck scale (a.k.a hierarchyproblem) has stirred up a variety of theoretical imagina-tions as to a new physics beyond the SM. Among severalideas addressing the issue, supersymmetry (SUSY) is themost promising one because the systematic cancellationamong the radiative corrections contributed by fermionicand bosonic degrees of freedom is inevitable consequencethereof.As such, SUSY asks for a way to communicate itsbreaking to the visible sector. And as a plausible way,the minimal gauge mediation (MGM) model is advanta-geous in that it is free of the SUSY flavor changing neu-tral current (FCNC) and CP problem [1]. In addition,in relatively low scale SUSY-breaking scenario inferredfrom gauge mediation, the less degree of fine-tuned can-cellation is required among F-term and R-breaking con-tributions to a scalar potential in the theory to producethe vanishingly small cosmological constant.In particular, for a supergravity model with the SUSY-breaking scale as low as O (1) − O (10)PeV, the mass ofthe gravitino ( m / ) becomes O (10)keV which is attrac-tive from a cosmological point of view. As a lightest su- ∗ [email protected] † [email protected] ‡ [email protected] persymmetric particle (LSP) that is stable, neutral andvery weakly interacting with MSSM particles, the grav-itino can serve as a warm dark matter (WDM) candidate.The gravitino becoming free after SUSY particles areintegrated-out in the MSSM thermal bath, the growth ofthe matter fluctuations at scales below its free-streaminglength is expected to be suppressed. This may help us ad-dress small scale problems that the concordance ΛCDMsuffers from [2–4].In this work, motivated by the aforesaid theoreticalmerits of the MGM implying a low SUSY-breaking scaleand cosmological winning attributes of having keV-scalegravitino WDM, we consider the MGM model with the O (10)keV LSP gravitino. There are noteworthy featuresof the model. Above all, the MGM model is featured bythe vanishing B -term at the messenger mass scale, whichgives rise to a suppressed B -term and a large tan β at theelectroweak scale [5]. Relying on this distinct property,for the range of m / enabling the gravitino to resolve thecosmological small scale problems, the model is shownto predict the masses of the second CP-even Higgs bo-son and CP-odd Higgs boson below 4 TeV which can betested in the future Large Hadron Collider (LHC) exper-iments. On top of this, the MGM model has a unstablevacuum with a life time greater than the age of the uni-verse. Of course one may consider other gauge-mediationmodels with a stable SUSY-breaking vacuum (see for ex-ample Ref. [6]). In those cases, however, gaugino massesnaturally become too small to survive against the LHCprobe for m / (cid:39) − a r X i v : . [ h e p - ph ] D ec cuss conditions for the gravitino to serve as a WDMcandidate and how those are parametrized by the MGMmodel parameters. In Sec. IV, we discuss correlation be-tween tan β at the electroweak scale and Higgs masses.Particularly we attend to an effect on prediction for Higgsmass in the MSSM in the large tan β limit. In Sec. V, wepresent the results of the analysis probing the parameterspace of the model. We will make explicit discussion forhow the assumption for the gravitino WDM is connectedto the prediction for relatively small masses of the sec-ond CP-even Higgs boson and CP-odd Higgs boson inthe MSSM. Furthermore, we discuss possible observablenature of the additional Higgs bosons at LHC, provid-ing their production cross sections and decay channels.Lastly in Sec. VI, we conclude this paper by presentinga compact summary of our assumptions, the noticeablestructure of the MGM model and phenomenological con-sequences. II. MINIMAL GAUGE MEDIATION MODEL
Assuming a spontaneous SUSY-breaking mechanism(e.g., O’Raifeartaigh model [7] or dynamical SUSY-breaking model with a vector-like asymptotically freegauge theory [8, 9]), for simplicity, we consider the follow-ing SUSY-breaking sector with a single chiral superfield Z of which vacuum expectation value (VEV) induces thespontaneous SUSY-breaking L (cid:24)(cid:24)(cid:24) SUSY = (cid:90) d θ (cid:20) Z † Z − ( Z † Z ) κ (cid:21) + (cid:90) d θ (cid:2) − µ Z Z (cid:3) + h.c. , (1)where κ and µ Z are dimensionful parameters given by adynamics of the SUSY-breaking sector. Based on Eq. (1),the VEV of Z is given by (cid:104) Z (cid:105) = 0 , (cid:104) F Z (cid:105) = µ Z . (2)Aside from the VEV given in Eq. (2), there exists an-other VEV which respects SUSY, i.e. (cid:104) Z (cid:105) = M mess /k and (cid:104) ΨΨ (cid:105) = µ Z /k . Thus, we notice that the vacuum with theVEV in Eq. (2) is meta-stable. Nonetheless, since we as-sume M >> k µ Z , the stability of the SUSY-breakingvacuum is guaranteed for the time length longer than theage of the current universe [10].For communicating SUSY-breaking to the MSSM sec-tor, we introduce N mess pairs of messenger fields (Ψ , Ψ)transforming as a fundamental and an anti-fundamentalrepresentation of SU (5) GUT . The messengers of the mass M mess are coupled to the SUSY-breaking field Z via W ⊃ ( kZ + M mess )ΨΨ . (3)Here we assumed a SU (5)-invariant mass M mess for mes-sengers. The gaugino and the soft SUSY-breaking scalarmasses are given by M a (cid:39) N mess α a π kµ Z M mess , (4) m (cid:39) (cid:88) a =1 N mess C ,a (cid:12)(cid:12)(cid:12)(cid:12) α a π kµ Z M mess (cid:12)(cid:12)(cid:12)(cid:12) , (5)where a specifies a SM gauge group, α a ≡ g a / (4 π ) isdefined, and C ,a is a quadratic Casimir of a SM gaugegroup. N mess denotes a number of messenger pairs andthe MGM model is characterized by N mess = 1.With all the couplings that hidden sector fields ( Z ,Ψand Ψ) enjoy specified above, we assume the Higgsinomass term ( µ -term) is just given as the symmetry-respecting marginal operator W ⊃ µH u H d , (6)where H u and H d are chiral superfields for the up-typeand the down-type Higgs respectively. This assumptionmakes the model featured by the suppressed B -term atthe scale of M mess . This is because Z -dependent terms ofthe wave function renormalization constants of the chiralsuperfields H u and H d vanish at the one-loop level. Notethat MSSM scalar trilinear coupling terms ( A -terms) arealso suppressed due to the same reason. Thanks tothis suppression of A -terms and the fact that soft scalarmasses in Eq. (5) are diagonal, the MGM model can avoidthe FCNC problem. As can be seen later, combined withthe electroweak symmetry breaking (EWSB) conditions,the resultant B (cid:39) β of order O (50), which is basicallythe essential point allowing for relatively small masses ofthe second CP-even Higgs and CP-odd Higgs.Regarding the potential CP problem, the complexphases of gaugino mass, Higgsino mass parameter, di-mensionful parameters in A -term and B -term are rele-vant. Above all, phases of gauginos can be rotated awayby applying U (1) R . Meanwhile, Higgsino mass can berendered real by PQ-like chiral rotation. Finally there isno concern for complex phases of parameters in A -termand B -term since those are zero at the scale of M mess and induced by gaugino loops. Hence, the MGM modelbecomes free of the potential CP problem.We end this section by clarifying the free parameters inthe MGM model. For the energy scale above a messengermass M mess , the model is described by the parameters µ Z in Eq. (1), and k and M mess in Eq. (3). After integrat-ing out the messengers, the gaugino masses in Eq. (4)and the soft SUSY-breaking scalar masses in Eq. (5) aregenerated. For the energy scale below M mess , we takeΛ ≡ kµ Z M mess , M mess , (7)as a set of free parameters for the model. The later dis-cussion in Sec. IV as to the gravitino cosmology andthe second Higgs mass is to be done in the plane of( M mess , Λ). In this work, we do not address the possible origin of the Higgsinomass term.
III. GRAVITINO WARM DARK MATTER
Along with the cosmological constant, the cold colli-sionless dark matter (CDM) has been successful in ac-counting for the lager scale structure of the universe andits time evolution. This paradigm known as ΛCDM is,however, being challenged by discrepancy between itsprediction for the small scale physics ( (cid:46) . (cid:46) λ FS (cid:46) . a dec ∼ a EW T EW O (1)TeV ∼ O (10 − ) , (8)where we used the scaling behavior of the MSSM ther-mal bath temperature, i.e. T ∼ /a , the tempera-ture of the MSSM thermal bath at the electroweak scale T EW ∼ a EW (cid:39) − .By using Eq. (8), one can make a numerical estimateof a free-streaming length of the gravitino with a mass m / by following integral λ FS = (cid:90) t t FS a dec (cid:113) ( a dec ) + m / a , (9)where F ( a ) ≡ (cid:113) Ω rad , + a Ω m , + a Ω Λ , and H is thecurrent Hubble expansion rate. At the time of decou-pling, the average energy of the highly relativistic keV-scale gravitino is almost identical to its average momen-tum < p / ( a dec ) > . Thus from the ratio of the energy density to the number density of the thermal gravitino,we obtain ∼ . T dec with T dec the MSSMthermal bath temperature at the decoupling.Applying the criterion 0 . (cid:46) λ FS (cid:46) . (cid:46) m / (cid:46) Given the gravitino mass m / = (cid:104) F Z (cid:105) / √ M P in terms of a SUSY-breaking scale and thereduced Planck mass M P (cid:39) . × GeV, this range of m / is converted into the following range of the SUSY-breaking scale √ M P GeV (cid:46) (cid:104) F Z (cid:105) (cid:46) √ M P GeV , (10)where (cid:104) F Z (cid:105) was defined in Eq. (2).We notice that avoiding too much relic abundance ofthe thermal gravitino of the mass m / (cid:46) T RH (cid:46) GeV [16]. Bearing in mind that the lowestpossible reheating temperature consistent with leptogen-esis (non-thermal one) is ∼ GeV [17, 18], we realizethat there must be a mechanism to dilute the relic abun-dance of the keV-scale gravitino WDM. In compliancewith this reasoning, as was suggested in Ref. [19], onemay consider the possibility where a late time entropyproduction is made by the decay of messenger particlesembedded in gauge-mediated SUSY-breaking models.For inducing the messenger decay to a pair MSSMparticles, we introduce the following term that mixesup the messenger and MSSM supermultiplets via theR-symmetry breaking constant term in the superpoten-tial [19] W ⊃ f i (cid:104) W (cid:105) M Ψ5 i = f i m / Ψ5 i , (11)where the subscript i is the generation index and f i isa dimensionless coefficient. As the consequence of themixing in Eq. (11), an operator for the messenger decayto H d and 10 i is induced with the coupling proportionalto f i m / /M mess . Thus the lightest scalar component ofthe messenger weak doublet can decay to the higgsinoand the SM lepton with the decay rateΓ Ψ (cid:39) π (cid:18) m τ v cos β (cid:19) (cid:18) f m / M mess (cid:19) M mess , (12)where v (cid:39) (cid:113) v u + v d (cid:39) v u ( v d )the up (down)-type Higgs VEV. We note that it is de-manded for the model to have large enough f i s so as For our purpose in this paper, it suffices to discuss 10keV asthe lower bound of m / without referring to the most recentmass constraint on the thermal WDM from the Lyman- α forestobservation. to complete Ψ-decay before the big bang nucleosynthe-sis (BBN) era begins. Otherwise, the primordial lightelements formed during BBN time can be destroyed bythe electrically charged high energy decay products ofΨ-decay and inconsistent deficit of the primordial lightelements is caused.By comparing the decay rate of the messenger inEq. (11) to the Hubble expansion rate during theradiation-dominated era, i.e. Γ Ψ ∼ H (cid:39) T /M P , wecan obtain the MSSM thermal bath temperature T Ψ atthe time when the messenger decays. Now the ratio ofthe energy density of radiation coming from the decayof the messenger to that of existing MSSM radiation isgiven by ∆ ≡ ρ Ψ /ρ rad = (4 / M mess Y mess /T Ψ ) where Y mess = 3 . × − ( M mess / GeV) is the comovingnumber density of the messenger [20]. For a large enough∆, the entropy of the universe is dominated by that ofthe messenger, which makes it possible to dilute overpro-duction of the thermal gravitino. For the gravitino to beWDM today, parameters controlling ∆ should satisfyΩ DM h = Ω / h ∆ (cid:39) . (cid:18) f . (cid:19) (cid:16) g ∗ , MSSM (cid:17) − (cid:18) tan β (cid:19) × (cid:16) m / (cid:17) (cid:18) GeV M mess (cid:19) , (13)where Ω / is the relic abundance for the thermal grav-itino, h is defined via H = 100 h (km / Mpc / sec) and g ∗ , MSSM = 228 . 75 is the effective degrees of freedom ofMSSM particles after the messengers decay. In Sec. V,we shall discuss whether Eq. (13) can be satisfied by(tan β, M mess ) of our interest for m / (cid:39) − IV. HIGGS MASSES IN THE MGM MODEL As was pointed out in Sec. II, B -term at the scale of M mess vanishes due to the set-up of the MGM model (seeEq. (6) and the associated text), and the B-term at alow energy scale is generated by gaugino masses throughradiative corrections. The renormalization group equa-tion (RGE) of the parameter B consists of contributionsproportional to gaugino masses and scalar trilinear cou-plings, where the scalar trilinear couplings are also ra-diatively generated from the gaugino masses. Since thesetwo contributions are loop suppressed and have oppositesigns, the MGM model tends to produce a smaller B value at the low energy scale than other models. When the µ -term has the positive sign, the B -term should bepositive at the low energy scale, which is achieved if the contri-butions proportional to the scalar trilinear couplings are largerthan the contributions proportional to the gaugino masses. Incontrast, for the negative µ case, the B -term should be negativeat the low energy scale, which is achieved if the contributionsproportional to the gaugino masses are larger. Together with soft masses for H u and H d , the pa-rameter B obtained at the electroweak scale via RGEcan determine values of tan β and µ -parameter withthe aid of the following two conditions for the EWSB( ∂V /∂H u = ∂V /∂H d = 0) m Z m H d + v d ∂ (∆ V ) ∂v d ) − ( m H u + v u ∂ (∆ V ) ∂v u ) tan β tan β − − µ , (14) Bµ (tan β + cot β )= m H u + 12 v u ∂ (∆ V ) ∂v u + m H d + 12 v d ∂ (∆ V ) ∂v d + 2 µ , (15)where m Z is the Z -boson mass, m H u ( m H d ) is the softmass for H u ( H d ) and ∆ V is a radiative correction to theHiggs potential. Note that the left hand side of Eq. (15)is nothing but CP-odd Higgs mass squared, i.e. m A = Bµ (tan β + cot β ).As is well known, the RGE for m H u is subject to thenegative contributions attributable to Yukawa couplings,which gives rise to the negative sign of m H u at the elec-troweak scale. For a large tan β case, simplification ofEq. (14) to remove m H d dependence and substitution ofthe resultant expression of 2 µ into Eq. (15) yields m A (cid:39) m H d + 12 v d ∂ (∆ V ) ∂v d − m H u − v u ∂ (∆ V ) ∂v u , (tan β >> β , we notice thatnot only m H u but also m H d could be negative at theelectroweak scale. This enables cancellation between thetwo results, allowing for the smaller m A value than thecase with a small tan β . Essentially this is attributed tolarge Yukawa couplings for the tau lepton and the bottomquark resulting from a large tan β .For eigenvalues of the mass matrix for the CP-evenHiggs ( m H and m h with m H > m h ), one obtains m A (cid:39) m H (17)when m A >> m Z is satisfied. This predicts a relativelylight second Higgs mass ( m H ) comparable to a relativelylight m A for a large tan β . In the next section, we shallsee how (1) large tan β could arise for the model parame-ter space producing keV-scale gravitino and (2) how light m H and m A could be for the resulting large tan β . V. RESULTS OF ANALYSIS In this section, we discuss the parameter space( M mess , Λ) of the MGM model yielding the consistent val-ues of the lightest CP-even Higgs mass ( m h ) and the cor-responding resultant second CP-even Higgs mass ( m H ).Λ was defined in Eq. (7). To this end, we perform theanalysis of solving the RGE equations for MSSM parame-ters and computing MSSM particle mass spectra with the km (GeV)100020003000400050006000 m A / m H ( G e V ) N mess = 1, m h = 125 GeV FIG. 1. The plot of the second CP even Higgs mass ( m H )and the CP odd Higgs mass ( m A ) as a function of km / .The green shaded region is excluded when the MGM modelproduces the gravitino WDM. aid of SOFTSUSY package [21]. Following this, the Higgsmass spectra is obtained by FeynHiggs 2.16.1 [22–29].Firstly, we show in Fig. 1 the second CP even Higgsmass ( m H ) and CP odd Higgs mass ( m A ) as a functionof km / where k was defined in Eq. (3). The regimeof km / smaller than shown in the horizontal axis isirrelevant since the messengers become tachyonic. Forthe two disconnected red lines, the left one correspondsto µ < µ > 0. The leftline is cut at km / (cid:39) × − GeV since EWSB fails tooccur for the larger km / .For km / (cid:46) ∼ µ -parameter. Remarkably, this im-plies that the MGM model predicts for the upperbound ∼ km / (cid:39) k (cid:46) 1. On theother hand, it can be observed that m H /m A below 3TeVis allowed in Fig. 1 and this region of m H /m A is crucialparticularly for the LHC search of the second Higgs bo-son. Therefore, a reasonable question can be how largea parameter space the model has for m H /m A (cid:46) µ > 0, below we probethe parameter space for µ > M mess , Λ). In the left and right panel, using blue and redlines, we show values of ( M mess , Λ) yielding the specifiedthe lightest CP even Higgs mass and the second CP evenHiggs mass, respectively. For both panels, each black lineshows a set of points yielding the specified tan β value.Note that the larger M mess enhances B -parameter valuevia logarithmic dependence arising from the RGE andthus corresponds to a smaller tan β for a fixed Λ. Alsodisplayed are three green lines corresponding to km / = TABLE I. The b -associated production cross-section of H/A for the MGM model with m h = 125GeV and µ > m H [TeV] σ [pb]2.2 0.00542.4 0.00282.6 0.00152.8 0.000803.0 0.000453.2 0.000263.4 0.00015TABLE II. Mass spectra in the MGM model with µ > I Point II Λ (TeV) 1300 1700 M mess (GeV) 5 . × . × Particles Mass (TeV) Mass (TeV)˜ g q t , b , e L,R τ , χ , µ χ ± h SM-like (GeV) 125 126 H/A (GeV) 2660 2220tan β km / (keV) 15.6 14.3 , , k = O (0 . − O (1), one can see that10keV (cid:46) m / (cid:46) km / = 100keV) and the left-most ( km / = 1keV) green lines. The right green shadedregion gives km / > m / > k (cid:46) 1. Inother words, the MGM model cannot have the gravitinoWDM resolving the small scale problems there. Boththe blue line for 125GeV and the right green shaded re-gion being taken into account together simultaneously,we realize that the upperbound of m H lies in 3 . − m A , tan β ) plane(95% C.L.) given in Ref. [30], we notice that m H (cid:46) m A (cid:46) β > 60 are excludeddue to the large production cross section (times branch-ing ratio). For the gray shaded region, EWSB does notoccur. In order to demonstrate that a non-minimal modelhas an interesting parameter space as large as the min-imal case, in Fig. 3, we show the result of analysis forthe case with N mess = 3. One can see that almost simi-lar interesting parameter space arises even for N mess = 3case.Finally, we notice that points shown in Fig. 2 are ac-companied by tan β (cid:39) − 60 and m / (cid:39) − FIG. 2. Parameter space of the model for µ > , Ψ) in (5,5) representation of SU (5) GUT respectively, we show values of the set of parameters ( M mess , Λ) producing the lightest CP-even Higgs mass 124GeV,125GeV and 126GeV (blue solid line). Λ ≡ k (cid:104) F Z (cid:105) /M mess was defined in Eq. (7). In the right panels, each red line is a setof points yielding the specified second CP-even Higgs mass m H . For both panels, the black lines are the group of points inthe parameter space corresponding to the specified tan β values. Each green line is the set of points giving km / = 1 , 10 and100keV from the left. For the gray shaded region, EWSB cannot happen. The left (right) green shaded region correspond to km / < km / > µ > , Ψ) in (5,5)representation of SU (5) GUT respectively, we show values of the set of parameters ( M mess , Λ) producing the lightest CP-evenHiggs mass 124GeV, 125GeV and 126GeV (blue solid line). Λ ≡ k (cid:104) F Z (cid:105) /M mess was defined in Eq. (7). In the right panels,each red line is a set of points yielding the specified second CP-even Higgs mass m H . For both panels, the black lines are thegroup of points in the parameter space corresponding to the specified tan β values. Each green line is the set of points giving km / = 1 , 10 and 100keV from the left. For the gray shaded region, EWSB cannot happen. The left (right) green shadedregion correspond to km / < km / > rent DM population requires f = O (10 − ) − O (10 − ).Note that Λ remains almost constant in the viable param-eter space in Fig. 2 and thus roughly Ω / h / ∆ becomesproportional to M − / . Hence, even if M mess changesby two orders of magnitude in Fig. 2 in the viable pa-rameter space, f needs to change only by one order ofmagnitude for DM relic density matching. We checkedthat this f is large enough to induce the decay of themessenger particle to the Higgsino and the SM lepton before the BBN era is reached. Therefore, the model canindeed produce the thermal warm gravitino dark matterfor the parameter space of ( M mess , Λ) of our interest.We conclude this section by discussing (1) informationuseful for experimental searching for the second Higgsand (2) particle mass spectra for points in viable param-eter space in Fig. 2. For the minimal MGM model with m h = 125GeV and µ > 0, using HDECAY [31], we findthat the branching ratios of the the second Higgs decayfor the main decay modes read BR( H → b + b ) (cid:39) . 78 andBR( H → τ + τ ) (cid:39) . 22. In addition, shown in Table. I isthe b -associated production cross-section of H/A for theMGM model with m h = 125GeV and µ > 0, which is ob-tained by using SusHi package [32, 33]. To help reader’sunderstanding, in Table. II, we display the particle massspectra of the model for two selective points in viableparameter space in Fig. 2. The point I ( II ) lies in theblue line of 125GeV (126GeV) in the left panel. For bothcases, one can see that stau becomes NLSP of the model.However, observation of the stau seems very challengingas can be seen in Ref. [34]. VI. CONCLUSION The minimal gauge mediation (MGM) model is appeal-ing in that it is free of FCNC and CP problems. More-over, µ -term is present in the model just as a marginal op-erator, which makes B -term vanish at a messenger massscale. Accordingly, the model is featured by a rathersmall B -term and a large tan β at the electroweak scale.The resultant large tan β , in turn, permits cancellationbetween m H u and m H d in Eq. (16), opening up the in-teresting possibility to have relatively small masses forsecond CP-even Higgs and CP-odd Higgs.This structure of the MGM model alone, neverthe-less, does not necessarily predict a light second CP-evenHiggs boson simply because there is no upper bound on aSUSY-breaking scale or a messenger mass scale. On theother hand, the small scale issues continue to challengeΛCDM model in the modern cosmology and assuming aWDM candidate can help us resolve the issue. Notablythe gravitino can play a role of the WDM provided itsmass lies in 10 − M mess , Λ) was shownto be able to be consistent with the observed Higgs mass125GeV when the mass of the lightest CP-even Higgs isidentified with 125GeV. For values of ( M mess , Λ) achiev-ing the consistency, we computed values of tan β andthe second CP-even Higgs mass m H . To our surprise,it turns out that the upper bounds of m H and m A are assmall as ∼ M mess = O (10 ) − O (10 )GeV , Λ = O (10 )GeV) in the MGM model. In particular, the massof the second Higgs (and the CP-odd Higgs) is as small as2-3 TeV for O (10) keV gravitino with k = 1, and the pre-dicted tan β is as large as 55-60. Due to the large tan β ,the Yukawa coupling to bottom quarks becomes ∼ 1, gen-erating the large production cross section for those Higgsbosons as shown in Table. I. Because of the above tworeasons we expect observation/exclusion of the additionalHiggs bosons in MSSM at the future LHC experimentsat 14 TeV run, providing us with the opportunity to testsupersymmetry with the warm gravitino dark matter. ACKNOWLEDGMENTS