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Comprehensive Analysis on the Light Higgs Boson Scenario

Comprehensive analysis on the light Higgs scenario of the minimal supersymmetric standard model is performed in the framework of the non-universal Higgs mass model (NUHM). The NUHM is known to be the simplest framework realizing the light Higgs scenario with the unifications of gaugino and sfermion masses at the scale of grand unified theory being consistent. All important constraints from collider experiments, flavor-changing processes, and cosmological observations are considered in order to clarify the allowed region of the model parameter space, where the Markov Chain Monte Carlo method is used to obtain the region. We show that the region is excluded by current LHC results for the SUSY particle searches. We discuss possible extensions to save the LHS. Even with such extensions, the measurement of B_s to mu^+ mu^- would enable us to test the LHS in the near future.


Introduction
In spite of tremendous successes of the standard model (SM), both hierarchy and dark matter (DM) problems require us to consider new physics beyond the SM. The hierarchy problem is essentially given rise to by quadratically divergent corrections to the Higgs mass term, so that the new physics is expected to appear at around or less than O(1) TeV scale. On the other hand, recent cosmological observations such as the Wilkinson Microwave Anisotropy Probe (WMAP) experiment [1,2] have established the existence of dark matter, which cannot be accounted for in the framework of the SM. Among several candidates for dark matter, the weakly interacting massive particle (WIMP) is one of the most promising candidates [3]. Since the mass of the WIMP is predicted to be less than O(1) TeV scale in order for its correct abundance observed today [1,2], the WIMP is also likely to be obtained by the new physics at the TeV scale.
The minimal supersymmetric standard model (MSSM) [4] is thought to be the most promising candidate for the new physics. It gives solutions to both hierarchy and dark matter problems simultaneously and enables us to unify SM gauge interactions. The direct search of the Higgs boson at the LEP experiment [5] severely constrains the parameter space of the MSSM, because the model predicts the mass of the lightest neutral Higgs boson h to be the same as or less than the Z boson mass at the tree level. In principle, there are two prescriptions to avoid the constraint. One is that h is composed mainly of the up-type Higgs. Radiative corrections from scalar top quarks then push up the mass of h [6]. This scenario has frequently been discussed so far. In the limit where superpartners are very heavy (although it needs fine-tunings), signals of the Higgs boson at collider experiments becomes very similar to that of the SM.
Another prescription is that h is composed mainly of the down-type Higgs. In this case, there are regions to evade the constraint though radiative corrections from scalar bottom quarks do not push up the h mass enough. The LEP constraint is evaded because the coupling of the ZZh interaction can be suppressed [7][8][9][10][11][12][13][14]. This is called the light Higgs boson scenario (LHS). The LHS can be tested at the LHC experiment because the difference from the SM inevitably exists; for example, the masses of other Higgs bosons in the LHS as well as that of h are predicted to be around the Z boson mass. The LHS has some attractive features. The LEP anomaly [9], which is the excess of the Higgs-like events with the Higgs mass being around 98 GeV reported by the LEP collaborations, can be explained. Severe fine-tunings on the supersymmetric (SUSY) little hierarchy problem is not required [10]. It is also worth noting that the LHS is consistent with a simple framework of grand unified theory (GUT) and seesaw mechanism [15]. The cosmological aspects have also studied [11,14,16,17]. Several recent papers have studied the expected parameter space in MSSM [18].
In the LHS, the charged Higgs boson significantly contributes to the b → sγ process, since their mass is very light as that of Z boson. This contribution turns out to be too large, because m H + > 350 GeV is required for the type II two Higgs doublet model [19]. This should be hence canceled out with SUSY contributions. Because of this reason, the LHS predicts that both chargino and stop are light enough. This fact inevitably leads to large Br(B s → µ + µ − ). As a result, the LHS must have large SUSY contribution to b → sγ, while small contribution to B s → µ + µ − . In order to calculate these processes accurately, the LHS should be studied comprehensively in the framework of a unification model. In this paper, we analyze the LHS in the framework of the non-universal Higgs masses model (NUHM), which is the simplest model realizing the LHS in the MSSM. Using the Markov Chain Monte Carlo (MCMC) method, we search the parameter region consistent with all important constraints from collider 1 experiments, flavor-changing processes, and cosmological observations. In particular, we impose the constraint of the dark matter relic abundance observed by WMAP [1] and the bound by direct detection experiments [20]. We find that the LHS in the framework of the NUHM is marginally inconsistent with SUSY particle search in the LHC experiments [21,22]. This paper is organized as follows. In the next section, we briefly introduce the LHS in the framework of the MSSM, and also explain how the NUHM realizes the LHS. In Section 3, the framework of our analysis for the comprehensive study of the LHS is shown in addition to some detailed explanations of experimental constrains used in our analysis. All of our physics results is presented in Section 4. Section 5 is devoted to summary. We also discuss some relaxations of experimental constraints and the boundary condition of the NUHM.

Light Higgs boson scenario
The MSSM predicts two Higgs doublets denoted by H u and H d . One Higgs doublet H u gives the masses of up-type quarks, while down-type quarks as well as charged leptons acquire their masses by interactions with H d . Five (real) scalar components in H u and H d remain as physical states; one CP-odd pseudo-scalar Higgs boson A, a pair of charged Higgs bosons H ± , and two CP-even Higgs bosons h (lighter) and H (heavier). In the basis of (Re H 0 d , Re H 0 u ) with H 0 d and H 0 u being neutral components of the two Higgs doublets, the mass matrix is given by Here, m A (m Z ) is the mass of the pseudo-scalar Higgs boson (the Z boson) and c β (s β ) ≡ cos β (sin β), where the ratio of two vacuum expectation values H u and H d defines the angle β as tan β ≡ H u / H d . Radiative corrections in each component of the mass matrix are summarized in ∆ ij , and their detailed expressions are found in Ref. [4]. The mass eigenstate of CP-even Higgs bosons h and H are obtained by diagonalizing this matrix. With the use of the mixing angle α, the mixing matrix to diagonalize the mass matrix is given by The LEP collaborations have given constraints on masses and couplings of the Higgs bosons not only in the SM but also in the MSSM [13]. According to their results, the lightest Higgs boson h in the MSSM can be lighter than 114.4 GeV , which is the lower bound on the SM Higgs boson, when the coupling between h and Z bosons g ZZh is suppressed enough. This is because the discovery of the Higgs boson at the LEP2 experiment relied on the process e + e − → Zh through the s-channel exchange of the Z boson. In the MSSM, this coupling is given by ZZh being the coupling between Z bosons and the Higgs boson of the SM. In order to obtain the Higgs boson h lighter than 114.4 GeV , a suppressed sin(β − α) is required. Since tan β is expected to be large enough to satisfy experimental constraints as shown in the following sections, a large mixing angle α ∼ π/2 is needed. On the other hand, the LEP bound on the SM Higgs boson is also applied to the heavier Higgs boson H. This constraint is complementary to that on h, because the coupling between H and Z bosons is given by g ZZH = g (SM) ZZh cos(β − α). Furthermore, the coupling between Z, A, and h is also proportional to cos(β − α), so that the search of Higgs bosons at the LEP2 experiment using the e + e − → Ah process should be paid attention to. Since the g ZAh coupling originates in a derivative coupling, this process does not provide a serious constraint due to P-wave suppression when m h > 90 GeV [13].
Since the constrained MSSM (CMSSM) [23,24] is too restricted to represent all degrees of freedom of the Higgs sector, the LHS is impossible to be realized. Particularly, the soft SUSY breaking masses of two Higgs doublet fields m 2 Hu and m 2 H d are fixed by other parameters such as universal supersymmetric particle masses m 0 and m 1/2 . As a result, when m h is lighter than 114.4 GeV , all other supersymmetric particles become also very light, and such a light Higgs boson has already been excluded. The mass spectrum of the Higgs bosons in the CMSSM is therefore inevitably the decoupled type, namely, the lightest Higgs boson is up-type (Re 3 Framework of our analysis

Markov Chain Monte Carlo
We use the MCMC method to clarify the allowed region of the LHS. The MCMC method is a random sampling algorithm, which gives a series of parameter sets called the Markov chain as output [25]. The samples of the chain obey the distribution which is proportional to a given distribution function. The distribution function of our interest is a posterior probability distribution function of model parameters x under experimental data D. Bayes' theorem tells us that the posterior probability distribution function P (x|D) satisfies the following equation, where P (x) is the prior probability function reflecting our knowledge about model parameters x, while P (D|x) represents the likelihood for the distribution function of experimental data D at given model parameters x. In our analysis, a linearly flat prior has been used for P (x), where P (x)dx gives a constant probability. For experimental data D, we have used constraints shown in the upper part of See the text [20] Anomalous magnetic moment of muon (∆a µ ) 9.5 < (∆a µ × 10 10 ) < 41.5 [26]

Constraints used in the MCMC calculation
Here, we explain some details of constraints used in our MCMC study. In addition to experimental constraints shown in Table 1, we also take a theoretical constraint into account in our study, which relates to the stability of the vacuum we are living. Constraint from the ρ parameter is not imposed because we found that it is always satisfied very well by our MCMC samples of the LHS where Higgs bosons have similar masses.
• Constraints on the ZZh and ZZH coupling constants The upper bound on (g ZZH1 /g ZZh ) 2 is obtained by the search for the e + e − → ZH 1 process in the LEP2 experiment [13]. This corresponds to the upper bounds on sin 2 (β − α) for H 1 = h and cos 2 (β − α) for H 1 = H in the MSSM. The bound at 95% C.L. is shown in Table 14 of Ref. [13] as a function of the H 1 mass. Bounds UB b and UB τ are available for the cases of Br(H 1 → bb) = 100 % and Br(H 1 → ττ ) = 100 %, respectively. For arbitrary values of Br(H 1 → bb) and Br(H 1 → ττ ), we use the upper bound UB tot (H 1 ) defined by 4 In order to satisfy the constraint for the lightest Higgs boson h, too small sin(β − α) is not required in the range 90 GeV < m h < 114.4 GeV , and, in fact, sin(β − α) 0.5 is enough. On the other hand, h can evade the LEP2 constraint even for m h < 90 GeV when sin(β − α) is smaller than 0.2. The constraint is so tight that such a region is very restricted as mentioned in the next section. It is to be noted that another coupling g ZZH ∝ cos(β − α) inevitably be large when the coupling g ZZh ∝ sin(β − α) is small. Hence, the heavier Higgs boson H tends to receive a severe constraint from the LEP2 experiment in the LHS.
• Constraints on the ZAh and ZAH coupling constants Upper bounds on the MSSM suppression factors (cos 2 (β − α) for H 1 = h and sin 2 (β − α) for H 1 = H) are obtained also by the search for e + e − → AH 1 in the LEP2 experiment, which are shown in Table 17 of Ref. [13]. Three bounds (we call UB bb , UB bτ , and UB τ τ ) are presented there. Bounds UB bb and UB τ τ are obtained for cases where both of Higgs bosons decay only into bb and ττ , respectively. The bound UB bτ is given for the case where one of Higgs bosons decays into bb and the other does into ττ . For arbitrary branching ratios of H 1 and A decays, we constrain cos 2 (β − α) and sin 2 (β − α) by using UB tot (H 1 , A) defined by Since the upper bounds come originally from the production cross section of two scalars with the center of mass energy of ∼ 200 GeV, we do not have a severe constraint on the coupling due to the P-wave suppression when m A + m H1 ∼ 200 GeV. Roughly speaking, the constraint gives a lower bound on m A as m A (200 GeV − m h ).
• Constraint on the dark matter abundance The lightest neutralino plays the role of the DM in the NUHM, and its relic abundance Ω DM h 2 is computed by solving the Boltzmann equation numerically. This abundance is constrained by the WMAP experiment [1,2]. We have used the five-year result [1] of the experiment (Ω DM h 2 = 0.1099 ± 0.0062) as shown in Table 1, which is obtained by the WMAP data alone and seems conservative. Recent result is found in Ref. [2] as Ω DM h 2 = 0.1120 ± 0.0056. The relic abundance is calculated with the use of the micrOMEGAs 2.2.CPC code [35] which allows us to deal with resonant and coannihilation processes appropriately.

• Constraints from direct detections of dark matter
Since all Higgs bosons are predicted to be light in the LHS, the spin-independent scattering cross section between the dark matter and a nucleon (a nucleus) is expected to be large. Direct detection experiments of the dark matter therefore give an important constraint on model parameters. The most severe constraint is now given by the XENON 100 experiment [20]. We have used the upper bound on the cross section as a function of the dark matter mass in our MCMC calculation. This constraint implicitly assumes that the local dark matter density around us is 0.3 GeV /cm 3 . This constraint is relaxed if we consider a non-smooth distribution of dark matter in our halo. We take into account this uncertainty on the dark matter density by relaxing the constraint of XENON 100 by factor 2 [36]. In our work, we set y = 0 for the strangeness component in a nucleon [37].
• Constraint on g µ − 2 The experimental result of the anomalous magnetic moment of muon is known to be deviated from the SM prediction. The deviation has been reported as ∆a µ = (25.5 ± 8.0) × 10 −10 [26] (See also Ref. [33]), and we use this value with the 2σ region, 9.5 < (∆a µ × 10 10 ) < 41.5, in order to constrain model parameters.
• Constraint on the b → sγ process In our MCMC calculation, we have used Br(b → sγ) = (3.52 ± 0.25) × 10 −4 [27]. Recent experimental bound, Br(b → sγ) = (3.55 ± 0.24 ± 0.09) × 10 −4 , is found in Ref. [34]. The SM prediction to the branching ratio is, on the other hand, Br(b → sγ) = (2.98 ± 0.26) × 10 −4 [38] which is somewhat smaller than the observed value. Interestingly, the LHS is possible to resolve the discrepancy, because the contribution of H ± makes Br(b → sγ) increased from the SM prediction [19]. Since all Higgs bosons are as light as the Z boson mass in the LHS, H ± contributes to Br(b → sγ) significantly and this contribution is, in fact, even too large. Actually, in the type II two Higgs doublet model, m H + > 350 GeV is required [19]. Hence, it should be canceled by SUSY contributions [10], for example, by large A-terms. We have computed the branching ratio with the SusyBSG code [39], in which the next-to-next-to-leading order calculation in QCD is involved as in the calculation of Ref. [40].
• Constraint on the B s → µ + µ − process Diagrams mediated by neutral Higgs bosons contribute to B s → µ + µ − , and their contributions are proportional to (tan β) 6 /m 4 A [41]. Since neutral Higgs boson masses are of the order of m Z in the LHS, the branching ratio of this process gives an important constraint on model parameters. Actually, the parameter region with tan β 10 is almost forbidden as can be seen in the next section. We use the current experimental bound Br(B s → µ + µ − ) < 1.08 × 10 −8 at 95% C.L.
[28] which is given by combined analysis of results in the LHCb and the CMS experiments.
• Constraint on the B ± → τ ± ν process The light charged Higgs boson H ± contributes to the B ± → τ ± ν process as in the case of B s → µ + µ − . The experimental result on the process is Br(B ± → τ ± ν) exp = (1.63 ± 0.39) × 10 −4 [27]. On the other hand, the SM prediction is Br(B ± → τ ± ν) SM = (1.01 ± 0.26) × 10 −4 [29], somewhat smaller than the experimental value. We define their ratio as r exp = Br(B ± → τ ± ν) exp /Br(B ± → τ ± ν) SM . As a result, the deviation between experimental and theoretical results is quantitatively given by r exp = 1.62 ± 0.57. Although the uncertainty still be large at present, the constraint is potentially important to restrict the LHS, because the additional 6 contribution from H ± increases the deviation [42]. We have used the formula [43][44][45][46], It is worth noting that r MSSM is always less than one. In our analysis, the experimental result with the 2σ region, 0.68 < r exp < 2.76, is applied as the cut for r MSSM .
• Constraints from sparticle searches In our analysis, we first derive the boundary condition at the GUT scale M G using tan β, µ, and m A at the m Z scale through the renormalization group running which is evaluated by the ISAJET 7.75 code [47]. Masses of sparticles are then obtained by running back from M G to m Z using the renormalization group running again. There are several lower bounds on the masses of sparticles as shown briefly in Table 1. More details on the bounds are as follows: The lightest chargino : Charged slepton : ml R > 73 GeV (ml R − m χ1 < 10 GeV) mẽ R > 99.9 GeV, mμ R > 94.9 GeV, mτ R > 86.6 GeV (else) Sneutrino : mν > 43 GeV (mν − m χ1 < 10 GeV) mν > 94 GeV (else) Stop and sbottom : mt 1 > 65 GeV (mt 1 − m χ1 < 10 GeV) mb 1 > 59 GeV (mb 1 − m χ1 < 10 GeV) mt 1 , mb 1 > 95 GeV (else).
Among those, the bound on m χ ± 1 is particularly important and gives a severe constraint on model parameters. This is because it determines the lower bound on m 1/2 , and then the lower bound on the dark matter mass is given through the GUT relation on gaugino masses. On the other hand, the bounds on squark masses also give important constraints on model parameters because large |A 0 | and/or µ tend to be favored in the LHS. Constraints on squark mass (mq > 379 GeV) and gluino mass (mg > 308 GeV) used in the MCMC study are pre-selections. We use further cuts, mq > 1100 GeV and mg > 750 GeV, for generated MCMC samples later in order to satisfy recent results at the ATLAS [21] and CMS [22] experiments.

• Constraint from the vacuum stability
Too large A-terms gives rise to charge and color breaking (CCB) minima. We therefore take the constraint |A t | 2 < 3(m 2 t1 + m 2 t2 + m 2 Hu + µ 2 ) into account in our MCMC calculation. The constraint guarantees that the vacuum we are living is the absolute minimum even if there are CCB minima [48]. Since m 2 Hu + µ 2 ∼ m 2 Z which are much smaller than stop masses m 2 t1 and m 2 t2 , the constraint gives an upper bound on |A t | in terms of the stop masses. The above constraint can be weaker when we also consider the case that our vacuum is metastable whose lifetime is much longer than the age of the universe. In such a case, the constraint is relaxed to be |A t | 2 + 3µ 2 < 7.5(m 2 tL + m 2 tR ) [49]. We adopt, however, the first one to constrain model parameters, because it is less ambiguous than the weaker constraint. It is also worth noting that the problem of CCB minima occurs when A 0 is much larger than m 0 , and such a parameter region is not favored in the view point of the naturalness. The extension of the allowed parameter region to larger A 0 is, hence, not so attractive.

LHS samples
With the use of the constraints discussed in Sec. 3.2, we have generated 4.0 × 10 5 samples in the range m h < 114.4 GeV through the MCMC algorithm. The distribution of the samples on the (m h , sin α)-plane is shown in Fig. 1. Two distinct regions can be seen in the figure. In the region around m h = 114.4 GeV, the lightest Higgs boson h is composed almost of the up-type Higgs boson Re H 0 u because of small | sin α|. Thus, the physics property in the region is essentially the same as the one frequently discussed in comprehensive analyses of the CMSSM [24]. On the other hand, in the region with large | sin α|, h consists dominantly of the down-type Higgs boson Re H 0 d , which is characteristic of the LHS as discussed in Sec. 2. We therefore use only 2.2 × 10 5 samples satisfying | sin α| > 0.9 in the next section.

Physics results
There are six parameters in the NUHM. However, only two constraints on Ω DM h 2 and Br(b → sγ) are used as Gaussian distributions in our MCMC calculation. Central values of those constraints are reproduced simultaneously not only at a point but in a wide parameter space. When we see number distributions of samples on two parameter space, the region of large number of samples does not always means better agreement with experimental data than other regions; the region of large number of samples may mean just the large spread of the acceptable region in the projected four parameter space. Therefore, it is better to use the likelihood in order to see the experimentally acceptable region of the parameter space. The likelihood quantifies the agreement with experiments, and we define it as the product of two Gaussian distribution functions for Ω DM h 2 and Br(b → sγ). The normalization is defined so that the set of the central values for them in Table 1 makes the likelihood unity. The larger value of the likelihood means the better agreement with experiments. The maximum value of the likelihood is calculated in each cell in two parameter space, and we show the distributions in this section. In the region where the maximum of the likelihood is close to unity in the two parameter space, experimental constraints can be satisfied very well by appropriate choices of other four parameters.

Parameter region
First, results for the parameter distribution in the NUHM are shown in Fig. 2  Other two parameter, m A and tan β, are distributed in restricted regions, m A ≃ 94 -107 GeV and tan β ≃ 7 -10, respectively. The yellow (bright) region is of a large value of likelihood maximum in a cell. The contribution to b → sγ from charged Higgs should be cancelled out by SUSY contribution, and hence allowed parameters have upper bounds. Allowed parameters in Fig. 2(a) are divided into two regions by dark matter annihilation processes. One around m 1/2 = 140 GeV is pseudo-Higgs mediation region, and the other is coannihilation region. In the former region, the relic abundance of the LSP is governed by the pseudo-Higgs mediated process,χ 0 1χ 0 1 → A → bb. In comparison between these two regions, the likelihood is somewhat better for the coannihilation region.
As seen in Fig. 2(b), smaller value of µ < 400 GeV are entirely rejected. This is due to the direct detection constraint by XENON 100. Since the LHS has the small Higgs masses and a large tan β, direct detection rate which is given by mediation of the lightest Higgs (down-type one) is predicted to be large. For a small µ, the rate becomes too large because of additional enhancement due to a significant mixing between bino and Higgsino.
In the coannihilation region, the appropriate relic abundance of the LSP is given by a tuning of the masses of the LSP and stau, i.e. a tuning of m 1/2 and m 0 . Since the relic abundance in the region is independent of the value of µ, a broad range of µ is allowed. In the pseudo-Higgs mediation region, the relic abundance depends on the value of the µ term as well as the mass difference between the LSP and pseudo-scalar Higgs boson mass because the coupling of the LSP to Higgs bosons depends on µ. Meanwhile the relic abundance is independent of m 0 . Large µ is favored in order to satisfy the constraint by direct detection experiments. On the other hand, as concerns little hierarchy, too large value of µ is not favorable because light Higgs masses require the cancellation between very large values of µ and m H d,u .
In Ref. [16], the authors pointed out that there is a region where the relic abundance is determined by the processesχ 0 1χ 0 1 → hA, HA, W ± H ∓ , Zh processes for lower value of µ ≃ 300 GeV. For this small µ, since the mixing between bino and Higgsino is significant, these processes can be dominant for the control of the relic abundance of the LSP. However, due to the smallness of µ, this region is excluded by the recent constraint of direct detection experiments.
In Fig. 2(c), the distribution of A 0 is shown. The relic abundance of the LSP is independent of A 0 , and hence the allowed region is almost determined by the b → sγ process and the condition to avoid dangerous vacua. The large negative A 0 is favored by the b → sγ process in order to cancel a large H + contribution while too large |A 0 | leads to dangerous charge and  color breaking minimum. As a result, the allowed range of A 0 has upper and lower bounds. In the coannihilation and the pseudo-Higgs mediation regions, |A 0 | becomes large for a large m 1/2 and m 0 , respectively. This trend compensates heavy SUSY particle masses for the contribution to b → sγ process. In the pseudo-Higgs mediation region, SUSY particles have larger masses than those in the coannihilation region. Therefore, a larger |A 0 | is required in the pseudo-Higgs mediation region than that in the the coannihilation region for m 1/2 ≃ 140 GeV.
In addition, the B s → µ + µ − process gives complementary condition to the b → sγ process and the relic abundance. The branching ratio of B s → µ + µ − is proportional to tan 6 β/m 4 A . Hence, this process is drastically enhanced in the LHS due to a small m A . As a result, large tan β 10 is ruled out by this constraint. On the other hand, the lower bound on tan β comes from the realization of the LHS [8]. The lower bound tan β 7 results in Br(B s → µ + µ − ) 7 × 10 −9 (See Fig. 3). Constraint on this process also excludes the region where lighter stop mass is very light as 200 -300 GeV due to a large |A 0 |. A light stop mass is favored for b → sγ, and however this inevitably enhances Br(B s → µ + µ − ).
Before the discussion about the LHC constraint on SUSY particles, we mention the distributions of Higgs boson masses. The SM-like Higgs search at the LHC constrains the mass of the heavier neutral Higgs in the LHS because it is composed mainly of the up-type Higgs. Although stop masses are not large in LHS, the SM-like Higgs mass is lifted up because of following two contributions; one is from the large A-term which is required to be consistent with the b → sγ constraint and the other is from the off-diagonal component of the neutral Higgs mass matrix. As a result, the mass is found to be distributed in the range of 114 GeV < m H < 122 GeV in our MCMC samples. Since the LHS requires a large | sin α| (namely, m A ∼ m Z ), the allowed region of m A is nearly the same range; 94 GeV m A 107 GeV. Because of m 2 H + = m 2 A + m 2 W at the tree level, m H + is restricted as 122 GeV m H + 133 GeV. The ranges of m H + and tan β are not constrained by present results of the charged Higgs search by collider experiments.
Finally, let us consider LHC constraints on SUSY particles. In the analysis, we use the NUHM as a benchmark model in which the constraint on boundary conditions of m Hu and m H d are relaxed as compared with CMSSM. But the constraints on the parameter space of (m 0 , m 1/2 ) would be exactly similar to the CMSSM case in the LHS region shown above. In these models, differences arise only from m Hu and m H d values which can be rewritten by m A and µ at the weak scale while the masses of gluino and first generation of squark are nearly the same. Additionally, the LSP mass is also nearly the same in the LHS region shown above because the Higgsino component of the LSP is small. Then, the production cross section of SUSY particles and collider signal of these events would be no change and the constraint on the CMSSM parameter space [21,22] excludes the whole LHS region shown above.

Summary and Discussions
We have analyzed comprehensively the LHS of the MSSM in the framework of the NUHM, which is the simplest one realizing the LHS in GUTs. We have taken into account, all important constraints from collider experiments, flavor-changing processes, and cosmological observations. To search allowed parameter sets, we have used the MCMC method. We have shown that there are two parameter regions consistent with these constraints except for LHC results. One is the coannihilation region and the other is the pseudo-Higgs mediation region. To reconcile the H ± contribution to the b → sγ process with experimental bounds, all SUSY particles remains light, as 1000 GeV. Hence, we have found that the current LHC results have already excluded the whole LHS region in the NUHM.
Finally, we discuss that some relaxation of our restriction or some extensions of our simplest model could have allowed LHS regions in the rest of this paper. First, we consider the relaxation of experimental constrains on Ω DM and muon g − 2. Other constraints are considered to be robust or have not reduced sample points significantly. Second, the restriction of the NUHM is relaxed. That is, the boundary condition at the GUT scale is relaxed, or the gauge group of the GUT is broken at an intermediate scale above the EW scale.
The relaxation of the restriction of the relic abundance of the LSP is expected to bring us the large number of allowed parameter sets. It is favorable that the thermal relic abundance of the LSP is consistent with the WMAP observed dark matter abundance. However, if this is not the case, the model is not excluded unless the relic abundance is larger than the WMAP abundance. The relic abundance can significantly be reduced for a smaller µ, since the Higgsinobino mixing is enhanced. The reduced abundance of the LSP relaxes the constraint from direct detection experiments. Then, a small µ which enhances the direct detection cross section is allowed in this relaxation condition. In m 1/2 -m 0 space (or mg-mq space), the extended allowed region of µ does not give new sample points at the neighborhood of the coannihilation regions where the relic abundance is independent of µ. However, a new allowed region with µ 400 GeV appears for m 1/2 300 GeV. In this region, the LSP is significantly annihilated byχ 0 1χ 0 1 → hA, HA, W ± H ∓ , Zh processes and larger values of m 0 and m 1/2 are allowed. Then, the squark and gluino masses can exceed the LHC constraints.
Next, we mention the relaxation of the muon g − 2 constraint. In pseudo-Higgs mediation region, mq is extended to mq ≃ 1.2 TeV by removing the g − 2 constraint. However, this relaxed region is still rejected by the LHC results because the gluino mass is light as 400 GeV. On the other hand, the coannihilation region is extended to m 1/2 ≃ 450 GeV. For this upper limit, the gluino mass is about 1000 GeV. This remains as the region excluded by the LHC results. Hence, the relaxation of the g − 2 constraint does not bring us new allowed parameter sets.
The relaxation of NUHM conditions changes the spectrum of the MSSM particles. If the gauge group of the GUT is broken into some gauge group (for example, SU(3) C × SU(2) L × SU(2) R × U(1) B−L of the left-right model) at an intermediate scale above the EW scale, the mass relation of the MSSM particles can be modified. In the modified spectrum of the MSSM particle masses, first and second generation squarks are required to be heavy because of the LHC constraints while stop should be light in order to have a sizable SUSY contribution to b → sγ. This might be realized through the renormalization running with a large |A 0 |. Another possibility might be that the third generation squarks are very distinguished from first and second generations as a boundary condition. Of course, the MSSM without any boundary conditions at the GUT scale would have enough parameter space to realize the LHS allowed by current experimental data. Detailed studies of each possibilities are beyond the scope of this paper. Even in new parameter sets obtained by such relaxations, Higgs bosons should be light in the LHS and they give large contributions to b → sγ and B s → µ + µ − . Gaugino and stop should also be light enough to cancel a large contribution of H ± to b → sγ. Furthermore, the charged Higgs boson production is enhanced because the coupling, g W ± H ∓ h , depends on cos(β − α) [8] (For the case with CP-violating scalar sector, see also Ref. [50] and references therein). As a result, the LHS can be judged in the near future by the LHC SUSY particle search, the LHCb search of B s → µ + µ − process and the search for H ± h production at the LHC experiments.
Note added: Updated constraint on B s → µ + µ − at the CMS just appeared as Br(B s → µ + µ − ) < 7.7 × 10 −9 at 95% C.L. [51] during finalization of this article. A small region in Fig. 3 satisfies the bound, and the LHS would be tested in near future without assuming mass spectrum of SUSY particles.