Footprints of Supersymmetry on Higgs Decay

Motivated by future collider proposals that aim to measure the Higgs properties precisely, we study the partial decay widths of the lightest Higgs boson in the minimal supersymmetric standard model with an emphasis on the parameter region where all superparticles and heavy Higgs bosons are not accessible at the LHC. Taking account of phenomenological constraints such as the Higgs mass, flavor constraints, vacuum stability, and perturbativity of coupling constants up to the grand unification scale, we discuss how large the deviations of the partial decay widths from the standard model predictions can be. These constraints exclude large fraction of the parameter region where the Higgs widths show significant deviation from the standard model predictions. Nevertheless, even if superparticles and the heavy Higgses are out of the reach of 14TeV LHC, the deviation may be large enough to be observed at future $e^+e^-$ collider experiments.


Introduction
The discovery of the Higgs boson at ATLAS and CMS experiments [1,2] made a revolutionary impact on the field of particle physics. It not only confirmed the so-called Higgs mechanism for the electroweak symmetry breaking, but also opened a new possibility to perform a precise test of the standard model (SM) by studying the properties of the Higgs boson. In the SM, the coupling constants of the Higgs boson with other particles are well understood using the fact that the masses of quarks, leptons, and weak bosons originate in the vacuum expectation value (VEV) of the Higgs field, resulting in the prediction of the partial decay widths of the Higgs boson into various particles.
In models with physics beyond the SM (BSM), measurements of the Higgs couplings provide even exciting possibilities. In large class of BSM models, there exist new particles at the electroweak to TeV scale, which affect the properties of the Higgs boson. Thus, with the detailed study of the Higgs properties at collider experiments, we have a chance to observe a signal of BSM physics. Such a study will be one of the major subjects in forthcoming collider experiments, i.e., the LHC and future e + e − colliders like ILC and TLEP [3].
Low energy supersymmetry (SUSY) is a well-motivated candidate of BSM physics. Compared to the SM, the particle content is enlarged in SUSY models. Even in the minimal setup, i.e., in the minimal SUSY SM (MSSM), there exist two Higgs doublets, H u and H d , as well as superparticles. The lightest Higgs boson h, which plays the role of the "Higgs boson" discovered by ATLAS and CMS, is a linear combination of the neutral components of H u and H d , while there exist other heavier Higgses. In the case where the mass scales of the heavier Higgses and the superparticles are high enough, the properties of h are close to those of the SM Higgs boson. On the contrary, if the heavier Higgses or superparticles are relatively light, deviations of the Higgs properties from the SM predictions may be observed by future collider experiments. With the precise measurement of the partial decay widths (or branching ratios) of the Higgs boson, information about the heavy Higgses and/or superparticles may be obtained even if those heavy particles can not be directly discovered.
In this paper, we discuss how low the mass scales of the heavier Higgs bosons and superparticles should be to observe a deviation. We evaluate the partial decay widths of the lightest Higgs boson in the MSSM, taking account of the following phenomenological constraints: Higgs mass, flavor constraints of the B mesons, stability of the electroweak (SM-like) vacuum against the transition to charge and color breaking (CCB) vacua, and perturbativity of coupling constants up to a high scale. These constraints exclude large fraction of the parameter region giving rise to a significant deviation. Even so, we will see that the deviations of the partial widths from the SM predictions can be of O(1) % for some of the decay modes in the parameter region allowed by the above-mentioned constraints. In particular, the deviations may be large enough to be observed by future future e + e − colliders like ILC and TLEP even if superparticles are so heavy that they would not be observed at the LHC.
The organization of this paper is as follows. In Sec. 2, we briefly overview the properties of the Higgs bosons in the MSSM. We also summarize the phenomenological constraints that are taken into account in our analysis. Then, in Sec. 3, we calculate the partial decay widths of the lightest Higgs boson in the MSSM and discuss how large the deviation from the SM prediction can be. Sec. 4 is devoted for conclusions and discussion. At the tree level, the lightest Higgs mass is predicted to be smaller than the Z-boson mass, while it is significantly pushed up by radiative corrections [4,5,6,7,8]. The mass matrix of the neutral CP-even Higgs bosons is denoted as where δM 2 ij represents radiative corrections. At the one-loop level, the top-stop contribution dominates the radiative correction to the lightest Higgs mass, and is approximated as where v ≃ 246 GeV is the SM Higgs VEV, m t is the top-quark mass, m 2 t ≡ mt 1 mt 2 (with mt 1 and mt 2 being the lighter and heavier stop masses, respectively), and X t = A t −µ cot β (with A t and µ being the tri-linear scalar couplings for stop and the SUSY invariant Higgsino mass parameter, respectively). #1 The top-stop contribution can significantly enhance the lightest Higgs mass. On the other hand, the bottom-sbottom contribution to the lightest Higgs mass becomes sizable when the bottom Yukawa coupling is large. It is likely to decrease the lightest Higgs mass.
When stop masses are O(1) TeV, there are up to four solutions for A t to satisfy the observed value of the Higgs mass m h , for which we use m h = 125.7 GeV [10]. Let us call these four solutions as #1 In this paper, we adopt the convention of the SLHA format [9].
• NS: negative A t with smaller |A t |, • NL: negative A t with larger |A t |, • PS: positive A t with smaller |A t |, • PL: positive A t with larger |A t |.
Assuming universal sfermion masses at the SUSY scale, the value of |A t | is typically a few times larger than the stop mass for NL and PL cases. Such a large value of |A t | has significant phenomenological implications, as we will discuss in the next section.
Since the one-loop correction to the Higgs mass is comparable to the tree-level value, higher order corrections are necessary to obtain reliable results. In particular, QCD correction, which appears at the two-loop level, and a large hierarchy between the SUSY scale and the electroweak scale require the resummation of the leading and sub-leading logarithms. We use FeynHiggs 2.10.2 [11,12,13,14,15] for the precise evaluation of the Higgs masses (as well as the mixing parameters and the partial decay widths of h).
At the tree level, H u (H d ) couples only to up-type quarks (down-type quarks as well as leptons). However, this is not the case once radiative corrections due to superparticles are taken into account. The Higgs couplings to bottom quark and tau lepton can be subject to sizable corrections even when SUSY breaking scale is very large. Let us parameterize the effective hbb and htt vertices including radiative corrections as [16,17,18,19,20,21] where b R , t R , and Q L are right-handed bottom, right-handed top, and third-generation quark-doublets, respectively. In addition, i, j and k are SU(2) L indices, while the color indices are omitted for simplicity. Here, ∆y b and ∆y t are non-holomorphic radiative corrections to the Yukawa coupling constants. #2 After the electroweak symmetry breaking, the Yukawa couplings are related to the quark masses as #3 where, at the leading order in the mass-insertion approximation, ∆ f is given by #2 For a detailed treatment of the non-holomorphic corrections, see Refs. [22,23]. #3 These relations hold at the SUSY breaking scale. Thus, the quark masses and Yukawa couplings in the formula should be understood as the running parameters at the scale.
with M 3 being the gluino mass. The loop integral is defined as Notice that ∆ b is enhanced when tan β is large, while ∆ t is suppressed by cot β.
The mass eigenstates of the Higgs bosons are given by linear combinations of H u and H d . The CP-even parts of their neutral components are related to the mass eigenstates as The mixing angle α depends on the pseudo-scalar mass m A , and shows a decoupling behaviour, i.e., cos(β − α) → 0 as m A → ∞. In this limit, h behaves as the SM Higgs boson. Using Eq. (2.1), we obtain [24] cos (2.11) In Fig. 1, we show the behavior of cos(β − α) as a function of m A with the masses of superparticles being fixed. Here, all the sfermion mass parameters, mQ, mŨ , mD, mL, and mẼ, are taken to be universal at the SUSY scale M SUSY , where these mass parameters are soft SUSY breaking masses of sfermions with gauge quantum numbers of (3, 2, 1 6 ), (3, 1, − 2 3 ), (3, 1, 1 3 ), (1, 2, − 1 2 ), and (1, 1, 1), respectively, with the numbers in the parenthesis being quantum numbers for SU(3) C , SU(2) L , and U(1) Y . Throughout our study, we take M SUSY = (mQmŨ ) 1/2 . In addition, the sfermion masses are assumed to be universal in generation indices. We can see that radiative corrections can enhance cos(β − α) by an order of magnitude when A t is large, while it is comparable to the tree-level value with the smaller |A t | solutions.
Denoting hf f coupling (with f being the SM fermions) as we obtain the hbb coupling constant as where the superscript "(SM)" is used for the SM prediction. When m A is relatively large, sin(β − α) is almost unity, while the second term proportional to cos(β − α) induces a sizable deviation from the SM value. Similar relation holds for hτ τ vertex, and ∆ τ is approximately given by with M 2 being the Wino mass. Quantitatively, ∆ τ is smaller than ∆ b in the parameter space of our study. As we will see later, Γ(h →bb) and Γ(h →τ τ ) may show sizable deviations from the SM predictions even if m A is above TeV. The htt coupling constant is obtained as The deviation from the SM value mainly comes from the second term in the bracket and is not enhanced by tan β, since ∆ t is proportional to cot β. The gauge-boson final states are also important. In order to calculate the partial decay widths of the Higgs to gauge bosons, we have modified FeynHiggs 2.10.2 package to properly take account of the effect of non-holomorphic correction to the Higgs interaction. #4 We have also modified the package to take the α eff approximation [28] for calculating the partial widths, in which the renormalization scale of the Higgs wave functions is set to be p 2 = 0, and the effects of radiative corrections are included in the mixing between the light and heavy Higgs bosons. Then, as we will see below, the partial decay widths show proper decoupling behavior in the large m A limit. #5 The processes induced by triangle loops, h → gg and γγ, have been important for the Higgs discovery at the LHC. They are also important in studying new particles that couple to the Higgs boson. In SUSY models, the stop and sbottom loops contribute to the hgg coupling. It is expressed by an approximate formula (cf. Refs. [25,26,27]) up to D-term and bottom-loop contributions. Here, In the right-hand side, the first term comes from the top loop, while the second term is given by the stop and sbottom loops. The correction is positive in the non-mixing limit (i.e., X f → 0), whereas it becomes negative when the mixing terms are sizable. We also note here that hV V couplings (with V V = W + W − and ZZ) are approximately given by g hV V /g . This ratio is very close to unity, and hence the deviations in these modes are very small.

Constraints
Before discussing the possibility of observing a deviation of the Higgs partial widths from the SM prediction at future colliders, we summarize phenomenological constraints on the MSSM parameter space which are adopted in our analysis.
In the SM, the flavor-changing decay, B s → µ + µ − , proceeds by virtual exchanges of the Z and W bosons. They are suppressed by the final-state helicity. In contrast, SUSY contributions can be enhanced considerably by large tan β, when virtual exchanges of the heavy Higgs boson contribute to the decay [30,31]. The branching ratio is expressed as [32,33,34,35] Br We use Eq. (2.13) for the hbb coupling to calculate the partial decay widths of h → γγ, γZ and gg. We found that the factor of (1 − ∆ b cot α cot β) was missing in FeynHiggs 2.10.2 package (see Eq. (2.13)). #5 In particular, the partial width Γ(h → gg) converges to the SM value in the limit of large m A and squark masses. This behavior looks inconsistent with the result shown in Ref. [29].
where G F is the Fermi constant, m Bs is the B s -meson mass, f Bs is the decay constant of B s , m µ is the muon mass, τ Bs is the lifetime of B s , and V ij is Cabbino-Kobayashi-Maskawa matrix element. In the above expression, C 10 , C Q1 and C Q2 are the Wilson coefficients of the effective operators, i are obtained by flipping chiralities, R ↔ L. Among them, the SM contribution appears only in C 10 . Including higher order contributions and taking account of effects of the B s -B s oscillation, the SM prediction becomes [36] Br (2.18) This can be compared with the LHCb measurements [37] Br where the first term in the right-hand side includes both the SUSY and SM contributions, the 95% C.L. bound is estimated as (2.20) We will adopt this constraint in our numerical analysis. Even when superparticles are heavy, they affect the branching ratio through non-holomorphic contributions to the heavy Higgs couplings. Including radiative corrections and diagonalizing the quark mass matrices, effective couplings of the heavy Higgs bosons to the down-type fermions become [30,31] The flavor-changing coupling is induced by ∆ FC as Here, soft scalar masses are assumed to be universal in generation, and ∆ ℓ is obtained by substitutingτ →l in Eq. (2.14). Then, the Wilson coefficients receive Higgs-mediated contributions, where m b is the bottom-quark mass, θ W is the Weinberg angle, and xt µ ≡ m 2 t /µ 2 . The non-holomorphic correction ∆ FC as well as ∆ b does not decouple even for very heavy superparticles. We will see later that the corrections to Br(B s → µ + µ − ) can be sizable and that the constraint excludes some part of the parameter space of our interest.
The branching ratio of the inclusive decay of b → sγ may also be sensitive to the nondecoupling contributions to the (charged) Higgs boson. In the numerical analysis, defining where the experimental value Br(b → sγ) exp = (3.43 ± 0.21 ± 0.07) × 10 −4 [38] and the SM prediction Br(b → sγ) SM = (3.15±0.23)×10 −4 [39] are combined. At the current accuracies, B s → µ + µ − imposes more stringent bound on the parameter space than b → sγ except when superparticles are light.
In the numerical analysis, SuperIso 3.4 [40] is used for evaluating the SUSY contributions to the branching ratios as well as the SM predictions. In our analysis, we assume that the squark masses are universal in generations. If the squark masses are non-universal, there are extra contributions to ∆ FC , and the flavor constraints are affected (see e.g., Ref. [41]). Such a non-universality is expected even in the model with the universal scalar masses at the GUT scale. Thus, it should be noted that the flavor constraints that we will show below are just for a particular choice of the squark-mass parameters and may change if the squark mass matrices have non-universal structures.

Vacuum stability
With sufficiently large |A t |, CCB vacua arise, and the minimum of the scalar potential with the correct EWSB becomes a false vacuum [42,43,44,45]. When |µ| ≪ |A t |, stop and the up-type Higgs fields acquire large VEVs at the CCB vacua, while the VEVs of other fields are relatively small. Recently, the decay rate of the SM-like vacuum has been studied in detail for such a case [46,47,48,49]. On the other hand, if µ is as large as the stop masses, the down-type Higgs boson also has a large VEV at the CCB vacua due to the tri-linear scalar coupling among stops and the down-type Higgs, which is proportional to y t µ. The vacuum stability condition is important in such a case because significant deviations of the Higgs partial widths from the SM prediction may occur. In order to study the SM-like vacuum stability, we consider the tree-level scalar potential in the field space involvingt L , t R , h u and h d (which are canonically normalized scalar fields embedded in the left-handed stop, right-handed stop, the up-type Higgs and the down-type Higgs, respectively).
The relevant part of the potential is given by When the CCB vacua become deeper than the SM-like vacuum, the latter is not stable. The vacuum decay rate per unit volume is calculated in the semi-classical approximation, and is expressed as where S E is the Euclidean action of the so-called bounce solution [50,51]. For the decay of the SM-like vacuum, C is estimated to be ∼ (100 GeV) 4 . In order for the lifetime of the SM-like vacuum to be longer than the present age of the universe, the Euclidean action is required to satisfy In our numerical analysis, CosmoTransition 2.0a1 [52] is used to find the bounce solution in the four-dimensional field space parameterized by h d , h u ,t L andt R , and to calculate the Euclidean bounce action S E . The calculation is done at zero temperature, and therefore thermal effects are not taken into account. The model parameters such as the tri-linear and top Yukawa couplings are evaluated at the SUSY scale M SUSY .
In Fig. 2 we show the contours of constant S E for the PS and PL solutions on m A vs. µ plane. Here, mQ = mŨ = M 3 = 5 TeV, while sfermion masses, mD, mL and mẼ, are taken to be max(mŨ , µ) (see Sec. 3). In each plot, the region between the solid (dashed) green lines satisfies S E > 400 (300). For the PS solution, these lines appear only for |µ| > 10 TeV. On the other hand, the upper bound on |µ| is comparable to the stop masses for the PL solution.
The tree-level potential Eq. (2.25) is used for our numerical analysis. Radiative corrections may change the scalar potential particularly around the SM-like vacuum. In order to study their effects, we also estimated the decay rate of the SM-like vacuum by including the top-stop and bottom-sbottom one-loop correction to the Higgs potential. #6 We found that the Euclidean action tends to increase by about 10 -20% for S E ∼ 400 from that of the tree-level potential. In order to see the sensitivity of S E to µ, we show the contours of S E = 300 in the same plot.
When both µ and tan β are large, we might have to consider other CCB vacua in the sbottom-Higgs direction, which is driven by the tri-linear coupling of y b (A b h d −µh u )b LbR / √ 2 (whereb L andb R are left-and right-handed sbottoms, respectively) (cf. Ref. [41]). If both µ and A b are large, the bottom Yukawa coupling can be enhanced not only by tan β but also by (1 + ∆ b ) −1 (see Eq. (2.31)). However, when the squark masses are universal, such #6 A better treatment would be to introduce full one-loop radiative corrections to the potential involving stop, sbottom and the Higgs boson so that the potential is stable against the renormalization scale at least at the one-loop level. a parameter region is already excluded by the other constraints discussed in this section. Therefore we do not consider the constraint coming from the CCB vacuum involving the sbottom sector in this paper.

Bottom Yukawa coupling
When tan β is very large, the bottom Yukawa coupling is sizable. We can impose an upper bound on tan β by requiring perturbativity of the Yukawa coupling constants up to, for instance, the GUT scale. In the MSSM, the bottom Yukawa coupling constant y b is proportional to (1 + ∆ b ) −1 , and hence, y b is enhanced when ∆ b is negative (see Eq. (2.4)). Consequently, the bound on tan β is more severe when ∆ b < 0. In our numerical analysis, we estimate the bottom Yukawa coupling constant using the following relation: #7 . (2.31) Then, we follow the evolutions of coupling constants by solving the renormalization group equations at the one-loop level and impose a condition that the bottom Yukawa coupling is perturbative up to the grand unified theory (GUT) scale M GUT . Numerically, we require In discussing the perturbative bound, we neglect holomorphic corrections to the Yukawa coupling constant, which are orders-of-magnitude smaller than the tree-level value of the Yukawa coupling constant for tan β ≫ 1.
where we take M GUT = 2×10 16 GeV. This constraint excludes a large tan β region especially when ∆ b < 0. Notice that ∆ b is a non-decoupling parameter, and hence, this constraint is important even in the limit of heavy superparticles.

Higgs Partial Decay Widths
In this section,we discuss the partial decay width of the lightest Higgs boson h. We define the ratio of the partial decay width of the lightest Higgs boson to that of the SM prediction: where F denotes a specific final state. In the following, we show how much R F can deviate from the SM prediction (R F = 1) for various final states. As we have mentioned, FeynHiggs 2.10.2 is used to calculate the partial decay widths of the Higgs boson, in which full oneloop contributions are taken into account for the fermionic final states. In the package, a resummation of the ∆ b corrections is also included in calculating the partial decay width for h →bb [53]. In our numerical calculation, we adopt (approximate) GUT relation among the SU(3) C , SU(2) L , and U(1) Y gaugino masses: All the phases in the MSSM parameters are assumed to be negligible, and we adopt the convention of M 3 > 0. For simplicity, we also assume that the sfermion masses are universal with respect to the generation indices. First, we show the soft mass dependence of R F without taking the phenomenological constraints into account. In Fig. 3, R F are shown for F = γγ, W + W − , gg,bb, andτ τ as functions of m soft , taking M 3 = m A = −µ = mQ = mŨ = mD = mL = mẼ ≡ m soft and tan β = 40. #8 By choosing negative µ with |µ| ∼ m soft , the correction to the bottom Yukawa coupling becomes significant.
Remarkably, even though m soft is around several TeV, the partial decay widths for F =bb, τ τ and gg deviate from the SM prediction by more than O(1) % for the PL solution. However, SUSY contributions to the other widths are smaller. Note that Rb b − 1 is as about twice as Rτ τ − 1 for the PL solution, which is due to the difference between ∆ b and ∆ τ . We also note here that the partial decay widths have an appropriate decoupling behaviour, i.e., Γ(h → F ) → Γ (SM) (h → F ) as the masses of superparticles and the heavy Higgs bosons become infinitely large.
Next, let us include the phenomenological constraints discussed in Sec. 2.2. In Fig. 4, contours of Rb b − 1 are shown on m A vs. tan β plane. Here, all the sfermion masses, M 3 , and |µ| are set to be 5 TeV. In the cases of the PS and PL solutions, the contours end at #8 We have checked that R W + W − ≃ R ZZ holds in the parameter region of our study.  merge into a single solution at certain tan β, which is the value where the top-stop contributions cannot raise the Higgs mass anymore for fixed m soft . Note that, ∆ b is negative when µ < 0 and A t > 0. In such a case, y b is significantly larger than the tree-level value, and thus, the bottom-sbottom loop contributions are enhanced.
In Fig. 4, the constraints from Br(B s → µ + µ − ), the vacuum stability, and the perturbativity of the bottom Yukawa couplings are shown. Wide parameter region is excluded when A t is large, i.e., in the PL and NL panels. The constraints from Br(B s → µ + µ − ) and the vacuum stability become weaker for large m A , while that from the perturbativity is not. For the PL solution, the constraints from Br(B s → µ + µ − ) and the vacuum stability change drastically as we vary tan β for tan β 30, where the bottom Yukawa coupling is much larger than the tree-level value.
Even if the masses of superparticles are relatively large (i.e., 5 TeV), Rb b −1 can be as large as O(1) %. Such a large deviation may be within the reach of future collider experiments. Expected accuracies at the future experiments have been discussed (see Table 1-16 of Ref. [3] and Ref. [54]); the numbers are summarized in Table 1. The accuracies of δΓ(h →bb) = 0.64%, δΓ(h →τ τ ) = 1.1%, and δΓ(h → gg) = 1.2% are claimed to be achievable at e + e − colliders ultimately. Therefore, it is found that, even if the superparticles are kinematically  Table 1: Expected accuracies of the determinations of the partial decay widths of the Higgs boson in units of percents [3,54]. The accuracies of the widths are assumed to be twice the accuracies of the determinations of couplings.
unaccessible at the LHC, we may observe the MSSM signal by studying the partial decay widths of h in detail.
For the PL solution, it is also found that the partial width of h →bb can deviate from the SM prediction by about 2% even for m A = 6 TeV. On the other hand, we checked that Rτ τ is smaller by about 1% than Rb b at the same parameter point, because the non-holomorphic correction |∆ b | is bigger than |∆ τ | at this model point. When tan β is smaller, the difference between Rb b and Rτ τ decreases, since ∆ b is approximately proportional to tan β; in such a case both Rb b and Rτ τ are well approximated by (sin α/ cos β) 2 .
Let us see how much Rb b and Rτ τ can change in the parameter space consistent with the phenomenological constraints. We have performed the scan in the following parameter space of the MSSM: • mQ = mŨ = M 3 = 2, 3, 4, and 5 TeV, • mD = mL = mẼ = max(mŨ , |µ|), • 0.8 TeV ≤ m A ≤ 6 TeV, Here, A t is determined to satisfy m h = 125.7 GeV. The other tri-linear couplings (e.g., A b and A τ ) are assumed to be equal to A t ; we checked that our numerical results are insensitive to this assumption. We take mQ = mŨ = M 3 , while sfermion masses other than mQ and mŨ are set to be equal to max(mŨ , |µ|). We have checked that Rb b and Rτ τ are almost insensitive to these scalar masses unless they are very small. However, when |µ| is much larger than mD, mL, and mẼ, bottom and stau mixings become sizable which causes additional complexity Figure 5: The ranges of Rb b (blue) and Rτ τ (red) with small |A t | solutions (i.e, the PS and NS solutions for A t ). We take mQ = mŨ = M 3 = 2, 3, 4 and 5 TeV. Other sfermion masses are set to be max(mQ, |µ|). The partial widths are constrained by Br(B s → µ + µ − ), the vacuum stability condition (S E > 400), and the perturbativity of the bottom Yukawa coupling (|y b (GUT)| < 1). The black region corresponds to the tree-level prediction for tan β = 5-50. The black dashed line is the SM prediction. to our parameter scan, and/or lighter stau become tachyonic. The values of mD, mL, and mẼ are chosen to avoid this problem.
If |µ| would become much larger than squark masses (e.g., |µ|/mŨ ≫ 5), the Higgs partial widths could deviate from the SM prediction sizably, since the Higgs mixing angle would be enhanced by radiative corrections. However, the vacuum stability condition excludes significant amount of the parameter region with large |µ|, as we have shown in Fig. 2. In our scan, we set |µ|/mŨ ≤ 5; this upper bound is large enough to cover the whole region allowed by the vacuum stability.
In Figs. 5 and 6, we show the minimal and maximal values of Rb b and Rτ τ as functions of m A and m soft , taking into account the phenomenological constraints discussed in the previous section. Fig. 5 corresponds to the small A t solutions (NS and PS). The largest and smallest values of Rb b and Rτ τ are achieved when |µ| and tan β are large and are marginally allowed by the phenomenological constraints. The deviations mainly come from radiative corrections to the Higgs mixing angle and ∆ b . In particular, the maximal value of Rb b is larger than that of Rτ τ for large mQ = mŨ . As discussed earlier, ∆ b becomes negative and sizable with µ < 0, which results in a significant enhancement of Rb b . Fig. 6 shows the minimal and maximal values of Rb b and Rτ τ , taking all the solutions of NS, PS, NL, and PL into consideration. They change significantly compared to Fig. 5. The partial widths become extremum when A t takes the NL or PL solution except the maximal value for small m A . For m A 1 TeV, the NS or PS solutions give the maximal value, since the phenomenological constraints, especially the vacuum stability condition, are too severe for the NL and PL solutions (see Fig. 4). As in the case of Fig. 5, the maximal value of Rb b is much larger than that of Rτ τ because |∆ b | ≫ |∆ τ |. The maximal value of Rb b has a non-trivial bump-like structure. When m A is small, medium (i.e., around the peak of the bump), and large, the partial decay width is bounded by the vacuum stability, flavor constraint, and the perturbativity of y b , respectively.
The measurement of the Higgs couplings may provide an evidence of BSM, because the values of Rb b and Rτ τ may show significant deviation from the SM prediction. In future experiments, the partial decay widths of h →bb and h →τ τ may be measured at the 1% level (see Table 1). In the case of the PS and NS solutions, Rb b − 1 can be as large as 2% (3%) if m A is 2.7 TeV (2.2 TeV) for m soft = 2 TeV, and 2.6 TeV (2.1 TeV) for m soft = 5 TeV. On the other hand, Rτ τ − 1 becomes larger than 2% (3%) when m A is smaller than 3.0 TeV In Fig. 7, we show the sfermion mass dependence of R F with taking the phenomenological constraints into account. The parameter scan is performed in the same setup as Fig. 6, but mŨ is varied. Here, m A is fixed to be 5 TeV. In the left plot, the minimal and maximal values of Rb b and Rτ τ are shown. They are achieved by the PL or NL solution. Since m A is fixed, the decoupling behaviour is not observed. Rather, the non-decoupling contribution to ∆ b enhances Rb b as mŨ increases, since larger |A t | is required to satisfy m h = 125.7 GeV for the solutions. On the other hand, it is found that radiative corrections to the Higgs mixing angle do not change so much even if mŨ increases.
In the right panel of Fig. 7, the minimal and maximal values of R gg are displayed. The maximal value occurs for the PS or NS solution, while the minimal value is achieved by the PL or NL solution. We can see the decoupling behavior, i.e., R gg approaches to unity as mŨ increases. The partial decay width of h → gg can deviate from the SM prediction by about 3% for mŨ = 2 TeV, while it decreases rapidly and becomes 1% for mŨ = 3.7 TeV. We have also checked that these values do not change so much for m A = 8 TeV. (However, they change significantly if m A is smaller, since the phenomenological constraints exclude the parameter space severely.) According to Table 1, the partial width of h → gg is expected to be measured at the 1.2% accuracy in future experiments. Thus, as far as superparticles are relatively light, we may observe a signal of the MSSM in the measurements of this partial width even if the heavier Higgses are out of the reach of the LHC.
Finally, we show how large fraction of the parameter space can be covered by future e + e − colliders. For this purpose, we define the δχ 2 F variable as where δΓ(h → F ) is the expected accuracies of the determinations of the Higgs partial decay widths at ILC with √ s = 1 TeV and dt L = 2500 fb −1 (see Table 1). Based on this quantity, we define the parameter region which is accessible with ILC at δχ 2 F ≥ 4. We perform a parameter scan and study if each model point is accessible with ILC and satisfies the phenomenological constraints. The MSSM parameters are scanned in the ranges of  800 GeV ≤ m A ≤ 5 TeV (with the step size of 100 GeV), 5 ≤ tan β ≤ 50 (with the step size of 1), and 0.5 ≤ |µ|/mŨ ≤ 2 (with the step size of 0.1). Thus, for each set of (m A , tan β), 384 model points are studied, taking account of positive and negative values of µ as well as all four solutions of A t . In addition, we adopt the relations mQ = M 3 = mŨ , and mD = mL = mẼ = max(mŨ , |µ|). Concentrating on the parameter space where the LHC will have a difficulty in finding superparticles, mŨ is taken to be 3, 4, and 5 TeV. In Fig. 8, the ILC coverage of the parameter space of our scan is displayed. In the upper-left panel, we show the distribution of the number of model points accessible with ILC by any of the decay modes of h as a function of m A . The green histogram is a distribution of the number of model points which satisfy δχ 2 F ≥ 4 and the phenomenological constraints, while the dotted one is that satisfying the phenomenological constraints without imposing δχ 2 F ≥ 4. We find that the number reduces drastically at m A ∼ 2 TeV. In the upper-right panel, we show the number of model points which survive the phenomenological constraints on m A vs. tan β plane. Here, δχ 2 F ≥ 4 is not imposed. Then, in the lower panels, we show the numbers of model points which can be accessed by the decay modes of h →bb (lower-left) and h →τ τ (lower-right) with taking account of the phenomenological constraints. They reduce significantly for m A 2 TeV irrespective of tan β. The accessible points for m A 2 TeV are mostly with PL or NL solution.

Conclusions and Discussion
In this paper, we have studied the partial decay widths of the lightest Higgs boson in the MSSM. Taking account of relevant phenomenological constraints, i.e., Higgs mass, flavor constraints, vacuum stability, and the perturbativity of coupling constants up to the GUT scale, we have calculated the expected deviations of the partial decay widths from the SM predictions.
The partial decay widths are enhanced if the µ-parameter is relatively large. However, such a choice may conflict with some of the phenomenological constraints. In particular, the vacuum-stability condition imposes a stringent constraint on the parameter space. We have found that, with too large |µ|, there show up CCB vacua where the down-type Higgs field as well as the up-type Higgs and stop fields acquire large VEVs; existence of such CCB vacua was not seriously considered in the previous studies. In addition, when µ tan β is large, non-holomorphic correction to the bottom Yukawa interaction becomes so large that the bottom Yukawa coupling constant becomes non-perturbative below the GUT scale. Large value of µ tan β may also cause too large flavor-violating decay of B-mesons. By taking these constraints into account, the maximal and minimal possible values of the Higgs partial widths are restricted.
We found that the deviations of the partial decay widths from the SM predictions can be of O(1) % for some of the decay modes. In particular, those of Γ(h →bb) and Γ(h →τ τ ) may show significant deviations even if the superparticles are out of the reach of 14TeV LHC. In addition, the deviation of Γ(h → gg) may also be sizable if the superparticles are relatively light. We emphasize that, although our scan is limited to some part of the MSSM parameter space, we have found the regions where the deviations from the SM predictions are within the reach of proposed e + e − colliders even if superparticles would not be observed at the LHC.