A 125 GeV Higgs and its di-photon signal in different SUSY models: a mini review

In this note we briefly review our recent studies on a 125 GeV Higgs and its di-photon signal rate in different low energy supersymmetric models, namely the minimal supersymmetric standard model (MSSM), the next-to-minimal supersymmetric standard model (NMSSM), the nearly minimal supersymmetric standard model (nMSSM) and the constrained MSSM. Our conclusion is: (i) In the allowed parameter space the SM-like Higgs boson can easily be 125 GeV in the MSSM, NMSSM and nMSSM, while it is hard to realize in the constrained MSSM; (ii) The di-photon Higgs signal rate in the nMSSM and constrained MSSM is suppressed relative to the prediction of the SM, while the signal rate can be enhanced in the MSSM and NMSSM; (iii) The NMSSM may allow for a lighter top-squark than the MSSM, which can thus ameliorate the fine-tuning problem.


I. INTRODUCTION
Considering the important role of the Higgs boson in particle physics, hunting for it has been one of the major tasks of the running Large Hadron Collider (LHC). Recently, both the ATLAS and CMS collaborations have reported some evidence for a light Higgs boson near 125 GeV [1,2] with a di-photon signal rate slightly above the SM prediction [3].
As is well known, in new physics beyond the SM model several Higgs bosons are predicted, among which the SM-like one may be near 125 GeV [4][5][6][7]. Recently, in our studies [6,7] we examined the mass of the SM-like Higgs boson in several supersymmetric (SUSY) models including the minimal supersymmetric standard model (MSSM) [8,9], the next-to-minimal supersymmetric standard model (NMSSM) [10,11] and the constrained MSSM [12,13]. At tree-level, these SUSY models are hard to predict a Higgs boson near 125 GeV, and sizable radiative corrections, which mainly come from the top and top-squark loops, are necessary to enhance the Higgs boson mass [14]. Due to the different properties of these SUSY models, the loop contributions to the Higgs boson mass are different for giving a 125 GeV Higgs boson. Therefore, different models have different lower bounds on the top-squark mass which is associated with the fine-tuning problem [15]. On the other hand, since the di-photon Higgs signal is the most promising discovery channel for a light Higgs boson at the LHC [16], in our recent study [17] we performed a comparative study for the di-photon Higgs signal in different SUSY models, namely the MSSM, NMSSM and the nearly mininal supersymmertric standard model (nMSSM) [18,19]. In this note we briefly review these stuides on a 125 GeV Higgs boson and its di-photon signal rate in different SUSY models. This note is organized as follows. In the next section we briefly describe the Higgs sector and the di-photon Higgs signal in these SUSY models. Then we present the numerical results and discussions in Sec. III. Finally, the conclusions are given in Sec. IV.

II. THE HIGGS SECTOR AND DI-PHOTON SIGNAL RATE IN SUSY MODELS
A. The Higgs sector in SUSY models Different from the SM, the Higgs sector in the supersymmetric models is usually extended by adding Higgs doublets and/or singlets. The most economical realization is the MSSM, which consists of two Higgs doublet H u and H d . In order to solve the µ−problem and the little hierarchy problem in the MSSM, the singlet extension of MSSM, such as the NMSSM [10] and nMSSM [18,19] has been intensively studied [20]. The differences between these models come from their superpotentials and the corresponding soft-breaking terms, which are given by: where W F is the MSSM superpotential without the µ term, λ, κ and ξ F are the dimensionless parameters andm u ,m d ,m S , B, A λ , A κ and ξ S M 3 n are soft-breaking parameters. Note that in the NMSSM and nMSSM the µ-term is replaced by the µ eff = λs when the singlet Higgs fieldŜ develops a VEV s. The differences between the NMSSM and nMSSM reflect the last term in the superpotential, where the cubic singlet term κŜ 3 in the NMSSM is replaced by a tadpole term ξ F M 2 nŜ in the nMSSM. This replacement in the superpotential makes the nMSSM has no discrete symmetry and thus free of the domain wall problem that the NMSSM suffers from. Actually, due to the tadpole term ξ F M 2 n does not induce any interaction, the nMSSM is identical to the NMSSM with κ = 0, except for the minimization conditions of the Higgs potential and the tree-level Higgs mass matrices.
With the superpotentials and the soft-breaking terms giving above, one can get the Higgs potentials of these SUSY models, and then can derive the Higgs mass matrics and eigenstates. At the minimum of the potential, the Higgs fields H u , H d and S are expanded with v = v 2 u + v 2 d = 174 GeV. By unitary rotation the mass eigenstates can be given by  where h 1 , h 2 , h 3 are physical CP-even Higgs bosons (m h 1 < m h 2 < m h 3 ), a, A are CP-odd Higgs bosons, H + is the charged Higgs boson, and G 0 , G + are Goldstone bosons eaten by Z and W + . Due to the absence of the singlet field S, the MSSM only has two CP-even Higgs bosons and one CP-odd Higgs bosons, as well as one pair of charged Higgs bosons.
At the tree-level, the Higgs masses in the MSSM are conventionally parameterized in terms of the mass of the CP-odd Higgs boson (m A ) and tan β ≡ v u /v d and the loop corrections typically come from top and stop loops due to their large Yukawa coupling. For small splitting between the stop masses, an approximate formula of the lightest Higgs boson mass is given by [23], where Different from the case in the MSSM, the Higgs sector in the NMSSM depends on the following six parameters, and in the nMSSM the input parameters in the Higgs sector are Because the coupling λĤ u ·Ĥ dŜ in the superpotential, the tree-level Higgs boson mass has an additional contribution in the NMSSM and nMSSM, In order to push up the tree-level Higgs boson mass, λ has to be as large as possible and tan β has to be small. The requirement of the absence of a landau singularity below the GUT scale implies that λ 0.7 at the weak scale, and the upper bound on λ at the weak scale depends strongly on tan β and grows with increasing tan β [24]. However, this can still lead to a larger tree-level Higgs boson mass than in the MSSM. Therefore, the radiative corrections to m 2 h may be reduced in the NMSSM and nMSSM, which may induce light top-squark and ameliorate the fine-tuning problem [21].
In the NMSSM and nMSSM, due to the mixing between the doublet Higgs fields and the singlet Higgs field, the SM-like Higgs boson h may either be the lightest CP-even Higgs boson or the next-to-lightest CP-even Higgs boson, which corresponds to the so-called pull-down case or the push-up case [6], respectively. Although the mass of the SM-like Higgs boson in the nMSSM is quite similar to that in the NMSSM, the Higgs signal is quite different. This is because the peculiarity of the neutralino sector in the nMSSM, where the lightest neutralinõ χ 0 1 as the lightest supersymmetric particle(LSP) acts as the dark matter candidate, and its mass takes the form [22] This expression implies thatχ 0 1 must be lighter than about 60 GeV for λ < 0.7 (perturbativity bound) and µ > 100GeV (from lower bound on chargnio mass). Andχ 0 1 must annihilate by exchanging a resonant light CP-odd Higgs boson to get the correct relic density. For such a light neutralino, the SM-like Higgs boson around 125GeV tends to decay predominantly into light neutralinos or other light Higgs bosons [19].

B. The di-photon Higgs signal
Considering the di-photon signal is of prime importance to searching for Higgs boson near 125 GeV, it is necessary to estimate its signal rate, and we define the normalized production rate as where C hgg and C hγγ are the couplings of Higgs to gluons and photons in SUSY with respect to their SM values, respectively. In SUSY, the hgg coupling arises mainly from the loops mediated by the third generation quarks and squarks, while the hγγ coupling has additional contributions from loops mediated by W-boson, charged Higgs boson, charginos and the third generation leptons and sleptons. Their decay widths are given by [9] Γ with τ i = m 2 h /(4m 2 i ), and where mf and m χ i are the mass of sfermion and chargino, respectively. In the limit τ i ≪ 1, the asymptotic behaviors of A h i are given by One can easily learn that the W-boson contribution to hγγ is by far dominant, however, for light stau or squarks with large mixing, the hγγ coupling can be enhanced. While light squarks with large mixing can suppress the hgg coupling. Therefore, light stau with large mixing may enhance the di-photon signal rate [23], while light squarks with large mixing have little effect on the di-photon signal rate.

III. NUMERICAL RESULTS AND DISCUSSIONS
In our work the Higgs boson mass are calculated by the package NMSSMTools [25], which include the dominant one-loop and leading logarithmic two-loop corrections. Considering the Higgs hints at the LHC, we focus on the Higgs boson mass between 123 GeV and 127 GeV, furthermore, we consider the following constraints: (1) The constraints from LHC experiment for the non-standard Higgs boson.
(2) The constraints from LEP and Tevatron on the masses of the Higgs boson and sparticles, as well as on the neutralino pair productions.  (4) The constraints from the muon g − 2: a exp µ − a SM µ = (25.5 ± 8.2) × 10 −10 [27]. We require SUSY to explain the discrepancy at 2σ level.
Note that most of the above constraints have been encoded in the package NMSSMTools.
Natural supersymmetry are usually characterized by a small superpotential parameter µ, and the third generation squarks with mass 0.5 −1.5 TeV [30]. Therefore, we only consider the case with For the case with λ < 0.2 in the NMSSM, the property of the NMSSM is similar to the case in the MSSM [6]. In order to distinguish the features between MSSM and NMSSM, we only consider the case with λ > m Z /v ≃ 0.53 in the NMSSM. We scan over the parameter space of the MSSM and NMSSM under the above experimental constraints and study the property of the Higgs boson for the samples surviving the constraints. In Fig.1 we display the surviving samples in the MSSM and NMSSM (with λ > 0.53), showing the correlation between the lighter top-squark mass and the ratio X t /M S with M S ≡ √ mt 1 mt 2 . From the figure we see that for a moderate lightt 1 , large X t is necessary to satisfy m h ∼ 125 GeV, and for large mt 1 , the ratio X t /M S decreases. In the MSSM, |X t /M S | > 1.6 for mt 1 < 1 TeV, i.e. no-mixing scenario(X t = 0) cannot survive, and the top-squark mass is usually larger than 300 GeV. This implies that a large top-squark mass or a near-maximal stop mixing is necessary to satisfy the Higgs mass near 125 GeV. However, the case is very different in the NMSSM, X t ≈ 0 may also survive, and the lighter top-squark mass can be as light as about 100 GeV, which may alleviate the fine-tuning problem and make the NMSSM seems more natural. In the case of light mt 1 , |X t /M S | is usually larger than √ 6, which corresponds to a large splitting between mt 1 and mt 2 , as the Fig.2 shown.
Due to the clean background, the di-photon signal is crucial for searching for the Higgs boson near 125 GeV. As discussed in the Sec.II, the signal rate is relevant with the coupling  Eq.(16) shown. As analyzed in the Sec.II, light stau may enhance the coupling C hγγ , so in Fig.4 we give the correlation between mτ 1 and the coupling C hγγ in the MSSM. The figure clearly shows that the coupling C hγγ can enhance to 1.25 for mτ 1 ∼ 100 GeV. Fig.4 also manifests that the enhancement of the coupling C hγγ corresponds to large µ tan β, which leads to large mixing. These results exactly verifies the discussions in the Sec.II.  Since h → bb is the main decay mode of the light Higgs boson, the total width of the SM-like Higgs boson may be affected by the effective hbb coupling C hbb , as discussed in [17].
Under the effect of the combination C hgg C hγγ /C hbb , the di-photon Higgs signal rate may be either enhanced or suppressed, as shown in Fig.5, which also manifest that for the signal rate larger than 1, the effective hbb coupling is enhanced a little in the MSSM, while it is suppressed significantly in the NMSSM. Therefore, we can conclude that the reason for the enhancement in the signal rate is very different between the MSSM and NMSSM. In the MSSM the enhancement of the signal is mainly due to the enhancement of the coupling C hγγ , while in the NMSSM it is mainly due to the suppression of the hbb coupling. Due to the presence of the singlet field in the NMSSM, the doublet component in the SM-like Higgs boson h may be different from the case in the MSSM, which will affect the coupling hbb, and accordingly affect the total width of h. At the tree-level, C hbb = S d / cos β.
In Fig.6 we show the dependence of the signal rate R γγ on S 2 d . Obviously, for the signal rate larger than 1, S 2 d is usually very small, which leads to large suppression on the reduced coupling hbb. The figure also shows that the push-up case is more effective to enhance the signal rate than the pull-down case. This is because the push-up case is easier to realize the large mixing between the singlet field and the doublet field [6].
As the case in the NMSSM, nMSSM can also accommodate a 125 GeV SM-like Higgs [17].
However, due to the peculiar property of the lightest neutralinoχ 0 1 in the nMSSM [22], the decay mode of the SM-like Higgs is very different from the case in the MSSM and NMSSM.
As discussed in the Sec.II, h →χ 0 1χ 0 1 may be dominant over h → bb in a major part of parameter space in the nMSSM [17,31], which induce a severe suppression on the di-photon Higgs signal. Although the Higgs mass can be easily reach to 125 GeV, the di-photon signal is not consistent with the LHC experiment. Therefore, the nMSSM may be excluded by LHC experiment. We also considered the SM-like Higgs boson mass and its di-photon signal in the constrained MSSM (including CMSSM and NUHM2) under various experimental constraints, especially the limits from B s → µ + µ − . Because Br(B s → µ + µ − ) ∝ tan 6 β/M 4 A , so it may provide a rather strong constraint on SUSY with large tan β. Considering the large theoretical uncertainties for the calculation of Br(B s → µ + µ − ), we use not only the LHCb data, but also the double ratio of the purely leptonic decays defined as R ≡ η η SM with η ≡ Br(Bs→µ + µ − )/Br(Bu→τ ντ ) Br(Ds→τ ντ )/Br(D→µνµ) . The surviving parameter space is plotted in Fig.7 for the CMSSM and Fig.8 for the NUHM2. It shows that both the CMSSM and NUHM2 are hard to realize a 125 GeV SM-like Higgs boson, and also the di-photon Higgs signal is suppressed relative to the SM prediction due to the enhanced hbb coupling. Therefore, the constrained MSSM may also be excluded by the LHC experiment.