A Higgs boson near 125 GeV with enhanced di-photon signal in the NMSSM

A natural region in the parameter space of the NMSSM can accomodate a CP-even Higgs boson with a mass of about 125 GeV and, simultaneously, an enhanced cross section times branching ratio in the di-photon channel. This happens in the case of strong singlet-doublet mixing, when the partial width of a 125 GeV Higgs boson into \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ b\overline b $\end{document} is strongly reduced. In this case, a second lighter CP-even Higgs boson is potentially also observable at the LHC.


Introduction
Based on the analysis of 5 fb −1 of data at the LHC, the ATLAS [1] and CMS [2] collaborations have presented evidence for a Higgs boson with a mass in the 125 GeV range. The relevant search channels are H → γ γ, H → Z Z * → 4l, H → W W * → 2l 2ν and to some extend (at CMS) H → τ τ . Interestingly, the best fit to the signal strength σ γγ = σ prod × BR(H → γ γ) in the γ γ search channel is by about one standard deviation larger than expected in the Standard Model (SM) for both collaborations: σ γγ /σ γγ SM ∼ 2 (ATLAS), and σ γγ /σ γγ SM ∼ 1.7 (CMS). Of course, the present evidence for a Higgs boson is not (yet?) sufficiently significant in order to consider its existence as assured, even less is the excess in the H → γ γ channel a proof for a non-SM-like Higgs boson.
A relatively light Higgs boson (with a mass not too far above the LEP bound of ∼ 114 GeV) is a genuine prediction of supersymmetric extensions of the SM which remain consistent up to a Grand Unification (GUT) scale of about 10 16 GeV, in particular in the Minimal Supersymmetric extension (MSSM) with a minimal Higgs sector consisting of two SU(2) doublets H u and H d . In fact, in the MSSM the solution of the fine tuning problem offered by supersymmetry works the better, the lighter is the mostly SM-like Higgs boson.
Still, the parameter space of the MSSM allows to describe a Higgs boson with a mass in the 125 GeV range if certain combinations of the stop masses, stop mixings, tan β and the parameter M A (essentially the heavy Higgs masses) are large enough [3][4][5][6][7][8][9][10][11][12]. This implies a fine tuning within the MSSM parameter space of the order of 1% [13], or extra matter [14]. An enhancement of σ γγ /σ γγ SM may be possible in the presence of light staus [8]. The Next-to-Minimal Supersymmetric Standard Model (NMSSM) [15][16][17][18][19] is the simplest supersymmetric (Susy) extension of the SM with a scale invariant superpotential, i.e. where the only dimensionful parameters are the soft Susy breaking terms. No supersymmetric Higgs mass term µ as in the MSSM is required, since it is generated dynamically by the vacuum expectation value (vev) of a gauge singlet superfield S and a coupling λSH u H d in the superpotential. Together with the neutral components of the two SU(2) doublet Higgs fields H u and H d of the MSSM, one finds three neutral CP-even Higgs states in this model. These three states mix in the form of a 3 × 3 mass matrix and, accordingly, JHEP03(2012)044 the physical eigenstates are superpositions of the neutral CP-even components of H u , H d and S. In general, the couplings of the physical states to gauge bosons, quarks and leptons can differ considerably from the corresponding couplings of a SM Higgs boson. The possible alleviation in the NMSSM of the "little fine tuning problem" in the Higgs sector of the MSSM has been studied in [21] in the light of 2 fb −1 of data at the LHC, and in the light the recent evidence for a Higgs mass of about 126 GeV in [13] (although mostly for large values of λ, implying new strong interactions below the GUT scale).
In most of the parameter space of the NMSSM, the physical Higgs spectrum contains a heavy CP-even state, a heavy CP-odd state and a charged Higgs boson which are nearly degenerate as in the MSSM with a common mass ∼ M A . However, the lighter doublet-like CP-even state (corresponding to the SM-like Higgs boson H SM ) can mix strongly with the real part of S and form eigenstates with reduced couplings to gauge bosons, quarks and leptons [15,16,19,20,[22][23][24][25][26][27][28][29][30]. In this case, possibly both eigenstates are visible at the LHC (see [30], where a second visible state with reduced couplings in the 140-150 GeV range has been studied, and refs. therein).
It is well known that, for small values of tan β, the coupling λSH u H d in the superpotential leads to a positive contribution to the mass squared of the SM-like Higgs boson H SM relative to the MSSM [15,16,19]. However, H SM −S mixing has an additional impact on the physical spectrum: if the diagonal mass term m 2 SS is larger than the one of H SM , the mixing reduces the mass of H SM ; if the diagonal mass term m 2 SS is smaller than the one of H SM , the mixing leads to an additional increase of the mass of H SM . In this latter case, the mass of the lighter eigenstate H 1 can be well below 114 GeV and compatible with constraints from LEP [31], if its reduced signal strength ξ 2 1 ≡ḡ 1 2 × BR(H 1 → bb) is small enough. (Hereḡ 1 is the reduced coupling of H 1 to the Z boson normalized with respect to the SM, and BR(H 1 → bb) is the branching ratio into bb normalized with respect to the SM.) In addition, H SM − S mixing can lead to an increase of the branching ratio BR(H i → γ γ) of one of the eigenstates H i with respect to the SM: if the coupling to bb and hence the partial decay width into bb (which is close to the total width Γ Tot ) is strongly reduced with respect to the SM, BR(H i → γ γ) = Γ(H i → γ γ)/Γ Tot is correspondingly enhanced. This phenomenon has been discussed in the context of the lighter eigenstate H 1 in [32], but is equally possible for the heavier eigenstate as will be discussed below. In view of the latest LHC results, the possible enhancement of BR(H i → γ γ) in the NMSSM was also discussed in [13], and a Higgs mass near 125 GeV in the constrained NMSSM -but without enhancement of BR(H i → γ γ) -in [33].
In the next section we will study a region of the parameter space of the NMSSM with a scale invariant superpotential, which leads naturally to an eigenstate H 2 after H SM − S mixing with a mass in the 124-127 GeV range. Its BR(H 2 → γ γ) is always enhanced with respect to the SM. The lighter eigenstate H 1 has a mass in the 70-120 GeV range, compatible with LEP constraints, and is potentially also observable at the LHC. In section 3 we conclude and summarize the possibilities allowing to distinguish this scenario from the SM and/or the MSSM.

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2 Implications of H SM − S mixing in the NMSSM in the light of recent and future LHC results The NMSSM differs from the MSSM due to the presence of the gauge singlet superfield S. In the simplest Z 3 invariant realisation of the NMSSM, the Higgs mass term µH u H d in the superpotential W MSSM of the MSSM is replaced by the coupling λ of S to H u and H d and a self-coupling κS 3 . Hence, in this simplest version the superpotential W NMSSM is scale invariant, and given by: where hatted letters denote superfields, and the dots denote the MSSM-like Yukawa couplings ofĤ u andĤ d to the quark and lepton superfields. Once the real scalar component ofŜ develops a vev s, the first term in W NMSSM generates an effective µ-term A constraint |µ eff | > ∼ 100 GeV follows from the non-observation of higgsino-like charginos at LEP. The soft Susy breaking terms consist of mass terms for the Higgs bosons H u , H d and S, and trilinear interactions (omitting squarks and sleptons) Expressions for the mass matrices of the physical CP-even and CP-odd Higgs statesafter H u , H d and S have assumed vevs v u , v d and s and including the dominant radiative corrections -can be found in [19] in will not be repeated here. As compared to two independent parameters in the Higgs sector of the MSSM at tree level (often chosen as tan β and M A ), the Higgs sector of the NMSSM is described by the six parameters Alternatively, the parameter A λ can be replaced by the MSSM-like parameter where B eff = A λ + κs. Subsequently we are interested in regions of the parameter space where the soft Susy breaking terms are not very large (in order to avoid large fine tuning), but they have to comply with the present non-observation of sparticles at the LHC. In the gaugino, squark and slepton sectors we make the following choice, motivated to a certain extend by the renormalization group running from the GUT scale down to the weak scale (although the precise values are not very important): bino, wino and gluino masses M 1 = 175 GeV, M 2 = 350 GeV and M 3 = 1000 GeV respectively, squark masses of 1200 GeV (but 800 GeV for the third generation), slepton masses of 300 GeV, A t = A b = −1000 GeV.

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In the Higgs sector we have to keep in mind that the soft Susy breaking masses m 2 Hu , m 2 H d and m 2 S are determined implicitely (through the minimization equations of the scalar potential) in terms of M Z , tan β and µ eff . Large values of m 2 Hu , m 2 H d and m 2 S are avoided if µ eff , M A and tan β are relatively small. (Large values of tan β require small tuned values for B eff in the NMSSM, unless |m 2 Hu | and/or |m 2 H d | are large.) Hence we choose µ eff = 140 GeV, M A = 300 GeV and 1.7 < tan β < 2 leading to A λ ∼ 140-200 GeV. Then, the interesting regions of the remaining parameters λ, κ and A κ are determined by the conditions that i) one of the physical eigenstates in the CP-even Higgs sector (actually always H 2 ) has a mass in the 124-127 GeV range, and ii) the lighter eigenstate H 1 is not in conflict with LEP constraints. The density of viable points is particularly large for 0.5 < λ < 0.6, 0.3 < κ < 0.4 and −250 GeV < A κ < −200 GeV. Of course, viable points outside this range exist as well, but these will not invalidade our subsequent conclusions.
A corresponding scan in parameter space is performed with the help of the code NMSSMTools [17,34]; we employed the version 3.0.2 which is includes radiative corrections to the Higgs sector from [35]. Only points respecting constraints on the Higgs sector from LEP and from B physics are retained. We find that about 50% of all points in this region of parameter space respect these phenomenological constraints, and ∼ 5-6% (∼ 550 out of 10000) lead to a Higgs boson H 2 with a mass in the 124-127 GeV range. (Of course, measurements always reduce the allowed regions in parameter space.) The couplings of the Higgs states depend on their decompositions into the CP-even weak eigenstates H d , H u and S, which are given by Then the reduced tree level couplings (relative to a SM-like Higgs boson) of H i to b quarks, τ leptons, t quarks and electroweak gauge bosons V are For the low values of tan β considered here, the couplings of Higgs bosons to gluons (relevant for their production) and to photons are induced by loop diagrams dominated by top-quark loops. As stated above, the branching ratios into two photons can be enhanced, if the coupling to b-quarks is reduced, which is the case if S i,d is small.
Subsequently we are interested in the signal strength σ γγ 2 = σ prod × BR(H 2 → γ γ) relative to the SM, R γγ 2 = σ γγ 2 /σ γγ SM . R γγ 2 is the product of two factors: i) the reduced coupling of H 2 to gluons, which is essentially given by g H 2 tt /g H SM tt (but contributions from non-SM particles in the loop are taken into account), and ii) the BR(H 2 → γ γ), the branching ratio of H 2 into γ γ normalized with respect to the corresponding branching ratio of a SM-like Higgs boson of the same mass. 1 BR(H 2 → γ γ) can be considerably larger than 1. In figure 1 we show R γγ 2 as function of S 2 2,d for ∼ 550 points in the region of the parameter space of the NMSSM described above, in which M H 2 is in the 124-127 GeV range. We see that R γγ 2 is always larger than 1.1, with an expected dependence on S 2 2,d . If one modifies somewhat the soft Susy breaking squark and slepton masses (and trilinear couplings A) at the weak scale, the parameters can be mapped to a semi-constrained version of the NMSSM together with non-universal soft Higgs masses at the GUT scale as studied in [33] 2 The fact that all 3 Yukawa couplings are close to (but just below) a Landau singularity at the GUT scale is intriguing.
Next we turn to the lighter Higgs boson H 1 in this scenario. Its mass is in the 70-120 GeV range. The most relevant search channels in this mass range are again the γ γ mode, but also H 1 → τ τ (with H 1 produced by vector boson fusion, VBF) and, to some extent, H 1 → bb with H 1 produced in association with W or Z bosons. The reduced signal strength in the γ γ mode, R γγ 1 = σ γγ 1 /σ γγ SM , can be obtained as above. The reduced signal strength in the τ τ mode and VBF, R τ τ 1 = σ τ τ 1 /σ τ τ SM , is the product of the reduced couplinḡ g 2 1 of H 1 to the electroweak gauge bosons, and the BR(H 1 → τ τ ), the branching ratio of H 1 into τ τ normalized with respect to the corresponding branching ratio of a SM-like Higgs boson of the same mass. (The reduced signal strength in the bb mode is practically the same as R τ τ 1 , since it is again proportional to the coupling to electroweak gauge bosons, and the branching ratio into bb remains proportional to the branching ratio into τ τ .) In figure 2 we show R γγ 1 and R τ τ 1 as function of M H 1 . We see that R γγ 1 is not enhanced, but mostly strongly reduced due to the small coupling of H 1 to two gluons, which is not compensated by an enhanced branching ratio into two photons in this case. Hence, except perhaps for M H 1 > ∼ 110 GeV, the prospects for a discovery of H 1 in this channel are not rosy. Likewise, R τ τ 1 (≃ R bb 1 ) is not enhanced, but not as small as R γγ 1 . Actually the upper bound on R τ τ 1 coincides with the upper LEP bound on ξ 2 1 ≡ḡ 1 2 × BR(H 1 → bb) as function of M H [31], which is not astonishing given that BR(H 1 → bb) ∼ BR(H 1 → τ τ ). Hence, although a discovery of H 1 in the τ τ channel (or bb mode) is not guaranteed, this is not excluded in particular after future high luminosity runs of the LHC or if its mass is in the 110-120 GeV range.

Conclusions
We have presented a natural region in the parameter space of the NMSSM, where the NMSSM-specific coupling λ and mixing effects push up the mass of a CP-even Higgs boson into the 124-127 GeV range without the need for excessive radiative corrections from heavy sparticles. The relative signal rate in the γ γ channel is always enhanced by a factor 1.1-1.8 with respect to a SM-like Higgs boson of the same mass. This Higgs boson complying with recent evidence from the ATLAS and CMS collaborations is accompagnied by a lighter CP-even neutral Higgs state.
Under the following circumstances it might be possible to distinguish this scenario from the SM and/or the MSSM: • sparticles are detected, and their masses turn out to be incompatible with the necessarily large radiative corrections to the Higgs mass in the MSSM; • the lighter CP-even state H 1 is discovered.
Of course, first of all the present evidence for a Higgs boson into the 124-127 GeV range should be confirmed by more data; then the same data can give us possible hints for non-SM-like properties of the Higgs sector along the lines discussed here.