Study of diphoton decays of the lightest scalar Higgs boson in the Next-to-Minimal Supersymmetric Standard Model

The CMS and ATLAS experiments at the LHC have announced the discovery of a Higgs boson with mass at approximately 125 $GeV/c^{2}$ in the search for the Standard Model Higgs boson via notably the $\gamma\gamma$ and $ZZ$ to four leptons final states. Considering the recent results on the Higgs boson searches from the LHC, we study the lightest scalar Higgs boson $h_{1}$ in the Next-to-Minimal Supersymmetric Standard Model by restricting the next-to-lightest scalar Higgs boson $h_{2}$ to be the observed 125 $GeV/c^{2}$ state. We perform a scan over the relevant NMSSM parameter space that is favoured by low fine-tuning considerations. Moreover, we also take the experimental constraints from direct searches, $B$-physics observables, relic density and anomalous magnetic moment of the muon measurements as well as the theoretical considerations into account in our specific scan. We find that the signal rate in the two-photon final state for the NMSSM Higgs boson $h_{1}$ with the mass range from approximately 80 $GeV/c^{2}$ to 122 $GeV/c^{2}$ can be enhanced by a factor up to 3.5, when the Higgs boson $h_{2}$ is required to be compatible with the excess from latest LHC results. This motivates the extension of the search at the LHC for the Higgs boson $h_{1}$ in the diphoton final state down to masses of 80 $GeV/c^{2}$ or lower, in particular with the upcoming proton-proton collision data to be taken at center-of-mass energies of 13-14 $TeV$.


Introduction
The Standard Model (SM) of particle physics has been very successful in explaining high-energy experimental results. One of the remaining questions is to find out what is the source of mass. The solution to this question in the SM is given by the mechanism introduced by Higgs, Englert and Brout [1][2][3] which introduces an additional scalar field whose quantum, the so-called Higgs boson, should be experimentally observable. In July 2012, a Higgs boson-like particle with mass at approximately 125 GeV /c 2 was announced to be discovered by the two experiments, ATLAS and CMS, independently at the LHC via notably the two most promising channels, H → γγ and H → ZZ * channel with a four-lepton final state [4][5][6][7]. Meanwhile, the Tevatron collaborations also announced their new Higgs boson search results, based mainly on V H associated production with H → bb decay channel [8], which supported the LHC ∼125 GeV /c 2 Higgs boson-like particle discovery results. More data should be accumulated in order to test, with higher precision, the consistency between the data analysis results and the SM predictions on the signal strength. If a significant offset were to appear with a more precise measurement in the future, it could provide a window to new physics beyond the Standard Model (BSM).
Supersymmetry (often abbreviated SUSY) [9][10][11][12] is one of the theoretical options for BSM physics. It introduces a new symmetry between fermions and bosons. The most common SUSY framework is the Minimal Supersymmetric Standard Model (MSSM) [13][14][15] which keeps the number of new fields and couplings to the minimum. In the MSSM, the Higgs sector contains two Higgs doublets, which leads to a spectrum including two CP-even, one CP-odd and two charged Higgs bosons. The Lagrangian of the MSSM contains a supersymmetric mass term, the µ-term. This mass term is invariant under supersymmetry and therefore it seems unrelated to the electroweak scale, although it is phenomenologically required to be in this scale. This leads to the well known "µ problem" [16,17] in the MSSM. The simplest solution to this problem is the so-called Next-to-Minimal Supersymmetric Standard Model (NMSSM) [18] by introducing a new gauge singlet superfield which only couples to the Higgs sector in a similar way as the Yukawa coupling and can give rise to an effective µ-term to solve the "µ problem". Meanwhile, this new singlet adds additional degrees of freedom to the NMSSM particle spectrum. In the CP conserving case, which is assumed in this paper, the states in the Higgs sector can be classified as three CP-even Higgs bosons h i (i = 1,2,3), two CP-odd Higgs bosons a j (j = 1,2) and two charged Higgs bosons h ± , for a total of seven observable states.
The extended parameter space of the NMSSM gives rise to a rich and interesting phenomenology, in particular related to the two lightest CP-even Higgs bosons h i (i = 1,2). Inspired by the discovery of the new particle with mass at approximately 125 GeV /c 2 from the LHC and also the small LEP excess (approximately 2σ) at about 98 GeV /c 2 in e + e − → Zh with h → bb [19], in this paper we study the lightest CP-even Higgs boson h 1 in the mass range down to approximately 80 GeV /c 2 , by assuming the next-to-lightest CP-even Higgs boson h 2 as the new particle at m ∼125 GeV /c 2 , along the lines of the studies of Belanger, Ellwanger, Gunion, Jiang, Kraml and Schwarz [20]. The third CP-even Higgs boson is out of reach of current experiments due to its low cross section in our scanned parameter ranges. To distinguish our study from many other NMSSM studies [21][22][23][24][25], we mainly focus on the regions of parameter space favoured by low fine-tuning [23,46] considerations, with tan β chosen small, µ ef f chosen positive with minimal variations and low soft SUSY-breaking masses of the stop sector M Q 3 and M t R . We choose an sbottom mass which is compatible with the SUSY search results at the LHC [26][27][28]. Furthermore, we perform our scan over the parameter space which can explain both dark matter [29], (g − 2) [30] and some other experimental constraints, described in section 3. For completely testing the compatibility of our chosen region of parameter space with the recent LHC results, we interface the package N M SSM T ools(version 4.1.0)[31 -38] with the newly public packages HiggsBounds−4 [40] and HiggsSignals−1 [41] * Additionally, we show in section 4 that the h 2 → XX (XX represents γγ, ZZ, W W , τ τ , or bb) signal strengths can be compatible with the current experimental results, that signal strengths for an h1 with a mass below 110 GeV /c 2 having higher values than currently predicted by the Standard Model are possible, and that the current sensitivities of the LHC experiments are such that the Higgs boson h1 could be detected.
The structure of this paper is organized as follows. In section 2, we briefly introduce the Higgs sector of the NMSSM. The details of the parameter ranges we choose for the scan in the NMSSM parameter space are described in section 3. Section 4 shows the results of our numerical study including the Higgs boson h 2 signal strength in each decay mode and the discussion on the lightest scalar Higgs boson h 1 . The summary and outlook are given in section 5.

Brief description of the NMSSM
The general NMSSM includes two Higgs superfieldŝ H u ,Ĥ d and one additional gauge singlet chiral super-fieldŜ. To start, we consider the NMSSM with a scale invariant superpotential W NM SSM and the corresponding soft SUSY-breaking masses and couplings L sof t , both of which are limited to the R-parity and CP-conserving case. The superpotential W NM SSM depending on the Higgs superfieldsĤ u ,Ĥ d andŜ is [18] In the right-hand side of the above formula, the first three terms are the Yukawa couplings of the quark and lepton superfields. The fouth term replaces the µ-term µĤ uĤd of the MSSM superpotential. The last term, cubic in the singlet superfield, is introduced to avoid the appearance of a Peccei-Quinn axion, tightly constrained by cosmological observation [18]. The corresponding soft SUSY-breaking masses and couplings are given in the SLHA2 [43] conventions by [18] −L sof t = m 2 In Eq.1 and Eq.2, clearly the non-zero vacuum expectation value s of the singletŜ of the order of the weak or SUSY-breaking scale gives rise to an effective µ-term with µ ef f = λs. (3) which solves the "µ problem" of the MSSM. Meanwhile, the three SUSY-breaking mass-squared terms for H u , H d and S appearing in L sof t can be expressed in terms of their VEVs (Vacuum Expectation Value) through the three minimization conditions of the scalar potential. Therefore, the Higgs sector of the NMSSM can be described by the following six parameters in which each pair of brackets denotes the VEV of the respective field inside them. In addition to these six parameters of the Higgs sector, during the scan as described below we need to specify the squark and slepton soft SUSY-breaking masses and the trilinear couplings as well as the gaugino soft SUSY-breaking masses to describe the model completely.

Signal strength of Higgs boson
As in the Standard Model (SM), the main Higgs boson production processes include gluon-gluon fusion, vector boson fusion, Higgs-strahlung and associated production with a vector boson or tt. The most dominant process is gluon-gluon fusion followed by vector boson fusion and the other two associated production modes. In this paper, we will take all four production processes into account.
We are interested in the Higgs boson signal strengths µ h i XX (XX = γγ, ZZ, W W , bb, τ τ ), which are the relative ratios of the cross section times branching ra- In the NMSSM framework, the couplings of the Higgs bosons h 1 , h 2 and h 3 depend on their decompositions into the CP-even weak eigenstates H d , H u and S, which are given by [46] where the coefficients S i,u , S i,d quantify the amount of up-(down-) likeness and S i,s is a measure for the singlet component of a Higgs mass eigenstate. Then the reduced tree-level couplings of h i (i = 1, 2) to b quarks, t quarks and electroweak gauge bosons V relative to the SM value are [47] In the NMSSM, the coupling of h 1 to photons relative to that in the Standard Model is increased due to contributions from non-SM particles in the inducing loop diagrams. These can be from stop squarks but also to an even larger extent from charged Higgsinos where they are proportional to λ [44,45]. In addition, the h 1 total width is smaller than that in the Standard Model due to a reduced coupling to b-quarks [47]. Both effects can serve to enhance the rate of the h 1 decay into two photons in some portions of parameter space.

Scans with constrained parameters
In the following, we will perform a specific scan in the NMSSM parameter space which favours a Higgs boson h2 with a mass close to 125 GeV /c 2 and with coupling strengths compatible with those measured at the LHC, and the Higgs boson masses, decay widths and branching ratios which will be used in this paper. Furthermore, the N M SSM T ools package applies the constraints from theory, constraints from direct Higgs boson searches at LEP [19], Tevatron[8] and LHC[55, 56], some bounds from direct searches of SUSY particles in LHC [26][27][28], relic density Ωh 2 [29], B-physics observables such as the mass mixings ∆M s , ∆M d [48][49][50][51], and anomalous magnetic moment of the muon (g − 2) constraints [30]. All these constraints are used to perform our scan. More details on the implementation of all these constraints in the package can be checked from the webpage of the N M SSM T ools program [31].
After careful study, we have chosen to use the following parameter ranges motivated by the theoretical and experimental considerations detailed below. We realize that these may not be unique; more general ranges consistent with these considerations could possibly be more efficiently obtained via techniques such as Markov Chain Monte Carlo (MCMC) as described in [39].
1. To keep large doublet-singlet mixing in the Higgs sector, we are more interested in large values of λ, κ (but small enough to avoid Landau pole below GUT scale), and to keep the fine-tuning as low as possible in a natural way, low values of tan β [46]. Considering the anomalous magnetic moment of the muon (g − 2) constraint, we keep µ ef f positive with minimal variations, in order to avoid finetuning [30]. Hence the 4 parameters are constrained in the following ranges 0.6 < λ < 0.75, 0.2 < κ < 0.3, 2. The soft SUSY-breaking trilinear couplings A λ and A κ are varied in the ranges [18] −100 GeV /c We remark that, constraining the parameters A λ and A κ in these ranges favor h 1 with higher signal strength as well as being in the mass range down to ∼80 GeV /c 2 .
3. In order to compare with the recent LHC search bounds [26][27][28], we conservatively set the left-handed soft SUSY-breaking masses of the squark sector (M Q 1,2 ) and right-handed soft SUSY-breaking sup masses (M u R and M c R ) to 2500 GeV /c 2 , both of which are in the first two generations. We take low values of soft SUSYbreaking masses of the slepton sector (M L 1,2,3 , M e R , M µ R and M τ R ) as 300 GeV /c 2 to follow the (g − 2) constraint [30]. Furthermore, we set the right-handed soft SUSY-breaking masses (M D R ) and the trilinear couplings (A D , A E and A U ) to 2500 GeV /c 2 and 1000 GeV /c 2 respectively. This results in light sbottom mass of approximately 400 GeV /c 2 < M b 1 < 1000 GeV /c 2 which is compatible with the recent LHC SUSY results. Hence we have , s, b), 4. The Higgs sector is strongly influenced by the stop sector via radiative corrections [52], and also for fine-tuning reasons we further need to specify the soft SUSY-breaking masses of the stop sector [46]. We modify the N M SSM T ools code in order to constrain them to be rather low. After studying the properties of these parameters, we vary them simultaneously within (Eqs.9 and 10 presuppose a SUSY scale.)

5.
Concerning the relic density constraint [29], the remaining gaugino soft SUSY-breaking masses are set to be within Then we perform our scan after the application of the constraints on the parameters as described above.

Numerical study
In section 2, we introduced the production processes and the signal strengths of the NMSSM Higgs bosons. In this section, we demonstrate that the constraints on the parameters as described in the above section can produce a next-to-lightest NMSSM scalar Higgs boson h 2 compatible with the observed state at the LHC with mass at approximately 125 GeV /c 2 . We concentrate our study on the lightest NMSSM scalar Higgs boson h 1 . Considering the relic density Ωh 2 , we will focus on two cases, Ωh 2 < 0.1102 (named case I) and 0.1102 < Ωh 2 < 0.1272 (the "WMAP" window [53], named case II). In all plots below, points for case I are represented by blue squares and case II by red triangles.

Mass distributions of the NMSSM Higgs bosons
Based on the constrained parameters, firstly we show the mass distributions of the two lightest NMSSM scalar Higgs bosons h 1 and h 2 in Fig. 1. As can be seen, most of the parameter points cluster around mass values centered around 125 GeV /c 2 for M h 2 in case I and case II. We conclude that the parameter ranges are correctly chosen to give a mass of the Higgs boson h 2 close to 125 GeV /c 2 . Considering the lightest NMSSM scalar Higgs boson h 1 , it is clear that its mass can lie in a wide range, approximately from 60 (70) GeV /c 2 to 122 GeV /c 2 for case I (case II). We point out that the excluded region below 114.7 GeV /c 2 at LEP [19] could still be allowed in the context of the NMSSM for points in the parameter phase space where the production rate of ee → Z * → Zh 1 with h 1 decaying into bb or τ τ (the channels searched for at LEP) are reduced or suppressed with respect to the SM.

Signal strengths of the NMSSM Higgs boson h 2
In the NMSSM framework, not all the µ h i XX (i=1, 2) are independent, for example, Only the reduced couplings are calculated by NMSSMTools, we use the absolute values in the Standard Model[54] to calculate the total signal strength including four production modes mentioned in section 2.2. We also check that the differences of weights of the production mode are quite negligible between 7 and 8 T eV with repect to experimental uncertainties. In order to further test whether a given point in our scanned parameter space is allowed or excluded by the recent LEP, Tevatron and LHC results at 95% confidence level (CL), two new public tools are utilised.
We use the public tool HiggsBounds−4 [40] to further compare Higgs sector predictions with existing exclusion limits of various search channels. The SLHA format files calculated by N M SSM T ools are used as the inputs for HiggsBounds−4. The main algorithm of HiggsBounds can be described in two steps. In the first step, HiggsBounds uses the expected experimental limits from LEP, Tevatron and the LHC [8, 19,55, 56] to determine which decay channel has the highest statistical sensitivity. In the second step, only for this particular channel the theory prediction is compared to the observed experimental limits in order to conclude whether this parameter point is allowed or excluded at 95% CL.
Then, compatibility with the measured mass and rates of the observed new state having a mass of ∼125 GeV /c 2 is imposed, using the public code HiggsSignals−1 [41]. HiggsSignals−1 takes the predictions of an arbitrary model (here the NMSSM) as an input, providing a quantitative answer to the statistical question of how compatible the model predictions are with the Higgs boson search experimental results, especially signal strengths and the measured mass, by evaluating a χ 2 calculation. The main results from HiggsSignals−1, which are used to further constrain our parameter space, are reported in the form of a χ 2 value and the associated p-value. We consider that the given parameter point is compatible with the experimental constraints only if the p-value given by HiggsSignals−1 is greater than 0.05. By using these two programs in parallel, we obtain the most complete test for the scanned NMSSM parameter space.
The allowed values for µ h 2 XX from the scan over the NMSSM parameter space are shown in Fig. 2, where all the constraints described in section 3 have been applied. The results are shown before and after applying the additional constraints from the above two programs. We first show µ h 2 γγ plotted versus µ h 2 XX (XX = ZZ, W W , bb, τ τ ). The points including error bars represent the latest LHC public results for the best fit values of the signal strengths µ h 2 XX with uncertainties in the different final states, reported by the CMS and ATLAS collaborations [55,56]. The values and errors are listed in Table 1 in the Appendix[6, 7, 55, 56]. It is clearly visible that the parameter points compatible with both HiggsBounds−4 and HiggsSignals−1 provide theory predictions which are consistent with the experimental results. In Fig. 2, by taking the di-photon final state as an example, we also show the signal strength µ h 2 γγ plotted against its mass in Fig. 2 (d). From the right-hand plot, the µ h 2 γγ values cover the range 0.5 to 1.8 while the mass is in the range 120 GeV /c 2 to 132 GeV /c 2 , both of which are consistent with the new observed state within errors. It is clearly visible that the NMSSM can produce rates compatible with both the CMS and ATLAS results for both relic density cases. The plots show that the relatively sizable enhancements with respect to the SM rates for the γγ and ZZ final states reported by ATLAS are possible in the vicinity of 125 GeV /c 2 in the NMSSM framework and also possible for the relatively suppressed rates reported by CMS.

Branching ratio and signal strength of the NMSSM Higgs boson h 1
We will now focus our discussion on the lightest NMSSM scalar Higgs boson h 1 by looking at the diphoton final state, and will further restrict ourselves to the case of CMS results only.
Over and above the constraints mentioned in previous sections, we now demand in addition that the mass of the NMSSM Higgs boson h 2 be explicitly compatible with that most recently measured by CMS for the newly-discovered boson, and that the h 2 signal strength in the diphoton channel be compatible at a more stringent level with that measured by CMS. In the most recent CMS results [7], the SM-like Higgs boson mass has been measured to be 125.7±0.3(stat.)±0.3(syst.) GeV /c 2 . Assuming 3σ error, where σ is taken as the linear sum of the above statistical and systematic uncertainties, the mass of h 2 is constrained within the range 123.9 GeV /c 2 < M h 2 < 127.5 GeV /c 2 .
We also demand that the signal strength µ h 2 γγ should be within 1σ (taking as σ the uncertainty shown in the Appendix) of the CMS measured value: Based on these additional constraints, Fig. 3 shows the allowed values for the branching ratio of the h 1 → γγ decay mode in the NMSSM. The cyan solid line shows the quantity including error bands evaluated in the SM for the same mass [54]. From the plots, most of the points show that an enhanced branching ratio relative to that in the SM is possible for both relic density cases. The theory explanation for this enhanced two-photons branching ratio has already been introduced in Section 2.2. In Fig. 4, we display the possible signal strengths µ h 1 γγ plotted against the Higgs boson h 1 mass. As seen from Fig. 4, the remaining points selected after application of all the conditions discussed in Sections 3 and 4.2 indicate the possibility of the h 1 mass lying in the range between 80 GeV /c 2 to 122 GeV /c 2 for both relic density cases.
Turning to the signal strength µ h 1 γγ , the figure shows that a sizable enhancement over the SM rate is possible for the Higgs boson h 1 for both relic density cases, reaching values as high as 3.5, corresponding to an h 1 mass of ∼ 90 GeV /c 2 . We note that, for the mass range between 100 GeV /c 2 and 110 GeV /c 2 , the allowed signal strengths µ h 1 γγ are lower, falling to ∼0.9. In order to compare with our signal strength µ h 1 γγ , we also superpose the official CMS public exclusion limit plot in Fig. 4. The yellow and green regions correspond to the uncertainties at 95% and 68% confidence interval respectively and the cyan solid line corresponds to the SM value. It is clearly seen that the NMSSM points above the solid black line (representing the CMS observed exclusion limit) are almost excluded by the CMS result in the mass range 110 GeV /c 2 to 122 GeV /c 2 . We note that there is a small interesting region that is favoured by a cluster of parameter points in the neighborhood of 120 GeV /c 2 . Especially, the points below 120 GeV /c 2 are already excluded by comparing with the solid black line. The remaining points lying between 120 and 122 GeV /c 2 could constitute a case of a so-called "degenerate Higgs boson"[57] which is outside the scope of our discussion in this paper. But it would be advisable to test this interesting scenario with the increased quantity of data and improved resolution in the future 13 and 14-T eV collisions at the LHC. To date, the present experimental results from the LHC do not cover the lower Higgs boson mass range between 80-110 GeV /c 2 in the H → γγ decay channel. In order to be able to make a conclusion for the NMSSM points in this mass range, which show the potential for sizeably enhanced signal strengths in the diphoton decay channel with respect to those predicted in the SM, a detailed analysis is needed taking into account especially the Z → ee background faking the diphoton signals. If the limit curve were to be extrapolated down to a mass of ∼ 80 GeV /c 2 and the measurement on the signal strength improved in the future experimental analysis, most of the NMSSM parameter space shown in Fig. 4 could be probed. Expected and observed exclusion limits on the signal strength from CMS[55] compared with the possible signal strengths of the process pp → h1 → γγ from the NMSSM parameter scan. Points for case I are represented by blue squares and case II by red triangles.The solid black line together with the black squares corresponds to the ratio of the CMS observed cross sections with respect to the SM predictions, and the dashed line is the expected ratio.

Summary and outlook
In this paper, we have performed a scan in the NMSSM, focusing on the regions of parameter space favoured by low fine-tuning considerations. We have studied the lightest scalar Higgs boson h 1 including the mass and the relative signal strength to the SM prediction, especially for Higgs bosons decaying into the di-photon final state, by assuming that the secondlightest scalar Higgs boson h 2 corresponds to the observed ∼125 GeV /c 2 state at the LHC. We find that a significant excess of the signal strength relative to that of the Standard Model in pp → h 1 → γγ up to a factor ∼3.5 is possible in the NMSSM, especially for the mass range below the LEP bound of 114.7 GeV /c 2 . We recommend that experiments extend the exclusion limit to this lowmass region in order to investigate the possibilities of the NMSSM in more detail.
With future LHC data, the best fit values of the signal strengths in each channel may evolve and the uncertainties improve, which may result in changes in the experimental results and reduced error bars in our plots. Additionally, the allowed regions in NMSSM parameter space for the interesting Higgs boson h 1 may also change. With the upcoming 13 and 14-T eV collisions at LHC, the signal for the low-mass NMSSM Higgs boson h 1 could still well be detected by the experiments due to the higher collision energy and integrated luminosity.