Light Higgs boson in the NMSSM confronted with the CMS diphoton and ditau excesses

In 2018, the CMS collaboration reported a di-photon excess around 95.3 GeV with a local significance of 2.8 $\sigma$. Interestingly, the CMS collaboration also reported a di-tau excess recently at 95 $\sim$ 100 GeV with a local significance of 2.6 $\sim$ 3.1 $\sigma$. Besides, a $b\bar{b}$ excess at 98 GeV with a 2.3 $\sigma$ local significance was reported with LEP data about twenty years ago. In this work, we consider interpreting these excesses together with a light Higgs boson in the next-to-minimal supersymmetric standard model (NMSSM). We conclude that in NMSSM the 95 $\sim$ 100 GeV excesses are difficult to be satisfied simultaneously (not possible globally at $1\sigma$ level, or simultaneously at $2\sigma$ level), and we analyze two partial-satisfied scenarios: the globally $2\sigma$ scenario and small di-photon scenario. An approximate equation of global fit to the three excesses is derived, and two representative types of surviving samples are analyzed in detail. Since the mass regions of these excesses are near the Z boson, we also consider checking the light Higgs boson in the $t\bar{t}$-associated channels. The detailed results may be useful for further checking the low-mass-region excesses in the future.


Introduction
In 2012, the ATLAS and CMS collaborations reported a new boson of around 125 GeV discovered at the LHC [1,2].Then it was proved to be the Standard Model (SM)-like Higgs boson, according to its spin, CP property, production and decay performances in Run I and Run II data globally [3][4][5].Higgs boson is related to the electroweak symmetry-breaking mechanism, hierarchy problem, and interesting phenomenology in many new physics models.Whether there are additional Higgs bosons is a natural, important, and unsolved question.So ten years after the 125-GeV Higgs boson was discovered, experimentalists are still making efforts to search for additional Higgs scalars, even if in the low-mass region.
In 2018, the CMS collaboration reported a di-photon excess around 95. 3 GeV with a local significance of 2.8 σ [6], with a signal strength of Interestingly, the CMS collaboration also reported a di-tau excess recently at 95 ∼ 100 GeV with a local significance of 2.6 ∼ 3.1 σ [7], with the signal strength of Besides, a b b excess around 98 GeV with a local significance of 2.3 σ was reported with the LEP data about twenty years ago [8], whose signal strength is σ ex (e + e − → Zϕ → Zbb) σ SM (e + e − → Zh → Zbb) = 0.117 ± 0.057 .
The rest of this paper is organized as follows.In Sec. 2 we introduce the Higgs sector in NMSSM and present the relevant analytic equations briefly.In Sec. 3 we show the numerical-calculation results and discussions.Finally, we draw our main conclusions in Sec. 4.

The Higgs sector in NMSSM
SUSY models are mainly determined by their superpotential and soft-breaking terms.In the NMSSM, they can be written as where W µ→λ Ŝ MSSM is the MSSM superpotential with the µ-term generated effectively by the Vacuum Expectation Value (VEV) of singlet field, and mHu , mH d , mS , A λ and A κ are soft-breaking parameters.Ĥu , Ĥd are the SU(2) doublet and Ŝ is the singlet Higgs superfields, and after getting VEVs the scalar fields can be expressed as then the parameter tan β ≡ v u /v d .The three gauge-eigenstate scalars {ϕ u , ϕ d , ϕ s } mix to form three CP-even mass-eigenstate Higgs scalars {h 1 , h 2 , h 3 }, with mass order m h 1 < m h 2 < m h 3 , and the mixing matrix The reduced couplings of h 1 to up-and down-type fermions, and massive gauge bosons, are given by While the loop-induced coupling to gluons c g is mainly determined by c t and light colored SUSY particles, and that of photon c γ are mainly by c t , c V , and light charged SUSY particles.

Numerical results and discussions
In the calculation, we first perform a scan over the parameter space of NMSSM with the public code NMSSMTools_5.6.1 [52][53][54] under a series of experimental and theoretical constraints * .The parameter space we consider are: Note that the NMSSM we consider in this work is GUT-scale constrained, where both Higgs and gaugino masses are considered non-universal.So M 0 and A 0 are the unified sfermion masses and trilinear couplings in the sfermion sector, and M 1,2,3 are the gaugino masses at the GUT scale.
While the three non-universal Higgs masses at the GUT scale are calculated from the minimization equations, with λ, κ and µ eff ≡ λv S at the SUSY scale as the input parameters.The µ eff parameter is chosen to be positive to interpret the muon g − 2 anomaly.One sign in three of M 1,2,3 can be absorbed in a field redefinition [35].The sign of M 3 can have other effects, e.g., as shown in Ref. [55].
To interpret the CMS di-photon and di-tau, and LEP b b excesses together, we also require a light Higgs boson of 95 ∼ 100 GeV.For the surviving samples, we define a chi-square quantity χ 2 γγ+τ τ +bb to describe its ability to interpret the three excesses globally. where with i = γγ, τ τ, bb, R i standing for the corresponding theoretical signal strength of our samples, Rex i and δR ex i for the corresponding experimental mean and error values, respectively.With χ 2 γγ+τ τ +bb ≤ 8.03, the surviving samples can interpret the three excesses globally at 2σ level, and are called 'globally 2σ samples', or simplified as '2σ samples' hereafter.We note that for surviving samples the minimum value of χ 2 γγ+τ τ +bb is 5.37, so there are no samples satisfying the three excesses at 1σ level globally (χ 2 γγ+τ τ +bb < 3.53).In Tab. 1 we list the parameter regions for the 2σ and all surviving samples respectively.In Fig. 1, we project the surviving samples on the signal strengths R γγ (gg → h 1 → γγ), R τ τ (gg → h 1 → τ τ ) and R bb (e + e − → Zh 1 → Zb b) versus width ratio R Γ (total decay width of h 1 divided by that of a SM Higgs of the same mass) planes, with colors denoting χ 2 γγ+τ τ +bb .From this figure, one can see that the low-mass excess data are powerful in distinguishing the surviving samples.For the 2σ samples, R Γ ≲ 0.1, 0.2 ≲ R γγ ≲ 0.8, and R τ τ , R bb ≲ 0.2.The surviving samples can be sorted into two regions obviously: the R Γ ≲ 0.1 region and the R γγ ≲ 0.2 region.Note that the 2σ samples can only locate in the former.Hereafter to compare with the 2σ samples in the former region, we also consider the 3σ samples, or samples with 8.03 ≲ χ 2 γγ+τ τ +bb ≲ 14.16, in the latter region, calling them small-R γγ samples.From the middle and right planes, one can see that for the 2σ samples 0.04 ≲ R τ τ , R bb ≲ 0.16, while for the small-R γγ samples 0.05 ≲ R τ τ , R bb ≲ 0.25.Combining with the experimental data, one can know that the 2σ samples mainly fit well with the CMS di-photon excess, and small-R γγ samples mainly fit well with the LEP Zb b excess.The CMS di-tau excess has so large uncertainty that it can not be dominant in our samples.
The signal strengths are related to the reduced couplings by So in Fig. 2, we project the surviving samples on the signal strengths versus reduced coupling planes, also with colors denoting χ 2 γγ+τ τ +bb .From this figure, one can see that the reduced couplings can be sorted into two classes: and c b ≈ c τ .And it can also be found that the width ratio is determined by c 2 b , for the dominant branching ratio of the light scalar is that to b b.Thus one can rewrite the signal strengths approximately to where one can see that small width ratio R Γ , or approximate c 2 b , can increase the di-photon rate, but can not increase the b b and di-tau rates.Then χ 2 γγ+τ τ +bb can be approximately written to From Fig. 2 one can also see that: for 2σ samples, the light scalar has negative reduced couplings to fermions and W/Z bosons, with 0.
From Fig. 3, one can also see that for the 2σ samples, |S 11 | ≫ |S 12 |, which means that the lightest Higgs boson is mainly mixed by the singlet and up-type doublet fields.Different from this, in the wrong sign limit [72,73] in the type-II two Higgs doublet model, the lighter Higgs boson is mixed by the up-and down-type doublets fields.And we also checked that the missing of c t ≳ c b case in Fig. 3 is because we choose positive µ eff , which is favored by the muon g − 2 constraint.For the down-type doublet-like Higgs boson in NMSSM needs to be much heavier than the other   two Higgs bosons to escape the constraints, S 12 , or the mixing between singlet and down-type doublet, should be very small compared with S 11 , thus the case of c t ≲ 0 but c b ≳ 0 is also not very favored.
Considering the mass region of excesses are close to the Z boson mass, we consider the scalar's production associated with a top quark pair to reduce the backgrounds, with the signal strengths written as In Fig. 4, we project the surviving samples on the planes of signal strengths of top-quark-pair associated channels versus these of three excess channels, respectively.From this figure one can see that There is a small difference, especially between top-pair-associated and gluon-gluon-fusion channels.For 2σ samples, the latter is slightly larger than the former; while for the small-R γγ samples, the former is slightly larger than the latter.The difference comes from the contributions of squarks, and they are positive or negative depending on c t , the reduced couplings to the top quark.And the difference is small because of the high mass bounds of squarks [74] from SUSY search results.As a comparison, new light colored particles can contribute much to the gluon-gluon-fusion channel [75].In Tab. 2, we list the detailed information of eight representative benchmark points for further study, where χ 2 125 and P 125 are the chi-square and P value from 125 GeV Higgs data of 111 groups (the number of degrees of freedom is 111).Note that for a SM Higgs of 125.09GeV, χ 2 125 = 89.7 and P 125 = 0.932.From this table, one can see that it is difficult to satisfy the 125 GeV Higgs data and the 95 ∼ 100 GeV excesses simultaneously at 2σ level.E.g., for Point P4, with the 95 ∼ 100 GeV excesses globally satisfied at 2σ level the In the end, we add discussions on dark matter, invisible Higgs decay, and electroweakino searches.
• For benchmark points P1-P7, the dark matter is bino-like, and the main annihilation mechanism is Z/h 2 funnel.The mass of dark matter is different from M 1 , because the parameters M 1,2,3 are defined at the GUT scale.There are correlations between parameters at GUT and SUSY scales, similar to those we present in Appendix A of our former work [55].
• We have considered the constraint of invisible Higgs decay with the code HiggsBounds, where the corresponding experimental data in Refs.[76,77] are included.For benchmark points P1-P7, the invisible Higgs decay Br(h 2 → χ0 1 χ0 1 ) are all smaller than about 1%, because the large invisible ratios are not favored by both 125 and 95 ∼ 100 GeV Higgs data.
• We have also imposed constraints from SUSY searches with the code SModelS.For benchmark point P8, the dark matter is Higgsino-like and the main annihilation mechanism is W ± /Z exchanges.This point can escape the constraints from searches for electroweakinos in Ref. [78] because of its compressed mass spectrum and multiple decay modes.In the low mass region, it has SUSY particles like Higgsino-like charginos and neutralinos of about 230 GeV, bino-like neutralino of 390 GeV, wino-like charginos and neutralinos of about 590 GeV, τ1 of 246 GeV, ντ of 353 GeV, μ1 of 478 GeV, νµ of 472 GeV, etc.

Conclusions
In this work, we consider a light Higgs boson in the NMSSM to interpret the CMS di-photon and di-tau excesses, and LEP b b excess, in the 95 ∼ 100 GeV mass region.We first scan the parameter space and consider a series of constraints, including these of Higgs, dark matter, SUSY searches, etc.Then for each surviving sample, we calculate a chi-square considering its global fit to the three excess data.And we focus on two respective kinds of samples: the 2σ samples and small-R γγ samples.Finally, we get the following conclusions: • In NMSSM it is difficult to satisfy the 95 ∼ 100 GeV excesses simultaneously (not possible globally at 1σ level, or simultaneously at 2σ level).
• The light Higgs boson's global fit to the three excesses is mainly determined by its couplings to up-and down-type fermions, which can be approximately written as in Eq. ( 13).
• The globally 2σ samples have negative reduced couplings to fermions and massive vector bosons, while they are positive for the small-R γγ ones.
• The globally 2σ samples have decay width smaller than one-tenth of the corresponding SM value, which can increase its di-photon rate but can not increase its di-tau rate.
• The small-R γγ samples can have Zb b signal right fit to the LEP b b excess, but have smaller di-photon and di-tau rates.
• The top-quark-pair associated signal strengths are nearly equal to these of the three exciting excesses, respectively.
To show the error level between the approximated chi-square and complete ones in Eq. ( 13), we give Fig. 5.In this figure, one can see clearly that c V ≈ c t is a very good approximation.For most samples, since the charged Higgs bosons and most SUSY particles are heavy, their contributions to the loop-induced couplings c g (c γ ) are much smaller than these of the SM particles top quark (and W boson).Thus c g (c γ ) are mainly determined by the top quark's coupling c t (and the W boson's coupling c V ).And since c V ≈ c t , for most samples we have c γ ≈ c V ≈ c t ≈ c g .From Fig. 5, one can also see that, the error level between the approximate chi-square and complete ones in Eq. ( 13) are below 5% for most samples, and at most about 15% for all ones.Therefore, Eqs. ( 13) and ( 12) are good approximations for most samples, and are interesting with only two variable quantities.

Fig. 4 .
Fig. 4. Same as in Fig. 1, but on the planes of signal strengths in top-quark-pair associated channels versus these of existing excess channels: R t tγγ versus Rγγ (left), R t tτ τ versus Rττ (middle) and R t tb b versus R bb (right) planes.

Table 2 .
Eight benchmark points for the surviving samples.