# Light Higgs bosons in the general NMSSM

## Abstract

Physics beyond the Standard Model (SM) may manifest itself as small deviations from the SM predictions for Higgs signal strengths at 125 GeV. Then, a plausible and interesting possibility is that the Higgs sector is extended and at the weak scale there appears an additional Higgs boson weakly coupled to the SM sector. Combined with the LEP excess in \(e^+e^-\rightarrow Z(h\rightarrow b\bar{b})\), the diphoton excess around 96 GeV recently reported by CMS may suggest such a possibility. We examine if those LEP and CMS excesses can be explained simultaneously by a singlet-like Higgs boson in the general next-to-minimal supersymmetric Standard Model (NMSSM). Higgs mixing in the NMSSM relies on the singlet coupling to the MSSM Higgs doublets and the higgsino mass parameter, and thus is subject to the constraints on these supersymmetric parameters. We find that the NMSSM can account for both the LEP and CMS excesses at 96 GeV while accommodating the observed 125 GeV SM-like Higgs boson. Interestingly, the required mixing angles constrain the heavy doublet Higgs boson to be heavier than about 500 GeV. We also show that the viable region of mixing parameter space is considerably modified if the higgsino mass parameter is around the weak scale, mainly because of the Higgs coupling to photons induced by the charged higgsinos.

## 1 Introduction

Although the Standard Model (SM) successfully describes the observed particle physics up to energy scales around TeV, it is clear that a more fundamental theory is needed to provide a complete description of nature. So far the LHC has seen no clear signal for physics beyond the SM, and the discovered 125 GeV Higgs boson has properties compatible with the SM predictions [1, 2]. Yet an interesting possibility is that the Higgs sector is extended to include an additional light Higgs boson which is accessible to collider experiments and result in some deviations of the 125 GeV Higgs boson from the SM predictions. The CMS has recently announced that Higgs searches in the diphoton final state show a local excess of about \(3\sigma \) at 96 GeV [3]. The results from the ATLAS do not show a relevant excess, but are well compatible with the CMS limit [4]. Combined with the \(2.3\sigma \) local excess observed in the LEP searches for \(e^+e^-\rightarrow Z(h \rightarrow b{\bar{b}})\) [5, 6], the CMS results provide a motivation to consider the possibility that the Higgs sector involves an additional scalar boson at 96 GeV, which has been studied recently in Refs. [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17].

In this paper we explore if the next-to-minimal supersymmetric SM (NMSSM) can explain the LEP and CMS excesses around 96 GeV while accommodating the observed 125 GeV Higgs boson. The NMSSM extends the Higgs sector to include a gauge singlet scalar which generates the higgsino mass parameter \(\mu \) via its coupling \(\lambda \) to the MSSM Higgs doublets [18, 19]. As noticed in Refs. [20, 21], there are intriguing relations between Higgs mixings and the model parameters \(\lambda \) and \(\mu \) that hold for the general NMSSM. Those relations are quite useful when examining how much Higgs mixings, which determine how the neutral Higgs bosons couple to SM particles [22, 23, 24, 25, 26, 27, 28, 29], are constrained by the requirements on the model such as radiative corrections to the Higgs masses, the perturbativity bound on \(\lambda \), and the chargino mass limit on \(\mu \). The viable region of mixing parameter space would be further constrained if one specifies singlet self-interactions. For instance, there are no tadpole and quadratic terms for the singlet in the \(Z_3\)-symmetric NMSSM,^{1} for which the mixing between the neutral singlet and doublet Higgs bosons has a dependence on the mass of the CP-odd singlet scalar.

Our analysis is based on the relations between Higgs mixings and the model parameters, and is performed for the general NMSSM without specifying singlet self-interactions. We first examine if a singlet-like Higgs boson at 96 GeV can be responsible for the LEP and CMS excesses within the range of mixing angles allowed by the current LHC data on the 125 GeV Higgs boson, under the assumption that the gauginos, squarks and sleptons are heavy enough, above TeV as indicated by the LHC searches for supersymmetry (SUSY), while the higgsinos can be significantly lighter. We then impose the constraints on \(\lambda \) and \(\mu \) to find the viable mixing angles. It turns out that the general NMSSM can accommodate the SM-like 125 GeV Higgs boson compatible with the current LHC data, and also a singlet-like 96 GeV Higgs boson explaining both the LEP and CMS excesses. The allowed range of mixing angles is considerably modified if \(\mu \) is around the weak scale because the charged higgsinos enhance the Higgs coupling to photons. Interestingly, if the excesses around 96 GeV are due to the singlet-like Higgs boson, the heavy doublet Higgs boson should be heavier than about 500 GeV.

This paper is organized as follows. In Sect. 2, we briefly discuss the effects of the neutral Higgs boson mixings on Higgs phenomenology and examine the relations between the mixing angles and the NMSSM parameters. The region of the mixing parameter space compatible with the current LHC data on the SM-like Higgs boson is presented in Sect. 3. Section 4 is devoted to our main results, which show the mixing angles required to explain the LEP and CMS excesses, while satisfying the various constraints on the NMSSM parameters. It is also shown that the allowed mixing angles constrain the heavy Higgs boson to have a mass in a certain range. The final section is for the summary and comments.

## 2 Higgs bosons in the general NMSSM

In this section we describe how the neutral Higgs boson mixings depend on the NMSSM parameters, in particular, on the singlet coupling \(\lambda \) to the MSSM Higgs doublets and the higgsino mass parameter \(\mu \). Such relations should be taken into account when examining the Higgs mixings consistent with the experimental constraints. We also discuss the properties of Higgs bosons within the low energy effective theory constructed by integrating out heavy superparticles under the assumption that the higgsinos can be light. Note that our approach is applicable to a general form of NMSSM.

### 2.1 Dependence of Higgs mixing on NMSSM parameters

*S*is the gauge singlet superfield. There are various types of NMSSM, depending on the form of the singlet superpotential

*f*(

*S*). Our subsequent discussion applies for a general form of

*f*(

*S*), but for simplicity we will assume no CP violation in the Higgs sector.

^{2}

*i.e.*for maximal stop mixing. On the other hand, the charged Higgs boson has a mass around the square-root of \({\hat{M}}^2_{22}\), and it should be heavier than about 350 GeV to avoid the experimental constraint associated with \(b\rightarrow s\gamma \) [31].

*U*can be parametrized as

### 2.2 Effective Higgs couplings to the SM sector

*f*denotes the SM fermions.

## 3 Mixing consistent with the 125 GeV Higgs boson

*h*has properties close to those of the SM Higgs boson. In this paper, we identify

*h*with the SM-like Higgs boson observed at the LHC and examine how the scalar mixing is constrained by the measured signal strengths. The SM-like Higgs boson has \(m_h=125\) GeV, and its couplings to the massive SM particles are given by

*r*is defined by

*r*and \(\tan \beta \).

*h*can be easily estimated by using the well-known decay properties of the hypothetical Higgs boson \(\phi _{125}\) of the minimal SM with mass 125 GeV:

*h*does not decay to non-SM particles, one also finds its total decay rate to be

*WW*/

*ZZ*channel, where \(\mathrm{Br}(h\rightarrow ii)\) is the branching ratio of the indicated mode. For other channels, one obtains

^{3}

*h*are determined by two mixing angles \(\theta _1\) and \(\theta _2\) for given values of

*r*and \(\tan \beta \). For instance, \(\mu ^{VV}_h=1\) is obtained if \(\theta _1\) and \(\theta _2\) satisfy

^{4}

*r*, which is relevant for the diphoton signal strength, is below about 1.2 because \(\lambda \) should be smaller than about 0.7 in order for the NMSSM to remain perturbative up to the GUT scale, and \(|\mu |\) should be larger than 104 GeV to satisfy the LEP bound on the chargino mass.

*r*is around unity, the charged higgsinos can significantly enhance the diphoton signal rate, excluding \(\theta _2\) in the range between about 0.4 and 0.8. We can understand this feature from the fact that the charged higgsinos induce a Higgs coupling to photons, \(\delta C^h_\gamma \approx -0.17 r \theta _2\) for small mixing angles, whereas the Higgs couplings to other SM particles only quadratically depend on \(\theta _2\).

## 4 LEP and CMS excesses around 96 GeV

*Z*-boson associated Higgs production (\(e^+ e^- \rightarrow Z \varphi \)) at LEP [5]. The signal strength is [35]

*s*, the signal strengths can be expressed by the effective couplings in Eq. (2.14) as follows,

*s*decays only into the SM particles. Here we have used HDECAY [36, 37] to calculate the decay widths of the 96 GeV Higgs boson with the SM couplings. The effective couplings of

*s*are written in terms of the mixing angles as

*s*is responsible for the LEP and CMS excesses. The LEP signal rate given in Eq. (4.4) is approximated by

*h*and

*s*are functions of the mixing angles, a combination of \(\lambda \) and \(\mu \), and \(\tan \beta \):

*r*. The above relations allow us to analyze the viable mixing angles as follows. We first examine the \((\theta _1, \, \theta _2)\) space to see in which region \(\mu ^{ii}_h\) are consistent with the current LHC data, and then continue to check if it is further possible to explain both \(\mu _{\mathrm{LEP}}\) and \(\mu _{\mathrm{CMS}}\).

^{5}Then it follows \(r\le 1.1\). Note that

*r*parameterizes the radiative effect of the charged higgsinos on Higgs decays. As benchmark points, we have taken \(r = 0.1,\,1\) for \(1.5\le \tan \beta \le 15\).

^{6}Each color in Fig. 3 represents how much the above constraints reduce the viable region. We note that the bound on \(m_0\) gets important when \(\tan \beta \) is small and

*r*is around 1 or above.

In the parameter region for the LEP and CMS excesses, the main effect of stop loop corrections \(\Delta m_{12}^2\) is to increase (decrease) the coupling \(\lambda \) if \(\Delta m_{12}^2\) is negative (positive), as can be deduced from the last relation in Eq. (2.11). This implies that the parameter space compatible with both excesses shrinks for larger negative \(\Delta m_{12}^2\) due to the perturbativity bound on \(\lambda \). In the parameter region with \(m_0\ge 115\) GeV, however, the dependence on \(\Delta m_{12}^2\) becomes quite weak because \(\lambda \) should be small in order to get \(m_h=125\) GeV. On the other hand, the LEP limit on the chargino mass given in Eq. (4.11) implies \(\lambda > 0.6r\), following from \(r\equiv \lambda v/|\mu |\). Taking this together with the perturbativity bound on \(\lambda \), one can find that \(\lambda \) would be more severely constrained at larger *r* when the stop correction \(\Delta m^2_{12}\) is negative. We have checked these features by taking analysis for nonzero values of \(\epsilon \) between \(-0.05\) and 0.05.

We close this section by pointing out that the LEP and CMS excesses can constrain the masses of the heavy Higgs boson and higgsinos, if they are due to the singlet-like Higgs boson. Eq. (2.11) enable us to extract the information on the region of \(\mu \) and \(m_H\) compatible with the Higgs signal strengths, \(\mu ^{ii}_h\), \(\mu _{\mathrm{LEP}}\) and \(\mu _\mathrm{CMS}\). Figure 4 shows the allowed region of \((\mu , m_H)\), where we have taken \(\tan \beta =2\) (left) and 5 (right) with \(0<r<1.1\). As discussed already, the \(m_0\) cut is relevant for small \(\tan \beta \). It is important to note that the CMS and LEP excesses put a lower and upper bound on \(m_H\). The lower bound turns out to be \(m_H\gtrsim 500\) GeV, nearly irrespectively of the values of \(\mu \) and \(\tan \beta \), while the upper bound depends on those parameters and is found to increase with \(\tan \beta \).

## 5 Summary

Extended with an additional gauge singlet scalar, the Higgs sector of the NMSSM offers a rich phenomenology to be explored at collider experiments. In particular, as experimentally allowed to be light, a singlet-like Higgs boson could be observable in the searches for \(e^+e^-\rightarrow Z(h\rightarrow b{\bar{b}})\) and \(pp\rightarrow h\rightarrow \gamma \gamma \) if it couples to the SM sector via the Higgs mixing. It is thus interesting to examine if the excesses reported by LEP and CMS in those channels can be interpreted as signals of a singlet-like Higgs boson with mass around 96 GeV within the NMSSM.

For the case that the gauginos, squarks and sleptons have masses above TeV, while the Higgsinos can be significantly lighter, which is perfectly consistent with the null results for SUSY searches at LHC so far, we have found that the general NMSSM can successfully accommodate such a light singlet-like Higgs boson explaining the LEP and CMS excesses simultaneously, as well as the 125 GeV Higgs boson compatible with the current LHC data. The range of mixing angles required to explain the 96 GeV excesses can be considerably modified if the higgsinos are around the weak scale, because the singlet-like Higgs coupling to photons is enhanced.

To examine a viable region of mixing parameter space, it should be taken into account that Higgs mixing is subject to various constraints on the NMSSM parameters. We have shown that, if a singlet-like Higgs boson is responsible for the LEP and CMS excesses, Higgs mixing is strongly constrained by the LEP bound on the charged higgsino mass and the perturbativity bound on the singlet coupling to the Higgs doublets. Interestingly, in the viable mixing space, the heavy doublet Higgs boson is found to be heavier than about 500 GeV.

The physics underlying the electroweak symmetry breaking may manifest itself as slight deviations from the SM predictions for the Higgs signal strengths at 125 GeV. It is then a plausible possibility that there exist additional light Higgs bosons weakly coupled to the SM sector, which would provide crucial information on how the Higgs sector is extended. The excesses reported by LEP and CMS, both of which are interestingly around 96 GeV, would thus deserve more attention.

## Footnotes

- 1.
The possibility of accommodating both LEP and CMS excesses in the \(Z_3\)-symmetric NMSSM was firstly explored in Ref. [13]. Our study considers the general NMSSM without any additional symmetry or matter.

- 2.The value of \({\hat{M}}^2_{33}\) is determined by the singlet superpotential
*f*(*S*) and the associated soft SUSY breaking terms*F*(*S*) according towhere all the terms are evaluated at the vacuum.$$\begin{aligned} {\hat{M}}^2_{33}= & {} (\partial _S^2 f)^2 + \left( \partial _S f - \frac{1}{2}\lambda v^2 \sin 2\beta \right) \left( \partial _S^3 f - \frac{ \partial _S^2 f }{ S} \right) \nonumber \\&+ \frac{1}{2} \lambda v^2 \frac{A_\lambda }{ S} \sin 2\beta + \left( \partial _S^2 F - \frac{ \partial _S F }{ S } \right) , \end{aligned}$$(2.5) - 3.
Both ATLAS and CMS collaborations have recently reported their analyses results on the Higgs coupling measurements using the LHC Run 2 data [1, 2]. The ATLAS analysis used the larger amount of the Run 2 data, up to the integrated luminosity of 80 \(\hbox {fb}^{-1}\). As there is no combined global fit for the full Run 2 data yet, we adopted only the ATLAS result in our study.

- 4.Although we will not pursue in this paper, a large value of \(\theta _1\) satisfyingcan also lead to \(\mu ^{VV}_h=1\). In this case, \(C^h_b\) is negative and thus leads to wrong sign Yukawa couplings [20], and one needs a large \(\lambda \) beyond the perturbativity bound [34].$$\begin{aligned} \theta _1 \approx \frac{2.1-1.9 s^2_{\theta _2} }{\tan \beta } \end{aligned}$$
- 5.
- 6.

## Notes

### Acknowledgements

We would like to thank S. Heinemeyer for useful comments on the manuscript. This work was supported by IBS under the project code, IBS-R018-D1 (KC and CBP), and by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (NRF-2018R1C1B6006061) (SHI and KSJ).

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