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,Footnote 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

Taking an appropriate redefinition of superfields, one can always write the superpotential of the general NMSSM as

$$\begin{aligned} W= \lambda S H_u H_d + f(S) + (\text {MSSM Yukawa terms}), \end{aligned}$$
(2.1)

with a canonical Kähler potential. Here \(H_u\) and \(H_d\) are the Higgs doublet superfields, and 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.

After the electroweak symmetry breaking, the CP-even neutral Higgs bosons

$$\begin{aligned} \begin{pmatrix} {\hat{h}} \\ {\hat{H}} \\ {\hat{s}} \end{pmatrix} = \sqrt{2} \begin{pmatrix} \sin \beta &{} \cos \beta &{} 0 \\ -\cos \beta &{} \sin \beta &{} 0 \\ 0 &{} 0 &{} 1 \end{pmatrix} \begin{pmatrix} {\text {Re}}H^0_u - v\sin \beta \\ {\text {Re}}H^0_d - v\cos \beta \\ {\text {Re}}S - \langle S \rangle \end{pmatrix}\nonumber \\ \end{aligned}$$
(2.2)

mix with each other due to the potential terms

$$\begin{aligned} V_{\mathrm{mix}}= & {} \lambda ^2 |S|^2 ( |H_u|^2 +|H_d|^2 ) + \Big ( A_\lambda \lambda S H_u H_d\nonumber \\&+ (\partial _S f)^* \lambda H_uH_d + \mathrm{h.c.} \Big ), \end{aligned}$$
(2.3)

where \(\langle H^0_u \rangle = v\sin \beta \) and \(\langle H^0_d \rangle = v\cos \beta \) with \(v=174\) GeV, and \(A_\lambda \) is the SUSY breaking trilinear coupling. With the above Higgs potential terms, the effective Higgs \(\mu \) and \(B\mu \) parameters are given by

$$\begin{aligned} \mu= & {} \lambda \langle S \rangle , \nonumber \\ B\mu= & {} A_\lambda \lambda \langle S \rangle + \lambda \langle \partial _S f \rangle . \end{aligned}$$
(2.4)

The supersymmetric parameters \(\lambda \) and \(\mu \), on which the Higgs mixing depends, are subject to the perturbativity bound and the chargino mass bound, respectively. Imposing the conditions for the electroweak symmetry breaking, the mass squared matrix for \(({\hat{h}},{\hat{H}},{\hat{s}})\) readsFootnote 2

$$\begin{aligned}&{\hat{M}}^2_{11} = m^2_0 + (\lambda ^2 v^2 -m^2_Z ) \sin ^2 2\beta ,\nonumber \\&{\hat{M}}^2_{12} = {\hat{M}}^2_{21} = \frac{1}{2} (m^2_Z - \lambda ^2 v^2) \sin 4\beta + \Delta m^2_{12},\nonumber \\&{\hat{M}}^2_{13} = {\hat{M}}^2_{31} = \lambda v (2 \mu -\Lambda \sin 2\beta ), \nonumber \\&{\hat{M}}^2_{22} = \frac{2 |B\mu |}{\sin 2\beta } - (\lambda ^2 v^2-m_Z^2) \sin ^2 2\beta + \Delta m_{22}^2, \nonumber \\&{\hat{M}}^2_{23} = {\hat{M}}^2_{32} = \lambda v \Lambda \cos 2\beta , \end{aligned}$$
(2.6)

with \(\Lambda \) defined by

$$\begin{aligned} \Lambda \equiv A_\lambda + \langle \partial _S^2 f \rangle . \end{aligned}$$
(2.7)

Here \({\hat{M}}^2_{11}\), \({\hat{M}}^2_{12}\), and \({\hat{M}}^2_{22}\) include radiative corrections, which can be sizable as arising from top and stop loops [30]:

$$\begin{aligned} m^2_0&\simeq m^2_Z + \frac{3v^2 y^4_t}{4\pi ^2}\left\{ \ln \left( \frac{M^2_S}{m^2_t} \right) + \frac{X^2_t}{M_S^2} \left( 1 - \frac{X^2_t}{12M_S^2} \right) \right\} , \nonumber \\ \Delta m^2_{12}&\simeq -\frac{3 v^2 y^4_t}{4\pi ^2 \tan \beta }\nonumber \\&\quad \left\{ \ln \left( \frac{M^2_S}{m^2_t} \right) + \frac{X_t (X_t + Y_t)}{2M_S^2} - \frac{X_t^3 Y_t}{12 M_S^4} \right\} , \nonumber \\ \Delta m_{22}^2&\simeq \frac{3 v^2 y^4_t}{4\pi ^2 \tan ^2\beta } \left\{ \ln \left( \frac{M^2_S}{m^2_t} \right) + \frac{X_t Y_t}{M_S^2} \left( 1 - \frac{X_t Y_t}{12M_S^2} \right) \right\} , \end{aligned}$$
(2.8)

with \(X_t= A_t -\mu \cot \beta \) and \(Y_t = A_t + \mu \tan \beta \), where \(M_S=\sqrt{m_{{\tilde{t}}_1}m_{{\tilde{t}}_2}}\) is the geometric mean of the two stop mass-eigenvalues, and \(A_t\) is the SUSY breaking trilinear coupling associated with the top quark Yukawa \(y_t = m_t/v\). Note that \(m_0\) corresponds to the SM-like Higgs boson mass at large \(\tan \beta \) in the decoupling limit of the MSSM. The LHC results constrain the stops to be heavier than TeV, and thus \(m_0\) cannot be lower than about 115 GeV as long as stop mixing has \(X^2_t \lesssim 10 M_S^2\) as is the case in the conventional mediation schemes of SUSY breaking. The stop loop corrections maximizes \(m_0\) at \(X_t = \pm \sqrt{6} M_S\), 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].

To find the mass eigenstates, one needs to diagonalize the mass squared matrix as

$$\begin{aligned} U{\hat{M}}^2 U^{\mathrm{T}} = \mathop {\mathrm {diag}}\nolimits (m^2_h, \, m^2_H, \, m^2_s), \end{aligned}$$
(2.9)

where the orthogonal mixing matrix U can be parametrized as

$$\begin{aligned} U = \begin{pmatrix} c_{\theta _1} c_{\theta _2} &{} - s_{\theta _1} &{} - c_{\theta _1} s_{\theta _2} \\ s_{\theta _1} c_{\theta _2} c_{\theta _3} - s_{\theta _2} s_{\theta _3} &{} c_{\theta _1} c_{\theta _3} &{} - c_{\theta _2} s_{\theta _3} - s_{\theta _1} s_{\theta _2} c_{\theta _3} \\ s_{\theta _1} c_{\theta _2} s_{\theta _3} + s_{\theta _2} c_{\theta _3} &{} c_{\theta _1} s_{\theta _3} &{} c_{\theta _2} c_{\theta _3} - s_{\theta _1} s_{\theta _2} s_{\theta _3} \end{pmatrix}\nonumber \\ \end{aligned}$$
(2.10)

with \(s_\theta \equiv \sin \theta \) and \(c_\theta \equiv \cos \theta \), for which the angles \(\theta _1\), \(\theta _2\) and \(\theta _3\) represent \({\hat{h}}\)–\({\hat{H}}\), \({\hat{s}}\)–\({\hat{h}}\) and \({\hat{s}}\)–\({\hat{H}}\) mixing, respectively. Obviously each matrix element of \({\hat{M}}^2\) can be expressed in terms of the mass eigenvalues \(\{m_h,\,m_H,\,m_s\}\) and the mixing angles \(\{\theta _1,\,\theta _2,\,\theta _3\}\). Among such relations, the following ones are particularly relevant for our subsequent discussion:

$$\begin{aligned} m^2_0= & {} m^2_h + U_{21}\Big ( U_{21} + U_{22} \tan 2\beta \Big ) (m^2_H - m^2_h) \nonumber \\&-\, U_{31}\Big (U_{31}+U_{32} \tan 2\beta \Big ) (m^2_h- m^2_s)\nonumber \\&- \Delta m^2_{12} \tan 2\beta , \nonumber \\ \lambda v \mu= & {} \frac{1}{2}U_{23}\Big ( U_{21} + U_{22}\tan 2\beta \Big ) (m^2_H - m^2_h)\nonumber \\&- \frac{1}{2}U_{33}\Big ( U_{31} + U_{32}\tan 2\beta \Big )(m^2_h - m^2_s), \nonumber \\ \lambda ^2v^2= & {} m^2_Z - \frac{2}{\sin 4\beta } \left( U_{21}U_{22}(m^2_H - m^2_h)\right. \nonumber \\&\left. - U_{31}U_{32}(m^2_h - m^2_s) - \Delta m^2_{12} \right) , \end{aligned}$$
(2.11)

because the constraints on the parameters \(\lambda \), \(\mu \) and \(m_0\) can be converted into those on the Higgs boson masses and mixing angles, and also vice versa. Here one should note that \(\Delta m^2_{12}\) in Eq. (2.11) is written

$$\begin{aligned} \Delta m^2_{12}= -\frac{m^2_0 - m^2_Z}{\tan \beta } + \epsilon v^2, \end{aligned}$$
(2.12)

with \(\epsilon \) given by

$$\begin{aligned} \epsilon = - \frac{3y^4_t}{8\pi ^2}\frac{\mu }{M_S}\left( 1 +\frac{1}{\tan ^2\beta } \right) \left( \frac{X_t}{M_S} - \frac{X^3_t}{6 M_S^3} \right) . \end{aligned}$$
(2.13)

The above shows that \(\epsilon \) vanishes at \(X_t = 0\) and \(X_t = \pm \sqrt{6} M_S\), and therefore \(\Delta m^2_{12}\) is tightly correlated with \(m_0\) near the regions of minimal and maximal stop mixing. The correlation of the stop corrections in large regions of parameter space has been noted in Ref. [30]. It is also easy to see that \(|\epsilon |\) is smaller than about 0.1 for \(X^2_t \lesssim 10 M_S^2\). For stop mixing with \(X_t\) far from 0 or \(\pm \sqrt{6} M_S\), the \(\epsilon \)-contribution to \(\Delta m^2_{12}\) can be sizable only when \(|\mu |\) is large, close to \(M_S\), and \(\tan \beta \) is low. Keeping this feature in mind, we shall neglect the \(\epsilon \)-contribution in our analysis unless stated otherwise.

2.2 Effective Higgs couplings to the SM sector

At energy scales around the electroweak scale, the properties of the Higgs bosons can be examined within an effective theory constructed by integrating out heavy sparticles. The LHC results on the Higgs sector and the searches for new physics indicate that SUSY, if exists, would be broken at a scale above TeV. Taking this into account, we assume that all gauginos, squarks and sleptons have masses above TeV, while the higgsinos and additional Higgs bosons can be lighter than TeV. Then the effective lagrangian describing how the neutral Higgs bosons interact with the SM fermions and gauge bosons is written as [32]

$$\begin{aligned} \mathcal{L}_{\mathrm{eff}}= & {} C_V^i \left( \frac{\sqrt{2}m^2_W}{v} \phi _i W^+_\mu W^{- \mu } + \frac{m^2_Z}{\sqrt{2} v} \phi _i Z_\mu Z^\mu \right) \nonumber \\&- C_f^i \frac{m_f}{\sqrt{2} v} \phi _i {\bar{f}} f \nonumber \\&+\, \Delta C_g^i \frac{\alpha _s}{12\sqrt{2}\pi v} \phi _i G^a_{\mu \nu } G^{a \mu \nu } \nonumber \\&+ \Delta C_\gamma ^i \frac{\alpha }{\sqrt{2}\pi v} \phi _i F_{\mu \nu } F^{\mu \nu }, \end{aligned}$$
(2.14)

where \((\phi _1,\,\phi _2,\,\phi _3)=(h,\,H,\,s)\) and f denotes the SM fermions.

At tree level, the Higgs couplings to massive SM particles are given by

$$\begin{aligned}&C_V^i = U_{i1}, \quad C_t^i = U_{i1} - U_{i2}\cot \beta , \nonumber \\&C_b^i = C^i_\tau = U_{1i} + U_{i2} \tan \beta . \end{aligned}$$
(2.15)

The Higgs couplings to gluons and photons are radiatively generated, which results in

$$\begin{aligned} \Delta C^i_g\simeq & {} A_{1/2}(\tau ^i_t) C^i_t + A_{1/2}(\tau ^i_b)C^i_b + \delta C^i_g, \nonumber \\ \Delta C^i_\gamma\simeq & {} \frac{2}{9} A_{1/2}(\tau ^i_t) C^i_t - \frac{7}{8} A_1(\tau ^i_W)C^i_V + \delta C^i_\gamma , \end{aligned}$$
(2.16)

where \(\delta C^i_g\) and \(\delta C^i_\gamma \) are additional contributions from sparticle loops, and the loop functions are given by

$$\begin{aligned} A_{1/2} (\tau )= & {} \frac{3}{2\tau ^2} \left\{ (\tau - 1) f(\tau ) + \tau \right\} , \nonumber \\ A_1 (\tau )= & {} \frac{1}{7\tau ^2} \left\{ 3 (2 \tau - 1) f (\tau ) + 3 \tau + 2 \tau ^2 \right\} , \end{aligned}$$
(2.17)

where \(\tau ^i_j \equiv m^2_\phi /(4m^2_j)\) and

$$\begin{aligned} f(\tau ) = {\left\{ \begin{array}{ll} \arcsin ^2 \sqrt{\tau } &{} \quad \mathrm{for}\,\, \tau \le 1 \\ -\frac{1}{4} \left( \ln \frac{1 + \sqrt{1 - \tau ^{-1}}}{1 - \sqrt{1 - \tau ^{-1}}} - i \pi \right) ^2 &{} \quad \mathrm{for}\,\, \tau > 1 \end{array}\right. }. \end{aligned}$$
(2.18)

For the case when the superpartners of SM particles are heavier than TeV, sparticles give negligible contributions to \(\Delta C^i_g\). However, if \(\mu \) is small, the charged higgsinos are light and can give a sizable contribution to \(\Delta C^i_\gamma \) through the following Higgs-higgsino couplings

$$\begin{aligned} \frac{\lambda }{\sqrt{2}} \sum _i U_{i3} \phi _i {\tilde{H}}^+_u \tilde{H}^-_d. \end{aligned}$$
(2.19)

Under the assumption that the gauginos are significantly heavier than the higgsinos, the Higgs coupling to photons induced by charginos can be approximated to be

$$\begin{aligned} \delta C^i_\gamma \simeq \frac{\lambda v}{6|\mu |} \left( 1 + \frac{7}{30} \frac{m^2_{\phi _i}}{4|\mu |^2} \right) U_{i3} \end{aligned}$$
(2.20)

for a Higgs boson with \(m^2_{\phi _i} \ll 4|\mu |^2\).

3 Mixing consistent with the 125 GeV Higgs boson

For small scalar mixing, 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

$$\begin{aligned}&C_V^h = c_{\theta _1} c_{\theta _2}, \quad C_t^h = c_{\theta _1} c_{\theta _2} + s_{\theta _1} \cot \beta , \nonumber \\&C_b^h = C_\tau ^h = c_{\theta _1} c_{\theta _2} - s_{\theta _1} \tan \beta , \end{aligned}$$
(3.1)

while the couplings to gluons and photons read

$$\begin{aligned} \Delta C^h_g\simeq & {} 0.97c_{\theta _1} c_{\theta _2} + (1.03\cot \beta + 0.06\tan \beta ) s_{\theta _1}, \nonumber \\ \Delta C^h_\gamma\simeq & {} -0.81c_{\theta _1} c_{\theta _2} + 0.23 s_{\theta _1} \cot \beta - \frac{r}{6}c_{\theta _1} s_{\theta _2}. \end{aligned}$$
(3.2)

Here r is defined by

$$\begin{aligned} r \equiv \frac{\lambda v}{|\mu |}, \end{aligned}$$
(3.3)

and measures the chargino contribution, which has been approximated by using the fact that it is non-negligible only when \(|\mu |\) is not far above the electroweak scale for \(\lambda \) below the perturbative bound, and the chargino search at LEP requires \(|\mu |>104\) GeV. One should note that the couplings of the SM-like Higgs boson are determined by four parameters: \(\theta _1\), \(\theta _2\), r and \(\tan \beta \).

The partial decay rates of the SM-like Higgs boson 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:

$$\begin{aligned} \frac{\Gamma (h\rightarrow bb)}{\Gamma (\phi _{125}\rightarrow bb)}= & {} \frac{\Gamma (h\rightarrow \tau \tau )}{\Gamma (\phi _{125} \rightarrow \tau \tau )} = |C^h_b|^2, \nonumber \\ \frac{\Gamma (h\rightarrow WW)}{\Gamma (\phi _{125}\rightarrow WW)}= & {} \frac{\Gamma (h\rightarrow ZZ)}{\Gamma (\phi _{125} \rightarrow ZZ)} = |C^h_V|^2, \nonumber \\ \frac{\Gamma (h\rightarrow gg)}{\Gamma (\phi _{125}\rightarrow gg)}= & {} \frac{|\Delta C^h_g|^2}{|\Delta C^{\phi _{125}}_g|^2}, \nonumber \\ \frac{\Gamma (h\rightarrow \gamma \gamma )}{\Gamma (\phi _{125}\rightarrow \gamma \gamma )}= & {} \frac{|\Delta C^h_\gamma |^2}{|\Delta C^{\phi _{125}}_\gamma |^2}. \end{aligned}$$
(3.4)

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

$$\begin{aligned} \frac{\Gamma _{\mathrm{tot}}(h)}{\Gamma _{\mathrm{tot}}(\phi _{125}) }\simeq & {} 0.64|C^h_b|^2 + 0.12 |C^h_t|^2 + 0.24 |C^h_V|^2, \end{aligned}$$
(3.5)

with \(\Gamma _{\mathrm{tot}}(\phi _{125})\) being the total decay width of \(\phi _{125}\). Here we have used the branching ratios of the SM Higgs boson listed in Ref. [33]. The production of the SM-like Higgs boson is dominated by the gluon fusion process, and the signal strength normalized by the SM value is given by

$$\begin{aligned} \mu ^{VV}_h= & {} \frac{\sigma (pp\rightarrow h)}{\sigma (pp \rightarrow \phi _{125})} \frac{\mathrm{Br}(h \rightarrow VV)}{\mathrm{Br}(\phi _{125}\rightarrow VV)} \nonumber \\\simeq & {} \frac{0.94 |\Delta C^h_g|^2|C^h_V|^2 + 0.12 |C^h_V|^4}{0.64|C^h_b|^2 + 0.12 |C^h_t|^2 + 0.24 |C^h_V|^2}, \end{aligned}$$
(3.6)

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

$$\begin{aligned} \frac{\mu ^{bb}_h}{\mu ^{VV}_h}= & {} \frac{\mu ^{\tau \tau }_h}{\mu ^{VV}_h} = \frac{|C^h_b|^2}{|C^h_V|^2}, \nonumber \\ \frac{\mu ^{\gamma \gamma }_h}{\mu ^{VV}_h}= & {} \frac{|\Delta C^h_\gamma |^2}{|\Delta C^{\phi _{125}}_\gamma |^2 |C^h_V|^2} \simeq \frac{1.52|\Delta C^h_\gamma |^2}{|C^h_V|^2}. \end{aligned}$$
(3.7)

It is obvious that one should have \(\mu ^{ii}_h=1\) if \(\theta _1=\theta _2=0\) and \(r=0\).

The ATLAS collaboration has recently updated the measurements on the Higgs signal strengths using the 13 TeV data [1]:Footnote 3

$$\begin{aligned} \mu ^{ZZ}_h = 1.13\pm 0.13, \end{aligned}$$
(3.8)

and for the other channels

$$\begin{aligned}&\frac{\mu ^{WW}_h}{\mu ^{ZZ}_h} = 0.84^{+0.18}_{-0.15}, \quad \frac{\mu _h^{\gamma \gamma }}{\mu _h^{ZZ}} = 0.87_{-0.12}^{+0.14}, \nonumber \\&\frac{\mu _h^{\tau \tau }}{\mu _h^{ZZ}} = 0.86_{-0.22}^{+0.26}, \quad \frac{\mu _h^{bb}}{\mu _h^{ZZ}} = 0.84_{-0.27}^{+0.38}. \end{aligned}$$
(3.9)

Note that the NMSSM leads to \(\mu ^{WW}_h=\mu ^{ZZ}_h\), which is within the \(1\sigma \) range.

Let us now examine how severely the Higgs mixing is constrained by the LHC experimental results on the Higgs boson at 125 GeV. The signal rates of 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 \(\theta _2\) satisfyFootnote 4

$$\begin{aligned} \theta _1 \approx \frac{\tan \beta }{1.4 \tan ^2\beta + 1.7}\,s^2_{\theta _2}, \end{aligned}$$
(3.10)

for which the branching ratio for \(h\rightarrow VV\) is suppressed compared to the case of the SM Higgs boson, but such effect is compensated by the enhancement of production rate via the gluon fusion process. Note that 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.

Fig. 1
figure 1

Mixing angles \((\theta _1,\theta _2)\) compatible with the current LHC data on the 125 GeV Higgs boson for \(r=0.1\) (left) and \(r=1\) (right), respectively. The shaded region is allowed for \(1.5\le \tan \beta \le 15\). The gray and yellow colors show how the allowed region changes with \(\tan \beta \)

Full size image

Figure 1 shows the \(2\sigma \) range of \((\theta _1,\theta _2)\) allowed by the current LHC data on the Higgs boson. The left panel is for \(r=0.1\), for which the Higgs coupling to photons is rarely affected by the charged higgsinos, and the right panel is for \(r=1\). Here we have taken \(1.5\le \tan \beta \le 15\) to see how the allowed region changes with \(\tan \beta \). From the figure, one can see that the allowed region of \(\theta _1\) gets smaller if \(\tan \beta \) increases, but a broad range of \(\theta _2\) is allowed insensitively to \(\tan \beta \). This is because the Higgs coupling \(C^h_b\) gets sensitive to \(\theta _1\) at large \(\tan \beta \) while the couplings to the top quark and gauge bosons do not. If 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 \(\theta _2\).

Fig. 2
figure 2

Mixing angles to explain the LEP and CMS excesses for \(r=0.1\) and \(r=1\) in the left and right panels, respectively, under the conditions \(\lambda <0.7\) and \(|\mu |>104\) GeV. The shaded region is compatible with the observed 125 GeV Higgs boson for \(1.5\le \tan \beta \le 15\) as noticed in Fig. 1. The singlet-like Higgs boson with mass 96 GeV can account for the LEP and CMS excesses simultaneously in the red region

Full size image

4 LEP and CMS excesses around 96 GeV

The CMS collaboration has recently reported a local excess of 2.8\(\sigma \) in the diphoton channel around \(m_{\gamma \gamma } =96\) GeV [3]. The signal strength amounts to

$$\begin{aligned} \mu _{\mathrm{CMS}} = \frac{\sigma (pp\rightarrow \varphi )}{\sigma (pp \rightarrow \phi _{96})} \frac{\mathrm{Br}(\varphi \rightarrow \gamma \gamma ) }{\mathrm{Br}(\phi _{96}\rightarrow \gamma \gamma )} \simeq 0.6\pm 0.2, \end{aligned}$$
(4.1)

where \(\phi _{96}\) denotes the hypothetical SM Higgs boson with mass 96 GeV, and \(\mathrm{Br}(\varphi \rightarrow \gamma \gamma )\) denotes the branching ratio for the diphoton channel [13]. Intriguingly, there is another 2.3\(\sigma \) local excess at the similar mass region from the Higgs searches in the Z-boson associated Higgs production (\(e^+ e^- \rightarrow Z \varphi \)) at LEP [5]. The signal strength is [35]

$$\begin{aligned} \mu _{\mathrm{LEP}}= & {} \frac{\sigma (e^+e^-\rightarrow Z\varphi )}{\sigma (e^+e^- \rightarrow Z\phi _{96})} \frac{\mathrm{Br}(\varphi \rightarrow b{\bar{b}}) }{\mathrm{Br}(\phi _{96}\rightarrow b{\bar{b}})} \nonumber \\\simeq & {} 0.117 \pm 0.057. \end{aligned}$$
(4.2)

It has long been known that the LEP excess can be explained by a light singlet-like Higgs boson in the NMSSM. At this stage, a naturally occurring question is whether this singlet-like Higgs boson can explain the CMS diphoton excess as well.

If both excesses were arisen due to the light singlet-like Higgs boson s, the signal strengths can be expressed by the effective couplings in Eq. (2.14) as follows,

$$\begin{aligned} \mu _{\mathrm{CMS}}= & {} \frac{\sigma (pp\rightarrow s)}{\sigma (pp \rightarrow \phi _{96})} \frac{\mathrm{Br}(s \rightarrow \gamma \gamma ) }{\mathrm{Br}(\phi _{96}\rightarrow \gamma \gamma )} \nonumber \\\simeq & {} \frac{|\Delta C^s_g|^2 |\Delta C^s_\gamma |^2}{0.42 |C^s_b|^2 + 0.05|C^s_t|^2}, \end{aligned}$$
(4.3)
$$\begin{aligned} \mu _{\mathrm{LEP}}= & {} \frac{\sigma (e^+e^-\rightarrow Zs)}{\sigma (e^+e^- \rightarrow Z\phi _{96})} \frac{\mathrm{Br}(s \rightarrow b{\bar{b}}) }{\mathrm{Br}(\phi _{96}\rightarrow b{\bar{b}})}\nonumber \\\simeq & {} \frac{|C^s_V|^2|C^s_b|^2}{0.89|C^s_b|^2+0.11|C^s_t|^2}, \end{aligned}$$
(4.4)

assuming that the CP-odd singlet scalar and the singlino are heavy enough so that 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

$$\begin{aligned} \begin{aligned} C_V^s&= s_{\theta _1} c_{\theta _2} s_{\theta _3} + s_{\theta _2} c_{\theta _3} , \\ C_t^s&= s_{\theta _1} c_{\theta _2} s_{\theta _3} + s_{\theta _2} c_{\theta _3} - c_{\theta _1} s_{\theta _3} \cot \beta ,\\ C_b^s&= C_\tau ^s = s_{\theta _1} c_{\theta _2} s_{\theta _3} + s_{\theta _2} c_{\theta _3} + c_{\theta _1} s_{\theta _3} \tan \beta , \end{aligned} \end{aligned}$$
(4.5)

and those to gluons and photons read

$$\begin{aligned} \begin{aligned} \Delta C_g^s&\simeq 1.02 C_t^s - 0.08 C_b^s ,\\ \Delta C_\gamma ^s&\simeq 0.23 C_t^s - 0.96 C_V^s + \frac{r}{6}U_{33}, \end{aligned} \end{aligned}$$
(4.6)

including the contribution from the loops of charged higgsinos.

Fig. 3
figure 3

Mixing angles to explain the LEP and CMS excesses for different values of r, continued from Fig. 2. Each color shows how the viable region of mixing parameter space is modified when the indicated constraint is imposed

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Before performing a numerical analysis, we present approximate analytic relations between mixing angles holding if s is responsible for the LEP and CMS excesses. The LEP signal rate given in Eq. (4.4) is approximated by

$$\begin{aligned} \mu _{\mathrm{LEP}} \approx s^2_{\theta _2}, \end{aligned}$$
(4.7)

for small \(\theta _2\), and thus the LEP excess is explained if \(s^2_{\theta _2} \sim 0.1\). On the other hand, the ratio between the CMS and LEP signal rates is given by

$$\begin{aligned} \frac{\mu _{\mathrm{CMS}}}{\mu _{\mathrm{LEP}}}\approx & {} \Big ( 1- k_{32} (1.1\cot \beta + 0.1\tan \beta ) \Big )^2 \nonumber \\&\times \left( \frac{ 1 + 0.32 k_{32} \cot \beta -0.24s^{-1}_{\theta _2} r }{ 1+k_{32} \tan \beta } \right) ^2, \end{aligned}$$
(4.8)

where \(k_{32}\equiv s_{\theta _3}/s_{\theta _2}\), and it should be around 6 to account for the LEP and CMS excesses. Here the first factor of the right-hand side represents the Higgs production dominated by gluon fusion, while the second one concerns the branching ratio for the diphoton mode. Note that both effects are enhanced if \(k_{32}\) is negative, and the charged higgsinos can further increase the branching ratio into photons for \(s_{\theta _2}<0\). These features help to understand the numerical analysis given below.

Let us explain in detail how to search the viable region of mixing angles in the parameter scan. The signal rates of h and s are functions of the mixing angles, a combination of \(\lambda \) and \(\mu \), and \(\tan \beta \):

$$\begin{aligned} \mu ^{ii}_h= & {} \mu ^{ii}_h(\theta _1,\theta _2,r,\tan \beta ), \nonumber \\ \mu _{\mathrm{LEP,\,CMS}}= & {} \mu _\mathrm{LEP,\,CMS}(\theta _1,\theta _2,\theta _3,r,\tan \beta ), \end{aligned}$$
(4.9)

for \(r\equiv \lambda v/|\mu |\), with \(m_h=125\) GeV and \(m_s=96\) GeV. On the other hand, the relations (2.11) exhibit how the parameters \(\lambda \), \(\mu \) and \(m_0\) change with the mixing angles

$$\begin{aligned} \lambda= & {} \lambda (\theta _1,\theta _2,\theta _3,m_H,\tan \beta ), \nonumber \\ \mu= & {} \mu (\theta _1,\theta _2,\theta _3,m_H,\tan \beta ), \nonumber \\ m_0= & {} m_0(\theta _1,\theta _2,\theta _3,m_H,\tan \beta ), \end{aligned}$$
(4.10)

where we have taken \(\epsilon =0\) in Eq. (2.12) since it is negligibly small in most of the parameter space of our interest. It is obvious that \(m_H\) is determined by \(\theta _{1,2,3}\) and \(\tan \beta \) once we fix 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 \(\mu _{\mathrm{CMS}}\).

Fig. 4
figure 4

Region of \((\mu ,m_H)\) compatible with the LEP and CMS excesses for \(\tan \beta =2\) (left) and 5 (right), respectively. Here we have taken r smaller than 1.1 for \(\lambda \) below the perturbativity bound. In each panel, the yellow band is excluded by the LEP results on chargino searches and \(m_0 < 115\) GeV in the lighter red shaded region. Note that the \(m_0\) cut is important for \(\tan \beta \lesssim 3\)

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In Fig. 2, the Higgs signal rates \(\mu ^{ii}_h\) are consistent with the measurements in the shaded region, and the excesses at 96 GeV are explained in the red shaded region. Figure 3 shows the region of \(\theta _3\) for the excesses at 96 GeV. The next thing that one has to examine is if the viable region above is consistent also with the various constraints on \(\lambda \), \(\mu \), and \(m_0\). Here we have imposed the conditions

$$\begin{aligned} \lambda < 0.7,\quad |\mu |>104\,\mathrm{GeV},\quad m_0\ge 115\,\mathrm{GeV}, \end{aligned}$$
(4.11)

as required by the perturbativity up to the GUT scale, the LEP limit on the chargino mass, and the radiative contribution to the Higgs mass from stops above TeV, respectively, provided that stop mixing is not too large as would be the case in the conventional mediation models of SUSY breaking.Footnote 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 \(1.5\le \tan \beta \le 15\).Footnote 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.