Phenomenology of the inflationinspired NMSSM at the electroweak scale
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Abstract
The concept of Higgs inflation can be elegantly incorporated in the NexttoMinimal Supersymmetric Standard Model (NMSSM). A linear combination of the two Higgsdoublet fields plays the role of the inflaton which is nonminimally coupled to gravity. This nonminimal coupling appears in the lowenergy effective superpotential and changes the phenomenology at the electroweak scale. While the field content of the inflationinspired model is the same as in the NMSSM, there is another contribution to the \(\mu \) term in addition to the vacuum expectation value of the singlet. We explore this extended parameter space and point out scenarios with phenomenological differences compared to the pure NMSSM. A special focus is set on the electroweak vacuum stability and the parameter dependence of the Higgs and neutralino sectors. We highlight regions which yield a SMlike \(125\,\hbox {GeV}\) Higgs boson compatible with the experimental observations and are in accordance with the limits from searches for additional Higgs bosons. Finally, we study the impact of the nonminimal coupling to gravity on the Higgs mixing and in turn on the decays of the Higgs bosons in this model.
1 Introduction
In the history of our universe, there has been a period in which the size of the universe exponentially increased. This short period is known as inflationary epoch, and many models have been developed in order to explain the inflation of the early universe. Unfortunately, most of these models of inflation cannot be tested directly in the laboratory; the observation of the universe is the only discriminator to disfavor or support such models. Therefore, testing the phenomenology of a particle physics model of inflation at the electroweak scale with colliders is of interest both from the point of view of particle physics and cosmology.
One possibility to describe inflation is the extension of a particle physics model by additional scalar fields which drive inflation but are removed from the theory afterwards. A more economical approach is the idea of using the Higgs field of the Standard Model (SM) as inflaton [1, 2, 3]. The simplest version, however, is under tension as it suffers from a finetuning and becomes unnatural [4]. A less minimal version of Higgsportal inflation with an additional complex scalar field can in addition solve further problems of the SM, see Refs. [5, 6]. Also the concept of critical Higgs inflation can raise the range of perturbativity to the Planck scale and solve further problems of the SM, see Refs. [7, 8, 9]. Other solutions are offered by scalefree extensions of the SM. A natural way of such an implementation can be realized in canonical superconformal supergravity (CSS) models as proposed by Refs. [10, 11] based on earlier work by Ref. [12].
The simplest implementation of a superconformal model which can accommodate the nonminimal coupling term \(\chi \,\hat{H}_u \cdot \hat{H}_d\) is the wellknown \(\mathbb {Z}_3\)invariant NMSSM augmented by an additional \(\mu \) term, which we call \(\mu \)extended NMSSM (\(\mu \)NMSSM) in the following. We neglect all additional \(\mathbb {Z}_3\)violating parameters in the superpotential at the tree level (see the discussion below). These terms are not relevant for the physics of inflation: the function X could potentially also contain an \(\hat{S}^2\) term, since it has the same structure as \(\hat{H}_u\cdot \hat{H}_d\) and is allowed by gauge symmetries. However, inflation driven by this term does not lead to the desired properties as pointed out in Ref. [12]. The other term, which is not present in the NMSSM, is a singlet tadpole proportional to \(\hat{S}\) that is not quadratic or bilinear in the chiral superfields and thus would need a dimensionful coupling to supergravity instead of the dimensionless \(\chi \).
In this work, we are going to study the lowenergy electroweak phenomenology of the model outlined in Refs. [10, 11] and Ref. [13], where previously the focus was put on the description of inflation and the superconformal embedding of the NMSSM into supergravity. We have generated a model file for FeynArts [16, 17], where SARAH [18, 19, 20, 21] has been used to generate the treelevel couplings of the \(\mu \)NMSSM, and we have implemented the oneloop counterterms. The loop calculations have been carried out with the help of FormCalc [22] and LoopTools [22]. In order to predict the Higgsboson masses, we have performed a oneloop renormalization of the Higgs sector of the \(\mu \)NMSSM which is compatible with the renormalization schemes that have been employed in Refs. [23, 24] for the cases of the MSSMand NMSSM, respectively. This allowed us to add the leading MSSMlike twoloop corrections which are implemented in FeynHiggs [25, 26, 27, 28, 29, 30, 31, 32] in order to achieve a stateoftheart prediction for the Higgs masses and mixing. The parameter space is checked for compatibility with the experimental searches for additional Higgs bosons using HiggsBounds version 5.1.0beta [33, 34, 35, 36, 37] and with the experimental observation of the SMlike Higgs boson via HiggsSignals version 2.1.0beta [38]. In addition, we check the electroweak vacuum for its stability under quantum tunneling to a nonstandard global minimum and for tachyonic Higgs states in the treelevel spectrum. Finally, we investigate some typical scenarios and study their collider phenomenology at the Large Hadron Collider (LHC) and a future electronpositron collider. For this purpose in some analyses we use SusHi [39, 40] for the calculation of neutral Higgsboson production crosssections. We emphasize the possibility of light \({\mathcal {CP}}\)even singlets in the spectrum with masses below \(100\,\text {GeV}\) that could be of interest in view of slight excesses observed in the existing data of the Large Electron–Positron collider (LEP) [41] and the Compact Muon Solenoid (CMS) [42] which are compatible with bounds from A Toroidal LHC ApparatuS (ATLAS) [43]. For one scenario that differs substantially from the usual NMSSM, we exemplarily discuss the total decay widths and branching ratios of the three lightest Higgs bosons and their dependence on the additional parameters of the \(\mu \)NMSSM.
The paper is organized as follows: we start with a description of our model and the theoretical framework in Sect. 2 by discussing analytically the phenomenological differences of the Higgs potential in the \(\mu \)NMSSM compared to the \(\mathbb {Z}_3\)invariant NMSSM. We study vacuum stability and the incorporation of higherorder corrections for the Higgs boson masses. Then, we derive the trilinear selfcouplings of the Higgs bosons and comment on the remaining sectors of the model which are affected by the additional \(\mu \) term. In Sect. 3, we focus on the parameter space of interest and investigate the Higgsboson masses as well as the stability of the electroweak vacuum numerically and also show the neutralino spectrum. Furthermore, we study the effect of the additional \(\mu \) parameter on Higgsboson production and decays. Lastly, we conclude in Sect. 4. In the Appendix we present the beta functions for the superpotential and some softbreaking parameters of the general NMSSM (GNMSSM) [44, 45, 46] including all \(\mathbb {Z}_3\)breaking terms.
2 Theoretical framework
In this section we introduce the model under consideration, the \(\mu \)NMSSM, which differs by an additional \(\mu \) term from the scaleinvariant NMSSM. We derive the Higgs potential and investigate vacuum stability and the prediction for the Higgsboson masses of the model. Furthermore, we discuss the trilinear selfcouplings of the Higgs bosons and comment on the electroweakinos – i.e. charginos and neutralinos – as well as on the sfermion sector. We constrain our analytical investigations in this section mostly to treelevel relations. Higherorder contributions, e.g. for the Higgsboson masses, are explained generically and are evaluated numerically in the subsequent phenomenological section.
2.1 Model description
In order to avoid the cosmological gravitino problem [48], where the light gravitino dark matter overcloses the universe [49, 50], one has to control the reheating temperature in order to keep the production rate of the light gravitinos low [51]. This potential problem may affect the model under consideration for gravitino masses in the range from MeV to GeV; it disappears for much heavier gravitinos (\(\mathord {\gtrsim }\,10\,\text {TeV}\)). In the latter case the inflationary \(\mu \) term would dominate over the NMSSMlike \(\mu _{\text {eff}}\) and drive the higgsino masses to very high values (unless \(\mu _{\text {eff}}\) is tuned such that the sum \((\mu +\mu _{\text {eff}})\) remains small). For gravitino masses \(m_{3/2} > 1 \,\text {GeV}\) it affects Big Bang Nucleosynthesis via photodeconstruction of light elements, see Ref. [48]. As discussed in Ref. [11], in the \(\mu \)NMSSM there is no strict constraint on the reheating temperature \(T_R\). We note that a reheating temperature below \(T_R \lesssim 10^8\)–\(10^9\,\text {GeV}\), as advocated in Ref. [52], avoids the gravitino problem. The rough estimate of \(m_{3/2} \sim 10\,\text {MeV}\) even needs \(T_R \lesssim 10^5\,\text {GeV}\) in order to not overclose the universe with thermally produced gravitinos after inflation [53, 54, 55, 56]. Interestingly, such low reheating temperatures preserve highscale global minima after inflation, see Ref. [57], and disfavor the preparation of the universe in a metastable state after the end of inflation [58]. In any case, the reheating temperature at the end of inflation is very model dependent and rather concerns the inflationary physics. A study to estimate the reheating temperature \(T_R\) is given in Ref. [59]. Therein, a relation is drawn between the decay width of the inflaton and \(T_R\). Interestingly, if we naïvely assume that this width at the end of inflation is equal to the SMlike Higgs width \(\Gamma _h \approx 4 \times 10^{3}\,\text {GeV}\), we can estimate a rather low reheating temperature \(T_R \sim \sqrt{\Gamma _h M_{\text {Pl}}} \approx 10^7\,\text {GeV}\) with the Planck mass \(M_{\text {Pl}}\approx 2.4 \times 10^{18}\,\text {GeV}\). For our studies below we assume that a reheating temperature as low as \(T_R\lesssim 10^9\,\text {GeV}\) can be achieved even with large couplings.
Because the \(\mathbb {Z}_3\) symmetry is broken (which avoids the typical domainwall problem of the NMSSM [60]), another symmetry at the high scale is required in order to solve the tadpole problem [61, 62, 63, 64, 65, 66]: without such a symmetry, Planckscale corrections could possibly induce large contributions to the tadpole term [67]. The superconformal embedding of the \(\mu \)NMSSM, where the \(\mu \) term is generated from the Kähler potential, serves as this symmetry. As pointed out in Ref. [67], other possibilities consist of discrete or continuous nongauge symmetries, socalled R symmetries. Imposing discrete \(\mathbb {Z}_4\) or \(\mathbb {Z}_8\) R symmetries as proposed in Refs. [45, 68, 69] provide a viable solution, since dimensionful linear and bilinear terms are forbidden as long as the symmetry is not broken.^{3}
Contrary to studies in the GNMSSM (see Refs. [44, 45, 46, 73]), where the MSSMlike \(\mu \) term can be easily shifted away and absorbed in a redefinition of the other parameters – especially the tadpole contribution – we cannot do so in the inflationinspired \(\mu \)NMSSM. First of all, the \(\mu \) term is introduced via the R symmetrybreaking nonminimal coupling to supergravity only. The other parameters in the singlet sector are not supposed to be generated by this breaking. Secondly, by redefining the parameters, we would introduce a tadpole term and shift the effect simply there. Note that the authors of Ref. [45] perform this shift in order to eliminate the linear (i.e. tadpole) term in the superpotential and keep \(\mu \), while others (e.g. Ref. [74]) shift the \(\mu \) term to zero and keep the tadpole and bilinear terms for the singlet in the superpotential. As discussed above, in the \(\mu \)NMSSM considered in this paper due to the superconformal symmetry breaking at the Planck scale solely the \(\mathbb {Z}_3\)breaking \(\mu \) term is present.
2.2 Higgs potential
 Higgs doublets

The massmatrix elements of the doublet fields in the upperleft \(\left( 2\times 2\right) \) block matrices of Eqs. (17a)–(17b) contain the abbreviation \(a_1^\prime \). From Eq. (20a) it is apparent that they are determined by SM parameters and \(m_{H^\pm }\), \(\lambda \) and \(t_\beta \) like in the NMSSM. Neglecting the mixing between the doublet and singlet sector, the mass of the light \({\mathcal {CP}}\)even doublet state has an upper bound of \(m_Z^2\,c^2_{2\beta }+\lambda ^2\,v^2\,s^2_{2\beta }\). In the limit \(m_{H^\pm }\gg m_Z\), the other two doublet fields decouple and obtain a mass close to \(m_{H^\pm }\). Smaller values of \(m_{H^\pm }\) increase the mixing of both \({\mathcal {CP}}\)even doublet fields. Also \(t_\beta \) needs to be close to one for large doublet mixing.
 Higgs singlets

The \(\left( 3,3\right) \) elements of \(\mathcal {M}_S\) and \(\mathcal {M}_P\) in Eqs. (17a) and (17b) set the mass scale of the Higgs singlets. They contain the terms \(a_4^\prime \) from Eq. (20c), \(a_5\) from Eq. (14e), and \(a_7\) from Eq. (14g). All \(\mathbb {Z}_3\)violating parameters besides \(\mu \) and \(B_\mu \) appear in these terms; in our later analysis we set these parameters besides \(\mu \) and \(B_\mu \) to zero, but for completeness we mention them in the following discussion of this section.
The parameter \(A_\kappa \) appears only in the term \(a_5\), whereas \(B_\nu \) only appears in \(a_7\). Thus it is obvious that the diagonal massmatrix elements for the singlet fields – and therefore their masses – can be controlled by these two quantities, without changing any other matrix element. If all \(\mathbb {Z}_3\)violating parameters except \(\mu \) and \(B_\mu \) were set to zero, we would rediscover the NMSSMspecific feature that \(A_\kappa \) is bound from below and above to avoid tachyonic singlet states at the tree level.
The ratio \(\kappa /\lambda \) which appears in both terms, \(a_5\) and \(a_7\), has sizable impact on the mass scale of the singlets. If \(\kappa \ll \lambda \) the \({\mathcal {CP}}\)even singlet entry is purely controlled by \(a_4^\prime \), which in turn is proportional to \(1/\mu _{\text {eff}}\); in the same limit, the \({\mathcal {CP}}\)odd singlet entry is controlled by \(a_4^\prime \) and the remainder of \(a_7\) which is \(B_\nu \,\nu \). Also note that \(a_4^\prime \) contains a term which is linear in \(\mu \). In the opposite case \(\kappa \gtrsim \lambda \), the term \(a_5\) is likely to dominate the \(\left( 3,3\right) \) matrix element for the \({\mathcal {CP}}\)even singlet due to the suppression of \(a_4^\prime \) by \(\mu _{\text {eff}}\) if it is of the order of a few \(100\,\hbox {GeV}\). The term \(a_5\) is proportional to \((\kappa /\lambda )^2\,\mu _{\text {eff}}^2\), such that the \({\mathcal {CP}}\)even singlet exhibits a strong dependence on \(\mu _{\text {eff}}\). On the other hand for \(\mu \gtrsim \mu _{\text {eff}}\), the term \(a_4'\) can balance the large \(\kappa \)enhanced contribution in \(a_5\); thus, possible upper bounds on \(\kappa \) as derived in Ref. [75] might be evaded.
For the case of the \({\mathcal {CP}}\)odd singlet, the terms in \(a_5\) and \(a_7\) that are quadratic in \(\mu _{\text {eff}}\) cancel each other. Then the size of the other parameters (especially \(A_\kappa \), \(\mu \) and \(\mu _{\text {eff}}\)) determines which contribution is dominant. For moderate values of \(\kappa \approx \lambda \gtrsim 0.1\) together with small \(A_\kappa \) the \({\mathcal {CP}}\)odd singlet develops a dependence on \(\mu /\mu _{\text {eff}}\), as we will discuss later. Lastly, we note that in the case of \(\kappa \gg \lambda \) and \(A_\kappa \ne 0\,\text {GeV}\) the \({\mathcal {CP}}\)even and \({\mathcal {CP}}\)odd singlet masses are controlled through \((\kappa /\lambda )^2\,\mu _{\text {eff}}^2\) and \((\kappa /\lambda )\,\mu _{\text {eff}}\,A_\kappa \), respectively. Later, this will allow us to present a rescaling procedure that keeps both singlet masses constant over a large parameter range.
 Doublet–singlet mixing

The masses of the doubletlike and the singletlike Higgs states can be significantly shifted by mixing between both sectors. The relevant matrix elements are the ones in the third columns of Eqs. (17a) and (17b). They contain the abbreviations \(a_2\), \(a_3^\prime \) and \(a_6^\prime \), see Eqs. (14b), (20b) and (20d), respectively. The mixing vanishes in the limit \(\lambda \rightarrow 0\) with constant \(\kappa /\lambda \), and it is enhanced for larger values of \(\lambda \). For fixed \(\lambda \) it is also strongly enhanced in the limit \(\mu _{\text {eff}}\rightarrow 0\,\text {GeV}\).
In the \({\mathcal {CP}}\)even sector, two terms contribute to the doublet–singlet mixing: \(a_2\) which depends on the sum \((\mu +\mu _{\text {eff}})\), and \(a_3^\prime \) which does not directly depend on \(\mu \), but only on the softbreaking term \(B_\mu \,\mu \). In the case of large \(\mu \) and \(\mu _{\text {eff}}\) of the same sign, \(a_2\) often dominates the mixing with the lighter doublet, eventually yielding a tachyonic singlet or doublet Higgs; this behavior can be avoided by choosing a proper value for \(B_\mu \) (or \(\xi \)) to cancel the large effect in \(a_2\) by \(a_3^\prime \). In the case of similar \(\mu \) and \(\mu _{\text {eff}}\) of opposite signs, \(a_3^\prime \) will always dominate the mixing. Again, the mixing strength can be adjusted by setting \(B_\mu \) (or \(\xi \)).
The doublet–singlet mixing in the \({\mathcal {CP}}\)odd sector contains only one term \(a_6^\prime \) which is similar to \(a_3^\prime \) with opposite sign. Furthermore, the \({\mathcal {CP}}\)odd mixing elements can be modified by nonzero \(\xi \) and \(\nu \). As indicated above, due to the dependences of \(a_3^\prime \) and \(a_6^\prime \) on \(1/\mu _{\text {eff}}\), a small \(\mu _{\text {eff}}\ll 100\,\hbox {GeV}\) yields a strong mixing between singlets and doublets.
2.3 Vacuum structure and vacuum stability bounds
The space of model parameters can be constrained using experimental exclusion limits and theoretical bounds. Those constraints can be applied to rule out certain parts of the parameter space. In this context, constraints from the stability of the electroweak vacuum appear to be very robust and theoretically well motivated. It has already been noticed in the early times of supersymmetry that constraints from the electroweak vacuum stability on the trilinear soft SUSYbreaking parameters can be important [76, 77, 78, 79, 80, 81, 82, 83, 84]. Recently they have been rediscussed in light of the Higgs discovery [85, 86, 87, 88, 89]. These constraints are usually associated with nonvanishing vacuum expectation values of sfermion fields (e.g. staus or stops) and thus known under the phrase “charge and colorbreaking minima”. Such minima can invalidate the electroweak vacuum and therefore lead to unphysical parameter configurations (see below).
However, the existence of charge and colorbreaking minima is only a necessary condition for the destabilization of the electroweak vacuum. Clearly one has to compare the value of the potential at this new minimum with the desired electroweak one, and only if the nonstandard vacuum is deeper the corresponding scenario is potentially excluded. In fact, some of the points with a deeper nonstandard vacuum may be valid when accepting metastable vacua under the condition that the transition time from the local electroweak vacuum to the global true vacuum appears to be longer than the age of the universe [90]. However, the possibility of the existence of metastable vacua is of limited practical relevance for our analysis: typically only parameter points in close neighborhood to the stable region are affected by such considerations; wellbeyond the boundary region, the false vacua become rather shortlived and thus are strictly excluded. In addition, there are thermal corrections in the early universe which give a sizable and positive contribution to the effective potential as the oneloop corrections are proportional to \(m^2(\phi )\,T^2\) for the fielddependent masses \(m(\phi )\). For finite temperature, they shift the ground state to the symmetric phase around \(\phi = 0\,\text {GeV}\) [91, 92]. We presume, however, that our inflationary scenario preselects a vacuum at field values different from zero and, thanks to the relatively low reheating temperatures in our scenario, gets caught in it, see Ref. [57]. Following the inflationary scenario of Ref. [11], the trajectory in field space lies at \(\beta = \pi /4\) with \(h_u^2 = h_d^2 = h^2\) and \(s = 0\,\text {GeV}\); the presence of the singlet field S is needed for the stabilization of the inflationary trajectory in order to not fall into the tachyonic direction as pointed out by Refs. [11, 13]. Inflation ends at field values \(h = \mathcal {O}(0.01)\) in units of the Planck mass. For small \(\lambda \sim 10^{2}\), the Dflat trajectory remains stable after inflation ends according to Ref. [11], and will change to \(\beta \ne \pi /4\) and \(s \ne 0\,\text {GeV}\) when the SUSYbreaking terms become important. NMSSMspecific effects like the relevance of singlet Higgs bosons and the additional contribution to the \(125\,\text {GeV}\) Higgs boson are usually connected to a large value of \(\lambda \). This is not necessarily the case in the \(\mu \)NMSSM, where striking differences also appear for small values of \(\mu _{\text {eff}}\). Moreover, we will take it as a working assumption that after inflation ends, even for larger values of \(\lambda \) the universe will remain in the state with the inflationary field direction until it settles down in a minimum closest to this direction. If it is the global minimum of the zerotemperature potential, reheating may not be sufficient to overcome the barrier and to select a false (and maybe metastable) vacuum. The thermal history of the universe plays then no role for the choice of the vacuum, and in this case the universe would remain in the global minimum. Accordingly, we adopt the prescription to exclude all points with a global minimum that does not coincide with the electroweak vacuum. This means that we do not consider metastable electroweak vacua as they are excluded by the selection rule. A similar discussion and argument has been given in Ref. [93], where a selection of the vacuum with the largest expectation values was promoted, irrespective whether or not it is the global minimum of the theory.
We will see that actually in most cases scenarios are excluded because of a tachyonic Higgs mass. Tachyonic masses are related to the fact that the electroweak point – around which the potential is expanded – is not a local minimum in the scalar potential, but rather resembles a saddle point or even local maximum, and the true vacuum lies at a deeper point along this tachyonic direction. Thus, the true vacuum has vevs different from the input values, and the electroweak breaking condition \(\mathcal {T}_S = \mathbf {0}\) in Eq. (15a) does not select a minimum.
The “desired” electroweak vacuum can be constructed by fulfilling the minimization conditions at the tree level, \(\mathcal {T}_S = \mathbf {0}\), with \(\mathcal {T}_S\) given by Eq. (15a). The vevs of the doublet fields are taken as fixed input parameters, whereas the value of \(\mu _{\text {eff}}\) is treated as variable similar to \(\mu \). These equations can be solved for the softbreaking masses \(m_{H_u}^2\), \(m_{H_d}^2\) and \(m_S^2\) according to Eq. (16).
In our analysis, we focus for clarity on constraints from the treelevel potential, considering the appearance of global nonstandard minima and, as discussed above, disregarding the possibility of metastable false vacua. Employing higherorder (i.e. oneloop) corrections does not necessarily give more accurate predictions of vacuum stability, see Ref. [103]. An approach to include oneloop effects using a certain numerical procedure has been implemented in the public code collection of Vevacious, see Ref. [104], including a tunneling calculation also at finite temperature using CosmoTransitions [105]. The treelevel evaluation is much faster and numerically more stable; moreover, it has been argued that the oneloop effective potential is problematic for tunneling rate calculations [106].
The second constraint, on \(A_\lambda \), follows from a nontachyonic charged Higgs mass, since a tachyonic mass (\(m^2 < 0\,\text {GeV}^2\) ) means that the potential has negative curvature at this stationary point derived by the minimization conditions. Thus, the true vacuum would have some nonzero vev for a charged Higgs component. Configurations like this are possible in the NMSSM, whereas they do not exist as global or local minima in the MSSM [82]. From the (treelevel) charged Higgs mass in Eq. (18), we get an indirect bound on \(A_\lambda \). Taking \(m_{H^\pm }\) as input value, we can eliminate \(A_\lambda \) as free parameter, see Eq. (19). Hence, we can ensure that \(m_{H^\pm }^2\) is always positive. Still, it is worth noticing that by this procedure \(A_\lambda \) gets strongly enhanced for small \(\mu _{\text {eff}}\) (compared to \(m_{H^\pm }\)) and thus drives tachyonic neutral Higgs bosons.
Constraints from CCB minima as given in Eq. (24), are less important in comparison to the MSSM for both, the NMSSM and the \(\mu \)NMSSM, even if large stop corrections are needed to shift the SMlike Higgs mass (as in the case for small \(\lambda \)). If the singletfield direction were neglected and the stop Dflat direction \({\tilde{t}}_R = {\tilde{t}}_L = \tilde{t}\) defined, one could directly apply Eq. (24) for the \(\mu \)NMSSM, keeping \(v_s \ne 0\,\text {GeV}\) and replacing \(\mu \rightarrow \mu + \mu _{\text {eff}}\). However, with the singlet as dynamical degree of freedom, the stability of the electroweak vacuum is improved as the only singlet–stop contribution is actually a quadrilinear term \(\lambda \,h_d\,s\,{{\tilde{t}}}^2\) and the occurrence of a true vacuum with \(\langle h_{u,d} \rangle \ne v_{u,d}\), \(\langle s \rangle \ne v_s\) and \(\langle {\tilde{t}} \rangle \ne 0\,\text {GeV}\) is disfavored.
Metastability and tunneling rates Lastly, we comment on vacuumtovacuum transitions in case of a local electroweak vacuum. It is in general of interest to see how long such a metastable state could survive compared with the lifetime of the universe. We have outlined some arguments why – in view of the inflationary history of the universe – we disregard metastable longlived vacua. We will see in Sect. 3.3 that totally stable points survive in a wide range of the parameter space.
For an estimate of the bounce action of the unstable configuration [112], we define an effectively singlefield scalar potential linearly interpolating between the electroweak local minimum and the true vacuum found by the numerical minimization of the scalar potential at different field values and apply an exact solution of the quartic potential given by Ref. [113]. See also Ref. [114] for the application of this method to the \(\mu \)NMSSM.
2.4 Higherorder corrections to Higgsboson masses and mixing
It is wellknown that perturbative corrections beyond the tree level alter the Higgs masses and mixing significantly in supersymmetric models. For instance, in the MSSM such large corrections are needed to lift the lightest \({\mathcal {CP}}\)even Higgs mass beyond the Zboson mass. On the other hand, in the NMSSM and similarly the \(\mu \)NMSSM there are scenarios where an additional treelevel term lowers the tension between the treelevel SMlike Higgs mass and the measured value of the SMlike Higgs boson at \(125\,\hbox {GeV}\). Still, since loop corrections to the Higgs spectrum have a large impact, in our phenomenological analysis we take into account contributions of higher order as described in the following.
In this work, a model file for FeynArts [16, 17] of the GNMSSM at the tree level has been generated with the help of SARAH [18, 19, 20, 21]. In addition, the oneloop counterterms for all vertices and propagators have been implemented, and a renormalization scheme which is consistent with Refs. [23, 24] for the cases of the MSSM and NMSSM has been set up. All \(\mathbb {Z}_3\)violating parameters are renormalized in the \(\overline{\text {DR}}\) scheme, see Appendix A for a list of the respective beta functions. The numerical input values of all \(\overline{\text {DR}}\)renormalized parameters are understood to be given at a renormalization scale which equals the topquark pole mass. The renormalized selfenergies of the Higgs bosons \({\hat{{\varvec{\Sigma }}}}_{hh}\) are evaluated with the help of FormCalc [22] and LoopTools [22] by taking into account the full contributions from the GNMSSM at the oneloop order. For other variations of the NMSSM, similar calculations of Higgsmass contributions up to the twoloop order have been performed in Refs. [118, 119, 120, 121, 122, 123, 124, 125, 126]. A comparison of results from public codes using different renormalization schemes can be found in Refs. [127, 128].
2.5 Trilinear Higgsboson selfcouplings
In order to discuss possible distinctions between the NMSSM and the \(\mu \)NMSSM, the Higgsboson selfcouplings are particularly relevant. Experimentally these selfcouplings can be probed through Higgs pair production or through decays of a heavier Higgs boson to two lighter ones. Through electroweak symmetry breaking there is also a strong correlation with Higgsboson decays into Higgs bosons and gauge bosons, e.g. \(A^0\rightarrow Zh^0\) or \(H^0\rightarrow Za_s\). For both, the Higgs mixing between singlets and doublets is essential. We take both types of decays into account when checking against experimental limits from Higgs boson searches, but only exemplify the parameter dependence for the decays involving only Higgs bosons in our numerical analysis below.
2.6 Neutralino and chargino masses
For the case where \(\kappa \) and \(\lambda \) are kept fixed, an interesting behavior can be observed for light higgsinos. For small \((\mu + \mu _{\text {eff}})\) huge cancellations may occur between the two contributions with large \(\mu > 0\,\text {GeV}\) and \(\mu _{\text {eff}}\) of the same size but opposite sign. As a consequence, the singlino state becomes much heavier compared to the case of the NMSSM (of the order of \(\mu _{\text {eff}}\)). Such a scenario is displayed in Fig. 1 where the neutralino–chargino spectrum is shown for the cases \(\mu \in \{0,200,1000\}\) GeV (\(\nu \) is set equal to zero). The left column with \(\mu =0\,\text {GeV}\) corresponds to the case of the NMSSM. The masses are obtained by diagonalizing the treelevel mass matrices in Eq. (35). With respect to the \(\mathbb {Z}_3\)invariant NMSSM, the most significant alteration is visible in the singlino component (blue): the mass shows an aboutlinear increase with \(\mu \) since the sum \((\mu +\mu _{\text {eff}})\) is kept fixed. Due to the varying mixing, some influence on the masses of the other two neutral higgsino states (orange) can be seen despite a constant higgsino mass parameter \((\mu +\mu _{\text {eff}})\); the impact on the gaugino states (red and purple) remains negligible. The chargino masses (rose) are not influenced by the different choices.
In a scenario as discussed above, with light higgsinos as well as large \(\mu \) and \(\mu _{\text {eff}}\) of opposite signs, the lightest neutralino is typically not the singlino state as the singlino mass is pushed up, see Fig. 1. The lightest supersymmetric particle (LSP), however, tends to be the gravitino, which is at risk to overclose the universe as dark matter candidate. In this case, the inflationary scenario has to be such that the reheating temperature stays below a certain value and gravitinos are not overproduced in the early universe, see our discussion in Sect. 2.1.
2.7 Sfermion masses
Therein we denote the fermion mass by \(m_f\), the bilinear softbreaking parameters by \(m_{\tilde{f}_{{\text {L,R}}}}\), the trilinear softbreaking parameter by \(A_f\), and the electric and weak charges by \(Q_f\) and \(T^{(3)}_f\).
In this sector we encounter the sum \((\mu +\mu _{\text {eff}})\) in the offdiagonal elements of the sfermion mass matrices as the only difference compared to the NMSSM or MSSM. If this sum becomes large, \(A_f/\theta _f\) needs to be adjusted in order to avoid tachyonic sfermions in particular for the third generation squarks. In that case, bounds from vacuum stability (see e.g. Eq. (24)) can also constrain the viable size of \(\left( \mu +\mu _{\text {eff}}\right) \).
3 Phenomenological analysis
In this section we investigate various scenarios of the \(\mu \)NMSSM with a particular focus on the \(\mu \) parameter. We will point out differences between the \(\mu \)NMSSM and the ordinary \(\mathbb {Z}_3\)preserving NMSSM, where the latter corresponds to the limit \(\mu = 0\,\text {GeV}\) of the \(\mu \)NMSSM. At first we qualitatively define the investigated scenarios, before we numerically analyze them.
3.1 Viable parameter space compatible with theoretical and experimental bounds
In the previous sections we have analytically discussed the relevant sectors of the \(\mu \)NMSSM with respect to effects of the inflationinspired \(\mu \) parameter. Before we provide a phenomenological analysis – including the higherorder effects specified in Sect. 2.4 – we discuss the viability of various parameter regions. As discussed in Sect. 2.1 we focus on scenarios with nonzero \(\mu \) and \(B_\mu \), but set all other \(\mathbb {Z}_3\)violating parameters in the superpotential (9) and softbreaking Lagrangian (10), i.e. \(\xi \), \(C_\xi \), \(\nu \) and \(B_\nu \), equal to zero.
The \(\mu \) parameter of the model is positive by construction in the inflationinspired model, see Eqs. (6) and (8). Furthermore, we only investigate scenarios with \(\mu \lesssim 2\,\hbox {TeV}\) to stay in the phenomenologically interesting region for the collider studies. Still, we point out that also much larger scales are viable from the inflationary point of view. As discussed in Sect. 2.1, \(\mu \simeq \frac{3}{2} \, m_{3/2} \, 10^5 \, \lambda \) implies that much larger values of \(\mu \) are possible depending on \(\lambda \) and \(m_{3/2}\). However, large values of \(\mu \) can cause tachyonic states as discussed in Sect. 2.2.
 small \({\varvec{\mu \simeq 1}}\,{{\mathbf {GeV}}}\)

in the case of small \(\mu \) also the softbreaking term \(B_\mu \,\mu \) becomes small. Since in addition we set all other \(\mathbb {Z}_3\)violating parameters to zero, we recover the standard NMSSM in this limit (see the discussion in Fig. 1). Thus, differences between the NMSSM and the \(\mu \)NMSSM can directly be deduced by comparing scenarios with zero and nonzero \(\mu \) parameter.
 large \({\varvec{\mu \pmb {\gtrsim } 1}}\,{\mathbf {TeV}}\) with

\({\varvec{\mu _{\text {eff}}\simeq \mu }}\) as discussed in Sect. 2.6, the higgsino masses depend only on the sum \(\left( \mu +\mu _{\text {eff}}\right) \) at the tree level. The same combination contributes to the sfermion mixing in combination with the trilinear soft SUSYbreaking terms. In order to keep these quantities small at a large value of \(\mu \), one can assign the same value with opposite sign to \(\mu _{\text {eff}}\); note, however, that the region \(\mu +\mu _{\text {eff}}\lesssim 100\,\text {GeV}\) is experimentally excluded by direct searches for charginos [151, 152]. An immediate consequence of large, opposite sign \(\mu _{\text {eff}}\) and \(\mu \) is that the singlino and the singletlike Higgs states receive large masses of the order of \(\mu _{\text {eff}}\) [see the (5, 5) entry in Eq. (35a) and the (3, 3) elements in Eqs. (17a) and (17b)], which provides a potential distinction from the standard NMSSM. Similar to the increase of the singlino mass, fixing \(\left( \mu +\mu _{\text {eff}}\right) \) together with an increase in \(\mu \) – and thus an increase in the absolute value of \(\mu _{\text {eff}}\) – lifts the masses of the singlet states also in the Higgs sector. In the neutralino sector these contributions can be absorbed by a rescaling of \(\kappa \), see Sect. 2.6; however, in the Higgs sector \(\mu _{\text {eff}}\) also appears in other combinations, thus leaving traces which can potentially distinguish the \(\mu \)NMSSM from the NMSSM.
 \({\varvec{\mu \,\pmb {\mathord {\gtrsim }\,} 100}}\,{\mathbf {GeV}}\) with

\({\varvec{\pmb {}\mu _{\text {eff}}\pmb {}\,\pmb {\mathord {\lesssim }\,}100}}\,{\mathbf {GeV}}\) if we allow for a large \(\mu \) parameter without constraining the sum \(\left( \mu +\mu _{\text {eff}}\right) \), the spectra of higgsinos, sfermions and Higgs bosons are changed at the same time. A large sum \(\left( \mu +\mu _{\text {eff}}\right) \) causes very large mixing between the singlet and doublet sectors (see discussion in Sect. 2.2), eventually driving one Higgs state tachyonic. In some part of the parameter space this can be avoided by tuning \(B_\mu \) accordingly. Another constraint arises from the sfermion sector, most notably the sbottoms and staus: a large \(\left( \mu +\mu _{\text {eff}}\right) \) induces large terms in the offdiagonal elements of the sfermion mass matrices (enhanced by \(\tan \beta \) for the case of downtype sfermions) which can potentially cause tachyons, also depending on the values of the trilinear softbreaking parameters \(A_f\). As discussed in Sect. 2.3, constraints from charge and colorbreaking minima induced by too large softbreaking trilinear parameters (see Eq. (24) with \(\mu \) promoted to \(\left( \mu +\mu _{\text {eff}}\right) \)), have a much smaller impact in the \(\mu \)NMSSM as compared to the MSSM [108].
A special case of this scenario is the possibility of having \(\mu \) at the electroweak scale in combination with an almost vanishing \(\mu _{\text {eff}}\ll \mu \). This implies that \(\left( \mu +\mu _{\text {eff}}\right) \) remains at the electroweak scale. In contrast to the standard NMSSM this scenario allows the occurrence of both, \(\kappa \gg \lambda \) and a light singlet sector. As discussed in Sect. 2.2, the mixing between singlets and doublets is in this case dominated by terms proportional to \(\mu _{\text {eff}}^{1}\). We will explicitly discuss such a scenario in Sect. 3.5.
The input parameters which are fixed throughout our numerical analysis (interpreted as onshell parameters if not specified otherwise). The gaugino mass parameters are denoted as \(M_i\) with \(i=1,2,3\). The trilinear softbreaking terms for the sfermions \(A_{f_g}\) carry the generation index \(g=1,2,3\). The charged Higgs mass \(m_{H^\pm }\) is fixed to the shown value, if not mentioned otherwise
\(m_{H^\pm } =800\,\text {GeV},\)  \(m_t = 173.2\,\text {GeV},\)  \(\alpha _s (m_Z) = 0.118,\) 
\(G_F=1.16637\cdot 10^{5}\,\text {GeV}^{2},\)  \(m_Z = 91.1876\,\text {GeV},\)  \(m_W = 80.385\,\text {GeV},\) 
\(M_3 = 2.5\,\text {TeV},\)  \(M_2 = 0.5\,\text {TeV}, \)  \(M_1 = \frac{5}{3}\frac{g_1^2}{g_2^2}M_2,\) 
\(m_{\tilde{f}_{\text {L}}} = 2\,\text {TeV},\)  \(m_{\tilde{f}_{\text {R}}} = 2\,\text {TeV},\)  \(A_{f_3} = 4\,\text {TeV}, \quad A_{f_{1,2}} = 0\,\text {TeV}\) 
There are more parameters that are relevant for the following phenomenological studies. We keep those fixed which behave similarly as in the MSSM and NMSSM. The choice of our constant input values is given in Table 1. Furthermore, we specify the values of \(t_\beta \), \(\kappa \), \(\lambda \), and \(A_\kappa \) directly at the respective places. Besides the analyses where we explicitly study the dependence on \(B_\mu \), we use \(B_\mu = 0\,\text {GeV}\) as default value.
As our analysis is focused on the impact of the inflation model, we are not going to discuss the influence of the sfermion parameters. If not mentioned otherwise, we use \(m_{{\tilde{f}}} \equiv m_{{\tilde{f}}_L} = m_{{\tilde{f}}_R}\) and \(A_{f_3} / m_{{\tilde{f}}} = 2\), which maximizes the prediction for the SMlike Higgsboson mass at \(\mu +\mu _{\text {eff}}=0\,\text {GeV}\). The gluino mass parameter \(M_3\) is set well above the squark masses of the third generation which is in accordance with the existing LHC bounds. For completeness, we also give the parameters of the SM which are most relevant for our numerical study in Table 1.
We have chosen \(m_{H^\pm }\) as an input parameter and adjust \(A_\lambda \) according to Eq. (19). If not denoted otherwise, we set \(m_{H^\pm } = 800\,\text {GeV}\). We use HiggsBounds version 5.1.0beta [33, 34, 35, 36, 37] in order to implement the constraints on the parameter space of each of our scenarios resulting from the search limits for additional Higgs bosons. In this context, the exclusion limits from \(H,A \rightarrow \tau \tau \) decays are particularly important. For relatively low values of \(\tan \beta \) the choice of \(m_{H^\pm }=800\,\text {GeV}\) is well compatible with these bounds. The code HiggsBounds determines for each parameter point the most sensitive channel and evaluates whether the parameter point is excluded at the \(95\%\) confidence level (C.L.). We use those exclusion bounds as a hard cut in the parameter spaces of our analyses.
We also indicate the regions of the parameter space which provide a Higgs boson that is compatible with the observed state at \(125\,\hbox {GeV}\). These regions are obtained with the help of HiggsSignals version 2.1.0beta [38]. The code HiggsSignals evaluates a total \(\chi ^2\) value, obtained as a sum of the \(\chi ^2\) values for each of the 85 implemented observables. Four more observables are added, which test the compatibility of the predicted Higgsboson mass with the observed value of \(125\,\hbox {GeV}\). This latter test includes a theoretical uncertainty on the predicted Higgsboson mass of about \(3\,\hbox {GeV}\), such that a certain deviation from the four measured mass values (from the two channels with either a \(\gamma \gamma \) or a \(ZZ^{(*)}\) final state from both experiments ATLAS and CMS) is acceptable. Thus, in total HiggsSignals tests 89 observables.
Since all our twodimensional figures include a region with a SMlike Higgs boson,^{9} we classify the compatibility with the observed state as follows: we determine the minimal value of \(\chi ^2\), denoted by \(\chi _m^2\), in the twodimensional plane and then calculate the deviation \(\Delta \chi ^2=\chi ^2\chi _m^2\) from the minimal value in each parameter point. We allow for a maximal deviation of \(\Delta \chi ^2 < 5.99\), which corresponds to the \(95\%\) C.L. region in the Gaussian limit. All parameter points that fall in this region \(\Delta \chi ^2<5.99\) are considered to successfully describe the observed SMlike Higgs boson.
Lastly, we note that HiggsBounds and HiggsSignals are operated through an effectivecoupling input. We will comment on the results of the two codes where appropriate.
For our implementation of the constraints from the electroweak vacuum stability we refer to Sect. 2.3. For informative reasons, we distinguish longlived vacua from shortlived ones in the numerical analysis. We do not explicitly enforce a perturbativity bound on \(\kappa \) and \(\lambda \), but discuss this issue below.
3.2 Higgsboson and neutralino mass spectra
In this section, we point out the differences of the Higgsboson and neutralino mass spectra in the \(\mu \)NMSSM with respect to the NMSSM. Similar to the case of the MSSM, the charged and the \({\mathcal {CP}}\)even heavy doublet as well as the MSSMlike \({\mathcal {CP}}\)odd Higgs bosons are (for sufficiently large \(m_{H^\pm }\gg M_Z\)) quasidegenerate.
In Fig. 2, we show the masses of the Higgs bosons for vanishing \(A_\kappa \) in the left, \(A_\kappa = 100\,\text {GeV}\) in the middle frame, and the masses of the neutralinos in the right frame. Each frame contains three different scenarios which are characterized by the three values \(\mu \in \{0, 200, 1000\}\,\text {GeV}\) while keeping all other parameters fixed: \(\mu + \mu _{\text {eff}}= 200\,\text {GeV}\), \(t_\beta =3.5\), \(\lambda =0.2\), \(\kappa =0.2\,\lambda \), and the other parameters as given in Table 1. The additional \(\mu \) term has the biggest influence on the singletlike states \(s^0\) and \(a_s\), as well as the singlinolike state \(\tilde{S}^0\). In analogy to the discussion in Fig. 1, the reason for this behavior is the fixed sum \(\left( \mu +\mu _{\text {eff}}\right) \): an increase in \(\mu \) causes a larger negative \(\mu _{\text {eff}}\) which primarily drives the singletmass terms in the (3, 3) elements of Eqs. (17a) and (17b), and the singlinomass term in the (5, 5) element of Eq. (35a) to large values. In the investigated parameter region, the mass of the \({\mathcal {CP}}\)odd singlet is also very sensitive to \(A_\kappa \): in order to avoid a tachyonic state \(a_s\) over a large fraction of the parameter space, it is essential to keep \(A_\kappa \) sufficiently large. However, in the left frame a scenario is shown where even a vanishing \(A_\kappa \) is possible. It generates a rather light \({\mathcal {CP}}\)odd singletlike state, whereas a sizable \(A_\kappa = 100\,\text {GeV}\) (middle) lifts this mass up. There is thus the potential for a distinction between the NMSSMlimit for \(\mu = 0\,\text {GeV}\) and the \(\mu \)NMSSM with a large \(\mu = 1\,\text {TeV}\). Note that in the middle frame for \(\mu = 200\,\text {GeV}\), the purple and blue lines are on top of each other.
As already mentioned in Sect. 2.6, the electroweakino sector alone, at least at the tree level, does not allow one to distinguish the \(\mu \)NMSSM from the NMSSM: one can keep the neutralino–chargino spectrum at the tree level invariant by identifying the sum \(\left( \mu +\mu _{\text {eff}}\right) \) with the \(\mu _{\text {eff}}\) term of the NMSSM, and rescaling \(\kappa \) according to Eq. (36). However, as pointed out above, the rescaling does have an impact on the Higgs spectrum. We show in Fig. 3 spectra for \(\mu \in \{0, 200, 1000\}\,\text {GeV}\) and \(A_\kappa \in \{0, 100\}\,\text {GeV}\) with fixed \(\mu +\mu _{\text {eff}}=200\,\text {GeV}\). The neutralino spectrum is shown in only one column in the very right frame. In analogy to Fig. 2, the left and middle frames show the Higgsboson masses for the two values of \(A_\kappa \) where one still can see the effect of a varying \(\mu \) term. While contributions to the mass matrices in Eqs. (17) which are proportional to \(\left( \mu +\mu _{\text {eff}}\right) \) or \(\kappa \,\mu _{\text {eff}}\) are kept constant, other terms \({\propto }\,\mu _{\text {eff}}^{1},\mu _{\text {eff}}^{2}\) induce variations. Accordingly, the singletlike Higgs masses in Fig. 3 are only slightly sensitive to \(\mu \), much less than the changes observed in Fig. 2. A rising \(\mu \) slightly increases the mass splitting between the singletlike and the SMlike Higgs state.
In Fig. 3 only the case \(A_\kappa =100\,\hbox {GeV}\) in combination with \(\mu =0\,\hbox {GeV}\) is allowed by HiggsBounds and HiggsSignals (\(\chi ^2=80.1\)), since the other scenarios are either ruled out by the decay of the SMlike Higgs into a pair of light \({\mathcal {CP}}\)odd singlets or by a too large deviation of the SMlike Higgsboson mass from \(125\,\hbox {GeV}\). In addition to our discussion above, we emphasize that in particular the latter exclusion can be easily avoided through a slight adjustment of the input parameters.
3.3 Parameter scan
In Figs. 4, 5, 6, 7, we present a selection of parameter regions. Before we discuss them individually, their common features are explained. The colored dots in the background display different states of the electroweak vacuum: we distinguish stable (blue), longlived metastable (purple), shortlived metastable (red), and tachyonic (rose). As discussed above, we regard not only tachyonic but also metastable regions as excluded in the context of this inflationary scenario, but nevertheless display long and shortlived metastable regions for illustration. Furthermore, we indicate those points that do not fulfill Eq. (23) and thus have no singlet vev (orange), although, as explained in Sect. 2.3, this constraint is not relevant for the \(\mu \)NMSSM. We overlay mass contours for the SMlike Higgs \(h^0\) (black), the \({\mathcal {CP}}\)even singletlike Higgs \(s^0\) (blue), and the \({\mathcal {CP}}\)odd singletlike Higgs \(a_s\) (red). The spectrum is calculated taking into account the full oneloop and the known MSSMlike twoloop contributions as described in Sect. 2.4. The assignment of the labels \(h^0\), \(s^0\) and \(a_s\) to the loopcorrected states is determined by the largest respective contribution in the mixing matrix \(Z^{{\mathrm{mix}}}_{ij}\). We emphasize again that the parameters of the stop sector specified in Table 1 for the given scale of SUSY masses maximize the SMlike Higgs mass for \(\mu +\mu _{\text {eff}}= 0\,\text {GeV}\); therefore, lower values for the SMlike Higgs mass could easily be obtained by reducing the mixing in the stop sector. Finally, we also indicate a naïve exclusion bound from direct searches for charginos by the grayshaded band: Ref. [152] reports a lower bound on the chargino mass of \(94\,\text {GeV}\) which translates into the requirement that \(\mu +\mu _{\text {eff}}\) must be above that value in the \(\mu \)NMSSM. Lastly, all Figs. 4, 5, 6, 7 show the region of parameter points that successfully passed HiggsBounds and HiggsSignals and thus, in particular, yield a SMlike Higgs boson compatible with the observed state at \(125\,\text {GeV}\). This region is represented through the larger, light green dots in the background. We refer to Sect. 3.1 for our statistical interpretation of the results obtained from the two codes.
A large part of the parameter region that is consistent with the measured SMlike Higgs mass is also in concordance with an absolutely stable electroweak vacuum. Small intersections between stable regions and regimes with tachyonic Higgs states exist, where there are metastable nonstandard vacua. The strongest constraints arise from the existence of tachyonic masses for one of the physical Higgs states at the tree level. In the remaining region only a small fraction of points has a global minimum which does not coincide with the electroweak vacuum whereas the majority has a true electroweak vacuum. For the shortlived metastable regions, the vacuum lifetime is longer than the age of the universe.
In Fig. 4 we indicate the Higgsmass contours and the constraints from vacuum stability in the plane of \(\left( \mu +\mu _{\text {eff}}\right) \) and \(\kappa /\lambda \) with fixed \(\mu \) and \(\lambda \). Note that for this choice of variables the treelevel doublet sector in Eqs. (17a), (17b) and (20a) remains constant; any structure visible in the prediction of the SMlike Higgs mass is thus induced by mixing with the singlet state, or by loop corrections. The chosen parameter values are indicated in the legends of the figures and in Table 1; in the left plot \(A_\kappa =0\,\text {GeV}\) is used, while in the right plot \(A_\kappa =100\,\text {GeV}\). The value of \(A_\kappa \) has an impact in particular on the mass scale of the \({\mathcal {CP}}\)odd singletlike Higgs which is much lighter on the lefthand side. In fact, for a light \({\mathcal {CP}}\)odd singletlike Higgs a parameter region opens up where decays of the SMlike Higgs into a pair of them become kinematically allowed. The \({\mathcal {CP}}\)even singletlike Higgs is also somewhat lighter for \(A_\kappa =0\,\text {GeV}\), while the SMlike Higgs is scarcely affected. The contour lines of the Higgs masses stop when one Higgs becomes tachyonic. The reason why this does not exactly coincide with the border between the blue and pink dotted regions are the loop corrections to the Higgs spectrum while the constraints from vacuum stability were investigated at the tree level. It can be seen that the boundaries at the left of the stable region are parallel to one of the displayed Higgsmass contours – the corresponding particle becomes tachyonic at this boundary. The boundary to the right of the stable region can be understood when comparing the right plots of Figs. 4 and 5, which differ from each other by the value of \(\mu \): in the right plot of Fig. 5 a contour for the SMlike Higgs mass which is parallel to the tachyonic border appears around \(\mu +\mu _{\text {eff}}=250\,\text {GeV}\) and \(\kappa /\lambda =0.5\). In Fig. 4 such a contour is not visible as this particular parameter region is excluded by a tachyonic SMlike state at the tree level. Note that the NMSSM and \(\mu \)NMSSMspecific oneloop contributions to the Higgs spectrum are particularly large in that region (about \(60\,\text {GeV}\) additional shift compared to the same scenario in the MSSMlimit with \(\lambda \rightarrow 0\) and \(\kappa /\lambda \) constant), see also Ref. [124]; a dedicated analysis taking into account twoloop effects beyond the MSSMlimit might be necessary for a robust prediction of the Higgs mass close to the right border of the stable region, see e.g. Ref. [123]. It should be noted that in Fig. 4 the region where the Higgs mass is close to the right border of the stable region is disfavored by the limits from chargino searches at LEP.
As expected, the region allowed by HiggsBounds and HiggsSignals is a subset of the region where the SMlike Higgs has a mass in the vicinity of \(125\,\hbox {GeV}\). In the greenmarked region, \(\Delta \chi ^2\) is at maximum 5.99. The minimal value \(\chi _m^2\) from HiggsSignals is 74.6 in both figures. One can see on the lefthand side of Fig. 4 that this region is split into two: in between the two regions the SMlike Higgs can decay into a pair of \({\mathcal {CP}}\)odd singletlike Higgs bosons \(h^0 \rightarrow a_sa_s\) with a branching ratio of up to \(90\,\%\); this behavior is not compatible with the observed signal strengths implying a limit on decays of the state at \(125\,\text {GeV}\) into nonSM particles. For a very light \({\mathcal {CP}}\)odd singlet, the admixture between the SMlike Higgs and the \({\mathcal {CP}}\)even singlet component is reduced, since the latter becomes heavier in this region. In the scenario under consideration, the decay \(h^0 \rightarrow a_sa_s\) is dominated by the coupling among the two singlet states, \(\lambda _{355}\) in Eq. (31q), such that a reduced admixture between \(h^0\) and \(s^0\) also closes the decay \(h^0 \rightarrow a_sa_s\). This is why – despite the very light \({\mathcal {CP}}\)odd Higgs \(a_s\) – the region at \(\mu +\mu _{\text {eff}}\simeq 300\,\hbox {GeV}\) and \(\kappa /\lambda \simeq 0.4\) is allowed by the constraints from both HiggsSignals and HiggsBounds.
In Fig. 5 we present scenarios similar to the righthand side of Fig. 4 with \(A_\kappa = 100 \, \text {GeV}\), but with different values of \(\mu \) (note the larger scale at the xaxis). Thus, the influence of this parameter that distinguishes the \(\mu \)NMSSM from the NMSSM can be seen directly. Obviously, the parameter region with a stable vacuum is enlarged: for a given value \((\mu +\mu _{\text {eff}})\) the tachyonic border moves to smaller ratios of \(\kappa /\lambda \) as \(\mu \) increases. Concerning the Higgs spectrum, the most notable difference is seen for the SMlike Higgs mass: for \(\mu = 1\,\text {TeV}\) a turning point at about \(\mu +\mu _{\text {eff}}=800\,\text {GeV}\) is visible, which moves to smaller values of \(\kappa /\lambda \) for \(\mu = 1.5\,\text {TeV}\). For the larger value of \(\mu \) one can see that the possibility emerges for scenarios with the correct SMlike Higgs mass but positive \((\mu +\mu _{\text {eff}})\). Again all tested points which yield a SMlike Higgs boson close to \(125\,\hbox {GeV}\) successfully pass the constraints implemented in HiggsBounds and HiggsSignals. The minimal values of \(\chi _m^2\) from HiggsSignals are 74.9 and 74.6 on the lefthand and on the righthand side of Fig. 5, respectively.
Figure 6 shows scenarios with larger \(\tan \beta \) and smaller \(\lambda \) compared to the previous figures. Like in Fig. 4 we set \(A_\kappa =0\,\hbox {GeV}\) on the left, and \(A_\kappa =100\,\hbox {GeV}\) on the righthand side, but \(\mu = 1\,\text {TeV}\) is used. We observe again that a larger value of \(A_\kappa \) widens the allowed parameter region, because the mass of the \({\mathcal {CP}}\)odd singlet is lifted up, giving rise to a drastic effect in this case. In fact, for \(A_\kappa =0\,\hbox {GeV}\) only a rather small area in the plane of \((\mu +\mu _{\text {eff}})\) and \(\kappa /\lambda \) is allowed, while the allowed region is very significantly enhanced for \(A_\kappa =100\,\hbox {GeV}\). In the plot on the righthand side one can see a (nearly) closed \(125\,\text {GeV}\) contour for the mass of the SMlike Higgs with even larger values in the enclosed area. Adjusting the parameters of the stop sector in order to obtain a smaller contribution to the SMlike Higgs mass can render a SMlike Higgs with a mass of about \(125\,\text {GeV}\) in the whole enclosed region. Close to the tachyonic borders we find larger regions with a longlived metastable vacuum (purple) than in Figs. 4 and 5. However, in this part of the plot the prediction for the mass of the SMlike Higgs is below the experimental value. On the righthand side of Fig. 6 a large region is allowed by the constraints from HiggsBounds and HiggsSignals. Only low values of \(\mu +\mu _{\text {eff}}<m_h/2\) are excluded by HiggsSignals due to the decay of the SMlike Higgs boson into a pair of higgsinos. However, this region is anyhow not compatible with the LEP bound on light charginos. The minimal values of \(\chi _m^2\) from HiggsSignals are 74.7 in both plots.
In Fig. 7 we change the parameter on the yaxis: \(B_\mu \) is varied and \(\kappa \) is kept fixed. We set \(A_\kappa =0\,\text {GeV}\) on the lefthand side, and \(A_\kappa =100\,\hbox {GeV}\) on the righthand side. One can see that nonzero values for \(B_\mu \) can have a significant impact on the predicted Higgs masses and might determine whether or not a scenario is excluded. For larger negative values of \(B_\mu \), one can see an area where the electroweak vacuum is metastable and longlived, while the area in the lower left corner of the plots indicates that the electroweak vacuum is unstable and shortlived. The effect of a larger \(A_\kappa \) mainly lifts the tachyonic boundary at the top so that values of \(B_\mu = 1\,\text {TeV}\) are allowed for \(A_\kappa = 100\,\text {GeV}\) and leaves the other regions invariant. However, towards the upper limit of \(B_\mu \), there is a small shortlived area. As a new feature, we find large regions with a metastable vacuum but a SMlike Higgs with a mass of \(125\,\text {GeV}\) for both values of \(A_\kappa \). Accordingly, scenarios with too large negative values of \(B_\mu \) are excluded due to a rapidly decaying vacuum despite providing a SMlike Higgs boson close to the observed mass. The constraints from HiggsBounds and HiggsSignals indicate that a large part of the region with the correct Higgs mass is compatible with the experimental data. For both plots HiggsSignals yields a minimal value of \(\chi _m^2 = 74.9\). Only in those scenarios where the decay channel \(h^0 \rightarrow a_sa_s\) is kinematically allowed – which happens in the plot for \(A_\kappa = 0\,\text {GeV}\) for \(\mu + \mu _{\text {eff}}\gtrsim 300\,\text {GeV}\) and \(\mu + \mu _{\text {eff}}\lesssim 700\,\text {GeV}\) – the parameter region is incompatible with the data on the detected Higgs boson.
We briefly summarize the observed features and give an outlook for the phenomenological studies in the following. The allowed parameter region is mainly constrained by configurations where one Higgs field is tachyonic at the tree level. It can be seen that the tachyonic boundaries follow the Higgs mass contours in the Figs. 4, 5, 6, 7; in addition, there are effects from \(\mu _{\text {eff}}^{1}\) terms as discussed in Sect. 2.2 which enhance the doublet–singlet mixing and eventually cause tachyons. This feature can be observed towards the right end of the Figs. 4, 5, 6. The experimental limits and constraints confine the allowed regions further around the region where the SMlike Higgs has a mass of about \(125\,\text {GeV}\) and exclude parameter regions where for instance the decay of the SMlike Higgs into a pair of light \({\mathcal {CP}}\)odd singlets has a large branching ratio. In this context, the singlet sector has a significant impact on the features discussed in Figs. 4, 5, 6, 7.
In the NMSSM, one usually expects to find the phenomenologically most interesting regions (accommodating a \(125\,\text {GeV}\) Higgs) for rather large values of \(\lambda \gtrsim 0.1\), since the NMSSM contribution to the SMlike Higgs mass at the tree level is enhanced. In addition, large \(\lambda \) enhances the doublet–singlet mixing. However, in the \(\mu \)NMSSM, there is another way to obtain a large doublet–singlet mixing also for small values of \(\lambda \): this is the region of low \(\mu _{\text {eff}}\) where terms proportional to \(\mu _{\text {eff}}^{1}\) become large, as discussed in Sect. 3.1. We will investigate this class of scenarios, which are not possible in the NMSSM but generic to the \(\mu \)NMSSM, in Sect. 3.5 in more detail.
Similar to the NMSSM, the chosen value of \(A_\kappa \) has a strong influence on the singletlike Higgs masses, which is relevant for the tachyonic regions. In a large part of the viable parameter space the relation \({\text {sign}}{(A_\kappa )} = {\text {sign}}{(\mu _{\text {eff}})}\) applies, where for \(A_\kappa = 0\,\text {GeV}\) both signs of \(\mu _{\text {eff}}\) are allowed in general. This dependence on the relative signs of \(A_\kappa \) and \(\mu _{\text {eff}}\) can be derived from the discussion in Sect. 2.2 about the Higgs singlets and especially the functional dependence of \(a_5\) in Eq. (14e) versus \(a_4'\) in Eq. (20c): large negative values of the sum \((a_4' + a_5)\) drive the \({\mathcal {CP}}\)even singlet tachyonic. In the investigated scenarios above, which have either \(A_\kappa = 0\,\text {GeV}\) or \(A_\kappa = 100\,\text {GeV}\), the sign of \(\mu _{\text {eff}}\) is negative in most of the viable parameter space. Accordingly, there is a preference for negative \((\mu + \mu _{\text {eff}})\). The allowed region with small positive values occurs where the negative value of \(\mu _{\text {eff}}\) is overcompensated by the positive value of \(\mu \). In Sect. 3.5 we will investigate a scenario where we keep \((\mu + \mu _{\text {eff}})\) fixed at a positive value, while for \(A_\kappa \) small negative and small positive values are used for \(\mu _{\text {eff}}> 0\,\text {GeV}\) and \(\mu _{\text {eff}}< 0\,\text {GeV}\), respectively. There we will also discuss the dependence of the singlet masses on \(\mu \) and \(\mu _{\text {eff}}\) in more detail.
3.4 Higgsboson and electroweakino production
Scenarios that yield a light \({\mathcal {CP}}\)even singletlike Higgs boson. The Higgs boson at about \(125\,\hbox {GeV}\) is SMlike. All other parameters are chosen in accordance to Table 1
Scenario  1  2  3  4 

\(\lambda \)  0.08  0.08  0.28  0.08 
\(\kappa \)  0.04  0.023  0.08  0.0085 
\(\tan \beta \)  12  12  2.5  2 
\((\mu +\mu _{\text {eff}})\,\hbox {[GeV]}\)  \(\)140  \(\)140  \(\)300  \(\)400 
\(\mu \,\hbox {[GeV]}\)  5  195  5  150 
\(B_\mu \,\hbox {[GeV]}\)  0  0  0  \(\)300 
\(m_{H^\pm }\,\hbox {[GeV]}\)  800  800  800  1000 
\(A_\kappa \,\hbox {[GeV]}\)  130  265  250  32 
\(A_{f}\,\hbox {[GeV]}\)  400  450  3200  4000 
\(m_{s^0}\,\hbox {[GeV]}\)  97.6  95.7  97.2  97.1 
\(m_{h^0}\,\hbox {[GeV]}\)  124.7  126.8  124.6  125.0 
\(m_{a^s}\,\hbox {[GeV]}\)  168.2  277.0  257.2  75.6 
\(\frac{\sigma {\left( e^+e^\rightarrow Zs^0\right) }\cdot \text {BR}{\left( s^0\rightarrow b\bar{b}\right) }}{\sigma ^{{\text {SM}}}{\left( e^+e^\rightarrow ZH\right) }\cdot \text {BR}^{{\text {SM}}}{\left( H\rightarrow b\bar{b}\right) }}\)  0.28  0.31  0.14  0.35 
\(\sigma {\left( gg\rightarrow s^0\right) }\,\hbox {[pb]}\)  25.3  28.1  14.4  31.5 
BR\({\left( s^0\rightarrow \gamma \gamma \right) }\)  0.0020  0.0016  0.0024  0.0005 
\(\chi ^2\)(HiggsSignals)  97  96  82  101 
In most parts of our numerical study, we make use of the approximation of SMnormalized effective couplings of a Higgs boson to gluons – calculated at leading order – which we insert into HiggsBounds for the evaluation of the Higgsproduction crosssections for the neutral Higgs bosons at the LHC. This treatment should be sufficiently accurate for determining the allowed regions in our scans over the parameter space. In the following, however, we will investigate to what extent the \(\mu \)NMSSM can accommodate the slight excesses in the data over the background expectation at a mass around 95–\(98\,\text {GeV}\) that have been reported recently by CMS [42] in the \(\gamma \gamma \) channel^{12} and earlier at LEP [41] in the \(b\bar{b}\) channel. For this purpose we use more sophisticated predictions for the Higgsproduction crosssections in order to compare with the experimental results. We obtain those predictions from SusHi [39, 40, 161, 162, 163, 164, 165, 166, 167], for which a dedicated version for the NMSSM exists [168]. The predictions include \(\hbox {N}^{3}\hbox {LO}\) QCD corrections for the topquark contribution of the light \({\mathcal {CP}}\)even Higgs bosons, while we have neglected contributions from heavy squarks and gluinos beyond the resummed contributions in the bottomYukawa coupling.
Crosssections for electroweakinos at an electron–positron collider for Scenario 1 defined in Table 2
Scenario 1  \({\tilde{\chi }}^0_1\)  \({\tilde{\chi }}^0_2\)  \({\tilde{\chi }}^0_3\)  \({\tilde{\chi }}_1^\pm \)  
Masses [GeV]  127.3  138.3  155.9  138.4  
\(\sigma (e^+e^\rightarrow {\tilde{\chi }}_i{\tilde{\chi }}_j)\) [fb] for \(\sqrt{s}=350\,\hbox {GeV}\)  \({\tilde{\chi }}^0_1{\tilde{\chi }}^0_2\)  \({\tilde{\chi }}^0_1{\tilde{\chi }}^0_3\)  \({\tilde{\chi }}^0_2{\tilde{\chi }}^0_3\)  \({\tilde{\chi }}^0_2{\tilde{\chi }}^0_2\)  \({\tilde{\chi }}^+_1{\tilde{\chi }}^_1\) 
Unpolarized  141  195  0.08  0.19  795 
Pol\((e^+,e^)=(+30\%,80\%)\)  208  287  0.12  0.28  1620 
Pol\((e^+,e^)=(30\%,+80\%)\)  142  196  0.08  0.19  352 
\(\sigma (e^+e^\rightarrow {\tilde{\chi }}_i{\tilde{\chi }}_j)\) [fb] for \(\sqrt{s}=500\,\hbox {GeV}\)  \({\tilde{\chi }}^0_1{\tilde{\chi }}^0_2\)  \({\tilde{\chi }}^0_1{\tilde{\chi }}^0_3\)  \({\tilde{\chi }}^0_2{\tilde{\chi }}^0_3\)  \({\tilde{\chi }}^0_2{\tilde{\chi }}^0_2\)  \({\tilde{\chi }}^+_1{\tilde{\chi }}^_1\) 
Unpolarized  74  109  0.12  0.22  459 
Pol\((e^+,e^)=(+30\%,80\%)\)  110  161  0.19  0.32  926 
Pol\((e^+,e^)=(30\%,+80\%)\)  75  110  0.13  0.22  212 
Of course, a large admixture of \(s^0\) with the SMlike Higgs boson in turn has an impact on the SMlike Higgs properties, visible through the increase in \(\chi ^2\) returned by HiggsSignals. In fact, from the listed scenarios only Scenario 3 with \(\chi ^2=82\) is compatible with the SMlike Higgs boson at the 95% C.L. The other scenarios have \(\chi ^2\) values outside of the 95% C.L. region, as they have a slightly larger mixing of the singlet state with the SMlike Higgs boson. The enhanced mixing increases the \(s^0\) crosssections, but on the other hand yields reduced relative couplings to fermions and gauge bosons for the SMlike Higgs boson \(h^0\). It is thus apparent that explaining the excesses through a singlet state that only couples to SM particles through its admixture with the SMlike Higgs boson is under a certain tension from the measured SMlike Higgsboson properties for both the \(\mu \)NMSSM and the NMSSM, if one requires signal rates that fully saturate the amount of deviation from the SM indicated by the excesses observed by LEP and CMS.
Scenarios with light singletlike Higgs bosons tend to have a light singlino. For Scenario 1 we provide the light electroweakino spectrum, i.e. the masses of \({\tilde{\chi }}^0_{1,2,3}\) and \({\tilde{\chi }}^\pm _1\), in Table 3. Due to \(\mu +\mu _{\text {eff}}=140\,\hbox {GeV}\) the scenario has light higgsinolike states, whereas the gauginos are close in mass to \(M_1=239\,\hbox {GeV}\) and \(M_2=500\,\hbox {GeV}\). The higgsinolike states are strongly admixed with the singlino, e.g. the singlinofraction of \({\tilde{\chi }}^0_2\) is 59%, the singlinofraction of \(\tilde{\chi }^0_3\) is 40%. It is apparent that the three lightest neutralinos and the light chargino are very close to each other in mass. At the LHC, ATLAS and CMS have only recently started to probe such compressed mass spectra by dedicated analyses, see e.g. Refs. [171, 172]. In fact, an electron–positron collider may be required to ultimately probe scenarios of this kind, see for instance Ref. [173] tackling such compressed higgsinolike scenarios at the International Linear Collider (ILC). For Scenario 1 we provide the crosssections for the two centerofmass energies \(\sqrt{s}=350\,\hbox {GeV}\) and \(\sqrt{s}=500\,\hbox {GeV}\), which are considered for Higgsboson and topquark precision studies at the ILC [174], in Table 3. Although in this scenario the LSP is the gravitino, the lightest neutralino \({\tilde{\chi }}_1^0\) has a lifetime of a few milliseconds such that it only gives rise to a missingenergy signature. Besides the possibility to tag \(e^+e^\rightarrow {\tilde{\chi }}_1^0{\tilde{\chi }}_1^0\) through initialstate radiation (ISR), the production of one or more heavier neutralinos or charginos results in detectable SM particles. The possibility to polarize the initial state is an important tool to enhance the signaltobackground ratio, and allows one to minimize systematic uncertainties. This capability is mandatory for performing precision measurements. In Table 3 we provide results for three different polarizations: an unpolarized initial state (as reference only), and polarizations of \(\pm 80\)% and \(\mp 30\)% for the initialstate electron and positron, respectively. Such polarizations are foreseen in the current baseline design of the ILC. As one can see from Table 3, polarized beams with \(\hbox {Pol}(e^+,e^)=(+30\%,80\%)\), corresponding to the socalled effective polarization [175] \(\hbox {Pol}_{\text {eff}}=89\,\%\), enhance the production crosssections of \({\tilde{\chi }}^0_1{\tilde{\chi }}^0_2\) and \({\tilde{\chi }}^0_1{\tilde{\chi }}^0_3\) by about a factor 1.5 as well as the one of \({\tilde{\chi }}^+_1{\tilde{\chi }}^_1\) by about a factor 2. The fact that the production crosssections of \({\tilde{\chi }}^0_2{\tilde{\chi }}^0_2\) and \({\tilde{\chi }}^0_2{\tilde{\chi }}^0_3\) are significantly smaller than the other quoted crosssections is due to a cancellation between the higgsino components \(\tilde{h}_d^0\) and \(\tilde{h}_u^0\).
As discussed above, the electroweakino spectrum of the \(\mu \)NMSSM is a priori indistinguishable from the NMSSM if one restricts the analysis to information from the electroweakino sector and employs treelevel predictions, see Sect. 3.2. Previous studies of the electroweakino sector, see e.g. Ref. [173] and Refs. [176, 177], discussed the ILC capabilities for distinguishing the MSSM from the NMSSM electroweakino sector. From such studies one can infer that a determination of the parameters of the electroweakino sector with an accuracy at the percent level is possible using the measurements of the light electroweakino masses and the corresponding production crosssections for different polarizations, see e.g. Refs. [178, 179] and references therein. This holds even if only the lightest electroweakinos are accessible. Based on earlier comprehensive studies where similar rates as in the scenarios of Table 3 were considered, the input parameters of the corresponding sector can be extracted: as an example, the values of \(M_1\) and \(M_2\) can be determined from the measurement of light gauginos, or the value of \((\mu +\mu _{\text {eff}})\) from the measurement of light higgsinos. In this regard beam polarization plays a crucial role: it allows one to even resolve scenarios where only a few light particles are kinematically accessible. Furthermore, the clean environment at an electron–positron collider allows the application of an ISR method [173] to detect and precisely measure scenarios where the light spectrum is close together in mass, as it is the case for instance for the compressed electroweakino spectrum in Scenario 1 leading to very soft decay characteristics. Complementing the particle spectrum via measuring additional heavier electroweakino masses and parts of the scalar and colored sector at the LHC would allow global fits of the model parameters, so that a model distinction between the \(\mu \)NMSSM, NMSSM and the MSSM might be feasible.
3.5 Higgsboson mixing and decays
For the discussion of the dependence of the masses of the light singlet states on \(\mu \) and \(\mu _{\text {eff}}\) we choose a scenario based on Table 1 and fix in addition \(\lambda =1/4\), \(\kappa =1/5\), \(\tan \beta =4\), \(A_\kappa =7\,\text {GeV}\). We vary \((\mu +\mu _{\text {eff}})\) between \(600\) and \(450\,\hbox {GeV}\) and \(\mu \) between 1 and \(1000\,\hbox {GeV}\). The lower end of the range of \(\mu \) values corresponds to the NMSSMlimit, \(\mu \rightarrow 0\,\text {GeV}\). The results are depicted in Fig. 8, where we show the masses of the three lightest neutral Higgs bosons as a function of \(\mu \) and \((\mu +\mu _{\text {eff}})\). The background colors indicate the constraints from vacuum stability and from the experimental results on the Higgs sector using the same color coding as in Figs. 4, 5, 6, 7. It is apparent that with increasing \(\mu \) the range in \((\mu +\mu _{\text {eff}})\) that is allowed by the constraints from vacuum stability is also increasing, which is in accordance to our observations in the previous section. Since the mixing between the light \({\mathcal {CP}}\)even doublet state and the \({\mathcal {CP}}\)even singletlike state is large in the parts of the displayed parameter plane where the masses of the two lightest \({\mathcal {CP}}\)even states are close to each other, we label the \({\mathcal {CP}}\)even states in Fig. 8 as the mass eigenstates \(h_1^0\) and \(h_2^0\) rather than as \(h^0\) and \(s^0\). However, for large values of \(\mu \) the two lightest \({\mathcal {CP}}\)even Higgs bosons are sufficiently separated in mass so that the light \({\mathcal {CP}}\)even doublet state \(h_1^0\) can be identified with \(h^0\) and exhibits only a mild dependence on \(\mu \), since such a dependence is only induced through the mixing with the singlet at treelevel. For smaller values \(200\,\text {GeV}\lesssim \mu \lesssim 300\,\hbox {GeV}\) the contour lines of \(m_{h_1^0}\) become very dense, and the mass of the singletlike state \(h_2^0\) approaches values below \(150\,\text {GeV}\), implying a large mixing between the \({\mathcal {CP}}\)even light doublet state and the singlet state. With further decreasing \(\mu \) the mass eigenstates \(h_1^0\) and \(h_2^0\) flip their role, i.e. \(h_1^0\) is singletlike, and \(h_2^0\) corresponds to the SMlike doublet state in this region.
We now focus on the region of large \(\mu \) where both states can be clearly separated: the region with a SMlike Higgs mass of \(m_{h_1^0}\sim 125\,\text {GeV}\) is strongly affected by the values of \(\lambda \) and/or \(\tan \beta \) through their impact on the NMSSMlike treelevel contribution to the doublet states, see the quantity \(a'_1\) in Eq. (20). In addition, it is wellknown that this state receives large radiative corrections that depend on the mass splitting in the stop sector, which is proportional to \(X_t = A_t  (\mu + \mu _{\text {eff}}) / \tan \beta \). As discussed above, we have chosen \(A_t\) in such a way that the contribution is maximized at \(\mu + \mu _{\text {eff}}= 0\,\hbox {GeV}\) and thus decreases to both directions. This behavior is visible in Fig. 8 for the contours displaying the mass of \(h_1^0\) at values of \(\mu \gtrsim 300\,\hbox {GeV}\). The behavior of the singletlike states is important for the phenomenology: as explained in Sect. 2.2, for a scenario with \(\kappa \sim \lambda \) the mass of the \({\mathcal {CP}}\)even singlet \(s^0\) is mainly controlled by \(a_5\), see Eq. (14e), and therefore proportional to \(\mu _{\text {eff}}\). As in our scenario \(A_\kappa \) is small, the mass of the \({\mathcal {CP}}\)odd singlet \(a_s\) is dominated by \(a_4^\prime \), see Eq. (20c), and therefore proportional to \(\sqrt{\mu /\mu _{\text {eff}}}\). Those mass dependences of the singletlike \(m_{h_2^0} \gtrsim 150\, \text {GeV}\) (blue) and \(m_{a_s}\) (red) can be clearly identified in Fig. 8: for \(m_{h_2^0}\), the lines are roughly diagonal, and thus the mass contours follow lines with constant \(\mu _{\text {eff}}\). For the mass of \(a_s\), the dependence on the square root \(\sqrt{\mu /\mu _{\text {eff}}}\) leads to the shape of the contours. We emphasize that the behavior displayed in Fig. 8 is specific to a small value of \(A_\kappa \ll \mu \) or \(\mu _{\text {eff}}\) for \(\kappa \sim \lambda \).
As above, the parameter range allowed by constraints from HiggsBounds and HiggsSignals is indicated by lightgreen dots in the background. For values of \(\mu \gtrsim 300\,\hbox {GeV}\), the light \({\mathcal {CP}}\)even Higgs \(h_1^0\) corresponds to the SMlike state. In this region, the decay \(h_1^0\) into higgsinos forbids low values of \(\mu +\mu _{\text {eff}}\), while the decay \(h_1^0\rightarrow a_sa_s\) is kinematically closed. For \(\mu \lesssim 500\,\text {GeV}\) the mixing between the \({\mathcal {CP}}\)even doublet state and the singlet state becomes larger which is not compatible with the observation of the properties of the SMlike Higgs. The minimal value of \(\chi ^2\) in this figure is \(\chi _m^2 = 76.4\) and thus slightly worse than the scenarios studied in Sect. 3.3. There is also a small allowed region with \(\Delta \chi ^2<5.99\) at low values of \(\mu \), where a SMlike Higgs boson is present. In this region the mass of the singlet state \(s^0\) has crossed the mass of the doublet state \(h^0\), and the states \(h_1^0\) and \(h_2^0\) have changed their character as discussed previously. The doublet–singlet mixing in this case yields a positive contribution to the mass of the SMlike state \(h_2^0\), lifting the treelevel value towards the experimentally allowed mass window (in the allowed region at low values of \(\mu \) the mass of \(h_2^0\) is about \(126\,\text {GeV}\)). In this region the decay \(h_2^0\rightarrow a_sa_s\) is kinematically open, but sufficiently suppressed to be in accordance with experimental observations.
We conclude that the additional \(\mu \) term of the \(\mu \)NMSSM lifts up the \({\mathcal {CP}}\)odd Higgs mass and enlarges the allowed parameter space compared to the NMSSM. Still, in particular due to the large admixture of the singlet and doublet states, such a scenario is difficult to distinguish from the standard NMSSM, if not all Higgs states are fully determined. As a consequence of the strong admixture of the Higgs bosons and the influence of their masses on the kinematics, all decay modes show a nontrivial dependence on the coupling structure. The decay rates of the heavy Higgs bosons \(H^0\) and \(A^0\) into any combination of the three light Higgs bosons remain small throughout the parameter plane, i.e. the branching ratios are below \(3\%\). The maximal branching ratios for \(A^0\rightarrow h^0 a_s\) and \(A^0\rightarrow s^0 a_s\) are reached at large \(\mu \) and \(\mu _{\text {eff}}\), i.e. in the lower right corner of Fig. 8. The two decays show a different dependence on \(\mu \) and \(\mu _{\text {eff}}\), which is in accordance with our discussion in Sect. 2.5. Whereas \(h_2^0\rightarrow a_s a_s\) is kinematically only allowed for very low \(\mu \) in this scenario, the decay \(h_2^0\rightarrow h_1^0h_1^0\) is – when kinematically open – strongly dependent on \(\mu _{\text {eff}}\). We will demonstrate below the dependence of the different decay modes on \(\mu \) and \(\mu _{\text {eff}}\) in a scenario with essentially fixed Higgsboson masses.
In the standard NMSSM a measurement of the masses of the whole neutralino and neutral Higgs spectrum would fix all free parameters, in particular \(\mu _{\text {eff}}\), \(\lambda \), \(\kappa \) and \(A_\kappa \). With these parameters also the Higgs mixing is completely determined (at the tree level). A small value of \(\lambda \) in any case implies a small mixing between the singlet and doublet states of the Higgs sector. This is not the case in the \(\mu \)NMSSM: we show our results in Figs. 9, 10, 11. As explained above, in the considered parameter region the Higgsboson masses are almost constant, see Fig. 9 on the lefthand side. The two heavy Higgs bosons \(H^0\) and \(A^0\) both have a mass very close to \(800\,\hbox {GeV}\) within a range of \(3\,\hbox {GeV}\). The neutralino masses are constant, in detail \(m_{{\tilde{\chi }}_i^0}=\{134.7,163.9,252.1,320.0,516.1\}\,\text {GeV}\), where the particle with mass \(m_{{\tilde{\chi }}_3^0}=m_{{\tilde{s}}}=320\,\hbox {GeV}\) corresponds to the singlinolike state with a purity of 99.9%. The two lightest neutralinos are higgsinolike states. Though the mixing in the neutralino sector remains constant, the mixing between the light \({\mathcal {CP}}\)even Higgs boson \(h^0\) and the singlet component \(s^0\) is strongly enhanced for \(\mu _{\text {eff}}\rightarrow 0\,\hbox {GeV}\). We depict the total widths for the three lightest Higgs bosons on the righthand side of Fig. 9. The enhancement of the total width of \(h^0\) for \(\mu _{\text {eff}}\rightarrow 0\,\hbox {GeV}\) is due to the decay \(h^0\rightarrow a_sa_s\). This is also apparent in the left plot of Fig. 10, where the branching ratio for the decay of the SMlike state \(h^0\) into a pair of light \({\mathcal {CP}}\)odd singlets is displayed. For \(s^0\) both the decays into \(h^0h^0\) and \(a_sa_s\) are of relevance, whereas other nonstandard decay modes – e.g. into a pair of higgsinos – have a small rate, see the righthand sides of Figs. 9 and 10. Apart from decays into Higgs bosons, \(s^0\) decays into massive SM gauge bosons. As mentioned above, all of the shown area in Fig. 10 is allowed by the constraints from HiggsBounds, i.e. both \(a_s\) and \(s^0\) are compatible with searches for additional Higgs bosons. However, for the state \(s^0\) the region around \(\mu _{\text {eff}}=\pm 10\,\hbox {GeV}\) is close to the boundary of the region that is excluded by the limits from Higgs searches, see below. As a result, we conclude that in this scenario with small \(\mu _{\text {eff}}\) the singlet \(s^0\) can again be directly produced at a hadron collider through its admixture with the two \({\mathcal {CP}}\)even doublets, see the discussion in Sect. 3.4. For \(\mu =\{150,170\}\,\hbox {GeV}\) the mass of the singlet is \(m_{s^0}=\{323.9,324.8\}\,\hbox {GeV}\), and the gluonfusion production crosssection is \(\sigma (gg\rightarrow s^0)=\{270,274\}\,\hbox {fb}\). The production rates through bottomquark annihilation is negligible. Given the large branching ratios \(\hbox {BR}(s^0\rightarrow a_sa_s)\sim 57\%\), \(\hbox {BR}(s^0\rightarrow h^0h^0)\sim 19\%\), \(\hbox {BR}(s^0\rightarrow W^+W^)\sim 15\%\) and \(\hbox {BR}(s^0\rightarrow ZZ)\sim 7\%\), the most sensitive searches are those with a decay into a pair of SMlike Higgs or gauge bosons. As an example, for \(m_X\sim 320\,\hbox {GeV}\) the upper limits \(\sigma (pp\rightarrow X\rightarrow h^0h^0)\lesssim 500\,\hbox {fb}\) [180] and \(\sigma (pp\rightarrow X\rightarrow ZZ)\lesssim 200\,\hbox {fb}\) [181] are already within a factor 10 of the signal rates that can be obtained at \(\mu =\{150,170\}\,\hbox {GeV}\). Lastly, also the decays of the heavy Higgs bosons – whose total decay widths only vary within 10% for the considered scenario – show potentially observable branching ratios into pairs of lighter Higgs bosons in the limit \(\mu _{\text {eff}}\rightarrow 0\,\hbox {GeV}\), see Fig. 11. At such low values of \(\tan \beta \) both heavy Higgs bosons \(H^0\) and \(A^0\) are not predominantly decaying into bottom quarks or tau leptons, but decay into a pair of top quarks with a branching ratio of about \(30\%\). Thus, decay modes into Higgs bosons could actually serve as discovery modes. However, note that our scenario includes light electroweakinos, into which heavy Higgs bosons tend to decay with large branching fractions. The branching ratios \(\hbox {BR}(A^0\rightarrow {\tilde{\chi }}_i{\tilde{\chi }}_j)\) and \(\hbox {BR}(H^0\rightarrow {\tilde{\chi }}_i{\tilde{\chi }}_j)\), shown in Fig. 11, both exceed 60% except for small values of \(\mu _{\text {eff}}\). Both branching ratios include all kinematically allowed decays into pairs of neutralinos and charginos. This adds to the motivation for dedicated searches for heavy Higgs bosons decaying either into a pair of lighter Higgs bosons or into supersymmetric particles, see also the discussion in Refs. [182, 183].
We conclude that a small value of \(\mu _{\text {eff}}\) in the discussed scenario strongly enhances the mixing among the Higgs bosons despite a low value of \(\lambda \), which makes both singlet states potentially accessible at colliders. We have demonstrated that the Higgsboson decays are not only controlled through the selfcoupling dependences given in Sect. 2.5 for gauge eigenstates, but are also strongly dependent on the mixing of the Higgs bosons. In the standard NMSSM, light singlet states are usually associated with \(\kappa <\lambda \), since the limits from chargino searches at LEP imply \(\mu _{\text {eff}}\gtrsim 120\,\hbox {GeV}\), and therefore \(v_s\gg 120\,\hbox {GeV}\). Accordingly, only small \(\kappa \ll \lambda <1\) results in two light singlet states. However, in the \(\mu \)NMSSM scenario that we have considered \(\kappa /\lambda \) and \(\mu \) can be large in combination with a small \(\mu _{\text {eff}}\). Whereas the \({\mathcal {CP}}\)odd singlet \(a_s\) can be as light as a few GeV, the \({\mathcal {CP}}\)even singlet \(s^0\) is usually in the ballpark of a few hundred GeV in such scenarios. This scenario is intrinsically different from the behavior of Higgs masses and mixing known in the NMSSM.
4 Conclusions
We have analyzed the phenomenology at the electroweak scale of an inflationinspired extension of the NexttoMinimal Supersymmetric Standard Model (NMSSM). We have put special emphasis on the spectra of additional, nonSMlike Higgs bosons and the branching ratios of their decays. This model has the same field content as the NMSSM, but at early times in the universe the Dflat direction of the Higgs doublet plays the role of the inflaton. Such a model can successfully describe inflation without the need of introducing a dedicated inflaton field. The singlet superfield \(\hat{S}\) of the NMSSM is needed to stabilize the inflationary direction at the origin of \(\hat{S} = 0\). Inflation occurs due to a nonminimal coupling of the doublet Higgs fields to gravity \(\mathord {\sim }\,\chi \,H_u \cdot H_d\), where the proportionality factor involves the gravitino mass \(m_{3/2}\) at low energies. Thus, this model is characterized by an MSSMlike \(\mu \) term, which is generated from the coupling \(\chi \) and involves \(m_{3/2}\), in addition to the usual effective \(\mu _{\text {eff}}\) term of the NMSSM. The latter arises since the scalar component of the singlet superfield acquires a vacuum expectation value as in the NMSSM. At low energies, i.e. the electroweak scale, this model differs from the NMSSM by the additional \(\mu \) term which breaks the accidental \(\mathbb {Z}_3\) symmetry of the NMSSM; we denote this model as the \(\mu \)NMSSM. The higgsinomass term in the \(\mu \)NMSSM is composed of the sum \((\mu + \mu _{\text {eff}})\). We have classified and discussed various scenarios regarding the prospects to distinguish the \(\mu \)NMSSM from the NMSSM, where the latter corresponds to the limit \(\mu = 0\,\hbox {GeV}\) of the \(\mu \)NMSSM. We have derived constraints on the model parameters from theoretical and phenomenological considerations. For that purpose, we have computed the SMlike Higgs mass at the oneloop order in the \(\mu \)NMSSM and added as approximation at the twoloop level the known twoloop results from the MSSM which are implemented in FeynHiggs. We have probed our scenarios against the rate measurements of the SMlike Higgs boson and the limits from searches for additional Higgs bosons at colliders with the codes HiggsBounds and HiggsSignals. Furthermore, we have checked whether the electroweak ground state of the Higgs potential corresponds to the absolute minimum of the theory, i.e. the true vacuum, or whether the Higgs potential has a deeper nonstandard minimum such that the electroweak vacuum eventually decays. In the inflationary scenario considered here, configurations with a metastable electroweak vacuum in general do not yield a viable phenomenology. In fact, the most stringent constraints arise from the possible appearance of tachyonic Higgs states at the tree level.
The additional freedom of varying \(\mu \) and \(B_\mu \) in the \(\mu \)NMSSM allows one to choose values for the parameters of the NMSSM which would otherwise be excluded. In this extended parameter space, we have focused on relatively small values of \(\tan \beta \), since in this case – like in the NMSSM– the light doubletlike Higgs mass squared is increased by a shift \({\propto }\,\lambda ^2\,v^2\); in this way the loop corrections which are required in order to acquire a SMlike Higgs at \(125\,\hbox {GeV}\) can be smaller. As expected, in particular the requirement of a SMlike Higgs boson at about \(125\,\hbox {GeV}\) yields important constraints on the parameter space. Concerning the constraints from vacuum stability, we find that the region with a phenomenologically viable Higgs spectrum is strongly correlated with the region of a stable electroweak vacuum, where the electroweak ground state corresponds to the true vacuum at the electroweak scale. An exception is the case where the soft SUSYbreaking \(B_\mu \,\mu \) term is large. We have demonstrated that large negative values of \(B_\mu \,\mu \) destabilize the vacuum.
For most of the numerical analyses in this paper we have fixed the sum \((\mu +\mu _{\text {eff}})\), since \(\mu \) enters at the tree level only in this combination in the mass matrices for the charginos and sfermions as well as in the MSSMlike part of the neutralino mass matrix. Accordingly, the particle spectrum of the \(\mu \)NMSSM in those sectors resembles the one of the NMSSM if the sum \((\mu +\mu _{\text {eff}})\) in the \(\mu \)NMSSM is identified with the \(\mu _{\text {eff}}\) term of the NMSSM. Moreover, we have pointed out the possibility to further reduce the influence of the nonminimal coupling to supergravity \(\mathord {\sim }\,\mu \) on the neutralino sector by a rescaling of the parameter \(\kappa \). This rescaling compensates the dependence of the singlino component of the neutralino mass matrix on \(\mu _{\text {eff}}\), so that the neutralino, chargino and sfermion sectors of the \(\mu \)NMSSM and the NMSSM become indistinguishable from each other at tree level. We have demonstrated that the dependence of the Higgs masses on \(\mu \) is significantly weakened after this transformation, but the individual dependences on \(\mu \) and \(\mu _{\text {eff}}\) still have a large impact on the Higgs mixing and thus the branching ratios of Higgs decays. The modified value of \(\kappa \) resulting from the rescaling can also have an important influence on Higgs phenomenology.
Since with the above parameter settings the neutralino sector of the \(\mu \)NMSSM is NMSSMlike, we have not performed a detailed numerical analysis of the neutralino sector – besides our discussion of Higgs decays into electroweakinos. In general, the gravitino is found to be the LSP since it is tightly connected to the size of \(\mu \). For phenomenological reasons in our scenarios it typically has a mass of \(\mathcal {O}(10\,\text {MeV})\). The NLSP, which is either singlino or binolike, tends to be sufficiently longlived such that it only gives rise to missingenergy signatures in collider searches. Accordingly, typical constraints from SUSY searches including missing energy apply without large modifications. The character of the NLSP is influenced by a variation of the corresponding parameters, i.e. \((\mu +\mu _{\text {eff}})\) for the higgsino mass and \(M_1\) or \(M_2\) for the bino or wino mass, respectively. Our choices for \(M_{1,2}\) are rather arbitrary in this context. Their impact could be scrutinized with a dedicated study of the neutralino phenomenology in the \(\mu \)NMSSM.
In some of our analyses we have kept \(\lambda \) large in order to lift up the mass of the SMlike Higgs boson at the tree level through genuine NMSSM effects, and in order to allow for sizable doublet–singlet mixing. However, we emphasize that large mixing between the doublet and singlet fields can also be achieved through small \(\lambda \) in combination with nearly vanishing \(\mu _{\text {eff}}\). Such a scenario is viable in the \(\mu \)NMSSM and gives rise to a phenomenology that significantly differs from the NMSSM.
A phenomenologically very interesting set of scenarios includes light singlet states. The direct production of these states at colliders suffers from their nature as gauge singlets: couplings to SM particles only emerge through the admixture with doubletlike Higgs states. Similarly, HiggstoHiggs decays involving doublet and singlet fields are strongly correlated with Higgs mixing. We have shown this effect exemplarily for decays of the SMlike Higgs boson into a pair of light \({\mathcal {CP}}\)odd singlets, which depends on the fraction of the \({\mathcal {CP}}\)even singlet component in the SMlike Higgs boson \(h^0\). In the \(\mu \)NMSSM, this mixing is not only controlled through \(\lambda \), but also depends sensitively on the values of \(\mu \) and \(\mu _{\text {eff}}\). We conclude that in order to distinguish the Higgs sectors of the \(\mu \)NMSSM and the NMSSM, the detection of singlet states in the Higgs spectrum and their couplings to other Higgs bosons and the SM particles will be crucial. We have discussed four scenarios that yield a light \({\mathcal {CP}}\)even singletlike Higgs around \(97\,\hbox {GeV}\), motivated by slight excesses in experimental searches performed with CMS and at LEP. These scenarios are associated with a compressed spectrum of light electroweakinos. We have pointed out that searches at a future electronpositron collider would provide complementary information to the results achievable at the LHC in scenarios of this kind.
Footnotes
 1.
The field content of the MSSM alone (without the Higgs singlet) is not sufficient to describe inflation successfully as pointed out in Ref. [12].
 2.
 3.
There is an interplay between discrete R symmetries, SUSY breaking and hence the gravitino mass in supergravity, which favors the \(\mathbb {Z}_4\) R symmetry [70]. Note, however, that our model at hand is fundamentally different from Ref. [70] as the inflaton is related to the Higgs fields of the NMSSM.
 4.
We treat the fields as “classical field values” in the sense of vacuumexpectation values. To avoid confusion with the true and desired electroweak vevs, we always keep the fields as commuting variables \(h_u\), \(h_d\) and s and interpret them as vacuumexpectation values only at the minima.
 5.
 6.
 7.
The state A differs from the mass eigenstate \(A^0\) that we defined in the previous section.
 8.
We do not set \(\mu \) exactly to zero for purely technical reasons: the MSSMlike twoloop contributions to the Higgs masses are taken from FeynHiggs where the \(\mu \) parameter of the \(\mu \)NMSSM is identified with the \(\mu \) parameter of the MSSM. In the limit \(\mu \rightarrow 0\,\text {GeV}\) numerical instabilities appear.
 9.
The minimal \(\chi ^2\) value obtained in our numerical analysis is \(\chi _m^2=74.6\). All subsequently discussed benchmark planes include a parameter region with \(\chi _m^2<80\). Further details are provided below.
 10.
In the GNMSSM, there are further possibilities of absorbing shifts in \(\mu _{\text {eff}}\) through a redefinition of other \(\mathbb {Z}_3\)violating parameters.
 11.
 12.
The results of ATLAS [43] are presented in a fiducial region and are compatible with both the SM expectation and the excess reported by CMS.
 13.
The results are given in ’t Hooft–Feynman gauge. For details we refer to Ref. [184].
Notes
Acknowledgements
The authors thank S. Abel, P. Basler, F. Domingo, K. SchmidtHoberg, T. Stefaniak, A. Westphal and J. Wittbrodt for helpful discussions, and I. BenDayan, A. Ringwald and A. Salvio for insights on Higgs inflation. This project has been supported by the Deutsche Forschungsgemeinschaft through a lump sum fund of the SFB 676 “Particles, Strings and the Early Universe”. S. P. acknowledges support by the ANR Grant “HiggsAutomator” (ANR15CE310002).
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