disposition [toc]toc.1toc.1ContentsContentstoc.1

#  arXiv:1809.07371, DESY-17-075, KA-TP-26-2018, TTP18–035
Phenomenology of the inflation-inspired NMSSM at the electroweak scale

Wolfgang Gregor Hollika,b,c Stefan Lieblerd Gudrid Moortgat-Picka,e
Sebastian Paßehrf Georg Weigleina
aDESY, Notkestraße 85, D-22607 Hamburg, Germany
bInstitute for Nuclear Physics (IKP),
Karlsruhe Institute of Technology, D-76021 Karlsruhe, Germany
cInstitute for Theoretical Particle Physics (TTP),
Karlsruhe Institute of Technology, D-76128 Karlsruhe, Germany
dInstitute for Theoretical Physics (ITP),
Karlsruhe Institute of Technology, D-76131 Karlsruhe, Germany
eII. Institut für Theoretische Physik, Universität Hamburg,
Luruper Chaussee 149, D-22761 Hamburg, Germany
fSorbonne Université, CNRS,
Laboratoire de Physique Théorique et Hautes Énergies (LPTHE),
4 Place Jussieu, F–75252 Paris CEDEX 05, France
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###### Abstract

The concept of Higgs inflation can be elegantly incorporated in the Next-to-
Minimal Supersymmetric Standard Model (NMSSM). A linear combination of the two
Higgs-doublet fields plays the role of the inflaton which is non-minimally
coupled to gravity. This non-minimal coupling appears in the low-energy
effective superpotential and changes the phenomenology at the electroweak
scale. While the field content of the inflation-inspired 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 SM-like $125$ 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 non-
minimal coupling to gravity on the Higgs mixing and in turn on the decays of
the Higgs bosons in this model.

###### Contents

  1. 1 Introduction
  2. 2 Theoretical framework
    1. 2.1 Model description
    2. 2.2 Higgs potential
    3. 2.3 Vacuum structure and vacuum stability bounds
    4. 2.4 Higher-order corrections to Higgs-boson masses and mixing
    5. 2.5 Trilinear Higgs-boson self-couplings
    6. 2.6 Neutralino and chargino masses
    7. 2.7 Sfermion masses
  3. 3 Phenomenological analysis
    1. 3.1 Viable parameter space compatible with theoretical and experimental bounds
    2. 3.2 Higgs-boson and neutralino mass spectra
    3. 3.3 Parameter scan
    4. 3.4 Higgs-boson and electroweakino production
    5. 3.5 Higgs-boson mixing and decays
  4. 4 Conclusions
  5. A Beta functions
  6. References

##  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 fine-tuning and
becomes unnatural [4]. A less minimal version of Higgs-portal 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 scale-
free 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 Higgs inflation in the supergravity framework is triggered by a non-
minimal coupling to Einstein gravity. For the supergravity Lagrangian this can
be achieved with an additional term $X(\hat{\Phi})\,R$ of chiral superfields
$\hat{\Phi}$ and the curvature multiplet $R$ (the supersymmetrized field
version of the Ricci scalar which contains the scalar curvature in the
Grassmannian coordinate $\theta^{2}$), following the notation of Ref. [12].
The Lagrangian then reads

$\mathcal{L}_{X}=-6\int\operatorname{d^{2}\theta}\mathcal{E}\left[R+X(\hat{\Phi})\,R-\frac{1}{4}\left({\bar{\mathcal{D}}}^{2}-8\,R\right)\hat{\Phi}^{\dagger}\,\hat{\Phi}+\mathcal{W}(\hat{\Phi})\right]+\text{h.\,c.}+\ldots,$
(1)

where $X(\hat{\Phi})$ as well as the Superpotential $\mathcal{W}(\hat{\Phi})$
are holomorphic functions of the (left) chiral superfields $\hat{\Phi}$,
$\mathcal{E}$ is the vierbein multiplet and $\bar{\mathcal{D}}$ a covariant
derivative. The ellipses encode further gauge terms. The only possible choice
of such a non-minimal coupling suitable for inflation is given by [12]

$X=\chi\,\hat{H}_{u}\cdot\hat{H}_{d},$ (2)

where $\chi$ is a dimensionless coupling and $\hat{H}_{d,u}$ contain the two
$SU(2)_{\text{L}}$ Higgs doublets of the Next-to-Minimal Supersymmetric
Standard Model (NMSSM).111The field content of the MSSM alone (without the
Higgs singlet) is not sufficient to describe inflation successfully as pointed
out in Ref. [12]. The extension by an additional scalar singlet like in the
NMSSM has been shown to be a viable model for inflation, although this version
suffers from a tachyonic instability [13]. In order to avoid this instability,
a stabilizer term has been introduced in Refs. [13, 11] that is suppressed at
low energies. The stabilizer term can be avoided in a model with minimal
supergravity couplings where the Kähler potential has a shift symmetry in the
doublet fields [14]; however, cosmological phenomenology and observations have
meanwhile ruled out this possibility [15].

The simplest implementation of a superconformal model which can accommodate
the non-minimal coupling term $\chi\,\hat{H}_{u}\cdot\hat{H}_{d}$ is the well-
known $\mathbb{Z}_{3}$-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 low-energy 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 tree-
level couplings of the $\mu$NMSSM, and we have implemented the one-loop
counterterms. The loop calculations have been carried out with the help of
FormCalc [22] and LoopTools [22]. In order to predict the Higgs-boson masses,
we have performed a one-loop 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 MSSM-like two-loop corrections which are
implemented in FeynHiggs [25, 26, 27, 28, 29, 30, 31, 32] in order to achieve
a state-of-the-art 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 SM-like Higgs boson via
HiggsSignals version 2.1.0beta [38]. In addition, we check the electroweak
vacuum for its stability under quantum tunneling to a non-standard global
minimum and for tachyonic Higgs states in the tree-level spectrum. Finally, we
investigate some typical scenarios and study their collider phenomenology at
the Large Hadron Collider (LHC) and a future electron-positron collider. For
this purpose in some analyses we use SusHi [39, 40] for the calculation of
neutral Higgs-boson production cross-sections. We emphasize the possibility of
light $\mathcal{CP}$-even singlets in the spectrum with masses below
$100\,\textrm{GeV}$ 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 Section 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 higher-order corrections for the Higgs boson masses. Then, we
derive the trilinear self-couplings of the Higgs bosons and comment on the
remaining sectors of the model which are affected by the additional $\mu$
term. In Section 3, we focus on the parameter space of interest and
investigate the Higgs-boson 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 Higgs-boson production
and decays. Lastly, we conclude in Section 4. In the Appendix we present the
beta functions for the superpotential and some soft-breaking 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 scale-invariant NMSSM. We
derive the Higgs potential and investigate vacuum stability and the prediction
for the Higgs-boson masses of the model. Furthermore, we discuss the trilinear
self-couplings 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 tree-level relations.
Higher-order contributions, e. g. for the Higgs-boson masses, are explained
generically and are evaluated numerically in the subsequent phenomenological
section.

###  2.1 Model description

For the Higgs sector of the NMSSM the superpotential is of the form222Compared
to Refs. [10, 11], we flip the sign of $\lambda$ to follow the conventions of
the NMSSM literature—see e. g. Ref. [44]—and thus have $\lambda>0$. As shown
in Ref. [10], the product of $\kappa$ and $\lambda$ needs to be positive for
that convention.

$\displaystyle\mathcal{W}_{\text{Higgs}}$
$\displaystyle=\lambda\,\hat{S}\,\hat{H}_{u}\cdot\hat{H}_{d}+\tfrac{1}{3}\,\kappa\,\hat{S}^{3}\,.$
(3)

where $\hat{H}_{u}$ and $\hat{H}_{d}$ are the well-known $SU(2)_{\text{L}}$
doublets of the MSSM, and $\hat{S}$ is the additional $SU(2)_{\text{L}}$
singlet. The $SU(2)_{\text{L}}$-invariant product
$\hat{H}_{u}\cdot\hat{H}_{d}$ is defined through
$\hat{H}_{u}\cdot\hat{H}_{d}=\sum_{a,b}\epsilon_{ab}\,\hat{H}_{d}^{a}\,\hat{H}_{u}^{b}$
with $\epsilon_{21}=1$, $\epsilon_{12}=-1$ and $\epsilon_{aa}=0$ with
$a,b\in\\{1,2\\}$. As outlined in Ref. [11], a Kähler transformation starting
from Jordan-frame supergravity introduces a correction in the superpotential,
which is of the form

$\displaystyle\mathcal{W}_{\text{Higgs}}$
$\displaystyle\rightarrow\mathcal{W}_{\text{Higgs}}+\tfrac{3}{2}\,m_{3/2}\,\chi\,\hat{H}_{u}\cdot\hat{H}_{d}\,.$
(4)

The parameter $m_{3/2}$ denotes the gravitino mass, and $\chi$ is the coupling
of Eq. (2). The scalar Higgs fields are denoted by $H_{u}$, $H_{d}$ and $S$ in
the following. During electroweak symmetry breaking, they receive the vacuum
expectation values (vevs) $v_{u}$, $v_{d}$ and $v_{s}$, respectively.
Expanding around the vevs, we decompose the fields as follows:

$\displaystyle H_{u}$ $\displaystyle\equiv\begin{pmatrix}h_{u}^{+}\\\
h_{u}\end{pmatrix}=\begin{pmatrix}\eta_{u}^{+}\\\
v_{u}+\tfrac{1}{\sqrt{2}}\left(\sigma_{u}+i\,\phi_{u}\right)\end{pmatrix},\qquad
H_{d}\equiv\begin{pmatrix}h_{d}\\\
h_{d}^{-}\end{pmatrix}=\begin{pmatrix}v_{d}+\tfrac{1}{\sqrt{2}}\left(\sigma_{d}+i\,\phi_{d}\right)\\\
\eta_{d}^{-}\end{pmatrix},$ (5a) $\displaystyle S$ $\displaystyle\equiv
v_{s}+\tfrac{1}{\sqrt{2}}\left(\sigma_{s}+i\,\phi_{s}\right)\,.$ (5b)

The additional bilinear contribution to the superpotential in Eq. (4)
generates a term which is analogous to the $\mu$ term of the MSSM, but with

$\displaystyle\mu$ $\displaystyle=\tfrac{3}{2}\,m_{3/2}\,\chi\,.$ (6)

When the singlet $S$ acquires its vev, an effective
$\mu_{\text{eff}}=\lambda\,v_{s}$ is dynamically generated. Often, the sum
$\left(\mu+\mu_{\text{eff}}\right)$ is the phenomenologically more relevant
parameter of the model. It takes the form

$\displaystyle\mu+\mu_{\text{eff}}$
$\displaystyle=\tfrac{3}{2}\,m_{3/2}\,\chi\,+\lambda\,v_{s}\,$ (7)

and corresponds to the MSSM-like higgsino mass term. In the following, we
consider both quantities $\mu$ and $\mu_{\text{eff}}$ as independent input
parameters, where $\mu$ is linearly dependent on the gravitino mass $m_{3/2}$.
In order to be a viable dark-matter candidate, the gravitino mass can range
from a few eV to multiple TeV, see e. g. Ref. [47]. The value of $\chi$ is a
priori not fixed; for cosmological reasons we adopt

$\displaystyle\chi\simeq 10^{5}\;\lambda$ (8)

according to Refs. [13, 11]. The additional contribution to the superpotential
in the $\mu$NMSSM is thus mainly steered by the gravitino mass, whereas
$v_{s}$ can be traded for $\mu_{\text{eff}}$. If we require a $\mu$ parameter
above the electroweak scale, $\mu\gtrsim 1$ TeV, and in addition a sizable
coupling $\lambda\gtrsim 0.1$, the typical gravitino mass turns out to be much
below the electroweak scale at $m_{3/2}\gtrsim 10$ MeV. However, if we allow
for very small values of $\lambda\ll 10^{-2}$ and very large values of $\mu\gg
1\,\textrm{TeV}$, the gravitino mass could as well be above the TeV scale. In
the latter case, the phenomenology of the $\mu$NMSSM is not necessarily
similar to the MSSM: the singlets only decouple for $\lambda\to 0$ with
$\kappa\propto\lambda$ and therefore $v_{s}\to\infty$. If the constraint
$\kappa\propto\lambda$ is dropped, interesting effects can occur; e. g. we
will discuss a scenario with small $\lambda$ and small $\mu_{\text{eff}}$ in
our numerical studies. In contrast to the NMSSM, the higgsino mass can be
generated by $\mu$ alone and thus even a vanishing $v_{s}$ is not in conflict
with experimental bounds.

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 ($\mathord{\gtrsim}\,10\,\textrm{TeV}$). In the
latter case the inflationary $\mu$ term would dominate over the NMSSM-like
$\mu_{\text{eff}}$ 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 $m_{3/2}>1\,\textrm{GeV}$ it affects Big Bang
Nucleosynthesis via photo-deconstruction 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
$T_{R}\lesssim 10^{8}$–$10^{9}\,\textrm{GeV}$, as advocated in Ref. [52],
avoids the gravitino problem. The rough estimate of $m_{3/2}\sim
10\,\textrm{MeV}$ even needs $T_{R}\lesssim 10^{5}\,\textrm{GeV}$ in order to
not overclose the universe with thermally produced gravitinos after inflation
[53, 54, 55, 56]. Interestingly, such low reheating temperatures preserve
high-scale global minima after inflation, see Ref. [57], and disfavor the
preparation of the universe in a meta-stable 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 SM-like Higgs width $\Gamma_{h}\approx 4\times
10^{-3}\,\textrm{GeV}$, we can estimate a rather low reheating temperature
$T_{R}\sim\sqrt{\Gamma_{h}M_{\text{Pl}}}\approx 10^{7}\,\textrm{GeV}$ with the
Planck mass $M_{\text{Pl}}\approx 2.4\times 10^{18}\,\textrm{GeV}$. For our
studies below we assume that a reheating temperature as low as $T_{R}\lesssim
10^{9}\,\textrm{GeV}$ can be achieved even with large couplings.

Since the bilinear $\mu$ term breaks the $\mathbb{Z}_{3}$ symmetry, additional
parameters are allowed compared to the NMSSM. In the general NMSSM
(GNMSSM)—including the bilinear singlet mass parameter $\nu$ and the singlet
tadpole coefficient $\xi$—the Higgs sector of the superpotential is given by

$\displaystyle\mathcal{W}_{\text{Higgs}}$
$\displaystyle=\lambda\,\hat{S}\,\hat{H}_{u}\cdot\hat{H}_{d}+\tfrac{1}{3}\,\kappa\,\hat{S}^{3}+\mu\,\hat{H}_{u}\cdot\hat{H}_{d}+\tfrac{1}{2}\,\nu\,\hat{S}^{2}+\xi\,\hat{S}\,.$
(9)

However, we assume that the non-minimal coupling of the Higgs doublets to
supergravity is the only source of superconformal and thus $\mathbb{Z}_{3}$
symmetry breaking—as outlined in Section 5 of Ref. [11]. In this case, all
other superpotential parameters that are forbidden by $\mathbb{Z}_{3}$
symmetry remain exactly zero at all scales: the beta functions for the
parameters of the superpotential are proportional to the respective parameter
itself and thus they cannot be generated radiatively.

Because the $\mathbb{Z}_{3}$ symmetry is broken (which avoids the typical
domain-wall 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, Planck-scale 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 non-gauge symmetries, so-called $R$
symmetries. Imposing discrete $\mathbb{Z}_{4}$ or $\mathbb{Z}_{8}$ $R$
symmetries as proposed in Refs. [68, 69, 45] provide a viable solution, since
dimensionful linear and bilinear terms are forbidden as long as the symmetry
is not broken.333There 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.

Furthermore, each parameter in the superpotential induces a corresponding
soft-breaking term; additional mass terms are allowed:

$\displaystyle\begin{split}-\mathcal{L}_{\text{soft}}&=\left[A_{\lambda}\,\lambda\,S\,H_{u}\cdot
H_{d}+\tfrac{1}{3}\,A_{\kappa}\,\kappa\,S^{3}+B_{\mu}\,\mu\,H_{u}\cdot
H_{d}+\tfrac{1}{2}\,B_{\nu}\,\nu\,S^{2}+C_{\xi}\,\xi\,S+\text{h.\,c.}\right]\\\
&\quad+m_{H_{d}}^{2}\,\lvert H_{d}\rvert^{2}+m_{H_{u}}^{2}\,\lvert
H_{u}\rvert^{2}+m_{s}^{2}\,\lvert S\rvert^{2}\,.\end{split}$ (10)

It should be noted that the beta functions for soft-breaking parameters are
not only proportional to themselves, but also receive contributions from the
other soft-breaking parameters. Thus, in contrast to the terms in the
superpotential, finite contributions may emerge even if a soft-breaking
parameter is set to zero at the tree level. The beta functions for the
parameters of the superpotential in Eq. (9) and its corresponding soft-
breaking parameters in Eq. (10) can be found in Refs. [71, 72, 44]; however,
since we employ different conventions we list them in Appendix A.

Contrary to studies in the GNMSSM (see Refs. [44, 45, 46, 73]), where the
MSSM-like $\mu$ 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 inflation-inspired $\mu$NMSSM. First of all, the $\mu$ term is introduced
via the $R$ symmetry-breaking non-minimal 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

With the superpotential of Eq. (9) and the soft-breaking Lagrangian of Eq.
(10), we derive the following Higgs potential, where we stick to real
parameters:

$\displaystyle\begin{split}V&=\left[m_{H_{d}}^{2}+\left(\mu+\lambda\,S\right)^{2}\right]\lvert
H_{d}\rvert^{2}+\left[m_{H_{u}}^{2}+\left(\mu+\lambda\,S\right)^{2}\right]\lvert
H_{u}\rvert^{2}+\left(m_{S}^{2}+B_{\nu}\,\nu\right)S^{2}\\\
&\quad+2\,C_{\xi}\,\xi\,S+\tfrac{2}{3}\,\kappa\,A_{\kappa}\,S^{3}+\left[\xi+\nu\,S+\kappa\,S^{2}+\lambda\,H_{u}\cdot
H_{d}\right]^{2}+2\left(B_{\mu}\,\mu+\lambda\,A_{\lambda}\,S\right)H_{u}\cdot
H_{d}\\\ &\quad+\tfrac{1}{8}\left(g_{1}^{2}+g_{2}^{2}\right)\left(\lvert
H_{d}\rvert^{2}-\lvert
H_{u}\rvert^{2}\right)^{2}+\tfrac{1}{2}\,g_{2}^{2}\,\lvert
H_{d}^{\dagger}\,H_{u}\rvert^{2}\,.\end{split}$ (11)

This potential can be expanded in the components of the Higgs fields in Eq.
(5). Defining the vectors in field space
$\mathcal{S}^{\text{T}}=\left(\sigma_{d},\sigma_{u},\sigma_{s}\right)$,
$\mathcal{P}^{\text{T}}=\left(\phi_{d},\phi_{u},\phi_{s}\right)$ and
$\mathcal{C}^{\text{T}}=\left(\phi_{d}^{-},\phi_{u}^{-}\right)=\left(\eta_{d}^{+},\eta_{u}^{+}\right)^{*}$,
it reads

$\displaystyle\begin{split}V&=\text{const}-\mathcal{T}_{S}^{\text{T}}\,\mathcal{S}-\mathcal{T}_{P}^{\text{T}}\,\mathcal{P}+\tfrac{1}{2}\,\mathcal{S}^{\text{T}}\,{\mathcal{M}}_{S}^{2}\,\mathcal{S}+\tfrac{1}{2}\,\mathcal{P}^{\text{T}}\,{\mathcal{M}}_{P}^{2}\,\mathcal{P}+\mathcal{C}^{\text{T}}\,{\mathcal{M}}_{C}^{2}\,\mathcal{C}^{*}\\\
&\quad+\sum\limits_{ijk\,=\,1}^{6}\tfrac{1}{\sqrt{2}}\,\lambda_{ijk}^{\prime}\left(\mathcal{S},\mathcal{P}\right)_{i}\left(\mathcal{S},\mathcal{P}\right)_{j}\left(\mathcal{S},\mathcal{P}\right)_{k}+\sum\limits_{i\,=\,1}^{6}\sum_{jk\,=\,1}^{2}\tfrac{1}{\sqrt{2}}\,\tilde{\lambda}_{ijk}^{\prime}\left(\mathcal{S},\mathcal{P}\right)_{i}\left(\mathcal{C}\right)_{j}\left(\mathcal{C}^{*}\right)_{k}+\cdots\,,\end{split}$
(12)

where the $\mathcal{CP}$-even and $\mathcal{CP}$-odd tadpole coefficients
$\mathcal{T}_{S}$ and $\mathcal{T}_{P}$, the $\mathcal{CP}$-even,
$\mathcal{CP}$-odd and charged squared mass matrices $\mathcal{M}_{S}^{2}$,
$\mathcal{M}_{P}^{2}$ and $\mathcal{M}_{C}^{2}$ are given below, and the
trilinear couplings $\lambda_{ijk}^{\prime}$ and
$\tilde{\lambda}_{ijk}^{\prime}$ are specified in Section 2.5, though in a
basis where the Goldstone mode corresponds to a mass eigenstate and does not
mix with the other states at lowest order. The ellipses denote quadrilinear
terms which are immaterial for the following.

We substitute the electroweak vevs $v_{u}$ and $v_{d}$ by their ratio
$\tan\beta=v_{u}/v_{d}$ and the sum of their squares $v^{2}\equiv
v_{u}^{2}+v_{d}^{2}=(174\,\textrm{GeV})^{2}$. The symbols $t_{\beta}$,
$c_{\beta}$ and $s_{\beta}$ denote $\tan\beta$, $\cos\beta$ and $\sin\beta$,
respectively. Furthermore, $g_{1}$ and $g_{2}$ are substituted by the $W$ and
$Z$ gauge-boson masses,

$\displaystyle m_{W}^{2}$ $\displaystyle=\tfrac{1}{2}\,g_{2}^{2}\,v^{2}\,,$
$\displaystyle m_{Z}^{2}$
$\displaystyle=\tfrac{1}{2}\left(g_{1}^{2}+g_{2}^{2}\right)v^{2}\,.$ (13)

Using the abbreviations

$\displaystyle a_{1}$
$\displaystyle=B_{\mu}\,\mu+\xi\,\lambda+\mu_{\text{eff}}\left(\nu+\frac{\kappa}{\lambda}\,\mu_{\text{eff}}+A_{\lambda}\right),$
(14a) $\displaystyle a_{2}$
$\displaystyle=2\,v\,\lambda\left(\mu+\mu_{\text{eff}}\right),$ (14b)
$\displaystyle a_{3}$
$\displaystyle=v\,\lambda\left(\nu+2\,\frac{\kappa}{\lambda}\,\mu_{\text{eff}}+A_{\lambda}\right),$
(14c) $\displaystyle a_{4}$
$\displaystyle=\frac{1}{\mu_{\text{eff}}}\left[v^{2}\,\lambda^{2}\,c_{\beta}\,s_{\beta}\left(\nu+\frac{\kappa}{\lambda}\,\mu_{\text{eff}}+A_{\lambda}\right)-v^{2}\,\lambda^{2}\,\mu-\xi\,\lambda\left(\nu+C_{\xi}\right)\right],$
(14d) $\displaystyle a_{5}$
$\displaystyle=4\left(\frac{\kappa}{\lambda}\right)^{2}\mu_{\text{eff}}^{2}+\frac{\kappa}{\lambda}\left[\mu_{\text{eff}}\left(A_{\kappa}+3\,\nu\right)-v^{2}\,\lambda^{2}\,c_{\beta}\,s_{\beta}\right],$
(14e) $\displaystyle a_{6}$
$\displaystyle=v\,\lambda\left(\nu+2\,\frac{\kappa}{\lambda}\,\mu_{\text{eff}}-A_{\lambda}\right),$
(14f) $\displaystyle a_{7}$
$\displaystyle=-6\left(\frac{\kappa}{\lambda}\right)^{2}\mu_{\text{eff}}^{2}+2\,\frac{\kappa}{\lambda}\left(\xi\,\lambda-4\,\nu^{2}\right)+B_{\nu}\,\nu\,,$
(14g)

we can write the explicit expressions for the tadpole coefficients
$\mathcal{T}_{S,P}$ as

$\displaystyle\mathcal{T}_{S}$
$\displaystyle=\begin{pmatrix}\sqrt{2}\,v\left\\{s_{\beta}\,a_{1}-c_{\beta}\left[m_{H_{d}}^{2}+\left(\mu+\mu_{\text{eff}}\right)^{2}+v^{2}\,\lambda^{2}\,s_{\beta}^{2}+\tfrac{1}{2}\,m_{Z}^{2}\,c_{2\beta}\right]\right\\}\\\\[6.45831pt]
\sqrt{2}\,v\left\\{c_{\beta}\,a_{1}-s_{\beta}\left[m_{H_{u}}^{2}+\left(\mu+\mu_{\text{eff}}\right)^{2}+v^{2}\,\lambda^{2}\,c_{\beta}^{2}-\tfrac{1}{2}\,m_{Z}^{2}\,c_{2\beta}\right]\right\\}\\\\[6.45831pt]
\sqrt{2}\,\frac{\mu_{\text{eff}}}{\lambda}\left[a_{4}-m_{S}^{2}-a_{5}-a_{7}-v^{2}\,\lambda^{2}-\left(\nu+2\,\mu_{\text{eff}}\,\frac{\kappa}{\lambda}\right)^{2}\right]\end{pmatrix},$
$\displaystyle\mathcal{T}_{P}$ $\displaystyle=\begin{pmatrix}0\\\ 0\\\
0\end{pmatrix}\equiv\mathbf{0}\,.$ (15)

The minimization of the Higgs potential requires all tadpole coefficients in
Eq. (15) to be equal to zero. With the conditions $\mathcal{T}_{S}=\mathbf{0}$
we choose to eliminate $m_{H_{d}}^{2}$, $m_{H_{u}}^{2}$ and $m_{S}^{2}$
according to

$\displaystyle m_{H_{d}}^{2}$
$\displaystyle=-\left(\mu+\mu_{\text{eff}}\right)^{2}-v^{2}\,\lambda^{2}\,s_{\beta}^{2}-\tfrac{1}{2}\,m_{Z}^{2}\,c_{2\beta}+a_{1}\,t_{\beta}\,,$
(16a) $\displaystyle m_{H_{u}}^{2}$
$\displaystyle=-\left(\mu+\mu_{\text{eff}}\right)^{2}-v^{2}\,\lambda^{2}\,c_{\beta}^{2}+\tfrac{1}{2}\,m_{Z}^{2}\,c_{2\beta}+\frac{a_{1}}{t_{\beta}}\,,$
(16b) $\displaystyle m_{S}^{2}$
$\displaystyle=a_{4}-a_{5}-a_{7}-v^{2}\,\lambda^{2}-\left(\nu+2\,\frac{\kappa}{\lambda}\,\mu_{\text{eff}}\right)^{2}\,.$
(16c)

Substituting these expressions in the symmetric mass matrices
${\mathcal{M}}_{S,P,C}$ we find

$\displaystyle\mathcal{M}_{S}^{2}$
$\displaystyle=\begin{pmatrix}m_{Z}^{2}\,c_{\beta}^{2}+a_{1}\,t_{\beta}&\left(2\,v^{2}\,\lambda^{2}-m_{Z}^{2}\right)c_{\beta}\,s_{\beta}-a_{1}&a_{2}\,c_{\beta}-a_{3}\,s_{\beta}\\\
\cdot&m_{Z}^{2}\,s_{\beta}^{2}+a_{1}/t_{\beta}&a_{2}\,s_{\beta}-a_{3}\,c_{\beta}\\\
\cdot&\cdot&a_{4}+a_{5}\end{pmatrix},$ (17a)
$\displaystyle\mathcal{M}_{P}^{2}$
$\displaystyle=\begin{pmatrix}a_{1}\,t_{\beta}&a_{1}&-a_{6}\,s_{\beta}\\\
\cdot&a_{1}/t_{\beta}&-a_{6}\,c_{\beta}\\\
\cdot&\cdot&a_{4}-3\,a_{5}-2\,a_{7}\end{pmatrix},$ (17b)
$\displaystyle\mathcal{M}_{C}^{2}$
$\displaystyle=\left[\left(m_{W}^{2}-v^{2}\,\lambda^{2}\right)\,c_{\beta}\,s_{\beta}+a_{1}\right]\begin{pmatrix}t_{\beta}&1\\\
\cdot&1/t_{\beta}\end{pmatrix}.$ (17c)

Diagonalizing Eq. (17c) yields zero for the massless charged Goldstone boson,
and the charged Higgs-boson mass $m_{H^{\pm}}$ at the tree level is given by

$\displaystyle m_{H^{\pm}}^{2}$
$\displaystyle=m_{W}^{2}-v^{2}\,\lambda^{2}+\frac{a_{1}}{c_{\beta}\,s_{\beta}}\,,$
(18)

which we employ as an input parameter. Inserting Eq. (14a) we can then
eliminate $A_{\lambda}$ via

$\displaystyle A_{\lambda}$
$\displaystyle=\frac{c_{\beta}\,s_{\beta}}{\mu_{\text{eff}}}\left(m_{H^{\pm}}^{2}-m_{W}^{2}+v^{2}\,\lambda^{2}\right)-\frac{1}{\mu_{\text{eff}}}\left(B_{\mu}\,\mu+\xi\,\lambda\right)-\left(\nu+\frac{\kappa}{\lambda}\,\mu_{\text{eff}}\right).$
(19)

Substituting $A_{\lambda}$ in the abbreviations of Eq. (14) yields ($a_{2}$,
$a_{5}$ and $a_{7}$ are not changed)

$\displaystyle a_{1}^{\prime}$
$\displaystyle=c_{\beta}\,s_{\beta}\left(m_{H^{\pm}}^{2}-m_{W}^{2}+v^{2}\,\lambda^{2}\right),$
(20a) $\displaystyle a_{3}^{\prime}$
$\displaystyle=v\,\lambda\left[\frac{\kappa}{\lambda}\,\mu_{\text{eff}}+\frac{1}{\mu_{\text{eff}}}\left(a_{1}^{\prime}-B_{\mu}\,\mu-\xi\,\lambda\right)\right],$
(20b) $\displaystyle a_{4}^{\prime}$
$\displaystyle=c_{\beta}\,s_{\beta}\left(\frac{v\,\lambda}{\mu_{\text{eff}}}\right)^{2}\left(a_{1}^{\prime}-B_{\mu}\,\mu-\xi\,\lambda\right)-\frac{1}{\mu_{\text{eff}}}\left[\mu\,v^{2}\,\lambda^{2}+\xi\,\lambda\left(\nu+C_{\xi}\right)\right],$
(20c) $\displaystyle a_{6}^{\prime}$
$\displaystyle=v\,\lambda\left[3\,\frac{\kappa}{\lambda}\,\mu_{\text{eff}}+2\,\nu-\frac{1}{\mu_{\text{eff}}}\left(a_{1}^{\prime}-B_{\mu}\,\mu-\xi\,\lambda\right)\right]=-a_{3}^{\prime}+2\,v\,\lambda\left(2\,\frac{\kappa}{\lambda}\,\mu_{\text{eff}}+\nu\right).$
(20d)

The tree-level masses of the three neutral $\mathcal{CP}$-even Higgs bosons
$m_{h_{1,2,3}}^{2}$ are determined by diagonalizing Eq. (17a). Analogously,
diagonalizing Eq. (17b) yields the masses $m_{a_{1,2}}^{2}$ of the
$\mathcal{CP}$-odd Higgs bosons at the tree level; the third eigenvalue is
equal to zero and belongs to the neutral Goldstone boson.

Higgs doublets:

The mass-matrix elements of the doublet fields in the upper-left
$\left(2\times 2\right)$ 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 mass-matrix
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 NMSSM-specific 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 $100$ 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 $a_{4}^{\prime}$ 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 $A_{\kappa}\neq 0$ 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 doublet-like and the singlet-like 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
$\lambda\to 0$ with constant $\kappa/\lambda$, and it is enhanced for larger
values of $\lambda$. For fixed $\lambda$ it is also strongly enhanced in the
limit $\mu_{\text{eff}}\to 0$ 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
soft-breaking 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 non-
zero $\xi$ and $\nu$. As indicated above, due to the dependences of
$a_{3}^{\prime}$ and $a_{6}^{\prime}$ on $1/\mu_{\text{eff}}$, a small
$\mu_{\text{eff}}\ll 100$ GeV yields a strong mixing between singlets and
doublets.

We subsequently discuss vacuum structure and vacuum stability bounds in the
$\mu$NMSSM around the electroweak scale. We do not discuss tachyonic
instabilities during inflation or the stabilization of the inflationary
direction, since they are not of relevance for our study (see e. g. Refs. [11,
13]).

###  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 SUSY-breaking 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
non-vanishing vacuum expectation values of sfermion fields (e. g. staus or
stops) and thus known under the phrase “charge- and color-breaking minima”.
Such minima can invalidate the electroweak vacuum and therefore lead to
unphysical parameter configurations (see below).

However, the existence of charge- and color-breaking 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 non-standard vacuum is deeper the
corresponding scenario is potentially excluded. In fact, some of the points
with a deeper non-standard vacuum may be valid when accepting meta-stable
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 meta-stable 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; well-beyond the boundary region, the false vacua become rather
short-lived 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 one-loop corrections are
proportional to $m^{2}(\phi)\,T^{2}$ for the field-dependent masses $m(\phi)$.
For finite temperature, they shift the ground state to the symmetric phase
around $\phi=0\,\textrm{GeV}$ [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 $s=0$ 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. [13, 11].
Inflation ends at field values $h=\mathcal{O}(0.01)$ in units of the Planck
mass. For small $\lambda\sim 10^{-2}$, the $D$-flat trajectory remains stable
after inflation ends according to Ref. [11], and will change to
$\beta\neq\pi/4$ and $s\neq 0$ GeV when the SUSY-breaking terms become
important. NMSSM-specific effects like the relevance of singlet Higgs bosons
and the additional contribution to the $125\,\textrm{GeV}$ 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 zero-temperature potential, reheating may not be sufficient to
overcome the barrier and to select a false (and maybe meta-stable) 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 meta-stable 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. (15)
does not select a minimum.

We briefly sketch how to get constraints on the relevant model parameters in
the (neutral) Higgs sector of the $\mu$NMSSM. Similar observations for the
NMSSM have been intensively discussed in the literature [94, 95]. Already the
presence of an additional Higgs singlet (see e. g. Refs. [96, 97, 98])
invalidates the well-known results that no charge-breaking Higgs vevs exist at
lowest order in the MSSM (see e. g. Refs. [99, 82]) and in two-Higgs-doublet
models (see e. g. Refs. [100, 101]). On the other hand, in the NMSSM the
inclusion of such charge-breaking minima has rather little impact on the
overall vacuum stability and gives no further information, see Ref. [102]. In
a similar manner, we neglect non-vanishing squark vevs (see discussion below)
and therefore we only have to deal with the following potential:

$\displaystyle\begin{split}V&=\kappa^{2}\,s^{4}+\tfrac{1}{8}\left(g_{1}^{2}+g_{2}^{2}\right)\left(h_{u}^{2}-h_{d}^{2}\right)^{2}+\left(\lambda^{2}\,s^{2}+2\,\lambda\,\mu\,s\right)\left(h_{u}^{2}+h_{d}^{2}\right)-2\,\lambda\left(\kappa\,s^{2}+A_{\lambda}\,s\right)h_{u}\,h_{d}\\\
&\quad+\lambda^{2}\,h_{u}^{2}\,h_{d}^{2}+\tfrac{2}{3}\,\kappa\,A_{\kappa}\,s^{3}+\left(m_{H_{u}}^{2}+\mu^{2}\right)h_{u}^{2}+\left(m_{H_{d}}^{2}+\mu^{2}\right)h_{d}^{2}+m_{S}^{2}\,s^{2}-2\,B_{\mu}\,\mu\,h_{u}\,h_{d}\,,\end{split}$
(21)

where we just presented the real fields as we do not consider spontaneous
$\mathcal{CP}$ violation.444We treat the fields as “classical field values” in
the sense of vacuum-expectation 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 vacuum-expectation values only
at the minima. Notice also that we do not consider the shifted theory with all
fields $\phi\to\phi-v_{\phi}$ expanded around the electroweak point,
$h_{u}=v_{u},h_{d}=v_{d},s=\mu_{\text{eff}}/\lambda$. In our case for the
stability analysis, the potential vanishes at the origin, and the electroweak
minimum is one of the minima not located at the origin. It is not necessarily
the global minimum. Furthermore, compared to Eq. (11), we neglect all
additional $\mathbb{Z}_{3}$-breaking terms besides the contributions of $\mu$
and $B_{\mu}\,\mu$ of the $\mu$NMSSM (see the discussion above).

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. (15). 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 soft-
breaking masses $m_{H_{u}}^{2}$, $m_{H_{d}}^{2}$ and $m_{S}^{2}$ according to
Eqs. (16).

The masses of the Higgs sector are determined in such a way that the desired
vacuum with $\langle h_{u}\rangle=v_{u}$, $\langle h_{d}\rangle=v_{d}$ and
$\langle s\rangle=\mu_{\text{eff}}/\lambda$ is a viable vacuum of the
potential $V$ in Eq. (21). However, one has to ensure that there is no deeper
minimum of $V$. This can only be achieved reasonably-well through a numerical
evaluation. For that purpose, we determine the stationary points of the
potential $V$ and then compare the corresponding values of $V$ at these points
with the desired minimum given by

$\displaystyle\begin{split}V_{\text{min}}^{\text{des}}&=-\frac{1}{8}\left(g_{1}^{2}+g_{2}^{2}\right)v^{4}\,c^{2}_{2\beta}-\frac{1}{4}\,\lambda^{2}\,v^{4}\,s^{2}_{2\beta}-v^{2}\,\mu_{\text{eff}}^{2}\left[1-\frac{\kappa^{2}}{\lambda^{2}}\,s_{2\beta}\right]\\\
&\quad-\frac{\kappa^{2}}{\lambda^{4}}\,\mu_{\text{eff}}^{4}-v^{2}\,\mu\,\mu_{\text{eff}}-\frac{1}{3}\frac{\kappa\,A_{\kappa}}{\lambda^{3}}\,\mu_{\text{eff}}^{3}+\frac{1}{2}\,v^{2}\,A_{\lambda}\,\mu_{\text{eff}}\,s_{2\beta}-B_{\mu}\,\mu\,v^{2}\,s_{2\beta}\,.\end{split}$
(22)

From the expression in Eq. (22), one can derive a few general results: (a) for
small values of $\lambda$ the desired minimum gets deeper and—as the singlet
contribution decouples from the rest of the potential—it becomes more
difficult for a non-standard vacuum to appear and to be deeper than the
desired minimum; (b) the ($\mu$)NMSSM potential at the desired minimum is
usually deeper than in the case of the MSSM555Compare Eq. (22) with the
desired minimum of the MSSM in Eq. (25) which is solely determined by the $D$
term and $M_{A}^{2}$. and is mainly driven by $\mu_{\text{eff}}$; (c) the
contribution of $A_{\lambda}$ plays a subdominant role compared to
$A_{\kappa}$ whose impact is strongly influenced by $\mu_{\text{eff}}$ and
$\lambda$; (d) parameter points with $V_{\text{min}}^{\text{des}}>0$ have to
be excluded because the trivial minimum at $\langle h_{u}\rangle=\langle
h_{d}\rangle=\langle s\rangle=0$ GeV is obviously deeper.

In our analysis, we focus for clarity on constraints from the tree-level
potential, considering the appearance of global non-standard minima and, as
discussed above, disregarding the possibility of meta-stable false vacua.
Employing higher-order (i. e. one-loop) corrections does not necessarily give
more accurate predictions of vacuum stability, see Ref. [103]. An approach to
include one-loop 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 tree-level evaluation is much faster and
numerically more stable; moreover, it has been argued that the one-loop
effective potential is problematic for tunneling rate calculations [106].

#### Constraints on the NMSSM parameters:

There are two main constraints known for the trilinear soft SUSY-breaking
parameters $A_{\kappa}$ and $A_{\lambda}$. The first constraint relies on the
existence of a non-vanishing singlet vev to generate $\mu_{\text{eff}}\neq 0$
GeV. This can be easily derived from the Higgs potential with only $s\neq 0$
GeV and is given by the requirement [75]

$A_{\kappa}^{2}>9\,m_{S}^{2}\,.$ (23)

This lower bound on $A_{\kappa}$ is inappropriate for the $\mu$NMSSM, as there
always exists a non-vanishing higgsino mass term from
$\mu=\tfrac{3}{2}\,m_{3/2}\,\chi$. As shown in Section 3, this constraint has
hardly any impact on our analyses. We simply keep it for illustrative reasons.

The second constraint, on $A_{\lambda}$, follows from a non-tachyonic charged
Higgs mass, since a tachyonic mass ($m^{2}<0\,\textrm{GeV}^{2}$ ) means that
the potential has negative curvature at this stationary point derived by the
minimization conditions. Thus, the true vacuum would have some non-zero 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 (tree-level) 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.

#### Charge and color breaking:

There exist quite strong constraints in the MSSM from the formation of non-
standard minima which break the electric and color charges, known as charge-
and color-breaking (CCB) minima. The famous “$A$-parameter bounds” read
traditionally [76, 80, 82, 107]

$\displaystyle A_{t}^{2}$
$\displaystyle<3\,\big{(}m_{H_{u}}^{2}+\mu^{2}+m_{\tilde{Q}}^{2}+m_{\tilde{t}}^{2}\big{)}\,,$
(24a) $\displaystyle A_{b}^{2}$
$\displaystyle<3\,\big{(}m_{H_{d}}^{2}+\mu^{2}+m_{\tilde{Q}}^{2}+m_{\tilde{b}}^{2}\big{)}\,,$
(24b)

where $m_{\tilde{Q}}^{2}$ and $m_{\tilde{t},\tilde{b}}^{2}$ are the soft SUSY-
breaking masses for the superpartners of the left-handed $SU(2)_{\text{L}}$
quark doublet, $\tilde{Q}$, and of the right-handed quark singlets,
$\tilde{t}$ and $\tilde{b}$. Several modifications and improvements of Ineqs.
(24) are present in the literature, see e. g. Refs. [82, 84, 90]. These
constraints follow from the “$D$-flat” directions in the scalar potential of
the MSSM, i. e. $h_{u}=\tilde{t}_{L}=\tilde{t}_{R}$ and
$h_{d}=\tilde{b}_{L}=\tilde{b}_{R}$, respectively. Thus the quartic terms
associated with squared gauge couplings vanish. In addition, one has to be
reminded that Ineqs. (24) are only necessary conditions for the formation of a
non-trivial minimum with non-vanishing squark vevs in that specific direction.
In the case of a violation of Ineqs. (24), one has to check that the generated
CCB vacuum is actually deeper than the electroweak minimum. In the MSSM the
desired minimum takes on a comparably small numerical value, only depending on
$c_{2\beta}$ (and the $B_{\mu}$ term which can be replaced by the
$\mathcal{CP}$-odd Higgs mass $M_{A}$):

$\displaystyle V_{\text{min}}^{\text{{MSSM}}}$
$\displaystyle=-\tfrac{1}{8}\left(g_{1}^{2}+g_{2}^{2}\right)v^{4}\,c^{2}_{2\beta}-\tfrac{1}{2}\,M_{A}^{2}\,v^{2}\,s^{2}_{2\beta}\,.$
(25)

In principle, the $A$-parameter bounds (24) can be simply transferred to the
$\mu$NMSSM, where $\mu$ has to be replaced by $(\mu+\mu_{\text{eff}})$, as
they can be transferred to the NMSSM [108]. The net effect is roughly the same
in the MSSM, NMSSM and $\mu$NMSSM; if $A_{t}$ fulfills Ineq. (24a), no CCB
will appear. Constraints on $\mu_{\text{eff}}$ alone may get weakened, because
the desired minimum also gets deeper for larger $\mu_{\text{eff}}$. Moreover,
the additional singlet direction stabilizes the potential with respect to CCB
minima since the $\mu_{\text{eff}}$ term originates from a quadrilinear scalar
coupling, and the vacuum with non-vanishing $\mu_{\text{eff}}$ or $v_{s}$ is
typically deeper than a CCB vacuum. Generically, constraints from the coupling
to the wrong Higgs doublet relating down-type sfermion vevs to the up-type
Higgs and vice versa, see Refs. [109, 110], are expected to be valid for
$(\mu+\mu_{\text{eff}})$ and not weakened if the singlet is fixed at its vev.
Similarly, there are bounds on $A_{t,b}$ not related to $D$-flat directions as
discussed in Ref. [111]. These can be reasonably-well determined only
numerically. Generically speaking, for the $\mu$NMSSM the risk of generating a
CCB vacuum is reduced because (a) the dependence of the desired minimum on
$\mu_{\text{eff}}$ drives the electroweak vevs to be more stable, and (b) not
as large values of $A_{t}$ are needed to raise the SM-like Higgs mass because
of the additional NMSSM-specific tree-level contribution.

Constraints from CCB minima as given in Ineqs. (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 SM-like Higgs mass (as in the case
for small $\lambda$). If the singlet-field direction were neglected and the
stop $D$-flat direction $\tilde{t}_{R}=\tilde{t}_{L}=\tilde{t}$ defined, one
could directly apply Ineqs. (24) for the $\mu$NMSSM, keeping $v_{s}\neq 0$ GeV
and replacing $\mu\to\mu+\mu_{\text{eff}}$. 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
$\langle h_{u,d}\rangle\neq v_{u,d}$, $\langle s\rangle\neq v_{s}$ and
$\langle\tilde{t}\rangle\neq 0$ GeV is disfavored.

#### Meta-stability and tunneling rates:

Lastly, we comment on vacuum-to-vacuum transitions in case of a local
electroweak vacuum. It is in general of interest to see how long such a meta-
stable state could survive compared with the life-time of the universe. We
have outlined some arguments why—in view of the inflationary history of the
universe—we disregard meta-stable long-lived vacua. We will see in Section 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 single-field 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 Higher-order corrections to Higgs-boson masses and mixing

It is well-known 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 $Z$-boson mass. On the other hand,
in the NMSSM and similarly the $\mu$NMSSM there are scenarios where an
additional tree-level term lowers the tension between the tree-level SM-like
Higgs mass and the measured value of the SM-like Higgs boson at $125$ 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.

The masses of the Higgs bosons are obtained from the complex poles of the full
propagator matrix. The inverse propagator matrix is a $(6\times 6)$ matrix
that reads

$\displaystyle\mathbf{\hat{\Delta}}^{-1}{\left(k^{2}\right)}=i\left[k^{2}\mathbf{1}-\begin{pmatrix}\mathcal{M}_{S}^{2}&0\\\
0&\mathcal{M}_{P}^{2}\end{pmatrix}+\begin{pmatrix}\mathbf{\hat{\Sigma}}_{S}{\left(k^{2}\right)}&0\\\
0&\mathbf{\hat{\Sigma}}_{P}{\left(k^{2}\right)}\end{pmatrix}\right].$ (26)

Here $\mathbf{\hat{\Sigma}}_{S}$ and $\mathbf{\hat{\Sigma}}_{P}$ denote the
matrices of the renormalized self-energy corrections to the neutral
$\mathcal{CP}$-even and $\mathcal{CP}$-odd Higgs fields. In the
$\mathcal{CP}$-conserving limit there are no transition elements between
$\mathcal{CP}$-even and $\mathcal{CP}$-odd degrees of freedom, which is why
Eq. (26) is block diagonal.

In principle, contributions from mixing with the longitudinal $Z$ boson have
to be considered as well. However, these contributions as well as those from
mixing with the Goldstone mode enter the mass predictions only at subleading
two-loop level [115, 116]. Since these contributions are numerically small
[117] we neglect them in the following and use a $(5\times 5)$ propagator
matrix. The $(5\times 5)$ matrices are denoted by the symbols
$\mathbf{\hat{\Delta}}_{hh}$ for the propagators and
$\mathbf{\hat{\Sigma}}_{hh}$ for the renormalized self-energies in the
following. The complex poles of the propagator are given by the values of the
squared external momentum $k^{2}$ for which the determinant of
$\mathbf{\hat{\Delta}}_{hh}^{-1}$ vanishes,

$\displaystyle\det{\big{[}\mathbf{\hat{\Delta}}^{-1}_{hh}{\left(k^{2}\right)}\big{]}_{k^{2}\;=\;M_{h_{i}}^{2}\;+\;i\,\Gamma_{h_{i}}\,M_{h_{i}}}}$
$\displaystyle\overset{!}{=}0\,.$ (27)

The real part, $M_{h_{i}}^{2}$, of each pole yields the loop-corrected mass of
the corresponding Higgs boson $h_{i}$.

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 one-loop 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 top-quark pole mass. The renormalized
self-energies of the Higgs bosons $\mathbf{\hat{\Sigma}}_{hh}$ are evaluated
with the help of FormCalc [22] and LoopTools [22] by taking into account the
full contributions from the GNMSSM at the one-loop order. For other variations
of the NMSSM, similar calculations of Higgs-mass contributions up to the two-
loop 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].

As an approximation, we have added the leading two-loop contributions in the
MSSM of $\mathcal{O}{\left(\alpha_{t}\alpha_{s}\right)}$ [129] and
$\mathcal{O}{\left(\alpha_{t}^{2}\right)}$ [130, 131] at vanishing external
momentum to their MSSM-like counterparts in the $\mu$NMSSM (for a discussion
of this approximation in the NMSSM see Ref. [124]). They are taken from their
current implementation in FeynHiggs [25, 26, 27, 28, 29, 30, 31,
32].666Additional contributions from the MSSM at the two-loop order or
beyond—e. g. further fixed-order results [132, 133] or resummation of large
logarithms for heavy sfermions [134, 30, 31]—are available. However, we will
confine our discussion in this paper to the leading two-loop contributions. We
thus have

$\displaystyle\mathbf{\hat{\Sigma}}_{hh}{\left(k^{2}\right)}\approx\left.\mathbf{\hat{\Sigma}}^{(\text{1L})}_{hh}{\left(k^{2}\right)}\right|^{\text{{GNMSSM}{}}}+\left.\mathbf{\hat{\Sigma}}^{(\text{2L})}_{hh}{\left(k^{2}\right)}\right|_{k^{2}=0}^{\text{{MSSM}{},
leading}}.$ (28)

We note that the two-loop contributions of
$\mathcal{O}{\left(\alpha_{b}\alpha_{s}\right)}$ to the MSSM-like Higgs self-
energies are not included in our calculation. However, in the definition of
the bottom-Yukawa coupling we employ a running $\overline{\text{DR}}$ bottom
mass at the scale $m_{t}$ [116] which enters
$\mathbf{\hat{\Sigma}}^{(\text{1L})}_{hh}{\left(k^{2}\right)}\big{|}^{\text{{GNMSSM}{}}}$,
and we take into account large $t_{\beta}$-enhanced contributions to the
bottom mass as discussed in Refs. [135, 136, 137, 138, 139, 140, 116]. We
expect that the missing two-loop piece of
$\mathcal{O}{\left(\alpha_{b}\alpha_{s}\right)}$ is numerically subleading
(for a discussion in the MSSM see [141, 142]).

Higher-order propagator-type corrections are not only needed for predicting
the Higgs-boson masses, but also for the correct normalization of $S$-matrix
elements involving Higgs bosons as external particles. The wave-function
normalization factors incorporating the effects of the mixing between the
different Higgs bosons can be written as a non-unitary matrix
$Z_{ij}^{\mbox{\tiny mix}}$. It is constructed from the Higgs self-energies
and their derivatives with respect to $k^{2}$, evaluated at the various
physical poles; for details we refer the reader to Refs. [143, 144, 145, 146,
24]. A recent application in the framework of the NMSSM can be found in Ref.
[147]. Here, we follow the setup outlined in Section $2.6$ of Ref. [24] and
determine the matrix elements of $Z_{ij}^{\text{\tiny mix}}$ from the
eigenvalue equation

$\displaystyle\Big{[}\text{diag}{\left(m_{h_{1}}^{2},m_{h_{2}}^{2},m_{h_{3}}^{2},m_{a_{1}}^{2},m_{a_{2}}^{2}\right)}-\mathbf{\hat{\Sigma}}_{hh}\big{|}_{k^{2}\;=\;M_{h_{i}}^{2}\;+\;i\,\Gamma_{h_{i}}\,M_{h_{i}}}\Big{]}_{kl}\,Z^{\mbox{\tiny
mix}}_{il}=\left(M_{h_{i}}^{2}+i\,\Gamma_{h_{i}}\,M_{h_{i}}\right)Z^{\mbox{\tiny
mix}}_{ik}\,.$ (29) The normalization of each eigenvector is fixed by
$\displaystyle\Bigg{[}\frac{\mathrm{d}\mathbf{\hat{\Delta}}^{-1}_{hh}}{\mathrm{d}k^{2}}\bigg{|}_{k^{2}\;=\;M_{h_{i}}^{2}\;+\;i\,\Gamma_{h_{i}}\,M_{h_{i}}}\Bigg{]}_{kl}\,Z^{\mbox{\tiny
mix}}_{ik}\,Z^{\mbox{\tiny
mix}}_{il}=\Bigg{[}\mathbf{1}+\frac{\mathrm{d}\mathbf{\hat{\Sigma}}_{hh}}{\mathrm{d}k^{2}}\bigg{|}_{k^{2}\;=\;M_{h_{i}}^{2}\;+\;i\,\Gamma_{h_{i}}\,M_{h_{i}}}\Bigg{]}_{kl}\,Z^{\mbox{\tiny
mix}}_{ik}\,Z^{\mbox{\tiny mix}}_{il}=1\,.$ (30)

In our numerical analysis we denote the three $\mathcal{CP}$-even mass
eigenstates $h_{i}$ as $h^{0}$, $H^{0}$ and $s^{0}$, and the two
$\mathcal{CP}$-odd mass eigenstates $a_{i}$ as $A^{0}$ and $a_{s}$. These
assignments become ambiguous as soon as loop corrections are included. In our
analysis we use the largest admixture to a loop-corrected mass state in order
to define the assignment. For this purpose we employ the previously discussed
loop-corrected mixing matrix $Z^{\text{\tiny mix}}_{ij}$. In this way $s^{0}$
denotes the dominantly singlet-like state. The light doublet-like state is
named $h^{0}$ and the heavy doublet-like state is $H^{0}$. The
$\mathcal{CP}$-odd Higgs bosons are the predominantly singlet-like state
$a_{s}$ and the doublet-like state $A^{0}$.

###  2.5 Trilinear Higgs-boson self-couplings

In order to discuss possible distinctions between the NMSSM and the
$\mu$NMSSM, the Higgs-boson self-couplings are particularly relevant.
Experimentally these self-couplings 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
Higgs-boson decays into Higgs bosons and gauge bosons, e. g. $A^{0}\to Zh^{0}$
or $H^{0}\to Za_{s}$. 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.

The Higgs self-couplings are introduced in Eq. (12). In order to simplify
their presentation in the neutral sector we define $\phi_{i}$ to be the $i$-th
component of $\Phi=\left(\sigma_{d},\sigma_{u},\sigma_{s},A,\phi_{s}\right)$,
where in the $\mathcal{CP}$-odd sector the Goldstone boson is in a basis where
it does not mix with the other Higgs bosons at lowest order (see discussion in
Section 2.2).777The state $A$ differs from the mass eigenstate $A^{0}$ that we
defined in the previous section. We denote the couplings as $\lambda_{ijk}$
for the interactions among three Higgs bosons $\phi_{i}\phi_{j}\phi_{k}$ in
the basis $\Phi$. For the couplings among the $\mathcal{CP}$-even
components—expressed in gauge couplings (see Eq. (13) for the relation to the
gauge-boson masses)—we obtain at the tree level

$\displaystyle\lambda_{111}$
$\displaystyle=-\tfrac{3}{2}\left(g_{1}^{2}+g_{2}^{2}\right)c_{\beta}\,v\,,$
$\displaystyle\lambda_{112}$
$\displaystyle=\tfrac{1}{2}\left(g_{1}^{2}+g_{2}^{2}-4\,\lambda^{2}\right)s_{\beta}\,v\,,$
(31a) $\displaystyle\lambda_{113}$
$\displaystyle=-2\,\lambda\left(\mu+\mu_{\text{eff}}\right)\,,$
$\displaystyle\lambda_{122}$
$\displaystyle=\tfrac{1}{2}\left(g_{1}^{2}+g_{2}^{2}-4\,\lambda^{2}\right)c_{\beta}\,v\,,$
(31b) $\displaystyle\lambda_{123}$
$\displaystyle=\lambda\left(\nu+A_{\lambda}+2\,\frac{\kappa}{\lambda}\,\mu_{\text{eff}}\right),$
$\displaystyle\lambda_{133}$
$\displaystyle=2\,\lambda\left(\kappa\,s_{\beta}\,v-\lambda\,c_{\beta}\,v\right),$
(31c) $\displaystyle\lambda_{222}$
$\displaystyle=-\tfrac{3}{2}\left(g_{1}^{2}+g_{2}^{2}\right)s_{\beta}\,v\,,$
$\displaystyle\lambda_{223}$
$\displaystyle=-2\,\lambda\left(\mu+\mu_{\text{eff}}\right),$ (31d)
$\displaystyle\lambda_{233}$
$\displaystyle=2\,\lambda\left(\kappa\,c_{\beta}\,v-\lambda\,s_{\beta}\,v\right),$
$\displaystyle\lambda_{333}$
$\displaystyle=-2\,\kappa\left(A_{\kappa}+3\,\nu\right)-12\,\frac{\kappa}{\lambda}\,\mu_{\text{eff}}.$
(31e) The couplings of $\mathcal{CP}$-even components to $\mathcal{CP}$-odd
components are given by $\displaystyle\lambda_{144}$
$\displaystyle=-\tfrac{1}{2}\left(g_{1}^{2}+g_{2}^{2}\right)c_{\beta}\,v\,,$
$\displaystyle\lambda_{244}$
$\displaystyle=\tfrac{1}{2}\left(g_{1}^{2}+g_{2}^{2}-4\,\lambda^{2}\right)s_{\beta}\,v\,,$
(31f) $\displaystyle\lambda_{344}$
$\displaystyle=-2\,\lambda\left(\mu+\mu_{\text{eff}}\right)\,,$
$\displaystyle\lambda_{345}$
$\displaystyle=-\lambda\left(\nu+A_{\lambda}+2\,\frac{\kappa}{\lambda}\,\mu_{\text{eff}}\right),$
(31g) $\displaystyle\lambda_{155}$
$\displaystyle=\tfrac{1}{2}\left(g_{1}^{2}+g_{2}^{2}-4\,\lambda^{2}\right)c_{\beta}\,v\,,$
$\displaystyle\lambda_{255}$
$\displaystyle=-\tfrac{1}{2}\left(g_{1}^{2}+g_{2}^{2}\right)s_{\beta}\,v,$
(31h) $\displaystyle\lambda_{355}$
$\displaystyle=-2\,\lambda\left(\mu+\mu_{\text{eff}}\right)\,.$ (31i)

Similarly we can write down the couplings $\tilde{\lambda}_{i}$ for the
interaction $\phi_{i}H^{+}H^{-}$ of the neutral Higgs bosons in the basis
$\Phi$ to the physical charged Higgs bosons (the Goldstone bosons are again in
a basis where they do not mix) as follows:

$\displaystyle\tilde{\lambda}_{1}$
$\displaystyle=\lambda^{2}\,s_{\beta}\,s_{2\beta}\,v+\tfrac{1}{2}\left(g_{1}^{2}+g_{2}^{2}\right)c_{\beta}\,c_{2\beta}\,v-g_{2}^{2}\,c_{\beta}\,v,$
(32a) $\displaystyle\tilde{\lambda}_{2}$
$\displaystyle=\lambda^{2}\,c_{\beta}\,s_{2\beta}\,v-\tfrac{1}{2}\left(g_{1}^{2}+g_{2}^{2}\right)s_{\beta}\,c_{2\beta}\,v-g_{2}^{2}\,s_{\beta}\,v,$
(32b) $\displaystyle\tilde{\lambda}_{3}$
$\displaystyle=-\lambda\left[2\left(\mu+\mu_{\text{eff}}\right)+\left(\nu+2\,\frac{\kappa}{\lambda}\,\mu_{\text{eff}}+A_{\lambda}\right)s_{2\beta}\right].$
(32c)

The remaining couplings which are not present above are equal to zero. Again
$s_{x}$ and $c_{x}$ are defined as $s_{x}=\sin(x)$ and $c_{x}=\cos(x)$. In
most of the cases when $\mu$ or $\mu_{\text{eff}}$ appear, the coupling
depends on the sum $\left(\mu+\mu_{\text{eff}}\right)$. For the interactions
of the neutral Higgs bosons, only a few couplings carry an (additional)
proportionality to $\mu_{\text{eff}}$ itself, see $\lambda_{123}$,
$\lambda_{345}$ and $\lambda_{333}$ which all involve the singlet state. This
dependence manifests itself for the former two couplings in the Higgs-to-Higgs
decays $s^{0}\to h^{0}\,h^{0}$, $H^{0}\to s^{0}\,h^{0}$ and $A^{0}\to
s^{0}\,a_{s}$. In the charged Higgs sector, the decay $s^{0}\to H^{+}\,H^{-}$
has a direct dependence on $\mu_{\text{eff}}$ at the tree level in addition to
$(\mu+\mu_{\text{eff}})$ for a dominantly singlet-like state $s^{0}$, as can
be seen in $\tilde{\lambda}_{3}$. For both cases a very pronounced mixing of
the singlet states with the Higgs doublets, and an individual dependence on
$\mu_{\text{eff}}$ and on the sum $(\mu+\mu_{\text{eff}})$ can also occur in
other Higgs-to-Higgs decays. We will emphasize later that Higgs mixing is
crucial for the observed dependences on $\mu_{\text{eff}}$ and $\mu$. We
consider the decays at the tree level, however, including the external
corrections to Higgs-boson masses and mixing as discussed in Section 2.4.
Though, we emphasize that higher-order contributions to Higgs-boson self-
couplings and Higgs-boson decays can be large, see Refs. [148, 149, 150, 147]
for corresponding calculations in the NMSSM.

###  2.6 Neutralino and chargino masses

We write the neutralino and chargino sector in the gauge-eigenstate bases

$\displaystyle\big{(}\psi^{0}\big{)}^{\text{T}}$
$\displaystyle=\big{(}\tilde{B}^{0},\tilde{W}_{3}^{0},\tilde{h}_{d}^{0},\tilde{h}_{u}^{0},\tilde{s}\big{)}\,,$
$\displaystyle\big{(}\psi^{+}\big{)}^{\text{T}}$
$\displaystyle=\big{(}\tilde{W}^{+},\tilde{h}_{u}^{+}\big{)}$ and
$\displaystyle\big{(}\psi^{-}\big{)}^{\text{T}}$
$\displaystyle=\big{(}\tilde{W}^{-},\tilde{h}_{d}^{-}\big{)}\,,$ (33)

which includes the bino component $\tilde{B}^{0}$, the neutral and charged
wino components $\tilde{W}_{3}^{0}$ and $\tilde{W}^{\pm}$, the neutral and
charged higgsino components $\tilde{h}_{u,d}^{0}$ and $\tilde{h}_{u,d}^{\pm}$,
and the singlino component $\tilde{s}^{0}$ in the form of Weyl spinors. Their
mass terms in the Lagrangian can be written in the form

$\displaystyle-\mathcal{L}_{\chi\text{-masses}}$
$\displaystyle=\tfrac{1}{2}\big{(}\psi^{0}\big{)}^{\text{T}}{\mathcal{M}}_{\chi}\,\psi^{0}+\tfrac{1}{2}\big{[}\big{(}\psi^{-}\big{)}^{\text{T}}{\mathcal{M}}_{\chi^{\pm}}\,\psi^{+}+\big{(}\psi^{+}\big{)}^{\text{T}}{\mathcal{M}}_{\chi^{\pm}}^{\text{T}}\,\psi^{-}\big{]}+\text{h.\,c.}\,.$
(34)

The symmetric mass matrix of the neutralinos and the mass matrix of the
charginos are given by

$\displaystyle{\mathcal{M}}_{\chi}$
$\displaystyle=\begin{pmatrix}M_{1}&0&-m_{Z}\,s_{\text{w}}\,c_{\beta}&m_{Z}\,s_{\text{w}}\,s_{\beta}&0\\\
\cdot&M_{2}&m_{Z}\,c_{\text{w}}\,c_{\beta}&-m_{Z}\,c_{\text{w}}\,s_{\beta}&0\\\
\cdot&\cdot&0&-\left(\mu+\mu_{\text{eff}}\right)&-\lambda\,v\,s_{\beta}\\\
\cdot&\cdot&\cdot&0&-\lambda\,v\,c_{\beta}\\\
\cdot&\cdot&\cdot&\cdot&2\,\tfrac{\kappa}{\lambda}\,\mu_{\text{eff}}+\nu\end{pmatrix},$
(35a) $\displaystyle{\mathcal{M}}_{\chi^{\pm}}$
$\displaystyle=\begin{pmatrix}M_{2}&\sqrt{2}\,m_{W}\,s_{\beta}\\\
\sqrt{2}\,m_{W}\,c_{\beta}&\mu+\mu_{\text{eff}}\end{pmatrix}.$ (35b)

The abbreviations $s_{\text{w}}=g_{2}/\sqrt{g_{1}^{2}+g_{2}^{2}}$ and
$c_{\text{w}}=g_{1}/\sqrt{g_{1}^{2}+g_{2}^{2}}$ denote the sine and cosine of
the weak-mixing angle, respectively. We see that the mass scale of the MSSM-
like higgsinos is given by the sum $(\mu+\mu_{\text{eff}})$, and the mass
scale of the singlino is controlled by
$(2\,\kappa/\lambda\,\mu_{\text{eff}}+\nu)$. If only the electroweakinos were
taken into account at the tree level, it is apparent that the $\mu$NMSSM would
be indistinguishable from the NMSSM, since any shift in masses and mixing
induced through $\mu$ could be compensated through shifts in
$\mu_{\text{eff}}$. However, such shifts will induce differences in the Higgs
sector.

Including the singlino elements (with $\nu=0$ GeV as discussed in Section
2.1), an NMSSM-like neutralino spectrum can be generated, where
$(\mu+\mu_{\text{eff}})$ serves as the NMSSM-like $\mu_{\text{eff}}$ term and
$\kappa$ is rescaled as

$\displaystyle\kappa$
$\displaystyle\to\tilde{\kappa}=\kappa\,\frac{\mu+\mu_{\text{eff}}}{\mu_{\text{eff}}}\,.$
(36)

This rescaling on the other hand affects the Higgs spectrum, thus giving a
possible handle to distinguish the $\mu$NMSSM from the NMSSM.

Figure 1:  The masses of the neutralinos and charginos are shown for different
values of $\mu$. The effective higgsino mass parameter is fixed at
$\mu+\mu_{\text{eff}}=-200\,\textrm{GeV}$, and the mass parameters for the
gauginos are set to $M_{1}=100\,\textrm{GeV}$ and $M_{2}=300\,\textrm{GeV}$.
The other relevant parameters are given in the legend of the figure. The
mostly bino- and wino-like states $\tilde{B}^{0}$ (purple) and $\tilde{W}^{0}$
(red) as well as the charginos $\tilde{\chi}^{\pm}$ (rose) have (nearly)
constant masses. The masses of the two mostly higgsino-like states
$\tilde{H}^{0}$ (orange) and the mostly singlino-like state $\tilde{S}^{0}$
(blue) vary visibly.

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 $\mu>0$ 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 $\mu=0$ GeV corresponds to the case of the NMSSM. The
masses are obtained by diagonalizing the tree-level 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
about-linear 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 Section 2.1.

###  2.7 Sfermion masses

The mass term for each charged sfermion—for which we distinguish the
superpartners of the left- and right-handed components by the notation
$\tilde{f}_{\text{L}}$ and $\tilde{f}_{\text{R}}$, respectively—takes the
following form in the Lagrangian

$\displaystyle-\mathcal{L}_{\tilde{f}\text{-masses}}=\left(\tilde{f}_{\text{L}}^{\dagger},\tilde{f}_{\text{R}}^{\dagger}\right){\mathcal{M}}^{2}_{\tilde{f}}\begin{pmatrix}\tilde{f}_{\text{L}}\\\
\tilde{f}_{\text{R}}\end{pmatrix},$ (37)

where the squared mass matrix reads

$\displaystyle{\mathcal{M}}^{2}_{\tilde{f}}$
$\displaystyle=\begin{pmatrix}m_{f}^{2}+m_{\tilde{f}_{\text{L}}}^{2}+m_{Z}^{2}\,c_{2\beta}\left(T^{(3)}_{f}-Q_{f}\,s_{\text{w}}^{2}\right)&m_{f}\left[A_{f}-\theta_{f}\left(\mu+\mu_{\text{eff}}\right)\right]\\\
m_{f}\left[A_{f}-\theta_{f}\left(\mu+\mu_{\text{eff}}\right)\right]&m_{f}^{2}+m_{\tilde{f}_{\text{R}}}^{2}+m_{Z}^{2}\,c_{2\beta}\,Q_{f}\,s_{\text{w}}^{2}\end{pmatrix},$
(38a) $\displaystyle\theta_{f}$
$\displaystyle=\left\\{\begin{matrix}[l]t_{\beta}\,,&f\in\\{e,\mu,\tau,d,s,b\\}\,,\\\
\frac{1}{t_{\beta}}\,,&f\in\\{u,c,t\\}\,.\end{matrix}\right.$ (38b)

Therein we denote the fermion mass by $m_{f}$, the bilinear soft-breaking
parameters by $m_{\tilde{f}_{\text{L,R}}}$, the trilinear soft-breaking
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 off-
diagonal 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. Ineqs. (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 $\mu=0$ 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 inflation-inspired $\mu$
parameter. Before we provide a phenomenological analysis—including the higher-
order effects specified in Section 2.4—we discuss the viability of various
parameter regions. As discussed in Section 2.1 we focus on scenarios with non-
zero $\mu$ and $B_{\mu}$, but set all other $\mathbb{Z}_{3}$-violating
parameters in the superpotential (9) and soft-breaking 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 inflation-
inspired model, see Eqs. (6) and (8). Furthermore, we only investigate
scenarios with $\mu\lesssim 2$ 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 Section 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 Section 2.2.

We characterize the scenarios in the following parameter regions: small values
of $\mu\simeq 1\,\textrm{GeV}$,888We do not set $\mu$ exactly to zero for
purely technical reasons: the MSSM-like two-loop 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 $\mu\to 0$ GeV
numerical instabilities appear. large values of $\mu\gtrsim 1\,\textrm{TeV}$
with $\mu_{\text{eff}}\simeq-\mu$, and values of $\mu$
$\mathord{\gtrsim}\,100\,\textrm{GeV}$ with moderate or small
$\lvert\mu_{\text{eff}}\rvert$ $\mathord{\lesssim}\,100\,\textrm{GeV}$.

small $\mu\simeq 1\,\textrm{GeV}$:

in the case of small $\mu$ also the soft-breaking 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 non-zero $\mu$
parameter.

large $\mu\boldsymbol{\gtrsim}1\,\textrm{TeV}$ with
$\mu_{\text{eff}}\simeq-\mu$:

as discussed in Section 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
SUSY-breaking 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
$\lvert\mu+\mu_{\text{eff}}\rvert\lesssim 100\,\textrm{GeV}$ 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 singlet-like Higgs states receive large masses of the order of
$\lvert\mu_{\text{eff}}\rvert$ [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
Section 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.

$\mu\,\boldsymbol{\mathord{\gtrsim}\,}100\,\textrm{GeV}$ with
$\boldsymbol{\lvert}\mu_{\text{eff}}\boldsymbol{\rvert}\,\boldsymbol{\mathord{\lesssim}\,}100\,\textrm{GeV}$:

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 Section 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 off-diagonal
elements of the sfermion mass matrices (enhanced by $\tan\beta$ for the case
of down-type sfermions) which can potentially cause tachyons, also depending
on the values of the trilinear soft-breaking parameters $A_{f}$. As discussed
in Section 2.3, constraints from charge- and color-breaking minima induced by
too large soft-breaking trilinear parameters (see Ineqs. (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
$\lvert\mu_{\text{eff}}\rvert\ll\mu$. 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 Section 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 Section 3.5.

Table 1: The input parameters which are fixed throughout our numerical
analysis (interpreted as on-shell parameters if not specified otherwise). The
gaugino mass parameters are denoted as $M_{i}$ with $i=1,2,3$. The trilinear
soft-breaking 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.

$\displaystyle m_{H^{\pm}}$ $\displaystyle=800\,\textrm{GeV},$ $\displaystyle
m_{t}$ $\displaystyle=173.2\,\textrm{GeV},$ $\displaystyle\alpha_{s}(m_{Z})$
$\displaystyle=0.118,$ $\displaystyle G_{F}$ $\displaystyle=1.16637\cdot
10^{-5}\,\textrm{GeV}^{-2},$ $\displaystyle m_{Z}$
$\displaystyle=91.1876\,\textrm{GeV},$ $\displaystyle m_{W}$
$\displaystyle=80.385\,\textrm{GeV},$ $\displaystyle M_{3}$
$\displaystyle=2.5\,\textrm{TeV},$ $\displaystyle M_{2}$
$\displaystyle=0.5\,\textrm{TeV},$ $\displaystyle M_{1}$
$\displaystyle=\frac{5}{3}\frac{g_{1}^{2}}{g_{2}^{2}}M_{2},$ $\displaystyle
m_{\tilde{f}_{\text{L}}}$ $\displaystyle=2\,\textrm{TeV},$ $\displaystyle
m_{\tilde{f}_{\text{R}}}$ $\displaystyle=2\,\textrm{TeV},$ $\displaystyle
A_{f_{3}}$ $\displaystyle=4\,\textrm{TeV},\quad A_{f_{1,2}}=0\,\textrm{TeV}.$

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 Tab. 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 $B_{\mu}=0\,\textrm{GeV}$ 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 SM-
like Higgs-boson mass at $\mu+\mu_{\text{eff}}=0\,\textrm{GeV}$. 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 Tab. 1.

The gaugino-mass parameters $M_{1}$ and $M_{2}$ do not play a big role in the
following analysis, but are necessary input parameters for the mass matrices
of the charginos and neutralinos in Eqs. (35). We set
$M_{2}=500\,\textrm{GeV}$ and fix $M_{1}$ via the usual GUT relation, see Tab.
1. Our phenomenological analysis is most sensitive to the neutralino and
chargino spectrum if a Higgs boson can decay into them. This is in particular
the case if the particle spectrum contains light higgsinos, whose masses are
controlled through $\left(\mu+\mu_{\text{eff}}\right)$. For a scenario with
light higgsinos and a light singlino we will later also discuss the
electroweakino phenomenology at a linear collider, see Section 3.4.

As we use $\mu\simeq\frac{3}{2}\,m_{3/2}\,10^{5}\,\lambda$ and focus on
$\mu\lesssim 2\,\textrm{TeV}$, we are considering scenarios where the
gravitino typically is the LSP. We do not specify the mediator mechanism of
SUSY breaking; however, we assume that such a light gravitino is always
possible. Although the gravitino is the Dark Matter candidate, traditional
collider searches for a neutralino LSP do apply in our case: for instance, if
the next-to LSP (NLSP) is gaugino-like, it can decay into a photon and the
gravitino, where the NLSP lifetime is typically so large that it can escape
the detector [153]. We roughly estimate the NLSP phenomenology via the
approximate partial decay width of the neutralino NLSP into a photon or $Z$
boson and gravitino $\psi_{3/2}$ according to Refs. [154, 155, 156]

$\displaystyle\Gamma_{\tilde{\chi}^{0}_{1}\to\gamma\psi_{3/2}}\simeq\frac{\left|N_{11}\,c_{\text{w}}+N_{12}\,s_{\text{w}}\right|^{2}}{48\,\pi\,M_{\text{Pl}}^{2}}\frac{m_{\tilde{\chi}_{1}^{0}}^{5}}{m_{3/2}^{2}}\,,\quad\Gamma_{\tilde{\chi}^{0}_{1}\to
Z\psi_{3/2}}\simeq\frac{\left|-N_{11}\,s_{\text{w}}+N_{12}\,c_{\text{w}}\right|^{2}}{48\,\pi\,M_{\text{Pl}}^{2}}\frac{m_{\tilde{\chi}_{1}^{0}}^{5}}{m_{3/2}^{2}}\left(1-\frac{m_{Z}^{2}}{m_{\tilde{\chi}_{1}^{0}}^{2}}\right)^{4},$
(39)

where we expanded in a small gravitino mass $m_{3/2}$ and use $s_{\text{w}}$
and $c_{\text{w}}$ for the sine and cosine of the weak mixing angle,
respectively. The neutralino mixing matrix elements $N_{ij}$ follow from the
diagonalization of Eq. (35a). As an example for the decay of the NLSP with
$m_{\tilde{\chi}_{1}^{0}}\simeq 100\,\textrm{GeV}$ and $m_{3/2}\simeq
10\,\textrm{MeV}$, we find a lifetime of $\tau\equiv
1/\Gamma=\mathcal{O}(1\,\mathrm{s})$. Thus, the NLSP decays outside of the
detector and is counted as missing energy. Nevertheless, such decays might be
of certain interest with respect to future experimental searches for long-
lived particles like the MATHUSLA experiment [157]. Note that for a higgsino-
like NLSP the decay into a $Z$ boson and the gravitino is obtained by
replacing the mixing factor in Eq. (39) by
$\lvert{-}N_{13}c_{\beta}+N_{14}s_{\beta}\rvert^{2}$. If kinematically open,
also the decay into a (singlet-like) $\mathcal{CP}$-even or $\mathcal{CP}$-odd
Higgs boson and the gravitino can occur (see Ref. [155]), but this decay mode
does not change the qualitative features described above.

We have chosen $m_{H^{\pm}}$ as an input parameter and adjust $A_{\lambda}$
according to Eq. (19). If not denoted otherwise, we set
$m_{H^{\pm}}=800\,\textrm{GeV}$. 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 $H,A\to\tau\tau$ decays are
particularly important. For relatively low values of $\tan\beta$ the choice of
$m_{H^{\pm}}=800\,\textrm{GeV}$ 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 $125$ 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 Higgs-
boson mass with the observed value of $125$ GeV. This latter test includes a
theoretical uncertainty on the predicted Higgs-boson mass of about $3$ 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 two-dimensional figures include a region with a SM-like Higgs
boson,999The 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.
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 two-
dimensional 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 SM-like Higgs boson.

Lastly, we note that HiggsBounds and HiggsSignals are operated through an
effective-coupling 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 Section 2.3. For informative reasons, we distinguish
long-lived vacua from short-lived ones in the numerical analysis. We do not
explicitly enforce a perturbativity bound on $\kappa$ and $\lambda$, but
discuss this issue below.

###  3.2 Higgs-boson and neutralino mass spectra

In this section, we point out the differences of the Higgs-boson 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 MSSM-like $\mathcal{CP}$-odd Higgs bosons are (for sufficiently
large $m_{H^{\pm}}\gg M_{Z}$) quasi-degenerate.

In Fig. 3, we show the masses of the Higgs bosons for vanishing $A_{\kappa}$
in the left, $A_{\kappa}=100\,\textrm{GeV}$ 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
$\mu\in\\{0,200,1000\\}\,\textrm{GeV}$ while keeping all other parameters
fixed: $\mu+\mu_{\text{eff}}=-200\,\textrm{GeV}$, $t_{\beta}=3.5$,
$\lambda=0.2$, $\kappa=0.2\,\lambda$, and the other parameters as given in
Tab. 1. The additional $\mu$ term has the biggest influence on the singlet-
like states $s^{0}$ and $a_{s}$, as well as the singlino-like 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
singlet-mass terms in the $(3,3)$ elements of Eqs. (17a) and (17b), and the
singlino-mass 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 singlet-like state, whereas a sizable
$A_{\kappa}=100\,\textrm{GeV}$ (middle) lifts this mass up. There is thus the
potential for a distinction between the NMSSM-limit for $\mu=0\,\textrm{GeV}$
and the $\mu$NMSSM with a large $\mu=1\,\textrm{TeV}$. Note that in the middle
frame for $\mu=200\,\textrm{GeV}$, the purple and blue lines are on top of
each other.

The masses of the neutralino sector do not depend on $A_{\kappa}$ at the tree
level. Concerning the Higgs sector, only the two cases in Fig. 3 with $\mu=0$
GeV and $A_{\kappa}\in\\{0,100\\}\,\textrm{GeV}$ yield a SM-like Higgs boson
that is compatible with the experimental data with $\chi^{2}$ values of
maximal $77$. These two cases are also compatible with searches for additional
Higgs bosons probed by HiggsBounds. The two cases with $\mu=1$ TeV and
$A_{\kappa}\in\\{0,100\\}\,\textrm{GeV}$ yield minimal $\chi^{2}$ values of
$82.6$ and $84.0$, respectively. The larger values of $\chi^{2}$ mainly arise
because the SM-like Higgs-boson mass is slightly below $122$ GeV. The large
variation with $\mu$ for the mass prediction of the mostly SM-like Higgs boson
is mainly induced by a large mixing with the $\mathcal{CP}$-even singlet. The
mixing for $\mu=200\,\textrm{GeV}$ in this scenario becomes very large for
both values of $A_{\kappa}$ such that these cases are outside the parameter
region that is compatible with the constraints by HiggsSignals. Note that the
apparent preference for $\mu=0\,\textrm{GeV}$ over
$\mu\in\\{200,1000\\}\,\textrm{GeV}$ in this scenario is purely accidental and
could be reversed by a slight shift in the input parameters, see the
discussion below.

Figure 2:  The loop-corrected Higgs-boson spectrum and the tree-level
neutralino spectrum are shown in the $\mu$NMSSM for scenarios with
$\mu\in\\{0,200,1000\\}\,\textrm{GeV}$ and
$\mu+\mu_{\text{eff}}=-200\,\textrm{GeV}$ fixed. The parameters are chosen
such that the state $h^{0}$ (black) that is mostly SM-like has a mass around
$125\,\textrm{GeV}$; the gray band shows a $3\,\textrm{GeV}$ interval around
the experimentally measured Higgs mass. Furthermore, the masses of the
$\mathcal{CP}$-even singlet-like state $s^{0}$ (blue), the $\mathcal{CP}$-odd
singlet-like state $a_{s}$ (purple), and the heavy $\mathcal{CP}$-even Higgs
doublet and MSSM-like $\mathcal{CP}$-odd components $H^{0},A^{0}$ with values
close to the input $m_{H^{\pm}}\sim 800\,\textrm{GeV}$ (red) are shown, where
the assignments are made according to the loop-corrected mixing matrix
$Z^{\text{\tiny mix}}_{ij}$ for the Higgs sector, see Section 2.4. For the
neutralino sector on the right, yellow lines show the dominantly bino-like
state $\tilde{B}^{0}$, and green lines the wino-like state $\tilde{W}^{0}$.
The singlino $\tilde{S}^{0}$ is shown in rose and the two (doublet) higgsinos
$\tilde{H}^{0}$ appear in orange. The assignments are determined by the tree-
level mixing matrix. The parameter values are given in the plot and in Tab. 1.

Figure 3:  In a similar manner as in Fig. 3, the spectra of Higgs bosons and
neutralinos are shown in the $\mu$NMSSM. The neutralino masses are invariant
under changes in $\mu$ by identifying the sum
$\left(\mu+\mu_{\text{eff}}\right)$ of the $\mu$NMSSM with the
$\mu_{\text{eff}}$ term of the NMSSM, and by rescaling $\kappa$ according to
Eq. (36). We set $\kappa=0.8\,\lambda$, and for $\mu=0$ GeV we assign
$\mu_{\text{eff}}=-200\,\textrm{GeV}$. The Higgs mass spectra are slightly
affected by the rescaling.

As already mentioned in Section 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
$\mu\in\\{0,200,1000\\}\,\textrm{GeV}$ and
$A_{\kappa}\in\\{0,100\\}\,\textrm{GeV}$ with fixed
$\mu+\mu_{\text{eff}}=-200\,\textrm{GeV}$. The neutralino spectrum is shown in
only one column in the very right frame. In analogy to Fig. 3, the left and
middle frames show the Higgs-boson 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 singlet-like Higgs masses in Fig. 3 are
only slightly sensitive to $\mu$, much less than the changes observed in Fig.
3. A rising $\mu$ slightly increases the mass splitting between the singlet-
like and the SM-like Higgs state.

Still, while the Higgs masses remain almost constant for not too small
$\mu_{\text{eff}}$, the doublet–singlet mixing can be strongly affected by
varying $\mu$ and $\mu_{\text{eff}}$ (but keeping their sum constant), in
particular if the doublet–singlet mixing almost vanishes at a certain choice
of $\mu$ and $\mu_{\text{eff}}$. In general, the mixing between the singlet
and doublet states is affected by a large $\lvert\mu_{\text{eff}}\rvert$.
However, by rescaling $\kappa$ according to Eq. (36) all contributions linear
in $\mu_{\text{eff}}$ are absorbed, while the contributions
${\propto}\,\mu_{\text{eff}}^{-1}$ depend on the values of $t_{\beta}$,
$M_{H^{\pm}}$ and $B_{\mu}$, see Eqs. (20a) and (20b).101010In the GNMSSM,
there are further possibilities of absorbing shifts in $\mu_{\text{eff}}$
through a redefinition of other $\mathbb{Z}_{3}$-violating parameters. In
Section 3.5 we will further investigate scenarios with very small
$\mu_{\text{eff}}$ and enhanced Higgs-boson mixing.

In Fig. 3 only the case $A_{\kappa}=100$ GeV in combination with $\mu=0$ GeV
is allowed by HiggsBounds and HiggsSignals ($\chi^{2}=80.1$), since the other
scenarios are either ruled out by the decay of the SM-like Higgs into a pair
of light $\mathcal{CP}$-odd singlets or by a too large deviation of the SM-
like Higgs-boson mass from $125$ 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

We have discussed above the dependence of the Higgs masses and of the
condition for the stability of the electroweak vacuum on the model parameters.
Apart from the fixed parameters in Tab. 1, we choose seven “free” parameters
that we vary in the following regimes for our analyses:

$\displaystyle\begin{aligned}
\mu_{\text{eff}}&\in[-2,2]\,\textrm{TeV}\,,&\mu&\in[0,2]\,\textrm{TeV}\,,&B_{\mu}&\in[-3,3]\,\textrm{TeV}\,,\\\\[-4.30554pt]
\lambda&\in[10^{-4},1]\,,&\kappa&\in[10^{-4},1]\,,&A_{\kappa}&\in\\{0,100\\}\,\textrm{GeV}\,,&\tan\beta&\in[1.5,3.5]\,,\end{aligned}$
(40)

where the largest values of $\lambda$ and $\kappa$ in the specified range of
(40) violate the approximate perturbativity bound
$\lambda^{2}+\kappa^{2}\lesssim 0.5$.111111This perturbativity bound was
explicitly derived for the NMSSM in Ref. [158]. According to the beta
functions for $\lambda$ and $\kappa$ (see appendix A) no additional scale-
dependent contribution is introduced by the $\mu$NMSSM at the one-loop order.
For the results presented in the following, this bound is always fulfilled and
lies outside the plot ranges. Values of $\tan\beta\gtrsim 4$ push the model
into the MSSM-like regime and are of less interest for studying the $\mu$NMSSM
effects.

We have performed a scan over the parameter space defined in (40) and
identified regions which are compatible with current observations concerning
the properties of the SM-like Higgs boson at $125\,\textrm{GeV}$ and the
limits from searches for additional Higgs bosons with HiggsBounds and
HiggsSignals as described above. In the following, we present a selection of
results from this scan; different regions of vacuum stability are illustrated,
and the experimental constraints from Higgs physics are indicated. While we
display some typical examples, it should be noted that similar observations
hold for other regions in the parameter space as well.

In Figs. 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), long-lived meta-stable (purple), short-lived meta-
stable (red), and tachyonic (rose). As discussed above, we regard not only
tachyonic but also meta-stable regions as excluded in the context of this
inflationary scenario, but nevertheless display long- and short-lived meta-
stable regions for illustration. Furthermore, we indicate those points that do
not fulfill Ineq. (23) and thus have no singlet vev (orange), although, as
explained in Section 2.3, this constraint is not relevant for the $\mu$NMSSM.
We overlay mass contours for the SM-like Higgs $h^{0}$ (black), the
$\mathcal{CP}$-even singlet-like Higgs $s^{0}$ (blue), and the
$\mathcal{CP}$-odd singlet-like Higgs $a_{s}$ (red). The spectrum is
calculated taking into account the full one-loop and the known MSSM-like two-
loop contributions as described in Section 2.4. The assignment of the labels
$h^{0}$, $s^{0}$ and $a_{s}$ to the loop-corrected states is determined by the
largest respective contribution in the mixing matrix $Z^{\text{\tiny
mix}}_{ij}$. We emphasize again that the parameters of the stop sector
specified in Tab. 1 for the given scale of SUSY masses maximize the SM-like
Higgs mass for $\mu+\mu_{\text{eff}}=0\,\textrm{GeV}$; therefore, lower values
for the SM-like 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 gray-shaded band: Ref. [152] reports a lower
bound on the chargino mass of $94\,\textrm{GeV}$ which translates into the
requirement that $\lvert\mu+\mu_{\text{eff}}\rvert$ must be above that value
in the $\mu$NMSSM. Lastly, all Figs. 6–7 show the region of parameter points
that successfully passed HiggsBounds and HiggsSignals and thus, in particular,
yield a SM-like Higgs boson compatible with the observed state at
$125\,\textrm{GeV}$. This region is represented through the larger, light
green dots in the background. We refer to Section 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 SM-
like 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 meta-stable non-standard 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
short-lived meta-stable regions, the vacuum lifetime is longer than the age of
the universe.

Figure 4: Contours for the SM-like Higgs mass (black) and the masses of the
two singlet-like states ($\mathcal{CP}$-even in blue and $\mathcal{CP}$-odd in
red) in the plane $\kappa/\lambda$ versus $(\mu+\mu_{\text{eff}})$, where
$\lambda=0.6$ and $\mu=500\,\textrm{GeV}$ are kept fixed and $\kappa$ and
$\mu_{\text{eff}}$ vary. In the left plot $A_{\kappa}=0$ GeV is used; in the
right one $A_{\kappa}=100\,\textrm{GeV}$. Furthermore, $\tan\beta=2.5$ is set
in both plots. The other relevant parameters are listed in Tab. 1. The few red
and purple points have a short- and long-lived meta-stable electroweak vacuum,
respectively, whereas blue points have a stable electroweak vacuum. Rose
points are excluded because of tachyonic tree-level masses. The orange points
cannot reproduce a non-vanishing $\mu_{\text{eff}}$ at the electroweak vacuum
via the constraint of Ineq. (23). With the gray vertical band we mark a naïve
direct experimental exclusion bound from the chargino mass
$m_{\chi^{\pm}}>94\,\textrm{GeV}$. Green areas are allowed by HiggsBounds and
HiggsSignals (indicated as “HBHS” in the legend).

Figure 5: The same as Fig. 6, except that $A_{\kappa}=100\,\textrm{GeV}$ is
used in both plots, and the parameter $\mu$ is set to $\mu=1000\,\textrm{GeV}$
(left) and $1500\,\textrm{GeV}$ (right).

Figure 6: The same as Fig. 6 but for $\mu=1000\,\textrm{GeV}$, $\tan\beta=3.5$
and $\lambda=0.3$.

In Fig. 6 we indicate the Higgs-mass 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 tree-level doublet sector in Eqs. (17a), (17b) and (20a) remains
constant; any structure visible in the prediction of the SM-like 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
Tab. 1; in the left plot $A_{\kappa}=0\,\textrm{GeV}$ is used, while in the
right plot $A_{\kappa}=100\,\textrm{GeV}$. The value of $A_{\kappa}$ has an
impact in particular on the mass scale of the $\mathcal{CP}$-odd singlet-like
Higgs which is much lighter on the left-hand side. In fact, for a light
$\mathcal{CP}$-odd singlet-like Higgs a parameter region opens up where decays
of the SM-like Higgs into a pair of them become kinematically allowed. The
$\mathcal{CP}$-even singlet-like Higgs is also somewhat lighter for
$A_{\kappa}=0$ GeV, while the SM-like 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 Higgs-mass 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 Fig. 6 and Fig. 6, which
differ from each other by the value of $\mu$: in the right plot of Fig. 6 a
contour for the SM-like Higgs mass which is parallel to the tachyonic border
appears around $\mu+\mu_{\text{eff}}=250\,\textrm{GeV}$ and
$\kappa/\lambda=0.5$. In Fig. 6 such a contour is not visible as this
particular parameter region is excluded by a tachyonic SM-like state at the
tree level. Note that the NMSSM\- and $\mu$NMSSM-specific one-loop
contributions to the Higgs spectrum are particularly large in that region
(about $60\,\textrm{GeV}$ additional shift compared to the same scenario in
the MSSM-limit with $\lambda\to 0$ and $\kappa/\lambda$ constant), see also
Ref. [124]; a dedicated analysis taking into account two-loop effects beyond
the MSSM-limit 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. 6 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 SM-like Higgs has a mass in the vicinity of $125$ GeV. In
the green-marked 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 left-hand side of Fig. 6 that this region is split into two: in between
the two regions the SM-like Higgs can decay into a pair of $\mathcal{CP}$-odd
singlet-like Higgs bosons $h^{0}\to a_{s}a_{s}$ 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 $125\,\textrm{GeV}$ into
non-SM particles. For a very light $\mathcal{CP}$-odd singlet, the admixture
between the SM-like 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 $h^{0}\to a_{s}a_{s}$ is dominated by the
coupling among the two singlet states, $\lambda_{355}$ in Eq. (31i), such that
a reduced admixture between $h^{0}$ and $s^{0}$ also closes the decay
$h^{0}\to a_{s}a_{s}$. This is why—despite the very light $\mathcal{CP}$-odd
Higgs $a_{s}$—the region at $\mu+\mu_{\text{eff}}\simeq-300$ GeV and
$\kappa/\lambda\simeq 0.4$ is allowed by the constraints from both
HiggsSignals and HiggsBounds.

In Fig. 6 we present scenarios similar to the right-hand side of Fig. 6 with
$A_{\kappa}=100\,\textrm{GeV}$, but with different values of $\mu$ (note the
larger scale at the $x$-axis). 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 SM-like Higgs mass: for
$\mu=1\,\textrm{TeV}$ a turning point at about
$\mu+\mu_{\text{eff}}=-800\,\textrm{GeV}$ is visible, which moves to smaller
values of $\kappa/\lambda$ for $\mu=1.5\,\textrm{TeV}$. For the larger value
of $\mu$ one can see that the possibility emerges for scenarios with the
correct SM-like Higgs mass but positive $(\mu+\mu_{\text{eff}})$. Again all
tested points which yield a SM-like Higgs boson close to $125$ 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 left-hand and on the right-hand side of Fig. 6, respectively.

Fig. 6 shows scenarios with larger $\tan\beta$ and smaller $\lambda$ compared
to the previous figures. Like in Fig. 6 we set $A_{\kappa}=0$ GeV on the left,
and $A_{\kappa}=100$ GeV on the right-hand side, but $\mu=1\,\textrm{TeV}$ 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 $A_{\kappa}=0$
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 $A_{\kappa}=100$ GeV. In the plot on the right-hand side one can
see a (nearly) closed $125\,\textrm{GeV}$ contour for the mass of the SM-like
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 SM-like
Higgs mass can render a SM-like Higgs with a mass of about $125\,\textrm{GeV}$
in the whole enclosed region. Close to the tachyonic borders we find larger
regions with a long-lived meta-stable vacuum (purple) than in Figs. 6 and 6.
However, in this part of the plot the prediction for the mass of the SM-like
Higgs is below the experimental value. On the right-hand side of Fig. 6 a
large region is allowed by the constraints from HiggsBounds and HiggsSignals.
Only low values of $\lvert\mu+\mu_{\text{eff}}\rvert<m_{h}/2$ are excluded by
HiggsSignals due to the decay of the SM-like 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 $y$-axis: $B_{\mu}$ is varied and
$\kappa$ is kept fixed. We set $A_{\kappa}=0\,\textrm{GeV}$ on the left-hand
side, and $A_{\kappa}=100$ GeV on the right-hand side. One can see that non-
zero 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 meta-stable and long-lived, while the area in the lower left corner of the
plots indicates that the electroweak vacuum is unstable and short-lived. The
effect of a larger $A_{\kappa}$ mainly lifts the tachyonic boundary at the top
so that values of $B_{\mu}=1\,\textrm{TeV}$ are allowed for
$A_{\kappa}=100\,\textrm{GeV}$ and leaves the other regions invariant.
However, towards the upper limit of $B_{\mu}$, there is a small short-lived
area. As a new feature, we find large regions with a meta-stable vacuum but a
SM-like Higgs with a mass of $125\,\textrm{GeV}$ 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
SM-like 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 $h^{0}\to a_{s}a_{s}$ is kinematically
allowed—which happens in the plot for $A_{\kappa}=0\,\textrm{GeV}$ for
$\mu+\mu_{\text{eff}}\gtrsim-300\,\textrm{GeV}$ and
$\mu+\mu_{\text{eff}}\lesssim-700\,\textrm{GeV}$—the parameter region is
incompatible with the data on the detected Higgs boson.

Figure 7:  Dependence of mass contours and vacuum stability, see Fig. 6 for an
explanation of the color code, on the $\mathbb{Z}_{3}$-breaking soft SUSY-
breaking $B_{\mu}$ term and $(\mu+\mu_{\text{eff}})$ for $\lambda=0.5$. On the
left-hand side, the value $A_{\kappa}=0\,\textrm{GeV}$ was chosen, while on
the right $A_{\kappa}=100\,\textrm{GeV}$.

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. 6–7; in addition, there are effects from
$\mu_{\text{eff}}^{-1}$ terms as discussed in Section 2.2 which enhance the
doublet–singlet mixing and eventually cause tachyons. This feature can be
observed towards the right end of the Figs. 6–6. The experimental limits and
constraints confine the allowed regions further around the region where the
SM-like Higgs has a mass of about $125\,\textrm{GeV}$ and exclude parameter
regions where for instance the decay of the SM-like 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.
6–7.

In the NMSSM, one usually expects to find the phenomenologically most
interesting regions (accommodating a $125\,\textrm{GeV}$ Higgs) for rather
large values of $\lambda\gtrsim 0.1$, since the NMSSM contribution to the SM-
like 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 Section
3.1. We will investigate this class of scenarios, which are not possible in
the NMSSM but generic to the $\mu$NMSSM, in Section 3.5 in more detail.

Similar to the NMSSM, the chosen value of $A_{\kappa}$ has a strong influence
on the singlet-like Higgs masses, which is relevant for the tachyonic regions.
In a large part of the viable parameter space the relation
$\operatorname{sign}{(A_{\kappa})}=-\operatorname{sign}{(\mu_{\text{eff}})}$
applies, where for $A_{\kappa}=0\,\textrm{GeV}$ 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 Section 2.2 about the Higgs singlets and especially the
functional dependence of $a_{5}$ in Eq. (14e) versus $a_{4}^{\prime}$ in Eq.
(20c): large negative values of the sum $(a_{4}^{\prime}+a_{5})$ drive the
$\mathcal{CP}$-even singlet tachyonic. In the investigated scenarios above,
which have either $A_{\kappa}=0\,\textrm{GeV}$ or
$A_{\kappa}=100\,\textrm{GeV}$, 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 Section 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 $\mu_{\text{eff}}>0$ GeV and $\mu_{\text{eff}}<0$ GeV,
respectively. There we will also discuss the dependence of the singlet masses
on $\mu$ and $\mu_{\text{eff}}$ in more detail.

###  3.4 Higgs-boson and electroweakino production

In this and the next section we discuss phenomenological features of Higgs-
boson mixing and thus consequences on Higgs-boson production and decays due to
the $\mu$ parameter of the $\mu$NMSSM. For vanishing $\mu$ the phenomenology
of the Higgs bosons equals the one of the NMSSM, for which typical benchmark
scenarios can be found in Ref. [159] (see also Ref. [160]). Naturally they
differ from MSSM-type benchmark scenarios through singlet states modifying the
phenomenology: since the singlet states $s^{0}$ and $a_{s}$ neither directly
couple to fermions nor to gauge bosons, but only through their admixture with
the doublet states, their direct production—both at a hadron collider and a
lepton collider—is negligible in many scenarios. However, besides their direct
production light singlet states can also be potentially observable via their
production in cascade decays of heavier Higgs bosons, as we will discuss in
the following.

In most parts of our numerical study, we make use of the approximation of SM-
normalized effective couplings of a Higgs boson to gluons—calculated at
leading order—which we insert into HiggsBounds for the evaluation of the
Higgs-production cross-sections 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
$95$–$98\,\textrm{GeV}$ that have been reported recently by CMS [42] in the
$\gamma\gamma$ channel121212The results of ATLAS [43] are presented in a
fiducial region and are compatible with both the SM expectation and the excess
reported by CMS. and earlier at LEP [41] in the $b\bar{b}$ channel. For this
purpose we use more sophisticated predictions for the Higgs-production cross-
sections in order to compare with the experimental results. We obtain those
predictions from SusHi [161, 162, 163, 39, 40, 164, 165, 166, 167], for which
a dedicated version for the NMSSM exists [168]. The predictions include N3LO
QCD corrections for the top-quark 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 bottom-
Yukawa coupling.

In the NMSSM, the observed excesses in the data around $95$–$98\,\textrm{GeV}$
can be interpreted in terms of a singlet-like state $s^{0}$, see Ref. [74] for
a discussion of the LEP result, and Ref. [169] for a discussion of the CMS
data. At first sight it seems to be non-trivial to describe both excesses
simultaneously, since accommodating the LEP excess would require a rather
large rate $s^{0}\to b\bar{b}$, which in turn would suppress the channel
$s^{0}\to\gamma\gamma$ that is employed in the interpretation of the CMS
excess. As it was pointed out in Ref. [147] based on a detailed analysis of
the Higgs mixing properties, this is nevertheless possible—albeit in a
relatively narrow region of the parameter space, which is somewhat enlarged if
the possibility of non-vanishing phases giving rise to
$\mathcal{CP}$-violating effects is taken into account. We investigate in the
following to which extent the additional freedom that is present in the
$\mu$NMSSM with respect to the possible values of the masses in combination
with the mixing properties has an impact regarding a possible interpretation
of the observed excesses. In Tab. 3 we present four scenarios with $s^{0}$
masses in the range $95$–$98$ GeV that have a phenomenology addressing the
excesses observed both at LEP and CMS. Scenarios 1 and 3 have a small value of
$\mu$ and are NMSSM-like (inspired by the scenarios investigated in Ref.
[147]), while Scenarios 2 and 4 both have $\mu$ values that significantly
differ from zero, and Scenario 4 furthermore has a non-zero value of
$B_{\mu}$. These two $\mu$NMSSM scenarios are intrinsically different from the
NMSSM. Similar scenarios could also be obtained by changing the signs of
$(\mu+\mu_{\text{eff}})$ and $A_{\kappa}$ simultaneously.

Table 2:  Scenarios that yield a light $\mathcal{CP}$-even singlet-like Higgs
boson. The Higgs boson at about $125$ GeV is SM-like. All other parameters are
chosen in accordance to Tab. 1.

Scenario | 1 | 2 | 3 | 4
---|---|---|---|---
$\lambda$ | $0.08$ | $0.08$ | $0.28$ | $0.08$
$\kappa$ | $0.04$ | $0.023$ | $0.08$ | $0.0085$
$\tan\beta$ | $12$ | $12$ | $2.5$ | $2$
$(\mu+\mu_{\text{eff}})$ [GeV] | $-140$ | $-140$ | $-300$ | $-400$
$\mu$ [GeV] | $5$ | $195$ | $5$ | $150$
$B_{\mu}$ [GeV] | $0$ | $0$ | $0$ | $-300$
$m_{H^{\pm}}$ [GeV] | $800$ | $800$ | $800$ | $1000$
$A_{\kappa}$ [GeV] | $130$ | $265$ | $250$ | $32$
$A_{f}$ [GeV] | $400$ | $450$ | $3200$ | $4000$
$m_{s^{0}}$ [GeV] | $97.6$ | $95.7$ | $97.2$ | $97.1$
$m_{h^{0}}$ [GeV] | $124.7$ | $126.8$ | $124.6$ | $125.0$
$m_{a^{s}}$ [GeV] | $168.2$ | $277.0$ | $257.2$ | $75.6$
$\frac{\sigma{\left(e^{+}e^{-}\to Zs^{0}\right)}\cdot\text{BR}{\left(s^{0}\to b\bar{b}\right)}}{\sigma^{\text{{SM}{}}}{\left(e^{+}e^{-}\to ZH\right)}\cdot\text{BR}^{\text{{SM}{}}}{\left(H\to b\bar{b}\right)}}$ | $0.28$ | $0.31$ | $0.14$ | $0.35$
$\sigma{\left(gg\to s^{0}\right)}$ [pb] | $25.3$ | $28.1$ | $14.4$ | $31.5$
BR${\left(s^{0}\to\gamma\gamma\right)}$ | $0.0020$ | $0.0016$ | $0.0024$ | $0.0005$
$\chi^{2}(\text{{HiggsSignals}})$ | $97$ | $96$ | $82$ | $101$

Table 3:  Cross-sections for electroweakinos at an electron–positron collider
for Scenario 1 defined in Tab. 3.

Scenario 1 | $\tilde{\chi}^{0}_{1}$ | $\tilde{\chi}^{0}_{2}$ | $\tilde{\chi}^{0}_{3}$ | $\tilde{\chi}_{1}^{\pm}$ |
---|---|---|---|---|---
Masses [GeV] | $127.3$ | $138.3$ | $155.9$ | $138.4$ |
$\sigma(e^{+}e^{-}\to\tilde{\chi}_{i}\tilde{\chi}_{j})$ [fb] for $\sqrt{s}=350$ 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($e^{+},e^{-})=(-30\%,+80\%)$ | $142$ | $196$ | $0.08$ | $0.19$ | $352$
$\sigma(e^{+}e^{-}\to\tilde{\chi}_{i}\tilde{\chi}_{j})$ [fb] for $\sqrt{s}=500$ 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($e^{+},e^{-})=(-30\%,+80\%)$ | $75$ | $110$ | $0.13$ | $0.22$ | $212$

Interpreting the LEP excess as the contribution of a singlet-like state
$s^{0}$ in the considered mass range yields a “signal strength” of

$\displaystyle\frac{\sigma{\left(e^{+}e^{-}\to
Zs^{0}\right)}\cdot\text{BR}{\left(s^{0}\to
b\bar{b}\right)}}{\sigma^{\text{{SM}{}}}{\left(e^{+}e^{-}\to
ZH\right)}\cdot\text{BR}^{\text{{SM}{}}}{\left(H\to b\bar{b}\right)}}$
$\displaystyle\simeq\text{$0.2$--$0.3$}\,,$ (41)

while a “signal rate” of $\sigma(pp\to s^{0}\to\gamma\gamma)\simeq 0.1$ pb
would be compatible with the CMS observation. As mentioned above, the cross-
section $gg\to s^{0}$ in our analysis is obtained from SusHi [161, 162] for
the $13$ TeV LHC at N3LO QCD. The renormalization- and factorization-scale
uncertainties amount to about $\pm 5\%$. Sizable values for the cross-sections
$gg\to s^{0}$ and $e^{+}e^{-}\to Zs^{0}$ as well as the branching ratio
BR$(s^{0}\to b\bar{b})$ arise if the admixture of $s^{0}$ with the SM-like
Higgs boson is sufficiently large. A sizable BR$(s^{0}\to\gamma\gamma)$ can
occur as a consequence of a significant $H_{u}$ component of the singlet state
$s^{0}$, whereas a small $H_{d}$ component suppresses the decay into
$b\bar{b}$. In all the listed scenarios the $\mathcal{CP}$-odd singlet-like
Higgs boson $a_{s}$ has a mass below $300\,\textrm{GeV}$. It should be noted
that the occurrence of the state $s^{0}$ at low masses in combination with a
very heavy $a_{s}$ state through a large value of $A_{\kappa}$ would usually
yield a meta-stable (long-lived) vacuum. The listed scenarios involve a
certain amount of tuning in the choice of $A_{\kappa}$ since an increase in
$A_{\kappa}$ by a few GeV yields a tachyonic $s^{0}$ state. It is well-known
from the NMSSM that a too large $A_{\kappa}$ yields a tachyonic
$\mathcal{CP}$-even singlet-like Higgs boson $s^{0}$, see Eq. (37) in Ref.
[158] or Eq. (26) in Ref. [170] for lower and upper bounds on $A_{\kappa}$.
Similarly, we have noted a very pronounced dependence of the masses of both
states, $s^{0}$ and $a_{s}$, on $A_{\kappa}$ for the $\mu$NMSSM scenarios
investigated here.