NMSSM Explanation for Excesses in the Search for
Neutralinos and Charginos and a 95 GeV Higgs Boson

Ulrich Ellwangera, Cyril Hugonieb, Stephen F. Kingc, Stefano Morettic,d
a IJCLab, CNRS/IN2P3, University Paris-Saclay, 91405 Orsay, France
b LUPM, UMR 5299, CNRS/IN2P3, Université de Montpellier, 34095 Montpellier,
France
c School of Physics and Astronomy, University of Southampton, Highfield,
Southampton, SO17 1BJ, United Kingdom
d Department of Physics & Astronomy, Uppsala University, Box 516, 75120
Uppsala, Sweden

The observed excesses in the search for neutralinos and charginos by ATLAS and
CMS can be fitted simultaneously in the minimal supersymmetric standard model
(MSSM) assuming a light higgsino mass, of magnitude less than about 250 GeV,
and a compressed higgsino dominated neutralino and chargino spectrum, with
$5-10\%$ mass splittings. However, light higgsinos as dark matter would have
far too large direct detection cross sections. We consider the next-to-MSSM
(NMSSM) with an additional singlino-like lightest supersymmetric particle
(LSP) a few GeV below the next-to-lightest supersymmetric particle (NLSP).
Sparticles prefer to decay first into the NLSP and remnants from the final
decay into the LSP are too soft to contribute to the observed signals. Co-
annihilation in the higgsino-sector can generate a relic density in the
WMAP/Planck window. The singlino-like LSP has automatically a direct detection
cross section below present and future sensitivities: a direct detection
signal in the near future would exclude this scenario. The singlet-like Higgs
scalar of the NMSSM can have a mass around 95 GeV and signal cross sections in
the $b\bar{b}$ channel at LEP and in the $\gamma\gamma$ channel at the LHC
compatible with the respective observations.

##  1 Introduction

One of the most important tasks of the ATLAS and CMS experiments at the LHC is
the search for deviations from predictions of the Standard Model (SM), which
could be signs for physics beyond the Standard Model (BSM). Particularly
relevant are such deviations when observed simultaneously by both experiments.

Signals for supersymmetry (SUSY) extensions of the SM include missing
transverse energy ($E_{T}^{\text{miss}}$), if R-parity is conserved and the
Lightest Supersymmetric Particle (LSP) is neutral and stable. Often the LSP is
the lightest neutralino111Neutralinos (and charginos) are the spin-1/2
superparticle mass eigenstates which are linear combinations of superpartners
of electroweak gauge bosons (gauginos) and Higgs bosons (Higgsinos), they are
present in all supersymmetric extensions of the SM.. In the minimal
supersymmetric standard model (MSSM), the lightest neutralino is a candidate
for the LSP and, being stable, is a welcome candidate for dark matter (DM).
The decays of heavier neutralinos and charginos into the LSP can proceed via
the emission of $Z$ or $W$ bosons, which can be detected via their leptonic
decays. Hence, interesting search channels are final states with
$E_{T}^{\text{miss}}$ and leptons which can be signals for the pair production
of neutralinos and charginos.

After the end of Run 2 of the LHC, both ATLAS and CMS have published their
corresponding SUSY search results in leptonic channels in [1, 2, 3, 4] and [5,
6, 7, 8], respectively. Interestingly, both experiments have observed (mild)
excesses of events with respect to the SM. These extra events correspond to
final states expected from chargino-neutralino pair production in the case
where chargino and neutralino decays proceed via off-shell $W$ and $Z$ bosons
leading to soft leptons.

The precise kinematic properties of these final states depend on the spectra
of the participating neutralinos and charginos. If the corresponding charginos
are dominantly superpartners of $W^{\pm}$-bosons, one expects a neutralino of
similar mass, the superpartner of the $W^{3}$-boson before electroweak mixing.
A candidate for the LSP is then a bino, the superpartner of the $B$-boson
before electroweak mixing. In contrast, if the chargino is dominantly the
superpartner of a charged Higgs boson (present in all supersymmetric
extensions of the SM), one expects two light neutralinos (dominantly
higgsinos), one heavier and one lighter than the lightest chargino (dominantly
higgsino), whose mass lies approximately halfway between the two light
neutralino masses, since all three lightest “ino” states are dominantly
arising from the higgsino triplet (two neutral Majorana higgsinos and one
charged Dirac higgsino). These two scenarios are denoted by wino/bino or
higgsino scenario, respectively, by the experimental collaborations. (The
wino/bino scenarios depend somewhat on the relative sign among the neutral
wino and bino masses, and are denoted by wino/bino+ and wino/bino-
correspondingly.)

The ATLAS and CMS collaborations interpret the above observed events in both
wino/bino and higgsino scenarios and derive observed limits on the
neutralino/chargino masses in both scenarios. Interestingly, due to a mild
excess of events both collaborations obtain observed limits below the expected
ones, although only at the $(1-2)\,\sigma$ level. An interpretation of the
deviations from the SM depends on the assumed masses in the
neutralino/chargino sector and on the assumed scenario. Within the wino/bino
scenario the preferred value (corresponding to an excess beyond $1\,\sigma$)
for the chargino-neutralino mass difference $\Delta m$ is $\Delta m<25$ GeV in
case of the ATLAS results, but $\Delta m>25$ GeV in case of the CMS results.
If the higgsino scenario is assumed, a value for the neutral heavy-light mass
difference $\Delta m$ in the $15-30$ GeV range explains simultaneously both
ATLAS and CMS excesses of events beyond the $1\,\sigma$ level.

The excesses of events have been interpreted in the framework of the MSSM in
the context of both wino/bino and higgsino scenarios in [9]. In their Fig. 1
the authors of [9] compare the resulting limits from ATLAS in [2] shown in
Fig. 16d and from CMS in [7] shown in Fig. 9a, which leads to the above
conclusions. The authors of [9] apply an experimental mass resolution of $\sim
5$ GeV; then a common region of chargino-neutralino mass differences
describing simultaneously minor excesses in both ATLAS and CMS data exists
also for the wino/bino scenario.

In the MSSM, the higgsino scenario is ruled out since DM in the form of light
higgsino LSPs as required here would have too large direct detection cross
sections in order to comply with the latest constraints from the LZ experiment
[10]. However, these constraints are easy to satisfy within the Next-to-MSSM
(NMSSM) [11]. The NMSSM contains two extra singlet-like Higgs bosons (a scalar
and a pseudo-scalar) as well as an extra singlet-like neutralino - the
singlino - which can well be the LSP. Singlino DM can have very small direct
detection cross sections, but can simultaneously generate a relic density
compatible with the WMAP/Planck value $\Omega_{\rm DM}h^{2}=0.1187$ [12, 13].
For these reasons the singlino has become an attractive DM candidate, at least
after the latest constraints from the LZ experiment [14, 15, 16, 17, 18, 19,
20, 21, 22]. In the context of excess events in searches for neutralinos and
charginos, the NMSSM with a singlino-like LSP has already been considered in
[23]. There, the authors considered simultaneously excesses of events with
monojets [24].

The experimental signatures for supersymmetric extensions of the SM can remain
nearly unchanged relative to the MSSM in the presence of the singlino of the
NMSSM with its possibly very small couplings to the fermions and bosons of the
MSSM. Then the MSSM-like superpartners of the SM do not decay directly into
the singlino, unless this is the only decay channel allowed by the
conservation of R-parity. Hence the decay chains of superpartners of the SM
have the form of the MSSM, and terminate provisionally with the Next-to-LSP
(NLSP). Only subsequently, the NLSP decays in a last step into the singlino
LSP plus a photon or an off-shell $Z$ boson. The NLSP can well be the lighter
neutral higgsino whose presence is assumed in the higgsino scenario which can
explain simultaneously both ATLAS and CMS excesses of events. If the singlino
is just a few GeV lighter than the higgsino NLSP, the decay products of the
higgsino-to-singlino decay will remain too soft to be observable at the LHC.

In this paper, then, we shall interpret the ATLAS and CMS SUSY search excesses
in the framework of the NMSSM, assuming the above scenario, namely the
“compressed” light higgsino-like triplet consisting of ($\chi_{3}^{0}$,
$\chi_{1}^{\pm}$, $\chi_{2}^{0}$) plus a nearby singlino LSP $\chi_{1}^{0}$,
with all masses less than about 250 GeV with of order $5-10\%$ mass
splittings. Note that this NMSSM scenario differs from the one considered in
[23]: there the higgsino-singlino mass splitting is assumed to be in the
$5-20$ GeV range, and the decay products of the higgsino-to-singlino decay are
part of the signal. Consequently the statements in [23] on the NMSSM in
connection with events with monojets are not applicable.

Another attractive feature of the NMSSM is the extra dominantly gauge singlet
scalar Higgs state which can naturally have a mass in the 95 GeV range. Hints
for such an additional Higgs boson have been observed by the CMS and ATLAS
experiments. A search at Run 1 by CMS showed a $\sim 2\,\sigma$ excess at 97
GeV [25], which was confirmed by CMS later in [26] and in [27] for a mass
hypothesis of 95.4 GeV. A somewhat less sensitive search in the diphoton
channel by ATLAS in [28] led to an upper limit on the fiducial cross section
which did not contradict the possible excess observed by CMS, but a recent
analysis by ATLAS in the diphoton channel in [29] showed a mild excess of
$1.7\,\sigma$ at 95 GeV. Actually already the combination of searches for the
SM Higgs boson by the ALEPH, DELPHI, L3 and OPAL experiments at LEP [30]
showed some mild excess of events in the $Z^{*}\to Z+b\bar{b}$ channel in the
mass region 95 GeV. The fact that such a state can well be described within
the NMSSM has already been widely discussed [31, 32, 33, 35, 34, 36, 38, 37,
39, 40, 41, 42, 43, 19, 20, 44, 45].

In Section 2 we present our conventions for the neutralino/chargino sector of
the NMSSM. In Section 3 we discuss the constraints that we will apply to the
parameter space of the NMSSM, which we scan employing the codes
NMSSMTools-6.0.3 [46, 47, 48] and micrOMEGAs$\\_$3 [49]. The results will be
presented in the form of figures and benchmark points in Section 4. Then, we
conclude with a summary in Section 5.

##  2 The Neutralino/Chargino Sector of the NMSSM

We consider the ${\mathbb{Z}}_{3}$ invariant NMSSM with the superpotential

$W_{\text{NMSSM}}=\lambda\hat{S}\hat{H}_{u}\cdot\hat{H}_{d}+\frac{\kappa}{3}\hat{S}^{3}+\dots$
(2.1)

where the dots denote the Yukawa couplings of the superfields $\hat{H}_{u}$
and $\hat{H}_{d}$ to the quarks and leptons as in the MSSM. Once the scalar
component of the superfield $\hat{S}$ develops a vacuum expectation value
$\left<S\right>\equiv s$, the first term in $W_{\text{NMSSM}}$ generates an
effective $\mu$-term with

$\mu_{\mathrm{eff}}=\lambda\,s\;.$ (2.2)

(Subsequently the index eff of $\mu$ will be omitted for simplicity.) Hence,
$\mu$ generates Dirac mass terms for the charged and neutral SU(2) doublet
higgsinos $\psi_{u}$ and $\psi_{d}$.

In the “decoupling” limit $\lambda,\kappa\to 0$ all components of the
superfield $\hat{S}$ decouple from all components of $\hat{H}_{u},\
\hat{H}_{d}$ and the matter superfields. However, since $s\sim M_{\rm
SUSY}/\kappa$ where $M_{\rm SUSY}$ denotes the scale of soft Susy breaking
masses and trilinear couplings, $\mu$ remains of ${\cal O}(M_{\rm SUSY})$ in
the decoupling limit provided $\lambda/\kappa\sim{\cal O}(1)$.

Including bino ($\widetilde{B}$) masses $M_{1}$ and wino ($\widetilde{W}^{3}$)
masses $M_{2}$, the symmetric $5\times 5$ neutralino mass matrix ${\cal
M}_{0}$ in the basis
$\psi^{0}=(-i\widetilde{B},-i\widetilde{W}^{3},\psi_{d}^{0},\psi_{u}^{0},\psi_{S})$
is given by [11]

${\cal
M}_{0}=\left(\begin{array}[]{ccccc}M_{1}&0&-\frac{g_{1}v_{d}}{\sqrt{2}}&\frac{g_{1}v_{u}}{\sqrt{2}}&0\\\
&M_{2}&\frac{g_{2}v_{d}}{\sqrt{2}}&-\frac{g_{2}v_{u}}{\sqrt{2}}&0\\\
&&0&-\mu&-\lambda v_{u}\\\ &&&0&-\lambda v_{d}\\\ &&&&2\kappa
s\end{array}\right),$ (2.3)

where $v_{u}^{2}+v_{d}^{2}=v^{2}\simeq(174\ \text{GeV})^{2}$ and
$\frac{v_{u}}{v_{d}}=\tan\beta$. The eigenvalues of ${\cal M}_{0}$ will be
denoted by $M_{\chi_{i}^{0}}$, the eigenstates by $\chi_{i}^{0}$, $i=1...5$
ordered in increasing mass. The lightest eigenstate $\chi_{1}^{0}$ is assumed
to represent the LSP (which could, in principle, also be represented by a
lighter sneutrino).

The charged $SU(2)$ gauginos are
$\lambda^{-}=\frac{1}{\sqrt{2}}\left(\lambda_{2}^{1}+i\lambda_{2}^{2}\right)$,
and
$\lambda^{+}=\frac{1}{\sqrt{2}}\left(\lambda_{2}^{1}-i\lambda_{2}^{2}\right)$,
which mix with the charged higgsinos $\psi_{u}^{+}$ and $\psi_{d}^{-}$.
Defining

$\psi^{+}=\left(\begin{array}[]{c}-i\lambda^{+}\\\
\psi_{u}^{+}\end{array}\right)\,,\qquad\psi^{-}=\left(\begin{array}[]{c}-i\lambda^{-}\\\
\psi_{d}^{-}\end{array}\right)\,,$ (2.4)

the corresponding mass terms in the Lagrangian can be written as

${\cal L}=-\frac{1}{2}(\psi^{+},\psi^{-})\left(\begin{array}[]{cc}0&X^{T}\\\
X&0\end{array}\right)\left(\begin{array}[]{c}\psi^{+}\\\
\psi^{-}\end{array}\right)+\mathrm{h.c.}$ (2.5)

with

$X=\left(\begin{array}[]{cc}M_{2}&g_{2}v_{u}\\\
g_{2}v_{d}&\mu\end{array}\right)\,.$ (2.6)

The diagonalization of the (not symmetric) $2\times 2$ chargino mass matrix
$X$ (2.6) in the basis $\psi^{-}$, $\psi^{+}$ requires different rotations of
$\psi^{-}$ and $\psi^{+}$ into the 2-component mass eigenstates $\chi^{-}$,
$\chi^{+}$ as

$\chi^{-}=U\psi^{-},\qquad\chi^{+}=V\psi^{+}$ (2.7)

with

$U=\left(\begin{array}[]{cc}\cos\theta_{U}&\sin\theta_{U}\\\
-\sin\theta_{U}&\cos\theta_{U}\end{array}\right)\,,\qquad
V=\left(\begin{array}[]{cc}\cos\theta_{V}&\sin\theta_{V}\\\
-\sin\theta_{V}&\cos\theta_{V}\end{array}\right)\,.$ (2.8)

The eigenvalues of the chargino mass matrix $X$ are $M_{\chi_{i}^{\pm}}$ and
the eigenstates are $\chi_{i}^{\pm}$, $i=1...2$ ordered in increasing mass.

##  3 Constraints on the Parameter Space of the NMSSM

We scan the parameter space over the NMSSM-parameters $\lambda$, $\kappa$,
$A_{\lambda}$, $A_{\kappa}$, $\mu_{\text{eff}}$, $\tan\beta$ in the Higgs
sector and non universal soft SUSY breaking terms in the gaugino/sfermion
sector; for the definitions of the latter we refer to the review in [11]. All
soft SUSY breaking terms are taken below 5 TeV. The aim is to reproduce the
“compressed” light higgsino-like spectrum consisting of ($\chi_{3}^{0}$,
$\chi_{1}^{\pm}$, $\chi_{2}^{0}$) plus the nearby singlino LSP $\chi_{1}^{0}$.

Of relevance for a potential description of the excesses observed by ATLAS and
CMS are the masses of the charged and neutral higgsino-like states as well as
of the singlino LSP. Since we focus on the higgsino scenario, we assume
$M_{1},M_{2}>\mu$ leading to heavier binos and winos than higgsinos. The mass
of the singlino is approximately given by $2\kappa s$ and we assume
$\lambda\sim{\cal O}(10^{-2})$ such that its mixings and couplings to other
particles are small. Its mass is assumed just (less than 5 GeV) below the mass
of the lighter neutral higgsino such that it corresponds to the LSP
$\chi_{1}^{0}$ while the neutral higgsinos are $\chi_{2}^{0}$ and
$\chi_{3}^{0}$.

The mass splittings $M_{\chi_{3}^{0}}-M_{\chi_{1}^{\pm}}$ and
$M_{\chi_{1}^{\pm}}-M_{\chi_{2}^{0}}$ among the charged and neutral higgsinos
are not exactly the same as assumed by the experimental collaborations for the
interpretation of excesses within the higgsino-scenario. In order to reproduce
these assumptions within the experimental mass resolution we require

$|M_{\chi_{1}^{\pm}}-(M_{\chi_{3}^{0}}+M_{\chi_{2}^{0}})/2|<5\,\text{GeV}\,.$
(3.1)

As discussed in the Introduction, a simultaneous fit to both excesses of
events observed by ATLAS and CMS is possible, more precisely assuming

$15\,\text{GeV}<\Delta m<30\,\text{GeV}\,,\qquad\Delta
m=M_{\chi_{3}^{0}}-M_{\chi_{2}^{0}},$ (3.2)

which we impose on our parameters, resulting in a “compressed” light higgsino-
like triplet of states ($\chi_{3}^{0}$, $\chi_{1}^{\pm}$, $\chi_{2}^{0}$). In
addition we will have a nearby singlino LSP $\chi_{1}^{0}$ with a mass a few
GeV below the one of $\chi_{2}^{0}$. In order to explain the excesses we
require all these masses to be less than about 250 GeV, as we now discuss.

The signal cross sections required to reproduce the excesses are estimated as
follows. First we employ the code Prospino2 at NLO [50] in order to compute
the higgsino production cross sections for the various relevant neutral and
charged higgsino-like masses $M_{\chi_{3}^{0}}$ and $M_{\chi_{1}^{\pm}}$.
Assuming 100% branching fractions for the decays into $\chi_{2}^{0}$ plus
$Z^{*}$ and $W^{*}$ bosons for the masses corresponding to the expected
limits, we obtain the expected upper limits on the higgsino pair production
cross sections. Comparing to the observed limits on the relevant charged
higgsino masses, we obtain the observed upper limits on the higgsino-like pair
production cross sections. The difference between the expected and observed
upper limits on the production cross sections gives an estimate for the
required signal cross sections. For the considered range of $\Delta m$ and
within the uncertainties for the expected cross sections we obtain estimates
for the required signal cross sections of $0.2-1$ pb, which coincide the
estimates in [9].

Prospino2 at NLO is also used to compute the higgsino-like pair production
cross sections for the relevant neutralino and chargino masses
$M_{\chi_{3}^{0}}$ and $M_{\chi_{1}^{\pm}}$ within the NMSSM. These are
multiplied by the branching fractions for the decays into $\chi_{2}^{0}$ plus
$Z^{*}$ and $W^{*}$ bosons using the code NMSDECAY [51] which lead to the
effective signal cross section
$\sigma_{\text{eff}}^{\text{signal}}$222NMSDECAY is based on the code SDECAY
[52], and for loop induced neutralino decays into photons SDECAY uses the
conventions for signs for chargino masses and the matrices $U$ and $V$ in eq.
(2.8) from [53]. These were not implemented in earlier versions of NMSSMTools,
hence an updated version NMSSMTools-6.0.3 has been employed..

The sum over the final decays $\chi_{2}^{0}\to\chi_{1}^{0}+Z^{*}$ or $\gamma$
proceeds with 100% probability, but they are considered as unobservable. Thus
we require

$0.2\,\text{pb}<\sigma_{\text{eff}}^{\text{signal}}<1.0\,\text{pb}\,.$ (3.3)

The final decays $\chi_{2}^{0}\to\chi_{1}^{0}+X$, $X=Z^{*}/\gamma$ are
unobservable if the mass difference
$\varepsilon=M_{\chi_{2}^{0}}-M_{\chi_{1}^{0}}$ is small enough. In an
expansion in powers of $\varepsilon$ one can derive from energy and momentum
conservation

$E_{X}^{2}+({p_{T}}_{X})^{2}\left(\frac{{p_{T}}_{\chi_{2}^{0}}}{M_{\chi_{2}^{0}}}\right)^{2}<\varepsilon^{2}$
(3.4)

which leads indeed to $E_{X}$ and/or ${p_{T}}_{X}$ below the applied cuts for
most naturally occuring values of ${p_{T}}_{\chi_{2}^{0}}$ and
$M_{\chi_{2}^{0}}$ if

$\varepsilon<5\ \text{GeV}\,.$ (3.5)

In principle, off-shell sleptons (selectrons, smuons) can also contribute to
the signal cross sections consisting in $E_{T}^{\text{miss}}$ and soft
leptons. In order to estimate these contributions, dedicated simulations and
comparisons with the excesses observed in the well-defined signal regions
would be necessary which is beyond the scope of the present work. In order to
avoid such contributions to the signal cross sections we assume all slepton
masses at or beyond 1 TeV.

Herewith we conclude the applied constraints related to the potential
description of the excesses observed by ATLAS and CMS. For the DM relic
density we require $\Omega_{\rm DM}h^{2}=0.1187$, allowing for $\pm 10\%$
theoretical uncertainty. Despite the weak couplings of the singlino LSP the
required reduction of its relic density is easy to achieve through co-
annihilation: by construction, the masses of the higgsino NLSP and chargino
are not far above the mass of the LSP. Hence, at a temperature in the early
universe around the LSP mass, the densities of the higgsino-like NLSP and
chargino are similar to the density of the singlino LSP. The higgsino-like
NLSP and chargino undergo annihilation processes which are not suppressed by
small couplings and reduce this way, through exchange of their densities with
the density of the LSP via equilibrium processes, also the latter. (These
processes are suppressed only for couplings $\lambda\;\raise
1.29167pt\hbox{$<$\kern-7.5pt\raise-4.73611pt\hbox{$\sim$}}\;10^{-5}$.) The
final LSP relic density can easily reach the required value above.

In order to make sure that the higgsino-like signal cross sections at the LHC
are not spoiled by direct decays into the singlino LSP, the Yukawa coupling
$\lambda$ in the neutralino mass matrix (2.3) has to be small enough, i.e. of
the order $\lambda\;\raise
1.29167pt\hbox{$<$\kern-7.5pt\raise-4.73611pt\hbox{$\sim$}}\;10^{-2}$ which
implies small mixing angles of the mostly singlino LSP with the neutral
higgsinos. Then the LSP direct detection cross sections satisfy automatically
the LZ constraints from [10].

The NMSSM also contains always an extra singlet-like scalar Higgs state whose
mass depends on the yet unspecified trilinear coupling $A_{\kappa}$. This
parameter can easily have a value such that the mass of the extra singlet-like
scalar is in the 95 GeV range. We will therefore note this state $H_{95}$ and
allowing for some theoretical uncertainty we require $M_{H_{95}}=95.4\pm 3$
GeV. The best fits for the diphoton signal of $H_{95}$ observed by CMS and
ATLAS were combined in [43]. The authors in [43] obtain

$\mu_{\gamma\gamma}^{\rm LHC}=\frac{\sigma(gg\to
H_{95}\to\gamma\gamma)}{\sigma(gg\to H^{\rm
SM}_{95}\to\gamma\gamma)}={0.24^{+0.09}_{-0.08}}\;.$ (3.6)

Here $H^{\rm SM}_{95}$ denotes a SM-like Higgs boson with a mass of $\sim 95$
GeV. The $\sim 2\,\sigma$ excess at LEP was quantified in [32]. Let us denote
the reduced coupling of $H_{95}$ to vector bosons $W^{\pm},Z$ (relative to the
coupling of a SM-like Higgs boson of corresponding mass) by $C_{V}(H_{95})$.
Then the authors in [32] define

$\mu^{\rm LEP}_{bb}\equiv C_{V}(H_{95})^{2}\times{BR}(H_{95}\to
b\bar{b})/{BR}(H^{\rm SM}_{95}\to b\bar{b})=0.117\pm 0.057\,.$ (3.7)

We require $\mu^{\rm LEP}_{bb}$ in the $2\,\sigma$ range of (3.7) and
$\mu_{\gamma\gamma}^{\rm LHC}$ in the $2\,\sigma$ range of (3.6).

Actually hints for $H_{95}$ have also been observed in the di-tau channel by
CMS in [54], where the observed excess corresponds to a cross section times
branching fraction

$\sigma(gg\to H_{95}\to\tau\tau)=7.8^{+3.9}_{-3.1}~{}\text{pb}\;.$ (3.8)

Relative to $H^{\rm SM}_{95}$ we obtain

$\mu_{\tau\tau}^{\rm LHC}=\frac{\sigma(gg\to H_{95}\to\tau\tau)}{\sigma(gg\to
H^{\rm SM}_{95}\to\tau\tau)}=1.38^{+0.69}_{-0.55}\;.$ (3.9)

Couplings of $H_{95}$ to gluons (required for its production at the LHC),
photons (required for its observation at the LHC) and $b$-quarks (required for
its observation at LEP) are induced by the mixing of $H_{95}$ with the SM-like
Higgs boson with mass near 125 GeV. For the SM-like Higgs boson we require
that the couplings in the $\kappa$-framework satisfy combined limits of CMS
[55] and ATLAS [56]. These limits require that the reduced couplings
$C_{V}(H_{\rm SM}$) of the SM-like Higgs boson to gauge fields satisfy
$C_{V}(H_{\rm SM}$) $>$$\sim$ 0.96, hence the mixing angle between $H_{95}$
and $H_{\rm SM}$ is limited such that the reduced coupling of $H_{95}$ to
$\tau\tau$ is bounded to below $25\%$. Then the (reduced) cross section
$\times$ BR($H_{95}\to\tau\tau$) can hardly exceed 0.1 pb in contrast to what
is desired within the $2\,\sigma$ range for the di-tau excess
$\mu_{\tau\tau}^{\rm LHC}$ in (3.9): its description within the $2\,\sigma$
range is therefore left aside since impossible for the type II Yukawa
couplings present in the (N)MSSM to provide a suitable excess (i.e., we have
to assume that the measured value represents a statistical fluctuation).

The constraint from the anomalous magnetic moment of the muon $a_{\mu}$ [57,
58] as in [14] cannot be satisfied if the soft SUSY-breaking smuon, bino and
wino masses are simultaneously large. According to the SM contribution to
$a_{\mu}$ on the lattice as in [59], the deviation of the measurements from
the SM value may even not be that large (beyond the $2\,\sigma$ level). Hence
we will not consider this constraint at present.

Constraints on the sparticle spectrum from the absence of signals at the LHC
are taken into account using the code SModels-2.2.0 [60, 61, 62, 63].
Constraints from recent searches for light neutralinos or charginos by ATLAS
and/or CMS are taken into account by construction whereas corresponding
constraints from LEP are build into the code NMSSMTools. Moreover, in the
presence of several light electroweakly interacting sparticles, heavy
sparticles (squarks, gluino) can undergo several distinct cascade decays such
that the probability for each of them is reduced. As a consequence such
scenarios are difficult to rule out.

Furthermore we impose constraints from $b$-physics and constraints from
searches for BSM Higgs bosons by ATLAS and CMS as implemented in
NMSSMTools-6.0.3. The references to constraints from BSM Higgs boson searches
and $b$-physics (of little relevance here) are listed on the web page
https://www.lupm.in2p3.fr/users/nmssm/history.html. Constraints from the
absence of a Landau singularity for the Yukawa couplings below the grand
unification theory (GUT) scale confine values of the NMSSM-specific coupling
$\lambda$ to $\lambda\;\raise
1.29167pt\hbox{$<$\kern-7.5pt\raise-4.73611pt\hbox{$\sim$}}\;0.7$, which is
satisfied automatically. Herewith we conclude the list of applied constraints
on the parameter space during the scan.

##  4 Results

In Table 1 we show the contributing range of parameters for the NMSSM specific
parameters and the soft SUSY-breaking masses $M_{1}$, $M_{2}$ and $M_{3}$
where the latter refer to the bino, wino and gluino masses, respectively. The
soft SUSY-breaking mass terms for scalars vary from 1 TeV (as imposed for
sleptons) and 272 GeV (for squarks) up to 5 TeV. The upper limits of $\pm 5$
TeV are imposed by hand. Only negative values for $\mu$ and $A_{\lambda}$ have
been found. Given the possibly complicated squark decay cascades, squarks can
be relatively light without contradicting constraints from searches at the
LHC. In any case, due to radiative corrections the pole squark masses can
differ considerably from the soft SUSY-breaking mass terms. The MSSM-like CP-
even, CP-odd and charged Higgs masses are in the $1.5-2.5$ TeV range.

$\lambda$ | $\kappa$ | $A_{\lambda}$ | $A_{\kappa}$ | $\mu_{\rm eff}$ | $\tan\beta$
---|---|---|---|---|---
$0.013-0.019$ | $0.0058-0.0086$ | -5000 $-$ -1820 | $93-362$ | -244 $-$ -148 | $3.61-10.9$
$M_{1}$ | $M_{2}$ | $M_{3}$ | $A_{t}$ | $M_{Q_{3}}$ | $M_{U_{3}}$
$178-290$ | $304-5000$ | $739-5000$ | -5000 $-142$ | $272-5000$ | $570-5000$
Table 1: Range of input parameters satisfying all constraints (dimensionful
parameters in GeV).

Next we present the range of masses $M_{\chi^{0}_{3}}$ for which all
constraints can be satisfied simultaneously in the form of Fig. 1 showing the
signal cross section $\sigma_{\text{eff}}^{\text{signal}}$ as function of
$M_{\chi^{0}_{3}}$. It allows to select allowed values of $M_{\chi^{0}_{3}}$
in case more restricted values for $\sigma_{\text{eff}}^{\text{signal}}$ are
preferred333The lower branch corresponds to a somewhat larger loop induced
BR$(\chi^{0}_{3}\to\chi^{0}_{2}+\gamma)$, which reduces the
BR$(\chi^{0}_{3}\to\chi^{0}_{2}+Z^{*})$ required for the signal..
Subsequently, Fig. 2 allows one to deduce the values for $M_{\chi^{\pm}_{1}}$
and $M_{\chi^{0}_{1}}$ from the range of $M_{\chi^{0}_{3}}$.

Figure 1: The values of $\sigma_{\text{eff}}^{\text{signal}}$ for points in
the NMSSM satisfying all constraints as function of $M_{\chi^{0}_{3}}$. The
coloured dots denote benchmark points whose details are given in the Tables
2-4 below.

Fig. 3 shows the points satisfy the constraint (3.1) on the masses of the
higgsinos as function of $\Delta m=M_{\chi_{3}^{0}}-M_{\chi_{2}^{0}}$ defined
in (3.2). We see that even a value close to 0 would not be difficult to
achieve in case this gives the best fit to the data. Again, as function of
$\Delta m$, we show in Fig. 3 also the mass difference between the higgsino
NLSP and singlino LSP, $M_{\chi^{0}_{2}}-M_{\chi^{0}_{1}}$. This mass
difference plays the role of $\varepsilon$ in eq. (3.5).

Fig. 4 show the signal rates $\mu^{\rm LEP}_{bb}$ (3.7) and
$\mu_{\gamma\gamma}^{\rm LHC}$ (3.6) as function of $M_{\chi^{0}_{3}}$.
Whereas $\mu^{\rm LEP}_{bb}$ is obtained within the $1\,\sigma$ level
$\mu_{\gamma\gamma}^{\rm LHC}$ is obtained only within the $2\,\sigma$ level.

|
---|---
Figure 2: $M_{\chi^{\pm}_{1}}$ (left) and $M_{\chi^{0}_{1}}$ (right) as function of of $M_{\chi^{0}_{3}}$.  |
---|---
Figure 3: The constraint (3.1) on the mass of the lighter chargino and the mass difference between higgsino NLSP and the singlino LSP as function of $\Delta m=M_{\chi_{3}^{0}}-M_{\chi_{2}^{0}}$. |
---|---
Figure 4: $\mu^{\rm LEP}_{bb}$ from (3.7) (left) and $\mu_{\gamma\gamma}^{\rm
LHC}$ from (3.6) (right) as function of $M_{\chi^{0}_{3}}$.

Fig. 5 show the DM spin-idependent (SI) and spin-dependent (SD) direct
detection rates $\sigma_{p}^{\rm SI}$ and $\sigma_{n}^{\rm SD}$ as function of
$M_{\chi^{0}_{3}}$ together with the projected sensitivity of the LZ
experiment from [64] in red and the neutrino floor from [65] in green. Note
that present constraints are satisfied automatically due to the singlet-nature
of the LSP: we find a few points within the projected sensitivity of the LZ
experiment but all points are above the neutrino floor.

|
---|---
Figure 5: $\sigma_{p}^{\rm SI}$ (left) and $\sigma_{n}^{\rm SD}$ (right) as
function of $M_{\chi^{0}_{3}}$, together with the projected sensitivity of the
LZ experiment from [64] in red and the neutrino floor from [65] in green.

The coloured dots in all figures indicate six benchmark points: BP1 (red), BP2
(green), BP3 (yellow), BP4 (violet), BP5 (pink) and BP6 (orange). In Tables
2-4 we show their properties: input parameters in Table 2 and lightest
neutralino and chargino masses in Table 3 while signal rates of the benchmark
points shown in Figs. 4-5 are given in Table 4.

| BP1 | BP2 | BP3 | BP4 | BP5 | BP6
---|---|---|---|---|---|---
$\lambda$ | $1.74\\!\times\\!10^{-2}$ | $1.44\\!\times\\!10^{-2}$ | $1.71\\!\times\\!10^{-2}$ | $1.52\\!\times\\!10^{-2}$ | $1.44\\!\times\\!10^{-2}$ | $1.53\\!\times\\!10^{-2}$
$\kappa$ | $7.48\\!\times\\!10^{-3}$ | $6.48\\!\times\\!10^{-3}$ | $7.57\\!\times\\!10^{-3}$ | $6.93\\!\times\\!10^{-3}$ | $6.58\\!\times\\!10^{-3}$ | $7.04\\!\times\\!10^{-3}$
$A_{\lambda}$ | -4287 | -3314 | -3624 | -3551 | -3416 | -4128
$A_{\kappa}$ | 124 | 149 | 240 | 252 | 339 | 336
$\mu_{\rm eff}$ | -155 | -161 | -197 | -200 | -234 | -235
$\tan\beta$ | 8.35 | 8.81 | 6.33 | 6.30 | 5.28 | 7.19
$M_{1}$ | 210 | 207 | 246 | 242 | 271 | 282
$M_{2}$ | 353 | 4853 | 391 | 1296 | 597 | 964
$M_{3}$ | 3822 | 3616 | 2264 | 2444 | 832 | 2259
Table 2: NMSSM specific input parameters and bino ($M_{1}$), wino ($M_{2}$) and gluino ($M_{3}$) masses for the six benchmark points. All dimensionful parameters are given in GeV. | BP1 | BP2 | BP3 | BP4 | BP5 | BP6
---|---|---|---|---|---|---
$M_{\chi^{0}_{1}}$ | 136 | 149 | 177 | 186 | 221 | 224
$M_{\chi^{0}_{2}}$ | 140 | 154 | 182 | 190 | 225 | 229
$M_{\chi^{0}_{3}}$ | 170 | 170 | 210 | 210 | 250 | 250
$M_{\chi^{\pm}_{1}}$ | 156 | 166 | 187 | 205 | 242 | 245
Table 3: Lightest neutralino and chargino masses (in GeV) for the six benchmark points. | BP1 | BP2 | BP3 | BP4 | BP5 | BP6
---|---|---|---|---|---|---
$\mu^{\rm LEP}_{bb}$ | $9.85\\!\times\\!10^{-2}$ | 0.112 | $9.48\\!\times\\!10^{-2}$ | 0.105 | $9.13\\!\times\\!10^{-2}$ | $9.97\\!\times\\!10^{-2}$
$\mu^{\rm LHC}_{\gamma\gamma}$ | $8.04\\!\times\\!10^{-2}$ | $9.66\\!\times\\!10^{-2}$ | $8.05\\!\times\\!10^{-2}$ | $9.27\\!\times\\!10^{-2}$ | $8.09\\!\times\\!10^{-2}$ | $8.98\\!\times\\!10^{-2}$
$\sigma_{p}^{\rm SI}$ | $3.09\\!\times\\!10^{-12}$ | $1.05\\!\times\\!10^{-12}$ | $3.13\\!\times\\!10^{-12}$ | $1.28\\!\times\\!10^{-12}$ | $2.35\\!\times\\!10^{-12}$ | $1.14\\!\times\\!10^{-12}$
$\sigma_{n}^{\rm SD}$ | $3.47\\!\times\\!10^{-8}$ | $3.07\\!\times\\!10^{-8}$ | $1.88\\!\times\\!10^{-8}$ | $1.99\\!\times\\!10^{-8}$ | $7.46\\!\times\\!10^{-9}$ | $1.29\\!\times\\!10^{-8}$
Table 4: The signal rates $\mu^{\rm LEP}_{bb}$, $\mu^{\rm LHC}_{\gamma\gamma}$
and DM direct detection cross sections $\sigma_{p}^{\rm SI}$ and
$\sigma_{n}^{\rm SD}$ (the latter in pb) for the six benchmark points.

##  5 Conclusions

Although the observed excesses in the search for neutralinos and charginos by
ATLAS and CMS can in principle be interpreted in the MSSM, assuming a light
higgsino mass and a compressed higgsino dominated neutralino and chargino
spectrum, such light higgsinos as DM would have far too large direct detection
cross sections. In this paper we have instead considered the NMSSM with an
additional singlino LSP a few GeV below the NLSP. Herein, sparticles
predominantly decay first into the NLSP and remnants from the final decay into
the LSP are too soft to contribute to the observed signals. In contrast to the
MSSM, a singlino LSP in the NMSSM allows to describe easily a DM relic density
in the WMAP/Planck window and is consistent with present limits on the DM
direct detection cross sections.

The considered NMSSM scenario consists of the “compressed” light higgsino-like
triplet made up by ($\chi_{3}^{0}$, $\chi_{1}^{\pm}$, $\chi_{2}^{0}$) plus the
nearby singlino LSP $\chi_{1}^{0}$, with all masses less than about 250 GeV
with of order $5-10\%$ mass splittings. In this scenario, if the LSP is very
close in mass to the NLSP, co-annihilation in the higgsino sector can generate
a relic density in the WMAP/Planck window. This allows us to describe the
excesses observed by the above ATLAS and CMS searches together with DM which
automatically satisfies present (and even future) limits on the DM direct
detection cross sections. Indeed, our scenario can be falsified by the
observation of direct detection cross sections.

Simultaneously, the additional singlet-like Higgs scalar can describe excesses
in the $b\bar{b}$ channel at LEP and in the $\gamma\gamma$ (but not
$\tau\tau$) channel at the LHC at around 95 GeV, with signal cross sections
compatible with the respective observations. Clearly, these and the excesses
in the search for neutralinos and charginos remain to be confirmed. If this is
the case, the NMSSM would provide a very promising framework for their common
explanation.

Finally, to invite exploration of possible solutions provided by this SUSY
framework, we have defined six benchmark points amenable to further
phenomenological investigation.

## Acknowledgements

SFK would like to thank CERN for hospitality and acknowledges the STFC
Consolidated Grant ST/L000296/1 and the European Union’s Horizon 2020 Research
and Innovation programme under Marie Sklodowska-Curie grant agreement HIDDeN
European ITN project (H2020-MSCA-ITN-2019//860881-HIDDeN). SM is supported in
part through the NExT Institute and the STFC Consolidated Grant ST/L000296/1.

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