aainstitutetext: HEP Division, Argonne National Laboratory, 9700 Cass Ave.,
Argonne, IL 60439, USAbbinstitutetext: Enrico Fermi Institute, Physics
Department, University of Chicago, Chicago, IL 60637, USAccinstitutetext:
Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL
60637, USA

# Dark Matter searches with photons at the LHC

Subhojit Roy a,b,c and Carlos E.M. Wagner<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We unveil blind spot regions in dark matter (DM) direct detection (DMDD), for
weakly interacting massive particles with a mass around a few hundred GeV that
may reveal interesting photon signals at the LHC. We explore a scenario where
the DM primarily originates from the singlet sector within the
$Z_{3}$-symmetric Next-to-Minimal Supersymmetric Standard Model (NMSSM). A
novel DMDD spin-independent blind spot condition is revealed for singlino-
dominated DM, in cases where the mass parameters of the higgsino and the
singlino-dominated lightest supersymmetric particle (LSP) exhibit opposite
relative signs (i.e., $\kappa<0$), emphasizing the role of nearby bino and
higgsino-like states in tempering the singlino-dominated LSP. Additionally,
proximate bino and/or higgsino states can act as co-annihilation partner(s)
for singlino-dominated DM, ensuring agreement with the observed relic
abundance of DM. Remarkably, in scenarios involving singlino-higgsino co-
annihilation, higgsino-like neutralinos can distinctly favor radiative decay
modes into the singlino-dominated LSP and a photon, as opposed to decays into
leptons/hadrons. In exploring this region of parameter space within the
singlino-higgsino compressed scenario, we study the signal associated with at
least one relatively soft photon alongside a lepton, accompanied by
substantial missing transverse energy ($\not{E}_{T}$) and a hard initial state
radiation jet at the LHC. In the context of singlino-bino co-annihilation, the
bino state, as the next-to-LSP, exhibits significant radiative decay into a
soft photon and the LSP, enabling the possible exploration at the LHC through
the triggering of this soft photon alongside large $\not{E}_{T}$ and
relatively hard leptons/jets resulting from the decay of heavier higgsino-like
states.

###### Keywords:

Supersymmetry Phenomenology, Dark Matter

††preprint: EFI 24-1

##  1 Introduction

Dark matter (DM) is a fundamental component of the present universe, playing a
vital role in the formation of structures and providing explanations for
phenomena like the rotation patterns of galaxies and various observations in
astrophysics and cosmology. In recent times, the discovery of the Higgs Boson
($h_{\rm SM}$) at the Large Hadron Collider (LHC) around 125 GeV ATLAS:2012yve
; CMS:2012qbp  was a breakthrough moment in our understanding of the laws of
nature, completing the particle content of the Standard Model (SM). However,
following this discovery, in the present understanding of particle physics,
the most crucial and compelling questions are the nature of the DM and the
origin of the weak scale, which is not protected under radiative corrections
caused by the heavy particles that interact with the Higgs boson. Low-scale
supersymmetry (SUSY) Nilles:1983ge ; Haber:1984rc ; Martin:1997ns  emerges as
a highly versatile and among the most popular beyond-the-Standard-Model (BSM)
scenarios since it not only offers a viable candidate for DM Goldberg:1983nd ;
Ellis:1983ew  and solution to the stability problem of the electroweak scale
but also addresses some other theoretical issues of the SM such as the
instability of the electroweak vacuum and the (non)unification of gauge
couplings.

In such a scenario, the lightest supersymmetric Particle (LSP) can potentially
serve as a suitable candidate for DM when a well-known discrete symmetry
called $R$-parity is conserved. In this study, we investigate the scenario
within the $Z_{3}$-symmetric Next-to-Minimal Supersymmetric Standard Model
(NMSSM) Ellwanger:2009dp  where the LSP is singlino-dominated neutralino,
i.e., originating primarily from the singlet sector. As it is neutral, stable,
and exhibits weak interactions, it behaves as a Weakly Interacting Massive
Particle (WIMP)-type DM candidate. Such a singlino-dominated LSP is tempered
by the other nearby neutralinos Arkani-Hamed:2006wnf , which are the
superpartners of the electroweak gauge and Higgs bosons, i.e., the bino, wino
and higgsinos. To avoid over-closing the universe and comply with the Planck
experiment Planck:2015fie ; Planck:2018vyg  reported DM relic density,
sufficient annihilation processes of the singlino-dominated DM in the early
universe is required. A pair of singlino-dominated DM can annihilate via the
usual resonant $s$-channel process exchanging the Z-boson, $h_{\rm SM}$, the
other doublet-like ($H,A$) and singlet-like ($h_{{}_{S}},a_{{}_{S}}$) scalars
and through $t$-channel processes via the exchange of a chargino or
neutralino. It can also co-annihilate with other nearby neutralino and
chargino states. Furthermore, since the singlino-dominated DM can have some
bino mixing, DM co-annihilation with the superpartners of leptons (sleptons)
is also possible. But for the present work, such a process is not possible as
we have fixed all the slepton masses at a few TeV.

In recent years, numerous dedicated searches have been conducted to detect DM
through various experiments focused on DM direct detection (DMDD) and in
colliders such as the LHC. The stringent and improved upper bounds on spin-
independent (SI) and spin-dependent (SD) DM-nucleus (elastic) scattering
cross-sections that are routinely arriving from various DMDD experiments
XENON:2019rxp ; XENON:2023cxc ; PandaX-4T:2021bab ; LZ:2022lsv ; PICO:2017tgi
; IceCube:2021xzo ; Fermi-LAT:2017bpc ; MAGIC:2016xys , put constraints on the
interaction rates of a pair of DM with $h_{\rm SM}$ and other Higgs bosons of
the model and the gauge boson $Z$. For values of the supersymmetric particle
masses of the order of the weak scale, these limits necessitate the
theoretical framework to be in close proximity to blind spots. In this work,
an involved set of DMDD-SI and -SD blind spot conditions for the singlino-
dominated DM are derived by considering $(4\times 4)$ bino-higgsino-singlino
neutralino sector. We consider the wino to be decoupled.

It is well known that at the minimal mixing between the SM-like Higgs boson
and the other Higgs bosons and considering only the singlino-higgsino
neutralino sector, DMDD-SI coupling blind spot condition arises exclusively
when the ratio $m_{{{}_{\chi}}_{{}_{1}}^{0}}/\mu_{\mathrm{eff}}>0$, i.e.,
$\kappa>0$ Badziak:2015exr ; Baum:2017enm ; Cheung:2014lqa ; Badziak:2017uto .
In this blind spot scenario, the effect of the $\lambda$ governed singlino-
higgsino-Higgs interaction term on the LSP becomes significantly suppressed.
In this study, we demonstrate that a new blind spot condition can emerge for
singlino-dominated DM due to the tempering with the bino and higgsino-like
neutralinos. This occurs when singlino mass parameter
$2\kappa\mu_{\mathrm{eff}}/\lambda$ and $\mu_{\mathrm{eff}}$ exhibit an
opposite relative sign, corresponding, for $\lambda>0$, to the $\kappa<0$
region in the parameter space. This unique scenario results from a
cancellation between the effects of the $g_{1}$ governed gaugino-higgsino-
Higgs interaction term and the $\lambda$ proportional singlino-higgsino-Higgs
interaction term on the LSP. Hence, in general, the blind spot condition
arises when there is the same (or opposite) relative sign between
$\mu_{\mathrm{eff}}$ and $M_{1}$ for $\kappa<0$ ($>0$). This opens up a new
region of parameter space characterized by a smaller DMDD-SI cross-section for
the singlino-dominated DM.

This newly found DMDD-SI blind spot region of parameter space of $\kappa<0$
can exhibit an interesting link on the observed discrepancy between the
experimental observations (from Fermilab and BNL) and the SM prediction of the
anomalous muon magnetic moment Muong-2:2006rrc ; Muong-2:2021ojo . Since the
experimentally measured value is larger than the SM prediction, a positive
contribution from new physics is required to explain this discrepancy. Note
that for large values of the Wino mass $M_{2}$, assumed in this work, the
DMDD-SI blind spot condition for $\kappa<0$ demands the same relative sign
between $M_{1}$ and $\mu_{\mathrm{eff}}$, which also provides a positive
contribution from the Bino-smuon loop to the anomalous muon magnetic moment
that can explain the observed discrepancy in the presence of light smuons. It
is important to remember that this discrepancy is based on the theoretical
estimation utilizing cross-section measurements data from ($e^{+}e^{-}\to{\rm
hadrons}$) to determine the hadronic vacuum polarization (HVP) contribution
through dispersion relations Aoyama:2020ynm . Presently, a disagreement exists
in the assessment of the HVP contribution between lattice calculations
Borsanyi:2020mff  and this data-driven determination (for a discussion of this
issue, see reference Coyle:2023nmi ). As this disparity remains unresolved and
awaits further scrutiny, our focus in this work is not directed towards
investigating this phenomenon in the NMSSM framework. Consequently, we treat
smuons as heavy in our analysis. However, if this discrepancy persists, it
would be intriguing to conduct a comprehensive exploration, incorporating
light smuons, within the context of DM phenomenology and the anomalous muon
magnetic moment in NMSSM.

In recent years, various searches have been conducted to discover new
particles at the LHC. Searches for electroweakinos are conducted from the
direct pair-production of the wino-like charginos and neutralinos
($\chi_{{}_{1}}^{\pm}\chi_{{}_{2}}^{0}$) at the LHC Liu:2020ctf  with the
assumption that they eventually decay into two modes:
$W^{\pm}Z\chi_{{}_{1}}^{0}$ and $W^{\pm}h_{\rm SM}\chi_{{}_{1}}^{0}$ and the
LSP ($\chi_{{}_{1}}^{0}$) is bino-like. Such decay modes are studied
considering final states characterized by $3\ell+\not{E}_{T}$ ATLAS:2020ckz ;
CMS:2017moi ; CMS:2018szt ; ATLAS:2018ojr ; ATLAS:2018eui ; ATLAS:2019wgx ;
CMS:2021edw , $1\ell+2b$-${\rm jet}+\not{E}_{T}$ CMS:2017moi ; CMS:2018szt ;
ATLAS:2018qmw ; CMS:2019pov ; ATLAS:2020pgy ; ATLAS:2020ckz  and
$2\ell+2$-${\rm jet}+\not{E}_{T}$ CMS:2017moi ; CMS:2018szt ; ATLAS:2018ojr ;
ATLAS:2018eui ; CMS:2021edw . The pair production of the charginos, i.e.,
$\chi_{{}_{1}}^{\pm}\chi_{{}_{1}}^{\mp}$, is investigated by considering their
decay into the $W^{\pm}W^{\mp}\chi_{{}_{1}}^{0}$ mode in the final state of
$2\ell+\not{E}_{T}$ ATLAS:2019lff ; CMS:2018xqw . Dedicated searches are
conducted considering final states comprising soft multi-leptons/jets,
$\not{E}_{T}$, and possible ISR jets to explore the compressed region of
parameter space. ATLAS:2019lng ; ATLAS:2018ojr ; ATLAS:2019lff ; CMS:2020bfa ;
ATLAS:2021moa ; CMS:2021cox ; CMS:2021few ; ATLAS:2021yqv ; ATLAS:2022zwa ;
CMS:2022sfi ; CMS:2019san . These searches are motivated assuming
$\chi_{{}_{2}}^{0}\rightarrow\chi_{{}_{1}}^{0}f\bar{f}$ decay mode where the
pair of soft leptons/jets ($f\bar{f}$) come from the off-shell $Z$ and $h_{\rm
SM}$. However, null results from these searches of electroweakinos at the LHC
put constraints on their masses. In this work, we focus on the compressed
region of parameter space where the next-to LSP (NLSP) is either higgsino-like
or bino-like neutralino that exhibits substantial radiative decay modes
characterized by larger branching fraction to a photon and the singlino-
dominated LSP ($\chi_{{}_{2}}^{0}\rightarrow\chi_{{}_{1}}^{0}\gamma$).
Furthermore, a novel parameter space region is discussed where both higgsino-
like neutralino states ($\chi_{{}_{2}}^{0}$ and $\chi_{{}_{3}}^{0}$) features
enhanced branching fraction in the
$\chi_{{}_{2}}^{0}\rightarrow\chi_{{}_{1}}^{0}\gamma$ and
$\chi_{{}_{3}}^{0}\rightarrow\chi_{{}_{2}}^{0}\gamma$ decay modes. These
distinct decay patterns may lead to intriguing collider signatures at the LHC.
Note that certain decay channels among these have not been explored in
previous LHC searches, resulting in a relaxation of mass bounds for
electroweakinos. This region of parameter space is primarily motivated by the
DMDD blind spots, where the DMDD-SI scattering cross-section either stays
below the latest bounds from the latest DMDD experiments or even falls below
the irreducible neutrino background. Therefore, it is crucial to explore these
regions of the DM parameter space at the LHC.

In the scenario of singlino-higgsino co-annihilation, the small mass gap
between higgsino-like states ($\chi_{{}_{2,3}}^{0}$) and the singlino-
dominated LSP leads to soft decaying objects from higgsino-like neutralinos
and charginos, arising from their associated production processes
$pp\rightarrow\chi_{{}_{2,3}}^{0}\chi_{{}_{1}}^{\pm}$. Therefore, suppressing
the SM backgrounds becomes difficult. In this work, we discuss, illustrating
various differential distributions of kinematic variables, that this scenario
can be addressed by analyzing the
$pp\rightarrow\chi_{{}_{2,3}}^{0}\chi_{{}_{1}}^{\pm}$ process with an initial-
state radiation jet Baum:2023inl , considering a signal with hard mono-jet,
significant missing energy, and at least one photon. Furthermore, we show that
the presence of a bino-like fourth neutralino state indirectly influences both
collider and DMDD phenomena. It tempers the neutralinos in such a way that
DMDD-SI cross-section of the singlino-dominated DM remains small and higgsino-
like neutralinos feature a larger branching fraction in the radiative photonic
decay mode.

In the singlino-bino co-annihilation scenario, the bino-like NLSP can undergo
a significant radiative decay Domingo:2018ykx ; Baer:2005jq , emitting a
photon along with the singlino-dominated DM. In this scenario, a bino-like
NLSP can emerge with a significant boost resulting from the decay of the
heavier higgsino-like neutralinos and charginos. Such a boost can prevent the
photon, originating from the decay of the NLSP, from remaining soft. The
leptons/ jets originating from the on-shell $Z$, $h_{\rm SM}$ and
$W^{\pm}$-bosons, that are coming from the cascades of the heavier higgsino-
like states, are relatively hard. The cascades of the produced higgsino-like
$\chi_{{}_{3,4}}^{0}$ and $\chi_{{}_{1}}^{\pm}$ via $\chi_{{}_{2}}^{0}$ could
lead to some final states such as $3\ell+\geq 1\gamma+\not{E}_{T}$ or
$1\ell+2b+\geq 1\gamma+\not{E}_{T}$ at the LHC. Note that in some situations,
the cascades of $\chi_{{}_{1}}^{\pm}$ and $\chi_{{}_{3,4}}^{0}$ via bino-like
$\chi_{{}_{2}}^{0}$ can lead to similar final states as when these higgsino-
like states directly decay to the singlino-dominated LSP, if the photon coming
from $\chi_{{}_{2}}^{0}$ remains soft and undetected. Therefore, observing
excess over the SM backgrounds in the signals with and without soft photons
with $3\ell+\not{E}_{T}$ and $1\ell+2b+\not{E}_{T}$ at the HL-LHC could
promptly point back to the scenario that includes singlino-bino co-
annihilation with relatively heavier higgsinos. Furthermore, utilizing
CheckMATE Dercks:2016npn , we point out that the singlino-bino co-annihilation
region of parameter space can be constrained from an analysis ATLAS:2020qlk
conducted by ATLAS. The analysis primarily aims to search for
neutralinos/charginos, considering a pair of photons originating from the
decay of the on-shell $h_{\rm SM}$, which arises from the decay of a heavier
neutralino. As the analysis involves photons, some of the signal regions can
overlap with this specific singlino-bino co-annihilation region of parameter
space of NMSSM.

The paper is organized as follows. In section 2, we describe the theoretical
framework by presenting the superpotential of NMSSM along with its key
aspects, including the electroweakino and scalar (Higgs) sectors. The
interplay of the Higgs sector and the electroweakino sector, involving
correlations in their interactions and masses, is subsequently explored,
considering LSP to be singlino-dominated. Furthermore, the phenomenology of
the singlino-dominated DM relic abundance and various DMDD-SI and DMDD-SD
blind spot conditions considering the singlino-higgsino-bino $(4\times 4)$
neutralino system are discussed. In section 3, we discuss various constraints,
which are relevant to the present work, coming from both the dark sector and
the LHC. The motivated region of NMSSM parameter space of the present work is
discussed in section 4. Furthermore, in section 5, we discuss the searches of
singlino-dominated DM with photons at the LHC. We present a few benchmark
scenarios that feature photons from the cascades of the heavier
electroweakinos to the DM and capture the salient aspects of the present work.
Further, a brief observation is made about the new search channels of those
benchmark points at the LHC. Finally, we conclude in section 6.

##  2 The theoretical scenario

The superpotential of the $Z_{3}$-symmetric NMSSM is given by Ellwanger:2009dp

${\cal W}={\cal
W}_{\mathrm{MSSM}}|_{\mu=0}+\lambda\widehat{S}\widehat{H}_{u}\cdot\widehat{H}_{d}+{\kappa\over
3}\widehat{S}^{3}\,,$ (1)

where ${\cal W}_{\mathrm{MSSM}}|_{\mu=0}$ is the MSSM superpotential including
the Yukawa interactions of the Higgs doublet superfields with the SM quark and
leptons superfields, but with no higgsino mass term (known as $\mu$-term),
$\widehat{H}_{u},\widehat{H}_{d}$ and $\widehat{S}$ are the $SU(2)$ Higgs
doublet superfields of the MSSM and the gauge singlet superfield of the
$Z_{3}$-symmetric NMSSM, respectively, and ‘$\lambda$’ and ‘$\kappa$’ are
dimensionless coupling constants. Assuming the conservation of $R$-parity,
this is the most general $Z_{3}$-symmetry superpotential. Adding the extra
superfield $\widehat{S}$ to the MSSM solves the so-called $\mu$-problem
Kim:1983dt  when it gets non-zero vacuum expectation value (vev)
$v_{\\!{}_{S}}$ and generates dynamically an effective $\mu$-term given by
$\mu_{\mathrm{eff}}=\lambda v_{\\!{}_{S}}$ from the second term in eq. (1).
The soft SUSY-breaking Lagrangian is given by

$-\mathcal{L}^{\rm soft}=-\mathcal{L_{\rm MSSM}^{\rm
soft}}|_{B\mu=0}+m_{S}^{2}|S|^{2}+(\lambda A_{\lambda}SH_{u}\cdot
H_{d}+\frac{\kappa}{3}A_{\kappa}S^{3}+{\rm h.c.})\,,$ (2)

where $m_{S}$ is the soft SUSY-breaking mass of the singlet scalar field,
‘$S$’ and $A_{\lambda}$ and $A_{\kappa}$ are the NSSM-specific trilinear soft
couplings with mass dimension one, and $\mathcal{L_{\rm MSSM}^{\rm
soft}}|_{B\mu=0}$ includes no $H_{u}H_{d}$ Higgs bilinear soft supersymmetry
breaking term . In the next subsections we briefly discuss the electroweakino
sector, the Higgs sector and their interactions.

###  2.1 The electroweakino sector

The electroweakino sector contains the neutralinos and the charginos. NMSSM
has one extra neutralino known as singlino ($\widetilde{S}$) state coming from
the gauge-singlet superfield appearing in the NMSSM superpotential in eq. (1)
compared to the MSSM. Thus, the symmetric $(5\times 5)$ neutralino mass
matrix, in the basis
$\psi^{0}=\\{\widetilde{B},~{}\widetilde{W}^{0},~{}\widetilde{H}_{d}^{0},~{}\widetilde{H}_{u}^{0},~{}\widetilde{S}\\}$,
is given by Ellwanger:2009dp

${\cal
M}_{0}=\left(\begin{array}[]{ccccc}M_{1}&0&-\dfrac{g_{1}v_{d}}{\sqrt{2}}&\dfrac{g_{1}v_{u}}{\sqrt{2}}&0\\\\[11.38092pt]
\ldots&M_{2}&\dfrac{g_{2}v_{d}}{\sqrt{2}}&-\dfrac{g_{2}v_{u}}{\sqrt{2}}&0\\\
\ldots&\ldots&0&-\mu_{\mathrm{eff}}&-\lambda v_{u}\\\
\ldots&\ldots&\ldots&0&-\lambda v_{d}\\\ \ldots&\ldots&\ldots&\ldots&2\kappa
v_{\\!{}_{S}}\end{array}\right)\,,$ (3)

where $g_{1}$ and $g_{2}$ are the $U(1)_{Y}$ and $SU(2)_{L}$ gauge couplings,
respectively, and $v_{u}=v\sin\beta$, $v_{d}=v\cos\beta$ such that
$v^{2}=v_{u}^{2}+v_{d}^{2}\approx(174$ GeV)2 and $\tan\beta=v_{u}/v_{d}$.
$M_{1}$ ($M_{2}$) is the soft SUSY-breaking masses for the $U(1)_{Y}$
($SU(2)_{L}$) gaugino, known as bino (wino). The (5,5) element of the
neutralino mass-matrix in eq. (3) is the singlino mass term,
$m_{{}_{\widetilde{S}}}=2\kappa v_{\\!{}_{S}}$.

The symmetric matrix ${\cal M}_{0}$ in the absence of $CP$-violation can be
diagonalized by an orthogonal $(5\times 5)$ matrix ‘$N$’, i.e.,

$\displaystyle N{\cal M}_{0}N^{T}={\cal M}_{D}={\rm
diag}(m_{{{}_{\chi}}_{{}_{1}}^{0}},m_{{{}_{\chi}}_{{}_{2}}^{0}},m_{{{}_{\chi}}_{{}_{3}}^{0}},m_{{{}_{\chi}}_{{}_{4}}^{0}},m_{{{}_{\chi}}_{{}_{5}}^{0}})\,.$
(4)

The neutralino mass-eigenstates ($\chi_{{}_{i}}^{0}$) are represented in terms
of the weak eigenstates by ($\psi_{j}^{0}$)

$\chi_{{}_{i}}^{0}=N_{ij}\psi_{j}^{0}\,,$ (5)

where $i$ and $j$ run from 1 to 5. This study considers a scenario in which
the wino mass value is decoupled from the other neutralino states. As a
result, the heaviest neutralino state becomes a wino-like state and
$N_{52}\sim 1$. The $(5\times 5)$ symmetric neutralino mass-matrix of eq. (3)
effectively reduces to a $(4\times 4)$ matrix contained the bino, the
higgsinos and the singlino states at the decoupled wino limit. The relations
among these components of a $j$-th neutralino state are given by:

$\displaystyle{N_{j5}\over N_{j1}}$ $\displaystyle=$
$\displaystyle{-{g^{2}_{1}\over 2}v^{2}(\mu_{\mathrm{eff}}\sin
2\beta+m_{{{}_{\chi}}_{j}^{0}})+(M_{1}-m_{{{}_{\chi}}_{j}^{0}})(\mu_{\mathrm{eff}}^{2}-m_{{{}_{\chi}}_{j}^{0}}^{2})\over{g_{1}\over\sqrt{2}}\lambda\,\mu_{\mathrm{eff}}\,(v^{2}_{u}-v^{2}_{d})}\,,$
(6) $\displaystyle{N_{j3}\over N_{j1}}$ $\displaystyle=$
$\displaystyle{{g^{2}_{1}\over
2}v^{2}v_{u}+(M_{1}-m_{{{}_{\chi}}_{j}^{0}})(v_{u}\,m_{{{}_{\chi}}_{j}^{0}}-v_{d}\,\mu_{\mathrm{eff}})\over{g_{1}\over\sqrt{2}}\,\mu_{\mathrm{eff}}\,(v^{2}_{u}-v^{2}_{d})}\,,$
(7) $\displaystyle{N_{j4}\over N_{j1}}$ $\displaystyle=$
$\displaystyle{{g^{2}_{1}\over
2}v^{2}v_{d}+(M_{1}-m_{{{}_{\chi}}_{j}^{0}})(v_{d}\,m_{{{}_{\chi}}_{j}^{0}}-v_{u}\,\mu_{\mathrm{eff}})\over{g_{1}\over\sqrt{2}}\,\mu_{\mathrm{eff}}\,(v^{2}_{u}-v^{2}_{d})}\,,\quad$
(8)

where $N_{j1}$, $N_{j5}$ are the bino and the singlino admixtures,
respectively, and $N_{j3}$, $N_{j4}$ are the higgsino components in the $j$-th
neutralino state. Various components can be obtained from the unitarity
relation, i.e., $N^{2}_{j1}+N^{2}_{j3}+N^{2}_{j4}+N^{2}_{j5}\simeq 1$ For
example, the bino component in the LSP is given by,

$\displaystyle N^{2}_{11}={Z^{2}\over
I(M_{1}-m_{{{}_{\chi}}_{{}_{1}}^{0}})^{2}\,(\mu_{\mathrm{eff}}^{2}-m_{{{}_{\chi}}_{{}_{1}}^{0}}^{2})}\,,$
(9)

where

$\displaystyle
Z={g_{1}\over\sqrt{2}}\,\lambda\,\mu_{\mathrm{eff}}\,(v^{2}_{u}-v^{2}_{d}),$
(10)

and

$\displaystyle I$ $\displaystyle=$
$\displaystyle\mu_{\mathrm{eff}}^{2}-m_{{{}_{\chi}}_{{}_{1}}^{0}}^{2}+(\lambda
v)^{2}{m_{{{}_{\chi}}_{{}_{1}}^{0}}^{2}+\mu_{\mathrm{eff}}^{2}-2m_{{{}_{\chi}}_{{}_{1}}^{0}}\mu_{\mathrm{eff}}\sin
2\beta\over{\mu_{\mathrm{eff}}^{2}-m_{{{}_{\chi}}_{{}_{1}}^{0}}^{2}}}+{\lambda^{2}g^{2}_{1}v^{4}(m_{{{}_{\chi}}_{{}_{1}}^{0}}-\mu_{\mathrm{eff}}\sin
2\beta)\over(M_{1}-m_{{{}_{\chi}}_{{}_{1}}^{0}})\,(\mu_{\mathrm{eff}}^{2}-m_{{{}_{\chi}}_{{}_{1}}^{0}}^{2})}$
(11) $\displaystyle-$ $\displaystyle{g_{1}^{2}v^{2}(\mu_{\mathrm{eff}}\sin
2\beta+m_{{{}_{\chi}}_{{}_{1}}^{0}})\over{M_{1}-m_{{{}_{\chi}}_{{}_{1}}^{0}}}}+{Z^{2}+g_{1}^{4}v^{4}\big{[}\lambda^{2}_{1}v^{2}+(\mu_{\mathrm{eff}}\sin
2\beta+m_{{{}_{\chi}}_{{}_{1}}^{0}})^{2}\big{]}\over(M_{1}-m_{{{}_{\chi}}_{{}_{1}}^{0}})^{2}\,(\mu_{\mathrm{eff}}^{2}-m_{{{}_{\chi}}_{{}_{1}}^{0}}^{2})}\,.\hskip
35.0pt\quad$

This is an extension of the relations obtained for the (3$\times$3) singlino-
higgsino and bino-higgsino systems Baum:2017enm ; Abdallah:2020yag  from eqs.
(6), (7), (8) via decoupling $M_{1}$ and $m_{{}_{\widetilde{S}}}$,
respectively. In this work, we will utilize these relations provided in eqs.
(6) to (11) to calculate various couplings of a pair of singlino-dominated LSP
with other particles.

The chargino sector in the NMSSM is exactly similar with the MSSM but for
$\mu$ $\rightarrow$ $\mu_{\mathrm{eff}}$. The ($2\times 2$) chargino mass
matrix in the bases
$\psi^{+}=\\{-i\widetilde{W}^{+},\,\widetilde{H}_{u}^{+}\\}$ and
$\psi^{-}=\\{-i\widetilde{W}^{-},\,\widetilde{H}_{d}^{-}\\}$, is given by
Ellwanger:2009dp

${\cal M}_{C}=\left(\begin{array}[]{cc}M_{2}&\quad g_{2}v_{u}\\\
g_{2}v_{d}&\quad\mu_{\mathrm{eff}}\end{array}\right).$ (12)

Similar to MSSM, ${\cal M}_{C}$ can be diagonalized by two ($2\times 2$)
unitary matrices ‘$U$’ and ‘$V$’:

$U^{*}{\cal
M}_{C}V^{\dagger}=\mathrm{diag}(m_{{{}_{\chi}}_{{}_{1}}^{\pm}},m_{{{}_{\chi}}_{{}_{2}}^{\pm}})\;;\quad\mathrm{with}\;\;m_{{{}_{\chi}}_{{}_{1}}^{\pm}}<m_{{{}_{\chi}}_{{}_{2}}^{\pm}}\,.$
(13)

###  2.2 The Higgs sector and the interactions with electroweakinos

The soft Lagrangian for the NMSSM Higgs sector from eq. (2) is given below:

$-{\cal L}^{\mathrm{soft}}\supset
m_{H_{u}}^{2}|H_{u}|^{2}+m_{H_{d}}^{2}|H_{d}|^{2}+m_{S}^{2}|S|^{2}+\left(\lambda
A_{\lambda}SH_{u}\cdot
H_{d}+\frac{\kappa}{3}A_{\kappa}S^{3}+\mathrm{h.c.}\right).$ (14)

The neutral Higgs fields $H_{d}^{0}$, $H_{u}^{0}$ and $S$ can be expanded
around their real vev’s i.e., $v_{d}$, $v_{u}$ and $v_{\\!{}_{S}}$. Thus,

$H_{d}^{0}=v_{d}+\frac{H_{dR}+iH_{dI}}{\sqrt{2}},\quad
H_{u}^{0}=v_{u}+\frac{H_{uR}+iH_{uI}}{\sqrt{2}},\quad
S=v_{\\!{}_{S}}+\frac{S_{R}+iS_{I}}{\sqrt{2}},$ (15)

where ‘$R$’ and ‘$I$’ correspond to the $CP$-even and the $CP$-odd states,
respectively. The $CP$-even scalars symmetric squared mass matrix (${\cal
M}_{S}^{2}$) in the basis $H_{jR}=\\{H_{dR},H_{uR},S_{R}\\}$ is given by
Ellwanger:2009dp

${\small{{\cal
M}_{S}^{2}=\left(\begin{array}[]{ccc}g^{2}v_{d}^{2}+\mu_{\mathrm{eff}}B_{\mathrm{eff}}\,\tan\beta&\>\;(2\lambda^{2}-g^{2})v_{u}v_{d}-\mu_{\mathrm{eff}}B_{\mathrm{eff}}&\>\;\lambda(2\mu_{\mathrm{eff}}\,v_{d}-(B_{\mathrm{eff}}+\kappa
v_{\\!{}_{S}})v_{u})\\\\[5.69046pt]
\ldots&\>\;g^{2}v_{u}^{2}+\mu_{\mathrm{eff}}B_{\mathrm{eff}}/\tan\beta&\>\;\lambda(2\mu_{\mathrm{eff}}\,v_{u}-(B_{\mathrm{eff}}+\kappa
v_{\\!{}_{S}})v_{d})\\\\[5.69046pt] \ldots&\ldots&\>\>\,\lambda
A_{\lambda}\frac{v_{u}v_{d}}{v_{\\!{}_{S}}}+\kappa
v_{\\!{}_{S}}(A_{\kappa}+4\kappa v_{\\!{}_{S}})\end{array}\right),}}$ (16)

where $B_{\mathrm{eff}}=A_{\lambda}+\kappa v_{\\!{}_{S}}$ and
$g^{2}=(g_{1}^{2}+g_{2}^{2})/2$. The $CP$-even Higgs bosons ($h_{i}$) mass
eigenstates are then given by

$\displaystyle h_{i}$ $\displaystyle=$ $\displaystyle
S_{ij}H_{jR},\qquad\mathrm{with}\quad{i,j=1,2,3}\,,$ (17)

where the matrix ‘$S$’ diagonalizes ${\cal M}_{S}^{2}$. The interaction
coupling of the $CP$-even scalars ($h_{i}$), a pair of neutralinos
$\chi_{{}_{j}}^{0}$ and $\chi_{{}_{k}}^{0}$ are given by Ellwanger:2009dp

$\displaystyle g_{{}_{h_{i}\chi_{{}_{j}}^{0}\chi_{{}_{k}}^{0}}}$
$\displaystyle=$
$\displaystyle{\lambda\over\sqrt{2}}(S_{i1}\Pi^{45}_{jk}+S_{i2}\Pi^{35}_{jk}+S_{i3}\Pi^{34}_{jk})-\sqrt{2}\,\kappa
S_{i3}N_{j5}N_{k5}$ (18) $\displaystyle+$ $\displaystyle{g_{1}\over
2}(S_{i1}\Pi^{13}_{jk}-S_{i2}\Pi^{14}_{jk})-{g_{2}\over
2}(S_{i1}\Pi^{23}_{jk}-S_{i2}\Pi^{24}_{jk}),$

where $\Pi^{ab}_{jk}=N_{ja}N_{kb}+N_{jb}N_{ka}$.

In a more convenient rotated basis, ($\hat{h},\widehat{H},\hat{s}$)
Miller:2003ay ; Badziak:2015exr  where
$\hat{h}=H_{dR}\cos{\beta}+H_{uR}\sin{\beta}$,
$\widehat{H}=H_{dR}\sin{\beta}-H_{uR}\cos{\beta}$ and $\hat{s}=S_{R}$,
$\hat{h}$ mimics the SM Higgs field where as $\widehat{H}$ resembles the MSSM
heavy doublet-like $CP$-even Higgs. The physical $CP$-even scalar states are
given by

$\displaystyle h_{i}$ $\displaystyle=$ $\displaystyle
E_{h_{i}\hat{h}}\hat{h}+E_{h_{i}\widehat{H}}\widehat{H}+E_{h_{i}\hat{s}}\hat{s}\,,$
(19)

where $E_{ab}$ is the diagonalizing matrix of the mass-squared matrix for the
$CP$-even scalars in the rotated basis. In this rotated basis
($\hat{h},\widehat{H},\hat{s}$) eq. (18) reduces to

$\displaystyle g_{{}_{h_{i}\chi_{{}_{j}}^{0}\chi_{{}_{k}}^{0}}}$
$\displaystyle=$
$\displaystyle\Bigg{[}{\lambda\over\sqrt{2}}\big{[}E_{h_{i}\hat{h}}N_{j5}(N_{k3}\sin\beta+N_{k4}\cos\beta)+E_{h_{i}\hat{H}}N_{j5}(N_{k4}\sin\beta-
N_{k3}\cos\beta)$ (20) $\displaystyle\hskip
25.0pt+\;E_{h_{i}\hat{s}}(N_{j3}N_{k4}-{\kappa\over\lambda}N_{j5}N_{k5})\big{]}+{1\over
2}\big{[}g_{1}N_{j1}-g_{2}N_{j2}\big{]}\big{[}E_{h_{i}\hat{h}}(N_{k3}\cos\beta-
N_{k4}\sin\beta)$ $\displaystyle\hskip
25.0pt+E_{h_{i}\hat{H}}(N_{k3}\sin\beta+N_{k4}\cos\beta)\big{]}\Bigg{]}+\Bigg{[}j\longleftrightarrow
k\Bigg{]}\,.$

For no mixing between the singlet and the doublet scalars ($E_{h_{\rm
SM}\hat{s}},E_{H\hat{s}}\sim 0$) and between the doublet Higgs states (i.e.,
$E_{h_{\rm SM}\hat{h}},E_{H\hat{H}}\sim 1$ and $E_{h_{\rm SM}\hat{H}}\sim 0$)
Badziak:2015exr , the coupling of a pair of LSP with the SM-like Higgs boson
is given by Abdallah:2020yag ,

$\displaystyle g_{{}_{h_{\rm SM}\chi_{{}_{1}}^{0}\chi_{{}_{1}}^{0}}}$
$\displaystyle\simeq$
$\displaystyle\sqrt{2}\lambda\,(N_{13}\sin\beta+N_{14}\cos\beta)N_{15}+g_{1}(N_{13}\cos\beta-
N_{14}\sin\beta)N_{11},$ (21)

where the $g_{1}$-proportional second term in (21) has pure MSSM origin
(gaugino-higgsino-Higgs interaction) and the first $\lambda$-proportional term
is only possible in the NMSSM due to the higgsino-singlino-Higgs interaction.
Using eqs. (6) to (11), eq. (21) reduces to

$\displaystyle\hskip-30.0ptg_{{}_{h_{\rm
SM}\chi_{{}_{1}}^{0}\chi_{{}_{1}}^{0}}}$ $\displaystyle\simeq$
$\displaystyle{\sqrt{2}\lambda^{2}v\over
I}\Bigg{[}m_{{{}_{\chi}}_{{}_{1}}^{0}}-\mu_{\mathrm{eff}}\sin
2\beta+{{g_{1}^{2}v^{2}}\over{M_{1}-m_{{{}_{\chi}}_{{}_{1}}^{0}}}}+{g_{1}^{4}v^{4}\over
4}{m_{{{}_{\chi}}_{{}_{1}}^{0}}+\mu_{\mathrm{eff}}\sin
2\beta\over(M_{1}-m_{{{}_{\chi}}_{{}_{1}}^{0}})^{2}\,(\mu_{\mathrm{eff}}^{2}-m_{{{}_{\chi}}_{{}_{1}}^{0}}^{2})}\Bigg{]}\,.$
(22)

This coupling holds particular importance as it directly influences the spin-
independent (SI) scattering cross-section for the direct detection of DM.
Subsequently, we will utilize it to obtain the“coupling blind spot” for the
singlino-dominated DMDD-SI cross-section.

On the other hand, the $CP$-odd scalars (3 $\times$ 3) symmetric squared mass
matrix in the basis $H_{jI}=\\{H_{dI},H_{uI},S_{I}\\}$ is given by
Ellwanger:2009dp

${\small{{\cal
M^{\prime}}_{P}^{2}=\left(\begin{array}[]{ccc}\mu_{\mathrm{eff}}B_{\mathrm{eff}}\,\tan\beta&\mu_{\mathrm{eff}}B_{\mathrm{eff}}&\lambda
v_{u}(A_{\lambda}-2\kappa v_{\\!{}_{S}})\\\\[5.69046pt]
\ldots&\mu_{\mathrm{eff}}B_{\mathrm{eff}}/\tan\beta&\lambda
v_{d}(A_{\lambda}-2\kappa v_{\\!{}_{S}})\\\\[5.69046pt]
\ldots&\ldots&\lambda(B_{\mathrm{eff}}+3\kappa
v_{\\!{}_{S}})\frac{v_{u}v_{d}}{v_{\\!{}_{S}}}-3\kappa
A_{\kappa}v_{\\!{}_{S}}\end{array}\right),}}$ (23)

Where $B_{\mathrm{eff}}=A_{\lambda}v_{s}$. Similar to the $CP$-even Higgs
sector, working on the rotated basis $\\{A,S_{I}\\}$ where $A=\cos\beta
H_{uI}+\sin\beta H_{dI}$, dropping the massless Nambu-Goldstone mode the (2
$\times$ 2) $CP$-odd (pseudoscalar) Higgs boson squared mass matrix is given
by,

${\small{\cal
M}_{P}^{2}=\left(\begin{array}[]{cc}m_{A}^{2}{}&~{}\lambda(A_{\lambda}-2\kappa
v_{\\!{}_{S}})\,v\\\\[5.69046pt] \lambda(A_{\lambda}-2\kappa
v_{\\!{}_{S}})\,v{}&~{}\lambda(A_{\lambda}+4\kappa
v_{\\!{}_{S}})\frac{v_{u}v_{d}}{v_{\\!{}_{S}}}-3\kappa
A_{\kappa}\,v_{\\!{}_{S}}\end{array}\right),}$ (24)

where, $m_{A}^{2}=2\mu_{\mathrm{eff}}B_{\mathrm{eff}}/\sin 2\beta$ is the
MSSM-like $CP$-odd Higgs squared mass. The $CP$-odd (pseudoscalar, $a_{k}$)
mass eigenstates, in terms of ${\cal M^{\prime}}_{P}^{2}$ diagonalization
matrix ‘${\cal O}$’, are given by

$\displaystyle a_{k}={\cal O}_{kA}A+{\cal
O}_{kS_{I}}S_{I},\qquad\mathrm{with}\quad{k=1,2},$ (25)

The relations among the elements of the ‘${\cal O}$’ and ‘$P$’ which
diagonalizes ${\cal M^{\prime}}_{P}^{2}$ are given by Ellwanger:2009dp

$P_{i1}=\sin\beta\,{\cal O}_{iA}\,,\qquad P_{i2}=\cos\beta\,{\cal
O}_{iA}\,,\qquad P_{i3}={\cal O}_{iS_{I}}\,.$ (26)

The interaction coupling of the $CP$-odd scalars, $a_{i}$ and a pair of
neutralinos, $\chi_{{}_{j}}^{0}$ and $\chi_{{}_{k}}^{0}$ is given by,

$\displaystyle g_{{}_{a_{i}\chi_{{}_{j}}^{0}\chi_{{}_{k}}^{0}}}$
$\displaystyle=$
$\displaystyle\Bigg{[}i\Big{(}{\lambda\over\sqrt{2}}\big{[}{\cal
O}_{iA}N_{j5}(N_{k4}\sin\beta+N_{k3}\cos\beta)+{\cal
O}_{iS_{I}}(N_{j3}N_{k4}-{\kappa\over\lambda}N_{j5}N_{k5})\big{]}$ (27)
$\displaystyle\hskip 25.0pt-\;{1\over
2}\big{[}g_{1}N_{j1}-g_{2}N_{j2}\big{]}\big{[}{\cal O}_{iA}(N_{k3}\sin\beta-
N_{k4}\cos\beta)\big{]}\Big{)}\Bigg{]}+\Bigg{[}j\longleftrightarrow
k\Bigg{]}.\quad$

For singlet-dominated $CP$-odd Higgs ${\cal O}_{iS_{I}}\sim 1$ and ${\cal
O}_{iA}\sim 0$. This indicates that the $CP$-odd singlet Higgs interaction
with a pair of singlino-dominated LSP mainly depends on the second term in the
first line of eq. (27). We will utilize these relationships later in our
discussion when estimating DM relic density and DMDD cross-sections.

The squared mass of the SM-like Higgs, at one-loop level, is given by
Ellwanger:2011sk ; Carena:2015moc

$m_{h_{\mathrm{SM}}}^{2}=m_{Z}^{2}\cos^{2}2\beta+\lambda^{2}v^{2}\sin^{2}2\beta+\Delta_{\mathrm{mix}}+\Delta_{\mathrm{rad.\,corrs.}}\,,$
(28)

where $\Delta_{\mathrm{mix}}$ is the contribution from the possible a singlet-
doublet mixing and $\Delta_{\mathrm{rad.\,corrs.}}$ is the MSSM-like one-loop
radiative corrections.

The tree-level squared mass of the $CP$-even singlet-like Higgs ($h_{s}$), at
the minimal mixing with the doublet-like states, is given by the (3,3)
component of ${\cal M}_{S}^{2}$, i.e.,

$m^{2}_{{}_{h_{S}}}\approx{\cal{M}}^{2}_{S,33}=\lambda
A_{\lambda}\frac{v_{u}v_{d}}{v_{\\!{}_{S}}}+\kappa
v_{\\!{}_{S}}(A_{\kappa}+4\kappa v_{\\!{}_{S}})\,.$ (29)

On the other hand, the tree-level singlet-like $CP$-odd Higgs squared mass (up
to some mixing with its doublet cousin) is given the (2 $\times$ 2) component
of the ${\cal M}_{P}^{2}$ matrix, i.e.,

$m^{2}_{a_{{}_{S}}}\approx{\cal{M}}^{2}_{P,22}=\lambda(A_{\lambda}+4\kappa
v_{\\!{}_{S}})\frac{v_{u}v_{d}}{v_{\\!{}_{S}}}-3\kappa
A_{\kappa}\,v_{\\!{}_{S}}\,.$ (30)

It can be seen from ${\cal M}_{S}^{2}$ and ${\cal M}_{P}^{2}$ that the masses
of various Higgs bosons have involved dependency on the six input parameters
$\tan\beta$, $\mu_{\mathrm{eff}}$, $\lambda$, $\kappa$, $A_{\lambda}$ and
$A_{\kappa}$. The electroweakino mass matrix ${\cal M}_{0}$ depends on the
first four input parameters with $M_{1}$ and $M_{2}$. It is expected that the
masses of the singlet-like Higgs bosons ($a_{s}$ and $h_{s}$) are connected
with the singlino mass ($m_{{}_{\widetilde{S}}}$).

The first term in eq. (30) could be ignored at small $\lambda$ and
$m^{2}_{a_{{}_{S}}}\approx|-3\kappa A_{\kappa}v_{\\!{}_{S}}|=|-{3\over
2}A_{\kappa}m_{{}_{\widetilde{S}}}|$. For a fixed $m_{{}_{\widetilde{S}}}$,
the mass of $a_{S}$ decreases with $A_{\kappa}$. In this study, we focus on
the small $A_{\kappa}$ region of parameter space for relatively lighter
$a_{S}$. For larger $m_{{}_{\widetilde{S}}}$,
$m_{{}_{h_{S}}}\sim|m_{{}_{\widetilde{S}}}|$ (from eq. 29). For smaller
$m_{{}_{\widetilde{S}}}$, $m_{{}_{h_{S}}}$ depends on various other parameters
and it could be even smaller than $m_{h_{\mathrm{SM}}}$. In this study, the
singlino is relatively light, which corresponds to a relatively lighter
$h_{{}_{S}}$ in the spectrum. The $CP$-even and $CP$-odd doublet-like heavy
doublet Higgs masses ($m_{H}$ and $m_{A}$, respectively) decouple at larger
$A_{\lambda}$ limit which could be observed from the (1,1) component of ${\cal
M}_{S}^{2}$ and ${\cal M}_{P}^{2}$. Larger $A_{\lambda}$ provides the
“alignment via decoupling” condition in the doublet Higgs sector. Before
ending this section, we would like to highlight the interaction coupling of
the $Z$-boson with a pair of neutralinos, as we will refer to it in future
discussions. The same coupling is given by

$g_{{}_{Z\chi_{{}_{j}}^{0}\chi_{{}_{k}}^{0}}}=\frac{g_{2}}{2\cos\theta_{W}}(N_{j3}N_{k3}-N_{j4}N_{k4})\,,$
(31)

where $\theta_{W}$ is the weak mixing (Weinberg) angle.

###  2.3 The dark matter sector

In the presence of $R$-parity-conserving SUSY models, the LSP (R-odd)
naturally produces a DM candidate of the universe Goldberg:1983nd ;
Ellis:1983ew . In this work, we choose singlino-dominated DM with singlino
content at least 90$\%$. We consider the Planck reported DM relic density
upper bound ($\Omega h^{2}\lesssim 0.120$) constraint (to avoid the
possibility of an over-closed universe). To meet this constraint, sufficient
annihilation of the singlino-dominated LSP in the early universe is needed.
The usual resonant $s$-channel annihilation for a pair of LSP is via the
exchange of $Z$-boson, $h_{\rm SM}$ and the singlet-like Higgs bosons $a_{S}$,
$h_{S}$ 111Note that constraints arising from searches for heavy doublet-like
Higgs bosons ($H$, $A$) at the LHC restrict the possibility of light DM (below
200 GeV) annihilation through their exchange.. A pair of DM may also
annihilate through $t$-channel processes via the exchange of a chargino or
neutralino. Furthermore, the singlino-dominated DM might co-annihilate with
bino-like or higgsino-like NLSP222As the DM has some bino mixing, DM co-
annihilation with sleptons is also possible. But for the present work, this DM
annihilation mechanism is not possible as we have fixed all the slepton masses
at few TeV.. It is crucial to emphasize that, for all these processes, the
mixing of the singlino-dominated state with the bino-like and the higgsino-
like states can play an important role. It can be seen from eq. (31) and (21)
that for the $Z$ and $h_{\rm SM}$ mediated processes, the singlino-dominated
LSP needs at least some higgsino admixtures. This singlino-higgsino mixing
significantly depends on $\lambda$, which controls the strength of some
important couplings of those processes. It can be observed from eqs. (6), (7)
and (8) that at smaller $\lambda$ region a significant bino-mixing (driven by
$g_{1}$ and the soft mass $M_{1}$) is possible in the singlino-dominated LSP.
In this scenario, the $g_{1}$ proportional second term in eq. (21), i.e., the
gauge interactions with the pair of LSP, can become significant in
$g_{{}_{h_{\rm SM}\chi_{{}_{1}}^{0}\chi_{{}_{1}}^{0}}}$. Furthermore, it can
also be noted from eqs. (7) and (8) that such bino mixing could exhibit a non-
trivial dependence on the $Z$-boson coupling with a pair of LSP, as described
in eq. (31) with $j,k=$ 1\. In a subsequent section, we will explicitly
demonstrate some intricate dependencies of these couplings on the $M_{1}$
parameter. The $a_{S}$ and $h_{S}$ mediated DM annihilation processes
significantly depend on $\kappa$ (via the singlino-singlino-singlet
interaction) and $\lambda$ (through the higgsino-higgsino-singlet interaction)
parameters.

The null results from numerous searches of DM at various DMDD experiments put
strong constrains on both the DM-nucleon SI and SD scattering cross-sections.
In the heavy squark limit, the $t$-channel three $CP$-even scalars ($h_{SM}$,
$h_{H}$ and $h_{S}$) mediated processes mostly contribute to the DMDD-SI
scattering cross-section. The interactions of these scalars with an LSP pair
are given by eq. (20) (with $j=k=0$) and with a pair of nucleons is given by,

$\displaystyle g_{{}_{h_{i}NN}}$ $\displaystyle=$
$\displaystyle\frac{m_{N}}{\sqrt{2}\,v}\left({S_{i1}\over\cos\beta}F^{(N)}_{d}+{S_{i2}\over\sin\beta}F^{(N)}_{u}\right),$
(32)

where $m_{N}$ is the mass of the nucleon, $F^{(N)}_{d,u}$ are the combinations
of various nucleon form factors, refs. Belanger:2013oya ; Badziak:2015nrb .
Note that this interaction only depends on the doublet admixtures of the
$CP$-even Higgs bosons. This implies that a pair of nucleons coupling with the
singlet-like $CP$-even Higgs boson is highly suppressed. The SI scattering
cross-section $(\sigma^{\rm SI}_{\chi_{{}_{1}}^{0}-(N)})$ is given by
Cao:2018rix

$\sigma^{\rm
SI}_{\chi_{{}_{1}}^{0}-(N)}=\frac{4\mu^{2}_{r}}{\pi}\left|f^{(N)}\right|^{2},\hskip
14.22636ptf^{(N)}=\sum_{i=1}^{3}{\frac{g_{{}_{h_{i}\chi_{{}_{1}}^{0}\chi_{{}_{1}}^{0}}}\,g_{{}_{h_{i}NN}}}{2m^{2}_{{}_{h_{i}}}}}\,,$
(33)

where $\mu_{r}$ is the DM-nucleon system reduced mass. We consider the
“$decoupling$ $limit$” condition in the Higgs sector. In this case,
$1/m_{H}^{4}$ effectively suppresses the contribution of this decoupled
$CP$-even Higgs boson to the SI cross-section.

In the basis of eq. (19), eq. (32) is given by Badziak:2015exr

$\displaystyle g_{{}_{h_{i}NN}}$ $\displaystyle=$
$\displaystyle\frac{m_{N}}{\sqrt{2}v}\Big{[}E_{h_{i}\hat{h}}(F^{(N)}_{d}+F^{(N)}_{u})+E_{h_{i}\hat{H}}\big{(}\tan\beta
F^{(N)}_{d}-\frac{1}{\tan\beta}F^{(N)}_{u}\big{)}\Big{]}\,.$ (34)

The second term in (34) suggests that for larger $\tan\beta$, the
$g_{{}_{HNN}}$ coupling is strengthened. In the larger $\tan\beta$ region of
parameter space, the mass suppression effect of the heavy doublet-like Higgs,
$H$ (with a mass of a few TeV), in the contribution to the DMDD-SI cross-
section can be compensated by this enhancement factor. In such cases, the
heavy Higgs contribution may become comparable to that of the SM-like Higgs
boson at 125 GeV. The contribution of the singlet-like Higgs boson, $h_{S}$,
is suppressed by its coupling with a pair of nucleons. On the other hand, for
the singlino-dominated LSP, the
$g_{{}_{h_{{}_{S}}\chi_{{}_{1}}^{0}\chi_{{}_{1}}^{0}}}$ coupling can become
relatively large, resulting in a considerable contribution to the SI cross-
section. Additionally, a lighter singlino state correlates to a lighter
$h_{{}_{S}}$ (even smaller than $h_{\rm SM}$), which can aid in increasing its
contribution to the SI cross-section.

The region of parameter space where the DMDD-SI scattering cross-section is
modest has been pushed by recent experimental limits. Nonetheless, there are
“blind spot” locations in the parameter space where the neutralino LSP SI
scattering cross-section (nearly) vanishes. A relatively small $g_{{}_{h_{\rm
SM}\chi_{{}_{1}}^{0}\chi_{{}_{1}}^{0}}}$ (the so-called ‘coupling blind spot’)
reduces the contribution of the $h_{\rm SM}$ mediated process to the SI cross-
section. On the other hand, another type of blind spot of the SI cross-section
could happen due to the destructive interference among the processes mediated
by various $CP$-even scalars. In the vicinity of such blind spot conditions,
the DMDD-SI cross-section could be brought down to the irreducible neutrino
background (the so-called ‘neutrino floor/fog’ $\sim
10^{-49}\,\mathrm{cm}^{2}$) Billard:2013qya ) of the DMDD experiments. When
the squarks are heavy, $Z$-boson-mediated t-channel process contributes to the
DMDD-SD scattering cross-section. A similar type of “SD blind spot” could
occur when $g_{{}_{Z\chi_{{}_{1}}^{0}\chi_{{}_{1}}^{0}}}$ becomes very small.
In the next subsections, we will discuss the conditions of blind spots of SI
and SD scattering cross-sections for the singlino-dominated DM (LSP).

####  2.3.1 The SI coupling blind spot

The SM-like $CP$-even Higgs, $h_{\rm SM}$, dominates the DMDD-SI cross-section
in the region of parameter space where the other $CP$-even Higgs bosons $H$
and $h_{{}_{S}}$ are decoupled and their contributions to the SI scattering
cross-section are negligible. A vanishing $g_{{}_{h_{\rm
SM}\chi_{{}_{1}}^{0}\chi_{{}_{1}}^{0}}}$ ($\approx 0$) causes a (coupling)
blind spot scenario, which indicates a significant drop in the contributions
to the DMDD-SI cross-section via the $h_{\rm SM}$-mediated process. The
required condition for the coupling blind spot is given by (from eq. 22),

$\displaystyle\Bigg{[}m_{{{}_{\chi}}_{{}_{1}}^{0}}+{{g_{1}^{2}v^{2}}\over{M_{1}-m_{{{}_{\chi}}_{{}_{1}}^{0}}}}+{g_{1}^{4}v^{4}\over
4}{m_{{{}_{\chi}}_{{}_{1}}^{0}}+\mu_{\mathrm{eff}}\sin
2\beta\over(M_{1}-m_{{{}_{\chi}}_{{}_{1}}^{0}})^{2}\,(\mu_{\mathrm{eff}}^{2}-m_{{{}_{\chi}}_{{}_{1}}^{0}}^{2})}\Bigg{]}{1\over\mu_{\mathrm{eff}}\sin
2\beta}=1\,.$ (35)

In the bino decoupling limit
($M_{1}>>\mu_{\mathrm{eff}},m_{{}_{\widetilde{S}}}$), eq. (35) reproduces the
“well-known” coupling blind spot condition for singlino-dominated LSP
Baum:2017enm  in a singlino-higgsino (3$\times$3) system of neutralinos, i.e.,

$\displaystyle{m_{{{}_{\chi}}_{{}_{1}}^{0}}\over\mu_{\mathrm{eff}}}\approx\sin
2\beta\,.$ (36)

This blind spot condition appears only when the $\lambda$ proportional first
term in eq. (21), i.e., the singlino-higgsino-Higgs interaction term, is
considered. Note that such a criterion of eq. (36) can only be satisfied if
$m_{{{}_{\chi}}_{{}_{1}}^{0}}$ and $\mu_{\mathrm{eff}}$ have the same relative
sign. Usually, the sign of a singlino-dominated LSP
($m_{{{}_{\chi}}_{{}_{1}}^{0}}$) depends on the sign of
$m_{{}_{\widetilde{S}}}$ (2$\kappa\mu_{\mathrm{eff}}/\lambda$). This implies
$\kappa>0$ to satisfy eq. (36). It is also worth noting that to meet the
aforementioned blind-spot requirement, $\tan\beta$ must grow with
$\mu_{\mathrm{eff}}$ for a fixed $m_{{{}_{\chi}}_{{}_{1}}^{0}}$.

The presence of a relatively light bino-like state (low $M_{1}$) can open up a
new region of parameter space where singlino-dominated DM exhibits a notably
low SI scattering cross-section in DMDD experiments. In comparison to the
second term, the third term on the left-hand side of eq. (35) is suppressed
for $|M_{1}|-|m_{{{}_{\chi}}_{{}_{1}}^{0}}|\lesssim g_{1}v$ and
$|\mu_{\mathrm{eff}}|-|m_{{{}_{\chi}}_{{}_{1}}^{0}}|\gg g_{1}v$. In this
scenario, it is more probable that $\chi_{{}_{2}}^{0}$ exhibits
characteristics of the bino-like state, while the higgsino-like neutralino
states correspond to $\chi_{{}_{3}}^{0}$ and $\chi_{{}_{4}}^{0}$. In this
limit, the coupling blind spot condition of eq. (35) can be approximated as,

$\displaystyle\bigg{(}m_{{{}_{\chi}}_{{}_{1}}^{0}}+{g_{1}^{2}v^{2}\over{M_{1}-m_{{{}_{\chi}}_{{}_{1}}^{0}}}}\bigg{)}{1\over\mu_{\mathrm{eff}}\sin
2\beta}\simeq 1\,.$ (37)

It is worth noting that the second term on the left-hand side of eq. (37) is
derived from the $g_{1}$-proportional term (i.e., the MSSM origin gaugino-
higgsino-Higgs interaction term) in eq. (21). A further inspection reveals
that when $m_{{{}_{\chi}}_{{}_{1}}^{0}}$ and $\mu_{\mathrm{eff}}$ carry a
relatively opposite sign between them, i.e. $\kappa<0$ region of parameter
space, such agreement of eq. (37) requires $M_{1}$ and $\mu_{\mathrm{eff}}$ to
have the same relative sign. This opens up a new parameter space for singlino-
dominated DM with a significantly small DMDD-SI cross-section. eq. (36) always
demands $\kappa>0$ whereas the criteria of eq. (37) can be fulfilled even in
$\kappa<0$ in the presence of light $M_{1}$ with same relative sign with
$\mu_{\mathrm{eff}}$. For $\kappa>0$ situation, i.e. same relative sign
between $m_{{{}_{\chi}}_{{}_{1}}^{0}}$ and $\mu_{\mathrm{eff}}$, condition of
eq. (37) can be satisfied irrespective of $M_{1}$ sign with
$\mu_{\mathrm{eff}}$. In this case, the preferred region for the DM to exhibit
a relatively low DMDD-SI scattering cross-section depends on the relative sign
and the relative mass splits between the neutralino states and the term
$\mu_{\mathrm{eff}}\sin\beta$.

As previously discussed, in order to satisfy eq. (36) with $\kappa>0$, for a
fixed mass of the singlino-dominated DM $\tan\beta$ must rise with
$\mu_{\mathrm{eff}}$. Thus, for relatively light DM exhibiting a considerably
low DMDD-SI cross-section, the parameter space favoring higher values of
$\tan\beta$ would also lean toward higher values of $\mu_{\mathrm{eff}}$.
However, the more generalized blind spot condition expressed in eq. (37)
indicates that DMDD-SI rates can become small at relatively low $\tan\beta$
for larger $\mu_{\mathrm{eff}}$ in the context of relatively light singlino-
dominated DM and $\kappa>0$. This requires relatively low $M_{1}$ and prefers
relative sign combinations between $\mu_{\mathrm{eff}}$ and $M_{1}$.

The DMDD-SI scattering blind spot conditions due to the destructive
interference between the $CP$-even Higgs bosons ($h_{\rm SM}$, $H$,
$h_{{}_{S}}$) mediated processes for singlino-dominated DM of a singlino-
higgsino (3$\times$3) neutralino system is well studied in the literature
Badziak:2015exr ; Badziak:2015nrb . However, the presence of a relatively
light bino state could alter the blind spot conditions and open up new regions
of parameter space of low DMDD-SI scattering cross-section. This requires a
more detailed study and we plan to explore it in a future work.

####  2.3.2 The SD blind spot

The $Z$-mediated LSP-nucleon DMDD-SD scattering cross-section $(\sigma^{\rm
SD}_{\chi_{{}_{1}}^{0}-(N)})$ approximately equals to

$\sigma^{\rm SD}_{\chi_{{}_{1}}^{0}-(N)}\simeq
C_{N}\times\bigg{(}\frac{g_{{}_{Z\chi_{{}_{1}}^{0}\chi_{{}_{1}}^{0}}}}{0.01}\bigg{)}^{2},$
(38)

where $C_{p}\simeq 2.9\times 10^{-41}$ $\text{cm}^{2}$ for proton and
$C_{n}\simeq 2.3\times 10^{-41}$ $\text{cm}^{2}$ for neutron. The SD
scattering cross-section depends on
$g_{{}_{Z\chi_{{}_{1}}^{0}\chi_{{}_{1}}^{0}}}$ which is proportional to
$N_{13}^{2}-N_{14}^{2}$ (from eq. 31 with $j=k=1$) Haber:1984rc . Using eqs. 8
to 11 one can find,

$\displaystyle N^{2}_{13}-N^{2}_{14}={\lambda^{2}v^{2}\over I}\cos
2\beta\Bigg{[}-1+{g_{1}^{2}v^{2}\over\bigg{(}{M_{{}_{1}}\over
m_{{{}_{\chi}}_{{}_{1}}^{0}}}-1\bigg{)}(\mu_{\mathrm{eff}}^{2}-m_{{{}_{\chi}}_{{}_{1}}^{0}}^{2})}+{g_{1}^{4}v^{4}\over
4(M_{1}-m_{{{}_{\chi}}_{{}_{1}}^{0}})^{2}(\mu_{\mathrm{eff}}^{2}-m_{{{}_{\chi}}_{{}_{1}}^{0}}^{2})}\Bigg{]}\,.$

At $\tan\beta=1$, $\cos 2\beta$ vanishes, and a well-known MSSM-like DMDD-SD
blind spot appears. Another new kind of DMDD-SD blind spot can happen in the
NMSSM if the terms inside the bracket of the above expression cancel each
other. This can happen most likely when $M_{1}$ and
$m_{{{}_{\chi}}_{{}_{1}}^{0}}\sim m_{{}_{\widetilde{S}}}$ have the same sign.
It indicates that the presence of a light bino state could degrade the DMDD-SD
scattering cross-section of singlino-dominated DM irrespective of the
$\tan\beta$ value. Note that, in the limit
$m_{{{}_{\chi}}_{{}_{1}}^{0}}^{2}\sim
m_{{}_{\widetilde{S}}}^{2}\ll\mu_{\mathrm{eff}}^{2},\,M_{1}^{2}$, eq. (38)
leads to $\sigma^{\rm SD}_{\chi_{{}_{1}}^{0}-(N)}\propto
1/\mu_{\mathrm{eff}}^{4}$. Thus, the DMDD-SD scattering cross-section
decreases as $\mu_{\mathrm{eff}}$ increases.

##  3 The spectra and constraints from various sectors

In our study, we take into consideration a variety of constraints from diverse
sectors, spanning both theoretical and experimental domains. The theoretical
constraints involve ensuring that the spectra remain free from tachyonic
states, preventing the scalar potential from developing an unphysical global
minimum, and ensuring that various relevant couplings within the theory do not
encounter Landau poles as energy varies, etc. On the experimental front, we
incorporate constraints derived from observations in the Higgs sector, DM
sector, flavor sectors, and various new physics searches at the colliders. We
use various publicly available packages like NMSSMTools (v5.5.3)
Ellwanger:2005dv ; Das:2011dg , HiggsBounds (v5.8.0) Bechtle:2020pkv ,
HiggsSignals (v2.5.0) Bechtle:2020uwn , SModelS (v2.1.3) Alguero:2021dig  and
CheckMATE (v2.0.37) Dercks:2016npn .

NMSSMTools is employed to compute the masses, mixings and decays of various
NMSSM excitations and constrain various relevant observables from the DM, the
flavor and the collider sectors. The above-mentioned theoretical constraints
are also checked using NMSSMTools. Various key observables in the DM sector
are obtained using micrOMEGAs-v4.3 Belanger:2006is ; Belanger:2008sj ;
Barducci:2016pcb , which is integrated within NMSSMTools. We consider the
Planck collaboration Planck:2018vyg  measured central value of the DM relic
abundance (i.e., $\Omega h^{2}=0.120$) and assume a theoretical uncertainty of
10% in the estimation of the DM (singlino-dominated LSP) relic density. The
most stringent bounds on the DMDD-SI scattering cross-section are provided by
the latest XENON-nT XENON:2019rxp ; XENON:2023cxc  and PANDA-4T
PandaX-4T:2021bab  results. In addition to this, the stringent constraints on
DMDD-SD scattering cross-section come from XENON-nT, LUX-ZEPLIN (LZ)
LZ:2022lsv , PICO-60 PICO:2017tgi  and IceCube IceCube:2021xzo . Indirect
searches for DM, conducted through experiments like Fermi-LAT Fermi-
LAT:2017bpc  and MAGIC MAGIC:2016xys , have led to constraints on the
individual thermal average annihilation cross-section of DM into the SM
charged pairs, denoted as $\langle\sigma v\rangle_{SS\to\psi\overline{\psi}}$,
where $\psi:={\mu,\tau,b,W}$.

Constraints coming from the Higgs sector are checked using HiggsBounds-v5.8.0
and HiggsSignals-v2.5.0. Direct searches of various BSM Higgs bosons at LEP,
Tevatron and recently at the LHC constraints are examined by the package
HiggsBounds. On the other hand, one of the $CP$-even Higgs bosons has to
correspond to the observed Higgs boson at the LHC and coincide its properties
with the various measurements of the observed Higgs boson by CMS and ATLAS.
This has been tested using the package HiggsSignals. Note that in our
analysis, we consider the mass of the observed SM-like Higgs boson to fall
within the range of $122$ GeV $<m_{h_{\mathrm{SM}}}<128$ GeV. This range
accounts for the estimated theoretical uncertainties in its prediction.

The direct searches for electroweakinos, particularly those of relatively
light mass, at the LHC are most effectively conducted through the examination
of final states characterized by multiple leptons and jets alongwith large
missing energy. These states arise from the direct production of
$\chi_{{}_{1}}^{\pm}\chi_{{}_{2}}^{0}$, with the assumption that both the
charged $\chi_{{}_{1}}^{\pm}$ and neutral $\chi_{{}_{2}}^{0}$ particles are
degenerate and wino-like, while the LSP is bino-like. These events
subsequently undergo cascades, manifesting in two distinct decay modes:
$WZ\chi_{{}_{1}}^{0}$ and $Wh_{\rm SM}\chi_{{}_{1}}^{0}$. In the former mode,
the final state is characterized by $3\ell+\not{E}_{T}$, with both $W^{\pm}$
and $Z$ bosons decaying leptonically CMS:2017moi ; CMS:2018szt ; ATLAS:2018ojr
; ATLAS:2018eui ; ATLAS:2019wgx ; ATLAS:2020ckz ; CMS:2021edw . Alternatively,
the cascade may result in the $2\ell+2$-${\rm jet}+\not{E}_{T}$ final state,
where $W^{\pm}$ bosons decay hadronically CMS:2017moi ; CMS:2018szt ;
ATLAS:2018ojr ; ATLAS:2018eui . Additionally in the later mode, another type
of final state characterized by $1\ell+2b$-${\rm jet}+\not{E}_{T}$ can occur,
where the $b$-jets originate from the decay of the Higgs boson ($h_{\rm SM}$)
CMS:2017moi ; CMS:2018szt ; ATLAS:2018qmw ; CMS:2019pov ; ATLAS:2020pgy ;
ATLAS:2020ckz . This scenario has further been studied considering the di-
photon decay mode of $h_{\rm SM}$ ATLAS:2020qlk . It is also possible to
search these electroweakinos from the pair production of the charginos, i.e.,
$\chi_{{}_{1}}^{\pm}\chi_{{}_{1}}^{\mp}$, in the final state of
$2\ell+\not{E}_{T}$ ATLAS:2019lff ; CMS:2018xqw . These investigations over
the years represent a critical aspect of the efforts to explore
electroweakinos and their properties at the LHC. However, null results from
various direct searches of electroweakinos at the LHC in these modes put
constraints on their masses.

In the scenario where the concerned chargino-neutralinos are higgsino-like
(wino-like states are considered to be very heavy), the corresponding mass-
bounds are expected to be relaxed given their smaller combined direct
production cross-section at the LHC. On top of that, within certain regions of
the parameter space, the heavier neutralinos exhibit suppressed branching
ratios to decay into $Z$ and $h_{\rm SM}$, which can further erode these
experimental constraints. Recently it has been shown within the framework of
the $Z_{3}$-symmetric NMSSM that the potential cascade decays of these
particles, involving new singlet-like scalars, singlino-dominated and bino-
like neutralinos, have the capacity to significantly reduce the sensitivities
of current searches at the LHC Abdallah:2019znp ; Abdallah:2020yag ;
Chatterjee:2022pxf ; Datta:2022bvg .

The mass bounds derived from the LHC collaborations exhibit a significant
reduction in the compressed region, wherein the mass difference between the
wino-like neutralino/chargino and the bino-like LSP falls below the mass of
the $Z$ boson. In this regime, the cascade decays involve off-shell $Z$,
$h_{\rm SM}$, and $W$ bosons, leading to subsequent decays yielding soft
leptons, jets, missing energy, etc ATLAS:2019lng ; ATLAS:2018ojr ;
ATLAS:2019lff ; CMS:2020bfa ; ATLAS:2021moa ; CMS:2021cox ; CMS:2021few ;
ATLAS:2021yqv ; ATLAS:2022zwa ; CMS:2022sfi ; CMS:2019san . This drop in the
mass bounds typically persists as the mass splitting decreases, resulting in
even softer final-state leptons and jets. In this study, our focused
investigation is aimed at specific regions within the parameter space,
primarily motivated by DMDD blind spots. Notably, these regions may also
emerge in the context of the compressed regime. Within this framework, we
identify regions where the NLSP, whether bino-like or higgsino-like, exhibits
significant decay branching fraction to a photon and the singlino-dominated
LSP ($\chi_{{}_{2}}^{0}\rightarrow\chi_{{}_{1}}^{0}\gamma$). Additionally, we
uncover a novel parameter space region where both higgsino-like neutralino
states ($\chi_{{}_{2}}^{0}$ and $\chi_{{}_{3}}^{0}$) exhibit considerable
radiative decay rates, characterized by larger branching fractions for the
$\chi_{{}_{2}}^{0}\rightarrow\chi_{{}_{1}}^{0}\gamma$ and
$\chi_{{}_{3}}^{0}\rightarrow\chi_{{}_{2}}^{0}\gamma$ decay modes. These
distinctive decay patterns give rise to potentially intriguing collider
signatures at the LHC. It is noteworthy that some of these channels have not
been explored in previous LHC searches, resulting in a notable relaxation of
the mass bounds discussed earlier. In particular, such relaxations for the
higgsino-like electroweakinos could significantly extend the parameter space
allowing lower values of $|\mu_{\mathrm{eff}}|$, thereby enhancing the
‘naturalness’ of the scenario to a considerable degree Barbieri:1987fn ;
Ellis:1986yg ; Giudice:2013yca . We delve into a detailed discussion of these
decay channels in the subsequent sections. In order to assess the viability of
our benchmark scenarios in passing all pertinent LHC analyses, we utilize
SModelS and CheckMATE packages for the recasting of numerous relevant LHC
analyses, including recent ones conducted with 139 fb-1 of data. Both the
recast packages calculate a ‘$r$’-value, where $r=(S-1.64\Delta S)/S95$, with
‘$S$’, $S95$ and $\Delta S$ indicating the predicted number of signal events
and the experimental limit on ‘$S$’ at 95% confidence level and the associated
Monte Carlo error, respectively. Nominally, $r>(<)1$ denotes the benchmark
point to be disallowed (allowed). To take into account the significant NLO+NLL
contributions, all the electroweakino production cross-sections have been
multiplied by a $k$-factor of 1.25 Fiaschi:2018hgm .

##  4 The motivated region of parameter space

The choice of the region within the $Z_{3}$-symmetric NMSSM parameter space
for our present work is influenced by several considerations that pertain to
both the phenomenology observed at the LHC and its relevance to the DM sector.
We concentrate on the singlino-dominated DM (LSP), which corresponds to the
parameter space region with $|\kappa|<\lambda/2$. This prompts our search for
the lower $\kappa$ region. As demonstrated in section 2.3, bino tempering
within singlino-dominated DM can uncover a novel parameter space region where
the DMDD scattering cross-section significantly decreases. We specifically
investigate a moderately low $\lambda~{}(\lesssim 0.2)$ region, where this
tempering can become substantial.

In the Higgs sector, our particular focus lies on the relatively light
singlet-like Higgs bosons and the observed SM-like doublet Higgs boson. The
MSSM-like Higgs bosons ($A$, $H$, $H^{\pm}$) are considered to be
significantly more massive and effectively decoupled from the rest of the
system. This decoupling is achieved by selecting a large value for
$A_{\lambda}$. Lighter singlet-like scalars can act as a funnel for the
moderately tempered light DM mutual annihilation when $m_{a_{{}_{S}}}$ or
$m_{{}_{h_{S}}}\sim 2m_{{{}_{\chi}}_{{}_{1}}^{0}}$. A lower singlino mass
($m_{{}_{\widetilde{S}}}=2\kappa\mu_{\mathrm{eff}}/\lambda$) results a
relatively lighter $h_{{}_{S}}$ (with $m^{2}_{{}_{h_{S}}}\sim
m^{2}_{{}_{\widetilde{S}}}$ as indicated in eq. (29)). For a relatively
lighter $a_{{}_{S}}$, it is favorable to have smaller $A_{\kappa}$ (with
$m^{2}_{a_{{}_{S}}}\sim A_{\kappa}m_{{}_{\widetilde{S}}}$ as described in
equation (30)). In the neutralino sector, a moderate singlino-bino mixing
necessitates a lighter bino state, which motivates to scan over a moderate
range of $M_{1}$. The latest constraints from LHC and DMDD experiments put the
lighter $\mu_{\mathrm{eff}}$ region of parameter space under tension. This
encourages us to scan $\mu_{\mathrm{eff}}$ from a smaller to a larger value.
For a relatively low $m_{{{}_{\chi}}_{{}_{1}}^{0}}$ and moderately large
$\mu_{\mathrm{eff}}$ (i.e., for low
$m_{{{}_{\chi}}_{{}_{1}}^{0}}/\mu_{\mathrm{eff}}$) a larger value of
$\tan\beta$ is required for the blind spot to work (eq. (36)), leading to an
acceptably small DMDD-SI rates. For smaller $\mu_{\mathrm{eff}}$ (i.e.,
relatively larger $m_{{{}_{\chi}}_{{}_{1}}^{0}}/\mu_{\mathrm{eff}}$), a larger
$\sin 2\beta$ value eventually requires and hence a smaller value of
$\tan\beta$ to satisfy the blind spot criteria of eq. (36). This encourages us
to look at the wide range of $\tan\beta$.

The Higgs sector of the NMSSM experiences additional tree-level contributions
to $m_{h_{\mathrm{SM}}}$ (as shown in eq. (28)) compared to those in the MSSM
Ellwanger:2009dp . This extra tree-level contribution becomes substantial at
smaller $\tan\beta$ and larger $\lambda$ regions. In such regions, it is
possible to attain the mass of the observed $h_{\rm SM}$
($m_{h_{\mathrm{SM}}}\sim$ 125 GeV) without the need for substantial radiative
corrections coming from the top squarks. Conversely, in regions with
relatively lower $\lambda$ and higher $\tan\beta$, substantial radiative
corrections are necessary to achieve $m_{h_{\mathrm{SM}}}\sim$ 125 GeV. This
motivates us to explore a broad range of larger values for $A_{\text{top}}$
and the masses of third-generation squarks ($m_{Q_{3}}$ and $m_{U_{3}}$).
Table 1 presents the scan ranges considered for various model parameters. All
input parameters of NMSSMTools are defined at the scale
$Q^{2}=(2m_{\tilde{Q}}^{2}+m_{\tilde{U}}^{2}+m_{\tilde{D}}^{2})/4$, except for
$\tan\beta$, which is defined at $m_{Z}$ Ellwanger:2005dv .

$\lambda$ | $|\kappa|$ | $\tan\beta$ |  $|\mu_{\mathrm{eff}}|$ (GeV) |  $|A_{\lambda}|$ (TeV) |  $|A_{\kappa}|$ (GeV) |  $|M_{1}|$ (GeV) |  $|A_{top}|$ (TeV) |  $M_{Q_{3}}$ (TeV) |  $M_{U_{3}}$ (TeV)
---|---|---|---|---|---|---|---|---|---
0.001–0.2 | $\leq 0.1$ | 1–60 | $\leq 750$ | $\leq 5$ | $\leq 300$ |  $\leq$600 | $\leq 5$ | 2–5 | 2–5
Table 1: Ranges of various NMSSM parameters adopted for scanning the parameter
space.

##  5 Probing DM with photons at the LHC

In this section, we focus into the investigation of various search channels of
DM characterized by final states comprising photons at the LHC. These channels
are contingent upon the intricate cascade decays of the heavier electroweakino
states and their interactions with the DM. One particularly effective
mechanism for DM annihilation involves co-annihilation with heavier sleptons,
neutralinos and/or charginos Ellis:1998kh ; Ellis:1999mm ; Buckley:2013sca ;
Han:2013gba ; Cabrera:2016wwr ; Baker:2018uox . The significance of these
annihilation processes of DM depends on the condition that the cross-section
for processes involving the heavier co-annihilating particle ($X_{2}$), with a
mass denoted as $m_{X_{2}}$, significantly exceeds that of the pair of DM
($\chi_{{}_{1}}^{0}$) annihilation processes. If the mass difference between
$X_{2}$ and $\chi_{{}_{1}}^{0}$ remains relatively modest, allowing $X_{2}$ to
persist within the thermal plasma during DM freeze-out, co-annihilation
processes can efficiently reduce the relic density of the DM, despite the
Boltzmann-suppression factor
($\exp\left[-\left(m_{X_{2}}-m_{{{}_{\chi}}_{{}_{1}}^{0}}\right)/T\right]$).
This factor accounts for the difference in number densities between $X_{2}$
and $\chi_{{}_{1}}^{0}$. In addition to this, “assisted co-annihilation” can
significantly enhance the DM annihilation cross-section Ellwanger:2009dp ;
Ellwanger:2016sur , particularly when the annihilation cross-section of the
pair of $X_{2}$ particles into the SM particles is notably large. For example,
this is often the case when $m_{X_{2}}$ closely approaches half of the mass of
one of the Higgs bosons, giving rise to a resonant annihilation process of a
pair of $X_{2}$ that can overcome the Boltzmann-suppression factor.
Nevertheless, a critical hallmark of the co-annihilation mechanism is the
presence of a small mass gap between the heavier particle and the DM. This
feature prevents the Boltzmann suppression factor from reaching excessively
high values and makes the scenario significantly compressed, i.e.,
$(m_{X_{2}}-m_{{{}_{\chi}}_{{}_{1}}^{0}})<m_{Z}$.

In our scenario, singlino-dominated DM (LSP) can co-annihilate with the bino-
like neutralino or the higgsino-like neutralinos/charginos. Various decay
branching fractions of the heavier states, like the bino-like neutralino or
the higgsino-like neutralinos/charginos, to the LSP depend on their couplings
and the available phase space. In the compressed regions, all the decay modes
of the second neutralino, $\chi_{{}_{2}}^{0}$, is kinematically suppressed,
and it can be parametrized by the well-known “mass splitting parameter”
Baum:2023inl

$\varepsilon\equiv\frac{m_{{{}_{\chi}}_{{}_{2}}^{0}}}{m_{{{}_{\chi}}_{{}_{1}}^{0}}}-1\,\,.$
(40)

It is interesting to note that tree-level decay processes of
$\chi_{{}_{2}}^{0}$, such as
$\chi_{{}_{2}}^{0}\rightarrow\chi_{{}_{1}}^{0}f\bar{f}$ mediated by off-shell
$Z$, $h_{\rm SM},H,A,h_{s},a_{s}$, experience a suppression of
$\varepsilon^{5}$, whereas radiative decay processes, such as
$\chi_{{}_{2}}^{0}\rightarrow\chi_{{}_{1}}^{0}\gamma$ which are induced by
triangle loops involving fermions, sfermions, charginos, neutralinos, charged
Higgs bosons, and $W^{\pm}$, are subject to a suppression of $\varepsilon^{3}$
Ambrosanio:1996gz . Therefore, in the compressed region, radiative decay of
the heavier neutralino states can play a pivotal role. The radiative decay
branching fraction of the heavier neutralinos, $\chi_{i}^{0}$, can experience
a further significant enhancement when the couplings
$g_{\chi_{i}^{0}\chi_{{}_{1}}^{0}Z}$ and
$g_{\chi_{i}^{0}\chi_{{}_{1}}^{0}h_{\rm SM}}$ become significantly small
and/or the effective coupling $g_{\chi_{i}^{0}\chi_{{}_{1}}^{0}\gamma}$
increases. This scenario corresponds to specific regions within the parameter
space where such conditions are met.

It has been demonstrated that the couplings
$g_{\chi_{i}^{0}\chi_{{}_{1}}^{0}Z}$ and
$g_{\chi_{i}^{0}\chi_{{}_{1}}^{0}h_{\rm SM}}$ in the singlino-higgsino
neutralino sector crucially depend on the value of $\lambda$ Abdallah:2020yag
. The coupling $\lambda$ not only drives the higgsino-singlino-Higgs boson
(scalar) interaction (see (eq. 18)) involved in the decay
$\chi_{i}^{0}(\tilde{H})\rightarrow h_{\rm SM}\chi_{1}^{0}(\tilde{S})$, but
regulates the mixing in the higgsino-singlino sector, exerting a crucial
influence on the associated couplings. Conversely, the dependence on $\lambda$
in the coupling of the $Z$-boson to the neutralino states solely appears
through mixing. It has been shown explicitly in reference Abdallah:2020yag
that these couplings increase with $\lambda$. Thus, the radiative decay
branching fraction of the higgsino-like neutralinos can increase via
decreasing $g_{\chi_{i}^{0},\chi_{{}_{1}}^{0}Z}$,
$g_{\chi_{i}^{0},\chi_{{}_{1}}^{0}h_{\rm SM}}$ couplings at the lower
$\lambda$ region of parameter space. Furthermore, it is intriguing to note
that the presence of a light bino-like state ($\chi_{{}_{4}}^{0}$) close to
the higgsino-like and singlino-dominated states can increase the radiative
decay branching fraction of the higgsino-like states ($\chi_{{}_{2,3}}^{0}$).
In addition to this, a light bino state in the proximity of the higgsino and
singlino-dominated states can reduce the DMDD scattering cross-section, as we
discussed in section 2.3. Thus, bino can play an important role in the search
for DM in both collider and DMDD experiments.

Turning to the relevant decays that shape the phenomenology at the LHC, one of
the interesting channels to search this singlino-higgsino co-annihilation
region of parameter space from the production of
$pp\rightarrow\chi_{{}_{2,3}}^{0}(\tilde{H})\chi_{{}_{1}}^{\pm}(\tilde{H})$
could be $1\ell+\geq 1\gamma+\not{E}_{T}$ where leptonic decay of $W^{\pm}$ is
considered and the higgsino-like $\chi_{{}_{2,3}}^{0}$ branching ratio to
radiative decay mode is significantly large. Due to the small mass gap between
higgsino-like states and the singlino-dominated LSP, it is expected that in
this channel, lepton and photons will be relatively soft. Note that, in
reference ATLAS:2020qlk , ATLAS has already conducted a search for the
neutralino/charginos, considering a similar final state at the LHC. However,
in this analysis, it is considered that the pair of photons originate from the
decay of the on-shell $h_{\rm SM}$, which arises from the decay of heavier
neutralino. Therefore, this analysis does not remain relevant to the region of
parameter space of the singlino-higgsino co-annihilation that we focus on in
this work. On the other hand, in scenarios where the decaying leptons/jets are
too soft to be detected, the production of higgsino-like states together with
an ISR jet or a photon can become relevant for the mono-jet/$\gamma$ \+
$\not{E}_{T}$ LHC analyses ATLAS:2018nud ; ATLAS:2017bfj ; ATLAS:2021kxv ;
CMS:2021far . This can impose reasonable constraints on the chargino-
neutralino sector in this compressed scenario. However, mono-jet/mono-photon
searches at the LHC remain insensitive to the production of a pair of
singlino-dominated LSP due to its inherently small production cross-section.

In the case of singlino-bino co-annihilation, NLSP is bino-like, and it could
also have a significant radiative branching fraction. Therefore, in this
scenario, cascades of the produced higgsino-like $\chi_{{}_{3,4}}^{0}$ and
$\chi_{{}_{1}}^{\pm}$ via $\chi_{{}_{2}}^{0}$ could lead to:
$pp\rightarrow\chi_{{}_{3,4}}^{0}(\widetilde{H})\chi_{{}_{1}}^{\pm}(\widetilde{H})\rightarrow$
$h_{\rm
SM}/Z+W^{\pm}+\chi_{{}_{2}}^{0}(\tilde{B})\,\big{[}\chi_{{}_{2}}^{0}\rightarrow\gamma\,\chi_{{}_{1}}^{0}(\tilde{S})\big{]}$
$\Rightarrow 3\ell+\geq 1\gamma+\not{E}_{T}$ or $1\ell+2b+\geq
1\gamma+\not{E}_{T}$ final states at the LHC. Similar to the previous
scenario, due to a small mass gap, photons are also anticipated to remain soft
in this context. But, for a larger mass gap between the higgsino-like states
and the bino-like NLSP, the NLSP can emerge with a substantial boost resulting
from the decay of those higgsino-like states. In such cases, the photons
originating from the NLSP decay no longer remain soft. On the other hand, the
leptons and jets coming from the on-shell $W^{\pm},h_{\rm SM},Z$ bosons are
expected to be relatively hard. In some cases, if the photons are soft and
remain undetected, cascades of $\chi_{{}_{3,4}}^{0}$ and $\chi_{{}_{1}}^{\pm}$
via bino-like $\chi_{{}_{2}}^{0}$ would lead to similar final states as when
these higgsino-like heavier neutralino states directly decay to the singlino-
dominated LSP. Thus, virtually, a large effective branching ratio to the LSP
for these states could be envisaged, and stringent experimental lower bounds,
which we discuss in section 3, will be applicable to their masses. Note that,
as pointed out earlier, since the ATLAS analysis ATLAS:2020qlk  is dedicated
to probing the neutralinos/charginos by examining the di-photon decay mode of
$h_{\rm SM}$ resulting from the decay of the heavier neutralino, certain
signal regions involving photons can become sensitive to the singlino-bino co-
annihilation scenario. We will delve into this in detail in Section 5.1.

Figure 1: Scattered points showing the variation of
BR[$\chi_{{}_{2}}^{0}\rightarrow\chi_{{}_{1}}^{0}\gamma$] with
$m_{{{}_{\chi}}_{{}_{2}}^{0}}-m_{{{}_{\chi}}_{{}_{1}}^{0}}$ (left plot) and
BR[$\chi_{{}_{3}}^{0}\rightarrow\gamma\chi_{{}_{2}}^{0}$] with
BR[$\chi_{{}_{2}}^{0}\rightarrow\chi_{{}_{1}}^{0}\gamma$] (right plot). The
higgsino content of the NLSP ($N_{23}^{2}+N_{24}^{2}$) is shown through the
palette in both plots. The color palettes in these two plots are truncated at
the higgsino content of 0.5 for enhanced color clarity, although these plots
include points with values below that threshold, i.e., points with
$N_{23}^{2}+N_{24}^{2}<0.5$ are represented by black points.

Since the radiative decays of the heavier electroweakinos are the focus of our
present study, we illustrate the variation of the decay branching fraction of
$\chi_{{}_{2}}^{0}\rightarrow\gamma\chi_{{}_{1}}^{0}$ process with the mass
difference between $\chi_{{}_{2}}^{0}$ and $\chi_{{}_{1}}^{0}$ in the left
plot of figure 1. Here, LSP is singlino-dominated, i.e., $N_{15}^{2}>0.9$. The
total higgsino component of the NLSP ($N_{23}^{2}+N_{24}^{2}$) is presented
through the palette. Points ranging from red to blue denote highly higgsino-
like NLSP, while black points represent mostly bino-like NLSP (i.e.,
$N_{21}^{2}>0.5$ and $N_{23}^{2}+N_{24}^{2}<0.5$). It is noteworthy that, as
expected, the BR[$\chi_{{}_{2}}^{0}\rightarrow\gamma\chi_{{}_{1}}^{0}$]
increases with the decrease of
$m_{{{}_{\chi}}_{{}_{2}}^{0}}-m_{{{}_{\chi}}_{{}_{1}}^{0}}$. The figure
reveals two distinct band patterns. For
BR$[\chi_{{}_{2}}^{0}\rightarrow\gamma\chi_{{}_{1}}^{0}]>0.9$, the mass gap
needs to be below 3 GeV in the lower band. In contrast, in the upper band,
this can be achievable even with a mass gap exceeding 15 GeV. It is observed
that the upper band is only possible for $\mu_{\mathrm{eff}}<0$. It can be
observed that in this upper band, for a fixed BR, the mass gap increases as
the higgsino component in the NLSP decreases. This happens due to the presence
of a nearby bino-like state (as $\chi_{{}_{4}}^{0}$), which modifies the
mixing in the higgsino-like NLSP in such a way that the radiative decay
involved effective coupling increases significantly, and because of that, the
BR remains large even at the relatively larger mass difference.

$\kappa$ | $\mu_{\mathrm{eff}}$ | $M_{1}$
---|---|---
$-$ | $+$ | $+$
$-$ | $-$
$+$ | $+$ | $-$
$-$ | $+$
Table 2: Preferred sign combinations among $\kappa$, $\mu_{\mathrm{eff}}$, and
$M_{1}$ based on scan results.

The decay patterns of the heavier electroweakinos are also crucial in the
context of their searches at the LHC. In the right plot of Figure 1, we
present the variation of
BR[$\chi_{{}_{2}}^{0}\rightarrow\gamma\chi_{{}_{1}}^{0}$] with
BR[$\chi_{{}_{3}}^{0}\rightarrow\gamma\chi_{{}_{2}}^{0}$] based on our scan
results. Once again, the LSP is singlino-dominated. The higgsino component of
the NLSP is displayed in the palette. It is noteworthy that in our scenario,
when $\chi_{{}_{2}}^{0}$ is higgsino-like, the nearly degenerate
$\chi_{{}_{3}}^{0}$ will also be higgsino-like, and $\chi_{{}_{4}}^{0}$ will
be bino-like. In this plot, an intriguing observation is the existence of a
region in parameter space where both
BR[$\chi_{{}_{2}}^{0}\rightarrow\gamma\chi_{{}_{1}}^{0}$] and
BR[$\chi_{{}_{3}}^{0}\rightarrow\gamma\chi_{{}_{2}}^{0}$] are significantly
large. This scenario corresponds to the singlino-higgsino co-annihilation. The
existing multi-lepton $+\not{E}_{T}$ searches for compressed higgsino-like
neutralinos at the LHC become less sensitive in this region of parameter
space. In this scenario, we also find that the higgsino-like
$\chi_{{}_{1}}^{\pm}$ state tends to prefer decaying to an off-shell $W$ boson
and $\chi_{{}_{2}}^{0}$ rather than $\chi_{{}_{1}}^{0}$. The significant
radiative decay branching fractions of higgsino-like electroweakinos
underscore the need for a crucial new detection channel at the LHC involving
triggered photons, essential for probing this compressed singlino-higgsino
scenario in the NMSSM parameter space.

To illustrate our scenario, in the next subsection, we will present a few
benchmark points (BP1 to BP4) featuring larger radiative decay branching
fractions of heavier electroweakinos and pass all the existing bounds that are
discussed in section 3. We also present one benchmark scenario (BP-D1), which
is excluded by the recent ATLAS analysis, to demonstrate some interesting
collider aspects. Before delving into the discussion of those benchmark
points, here we will explore the implications of the favored sign combinations
among $\kappa$, $\mu_{\mathrm{eff}}$, and $M_{1}$. These combinations shed
light on the regions of the parameter space that are more favored,
particularly in the context of the latest experimental constraints. The
parameter space encompassing the sign combinations listed in Table 2
constitutes the majority of the allowed points based on our comprehensive scan
results. When $\kappa<0$, both $\mu_{\mathrm{eff}}$ and $M_{1}$ tend to share
the same relative sign, while for $\kappa>0$ they show a preference for
opposite relative signs. It is noteworthy that this favoured pattern can be
comprehended through the unique relationship that we introduce for the first
time in this work in eq. (37). Such sign combinations are required for the
blind spot to work for the singlino-dominated DM, leading to an acceptably
small DMDD-SI scattering cross-section.

Previous studies on the DMDD-SI blind spot criterion for singlino-dominated DM
only discussed the same-sign condition between $m_{{{}_{\chi}}_{{}_{1}}^{0}}$
and $\mu_{\mathrm{eff}}$, i.e., $\kappa>0$ (eq. (36)). In contrast, in this
work we introduce a more general blind spot criterion (eq. (37)) which points
out that for $\kappa>0$, $\mu_{\mathrm{eff}}$ and $M_{1}$ should have opposite
relative signs. Furthermore, this generalized condition reveals a new blind
spot region, i.e., $\kappa<0$, which was not previously addressed in the
literature. It is worth noting that this new region may have significant
implications for explaining the discrepancy of the anomalous Muon magnetic
moment ($a_{\mu}=(g_{\mu}-2)/2$) between the experimental observations (from
Fermilab and BNL) and the SM prediction Muong-2:2006rrc ; Muong-2:2021ojo .
The measured value of $a_{\mu}$ is larger than the predictions from the SM.
Therefore, a positive contribution from the new physics is required to explain
this discrepancy. In this scenario, it is well-known that the most important
contributions arise from a Bino-smuon and a chargino-(muon-sneutrino) loop.
Interestingly, the sign of these loop contributions depends on the sign of
$(M_{1}\times\mu_{\mathrm{eff}})$ and $(M_{2}\times\mu_{\mathrm{eff}})$,
respectively. They contribute positively when their signs are positive.
Therefore, we find that in the case of singlino-dominated DM, the region with
$\kappa<0$ prefers the DMDD-SI blind spot scenario if $M_{1}$ and
$\mu_{\mathrm{eff}}$ have the same relative sign which also provides a
positive contribution from the Bino-smuon loop to $a_{\mu}$. This indicates an
intriguing correlation in explaining the observed discrepancy of the anomalous
muon magnetic moment and the DMDD-SI blind spot conditions. However, as
emphasized before, in this work we do not study this issue as we consider
smuons to be heavy. We reserve this study for future work.

###  5.1 Benchmark scenarios

Three broad classes of benchmark scenarios, BP-D1 and BP1 to BP4, are
presented in table 3 based on the mode of annihilation of the highly singlino-
dominated DM. These include (i) singlino-bino co-annihilation (BP-D1 and BP1),
(ii) the $h_{\rm SM}$ funnel (BP2) and (iii) singlino-higgsino co-annihilation
(BP3 nad BP4). Note that all the benchmark points feature photons from the
cascades of the heavier electroweakinos to the DM.

In BP-D1, an efficient singlino-dominated DM annihilation with DM mass $\sim
55$ GeV is achieved via co-annihilation with a bino-like NLSP with mass $\sim
66$ GeV. The resonant s-channel process of
$\chi_{{}_{1}}^{0}\chi_{{}_{2}}^{0}\rightarrow h_{\rm SM}$ provides the most
efficient DM annihilation process to comply with the Planck-observed upper
bound on the DM relic abundance. As described in section 2.3, the tempering of
the singlino-dominated DM by a light bino-like state provides new mechanisms
in moderating $g_{h_{\rm SM}\chi_{{}_{1}}^{0}\chi_{{}_{1}}^{0}}$ and
$g_{Z\chi_{{}_{1}}^{0}\chi_{{}_{1}}^{0}}$ couplings in such ways that the
DMDD-SI and DMDD-SD rates, respectively, could be tamed to comply with their
experimentally allowed values. In addition to this, because of the small mass
gap between $\chi_{{}_{2}}^{0}$ and $\chi_{{}_{1}}^{0}$
($m_{{{}_{\chi}}_{{}_{2}}^{0}}-m_{{{}_{\chi}}_{{}_{1}}^{0}}\sim 10$ GeV), the
bino-like NLSP decays almost 100% to the LSP with a photon. As $\lambda<0.1$,
BR[$\tilde{H}\rightarrow\tilde{S}\,\,W/Z/h_{\rm SM}$] is much smaller than the
BR[$\tilde{H}\rightarrow\tilde{B}\,\,W/Z/h_{\rm SM}$]. This is since
interactions of $\tilde{S}$-$\tilde{H}$-$Z/h_{\rm SM}/W$ decreases with the
decrease of $\lambda$ Abdallah:2020yag . Therefore, the photon count is
enhanced from the cascades of the higgsino-like states, as they prefer to
decay to the bino-like NLSP, and subsequently, the NLSP undergoes a radiative
photon decay. If photons are sufficiently energetic to be detected, this could
signify a substantial increase in yields from the production process of
$pp\rightarrow\chi_{{}_{3,4}}^{0}\chi_{{}_{1}}^{\pm}$ in the final states
$3\ell+\not{E}_{T}$ and $1\ell+2b+\not{E}_{T}$, when photons are incorporated,
as opposed to states without the inclusion of photons.

On the collider front, a dedicated CheckMATE analysis rules out BP-D1 (with
$r$=$2.87$) via an ATLAS analysis ATLAS:2020qlk  of 139 fb-1 worth data for
the search of chargino-neutralinos by studying the di-photon decay channel of
the on-shell $h_{\rm SM}$, which results from the decay of a heavier
neutralino. It is not primarily motivated to explore this specific co-
annihilation scenario. However, some signal regions overlap, as the analysis
encompasses final states involving leptons, jets, and photons combined with
missing energy. This analysis focuses on the significant mass gap between the
heavier neutralino/chargino and the LSP regions, ensuring that the decaying
objects (jets, leptons, photons) maintain sufficiently larger $P_{T}$. In this
benchmark point, due to a larger mass gap ($\sim 350$ GeV) between
$\mu_{\mathrm{eff}}$ and $M_{1}$, bino-like NLSP emerges with a boost. This
induces some boost in the photon originating from the decay of the NLSP. As a
result, the tail of the invariant mass of the two photons from the process
$pp\rightarrow\chi_{{}_{3,4}}^{0}\chi_{{}_{1}}^{\pm}$ broadens relatively and
lies around the mass window of $h_{\rm SM}$, which is considered in the
selection cuts of this ATLAS analysis. The signal regions, particularly
‘Category-12’ ($N_{j}\geq 2$) and ‘Category-4’ ($N_{l}\geq 1$), gain
sensitivity and begin to exclude specific regions in the parameter space of
the singlino-bino co-annihilation scenario as they inclusively consider
leptons/jets in final states of their analyses. To demonstrate this collider
aspect of this scenario, we retain this point as a benchmark.

As emphasised above, we present another singlino-bino co-annihilation
benchmark point, BP1, featuring similar characteristics of BP-D1, except
$|\mu_{\mathrm{eff}}|$ is now on the higher side $\sim 700$ GeV. Larger
$\mu_{\mathrm{eff}}$ reduces the yields by reducing the cross-section of the
process $pp\rightarrow\chi_{{}_{3,4}}^{0}\chi_{{}_{1}}^{\pm}$ at the LHC,
consequently lowering the sensitivity of the aforementioned ATLAS analysis
ATLAS:2020qlk . A dedicated CheckMATE analysis, taking into account all
available analyses in its repository, indicates that BP1 is allowed with
$r=0.68$. Note that, similar to the previous benchmark point as $\lambda$ is
small, BR[$\tilde{H}\rightarrow\tilde{S}\,\,W/Z/h_{\rm SM}$] is much smaller
than the BR[$\tilde{H}\rightarrow\tilde{B}\,\,W/Z/h_{\rm SM}$]. Furthermore,
here BR$[\chi_{{}_{2}}^{0}\rightarrow\chi_{{}_{1}}^{0}\gamma]$ is around 88%.
Therefore, similar to BP-D1, since the higgsino-like states prefer to decay
mostly to the NLSP, there is a significant suppression of the yields from the
production process of $pp\rightarrow\chi_{{}_{3,4}}^{0}\chi_{{}_{1}}^{\pm}$ in
the canonical final states $3\ell+\not{E}_{T}$ and $1\ell+2b+\not{E}_{T}$ with
no involvement of photons, which are usually considered in the LHC analyses.
On the other hand, as we mentioned earlier, if some of the photons that
originated from $\chi_{{}_{2}}^{0}$ remain soft to be detected, the cascades
of $\chi_{{}_{1}}^{\pm}$ and $\chi_{{}_{3,4}}^{0}$ via bino-like
$\chi_{{}_{2}}^{0}$ lead to similar final states as when these heavier
higgsino-like electroweakinos directly decay to the singlino-dominated LSP. In
this scenario, leptons and jets originating from on-shell $W,Z,h_{\rm SM}$ are
relatively hard. Therefore, observing excess over the SM backgrounds in both
the signals with and without soft photons with $3\ell+\not{E}_{T}$ and
$1\ell+2b+\not{E}_{T}$ at the HL-LHC may point back to the scenario that
includes singlino-bino co-annihilation with relatively heavier higgsinos. In
the next section, we will discuss this by presenting distributions of some
kinematic variables of this scenario.

Input/Observables |  BP-D1 |  BP1 |  BP2 |  BP3 |  BP4
---|---|---|---|---|---
$\lambda$ |  $0.0964$ |  $0.0964$ |  $0.2086$ |  $0.067$ |  $0.018$
$\kappa$ |  $0.0062$ |  $0.0038$ |  $0.0118$ |  $0.0316$ |  $-0.0083$
$\tan\beta$ |  10.06 |  7.01 |  6.2 |  6.07 |  8.76
$A_{\lambda}$ ( GeV) |  -$3885.3$ |  $-4995.3$ |  $-3192.4$ |  $-1414.6$ |  $-3343.9$
$A_{\kappa}$ (GeV) |  31.1 |  61.1 |  40.5 |  296.4 |  $-25.2$
$\mu_{\mathrm{eff}}$ (GeV) |  $-418.5$ |  $-700.5$ |  $-526.9$ |  $-307.2$ |  $-198.7$
$M_{1}$ (GeV) |  66.4 |  66.2 |  $-91.67$ |  509.2 |  $-350.1$
$M_{2}$ (GeV) |  2500 |  2500 |  2500 |  2500 |  2000
$M_{\widetilde{Q}_{3}}$ (GeV) |  4456.1 |  2556.1 |  2560.8 |  2217.0 |  2302.8
$M_{\widetilde{t}_{R}}$ (GeV) |  4916.5 |  4916.5 |  4456.0 |  4182.1 |  2569.7
$A_{t}$ (GeV) |  4945.9 |  4945.9 |  4526.3 |  5126.4 |  4868.9
$m_{{{}_{\chi}}_{{}_{1}}^{0}}$ (GeV) |  $-55.5$ |  $-56.8$ |  $-61.7$ |  $-295.9$ |  187.9
$m_{{{}_{\chi}}_{{}_{2}}^{0}}$ (GeV) |  66.0 |  65.8 |  $-92.2$ |  312.2 |  $-199.5$
$m_{{{}_{\chi}}_{{}_{3}}^{0}}$ (GeV) |  433.08 |  716.46 |  542.0 |  $-321.2$ |  205.7
$m_{{{}_{\chi}}_{{}_{4}}^{0}}$ (GeV) |  $-435.8$ |  $-719.4$ |  $-544.3$ |  509.9 |  $-356.5$
$m_{\chi_{5}^{0}}$ (GeV) |  2553.0 |  2545.4 |  2541.6 |  2533.4 |  2045.2
$m_{{{}_{\chi}}_{{}_{1}}^{\pm}}$ (GeV) |  432.0 |  716.5 |  540.6 |  $-316.7$ |  $-205.7$
$m_{h_{1}}$ (GeV) |  48.8 |  50.4 |  70.7 |  124.1 |  125.3
$m_{h_{2}}$ (GeV) |  124.2 |  124.6 |  123.9 |  202.0 |  175.9
$m_{a_{{}_{S}}}$ (GeV) |  50.6 |  71.4 |  64.0 |  35.9 |  82.8
$m_{H^{\pm}}$ (GeV) |  4074.5 |  5026.3 |  3308.4 |  1763.5 |  2419.0
$N_{11}$ |  $0.99$ |  $0.99$ |  $0.99$ |  $0.96$ |  $0.99$
$N_{21}$ |  $-0.99$ |  $-0.99$ |  $0.99$ |  $-0.1$ |  $-0.2$
$\Omega h^{2}$ |  0.121 |  0.118 |  0.124 |  0.119 |  0.122
$\sigma^{\rm SI}_{\chi^{0}_{1}-p(n)}$ (cm2) |  $2.9(2.8)\times 10^{-48}$ |  $1.2(1.3)\times 10^{-48}$ |  $5.6(6.0)\times 10^{-48}$ |  $8.5(8.6)\times 10^{-47}$ |  $3.15(3.2)\times 10^{-48}$
$\sigma^{\rm SD}_{\chi^{0}_{1}-p(n)}$ (cm2) |  $9.03(-6.9)\times 10^{-44}$ |  $1.1(0.9)\times 10^{-44}$ |  $7.5(5.7)\times 10^{-43}$ |  $3.4(2.6)\times 10^{-42}$ |  $8.12(-6.2)\times 10^{-44}$
BR($\chi^{\pm}_{1}\to\chi_{1}^{0}W^{\pm}$) |  0.13 |  0.13 |  0.43 |  0.00 |  0.00
BR($\chi^{\pm}_{1}\to\chi_{2}^{0}W^{\pm}$) |  0.87 |  0.87 |  0.57 |  0.00 |  0.00
BR($\chi^{\pm}_{1}\to\chi_{1}^{0}f\bar{f}$) |  0.00 |  0.00 |  0.00 |  0.99 |  0.80
BR($\chi^{\pm}_{1}\to\chi_{2}^{0}f\bar{f}$) |  0.00 |  0.00 |  0.00 |  0.01 |  0.20
BR($\chi^{0}_{2}\to\chi_{1}^{0}f\bar{f}$) |  0.005 |  0.12 |  0.28 |  0.37 |  0.23
BR($\chi^{0}_{2}\to\chi_{1}^{0}\gamma$) |  0.995 |  0.88 |  0.72 |  0.63 |  0.73
BR($\chi^{0}_{3}\to\chi_{1}^{0}Z$) |  0.06 |  0.05 |  0.18 |  0.00 |  0.00
BR($\chi^{0}_{3}\to\chi_{1}^{0}\,h_{\rm SM}$) |  0.06 |  0.07 |  0.22 |  0.00 |  0.00
BR($\chi^{0}_{3}\to\chi_{2}^{0}Z$) |  0.42 |  0.48 |  0.30 |  0.00 |  0.00
BR($\chi^{0}_{3}\to\chi_{2}^{0}\,h_{\rm SM}$) |  0.45 |  0.40 |  0.28 |  0.00 |  0.00
BR($\chi^{0}_{3}\to\chi_{2}^{0}\,f\bar{f}$) |  0.00 |  0.00 |  0.00 |  0.12 |  0.078
BR($\chi^{0}_{3}\to\chi_{2}^{0}\gamma$) |  0.00 |  0.00 |  0.00 |  0.86 |  0.92
BR($\chi^{0}_{4}\to\chi_{1}^{0}Z$) |  0.08 |  0.08 |  0.26 |  0.02 |  0.006
BR($\chi^{0}_{4}\to\chi_{1}^{0}\,h_{\rm SM}$) |  0.05 |  0.05 |  0.16 |  0.002 |  0.0001
BR($\chi^{0}_{4}\to\chi_{2}^{0}Z$) |  0.49 |  0.38 |  0.29 |  0.04 |  0.02
BR($\chi^{0}_{4}\to\chi_{2}^{0}\,h_{\rm SM}$) |  0.37 |  0.49 |  0.28 |  0.23 |  0.2
BR($\chi^{0}_{4}\to\chi_{3}^{0}\,Z$) |  0.00 |  0.00 |  0.00 |  0.18 |  0.21
BR($\chi^{0}_{4}\to\chi_{3}^{0}\,h_{\rm SM}$) |  0.00 |  0.00 |  0.00 |  0.13 |  0.003
BR($\chi^{0}_{4}\to\chi_{{}_{1}}^{\pm}\,W^{\mp}$) |  0.00 |  0.00 |  0.00 |  0.52 |  0.54
$\sigma_{pp\rightarrow\chi_{{}_{2,3,4}}^{0}\chi_{{}_{1}}^{\pm}}$ (pb) |  0.0418 |  0.00425 |  0.01577 |  0.140 |  0.743
CheckMATE result |  Excluded |  Allowed |  Allowed |  Allowed |  Allowed
$r$-value |  2.87 |  0.68 |  0.61 |  0.07 |  0.12
Analysis ID |  atlas$\\_$2004$\\_$10894 ATLAS:2020qlk  |  atlas$\\_$2004$\\_$10894 ATLAS:2020qlk  |  atlas$\\_$2004$\\_$10894 ATLAS:2020qlk  |  atlas$\\_$conf$\\_$2017$\\_$060 ATLAS:2017dnw  |  atlas$\\_$conf$\\_$2020$\\_$048 ATLAS:2020wzf
Signal region ID |  Cat12 |  Cat12 |  Cat12 |  EM7 |  EM09
Table 3: Benchmark scenarios allowed (BP1 to BP4) by all relevant theoretical
and experimental constraints and disallowed (BP-D1) by the LHC searches for
the electroweakinos. Various input parameters and the resulting masses,
mixings, and branching fractions of the relevant states, along with the values
of various DM observables, are provided. The most sensitive LHC analyses with
their signal regions that are studied using CheckMATE are presented at the
end.

In contrast to the co-annihilation scenario in BP1, BP2 features a scenario
where singlino-dominated DM annihilation through the resonant s-channel
$h_{\rm SM}$ exchange, leading to a rapid reduction in the DM relic density.
This resonance phenomenon helps comply with the Planck-observed upper bound on
the DM relic abundance. While the nearby bino state has a limited impact on
estimating relic abundance, it significantly influences the DMDD-SI cross-
section and collider searches for DM. The presence of $M_{1}\sim-90$ GeV
tempers the singlino-higgsino-bino system, reducing the DMDD-SI cross-section
of the singlino-dominated DM to approximately $10^{-48}\text{cm}^{2}$.
Additionally, beyond its impact on the DMDD-SI cross-section, such a bino
state leaves its imprint on collider searches for DM.
BR$[\chi_{{}_{2}}^{0}\rightarrow\chi_{{}_{1}}^{0}\gamma]$ is around 72%. The
scenario parallels BP1, where higgsino-like $\chi_{{}_{3,4}}^{0}$ and
$\chi_{{}_{1}}^{\pm}$ undergo significant cascade decay through
$\chi_{{}_{2}}^{0}$. However, given that the mass difference
$m_{{{}_{\chi}}_{{}_{2}}^{0}}-m_{{{}_{\chi}}_{{}_{1}}^{0}}\sim 30$ GeV is
relatively larger than in the BP1 scenario, the radiated photon here is
expected to be relatively hard compared to BP1. On the other hand, note that
$\mu_{\mathrm{eff}}$ in BP2 is much smaller than that of BP1, resulting in the
production cross-section of the process
$pp\rightarrow\chi_{{}_{3,4}}^{0}\chi_{{}_{1}}^{\pm}$ being almost four times
larger than BP1. Nevertheless, a CheckMATE analysis allows this benchmark
point with a smaller $r$ ($=0.61$) value compared to BP1. This can be
understood from the fact that, in BP2, $\lambda$ is almost twice the BP1,
which reduces the branching fractions of higgsino-like states to decay to the
bino-like NLSP, resulting in a reduction in the yields of final states with
photons. This reduces the sensitivity of the ATLAS analysis ATLAS:2020qlk  for
this benchmark point.

We present another benchmark point, BP3, where the observed relic abundance of
DM is realized via singlino higgsino co-annihilation. The relative
contribution to the relic density of the annihilation processes of
$\\{\chi_{{}_{1}}^{\pm}\chi_{{}_{2,3}}^{0}\rightarrow\text{SM}\,\,\text{SM}\\}$
and
$\\{\chi_{{}_{1}}^{\pm}\chi_{{}_{1}}^{\mp}\rightarrow\text{SM}\,\,\text{SM}\\}$
are much larger than the co-annihilation process involving LSP, such as,
$\\{\chi_{{}_{1}}^{\pm}\chi_{{}_{1}}^{0}\rightarrow\text{SM}\,\,\text{SM}\\}$.
Thus, in this scenario, “assisted co-annihilation” Ellwanger:2009dp  plays a
very important role in reconciling the relic density of the singlino-dominated
DM with the Planck measurement. The spectrum of the singlino higgsino sector
corresponds to a compressed scenario. Note that, here singlino-dominated DM
and the higgsino-like chargino/neutralinos are around 300 GeV, which seems to
be much above the latest constraints from LHC analyses ATLAS:2021moa . It is
important to note that, in this scenario, such experimental bounds would be
relaxed significantly due to the smaller production cross-section (smaller by
50%–60%) of higgsino-like $\chi_{{}_{2,3}}^{0}\chi_{{}_{1}}^{\pm}$ when
compared to that of wino-like $\chi_{{}_{2}}^{0}\chi_{{}_{1}}^{\pm}$ assumed
in the LHC analyses. Furthermore, these experimental analyses consider the
wino-like states decay via off-shell $W,Z,h_{\rm SM}$ states whereas in this
benchmark point $\chi_{{}_{2,3}}^{0}$ exhibit significant amount of radiative
decays which significantly suppress the reach of the multi-lepton search
analyses at the LHC.

Here, BR$[\chi_{{}_{2}}^{0}\rightarrow\chi_{{}_{1}}^{0}\gamma]$ and
BR$[\chi_{{}_{3}}^{0}\rightarrow\chi_{{}_{2}}^{0}\gamma]$ are around 63% and
86%, respectively. One photon would appear from the decay of
$\chi_{{}_{2}}^{0}$, and two photons would appear from the cascades of
$\chi_{{}_{3}}^{0}$. Note that,
$m_{{{}_{\chi}}_{{}_{2}}^{0}}-m_{{{}_{\chi}}_{{}_{1}}^{0}}\sim 16$ GeV,
$m_{{{}_{\chi}}_{{}_{3}}^{0}}-m_{{{}_{\chi}}_{{}_{1}}^{0}}\sim 25$ GeV and
$m_{{{}_{\chi}}_{{}_{3}}^{0}}-m_{{{}_{\chi}}_{{}_{2}}^{0}}\sim 9$ GeV. Thus,
in this scenario, the photons coming from the higgsino-like neutralinos and
other visible decay products of the higgsino-like chargino are expected to be
relatively soft to be detected. The current investigations at the LHC do not
examine this specific area of parameter space, mostly because of the large
radiative branching ratios of higgsino-like heavier neutralinos. This impedes
the decay into charged-lepton final states, which are predominantly considered
in the current LHC searches. Therefore, it indicates the possibility of
utilizing a new detection channel at the LHC more effectively to explore this
region of the NMSSM parameter space. To augment the distinctive features of
the radiative decay channel involving a soft photon and $\not{E}_{T}$ at the
LHC, one may explore events where higgsino pairs
($\chi_{{}_{2,3}}^{0}+\chi_{{}_{1}}^{\pm}$) are generated alongside an initial
state radiation (ISR) jet (j) which is relatively hard Baum:2023inl . Despite
the limited impact of the ISR jet on the $P_{T}$ distribution of the photon,
it actively recoils against the ($\chi_{{}_{2,3}}^{0}+\chi_{{}_{1}}^{\pm}$)
system, which leads to a sizable increase $\not{E}_{T}$ of such events. A
comprehensive collider simulation and discussion on this is presented in the
following subsection.

At this benchmark point, the bino-like fourth neutralino state indirectly
impacts both collider and DMDD phenomena. A mass parameter $M_{1}\sim 509$ GeV
introduces a splitting of approximately 10 GeV between the higgsino-like
states $\chi_{{}_{2}}^{0}$ and $\chi_{{}_{3}}^{0}$. This splitting leads to a
notable decay branching fraction of $\chi_{{}_{3}}^{0}$ into
$\chi_{{}_{2}}^{0}$ and $\gamma$. It is noteworthy that the sign of $M_{1}$
plays a crucial role. Such a splitting between the higgsino-like states
$\chi_{{}_{2}}^{0}$ and $\chi_{{}_{3}}^{0}$ is unattainable for
$M_{1}\sim-509$ GeV, leading to a shift in the decay patterns of both
$\chi_{{}_{2}}^{0}$ and $\chi_{{}_{3}}^{0}$ with a significant reduction in
radiative decay branching fractions. The DMDD-SI cross-section remains below
the latest experimental bounds due to $M_{1}>0$. If we consider $M_{1}<0$ with
the same absolute value, the DMDD-SI cross-section would be an order of
magnitude larger, leading to exclusion from the latest DMDD-SI constraints.
Hence, as previously discussed, the bino state, serving as the fourth
neutralino, continues to impact the exploration of singlino-dominated DM in
both collider and DMDD experiments.

Benchmark point BP4 shares similarities with BP3 in its singlino-higgsino co-
annihilation scenario but stands out with a relatively small
$\mu_{\text{eff}}$ of about $200$ GeV, leading to a significantly augmented
production cross-section of higgsino-like electroweakinos at the LHC. Here
also, both the higgsino-like neutralinos ($\chi_{{}_{2,3}}^{0}$) exhibit large
radiative decay modes in the singlino-dominated LSP and a photon rather than
decay into leptons/hadrons. Thus,
BR$[\chi_{{}_{2}}^{0}\rightarrow\chi_{{}_{1}}^{0}\gamma]$ and
BR$[\chi_{{}_{3}}^{0}\rightarrow\chi_{{}_{2}}^{0}\gamma]$ are around 73% and
92%, respectively. However, the mass gaps between the singlino-like LSP and
the higgsino-like neutralinos are relatively small compared to BP3.
Consequently, it is expected that the photons arising from the decay of
higgsino-like states would be relatively soft compared to those in BP3. In
this benchmark point, since the $\kappa<0$, $M_{1}$ and $\mu_{\mathrm{eff}}$
have same relative sign, i.e., both are $-$ve, which reduces the DMDD-SI
cross-section below the latest experimental constraints. Concerning collider
studies, an analysis using CheckMATE, which includes all available analyses in
its repository, reveals an ’r’ value significantly below 1 for points BP3 and
BP4. This indicates a notable insensitivity to LHC searches, implying that
these points are allowed.

It is interesting to note that both ATLAS ATLAS:2021moa ; ATLAS:2019lng  and
CMS CMS:2021cox ; CMS:2021edw  collaborations observe small excess in searches
for the chargino-neutralinos in the compressed regions in the soft di-lepton
channels. This excess can be interpreted as the production of higgsino-like
states where the mass splitting between the chargino and the LSP is around
$5–20$ GeV 333Recently, in reference Agin:2023yoq , it has been claimed that
the excess can also be realised from the monojet searches ATLAS:2021kxv ;
CMS:2021far  in a similar region of parameter space in the context of light-
compressed higgsinos after recasting those analyses in Madanalysis5
Araz:2020lnp .. Note that the excess in the soft lepton channels can be
explained within the context of singlino-higgsino co-annihilation scenarios
discussed in this work. As we pointed out, such a co-annihilation scenario can
also indicate another possible detection channel involving photons. Therefore,
if the LHC conducts a dedicated analysis using existing Run-2 data to probe
this scenario with photons, it would be intriguing to observe whether the same
excess is present. Nevertheless, these compressed regions will undergo further
investigation in the upcoming runs of the LHC, and it will be interesting to
see whether the excess persists.

###  5.2 Possible new search channels at the LHC

In this section, we take a closer look at the differential distributions of
various kinematic variables and their correlations in order to differentiate
signal events from backgrounds. To analyze the kinematics of our signal events
of the benchmark points, we simulate
$pp\to\chi_{{}_{2,3,4}}^{0}\chi_{{}_{1}}^{\pm}$ events for all benchmark
points at the lowest order (LO) in the perturbation theory using
<EMAIL_ADDRESS>Alwall:2014hca  for the 14 TeV HL-LHC run. The
default parton distribution function (‘nn23lo1’ Ball:2010de ) is utilized with
a dynamic factorization scale selection defined as
$Q^{2}=\frac{1}{2}\sum_{i}M_{i}^{T^{2}}=\frac{1}{2}\sum_{i}(m_{i}^{2}+p_{Ti}^{2})$,
where $M^{T}_{i}$ represents the transverse mass of the ’$i$’-th final state
particle. We focus on studying the distribution of kinematical variables such
as $\not{E}_{T}$, $P_{T}$ of the leading jet $(P_{T}^{\text{jet}})$ and photon
$(P_{T}^{\gamma})$. Particularly for BP3 and BP4, we also simulate
$pp\to\chi_{{}_{2,3,4}}^{0}\chi_{{}_{1}}^{\pm}j$ process to study the effect
of the ISR on distributions of these kinematic variables. To generate signal
events, the interface between MadGraph5 and the NMSSM is facilitated by
incorporating the UFO model file Degrande:2011ua  from the FeynRules package
Alloul:2013bka .

Calculations for the decay kinematics of unstable excitations, as well as the
subsequent processes of showering and hadronization, are executed employing
PYTHIA8-v8.3 Sjostrand:2006za ; Sjostrand:2007gs . The default PYTHIA8 cards
are employed for these simulations. As mentioned earlier, various masses and
decay branching fractions of NMSSM excitations of those benchmark points are
generated via NMSSMTools and PYTHIA8 reads those from the NMSSMTools-generated
SLHA Skands:2003cj  files. Finally, the effects of the detector response are
incorporated using DELPHES-v3.4.2 deFavereau:2013fsa . Utilizing Prospino 2.1
Beenakker:1996ed , we compute the NLO corrections for electroweakino
production, projecting a 20-30% increase in the LHC cross-sections within the
region of parameter space of present interest. To accommodate this, we apply a
common K-factor of 1.25 Fiaschi:2018hgm .

Figure 2: Differential event distributions of $\not{E}_{T}$,
$P_{T}^{\text{jet}}$, and $P_{T}^{\gamma}$ from top to bottom in the left
panel for BP1 (in black) and BP2 (in red). On the middle and the right panel,
differential event distributions for similar kinematic variables are presented
for BP3 and BP4, respectively, where black (red) indicates distributions for
the process $pp\to\chi_{{}_{2,3,4}}^{0}\chi_{{}_{1}}^{\pm}$
($pp\to\chi_{{}_{2,3,4}}^{0}\chi_{{}_{1}}^{\pm}j$).

As discussed in the preceding section, for BP1 and BP2, the production cross-
section of the process $pp\rightarrow\chi_{{}_{2}}^{0}\chi_{{}_{1}}^{\pm}$ is
significantly smaller, given the fact that $\chi_{{}_{2}}^{0}$ is a bino-
dominated neutralino, in contrast to the production process
$pp\rightarrow\chi_{{}_{3,4}}^{0}\chi_{{}_{1}}^{\pm}$, where
$\chi_{{}_{3,4}}^{0}$ are higgsino-dominated neutralinos. In the left panel of
figure 2, the top-to-bottom sequence displays the distributions of
$\not{E}_{T}$, $P_{T}^{\text{jet}}$ and $P_{T}^{\gamma}$, respectively. The
distributions in black (red) are for BP1 (BP2). Relatively lighter higgsino-
like states (smaller $\mu_{\mathrm{eff}}$) in BP2 result in a larger
production cross-section compared to BP1, leading to significantly larger
amplitudes of events in its differential distributions of those kinematic
variables. Since both the benchmark points exhibit a similar type of hierarchy
among the states, the distribution of $\not{E}_{T}$ and $P_{T}^{\text{jet}}$
peak around similar ranges. On the other hand, as expected, due to a
relatively larger mass gap between $\chi_{{}_{2}}^{0}$ and
$\chi_{{}_{1}}^{0}$, the $P_{T}^{\gamma}$ peaks at a relatively larger value
for BP2 compared with BP1. Therefore, in order to suppress the SM background,
one can demand a selection cut on a relatively larger value of $\not{E}_{T}$
($\gtrsim 150$ GeV) along with a similarly relatively larger $P_{T}$ of the
leading jets/leptons coming from on-shell $W,Z,h_{\rm SM}$-bosons resulting
from the decay of heavier higgsino-like states
($\chi_{{}_{3,4}}^{0},\chi_{{}_{1}}^{\pm}$). Additionally, implementing event
selection cuts on the $P_{T}$ of the relatively soft leading photon ($\gtrsim
10$ GeV) can further enhance the suppression of the SM background. Therefore,
as we mentioned in the previous section, integrating selection criteria that
include hard jets/leptons, large missing energy, and a relatively soft photon
in the analysis would facilitate the search of the singlino-bino co-
annihilation scenario with relatively large higgsino-like states.

In the middle and the right panel of Figure 2, the sequence from top to bottom
illustrates distributions of the same kinematic variables for BP3 and BP4,
respectively. The black and red curves in these plots represent the processes
$pp\to\chi_{{}_{2,3,4}}^{0}\chi_{{}_{1}}^{\pm}$ and
$pp\to\chi_{{}_{2,3,4}}^{0}\chi_{{}_{1}}^{\pm}j$, respectively. Relatively
smaller $\mu_{\text{eff}}$ in BP4 results in a greater production cross-
section of the higgsino-like states compared to BP3. This, in turn, leads to
significantly larger amplitudes of events in the differential distributions of
those kinematic variables. In the overall production cross-section for both
BP3 and BP4, the process involving $\chi_{{}_{4}}^{0}$, being bino-like, is
subdominant. As discussed in the previous section, unlike BP1 and BP2, BP3 and
BP4 feature singlino-higgsino compressed scenarios where higgsino-like
neutralinos ($\chi_{{}_{2,3}}^{0}$) prominently exhibit radiative decay modes,
favoring emissions involving the singlino-dominated LSP and a photon rather
than decay into leptons/hadrons. Additionally, due to the limited available
phase space, the visible decay productions from the higgsino-like states are
expected to be relatively soft. In order to suppress the SM background, events
featuring the production of higgsino-like states
($\chi_{{}_{2,3}}^{0}\chi_{{}_{1}}^{\pm}$) in conjunction with a hard ISR jet
can be considered. In the absence of an ISR jet, $\chi_{{}_{2,3}}^{0}$ and
$\chi_{{}_{1}}^{\pm}$ would primarily be produced at the LHC with equal and
opposite $P_{T}$. However, in the presence of the ISR jet, the
($\chi_{{}_{2,3}}^{0}\chi_{{}_{1}}^{\pm}$) system recoils against the ISR jet
in the transverse plane. Due to the small mass difference between the LSP and
the higgsino-like states, a significant portion of the $P_{T}$ of the
higgsino-like $\chi_{{}_{2,3}}^{0}$ and $\chi_{{}_{1}}^{\pm}$ is transferred
to the LSP, contributing to event $\not{E}_{T}$ that approximately balances
with $P_{T}$ of the ISR jet. The plots reveal that the peak of $\not{E}_{T}$
occurs at a relatively higher value for the process involving the ISR jet.
Additionally, a broad high $\not{E}_{T}$ tail is observed for events
containing one ISR jet. This characteristic allows for more aggressive
selection cuts on $\not{E}_{T}$ in the analysis, effectively rejecting a
significant amount of the SM backgrounds at a moderate cost in losing signal
events.

It is noteworthy that a similar broader high $P_{T}$ tail of the leading jet
is also observed in events containing one ISR jet. Given the correlation
between $P_{T}^{\text{jet}}$ and $\not{E}_{T}$ in events with one ISR jet,
imposing a stringent cut on $\not{E}_{T}$ ($\gtrsim$ 100 GeV) ensures that
most signal events have substantially larger $P_{T}^{\text{jet}}$ ($\gtrsim$
100 GeV). Consequently, incorporating hard cuts on both $\not{E}_{T}$ and
$P_{T}^{\text{jet}}$ enables a significant reduction in the SM background
while minimally sacrificing signal events. On the other hand, distributions of
the $P_{T}$ of the leading photon for events with and without an ISR jet
appear nearly identical. Due to the comparatively larger available phase space
for the decay of $\chi_{{}_{2,3}}^{0}$ to $\chi_{{}_{1}}^{0}$, $P_{T}$
distribution of the leading photon exhibits a relatively broader range, with
the peak occurring at a higher value in BP3 as compared to BP4.

Figure 3: $P_{T}$ distribution of the leading photon for BP3 (left) and BP4
(right) considering $\not{E}_{T}$ and $P_{T}$ of the leading jet more than 100
GeV. Black and red denote event distributions for
$pp\to\chi_{{}_{2,3,4}}^{0}\chi_{{}_{1}}^{\pm}$ and
$pp\to\chi_{{}_{2,3,4}}^{0}\chi_{{}_{1}}^{\pm}j$ processes, respectively.

In figure 3, we present the event distribution of $P_{T}^{\gamma}$ for BP3
(left) and BP4 (right) considering the two above mentioned cuts, i.e.,
$\not{E}_{T}>100$ GeV and $P_{T}^{\text{jet}}>100$ GeV. The presence of a
single ISR jet in the events under those specified cuts leads to a notable
increase in the number of events at the peak of the distribution and a
broadening of the high $P_{T}^{\gamma}$ tail. This phenomenon arises due to a
substantial drop in the cross-section of the process in the absence of any ISR
jet under such cuts. Moreover, the distribution exhibits a peak at a slightly
higher $P_{T}^{\gamma}$ when the ISR jet is considered, suggesting an overall
transverse boost for the photon in the context of
$pp\to\chi_{{}_{2,3,4}}^{0}\chi_{{}_{1}}^{\pm}j$, particularly for a
relatively hard ISR jet. This can be understood from the fact that if the
decaying photon from $\chi_{{}_{2,3}}^{0}$ originated in the same direction in
which $\chi_{{}_{2,3}}^{0}$ are produced and boosted due to large $P_{T}$ of
the ISR jet in the event. These observations imply a significant influence of
the ISR jet on both the yield of events and the kinematics of the final-state
photon. The enhanced $P_{T}^{\gamma}$ peak and the extended high
$P_{T}^{\gamma}$ tail underscore the importance of accounting for a hard ISR
jet in the process. Consequently, triggering on a photon with $P_{T}>10$ GeV
in such a scenario would capture a significantly larger number of signal
events in $pp\to\chi_{{}_{2,3,4}}^{0}\chi_{{}_{1}}^{\pm}j$ compared to
$pp\to\chi_{{}_{2,3,4}}^{0}\chi_{{}_{1}}^{\pm}$ under the aforementioned cuts
on $\not{E}_{T}$ and $P_{T}^{\text{jet}}$.

In this section, we present differential distributions of some kinematic
variables and propose some signal regions in terms of these variables for BP1,
BP2, BP3 and BP4. However, more dedicated analyses are essential for the
proposed signals and the SM background to explore novel signatures at the LHC,
thereby discerning the most promising signal topologies and optimal event
selection strategies. Conducting multivariate analyses with an expanded set of
kinematic variables for all visible objects resulting from the decays of
heavier neutralinos (including leading jets, leptons, photons, and missing
energy) can become essential in some scenarios. This strategy can be
instrumental in significantly reducing the impact of the SM background and
improving signal efficiency. We preserve this for our future work. Note that a
recent work Datta:2022bvg  demonstrates that relatively not-so-heavy top
squarks in the NMSSM can probe a unique parameter space of the higgsino-
singlino-bino system involving a relatively light singlet-like scalar. A
relatively larger production cross-section of top squark pairs at the LHC is
instrumental in enabling the exploration of this specific parameter space,
which, remarkably, has not yet been investigated by experimental
collaborations. In the context of our specific region of interest,
characterized by substantial radiative decay modes of heavier neutralinos that
involve photon emission, the production of moderately heavy top squarks at the
LHC could also yield intriguing collider signals, in a way similar to an
analysis in reference Baer:2005jq . Higgsino-like or bino-like neutralinos can
arise from the decay cascades of top squarks, and these neutralinos can emit a
photon. In this context, the photon could exhibit significant boosting
compared to the scenario we are discussing, especially if a substantial mass
gap exists between the top squark and the decaying neutralino state, resulting
in a boosted neutralino. A dedicated analysis can be undertaken to investigate
such compelling collider signatures arising from the region of interest
delineated in this study.

##  6 Conclusion

Understanding the nature of the DM stands out as a prominent challenge in
theoretical particle physics and cosmology, necessitating the exploration of
novel physics that extends beyond the confines of the SM. In this work, we
investigate the electroweakino sector of the NMSSM, which can provide a cold
DM candidate of the Universe. We focus on the singlino-dominated LSP DM,
tempered by the nearby higgsino-like and bino-like neutralinos. An involved
set of blind spot conditions of DMDD-SI and DMDD-SD scattering cross-sections
for the singlino-dominated DM are derived considering $(4\times 4)$ bino-
higgsino-singlino neutralino sector. At the minimal mixing between the SM-like
Higgs boson and the other Higgs bosons and considering only the singlino-
higgsino neutralino sector, a spin-independent blind spot condition arises
exclusively when the ratio
$m_{{{}_{\chi}}_{{}_{1}}^{0}}/\mu_{\mathrm{eff}}>0$, i.e., $\kappa>0$. It has
been shown in this work that due to the tempering of the bino-like neutralino,
a novel blind spot condition can appear for the singlino-dominated DM when
$m_{{{}_{\chi}}_{{}_{1}}^{0}}$ and $\mu_{\mathrm{eff}}$ carry an opposite
relative sign, i.e., $\kappa<0$ region of parameter space. Such a scenario
arises due to a cancellation between the $g_{1}$ proportional gaugino-
higgsino-Higgs interaction term and the $\lambda$ proportional singlino-
higgsino-Higgs interaction term. It demands the same (opposite) relative sign
between $\mu_{\mathrm{eff}}$ and $M_{1}$ for $\kappa<0$ $(>0)$. This opens up
a new region of parameter space that exhibits a smaller DMDD-SI cross-section
of the singlino-dominated DM. On the other hand, the spin-dependent rate
(proportional with $g_{\chi_{{}_{1}}^{0}\chi_{{}_{1}}^{0}Z}$ $\sim
N_{13}^{2}-N_{14}^{2}$, i.e., the difference between the higgsino components
of the LSP) is suppressed in the MSSM when $\tan\beta$ is small (approaching
zero in the limit of $\tan\beta\rightarrow 1$ or very large values of
$\mu_{\mathrm{eff}}$), could now have new compensating terms in its expression
in the presence of a light bino-like state. These can then play their roles in
suppressing the relevant coupling even for larger $\tan\beta$ values.

In addition to highlighting the role of light bino-like and higgsino-like
states in tempering the singlino-dominated LSP on the DMDD rates, we explore
scenarios where $M_{1}$ or $\mu_{\mathrm{eff}}$ is close to
$m_{{}_{\widetilde{S}}}$ and as a result the bino or the higgsino states can
act as co-annihilation partners for singlino-dominated DM, aiming to ensure
compliance with the observed DM relic abundance of the Universe. In the case
of singlino-bino co-annihilation, NLSP is bino-like, and it can exhibit a
significant radiative decay into a soft photon and the singlino-dominated DM.
In this scenario, cascades of the produced higgsino-like $\chi_{{}_{3,4}}^{0}$
and $\chi_{{}_{1}}^{\pm}$ via $\chi_{{}_{2}}^{0}$ could lead to:
$pp\rightarrow\chi_{{}_{3,4}}^{0}(\widetilde{H})\chi_{{}_{1}}^{\pm}(\widetilde{H})\rightarrow$
$h_{\rm
SM}/Z+W^{\pm}+\chi_{{}_{2}}^{0}(\tilde{B})\,\big{[}\chi_{{}_{2}}^{0}\rightarrow\gamma\,\chi_{{}_{1}}^{0}(\tilde{S})\big{]}$
$\Rightarrow 3\ell+\geq 1\gamma+\not{E}_{T}$ or $1\ell+2b+\geq
1\gamma+\not{E}_{T}$ final states at the LHC. If the photon from
$\chi_{{}_{2}}^{0}$ remains soft and undetected, the cascades of
$\chi_{{}_{1}}^{\pm}$ and $\chi_{{}_{3,4}}^{0}$ via bino-like
$\chi_{{}_{2}}^{0}$ lead to similar final states as when these higgsino-like
states directly decay to the singlino-dominated LSP. In this scenario, leptons
and jets originating from on-shell $W,Z,h_{\rm SM}$ are relatively hard.
Observing excess over the SM backgrounds in the signals with and without soft
photons with $3\ell+\not{E}_{T}$ and $1\ell+2b+\not{E}_{T}$ at the HL-LHC
would promptly point back to the scenario that includes singlino-bino co-
annihilation with relatively heavier higgsinos. We also point out that, not
only in the co-annihilation scenario, a similar signal at the LHC could appear
from the singlino-dominated DM, which is mainly annihilated through the
$h_{\rm SM}$ resonance funnel.

It is further pointed out that in the DMDD blind spot regions, higgsino-like
neutralinos, being the $\chi_{{}_{2}}^{0}$ and $\chi_{{}_{3}}^{0}$, can
exhibit a significant large radiative decay mode into photon in the singlino-
higgsino co-annihilation scenario. The bino-like fourth neutralino state
indirectly impacts both collider and DMDD phenomena. It tempers the singlino-
like LSP in such a way that the DMDD-SI cross-section remains small and
modifies the compositions of various neutralinos in such a way that higgsino-
like neutralinos prefer to exhibit radiative photonic decay mode. We observe
that in such a compressed scenario, the decaying objects from higgsino-like
neutralino ($\chi_{{}_{2,3}}^{0}$) and charginos ($\chi_{{}_{1}}^{\pm}$) could
remain mostly soft from their associated production processes, and it is
difficult to suppress the SM backgrounds. On the contrary, considering an
initial state radiation jet associated with the
$pp\rightarrow\chi_{{}_{2,3}}^{0}\chi_{{}_{1}}^{\pm}$ process can be analysed
by considering a hard mono-jet with significant missing energy and at least
one photon in the signal. We provide the signal cross-sections and kinematic
distributions of these objects, which indicate that the process of
$pp\rightarrow\chi_{{}_{2,3}}^{0}\chi_{{}_{1}}^{\pm}$ with an ISR would be a
more efficient one to suppress the SM backgrounds. It would be important for
the ATLAS/CMS collaborations to conduct an investigation into this potential
signal. Exploring radiative photonic decays of higgsino-like neutralinos
through such a search could facilitate a more thorough examination of
compressed scenarios and the blind spot regions for DMDD within the NMSSM.

Acknowledgements

C.W. would like to thank the Aspen Center for Physics, which is supported by
National Science Foundation grant No. PHY-1607611, where part of this work has
been done. We would like to thank S. Baum, M. Carena, A. Datta, C. Hugonie, T.
Ou, D. Rocha and N. Shah for useful discussions and comments. S.R. and C.W.
have been partially supported by the U.S. Department of Energy under contracts
No. DEAC02- 06CH11357 at Argonne National Laboratory. The work of C.W. at the
University of Chicago has also been supported by the DOE grant DE-SC0013642.

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