# Implications of Sgr A∗ on the $\gamma$-rays searches of Bino Dark Matter
with $(g-2)_{\mu}$

Utpal Chattopadhyay<EMAIL_ADDRESS>School of Physical Sciences, Indian
Association for the Cultivation of Science, Jadavpur, Kolkata 700032, India
Debottam Das<EMAIL_ADDRESS>Institute of Physics, Sachivalaya Marg,
Bhubaneswar, 751005, India  Homi Bhabha National Institute, Training School
Complex, Anushakti Nagar, Mumbai 400094, India  Sujoy Poddar
<EMAIL_ADDRESS>Department of Physics, Diamond Harbour Women’s University,
Diamond Harbour Road, Sarisha, South 24 Parganas, West Bengal — 743368, India
Rahul Puri<EMAIL_ADDRESS>Institute of Physics, Sachivalaya Marg,
Bhubaneswar, 751005, India  Homi Bhabha National Institute, Training School
Complex, Anushakti Nagar, Mumbai 400094, India  Abhijit Kumar Saha
<EMAIL_ADDRESS>Institute of Physics, Sachivalaya Marg, Bhubaneswar,
751005, India  Homi Bhabha National Institute, Training School Complex,
Anushakti Nagar, Mumbai 400094, India

###### Abstract

Abstract: We analyse the impact of dark matter density spike around the Milky
Way’s supermassive black hole (SMBH), Sgr A∗, in probing the Bino-dominated
neutralino dark matter (DM) $\tilde{\chi}_{1}^{0}$ within the MSSM, which
typically produces relatively faint signals in the conventional DM halos. In
particular, we explore the indirect search prospects of sub-TeV Bino-Higgsino
and Bino-Wino-Higgsino DM in the MSSM, consistent with the supersymmetric
predictions required to explain the anomalous magnetic moment of the muon.
Typical over-abundance of Bino DM is ameliorated with slepton and/or Wino
coannihilations. The lightest neutralino, thus may be associated with a
compressed supersymmetric particle spectrum, which, in general, is difficult
to probe at conventional LHC searches. Similarly, for a rather tiny Higgsino
mixing, $\tilde{\chi}_{1}^{0}$ does not offer much prospect to assess its
predictions at dark matter direct detection searches. Accommodating the
inclusive effects of density spike, here, we present the requisite boost
factor to facilitate $\gamma-$ray searches of Bino-dominated DM in the MSSM,
especially focusing on the Fermi-LAT and HESS observations.

##  I Introduction

For decades, it has been well recognized that the Minimal Supersymmetric
Standard Model (MSSM) [1, 2, 3, 4, 5] offers a natural candidate for dark
matter (DM) [6, 7, 8] in the Universe if the $R$-parity is conserved [9]. The
neutral supersymmetric (SUSY) partners of the Higgs (Higgsinos) and
electroweak gauge bosons (gauginos) can be assembled into four neutralinos,
$\tilde{\chi}_{1,2,3,4}^{0}$, with $\tilde{\chi}_{1}^{0}$ referring to the
lightest among them. In MSSM, $\tilde{\chi}_{1}^{0}$ can be considered to be
the lightest supersymmetric particle (LSP) to serve the role of a viable DM
candidate [8, 7]. Depending on the individual compositions of the gauginos and
the Higgsinos, the neutralino DM could become Bino, Wino, or Higgsino-
dominated or any suitable admixtures of them.

On the experimental front, the LHC null observation of coloured sparticles has
excluded the masses of squarks and gluinos smaller than a few TeV [10, 11].
The uncoloured particles, such as elctroweakinos and sleptons, can still be in
the sub-TeV range since LHC constraints are relatively weaker due to smaller
production cross-sections [12] (see, also [13]). Importantly, lighter
electroweakinos and sleptons serve a pivotal role in reconciling the
difference observed between the measured and the theoretically predicted
values of the muon’s anomalous magnetic moment, $a_{\mu}=(g-2)_{\mu}/2$.
Specifically, at one-loop level, loops containing electroweakinos and lighter
sleptons e.g., $\tilde{\chi}^{-}-\tilde{\nu}_{\mu}$ and
$\tilde{\chi}^{0}-\tilde{\mu}$ contribute to $\delta a_{\mu}=a_{\mu}^{\rm
Exp}-a_{\mu}^{\rm SM}$ [14, 15, 16, 17]. The disagreement over $a_{\mu}$ value
was previously reported by Brookhaven National Laboratory (BNL) E821
experiment [18]. Combining with Fermilab Muon g-2 experiments [19], it
establishes a deviation of $4.2\sigma$ from the value predicted by the SM. The
recent Fermilab E989 experimental data from Run-2 and Run-3 have also
confirmed the discrepancy, giving a new world average of $5.1\sigma$ [20].

In the past, the MSSM parameter space analyses combining neutralino DM and
$\delta a_{\mu}$ were thoroughly performed in several works [21, 22, 14, 23,
15, 24, 16, 25, 26, 27, 28, 13, 29, 30, 31, 32, 33, 34, 35, 36]. A dominantly
Bino-like light $\tilde{\chi}_{1}^{0}$ along with co-annihilating sleptons (in
particular, $\tilde{\mu}$, $\tilde{\nu}_{\mu}$) are favoured by $\delta
a_{\mu}$, especially if the observed DM abundance has to come entirely from
the lightest neutralino having masses $\lesssim 500$ GeV. To note, Bino-
slepton coannihilation $\tilde{\chi}_{1}^{0}\tilde{l}\xrightarrow{l}{l\gamma}$
does not need any tuning in the neutralino mixing matrix to produce the
correct relic abundance. This may not be true for a significantly mixed
$\tilde{\chi}_{1}^{0}$ in the form of Bino-Higgsino-like DM, which can also
reconcile the observed DM abundance and the discrepancy in $\delta a_{\mu}$.
In this case, apart from the fact that fine-tuning the different elements in
the mixing matrices needs to be adhered to, the null results in the Spin-
Independent (SI) dark matter direct detection experiments (DD) [37, 38, 39,
40, 41, 42, 43, 44, 45, 46] may severely restrict the MSSM parameter space
[47, 48, 49, 50]. In fact, SI-DD searches directly curb the Bino-Higgsino-
Higgs coupling. On the other hand, in the presence of a minimal Wino
composition in $\tilde{\chi}_{1}^{0}$ (assuming the Higgsino fraction to be
negligibly small), the SI-DD rates will never reach the sensitivity of the
present or future experiments for a dominantly Bino-like DM. It follows from
the fact that the Higgs coupling to the LSP pair is proportional to the
product of their appropriate mixing matrix elements corresponding to the
Higgsino and gaugino components [51, 5].

In recent times, Ref. [50, 36] explore the regions of the allowed MSSM
parameter space, compatible with the $a_{\mu}$ anomaly, DM relic density, DD
limits, and the latest LHC Run-2 data. It turns out that two specific regions
will be of interest for future searches : (1) a dominantly Bino-like light
$\tilde{\chi}_{1}^{0}$ with minimal Higgsino contributions ($M_{1}\ll\mu$),
and (2) a compressed scenario of Bino, Wino, a minimal Higgsino
($M_{1}\lesssim M_{2}\ll\mu$) components. On both occasions, masses of
sleptons are not far from that of the LSP. In the subsequent sections, we
refer them as $\tilde{B}_{\tilde{H}}$ and $\tilde{B}_{\tilde{W}\tilde{H}}$
zones of the MSSM parameter space. The SI-DD search prospects of the DM can be
improved to some extent if one incorporates all the electroweak vertex
corrections in the computations of SI-DD of the $\tilde{\chi}_{1}^{0}$ DM [52,
36]. Even then, a substantial part of the parameter space, specifically with
higher $\mu$ values, will be out of the periphery of future SI-DD searches.
Our aim in this work would be to survey the $\tilde{B}_{\tilde{H}}$ and
$\tilde{B}_{\tilde{W}\tilde{H}}$ regimes of the MSSM parameter space through
the possible indirect searches of $\gamma$-ray observations, which otherwise
may escape the LHC limits of the direct production of the SUSY particles.
Specifically, we will use Fermi-LAT and HESS observations for scrutinizing the
parameter space [53, 54, 55, 56].

It is well-known that the $\gamma$-ray flux from the DM pair annihilations
completely depends on the Wino or Higgsino components of the LSP. Naturally, a
high value for either one of the components or both is desirable for a
potential rise in the $\gamma-$ray flux. However, the observed abundance of DM
along with the compatibility with $\delta a_{\mu}$ and the SI-DD bounds
strongly restrict the amount of Wino or Higgsino components in the LSP that
belong to the $\tilde{B}_{\tilde{H}}$ and $\tilde{B}_{\tilde{W}\tilde{H}}$
regimes. This, in turn, may result in suppression of the $\gamma$-ray spectra
from the DM-DM annihilations inside the conventional DM halos, thus posing a
challenge to the standard indirect searches of the dark matter. The dark
matter signal enhancement in the galactic halos is possible in the presence of
a boost factor. Different proposals explaining the boost factor, such as
Sommerfield enhancement [57, 58, 59], Breit-Wigner resonance [60] and
enhancement in the local DM density, are prevalent in the literature. However,
for a dominantly Bino-like DM, away from the resonance, the rise in the local
DM density may become the only possibility to produce a much-needed boost in
the $\gamma$-ray signals of the DM coming from the Galactic Center (GC) or any
dwarf galaxies.

A boost such as that mentioned above can potentially be realized through the
DM density spikes around the supermassive black hole (SMBH) at the center of
the Milky Way [61]. Recently, the indirect evidence of a dark matter density
spike around two stellar-mass black holes has been reported [62]. One can find
that the high density of DM in the form of a spike within the gravitational
zone of SMBH can magnify the DM annihilation rates to some extent, resulting
in a bright source to the $\gamma$-ray telescope. Earlier works [63, 64, 65,
66, 67, 68, 69, 70, 71, 72, 73, 74] in this direction have anticipated that
such magnification in the DM annihilation signals can indeed offer a unique
probe of the dark matter models. Considering Milky Way’s SMBH Sgr A∗ [69] at
the backdrop of our analysis, we exemplify the DM density profile and
subsequently calculate the observable $\gamma$-ray flux in the MSSM for both
$\tilde{B}_{\tilde{H}}$ and $\tilde{B}_{\tilde{W}\tilde{H}}$ cases. The
predictions obtained will further be substantiated by the public results of
Fermi-LAT [53] and HESS telescopes [56] that consider GC as source.

We organize our paper as follows. Sec. II briefly discusses the properties of
$\tilde{\chi}_{1}^{0}$. The dependence of $\gamma$-ray flux on the
astrophysical and particle physics inputs has also been summarized. Here, we
detail the astrophysical model, which can lead to a spike in the DM density
distribution within the gravitational influence zone of SMBH. In Sec. III, we
summarize the precision observables, e.g., supersymmetric contributions to
anomalous magnetic moments of muon ($a^{\rm SUSY}_{\mu}$) and other
constraints like the limits from the SUSY searches at the collider
experiments. The constraints from the flavour physics and non-accelerator-
based experiments are also noted. Sec. IV presents the numerical results of
the $\gamma$-ray flux for $\tilde{B}_{\tilde{H}}$ and
$\tilde{B}_{\tilde{W}\tilde{H}}$ regimes in the MSSM. Finally, we conclude in
Sec. V.

##  II Photon spectra of Neutralino dark matter in the presence of the Milky
Way’s supermassive black hole

In the MSSM, the supersymmetric partner of the $U(1)_{Y}$ gauge boson,
$\tilde{B}$, $SU(2)_{L}$ neutral gauge boson, $\tilde{W}^{0}$, mix with the
supersymmetric partners of the two MSSM Higgs bosons, known as down and up
type Higgsinos $\tilde{H}_{d}^{0}$ and $\tilde{H}_{u}^{0}$ respectively. The
mass matrix of the neutralino sector in the basis
$(\tilde{B},\tilde{W}^{0},{\tilde{H}}^{0}_{u},{\tilde{H}}^{0}_{d})$ is given
by:

$\displaystyle\mathbb{M}=\left(\begin{array}[]{c c c
c}M_{1}&0&-M_{Z}s_{W}c_{\beta}&M_{Z}s_{W}s_{\beta}\\\\[6.0pt]
0&M_{2}&M_{Z}c_{W}c_{\beta}&-M_{Z}c_{W}s_{\beta}\\\\[6.0pt]
-M_{Z}s_{W}c_{\beta}&M_{Z}c_{W}c_{\beta}&0&-\mu\\\\[6.0pt]
M_{Z}s_{W}s_{\beta}&-M_{Z}c_{W}s_{\beta}&-\mu&0\end{array}\right),$ (5)

where $M_{1}$ and $M_{2}$ are the Bino and Wino masses, respectively, $\mu$ is
the Higgs mixing parameter, $c_{\beta}=\cos\beta$, $s_{\beta}=\sin\beta$ with
$\tan\beta=\frac{v_{2}}{v_{1}}$, $v_{2}$ and $v_{1}$ being the vacuum
expectation values of the two Higgs doublets ($H_{u}$ and $H_{d}$). Also,
$c_{W}=\cos\theta_{W}$, $s_{W}=\sin\theta_{W}$ where $\theta_{W}$ is the weak
mixing angle.

The lightest neutralino, widely considered as a potential DM candidate in an
R-parity conserving MSSM can be expressed as,

$\displaystyle\tilde{\chi}_{1}^{0}=N_{11}{\tilde{B}}+N_{12}{\tilde{W}^{0}}+N_{13}\tilde{H}^{0}_{u}+N_{14}{\tilde{H}}^{0}_{d},$
(6)

where $N$ is $4\times 4$ unitary matrix that diagonalizes $\mathbb{M}$ in
Eq.(5). The LSP could be of different natures in different parts of the MSSM
parameter space depending on the relative mass values of the gaugino and
Higgsino mass parameters. For example, if $M_{1}\ll M_{2},\mu$, it turns out
that $N_{11}\gg N_{1i}$ $(i=2$ \- $4)$ and the LSP becomes Bino-like. In
another case, when $M_{1}\lesssim\mu\ll M_{2}$, the DM becomes Bino-Higgsino-
like ($\tilde{B}_{\tilde{H}}$). It is also possible that $M_{1}\sim
M_{2}\lesssim\mu$; in that case, the LSP will be Bino-Wino-Higgsino
($\tilde{B}_{\tilde{W}\tilde{H}}$) type. However, following the strong limits
of the SI-DD searches, in both scenarios, $\mu\gg M_{1}$ is assumed.

We now begin our journey with the prerequisites to calculate photon flux in
the DM halo. Subsequently, we will elaborate on the astrophysical model, which
may lead to a spike in the DM density distribution around an SMBH.

Photon Flux and Astrophysical Model

The indirect detection techniques are based on the detection of secondary
particles that come out of dark matter (DM) annihilations in the over-dense
region of the universe [75, 76, 77]. The observable signals in the form of
photons ($\gamma$), positrons ($e^{+}$), antiprotons ($\overline{p}$),
neutrinos ($\nu$), etc may be searched for the quest of the DM sources.
Neutral particles ($\gamma,\nu$, etc.) travel undisturbed through the
interstellar medium and are more accurate in describing their origin. On the
other hand, charged particles encounter several interactions, such as the
galactic winds, interstellar magnetic field, etc, before reaching the
observatory and thus are subject to cosmic-rays (CR) propagation models [78,
79, 80, 81]. The differential photon flux originating from the DM-DM
annihilations in the galactic halo can be calculated as [75, 76]:

$\displaystyle\dfrac{d\Phi}{dE_{\gamma}}$
$\displaystyle=\frac{1}{4\pi}\dfrac{\langle\sigma
v\rangle}{2m_{\tilde{\chi}_{1}^{0}}^{2}}\dfrac{dN}{dE_{\gamma}}\int_{\Delta\Omega}d\Omega\int_{\rm
LOS}d\ell\;\rho_{\tilde{\chi}_{1}^{0}}^{2}(r)$ (7)
$\displaystyle=\dfrac{\langle\sigma
v\rangle}{2m_{\tilde{\chi}_{1}^{0}}^{2}}\dfrac{dN}{dE_{\gamma}}\times J_{\rm
halo}[\rho_{\tilde{\chi}_{1}^{0}}(r)],$ (8)

where $\langle\sigma v\rangle$ is the velocity averaged DM-DM annihilation
cross-section with $v\sim 0.001$, $m_{\tilde{\chi}_{1}^{0}}$ is the mass of
the DM, and ${dN}/{dE_{\gamma}}$ is the differential photon spectrum per DM-DM
annihilation. The latter can be obtained by summing the spectra of individual
annihilation channels, each weighted by the corresponding branching ratio
[82], such that,

$\displaystyle\frac{dN}{dE_{\gamma}}=\sum_{i}{\rm
Br(\tilde{\chi}_{1}^{0}\,\tilde{\chi}_{1}^{0}}\to
f_{i})\,\frac{dN_{i}}{dE_{\gamma}},$ (9)

where $f_{i}$ are the final state particles in the $i$-th annihilation
channel. The remaining term in Eq.(8), known as the $J-$factor, is of
astrophysical origin and is a functional of the DM density profile. In some
specific cases, it may also depend on particle physics inputs such as the DM
mass and DM annihilation cross-section (see e.g. [64, 69]). The functional
form of DM density profile $\rho(r)$ has many proposals based upon different
considerations [83, 84, 85, 86, 87, 88].

Around an SMBH (e.g. Sgr A∗), a spike is expected to appear due to DM’s local
overdensity within its gravitational influence zone. The radius of the
gravitational influence of the SMBH is $r_{\rm h}=GM/v_{0}^{2}$. Here, $M$ is
the mass of the SMBH, $G$ is Newton’s universal gravitational constant, and
$v_{0}$ is the velocity dispersion of DM in the halo outside the spike. This,
in turn, may enhance the detection possibility of a DM candidate, even with a
suppressed annihilation rate. Following the references [63, 89, 90], the
possible DM density spike can be modelled by various connected power laws with
different parameters in varying regions around the SMBH. The multiple regions
of the profile are described as follows.

  * •

Outside the gravitational influence region of the SMBH, the DM halo can be
characterised as a generalized Navarro-Frenk-White (NFW) profile [84]. This
profile can be well approximated by a power-law distribution, expressed as
$\rho(r)=\rho(r_{0})(r_{0}/r)^{\gamma_{c}}$, where $r$ is the distance from
the GC. Here, we utilize the local DM density in the solar system,
$\rho(r_{\odot})=0.3\,{\rm GeV/cm^{3}}$ [91], at a distance
$r_{\odot}=8.46\,{\rm kpc}$ [92] from the GC, to define the reference density,
$\rho(r_{0})=\rho(r_{\odot})\left({r_{\odot}}/{r_{0}}\right)^{\gamma_{c}}$.
The value of cusp parameter $\gamma_{c}$ is typically determined through
numerical simulations. Simulations focusing on DM distribution reveal a range
for $\gamma_{c}$ which is $0.9\lesssim\gamma_{c}\lesssim 1.2$ [93, 94].
$\gamma_{c}$ can be even larger in case of adiabatic contraction of galactic
halos due to baryonic infall [95, 96, 97].

  * •

The spike in DM density appears at or below $r_{b}$ ($r\lesssim r_{b}$) where
$r_{b}\simeq 0.2$ pc, $r_{\bf h}=0.3$ pc [98, 99]. Here we have considered
$M=4\times 10^{6}M_{\odot}$ for Sgr A∗ and $v_{0}=105$ km s-1 to evaluate
$r_{\bf{h}}$. The spike is well described by the power law $\rho_{\rm
sp}(r)=\rho(r_{b})(r_{b}/r)^{\gamma_{\rm sp}}$ where
$\rho(r_{b})=\rho(r_{0})(r_{0}/r_{b})^{\gamma_{c}}$. The value of spike
parameter $\gamma_{\rm sp}$ depends on the formation history of SMBH and spike
[61, 100, 101, 98, 102, 103, 104]. In case of an adiabatic growth of the cuspy
DM halo density around an SMBH, the $\gamma_{\rm sp}$ is usually large and
takes value $\gamma_{\rm sp}^{\rm
ad}=\left(\frac{9-2\gamma_{c}}{4-\gamma_{c}}\right)$ [61, 101]. A shallower
spike implies a relatively smaller $\gamma_{\rm sp}$, which is also possible
in the presence of gravitational heating by the nuclear star cluster
surrounding the SMBH [102, 104, 74, 103].

  * •

As we approach the SMBH from the tail of the spike, the DM density $\rho_{\rm
sp}$ increases until it reaches a critical point at $r=r_{\rm in}$. At this
critical point $\rho_{\rm sp}\sim\rho_{\rm ann}$ with $\rho_{\rm
ann}={m_{\tilde{\chi}_{1}^{0}}}/{\langle\sigma v\rangle\tau}$ being the
“annihilation plateau” density. This forms due to a strong DM annihilation
rate, triggered by very high DM density at the innermost region [61]. Here,
$\tau\sim 10^{10}$ yrs indicates the age of the DM spike around the SMBH. This
critical density is attained at a distance $r_{\rm
in}=r_{b}\cdot(\rho_{b}/\rho_{\rm ann})^{\nicefrac{{1}}{{\gamma_{\rm sp}}}}$
for $\rho_{b}=\rho(r_{b})$, and for $r<r_{\rm in}$, the free rise of DM
density spike towards the centre of SMBH is essentially restricted. In
particular, the shape of DM density gets flattened for $r<r_{\rm in}$ as
characterized by a mild power law, $\rho_{\rm in}(r)=\rho_{\rm
ann}\cdot(r_{\rm in}/r)^{\gamma_{\rm in}}$ with $\gamma_{\rm in}=0.5$ [105,
64].

Figure 1: Shape of the $\rho(r)$, considering different values for the age
($\tau$) of DM spike. The values of $\tau$ are in increasing order
($\tau_{1}<\tau_{2}<\tau_{3}<\tau_{4}$)

.

  * •

Within distances less than $4GM$ (region inside marginally bound orbits), the
gravitational influence of the SMBH is significant enough to capture all the
dark matter. Consequently, little or no dark matter is present inside the
sphere of radius $4GM$.

Putting everything together, the DM density profile takes the following form:

$\displaystyle\rho(r)=\begin{cases}0&(r<4GM),\\\ \dfrac{\rho_{\rm
sp}(r)\rho_{\rm in}(r)}{\rho_{\rm sp}(r)+\rho_{\rm in}(r)}&(4GM\leq r\leq
r_{b}),\\\\[13.0pt]
\rho_{b}\left(\dfrac{r_{b}}{r}\right)^{\gamma_{c}}&(r_{b}<r\leq
r_{\odot}),\end{cases}$ (10)

where $\rho_{\rm sp}(r)=\rho(r_{b})(r_{b}/r)^{\gamma_{\rm sp}}$, $\rho_{\rm
in}(r)=\rho_{\rm ann}\cdot(r_{\rm in}/r)^{\gamma_{\rm in}}$ and
$\rho_{b}=\rho(r_{b})$ as defined earlier. The spike profile is defined in
such a way that a smooth transition in the DM density occurs between $r<r_{\rm
in}$ and $r_{\rm in}\leq r<r_{b}$ regions [90, 63, 65], which however is not
always a strict requirement for proposition of Eq.(10) [62, 69].

In Fig. 1, we depict the shape of $\rho(r)$ (see Eq.(10)) considering
different values for the age of the DM spike. For representative purpose, we
set $\langle\sigma v\rangle=10^{-28}$ cm3/s, $m_{\tilde{\chi}_{1}^{0}}=100$
GeV, $\gamma_{c}=1.2$ and $\gamma_{\rm sp}=2.1$. Earlier we mentioned that
$\rho_{\rm ann}\propto{1}/{\tau}$ implying $r_{\rm
in}\sim\tau^{\nicefrac{{1}}{{\gamma_{\rm sp}}}}$ and $\rho_{\rm
in}(r)\sim\tau^{\nicefrac{{\gamma_{\rm in}}}{{\gamma_{\rm sp}}}-1}$. These
distinct features are highlighted in Fig. 1, where we find a larger $r_{\rm
in}$, corresponding to a larger $\tau$, thus leading to the early beginning of
flattening of $\rho(r)$. In addition, $\rho_{\rm in}$ is found to decrease
with the increase of $\tau$ across its domain.

Next, in Fig. 2, we discuss the sensitivity of $\gamma_{\rm sp}$ (left) and
$\gamma_{c}$ (right) to $\rho(r)$. Once again, we consider the same
representative choices of $m_{\tilde{\chi}_{1}^{0}}$ and $\langle\sigma
v\rangle$ for the purpose. Additionally, we fix $\gamma_{c}$ in the left plot
and $\gamma_{\rm sp}$ in the right plot and consider $\tau=10^{10}$ yrs. We
find that a larger $\gamma_{\rm sp}$ results in a steeper spike, specifically,
in the intermediate region $r_{\rm in}<r<r_{b}$. Similarly, a larger
$\gamma_{c}$ leads to an enhancement of $\rho(r)$ in the whole region of
interest ($4GM\leq r\leq r_{\odot}$). Since in this study, we are aiming for a
possible amplification in the $\gamma$-ray flux, resulting from DM
annihilation, we set $\gamma_{c}$ to 1.2 (maximum allowed value from DM only
simulations [93, 94]) throughout the analysis and treat $\gamma_{\rm sp}$ as a
free parameter.

Figure 2: $\rho(r)$-$r$ behavior for varying $\gamma_{\rm sp}$ (left) and
$\gamma_{c}$ (right). Here, $\rho_{\rm ann}$ is evaluated considering
$\tau\sim 10^{10}$ yrs, the typical age of the DM spike associated with the
SMBH.

In the present work, the resulting $\gamma$-ray flux from DM annihilation
dominantly originates from a narrow region containing the density spike [65].
Here, we will treat the DM spike as a $\gamma$-ray point source. The
contribution from the spike to the $J_{\rm halo}$ is given as [65],

$\displaystyle J[\rho(r)]$
$\displaystyle=\frac{1}{r_{\odot}^{2}}\int_{4M}^{r_{b}}dr\,\rho(r)^{2}r^{2}.$
(11)

One finds that, for the relevant region of interest, the $J-$factor from the
spike, appearing in Eq.(11), is 3 to 6 orders larger than the entire $J_{\rm
halo}$ in absence of the spike, depending upon
$m_{\tilde{\chi}_{1}^{0}},\langle\sigma v\rangle$ and $\gamma_{\rm sp}$. Thus,
the differential photon flux in the presence of the DM spike in Eq.(8) can be
approximated as [65],

$\displaystyle\dfrac{d\Phi}{dE_{\gamma}}$
$\displaystyle\simeq\dfrac{\langle\sigma
v\rangle}{2m_{\tilde{\chi}_{1}^{0}}^{2}}\dfrac{dN}{dE_{\gamma}}\times
J\left[\rho(r)\right].$ (12)

Unlike the case of a conventional DM (e.g. NFW) halo profile where the
$J$-factor can be segregated from the particle physics inputs, interestingly,
for the DM density profile of Eq.(10), the $J$-factor turns sensitive to the
DM annihilation cross-section and the DM mass. This is demonstrated in Fig. 3
where the different contours of $J$-factor are plotted in
$m_{\tilde{\chi}_{1}^{0}}-\langle\sigma v\rangle$ plane. A rise in the $J$
values can be observed with a low $\langle\sigma v\rangle$ and for a high DM
mass.

Figure 3: Contours of $J$-factor values in
$m_{\tilde{\chi}_{1}^{0}}\,-\,\langle\sigma v\rangle$ plane for the DM density
profile of Eq.(10) with $\gamma_{\rm sp}=2.1$ and $\gamma_{c}=1.2$ .

##  III Precision Observables and Constraints from colliders and DM Searches

We outline here the necessary steps followed to evaluate the differential
$\gamma$-ray flux ($d\Phi/dE_{\gamma}$) and the required spike for the
$\tilde{\chi}_{1}^{0}$ DM considering the density profile as given in Eq.(10).
We use the publicly available code SARAH 4.15.2 [106, 107] for generating the
MSSM model files for both SPheno [108, 109] and MicrOMEGAs [110, 111, 112,
113, 114]. Next, using SPheno 4.0.5 [108, 109], we calculate the particle
spectrum that is fed into MicrOMEGAs 6.0 [114] which further computes the DM
relic abundance and SI-DD cross-section. The quantities, ${dN}/{dE_{\gamma}}$
and $\langle\sigma v\rangle$, relevant for computing $\gamma$-ray flux in
Eq.(12), are also subsequently obtained from MicrOMEGAs 6.0 [114]. The
$J$-factor is calculated using Eq.(11). Finally, we follow the steps discussed
in Sec. II to calculate the photon flux for the MSSM parameter space.

The validity of each point is tested through the relevant observables related
to precision and collider phenomenology. If the SUSY states are sufficiently
light, they would impact various precision observables. Here, we enlist the
constraints from different precision experiments relevant to our analysis.
Unless otherwise stated, all these observables have been calculated using
SPheno 4.0.5 [108, 109].

$\bullet$ The light, neutral, CP-even Higgs boson resembles the SM Higgs with
a theoretical uncertainty of 3 GeV around the central value of 125 GeV [115,
116]. Thus the range for $m_{h}$ can be read as,

$122\,{\rm GeV}<m_{h}<128\,\rm GeV.$ (13)

Additionally, it is crucial to test the branching ratios of the Higgs scalar
predicted by the model against the limits from the LHC. We use HiggsTools
[117, 118, 119, 120, 121] to check for such constraints. The package includes
HiggsBounds - 5.10.2 [118, 119] and HiggsSignals - 2.6.2 [120, 121]

$\bullet$ As noted already, the recent experiment, E989 at the Fermilab,
provides new results from the Run-2 and Run-3 data analyses, providing the new
deviation of the anomalous magnetic moment of the muon, [19, 122],

$\displaystyle\delta a_{\mu}=a_{\mu}^{\rm exp}-a_{\mu}^{\rm SM}=(249\pm
48)\times 10^{-11},$ (14)

compared to that of the SM value. If the entire discrepancy stems from SUSY,
then for a valid parameter space point $a_{\mu}^{\rm SUSY}$ needs to reconcile
with $\delta a_{\mu}$. In the above $a_{\mu}^{\rm SM}=(116591810\pm 43)\times
10^{-11}$ [123] has been considered (For the SM calculations, see also [124,
125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139,
140, 141, 142, 143]). The combined experimental result for the value of
$a^{\rm exp}_{\mu}$ deviates from the SM predicted value by $5.1\sigma$. The
Standard Model’s predicted value, $a_{\mu}^{\rm SM}$, is subject to
theoretical uncertainty primarily due to the leading-order contribution from
Hadronic Vacuum Polarization (HVP). In this context, $a_{\mu}^{\rm SM}$ is
determined using HVP contributions derived from $e^{+}e^{-}\rightarrow$
hadrons data through dispersive techniques [123] 111However, utilizing HVP
contributions from lattice-QCD results by the BMW collaboration reduces the
discrepancy in $a_{\mu}$ to $1.6\sigma$, Reference [144] discusses this
tension without considering the recent CMD-3 result. Reference [145] addresses
this conflict by discussing the role of Euclidean time windows in lattice-QCD.
More recent work, ref. [146], calculates $(g-2)_{\mu}$ using a data-driven
approach for HVP contributions, incorporating CMD-3 data, but the result does
not agree with previous calculations. .

In MSSM, the leading contribution to $a^{\rm SUSY}_{\mu}$ comes from the
neutralino-smuon and chargino-sneutrino loops at one loop level. Under the
assumption that all sparticles have a uniform mass $M_{\rm SUSY}$, an
approximated form of $a^{\rm SUSY}_{\mu}$ at one-loop is given by [14, 15],

$\displaystyle a^{\rm
SUSY}_{\mu}=\frac{1}{192\pi^{2}}\frac{m_{\mu}^{2}\tan\beta}{M_{\rm
SUSY}^{2}}(5g_{2}^{2}+g_{1}^{2}),$ (15)

where $g_{2}$ and $g_{1}$ are the gauge coupling constants corresponding to
$SU(2)_{L}$ and $U(1)_{Y}$ respectively. The Eq.(15) gives the impression that
$a^{\rm SUSY}_{\mu}$ contribution in MSSM can be enhanced with a relatively
larger $\tan\beta$ and lighter SUSY particles, specifically having the weak
interaction.

We note that $a_{\mu}^{\rm SUSY}$ may also receive contributions from the two-
loop (mainly Barr-Zee type) diagrams involving fermion/sfermion in the loop
[147, 148]. Hence, we use the publicly available package GM2Calc [17, 149,
150, 151, 148, 147], which computes the $a_{\mu}^{\rm SUSY}$ value up to two
loops. We generate the input file using SPheno 4.0.5 and pass the tree-level
masses to GM2Calc [17, 149, 150, 151, 148, 147] for obtaining $a_{\mu}^{\rm
SUSY}$.

$\bullet$ Assuming the $\tilde{\chi}_{1}^{0}$ to be the only source for DM,
the acceptable value of the relic abundance data [152, 153] at $1\sigma$
confidence level is read as

$\Omega_{\rm DM}h^{2}=0.1198\pm 0.0012.$ (16)

We consider a $2\sigma$ level of allowance of $\Omega_{\rm DM}h^{2}$ in
Eq.(16), thus accept the parameter points of both $\tilde{B}_{\tilde{H}}$ and
$\tilde{B}_{\tilde{W}\tilde{H}}$ types that satisfy $0.1174<\Omega_{\rm
DM}h^{2}<0.1222$.222We assume a conventional radiation-dominated universe and
adopt the standard thermal freeze-out mechanism for the DM relic abundance.

The spin-independent scattering of neutralino DM and the light quark and
gluons can be read from the following effective Lagrangian [154, 155, 156,
157, 158]

$\displaystyle\mathcal{L}_{\rm SI}^{\rm
eff}=f_{q}\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}\overline{q}q+f_{G}{\tilde{\chi}}^{0}_{1}\tilde{\chi}^{0}_{1}G^{a}_{\mu\nu}G^{a\,\mu\nu}+\dots$
(17)

where the effective couplings $f_{q}$ and $f_{G}$ include the effects of the
SI scattering, mediated by the light quarks and gluons, respectively. The
first term in the R.H.S. receives contributions mainly from the exchange of
CP-even Higgses via $t$-channel. The squark exchange contribution is
subdominant and can be neglected in this analysis. The twist-2 operators
(traceless part of the energy-momentum tensor) for the quarks and gluons may
also be present in Eq.(17) as indicated by the ellipsis. However, we do not
include any new twist-2 contributions except those that are already present in
MicrOMEGAs [113].

The SI scattering cross-section of the neutralino with target nuclei can be
expressed as [8, 159, 160, 161, 162, 163, 164, 155]

$\displaystyle\sigma_{\rm
SI}=\frac{4}{\pi}\left(\frac{m_{\tilde{\chi}^{0}_{1}}M_{A}}{m_{\tilde{\chi}^{0}_{1}}+M_{A}}\right)^{2}\Bigg{[}\\{Zf_{p}+(A-Z)f_{n}\\}^{2}\Bigg{]},$
(18)

where $Z$ and $A$ represent its atomic and mass numbers, respectively. The
form factor $f_{N}$ ($N=p,n$) in Eq.(18) includes spin-independent coupling of
the neutralino with nucleon (of mass $m_{N}$) [154, 155, 156, 157, 158]. In
the numerical representation, we have used the recent results of LUX-ZEPLIN
(LZ) collaboration [165] on the SI-DD cross-section, which gives a stronger
upper limit than the previous XENON1T [39], PandaX-4T [166], LUX [167]
experiments to constrain our model parameter space.

$\bullet$ The presence of a light SUSY state can significantly alter the
branching fractions (Br) of certain flavour-violating processes, e.g. $B\to
X_{s}\gamma$, $B_{s}\to\mu^{+}\mu^{-}$, $B^{+}\to\tau^{+}\nu_{\tau}$ and $B\to
X_{s}\nu\,\overline{\nu}$ compared to the SM predictions. We consider the
following bounds for our analysis,

$\displaystyle{\rm Br}(B\to X_{s}\gamma)\,=\,(3.49\pm 0.38)\times 10^{-4},$
(19) $\displaystyle{\rm Br}(B_{s}^{0}\to\mu^{+}\mu^{-})\,=\,(3.45\pm
0.58)\times 10^{-9},$ (20) $\displaystyle{\rm
Br}(B^{+}\to\tau^{+}\nu_{\tau})\,=(1.09\pm 0.4)\times 10^{-4}\,,$ (21)

at 95$\%$ C.L. [168, 169]. Similarly, we used

$\displaystyle{\rm Br}(B\to X_{s}\nu\,\overline{\nu})\,<\,6.4\times 10^{-4},$
(22)

in this analysis [170].

$\bullet$ Searches for the lighter charginos, lighter neutralinos, and
sleptons at a center-of-mass energy of $\sqrt{s}=13$ TeV through various final
states like $2l+jets+\mbox{${E\\!\\!\\!\\!/_{T}}$}$,
$3l+jets+\mbox{${E\\!\\!\\!\\!/_{T}}$}$, etc. (here, $l$ stands for leptons of
flavours $e$ or $\mu$) have been conducted by both the ATLAS [171, 172, 173,
174, 175, 176, 177, 178, 179, 180, 181] and the CMS [182, 183, 184, 185, 186,
187] collaborations of the LHC in the context of simplified models. Although
in this work only compressed-type scenarios [174] are relevant, for the sake
of completeness, we also mention the mass bounds at LHC Run-II for non-
compressed models. Reference [188] includes a more recent review on
electroweakinos at the LHC.

In reference [174] ATLAS collaboration discussed the searches for lighter
electroweakinos and sleptons in compressed-type scenarios in which the mass
gaps between electoweakinos/sleptons and the LSP are very small. The bounds
are drastically relaxed in such cases as the final state products are very
soft, making their experimental detection more challenging. A brief discussion
on LHC searches for such sparticles through various lepton multiplicity
channels is given below :

  * •

The final state comprising $2l+jets+\mbox{${E\\!\\!\\!\\!/_{T}}$}$ [173]
arising from the production process
$pp\rightarrow\tilde{\chi}_{1}^{\pm}\tilde{\chi}_{1}^{\mp}$ at LHC Run-II for
an integrated luminosity of 139 fb-1 put bounds on
$m_{\tilde{\chi}_{1}^{\pm}}$ around 420 GeV for negligible LSP mass in which
$\tilde{\chi}_{1}^{\pm}$ decays via on-shell SM gauge bosons. The bound
becomes 1 TeV when lighter chargino decays via sleptons. In both cases, the
lighter charginos are Wino-like. For pair production of sleptons, the bound is
about 700 GeV for negligible LSP mass. For details see ref. [173],

  * •

At LHC Run-II with an integrated luminosity of 139 fb-1, the ATLAS
collaboration[174] has obtained bound on the mass of
$\tilde{\chi}_{1}^{\pm}/\tilde{\chi}_{2}^{0}$ around 640 GeV for negligible
LSP mass through the associated production process,
$pp\rightarrow\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\pm}$, which results in
the final state containing $3l+jets+\mbox{${E\\!\\!\\!\\!/_{T}}$}$. Here,
$\tilde{\chi}_{1}^{\pm}/\tilde{\chi}_{2}^{0}$ decays via on-shell SM gauge
bosons ($W^{\pm}/Z$). For off-shell SM gauge bosons ($W^{\pm}/Z$), the
corresponding bound is 300 GeV. If $\tilde{\chi}_{2}^{0}$ decays into the
Higgs boson and the LSP, the corresponding bound on
$\tilde{\chi}_{2}^{0}/\tilde{\chi}_{1}^{\pm}$ reduces to 190 GeV. In all these
cases, $\tilde{\chi}_{2}^{0}/\tilde{\chi}_{1}^{\pm}$ are assumed to be Wino-
like. However, for Higgsino-type scenarios the corresponding bound on
$m_{\tilde{\chi}_{2}^{0}}$ is 210 GeV for almost degenerate
$\tilde{\chi}_{1}^{0}$,$\tilde{\chi}_{2}^{0}$ and$\tilde{\chi}_{1}^{\pm}$. For
details, see the refs.[177].

  * •

For compressed-type scenarios (pertinent for this work) the ATLAS
collaboration [174] puts bounds on masses of lighter electoweakinos and
sleptons through the productions,
$pp\rightarrow\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\pm}j$ and
$pp\rightarrow{\tilde{l}}\,{\tilde{l}}^{*}j$ respectively at LHC Run-II where,
$j$ stands for $jet$. These processes finally lead to a signal,
$2l+jet+\mbox{${E\\!\\!\\!\\!/_{T}}$}$, with the opposite sign, the same
flavour leptons having soft transverse momentum ($p_{T}$) with a boosted jet.
The production of lighter electroweakinos
($\tilde{\chi}_{1}^{\pm},\tilde{\chi}_{2,3}^{0}$) in association with two
boosted forward jets via vector boson fusion processes has a low cross-
section, but due to the presence of boosted jets, it has improved signal-to-
background ratio, signal isolation and identification power. These
improvements lower the systematic uncertainties while enhancing the signal
sensitivity.

  * •

For Higgsino-type models, the bound on lighter electroweakinos are around 193
GeV for a mass splitting of 9.3 GeV between
$\tilde{\chi}_{2}^{0}/\tilde{\chi}_{1}^{\pm}$ and the LSP. For Wino-type
scenarios, the corresponding mass bound is 240 GeV for the mass splitting of 7
GeV of the same set of sparticles. In compressed scenarios, the bound on
masses of sleptons is around 251 GeV for 10 GeV mass splitting between
sleptons and the LSP. In this analysis, events consistent with the production
of electroweakinos through vector-boson fusion processes are used to constrain
Wino and Higgsino models via $q\bar{q}$ fusion production processes. Here,
additional hadronic initial-state radiation(ISR) jets are required at the
production level to enhance the sensitivity of the signal. For details, see
the ref. [174].

  * •

Electroweakino searches have also been performed at the LHC Run-II experiment
in the $1l+2\,b\,\text{-}jets+\mbox{${E\\!\\!\\!\\!/_{T}}$}$ channel [175] for
the production process
$pp\rightarrow\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{\pm}$. In this analysis,
they assumed the BR of $\tilde{\chi}_{2}^{0}\rightarrow\tilde{\chi}_{1}^{0}h$
to be 100$\%$ where $h$ decays into a pair of $b$-quarks, finally leading to
one lepton signal. Here, a Wino-type scenario is considered, and the mass
bound obtained is around 740 GeV for negligible LSP mass.

  * •

The refs. [189, 173, 174, 171, 184] provide results on searches for sleptons
and lighter electroweakinos which result in final states comprising of charged
leptons ($e^{\pm},\mu^{\pm}$) and $\tilde{\chi}_{1}^{0}$. With $\Delta
m(\tilde{l},\tilde{\chi}_{1}^{0})=m_{\tilde{l}}-m_{\tilde{\chi}_{1}^{0}}$ and
$\Delta
m(\tilde{\chi}_{1}^{\pm},\tilde{\chi}_{1}^{0})=m_{\tilde{\chi}_{1}^{\pm}}-m_{\tilde{\chi}_{1}^{0}}$,
Ref. [189] investigated the ’moderately compressed and experimentally
challenging mass regions $\Delta m(\tilde{l},\tilde{\chi}_{1}^{0})$ and
$\Delta m(\tilde{\chi}_{1}^{\pm},\tilde{\chi}_{1}^{0})$ that are close to the
$W$-boson mass. The slepton masses up to 150 GeV was excluded for $\Delta
m(\tilde{l},\chi_{1}^{0})$ to be around 50 GeV. For lighter charginos, the
mass bound obtained is 140 GeV for $\Delta
m(\tilde{\chi}_{1}^{\pm},\tilde{\chi}_{1}^{0})$ to be around 100 GeV.

The latest package SModelS 2.3.3 [190, 191, 192, 193, 194] is used to test the
set of parameters against these constraints.

##  IV Results

As mentioned earlier, in the present work, we explore two light LSP scenarios
$\tilde{B}_{\tilde{H}}$ and $\tilde{B}_{\tilde{W}\tilde{H}}$ and probe the
$\gamma$-ray search prospects around an SMBH that can also accommodate the
recent $a_{\mu}$ data and satisfy other relevant experimental constraints,
including the SI-DD bound. We begin by presenting six distinct benchmark
points (BMPs) in Table 1 for illustration. The first three points pertain to
the $\tilde{B}_{\tilde{H}}$ case, and the remaining three represent the
$\tilde{B}_{\tilde{W}\tilde{H}}$ scenario. All the BMPs can accommodate the
correct relic abundance (via Eq.(16)) and the desired SUSY contribution to
$\delta a_{\mu}$ in agreement with Eq.(14). The chosen BMPs are also
consistent with the bound on flavour violating processes, LHC constraints,
limits on SI-DD cross-section and the observables that check the mass and the
properties of an SM-like Higgs scalar. Table 1 shows the values of a few such
observables corresponding to each of the BMPs.

| $\tilde{B}_{\tilde{H}}$ | $\tilde{B}_{\tilde{W}\tilde{H}}$
---|---|---
BMP 1 | BMP 2 | BMP 3 | BMP 4 | BMP 5 | BMP 6
$M_{1}$ [GeV] | 200 | 300 | 350 | 200 | 300 | 600
$M_{2}$ [GeV] | 1500 | 1500 | 1500 | 230 | 302 | 582
$\mu$ [GeV] | 810 | 800 | 800 | 810 | 900 | 1200
$\tan\beta$ | 16 | 47 | 45 | 16 | 25 | 55
$m^{\rm in}_{\tilde{e}_{L}},m^{\rm in}_{\tilde{e}_{R}}$ [GeV] | 221 | 335 | 379 | 221 | 350 | 624
$m^{\rm in}_{\tilde{\mu}_{L}},m^{\rm in}_{\tilde{\mu}_{R}}$ [GeV] | 225 | 340 | 381 | 225 | 357 | 635
$m_{A}$ | 3000 | 4000 | 4000 | 3000 | 4000 | 4200
$m_{h}$ [GeV] | 122.7 | 123.3 | 123.3 | 123.6 | 123.8 | 123.7
$m_{\tilde{\chi}_{1}^{0}}$ [GeV] | 199.4 | 300.3 | 350.5 | 199.40 | 300.4 | 603.5
$m_{\tilde{\chi}_{2}^{0}}$ [GeV] | 831.9 | 829.5 | 829.1 | 241.2 | 323.0 | 618.5
$m_{\tilde{\chi}_{1}^{\pm}}$ [GeV] | 831.7 | 829.2 | 828.6 | 245.3 | 323.1 | 618.4
$m_{\tilde{e}_{R}}$ [GeV] | 206 | 303 | 351 | 206 | 319 | 606
$m_{\tilde{e}_{L}}$ [GeV] | 244 | 363 | 405 | 239 | 373 | 643
$m_{\tilde{\mu}_{R}}$ [GeV] | 210 | 308 | 353 | 210 | 327 | 617
$m_{\tilde{\mu}_{L}}$ [GeV] | 248 | 369 | 407 | 243 | 380 | 655
$m_{\tilde{\nu}_{e}}$ [GeV] | 231 | 354 | 397 | 226 | 364 | 638
$m_{\tilde{\nu}_{\mu}}$ [GeV] | 235 | 359 | 399 | 230 | 371 | 649
${\rm Br}(B\rightarrow X_{s}\gamma)\times 10^{4}$ | 3.19 | 3.14 | 3.15 | 3.18 | 3.16 | 3.13
${\rm Br}(B^{+}\to\tau^{+}\,\nu_{\tau})\times 10^{4}$ | 1.25 | 1.24 | 1.24 | 1.24 | 1.24 | 1.24
${\rm Br}(B_{s}\to\mu^{+}\mu^{-})\times 10^{9}$ | 3.17 | 3.13 | 3.13 | 3.17 | 3.17 | 3.11
${\rm Br}(B\to X_{s}\,\nu\,\overline{\nu})\times 10^{5}$ | 3.98 | 3.98 | 3.98 | 3.98 | 3.98 | 3.98
$a_{\mu}^{\rm SUSY}$ $\times 10^{9}$ | 1.77 | 2.14 | 1.59 | 2.82 | 1.94 | 1.51
$\Omega_{\rm DM}h^{2}$ | 0.119 | 0.122 | 0.121 | 0.121 | 0.119 | 0.118
$\sigma_{\rm SI}^{p}$ [pb]$\times 10^{11}$ | 2.40 | 3.77 | 5.46 | 2.95 | 4.16 | 8.88
$\langle\sigma_{\rm ann}v\rangle$ [cm3/s]$\times 10^{28}$ | 0.95 | 1.19 | 1.30 | 1.08 | 1.31 | 1.73
Table 1: A few BMPs that we consider here to illustrate the indirect search
prospects of the $\tilde{B}_{\tilde{H}}$ and $\tilde{B}_{\tilde{W}\tilde{H}}$
dark matter regimes in the presence of an SMBH. All BMPs yield correct relic
abundance and accommodate $\delta a_{\mu}$ through $a^{\rm SUSY}_{\mu}$. All
the relevant experimental constraints e.g. as listed in sec. III are
respected. Apart from the parameters mentioned above, we set the gluino mass,
$M_{3}=4$ TeV, stau masses $m_{\tilde{\tau}_{L,R}}=1.3$ TeV, squark soft
masses at $5$ TeV.

Among the chosen points, we find that the co-annihilation with sleptons
dominates for BMP 1 - BMP 3, which are within the $\tilde{B}_{\tilde{H}}$ set.
The remaining BMPs (BMP 4 - BMP 6) come under $\tilde{B}_{\tilde{W}\tilde{H}}$
category, and in this case, co-annihilation with charginos additionally become
important. For BMP 4, the lightest chargino is heavier than the next-to-
lightest SUSY particle (NLSP), the lightest slepton in contrast to BMP 5 and
BMP 6. Thus, the slepton coannihilation is still one of the dominant channels
for BMP 4. In BMP 5 and BMP 6, the mass difference between the LSP and
charginos are relatively smaller, thus making chargino coannihilations the
leading contributing processes. In both of these sets, the annihilation cross-
sections remain relatively suppressed and negligibly contribute to the DM
relic abundance. Additionally, the DM annihilation cross-section in the
present time is not sufficiently large to yield observable $\gamma$-ray flux
from the galactic halo. This is evident from Fig. 4, where we present the
total $\langle\sigma v\rangle$ against the available experimental upper limits
on the DM pair annihilation cross-sections to different final state particles.

Figure 4: Comparison of total $\langle\sigma v\rangle$ values for BMP 1 - BMP
6 with experimental upper limits on DM pair annihilation cross-section to
different final state particles. The Fermi-LAT upper bounds (for the NFW
profile) to different final state particles assuming $100\%$ branching ratio
are from the observations of dSphs [195], and the indicated HESS bounds (for
the Einasto profile) are from observations of GC [196].

Before we proceed to discuss the $\gamma$-ray flux, a few comments on the
dominant decay channels of the EW particles are in order. This will be
consistent with the non-observations of any LHC signatures so far. For BMPs 1
- 3, ($\tilde{B}_{\tilde{H}}$-case) $\tilde{\chi}_{2}^{0}$ and
$\tilde{\chi}_{3}^{0}$ are dominated by Higgsinos with masses more than 800
GeV and decay to $\tilde{\chi}_{1}^{0}$ and $h$ or $Z$ boson. Naturally, the
production of $\tilde{\chi}_{1}^{\pm}\tilde{\chi}_{2}^{0}$ would be
suppressed. Similarly, the direct production of
$\tilde{e}_{L}^{\star}\tilde{e}_{L}$ or
$\tilde{\mu}_{L}^{\star}\tilde{\mu}_{L}$ also agrees to the necessary mass
splitting between $\tilde{e}_{L}(\tilde{\mu}_{L})-\tilde{\chi}_{1}^{0}$. On
the other hand, for BMPs 4 - 6, ($\tilde{B}_{\tilde{W}\tilde{H}}$-case )
$\tilde{\chi}_{2}^{0}$ is Wino-like with masses very close to
$m_{\tilde{\chi}_{1}^{0}}$. Here $\tilde{\chi}_{2}^{0}$ decays dominantly to
$\nu_{e,\mu}\tilde{\nu}_{e,\mu}$ for BMP 4 and $e^{\pm}\tilde{e}_{L}^{\mp}$
for BMP 5 and BMP 6. On the other hand, the lightest chargino mainly decays to
an electron, and the left-handed electron sneutrino for BMP 4. In the case of
BMP 5, the decay favours a three-body process via an off-shell $W$ to
$\tilde{\chi}_{1}^{0},u(c),\overline{d}(\overline{s})$. For all the BMPs,
$\tilde{e},\tilde{\mu}$ are consistent with the experimental limits as
described in Ref. [184, 189].

Since the compressed nature of SUSY spectra does not offer the required handle
to probe the BMP 1-6 through collider and DM searches, which otherwise
satisfies all other experimental and theoretical constraints, we now like to
see whether Milky Way’s SMBH Sgr A∗ can provide the desired boost in the DM
indirect searches to test these specific zones of the $\tilde{B}_{\tilde{H}}$
and $\tilde{B}_{\tilde{W}\tilde{H}}$ regimes. In particular, we estimate the
$\gamma$-ray spectra resulting from the DM pair annihilations in the presence
of the DM density spike (Eq.(10)) for all the six benchmark points listed in
Table 1. In Fig. 5, we compare our prediction with the observation of Fermi
point source 3FGL J1745.6 - 2859c [53] within the energy range,
$100\,\text{MeV}\leq E_{\gamma}\leq 100\,\text{GeV}$ and HESS source J1745 -
290 [56] for $180\,\text{GeV}\leq E_{\gamma}\leq 79\,\text{TeV}$ associated
with Sgr A∗. To compare our results with an available experimental bound, we
evaluate the energy flux prediction in each bin used in the corresponding
experimental analysis. The photon flux $\Phi_{i}$ for the $i$-th bin is
assessed as:

$\displaystyle(\Phi)_{i}$ $\displaystyle=\int_{E_{\rm min}^{i}}^{E_{\rm
max}^{i}}dE_{\gamma}\frac{d\Phi_{\gamma}}{dE_{\gamma}},$ (23)

where, $E^{i}_{\rm max}-E^{i}_{\rm min}$ is the span of the $i$-th bin. The
energy flux of each bin can be obtained by multiplying the total photon flux
of the bin to its median energy given by $(E_{\rm med})_{i}=\sqrt{E_{\rm
min}^{i}E_{\rm max}^{i}}$. Thus, the total energy flux for $i$-th bin can be
evaluated as:

$\displaystyle(E_{\rm med}\Phi)_{i}=\sqrt{E_{\rm min}^{i}E_{\rm
max}^{i}}\int_{E_{\rm min}^{i}}^{E_{\rm
max}^{i}}dE_{\gamma}\frac{d\Phi_{\gamma}}{dE_{\gamma}}.$ (24)

Subsequently, we consider three different $\gamma_{\rm sp}$ values when
computing the $\gamma$-ray energy flux for the BMPs in Fig. 5. We notice that
for any of the BMPs, the amount of $\gamma$-ray flux increases with the rise
of $\gamma_{\rm sp}$, as a relatively larger $\gamma_{\rm sp}$ amplifies the
amount of spike in the DM density as indicated in Fig. 2. If, for a particular
BMP, the calculated $\gamma$-ray flux exceeds the observed flux from either
the Fermi or the HESS data in any of the energy bins, it will be considered as
disfavored. For both $\tilde{B}_{\tilde{H}}$ and
$\tilde{B}_{\tilde{W}\tilde{H}}$ regimes, the choice of $\gamma_{\rm sp}=2.3$
essentially rules out all the BMPs as viable points that can address DM relic
abundance and $\delta a_{\mu}$ simultaneously. With a relatively small value
for $\gamma_{\rm sp}$, the computed $\gamma$-ray flux values fall below the
allowed limits for at least some of the BMPs that we have considered for the
$\tilde{B}_{\tilde{H}}$ and $\tilde{B}_{\tilde{W}\tilde{H}}$ zones of MSSM.
Thus the above exercise establishes the importance of DM density spikes around
Milky Way’s SMBH Sgr A∗.In particular, it may effectively provide the
necessary boost to the $\gamma$-ray flux observable to probe the
$\tilde{B}_{\tilde{H}}$ and $\tilde{B}_{\tilde{W}\tilde{H}}$ regimes of MSSM.

The behaviour of photon energy flux $E_{\rm med}\Phi$ in Fig. 5 at different
energies can be traced to the differential spectra ($dN/dE_{\gamma}$) of
individual annihilation channels (see Eq.(9)). The dominant annihilation modes
of the DM for the BMPs at the present epoch are,

$\displaystyle\tilde{B}_{\tilde{H}}:~{}~{}~{}$
$\displaystyle\text{BMP\,1}:~{}~{}~{}\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}\to\overline{t}t,\,\gamma
e^{+}e^{-},\,\gamma\mu^{+}\mu^{-},$ (25)
$\displaystyle\text{BMP\,2}:~{}~{}~{}\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}\to\overline{t}t,\,\gamma
e^{+}e^{-},\,\gamma\mu^{+}\mu^{-},$ (26)
$\displaystyle\text{BMP\,3}:~{}~{}~{}\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}\to\overline{t}t,\,\gamma
e^{+}e^{-},\,\gamma\mu^{+}\mu^{-},$ (27)
$\displaystyle\tilde{B}_{\tilde{W}\tilde{H}}:~{}~{}~{}$
$\displaystyle\text{BMP\,4}:~{}~{}~{}\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}\to\overline{t}t,\,\gamma
e^{+}e^{-},\,\gamma\mu^{+}\mu^{-},$ (28)
$\displaystyle\text{BMP\,5}:~{}~{}~{}\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}\to\overline{t}t,\,W^{+}W^{-},\,\gamma
e^{+}e^{-},$ (29)
$\displaystyle\text{BMP\,6}:~{}~{}~{}\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}\to
W^{+}W^{-},\,\overline{t}t,\,b\overline{b}.$ (30)

The channels with leptonic final states produce fewer high energy photons with
the corresponding $dN_{i}/dE_{\gamma}$ peaked at energies close to the DM
mass. On the other hand, quarks, as final states, produce a copious number of
photons during hadronization with relatively lower energies. The spectrum
$dN_{i}/dE_{\gamma}$ from these channels is expected to exhibit peaks at lower
energies [81, 76]. The spectrum of the channels having $W^{\pm}$ as final
states also shows similar behaviour as in the case of hadrons as the produced
$W^{\pm}$ bosons from DM annihilation subsequently decay to quarks with
significant branching ratios. The overall differential photon spectrum thus
usually features two maxima, one at an energy close to the DM mass
($E_{\gamma}\simeq m_{\tilde{\chi}_{1}^{0}}$) and another at a relatively
lower energy. The appearance of two peaks is evident from Fig. 6 where we have
shown the variation of differential photon spectra
$E_{\gamma}{dN}/{dE_{\gamma}}$ as a function of $E_{\gamma}$ for BMP 3 and BMP
6. The amplitude of each of the peaks crucially depends on the branching ratio
of DM annihilation modes having quarks and leptons in the final states. In the
BMPs 1-3, the leptonic final states ($\gamma e^{+}e^{-}$ and
$\gamma\mu^{+}\mu^{-}$) are among the first three dominantly contributing DM
annihilation processes. The dominant presence of 3-body leptonic final states
leads to a prominent second peak in the energy range covered by HESS.

Figure 5: Comparison between predictions of $\gamma$-ray energy flux at each
bin due to the density spike around Sgr A∗ for different BMPs mentioned in
Table 1 and the observed ones from Fermi point source 3FGL J1745.6 - 2859c
[53] and HESS source J1745 - 290 [56] associated with Sgr A∗. Different
$\gamma_{\rm sp}$ values have been considered with $\gamma_{c}$ set at 1.2.
Here, $E_{\rm med}$ is the central value of the energy of the bin on the
logarithmic scale.

In the BMP 4, which although belongs to $B_{\tilde{W}\tilde{H}}$ category,
$M_{2}$ is still far from $M_{1}$. Thus, the Wino component in the DM is still
small, which results in the dominant DM annihilation modes and energy flux
similar to the $\tilde{B}_{\tilde{H}}$ case. On the contrary, in BMP 5, the
dark matter has a relatively larger Wino component, making
$\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}\to W^{+}W^{-}$ channel dominating
over the ones with leptonic final states. In BMP 6, which has even more
significant Wino mixing in DM, the branching ratios for
$\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}\to W^{+}W^{-}$ further increases,
and the leptonic final states contribute even less. In general, a larger Wino
component reduces the branching ratio for leptonic final states, leading to
suppression of the second peak at the photon flux spectrum as can be seen in
Fig. 6. Hence, for the BMPs (e.g. BMP 6) that mildly contribute to leptonic
final states, the Fermi-LAT data may also become relevant alongside the HESS
data for exclusion as we increase the spike parameter. It is important to note
that the two-body processes with leptonic final states,
$\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}\to e^{+}e^{-},\mu^{+}\mu^{-}$, are
helicity suppressed by a factor of
$m_{e,\mu}^{2}/m^{2}_{\tilde{\chi}_{1}^{0}}$ since the DM is a non-
relativistic Majorana particle [197, 198, 199], thereby allowing the three-
body processes to contribute predominantly to the photon flux at
$E_{\gamma}\simeq m_{\tilde{\chi}_{1}^{0}}$.

Figure 6: Differential photon spectra, $E_{\gamma}\frac{dN}{dE_{\gamma}}$ as a
function of $E_{\gamma}$ for BMP 3 and BMP 6.

We will now study the parametric dependence of $\Phi_{\gamma}$ on the choice
of the different MSSM and astrophysical inputs. Again, we focus on the
quantity $(E_{\rm med}\Phi)_{i}$, defined earlier and consider both
$\tilde{B}_{\tilde{H}}$ and $\tilde{B}_{\tilde{W}\tilde{H}}$ regimes in this
regard. To begin with, we demonstrate the dependency of photon flux on the
$\mu$ parameter in the top panel of Fig. 7 in the context of
$\tilde{B}_{\tilde{H}}$ DM. All the input parameters are set in accordance
with BMP 2 except for the parameters which are varied 333The choice of BMP 2
and BMP 5 (which will subsequently be presented to illustrate the parametric
dependence of $\gamma$-ray flux in the $\tilde{B}_{\tilde{W}\tilde{H}}$ region
in Fig. 8) is motivated so as to allow relative comparisons for photon flux at
a given $m_{\tilde{\chi}_{1}^{0}}$.. In Fig. 7, it is apparent that the
quantity $E_{\rm med}\Phi$ rises as $\mu$ decreases in the energy range probed
by Fermi. On the contrary, the behaviour is opposite for the energy range
relevant to the HESS experiment, where $E_{\rm med}\Phi$ gets reduced as we
decrease $\mu$. The behaviour, seemingly quite intriguing, can be traced once
again to the differential gamma-ray yield from the spectra of individual
annihilation channels as noted in Eq.(9) 444the other inputs, e.g., the
$J$-factor (see Eq.12) does not show any meaningful effect with variations in
$\mu$.. The dominant DM annihilation channels of DM in this case remain
similar as specified in Eq.(26). A larger $\mu$ leads to a smaller Higgsino
mixing in the DM, thereby reducing the branching ratios of processes with
hadronic final states and increasing the contribution of modes with three body
leptonic final states. Consequently, the peak at energies near the dark matter
mass is enhanced, while the peak at relatively lower energies is diminished.

Figure 7: Results for a $\tilde{B}_{\tilde{H}}$ scenario. Top panel: Variation
of $\gamma$-ray energy flux with energy for varying $\mu$ considering
$\gamma_{\rm sp}=2.2$. Except $\mu$, other relevant parameters are set as in
BMP 2 of Table 1. Bottom left panel: Excluded (coloured) and allowed regions
(white region between the dashed lines) in the $\gamma_{\rm sp}\,$-$\,\mu$
plane considering relic density, SI-DD, HESS and Fermi-LAT data. Bottom right
panel: Results of varying $\tan\beta$ and $\mu$ with other parameters set as
in BMP 2. Excluded regions are shown in solid colours. The scattered circles
are graded with the colours of their $\gamma_{\rm sp}^{\rm min}$ values. The
$\gamma_{\rm sp}^{\rm min}$ values are spread over a small range.
Additionally, as depicted, the same circles satisfy constraints due to relic
density, SI-DD, $\delta a_{\mu}$, and all the other experimental constraints
mentioned earlier. Thus each viable point with a $\gamma_{\rm sp}$ value
smaller than $\gamma_{\rm sp}^{\rm min}$ would be allowed by both Fermi-LAT
and HESS data. This, as well as all the subsequent figures, refer to
$\gamma_{c}=1.2$ .

Figure 8: Same as in Fig 7, but for depicting the $\gamma$-ray flux signatures
of $\tilde{B}_{\tilde{W}\tilde{H}}$ regime. Here, the input parameters that
are held fixed refer to BMP 5.

The distinct behaviour exhibited by $E_{\rm med}\Phi$ with respect to
$E_{\gamma}$ (top panel of Fig. 7) renders the HESS data particularly relevant
for higher $\mu$ values which cause the enhancement of the $\gamma$-ray flux
at $E_{\gamma}\geq 180\,{\rm GeV}$. On the contrary, Fermi data is more likely
to impose constraints on lower $\mu$ values for a given $\gamma_{\rm sp}$. The
above is evident in the bottom-left panel of Fig. 7 that identifies allowed
and disallowed regions in the plane of $\mu-\gamma_{\rm sp}$ based on the HESS
data. The zones are separated by a solid blue line. Thus, for a given value of
$\mu$, the excluded region by HESS data falls on the right of the blue line.
The above corresponds to the minimum possible value of $\gamma_{\rm sp}$
(=$\gamma_{\rm sp}^{\rm min}$) above which one obtains a HESS-disallowed zone.
Notably, a higher $\mu$ necessitates a lower $\gamma_{\rm sp}^{\rm min}$ so as
to avoid HESS exclusion (see Fig. 7). However, as shown in Fig. 7, SI-DD is
more restrictive and rules out the parameter space for $\mu\lesssim 720$ GeV.
Furthermore, the observed DM relic density (via Eq.(16)) constrains $\mu$ to
vary within a narrow range. This permissible range of $\mu$ is expected to
shift if one of the other pertinent parameters, such as $\tan\beta$, deviates
from the value as in BMP 2. The figure also shows a region discarded by Fermi-
LAT data that appears for small $\mu$ and large values of $\gamma_{\rm sp}$.
In the bottom-right panel of Fig. 7, we depict the allowed parameter points as
coloured circles in the $\mu$-$\tan\beta$ plane consistent with the observed
DM relic density, $\delta a_{\mu}$, and the SM-like Higgs mass while
simultaneously satisfying the SI-DD constraint on $\tilde{\chi}_{1}^{0}$ DM.
The scattered circles are graded with colours for their $\gamma_{\rm sp}^{\rm
min}$ values. As evident, the $\gamma_{\rm sp}^{\rm min}$ values are spread in
a small range, irrespective of $\tan\beta$.

Figure 9: Variation of $\gamma$-ray energy flux with $M_{2}$ is highlighted in
top panel, considering $\gamma_{\rm sp}=2.2$ while the other relevant
parameters are set as in BMP 5 of Table 1. In the bottom left panel, the HESS
excluded region is shown in $\gamma_{\rm sp}\,$-$\,M_{2}$ plane. In the bottom
right panel, we consider a deviation of $m_{\tilde{l}_{1}}$ from the one in
BMP 5 and highlight the viable points in the $M_{2}$-$m_{\tilde{l}_{1}}$ plane
that provides the desired value of DM relic and $\delta a_{\mu}$ and also
consistent with all other relevant experimental constraints.

We also perform a comparable examination for the
$\tilde{B}_{\tilde{W}\tilde{H}}$ scenario, as depicted in Fig. 8, showing how
$\mu$ correlates with the photon energy flux. Here, we adhere to BMP 5 for
choosing the input parameters except for those under variations. The results
presented in Fig. 8 echo similar observations discussed earlier in Fig. 7.
Noticeably, the Fermi-LAT exclusion zone in this case is larger than the
$\tilde{B}_{\tilde{H}}$ case. This is due to $M_{2}$ being close to $M_{1}$ in
BMP 5, which, as explained earlier, enhances the flux in the lower energy
range relevant to Fermi-LAT data.

Focusing on the same $\tilde{B}_{\tilde{W}\tilde{H}}$ scenario, we now explore
how a variation of $M_{2}$ would affect the dark matter sector as shown in
Fig. 9. $M_{2}$ influences the DM relic abundance and $E_{\rm med}\Phi$ at any
energy bin. Reducing $M_{2}$ would lead to a rise in the Wino component in the
LSP, which in turn would lower the relic abundance via Bino-Wino
coannihilations. The Higgsino mixing parameter $\mu$ set at 900 GeV (BMP 5)
certainly has a milder influence on the relic density. However, regarding
photon flux, the behaviour for varying $M_{2}$ and $\mu$ as evident from the
top panels of Fig. 9 and Fig. 8 are not so different. The dominant
annihilation modes of the DM at the present epoch are given in Eq.(29). Once
again, the HESS data emerges as the pertinent factor in constraining $M_{2}$,
especially for the higher values of the latter. The bottom left panel of Fig.
9 shows that one requires larger $\gamma_{\rm sp}$ that would likely eliminate
the smaller $M_{2}$ region by the HESS data. Both the figures (i.e. the top
and the bottom-left panels) affirm that for smaller $M_{2}$ values, it becomes
easier to evade the HESS constraint, necessitating a higher value of
$\gamma_{\rm sp}^{\rm min}$ for the exclusion.555Since the LSP,
$\tilde{\chi}_{1}^{0}$, is predominantly a Bino with only a little admixture
of Higgsino and Wino, $\tan\beta$ and slepton masses only have negligible
impacts on the photon flux, so we refrain from considering them further. The
bottom-right panel of Fig. 9 shows the relic density satisfied regions arising
out of Bino-slepton or Bino-Wino coannihilations. For our choice of
parameters, the relic density is satisfied via Bino-Wino coannihilations in a
nearly vertical strip around $M_{2}\sim 300~{}$GeV. As Wino has a higher
isospin coupling compared to Higgsino, we find that varying $M_{2}$ will cause
a significant deviation in the relic density. However, an enlarged parameter
space accommodating the correct amount of DM relic and the required $\delta
a_{\mu}$ can be generated by varying one of the slepton masses, such as
$m_{\tilde{\mu}}$ (or $m_{\tilde{e}}$) and $M_{2}$, around their respective
values for BMP 5. When $M_{2}$ is large, reducing $m_{\tilde{\mu}}$ (and/or
$m_{\tilde{e}}$) can assist in achieving the correct DM relic by boosting the
co-annihilations involving slepton. On the other hand, the parameter space
corresponding to Fig. 9 is entirely compliant with SI-DD limits. This is
because of a reduced Higgsino fraction in the LSP arising out of an adequately
large $\mu$ relative to $M_{1}$.

Figure 10: Viable points out of random scanning for the
$\tilde{B}_{\tilde{H}}$ scenario. Left panel: Scatter plot in the
$M_{1}-m_{\tilde{e}_{L,R}}$ plane with points represented by colour-graded
circles for $\tan\beta$, the latter shown as a colour bar. Right panel:
Scatter plot in the same plane with points shown as colour-graded circles for
$\gamma_{\rm sp}^{\rm min}$, the latter shown as a colour bar.

Guided by the previous lessons, considering both $\tilde{B}_{\tilde{H}}$ and
$\tilde{B}_{\tilde{W}\tilde{H}}$ scenarios, we now proceed for random scans of
the relevant parameters in the specified ranges as listed below:

$\bullet$ Bino-Higgsino : Here we fix $M_{2}=1.5~{}$TeV. The other relevant
parameters are varied in the following ranges.

$~{}\begin{array}[]{r@{\,}r@{\,}c@{\,}ll}100\,{\rm
GeV}&\leq&~{}M_{1}{}&\leq&700\,{\rm GeV},\\\ 2.0\,{\rm
TeV}&\leq&~{}M_{A}{}&\leq&4.5\,{\rm TeV},\\\ 500\,{\rm
GeV}&\leq&~{}\mu{}&\leq&1500\,{\rm GeV},\\\ 100\,{\rm
GeV}&\leq&~{}m_{\tilde{l}_{L,R}}{}&\leq&1\,{\rm TeV},\\\
5&~{}\leq&~{}\tan\beta{}&\leq&55.\end{array}$ (31)

Apart from the parameters mentioned above, we set the gluino mass, $M_{3}=4$
TeV, $m_{\tilde{\tau}_{L,R}}=1.3$ TeV, squark soft masses at $5$ TeV and all
the trilinear soft SUSY breaking parameters $T_{i}=-2$ TeV. In the above, we
have assumed $\tilde{l}=\\{\tilde{e},\tilde{\mu}\\}$. During the scan, we have
omitted the points for which the lightest neutralino carries a predominant
fraction of Higgsino.

The results are shown in Fig. 10. The left panel shows valid parameter points
where DM relic density is satisfied via Bino-slepton coannihilations for
varying $\tan\beta$ shown in the colour bar. In addition to the relic density
limits, the coloured points satisfy the $\delta a_{\mu}$ data as well as all
other relevant constraints as detailed in section III. In the right panel, we
have provided the guiding parameter $\gamma_{\rm sp}^{\rm min}$ so that for
$\gamma_{\rm sp}<\gamma_{\rm sp}^{\rm min}$ one finds the valid parameter
points allowed by both the Fermi-LAT and the HESS data. Notably, in the
$\tilde{B}_{\tilde{H}}$ case, the neutralino mass is strictly restricted in
the regime: $120\,{\rm GeV}\lesssim m_{\tilde{\chi}_{1}^{0}}\lesssim 360\,{\rm
GeV}$ beyond which it produces an overabundance of DM. The irregularity in the
behaviour of $\gamma_{\rm sp}^{\rm min}$ in the range 100 GeV $\lesssim
M_{1}\lesssim$ 200 GeV is due to the 80 GeV gap between the observed photon
energy ranges of Fermi-LAT ($E_{\gamma}\leq 100\,\rm GeV$) and HESS
($E_{\gamma}\geq 180\,\rm GeV$) experiments. For DM mass in this energy gap,
the peak in photon energy flux, which lies at $E_{\gamma}\sim
m_{\tilde{\chi}_{1}^{0}}$, can not be probed by either experiment, requiring a
larger $\gamma_{\rm sp}$ value to exclude the parameter set.

Figure 11:  Viable points out of random scanning for the
$\tilde{B}_{\tilde{W}\tilde{H}}$ scenario. Top left panel: Scatter plot in the
$M_{1}$ \- $M_{2}$ plane with points represented by color graded circles for
$\tan\beta$, the latter shown as a colour bar. Top right panel: Scatter plot
in the same plane with points represented by colour-graded circles for
$m_{\tilde{l}_{1}}$, the latter shown as a colour bar. Bottom panel: Scatter
plot in the same plane with colour-graded circles for $\gamma_{\rm sp}^{\rm
min}$, the latter shown as a colour bar.

$\bullet$ Bino-Wino-Higgsino : Here $M_{2}$ has also been varied. The ranges
of the variations are the same as before, except for the parameter $\mu$ that
can now assume relatively higher values.

$\displaystyle~{}\begin{array}[]{r@{\,}r@{\,}c@{\,}ll}100\,{\rm
GeV}&\leq&M_{1}&\leq&700\,{\rm GeV},\\\ 100\,{\rm GeV}&\leq&M_{2}&\leq&1\,{\rm
TeV},\\\ 2.0\,{\rm TeV}&\leq&M_{A}&\leq&4.5\,{\rm TeV},\\\ 500\,{\rm
GeV}&\leq&\mu&\leq&2.0\,{\rm TeV},\\\ 100\,{\rm
GeV}&\leq&m_{\tilde{l}_{L,R}}&\leq&1\,{\rm TeV}.\\\
5&~{}\leq&~{}\tan\beta{}&\leq&55,\end{array}$ (33)

All the other parameters are the same as in the previous case. Again, we have
filtered out the parameter points associated with bino-dominated LSP only
while discarding other possibilities. The result for
$\tilde{B}_{\tilde{W}\tilde{H}}$ case is depicted in Fig. 11. Here, additional
co-annihilation processes involving $\tilde{\chi}_{2}^{0}$ and charginos are
present, as mentioned earlier. This introduces more flexibility to the MSSM
parameter space in order to achieve the correct value of DM relic abundance.
Unlike the $\tilde{B}_{\tilde{H}}$ case, the close proximity of
$m_{\tilde{l}_{L,R}}$ and $M_{1}$ is no longer an absolute necessity for
satisfying the relic abundance. Indeed, the relative variations between
$M_{2}$ and $m_{\tilde{l}_{L,R}}$ result in a relatively broader space where
relic abundance can be satisfied in the $M_{1}$ and $M_{2}$ plane, as
illustrated in Fig. 11. We also find a relatively broader mass window for
neutralino DM ($120\,{\rm GeV}\lesssim m_{\tilde{\chi}_{1}^{0}}\lesssim
530\,{\rm GeV}$) in this case. The requisite value of $\gamma_{\rm sp}^{\rm
min}$ value for each feasible point can be found from the bottom panel of Fig.
11. The irregularity in the range 100 GeV $\lesssim M_{1}\lesssim$ 180 GeV is
again due to the gap in observation energies of the experiments, as explained
earlier.

##  V Summary and Conclusion

In the Minimal Supersymmetric Standard Model (MSSM), a dominantly Bino-like
neutralino with a minimal mixing of (i) Higgsino and (ii) Higgsino-Wino
components can effectively explain both the observed relic density of DM and
the current data on the muon magnetic moment anomaly, denoted as $a_{\mu}$. In
the $\tilde{B}_{\tilde{H}}$ scenario, a precise adjustment of the slepton mass
is essential to facilitate effective slepton coannihilation processes in order
to yield the DM abundance in the right ballpark. On the other hand, the
$\tilde{B}_{\tilde{W}\tilde{H}}$ case involves chargino coannihilation in
addition to the slepton coannihilations. Although the acceptability of these
scenarios as a potential candidate beyond the standard model is irrefutable,
the direct and indirect searches of these models may remain challenging in
specific parts of the parameter space. In particular, with compressed SUSY
spectra, e.g. close proximity of $m_{\tilde{\chi}_{1}^{0}}$ \-
$m_{\tilde{l}_{L,R}}$ or $m_{\tilde{\chi}_{1}^{0}}$ \-
$m_{\tilde{\chi}_{1}^{\pm}}$, one finds it difficult to probe the scenarios at
the LHC. Similarly, with minimal mixings with Higgsinos, the DM direct
detections may not be quite promising, even with the future generations of
experiments. On the other hand, the present-day annihilation strength of the
DM in galactic halos turns out to be very weak, resulting in faint indirect
search signals in the conventional DM halo profile.

However, the situation, especially in regard to indirect searches, can be
improved drastically in the presence of a density spike around an SMBH (e.g.
Sgr A∗) in the centre of the Milky Way. A sufficient density spike can provide
an effective boost to the $\gamma$-ray flux that might be detectable in DM
indirect search experiments. The spike profile is primarily characterized by
three parameters, namely $\gamma_{\rm sp}$, $\gamma_{c}$ and $\rho_{\rm ann}$.
An increase in either of the first two parameters results in a steeper DM
spike around the SMBH and can lead to the desired boost to the DM annihilation
rate in the GC. In our analysis, we have kept $\gamma_{c}$ fixed at 1.2 (i.e.
the maximum possible value) and considered $\gamma_{\rm sp}$ as a free
parameter. On the other hand, $\rho_{\rm ann}$ depends on the DM mass and
annihilation cross-section, thereby making $\rho(r)$, and consequently the
$J$-factor, dependent on particle physics parameters. Turning to the low
energy data, along with the $a_{\mu}$ range, we undertake several constraints
spanning from bounds on Higgs observables, SUSY particle searches at LHC, DM
relic and SI-DD bounds to flavour constraints. We begin with portraying a few
benchmark points, which indicate that it is indeed possible to reach the
present sensitivity of the Fermi and HESS data for $\gamma$-ray searches
associated with Sgr A∗ in the presence of an appropriate level of boost. We
find that tuning $\gamma_{\rm sp}$ is sufficient to obtain the desired boost.
Subsequently, we make use of a few of the BMPs to unfold the role of some of
the relevant parameters in the estimate of $\gamma$-ray energy flux.
Specifically, the parameter $\mu$, which predominantly determines the Higgsino
fraction in both $\tilde{B}_{\tilde{H}}$ and $\tilde{B}_{\tilde{W}\tilde{H}}$
DM is found to play a crucial role in controlling the $\gamma$-ray flux for a
particular DM mass. In the $\tilde{B}_{\tilde{W}\tilde{H}}$ scenario, the
parameter $M_{2}$ also influences the $\gamma$-ray flux essentially in a
similar manner.

Finally, we present the results of random scans showing the allowed ranges of
DM mass for each category of $\tilde{B}_{\tilde{H}}$ and
$\tilde{B}_{\tilde{W}\tilde{H}}$. We depict the minimum required value of
$\gamma_{\rm sp}$ necessary to test the feasibility of each point based on
indirect search data of Fermi and HESS. The preferred mass windows for
$\tilde{B}_{\tilde{H}}$ and $\tilde{B}_{\tilde{W}\tilde{H}}$ are found to be
$120\,{\rm GeV}\lesssim m_{\tilde{\chi}_{1}^{0}}\lesssim 360\,{\rm GeV}$ and
$120\,{\rm GeV}\lesssim m_{\tilde{\chi}_{1}^{0}}\lesssim 530\,{\rm GeV}$
respectively. The requisite $\gamma_{\rm sp}^{\rm min}$ to eliminate the
viable parameter space remains close to its adiabatic value, $\gamma_{\rm
sp}\sim 2.35$.

## Acknowledgement

The computations in this project were partially supported by SAMKHYA, the
high-performance computing (HPC) facility provided by the Institute of
Physics, Bhubaneswar (IOPB). The authors acknowledge S. Bisal, B.T. Chiang, S.
Rout, A. Tapadar and F. Varsi for valuable discussions.

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