# Neutron Star Heating in Dark Matter Models for Muon g-2 with Scalar Lepton
Partners up to the TeV Scale

Jan Tristram Acuña<EMAIL_ADDRESS>Scuola Internazionale Superiore
di Studi Avanzati (SISSA), via Bonomea 265, 34136 Trieste, Italy  INFN,
Sezione di Trieste, via Valerio 2, 34127 Trieste, Italy  Institute for
Fundamental Physics of the Universe (IFPU), via Beirut 2, 34151 Trieste, Italy
National Institute of Physics, University of the Philippines Diliman, Quezon
City 1101, Philippines  Department of Physics, National Tsing Hua University,
Hsinchu, 30013, Taiwan  Patrick Stengel<EMAIL_ADDRESS>Scuola
Internazionale Superiore di Studi Avanzati (SISSA), via Bonomea 265, 34136
Trieste, Italy  INFN, Sezione di Trieste, via Valerio 2, 34127 Trieste, Italy
Institute for Fundamental Physics of the Universe (IFPU), via Beirut 2, 34151
Trieste, Italy  Piero Ullio<EMAIL_ADDRESS>Scuola Internazionale Superiore di
Studi Avanzati (SISSA), via Bonomea 265, 34136 Trieste, Italy  INFN, Sezione
di Trieste, via Valerio 2, 34127 Trieste, Italy  Institute for Fundamental
Physics of the Universe (IFPU), via Beirut 2, 34151 Trieste, Italy

###### Abstract

We investigate the kinetic heating of neutron stars due to dark matter
scattering in minimal models with scalar lepton partners mediating
interactions which can account for the anomalous magnetic moment of the muon.
In particular, such models in which the dark matter is a thermally produced
Majorana fermion can be extremely difficult to detect at conventional
searches. We demonstrate that future infrared observations of an old neutron
star population could definitively probe the entire parameter space of this
scenario.

The hint for new physics indicated by recent measurements of the anomalous
magnetic dipole moment of muon, $g_{\mu}-2$, Abi et al. (2021) provides a
starting point for the consideration of beyond-the-Standard Model (BSM)
frameworks potentially accounting for the $g_{\mu}-2$ and embedding a dark
matter (DM) candidate. Some minimal scenarios address these two aspects in
terms of the very same BSM states; even in such setups, however, it has been
difficult to single out other observables to confirm or reject the model. In
this Letter, we consider the effect of kinetic heating of neutron stars (NSs)
by DM particles in the galactic halo Baryakhtar et al. (2017); Raj et al.
(2018); Bell et al. (2018); Acevedo et al. (2020), and propose it as a generic
tool to test minimal BSM frameworks in which the DM state, a thermal relic
from the early Universe, is invoked, together with another state carrying
electric and muonic lepton charges, to provide the required extra 1-loop
contribution to the $g_{\mu}-2$.

To prove this connection, as a reference minimal scenario, we will consider a
model in which the DM is a SM singlet Majorana fermion with DM-SM interactions
mediated by chirally mixed charged scalar lepton partners. While this choice
selects a small subset of the particle content of the Minimal Supersymmetric
Standard Model (MSSM), respectively the bino and two sleptons (MSSM jargon
will be borrowed hereafter), most MSSM realizations rely on several extra
ingredients, with different DM and $g_{\mu}-2$ phenomenologies, and a feeble
link between the two. The model at hand (see Ref. Ghosh et al. (2022) for a
recent overview) displays regions in parameter space representative of the two
possible effects driving the relic density for leptophilic DM models: an
enhanced bino annihilation rate when the left-right slepton mixing is sizable,
as for the so-called Incredible Bulk scenario Fukushima et al. (2014), and
large co-annihilations for a relatively small mass splittings between the bino
and lightest slepton (e.g. see Ref. Acuña et al. (2022), hereafter referred to
as ASU).

Leptophilic models have DM-nucleon interactions generated at 1-loop level,
reducing the prospects for direct detection. Most relevant for NS heating are
tree-level DM-lepton scatterings, since, although neutrons are the primary
component of NSs, the transfer of kinetic energy can also occur by DM
scattering off subdominant proton, electron and muon constituents Bell et al.
(2019); Garani and Heeck (2019); Joglekar et al. (2020a, b). Moreover, due to
the gravitational acceleration of DM to semi-relativistic speeds prior to
scattering off NS constituents, interactions which are typically suppressed by
the velocity or momentum of the DM in terrestrial direct detection experiments
can potentially be relevant for NSs.

The DM-induced NS temperatures are potentially detectable by infrared
telescopes such as the James Webb Space Telescope (JWST) Gardner et al.
(2006). Recent work in Ref. Hamaguchi et al. (2022) has investigated NS
heating in similar DM models which satisfy $g_{\mu}-2$, including benchmark
models with slepton-mediated bino interactions. Our work studies a more
comprehensive parameter space encompassing such models, thus allowing more
general conclusions: We demonstrate that, for both mechanisms responsible for
the bino relic abundance, the scenario could be confirmed with future
observations of heated NSs by telescopes such as the JWST, or alternatively,
the lack of such observations could definitively exclude the full broad class
of slepton mediator models.

Dark matter model.—We define the interactions between the bino
$\tilde{B}^{0}$, SM leptons and the associated scalar lepton partners
$\tilde{\ell}$,

$\displaystyle\mathcal{L}_{\rm int}$ $\displaystyle=$
$\displaystyle-\lambda_{\tilde{\mu}_{R}}e^{-\imath\phi_{\tilde{\mu}}/2}\tilde{\mu}_{R}^{*}\bar{\tilde{B}}^{0}P_{R}\mu-\lambda_{\tilde{\mu}_{L}}e^{\imath\phi_{\tilde{\mu}}/2}\tilde{\mu}_{L}^{*}\bar{\tilde{B}}^{0}P_{L}\mu$
(1)
$\displaystyle-\lambda_{\tilde{\nu}}e^{\imath\phi_{\tilde{\mu}}/2}\tilde{\nu}_{\mu}^{*}\bar{\tilde{B}}^{0}P_{L}\nu_{\mu}+{\rm
h.c.}$

where $P_{R,L}$ are the right- and left-handed projectors. We have introduced
the $SU(2)_{L}$ singlet scalar muon partner $\tilde{\mu}_{R}$ and the
corresponding $SU(2)_{L}$ doublet
$\tilde{\ell}_{L}=(\tilde{\nu}_{\mu},\tilde{\mu}_{L})^{T}$. All BSM states are
charged under a $\mathbb{Z}_{2}$ symmetry, which stabilizes the lightest such
state, assumed to be the bino. For simplicity, although in principle couplings
can be independent, we assume MSSM-like relations, with
$\lambda_{\tilde{\nu}}=\lambda_{\tilde{\mu}_{L}}$ and
$\lambda_{\tilde{\mu}_{R,L}}=\bar{\lambda}\sqrt{2}g^{\prime}Y_{R,L}$, where
$Y_{R,L}$ are the SM hypercharge and $g^{\prime}$ is the hypercharge coupling,
and the extra overall factor $\bar{\lambda}$ would be equal to 1 in the MSSM.
We also allow for the coupling constants to have a relative $CP$-violating
phase $\phi_{\tilde{\mu}}$.

In order for the model described above to satisfy $g_{\mu}-2$, the left- and
right-handed chiral smuon eigenstates must mix. The smuon mass eigenstates are
related to the chiral eigenstates by the left-right mixing angle
$\theta_{\tilde{\mu}}$,

$\begin{pmatrix}\tilde{\mu}_{1}\\\
\tilde{\mu}_{2}\end{pmatrix}=\begin{pmatrix}\cos\theta_{\tilde{\mu}}&-\sin\theta_{\tilde{\mu}}\\\
\sin\theta_{\tilde{\mu}}&\cos\theta_{\tilde{\mu}}\end{pmatrix}\begin{pmatrix}\tilde{\mu}_{L}\\\
\tilde{\mu}_{R}\end{pmatrix}\,,$ (2)

where we adopt the convention in which $\tilde{\mu}_{1}$ is always lighter
than $\tilde{\mu}_{2}$ and the left-right mixing angle is defined on the
interval $[-\pi/2,\pi/2)$. Also, to characterize the off-diagonal term in the
corresponding smuon mass matrix, it is convenient to define the smuon mass-
splitting parameter

$y\equiv\left(m_{\tilde{\mu}_{2}}^{2}-m_{\tilde{\mu}_{1}}^{2}\right)\sin\left(2\theta_{\tilde{\mu}}\right)/\left(4m_{W}^{2}\right)\,,$
(3)

where $m_{W}$ is the W-boson mass. In models which assume minimal flavor
violation such as typical realizations of the MSSM, $y\ll 1$ since the off-
diagonal term in smuon mass (squared) matrix is $\propto
m_{\mu}^{2}/m_{W}^{2}$. We take the sneutrino mass to be defined as in the the
MSSM,

$m_{\tilde{\nu}_{\mu}}^{2}\equiv
m_{\tilde{\mu}_{1}}^{2}-m_{W}^{2}+\left(m_{\tilde{\mu}_{2}}^{2}-m_{\tilde{\mu}_{1}}^{2}\right)\left[1-\cos\left(2\theta_{\tilde{\mu}}\right)\right]/2\,.$
(4)

Figure 1: Parameter space for bino annihilation models which satisfy the DM
relic density for fixed left-right smuon mixing angle
$\theta_{\tilde{\mu}}=-\pi/4$ and smuon mass ratio
$m_{\tilde{\mu}_{2}}/m_{\tilde{\mu}_{1}}=1.25$. Each colored curve corresponds
to a different multiple, $\bar{\lambda}$, of the bino-slepton-lepton coupling
in the MSSM indicated in the legend and terminates at small relative mass
splittings where bino annihilation contributes $\lesssim 80\%$ of the
effective annihilation rate. Collider constraints on the production of two
mass degenerate chiral smuons with relatively large mass splittings from the
bino Aad et al. (2020); ATL (2022) and a single chiral smuon for compressed
spectra ALEPH, DELPHI, L3, OPAL Experiments (2004); Aad et al. (2020) are show
by the black dashed lines and dot-dashed, respectively. Assuming the left-
right smuon mixing arises from a trilinear interaction with the SM Higgs
boson, the dotted line shows the upper limit on the relative mass splitting
for a given bino mass from EW vacuum stability corresponding to a smuon mass
splitting parameter of $y\simeq 23$.

As discussed in ASU, the origin of the off-diagonal term in the smuon mass
matrix which yields the left-right mixing need not be specified in order to
have models with $\sim{\rm TeV}$ sleptons satisfying both the DM relic density
and $g_{\mu}-2$. However, if we associate the left-right mixing with
electroweak (EW) symmetry breaking in the SM, the corresponding trilinear
interaction between chiral smuon eigenstates and the Higgs boson can have some
impact on the phenomenology of the model. In particular, a large trilinear
coupling can impact the calculation of the relic density in models where the
effects of co-annihilation are important. In addition, large trilinear
couplings in the scalar potential can destabilize the EW vacuum, yielding
constraints from the lifetimes of the metastable EW vacua. In a detailed
analysis of our model in ASU, we determined that in general a smuon mass
splitting parameter of $y\gtrsim 23$ yields an EW vacuum metastable on
timescales too short relative to the age of the Universe.

Figure 2: Total NS heating efficiency for bino annihilation models shown in
Fig. 1 (left, same color scheme) and smuon co-annihilation models along the
RHB shown in Figs. 2 and 3 of ASU. Heating efficiency is normalized to the
geometric capture rate, such that every DM particle incident on the NS would
be captured after a single scatter for $f>1$. The NS equation of state is
assumed to be BSk24 Pearson et al. (2018) with NS mass of $1.5\,M_{\odot}$ and
radius of $12.6\,{\rm km}$.

We can separate the parameter space of the above DM model into two distinct
regions characterized by the dominant processes which deplete the relic
density. For the region of parameter space studied in ASU, the bino is nearly
degenerate in mass with at least one of the sleptons such that co-annihilation
processes control the DM relic density. In the ASU parameter space, we only
consider coupling constants identical to the bino-slepton-lepton couplings in
the MSSM (i.e. $\bar{\lambda}=1$) and assume the $CP$-violating phase
vanishes. Accounting for $g_{\mu}-2$ requires the smuon mixing to be small for
lighter $m_{B}$, yielding models along either a left- or right-handed branch
(RHB) of the ASU parameter space. Satisfying the relic density with slepton
co-annihilation then fixes the relative mass splitting between the lightest
slepton and the bino along curves with either $y$ or $\theta_{\tilde{\mu}}$
held constant.

For the parameter space in which the depletion of the relic density is
controlled by bino annihilation, we consider both $\bar{\lambda}\gtrsim 1$ and
$\phi_{\tilde{\mu}}\neq 0$. We assume $\theta_{\tilde{\mu}}=-\pi/4$ since the
$s$-wave contribution to the smuon-mediated bino annihilation to muons is
maximized for larger left-right mixing. We also assume a fixed ratio of smuon
masses $m_{\tilde{\mu}_{2}}/m_{\tilde{\mu}_{1}}=1.25$. Since the bino
annihilation cross section is largely independent of $\phi_{\tilde{\mu}}$, we
can then determine the smuon mass necessary to yield the correct relic density
for a choice of bino mass and $\lambda_{\tilde{\mu}_{R,L}}$. The $g_{\mu}-2$
contribution in this parameter space is typically too large to match the
observed value without a near-maximal $CP$-violating phase, which suppresses
the anomalous magnetic moment of the muon $\propto\cos\phi_{\tilde{\mu}}$. For
each model point in the bino annihilation parameter space we fix
$\phi_{\tilde{\mu}}$ to a value which yields the central observed value of
$g_{\mu}-2$.

Using the DarkSUSY package Bringmann et al. (2018), we determine the parameter
space where bino annihilation can satisfy the observed DM relic density. We
show the relative mass splittings between the lightest smuon and the bino as a
function of bino mass for several benchmark curves with fixed values of
$\lambda_{\tilde{\mu}_{R,L}}$ in Fig. 1. In order for the bino annihilation
cross section to match that of thermal relic DM, heavier bino masses must be
compensated by either a larger coupling or a smaller relative mass splitting
between the lightest smuon and the bino. For the largest values of
$\lambda_{\tilde{\mu}_{R,L}}$, we see models where the relic density can be
depleted to the observed value through bino annihilation mediated by $\sim{\rm
TeV}$ smuons. For smaller relative mass splittings, $\lesssim 0.2$, we see the
effects of co-annihilation start to allow for models with even heavier bino
masses to satisfy the relic density.

Although a detailed interpretation of collider constraints on slepton masses
Dutta et al. (2015); Han and Liu (2015); Dutta et al. (2017) for our
particular model is beyond the scope of this work, we also show the
constraints from LEP and LHC in Fig. 1 assuming the production of either a
single right ALEPH, DELPHI, L3, OPAL Experiments (2004) or left Aad et al.
(2020) chiral smuon or two mass degenerate chiral smuons Aad et al. (2020);
ATL (2022), each decaying to a bino and a muon. Even though these constraints
should only be considered indicative, we can see that all of the model curves
except for the MSSM benchmark coupling can yield viable parameter points
satisfying the DM relic density. Also, note that indirect detection searches
can in principal constrain DM models with $m_{B}\lesssim 100\,{\rm GeV}$ and a
thermal WIMP annihilation cross-section, however WIMP annihilation to muons
can be consistent with multimessenger constraints for $m_{B}\gtrsim 40\,{\rm
GeV}$ Di Mauro and Winkler (2021); Abdughani et al. (2022). Regarding slepton
co-annihilation models, neither current collider searches nor indirect
detection experiments are sensitive to parameter space considered in ASU.
However, a future lepton collider with a $\gtrsim 500\,{\rm GeV}$ center of
mass energy de Blas et al. (2018); Berggren (2013); Baum et al. (2020) would
be a complementary probe, potentially able to distinguish between the slepton
mediator models explored in this Letter and other DM models which can be
observed by the kinetic heating of NSs.

Conventional direct detection experiments designed to detect weak scale dark
matter scattering off nuclei have limited sensitivity to Majorana singlet DM
candidates which only couple to the SM at tree-level through muons Sandick et
al. (2016); Acuña et al. (2022); Hamaguchi et al. (2022). More specifically,
the leading contribution to nucleon scattering typically arises from the
anapole moment. In addition to being loop-suppressed, the anapole moment only
yields proton scattering rates large enough to be detected at next generation
direct detection experiments in scenarios with very small mass splittings
between the lightest smuon and the bino.

We have checked that none of the models shown in Fig. 1 would be accessible to
an experiment with sensitivity similar to what has been proposed for DARWIN
Aalbers et al. (2016). While the parameter space of the co-annihilation models
considered in ASU can feature sufficiently compressed spectra to enhance the
anapole moment, a future experiment similar to DARWIN would only be sensitive
to a small subset models with MSSM-like $\lambda_{\tilde{\mu}_{R,L}}$ and
$m_{B}\simeq m_{\tilde{\mu}_{1}}\lesssim 400\,{\rm GeV}$ Acuña et al. (2022).

Neutron star kinetic heating.—As an alternative to conventional direct
detection experiments, the effects of DM scattering can also be detected in
observations of old NSs. More specifically, the transfer of kinetic energy
from the incident DM flux to the neutron, proton, muon and electron NS
constituents can result in an elevated NS temperature which could be measured
by JWST. Following Refs. Joglekar et al. (2020b); Bell et al. (2020, 2021), we
assume the BSk24 family of configurations for the NS equations of state (EoS)
Pearson et al. (2018). We calculate the associated DM capture rate for all
potentially significant NS constituents given the model with bino-slepton-
lepton interactions described by Eq. 1 and the subsequent discussion of the
parameter space in which both the DM relic density and $g_{\mu}-2$ are
satisfied.

We calculate the DM capture rate summed over the scattering contributions from
all NS constituents (see Supplementary Material for discussion of
contributions from scattering off different targets) for the models along the
curves in Fig. 1 and on the RHB of the models satisfying the relic density and
$g_{\mu}-2$ in Figs. 2 and 3 of ASU. We characterize the kinetic heating of
NSs by the heating efficiency, $f$, which is the ratio of the capture rate to
the ‘geometric capture rate’ for a $1.5\,M_{\odot}$ NS with a radius of
$12.6\,{\rm km}$ (BSk24-2 EoS from Refs. Bell et al. (2020, 2021)). For $f>1$
all DM particles incident on the NS will be captured after a single scatter,
while $f<1$ gives the probability for incident DM particles to be captured
Joglekar et al. (2020b). Models with $f>1$ would heat NSs to $\sim 1600\,{\rm
K}$ over $\sim 10^{9}$ years given a dark matter density equivalent to that at
the solar radius, $\sim 0.4\,{\rm GeV}/{\rm cm^{3}}$. NS surface temperatures
$\gtrsim 1000\,{\rm K}$ are potentially detectable by JWST and significantly
higher than the $\sim 100\,{\rm K}$ expected from the cooling of a old NS
Yakovlev and Pethick (2004); Page et al. (2004). The smallest NS temperature
conceivably detectable in future infrared observations of ${\cal O}(100)$ old
NSs corresponds to $f\sim 10^{-2}$ Baryakhtar et al. (2017); Raj et al.
(2018).

In the left panel of Fig. 2, we see that the lower slepton and bino masses
associated with smaller $\lambda_{\tilde{\mu}_{R,L}}$ in bino annihilation
models can yield $f\gg 1$ even in the parameter space where the relative mass
splitting between the bino and lightest smuon is large. For increasingly
larger $\lambda_{\tilde{\mu}_{R,L}}$ towards the perturbative unitarity limit,
the heating efficiency decreases as $m_{B}$ increases past the $\sim{\rm TeV}$
scale but $f\gtrsim 1$ for all models investigated. We also see that the
heating efficiency for all of the benchmark $\lambda_{\tilde{\mu}_{R,L}}$
curves becomes strongly enhanced as the bino and lightest smuon become mass
degenerate. Although $\phi_{\tilde{\mu}}\sim\pi/2$ for these models to satisfy
$g_{\mu}-2$, the dominant contributions to NS heating are virtually
independent of the $CP$-violating phase.

In the right panel of Fig. 2, the smuon co-annihilation model benchmarks with
fixed $y$ demonstrate the transition from cases more similar to bino
annihilation, in which the relative mass splitting must decrease to compensate
for increasing $m_{B}$, to cases with larger $y$, in which the relative mass
splitting must remain constant or increase to compensate for the associated
enhancements to the effective annihilation rate. In particular, we see the NS
heating efficiency fall with $m_{B}$ in the cases with larger $y$. These
effects in the smuon co-annihilation model parameter space are most clearly
demonstrated for the benchmark curves with fixed $\theta_{\tilde{\mu}}$, for
which we see the heating efficiency falling rapidly as $y$ increases along the
curves towards larger $m_{B}$. However, we see that all models along the RHB
should heat NSs to temperatures sufficient to be detected by future infrared
observations.

Figure 3: Total NS capture efficiency for bino annihilation models with
$\bar{\lambda}=7$ from Fig. 1 (solid) and smuon co-annihilation models with
fixed $\theta_{\tilde{\mu}}=-\pi/2+\pi/8$ from ASU (dashed) given different
EoS configurations BSk24-1 (red), BSk24-2 (green) and BSk24-4 (blue), as
described in the text.

In Fig. 3, we show the total capture efficiency for bino annihilation models
with the largest $\bar{\lambda}=7$ in Fig. 1 and smuon co-annihilation models
with fixed $\theta_{\tilde{\mu}}=-\pi/2+\pi/8$ given different NS EoS
configurations. In addition to BSk24-2 which we assume as our benchmark EoS
configuration, BSk24-1 and BSk24-4 from Refs. Bell et al. (2020, 2021)
respectively correspond to a mass of $1.0\,M_{\odot}$ with a radius of
$12.22\,{\rm km}$, and a mass of $2.16\,M_{\odot}$ with a radius of
$11.97\,{\rm km}$. We can see that, even under the assumption of the BSk24-1
configuration yielding the least efficient DM capture, the model curves with
the smallest capture efficiencies in Fig. 2 would heat NSs to sufficiently
high temperatures to potentially be detected.

Conclusions.—In this Letter, we have studied dark matter models which can
satisfy $g_{\mu}-2$ with interactions mediated by scalar lepton partners. We
investigate a representative parameter space of models with Majorana singlet
dark matter produced thermally in the early Universe either by dark matter
annihilation or co-annihilation processes involving the scalar leptons. For
such models with dark matter interactions mediated by scalar muon partners,
the only tree-level interactions are with muons and are notoriously difficult
to probe at either indirect detection or collider searches. In contrast to
similarly challenging searches at conventional direct detection experiments,
the presence of leptonic constituents makes the kinetic heating of neutron
stars particularly appealing.

For nearly the entire parameter space investigated, we show that scattering is
efficient enough for the entire dark matter flux incident on the neutron star
to be captured. Geometric capture efficiency of dark matter in the local
galactic halo implies the kinetic heating of a population of old neutron stars
to temperatures which should observable by future infrared telescopes, such as
JWST. The observation of an old population of neutron stars with temperatures
$\gtrsim 1000\,{\rm K}$ would thus be consistent with scalar muon mediated
interactions, but without a way to distinguish from other dark matter models
which also have sufficiently strong interactions with neutron star
constituents. On the other hand, non-observation of neutron stars which have
undergone kinetic heating would definitively rule out Majorana singlet dark
matter models satisfying $g_{\mu}-2$ with scalar muon partners up to the
$\sim{\rm TeV}$ scale.

###### Acknowledgements.

JTA would like to thank IFPU, Trieste and the National Institute of Physics,
UP Diliman, where most of this work was carried out, for the hospitality and
support. This work was supported by the research grant “The Dark Universe: A
Synergic Multi-messenger Approach” number 2017X7X85K under the program PRIN
2017 funded by the The Italian Ministry of Education, University and Research
(MIUR), and by the European Union’s Horizon 2020 research and innovation
program under the Marie Sklodowska-Curie grant agreement No 860881-HIDDeN.
Partial funding is also provided by the National Science and Technology
Council of the Republic of China (formerly the Ministry of Science and
Technology), with grant number MOST 111-2811-M-007-018-MY2.

## References

  * Abi et al. (2021) Muon g-2 Collaboration, B. Abi et al., “Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm”, Phys. Rev. Lett. 126 (2021), no. 14, 141801, arXiv:2104.03281.
  * Baryakhtar et al. (2017) M. Baryakhtar, J. Bramante, S. W. Li, T. Linden, and N. Raj, “Dark Kinetic Heating of Neutron Stars and An Infrared Window On WIMPs, SIMPs, and Pure Higgsinos”, Phys. Rev. Lett. 119 (2017), no. 13, 131801, arXiv:1704.01577.
  * Raj et al. (2018) N. Raj, P. Tanedo, and H.-B. Yu, “Neutron stars at the dark matter direct detection frontier”, Phys. Rev. D 97 (2018), no. 4, 043006, arXiv:1707.09442.
  * Bell et al. (2018) N. F. Bell, G. Busoni, and S. Robles, “Heating up Neutron Stars with Inelastic Dark Matter”, JCAP 09 (2018) 018, arXiv:1807.02840.
  * Acevedo et al. (2020) J. F. Acevedo, J. Bramante, R. K. Leane, and N. Raj, “Warming Nuclear Pasta with Dark Matter: Kinetic and Annihilation Heating of Neutron Star Crusts”, JCAP 03 (2020) 038, arXiv:1911.06334.
  * Ghosh et al. (2022) T. Ghosh, C. Kelso, J. Kumar, P. Sandick, and P. Stengel, “Simplified dark matter models with charged mediators”, in “2022 Snowmass Summer Study”.  3 2022.  arXiv:2203.08107.
  * Fukushima et al. (2014) K. Fukushima, C. Kelso, J. Kumar, P. Sandick, and T. Yamamoto, “MSSM dark matter and a light slepton sector: The incredible bulk”, Phys. Rev. D 90 (2014), no. 9, 095007, arXiv:1406.4903.
  * Acuña et al. (2022) J. T. Acuña, P. Stengel, and P. Ullio, “Minimal dark matter model for muon g-2 with scalar lepton partners up to the TeV scale”, Phys. Rev. D 105 (2022), no. 7, 075007, arXiv:2112.08992.
  * Bell et al. (2019) N. F. Bell, G. Busoni, and S. Robles, “Capture of Leptophilic Dark Matter in Neutron Stars”, JCAP 06 (2019) 054, arXiv:1904.09803.
  * Garani and Heeck (2019) R. Garani and J. Heeck, “Dark matter interactions with muons in neutron stars”, Phys. Rev. D 100 (2019), no. 3, 035039, arXiv:1906.10145.
  * Joglekar et al. (2020a) A. Joglekar, N. Raj, P. Tanedo, and H.-B. Yu, “Relativistic capture of dark matter by electrons in neutron stars”, Phys. Lett. B (2020)a 135767, arXiv:1911.13293.
  * Joglekar et al. (2020b) A. Joglekar, N. Raj, P. Tanedo, and H.-B. Yu, “Dark kinetic heating of neutron stars from contact interactions with relativistic targets”, Phys. Rev. D 102 (2020)b, no. 12, 123002, arXiv:2004.09539.
  * Gardner et al. (2006) J. P. Gardner et al., “The James Webb Space Telescope”, Space Sci. Rev. 123 (2006) 485, astro-ph/0606175.
  * Hamaguchi et al. (2022) K. Hamaguchi, N. Nagata, and M. E. Ramirez-Quezada, “Neutron Star Heating in Dark Matter Models for the Muon $g-2$ Discrepancy”, arXiv:2204.02413.
  * Aad et al. (2020) ATLAS Collaboration, G. Aad et al., “Search for electroweak production of charginos and sleptons decaying into final states with two leptons and missing transverse momentum in $\sqrt{s}=13$ TeV $pp$ collisions using the ATLAS detector”, Eur. Phys. J. C 80 (2020), no. 2, 123, arXiv:1908.08215.
  * ATL (2022) ATLAS Collaboration Collaboration, “Search for direct pair production of sleptons and charginos decaying to two leptons and neutralinos with mass splittings near the $W$ boson mass in ${\sqrt{s}=13\,}$TeV $pp$ collisions with the ATLAS detector”, Technical Report, CERN, Geneva, 2022\.
  * ALEPH, DELPHI, L3, OPAL Experiments (2004) ALEPH, DELPHI, L3, OPAL Experiments, “Combined LEP Selectron/Smuon/Stau Results, 183-208 GeV”. LEPSUSYWG/04-01.1, 2004.
  * Aad et al. (2020) ATLAS Collaboration, G. Aad et al., “Searches for electroweak production of supersymmetric particles with compressed mass spectra in $\sqrt{s}=$ 13 TeV $pp$ collisions with the ATLAS detector”, Phys. Rev. D 101 (2020), no. 5, 052005, arXiv:1911.12606.
  * Pearson et al. (2018) J. M. Pearson, N. Chamel, A. Y. Potekhin, A. F. Fantina, C. Ducoin, A. K. Dutta, and S. Goriely, “Unified equations of state for cold non-accreting neutron stars with Brussels–Montreal functionals – I. Role of symmetry energy”, Mon. Not. Roy. Astron. Soc. 481 (2018), no. 3, 2994–3026, arXiv:1903.04981, [Erratum: Mon.Not.Roy.Astron.Soc. 486, 768 (2019)].
  * Bringmann et al. (2018) T. Bringmann, J. Edsjö, P. Gondolo, P. Ullio, and L. Bergström, “Darksusy 6: an advanced tool to compute dark matter properties numerically”, Journal of Cosmology and Astroparticle Physics 2018 (2018), no. 07, 033.
  * Dutta et al. (2015) B. Dutta, T. Ghosh, A. Gurrola, W. Johns, T. Kamon, P. Sheldon, K. Sinha, K. Wang, and S. Wu, “Probing Compressed Sleptons at the LHC using Vector Boson Fusion Processes”, Phys. Rev. D 91 (2015), no. 5, 055025, arXiv:1411.6043.
  * Han and Liu (2015) Z. Han and Y. Liu, “MT2 to the Rescue – Searching for Sleptons in Compressed Spectra at the LHC”, Phys. Rev. D 92 (2015), no. 1, 015010, arXiv:1412.0618.
  * Dutta et al. (2017) B. Dutta, K. Fantahun, A. Fernando, T. Ghosh, J. Kumar, P. Sandick, P. Stengel, and J. W. Walker, “Probing Squeezed Bino-Slepton Spectra with the Large Hadron Collider”, Phys. Rev. D 96 (2017), no. 7, 075037, arXiv:1706.05339.
  * Di Mauro and Winkler (2021) M. Di Mauro and M. W. Winkler, “Multimessenger constraints on the dark matter interpretation of the Fermi-LAT Galactic center excess”, Phys. Rev. D 103 (2021), no. 12, 123005, arXiv:2101.11027.
  * Abdughani et al. (2022) M. Abdughani, Y.-Z. Fan, C.-T. Lu, T.-P. Tang, and Y.-L. S. Tsai, “Muonphilic dark matter explanation of gamma-ray galactic center excess: a comprehensive analysis”, JHEP 07 (2022) 127, arXiv:2111.02946.
  * de Blas et al. (2018) J. de Blas et al., “The CLIC Potential for New Physics”, arXiv:1812.02093.
  * Berggren (2013) M. Berggren, “Simplified SUSY at the ILC”, in “Community Summer Study 2013: Snowmass on the Mississippi”.  8 2013.  arXiv:1308.1461.
  * Baum et al. (2020) S. Baum, P. Sandick, and P. Stengel, “Hunting for scalar lepton partners at future electron colliders”, Phys. Rev. D 102 (2020), no. 1, 015026, arXiv:2004.02834.
  * Sandick et al. (2016) P. Sandick, K. Sinha, and F. Teng, “Simplified Dark Matter Models with Charged Mediators: Prospects for Direct Detection”, JHEP 10 (2016) 018, arXiv:1608.00642.
  * Aalbers et al. (2016) J. Aalbers, F. Agostini, M. Alfonsi, F. Amaro, C. Amsler, E. Aprile, L. Arazi, F. Arneodo, P. Barrow, L. Baudis, et al., “Darwin: towards the ultimate dark matter detector”, Journal of Cosmology and Astroparticle Physics 2016 (2016), no. 11, 017.
  * Bell et al. (2020) N. F. Bell, G. Busoni, S. Robles, and M. Virgato, “Improved Treatment of Dark Matter Capture in Neutron Stars”, JCAP 09 (2020) 028, arXiv:2004.14888.
  * Bell et al. (2021) N. F. Bell, G. Busoni, S. Robles, and M. Virgato, “Improved Treatment of Dark Matter Capture in Neutron Stars II: Leptonic Targets”, JCAP 03 (2021) 086, arXiv:2010.13257.
  * Yakovlev and Pethick (2004) D. G. Yakovlev and C. J. Pethick, “Neutron star cooling”, Ann. Rev. Astron. Astrophys. 42 (2004) 169–210, astro-ph/0402143.
  * Page et al. (2004) D. Page, J. M. Lattimer, M. Prakash, and A. W. Steiner, “Minimal cooling of neutron stars: A New paradigm”, Astrophys. J. Suppl. 155 (2004) 623–650, astro-ph/0403657.
  * Berlin et al. (2016) A. Berlin, D. S. Robertson, M. P. Solon, and K. M. Zurek, “Bino variations: Effective field theory methods for dark matter direct detection”, Phys. Rev. D 93 May (2016) 095008.
  * Bishara et al. (2017) F. Bishara, J. Brod, B. Grinstein, and J. Zupan, “From quarks to nucleons in dark matter direct detection”, JHEP 11 (2017) 059, arXiv:1707.06998.

SUPPLEMENTARY MATERIAL

Figure 4: Capture efficiency contributions from muon (solid), proton (dashed)
and neutron (dotted) scattering for model curves with fixed $y=10$ (red),
$y=50$ (blue) and $\theta_{\tilde{\mu}}=-\pi/2+\pi/8$ (green) from the right
panel of Fig. 2.

In contrast to the anapole moment, loop-induced couplings of the bino to
nucleons through both Higgs and Z-boson exchange are less sensitive to the
mass splitting between the lightest smuon and the bino, but only the former
can become important at conventional direct detection experiments in models
with a sufficiently large trilinear coupling between the Higgs and the smuons
(see discussion in ASU). For the kinetic heating of NSs in our slepton
mediator model, we demonstrate that the contribution from nucleon scattering
at loop-level is always subdominant to that of tree-level muon scattering.
Note that all nucleon scattering contributions are calculated in an effective
contact operator framework with Wilson coefficients defined in Ref. Berlin et
al. (2016) and nucleon form factors from Ref. Bishara et al. (2017).

In Fig. 4, we show the contributions to NS heating from DM scattering off
different NS constituents for several model curves from the right panel of
Fig. 2. For $y=10$, we see that the NS heating contribution from muon
scattering dominates over the nucleon scattering contributions by several
orders of magnitude despite the number density of the muons within the NS
being less than that of the protons and much less than that of the neutrons.
In fact, with the proton scattering contributions also much larger than the
neutron scattering contributions due to the enhanced anapole moment, we see
there the hierarchy of NS heating contributions is actually the reverse of
that of the number densities of NS constituents Bell et al. (2019); Garani and
Heeck (2019); Joglekar et al. (2020a, b). A similar ordering of NS heating
contributions can be seen across most of the parameter space of bino
annihilation models, although the ratios of respective contributions are not
as large as in the smuon co-annihilation models due to relatively uncompressed
spectra.

While the reverse hierarchy of NS heating contributions holds through most of
the parameter space for both bino annihilation and smuon co-annihilation
models, we can see the nucleon scattering become more significant as the smuon
mass splitting parameter becomes large. In particular, the hierarchy of NS
heating contributions begin to change for smuon co-annihilation models with
fixed $y=50$, as the neutron scattering contribution becomes similar to the
proton scattering contribution at $m_{B}\gtrsim 1\,{\rm TeV}$. The mass
splitting between the bino and lightest smuons can remain relatively large for
these points, thus suppressing the anapole moment and the muon scattering
contributions. When $y>50$ along the model curve with fixed
$\theta_{\tilde{\mu}}=-\pi/2+\pi/8$, we can see the neutron scattering
contribution becomes larger than the proton scattering contribution and within
an order of magnitude of the muon scattering contribution.