# The Tiny (g-2) Muon Wobble from Small-$\boldsymbol{\mu}$ Supersymmetry

Sebastian Baum<EMAIL_ADDRESS>Stanford Institute for Theoretical Physics,
Department of Physics, Stanford University, Stanford, CA 94305, USA  Marcela
Carena<EMAIL_ADDRESS>Fermi National Accelerator Laboratory, P. O. Box 500,
Batavia, IL 60510, USA  Enrico Fermi Institute and Kavli Institute for
Cosmological Physics, Department of Physics, University of Chicago, Chicago,
IL 60637, USA  Nausheen R. Shah<EMAIL_ADDRESS>Department of Physics
& Astronomy, Wayne State University, Detroit, MI 48201, USA  Carlos E. M.
Wagner<EMAIL_ADDRESS>HEP Division, Argonne National Laboratory, 9700
Cass Ave., Argonne, IL 60439, USA  Enrico Fermi Institute and Kavli Institute
for Cosmological Physics, Department of Physics, University of Chicago,
Chicago, IL 60637, USA

###### Abstract

A new measurement of the muon anomalous magnetic moment, $g_{\mu}-2$, has been
reported by the Fermilab Muon g-2 collaboration and shows a $4.2\,\sigma$
departure from the most precise and reliable calculation of this quantity in
the Standard Model. Assuming that this discrepancy is due to new physics, we
concentrate on a simple supersymmetric model that also provides a dark matter
explanation in a previously unexplored region of supersymmetric parameter
space. Such interesting region can realize a Bino-like dark matter candidate
compatible with all current direct detection constraints for small to moderate
values of the Higgsino mass parameter $|\mu|$. This in turn would imply the
existence of light additional Higgs bosons and Higgsino particles within reach
of the high-luminosity LHC and future colliders. We provide benchmark
scenarios that will be tested in the next generation of direct dark matter
experiments and at the LHC.

††preprint: EFI-21-3††preprint: FERMILAB-PUB-21-184-T††preprint: WSU-HEP-2101

##  I Introduction

The Standard Model (SM) of particle physics has built its reputation on
decades of measurements at experiments around the world that testify to its
validity. With the discovery of the Higgs boson almost a decade ago Aad  _et
al._ (2012); Chatrchyan  _et al._ (2012) all SM particles have been observed
and the mechanism that gives mass to the SM particles, with the possible
exception of the neutrinos, has been established. Nonetheless, we know that
physics beyond the SM (BSM) is required to explain the nature of dark matter
(DM) and the source of the observed matter-antimatter asymmetry. Furthermore,
an understanding of some features of the SM such as the hierarchy of the
fermion masses or the stability of the electroweak vacuum is lacking.

The direct discovery of new particles pointing towards new forces or new
symmetries in nature will be the most striking and conclusive evidence of BSM
physics. However, it may well be the case that BSM particles lie beyond our
present experimental reach in mass and/or interaction strength, and that clues
for new physics may first come from results for precision observables that
depart from their SM expectations. With that in mind, since the discovery of
the Higgs boson, we are straining our resources and capabilities to measure
the properties of the Higgs boson to higher and higher accuracy, and flavor
and electroweak physics experiments at the LHC and elsewhere are pursuing a
complementary broad program of precision measurements. Breakthroughs in our
understanding of what lies beyond the SM could occur at any time.

Recently, new results of measurements involving muons have been reported. The
LHCb experiment has reported new values of the decay rate of $B$-mesons to a
kaon and a pair of muons compared to the decay into a kaon and electrons Aaij
_et al._ (2021a), providing evidence at the $3\,\sigma$-level of the violation
of lepton universality. This so-called $R_{K}$ anomaly joins the ranks of
previously reported anomalies involving heavy-flavor quarks such as the bottom
quark forward-backward asymmetry at LEP Schael  _et al._ (2006); ALEPH (2010),
and measurements of meson decays at the LHC and $B$-factories such as
$R_{K^{*}}$ Aaij  _et al._ (2017a, 2020, 2021b) and $R_{D^{(*)}}$ Lees  _et
al._ (2012); Aaij  _et al._ (2015); Huschle  _et al._ (2015); Hirose  _et al._
(2017); Aaij  _et al._ (2018); Abdesselam  _et al._ (2019). The Fermilab Muon
(g-2) experiment has just reported a new measurement of the anomalous magnetic
moment of the muon, $a_{\mu}\equiv\left(g_{\mu}-2\right)/2$. The SM prediction
of $a_{\mu}$ is known with the remarkable relative precision of $4\times
10^{-8}$, $a_{\mu}^{\rm SM}=116~{}591~{}810(43)\times 10^{-11}$ Aoyama  _et
al._ (2020, 2012, 2019); Czarnecki  _et al._ (2003); Gnendiger  _et al._
(2013); Davier  _et al._ (2017); Keshavarzi  _et al._ (2018); Colangelo  _et
al._ (2019); Hoferichter  _et al._ (2019); Davier  _et al._ (2020); Keshavarzi
_et al._ (2020a); Kurz  _et al._ (2014); Melnikov and Vainshtein (2004);
Masjuan and Sanchez-Puertas (2017); Colangelo  _et al._ (2017); Hoferichter
_et al._ (2018); Gérardin  _et al._ (2019); Bijnens  _et al._ (2019);
Colangelo  _et al._ (2020); Blum  _et al._ (2020); Colangelo  _et al._ (2014).
From the new Fermilab Muon (g-2) experiment, the measured value is
$a_{\mu}^{\rm exp,~{}FNAL}=116~{}592~{}040(54)\times 10^{-11}$ Abi  _et al._
(2021), which combined with the previous E821 result $a_{\mu}^{\rm
exp,~{}E821}=116~{}592~{}089(63)\times 10^{-11}$ Bennett  _et al._ (2006),
yields a value $a_{\mu}^{\rm exp}=116~{}592~{}061(41)\times 10^{-11}$.

An important point when considering the tension between experimental results
and the SM predictions are the current limitations on theoretical tools in
computing the hadronic vacuum polarization (HVP) contribution to $a_{\mu}^{\rm
SM}$, which is governed by the strong interaction and is particularly
challenging to calculate from first principles. The most accurate result of
the HVP contribution is based on a data-driven result, extracting its value
from precise and reliable low-energy ($e^{+}e^{-}\to{\rm hadrons}$) cross
section measurements via dispersion theory. Assuming no contribution from new
physics to the low energy processes and conservatively accounting for
experimental errors, this yields a value $a_{\mu}^{\rm HVP}=685.4(4.0)\times
10^{-10}$ Aoyama  _et al._ (2020); Davier  _et al._ (2017); Keshavarzi  _et
al._ (2018); Colangelo  _et al._ (2019); Hoferichter  _et al._ (2019); Davier
_et al._ (2020); Keshavarzi  _et al._ (2020a); Kurz  _et al._ (2014), implying
an uncertainty of 0.6 % in this contribution.111The HVP contribution has
recently been computed in lattice QCD, yielding a higher value of
$a_{\mu}^{\rm HVP}=708.7(5.3)\times 10^{-10}$ Borsanyi  _et al._ (2021). Given
the high complexity of this calculation, independent lattice calculations with
commiserate precision are needed before confronting this result with the well
tested data-driven one. We stress that if a larger value of the HVP
contribution were confirmed, which would (partially) explain the $(g_{\mu}-2)$
anomaly, new physics contributions will be needed to bring theory and
measurements of ($e^{+}e^{-}\to{\rm hadrons}$) in agreement Lehner and Meyer
(2020); Crivellin  _et al._ (2020); Keshavarzi  _et al._ (2020b); de Rafael
(2020). The SM prediction for the anomalous magnetic moment of the muon and
the measured value then differ by $4.2\,\sigma$,

$\Delta a_{\mu}\equiv(a_{\mu}^{\rm exp}-a_{\mu}^{\rm SM})=\left(251\pm
59\right)\times 10^{-11}\,.$ (1)

It is imperative to ask what these anomalies may imply for new physics. The
most relevant questions that come to mind are: Can the $a_{\mu}$ and
$R_{K^{(*)}}$ anomalies be explained by the same BSM physics? Can they give
guidance about the nature of DM? Are they related to cosmological
discrepancies? How constrained are the possible solutions by other
experimental searches? What are future experimental prospects for the possible
solutions?

In Sec. II we provide a brief overview of the many models which have been
previously proposed in the literature to explain the $(g_{\mu}-2)$ anomaly and
consider their impact on other possible anomalies and on unresolved questions
of the SM. Then, in Sec. III, we discuss a supersymmetric solution in the most
simplistic supersymmetric model at hand, the Minimal Supersymmetric Standard
Model (MSSM). We focus on a region of the parameter space of the MSSM where
the $(g_{\mu}-2)$ anomaly can be realized simultaneously with a viable DM
candidate. We show that in the region of moderate $\left|\mu\right|$ and
moderate-to-large values of $\tan\beta$, a Bino-like DM candidate can be
realized in the proximity of blind spots (that require $\mu\times M_{1}<0$)
for spin-independent direct detection (SIDD) experiments Huang and Wagner
(2014). In this way, our MSSM scenario explores a different region of
parameter space than the one considered in the study of Refs. Chakraborti  _et
al._ (2020, 2021), which considers regions of large $\mu$ as a way to
accommodate current SIDD bounds. We summarize and conclude in Sec. IV. In
Appendix A, we give details about the LHC constraints on these scenarios.

##  II $\boldsymbol{(g_{\mu}-2)}$ connections to cosmic puzzles and the LHC

In order to bridge the gap between the SM prediction and the measured value
for the anomalous magnetic moment of the muon, a BSM contribution of order
$\Delta a_{\mu}=(20$–$30)\times 10^{-10}$ is needed. Taking the $a_{\mu}$
anomaly as a guidance for new physics, it is natural to ask how it can be
connected to other anomalies, specially those in the muon sector, or to
solving puzzles of our universe’s early history. There are two broad classes
of solutions to the ($g_{\mu}-2$) anomaly that may be considered in the light
of the above:

  * •

New relatively light particles with small couplings to muons, typically
featuring particles with $\mathcal{O}(100)\,$MeV masses and
$\mathcal{O}(10^{-3})$ couplings to muons. Examples of such models we will
discuss here are new (light) scalars and new (light) ($Z^{\prime}$) vector
bosons. These new light particles may have left important clues in the cosmos.

  * •

New heavy fermions or scalars (possibly accompanied by additional new
particles), as well as leptoquark particles, with larger couplings to muons.
Similar solutions appear also in supersymmetric extensions of the SM that we
shall discuss separately in some detail in Sec. III. In addition, new gauge
symmetries, spontaneously broken at low energies, can induce $Z^{\prime}$
vector bosons with masses comparable to the electroweak scale and
$\mathcal{O}(1)$ couplings to muons. These types of new particles can be
sought for at the LHC and other terrestrial experiments.

The most recent LHCb measurement Aaij  _et al._ (2021a), $R_{K}={\rm
BR}(B\rightarrow K\mu^{+}\mu^{-})/{\rm BR}(B\rightarrow
Ke^{+}e^{-})=0.846^{+0.044}_{-0.041}$ in the kinematic regime of $1.1\,{\rm
GeV}^{2}\leq q^{2}\leq 6.0\,{\rm GeV}^{2}$ implies a violation of lepton
universality and differs from the SM expectation at the $3.1\,\sigma$ level.
Since $R_{K}$ also involves muons, it naturally appears related to the
$(g_{\mu}-2)$ anomaly. However, as we shall discuss, it is hard to
simultaneously fit both $R_{K}$ and $(g_{\mu}-2)$.

Scalar solutions: This is perhaps the simplest scenario for the explanation of
the observed $\Delta a_{\mu}$. A scalar particle, with mass $\lesssim
200\,$MeV and couplings to muons of similar size as the corresponding SM-Higgs
coupling, can lead to a satisfactory explanation of $\Delta a_{\mu}$ Zhou and
Wu (2003); Barger  _et al._ (2011); Chen  _et al._ (2016); Batell  _et al._
(2017); Davoudiasl and Marciano (2018); Liu  _et al._ (2019). One can
construct models with such a scalar particle and suppressed couplings to other
leptons or quarks in a straightforward way Liu  _et al._ (2019).
Alternatively, one can construct models with appropriate values of the
couplings of the new scalar to quarks to lead to an explanation of some flavor
anomalies, for example the KOTO anomaly Ahn  _et al._ (2021), but the
constraints tend to be more severe and the model-building becomes more
involved Liu  _et al._ (2020a). It is important to stress that it proves
impossible to fully explain the $R_{K}$ anomaly with scalars without violating
$B_{s}\rightarrow\mu^{+}\mu^{-}$ measurements Aaij  _et al._ (2017b); see, for
example, Ref. Altmannshofer and Stangl (2021).

A pseudoscalar particle may also lead to an explanation of $\Delta a_{\mu}$,
provided it couples not only to muons, but also to photons. The typical
example are axion-like particles Marciano  _et al._ (2016); Bauer  _et al._
(2020), although obtaining the proper $\Delta a_{\mu}$ requires a delicate
interplay between the muon and photon couplings.222A similar mechanism applies
for $(g_{e}-2)$ in the case of the QCD axion; see, for instance, Ref. Liu  _et
al._ (2021) Alternatively, a positive contribution to $a_{\mu}$ can arise from
a two loop Barr-Zee diagram mediated by the pseudoscalar couplings to heavier
quarks and leptons Gunion (2009); Wang  _et al._ (2019).

Fermionic solutions: Another interesting solution occurs in the case of
vector-like leptons, which may induce a contribution to $a_{\mu}$ via gauge
boson and Higgs mediated interactions Kannike  _et al._ (2012); Dermisek and
Raval (2013). Note that the mixing between the SM leptons and the new heavy
leptons must be carefully controlled to prevent dangerous flavor-changing
neutral currents in the lepton sector. A recent analysis shows that
consistency with the measured values of $\Delta a_{\mu}$ may be obtained for
vector-like leptons with masses of the order of a few TeV Dermisek  _et al._
(2021).333See Ref. Megias  _et al._ (2017) for an attempt to adress both
$a_{\mu}$ and $R_{K^{*}}$ in a vector-like lepton model with extra dimensions.

Leptoquark solutions: This is one of the most interesting solutions to $\Delta
a_{\mu}$, since it can also lead to an explanation of the $R_{K}$ anomaly;
see, for example, Refs. Bauer and Neubert (2016); Crivellin  _et al._ (2021a,
b); Hiller  _et al._ (2021). A directly related and particularly attractive
realization arises in R-parity violating supersymmetry, which enables the same
type of interactions as a leptoquark theory; see, for example, Ref.
Altmannshofer  _et al._ (2020). This solution requires the scalar partner of
the right-handed bottom quark to have masses of a few TeV, which may be tested
at future LHC runs. Similar to the vector-like lepton scenarios, a careful
choice of the leptoquark couplings is necessary to avoid flavor-changing
neutral currents. This tuning is perhaps the least attractive feature of such
scenarios, although it may be the result of symmetries Hiller  _et al._
(2021).

Gauge boson solutions: New gauge bosons coupled to muons are an attractive
solution to the $a_{\mu}$ anomaly, since they can be incorporated in an
anomaly-free framework that can also lead to an explanation of the
$R_{K^{(*)}}$ anomalies. Of particular interest is the gauged
$(L_{\mu}-L_{\tau})$ scenario Heeck and Rodejohann (2011), since it avoids the
coupling to electrons.444Models with $(L_{\mu}+L_{\tau})$ give an intriguing
connection to a novel mechanism of electroweak baryogenesis with CP-violation
triggered in a dark sector that allows for a suitable DM candidate Carena  _et
al._ (2019, 2020). Unfortunately, solutions to $(g_{\mu}-2)$ in this appealing
scenario are ruled out by $(B\rightarrow K\mu^{+}\mu^{-})$ constraints due to
contributions from the anomalous $WWZ^{\prime}$ coupling. The $R_{K^{(*)}}$
anomalies may be explained by the addition of vector-like quarks that mix with
the second and third generation SM quarks Altmannshofer  _et al._ (2014a,
2016a, 2016b), connecting the ($L_{\mu}-L_{\tau}$) gauge boson to baryons. A
common explanation of both $R_{K^{(*)}}$ and $a_{\mu}$ is, however, strongly
constrained by neutrino trident bounds on $Z^{\prime}$ bosons coupled to muons
Altmannshofer  _et al._ (2014b); Geiregat  _et al._ (1990); Mishra  _et al._
(1991).555There are also bounds from Coherent $\nu$-Nucleus Scattering
(CE$\nu$NS), although these are not yet competitive with the bounds from
neutrino trident processes Abdullah  _et al._ (2018); Amaral  _et al._ (2020).
In addition, bounds from BaBar Lees  _et al._ (2016) and CMS Sirunyan  _et
al._ (2019) from
[$e^{+}e^{-}/pp\to\mu^{+}\mu^{-}+(Z^{\prime}\to\mu^{+}\mu^{-})$] rule out the
values of the new gauge coupling which could explain the observed value of
$a_{\mu}$ for $m_{Z^{\prime}}\geq 2m_{\mu}\simeq 210\,$MeV. Due to these
experimental constraints, explaining the $\Delta a_{\mu}$ anomaly with a light
new gauge boson requires $m_{Z^{\prime}}\lesssim 200\,$MeV. Explanations of
the flavor anomalies require larger gauge boson masses, preventing
simultaneous explanations of $R_{K^{(*)}}$ and $a_{\mu}$.

It is interesting to note that explanations of the $(g_{\mu}-2)$ anomaly via
gauged $(L_{\mu}-L_{\tau})$ may have a relation to some of the cosmological
puzzles, in particular the tensions of the late and early time determinations
of the Hubble constant, $H_{0}$ Escudero  _et al._ (2019); Amaral  _et al._
(2020). In the $m_{Z^{\prime}}\sim 10\,$MeV region, the effective number of
degrees of freedom can be enhanced by $\Delta N_{\rm eff}\approx 0.2$,
alleviating the $H_{0}$-tension. Note that constraints from solar neutrino
scattering in Borexino Harnik  _et al._ (2012); Bilmis  _et al._ (2015);
Amaral  _et al._ (2020) and $\Delta N_{\rm eff}$ bounds Escudero  _et al._
(2019) rule out the couplings preferred by the $a_{\mu}$ anomaly for
$m_{Z^{\prime}}\lesssim 5\,$MeV.

Before considering minimal supersymmetric scenarios for the $(g_{\mu}-2)$
anomaly in some detail, let us summarize the discussion above as follows: 1)
All the above solutions, with a broad range of masses and couplings of the new
particles, can readily explain the $(g_{\mu}-2)$ anomaly, but it is difficult
to simultaneously accommodate the $R_{K^{(*)}}$ anomalies. This difficulty
mainly arises from experimental constraints.666See also Refs. Yin and
Yamaguchi (2020); Capdevilla  _et al._ (2021) for prospects of probing models
adressing the ($g_{\mu}-2$) anomaly at high energy muon colliders. In the rare
examples of models where both solutions can be accommodated simultaneously, it
is only possible at the cost of significant tuning of the parameters. 2) In
most scenarios, a DM candidate can be included in the model (with different
levels of complexity). However, there does not appear to be a compelling
connection offering a unique guidance for model building. On the other hand,
in low-energy SUSY models with R-parity conservation, an explanation of the
$(g_{\mu}-2)$ anomaly is naturally connected to the presence of a DM candidate
and other new particles within the reach of the (HL-)LHC and future colliders.
We explore this possibility in its simplest realization in the next section.

##  III Tiny ($\boldsymbol{g_{\mu}-2}$) Muon Wobble with Small
$\boldsymbol{|\mu|}$ in the MSSM

Supersymmetric extensions of the SM remain among the most compelling BSM
scenarios Nilles (1984); Haber and Kane (1985); Martin (1998), not least
because the stability of the Higgs mass parameter under quantum corrections
can be ensured. In minimal supersymmetric extensions of the SM, the SM-like
Higgs is naturally light Casas  _et al._ (1995); Carena  _et al._ (1995,
1996); Haber  _et al._ (1997); Degrassi  _et al._ (2003); Bagnaschi  _et al._
(2014); Draper  _et al._ (2014); Lee and Wagner (2015); Pardo Vega and
Villadoro (2015); Bahl  _et al._ (2018); Slavich  _et al._ (2021) and the
corrections to electroweak precision as well as flavor observables tend to be
small, leading to good agreement with observations. Supersymmetric extensions
can also lead to gauge coupling unification and provide a natural DM
candidate, namely the lightest neutralino.

In this section, we propose simultaneous ($g_{\mu}-2$) and DM solutions in the
Minimal Supersymmetric Standard Model (MSSM) Nilles (1984); Haber and Kane
(1985); Martin (1998) which have not been explored before. Related recent (but
prior to the publication of the Fermilab Muon (g-2) result) studies can, for
example, be found in Refs. Okada and Tran (2016); Cox  _et al._ (2018); Carena
_et al._ (2018); Endo and Yin (2019); Badziak and Sakurai (2019); Abdughani
_et al._ (2019); Chakraborti  _et al._ (2020, 2021). One crucial difference
between our study and the very recent work in Refs. Chakraborti  _et al._
(2020, 2021) is that the spin-independent direct detection (SIDD) cross
section is suppressed not by decoupling the Higgsino and heavy Higgs
contributions, but by a partial cancellation between the amplitudes mediated
by the two neutral CP-even Higgs boson mass eigenstates. This cancellation
requires opposite signs of the Higgsino and the Bino mass parameters,
$(\mu\times M_{1})<0$ Huang and Wagner (2014). Demonstrating that one can
explain the $a_{\mu}$ anomaly in this region of parameter space is non-
trivial, as this combination of the Higgsino and Bino mass parameters renders
the contribution of the neutralino-smuon loop to $a_{\mu}$ negative, while the
experimentally observed value is larger than the SM prediction. Explaining the
experimental measurement is only possible if the chargino-sneutrino
contribution to $a_{\mu}$ is positive and has larger absolute magnitude than
the neutralino-smuon contribution, and if the values of the individual
contributions are such that the observed anomaly, $\Delta
a_{\mu}=(20-30)\times 10^{-10}$, can be explained. Moreover, this can only be
achieved for moderate (absolute) values of the Higgsino mass parameter
$|\mu|\lesssim 500\,$GeV, and values of the heavy Higgs boson masses than are
not far away from the current experimental limit coming from direct searches.

###  III.1 $\boldsymbol{\Delta a_{\mu}}$ and Direct Dark Matter Detection
Constraints

The MSSM contributions to $a_{\mu}$ have been discussed extensively in the
literature, see, for example, Refs. Barbieri and Maiani (1982); Ellis  _et
al._ (1982); Kosower  _et al._ (1983); Moroi (1996); Carena  _et al._ (1997);
Feng and Matchev (2001); Martin and Wells (2001); Badziak and Sakurai (2019).
The most important contributions arise via chargino-sneutrino and neutralino-
smuon loops, approximately described by Badziak and Sakurai (2019)

$\displaystyle a_{\mu}^{\widetilde{\chi}^{\pm}-\widetilde{\nu}_{\mu}}$
$\displaystyle\simeq\frac{\alpha m_{\mu}^{2}\mu
M_{2}\tan\beta}{4\pi\sin^{2}\theta_{W}m_{\widetilde{\nu}_{\mu}}^{2}}\left[\frac{f_{\chi^{\pm}}\left(M_{2}^{2}/m_{\widetilde{\nu}_{\mu}}^{2}\right)-f_{\chi^{\pm}}\left(\mu^{2}/m_{\widetilde{\nu}_{\mu}}^{2}\right)}{M_{2}^{2}-\mu^{2}}\right]\;,$
(2) $\displaystyle a_{\mu}^{\widetilde{\chi}^{0}-\widetilde{\mu}}$
$\displaystyle\simeq\frac{\alpha m_{\mu}^{2}M_{1}\left(\mu\tan\beta-
A_{\mu}\right)}{4\pi\cos^{2}\theta_{W}\left(m_{\widetilde{\mu}_{R}}^{2}-m_{\widetilde{\mu}_{L}}^{2}\right)}\left[\frac{f_{\chi^{0}}\left(M_{1}^{2}/m_{\widetilde{\mu}_{R}}^{2}\right)}{m_{\widetilde{\mu}_{R}}^{2}}-\frac{f_{\chi^{0}}\left(M_{1}^{2}/m_{\widetilde{\mu}_{L}}^{2}\right)}{m_{\widetilde{\mu}_{L}}^{2}}\right]\;,$
(3)

where $M_{2}$ is the Wino mass parameter and $m_{\widetilde{f}}$ are the
scalar particle $\widetilde{f}$ masses, with the loop functions

$\displaystyle f_{\chi^{\pm}}(x)$
$\displaystyle=\frac{x^{2}-4x+3+2\ln(x)}{\left(1-x\right)^{3}}\;,$ (4)
$\displaystyle f_{\chi^{0}}(x)$
$\displaystyle=\frac{x^{2}-1-2x\ln(x)}{\left(1-x\right)^{3}}\;;$ (5)

see Refs. Moroi (1996); Martin and Wells (2001) for the full (one-loop)
expressions. It is interesting to note that these two contributions can be of
the same order of magnitude: The chargino-sneutrino contribution is
proportional to Higgsino-Wino mixing which can be sizeable, but suppressed by
the smallness of the Higgsino-sneutrino-muon coupling which is proportional to
the muon Yukawa coupling, $\propto m_{\mu}\tan\beta/v$, with the SM Higgs
vacuum expectation value $v$. The neutralino-smuon contribution, on the other
hand, arises via muon-smuon-neutralino vertices which are proportional to the
gauge couplings, but is suppressed by the small smuon left-right mixing,
$\propto m_{\mu}(\mu\tan\beta-
A_{\mu})/(m_{\widetilde{\mu}_{R}}^{2}-m_{\widetilde{\mu}_{L}}^{2})$. Regarding
corrections beyond one-loop Marchetti  _et al._ (2009); Athron  _et al._
(2016), the most relevant contribution is associated with corrections to the
muon Yukawa coupling, $\Delta_{\mu}$. These corrections become relevant at
large values of $\mu\tan\beta$ and can be re-summed at all orders of
perturbation theory Carena  _et al._ (2000). While these corrections lead to
small modifications of $a_{\mu}$, they do not change the overall dependence of
$\Delta a_{\mu}$ on the masses of the supersymmetric particles.

From Eqs. (2)–(3) we can observe that the signs of the MSSM contributions to
$a_{\mu}$ depend sensitively on the relative signs of the gaugino masses
$M_{1}$ and $M_{2}$ and the Higgsino mass parameter $\mu$. As emphasized
before, a DM candidate compatible with the current null-results from direct
detection experiments can be realized for $|\mu|\lesssim 500$ GeV if $M_{1}$
and $\mu$ have opposite signs. For this combinations of signs, the
contribution from the neutralino-smuon loop to $a_{\mu}$ will be negative,
$a_{\mu}^{\widetilde{\chi}^{0}-\widetilde{\mu}}<0$. Since the measured value
of $a_{\mu}$ is larger than the SM prediction by $\Delta a_{\mu}\simeq
25\times 10^{-10}$, we require the chargino-sneutrino contribution to be
positive and larger than the neutralino-smuon contribution. This can be
realized if $M_{2}$ has the same sign as $\mu$ and if $|M_{2}|$ is of similar
size as $|\mu|$ and the soft smuon masses. In the regime of moderate or large
values of $\tan\beta$, and assuming all weakly interacting sparticles have
masses of the same order, $\widetilde{m}$, one obtains approximately

$\Delta a_{\mu}\simeq 1.3\times 10^{-9}\tan\beta\times\left(\frac{100\,{\rm
GeV}}{\widetilde{m}}\right)^{2}\;.$ (6)

The factor 1.3 reduces to values closer to 1 if $M_{1}$ and $M_{2}$ have
opposite signs. This implies that for values of $\tan\beta\simeq 10$,
sparticles with masses $\widetilde{m}\sim 200\,$GeV can lead to an explanation
of the observed $\Delta a_{\mu}$ anomaly, while for $\tan\beta=60$, the
characteristic scale of the weakly interacting sparticle masses may be as
large as $\widetilde{m}\sim 500$ GeV.

The range of $\tan\beta$ and of sparticle masses consistent with the observed
$\Delta a_{\mu}$ has implications on the DM properties. We will concentrate on
DM candidates with masses comparable to the weak scale, such that the thermal
DM relic density reproduces the observed value. In the MSSM, DM candidates in
this mass range can be realized if the lightest supersymmetric particle is an
almost-pure Bino, $m_{\chi}\simeq|M_{1}|$.

For the moderate-to-large values of $\tan\beta$ required to explain the
$(g_{\mu}-2)$ anomaly, the SIDD amplitude for the scattering of DM with nuclei
($N$) is proportional to

${\mathcal{M}}_{p}^{\rm SI}\propto\frac{v}{\mu^{2}}\left[2\frac{(M_{1}+\mu\sin
2\beta)}{m_{h}^{2}}-\frac{\mu\cos 2\beta}{m_{H}^{2}}\tan\beta\right],$ (7)

where $m_{h}$ and $m_{H}$ are the masses of the SM-like and the new heavy
neutral Higgs boson, respectively. We see that the SIDD amplitude depends in a
crucial way on the sizes and signs of $M_{1}$ and $\mu$. There are two options
to lower the SIDD amplitude: For large values of $|\mu|$, the Higgsino
components of the DM candidate become small and the SIDD amplitude is
suppressed. Alternatively, the light and heavy CP-even Higgs contributions
(first and second terms inside the brackets in Eq. (7)) may interfere
destructively, leading to a suppression of the SIDD amplitude. The latter
option is particularly interesting since it allows $|\mu|$ to remain of the
order of the electroweak scale; see, for example Ref. Drees and Ghaffari
(2021) for a recent discussion of naturalness and the connection with direct
detection bounds.

Regarding the first term in Eq. (7), if $M_{1}\simeq-\mu\sin 2\beta$, the
contributions of the Higgsino-up and the Higgsino-down admixtures to the
$(\chi\chi h)$ interaction cancel. The second term is the contribution to the
$(\chi N\to\chi N)$ amplitude arising from the $t$-channel exchange of the
non-SM-like heavy Higgs boson $H$. The generalized blind spot condition for
the SIDD cross section of a Bino-like DM candidate is then Huang and Wagner
(2014)

$\frac{2\left(M_{1}+\mu\sin
2\beta\right)}{m_{h}^{2}}\approx\frac{\mu\tan\beta\cos 2\beta}{m_{H}^{2}}\;.$
(8)

If the condition in Eq. (8) is satisfied, the amplitudes mediated by $h$ and
by $H$ exchange interfere destructively, suppressing the SIDD cross section; a
property that also holds at the one-loop level Han  _et al._ (2019). In
general, even if one is not in the proximity of the blind spot solution, if
the neutralino is mostly Bino-like, for a given value of $|\mu|$ and $M_{1}$,
the cross section is suppressed (enhanced) if $\mu$ and $M_{1}$ have opposite
(the same) sign.777Note that
$\cos(2\beta)=(1-\tan^{2}\beta)/(1+\tan^{2}\beta)\simeq-1$ for moderate-to-
large values of $\tan\beta$.

The mass of the heavy Higgs boson plays an important role in the blind-spot
cancellation. In the presence of light electroweakinos, the current LHC bounds
on $m_{H}$ coming from searches for heavy Higgs bosons decaying into tau-
leptons Aaboud  _et al._ (2018a); Sirunyan  _et al._ (2018a, 2020a); Aad  _et
al._ (2020a) can be approximated by

$m_{H}\gtrsim 250\,{\rm GeV}\times\sqrt{\tan\beta}\sim
2~{}m_{h}\sqrt{\tan\beta}~{}.$ (9)

For values of $m_{H}$ close to this bound, the SIDD amplitude is proportional
to

${\mathcal{M}}_{p}^{\rm
SI}\propto\frac{M_{1}v}{\mu^{2}}\left[1+\frac{\mu}{2M_{1}}\left(\frac{4}{\tan\beta}+\frac{1}{4}\right)\right].$
(10)

To exemplify the relevance of the relative sign and size of $\mu$ and $M_{1}$,
consider ${\mathcal{M}}_{p}^{\rm SI}$ for $\tan\beta=16$. As a reference value
for the SIDD amplitude, let us set $\mu\simeq-M_{1}$. Keeping $M_{1}$ fixed,
but increasing the value of $\left|\mu\right|$ to $\mu\simeq-2M_{1}$, the
value of ${\mathcal{M}}_{p}^{\rm SI}$ becomes a factor of $\approx 1/6$
smaller. Let us compare this to the situation for which $\mu$ and $M_{1}$ have
the same sign. First, we can note that for $\mu=M_{1}$, the SIDD amplitude is
almost a factor 2 larger than for $\mu=-M_{1}$. Furthermore, in order to
obtain a reduction of ${\mathcal{M}}_{p}^{\rm SI}$ by a factor of $1/6$, one
would have to raise the value of $\left|\mu\right|$ from $\mu\sim M_{1}$ to
$\mu\sim 4M_{1}$. This exemplifies that obtaining SIDD cross sections
compatible with experimental limits either requires $(\mu M_{1})<0$ (blind
spot solution) or, to compensate for a positive sign of this product, one must
sufficiently enhance the ratio $\mu/M_{1}$ (large-$\mu$ solution).

The spin dependent (SD) interactions are instead dominated by $Z$-exchange,
and can only be suppressed by lowering the Higgsino component of the lightest
neutralino. At moderate or large values of $\tan\beta$, the amplitude for SD
interactions is proportional to Carena  _et al._ (2018)

${\cal{M}}^{\rm SD}\propto\left(\frac{v}{\mu}\right)^{2}\cos 2\beta\;.$ (11)

Comparison with the results from direct detection experiments Behnke  _et al._
(2017); Fu  _et al._ (2017); Aprile  _et al._ (2019a); Amole  _et al._ (2019)
leads to an approximate bound on $\mu$,

$|\mu|\gtrsim 300\,{\rm GeV}\;,$ (12)

with a mild dependence on $M_{1}$.

Figure 1: The colored shades show approximate regions in the $\mu$–$M_{1}$
parameter plane allowed by current DM direct detection constraints on the
spin-dependent WIMP-proton, spin-dependent WIMP-neutron, and spin-independent
WIMP-nucleon cross section for $\tan\beta=20$ and values of the slepton, Higgs
and Wino mass parameters as indicated in the plot. In the gray areas bounded
by the dashed black lines we find a MSSM contribution $\Delta a_{\mu}=(25.1\pm
5.9)\times 10^{-10}$, explaining the value observed by the Fermilab and
Brookhaven Muon (g-2) experiments. The dash-dotted purple lines indicate
constraints arising from tau-leptons $+{\rm
missing~{}transverse~{}energy}(+{\rm jet})$ searches at the LHC, applicable if
the mass of the lightest stau is approximately in the middle of the lightest
chargino and neutralino masses Sirunyan  _et al._ (2020b).  Figure 2: Same as
Fig. 1 but for $\tan\beta=30$ and $m_{H}=1500$ GeV.

To summarize this discussion, we show the qualitative behavior of the direct
detection cross sections in Figs. 1 and 2 in the $M_{1}$–$\mu$ plane. We use
approximate analytic expressions for the cross sections and set the masses of
the heavy Higgs boson and $\tan\beta$ to characteristic values. The values of
$M_{2}$ and the slepton masses have been chosen to lead to a compressed
spectrum, alleviating constraints from slepton and chargino searches at the
LHC, see, for example, Refs. Aad  _et al._ (2014a, 2020b); Sirunyan  _et al._
(2020c); Aad  _et al._ (2020c, d); Aaboud  _et al._ (2018b); Sirunyan  _et
al._ (2018b); Aaboud  _et al._ (2018c, d, 2019a); Collaboration (2019a); Aad
_et al._ (2021a). The regions shaded in the different colors denote the region
allowed by current direct detection constraints on the SD-proton Behnke  _et
al._ (2017); Amole  _et al._ (2019), SD-neutron Fu  _et al._ (2017); Aprile
_et al._ (2019a), and SI Angloher  _et al._ (2016); Agnes  _et al._ (2018);
Aprile  _et al._ (2018, 2019b) scattering cross section. We see that whereas
the SD constraints provide an approximately symmetric lower bound on $\mu$,
due to the SIDD constraints, the values of $|\mu|$ need to be significantly
larger for positive $\mu\times M_{1}$ than for negative $\mu\times M_{1}$. We
show the region where the MSSM contribution explains the $(g_{\mu}-2)$ anomaly
in Figs. 1 and 2 with the gray shade bounded by the dashed black line. The
shape of the region preferred by $\Delta a_{\mu}$ may be understood from the
interplay between the Bino- and Wino-mediated contributions. For large values
of $|\mu|$, the Bino contribution tends to be the most relevant one. If one
considers positive values of $\mu\times M_{1}$, the Bino-smuon loop gives a
positive contribution to $\Delta a_{\mu}$ which can account for the
$(g_{\mu}-2)$ anomaly for sufficiently large values of $\tan\beta$. However,
for smaller values of $|\mu|$ and negative values of $\mu\times M_{1}$, as
required by the blind spot solution, the Bino contribution tends to be
subdominant and neither has the sign nor the magnitude to account for the
$(g_{\mu}-2)$ anomaly. If anything, depending on the sign of $M_{1}\times
M_{2}$, it will partially cancel the Wino contribution to $\Delta a_{\mu}$.
For smaller values of $|\mu|$, an explanation for the $(g_{\mu}-2)$ anomaly
requires a Wino-mediated contribution enabled by $\mu\times M_{2}>0$ and
moderate values of $M_{2}$.

For the LHC constraints indicated in Figs. 1 and 2 ($\left|M_{1}\right|\gtrsim
240\,$GeV, shown with the purple dash-dotted line) we have assumed that the
lightest stau is the next-to-lightest supersymmetric particle, with a mass
such that the proper relic density is obtained by co-annihilation of the
lightest stau with the lightest neutralino. In such a case, the Wino-like
chargino and neutralino have sizable branching ratios into staus, increasing
the stau production rate. In order to estimate the LHC limits, we use a recent
analysis Sirunyan  _et al._ (2020b) searching for tau-lepton final states,
which assumed that the mass gap between the lightest chargino and neutralino
is 50 GeV and the lightest stau mass lies in the middle of the lightest
chargino and neutralino masses, which is close to the situation found under
our assumptions. This shows that the LHC is already putting strong constraints
on the realization of this scenario. Note that we chose the Wino- ($M_{2}$)
and the first and second generation slepton ($M_{L}^{1,2}$, $M_{R}^{1,2}$)
mass parameters to be approximately degenerate ($M_{2}\approx
M_{L}^{1,2}\approx M_{R}^{1,2}\approx\left|M_{1}\right|+50\,$GeV) such that
current LHC limits for direct slepton searches are avoided for slepton masses
above $\sim 200\,$GeV Aad  _et al._ (2014a, 2020b); Sirunyan  _et al._
(2020c); Aad  _et al._ (2020c, d); Aaboud  _et al._ (2018b); Sirunyan  _et
al._ (2018b); Aaboud  _et al._ (2018c, d, 2019a); Collaboration (2019a); Aad
_et al._ (2021a).

Additional constraints from LHC searches with charged leptons in the final
states can arise from production of the Higgsino-like neutralino and chargino
states. These states decay into gauge and Higgs bosons and lighter charginos
and neutralinos. If $\mu$ is large compared to $M_{2}$, Higgsino production
and their decays can lead to relevant signals at the LHC despite Higgsinos
having much smaller production cross section that Winos, because for such a
choice of parameters, the Higgsino-decays lead to final states with much
harder leptons than the leptons arising from Wino or slepton production. We
have checked a number of example points from the preferred regions of Figs. 1
and 2 (i.e., where all direct detection constraints are satisfied, where
$\left|M_{1}\right|\gtrsim 240\,$GeV, avoiding the LHC constrain indicated in
the figures, and where the $\Delta a_{\mu}$ contribution explains the observed
value) using checkmate2 to check that no additional LHC constraints arise from
slepton and electroweakino (including Higgsino) production.

###  III.2 Benchmark Points Explaining $\boldsymbol{\Delta a_{\mu}}$, Dark
Matter and Avoiding LHC Constraints

For a Bino-like DM candidate with mass in the few-hundred GeV range, the
observed relic density can be realized via thermal production through
different mechanisms, such as co-annihilation with sleptons or charginos Ellis
_et al._ (1998, 2000); Buckley  _et al._ (2013); Han  _et al._ (2013); Cabrera
_et al._ (2016); Baker and Thamm (2018), $t$-channel annihilation via light
left-right mixed staus Pierce  _et al._ (2013) or smuons Fukushima  _et al._
(2014), or resonant $s$-channel annihilation Han  _et al._ (2013); Cabrera
_et al._ (2016). In Table 1 we present a few benchmark scenarios which
simultaneously accommodate the ($g_{\mu}-2$) anomaly and a viable DM
candidate. All of them are consistent with the observed relic density, the
observed value of $\Delta a_{\mu}$, and satisfy LHC constraints as well as
constraints from direct detection. For all benchmark points, we set the
parameters in the squark and gluino sectors such that experimental bounds are
satisfied and that the observed mass of the SM-like Higgs boson is reproduced.
In general, the supersymmetric partners of the color-charged particles must
have masses of the order of a few TeV to satisfy current experimental bounds
(see, for instance, Refs. Vami (2019); Aad  _et al._ (2021b, c, d)). Note that
in some of our benchmark scenarios, the hierarchy between the gluino and the
weak gaugino masses is larger than the hierarchy induced by the running of the
gaugino masses from (approximately) universal values at the Grand Unification
scale. While a Grand Unified Theory (GUT) is theoretically attractive, we do
not know if any GUT is realized in nature. The symmetries of the low energy
theory do not impose any constraint on the hierarchy between the gluino and
the weak gaugino masses. Furthermore, even in GUT models, higher order
operators at the GUT scale can lead to departures from universal gaugino
masses Martin (2009). A somewhat related point is that the
small-$\left|\mu\right|$ region we are interested in here tends to require
particular choices for the soft supersymmetry breaking masses of the Higgs
doublets. As is well known, the large stop masses required to reproduce a 125
GeV SM-like Higgs boson result in large radiative corrections to
$m_{H_{u}}^{2}$. Starting from universal soft parameters
$m_{H_{u}}^{2}=m_{H_{d}}^{2}$ at high energy scales, $m_{H_{u}}^{2}$ is driven
to large negative values at the electroweak scale, while $m_{H_{d}}^{2}$
receives much smaller radiative corrections. The correct electroweak symmetry
breaking (and, in particular, the correct mass of the electroweak gauge
bosons) is then achieved for $-m_{H_{u}}^{2}\approx\left|\mu\right|^{2}\gg
m_{H_{d}}^{2}$ at the electroweak scale. We leave a dedicated investigation of
the required values of the soft parameters to achieve the correct electroweak
symmetry breaking pattern for future work, however, let us note that
(radiative) electroweak symmetry breaking as well as the correct mass of the
electroweak gauge bosons are straightforward to achieve regardless of the
value of $|\mu|$ if one allows for different values of the soft supersymmetry
breaking parameters $m_{H_{u}}^{2}$ and $m_{H_{d}}^{2}$ at high energy scales.

The constraints from Higgsino and Wino pair production depend on a careful
consideration of the production cross sections and decay branching ratios Liu
_et al._ (2020b, c). Here, we consider a compressed spectrum, for which the
electroweakino and slepton constraints are weakened. The results for the
spectrum, $\Delta a_{\mu}$, the relic density, as well as the SI and SD cross
sections have been obtained with Micromegas 5.2.7.a Belanger  _et al._ (2009);
Bélanger  _et al._ (2018, 2021). We use SUSY-HIT 1.5 Djouadi  _et al._ (2007)
to compute branching ratios relevant for checking the electroweakino and
slepton constraints. One problem in the analysis of the LHC limits is that, in
many cases, signals can be obtained from the chain decay of many different
electroweak particles, and therefore it is difficult to directly apply the
bounds from LHC analyses which are typically presented in terms of simplified
models. In order to solve this problem, we use checkmate2 Dercks  _et al._
(2017); Alwall  _et al._ (2014); Sjöstrand  _et al._ (2015); de Favereau  _et
al._ (2014), that uses Monte Carlo event generation to compare all production
and decay channels for the neutralinos, charginos and sleptons with the
current LHC analyses Aad  _et al._ (2013a, b, 2014b, 2014c, 2014d, 2014e,
2014f, 2014g, 2014h, 2014i, 2014j, 2015a, 2015b, 2015c, 2015d, 2015e, 2015f,
2015g); Collaboration (2012, 2013a, 2013b, 2013c, 2013d, 2015a); Chatrchyan
_et al._ (2013); Khachatryan  _et al._ (2015a, b, c, 2016); Aaboud  _et al._
(2016); Aad  _et al._ (2016); Aaboud  _et al._ (2017a, b, c, 2018e, d, 2018f,
2018g, 2018h, 2018i, 2019b); Aad  _et al._ (2019, 2020e, 2020f, 2020g, 2020h,
2020i, 2020j, 2021e, 2021f, 2021g); Collaboration (2015b, 2016a, 2016b, 2016c,
2016d, 2016e, 2016f, 2017, 2018a, 2019b, 2019c, 2020); Sirunyan  _et al._
(2018c). Although most of the relevant LHC analyses have been included in
checkmate2, a few of the most recent analyses are not yet implemented in this
code. In these cases, we check the compatibility of our points by using
conservative estimates of the particle contributions to the different search
signals, as explained in Appendix A.

The scenarios presented below correspond to different origins of the observed
DM relic density and should serve as a guidance for experimental probes of the
supersymmetric explanation of the muon $(g-2)$.

| BMSM | BMST | BMW | BMH
---|---|---|---|---
$M_{1}$ [GeV] | -352 | -258 | -274 | 63
$M_{2}$ [GeV] | 400 | 310 | 310 | 700
$\mu$ [GeV] | 690 | 475 | 500 | 470
$M_{L}^{1,2}$ [GeV] | 360 | 320 | 350 | 750
$M_{L}^{3}$ [GeV] | 500 | 320 | 350 | 750
$M_{R}^{1,2}$ [GeV] | 360 | 320 | 350 | 750
$M_{R}^{3}$ [GeV] | 500 | 320 | 350 | 750
$M_{A}$ [GeV] | 2000 | 1800 | 1600 | 3000
$\tan\beta$ | 60 | 40 | 35 | 65

| BMSM | BMST | BMW | BMH
---|---|---|---|---
$m_{\chi}$ [GeV] | 350.2 | 255.3 | 271.4 | 61.0 (124.9)
$m_{\tilde{\tau}_{1}}$ [GeV] | 414.4 | 264.2 | 305.3 | 709.5
$m_{\tilde{\mu}_{1}}$ [GeV] | 362.7 | 323.0 | 352.8 | 751.3
$m_{\tilde{\nu}_{\tau}}$ [GeV] | 496.0 | 313.7 | 344.2 | 747.3
$m_{\tilde{\nu}_{\mu}}$ [GeV] | 354.4 | 313.7 | 344.2 | 747.3
$m_{\chi^{\pm}_{1}}$ [GeV] | 392.3 | 296.2 | 297.9 | 469.6
$\Delta a_{\mu}$ [$10^{-9}$] | 2.10 | 2.89 | 2.35 | 1.93
$\Omega_{\rm DM}h^{2}$ | 0.121 | 0.116 | 0.124 | 0.121
$\sigma_{p}^{\rm SI}$ [$10^{-10}\,$pb] | 0.645 | 1.58 | 1.42 | 0.315
$\sigma_{p}^{\rm SD}$ [$10^{-6}\,$pb] | 1.03 | 5.11 | 4.23 | 3.01
$\sigma_{n}^{\rm SI}$ [$10^{-10}\,$pb] | 0.632 | 1.57 | 1.41 | 0.330
$\sigma_{n}^{\rm SD}$ [$10^{-6}\,$pb] | 0.882 | 4.10 | 3.42 | 2.34

Table 1: Values of the MSSM parameters, mass spectrum and quantities relevant
for DM and $(g_{\mu}-2)$ for the case of Bino-like DM co-annihilating with
light sleptons (BMSM), co-annihilating with a light stau (BMST), co-
annihilating with a Wino (BMW) and resonant $s$-channel annihilation via the
SM-like Higgs boson (BMH). For BMH we also provide the mass of the SM-like
Higgs boson $m_{h}$ between brackets.

  * •

BMSM: A DM production scenario closely related to the relatively low masses of
the muon (neutrino) superpartners required to address the $a_{\mu}$-anomaly is
co-annihilation of the lightest neutralino with the light slepton states. The
benchmark BMSM gives a representation of this possibility, where we set the
masses of the tau-lepton superpartners to be larger than those of the first
and second generation sleptons. Since multiple production channels contribute
to final states containing leptons at the LHC, current searches strongly
constrain the presence of light electroweak interacting particles in this
scenario. In order to be compatible with $\Delta a_{\mu}$, DM phenomenology
and LHC searches, BMSM features the largest values of $|\mu|$ and $\tan\beta$
of the benchmark points presented in this article.

  * •

BMST: A similar solution to BMSM is associated with the co-annihilation of a
light stau with the lightest neutralino. For universal soft slepton masses,
this happens naturally at large values of $\tan\beta$, where the lightest stau
is pushed to masses lower than those of the sneutrinos. BMST gives a
representative example of this possibility.

  * •

BMW: The lightest neutralino may co-annihilate with the lightest chargino. The
benchmark BMW represents such a possibility.

  * •

BMH: The lightest neutralino can acquire the proper relic density via resonant
$s$-channel annihilation via the SM-like Higgs boson. BMH represents such a
possibility.

Although the mechanisms controlling the relic density are different for the
different benchmark points, they share many characteristics. They feature
masses of weakly interacting sparticles masses lower than about 500 GeV and
values of $\tan\beta$ of the order of a few 10’s, leading to values of $\Delta
a_{\mu}$ in the desired range. Apart from BMH, which we will discuss further
below, all benchmark points in Table 1 have negative values of $\mu\times
M_{1}$ and positive values of $\mu\times M_{2}$.

The observed relic density for a Bino-like DM candidate may also be obtained
via resonant $s$-channel annihilation via the heavy Higgs boson $A$ and $H$.
However, for the values of $\tan\beta$ necessary to enhance $\Delta a_{\mu}$,
LHC bounds on the heavy Higgs bosons become very strong, implying a heavy
spectrum. Using the bound on $m_{H}$ provided in Eq. (9), the approximate
expression for $\Delta a_{\mu}$ in Eq. (6), and assuming that all the weakly
interacting sparticles have masses close to $m_{H}/2$, the maximal value for
$\Delta a_{\mu}$ that may be obtained is

$\Delta a_{\mu}\simeq 10^{-9}\tan\beta\frac{4}{m_{H}^{2}}(100\,{\rm
GeV})^{2}\lesssim 7\times 10^{-10},$ (13)

which is a factor of a few smaller than the observed anomaly. Therefore, we
shall not discuss this particular solution further.

Resonant $s$-channel annihilation via the $Z$-boson presents similar
characteristics to resonant annihilation mediated by the SM-like Higgs, $h$.
Thus, we present only an example of the latter case here, BMH. For such small
values of $M_{1}\simeq m_{h}/2=62.5$ GeV, values of $|\mu|\sim 500$ GeV may
lead to the desired suppression of the SIDD cross section for either sign of
$\mu$. This follows, for instance, from Eq. (7), where we also observe that
for positive values of $\mu\times M1$, values of $m_{H}$ significantly larger
than the current experimental bounds are preferred. Note that, for BMH, we
chose the sleptons and the Winos to be heavy to avoid the bounds from the LHC.
Hence, obtaining the proper value of $\Delta a_{\mu}$ requires relatively
large values of $\tan\beta$.

###  III.3 Future Prospects

The benchmark points presented above are compatible with current experimental
limits, but will be tested in the near future in several ways.

First, all four benchmark points will be probed by the next generation of
direct detection experiments: The SIDD cross sections of all four benchmark
points are within the projected sensitivities of the LZ and XENONnT
experiments Akerib  _et al._ (2020); Aprile  _et al._ (2020). More generally,
for $\mu\times M_{1}<0$, and for fixed values of $M_{1}$, $\mu$ and
$\tan\beta$, the smallest possible value of the SIDD cross section is
associated with the smallest allowed value of the heavy Higgs mass, see Eq.
(7). For masses $200\,{\rm GeV}\lesssim|M_{1}|\lesssim 500$ GeV, a hierarchy
$1\lesssim|\mu/M_{1}|\lesssim 3$, and $\tan\beta\gtrsim 20$, compatible with
collider physics, muon $(g-2)$, and Dark Matter relic density constraints, the
smallest possible SIDD cross section is (see Eq. (10))

$\sigma_{p}^{\rm SI}>\mathcal{O}\left(10^{-10}\right)\,{\rm
pb}\times\left(\frac{M_{1}}{250\,{\rm
GeV}}\right)^{2}\times\left(\frac{500\,{\rm GeV}}{\mu}\right)^{4}\;.$ (14)

The LZ and XENONnT experiments will probe cross sections as small as
$\sigma_{p}^{\rm SI}\sim\mathcal{O}\left(10^{-12}\right)\,$pb for
$\left|M_{1}\right|\sim 40\,$GeV, growing to $\sigma_{p}^{\rm
SI}\sim\mathcal{O}\left(10^{-11}\right)\,$pb for $\left|M_{1}\right|\sim
500\,$GeV, implying full coverage of this representative region of parameters.

Furthermore, the spin-dependent WIMP-neutron cross sections can be probed by
LZ and XENONnT, while the next generation of the PICO experiment will probe
the spin-dependent WIMP-proton cross sections Amole  _et al._ (2015). From Eq.
(11) we can see that the spin-dependent WIMP-nucleon cross sections are

$\sigma_{n}^{\rm SD}\sim\sigma_{p}^{\rm
SD}>\mathcal{O}\left(10^{-6}\right)\,{\rm pb}\times\left(\frac{500\,{\rm
GeV}}{\mu}\right)^{4}\;,$ (15)

with a mild dependence on $M_{1}$. The future sensitivities of LZ/XENONnT on
$\sigma_{n}^{\rm SI}$ move from a few times $10^{-7}\,$pb for
$\left|M_{1}\right|\sim 100\,$GeV to $\sim 10^{-6}\,$pb for
$\left|M_{1}\right|\sim 500\,$GeV, while PICO-500 will probe $\sigma_{p}^{\rm
SD}\sim 10^{-6}\,$pb for $\left|M_{1}\right|\sim 100\,$GeV and
$\sigma_{p}^{\rm SD}\sim 5\times 10^{-6}\,$pb for $\left|M_{1}\right|\sim
500\,$GeV. Hence, these experiments will probe the region of parameter space
where $\left|\mu\right|\lesssim 500\,$GeV. In particular, LZ, XENONnT and
PICO-500 will probe the spin-dependent cross sections of the benchmark points
BMST, BMW, BMH, while BMSM has spin-dependent interactions smaller than the
projected sensitivities of these experiments.

Second, for all benchmark points with $M_{1}\times\mu<0$ (BMSM, BMST, and
BMW), the SIDD cross section is suppressed below current experimental limits
due to the destructive interference between the amplitudes mediated by the SM-
like and the heavy Higgs bosons discussed above. For this suppression to be
effective, the masses of the non-SM-like Higgs bosons must be low enough to
within the reach (see, for example, Ref. Bahl  _et al._ (2020)) of future runs
of the LHC: The high-luminosity LHC will be sensitive to Higgs bosons with
masses of about a factor 1.5 larger than current exclusion limits (keeping all
other parameters, in particular $\tan\beta$, fixed). From the expression of
the SIDD cross section, Eq. (7) we see that increasing $m_{H}\to 1.5\ m_{H}$
corresponds to a factor 2-3 increase of the SIDD cross section. Such SIDD
cross sections would be in conflict with current experimental constraints, or
conversely, values of the heavy Higgs mass allowed by current direct detection
bounds will be efficiently probed by the high-luminosity LHC. For BMSH, on the
other hand, the SIDD cross section is suppressed by a large hierarchy between
the Higgsino and Bino mass parameters, $|\mu|\gg|M_{1}|$. Such “large
$\left|\mu\right|$” solutions to suppressing the SIDD cross sections allow for
heavy Higgs masses beyond the projected reach of the high-luminosity LHC.

Last but not least, our benchmark scenarios are also testable in searches for
electroweakly interacting particles at future runs of the LHC, see, for
example, Refs. Collaboration (2018b, c). We note that some of these
projections have already been surpassed by innovative searches with current
LHC data, like those presented in Ref. Aad  _et al._ (2021h), further
bolstering the prospects of probing our benchmark points and similar scenarios
in the upcoming runs of the LHC. The extrapolation of these conclusions to the
whole region of parameters analyzed in this article should be the object of an
independent dedicated study, that we plan to perform but is beyond the scope
of the current article. Let us also emphasize that future lepton colliders
play an important role to probe sleptons and charginos, especially for
(semi-)compressed spectra, see Refs. Freitas  _et al._ (2004); Berggren
(2013); Fujii  _et al._ (2015); Collaboration (2018d); Dong  _et al._ (2018);
Franceschini  _et al._ (2018); Cid Vidal  _et al._ (2019); Ellis  _et al._
(2019); Baer  _et al._ (2020); Habermehl  _et al._ (2020); Berggren (2020);
Baum  _et al._ (2020); Natalia  _et al._ (2021).

##  IV Summary and Conclusions

A wide range of possible extensions of the Standard Model (SM) can lead to an
explanation of the value of $\Delta a_{\mu}$ measured at the Fermilab and
Brookhaven experiments. While arguably the simplest explanation is the
addition of a scalar particle, one can also rely on new gauge bosons, vector-
like fermions or leptoquark models. The leptoquark (or R-parity violating
supersymmetry) solution seems to be interesting since it can accommodate not
only the values of $\Delta a_{\mu}$, but can also lead to an explanation of
the flavor anomalies, although at the prize of a delicate choice of the
couplings of the leptoquarks.

In this work, we explore a solution based on the (R-parity conserving) Minimal
Supersymmetric extension of the SM, in which, although one cannot address the
flavor anomalies, one can find solutions leading to a compelling DM
explanation. In particular, we discuss the conditions that are required to be
consistent with the observed $\Delta a_{\mu}$, existing direct dark matter
(DM) detection constraints, and the bounds from the LHC on new Higgs bosons
and supersymmetric particles. We look for solution in which direct DM
detection constraints are fulfilled by a partial cancellation of the light and
heavy CP-even Higgs mediated contributions which significantly differ from
previous studies relying on very heavy Higgs and Higgsino particles. This
cancellation requires negative values of $\mu\times M_{1}$. Since the observed
value of $a_{\mu}$ is larger than the SM prediction, the Bino contribution to
$a_{\mu}$, which is proportional to $\mu\times M_{1}$, must be subdominant.
This can only be realized for small-to-moderate values of $|\mu|$. We present
corresponding benchmark scenarios associated with different DM production
mechanisms to achieve the observed relic density, including co-annihilation
with sleptons, or resonant $s$-channel annihilation mediated by the SM-like
Higgs or $Z$ bosons.

The corresponding spectra have a number of interesting consequences: 1) The
relatively small values of the Higgsino mass parameter lead to a more natural
model, in terms of the electroweak hierarchy problem. 2) The DM candidates
have spin-independent and spin-dependent direct detection cross sections which
can be probed in the next generation of direct detection experiments. 3)
Explaining the anomalous magnetic moment of the muon for negative values of
$\mu\times M_{1}$ requires light sleptons and electroweakinos, which should be
probed at run 3 of the LHC, the HL-LHC, or, ultimately, at future lepton
colliders. 4) The suppression of the direct detection cross section is only
possible for relatively light non-SM-like Higgs bosons in the MSSM, which can
be probed at run 3 of the LHC and the HL-LHC.

###### Acknowledgements.

SB is supported in part by NSF Grant PHY-2014215, DOE HEP QuantISED award
#100495, and the Gordon and Betty Moore Foundation Grant GBMF7946. Fermilab is
operated by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359
with the U.S. Department of Energy. The work of CW at the University of
Chicago has been also supported by the DOE grant DE-SC0013642. Work at ANL is
supported in part by the U.S. Department of Energy (DOE), Div. of HEP,
Contract DE-AC02-06CH11357. This work was supported in part by the DOE under
Task TeV of contract DE-FGO2-96-ER40956. NRS is supported by U.S. Department
of Energy under Contract No.DE-SC0021497. This work was performed in part at
Aspen Center for Physics, which is supported by National Science Foundation
grant PHY-1607611.

##  Appendix A LHC constraints from chargino and slepton searches

In this appendix, we discuss the constraints from chargino and slepton
searches on our benchmark points presented in Table 1. The most severe
chargino constraints tend to stem from production of the lightest chargino
($\widetilde{\chi}_{1}^{\pm}$) and the next-to-lightest neutralino
($\widetilde{\chi}_{2}^{0}$) at the LHC,
$pp\to\widetilde{\chi}_{1}^{\pm}\widetilde{\chi}_{2}^{0}$. Note that for all
of our benchmark points, the lightest neutralino is Bino-like, while
$\widetilde{\chi}_{2}^{0}$ is Wino-like (for BMSM, BMST and BMW) or Higgsino-
like (for BMH) depending on the hierarchy of $\left|\mu\right|$ and
$\left|M_{2}\right|$. Hence, $\widetilde{\chi}_{2}^{0}$ and
$\widetilde{\chi}_{1}^{\pm}$ will typically be mass degenerate. All the
benchmark points presented in this article fulfill the current LHC constraints
Aad  _et al._ (2013a, b, 2014b, 2014c, 2014d, 2014e, 2014f, 2014g, 2014h,
2014i, 2014j, 2015a, 2015b, 2015c, 2015d, 2015e, 2015f, 2015g); Collaboration
(2012, 2013a, 2013b, 2013c, 2013d, 2015a); Chatrchyan  _et al._ (2013);
Khachatryan  _et al._ (2015a, b, c, 2016); Aaboud  _et al._ (2016); Aad  _et
al._ (2016); Aaboud  _et al._ (2017a, b, c, 2018e, d, 2018f, 2018g, 2018h,
2018i, 2019b); Aad  _et al._ (2019, 2020e, 2020f, 2020g, 2020h, 2020i, 2020j,
2021e, 2021f, 2021g); Collaboration (2015b, 2016a, 2016b, 2016c, 2016d, 2016e,
2016f, 2017, 2018a, 2019b, 2019c, 2020); Sirunyan  _et al._ (2018c)
implemented in checkmate2 Dercks  _et al._ (2017); Alwall  _et al._ (2014);
Sjöstrand  _et al._ (2015); de Favereau  _et al._ (2014). We also check
compatibility with very recent LHC searches which are not yet implemented in
checkmate2 by using conservative estimates of the particle contribution to
these search channels.

In order to gain a physical intuition of how the benchmark points avoid the
LHC constraints, we provide a brief discussion of their properties.

BMSM: The lightest neutralino has a mass
$m_{\widetilde{\chi}_{1}^{0}}=350\,$GeV, and the Wino-like next-to-lightest
neutralino and lightest chargino have masses
$m_{\widetilde{\chi}_{2}^{0}}=m_{\widetilde{\chi}_{1}^{\pm}}=392\,$GeV. We
have computed the $\widetilde{\chi}_{2}^{0}+\widetilde{\chi}_{1}^{\pm}$
production cross section at the 13 TeV LHC with MadGraph5_v3.1.1 Alwall  _et
al._ (2014), finding
$\sigma(pp\to\widetilde{\chi}_{2}^{0}+\widetilde{\chi}_{1}^{\pm})=0.08\,$pb .
Comparing this to the upper limit from Ref. Aad  _et al._ (2021a)
$\sigma(pp\to WZ+2\widetilde{\chi}_{1}^{0})\lesssim 0.6\,$pb at these masses
(this search is not yet implemented in checkmate2; we have taken the limit
from the supplementary material of Ref. Aad  _et al._ (2021a) accessible via
HEPdata or the CERN Document Server), we see that this benchmark point is not
constrained by this search even before taking into account that the
$\widetilde{\chi}_{2}^{0}$ and $\widetilde{\chi}_{1}^{\pm}$ decay branching
ratios into gauge bosons are small in this scenario. For this benchmark point,
however, the Wino- and Higgsino-like neutralinos and charginos can undergo
cascade decays involving the light sleptons, giving rise to potentially
detectable signatures in searches for charged leptons and missing energy at
the LHC. Due to BMSM’s mass spectrum, the production of the Wino-like
$\widetilde{\chi}_{2}^{0}$ and $\widetilde{\chi}_{1}^{\pm}$ gives rise to
relatively soft leptons. The most sensitive search corresponding to this final
state currently implemented in checkmate2 is Ref. Aad  _et al._ (2020h), for
which we find a signal strength of $r\sim 0.8$. The signal strength here is
defined as the ratio of the number of events predicted for the model point and
the observed limits on the number of events in the most constraining signal
region of any given LHC search. Production of the Higgsino-like
$\widetilde{\chi}_{3}^{0}$, $\widetilde{\chi}_{4}^{0}$, and
$\widetilde{\chi}_{2}^{\pm}$, on the other hand, leads to final states with
much harder charged leptons and larger missing transverse energy. The most
sensitive LHC search currently implemented in checkmate2 for such signatures
is Ref. Sirunyan  _et al._ (2018c), for which we find a signal strength of
$r\sim 0.8$. Regarding direct slepton searches, this benchmark point features
approximately mass degenerate left- and right-handed selectrons and smuons
with $m_{\widetilde{\ell}^{\pm}}=313\,$GeV. Such compressed spectra with
$m_{\widetilde{\ell}^{\pm}}-m_{\widetilde{\chi}_{1}^{0}}=10\,$GeV are not
constrained by current LHC searches for direct slepton production, see, for
example, Ref. Collaboration (2019a).

BMST: The lightest neutralino has mass
$m_{\widetilde{\chi}_{1}^{0}}=255\,$GeV, and the Wino-like next-to-lightest
neutralino and lightest chargino have masses
$m_{\widetilde{\chi}_{2}^{0}}=m_{\widetilde{\chi}_{1}^{\pm}}=296\,$GeV. Both
the next-to-lightest neutralino and the lightest chargino decay into staus for
this benchmark point, ${\rm
BR}(\widetilde{\chi}_{2}^{0}\to\widetilde{\tau}_{1}^{\pm}+\tau^{\mp})={\rm
BR}(\widetilde{\chi}_{1}^{\pm}\to\widetilde{\tau}_{1}^{\pm}+\nu_{\tau})=100\,\%$
from our SUSY-HIT results. The staus in turn decay into tau-leptons, ${\rm
BR}(\widetilde{\tau}^{\pm}\to\tau^{\pm}+\widetilde{\chi}_{1}^{0})=100\,\%$,
leading to tau-leptons + missing transverse energy final states from
$\widetilde{\chi}_{2}^{0}$ and $\widetilde{\chi}_{1}^{\pm}$ production at the
LHC. Although the corresponding searches are quite challenging, studies with
initial state radiation jets in a compressed region with chargino-neutralino
mass gap $m_{\widetilde{\chi}_{1}^{\pm}}-m_{\widetilde{\chi}_{1}^{0}}\approx
50\,$GeV and stau masses in the middle between the chargino and neutralino,
$(m_{\widetilde{\chi}_{1}^{\pm}}+m_{\widetilde{\chi}_{1}^{0}})/2=m_{\widetilde{\tau}}$,
constrain the chargino mass to $m_{\widetilde{\chi}_{1}^{\pm}}\gtrsim 290$ GeV
Sirunyan  _et al._ (2020b). We have arranged the spectrum of BMST such that
this bound is approximately applicable, and accordingly, we chose the masses
of the Wino-like next-to-lightest neutralino and the lightest chargino to be
larger than 290 GeV. Regarding the slepton searches, the selectrons and smuons
have masses $m_{\widetilde{\ell}^{\pm}}=323\,$GeV for BMST. Hence,
$m_{\widetilde{\ell}^{\pm}}-m_{\widetilde{\chi}_{1}^{0}}=68\,$GeV, which is
below the mass gaps excluded by current LHC searches Collaboration (2019a). We
note that out of the searches implemented in checkmate2, BMST has the largest
signal strength ($r\sim 0.3$) for the search in Ref. Aad  _et al._ (2020i).

BMW: The lightest neutralino has mass $m_{\widetilde{\chi}_{1}^{0}}=271\,$GeV,
and the Wino-like next-to-lightest neutralino and lightest chargino have
masses $m_{\widetilde{\chi}_{2}^{0}}=m_{\widetilde{\chi}_{1}^{\pm}}=298\,$GeV.
For the $\widetilde{\chi}_{2}^{0}+\widetilde{\chi}_{1}^{\pm}$ production cross
section, we find
$\sigma(pp\to\widetilde{\chi}_{2}^{0}+\widetilde{\chi}_{1}^{\pm})=0.26\,$pb.
While this point is not constrained by any of the analyses included in
checkmate2, it may be constrained by the recent bounds coming from the multi-
lepton final state analyses in Ref. Aad  _et al._ (2021a), which is not yet
implemented in checkmate2. Note that the dominant production mechanism of
charged lepton final states from the charginos and neutralinos in BMS is via
tau-leptons. In Ref. Aad  _et al._ (2021a), however, the limits are obtained
assuming decays of the charginos and neutralinos into gauge bosons and the
lightest neutralinos and hence the limits are not directly applicable to this
case. In order to make a conservative comparison to the upper limit
$\sigma(pp\to WZ+2\widetilde{\chi}_{1}^{0})\lesssim 0.9\,$pb Aad  _et al._
(2021a) at these masses, we can note that, including the dominant contribution
coming from $\tau$ lepton decays, the total leptonic branching ratio from
$\widetilde{\chi}_{2}^{0}+\widetilde{\chi}_{1}^{\pm}$ production888Here, we
define the leptonic branching ratio as the sum of the branching ratios of
$\widetilde{\chi}_{2}^{0}$/$\widetilde{\chi}_{1}^{\pm}$ involving
$Z$/$W^{\pm}$-bosons multiplied by their leptonic branching ratios and the
(stau-mediated) decays into tau-lepton(s) multiplied with the leptonic
branching ratio of the taus. for this benchmark points is 4.5 %, while Ref.
Aad  _et al._ (2021a) assumed ${\rm
BR}(\widetilde{\chi}_{2}^{0}\to\widetilde{\chi}_{1}^{0}Z)={\rm
BR}(\widetilde{\chi}_{1}^{\pm}\to\widetilde{\chi}_{1}^{0}W^{\pm})=100\,\%$,
corresponding to a total leptonic branching ratio of $1.9\,\%$, leading to an
estimate of the signal strength of $r=(0.26\,{\rm pb}/0.9\,{\rm
pb})\times(4.5\,\%/1.9\,\%)\sim 0.7$. Note that the true signal strength is
most likely significantly lower since leptons coming from tau-lepton decays
are softer than those coming from direct lepton production.

Regarding direct slepton searches, the lightest charged sleptons for this
benchmark point are the staus, $m_{\widetilde{\tau}_{1}}=305\,$GeV, followed
by the selectrons and smuons with $m_{\widetilde{\ell}^{\pm}}=353\,$GeV. Such
mass gaps, $m_{\widetilde{\ell}^{\pm}}-m_{\widetilde{\chi}_{1}^{0}}=82\,$GeV,
are not constrained by current LHC bounds even under the assumption of ${\rm
BR}(\widetilde{\ell}^{\pm}\to\ell^{\pm}+\chi_{1}^{0})=100\,\%$ for all four of
the left- and right-handed selectron and smuon states, with stau bounds being
even weaker. For this benchmark point, the left-handed charged sleptons decay
preferentially into charginos, ${\rm
BR}(\widetilde{\ell}_{L}^{\pm}\to\nu_{\ell}+\chi_{1}^{\pm})=53\,\%$, and have
sizeable branching ratios into the next-to-lightest (Wino-like) neutralino,
${\rm BR}(\widetilde{\ell}_{L}^{\pm}\to\ell^{\pm}+\chi_{2}^{0})=28\,\%$. The
reduced decay branching ratios into the lightest neutralino implies softer
spectra of visible decay products at the LHC and hence even weaker bounds.
Moreover, due to the compressed chargino and neutralino spectrum, no relevant
additional constraints emerge from the decay of the sleptons into the Wino-
like states.

We note that out of the searches implemented in checkmate2, BMW has the
largest signal strength ($r\sim 0.4$) for the search in Ref. Aad  _et al._
(2020i).

BMH: The lightest neutralino has mass $m_{\widetilde{\chi}_{1}^{0}}=61\,$GeV,
and the next-to-lightest neutralino and lightest chargino have masses
$m_{\widetilde{\chi}_{2}^{0}}=m_{\widetilde{\chi}_{1}^{\pm}}=470\,$GeV. Unlike
for all of the other benchmark points, $m_{\widetilde{\chi}_{2}^{0}}$ and
$m_{\widetilde{\chi}_{1}^{\pm}}$ are Higgsino-like, leading to a relatively
small $\widetilde{\chi}_{2}^{0}+\widetilde{\chi}_{1}^{\pm}$ production cross
section of
$\sigma(pp\to\widetilde{\chi}_{2}^{0}+\widetilde{\chi}_{1}^{\pm})=0.013\,$pb
at the 13 TeV LHC. This is significantly below the upper limit from Ref. Aad
_et al._ (2021a) at theses masses, $\sigma(pp\to
WZ+2\widetilde{\chi}_{1}^{0})\lesssim 0.02\,$pb, even before taking the
branching ratios of $\widetilde{\chi}_{2}^{0}$ and
$\widetilde{\chi}_{1}^{\pm}$ into account [${\rm
BR}(\widetilde{\chi}_{2}^{0}\to\widetilde{\chi}_{1}^{0}+h)=62\,\%$ and ${\rm
BR}(\widetilde{\chi}_{3}^{0}\to\widetilde{\chi}_{1}^{0}+h)=34\,\%$ for this
point]. The masses of the Wino-like state are
$m_{\widetilde{\chi}_{4}^{0}}=m_{\widetilde{\chi}_{2}^{\pm}}=745\,$GeV, beyond
the current limit on these states Aaboud  _et al._ (2019a, 2018i); Tumasyan
_et al._ (2021), even before accounting for the decay patterns of the heavy
Winos. The Wino-like states dominantly decay into the intermediate Higgsino-
like states,
$\widetilde{\chi}_{4}^{0}/\widetilde{\chi}_{2}^{\pm}\to\widetilde{\chi}_{3}^{0}/\widetilde{\chi}_{2}^{0}/\widetilde{\chi}_{1}^{\pm}+W^{\pm}/Z/h$.
Thus, production of Wino-like states at the LHC will mostly lead to cascade
decays with softer visible final states than if the Wino-like states would
directly decay into the lightest neutralino,
$\widetilde{\chi}_{4}^{0}/\widetilde{\chi}_{2}^{\pm}\to\widetilde{\chi}_{1}^{0}+W^{\pm}/Z/h$,
complicating experimental searches. Furthermore, let us stress that the bounds
on Wino production presented by the experimental collaborations assume that
the squarks are decoupled, and accordingly ignore the important $t$-channel
squark-mediated contributions to the Wino production cross section which can
lower the cross section by order one factors depending on the exact squark
masses Liu  _et al._ (2020c). These arguments apply to very recent searches
for Winos in hadronic final states Aad  _et al._ (2021h); Collaboration (2021)
that would rule out this scenario in the absence of cascade decays and the
$t$-channel squark contributions. Nonetheless, these impressive searches
clearly show the potential of the experimental collaborations to test the
regions of parameters represented by our scenarios in future runs of the LHC.
Regarding the slepton searches, the lightest charged sleptons for this
benchmark point are the staus, $m_{\widetilde{\tau}_{1}}=710\,$GeV, and the
selectrons and smuons have masses $m_{\widetilde{\ell}^{\pm}}=751\,$GeV,
beyond the reach of current LHC searches Collaboration (2019a). We note that
out of the searches implemented in checkmate2, BMH has the largest signal
strength ($r\sim 0.4$) for the search in Ref. Aad  _et al._ (2020i).

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