IP-BBSR-2019-12

A Minimal Model of Torsion Mediated Dark Matter

Basabendu<EMAIL_ADDRESS>Tapobroto
<EMAIL_ADDRESS>Debottam<EMAIL_ADDRESS>Debaprasad<EMAIL_ADDRESS>
a Department of Physics, Indian Institute of Technology Guwahati, Assam
781039, India
bInstitute of Physics, Bhubhaneswar, Odhisha- 751005, India & Homi Bhabha
National Institute, Training School Complex, Anushakti Nagar, Mumbai 400085,
India

Abstract

###### Abstract

We present a minimal model of fermionic dark matter (DM), where a singlet
Dirac fermion can interact with the Standard Model (SM) particles via the
torsion field of gravitational origin. In general, torsion can be realized as
an antisymmetric part of the affine connection associated with the spacetime
diffeomorphism symmetry and thus can be thought of as a massive axial vector
field. Because of its gravitational origin, the torsion field couples to all
the fermion fields including the DM with equal strength, which makes the model
quite predictive. The DM is naturally stable without any imposition of ad-hoc
symmetry e.g., $\mathcal{Z}_{2}$. Apart from producing the correct thermal
abundance, singlet fermion can easily evade the stringent bounds on the spin-
independent DM-nucleon direct detection cross-section due to its axial nature.
However, in the allowed parameter space, strong bounds can be placed on the
torsion mass and its couplings to fermions from the recent LHC searches.
Assuming a non universal torsion-DM and torsion-SM coupling, smaller values of
torsion masses may become allowed. In both cases we also study the reach of
spin-dependent direct detection searches of the DM.

##  I Introduction

The Standard Model (SM) of particle physics in its present form has been
extremely successful in accommodating fundamental interactions of nature.
However, the model is regarded as incomplete when it comes to explain
observations e.g., dark matter (DM) abundance or the matter-anti matter
asymmetry in the Universe. In “Beyond Standard Model” (BSM) physics, one may
explain most of these puzzles in terms of new particles and their couplings
with the SM particles. But null results from different accelerator/non-
accelerator based experiments to test these new particles have pushed their
masses above TeV scale.

The presence of DM in the universe has been established via a number of
successful experiments through its gravitational impact Kolb and Turner (1990)
(for reviews see e.g, Jungman  _et al._ (1996); Bertone  _et al._ (2005); Feng
(2010); Bertone and Hooper (2018); Lin (2019)), though its nature is yet to be
known. One of the most popular and accepted DM candidate is considered to be
WIMP (Weakly Interacting Massive Particle), obtained naturally in most of the
well motivated BSM scenarios like Supersymmetric (SUSY) theories or in the
extra-dimensional models. However, no evidence of this prototype DM candidate
has so far been observed in accelerator based experiments like the LHC or in
the direct detection (DD) experiments like XENON100 Aprile  _et al._ (2012),
LUX Akerib  _et al._ (2017), PandaX-II Cui  _et al._ (2017) or Xenon-1T Aprile
_et al._ (2018) which search for DM scatterings off nuclei in detectors. All
these searches have essentially put stringent constraints on the parameter
space of some of the extensions of the SM (for a review see Arcadi  _et al._
(2018a)) like $Z$-portal Arcadi  _et al._ (2015); Hamaguchi and Ishikawa
(2016); Escudero  _et al._ (2016); Kearney  _et al._ (2017); Ellis  _et al._
(2018), Higgs-portal Lebedev  _et al._ (2012); Djouadi  _et al._ (2012); Gross
_et al._ (2015); Casas  _et al._ (2017); Hoferichter  _et al._ (2017); Hardy
(2018); Arcadi  _et al._ (2019)111In the context of SUSY, lightest neutralino
DM can efficiently annihilate through Higgs states to accommodate the relic
abundance Drees and Nojiri (1993); Nath and Arnowitt (1993); Baer and Brhlik
(1998); Ellis  _et al._ (2003); Chattopadhyay and Das (2009); Chattopadhyay
_et al._ (2010); Das  _et al._ (2010)(for review see Jungman  _et al._ (1996);
Bertone  _et al._ (2005); Feng (2010)) $Z^{\prime}$-portal Mambrini (2010);
Dudas  _et al._ (2013); Alves  _et al._ (2014); Lebedev and Mambrini (2014);
Hooper (2015); Alves  _et al._ (2015a, b); Allanach  _et al._ (2016); Alves
_et al._ (2017), or pseudoscalar portals Berlin  _et al._ (2015); Baek  _et
al._ (2017); Bauer  _et al._ (2018). Among these models, $Z^{\prime}$-portal
with only axial coupling, or pseudoscalar portal models are somewhat preferred
for a fermionic DM as non observation of any signal, specially in the context
of spin-independent (SI) DM-nucleon scattering can be easily accommodated.
Driven by the similar quest, here we explore the possibility of DM-SM
interactions through the torsion portal which is of geometric origin, thus
completely independent of any representation of the grand unified or the SM
gauge group.

Due to its geometric origin, the interaction of torsion field with SM has some
interesting features. In general, torsion field can be identified as one of
the irreducible representations of the antisymmetric part of the affine
connection associated with the spacetime diffeomorphism symmetry. Because of
its anti-symmetric property, the torsion field behaves like a massive axial
vector field when it interacts with the SM fields. In a natural extension, one
may consider minimal coupling prescription Shapiro (2002) which reduces the
dimension of the parameter space of the theory under consideration222As a
matter of fact, superstring theories predict a non-minimal coupling of torsion
with the matter fields, such that the compactification of extra-dimensions
results in a general theory containing the modified version of the non-minimal
couplings at low energy scale.. In recent times, a torsion field with masses
around TeV scale has drawn attention in the context of low energy
phenomenology Belyaev and Shapiro (1998, 1999); de Berredo-Peixoto  _et al._
(2000); Das  _et al._ (2002); Mahanta and Raychaudhuri (2003); Belyaev  _et
al._ (2007, 2017); De Almeida  _et al._ (2018) (for a review see Shapiro
(2002)). One of this works also considers a possibility of having torsion as a
DM candidate Belyaev  _et al._ (2017) where the DMs are pair annihilated
through Higgs portal. Their analysis require vanishingly small coupling
between the torsion and matter field.

In the present work, we consider a minimal model of a fermionic DM in presence
of external propagating torsion field, where torsion plays the role of a
mediator between the dark sector and the visible sector. We ignore any
coupling between gauge fields and torsion fields Shapiro (2002) as it may
spoil gauge invariance. Since torsion appears in the covariant derivative, the
coupling constants with the DM or with the SM fields could be identical and
independent of their intrinsic properties such as mass, color, flavor, charge,
etc. This particular property makes our model economical though tightly
constrained from different experimental observations, particularly from the
LHC searches.333However the universality in coupling constant can be broken if
we consider a High scale model involving torsion. One may need to use
Renormalization Group (RG) equations to calculate the coupling constants at
the electroweak scale which may be different and may provide with rich
phenomenology. Assuming a minimal deviation, i.e., allowing a non equal
torsion-DM coupling compared to the SM fermions, we find that LHC constraints
can be easily accommodated. To this end we may note a few important points:

  * •

Stability of the DM does not require imposition of any ad-hoc symmetry (like
${\mathcal{Z}_{2}}$ or ${\mathcal{Z}_{3}}$ symmetry in general). Since there
is no renormalizable coupling between DM-SM, the decay of DM to SM particles
is automatically forbidden.

  * •

The presence of the axial coupling in the DM-SM interaction ensures spin-
dependent (SD) direct search cross-section. In this sense, stringent limits
from spin independent direct detection experiments on the DM scattering cross-
section can naturally be evaded.

  * •

The universal coupling $\eta$ does not require any fine tuning, and can assume
$\sim O(1)$ value which is compatible with the observed DM relic abundance. In
contrast,
$\eta\mbox{\raisebox{-2.58334pt}{~{}$\stackrel{{\scriptstyle<}}{{\sim}}$~{}}}10^{-20}$
has been assumed for torsion DM Belyaev  _et al._ (2017) scenario in order to
make it stable over the cosmological time scale. This forbids any chance to
produce such massive vector field at the LHC.

Our present paper is organized as follows. In Sec. II, we recapitulate the
essentials of the theory of torsion and their interactions with the fermion
fields. Our Sec. III is devoted to the discussion on the experimental and
theoretical constraints on the parameter space. In Sec. IV, we study the DM
phenomenology of our model with torsion as a mediator between the dark and SM
sector. In this analysis we particularly maintain the universality of the
torsion-fermion (this includes the fermionic DM as well) coupling parameter
$\eta$, and discuss the relevant bounds on the torsion mass and its couplings
to SM fermions. We further explore the prospects of DM direct detection via a
detailed scan over the relic density allowed parameter space. Direct
production of the heavy resonances at the LHC may put stringent limits on the
universal coupling, which can be evaded if one assumes torsion-DM coupling is
different than torsion-SM fermions couplings. It is expected, as the LHC
limits are mainly defined for latter interactions. A brief discussion along
with changes in DM phenomenology would be presented in Sec. V. Finally, we
summarize our key results in Sec. VI.

##  II  Origin of Torsion and its coupling with matter

We begin this section with a brief introduction to torsion. Here we closely
follow Refs.Shapiro (2002); Belyaev and Shapiro (1998, 1999); de Berredo-
Peixoto  _et al._ (2000); Belyaev  _et al._ (2007, 2017). In the general
theory of relativity, the covariant derivative of a vector field $A^{\nu}$ is
a tensor and is defined as:

$\displaystyle\nabla_{\mu}\,A^{\nu}=\partial_{\mu}\,A^{\nu}+\Gamma^{\nu}_{\,\,\mu\lambda}\,A^{\lambda},$
(1)

where the last term $\Gamma^{\lambda}_{\,\,\mu\nu}$ is known as the affine
connection. The simplest form of this affine connection in general relativity
is known as the Christoffel symbol and is based on the two following special
requirements

  * •

it is symmetric in nature,
$\Gamma^{\lambda}_{\,\,\mu\nu}=\Gamma^{\lambda}_{\,\,\nu\mu}$,

  * •

the metricity condition i.e., $\nabla_{\alpha}\,g_{\mu\nu}=0$,

where $g_{\mu\nu}$ is the metric tensor of the corresponding spacetime we are
working in. When the above two conditions are fulfilled we have an expression
of the Christoffel symbol as,

$\displaystyle\Gamma^{\lambda}_{\,\,\mu\nu}=\frac{1}{2}\,g^{\lambda\alpha}\,\big{(}\partial_{\mu}\,g_{\nu\alpha}+\partial_{\nu}\,g_{\mu\alpha}-\partial_{\alpha}\,g_{\mu\nu}\big{)}.$
(2)

However, in the Einstein-Cartan extension of general relativity, the said
conditions can be relaxed so that the affine connection can have an anti-
symmetric part defined as:

$\displaystyle{\tilde{\Gamma}}^{\lambda}_{\,\,\mu\nu}={\Gamma}^{\lambda}_{\,\,\mu\nu}+C^{\lambda}_{\,\,\mu\nu},$
(3)

where $C^{\lambda}_{\,\,\mu\nu}$ is a mixed tensor field. Thus, new degrees of
freedom are introduced into the system which cannot be fixed by the background
metric anymore. Therefore, even in the flat space limit, this field will
survive. A particular irreducible representation of this tensor field may play
the role for a torsion field.

For better understanding, we may define the general torsion tensor as

$\displaystyle
T^{\lambda}_{\,\,\mu\nu}~{}=~{}{\tilde{\Gamma}}^{\lambda}_{\,\,\mu\nu}-{\tilde{\Gamma}}^{\lambda}_{\,\,\nu\mu},$
(4)

which is antisymmetric in its lower indices. The above torsion tensor can be
re-written in terms of irreducible representation:

$\displaystyle
T_{\mu\nu\lambda}=\frac{1}{3}\,\big{(}T_{\nu}\,g_{\mu\lambda}-T_{\lambda}\,g_{\mu\nu}\big{)}-\frac{1}{6}\,\varepsilon_{\mu\nu\lambda\sigma}\,S^{\sigma}+q_{\mu\nu\lambda},$
(5)

where $S^{\lambda}=\varepsilon^{\mu\nu\alpha\lambda}\,T_{\mu\nu\alpha}$ is
torsion trace axial vector mode, $q_{\mu\nu\alpha}$ is trace free tensor mode
satisfying the conditions $q^{\mu}_{\,\,\nu\mu}=0$ and
$\varepsilon^{\mu\nu\alpha\sigma}\,q_{\mu\nu\alpha}=0$. The trace free vector
mode is defined as $T_{\mu}=T^{\alpha}_{\,\,\mu\alpha}$. Because of its
geometric nature and specific parity properties, different irreducible torsion
components will have specific type of couplings with the fermions. The general
action of a Dirac spinor coupled to torsion field reads:

$\displaystyle{\cal S}_{D}\,=\,\int
d^{4}x\,\left\\{\,i\bar{f}\gamma^{\mu}\big{(}\partial_{\mu}-ie\,A_{\mu}-i\eta_{1}\,\gamma^{5}\,S_{\mu}+i\eta_{2}\,T_{\mu}\big{)}f-m\,\bar{f}\,f\right\\},$
(6)

where $\eta_{1}$, $\eta_{2}$ are non-minimal coupling parameters. Simply from
the symmetry properties of various terms of the torsion fields, in the minimal
coupling scenario, these parameters may assume fixed values:
$\eta_{1}=\frac{1}{8}$ and $\eta_{2}=0$. However, if we consider dynamical
torsion interactions with the matter fields, $\eta_{1}$ and $\eta_{2}$ are
expected to change under the renormalization group (RG) flow (see e.g.,Shapiro
(2002) and references therein). For phenomenological purposes we choose them
as free parameters at low energy scale. Moreover, keeping the simplicity in
mind, we will set all the other irreducible components of the torsion field to
be zero except the traceless axial vector component $S_{\mu}$.

We consider massive fermion fields in the action mentioned above. In the
fermion mass limit $m=0$, the action enjoys an additional gauge symmetry
transformation related to torsion:

$\displaystyle f\to
f^{\prime}=f\,e^{\gamma_{5}\beta(x)},\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\bar{f}}\to{\bar{f}}^{\prime}={\bar{f}}\,e^{\gamma_{5}\beta(x)},\,\,\,\,\,\,\,\,\,\,\,\,\,\,S_{\mu}\to
S_{\mu}^{\prime}=S_{\mu}-{\eta}^{-1}\,\partial_{\mu}\beta(x)\,.$ (7)

Massive fermion terms may be explained if the said symmetry is softly broken.
This in turn will automatically generate the torsion mass term at the loop
level, and will be proportional the fermionic mass term. This fact prompts one
to consider a massive dynamical torsion field $S_{\mu}$ with the following
simplest action (see Shapiro (2002), for details)

$\displaystyle{\cal S}_{torsion}=\,\int
d^{4}x\,\Big{\\{}\,-\frac{1}{4}\,S_{\mu\nu}S^{\mu\nu}+\frac{1}{2}\,m_{s}^{2}\,S_{\mu}S^{\mu}\,\Big{\\}}\,,$
(8)

where $S_{\mu\nu}=\partial_{\mu}\,S_{\nu}-\partial_{\nu}\,S_{\mu}$ and $m_{s}$
is the torsion mass. The first term is the usual abelian kinetic term which
naturally satisfies the above gauge conditions in Eq.7.

Inclusion of a DM, specifically a SM-singlet Dirac fermion $\psi$, is possible
in a similar way since, by construction, the new state interacts with torsion
field in the same way like that of a SM fermion. Importantly, the singlet
Dirac state cannot have any renormalizable couplings to any SM particles. Thus
DM-SM interaction can be only realized through $S_{\mu}$ portal, which reads:

$\displaystyle{\cal S}_{DM}\,=\,\int
d^{4}x\,\left\\{\,i\bar{\psi}\gamma^{\mu}\big{(}\partial_{\mu}-i\eta\,\gamma^{5}\,S_{\mu}\big{)}\psi-
m_{\psi}\,\bar{\psi}\,\psi\right\\}.$ (9)

Finally, the relevant parts of the action which includes the new interactions
can be written as:

$\displaystyle{\cal S}_{New}$ $\displaystyle=$ $\displaystyle{\cal
S}_{torsion}+{\cal S}_{D}+{\cal S}_{DM}=\,\int
d^{4}x\,\Big{\\{}\,-\frac{1}{4}\,S_{\mu\nu}S^{\mu\nu}+\frac{1}{2}\,m^{2}_{s}\,S_{\mu}S^{\mu}\,\Big{\\}}$
(10) $\displaystyle+$ $\displaystyle\int
d^{4}x\,\big{[}\left\\{\,i\bar{f}\gamma^{\mu}\big{(}\partial_{\mu}-i\eta_{1}\,\gamma^{5}\,S_{\mu}\big{)}f\right\\}+\left\\{\,i\bar{\psi}\gamma^{\mu}\big{(}\partial_{\mu}-i\eta\,\gamma^{5}\,S_{\mu}\big{)}\psi-
m_{\psi}\,\bar{\psi}\,\psi\right\\}\big{]}.$

In our analysis we will mainly consider two scenarios:

  * •

Universal coupling scenario ($\eta=\eta_{1}$): Both SM and DM can couple to
the torsion field with a universal coupling $\eta$, which is a natural outcome
of torsion portal. The model, thus naturally becomes very predictive as only a
minimal set of input parameters are required to compute the DM phenomenology.

  * •

Non-universal coupling (NU) scenario ($\eta\neq\eta_{1}$) : Though the
universal coupling scenario is natural, but can be highly constrained in terms
of the available parameter space. We shall illustrate this in detail in the
subsequent sections. The situation may be considerably improved if one relaxes
the universality condition, and set only the DM-torsion coupling $\eta$ as
free parameter. We shall also briefly discuss the outcome of such a scenario
where $\eta$ is different from torsion-SM coupling $\eta_{1}$.

##  III Constraints on the model parameters

In this section we will discuss the constraints arising from various
experimental and theoretical bounds. Under the universal coupling scenario
(this is always be the case unless otherwise stated), our model can be
characterized by only three free parameters, namely:

$\displaystyle\\{m_{s},m_{\psi},\eta\\},$ (11)

which respectively are the torsion mass, the DM mass and the torsion-matter
universal coupling. The constraints on the DM mass come from the requirement
of obtaining observed relic abundance and satisfying the direct detection
bounds. We discuss these in detail in the next section. Here we will mainly
summarize the experimental constraints on the torsion mass and its coupling
focusing on the recent LHC results.

  * •

Before the advent of LHC, torsion mass and couplings were not entirely free
parameters, rather significantly constrained from different analysis,
including accelerator searches. Assuming the torsion mass to be considerably
heavier than other particles present in the action (Eq. 10), torsion
interaction with fermion fields can be effectively recasted in terms of four-
fermion contact interaction Belyaev and Shapiro (1998, 1999).

$\displaystyle\mathcal{L}_{int}=-\frac{\eta^{2}}{m_{s}^{2}}(\bar{f}\gamma^{\mu}\gamma^{5}f)(\bar{f}\gamma^{\mu}\gamma^{5}f).$
(12)

Now, such contact interactions can be constrained from several experiments as
elaborated in Belyaev and Shapiro (1998, 1999). From global analysis one can
put a limit on the contact axial $eeqq$ interaction, which then translates
into a bound on the torsion mass and its coupling Belyaev and Shapiro (1998);
Shapiro (2002):

$\displaystyle\frac{m_{s}}{\eta}>1.7~{}{\text{TeV}}.$ (13)

For a light torsion with mass below 1 TeV, accelerator searches from LEP or
TEVATRON collaborations became important. Analysis of the data constraints the
coupling $\eta$ in the range
$0.02\mbox{\raisebox{-2.58334pt}{~{}$\stackrel{{\scriptstyle<}}{{\sim}}$~{}}}\eta\mbox{\raisebox{-2.58334pt}{~{}$\stackrel{{\scriptstyle<}}{{\sim}}$~{}}}0.1$
Belyaev and Shapiro (1998, 1999), depending upon torsion mass. A combined
analysis from LEP-2 and TEVATRON data excludes torsion mass upto 600 GeV
unless the torsion-fermion coupling drops below 0.1. Also, for torsion
coupling ($\sim 0.3-0.6$), the torsion mass is pushed well above TeV scale.

In recent years both ATLAS and CMS collaborations have searched for high-mass
resonances through their direct production and subsequent decays to dilepton,
dijet or mono-$X$ final states Aad  _et al._ (2012); Sirunyan  _et al._
(2017a, b); Aaboud  _et al._ (2018, 2017); Sirunyan  _et al._ (2018);
Goldouzian (2018); White (2019); Aad  _et al._ (2019); Sirunyan  _et al._
(2019). Dilepton final states are of special interests as they provide the
most stringent bounds, though other searches namely dijet and mono-jet limits
Sirunyan  _et al._ (2017a); Aaboud  _et al._ (2017, 2018); Sirunyan  _et al._
(2017b) may also be of importance Drees and Zhang (2019). Based on the
dilepton searches, that includes the recent CMS analysis at 13 TeV center of
mass (CM) energy with integrated luminosity 140 fb-1 CMS (2019), a bound on
$m_{s}\mbox{\raisebox{-2.58334pt}{~{}$\stackrel{{\scriptstyle>}}{{\sim}}$~{}}}4.5$
TeV can be placed assuming fermion coupling to torsion is different than the
respective couplings with the $Z$ boson. In fact, a similar limit for
$\eta\sim 0.1$ was earlier presented in Belyaev  _et al._ (2007) where authors
analyzed LHC discovery reach in the $\eta-m_{s}$ plane at 14 TeV CM energy
with integrated luminosity of 100 fb-1. However, they considered only SM
fermions in their analysis. Torsion specific analysis has also been performed
by the ATLAS collaboration at $\sqrt{s}=7$ TeV, for the same $e^{-}$ and
$\mu^{-}$ final states and the bound is much relaxed compared to the present
one. One may also refer to the latest ATLAS $t\bar{t}$ production results to
set limits on torsion parameters considering torsion-top quark couplings
$O(1)$ De Almeida  _et al._ (2018). Here a much weaker constraint
$m_{s}\mbox{\raisebox{-2.58334pt}{~{}$\stackrel{{\scriptstyle>}}{{\sim}}$~{}}}3$
TeV can be observed.

In this analysis, our primary objective is to study the interplay of different
input parameters in DM phenomenology considering universal coupling scenario
in particular. So, on the very outset, we do not impose any collider
constraint on the parameter space. After investigating the DM phenomenology,
limits are placed to depict the net available parameter space. In applying the
LHC bounds, one should keep in mind that such limits on the torsion mass have
been derived assuming narrow width approximation (NWA) while studying the
distribution of the invariant mass of the decay products. However, this
approximation does not hold good in the limit $\Gamma_{s}/m_{s}\geq 0.5$
Abdallah  _et al._ (2014); Alves  _et al._ (2015a), where perturbative limits
break down. In this regime direct application of LHC limits may not be
completely justified. Keeping this in mind we indicate the part of the
parameter space where indeed one may have $\Gamma_{s}/m_{s}\geq 0.5$.

  * •

The propagating axial vector field $S_{\mu}$ may also contribute to the muon
anomalous magnetic moment ($a_{\mu}$) at one loop, though the contribution is
found to be negative:
$a^{S}_{\mu}(\rm{m_{s}>>m_{\mu}})\sim-\frac{\eta^{2}}{4\pi^{2}}\frac{5m_{\mu}^{2}}{3m_{s}^{2}}$
Das  _et al._ (2002); Queiroz and Shepherd (2014). From the recent
measurements of $a_{\mu}$ Tanabashi and Hagiwara (2018) one may observe the
discrepancy in $\Delta a_{\mu}$ is positive: $\left(268\pm 76.27\right)\times
10^{-11}$ which is an excess of $\sim 3.51\sigma$ compared to the SM
expectation. Thus, only a positive contribution from the BSM physics can match
the difference which is not favoured in the present case.

  * •

A theoretical constraint can arise from the requirement of the preservation of
unitarity of the theory due to quantum corrections Belyaev and Shapiro (1999,
1998); de Berredo-Peixoto  _et al._ (2000); Belyaev  _et al._ (2017). This
demands:

$\displaystyle\frac{m_{s}^{2}}{\eta}>>m_{f}^{2},$ (14)

where $m_{f}$ is the mass of any fermion field that couples to the torsion.
This constraint, however, is satisfied for our entire parameter space.

##  IV Dark Matter through Torsion portal: WIMP scenario

As explained earlier, we consider the DM phenomenology of a singlet Dirac
fermion $\psi$ in presence of a torsion field where interactions between the
SM fermions or the DM with the torsion is parameterized by a universal
coupling parameter $\eta$. We remind that the DM is stable over the time scale
of the age of the universe without requiring any ad-hoc symmetry. This is also
true in the parameter space where DM is heavier than the torsion field.

Figure 1: Pair-annihilation of the DM $\psi$ to the SM particles via torsion-
portal (left) and to the torsion pairs via $t$-channel process (right).

In the WIMP paradigm, the DM particles are assumed to be in thermal
equilibrium with the SM in the early universe. In the present set-up, the
thermal equilibrium is maintained through the torsion mediated $s$-channel and
$t$-channel processes as shown in Fig:1. At some later time when the rate of
interaction becomes smaller than the Hubble rate of expansion the DM falls out
of the equilibrium, and this defines freeze out Kolb and Turner (1990);
Jungman  _et al._ (1996) of the DM:

$\displaystyle\Gamma_{A}<H,$ (15)

where $\Gamma_{A}$ is the interaction rate of the DM pair annihilations and it
is given by:

$\displaystyle\Gamma_{A}=n_{\psi}\langle\sigma v\rangle,$ (16)

where $n_{\psi}$ is the DM number density and $H$ is the Hubble constant. The
present number density of the DM is decided by its number density at the time
of freeze-out, which can be computed by using the standard prescription of
Boltzmann equation Gondolo and Gelmini (1991) for a single component DM
candidate. Note that, the relic abundance of the DM is solely controlled by
the $2\to 2$ annihilation channels, where the annihilation cross-sections are
dependent on the parameters in Eq. 11. In the present case, dominant non-
relativistic form of the thermally-averaged annihilation cross-section for the
$s$ channel ($\psi\bar{\psi}\to\bar{f}f$) and $t$ channel ($\psi\bar{\psi}\to
S_{\mu}S_{\mu}$ when kinematically allowed) processes can be expressed as:

$\displaystyle\langle\sigma
v\rangle_{\bar{f}f}\simeq\sum_{f}N_{c}^{f}\frac{\eta^{4}}{2\pi}\frac{m_{f}^{2}}{m_{s}^{4}}\sqrt{1-\frac{m_{f}^{2}}{m_{\psi}^{2}}}\left(\frac{\left(m_{s}^{2}-4m_{\psi}^{2}\right)^{2}}{\left(m_{s}^{2}-4m_{\psi}^{2}\right)^{2}+\Gamma_{s}^{2}m_{s}^{2}}\right)+\mathcal{O}\left(v^{2}\right),$
(17)

$\displaystyle\langle\sigma v\rangle_{ss}\simeq\frac{\eta^{4}}{16\pi
m_{\psi}^{2}}\left(1-\frac{m_{s}^{2}}{m_{\psi}^{2}}\right)^{3/2}\left(1-\frac{m_{s}^{2}}{2m_{\psi}^{2}}\right)^{-2}+\mathcal{O}\left(v^{2}\right),$
(18)

where $v$ is the relative velocity of the annihilating DM pair, $m_{f}$
corresponds to the mass of all SM fermions that couple to the torsion and
$N^{f}_{c}$ refers the number of colors for final state fermions (= 3(1) for
quarks (leptons)). $\Gamma_{s}$ is the total decay width of the torsion to all
fermions given by (in the rest frame of the torsion):

$\displaystyle\Gamma_{s}=\frac{N_{c}^{f}\eta^{2}m_{s}}{12\pi}\left[1-4(\frac{m_{f}}{m_{s}})^{2}\right]^{\frac{3}{2}}+\eta^{2}\frac{m_{s}}{12\pi}\left[1-4(\frac{m_{\psi}}{m_{s}})^{2}\right]^{\frac{3}{2}},$
(19)

where the last term is relevant for $m_{s}>2m_{\psi}$. The cross-section has
been computed for $s=4m_{\psi}^{2}$ ($s$ is the CM energy).
$\mathcal{O}\left(v^{2}\right)$ refers to the velocity dependent $p$-wave
terms, which are sub-dominant in presence of $s$-wave. It is intriguing to
note that for $s$-channel process, assuming $m_{\psi}\sim m_{s}>>m_{f}$, the
dominant $v^{2}$ contribution involves
$\frac{\eta^{4}m_{\psi}^{2}}{\left(4m_{\psi}^{2}-m_{s}^{2}\right)^{2}}$, which
carries the resonance effect Berlin  _et al._ (2014); Arcadi  _et al._
(2018b). Interestingly the leading order term in Eq.17 may not be sensitive to
Breit-Wigner type narrow width resonance. The $p$-wave terms can also lead to
important contributions for the relic density calculations, specially when the
DM is lighter than top quark. Then the contribution of the leading order i.e,
the velocity independent term in the $s$ channel (see Eq.17) is extremely
suppressed $\sim\frac{m_{f}^{2}}{m_{s}^{4}}$, where $f$ refers any fermions
except the top quark. The $p$-wave term becomes significant as it goes as
$\sim\frac{1}{m^{2}_{\psi}}$ for $m_{\psi}\sim m_{s}$. Similarly $t$-channel
contributions, free from $v^{2}$ suppression, are also important for
$m_{s}\mbox{\raisebox{-2.58334pt}{~{}$\stackrel{{\scriptstyle<}}{{\sim}}$~{}}}m_{\psi}$.
For numerical calculations, we use the general expressions for $\langle\sigma
v\rangle$. This has been done by implementing the model in CalcHEP Belyaev
_et al._ (2013) and then the model output has been fed to public code
MicrOmegas Belanger  _et al._ (2007, 2009, 2014); Barducci  _et al._ (2018).
The valid parameter space should comply with the observed relic abundance data
Jarosik  _et al._ (2011); Aghanim  _et al._ (2018):

$\displaystyle\Omega_{\psi}^{obs}h^{2}=0.1199\pm 0.0022.$ (20)

Figure 2: Left: Variation of relic abundance of $\psi$ with $m_{s}$ for
different choices of the couplings $\eta:\\{0.1,0.3,0.5,0.7\\}$, are shown in
red, green, blue and black respectively. Here the DM mass is set at 500 GeV.
Right: Same for a fixed torsion mass $m_{s}=3~{}\rm TeV$ for three different
choices of $\eta:\\{0.3,0.5,0.7\\}$ respectively in red, green and blue. In
each figure the black dashed line corresponds to the PLANCK observed relic
abundance.

For illustration we first fixed the DM mass $m_{\psi}=500$ GeV and plotted the
variation of relic abundance with $m_{s}$ for four different choices of the
$\eta:\\{0.1,0.3,0.5,0.7\\}$ respectively which are shown by red, green, blue
and black curves on the left hand side (LHS) of Fig. 2. As expected, in each
case we see a resonance at $m_{\psi}\simeq\frac{m_{s}}{2}$ where the observed
relic abundance Aghanim  _et al._ (2018) is satisfied. The relic abundance
increases for smaller $\eta$, and at resonance it is difficult to produce an
acceptable $\Omega h^{2}$ for $\eta<0.1$. The resonance regions may become
flat for a larger $\eta$ when the Breit-Wigner decay width $\Gamma_{s}$ is
large for a given torsion mass $m_{s}$. The situation is different for a
lighter $S_{\mu}$ hence for a lighter DM, as both $s$ and $t$ channel
processes may be relevant here. As an example, for our chosen DM mass,
$\langle\sigma v\rangle_{\bar{\psi}\psi\to\bar{t}t}$ can be sufficiently large
even for a moderate value of $\eta$, where right relic abundance or even a
under abundant region can be observed. Here annihilations are facilitated to
all possible final states. The viable parameter space may also include the
region where
$m_{s}\mbox{\raisebox{-2.58334pt}{~{}$\stackrel{{\scriptstyle<}}{{\sim}}$~{}}}m_{\psi}$.
Next, we fix the torsion mass $m_{s}=3~{}\rm TeV$ and choose three different
couplings $\eta:\\{0.3,0.5,0.7\\}$ as shown on the right hand side (RHS) of
Fig. 2 through red, green and blue curves respectively. Here the relic density
is satisfied only at the resonance region while larger width can be observed
for a larger $\eta$.

Figure 3: Left: Relic density satisfied parameter space in $m_{\psi}$-$\eta$
plane, where different colours show torsion masses in the range
$\\{10-100\\}~{}\rm GeV$ in red, $\\{100-500\\}~{}\rm GeV$ in green,
$\\{500-1000\\}~{}\rm GeV$ in blue and $\\{1000-5000\\}~{}\rm GeV$ in black.
Right: Parameter space satisfying relic abundance in $m_{\psi}$-$m_{s}$ plane,
where the colour codes corresponding to different values of the coupling
$\eta$: {0.01-0.05} in red, {0.05-0.1} in green, {0.1-0.5} in blue and
{0.5-1.0} in black. In each plot the orange region corresponds to
$\Gamma_{s}/m_{s}\geq 0.5$ limit.

Now, in order to explore the parameter space satisfying the observed relic
abundance (via Eq.20) we scan over the parameters listed in Eq.11 in the
following range:

$\displaystyle
m_{\psi}:\\{10-5000~{}{\text{GeV}}\\};m_{s}:\\{10-10^{4}~{}{\text{GeV}}\\};\eta:\\{0.01-1.0\\}.$
(21)

The corresponding parameter space is shown in Fig. 3. On the LHS we show the
parameter space satisfying relic abundance in $m_{\psi}$-$\eta$ plane for
different ranges of the torsion mass $m_{s}$ indicated by different colours.
Here both DM and the torsion can take small mass values $\sim$10 GeV in
different parts of the parameter space specified by $\eta$, where relic
abundance can be satisfied either through annihilations to $f\bar{f}$ or via
$t$ channel process.

In the RHS of Fig. 3, relic density allowed region is shown in
$m_{\psi}-m_{s}$ plane where different colours correspond to different values
of the coupling $\eta$. We see here, smaller values of $\eta$ (in red) favours
$m_{s}\mbox{\raisebox{-2.58334pt}{~{}$\stackrel{{\scriptstyle<}}{{\sim}}$~{}}}100~{}\rm
GeV$, while a larger $m_{s}$ is required for larger $\eta$ (in blue and black)
to satisfy the relic abundance. This is expected as the rise in the torsion
mass reduces $\langle\sigma v\rangle$. Hence larger $\eta$ is expected to make
relic density compatible with the observed abundance. In each plot highlighted
orange regions depict $\Gamma_{s}/m_{s}\geq 0.5$, where LHC constraints may
not be directly applied to disqualify a parameter space point as discussed in
Sec. III.

###  IV.1 Indirect detection of the Dark Matter

The relic density allowed parameter space for the DM can also be probed via
different indirect detection experiments that look for annihilation and/or
decay products of the DM that produces SM particles. Residual DM
annihiliations may still occur at late universe where the relative velocity
takes a value $\sim 10^{-3}$. Among these final states, photon and neutrinos,
being neutral and stable, can reach the detectors without getting much
affected by the intermediate space. In case of high energy $\gamma$-rays, one
considers hadronization and production of neutral pions and its subsequent
decays to two photons with varying energies and also final state radiations.
Detectable gamma-rays can be tested at the instruments like the Fermi Large
Area Telescope (LAT)444http://fermi.gsfc.nasa.gov in space or the ground-based
Cherenkov telescopes like MAGIC 555https://magic.mpp.mpg.de/. Since no signal
has so far been observed, we would like to see the status of the relic
abundance satisfying parameter space from the bounds arising from a joint
global analysis from the MAGIC and the Fermi-LAT which search for gamma-ray
signals from DM annihilation in dwarf spheroidal galaxies (dSphs) Ahnen  _et
al._ (2016).

Figure 4: DM annihilations into $\mu^{+}\mu^{-}$ (top left),
$\tau^{+}\tau^{-}$ (top right) and $b\bar{b}$ (bottom) compared against
indirect detection bounds from the combined observation of MAGIC and Fermi-
LAT. In each figure regions corresponding different couplings are shown in red
{0.01-0.1}, green {0.1-0.5}, blue {0.5-1.0}. The thick orange curve represents
bound from non observations of $\gamma$-rays corresponding to particular final
state.

The differential gamma ray flux produced due to WIMP-like DM annihilation in a
given region of the sky $\left(\Delta\Omega\right)$ and observed at the Earth
reads Cirelli  _et al._ (2011); Ibarra  _et al._ (2013); Gaskins (2016);
Slatyer (2018); Hooper (2019):

$\displaystyle\frac{d\Phi}{dE}\left(\Delta\Omega\right)=\frac{1}{4\pi}\frac{\langle\sigma
v\rangle J\left(\Delta\Omega\right)}{2m_{\psi}^{2}}\frac{dN}{dE},$ (22)

where $dN/dE$ is the average gamma-ray spectrum per annihilation and,

$\displaystyle
J\left(\Delta\Omega\right)=\int_{\Delta\Omega}d\Omega^{{}^{\prime}}\int_{LOS}dl\rho^{2}\left(l,\Omega^{{}^{\prime}}\right),$
(23)

is the well-known $J$-factor with $\rho$ being the DM density profile, and the
integrals running over $\Delta\Omega$ and the line-of-sight (LOS) through the
DM distribution. Here we consider a Navarro-Frenk-While (NFW) profile for the
computation Navarro  _et al._ (2010). Since the DM $\psi$ can annihilate into
charged fermion final states: $\mu^{+}\mu^{-},\tau^{+}\tau^{-},b\bar{b}$ etc,
all such final states can give rise to gamma rays. Thus, one can constrain the
DM annihilations to different charged final states from non observations of
gamma rays. In Fig. 4 we bounded our relic density allowed parameter space
from MAGIC and Fermi-LAT data Ahnen  _et al._ (2016) for muon (top left),
tauon (top right) and $b$-quark (bottom) final states. We see that for our
choice of the coupling $\eta$ and torsion mass $m_{s}$, the parameter space is
completely unconstrained. This is not completely unexpected, as the velocity
dependent contributions in Eq. 17 which may show the resonance behaviour is
largely suppressed at late times.

###  IV.2 Direct Detection of the Dark Matter

Since the DM can annihilate into the SM particles via $s$-channel mediation of
the torsion producing right relic abundance, the same interaction can give
rise to DM-nucleon elastic scattering. This will lead to the direct detection
of the DM in present and future experiments. In the present case, driven by a
purely axial coupling, the DM-nucleon scattering would give rise to only spin-
dependent interaction. Note that, the SD cross section involves the spin
content of the nucleus i.e., there is a $J(J+1)$ enhancement from the nuclear
spin $J$. This is highly welcome since a weak-scale Dirac dark matter
accompanied by vector like interactions with the some mediator $M_{\mu}$ i.e,
$(\psi\gamma^{\mu}\psi)M_{\mu}$ is either ruled out or very tightly
constrained. The relevant Feynman graph for the spin-dependent scattering is
depicted in Fig. 5. For such a process, the spin-dependent direct search
cross-section is proportional to Belanger  _et al._ (2009):

$\displaystyle\sigma_{SD}\sim\frac{\eta^{4}}{m_{s}^{4}}\frac{16}{\pi}\mu^{2}\left(\frac{J+1}{J}\right)\left(\xi_{p}\langle
s_{p}\rangle+\xi_{n}\langle s_{n}\rangle\right)^{2},$ (24)

Figure 5: DM-nucleon elastic scattering mediated by the torsion.

Figure 6: Top Left: Relic density satisfied parameter space in direct search
plane, where different colours correspond to different choices of $\eta$:
{0.01-0.1} in red, {0.1-0.5} in green, {0.5-0.8} in blue and {0.8-1.0} in
black. The black dashed line shows the exclusion limit from spin-dependent
direct search for neutron-DM scattering. Similarly, the magenta line refers
future LZ sensitivity. Top Right: Same but here the exclusion limits refer
spin-dependent direct search for proton-DM scattering. In both plots orange
region corresponds to $\Gamma_{s}/m_{s}\geq 0.5$. Bottom Left: Net parameter
space satisfying relic abundance and spin-dependent direct search in
$m_{\psi}$-$m_{s}$ plane, where the top panel shows the variation of the
coupling $\eta$. Bottom Right: Net parameter space within the perturbative
limit
$\Gamma_{s}/m_{s}\mbox{\raisebox{-2.58334pt}{~{}$\stackrel{{\scriptstyle<}}{{\sim}}$~{}}}0.5$.

where $\mu=\frac{m_{\psi}m_{N}}{m_{\psi}+m_{N}}$ is the DM-nucleon reduced
mass with $N\in\\{n,p\\}$ i.e, neutron or proton. $\langle s_{p,n}\rangle$ are
obtained from nuclear calculations or from simple nuclear models, such as the
odd-group model Belanger  _et al._ (2009) and they are estimated to be
$\approx 0.5$ for a nuclei with an odd number of protons or neutrons, while
zero for nuclei with even number of protons or neutrons. $\xi_{p,n}$ are
obtained from lattice data Airapetian  _et al._ (2007); Ageev  _et al._
(2007); Belanger  _et al._ (2009). We consider the most recent limit on spin-
dependent WIMP-proton or WIMP-neutron cross-section from the run-3 data of the
LUX experiment Akerib  _et al._ (2016) and constrain our relic density
satisfying parameter space. In both the plots DM and torsion mass span over a
few TeV mass scale and $\eta:\\{0.01-1.0\\}$. From the top panel in Fig. 6 we
see that the relic density allowed parameter space lies well below the present
bound from LUX. This is due to the additional suppression coming from the
torsion mass in Eq. 24. Similarly, smaller coupling results in even smaller
cross-section. In the plots, we also show projected sensitivity of the next
generation LUX-ZEPLIN (LZ) direct detection experiments which may in future
probe some parts of the parameter space Akerib  _et al._ (2018). Clearly, SD
neutron-DM scattering offers a better prospect which can be attributed to even
number of protons present in the xenon-based detectors like LUX Schnee (2011).
Finally in the bottom left panel of Fig. 6 we show the net parameter space
satisfying relic abundance and direct search exclusion limit. We find that if
we strictly follow
$\Gamma_{s}/m_{s}\mbox{\raisebox{-2.58334pt}{~{}$\stackrel{{\scriptstyle<}}{{\sim}}$~{}}}0.5$,
the parameter space is shrinked a bit in the high torsion mass region with
$\eta\mbox{\raisebox{-2.58334pt}{~{}$\stackrel{{\scriptstyle<}}{{\sim}}$~{}}}0.8$.

##  V Limits on the universal coupling scenario and non universal torsion
matter coupling

Based on our discussion in Sec. III, it is quite expected that our minimal
coupling scenario would be tightly bounded from the direct production of
torsion and its subsequent decays to charged leptons: $e,\mu$. For a
qualitative understanding, we calculated effective cross-section for dilepton
productions $\sigma(pp\to S_{\mu})\times Br(S_{\mu}\to l^{+}l^{-})$ ($l\in
e,\mu$) and plotted the variation with $m_{s}$ for different sets of torsion-
matter couplings. Torsion production cross-section $\sigma(pp\to S_{\mu})$ at
the LHC has been calculated using MadGraph-v2.6.6 Alwall  _et al._ (2011) at a
CM energy of 13TeV with PDF choice CTEQ6L Alwall  _et al._ (2011, 2014);
Placakyte (2011); Ball  _et al._ (2013, 2015) while the branching fraction has
been determined from Eq. 19. The result is presented in Fig. 7, where the
exclusion limit is from Goldouzian (2018). The bound would be relaxed if one
considers finite width of the mediator White (2019). Torsion mass in the range
$m_{s}\sim\\{3-5\\}$ TeV is typically allowed modulo the universal coupling
$\eta$ is small $\eta:\\{0.01-0.1\\}$. However, in this range, satisfying the
PLANCK data will be extremely difficult as one requires relatively light
$S_{\mu}$ to compensate the coupling suppression in order to produce relic
abundance in the right ballpark. In other words, the typical $m_{s}$ value
which can satisfy the observed abundance for a given $\eta$ in the aforesaid
range has been mostly ruled out by the LHC results. On the higher side of the
coupling $\eta\sim 1$, one mostly finds $\frac{\Gamma}{m_{s}}>0.5$, thus LHC
results cannot be used to rule out any parameter space point.

Figure 7: $\sigma(pp\to S_{\mu})\times Br(S_{\mu}\to l^{+}l^{-})$ ($l\in
e,\mu$) vs. $m_{s}$ in the universal coupling scenario.  Figure 8: Left:
$\sigma(pp\to S_{\mu})\times Br(S_{\mu}\to l^{+}l^{-})$ ($l\in e,\mu$) vs.
$m_{s}$ in the non minimal coupling scenario. Here torsion-SM fermion
couplings are set at $\eta_{1}=0.01$. In the minimal coupling case, the bound
on $m_{s}$ reads as $m_{s}>3$ TeV. But with a different torsion-DM coupling,
torsion’s branching fraction to leptonic final states will be reduced further
to allow even lighter $S_{\mu}$. Assuming $\eta=0.05$, one finds $m_{s}>1.4$
TeV using LHC dilepton searches. Right: Allowed parameter space complied with
PLANCK data in $m_{s}-m_{\psi}$ plane in the NU coupling scenario. Figure 9:
Left:DM-neutron SD scattering that satisfies PLANCK data on relic abundance
where limits from LUX and proposed LUX-ZEPLIN have been shown. Right: Same but
for DM-proton SD scattering.

The situation can be improved a lot when one assumes a different value of the
coupling parameter for DM-torsion ($\eta$) relative to the SM fermions-torsion
which is parameterized by $\eta_{1}$, as introduced in Eq. 10. As discussed,
we refer this scenario as non-universal torsion matter coupling (NU) model. In
this case, we always keep $\eta_{1}\sim 0.01$ which allows $m_{s}>3$ TeV to
comply with the LHC dilepton results, while $\eta$ has been varied within
$\\{0.1-1\\}$. With this choice of torsion-matter couplings, $Br(S_{\mu}\to
l^{+}l^{-})$ would be further reduced, thanks to rise in invisible branching
fraction of the torsion. In the left side of Fig. 9 we place bounds on $m_{s}$
for different choices of $\eta=0.05,0.5,1$ keeping $\eta_{1}=0.01$. For
example for $\eta=0.05$, the bounds on $m_{s}$ reduced to $m_{s}>1.4$ TeV from
dilepton searches. Further rise in $\eta$ would make torsion mass even less
constraint due to increase in the invisible branching fractions. Similar to
dilepton searches, dijet bounds would not be of much significance. The
relevant limits may come from the monojet searches which is induced by the
axial vector couplings Aaboud  _et al._ (2018); Sirunyan  _et al._ (2017b).
According to an analysis in Sirunyan  _et al._ (2017b), which corresponds to
13 TeV CM energy and 12.9 $fb^{-1}$ integrated luminosity, torsion mass upto
$\sim$ 2 TeV has been ruled out if one assumes a coupling strength of 0.25
between the mediators and the SM fermions while a coupling strength of 1.0
between the mediators and the DM particles. Similar limit $m_{s}>1.55$TeV can
be set corresponding to an integrated luminosity of 36.1$fb^{-1}$ Aaboud  _et
al._ (2018). For our chosen input parameters $\eta_{1}$ and $\eta$, the limits
should get much less stringent due to strong suppression in the production
cross-section for $S_{\mu}$. In the RHS of Fig.9, we depict the allowed
parameter points by PLANCK data. Acceptable relic abundance can be obtained
along two regions: (i) resonance annihilations, which pass almost through the
middle of the plane and (ii) $s$ and $t$-channel processes when the torsion is
lighter than the DM. A detailed study of mono-jet searches on the parameter
space may further constrain the parameter space. However, this is beyond the
scope of this paper.

Finally, we also explore the parameter space via the SD direct detections in
the plane of neutron-DM and proton-DM plane. Like our previous results in the
universal coupling scenario, unpaired neutrons in Xenon based detectors would
help to make neutron-DM scattering more effective to constrain the parameter
space. Part of the parameter space, specially the resonance annihilation
regions, can be probed in the near future via LZ experiments.

##  VI Conclusions

In this paper we have explored the possibility of having a fermionic dark
matter (DM) in an extension of the SM, where torsion field ($S_{\mu}$) plays
the role of portal. Guided by its geometric origin, the torsion field can
naturally be realized as a massive axial vector field. The dark sector is
comprised only of the SM-singlet Dirac fermion ($\psi$) that makes up all of
the observed DM relic abundance. The DM is naturally stable over the
cosmological time scale (i.e., non-decaying) without the imposition of any ad-
hoc symmetry or kinematical condition to forbid its decay (in other words, the
DM need not to be the lightest stable particle (LSP) in this set-up). Thus the
proposed model is a minimal extension of the SM. We consider two scenarios to
parametrize the torsion-matter (includes both DM and SM particles) couplings.
In the first case both the torsion-DM and torsion-SM couplings are assumed to
be equal (dubbed as universal coupling scenario) at the low energy scale. It
leads to a simplified DM phenomenology involving only three free parameters:
the universal torsion-DM coupling $\eta$, DM mass $m_{\psi}$ and the torsion
mass $m_{s}$. We consider the DM to be WIMP-like, and via a detailed parameter
space scan we find that correct relic abundance can be achieved over a large
parameter space for DM mass $m_{\psi}:\\{1-5000\\}~{}\rm GeV$, with torsion
mass lying in the same range and for $\eta\sim\\{0.1-1\\}$. The DM
phenomenology is somewhat equivalent to popular $Z^{{}^{\prime}}$-mediated DM
models but unlike $Z^{{}^{\prime}}$, the torsion can only accommodate axial
couplings. However, a significant difference lies in the fact that in this
set-up the dark sector and the visible sector fermions can naturally enjoy the
same coupling with the mediator. Parameter space of such a simple model is
highly constrained from the recent LHC data. Thus, stringent bounds can be
derived from the dilepton searches at the LHC similar to massive (axial)
$Z^{{}^{\prime}}$ searches. From our computation of the observable
$\sigma(pp\to S_{\mu})\times Br(S_{\mu}\to l^{+}l^{-})$ ($l\in e,\mu$), we
find that
$\eta\mbox{\raisebox{-2.58334pt}{~{}$\stackrel{{\scriptstyle<}}{{\sim}}$~{}}}0.01(0.1)$
is required for
$m_{s}\mbox{\raisebox{-2.58334pt}{~{}$\stackrel{{\scriptstyle<}}{{\sim}}$~{}}}3(5)$
TeV which places very tight constraint on the viable DM parameter space. In
fact, the whole parameter space is mostly excluded except for $\eta\sim 1$,
where large decay width of the torsion may invalidate the LHC limits. Then we
move to the non universal coupling scenario where torsion-DM interaction
strength is assumed to be different from that of torsion-SM fermion (dubbed as
non-universal coupling). Since LHC constraints are mainly determined by
torsion-quark couplings, we set them at 0.01 and vary the DM-torsion couplings
in the range {0.1-1}. Here $Br(S_{\mu}\to l^{+}l^{-})$ is reduced due to rise
in the invisible decay of the torsion, which essentially helps to have even
lighter $S_{\mu}<3$ TeV compared to the universal scenario. The PLANCK-
observed relic abundance is satisfied along the resonance region and also for
$m_{s}\leq m_{\psi}$. In both universal and non-universal scenarios direct
detection of the DM has been computed. Here, the presence of the axial-vector
interaction turns out to be beneficial as this gives rise only to a spin-
dependent direct detection cross-section. We find that almost all of the relic
density allowed parameter space lies comfortably below the present exclusion
limit of direct search experiments. However, future spin-dependent direct
search experiments e.g., LUX-ZEPLIN with higher sensitivity can probe some
parts of the parameter space.

##  VII Acknowledgments

The computation was supported in part by the SAMKHYA: High Performance
Computing Facility provided by Institute of Physics, Bhubaneswar. TB
acknowledges the local hospitality at IOP, Bhubaneswar where this work was
started. DD acknowledges discussion with Sujoy Poddar.

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