# Improved White Dwarves Constraints on Inelastic Dark Matter and Left–Right
Symmetric Models

Anirban Biswas  Arpan Kar  Hyomin Kim  Stefano Scopel  Liliana Velasco-Sevilla

###### Abstract

Weakly Interacting Massive Particles (WIMPs) can be captured in compact stars
such as white dwarves (WDs) if they are in a Dark Matter (DM)–rich
environment, leading to an increase in the star luminosity through their
annihilation process. N–body simulations suggest that the core of the Messier
4 globular cluster (where plenty of WDs are observed) is rich of Dark Matter.
Assuming this is the case, we use a recent improvement in the calculation of
the WD equation of state to show that when the WIMP interacts with the nuclear
targets within the WD through inelastic scattering, and its mass exceeds a few
tens GeV, the data on low–temperature large-mass WDs in M4 can probe values of
the mass splitting as large as $\delta\mathrel{\mathop{\kern 0.0pt\hbox
to0.0pt{\raise 0.86108pt\hbox{$<$}\hss}}\lower
3.87495pt\hbox{\kern-1.90002pt$\sim$}}$ 40 MeV. Such value largely exceeds
those ensuing from direct detection and from solar neutrino searches. We apply
such improved constraint to the specific DM scenario of a self–conjugate bi-
doublet in the Left–Right Symmetric Model (LRSM), where the standard
$SU(2)_{L}$ group with coupling $g_{L}$ is extended by an additional
$SU(2)_{R}$ with coupling $g_{R}$ in order to explain maximal parity violation
in Weak Interactions. We show that bounds from WDs significantly reduce the
cosmologically viable parameter space of such scenario, in particular
requiring $g_{R}>g_{L}$. For instance, for $g_{R}/g_{L}$ = 1.8 we find the two
viable mass ranges 1.2 TeV $\mathrel{\mathop{\kern 0.0pt\hbox to0.0pt{\raise
0.86108pt\hbox{$<$}\hss}}\lower
3.87495pt\hbox{\kern-1.90002pt$\sim$}}m_{\chi}\mathrel{\mathop{\kern
0.0pt\hbox to0.0pt{\raise 0.86108pt\hbox{$<$}\hss}}\lower
3.87495pt\hbox{\kern-1.90002pt$\sim$}}$ 3 TeV and 5 TeV
$\mathrel{\mathop{\kern 0.0pt\hbox to0.0pt{\raise
0.86108pt\hbox{$<$}\hss}}\lower
3.87495pt\hbox{\kern-1.90002pt$\sim$}}m_{\chi}\mathrel{\mathop{\kern
0.0pt\hbox to0.0pt{\raise 0.86108pt\hbox{$<$}\hss}}\lower
3.87495pt\hbox{\kern-1.90002pt$\sim$}}$ 10 TeV, when the charged $SU(2)_{R}$
gauge boson mass $M_{W_{2}}$ is lighter than $\simeq$ 12 TeV. We also discuss
the ultraviolet completion of the LRSM model, when the latter is embedded in a
Grand Unified Theory. We show that such low–energy parameter space and
compatibility to proton-decay bounds require a non–trivial extension of the
particle content of the minimal model. We provide a specific example where
$M_{W_{2}}\mathrel{\mathop{\kern 0.0pt\hbox to0.0pt{\raise
0.86108pt\hbox{$<$}\hss}}\lower 3.87495pt\hbox{\kern-1.90002pt$\sim$}}$ 10 TeV
is achieved by extending the LRSM at high energy with color triplets that are
singlets under all other groups, and $g_{R}/g_{L}>$1 is obtained by
introducing $SU(2)_{L}$ triplets with no $SU(2)_{R}$ counterparts, i.e. by
breaking the symmetry between the multiplets of $SU(2)_{L}$ and $SU(2)_{R}$.

CQUeST-2022-0691, KIAS P22038

##  1 Introduction

The discovery of the nature of Cold Dark Matter (CDM) is one of the most
urgent pending issues of modern physics. Weakly Interacting Massive Particles
(WIMPs) are the most popular CDM candidates: their mass $m_{\chi}$ falls in
the GeV–TeV range and they are expected to have only weak-type interactions
with ordinary matter. Such small but non–vanishing interactions keep them in
thermal equilibrium in the Early Universe, providing an expected thermal relic
abundance in agreement with observation. They also provide promising signals
to detect them: in direct detection (DD) WIMP scattering events off nuclear
targets are searched for through the measurement of the ensuing nuclear
recoils in low-background detectors (see for instance [1, 2]); in accelerators
WIMPs can be created from the scattering of Standard Model (SM) initial states
(see for instance [3, 4, 5]); finally, in indirect detection (ID) the products
of the annihilations of DM particles in the halo of our Galaxy or that of
dwarf galaxies are searched for both in ground–based and in satellite
telescopes (see for example [6, 7, 8, 9]).

Annihilation is enhanced wherever the WIMP density is large, so that a
peculiar ID signal is provided by celestial bodies such as the Sun or the
Earth, that, through the same WIMP–nucleus scattering processes that are
searched for in DD, can capture WIMPs developing a high–density population of
them in their interior [10, 11, 12, 13, 14, 15, 16]. The opacity of the
celestial body plasma implies that among the WIMP annihilation products only
neutrinos can escape from its surface, with energies that can reach about one
third of the WIMP mass [17, 18, 19, 20]. The detection of neutrinos coming out
from the Sun or the Earth with energies ranging from the GeV to the TeV scale
would represent a smoking gun of an exotic process, since it could not be
explained by any known mechanism [21, 22]. The direction of the muons produced
by the conversion of such neutrinos in the vicinity of a Cherenkov detector
such as Super–Kamiokande or IceCube is strongly correlated to that of the
source, providing a powerful background subtraction method [23, 24, 25].

Both DD and capture are substantially modified in the scenario of Inelastic
Dark Matter (IDM). In this class of models a DM particle $\chi$ of mass
$m_{\chi}$ interacts with atomic nuclei exclusively by up–scattering to a
second heavier state $\chi^{\prime}$ with mass $m_{\chi^{\prime}}$ =
$m_{\chi}$ \+ $\delta$. A peculiar feature of IDM is that there is a minimal
WIMP incoming speed in the target frame matching the kinematic threshold for
inelastic upscatters and given by:

$v^{*}_{min}=\sqrt{\frac{2\delta}{\mu_{\chi N}}},$ (1.1)

with $\mu_{\chi N}$ the WIMP–nucleus reduced mass. The WIMP speed in the halo
of our Galaxy is bounded by the escape velocity $u_{esc}\simeq$ 550 km/s and
for both DD and capture in the Sun or the Earth the targets move in the
Galactic rest frame with a relative speed $v_{0}\simeq$ 220 km/s corresponding
to the rotational curve at the solar system position [26]. As a consequence
the WIMP-scattering process in DD vanishes if
$v^{*}_{min}\mathrel{\mathop{\kern 0.0pt\hbox to0.0pt{\raise
0.86108pt\hbox{$>$}\hss}}\lower
3.87495pt\hbox{\kern-1.90002pt$\sim$}}v_{max}=u_{esc}+v_{0}\simeq$ 800 km/s,
because it becomes kinematically not accessible, the incoming WIMP speed being
too slow to overcome the inelasticity threshold. Such kinematic bound implies
that the largest values of $\delta$ that DD can probe are reached by the
heaviest targets (xenon, iodine, tungsten) employed in existing experiments
and cannot exceed $\delta\simeq$ 200 keV [27].

A trivial way to extend the sensitivity to the $\delta$ parameter beyond DD is
to look for processes where the WIMP scatters off nuclei at higher speeds. For
instance, the gravitational potential accelerates the WIMP particles before
they scatter off the nuclei of a celestial body, so that their speed can be
larger than $v_{max}$. Indeed, it can reach 1600 km/s in the center of the
Sun, implying that the values of $\delta$ that can be probed in IDM by capture
can reach $\simeq$ 600 keV [28]. As a consequence, for these scenarios the
solar capture bounds are stronger than those from DD.

Clearly, even stronger bounds can in principle be obtained from WIMP capture
in celestial bodies with a gravitational potential stronger than that of the
Sun, such as white dwarves (WDs) [29, 30, 31] and neutron stars (NSs) [32,
33]. Indeed, in such cases the speed of the scattering WIMP can reach $\simeq$
a few $10^{4}$ km/s (for WDs) [34] or a few $10^{5}$ km/s (for NSs) [35],
potentially extending the values of $\delta$ that can be probed up to several
tens of MeV for WDs and hundreds of MeV for NSs. WDs have already been used in
the literature to put constraints on other DM scenarios [36].

In Ref. [34] the WD equation of state improved in [37] was used to constrain
the WIMP–nucleus cross-section in the case of elastic scattering. The aim of
our paper is to extend the analysis of Ref. [34] to the case of inelastic
scattering, showing how this can have dramatic consequences on a specific
realization of the IDM scenario, bi–doublet DM in Left-Right symmetric models
(LRSM) [38].

In such scenario the DM particle has a non–vanishing hypercharge leading to a
coupling to the Z boson whose size is several orders of magnitude larger than
that excluded by DD experiments in the case of elastic scattering. However it
can be reconciled to DD constraints by selecting the parameter space for which
the mass splitting $\delta\mathrel{\mathop{\kern 0.0pt\hbox to0.0pt{\raise
0.86108pt\hbox{$>$}\hss}}\lower 3.87495pt\hbox{\kern-1.90002pt$\sim$}}$ 200
keV so that the nuclear scattering process is kinematically forbidden [39]. In
particular for bi–doublet DM in the LRSM only DD constraints have been discuss
so far. In Section 3 we will update its phenomenological analysis including
the improved bound from WDs.

The paper is organized as follows. In Section 2 we extend the WD bounds
discussed in [34] to IDM. In particular in Section 2.1 we review the present
knowledge on the amount of DM in the globular clusters (GCs) of our Galaxy and
specifically in Messier 4 (M4) showing that N–body simulations suggest that it
can be large in their core. Based on this assumption, in Section 2.2 we will
use the data collected by the Hubble Space Telescope on low–temperature WDs in
M4 to constraint IDM, showing that indeed it is possible to probe in this way
values of $\delta$ as large as 30 or 40 MeV. At the end of the same Section we
also provide a short discussion on how the detection of low–temperature
neutron stars could further improve the bounds from WDs to values of the mass
splitting $\delta$ as large as $\simeq$ 300 MeV. In Section 3 we summarize the
main features of the bi–doublet LRSM model and show how the results of Section
2.2 substantially reduce the range of variation of its parameters, in
particular requiring a significant splitting at low energy between the
couplings $g_{L}$ and $g_{R}$ of the two groups $SU(2)_{L}$ and $SU(2)_{R}$ in
order to yield a thermal relic abundance in agreement to observation. In
Section 3.3 we discuss the consequences of the results of Section 3 on the
running of $g_{L}$ and $g_{R}$ and on their unification scale when the LRSM is
embedded in a Grand Unified Theory (GUT) and provide a specific realization of
such scenario, whose features are given in Appendix A. Finally, Section 4
contains our Conclusions.

##  2 Constraints on Inelastic Dark Matter from White Dwarves

###  2.1 The Dark Matter Content of the M4 Globular Cluster

The composition of massive White Dwarves is a matter of debate. In particular,
recent computations of their interior suggest that their core could be
composed of neon [40]. Depending on the metallicity of their surrounding
medium their chemical composition can also include Carbon and Oxygen.

They are the most abundant stellar remnants, providing a promising means of
probing DM interactions, complementary to terrestrial searches [41, 34]. In
particular the core temperature of a WD is too low to ignite nuclear fusion
reactions, and as a result, WDs have no internal energy source. On the other
hand, compared to the Sun their interior density can be eight orders of
magnitude larger [34], while their local environment can be richer of DM
particles with a lower velocity dispersion, as is the case for the M4 globular
cluster, where low–temperature WDs have been observed [29]. All these features
imply that indeed WDs in M4 can capture WIMPs more efficiently than the Sun,
and that the DM annihilation process inside the WD can easily dominate its
total luminosity, driving its temperature beyond the observed ones. As a
consequence, as shown in Section 2.2, the observation of low–temperature WDs
in M4 can yield to constraints that are stronger compared to those obtained
from WIMP capture in the Sun.

In order to set such limits it is necessary to estimate the DM density
surrounding the WDs in the inner region of M4. Unfortunately, direct
measurements of the velocity dispersion of stars within globular clusters [42]
do not allow to reconstruct their gravitational potential with enough
precision to obtain their DM content. As a consequence, one needs to rely on
models of GC formation based on $N$–body simulations [43, 29]. This clearly
represents the major source of uncertainty in our analysis and other similar
ones in the literature. It is fair to say that, for instance, in [44], the
presence of DM in GC is referred to as a “controversial” issue, mainly due to
the effect of tidal disruption of the host galaxy. However, several analyses
suggest that the cores of DM halos in globular clusters should indeed survive
successive tidal interactions with the host galaxy. The analysis of the
following Sections will show that if indeed M4 is rich of DM this can have
dramatic consequences on some classes of particle–physics models of Dark
Matter. So, following previous phenomenological analyses [29, 41, 34] we will
assume this possibility. In the present paper we will use the results of [29].
We summarize such results below, addressing the reader to the original
literature for further details.

An estimate of the DM content in the M4 GC is obtained in [29] by modeling the
GC formation starting from an original DM subhalo of mass $M_{\rm DM}\simeq
10^{7}M_{\odot}$ that falls into our host galactic halo [45]. Once the GC
falls into the host halo, mass loss occurs continually through tidal
stripping, and the orbit of the subhalo decays down toward the center. Such
mass loss can be significant, resulting in only a few percent of the DM of the
original subhalo surviving inside the final GC, with a baryon-dominated core
with a small mass-to-light ratio that actually resembles those of observed
globular clusters. The presence of baryons in the GC must also be taken into
account: on the one hand, in the stellar core it can gravitationally enhance
the DM density, while on the other it can reduce it, as a consequence of the
heating due to the interaction of DM particles with stars (albeit this latter
effect is estimated to be negligible at the radii where WDs are observed
within M4). Moreover, when the gas in the original halo which eventually forms
the globular cluster loses energy and falls into the core conservation of
angular momentum is expected to lead to a contraction of the DM core: also
such effect turns out to be negligible at the position where the WDs are
observed in M4. Finally, the effect of tidal stripping is taken into account
by truncating the density distributions of stars and DM at the tidal radius of
the GC. The baryon and DM density profiles within M4 ensuing from such
procedure are shown in Fig. 1 of [29], which implies that DM makes up less
then $\simeq$ 40% of the total mass of the cluster, in agreement to
observations that indicate a lack of DM in globular clusters. In this sense
the authors of [29] consider such estimation to be conservative. In particular
such value corresponds to less than 1% of the original $10^{7}M_{\odot}$ halo.
Moreover in Ref. [29] the value of the DM density at the largest radius where
WDs within M4 are observed ($\simeq$ 2.3 pc, well inside the tidal radius) is
estimated to be $\rho_{\rm DM}\simeq$21 $M_{\odot}$ pc-3 = 798 GeV cm-3.
Furthermore, using the baryon and DM density profiles estimated with such
procedure a WIMP velocity dispersion of $\simeq 8$ km/s can be obtained by
assuming hydrostatic equilibrium and spherical symmetry.

The quantitative estimations of Ref. [29] and summarized in this Section will
be used in Section 2.2 to evaluate the effect of DM capture on the WDs within
M4.

###  2.2 WIMP Capture in White Dwarves

Signals generated by pair annihilations of WIMPs captured and accumulated
inside the Earth and the Sun have been discussed for a long time [10, 11, 12,
13].

In the case of the Earth and the Sun only neutrinos escape from the celestial
body, that can be searched for in the up-going muons flux generated by their
conversion in the rock close to a Cherenkov detector. Such signals are
sensitive to the yield of the neutrinos produced above the detector’s
threshold by the particles produced in the WIMP annihilation, typically
fermions or gauge bosons, that are boosted to an energy equal to the WIMP mass
before decaying. An exception to this mechanism is provided by light quarks,
which are stopped inside the medium before decaying, producing below–threshold
neutrinos that are not detectable.

In compact stars such as white dwarves or neutron stars the capture process is
analogous. However, in this case the observable effect is an increase of the
total luminosity or of the temperature of the star, which depends only on the
total electromagnetic energy $E_{\gamma}$ injected by the annihilation
process. In particular, for the high energies corresponding to the values of
the WIMP mass discussed in Section 3, also the the neutrinos produced in the
annihilation process are converted into electromagnetic radiation, so that the
signal can be practically considered as bolometric and $E_{\gamma}$ = 2
$m_{\chi}$.

While crossing a compact star DM particles may get trapped in its
gravitational potential, continue to lose energy by scattering against its
constituents and get ultimately captured. A DM particle $\chi$ (of mass
$m_{\chi}$) can undergo an inelastic transition to a second heavier state
(say, $\chi^{\prime}$) by scattering against a nucleus (of mass $m_{N}$)
present inside the star if the total kinetic energy in the DM-nucleus center
of mass frame is larger than the mass difference $\delta$ between the DM
particle and the heavier state [46, 47, 48, 28], i.e.,

$\frac{1}{2}\mu_{\chi N}w^{2}>\delta,$ (2.1)

which is equivalent to Eq. (1.1). The quantity $w$, which is the speed of the
DM particle at a radial distance $r$ from the centre of the star, is given by:

$w(r)=\sqrt{u^{2}+v_{esc}(r)^{2}},$ (2.2)

where $v_{esc}(r)$ is the local escape velocity from the star at radius $r$
and $u$ is the asymptotic speed of the DM particles far away from the star.

Considering the WD stars in a nearby globular cluster such as M4, a common
choice for the distribution of the speed $u$ in the cluster is a boosted
Maxwell Boltzmann (MB) one, which in the reference frame of a WD can be
expressed as:

$f(u)=\frac{u}{v_{d}v_{*}}\sqrt{\frac{3}{2\pi}}\left({\rm
exp}\left[-\frac{3}{2v^{2}_{d}}(u-v_{*})^{2}\right]-{\rm
exp}\left[-\frac{3}{2v^{2}_{d}}(u+v_{*})^{2}\right]\right),$ (2.3)

where $v_{*}$ and $v_{d}$ are the velocity of the WD and the velocity
dispersion of DM inside the globular cluster, respectively. Following [29,
34], we have assumed conservative values for $v_{*}$ and $v_{d}$, which are
$v_{*}=20$ km/s and $v_{d}=8$ km/s for the WDs in M4.

Given the density profiles of nuclear elements in a WD, one can estimate the
corresponding escape velocity profile $v_{esc}(r)$. In particular, the authors
of Ref. [34] have used an improved equation of state for WDs and calculated
the density profiles of thermal electrons and nuclear ions for WDs with
different configurations that are made of elements like carbon, oxygen, etc.
In particular assuming that the vast majority of heavier WDs are made of
carbon, oxygen or neon, carbon provides the lowest rate for a spin–independent
(SI) cross section that scales with the atomic mass number squared. Indeed,
this is the most common type of interaction expected for WIMPs with nuclei,
and it arises in the specific scenario analyzed in Section 3, so in the
following we will assume that the WDs are entirely made of carbon to get
conservative bounds. Among the various benchmarks for carbon WD tabulated in
[34] we take the heaviest WDs with masses $M_{*}=1.38$ $M_{\odot}$ and 1.25
$M_{\odot}$ and extract the associated radial profiles for the nuclear density
and the escape velocity. The corresponding radii of these two benchmark WDs
are $R_{*}\simeq 1.25\times 10^{3}$ km and $R_{*}\simeq 3.29\times 10^{3}$ km,
respectively.

For the heaviest WD considered here, the escape velocity $v_{esc}$ in the
interior of the star can reach a very high value $\sim 0.1$ $c$ [34] which in
turn extends the kinematic upper limit (shown in Eq. (2.1)) on $\delta$ up to
several tens of MeV. Due to this reason, using heavier WDs it is possible to
probe a high value of $\delta$, as will be described in detail in this sub-
section.

In the optically thin limit, DM particles crossing a WD can become
gravitationally bound to the star after a single scattering. In this case, the
rate of DM capture through inelastic scattering is given by [28, 48, 47, 46]:

$C_{\rm opt-thin}=\frac{\rho_{\chi}}{m_{\chi}}\int_{0}^{R_{*}}dr\hskip
1.42262pt4\pi r^{2}\int_{0}^{\infty}du\hskip 1.42262pt\frac{f(u)}{u}\hskip
1.42262ptw\hskip 1.42262pt\Omega(w,r)\hskip
2.84526pt\Theta\left(\frac{1}{2}\mu_{\chi N}w^{2}-\delta\right).$ (2.4)

Here $\rho_{\chi}$ is the density of DM at the location of the WD inside the
globular cluster. As discussed in Section 2.1 in the local environment of the
WDs inside M4 we take $\rho_{\chi}$ = 798 GeV cm-3. Note that the $\Theta$
function in Eq. (2.4) ensures that the capture rate goes to zero if the
condition given in Eq. (2.1) is violated. The interaction rate $\Omega(w,r)$
can be obtained using the following expression:

$\Omega(w,r)=\eta_{N}(r)\hskip 1.42262ptw\hskip 1.42262pt\Theta(E_{\rm
max}-E_{\rm cap})\int_{E_{\rm min}}^{E_{\rm max}}dE\hskip
1.42262pt\frac{d\sigma[\chi+N\rightarrow\chi^{\prime}+N]}{dE}\hskip
2.84526pt\Theta(E-E_{\rm cap}),$ (2.5)

where $\eta_{N}(r)$ is the number density distribution (extracted from [34]
for the two benchmark WDs considered) of the target nuclear ion (in our case
${}^{12}C$) in the WD, and
$\frac{d\sigma[\chi+N\rightarrow\chi^{\prime}+N]}{dE}$ is the differential
cross-section (as a function of the recoil energy $E$) for the DM-nucleus
inelastic scattering $\chi+N\rightarrow\chi^{\prime}+N$. The minimum and the
maximum recoil energies (i.e., $E_{\rm min}$ and $E_{\rm max}$) corresponding
to this scattering process and the minimum energy transfer $E_{\rm cap}$
required for the capture (i.e., to scatter a DM particle from $w$ to a
velocity less than $v_{esc}(r)$) are:

$\displaystyle E_{\rm min,\rm max}$ $\displaystyle=$
$\displaystyle\frac{1}{2}m_{\chi}w^{2}\left[1-\frac{\mu^{2}_{\chi
N}}{m^{2}_{N}}\left(1\pm\frac{m_{N}}{m_{\chi}}\sqrt{1-\frac{\delta}{\mu_{\chi
N}w^{2}/2}}\right)^{2}\right]-\delta,$ $\displaystyle E_{\rm cap}$
$\displaystyle=$ $\displaystyle\frac{1}{2}m_{\chi}u^{2}-\delta.$ (2.6)

In Eq. (2.5) the first Heaviside step function $\Theta$ imposes the condition
$E_{\rm max}>E_{\rm cap}$ which is required for the capture to occur, while
the second one sets the lower limit of the energy integration to ${\rm
max}[E_{\rm min},E_{\rm cap}]$.

Notice that, after the first up-scattering of the DM particle $\chi$, there
are two possible situations that may arise depending on the lifetime of the
heavier state $\chi^{\prime}$. If the lifetime of $\chi^{\prime}$ is very
short, it quickly decays back to $\chi$ along with other particle(s) which are
much lighter and carry away most of the energy produced in the decay, so that
no significant kinetic energy is imparted to $\chi$. As a consequence Eqs.
(2.4, 2.5) can be used to calculate the capture rate because a single
scattering with $E>E_{\rm cap}$ is sufficient to keep the $\chi$ particle
gravitationally bound during the lifetime of the star eventually driving it to
the core of the celestial body after subsequent interactions with its nuclear
targets. We will see that this is the situation which occurs in the types of
DM models discussed in the next Section. On the other hand, if the lifetime of
$\chi^{\prime}$ is large enough it eventually down-scatters off a WD nuclear
target back to a $\chi$ through an exothermic process that can in principle
eject the outgoing DM particle to a non–bound orbit. In this case Eqs. (2.4,
2.5) cannot be used to describe the capture process. However, if $\chi$ is
much heavier than the target nucleus, the latter carries away most of the
energy, leading to a situation where the DM particle does not have sufficient
kinetic energy to escape from the star’s gravitational potential and thus also
in this case it is eventually captured [47]. Actually, in the present work we
will be interested in the capture of TeV scale DM particles, so they would be
captured in the WD star after the first inelastic scattering with $E>E_{\rm
cap}$ even for $\chi^{\prime}$ lifetimes longer than those discussed in
Section 3.2.

The optically thin approximation holds when the scattering cross-section
(i.e., the integration of $\frac{d\sigma}{dE}$ over $E$ in Eq. (2.5)) is small
enough so that the effect of multiple scatterings can be neglected. If the
cross-section becomes very large, each DM particle that traverses the WD is
captured and the capture rate saturates to its maximum value (known as the
geometric limit) which is independent of the DM-nucleus interaction [34, 49,
30, 44, 50]:

$C_{\rm geom}=\pi
R^{2}_{*}\left(\frac{\rho_{\chi}}{m_{\chi}}\right)\int_{0}^{\infty}du\hskip
1.42262pt\frac{f(u)}{u}\hskip 1.42262ptw^{2}(R_{*}).$ (2.7)

Finally, considering both regimes (i.e., optically thin and geometric
regimes), the capture rate of DM in a WD star through inelastic scattering can
be estimated as [31, 49, 51, 44]:

$C_{*}={\rm min}[C_{\rm opt-thin},C_{\rm geom}].$ (2.8)

After being captured, the DM particles continue to scatter with the
constituents of the WD and eventually thermalise and settle down to the WD
interior where they can annihilate in pairs to produce various SM particles.
The thermalisation time-scale is expected to be several orders of magnitude
smaller than the typical age of the old and cold WDs (the types of which have
been considered here) in the M4 globular cluster [34]. The evolution of the
number of DM particles ($N_{\chi}$) in the WD interior can be expressed as
[34],

$\frac{dN_{\chi}}{dt}=C_{*}-AN^{2}_{\chi},$ (2.9)

where $A$ is the annihilation coefficient (which is proportional to the
averaged annihilation cross-section $\langle\sigma v\rangle$) and is related
to the annihilation rate as,

$\Gamma_{\rm ann}=\frac{1}{2}AN^{2}_{\chi}.$ (2.10)

The primary SM particles injected by DM annihilation in a WD interior lead to
further cascades and give rise to secondary particles which quickly thermalise
in the medium of the star and thus end up in heating the star. Assuming that
the total energy emitted in the DM annihilation process contributes to heat up
the star, the resulting luminosity can be estimated as:

$L_{\chi}=2m_{\chi}\Gamma_{\rm ann}.$ (2.11)

Note that, in reality, some percentage of the energy produced in DM
annihilation inside the WD may be carried away by neutrinos which escape the
star without heating it up. However, we have checked that, for the WDs
considered here, neutrinos produced in the star core with energies above a GeV
cannot come out from the star as their mean free path, estimated using the
number density of the WD constituents (obtained in [34] considering an
improved analysis) and the interaction cross-section of neutrinos with those
constituents [52], is much smaller than the typical size of the WD star. The
energetic neutrinos that cannot escape the star eventually thermalise in the
star and are converted into electromagnetic radiation. In case of the
annihilation of a TeV scale DM (which is our present interest), the fraction
of energy carried away by neutrinos with energies below 1 GeV (that are
produced in the cascade of various primary annihilation states such as
$W^{+}W^{-},ZZ,b\bar{b}$) is less than 1$-$2% (estimated using [53]). Thus, we
can assume that almost all of the energy produced in the annihilation
contributes to the luminosity of the star and use Eq. (2.11) to calculate it.
Notice that this implies that the ensuing bounds do not depend on assumptions
on the primary SM states the DM particles annihilate to. This is at variance
with capture in the Sun or the Earth that may lead to a neutrino flux below
the experimental energy threshold when the DM particles annihilate only to
light states that are stopped before decaying.

The capture and annihilation processes inside the WD star equilibrate (so that
$\frac{dN_{\chi}}{dt}$ becomes zero) over a time-scale $\tau_{\rm
eq}=(C_{*}A)^{-1/2}$. A conservative estimate (following [31]) with the
assumptions that $\langle\sigma v\rangle=\langle\sigma v\rangle_{\rm thermal}$
and $C_{*}=C_{\rm geom}$, the equilibrium time-scale $\tau_{\rm eq}$ (for DM
particles of mass $m_{\chi}=1$ TeV) turns out to be of the order of $\sim
10^{-6}$ Gyr for our benchmark WDs. This time-scale is clearly several orders
of magnitude lower than the typical ages of the old and faint WDs in M4, which
are expected to be at least a few Gyrs [34]. Hence, one can in general assume
equilibrium between the capture and annihilation processes inside these WDs in
M4, and specifically in the model discussed in Section 3. In that case the DM
annihilation rate is $\Gamma_{\rm ann}=C_{*}/2$ and the corresponding
luminosity (given by Eq. (2.11)) becomes [30, 41, 34]:

$L_{\chi}=m_{\chi}C_{*}.$ (2.12)

Figure 1: DM induced luminosities in both optically thin (blue and red lines)
and geometric (cyan lines) limits as functions of $\delta$, considering two
benchmark carbon WDs in M4 with masses $M_{*}=1.38$ $M_{\odot}$ (left panel)
and $M_{*}=1.25$ $M_{\odot}$ (right panel). The blue and the red lines are
corresponding to the capture of 500 GeV and 5 TeV DM particles, respectively,
assuming a spin-independent DM-nucleus interaction. In both panels the black
horizontal lines indicate the luminosities (taken from [34]) that are observed
for the considered WDs, while the cyan horizontal lines show the luminosities
of the two benchmark WDs when DM capture is calculated in the geometric limit.

In Fig. 1, we show, as a function of the mass-splitting $\delta$, the
luminosities that are induced by the annihilation of inelastically captured DM
particles in carbon WDs (located in M4) with masses and radii $M_{*}=1.38$
$M_{\odot}$, $R_{*}\simeq 1.25\times 10^{3}$ km (left panel) and $M_{*}=1.25$
$M_{\odot}$, $R_{*}\simeq 3.29\times 10^{3}$ km (right panel). The blue and
the red lines in each panel represent the luminosities corresponding to the
capture of DM with masses $m_{\chi}=500$ GeV and 5 TeV, respectively, assuming
a standard SI cross-section with an exponential (Helm) nuclear form factor
[14, 48] for the WIMP scattering events off carbon (${}^{12}C$) in the
optically thin limit (Eq. (2.4)). In particular, two illustrative values of
the DM-nucleon cross-section $\sigma_{\mathcal{N}}$ have been used. The cyan
horizontal lines in both panels of Fig. 1 show the luminosities obtained for
the two benchmark WDs considering the capture of DM in the geometric limit
(Eq. (2.7)). Note that, since the capture rate falls as the inverse of
$m_{\chi}$ in the geometric limit as well as in the optically thin limit (for
$m_{\chi}\lesssim{\cal O}(10^{5})$ GeV [34] and much larger than the target
nucleus mass), the estimated luminosities in both regimes are independent of
the DM mass.

The additional black lines shown in Fig. 1 represent the luminosities for the
two considered WDs in M4 observed by the Hubble Space Telescope [54, 55] and
taken from [34]. These two WDs are among the faintest WDs observed in M4.
Therefore, if the DM predicted contribution to the total luminosity of these
WDs is less than or at most equal to the observed ones, then there will be no
contradiction between the DM prediction and the observations.

There are a few important points to notice in Fig. 1. The first one is that
the DM luminosities calculated for the selected WDs in the optically thin
limit are largely independent of $\delta$ up to a certain value of this
parameter, beyond which the capture rates (hence the luminosities) start
falling rapidly and ultimately go to zero when $\delta$ exceeds several tens
of MeV. The second point is that the maximally achievable DM luminosity (i.e.,
the luminosity obtained in the geometric limit) for each WD is above the
observed limit. Based on these two points, a comparison between the WD data
and the DM luminosity (obtained using Eqs. (2.8) and (2.12)) enables one to
constrain the DM interaction strength (for example, the DM-nucleon cross-
section $\sigma_{\mathcal{N}}$ in the case of a SI interaction) up to a
maximum value of $\delta$ (a few tens of MeV for the WDs considered here),
above which the DM luminosity goes below the observation. The maximum value of
$\delta$ depends on the WD parameters and is larger for heavier WDs, since
heavier and compact WDs have stronger gravitational potential which increases
the kinematic upper limit on $\delta$ (see Eq. (2.1)). Notice that the mass of
the heaviest of our benchmark WDs almost saturates the Chandrasekhar limit
$M_{*}\mathrel{\mathop{\kern 0.0pt\hbox to0.0pt{\raise
0.86108pt\hbox{$<$}\hss}}\lower 3.87495pt\hbox{\kern-1.90002pt$\sim$}}$ 1.4
$M_{\odot}$ [56, 57], and so in practice it kinematically saturates the best
achievable bound on $\delta$ using WDs.

Figure 2: Constraints in the $\sigma_{\mathcal{N}}$ vs. $\delta$ plane (for a
SI interaction) obtained using the observed luminosities for the two selected
WDs in M4. The dashed green and red vertical lines show the maximum values of
$\delta$ that can be probed in present direct-detection experiments and
through solar neutrino searches, respectively.

In order to illustrate the above discussion, we show in Fig. 2 the regions of
the $\sigma_{\mathcal{N}}$ vs. $\delta$ plane (for a SI interaction) that are
excluded using the luminosity data of two selected faint WDs located in M4,
assuming that they are made of carbon. The DM mass is taken to be $m_{\chi}=1$
TeV, although, as mentioned earlier, for a sufficiently heavy DM (much heavier
than the target nucleus) these constraints do not depend on $m_{\chi}$. In the
same figure the green and the red dashed lines indicate the maximum values of
$\delta$ that can be probed in direct-detection experiments [27] and in solar
neutrino searches [28], respectively. Fig. 2 shows that WDs can improve the
pre–existing bounds on $\delta$ by more than one order of magnitude ruling out
$\delta$ up to a few tens of MeV for
$\sigma_{\mathcal{N}}\mathrel{\mathop{\kern 0.0pt\hbox to0.0pt{\raise
0.86108pt\hbox{$>$}\hss}}\lower 3.87495pt\hbox{\kern-1.90002pt$\sim$}}$
3$\times$ 10-43 cm2.

The constraints that we obtain here using the observations of faint and heavy
WDs in M4 assume a spin-independent interaction between DM and the target
nucleus. WDs are expected to be mostly composed of spinless targets
(${}^{12}C$, ${}^{16}O$ or ${}^{20}Ne$)111The abundance of targets with spin
such as ${}^{13}C$ and ${}^{19}O$ in WDs is however subject to large
uncertainties and strongly depends on the WD progenitor [58]., so that in
order to derive constraints on spin-dependent (SD) type interactions one needs
to consider DM capture in other celestial objects such as the Sun. Notice
that, as already pointed out ,the reach on $\delta$ in the case of the Sun is
considerably smaller (up to $\simeq$ 300 keV for SD) [48, 28].

Apart from WDs, one can also consider capture of DM particles in more compact
objects such as neutron stars (NSs) [59, 32, 35, 33] where the gravitational
potential is so strong that the infalling DM particles are accelerated to
quasi-relativistic speeds. When such a high speed DM particle collides with
the NS constituents it can up-scatter to the heavier state for a value of the
inelastic splitting $\delta$ that can be as large as $\sim 300$ MeV [32, 33].
The capture process of DM particles and their subsequent annihilation at the
star core can heat a NS maximally up to a temperature $T^{\infty}$ (measured
at a large distance from the NS) which is given by [32, 60, 61]:

$T^{\infty}\simeq 2300\hskip 1.70717pt{\rm
K}\left(\frac{\rho_{\chi}}{0.4\hskip 1.70717pt{\rm
GeVcm^{-3}}}\right)^{1/4}F\left(\frac{v_{*}}{230\hskip 1.70717pt{\rm km\hskip
1.42262pts^{-1}}}\right),$ (2.13)

where $\rho_{\chi}$ is the DM density at the location of the NS, $v_{*}$ is
the velocity of the NS in the halo and the function $F$ is defined as,

$F\left(x\right)=\left[\frac{{\rm Erf}(x)}{x\hskip 1.70717pt{\rm
Erf}(1)}\right]^{1/4}.$ (2.14)

Eq. (2.13) corresponds to a temperature $\simeq$ a few$\times 10^{3}$ K for
nearby NSs. The observation of old NSs with such a low temperature is beyond
the reach of existing telescopes but could be achievable by forthcoming
infrared telescopes such as the James Webb Space Telescope (JWST), the Thirty
Meter Telescope (TMT) and the European Extremely Large Telescope (E-ELT) [62,
61, 63].

##  3 The case of Bi–doublet DM in Left–Right Symmetric Models

The phenomenological discussion of Section 2 shows how WDs can provide bounds
on the IDM scenario that are significantly stronger compared to those from DD.
In this Section we wish to illustrate the impact that such enhanced bounds can
have on a specific IDM model, the fermion bi-doublet extension of the
Left–Right symmetric model (LRSM).

The main motivation of the LRSM scenario is to explain one of the most
puzzling features of the Standard Model, i.e. the observed maximal parity
violation in its weak sector that forces to arrange the left–handed chiral
components of fermions in $SU(2)$ electroweak doublets and their right–handed
counterparts in singlets. In the LRSM scenario the SM gauge group is enlarged
in order to contain an $SU(2)_{L}$ and an $SU(2)_{R}$, so that the doublets of
$SU(2)_{L}$ are singlets of $SU(2)_{R}$ and the doublets of $SU(2)_{R}$ are
singlets of $SU(2)_{L}$. Interestingly, such scenarios can be easily extended
by a multiplet that contains a Dark Matter candidate that is automatically
stable [64, 65] and whose mass is the only required additional free parameter.
A detailed description of the LRSM scenario can be found in [66], and a
discussion of the phenomenology of its DM extensions in [38]. In Section 3.1
we will summarize the specific scenario where the LRSM model is extended with
a fermion bi-doublet that contains an IDM candidate, and in Section 3.2 we
apply the results of Section 2.2 to constrain its parameter space. Moreover,
in Section 3.3 we discuss how the low-energy LRSM parameter space compatible
with observation can be obtained when the LRSM model is embedded in a Grand
Unified Theory at high energy.

###  3.1 Model Description

The gauge symmetry of LRSM is $SU(3)\times
SU(2)_{L}\times{SU}(2)_{R}\times{U(1)}_{B-L}$, with $B$ and $L$ the baryon and
lepton number. The representations under the different gauge groups of the
fermions and scalars contained in the model are listed in Tab. 2 of Appendix A
for both its minimal realization and for various extension that will be
discussed in Section 3.3. In its minimal realization the LRSM preserves D
parity, i.e. it is invariant by the interchange of any multiplet of
$SU(2)_{L}$ into the corresponding $SU(2)_{R}$ multiplet. In particular, in
one of the most popular variants of minimal LRSM the scalar sector contains an
$SU(2)_{L}$ triplet $T_{L}$, an $SU(2)_{R}$ triplet $T_{R}$ and a scalar bi-
doublet $\Phi$. The LR-symmetry of the model is broken spontaneously down to
the SM in two steps. First, the $SU(2)_{R}\times U(1)_{B-L}$ symmetry is
broken spontaneously to $U(1)_{Y}$ at a high scale $M_{R}$ by the vacuum
expectation value (VEV) $v_{R}$ of $T_{R}$, and the masses of the $SU(2)_{R}$
gauge bosons $W_{R}$ and $Z_{R}$ are generated. This breaks the parity
invariance of the model and implies the relation
$Y=T^{3}_{R}+\dfrac{1}{2}(B-L)$ between the generator of $U(1)_{Y}$ and those
of $U(1)_{B-L}$ and $SU(2)_{R}$. In particular $Z_{R}$ is a linear combination
of $W^{3}_{R}$ (the gauge boson corresponding to the generator $T^{3}_{R}$)
and $B$ (the gauge boson for $U(1)_{B-L}$), while the orthogonal combination
$B_{Y}$ (the gauge boson corresponding to the remnant $U(1)_{Y}$ symmetry)
remains massless at this stage. Notice that the breaking of $SU(2)_{R}\times
U(1)_{B-L}$ also generates large Majorana masses for right–handed neutrinos.
The second stage of symmetry breaking happens at the electroweak scale, when
the scalar bi-doublet $\Phi$ gets VEVs $v_{1}/\sqrt{2}$ and $v_{2}/\sqrt{2}$,
and the intermediate $SU(2)_{L}\times U(1)_{Y}$ breaks down to the $U(1)_{em}$
symmetry, leading to the usual relation $Q=T^{3}_{L}+Y$, so that
$Q=T^{3}_{L}+T^{3}_{R}+\dfrac{1}{2}(B-L)$. This leads to the mass generation
of all the known SM particles. Moreover, since $\Phi$ is a bi-doublet, its
VEVs generate mixings among left–handed and right–handed gauge bosons. This
also relates the gauge couplings in the following manner222When models are
embedded into a Grand Unified Theory, the normalization of $g_{R}$ and
$g_{B-L}$ is that of Eqs. (A-A.3).

$\displaystyle\dfrac{1}{g_{Y}^{2}}$ $\displaystyle=$
$\displaystyle\dfrac{1}{g^{2}_{R}}+\dfrac{1}{g^{2}_{B-L}}\,,$ (3.1)
$\displaystyle\dfrac{1}{e^{2}}$ $\displaystyle=$
$\displaystyle\dfrac{1}{g^{2}_{L}}+\dfrac{1}{g^{2}_{Y}}\,,$ (3.2)

where $g_{\alpha}$ is the gauge coupling of $U(1)_{\alpha}$
($\alpha=B-L,\,Y$). The $SU(2)_{L(R)}$ gauge coupling is denoted by $g_{L(R)}$
while $e$ is the gauge coupling of the residual $U(1)_{em}$. The three mixing
angles among the neutral gauge bosons i.e. $\theta_{R}$ (mixing between
$W^{3}_{R}$ and $B$), $\theta_{W}$ (mixing between $W^{3}_{L}$ and $B_{Y}$,
the Weinberg angle) and $\phi$ (mixing between $Z_{R}$ and $Z_{L}$) are given
by [38]:

$\theta_{W}=\tan^{-1}\left(\dfrac{g_{Y}}{g_{L}}\right),\,\,\theta_{R}=\sin^{-1}\left(\dfrac{g_{L}}{g_{R}}\,\tan\theta_{W}\right),\,\,\phi\simeq\dfrac{1}{2}\tan^{-1}\left(-2\,\cos\theta_{W}\cos\theta_{R}\dfrac{g_{R}}{g_{L}}\dfrac{M^{2}_{Z_{L}}}{M^{2}_{Z_{R}}}\right),$
(3.3)

where:

$\displaystyle M^{2}_{Z_{L}}\simeq
M^{2}_{Z_{1}}=\dfrac{g^{2}_{L}}{4\,\cos^{2}\theta_{W}}\,v^{2}=M^{2}_{Z},\,\,\,M^{2}_{Z_{R}}\simeq
M^{2}_{Z_{2}}=\dfrac{g^{2}_{R}}{\cos^{2}\theta_{R}}v^{2}_{R}+\dfrac{g^{2}_{R}}{4}\cos^{2}\theta_{R}\,v^{2}.$
(3.4)

In Eq. (3.4) the mixing $\phi$ is neglected since $M_{Z_{R}}\gg M_{Z_{L}}$,
$Z$ is the Standard Model massive neutral gauge boson and
$v=\sqrt{v^{2}_{1}+v^{2}_{2}}=246$ GeV. The expression of $\phi$ in Eq. (3.3)
is valid in the phenomenological limit where $v_{R}>>v_{1},\,v_{2}$ and the
VEV of $T_{L}$ ($v_{L}$) $\rightarrow 0$. Similarly, when $v_{1}$ and $v_{2}$
are real the mixing angle between $W^{\pm}_{R}$ and $W^{\pm}_{L}$ is given, in
the phenomenological limit, by [38]:

$\displaystyle\xi$ $\displaystyle\simeq$
$\displaystyle\dfrac{1}{2}\tan^{-1}\left(-4\,\dfrac{g_{R}}{g_{L}}\dfrac{M^{2}_{W_{L}}}{M^{2}_{W_{R}}}\dfrac{v_{1}v_{2}}{v^{2}}\right)\,,$
(3.5)

with:

$\displaystyle M^{2}_{W_{L}}\simeq
M^{2}_{W_{1}}=\dfrac{g^{2}_{L}}{4}\,v^{2}=M^{2}_{W},\,\,\,\,\,\,\,\,\,\,M^{2}_{W_{R}}\simeq
M^{2}_{W_{2}}=\dfrac{1}{4}g^{2}_{R}\left(v^{2}+2v^{2}_{R}\right)\,,$ (3.6)

where $M_{W}$ is the mass of the electroweak charged gauge boson of the
Standard Model.

In the DM scenario that we wish to discuss the LRSM is minimally extended by
adding a self–conjugate fermionic bi-doublet $\Psi$ (with
$\tilde{\Psi}\equiv-\sigma_{2}\Psi^{c}\sigma_{2}=\Psi$) having zero $B-L$
charge. In $2\times 2$ matrix representation, the bi-doublet $\Psi$ can be
written explicitly in terms of two Dirac field $\psi^{\pm}$ and $\psi^{0}$ as
[38]:

$\displaystyle\Psi=\begin{bmatrix}\psi^{0}&\psi^{+}\\\ \\\
\psi^{-}&-\left(\psi^{0}\right)^{c}\end{bmatrix}.$ (3.7)

The Lagrangian for $\Psi$ is given by

$\displaystyle\mathcal{L}_{\rm BD}=\dfrac{1}{2}{\rm
Tr}\left[\overline{\Psi}i\not{D}\Psi\right]-\dfrac{1}{2}M_{\Psi}{\rm
Tr}\left[\overline{\Psi}\Psi\right]\,,$ (3.8)

with covariant derivative:

$\displaystyle
D_{\mu}\Psi=\partial_{\mu}\Psi-i\dfrac{g_{L}}{2}\sigma_{a}{W_{L}^{a}}_{\mu}\Psi+i\dfrac{g_{R}}{2}\Psi\,\sigma_{a}{W_{R}^{a}}_{\mu}\,.$
(3.9)

Substituting, the expressions of $D_{\mu}$ and $\Psi$ in Eq. (3.8), the
Lagrangian in terms of the component fields reads:

$\displaystyle\mathcal{L}_{\rm BD}$ $\displaystyle=$ $\displaystyle
i\overline{\psi^{0}}\not{\partial}\psi^{0}+i\overline{\psi^{-}}\not{\partial}\psi^{-}+\dfrac{g_{L}}{2}\left(\overline{\psi^{0}}\not{W}^{3}_{L}{\psi^{0}}-\overline{\psi^{-}}\not{W}^{3}_{L}{\psi^{-}}+\sqrt{2}\,\,\overline{\psi^{0}}\not{W}^{+}_{L}\psi^{-}+\sqrt{2}\,\,\overline{\psi^{-}}\not{W}^{-}_{L}\psi^{0}\right)$
(3.10)
$\displaystyle~{}~{}~{}~{}~{}-\dfrac{g_{R}}{2}\left(\overline{\psi^{0}}\not{W}^{3}_{R}{\psi^{0}}+\overline{\psi^{-}}\not{W}^{3}_{R}{\psi^{-}}+\sqrt{2}\,\,\overline{\psi^{0}}{\not{W}^{-}_{R}}\psi^{+}+\sqrt{2}\,\,\overline{\psi^{+}}{\not{W}^{+}_{R}}\psi^{0}\right)$
$\displaystyle~{}~{}~{}~{}~{}-M_{\Psi}\overline{\psi^{0}}\psi^{0}-M_{\Psi}\overline{\psi^{-}}\psi^{-}\,.$

Although in Eq. (3.10) $\psi^{\pm}$ and $\psi^{0}$ have the degenerate tree
level mass $M_{\Psi}$, a mass splitting

$\displaystyle M_{\psi^{Q}}-M_{\psi^{0}}$ $\displaystyle\simeq$ $\displaystyle
Q\left(Q+\dfrac{2Y}{\cos\theta_{W}}\right)\Delta{M}\,,$ $\displaystyle{\rm
with}$ $\displaystyle\Delta{M}$ $\displaystyle=$
$\displaystyle\dfrac{g^{2}_{L}}{4\pi}M_{W_{L}}\sin^{2}\frac{\theta_{W}}{2}\simeq
166\,\,{\rm MeV}\,,$

is generated between the neutral and charged components by radiative
corrections [64]. Due to such mass splitting the charged component
$\psi^{\pm}$ is unstable and decays into $\psi^{0}$ and $\pi^{\pm}$ or a pair
of light leptons on very short timescales $\mathcal{O}(10^{-11})$ s [38].
Notice that the specific representation of $\Psi$ in Eq. (3.7) only allows the
terms in Eq. (3.8), automatically preventing the decay of its lightest
component $\psi_{0}$, that is then a natural DM candidate.

Moreover the Dirac nature of $\psi^{0}$ is only protected by the
$SU(2)_{L}\times SU(2)_{R}$ symmetry, which is no longer a good symmetry after
the bi-doublet $\Phi$ acquires non–vanishing VEVs. Specifically, the
$W^{\pm}_{R}-W^{\pm}_{L}$ mixing induces a transition at the one–loop level
between $\psi^{0}$ and $(\psi^{0})^{c}$ (with either $W^{\pm}_{1}$ or
$W^{\pm}_{2}$ and $\psi^{\mp}$ running in the loop) that generates a tiny
($\sim\mathcal{O}$(keV–MeV)) off-diagonal Majorana mass term $\delta{M}$
between $\psi^{0}$ and $(\psi^{0})^{c}$ proportional to $\sin(2\xi)$ [38].
Consequently, the Dirac fermion splits into two quasi-degenerate Majorana
fermions $\chi_{{}_{1,\,2}}$ with opposite CP given by:

$\displaystyle\chi_{{}_{1,\,2}}$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{2}}\left(\psi^{0}\mp(\psi^{0})^{c}\right)\,,$
$\displaystyle{\rm with}$ $\displaystyle M_{\chi_{1,\,2}}$ $\displaystyle=$
$\displaystyle M_{\Psi}\mp\delta{M}\,,$ (3.13)

providing in a natural way a specific realization of the IDM scenario that we
discussed at the phenomenological level in Section 2. The mass splitting
between $\chi_{1}$ and $\chi_{2}$ is given by [38]:

$\delta=2\hskip
1.70717pt\delta{M}=\frac{g_{L}^{2}}{16\pi^{2}}\frac{g_{R}}{g_{L}}\sin(2\xi)M_{\Psi}\left[f(r_{W_{1}})-f(r_{W_{2}})\right],$
(3.14)

where $r_{V}=M_{V}/M_{\Psi}$ and the loop function $f(r_{V})$ given by:

$f(r_{V})=2\int_{0}^{1}dx\hskip
1.70717pt(1+x)\log\left[x^{2}+(1-x)r^{2}_{V}\right].$ (3.15)

In particular, in the phenomenological analysis of Section 3.2 we will assume
$v_{1}=v_{2}$ in order to maximize the splitting $\delta$ at fixed $M_{W_{2}}$
and $g_{R}$ (see Eq. (3.5)).

Finally, in terms of the physical states $\chi_{1}$ and $\chi_{2}$ Eq. (3.10)
is given by:

$\displaystyle\mathcal{L}_{\rm BD}$ $\displaystyle=$
$\displaystyle\dfrac{i}{2}\overline{\chi_{1}}\not{\partial}\chi_{1}+\dfrac{i}{2}\overline{\chi_{2}}\not{\partial}\chi_{2}+i\overline{\psi^{-}}\not{\partial}\psi^{-}-\dfrac{M_{\chi_{1}}}{2}\overline{\chi_{1}}\chi_{1}-\dfrac{M_{\chi_{2}}}{2}\overline{\chi_{2}}\chi_{2}-M_{\psi^{-}}\overline{\psi^{-}}\psi^{-}$
(3.16)
$\displaystyle+\dfrac{1}{2}\overline{\chi_{1}}\left(g_{L}\not{W}^{3}_{L}-g_{R}\not{W}^{3}_{R}\right)\chi_{2}-\dfrac{1}{2}\overline{\psi^{-}}\left(g_{L}\not{W}^{3}_{L}+g_{R}\not{W}^{3}_{R}\right)\psi^{-}$
$\displaystyle+\left\\{\dfrac{1}{2}\overline{\chi_{1}}\left(g_{L}\not{W}^{+}_{L}-g_{R}\not{W}^{+}_{R}\right)\psi^{-}+\dfrac{1}{2}\overline{\chi_{2}}\left(g_{L}\not{W}^{+}_{L}+g_{R}\not{W}^{+}_{R}\right)\psi^{-}+{\rm
h.c.}\right\\},$

where, $M_{\psi^{-}}\simeq M_{\Psi}+(1+\sec\theta_{W})\Delta{M}$ from Eq.
(3.1).

Depending on the mass splitting $\delta$ the heavier bi-doublet state
$\chi_{2}$ decays into the lighter state $\chi_{1}$ via different channels,
with the radiative decay $\chi_{2}\rightarrow\chi_{1}\gamma$ always
kinematically allowed having a lifetime $\simeq 10^{-3}\hskip 1.42262pt{\rm
s}\left(\frac{1\hskip 1.42262pt{\rm MeV}}{\delta}\right)^{3}$ [38] (such
expression is valid when $M_{\Psi}\simeq{\cal O}$(TeV) and $\delta\ll
M_{\Psi}$). As a consequence, for the choice of parameters in the next
Section, by the present time all the $\chi_{2}$ states have decayed and
$\chi_{1}$ is the DM candidate in our Universe.

###  3.2 Bounds from White Dwarves on LRSM Bi–doublet Dark Matter

In this Section we will identify $\chi$ and $\chi^{\prime}$ with $\chi_{1}$
and $\chi_{2}$. The process of inelastic scattering of $\chi$ against a WD
nucleus occurs dominantly through the exchange of the light $Z$-boson. The
corresponding interaction Lagrangian (involving $\chi$, $\chi^{\prime}$ and
$Z$) can be written as:

$\mathcal{L}_{\chi\chi^{\prime}Z}=g_{\chi}\hskip
1.42262pt\overline{\chi}\not{Z}\chi^{\prime},$ (3.17)

where

$g_{\chi}=\frac{1}{2}\left(g_{L}\cos\theta_{W}\cos\phi+g_{R}\sin\theta_{W}\sin\theta_{R}\cos\phi+g_{R}\cos\theta_{R}\sin\phi\right).$
(3.18)

The coupling $g_{\chi}$ is obtained by expressing $W^{3}_{L}$ and $W^{3}_{R}$
in Eq. (3.16) in terms of the mass eigenstate $Z$, using the neutral gauge
boson mixing matrix given in the appendix of [38]. Since the typical values of
the momentum transfer in such a process are always much lower than $M_{Z}$,
one can express the interaction of $\chi$ and $\chi^{\prime}$ with SM quarks
in terms of the following dimension-6 operators:

$\mathcal{O}^{(6)}_{V,q}=C^{(6)}_{V,q}\left(\overline{\chi}\gamma_{\mu}\chi^{\prime}\right)\left(\overline{q}\gamma^{\mu}q\right),\hskip
11.38109pt\mathcal{O}^{(6)}_{A,q}=C^{(6)}_{A,q}\left(\overline{\chi}\gamma_{\mu}\chi^{\prime}\right)\left(\overline{q}\gamma^{\mu}\gamma_{5}q\right),$
(3.19)

with:

$C^{(6)}_{V,q}=\frac{g_{\chi}g_{L}g^{q}_{V}}{M^{2}_{Z}\cos\theta_{W}},\hskip
11.38109ptC^{(6)}_{A,q}=\frac{g_{\chi}g_{L}g^{q}_{A}}{M^{2}_{Z}\cos\theta_{W}}\,,$
(3.20)

where $g^{q}_{V}$ and $g^{q}_{A}$ are defined in [67].

In the non–relativistic limit the above interaction terms lead to a
WIMP–nucleon interaction driven by the effective Hamiltonian:

${\cal H}=\sum_{\mathcal{N}=p,n}(c_{1}^{\mathcal{N}}{\cal
O}_{1}^{\mathcal{N}}+c_{7}^{\mathcal{N}}{\cal
O}_{7}^{\mathcal{N}}+c_{9}^{\mathcal{N}}{\cal O}_{9}^{\mathcal{N}}),$ (3.21)

(in the operator base of [68]), with $p,n$ indicating proton and neutron. The
Wilson coefficients $c_{1}^{\mathcal{N}}$, $c_{7}^{\mathcal{N}}$ and
$c_{9}^{\mathcal{N}}$ in terms of the $C^{(6)}_{V,q}$ and $C^{(6)}_{A,q}$
quantities can be obtained for instance, from [69]. In particular, the
effective operator ${\cal O}_{1}$ leads to a spin–independent WIMP–nucleus
cross-section proportional to the square of the atomic mass number of the
target, while the ${\cal O}_{7}$ and ${\cal O}_{9}$ operators, which are
velocity and momentum suppressed, vanish in the case of the spinless targets
(${}^{12}C$, ${}^{16}O$ or ${}^{20}Ne$) that constitute the interior of a WD.
Specifically, for the operator $\mathcal{O}^{(6)}_{V,q}$ one has
$c_{1}^{\mathcal{N}}$ = $\sum_{q=u,d}C^{(6)}_{V,q}F_{\mathcal{N}}^{(q)}$ with:

$\displaystyle F^{(u)}_{p}$ $\displaystyle=$ $\displaystyle 2,\hskip
1.42262ptF^{(d)}_{p}=1,$ (3.22) $\displaystyle F^{(u)}_{n}$ $\displaystyle=$
$\displaystyle 1,\hskip 1.42262ptF^{(d)}_{n}=2,$ (3.23)

while the WIMP–nucleon cross-section $\sigma_{\mathcal{N}}$ introduced in
Section 2 is given by
$\sigma_{\mathcal{N}}=(c_{1}^{\mathcal{N}})^{2}\mu_{\chi\mathcal{N}}^{2}/\pi$.
In order to evaluate the WD capture rate we have assumed that the WDs observed
in M4 are made of carbon and we have calculated the differential cross-section
$\frac{d\sigma[\chi+N\rightarrow\chi^{\prime}+N]}{dE}$ in Eq. (2.5) using the
`WimPyDD` code [70, 71].

Figure 3: Bi-doublet DM induced luminosities in optically thin (solid blue and
red dashed lines) and geometric (horizontal cyan lines) limits as functions of
the mass splitting $\delta$, considering two benchmark carbon WDs in M4 with
masses $M_{*}=1.38$ $M_{\odot}$ (left panel) and $M_{*}=1.25$ $M_{\odot}$
(right panel). The solid blue line and the red dashed line correspond to the
capture of 500 GeV and 5 TeV DM particles, respectively. The black dotted
lines in both panels represent the luminosities observed for the considered
WDs [34]. The vertical gray lines indicate the maximum values of $\delta$ that
are possible to probe for these WDs.

In Fig. 3 the solid blue line and the dashed red line represent the result of
the evaluation of the expected luminosity of Eq. (2.12) for the capture of
LRSM bi–doublet DM by WDs in the optically–thin limit as a function of the
mass splitting parameter $\delta$ for DM masses $m_{\chi}$ = 500 GeV and
$m_{\chi}$ = 5 TeV, respectively, for $g_{L}=g_{R}$ and for the the same
carbon WDs of Fig. 1. In both plots the cyan and black dotted horizontal lines
are the same of Fig. 1 and represent, respectively, the geometrical limit of
the capture rate and the corresponding observed luminosities [34]. As
expected, unless the capture process in the WD is kinematically forbidden the
large WIMP–nucleus cross-section implied by the non–vanishing hypercharge of
the bi–doublet saturates the geometrical capture upper bound and exceeds the
observation. This is achieved for $\delta\mathrel{\mathop{\kern 0.0pt\hbox
to0.0pt{\raise 0.86108pt\hbox{$>$}\hss}}\lower
3.87495pt\hbox{\kern-1.90002pt$\sim$}}$ 40 MeV (17 MeV) for $M_{*}=1.38$
$M_{\odot}$ ($M_{*}=1.25$ $M_{\odot}$).

Notice that, as described in Section 3.1, the heavier bi-doublet state
$\chi^{\prime}$ decays to the DM state $\chi$ within a time-scale that is
extremely small compared to the typical age of the WDs in M4, expected to be a
few Gyrs. In such a case, the expressions of the capture rate given in Section
2.2 apply, and the final population of DM particles in the WD core consists of
only $\chi$. The pair-annihilation of these particles, which is driven by
t-channel processes, injects electroweak gauge bosons whose cascade products
thermalise in the WD medium and heat up the star.

Figure 4: The region of the bi-doublet parameter space (spanned by $M_{W_{2}}$
and the bi-doublet DM mass $m_{\chi}$) that are excluded by the observation of
the faintest and heaviest ($M_{*}=1.38$ $M_{\odot}$) WD (assuming it is made
of carbon) in M4 is shown by the blue shaded area, considering $g_{R}=g_{L}$.
The pink, green and red shaded areas correspond to the regions excluded by LHC
search, direct detection (DD) experiments and solar neutrino search,
respectively. The black band consists of parameter points where the calculated
thermal relic density is in the observed range.

Figure 5: Same as Fig. 4, but considering scenarios with $g_{R}/g_{L}=1.2$
(left panel) and $g_{R}/g_{L}=1.8$ (right panel).

As already pointed out in Section 2, the bound from WDs is much more
constraining than that expected from direct detection and from solar neutrino
searches. The impact from such bound on the $m_{\chi}$–$M_{W_{2}}$ parameter
space of the LRSM bi-doublet Dark Matter is shown in Fig. 4 for $g_{R}=g_{L}$,
which represents the most commonly used value in phenomenological and
experimental analyses. In the upper–left corner the green–shaded area shows
the part of parameter space excluded by direct detection, which can probe
$\delta\mathrel{\mathop{\kern 0.0pt\hbox to0.0pt{\raise
0.86108pt\hbox{$<$}\hss}}\lower 3.87495pt\hbox{\kern-1.90002pt$\sim$}}$ 200
keV [27], while the red–shaded one the part of parameter space excluded by the
solar neutrino searches, which can probe $\delta\mathrel{\mathop{\kern
0.0pt\hbox to0.0pt{\raise 0.86108pt\hbox{$<$}\hss}}\lower
3.87495pt\hbox{\kern-1.90002pt$\sim$}}$ 600 keV [28]. On the other hand, the
pink–shaded region is excluded by the non observation of the $W_{2}$ boson at
the LHC ($M_{W_{2}}<$ 3.8 TeV [72]), and the black band represents the region
where the thermal relic density is in the observed range (0.11$\leq\Omega
h^{2}\leq$0.13). As shown in such figure, when only the direct detection,
solar neutrino searches and the LHC bounds are considered (as done in the
literature so far [38]) the LRSM bi–doublet is a viable DM candidate in a wide
range of masses, 1.1 TeV $\mathrel{\mathop{\kern 0.0pt\hbox to0.0pt{\raise
0.86108pt\hbox{$<$}\hss}}\lower
3.87495pt\hbox{\kern-1.90002pt$\sim$}}m_{\chi}\mathrel{\mathop{\kern
0.0pt\hbox to0.0pt{\raise 0.86108pt\hbox{$<$}\hss}}\lower
3.87495pt\hbox{\kern-1.90002pt$\sim$}}$ 30 TeV and for $g_{R}=g_{L}$. However,
when the bound from WD is considered this changes considerably: in Fig. 4 the
blue shaded area, that lies above the dashed blue line for $\delta$ = 40 MeV,
represents the parameter space excluded by WDs. Indeed, for $g_{R}=g_{L}$ the
full cosmologically viable parameter space now lies in such region and is
ruled out. As pointed out below Eq. (3.14) in our calculation we have
maximized $\delta$ at fixed $m_{\chi}$, $g_{R}$ and $M_{W_{2}}$ in terms of
the other parameters of the model. As a consequence, for other choices of such
parameters the $M_{W_{2}}$ values excluded by WDs would be even lower, so that
the excluded region of Fig. 4 should be considered as a conservative one.

In order to evade the WD constraint one needs to increase further the value of
the mass splitting parameter $\delta$ at fixed $m_{\chi}$ and $M_{W_{2}}$. As
shown in Eq. (3.14), the only way to achieve this within the available
parameter space is to chose $g_{R}>g_{L}$. This is done in Fig. 5, where
$g_{R}/g_{L}$ =1.2 in the left–hand plot, and $g_{R}/g_{L}$ =1.8 in the
right–hand plot. Indeed, now in both cases some cosmologically viable
parameter space is recovered. In particular, for $g_{R}/g_{L}$ =1.2 this
requires $M_{W_{2}}\mathrel{\mathop{\kern 0.0pt\hbox to0.0pt{\raise
0.86108pt\hbox{$<$}\hss}}\lower 3.87495pt\hbox{\kern-1.90002pt$\sim$}}$ 6 TeV,
and 3.5 TeV $\mathrel{\mathop{\kern 0.0pt\hbox to0.0pt{\raise
0.86108pt\hbox{$<$}\hss}}\lower
3.87495pt\hbox{\kern-1.90002pt$\sim$}}m_{\chi}\mathrel{\mathop{\kern
0.0pt\hbox to0.0pt{\raise 0.86108pt\hbox{$<$}\hss}}\lower
3.87495pt\hbox{\kern-1.90002pt$\sim$}}$ 5 TeV, while for $g_{R}/g_{L}$ =1.8
$M_{W_{2}}\mathrel{\mathop{\kern 0.0pt\hbox to0.0pt{\raise
0.86108pt\hbox{$<$}\hss}}\lower 3.87495pt\hbox{\kern-1.90002pt$\sim$}}$ 12 TeV
and two bi-doublet allowed mass ranges are found: 1.2 TeV
$\mathrel{\mathop{\kern 0.0pt\hbox to0.0pt{\raise
0.86108pt\hbox{$<$}\hss}}\lower
3.87495pt\hbox{\kern-1.90002pt$\sim$}}m_{\chi}\mathrel{\mathop{\kern
0.0pt\hbox to0.0pt{\raise 0.86108pt\hbox{$<$}\hss}}\lower
3.87495pt\hbox{\kern-1.90002pt$\sim$}}$ 3 TeV and 5 TeV
$\mathrel{\mathop{\kern 0.0pt\hbox to0.0pt{\raise
0.86108pt\hbox{$<$}\hss}}\lower
3.87495pt\hbox{\kern-1.90002pt$\sim$}}m_{\chi}\mathrel{\mathop{\kern
0.0pt\hbox to0.0pt{\raise 0.86108pt\hbox{$<$}\hss}}\lower
3.87495pt\hbox{\kern-1.90002pt$\sim$}}$ 10 TeV. In Section 3.3 we will discuss
how $g_{R}/g_{L}>1$ can be achieved at low energy when the LRSM model is
embedded at high scale in a Grand Unified Theory, and provide an explicit
example with $g_{R}/g_{L}\simeq$ 1.2. On the other hand we consider
$g_{R}/g_{L}$ =1.8 a reasonable estimation of the maximal allowed value of
such ratio since it roughly coincides to the perturbative upper bound
discussed in [73]333See Fig. 7 of [73]. However, at variance with Fig. 5, in
the discussion of Ref. [73] $g_{R}/g_{L}$=1.8 is correlated to a high
$M_{W_{2}}$ mass, $M_{W_{2}}\mathrel{\mathop{\kern 0.0pt\hbox to0.0pt{\raise
0.86108pt\hbox{$>$}\hss}}\lower 3.87495pt\hbox{\kern-1.90002pt$\sim$}}$ 90
TeV..

For the calculation of the bi–doublet relic abundance in Figs. 4 and 5 we have
modified the `FeynRules` [74] implementation of the LRSM model of Ref. [75] in
order to allow $g_{R}\neq g_{L}$, and used it to produce a `CalcHEP` [76]
output file, that we have then modified implementing the Lagrangian of Eq.
(3.10) 444The relic abundance can be directly calculated in terms of the
$\Psi^{0}$ and $\psi^{\pm}$ Dirac fermions of the bi-doublet of Eq. (3.7),
since in the early Universe the effect of the mass splitting is negligible in
the calculation of the annihilation cross-section.. Finally, we have used such
modified `CalcHEP` file in `MicrOMEGAs` [77].

###  3.3 Consequences at High Scale

The implementation of the bound from WDs in the phenomenological analysis of
Section 3.2 singles out a relatively well defined parameter space in order for
the LRSM bi–doublet scenario to provide a viable DM candidate, characterized
by a low mass for the $W_{2}$ boson ($M_{W_{2}}\mathrel{\mathop{\kern
0.0pt\hbox to0.0pt{\raise 0.86108pt\hbox{$<$}\hss}}\lower
3.87495pt\hbox{\kern-1.90002pt$\sim$}}$ 10 TeV) and $g_{R}>g_{L}$. In this
Section we wish to discuss whether such parameter space can be obtained, and
how naturally, when the LRSM model is embedded in a Grand Unified Theory,
which represents its most popular ultraviolet completion.

The LRSM scenario naturally arises in Grand Unified Theories (GUT). Indeed,
many scenarios have been discussed in the literature [78, 79, 80, 81, 82, 83],
where the LR group $SU(2)_{L}\times SU(2)_{R}$ is embedded into a GUT SO(10).
Specific examples include: (i) SO(10)$\supset SU(2)_{L}\times SU(2)_{R}\times
SU(4)_{C}\times{\rm D}$; (ii) SO(10)$\supset SU(2)_{L}\times SU(2)_{R}\times
SU(4)_{C}$, (iii) SO(10)$\supset{\rm LRSMD}\equiv SU(2)_{L}\times
SU(2)_{R}\times SU(3)_{C}\times U(1)_{B-L}\times{\rm D}$ and (iv)
SO(10)$\supset{\rm LRSM}\cancel{\rm D}\equiv SU(2)_{L}\times SU(2)_{R}\times
SU(3)_{C}\times U(1)_{B-L}$. Here D stands for the parity symmetry that leaves
the theory invariant by the interchange of any multiplet of $SU(2)_{L}$ into
the corresponding $SU(2)_{R}$ multiplet. The low–energy scenario discussed in
Sections 3.1 and 3.2 corresponds to either LRSMD or LRSM$\cancel{\rm D}$
extended with the DM bi–doublet $\Psi$ (LRSMD + $\Psi$ and LRSM$\cancel{\rm
D}$ \+ $\Psi$ in the following).

In the analysis of Section 3.2 we have used WDs to constrain the bi–doublet
LRSM parameter space at the phenomenological level, i.e. taking the low–energy
parameters of the model introduced in Section 3.1 ($M_{\chi_{1}}$ =
$m_{\chi}$, $M_{W_{2}}$, $g_{R}$ and $v_{1}/v_{2}$) as independent without
assuming any specific ultraviolet completion. In a GUT theory context such
parameters are expected to be correlated. In the literature most often the so
called _minimal_ version of the LRSM assumes D parity. However, in the
following we will consider both cases, conserved D parity and broken D parity.

The most important appeal of GUT is the unification of gauge couplings at
high-energy, which comes together with the salient prediction of proton decay.
Therefore, once the embedding of the LR models in a GUT framework comes into
play we need to pay attention to the GUT scale and to the proton decay rate.
In particular, in some of its realizations LRSM models embedded in SO(10)
typically achieve a unification scale around $10^{15}$ GeV, which leads to
proton decay rates incompatible to the experimental constraints. Specifically,
for the minimal particle content of Tab. 2 and the embedding (i) the breaking
of the $SU(2)_{R}$ is achieved at around $\>M_{R}\approx 10^{13}$ GeV and the
GUT scale is $\>M_{\text{GUT}}\approx 10^{15}$ GeV. On the other hand, for
(ii) $\>M_{R}$ is around $10^{11}$ GeV, while $\>M_{\text{GUT}}\approx
10^{16}$ GeV. For (iii) LRSMD, $\>M_{R}\approx 10^{10}$ GeV, while the
unification scale is achieved around $\>M_{\text{GUT}}\approx 10^{15}$ GeV.
Finally for (iv) LRSM~~D~~ , $\>M_{R}\approx 10^{9}$ GeV and
$\>M_{\text{GUT}}\approx 10^{16}$ GeV [84] 555For the exact values of
$\>M_{R}$ and $\>M_{\text{GUT}}$ in each case check the original references
and the updates of [85, 86].

These basic scenarios hence do not offer an explanation of why $v_{R}\simeq
M_{W_{2}}\mathrel{\mathop{\kern 0.0pt\hbox to0.0pt{\raise
0.86108pt\hbox{$<$}\hss}}\lower 3.87495pt\hbox{\kern-1.90002pt$\sim$}}10^{4}$
GeV, as emerging from Fig. 4 (notice that, depending on the specific mechanism
of symmetry breaking, it is natural to expect that $v_{R}$ also determines the
WIMP mass $m_{\chi}$).

Extending the LRSMD model by adding only the DM bi–doublet fermion (the LRSMD
+ $\Psi$ case) actually slightly increases the value $\>M_{R}\simeq v_{R}$ of
the scale at which $SU(2)_{R}$ is broken and increases the proton decay (PD)
lifetime (see Tab. 1) 666It is well known that in non-supersymmetric theories
the dominant decay channel is $p\rightarrow\pi^{0}e^{+}$. For specific
formulas see for example Appendix D of [85] and Sections 3 and 4.2 of [87]..
So although adding $\Psi$ to the model can increase the PD lifetime (albeit
not enough in most cases) it is of no help to reduce $\>M_{R}$. This case is
shown in the top left plot of Fig. 6, where the running of the coupling
constants is plotted as a function of the energy scale $\mu$. As a
consequence, in order to lower $\>M_{R}$ one needs to add extra-matter (in
particular, all the additional degrees of freedom discussed in the following
examples are listed in Tab. 2). In [88] it was found that adding color
multiplets is key to lower $\>M_{R}$ down to $10^{5}$ GeV or $10^{4}$ GeV.
Notice that such additional high–scale degrees of freedom do not affect the
phenomenology discussed in Section 3.2. In the top-right plot of Fig. 6 we
show a specific example of this, where the MLRSM model is extended to include,
besides the DM fermion bi-doublet $\Psi$, four fermions $F_{c}^{\prime}$,
which are triplets under $SU(3)_{C}$ and singlets under all other interactions
except for the D parity charge, and one fermion $F_{d}$, which is a bi-doublet
under $SU(2)_{R}$ and $SU(2)_{L}$ and can have a non-zero D parity charge. We
can see that now the scale $\>M_{R}$ is lowered. However, the ratio
$g_{R}/g_{L}$ is still smaller than one at the electroweak scale $M_{\rm
EW}\simeq 10^{2}$ GeV.

The feature $g_{R}(\>M_{\text{EW}})/g_{L}(\>M_{\text{EW}})<1$ (that for
simplicity will be just indicated as $g_{R}/g_{L}$ in the discussion below) is
always present in D parity conserving models since the running of $g_{R}$ and
$g_{L}$ only splits below $M_{R}$, where $g_{R}$ practically stops evolving
while the value of $g_{L}$ gets larger as the energy scale is decreased
(notice that in Fig. 6 inverse couplings are plotted). So the only way to
obtain $g_{R}/g_{L}>$1 is to break D parity at high scale, i.e. to split the
running of $g_{R}$ and $g_{L}$ for $M_{R}<\mu<M_{\rm GUT}$, so that
$g_{R}>g_{L}$ already at $M_{R}$. This can be achieved in different ways which
are anomaly–free [88]. The most straightforward is by simply omitting the
left-handed triplet $T_{L}$ [73] (also this does not affect the
phenomenology). This case, indicated by LRSM$\cancel{\rm D}$ \+ $\Psi$, is
shown in the left–bottom plot of Fig. 6. Since the scale $\>M_{R}$ is lower
than for the LRSMD + $\Psi$ case the ratio $g_{R}/g_{L}$ turns out to be
slightly bigger but nevertheless less than 1 (notice also that in this case PD
is not consistent with observation). For this reason, in the final example
shown in the bottom–right plot of Fig. 6 we add more fermions transforming
under $SU(2)_{L}$ only than under $SU(2)_{R}$, so that the slope of $g_{L}$
with respect to the energy scale $\mu$ gets steeper compared to $g_{R}$. In
such scenario, indicated with LRSM$\cancel{\rm D}$ \+ $\Psi$ \+ 4 $F_{a}$ \+
$F_{b}$ \+ 4 $F_{c}$, we add four fermions $F_{a}$ which are doublets of
$SU(2)_{L}$, triplets of $SU(3)_{C}$ and singlets of the other groups, one
fermion $F_{b}$, which is a doublet of $SU(2)_{R}$, triplet of $SU(3)_{C}$ and
singlet of the other groups, and four fermions $F_{c}$ which are triplets of
$SU(3)_{c}$ and singlets under the rest of the groups (see Tab. 2). Indeed,
now $g_{R}/g_{L}\simeq$1.2 is achieved. This case then provides an example
where the value of $\>M_{R}$ can be explained to be around $10^{4}$ GeV
without any fine tuning, $M_{\rm GUT}\mathrel{\mathop{\kern 0.0pt\hbox
to0.0pt{\raise 0.86108pt\hbox{$>$}\hss}}\lower
3.87495pt\hbox{\kern-1.90002pt$\sim$}}10^{16}$ GeV so that the PD lifetime can
be acceptable (see Tab. 1), and $g_{R}/g_{L}>$1.

The values of the ratio $g_{R}/g_{L}$ are provided for all the models
discussed above in Fig. 6. To get the plots of Fig. 6 we obtained the
corresponding beta functions, given in Tab. 3, using the code `pyr@te` [89]
and used them in a private Runge-Kutta code.

Figure 6: Running of the inverse of the gauge couplings $\alpha_{i}$ of the
Left Right Symmetric models we consider. Top panels: D parity conserving
scenarios. Top-Left: LRSMD $+\,\Psi$, Minimal Model plus Dark Matter bi-
doublet. Top-Right: LRSMD $+\,\Psi\,+4\,F_{c}^{\prime}+F_{d}$. Bottom panels:
scenarios with broken D parity. Bottom-Left: LRSM~~D~~ $+\,\Psi$, Minimal
Model plus Dark Matter bi-doublet. Bottom-Right: LRSM~~D~~
$+\,\Psi\,+4\,F_{a}+F_{b}+4\,F_{c}$. The definition of the fermions $F_{i}$ is
given in Tab. 2. The value of $\>M_{R}$ ($M_{\rm GUT}$) corresponding to each
case is indicated by the black solid (dotted) line. Table 1:  Proton decay
lifetime $\tau(p\rightarrow\pi^{0}\,e^{+})$[yrs]. The 1 $\sigma$ errors are
extracted from a chi–square with the observables at the electroweak scale.
Model Predictions
---
LRSMD | $(1.3\pm 0.9)\times 10^{32}$
LRSMD + $\Psi$ | $(2.9\pm 1.9)\times 10^{32}$
LRSMD + $\Psi$ \+ 4 $F_{c}^{\prime}$ \+ $F_{d}$ | $(5.4\pm 3.7)\times 10^{36}$
LRSM~~D~~ | $(3.8\pm 2.5)\times 10^{36}$
LRSM~~D~~ \+ $\Psi$ | $(7.6\pm 5.2)\times 10^{32}$
LRSM~~D~~ \+ $\Psi$ \+ 4 $F_{a}$ \+ $F_{b}$ \+ 4 $F_{c}$ | $(2.0\pm 1.3)\times 10^{37}$
Experimental bounds/discovery
Current Bound: $1.6\times 10^{34}$ at $95\%$ C.L [90, 91] | Projected Discovery: $6.3\times 10^{34}$ [92]
Projected Bound: $7.8\times 10^{34}$ $90\%$ C.L [92] |

##  4 Conclusions

Models of structure formation suggest that the inner part of globular clusters
may have a DM density that largely exceeds that estimated in the solar
neighborhood. Assuming this is so and that the DM is a WIMP particle, compact
stars such as white dwarves can capture WIMPs through a process similar to
that studied in the case of the Sun or the Earth, leading to an increase of
the star luminosity through their annihilation process. As a consequence the
issue of the amount of DM in globular clusters, and specifically in M4, has
drawn increasing interest in the literature [29, 41, 34]. We have shown that
if the WIMP interacts with the nuclear targets within the WD through inelastic
scattering the data on low–temperature large-mass WDs in the Messier 4
globular cluster can lead to a constraint on the mass splitting
$\delta\mathrel{\mathop{\kern 0.0pt\hbox to0.0pt{\raise
0.86108pt\hbox{$<$}\hss}}\lower 3.87495pt\hbox{\kern-1.90002pt$\sim$}}$ few
tens of MeV that largely exceeds that ensuing from Direct Detection in
terrestrial underground experiments, $\delta\mathrel{\mathop{\kern 0.0pt\hbox
to0.0pt{\raise 0.86108pt\hbox{$<$}\hss}}\lower
3.87495pt\hbox{\kern-1.90002pt$\sim$}}$ 200 keV, and from solar neutrino
searches, $\delta\mathrel{\mathop{\kern 0.0pt\hbox to0.0pt{\raise
0.86108pt\hbox{$<$}\hss}}\lower 3.87495pt\hbox{\kern-1.90002pt$\sim$}}$ 600
keV, for a generic WIMP with a spin–independent cross-section off nucleons
larger than $\simeq$ a few $\times$ 10-43 cm2, assuming conservatively that
the WD is entirely composed of carbon.

We have applied such improved constraint to the specific DM scenario of a
self–conjugate bi-doublet in the Left–Right Symmetric Model (LRSM), where the
standard $SU(2)_{L}$ group with coupling $g_{L}$ is extended by an additional
$SU(2)_{R}$ with coupling $g_{R}$ in order to explain maximal parity violation
in Weak Interactions. The ensuing bounds significantly reduce the
cosmologically viable parameter space of such scenario, in particular
requiring $g_{R}>g_{L}$. For instance, for $g_{R}/g_{L}$ = 1.8 we have found
the two viable mass ranges for the bi–doublet DM candidate, 1.2 TeV
$\mathrel{\mathop{\kern 0.0pt\hbox to0.0pt{\raise
0.86108pt\hbox{$<$}\hss}}\lower
3.87495pt\hbox{\kern-1.90002pt$\sim$}}m_{\chi}\mathrel{\mathop{\kern
0.0pt\hbox to0.0pt{\raise 0.86108pt\hbox{$<$}\hss}}\lower
3.87495pt\hbox{\kern-1.90002pt$\sim$}}$ 3 TeV and 5 TeV
$\mathrel{\mathop{\kern 0.0pt\hbox to0.0pt{\raise
0.86108pt\hbox{$<$}\hss}}\lower
3.87495pt\hbox{\kern-1.90002pt$\sim$}}m_{\chi}\mathrel{\mathop{\kern
0.0pt\hbox to0.0pt{\raise 0.86108pt\hbox{$<$}\hss}}\lower
3.87495pt\hbox{\kern-1.90002pt$\sim$}}$ 10 TeV, when the charged $SU(2)_{R}$
gauge boson mass $M_{W_{2}}$ is less than $\simeq$ 12 TeV.

In Section 3.3 we have provided a short discussion on how such
phenomenological parameter space at low energy can be achieved when the LRSM
model is embedded at high scale in a Grand Unified Theory (GUT), showing that
in the minimal scenario of LRSM + DM bi–doublet one naturally gets
$M_{W_{2}}$, $m_{\chi}\mathrel{\mathop{\kern 0.0pt\hbox to0.0pt{\raise
0.86108pt\hbox{$>$}\hss}}\lower 3.87495pt\hbox{\kern-1.90002pt$\sim$}}$ 1010
GeV, and $g_{R}<g_{L}$, with a GUT scale $M_{\rm GUT}\simeq 10^{15}$ GeV
incompatible to proton decay bounds. As a consequence, one needs to extend the
ultraviolet completion of the LRSM model with additional multiplets besides
the DM bi–doublet in order for the latter to explain the DM in the Universe in
agreement with the WD bounds discussed in Section 3.2. Specifically, we found
that adding color triplets that are singlets under all other groups and carry
zero B-L charges can naturally yield $M_{W_{2}},m_{\chi}\mathrel{\mathop{\kern
0.0pt\hbox to0.0pt{\raise 0.86108pt\hbox{$<$}\hss}}\lower
3.87495pt\hbox{\kern-1.90002pt$\sim$}}$ 10 TeV. On the other hand, the only
way to obtain $g_{R}>g_{L}$ is to split the running of $g_{R}$ and $g_{L}$ by
assuming that D parity is broken at high scale, i.e. by introducing an
asymmetry between the multiplets of $SU(2)_{L}$ and $SU(2)_{R}$. We provided a
specific example of such scenario in Fig. 6, with the particle content of Tab.
2.

We conclude by reminding that the amount of DM in M4 is still controversial.
Our analyses motivates further studies to settle this issue. Moreover, at the
end of Section 2.2 we pointed out that the WDs bounds discussed in the present
paper could be extended using more dense stellar object such as neutron stars,
probing values of the inelastic splitting $\delta$ as large as $\sim 300$ MeV.
In particular, a future observation of neutron stars with temperatures
$T\mathrel{\mathop{\kern 0.0pt\hbox to0.0pt{\raise
0.86108pt\hbox{$<$}\hss}}\lower 3.87495pt\hbox{\kern-1.90002pt$\sim$}}$ a few
thousand Kelvin would rule out the full parameter space of LRSM bi–doublet DM.

## Acknowledgements

We thank Alexander Pukhov and Julian Heeck for useful discussions. We thank
Lohan Sartore for helping us with running `pyr@te`. This research was
supported by the National Research Foundation of Korea(NRF) funded by the
Ministry of Education through the Center for Quantum Space Time (CQUeST) with
grant number 2020R1A6A1A03047877 and by the Ministry of Science and ICT with
grant number 2021R1F1A1057119. L. V.-S. acknowledges the KIAS warm hospitality
and stimulating environment in these challenging times.

##  Appendix A Ultraviolet Completions of the LRSM Model and Beta Functions

Table 2: The matter content of the Minimal Left Right Symmetric Model and its
extensions discussed in Section 3.3. The LRSM with broken D parity does not
contain $T_{L}$, while the one with D parity conservation does.  Minimal Left
Right Symmetric Model
---
Matter | Generations | $SU(2)_{L}\times SU(2)_{R}\times U(1)_{B-L}\times SU(3)_{C}$
Fermions |  |
$L_{L}$ | 3 | ($\mathbf{2}$, $\mathbf{1}$, $-1$, $\mathbf{1}$)
$L_{R}$ | 3 | ($\mathbf{1}$, $\mathbf{2}$, $-1$, $\mathbf{1}$)
$Q_{L}$ | 3 | ($\mathbf{2}$, $\mathbf{1}$, $+\frac{1}{3}$, $\mathbf{3}$)
$Q_{R}$ | 3 | ($\mathbf{1}$, $\mathbf{2}$, $+\frac{1}{3}$, $\mathbf{3}$)
Scalars |  |
$\Phi$ | 1 | ($\mathbf{2}$, $\overline{\mathbf{2}}$, $0$, $\mathbf{1}$)
$T_{R}$ | 1 | ($\mathbf{1}$, $\mathbf{3}$, $+2$, $\mathbf{1}$)
$T_{L}$ | 1 | ($\mathbf{3}$, $\mathbf{1}$, $+2$, $\mathbf{1}$)
DM Candidates
Fermion |  |
$\Psi$ | 1 | ($\mathbf{2}$, $\mathbf{2}$, $0$, $\mathbf{1}$)
Additional Matter for conserving D parity
Fermions |  |
$F_{c}^{\prime}$ | 4 | ($\mathbf{1}$, $\mathbf{1}$, $0$ $\mathbf{3}$)L $\oplus$ ($\mathbf{1}$, $\mathbf{1}$, $0$, $\mathbf{3}$)R
$F_{d}$ | 1 | ($\mathbf{2}$, $\mathbf{2}$, $0$ $\mathbf{1}$)L $\oplus$ ($\mathbf{2}$, $\mathbf{2}$, $0$, $\mathbf{1}$)R
Additional Matter for ~~D~~ parity
Fermions |  |
$F_{a}$ | 4 | ($\mathbf{2}$, $\mathbf{1}$, $0$ $\mathbf{3}$)
$F_{b}$ | 1 | ($\mathbf{1}$, $\mathbf{2}$, $0$, $\mathbf{3}$)
$F_{c}$ | 4 | ($\mathbf{1}$, $\mathbf{1}$, $0$, $\mathbf{3}$)

#### Beta functions

The beta functions of the gauge couplings at two loops determine to a pretty
good accuracy the unification and breaking of the $SU(2)_{R}$ scales. At two
loops, the gauge couplings are only affected by their own running and the
running of the top Yukawa coupling and hence one can consider the running of
these quantities, without making any assumptions about the running of all of
the other parameters of the theory. Obviously, for a specific model one must
perform a complete running of all the parameters, but $\>M_{R}$ and
$\>M_{\text{GUT}}$ can be determined to a great accuracy. For the beta
functions of the gauge couplings of the models we present, we adopt the
convention

$\displaystyle\frac{dg_{i}}{dt}=\frac{b_{i}}{16\pi^{2}}g_{i}^{3}+\frac{g_{i}^{3}}{(16\pi^{2})^{2}}\left[\sum_{j=1}^{j=3}b_{ij}g_{j}^{2}-c_{i}y_{t}^{2}\right],$
(A.1)

where $g_{i}$ are the gauge couplings of each $i$ factor and $y_{t}$ is the
gauge couplings of top Yukawa coupling. The coefficients $b_{i}$, $b_{ij}$ and
$c_{i}$ are determined by group invariants and the matter content of the
theory. The beta functions of the basic LR symmetric model are well known for
the D parity conserving case, e.g. at one loop [88] and at two loops [86, 93]
and can be easily checked with pyr@te [89]. For the one loop beta functions of
other models we use [88] and pyr@te [89] for cross checking and two loops
computations. The coefficients of the beta functions for all the models we
present in Section 3.3 are given in Tab. 3.

Table 3: Beta functions of the models we consider. The order of the indices is $i=L,R,BL,3$.  Model | $b_{i}^{(1)}$ | $b_{ij}^{(2)}$
---|---|---
LRSMD + $\Psi$ | $\left(\begin{array}[]{c}-5/3\\\ -5/3\\\ 7\\\ -7\\\ \end{array}\right)$ | $\left(\begin{array}[]{cccc}209/6&9/2&27/2&12\\\ 9/2&209/6&27/2&12\\\ 81/2&81/2&115/2&4\\\ 9/2&9/2&1/2&-26\end{array}\right)$
LRSMD + $\Psi$ \+ $4\,F_{c}^{\prime}+$ $\,F_{d}$ | $\left(\begin{array}[]{c}-1/3\\\ -1/3\\\ 7\\\ -13/3\\\ \end{array}\right)$ | $\left(\begin{array}[]{cccc}307/6&15/2&27/2&12\\\ 15/2&307/6&27/2&12\\\ 81/2&81/2&115/3&2\\\ 9/2&9/2&1/2&74/3\end{array}\right)$
LRSM~~D~~ \+ $\Psi$ | $\left(\begin{array}[]{c}-7/3\\\ -5/3\\\ 11/2\\\ -7\\\ \end{array}\right)$ | $\left(\begin{array}[]{cccc}97/6&9/2&3/2&12\\\ 9/2&209/6&27/2&12\\\ 9/2&81/2&61/2&4\\\ 9/2&9/2&1/2&-26\end{array}\right)$
LRSM~~D~~ \+ $\Psi+\,4\,F_{a}+\,F_{b}+\,4\,F_{c}$ | $\left(\begin{array}[]{c}5/3\\\ -2/3\\\ 11/2\\\ -7/3\\\ \end{array}\right)$ | $\left(\begin{array}[]{cccc}391/6&9/2&3/2&28\\\ 9/2&565/12&27/2&16\\\ 9/2&81/2&61/3&-1\\\ 21/2&6&1/2&188/3\end{array}\right)$

Due to GUT embedding the matching of the $LR$ gauge couplings to the SM gauge
couplings is given by:

$\displaystyle g^{LR}_{B-L}(\>M_{R})$ $\displaystyle=$ $\displaystyle
z\sqrt{\left(\frac{2}{5\,z^{2}}+\frac{3}{5}\right)}\,g^{SM}_{1}(\>M_{R}),$
$\displaystyle g^{LR}_{L}(\>M_{R})$ $\displaystyle=$ $\displaystyle
g^{SM}_{2}(\>M_{R}),$ $\displaystyle g^{LR}_{C}(\>M_{R})$ $\displaystyle=$
$\displaystyle g^{SM}_{3}(\>M_{R}),$ (A.2)

where

$\displaystyle g^{LR}_{B-L}(\>M_{R})=z\,g^{LR}_{R}(\>M_{R}).$ (A.3)

Here $z$ is a real number of $\mathcal{O}(1)$ that is determined through the
constraint of unification at $\>M_{\text{GUT}}$. The gauge couplings obtained
in the plots in Fig. 6 correspond to the last running of the program that
looks for convergence of the up and down running of the gauge couplings with
the boundary conditions at $M_{EW}$ and gauge unification at $M_{\rm GUT}$.
The matching to the SM model is given by:

$\displaystyle\frac{1}{g^{2}_{1}(\>M_{R})}$ $\displaystyle=$
$\displaystyle\frac{2}{5}\frac{1}{g^{2}_{B-L}(\>M_{R})}+\frac{3}{5}\frac{1}{g^{2}_{R}(\>M_{R})},$
(A.4) $\displaystyle g^{SM}_{2}(\>M_{R})$ $\displaystyle=$ $\displaystyle
g^{LR}_{L}(\>M_{R}),$ (A.5) $\displaystyle g^{SM}_{3}(\>M_{R})$
$\displaystyle=$ $\displaystyle g^{LR}_{3}(\>M_{R}).$ (A.6)

We have used the input parameters given in Table 1 of [87].

#### Understanding the Need of Additional Matter

In order to understand why the addition of color triplets can drastically
lower the scale $\>M_{R}$, we consider the one loop beta function coefficients
of the Minimal LRSM models (both D conserving and D breaking). These
coefficients can be written as:

$\displaystyle\left(\begin{array}[]{c}b_{L}\\\ b_{R}\\\ b_{B-L}\\\
b_{3}\end{array}\right)=\left(\begin{array}[]{c}-6\\\ -6\\\ 4/3\\\
-29/3\end{array}\right)+N_{HBD}\left(\begin{array}[]{c}1/3\\\ 1/3\\\ 0\\\
0\end{array}\right)+N_{HT_{L}}\left(\begin{array}[]{c}0\\\ 2/3\\\ 3\\\
0\end{array}\right)+N_{HT_{L}}\left(\begin{array}[]{c}2/3\\\ 0\\\ 3\\\
0\end{array}\right)\\!,$ (A.27)

where $N_{HBD}$ is the number of Higgs bi-doublets, $N_{HT_{L}}$ and
$N_{HT_{R}}$ is the number of Higgs Left-Triplets and Right-Triplets
respectively. The first term on the right–hand side of Eq. (A.27) corresponds
to the one-loop beta functions from gauge interactions and the three families
of fermions of the LRSM. In the D parity conserving case both the left- and
right-handed triplets $T_{L}$ and $T_{R}$ are present and $b_{L}=b_{R}=-5/3$.
On the other hand, when only $T_{R}$ is included D parity is broken and
$b_{R}>b_{L}=-7/2$.

In order to obtain a phenomenologically viable scenario at low energy one
needs to push the $M_{R}$ and $M_{\rm GUT}$ scales further apart compared to
the minimal cases LRSMD + $\Psi$ and LRSM$\cancel{\rm D}$ \+ $\Psi$. In Fig. 6
$b_{i}<0$ implies a positive slope for $1/\alpha_{i}$. Indeed, the addition of
color triplets makes $b_{3}$ less negative flattening the running of
$1/\alpha_{3}$ and shifting $M_{\rm GUT}$ to higher scales and $M_{R}$ to
lower scales. However adding only color triplets is not enough because
multiplets of all the other groups are required in order to ensure that
$g_{L}$ and $g_{R}$ unify at high scale. In particular, in order to increase
the $g_{R}/g_{L}$ ratio more multiplets of $SU(2)_{L}$ than of $SU(2)_{R}$ are
needed in order to keep the one loop beta function $b_{R}$ as negative as
possible so that $g_{R}(\>M_{R})>g_{L}(\>M_{R})$ already at the scale
$\>M_{R}$ (moreover, to overcome the running of $g_{L}$ the hierarchy between
the two couplings needs to be strong enough).

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