# Strange Stars within Bosonic and Fermionic Admixed Dark Matter

Luiz L. Lopes  H. C. Das 11footnotetext: Corresponding author.

###### Abstract

In this work, we study dark matter (DM) admixed strange quark stars exploring
the different possibilities about the nature of the DM and their effects on
the macroscopic properties of strange stars, such as maximum masses, radii, as
well the dimensionless tidal parameter. We observe that the DM significantly
affects the macroscopic properties that depend on its mass, type, and fraction
inside the star.

##  1 Introduction

Recently, it was suggested that quarks probably appear inside the core of a
massive neutron star (NS) due to a very high density, where hadronic matter
undergoes phase transitions to a new phase of quarks and gluons [1].
Furthermore, several exotic particles, such as hyperons production, dark
matter accretions, kaon condensations, and so on, appeared primarily in the
core of the NS. As a result, in those regimes, the equation of state (EoS) is
the fundamental component that can describe both micro/macroscopic properties.
Another possibility is that at least some of the observed pulsars are indeed
stable quark stars or strange stars. Several theoretical works have proposed
the existence of strange quark stars (SQSs) [2], which are made up of $u$,
$d$, and $s$ quarks in equilibrium in terms of weak interactions. The Bodmen-
Witten conjecture states that strange quark matter (SQM) can have a lower
energy per baryon than pure nucleons because the exclusion principle may be
dominant at absolute zero pressure and temperature [3, 4]. Hence, the SQM
might be the true ground state of the hadronic matter. Hence, it stands to
reason that the SQS must be more stable than the ordinary NS.

Various phenomenological models have been proposed to explore the SQSs
properties. Among them, the MIT bag model and Nambu-Jona-Lasinio (NJL) have
been widely used. In this study, we use the vector MIT (vMIT) bag model to
describe the quark matter interactions [5, 6]. In the MIT bag model, it has
been assumed that the quarks are bound in a bag of finite dimensions. In
contrast to their absolute mass, which is very high, it is hypothesized that
quarks inside the bag have a very low mass. The system is given a bag constant
$B$ as a constant energy density to balance the bag’s behavior and determine
its size. The inward pressure at the bag’s surface counterbalances the outward
pressure the quarks cause, which means the pressure between the true and
perturbative vacuum. Consequently, as $B$ increases, the quark pressure
lowers, which impacts the star’s structure. The value of $B$ relies on the
mass of the strange quark when $u$ and $d$ quarks have very low masses. The
values of $B$ still need to be established and are fully model-dependent. One
can constrain its values with the help of observational results. For example,
in the observational limit of GW170817, the predicted values of
$B^{1/4}=134.1-141.4$ MeV with low-spin prior and $B^{1/4}=126.1-141.4$ MeV
with high spin prior for SQSs [7]. In Ref. [8], they have predicted the range
of $B^{1/4}=133.68-222.5$ MeV for SQSs. However, in the vMIT bag model [5, 6],
the value of $B$ can be obtained by including the stability window, as
mentioned in Refs. [3, 4].

Moreover, there are different phenomenological and macroscopic studies
suggesting that the quark phases inside the compact stars can undergo a phase
transition into a color superconducting state of 2-flavour superconducting
(2SC), and color-flavor locked (CFL) [9, 10]. They form Cooper pairs at high
density and low temperature [11]. The gap parameter ($\Delta$) determines the
pairing strength of Cooper pairs influence the formation of pure CFL stars
[12, 13, 14, 15, 16, 17] and CFL magnetars [18, 19]. Recently, it has been
suggested that with the proper choice of $\Delta$ and bag pressure $B$, the
CFL stars and their EoS can successfully reproduce various observational
constraints such as GW and NICER results [20, 21, 22]. In this study, we want
to explore the dark matter (DM) effects on the strange stars with and without
CFL phases and try to constrain the macroscopic properties with various
observational data.

Compact objects such as NS, and white dwarfs may capture some amount of DM
inside them in their evolving time due to their immense gravitational
potential. Various theoretical predictions provide us with the unknown nature
of DM. Still, numerous work has been fully dedicated to explaining its
properties by applying it to different systems such as white dwarf [23], NS
[24, 25, 26, 27, 28], and even our earth [29]. In the present study, we assume
that the SQSs might contain a certain amount of DM in their lifetime. The
types of DM particles may be either bosonic or fermionic, and also the
percentage of DM depends on the (i) evolution time and (ii) types of
accretions. However, the accreted DM particles interact directly or indirectly
with hadrons by exchanging other bosonic particles, mainly depending on the
model used. Here, we take different types of possible scenarios for DM admixed
SQS.

The direct detection experiments have already been established, such as
XENON100 [30], XENON1T[31], CDMS [32], LUX [33], PANDAX-II [34] etc. to
measure the scattering cross-section of the DM and nucleons. Although, they
provided some exclusions bound to the scattering cross-section. Still, the
null results provided by the experiments alluded to an inconclusive nature of
DM. However, the exclusion bounds prescribed by such direct detection
experiments depend on the local DM density around the solar neighborhood,
which does not affect the density of DM in the NS/SQS environment. After the
accretion of DM inside NS/SQS, it collides with nucleons or quarks by losing
its kinetic energy, and eventually, it is bound inside the star. When the
accretion ends, the DM particles finally reach thermal equilibrium with one
another due to their internal interactions. This explains why NSs with admixed
DM have essentially constant DM particle densities [27, 25, 35, 28].
Therefore, the accreted DM particles are restricted to a narrow radius area
inside the star. In this study, we choose two types of DM and see their
effects on the SQS properties with the vMIT bag model and a model with
superconducting phases.

Recently, the fastest and heaviest Galactic NS named PSR J0952-0607 (black
widow) in the disk of the Milky Way has been detected to have mass $M=2.35\pm
0.17\ M_{\odot}$ in continuation of the pulsars PSR J0740+6620 ($M=2.08\pm
0.07\ M_{\odot}$ [36, 37]). The simultaneous measurements of the $M$ and $R$
for NS are done by neutron star interior composition explorer (NICER) [38, 39]
while the limit on the dimensionless tidal deformability of
$\Lambda_{1.4}=190_{-120}^{+390}$ was provided in GW170817 event [40]. We
calculate the mass, radius, and tidal deformability for the DM admixed SQS and
put constraints using the observational results obtained from different
x-ray/pulsars data, GW170817 data.

##  2 Formalism

###  2.1 Vector MIT bag model

We use the thermodynamic consistent vector MIT bag model introduced in Ref.
[5, 6] to describe the quark matter. In this model, the quark interaction is
mediated by the vector channel $V^{\mu}$, analogous to the $\omega$ meson in
QHD [41]. Its Lagrangian reads:

$\displaystyle\mathcal{L}_{\rm vMIT}$ $\displaystyle=$
$\displaystyle\bigg{\\{}\bar{\psi}_{q}\big{[}\gamma^{\mu}(i\partial_{\mu}-g_{qV}V_{\mu})-m_{q}\big{]}\psi_{q}$
(2.1)
$\displaystyle-B+\frac{1}{2}m_{V}^{2}V^{\mu}V_{\mu}\bigg{\\}}\Theta(\bar{\psi}_{q}\psi_{q}),$

where $m_{q}$ is the mass of the quark $q$ of flavor $u$, $d$ or $s$,
$\psi_{q}$ is the Dirac quark fleld, $B$ is the constant vacuum pressure, and
$\Theta(\bar{\psi}_{q}\psi_{q})$ is the Heaviside step function to assure that
the quarks exist only conflned to the bag. Applying Euler-Lagrange, we obtain
the energy eigenvalue ($E_{q}$), which at $T=0$ K, is also the chemical
potential ($\mu_{q}$):

$E_{q}=\mu_{q}=\sqrt{m_{q}^{2}+k^{2}}+g_{qV}V_{\mu},$ (2.2)

now, using Fermi-Dirac statistics, we can obtain the EoS in mean field
approximation. The number density ($n_{q})$ and the energy density
($\epsilon_{q}$) of the quarks are:

$\displaystyle n_{q}=\frac{N_{c}}{3\pi^{2}}k_{f}^{3},$
$\displaystyle\epsilon_{q}=\frac{N_{c}}{\pi^{2}}\int_{0}^{k_{f}}E_{q}k^{2}d^{3}k,$
(2.3)

where $N_{c}=3$ is the number of colors and $k_{f}$ is the Fermi momentum. The
contribution of the bag and the mesonic mass term is obtained with the
Hamiltonian: $\mathcal{H}$ = $-\langle\mathcal{L}\rangle$. The total number
density ($n$) and energy density ($\epsilon$) of the quarks read:

$\displaystyle n=\sum_{q}n_{q},$
$\displaystyle\epsilon=\sum_{q}\epsilon_{q}+B-\frac{1}{2}m_{v}^{2}V_{0}^{2}.$
(2.4)

To construct an electrically neutral, beta-stable matter, leptons are added as
a free Fermi gas. The pressure is obtained via the relation: $p=\sum\mu
n-\epsilon$, where the sum runs over all the fermions.

The parameters utilized in this work are the same as presented in Ref. [5]. We
use $m_{u}=m_{d}$ = 4 MeV, and $m_{s}$ = 95 MeV. We also assume a universal
coupling of quarks with the vector meson, i.e., $g_{uV}=g_{dV}=g_{sV}=g_{V}$,
and use a value of $G_{V}$ = 0.3 fm2 as defined below:

$G_{V}=\bigg{(}\frac{g_{V}}{m_{V}}\bigg{)}^{2}=0.3\;\mbox{fm}^{2}.$ (2.5)

Now, the value of $G_{V}$ is somewhat arbitrary. To reproduce stable strange
matter, the value of $G_{V}$ combined with the bag must lie in the range known
as the stability window. The stability window is related to the so-called
Bodmer-Witten conjecture [3, 4], which states that the true ground state of
the strongly interacting matter is not protons and neutrons but consists of
strange quark matter, which in turn is composed of deconfined up, down, and
strange quarks. For the SQM hypothesis to be accurate, the energy per baryon
of the deconfined phase (for $p=0$ and $T=0$) is lower than the nonstrange
infinite baryonic matter [5, 3, 4].

$E_{uds}/A<930\;\mbox{MeV},$ (2.6)

at the same time, the nonstrange matter still needs to have an energy per
baryon higher than nonstrange infinite baryonic one; otherwise, protons and
neutrons would decay into $u$ and $d$ quarks:

$E_{ud}/A>930\;\mbox{MeV}.$ (2.7)

Therefore, both, Eqs. 2.6 and 2.7 must simultaneously satisfied. For $G_{V}$ =
0.3 fm2 used in this work, the stability window lies between 139 MeV
$<B^{1/4}<$ 146 MeV [5]. Here, we assume the maximum allowed value: $B^{1/4}$
= 146 MeV, as it will produce the lower radius for the canonical star, as well
the lower value of the dimensionless tidal parameter $\Lambda$, while still
producing very massive strange quark stars, $M>2.40\,M_{\odot}$.

###  2.2 Superconducting CFL quark matter via analytical approximation

Due to the low temperature and high densities reached in the strange star
interiors, the quark matter may be a color superconductor, which is a
degenerate Fermi gas of quarks with a condensate of Cooper pairs near the
Fermi surface that induces color Meissner effects [11]. Among the various
possible configurations of superconducting matter, we can cite two
possibilities: The two-flavor color-superconducting phase, where quarks with
two out of three colors and two out of three flavors pair in the standard BCS
fashion. The flavors with the most phase space near their Fermi surfaces,
namely, $u$ and $d$, are the ones that pair, leaving the strange quark and the
remaining color unpaired. Such phase is expected at densities around 2
$<n/n_{0}<$ 4 [42]. Another one is the color-flavor locked phase, where the
up, down, and strange quarks can be treated on an equal footing, and the
disruptive effects of the strange quark mass can be neglected. In this phase,
quarks of all three colors and all three flavors form conventional spinless
Cooper pairs. The CFL phase is expected at $n>4n_{0}$ [42]. For additional
discussion about 2SC, CFL, and other color superconducting phases, see Ref.
[11] and the references therein.

The 2SC and the CFL phases were explored within the NJL model in Ref. [43],
while in Ref. [42], the authors show that the color superconducting NJL EoS is
very well fitted by an analytical approximation, called constant-sound-speed
(CSS) parameterization, whose EoS and total number density read [42, 44, 45]:

$\displaystyle p=a(\epsilon-\epsilon_{*}),$ $\displaystyle
n=n_{*}[(1+a)p/(a\epsilon_{*})]^{1/(1+a)}.$ (2.8)

We have, therefore, three free parameters, the square of the speed of sound
($v_{s}^{2}=a$), the energy density at $p=0$ ($\epsilon_{*}$), which plays a
role similar to the bag in the MIT base models, and the number density at
$p=0$ ($n_{*}$), which in turn, plays the role of the saturation density
($n_{0}$) of the MIT based models. In Ref. [42], the authors freely vary the
value of $a$ in the range $0.2<a<0.8$ and found that - depending on the NJL
parametrization - the 2SC phase is well described by $a<0.33$ while the CFL
phase is described by $a>0.35$. On the other hand, Ref. [44] uses the extreme
case $a=1$. Here we consider that the quark matter is in the CFL phase and use
an intermediate value, $a=0.6$ (see the text and Fig. 4 from Ref. [42], as
well Ref. [45]). The value of $\epsilon_{*}$ is chosen as 203 MeV/fm3 to match
the value coming from the vector MIT bag model. Finally, $n_{*}$ has to be
constrained, as we still need to reproduce strange quark stars in accordance
with the Bodmer-Witten conjecture. We choose $n_{*}$ = 0.24 fm-3, which is
very close to $n_{0}$ = 0.23 fm-3 coming from the vector MIT. Within this
value, we have $E/A$ = 906 MeV, with implies that the analytical approximation
of the CFL satisfies Eq. 2.6 and, therefore, the Bodmer-Witten conjecture.

##  3 Results and Discussions

###  3.1 Bosonic DM

This section briefly reviews the formalism of a bosonic DM model initially
proposed in Refs. [46, 47]. At very low temperatures, all particles in a
dilute Bose gas condense to the same quantum ground state, forming a Bose-
Einstein Condensate (BEC). Particles become correlated when their wavelengths
overlap; that means the thermal wavelength is greater than the mean inter-
particle distance. Assuming $T=0$ K approximation, almost all the DM particles
are in the condensate. Only binary collisions at low energy are relevant in a
dilute and cold gas. These collisions are characterized by a single parameter,
the s-wave scattering length $l_{a}$, independently of the details of the two-
body potential. Therefore, one can replace the interaction potential with an
effective repulsive interaction:

$V(\vec{r}-\vec{r^{\prime}})=\frac{4\pi
l_{a}}{m_{x}}\delta(\vec{r}-\vec{r^{\prime}}),$ (3.1)

where $m_{x}$ is the mass of the bosonic DM.

The ground state properties of the DM are described by the mean-field Gross-
Pitaevskii (GP) equation, and the equation of the state (EoS) has the form
[46, 47, 27, 48]:

$p_{x}=\frac{2\pi l_{a}}{m_{x}^{3}}\epsilon_{x}^{2}..$ (3.2)

The scattering length $l_{a}$ is assumed equal to 1 fm, as in the Ref. [46,
47, 27, 48]. Moreover, the pressure strongly depends on the bosonic DM’s mass
due to the cubic dependence. Therefore this parameter must be taken with care.
Based on the self-interaction cross-section of the DM constraint (see Refs.
[27, 48], the DM mass in the range 50 MeV $<m_{x}<$ 160 MeV. However, the
original works from Ref. [46, 47] suggest a mass of around 1 GeV. It is worth
emphasizing that a mass ten times larger implies in pressure 1000 times lower.
In Ref. [49], the authors use a slightly different model of bosonic DM, where
the self-interaction is based on a scalar quartic term in the potential. They
use the same constraint based on the self-interaction cross-section of the DM
and suggest a mass of 400 MeV. To explore the ambiguity relative to the mass
of the bosonic DM, we use here two values: 100 MeV, which agrees with Ref.
[27, 48] and 400 MeV, which is in agreement with Ref. [49], and it is not so
far from 1 GeV as suggested in Ref. [46, 47]. With these settings, the
pressure for $m_{x}$ = 400 MeV is 64 times lower than for $m_{x}$ = 100 MeV.

The total EoS of the strange star is, therefore, the sum of the contribution
of the ordinary quark matter and the DM:

$p=p_{q}+p_{x},\quad\mbox{and}\quad\epsilon=\epsilon_{q}+\epsilon_{x}.$ (3.3)

Another important quantity is the fraction of the DM. To solve the TOV
equations [50], we need to specify the central values both for normal matter
and for DM: $p_{q}(0)$, $p_{x}(0)$ respectively. Here, we follow Ref. [27, 48]
and define:

$f_{x}=\frac{p_{x}(0)}{p_{q}(0)+p_{x}(0)},$ (3.4)

and use three different values for $f_{x}=0.05,0.075$ and $0.10$. As pointed
out in Ref. [27, 48], these values agree with the current DM constraints
obtained from stars like the Sun.

### 1\. Bosonic DM within vector MIT bag model

Figure 1: Mass-radius relation for bosonic DM admixed strange stars with
$m_{x}$ =100 MeV (left) and $m_{x}$ = 400 MeV (right).

In Fig. 1, we plot the TOV solution for bosonic DM admixed strange stars with
the mass of 100 MeV and 400 MeV.

As can be seen, for a bosonic DM mass of 100 MeV, we have an increase in the
maximum mass with the increase of the fraction of DM. This result is coherent
with those presented in Ref. [27, 48] for the original, massless MIT.
Moreover, as in the case of the original massless MIT, with the massive vector
MIT, we also see that the presence of DM affects only massive stars. Strange
stars with $M<1.5\,M_{\odot}$ reproduced essentially the same radii. The
maximum masses vary from 2.41 $M_{\odot}$ for pure strange stars to
2.51$M_{\odot}$ for bosonic DM admixed with a fraction of 0.10. This indicates
that the PSR J0740+6620 with a gravitational mass of 2.08 $\pm$ 0.07
$M_{\odot}$ [37] can indeed be a stable strange star with or without admixed
bosonic DM. Even the possible mass of 2.35 $\pm$ 0.17 $M_{\odot}$ of the black
widow pulsar PSR J0952-0607 [51] can be explained as bosonic DM matter admixed
strange star. On the other hand, the radius of the canonical star is in the
narrow range of 11.37 km to 11.40 km. In the literature, there is no consensus
about the true value of the radius of the canonical star. For instance, in
ref. [52], the constraint on the radius of the canonical star is $10.1-11.1$
km, which provides a very narrow range. If this is true, neither of our
results can fulfill such tight constraints. In Ref. [53], an upper limit of
11.9 km was provided. In this case, our results are in full agreement.
However, recent results from the NICER x-ray telescope point that the radius
of the canonical star is between 11.52 km and 13.85 km [39] and between 11.96
km and 14.26 km as given in Ref. [38]. In these cases, our radii are too
small.

Now, we have opposite results for a mass $m_{x}$ = 400 MeV. First, the maximum
mass decrease with the increase of DM fraction, dropping from
$2.41\,M_{\odot}$ to $2.29\,M_{\odot}$ for a fraction $f_{x}$ of 0.10.
However, all values agree with the mass of the PSR J0740+6620 [37] and the PSR
J0952-0607 [51]. Secondly, we see that even low-mass strange stars are already
affected by the DM and are significantly more compact. The radius of the
$1.4\,M_{\odot}$ strange star can reach a value as low as 11.08 km. Therefore,
this result is in agreement with both Refs. [52, 53]. The polytropic EoS of
Eq. 3.2 can easily explain these results. A four times higher DM matter mass
produces sixty-four times smaller pressure. The reduction of the pressure
causes the reduction of the maximum mass and increases the star compression.

Figure 2: Dimensionless tidal parameter $\Lambda$ for bosonic DM admixed
strange stars with $m_{x}=100$ MeV (left) and $m_{x}=400$ MeV (right).

Another essential quantity and constraint is the so-called dimensionless tidal
deformability parameter $\Lambda$. If we put an extended body in an
inhomogeneous external field, it will experience different forces throughout
its surface. The result is a tidal interaction. The tidal deformability of a
compact object is a single parameter $\lambda$ that quantifies how easily the
object is deformed when subjected to an external tidal field. Larger tidal
deformability indicates that the object is easily deformed. Conversely, a
compact object with a small tidal deformability parameter is more compact and
more difficult to deform. The tidal deformability is defined as:

$\Lambda~{}\equiv~{}\frac{\lambda}{M^{5}}~{}\equiv~{}\frac{2k_{2}}{3C^{5}},$
(3.5)

where $M$ is the compact object mass and $C=GM/R$ is its compactness. The
parameter $k_{2}$ is called the second (order) Love number (see the appendix).
Nevertheless, as pointed out in Refs. [54, 22], the value of $y_{R}$ must be
corrected since strange stars are self-bound and present a discontinuity at
the surface. Therefore we must have

$y_{R}\rightarrow y_{R}-\frac{4\pi R^{3}\Delta\epsilon_{S}}{M},$ (3.6)

where $R$ and $M$ are the star radius and mass, respectively, and
$\Delta\epsilon_{S}$ is the difference between the energy density at the
surface ($p=0$) and the star’s exterior (which implies $\epsilon=0$). The
results for the dimensionless tidal parameter are displayed in Fig. 2.

As we can be seen, some features present in the mass-radius relation are also
present here. For instance, for a mass $m_{x}$ = 100 MeV, the low masses of
strange stars have similar tidal parameters, despite their DM fraction. The
tidal parameter for the canonical mass lies between 638 and 644. These values
are in agreement with the constraint $\Lambda<$ 800 [55], but fail to fulfill
the constraint 70 $<\Lambda<$ 580 [40].

In the case of $m_{x}$ = 400 MeV, the strange stars’ huge compression due to
an increase in the DM fraction reduces the tidal parameter $\Lambda$. The
tidal parameter now lies around 500. This indicates that for $m_{x}$ = 400
MeV, we are able to explain very massive neutron stars as the PSR J0952-0607
[51], and simultaneously fulfills the constraints of $\Lambda<$ 800 [55] and
70 $<\Lambda<$ 580 [40]. We summarize the results of this section in Tab 1.

Table 1: Macroscopic properties of bosonic DM admixed strange stars  $m_{x}$ (MeV) | $f_{x}$ | $M/M_{\odot}$ | R (km) |  $R_{1.4}$ (km) | $\Lambda_{1.4}$
---|---|---|---|---|---
100 | 0.000 | 2.41 | 11.86 | 11.37 | 644
100 | 0.050 | 2.46 | 12.01 | 11.37 | 638
100 | 0.075 | 2.48 | 12.06 | 11.38 | 645
100 | 0.100 | 2.51 | 12.08 | 11.40 | 652
400 | 0.000 | 2.41 | 11.86 | 11.37 | 644
400 | 0.050 | 2.31 | 11.42 | 11.16 | 526
400 | 0.075 | 2.30 | 11.38 | 11.12 | 497
400 | 0.100 | 2.29 | 11.31 | 11.08 | 480

### 2\. Bosonic DM within CFL quark matter

In order to better understand the effects of the DM in strange stars, we now
assume that the quark matters are in the CFL superconducting phase via the
analytical approximation EoS in Eq. 2.8. The mass-radius relations are
presented in Fig. 3.

Figure 3: Mass-radius relation for bosonic DM admixed CFL strange stars with
$m_{x}$ =100 MeV (left) and $m_{x}$ = 400 MeV (right).

As can be seen, for a bosonic DM mass of 100 MeV, we have an increase in the
maximum mass with the increase of the fraction of DM. The qualitative results
for CFL superconducting quark stars are analogous to both the original,
massless MIT as showed in Ref. [27, 48], as well for the massive vector MIT
bag model as presented in the last section. This indicates a possible model-
independent behavior about the effect of the bosonic DM. Moreover, as in the
case of the original massless MIT and the massive vector MIT, in the CFL
phase, we also see that the presence of DM affects only massive stars. CFL
strange stars with $M<1.8\,M_{\odot}$ reproduced essentially the same radii.
The maximum masses vary from 2.81 $M_{\odot}$ for pure strange stars to
2.88$M_{\odot}$ for bosonic DM admixed with a fraction of 0.10. This indicates
that the PSR J0740+6620 with $M$ = 2.08 $\pm$ 0.07 $M_{\odot}$ [37] can be a
stable CFL strange star with or without admixed bosonic DM. Even the possible
mass of 2.35 $\pm$ 0.17 $M_{\odot}$ of the pulsar PSR J0952-0607 [51] can be
explained as bosonic DM matter admixed strange star. On the other hand, the
radius of the canonical star presents almost no variation and is fixed at
around 11.57 km. Such a value is too low to reproduce the constraint range of
$10.1-11.1$ km, shown in Ref. [52] while agreeing with Ref. [53], whose upper
limit is 11.9 km. About the NICER x-ray telescope, the constraint between
11.52 km and 13.85 km pointed in Ref. [39] is fulfilled, but the bound in the
range between 11.96 km and 14.26 km (Ref. [38]) is not.

For a mass $m_{x}$ = 400 MeV, the results for CLF superconducting strange
stars are analogous to the massive MIT bag model discussed in the last
section. The maximum mass decrease with the increase of DM fraction, dropping
from $2.81\,M_{\odot}$ to $2.61\,M_{\odot}$ for a fraction $f_{x}$ of 0.10.
However, all values agree with the mass of the PSR J0740+6620 [37] and the
black widow pulsar PSR J0952-0607 [51]. Secondly, we see that even low-mass
strange stars are already affected by the DM and are significantly more
compact. The radius of the $1.4\,M_{\odot}$ for $f_{x}=$ 0.10 is about 11.29
km. Such a low radius fails to fulfill both NICER constraints [39, 38], but is
in agreement with Capano et al. [53]. The reduction of the CFL strange star
and its compression can again be explained by the polytropic EoS of Eq. 3.2. A
four times higher DM matter mass produces sixty-four times smaller pressure.
The reduction of the pressure causes the reduction of the maximum mass and
increases the star compression.

Figure 4: Dimensionless tidal parameter $\Lambda$ for bosonic DM admixed CFL
superconducting strange stars with $m_{x}=100$ MeV (left) and $m_{x}=400$ MeV
(right).

We also calculate the dimensionless tidal parameter $\Lambda$ for the CFL
superconducting strange stars. The results are presented in Fig. 4. As we can
be seen, the results are analogous to the vector MIT bag model. As in the case
of the mass-radius relation, for low mass stars there is very low variation in
the $\Lambda$. For instance, for a mass $m_{x}$ = 100 MeV, the low masses
strange stars have similar tidal parameters, despite their DM fraction. The
tidal parameter for the canonical mass lies between 709 and 721. These values
are in agreement with the constraint $\Lambda<$ 800 [55], but fail to fulfill
the constraint 70 $<\Lambda<$ 580 [40].

In the case of $m_{x}$ = 400 MeV, the results for CFL superconducting strange
stars are again analogous to vector MIT strange stars. The stars’ huge
compression as the DM fraction increases reduce the tidal parameter $\Lambda$.
The tidal parameter now lies around 550. This indicates that for $m_{x}$ = 400
MeV, we are able to explain very massive neutron stars as the PSR J0952-0607
[51], and simultaneously fulfills the constraints of $\Lambda<$ 800 [55] and
70 $<\Lambda<$ 580 [40]. We summarize the results of this section in Tab 2.

Table 2: Macroscopic properties of bosonic DM admixed CFL superconducting strange stars $m_{x}$ (MeV) | $f_{x}$ | $M/M_{\odot}$ | R (km) |  $R_{1.4}$ (km) | $\Lambda_{1.4}$
---|---|---|---|---|---
100 | 0.000 | 2.81 | 12.89 | 11.57 | 721
100 | 0.050 | 2.83 | 12.84 | 11.57 | 709
100 | 0.075 | 2.86 | 12.96 | 11.58 | 717
100 | 0.100 | 2.88 | 13.00 | 11.58 | 717
400 | 0.000 | 2.81 | 12.89 | 11.57 | 721
400 | 0.050 | 2.63 | 12.30 | 11.37 | 570
400 | 0.075 | 2.62 | 12.22 | 11.32 | 545
400 | 0.100 | 2.61 | 12.13 | 11.29 | 531

###  3.2 Fermionic DM

The Lagrangian of the fermionic DM reads [25, 35, 22]:

$\displaystyle\mathcal{L}_{\rm DM}$ $\displaystyle=$
$\displaystyle\bar{\chi}(i\gamma^{\mu}\partial_{\mu}-(m_{x}-g_{H}h))\chi+\frac{1}{2}(\partial^{\mu}h\partial_{\mu}h-m_{H}^{2}h^{2}).$
(3.7)

Here, we assume a dark fermion represented by the Dirac field $\chi$ that
self-interacts through the exchange of the Higgs boson, whose mass is $m_{H}$
= 125 GeV. The coupling constant is assumed to be $g_{H}=0.1$, which agrees
with the constraints in Refs. [27, 25]. Within this prescription, the DM self-
interaction is very feeble and behaves as a free Fermi gas. More explicitly,
the strength of the interaction is:

$G_{H}=\bigg{(}\frac{g_{H}}{m_{H}}\bigg{)}^{2}=2.492\times
10^{-8}\quad\mbox{fm}^{2}.$ (3.8)

Figure 5: Mass-radius relation for fermionic DM admixed strange stars with
$m_{x}=200$ GeV (left) and $m_{x}=60$ GeV (right).

The EoS is easily obtained in mean field approximation, completely analogous
to the QHD model [41]. The fermionic DM is assumed to be the lightest
neutralino, with $m_{x}$ = 200 GeV, as done in Ref. [35, 25]. However, as
pointed out in Ref. [56], the lower limit for weakly interacting massive
particles (WIMP) is 60 GeV. Therefore we also use $m_{x}$ = 60 GeV to better
study the influence of the DM mass. As in the case of the bosonic DM, we must
fix the DM fraction. As we are dealing here with fermionic DM, we follow ref.
[25, 28, 35] and use the Fermi momentum to fix the DM fraction, using three
different values: $k_{f}^{\rm DM}$ = 0.02 GeV, 0.04 GeV, and 0.06 GeV.

### 1\. Fermionic DM within vector MIT bag model

We display in Fig. 5 the TOV solution for a fermionic DM with a mass of 200
GeV and 60 GeV within the vector MIT bag model. As can be seen, the results
for fermionic DM are significantly different when compared with bosonic DM.
The maximum masses are always reduced, and the star compression always
increases, even for very low masses. Also, different DM fractions always
produce different mass-radius relations, affecting all the strange star
families, unlike the bosonic case, where we have very similar stars for
different DM fractions, which is easily understood by the different criteria
of the DM fraction. In the case of bosonic DM, the DM fraction is dependent on
the quark EoS via Eq. 3.4. In the case of fermionic DM, the Fermi momentum is
fixed and independent of the quark EoS.

Qualitatively, the results for $m_{x}$ = 200 GeV and 60 GeV are the same.
Increasing the DM fraction compress the star and reduces the maximum mass.
Quantitatively, we see that a higher DM mass has a strong influence once it
has a higher increase in the energy density, and at the same time, that
produces a lower contribution to the pressure. The maximum mass drops from
2.41 $M_{\odot}$ for $k_{f}^{\rm DM}$ = 0.00 to only 1.80 $M_{\odot}$ for
$k_{f}^{\rm DM}$ = 0.06 GeV in the case of $m_{x}$ = 200 GeV and to 2.16
$M_{\odot}$ for $m_{x}$ = 60 GeV. In the same sense, the radius of the
canonical star drops from 11.37 km for $k_{f}^{\rm DM}$ = 0.00 to only 8.95 km
for $k_{f}^{\rm DM}$ = 0.06 GeV in the case of $m_{x}$ = 200 GeV, and to 10.42
km for $m_{x}$ = 60 GeV. As can be seen, the results for $k_{f}^{\rm DM}$ =
0.06 GeV with $m_{x}$ = 200 GeV can be ruled out once it has a very low
maximum mass in disagreement with the NICER result of the PSR J0740+6620 with
a gravitational mass of 2.08 $\pm$ 0.07 $M_{\odot}$ [37], and also a very low
radius for the canonical star, in disagreement even with the low limit of 10.1
km presented in Ref. [52].

We plot in Fig. 6 the dimensionless parameter $\Lambda$ for fermionic DM
admixed strange stars with $m_{x}$ = 200 GeV and $m_{x}$ = 60 GeV within the
vector MIT bag model. As we can see, the strong compression due to the
fermionic DM contribution reduces the tidal parameter significantly. In the
case with $m_{x}$ = 200 GeV and $k_{f}^{\rm DM}$ = 0.06 GeV, the tidal
parameter drops to only 108, which is six times lower than for $k_{f}^{\rm
DM}$ = 0.00, although, as we pointed out before, such parametrization must be
ruled out.

Figure 6: Dimensionless tidal parameter $\Lambda$ for fermionic DM admixed
strange stars with $m_{x}=200$ GeV (left) and $m_{x}=60$ GeV (right).

As can be seen, most of the parametrizations are able to fulfill the main
constraints for pulsar observations, i.e., $M>2.01M_{\odot}$ and
$70~{}<~{}\Lambda~{}<580$. Indeed, the presence of DM improves the theoretical
prediction and the observational constraints, although it can be some debate
about the radius of the canonical star. They do not fulfill NICER results [38,
39] but agree with Ref. [53].

It is also worth to point the existence of almost degenerate results. As can
be seen, for $m_{x}$ = 200 GeV with $k_{f}^{\rm DM}$ = 0.04 GeV, the
macroscopic are essentially the same for the $m_{x}$ = 60 GeV and $k_{f}^{\rm
DM}$ = 0.06 GeV. The main results are summarized in Tab. 3.

Table 3: Macroscopic properties of fermionic DM admixed strange stars $m_{x}$ (GeV) |  $k_{f}^{\rm DM}$ (GeV) | $M/M_{\odot}$ | R (km) |  $R_{1.4}$ (km) | $\Lambda_{1.4}$
---|---|---|---|---|---
200 | 0.000 | 2.41 | 11.86 | 11.37 | 644
200 | 0.02 | 2.37 | 11.75 | 11.22 | 586
200 | 0.04 | 2.16 | 10.70 | 10.39 | 346
200 | 0.06 | 1.80 | 8.72 | 8.95 | 108
60 | 0.000 | 2.41 | 11.86 | 11.37 | 644
60 | 0.02 | 2.40 | 11.84 | 11.30 | 625
60 | 0.04 | 2.33 | 11.46 | 11.05 | 524
60 | 0.06 | 2.16 | 11.31 | 10.42 | 351

### 2\. Fermionic DM within CFL quark matter

We now study the effect of Fermionic DM in CFL superconducting matter
described by the analytical approximation of Eq. 2.8.

Figure 7: Mass-radius relation for fermionic DM admixed CFL superconducting
strange stars with $m_{x}=200$ GeV (left) and $m_{x}=60$ GeV (right).

We display in Fig. 7 the TOV solution for a fermionic DM with a mass of 200
GeV and 60 GeV. As in the case of the vector MIT bag model, for CFL
superconducting quark matter, the results for fermionic DM are significantly
different when compared with bosonic DM. And again, the qualitative effect of
fermionic DM is the same for CFL as it is for the vector MIT bag model. The
maximum masses are always reduced, and the star compression always increases,
even for very low masses. Again, different DM fractions always produce
different mass-radius relations, affecting all the strange star families.

From the quantitative point of view, the maximum mass drops from 2.81
$M_{\odot}$ for $k_{f}^{\rm DM}$ = 0.00 to 2.04 $M_{\odot}$ for $k_{f}^{\rm
DM}$ = 0.06 GeV in the case of $m_{x}$ = 200 GeV and to 2.49 $M_{\odot}$ for
$m_{x}$ = 60 GeV. In the same sense, the radius of the canonical star drops
from 11.57 km for $k_{f}^{\rm DM}$ = 0.00 to 9.21 km for $k_{f}^{\rm DM}$ =
0.06 GeV in the case of $m_{x}$ = 200 GeV, and 10.66 km for $m_{x}$ = 60 GeV.
Now, unlike the case of the vector MIT, none of the CFL superconducting
strange stars can be ruled out in the light of the PSR J0740+6620, $M~{}=$
2.08 $\pm$ 0.07 $M_{\odot}$ [37], although for $k_{f}^{\rm DM}$ = 0.06 GeV and
$m_{x}$ = 200 GeV the radius of the canonical star is below the lower limit of
10.1 km presented in Ref. [52].

We plot in Fig. 8 the dimensionless parameter $\Lambda$ for fermionic DM
admixed superconducting strange stars with $m_{x}$ = 200 GeV and $m_{x}$ = 60
GeV. The results are completely analogous to the case of the vector MIT bag
model; however, the value of $\Lambda$ here is always higher. The compression
due to the fermionic DM contribution reduces the tidal parameter. In the case
with $m_{x}$ = 200 GeV and $k_{f}^{\rm DM}$ = 0.06 GeV, the tidal parameter
drops from 721 to 151.

Figure 8: Dimensionless tidal parameter $\Lambda$ for fermionic DM admixed CFL
strange stars with $m_{x}=200$ GeV (left) and $m_{x}=60$ GeV (right).

It is also worth noting that some parametrizations can fulfill the main
constraints for pulsar observations, 70 $<\Lambda<580$, and yet produce a very
high maximum mass, sometimes reaching 2.50 $M_{\odot}$. The presence of DM
again improves the theoretical prediction and the observational constraints,
although it can be some debate about the radius of the canonical star. They do
not fulfill NICER results [39, 38], but agree with Ref. [53]. Moreover, most
parametrizations can explain even the black widow pulsar PSR J0952-0607 [51].

Finally, even when we use a different model for the quark matter, the
existence of almost degenerate results is still present: for $m_{x}$ = 200 GeV
with $k_{f}^{\rm DM}$ = 0.04 GeV and $m_{x}$ = 60 GeV with $k_{f}^{\rm DM}$ =
0.06 GeV. The main results are summarized in Tab. 4.

Table 4: Macroscopic properties of fermionic DM admixed color superconducting quark stars $m_{x}$ (GeV) |  $k_{f}^{\rm DM}$ (GeV) | $M/M_{\odot}$ | R (km) |  $R_{1.4}$ (km) | $\Lambda_{1.4}$
---|---|---|---|---|---
200 | 0.000 | 2.81 | 12.89 | 11.57 | 721
200 | 0.02 | 2.75 | 12.65 | 11.43 | 653
200 | 0.04 | 2.50 | 11.50 | 10.67 | 421
200 | 0.06 | 2.04 | 9.38 | 9.21 | 151
60 | 0.000 | 2.81 | 12.89 | 11.57 | 721
60 | 0.02 | 2.78 | 12.75 | 11.53 | 694
60 | 0.04 | 2.69 | 12.45 | 11.28 | 610
60 | 0.06 | 2.49 | 11.53 | 10.66 | 422

### 3\. Fermionic DM with a vector channel

Now we study if the presence of a dark, repulsive vector channel affects the
macroscopic properties of the fermionic DM admixed strange stars. The new
Lagrangian is the Lagrangian in Eq. 3.7 plus the repulsive channel and the
respective meson mass, and reads [28]:

$\mathcal{L}_{\rm
VDM}=g_{\xi}\bar{\chi}(\gamma^{\mu}\xi_{\mu})\chi+\frac{1}{2}m_{\xi}^{2}\xi_{\mu}\xi^{\mu}-\frac{1}{4}V^{\mu\nu}V_{\mu\nu}.$
(3.9)

The Lagrangian of Eq. 3.9 is analogous to the $\omega$ contribution to the QHD
Lagrangian [5, 41]. Indeed, the junction of Eq. 3.7 and Eq. 3.9 makes this
model of DM fully analogous to the original $\sigma-\omega$ model of the QHD
[41]. The coupling constant $g_{\xi}=0.1$ is fixed, and it is equal to the
$g_{H}$, while the mass of the vector of the dark meson is assumed to be 34
MeV, following Ref. [28]. As the mass of the dark vector meson is 3000 times
smaller than the mass of the Higgs boson, the repulsive channel is much
stronger than the attractive one. Indeed, we have

$G_{\xi}=\bigg{(}\frac{g_{\xi}}{m_{\xi}}\bigg{)}^{2}=0.337\quad\mbox{fm}^{2},$
(3.10)

which is stronger than the quark repulsion and millions of times higher than
the DM scalar coupling (Eq. 3.8). Nevertheless, despite the strong self-
repulsion of the fermionic DM, the numerical results are barely affected by
the repulsive channel. For the vector MIT bag model, the only noticeable
difference appears for $m_{x}$ = 200 GeV and $k_{f}^{\rm DM}$ = 0.04 GeV. In
this case, the maximum mass increase from 2.16 $M_{\odot}$ to 2.17
$M_{\odot}$. The radius of the canonical star also grows from 11.39 km to
11.46 km. The tidal parameter $\Lambda_{1.4}$ also increases from 346 to 358.
It is worth noticing that all these variations are far beyond the precision
with which experimental measurements are made. All the other parametrizations
present even lower (or none) differences. Herefore, we do not provide any
figures in this section since they would be visually indistinguishable from
those in the last paragraph. We only display the main results in Tab. 5. In
the case of CFL superconducting quark matter, the differences are even
smaller.

The nature of the vector coupling can explain why the differences are so
small. The vector mesons couple to the number density, and we are dealing with
a very low-density regime. Indeed, even $k_{f}^{\rm DM}$ = 0.06 GeV implies a
number density is around 9.6 $\times~{}10^{-4}$ fm-3. Of course, we could
increase the repulsion of the dark vector boson, but we believe this would be
very unrealistic since DM was proposed to explain higher attraction in galaxy
curves [57].

Table 5: Macroscopic properties of dark vector boson fermionic DM admixed strange stars within the vector MIT bag model. The only significant differences are for $k_{f}^{\rm DM}$ = 0.04 GeV  $m_{x}$ (GeV) |  $k_{f}^{\rm DM}$ (GeV) | $M/M_{\odot}$ |  $R$ (km) |  $R_{1.4}$ (km) | $\Lambda_{1.4}$
---|---|---|---|---|---
200 | 0.000 | 2.41 | 11.86 | 11.37 | 644
200 | 0.02 | 2.37 | 11.75 | 11.22 | 586
200 | 0.04 | 2.17 | 10.70 | 10.46 | 358
200 | 0.06 | 1.80 | 8.72 | 9.01 | 112
60 | 0.000 | 2.41 | 11.86 | 11.37 | 644
60 | 0.02 | 2.40 | 11.84 | 11.30 | 625
60 | 0.04 | 2.33 | 11.46 | 11.10 | 532
60 | 0.06 | 2.16 | 11.31 | 10.43 | 353

##  4 Conclusions

In this work, we calculate the properties for the DM admixed for strange quark
stars. We use two different models for the quark model: the vector MIT bag
model, as presented in Refs. [5, 6] and the CFL color superconducting quark
matter via an analytical approximation, as discussed in Refs. [42, 44, 45];
and two different kinds of dark matter: a bosonic as discussed in Refs. [46,
47, 27, 48] and for fermionic [35, 25, 22]. For each kind of DM, we use two
different mass values, and the strange stars always agree with the Bodmer-
Witten conjecture [3, 4]. Our main conclusions can be summarized as follows:

  * •

The qualitative results for DM admixed strange stars are independent of the
quark model utilized. This is true for both bosonic and fermionic, as well it
is independent of the DM mass.

  * •

For a bosonic DM with a mass of $m_{x}$ = 100 MeV, we have an increase of the
maximum mass, while the properties of low-mass strange stars are not
significantly affected. This is the only case in that we have an increase in
the star’s mass. Such a situation happens for the vector MIT, the CFL
superconducting quark matter, and also for the massless MIT, as pointed out in
Refs. [48, 27].

  * •

For a bosonic DM with a mass of $m_{x}$ = 400 MeV, we have a decrease of the
maximum mass, whilst the radii of the low-mass strange stars, in this case,
are also affected.

  * •

For a fermionic DM, the maximum mass always decreases. The higher the DM
fraction, the lower the maximum mass, and the smaller the radii. Also, the
higher the DM mass, the higher the stellar compression and the lower the
maximum mass.

  * •

Although we introduce a repulsive dark vector field with a mass 3000 times
smaller than the attractive scalar field, we do not find significant variation
in the stellar macroscopic properties.

  * •

There are almost degenerate results both for $m_{x}$ = 200 GeV with
$k_{f}^{\rm DM}$ = 0.04 GeV and $m_{x}$ = 60 GeV with $k_{f}^{\rm DM}$ = 0.06
GeV, the maximum mass, as well the properties of the canonical star are
essentially the same.

  * •

About the observational constraints, we can see that the mass of the PSR
J0740+6620 pulsar, $M~{}=$ 2.08 $\pm$ 0.07 $M_{\odot}$ [37] is easily
obtained. Even the mass range of 2.35 $\pm$ 0.17 $M_{\odot}$ of the black
widow pulsar PSR J0952-0607 [51] can be reached for some parametrization.

  * •

The radius of the canonical star is still a matter of debate. Most of our
results point to a radius between 11.0 km to 11.5 km. In general, our results
are in agreement with Ref. [53] but are too low to reproduce the NICER results
[39, 38] whilst at the same time are too high to agree with Ref. [52].

  * •

Except for bosonic DM with a mass of $m_{x}$ = 100 MeV, in all other cases,
the presence of the DM reduces the dimensionless tidal parameter $\Lambda$. In
most of these cases, the constraint 70 $<\Lambda<580$ [40] is easily
fulfilled.

##  Appendix A The Tidal deformability parameter

Let us start with the well known TOV equations [50]:

$\displaystyle\frac{dp}{dr}$ $\displaystyle=$
$\displaystyle\frac{-GM(r)\epsilon(r)}{r^{2}}\bigg{[}1+\frac{p(r}{\epsilon(r)}\bigg{]}\times\bigg{[}1+\frac{4\pi
p(r)r^{3}}{M(r)}\bigg{]}\bigg{[}1-\frac{2GM(r)}{r}\bigg{]}^{-1},$
$\displaystyle\frac{dM}{dr}$ $\displaystyle=$ $\displaystyle 4\pi
r^{2}\epsilon(r).$ (A.1)

The dimensionless tidal deformability parameter $\Lambda$ is defined as:

$\Lambda~{}\equiv~{}\frac{\lambda}{M^{5}}~{}\equiv~{}\frac{2k_{2}}{3C^{5}},$
(A.2)

where $M$ is the compact object mass and $C=GM/R$ is its compactness. The
parameter $k_{2}$ is called the second (order) Love number:

$\displaystyle k_{2}$ $\displaystyle=$
$\displaystyle\frac{8C^{5}}{5}(1-2C)^{2}\times[2+2C(y_{R}-1)-y_{R}]\\{2C[6-3y_{R}+3C(5y_{R}-8)]$
$\displaystyle+$ $\displaystyle
4C^{3}[13-11y_{R}+C(3y_{R}-2)+2C^{2}(1+y_{R})]+3(1-2C)^{2}$
$\displaystyle[2-y_{R}+2C(y_{R}-1))]\ln(1-2C)\\}^{-1},$

where $y_{R}=y(r=R)$ and $y(r)$ is obtained by solving:

$r\frac{dy}{dr}+y^{2}+yF(r)+r^{2}Q(r)=0.$ (A.4)

Eq. (A.4) must be solved coupled with the TOV equations, Eq. (A.1). The
coefficients $F(r)$ and $Q(r)$ are given by:

$F(r)=\frac{1-4\pi Gr^{2}[\epsilon(r)-p(r)]}{E(r)},$ (A.5)

$\displaystyle Q(r)=\frac{4\pi
G}{E(r)}\bigg{[}5\epsilon(r)+9p(r)+\frac{\epsilon(r)+p(r)}{\partial
p/\partial\epsilon}-\frac{6}{4\pi Gr^{2}}\bigg{]}-4\bigg{[}\frac{G[M(r)+4\pi
r^{3}p(r)]}{r^{2}E(r)}\bigg{]}^{2},$ (A.6)

where $E(r)=(1-2GM(r)/r)$. Additional discussion about the theory of tidal
deformability and the tidal Love numbers are beyond the scope of this work and
can be found in Refs. [55, 40, 58, 59, 54, 22, 60, 61] and references therein.

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