# Structure of magnetized strange quark star in perturbative QCD

J. Sedaghat1111email address<EMAIL_ADDRESS>S. M. Zebarjad1,2222
email address<EMAIL_ADDRESS>G. H. Bordbar1,3333email address:
<EMAIL_ADDRESS>and B. Eslam Panah4,5,6444email address:
<EMAIL_ADDRESS>1 Department of Physics, Shiraz University, Shiraz 71454,
Iran
2 Department of Physics, University of California at San Diego,
La Jolla, CA 92093, USA
3 Department of Physics and Astronomy, University of Waterloo,
200 University Avenue West, Waterloo, Ontario, N2L 3G1, Canada
4 Department of Theoretical Physics, Faculty of Science, University of
Mazandaran, P. O. Box 47416-95447, Babolsar, Iran
5 ICRANet-Mazandaran, University of Mazandaran, P. O. Box 47416-95447,
Babolsar, Iran
6 ICRANet, Piazza della Repubblica 10, I-65122 Pescara, Italy

###### Abstract

We have performed the leading order perturbative calculation to obtain the
equation of state (EoS) of the strange quark matter (SQM) at zero temperature
under the magnetic field $B=10^{18}\ G$. The SQM comprises two massless quark
flavors (up and down) and one massive quark flavor(strange). Consequently, we
have used the obtained EoS to calculate the maximum gravitational mass and the
corresponding radius of the magnetized strange quark star (SQS). We have
employed two approaches, including the regular perturbation theory (RPT) and
the background perturbation theory (BPT). In RPT the infrared (IR) freezing
effect of the coupling constant has not been accounted for, while this effect
has been included in BPT. We have obtained the value of the maximum
gravitational mass to be more than three times the solar mass. The validity of
isotropic structure calculations for SQS has also been investigated. Our
results show that the threshold magnetic field from which an anisotropic
approach begins to be significant lies in the interval $2\times
10^{18}G<B<3\times 10^{18}G$. Furthermore, we have computed the redshift,
compactness and Buchdahl-Bondi bound of the SQS to show that this compact
object cannot be a black hole.

##  I Introduction

In the last stage of stellar evolution, due to the completion of the fusion
reactions in a luminous star, the gravity pressure dominates the fusion
pressure. This can lead to the birth of a highly dense matter star known as
compact star shapiro2008black ; schmitt2010dense ; glendenning2012compact ;
karttunen2016fundamental . Having large compactness (the ratio of mass to the
radius of a star) is a feature that distinguishes compact stars from ordinary
stars. Based on the range of the compactness, compact stars are classified
into three groups; white dwarfs, neutron stars, and stellar black holes
camenzind2007compact ; rezzolla2018physics . After Murray Gell-Mann
gell2010schematic  and George Zweig zweig1964_3 , proposed quarks and gluons
as confined constituents of hadrons and also after the first experimental
evidence for this postulate in SLAC friedman1972deep , Ivanenko, Kurdgelaidze
and then Itoh discussed a new phase denser than nuclear matter. Such dense
matter can be created by squeezing nucleons at the core of neutron stars. This
phase in which the quarks are considered as the degrees of freedom
ivanenko1965hypothesis ; itoh1970hydrostatic ; haensel1986strange  was named
quark matter. If the quark matter includes strange quarks, it is called
strange quark matter. Bodmer and Witten independently discussed an interesting
feature of SQM in Refs. bodmer1971collapsed ; witten1984cosmic ;
farhi1984strange . They claimed that SQM is the true ground state of QCD which
means that the minimum energy per baryon in SQM is lower than that of the most
stable nuclear matter (${}^{56}Fe\sim 930\ MeV$). Since the strange hadrons
are so unstable, to find a stable SQM from nuclear matter, the conversion rate
should be so that a large number of quarks change their flavor to strange by
weak interactions in a short time. This situation can only be available in the
bulk size and extreme pressure, which is accessible at the core of the neutron
stars farhi1986physics . For this reason, the possibility of the existence of
SQSs has been greatly discussed in literature alcock1986strange ;
chakrabarty1991equation ; madsen1999physics ; weber2005strange . However, in
Ref. holdom2018quark , there is a discussion on a phenomenological model
including scalar and pseudoscalar nonets which results in finding a two flavor
quark matter to be more stable than SQM for baryon number $A>300$.

Nowadays, the quark matter is proposed to exist in two new classes of compact
stars: i) hybrid stars with quark core surrounded by hadronic shell
schertler1998influence ; blaschke2001cooling ; burgio2003maximum ;
alford2005hybrid  and ii) pure quark stars or strange stars in which surface
density exceeds nuclear saturation density michel1988quark ; drago2001quark ;
bombaci2008quark . The behavior of the mass-radius diagram in pure quark stars
is different from that of neutron stars. It is originated from their different
EoSs farhi1986physics ; greiner2013nuclear ; panah2019contraction ;
sedaghat2021compact .

Until now, it has been confirmed observationally that pulsars and magnetars
have a strong magnetic field at their surface as large as $10^{12}-10^{13}\ G$
and $10^{14}-10^{16}\ G$, respectively kouveliotou1998x ; turolla2015magnetars
. However, theoretically, the maximum strength of the magnetic field in a
magnetar is of the order of $10^{18}\ G$ lai1991cold ; chanmugam1992magnetic ;
lai2001matter  and for self-bound quark stars, this limit may increase up to
$10^{20}\ G$ ferrer2010equation . The origin of such strong magnetic fields
has not been clarified. They might be due to the fossil magnetic field of the
progenitor star chanmugam1992magnetic ; sotani2015massive . In rotating
protoneutron stars in which the period of rotation is less than $3ms$, a weak
magnetic field can be intensified up to $10^{15}\ G$ duncan1992formation ;
sotani2015massive . In such strong magnetic fields, the Landau quantization of
the transverse momenta in stellar structure is significant landau1956zhetf ;
landau1977quantum ; kayanikhoo2020influence .

The magnetic field geometry in the star is still unknown. Many authors have
considered a uniform magnetic field in the star chakrabarty1996quark ;
sotani2015massive ; fogacca2016compact , while others have used a density-
dependent magnetic field dexheimer2017magnetic ; sotani2017quark ;
kayanikhoo2020influence . The magnetic fields used in these references are in
the range from $5\times 10^{13}\ G$ to $10^{20}\ G$. For example, in the Ref.
chakrabarty1996effect , by using the MIT bag model, the maximum gravitational
mass of the SQS in the presence of a uniform magnetic field of the order of
$10^{18}\ G$, is obtained $1.5M_{\odot}$. In the Ref. sotani2015massive , the
maximum mass of an isotropic massive hybrid star in the presence of the
magnetic field $B=10^{19}\ G$, has been evaluated $2.8M_{\odot}$. In Ref.
kayanikhoo2020influence , the maximum mass of the SQS under a Gaussian
dependent magnetic field with the values $B=5\times 10^{18}\ G$ for interior
and $B=5\times 10^{13}\ G$ for the surface of the star has been calculated
$1.44M_{\odot}$. Moreover, in the Ref. sotani2017quark , using the bag model
with a density-dependent magnetic field, the maximum mass of anisotropic
hybrid star reaches to $2.2M_{\odot}$ for magnetic fields greater than
$B=6.86\times 10^{19}\ G$ at the core of the star.

The anisotropic effect in the calculation of the structural properties of SQS
is usually considered at strong magnetic fields. In a strong magnetic field,
the rotational symmetry is broken due to the difference between the
longitudinal pressure, $P_{\lVert}$ (which is parallel to the magnetic field)
and the transverse pressure, $P_{\bot}$ (which is perpendicular to the
magnetic field) ferrer2010equation ; chu2018quark . If this difference is
negligible, the isotropic approximation can be justified; otherwise,
considering the anisotropic structure for the star is significant.

In Ref. andreichikov2013asymptotic , it has been shown that the running
coupling constant of QCD has asymptotic freedom in strong magnetic fields.
Therefore, considering the possibility of performing perturbative calculations
due to this feature of coupling constant in strong magnetic fields would be
interesting. Thus we use the perturbative QCD calculation to obtain the
maximum gravitational mass and the corresponding radius of SQS in the presence
of the magnetic field $B=10^{18}\ G$ (equipartition limit). In order to
simplify the perturbative QCD calculations, we consider a uniform magnetic
field. Employing QCD perturbation theory is one of the approaches to obtain
the EoS of the SQM in the absence of a magnetic field. Previously, it has been
done in Refs. fraga2005role  and kurkela2010cold , of the order of
$\alpha_{s}$ and $\alpha_{s}^{2}$, respectively, in the absence of a magnetic
field. In this paper, we consider the interaction between quarks and strong
magnetic field non-perturbatively and QCD interaction of quarks perturbatively
of the order of $O(\alpha_{s})$. By using the behavior of the running
$\alpha_{s}$ in the presence of a constant strong magnetic field
andreichikov2013asymptotic , we calculate the EoS of a stable strange quark
star exposed to a uniform strong magnetic field as large as $10^{18}\ G$ at
zero temperature. We then evaluate the structural properties of this star.

In this paper, we study a pure SQS. However, in a realistic description of
SQS, one should consider a hadronic layer at the star’s surface. An electric
dipole layer is formed in the gap between the SQM and hadronic matter. This
electric dipole gap can create a strong electric field to enable the SQM to
carry the hadronic layer. A discontinuity is expected between the SQM phase
and the hadronic layer across the electric dipole gap weber2005strange . The
outline of this paper is as follows. In the next section, the Landau levels
are considered as the energy states emerging in solving the Dirac equation for
a charged particle moving in a uniform magnetic field. In section Ref sec 3,
to perform perturbative calculations, the Feynman rules for quark propagator
and running coupling in the presence of uniform magnetic field are presented.
Then, the thermodynamic potential of SQM is computed numerically up to the
leading order of coupling constant. Also, the stability conditions to obtain
the EoS of the SQS in our different approaches are explained. Two models
including regular perturbation theory (RPT) and background perturbation theory
(BPT) are considered. In section Ref Thermodynamic properties, our results
from perturbative calculations for the EoS and thermodynamic properties of the
SQS in RPT and BPT are presented. Then, by numerically solving the Tolman-
Oppenhaimer-Volkoff (TOV) equations, the maximum gravitational mass and the
corresponding radius of SQS in RPT and BPT are obtained. Furthermore, the
redshift, compactness, and Buchdahl-Bondi bound of the SQS are evaluated to
show that the compact object under study cannot be a black hole. To evaluate
the validity of isotropic approximation in the calculation of the structure of
SQS, the effect of anisotropy on the structure of SQS for different magnetic
fields is investigated in section Ref anisotropic structure. We finish our
paper with some concluding remarks.

##  II Landau levels

Since we are working on a QCD dense matter in a uniform strong magnetic field,
we can neglect the QED interaction between quarks. In this case, there are two
significant interactions, including the interaction between quarks and
magnetic field and the QCD interaction between quarks. The first interaction
is not perturbative, while the QCD interaction is perturbative in the high
energy limit. It is natural to solve the Dirac equation for a single particle
with charge $e_{f}$ and mass $m_{f}$ moving in a uniform magnetic field along
the _z_ axis. The energy spectrum is given by

$E_{j}(p_{z},B)=\sqrt{p_{z}^{2}+m_{f}^{2}+2j\lvert e_{f}B\rvert},$ (1)

where numbers, $j=0,1,2,{...}$, are referred to as Landau levels and $f$
denotes the quark flavor. It is shown that except for the lowest Landau level
(LLL), the solutions are twofold degenerate for a particle with spins up and
down bhattacharya2007solution . While the orbital motion of the quarks in
transverse direction, $p_{\perp}^{2}=p_{x}^{2}+p_{y}^{2}$, is quantized, the
longitudinal component of the momentum, $p_{\lvert\lvert}=p_{z}$, is
unaffected by the magnetic field and remains free shovkovy2013magnetic . The
splitting gap between levels is of the order of $\sqrt{2\lvert e_{f}B\rvert}$
kojo2013renormalization . If $\mu$ is the chemical potential of the fermion,
the highest Landau level is obtained by fogacca2016compact

$j_{max}=\left[\dfrac{\mu^{2}-m_{f}^{2}}{2e_{f}B}\right],$ (2)

where $[x]$ is the integer part of $x$.

##  III Fermion propagator and QCD running coupling in uniform magnetic field

To calculate the perturbative contribution of the Feynman diagrams, we need to
know fermion and gluon propagators in the presence of a strong magnetic field.
It is more convenient to work in Euclidean space at the finite chemical
potential. So the Euclidean metric $(g_{\mu\nu}=\delta_{\mu\nu})$ must be
used. This makes no difference between the covariant and contravariant
components of the vectors. In Euclidean space, the propagator of a quark with
charge $e_{f}$ moving in a uniform magnetic field is given in terms of
generalized Laguerre polynomials labeling the $j$th Landau levels
miransky2015quantum .

$S(p)=-i\sum_{j=0}^{\infty}\left(-1\right)^{j}e^{\frac{-p_{\perp}^{2}}{{\lvert
e_{f}B\rvert}}}\dfrac{D_{j}\left(\lvert
e_{f}B\rvert,p\right)}{p_{0}^{2}+\left(p_{3}^{2}+m_{f}^{2}+2j\lvert
e_{f}B\rvert\right)},$ (3)

where,

$\displaystyle D_{j}\left(\lvert e_{f}B\rvert,p\right)=$
$\displaystyle(m_{f}-\gamma_{0}p_{0}-\gamma_{3}p_{3})$
$\displaystyle\times\left[(1+\mathcal{K})L_{j}\left(\mathcal{X}\right)-(1-\mathcal{K})L_{j-1}\left(\mathcal{X}\right)\right]$
$\displaystyle+4\left(\gamma_{1}p_{1}+\gamma_{2}p_{2}\right)L_{j-1}^{(1)}\left(\dfrac{2p_{\bot}^{2}}{e_{f}B}\right).$
(4)

where $\gamma_{i}$s are Dirac gamma matrices in Euclidean space. Also,
$\mathcal{K}=$ $i\gamma_{1}\gamma_{2}$sign$(e_{f}B)$ and
$\mathcal{X=}\dfrac{2p_{\bot}^{2}}{e_{f}B}$. At finite chemical potential, the
zero components of the fermionic momenta are shifted by $p_{0}\longrightarrow
p_{0}+i\mu$ kurkela2010cold . Since gluon has no interaction with magnetic
field, its propagator has the same regular form as in absence of magnetic
field. Thus in Feynman gauge, the gluon propagator is

$D_{\mu\nu}(k)=\dfrac{\delta_{\mu\nu}}{k^{2}}.$ (5)

The running coupling of QCD at one-loop level (one gluon exchange
approximation) in the presence of uniform magnetic field has been given in
andreichikov2013asymptotic

$\alpha(Q,B)=\frac{\alpha(\Lambda_{0})}{1+\alpha(\Lambda_{0})\left[\dfrac{11N_{c}}{12\pi}\ln\left(\dfrac{Q^{2}+M_{B}^{2}}{\Lambda_{0}^{2}}\right)+F(Q,B)\right]},$
(6)

where

$\displaystyle F(Q,B)$ $\displaystyle=$
$\displaystyle\dfrac{1}{Q^{2}}\sum_{f}\dfrac{\lvert
e_{f}B\lvert}{\pi}\exp\left(\dfrac{-q_{\bot}^{2}}{2e_{f}B}\right)T\left(\dfrac{Q^{2}}{4m_{f}^{2}}\right),$
(7) $\displaystyle\alpha(\Lambda_{0})$ $\displaystyle=$
$\displaystyle\dfrac{12\pi}{11N_{c}\ln\left(\dfrac{\Lambda_{0}^{2}+M_{B}^{2}}{\Lambda_{V}^{2}}\right)},$
(8)

and

$T(z)=-\dfrac{\ln(\sqrt{1+z}+\sqrt{z})}{\sqrt{z(z+1)}}.$ (9)

With an accuracy better than $10\%$, $T(z)$ becomes vysotsky2010atomic ;
andreichikov2013asymptotic

$T(z)=\dfrac{2z}{3+2z}.$ (10)

In Eqs. (6) and (8), $N_{c}$ is the number of colors and $M_{B}$ is originated
from background perturbation theory (BPT) for explaining infrared (IR)
freezing effect simonov2011asymptotic ; deur2016qcd . According to this effect
which is represented by some experiments such as colliding polarized electrons
to protons deur2008determination  or hadronic decays of the $\tau$ lepton
ackerstaff1999measurement , the running coupling has slowly varying behavior
in the IR region of energy. IR freezing effect can be explained in such a way
that the running behavior of the coupling constant is originated from
particle-antiparticle loop corrections. Confinement implies that quarks and
anti-quarks cannot have a wavelength more than the size of the hadron. Thus
the loop corrections for the coupling constant are suppressed in IR scale and
$\alpha_{s}$ is expected to lose its energy dependence brodsky2008maximum ;
deur2016qcd . We have chosen two approaches with and without considering the
IR freezing effect in running coupling to check the impact of this QCD effect
on the star structure. These two approaches are as follows: i) without
considering IR freezing effect in which $M_{B}=0$ (RPT) and ii) with
considering IR freezing effect in which $M_{B}$ $\cong 1\ GeV$ interpreted as
the ground-state mass of two gluons connected by the fundamental string, with
string tension $\sigma=0.18\ GeV^{2}$ (BPT) andreichikov2013asymptotic ;
ferrer2015quark . $\Lambda_{V}$ is the QCD scale parameter which is considered
$0.38\ GeV$ and $0.48\ GeV$ in RPT and BPT, respectively badalian2019radial .
We have used $\Lambda_{0}=2\ GeV$ for both approaches and considered the
strange quark as a massive particle while the masses of up and down quarks are
negligible. The running mass of strange quark up to one-loop correction in the
absence of a magnetic field is given by vermaseren19974

$m_{s}(Q)=m_{s}(2GeV)\left[\dfrac{\alpha_{s}(Q)}{\alpha_{s}(2GeV)}\right]^{\dfrac{\gamma_{0}}{\beta_{0}}},$
(11)

where $m_{s}(2\ GeV)\cong 0.1\ GeV$ tanabashi2018review , and $\beta_{0}$ and
$\gamma_{0}$ are beta function and anomalous dimension, respectively in one-
loop order which are given as

$\displaystyle\beta_{0}$ $\displaystyle=$
$\displaystyle\dfrac{1}{3}\left(11N_{c}-2N_{f}\right),$
$\displaystyle\gamma_{0}$ $\displaystyle=$ $\displaystyle 3C_{F},$ (12)

where $N_{f}$ is the number of flavors and $C_{F}$ is given as

$\displaystyle C_{F}=\dfrac{N_{c}^{2}-1}{2N_{c}}.$ (13)

By inserting Eq. (6) in Eq. (11), and assuming the same initial conditions, we
can obtain the running mass of strange quark in both RPT and BPT which is
shown in Fig. 1. As we can see from Fig. 1, the running mass of the strange
quark in BPT is lower than that of RPT below $Q=2\ GeV$. In energies more than
this value, the running mass in RPT is lower than that of BPT.

Figure 1:  Running mass of strange quark versus energy scale ($Q$) in presence
of magnetic field $B=10^{18}\ G$ in RPT (full curve) and BPT (dashed curve)
approaches.

Fig. 2 shows the comparison of the running coupling constant in the presence
of $B=10^{18}$ G in both RPT and  BPT approaches. From this figure, we see
that in all energy scales, the coupling constant in BPT is lower than that of
RPT. Furthermore, the running coupling in BPT approach changes smoothly due to
IR freezing effect.

Figure 2: The comparison of QCD running coupling constant in presence of
magnetic field $B=10^{18}\ G$ in RPT (dashed curve) and  BPT (dotted curve)
approaches.

##  IV Thermodynamic potential up to the leading order

In grand canonical ensemble, the grand potential $\Omega$ at zero temperature
is defined by

$\dfrac{\Omega}{V}=\epsilon-{\mu}_{f}n_{f}-{\mu}_{e}n_{e},$ (14)

where $V$ and $\epsilon$ are the volume and energy density of the system,
respectively. $\mu_{f}$ and $n_{f}$ are quark’s chemical potential and density
with flavor $f$. $\mu_{e}$ and $n_{e}$ are the electron chemical potential and
electron number density, respectively. The quark number density, electron
number density, longitudinal pressure and transverse pressure can be evaluated
from the grand potential by

$\displaystyle n_{i,e}$ $\displaystyle=$
$\displaystyle-\dfrac{1}{V}\dfrac{\partial\Omega}{\partial\mu_{i,e}},$ (15)
$\displaystyle P_{\lVert}$ $\displaystyle=$
$\displaystyle-\mathfrak{B}-\dfrac{\Omega}{V}+P_{CSC}-\dfrac{B^{2}}{8\pi},$
(16) $\displaystyle P_{\bot}$ $\displaystyle=$
$\displaystyle-\mathfrak{B}-\dfrac{\Omega}{V}+P_{CSC}-M.B+\dfrac{B^{2}}{8\pi},$
(17)

where $\mathfrak{B}$ is a free parameter interpreted as non-perturbative
effects not included in perturbative expansion kurkela2010cold . To describe a
more realistic quark matter, we have considered the color superconductivity
pressure, $P_{CSC}$. This pressure is due to an appearing gap in the energy of
the Cooper pairs in the Color-Flavor-Locked phase which is given as
$P_{CSC}=\dfrac{\Delta^{2}(\mu_{u}+\mu_{d}+\mu_{s})^{2}}{3\pi^{2}}$,
alford1999color  where $\Delta$ is the gap parameter. The gap parameter
depends on the magnetic field. Therefore, we consider different values for
$\Delta$ ranging from $0$ to $100MeV$ in appendix B. This appendix shows that
the maximum gravitational mass of SQS increases by increasing $\Delta$. In the
following, we only obtain our results for $\Delta=100MeV$. In Eq. (17), $M$ is
the magnetization of the system, which is calculated by
$-\dfrac{\partial\Omega}{\partial B}$. To obtain the grand potential, we use
the fact that $\Omega$ for each quark flavor can be divided into two
perturbative and non-perturbative parts

$\Omega=\Omega_{free}+\Omega_{2L},$ (18)

where the non-perturbative part, $\Omega_{free}$, is the thermodynamic the
potential of a free fermion moving in a uniform magnetic field which is given
by

$\displaystyle\frac{\Omega_{free}}{V}=$
$\displaystyle-\,\sum_{f,j=0}^{\infty}\frac{|e_{f}B|}{2\pi^{2}}\bigg{\\{}3(2-\delta_{j0})$
$\displaystyle\times\int_{0}^{(k_{z}^{f})_{F}}dk_{z}\frac{k_{z}{}^{2}}{\sqrt{m_{f}{}^{2}+k_{z}{}^{2}+2j|e_{f}B|}}\bigg{\\}}$
$\displaystyle-\,\sum_{j=0}^{\infty}\frac{|eB|}{2\pi^{2}}\bigg{\\{}(2-\delta_{j0})$
$\displaystyle\times\int_{0}^{(k_{z}^{e})_{F}}dk_{z}\frac{k_{z}{}^{2}}{\sqrt{m_{e}{}^{2}+k_{z}{}^{2}+2j|eB|}}\bigg{\\}}$
(19)

In above equation, $\left(k_{z}^{f}\right)_{F}$ denotes the quark fermi
momentum with flavor $f$ which is $\sqrt{\mu_{f}^{2}-m_{f}^{2}-2j|e_{f}B|}$.
$\left(k_{z}^{e}\right)_{F}$ denotes the electron fermi momentum with which is
$\sqrt{\mu_{e}^{2}-m_{e}^{2}-2j|eB|}$. The perturbative part of the grand
potential, $\Omega_{2L}$, is the QCD contribution at leading order for a
system of quarks and gluons under strong magnetic field which is depicted by a
two-loop diagram in Fig. 3.

Figure 3:  Two-loop diagram contributing to perturbative part of the grand
potential at the leading order.

Appendix A shows the procedure to calculate the two-loop diagram at finite
chemical potential. The masses of up and down quarks are negligible. For the
mass of strange quark, we use the running mass from Eq. (11). The result of
two-loop diagram (Eq. (A)), depends on the renormalization scale $Q$ that
appears in mass and coupling constant through Eqs. (6) and (11). In the
absence of strong magnetic field, phenomenological models propose
$Q=2\pi\sqrt{T^{2}+\frac{\mu^{2}}{\pi^{2}}}$ for a massless quark with
chemical potential $\mu$ at temperature $T$. At zero temperature, for a system
composed of $N_{f}$ flavors, $Q=2\left({\sum_{f}\mu_{f}/3}\right)\equiv
2\overline{\mu}$ which can have a variation by a factor of 2 with respect to
its central value schneider2003qcd ; bandyopadhyay2019pressure ;
karmakar2019anisotropic .

Fig. 4 shows the ratio of $\dfrac{P_{\bot}-P_{\lVert}}{P_{\lVert}}$ versus
baryon density, $n_{B}$, which indicates that the difference between
longitudinal and transverse pressures decreases by increasing baryon number
density. In section VIII, we show that the maximum gravitational masses due to
these pressures ($P_{\lVert},P_{\bot}$) result in the same values
approximately. Therefore our assumption for having an isotropic structure for
the system under study is justified.

Fig. 5 shows the validity of our perturbative calculations. In this figure,
the ratio of calculated pressure up to the leading order in $\alpha_{s}$ to
the pressure of free quarks is presented. As the figure shows, this ratio
tends to unity by increasing the renormalization scale $Q$ for three choices
of renormalization scale in RPT approach. Such an asymptotic behavior is
necessary for any perturbative method, which shows that the perturbative part
is lower than the non-perturbative part. Also, the non-perturbative
contribution decreases by increasing the energy due to the asymptotic freedom
of the coupling constant. Furthermore, we have imposed the following
constraints in our perturbative calculations:

i) To establish the causality, the ratio of the speed of sound to the speed of
light in the vacuum should be lower than unity.

ii) To establish dynamical stability, the adiabatic index should be lower than
$\dfrac{4}{3}$ panah2019contraction .

iii) The relation between pressure and energy density is in such a way that
$\epsilon+P\geq 0$ and $\epsilon\geq\mid P\mid$ roupas2021qcd .

iv) To describe a realistic SQS, the minimum value of the baryon number
density should be higher than the nuclear saturation density
$(n_{B}=0.16fm^{-3})$.

Figure 4: The ratio of $\dfrac{P_{\bot}-P_{\lVert}}{P_{\lVert}}$ versus baryon
number density.  Figure 5: The ratio of calculated pressure up to leading
order in $\alpha_{s}$ to the pressure of the free quarks versus
renormalization scale $Q$ for three choices of renormalization scale in RPT
approach.

##  V Equation of state and stability conditions

In this section, we use the numerical results from the grand potential
presented in the previous section to derive the EoS of a stable SQS. We derive
the relationship between the total energy density and pressure (EoS) to
evaluate the star’s maximum mass and corresponding radius. Therefore, we can
derive the EoS using Eqs. (15-IV) and (A) from the following relation

$\varepsilon=-P_{\lVert}+\mu_{u}n_{u}+\mu_{d}n_{d}+\mu_{s}n_{s}.$ (20)

It should be noted that in deriving the EoS, we have considered beta
equilibrium and charge neutrality conditions blaschke2001physics . For beta
equilibrium, we have the following weak processes.

$d\longrightarrow u+e+\bar{\nu}_{e},~{}~{}~{}\&~{}~{}~{}u+e\longrightarrow
d+\nu_{e},$ $s\longrightarrow
u+e+\bar{\nu}_{e},~{}~{}~{}\&~{}~{}~{}u+e\longrightarrow s+\nu_{e},$
$s+u\longleftrightarrow d+u,$ (21)

which lead to the following conditions

$\mu_{s}=\mu_{d}\equiv\mu,~{}~{}~{}\&~{}~{}~{}\mu_{u}=\mu-\mu_{e}.$ (22)

The mean free path of neutrinos is considerably larger than the size of the
star, and therefore their chemical potential can be neglected in the
calculations. For charge neutrality, the following condition must be imposed.

$\frac{2}{3}n_{u}-\frac{1}{3}n_{d}-\frac{1}{3}n_{s}-n_{e}=0,$ (23)

where $n_{e}$, the electron number density, is equal to

$n_{e}=\dfrac{eB}{(2\pi)^{2}}\mu_{e}^{2}.$ (24)

Furthermore, for the existence of SQS, another condition must be implemented,
which implies that the minimum energy per baryon should be lower than that of
the most stable nuclei $\left({}^{56}Fe\right)$ weber2005strange

$\dfrac{\epsilon}{n_{B}}\leq 0.93GeV,$ (25)

where $n_{B}$, the baryon number density, is obtained by quark number
densities as

$n_{B}=\dfrac{n_{u}+n_{d}+n_{s}}{3}.$ (26)

##  VI Thermodynamic properties of strange quark matter in strong magnetic
field

In this section, we investigate our results for thermodynamic properties such
as baryon density, speed of sound, adiabatic index, and also the EoS of SQM in
RPT and BPT, respectively.

###  VI.1 Results in RPT

Using Eqs. (15), (IV) and (A), the quark number densities as functions of
quark chemical potentials are obtained. Then, we use Eqs. (22), (23), (24) and
(26) to get the allowed values of chemical potentials for the quarks. The
values of $\mathfrak{B}$ in Eq. (16) are chosen in such a way to get zero
total pressure at the surface of the star where the baryon density is minimum.
If we neglect interaction between quarks, $\mathfrak{B}$ has the role of MIT
bag constant ranging from $41.58\ MeV/fm^{3}$ to $319.31\ MeV/fm^{3}$
aziz2019constraining . The EoS of SQM for different choices of renormalization
scale is shown in Fig. 6. The minimum value for the renormalization scale
starts at a point from which the perturbative calculation is valid.
Furthermore, for renormalization scales more than $Q=3.25\overline{\mu}$, the
results do not change significantly. From this figure, it is understood that
the EoS becomes stiffer by increasing the renormalization scale. To establish
causality condition, the ratio of the speed of sound to the speed of light in
vacuum $\left(\sqrt{dP/d\epsilon}\right)$ must be lower than unity. Fig. 7
shows that our results satisfy the causality condition for different choices
of renormalization scale.

Figure 6: Pressure versus energy density (EoS) for different choices of
renormalization scale in RPT. Figure 7:  Squared speed of sound versus energy
density for different choices of renormalization scale in RPT.

It is known that for dynamical stability the adiabatic index
$\Gamma=dP/d\epsilon\dfrac{(P+\epsilon)}{P}$ must be higher than $4/3$
bardeen1966catalogue ; knutsen1988stability ; mak2013isotropic ;
panah2019white ; panah2019contraction . The adiabatic index has been presented
as a function of energy density in Fig. 8. As one can see from this figure,
the condition  $\Gamma>4/3$ is satisfied for all renormalization scales.

Figure 8: Adiabatic index versus the ratio of baryon density to nuclear
saturation density for different choices of renormalization scale in RPT.

###  VI.2 Results in BPT

Now we perform the calculations for BPT the same as we did for RPT. Here,
$M_{B}$ in Eq. (6) is equal to $\sqrt{2\pi\sigma}$. The parameter $\sigma$ is
the string tension which is equal to $0.18\ GeV^{2}$ deur2016qcd . Assuming
the same constraints presented in section V, we obtain the EoS of the SQM for
different renormalization scales. For the values $Q>2.75\overline{\mu}$, the
results for the structural properties of the star do not change significantly.

Figure 9: Pressure versus energy density (EoS) for different choices of
renormalization scale in BPT.

The diagrams of the speed of sound and adiabatic index in BPT are shown in
Figs. 10 and 11. These diagrams indicate that the constraints related to the
speed of sound and adiabatic indices are satisfied well by our obtained EoSs.

Figure 10: Squared speed of sound versus energy density for different choices
of renormalization scale in BPT.  Figure 11:  Adiabatic index versus the ratio
of baryon density to nuclear saturation density for different choices of
renormalization scale in BPT.

##  VII Magnetized strange quark star structure

After computing the EoS, the structure of a hydrostatic and non-rotating
strange stars can be obtained by general relativistic equations of
hydrostatics. Here, the relation between mass and radius of the star is found
by solving TOV equations oppenheimer1939massive ,

$\displaystyle\ dM(r)$ $\displaystyle=$ $\displaystyle 4\pi
r^{2}\epsilon(r)dr,$ (27) $\displaystyle\frac{dP(r)}{dr}$ $\displaystyle=$
$\displaystyle\dfrac{G\left[P(r)+\epsilon(r)\right]\left[M(r)+4\pi
r^{3}P(r)\right]}{r\left(2GM(r)-r\right)},$ (28)

where $G$ and $r$ are the Newton gravitational constant and radial coordinate
of the star, respectively. For different central pressure values, different
gravitational masses and radii are obtained. In the following, the results for
different values of the renormalization scale in RPT and BPT approaches are
discussed.

Our results for the structural properties of SQS in RPT have been presented in
Figs. 12 and 13. The limiting behavior of mass in Fig. 12 shows the maximum
gravitational mass of SQS. From this figure, we see that the maximum
gravitational mass increases by increasing the renormalization scale, $Q$,
which corresponds to stiffer EoSs (Fig. 9). This behavior indicates that
stiffer EoS leads to higher gravitational mass for SQS in our model. Our
results for maximum gravitational mass and corresponding radius in RPT are
given in Table 1. Previously

Figure 12:  Mass versus energy density for different choices of
renormalization scale in RPT. Figure 13:  Mass-radius diagram for different
choices of renormalization scale in RPT.

the maximum gravitational mass of the SQS by perturbative calculation in the
absence of magnetic field was obtained in Refs. fraga2005role ;
kurkela2010cold . In Ref. kurkela2010cold , by using regular perturbation
theory up to second order of coupling constant, the maximum mass of a pure
strange quark star was obtained to be $2.75M_{\odot}$. From Table. 1, we see
that in RPT model, the maximum mass of SQS can reach the value $M\simeq
3.17M_{\odot}$ which is considerably higher than that of in the absence of
magnetic field calculated in Refs. kurkela2010cold ; fraga2005role .

Table 1: maximum gravitational mass and the corresponding radius of SQS for different renormalization scales in RPT. $Q$/$\overline{\mu}$ | $\dfrac{M}{M_{\odot}}$ | $R(km)$ | $R_{Sch}(km)$ | $z$ | $\dfrac{M_{BB}}{{M}_{\odot}}$ | $\sigma(10^{-1})$
---|---|---|---|---|---|---
2.25 | 1.72 | 10.56 | 5.07 | 0.38 | 3.18 | 4.80
2.75 | 2.09 | 11.98 | 6.16 | 0.43 | 3.61 | 5.14
3.25 | 2.97 | 15.67 | 8.76 | 0.50 | 4.72 | 5.59
3.75 | 3.17 | 16.53 | 9.34 | 0.52 | 4.98 | 5.65

Now we discuss the maximum gravitational mass and the corresponding radius of
the SQS in BPT. Figs. 14 and 15 show that by increasing $Q$, the maximum mass
increases. Our

Figure 14:  Mass versus energy density for different choices of
renormalization scale in BPT. Figure 15: Mass-radius diagram for different
choices of renormalization scale in BPT.

results for the maximum gravitational mass and the corresponding radius in BPT
are given in Table 2. By comparing our results in two approaches from Tables 1
and 2, we can conclude that the gravitational mass of strange quark star in
BPT is higher than that of RPT. It seems that it is originated from the
behavior of the coupling constant. The coupling constant in BPT is smaller and
runs slower than that in RPT.

Table 2: The values of the mass and the corresponding radius of the star in different renormalization scales in BPT $Q$/$\overline{\mu}$ | $\dfrac{M}{M_{\odot}}$ | $R(Km)$ | $R_{Sch}(km)$ | $z$ | $\dfrac{M_{BB}}{{M}_{\odot}}$ | $\sigma(10^{-1})$
---|---|---|---|---|---|---
1.75 | 2.11 | 12.22 | 6.22 | 0.43 | 3.68 | 5.09
2.25 | 2.77 | 14.96 | 8.17 | 0.48 | 4.51 | 5.46
2.75 | 3.31 | 17.26 | 9.76 | 0.52 | 5.20 | 5.65

###  VII.1 Compactness

Compactness is a quantity that gives us information about the strength of
gravity of compact objects. The compactness of a spherical object is usually
defined in the following form

$\sigma=\frac{R_{Sch}}{R},$ (29)

where $R_{Sch}$ and $R$ are the Schwarzschild radius and radius of compact
object, respectively. Our results for the compactness are presented in Tables.
1 and 2, which indicate that by increasing renormalization scale ($Q$), the
compactness from the perspective of a distant observer (or an observer outside
the compact object) increases.

###  VII.2 Gravitational Redshift

Another quantity that gives us information on the strength of gravity is
related to the gravitational redshift. The gravitational redshift is given by

$z=\frac{1}{\sqrt{1-\frac{2GM}{c^{2}R}}}-1,$ (30)

where $M$ and $R$ are related to the compact star’s mass and radius,
respectively. The results related to the gravitational redshift are given in
Tables. 1 and 2. These results indicate that the gravitational redshift
increases by increasing the renormalization scale. But this quantity is less
than $1$ for magnetized SQSs.

###  VII.3 Buchdahl-Bondi Bound

Here, we want to study the upper mass limit of a static spherical quark star
with uniform density in GR, the so-called Buchdahl theorem buchdahl1959general
. The GR compactness limit is given by buchdahl1959general

$M\leq M_{BB}=\frac{4c^{2}R}{9G},$ (31)

in which the upper mass limit is $M_{\max}=\frac{4c^{2}R}{9G}$. Our numerical
results in Tables. 1 and 2, confirm that the obtained masses of magnetized
SQSs in GR respects the equation (31).

Our calculations from the redshift ($z$) and Buchdahl-Bondi bound ($M_{BB}$)
of magnetized SQSs confirm that these compact objects cannot be black holes,
because the values of redshift for these compact objects are finite ($z<1$),
and also the maximum mass of these compact objects are less than Buchdahl-
Bondi bound ($M_{\max}<M_{BB}$).

##  VIII The effect of the anisotropy on the structure of the magnetized SQS

In this section, we consider the magnetic fields with the values $10^{18}\ G$,
$2\times 10^{18}\ G$ and $3\times 10^{18}\ G$ and calculate the maximum mass
of the SQS by considering longitudinal pressure, $P_{\lVert}$ and the
transverse pressure, $P_{\bot}$, in TOV equations separately. The maximum
gravitational mass and the corresponding radius evaluated by $P_{\lVert}$, are
denoted by $M_{\lVert}$ and $R_{\lVert}$ and the ones obtained by $P_{\bot}$,
are denoted by $M_{\bot}$ and $R_{\bot}$, respectively. Fig. 16 and Table. 3
show our results in RPT by assuming the renormalization scale
$Q=3.75\overline{\mu}$. As we can see from the results, there is no
distinguishable difference between $M_{\lVert}$ and $M_{\bot}$ for the
magnetic field $10^{18}\ G$ and $2\times 10^{18}\ G$. But for $B=3\times
10^{18}\ G$ this difference is obvious and the isotropic structure is not a
good approximation. We have defined the $\delta$ parameter as the measure of
the anisotropy of the star which is equal to
$\dfrac{M_{\lVert}-M_{\bot}}{\left(M_{\lVert}+M_{\bot}\right)/2}$ chu2018quark
. The values of the $\delta$ parameter for different magnetic fields have been
presented in Table. 3. As we can see from this table, this parameter is
negligible for the magnetic fields less than about $2\times 10^{18}\ G$. The
threshold magnetic field from which an anisotropic approach begins to be
significant lies in the interval $2\times 10^{18}G<B<3\times 10^{18}G$. Such
behavior can also be seen in Ref. chu2018quark  within the extended confined
isospin-density-dependent mass model. Therefore, it seems that considering the
isotropic structure for a SQS under the magnetic fields $B<2\times 10^{18}G$
is reasonable in our models.

Figure 16:  Mass-radius diagram for different magnetic fields in RPT and $Q=3.75\overline{\mu}$. $M_{\lVert}$ and $M_{\bot}$ are the maximum mass calculated by the pressures $P_{\lVert}$ and $P_{\bot}$, respectively. Table 3: The values of the maximum mass and the corresponding radius of the SQS for different magnetic fields in RPT and $Q=3.75\overline{\mu}$. $M_{\lVert}$ and $M_{\bot}$ are the maximum mass calculated by the pressures $P_{\lVert}$ and $P_{\bot}$, respectively. $R_{\lVert}$ and $R_{\bot}$ are their corresponding radii. $B(G)$ | $\dfrac{M_{\lVert}}{M_{\odot}}$ | $R_{\lVert}(Km)$ | $\dfrac{M_{\bot}}{M_{\odot}}$ | $R_{\bot}(Km)$ | $\delta(\%)$
---|---|---|---|---|---
$10^{18}$ | 3.17 | 16.53 | 3.13 | 16.35 | 1.27
$2\times 10^{18}$ | 3.11 | 16.16 | 3.01 | 15.67 | 3.27
$3\times 10^{18}$ | 2.95 | 15.57 | 2.49 | 13.40 | 17

##  IX Conclusions and Outlook

In this paper, we first introduced QCD Feynman rules in the presence of a
strong magnetic field. Then we performed perturbative calculations of the
thermodynamic potential of a strange quark matter (SQM) up to
_O_($\alpha_{s}$) in the presence of magnetic field $B=10^{18}\ G$. We used
two approaches in our perturbative calculations: i) QCD without IR freezing
effect in regular perturbation theory (RPT) and ii) QCD with IR freezing
effect in background perturbation theory (BPT). The thermodynamic potential of
SQM was computed by separating it into non-perturbative and perturbative
parts. The non-perturbative part contained the interaction of quarks with the
magnetic field, and the perturbative part included QCD interaction between
quarks. The perturbative part contained a two-loop diagram added to the
thermodynamic potential of a moving fermion in a uniform magnetic field as the
non-perturbative part. Using the obtained thermodynamic potential and also
considering the stability conditions of SQM, thermodynamic properties such as
quark number densities, speed of sound, adiabatic index, and also the EoSs of
strange quark stars (SQSs) were obtained in RPT and BPT models. Then by
employing the obtained EoSs, we calculated the structure properties of SQSs.
Moreover, the redshift, compactness, and Buchdahl-Bondi bound of the SQSs were
calculated to show that these compact objects cannot be black holes. Then, the
anisotropic effect on the structure of SQS in our models under different
magnetic fields was calculated. it was shown that the threshold magnetic field
from which an anisotropic approach begins to be significant lies in the
interval $2\times 10^{18}G<B<3\times 10^{18}G$. The maximum mass of the SQS
under the magnetic field $B=10^{18}\ G$ was obtained $3.17M_{\odot}$ and
$3.31M_{\odot}$ in RPT and BPT, respectively, which is considerably higher
than those of Bag models and NJL theories. Furthermore, it is notable that
some of our results for the maximum gravitational mass are outside the limits
obtained by observational constraints for neutron stars. Therefore an observed
violation of those limits might be a piece of evidence for SQS. For example
the compact object with the mass $2.5-2.67M_{\odot}$ in GW190814
miao2021bayesian  or the remnant mass of GW190425 sedaghat2021compact  might
be SQSs.

## Acknowledgements

We wish to thank Shiraz University Research Council. S. M. Zebarjad thanks
Physics Department of UCSD for hospitality during his sabbatical. B. Eslam
Panah thanks the University of Mazandaran. We are indebted to Yu.A. Simonov
for his good comments and discussions, and A. Manohar for reading the
manuscript and his helpful comment.

##  X Appendix

##  Appendix A Calculation of the thermodynamic potential of the two-loop
diagram

By using the Feynman rules introduced in section III we have

$\displaystyle\dfrac{\Omega_{2L}}{V}=$ $\displaystyle\dfrac{g^{2}d_{A}}{2}$
$\displaystyle\times\sum_{n,j=0}^{\infty}\int\dfrac{d^{4}p}{(2\pi)^{4}}\int\dfrac{d^{4}k}{(2\pi)^{4}}\dfrac{(-1)^{j+n}e^{-(p_{\perp}^{2}+k_{\perp}^{2})/{\lvert
e_{f}B\rvert}}}{(p-k)^{2}}$
$\displaystyle\times\dfrac{Tr\left[\gamma^{\mu}D_{j}(\lvert
e_{f}B\rvert,p)\gamma_{\mu}D_{n}(\lvert
e_{f}B\rvert,k)\right]}{\mathcal{A}_{1}\mathcal{A}_{2}},$ (32)

where $\mathcal{A}_{1}=(p_{0}+i\mu)^{2}+(p_{3}^{2}+m_{f}^{2}+2j\lvert
e_{f}B\rvert)$ and
$\mathcal{A}_{2}=(k_{0}+i\mu)^{2}+(k_{3}^{2}+m_{f}^{2}+2n\lvert
e_{f}B\rvert)$. Also, $d_{A}$ is $N_{c}^{2}-1$. The summation is over Landau
levels, and $D_{j}(\lvert e_{f}B\rvert,p)$ and $D_{n}(\lvert e_{f}B\rvert,k)$
are functions in terms of generalized Laguerre polynomials and Dirac Gamma
matrices in Euclidean space. To obtain the contribution (A), we use a very
useful method derived in Ref. ghicsoiu2017high  which is referred to as
cutting rules. Based on this method, the evaluation of the Feynman integrals
at zero temperature and finite chemical potential is reduced to the evaluation
of three-dimensional phase space integrals over on-shell amplitudes at
$T=\mu=0$. Indeed, the one point irreducible _N_ -loop _n_ -point Feynman
diagram, $F(P_{k},\mu)$, can be simplified as:

$\displaystyle F(P_{k},\mu)=$ (33) $\displaystyle
F_{0-cut}(P_{k})+F_{1-cut}(P_{k},\mu)+...+F_{N-cut}(P_{k},\mu),$

where $F_{0-cut}(P_{k})$ is the original diagram computed at zero chemical
potential, and the other terms are contributions resulting from the cutting
procedure. Indeed, $F_{j-cut}(P_{k},\mu)$ represents the sum of all diagrams
in which the _j_ number of internal fermion propagators have been cut from the
original graph. In the following, we explain the steps in the cutting
procedure:

1\. The cut propagators are removed from the original

graph

2\. The remaining $N-j$\- loop $n+2j$\- point amplitude

is computed by considering the point that the all

external momenta are real-valued.

3\. The cut momenta $p_{i}$ is set on-shell by inserting

$({p_{0}})_{i}=iE_{i}$.

4\. The remaining expression is integrated over the cut

three-dimensional momenta $p_{i}$ with the weight

$\dfrac{-\theta(\mu-E_{i})}{2E_{i}}$, where for a quark moving in a uniform

magnetic field $B$ at the $j$th Landau level, $E_{p}$ is equal

to $\sqrt{p_{3}^{2}+m_{f}^{2}+2j\lvert e_{f}B\rvert}$.

Notably, $F_{0-cut}(P_{k})$ has no dependence on chemical potential, and
according to Eq. (15), it has no contribution in quark number densities.
Therefore, it is has been removed from calculations. By following the cutting
procedure and Eq. (A), we get

$\dfrac{\Omega_{2L}}{V}=F_{1-cut}+F_{2-cut},$ (34)

where $F_{1-cut}$ and $F_{2-cut}$ are given as

$\displaystyle F_{1-cut}$
$\displaystyle=-2\times\dfrac{g^{2}d_{A}}{2}\bigg{\\{}\sum_{n,j=0}^{\infty}\int\dfrac{\theta(\mu-
E_{p})}{2E_{p}}\dfrac{d^{3}\mathbf{p}}{(2\pi)^{3}}$
$\displaystyle\times\int\dfrac{d^{4}k}{(2\pi)^{4}}I_{1}\bigg{\\}}_{p_{0}=iE_{p}},$
(35) $\displaystyle F_{2-cut}$
$\displaystyle=\dfrac{g^{2}d_{A}}{2}\bigg{\\{}\sum_{n,j=0}^{\infty}\int\dfrac{\theta(\mu-
E_{p})}{2E_{p}}\dfrac{d^{3}\mathbf{p}}{(2\pi)^{3}}$
$\displaystyle\times\int\dfrac{\theta(\mu-
E_{k})}{2E_{k}}\dfrac{d^{3}\mathbf{k}}{(2\pi)^{3}}I_{2}\bigg{\\}}_{p_{0}=iE_{p},k_{0}=iE_{k}},$
(36)

where, $I_{1}$ and $I_{2}$ are defined as

$\displaystyle I_{1}$
$\displaystyle=(-1)^{j+n}e^{-(p_{\perp}^{2}+k_{\perp}^{2})/{\lvert
e_{f}B\rvert}}$ $\displaystyle\times\dfrac{Tr\left[\gamma^{\mu}D_{j}(\lvert
e_{f}B\rvert,p)\gamma_{\mu}D_{n}(\lvert
e_{f}B\rvert,k)\right]}{({k_{0}}^{2}+{E_{k}}^{2})((p_{0}-k_{0})^{2}+(\mathbf{p}-\mathbf{k})^{2})},$
(37) $\displaystyle I_{2}$
$\displaystyle=(-1)^{j+n}e^{-(p_{\perp}^{2}+k_{\perp}^{2})/{\lvert
e_{f}B\rvert}}$ $\displaystyle\times\dfrac{Tr\left[\gamma^{\mu}D_{j}(\lvert
e_{f}B\rvert,p)\gamma_{\mu}D_{n}(\lvert
e_{f}B\rvert,k)\right]}{(p_{0}-k_{0})^{2}+(\mathbf{p}-\mathbf{k})^{2}}.$ (38)

Due to the Landau quantization of transverse momentums at the $j$th Landau
level, we set $\sqrt{2j\lvert e_{f}B\rvert}$ and $\sqrt{2(j+1)\lvert
e_{f}B\rvert}$ for the lower and upper limits of the transverse momentum
integrals, respectively. To calculate Eq. (34), we need to sum over all the
terms corresponding to $n$th and $j$th Landau levels. Indeed for each pair
$(n,j)$ which appear as Landau levels in propagators of quarks, there is an
integral which should be computed. It should be noted that by multiplying a
factor $-1/2$ in Eq. (34), originated from perturbative expansion
kapusta2006finite , we obtain Eq. (A).

$\displaystyle\dfrac{\Omega_{2L}}{V}=\dfrac{g^{2}d_{A}}{2}\sum_{n,j=0}^{\infty}\bigg{\\{}\int\dfrac{d^{4}p}{(2\pi)^{4}}\int\dfrac{d^{4}k}{(2\pi)^{4}}I_{1}(-\theta(\mu-
E_{q}))$
$\displaystyle-\dfrac{1}{2}\int\dfrac{d^{4}p}{(2\pi)^{4}}\int\dfrac{d^{4}k}{(2\pi)^{4}}I_{2}(-\theta(\mu-
E_{q}))(-\theta(\mu-E_{p}))\bigg{\\}},$ (39)

##  Appendix B Investigating the effect of the gap parameter in maximum
gravitational mass of SQS

By using different values of $\Delta$ in Eq. (16), we have plotted mass-radius
diagrams for SQS under the magnetic field $B=10^{18}G$. As the Table 4 and
Figures 17 and 18 show, the maximum gravitational mass of SQS increases by
increasing $\Delta$ for both RPT and BPT models. Furthermore, it is notable
that our calculations show that the impact of $\Delta$ on the structure of SQS
begins to be significant from $\Delta>25MeV$.

Figure 17:  Mass-radius diagram for different values of $\Delta$ parameter in RPT ($Q=3.75\overline{\mu}$) model. Figure 18:  Mass-radius diagram for different values of $\Delta$ parameter in BPT ($Q=2.75\overline{\mu}$) model. Table 4: The values of the maximum mass and the corresponding radius of SQS for different values of $\Delta$ in RPT($Q=3.75\overline{\mu}$) and BPT($Q=2.75\overline{\mu}$) models. | RPT | RPT | BPT | BPT
---|---|---|---|---
$\Delta(MeV)$ | $\dfrac{M_{\lVert}}{M_{\odot}}$ | $R_{\lVert}(Km)$ | $\dfrac{M_{\lVert}}{M_{\odot}}$ | $R_{\lVert}(Km)$
$0$ | 2.71 | 14.81 | 2.84 | 15.80
$25$ | 2.75 | 15.00 | 2.87 | 15.89
$50$ | 2.86 | 15.65 | 2.95 | 16.04
$100$ | 3.17 | 16.53 | 3.31 | 17.26

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