# Magnetized strangelets with anomalous magnetic moment and Coulomb
interactions

Huai-Min Chen,1<EMAIL_ADDRESS>Xiao-Wei Li,3
<EMAIL_ADDRESS>Cheng-Jun Xia4<EMAIL_ADDRESS>Jing-Tao Wang2
Guang-Xiong Peng2,5,6<EMAIL_ADDRESS>1School of Mechanical and Electrical
Engineering, Wuyi University, Wuyishan 354300, China
2School of Nuclear Science and Technology, University of Chinese Academy of
Sciences, Beijing 100049, China
3Shenzhen Institute for Quantum Science and Engineering, Southern University
of Science and Technology, Shenzhen 518055, China
4Center for Gravitation and Cosmology, College of Physical Science and
Technology, Yangzhou University, Yangzhou 225009, China
5Theoretical Physics Center for Science Facilities, Institute of High Energy
Physics, P.O. Box 918, Beijing 100049, China
6Synergetic Innovation Center for Quantum Effects and Application, Hunan
Normal University, Changsha 410081, China

###### Abstract

We study the magnetized strangelets in the baryon density-dependent quark mass
model, including the effects of both confinement and lead-order perturbation
interactions. The properties of magnetized strangelets are investigated under
the the field strength $2\times 10^{17}\ \mathrm{G}$, where the anisotropy
caused by the strong magnetic field is insignificant can be treated
approximately as an isotropic system. The consideration of anomalous magnetic
moments in the energy spectrum naturally solves the difficulty of infrared
divergence encountered in integrating the density of states. The Coulomb
interaction is accounted for a self-consistent treatment. The energy per
baryon, mechanically stable radius, strangeness and electric charge of
magnetized strangelets are presented, where their dependence on the field
strength and parameter of confinement and perturbation are investigated.

##  I Introduction

Strange quark matter(SQM) is one type of matter comprised of deconfined up,
down and strange quarks, which could be the ground state of QCD, rather than
${}^{56}\mathrm{Fe}$ Madsen_-162 . This idea is first suggested by Bodmer in
1971 Bodmer1971_PRD4-1601 , and developed by Witten in 1984
Witten1984_PRD30-272 . The small lumps of SQM whose baryon number $A\lesssim
10^{7}$ is called strangelets, named by Berger and Jaffe Berger1987_PRC35-213
, which is so small that electron can not exist in its interior, as the
electron Compton’s wavelength is much larger than the radius of strangelets
Madsen_-162 .

It is possible that strangelets can be produced by high-energy cosmic rays
collide with each other or the core of neutron stars Finch2006_JPG32-s251 ;
Biswas2017_PRC95-045201 ; Annala2020 ; Xia_PhysRevD.102.023031 , strange star
collisions could release strangelets as part of the energetic cosmic rays
Banerjee_JPG.25.L15 ; Banerjee_PhysRevL.85.1384 ; Madsen_PhysRevL.90.121102 ;
Madsen_PhysRevD.71.014026 ; Shaulov_EPJWebC.52.04010 . In addition,
strangelets could also be found as the products of high energy heavy ion
collision Schaffner-Bielich1997_PRC55-3038 ; Alimena2020_JPG47-090501 ;
Parvu2021_JCAP11-040 .

The strong magnetic field could exist in heavy-ion collisions and compact
stars, e.g., pulsars and magnetars. For pulsars, the typical magnitudes of
surface magnetic fields are about $10^{14}\mathrm{G}$ Duncan1992_AJL392-L9 ;
Taylor1993_AJSS88-529 ; Sob'yanin2023_PRD107-L081301 , and the observed
magnetic field strength on the surface of magnetars could be $10^{14}\sim
10^{15}\mathrm{G}$ Kaspi2017_55-261 . Moreover, the strong magnetic fields
might be generated in non-central high-energy heavy ion collisions, reaching
the value of $10^{19}\mathrm{G}$ Mizher2010_PRD82-105016 .

Due to the difficulties of perturbation calculations at low energy and high
density region and the sign problem in lattice point simulations, the study of
quark matter depends on various phenomenological models, e.g., the magnetized
strange quark matter (MSQM) with bag model CHAKRABARTY1994_09-3611 ;
Felipe2008_PRC77-015807 ; Tomoki2020_JPCS , the Nambu-Jona-Lasinio (NJL) model
Mizher2010_PRD82-105016 ; Li2016_PRD93-054005 ; Maruyama2019_JCP , the mass-
density-dependent model Hou2015_CPC39-015101 , the quark quasiparticle model
Wen2013a , the confined-isospin-density-dependent mass model
Chu2018_PLB778-447 , and the magnetized strangelets with the bag model
Felipe2012_39-045006 ; Ding2014_62-859 . At present, the MIT bag model has
been widely used and obtained some important results. Madsen considered the
finite size effects of strangelets Madsen_PhysRevLett.70.391 . Lugones and
Grunfeld studied the effects of surface tension and vector interactions on the
existence of a strangelet crust in the MIT Bag model GL2021prc103 ;
GL2021prd104 . Recently, they have revisit the quark-mass density-dependent
model and show that thermodynamic inconsistencies that have plagued the model
for decades can be solved if the model is formulated in the canonical ensemble
instead of the grand canonical one GL_PhysRevD.107.043025 . Presently, we
adopt a thermodynamically self-consistent mass-density-dependent model, so
called baryon density-dependent quark mass model Chen_PhysRevD.105.014011 ;
Chen_CPC.46.055102 , to investigate the properties of magnetized strangelets
at zero temperature.

As is shown in Ref. Ferrer2010_PRC82-065802 , due to the existence of uniform
magnetic field, the $\emph{O}(3)$ rotation symmetry breaks into $\emph{O}(2)$
symmetry, and therefore distinction between longitudinal- and transverse-to-
the-field pressures is caused. Nevertheless, we assume a spherical symmetry of
strangelets when magnetic field strength $B\lesssim 2\times
10^{17}\mathrm{G}$. Using the method adopted in Ref. Ferrer2010_PRC82-065802 ;
Felipe2012_39-045006 , we calculate the quantity
$\delta=|P_{\parallel}-P_{\perp}|/|P(B=0)|$, and use it as a measure to
distinguish between isotropic and anisotropic regions Felipe2012_39-045006 .

The electric charge of strangelets is an important property of strangelets. In
the absence of electron, generally, strangelets carry slight charge. Some
authors propose that positively charged strangelets would repel nuclei, while
negatively charged ones would attract them, leading to catastrophic transition
from normal matter into SQM Madsen_-162 . We treat the Coulomb interaction in
a thermodynamicly self-consistent manner, where the chemical potential
contributed by the Coulomb energy is included.

The present paper is organized as follow. In Sec. II, we give the
thermodynamic treatment of strangelets in a strong magnetic field with baryon
density-dependent quark mass model, while, the contribution of Coulomb
interaction is accounted for in a thermodynamically self-consistent approach.
In Sec. III, we adopt the quark mass scaling at zero temperatures, where both
the confinement interaction and perturbation interaction are considered with
the equivalent mass of quarks. In Sec. IV, the numerical results of the
magnetized strangelets are presented, e.g., their energy per baryon, radius
and charges, as well as the dependence of energy per baryon and radius on the
parameters of mass scaling. Finally, a short summary is given in Sec. V.

##  II Thermodynamic treatment in a strong magnetic field

###  II.1 Thermodynamic framework

The thermodynamic potential density $\Omega_{0}$ at zero temperature is given
for a free quasiparticle system due to its dependence on $\mu^{*}$, i.e.,

$\Omega_{0}=\sum_{i}\int\Big{(}\sqrt{p^{2}+m_{i}^{2}}-\mu_{i}^{*}\Big{)}\rho_{i}(p)\mathrm{d}^{3}p,$
(1)

where index $i$ goes over the particle types of system, and $\rho_{i}$ is the
density of state. In previous quark mass-density-dependent model, quark acts
like a free particle with a density-dependent mass
$m_{i}=m_{0,i}+m_{I}(n_{b})$, where $n_{b}=\frac{1}{3}\sum_{i}n_{i}$ and
$m_{0,i}$ are baryon number density and the quark current mass with $m_{u0}=5\
\mathrm{MeV}$, $m_{d0}=10\ \mathrm{MeV}$, and $m_{s0}=100\ \mathrm{MeV}$. In
terms of the multiple reflection expansion method Madsen_PhysRevLett.70.391 ;
Balian1970_AoP60-401 , the function $\rho_{i}$ of strangelets can be given by

$\rho_{i}=\frac{g_{i}}{(2\pi)^{3}V}\Big{[}V+\frac{S}{p}f_{S}\big{(}\frac{m_{i}}{p}\big{)}+\frac{C_{r}}{p^{2}}f_{C}\big{(}\frac{m_{i}}{p}\big{)}\Big{]},$
(2)

where $V$, $S$ and $C_{r}$ are volume, area and extrinsic curvature of
strangelets, and $g_{i}$ is the corresponding degree of freedom. For a
spherical system, we have $V=4\pi R^{3}/3$, $S=4\pi R^{2}$, $C_{r}=8\pi R$.
The functions $f_{S}$ and $f_{C}$ are given by Berger1987_PRC35-213 ;
Madsen1994_PRD50-3328

$\displaystyle f_{S}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\mathrm{arctan}\big{(}\frac{m_{i}}{p}\big{)},$ (3)
$\displaystyle f_{C}$ $\displaystyle=$
$\displaystyle\frac{1}{6}\Big{[}1-\frac{3p}{2m_{i}}\mathrm{arctan}\big{(}\frac{m_{i}}{p}\big{)}\Big{]}.$
(4)

Note that the density of states in Eq. (2) becomes negative and is hence
unphysical at small radii and momenta. An infrared cutoff
Neergaard1999_PRD60-054011  is usually introduced to treat the unphysical
contributions, which is not included here since their contributions are small
and the natural truncation caused by anomalous magnetic moments has
considered.

At the zero temperature, within the framework of an equivalent mass model
Peng_PhysRevC.62.025801 , the particle mass depends on baryon number density
$n_{b}$. Given that the baryon number density is defined as
$n_{b}\equiv\sum_{i}N_{i}/(3V)$, this implies that the particle mass becomes a
function of volume and particle numbers. In such a model, if we adopt the
expressions of free particle for the thermodynamic potential density and
chemical potential as presented in Eq. (1) to represent the actual values for
the system, it would lead to thermodynamic inconsistencies
Xia2014_PRD89-105027 .

To avoid this issue, it is necessary to incorporate the corrections arising
from the dependence of mass on the state variables. Based on the
consideration, the analysis starts with the energy
$\overline{E}(V,\\{N_{i}\\})=V\int\sqrt{p^{2}+m_{i}^{2}}\rho_{i}\text{d}^{3}p$,
which is a function of the state variables such as volume and particle number.
At the zero temperature, the energy is given by

$\overline{E}(V,\\{N_{i}\\},\\{m_{i}\\})=\Omega_{0}V+\sum_{i}\mu_{i}^{*}N_{i}.$
(5)

Considering mass as an intermediate variable, the differential of Eq. (5) is
given by

$\displaystyle\text{d}\overline{E}$ $\displaystyle=$
$\displaystyle\left[\Omega_{0}+V\left(\frac{\partial\Omega_{0}}{\partial
V}\right)_{\\{m_{i}\\},\\{N_{i}\\}}\right]\text{d}V+\sum_{i}\mu_{i}^{*}\text{d}N_{i}$
(6) $\displaystyle+$
$\displaystyle\sum_{i}V\left(\frac{\partial\Omega_{0}}{\partial
m_{i}}\right)_{\\{N_{i}\\},V}\text{d}m_{i}.$

The differential of $m_{i}$ is as follows

$\displaystyle\text{d}m_{i}$ $\displaystyle=\frac{\partial m_{i}}{\partial
n_{b}}\left[\left(\frac{\partial n_{b}}{\partial
V}\right)_{N_{i}}\text{d}V+\sum_{j}\left(\frac{\partial n_{b}}{\partial
N_{j}}\right)_{V}\text{d}N_{j}\right]$ (7)
$\displaystyle=\frac{1}{3V^{2}}\sum_{j}\frac{\partial m_{i}}{\partial
n_{b}}(V\text{d}N_{j}-N_{j}\text{d}V).$

The fundamental thermodynamic differential relations of energy is

$\text{d}\overline{E}=-P\text{d}V+\sum_{i}\mu_{i}\text{d}N_{i},$ (8)

where $P$, $V$, $\mu_{i}$ and $N_{i}$ correspond to the pressure, volume,
chemical potential and particle number of the system, respectively.

For a uniform system, the energy density $E=\overline{E}/V$, and particle
number density $n_{i}=N_{i}/V$. Comparing Eq. (6) with Eq. (8), we can derive
the following thermodynamic relationships

$\displaystyle P$ $\displaystyle=$
$\displaystyle-\Omega_{0}-V\left(\frac{\partial\Omega_{0}}{\partial
V}\right)_{\\{m_{i}\\},\\{N_{i}\\}}$ (9) $\displaystyle+$ $\displaystyle
n_{b}\sum_{j}\frac{\partial m_{j}}{\partial
n_{b}}\left(\frac{\partial\Omega_{0}}{\partial m_{j}}\right)_{\\{N_{i}\\},V},$
$\displaystyle\mu_{i}$ $\displaystyle=$
$\displaystyle\mu_{i}^{*}+\frac{1}{3}\sum_{j}\frac{\partial m_{j}}{\partial
n_{b}}\left(\frac{\partial\Omega_{0}}{\partial m_{j}}\right)_{\\{N_{i}\\},V}.$
(10)

Based on $E=\Omega_{0}+\sum_{i}\mu^{*}_{i}n_{i}=\Omega+\sum_{i}\mu_{i}n_{i}$,
we have

$\displaystyle\Omega=\Omega_{0}-n_{b}\sum_{j}\frac{\partial m_{j}}{\partial
n_{b}}\left(\frac{\partial\Omega_{0}}{\partial m_{j}}\right)_{\\{N_{i}\\},V},$
(11)

where the second term is under a fixed volume, i.e.,
$\partial\Omega_{0}/\partial V=\partial\Omega/\partial V$. By Eqs. (9)-(11),
we obtained

$\displaystyle E$ $\displaystyle=$
$\displaystyle-P+\sum_{i}\mu_{i}n_{i}-V\left(\frac{\partial\Omega_{0}}{\partial
V}\right)_{\\{m_{i}\\},\\{N_{i}\\}}$ (12) $\displaystyle=$
$\displaystyle-P+\sum_{i}\mu_{i}n_{i}-V\left(\frac{\partial\Omega}{\partial
V}\right)_{\\{m_{i}\\},\\{N_{i}\\}},$ $\displaystyle P$ $\displaystyle=$
$\displaystyle-\Omega-V\left(\frac{\partial\Omega_{0}}{\partial
V}\right)_{\\{m_{i}\\},\\{N_{i}\\}}$ (13) $\displaystyle=$
$\displaystyle-\Omega-V\left(\frac{\partial\Omega}{\partial
V}\right)_{\\{m_{i}\\},\\{N_{i}\\}}.$

Because of the presence of the surface and curvature terms, the thermodynamic
potential density needs to be derived for the volume. When $R$ is large
enough, the surface and curvature terms tend to vanish, i.e.,
$\partial\Omega/\partial V=0$. The system transitions from strangelets to
strange quark matter in such large $R$, thus $E=-P+\sum_{i}\mu_{i}n_{i}$ and
$P=-\Omega$. The Euler equation at zero temperature is validated.

By Eq. (8), we have

$\displaystyle P=-\left.\frac{d(VE)}{\mbox{d}V}\right|_{\\{N_{i}\\}}.$ (14)

i.e.,

$\displaystyle P$ $\displaystyle=$
$\displaystyle-\frac{d(VE)}{\mbox{d}n_{\mathrm{b}}}\left.\frac{\mbox{d}n_{\mathrm{b}}}{dV}\right|_{\\{N_{i}\\}}$
(15) $\displaystyle=$
$\displaystyle-\frac{N\mbox{d}(E/n_{\mathrm{b}})}{\mbox{d}n_{\mathrm{b}}}\left(-\frac{N}{V^{2}}\right)$
$\displaystyle=$
$\displaystyle\frac{N^{2}}{V^{2}}\frac{\mbox{d}(E/n_{\mathrm{b}})}{\mbox{d}n_{\mathrm{b}}}$
$\displaystyle=$ $\displaystyle
n_{\mathrm{b}}^{2}\frac{\mbox{d}(E/n_{\mathrm{b}})}{\mbox{d}n_{\mathrm{b}}}.$

It can be easily seen from the relationship that the pressure must be zero at
the minimum energy per baryon, which provides a simple and intuitive method
for testing thermodynamic self consistency.

Based on
$E(V,\\{n_{i}\\},\\{m_{i}\\})=\Omega_{0}+\sum_{i}\mu^{*}_{i}n_{i}=\Omega+\sum_{i}\mu_{i}n_{i}$,
we have $\partial
E/\partial\mu_{i}^{*}=\partial\Omega_{0}/\partial\mu_{i}^{*}+n_{i}=0$. The
particle number densities is

$\displaystyle n_{i}=-\frac{\partial\Omega_{0}}{\partial\mu_{i}^{*}}.$ (16)

This is the same as the formula derived from the equivalent particle model
Xia2014_PRD89-105027 .

###  II.2 Charged fermion in a uniform magnetic field

In the non-relativistic case, the energy levels of charged particles under a
uniform magnetic field were given by Landau in 1930. Transitioning from
classical theory to quantum mechanics, taking into account the spin of charged
particles, the Hamiltonian operator in electromagnetic fields is

$\hat{H}=\frac{1}{2m}\Big{(}\hat{{\bf
p}}-q{\bm{A}}\Big{)}^{2}-\hat{{\bm{\mu}}}\cdot{\bm{B}}+q\phi,$ (17)

where ${\bm{B}}$ is the magnetic field strength (in the natural system of
units, taking into account that the vacuum permeability is taken to be 1, the
magnetic field strength is equal to the magnetic induction strength
${\bm{H}}={\bm{B}}$), ${\bm{A}}$ is the magnetic vector potential, and $\phi$
is the electric potential. The intrinsic magnetic moment
$\hat{{\bm{\mu}}}=\mu\hat{{\bm{s}}}/s$, where $\mu$ is the magnetic moment
value, $\hat{{\bm{s}}}$ is the particle spin operator, and $s$ is the spin
quantum number. Let the uniform magnetic field ${\bm{B}}=(0,0,B)$, then the
vectorial potential can be taken as

$A_{x}=-By,\quad A_{y}=A_{z}=0.$ (18)

The electric potential is taken to be 0 and the Hamiltonian operator becomes

$\hat{H}=\frac{1}{2m}\Big{(}\hat{p_{x}}+qBy\Big{)}^{2}+\frac{\hat{p}_{y}^{2}+\hat{p}_{z}^{2}}{2m}-\frac{\mu}{s}\hat{s}_{z}B.$
(19)

Noting that $\hat{s}_{z}$ is commutative with the Hamiltonian, this operator
can be replaced by the eigenvalue $\sigma$, then the spin part of the wave
function does not matter, and the Schr$\ddot{\mathrm{o}}$dinger equation is
taken to be

$\frac{1}{2m}\Big{[}\Big{(}\hat{p_{x}}+qBy\Big{)}^{2}+\hat{p}_{y}^{2}+\hat{p}_{z}^{2}\Big{]}\psi-\frac{\mu}{s}\sigma
B\psi=E\psi.$ (20)

Further note that the Hamiltonian is commutative with $\hat{p}_{x}$ and
$\hat{p}_{z}$ and $\psi$ can be taken as
$\psi=\mathrm{exp}(i(p_{x}x+p_{z}z))\chi(y)$. The $\chi$ satisfies

$\chi^{\prime\prime}+2m\Big{[}\Big{(}E+\frac{\mu\sigma
B}{s}-\frac{p_{z}^{2}}{2m}\Big{)}-\frac{1}{2}m\omega^{2}(y-y_{0})^{2}\Big{]}\chi=0,$
(21)

where $y_{0}=-p_{x}/qB$, $\omega=|q|B/m$. Let $E^{\prime}=E+\mu\sigma
B/s-p_{z}^{2}/2m$, then Eq. (21) is equivalent to the equation of a linear
harmonic oscillator with energy $E^{\prime}$ and frequency $\omega$. The
energy eigenvalues of the linear harmonic oscillator are
$E^{\prime}=(n+1/2)\omega$, $n=0,1,2,\cdots$.

As a result, the energy level of a charged particle in a uniform magnetic
field is

$E_{n}=\Big{(}n+\frac{1}{2}\Big{)}\omega+\frac{p_{z}^{2}}{2m}-\frac{\mu\sigma
B}{s}.$ (22)

The first term in the above equation corresponds to the motion in the
transverse plane, and these energy levels $n$ are known as Landau energy
levels. For charged fermions, $\mu/s=\mathrm{sgn}(q)|q|/m$, where
$\mathrm{sgn}(q)$ is the sign function. Thereby, Eq. (22) can be rewritten as

$E_{p,q}=\frac{p_{z}^{2}}{2m}+\frac{e|q|B}{2m}\Big{[}2n+1-2\sigma\mathrm{sgn}(q)\Big{]}=\frac{p^{2}}{2m}.$
(23)

In the case of relativity, the energy of particles

$E_{p}=\sqrt{p^{2}+m^{2}}.$ (24)

Taking $\eta=2\sigma\mathrm{sgn}(q)$, it can be seen from (23) that the
transverse momentum is
$p_{\perp}=\sqrt{p_{x}^{2}+p_{y}^{2}}=\sqrt{|q|B(2n+1-\eta)}$. Thus, the
energy of a charged particle in a uniform magnetic field in the relativistic
case should be

$E_{p_{z},q}=\sqrt{p_{z}^{2}+e|q|B(2n+1-\eta)+m^{2}}.$ (25)

For multi-body systems, to determine the state in which the system is in, it
is usually necessary to integrate in momentum space. Notice that the
transverse momentum is discretized, and the integration in the transverse
direction should become a summation. Such that $2\nu=2n+1-\eta$. The rules for
substituting integrals for summation given by Chakrabarty1996_PRD54-1306

$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\mathrm{d}p_{x}\mathrm{d}p_{y}\longrightarrow
2\pi e|q_{i}|B\sum_{\nu=0}^{\infty}(2-\delta_{\nu,0}),$ (26)

In addition to having an impact on particle energy levels and momentum, a
strong magnetic field can also cause the pressure distribution of the system
to be anisotropic. The pressure also split in $P_{\parallel}$ and $P_{\perp}$,
which denote parallel and transverse to the magnetic field direction
Ferrer2010_PRC82-065802 . The expressions of $P_{\parallel}$ and $P_{\perp}$
for a magnetized fermion system can be written as

$\displaystyle P_{\parallel}$ $\displaystyle=$
$\displaystyle\sum_{i}\mu_{i}n_{i}-E-V\left(\frac{\partial\Omega}{\partial
V}\right),$ (27) $\displaystyle P_{\perp}$ $\displaystyle=$
$\displaystyle\sum_{i}\mu_{i}n_{i}-E-V\left(\frac{\partial\Omega}{\partial
V}\right)+B^{2}-\mathcal{M}B.$ (28)

where $\mathcal{M}$ is the magnetization of the system, which is given by

$\mathcal{M}=-\frac{\partial\Omega}{\partial B}.$ (29)

Details of the derivation of the parallel and transverse pressures from the
energy-momentum tensor are provided in Appendix A. It can been see that the
parallel pressure $P_{\parallel}$ satisfies the Hugenholtz-Van Hove (HVH)
theorem HUGENHOLTZ1958363 , while the transverse pressure $P_{\perp}$ has
extra contributions from the magnetic field, which will result in the zero-
pressure point density being consistent with the density at the minimum of the
energy per baryon for $P_{\parallel}$ but not for $P_{\perp}$
Chu_PhysRevD.90.063013 .

Considering a strong magnetic field, the energy density at zero temperature is
given by Ferrer2010_PRC82-065802

$\displaystyle E$ $\displaystyle=$
$\displaystyle\Omega_{0}+\sum_{i}\mu_{i}^{*}n_{i}+\frac{B^{2}}{2}$ (30)
$\displaystyle=$ $\displaystyle\Omega+\sum_{i}\mu_{i}n_{i}+\frac{B^{2}}{2}.$

Due to $\partial\Omega_{0}/\partial V=\partial\Omega/\partial V$, the parallel
and transverse pressures in a strong magnetic field are given by

$\displaystyle P_{\parallel}$ $\displaystyle=$
$\displaystyle-\Omega-V\frac{\partial\Omega}{\partial V}-\frac{B^{2}}{2}$ (31)
$\displaystyle=$ $\displaystyle-\Omega-V\frac{\partial\Omega_{0}}{\partial
V}-\frac{B^{2}}{2},$ $\displaystyle P_{\perp}$ $\displaystyle=$
$\displaystyle-\Omega-V\frac{\partial\Omega}{\partial
V}+\frac{B^{2}}{2}-\mathcal{M}B$ (32) $\displaystyle=$
$\displaystyle-\Omega-V\frac{\partial\Omega_{0}}{\partial
V}+\frac{B^{2}}{2}-\mathcal{M}B.$

We can use a quantity $\delta$ to quantify the anisotropy of SQM, which is
defined by

$\delta=|\frac{P_{\parallel}-P_{\perp}}{P(B=0)}|.$ (33)

We assume the spherical shape for strangelets and $P_{\parallel}\simeq
P_{\perp}$ when $\delta\lesssim 0.1$. For SQM, it up to magnetic field values
approximated of $2\times 10^{17}\ \mathrm{G}$ Felipe2012_39-045006  when
$\mu_{d}^{*}=\mu_{s}^{*}=\mu_{u}^{*}+\mu_{e}=400\mathrm{MeV}$. In our
calculation, we adopt that the mechanical stability condition is
$P_{\parallel}=0$ for $B\leq 2\times 10^{17}\ \mathrm{G}$ since the
strangelets is a self bound system, which is equivalent to minimize the total
free energy at fixed $N_{i}$ as usually adopted in the literature
Madsen_PhysRevD.50.3328 ; Lugones2011_PRD84-085003 .

###  II.3 Anomalous magnetic moment

Strictly speaking, the energy formula under the relativistic case should be
given by the Dirac equation, and, in addition to the intrinsic magnetic
moments, the anomalous magnetic moments of charged particles should be taken
into account. In 1950, Johnson and Lippman first considered the anomalous
moments in the Dirac equation Johnson1950_PR77-702 . However, the formula they
gave was not covariant. Bjorken and Drell provided a formula for covariation

$\Big{[}\gamma^{\mu}\Big{(}\partial_{\mu}+i|q|A_{\mu}\Big{)}-\frac{i\mu}{2}F_{\mu\nu}\gamma^{\mu}\gamma^{\nu}+m\Big{]}\psi=0.$
(34)

Then, the energy of charged particles containing anomalous magnetic moments
can be obtained

$E=\sqrt{p_{z}^{2}+\Big{(}\sqrt{e|q|B(2n+1-\eta)+m^{2}}-\eta|Q|B\Big{)}^{2}}.$
(35)

It should be noted that $(n=\nu,\eta=1)$ and $(n=\nu-1,\eta=-1)$ no longer
equivalent due to the existence of anomalous magnetic moment $Q$, then the
integral replacement rule Eq. (26) becomes

$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\mathrm{d}p_{x}\mathrm{d}p_{y}\longrightarrow
2\pi e|q_{i}|B\sum_{\eta=\pm 1}\sum_{n=0}^{\infty}.$ (36)

Considering that the components of the quark momentum must be real and that
the Fermi momentum $\nu_{q}=\sqrt{\mu_{q}^{*2}-m_{q}^{2}}$ is an upper limit
to the quark momentum at zero temperature, there must be

$\mu_{q}^{*2}-m_{q}^{2}-p_{\perp}^{2}\geq 0.$ (37)

By Eq. (37), we have

$n\leq\frac{\mu_{q}^{*2}-m_{q}^{2}}{2e|q_{q}|B}+\frac{\eta-1}{2}.$ (38)

Thus, the upper limit for the sum of Landau energy levels in Eq. (36) is

$n_{\mathrm{max}}=\mathrm{int}\Big{[}\frac{\mu_{q}^{*2}-m_{q}^{2}}{2e|q_{q}|B}\Big{]}+\frac{\eta-1}{2},$
(39)

where $\mathrm{int}[x]$ is an integer function that means taking the integer
part of argument $x$.

The anomalous magnetic moments of electrons and quarks are given

$\begin{split}Q_{e}=0.0016\mu_{B},\quad&Q_{u}=1.85\mu_{N},\\\
Q_{d}=-0.97\mu_{N},\quad&Q_{s}=-0.58\mu_{N},\end{split}$ (40)

where $\mu_{B}\simeq 5.79\times 10^{-15}\ \mathrm{MeV/G}$, $\mu_{N}\simeq
3.15\times 10^{-18}\ \mathrm{MeV/G}$. We can see that these anomalous magnetic
moments are very small, and unless in extremely strong magnetic field
environments, the contribution of anomalous magnetic moments can be completely
ignored. Taking $Q=0$, the Eq. (35) returns to Eq. (25). The contribution of
anomalous magnetic moments is not significant, but they play a crucial role in
the study of the properties of strangelets under strong magnetic fields.

If the anomalous magnetic moment is not considered, the integration of the
density of states under a magnetic field will encounter difficulties in
infrared divergence due to the presence of surface and curvature terms. When
$n=0$ and $\eta=1$, there are two infrared divergences
$\mathrm{ln}(p_{\perp,i})$ and $\mathrm{ln}(p_{\perp,i})-1/p_{\perp,i}$ for
density of states under magnetic field, caused by the second and third term of
Eq. (2), so called surface term and curvature term, respectively.

To address this issue, we introduce an infrared cutoff for $p_{z}$. The
anomalous magnetic moment provides a natural cutoff $(2m-|Q|B)|Q|B$
Felipe2008_PRC77-015807 , meaning that when $n=0$ and $\eta=1$, the lower
limit of the integral for $p_{z}$ becomes $(2m-|Q|B)|Q|B$. The natural cutoff
arises due to the mass-shell condition

$p^{2}=E^{2}-m^{2}\geq 0.$ (41)

When $n=0$ and $\eta=1$, Eq. (35) gives us

$E=\sqrt{p_{z}^{2}+(m-|Q|B)^{2}}.$ (42)

Substituting Eq. (42) into Eq. (41), we obtain

$p_{z}^{2}+m^{2}-2m|Q|B+|Q|^{2}B^{2}-m^{2}\geq 0,$ (43)

i.e.,

$p_{z}\geq\sqrt{(2m-|Q|B)|Q|B}.$ (44)

In essence, under the condition $n=0$ and $\eta=1$, the minimum allowed value
of $p_{z}$ is determined by the “natural cutoff” derived from the interplay
between the particle’s effective mass $m$ and the external magnetic field $B$.
This “natural cutoff” circumvents infrared divergences arising from surface
and curvature terms.

Considering the anisotropy caused by a strong magnetic field and the anomalous
magnetic moment of quarks, the Eq. (1) becomes

$\Omega_{0}=\sum_{i}\sum_{\eta=\pm 1}\sum_{n=0}^{n_{max}}2\pi
e|q_{i}|B\int_{-p_{i}^{(n,\eta)}}^{p_{i}^{(n,\eta)}}(E_{p,i}-\mu_{i}^{*})\rho_{i}\mathrm{d}p_{z},$
(45)

where $p_{i}^{(n,\eta)}=\sqrt{p_{F,i}^{2}-p_{\perp}^{2}}$,
$p_{F,i}=\sqrt{\mu_{i}^{*2}-m_{i}^{2}}$ is the Fermi momenta of quark with
flavor $i$. Due to the very small value of the anomalous magnetic moment, the
contribution of the anomalous magnetic moment to the energy levels is
negligible. Taking $Q=0$ in Eq. (35), the energy spectrum $E_{p,i}$ of the
quark with flavor $i$ is

$E_{p,i}=\sqrt{p_{z}^{2}+p_{\perp}^{2}+m_{i}^{2}},$ (46)
$p_{\perp}=m_{i}\sqrt{\frac{B}{B_{i}^{c}}(2n+1-\eta)},\qquad
B_{i}^{c}=\frac{m_{i}^{2}}{e|q_{i}|},$ (47)

i.e.,

$p_{\perp}=\sqrt{e|q_{i}|B(2n+1-\eta)}.$ (48)

Here, $m_{i}$ are the quark masses, $n$ indexes the Landau level,
$\eta=2\mathrm{sgn}(q_{i})s=\pm 1$ correspond to the orientations of the
particle magnetic moment parallel or antiparallel to the magnetic field. The
sign function $\mathrm{sgn}(x)$ equals to $1$ with a positive argument and to
$-1$ with a negative argument, $s=\pm\frac{1}{2}$ are spin projections onto
the magnetic field direction. $B_{i}^{c}$ are the critical magnetic field
Felipe2008_PRC77-015807 , $q_{i}$ denote the quark electric charges(e.g.,
$q_{u}=2/3$, $q_{d}=q_{s}=-1/3$), $e$ represents the elementary charge. Note
that the degeneracy breaking down caused by magnetic field, the statistical
weight $g_{i}=3$ for color of quarks.

Consequently, the number densities of quarks are given by

$\displaystyle n_{i}$ $\displaystyle=$
$\displaystyle-\frac{\partial\Omega_{0}}{\partial\mu_{i}^{*}}$ (49)
$\displaystyle=$ $\displaystyle\sum_{\eta=\pm 1}\sum_{n=0}^{n_{max}}2\pi
e|q_{i}|B\int_{-p_{i}^{(n,\eta)}}^{p_{i}^{(n,\eta)}}\rho_{i}\mathrm{d}p_{z}.$

###  II.4 Coulomb interaction

The size of strangelets with $A\ll 10^{7}$ is smaller than the electron
Compton wavelength, thus electrons can not coexist with quarks in strangelets,
in present calculations, the electron is ignored. Therefore, generally
strangelets are not electric neutrality, this leads to a small Coulomb energy,
in addition, the chemical potential of electron is treat as zero. Thus,
combining with Eq. (30), the energy density and energy per baryon of
strangelets is given by

$\displaystyle E$ $\displaystyle=$
$\displaystyle\Omega+\sum_{i}\mu_{i}n_{i}+E_{C}+\frac{B^{2}}{2}$ (50)
$\displaystyle=$
$\displaystyle\Omega_{0}+\sum_{i}\mu_{i}^{*}n_{i}+E_{C}+\frac{B^{2}}{2},$
$\displaystyle\frac{E}{n_{b}}$ $\displaystyle=$
$\displaystyle\frac{VE}{A}=\frac{4\pi R^{3}E}{3A}.$ (51)

where $E_{C}$ is the Coulomb energy density, which is given by Xia2014

$E_{C}=\frac{2}{15}\pi R^{2}\alpha(Q_{V}^{2}+5Q^{2}),$ (52)

where $\alpha\approx 1/137$ is the fine structure constant and $Q_{V}$ is the
volume term of the total electric charge density $Q$, i.e.,
$Q=\sum_{i}q_{i}n_{i}$ and $Q_{V}=\sum_{i}q_{i}n_{i,V}$. Here $n_{i,V}$ is
given by

$\displaystyle n_{i,V}$ $\displaystyle=$
$\displaystyle\sum_{\eta=\pm}\sum_{n=0}^{n_{max}}2\pi
e|q_{i}|B\int_{-p_{i}^{(n,\eta)}}^{p_{i}^{(n,\eta)}}\frac{g_{i}}{(2\pi)^{3}}\mathrm{d}p_{z}$
(53) $\displaystyle=$
$\displaystyle\sum_{\eta=\pm}\sum_{n=0}^{n_{max}}\frac{e|q_{i}|Bg_{i}}{2\pi^{2}}p_{i}^{(n,\eta)}.$

For strangelets, taking account into the contribution of Coulomb interaction,
the parallel and transverse pressures is given by

$\displaystyle P_{\parallel}$ $\displaystyle=$
$\displaystyle-\Omega-V\frac{\partial\Omega_{0}}{\partial
V}+P_{C}-\frac{B^{2}}{2}$ (54) $\displaystyle=$
$\displaystyle-\Omega_{0}-\frac{R}{3}\frac{\partial\Omega_{0}}{\partial
R}-\sum_{i}n_{i}\mu_{I}+P_{C}-\frac{B^{2}}{2},$ $\displaystyle
P_{\perp}=-\Omega-V\frac{\partial\Omega_{0}}{\partial
V}+P_{C}+\frac{B^{2}}{2}-\mathcal{M}B.$ (55)

In terms of the basic thermodynamic differential relation related to Coulomb
interaction at zero temperature, i.e.,

$\mathrm{d}(VE_{C})=-P_{C}\mathrm{d}V+\sum_{i}\mu_{C,i}\mathrm{d}(n_{i}V),$
(56)

the pressure and chemical potential contributions of Coulomb interaction are
given by Xia2014

$\displaystyle P_{C}$ $\displaystyle=$ $\displaystyle\frac{2}{9}\pi
R^{2}\alpha\Big{[}\frac{3Q_{V}}{5\pi^{2}}\sum_{i}\frac{q_{i}g_{i}p_{F,i}^{2}}{n^{\prime}_{i}}$
(57) $\displaystyle\times$
$\displaystyle\Big{(}n_{i}+\frac{R}{3}\frac{\partial n_{i}}{\partial
R}-\frac{\partial n_{i}}{\partial
m_{i}}\frac{\mathrm{d}n_{i}}{\mathrm{d}n_{b}}n_{b}\Big{)}+Q^{2}-Q_{V}^{2}\Big{]},$
$\displaystyle\mu_{C,i}$ $\displaystyle=$ $\displaystyle\frac{4}{15}\pi
R^{2}\alpha
q_{i}\Big{(}5Q+\frac{g_{i}p_{F,i}^{2}Q_{V}}{2\pi^{2}n^{\prime}_{i}}\Big{)}$
(58) $\displaystyle-$ $\displaystyle\frac{2\alpha
R^{2}Q_{V}}{45\pi}\sum_{j}\frac{q_{j}g_{j}p_{F,j}^{2}}{n^{\prime}_{j}}\frac{\partial
n_{j}}{\partial m_{j}}\frac{\mathrm{d}m_{j}}{\mathrm{d}n_{b}}.$

where $n^{\prime}_{i}$ is defined by

$n^{\prime}_{i}\equiv\frac{\partial n_{i}}{\partial p_{F,i}}.$ (59)

Thus, the actual chemical potential of quark with flavor $i$ is given by

$\mu_{i}=\mu_{i}^{*}-\mu_{I}+\mu_{C,i}.$ (60)

##  III Density and/or temperature dependent particle masses

Originally, the quark mass in quark mass-density-dependent model is given by
Chakrabarty1989_PLB229-112

$m_{i}=m_{0,i}+\frac{B_{bag}}{3n_{b}},$ (61)

where $i$ goes over $u$, $d$ and $s$, $m_{0,i}$ is the current mass of quark
with flavor $i$, $B_{bag}$ is the bag constant. Based on the in-medium chiral
condensates and linear confinement, a cubic root scaling was derived
Peng1999_PRC61-015201

$m_{i}=m_{0,i}+\frac{D}{n_{b}^{1/3}},$ (62)

where $D$ is the confinement parameter, which are determined by stability
arguments of Witten-Bodmer hypothesis. On this foundation, quark mass scaling
containing the effects of linear confinement and one-gluon-exchange
interaction was obtained Chen2012

$m_{i}=m_{0,i}+\frac{D}{n_{b}^{1/3}}-Cn_{b}^{1/3},$ (63)

where $D$ is still the confinement parameter, and $C$ represents the effect of
one-gluon-exchange interaction. Xia and Peng noted that the $m_{I}$ in Eq.
(63) can be understood as a Laurent series of Fermi momentum. From this point
of view, considering the perturbative effect, a new mass scaling was
formulated as Xia2014_PRD89-105027

$m_{i}=m_{0,i}+\frac{D}{n_{b}^{1/3}}+C^{{}^{\prime}}n_{b}^{1/3},$ (64)

where $C^{{}^{\prime}}$ represents the leading order perturbative interaction.

To facilitate the comparison of the effects of one-gluon-exchange and
perturbative effect on the properties of strangelets, we will unify formulas
Eq. (63) and Eq. (64) in writing

$m_{i}=m_{0,i}+\frac{D}{n_{b}^{1/3}}+Cn_{b}^{1/3},$ (65)

where it corresponds to the perturbation and one-gluon-exchange effect
respectively when $C>0$ and $C<0$.

##  IV Properties of Magnetized Strangelets

At a given baryon number, $A$, with the quark number density $n_{i}$ given in
Eq. (49), the baryon number conservation is given by

$A=\frac{1}{3}\sum_{i}n_{i}V.$ (66)

In the absence of electron, the beta equilibrium condition of strangelets
should be replaced by

$\mu_{u}=\mu_{d}=\mu_{s}.$ (67)

In this section, we solve numerically the equation of mechanical equilibrium,
$P_{||}=0$, with baryon number conservation conditions and the beta
equilibrium, i.e., Eqs. (66)-(67). We use $Z$ to denote the electric charge(in
units of $e$), then we have

$Z=\sum_{i}n_{i}q_{i}V.$ (68)

In present calculations, we use the quark mass scaling given by Eq. (65). Note
that for a negative parameter $C$, the quark mass scaling Eq. (65) is
equivalent to the mass scaling considering one-gluon-exchange effect, i.e.,
Eq. (63).

In Fig. 1, we present the anisotropy of SQM as a function of the field
strength $B$ for fixed parameters $C$ and $D$. The dashed, dotted, dashed
dotted and solid curves correspond, respectively, to the baryon number density
$n_{b}=4n_{0}$, $5n_{0}$, $6n_{0}$ and $\delta=0.1$, where $n_{0}=0.165\
\mathrm{fm^{-3}}$ is the nuclear saturation density. It can be seen from the
Fig. 1 that the anisotropy of SQM increases with the increase of magnetic
field strength $B$, but the anisotropy is not significant for the pressure
when the field strength $B\leq 2\times 10^{17}\ \mathrm{G}$. In addition, the
anisotropy of SQM decreases with increasing baryon number density. This is
consistent with the conclusion of previous studies Ferrer2010_PRC82-065802 ;
Cui_NST2015.26.040503 . For different values of parameter $C$, we notice that
the anisotropy of SQM take the maximum value when $C=0$. When $n_{b}$ is
large, the anisotropy of SQM of $C>0$ is more clearly larger than that of
$C<0$. There are some interesting points, which we explain in combination with
Fig. 2. From Fig. 2, we can see that this is mainly because the pressure
$P_{0}$ ($B=0$) of strange quark matter is minimum at $C=0$, centered at
$C>0$, and maximum at $C<0$, respectively. At the same time, there is not much
difference in the degree of deviation between $P_{\parallel}$ and $P_{\perp}$
from $P_{0}$.

Figure 1: The anisotropy of SQM as a function of magnetic field strength,
where $B_{0}=10^{17}\ \mathrm{G}$.  Figure 2: The parallel and transverse
pressures of SQM as a function of magnetic field strength at $n_{b}=4n_{0}$.
The corresponding pressures $P_{0}$ at $B=0$ are included for comparison.
Figure 3: The energy per baryon as a function of baryon number in both $B=0$
and $B=2\times 10^{17}\ \mathrm{G}$. Figure 4: The each component’s energy per
baryon $E_{i}/n_{b}$ as a function of baryon number in $B=2\times 10^{17}\
\mathrm{G}$.

Figures 3 shows the dependence of energy per baryon $E/n_{b}$ given by Eq.
(51) on the baryon number $A$ in both $B=2\times 10^{17}\ \mathrm{G}$ and
$B=0$ for fixed parameters $C$ and $D$. The solid, dashed and dotted curves
correspond, respectively, to $B=2\times 10^{17}$, $0$ $\mathrm{G}$ and
${}^{56}\mathrm{Fe}$. The magenta, orange, blue, and red curves correspond,
respectively, to the parameter sets ($C$, $\sqrt{D}$ in $\mathrm{MeV}$):
$(0.1,160)$, $(0,160)$, $(-0.1,160)$, and $(-0.3,140)$. It can be seen that
the strangelets under $(-0.3,140)$ are absolutely stable at zero magnetic
field strength, while the strangelets under $(-0.1,160)$ is metastable at
$B=2\times 10^{17}\ \mathrm{G}$ when $A>8000$. The energy per baryon decrease
with increasing baryon number $A$. This is consistent with Jensen and Madsen’s
results at $B=0$ JDM53prd4719 , and the value is slightly smaller. This
indicate that strangelets with large baryon numbers could be absolutely stable
under strong magnetic field strength, which opens up the possibility of
detecting strangelets from the particles emitted by unknown astrophysical
sources that may have strong magnetic fields and high baryon numbers. In
addition, it is found that the first-order perturbation interaction($C>0$)
increase energy per baryon, while the one-gluon-exchange interaction($C<0$)
decreases energy per baryon, and the stronger one-gluon-exchange interaction,
the lower energy per baryon.

Figure 5: The mechanically stable radius as a function of baryon number in
both $B=0$ and $B=2\times 10^{17}\ \mathrm{G}$. Figure 6: The ratios of the
strange quark number to tripling in baryon number $\rho_{s}$ as a function of
baryon number in both $B=0$ and $B=2\times 10^{17}\ \mathrm{G}$.

The dependence of each component’s energy per baryon on the baryon number is
depicted in Fig. 4 in $B=2\times 10^{17}\ \mathrm{G}$ for fixed parameters $C$
and $D$. The black, magenta, red, green, and blue curves correspond,
respectively, to the total energy per baryon $E_{total}/n_{b}$, lowest Landau
level (LLL) energy per baryon $E_{LLL}/n_{b}$, higher Landau level (HLL)
energy per baryon $E_{HLL}/n_{b}$, magnetic energy per baryon $E_{B}/n_{b}$
and Coulomb energy per baryon $E_{C}/n_{b}$, which the lowest Landau level
energy $E_{LLL}$ and higher Landau level energy $E_{HLL}$ correspond to the
$n=0$ and $n>0$ parts of the first two terms of Eq. (50), and the magnetic
energy $E_{B}$ and Coulomb energy $E_{C}$ correspond to the third and fourth
terms of Eq. (50). It can be seen that the higher Landau level energy
$E_{HLL}$ accounts for the majority of the total energy and decreases as the
number of baryons increases, however, the lowest Landau level energy $E_{LLL}$
decreases first and then increases, and tends to be stable. The magnetic
energy $E_{B}$ decreases as the number of baryons increases. In addition, the
Coulomb energy $E_{C}$ increases first and then decreases, and eventually
tends to be zero. This is consistent with the electrically neutral of strange
quark matter.

Figure 7: The charge per baryon $Z/A$ as a function of baryon number. Figure
8: The energy per baryon as a function of perturbative parameter with
different confinement parameters.

The dependence of mechanically stable radius of strangelets on the baryon
number at $B=2\times 10^{17}\ \mathrm{G}$ and $B=0$ is shown in Fig. 5 by
solving numerically the equation of mechanical equilibrium $P_{||}=0$ in Eq.
(54), with baryon number conservation conditions and the beta equilibrium,
i.e., Eqs. (66)-(67). The red and blue curves correspond, respectively, to the
field strength $B=2\times 10^{17}\ \mathrm{G}$ and $B=0$. The solid and dashed
curves correspond, respectively, to the parameter sets ($C$, $\sqrt{D}$ in
$\mathrm{MeV}$): $(0.1,160)$ and $(-0.1,160)$. It can be seen that the ratio
$R/A^{1/3}$ decreases with baryon number $A$ and tends to be a constant as $A$
approaches infinity, corresponding to an equation $R=r_{0}A^{1/3}$ with a
constant $r_{0}$. This is consistent with the conclusion of previous model
studies such as NJL model KO2005prd72 .

We use the ratio of the strange quark number to tripling of baryon number
$\rho_{s}=N_{s}/3A$ to express the strangeness of strangelets, which is shown
in Fig. 6. It is increasing with the baryon number, and tend to constant
values at a large baryon number. Moreover, one-gluon-exchange interaction
compared to perturbative interactions give a larger $\rho_{s}$. The
strangeness of strangelets under a strong magnetic field is greater than that
without a magnetic field.

Figure 9: The charge per baryon $Z/A$ as a function of perturbative parameter
with different confinement parameters. Figure 10: The mechanically stable
radius as a function of perturbative parameter with different confinement
parameters. Figure 11: The mechanically stable radius and energy per baryon as
a function of magnetic field strength, where $B_{0}=10^{17}\mathrm{G}$. Figure
12: The each component’s energy per baryon $E_{i}/n_{b}$ as a function of
magnetic field strength, where $B_{0}=10^{17}\mathrm{G}$.

The ratios of electric charge to the baryon number $Z/A$ as a function of
baryon number $A$ is shown in Fig. 7. For a larger baryon number, generally,
the strangelets and magnetized strangelets tend to be electrically neutral,
while the finite size effect becomes weaker, i.e., they tend to be the SQM and
MSQM. At a small baryon number, the radio $Z/A$ of magnetized strangelets is
larger than the strangelets, however, with the increasing of $A$, they become
close.

In Fig. 8, we note that the energy per baryon of strangelets and magnetized
strangelets is a monotone increasing function of the perturbative parameter
$C$ when $A$ is 100. Furthermore, we notice that the larger confinement
interaction and strong magnetic field increases the energy per baryon of
strangelets for $A=100$.

Next, the relationship between the charge per baryon of strangelets and
magnetized strangelets and perturbative parameter $C$ is studied in Fig. 9. It
can be seen that the perturbative parameter $C$ increases the charge per
baryon of strangelets in $B=0$ and $B=2\times 10^{17}\ \mathrm{G}$, and the
rate of increase of magnetized strangelets is faster than strangelets.

The perturbative parameter $C$ dependence of the mechanically stable radii of
strangelets and magnetized strangelets are presented in Fig. 10. The
mechanically stable radius is increase with the increasing of the perturbative
parameter $C$, both in $B=0$ and $B=2\times 10^{17}\ \mathrm{G}$, and a strong
magnetic field will reduce the mechanically stable radii of the strangelets.
This is consistent with the conclusion of previous model studies such as the
MIT bag model Ding2014_62-859 . In the model of the article, that also means
the perturbative effects lead to larger mechanically stable radii of both
strangelets and magnetized strangelets than the one-gluon-exchange effects. In
addition, it is found that the stronger confinement interaction reduces the
mechanically stable radii of strangelets and magnetized strangelets.

Figure 13: The particle abundances $N_{i}/3A$ of quarks of the strangelets
under fixed parameter set as a function of the magnetic field strength.

The dependence of energy per baryon and the mechanically stable radius of
magnetized strangelets on the magnetic field strength are depicted in Fig. 11.
A parameter set $C=0.1$, $D^{1/2}=160\mathrm{MeV}$ is choosen, where the
energy per baryon increases with field strength, inversely, the mechanically
stable radii decreases with field strength. This is consistent with the result
of the energy per baryon and mechanically stable radii in Fig. 8 and Fig. 10.

The dependence of each component’s energy per baryon on the magnetic field
strength is depicted in Fig. 12 for fixed parameters $C$ and $D$. It is found
that the higher Landau level energy $E_{HLL}$ accounts for the majority of the
total energy and decreases as magnetic field strength increases, while the
lowest Landau level energy $E_{LLL}$ is the opposite. This could be visualized
from energy per baryon of strangelets at $n=0$ in Eq. (50) and the reduction
of energy levels in Eq. (39), respectively. The magnetic energy $E_{B}$
increases as the magnetic field strength increases. In addition, the Coulomb
energy $E_{C}$ decreases with magnetic field strength, which is consistent
with the phenomenon of the mechanical stability radius decreasing with
magnetic field strength in Fig. 11.

Figure 13 shows the dependence of the particle abundances of quarks of the
strangelets on the magnetic field strength. The solid curves correspond to the
parameter set $C=0.1$, $D^{1/2}=160\mathrm{MeV}$. The black, red and blue
curves correspond, respectively, to the quark $u$, $d$ and $s$. As the
magnetic field strength increases, the particle abundance of $u$ and $d$
quarks decreases, while that of $s$ quarks increases because $u$ and $d$
quarks are converted to $s$ quarks under strong magnetic field strength.

##  V Summary

We have investigated the properties of magnetized strangelets by the baryon
density-dependent quark mass model with a quark mass scaling contain
confinement and perturbative effects. The contribution of Coulomb interaction
has been treated in a thermodynamic self-consistent way, where the
contribution of Coulomb interaction to the chemical potential and pressure is
accounted for. Considering the anisotropy caused by a strong magnetic field
and the anomalous magnetic moment of quarks, the thermodynamic quantities of
the magnetized strangelets that satisfy thermodynamic consistency are
obtained.

It is found that the dependences of energy and charge per baryon and
mechanically stable radius of magnetized strangelets on the baryon number and
perturbative parameter are similar to the strangelets in zero magnetic field.
The electric charge per baryon of strangelets and magnetized strangelets
become close at a large baryon number, and both tend to electric neutrality.
However, the strong magnetic field leads to larger energy and charge per
baryon, larger strangeness, and smaller mechanically stable radii of
strangelets for fixed parameters, compared with the zero magnetic field one.
For fixed confinement parameters, the energy and charge per baryon and
mechanically stable radius increase as perturbative parameter $C$ increases,
and the rate of charge per baryon of magnetized strangelets is faster than
strangelets. In addition, the dependence of energy per baryon and mechanically
stable radius on magnetic field strength are obtained, where the strong
magnetic field lead to larger energy and smaller stable radius of strangelets
with same baryon number.

##  Appendix A Thermodynamics of quark matter in a uniform magnetic field

In the presence of electromagnetic fields, the energy-momentum tensor can be
decomposed into matter and field parts, i.e.,

$T^{\mu\nu}=T^{\mu\nu}_{\rm matter}+T^{\mu\nu}_{\rm fields},$ (69)

where the field part is fixed by

$\displaystyle T^{00}_{\rm fields}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(E^{2}+B^{2}),$ (70) $\displaystyle T^{0i}_{\rm
fields}$ $\displaystyle=$ $\displaystyle T^{i0}_{\rm
fields}=(\vec{E}\times\vec{B})_{i},$ (71) $\displaystyle T^{ij}_{\rm fields}$
$\displaystyle=$
$\displaystyle\frac{1}{2}(E^{2}+B^{2})\delta_{ij}-E_{i}E_{j}-B_{i}B_{j}.$ (72)

To obtain the matter part of the energy-momentum tensor $T^{\mu\nu}_{\rm
matter}\equiv{\cal T}^{\mu\nu}$ in the local rest frame of the system, we
first write the Lagrangian density of the equivparticle model for quark
matter, i.e.,

$\mathcal{L}=\sum_{i}\bar{\psi}_{i}\left[i\gamma^{\mu}D_{\mu}-m_{i}(n_{\mathrm{b}})+\frac{1}{2}Q_{i}\sigma^{\mu\nu}F_{\mu\nu}\right]\psi_{i},$
(73)

where $q_{i}$ is the charge of quark $i$, $Q_{i}$ the anomalous magnetic
moment, $D_{\mu}=\partial_{\mu}+iq_{i}A_{\mu}$,
$m_{i}=m_{0i}+m_{\mathrm{I}}(n_{\mathrm{b}})$, and
$\sigma^{\mu\nu}=i[\gamma^{\mu},\gamma^{\nu}]/2$. The baryon number density is
obtained with
$n_{\mathrm{b}}=\sum_{i}\langle\bar{\psi}_{i}\gamma^{0}\psi_{i}\rangle/3$. The
equations of motion for $\psi_{i}$ can be determined by Euler-Lagrange
equation, i.e.,

$\frac{\partial{\cal
L}}{\partial\bar{\psi}_{i}}-\partial_{\mu}\left(\frac{\partial{\cal
L}}{\partial(\partial_{\mu}\bar{\psi}_{i})}\right)=0.$ (74)

This gives the Dirac equation for quarks, i.e.,

$\left[i\gamma^{\mu}D_{\mu}-V_{V}\gamma^{0}-m_{i}(n_{\mathrm{b}})+\frac{1}{2}Q_{i}\sigma^{\mu\nu}F_{\mu\nu}\right]\psi_{i}=0,$
(75)

where the term $V_{V}$ arises from the density dependent quark masses, i.e.,

$V_{V}=\frac{1}{3}\frac{\mbox{d}m_{\mathrm{I}}}{\mbox{d}n_{\mathrm{b}}}\sum_{i}\langle\bar{\psi}_{i}\psi_{i}\rangle.$
(76)

The energy energy-momentum tensor is then obtained with

$\displaystyle{\cal T}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{i}\bar{\psi}_{i}\biggl{[}i\left(\gamma^{\mu}D^{\nu}+\gamma^{\nu}D^{\mu}\right)\biggr{]}\psi_{i}$
(77) $\displaystyle+$
$\displaystyle\frac{1}{2}\sum_{i}\bar{\psi}_{i}\biggl{[}Q_{i}\left(\sigma^{\mu\alpha}{F^{\nu}}_{\alpha}+\sigma^{\nu\alpha}{F^{\mu}}_{\alpha}\right)\biggr{]}\psi_{i}$
$\displaystyle-$ $\displaystyle 3n_{\mathrm{b}}V_{V}g^{\mu\nu}.$

Note that we have adopted the relation ${\cal L}=3n_{\mathrm{b}}V_{V}$ during
derivation using the Minkowski space metric $g^{\mu\nu}={\rm
diag}(1,-1,-1,-1)$, which is obtained multiplying Eq. (75) by
$\bar{\psi}_{i}$.

If the magnetic field is uniform and pointing to the $z$-direction, we can
adopt the vector potential $A^{\mu}=(0,-By,0,0)$, which gives
$F^{\mu\nu}=B(\delta^{\mu x}\delta^{\nu y}-\delta^{\nu x}\delta^{\mu y})$ and
consequently

$\frac{1}{2}Q_{i}\sigma^{\mu\nu}F_{\mu\nu}=iQ_{i}B\gamma^{x}\gamma^{y}=Q_{i}B\left(\begin{array}[]{cc}\sigma_{3}&0\\\
0&\sigma_{3}\end{array}\right)\equiv Q_{i}B{\cal S}_{3}.$ (78)

The Dirac equation for stationary states $\psi=e^{-i\varepsilon
t}\Psi_{i}(\vec{x})$ can then obtained with

$\displaystyle\varepsilon\Psi_{i}$ $\displaystyle=$
$\displaystyle\left[-\vec{\alpha}\cdot\left(i\vec{\nabla}-q_{i}\vec{A}\right)+q_{i}A_{0}\right]\Psi_{i}$
(79) $\displaystyle+$ $\displaystyle\left[q_{i}A_{0}+V_{V}+\beta
m_{i}-Q_{i}B\beta{\cal S}_{3}\right]\Psi_{i}.$

Here
$\vec{\alpha}=\gamma^{0}\vec{\gamma}=\left(\begin{array}[]{cc}0&\vec{\sigma}\\\
\vec{\sigma}&0\\\ \end{array}\right)$ and
$\beta=\gamma^{0}=\left(\begin{array}[]{cc}I&0\\\ 0&-I\\\ \end{array}\right)$.
The diagonal components of ${\cal T}^{\mu\nu}$ in a constant magnetic field
are then given by

$\displaystyle{\cal T}^{00}$ $\displaystyle=$
$\displaystyle\sum_{i}\bar{\psi}_{i}\left(i\gamma^{0}D^{0}\right)\psi_{i}-3n_{\mathrm{b}}V_{V},$
(80) $\displaystyle{\cal T}^{xx}$ $\displaystyle=$
$\displaystyle\sum_{i}\bar{\psi}_{i}\left(i\gamma^{x}D^{x}-Q_{i}B\sigma^{xy}\right)\psi_{i}+3n_{\mathrm{b}}V_{V},$
(81) $\displaystyle{\cal T}^{yy}$ $\displaystyle=$
$\displaystyle\sum_{i}\bar{\psi}_{i}\left(i\gamma^{y}D^{y}-Q_{i}B\sigma^{xy}\right)\psi_{i}+3n_{\mathrm{b}}V_{V},$
(82) $\displaystyle{\cal T}^{zz}$ $\displaystyle=$
$\displaystyle\sum_{i}\bar{\psi}_{i}\left(i\gamma^{z}D^{z}\right)\psi_{i}+3n_{\mathrm{b}}V_{V}.$
(83)

For uniform matter comprised of $u$, $d$, and $s$ quarks, the electric field
vanish. The energy energy-momentum tensor for a uniform magnetic field $B$ in
parallel to the $z$-axis is determined by

$T^{\mu\nu}_{\rm fields}=\frac{B^{2}}{2}{\rm diag}(1,1,1,-1)$ (84)

The Dirac equation (79) can be solved by assuming
$\Psi_{i}(\vec{x})=e^{ip_{x}x}e^{ip_{z}z}u^{(s)}_{n}(y)$ with

$u^{(s)}_{n}(y)=\left(\begin{array}[]{l}c_{1}\phi_{\nu}(y)\\\
c_{2}\phi_{\nu-1}(y)\\\ c_{3}\phi_{\nu}(y)\\\ c_{4}\phi_{\nu-1}(y)\\\
\end{array}\right)\ \ {\rm and}\ \
\nu=n+\frac{1}{2}-\frac{s}{2}\frac{q_{i}}{|q_{i}|},$ (85)

where $n=0,1,2,\cdots$ and the spin $s=\pm 1$. The function $\phi_{m}$ is
determined by

$\phi_{m}(y)=N_{m}e^{-\xi^{2}/2}H_{m}(\xi)\ \ {\rm with}\ \
\xi=\sqrt{|q_{i}|B}\left(y+\frac{p_{x}}{q_{i}B}\right),$ (86)

where $m\geq 0$ is an integer, $H_{m}$ is a Hermite polynomial, and
$N_{m}=(q_{i}B)^{1/4}(\sqrt{\pi}2^{m}m!)^{-1/2}$ is a normalization constant
which ensures $\int_{-\infty}^{\infty}dy\,\phi_{n}^{2}(y)~{}=~{}1$. Inserting
this to the Dirac equation (79) gives

$\left(\begin{array}[]{cccc}m-QB&0&p_{z}&p_{\nu}\\\ 0&m+QB&p_{\nu}&-p_{z}\\\
p_{z}&p_{\nu}&-m+QB&0\\\ p_{\nu}&-p_{z}&0&-m-QB\\\
\end{array}\right)\chi=E\chi,$ (87)

where $\chi=(c_{1}\;c_{2}\;c_{3}\;c_{4})^{\rm T}$,
$p_{\nu}=\sqrt{2|q_{i}|B\nu}$, and $E=\varepsilon-q_{i}A_{0}-V_{V}$. The
eigenstates for particles are

$\chi_{s}=\frac{1}{\sqrt{2\lambda\alpha_{s}\beta_{s}}}\left(\begin{array}[]{c}s\alpha_{s}\beta_{s}\\\
-p_{z}p_{\nu}\\\ s\beta_{s}p_{z}\\\ \alpha_{s}p_{\nu}\end{array}\right),$ (88)

where $\alpha_{s}\equiv E_{s}-Q_{i}B+s\lambda$ and
$\beta_{s}\equiv\lambda+sm_{i}$. The corresponding eigenvalues are

$E_{s}=\sqrt{p_{z}^{2}+(\lambda-sQ_{i}B)^{2}},$ (89)

with $\lambda\equiv\sqrt{m_{i}^{2}+p_{\nu}^{2}}$. The general positive energy
states can be constructed by

$\psi_{i}(x)=\sum_{s=\pm
1}\frac{|q|B}{2\pi}\sum_{n}\int_{-\infty}^{\infty}\frac{dp_{z}}{2\pi}\frac{1}{\sqrt{2E_{s}}}b_{s}({\bf
p})u^{(s)}({\bf p})e^{i\tilde{p}_{\mu}x^{\mu}},$ (90)

where $b_{s}({\bf k})$ obeys following relation

$\\{b_{r}({\bf p}),b_{s}^{\dagger}({\bf
k})\\}=(2\pi)\delta_{rs}\delta_{mn}\delta(p_{z}-k_{z}),$ (91)

${\bf p}=(n,k_{z})$ with $n=0,1,2,\cdots$, $\tilde{p}=(E_{s},p_{x},0,p_{z})$.
For details, see Ref. Strickland2012PRD86-125032 .

The energy density for infinite quark matter is then

$\displaystyle{\cal E}$ $\displaystyle=$ $\displaystyle\langle{\cal
T}^{00}\rangle=\sum_{i}\frac{|q_{i}|B}{2\pi}\sum_{s=\pm
1}\sum_{n}\int_{-\infty}^{\infty}\frac{dp_{z}}{2\pi}$ (92)
$\displaystyle\times E_{s}f_{+}(E_{s},T,\mu_{i}^{*}),$

with the Fermi-Dirac distribution of particles fixed by

$\langle b_{s}^{\dagger}({\bf p})b_{s}({\bf
p})\rangle=f_{+}(E_{s},T,\mu_{i}^{*})=\frac{1}{e^{(E_{s}-\mu_{i}^{*})/T}+1}.$
(93)

Note that we have adopted the relation
$E_{s}=\varepsilon_{s}-q_{i}A_{0}-V_{V}$ in the deriving Eq. (92). The number
density can be determined by

$n_{i}=\langle\bar{\psi}_{i}\gamma^{0}\psi_{i}\rangle=\frac{|q_{i}|B}{2\pi}\sum_{s=\pm
1}\sum_{n}\int_{-\infty}^{\infty}\frac{dp_{z}}{2\pi}f_{+}(E_{s},T,\mu_{i}^{*}),$
(94)

while the baryon number density is $n_{\mathrm{b}}=\sum_{i}n_{i}/3$. According
to Eq. (79), we have

$\displaystyle(\varepsilon-
q_{i}A_{0}-V_{V})\Psi_{i}^{\dagger}\Psi_{i}=\Psi_{i}^{\dagger}\left[-\vec{\alpha}\cdot\left(i\vec{\nabla}-q_{i}\vec{A}\right)\right]\Psi_{i}$
(95) $\displaystyle+$ $\displaystyle\Psi_{i}^{\dagger}\left[\beta
m_{i}-Q_{i}B\beta{\cal S}_{3}\right]\Psi_{i}.$

Upon taking the derivative of the equation above with respect to mass, we
obtain the following result:

$\Psi_{i}^{\dagger}\beta\Psi_{i}=\frac{\partial E}{\partial
m_{i}}\Psi_{i}^{\dagger}\Psi_{i},$ (96)

where $E=\varepsilon-q_{i}A_{0}-V_{V}$. Note that
$\bar{\psi}_{i}\psi_{i}=\psi_{i}^{\dagger}\gamma^{0}\psi_{i}$ and
$\beta=\gamma^{0}$, we obtain

$\displaystyle\langle\bar{\psi}_{i}\psi_{i}\rangle$ $\displaystyle=$
$\displaystyle\frac{|q_{i}|B}{2\pi}\sum_{s=\pm}\sum_{n}\int^{\infty}_{-\infty}\frac{dp_{z}}{2\pi}\frac{\partial
E_{s}}{\partial m_{i}}\langle b_{s}^{\dagger}({\bf p})b_{s}({\bf p})\rangle$
$\displaystyle=$
$\displaystyle\frac{|q_{i}|B}{2\pi}\sum_{s=\pm}\sum_{n}\int^{\infty}_{-\infty}\frac{dp_{z}}{2\pi}\frac{\partial
E_{s}}{\partial m_{i}}f_{+}(E_{s},T,\mu^{*}_{i}).$

Thus, for vanishing temperatures with $T=0$, the expression of $V_{V}$ is
given by

$V_{V}=\frac{1}{3}\frac{\text{d}m_{I}}{\text{d}n_{b}}\sum_{i}\langle\bar{\psi}_{i}\psi_{i}\rangle=\frac{1}{3}\frac{\text{d}m_{I}}{\text{d}n_{b}}\sum_{i}\frac{\partial\mathcal{E}}{\partial
m_{i}}.$ (98)

The longitudinal pressure along magnetic field lines is

$\displaystyle P_{\parallel}$ $\displaystyle=$ $\displaystyle\langle{\cal
T}^{zz}\rangle$ (99) $\displaystyle=$
$\displaystyle\sum_{i}\frac{|q_{i}|B}{2\pi}\sum_{s=\pm
1}\sum_{n}\int_{-\infty}^{\infty}\frac{dp_{z}}{2\pi}\frac{p_{z}^{2}}{E_{s}}f_{+}(E_{s},T,\mu_{i}^{*})$
$\displaystyle+$ $\displaystyle 3n_{\mathrm{b}}V_{V}.$

The transverse pressure is fixed by

$\displaystyle P_{\perp}$ $\displaystyle=$ $\displaystyle\langle{\cal
T}^{yy}\rangle=\langle{\cal T}^{xx}\rangle$ $\displaystyle=$
$\displaystyle\sum_{i}\frac{|q_{i}|B^{2}}{2\pi^{2}}\sum_{s=\pm
1}\sum_{n}\int_{-\infty}^{\infty}dp_{z}\frac{1}{E_{s}}f_{+}(E_{s},T,\mu_{i}^{*})$
$\displaystyle\times\Biggl{[}\frac{|q_{i}|\nu\bar{m}_{i}(\nu)}{\sqrt{m_{i}^{2}+2\nu|q_{i}|B}}-sQ_{i}\bar{m}_{i}(\nu)\Biggr{]}+3n_{\mathrm{b}}V_{V},$

where $\bar{m}_{i}(\nu)\equiv\sqrt{m_{i}^{2}+2\nu|q_{i}|B}-sQ_{i}B$.

For vanishing temperatures with $T=0$, the thermodynamic potential density is
then fixed by $\Omega={\cal E}-\mu_{i}n_{i}=-P_{\parallel}$ with
$\mu_{i}=\mu_{i}^{*}+V_{V}$. A relationship between $P_{\parallel}$ and
$P_{\perp}$ obtained according to the magnetization
$\mathcal{M}\equiv-\partial\Omega/\partial B=\partial P_{\parallel}/\partial
B$, which gives $P_{\perp}=P_{\parallel}-\mathcal{M}B$. These relations can be
derived through direct integration, as detailed in Ref.
Strickland2012PRD86-125032 .

If we further include the field contributions to energy-momentum tensor, the
energy density, longitudinal and transverse pressures at $T=0$ are fixed by

$\displaystyle{\cal E}$ $\displaystyle=$
$\displaystyle\sum_{i}\frac{|q_{i}|B}{2\pi}\sum_{s=\pm
1}\sum_{n}\int_{-\infty}^{\infty}\frac{dp_{z}}{2\pi}E_{s}\Theta(\mu_{i}^{*}-E_{s})$
(101) $\displaystyle+$ $\displaystyle\frac{1}{2}B^{2},$ $\displaystyle
P_{\parallel}=-\Omega-\frac{1}{2}B^{2},$ (102) $\displaystyle
P_{\perp}=-\Omega-\mathcal{M}B+\frac{1}{2}B^{2}=P_{\parallel}+B^{2}-\mathcal{M}B,$
(103)

where

$\displaystyle\Omega$ $\displaystyle=$
$\displaystyle\sum_{i}\frac{|q_{i}|B}{2\pi}\sum_{s=\pm
1}\sum_{n}\int_{-\infty}^{\infty}\frac{dp_{z}}{2\pi}(E_{s}-\mu_{i})$ (104)
$\displaystyle\times\Theta(\mu_{i}^{*}-E_{s}),$
$\displaystyle\mu_{i}=\mu_{i}^{*}+V_{V},$ (105)
$\displaystyle\mathcal{M}=-\frac{\partial\Omega}{\partial B}.$ (106)

Due to the independence of chemical potential $\mu_{i}$ on mass $m_{i}$,
combining with Eq. (98), we derive the relationship between $V_{V}$ and the
thermodynamic potential density as follows

$\displaystyle
V_{V}=\frac{1}{3}\frac{\mathrm{d}m_{I}}{\mathrm{d}n_{b}}\sum_{i}\frac{\partial\Omega}{\partial
m_{i}}.$ (107)

For a finite system, the contributions from surface corrections and Coulomb
interactions need to be considered, which introduce additional terms to the
energy density and pressures.

## References

  * (1) J. Madsen, Lect. Notes Phys. 516, 162 (1999).
  * (2) A. R. Bodmer, Phys. Rev. D 4, 1601 (1971).
  * (3) E. Witten, Phys. Rev. D 30, 272 (1984).
  * (4) M. S. Berger and R. L. Jaffe, Phys. Rev. C 35, 213 (1987).
  * (5) E. Finch, J. Phys. G 32, S215 (2006).
  * (6) S. Biswas and J. N. De and P. S. Joarder and S. Raha and D. Syam, Phys. Rev. C 95, 045201 (2017).
  * (7) E. Annala and T. Gorda and A. Kurkela and J. Nättilä and A. Vuorinen, Nat. Phys. 15, 907 (2020).
  * (8) C. J. Xia and T. Maruyama and N. Yasutake and T. Tatsumi and H. Shen and H. Togashi, Phys. Rev. D 102, 023031 (2020).
  * (9) S. Banerjee and S. K. Ghosh and S. Raha et al., J. Phys. G: Nucl. Part. Phys. 25, L15 (1999).
  * (10) S. Banerjee and S. K. Ghosh and S. Raha and D. Syam, Phys. Rev. Lett. 85, 1384 (2000).
  * (11) J. Madsen and J. M. Larsen, Phys. Rev. Lett. 90, 121102 (2003).
  * (12) J. Madsen, Phys. Rev. D 71, 014026 (2005).
  * (13) S. Shaulov and S. Bezshapov, EPJ Web Conf. 52, 04010 (2013).
  * (14) J. Schaffner-Bielich and C. Greiner and A. Diener and H. Stocker, Phys. Rev. C 55, 3038 (1997).
  * (15) J. Alimena and J. Beacham and M. Borsato and Y. Cheng and X. Cid Vidal and G. Cottin and A. De Roeck and N. Desai and D. Curtin and J. A. Evans, J. Phys. G 47, 090501 (2020).
  * (16) M. Parvu and I. Lazanu, JCAP 11, 040 (2021).
  * (17) R. C. Duncan and C. Thompson, Astrophys. J. Lett. 392, L9 (1992).
  * (18) J. H. Taylor and R. N. Manchester and A. G. Lyne, Astrophys. J. Suppl. Ser. 88, 529 (1993).
  * (19) D. N. Sob’yanin, Phys. Rev. D 107, L081301 (2023).
  * (20) V. M. Kaspi and A. M. Beloborodov, Ann. Rev. Astron. Astrophys. 55, 261 (2017).
  * (21) A. J. Mizher and M. N. Chernodub and E. S. Fraga, Phys. Rev. D 82, 105016 (2010).
  * (22) S. Chakrabarty and A. Goyal, Mod. Phys. Lett. A 09, 3611 (1994).
  * (23) R. G. Felipe and A. P. Martínez and H. P. Rojas and M. Orsaria, Phys. Rev. C 77, 015807 (2008).
  * (24) T. Endo and H. Sotani and T. Tatsumi, J. Phys. Conf. Ser. 1468, 012087 (2020).
  * (25) C. F. Li and L. Yang and X. J. Wen and G. X. Peng, Phys. Rev. D 93, 054005 (2016).
  * (26) T. Maruyama and T. Tatsumi, JPS Conf. Proc. 26, 024020 (2019).
  * (27) J. X. Hou and G. X. Peng and C. J. Xia and J. F. Xu, Chin. Phys. C. 39, 015101 (2015).
  * (28) X. J. Wen, Phys. Rev. D 88, 034031 (2013).
  * (29) P. C. Chu and X. H. Li and H. Y. Ma and B. Wang and Y. M. Dong and X. M. Zhang, Phys. Lett. B 778, 447 (2018).
  * (30) R G. Felipe and E L. Fune and D M. Paret and A P. Martínez, J. Phys. G: Nucl. Part. Phys. 39, 045006 (2012).
  * (31) M. Ding and X. J. Wen and F. H. Liu and B. C. Li, Commun. Theor. Phys. 62, 859 (2014).
  * (32) J. Madsen, Phys. Rev. Lett. 70, 391 (1993).
  * (33) G. Lugones and A. G. Grunfeld, Phys. Rev. C 103, 035813 (2021).
  * (34) G. Lugones and A. G. Grunfeld, Phys. Rev. D 104, L101301 (2021).
  * (35) G. Lugones and A. G. Grunfeld, Phys. Rev. D 107, 043025 (2023).
  * (36) H. M. Chen and C. J. Xia and G. X. Peng, Phys. Rev. D 105, 014011 (2022).
  * (37) H. M. Chen and C. J. Xia and G. X. Peng, Chin. Phys. C 46, 055102 (2022).
  * (38) E. J. Ferrer and V. delaIncera and J. P. Keith and I. Portillo and P. L. Springsteen, Phys. Rev. C 82, 065802 (2010).
  * (39) R. Balian and C. Bloch, Ann. Phys. 60, 401 (1970).
  * (40) J. Madsen, Phys. Rev. D 50, 3328 (1994).
  * (41) G. Neergaard and J. Madsen, Phys. Rev. D 60, 054011 (1999).
  * (42) G. X. Peng and H. C. Chiang and B. S. Zou and P. Z. Ning and S. J. Luo, Phys. Rev. C 62, 025801 (2000).
  * (43) C. J. Xia and G. X. Peng and S. W. Chen and Z. Y. Lu and J. F. Xu, Phys. Rev. D 89, 105027 (2014).
  * (44) S. Chakrabarty, Phys. Rev. D 54, 1306 (1996).
  * (45) N. M. Hugenholtz and L. Van Hove, Physica (Amsterdam) 24, 363 (1958).
  * (46) P. C. Chu and L. W. Chen and X. Wang, Phys. Rev. D 90, 063013 (2014).
  * (47) J. Madsen, Phys. Rev. D 50, 3328 (1994).
  * (48) G. Lugones and A. G. Grunfeld, Phys. Rev. D 84, 085003 (2011).
  * (49) M. H. Johnson and B. A. Lippmann, Phys. Rev. 77, 702 (1950).
  * (50) C. J. Xia and S. W. Chen and G. X. Peng, Sci. China Phys. Mech. Astron. 57, 1304 (2014).
  * (51) S. Chakrabarty and S. Raha and B. Sinha, Phys. Lett. B 229, 112 (1989).
  * (52) G. X. Peng and H. C. Chiang and J. J. Yang and L. Li and B. Liu, Phys. Rev. C 61, 015201 (1999).
  * (53) S. W. Chen and L. Gao and G. X. Peng, Chin. Phys. C. 36, 947 (2012).
  * (54) S. S. Cui and G. X. Peng and Z. Y. Lu and C. Peng and J. F. Xu, Nucl. Sci. Tech. 26, 040503 (2015).
  * (55) D. M. Jensen, and J. Madsen, Phys. Rev. D 53, R4719 (1996).
  * (56) O. Kiriyama, Phys. Rev. D 72, 054009 (2005).
  * (57) M. Strickland and V. Dexheimer and D. P. Menezes, Phys. Rev. D 86, 125032 (2012).