# Shear viscosity coefficient of magnetized QCD medium with anomalous magnetic
moments near chiral phase transition

Yi-Wei Qiu  College of Science, China Three Gorges University, Yichang 443002,
China  Sheng-Qin Feng  Corresponding author<EMAIL_ADDRESS>College of
Science, China Three Gorges University, Yichang 443002, China  Center for
Astronomy and Space Sciences and Institute of Modern Physics, China Three
Gorges University, Yichang 443002, China  Key Laboratory of Quark and Lepton
Physics (MOE) and Institute of Particle Physics,
Central China Normal University, Wuhan 430079, China  Xue-Qiang Zhu  Key
Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle
Physics,
Central China Normal University, Wuhan 430079, China

###### Abstract

Abstract: We study the properties of the shear viscosity coefficient of quark
matter near the chiral phase transition at finite temperature and chemical
potential, and the kinds of high temperature, high density and strong magnetic
field background might be generated by high-energy heavy ion collisions. The
strong magnetic field induces anisotropy, that is, the quantization of Landau
energy levels in phase space. If the magnetic field is strong enough, it will
interfere with significant QCD phenomena, such as the generation of dynamic
quark mass, which may affect the transport properties of quark matter. The
inclusion of the anomalous magnetic moments (AMM) of the quarks at finite
density into the NJL model gives rise to additional spin polarization magnetic
effects. As the inclusion of AMM of the quarks leads to inverse magnetic
catalysis around the transition temperature, we will systematically study the
thermodynamic phase transition characteristics of shear viscosity coefficient
in QCD media near the phase boundary. The shear viscosity coefficient of the
dissipative fluid system can be decomposed into five different components as
the strong magnetic field exists. The influences of the order of chiral phase
transition and the critical endpoint on dissipative phenomena in such a
magnetized medium are quantitatively investigated. It is found that
${\eta}_{1}$, ${\eta}_{2}$, ${\eta}_{3}$, and ${\eta}_{4}$ all increase with
temperature. For first-order phase transitions, ${\eta}_{1}$, ${\eta}_{2}$,
${\eta}_{3}$, and ${\eta}_{4}$ exhibit discontinuous characteristics.

##  I Introduction

The study of the strong interaction under the influence of different external
agents including strong magnetic field Bzdak and Skokov (2012); Skokov  _et
al._ (2009); Deng and Huang (2012); Kharzeev  _et al._ (2008); Zhong  _et al._
(2014); Mo  _et al._ (2013); Feng  _et al._ (2018) and large angular momentum
rotation Liang and Wang (2005); Gao  _et al._ (2008); Guo  _et al._ (2019);
Becattini  _et al._ (2008) will eventually shed light on the complexity of its
phase diagram and transport properties, or could reveal new unknown features.
Especially, due to the complexity of basic theories, the study of low energy
states relies on specific tools such as lattice simulations and effective
models. The Nambu-Jona-Lasinio (NJL) model has been proven to be a useful
conceptual tool for solving different problems Miransky and Shovkovy (2015);
Andersen  _et al._ (2016). As a special chapter of this method, one can
mention the study of quarks interacting with external magnetic fields, where a
variety of issues have been studied such as magnetic catalysis Gusynin  _et
al._ (1995), quark matter transport characteristics Zhu and Feng (2023); Ghosh
_et al._ (2019), magnetic oscillations Ebert  _et al._ (2000), vector Denke
and Pinto (2013) and tensor Ferrer  _et al._ (2014); Qiu and Feng (2023)
additional couplings.

In a magnetic field environment, it is naturally believed that the chiral
critical temperature will increase with the magnetic field, which is known as
the magnetic catalytic (MC) effect. However, lattice QCD results Bali  _et
al._ (2012a); Bornyakov  _et al._ (2014) detected the feature of inverse
magnetic catalytic effect (IMC) near the critical temperature. Recent studies
Qiu and Feng (2023); Xu  _et al._ (2021); Chaudhuri  _et al._ (2019) have
shown that the addition of AMM can also exhibit IMC properties, consistent
with the lattice QCD results. The remaining IMC puzzle and recently discovered
properties of magnetized matter attract our renewed interest to revisit QCD
vacuum and matter under external magnetic field and to find the underlying
mechanism for these properties. It is known that the dynamical chiral symmetry
broken is one of the most significant features of QCD, where quark obtains a
dynamical mass. It has been confirmed that the anomalous magnetic moments
(AMM) of quarks can also be generated dynamically like dynamical quark mass
Ferrer  _et al._ (2014); Chang  _et al._ (2011); Ferrer and de la Incera
(2009) because of spin polarization in the presence of the magnetic field.

The magnetic field also impacts on the radiation correction of fermion self-
energy, which relates to the AMM coupling between the magnetic field and the
fermion spin polarization Chang  _et al._ (2011); Ferrer and de la Incera
(2009); Schwinger (1948); Brekke and Rosner (1988); Bicudo  _et al._ (1999);
Mohr  _et al._ (2008); Ferrer  _et al._ (2015a). This generates a new term
$\frac{1}{2}q_{f}{\kappa}_{f}{\sigma}_{\mu\nu}{F}_{\mu\nu}$ in the Dirac
Hamiltonian, where
${\sigma}_{\mu\nu}=\frac{i}{2}\left[{\gamma}_{\mu},{\gamma}_{\nu}\right]$,
${F}_{\mu\nu}$ and ${q}_{f}$ are the spin tensor, field tensor and quark
charge, respectively. The AMM term in the Hamiltonian alters the energy
spectrum of fermions by eliminating spin degeneracy and influences the
properties of the magnetized QCD system Ferrer  _et al._ (2014); Xu  _et al._
(2021); Ferrer  _et al._ (2015b); Mei and Mao (2020); Fayazbakhsh and Sadooghi
(2014); Aguirre (2020); Blaizot  _et al._ (2013); Ferrer and de la Incera
(2010), where the coefficient ${\kappa}_{f}$ is identified as the fermion AMM.
Quark AMM reduces the effective quark mass with
${m}_{f}=m-{\kappa}_{f}\left|{q}_{f}B\right|$ in the lowest Landau energy
level approximation, and it takes an IMC in the chiral restoration and
deconfinement phase transitions.

Modifications related to the QCD phase diagram may also have some impact on
the transport characteristics of the QCD medium with the magnetic field
generated by relativistic heavy ion collisions (HIC). Different transport
coefficients of quark matter have recently been investigated, such as shear
viscosity Huang  _et al._ (2010, 2011); Tuchin (2012); Nam and Kao (2013);
Alford  _et al._ (2014); Tawfik  _et al._ (2016); Hattori  _et al._ (2017); Li
and Yee (2018), bulk viscosity Agasian (2012, 2013) in the presence of the
magnetic field. The simulation of magnetic fluid mechanics Roy  _et al._
(2015); Pu  _et al._ (2016) and the transport simulation of the external
magnetic field Das  _et al._ (2017) may need these transport coefficients
related to temperature and chemical potential for future study.

The shear coefficient $\eta$ and bulk viscosity coefficient $\zeta$ are
important parameters for quantifying the dissipation process in QCD fluid
dynamics evolution under the external magnetic field background. It is
particularly noteworthy that: $\eta$ and $\zeta$ are very sensitive to the QCD
phase transition characteristics in a magnetized media Sasaki and Redlich
(2010); Mykhaylova and Sasaki (2021); Soloveva  _et al._ (2022). Of particular
interest in QCD is the properties of the transport coefficient near the
critical end point, where deconfinement and chiral phase transition begin.
This is because any changes for shear viscosity and bulk viscosity near
${T}_{c}$ will alter the characteristics of dynamics evolution of QCD medium
and affect the observations characterizing its expansion kinetics. Obviously,
during the evolution of QCD media, its transport properties are changing as it
approaches the phase boundary. Therefore, it is extremely important to
quantitatively study the relationship between the shear viscosity coefficient
near the expected phase transition in the magnetized QCD media and the
variation of thermal parameters.

It was found that Refs. Nam and Kao (2013); Alford  _et al._ (2014); Tawfik
_et al._ (2016) had not yet explored the components generated by the magnetic
fields in the early calculations of the shear viscosity of magnetized
materials Huang  _et al._ (2010, 2011); Tuchin (2012); Nam and Kao (2013);
Alford  _et al._ (2014); Tawfik  _et al._ (2016); Hattori  _et al._ (2017);
Mohanty  _et al._ (2019). The decompositions of the components of shear
viscosity caused by anisotropy generated by external magnetic fields or other
sources in Refs. Huang  _et al._ (2010, 2011); Tuchin (2012); Mohanty  _et
al._ (2019) have been well studied in the direction of gauge/gravity duality
(see references Critelli  _et al._ (2014); Jain  _et al._ (2015) and their
references). The general structure of the shear viscosity coefficients of the
five components near the phase boundary is crucial for studying phase
transition characteristics.

In this work, we will use the dynamical NJL model Hatsuda and Kunihiro (1994);
Buballa (2005) to study the transport properties of magnetized quark matter
near the chiral phase transition. Most of our research on the low energy state
of QCD, especially the QCD phase diagram, is based on the quasiparticle model.
The effective quark mass is determined by the condensed state of quarks in the
medium. As mentioned above, a strong magnetic field can generate AMM that
alter quark condensation. This effect is taken into account for calculating
the shear viscosity coefficient in this paper. The results of NJL models with
AMM are qualitatively consistent with the lattice simulation results Aguirre
(2020); Blaizot  _et al._ (2013); Bali  _et al._ (2012b), providing the IMC
characteristics of the chiral phase transition, which is also qualitatively
consistent with the QCD coupling characteristics. Then, we use the Boltzmann
equation’s dynamic model and relaxation time approximation to study the
characteristics of the viscosity coefficient components in the first-order
phase transition and critical endpoint (CEP) phase transition regions under
the strong magnetic field background.

This paper is organized as follows. In Sec. II we introduce the description of
the NJL model with AMM for the magnetized QCD matter. After considering the
influence of IMC on thermodynamic physical quantities, we study the chiral
phase transition and some thermodynamic properties. Then, the Boltzmann
equation’s dynamic model and relaxation time approximation are utilized to
study the characteristics of the viscosity coefficient components in Sec. III.
The characteristics of the shear viscosity coefficient component of a
magnetized medium system with first-order phase transitions and critical
endpoint transitions near CEP are investigated in Sec.IV. We summarized and
concluded in Sec. V.

##  II THE TWO FLAVORS NJL MODEL UNDER A MAGNETIC FIELD

Let’s study the Lagrangian density within two-flavor NJL model with AMM in the
presence of an external magnetic field, which is given by Fayazbakhsh and
Sadooghi (2014):

$\begin{split}\mathcal{L}=\bar{\psi}\left(i{{\gamma}^{\mu}}{{D}_{\mu}}+{{\gamma}^{0}}\mu-m+\frac{1}{2}q_{f}\kappa_{f}{\sigma^{\mu\nu}}{F_{\mu\nu}}\right)\psi+{G}\left[{{\left(\bar{\psi}{{\lambda}_{a}}\psi\right)}^{2}}+{{\left(\bar{\psi}i{{\gamma}^{5}}{{\lambda}_{a}}\psi\right)}^{2}}\right],\\\
\end{split}$ (1)

where the quark field $\psi$ carries two flavors ($f=u,d$) and three colors
($c=r,g,b$ ). Current quark mass $m$ is considered as ${{m}_{u}}={{m}_{d}}$
for maintaining isospin symmetry, and covariant derivative
${{D}_{u}}={{\partial}_{\mu}}+\operatorname{i}QA_{\mu}^{\operatorname{ext}}$
and charge matrix in flavor space
$q_{f}={\rm{diag}}\left({{q_{u}},{q_{d}}}\right)={\rm{diag}}\left({\frac{2}{3}e,-\frac{2}{3}e}\right)$
are introduced by considering the magnetic field. If one selects the gauge
field $A_{\mu}^{ext}=\left({0,0,B{x_{1}},0}\right)$, the corresponding
constant magnetic field should point at the z-direction. By using mean-field
approximation, one can obtain the one-loop effective potential as

$\begin{split}\Omega=&\frac{{{\sigma^{2}}}}{{4G}}-\frac{{{N_{c}}\mathop{\sum}\nolimits_{l=0}^{\infty}{\alpha_{l}}\mathop{\sum}\nolimits_{f=u}^{d}|{q_{f}}B|}}{{2\pi}}\mathop{\int}\nolimits_{-\infty}^{\infty}\frac{{d{p_{z}}}}{{2\pi}}\bigg{\\{}{\varepsilon_{l,\eta}}+T\ln\left[{1+\exp\left({\frac{{-{\varepsilon_{l,\eta}}-\mu}}{T}}\right)}\right]\\\
&+T\ln\left[{1+\exp\left({\frac{{-{\varepsilon_{l,\eta}}+\mu}}{T}}\right)}\right]\bigg{\\}},\end{split}$
(2)

where
${E_{{}_{f,l,t}}}=\sqrt{p_{z}^{2}+{{\left({{{\left({{M_{f}}^{2}+2|{q_{f}}B|l}\right)}^{1/2}}-\eta{\kappa_{f}}{q_{f}}eB}\right)}^{2}}}$
is the energy spectrum with different Landau energy levels, which $l$ = 0, 1,
2 … represents the quantum number of Landau level and $\eta=\pm 1$ corresponds
to the two kinds of spin direction of quark-antiquark pair. Contribution of
non-degenerate particles due to spin difference at non-lowest Landau energy
levels (non-LLL) should be taken into account with the definition of this
operator ${\alpha_{l}}={\delta_{0,l}}+\Delta\left(l\right)\sum_{{\eta=\pm
1}}$, where the step function $\Delta\left(l\right)$ is taken as
$\Delta\left({l=0}\right)=0$ and $\Delta\left({l>0}\right)=1$, respectively.
One can obtain two coupling gap equations for each order parameter as

$\frac{{\partial{\Omega_{{\rm{AMM}}}}}}{{\partial{M_{f}}}}=0,$ (3)

where $f=u,{\rm{}}d$ for the two different flavors. Then, one can obtain two
dynamical quark mass

${M_{f}}={m_{0}}-2G{\left\langle{\bar{\psi}\psi}\right\rangle_{f}}.$ (4)

By quantitatively investigating the contribution of different Landau energy
levels to the chiral condensate, We then make the calculations as

${\left\langle{\bar{\psi}\psi}\right\rangle_{f}}=\sum\limits_{{\rm{LL=0}}}^{\infty}{{{\left\langle{\bar{\psi}\psi}\right\rangle}_{{\rm{LL}}}}}=\sum\limits_{l=0}^{\infty}{\sum\limits_{f=u,d}{{\varphi_{f,l}}}},$
(5)

where

${\varphi_{f,l}}=\frac{{{N_{c}}{G_{s}}}}{{2\pi}}{\alpha_{l}}|{q_{f}}B|\int_{-\infty}^{+\infty}{{f_{\Lambda}}(p)\frac{{d{p_{z}}}}{{2\pi}}\frac{{{M_{f}}}}{{{\varepsilon_{f,l,t}}}}\left({1-\frac{{s{\kappa_{f}}{q_{f}}B}}{{{{\hat{M}}_{f,l,t}}}}}\right)\left\\{{1-\frac{1}{{{e^{\frac{{{\varepsilon_{f,l,t}}+\mu}}{T}}}+1}}-\frac{1}{{{e^{\frac{{{\varepsilon_{f,l,t}}-\mu}}{T}}}+1}}}\right\\}},$
(6)

corresponds to chiral condensation of different quark flavors ($f=u,d$). As is
well known, the NJL model is nonrenormalized, so the regularization scheme is
necessary for extracting finite numerical results. A soft truncation by
introducing a smooth truncation function ${f_{\Lambda}}(p)$ Ferrer  _et al._
(2015a) is adopted to ensure its convergence during the momentum integration
as

${f_{\Lambda}}(p)={\left({1+\exp\left({\frac{{\sqrt{{p^{2}}+2l\left|{{q_{f}}eB}\right|}-\Lambda}}{{0.05\Lambda}}}\right)}\right)^{-1}}.$
(7)

Figure 1:  The dependence of dynamical quark mass ($M$) on temperature (T) for
four different magnetic fields ($eB=$0.02, 0.10, 0.20, and 0.30
${\rm{Ge}}{{\rm{V}}^{\rm{2}}}$) for standard NJL model without AMM. (a) is for
$\mu=$0.00 GeV, and (b) is for $\mu=$0.225 GeV.

It is well known that the dynamical mass or the quark condensate plays as an
order parameter for the chiral-phase transition. Chiral restoration occurs at
high temperatures and/or high chemical potentials. In Figs. 1(a) and 1(b), the
dynamical quark mass ${M}$ of $u$ and $d$ quarks without considering AMM are
manifested as decreasing smooth functions of temperatures at $\mu=$ 0 GeV and
$\mu=$ 0.225 GeV, which indicates a chiral crossover. Since we have considered
nonvanishing current quark mass, the chiral symmetry is never restored fully.
Since the dynamical mass is proportional to chiral condensate, it can be seen
from Fig. 1 that the larger the magnetic field is, the larger the
corresponding chiral condensation is. This phenomenon is manifested as
magnetic catalysis Kharzeev  _et al._ (2008); Gusynin  _et al._ (1999, 1996,
1994), which accounts for the magnetic field has a strong trend to lift (or
catalyze) spin-zero $q\bar{q}$ condensates.

Figure 2 displays the dependence of dynamical quark mass (${M}$) on
temperature (T) for four different magnetic fields ($eB=$ 0.02, 0.10, 0.20,
and 0.30 ${\rm{Ge}}{{\rm{V}}^{\rm{2}}}$) by considering the AMM. Figures 2(a)
and 2(b) are for $\mu=$ 0.00 GeV and 0.225 GeV, respectively. Contrary to the
behavior of the without AMM shown in Fig. 1, the dynamical mass-decreasing
behavior of $u$ and $d$ quarks in the chiral restoration is not a smooth
function but a sudden drop, which indicates the existence of a first-order
transition. However, the smooth slope of the dynamical mass for the crossover
can be still present in the weak field $eB=$ 0.02
${\rm{Ge}}{{\rm{V}}^{\rm{2}}}$ . It is found that the dynamical quark mass of
$u$ and $d$ quarks have the characteristics of IMC in the chiral restoration
phase ($T\geq{T_{C}}$) by inducing AMM.

Figure 2:  The dependence of dynamical quark mass ($M$) on temperature (T) for
four different magnetic fields ($eB=$ 0.02, 0.10, 0.20, and 0.30
${\rm{Ge}}{{\rm{V}}^{\rm{2}}}$) with AMM. (a) is for $\mu=$ 0.00 GeV, and (b)
is for $\mu=$ 0.225 GeV.

The contribution ratios from different Landau levels to chiral condensation
under strong magnetic fields are manifested in Fig. 3. The contribution ratios
for considering AMM shown in Fig. 3(a, b) are compared to the ratios for
without considering AMM shown in Fig. 3 (c, d). It is found that the
contribution ratio of each Landau level of the NJL model without inducing AMM
to the chiral condensation does not change significantly with the increasing
of temperature. Even when the temperature rises to the chiral phase transition
temperature where the chiral symmetry is restored, the contribution of each
different Landau level to the chiral condensation is roughly unchanged.

Figure 3:  The temperature dependence of the contribution ratio of quark
condensation with different Landau levels under various magnetic fields in
zero chemical potential. Fig. 3(a) and (b) by considering AMM are for $eB=$
0.1 ${\rm{Ge}}{{\rm{V}}^{\rm{2}}}$ and $eB=$ 0.3
${\rm{Ge}}{{\rm{V}}^{\rm{2}}}$, respectively. While Fig.3 (c) and (d) is same
as Fig. 3(a) and (b), but without considering AMM.

As following, we will focus on studying the contribution ratio of NJL model by
inducing AMM. It is found that as the temperature increases, after reaching
the chiral phase transition temperature, the contribution ratio of the lowest
Landau level to chiral condensation decreases significantly with temperature,
but the contribution ratio of the excited Landau level increases with
temperature accordingly as shown in Fig. 3(a) and (b). From there analysis, we
can understand it as that the effect of AMM can make particles with the lowest
Landau level more easily to be excited to higher excited energy level at high
temperatures. And as the magnetic field increases, this effect becomes more
pronounced. It is believed that this may be the reason for the formation of
IMC at high temperatures after the introduction of AMM.

The dependence of the renormalized critical temperature $T_{c}/T_{0}$ on the
magnetic field $eB$ is given shown in Fig. 4. $T_{0}$ is the critical
temperature when the magnetic field is zero. It can be seen that the NJL model
with AMM has obvious IMC effect, that is, with the increase of magnetic field,
the critical temperature $T_{c}$ gradually decreases with magnetic field. On
the other hand, for the NJL model without AMM, it exhibits a MC
characteristic, where the critical temperature increases with the increase of
the magnetic field. Our explanation for this phenomenon is that the
dimensional reduction in the magnetic field background cause particles to
occupy different excited Landau levels. Although the contribution of Landau
levels in the low-temperature region is less affected by temperature, the
effect of the inducing AMM makes the contribution of higher Landau levels
becoming great near the critical temperature, as shown in Fig. 3. Without
considering AMM, it would have led us to overestimate the effective kinetic
mass of quarks, resulting in the MC effect. At high temperatures, the effect
of more particles exciting to higher Landau levels will counteract some of the
MC effect and display IMC.

Figure 4:  The dependence of renormalized critical temperature on magnetic
field for NJL model with or without the consideration of AMM at zero and
finite chemical potentials. Figure 5:  Phase diagrams in the presence of
strong magnetic fields $eB=$ 0.1, 0.2, and 0.3 ${\rm{Ge}}{{\rm{V}}^{\rm{2}}}$,
respectively. The solid lines correspond to first-order transition, the dashed
lines correspond to crossover phase transition, and the full dots correspond
to CEP.

The phase diagrams by introducing AMM with magnetic fields of $eB$ = 0.10 ,0.2
and $eB$ = 0.30 ${\rm{Ge}}{{\rm{V}}^{\rm{2}}}$ are published in Fig. 5. It is
found that when the magnetic field is as low as $eB$ = 0.1
${\rm{Ge}}{{\rm{V}}^{\rm{2}}}$, the phase transition is crossover transition,
and no first-order phase transition occurs. When the magnetic field rises from
$eB=$ 0.2 to 0.3 ${\rm{Ge}}{{\rm{V}}^{\rm{2}}}$, the crossover occurs at high
temperature and small chemical potentials $\mu$, while the first-order phase
transition happens at low temperatures $T$ and large chemical potential $\mu$.
The results are consistent with those of Refs. Zhu and Feng (2023); Su and Wen
(2021). The phase transition between the crossover and first order transition
is separated by the critical end point (CEP) in $T-\mu$ plane. The existence
of CEP is an important topic in QCD phase diagram, and the location of CEP has
been investigated by lattice QCD and some effective models Zhu and Feng
(2023); Fodor and Katz (2004); Costa  _et al._ (2015). Fig. 5 displays the
locations of CEP for $eB$ = 0.2 ${\rm{Ge}}{{\rm{V}}^{\rm{2}}}$ and $eB$ = 0.3
${\rm{Ge}}{{\rm{V}}^{\rm{2}}}$, respectively. When $eB$ = 0.2
${\rm{Ge}}{{\rm{V}}^{\rm{2}}}$ and $eB$ = 0.3 ${\rm{Ge}}{{\rm{V}}^{\rm{2}}}$,
the CEP are located at ($T$, $\mu$) = (47.3, 245.7) MeV and ($T$, $\mu$) =
(76.1, 209.7) MeV, respectively. The temperature of CEP will increase with the
magnetic field, while the chemical potential of the CEP decreases with the
temperature of CEP.

##  III Boltzmann equation in relaxation time approximation and shear
viscosity coefficient

The produced medium can be supposed to be a kind of near equilibrium thermal
system. Therefore, we can investigate the transport properties of the nearly
perfect fluid impacted by QCD phase transition. The shear viscosity of the
medium can be defined by the Boltzmann equation of the relativistic fluid, and
the relaxation time of quarks can be calculated from the thermally averaged
total cross section of elastic scattering. The local equilibrium distribution
function for the quarks is given as

$f(x,p)={f^{0}}(x,p)+{f^{1}}(x,p),$ (8)

where ${f^{0}}(x,p)$ is the equilibrium distribution function as

${f^{0}}(x,p)={\left[{\exp\left({\beta\left(x\right)\left({{u_{\mu}}\left(x\right){p^{\mu}}\mp\mu\left(x\right)}\right)}\right)+1}\right]^{-1}},$
(9)

where ${u_{\mu}}\left(x\right)$ is the four-velocity of the fluid, and
$\mu\left(x\right)$ is the chemical potential. The covariant Boltzmann
equation of motion can be taken as

$\frac{{df}}{{dt}}=\frac{{{p^{\mu}}}}{E}{\partial_{\mu}}f-\frac{{\partial
E}}{{\partial{x^{i}}}}\frac{{\partial
f}}{{\partial{p^{i}}}}=-\frac{{{f^{1}}}}{\tau},$ (10)

where $\tau$ is the relaxation time. The total energy-momentum tensor of the
fluid is given as ${T_{\mu\nu}}=T_{\mu\nu}^{0}+T_{\mu\nu}^{D}$, where the
$T_{\mu\nu}^{0}=-p{g^{\mu\nu}}+w{u^{\mu}}{u^{\nu}}$ is the ideal part, and the
dissipative part is given as

$T_{\mu\nu}^{D}=\eta\left({{D^{\mu}}{u^{\nu}}+{D^{\mu}}{u^{\nu}}+\frac{2}{3}{\Delta^{\mu\nu}}{\partial_{\alpha}}{u^{\alpha}}}\right)-\zeta{\kern
1.0pt}{\partial_{\alpha}}{u^{\alpha}},$ (11)

where $\eta$ is the shear viscosity coefficient and $\zeta$ is the bulk
viscosity coefficient. The total energy-momentum tensor can be expressed as

${T_{\mu\nu}}=\int{\frac{{{d^{3}}k}}{{{{(2\pi)}^{3}}}}}\frac{{{k^{\mu}}{k^{\nu}}}}{\omega}\left({{f^{0}}+\Phi{\kern
1.0pt}{f^{0}}\left({1-{f^{0}}}\right)}\right).$ (12)

The shear viscosity coefficient $\eta$ can be obtained by comparing the
dissipative part of the energy-momentum tensor and possess five independent
components in the presence of an external magnetic field Tuchin (2012),
wherein the viscosity coefficient of ${\eta}_{0}$ is exactly the same as the
viscosity coefficient when the magnetic field is zero. These remaining four
components can be written as

${\eta_{n(n=1,2,3,4)}}=\frac{1}{{15T}}\sum\limits_{a}{{g_{a}}}\int{\frac{{{d^{3}}{{\vec{k}}_{a}}}}{{{{(2\pi)}^{3}}}}}\frac{{{{\vec{k}}_{a}}^{4}}}{{{E_{a}}^{2}}}\tau_{a,n}^{eff}\left({f_{a}^{0}\left({1-f_{a}^{0}}\right)}\right),$
(13)

where ${g_{a}}$ is the degeneracy of collisional particle type $a$, the
$\tau_{a,n}^{eff}$ is the effective relaxation time, which $\tau_{a}^{B}$ is
related to the magnetic relaxation time and collision relaxation time
$\tau_{a}^{c}$. The approximation of the field-induced relaxation time
$\tau_{a}^{B}=E{}_{a}/({q_{a}}B)$ has been applied in the strong magnetic
field since the deviation of equilibrium state mostly result from the field
rather the particle collisions. The ansatz of effective relaxations
$\tau_{a,n}^{eff}$ Ghosh  _et al._ (2019); Critelli  _et al._ (2014) is taken
as

$\begin{array}[]{l}\tau_{a,1}^{eff}=\tau_{a}^{c}\frac{1}{{4\\{\frac{1}{4}+{{(\tau_{a}^{c}/\tau_{a}^{B})}^{2}}\\}}},\quad\tau_{a,2}^{eff}=\tau_{a}^{c}\frac{1}{{\\{1+{{(\tau_{a}^{c}/\tau_{a}^{B})}^{2}}\\}}}\\\
\tau_{a,3}^{eff}=\tau_{a}^{c}\frac{{\tau_{a}^{c}/\tau_{a}^{B}}}{{2\\{\frac{1}{4}+{{(\tau_{a}^{c}/\tau_{a}^{B})}^{2}}\\}}},\quad\tau_{a,4}^{eff}=\tau_{a}^{c}\frac{{\tau_{a}^{c}/\tau_{a}^{B}}}{{\\{1+{{(\tau_{a}^{c}/\tau_{a}^{B})}^{2}}\\}}}.\end{array}$
(14)

By considering the elastic scattering of two-body for this two-flavor system,
one can obtain 12 different collision processes. In order to determine the
collision relaxation time of the system, the elastic scattering theory of the
two-body collision $a+b\to c+d$ is utilized in our study. The constituent
particles of the system medium are mainly $u$ and $d$ quarks and their
antiquarks near the phase transition temperature. A total of 12 collision
scenarios are taken as

$\begin{array}[]{*{20}{c}}{uu\to uu,}&{ud\to ud,}&{u\bar{u}\to
u\bar{u},}&{u\bar{u}\to d\bar{d},}\\\ {u\bar{d}\to u\bar{d},}&{dd\to
dd,}&{\bar{u}d\to\bar{u}d,}&{d\bar{d}\to d\bar{d},}\\\ {d\bar{d}\to
u\bar{u},}&{\bar{u}\bar{u}\to\bar{u}\bar{u},}&{\bar{u}\bar{d}\to\bar{u}\bar{d},}&{\bar{d}\bar{d}\to\bar{d}\bar{d}.}\end{array}$
(15)

We assume that particle $a$ is a probe-particle, the collision relaxation time
of the system could be evaluated $\tau_{a}^{c}=\frac{1}{{\Gamma_{a}^{c}}}$.
The collision width $\Gamma_{a}^{c}$ is given as

$\begin{array}[]{*{20}{c}}\Gamma_{a}^{c}=\frac{{\int{\frac{{{d^{3}}{p_{a}}}}{{{{\left({2\pi}\right)}^{3}}}}\Gamma_{a}^{c}({p_{a}})}{f_{a}}}}{{\int{\frac{{{d^{3}}{p_{a}}}}{{{{\left({2\pi}\right)}^{3}}}}}{f_{a}}}},\end{array}$
(16)

where the $\Gamma_{a}^{c}({p_{a}})$ is taken as

$\Gamma_{a}^{c}({p_{a}})=\sum\limits_{a}{\int{\frac{{{d^{3}}{p_{b}}}}{{{{\left({2\pi}\right)}^{3}}}}{\sigma_{ab}}\left({{p_{a}},{p_{b}}}\right)}}{\nu_{ab}}\left({{p_{a}},{p_{b}}}\right){f_{b}}.$
(17)

where ${\sigma_{ab}}$ and ${\nu_{ab}}$ are the collision cross section and the
relative velocity, which are given as

${\sigma_{ab}}\left({{p_{a}},{p_{b}}}\right)=\frac{1}{{16\pi
s}}{\left|{{{\tilde{M}}_{ab}}}\right|^{2}},$ (18)

and

${\nu_{ab}}\left({{p_{a}},{p_{b}}}\right)=\frac{{\left({{E_{a}}+{E_{b}}}\right)\sqrt{{s^{2}}-4{M^{2}}}}}{{2{E_{a}}{E_{b}}}},$
(19)

where $s={\left({{E_{a}}+{E_{b}}}\right)^{2}}$ is the Mandelstam variable, and
the scattering matrix ${\left|{{{\tilde{M}}_{ab}}}\right|^{2}}$ in Eq. (18) is
given as

${\left|{{{\tilde{M}}_{ab}}}\right|^{2}}=\frac{1}{{2\times
2}}{G^{2}}16{\left({\frac{s}{2}}\right)^{2}}={G^{2}}{s^{2}}.$ (20)

Figure 6:  The temperature dependencies of collision relaxation time with
different magnetic fields. Fig.6 (a, b) is for relaxation time by considering
AMM, and (c, d) is for relaxation time without considering AMM. (a, c) is for
zero chemical potential, and (b, d) is for finite chemical potential $\mu$ =
0. 225 GeV.

The dependences of the collision relaxation time ${\tau_{c}}$ on temperature
with different magnetic fields with AMM Fig.6 (a, b) and without AMM Fig.6 (c,
d) at chemical potential $\mu$ = 0 (Fig. 6(a, c)) and $\mu$ = 0.225 GeV (Fig.
6(b, d)) are, respectively, displayed in Fig. 6. We compared the relaxation
time with and without the introduction of AMM. From the curve of relaxation
time changing with temperature, it can be seen that the relaxation time
decreases rapidly with the increasing temperature at low temperature. It is
found that adding AMM can change the dependence of the relaxation time on the
magnetic field at low temperatures. When AMM is added, at the same
temperature, the larger the magnetic field, the smaller the relaxation time.
However, without considering AMM, the larger the magnetic field, the larger
the relaxation time. It is observed that when the temperature is near the
phase transition temperature 150-180 MeV at $\mu$ = 0 with AMM, the
corresponding relaxation time is between 10 fm and 30 fm. When the chemical
potential is $\mu$ = 0.225 GeV, the relaxation time decreases faster with
temperature than that of the zero-chemical potential, especially it almost
drops to zero when the magnetic field become large with $eB$ = 0.3
${\rm{Ge}}{{\rm{V}}^{\rm{2}}}$. But the relaxation time hardly changes with
increasing temperature when the temperature exceeds 0.04 GeV. In the
following, we will use Eq. (13) to study shear viscosities near chiral phase
transition of the magnetized QCD medium.

##  IV The calculated results of the shear viscosities near chiral phase
transition

The dependencies of the ratio of shear viscosity coefficient to entropy
density $\eta_{(1,2,3,4)}/s$ on temperature with magnetic field $eB$ = 0.1
${\rm{Ge}}{{\rm{V}}^{\rm{2}}}$ at $\mu$ = 0 (Fig.7 (a)) and $\mu$ = 0.225 GeV
(Fig.7 (b)) are, respectively, displayed in Fig. 7. One finds that the ratios
of shear viscosity coefficient to entropy density $\eta_{(1,2,3,4)}/s$
increase with temperature. It is found that this ratio is an increasing
function of temperature while the relaxation time decrease with temperature.
Fig. 6, shows that the collision relaxation time $\tau$ is a rapidly
decreasing function of temperature. However, $\tau$ appears on the denominator
in Eq. (14), which makes the effective relaxation time rise overall.
Therefore, $\eta_{(1,2,3,4)}/s$ increase with temperature mainly due to the
change of effective relaxation time when magnetic field is applied.

Figure 7:  The temperature dependence of four components of shear viscosity
for the quarks in $eB$ = 0.1 ${\rm{Ge}}{{\rm{V}}^{\rm{2}}}$. (a) is for zero
chemical potential and (b) is for finite chemical potential $\mu$ = 0. 225
GeV. Figure 8:  The dependencies of the dynamic quark mass $M$ on temperature
of the crossover, the first order phase transition and the CEP transition,
respectively. The yellow solid line corresponds to the first$-$order phase
transition, the red dashed$-$line corresponds to the CEP phase transition and
the blue dotted line corresponds to the crossover phase transition,
respectively.

Figure 8 shows the dependence of the dynamic quark mass $M$ on temperature of
the crossover phase transition, the first order phase transition and the
critical endpoint (CEP) phase transition respectively. For the crossover phase
transition, the thermodynamic potential of the system has only one extreme
point, which corresponding to that the dynamic mass varies continuously with
temperature. The first-order phase transition, which refers to the phase
transition from the hadron phase to the quark gluon plasma phase, has clear
phase boundaries. For the first order phase transition, its dynamic mass $M$
has a discontinuous change at the phase transition point, which is called
abrupt change. For a first-order phase transition, the dynamic mass $M$
undergoes a discontinuous change at the phase transition point, which is
called a sudden change. The so-called CEP phase transition is the connection
point between the first-order phase transition and the crossover phase
transition on the phase diagram, and also the endpoint of the first-order
phase transition.

Figure 9:  The dependencies of renormalized shear viscosity coefficient
components (${\eta_{1,2,3,4}}/{\Lambda^{3}}$) on temperature of CEP phase
transition and first order phase transition respectively with magnetic field
($eB$ = 0.2 ${\rm{Ge}}{{\rm{V}}^{\rm{2}}}$).

The dependencies of renormalized shear viscosity coefficient components
(${\eta_{1,2,3,4}}/{\Lambda^{3}}$) on temperature of CEP phase transition and
first order phase transition with $eB$ = 0.2 ${\rm{Ge}}{{\rm{V}}^{\rm{2}}}$
are shown in Fig. 9. The chemical potential $\mu={\mu_{{\rm{CEP}}}}$ is fixed
to study the shear viscosity coefficient components for the CEP transition.
And then, the chemical potential $\mu={\mu_{{\rm{C}}}}$ is fixed to study the
shear viscosity coefficient components for the first order transition. It is
found that ${\eta_{1,2,3,4}}/{\Lambda^{3}}$ all increases with the temperature
from low temperature to high temperature, and then reach the first order phase
transition point. Each of ${\eta_{1,2,3,4}}/{\Lambda^{3}}$ has a discontinuous
at the phase transition point, an upward jump from chiral broken phase to
chiral restoration phase is shown in Fig. 9. ${\eta_{1,2,3,4}}/{\Lambda^{3}}$
increasing with the temperature are very similar to that of first order phase
transition, but ${\eta_{1,2,3,4}}/{\Lambda^{3}}$ of CEP at the phase
transition point continuously changes with temperature.

Figure 10:  The dependencies of renormalized shear viscosity coefficient
components (${\eta_{1,2,3,4}}/{\Lambda^{3}}$) on temperature of first order
phase transition with the magnetic fields $eB$ = 0.2
${\rm{Ge}}{{\rm{V}}^{\rm{2}}}$ and $eB$ = 0.3 ${\rm{Ge}}{{\rm{V}}^{\rm{2}}}$,
respectively.

The dependencies of shear viscosity coefficient components
(${\eta_{1,2,3,4}}/{\Lambda^{3}}$) on temperature of first order phase
transition with magnetic fields $eB$ = 0.2 ${\rm{Ge}}{{\rm{V}}^{\rm{2}}}$ and
$eB$ = 0.3 ${\rm{Ge}}{{\rm{V}}^{\rm{2}}}$ are respectively displayed in Fig.
10. The following characteristics have been displayed as: (1) Discontinuities
of ${\eta_{1,2,3,4}}/{\Lambda^{3}}$ for the first order phase transition
point, and there is an upward discontinuous jump from the chiral broken phase
to chiral restoration phase of the high temperature; (2)
${\eta_{2}}/{\Lambda^{3}}$ and ${\eta_{4}}/{\Lambda^{3}}$, their jump values
at the first order phase transition point increase with magnetic field; (3)
Comparing ${\eta_{1,2,3,4}}/{\Lambda^{3}}$, it is obvious that at the same
temperature, shear viscosity coefficient components satisfy the following
relationship ${\eta_{4}}>{\eta_{3}}>{\eta_{2}}>{\eta_{1}}$.

##  V SUMMARY AND CONCLUSIONS

Noncentral heavy ion collisions of RHIC and LHC can produce a strong magnetic
field. The phase transition under such a magnetic field during the evolution
of QGP to hadron gas depends mainly on transport coefficients like shear
viscosity. The dissipative coefficients in the presence of magnetic field are
also important and are essential ingredients for the magnetohydrodynamic
evolution of the strongly interacting medium. When the magnetic field exists,
the shear viscosity coefficient of the dissipative fluid system can be
decomposed into five different components $\eta_{0}$, $\eta_{1}$, $\eta_{2}$,
$\eta_{3}$, and $\eta_{4}$. As we know that $\eta_{0}$ is the same as the
shear viscosity coefficient with zero magnetic field. We conducted a detailed
study of the dependencies of $\eta_{1}$, $\eta_{2}$, $\eta_{3}$, and
$\eta_{4}$ on temperature and magnetic field for the first order phase
transition and critical endpoint transition.

We have used the formalism of the NJL model with AMM in the presence of
magnetic field to describe the magnetothermodynamics of quark matter, and we
get a temperature and magnetic field dependent quark mass, which will be
accessed to the phase space factors of $\eta_{1}$, $\eta_{2}$, $\eta_{3}$, and
$\eta_{4}$. For strongly coupled RHIC or LHC collisions, the expected quark
collision width $\Gamma_{c}$ will take a larger value. To describe that zone,
we need a general structure of $\eta_{1}$, $\eta_{2}$, $\eta_{3}$, and
$\eta_{4}$. A detailed study of the dependencies of $\eta_{1}$, $\eta_{2}$,
$\eta_{3}$, and $\eta_{4}$ on temperature and magnetic field for the first
order phase transition, critical endpoint transition and crossover phase
transition are respectively investigated.

The temperature dependence of relaxation time from a simple contact diagram of
$2\to 2$ scattering processes are analyzed by using the interaction Lagrangian
density of the NJL model with AMM. We compared the relaxation time with and
without the introduction of AMM. It is found that adding AMM can change the
relationship between the magnetic field and the relaxation time at low
temperatures. It is found that the larger the magnetic field is, the smaller
the relaxation time is at the same temperature when AMM is added.

The shear viscosity coefficient components $\eta_{1}$, $\eta_{2}$, $\eta_{3}$,
and $\eta_{4}$ all increase with temperature. Discontinuities of $\eta_{1}$,
$\eta_{2}$, $\eta_{3}$, and $\eta_{4}$ for the first order phase transition
point, and there is an upward discontinuous jump from the chiral broken phase
of the low temperature to the chiral restoration phase of the high
temperature. This implies that the shear viscosity coefficient of magnetized
QCD medium is not a function of smooth transition for the first order phase
transition.

In this paper, we only discuss the viscosity coefficient by using the two-
flavors NJL model with AMM. Can we expand to the Polyakov loop (Polyakov NJL)
to discuss the transport coefficient near the confinement-deconfinement phase
transition boundary? The answer is yes. In the near future, we are intending
to establish a better interaction picture through Feynmann diagram
calculations in presence of magnetic field.

## Acknowledgments

This work was supported by National Natural Science Foundation of China
(Grants No. 11875178, No. 11475068,No. 11747115).

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