# Effects of tensor spin polarization on the chiral restoration and
deconfinement phase transitions

Yan-Ru Bao  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

###### Abstract

Abstract: Effects of tensor spin polarization (TSP) on the chiral restoration
and deconfinement phase transitions are studied in Polyakov loop extended
Nambu-Jona-Lasinio (PNJL) model. For chiral phase transition, the higher the
polarized degree of quark-antiquark pairs under the strong magnetic field, the
higher the phase transition temperature. The TSP corrects the position of the
critical end point (CEP). The small impact of TSP on the phase transition
temperature is found for the deconfinement phase transition. On the other
hand, we divide the phase space into three ranges based on the phase diagram
obtained from the PNJL model: the confinement phase with chiral symmetry
broken, the deconfinement phase with restored chiral symmetry, and the
confinement phase with restored chiral symmetry (quarkonic phase). It is found
that TSP has only a very small effect on the anisotropic pressure in the
deconfined phase with chiral symmetry restored and the quarkyonic phase, but
it has a very strong effect on the anisotropic pressure in the confined phase
with chiral symmetry broken. This is because TSP is closely related to chiral
symmetry. The restoration of chiral symmetry means the dissociation of spin
polarization condensate.

##  I Introduction

The properties of QCD mater under strong magnetic field background have
attracted widespread interest among researchers Menezes  _et al._ (2009);
Miransky and Shovkovy (2015); Andersen  _et al._ (2016); Huang (2016);
Kharzeev  _et al._ (2016); Bzdak  _et al._ (2020); Qiu and Feng (2023); Qiu
_et al._ (2023). In the early universe, the magnetic field may reach the order
of $10^{22}\mathrm{G}$ Joyce and Shaposhnikov (1997); Brandenburg  _et al._
(2021). The surface magnetic field of magnetars can reach
$10^{14}$-$10^{15}\mathrm{G}$, while its internal magnetic field can reach
$10^{18}$-$10^{20}\mathrm{G}$ Duncan and Thompson (1992); Thompson and Duncan
(1993). And the extremely high magnetic fields can be generated in noncentral
heavy ion collision experiments. The magnetic field created in RHIC
experiments can reach $\sqrt{eB}\sim$ 0.1 GeV, while in LHC, the magnetic
field intensity can reach $\sqrt{eB}\sim$ 0.5 GeV. Although the magnetic field
is an external field with a short lifetime (of the order of 1 fm/c), the
existence of quark-gluon plasma (QGP) medium response effect is significant,
it greatly delays the decay of these time-dependent magnetic fields Gürsoy
_et al._ (2014); Chen  _et al._ (2018); She  _et al._ (2018); Chen and Feng
(2020). Therefore, it seems reasonable to assume the existence of a constant
external magnetic field in many cases. The addition of strong magnetic field
will make the phase diagram and phase structure of QCD more complex and
interesting, leading to the emergence of many new phenomena, such as chiral
magnetic effect (CME) Fukushima  _et al._ (2008); Kharzeev  _et al._ (2008);
Guo  _et al._ (2019); Deng and Feng (2022), magnetic catalysis in vacuum (MC)
Klevansky and Lemmer (1989); Gusynin  _et al._ (1996, 1999), and inverse
magnetic catalysis (IMC) Bali  _et al._ (2012a, b, 2013); Fayazbakhsh and
Sadooghi (2014); D’Elia  _et al._ (2018); Mao (2022) around chiral phase
transitions.

The magnetic field also induces spin polarization, which is the condensate of
quark antiquark pairs with parallel spins. As shown in reference Ferrer  _et
al._ (2014), the tensor type interaction
$(\bar{\psi}\Sigma^{3}\psi)^{2}+(\bar{\psi}i\gamma^{5}\Sigma^{3}\psi)^{2}$
induces spin polarization $<\bar{\psi}i\gamma^{1}\gamma^{2}\psi>$ which is
analogous to the form of an anomalous magnetic moment (AMM) Ferrer  _et al._
(2014); Mao and Rischke (2019) of quarks developed in the presence of a
magnetic field. Note that the tensor polarization operator
$<\bar{\psi}\sigma^{12}\psi>$ is also known as the tensor spin polarization
(TSP) operator or spin density, because
$<\bar{\psi}\sigma^{12}\psi>=<\psi^{\dagger}\gamma^{0}\Sigma^{3}\psi>$ and
with ${\Sigma}^{3}=\Bigl{(}\begin{smallmatrix}{\sigma}^{3}&0\\\
0&{\sigma}^{3}\end{smallmatrix}\Bigr{)}$ ,
$\sigma^{3}=-i\sigma^{1}\sigma^{2}$. By projecting the quark spinors $\psi$
into the spin subspace $\psi=\psi_{\uparrow}+\psi_{\downarrow}$, one can
obtain
$\bar{\psi}\sigma^{12}\psi\sim\langle\bar{\psi}_{\uparrow}\psi_{\uparrow}\rangle-\langle\bar{\psi}_{\downarrow}\psi_{\downarrow}\rangle$,
which measures the difference between spin up quark pairing and spin down
quark pairing Qiu and Feng (2023); Lin  _et al._ (2022).

The effects of TSP on the dynamic generation of quark magnetic moments in non-
equilibrium quark matter, the magnetic properties of QCD matter, pion mass,
and the chiral phase transition in the (2+1)-flavor Nambu-Jona-Lasinio (NJL)
model have been investigated Qiu and Feng (2023); Mao and Rischke (2019); Lin
_et al._ (2022). We know that the NJL model can only discuss issues such as
chiral symmetry breaking and restoration phase transition, while the Polyakov
loop extended Nambu-Jona-Lasinio (PNJL) model can analyze both chiral and
deconfinement phase transition simultaneously.

In recent years, the anisotropy induced by magnetic fields has also been
widely studied Chaudhuri  _et al._ (2022); Goswami  _et al._ (2023). The
destruction of rotational symmetry by magnetic fields leads to the anisotropy
of energy momentum tensors (EMTs). If the spatial elements of EMT are
interpreted as the pressure generated by the response of the thermodynamic
potential of the system to compression in the corresponding direction, there
exists a difference caused by the orientation of the magnetic field in the
local rest framework Chatterjee  _et al._ (2015). It has been proven that the
derivative of the partition function obtained under constant magnetic flux
corresponds to spatial elements where the directional difference of EMT
becomes apparent. Usually, these different elements are referred to as
longitudinal ($P_{\parallel}$) and transverse ($P_{\bot}$) pressures Bali  _et
al._ (2013). These quantities will affect the equation of state of strongly
interacting substances.

Once the anisotropy of pressure under magnetic field background is considered,
many studies on the equation of state (EOS) of dense stars will yield new
results Chatterjee  _et al._ (2015); Canuto and Chiu (1968); Martínez  _et
al._ (2003); Noronha and Shovkovy (2007); Ferrer  _et al._ (2010); Huang  _et
al._ (2010); Strickland  _et al._ (2012); Sinha  _et al._ (2013); Ferrer  _et
al._ (2015a); Menezes  _et al._ (2015); Menezes and Lopes (2016); Avancini
_et al._ (2018); Ferrer and Hackebill (2019). Due to the fact that both TSP
and anisotropic pressure are caused by magnetic fields, it is of great
research significance to focus on the influence of TSP on anisotropic
pressure. In addition, under the magnetic field background, the original
isotropic fermion vertices split into longitudinal and transverse fermions in
the new tensor channel D’Elia  _et al._ (2018), and it is expected that this
anisotropy will be reflected in pressure, and TSP will promote the anisotropy
of pressure.

The impacts of TSP on the chiral restoration phase transition and
deconfinement phase transition in the PNJL model are investigated in the
paper. And then, the impacts of TSP on the anisotropic pressure under three
different phase ranges are studied. This paper is organized as follows. The
two flavor PNJL model with tensor channel is introduced in Sec. II. In Sec.
III, we will study the distributions of TSP under different backgrounds, the
influences of TSP on chiral restoration and deconfinement phase transition,
and the effect of TSP on the anisotropy of pressure under three different
phases. Finally, we make the summaries and conclusions in Sec. IV.

##  II The 2 -flavors PNJL model with TSP

The destruction of rotational symmetry by uniform magnetic field leads to the
separation of longitudinal and transverse-fermion modes along the direction of
the magnetic field D’Elia  _et al._ (2018); Ferrer  _et al._ (2014). This
separation leads to an effective splitting of the coupling in the gluon
exchange interaction which the NJL model are usually based on. Therefore, this
splitting can be reflected in the four-fermion coupling of the QCD effective
field NJL model. By using Fierz identities Ferrer  _et al._ (2014); Lin  _et
al._ (2022); Ferrer  _et al._ (2015b) in a magnetic field, we can obtain the
Lagrangian of the scalar and tensor interactions of the 2-flavor PNJL model as

$\displaystyle\mathcal{L}=$
$\displaystyle\bar{\psi}(i\gamma^{\mu}D_{\mu}-m_{0}+\gamma^{0}\mu_{q})\psi+G_{s}[(\bar{\psi}\psi)^{2}+(\bar{\psi}i\gamma^{5}\psi)^{2}]$
(1)
$\displaystyle+G_{t}[(\bar{\psi}\Sigma^{3}\psi)^{2}+(\bar{\psi}i\gamma^{5}\Sigma^{3}\psi)^{2}]-\mathcal{U}(\Phi,\bar{\Phi}),$

where $\psi=(u,d)^{T}$ is two-flavor quark field with
$\hat{m}=\textrm{diag}(m_{u},m_{d})$. Due to the spin symmetry of light
quarks, the current quark mass is $m_{0}=m_{u}=m_{d}$.
$\Sigma^{3}=\frac{i}{2}[\gamma^{1},\gamma^{2}]=i\gamma^{1}\gamma^{2}$ is the
spin operator. In addition, covariant derivative
$D_{\mu}=\partial_{\mu}-iq_{f}A_{\mu}$ contains charge matrix
$q_{f}=\textrm{diag}(q_{u},q_{d})=\textrm{diag}(\frac{2}{3},-\frac{1}{3})$. In
the external electromagnetic field $A_{\mu}=(0,0,-Bx_{1},0)$, a constant and
homogenous magnetic field of magnitude B points towards the $x_{3}$-direction.

The second term in Eq. (1) is the traditional scalar channel, which produces a
dynamical quark mass. The third term of Eq. (1) is the tensor channel, which
preserves chiral symmetry and rotational symmetry along the direction of the
magnetic field. Tensor channel is closely related to spin interactions, and
induces spin polarization. In the magnetic field background, the running
coupling constant is divided into longitudinal ($g_{\parallel}$) and
transverse ($g_{\bot}$) components Ferrer  _et al._ (2014). The coupling
coefficients $G_{s}$ and $G_{t}$ of the NJL interaction related to quark gluon
vertex coupling can be determined by
$G_{s}=(g_{\parallel}^{2}+g_{\bot}^{2})/\Lambda^{2}$ and
$G_{t}=(g_{\parallel}^{2}-g_{\bot}^{2})/\Lambda^{2}$. By setting
$G_{t}/G_{s}=\alpha$, one can obtain $g_{\parallel}=g_{\bot}(\alpha=0)$ at
zero magnetic field, and $g_{\parallel}\gg g_{\bot}(\alpha\to 1)$ as
$eB\to\infty$. In the following, we will choose the cases of $\alpha=0$,
$\alpha=1/2$ and $\alpha=1$.

By using the mean-field approximation, one can obtain the Lagrangian density
as

$\mathcal{L}_{MF}=\bar{\psi}(i\gamma^{\mu}D_{\mu}-M+\gamma^{0}\mu_{q}-i\xi\gamma^{1}\gamma^{2})\psi-\frac{\sigma^{2}}{4G_{s}}-\frac{\xi^{2}}{4G_{t}}-\mathcal{U}(\Phi,\bar{\Phi}),$
(2)

where $M=m_{0}+\sigma$ is the dynamical quark mass, and
$\sigma=-2G_{s}\left\langle\bar{\psi}\psi\right\rangle$ is the chiral
condensate. As mentioned earlier, tensor channels are closely related to spin-
spin interactions. Under the background of a magnetic field, quark-antiquark
pairs with opposite spin and opposite charge undergo orderly arrangement,
resulting in tensor spin polarization condensate

$\xi=-2G_{t}\left\langle\bar{\psi}\Sigma^{3}\psi\right\rangle.$ (3)

The fourth term of Eq. (1) is the Polyakov potential
$\mathcal{U}(\Phi,\bar{\Phi})$ associated with the deconfinement phase
transition Rößner  _et al._ (2007), where $\Phi$ is the order parameter
describing the deconfinement phase transition. When $\Phi\to 0$, the system is
considered to be in the confinement phase, while when $\Phi\to 1$, the system
is considered to be in the deconfinement phase. The Polyakov potential
$\mathcal{U}(\Phi,\bar{\Phi})$ is given

$\frac{\mathcal{U}(\Phi,\bar{\Phi})}{T^{4}}=-\frac{1}{2}A(T)\bar{\Phi}\Phi+B(T)\ln\left\\{{1-6\bar{\Phi}\Phi+4(\bar{\Phi}^{3}+\Phi^{3})-3(\bar{\Phi}\Phi)^{2}}\right\\},$
(4)

where the Polyakov potential $\mathcal{U}(\Phi,\bar{\Phi})$ is related to the
Z(3) center symmetry. By simulating the deconfinement at finite temperature,
one can obtain Meisinger  _et al._ (2002) the coefficients as

$A(T)=a_{0}+a_{1}(\frac{T_{0}}{T})+a_{2}(\frac{T_{0}}{T})^{2},$ (5)
$B(T)=b_{3}(\frac{T_{0}}{T})^{3}.$ (6)

The different parameters Meisinger  _et al._ (2002) of Eq. (5) and Eq. (6) are
given in Table I.

Table 1: Parameters set for polyakov potential. $a_{0}$ | $a_{1}$ | $a_{2}$ | $b_{3}$ |  $T_{0}$ (MeV)
---|---|---|---|---
$3.51$ | $-2.47$ | $15.2$ | $-1.75$ | $270$

The effective potential at finite temperature and chemical potential obtained
by the standardized process is

$\displaystyle\Omega=$
$\displaystyle\frac{\sigma^{2}}{4G_{s}}+\frac{\xi^{2}}{4G_{t}}+\mathcal{U}(\Phi,\bar{\Phi})-3\sum_{n,f,s}\frac{\left|q_{f}B\right|}{2\pi}\int_{-\infty}^{\infty}\frac{dp_{z}}{2\pi}\epsilon_{n,f,s}$
(7)
$\displaystyle-T\sum_{n,f,s}\frac{\left|q_{f}B\right|}{2\pi}\int_{-\infty}^{\infty}\frac{dp_{z}}{2\pi}[T\ln(1+g^{-})+T\ln(1+g^{+})],$

where

$g^{-}(\Phi,\bar{\Phi})=1+3(\Phi+\bar{\Phi}\exp(\frac{-E_{n,f,s}^{(-)}}{T}))\exp(\frac{-E_{n,f,s}^{(-)}}{T})+\exp(\frac{-3E_{n,f,s}^{(-)}}{T}),$
(8)
$g^{+}(\Phi,\bar{\Phi})=1+3(\bar{\Phi}+\Phi\exp(\frac{-E_{n,f,s}^{(+)}}{T}))\exp(\frac{-E_{n,f,s}^{(+)}}{T})+\exp(\frac{-3E_{n,f,s}^{(+)}}{T}),$
(9)

where $E_{n,f,s}^{(\pm)}=\epsilon_{n,f,s}\pm\mu_{q}$, and dispersion relation
of quarks with TSP Mao and Rischke (2019); Lin  _et al._ (2022) is given by:

$\epsilon_{n,f,s}^{2}=\begin{cases}p_{z}^{2}+(\sqrt{M^{2}+2n\left|q_{f}\right|B}-s\xi)^{2},n\geq
1,\\\ p_{z}^{2}+(M+\xi)^{2},n=0,\end{cases}$ (10)

where the summation of $n$ is taken over all Landau levels. $f=u,d$
corresponds to flavor quantum number, and the $s=\pm 1$ corresponds to the
different spin projections. Note that since the fermion in the lowest Landau
level has only one spin projection, no splitting is present in the $n$ = 0
case. But for the excited Landau levels $n\geq 1$, the spectrum of the quasi-
quarks exhibits a Zeeman splitting ($s=\pm 1$) due to the tensor spin
condensation $\xi$. One can obtain the gap equations as

$\frac{\partial\Omega}{\partial
M}=\frac{\partial\Omega}{\partial\xi}=\frac{\partial\Omega}{\partial\Phi}=\frac{\partial\Omega}{\partial\bar{\Phi}}=0.$
(11)

In order to ensure that the thermodynamic potential in vacuum is zero, we
define the normalized thermodynamic potential as the effective potential Qiu
and Feng (2023)

$\Omega_{eff}(T,\mu,eB)=\Omega(T,\mu,eB)-\Omega(0,0,eB).$ (12)

The expressions Chaudhuri  _et al._ (2022) of transverse pressure and
longitudinal pressure are

$P_{\parallel}(T,\mu,eB)=-\Omega_{eff}(T,\mu,eB),$ (13)
$P_{\bot}(T,\mu,eB)=P_{\parallel}(T,\mu,eB)-(\mathcal{M})eB,$ (14)

where the magnetization
$\mathcal{M}=-\frac{\partial\Omega_{eff}}{\partial(eB)}$.

It can be found that the energy integral term in the thermodynamic potential
Eq. (7) is ultraviolet divergent. The renormalization scheme cannot be used to
eliminate the divergence because of the dot-like interaction between quarks.
Therefore, it is necessary to use appropriate regularization scheme to
eliminate UV divergence. We use the Pauli-Villars regularization scheme with
gauge covariance to eliminate the divergence in this paper. The key of Pauli-
Villars regularization scheme is to replace integration with summation after
introducing the normalized energy. The normalized energy is given Mao (2019)
as

$\epsilon_{n,f,s,i(PV)}^{2}=\begin{cases}P_{z}^{2}+(\sqrt{M^{2}+2n\left|q_{f}\right|B+a_{i}\Lambda^{2}}-s\xi)^{2},n\geq
1,\\\ P_{z}^{2}+(\sqrt{M^{2}+a_{i}\Lambda^{2}}+\xi)^{2},n=0,\end{cases}$ (15)
$\sum_{n,f,s}\int_{-\infty}^{\infty}\frac{dp_{z}}{2\pi}\epsilon_{n,f,s}\to\sum_{n,f,s}\int_{-\infty}^{\infty}\frac{dp_{z}}{2\pi}\sum_{i=0}^{N}c_{i}\epsilon_{n,f,s,i(PV)},$
(16)

the parameters of regularization scheme are given as $\mathit{N}$=3,
$a_{i}=\left\\{0,1,2,3\right\\}$ and $c_{i}=\left\\{1,-3,3,-1\right\\}$. By
fitting the vacuum values such as pion-decay constant $f_{\pi}=93$ MeV and
chiral condensation $\left\langle\bar{\psi}\psi\right\rangle=(-250\
\mathrm{MeV})^{3}$, one can obtain Mao (2019) the relevant parameters such as
$G_{s}=3.44\ \mathrm{GeV}^{-2}$, $\Lambda=1127\ \mathrm{MeV}$ and $m_{0}=5\
\mathrm{MeV}$ at $G_{t}=0$.

##  III NUMERICAL RESULTS

###  III.1 Tensor spin polarization and chiral condensation

Figure 1:  The dynamical quark mass $M$ and Polyakov loop $\Phi$ as functions
of temperature $T$ with different external magnetic field ($eB$ = 0.20 GeV2
and $eB$ = 0.40 GeV2) and different set of spin-spin interaction coupling
constants ($G_{t}$ = 0 $G_{s}$ and $G_{t}$ = 0.5 $G_{s}$ for $\mu$ = 0 GeV and
$\mu$ = 0.20 GeV).

Firstly, we will investigate the effects of TSP on the chiral and
deconfinement phase transitions. The temperature dependence of order
parameters $M$ and $\Phi$ under different magnetic fields and chemical
potentials are shown in Fig. 1. As mentioned earlier, when $G_{t}$ is zero,
TSP contribution is zero, but when $G_{t}=0.5\ G_{s}$ is not zero, TSP
contribution is not zero. The temperature dependences of chiral order
parameters $M$ in the case of $eB=0.20\ \mathrm{GeV}^{2}$ and $eB=0.40\
\mathrm{GeV}^{2}$ are manifested in Fig. 1(a) and Fig. 1(b). The chiral
symmetry is broken with $M\neq 0$ at low temperature, and the chiral symmetry
is restored with $M\to 0$ at high temperature. It is found that the chiral
phase transition temperature increases by considering the influence of TSP.
Comparing Fig. 1 (a) and Fig. 1 (b), we find that the increase of the phase
transition temperature is more significant under higher magnetic fields
($eB=0.40\ \mathrm{GeV}^{2}$), which also means that the spin polarization
effect is more significant under higher magnetic fields. The deconfinement
phase transitions at $\mu=0\ \mathrm{GeV}$ and $\mu=0.20\ \mathrm{GeV}$ are
shown in Fig. 1 (c) and Fig. 1 (d), respectively. It is found that the impact
of TSP increases with the magnetic field, but slightly enhances the
deconfinement phase transition temperature.

In the $T-eB$ plane of Fig. 2, the corresponding temperature range is $0.01\
\mathrm{GeV}\leq T\leq 0.25\ \mathrm{GeV}$, and the magnetic field range is
within $0.01\ \mathrm{GeV}^{2}\leq eB\leq 0.40\ \mathrm{GeV}^{2}$. Figs. 2(a)
and 2(b) display the contour plots of the spin polarization condensation
$\xi=-2G_{t}\left\langle\bar{\psi}\Sigma^{3}\psi\right\rangle$ with $\mu=0\
\mathrm{GeV}$ and $\mu=0.20\ \mathrm{GeV}$, respectively.

Figure 2:  (a), (b) show contour plots of the distribution of spin
polarization condensate $\xi$ for $\mu$ = 0 GeV and $\mu$ = 0.20 GeV in the
$T-eB$ plane. Figure 3:  (a), (b) show contour plots of the distribution of
spin polarization condensate $\xi$ and dynamical mass $M$ for $eB$ = 0.20 GeV2
in the $T-\mu$ plane.

The spin polarization condensate $\xi$ decreases with the increase of
temperature in the $T-eB$ plane shown in Fig. 2(a) and 2(b). This shows that
the thermal background is not conducive to the formation of quark-antiquark
pairs, which leads to the inhibition of TSP production. Fig. 2(a) - (b) show
that $\xi$ increases with the magnetic field under different chemical
potentials. this is because the charged quark-antiquark pairs are easier to be
polarized under strong magnetic field.

Figure 3(a) and 3(b) show the distribution plots of spin polarization
condensate $\xi$ and dynamical quark mass $M$ with $eB=0.20\ \mathrm{GeV}^{2}$
in the $T-\mu$ plane. It is worth noting that due to the close relationship
between spin polarization condensate $\xi$ (also called dynamical quark
moment) and dynamical quark mass $M$ Ferrer  _et al._ (2014); Mao and Rischke
(2019), the $T-\mu$ distribution diagrams of $M$ and $\xi$ are very similar.
The distribution of $\xi$ shows a continuous change at low chemical potentials
and a sharp drop at high chemical potentials, which is consistent with the
behavior of the order parameter $M$ during chiral phase transition. Once
quarks obtain a dynamical mass, they should also obtain a tensor spin
polarization. This effect has also been reported in massless QED and in a one-
flavor NJL model Ferrer  _et al._ (2014); Ferrer and de la Incera (2009,
2010). From the view of symmetry, once the chiral symmetry is dynamically
broken, there is no symmetry protecting the TSP, because a nonvanishing value
of the latter breaks exactly the same symmetry.

###  III.2 Phase diagram

The $T-\mu$ phase diagram of chiral and deconfinement phase transition with
and without TSP ($G_{t}=0\ G_{s}$ and $G_{t}=0.5\ G_{s}$) under different
magnetic fields are manifested in Fig. 4. It is found that 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 influences of TSP on the phase diagrams of the
deconfinement phase transition and chiral phase transition can be summarized
as follows: (1) In general, considering TSP, it has little effect on the
deconfinement phase diagram, no matter whether it is a large magnetic field
($eB=0.40\ \mathrm{GeV}^{2}$) or a small magnetic field ($eB=0.20\
\mathrm{GeV}^{2}$). (2) TSP has great influence on the chiral phase diagram.
In our chiral phase diagram, there is a crossover phase transition region and
a first-order phase transition region. Considering the contribution of TSP,
the influence on the first-order phase transition is larger and the influence
on the crossover phase transition is smaller. However, with the increasing of
magnetic field, the influence of TSP on the phase diagram will increase. When
the magnetic field rises from 0.2 to $0.4\ \mathrm{GeV}^{2}$, the line of the
first order phase transition line becomes longer as the magnetic fields become
stronger. The results are consistent with those of Ref.de A. Bicudo  _et al._
(1999).

Figure 4:  $T-\mu$ phase diagram for chiral and deconfinement transition at
different magnetic fields $eB$ = 0.20 GeV2 and $eB$ = 0.40 GeV2 for different
spin polarization coupling constants $G_{t}$ = 0 $G_{s}$ and $G_{t}$ = 0.5
$G_{s}$. The solid lines correspond to chiral first-order transition, the dash
dot lines correspond to chiral crossover phase transition, the full dots
correspond to CEP, and the dot lines correspond to deconfinement crossover
phase transition.

In addition, according to the specific location of the critical end point
(CEP) in Fig. 4, we list the location of the CEP in different cases in Table
II. Compared with the case without TSP, we find that the introduction of TSP
makes the position of CEP move slightly. The temperature and chemical
potential of CEP are increased.

Table 2: CEP position under different magnetic fields and spin polarization coupling constant $G_{t}$. | $eB=0.20\,\text{GeV}^{2}$ | $eB=0.40\,\text{GeV}^{2}$
---|---|---
$G_{t}=0\,G_{s}$ | $\\{T_{E},\mu_{E}\\}=\\{0.087,0.222\\}$ | $\\{T_{E},\mu_{E}\\}=\\{0.148,0.225\\}$
$G_{t}=0.5\,G_{s}$ | $\\{T_{E},\mu_{E}\\}=\\{0.102,0.227\\}$ | $\\{T_{E},\mu_{E}\\}=\\{0.154,0.233\\}$

###  III.3 Anisotropic pressure

The phase diagrams of chiral phase transition and deconfinement phase
transition using the PNJL model are shown in Fig. 4. According to this phase
diagram, the phase space can be divided into three ranges: (1) The confinement
phase with chiral symmetry broken, (2) The deconfinement phase with chiral
symmetry restoration, (3) The confinement phase with chiral symmetry
restoration (also known as quarkyonic phase). Next, we study the dependences
of the normalized pressures $P_{\parallel}$ and $P_{\bot}$ on the magnetic
field after the introduction of TSP in these three ranges, where
$P_{\parallel}$ is the pressure parallel to the magnetic field direction, and
$P_{\bot}$ is the pressure perpendicular to the magnetic field direction.

####  III.3.1 Anisotropic pressure in the confinement phase with chiral
symmetry broken

By considers the influence of TSP, we study the dependence of the transverse
pressure and longitudinal pressure on the magnetic field in the confinement
phase with chiral symmetry broken in Fig. 5. Fig. 5(a) corresponds to $T=0.15\
\mathrm{GeV},\mu=0.0\ \mathrm{GeV}$. The results show that at small magnetic
field $eB=0.01\ \mathrm{GeV}^{2}$, the transverse pressure and longitudinal
pressure coincide. When $G_{t}=0$, the spin polarization effect is not
considered, $P_{\parallel}$ and $P_{\bot}$ almost do not change with the
increase of the magnetic field, and $P_{\parallel}$ and $P_{\bot}$ are almost
the same without splitting. When considering TSP ($G_{t}=0.5\ G_{s}$ and
$G_{t}=G_{s}$), the longitudinal pressure $P_{\parallel}$ increases not only
with the increase of magnetic field, but also with the increase of $G_{t}$.
While the transverse pressure $P_{\bot}$ decreases not only with the increase
of magnetic field, but also with the increase of $G_{t}$. That is to say, the
splitting of $P_{\parallel}$ and $P_{\bot}$ not only with the increase of
magnetic field, but also with the increase of $G_{t}$.

Fig. 5(b) corresponds to $T=0.15\ \mathrm{GeV},\ \mu=0.10\ \mathrm{GeV}$,
which is generally similar to that of Fig. 5(a). One finds that when $eB=0.01\
\mathrm{GeV}^{2}$, the values of $P_{\parallel}$ and $P_{\bot}$ with
$\mu=0.10\ \mathrm{GeV}$ are larger than that of $\mu=0\ \mathrm{GeV}$, which
is the natural result of more quark degrees of freedom Su and Wen (2021). On
the other hands, it is surprised to find that when considering TSP, the
transverse pressure $P_{\bot}$ produces great oscillation with the increase of
magnetic field.

Note that in the confinement phase with chiral symmetry broken, the influence
of TSP on the longitudinal and transverse pressure is very significant. In
conclusion, considering the contribution of TSP increases the anisotropic
characteristics of pressure in the magnetic field background.

Figure 5:  Normalized longitudinal ($P_{\parallel}$) and transverse
($P_{\bot}$ ) pressure as a function of $eB$ or the confinement phase with
chiral symmetry broken for different spin polarization coupling constant
$G_{t}$ with different chemical potential $\mu$ = 0 GeV (a) and $\mu$ = 0.2
GeV (b), respectively.

####  III.3.2 Anisotropic pressure in the deconfinement phase with chiral
symmetry restored

By considers the influence of TSP, we study the dependence of the transverse
pressure $P_{\bot}$ and longitudinal pressure $P_{\parallel}$ on the magnetic
field in the deconfinement phase with chiral symmetry restored shown in Fig.
6. It is found that the longitudinal pressure $P_{\parallel}$ increases with
the increase of the magnetic field, and transverse pressures $P_{\bot}$
decrease with magnetic field. An obvious difference from the confinement phase
with chiral symmetry broken is that the effect of TSP on anisotropic pressure
is very small, and the effect of TSP can be almost ignored. With the
restoration of chiral symmetry at high temperature $T=0.25\ \mathrm{GeV}$, the
effect of TSP tends to zero.

Figure 6:  Normalized longitudinal ($P_{\parallel}$) and transverse
($P_{\bot}$ ) pressure as a function of $eB$ at deconfinement phase with
restored chiral symmetry for different chemical potential $\mu$ = 0 GeV (a)
and $\mu$ = 0.2 GeV (b), respectively.

####  III.3.3 Anisotropic pressure in the quarkyonic phase

Fig. 7 shows the dependences of longitudinal and transverse pressure on
magnetic field in quarkyonic phase. Quarkyonic phase is a new phase of QCD, in
which the chiral symmetry has been restored, but it is still in the
confinement phase Chaudhuri  _et al._ (2022); Abu-Shady (2014); Fukushima
(2008); Abuki  _et al._ (2008); Buisseret and Lacroix (2012); Chaudhuri  _et
al._ (2020); Hidaka  _et al._ (2008); McLerran and Pisarski (2007). When
$T=0.15\ \mathrm{GeV},\mu=0.35\ \mathrm{GeV}$, the dynamic quark mass is close
to the current quark mass and $\Phi<0.2$, which corresponds to the confinement
phase with restored chiral symmetry, meaning that the system at quarkyonic
phase. In this region, the effect of spin polarization condensate on the
longitudinal and transverse pressures is very slight, similar to the
deconfinement phase with restored chiral symmetry.

The results show that TSP has only a very small effect on the anisotropic
pressure in the deconfined phase and the quarkyonic phase with chiral symmetry
restored, but it has a very strong effect on the anisotropic pressure in the
confined phase with chiral symmetry broken. This is also the result that TSP
is closely related to chiral symmetry. The restoration of chiral symmetry
means the dissociation of spin polarization condensate. Therefore, attention
should be paid to the role of TSP in chiral symmetry breaking. TSP is closely
related to the polarization caused by the magnetic field, and the pressure is
also affected by the magnetic field. The pressures are slitted into the
direction along the magnetic field and perpendicular to the magnetic field,
which is a sign of anisotropy. Therefore, when considering the anisotropy of
pressure in chiral symmetry breaking phase, TSP will promote anisotropy.

Figure 7:  Normalized longitudinal ($P_{\parallel}$) and transverse
($P_{\bot}$ ) pressure as a function of $eB$ at quarkyonic phase.

##  IV SUMMARY AND CONCLUSIONS

The generation mechanism of TSP is that quark and antiquark are polarized in
the magnetic field background due to the opposite charge and spin, inducing a
magnetic moment along the direction of the magnetic field. This new
condensation formed by spin polarization interaction is called spin
polarization condensate.

The introduction of TSP leads to an increase in the dynamical quark mass at
the lowest Landau level in the magnetic field background, leading to an
increase in the phase transition temperature. That is to say, for chiral phase
transitions, the degree of polarization of quark-antiquark pairs increases
with the magnetic field, resulting in an increase in the chiral phase
transition temperature. The TSP will correct the position of the CEP. For the
deconfinement phase transition, the effect of TSP on the phase transition
temperature is very small, and it almost disappears at a large chemical
potential.

Because the quark-antiquark pair produces magnetic moments along the direction
of the magnetic field, the polarization of the quark-antiquark pair increases
with the increase of the magnetic field, resulting in a larger TSP. In the
high temperature QGP background, the pairing of quark and antiquark is
blocked, and TSP decreases with the increase of temperature and chemical
potential. In addition, since the generation of TSP and chiral condensation
depends on the pairing of quark and antiquark, the distribution of TSP is
closely related to the chiral phase diagram. Under the same background, the
phase transition temperature of TSP is basically the same as the chiral phase
transition temperature.

It is found that TSP has only a very small effect on the anisotropic pressure
in the deconfined phase with chiral symmetry restored and the quarkyonic
phase, but it has a very strong effect on the anisotropic pressure in the
confined phase with chiral symmetry broken. This is also the result that TSP
is closely related to chiral symmetry. The restoration of chiral symmetry
means the dissociation of spin polarization condensate. Therefore, attention
should be paid to the role of TSP in chiral symmetry breaking.

## Acknowledgments

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

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