# Deconfinement and chiral restoration phase transition under rotation from
holography in an anisotropic gravitational background

Yidian Chen<EMAIL_ADDRESS>School of Physics, Hangzhou Normal
University, Hangzhou, 311121, P.R. China  Xun Chen<EMAIL_ADDRESS>School
of Nuclear Science and Technology, University of South China, Hengyang 421001,
China  Danning Li<EMAIL_ADDRESS>Department of Physics and Siyuan
Laboratory, Jinan University, Guangzhou 510632, P.R. China  Mei Huang
<EMAIL_ADDRESS>School of Nuclear Science and Technology, University of
Chinese Academy of Sciences, Beijing 100049, China

###### Abstract

We investigate the effects of rotation on deconfinement and chiral phase
transitions in the framework of dynamical holographic QCD model. Instead of
transforming to the rotating system by Lorentz boost, we construct an
anisotropic gravitational background by incorporating the rotating boundary
current. We firstly investigate the pure gluon system under rotation to
extract deconfinement phase transition from the Polyakov loop then add
2-flavor probe for chiral restoration phase transition from the chiral
condensate. It is observed that at low chemical potentials, the deconfinement
phase transition of pure gluon system is of first order and the chiral phase
transition of 2-flavor system is of crossover. Both the critical temperatures
of deconfinement and chiral phase transitions decrease/increase with
imaginary/real angular velocity ($\Omega_{I}/\Omega$) as $T/T_{c}\sim
1-C_{2}\Omega_{I}^{2}$ and $T/T_{c}\sim 1+C_{2}\Omega^{2}$, which is
consistent with lattice QCD results. In the temperature-chemical potential
$T-\mu$ phase diagram, the critical end point (CEP) moves towards regions of
higher temperature and chemical potential with real angular velocity.

##  I Introduction

In non-central ultra-relativistic heavy ion collisions, the strength of
magnetic field can reach the magnitude of $eB\sim 10^{18-20}\ \text{Gauss}$
[1, 2] and the angular momentum can reach the strength of
$\mathcal{O}(\text{10}^{4}-\text{10}^{5})\hbar$ [3] with local angular
velocity in the range of $0.01\sim 0.1$ GeV and the angular velocity of order
$\omega\sim 10^{21}\text{s}^{-1}$ [4]. QCD matter under strong magnetic field
and fast rotation has attracted lots of interests in recent decades [5, 6, 7,
8, 9, 10, 11, 12].

It was predicted that the global spin polarization for the final state
hyperons [11] as well as the vector meson spin alignment [12] would occur due
to the spin-orbit coupling of quantum chromodynamics (QCD). The global
polarization of $\Lambda$ and $\bar{\Lambda}$ hyperons was observed in non-
central heavy-ion collisions in 2017 [13, 14, 15]. The spin alignment was
reported with $\rho_{00}<1/3$ by the ALICE Collaboration for both $\phi$ and
$K^{*0}$ vector mesons at small transverse momenta [16] in 2020, while STAR
Collaboration gave $\rho_{00}>1/3$ for $\phi$ meson and $\rho_{00}\approx 1/3$
for $K^{*0}$ [17] in 2023. The inconsistent results attracted lots of
discussions [18, 19, 20, 21, 22, 23, 24, 25, 26, 26, 27, 28, 29, 30, 31].

QCD matter and the chiral restoration and deconfinement phase transitions
under rotation have also attracted much attention in recent years. It was
found in effective QCD models that the critical temperature of chiral phase
transition is suppressed by the rotation in Refs. [32, 33, 34, 35, 36, 37, 38,
39, 40]. In the framework of holographic QCD in Ref. [41], it was found that
the critical temperature of deconfinement phase transition decreases with
angular velocity, which is confirmed by other holographic studies [42, 43, 44,
45, 46, 47, 48, 49, 50, 51, 52]. However, lattice QCD simulations [53, 54, 55]
showed that the critical temperature of the deconfinement phase transition in
gluon dynamics as well as the chiral restoration phase transition in 2+1
flavor system increases with angular velocity, which is opposite to above
results obtained in effective and holographic models.

The contradiction between the effective and holographic models calculations
and lattice results on rotation attracted much attention and comprehensive
analysis [56, 57, 58, 59, 60]. Recently in the Polarized-Polyakov-loop Nambu-
Jona-Lasinio (PPNJL) model shown in Ref. [61], which takes into account the
rotating gluon background, it can be realized that both critical temperatures
of deconfinement and chiral phase transition increase with the angular
velocity in consistent with lattice results. This confirms that the
gluodynamics is sensitive to rotation as implied from the Zeeman-like behavior
of spin-1 vector meson under rotation observed in [62].

The lesson learnt from Ref. [61] is that the rotating gluon background has to
be properly taken into account. Therefore, in this work we reanalyze the
rotating gluodynamics in the framework of holographic QCD model then add
flavor probes on this background. In previous holographic studies, many
efforts have been made in understanding the rotating effect on QCD phase
transitions, besides studies in the Kerr black hole background [63, 64, 65,
66, 67]. One common way to introduce angular velocity is to do a simple
coordinate transformation to the dual gravity backgrounds [68, 69, 70, 71,
41], which have been carefully tested in non-rotational systems. In this
approximation, the rotating effect can be partially encoded in the holographic
approach. It is shown that the transition temperature of deconfining phase
transiton decreases with increasing angular velocity, as discussed above. But
the effect on the scalar operators, like the chiral condensation, which is the
order parameter of chiral phase transtion, has not been studied. The reason is
that the scalar operator is invariant under the coordinate transformation.

In Ref. [47], an alternative way was applied to consider the effects on the
chiral phase transtion. By introducing rotating boundary currents and solving
the soft-wall model in the probe limit (as also in Refs. [72, 73, 74, 75,
76]), the rotational effect on chiral condensation can be obtained. It gives a
similar effect with that on the deconfining transition, i.e. decreasing
transition temperatures with increasing angular velocities. However, since the
analysis is done in the probe limit, the corrections from the rotating
boundary currents can not be encoded in the gravity background, and thus the
effects on the gluodynamics have not been incorporated in this way. Obviously,
if one tries to go beyond the probe limit, it is possible to get the
rotational effects on both gluo-dynamics and chiral dynamics simultaneously.
However, to calculate the full gravity background dual to a rotating system is
not an easy task. Hence, in this work, we will focus on the near-centre region
of the rotating system, in which we can take a simple ansatz on the gravity
background for the rotating system. After solving the system, we will try to
extract the rotating effects on the dynamics of both the two sectors.

The organization of the paper is as follows. In Sec. II, the dynamical
holographic QCD model is reviewed and extended to the anisotropic case.
Thermodynamic quantities under rotation are explored in Sec. III.
Deconfinement and chiral phase transitions with angular velocity are studied
in Sec. IV and V, respectively. Finally, a summary and discussion are provided
in Sec. VI.

##  II Model setup

The holographic QCD method, based on the Anti-de Sitter/conformal field theory
(AdS/CFT) correspondence [77, 78, 79], has become one of the important
nonperturbative methods to understand the properties of QCD and its phase
transitions. In the past, the holographic method has been successfully applied
in various aspects, including transport coefficients [80, 81, 82],
thermodynamical quantities [83, 84, 85, 86, 87, 88], hadronic physics [89, 90,
91], and so on. Building on these studies, the dynamical holographic QCD model
(DHQCD) [92] has further simulated the dynamical behavior of QCD and
successfully described the confinement/deconfinement and chiral phase
transitions [93, 94, 95, 96]. However, to fully understand the behavior of QCD
under extreme conditions, especially the impact of rotational effects, it is
necessary to extend the DHQCD model. In this section, we review the DHQCD
model and present a new extension to the model which allows it to be applied
to the rotating case.

###  II.1 The Einstein-Maxwell-Dilaton system

From Ref. [96], the quenched DHQCD model with chemical potential is described
by the Einstein-Maxwell-dilaton system. The action in the string frame can be
written as

$\displaystyle S_{\rm tot}^{s}$ $\displaystyle=$ $\displaystyle
S_{G}^{s}+S_{M}^{s},$ (1) $\displaystyle S_{G}^{s}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi G_{5}}\int
d^{5}x\sqrt{-g^{s}}e^{-2\Phi}\left[R^{s}+4\partial_{M}\Phi\partial^{M}\Phi-V^{s}(\Phi)-\frac{h(\Phi)}{4}e^{\frac{4\Phi}{3}}F_{MN}F^{MN}\right],$
(2) $\displaystyle S_{M}^{s}$ $\displaystyle=$ $\displaystyle-\int
d^{5}x\sqrt{-g^{s}}e^{-\Phi}{\rm
Tr}\left[\nabla_{M}X^{\dagger}\nabla^{M}X+V_{X}(|X|,F_{MN}F^{MN})\right],$ (3)

with lowercase $s$ for string frame and 5D Newton constant $G_{5}$. In the
DHQCD model, the total action $S_{\rm tot}^{s}$ can be decomposed into
$S_{G}^{s}$ describing the gluo-dynamics and $S_{M}^{s}$ representing the
quark dynamics. The $\Phi$ and $V^{s}(\Phi)$ represent the dilaton field and
its potential. The strength tensor of the gauge field $A_{M}$ dual to the
baryon number current is denoted by $F$. The coupling between the dilaton and
the gauge field is given by $h(\Phi)$. In the presence of a non-zero baryon
chemical potential ($\mu_{B}\neq 0$), the only non-trivial component of the
$U(1)$ gauge field is $A_{t}$, i.e. $A=A_{t}dt$. The scalar field $X$ is
associated with the quark condensate $\langle\bar{q}q\rangle$, and $V_{X}$
represents its potential, which is coupled to the gauge field strength. The
EMD system describes the gluon dynamics quite well and enables to obtain the
QCD phase transition [93, 97, 98] and the glueball spectra [92] in the pure
gluon system.

The equations of motion and thermodynamics are more conveniently solved in the
Einstein frame, so the background action is rewritten as

$\displaystyle S_{G}^{e}=\frac{1}{16\pi G_{5}}\int
d^{5}x\sqrt{-g^{e}}\left[R^{e}-\frac{1}{2}\partial_{M}\Phi\partial^{M}\Phi-V^{e}(\Phi)-\frac{h(\Phi)}{4}F_{MN}F^{MN}\right],$
(4)

with lowercase $e$ for the Einstein frame. Here, the metric and potential
satisfy $g^{s}_{MN}=e^{\frac{4\Phi}{3}}g^{e}_{MN}$ and
$V^{e}=e^{\frac{4\Phi}{3}}V^{s}$, respectively. In Einstein’s framework, the
metric can be written as

$\displaystyle
ds^{2}=\frac{L^{2}e^{2A_{e}(z)}}{z^{2}}\bigg{[}-f(z)dt^{2}+\frac{dz^{2}}{f(z)}+d\vec{x}^{2})\bigg{]},$
(5)

where the warp factors $A_{e}(z)$ and $A_{s}(z)$ of the two frames satisfy
$A_{e}(z)=A_{s}(z)-\frac{2}{3}\Phi(z)$. Without loss of generality, the AdS
radius is set to $L=1$.

There are generally two ways to solve the equations of motion. The first is
through fixing $h(\Phi)$ and the explicit form of the potential $V(\Phi)$. The
second, known as the “potential reconstruction method”, is to resolve the
dilaton potential by inputting function $\Phi$ or $A_{e}$ and $h(\Phi)$. It
should be emphasized that the coefficients of the potential $V(\Phi)$ of this
method are temperature and chemical potential dependent, which is in a sense
consistent with the spirit of finite temperature field theory. This method is
easier to handle and is a good approximation to the fixed potential method and
is employed in this paper. As in Ref. [93], the dilaton field is required to
have the following form

$\displaystyle\Phi(z)=\mu_{G}^{2}z^{2}\tanh(\mu_{G^{2}}^{4}z^{2}/\mu_{G}^{2}).$
(6)

In the UV region, the dilaton field has $\Phi\stackrel{{\scriptstyle z\to
0}}{{\longrightarrow}}\mu_{G^{2}}^{4}z^{4}$, which is dual to the gauge-
invariant gluon condensation ${\rm Tr}\langle G^{2}\rangle$. In the IR region,
the dilaton field has $\Phi\stackrel{{\scriptstyle
z\to\infty}}{{\longrightarrow}}\mu_{G}^{2}z^{2}$ in order to satisfy the
linear Regge trajectory of mesons spectra.

As shown in the Tab. 1, there are four parameters $\mu_{G}$, $\mu_{G^{2}}$,
$G_{5}$, and $h(\Phi)$ in the model. For the first three parameters, the
values are taken from Ref. [93]. For the coupling $h(\Phi)$ of the dilaton and
gauge field, it is chosen as $h=4$, in order to realize the deconfinement
phase transition in the $T-\mu$ plane with critical endpoint (CEP)
$(T_{CEP},\mu_{CEP})=(0.271,0.14)$GeV. The $T-\mu$ phase diagram is consistent
with the deconfinement phase diagram for heavy quarks.

Parameters |  $\mu_{G}$ (GeV) |  $\mu_{G^{2}}$ (GeV) |  $G_{5}$ |  $h$ |  $\mu_{\Omega}$ (GeV-1)
---|---|---|---|---|---
Values |  0.75 |  3 |  1.25 |  4 |  122
Table 1:  The parameters of the DHQCD model, where the first three parameters
were selected from Ref. [93]. The last parameter $\mu_{G}$ is set as described
in Sec. II.2.

###  II.2 Anisotropic background under rotation

To investigate the QCD phase transition under rotation, one is faced with the
challenge of solving the EMD system under a rotating black hole. The
complexity of the problem is that in the cylindrical coordinate system, the
metric, dilaton, and gauge fields all depend on five-dimensional coordinates
$z$ and the radial direction $r$. As a result, different methods have been
used to study the effects of rotation.

In Ref. [41], the effect of rotation on the deconfinement phase transition has
been explored through the coordinate transformation. Alternatively, some have
analysed inhomogeneous chiral phase transitions under the probe approximation
[47]. However, the results of the holographic model suggest that rotation
reduces the critical temperature of the phase transition, which is
inconsistent with the findings of lattice QCD [53, 55].

In order to reconstruct the results of lattice QCD, the anisotropic background
is considered in this paper. As shown in Ref. [47], the scalar field, which is
dual to the condensation, does not depend significantly on the radial
coordinate near the center of rotation. Therefore, it can be assumed that the
metric and dilaton fields are only the function of the fifth-dimensional
coordinate near the center. Furthermore, an additional anisotropic function
$B(z)$ needs to be taken into account in the metric since the effect of
rotation. So, the background metric of the Einstein frame in the cylindrical
coordinate system is

$\displaystyle
ds^{2}=\frac{L^{2}e^{2A_{e}(z)}}{z^{2}}[-f(z)dt^{2}+\frac{dz^{2}}{f(z)}+e^{B(z)}dr^{2}+r^{2}e^{B(z)}d\theta^{2}+e^{-2B(z)}dx_{3}^{2}],$
(7)

where $r$ denotes the radial direction, $\theta$ is the direction of the polar
angle, and $x_{3}$ is the direction of the symmetry axis. According to the
holographic principle, the temperature of the system is given by the Hawking
temperature, i.e.,

$\displaystyle T=\frac{|f^{\prime}(z_{h})|}{4\pi},$ (8)

with the horizon $z_{h}$ of black brane.

As in Ref. [47], the rotation is introduced by a non-zero polar angle
component $A_{\theta}$ of the gauge field. It can be expanded as
$A_{\theta}\sim\Omega r^{2}+\rho_{\theta}(r,z)$. Near the center, the current
$\rho_{\theta}$ can be neglected and the field $A_{M}$ is approximated as

$\displaystyle A_{M}=(A_{t},0,0,A_{\theta},0),\qquad A_{\theta}=\Omega r^{2},$
(9)

with $A_{z}=0$ gauge and angular velocity $\Omega$. Although the gauge field
$A_{\theta}$ depends on the radial coordinates $r$, the strength $F^{2}$ is in
fact independent of $r$. Therefore, both the metric and dilaton fields depend
only on the fifth-dimensional coordinate $z$. Furthermore, It should be noted
that this choice naturally satisfies Maxwell’s equation in the
$\theta$-direction.

It is important to note that Eqs. (7) and (9) hold only under the near-centre
approximation. If the system is far from the center of the rotation axis, then
the radial direction dependence and the current $\rho_{\theta}$ are not
negligible. If near the edge of the system, the boundary has a significant
effect on the profile of the fields and the finite size effect becomes
indispensable. Additionally, the non-trivial current $\rho_{\theta}$ induces a
non-diagonal term $g_{\theta j}$ to the metric.

Under the near-centre approximation, i.e., Eqs. (7) and (9), the equations of
motion of the EMD system can be obtained as

$\displaystyle\frac{z^{2}h(\Phi)e^{-2A_{e}}\left(A_{t}^{\prime
2}+4e^{-2B}\Omega^{2}\right)}{4f}+\frac{3f^{\prime}}{2f}\left(A_{e}^{\prime}-\frac{1}{z}\right)+\frac{e^{2A_{e}}V^{e}(\Phi)}{2z^{2}f}$
$\displaystyle-\frac{12A_{e}^{\prime}}{z}+6A_{e}^{\prime 2}-\frac{3B^{\prime
2}}{4}-\frac{2\Phi^{\prime 2}}{3}+\frac{6}{z^{2}}$ $\displaystyle=0,$ (10a)
$\displaystyle-z^{2}e^{-2A_{e}}h(\Phi)\left(A_{t}^{\prime
2}+\frac{8}{3}e^{-2B}\Omega^{2}\right)+3f^{\prime}\left(A_{e}^{\prime}-\frac{1}{z}\right)+f^{\prime\prime}$
$\displaystyle=0,$ (10b) $\displaystyle\frac{B^{\prime
2}}{2}+\frac{4\Phi^{\prime 2}}{9}+\frac{2A_{e}^{\prime}}{z}-A_{e}^{\prime
2}+A_{e}^{\prime\prime}$ $\displaystyle=0,$ (10c)
$\displaystyle\frac{4\Omega^{2}z^{2}h(\Phi)e^{-2\left(A_{e}+B\right)}}{3f}+B^{\prime}\left(3A_{e}^{\prime}+\frac{f^{\prime}}{f}-\frac{3}{z}\right)+B^{\prime\prime}$
$\displaystyle=0,$ (10d) $\displaystyle
A_{t}^{\prime}\left(A_{e}^{\prime}+\frac{h^{\prime}}{h}-\frac{1}{z}\right)+A_{t}^{\prime\prime}$
$\displaystyle=0,$ (10e)
$\displaystyle\frac{z^{2}e^{-2A_{e}}\partial_{\Phi}h(\Phi)\left(A_{t}^{\prime
2}-4e^{-2B}\Omega^{2}\right)}{2f}-\frac{e^{2A_{e}}\partial_{\Phi}V_{e}(\Phi)}{z^{2}f}+\Phi^{\prime}\left(3A_{e}^{\prime}+\frac{f^{\prime}}{f}-\frac{3}{z}\right)+\Phi^{\prime\prime}$
$\displaystyle=0,$ (10f)

with the derivative ′ with respect to $z$.

As shown in Ref. [61], gluo-dynamics is sensitive to rotation, so the
coefficients $\mu_{G}$ and $\mu_{G^{2}}$ of the dilaton field should depend on
the angular velocity $\Omega$. In order to realize rotation-dependent gluo-
dynamics, the following simple form of the dilaton field is considered

$\displaystyle\Phi=(\mu_{G}+\mu_{\Omega}\Omega^{2})^{2}z^{2}\tanh(\mu_{G^{2}}^{4}z^{2}/(\mu_{G}+\mu_{\Omega}\Omega^{2})^{2}).$
(11)

In this paper, only the parameter $\mu_{G}$ is taken into account for the
angular velocity $\Omega$ dependence. Thus a new parameter $\mu_{\Omega}$ is
introduced in the profile of the dilaton field. In addition, the angular
velocity in quadratic form here is considered for the case where it can be
extended to complex plane, i.e. to imaginary angular velocity
$\Omega_{I}=-i\Omega$. Of course more complicated forms of $\mu_{G}(\Omega)$
can be considered, such as
$\mu_{G}+\mu_{\Omega}\Omega^{2}+\mu_{\Omega_{2}}\Omega^{4}+...$ . When the
angular velocity is not very high $\Omega\leq 0.1$ GeV, the effect of the
leading order is considered.

The appropriate boundary conditions need to be chosen in order to solve the
equations of motion Eqs. (10b-10e). At the ultraviolet (UV) boundary, the
background geometry is required to be asymptotically AdS5 spacetime, so

$\displaystyle f(0)=1,\qquad A_{e}(0)=0,\qquad B(0)=0.$ (12)

At the infrared (IR) boundary, $A_{e}$ and $B$ are required to satisfy the
natural boundary conditions, while the blackening factor has $f(z_{h})=0$.
According to the holographic principle, the gauge field $A_{t}(z)$ is required
to have the following boundary conditions

$\displaystyle A_{t}(0)=\mu,\qquad A_{t}(z_{h})=0.$ (13)

##  III Thermodynamics of the pure gluon system under rotation

In the previous section, we have successfully extended the DHQCD model to
anisotropic backgrounds. This extension allows us to describe rotating QCD
matter. In this section, we applied the holographic method to calculate the
thermodynamic quantities of the system. Specifically, we calculate the free
energy, entropy and specific heat capacity, and so on. Further, using the free
energy obtained, the model is able to fit the relationship between the phase
transition temperature $T_{c}(\Omega)$ and the angular velocity $\Omega$ of
lattice QCD, by tuning the new parameter $\mu_{\Omega}$.

In this paper, thermodynamic quantities are calculated by integral of entropy
density. According to the Bekenstein-Hawking formula, the entropy density of
the black brane can be obtained as

$\displaystyle s=\frac{e^{3A_{e}(z)}}{4G_{5}z^{3}}|_{z=z_{h}},$ (14)

with five-dimensional Newton constant $G_{5}$ listed in Tab. 1. From
thermodynamics, the total differential of Landau free energy density
$\mathcal{f}$ of the giant canonical system is

$\displaystyle d\mathcal{f}=-sdT-\rho d\mu-jd\Omega.$ (15)

with energy density $\epsilon$, baryon number density $\rho$, and angular
momentum density $j$. The free energy density can be obtained by integrating
the entropy density when the chemical potential and angular velocity are
fixed, which contains an integration constant. For pure glue system, as shown
in Ref. [99], $\mathcal{f}(z_{h}\to\infty)=0$ is required to satisfy, then the
free energy density can be written as

$\displaystyle\mathcal{f}=\int_{z_{h}}^{\infty}s\frac{dT}{dz_{h}}dz_{h}.$ (16)

For the pressure $P$ and energy density $\epsilon$, they can be given by the
following relation

$\displaystyle P=-\mathcal{f},\qquad\epsilon=\mathcal{f}+Ts+\mu\rho+\Omega j.$
(17)

In addition, the speed of sound $C_{s}$ and latent heat $C_{V}$ of the system
can also show messages about phase transition. Both these thermodynamic
quantities can be obtained from above quantities as

$\displaystyle C_{s}^{2}=\frac{dP}{d\epsilon}\bigg{|}_{\mu,\Omega},\qquad
C_{V}=\frac{d\epsilon}{dT}\bigg{|}_{\mu,\Omega}.$ (18)

Besides the method employed in this paper, another approach to obtain the free
energy density and other thermodynamic quantities is through the
renormalization on-shell action. In principle, the two approaches are
equivalent.

With the free energy density given above, the phase transition temperature
$T_{c}$ can be obtained for different angular velocities. The only parameter
$\mu_{\Omega}$ to be determined in the DHQCD model is based on the
relationship between the phase transition temperature $T_{c}(\Omega_{I})$ and
the imaginary angular velocity $\Omega_{I}$ predicted by lattice QCD [53].
Specifically, this relationship is given by a quadratic function

$\displaystyle T_{c}(\Omega_{I})/T_{c}(0)=1-C_{2}\Omega_{I}^{2}$ (19)

with $C_{2}\simeq 150\text{GeV}^{-2}$ [53]. Here $T_{c}(0)$ represents the
phase transition temperature without rotation. In Ref. [53], three different
boundary conditions are considered and similar results are observed. Without
loss of generality, the result from periodic boundary condition is chosen to
fit the parameter in this paper. The results of fitting are displayed in Fig.
1. The black points in the figure are the model’s results, the red and blue
points are the results of lattice QCD, and all the dashed lines indicate the
quadratic fitting curves. It should be noted that the red and blue points
correspond to different lattice size. Since the infinite volume limit cannot
be chosen for the rotating system, i.e., the finite volume effect is not
negligible, the relationship between the transition temperature and imaginary
angular velocity depends on the size. In the DHQCD model, the near-center
limit is taken into account, so such phenomena found in lattice QCD cannot be
observed. However, in Ref. [47], the holographic model indeed found the
dependence of the phase transition on the linear velocity. In the study, blue
dots have been chosen to fit the model’s parameter to enhance the effect of
rotation on the phase transition. The value of the parameter $\mu_{\Omega}$ is
shown in Tab. 1. As can be seen in Fig. 1, the results of the model agree well
with the quadratic function. This is comprehensible because the effect of
angular velocity in the dilaton field is the quadratic.

Figure 1:  The phase transition temperature as a function of the imaginary
angular velocity. The $T_{c}(0)$ represents the phase transition temperature
without rotation. The red and blue points in the figure show the lattice QCD
results from Ref. [53], and the black points show the calculations for the
model. The dashed lines indicate the quadratic fitting curves.

Within the DHQCD model, once the model parameter $\mu_{\Omega}$ is fixed, we
are able to calculate thermodynamic quantities under rotating. The results are
presented in Figs. 2 and 3. In these figures, the angular velocity $\Omega$ is
set to 0, 0.03, and 0.06 GeV, and we observe thermodynamic quantities such as
entropy density, pressure, energy density, trace anomaly, square of the speed
of sound, and latent heat. The red markers represent the results from lattice
QCD for the non-rotating case, as referenced in [100].

From Fig. 2, it can be observed that in regions close to and far from the
phase transition temperature, the thermodynamic quantities for the non-
rotating case show good agreement with lattice QCD data. For the entropy
density $s/T^{3}$ and pressure $P/T^{4}$, the effect of rotation is primarily
a translation of the curves, with little change in the overall shape. At an
angular velocity of 0.03 GeV, the translation of the curves is approximately
16$\%$. Furthermore, we observe that in the low-temperature region,
thermodynamic quantities are more sensitive to the angular velocity, and as
the temperature increases, thermodynamic quantities at different angular
velocities tend to converge to a common value. This is qualitatively different
from the results obtained in the coordinate transformation approach [41] but
similar with that obtained in previous NJL calculation [37].

Further, if we extend the relationship between the phase transition
temperature and the imaginary angular velocity to the real angular velocity,
that is, the relationship $T_{c}(\Omega)/T_{c}(0)=1+C_{2}\Omega^{2}$. We can
find that for an angular velocity of 0.03 GeV, the shift in the phase
transition temperature satisfies $C_{2}\times 0.03^{2}\simeq 14\%$. Thus,
these two quantities can be approximated as follows

$\displaystyle\frac{s}{T^{3}}(T,\Omega)\simeq\frac{s}{T^{3}}\bigg{(}T(1+C_{2}\Omega^{2}),0\bigg{)},\qquad\frac{P}{T^{4}}(T,\Omega)\simeq\frac{P}{T^{4}}\bigg{(}T(1+C_{2}\Omega^{2}),0\bigg{)}.$
(20)

As can be seen in Fig. 3, for the square of the speed of sound and the latent
heat, the behavior is almost the same as in $s/T^{3}$. Similarly, a
approximation similar to Eq. (20) above can be written as

$\displaystyle C_{s}^{2}(T,\Omega)\simeq
C_{s}^{2}\bigg{(}T(1+C_{2}\Omega^{2}),0\bigg{)},\qquad C_{V}(T,\Omega)\simeq
C_{V}\bigg{(}T(1+C_{2}\Omega^{2}),0\bigg{)}.$ (21)

However, for the energy density, the rotational effect causes the curve to
become smoother within the temperature range of 1-3 $T_{c}(0)$, which is
different from the observations made for the entropy density and pressure.
This suggests that the rotational effect may manifest differently across
various thermodynamic quantities. For the trace anomaly, we observe that as
the angular velocity increases, the peak of the curve is suppressed. This
trend may be related to the anisotropy of the system due to rotation.

(a) entropy density

(b) pressure

(c) energy density

(d) trace anomaly

Figure 2:  The entropy density, pressure, energy density and trace anomaly
under rotation. The red dots in the figure are lattice QCD results from Ref.
[100].

(a) sound velocity

(b) specific heat

Figure 3:  The square of the speed of sound and latent heat under rotation.
The red dots in the figure are lattice QCD results from Ref. [100].

##  IV Polyakov loop and Deconfinement phase transition of the pure gluon
system under rotation

In the previous section, we have utilized lattice QCD data to fix all model
parameters and calculated the free energy density and other thermodynamic
quantities under rotation. In this section, using the results from above, we
further explore the Polyakov loop the anisotropic background and plot the
$T-\Omega$ and $T-\mu$ phase diagrams of the deconfinement phase transition.

The Polyakov loop $\langle L\rangle$ is often used as the order parameter for
the deconfinement phase transition. A nonzero $\langle L\rangle$ implies a
deconfinement phase, while the vanishing $\langle L\rangle$ indicates
confinement. In the holographic duality, the Polyakov loop is given by the
Nambu-Goto action of the string. In gauge $\eta_{1}=t$ and $\eta_{2}=z$, the
Nambu-Goto action on the world sheet can be written as

$\displaystyle S_{NG}=\frac{g_{p}}{\pi
T}\int_{0}^{z_{h}}dz\frac{e^{2A_{s}}}{z^{2}}\sqrt{1+f(z)(\vec{x}^{\prime})^{2}},$
(22)

with world sheet coordinate $\eta_{1}$ and $\eta_{2}$. With the on-shell
action, the expectation value of the Polyakov loop is

$\displaystyle\langle L(T)\rangle=e^{C_{p}-S_{0}^{\prime}},$ (23)

where

$\displaystyle S_{0}^{\prime}=\frac{g_{p}}{\pi
T}\int_{0}^{z_{h}}dz(\frac{e^{2A_{s}}}{z^{2}}-\frac{1}{z^{2}}),$ (24)

with normalization constant $C_{p}$ and string tension $g_{p}$. The
expectation value of Polyakov loop for different real and imaginary angular
velocities can be got by the above equations. Here, the constants $C_{p}$ and
$g_{p}$ are fixed relying on lattice QCD data. It should be noted that these
constants are not fixed by the results without rotation. For example, in Ref.
[88], lattice data [101] was used to fix the constants. However, in rotating
gluodynamics, the infinite volume limit cannot be chosen, and therefore the
expectation value of $\langle L\rangle$ in data [101] and date [53] cannot be
compared directly. For this reason, only lattice data [101] have been chosen
to fit the constants, ensuring their values are consistent for both rotation
and non-rotation.

The results of the model calculations and lattice data under real and
imaginary rotation are depicted in Fig. 4. The triangles represent the lattice
QCD data, while the solid lines indicate the results from the model
calculations. The various colors in the diagram correspond to different
angular velocities, whether real or imaginary. It is important to note that
the phase transition in the lattice data is not first order, that comes from
finite size effects. Consequently, the data extracted from the lattice data
contains information about the finite volume. The blue data points at
$\Omega=\Omega_{I}=0$ were used to fit the values of $C_{p}$ and $g_{p}$,
resulting in $(C_{p},g_{p})=(1.04,1.01)$.

As shown in Fig. 4(a), the model results align well with the lattice QCD data
in the vicinity of the phase transition temperature. However, at a temperature
of approximately $1.2T_{c}(0)$, there are some discrepancies between the model
and the lattice data. Within this temperature range, the results from the
model calculations are more dispersed, whereas the lattice data are more
clustered. This could be attributed to the effects of finite size.
Additionally, it is observed that the phase transition temperatures predicted
by the model differ slightly from those of the lattice QCD data. That’s
because the data has some deviation from relation
$T_{c}(\Omega_{I})/T_{c}(0)=1-C_{2}\Omega_{I}^{2}$. For real angular
velocities, the model’s predictions exhibit an inverse trend compared to those
under imaginary angular velocities, as illustrated in Fig. 4(b).

(a) imaginary rotation

(b) real rotation

Figure 4:  The expectation value of the Polyakov loop $\langle L\rangle$
obtained from the DHQCD model for pure gluon system and lattice as a function
of temperature under both real and imaginary rotation. The triangles in the
figure represent the lattice QCD data [101] and the solid lines are the model
calculations.

The $T-\Omega$ and $T-\mu$ phase diagrams of the deconfinement phase
transition can be plotted using the free energy densities obtained in the
previous section, as shown in Fig. 5. In these diagrams, the dashed lines,
solid lines, and points represent crossover, first-order phase transitions,
and CEP, respectively. The pseudo-transition temperature at crossover can be
defined by the second derivative, that is, $\frac{d^{2}L}{dT^{2}}=0$ or
$\frac{d^{2}L}{d\Omega^{2}}=0$.

From Fig. 5, it can be observed that as the angular velocity $\Omega$
increases, the transition temperature $T_{c}$ also rises. Similar to the case
of imaginary angular velocity, the relationship between the temperature
$T_{c}$ and the angular velocity $\Omega$ approximates a quadratic function
$T_{c}(\Omega)/T_{c}(0)=1+C_{2}\Omega^{2}$, which is expected. Specifically,
when the angular velocity increases to 0.1 GeV, the transition temperature is
approximately 2.6 to 3 times higher than that without rotation. Furthermore,
the finite chemical potential can amplify the coefficient $C_{2}$.

As the angular velocity increases, the order of phase transition changes from
crossover to first-order, and a CEP appears in the phase diagram. For
instance, when the angular velocity increases to about 0.015 GeV, the system
undergoes a transition from crossover to first-order phase transition at the
chemical potential of 0.2 GeV. The CEP induced by the angular velocity is
located at $(T_{CEP},\Omega_{CEP})=(0.240,0.015)$ GeV.

Fig. 5 also reveals that the phase lines move towards the high-temperature
region with increasing angular velocity, and the CEPs continuously shift
towards the high chemical potential region. When the angular velocity
increases from 0 to 0.05 GeV, in the $T-\mu$ phase diagram, the CEP moves from
$(T_{CEP},\mu_{CEP})=(0.237,0.18)$ GeV to $(0.324,0.27)$ GeV.

Therefore, it can be concluded that the effect of angular velocity on the
deconfinement phase transition is opposite compared to the chemical potential.
The finite chemical potential decreases the phase transition temperature and
makes the phase from first-order to crossover. With rotation, the phase
transition temperature increases and allows the phase from crossover to first-
order.

Figure 5:  The $T-\Omega$ and $T-\mu$ phase diagrams of deconfinement phase
transition for the pure gluon system.

##  V Chiral phase transition of the 2-flavor system under rotation

In the previous sections, we have explored the phenomenon of first-order
deconfinement phase transitions under rotation. The non-zero gauge field
$A_{\theta}$ distorts the background geometry and causes the critical
temperature of phase transition to increase with the rotation. This effect is
crucial for a comprehensive understanding of the phase diagram of QCD matter.

This section turns to investigate the chiral phase transition. To realize
chiral phase transition in DHQCD model, we introduce 2-flavor probe brane in
the anisotropic background. This model provides insights into the chiral
symmetry breaking and restoration in QCD. Through this approach, we can
analyze the properties of chiral phase transitions affected by the rotation.
Although lattice QCD indicate that in 2-flavor case, the deconfinement phase
transition is a crossover under zero chemical potential. This does not affect
the qualitative conclusions of our study. Therefore, this study not alter the
geometric structure of the background but further explore the chiral phase
transition using the probe brane within the anisotropic background.

The action of the probe brane is shown in Eq. (3). The non-zero vacuum
expectation value $X_{0}=\frac{\chi(z)}{2}I_{2\times 2}$ of the scalar field
$X$ breaks the chiral symmetry $SU(2)_{L}\times SU(2)_{R}$ to subgroup
$SU(2)_{V}$, with identity matrix $I_{2\times 2}$. According to the
holographic principle, the asymptotic expansion of the scalar field $\chi$ at
UV can be written as $\chi\to m_{q}\zeta z+...+\sigma z^{3}/\zeta+..$ with
quark mass $m_{q}$, chiral condensate $\sigma$ and constant $\zeta=\sqrt{3}/2$
[102]. In this paper, the quark mass is fixed to $m_{q}=5/1000$ GeV. As in
Ref. [96], the potential of Eq. (3) is considered as the following form

$\displaystyle
V_{X}(|X|,F_{MN}F^{MN})=(\frac{m_{5}^{2}}{2}+\lambda_{2}F^{2})\chi^{2}+(\upsilon_{4}+\lambda_{4}F^{2})\chi^{4}+(\upsilon_{6}+\lambda_{6}F^{2})\chi^{6},$
(25)

with free parameters $\upsilon_{4}$, $\upsilon_{6}$, $\lambda_{2}$,
$\lambda_{4}$, and $\lambda_{6}$. According to the AdS/CFT correspondence, the
square of the five-dimensional mass of the field $X$ is
$m_{5}^{2}=\Delta(\Delta-4)=-3$ with dimension $\Delta=3$ operator
$\langle\bar{q}q\rangle$. From Eqs. (3) and (25), the equation of motion of
scalar field can be obtained as

$\displaystyle-$
$\displaystyle\frac{8e^{-2(A_{s}+B)}}{f}\chi\Omega^{2}z^{2}\left(\lambda_{2}+\lambda_{4}\chi^{2}+\lambda_{6}\chi^{4}\right)+\frac{2e^{-2A_{s}}}{f}\chi
A_{t}^{\prime
2}z^{2}\left(\lambda_{2}+\lambda_{4}\chi^{2}+\lambda_{6}\chi^{4}\right)$ (26)
$\displaystyle-$
$\displaystyle\frac{e^{2A_{s}}}{fz^{2}}\chi\left(-3+4\upsilon_{4}\chi^{2}+6\upsilon_{6}\chi^{4}\right)+\left(3A_{s}^{\prime}+\frac{f^{\prime}}{f}-\Phi^{\prime}-\frac{3}{z}\right)\chi^{\prime}+\chi^{\prime\prime}=0,$

where the fields of $A_{s}$, $f$, $B$, and $A_{t}$ are solved by the equations
of motion Eqs. (10b-10e) from the gravitational background. The suitable
boundary conditions need to be set for solving Eq. (26). The scalar field
$\chi$ is required to satisfy the expansion $\chi(0)=m_{q}\zeta
z+\mathcal{O}(z)=0$ at UV, and a natural boundary condition at IR. The
parameters in the potential is fixed by the phase diagram of $T-\mu$ plane. As
in Ref. [96], these parameters are chosen as shown in Tab. 2. With the
selection of parameters, the CEP of the chiral phase transition is located in
$(T_{CEP},\mu_{CEP})=(0.265,0.16)$GeV. The temperature of chiral phase
transition is higher than the results of other effective models since the
calculations are performed in the probe approximation at the quenched limit.

Parameters |  $\upsilon_{4}$ |  $\upsilon_{6}$ |  $\lambda_{2}$ |  $\lambda_{4}$ |  $\lambda_{6}$
---|---|---|---|---|---
Values |  -2 |  100 |  -25 |  200 |  $-\frac{1000}{3}$
Table 2:  The parameters of the potential $V_{X}$. They are selected from Ref.
[96].

It should be noted that a similar approach has been used in Ref. [47], except
that the scalar field $X$ describing the chiral condensate is coupled to the
gauge field through the covariant derivative $D_{M}=\partial_{M}-iA_{M}$.
Consequently, the phase transition temperature decreases with increasing
rotation. However, as shown in the Ref. [96], additional terms in the
potential are necessary for the consistency with the QCD phase diagram of
$T-\mu$ plane. Therefore, in contrast to Ref. [47], the covariant derivative
is not considered in this paper and is replaced by the potential
$V_{X}(|X|,F_{MN}F^{MN})$. Moreover, this choice brings another advantage that
the scalar field $\chi$ can be considered as only depends on the fifth
dimensional coordinate $z$, i.e., the near-centre approximation. Otherwise,
the inclusion of covariant derivatives would leave the equation of motion of
scalar field without solutions, except for the trivial one.

As can be seen from Eq. (26), the rotational and chemical potential terms in
the equation are very similar, except for the opposite sign. Therefore, it can
be expected that the modification of chiral condensate by rotation should be
opposite to the chemical potential. That is, when the rotation is increased,
the system transforms from chiral symmetry restoration to chiral symmetry
breaking.

Using the equation of motion Eq. (26), we can explore the chiral condensate as
a function of temperature at a chemical potential of $\mu=0.12$ GeV for
different real and imaginary angular velocities, as summarized in Fig. 6. The
$\sigma_{0}$ shown in the figure represents the value of the maximum
condensate, and $T_{c}(0)$ indicates the critical temperature without
rotation. Additionally, the dashed lines in the figure delineate the
boundaries of metastable and unstable solutions.

Observing Fig. 6 reveals that the chiral condensate undergoes sharp changes in
two distinct temperature regions, corresponding to crossover and first-order
phase transitions. Specifically, only the crossover transition is considered
as chiral phase transition, while the other is driven by the first-order
deconfinement phase transition in the background field, which can be verified
by the transition temperature and the value of condensate. Near the crossover
temperature, the condensate value grows rapidly from near 0 to
$0.9\sigma_{0}$; in the region of the first-order phase transition, the
condensate value jumps from $0.9\sigma_{0}$ to $\sigma_{0}$, with a
$0.1\sigma_{0}$ change. The first-order phase transition occurs within the
temperature of deconfinement phase transition, unlike the crossover. This
phenomenon further confirms that the region of significant change indeed
corresponds to the true chiral phase transition.

From Fig. 6(a), it can be observed that the critical temperature rises with an
increase in angular velocity. Fig. 6(b) illustrates the suppressive effect of
imaginary angular velocity on chiral condensate, which is in agreement with
the findings from lattice QCD [55]. It is worth noting that when the imaginary
angular velocity reaches 0.06 GeV, the value of chiral condensate is quite
close to 0. It can be suggested that the system is maintained in the chiral
symmetry restoration phase in this case.

(a) real rotation

(b) imaginary rotation

Figure 6:  The chiral condensate for 2-flavor system as a function of
temperature at different angular velocities and imaginary angular velocities
with chemical potential $\mu=0.12$ GeV. In the figure $\sigma_{0}$ is the
maximum value of condensation and $T_{c}(0)$ is the critical temperature with
zero angular velocity.

Based on the previous discussion, we can infer that the critical temperature
varies with rotation. By solving Eq. (26), we have obtained the $T-\Omega$ and
$T-\mu$ phase diagrams for chiral phase transition, as shown in Fig. 7. In
these diagrams, the dashed lines represent crossover, the solid lines indicate
first-order phase transition, and the dots mark the location of the CEP. The
temperature for crossover can be determined by the definition of the second
derivative, that is, $\frac{d^{2}\sigma}{dT^{2}}=0$ or
$\frac{d^{2}\sigma}{d\Omega^{2}}=0$.

From Fig. 7, it can be observed that as the angular velocity increases, the
transition temperature also rises. Similar to the deconfinement phase
transition, the relationship between the temperature $T_{c}$ and the angular
velocity $\Omega$ is approximately quadratic
$T_{c}(\Omega)/T_{c}(0)=1+C_{2}\Omega^{2}$. Therefore, when the angular
velocity reaches 0.1 GeV, the transition temperature is about 2.6 to 4.6 times
higher than that without rotation. Moreover, the finite chemical potential
amplifies the coefficient $C_{2}$. As the angular velocity increases, the
phase transition changes from first-order to crossover, and a CEP is induced
in the phase diagram. Specifically, at a chemical potential of 0.2 GeV and a
angular velocity of 0.065 GeV, the system transition from a first-order to
crossover. In this case, the CEP appears at the position
$(T_{CEP},\Omega_{CEP})=(0.468,0.065)$ GeV.

Fig. 7 also exhibits similar behavior, where the phase boundary moves towards
a higher temperature region as the angular velocity increases, and the CEP
continuously moves towards the region of higher chemical potential. When the
angular velocity increases to 0.1 GeV, in the $T-\mu$ phase diagram, the CEP
moves to $(T_{CEP},\mu_{CEP})=(0.732,0.28)$ GeV.

Figure 7:  The $T-\Omega$ and $T-\mu$ phase diagrams of chiral phase
transition for 2-flavor system.

As a consequence, it can be concluded that the effect of angular velocity on
the chiral phase transition is opposite to that of the chemical potential,
similar to the observation in deconfinement phase transition. With finite
chemical potential, the phase transition temperature is decreased and allows
the phase change from crossover to first-order. As the angular velocity
increases, the phase transition temperature increases and the phase crosses
from first-order to crossover.

##  VI Conclusion and discussion

In this paper, we use a new approach to investigate the effect of rotation on
deconfinement and chiral phase transitions in the dynamical holographic QCD
model. Our holographic calculations are consistent with lattice QCD data [53]
and [55].

By utilizing the non-trivial term $A_{\theta}=\Omega r^{2}$ in the gauge field
and the dilaton field influenced by rotation, we have constructed an
anisotropic gravitational background. This background can describe rotating
QCD matter under the near-centre approximation. The new parameter
$\mu_{\Omega}$ in the model is determined by the relationship between the
transition temperature and the imaginary angular velocity $T/T_{c}\sim
1-C_{2}\Omega_{I}^{2}$ from lattice data [53]. Additionally, we also
introduced a 2-flavor probe brane in this background to qualitatively analyze
chiral phase transition under rotation. It is observed that at low chemical
potentials, the deconfinement phase transition of pure gluon system is of
first order and the chiral phase transition of 2-flavor system is of
crossover.

Using the holographic method, we calculated various thermodynamic quantities
$Q$ for different angular velocities. We found that the effect of rotation on
the entropy density, pressure, square of the speed of sound, and latent heat
can be approximated by the relationship
$Q(T,\Omega)=Q(T(1+C_{2}\Omega^{2}),0)$. However, the energy density and trace
anomaly do not follow this pattern. In particular, rotation smooths the energy
density curve within the temperature range of $1-3T_{c}(0)$, and the peak of
the trace anomaly is suppressed as the angular velocity increases.

The deconfinement phase transition in the pure gluon system under rotation is
extracted from the Polyakov loop and the chiral restoration phase transition
is extracted from the chiral condensate in 2-flavor probe brane. In the DHQCD
model, the effect of rotation on the two phase transitions shows consistency.
Both the critical temperatures decrease/increase with imaginary/real angular
velocity ($\Omega_{I}/\Omega$) as $T/T_{c}\sim 1-C_{2}\Omega_{I}^{2}$ and
$T/T_{c}\sim 1+C_{2}\Omega^{2}$, which is consistent with lattice QCD results.

Furthermore, in the temperature-chemical potential $T-\mu$ phase diagrams, the
critical end point (CEP) moves towards regions of higher temperature and
chemical potential with real angular velocity. Specifically, when the angular
velocity increases from 0 to 0.05 GeV, the CEP of the deconfinement phase
transition moves from $(T_{CEP},\mu_{CEP})=(0.237,0.18)$ GeV to $(0.324,0.27)$
GeV, and the CEP of the chiral phase transition moves from
$(T_{CEP},\mu_{CEP})=(0.271,0.14)$ GeV through $(0.387,0.18)$ GeV to
$(0.732,0.28)$ GeV as the angular velocity increases from 0 to 0.1 GeV.

This study differs from previous ones [41] and [47]. The Ref. [41] introduced
rotation through a coordinate transformation and yielded results opposite to
this study. We believe that physical results should not depend on the choice
of coordinate system, so the discrepancy between the two methods needs further
investigation. In the Ref. [47] a different type of coupling is used, i.e. via
covariant derivative $D_{M}=\partial_{M}-iA_{M}$, whereas $V_{X}(X,F^{2})$ is
used in this study. Different couplings can lead to different results. If both
couplings are considered simultaneously, it may result in non-monotonic
outcomes.

In this work, we only considered results under the near-center approximation,
which is a preliminary analysis of rotation effects and does not include
finite volume effect and inhomogeneity. In future work, we will consider the
finite-sized rotating black brane and radial dependence in the DHQCD model to
reveal more information about the rotating quark-gluon plasma and the QCD
phase structure. Additionally, the 2-flavor system is treated as a probe in
the current work. Therefore, the critical temperature of chiral phase
transition deviates much from lattice results. In the future, the full
solution including the back-reaction from the probe action to the background
should be taken into account.

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
(NSFC) Grant Nos: 12305136, 12235016, 12275108, 12221005, and the start-up
funding of Hangzhou Normal University under Grant No. 4245C50223 204075, and
the Strategic Priority Research Program of Chinese Academy of Sciences under
Grant No XDB34030000, and the Fundamental Research Funds for the Central
Universities. Xun Chen is supported by the Research Foundation of Education
Bureau of Hunan Province, China(Grant No. 21B0402) and the Natural Science
Foundation of Hunan Province of China under Grants No.2022JJ40344.

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