# The properties of strange quark matter under strong rotation

Fei Sun1,2<EMAIL_ADDRESS>Anping Huang2,3<EMAIL_ADDRESS>1Department of Physics, China Three Gorges University, Yichang, 443002, China
2Physics Department and Center for Exploration of Energy and Matter, Indiana
University, 2401 N Milo B. Sampson Lane, Bloomington, Indiana 47408, USA
3School of Nuclear Science and Technology, University of Chinese Academy of
Sciences, Beijing 100049, China

###### Abstract

We investigate the rotating quark matter in the three-flavor Nambu and Jona-
Lasinio (NJL) model. The chiral condensation, spin polarization and number
susceptibility of the light and strange quarks are carefully studied at finite
temperature without or with finite chemical potential in this model. We find
that the rotation suppresses the chiral condensation and enhances the first-
order quark spin polarization, however for the second-order quark spin
polarization and quark number susceptibility the effect is complicated and
interesting. When extending to the situation with finite chemical potential,
we find the angular velocity also plays a crucial role, at small angular
velocity the chemical potential enhances the susceptibility, however in the
middle region of angular velocity the effect of the chemical potential is
suppressed by the angular velocity and susceptibility can be changed
considerably, it can be observed that at very low temperature in the presence
of quark chemical potential the quark number susceptibility has two maxima
with increasing angular velocity. Furthermore, it is found that at
sufficiently large angular velocity the contributions played by light quark
and strange quark to these phenomena are almost equal. we also explored the
phase diagram in the $T$-$\omega$ plane, we observe that there exist first
order phase transitions for the rotating system and the first order phase
transition lines move toward a higher temperature for decreasing angular
velocity. It is also found that the different chemical potentials change the
boundary of phase diagram, and that a larger chemical potential shifts down
the critical temperature. We expect these studies to be used to understand the
chiral symmetry breaking and restoration as well as probe the QCD phase
transition.

##  I Introduction

QCD thermodynamics has always been the subject of intense investigations for
many years, which motivates various works to try to understand it better, and
we know many about the equation of state of strongly interacting matter as a
function of temperature $T$ and in a limited range of quark chemical potential
$\mu$. Recently, QCD matter under rotation is of particular interest, there
exist some interesting phenomena in rotating QCD matter, such as chiral
vortical effect or chiral vortical wave D. Kharzeev ; D. T. Son ; D. E.
Kharzeev ; Y. Jiang , which is a key ingredient in theories that predict
observable effects associated with chiral symmetry restoration and the
production of false QCD vacuum states Kharzeev . Many works can be
investigated in various rotation-related phenomena, such as noncentral heavy-
ion collisions in high energy nuclear physics L. P. Csernai ; F. Becattini3 ;
Y. Jiang2 ; S. Shi ; W.T. Deng ; L. G. Pang ; L. Adamczyk ; F. Becattini4 ; X.
L. Xia , the mesonic condensation of isospin matter with rotation in hadron
physics Hui Zhang , the trapped non-relativistic bosonic cold atoms in
condensed matter physics A. L. Fetter ; Urban ; Iskin ; R. Takahashi ; J.
Gooth , the rapidly spinning neutron stars in astrophysics A. Vilenkin1 ; A.
Vilenkin2 ; A. Vilenkin3 ; M. Kaminski ; N. Yamamoto ; E. Shaverin ; A. L.
Watts ; I. A. Grenier ; E. Berti . Quark matter under rotation has been
studied in ultra-relativistic heavy-ion off-central collisions performed at
Relativistic Heavy Ion Collider (RHIC) or Large Hadron Collider (LHC) as well
as lattice simulation. It is known that for the region with very low
temperature and very large chemical potential there exists uncertainty in
lattice QCD due to the ``sign problem", Ref. A. Yamamoto  has calculate the
angular momenta of gluons and quarks in the rotating QCD vacuum, which would
be very important for future theoretical research.

One interesting phenomenon is the quark spin polarization in noncentral
collisions of heavy ions, where quark-gluon plasma (QGP) can be generated. And
the global quark polarization could occur in the QGP due to the large angular
momentum carried by two colliding nuclei, spin-orbit coupling can generate a
spin alignment (polarization) along the direction of the system angular
momentum. This polarization provides very valuable information about the QGP
properties and can be measured experimentally with hyperons via parity-
violating weak decays Z.T. Liang ; Sergei A. Voloshin ; F. Becattini5 ; Z.T.
Liang ; X.G. Huang ; X.G. Huang1 ; F. Becattini ; F. Becattini1 ; F.
Becattini2 ; A. Aristova ; W.T. Deng ; Shu Ebihara . Experimental measurements
of the $\Lambda$ hyperon polarization have been investigated at RHIC and LHC,
which would be very helpful in the study of the hottest, least viscous and
most vortical-fluid ever produced both for theoretical physics and
experimental physics. Recently, the global spin polarizations of $\Lambda$ and
$\overline{\Lambda}$ have been measured by the STAR collaboration in Au+Au
collisions over a wide range of beam energies $\sqrt{s_{NN}}$ = 7.7 - 200 GeV
and by ALICE collaboration in Pb+Pb collisions at 2.76 TeV and 5.02 TeV L.
Adamczyk et al ; J. Adam et al ; S. Acharya et al . On the other hand,
theoretical research of spin polarization in the quark matter has been
explored Y. Tsue1 ; Y. Tsue2 ; Y. Tsue3 ; H. Matsuoka1 ; H. Matsuoka2 ; X. G.
Huang4 ; F. Becattini6 ; Yu Guo , which plays an important role to explain the
origin of strong magnetic field in the magnetar as well as in changing the
dynamical mass and some other phenomenon related to the chiral symmetric phase
transition. It would be very interesting to take into account the influence of
the rotating effect to the quark spin polarization, especially, in the case of
$s$ quark matter under rotation. The study of quark spin polarization which is
linked to the vorticity may help us understand the vortical nature of QGP and
the chiral dynamics of the system.

Another interesting phenomenon in non-central collisions of heavy ions is the
fluctuations and correlations of conserved charges as quantified by the
corresponding susceptibilities, which are sensitive observable quantity in
relativistic heavy-ion collisions and also considered as a useful probe for
QGP S. Jeon and V. Koch ; V. Koch ; A. Bzdak1 ; STAR collaboration1 ; S. Gupta
; A. Bzdak2 ; A. Bazavov et al. ; X.F. Luo1 ; X. Luo ; STAR collaboration2 ;
Shuzhe Shi . In this paper we mainly focus on the baryon number fluctuation
which is also simply related the quark number susceptibility (QNS). QNS serves
as a signature for the QGP formation in ultra relativistic heavy-ion collision
and also plays an important factor to probe that the QCD phase transition as
well as the equation of state (EOS) of strongly interacting matter R. V. Gavai
; R. V. Gavai1 ; S. Gupta10 . Experimentally, various cumulants of net-kaon
multiplicity distributions of Au+Au collisions at $\sqrt{s_{NN}}$ = 7.7 - 200
GeV have been measured by the STAR experiment L. Adamczyk1 ; L. Adamczyk2 ; X.
Luo2 , which are related to the thermodynamic susceptibilities. The study of
QNS in lattice QCD has been interesting R. Gavai and S. Gupta ; S. Borsanyi1 ;
S. Borsanyi2 ; R. Bellwied ; H. T. Ding . Although susceptibilities have been
studied in the past T. Kunihiro ; H. Fujii and M. Ohtani ; Y. Hatta and T.
Ikeda ; B. J. Schaefer and J. Wambach , it has not been checked what is the
influence of considering the contribution from the strange quark matter A. R.
Bodmer ; E. Witten ; C. Alcock ; C. Alcock1 ; J. Madsen ; N. K. Glendenning1 ;
N. K. Glendenning2 ; N. K. Glendenning3 ; H.Terazawa ; H.Terazawa1 ;
H.Terazawa2 ; H.Terazawa3  in the rotation system.

In this paper, encouraged by the successful description of two-flavor QCD
under rotation Y. Jiang1  in the Nambu-Jona-Lasinio (NJL) model, which
embodies the spontaneous breaking of chiral symmetry via effective
interactions between quarks, we further study the three-flavor NJL model in
the framework of quark under rotation. Since there are several flavors and
colors of quarks, several pairings are possible, which probably lead to a
great variety of interesting phenomena. And the questions we are going to
address are how the chiral condensate, quark spin polarization and quark
number susceptibility are influenced by the rotation.

Our work is organized as follows. We first discuss the formalism of three-
flavor NJL model in the presence of rotation in Section II, by using mean-
field approach and the finite temperature field methods we obtain the grand
potential of the fermions with rotation, and the corresponding analytical
expressions for the gap equation, spin polarization and susceptibility of the
quarks are given. Section III presents numerical results and discussions in
detail. Section IV summarizes and concludes the paper. A brief description of
fermions under rotation is given in Appendix A, which discuss the complete set
of commuting operators in cylindrical coordinates and derive the eigenstates
of these operators.

##  II Formalism

We start from the NJL model, the simplest three-flavor NJL Lagrangian for
fermions without rotation is given by M. Buballa

$\displaystyle{\cal
L}=\bar{\psi}\left({i\partial_{\mu}\gamma^{\mu}-m}\right)\psi+G\sum\limits_{a=0}^{8}{\left({\bar{\psi}\lambda^{a}\psi}\right)^{2}}+{\cal
L}_{\rm det},$ (1)

here, $\psi$ is the quark field, $m$ is the bare quark mass matrix,
$\lambda^{a}(a=1,...8)$ are the Gell-Mann matrices in flavor space with
$\lambda^{0}=\sqrt{\frac{2}{3}}\bm{1}$ where $\bm{1}$ is the unit matrix in
the three-flavor space, and ${\cal L}_{\rm det}$ are given by

$\displaystyle{\cal
L}_{det}=-K\\{\rm{det}[\bar{\psi}(1+\gamma^{5})\psi]+\rm{det}[\bar{\psi}(1-\gamma^{5})\psi]\\},$
(2)

which is known as six-quark 't Hooft term, here the $\rm det$ runs over the
flavor degrees of freedom and indicates the favors become connected.

Using the mean field approximation which means that fluctuations are assumed
to be small, the chiral condensates
$\left\langle{\bar{\psi}_{i}\psi_{j}}\right\rangle$ with $i\neq j$ vanishes
since it is assumed that flavor is conserved and after expanding the operators
around their expectation values and neglecting the higher order fluctuations,
we obtain

$\displaystyle\left({\bar{\psi}_{i}\psi_{j}}\right)^{2}=-\left\langle{\bar{\psi}_{i}\psi_{j}}\right\rangle^{2}+2\left\langle{\bar{\psi}_{i}\psi_{j}}\right\rangle\bar{\psi}_{i}\psi_{j},\left({i=j}\right),$
(3)

and the Lagrangian can reads

$\displaystyle\begin{array}[]{l}{\cal
L}=\bar{\psi}\left({i\partial_{\mu}\gamma^{\mu}-M}\right)\psi-2G\left({\left\langle{\bar{u}u}\right\rangle^{2}+\left\langle{\bar{d}d}\right\rangle^{2}+\left\langle{\bar{s}s}\right\rangle^{2}}\right)+4K\left\langle{\bar{u}u}\right\rangle\left\langle{\bar{d}d}\right\rangle\left\langle{\bar{s}s}\right\rangle,\\\
\end{array}$ (5)

where we have defined the dynamical quark mass $M$ as follows

$\displaystyle
M_{f_{i}}=m_{f_{i}}-4G\left\langle{\bar{\psi_{f_{i}}}\psi_{f_{i}}}\right\rangle+2K\left\langle{\bar{\psi_{f_{j}}}\psi_{f_{j}}}\right\rangle\left\langle{\bar{\psi_{f_{k}}}\psi_{f_{k}}}\right\rangle~{}~{}~{}~{}~{}~{}~{}~{}~{}(i\neq
j\neq k),$ (6)

where $i,j,k$ denote the flavor of the quarks and take $u,d$ for two light
flavors while $s$ for strange quark.

The condensation of the QCD matter under the presence of rotation in the
2-flavor NJL model has been investigated in Ref. Y. Jiang1 , which exhibit
interesting behavior for the paring phenomena. In the present work we will
extend to study the properties of the quark matter under rotation in 3-flavor
NJL model with finite quark chemical potential. And the Lagrangian for spinor
with rotation can be written in the following way:

$\displaystyle\begin{array}[]{l}{\cal
L}=\sum\bar{\psi}_{f}\left({i\bar{\gamma}^{\mu}(\partial_{\mu}+\Gamma_{\mu})-m+\gamma^{0}\mu}\right)\psi_{f}+G\sum\limits_{a=0}^{8}{\left({\bar{\psi}\lambda^{a}\psi}\right)^{2}}+{\cal
L}_{det},\\\ \end{array}$ (8)

here $\bar{\gamma}^{\mu}=e_{a}^{\ \mu}\gamma^{a}$ with $e_{a}^{\ \mu}$ the
tetrads for spinors,
$\Gamma_{\mu}=\frac{1}{4}\times\frac{1}{2}[\gamma^{a},\gamma^{b}]\
\Gamma_{ab\mu}$ which is the spinor connection, where
$\Gamma_{ab\mu}=\eta_{ac}(e^{c}_{\ \sigma}G^{\sigma}_{\ \mu\nu}e_{b}^{\
\nu}-e_{b}^{\ \nu}\partial_{\mu}e^{c}_{\ \nu})$, and $G^{\sigma}_{\ \mu\nu}$
is the affine connection determined by $g^{\mu\nu}$. Considering the system
with an angular velocity along the fixed $z$-axis, then
$\vec{v}=\vec{\omega}\times\vec{x}$ and choosing $e^{a}_{\ \mu}=\delta^{a}_{\
\mu}+\delta^{a}_{\ i}\delta^{0}_{\ \mu}\,v_{i}$ and $e_{a}^{\
\mu}=\delta_{a}^{\ \mu}-\delta_{a}^{\ 0}\delta_{i}^{\ \mu}\,v_{i}$ (details
can be found in Ref.), then we can expand the Lagrangian to the first order of
angular velocity, finally, the Lagrangian is given by

$\displaystyle{\cal
L}=\bar{\psi}\left[{i\gamma^{\mu}\partial_{\mu}-m+\gamma^{0}\mu+\left({\gamma^{0}}\right)^{-1}\left({\left({\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{\omega}}\times\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{x}}}\right).\left({-i\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{\partial}}}\right)+\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{\omega}}.\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{S}}_{4\times
4}}\right)}\right]\psi+G\sum\limits_{a=0}^{8}{\left({\bar{\psi}\lambda^{a}\psi}\right)^{2}}+{\cal
L}_{det},$ (9)

where we can see as a result of rotation the Dirac operator includes the
orbit-rotation coupling term and the spin-rotation coupling term.

The general definition of the partition function can be written as

$\displaystyle{\cal Z}=\int{D[\bar{\psi}]}D[\psi]e^{iS},$ (10)

here, $S$ denotes the quark action, which is the integration of the Lagrangian
density $\mathcal{L}$. After by using the mean field approximation and
carrying out the general approach of the path integral formulation for
Grassmann variables, we are now able to exactly integrate out the fermionic
fields and obtain

$\displaystyle\begin{array}[]{l}\log{\cal
Z}=-\frac{1}{T}\int{d^{3}x}\left(-2G\left({\left\langle{\bar{u}u}\right\rangle^{2}+\left\langle{\bar{d}d}\right\rangle^{2}+\left\langle{\bar{s}s}\right\rangle^{2}}\right)+4K\left\langle{\bar{u}u}\right\rangle\left\langle{\bar{d}d}\right\rangle\left\langle{\bar{s}s}\right\rangle\right)+\sum\limits_{f}\log\det\frac{{D_{f}^{-1}}}{T}\\\
\end{array},$ (12)

here

$\displaystyle
D^{-1}=\gamma^{0}\left({-i{\omega_{l}}+\left({n+\frac{1}{2}}\right)\omega+\mu}\right)-M-\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{\gamma}}.\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{p}},$ (13)

which is the inverse of propagator for quarks, and

$\displaystyle\log\det\frac{{\hat{D}^{-1}}}{T}=tr\log\frac{{\hat{D}^{-1}}}{T}=\int{d^{3}x}\int{\frac{{d^{3}p}}{{\left({2\pi}\right)^{3}}}}\left\langle{\psi_{p}\left(x\right)\left|{\log\hat{D}^{-1}}\right|\psi_{p}\left(x\right)}\right\rangle.$
(14)

The Dirac fields can be defined in the terms of the wave functions $u(x),v(x)$

$\displaystyle\psi_{p}\left(x\right)=\sum\limits_{E,n,s,p}{\left({u\left(x\right)+v\left(x\right)}\right)}.$
(15)

In order to find solutions of the Dirac equation, we should firstly choose the
complete set of commutating operators consists of $\hat{H}$ which can be
obtained from three-flavor NJL Lagrangian, the momentum in the $z$-direction
$\hat{p}_{z}$, the square of transverse momentum $\hat{p}_{t}^{2}$, the
$z$-component of the total angular momentum $\hat{J}_{z}$ and the transverse
helicity $\hat{h}_{t}$. The positive and negative energy solutions of the
Dirac field can be determined by calculating these eigenvalue equations of the
complete set of commutating operators {$\hat{H}$, $\hat{p}_{z}$,
$\hat{p}_{t}^{2}$,$\hat{J}_{z}$, $\hat{h}_{t}$}, here, for the derivation and
detailed expression of the spinor $u,v$ can refer to the Appendix A, by
substituting Eq. (15) to Eq. (14) we have

$\displaystyle\log\det\frac{{\hat{D}^{-1}}}{T}=\sum\limits_{E,n,s,p}{tr\log\frac{{D_{u\left(x\right)}^{-1}}}{T}\int{d^{3}x}}\int{\frac{{d^{3}p}}{{\left({2\pi}\right)^{3}}}}\left({\left\langle{u\left(x\right)\left|{u\left(x\right)}\right.}\right\rangle}\right)$
(16) $\displaystyle{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
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1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern
1.0pt}+\sum\limits_{E,n,s,p}{tr\log\frac{{D_{v\left(x\right)}^{-1}}}{T}\int{d^{3}x}}\int{\frac{{d^{3}p}}{{\left({2\pi}\right)^{3}}}}\left({\left\langle{v\left(x\right)\left|{v\left(x\right)}\right.}\right\rangle}\right),$
(17)

here, the concrete form of the ${D_{u}^{-1}}$ that has been considered the
rotation and nonzero chemical potential is

$\displaystyle
D_{u\left(x\right)}^{-1}=\left({\begin{array}[]{*{20}c}{\left({-i\omega_{l}+\left({n+\frac{1}{2}}\right)\omega+\mu}\right)-M}&{-\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{\sigma}}.\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{p}}}\\\
{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{\sigma}}.\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{p}}}&{-\left({-i\omega_{l}+\left({n+\frac{1}{2}}\right)\omega+\mu}\right)-M}\\\
\end{array}}\right),$ (20)

which corresponding to the positive energy solution, and the concrete form for
the ${D_{v}^{-1}}$ is

$\displaystyle
D_{v\left(x\right)}^{-1}=\left({\begin{array}[]{*{20}c}{\left({i\omega_{l}-\left({n+\frac{1}{2}}\right)\omega+\mu}\right)-M}&{-\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{\sigma}}.\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{p}}}\\\
{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{\sigma}}.\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{p}}}&{-\left({i\omega_{l}-\left({n+\frac{1}{2}}\right)\omega+\mu}\right)-M}\\\
\end{array}}\right),$ (23)

which corresponding to the negative energy solution. Here, in order to study
the rotating system at finite density, we have introduced quark chemical
potential $\mu$ and note that the term $\left({n+\frac{1}{2}}\right)\omega$ in
above expressions denotes the rotational polarization energy, which are very
useful when we study the polarization in the following sections. By using the
general methods in the finite temperature fields J. I. Kapusta , we obtain

$\displaystyle\begin{array}[]{l}\log\det\frac{{D_{u}^{-1}}}{T}=\beta\left({\sqrt{M^{2}+{\rm{p}}_{t}^{2}+{\rm{p}}_{z}^{2}}+\left({n+\frac{1}{2}}\right)\omega}\right)\\\
\\\ {\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
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1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}+\log\left({e^{\beta\left({\sqrt{M^{2}+{\rm{p}}_{t}^{2}+{\rm{p}}_{z}^{2}}+\left({\left({n+\frac{1}{2}}\right)\omega-\mu}\right)}\right)}+1}\right)\\\
\\\ {\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}+\log\left({e^{\beta\left({\sqrt{M^{2}+{\rm{p}}_{t}^{2}+{\rm{p}}_{z}^{2}}+\left({\left({n+\frac{1}{2}}\right)\omega+\mu}\right)}\right)}+1}\right),\\\
\end{array}$ (29)

and

$\displaystyle\begin{array}[]{l}\log\det\frac{{D_{v}^{-1}}}{T}=\beta\left({-\sqrt{M^{2}+{\rm{p}}_{t}^{2}+{\rm{p}}_{z}^{2}}-\left({n+\frac{1}{2}}\right)\omega}\right)\\\
\\\ {\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
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1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}+\log\left({e^{\beta\left({-\sqrt{M^{2}+{\rm{p}}_{t}^{2}+{\rm{p}}_{z}^{2}}+\left({-\left({n+\frac{1}{2}}\right)\omega-\mu}\right)}\right)}+1}\right)\\\
\\\ {\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
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1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}+\log\left({e^{\beta\left({-\sqrt{M^{2}+{\rm{p}}_{t}^{2}+{\rm{p}}_{z}^{2}}+\left({-\left({n+\frac{1}{2}}\right)\omega+\mu}\right)}\right)}+1}\right),\\\
\end{array}$ (35)

for each flavor, here $\beta$ is the inverse temperature and the inner
products of $\left\langle{u_{n,s}\left|{u_{n,s}}\right.}\right\rangle$,
$\left\langle{v_{n,s}\left|{v_{n,s}}\right.}\right\rangle$ are derived with
very simple expressions as follows,

$\displaystyle\int{\frac{{d^{3}p}}{{\left({2\pi}\right)^{3}}}}\left\langle{u_{n,s}\left|{u_{n,s}}\right.}\right\rangle=\frac{1}{2}\left({J_{n}\left({p_{t}r}\right)^{2}+J_{n+1}\left({p_{t}r}\right)^{2}}\right),$
(36)
$\displaystyle\int{\frac{{d^{3}p}}{{\left({2\pi}\right)^{3}}}}\left\langle{v_{n,s}\left|{v_{n,s}}\right.}\right\rangle=\frac{1}{2}\left({J_{n}\left({p_{t}r}\right)^{2}+J_{n+1}\left({p_{t}r}\right)^{2}}\right).$
(37)

Combining the Eqs. (12), (29), (35), (36) and (37) one could now derive the
expression of the grand potential for strange quark when the momentum
summation turns into the integration

$\displaystyle\sum\limits_{p}{\to
V\int{\frac{{d^{3}p}}{{\left({2\pi}\right)^{3}}}}},$ (38)

and the energy summation performs over Matsubara frequency. Then the
thermodynamic grand potential $\Omega=-\frac{T}{V}\log{\cal Z}$ has the
following expression,

$\displaystyle\begin{array}[]{l}\Omega{\rm{=}}2G\left({2{{\left\langle{\bar{u}u}\right\rangle}^{2}}+{{\left\langle{\bar{s}s}\right\rangle}^{2}}}\right)-4K{\left\langle{\bar{u}u}\right\rangle^{2}}\left\langle{\bar{s}s}\right\rangle\\\
\\\
-\frac{{{N_{f}}{N_{c}}}}{{8{\pi^{2}}}}\sum\limits_{n=-\infty}^{\infty}{\int_{0}^{\Lambda}{{{\rm{p}}_{t}}d{{\rm{p}}_{t}}\int_{-\sqrt{{\Lambda^{2}}-p_{t}^{2}}}^{\sqrt{{\Lambda^{2}}-p_{t}^{2}}}{d{p_{z}}}}\left({\left({{J_{n+1}}{{({{\rm{p}}_{t}}r)}^{2}}+{J_{n}}{{({{\rm{p}}_{t}}r)}^{2}}}\right)}\right.}\times\\\
\\\
T\left\\{{\log\left({{e^{-\frac{{-{\mu_{u}}+\sqrt{M_{u}^{2}+{\rm{p}}_{t}^{2}+{\rm{p}}_{z}^{2}}-\left({n+\frac{1}{2}}\right)\omega}}{T}}}+1}\right)+\log\left({{e^{\frac{{-{\mu_{u}}+\sqrt{M_{u}^{2}+{\rm{p}}_{t}^{2}+{\rm{p}}_{z}^{2}}-\left({n+\frac{1}{2}}\right)\omega}}{T}}}+1}\right)}\right.\\\
\\\
\left.{+\log\left({{e^{-\frac{{{\mu_{u}}+\sqrt{M_{u}^{2}+{\rm{p}}_{t}^{2}+{\rm{p}}_{z}^{2}}-\left({n+\frac{1}{2}}\right)\omega}}{T}}}+1}\right)+\log\left({{e^{\frac{{{\mu_{u}}+\sqrt{M_{u}^{2}+{\rm{p}}_{t}^{2}+{\rm{p}}_{z}^{2}}-\left({n+\frac{1}{2}}\right)\omega}}{T}}}+1}\right)}\right\\}\\\
\\\
-\frac{{{N_{c}}}}{{8{\pi^{2}}}}\sum\limits_{n=-\infty}^{\infty}{\int_{0}^{\Lambda}{{{\rm{p}}_{t}}d{{\rm{p}}_{t}}\int_{-\sqrt{{\Lambda^{2}}-p_{t}^{2}}}^{\sqrt{{\Lambda^{2}}-p_{t}^{2}}}{d{p_{z}}}}\left({\left({{J_{n+1}}{{({{\rm{p}}_{t}}r)}^{2}}+{J_{n}}{{({{\rm{p}}_{t}}r)}^{2}}}\right)}\right.}\times\\\
\\\
T\left\\{{\log\left({{e^{-\frac{{-{\mu_{s}}+\sqrt{M_{s}^{2}+{\rm{p}}_{t}^{2}+{\rm{p}}_{z}^{2}}-\left({n+\frac{1}{2}}\right)\omega}}{T}}}+1}\right)+\log\left({{e^{\frac{{-{\mu_{s}}+\sqrt{M_{s}^{2}+{\rm{p}}_{t}^{2}+{\rm{p}}_{z}^{2}}-\left({n+\frac{1}{2}}\right)\omega}}{T}}}+1}\right)}\right.\\\
\\\
\left.{+\log\left({{e^{-\frac{{{\mu_{s}}+\sqrt{M_{s}^{2}+{\rm{p}}_{t}^{2}+{\rm{p}}_{z}^{2}}-\left({n+\frac{1}{2}}\right)\omega}}{T}}}+1}\right)+\log\left({{e^{\frac{{{\mu_{s}}+\sqrt{M_{s}^{2}+{\rm{p}}_{t}^{2}+{\rm{p}}_{z}^{2}}-\left({n+\frac{1}{2}}\right)\omega}}{T}}}+1}\right)}\right\\}.\end{array}$
(52)

Here, the isospin symmetry has been considered for the light quarks, so
$m_{d}=m_{u},\mu_{d}=\mu_{u}$, $N_{f}=2,N_{c}=3$ and $\Lambda$ is the three-
momentum. And $\mu_{u}$ is the chemical potential for the $u$ or $d$ quark as
well as $\mu_{s}$ is that for $s$ quark. When the isospin symmetry is
satisfied, the dynamical quark masses are simplified to give:

$\displaystyle{M_{u}}={m_{u}}+\left({2K\left\langle{\bar{s}s}\right\rangle-4G}\right)\left\langle{\bar{u}u}\right\rangle,$
(53)
$\displaystyle{M_{s}}={m_{s}}-4G\left\langle{\bar{s}s}\right\rangle+2K{\left\langle{\bar{u}u}\right\rangle^{2}}\;.$
(54)

We have discussed the evaluating of grand potential of quark under rotation in
detail in the previous section. In this section, we list our final analytical
expressions of the gap equation, quark spin polarization and quark number
susceptibility in the situation with rotation. Firstly, we consider the gap
equation which will be required to minimize the grand potential, the values
are determined by solving the stationary condition, namely,
$\frac{{\partial\Omega}}{{\partial\left\langle{\bar{u}u}\right\rangle}}=\frac{{\partial\Omega}}{{\partial\left\langle{\bar{s}s}\right\rangle}}=0$,
and the detailed expressions for the stationary condition are listed as
follows,

$\displaystyle\begin{array}[]{l}8G\left\langle{\bar{u}u}\right\rangle-8K\left\langle{\bar{u}u}\right\rangle\left\langle{\bar{s}s}\right\rangle-\frac{{{N_{c}}}}{{8{\pi^{2}}}}\sum\limits_{n=-\infty}^{\infty}{\int_{0}^{\Lambda}{{{\rm{p}}_{t}}d{{\rm{p}}_{t}}\int_{-\sqrt{{\Lambda^{2}}-p_{t}^{2}}}^{\sqrt{{\Lambda^{2}}-p_{t}^{2}}}{d{p_{z}}}}\left({\left({{J_{n+1}}{{({{\rm{p}}_{t}}r)}^{2}}+{J_{n}}{{({{\rm{p}}_{t}}r)}^{2}}}\right)}\right.}\times\\\
\\\
\left\\{{\frac{{4{N_{f}}{e^{\frac{{{\mu_{u}}}}{T}}}\left({{e^{\frac{{2n\omega+\omega}}{T}}}-{e^{\frac{{2\sqrt{M_{u}^{2}+p_{t}^{2}+p_{z}^{2}}}}{T}}}}\right)\left({2G-K\left\langle{\bar{s}s}\right\rangle}\right){M_{u}}}}{{\sqrt{M_{u}^{2}+p_{t}^{2}+p_{z}^{2}}\left({{e^{\frac{{\sqrt{M_{u}^{2}+p_{t}^{2}+p_{z}^{2}}}}{T}}}+{e^{\frac{{2n\omega+2{\mu_{u}}+\omega}}{{2T}}}}}\right)\left({{e^{\frac{{\sqrt{M_{u}^{2}+p_{t}^{2}+p_{z}^{2}}+{\mu_{u}}}}{T}}}+{e^{\frac{{\left({n+\frac{1}{2}}\right)\omega}}{T}}}}\right)}}}\right.\\\
\\\
\left.{-\frac{{8K{e^{\frac{{{\mu_{s}}}}{T}}}\left({{e^{\frac{{2n\omega+\omega}}{T}}}-{e^{\frac{{2\sqrt{M_{s}^{2}+p_{t}^{2}+p_{z}^{2}}}}{T}}}}\right)\left\langle{\bar{u}u}\right\rangle{M_{s}}}}{{\sqrt{M_{s}^{2}+p_{t}^{2}+p_{z}^{2}}\left({{e^{\frac{{\sqrt{M_{s}^{2}+p_{t}^{2}+p_{z}^{2}}}}{T}}}+{e^{\frac{{2n\omega+2{\mu_{s}}+\omega}}{{2T}}}}}\right)\left({{e^{\frac{{\sqrt{M_{s}^{2}+p_{t}^{2}+p_{z}^{2}}+{\mu_{s}}}}{T}}}+{e^{\frac{{\left({n+\frac{1}{2}}\right)\omega}}{T}}}}\right)}}}\right\\}=0,\end{array}$
(60)

$\displaystyle\begin{array}[]{l}4G\left\langle{\bar{s}s}\right\rangle-4K{\left\langle{\bar{u}u}\right\rangle^{2}}-\frac{{{N_{c}}}}{{8{\pi^{2}}}}\sum\limits_{n=-\infty}^{\infty}{\int_{0}^{\Lambda}{{{\rm{p}}_{t}}d{{\rm{p}}_{t}}\int_{-\sqrt{{\Lambda^{2}}-p_{t}^{2}}}^{\sqrt{{\Lambda^{2}}-p_{t}^{2}}}{d{p_{z}}}}\left({\left({{J_{n+1}}{{({{\rm{p}}_{t}}r)}^{2}}+{J_{n}}{{({{\rm{p}}_{t}}r)}^{2}}}\right)}\right.}\times\\\
\\\
\left\\{{-\frac{{4K{N_{f}}{e^{\frac{{{\mu_{u}}}}{T}}}\left({{e^{\frac{{2n\omega+\omega}}{T}}}-{e^{\frac{{2\sqrt{M_{u}^{2}+p_{t}^{2}+p_{z}^{2}}}}{T}}}}\right)\left\langle{\bar{u}u}\right\rangle{M_{u}}}}{{\sqrt{M_{u}^{2}+p_{t}^{2}+p_{z}^{2}}\left({{e^{\frac{{\sqrt{M_{u}^{2}+p_{t}^{2}+p_{z}^{2}}}}{T}}}+{e^{\frac{{2n\omega+2{\mu_{u}}+\omega}}{{2T}}}}}\right)\left({{e^{\frac{{\sqrt{M_{u}^{2}+p_{t}^{2}+p_{z}^{2}}+{\mu_{u}}}}{T}}}+{e^{\frac{{\left({n+\frac{1}{2}}\right)\omega}}{T}}}}\right)}}}\right.\\\
\\\
\left.{\frac{{8G{e^{\frac{{{\mu_{s}}}}{T}}}\left({{e^{\frac{{2n\omega+\omega}}{T}}}-{e^{\frac{{2\sqrt{M_{s}^{2}+p_{t}^{2}+p_{z}^{2}}}}{T}}}}\right){M_{s}}}}{{\sqrt{M_{s}^{2}+p_{t}^{2}+p_{z}^{2}}\left({{e^{\frac{{\sqrt{M_{s}^{2}+p_{t}^{2}+p_{z}^{2}}}}{T}}}+{e^{\frac{{2n\omega+2{\mu_{s}}+\omega}}{{2T}}}}}\right)\left({{e^{\frac{{\sqrt{M_{s}^{2}+p_{t}^{2}+p_{z}^{2}}+{\mu_{s}}}}{T}}}+{e^{\frac{{\left({n+\frac{1}{2}}\right)\omega}}{T}}}}\right)}}}\right\\}=0.\end{array}$
(66)

Here we are going to study the quark spin polarization which can be defined as
taking the partial derivative of minus grand potential with respect to angular
velocity, and we introduce the following quark spin polarization as in Ref. F.
Becattinia

$\displaystyle\mathcal{P}_{1}=\frac{\partial(\frac{-\Omega}{T^{4}})}{\partial(\frac{\omega}{T})},$
(67)
$\displaystyle\mathcal{P}_{2}=\frac{\partial^{2}(\frac{-\Omega}{T^{4}})}{\partial(\frac{\omega}{T})^{2}},$
(68)

such definition ensures dimensionless polarization, then we list the detailed
expression of the first-order polarization and second-order polarization as
follows,

$\displaystyle\begin{array}[]{l}{{\cal
P}_{1}}=\frac{{{N_{c}}}}{{8{\pi^{2}}{T^{3}}}}\sum\limits_{n=-\infty}^{\infty}{\int_{0}^{\Lambda}{{{\rm{p}}_{t}}d{{\rm{p}}_{t}}\int_{-\sqrt{{\Lambda^{2}}-p_{t}^{2}}}^{\sqrt{{\Lambda^{2}}-p_{t}^{2}}}{d{p_{z}}}}\left({\left({{J_{n+1}}{{({{\rm{p}}_{t}}r)}^{2}}+{J_{n}}{{({{\rm{p}}_{t}}r)}^{2}}}\right)}\right.}(2n+1)\times\\\
\\\
\left\\{{\frac{{{N_{f}}\sinh\left({\frac{{-2\sqrt{M_{u}^{2}+{\rm{p}}_{t}^{2}+{\rm{p}}_{z}^{2}}+2n\omega+\omega}}{{2T}}}\right)}}{{\left({\cosh\left({\frac{{-2\sqrt{M_{u}^{2}+{\rm{p}}_{t}^{2}+{\rm{p}}_{z}^{2}}+2n\omega+\omega}}{{2T}}}\right)+\cosh\left({\frac{{{\mu_{u}}}}{T}}\right)}\right)}}+\frac{{\sinh\left({\frac{{-2\sqrt{M_{s}^{2}+{\rm{p}}_{t}^{2}+{\rm{p}}_{z}^{2}}+2n\omega+\omega}}{{2T}}}\right)}}{{\left({\cosh\left({\frac{{-2\sqrt{M_{s}^{2}+{\rm{p}}_{t}^{2}+{\rm{p}}_{z}^{2}}+2n\omega+\omega}}{{2T}}}\right)+\cosh\left({\frac{{{\mu_{s}}}}{T}}\right)}\right)}}}\right\\},\end{array}$
(72) $\displaystyle\begin{array}[]{l}{{\cal
P}_{2}}=\frac{{{N_{c}}}}{{64{\pi^{2}}{T^{3}}}}\sum\limits_{n=-\infty}^{\infty}{\int_{0}^{\Lambda}{{{\rm{p}}_{t}}d{{\rm{p}}_{t}}\int_{-\sqrt{{\Lambda^{2}}-p_{t}^{2}}}^{\sqrt{{\Lambda^{2}}-p_{t}^{2}}}{d{p_{z}}}}\left({\left({{J_{n+1}}{{({{\rm{p}}_{t}}r)}^{2}}+{J_{n}}{{({{\rm{p}}_{t}}r)}^{2}}}\right)}\right.}{(2n+1)^{2}}\times\\\
\\\
\left\\{{{N_{f}}\left({{\rm{sec}}{{\rm{h}}^{2}}\left({\frac{{2{\mu_{u}}-2\sqrt{M_{u}^{2}+{\rm{p}}{{\rm{t}}^{2}}+{\rm{p}}{{\rm{z}}^{2}}}+2n\omega+\omega}}{{4T}}}\right)+{\rm{sec}}{{\rm{h}}^{2}}\left({\frac{{-2{\mu_{u}}-2\sqrt{M_{u}^{2}+{\rm{p}}{{\rm{t}}^{2}}+{\rm{p}}{{\rm{z}}^{2}}}+2n\omega+\omega}}{{4T}}}\right)}\right)}\right.\\\
\\\
\left.{+\left({{\rm{sec}}{{\rm{h}}^{2}}\left({\frac{{2{\mu_{s}}-2\sqrt{M_{s}^{2}+{\rm{p}}{{\rm{t}}^{2}}+{\rm{p}}{{\rm{z}}^{2}}}+2n\omega+\omega}}{{4T}}}\right)+{\rm{sec}}{{\rm{h}}^{2}}\left({\frac{{-2{\mu_{s}}-2\sqrt{M_{s}^{2}+{\rm{p}}{{\rm{t}}^{2}}+{\rm{p}}{{\rm{z}}^{2}}}+2n\omega+\omega}}{{4T}}}\right)}\right)}\right\\}.\end{array}$
(78)

Next, we will investigate how much the rotation affects the baryon number
fluctuations, these fluctuations can be quantified by the susceptibilities and
the QNS is defined through the Taylor expansion coefficients of the pressure
over the chemical potential R.V. Gavai and S. Gupta1 ; R.V. Gavai and S.
Gupta2 ; R.V. Gavai and S. Gupta3 ; C.R. Allton et al. ; M. Cheng et al. ; S.
Bors anyi et al. ; HotQCD collaboration :

$\displaystyle\chi_{n}=\frac{\partial^{n}(\frac{P}{T^{4}})}{\partial(\frac{\mu}{T})^{n}},$
(79)

here we focus on the second order derivative of pressure with respect to $\mu$
, due to symmetry all the odd susceptibilities vanishes when $\mu\rightarrow
0$ (note, in the context of lattice calculations the susceptibilities are
often defined as dimensionless quantities), using the relation of pressure
$P=-\Omega$, the actual calculation of the susceptibilities is straightforward
and the detailed result is

$\displaystyle\begin{array}[]{l}\chi_{2}^{f}=\frac{{{N_{c}}}}{{16{\pi^{2}}{T^{3}}}}\sum\limits_{n=-\infty}^{\infty}{\int_{0}^{\Lambda}{{{\rm{p}}_{t}}d{{\rm{p}}_{t}}\int_{-\sqrt{{\Lambda^{2}}-p_{t}^{2}}}^{\sqrt{{\Lambda^{2}}-p_{t}^{2}}}{d{p_{z}}}}\left({\left({{J_{n+1}}{{({{\rm{p}}_{t}}r)}^{2}}+{J_{n}}{{({{\rm{p}}_{t}}r)}^{2}}}\right)}\right.}\times\\\
\\\
\left({{\rm{sec}}{{\rm{h}}^{2}}\left({\frac{{2{\mu_{f}}-2\sqrt{M_{f}^{2}+{\rm{p}}{{\rm{t}}^{2}}+{\rm{p}}{{\rm{z}}^{2}}}+2n\omega+\omega}}{{4T}}}\right)+{\rm{sec}}{{\rm{h}}^{2}}\left({\frac{{-2{\mu_{f}}-2\sqrt{M_{f}^{2}+{\rm{p}}{{\rm{t}}^{2}}+{\rm{p}}{{\rm{z}}^{2}}}+2n\omega+\omega}}{{4T}}}\right)}\right),\end{array}$
(83)

here $f=u,d,s$. With the analytical expressions given above, we show our
numerical results in the next section.

##  III Numerical results and discussions

In this section we will present our numerical results for the gap equation,
quark spin polarization and quark number susceptibility. In our previous
analytic expressions, the $z$-angular-momentum quantum number $n=0,\pm 1,\pm
2...$ in principle we should sum all the values of $n$, fortunately, these
expressions converge so fast that it is enough for us to sum $n$ from $-5$ to
$5$, it should be noted that in order that the causality of a rigidly rotating
system is maintained we should make sure that the local velocity is smaller
than the light velocity, namely, the condition $\omega r<1$ should be
considered in all the calculations, and for simplicity, we take the same value
of $r$ as in Ref. Y. Jiang1 . Since any uniformly rotating system should be
spatially bounded, it has been expected that the presence of boundaries can
modify the properties of the rotating system H. L. Chen ; S. Ebihara ; M. N.
Chernodub1 ; M. N. Chernodub2 ; M. N. Chernodub3 , indeed, this is only true
when the angular velocity $\omega$ is much smaller than the inverse of the
system's size Xinyang Wang  as well as our discussion is mainly devoted to the
bulk properties of rotating system, so in our analytic derivation we ignore
the finite volume boundary effect and we leave it as our further study. In our
calculations, for simplicity, we have let the chemical potential
$\mu_{u/d}=\mu_{s}=\mu$, and the input parameters in the NJL are the coupling
constants $G$, $K$, the quark masses $m_{u/d}$, $m_{s}$ and the three-momentum
cutoff $\Lambda$. Then, in this context we use the second case in Table 1. of
Ref. H. Kohyama , that
$m_{u}=m_{d}=0.005~{}{\rm{GeV}},~{}m_{s}=0.1283~{}{\rm{GeV}},~{}G=3.672~{}{\kern
1.0pt}{\rm{GeV}}^{-2},~{}K=59.628~{}{\kern
1.0pt}{\rm{GeV}}^{-5},~{}\Lambda=0.6816~{}{\rm{GeV}}$, which have been
estimated by the fitting in light of the following observations: $m_{\pi}=138$
MeV, $f_{\pi}=92$ MeV, $m_{K}=495$ MeV and $m_{\eta^{\prime}}=958$ MeV.

Let us first discuss the results at chemical potential equals zero. We
investigate the chiral condensation in rotating matter under the three-flavor
NJL model, especially we consider the contribution from $s$ quark. It is
equivalent to study the gap equations since the gap equations correspond to
the coupling of the quarks naked masses with the associated chiral
condensates, firstly, in the plot of Fig. 1, we show the differences of
squared gap masses between the case at nonzero angular velocity to those at
zero angular velocity, $M^{2}(\omega)-M^{2}(\omega=0)$ with zero temperature
and zero chemical potential for the up quark and strange quark, respectively,
we found that the squared gap mass differences of all quarks decrease with
increasing angular velocity, and at large angular velocity there is a sudden
drop down for the squared gap mass differences, it obvious that the squared
gap mass differences of lighter quark is more affected by the angular
velocity, which decreases faster while that of strange quark decreases slower
with angular velocity, this can be interpreted as that the quark with large
current mass is less affected by the rotation. Then we plot the gap equation
as a function of $\omega$ with different values of $T$ in Fig. 2 respectively
for $u$ and $s$ quark. As one can see, both $u$ quark and $s$ quark gap
equations show that the rotation has a suppression effect for the chiral
condensation. It is clearly seen that at all temperatures the gap equations of
both $u$ and $s$ decrease with increasing $\omega$ and at very low temperature
the chiral condensate experiences a first-order transition when $\omega$
exceeds a certain value. It is very interesting when comparing the different
flavor situation, in the left panel of Fig. 2 shows that ${M_{u}}$ experiences
a first-order transition around $\omega=0.6$ GeV at $T=0.01$ GeV, in the right
panel of Fig. 2 we observe that ${M_{s}}$ also experiences a sharp change
around $\omega=0.6$ GeV but changes not so much their mass compared to the
light quark situation, due to the coupling between the different flavors. In
addition, we also found that for the strange quark there is a fast drop from a
high value for the quark effective mass to a small value around $\omega=1.0$
GeV. From the figure we can see the role of the $\omega$ as well as $T$ are
very important parameters for crossover or first-order transition. For the
high temperatures the chiral condensate vanishes with the increasing $\omega$
via a smooth crossover, and the temperature effect becomes weak with
increasing the value of $\omega$. As $\omega$ further increases the gap
equations for the both decrease more slowly, and both approach their naked
masses. This can be interpreted as a hint that the chiral phase transitions
for the $s,u$ do not occur at the same angular velocity with the same
temperature. It is found that the chiral condensation of $u$ quark has
produced results almost in agreement with those suggested in literature Y.
Jiang , but with a slight difference due to they adopted the parameters in
Ref. S. P. Klevansky .

Figure 1: (Color online) Differences of squared gap masses between the case at
$\omega\neq 0$ and $\omega=0$ with both $\mu=0$ and $T=0$ for up quark and
strange quark as a function of $\omega$. Figure 2: (Color online) The mean
field mass gap $M_{u}$ and $M_{s}$ as a function of $\omega$ for several
values of $T$ with $\mu=0$. Figure 3: (Color online) The mean field mass gap
$M_{u}$ and $M_{s}$ as a function of $T$ for several values of $r$ and
$\omega$ with $\mu=0$.

We now turn to a more realistic physical environment, in Fig. 3 we plot the
gap equation as a function of $T$ with fixed angular velocity $\omega$ and
radius $r$ for $u$ and $s$ quark, respectively. In order to enable a more
realistic comparison to experimental data in the future, here without losing
generality, we assume that $\omega=0.01,0.1$ GeV and $r=1,1.6$ fm. From the
figure we can see clearly that there exist some interesting behaviors in the
range of $T=0.1\sim 0.4$ GeV. The left hand side of the Fig. 3 shows that the
$u$ quark gap mass decreases with increasing temperature, the mass gap $M_{u}$
falls sharply in the range of $T=0.15\sim 0.2$ GeV, however, it is a smooth
behavior which means that at low $T$, small $\omega$ and large radius the
quark condensate experiences a fast crossover transition. Therefore, in this
region ($T=0.15\sim 0.2$ GeV), there occurs not a true phase transition with
corresponding critical temperature, but rather a pseudo-phase transition
(cross-over). The right hand side of the Fig. 3 shows that for the chosen
parameters the $s$ quark gap mass decreases with increasing temperature via a
crossover transition and the chiral condensate gradually vanishes with
increasing temperature. From the Fig. 3 one can also see clearly that, for a
fixed value of $r$ the effect of rotation suppresses the effective quark mass
and for a fixed value of small $\omega$ we can observe that the effective
quark mass becomes smaller at larger radius. Here we just want to capture the
essential physics of QCD matter under rotation, indeed, in realistic physical
environment requires a more detailed investigation.

Figure 4: (Color online) Study of the first-order spin polarizations of the
rotating system according to $\omega$ for several temperature $T$ with
$\mu=0$, here the result of $T=0.01$ GeV is divided by 100.

The first-order spin polarizations of the rotating system as a function of
$\omega$ for several temperatures with chemical potential equals zero are
shown in Fig. 4, to be clear, the result of $T=0.01$ GeV is divided by 100. It
is very clear to see that the angular velocity has a strong influence on the
quark first-order spin polarization as well as the temperature is also
important to the polarization of the quark. From the figure it is observed
that the rotation system may induce a large polarization. At all temperatures
the quark spin polarization increases with increasing angular velocity for all
quarks. At low temperature the quark first-order spin polarization increases
very rapidly in a certain angular velocity window and then increases very
slowly, while at high temperature the quark first-order spin polarization
increases almost linearly. And at very low temperature an interesting
phenomenon of the jump of the quark first-order spin polarization can be
observed around $\omega\sim 0.6$ GeV for the rotating system in the figure.
This jump of the first-order spin polarization is a hint for the first order
phase transition to occur, and this distinguishing feature of the spin
polarization may provide valuable insights for predicting the first order
phase transition in experiments.

Figure 5: (Color online) Study of the second-order spin polarizations of the
rotating system according to $\omega$ for several temperature $T$ with
$\mu=0$. Figure 6: (Color online) Susceptibilities of $u$ and $s$ quark as a
function of $\omega$ for several values of $T$ with $\mu=0$, here the result
of $T=0.01$ GeV is divided by 10.

In order to have a better understanding for the spin polarization of quark
with rotation, we plot the second-order spin polarizations of the rotating
system as a function of $\omega$ for several temperatures with the chemical
potential equals zero in Fig. 5. In the case of the temperature is very low,
the second-order spin polarization begins to occur at a certain value of the
angular velocity and increasing with the angular velocity increases, until it
reaches the highest value, when the angular velocity is increased further the
second-order spin polarization becomes weaker and finally disappears, the
reason why the second-order quark spin polarization disappears at large
angular velocity is that when the angular velocity becomes larger, the
dynamical mass of the quark becomes smaller and the chiral symmetry is
gradually restored so that the second-order quark spin polarization does not
occur.

On the other hand, in order to understand the properties of the quark matter
under the rotation system better, it is helpful to study the behavior of
baryon number susceptibility. We now turn to study the QNS and take into
account the influence of angular velocity on QNS. Firstly, we consider the
case of zero chemical potential, in Fig. 6 we plot the susceptibilities of $u$
and $s$ quark as a function of angular velocity for several fixed values of
temperatures with zero chemical potential, to be clear, the result of $T=0.01$
GeV is divided by 10. It is evident from the figure that the susceptibilities
of $u$ and $s$ quark increase as the $\omega$ increases when $\omega$ is
smaller than a certain value for all the temperatures, while the
susceptibilities decrease as the $\omega$ increases when $\omega$ is larger
than another certain value. And it also can be seen from the figure that the
temperature and the angular velocity play an important role in the
susceptibility, at low temperature the quark chiral symmetry is broken
spontaneously, however with the increasing of the angular velocity the chiral
symmetry is restored, so we can see at low temperature and low angular
velocity the susceptibility is very small and at low temperature and large
angular velocity the susceptibility disappears, however if the temperature is
high the susceptibility always occurs and the angular velocity plays a
slightly effect on the rotating matter.

It is fairly clear that the susceptibility is a finite quantity that is
furthermore sensitive to the mass of the quark, so next we will discuss the
differences of the $u$ and $s$ quark number susceptibilities under the
rotation. When angular velocity is small and temperature is high we find that
the susceptibility of $u$ is larger than the susceptibility of $s$ and when
angular velocity and temperature are both small we find that it is easy for
the susceptibility of $u$ quark to occur, for instance, at $T=0.01$ GeV, the
$u$ quark number susceptibility starts to occur at $\omega=0.5$ GeV while for
$s$ quark number susceptibility starts to occur at $\omega=0.9$ GeV. As we can
see from the figure that the curves have the highest values only for some
certain regions of the angular velocity, and at low temperature we find that
there exist a narrower region obvious changes the QNS when comparing that of
$s$ quark with that of $u$ quark at $T=0.01$ GeV, it is clearly can be seen
from the curves that QNS changes little when the angular velocity is changed
below 0.9 GeV or above 1.5 GeV for the $s$ quark, however for that of $u$
quark the angular velocity is below 0.5 GeV or above 1.5 GeV. It is very clear
from the figure that the contribution from the angular velocity becomes
dominant when $\omega\geq 1.1$ and the peaks of the susceptibilities appear at
almost the same angular velocity. At all values of the chemical potential the
behaviors of the $u$ quark and $s$ quark number susceptibility are very
similar with increasing the angular velocity reveal that when the angular
velocity is large the role played by the mass of different quarks becoming
weaker and weaker and finally almost can be ignored.

Figure 7: (Color online) The mean field mass gap of $u$ and $s$ quark as a
function of $\omega$ for several fixed values of $\mu$ at $T=0.01$ GeV. Figure
8: (Color online) First-order spin polarizations of the rotating system as a
function of $\omega$ for several fixed nonzero values of $\mu$ at $T=0.01$
GeV.

Let us now discuss the behavior of mean field mass gap of the quark at very
low temperature with nonzero chemical potential, in Fig. 7 we plot $M_{u}$ and
$M_{s}$ as a function of $\omega$ for a variety of values of $\mu$ at $T=0.01$
GeV, respectively. Comparing with the Fig. 2, it is clear that a nonzero value
of the chemical potential affects the first-order phase transition, at
$T=0.01$ GeV, there does not exist a sudden drop for the mean field mass gap
when $\mu$ is large both for $u$ and $s$ quark, which indicating there exists
a suppression effect for the chemical potential on the phase transition. At
large chemical potential the chiral condensate vanishes with increasing
$\omega$ via a smooth crossover. From the figure we can also see that there is
a different behavior between $u$ and $s$ quark, the $u$ quark is more affected
by the presence of the chemical potential and angular velocity than $s$ quark
because the $s$ quark has a substantial mass even after the chiral phase
transition.

In Fig. 8, the plots of the first-order spin polarization of the rotating
system as a function of angular velocity at $T=0.01$ GeV are presented for
nonzero chemical potential $\mu=0.1,0.2,0.3,0.4$ GeV, respectively. As is
clearly evident from the figure, the first-order spin polarizations of the
rotating system increase with increasing angular velocity. And it can be seen
that the first-order spin polarizations are affected obviously by the nonzero
quark chemical potential. With increasing angular velocity, the first-order
spin polarization of the system will first take place at larger quark chemical
potential, for example, the first-order spin polarization of $u$ start to
occur aound $\omega=0.2$ GeV and $\omega=0.4$ GeV for $\mu=0.2$ GeV and
$\mu=0.1$ GeV, respectively. Fig. 8 also demonstrates that at very large quark
chemical potential the first-order spin polarization of the system first
quickly reach a certain value then is only relatively slowly varying with
angular velocity, this can be understood by noting that at large value of
quark chemical potential the chiral symmetry restored quickly.

Fig. 9 displays the results of the second-order polarizations of the system as
a function of $\omega$ for several fixed nonzero values of chemical potential
at $T=0.01$ GeV. In particular, a non-monotonic is identified, with the
second-order polarizations to first increase, reach a maximum, and then
decrease sharply. This behaviour is the combined effect of the suppression in
both the angular velocity itself and the chemical potential at very low
temperature.

Figure 9: (Color online) Second-order spin polarizations of the rotating
system as a function of $\omega$ for several fixed nonzero values of $\mu$ at
$T=0.01$ GeV. Figure 10: (Color online) Susceptibilities of $u$ and $s$ quark
as a function of $\omega$ for several fixed nonzero values of $\mu$ at
$T=0.01$ GeV.

Next, we will analyze the patterns of second-order susceptibility of quark
with rotation at nonzero chemical potential. let us first consider the effect
of angular velocity $\omega$ on second-order susceptibility of the quarks at
very low temperature. We plot the second-order susceptibilities of $u$ and $s$
quark as a function of $\omega$ for several fixed nonzero values of chemical
potential at $T=0.01$ GeV in Fig. 10, respectively. From the figure we find
the susceptibility have similar behavior with respect to angular velocity. we
can see if one considers the case of nonzero chemical potential, the behavior
of QNS changes considerably and is quite dependent on the angular velocity. As
shown in the figure one can see the dependence of QNS on angular velocity is
complicated that the QNS increases with the increasing $\omega$ when $\omega$
is smaller than a certain value while the QNS decreases with the increasing
$\omega$ when $\omega$ excesses another certain value, which indicates that
the rotation matter may provide some new and helpful results to study the
phase transition. And there are some interesting changes compared with the
situation of zero chemical potential, at $T=0.01$ GeV the curves of
susceptibility of the quark have two peaks, which are very different compared
with that in Fig. 6, the curves of QNS have such behavior because the gap mass
with nonzero chemical potential are different from that the case with zero
chemical potential and for any angular velocity which should satisfy the gap
equation, whose constraint will have an effect to the susceptibility. In
addition, a plausible explanation for this phenomenon is that the rotational
velocity serves as an effective chemical potential and exhibits a non-trivial
behavior such that the competition between the chemical potential and angular
velocity renders the quark number susceptibility to reach its maximum in such
manner.

Figure 11: (Color online) The mean field mass gap $M_{u}$ and $M_{s}$ as a
function of $\omega$ for several fixed values of $\mu$ at $T=0.2$ GeV. Figure
12: (Color online) $M_{u}(\omega)/M_{u}(\omega=0)$ and
$M_{s}(\omega)/M_{s}(\omega=0)$ as a function of $\omega$ with several nonzero
chemical potentials at $T=0.2$ GeV.

It may need a bit more explanation about the part played by the angular
velocity $\omega$ with nonzero chemical potential at high temperature. Fig. 11
shows the mean field mass gap $M_{u}$ and $M_{s}$ versus $\omega$ with several
fixed values of $\mu$ at $T=0.2$ GeV, obviously from the figure we can see
there is a generally rotational suppression effect on the chiral condensate
for the system at high temperature with nonzero chemical potential. In order
to have better understanding in Fig. 12 we plot
$M_{u}(\omega)/M_{u}(\omega=0)$ and $M_{s}(\omega)/M_{s}(\omega=0)$ as a
function of $\omega$ with several nonzero chemical potentials at $T=0.2$ GeV,
respectively. We can see the influence of the angular velocity to the light
quark and strange quark is different. The figure shows that $M_{u}$ is much
affected due to the current mass of the $u$ quark is very small and whose
chiral symmetry can be easily restored compared to that of $s$ quark.

Figure 13: (Color online) First-order spin polarizations of the rotating
system as a function of $\omega$ for several fixed values of $\mu$ at $T=0.2$
GeV. Figure 14: (Color online) Second-order spin polarizations of the rotating
system as a function of $\omega$ for several fixed nonzero values of $\mu$ at
$T=0.2$ GeV.

In Fig. 13, we plot the first-order spin polarization of the system as a
function of $\omega$ for several fixed values of $\mu$ with $T=0.2$ GeV. From
Fig. 13 we find that at this temperature the first-order quark spin
polarization of the system always occurs with increasing the angular velocity
for all the nonzero chemical potential, at large chemical potential ($\mu=0.4$
GeV), the first-order spin polarization of the system increase almost linearly
with increasing the angular velocity. We now consider the effects of angular
velocity $\omega$ on second-order spin polarization the system at $T=0.2$ GeV
with chemical potential $\mu=0.1,0.2,0.3,0.4$ GeV, respectively. From the Fig.
14 one immediately makes the following observations, in the case of high
temperature the variation of second-order spin polarization of the system with
angular velocity is complicated, we find that at small chemical potential the
second-order spin polarization of the system have a peak structure with the
increase of angular velocity, when $\omega<1.0$ GeV at small chemical
potential ($\mu=0.1$ GeV) the second-order spin polarization of the system
increases with increase in angular velocity, while at large chemical potential
($\mu=0.4$ GeV) that decreases with increase in angular velocity.

Next, we would like to consider the effects of angular velocity $\omega$ on
second-order susceptibility of the quarks at high temperature with several
fixed nonzero values of chemical potential and the numerical results are shown
in Fig. 15. It can be found that at such high temperature the second-order
susceptibility of the quarks can always occur, although the maximum value of
susceptibility is very small compared to the situation in the Fig. 10, which
means that the contribution of angular velocity to the susceptibility is
suppressed by high temperature. From Fig. 15 one could infer the dependence of
the second-order susceptibility of the quarks on the quark current mass when
$\omega<0.5$ GeV, the values of the susceptibility of the strange quark is
smaller compared to that of light quarks for the same quark chemical
potential. However, with the increase of $\omega$, the behavior of the both
quarks is very similar which means that at high temperature and large chemical
potential the large angular velocity takes the predominant role compared to
different current mass of the quarks. From Fig. 15 one could also infer the
dependence of susceptibility on the chemical potential at high temperature, at
small $\omega$, the susceptibility increases with the increasing $\mu$. at
large $\omega$, the susceptibility decreases with the increasing $\mu$.

Let us move on to the topic of QCD phase diagram. The investigation of the
phase diagram of QCD has been an active subject for many years. There has been
much progress on the study of the QCD phase diagram with lattice QCD (LQCD)
simulations, however, at large chemical potential the predictions made by the
LQCD are not very reliable due to the sign problem of lattice gauge theory
P.de Forcrand . So, in order to investigate the QCD phase diagram many
effective models have been proposed such as Nambu and Jona-Lasinio (NJL)
models, quark-meson (QM) models, holographic QCD models, functional
renormalization group (fRG), Dyson Schwinger equations (DSE) as well as some
extending modes of these S. P. Klevansky ; H. Kohyama ; C. D. Roberts ; R.
Alkofer ; C. S. Fischer ; I. C. ; T.M. Schwarz ; P. Zhuang ; J.-W. Chen ;
J.-W. Chen1 ; W. Fan ; W. Fan1 ; W.-J. Fu ; E.S. Bowman ; H. Mao ; B.J.
Schaefer ; B.-J. Schaefer1 ; S.-X. Qin ; J. Luecker ; W.-J. Fu1 ; M. Buballa ;
Luis A. H. Mamani . It is believed that the addition of external influences or
new parameter ranges yield an increasing number of interesting results to the
phase diagram J.N. Guenther .

Figure 15: (Color online) Second-order susceptibilities of $u$ and $s$ quark
as a function of $\omega$ for several fixed nonzero values of $\mu$ at $T=0.2$
GeV. Figure 16: (Color online) The first order phase transitions of the system
on the $T$-$\omega$ plane.

And investigating the QCD matter under rotation is a fascinating topic, apart
from the chiral condensation, spin polarization and number susceptibility of
these matter, it is also of significant interest to explore the effects of
rotation on the phase transitions. We now first explore the phase diagram in
the $T$-$\omega$ plane, it is obvious that from the Fig. 2 there exist first
order phase transitions for the rotating system. Here our main focus is on the
first order phase transition and its associated critical end point with
rotation in the presence of small chemical potential, in Fig. 16 we show the
first order phase diagram of the system in the $T$-$\omega$ plane with
different chemical potentials $\mu=0.02,0.03$ and $0.04$ GeV, respectively. We
observe that the appearance of the first order phase transition line starting
at $T\sim 0$ GeV and terminating at the critical end point identified with a
full dot and these first order phase transition lines move toward a higher
temperature for decreasing $\omega$. It is also found that the different
chemical potentials change the boundary of phase diagram, and that a larger
chemical potential shifts down the critical temperature. It is no doubt that
there are many other discussions need to be done for the phase transition in
the rotation system, here is just a beginning and this particular topic will
be discussed in detail in future work.

##  IV Conclusions and outlook

Finally, we want to summarize our results and give a brief outlook. In this
paper, we have presented detailed analytic formulae for the quark matter under
rotation in three-flavor NJL model and related topics have been investigated.
In order to have a better understanding of the rotating system with finite
density we have also introduced the chemical potential. We studied the quark
fields in cylindrical coordinates as well as investigated the effect of the
rotating on the quark chiral condensate, quark spin polarization and quark
number susceptibility at finite temperature with or without finite chemical
potential in this model. We found that the angular velocity plays a very
crucial role in these topics, at low temperature, small chemical potential and
small angular velocity the chiral symmetry is broken spontaneously, while at
large enough angular velocity the chiral symmetry is restored. Our numerical
analysis shows that the rotation suppresses the chiral condensation and
enhances the first-order quark spin polarization, however for the second-order
quark spin polarization and quark number susceptibility the effect is very
complicated, which can be found to have a peak structure.

We have also explicitly computed these quantities in the rotation system in
the presence of chemical potential, we found that the nonzero chemical
potential affects and makes the chiral condensate, quark spin polarization and
quark number susceptibility have different behaviors. At very low temperature
the chiral condensate experiences a first order transition when $\omega$
exceeds a certain value with zero chemical potential, while at larger chemical
potential the first order transition is suppressed and changed to a crossover
transition. It can be also observed that at very low temperature the quark
number susceptibilities have two maxima, a plausible explanation for this
phenomenon is that the rotational velocity serves as an effective chemical
potential and exhibits a non-trivial behavior such that the competition
between the chemical potential and angular velocity renders the quark number
susceptibility to reach its maximum in such manner. In this paper we
especially considered the contributions from $s$ quark and made some
comparisons between $u$ quark and $s$ quark and found that at small angular
velocity the part played by current mass to these phenomena is important,
however, at sufficiently large angular velocity, the contributions played by
different quarks to these phenomena are almost equal. In addition, we explored
the phase diagram in the $T$-$\omega$ plane, we observe that there exist first
order phase transitions for the rotating system and the first order phase
transition lines move toward a higher temperature for decreasing angular
velocity. It is also found that the different chemical potentials change the
boundary of phase diagram, and that a larger chemical potential shifts down
the critical temperature. Based on the interpretations made above, it would be
possible to judge and forecast these phenomena of quark matter under rotation
if we jointly take angular velocity, chemical potential and temperature
factors into consideration in the three-flavor NJL model. We expect these
studies to play an important role in help understanding the properties of
strongly interacting rotating matter.

The theoretical interest in relativistic rotating systems is being revived in
many different physical environments, which is undoubtedly calls for more
investigation. For instance, the related effects of rotating fermions inside a
cylindrical boundary Victor E. Ambrus , the investigation of a possible phase
structure under rotation including the $s$ quark, especially the exploration
to those regions of the phase diagram that cannot be reached on the lattice
yet. The NJL model describes only quarks and antiquarks and neglects the
gluons, so it is also very worth to extend the rotation system to the Polyakov
extended Nambu and Jona-Lasinio (PNJL) model C. Ratti , which will consider
the complex interactions between quarks and gluons and that the chiral
symmetry restoration as well as the effect of quark confinement in PNJL under
rotation may provide needed insight into the QCD. This model will have a
clearer picture considering the constraint from the experimental related to
rotation conducted at numerous research facilities worldwide as the Brookhaven
National Laboratory (BNL), the European Organization for Nuclear Research
(CERN) and the GSI Helmholtz Centre for Heavy Ion Research (GSI), due to one
can see that some limits can be made for the parameter space under the given
assumptions. It would be interesting to use the results obtained in this paper
to investigate these topics discussed here, and we leave these as our further
study.

## Acknowledgements

We specially thank Jinfeng Liao for the early involvement in the work and
enlightening discussions, and thank Hui Zhang, Akira Watanabe for discussions,
corrections and comments. We also thank Shengqin Feng and Yafei Shi for their
encouragement and discussions. The work has been supported by the National
Natural Science Foundation of China (NSFC) under Grant No. 11647174, No.
11875178, No. 11801311, the Key Laboratory of Quark and Lepton Physics
Contracts under Grant No. QLPL201905 and the Science Research Foundation of
China Three Gorges University under Grant No. KJ2015A007. AH acknowledges
support by the NSF Grant No. PHY1913729 and by the U.S. Department of Energy,
Office of Science, Office of Nuclear Physics, within the framework of the Beam
Energy Scan Theory (BEST) Topical Collaboration. AH is also grateful to the
Fundamental Research Funds for the Central Universities.

##  Appendix A The brief description of fermions under rotation

The properties of fermions under global rotation are relevant to a number of
problems as discussed above, so it is important to choose an appropriate
complete set of commuting operators in the cylindrical coordinates. In this
section, we will start from the Dirac equation in the rotating frame, then we
will derive the eigenvectors of the those complete set of commuting operators.

The general Lagrangian of the rotating fermions is written in the following
way A. L. Fetter ; Y. Jiang1 ; A. Yamamoto

$\displaystyle{\cal
L}=\bar{\psi}\left[{i\gamma^{\mu}\partial_{\mu}-m+\left({\gamma^{0}}\right)^{-1}\left({\left({\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{\omega}}\times\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{x}}}\right).\left({-i\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{\partial}}}\right)+\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{\omega}}.\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{S}}_{4\times
4}}\right)}\right]\psi,$ (84)

where $\psi$ is the quark field, $\omega$ is the angular velocity and $m$ is
the bare quark mass matrix, as a result of rotation, we can see the Dirac
operator includes the orbit-rotation coupling term and the spin-rotation
coupling term, and we have defined

$\displaystyle\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{S}}_{4\times
4}=\frac{1}{2}\left({\begin{array}[]{*{20}c}{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{\sigma}}}&0\\\
0&{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{\sigma}}}\\\
\end{array}}\right),$ (87)

whose z-component related to the spin polarization of the quark. The
corresponding Hamiltonian of Eq. (84) in momentum space reads

$\displaystyle\hat{\cal H}=\gamma^{0}\left({\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{\gamma}}.\hat{\vec{p}}+m}\right)-\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{\omega}}.\left({\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{x}}\times\hat{\vec{p}}+\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{S}}_{4\times
4}}\right)=\hat{\cal H}_{0}+\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{\omega}}.\hat{\vec{J}},$
(88)

here

$\displaystyle\hat{\vec{J}}=\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{x}}\times\hat{\vec{p}}+\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{S}}_{4\times 4},$ (89)

and the first term is the contribution of the angular momentum, the second
term is the contribution of the spin angular momentum.

Now considering the energy eigenvalue equation

$\displaystyle\hat{\cal H}\psi=E\psi,$ (90)

here $\psi$ is the four-component spinors and can be written in terms of two-
component spinors as

$\displaystyle\psi=\left({\begin{array}[]{*{20}c}\phi\\\ \chi\\\
\end{array}}\right),$ (93)

substituting the Hamiltonian above to the energy eigenvalue equation, then Eq.
(90) transforms simply as

$\displaystyle\left\\{\begin{array}[]{l}\left({E-m+\omega_{z}J_{z}}\right)\phi=\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{\sigma}}.{\hat{\vec{p}}}\chi\\\
\left({E+m+\omega_{z}J_{z}}\right)\chi=\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{\sigma}}.{\hat{\vec{p}}}\phi\\\
\end{array}\right.$ (96)

here,

$\displaystyle J_{z}=L_{z}+\frac{1}{2}\sum_{z},$ (97)

which is the z-component of total angular momentum, then, we consider the
$z$-component angular momentum eigenvalue equation

$\displaystyle J_{z}\psi{\rm{=}}\left({n+\frac{1}{2}}\right)\psi,$ (98)

after some derivations, we can get the following equation

$\displaystyle\left({E-m+\omega_{z}\left({n+\frac{1}{2}}\right)}\right)\left({E+m+\omega_{z}\left({n+\frac{1}{2}}\right)}\right)\phi=\left(\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{\sigma}}.{\hat{\vec{p}}}\right)^{2}\phi,$
(99)

it is very convenient to make the transform Cartesian coordinate to the
Cylindrical coordinate, here separation variable method is applied and $\phi$
takes the form of

$\displaystyle\phi=f\left(\theta\right)g\left(r\right)h\left(z\right),$ (100)

and the solution for two-component spinors $\varphi$ in Eqs. (98) has the form

$\displaystyle f(\theta)=\left({\begin{array}[]{*{20}c}{e^{in\theta}}\\\
{e^{i\left({n+1}\right)\theta}}\\\ \end{array}}\right),$ (103)

substituting Eq. (103) into the Eq. (99) after some tedious calculations, we
find that $g\left(r\right)$ satisfies the Bessel-type equation as follows,

$\displaystyle r^{2}\frac{{\partial^{2}g\left(r\right)}}{{\partial
r^{2}}}+r\frac{{\partial g\left(r\right)}}{{\partial
r}}+\left({r^{2}p_{t}^{2}-n^{2}}\right)g\left(r\right)=0,$ (104)

$\displaystyle r^{2}\frac{{\partial^{2}g\left(r\right)}}{{\partial
r^{2}}}+r\frac{{\partial g\left(r\right)}}{{\partial
r}}+\left({r^{2}p_{t}^{2}-\left({n+1}\right)^{2}}\right)g\left(r\right)=0,$
(105)

the solutions of Eqs. (104) and (105) have the following form, respectively,

$\displaystyle
g\left(r\right)=J_{n}\left({p_{t}r}\right),J_{n+1}\left({p_{t}r}\right),$
(106)

where $J$ is the Bessel function. In order to commute with other operators we
must define the helicity operator, the general helicity operator has the
following form

$\displaystyle h_{t}=\gamma^{5}.\gamma^{3}\frac{{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{\sum}}.\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{p}}_{t}}}{{\left|{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{p}}_{t}}\right|}}=\frac{1}{i{\left|{\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{p}}_{t}}\right|}}\left({\begin{array}[]{*{20}c}0&{-P_{-}}&0&0\\\
{P_{+}}&0&0&0\\\ 0&0&0&{P_{-}}\\\ 0&0&{-P_{+}}&0\\\ \end{array}}\right),$
(111)

here, $p_{t}$ is the transverse momentum, $P_{+}=\hat{p}_{x}+i\hat{p}_{y}$,
$P_{-}=\hat{p}_{x}-i\hat{p}_{y}$ and in Cylindrical coordinates they have such
forms

$\displaystyle P_{+}=-ie^{i\theta}\left({\frac{\partial}{{\partial
r}}+i\frac{1}{r}\frac{\partial}{{\partial\theta}}}\right),$ (112)
$\displaystyle P_{-}=-ie^{-i\theta}\left({\frac{\partial}{{\partial
r}}-i\frac{1}{r}\frac{\partial}{{\partial\theta}}}\right),$ (113)

which like shift operators when act on the terms including angular momentum
quantum number $e^{in\theta}J_{n}(p_{t}r)$,
$e^{i\left({n+1}\right)\theta}J_{n+1}(p_{t}r)$, respectively, they satisfy the
following relationship

$\displaystyle
P_{+}e^{in\theta}J_{n}(p_{t}r)=ip_{t}e^{i\left({n+1}\right)\theta}J_{n+1}(p_{t}r),$
(114) $\displaystyle
P_{-}e^{i\left({n+1}\right)\theta}J_{n+1}(p_{t}r)=-ip_{t}e^{in\theta}J_{n}(p_{t}r).$
(115)

Reconsidering the transverse helicity equation and the generalized
orthogonality relation

$\displaystyle h_{t}\psi=s\psi,$ (116)

$\displaystyle\sum\limits_{n=-\infty}^{\infty}{\psi^{\dagger}\psi=1},$ (117)

here, $s=\pm 1$ represent the transverse helicity, the solutions of the
positive energy eigenvalues are obtained as follows

$\displaystyle
u=\frac{1}{{\sqrt{2}}}\left({\begin{array}[]{*{20}c}{e^{ip_{z}z}e^{i{\rm{n}}\theta}J_{n}\left({p_{t}r}\right)}\\\
{se^{ip_{z}z}e^{i\left({n+1}\right)\theta}J_{n+1}\left({p_{t}r}\right)}\\\
{e^{ip_{z}z}e^{i{\rm{n}}\theta}J_{n}\left({p_{t}r}\right)}\\\
{se^{ip_{z}z}e^{i\left({n+1}\right)\theta}J_{n+1}\left({p_{t}r}\right)}\\\
\end{array}}\right),$ (122)

where, $u$ is a four-component spinor which must satisfy the Dirac equation

$\displaystyle(i\gamma^{\mu}\partial_{\mu}-m)u=0,$ (123)

substituting $u$ into the Dirac equation gives

$\displaystyle\left({\begin{array}[]{*{20}c}{E-m}&{-\sigma.p}\\\
{\sigma.p}&{-E-m}\\\
\end{array}}\right)\left({\begin{array}[]{*{20}c}{c_{A}u_{A}}\\\
{c_{B}u_{B}}\\\ \end{array}}\right)=0,$ (128)

here $u_{A}$, $u_{B}$ are two-component spinors and $c_{A}$, $c_{B}$ are
normalization constants, after some calculations we get

$\displaystyle
c_{B}u_{B}=c_{A}\left({\begin{array}[]{*{20}c}{\frac{{\left({p_{z}-isp_{t}}\right)}}{{E+m}}e^{ip_{z}z}e^{i{\rm{n}}\theta}J_{n}\left({p_{t}r}\right)}\\\
{\frac{{\left({-sp_{z}+ip_{t}}\right)}}{{E+m}}e^{ip_{z}z}e^{i\left({n+1}\right)\theta}J_{n+1}\left({p_{t}r}\right)}\\\
\end{array}}\right),$ (131)

imposing the generalized completeness relation

$\displaystyle\sum\limits_{n=-\infty}^{\infty}{u^{\dagger}u=1},$ (132)

these constant factors can be determined and finally we obtain the positive
energy particle solutions with positive and negative helicity in the Dirac
representation, which take the following explicit form

$\displaystyle
u=\frac{1}{2}\sqrt{\frac{{E+m}}{E}}\left({\begin{array}[]{*{20}c}{e^{ip_{z}z}e^{i{\rm{n}}\theta}J_{n}\left({p_{t}r}\right)}\\\
{se^{ip_{z}z}e^{i\left({n+1}\right)\theta}J_{n+1}\left({p_{t}r}\right)}\\\
{\frac{{p_{z}-isp_{t}}}{{E+m}}e^{ip_{z}z}e^{i{\rm{n}}\theta}J_{n}\left({p_{t}r}\right)}\\\
{\frac{{-sp_{z}+ip_{t}}}{{E+m}}e^{ip_{z}z}e^{i\left({n+1}\right)\theta}J_{n+1}\left({p_{t}r}\right)}\\\
\end{array}}\right),$ (137)

in exactly the same way, the negative-energy antiparticle solutions are listed
below

$\displaystyle
v=\frac{1}{2}\sqrt{\frac{{E+m}}{E}}\left({\begin{array}[]{*{20}c}{\frac{{p_{z}-isp_{t}}}{{E+m}}e^{-ip_{z}z}e^{i{\rm{n}}\theta}J_{n}\left({p_{t}r}\right)}\\\
{\frac{{-sp_{z}+ip_{t}}}{{E+m}}e^{-ip_{z}z}e^{i\left({n+1}\right)\theta}J_{n+1}\left({p_{t}r}\right)}\\\
{e^{-ip_{z}z}e^{i{\rm{n}}\theta}J_{n}\left({p_{t}r}\right)}\\\
{-se^{-ip_{z}z}e^{i\left({n+1}\right)\theta}J_{n+1}\left({p_{t}r}\right)}\\\
\end{array}}\right).$ (142)

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