††thanks<EMAIL_ADDRESS>first author††thanks<EMAIL_ADDRESS>co-correspondence author††thanks: huangmei@ucas.ac.cn,co-correspondence author

# Mass splitting of vector meson and spontaneous spin polarization under
rotation

Minghua Wei1,2 Yin Jiang3 Mei Huang1 1 School of Nuclear Science and
Technology, University of Chinese Academy of Sciences, Beijing 100049, China
2 Institute of High Energy Physics, Chinese Academy of Sciences, Beijing
100049, P.R. China  3 Department of Physics, BeiHang University, Beijing
100191, P.R. China

###### Abstract

In the present paper, we study the effect of the rotation on the masses of
scalar meson as well as vector meson in the framework of 2-flavor Nambu–Jona-
Lasinio model. The existence of rotation causes a tedious quark propagator and
corresponding polarization function. Applying the random phase approximation,
the meson mass is calculated numerically. It is found that the behavior of
scalar and pseudoscalar meson masses under the angular velocity $\omega$ is
similar to that at finite chemical potential, both rely on the behavior of
constituent quark mass and reflect the property related to the chiral
symmetry. However, masses of vector meson $\rho$ have more profound relation
with rotation. After tedious calculation, it turns out that at low temperature
and small chemical potenial, the mass for spin component $s_{z}=0,\pm 1$ of
vector meson under rotation shows very simple mass splitting relation
$m_{\rho}^{s_{z}}(\omega)=m_{\rho}(\omega=0)-\omega s_{z}$, similar to the
Zeeman splitting of charged meson under magnetic fields. Especially it is
noticed that the mass of spin component $s_{z}=1$ vector meson $\rho$
decreases linearly with $\omega$ and reaches zero at
$\omega_{c}=m_{\rho}(\omega=0)$, this indicates the system will develop
$s_{z}=1$ vector meson condensation and the system will be spontaneously spin
polarized under rotation.

###### pacs:

12.38.Mh, 25.75.Nq, 25.75.-q

##  I Introduction

In non-central heavy-ion collision(HIC), large vorticity and strong magnetic
field are expected to be generated in extremely hot quark gluon plasma(QGP).
Straightforward electromagnetical(EM) computation shows the magnetic field
would reach about $O(10^{14})T$Kharzeev:2007jp  in the early stage of HIC,
while kinetic and hydrodynamic simulationsBecattini:2007sr ; Jiang:2016woz
indicate the local vorticity would exceed $0.5fm^{-1}$ with the total angular
momentum of QGP at a range of $O(10^{4})-O(10^{5})\hbar$. Known as the Barnett
and magnetization effects, spin particles are polarized by these pseudo vector
field and thus distribute differently from the normal thermal distributions.
Besides chiral effects induced by such pseudo vector fieldsKharzeev:2007tn ;
Son:2009tf ; Kharzeev:2010gr , studies on these distribution modifications
would be helpful to understand the hadronization mechanism of the strong
interaction as well. Inspired by the large amplitude and retention by the
angular momentum conservation, vorticity has attracted more and more interests
recently.

Comparing with magnetic field effects, the rotation-related effects are
electric charge blind, and only involve kinetic properties of the QGP and
strong interaction which we are mostly interested in. Experimentally, in order
to screen out the EM effects neutral particles with finite spin numbers are
chosen as carriers of the vorticity polarization effects. As it is difficult
to detect the chargeless particle directly the distribution of its charged
daughter particle serves as an alternative observable for the global
polarization effect. With the help of the $\Lambda$ hyperon the average
magnitude of the vorticity of QGP has been extracted by the STAR
collaborationSTAR:2017ckg . In these measurements the expectation of $\Lambda$
polarization as well as the vorticity behavior of collision energy have been
confirmed as well. All the results seem to be understandable by considering
the energy shift induced by the voriticity polarization to spins. However the
theory became a little vague when the $K^{*0}$ and $\phi$ mesons’ measurements
were presented in Acharya:2019vpe . The mismatch between these measurement
indicates the fine structure of hadrons may play an non-negligible role in
polarization processes.

The mass is one of the most fundamental attributes of a hadron. For a
composite particle it will be modified by the single-particle dispersion
relation of the fundamental degree of freedom as well as the interaction among
them. Studies of hadron masses would help us to discover many clues of the
environment where hadrons are born. As a well-known example, $\sigma$ meson
and pion masses would change with the growing temperature and chemical
potential because of chiral restoration Klevansky:1992qe . And recently people
have studied the vector meson $\rho$ mass in external magnetic field as well
by taking the polarization effect on quarks into account. A lattice
calculation demonstrates that charged $\rho$ meson mass decreases firstly and
increases finally, leaving a minimum around $eB\simeq 1GeV^{2}$ Hidaka:2012mz
. And by using effective models, such as the Nambu–Jona-Lasinio(NJL) model
with vector channel, the $\rho$ meson with different spin components have been
studied Liu:2014uwa ; Liu:2018zag . Therefore a natural question is what about
the mass behavior under a background vorticity field which is a little like
the magnetic case at the first sight. For the rotating effect, the co-rotating
frame Yamamoto:2013zwa  is usually adopted and a nontrivial spin connection
term will be introduced Matsuo:2015 , which serves as a polarization term for
angular momentums. With this extended NJL model it is suggested that chiral
phase transition would take place as angular velocity increasing Jiang:2016wvv
. Furthermore, people have established more complicated phase diagrams which
combine rotation and other physical conditions such as chemical potential,
isospin and magnetic field Wang:2018sur ; Zhang:2018ome ; Chen:2015hfc . In
those NJL models, at the quark level, the rotation always behaves as an
effective chemical potential. This analogy has been understood with a
Hamiltonian shifting $\hat{H}\rightarrow\hat{H}-\vec{\omega}\cdot\hat{J}$ and
the latter term may be corresponding to an effective chemical potential
Chen:2015hfc ; Matsuo:2012wv . At the same quark level, holographic models
also contribute to elaborate the property of rotating quark matter by setting
up a four-dimensional AdS-Kerr-Newman black hole to construct a rotation-
magnetism analogy McInnes:2016dwk . While for the composite hadrons, such as
vector mesons, there have been few works on the mass behaviors.

In this paper, we focus on the scalar and the vector meson and investigate
their masses under the rotation at finite chemical potential. In Sec. II, in
order to deal with both the finite temperature and density cases, we introduce
the two- flavor NJL model with vector channel in the co-rotating frame. In
this framework, we generate the dynamical quark mass with chiral symmetry
spontaneous breaking and construct scalar and vector masons with the dressed
quark propagator and extract the corresponding masses with the well-known
random phase approximation(RPA) in Sec. III and show their numerical results
in Sec. IV. Because of the rich phase structure at large chemical potential we
only study the range of $\mu_{q}<200$MeV in this work and leave the discussion
of the rotating color superconductivity in our following works. We have found
that masses of scalar mesons are controlled by the chiral phase transition
which could be driven by temperature, density and rotation. While the vector
meson, which carries net angular momentum, is governed by the polarization
effect on the total angular momentum before chiral symmetry restoration. At
large angular velocity the mass of spin component $s_{z}=1$ for the vector
meson vanishes. This indicates the macroscopic condensate of spin component
$s_{z}=1$ of vector meson $\langle\rho^{s_{z}=1}\rangle$ thus spontaneous spin
polarization would be induced in the ultra-fast rotating system. In Sec. V, we
summarize our main results and give an outlook.

##  II NJL model in co-rotating frame

NJL model is an effective model with 4-fermion interaction which is widely
used to study quark-quark and quark-antiquark pairing which corresponding to
chiral phase transition, superfluidity and superconductivity and so on.
Besides the usual scalar channels we take account the vector channels in order
to construct the vector $\rho$ mesons. The Lagrangian of the two-flavor NJL
model in the co-rotating frame is given by Wang:2018sur ; Bernard:1988db :

$\mathcal{L}=\bar{\psi}[i\bar{\gamma}^{\mu}(\partial_{\mu}+\Gamma_{\mu})-m]\psi+G_{S}[(\bar{\psi}\psi)^{2}+(\bar{\psi}i\gamma_{5}\vec{\tau}\psi)^{2}]-G_{V}[(\bar{\psi}\gamma_{\mu}\psi)^{2}+(\bar{\psi}\gamma_{\mu}\gamma_{5}\psi)^{2}],$
(1)

where $m$ is the current quark mass. $G_{S}$ and $G_{V}$ are the coupling
constants in the scalar and vector channels, respectively. In the curved co-
rotating frame the gamma matrices $\bar{\gamma}^{\mu}$ should be defined
according to the corresponding Clifford algebra. The curved gamma matrices are
connected with the flat ones with the vierbein as $\bar{\gamma}^{\mu}=e_{a}^{\
\mu}\gamma^{a}$ and where $e_{a}^{\ \mu}$ should be chosen to satisfy
$g_{\mu\nu}=\eta_{ab}e^{a}_{\ \mu}e^{b}_{\ \nu}$, where $\eta_{ab}$ is the
metric of flat space-time and $\gamma^{a}$ is flat gamma matrices. In our case
a simple enough choice is $e^{a}_{\ \mu}=\delta^{a}_{\ \mu}+\delta^{a}_{\
i}\delta^{0}_{\ \mu}\,v_{i}$ and $e_{a}^{\ \mu}=\delta_{a}^{\
\mu}-\delta_{a}^{\ 0}\delta_{i}^{\ \mu}\,v_{i}$, where $v_{i}$ is the linear
velocity $\vec{v}=\vec{\omega}\times\vec{x}$ under the presence of a constant
angular velocity $\vec{\omega}$. The so-called spinor connection is given by
$\Gamma_{\mu}=\frac{1}{4}\times\frac{1}{2}[\gamma^{a},\gamma^{b}]\,\Gamma_{ab\mu}$,
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 usual Christoffel connection determined by $g_{\mu\nu}$ Yamamoto:2013zwa ;
Jiang:2016wvv ; Matsuo:2015 . In the slow velocity limit
$|\vec{\omega}\times\vec{x}|\ll c$ we could only keep the $O(\vec{v})$ terms
which can be reduced to the ordinary polarization form as
$\vec{\omega}\cdot\vec{J}$, where $\vec{J}=\vec{x}\times\vec{p}+\vec{S}$ is
the total angular momentum Jiang:2016wvv ; Matsuo:2015  and
$\vec{S}=\frac{1}{2}\left(\begin{array}[]{cc}\vec{\sigma}&0\\\
0&\vec{\sigma}\\\ \end{array}\right)$ is the spin operator.

Applying the mean field approximation and choosing the direction of rotation
as the $z$-axis, the bilinear part of the Lagrangian at finite chemical
potential is given by Wang:2018sur

$\mathcal{L}=\bar{\psi}[i\gamma^{\mu}\partial_{\mu}+\gamma^{0}(\omega\hat{J_{z}}+\mu)-M]\psi-\frac{(M-m)^{2}}{4G_{S}},$
(2)

where $J_{z}$ is the third component of total angular momentum $\vec{J}$, and
$\mu$ is the quark chemical potential, it is seen that the angular velocity
plays similar role as the chemical potential, and $M$ is the constituent quark
mass which is given by the chiral condensate as
$M=m-2G_{S}\left<\bar{\psi}\psi\right>$. The general grand potential is given
by Jiang:2016wvv ; Wang:2018sur :

$\displaystyle\Omega(T,\mu,M,\omega)$ $\displaystyle=$ $\displaystyle\int
d^{3}\mathbf{r}~{}\bigg{\\{}\frac{(M-m)^{2}}{4G_{S}}$
$\displaystyle-\frac{N_{c}N_{f}}{16\pi^{2}}T\sum_{n}\int dk_{t}^{2}\int
dk_{z}[J_{n}(k_{t}r)^{2}+J_{n+1}(k_{t}r)^{2}]\left[\ln(1+e^{(E_{k}-(n+\frac{1}{2})\omega-\mu)/T})\right.$
$\displaystyle+\left.\ln(1+e^{-(E_{k}-(n+\frac{1}{2})\omega-\mu)/T})+\ln(1+e^{-(E_{k}+(n+\frac{1}{2})\omega+\mu)/T})+\ln(1+e^{(E_{k}+(n+\frac{1}{2})\omega+\mu)/T})\right]\bigg{\\}}.$

where $E_{k}=\sqrt{k_{t}^{2}+k_{z}^{2}+M^{2}}$ and $k_{t,z}$ are the
transverse and longitudinal momentum respectively. Obviously the local
potential approximation $\partial_{r}M(r)\simeq 0$ has been adopted during
solving the eigen modes. In the following computation we choose $N_{c}=3$ and
$N_{f}=2$. In this work we will neglect the four-fermion contributions to the
ground state, which means the chiral condensate is completely computed by the
gap equation as $\frac{\partial\Omega}{\partial M}=0$ with the constraint
$\frac{\partial^{2}\Omega}{\partial M^{2}}>0$. In Sec. IV, we will show the
numerical result of the constituent quark mass $M$. It serves as the
environment where mesons are given birth, and thus modifies their masses. In
the mean field approximation the gap equation is just the one-loop diagram of
the quark propagator which reads as

$\displaystyle S(\tilde{r};\tilde{r^{\prime}})$
$\displaystyle=\frac{1}{(2\pi)^{2}}\sum_{n}\int\frac{dk_{0}}{2\pi}\int
k_{t}dk_{t}\int
dk_{z}\frac{e^{in\left(\phi-\phi^{\prime}\right)}e^{-ik_{0}\left(t-t^{\prime}\right)+ik_{z}\left(z-z^{\prime}\right)}}{[k_{0}+(n+\frac{1}{2})\omega]^{2}-k_{t}^{2}-k_{z}^{2}-M^{2}+i\epsilon}$
(4)
$\displaystyle\times\\{[[k_{0}+(n+\frac{1}{2})\omega]\gamma^{0}-k_{z}\gamma^{3}+M][J_{n}(k_{t}r)J_{n}(k_{t}r^{\prime})\mathcal{P}_{+}+e^{i(\phi-\phi)^{\prime}}J_{n+1}(k_{t}r)J_{n+1}(k_{t}r^{\prime})\mathcal{P}_{-}]$
$\displaystyle-i\gamma^{1}k_{t}e^{i\phi}J_{n+1}(k_{t}r)J_{n}(k_{t}r^{\prime})\mathcal{P}_{+}-\gamma^{2}k_{t}e^{-i\phi^{\prime}}J_{n}(k_{t}r)J_{n+1}(k_{t}r^{\prime})\mathcal{P}_{-}\\},$

where $\mathcal{P}_{\pm}=\frac{1}{2}(1\pm i\gamma^{1}\gamma^{2})$ are
projection operators and $\tilde{r}=(t,r,\theta,\phi)$ are the coordinates in
the cylindrical frame.

##  III Scalar and vector meson mass under rotation

###  III.1 The scalar meson

In the NJL model, meson is regarded as $q\bar{q}$ bound states or resonances,
which can be obtained from the quark-antiquark scattering amplitude
Buballa:2003qv ; He:1997gn ; Rehberg:1995nr . In the random phase
approximation (RPA), the full propagator of $\sigma$ meson $D_{\sigma}(q^{2})$
can be expressed to leading order in $1/N_{c}$ as an infinite sum of quark-
loop chains:

$D_{\sigma}(q^{2})=\frac{2G_{S}}{1-2G_{S}\Pi_{s}(q^{2})},$ (5)

where $\Pi_{s}(q^{2})$ is the quark one-loop polarization function and takes
the form of

$\displaystyle\Pi_{s}(q)=-i\int
d^{4}\tilde{r}Tr_{sfc}[iS(0;\tilde{r})iS(\tilde{r};0)]e^{iq\cdot\tilde{r}},$
(6)

where $Tr_{sfc}$ means trace in spin, flavor and color space. After a tedious
calculation in Appendix. A, the polarization function could be simplified as
this form

$\displaystyle\Pi_{s}(q^{2})$
$\displaystyle=-2iN_{f}N_{c}\int\frac{d^{4}p}{(2\pi)^{4}}$ (7)
$\displaystyle\times\left\\{\frac{\left(p_{0}+q_{0}+\frac{1}{2}\omega\right)\left(p_{0}+\frac{1}{2}\omega\right)+M^{2}-(\vec{p}+\vec{q})\cdot\vec{p}}{\left[\left(p_{0}+q_{0}+\frac{1}{2}\omega\right)^{2}-(\vec{p}+\vec{q})^{2}-M^{2}\right]\left[\left(p_{0}+\frac{1}{2}\omega\right)^{2}-\vec{p}^{2}-M^{2}\right]}\right.$
$\displaystyle+\left.\frac{\left(p_{0}+q_{0}-\frac{1}{2}\omega\right)\left(p_{0}-\frac{1}{2}\omega\right)+M^{2}-(\vec{p}+\vec{q})\cdot\vec{p}}{\left[\left(p_{0}+q_{0}-\frac{1}{2}\omega\right)^{2}-(\vec{p}+\vec{q})^{2}-M^{2}\right]\left[\left(p_{0}-\frac{1}{2}\omega\right)^{2}-\vec{p}^{2}-M^{2}\right]}\right\\}.$

If we use finite temperature theory with chemical potential Kapusta , the
polarization function will be:

$\Pi_{s}(\vec{q},i\nu_{n})=2N_{f}N_{c}T\sum_{s=\pm}\sum_{N}\int\frac{d^{3}\vec{p}}{(2\pi)^{3}}\frac{[(i\tilde{\omega}_{N}+i\nu_{n})+\frac{1}{2}s\omega+\mu][i\tilde{\omega}_{N}+\frac{1}{2}s\omega+\mu]+M^{2}-(\vec{p}+\vec{q})\cdot\vec{p}}{[(i\tilde{\omega}_{N}+i\nu_{n}+\frac{1}{2}s\omega+\mu)^{2}-(\vec{p}+\vec{q})^{2}-M^{2}][(i\tilde{\omega}_{N}+\frac{1}{2}s\omega+\mu)^{2}-\vec{p}^{2}-M^{2}]},$
(8)

where $\tilde{\omega}_{N}=(2N+1)\pi T$ is Matsubara frequency. Considered
analytic continuation
$\Pi_{s}(\vec{q},\tilde{\nu})=\Pi_{s}(\vec{q},i\nu_{n})|_{\tilde{\nu}+i\eta}$
and set $\vec{q}=0$, an explicit form of $\Pi_{s}(0,\tilde{\nu})$ is shown in
Appendix A

From the pole of above propagator in Eq.(5), the $\sigma$ mass can be obtained
by solving:

$1-2G_{S}\Pi_{s}(0,\tilde{\nu})=0,$ (9)

We have similar operation for pseudoscalar meson $\pi$. The operators in
polarization functions are defined as
$\tau^{\pm}=\frac{1}{\sqrt{2}}(\tau_{1}\pm i\tau_{2})$ where $\tau_{i}$ are
Pauli Matrice. In polarization functions, we choose
$\tau^{a}=\tau^{3},\tau^{b}=\tau^{3}$ for neutral pion and
$\tau^{a}=\tau^{+},\tau^{b}=\tau^{-}$ for charged pion. However, polarization
functions have the same form for different charged mesones.

$\displaystyle\Pi_{ps}(q^{2})$ $\displaystyle=-i\int
d^{4}\tilde{r}Tr_{sfc}[i\gamma^{5}\tau^{a}iS(0;\tilde{r})i\gamma^{5}\tau^{b}iS(\tilde{r};0)]e^{iq\cdot\tilde{r}}$
(10) $\displaystyle=4iN_{f}N_{c}\int\frac{d^{4}p}{(2\pi)^{4}}$
$\displaystyle\times\left\\{\frac{\left(p_{0}+q_{0}+\frac{1}{2}\omega\right)\left(p_{0}+\frac{1}{2}\omega\right)-M^{2}-(\vec{p}+\vec{q})\cdot\vec{p}}{\left[\left(p_{0}+q_{0}+\frac{1}{2}\omega\right)^{2}-(\vec{p}+\vec{q})^{2}-M^{2}\right]\left[\left(p_{0}+\frac{1}{2}\omega\right)^{2}-\vec{p}^{2}-M^{2}\right]}\right.$
$\displaystyle+\left.\frac{\left(p_{0}+q_{0}-\frac{1}{2}\omega\right)\left(p_{0}-\frac{1}{2}\omega\right)-M^{2}-(\vec{p}+\vec{q})\cdot\vec{p}}{\left[\left(p_{0}+q_{0}-\frac{1}{2}\omega\right)^{2}-(\vec{p}+\vec{q})^{2}-M^{2}\right]\left[\left(p_{0}-\frac{1}{2}\omega\right)^{2}-\vec{p}^{2}-M^{2}\right]}.\right\\}$

For finite temperature formalism with chemical potential, the polarization
function will be:

$\Pi_{ps}(\vec{q},i\nu_{n})=-4N_{f}N_{c}T\sum_{s=\pm}\sum_{N}\int\frac{d^{3}\vec{p}}{(2\pi)^{3}}\frac{[(i\tilde{\omega}_{N}+i\nu_{n})+\frac{1}{2}s\omega+\mu][i\tilde{\omega}_{N}+\frac{1}{2}s\omega+\mu]-M^{2}-(\vec{p}+\vec{q})\cdot\vec{p}}{[(i\tilde{\omega}_{N}+i\nu_{n}+\frac{1}{2}s\omega+\mu)^{2}-(\vec{p}+\vec{q})^{2}-M^{2}][(i\tilde{\omega}_{N}+\frac{1}{2}s\omega+\mu)^{2}-\vec{p}^{2}-M^{2}]}.$
(11)

Considered analytic continuation
$\Pi_{ps}(\vec{q},\tilde{\nu})=\Pi_{ps}(\vec{q},i\nu_{n})|_{\tilde{\nu}+i\eta}$
and set $\vec{q}=0$, an explicit form of $\Pi_{ps}(0,\tilde{\nu})$ is shown in
Appendix A

From the pole of above propagator, the pion mass can be obtained by solving:

$1-2G_{S}\Pi_{ps}(0,\tilde{\nu})=0.$ (12)

###  III.2 The $\rho$ meson

Following the Ref. Liu:2014uwa , we construct the vector meson in a similar
way with the rotation-modified quark propagators. For the 2-flavor model we
take the vector $\rho$ meson for example, its 1-loop polarization function
reads as

$\Pi^{\mu\nu,ab}(q)=-i\int
d^{4}\tilde{r}Tr_{sfc}[i\gamma^{\mu}\tau^{a}S(0;\tilde{r})i\gamma^{\nu}\tau^{b}S(\tilde{r};0)]e^{iq\cdot\tilde{r}}.$
(13)

As there is no isospin breaking in the quark propagators $S(0;\tilde{r})$, the
polarization functions of charged and neutral $\rho$ mesons are supposed to be
the same under rotation. Nonzero elements of the matrix reads as

$\Pi^{\mu\nu}_{\rho}=\left(\begin{array}[]{cccc}0&0&0&0\\\
0&\Pi^{11}&\Pi^{12}&0\\\ 0&\Pi^{21}&\Pi^{22}&0\\\ 0&0&0&\Pi^{33}\\\
\end{array}\right).$ (14)

The explicit expressions of matrix elements are shown in Appendix B. The
analysis of the Lorentz structure suggests the tensor can be decomposed
according to its polarization directions as follows

$\Pi^{\mu\nu}_{\rho}=A_{1}^{2}P^{\mu\nu}_{1}+A_{2}^{2}P^{\mu\nu}_{2}+A_{3}^{2}L^{\mu\nu}+A_{4}^{2}u^{\mu}u^{\nu},$
(15)

where $u^{\mu}$ is the four momentum in the rest frame. $u^{\mu}=(1,0,0,0)$ is
a unit vector. And the projection operators are given as:

$\displaystyle P^{\mu\nu}_{1}$
$\displaystyle=-\epsilon^{\mu}_{1}\epsilon^{\nu}_{1},(s_{z}=-1\text{ for
}\rho\text{ meson }),$ (16) $\displaystyle P^{\mu\nu}_{2}$
$\displaystyle=-\epsilon^{\mu}_{2}\epsilon^{\nu}_{2},(s_{z}=+1\text{ for
}\rho\text{ meson }),$ $\displaystyle L^{\mu\nu}$
$\displaystyle=-b^{\mu}b^{\nu},(s_{z}=0\text{ for }\rho\text{ meson }).$

where in flat frame $\epsilon^{\mu}_{1}=\frac{1}{\sqrt{2}}(0,1,i,0)$ and
$\epsilon^{\mu}_{2}=\frac{1}{\sqrt{2}}(0,1,-i,0)$ are the right and left-hand
polarization vectors respectively. And $b^{\mu}=(0,0,0,1)$ is the direction of
rotation. As a result the $\rho$ meson propagator can be decomposed in the
similar way as:

$D^{\mu\nu}_{\rho}(q^{2})=D_{1}(q^{2})P^{\mu\nu}_{1}+D_{2}(q^{2})P^{\mu\nu}_{2}+D_{3}(q^{2})L^{\mu\nu}+D_{4}(q^{2})u^{\mu}u^{\nu},$
(17)

where coefficients $D_{i}$ have the RPA summation forms as:

$D_{i}(q^{2})=\frac{2G_{V}}{1+2G_{V}A_{i}^{2}}.$ (18)

Again the momentum poles here are corresponding to masses of vector $\rho$
mesons which are solutions to equations:

$1+2G_{V}A_{i}^{2}=0,$ (19)

where

$\displaystyle A_{1}^{2}$ $\displaystyle=-(\Pi_{11}-i\Pi_{12}),(s_{z}=-1\text{
for }\rho\text{ meson }),$ (20) $\displaystyle A_{2}^{2}$
$\displaystyle=-\Pi_{11}-i\Pi_{12},(s_{z}=+1\text{ for }\rho\text{ meson }),$
$\displaystyle A_{3}^{2}$ $\displaystyle=-\Pi_{33},(s_{z}=0\text{ for
}\rho\text{ meson }).$

##  IV Numerical results and discussion

In order to evaluate the mass of $\rho$ meson at finite chemical potential and
relatively large vorticity, we choose the soft cut-off scheme to avoid the
leakage of the energy scale. The cut-off function is:

$f_{\Lambda}(\bm{p})=\frac{\Lambda^{10}}{\Lambda^{10}+\bm{p}^{10}},$ (21)

where $\Lambda=582$MeV. In numerical calculation, Momentum integrals are
understood as follows Frasca:2011zn

$\int\frac{d\bm{p}}{2\pi}\rightarrow\int\frac{d\bm{p}}{2\pi}f_{\Lambda}(\bm{p}).$
(22)

The other parameters are chosen as those in Ref Liu:2014uwa  , i.e.
$G_{S}\Lambda^{2}=2.388$ and $G_{V}\Lambda^{2}=1.73$ and the current quark
mass $m_{0}=5$MeV.

By neglecting mesons’ fluctuations it is easy to solve the gap equation of
chiral condensate at finite temperature as well as chemical potential under
rotation. As the phase diagram shown in Ref. Jiang:2016wvv ; Wang:2018sur ;
Chen:2015hfc  the vorticity serves as another kind of chemical potential which
would weaken the chiral condensate at finite temperature case and complement
the chemical potential at finite density case. As shown in Fig.(2b) there is a
crossover at medium temperature along the angular velocity. While at low
temperature the increase of chemical potential will change the 1st order
chiral restoration to a crossover in Fig.(2a), (2c) and (2d). As the phase
structure determines the macroscopic properties of the system it is reasonable
to expect that the dependence of meson masses on the angular velocity would be
smooth at medium temperature and density systems, while kinked at the 1st
order point for the low density systems.

###  IV.1 The scalar meson

Because of carrying no net angular momentum, the profile of scalar meson mass
is completely determined by the chiral symmetry in our model. For the zero
chemical potential case shown in Fig.(2a) and (2b), as angular velocity
increases the chiral condensate behaves the same as that in the Jiang:2016wvv
. At extremely low temperature the chiral restoration is 1st order and thus
the masses keep invariant and then jump together at the critical angular
velocity. While in hot matter the condensate keeps melting slowly until the
crossover range $\omega\sim 0.6$GeV. As the consequence, $\sigma$ meson mass
stays almost static and pions serve as Goldstone particles in the chiral
breaking phase. When the $\omega$ close to the crossover range they approach
each other and eventually become almost degenerate because of the chiral
symmetry restoration. The behavior at finite density could be understood with
chiral symmetry as well by noticing the order of phase transition. As
Fig.(2a), (2c) and (2d) shown, at low density, i.e. $\mu<100$MeV and zero
temperature, there is a 1st order gap at $\omega\simeq 0.8$GeV for the
dependence of the chiral condensate on angular velocity. After that the pion
would break the constraint of Goldstone theorem, that is the mass increases to
meet that of $\sigma$ meson which driven by the chiral symmetry. As the
chemical potential increase further, the phase transition would be weaken into
the crossover, and the mass dependence on the angular velocity would become
more and more smooth as shown in Fig.(2c) and (2d).

Figure 1: The constituent quark mass as a function of angular velocity for
different chemical potentials.

The constituent quark mass calculated from
$M=m-2G_{S}\left<\bar{\psi}\psi\right>$ as a function of angular velocity is
shown in Fig.1 for different chemical potentials. It is seen that the chiral
condensate shows 1st order phase transition at large angular velocity for
small chemical potentials and at small angular velocity for large chemical
potentials, this is in agreement with the results in Wang:2018sur , where it
has been observed that the 1st order phase transition shows up in two corners
of the 3D $T-\mu-\omega$ phase diagram.

(a) scalar meson mass as a function of angular velocity at $\mu=0MeV$

(b) scalar meson mass as a function of angular velocity at $T=150MeV$

(c) scalar meson mass as a function of angular velocity at $\mu=100MeV$

(d) scalar meson mass as a function of angular velocity at $\mu=200MeV$

Figure 2: scalar meson mass as a function of angular velocity at different
chemical potential and temperature.

From the numerical result it is clear that the angular velocity and chemical
potential are complementary to each other when driven the chiral restoration.
At low chemical potential the critical/crossover angular velocity is larger
and become smaller when the chemical potential is larger. However it is
obvious that the chemical potential and angular velocity are not exactly
equivalent to each other. Because physically the chemical potential is the
energy shift from the difference between particle and anti-particle, while the
shift induced by the rotation polarization is from the spin up and down
difference. From this aspect the $\pm\frac{1}{2}\omega$ could be treated as
the spin chemical potential. Analytically the difference could explicitly
observed in the gap equation and the polarization functions as follows

$\displaystyle\Pi_{s}(0,i\nu_{n})=$ $\displaystyle
N_{f}N_{c}\sum_{s=\pm}\int\frac{d^{3}\vec{p}}{(2\pi)^{3}}\left[Res1(\vec{p},\nu_{n})\theta\left(-\mu-\frac{s\omega}{2}+E_{p}\right)n_{f}\left(E_{p}-\mu-\frac{s\omega}{2},T\right)\right.$
(23)
$\displaystyle+Res3(\vec{p},\nu_{n})\theta\left(-\mu-\frac{s\omega}{2}+E_{p}\right)n_{f}\left(E_{p}-\mu-\frac{s\omega}{2},T\right)$
$\displaystyle-
Res1(\vec{p},\nu_{n})\theta\left(\mu+\frac{s\omega}{2}-E_{p}\right)n_{f}\left(-E_{p}+\mu+\frac{s\omega}{2},T\right)-Res2(\vec{p},\nu_{n})n_{f}\left(E_{p}+\mu+\frac{s\omega}{2},T\right)$
$\displaystyle-
Res3(\vec{p},\nu_{n})\theta\left(\mu+\frac{s\omega}{2}-E_{p}\right)n_{f}\left(-E_{p}+\mu+\frac{s\omega}{2},T\right)-Res4(\vec{p},\nu_{n})n_{f}\left(E_{p}+\mu+\frac{s\omega}{2},T\right)$
$\displaystyle+Res1(\vec{p},\nu_{n})\theta\left(\mu+\frac{s\omega}{2}-E_{p}\right)+Res3(\vec{p},\nu_{n})\theta\left(\mu+\frac{s\omega}{2}-E_{p}\right)$
$\displaystyle-
Res1\left(\vec{p},\nu_{n}\right)-Res3\left(\vec{p},\nu_{n}\right)\Big{]},$

where $Res1(\vec{p},\nu_{n}),Res2(\vec{p},\nu_{n}),Res3(\vec{p},\nu_{n})$ and
$Res4(\vec{p},\nu_{n})$ are residues in Eq.(39).

And the polarization function $\Pi_{ps}$ is given as:

$\displaystyle\Pi_{ps}(0,i\nu_{n})=$ $\displaystyle
N_{f}N_{c}\sum_{s=\pm}\int\frac{d^{3}\vec{p}}{(2\pi)^{3}}\left[Res1^{\prime}(\vec{p},\nu_{n})\theta\left(-\mu-\frac{s\omega}{2}+E_{p}\right)n_{f}\left(E_{p}-\mu-\frac{s\omega}{2},T\right)\right.$
(24)
$\displaystyle+Res3^{\prime}(\vec{p},\nu_{n})\theta\left(-\mu-\frac{s\omega}{2}+E_{p}\right)n_{f}\left(E_{p}-\mu-\frac{s\omega}{2},T\right)$
$\displaystyle-
Res1^{\prime}(\vec{p},\nu_{n})\theta\left(\mu+\frac{s\omega}{2}-E_{p}\right)n_{f}\left(-E_{p}+\mu+\frac{s\omega}{2},T\right)-Res2^{\prime}(\vec{p},\nu_{n})n_{f}\left(E_{p}+\mu+\frac{s\omega}{2},T\right)$
$\displaystyle-
Res3^{\prime}(\vec{p},\nu_{n})\theta\left(\mu+\frac{s\omega}{2}-E_{p}\right)n_{f}\left(-E_{p}+\mu+\frac{s\omega}{2},T\right)-Res4^{\prime}(\vec{p},\nu_{n})n_{f}\left(E_{p}+\mu+\frac{s\omega}{2},T\right)$
$\displaystyle+Res1^{\prime}(\vec{p},\nu_{n})\theta\left(\mu+\frac{s\omega}{2}-E_{p}\right)+Res3^{\prime}(\vec{p},\nu_{n})\theta\left(\mu+\frac{s\omega}{2}-E_{p}\right)$
$\displaystyle-
Res1^{\prime}(\vec{p},\nu_{n})-Res3^{\prime}(\vec{p},\nu_{n})\Big{]}.$

where
$Res1^{\prime}(\vec{p},\nu_{n}),Res2^{\prime}(\vec{p},\nu_{n}),Res3^{\prime}(\vec{p},\nu_{n})$
and $Res4^{\prime}(\vec{p},\nu_{n})$ are residues in Eq.(42).

It is clear that the functions depend on both the $\mu\pm\omega/2$
combinations. However it is also reasonable that the critical behavior would
take place at one of the angular velocitys which satisfy
$E_{p}-\mu\pm\frac{\omega}{2}=0$. If we choose both the chemical potential and
angular velocity positive, the $\omega=2(m-\mu)$ part would dominate the
critical behavior. Hence the chemical potential and angular velocity appear to
be complementary to each other on the determination of the critical point.

###  IV.2 The $\rho$ meson

Taking the direction of rotation as the $z$-axies, and the three components of
a massive vector meson can be represented as $s_{z}=\pm 1$ and $s_{z}=0$. And
the nonzero spin ones would be polarized by the so-call Barnett effect which
introduces the shift as $-{\vec{\omega}}\cdot{\vec{S}}$ to the energy levels
under rotation. In our 2-flavor model we take the $\rho$ meson for example to
explore the rotation-induced energy shift with the self-consistent numerical
calculations at the quark level. Fig.3 shows the numerical results for $\rho$
masses with $s_{z}=\pm 1$ and $s_{z}=0$ as functions of angular velocity at
temperature $T=10$MeV. It is obvious that the splitting mass curves have shown
the different influence of rotation. For the $s_{z}=0$ case there is no net
angular momentum for the particle polarization by the rotation. This makes the
mass dependence on the rotation is almost the same as the scalar case which
stay invariant as the chiral condensate below the critical angular velocity.
While for the $s_{z}=\pm 1$ cases the rotation polarization would generate the
energy shift $\mp\omega$ to the corresponding masses. This is confirmed by the
numerical results in Fig.3. The mass dependence on the angular velocity is two
straight lines for the $s_{z}=\pm 1$ components. The behavior could be
analytically proven with explicit form of the polarization functions. In the
pole approximation the masses are determined by the pole of the meson
propagators as Eq.(19). With straightforward computation in the Appendix the
polarization functions of vector meson satisfy

$\frac{1}{2}A_{1}^{2}(m_{\rho}+\omega)+\frac{1}{2}A_{2}^{2}(m_{\rho}-\omega)=A_{3}^{2}(m_{\rho}).$
(25)

This means the $m_{\rho}(\omega=0)-s_{z}\omega$ are exactly the masses of
$s_{z}=0,\pm 1$ components. The mass of $s_{z}=1$ spin component decreases
linearly with the angular velocity, and reaches zero at the critical angular
velocity $\omega_{c}=m_{\rho}(\omega=0)$. Beyond the critical angular velocity
$\omega_{c}$, the $s_{z}=1$ spin component of vector meson will develop
condensation in the vacuum and this indicates that the system will be
spontaneously spin polarized under strong rotation.

Figure 3: $\rho$ meson masses as a function of angular velocity at temperature
$T=10MeV$.

##  V Conclusion

Using the NJL model with vector channel interaction we have calculated the
scalar, pseudoscalar and vector mesons’ masses at finite temperature, chemical
potential and angular velocity. In the RPA and pole approximation the mesons
are treated as the effective degree of freedoms which transmit the interaction
between quarks. And the masses are determined by the polarization functions.
This approximation could preserve the Goldstone theorem explicitly although
the back reaction of meson to the phase transition is neglected. Because of
the four-fermion point interaction and pole approximation the microscopic
details of mesons have been lost. And all of them behave as fundamental
particles which are polarized by rotation according to their net spin angular
momentum. For the scalar and pseudoscalar cases the mass spectra are
controlled by the chiral condensate which is the main mechanism generating the
hadron mass in NJL model. At low temperature and chemical potential the chiral
restoration is 1st order which make the meson masses a sudden jump at the
critical angular velocity. While as the temperature or chemical potential
increasing the phase transition would degenerate to crossovers which also
smoothen the mass curves of mesons along the angular velocity. It is easy to
expect that at large enough angular velocity the vector condensate vacuum
would be preferred and the corresponding effective mass should be zero. That
is why we have only studied the vector meson’s mass behavior below the
$\omega=m_{\rho}$. It is found that although the polarization function
computation is complicated masses of the three components $s_{z}=0,\pm 1$ are
the same as the result by treating them as the fundamental particles, that is
$m_{\rho}$ and $m_{\rho}\pm\omega$. Once the chiral restored the vector
condensate would emerge simultaneously which will be studied in our next work.

In non-central heavy-ion collisions, the created system carries large angular
momentum. The properties of particles will be changed under rotating medium.
In this paper, we investigated the behavior of scalar and vector meson mass
under the rotation. It is found that the behavior of scalar and pseudoscalar
meson masses under the angular velocity $\omega$ is similar to that at finite
chemical potential, both rely on the behavior of constituent quark mass and
reflect the property related to the chiral symmetry. However, masses of vector
meson have more profound relation with rotation. After tedious calculation, it
turns out that at low temperature and small chemical potenial, the mass for
spin component $s_{z}=0,\pm 1$ of vector meson under rotation shows very
simple mass splitting relation
$m_{\rho}^{s_{z}}(\omega)=m_{\rho}(\omega=0)-\omega s_{z}$, similar to the
Zeeman splitting of charged meson under magnetic fields. Especially it is
noticed that the mass of spin component $s_{z}=1$ vector meson $\rho$
decreases linearly with $\omega$ and reaches zero at
$\omega_{c}=m_{\rho}(\omega=0)$, this indicates the system will develop
$s_{z}=1$ vector meson condensation and the system will be spontaneously spin
polarized under rotation. It deserves further study to compare the spin
polarization with $s_{z}=1$ vector meson condensation and the spin
polarization defined by the condensation of
$<{\bar{\psi}}i\sigma^{\mu\nu}\psi>$ proposed in Tatsumi-spinpolarization .

###### Acknowledgements.

We thank Kun Xu for useful discussion. M.H.is supported by the NSFC under
Grant Nos. 11725523 and 11735007, Chinese Academy of Sciences under Grant No.
XDPB09, the start-up funding from University of Chinese Academy of
Sciences(UCAS), and the Fundamental Research Funds for the Central
Universities . Y.J. is supported by NSFC under Grant No.11875002 and the
Zhuobai Program of Beihang University.

##  Appendix A The polarization function under rotation

For the $\sigma$ meson, substituting the Eq. 4 into the definition of
polarization function, the scalar proper polarization function under rotation
is given as

$\displaystyle\Pi_{s}(q)$ $\displaystyle=-i\int
d^{4}\tilde{r}Tr_{sfc}\left[iS(0;\tilde{r})iS(\tilde{r};0)\right]e^{iq\cdot\tilde{r}}$
(26)
$\displaystyle=-iN_{f}N_{c}\sum_{n=-\infty}^{+\infty}\sum_{l=-\infty}^{+\infty}\int
d^{4}\tilde{r}\int\frac{dk_{0}dk_{z}}{(2\pi)^{2}}\int_{0}^{+\infty}\frac{k_{t}dk_{t}}{2\pi}\int\frac{dp_{0}dp_{z}}{(2\pi)^{2}}\int_{0}^{+\infty}\frac{p_{t}dp_{t}}{2\pi}$
$\displaystyle\times
Tr(A_{n}B_{l})\times\frac{e^{-ik_{0}t+ik_{z}z}e^{n\phi}}{[k_{0}+(n+\frac{1}{2})\omega]^{2}-k_{t}^{2}-k_{z}^{2}-M^{2}+i\epsilon}\times\frac{e^{ip_{0}t-ip_{z}z}e^{-l\phi}}{[p_{0}+(l+\frac{1}{2})\omega]^{2}-p_{t}^{2}-p_{z}^{2}-M^{2}+i\epsilon}\times
e^{iq\cdot\tilde{r}},$

where

$\displaystyle A_{n}$
$\displaystyle=\tiny\begin{pmatrix}\begin{smallmatrix}\left(k_{0}+M+\left(n+\frac{1}{2}\right)\omega\right)J_{n}(k_{t}r)J_{n}(0)&0&-k_{z}J_{n}(k_{t}r)J_{n}(0)&ik_{t}J_{n}(k_{t}r)J_{n+1}(0)\\\
0&\left(k_{0}+M+\left(n+\frac{1}{2}\right)\omega\right)e^{i\phi}J_{n+1}(k_{t}r)J_{n+1}(0)&-ie^{i\phi}k_{t}J_{n+1}(k_{t}r)J_{n}(0)&k_{z}e^{i\phi}J_{n+1}(k_{t}r)J_{n+1}(0)\\\
k_{z}J_{n}(k_{t}r)J_{n}(0)&-ik_{t}J_{n}(k_{t}r)J_{n+1}(0)&-\left(k_{0}-M+\left(n+\frac{1}{2}\right)\omega\right)J_{n}(k_{t}r)J_{n}(0)&0\\\
ie^{i\phi}k_{t}J_{n+1}(k_{t}r)J_{n}(0)&-k_{z}e^{i\phi}J_{n+1}(k_{t}r)J_{n+1}(0)&0&-\left(k_{0}-M+\left(n+\frac{1}{2}\right)\omega\right)e^{i\phi}J_{n+1}(k_{t}r)J_{n+1}(0)\\\
\end{smallmatrix}\end{pmatrix},$ (27) $\displaystyle B_{l}$
$\displaystyle=\tiny\begin{pmatrix}\begin{smallmatrix}\left(M+p_{0}+\left(l+\frac{1}{2}\right)\omega\right)J_{l}(p_{t}r)J_{l}(0)&0&-p_{z}J_{l}(p_{t}r)J_{l}(0)&ip_{t}e^{-i\phi}J_{l+1}(p_{t}r)J_{l}(0)\\\
0&\left(M+p_{0}+\left(l+\frac{1}{2}\right)\omega\right)e^{-i\phi}J_{l+1}(p_{t}r)J_{l+1}(0)&-ip_{t}J_{l}(p_{t}r)J_{l+1}(0)&p_{z}e^{-i\phi}J_{l+1}(p_{t}r)J_{l+1}(0)\\\
p_{z}J_{l}(p_{t}r)J_{l}(0)&-ip_{t}e^{-i\phi}J_{l+1}(p_{t}r)J_{l}(0)&-\left(-M+p_{0}+\left(l+\frac{1}{2}\right)\omega\right)J_{l}(p_{t}r)J_{l}(0)&0\\\
ip_{t}J_{l}(p_{t}r)J_{l+1}(0)&-p_{z}e^{-i\phi}J_{l+1}(p_{t}r)J_{l+1}(0)&0&-\left(-M+p_{0}+\left(l+\frac{1}{2}\right)\omega\right)e^{-i\phi}J_{l+1}(p_{t}r)J_{l+1}(0)\\\
\end{smallmatrix}\end{pmatrix}.$

Here we give a matrix form instead of the summation of projection operators in
Eq. 4. The symbol ”$Tr_{sfc}$” stands for evaluate the trace on spinor, flavor
and color space. After a tedious calculation, we get

$\displaystyle Tr(A_{n}B_{l})$
$\displaystyle=[\frac{1}{2}(2k_{0}+2n\omega+\omega)(2l\omega+2p_{0}+\omega)+2M^{2}-2k_{z}p_{z}]J_{l}(0)J_{n}(0)J_{n}(k_{t}r)J_{l}(p_{t}r)$
(28)
$\displaystyle+[\frac{1}{2}(2k_{0}+2n\omega+\omega)(2l\omega+2p_{0}+\omega)+2M^{2}-2k_{z}p_{z}]J_{l+1}(0)J_{n+1}(0)J_{n+1}(k_{t}r)J_{l+1}(p_{t}r)$
$\displaystyle-2k_{t}p_{t}J_{l+1}(0)J_{n+1}(0)J_{n}(k_{t}r)J_{l}(p_{t}r)$
$\displaystyle-2k_{t}p_{t}J_{l}(0)J_{n}(0)J_{n+1}(k_{t}r)J_{l+1}(p_{t}r).$

When $n\neq 0$, it’s obvious that $J_{n}(0)=0$ and $J_{0}(0)=1$. As a
consequence, the result of the summation will have finite terms

$\displaystyle\Pi_{s}(q)$ $\displaystyle=-i\int
d^{4}\tilde{r}Tr_{sfc}\left[S(0;\tilde{r})S(\tilde{r};0)\right]e^{iq\cdot\tilde{r}}$
(29) $\displaystyle=-iN_{f}N_{c}\int
d^{4}\tilde{r}\int\frac{dk_{0}dk_{z}}{(2\pi)^{2}}\int_{0}^{+\infty}\frac{k_{t}dk_{t}}{2\pi}\int\frac{dp_{0}dp_{z}}{(2\pi)^{2}}\int_{0}^{+\infty}\frac{p_{t}dp_{t}}{2\pi}$
$\displaystyle\times\left\\{\left[\frac{1}{2}(2k_{0}+\omega)(2p_{0}+\omega)+2M^{2}-2k_{z}p_{z}\right]\times\frac{J_{0}(k_{t}r)J_{0}(p_{t}r)e^{-ik_{0}t+ik_{z}z}e^{ip_{0}t-ip_{z}z}}{\left[(k_{0}+\frac{1}{2}\omega)^{2}-k_{t}^{2}-k_{z}^{2}-M^{2}+i\epsilon\right]\left[(p_{0}+\frac{1}{2}\omega)^{2}-p_{t}^{2}-p_{z}^{2}-M^{2}+i\epsilon\right]}\right.$
$\displaystyle+\left[\frac{1}{2}(2k_{0}-\omega)(2p_{0}-\omega)+2M^{2}-2k_{z}p_{z}\right]\times\frac{J_{0}(k_{t}r)J_{0}(p_{t}r)e^{-ik_{0}t+ik_{z}z}e^{ip_{0}t-ip_{z}z}}{\left[(k_{0}-\frac{1}{2}\omega)^{2}-k_{t}^{2}-k_{z}^{2}-M^{2}+i\epsilon\right]\left[(p_{0}-\frac{1}{2}\omega)^{2}-p_{t}^{2}-p_{z}^{2}-M^{2}+i\epsilon\right]}$
$\displaystyle-2k_{t}p_{t}\times\frac{J_{1}(k_{t}r)J_{1}(p_{t}r)e^{-ik_{0}t+ik_{z}z}e^{ip_{0}t-ip_{z}z}}{\left[(k_{0}+\frac{1}{2}\omega)^{2}-k_{t}^{2}-k_{z}^{2}-M^{2}+i\epsilon\right]\left[(p_{0}+\frac{1}{2}\omega)^{2}-p_{t}^{2}-p_{z}^{2}-M^{2}+i\epsilon\right]}$
$\displaystyle-2k_{t}p_{t}\times\left.\frac{J_{-1}(k_{t}r)J_{-1}(p_{t}r)e^{-ik_{0}t+ik_{z}z}e^{ip_{0}t-ip_{z}z}}{\left[(k_{0}-\frac{1}{2}\omega)^{2}-k_{t}^{2}-k_{z}^{2}-M^{2}+i\epsilon\right]\left[(p_{0}-\frac{1}{2}\omega)^{2}-p_{t}^{2}-p_{z}^{2}-M^{2}+i\epsilon\right]}\right\\}\times
e^{iq\cdot\tilde{r}}.$

Applying the integral representation of Bessel functions, the polarization
function can be simplified. In integral representation, Bessel functions are
expressed as:

$\displaystyle J_{n}(r)$
$\displaystyle=\frac{1}{2\pi}\int_{-\pi}^{+\pi}e^{i(r\sin\theta-n\theta)}d\theta,$
(30) $\displaystyle J_{0}(r)$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}e^{\pm ir\cos\theta}d\theta,$
$\displaystyle J_{1}(r)$ $\displaystyle=\frac{1}{2\pi
i}\int_{0}^{2\pi}e^{ir\cos\theta\pm i\theta}d\theta,$ $\displaystyle J_{1}(r)$
$\displaystyle=-\frac{1}{2\pi i}\int_{0}^{2\pi}e^{-ir\cos\theta\pm
i\theta}d\theta.$

Let $\vec{k_{t}}=(k_{t},\phi+\theta)=(k_{x},k_{y}),\vec{r}=(r,\phi)=(x,y)$,
and then we have the transformation formulae:

$\int_{0}^{\infty}\frac{k_{t}dk_{t}}{2\pi}\int_{0}^{2\pi}\frac{d\theta}{2\pi
i}ik_{t}e^{i\phi}e^{ik_{t}r\cos\theta+i\theta}=\int\frac{dk_{x}dk_{y}}{(2\pi)^{2}}(k_{x}+ik_{y})e^{i\vec{k_{t}}\cdot\vec{r}},$
(31)
$\int_{0}^{\infty}\frac{k_{t}dk_{t}}{2\pi}\int_{0}^{2\pi}\frac{d\theta}{2\pi
i}ik_{t}e^{-i\phi}e^{-ik_{t}r\cos\theta-i\theta}=\int\frac{dk_{x}dk_{y}}{(2\pi)^{2}}(k_{x}-ik_{y})e^{-i\vec{k_{t}}\cdot\vec{r}}.$
(32)

Applied the transformation formulae, the polarization function for scalar
meson can be expressed without Bessel function. It will be more efficient for
numerical calculation.

$\displaystyle\Pi_{s}(q)$ $\displaystyle=-iN_{f}N_{c}\int
d^{4}\tilde{r}\int\frac{d^{4}k}{(2\pi)^{4}}\int\frac{d^{4}p}{(2\pi)^{4}}$ (33)
$\displaystyle\times\left\\{\left[\frac{1}{2}(2k_{0}+\omega)(2p_{0}+\omega)+2M^{2}-2k_{z}p_{z}\right]\times\frac{e^{-ik\cdot\tilde{r}}e^{ip\cdot\tilde{r}}}{\left[\left(k_{0}+\frac{1}{2}\omega\right)^{2}-\vec{k}^{2}-M^{2}+i\epsilon\right]\left[\left(p_{0}+\frac{1}{2}\omega\right)^{2}-\vec{p}^{2}-M^{2}+i\epsilon\right]}\right.$
$\displaystyle+\left[\frac{1}{2}(2k_{0}-\omega)(2p_{0}-\omega)+2M^{2}-2k_{z}p_{z}\right]\times\frac{e^{-ik\cdot\tilde{r}}e^{ip\cdot\tilde{r}}}{\left[(k_{0}-\frac{1}{2}\omega)^{2}-\vec{k}^{2}-M^{2}+i\epsilon\right]\left[\left(p_{0}-\frac{1}{2}\omega\right)^{2}-\vec{p}^{2}-M^{2}+i\epsilon\right]}$
$\displaystyle-2(k_{x}+ik_{y})(p_{x}-ip_{y})\times\frac{e^{-ik\cdot\tilde{r}}e^{ip\cdot\tilde{r}}}{\left[\left(k_{0}+\frac{1}{2}\omega\right)^{2}-\vec{k}^{2}-M^{2}+i\epsilon\right]\left[\left(p_{0}+\frac{1}{2}\omega\right)^{2}-\vec{p}^{2}-M^{2}+i\epsilon\right]}$
$\displaystyle-2(k_{x}+ik_{y})(p_{x}-ip_{y})\times\frac{e^{-ik\cdot\tilde{r}}e^{ip\cdot\tilde{r}}}{\left[\left(k_{0}-\frac{1}{2}\omega\right)^{2}-\vec{k}^{2}-M^{2}+i\epsilon\right]\left[\left(p_{0}-\frac{1}{2}\omega\right)^{2}-\vec{p}^{2}-M^{2}+i\epsilon\right]}\\}\times
e^{iq\cdot\tilde{r}}.$

Furthermore, integrating the $\tilde{r}$ and $k$ analytically, we can get the
polarization function as following:

$\displaystyle\Pi_{s}(q)$
$\displaystyle=-iN_{f}N_{c}\int\frac{d^{4}p}{(2\pi)^{4}}$ (34)
$\displaystyle\times\left\\{\frac{\left[2\left(p_{0}+q_{0}+\frac{1}{2}\omega\right)\left(p_{0}+\frac{1}{2}\omega\right)+2M^{2}-2(p_{z}+q_{z})p_{z}\right]-2\left[(p_{x}+q_{x})+i(p_{y}+q_{y})\right](p_{x}-ip_{y})}{\left[\left(p_{0}+q_{0}+\frac{1}{2}\omega\right)^{2}-(\vec{p}+\vec{q})^{2}-M^{2}+i\epsilon\right]\left[\left(p_{0}+\frac{1}{2}\omega\right)^{2}-\vec{p}^{2}-M^{2}+i\epsilon\right]}\right.$
$\displaystyle+\left.\frac{\left[2\left(p_{0}+q_{0}-\frac{1}{2}\omega\right)\left(p_{0}-\frac{1}{2}\omega\right)+2M^{2}-2(p_{z}+q_{z})p_{z}\right]-2\left[(p_{x}+q_{x})+i(p_{y}+q_{y})\right](p_{x}-ip_{y})}{\left[\left(p_{0}+q_{0}-\frac{1}{2}\omega\right)^{2}-(\vec{p}+\vec{q})^{2}-M^{2}+i\epsilon\right]\left[\left(p_{0}-\frac{1}{2}\omega\right)^{2}-\vec{p}^{2}-M^{2}+i\epsilon\right]}\right\\}.$

Due to symmetric analysis for integration, the expression can be simplified as
following:

$\displaystyle\Pi_{s}(q^{2})$
$\displaystyle=-2iN_{f}N_{c}\int\frac{d^{4}p}{(2\pi)^{4}}$ (35)
$\displaystyle\times\left\\{\frac{\left(p_{0}+q_{0}+\frac{1}{2}\omega\right)\left(p_{0}+\frac{1}{2}\omega\right)+M^{2}-(\vec{p}+\vec{q})\vec{p}}{\left[\left(p_{0}+q_{0}+\frac{1}{2}\omega\right)^{2}-(\vec{p}+\vec{q})^{2}-M^{2}\right]\left[\left(p_{0}+\frac{1}{2}\omega\right)^{2}-\vec{p}^{2}-M^{2}\right]}\right.$
$\displaystyle+\left.\frac{\left(p_{0}+q_{0}-\frac{1}{2}\omega\right)\left(p_{0}-\frac{1}{2}\omega\right)+M^{2}-(\vec{p}+\vec{q})\vec{p}}{\left[\left(p_{0}+q_{0}-\frac{1}{2}\omega\right)^{2}-(\vec{p}+\vec{q})^{2}-M^{2}\right]\left[\left(p_{0}-\frac{1}{2}\omega\right)^{2}-\vec{p}^{2}-M^{2}\right]}\right\\}.$

For finite temperature formalism:

$p_{0}\rightarrow i\tilde{\omega}_{N},\hskip 10.0ptq_{0}\rightarrow
i\nu_{n},\hskip 10.0pt\int\frac{p_{0}}{2\pi}\rightarrow iT\sum_{N},\hskip
10.0pt\tilde{\omega}_{N}=(2N+1)\pi T.$ (36)

The polarization function at finite temperature and chemical potential under
rotation can be rewritten as:

$\Pi_{s}(\vec{q},i\nu_{n})=2N_{f}N_{c}T\sum_{s=\pm}\sum_{N}\int\frac{d^{3}\vec{p}}{(2\pi)^{3}}\frac{\left[(i\tilde{\omega}_{N}+i\nu_{n})+\frac{1}{2}s\omega+\mu\right]\left(i\tilde{\omega}_{N}+\frac{1}{2}s\omega+\mu\right)+M^{2}-(\vec{p}+\vec{q})\cdot\vec{p}}{\left[\left(i\tilde{\omega}_{N}+i\nu_{n}+\frac{1}{2}s\omega+\mu\right)^{2}-(\vec{p}+\vec{q})^{2}-M^{2}\right]\left[\left(i\tilde{\omega}_{N}+\frac{1}{2}s\omega+\mu\right)^{2}-\vec{p}^{2}-M^{2}\right]}.$
(37)

Setting $\vec{q}=0$, Matsubara Summation will give us a result in term of
residue theorem:

$\displaystyle\Pi_{s}(0,i\nu_{n})=$ $\displaystyle
N_{f}N_{c}\sum_{s=\pm}\int\frac{d^{3}\vec{p}}{(2\pi)^{3}}\left[Res1(\vec{p},\nu_{n})\theta\left(-\mu-\frac{s\omega}{2}+E_{p}\right)n_{f}\left(E_{p}-\mu-\frac{s\omega}{2},T\right)\right.$
(38)
$\displaystyle+Res3(\vec{p},\nu_{n})\theta\left(-\mu-\frac{s\omega}{2}+E_{p}\right)n_{f}\left(E_{p}-\mu-\frac{s\omega}{2},T\right)$
$\displaystyle-
Res1(\vec{p},\nu_{n})\theta\left(\mu+\frac{s\omega}{2}-E_{p}\right)n_{f}\left(-E_{p}+\mu+\frac{s\omega}{2},T\right)-Res2(\vec{p},\nu_{n})n_{f}\left(E_{p}+\mu+\frac{s\omega}{2},T\right)$
$\displaystyle-
Res3(\vec{p},\nu_{n})\theta\left(\mu+\frac{s\omega}{2}-E_{p}\right)n_{f}\left(-E_{p}+\mu+\frac{s\omega}{2},T\right)-Res4(\vec{p},\nu_{n})n_{f}\left(E_{p}+\mu+\frac{s\omega}{2},T\right)$
$\displaystyle+Res1(\vec{p},\nu_{n})\theta\left(\mu+\frac{s\omega}{2}-E_{p}\right)+Res3(\vec{p},\nu_{n})\theta\left(\mu+\frac{s\omega}{2}-E_{p}\right)$
$\displaystyle-
Res1\left(\vec{p},\nu_{n}\right)-Res3\left(\vec{p},\nu_{n}\right)\Big{]},$

where $n_{f}(x,T)=\frac{1}{e^{x/T}+1}$ is distribution function, and four
residues are given as following:

$\displaystyle Res1(\vec{p},\nu_{n})=$
$\displaystyle\frac{-i\nu_{n}E_{p}+E_{p}^{2}+M^{2}-\vec{p}^{2}}{E_{p}\left(-\nu_{n}^{2}-2i\nu_{n}E_{p}\right)},$
(39) $\displaystyle Res2(\vec{p},\nu_{n})=$
$\displaystyle\frac{-i\nu_{n}E_{p}-E_{p}^{2}-M^{2}+\vec{p}^{2}}{E_{p}\left(-\nu_{n}^{2}+2i\nu_{n}E_{p}\right)},$
$\displaystyle Res3(\vec{p},\nu_{n})=$
$\displaystyle\frac{+i\nu_{n}E_{p}+E_{p}^{2}+M^{2}-\vec{p}^{2}}{E_{p}\left(-\nu_{n}^{2}+2i\nu_{n}E_{p}\right)},$
$\displaystyle Res4(\vec{p},\nu_{n})=$
$\displaystyle\frac{i\nu_{n}E_{p}-E_{p}^{2}-M^{2}+\vec{p}^{2}}{E_{p}\left(-\nu_{n}^{2}-2i\nu_{n}E_{p}\right)}.$

We should notice that $E_{\bm{p}}=\sqrt{\bm{p}^{2}+M^{2}}$ and quark mass $M$
is a function of angular velocity $\omega$. For psuadoscalar meson, the finite
temperature version polarization function is:

$\Pi_{ps}(\vec{q},i\nu_{n})=-4N_{f}N_{c}T\sum_{s=\pm}\sum_{N}\int\frac{d^{3}\vec{p}}{(2\pi)^{3}}\frac{[(i\tilde{\omega}_{N}+i\nu_{n})+\frac{1}{2}s\omega+\mu][i\tilde{\omega}_{N}+\frac{1}{2}s\omega+\mu]-M^{2}-(\vec{p}+\vec{q})\cdot\vec{p}}{[(i\tilde{\omega}_{N}+i\nu_{n}+\frac{1}{2}s\omega+\mu)^{2}-(\vec{p}+\vec{q})^{2}-M^{2}][(i\tilde{\omega}_{N}+\frac{1}{2}s\omega+\mu)^{2}-\vec{p}^{2}-M^{2}]}.$
(40)

Seting $\vec{q}=0$, Matsubara Summation gives:

$\displaystyle\Pi_{ps}(0,i\nu_{n})=$ $\displaystyle
N_{f}N_{c}\sum_{s=\pm}\int\frac{d^{3}\vec{p}}{(2\pi)^{3}}\left[Res1^{\prime}(\vec{p},\nu_{n})\theta\left(-\mu-\frac{s\omega}{2}+E_{p}\right)n_{f}\left(E_{p}-\mu-\frac{s\omega}{2},T\right)\right.$
(41)
$\displaystyle+Res3^{\prime}(\vec{p},\nu_{n})\theta\left(-\mu-\frac{s\omega}{2}+E_{p}\right)n_{f}\left(E_{p}-\mu-\frac{s\omega}{2},T\right)$
$\displaystyle-
Res1^{\prime}(\vec{p},\nu_{n})\theta\left(\mu+\frac{s\omega}{2}-E_{p}\right)n_{f}\left(-E_{p}+\mu+\frac{s\omega}{2},T\right)-Res2^{\prime}(\vec{p},\nu_{n})n_{f}\left(E_{p}+\mu+\frac{s\omega}{2},T\right)$
$\displaystyle-
Res3^{\prime}(\vec{p},\nu_{n})\theta\left(\mu+\frac{s\omega}{2}-E_{p}\right)n_{f}\left(-E_{p}+\mu+\frac{s\omega}{2},T\right)-Res4^{\prime}(\vec{p},\nu_{n})n_{f}\left(E_{p}+\mu+\frac{s\omega}{2},T\right)$
$\displaystyle+Res1^{\prime}(\vec{p},\nu_{n})\theta\left(\mu+\frac{s\omega}{2}-E_{p}\right)+Res3^{\prime}(\vec{p},\nu_{n})\theta\left(\mu+\frac{s\omega}{2}-E_{p}\right)$
$\displaystyle-
Res1^{\prime}(\vec{p},\nu_{n})-Res3^{\prime}(\vec{p},\nu_{n})\Big{]},$

where

$\displaystyle Res1^{\prime}(\vec{p},\nu_{n})=$
$\displaystyle\frac{i\nu_{n}E_{p}+E_{p}^{2}+M^{2}+\vec{p}^{2}}{E_{p}\left(-\nu_{n}^{2}-2i\nu_{n}E_{p}\right)},$
(42) $\displaystyle Res2^{\prime}(\vec{p},\nu_{n})=$
$\displaystyle\frac{i\nu_{n}E_{p}+E_{p}^{2}-M^{2}-\vec{p}^{2}}{E_{p}\left(-\nu_{n}^{2}+2i\nu_{n}E_{p}\right)},$
$\displaystyle Res3^{\prime}(\vec{p},\nu_{n})=$
$\displaystyle\frac{-i\nu_{n}E_{p}-E_{p}^{2}+M^{2}+\vec{p}^{2}}{E_{p}\left(-\nu_{n}^{2}+2i\nu_{n}E_{p}\right)},$
$\displaystyle Res4^{\prime}(\vec{p},\nu_{n})=$
$\displaystyle\frac{-i\nu_{n}E_{p}+E_{p}^{2}-M^{2}-\vec{p}^{2}}{E_{p}\left(-\nu_{n}^{2}-2i\nu_{n}E_{p}\right)}.$

##  Appendix B The polarization function for vector meson under rotation

For $\rho$ meson, the polarization function with one loop contribution can be
expressed as

$\Pi^{\mu\nu,ab}=-i\int
d^{4}\tilde{r}Tr_{sfc}\left[i\gamma^{\mu}\tau^{a}S(0;\tilde{r})i\gamma^{\nu}\tau^{b}S(\tilde{r};0)\right]e^{iq\cdot\tilde{r}}.$
(43)

Using the approach introduced in Appendix A. It is obvious that the charge of
$\rho$ meson will make on difference with polarization function under
rotation. We can get the nonzero elements of the matrix

$\Pi^{\mu\nu}_{\rho}=\left(\begin{array}[]{cccc}0&0&0&0\\\
0&\Pi^{11}&\Pi^{12}&0\\\ 0&\Pi^{21}&\Pi^{22}&0\\\ 0&0&0&\Pi^{33}\\\
\end{array}\right).$ (44)

Using the same method in Appendix A and setting $\vec{q}=0$, we will get the
nonzero elements which is given by:

$\displaystyle\Pi^{11}(q_{0})$
$\displaystyle=N_{f}N_{c}\int\frac{d^{4}p}{(2\pi)^{4}}$ (45)
$\displaystyle\times\left\\{-\frac{2M^{2}-2\left(p_{0}+\frac{\omega}{2}\right)\left(p_{0}+q_{0}-\frac{\omega}{2}\right)-2p_{x}^{2}+2p_{y}^{2}+2p_{z}^{2}}{[\left(p_{0}+\frac{\omega}{2}\right)^{2}-\vec{p}^{2}-M^{2}][\left(p_{0}+q_{0}-\frac{\omega}{2}\right)^{2}-\vec{p}^{2}-M^{2}]}-\frac{2M^{2}-2\left(p_{0}-\frac{\omega}{2}\right)\left(p_{0}+q_{0}+\frac{\omega}{2}\right)-2p_{x}^{2}+2p_{y}^{2}+2p_{z}^{2}}{[\left(p_{0}-\frac{\omega}{2}\right)^{2}-\vec{p}^{2}-M^{2}][\left(p_{0}+q_{0}+\frac{\omega}{2}\right)^{2}-\vec{p}^{2}-M^{2}]}\right\\},$

$\displaystyle\Pi^{12}(q_{0})$
$\displaystyle=-iN_{f}N_{c}\int\frac{d^{4}p}{(2\pi)^{4}}$ (46)
$\displaystyle\times\left\\{\frac{2M^{2}-2\left(p_{0}+\frac{\omega}{2}\right)\left(p_{0}+q_{0}-\frac{\omega}{2}\right)-2p_{x}^{2}+2p_{y}^{2}+2p_{z}^{2}}{[\left(p_{0}+\frac{\omega}{2}\right)^{2}-\vec{p}^{2}-M^{2}][\left(p_{0}+q_{0}-\frac{\omega}{2}\right)^{2}-\vec{p}^{2}-M^{2}]}-\frac{2M^{2}-2\left(p_{0}-\frac{\omega}{2}\right)\left(p_{0}+q_{0}+\frac{\omega}{2}\right)-2p_{x}^{2}+2p_{y}^{2}+2p_{z}^{2}}{[\left(p_{0}-\frac{\omega}{2}\right)^{2}-\vec{p}^{2}-M^{2}][\left(p_{0}+q_{0}+\frac{\omega}{2}\right)^{2}-\vec{p}^{2}-M^{2}]}\right\\},$

$\displaystyle\Pi^{21}(q_{0})$
$\displaystyle=iN_{f}N_{c}\int\frac{d^{4}p}{(2\pi)^{4}}$ (47)
$\displaystyle\times\left\\{\frac{2M^{2}-2\left(p_{0}+\frac{\omega}{2}\right)\left(p_{0}+q_{0}-\frac{\omega}{2}\right)+2p_{x}^{2}-2p_{y}^{2}+2p_{z}^{2}}{[\left(p_{0}+\frac{\omega}{2}\right)^{2}-\vec{p}^{2}-M^{2}][\left(p_{0}+q_{0}-\frac{\omega}{2}\right)^{2}-\vec{p}^{2}-M^{2}]}-\frac{2M^{2}-2\left(p_{0}-\frac{\omega}{2}\right)\left(p_{0}+q_{0}+\frac{\omega}{2}\right)+2p_{x}^{2}-2p_{y}^{2}+2p_{z}^{2}}{[\left(p_{0}-\frac{\omega}{2}\right)^{2}-\vec{p}^{2}-M^{2}][\left(p_{0}+q_{0}+\frac{\omega}{2}\right)^{2}-\vec{p}^{2}-M^{2}]}\right\\},$

$\displaystyle\Pi^{22}(q_{0})$
$\displaystyle=-N_{f}N_{c}\int\frac{d^{4}p}{(2\pi)^{4}}$ (48)
$\displaystyle\times\left\\{\frac{2M^{2}-2\left(p_{0}+\frac{\omega}{2}\right)\left(p_{0}+q_{0}-\frac{\omega}{2}\right)-2p_{x}^{2}+2p_{y}^{2}+2p_{z}^{2}}{[\left(p_{0}+\frac{\omega}{2}\right)^{2}-\vec{p}^{2}-M^{2}][\left(p_{0}+q_{0}-\frac{\omega}{2}\right)^{2}-\vec{p}^{2}-M^{2}]}+\frac{2M^{2}-2\left(p_{0}-\frac{\omega}{2}\right)\left(p_{0}+q_{0}+\frac{\omega}{2}\right)-2p_{x}^{2}+2p_{y}^{2}+2p_{z}^{2}}{[\left(p_{0}-\frac{\omega}{2}\right)^{2}-\vec{p}^{2}-M^{2}][\left(p_{0}+q_{0}+\frac{\omega}{2}\right)^{2}-\vec{p}^{2}-M^{2}]}\right\\},$

$\displaystyle\Pi^{33}(q_{0})$
$\displaystyle=-N_{f}N_{c}\int\frac{d^{4}p}{(2\pi)^{4}}$ (49)
$\displaystyle\times\left\\{\frac{2M^{2}-2\left(p_{0}-\frac{\omega}{2}\right)\left(p_{0}+q_{0}-\frac{\omega}{2}\right)+2p_{x}^{2}+2p_{y}^{2}-2p_{z}^{2}}{[\left(p_{0}-\frac{\omega}{2}\right)^{2}-\vec{p}^{2}-M^{2}][\left(p_{0}+q_{0}-\frac{\omega}{2}\right)^{2}-\vec{p}^{2}-M^{2}]}+\frac{2M^{2}-2\left(p_{0}+\frac{\omega}{2}\right)\left(p_{0}+q_{0}+\frac{\omega}{2}\right)+2p_{x}^{2}+2p_{y}^{2}-2p_{z}^{2}}{[\left(p_{0}+\frac{\omega}{2}\right)^{2}-\vec{p}^{2}-M^{2}][\left(p_{0}+q_{0}+\frac{\omega}{2}\right)^{2}-\vec{p}^{2}-M^{2}]}\right\\}.$

We rewrite the relation in Eq.(20):

$\displaystyle A_{1}^{2}$ $\displaystyle=-(\Pi_{11}-i\Pi_{12}),(s_{z}=-1\text{
for }\rho\text{ meson }),$ (50) $\displaystyle A_{2}^{2}$
$\displaystyle=-\Pi_{11}-i\Pi_{12},(s_{z}=+1\text{ for }\rho\text{ meson }),$
$\displaystyle A_{3}^{2}$ $\displaystyle=-\Pi_{33},(s_{z}=0\text{ for
}\rho\text{ meson }).$

The explicit form of coefficients can be given by:

$\displaystyle
A^{2}_{1}(q_{0})=2N_{f}N_{c}\int\frac{d^{4}p}{(2\pi)^{4}}\frac{2M^{2}-2\left(p_{0}+\frac{\omega}{2}\right)\left(p_{0}+q_{0}-\frac{\omega}{2}\right)-2p_{x}^{2}+2p_{y}^{2}+2p_{z}^{2}}{\left[\left(p_{0}+\frac{\omega}{2}\right)^{2}-\vec{p}^{2}-M^{2}\right]\left[\left(p_{0}+q_{0}-\frac{\omega}{2}\right)^{2}-\vec{p}^{2}-M^{2}\right]},$
(51)

$\displaystyle
A^{2}_{2}(q_{0})=2N_{f}N_{c}\int\frac{d^{4}p}{(2\pi)^{4}}\frac{2M^{2}-2\left(p_{0}-\frac{\omega}{2}\right)\left(p_{0}+q_{0}+\frac{\omega}{2}\right)-2p_{x}^{2}+2p_{y}^{2}+2p_{z}^{2}}{\left[\left(p_{0}-\frac{\omega}{2}\right)^{2}-\vec{p}^{2}-M^{2}\right]\left[\left(p_{0}+q_{0}+\frac{\omega}{2}\right)^{2}-\vec{p}^{2}-M^{2}\right]},$
(52)

$\displaystyle A^{2}_{3}(q_{0})=N_{f}N_{c}\int\frac{d^{4}p}{(2\pi)^{4}}$
$\displaystyle\left\\{\frac{2M^{2}-2\left(p_{0}-\frac{\omega}{2}\right)\left(p_{0}+q_{0}-\frac{\omega}{2}\right)+2p_{x}^{2}+2p_{y}^{2}-2p_{z}^{2}}{\left[\left(p_{0}-\frac{\omega}{2}\right)^{2}-\vec{p}^{2}-M^{2}\right]\left[\left(p_{0}+q_{0}-\frac{\omega}{2}\right)^{2}-\vec{p}^{2}-M^{2}\right]}\right.$
(53)
$\displaystyle+\left.\frac{2M^{2}-2\left(p_{0}+\frac{\omega}{2}\right)\left(p_{0}+q_{0}+\frac{\omega}{2}\right)+2p_{x}^{2}+2p_{y}^{2}-2p_{z}^{2}}{\left[\left(p_{0}+\frac{\omega}{2}\right)^{2}-\vec{p}^{2}-M^{2}\right]\left[\left(p_{0}+q_{0}+\frac{\omega}{2}\right)^{2}-\vec{p}^{2}-M^{2}\right]}\right\\}.$

Now, it is obvious that

$\frac{1}{2}A_{1}^{2}(m_{\rho}+\omega)+\frac{1}{2}A_{2}^{2}(m_{\rho}-\omega)=A_{3}^{2}(m_{\rho}).$
(54)

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