# Relativistic spin dynamics for vector mesons

Xin-Li Sheng  Key Laboratory of Quark and Lepton Physics (MOE) and Institute
of Particle Physics, Central China Normal University, Wuhan, 430079,China
Lucia Oliva  Department of Physics and Astronomy "Ettore Majorana", University
of Catania, Via S. Sofia 64, I-95123 Catania, Italy  INFN Sezione di Catania,
Via S. Sofia 64, I-95123 Catania, Italy  Zuo-Tang Liang  Key Laboratory of
Particle Physics and Particle Irradiation (MOE), Institute of Frontier and
Interdisciplinary Science, Shandong University, Qingdao, Shandong 266237,
China  Qun Wang  Peng Huanwu Center for Fundamental Theory and Department of
Modern Physics, University of Science and Technology of China, Hefei, Anhui
230026, China  Xin-Nian Wang  Nuclear Science Division, MS 70R0319, Lawrence
Berkeley National Laboratory, Berkeley, California 94720, USA

###### Abstract

We propose a relativistic theory for spin density matrices of vector mesons
based on Kadanoff-Baym equations in the closed-time-path formalism. The theory
puts the calculation of spin observables such as the spin density matrix
element $\rho_{00}$ for vector mesons on a solid ground. Within the theory we
formulate $\rho_{00}$ for $\phi$ mesons into a factorization form in
separation of momentum and space-time variables. We argue that the main
contribution to $\rho_{00}$ at lower energies should be from the $\phi$ fields
that can polarize the strange quark and antiquark in the same way as
electromagnetic fields. The key observation is that there is correlation
inside the $\phi$ meson wave function between the $\phi$ field that polarizes
the strange quark and that polarizes the strange antiquark. This is reflected
by the fact that the contributions to $\rho_{00}$ are all in squares of fields
which are nonvanishing even if the fields may strongly fluctuate in space-
time. The fluctuation of strong force fields can be extracted from $\rho_{00}$
of quarkonium vector mesons as links to fundamental properties of quantum
chromodynamics.

##  I Introduction

There is an intrinsic connection between rotation and spin polarization since
they are related to the conservation of total angular momentum and can be
converted from one to another, as demonstrated in the Barnett effect (Barnett,
1935) and the Einstein-de Haas effect (Einstein and de Haas, 1915) in
materials. A recent example is the observation of a spin-current from the
vortical motion in a liquid metal (Takahashi et al., 2016). The same effects
also exist in high energy heavy-ion collisions (HIC) in which the huge orbital
angular momentum (OAM) along the direction normal to the reaction plane can be
partially converted to the global spin polarization of hadrons (Liang and
Wang, 2005a, b; Voloshin, 2004; Betz et al., 2007; Becattini et al., 2008; Gao
et al., 2008) (see, e.g. (Wang, 2017; Florkowski et al., 2019; Becattini and
Lisa, 2020; Gao et al., 2021; Huang et al., 2020a), for recent reviews). The
global spin polarization of $\Lambda$ (including $\overline{\Lambda}$
hereafter) has been measured through their weak decays in Au+Au collisions at
$\sqrt{s_{NN}}=7.7-200$ GeV (Adamczyk et al., 2017; Adam et al., 2018).

As spin-one particles, vector mesons can also be polarized in heavy ion
collisions in the same way as hyperons. Normally the spin states of vector
mesons are described by the spin density matrix element
$\rho_{\lambda_{1}\lambda_{2}}$ with $\lambda_{1},\lambda_{2}=0,\pm 1$
labeling spin states along the spin quantization direction. The vector mesons
mainly decay through strong interaction that respects parity symmetry. So
their spin polarization proportional to $\rho_{11}-\rho_{-1,-1}$ cannot be
measured through their decays. Instead, $\rho_{00}$ can be measured through
the angular distribution of its decay daughters (Liang and Wang, 2005b; Yang
et al., 2018; Tang et al., 2018; Gonçalves and Torrieri, 2022). If
$\rho_{00}=1/3$, the spin states are equally populated in the three spin
states implying that the vector meson is not polarized. If $\rho_{00}\neq
1/3$, the three spin states are not equally populated, so the spin of the
vector meson is aligned either in the direction of the spin quantization or of
the transverse direction perpendicular to it. In 2008, the STAR collaboration
measured $\rho_{00}$ for the vector meson $\phi(1020)$ in Au+Au collisions at
200 GeV, but the result is consistent with $1/3$ within errors due to
statistics (Abelev et al., 2008). Recently STAR has measured the $\phi$
meson’s $\rho_{00}$ at lower energies which shows a significant deviation from
$1/3$ (Abdallah et al., 2022). It can hardly be explained by conventional
mechanism (Yang et al., 2018; Xia et al., 2021; Gao, 2021; Müller and Yang,
2022). In Ref. (Sheng et al., 2020a), some of us proposed that a large
deviation of $\rho_{00}$ from 1/3 for the $\phi$ meson may possibly arise from
the $\phi$ field, a strong force field with vacuum quantum number in
connection with pseudo-Goldstone bosons and vacuum properties of quantum
chromodynamics. Such a proposal is based on a nonrelativistic quark
coalescence model for the spin density matrix of vector mesons (Yang et al.,
2018; Sheng et al., 2020b).

In this paper we will present a relativistic theory for the spin density
matrix of vector mesons from the Kadanoff-Baym (KB) equation (Kadanoff and
Baym, 1962) in the closed-time-path (CTP) formalism (Martin and Schwinger,
1959; Keldysh, 1964) (for reviews of the KB equation and the CTP formalism, we
refer the readers to Refs. (Chou et al., 1985; Blaizot and Iancu, 2002;
Berges, 2004; Cassing, 2009)). Then we can derive the spin Boltzmann equation
for vector mesons with their spin density matrices being expressed in terms of
the matrix valued spin dependent distributions (MVSD) of the quarks and
antiquarks (Sheng et al., 2021). This puts the calculation of spin observables
such as $\rho_{00}$ for vector mesons on a solid ground.

The paper is organized as follows. In Sec. II we will give an introduction to
Green functions on the CTP for vector mesons which can be expressed in MVSD.
In Sec. III the KB equations for vector mesons are derived. In Sec. IV the
spin density matrices for vector mesons will be formulated from the spin
Boltzmann equations. In Sec. V the spin density matrices for $\phi$ mesons
will be evaluated. Discussions on the main results and conclusions are given
in the final section, Sec. VI.

We adopt the sign convention for the metric tensor
$g^{\mu\nu}=g_{\mu\nu}=\mathrm{diag}(1,-1,-1,-1)$ where $\mu,\nu=0,1,2,3$. The
sign convention for the Levi-Civita symbol is
$\epsilon^{0123}=-\epsilon_{0123}=1$. We can write the space-time coordinate
as $x=x^{\mu}=(x^{0},\mathbf{x})=(t,\mathbf{x})$ and
$x_{\mu}=(x_{0},-\mathbf{x})$ with $x_{0}=x^{0}=t$. The four-momentum for a
particle is denoted as $p=p^{\mu}=(p^{0},\mathbf{p})$ or
$p_{\mu}=(p_{0},-\mathbf{p})$, if it is on-shell we have
$p_{0}=p^{0}=\sqrt{\mathbf{p}^{2}+m^{2}}\equiv E_{p}=E_{\mathbf{p}}$. Normally
we use Greek letters to denote four-dimensional indices of four-vectors and
four-tensors and Latin letters to denote their spatial components.

##  II Green functions on CTP for vector mesons

The massive spin-1 particle, such as the vector meson with the mass $m_{V}$,
can be described by the vector field $A_{V}^{\mu}(x)$ with the classical
Lagrangian density

$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}^{V}F_{V}^{\mu\nu}+\frac{m_{V}^{2}}{2}A_{\mu}^{V}A_{V}^{\mu}-A_{\mu}^{V}j^{\mu}.$
(1)

where $j^{\mu}$ is the source current,
$F_{V}^{\mu\nu}=\partial^{\mu}A_{V}^{\nu}-\partial^{\nu}A_{V}^{\mu}$ is the
field strength tensor, and $A_{V}^{\mu}$ is assumed to be the real classical
field for the charge (including flavor) neutral particle. From $\mathcal{L}$
one can obtain the Proca equation (Proca, 1936; Itzykson and Zuber, 1980)

$L^{\mu\nu}(x)A_{\nu}^{V}(x)=j^{\mu}(x),$ (2)

where the differential operator is defined as

$L^{\mu\nu}(x)=\left(\partial_{x}^{2}+m_{V}^{2}\right)g^{\mu\nu}-\partial_{x}^{\mu}\partial_{x}^{\nu}.$
(3)

A constraint equation can be derived by contracting the above equation with
$\partial_{\mu}$ as

$\partial_{\mu}A_{V}^{\mu}(x)=\frac{1}{m_{V}^{2}}\partial_{\mu}j^{\mu}(x)=0,$
(4)

if the source current is conserved $\partial_{\mu}j^{\mu}=0$. The above
equation means that the longitudinal component of $A_{V}^{\mu}(x)$ is
vanishing for the conserved current.

The free vector field can be quantized as

$\displaystyle A_{V}^{\mu}(x)$ $\displaystyle=$
$\displaystyle\sum_{\lambda=0,\pm
1}\int\frac{d^{3}k}{(2\pi\hbar)^{3}}\frac{1}{2E_{k}^{V}}$ (5)
$\displaystyle\times\left[\epsilon^{\mu}(\lambda,{\bf k})a_{V}(\lambda,{\bf
k})e^{-ik\cdot x/\hbar}+\epsilon^{\mu\ast}(\lambda,{\bf
k})a_{V}^{\dagger}(\lambda,{\bf k})e^{ik\cdot x/\hbar}\right],$

where $E_{k}^{V}=\sqrt{\mathbf{k}^{2}+m_{V}^{2}}$ and $\lambda$ denote the
energy and the spin state in the spin quantization direction respectively, the
creation and annihilation operators $a_{V}(\lambda,{\bf k})$ and
$a_{V}^{\dagger}(\lambda,{\bf k})$ satisfy the commutator

$\left[a_{V}(\lambda,{\bf k}),a_{V}^{\dagger}(\lambda^{\prime},{\bf
k}^{\prime})\right]=\delta_{\lambda\lambda^{\prime}}2E_{{\bf
k}}(2\pi\hbar)^{3}\delta^{(3)}({\bf k}-{\bf k}^{\prime}),$ (6)

and the polarization vector $\epsilon^{\mu}(\lambda,{\bf k})$ obeys

$\displaystyle k_{\mu}\epsilon^{\mu}(\lambda,{\bf k})=0,$
$\displaystyle\epsilon(\lambda,{\bf k})\cdot\epsilon^{*}(\lambda^{\prime},{\bf
k})=-\delta_{\lambda\lambda^{\prime}},$
$\displaystyle\sum_{\lambda}\epsilon^{\mu}(\lambda,{\bf
k})\epsilon^{\nu*}(\lambda,{\bf
k})=-\left(g^{\mu\nu}-\frac{k^{\mu}k^{\nu}}{m_{V}^{2}}\right).$ (7)

In above relations, the first one follows the constraint (4) and
$k^{\mu}=(E_{k}^{V},\mathbf{k})$ denotes the on-shell four-momentum for the
vector meson. By the field quantization in (5), one can check that
$A_{V}^{\mu}(x)$ is Hermitian, i.e. $A_{V}^{\mu\dagger}(x)=A_{V}^{\mu}(x)$.

Figure 1: The closed-time path and four components of the two-point Green
function on CTP. The positive and negative time-branches are denoted as
$C_{+}$ and $C_{-}$ respectively. (a) $x_{1}^{0}=t_{1}\in C_{+}$,
$x_{2}^{0}=t_{2}\in C_{+}$; (b) $x_{1}^{0}=t_{1}\in C_{+}$,
$x_{2}^{0}=t_{2}\in C_{-}$; (c) $x_{1}^{0}=t_{1}\in C_{-}$,
$x_{2}^{0}=t_{2}\in C_{+}$; (d) $x_{1}^{0}=t_{1}\in C_{-}$,
$x_{2}^{0}=t_{2}\in C_{-}$.

One can define the two-point Green function for the vector meson on the CTP

$G_{\mathrm{CTP}}^{\mu\nu}(x_{1},x_{2})\equiv\left\langle
T_{C}A_{V}^{\mu}(x_{1})A_{V}^{\nu\dagger}(x_{2})\right\rangle,$ (8)

where $x_{1}$ and $x_{2}$ are two space-time points whose time components are
defined on the CTP contour and $T_{C}$ represents the time-ordering on the CTP
contour. We can write $G_{\mu\nu}^{\mathrm{CTP}}(x_{1},x_{2})$ in a matrix
form

$G_{\mu\nu}^{\mathrm{CTP}}(x_{1},x_{2})=\left(\begin{array}[]{cc}G_{\mu\nu}^{++}(x_{1},x_{2})&G_{\mu\nu}^{+-}(x_{1},x_{2})\\\
G_{\mu\nu}^{-+}(x_{1},x_{2})&G_{\mu\nu}^{--}(x_{1},x_{2})\end{array}\right)=\left(\begin{array}[]{cc}G_{\mu\nu}^{F}(x_{1},x_{2})&G_{\mu\nu}^{<}(x_{1},x_{2})\\\
G_{\mu\nu}^{>}(x_{1},x_{2})&G_{\mu\nu}^{\overline{F}}(x_{1},x_{2})\end{array}\right).$
(9)

The ’$++$’ component of $G_{\mu\nu}^{\mathrm{CTP}}$ with both $t_{1}$ and
$t_{2}$ (time components of $x_{1}$ and $x_{2}$) on the positive time-branch
is just the Feynman propagator $G_{\mu\nu}^{F}(x_{1},x_{2})$ as shown in Fig.
1(a). The ’$+-$’ component with $t_{1}$ on the positive time-branch while
$t_{2}$ on the negative time-branch is denoted as
$G_{\mu\nu}^{<}(x_{1},x_{2})$ meaning that $t_{2}$ is always later than
$t_{1}$ on the CTP contour as shown in Fig. 1(b). Analogously,
$G_{\mu\nu}^{>}(x_{1},x_{2})$ denotes the ’$-+$’ component with $t_{1}$ on the
negative time-branch and $t_{2}$ on the positive time-branch as shown in Fig.
1(c), while $G_{\mu\nu}^{\overline{F}}(x_{1},x_{2})$ denotes the ’$--$’
component with both $t_{1}$ and $t_{2}$ on the negative time-branch as shown
in Fig. 1(d).

The Wigner function can be defined from $G_{\mu\nu}^{<}(x_{1},x_{2})$ by
taking a Fourier transform with respect to the relative position
$y=x_{1}-x_{2}$,

$\displaystyle G_{\mu\nu}^{<}(x,p)$ $\displaystyle\equiv$ $\displaystyle\int
d^{4}y\,e^{ip\cdot y/\hbar}G_{\mu\nu}^{<}(x_{1},x_{2})$ (10) $\displaystyle=$
$\displaystyle\int d^{4}y\,e^{ip\cdot y/\hbar}\left\langle
A_{\nu}^{\dagger}(x_{2})A_{\mu}(x_{1})\right\rangle.$

Inserting the quantized field (5) into the definition of the Wigner function
(10), we obtain

$\displaystyle G_{\mu\nu}^{<}(x,p)$ $\displaystyle=$ $\displaystyle
2\pi\hbar\sum_{\lambda_{1},\lambda_{2}}\delta\left(p^{2}-m_{V}^{2}\right)$
(11)
$\displaystyle\times\left\\{\theta(p^{0})\epsilon_{\mu}\left(\lambda_{1},{\bf
p}\right)\epsilon_{\nu}^{\ast}\left(\lambda_{2},{\bf
p}\right)f_{\lambda_{1}\lambda_{2}}(x,{\bf p})\right.$
$\displaystyle+\theta(-p^{0})\epsilon_{\mu}^{\ast}\left(\lambda_{1},-{\bf
p}\right)\epsilon_{\nu}\left(\lambda_{2},-{\bf p}\right)$
$\displaystyle\left.\times\left[\delta_{\lambda_{2}\lambda_{1}}+f_{\lambda_{2}\lambda_{1}}(x,-{\bf
p})\right]\right\\},$

where the gradient expansion has been taken with spatial gradient terms being
dropped, and the MVSD for the vector meson is defined as

$\displaystyle f_{\lambda_{1}\lambda_{2}}(x,{\bf p})$ $\displaystyle\equiv$
$\displaystyle\int\frac{d^{4}u}{2(2\pi\hbar)^{3}}\delta(p\cdot u)e^{-iu\cdot
x/\hbar}$ (12) $\displaystyle\times\left\langle
a_{V}^{\dagger}\left(\lambda_{2},{\bf p}-\frac{{\bf
u}}{2}\right)a_{V}\left(\lambda_{1},{\bf p}+\frac{{\bf
u}}{2}\right)\right\rangle.$

One can check that $f_{\lambda_{1}\lambda_{2}}(x,{\bf p})$ is a Hermitian
matrix, i.e. $f_{\lambda_{1}\lambda_{2}}^{*}(x,{\bf
p})=f_{\lambda_{2}\lambda_{1}}(x,{\bf p})$. Similarly we can define another
Wigner function from $G_{\mu\nu}^{>}(x_{1},x_{2})$

$\displaystyle G_{\mu\nu}^{>}(x,p)$ $\displaystyle\equiv$ $\displaystyle\int
d^{4}y\,e^{ip\cdot y/\hbar}\left\langle
A_{\mu}(x_{1})A_{\nu}^{\dagger}(x_{2})\right\rangle.$ (13) $\displaystyle=$
$\displaystyle
2\pi\hbar\sum_{\lambda_{1},\lambda_{2}}\delta\left(p^{2}-m_{V}^{2}\right)$
$\displaystyle\times\left\\{\theta(p^{0})\epsilon_{\mu}\left(\lambda_{1},{\bf
p}\right)\epsilon_{\nu}^{\ast}\left(\lambda_{2},{\bf
p}\right)\left[\delta_{\lambda_{1}\lambda_{2}}+f_{\lambda_{1}\lambda_{2}}(x,{\bf
p})\right]\right.$
$\displaystyle\left.+\theta(-p^{0})\epsilon_{\mu}^{\ast}\left(\lambda_{1},-{\bf
p}\right)\epsilon_{\nu}\left(\lambda_{2},-{\bf
p}\right)f_{\lambda_{2}\lambda_{1}}(x,-{\bf p})\right\\}.$

Note that $G_{\mu\nu}^{>}(x,p)$ can be obtained by replacing
$f_{\lambda_{1}\lambda_{2}}\rightarrow\delta_{\lambda_{1}\lambda_{2}}+f_{\lambda_{1}\lambda_{2}}$
and $\delta_{\lambda_{2}\lambda_{1}}+f_{\lambda_{2}\lambda_{1}}\rightarrow
f_{\lambda_{2}\lambda_{1}}$ from $G_{\mu\nu}^{<}(x,p)$.

##  III Kadanoff-Baym equations for vector mesons

The Wigner functions for massless vector particles such as gluons and photons
(Elze et al., 1986; Blaizot and Iancu, 2002; Wang et al., 2002; Huang et al.,
2020b; Hattori et al., 2021; Müller and Yang, 2022) have been studies for many
years, but to our knowledge there are few works about Wigner functions for
massive vector mesons in the context of spin polarization (see Ref.
(Weickgenannt et al., 2022) for a recent one). In this section, we will derive
the Boltzmann equation for vector mesons’ Wigner functions from two-point
Green functions on the CTP. The starting point is the KB equations

$\displaystyle L_{\eta}^{\mu}(x_{1})G^{<,\eta\nu}(x_{1},x_{2})$ (14)
$\displaystyle=$ $\displaystyle-\frac{i\hbar}{2}\int
d^{4}x^{\prime}\left[\Sigma_{\ \ \ \
\alpha}^{<,\mu}(x_{1},x^{\prime})G^{>,\alpha\nu}(x^{\prime},x_{2})-\Sigma_{\ \
\ \
\alpha}^{>,\mu}(x_{1},x^{\prime})G^{<,\alpha\nu}(x^{\prime},x_{2})\right],$

and

$\displaystyle L_{\eta}^{\nu}(x_{2})G^{<,\mu\eta}(x_{1},x_{2})$ (15)
$\displaystyle=$ $\displaystyle-\frac{i\hbar}{2}\int d^{4}x^{\prime}\left[G_{\
\ \ \
\alpha}^{<,\mu}(x_{1},x^{\prime})\Sigma^{>,\alpha\nu}(x^{\prime},x_{2})-G_{\ \
\ \
\alpha}^{>,\mu}(x_{1},x^{\prime})\Sigma^{<,\alpha\nu}(x^{\prime},x_{2})\right].$

Equations (14) and (15) are the result of the quasiparticle approximation
(Sheng et al., 2021). Note that the integrations over $x$ in Eqs. (14) and
(15) are ordinary ones.

Figure 2: The self-energies $\Sigma^{<,\mu\nu}$ and $\Sigma^{>,\mu\nu}$ of
vector mesons from quark loops in the quark-meson model. Two quark propagators
in the loop may have different flavors corresponding to the vector meson that
is not flavor neutral.

We consider the coupling between the vector meson and the quark-antiquark in
the quark-meson model (Manohar and Georgi, 1984; Fernandez et al., 1993; Li et
al., 1997; Zhao et al., 1998; Zacchi et al., 2015, 2017). Then at lowest order
in the coupling constant, the self-energies are from quark loops as shown in
Fig. 2

$\displaystyle\Sigma^{<,\mu\nu}(x_{1},x_{2})$ $\displaystyle=$
$\displaystyle-\text{Tr}\left[i\Gamma^{\mu}S^{<}(x_{1},x_{2})i\Gamma^{\nu}S^{>}(x_{2},x_{1})\right],$
$\displaystyle\Sigma^{>,\mu\nu}(x_{1},x_{2})$ $\displaystyle=$
$\displaystyle-\text{Tr}\left[i\Gamma^{\mu}S^{>}(x_{1},x_{2})i\Gamma^{\nu}S^{<}(x_{2},x_{1})\right],$
(16)

where $S^{>}(x_{1},x_{2})$ and $S^{<}(x_{1},x_{2})$ are two-point Green
functions of quarks, $i\Gamma^{\mu}$ denotes the vertex of the vector meson
and quark-antiquark, and the overall minus signs arise from quark loops.
Inserting the self-energies (16) into Eq. (14) and taking a Fourier transform
with respect to the difference $y=x_{1}-x_{2}$, we obtain the KB equation for
the Wigner function as

$\displaystyle\left\\{g_{\eta}^{\mu}\left[-\left(p^{2}-m_{V}^{2}-\frac{\hbar^{2}}{4}\partial_{x}^{2}\right)-i\hbar
p\cdot\partial_{x}\right]\right.$ (17)
$\displaystyle\left.-\frac{\hbar^{2}}{4}\partial_{x}^{\mu}\partial_{\eta}^{x}+p^{\mu}p_{\eta}+\frac{1}{2}i\hbar\left(p_{\eta}\partial_{x}^{\mu}+p^{\mu}\partial_{\eta}^{x}\right)\right\\}G^{<,\eta\nu}(x,p)$
$\displaystyle=$
$\displaystyle-\frac{i\hbar}{2}\int\frac{d^{4}p^{\prime}}{(2\pi\hbar)^{4}}\left\\{\text{Tr}\left[\Gamma^{\mu}S^{<}\left(x,p+p^{\prime}\right)\Gamma_{\alpha}S^{>}\left(x,p^{\prime}\right)\right]G^{>,\alpha\nu}\left(x,p\right)\right.$
$\displaystyle\left.-\text{Tr}\left[\Gamma^{\mu}S^{>}\left(x,p+p^{\prime}\right)\Gamma_{\alpha}S^{<}\left(x,p^{\prime}\right)\right]G^{<,\alpha\nu}\left(x,p\right)\right\\}$
$\displaystyle-\frac{\hbar^{2}}{4}\int\frac{d^{4}p^{\prime}}{(2\pi\hbar)^{4}}\left[\left\\{\text{Tr}\left[\Gamma^{\mu}S^{<}\left(x,p+p^{\prime}\right)\Gamma_{\alpha}S^{>}\left(x,p^{\prime}\right)\right],G^{>,\alpha\nu}\left(x,p\right)\right\\}_{\text{P.B.}}\right.$
$\displaystyle\left.-\left\\{\text{Tr}\left[\Gamma^{\mu}S^{>}\left(x,p+p^{\prime}\right)\Gamma_{\alpha}S^{<}\left(x,p^{\prime}\right)\right],G^{<,\alpha\nu}\left(x,p\right)\right\\}_{\text{P.B.}}\right],$

where the Poisson bracket involves space-time and momentum gradients and is
defined as

$\left\\{A,B\right\\}_{\text{P.B.}}\equiv(\partial_{x}^{\mu}A)(\partial_{\mu}^{p}B)-(\partial_{p}^{\mu}A)(\partial_{\mu}^{x}B).$
(18)

In the same way, we obtain from Eq. (15) another KB equation for the Wigner
function

$\displaystyle\left\\{g_{\eta}^{\nu}\left[-\left(p^{2}-m_{V}^{2}-\frac{\hbar^{2}}{4}\partial_{x}^{2}\right)+i\hbar
p\cdot\partial_{x}\right]\right.$ (19)
$\displaystyle\left.-\frac{\hbar^{2}}{4}\partial_{x}^{\nu}\partial_{\eta}^{x}+p^{\nu}p_{\eta}-\frac{1}{2}i\hbar\left(p_{\eta}\partial_{x}^{\nu}+p^{\nu}\partial_{\eta}^{x}\right)\right\\}G^{<,\mu\eta}(x,p)$
$\displaystyle=$
$\displaystyle-\frac{i\hbar}{2}\int\frac{d^{4}p^{\prime}}{(2\pi\hbar)^{4}}\left\\{G_{\
\ \ \
\alpha}^{<,\mu}(x,p)\text{Tr}\left[\Gamma^{\alpha}S^{>}\left(x,p+p^{\prime}\right)\Gamma^{\nu}S^{<}\left(x,p^{\prime}\right)\right]\right.$
$\displaystyle\left.-G_{\ \ \ \
\alpha}^{>,\mu}\left(x,p\right)\text{Tr}\left[\Gamma^{\alpha}S^{<}\left(x,p+p^{\prime}\right)\Gamma^{\nu}S^{>}\left(x,p^{\prime}\right)\right]\right\\}$
$\displaystyle-\frac{\hbar^{2}}{4}\int\frac{d^{4}p^{\prime}}{(2\pi\hbar)^{4}}\left[\left\\{G_{\
\ \ \
\alpha}^{<,\mu}(x,p),\text{Tr}\left[\Gamma^{\alpha}S^{>}\left(x,p+p^{\prime}\right)\Gamma^{\nu}S^{<}\left(x,p^{\prime}\right)\right]\right\\}_{\text{P.B.}}\right.$
$\displaystyle\left.-\left\\{G_{\ \ \ \
\alpha}^{>,\mu}\left(x,p\right),\text{Tr}\left[\Gamma^{\alpha}S^{<}\left(x,p+p^{\prime}\right)\Gamma^{\nu}S^{>}\left(x,p^{\prime}\right)\right]\right\\}_{\text{P.B.}}\right].$

Taking the difference between Eq. (17) and (19), we are able to derive the
Boltzmann equation for the Wigner function at the leading order

$\displaystyle
p\cdot\partial_{x}G^{<,\mu\nu}(x,p)-\frac{1}{4}\left[p^{\mu}\partial_{\eta}^{x}G^{<,\eta\nu}(x,p)+p^{\nu}\partial_{\eta}^{x}G^{<,\mu\eta}(x,p)\right]$
(20) $\displaystyle=$
$\displaystyle\frac{1}{4}\int\frac{d^{4}p^{\prime}}{(2\pi\hbar)^{4}}\left\\{\text{Tr}\left[\Gamma^{\mu}S^{<}\left(x,p+p^{\prime}\right)\Gamma_{\alpha}S^{>}\left(x,p^{\prime}\right)\right]G^{>,\alpha\nu}\left(x,p\right)\right.$
$\displaystyle\left.-\text{Tr}\left[\Gamma^{\mu}S^{>}\left(x,p+p^{\prime}\right)\Gamma_{\alpha}S^{<}\left(x,p^{\prime}\right)\right]G^{<,\alpha\nu}\left(x,p\right)\right\\}$
$\displaystyle+\frac{1}{4}\int\frac{d^{4}p^{\prime}}{(2\pi\hbar)^{4}}\left\\{G_{\
\ \ \
\alpha}^{>,\mu}\left(x,p\right)\text{Tr}\left[\Gamma^{\alpha}S^{<}\left(x,p+p^{\prime}\right)\Gamma^{\nu}S^{>}\left(x,p^{\prime}\right)\right]\right.$
$\displaystyle\left.-G_{\ \ \ \
\alpha}^{<,\mu}(x,p)\text{Tr}\left[\Gamma^{\alpha}S^{>}\left(x,p+p^{\prime}\right)\Gamma^{\nu}S^{<}\left(x,p^{\prime}\right)\right]\right\\},$

where we have neglected terms with Poisson brackets and those proportional to
$p_{\eta}$ in the left-hand-side since their contraction with the leading-
order $G^{<,\eta\nu}(x,p)$ and $G^{<,\mu\eta}(x,p)$ is vanishing. In the next
section we will rewrite the above Boltzmann equation in terms of MVSDs for
vector mesons, quarks and antiquarks.

##  IV Spin density matrix for quark coalescence and dissociation

The two-point Green functions $S^{>}\left(x,p\right)$ and
$S^{<}\left(x,p\right)$ for quarks are given by (Sheng et al., 2021)

$\displaystyle S^{<}(x,p)$ $\displaystyle=$
$\displaystyle-(2\pi\hbar)\theta(p_{0})\delta(p^{2}-m_{q}^{2})\sum_{r,s}u(r,\mathbf{p})\overline{u}(s,\mathbf{p})f_{rs}^{(+)}(x,\mathbf{p})$
$\displaystyle-(2\pi\hbar)\theta(-p_{0})\delta(p^{2}-m_{q}^{2})\sum_{r,s}v(s,-\mathbf{p})\overline{v}(r,-\mathbf{p})\left[\delta_{rs}-f_{rs}^{(-)}(x,-\mathbf{p})\right],$
$\displaystyle S^{>}(x,p)$ $\displaystyle=$
$\displaystyle(2\pi\hbar)\theta(p_{0})\delta(p^{2}-m_{q}^{2})\sum_{r,s}u(r,\mathbf{p})\overline{u}(s,\mathbf{p})\left[\delta_{rs}-f_{rs}^{(+)}(x,\mathbf{p})\right]$
(21)
$\displaystyle+(2\pi\hbar)\theta(-p_{0})\delta(p^{2}-m_{q}^{2})\sum_{r,s}v(s,-\mathbf{p})\overline{v}(r,-\mathbf{p})f_{rs}^{(-)}(x,-\mathbf{p}),$

where $f_{rs}^{(+)}$ and $f_{rs}^{(-)}$ are MVSD for quarks and antiquarks
respectively. We can parameterize them as

$\displaystyle f_{rs}^{(+)}(x,\mathbf{p})$ $\displaystyle=$
$\displaystyle\frac{1}{2}f_{q}(x,\mathbf{p})\left[\delta_{rs}-P_{\mu}^{q}(x,\mathbf{p})n_{j}^{(+)\mu}(\mathbf{p})\tau_{rs}^{j}\right],$
$\displaystyle f_{rs}^{(-)}(x,-\mathbf{p})$ $\displaystyle=$
$\displaystyle\frac{1}{2}f_{\overline{q}}(x,-\mathbf{p})\left[\delta_{rs}-P_{\mu}^{\overline{q}}(x,-\mathbf{p})n_{j}^{(-)\mu}(\mathbf{p})\tau_{rs}^{j}\right],$
(22)

where $f_{q}(x,\mathbf{p})$ and $f_{\overline{q}}(x,-\mathbf{p})$ are MVSDs
for quarks and antiquarks respectively, and $P_{q}^{\mu}(x,\mathbf{p})$ and
$P_{\overline{q}}^{\mu}(x,-\mathbf{p})$ are polarization four-vectors for
quarks and antiquarks respectively. The spin direction four-vectors for quarks
and antiquarks are given by

$\displaystyle n_{j}^{(+)\mu}(\mathbf{p})$ $\displaystyle\equiv$
$\displaystyle
n^{\mu}(\mathbf{n}_{j},\mathbf{p},m_{q})=\left(\frac{\mathbf{n}_{j}\cdot{\bf
p}}{m_{q}},\mathbf{n}_{j}+\frac{(\mathbf{n}_{j}\cdot{\bf p}){\bf
p}}{m_{q}(E_{{\bf p}}^{q}+m_{q})}\right),$ $\displaystyle
n_{j}^{(-)\mu}(\mathbf{p})$ $\displaystyle\equiv$ $\displaystyle
n^{\mu}(\mathbf{n}_{j},-\mathbf{p},m_{\overline{q}})=\left(-\frac{\mathbf{n}_{j}\cdot{\bf
p}}{m_{\overline{q}}},\mathbf{n}_{j}+\frac{(\mathbf{n}_{j}\cdot{\bf p}){\bf
p}}{m_{\overline{q}}(E_{{\bf p}}^{\overline{q}}+m_{\overline{q}})}\right),$
(23)

where $\mathbf{n}_{j}$ for $j=1,2,3$ are three basis unit vectors that form a
Cartesian coordinate system in the particle’s rest frame with $\mathbf{n}_{3}$
being the spin quantization direction, and $n_{j}^{(+)\mu}$ and
$n_{j}^{(-)\mu}$ are the Lorentz transformed four-vectors of $\mathbf{n}_{j}$
for quarks and antiquarks respectively which obey the sum rules

$\displaystyle n_{j}^{(+)\mu}(\mathbf{p})n_{j}^{(+)\nu}(\mathbf{p})$
$\displaystyle=$
$\displaystyle-\left(g^{\mu\nu}-\frac{p^{\mu}p^{\nu}}{m_{q}^{2}}\right),$
$\displaystyle n_{j}^{(-)\mu}(\mathbf{p})n_{j}^{(-)\nu}(\mathbf{p})$
$\displaystyle=$
$\displaystyle-\left(g^{\mu\nu}-\frac{\overline{p}^{\mu}\overline{p}^{\nu}}{m_{\overline{q}}^{2}}\right),$
(24)

where $p^{\mu}=(E_{p}^{q},\mathbf{p})$ and
$\overline{p}^{\mu}=(E_{p}^{\overline{q}},-\mathbf{p})$. We note that
$f_{rs}^{(+)}(x,\mathbf{p})$ and $f_{rs}^{(-)}(x,-\mathbf{p})$ are actually
the transpose of those MVSDs defined in Eqs. (117)-(118) of Ref. (Sheng et
al., 2021) in spin indices. We can flip the sign of the three-momentum,
$\mathbf{p}\rightarrow-\mathbf{p}$, in $f_{rs}^{(-)}(x,-\mathbf{p})$ to obtain

$f_{rs}^{(-)}(x,\mathbf{p})=\frac{1}{2}f_{\overline{q}}(x,\mathbf{p})\left[\delta_{rs}-P_{\mu}^{\overline{q}}(x,\mathbf{p})n_{j}^{(-)\mu}(-\mathbf{p})\tau_{rs}^{j}\right],$
(25)

where $n_{j}^{(-)\mu}(-\mathbf{p})$ has the same form as
$n_{j}^{(+)\mu}(\mathbf{p})$ except the quark mass. Note that in the self-
energy (16) of the vector meson that is not flavor neutral, $S^{<}(x,p)$ and
$S^{>}(x,p)$ may involve different flavors of quarks and antiquarks.

Inserting $S^{<}(x,p)$, $S^{>}(x,p)$, $G^{<,\mu\nu}(x,p)$, and
$G^{>,\mu\nu}(x,p)$ in Eqs. (21), (11) and (13) into Eq. (20), the Boltzmann
equation can be put into the following form

$\displaystyle
p\cdot\partial_{x}G^{<,\mu\nu}(x,p)-\frac{1}{4}\left[p^{\mu}\partial_{\eta}^{x}G^{<,\eta\nu}(x,p)+p^{\nu}\partial_{\eta}^{x}G^{<,\mu\eta}(x,p)\right]$
(26) $\displaystyle=$ $\displaystyle\frac{1}{4(2\pi\hbar)}\int
d^{4}p^{\prime}\delta(p^{\prime
2}-m_{\overline{q}}^{2})\delta\left[(p+p^{\prime})^{2}-m_{q}^{2}\right]\delta(p^{2}-m_{V}^{2})$
$\displaystyle\times\left\\{\theta(p_{0}^{\prime})\theta\left(p_{0}+p_{0}^{\prime}\right)\theta(p_{0})I_{+++}\right.$
$\displaystyle+\theta(p_{0}^{\prime})\theta\left(p_{0}+p_{0}^{\prime}\right)\theta(-p_{0})I_{++-}$
$\displaystyle+\theta(p_{0}^{\prime})\theta\left(-p_{0}-p_{0}^{\prime}\right)\theta(-p_{0})I_{+--}$
$\displaystyle+\theta(-p_{0}^{\prime})\theta\left(p_{0}+p_{0}^{\prime}\right)\theta(p_{0})I_{-++}$
$\displaystyle+\theta(-p_{0}^{\prime})\theta\left(-p_{0}-p_{0}^{\prime}\right)\theta(p_{0})I_{--+}$
$\displaystyle\left.+\theta(-p_{0}^{\prime})\theta\left(-p_{0}-p_{0}^{\prime}\right)\theta(-p_{0})I_{---}\right\\}.$

The terms $I_{ijk}$, with $i,j,k=\pm$ representing the positive/negative
energy, correspond to all possible processes at lowest order in the coupling
constant, as shown in Table 1. In Eq. (26), $I_{-+-}$ and $I_{+-+}$ are absent
due to incompatibility of theta functions, and $I_{-++}$ and $I_{+--}$ contain
the coalescence of quark-antiquark to the vector meson and vice versa, but
$I_{-++}$ corresponds to the positive energy sector of the two-point function
for the vector meson while $I_{+--}$ corresponds to the negative energy
sector. All terms except $I_{-++}$ and $I_{+--}$ are vanishing for on-shell
quarks, antiquarks and mesons at the one-loop level of the selfenergy. We
distinguish $m_{q}$ from $m_{\overline{q}}$ in Eq. (26) since the quark and
antiquark may have different flavors for the vector meson that is not flavor
neutral so the meson and its antiparticle are not the same particle.

$I_{+++}$ | $I_{++-}$ | $I_{+--}$ | $I_{-++}$ | $I_{--+}$ | $I_{---}$
---|---|---|---|---|---
$q\rightarrow q+M$ | $q+\overline{M}\rightarrow q$ | $\overline{M}\rightarrow q+\overline{q}$ | $q+\overline{q}\rightarrow M$ | $\overline{q}\rightarrow\overline{q}+M$ | $\overline{q}+\overline{M}\rightarrow\overline{q}$
$q+M\rightarrow q$ | $q\rightarrow q+\overline{M}$ | $q+\overline{q}\rightarrow\overline{M}$ | $M\rightarrow q+\overline{q}$ | $\overline{q}+M\rightarrow\overline{q}$ | $\overline{q}\rightarrow\overline{q}+\overline{M}$
Table 1: Collision terms in the Boltzmann equation. All terms except $I_{-++}$
and $I_{+--}$ are vanishing for on-shell quarks, antiquarks and mesons at the
one-loop level of the selfenergy.

In this paper, we are interested in the contribution from the coalescence and
dissociation processes corresponding to $I_{-++}$. The coalescence is regarded
as one of the main processes for particle production in heavy-ion collisions
(Greco et al., 2003a; Fries et al., 2003a; Greco et al., 2003b; Fries et al.,
2003b; Greco et al., 2004; Zhao et al., 2020). So the spin Boltzmann equation
for the vector meson’s MVSD reads

$\displaystyle p\cdot\partial_{x}f_{\lambda_{1}\lambda_{2}}(x,\mathbf{p})$
(27) $\displaystyle=$
$\displaystyle\frac{1}{16}\sum_{r_{1},s_{1},r_{2},s_{2},\lambda_{1}^{\prime},\lambda_{2}^{\prime}}\int\frac{d^{3}\mathbf{p}^{\prime}}{(2\pi\hbar)^{3}}\frac{1}{E_{p^{\prime}}^{\overline{q}}E_{{\bf
p}-{\bf
p}^{\prime}}^{q}}2\pi\hbar\delta\left(E_{p}^{V}-E_{p^{\prime}}^{\overline{q}}-E_{{\bf
p}-{\bf p}^{\prime}}^{q}\right)$
$\displaystyle\times\left\\{\delta_{\lambda_{2}\lambda_{2}^{\prime}}\epsilon_{\mu}^{\ast}(\lambda_{1},{\bf
p})\epsilon^{\alpha}\left(\lambda_{1}^{\prime},{\bf
p}\right)\text{Tr}\left[\Gamma_{\alpha}v(s_{1},{\bf
p}^{\prime})\overline{v}(r_{1},{\bf
p}^{\prime})\Gamma^{\mu}u(r_{2},\mathbf{p}-{\bf
p}^{\prime})\overline{u}(s_{2},\mathbf{p}-{\bf p}^{\prime})\right]\right.$
$\displaystyle+\left.\delta_{\lambda_{1}\lambda_{1}^{\prime}}\epsilon_{\nu}(\lambda_{2},{\bf
p})\epsilon_{\alpha}^{\ast}\left(\lambda_{2}^{\prime},{\bf
p}\right)\text{Tr}\left[\Gamma^{\nu}v(s_{1},{\bf
p}^{\prime})\overline{v}(r_{1},{\bf
p}^{\prime})\Gamma^{\alpha}u(r_{2},\mathbf{p}-{\bf
p}^{\prime})\overline{u}(s_{2},\mathbf{p}-{\bf p}^{\prime})\right]\right\\}$
$\displaystyle\times\left\\{f_{r_{1}s_{1}}^{(-)}(x,{\bf
p}^{\prime})f_{r_{2}s_{2}}^{(+)}(x,\mathbf{p}-\mathbf{p}^{\prime})\left[\delta_{\lambda_{1}^{\prime}\lambda_{2}^{\prime}}+f_{\lambda_{1}^{\prime}\lambda_{2}^{\prime}}(x,\mathbf{p})\right]\right.$
$\displaystyle\left.-\left[\delta_{r_{1}s_{1}}-f_{r_{1}s_{1}}^{(-)}(x,{\bf
p}^{\prime})\right]\left[\delta_{r_{2}s_{2}}-f_{r_{2}s_{2}}^{(+)}(x,\mathbf{p}-\mathbf{p}^{\prime})\right]f_{\lambda_{1}^{\prime}\lambda_{2}^{\prime}}(x,\mathbf{p})\right\\},$

where $\lambda_{1}$, $\lambda_{2}$, $\lambda_{1}^{\prime}$ and
$\lambda_{2}^{\prime}$ denote the spin states along the spin quantization
direction, and $\Gamma^{\alpha}$ is the $q\overline{q}V$ vertex given by

$\Gamma^{\alpha}\approx
g_{V}B(\mathbf{p}-\mathbf{p}^{\prime},\mathbf{p}^{\prime})\gamma^{\alpha},$
(28)

where $g_{V}$ is the coupling constant of the vector meson and quark-
antiquark, and $B(\mathbf{p}-\mathbf{p}^{\prime},\mathbf{p}^{\prime})$ denotes
the Bethe-Salpeter wave function of the $\phi$ meson (Xu et al., 2019, 2021)
in the following parametrization form

$B(\mathbf{p}-\mathbf{p}^{\prime},\mathbf{p}^{\prime})=\frac{1-\exp\left\\{-\left[(E_{\mathbf{p}-\mathbf{p}^{\prime}}^{s}-E_{\mathbf{p}^{\prime}}^{\overline{s}})^{2}-(\mathbf{p}-2\mathbf{p}^{\prime})^{2}\right]/\sigma^{2}\right\\}}{\left[(E_{\mathbf{p}-\mathbf{p}^{\prime}}^{s}-E_{\mathbf{p}^{\prime}}^{\overline{s}})^{2}-(\mathbf{p}-2\mathbf{p}^{\prime})^{2}\right]/\sigma^{2}},$
(29)

with $\sigma\approx 0.522$ GeV being the width parameter of the wave function.
The derivation of Eq. (27) is given in Appendix A. We see that there is a gain
term and a loss term in Eq. (27). In heavy ion collisions, the distribution
functions are normally much less than 1,
$f_{\lambda_{1}\lambda_{2}}(x,\mathbf{p})\sim f_{rs}^{(+)}\sim f_{rs}^{(-)}\ll
1$, so Eq. (27) can be approximated as

$\frac{p}{E_{p}^{V}}\cdot\partial_{x}f_{\lambda_{1}\lambda_{2}}(x,\mathbf{p})\approx
R_{\lambda_{1}\lambda_{2}}^{\mathrm{coal}}(\mathbf{p})-R^{\mathrm{diss}}(\mathbf{p})f_{\lambda_{1}\lambda_{2}}(x,\mathbf{p}),$
(30)

where $R_{\lambda_{1}\lambda_{2}}^{\mathrm{coal}}$ and
$R_{\lambda_{1}\lambda_{2}}^{\mathrm{diss}}$ denote the coalescence and
dissociation rates for the vector meson, i.e. the rates of
$q+\overline{q}\rightarrow M$ and $M\rightarrow q+\overline{q}$ respectively,
defined as

$\displaystyle R_{\lambda_{1}\lambda_{2}}^{\mathrm{coal}}(\mathbf{p})$
$\displaystyle=$
$\displaystyle\frac{1}{8(2\pi\hbar)^{2}}\sum_{r_{1},s_{1},r_{2},s_{2}}\int
d^{3}\mathbf{p}^{\prime}\frac{1}{E_{p^{\prime}}^{\overline{q}}E_{{\bf p}-{\bf
p}^{\prime}}^{q}E_{p}^{V}}$ (31)
$\displaystyle\times\delta\left(E_{p}^{V}-E_{p^{\prime}}^{\overline{q}}-E_{{\bf
p}-{\bf p}^{\prime}}^{q}\right)\epsilon_{\alpha}^{\ast}(\lambda_{1},{\bf
p})\epsilon_{\beta}(\lambda_{2},{\bf p})$
$\displaystyle\times\text{Tr}\left[\Gamma^{\beta}v(s_{1},{\bf
p}^{\prime})\overline{v}(r_{1},{\bf
p}^{\prime})\Gamma^{\alpha}u(r_{2},\mathbf{p}-{\bf
p}^{\prime})\overline{u}(s_{2},\mathbf{p}-{\bf p}^{\prime})\right]$
$\displaystyle\times f_{r_{1}s_{1}}^{(-)}(x,{\bf
p}^{\prime})f_{r_{2}s_{2}}^{(+)}(x,\mathbf{p}-{\bf p}^{\prime}),$
$\displaystyle R^{\mathrm{diss}}(\mathbf{p})$ $\displaystyle=$
$\displaystyle-\frac{1}{12(2\pi\hbar)^{2}}\sum_{r_{1},r_{2}}\int
d^{3}\mathbf{p}^{\prime}\frac{1}{E_{p^{\prime}}^{\overline{q}}E_{{\bf p}-{\bf
p}^{\prime}}^{q}E_{p}^{V}}$ (32)
$\displaystyle\times\delta\left(E_{p}^{V}-E_{p^{\prime}}^{\overline{q}}-E_{{\bf
p}-{\bf
p}^{\prime}}^{q}\right)\left(g_{\alpha\beta}-\frac{p_{\alpha}p_{\beta}}{m_{V}^{2}}\right)$
$\displaystyle\times\text{Tr}\left\\{\Gamma^{\beta}\left(p^{\prime}\cdot\gamma-
m_{\overline{q}}\right)\Gamma^{\alpha}\left[(p-p^{\prime})\cdot\gamma+m_{q}\right]\right\\}.$

Note that $R^{\mathrm{diss}}(\mathbf{p})$ does not depend on the MVSDs of the
quark, antiquark and the vector meson, therefore it is independent of the
quark polarization. Schematically the formal solution to Eq. (30) reads

$\displaystyle f_{\lambda_{1}\lambda_{2}}(x,\mathbf{p})$ $\displaystyle\sim$
$\displaystyle\frac{R_{\lambda_{1}\lambda_{2}}^{\mathrm{coal}}(\mathbf{p})}{R^{\mathrm{diss}}(\mathbf{p})}\left[1-\exp\left(-R^{\mathrm{diss}}(\mathbf{p})\Delta
t\right)\right]$ (33) $\displaystyle\sim$
$\displaystyle\begin{cases}R_{\lambda_{1}\lambda_{2}}^{\mathrm{coal}}(\mathbf{p})\Delta
t,&\mathrm{for}\;\Delta t\ll 1/R^{\mathrm{diss}}(\mathbf{p})\\\
\frac{R_{\lambda_{1}\lambda_{2}}^{\mathrm{coal}}(\mathbf{p})}{R^{\mathrm{diss}}(\mathbf{p})},&\mathrm{for}\;\Delta
t\gg 1/R^{\mathrm{diss}}(\mathbf{p})\end{cases}$

if $f_{\lambda_{1}\lambda_{2}}(x,\mathbf{p})$ for the vector meson at the
initial time is assumed to be zero, where $\Delta t$ is the formation time of
the vector meson.

The spin density matrix element $\rho_{\lambda_{1}\lambda_{2}}^{V}$ is assumed
to be proportional to $f_{\lambda_{1}\lambda_{2}}(x,\mathbf{p})$ which is
$R_{\lambda_{1}\lambda_{2}}^{\mathrm{coal}}(\mathbf{p})\Delta t$ if $\Delta
t\ll 1/R^{\mathrm{diss}}(\mathbf{p})$ or
$R_{\lambda_{1}\lambda_{2}}^{\mathrm{coal}}(\mathbf{p})/R^{\mathrm{diss}}(\mathbf{p})$
if $\Delta t\gg 1/R^{\mathrm{diss}}(\mathbf{p})$. In both cases,
$\rho_{\lambda_{1}\lambda_{2}}^{V}$ is proportional to
$R_{\lambda_{1}\lambda_{2}}^{\mathrm{coal}}(\mathbf{p})$ times a constant
independent of the spin states of the vector meson. Here we assume that the
coalescence of the vector meson takes place in a relatively short time, so we
have

$\displaystyle\rho_{\lambda_{1}\lambda_{2}}^{V}(x,{\bf p})$
$\displaystyle\approx$ $\displaystyle\frac{\Delta
t}{8}\sum_{r_{1},s_{1},r_{2},s_{2}}\int\frac{d^{3}\mathbf{p}^{\prime}}{(2\pi\hbar)^{3}}\frac{1}{E_{p^{\prime}}^{\overline{q}}E_{{\bf
p}-{\bf p}^{\prime}}^{q}E_{p}^{V}}$ (34) $\displaystyle\times
2\pi\hbar\delta\left(E_{p}^{V}-E_{p^{\prime}}^{\overline{q}}-E_{{\bf p}-{\bf
p}^{\prime}}^{q}\right)\epsilon_{\alpha}^{\ast}(\lambda_{1},{\bf
p})\epsilon_{\beta}(\lambda_{2},{\bf p})$
$\displaystyle\times\text{Tr}\left[\Gamma^{\beta}v(s_{1},{\bf
p}^{\prime})\overline{v}(r_{1},{\bf
p}^{\prime})\Gamma^{\alpha}u(r_{2},\mathbf{p}-{\bf
p}^{\prime})\overline{u}(s_{2},\mathbf{p}-{\bf p}^{\prime})\right]$
$\displaystyle\times f_{r_{1}s_{1}}^{(\overline{q})}(x,{\bf
p}^{\prime})f_{r_{2}s_{2}}^{(q)}(x,\mathbf{p}-{\bf p}^{\prime}),$

where we have changed the notation to $f_{rs}^{(q/\overline{q})}$ from
$f_{rs}^{(\pm)}$. The spin density matrix element (34) can be put into a
compact form with an explicit dependence on the polarization vector of the
quark and antiquark

$\displaystyle\rho_{\lambda_{1}\lambda_{2}}^{V}(x,{\bf p})$ $\displaystyle=$
$\displaystyle\frac{\Delta
t}{32}\int\frac{d^{3}\mathbf{p}^{\prime}}{(2\pi\hbar)^{3}}\frac{1}{E_{p^{\prime}}^{\overline{q}}E_{{\bf
p}-{\bf
p}^{\prime}}^{q}E_{p}^{V}}f_{\overline{q}}(x,\mathbf{p}^{\prime})f_{q}(x,\mathbf{p}-\mathbf{p}^{\prime})$
(35) $\displaystyle\times
2\pi\hbar\delta\left(E_{p}^{V}-E_{p^{\prime}}^{\overline{q}}-E_{{\bf p}-{\bf
p}^{\prime}}^{q}\right)\epsilon_{\alpha}^{\ast}(\lambda_{1},{\bf
p})\epsilon_{\beta}(\lambda_{2},{\bf p})$
$\displaystyle\times\text{Tr}\left\\{\Gamma^{\beta}\left(p^{\prime}\cdot\gamma-
m_{\overline{q}}\right)\left[1+\gamma_{5}\gamma\cdot
P^{\overline{q}}(x,\mathbf{p}^{\prime})\right]\Gamma^{\alpha}\right.$
$\displaystyle\times\left.\left[(p-p^{\prime})\cdot\gamma+m_{q}\right]\left[1+\gamma_{5}\gamma\cdot
P^{q}(x,\mathbf{p}-\mathbf{p}^{\prime})\right]\right\\},$

where $p^{\mu}=(E_{p}^{q},\mathbf{p})$ and
$p^{\prime\mu}=(E_{p^{\prime}}^{\overline{q}},\mathbf{p}^{\prime})$. The
derivation of the expression inside the trace is given in Appendix B. The
contraction of $\epsilon_{\alpha}^{\ast}(\lambda_{1},{\bf p})$ and
$\epsilon_{\beta}(\lambda_{2},{\bf p})$ with the trace can be worked out and
the result is given by Eq. (86). The normalized
$\rho_{\lambda_{1}\lambda_{2}}^{V}(x,{\bf p})$ is defined as

$\overline{\rho}_{\lambda_{1}\lambda_{2}}^{V}(x,{\bf
p})=\frac{\rho_{\lambda_{1}\lambda_{2}}^{V}(x,{\bf
p})}{\mathrm{Tr}(\rho_{V})},$ (36)

where $\mathrm{Tr}(\rho_{V})$ is the trace of the spin density matrix and is
evaluated using Eq. (87) and $\rho_{\lambda_{1}\lambda_{2}}^{V}(x,{\bf p})$ is
evaluated using Eq. (86).

For quarkonium vector mesons such as $\phi$ mesons with
$m_{q}=m_{\overline{q}}$, $\rho_{\lambda_{1}\lambda_{2}}^{V}(x,{\bf p})$ and
$\mathrm{Tr}(\rho_{V})$ can be simplified as

$\displaystyle\rho_{\lambda_{1}\lambda_{2}}^{V}(x,{\bf p})$ $\displaystyle=$
$\displaystyle-\frac{\Delta
t}{8}g_{V}^{2}\int\frac{d^{3}\mathbf{p}^{\prime}}{(2\pi\hbar)^{3}}\frac{1}{E_{p^{\prime}}^{\overline{q}}E_{{\bf
p}-{\bf
p}^{\prime}}^{q}E_{p}^{V}}f_{\overline{q}}(x,\mathbf{p}^{\prime})f_{q}(x,\mathbf{p}-\mathbf{p}^{\prime})$
(37) $\displaystyle\times
2\pi\hbar\delta\left(E_{p}^{V}-E_{p^{\prime}}^{\overline{q}}-E_{{\bf p}-{\bf
p}^{\prime}}^{q}\right)B^{2}(\mathbf{p}-\mathbf{p}^{\prime},\mathbf{p}^{\prime})\epsilon_{\alpha}^{\ast}(\lambda_{1},{\bf
p})\epsilon_{\beta}(\lambda_{2},{\bf p})$
$\displaystyle\times\left\\{\left(p^{\prime\alpha}P_{\overline{q}}^{\beta}+p^{\prime\beta}P_{\overline{q}}^{\alpha}\right)(p^{\prime}\cdot
P_{q})-\left(p^{\prime\alpha}P_{q}^{\beta}+p^{\prime\beta}P_{q}^{\alpha}\right)(p\cdot
P_{\overline{q}})\right.$
$\displaystyle+2p^{\prime\alpha}p^{\prime\beta}(1-P_{\overline{q}}\cdot
P_{q})+g^{\alpha\beta}\left[p^{\prime}\cdot p+(p^{\prime}\cdot P_{q})(p\cdot
P_{\overline{q}})\right]$ $\displaystyle+(p\cdot
p^{\prime})\left(P_{\overline{q}}^{\alpha}P_{q}^{\beta}+P_{q}^{\alpha}P_{\overline{q}}^{\beta}-g^{\alpha\beta}P_{\overline{q}}\cdot
P_{q}\right)$
$\displaystyle\left.-im_{q}\varepsilon^{\alpha\beta\mu\nu}p_{\mu}(P_{\nu}^{q}+P_{\nu}^{\overline{q}})\right\\},$
$\displaystyle\mathrm{Tr}(\rho_{V})$ $\displaystyle=$
$\displaystyle\frac{\Delta
t}{8}g_{V}^{2}\int\frac{d^{3}\mathbf{p}^{\prime}}{(2\pi\hbar)^{3}}\frac{1}{E_{p^{\prime}}^{\overline{q}}E_{{\bf
p}-{\bf
p}^{\prime}}^{q}E_{p}^{V}}f_{\overline{q}}(x,\mathbf{p}^{\prime})f_{q}(x,\mathbf{p}-\mathbf{p}^{\prime})$
(38) $\displaystyle\times
2\pi\hbar\delta\left(E_{p}^{V}-E_{p^{\prime}}^{\overline{q}}-E_{{\bf p}-{\bf
p}^{\prime}}^{q}\right)B^{2}(\mathbf{p}-\mathbf{p}^{\prime},\mathbf{p}^{\prime})$
$\displaystyle\times\left[-2m_{q}^{2}(P_{\overline{q}}\cdot
P_{q})+m_{V}^{2}+2m_{q}^{2}\right],$

where we have used the short-hand notation $P_{q}\equiv
P_{q}(x,\mathbf{p}-\mathbf{p}^{\prime})$ and $P_{\overline{q}}\equiv
P_{\overline{q}}(x,\mathbf{p}^{\prime})$. Equations (37) and (38) will be used
in the next section for evaluating spin density matrix elements for $\phi$
mesons.

##  V Spin density matrix elements for $\phi$ mesons

Now we consider the vector meson made of a quark and its antiquark, the so-
called quarkonium. For the quarkonium vector meson such as the $\phi$ meson,
the polarization distributions in phase space in Eq. (35) are given by
(Becattini et al., 2013, 2017; Fang et al., 2016; Yang et al., 2018;
Weickgenannt et al., 2019)

$\displaystyle P_{s}^{\mu}(x,{\bf p})$ $\displaystyle=$
$\displaystyle\frac{1}{4m_{s}}\epsilon^{\mu\nu\rho\sigma}\left(\omega_{\rho\sigma}+\frac{g_{\phi}}{E_{p}^{s}T_{\mathrm{eff}}}F_{\rho\sigma}^{\phi}\right)p_{\nu}\left[1-f_{s}(x,{\bf
p})\right],$ $\displaystyle P_{\overline{s}}^{\mu}(x,{\bf p})$
$\displaystyle=$
$\displaystyle\frac{1}{4m_{s}}\epsilon^{\mu\nu\rho\sigma}\left(\omega_{\rho\sigma}-\frac{g_{\phi}}{E_{p}^{\overline{s}}T_{\mathrm{eff}}}F_{\rho\sigma}^{\phi}\right)p_{\nu}\left[1-f_{\overline{s}}(x,{\bf
p})\right],$ (39)

where $p^{\mu}=(E_{p}^{s},{\bf p})$ and $p^{\mu}=(E_{p}^{\overline{s}},{\bf
p})$ denote the four-momenta of the strange quark $s$ and antiquark
$\overline{s}$ respectively, with
$E_{p}^{s}=E_{p}^{\overline{s}}=\sqrt{|\mathbf{p}|^{2}+m_{s}^{2}}$ and
$m_{\overline{s}}=m_{s}$. We have assumed that $s$ and $\overline{s}$ are
polarized by the thermal vorticity (tensor) field
$\omega_{\rho\sigma}=(1/2)[\partial_{\rho}(\beta
u_{\sigma})-\partial_{\sigma}(\beta u_{\rho})]$ and $\phi$ field strength
tensor
$F_{\rho\sigma}^{\phi}=\partial_{\rho}A_{\sigma}^{\phi}-\partial_{\sigma}A_{\rho}^{\phi}$
(Sheng et al., 2020a), where $u_{\sigma}$ is the fluid velocity,
$\beta=1/T_{\mathrm{eff}}$ is the inverse effective temperature, and
$A_{\sigma}^{\phi}$ is the vector potential of the $\phi$ field. Note that in
some literature the definition of $\omega_{\rho\sigma}$ may differ by a sign
(Becattini et al., 2013, 2017; Huang et al., 2020a). In Eq. (39) $f_{s}(x,{\bf
p})$ and $f_{\overline{s}}(x,{\bf p})$ are unpolarized phase space
distributions of $s$ and $\overline{s}$ respectively and given by the Fermi-
Dirac distribution

$f_{s/\overline{s}}(x,{\bf p})=\frac{1}{1+\exp(\beta
E_{p}^{s/\overline{s}}\mp\mu_{s})}$ (40)

where $\mu_{s}$ is the chemical potential for $s$ ($-\mu_{s}$ for
$\overline{s}$). In most cases $f_{s/\overline{s}}$ are negligible relative to
1 in $P_{s/\overline{s}}^{\mu}$ in Eq. (39). The spin-field coupling in (39)
can be derived from the Wigner functions for massive fermions (Weickgenannt et
al., 2019) and has a clear physical meaning: one contribution is from the
magnetic field through the magnetic moment and the other contribution from the
electric field through the spin-orbit coupling, the former is always there
while the latter is only present for moving fermions. The mean field effects
of vector mesons have been studied in the context of spin polarization of
$\Lambda$ hyperons (Csernai et al., 2019) and different elliptic flows between
hadrons of some species and their antiparticles (Xu et al., 2012) in heavy-ion
collisions.

The spin direction four-vector for the $\phi$ meson is given by

$\epsilon^{\mu}(\lambda,{\bf p})=\left(\frac{{\bf
p}\cdot\boldsymbol{\epsilon}_{\lambda}}{m_{\phi}},\boldsymbol{\epsilon}_{\lambda}+\frac{{\bf
p}\cdot\boldsymbol{\epsilon}_{\lambda}}{m_{\phi}(E_{p}^{\phi}+m_{\phi})}{\bf
p}\right),$ (41)

where $E_{p}^{\phi}\equiv\sqrt{m_{\phi}^{2}+{\bf p}^{2}}$ is the energy of the
$\phi$ meson, $\lambda=0,\pm 1$ denotes the spin states, and
$\boldsymbol{\epsilon}_{\lambda}$ denotes the three-vector of the spin state
(spin vector) in the $\phi$ meson’s rest frame. In order to calculate the spin
alignment along the direction of the global orbital angular momentum (the
$y$-direction) in heavy-ion collisions, we choose the $y$-direction as the
spin quantization direction. So the corresponding spin vectors are

$\displaystyle\boldsymbol{\epsilon}_{0}$ $\displaystyle=$
$\displaystyle\left(0,1,0\right),$ $\displaystyle\boldsymbol{\epsilon}_{+1}$
$\displaystyle=$ $\displaystyle-\frac{1}{\sqrt{2}}\left(i,0,1\right),$
$\displaystyle\boldsymbol{\epsilon}_{-1}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(-i,0,1\right).$ (42)

The 00-component of the spin density matrix is what can be measured in
experiments which concerns the real vector $\boldsymbol{\epsilon}_{0}$
satisfying $\boldsymbol{\epsilon}_{0}=\boldsymbol{\epsilon}_{0}^{*}$.

Substituting Eq. (39) into Eq. (35), we obtain

$\displaystyle\rho_{\lambda_{1}\lambda_{2}}^{\phi}$ $\displaystyle=$
$\displaystyle\rho_{\lambda_{1}\lambda_{2}}^{\phi}(0)+\rho_{\lambda_{1}\lambda_{2}}^{\phi}(\omega^{1})+\rho_{\lambda_{1}\lambda_{2}}^{\phi}(F_{\phi}^{1})$
(43)
$\displaystyle+\rho_{\lambda_{1}\lambda_{2}}^{\phi}(\omega^{2})+\rho_{\lambda_{1}\lambda_{2}}^{\phi}(F_{\phi}^{2}),$

where $\omega^{i}$ and $F_{\phi}^{i}$ with $i=0,1,2$ denote the zeroth, first,
and second order terms in the vorticity and $\phi$ field respectively. The
zeroth order term $\rho_{\lambda_{1}\lambda_{2}}^{\phi}(0)$ represents the
unpolarized contribution. In (43), we neglected mixing terms of
$\omega_{\mu\nu}$ and $F_{\mu\nu}^{\phi}$ since we assume that there is no
correlation between them in space-time so these terms are vanishing after
taking a space-time average of $\rho_{\lambda_{1}\lambda_{2}}^{\phi}$. For
$\lambda_{1}=\lambda_{2}=0$, $\epsilon_{\alpha}^{\ast}(0,{\bf
p})\epsilon_{\beta}(0,{\bf p})=\epsilon_{\alpha}(0,{\bf
p})\epsilon_{\beta}(0,{\bf p})$ is symmetric in $\alpha$ and $\beta$, then one
can verify that the first order terms $\rho_{00}^{\phi}(\omega^{1})$ and
$\rho_{00}^{\phi}(F_{\phi}^{1})$ are vanishing. The zeroth order term
$\rho_{00}^{\phi}(0)$ is given by

$\displaystyle\rho_{00}^{\phi}(0)$ $\displaystyle=$ $\displaystyle\frac{\Delta
t}{8}g_{\phi}^{2}\int\frac{d^{3}\mathbf{p}^{\prime}}{(2\pi\hbar)^{3}}\frac{1}{E_{p^{\prime}}^{\overline{s}}E_{{\bf
p}-{\bf
p}^{\prime}}^{s}E_{p}^{\phi}}f_{\overline{s}}(\mathbf{p}^{\prime})f_{s}(\mathbf{p}-\mathbf{p}^{\prime})B^{2}(\mathbf{p}-\mathbf{p}^{\prime},\mathbf{p}^{\prime})$
(44) $\displaystyle\times
2\pi\hbar\delta\left(E_{p}^{\phi}-E_{p^{\prime}}^{\overline{s}}-E_{{\bf
p}-{\bf p}^{\prime}}^{s}\right)\left\\{(p^{\prime}\cdot
p)-2[p^{\prime}\cdot\epsilon(0,{\bf p})]^{2}\right\\},$

where we have used the second relation of Eq. (7). The second order terms
$\rho_{\lambda_{1}\lambda_{2}}^{\phi}(\omega^{2})$ and
$\rho_{\lambda_{1}\lambda_{2}}^{\phi}(F_{\phi}^{2})$ read

$\displaystyle\rho_{\lambda_{1}\lambda_{2}}^{\phi}(\omega^{2})$
$\displaystyle\approx$ $\displaystyle-\frac{\Delta
t}{32}\frac{1}{4m_{s}^{2}}g_{\phi}^{2}\int\frac{d^{3}\mathbf{p}^{\prime}}{(2\pi\hbar)^{3}}\frac{1}{E_{p^{\prime}}^{\overline{s}}E_{{\bf
p}-{\bf
p}^{\prime}}^{s}E_{p}^{\phi}}B^{2}(\mathbf{p}-\mathbf{p}^{\prime},\mathbf{p}^{\prime})$
(45) $\displaystyle\times
f_{\overline{s}}(\mathbf{p}^{\prime})f_{s}(\mathbf{p}-\mathbf{p}^{\prime})2\pi\hbar\delta\left(E_{p}^{\phi}-E_{p^{\prime}}^{\overline{s}}-E_{{\bf
p}-{\bf p}^{\prime}}^{s}\right)$
$\displaystyle\times\epsilon_{\alpha}^{\ast}(\lambda_{1},{\bf
p})\epsilon_{\beta}(\lambda_{2},{\bf
p})\widetilde{\omega}_{\rho\xi}(x)\widetilde{\omega}_{\sigma\gamma}(x)p^{\prime\xi}(p-p^{\prime})^{\gamma}$
$\displaystyle\times\text{Tr}\left\\{\gamma^{\beta}\left(p^{\prime}\cdot\gamma+m_{s}\right)\gamma^{\rho}\gamma^{\alpha}\left[(p-p^{\prime})\cdot\gamma+m_{s}\right]\gamma^{\sigma}\right\\},$

and

$\displaystyle\rho_{\lambda_{1}\lambda_{2}}^{\phi}(F_{\phi}^{2})$
$\displaystyle\approx$ $\displaystyle\frac{\Delta
t}{32}\frac{1}{4m_{s}^{2}T_{\mathrm{eff}}^{2}}g_{\phi}^{4}\int\frac{d^{3}\mathbf{p}^{\prime}}{(2\pi\hbar)^{3}}\frac{1}{(E_{p^{\prime}}^{\overline{s}})^{2}(E_{{\bf
p}-{\bf
p}^{\prime}}^{s})^{2}E_{p}^{\phi}}B^{2}(\mathbf{p}-\mathbf{p}^{\prime},\mathbf{p}^{\prime})$
(46) $\displaystyle\times
f_{\overline{s}}(\mathbf{p}^{\prime})f_{s}(\mathbf{p}-\mathbf{p}^{\prime})2\pi\hbar\delta\left(E_{p}^{\phi}-E_{p^{\prime}}^{\overline{s}}-E_{{\bf
p}-{\bf p}^{\prime}}^{s}\right)$
$\displaystyle\times\epsilon_{\alpha}^{\ast}(\lambda_{1},{\bf
p})\epsilon_{\beta}(\lambda_{2},{\bf
p})\widetilde{F}_{\rho\xi}^{\phi}(x)\widetilde{F}_{\sigma\gamma}^{\phi}(x)p^{\prime\xi}(p-p^{\prime})^{\gamma}$
$\displaystyle\times\text{Tr}\left\\{\gamma^{\beta}\left(p^{\prime}\cdot\gamma+m_{s}\right)\gamma^{\rho}\gamma^{\alpha}\left[(p-p^{\prime})\cdot\gamma+m_{s}\right]\gamma^{\sigma}\right\\}.$

In Eqs. (45) and (46) we have used
$\widetilde{\omega}_{\rho\xi}=(1/2)\epsilon_{\rho\xi\alpha\beta}\omega^{\alpha\beta}$,
$\widetilde{F}_{\rho\xi}^{\phi}=(1/2)\epsilon_{\rho\xi\alpha\beta}F_{\phi}^{\alpha\beta}$,
and neglected $f_{s/\overline{s}}$ relative to 1 in
$P_{s/\overline{s}}^{\mu}$. The tensor part of
$\rho_{\lambda_{1}\lambda_{2}}^{\phi}(\omega^{2})$ and
$\rho_{\lambda_{1}\lambda_{2}}^{\phi}(F_{\phi}^{2})$ that is contracted with
$\epsilon_{\alpha}^{\ast}\epsilon_{\beta}\widetilde{\omega}_{\rho\xi}\widetilde{\omega}_{\sigma\gamma}$
and
$\epsilon_{\alpha}^{\ast}\epsilon_{\beta}\widetilde{F}_{\rho\xi}^{\phi}\widetilde{F}_{\sigma\gamma}^{\phi}$
respectively can be evaluated as

$\displaystyle I^{\alpha\beta;\rho\xi;\sigma\gamma}$ $\displaystyle=$
$\displaystyle
p^{\prime\xi}(p-p^{\prime})^{\gamma}\text{Tr}\left\\{\gamma^{\beta}\left(p^{\prime}\cdot\gamma+m_{s}\right)\gamma^{\rho}\gamma^{\alpha}\left[(p-p^{\prime})\cdot\gamma+m_{s}\right]\gamma^{\sigma}\right\\}$
(47) $\displaystyle=$ $\displaystyle
2p^{\prime\xi}p^{\gamma}\left[m_{\phi}^{2}\left(g^{\beta\rho}g^{\alpha\sigma}-g^{\alpha\beta}g^{\rho\sigma}+g^{\beta\sigma}g^{\alpha\rho}\right)\right.$
$\displaystyle+2p^{\rho}\left(g^{\alpha\beta}p^{\prime\sigma}-g^{\alpha\sigma}p^{\prime\beta}-g^{\beta\sigma}p^{\prime\alpha}\right)$
$\displaystyle\left.+2\left(g^{\beta\rho}p^{\prime\alpha}p^{\prime\sigma}+g^{\alpha\rho}p^{\prime\beta}p^{\prime\sigma}-2g^{\rho\sigma}p^{\prime\alpha}p^{\prime\beta}\right)\right]$
$\displaystyle-2p^{\prime\xi}p^{\prime\gamma}\left[m_{\phi}^{2}\left(g^{\beta\rho}g^{\alpha\sigma}-g^{\alpha\beta}g^{\rho\sigma}+g^{\beta\sigma}g^{\alpha\rho}\right)\right.$
$\displaystyle\left.-2p^{\rho}\left(g^{\alpha\sigma}p^{\prime\beta}+g^{\beta\sigma}p^{\prime\alpha}\right)-4g^{\rho\sigma}p^{\prime\alpha}p^{\prime\beta}\right].$

With the above tensor and the quantity inside the curly brackets in (44),
$\rho_{00}^{\phi}(0)$, $\rho_{\lambda_{1}\lambda_{2}}^{\phi}(\omega^{2})$ and
$\rho_{\lambda_{1}\lambda_{2}}^{\phi}(F^{2})$ involve following moments of
momenta

$\displaystyle\left\\{I_{0},I_{0}^{\mu},I_{0}^{\mu\nu},I_{0}^{\mu\nu\rho},I_{0}^{\mu\nu\rho\sigma}\right\\}$
$\displaystyle=$
$\displaystyle\int\frac{d^{3}\mathbf{p}^{\prime}}{(2\pi\hbar)^{3}}\frac{1}{E_{p^{\prime}}^{\overline{s}}E_{{\bf
p}-{\bf
p}^{\prime}}^{s}}B^{2}(\mathbf{p}-\mathbf{p}^{\prime},\mathbf{p}^{\prime})$
(48) $\displaystyle\times
f_{\overline{s}}(\mathbf{p}^{\prime})f_{s}(\mathbf{p}-\mathbf{p}^{\prime})2\pi\hbar\delta\left(E_{p}^{\phi}-E_{p^{\prime}}^{\overline{s}}-E_{{\bf
p}-{\bf p}^{\prime}}^{s}\right)$
$\displaystyle\times\left\\{1,p^{\prime\mu},p^{\prime\mu}p^{\prime\nu},p^{\prime\mu}p^{\prime\nu}p^{\prime\rho},p^{\prime\mu}p^{\prime\nu}p^{\prime\rho}p^{\prime\sigma}\right\\},$
$\displaystyle\left\\{I_{F}^{\mu},I_{F}^{\mu\nu},I_{F}^{\mu\nu\rho},I_{F}^{\mu\nu\rho\sigma}\right\\}$
$\displaystyle=$
$\displaystyle\int\frac{d^{3}\mathbf{p}^{\prime}}{(2\pi\hbar)^{3}}\frac{1}{(E_{p^{\prime}}^{\overline{s}})^{2}(E_{{\bf
p}-{\bf
p}^{\prime}}^{s})^{2}}B^{2}(\mathbf{p}-\mathbf{p}^{\prime},\mathbf{p}^{\prime})$
(49) $\displaystyle\times
f_{\overline{s}}(\mathbf{p}^{\prime})f_{s}(\mathbf{p}-\mathbf{p}^{\prime})2\pi\hbar\delta\left(E_{p}^{\phi}-E_{p^{\prime}}^{\overline{s}}-E_{{\bf
p}-{\bf p}^{\prime}}^{s}\right)$
$\displaystyle\times\left\\{p^{\prime\mu},p^{\prime\mu}p^{\prime\nu},p^{\prime\mu}p^{\prime\nu}p^{\prime\rho},p^{\prime\mu}p^{\prime\nu}p^{\prime\rho}p^{\prime\sigma}\right\\}.$

The tensors in (48) with the subscript ’0’ are those in $\rho_{00}^{\phi}(0)$
and $\rho_{\lambda_{1}\lambda_{2}}^{\phi}(\omega^{2})$, and the tensors in
(49) with the subscript ’F’ are those in
$\rho_{\lambda_{1}\lambda_{2}}^{\phi}(F_{\phi}^{2})$. The difference between
Eq. (48) and Eq. (49) is in the powers of $E_{p^{\prime}}^{\overline{s}}$ and
$E_{{\bf p}-{\bf p}^{\prime}}^{s}$ in the denominators. Note that all above
tensors with 2 or more indices are symmetric with respect to the interchange
of any two indices.

Using Eqs. (47)-(49), the zeroth and second order terms of the spin density
matrix in (44), (45) and (46) can be expressed in terms of moments of momenta

$\rho_{00}^{\phi}(0)=\frac{\Delta
t}{16}g_{\phi}^{2}m_{\phi}^{2}\frac{1}{E_{p}^{\phi}}I_{0}\left[1-4\epsilon_{\alpha}(0,{\bf
p})\epsilon_{\beta}(0,{\bf
p})\frac{I_{0}^{\alpha\beta}}{m_{\phi}^{2}I_{0}}\right],$ (50)
$\displaystyle\rho_{00}^{\phi}(\omega^{2})$ $\displaystyle=$
$\displaystyle-\frac{\Delta
t}{64}\frac{g_{\phi}^{2}}{m_{s}^{2}}\frac{1}{E_{p}^{\phi}}\epsilon_{\alpha}(0,{\bf
p})\epsilon_{\beta}(0,{\bf
p})\widetilde{\omega}_{\rho\xi}(x)\widetilde{\omega}_{\sigma\gamma}(x)$ (51)
$\displaystyle\times\left[p^{\gamma}m_{\phi}^{2}\left(g^{\beta\rho}g^{\alpha\sigma}-g^{\alpha\beta}g^{\rho\sigma}+g^{\beta\sigma}g^{\alpha\rho}\right)I_{0}^{\xi}\right.$
$\displaystyle+2p^{\gamma}p^{\rho}\left(g^{\alpha\beta}I_{0}^{\xi\sigma}-g^{\alpha\sigma}I_{0}^{\xi\beta}-g^{\beta\sigma}I_{0}^{\xi\alpha}\right)$
$\displaystyle+2p^{\gamma}\left(g^{\beta\rho}I_{0}^{\xi\sigma\alpha}+I_{0}^{\xi\sigma\beta}g^{\alpha\rho}-2g^{\rho\sigma}I_{0}^{\xi\alpha\beta}\right)$
$\displaystyle-
m_{\phi}^{2}\left(g^{\beta\rho}g^{\alpha\sigma}-g^{\alpha\beta}g^{\rho\sigma}+g^{\beta\sigma}g^{\alpha\rho}\right)I_{0}^{\xi\gamma}$
$\displaystyle\left.+2p^{\rho}\left(g^{\alpha\sigma}I_{0}^{\xi\beta\gamma}+g^{\beta\sigma}I_{0}^{\xi\alpha\gamma}\right)+4g^{\rho\sigma}I_{0}^{\xi\alpha\beta\gamma}\right],$
$\displaystyle\rho_{00}^{\phi}(F_{\phi}^{2})$ $\displaystyle=$
$\displaystyle\frac{\Delta
t}{64}\frac{g_{\phi}^{4}}{m_{s}^{2}T_{\mathrm{eff}}^{2}}\frac{1}{E_{p}^{\phi}}\epsilon_{\alpha}(0,{\bf
p})\epsilon_{\beta}(0,{\bf
p})\widetilde{F}_{\rho\xi}^{\phi}(x)\widetilde{F}_{\sigma\gamma}^{\phi}(x)$
(52)
$\displaystyle\times\left[p^{\gamma}m_{\phi}^{2}\left(g^{\beta\rho}g^{\alpha\sigma}-g^{\alpha\beta}g^{\rho\sigma}+g^{\beta\sigma}g^{\alpha\rho}\right)I_{F}^{\xi}\right.$
$\displaystyle+2p^{\gamma}p^{\rho}\left(g^{\alpha\beta}I_{F}^{\xi\sigma}-g^{\alpha\sigma}I_{F}^{\xi\beta}-g^{\beta\sigma}I_{F}^{\xi\alpha}\right)$
$\displaystyle+2p^{\gamma}\left(g^{\beta\rho}I_{F}^{\xi\sigma\alpha}+I_{F}^{\xi\sigma\beta}g^{\alpha\rho}-2g^{\rho\sigma}I_{F}^{\xi\alpha\beta}\right)$
$\displaystyle-
m_{\phi}^{2}\left(g^{\beta\rho}g^{\alpha\sigma}-g^{\alpha\beta}g^{\rho\sigma}+g^{\beta\sigma}g^{\alpha\rho}\right)I_{F}^{\xi\gamma}$
$\displaystyle\left.+2p^{\rho}\left(g^{\alpha\sigma}I_{F}^{\xi\beta\gamma}+g^{\beta\sigma}I_{F}^{\xi\alpha\gamma}\right)+4g^{\rho\sigma}I_{F}^{\xi\alpha\beta\gamma}\right].$

From Eq. (38), the trace of the spin density matrix for the $\phi$ meson reads

$\displaystyle\mathrm{Tr}(\rho_{\phi})$ $\displaystyle=$
$\displaystyle\frac{\Delta
t}{8}g_{\phi}^{2}(m_{\phi}^{2}+2m_{s}^{2})\frac{1}{E_{p}^{\phi}}I_{0}$ (53)
$\displaystyle\times\left[1-\frac{1}{2(m_{\phi}^{2}+2m_{s}^{2})}\widetilde{\omega}_{\rho\xi}(x)\widetilde{\omega}_{\sigma\gamma}(x)g^{\rho\sigma}\frac{1}{I_{0}}\left(p^{\gamma}I_{0}^{\xi}-I_{0}^{\xi\gamma}\right)\right.$
$\displaystyle\left.+\frac{g_{\phi}^{2}}{2(m_{\phi}^{2}+2m_{s}^{2})T_{\mathrm{eff}}^{2}}\widetilde{F}_{\rho\xi}^{\phi}(x)\widetilde{F}_{\sigma\gamma}^{\phi}(x)g^{\rho\sigma}\frac{1}{I_{0}}\left(p^{\gamma}I_{F}^{\xi}-I_{F}^{\xi\gamma}\right)\right].$

Here we have neglected mixing terms of $\omega_{\mu\nu}$ and
$F_{\mu\nu}^{\phi}$ since we assume that there is no correlation in space-time
between them.

From Eqs. (50)-(53) we obtain the 00-component of the normalized spin density
matrix for the $\phi$ meson defined in (36)

$\overline{\rho}_{00}^{\phi}(x,{\bf p})=c_{0}({\bf p})+c_{\omega}(x,{\bf
p})+c_{F}(x,{\bf p}),$ (54)

where $c_{0}$, $c_{\omega}$ and $c_{F}$ are given by

$c_{0}({\bf
p})=\frac{m_{\phi}^{2}}{2(m_{\phi}^{2}+2m_{s}^{2})}\left[1-4\epsilon_{\alpha}(0,{\bf
p})\epsilon_{\beta}(0,{\bf
p})\frac{I_{0}^{\alpha\beta}}{m_{\phi}^{2}I_{0}}\right],$ (55) $\displaystyle
c_{\omega}(x,{\bf p})$ $\displaystyle=$
$\displaystyle-\frac{1}{8m_{s}^{2}(m_{\phi}^{2}+2m_{s}^{2})}\epsilon_{\alpha}(0,{\bf
p})\epsilon_{\beta}(0,{\bf
p})\widetilde{\omega}_{\rho\xi}(x)\widetilde{\omega}_{\sigma\gamma}(x)$ (56)
$\displaystyle\times\frac{1}{I_{0}}\left[p^{\gamma}m_{\phi}^{2}\left(g^{\beta\rho}g^{\alpha\sigma}-g^{\alpha\beta}g^{\rho\sigma}+g^{\beta\sigma}g^{\alpha\rho}\right)I_{0}^{\xi}\right.$
$\displaystyle+2p^{\gamma}p^{\rho}\left(g^{\alpha\beta}I_{0}^{\xi\sigma}-g^{\alpha\sigma}I_{0}^{\xi\beta}-g^{\beta\sigma}I_{0}^{\xi\alpha}\right)$
$\displaystyle+2p^{\gamma}\left(g^{\beta\rho}I_{0}^{\xi\sigma\alpha}+I_{0}^{\xi\sigma\beta}g^{\alpha\rho}-2g^{\rho\sigma}I_{0}^{\xi\alpha\beta}\right)$
$\displaystyle-
m_{\phi}^{2}\left(g^{\beta\rho}g^{\alpha\sigma}-g^{\alpha\beta}g^{\rho\sigma}+g^{\beta\sigma}g^{\alpha\rho}\right)I_{0}^{\xi\gamma}$
$\displaystyle\left.+2p^{\rho}\left(g^{\alpha\sigma}I_{0}^{\xi\beta\gamma}+g^{\beta\sigma}I_{0}^{\xi\alpha\gamma}\right)+4g^{\rho\sigma}I_{0}^{\xi\alpha\beta\gamma}\right]$
$\displaystyle+\frac{c_{0}({\bf
p})}{2(m_{\phi}^{2}+2m_{s}^{2})}\widetilde{\omega}_{\rho\xi}(x)\widetilde{\omega}_{\sigma\gamma}(x)g^{\rho\sigma}\frac{1}{I_{0}}\left(p^{\gamma}I_{0}^{\xi}-I_{0}^{\xi\gamma}\right),$

and

$\displaystyle c_{F}(x,{\bf p})$ $\displaystyle=$
$\displaystyle\frac{1}{8m_{s}^{2}(m_{\phi}^{2}+2m_{s}^{2})}\frac{g_{\phi}^{2}}{T_{\mathrm{eff}}^{2}}\epsilon_{\alpha}(0,{\bf
p})\epsilon_{\beta}(0,{\bf
p})\widetilde{F}_{\rho\xi}^{\phi}(x)\widetilde{F}_{\sigma\gamma}^{\phi}(x)$
(57)
$\displaystyle\times\frac{1}{I_{0}}\left[p^{\gamma}m_{\phi}^{2}\left(g^{\beta\rho}g^{\alpha\sigma}-g^{\alpha\beta}g^{\rho\sigma}+g^{\beta\sigma}g^{\alpha\rho}\right)I_{F}^{\xi}\right.$
$\displaystyle+2p^{\gamma}p^{\rho}\left(g^{\alpha\beta}I_{F}^{\xi\sigma}-g^{\alpha\sigma}I_{F}^{\xi\beta}-g^{\beta\sigma}I_{F}^{\xi\alpha}\right)$
$\displaystyle+2p^{\gamma}\left(g^{\beta\rho}I_{F}^{\xi\sigma\alpha}+I_{F}^{\xi\sigma\beta}g^{\alpha\rho}-2g^{\rho\sigma}I_{F}^{\xi\alpha\beta}\right)$
$\displaystyle-
m_{\phi}^{2}\left(g^{\beta\rho}g^{\alpha\sigma}-g^{\alpha\beta}g^{\rho\sigma}+g^{\beta\sigma}g^{\alpha\rho}\right)I_{F}^{\xi\gamma}$
$\displaystyle\left.+2p^{\rho}\left(g^{\alpha\sigma}I_{F}^{\xi\beta\gamma}+g^{\beta\sigma}I_{F}^{\xi\alpha\gamma}\right)+4g^{\rho\sigma}I_{F}^{\xi\alpha\beta\gamma}\right]$
$\displaystyle-\frac{g_{\phi}^{2}c_{0}({\bf
p})}{2(m_{\phi}^{2}+2m_{s}^{2})T_{\mathrm{eff}}^{2}}\widetilde{F}_{\rho\xi}^{\phi}(x)\widetilde{F}_{\sigma\gamma}^{\phi}(x)g^{\rho\sigma}\frac{1}{I_{0}}\left(p^{\gamma}I_{F}^{\xi}-I_{F}^{\xi\gamma}\right).$

We see in Eqs. (55)-(57) that the momentum moments always come with the factor
$1/I_{0}$, so they can be understood as normalized moments by $I_{0}$, a kind
of momentum averages.

| $I^{\mu}$ | $I^{00}$ | $I^{aa}$ | $I^{000}$ | $I^{0aa}$ | $I^{0000}$ | $I^{00aa}$ | $I^{aabb}$ | $I^{aaaa}$
---|---|---|---|---|---|---|---|---|---
$\omega$ | $(m_{\phi}/2)g^{\mu 0}$ | $m_{\phi}^{2}/4$ | $d_{0}m_{\phi}^{2}/4$ | $m_{\phi}^{3}/8$ | $d_{0}m_{\phi}^{3}/8$ | $m_{\phi}^{4}/16$ | $d_{0}m_{\phi}^{4}/16$ | $d_{0}^{2}m_{\phi}^{4}/80$ | $3d_{0}^{2}m_{\phi}^{4}/80$
$F_{\phi}$ | $(2/m_{\phi})g^{\mu 0}$ | $1$ | $d_{0}$ | $m_{\phi}/2$ | $d_{0}m_{\phi}/2$ | $m_{\phi}^{2}/4$ | $d_{0}m_{\phi}^{2}/4$ | $d_{0}^{2}m_{\phi}^{2}/20$ | $3d_{0}^{2}m_{\phi}^{2}/20$
Table 2: All nonvanishing moments of momenta normalized by $I_{0}$ in
$\overline{\rho}_{00}^{\phi}$ from contributions of the vorticity and the
$\phi$ field, which are evaluated in the rest frame of the vector meson. Note
that $I$ represents either $I_{0}$ or $I_{F}$. The definition for some
quantities are $I^{aa}\equiv I^{11}+I^{22}+I^{33}$, $I^{0aa}\equiv
I^{011}+I^{022}+I^{033}$, $I^{00aa}\equiv I^{0011}+I^{0022}+I^{0033}$,
$I^{aabb}\equiv I^{1122}+I^{2233}+I^{3311}$, and $I^{aaaa}\equiv
I^{1111}+I^{2222}+I^{3333}$. The constant $d_{0}$ is defined as $d_{0}\equiv
1-4m_{s}^{2}/m_{\phi}^{2}$.

We see in Eqs. (55)-(57) that $c_{0}$, $c_{\omega}$ and $c_{F}$ are all
Lorentz scalars, so it is convenient to evaluate them in the rest frame of the
vector meson. All nonvanishing moments of momenta in Eqs. (55)-(57) that are
evaluated in the rest frame of the vector meson are listed in Table 2. Finally
the result for $\overline{\rho}_{00}^{\phi}$ is

$\displaystyle\overline{\rho}_{00}^{\phi}(x,{\bf p})$ $\displaystyle\approx$
$\displaystyle\frac{1}{3}+C_{1}\left[\frac{1}{3}\left(\boldsymbol{\omega}^{\prime}\cdot\boldsymbol{\omega}^{\prime}-\frac{4g_{\phi}^{2}}{m_{\phi}^{2}T_{\mathrm{eff}}^{2}}{\bf
B}_{\phi}^{\prime}\cdot{\bf
B}_{\phi}^{\prime}\right)-(\boldsymbol{\epsilon}_{0}\cdot\boldsymbol{\omega}^{\prime})^{2}+\frac{4g_{\phi}^{2}}{m_{\phi}^{2}T_{\mathrm{eff}}^{2}}\left(\boldsymbol{\epsilon}_{0}\cdot{\bf
B}_{\phi}^{\prime}\right)^{2}\right]$ (58)
$\displaystyle+C_{2}\left[\frac{1}{3}\left(\boldsymbol{\varepsilon}^{\prime}\cdot\boldsymbol{\varepsilon}^{\prime}-\frac{4g_{\phi}^{2}}{m_{\phi}^{2}T_{\mathrm{eff}}^{2}}{\bf
E}_{\phi}^{\prime}\cdot{\bf
E}_{\phi}^{\prime}\right)-(\boldsymbol{\epsilon}_{0}\cdot\boldsymbol{\varepsilon}^{\prime})^{2}+\frac{4g_{\phi}^{2}}{m_{\phi}^{2}T_{\mathrm{eff}}^{2}}(\boldsymbol{\epsilon}_{0}\cdot{\bf
E}_{\phi}^{\prime})^{2}\right],$

where the fields with primes are in the rest frame of the vector meson,
$\boldsymbol{\varepsilon}$ and $\boldsymbol{\omega}$ denote the electric and
magnetic part of the vorticity tensor $\omega^{\mu\nu}$ respectively, ${\bf
E}_{\phi}$ and ${\bf B}_{\phi}$ denote the electric and magnetic part of the
$\phi$ field tensor $F_{\phi}^{\mu\nu}$ respectively, and $C_{1}$ and $C_{2}$
are two coefficients depending on masses of the quark and vector meson defined
as

$\displaystyle C_{1}$ $\displaystyle=$
$\displaystyle\frac{8m_{s}^{4}+16m_{s}^{2}m_{\phi}^{2}+3m_{\phi}^{4}}{120m_{s}^{2}(m_{\phi}^{2}+2m_{s}^{2})},$
$\displaystyle C_{2}$ $\displaystyle=$
$\displaystyle\frac{8m_{s}^{4}-14m_{s}^{2}m_{\phi}^{2}+3m_{\phi}^{4}}{120m_{s}^{2}(m_{\phi}^{2}+2m_{s}^{2})}.$
(59)

The result for $\overline{\rho}_{00}^{\phi}(x,{\bf p})$ in Eq. (58) is
rigorous and remarkable since all contributions are in squares of the fields.
This is a clear piece of evidence that there exists in the $\phi$ meson an
exact correlation between the strong force field coupled to the $s$ quark and
that coupled to the $\overline{s}$ quark. This feature makes $\rho_{00}$ for
quarkonium vector mesons very different from that for other vector mesons
carrying net charges or flavors.

One can approximate $\overline{\rho}_{00}^{\phi}$ by expanding $C_{1}$ and
$C_{2}$ in terms of the average quark momentum inside the vector meson as

$\displaystyle C_{1}$ $\displaystyle\approx$
$\displaystyle\frac{1}{6}+\frac{1}{9}d_{0}+O(d_{0}^{2}),$ $\displaystyle
C_{2}$ $\displaystyle\approx$ $\displaystyle\frac{1}{18}d_{0}+O(d_{0}^{2}),$
(60)

with $d_{0}\equiv 1-4m_{s}^{2}/m_{\phi}^{2}$, the result is

$\displaystyle\overline{\rho}_{00}^{\phi}(x,{\bf p})$ $\displaystyle\approx$
$\displaystyle\frac{1}{3}+\left(\frac{1}{6}+\frac{1}{9}d_{0}\right)\left\\{\frac{1}{3}\left(\boldsymbol{\omega}^{\prime}\cdot\boldsymbol{\omega}^{\prime}-\frac{4g_{\phi}^{2}}{m_{\phi}^{2}T_{\mathrm{eff}}^{2}}{\bf
B}_{\phi}^{\prime}\cdot{\bf B}_{\phi}^{\prime}\right)\right.$ (61)
$\displaystyle\left.-(\boldsymbol{\epsilon}_{0}\cdot\boldsymbol{\omega}^{\prime})^{2}+\frac{4g_{\phi}^{2}}{m_{\phi}^{2}T_{\mathrm{eff}}^{2}}\left(\boldsymbol{\epsilon}_{0}\cdot{\bf
B}_{\phi}^{\prime}\right)^{2}\right\\}$
$\displaystyle+\frac{1}{18}d_{0}\left\\{\frac{1}{3}\left(\boldsymbol{\varepsilon}^{\prime}\cdot\boldsymbol{\varepsilon}^{\prime}-\frac{4g_{\phi}^{2}}{m_{\phi}^{2}T_{\mathrm{eff}}^{2}}{\bf
E}_{\phi}^{\prime}\cdot{\bf E}_{\phi}^{\prime}\right)\right.$
$\displaystyle\left.-(\boldsymbol{\epsilon}_{0}\cdot\boldsymbol{\varepsilon}^{\prime})^{2}+\frac{4g_{\phi}^{2}}{m_{\phi}^{2}T_{\mathrm{eff}}^{2}}(\boldsymbol{\epsilon}_{0}\cdot{\bf
E}_{\phi}^{\prime})^{2}\right\\}+O(d_{0}^{2}).$

The above result can be compared with that in the nonrelativistic limit (see
Appendix C). In order to recover the momentum dependence, one can express
$\overline{\rho}_{00}^{\phi}$ in terms of lab-frame fields. The transformation
of the fields between the lab and rest frame reads

$\displaystyle{\bf B}_{\phi}^{\prime}$ $\displaystyle=$
$\displaystyle\gamma{\bf B}_{\phi}-\gamma{\bf v}\times{\bf
E}_{\phi}+(1-\gamma)\frac{{\bf v}\cdot{\bf B}_{\phi}}{v^{2}}{\bf v},$
$\displaystyle{\bf E}_{\phi}^{\prime}$ $\displaystyle=$
$\displaystyle\gamma{\bf E}_{\phi}+\gamma{\bf v}\times{\bf
B}_{\phi}+(1-\gamma)\frac{{\bf v}\cdot{\bf E}_{\phi}}{v^{2}}{\bf v},$
$\displaystyle\boldsymbol{\omega}^{\prime}$ $\displaystyle=$
$\displaystyle\gamma\boldsymbol{\omega}-\gamma{\bf
v}\times\boldsymbol{\varepsilon}+(1-\gamma)\frac{{\bf
v}\cdot\boldsymbol{\omega}}{v^{2}}{\bf v},$
$\displaystyle\boldsymbol{\varepsilon}^{\prime}$ $\displaystyle=$
$\displaystyle\gamma\boldsymbol{\varepsilon}+\gamma{\bf
v}\times\boldsymbol{\omega}+(1-\gamma)\frac{{\bf
v}\cdot\boldsymbol{\varepsilon}}{v^{2}}{\bf v},$ (62)

where $\gamma=E_{{\bf p}}^{\phi}/m_{\phi}$ is the Lorentz factor and ${\bf
v}=\mathbf{p}/E_{{\bf p}}^{\phi}$ is the velocity of the $\phi$ meson. Taking
the $y$-direction as the spin quantization direction,
$\boldsymbol{\epsilon}_{0}=(0,1,0)$, we obtain $\overline{\rho}_{00}^{\phi}$
in terms of the fields in the lab frame

$\displaystyle\overline{\rho}_{00}^{\phi}(x,{\bf p})$ $\displaystyle\approx$
$\displaystyle\frac{1}{3}+\frac{1}{3}\sum_{i=1,2,3}I_{B,i}({\bf
p})\frac{1}{m_{\phi}^{2}}\left[\boldsymbol{\omega}_{i}^{2}-\frac{4g_{\phi}^{2}}{m_{\phi}^{2}T_{\mathrm{eff}}^{2}}(\mathbf{B}_{i}^{\phi})^{2}\right]$
(63) $\displaystyle+\frac{1}{3}\sum_{i=1,2,3}I_{E,i}({\bf
p})\frac{1}{m_{\phi}^{2}}\left[\boldsymbol{\varepsilon}_{i}^{2}-\frac{4g_{\phi}^{2}}{m_{\phi}^{2}T_{\mathrm{eff}}^{2}}(\mathbf{E}_{i}^{\phi})^{2}\right],$

where the coefficients are given by

$\displaystyle I_{B,x}({\bf p})$ $\displaystyle=$ $\displaystyle
C_{1}\left[(E_{{\bf
p}}^{\phi})^{2}-\left(1+\frac{3p_{y}^{2}}{(m_{\phi}+E_{{\bf
p}}^{\phi})^{2}}\right)p_{x}^{2}\right]+C_{2}(p_{y}^{2}-2p_{z}^{2}),$
$\displaystyle I_{E,x}({\bf p})$ $\displaystyle=$ $\displaystyle
C_{1}(p_{y}^{2}-2p_{z}^{2})+C_{2}\left[(E_{{\bf
p}}^{\phi})^{2}-\left(1+\frac{3p_{y}^{2}}{(m_{\phi}+E_{{\bf
p}}^{\phi})^{2}}\right)p_{x}^{2}\right],$ $\displaystyle I_{B,y}({\bf p})$
$\displaystyle=$ $\displaystyle C_{1}\left[6\frac{E_{{\bf
p}}^{\phi}}{m_{\phi}+E_{{\bf p}}^{\phi}}p_{y}^{2}-2(E_{{\bf
p}}^{\phi})^{2}-p_{y}^{2}-\frac{3p_{y}^{4}}{(m_{\phi}+E_{{\bf
p}}^{\phi})^{2}}\right]+C_{2}(p_{x}^{2}+p_{z}^{2}),$ $\displaystyle
I_{E,y}({\bf p})$ $\displaystyle=$ $\displaystyle
C_{1}(p_{x}^{2}+p_{z}^{2})+C_{2}\left[6\frac{E_{{\bf
p}}^{\phi}}{m_{\phi}+E_{{\bf p}}^{\phi}}p_{y}^{2}-2(E_{{\bf
p}}^{\phi})^{2}-p_{y}^{2}-\frac{3p_{y}^{4}}{(m_{\phi}+E_{{\bf
p}}^{\phi})^{2}}\right],$ $\displaystyle I_{B,z}({\bf p})$ $\displaystyle=$
$\displaystyle C_{1}\left[(E_{{\bf
p}}^{\phi})^{2}-\left(1+\frac{3p_{y}^{2}}{(m_{\phi}+E_{{\bf
p}}^{\phi})^{2}}\right)p_{z}^{2}\right]+C_{2}(p_{y}^{2}-2p_{x}^{2}),$
$\displaystyle I_{E,z}({\bf p})$ $\displaystyle=$ $\displaystyle
C_{1}(p_{y}^{2}-2p_{x}^{2})+C_{2}\left[(E_{{\bf
p}}^{\phi})^{2}-\left(1+\frac{3p_{y}^{2}}{(m_{\phi}+E_{{\bf
p}}^{\phi})^{2}}\right)p_{z}^{2}\right].$ (64)

The result in Eq. (63) is remarkable in its factorization form: the momentum
functions are separated from space-time functions. This has an advantage that
the momentum functions can be determined by experimental data on momentum
spectra while unknown space-time functions can be extracted from data on
$\overline{\rho}_{00}^{\phi}$.

One can take an average of $\overline{\rho}_{00}^{\phi}(x,{\bf p})$ over the
local space-time volume in which the vector meson is formed as

$\displaystyle\left\langle\overline{\rho}_{00}^{\phi}(x,{\bf
p})\right\rangle_{x}$ $\displaystyle\approx$
$\displaystyle\frac{1}{3}+\frac{1}{3}\sum_{i=1,2,3}I_{B,i}({\bf
p})\frac{1}{m_{\phi}^{2}}\left[\left\langle\boldsymbol{\omega}_{i}^{2}\right\rangle-\frac{4g_{\phi}^{2}}{m_{\phi}^{2}T_{\mathrm{eff}}^{2}}\left\langle(\mathbf{B}_{i}^{\phi})^{2}\right\rangle\right]$
(65) $\displaystyle+\frac{1}{3}\sum_{i=1,2,3}I_{E,i}({\bf
p})\frac{1}{m_{\phi}^{2}}\left[\left\langle\boldsymbol{\varepsilon}_{i}^{2}\right\rangle-\frac{4g_{\phi}^{2}}{m_{\phi}^{2}T_{\mathrm{eff}}^{2}}\left\langle(\mathbf{E}_{i}^{\phi})^{2}\right\rangle\right].$

These averaged field squares can play as parameters and be determined by
comparing $\left\langle\overline{\rho}_{00}^{\phi}(x,{\bf
p})\right\rangle_{x}$ with the data of $\rho_{00}^{\phi}$ as functions of
transverse momenta. One can further take a momentum average of
$\left\langle\overline{\rho}_{00}^{\phi}(x,{\bf p})\right\rangle_{x}$ and
compare with the data as functions of collision energies,

$\displaystyle\left\langle\overline{\rho}_{00}^{\phi}(x,{\bf
p})\right\rangle_{x,\mathbf{p}}$ $\displaystyle\approx$
$\displaystyle\frac{1}{3}+\frac{1}{3}\sum_{i=1,2,3}\left\langle I_{B,i}({\bf
p})\right\rangle\frac{1}{m_{\phi}^{2}}\left[\left\langle\boldsymbol{\omega}_{i}^{2}\right\rangle-\frac{4g_{\phi}^{2}}{m_{\phi}^{2}T_{\mathrm{eff}}^{2}}\left\langle(\mathbf{B}_{i}^{\phi})^{2}\right\rangle\right]$
(66) $\displaystyle+\frac{1}{3}\sum_{i=1,2,3}\left\langle I_{E,i}({\bf
p})\right\rangle\frac{1}{m_{\phi}^{2}}\left[\left\langle\boldsymbol{\varepsilon}_{i}^{2}\right\rangle-\frac{4g_{\phi}^{2}}{m_{\phi}^{2}T_{\mathrm{eff}}^{2}}\left\langle(\mathbf{E}_{i}^{\phi})^{2}\right\rangle\right],$

where the momentum average is defined as

$\left\langle O(\mathbf{p})\right\rangle=\frac{\int
d^{3}\mathbf{p}O(\mathbf{p})f_{\phi}(\mathbf{p})}{\int
d^{3}\mathbf{p}f_{\phi}(\mathbf{p})},$ (67)

with $f_{\phi}(\mathbf{p})$ being the momentum distribution of the $\phi$
meson which can be determined by experimental data. The theoretical results
for $\rho_{00}^{\phi}$ as functions of transverse momenta, collision energies
and centralities are presented in Ref. (Sheng et al., 2022), which are in a
good agreement with recent STAR data (Abdallah et al., 2022).

##  VI Discussions and conclusions

In this section we will discuss about the main results as well as
approximations or assumptions that have been made in this paper.

The Lagrangian (1) is for real vector fields since we are concerned about the
charge or flavor neutral particles such as quarkonia made of a quark and its
antiquark. To describe those particles that carry net charge or flavor, we
have to consider complex vector fields. The generalization of the formalism to
complex vector fields is straightforward.

The vector fields that polarize $s$ and $\overline{s}$ are assumed to be the
$\phi$ fields, the effective (color singlet) modes of the strong force that
carry vacuum quantum number. As an input to the general formula (35) we assume
that $P_{q}^{\mu}$ and $P_{\overline{q}}^{\mu}$ have the linear form in Eq.
(39) in the vorticity and $\phi$ fields. The coupling between the spin and
fluid velocity field is assumed to be through the vorticity. One can also
introduce other coupling forms such as spin-shear couplings (Liu and Yin,
2021; Becattini et al., 2021a; Fu et al., 2021; Becattini et al., 2021b). The
spin coupling to the $\phi$ field is assumed to have a covariant form
$\sim\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}^{\phi}p_{\nu}$. This is one
of our main assumptions. Of course one can use other forms of spin-field
couplings or add more terms to Eq. (39). An alternative choice is to use the
coupling of the spin and gluon field as in the nonrelativistic quantum
chromodynamics (NRQCD) (Bodwin et al., 1995; Braaten, 1996). But the
Hamiltonian of NRQCD is not covariant at all and may be different from the
covariant from $\sim\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}^{c}p_{\nu}$
where $F_{\alpha\beta}^{c}$ is the gluon field with adjoint color $c$. In this
case the final result may be different from the result in this paper.

If we use gluon fields that are coupled to the spin, the MVSD for the quark
and antiquark in (22) and the polarization vectors in (39) have to be modified
to include color indices. The effect to $\rho_{\lambda_{1}\lambda_{2}}^{V}$ in
(35) is to replace $P_{q}^{\mu}\rightarrow P_{q}^{\mu,c}$ and
$P_{\overline{q}}^{\mu}\rightarrow P_{\overline{q}}^{\mu,c}$ with $c$ denoting
the adjoint color of the gluon and to sum over the gluon color $c$ to
accommodate the color singlet requirement. Correspondingly, in Eq. (39), we
replace $P_{s}^{\mu}\rightarrow P_{s}^{\mu,c}$,
$P_{\overline{s}}^{\mu}\rightarrow P_{\overline{s}}^{\mu,c}$, and
$F_{\rho\sigma}^{\phi}\rightarrow F_{\rho\sigma}^{c}\lambda^{c}/2$, where
$\lambda^{c}$ are Gell-mann matrices. The effect to
$\rho_{00}^{\phi}(F_{\phi}^{2})$ in (46) is to replace
$\widetilde{F}_{\rho\xi}^{\phi}(x)\widetilde{F}_{\sigma\gamma}^{\phi}(x)\rightarrow\sum_{c}\widetilde{F}_{\rho\xi}^{c}(x)\widetilde{F}_{\sigma\gamma}^{c}(x)$
up to a color factor, which results in the replacements
$(\mathbf{E}_{i}^{\phi})^{2}\rightarrow\sum_{c}(\mathbf{E}_{i}^{c})^{2}$ and
$(\mathbf{B}_{i}^{\phi})^{2}\rightarrow\sum_{c}(\mathbf{B}_{i}^{c})^{2}$ in
the final results in Eqs. (63)-(66), where $\mathbf{E}^{c}$ and
$\mathbf{B}^{c}$ are the chromoelectric and chromomagnetic fields
respectively.

In evaluating the integrals in momentum moments in the rest frame of the
vector meson, we assume a simple form for the fluid four-velocity
$u^{\prime\mu}=(1,\mathbf{0})$ so that the quark and antiquark distributions
depend only on energies. Then we obtain the simple form of
$\overline{\rho}_{00}^{\phi}(x,{\bf p})$ in Eq. (58) with $C_{1}$ and $C_{2}$
depending only on masses as shown in (59). In general the fluid four-velocity
has also a spatial component or three-velocity, in this case
$\overline{\rho}_{00}^{\phi}(x,{\bf p})$ should have much more complicated
form than Eq. (58) where the coefficients also depend on the three-velocity of
the fluid in a more sophisticated way.

According to the chiral quark model in Ref. (Manohar and Georgi, 1984) the
local averaged field squares
$\left\langle(\mathbf{B}_{i}^{\phi})^{2}\right\rangle$ and
$\left\langle(\mathbf{E}_{i}^{\phi})^{2}\right\rangle$ are related to the
fields of psuedo-Goldstone bosons. They are also related to gluon fluctuation
of instantons (Shuryak, 1982a, b) according to the quark model based on
instanton vacuum (Diakonov and Petrov, 1986). If quarks and antiquarks are
polarized by gluon fields, the local averaged field squares are related to the
gluon condensate which contributes to the trace anomaly of the energy momentum
tensor. Therefore the local averaged field squares are in connection with
fundamental properties of the QCD vacuum which play an important role in
hadron structures (Shifman et al., 1979a, b).

In summary, a relativistic theory for the spin density matrix of vector mesons
is constructed based on Kadanoff-Baym (KB) equations from which the spin
Boltzmann equations are derived. With the spin Boltzmann equations we
formulate the spin density matrix element $\rho_{00}$ for $\phi$ mesons. The
dominant contributions to $\rho_{00}^{\phi}$ at lower energies are assumed to
come from the $\phi$ field, a kind of the strong force field that can polarize
the strange quark and antiquark in the same way as the electromagnetic field.
The key observation is that there is correlation inside the $\phi$ meson wave
function between the $\phi$ field that polarizes the strange quark and that
polarizes the strange antiquark. This is reflected by the fact that the
contributions to $\rho_{00}^{\phi}$ are all in squares of the fields which are
nonvanishing even if the fields may strongly fluctuate. Then the fluctuations
of strong force fields can be extracted from $\rho_{00}$ of quarkonium vector
mesons as links to fundamental properties of QCD.

## Acknowledgement

The authors thank C.D. Roberts for providing us with the Bethe-Salpeter wave
function of the $\phi$ meson. The authors thank X. G. Huang, J. F. Liao, S.
Pu, A. H. Tang, D. L. Yang and Y. Yin for helpful discussion. This work is
supported in part by the National Natural Science Foundation of China (NSFC)
under Grants No. 12135011, 11890713 (a subgrant of 11890710), the Strategic
Priority Research Program of the Chinese Academy of Sciences (CAS) under Grant
No. XDB34030102, and by the Director, Office of Energy Research, Office of
High Energy and Nuclear Physics, Division of Nuclear Physics, of the U.S.
Department of Energy under Contract No. DE-AC02-05CH11231.

##  Appendix A Collision terms for coalescence and dissociation

In this appendix, we will derive the collision term for the coalescence
process of the quark and antiquark into the vector meson corresponding to
$I_{-++}$ in Eq. (26).

The explicit form of $I_{-++}$ is

$\displaystyle I_{-++}$ $\displaystyle=$
$\displaystyle\sum_{r_{1},s_{1},r_{2},s_{2},\lambda_{1}^{\prime},\lambda_{2}^{\prime}}\left\\{\epsilon^{\alpha}\left(\lambda_{1}^{\prime},{\bf
p}\right)\epsilon^{\nu\ast}\left(\lambda_{2}^{\prime},{\bf p}\right)\right.$
(68) $\displaystyle\times\text{Tr}\left[\Gamma_{\alpha}v(s_{1},-{\bf
p}^{\prime})\overline{v}(r_{1},-{\bf
p}^{\prime})\Gamma^{\mu}u(r_{2},\mathbf{p}+\mathbf{p}^{\prime})\overline{u}(s_{2},\mathbf{p}+\mathbf{p}^{\prime})\right]$
$\displaystyle+\epsilon^{\mu}\left(\lambda_{1}^{\prime},{\bf
p}\right)\epsilon_{\alpha}^{\ast}\left(\lambda_{2}^{\prime},{\bf p}\right)$
$\displaystyle\times\left.\text{Tr}\left[\Gamma^{\nu}v(s_{1},-{\bf
p}^{\prime})\overline{v}(r_{1},-{\bf
p}^{\prime})\Gamma^{\alpha}u(r_{2},\mathbf{p}+\mathbf{p}^{\prime})\overline{u}(s_{2},\mathbf{p}+\mathbf{p}^{\prime})\right]\right\\}$
$\displaystyle\times\left\\{\left[\delta_{r_{1}s_{1}}-f_{r_{1}s_{1}}^{(-)}(x,-{\bf
p}^{\prime})\right]\left[\delta_{r_{2}s_{2}}-f_{r_{2}s_{2}}^{(+)}(x,\mathbf{p}+\mathbf{p}^{\prime})\right]f_{\lambda_{1}^{\prime}\lambda_{2}^{\prime}}(x,\mathbf{p})\right.$
$\displaystyle\left.-f_{r_{1}s_{1}}^{(-)}(x,-{\bf
p}^{\prime})f_{r_{2}s_{2}}^{(+)}(x,\mathbf{p}+\mathbf{p}^{\prime})\left[\delta_{\lambda_{1}^{\prime}\lambda_{2}^{\prime}}+f_{\lambda_{1}^{\prime}\lambda_{2}^{\prime}}(x,\mathbf{p})\right]\right\\}.$

The corresponding collision term reads

$\displaystyle C_{\text{coalescence}}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{4(2\pi\hbar)}\sum_{r_{1},s_{1},r_{2},s_{2},\lambda_{1}^{\prime},\lambda_{2}^{\prime}}\int
d^{3}\mathbf{p}^{\prime}\frac{1}{8E_{p^{\prime}}^{\overline{q}}E_{{\bf p}-{\bf
p}^{\prime}}^{q}E_{p}^{V}}\delta\left(E_{p}^{V}-E_{p^{\prime}}^{\overline{q}}-E_{{\bf
p}-{\bf p}^{\prime}}^{q}\right)\delta(p^{0}-E_{p}^{V})$ (69)
$\displaystyle\times\left\\{\epsilon^{\alpha}\left(\lambda_{1}^{\prime},{\bf
p}\right)\epsilon^{\nu\ast}\left(\lambda_{2}^{\prime},{\bf
p}\right)\text{Tr}\left[\Gamma_{\alpha}v(s_{1},{\bf
p}^{\prime})\overline{v}(r_{1},{\bf
p}^{\prime})\Gamma^{\mu}u(r_{2},\mathbf{p}-{\bf
p}^{\prime})\overline{u}(s_{2},\mathbf{p}-{\bf p}^{\prime})\right]\right.$
$\displaystyle+\left.\epsilon^{\mu}\left(\lambda_{1}^{\prime},{\bf
p}\right)\epsilon_{\alpha}^{\ast}\left(\lambda_{2}^{\prime},{\bf
p}\right)\text{Tr}\left[\Gamma^{\nu}v(s_{1},{\bf
p}^{\prime})\overline{v}(r_{1},{\bf
p}^{\prime})\Gamma^{\alpha}u(r_{2},\mathbf{p}-{\bf
p}^{\prime})\overline{u}(s_{2},\mathbf{p}-{\bf p}^{\prime})\right]\right\\}$
$\displaystyle\times\left\\{\left[\delta_{r_{1}s_{1}}-f_{r_{1}s_{1}}^{(-)}(x,{\bf
p}^{\prime})\right]\left[\delta_{r_{2}s_{2}}-f_{r_{2}s_{2}}^{(+)}(x,\mathbf{p}-\mathbf{p}^{\prime})\right]f_{\lambda_{1}^{\prime}\lambda_{2}^{\prime}}(x,\mathbf{p})\right.$
$\displaystyle\left.-f_{r_{1}s_{1}}^{(-)}(x,{\bf
p}^{\prime})f_{r_{2}s_{2}}^{(+)}(x,\mathbf{p}-\mathbf{p}^{\prime})\left[\delta_{\lambda_{1}^{\prime}\lambda_{2}^{\prime}}+f_{\lambda_{1}^{\prime}\lambda_{2}^{\prime}}(x,\mathbf{p})\right]\right\\},$

where we have changed the sign of the antiquark’s three-momentum as ${\bf
p}^{\prime}\rightarrow-{\bf p}^{\prime}$ in the integral, used Eq. (7) and the
relation

$\displaystyle\delta(p^{\prime
2}-m_{q}^{2})\delta\left[(p+p^{\prime})^{2}-m_{q}^{2}\right]\delta(p^{2}-m_{V}^{2})\theta(-p_{0}^{\prime})\theta\left(p_{0}+p_{0}^{\prime}\right)\theta(p_{0})$
(70) $\displaystyle=$
$\displaystyle\frac{1}{8E_{p^{\prime}}^{\overline{q}}E_{{\bf p}+{\bf
p}^{\prime}}^{q}E_{p}^{V}}\delta(p_{0}^{\prime}+E_{p^{\prime}}^{\overline{q}})\delta\left(p_{0}+p_{0}^{\prime}-E_{{\bf
p}+{\bf p}^{\prime}}^{q}\right)\delta(p_{0}-E_{p}^{V})$ $\displaystyle=$
$\displaystyle\frac{1}{8E_{p^{\prime}}^{\overline{q}}E_{{\bf p}+{\bf
p}^{\prime}}^{q}E_{p}^{V}}\delta(p_{0}^{\prime}+E_{p^{\prime}}^{\overline{q}})\delta\left(E_{p}^{V}-E_{p^{\prime}}^{\overline{q}}-E_{{\bf
p}+{\bf p}^{\prime}}^{q}\right)\delta(p_{0}-E_{p}^{V}).$

From (11), the particle sector of $p\cdot\partial_{x}G^{<,\mu\nu}(x,p)$ in the
left-hand-side of Eq. (26) becomes

$\displaystyle p\cdot\partial_{x}G^{<,\mu\nu}(x,p)$ $\displaystyle=$
$\displaystyle 2\pi\hbar\frac{1}{2E_{p}^{V}}\delta(p_{0}-E_{p}^{V})$ (71)
$\displaystyle\times\sum_{\lambda_{1}^{\prime},\lambda_{2}^{\prime}}\epsilon^{\mu}\left(\lambda_{1}^{\prime},{\bf
p}\right)\epsilon^{\nu\ast}\left(\lambda_{2}^{\prime},{\bf
p}\right)p\cdot\partial_{x}f_{\lambda_{1}^{\prime}\lambda_{2}^{\prime}}(x,\mathbf{p}).$

Using Eqs. (69) and (71) into Eq. (26), taking a contraction of the resulting
equation with $\epsilon_{\mu}^{\ast}(\lambda_{1},{\bf p})$ and
$\epsilon_{\nu}(\lambda_{2},{\bf p})$, and using the first identity in (7), we
obtain

$\displaystyle p\cdot\partial_{x}f_{\lambda_{1}\lambda_{2}}(x,\mathbf{p})$
(72) $\displaystyle=$
$\displaystyle\frac{1}{16}\sum_{r_{1},s_{1},r_{2},s_{2},\lambda_{1}^{\prime},\lambda_{2}^{\prime}}\int\frac{d^{3}\mathbf{p}^{\prime}}{(2\pi\hbar)^{3}}\frac{1}{E_{p^{\prime}}^{\overline{q}}E_{{\bf
p}-{\bf
p}^{\prime}}^{q}}2\pi\hbar\delta\left(E_{p}^{V}-E_{p^{\prime}}^{\overline{q}}-E_{{\bf
p}-{\bf p}^{\prime}}^{q}\right)$
$\displaystyle\times\left\\{\delta_{\lambda_{2}\lambda_{2}^{\prime}}\epsilon_{\mu}^{\ast}(\lambda_{1},{\bf
p})\epsilon^{\alpha}\left(\lambda_{1}^{\prime},{\bf
p}\right)\text{Tr}\left[\Gamma_{\alpha}v(s_{1},{\bf
p}^{\prime})\overline{v}(r_{1},{\bf
p}^{\prime})\Gamma^{\mu}u(r_{2},\mathbf{p}-{\bf
p}^{\prime})\overline{u}(s_{2},\mathbf{p}-{\bf p}^{\prime})\right]\right.$
$\displaystyle+\left.\delta_{\lambda_{1}\lambda_{1}^{\prime}}\epsilon_{\nu}(\lambda_{2},{\bf
p})\epsilon_{\alpha}^{\ast}\left(\lambda_{2}^{\prime},{\bf
p}\right)\text{Tr}\left[\Gamma^{\nu}v(s_{1},{\bf
p}^{\prime})\overline{v}(r_{1},{\bf
p}^{\prime})\Gamma^{\alpha}u(r_{2},\mathbf{p}-{\bf
p}^{\prime})\overline{u}(s_{2},\mathbf{p}-{\bf p}^{\prime})\right]\right\\}$
$\displaystyle\times\left\\{f_{r_{1}s_{1}}^{(-)}(x,{\bf
p}^{\prime})f_{r_{2}s_{2}}^{(+)}(x,\mathbf{p}-\mathbf{p}^{\prime})\left[\delta_{\lambda_{1}^{\prime}\lambda_{2}^{\prime}}+f_{\lambda_{1}^{\prime}\lambda_{2}^{\prime}}(x,\mathbf{p})\right]\right.$
$\displaystyle\left.-\left[\delta_{r_{1}s_{1}}-f_{r_{1}s_{1}}^{(-)}(x,{\bf
p}^{\prime})\right]\left[\delta_{r_{2}s_{2}}-f_{r_{2}s_{2}}^{(+)}(x,\mathbf{p}-\mathbf{p}^{\prime})\right]f_{\lambda_{1}^{\prime}\lambda_{2}^{\prime}}(x,\mathbf{p})\right\\},$

which reproduces Eq. (27). Note that the terms proportional to $p^{\mu}$ and
$p^{\nu}$ in the left-hand-side of Eq. (26) do not contribute since their
contraction with $\epsilon_{\mu}^{\ast}(\lambda_{1},{\bf p})$ and
$\epsilon_{\nu}(\lambda_{2},{\bf p})$ is vanishing.

We consider the coalescence process in heavy ion collisions in which the MVSDs
of quarks, antiquarks and vector mesons are assumed to be much smaller than
unity. So the term with $\delta_{\lambda_{1}\lambda_{2}}$ dominates the gain
term which can be simplified as

$\displaystyle\mathrm{gain}$ $\displaystyle\approx$
$\displaystyle\frac{1}{8(2\pi\hbar)^{2}}\sum_{r_{1},s_{1},r_{2},s_{2}}\int
d^{3}\mathbf{p}^{\prime}\frac{1}{E_{p^{\prime}}^{\overline{q}}E_{{\bf p}-{\bf
p}^{\prime}}^{q}}\delta\left(E_{p}^{V}-E_{p^{\prime}}^{\overline{q}}-E_{{\bf
p}-{\bf p}^{\prime}}^{q}\right)$ (73)
$\displaystyle\times\epsilon_{\mu}^{\ast}\left(\lambda_{1},{\bf
p}\right)\epsilon_{\alpha}\left(\lambda_{2},{\bf
p}\right)\text{Tr}\left[\Gamma^{\alpha}v(s_{1},{\bf
p}^{\prime})\overline{v}(r_{1},{\bf
p}^{\prime})\Gamma^{\mu}u(r_{2},\mathbf{p}-{\bf
p}^{\prime})\overline{u}(s_{2},\mathbf{p}-{\bf p}^{\prime})\right]$
$\displaystyle\times f_{r_{1}s_{1}}^{(-)}(x,{\bf
p}^{\prime})f_{r_{2}s_{2}}^{(+)}(x,\mathbf{p}-\mathbf{p}^{\prime}),$

which gives Eq. (31). The loss term can be simplified as

$\displaystyle\mathrm{loss}$ $\displaystyle\approx$
$\displaystyle-\frac{1}{16(2\pi\hbar)^{2}}\sum_{r_{1},s_{1},r_{2},s_{2}}\int
d^{3}\mathbf{p}^{\prime}\frac{1}{E_{p^{\prime}}^{\overline{q}}E_{{\bf p}-{\bf
p}^{\prime}}^{q}}\delta\left(E_{p}^{V}-E_{p^{\prime}}^{\overline{q}}-E_{{\bf
p}-{\bf p}^{\prime}}^{q}\right)$ (74)
$\displaystyle\times\left\\{\delta_{\lambda_{2}\lambda_{2}^{\prime}}\epsilon_{\mu}^{\ast}\left(\lambda_{1},{\bf
p}\right)\epsilon^{\alpha}\left(\lambda_{1}^{\prime},{\bf
p}\right)\text{Tr}\left[\Gamma_{\alpha}v(s_{1},{\bf
p}^{\prime})\overline{v}(r_{1},{\bf
p}^{\prime})\Gamma^{\mu}u(r_{2},\mathbf{p}-{\bf
p}^{\prime})\overline{u}(s_{2},\mathbf{p}-{\bf p}^{\prime})\right]\right.$
$\displaystyle+\left.\delta_{\lambda_{1}\lambda_{1}^{\prime}}\epsilon_{\nu}\left(\lambda_{2},{\bf
p}\right)\epsilon_{\alpha}^{\ast}\left(\lambda_{2}^{\prime},{\bf
p}\right)\text{Tr}\left[\Gamma^{\nu}v(s_{1},{\bf
p}^{\prime})\overline{v}(r_{1},{\bf
p}^{\prime})\Gamma^{\alpha}u(r_{2},\mathbf{p}-{\bf
p}^{\prime})\overline{u}(s_{2},\mathbf{p}-{\bf p}^{\prime})\right]\right\\}$
$\displaystyle\times\delta_{r_{1}s_{1}}\delta_{r_{2}s_{2}}f_{\lambda_{1}^{\prime}\lambda_{2}^{\prime}}(x,\mathbf{p}),$
$\displaystyle=$ $\displaystyle-\frac{1}{16(2\pi\hbar)^{2}}\int
d^{3}\mathbf{p}^{\prime}\frac{1}{E_{p^{\prime}}^{\overline{q}}E_{{\bf p}-{\bf
p}^{\prime}}^{q}}\delta\left(E_{p}^{V}-E_{p^{\prime}}^{\overline{q}}-E_{{\bf
p}-{\bf p}^{\prime}}^{q}\right)$
$\displaystyle\times\left\\{\sum_{\lambda_{1}^{\prime}}f_{\lambda_{1}^{\prime}\lambda_{2}}(x,\mathbf{p})\epsilon_{\mu}^{\ast}\left(\lambda_{1},{\bf
p}\right)\epsilon^{\alpha}\left(\lambda_{1}^{\prime},{\bf
p}\right)\text{Tr}\left[\Gamma_{\alpha}\left(p^{\prime}\cdot\gamma-
m_{\overline{q}}\right)\Gamma^{\mu}\left((p-p^{\prime})\cdot\gamma+m_{q}\right)\right]\right.$
$\displaystyle+\left.\sum_{\lambda_{2}^{\prime}}f_{\lambda_{1}\lambda_{2}^{\prime}}(x,\mathbf{p})\epsilon_{\nu}\left(\lambda_{2},{\bf
p}\right)\epsilon_{\alpha}^{\ast}\left(\lambda_{2}^{\prime},{\bf
p}\right)\text{Tr}\left[\Gamma^{\nu}\left(p^{\prime}\cdot\gamma-
m_{\overline{q}}\right)\Gamma^{\alpha}\left((p-p^{\prime})\cdot\gamma+m_{q}\right)\right]\right\\}$
$\displaystyle=$ $\displaystyle-\frac{1}{16(2\pi\hbar)^{2}}\int
d^{3}\mathbf{p}^{\prime}\frac{1}{E_{p^{\prime}}^{\overline{q}}E_{{\bf p}-{\bf
p}^{\prime}}^{q}}\delta\left(E_{p}^{V}-E_{p^{\prime}}^{\overline{q}}-E_{{\bf
p}-{\bf p}^{\prime}}^{q}\right)$
$\displaystyle\times\left[\sum_{\lambda_{1}^{\prime}}f_{\lambda_{1}^{\prime}\lambda_{2}}(x,\mathbf{p})\epsilon_{\mu}^{\ast}\left(\lambda_{1},{\bf
p}\right)\epsilon_{\alpha}\left(\lambda_{1}^{\prime},{\bf
p}\right)+\sum_{\lambda_{2}^{\prime}}f_{\lambda_{1}\lambda_{2}^{\prime}}(x,\mathbf{p})\epsilon_{\mu}^{\ast}\left(\lambda_{2}^{\prime},{\bf
p}\right)\epsilon_{\alpha}\left(\lambda_{2},{\bf p}\right)\right]$
$\displaystyle\times\text{Tr}\left\\{\Gamma^{\alpha}\left(p^{\prime}\cdot\gamma-
m_{\overline{q}}\right)\Gamma^{\mu}\left[(p-p^{\prime})\cdot\gamma+m_{q}\right]\right\\},$

where we have neglected $f_{r_{1}s_{1}}^{(-)}$ and $f_{r_{2}s_{2}}^{(+)}$
relative to $\delta_{r_{1}s_{1}}$ and $\delta_{r_{2}s_{2}}$ respectively.
After completing the integral over $\mathbf{p}^{\prime}$ in the vector meson’s
rest frame, one can prove

$\displaystyle\epsilon_{\mu}^{\ast}\left(\lambda_{1},{\bf
p}\right)\epsilon_{\alpha}\left(\lambda_{1}^{\prime},{\bf p}\right)\int
d^{3}\mathbf{p}^{\prime}\frac{1}{4E_{p^{\prime}}^{\overline{q}}E_{{\bf p}-{\bf
p}^{\prime}}^{q}}\delta\left(E_{p}^{V}-E_{p^{\prime}}^{\overline{q}}-E_{{\bf
p}-{\bf p}^{\prime}}^{q}\right)$
$\displaystyle\times\text{Tr}\left\\{\Gamma^{\alpha}\left(p^{\prime}\cdot\gamma-
m_{\overline{q}}\right)\Gamma^{\mu}\left[(p-p^{\prime})\cdot\gamma+m_{q}\right]\right\\}\propto\delta_{\lambda_{1}\lambda_{1}^{\prime}},$
(75)

so we can replace

$\displaystyle\sum_{\lambda_{1}^{\prime}}f_{\lambda_{1}^{\prime}\lambda_{2}}(x,\mathbf{p})\epsilon_{\mu}^{\ast}\left(\lambda_{1},{\bf
p}\right)\epsilon_{\alpha}\left(\lambda_{1}^{\prime},{\bf
p}\right)+\sum_{\lambda_{2}^{\prime}}f_{\lambda_{1}^{\prime}\lambda_{2}}(x,\mathbf{p})\epsilon_{\mu}^{\ast}\left(\lambda_{2},{\bf
p}\right)\epsilon_{\alpha}(\lambda_{2}^{\prime},{\bf p})$ (76)
$\displaystyle\rightarrow$
$\displaystyle-\frac{2}{3}f_{\lambda_{1}\lambda_{2}}(x,\mathbf{p})\left(g_{\mu\alpha}-\frac{p_{\mu}p_{\alpha}}{m_{V}^{2}}\right),$

then the loss term becomes

$\displaystyle\mathrm{loss}$ $\displaystyle\approx$
$\displaystyle\frac{1}{12(2\pi\hbar)^{2}}f_{\lambda_{1}\lambda_{2}}(x,\mathbf{p})\left(g_{\mu\alpha}-\frac{p_{\mu}p_{\alpha}}{m_{V}^{2}}\right)\int
d^{3}\mathbf{p}^{\prime}\frac{1}{E_{p^{\prime}}^{\overline{q}}E_{{\bf p}-{\bf
p}^{\prime}}^{q}}\delta\left(E_{p}^{V}-E_{p^{\prime}}^{\overline{q}}-E_{{\bf
p}-{\bf p}^{\prime}}^{q}\right)$ (77)
$\displaystyle\times\text{Tr}\left\\{\Gamma^{\alpha}\left(p^{\prime}\cdot\gamma-
m_{\overline{q}}\right)\Gamma^{\mu}\left[(p-p^{\prime})\cdot\gamma+m_{q}\right]\right\\}.$

This gives Eq. (32).

##  Appendix B Collision kernel

The spin density matrix element for vector mesons is given by Eq. (35). In
this appendix we evaluate the collision kernel in Eq. (35)

$\displaystyle I_{\lambda_{1}\lambda_{2}}({\bf p},{\bf p}^{\prime})$
$\displaystyle=$ $\displaystyle I^{\alpha\beta}({\bf p},{\bf
p}^{\prime})\epsilon_{\alpha}^{\ast}(\lambda_{1},{\bf
p})\epsilon_{\beta}(\lambda_{2},{\bf p}),$ (78)

where $I^{\alpha\beta}({\bf p},{\bf p}^{\prime})$ is defined as

$\displaystyle I^{\alpha\beta}({\bf p},{\bf p}^{\prime})$
$\displaystyle\equiv$
$\displaystyle\text{Tr}\left[\Gamma^{\beta}v(s_{1},\mathbf{p}^{\prime})\overline{v}(r_{1},\mathbf{p}^{\prime})\Gamma^{\alpha}u(r_{2},\mathbf{p}-\mathbf{p}^{\prime})\overline{u}(s_{2},\mathbf{p}-\mathbf{p}^{\prime})\right]$
(79) $\displaystyle\times
f_{r_{1}s_{1}}^{(\overline{q})}(x,\mathbf{p}^{\prime})f_{r_{2}s_{2}}^{(q)}(x,\mathbf{p}-\mathbf{p}^{\prime}).$

Now we use following formula to simplify $I^{\alpha\beta}$. For quark spinors
of particles and antiparticles we have

$\displaystyle u(r,\mathbf{p})\overline{u}(s,\mathbf{p})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(m_{q}+\gamma^{\mu}p_{\mu}\right)\delta_{rs}+\frac{1}{2}m_{q}\gamma^{5}\gamma^{\mu}n_{\mu}(\mathbf{n}_{sr},\mathbf{p},m_{q})$
$\displaystyle-\frac{1}{4}\epsilon_{\mu\nu\alpha\beta}\sigma^{\mu\nu}p^{\alpha}n^{\beta}(\mathbf{n}_{sr},\mathbf{p},m_{q}),$
$\displaystyle\sum_{r,s}u(r,\mathbf{p})\overline{u}(s,\mathbf{p})(\tau_{j})_{rs}$
$\displaystyle=$ $\displaystyle
m_{q}\gamma^{5}\gamma^{\mu}n_{\mu}(\mathbf{n}_{j},\mathbf{p},m_{q})-\frac{1}{2}\epsilon_{\mu\nu\alpha\beta}\sigma^{\mu\nu}p^{\alpha}n^{\beta}(\mathbf{n}_{j},\mathbf{p},m_{q}),$
$\displaystyle v(r,\mathbf{p})\overline{v}(s,\mathbf{p})$ $\displaystyle=$
$\displaystyle\frac{1}{2}(-m_{\overline{q}}+p^{\mu}\gamma_{\mu})\delta_{rs}-\frac{1}{2}m_{\overline{q}}\gamma_{5}\gamma_{\mu}n^{\mu}(\mathbf{n}_{sr}^{*},\mathbf{p},m_{\overline{q}})$
$\displaystyle-\frac{1}{4}\epsilon^{\mu\nu\alpha\beta}\sigma_{\mu\nu}p_{\alpha}n_{\beta}(\mathbf{n}_{sr}^{*},\mathbf{p},m_{\overline{q}}),$
$\displaystyle\sum_{r,s}v(r,\mathbf{p})\overline{v}(s,\mathbf{p})(\tau_{j})_{sr}$
$\displaystyle=$ $\displaystyle-
m_{\overline{q}}\gamma_{5}\gamma_{\mu}n^{\mu}(\mathbf{n}_{j},\mathbf{p},m_{\overline{q}})-\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}\sigma_{\mu\nu}p_{\alpha}n_{\beta}(\mathbf{n}_{j},\mathbf{p},m_{\overline{q}}),$
(80)

where we have used
$\mathbf{n}_{sr}^{*}=\mathbf{n}_{rs}=\mathbf{n}_{i}(\tau_{i})_{rs}$. Inserting
Eq. (22) into Eq. (79) gives

$\displaystyle I^{\alpha\beta}({\bf p},{\bf p}^{\prime})$ $\displaystyle=$
$\displaystyle\text{Tr}\left[\Gamma^{\beta}v(s_{1},\mathbf{p}^{\prime})\overline{v}(r_{1},\mathbf{p}^{\prime})\Gamma^{\alpha}u(r_{2},\mathbf{p}-\mathbf{p}^{\prime})\overline{u}(s_{2},\mathbf{p}-\mathbf{p}^{\prime})\right]$
(81)
$\displaystyle\times\frac{1}{2}f_{\overline{q}}(x,\mathbf{p}^{\prime})\left[\delta_{r_{1}s_{1}}-P_{\mu}^{\overline{q}}(x,\mathbf{p}^{\prime})n_{j}^{(-)\mu}(-\mathbf{p}^{\prime})\tau_{r_{1}s_{1}}^{j}\right]$
$\displaystyle\times\frac{1}{2}f_{q}(x,\mathbf{p}-\mathbf{p}^{\prime})\left[\delta_{r_{2}s_{2}}-P_{\mu}^{q}(x,\mathbf{p}-\mathbf{p}^{\prime})n_{j}^{(+)\mu}(\mathbf{p}-\mathbf{p}^{\prime})\tau_{r_{2}s_{2}}^{j}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{4}f_{\overline{q}}(x,\mathbf{p}^{\prime})f_{q}(x,\mathbf{p}-\mathbf{p}^{\prime})$
$\displaystyle\times\text{Tr}\left\\{\Gamma^{\beta}\left(p^{\prime}\cdot\gamma-
m_{\overline{q}}\right)\left[1+\gamma_{5}\gamma\cdot
P^{\overline{q}}(x,\mathbf{p}^{\prime})\right]\Gamma^{\alpha}\right.$
$\displaystyle\times\left.\left[(p-p^{\prime})\cdot\gamma+m_{q}\right]\left[1+\gamma_{5}\gamma\cdot
P^{q}(x,\mathbf{p}-\mathbf{p}^{\prime})\right]\right\\},$

where we have used

$\displaystyle
v(s_{1},\mathbf{p}^{\prime})\overline{v}(r_{1},\mathbf{p}^{\prime})\left[\delta_{r_{1}s_{1}}-P_{\mu}^{\overline{q}}(x,\mathbf{p}^{\prime})n_{j}^{(-)\mu}(-\mathbf{p}^{\prime})\tau_{r_{1}s_{1}}^{j}\right]$
(82) $\displaystyle=$ $\displaystyle\left(p^{\prime}\cdot\gamma-
m_{\overline{q}}\right)\left[1+\gamma_{5}\gamma\cdot
P^{\overline{q}}(x,\mathbf{p}^{\prime})\right],$

and

$\displaystyle
u(r_{2},\mathbf{p}-\mathbf{p}^{\prime})\overline{u}(s_{2},\mathbf{p}-\mathbf{p}^{\prime})$
(83)
$\displaystyle\times\left[\delta_{r_{2}s_{2}}-P_{\mu}^{q}(x,\mathbf{p}-\mathbf{p}^{\prime})n_{j}^{(+)\mu}(\mathbf{p}-\mathbf{p}^{\prime})\tau_{r_{2}s_{2}}^{j}\right]$
$\displaystyle=$
$\displaystyle\left[(p-p^{\prime})\cdot\gamma+m_{q}\right]\left[1+\gamma^{5}\gamma\cdot
P^{q}(x,\mathbf{p}-\mathbf{p}^{\prime})\right].$

In deriving (82) and (83) we have used

$\displaystyle\frac{1}{2}\epsilon^{\alpha\beta\mu\nu}\sigma_{\mu\nu}$
$\displaystyle=$
$\displaystyle\gamma_{5}\gamma^{\alpha}\gamma^{\beta}-g^{\alpha\beta}\gamma_{5},$
$\displaystyle n_{j}^{(-)\mu}(-\mathbf{p}^{\prime})$ $\displaystyle=$
$\displaystyle n^{\mu}(\mathbf{n}_{j},\mathbf{p}^{\prime},m_{\overline{q}}),$
$\displaystyle p^{\prime\mu}P_{\mu}^{\overline{q}}(x,\mathbf{p}^{\prime})$
$\displaystyle=$
$\displaystyle(p-p^{\prime})^{\mu}P_{\mu}^{q}(x,\mathbf{p}-\mathbf{p}^{\prime})=0.$
(84)

Inserting (81) into (78) we obtain

$\displaystyle I_{\lambda_{1}\lambda_{2}}$ $\displaystyle=$
$\displaystyle\epsilon_{\alpha}^{\ast}(\lambda_{1},{\bf
p})\epsilon_{\beta}(\lambda_{2},{\bf p})$ (85)
$\displaystyle\times\text{Tr}\left\\{\Gamma^{\beta}\left(p^{\prime}\cdot\gamma-
m_{\overline{q}}\right)\left[1+\gamma_{5}\gamma\cdot
P_{\overline{q}}(x,\mathbf{p}^{\prime})\right]\Gamma^{\alpha}\right.$
$\displaystyle\times\left.\left[(p-p^{\prime})\cdot\gamma+m_{q}\right]\left[1+\gamma_{5}\gamma\cdot
P_{q}(x,\mathbf{p}-\mathbf{p}^{\prime})\right]\right\\}.$

From Eq. (85) one arrives at Eq. 35. Using (28), the trace in (85) can be
worked out and the result of $I_{\lambda_{1}\lambda_{2}}$ is

$\displaystyle I_{\lambda_{1}\lambda_{2}}$ $\displaystyle=$
$\displaystyle-4g_{V}^{2}B^{2}(\mathbf{p}-\mathbf{p}^{\prime},\mathbf{p}^{\prime})\epsilon_{\alpha}^{\ast}(\lambda_{1})\epsilon_{\beta}(\lambda_{2})$
(86)
$\displaystyle\times\left\\{\left(p^{\prime\alpha}P_{\overline{q}}^{\beta}+p^{\prime\beta}P_{\overline{q}}^{\alpha}\right)(p^{\prime}\cdot
P_{q})-\left(p^{\prime\alpha}P_{q}^{\beta}+p^{\prime\beta}P_{q}^{\alpha}\right)(p\cdot
P_{\overline{q}})\right.$
$\displaystyle+2p^{\prime\alpha}p^{\prime\beta}(1-P_{\overline{q}}\cdot
P_{q})+g^{\alpha\beta}\left[p^{\prime}\cdot p+(p^{\prime}\cdot P_{q})(p\cdot
P_{\overline{q}})\right]$
$\displaystyle+\left[(m_{q}-m_{\overline{q}})m_{\overline{q}}+p\cdot
p^{\prime}\right]\left(P_{\overline{q}}^{\alpha}P_{q}^{\beta}+P_{q}^{\alpha}P_{\overline{q}}^{\beta}-g^{\alpha\beta}P_{\overline{q}}\cdot
P_{q}\right)$
$\displaystyle-(m_{\overline{q}}-m_{q})m_{\overline{q}}g^{\alpha\beta}-i(m_{q}-m_{\overline{q}})\varepsilon^{\alpha\beta\mu\nu}p_{\mu}^{\prime}(P_{\nu}^{q}+P_{\nu}^{\overline{q}})$
$\displaystyle\left.-im_{\overline{q}}\varepsilon^{\alpha\beta\mu\nu}p_{\mu}(P_{\nu}^{q}+P_{\nu}^{\overline{q}})\right\\},$

where we have used shorthand notations
$\epsilon(\lambda)\equiv\epsilon(\lambda,\mathbf{p})$, $P_{q}\equiv
P_{q}(x,\mathbf{p}-\mathbf{p}^{\prime})$, and $P_{\overline{q}}\equiv
P_{\overline{q}}(x,\mathbf{p}^{\prime})$. We can take the sum of
$I_{\lambda\lambda}$ over $\lambda$ as

$\displaystyle\sum_{\lambda}I_{\lambda\lambda}$ $\displaystyle=$
$\displaystyle
2g_{V}^{2}B^{2}(\mathbf{p}-\mathbf{p}^{\prime},\mathbf{p}^{\prime})$ (87)
$\displaystyle\times\left\\{\left[-\left(m_{q}+m_{\overline{q}}\right)^{2}+\frac{1}{m_{V}^{2}}\left(m_{\overline{q}}^{2}-m_{q}^{2}\right)^{2}\right](P_{\overline{q}}\cdot
P_{q})\right.$
$\displaystyle+\frac{2}{m_{V}^{2}}\left(m_{q}-m_{\overline{q}}\right)^{2}(p\cdot
P_{q})(p\cdot P_{\overline{q}})$
$\displaystyle\left.+2m_{V}^{2}+6m_{q}m_{\overline{q}}-m_{\overline{q}}^{2}-m_{q}^{2}-\frac{1}{m_{V}^{2}}\left(m_{\overline{q}}^{2}-m_{q}^{2}\right)^{2}\right\\}.$

From Eq. (87) one can obtain $\mathrm{Tr}(\rho_{V})$ in Eq. (36).

##  Appendix C Spin density matrix in nonrelativistic limit

We consider $m_{q}=m_{\overline{q}}$ and assume $m_{V}\approx 2m_{q}$. In non-
relativistic limit, we can approximate

$\displaystyle p^{\mu}$ $\displaystyle\approx$ $\displaystyle(m_{V},{\bf 0}),$
$\displaystyle p^{\prime\mu}$ $\displaystyle\approx$
$\displaystyle(m_{\overline{q}},{\bf 0})\approx(m_{q},{\bf 0}),$
$\displaystyle p^{\mu}-p^{\prime\mu}$ $\displaystyle\approx$
$\displaystyle(m_{V}-m_{\overline{q}},0)\approx(m_{q},0),$ $\displaystyle
P_{\overline{q}}^{\mu}(x,\mathbf{p}^{\prime})$ $\displaystyle\approx$
$\displaystyle(0,\mathbf{P}_{\overline{q}}(x,\mathbf{p}^{\prime})),$
$\displaystyle P_{q}^{\mu}(x,\mathbf{p}-\mathbf{p}^{\prime})$
$\displaystyle\approx$
$\displaystyle(0,\mathbf{P}_{q}(x,\mathbf{p}-\mathbf{p}^{\prime})),$
$\displaystyle\epsilon^{\mu}(\lambda_{1})$ $\displaystyle\approx$
$\displaystyle(0,\boldsymbol{\epsilon}(\lambda_{1})),$
$\displaystyle\epsilon^{\mu}(\lambda_{2})$ $\displaystyle\approx$
$\displaystyle(0,\boldsymbol{\epsilon}(\lambda_{2})),$ (88)

which leads to

$\displaystyle p^{\prime}\cdot\epsilon^{\ast}(\lambda_{1})$
$\displaystyle\approx$ $\displaystyle 0,$ $\displaystyle
p^{\prime}\cdot\epsilon(\lambda_{2})$ $\displaystyle\approx$ $\displaystyle
0,$ $\displaystyle p^{\prime}\cdot p$ $\displaystyle\approx$ $\displaystyle
m_{V}m_{q},$
$\displaystyle\epsilon^{\ast}(\lambda_{1})\cdot\epsilon(\lambda_{2})$
$\displaystyle\approx$
$\displaystyle-\boldsymbol{\epsilon}^{\ast}(\lambda_{1})\cdot\boldsymbol{\epsilon}(\lambda_{2}).$
(89)

In Eqs. (88) and (89) we have used the shorthand notation
$\epsilon(\lambda)\equiv\epsilon(\lambda,\mathbf{p})$. Using (88) and (89),
$I_{\lambda_{1}\lambda_{2}}$ in Eq. (86) has a simple form

$\displaystyle I_{\lambda_{1}\lambda_{2}}$ $\displaystyle=$ $\displaystyle
4g_{V}^{2}m_{V}m_{q}\left\\{\boldsymbol{\epsilon}^{\ast}(\lambda_{1})\cdot\boldsymbol{\epsilon}(\lambda_{2})\left(1+\mathbf{P}_{\overline{q}}\cdot\mathbf{P}_{q}\right)\right.$
(90)
$\displaystyle-[\mathbf{P}_{\overline{q}}\cdot\boldsymbol{\epsilon}^{\ast}(\lambda_{1})][\mathbf{P}_{q}\cdot\boldsymbol{\epsilon}(\lambda_{2})]-[\mathbf{P}_{q}\cdot\boldsymbol{\epsilon}^{\ast}(\lambda_{1})][\mathbf{P}_{\overline{q}}\cdot\boldsymbol{\epsilon}(\lambda_{2})]$
$\displaystyle\left.-i[\boldsymbol{\epsilon}^{\ast}(\lambda_{1})\times\boldsymbol{\epsilon}(\lambda_{2})]\cdot(\mathbf{P}_{q}+\mathbf{P}_{\overline{q}})\right\\}.$

One can verify $I_{\lambda_{2}\lambda_{1}}=I_{\lambda_{1}\lambda_{2}}^{*}$.

From (35), the spin density matrix for the vector meson in the non-
relativistic limit is given by

$\displaystyle\rho_{\lambda_{1}\lambda_{2}}^{V}(x,{\bf p})$ $\displaystyle=$
$\displaystyle\frac{\Delta
t}{8}g_{V}^{2}m_{V}m_{q}\int\frac{d^{3}\mathbf{p}^{\prime}}{(2\pi\hbar)^{3}}\frac{1}{E_{p^{\prime}}^{\overline{q}}E_{{\bf
p}-{\bf p}^{\prime}}^{q}E_{p}^{V}}$ (91) $\displaystyle\times
f_{\overline{q}}(x,\mathbf{p}^{\prime})f_{q}(x,\mathbf{p}-\mathbf{p}^{\prime})2\pi\hbar\delta\left(E_{p}^{V}-E_{p^{\prime}}^{\overline{q}}-E_{{\bf
p}-{\bf p}^{\prime}}^{q}\right)$
$\displaystyle\times\left\\{\boldsymbol{\epsilon}^{\ast}(\lambda_{1})\cdot\boldsymbol{\epsilon}(\lambda_{2})\left[1+{\bf
P}_{q}(x,{\bf p}-{\bf p}^{\prime})\cdot{\bf P}_{\overline{q}}(x,{\bf
p}^{\prime})\right]\right.$ $\displaystyle-\left[\mathbf{P}_{q}(x,{\bf p}-{\bf
p}^{\prime})\cdot\boldsymbol{\epsilon}(\lambda_{2})\right]\left[\mathbf{P}_{\overline{q}}(x,{\bf
p}^{\prime})\cdot\boldsymbol{\epsilon}^{\ast}(\lambda_{1})\right]$
$\displaystyle-\left[\mathbf{P}_{q}(x,{\bf p}-{\bf
p}^{\prime})\cdot\boldsymbol{\epsilon}^{\ast}(\lambda_{1})\right]\left[\mathbf{P}_{\overline{q}}(x,{\bf
p}^{\prime})\cdot\boldsymbol{\epsilon}(\lambda_{2})\right]$
$\displaystyle\left.-i\left[\boldsymbol{\epsilon}^{\ast}(\lambda_{1})\times\boldsymbol{\epsilon}(\lambda_{2})\right]\cdot\left[\mathbf{P}_{q}(x,{\bf
p}-{\bf p}^{\prime})+\mathbf{P}_{\overline{q}}(x,{\bf
p}^{\prime})\right]\right\\}.$

We can simplify the above formula by using the shorthand notation

$\displaystyle Dp^{\prime}$ $\displaystyle\equiv$ $\displaystyle\frac{\Delta
t}{8}g_{V}^{2}m_{V}m_{q}\int\frac{d^{3}\mathbf{p}^{\prime}}{(2\pi\hbar)^{3}}\frac{1}{E_{p^{\prime}}^{\overline{q}}E_{{\bf
p}-{\bf p}^{\prime}}^{q}E_{p}^{V}}$ (92) $\displaystyle\times
f_{\overline{q}}(x,\mathbf{p}^{\prime})f_{q}(x,\mathbf{p}-\mathbf{p}^{\prime})2\pi\hbar\delta\left(E_{p}^{V}-E_{p^{\prime}}^{\overline{q}}-E_{{\bf
p}-{\bf p}^{\prime}}^{q}\right).$

We can put $\rho_{\lambda_{1}\lambda_{2}}^{V}$ into a matrix form

$\displaystyle\rho^{V}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\rho_{11}&\rho_{10}&\rho_{1,-1}\\\
\rho_{1,0}^{*}&\rho_{00}&\rho_{0,-1}\\\
\rho_{1,-1}^{*}&\rho_{0,-1}^{*}&\rho_{-1,-1}\end{array}\right).$ (96)

Note that $\rho^{V}$ is a Hermitian matrix and we have suppressed the index
’V’ in all elements.

For a given spin quantization direction ${\bf n}_{3}$, we can construct
$\boldsymbol{\epsilon}(\lambda)$ as follows

$\displaystyle\boldsymbol{\epsilon}(0)$ $\displaystyle=$ $\displaystyle{\bf
n}_{3},$ $\displaystyle\boldsymbol{\epsilon}(1)$ $\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{2}}\left({\bf n}_{1}+i{\bf n}_{2}\right),$
$\displaystyle\boldsymbol{\epsilon}(-1)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left({\bf n}_{1}-i{\bf n}_{2}\right),$ (97)

where ${\bf n}_{1}$, ${\bf n}_{2}$ and ${\bf n}_{3}$ form orthogonal basis
vectors in the rest frame of the vector meson. From (91) we obtain

$\displaystyle\rho_{11}$ $\displaystyle=$ $\displaystyle\int
Dp^{\prime}\left(1+{\bf n}_{3}\cdot{\bf P}_{q}\right)\left(1+{\bf
n}_{3}\cdot{\bf P}_{\overline{q}}\right),$ $\displaystyle\rho_{10}$
$\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2}}\int
Dp^{\prime}\left\\{\left[({\bf n}_{1}-i{\bf n}_{2})\cdot{\bf
P}_{q}\right]\left(1+{\bf n}_{3}\cdot{\bf P}_{\overline{q}}\right)\right.$
$\displaystyle\left.+\left[({\bf n}_{1}-i{\bf n}_{2})\cdot{\bf
P}_{\overline{q}}\right]\left(1+{\bf n}_{3}\cdot{\bf P}_{q}\right)\right\\},$
$\displaystyle\rho_{1,-1}$ $\displaystyle=$ $\displaystyle\int
Dp^{\prime}\left[({\bf n}_{1}-i{\bf n}_{2})\cdot{\bf P}_{q}\right]\left[({\bf
n}_{1}-i{\bf n}_{2})\cdot{\bf P}_{\overline{q}}\right],$
$\displaystyle\rho_{00}$ $\displaystyle=$ $\displaystyle\int
Dp^{\prime}\left\\{1+{\bf P}_{q}\cdot{\bf P}_{\overline{q}}-2\left({\bf
n}_{3}\cdot{\bf P}_{q}\right)\left({\bf n}_{3}\cdot{\bf
P}_{\overline{q}}\right)\right\\}$ $\displaystyle\rho_{-1,0}$ $\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{2}}\int Dp^{\prime}\left\\{\left[({\bf
n}_{1}+i{\bf n}_{2})\cdot{\bf P}_{q}\right]\left(1-{\bf n}_{3}\cdot{\bf
P}_{\overline{q}}\right)\right.$ $\displaystyle\left.+\left[({\bf n}_{1}+i{\bf
n}_{2})\cdot{\bf P}_{\overline{q}}\right]\left(1-{\bf n}_{3}\cdot{\bf
P}_{q}\right)\right\\},$ $\displaystyle\rho_{-1,-1}$ $\displaystyle=$
$\displaystyle\int Dp^{\prime}\left(1-{\bf n}_{3}\cdot{\bf
P}_{q}\right)\left(1-{\bf n}_{3}\cdot{\bf P}_{\overline{q}}\right),$ (98)

where we have used shorthand notations ${\bf P}_{q}\equiv{\bf P}_{q}(x,{\bf
p}-{\bf p}^{\prime})$ and ${\bf P}_{\overline{q}}\equiv{\bf
P}_{\overline{q}}(x,{\bf p}^{\prime})$. The 00-element of the normalized
density matrix is given by

$\displaystyle\overline{\rho}_{00}$ $\displaystyle=$
$\displaystyle\frac{\rho_{00}}{\rho_{11}+\rho_{00}+\rho_{-1,-1}}$ (99)
$\displaystyle=$ $\displaystyle\frac{\int Dp^{\prime}\left[1+{\bf
P}_{q}\cdot{\bf P}_{\overline{q}}-2\left({\bf n}_{3}\cdot{\bf
P}_{q}\right)\left({\bf n}_{3}\cdot{\bf P}_{\overline{q}}\right)\right]}{\int
Dp^{\prime}\left(3+{\bf P}_{q}\cdot{\bf P}_{\overline{q}}\right)},$

where $N\equiv\int Dp^{\prime}$ is the normalization constant. If the
magnitude of the polarization is much smaller than 1, we can make a Taylor
expansion in it and obtain

$\displaystyle\overline{\rho}_{00}(x,\mathbf{p})$ $\displaystyle\simeq$
$\displaystyle\frac{1}{3}+\frac{2}{9N}\int Dp^{\prime}\left[\left({\bf
n}_{1}\cdot{\bf P}_{q}\right)\left({\bf n}_{1}\cdot{\bf
P}_{\overline{q}}\right)+\left({\bf n}_{2}\cdot{\bf P}_{q}\right)\left({\bf
n}_{2}\cdot{\bf P}_{\overline{q}}\right)\right.$ (100)
$\displaystyle\left.-2\left({\bf n}_{3}\cdot{\bf P}_{q}\right)\left({\bf
n}_{3}\cdot{\bf P}_{\overline{q}}\right)\right].$

If we assume ${\bf P}_{q}$ and ${\bf P}_{\overline{q}}$ are only in the
direction of ${\bf n}_{3}$, i.e.

${\bf n}_{x}\cdot{\bf P}_{q}={\bf n}_{y}\cdot{\bf P}_{q}={\bf n}_{x}\cdot{\bf
P}_{\overline{q}}={\bf n}_{y}\cdot{\bf P}_{\overline{q}}=0,$ (101)

we obtain

$\overline{\rho}_{00}(x,\mathbf{p})\approx\frac{1}{3}-\frac{4}{9N}\int
Dp^{\prime}\left[{\bf n}_{3}\cdot{\bf P}_{q}(x,{\bf p}-{\bf
p}^{\prime})\right]\left[{\bf n}_{3}\cdot{\bf P}_{\overline{q}}(x,{\bf
p}^{\prime})\right].$ (102)

which recovers the similar form to the previous result but expressed in terms
of the weighted integral $Dp^{\prime}$ in (92).

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