# Vector meson’s spin alignments in high energy reactions

Jin-Hui Chen<EMAIL_ADDRESS>Key Laboratory of Nuclear Physics and
Ion-beam Application (MoE), Institute of Modern Physics, Fudan University,
Shanghai 200433, China  Zuo-Tang Liang<EMAIL_ADDRESS>Institute of Frontier
and Interdisciplinary Science, Key Laboratory of Particle Physics and Particle
Irradiation (MOE), Shandong University, Qingdao, Shandong 266237, China  Yu-
Gang Ma<EMAIL_ADDRESS>Key Laboratory of Nuclear Physics and Ion-beam
Application (MoE), Institute of Modern Physics, Fudan University, Shanghai
200433, China  Xin-Li Sheng<EMAIL_ADDRESS>INFN-Firenze, Via Giovanni
Sansone, 1, 50019 Sesto Fiorentino FI, Italy  Qun Wang<EMAIL_ADDRESS>Department of Modern Physics, University of Science and Technology of China,
Hefei, Anhui 230026, China  School of Mechanics and Physics, Anhui University
of Science and Technology, Huainan, Anhui 232001, China

###### Abstract

The global spin alignment of vector mesons has been observed by the STAR
collaboration at the Relativistic Heavy Ion Collider (RHIC) at Brookhaven
National Laboratory (BNL). It provides a unique opportunity to probe the
correlation between the polarized quark and antiquark in the strongly coupled
quark-gluon plasma (sQGP) produced in relativistic heavy ion collisions,
opening a new window to explore the properties of sQGP. In addition, spin
alignments of vector mesons have also been observed in other high-energy
particle collisions. The results seem to be strongly dependent on the
hadronization mechanism, so comprehensive studies are needed. In this article,
we present a brief review of theoretical and experimental advances in the
study of vector meson’s spin alignments in a variety of high-energy particle
collisions, with emphasis on hadronization mechanisms.

##  I Introduction

Quantum Chromodynamics (QCD) is the fundamental theory for strong interaction
that is responsible for binding quarks and gluons together to form protons,
neutrons, and other hadrons. Relativistic Heavy Ion Collisions (RHIC) are a
powerful tool for studying Quantum Chromodynamics (QCD) and properties of
strongly interacting matter under extreme conditions Braun-Munzinger:2015hba ;
Florkowski:2018fap ; Ko ; Rapp ; Zhang1 ; ZhaoJ . In these collisions, heavy
nuclei are accelerated to nearly light speed and then collide to achieve
immense energy density that recreates conditions similar to those just
microseconds after the Big Bang, allowing scientists to study quark-gluon
plasma (QGP) - a state of matter in which quarks and gluons are decoupled from
hadrons Pandav:2022xxx ; HeWB ; Chen:2023mel ; MaYG ; PangLG ; Sun1 ; Chen1 .

Spin is an intrinsic form of angular momentum carried by elementary particles,
and it is a (pseudo-)vector quantity that can point to different directions.
As elementary particles of strong interaction, quarks and gluons carry spins.
As composite particles of strong interaction, many hadrons also carry spins.
The spin polarization refers to the alignment of the spin along a specific
direction (called spin quantization direction). Spin degrees of freedom have
been playing an important role in the development of modern physics since its
discovery in 1925. The global polarization effect (GPE) of the quark gluon
plasma (QGP) produced in high energy heavy-ion collisions is a new spin effect
in particle and nuclear physics. The theoretical prediction Liang:2004ph ;
Liang:2004xn  was made almost two decades ago and attracted immediate
attention Voloshin:2004ha ; Betz:2007kg ; Ipp:2007ng ; Becattini:2007sr ;
Liang:2007ma ; Chen:2007zzq ; Gao:2007bc ; Selyuzhenkov:2007ab ; STAR:2008lcm
; STAR:2007ccu . However the enthusiasm was soon dampened by STAR’s earlier
attempts Chen:2007zzq ; Selyuzhenkov:2007ab ; STAR:2008lcm ; STAR:2007ccu
that gave null results for global $\Lambda$-hyperon polarization as well as
global spin alignments of vector mesons due to limited statistics in data at
that time. The earlier results show that the GPE, even if it exists, could be
very small, so it is unclear if it is within the scope and resolution of
current experiments. The enthusiasm was aroused by non-vanishing results for
the GPE for $\Lambda$ hyperons in STAR’s beam-energy-scan experiments
STAR:2017ckg , which showed that the GPE decreases monotonically with
collision energies. Now the GPE has grown to be a sub-field in heavy-ion
collisions, see Refs. Wang:2017jpl ; Liang:2019clf ; Gao:2020lxh ;
Huang:2020dtn ; Gao:2020vbh ; Becattini:2020ngo ; Becattini:2022zvf ;
Becattini:2024uha  for recent reviews.

Another surprise came five years later from STAR’s measurements on the global
spin alignment of $\phi$ mesons with high statistics STAR:2022fan . The STAR’s
results STAR:2022fan  seem to conflict with hyperon’s polarization as
predicted in Ref. Liang:2004xn  that the vector meson’s spin alignment is
proportional to quark polarization squared and thus should be much smaller
than that observed in STAR’s experiment STAR:2022fan . So STAR’s results are
definitely non-trivial and may have deep implication for hadronization
mechanism and properties of QCD Yang:2017sdk ; Sheng:2019kmk ; Sheng:2022ffb ;
Sheng:2022wsy . When a system of particles exhibits spin polarization, it
means that the spins of these particles are aligned more often in a particular
direction than randomly oriented. Now the global spin alignment of vector
mesons has attracted a broad interest in experimental and theoretical
communities Wang:2023fvy ; Chen:2023hnb ; Xin-Li:2023gwh ; Jian-Hua:2023cna ;
Li-Juan:2023bws ; Chen:2024bik . Spin polarization in the context of QCD is a
critical aspect of understanding the internal structure and dynamics of
hadrons. Through both theoretical and experimental approaches, physicists aim
to unravel the complexities of how quarks and gluons contribute to the spin
and other properties of hadrons. Several theoretical interpretations have been
proposed and new measurements are underway.

In addition to relativistic heavy-ion collisions, the vector meson’s spin
alignment has also been studied in other high energy processes such as
$e^{+}e^{-}$, $e^{-}p$ and $pp$ collisions DELPHI:1997ruo ; OPAL:1997vmw ;
OPAL:1997nwj ; OPAL:1999hxs ; NOMAD:2006kuc . The results show quite different
features and theoretical efforts have also been made to describe them
Anselmino:1997ui ; Anselmino:1998jv ; Anselmino:1999cg ; Xu:2001hz ; Xu:2003fq
; Chen:2016moq ; Chen:2016iey ; Chen:2020pty . It is therefore desirable to
summarize these experimental and theoretical results and make comparison among
different high energy reactions as guidance for future studies.

Firstly, we briefly review spin density matrices for vector mesons as well as
measurement methods for the spin alignment, and then review experimental
results in different high energy collisions. Secondly, we summarize
theoretical approaches in two different hadronization mechanisms as well as
the linear response theory for the spin alignment of vector mesons in
thermalized QGP. Finally, we present a short summary and outlook.

##  II Spin density matrices and measurement methods for spin alignment

###  II.1 Spin density matrices

The spin polarization of particles produced in high energy reaction can be
described by the spin density matrix $\hat{\rho}$. For particles with
spin-$1/2$ such as quarks and anti-quarks, $\hat{\rho}_{q}$ is a $2\times 2$
Hermitian matrix which can be expanded as

$\displaystyle\hat{\rho}_{q}=\frac{1}{2}(1+\vec{P}_{q}\cdot\vec{\sigma}),$ (1)

where $\vec{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$ are Pauli matrices,
and $\vec{P}_{q}={\rm Tr}(\vec{\sigma}\hat{\rho}_{q})$ is the mean spin
polarization vector in the quark’s rest frame.

For spin-1 particles, the spin density matrix is a $3\times 3$ Hermitian
matrix. With the spin quantization axis specified, the spin states are denoted
as $|jm\rangle$ and the spin density matrix can be put into the form
Bacchetta:2000jk

$\displaystyle\hat{\rho}_{V}=$
$\displaystyle\left(\begin{array}[]{ccc}\rho_{11}&\rho_{10}&\rho_{1-1}\\\
\rho_{01}&\rho_{00}&\rho_{0-1}\\\
\rho_{-11}&\rho_{-10}&\rho_{-1-1}\end{array}\right)$ (5) $\displaystyle=$
$\displaystyle\frac{1}{3}+\frac{1}{2}P_{i}\Sigma_{i}+T_{ij}\Sigma_{ij},$ (6)

where $i,j$=1,2,3, and $\Sigma_{i}$ and $\Sigma_{ij}$ are 3$\times$3 traceless
matrices defined as Xin-Li:2023gwh ; Becattini:2024uha

$\displaystyle\Sigma_{1}=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}0&1&0\\\ 1&0&1\\\
0&1&0\end{array}\right),$ (10) $\displaystyle\Sigma_{2}=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}0&-i&0\\\ i&0&-i\\\
0&i&0\end{array}\right),$ (14) $\displaystyle\Sigma_{3}=$
$\displaystyle\left(\begin{array}[]{ccc}1&0&0\\\ 0&0&0\\\
0&0&-1\end{array}\right),$ (18) $\displaystyle\Sigma_{ij}=$
$\displaystyle\frac{1}{2}(\Sigma_{i}\Sigma_{j}+\Sigma_{j}\Sigma_{i})-\frac{2}{3}\delta_{ij}.$
(19)

The polarization vector $\vec{\varepsilon}_{m}$ in the rest frame of the
spin-1 particle in the spin state $|1m\rangle$ can be defined as

$\displaystyle\vec{\varepsilon}_{m=0}=\hat{z},$ (20)
$\displaystyle\vec{\varepsilon}_{m=\pm 1}=\mp\frac{1}{\sqrt{2}}(\hat{x}\pm
i\hat{y}),$ (21)

where the spin quantization direction is assumed to be $\hat{z}$ direction.

The spin vector $\vec{S}$ is related to the polarization vector
$\vec{\varepsilon}$ by $\vec{S}={\rm
Im}({\vec{\varepsilon}_{m}}^{\,*}\times\vec{\varepsilon}_{m})$. If the vector
meson is in the pure spin state with $m=0$, the spin vector is vanishing,
$\vec{S}=\vec{\varepsilon}_{0}\times\vec{\varepsilon}_{0}=0$. If the vector
meson is in the pure spin state with $m=\pm 1$ we have $\vec{S}={\rm
Im}({\vec{\varepsilon}_{\pm 1}}^{\,*}\times\vec{\varepsilon}_{\pm
1})=\pm\hat{z}$, which is parallel or anti-parallel to the spin quantization
direction.

###  II.2 Measurement of spin density matrix through angular distribution of
decay product

In high energy reactions, the spin polarization of a particle $A$ is mainly
measured through the angular distribution of its decay product in two-body
decay $A\to 1+2$ in the rest frame of $A$. The corresponding formulae are
derived using symmetry properties and conservation laws in the decay process.

In the rest frame of $A$, the momenta of the particle $1$ and $2$ are denoted
as $\vec{p}=\vec{p}_{1}=-\vec{p}_{2}$, and their spin states are labeled by
their helicities $\lambda_{1}$ and $\lambda_{2}$ respectively. The decay
amplitude is given by,

$\displaystyle A_{m}(\vec{p};\lambda_{1}\lambda_{2})=$
$\displaystyle\langle\vec{p};\lambda_{1}\lambda_{2}|\hat{U}|j_{A}m_{A}\rangle,$
(22)

where $\hat{U}$ stands for the transition operator, $|j_{A}m_{A}\rangle$
denotes the spin state of $A$, and $|\vec{p};\lambda_{1}\lambda_{2}\rangle$ is
the helicity state of decay daughters.

We can insert the completeness identity for the helicity states of the two-
particle system $|E;jm;\lambda_{1}\lambda_{2}\rangle$ with fixed $E$ (the
energy of the system), $j$ (total angular momentum quantum number) and $m$
(the eigenvalue of $j_{z}$). From energy and angular momentum conservation
laws in the decay process in $A$’s rest frame, we have $E=M_{A}$ (the mass of
$A$), $j=j_{A}$ and $m=m_{A}$. So we obtain,

$\displaystyle A_{m}(\vec{p};\lambda_{1}\lambda_{2})=$
$\displaystyle\langle\vec{p};\lambda_{1}\lambda_{2}|M_{A};j_{A}m_{A};\lambda_{1}\lambda_{2}\rangle$
$\displaystyle\times\langle
M_{A};j_{A}m_{A};\lambda_{1}\lambda_{2}|\hat{U}|j_{A}m_{A}\rangle.$ (23)

The space rotation invariance demands that the helicity amplitude

$\displaystyle\langle
M_{A};j_{A}m_{A};\lambda_{1}\lambda_{2}|\hat{U}|j_{A}m_{A}\rangle\equiv
H_{A}(\lambda_{1},\lambda_{2}),$ (24)

is independent of $m_{A}$. So we obtain,

$\displaystyle A_{m}(\vec{p};\lambda_{1}\lambda_{2})$
$\displaystyle=\langle\vec{p};\lambda_{1}\lambda_{2}|M_{A};j_{A}m_{A};\lambda_{1}\lambda_{2}\rangle
H_{A}(\lambda_{1},\lambda_{2}).$ (25)

We see that the angular dependence is solely given by the inner product
$\langle\vec{p};\lambda_{1}\lambda_{2}|M_{A};j_{A}m_{A};\lambda_{1},\lambda_{2}\rangle$,
which we calculate as follows. First, we calculate it for $\vec{p}$ in
$z$-direction, i.e., $\langle
p,0,0;\lambda_{1}\lambda_{2}|M_{A};j_{A}m_{A};\lambda_{1}\lambda_{2}\rangle$,
which gives a constant $\sqrt{(2j_{A}+1)/4\pi}$ independent of $m_{A}$. We
then transform $|p,0,0;\lambda_{1}\lambda_{2}\rangle$ to
$|p,\theta,\varphi;\lambda_{1}\lambda_{2}\rangle$ in the direction of
$\vec{p}$ by a spatial rotation. The rotation can be achieved by three
successive rotations in Euler angles
$(\alpha,\beta,\gamma)=(\varphi,\theta,-\varphi)$ that rotate
$\vec{p}=(p,0,0)$ to $\vec{p}=(p,\theta,\varphi)$ with the rotation operator
Jacob:1959at

$\hat{R}(\varphi,\theta,-\varphi)=e^{-i\varphi\hat{J}_{z}}e^{-i\theta\hat{J}_{y}}e^{i\varphi\hat{J}_{z}}.$
(26)

Hence, we obtain the inner product as

$\displaystyle\langle\vec{p};\lambda_{1}\lambda_{2}|M_{A};j_{A}m_{A};\lambda_{1}\lambda_{2}\rangle$
$\displaystyle=\sqrt{\frac{2j_{A}+1}{4\pi}}d_{m_{A}\lambda}^{j_{A}*}(\theta)e^{i(m_{A}-\lambda)\varphi},$
(27)

where $d_{mm^{\prime}}^{j}(\theta)=\langle
jm|e^{-i\theta\hat{J}_{y}}|jm^{\prime}\rangle$ is the element of the Wigner
rotation matrix and $\lambda=\lambda_{1}-\lambda_{2}$. We note that the
definition of the rotation operator $\hat{R}$ in (26) introduces an additional
rotation $\gamma=-\varphi$ relative to that in Ref. Chung:1971ri .

The spin density matrix of the system of particles 1 and 2 can be defined as

$\displaystyle\hat{\rho}_{12}=\hat{U}\hat{\rho}^{A}\hat{U}^{\dagger},$ (28)

where $\rho^{A}$ is the spin density matrix of A. Then the angular
distribution is given by

$\displaystyle W(\theta,\varphi)=$ $\displaystyle
N\sum_{\lambda_{1},\lambda_{2}}\langle\vec{p};\lambda_{1},\lambda_{2}|\hat{\rho}_{12}|\vec{p};\lambda_{1},\lambda_{2}\rangle$
$\displaystyle=$ $\displaystyle
N\sum_{\lambda_{1},\lambda_{2};m_{A},m^{\prime}_{A}}|H_{A}(\lambda_{1},\lambda_{2})|^{2}\rho^{A}_{m_{A}m^{\prime}_{A}}$
$\displaystyle\times
e^{i(m_{A}-m^{\prime}_{A})\varphi}d_{m_{A}\lambda}^{j_{A}*}(\theta)d_{m^{\prime}_{A}\lambda}^{j_{A}}(\theta),$
(29)

where $\rho^{A}_{m_{A}m^{\prime}_{A}}=\langle
m_{A}|\hat{\rho}^{A}|m^{\prime}_{A}\rangle$ denotes elements of
$\hat{\rho}^{A}$.

Since the helicity amplitude $H_{A}(\lambda_{1},\lambda_{2})$ is generally
unknown, only in some special cases one can use $W(\theta,\varphi)$ to
determine elements of $\hat{\rho}^{A}$. Here are three such cases.

(1) $j_{A}=1/2$, $\lambda_{1}=\pm 1/2$, $\lambda_{2}=0$, as in a spin-1/2
baryon’s decay into a spin-1/2 baryon and a pion, $H\to B\pi$. In this case,
we have two independent helicity amplitudes $H_{A}(\pm 1/2,0)$ and
$W(\theta,\varphi)$ is given by

$\displaystyle W(\theta,\varphi)=$
$\displaystyle\frac{1}{4\pi}\left(1+\alpha_{A}\vec{P}_{A}\cdot\vec{n}\right),$
(30)

where $\vec{P}_{A}={\rm Tr}(\vec{\sigma}\hat{\rho}_{A})$ is the polarization
vector of $A$, $\vec{n}=\vec{p}/|\vec{p}~{}|$ is the momentum direction and

$\alpha_{A}=\frac{H_{A}(1/2,0)-H_{A}(-1/2,0)}{H_{A}(1/2,0)+H_{A}(-1/2,0)},$
(31)

is the decay parameter for $A\to 1+2$. We also see that if parity is conserved
in the decay process so that $H_{A}(1/2,0)=H_{A}(-1/2,0)$, we then have
$\alpha_{A}=0$ and the isotropic distribution $W(\theta,\varphi)={1}/{4\pi}$.
So Eq. (30) can be used to determine $\vec{P}_{A}$ only in weak decays.

(2) $j_{A}=1$, $\lambda_{1}=\lambda_{2}=0$, such as in the vector meson’s
decay into two pions, $V\to\pi\pi$. In this case, the helicity amplitude
$H_{A}(0,0)$ is a trivial constant that can be absorbed into the normalization
constant, so that

$\displaystyle W(\theta,\varphi)=$
$\displaystyle\frac{3}{8\pi}\Bigl{\\{}(1-\rho^{A}_{00})+(3\rho^{A}_{00}-1)\cos^{2}\theta$
$\displaystyle-{2}\sin^{2}\theta\bigl{[}\cos 2\varphi{\rm
Re}\rho^{A}_{1-1}-\sin 2\varphi{\rm Im}\rho^{A}_{1-1}\bigr{]}$
$\displaystyle-\sqrt{2}\sin 2\theta\bigl{[}\cos\varphi({\rm
Re}\rho^{A}_{10}-{\rm Re}\rho^{A}_{-10})$ $\displaystyle-\sin\varphi({\rm
Im}\rho^{A}_{10}+{\rm Im}\rho^{A}_{-10})\bigr{]}\Bigr{\\}},$ (32)

If we integrate over $\varphi$, we obtain,

$\displaystyle W(\theta)=\frac{3}{4}$
$\displaystyle\Bigl{\\{}(1-\rho^{A}_{00})+(3\rho^{A}_{00}-1)\cos^{2}\theta\Bigr{\\}}.$
(33)

We see that by measuring the distribution in $\theta$ one can extract the
value of $\rho^{A}_{00}$.

(3) $j_{A}=1$, $\lambda_{1}=\pm 1/2,\lambda_{2}=\pm 1/2$, such as in the
vector meson’s decay into a dilepton pair, $V\to e^{+}e^{-}$. Here, we have
four combinations of $\lambda_{1}$ and $\lambda_{2}$. In this case, only if
parity and helicity conservation are valid so that
$H_{A}(-\lambda_{1},-\lambda_{2})=H_{A}(\lambda_{1},\lambda_{2})$ and
$H_{A}(\lambda_{1},\lambda_{2})\not=0$ only for $\lambda_{1}=-\lambda_{2}$,
only one non-vanishing helicity amplitude is left and can be absorbed into the
normalization constant. We then obtain the angular distribution as

$\displaystyle W(\theta,\varphi)=$
$\displaystyle\frac{3}{8\pi(1+\rho^{A}_{00})}\Bigl{\\{}1+\lambda_{\theta}\cos^{2}\theta$
$\displaystyle+\lambda_{\varphi}\sin^{2}\theta\cos
2\varphi+\lambda_{\theta\varphi}\sin 2\theta\sin 2\varphi$
$\displaystyle+\lambda^{\perp}_{\varphi}\sin^{2}\theta\sin
2\varphi+\lambda^{\perp}_{\theta\varphi}\sin 2\theta\sin\varphi\Bigr{\\}},$
(34)

where the $\lambda$ coefficients are related to elements of $\rho^{A}$ by

$\displaystyle\lambda_{\theta}=\frac{1-3\rho^{A}_{00}}{1+\rho^{A}_{00}},$ (35)
$\displaystyle\lambda_{\varphi}=\frac{2{\rm
Re}\rho^{A}_{1-1}}{1+\rho^{A}_{00}},$ (36)
$\displaystyle\lambda_{\theta\varphi}=\frac{\sqrt{2}{\rm
Re}(\rho^{A}_{10}-\rho^{A}_{-10})}{1+\rho^{A}_{00}},$ (37)
$\displaystyle\lambda_{\varphi}^{\perp}=-\frac{2{\rm
Im}\rho^{A}_{1-1}}{1+\rho^{A}_{00}},$ (38)
$\displaystyle\lambda_{\theta\varphi}^{\perp}=-\frac{\sqrt{2}{\rm
Im}(\rho^{A}_{10}+\rho^{A}_{-10})}{1+\rho^{A}_{00}}.$ (39)

In principle, one can extract elements of $\rho^{A}$ from $W(\theta,\varphi)$
through these coefficients.

Measurements on hyperon polarization and vector meson’s spin alignment have
been carried out in different high energy reactions using Eqs. (30-34) (see
e.g. Refs. Lesnik:1975my ; Bunce:1976yb ; Bensinger:1983vc ; Gourlay:1986mf ;
TASSO:1984nda ; ALEPH:1996oew ; OPAL:1997oem , DELPHI:1997ruo ; OPAL:1997vmw ;
OPAL:1997nwj ; OPAL:1999hxs ; NOMAD:2006kuc  and Chen:2007zzq ;
Selyuzhenkov:2007ab ; STAR:2008lcm ; STAR:2007ccu ). In this brief review, we
concentrate on spin alignment of vector mesons with light flavors in
comparison with hyperon polarization, thus only Eqs. (30) and (32) are
involved.

##  III Experimental results

###  III.1 Vector meson’s spin alignments in $e^{+}e^{-}$-annihilations

The earliest measurements of vector meson’s spin alignments in high energy
reactions might be made at the Large Electron-Position collider (LEP) at
European Organization for Nuclear Research (CERN) in the 1990s DELPHI:1997ruo
; OPAL:1997vmw ; OPAL:1997nwj ; OPAL:1999hxs , where one of the popular
polarization axis was the helicity axis defined by the individual particle’s
momentum direction. Measurements have been carried out by DELPHI and OPAL
Collaborations for $K^{*0}$ and $\phi$ mesons and even for heavy flavor meson
such $B$ and $D^{*}$ DELPHI:1997ruo ; OPAL:1997vmw ; OPAL:1997nwj ;
OPAL:1999hxs . They also measured the off-diagonal element such as
$\rho_{1,-1}$. As an example, we show the results from Ref. OPAL:1997vmw  in
Fig. 1.

Figure 1: Experimental data obtained by OPAL Collaboration at LEP for the spin
alignment $\rho_{00}$ and off-diagonal element $\rho_{1-1}$ of $K^{*0}$. This
figure is taken from Ref.OPAL:1997vmw .

From these measurements DELPHI:1997ruo ; OPAL:1997vmw ; OPAL:1997nwj ;
OPAL:1999hxs  in $e^{+}e^{-}$ collisions, we see clearly that $\rho_{00}$ is
significantly larger than 1/3 in the fragmentation region $x_{p}>0.3$
($x_{p}\equiv p/p_{\rm beam}$) which show that the spin of vector mesons is
significantly aligned in that region. However, at small fractional momenta
$x_{p}\leq 0.3$, null results for $K^{*0}$ and $\phi$ mesons were reported in
$e^{+}e^{-}$ collisions DELPHI:1997ruo ; OPAL:1997vmw .

The data have attaracted much theoretical attention Anselmino:1997ui ;
Anselmino:1998jv ; Anselmino:1999cg ; Xu:2001hz ; Xu:2003fq ; Chen:2016moq ;
Chen:2016iey ; Chen:2020pty  and we will come back to this later in Sec. VI.

###  III.2 Global spin alignments of vector mesons in heavy-ion collisions

Measurements of the vector meson’s spin alignment in heavy-ion collisions are
different from conventional studies. A new polarization axis along the
nucleus-nucleus system’s orbital angular momentum can be defined Liang:2004ph
; Liang:2004xn .

Figure 2: Measurements of the vector meson’s spin density matrix element
$\rho_{00}$ as functions of transverse momentum ($p_{\rm T}$) for $K^{*0}$ and
$\phi$ mesons in Au+Au collisions at different energies. Data points are
STAR’s measurements STAR:2022fan , bands are theoretical calculations for
$\phi$ mesons Sheng:2022wsy .

In 2008 the STAR Collaboration performed a measurement of the vector meson’s
global spin alignment in Au+Au collisions at $\sqrt{s_{\rm NN}}$ = 200 GeV.
Due to limited statistics and only covered top RHIC collision energies, no
significant results were reported STAR:2008lcm . Since 2010, the collaboration
has been collecting and analyzing data of higher statistics and at lower
collision energies, including data in the Beam Energy Scan Phase I (BES-I)
runs and high statistics Au+Au runs at $\sqrt{s_{\rm NN}}$ = 200 GeV. The
analysis was focused on mid-central collisions (20-60%) where larger system
angular momenta are expected in comparison with the values in central or
peripheral collisions. Fig. 2 presents the transverse momentum dependence of
$\rho_{00}$ for $K^{*0}$ and $\phi$ in 20-60% central Au+Au collisions at
$\sqrt{s_{\rm NN}}$ = 11.5, 19.6, 27, 39, 62.4 (54.4), 200 GeV. The
$\rho_{00}$ results show a non-trivial $p_{\rm T}$ dependence. For $K^{*0}$
mesons at $\sqrt{s_{\rm NN}}$ = 54.4 and 200 GeV, they are larger than 1/3
with about $2\sigma$ significance at intermediate $p_{\rm T}$. At low beam
energies the statistics is not sufficient to observe any significant deviation
from 1/3. From the calculation in Ref. Liang:2004xn , one naively expect
$\rho_{00}$ to be smaller than 1/3 due to hadronization of the polarized quark
and antiquark via quark combination, and larger than 1/3 due to fragmentation
of the quark and antiquark. On the other hand, for $\phi$ mesons for all
energies considered, we see that the departure of $\rho_{00}$ from 1/3 mainly
occurs at $p_{\rm T}\sim$ 1.0 - 2.4 GeV/c, while at higher $p_{\rm T}$ the
result can be regarded as being consistent with 1/3 within $\sim 2\sigma$ or
less significance.

The STAR collaboration also studied the collision energy dependence by
integrating $\rho_{00}(p_{\rm T})$ with the weight 1/(stat. error)2. Fig. 3
presents $\rho_{00}$ for $K^{*0}$ and $\phi$ mesons in 20-60% central Au+Au
collisions at collision energies ranging from $\sqrt{s_{\rm NN}}$ = 11.5 to
200 GeV STAR:2022fan . We see that $\rho_{00}$ for $\phi$ mesons increases
with decreasing the collision energy, while $\rho_{00}$ for $K^{*0}$ mesons
fluctuates around 1/3 with the collision energy. To quantify the effect, the
average of $\rho_{00}$ is taken over lower collision energies for $K^{*0}$ and
$\phi$. The $\rho_{00}$ for $\phi$ mesons, averaged over beam energies between
11.5 and 62.4 GeV, is 0.3512 $\pm$ 0.0017 (stat.) $\pm$ 0.0017 (syst.). Taking
the total uncertainties as the sum in quadrature of statistical and systematic
uncertainties, the result indicates that $\rho_{00}$ for $\phi$ mesons is
above 1/3 with a significance of $7.4\sigma$, representing the first
observation of the global spin alignment. The $\rho_{00}$ for $K^{*0}$ mesons,
averaged over beam energies between 11.5 and 54.4 GeV, is 0.3356 $\pm$ 0.0034
(stat.) $\pm$ 0.0043 (syst.) and is consistent with 1/3. The measurements of
the ALICE collaboration in Pb+Pb collisions at 2.76 TeV ALICE:2019aid , taken
from the closest data points to the mean $p_{\rm T}$ for the $p_{\rm T}$ range
used in STAR’s measurements, are also shown for comparison in Fig. 3. The
$\rho_{00}$ data point for $K^{*0}$ and $\phi$ mesons from ALICE collaboration
is more or less consistent with 1/3 with large uncertainties.

Figure 3: Measurements of $\rho_{00}$ with respect to the system orbital
angular momentum in high energy heavy-ion collisions. Stars represent the data
for $\phi$ mesons STAR:2022fan ; ALICE:2019aid , circles represent the data
for $K^{*0}$ mesons STAR:2022fan ; ALICE:2019aid . The solid red line is the
prediction from a non-relativistic coalescence model with strong force fields
Sheng:2019kmk . More sophisticated and complete calculations using the
relativistic coalescence model and spin kinetic equation Sheng:2022ffb  were
presented in Ref. Sheng:2022wsy ; Sheng:2023urn .

According to various studies, there are many sources that contribute to the
global spin alignment of $\phi$ mesons including vortical flows Yang:2017sdk ,
electromagnetic fields Yang:2017sdk  generated by the electric currents
carried by the colliding nuclei, local spin alignment Xia:2020tyd , quark
polarization along the direction of its momentum (helicity polarization)
Gao:2021rom , the spin alignment by fragmentation Liang:2004xn  of polarized
quarks, and the shear stress tensor Li:2022vmb ; Wagner:2022gza ; Dong:2023cng
. However, these conventional mechanisms are not sufficient to account for the
observed $\rho_{00}$ for $\phi$ mesons. It was also proposed that local
correlations or fluctuations in turbulent color fields Muller:2021hpe  and
glasma fields Kumar:2023ghs  can also generate a significant contribution to
$\rho_{00}$. A recent theoretical development based on local correlations or
fluctuations of $\phi$ vector fields can describe the experimental data. The
$\phi$ vector field is the ’33’ component of the SU(3) vector multiplet
induced by currents of pseudo-Goldstone bosons Manohar:1983md , and it can
polarize $s$ and $\bar{s}$ quarks in the same way as the electromagnetic field
does. The solid curve of Fig. 3 is such a fit to the data. We see that the
calculation describes the data reasonably well. The correlation of the $\phi$
vector field can be quantified by $\langle{\vec{E}}_{\phi}^{2}\rangle$ and
$\langle{\vec{B}}_{\phi}^{2}\rangle$ and can be extracted by fitting data,
where ${\vec{E}}_{\phi}$ and ${\vec{B}}_{\phi}$ are electric and magnetic
components of the $\phi$ vector field respectively. We will discuss more
details about the theoretical model in Sect. IV.2.

##  IV Vector meson’s spin alignments in quark combination models

It was well-known that huge angular momenta are generated in non-central high
energy heavy-ion collisions Liang:2004ph ; Gao:2007bc . Due to spin-orbit
couplings in QCD interaction, such huge angular momenta lead to the global
polarization in quark scatterings in the form of the hyperon polarization
Liang:2004ph ; Gao:2007bc  and vector meson’s spin alignment Liang:2004xn ,
the so-called GPE.

The GPE depends not only on the global quark polarization but also on the
hadronization mechanism. In relativistic heavy-ion collisions, it is envisaged
that a system of decomfined quarks and anti-quarks is created in the central
rapidity and moderate transverse momentum region. Various aspects of
experimental data suggest that hadronization of the system proceeds via
combination of quarks and anti-quarks, a mechanism phrased as “quark
recombination”, “quark coalescence” or simply as “quark combination”. We
simply refer to it as “quark combination model” or “quark coalescence model”
in this paper. We will summarize the results of vector meson’s spin alignments
by the quark combination model in this section.

###  IV.1 Vector meson’s spin alignments in non-relativistic quark combination
models

In a non-relativistic quark combination model, the physics in the process that
$q$ (quark) and $\bar{q}$ (antiquark) combine to $M$ (meson) can be
demonstrated in a clear way. Here, it is envisaged that in the combination
process the vector meson’s spin is just the sum over the quark’s and anti-
quark’s spins. Hence the spin density matrix and the spin alignment of the
vector meson can be calculated from that of the quark and anti-quark. Such a
calculation is straightforward and was carried out in Ref. Liang:2004xn ,
which we will summarize in this subsection.

The global quark polarization was taken as a constant so that the spin density
matrix for the quark and antiquark takes the diagonal form Liang:2004xn

$\displaystyle\hat{\rho}^{q}=\frac{1}{2}\left(\begin{array}[]{cc}1+P_{q}&0\\\
0&1-P_{q}\end{array}\right).$ (42)

The spin density matrix of the $q_{1}\bar{q}_{2}$ system was taken as a direct
product of the quark’s and anti-quark’s,

$\displaystyle\hat{\rho}^{q_{1}\bar{q}_{2}}=\hat{\rho}^{q_{1}}\otimes\hat{\rho}^{\bar{q}_{2}}.$
(43)

The elements of the spin density matrix $\hat{\rho}^{V}$ for the vector meson
is obtained from $\hat{\rho}^{q_{1}\bar{q}_{2}}$ as

$\displaystyle\rho^{V}_{m^{\prime}m}=\langle
jm^{\prime}|\hat{\rho}^{q_{1}\bar{q}_{2}}|jm\rangle,$ (44)

which leads to

$\displaystyle\rho^{V}_{m^{\prime}m}=$
$\displaystyle\sum_{m_{1},m_{2},m_{1}^{\prime},m_{2}^{\prime}}\rho^{q_{1}\bar{q}_{2}}_{m_{1}^{\prime}m_{2}^{\prime},m_{1}m_{2}}$
$\displaystyle\times\langle
j_{V}m^{\prime}|m^{\prime}_{1}m^{\prime}_{2}\rangle\langle
m_{1}m_{2}|j_{V}m\rangle,$ (45)

where $|j_{V}m\rangle$ is the vector meson’s spin state in the constituent
quark model with $j_{V}=1$ and $m=0,\pm 1$, and $\langle
j_{V}m|m_{1}m_{2}\rangle$ are Clebsch-Gordan coefficients. After a
straightforward calculation, we obtain the normalized spin alignment
$\rho^{V}_{00}$ as Liang:2004xn ,

$\rho^{V}_{00}=\frac{1-P_{q_{1}}P_{\bar{q}_{2}}}{3+P_{q_{1}}P_{\bar{q}_{2}}}.$
(46)

If we take $P_{q_{1}}=P_{\bar{q}_{2}}=P_{q}$ (flavor blind for quarks and
antiquarks), we simply obtain

$\rho^{V}_{00}=\frac{1-P_{q}^{2}}{3+P_{q}^{2}}.$ (47)

In exact the same way, we obtain the global hyperon polarization $P_{H}=P_{q}$
for $\Lambda$, $\Sigma$, and $\Xi$ Liang:2004ph .

From Eqs. (46) and (47), we see that, in contrast to $P_{H}$, $\rho^{V}_{00}$
is quadratic in $P_{q}$ and should be less than $1/3$. We emphasize that the
case considered in Ref. Liang:2004xn  is over simplified in the fact that only
the spin degree of freedom for the quark and antiquark is considered,
neglecting other degree(s) of freedom and the correlation among the quark’s
and antiquark’s polarization. So it might not be a surprise that the STAR data
STAR:2017ckg ; STAR:2022fan  for the global spin alignment of $\phi$ mesons
show a large deviation from 1/3, far beyond the value estimated from
$P_{\Lambda}$’s data by its square. In fact, if we make a step forward by
considering the dependence of $P_{q}$ on other degree of freedom Sheng:2019kmk
; Sheng:2020ghv ; Sheng:2022ffb ; Sheng:2022wsy  $P_{q_{1}}P_{\bar{q}_{2}}$ in
Eq. (46) should be replaced by $\langle P_{q_{1}}P_{\bar{q}_{2}}\rangle$, so
we have

$\rho^{V}_{00}=\frac{1-\langle P_{q_{1}}P_{\bar{q}_{2}}\rangle}{3+\langle
P_{q_{1}}P_{\bar{q}_{2}}\rangle}.$ (48)

The STAR data STAR:2017ckg ; STAR:2022fan  indicate

$\langle P_{q_{1}}P_{\bar{q}_{2}}\rangle\not=\langle P_{q_{1}}\rangle\langle
P_{\bar{q}_{2}}\rangle.$ (49)

This means that there should be a strong correlation between the polarization
of the quark and antiquark. Hence, the study of the global spin alignment for
the vector meson in heavy-ion collisions provides a unique opportunity for
exploring the local correlation in the quark’s and antiquark’s polarization, a
new window for the study of properties of QGP.

It is also emphasized in Ref. Liang:2023talk  that the average in Eq. (48) is
two-folded: the first average is taken inside the vector meson’s wave function
(local correlation) and the second average is taken in the range outside the
vector meson’s wave function (long-range correlation). More measurements are
needed to tell the difference between two types of correlation Liang:2023talk
. A systematic formulation has been presented in Lv:2024uev . It has bee
extended to spin 3/2 baryons in Zhang:2024hyq .

###  IV.2 Vector meson’s spin alignments in quark combination models with
phase space dependence

####  IV.2.1 Non-relativistic model

In this subsection, we consider the non-relativistic quark combination model
with phase space dependence Sheng:2020ghv ; Xin-Li:2023gwh . We extend the
density operator (1) to the phase space as

$\displaystyle\hat{\rho}^{q}=$
$\displaystyle\sum_{rs}\int[d^{3}{\vec{p}}][d^{3}{\vec{q}}]\int
d^{3}{\vec{x}}e^{-i{\vec{q}}\cdot{\vec{x}}}$ $\displaystyle\times
w_{rs}^{q}({\vec{x}},{\vec{p}})\left|r,{\vec{p}}+\frac{{\vec{q}}}{2}\right\rangle\left\langle
s,{\vec{p}}-\frac{{\vec{q}}}{2}\right|\,,$ (50)

where $r,s=\pm$ denote spin states, $\left|r,{\vec{p}}\right\rangle$ denotes
the spin-momentum state, and the momentum measure is defined as
$[d^{3}{\vec{p}}]\equiv d^{3}{\vec{p}}/(2\pi)^{3}$. The weight function
$w_{rs}^{q}({\vec{x}},{\vec{p}})$ is the Wigner function in the spin-phase
space and can be obtained as

$w_{rs}^{q}({\vec{x}},{\vec{p}})=\int[d^{3}{\vec{q}}]e^{i{\vec{q}}\cdot{\vec{x}}}\left\langle
r,{\vec{p}}+\frac{{\vec{q}}}{2}\right|\hat{\rho}^{q}\left|s,{\vec{p}}-\frac{{\vec{q}}}{2}\right\rangle\,.$
(51)

It is a $2\times 2$ Hermitian matrix in the spin space and can be
parameterized as

$w_{rs}^{q}({\vec{x}},{\vec{p}})=\frac{1}{2}f_{q}({\vec{x}},{\vec{p}})\left[\delta_{rs}+{\vec{\sigma}}_{rs}\cdot{\vec{P}}_{q}({\vec{x}},{\vec{p}})\right]\,.$
(52)

Here $f_{q}({\vec{x}},{\vec{p}})$ is the un-polarized distribution function,
and ${\vec{P}}_{q}({\vec{x}},{\vec{p}})$ is the spin polarization in phase
space. The density operator for the anti-quark can be defined in the same way.
The density operator for a quark-antiquark pair $q_{1}\overline{q}_{2}$ is
defined from the direct product of quark’s and anti-quark’s one as
$\hat{\rho}^{q_{1}\bar{q}_{2}}\equiv\hat{\rho}^{q_{1}}\otimes\hat{\rho}^{\bar{q}_{2}}$.
The Wigner function can be normalized as

$\sum_{s}\int
d^{3}{\vec{x}}\int[d^{3}{\vec{p}}]w_{ss}({\vec{x}},{\vec{p}})=1.$ (53)

The quark’s spin quantization direction can be any direction, e.g., the
$z$-direction, without loss of generality.

The elements of the vector meson’s density matrix are obtained as

$\displaystyle\rho_{m^{\prime}m}^{V}({\vec{x}},{\vec{p}})$ (54)
$\displaystyle=$
$\displaystyle\frac{1}{N({\vec{x}},{\vec{p}})}\int[d^{3}{\vec{q}}]e^{i{\vec{q}}\cdot{\vec{x}}}$
$\displaystyle\times\left\langle
j_{V}m^{\prime};{\vec{p}}+\frac{{\vec{q}}}{2}\right|\hat{\rho}^{q_{1}}\otimes\hat{\rho}^{\bar{q}_{2}}\left|j_{V}m;{\vec{p}}-\frac{{\vec{q}}}{2}\right\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{N({\vec{x}},{\vec{p}})}\int
d^{3}{\vec{x}}_{b}[d^{3}{\vec{p}}_{b}][d^{3}{\vec{q}}_{b}]\exp(-i{\vec{q}}_{b}\cdot{\vec{x}}_{b})$
$\displaystyle\times\varphi_{V}^{\ast}\left({\vec{p}}_{b}+\frac{{\vec{q}}_{b}}{2}\right)\varphi_{V}\left({\vec{p}}_{b}-\frac{{\vec{q}}_{b}}{2}\right)$
$\displaystyle\times\sum_{r_{1},r_{2},s_{1},s_{2}}w_{m_{1}^{\prime}m_{1}}^{q_{1}}({\vec{x}}_{1},{\vec{p}}_{1})w_{m_{2}^{\prime}m_{2}}^{\bar{q}_{2}}({\vec{x}}_{2},{\vec{p}}_{2})$
$\displaystyle\times\left\langle j_{V}m^{\prime}\mid
m_{1}^{\prime}m_{2}^{\prime}\right\rangle\left\langle m_{1}m_{2}\mid
j_{V}m\right\rangle\,,$

where ${\vec{x}}_{1,2}\equiv{\vec{x}}\pm{\vec{x}}_{b}/2$ and
${\vec{p}}_{1,2}\equiv{\vec{p}}/2\pm{\vec{p}}_{b}$ are positions and momenta
for the quark $q_{1}$ and the antiquark $\overline{q}_{2}$ respectively, and
$N({\vec{x}},{\vec{p}})$ is the normalization factor that ensures
$\sum_{m=0,\pm 1}\rho_{mm}^{V}({\vec{x}},{\vec{p}})=1$. Since we work in the
non-relativistic limit, the spin and momentum can be decoupled. Therefore
Clebsch-Gordan coefficients $\left\langle m_{1}m_{2}\mid j_{V}m\right\rangle$
and $\left\langle j_{V}m^{\prime}\mid
m_{1}^{\prime}m_{2}^{\prime}\right\rangle$ as well as the meson’s wave
function $\varphi_{V}$ appear in Eq. (54). By further neglecting the phase-
space dependence of unpolarized distribution functions and choosing
$\varphi_{V}$ as a Gaussian distribution with the width $a_{V}$ in the
momentum space, we derive the spin alignment and other parameters in the
angular distribution (32),

$\displaystyle\rho_{00}^{V}({\vec{x}},{\vec{p}})\approx\frac{1}{3}-\frac{2}{3}\left\langle
P_{q_{1}}^{z}P_{\bar{q}_{2}}^{z}\right\rangle_{V}+\frac{2}{9}\left\langle{\vec{P}}_{q_{1}}\cdot{\vec{P}}_{\bar{q}_{2}}\right\rangle_{V}\,,$
$\displaystyle\text{Re}\rho_{1,-1}^{V}({\vec{x}},{\vec{p}})\approx\frac{1}{3}\left\langle
P_{q_{1}}^{x}P_{\bar{q}_{2}}^{x}-P_{q_{1}}^{y}P_{\bar{q}_{2}}^{y}\right\rangle_{V},$
$\displaystyle-\text{Im}\rho_{1,-1}^{V}({\vec{x}},{\vec{p}})\approx\frac{1}{3}\left\langle
P_{q_{1}}^{x}P_{\bar{q}_{2}}^{y}+P_{q_{1}}^{y}P_{\bar{q}_{2}}^{x}\right\rangle_{V},$
$\displaystyle\text{Re}\left[\rho_{1,0}-\rho_{-1,0}\right]({\vec{x}},{\vec{p}})\approx\frac{\sqrt{2}}{3}\left\langle
P_{q_{1}}^{z}P_{\bar{q}_{2}}^{x}+P_{q_{1}}^{x}P_{\bar{q}_{2}}^{z}\right\rangle_{V},$
$\displaystyle-\text{Im}\left[\rho_{1,0}+\rho_{-1,0}\right]({\vec{x}},{\vec{p}})\approx\frac{\sqrt{2}}{3}\left\langle
P_{q_{1}}^{y}P_{\bar{q}_{2}}^{z}+P_{q_{1}}^{z}P_{\bar{q}_{2}}^{y}\right\rangle_{V},$

where the correlation between quark and antiquark’s polarizations is defined
as

$\displaystyle\left\langle
P_{q_{1}}^{i}P_{\bar{q}_{2}}^{j}\right\rangle_{V}\equiv$
$\displaystyle\frac{1}{\pi^{3}}\int
d^{3}{\vec{x}}_{b}d^{3}{\vec{p}}_{b}\exp\left(-\frac{{\vec{p}}_{b}^{2}}{a_{V}^{2}}-a_{V}^{2}{\vec{x}}_{b}^{2}\right)$
$\displaystyle\times
P_{q_{1}}^{i}({\vec{x}}_{1},{\vec{p}}_{1})P_{\bar{q}_{2}}^{j}({\vec{x}}_{2},{\vec{p}}_{2})\,.$
(56)

We note that in Eq. (LABEL:eq:tensor-polar) the spin quantization direction is
set to the $z$-direction.

####  IV.2.2 Relativistic model in quantum kinetic approach

For a relativistic quark or antiquark, its polarization four-vector is always
perpendicular to its four-momentum, implying that the spin with momentum
cannot be decoupled as in the non-relativistic case. Consequently, the vector
meson’s spin cannot be obtained by the constituent quark’s and anti-quark’s
spins through the angular momentum coupling in non-relativistic quantum
mechanics. Based on the Kadanoff-Baym equation in the closed-time-path
formalism, a relativistic spin kinetic theory for vector mesons has been
constructed Sheng:2022ffb  to explain experimental data on the spin alignment
STAR:2017ckg .

In order to describe spin transport phenomena, we use the matrix-valued spin-
dependent distribution (MVSD) $f_{m_{1}m_{2}}^{V}(x,{\vec{p}})$ for the vector
meson. The spin Boltzmann equation with coalescence and dissociation collision
terms reads,

$\displaystyle p\cdot\partial_{x}f_{m_{1}m_{2}}^{V}(x,{\vec{p}})$ (57)
$\displaystyle=$
$\displaystyle\frac{1}{16}\sum_{m_{1}^{\prime},m_{2}^{\prime}}\left[\epsilon_{\mu}^{\ast}(m_{1},{\vec{p}})\epsilon_{\nu}(m_{1}^{\prime},{\vec{p}})\delta_{m_{2}m_{2}^{\prime}}\right.$
$\displaystyle\left.+\delta_{m_{1}m_{1}^{\prime}}\epsilon_{\mu}^{\ast}(m_{2}^{\prime},{\vec{p}})\epsilon_{\nu}(m_{2},{\vec{p}})\right]$
$\displaystyle\times\mathcal{C}_{m_{1}^{\prime}m_{2}^{\prime}}^{\mu\nu}(x,{\vec{p}}),$

where $\epsilon^{\mu}(m,{\vec{p}})$ denotes the meson’s normalized
polarization vector perpendicular to $p^{\mu}$, and $m_{1}$, $m_{2}$,
$m_{1}^{\prime}$, and $m_{2}^{\prime}$ label the vector meson’s spin states
along the spin quantization direction with three values 0 and $\pm 1$. Here we
have assumed that the vector meson is on its mass-shell
$p^{0}=\sqrt{m_{V}^{2}+{\vec{p}}^{2}}$. The gain and loss terms in
$\mathcal{C}_{m_{1}^{\prime}m_{2}^{\prime}}^{\mu\nu}$ correspond to the
coalescence and dissociation processes for the quark and antiquark,
respectively. In the dilute limit when distribution functions of the meson,
the constituent quark and antiquark are much less than unity, Eq. (57) is
simplified as

$\displaystyle p\cdot\partial_{x}f_{m_{1}m_{2}}^{V}(x,{\vec{p}})=$
$\displaystyle\frac{1}{8}\left[\epsilon_{\mu}^{\ast}(m_{1},{\vec{p}})\epsilon_{\nu}(m_{2},{\vec{p}})\mathcal{C}_{\text{coal}}^{\mu\nu}(x,{\vec{p}})\right.$
$\displaystyle\left.-\mathcal{C}_{\text{diss}}({\vec{p}})f_{m_{1}m_{2}}^{V}(x,{\vec{p}})\right],$
(58)

where the contribution from the dissociation is proportional to the MVSD with
the coefficient $\mathcal{C}_{\text{diss}}$ being independent of the MVSD. The
coalescence kernel $\mathcal{C}_{\text{coal}}^{\mu\nu}$ can be obtained from
the Kadanoff-Baym equation,

$\displaystyle\mathcal{C}_{\text{coal}}^{\mu\nu}(x,{\vec{p}})$
$\displaystyle=$
$\displaystyle\int\frac{d^{3}{\vec{p}}^{\prime}}{(2\pi)^{2}}\frac{\delta(E_{{\vec{p}}}^{V}-E_{{\vec{p}}^{\prime}}^{\bar{q}}-E_{{\vec{p}}-{\vec{p}}^{\prime}}^{q})}{E_{{\vec{p}}^{\prime}}^{\bar{q}}E_{{\vec{p}}-{\vec{p}}^{\prime}}^{q}}$
(59) $\displaystyle\times\text{Tr}\left\\{\Gamma^{\nu}(p^{\prime}\cdot\gamma-
M_{\bar{q}})\right.$ $\displaystyle\times[1+\gamma_{5}\gamma\cdot
P_{\bar{q}}(x,{\vec{p}}^{\prime})]$
$\displaystyle\times\Gamma^{\mu}[(p-p^{\prime})\cdot\gamma+M_{q}]$
$\displaystyle\times\left.[1+\gamma_{5}\gamma\cdot
P_{q}(x,{\vec{p}}-{\vec{p}}^{\prime})]\right\\}$ $\displaystyle\times
f_{\bar{q}}(x,{\vec{p}}^{\prime})f_{q}(x,{\vec{p}}-{\vec{p}}^{\prime})\,,$

where $M_{q}$ and $M_{\bar{q}}$ are quark and antiquark masses respectively,
$E_{{\vec{p}}}^{V}$, $E_{{\vec{p}}-{\vec{p}}^{\prime}}^{q}$, and
$E_{{\vec{p}}^{\prime}}^{\bar{q}}$ are energies for the vector meson, the
constituent quark and antiquark respectively, and $\Gamma^{\mu}$ is the
effective vertex of quark-antiquark-meson. Here $f_{q/\bar{q}}$ and
$P_{q/\bar{q}}^{\mu}$ are the unpolarized and spin polarization distributions
for the quark/antiquark respectively. By neglecting the space dependence (the
distributions are homogeneous in space) and setting $f_{m_{1}m_{2}}^{V}=0$
before the hadronization stage in heavy-ion collisions, one can obtain a
formal solution to Eq. (58). Spin density matrix elements can be derived as
Sheng:2022wsy ,

$\rho_{m_{1}m_{2}}^{V}(x,{\vec{p}})=\frac{\epsilon_{\mu}^{\ast}(m_{1},{\vec{p}})\epsilon_{\nu}(m_{2},{\vec{p}})\mathcal{C}_{\text{coal}}^{\mu\nu}(x,{\vec{p}})}{\sum_{m=0,\pm
1}\epsilon_{\alpha}^{\ast}(m,{\vec{p}})\epsilon_{\beta}(m,{\vec{p}})\mathcal{C}_{\text{coal}}^{\alpha\beta}(x,{\vec{p}})}.$
(60)

This is a relativistic formula for the vector meson’s spin density matrix
built from the coalescence kernel (59) encoding the polarization distributions
of the constituent quark and antiquark. Clearly the vector meson’s spin
density matrix has non-trivial dependence on the spin polarization and momenta
of the quark and antiquark.

We consider that $s$ and $\bar{s}$ are polarized in a thermal medium by an
effective vector field called the $\phi$ field Sheng:2019kmk ; Sheng:2022wsy ;
Sheng:2022ffb  induced by currents of pseudo-Goldstone bosons Manohar:1983md .
The spin polarization of $s$ and $\bar{s}$ is then given by

$P_{q/\bar{q}}^{\mu}(x,{\vec{p}})\approx\pm\frac{g_{\phi}}{4M_{s}(u\cdot
p)T_{\text{h}}}\epsilon^{\mu\nu\alpha\beta}p_{\nu}F_{\alpha\beta}^{\phi}(x),$
(61)

where $u^{\mu}$ denotes the velocity of the thermal background at the
effective hadronization temperature $T_{\text{h}}$, $g_{\phi}$ is the
effective coupling constant of the $s\bar{s}\phi$ vertex, and
$F_{\alpha\beta}^{\phi}(x)$ is the $\phi$ field’s strength tensor.
Substituting (61) into Eq. (59) and assuming $u^{\mu}=(1,0,0,0)$ in the
meson’s rest frame, we obtain the spin alignment of the $\phi$ meson

$\displaystyle\rho_{00}^{\phi}(x,{\vec{p}})$ $\displaystyle\approx$
$\displaystyle\frac{1}{3}-\frac{4g_{\phi}^{2}}{M_{\phi}^{2}T_{\text{h}}}C_{1}\left[\frac{1}{3}{\vec{B}}_{\phi}^{\prime}\cdot{\vec{B}}_{\phi}^{\prime}-({\vec{\epsilon}}_{0}\cdot{\vec{B}}_{\phi}^{\prime})^{2}\right]$
$\displaystyle-\frac{4g_{\phi}^{2}}{M_{\phi}^{2}T_{\text{h}}}C_{2}\left[\frac{1}{3}{\vec{E}}_{\phi}^{\prime}\cdot{\vec{E}}_{\phi}^{\prime}-({\vec{\epsilon}}_{0}\cdot{\vec{E}}_{\phi}^{\prime})^{2}\right],$
(62)

where $M_{\phi}$ is the mass of the $\phi$ meson, and
${\vec{B}}_{\phi}^{\prime}$ and ${\vec{E}}_{\phi}^{\prime}$ are electric and
magnetic components of the $\phi$ field in the meson’s rest frame
respectively. Here ${\vec{\epsilon}}_{0}$ denotes the spin quantization
direction. The coefficients $C_{1}$ and $C_{2}$ depend only on quark’s and
meson’s masses,

$\displaystyle C_{1}$ $\displaystyle=$
$\displaystyle\frac{8M_{s}^{4}+16M_{s}^{2}M_{\phi}^{2}+3M_{\phi}^{4}}{120M_{s}^{2}(M_{\phi}^{2}+2M_{s}^{2})},$
$\displaystyle C_{2}$ $\displaystyle=$
$\displaystyle\frac{8M_{s}^{4}-14M_{s}^{2}M_{\phi}^{2}+3M_{\phi}^{4}}{120M_{s}^{2}(M_{\phi}^{2}+2M_{s}^{2})}.$
(63)

We note that the momentum dependence can be obtained by expressing
${\vec{B}}_{\phi}^{\prime}$ and ${\vec{E}}_{\phi}^{\prime}$ in Eq. (62) in
terms of laboratory frame fields through Lorentz transformation.

$\displaystyle{\vec{B}}_{\phi}^{\prime}$ $\displaystyle=$
$\displaystyle\gamma{\vec{B}}_{\phi}-\gamma{\vec{v}}\times{\vec{E}}_{\phi}+(1-\gamma)\frac{{\vec{v}}\cdot{\vec{B}}_{\phi}}{v^{2}}{\vec{v}},$
$\displaystyle{\vec{E}}_{\phi}^{\prime}$ $\displaystyle=$
$\displaystyle\gamma{\vec{E}}_{\phi}+\gamma{\vec{v}}\times{\vec{B}}_{\phi}+(1-\gamma)\frac{{\vec{v}}\cdot{\vec{E}}_{\phi}}{v^{2}}{\vec{v}},$
(64)

where $\gamma=E_{{\vec{p}}}^{\phi}/M_{\phi}$ is the Lorentz factor and
${\vec{v}}={\vec{p}}/E_{{\vec{p}}}^{\phi}$ is the velocity of the $\phi$
meson. Then $\rho_{00}^{\phi}$ can be rewritten as

$\displaystyle\rho_{00}^{\phi}(x,{\vec{p}})$ $\displaystyle\approx$
$\displaystyle\frac{1}{3}-\frac{4g_{\phi}^{2}}{3M_{\phi}^{4}T_{\mathrm{h}}^{2}}\sum_{i=1,2,3}I_{B,i}({\vec{p}})({\vec{B}}_{i}^{\phi})^{2}$
(65)
$\displaystyle-\frac{4g_{\phi}^{2}}{3M_{\phi}^{4}T_{\mathrm{h}}^{2}}\sum_{i=1,2,3}I_{E,i}({\vec{p}})({\vec{E}}_{i}^{\phi})^{2},$

where $I_{B,i}({\vec{p}})$ and $I_{E,i}({\vec{p}})$ are momentum functions.
One can further take the space and momentum average of
$\rho_{00}^{\phi}(x,{\vec{p}})$

$\displaystyle\left\langle\rho_{00}^{\phi}(x,{\vec{p}})\right\rangle_{x,{\vec{p}}}$
$\displaystyle\approx$
$\displaystyle\frac{1}{3}-\frac{4g_{\phi}^{2}}{3M_{\phi}^{4}T_{\mathrm{h}}^{2}}\sum_{i=1,2,3}\left\langle
I_{B,i}({\vec{p}})\right\rangle_{{\vec{p}}}\left\langle({\vec{B}}_{i}^{\phi})^{2}\right\rangle_{x}$
$\displaystyle-\frac{4g_{\phi}^{2}}{3M_{\phi}^{4}T_{\mathrm{h}}^{2}}\sum_{i=1,2,3}\left\langle
I_{E,i}({\vec{p}})\right\rangle_{{\vec{p}}}\left\langle({\vec{E}}_{i}^{\phi})^{2}\right\rangle_{x},$
(66)

where the momentum average is defined as

$\left\langle O({\vec{p}})\right\rangle_{\vec{p}}=\frac{\int
d^{3}{\vec{p}}(E_{{\vec{p}}}^{\phi})^{-1}O({\vec{p}})f_{\phi}({\vec{p}})}{\int
d^{3}{\vec{p}}(E_{\vec{p}}^{\phi})^{-1}f_{\phi}({\vec{p}})},$ (67)

if we want to obtain momentum-integrated data for $\rho_{00}^{\phi}$. Here
$f_{\phi}({\vec{p}})$ is its momentum distribution which may contain
information about collective flows such as $v_{1}$ and $v_{2}$. If we want to
obtain the transverse momentum spectra of $\rho_{00}^{\phi}$, we have to
integrate over the azimuthal angle and rapidity and keep $p_{T}$, i.e. to
replace $d^{3}{\vec{p}}/E_{p}^{\phi}$ in (67) by $dyd\varphi$. We see in Eq.
(67) that the average $\rho_{00}^{\phi}$ depends on the space average field
squared which quantifies the field fluctuation. The theoretical results for
$\left\langle\rho_{00}^{\phi}\right\rangle$ as functions of transverse
momenta, collision energies and centralities are presented in Ref.
Sheng:2022wsy , which are in a good agreement with recent STAR data
STAR:2022fan .

##  V Linear response theory for spin alignment of vector mesons in thermal
media

In this section, we will show how to calculate the spin alignment of the
vector meson from the Kubo formula in linear response theory
zubarev1996statistical ; zubarev1997statistical ; Zubarev_1979 ;
Kapusta:2006pm  in thermalized QGP. The detailed discussion of this topic is
in Ref. Dong:2023cng .

The Closed-Time-Path (CTP) formalism is a field-theory based method for many-
particle systems in off-equilibrium as well in equilibrium Chou:1984es ;
Blaizot:2001nr ; Wang:2001dm ; Berges:2004yj ; Cassing:2008nn ;
Crossley:2015evo . When it is used for systems in equilibrium, it is actually
the real time formalism of the thermal field theory Kapusta:2006pm ;
Kapusta:2023eix . Wigner functions can be obtained from two-point Green’s
functions on the CTP Sheng:2021kfc ; Sheng:2022ffb ; Hidaka:2022dmn ;
Dong:2023cng ; Becattini:2024uha . The on-shell Wigner function for the vector
meson is related to its MVSD (proportional to the spin density matrix) and
defined as

$\displaystyle W^{\mu\nu}(x,p_{\mathrm{on}})=$
$\displaystyle\frac{E_{p}}{\pi}\int_{0}^{\infty}dp_{0}G_{<}^{\mu\nu}(x,p)$
$\displaystyle=$
$\displaystyle\sum_{m_{1},m_{2}}\epsilon^{\mu}\left(m_{1},{\vec{p}}\right)\epsilon^{\nu\ast}\left(m_{2},{\vec{p}}\right)f_{m_{1}m_{2}}^{V}(x,{\vec{p}}),$
(68)

where $G_{<}^{\mu\nu}(x,p)$ is the “$+-$” component of two-point Green’s
functions on the CTP, and $f_{V}=\mathrm{Tr}(f_{V})\hat{\rho}^{V}$ with
$f_{V}\equiv f_{m_{1}m_{2}}^{V}$ being the short-hand notation for the MVSD,
$\mathrm{Tr}(f_{V})=\sum_{m}f_{mm}^{V}$ being the trace of the MVSD, and
$\hat{\rho}_{V}$ being the spin density matrix in Eq. (6). One can check that
$W^{\mu\nu}(x,p_{\mathrm{on}})$ is always transverse to the on-shell momentum,
$p_{\mu}^{\mathrm{on}}W^{\mu\nu}(x,p_{\mathrm{on}})=0$. The on-shell Wigner
function can be decomposed into the scalar ($\mathcal{S}$), polarization
($W^{[\mu\nu]}$) and tensor polarization ($\mathcal{T}^{\mu\nu}$) parts as
Sheng:2023chinphyb ; Becattini:2024uha

$\displaystyle W^{\mu\nu}(x,p_{\mathrm{on}})=$ $\displaystyle
W^{[\mu\nu]}+W^{(\mu\nu)}$ $\displaystyle=$
$\displaystyle-\frac{1}{3}\Delta^{\mu\nu}(p_{\mathrm{on}})\mathcal{S}+W^{[\mu\nu]}+\mathcal{T}^{\mu\nu},$
(69)

where each part is defined as

$\displaystyle W^{[\mu\nu]}\equiv$
$\displaystyle\frac{1}{2}(W^{\mu\nu}-W^{\nu\mu}),$ $\displaystyle
W^{(\mu\nu)}\equiv$ $\displaystyle\frac{1}{2}(W^{\mu\nu}+W^{\nu\mu}),$
$\displaystyle\mathcal{T}^{\mu\nu}\equiv$ $\displaystyle
W^{(\mu\nu)}+\frac{1}{3}\Delta^{\mu\nu}(p_{\mathrm{on}})\mathcal{S}.$ (70)

With Eq. (69) one can show that both $W^{[\mu\nu]}$ and $\mathcal{T}^{\mu\nu}$
are traceless, $g_{\mu\nu}W^{[\mu\nu]}=g_{\mu\nu}\mathcal{T}^{\mu\nu}=0$.
Using Eqs. (6) and (68), we obtain

$\displaystyle\mathcal{S}=$
$\displaystyle\mathrm{Tr}(f_{V})=-\Delta^{\mu\nu}(p_{\mathrm{on}})W_{\mu\nu},$
$\displaystyle W^{[\mu\nu]}=$
$\displaystyle\frac{1}{2}\mathrm{Tr}(f_{V})\sum_{\lambda_{1},\lambda_{2}}\epsilon^{\mu}\left(\lambda_{1},{\vec{p}}\right)\epsilon^{\nu\ast}\left(\lambda_{2},{\vec{p}}\right)P_{i}\Sigma_{\lambda_{1}\lambda_{2}}^{i},$
$\displaystyle\mathcal{T}^{\mu\nu}=$
$\displaystyle\mathrm{Tr}(f_{V})\sum_{\lambda_{1},\lambda_{2}}\epsilon^{\mu}\left(\lambda_{1},{\vec{p}}\right)\epsilon^{\nu\ast}\left(\lambda_{2},{\vec{p}}\right)T_{ij}\Sigma_{\lambda_{1}\lambda_{2}}^{ij}.$
(71)

We see that $W^{[\mu\nu]}$ is related to $P_{i}$ while $\mathcal{T}^{\mu\nu}$
is related to $T_{ij}$.

We can extract $f_{00}\propto\rho_{00}$ by projecting

$L^{\mu\nu}(p_{\mathrm{on}})=\epsilon^{\mu,*}\left(0,{\vec{p}}\right)\epsilon^{\nu}\left(0,{\vec{p}}\right)+\frac{1}{3}\Delta^{\mu\nu}(p_{\mathrm{on}}),$
(72)

onto $W^{\mu\nu}$ in Eq. (68) as

$\displaystyle L_{\mu\nu}(p_{\mathrm{on}})W^{\mu\nu}$ $\displaystyle=$
$\displaystyle\sum_{\lambda_{1},\lambda_{2}}L_{\mu\nu}(p_{\mathrm{on}})\epsilon^{\mu}\left(\lambda_{1},{\vec{p}}\right)\epsilon^{\nu\ast}\left(\lambda_{2},{\vec{p}}\right)f_{\lambda_{1}\lambda_{2}}^{V}(x,{\vec{p}})$
$\displaystyle=$ $\displaystyle
f_{00}^{V}(x,{\vec{p}})+\frac{1}{3}\sum_{\lambda_{1},\lambda_{2}}\epsilon^{\mu}\left(\lambda_{1},{\vec{p}}\right)\epsilon_{\mu}^{\ast}\left(\lambda_{2},{\vec{p}}\right)f_{\lambda_{1}\lambda_{2}}^{V}(x,{\vec{p}})$
$\displaystyle=$ $\displaystyle
f_{00}^{V}(x,{\vec{p}})-\frac{1}{3}\mathrm{Tr}(f_{V}).$ (73)

In (72), $\epsilon^{\mu}(0,{\vec{p}})$ is the polarization vector along the
spin quantization direction. With the first line of Eq. (71) and Eq. (73), we
obtain

$\frac{L_{\mu\nu}(p_{\mathrm{on}})W^{\mu\nu}}{-\Delta^{\mu\nu}(p_{\mathrm{on}})W_{\mu\nu}}=\frac{f_{00}^{V}(x,{\vec{p}})}{\mathrm{Tr}[f_{V}(x,{\vec{p}})]}-\frac{1}{3}=\rho_{00}-\frac{1}{3}.$
(74)

The above formula relates the Wigner function to $\rho_{00}$.

The medium effects can be described by retarded and advanced two-point Green’s
functions through spectral functions. For vector mesons interacting with
thermal quarks, the spectral function can be defined through the imaginary
part of the retarded two-point Green’s function (propagator in momentum space)
as

$\mathrm{Im}\widetilde{G}_{R}^{\mu\nu}(p)=\Delta_{T}^{\mu\nu}\rho_{T}(p)+\Delta_{L}^{\mu\nu}\rho_{L}(p).$
(75)

The definition of two-point Green’s functions $G$ and $\Sigma$ differs by a
factor $i=\sqrt{-1}$ from the usual one in quantum field theory, which are
related by $G=i\widetilde{G}$ and $\Sigma=i\widetilde{\Sigma}$. In Eq. (75),
we defined

$\displaystyle\Delta_{T}^{\mu\nu}=$ $\displaystyle-g^{\mu 0}g^{\nu
0}+g^{\mu\nu}+\frac{\mathbf{p}^{\mu}\mathbf{p}^{\nu}}{|{\vec{p}}|^{2}},$
$\displaystyle\Delta_{L}^{\mu\nu}=$
$\displaystyle\Delta^{\mu\nu}-\Delta_{T}^{\mu\nu}\equiv
g^{\mu\nu}-\frac{p^{\mu}p^{\nu}}{p^{2}}-\Delta_{T}^{\mu\nu},$ (76)

as the transverse and longitudinal projector respectively with
$\mathbf{p}^{\mu}=(0,{\vec{p}})$, and $\rho_{T,L}$ are spectral functions in
the transverse and longitudinal directions given by Dong:2023cng

$\displaystyle\rho_{T}(p)=$
$\displaystyle-\mathrm{Im}\frac{1}{p^{2}-m^{2}+\widetilde{\Sigma}_{\perp}(p)+i\,\mathrm{sgn}(p_{0})\varepsilon},$
$\displaystyle\rho_{L}(p)=$
$\displaystyle-\mathrm{Im}\frac{1}{p^{2}-m^{2}+\frac{p^{2}}{|\mathbf{p}|^{2}}\widetilde{\Sigma}_{00}(p)+i\,\mathrm{sgn}(p_{0})\varepsilon},$
(77)

where $\widetilde{\Sigma}_{\perp}$ and $\widetilde{\Sigma}_{00}$ are from
$\widetilde{\Sigma}_{R}^{\mu\nu}$:
$\widetilde{\Sigma}_{\perp}\equiv-(1/2)\Delta_{\mu\nu}^{T}\widetilde{\Sigma}_{R}^{\mu\nu}$
and $\widetilde{\Sigma}_{00}=\widetilde{\Sigma}_{R}^{00}$,
$\mathrm{sgn}(p_{0})$ is the sign of $p_{0}$, and $\varepsilon$ is an
infinitesimal positive number. One can check in Eq. (76) that
$p_{\mu}\Delta_{T}^{\mu\nu}=p_{\mu}\Delta_{L}^{\mu\nu}=0$. In Eq. (77), one
can verify that the real parts of $\widetilde{\Sigma}_{\perp}$ and
$\widetilde{\Sigma}_{00}$ contribute to the mass correction while the
imaginary parts of $\widetilde{\Sigma}_{\perp}$ and $\widetilde{\Sigma}_{00}$
determines the width or life-time of the quasi-particle mode.

The two-point function $G_{<}^{\mu\nu}$ is related to $G_{A}^{\mu\nu}$ and
$G_{R}^{\mu\nu}$ through fetter2003quantum ; zubarev1997statistical

$\displaystyle G_{<}^{\mu\nu}(p)$ $\displaystyle=$ $\displaystyle
in_{B}(p_{0})\left[\widetilde{G}_{R}^{\mu\nu}(p)-\widetilde{G}_{A}^{\mu\nu}(p)\right]$
(78) $\displaystyle=$
$\displaystyle-2n_{B}(p_{0})\mathrm{Im}\widetilde{G}_{R}^{\mu\nu}(p),$

where $n_{B}(p_{0})=1/(e^{\beta p_{0}-\beta\mu_{V}}-1)$ is the Bose-Einstein
distribution with the inverse temperature $\beta\equiv 1/T$ and the vector
meson’s chemical potential $\mu_{V}$. Inserting Eq. (78) into (68) we obtain
the on-shell Wigner function $W^{\mu\nu}(p_{\mathrm{on}})$ from which the spin
alignment $\rho_{00}$ can be extracted through Eq. (74).

For free vector mesons, the spectral functions are
$\rho_{T}^{(0)}=\rho_{L}^{(0)}=\pi\mathrm{sgn}(p_{0})\delta(p^{2}-m^{2})$,
which give $G_{<}^{\mu\nu}(p)$ and $\mathrm{Im}\widetilde{G}_{R}^{\mu\nu}(p)$
for the free vector meson following Eqs. (75) and (78).

The non-equilibrium correction to $G_{<}^{\mu\nu}(p)$ can be calculated
through the Kubo formula in linear response theory. The Kubo formula has been
derived in Zubarev’s approach to non-equilibrium density operator Zubarev_1979
; Hosoya:1983id ; Becattini:2019dxo . Detailed derivation is given in Ref.
Becattini:2019dxo . Considering the thermal shear tensor
$\xi_{\mu\nu}=1/2(\partial_{\mu}\beta_{\nu}+\partial_{\nu}\beta_{\mu})$ as a
perturbation from local equilibrium, the next-to-leading order correction of
$G_{<}^{\mu\nu}$ can be written as Dong:2023cng

$\displaystyle\delta G_{<}^{\mu\nu}(x,p)$ $\displaystyle=$ $\displaystyle
4T\lim_{K^{\mu}\rightarrow 0}\frac{\partial}{\partial
K_{0}}\mathrm{Im}\int\frac{dp_{1}^{0}dp_{2}^{0}}{2\pi}$
$\displaystyle\times\frac{n_{B}(p_{1}^{0})-n_{B}(p_{2}^{0})}{p_{1}^{0}-p_{2}^{0}+K^{0}+i\epsilon}\delta\left(p^{0}-\frac{p_{1}^{0}+p_{2}^{0}}{2}\right)\xi_{\gamma\lambda}$
$\displaystyle\times\sum_{a,b=L,T}\rho_{a}(p_{1})\rho_{b}(p_{2})I_{ab}^{\mu\nu\gamma\lambda}(p_{1},p_{2}),$
(79)

where
$p_{1}=(p_{1}^{0},{\vec{p}}-{\vec{K}}/2),p_{2}=(p_{2}^{0},{\vec{p}}+{\vec{K}}/2)$.
The tensor $I_{ab}^{\mu\nu\gamma\lambda}(p_{1},p_{2})$ is expressed as
Dong:2023cng

$\displaystyle I_{ab}^{\mu\nu\gamma\lambda}(p_{1},p_{2})$ $\displaystyle=$
$\displaystyle(p_{1}^{\lambda}p_{2}^{\gamma}+p_{1}^{\gamma}p_{2}^{\lambda})\Delta_{a,\alpha}^{\nu}(p_{1})\Delta_{b}^{\mu\alpha}(p_{2})$
$\displaystyle+(p_{1,\alpha}p_{2}^{\alpha}-m_{V}^{2})$
$\displaystyle\times\left[\Delta_{a}^{\gamma\nu}(p_{1})\Delta_{b}^{\mu\lambda}(p_{2})+\Delta_{a}^{\lambda\nu}(p_{1})\Delta_{b}^{\mu\gamma}(p_{2})\right]$
$\displaystyle-\left[p_{1}^{\gamma}p_{2}^{\alpha}\Delta_{a,\alpha}^{\nu}(p_{1})\Delta_{b}^{\mu\lambda}(p_{2})\right.$
$\displaystyle\left.+p_{2}^{\gamma}p_{1}^{\alpha}\Delta_{a}^{\lambda\nu}(p_{1})\Delta_{b,\alpha}^{\mu}(p_{2})\right]$
$\displaystyle-\left[p_{1,\alpha}p_{2}^{\lambda}\Delta_{a}^{\gamma\nu}(p_{1})\Delta_{b}^{\mu\alpha}(p_{2})\right.$
$\displaystyle\left.+p_{1}^{\lambda}p_{2,\alpha}\Delta_{a}^{\alpha\nu}(p_{1})\Delta_{b}^{\mu\gamma}(p_{2})\right]$
$\displaystyle-g^{\gamma\lambda}\left[g_{\beta\alpha}(p_{2,\rho}p_{1}^{\rho}-m_{V}^{2})-p_{1,\beta}p_{2,\alpha}\right]$
$\displaystyle\times\Delta_{a}^{\alpha\nu}(p_{1})\Delta_{b}^{\mu\beta}(p_{2}),$
(80)

where $\Delta_{T}^{\mu\nu}(p)$ and $\Delta_{L}^{\mu\nu}(p)$ are given by Eq.
(76). Then the spin alignment of vector mesons can be obtained from Eq. (74)
through Eqs. (68) and (79).

Under the quasi-particle approximation (QPA), the self-energies and spectral
functions can be calculated analytically by expanding $p_{0}$ in powers of
$\delta p_{0}=p_{0}-E_{p}^{V}$. This expansion requires $\Delta
E/E_{p}^{V}\sim\Gamma/E_{p}^{V}\sim\epsilon\ll 1$, where $\Delta E$ and
$\Gamma$ are the mass shift and width of the vector meson respectively. Then
the spin alignment can be expressed as

$\delta\rho_{00}({\vec{p}})=\delta\rho_{00}^{(\xi=0)}({\vec{p}})+\xi_{\mu\nu}C^{\mu\nu}({\vec{p}}),$
(81)

where $\delta\rho_{00}^{(\xi=0)}$ is the spin alignment independent of the
shear stress tensor and $C^{\mu\nu}$ is the dimensionless tensor coefficient
in the shear stress term. The results under the QPA are compared with full
numerical results are shown in Fig. (4). For the values of parameters we
choose, the magnitudes of $\delta\rho_{00}^{(\xi=0)}$ and
$C^{\mu\nu}({\vec{p}})$ are $\mathcal{O}(10^{-3})$ and
$\mathcal{O}(10^{-2}\sim 10^{-3})$, respectively. As a consequence, if the
thermal shear tensor is $\mathcal{O}(10^{-2})$, the contribution from the
thermal shear tensor to the $\phi$ meson’s spin alignment is
$\mathcal{O}(10^{-4}\sim 10^{-5})$, which is much smaller than the
contribution from the strong force fields Sheng:2022wsy .

Figure 4: The full numerical and QPA results for $C^{\mu\nu}$ as functions of
vector mesons’ momenta. In the upper panel the momentum is perpendicular to
the spin quantization direction $z$, while in the lower panel the momentum is
parallel to the spin quantization direction.

##  VI Vector meson’s spin alignments in quark fragmentation

In high energy reactions, hadron production is normally described by
fragmentation functions (FFs). The hadron‘s Polarization can be realized in
terms of polarized FFs Chen:2016moq ; Chen:2016iey ; Chen:2020pty . These FFs
are defined via quark-quark correlators. Among different high energy
reactions, the clean and ideal place to study FFs is in high energy
$e^{+}e^{-}$ collisions. In Ref. Chen:2016moq , the complete decomposition of
quark-quark correlators for spin-1 hadrons with systematic results for hadron
polarization in terms of FFs up to twist-3 in $e^{+}e^{-}\to V\pi X$ have been
presented. Experimental measurements have been carried out many years ago for
the longitudinal polarization of $\Lambda$ hyperons ALEPH:1996oew ;
OPAL:1997oem  and for the spin alignments of vector mesons DELPHI:1997ruo ;
OPAL:1997vmw ; OPAL:1997nwj ; OPAL:1999hxs  in the inclusive production
process $e^{+}e^{-}\to hX$ at LEP. Significant effects have been observed.
These experimental data have attracted many phenomenological studies
Anselmino:1997ui ; Anselmino:1998jv ; Anselmino:1999cg ; Xu:2001hz ; Xu:2003fq
; Chen:2016moq ; Chen:2016iey ; Chen:2020pty  and parameterization of
corresponding FFs has been obtained Chen:2016moq ; Chen:2016iey ; Chen:2020pty
. In this section we will summarize these results.

###  VI.1 FFs from quark-quark correlator

For the fragmentation of the quark (or anti-quark), the quark-quark correlator
as a $4\times 4$ matrix in Dirac indices is defined by

$\displaystyle\hat{\Xi}(k;p,S)=$
$\displaystyle\sum_{X}\int\frac{d^{4}\xi}{2\pi}e^{-ik\xi}\langle
hX|\bar{\psi}(\xi)\mathcal{L}(\xi;\infty)|0\rangle$
$\displaystyle\times\langle
0|\mathcal{L}^{\dagger}(0;\infty)\psi(0)|hX\rangle,$ (82)

where $k$ and $p$ are four-momenta of the quark and hadron respectively, $S$
denotes the hadron’s spin, and $\mathcal{L}(\xi;\infty)$ is the gauge link
defined as

$\mathcal{L}(\xi;\infty)=P\mathrm{exp}\left[ig\int_{\xi^{-}}^{\infty}d\eta^{-}A^{+}(\eta^{-};\xi^{+},\vec{\xi}_{\perp})\right],$
(83)

which guarantees the gauge invariance of FFs. In the rest part of the paper,
for notational simplicity, we suppress the gauge link in forthcoming
equations. In this case, the field operator $\psi(\xi)$ just stands for
$\mathcal{L}^{\dagger}(\xi;\infty)\psi(\xi)$.

The FFs are obtained from $\hat{\Xi}(k;p,S)$ in the following way. First, we
expand $\hat{\Xi}(k;p,S)$ in terms of $\Gamma$ matrices as

$\displaystyle\hat{\Xi}(k;$ $\displaystyle
p,S)=\frac{1}{2}\Bigl{[}\Xi(k;p,S)+i\gamma_{5}\tilde{\Xi}(k;p,S)$
$\displaystyle+\gamma^{\alpha}\Xi_{\alpha}(k;p,S)+\gamma_{5}\gamma^{\alpha}\tilde{\Xi}_{\alpha}(k;p,S)$
$\displaystyle+i\sigma^{\alpha\beta}\gamma_{5}\Xi_{\alpha\beta}(k;p,S)\Bigr{]},$
(84)

where coefficients are all real functions and are Lorentz scalar, pseudo-
scalar, vector, axial-vector and tensor respectively. Then we expand these
coefficient functions according to their respective Lorentz transformation
properties in terms of basic Lorentz covariants multiplied by scalar functions
which are constructed from available variables. These scalar functions are the
FFs.

Clearly, basic Lorentz covariants to be constructed depend strongly on what
variables that we have at hand. Besides $p$ and $k$, we have variables for
spin states which are different for spin-0, 1/2 or 1 hadrons. For this
purpose, we need to decompose the spin density matrix in terms of known
operators multiplied by Lorentz covariants. For spin-1/2 hadrons, the spin
density matrix $\rho$ is decomposed as in Eq. (1), but $\vec{S}$ in the rest
frame of the hadron should be generalized to a covariant form $S=(0,\vec{S})$.
For spin-1 hadrons, the $3\times 3$ density matrix $\hat{\rho}$ is usually
decomposed as in Eq. (6). The tensor polarization $T^{ij}={\rm
Tr}(\rho\Sigma^{ij})$ can be parameterized as

$\displaystyle[T]=\frac{1}{2}\left(\begin{array}[]{ccc}-\frac{2}{3}S_{LL}+S_{TT}^{xx}&S_{TT}^{xy}&S_{LT}^{x}\\\
S_{TT}^{xy}&-\frac{2}{3}S_{LL}-S_{TT}^{xx}&S_{LT}^{y}\\\
S_{LT}^{x}&S_{LT}^{y}&\frac{4}{3}S_{LL}\end{array}\right).$ (88)

We see that, for spin-1 hadrons, besides the polarization vector $S$, there is
also a tensor polarization part. The polarization vector is similar to that
for spin-1/2 hadrons. The tensor polarization part has five components, namely
a Lorentz scalar $S_{LL}$, a Lorentz vector
$S_{LT}^{\mu}=(0,S_{LT}^{x},S_{LT}^{y},0)$ and a Lorentz tensor
$S_{TT}^{\mu\nu}$ that has two components $S_{TT}^{xx}=-S_{TT}^{yy}$ and
$S_{TT}^{xy}=S_{TT}^{yx}$. The vector meson’s spin alignment $\rho_{00}$ is
only related to $S_{LL}$ through $\rho_{00}=(1-2S_{LL})/3$.

The quark-quark correlator given by Eq. (82) is un-integrated, i.e. it depends
on the four-momentum $k$. If we consider three- or one-dimensional FFs, we
need to integrate it over $k^{-}$ and $\vec{k}_{\perp}$. We take one-
dimensional FFs as an example. In this case, after integrating over $k^{-}$
and $\vec{k}_{\perp}$, we obtain,

$\displaystyle\hat{\Xi}(z;p,S)=$
$\displaystyle\sum_{X}\int\frac{d\xi^{-}}{2\pi}e^{-ik^{+}\xi^{-}}$
$\displaystyle\times\langle hX|\bar{\psi}(\xi^{-})|0\rangle\langle
0|\psi(0)|hX\rangle,$ (89)

where $z\equiv p^{+}/k^{+}$. After Lorentz decomposition, we obtain terms
related to $S_{LL}$ as,

$\displaystyle z\Xi_{\alpha}(z;p,S)=$ $\displaystyle
p^{+}\bar{n}_{\alpha}[D_{1}(z)+S_{LL}D_{1LL}(z)]+$ $\displaystyle+{\rm
power~{}suppressed~{}terms}.$ (90)

We can obtain the expression for $D_{1}(z)+S_{LL}D_{1LL}(z)$ by reversely
solving Eqs. (89) and (90),

$\displaystyle D_{1}(z)+$ $\displaystyle
S_{LL}D_{1LL}(z)=\sum_{X}\int\frac{zd\xi^{-}}{2\pi p^{+}}e^{-ik^{+}\xi^{-}}$
$\displaystyle\times\langle hX|\bar{\psi}(\xi^{-})\gamma^{+}|0\rangle\langle
0|\psi(0)|hX\rangle,$ (91)

For comparison, we present the corresponding equations for $G_{1L}$ that
describes the longitudinal spin transfer in the fragmentation process

$\displaystyle z\tilde{\Xi}_{\alpha}(z;p,S)=$ $\displaystyle\lambda
p^{+}\bar{n}_{\alpha}G_{1L}(z)+{\rm power~{}suppressed~{}terms}$
$\displaystyle\lambda G_{1L}(z)=$
$\displaystyle\sum_{X}\int\frac{zd\xi^{-}}{2\pi p^{+}}e^{-ik^{+}\xi^{-}}$
$\displaystyle\times\langle
hX|\bar{\psi}(\xi^{-})\gamma^{+}\gamma_{5}|0\rangle\langle
0|\psi(0)|hX\rangle.$ (92)

The difference between $D_{1}(z)+S_{LL}D_{1LL}(z)$ in Eq. (91) and $\lambda
G_{1L}$ in Eq. (92) is that $\gamma^{+}$ is involved in the former while
$\gamma^{+}\gamma_{5}$ is involved in the latter.

###  VI.2 Vector meson’s spin alignments from FFs

From Eqs. (91), we see clearly that, similar to the well-known unpolarized FF
$D_{1}(z)$, $D_{1LL}$ is independent of the quark’s polarization because it is
a sum over the quark’s spin states

$\displaystyle D_{1}(z)+S_{LL}D_{1LL}(z)=\sum_{X}\int\frac{zd\xi^{-}}{2\pi
p^{+}}e^{-ik^{+}\xi^{-}}$ $\displaystyle\times\sum_{\lambda_{q}=L,R}\langle
hX|\bar{\psi}_{\lambda_{q}}(\xi^{-})\gamma^{+}|0\rangle\langle
0|\psi_{\lambda_{q}}(0)|hX\rangle,$ (93)

where $\lambda_{q}=L,R$ denotes the quark’s helicity (or chirality), and
$\psi_{L/R}=(1\pm\gamma_{5})\psi/2$. In contrast, the result for $G_{1L}$
gives

$\displaystyle\lambda G_{1L}(z)=$
$\displaystyle\sum_{X}\int\frac{zd\xi^{-}}{2\pi p^{+}}e^{-ik^{+}\xi^{-}}$
$\displaystyle\times$ $\displaystyle[\langle
hX|\bar{\psi}_{L}(\xi^{-})\gamma^{+}|0\rangle\langle 0|\psi_{L}(0)|hX\rangle$
$\displaystyle-\langle hX|\bar{\psi}_{R}(\xi^{-})\gamma^{+}|0\rangle\langle
0|\psi_{R}(0)|hX\rangle],$ (94)

which depends on the quark’s spin explicitly.

We can draw an important conclusion from Eq. (93) for the spin alignment
$\rho_{00}=(1-2S_{LL})/3$ for the vector meson produced in the fragmentation
process $q\rightarrow VX$: it is determined by $D_{1LL}$ and independent of
the initial polarization of the quark. The conclusion is rather unexpected
because the vector meson’s spin alignment in high energy reactions was first
observed in $e^{+}e^{-}\rightarrow VX$ at LEP DELPHI:1997ruo ; OPAL:1997vmw ;
OPAL:1997nwj ; OPAL:1999hxs  where initial quarks and anti-quarks are
longitudinally polarized. However, this is consistent with space reflection in
fragmentation processes where $\rho_{00}$ is space reflection invariant while
the helicity of the initial quark changes the sign. This conclusion is rather
solid since it follows from the general principle of QCD. It can also be
tested easily in experiments. In the following we present numerical results
for $e^{+}e^{-}\rightarrow VX$ and $pp\rightarrow VX$ as examples.

###  VI.3 Vector meson’s spin alignments in $e^{+}e^{-}\rightarrow VX$

Suppose that the quark fragmentation mechanism dominates hadron production in
$e^{+}e^{-}$ annihilation at high energies, we obtain the vector meson’s
alignment in $e^{+}e^{-}\rightarrow VX$ as

$\displaystyle\langle
S_{LL}\rangle(z,y)=\frac{\sum_{q}W_{q}(y)D_{1LL}(z)}{2\sum_{q}W_{q}(y)D_{1}(z)},$
(95)
$\displaystyle\langle\lambda\rangle(z,y)=\frac{\sum_{q}P_{q}(y)W_{q}(y)G_{1L}(z)}{\sum_{q}W_{q}(y)D_{1}(z)},$
(96)

where we also show the result of hyperon polarization for comparison. In Eqs.
(95) and (96), $P_{q}(y)$ and $W_{q}(y)$ are the quark polarization and
production weight at the vertex of $e^{+}e^{-}$ annihilation respectively
which are determined by the quark’s electric charge, $c_{V}^{e}$ and
$c_{A}^{e}$ in the weak interaction current, and so on (see e.g. Chen:2016iey
for the detailed expressions), $y=l_{2}\cdot p_{q}/[(l_{1}+l_{2})\cdot
p_{q}]$, where $l_{1}$ and $l_{2}$ are the four-momenta of the incident
$e^{-}$ and $e^{+}$ respectively, and $p_{q}$ is the four-momentum of the
produced quark.

The FFs are extracted from data available at a given scale and their scale
dependence is determined by the QCD evolution equation, the DGLAP equation
Dokshitzer:1977sg ; Gribov:1972ri ; Altarelli:1977zs ; Owens:1978qz ;
Georgi:1977mg ; Uematsu:1978yw ; Ravindran:1996ri ; Ravindran:1996jd . In Ref.
Chen:2016iey ; Chen:2020pty , such parameterizations of $D_{1LL}$ for vector
mesons have been obtained for the first time by fitting available data
DELPHI:1997ruo ; OPAL:1997vmw ; OPAL:1997nwj . As comparison,
parameterizations of $G_{1L}$ for $\Lambda$ are also given therein
Chen:2016iey ; Chen:2020pty  by fitting the LEP data ALEPH:1996oew ;
OPAL:1997oem . Here, we show the fitted results obtained there Chen:2016iey ;
Chen:2020pty  in Figs. 5 and 6.

Figure 5: Longitudinal polarization of $\Lambda$ in $e^{+}e^{-}\to\Lambda X$
at high energies. The LEP data are taken from Refs. ALEPH:1996oew ;
OPAL:1997oem . The solid line is the fit obtained in Ref. Chen:2016iey  at the
LEP energy while those at other energies are calculated results using the
DGLAP equation for FFs and energy dependence of $P_{q}$. The figure is taken
from Ref. Chen:2016iey .  Figure 6: The spin alignments of $K^{*0}$ and
$\rho^{0}$ in $e^{+}e^{-}\to VX$ at the $Z$-pole fitted in Ref. Chen:2020pty
compared with experimental data DELPHI:1997ruo ; OPAL:1997vmw . The solid line
is the fit in Ref. Chen:2020pty  at the LEP energy while those at other
energies are the results using the DGLAP equation for FFs. The figure is taken
from Ref. Chen:2016iey .

From Figs. 5 and 6, we see a distinct feature that there is a strong energy
dependence for $P_{L\Lambda}$, while the energy dependence for
$\rho_{00}^{K^{*}}$ is weak. The former comes mainly from the energy
dependence of $P_{q}$ while the latter comes mainly from the QCD evolution of
FFs. The relatively rapid change in the energy region around the mass of $Z$
boson comes from the influence of $W_{q}$. This feature can be tested in
future experiments.

###  VI.4 Vector meson’s spin alignments in $pp\rightarrow VX$

Assuming the universality of FFs, one can calculate vector meson’s spin
alignments in other high energy reactions where the fragmentation mechanism
dominates. This applies to hadron production at high transverse momentum in
$pp$ collisions and deeply inelastic scatterings. In Ref. Chen:2020pty , such
calculations have been carried out in $pp$ collisions. As examples, we show in
Fig. 7 the results obtained there for $K^{*0}$ and $\rho^{0}$ in two rapidity
regions as functions of $p_{T}$.

Figure 7: Spin alignments of vector mesons in $pp$ collisions at
$\sqrt{s}=200$ GeV and $\sqrt{s}=5.02$ TeV for $K^{*0}$ and $\rho^{0}$ in two
rapidity regions as functions of $p_{T}$. This figure is taken from Ref.
Chen:2020pty .

We see that $\rho^{V}_{00}$ for vector mesons deviate from $1/3$ significantly
showing large spin alignments. Such results can be tested in experiments at
RHIC and LHC.

##  VII Summary and outlook

Spin alignments for vector mesons have been observed in high energy reactions
STAR:2022fan ; DELPHI:1997ruo ; OPAL:1997vmw ; OPAL:1997nwj ; OPAL:1999hxs ;
NOMAD:2006kuc . There are two types of spin alignments: global spin alignments
in heavy-ion collisions STAR:2022fan  and helicity-basis spin alignments in
$e^{+}e^{-}$ collisions DELPHI:1997ruo ; OPAL:1997vmw ; OPAL:1997nwj ;
OPAL:1999hxs . These measurements show different features of spin alignments
arising from different hadronization mechanisms in collisions of heavy-ion and
$e^{+}e^{-}$.

In high energy heavy-ion collisions in the region of central rapidity and
small transverse momentum, hadron production is mainly through quark
combination or coalescence. The global spin alignment in this case not only
depends on the polarization of quarks and anti-quarks but is also sensitive to
the local correlation between the polarization of quarks and that of anti-
quarks. Measurements of the global spin alignment in this case provide a good
opportunity to study the polarization correlation in the sQGP produced in
heavy-ion collisions. The STAR’s measurements of $\phi$ meson’s spin
alignments STAR:2022fan  is consistent with the theoretical description based
on $\phi$ vector fields that lead to a strong local correlation between
$P_{s}$ and $P_{\bar{s}}$ Yang:2017sdk ; Sheng:2020ghv ; Sheng:2022ffb ;
Sheng:2022wsy . The data were used to extract the strength of the local
fluctuation in $\phi$ vector fields Sheng:2022ffb ; Sheng:2022wsy .

In $e^{+}e^{-}$ and $pp$ collisions at high energies, hadron production is
dominated by fragmentation with fragmentation functions. It has been shown
that the spin alignment of vector mesons in the helicity basis is independent
of the polarization of the initial quark in the quark fragmentation process.
With the parameterization of corresponding fragmentation functions the
prediction on spin alignments of vector mesons in high energy $e^{+}e^{-}$ and
$pp$ collisions has been made Chen:2016moq ; Chen:2016iey ; Chen:2020pty .

It is impressive and interesting to see the distinction between the global
spin alignments in high energy heavy-ion collisions and in $e^{+}e^{-}$ or
$pp$ collisions which are dominated by different hadronization mechanisms. It
is important to extend the measurements to different high energy reactions at
different energies to test the universality of these properties. Obviously
such studies provide new insights into properties of sQGP and hadronization
mechanisms.

## Acknowledgements

This work was supported in part by the National Key Research and Development
Program of China under Contract No. 2022YFA1604900, by the National Natural
Science Foundation of China (NSFC) under Contract Nos. 12025501, 11890710,
11890713, 11890714, 12147101 and 12135011, and by the Strategic Priority
Research Program of the Chinese Academy of Sciences (CAS) under Grant No.
XDB34030102.

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