# Impact of globally spin-aligned vector mesons on the search for the chiral
magnetic effect in heavy-ion collisions

Diyu Shen, Jinhui Chen  Key Laboratory of Nuclear Physics and Ion-beam
Application (MOE), Institute of Modern Physics, Fudan University, Shanghai
200433, China  Shanghai Research Center for Theoretical Nuclear Physics, NSFC
and Fudan University, Shanghai 200438, China  Aihong Tang  Brookhaven National
Laboratory, Upton, New York 11973<EMAIL_ADDRESS>Gang Wang  Department of
Physics and Astronomy, University of California, Los Angeles, California
90095, USA

###### Abstract

In high-energy heavy-ion collisions, the chiral magnetic effect (CME) is
predicted to arise from the interplay between the chirality imbalance of
quarks in the nuclear medium and the intense magnetic field, and will cause a
charge separation along the magnetic field direction. While the search for the
CME is still ongoing in experiments at Relativistic Heavy Ion Collider (RHIC)
and the Large Hadron Collider (LHC), the CME-sensitive observables need to be
scrutinized to exclude the non-CME contributions. In this work, we examine the
influence of globally spin-aligned $\rho$ mesons on the $\gamma_{112}$
correlator, the $R_{\Psi_{2}}(\Delta S)$ correlator, and the signed balance
functions, via a toy model and a multiphase transport model (AMPT). The global
spin alignment of vector mesons could originate from non-CME mechanisms in
heavy-ion collisions, and is characterized by the 00-component of the spin
density matrix, $\rho_{00}$. We find that the CME observables show similar
dependence on $\rho_{00}$, and could receive a positive (negative)
contribution from $\rho$-decay pions, if the $\rho_{00}$ of $\rho$ mesons is
larger (smaller) than 1/3. Since pions are the most abundant particles in such
collisions, the $\rho_{00}$ measurements for $\rho$ mesons become crucial to
the interpretation of the CME data.

##  1 Introduction

Heavy-ion collision experiments provide a unique test ground for quantum
chromodynamics (QCD), as the creation of the quark gluon plasma (QGP) brings
about many novel phenomena in the strong interaction. In this paper, we focus
on two such phenomena, the chiral magnetic effect (CME) and the global spin
alignment of vector mesons, and study how the latter could affect the
experimental search for the former. The CME postulates that in non-central
collisions, chiral quarks in the QGP interact with the intense magnetic field,
and form an electric current along the magnetic field direction [1], where the
chirality imbalance could emerge from the chiral anomaly in QCD [2, 3, 4], and
the magnetic field is generated by the energetic nuclear fragments [5, 6]. The
confirmation of the CME would reveal the topological structure of QCD, and
indicate ${\cal P}$ and ${\cal CP}$ violation in the strong interaction [2, 3,
4]. Experiments at RHIC and the LHC have been searching for the CME over the
past two decades without a definite conclusion [7, 8, 9, 10, 11, 12, 13, 14,
15], because of the incomplete understanding of the underlying backgrounds
[16, 17, 18, 19, 20, 21].

The major CME-sensitive observables include the $\gamma_{112}$ correlator
[22], the $R_{\Psi_{2}}(\Delta S)$ correlator [23], and the signed balance
functions [20]. A recent study [24] has verified the equivalence in the core
components among these methods, and ascertains that they have similar
sensitivities to the true CME signal, as well as the background due to the
collective motion (elliptic flow) of the collision system. Besides the flow-
related background, Ref. [20] points out that the signed balance functions
could acquire non-trivial contributions from globally spin-aligned vector
mesons, which heralds the impact of the global spin alignment of resonances on
all these observables.

The global spin alignment of vector mesons originates from the polarization of
constituent quarks, which can be caused by the spin-orbital coupling [25, 26,
27, 28], strong electromagnetic fields [29, 28], the local spin alignment [30,
31], locally fluctuating axial charge currents [32], and strong vector meson
fields [29, 33, 34, 35, 36]. Polarized quarks can be manifested in both the
global polarization of hyperons and the global spin alignment of vector
mesons. Non-trivial hyperon global polarization values have been reported at
RHIC and the LHC for $\Lambda$($\bar{\Lambda}$) [37, 38, 39, 40], $\Xi$ [41],
and $\Omega$ [41]. The global spin alignment has also been measured for
$K^{*0}$, $\phi$ [42, 43, 44] and $J/\psi$ [45]. In particular, unexpectedly
large $\rho_{00}$ values for $\phi$ mesons have been disclosed by the STAR
Collaboration [44].

In the following sections, we will demonstrate that globally spin-aligned
vector mesons could provide a finite contribution to each of the CME-sensitive
observables under study. The effect can be qualitatively understood with the
aid of analytical derivations, and is further investigated with a toy model
and a multiphase transport model. This work suggests that the global spin
alignment of $\rho$ mesons is a crucial component in the background estimation
for the CME measurements involving pions.

##  2 The $\gamma_{112}$ correlator

The CME-induced charge separation as well as the other modes of collective
motion is usually studied with the azimuthal angle distribution of final-state
particles in the momentum space [22],

$\displaystyle
E\frac{d^{3}N}{d^{3}p}=\frac{1}{2\pi}\frac{d^{2}N}{p_{T}dp_{T}d\mathsf{y}}\left(1+2a_{1}\sin\Delta\phi+\sum_{n=1}^{\infty}2v_{n}\cos
n\Delta\phi\right),$ (1)

where $p_{T}=\sqrt{p_{x}^{2}+p_{y}^{2}}$ is transverse momentum, $\mathsf{y}$
is rapidity, and $\Delta\phi$ is the azimuthal angle of a particle relative to
the reaction plane (spanned by the impact parameter and the beam momenta).
$a_{1}$ characterizes the charge separation perpendicular to the reaction
plane, and the $a_{1}$ values for positive and negative charges bear opposite
signs, i.e., $a_{1,+}=-a_{1,-}$ in a charge-symmetric system. $v_{n}$ denotes
the $n^{\rm th}$-harmonic flow coefficient of final-state particles, and
conventionally $v_{2}$ is called elliptic flow.

The CME-sensitive observable $\gamma_{112}$ [22] is defined as

$\displaystyle\gamma_{112}\equiv\left\langle\cos(\phi_{\alpha}+\phi_{\beta}-2\Psi_{\rm
RP})\right\rangle,$ (2)

where $\phi_{\alpha}$ and $\phi_{\beta}$ are the azimuthal angles of particles
$\alpha$ and $\beta$, respectively, and $\Psi_{\rm RP}$ represents the
reaction plane. The bracket means averaging over all particles and all events.
The difference in $\gamma_{112}$ between opposite-sign (OS) and same-sign (SS)
pairs is supposed to reflect the CME signal, i.e.,

$\displaystyle\Delta\gamma_{112}\equiv\gamma_{112}^{\rm{OS}}-\gamma_{112}^{\rm{SS}}\approx
2a_{1}^{2}.$ (3)

However, $\Delta\gamma_{112}$ is contaminated with backgrounds, e.g., decay
daughters of flowing resonances [22]. As a result, the expansion of
$\gamma_{112}^{\rm{OS}}$ contains finite covariance terms. For example, the
$\rho$ mesons decay to charged pions:

$\displaystyle\gamma_{112}^{\rm{OS}}=\left\langle\cos(\phi_{+}+\phi_{-}-2\Psi_{\rm
RP})\right\rangle$
$\displaystyle=\left\langle\cos\Delta\phi_{+}\right\rangle\left\langle\cos\Delta\phi_{-}\right\rangle+\frac{N_{\rho}}{N_{+}N_{-}}\rm{Cov}(\cos\Delta\phi_{+},\cos\Delta\phi_{-})$
$\displaystyle-\left\langle\sin\Delta\phi_{+}\right\rangle\left\langle\sin\Delta\phi_{-}\right\rangle-\frac{N_{\rho}}{N_{+}N_{-}}\rm{Cov}(\sin\Delta\phi_{+},\sin\Delta\phi_{-}),$
(4)

where Cov$(a,b)$ denotes the covariance of variables $a$ and $b$. $N_{\rho}$
is the yield of $\rho$ mesons, and $N_{+}$ and $N_{-}$ are the numbers of
$\pi^{+}$ and $\pi^{-}$, respectively. Note that the covariance terms in
$\gamma_{112}^{\rm{SS}}$ could also be finite, owing to mechanisms such as
transverse momentum conservation [17], but this effect should be cancelled in
$\Delta\gamma_{112}$.

Figure 1:  Illustration of pion emission (black solid arrow) in the rest frame
of the parent $\rho$ meson. The dashed arrow shows the projection in the
$x$-$y$ plane.

In addition to elliptic flow, another property of vector mesons, the global
spin alignment, could also contribute a background to $\Delta\gamma_{112}$. In
the decay of $\rho\rightarrow\pi^{+}+\pi^{-}$, the emission angle of
$\pi^{\pm}$ can be expressed as [46]

$\displaystyle\frac{dN}{d\cos\theta^{*}}=\frac{3}{4}\left[(1-\rho_{00})+(3\rho_{00}-1)\cos^{2}\theta^{*}\right],$
(5)

where $\theta^{*}$ is the angle between the pion momentum in the $\rho$ rest
frame and the angular momentum $\hat{L}$ of the collision system. In this
work, we assume the directions of $\hat{L}$ and the magnetic field are both
perpendicular to the reaction plane. $\rho_{00}$ is the 00-component of the
spin density matrix, and quantifies the global spin alignment. Projected to
the transverse ($x$-$y$) plane, Eq. (5) becomes

$\displaystyle\frac{dN}{d\phi^{*}}=\frac{1}{2\pi}\left[1-\frac{1}{2}(3\rho_{00}-1)\cos
2\phi^{*}\right],$ (6)

where $\phi^{*}$ is the azimuthal angle of the decay product in the $\rho$
rest frame as sketched in Fig. 1. Then, the covariance terms in Eq. (4) can be
calculated in the $\rho$ rest frame as

$\displaystyle\rm{Cov}(\cos\phi_{+}^{*},\cos\phi_{-}^{*})=-\left\langle\cos^{2}\phi_{+}^{*}\right\rangle+\left\langle\cos\phi_{+}^{*}\right\rangle^{2}$
$\displaystyle=-\frac{1}{2}+\frac{1}{8}(3\rho_{00}-1),$ (7)
$\displaystyle\rm{Cov}(\sin\phi_{+}^{*},\sin\phi_{-}^{*})=-\left\langle\sin^{2}\phi_{+}^{*}\right\rangle+\left\langle\sin\phi_{+}^{*}\right\rangle^{2}$
$\displaystyle=-\frac{1}{2}-\frac{1}{8}(3\rho_{00}-1).$ (8)

Here we use $\phi_{-}^{*}=\phi_{+}^{*}+\pi$. In the absence of the CME,
charged pions have
$\left\langle\cos\phi_{+}^{*}\right\rangle=\left\langle\cos\phi_{-}^{*}\right\rangle$
and
$\left\langle\sin\phi_{+}^{*}\right\rangle=\left\langle\sin\phi_{-}^{*}\right\rangle$,
and thus the $\Delta\gamma_{112}$ in the $\rho$ rest frame can be written as

$\displaystyle\Delta\gamma_{112}^{*}=\frac{N_{\rho}}{N_{+}N_{-}}\frac{3\rho_{00}-1}{4},$
(9)

which represents a finite background when $\rho_{00}$ deviates from $1/3$.

Essentially, Eq. (6) manifests the elliptic flow of decay products in the
$\rho$ rest frame, $v_{2}^{*}=-(3\rho_{00}-1)/4$, and therefore Eq. (9) can be
viewed as $\Delta\gamma_{112}^{*}\propto-v_{2}^{*}$. In the laboratory frame,
Eqs. (7) and (8) should be scaled by factors of $f_{c}$ and $f_{s}$,
respectively, due to the Lorentz boost of the $\rho$ meson. In general,
$f_{c}$ and $f_{s}$ are different because of the anisotropic motion
($v_{2}^{\rho}$) of $\rho$ mesons. One can thus expand $f_{c}$ and $f_{s}$
with $v_{2}^{\rho}$,

$\displaystyle f_{c}=f_{0}+\sum_{n=1}^{\infty}c_{n}(v_{2}^{\rho})^{n},$ (10)
$\displaystyle f_{s}=f_{0}+\sum_{n=1}^{\infty}s_{n}(v_{2}^{\rho})^{n},$ (11)

where $f_{0}$, $c_{n}$ and $s_{n}$ depend on the spectra of $\rho$ mesons.
Hence, the contribution of decay pions to $\Delta\gamma_{112}$ in the
laboratory frame can be expressed as

$\displaystyle\Delta\gamma_{112}=\frac{N_{\rho}}{N_{+}N_{-}}\left[\frac{1}{8}(f_{c}+f_{s})(3\rho_{00}-1)-\frac{1}{2}(f_{c}-f_{s})\right]$
$\displaystyle=\frac{N_{\rho}}{8N_{+}N_{-}}\left[2f_{0}+\sum_{n=1}^{\infty}(c_{n}+s_{n})(v_{2}^{\rho})^{n}\right](3\rho_{00}-1)$
$\displaystyle-\frac{N_{\rho}}{2N_{+}N_{-}}\sum_{n=1}^{\infty}(c_{n}-s_{n})(v_{2}^{\rho})^{n}.$
(12)

At a given $v_{2}^{\rho}$, we expect the $\Delta\gamma_{112}$ measurement
involving $\rho$-decay pions to have a linear dependence on the $\rho_{00}$ of
$\rho$ mesons.

Figure 2: Toy model simulations of the $\pi$-$\pi$ $\Delta\gamma_{112}$
correlation as a function of $\rho$-meson $\rho_{00}$ with various inputs of
$v_{2}^{\rho}$. Linear fits are applied to guide eyes.

We first test the aforementioned idea with toy model simulations without the
CME. Each event contains 195 $\pi^{+}$ and 195 $\pi^{-}$, with 33 pairs of
them from $\rho$ decays. For simplicity, the $v_{2}$ and $v_{3}$ values of
primordial pions are set to zero. The spectrum of primordial pions obeys the
Bose-Einstein distribution,

$\displaystyle\frac{dN_{\pi^{\pm}}}{dm_{T}^{2}}\propto\frac{1}{e^{m_{T}/T_{\rm{BE}}}-1},$
(13)

where $T_{\rm{BE}}=212$ MeV is set to match the experimentally observed
$\left\langle p_{T}\right\rangle=$ 400 MeV [47]. The spectrum of $\rho$ mesons
follows

$\displaystyle\frac{dN_{\rho}}{dm_{T}^{2}}\propto\frac{e^{-(m_{T}-m_{\rho})/T}}{T(m_{\rho}+T)},$
(14)

where $T=317$ MeV is set to match its $\left\langle p_{T}\right\rangle$ of 830
MeV as observed in data [48]. Pseudorapidity (Rapidity) is uniformly
distributed in the range of $[-1,1]$ for primordial pions ($\rho$ mesons).
$\rho$-meson decays are implemented with PYTHIA6 [49], and the spin alignment
effect is simulated by sampling the decay products according to Eq. (5).

Figure 2 shows the simulation results of the $\pi$-$\pi$ $\Delta\gamma_{112}$
correlation from the toy model. Each marker denotes a different input of
$v_{2}^{\rho}$. It is confirmed that at a given $v_{2}^{\rho}$,
$\Delta\gamma_{112}$ increases linearly with $\rho_{00}$. On the other hand,
$\Delta\gamma_{112}$ also increases with $v_{2}^{\rho}$ at a fixed
$\rho_{00}$, exhibiting the convolution of $v_{2}^{\rho}$ and $\rho_{00}$ in
the background contribution from $\rho$ mesons to $\Delta\gamma_{112}$. In the
case of $v_{2}^{\rho}=0$ and $\rho_{00}=1/3$, $\Delta\gamma_{112}$ is zero, as
expected by Eq. (12). Note that the global spin alignment effect could give a
negative contribution to the $\Delta\gamma_{112}$ measurement if $\rho_{00}$
is smaller than $1/3$.

We also study this effect with a more realistic model, a multiphase transport
(AMPT) model, without the CME and with the spin alignment implemented by
redistributing the momenta of decay products according to Eq. (5). The details
of this model can be found in Ref. [50, 51]. The selected decay channel is
$\rho\rightarrow\pi^{+}+\pi^{-}$. As a qualitative investigation, we only
simulate Au+Au collisions at $\sqrt{s_{NN}}=200$ GeV with the impact parameter
of 8 fm, and pions are analyzed without any kinematic cut to increase
statistics. The $\rho_{00}$ values are set to be $0.18$, $1/3$ and $0.43$,
respectively, with a million events for each case. Figure 3 shows the AMPT
calculations of $\pi$-$\pi$ $\Delta\gamma_{112}$ as a function of $\rho$-meson
$\rho_{00}$. $\Delta\gamma_{112}$ increases linearly with $\rho_{00}$, similar
to the toy model simulations. At $\rho_{00}=1/3$, the positive
$\Delta\gamma_{112}$, which is a non-CME background, may come from the
positive $v_{2}^{\rho}$ and transverse momentum conservation. The slope,
$d\Delta\gamma_{112}/d\rho_{00}$, could be different between the toy model and
the AMPT model, because of the different $\rho$-meson spectra.

Figure 3: AMPT calculations of the $\pi$-$\pi$ $\Delta\gamma_{112}$
correlation as a function of $\rho$-meson $\rho_{00}$ in Au+Au collisions at
$\sqrt{s_{NN}}=200$ GeV with the impact parameter of 8 fm. Solid line
represents a linear fit of calculations to guide eyes.

##  3 The $R_{\Psi_{2}}(\Delta S)$ correlator

Another CME-sensitive observable, the $R_{\Psi_{2}}(\Delta S)$ correlator
[23], is defined as a double ratio of four distributions,

$\displaystyle R_{\Psi_{2}}(\Delta S)\equiv\frac{N(\Delta S_{{\rm
real}})}{N(\Delta S_{{\rm shuffled}})}/\frac{N(\Delta S^{\perp}_{{\rm
real}})}{N(\Delta S^{\perp}_{{\rm shuffled}})},$ (15)

where

$\displaystyle\Delta
S=\left\langle\sin\Delta\phi_{+}\right\rangle-\left\langle\sin\Delta\phi_{-}\right\rangle,$
(16) $\displaystyle\Delta
S^{\perp}=\left\langle\cos\Delta\phi_{+}\right\rangle-\left\langle\cos\Delta\phi_{-}\right\rangle,$
(17)

and $\Delta\phi=\phi-\Psi_{2}$. $\Psi_{2}$ denotes the $2^{\rm nd}$-order
event plane, an approximation of the reaction plane using the $v_{2}$
information of produced particles. The bracket means averaging over all
particles of interest in an event. The subscripts “real” and “shuffled”
represent real events and charge shuffled events, respectively. Ideally, the
CME should cause a concave shape in $R_{\Psi_{2}}(\Delta S)$, which can be
quantified by the width of a Gaussian fit, $\sigma_{R}$. Analytically,
$\sigma_{R}$ is related to the widths of the four initial distributions,

$\displaystyle\frac{S_{\rm
concavity}}{\sigma_{R}^{2}}=\frac{1}{\sigma^{2}(\Delta S_{\rm
real})}-\frac{1}{\sigma^{2}(\Delta S_{\rm
shuffled})}-\frac{1}{\sigma^{2}(\Delta S_{\rm real}^{\perp})}$
$\displaystyle+\frac{1}{\sigma^{2}(\Delta S_{\rm shuffled}^{\perp})}.$ (18)

$S_{\rm concavity}$ is 1 (-1) when the $R_{\Psi_{2}}(\Delta S)$ distribution
is convex (concave).

Similar to the case of $\Delta\gamma_{112}$, we first evaluate each term in
Eq. (18) in the $\rho$ rest frame, and then use parameters $f_{c}$ and $f_{s}$
to describe the Lorentz effect along and perpendicular to the reaction plane,
respectively, in the laboratory frame. For example, $\sigma^{2}(\Delta S)$ can
be expressed as

$\displaystyle\sigma^{2}(\Delta
S)=\sigma^{2}(a_{1}^{+})+\sigma^{2}(a_{1}^{-})-2{\rm
Cov}(a_{1}^{+},a_{1}^{-})$
$\displaystyle=\frac{\sigma^{2}(\sin\Delta\phi_{+})}{N_{+}}+\frac{\sigma^{2}(\sin\Delta\phi_{-})}{N_{-}}$
$\displaystyle-\frac{2N_{\rho}{\rm
Cov}(\sin\Delta\phi_{+},\sin\Delta\phi_{-})}{N_{+}N_{-}},$ (19)

where $a_{1}^{+}$ and $a_{1}^{-}$ denote $\langle\sin\Delta\phi_{+}\rangle$
and $\langle\sin\Delta\phi_{-}\rangle$, respectively, in an event. After
applying Eq. (6), we convert each term into the laboratory frame,

$\displaystyle\sigma^{2}(\Delta S_{\rm
real})=f_{s}\left[\sigma_{s}^{2}+\frac{N_{\rho}}{N_{+}N_{-}}(1+\frac{3\rho_{00}-1}{4})\right],$
(20) $\displaystyle\sigma^{2}(\Delta S_{\rm shuffled})=f_{s}\sigma_{s}^{2},$
(21) $\displaystyle\sigma^{2}(\Delta S_{\rm
real}^{\perp})=f_{c}\left[\sigma_{c}^{2}+\frac{N_{\rho}}{N_{+}N_{-}}(1-\frac{3\rho_{00}-1}{4})\right],$
(22) $\displaystyle\sigma^{2}(\Delta S_{\rm
shuffled}^{\perp})=f_{c}\sigma_{c}^{2},$ (23)

where

$\displaystyle\sigma_{s}^{2}=\frac{\sigma^{2}(\sin\phi_{+}^{*})}{N_{+}}+\frac{\sigma^{2}(\sin\phi_{-}^{*})}{N_{-}},$
(24)
$\displaystyle\sigma_{c}^{2}=\frac{\sigma^{2}(\cos\phi_{+}^{*})}{N_{+}}+\frac{\sigma^{2}(\cos\phi_{-}^{*})}{N_{-}}.$
(25)

The sign of $S_{\rm concavity}$ in Eq. (18) is not instantly clear, because
$\sigma_{c}$ and $\sigma_{s}$, as well as $f_{c}$ and $f_{s}$, could be
different due to elliptic flow of primordial pions and $\rho$ mesons. Assuming
$v_{2}^{\pi}$ and $v_{2}^{\rho}$ are both zero, we have

$\displaystyle{\rm Sign}(S_{\rm concavity})={\rm
Sign}\left[-\frac{N_{\rho}}{2N_{+}N-}(3\rho_{00}-1)\right].$ (26)

In this case, if $\rho_{00}$ is smaller (larger) than $1/3$, $S_{\rm
concavity}$ becomes 1 (-1), and the $R_{\Psi_{2}}(\Delta S)$ distribution
becomes convex (concave).

Figure 4: (Left) toy model simulations of the $R_{\Psi_{2}}(\Delta
S^{\prime\prime})$ distribution for events with zero $v_{2}^{\rho}$ and
different $\rho_{00}$ inputs. The distributions are symmetrized around $\Delta
S^{\prime\prime}=$0\. (Right) $S_{\rm concavity}/\sigma_{R}^{2}$ extracted
using Gaussian fits to $R_{\Psi_{2}}(\Delta S^{\prime\prime})$ for different
$v_{2}^{\rho}$ and $\rho_{00}$ inputs. A positive value indicates the convex
shape, and a negative one represents the concave shape.

In the following simulations, the generated events and the analysis cuts are
the same as in the previous section. We take the same procedure as in Ref.
[23] to correct the $R_{\Psi_{2}}(\Delta S)$ correlator for the particle
number fluctuations, i.e., $\Delta S^{\prime\prime}=\Delta S/\sigma_{\rm sh}$,
where $\sigma_{\rm sh}$ is the width of $N(\Delta S_{\rm shuffuled})$. Figure
4 shows $R_{\Psi_{2}}(\Delta S^{\prime\prime})$ as a function of $\rho_{00}$
from the toy model simulations with zero $v_{2}^{\rho}$. The
$R_{\Psi_{2}}(\Delta S^{\prime\prime})$ shapes are concave (convex) for
$\rho_{00}>1/3$ ($\rho_{00}<1/3)$, indicating a finite background from spin-
aligned vector mesons. Figure 4 also shows the $S_{\rm
concavity}/\sigma_{R}^{2}$ extracted using Gaussian fits for different
$v_{2}^{\rho}$ and $\rho_{00}$ inputs. The red circles represent the case with
zero $v_{2}^{\rho}$, and corroborate Eq. (26). At a given $v_{2}^{\rho}$,
$S_{\rm concavity}/\sigma_{R}^{2}$ displays a decreasing trends as a function
of $\rho_{00}$. On the other hand, at a given $\rho_{00}$, $S_{\rm
concavity}/\sigma_{R}^{2}$ also decreases with increased $v_{2}^{\rho}$, which
is consistent with findings in Ref. [19].

Figure 5: (Left) AMPT calculations of $R_{\Psi_{2}}(\Delta S^{\prime\prime})$
with $\rho_{00}=$ 0.18 (red cross), 1/3 (gray square) and 0.43 (blue circle)
in Au+Au collisions at 200 GeV with the impact parameter of 8 fm. The
distributions are symmetrized around $\Delta S^{\prime\prime}=$0\. (Right)
$S_{\rm concavity}/\sigma_{R}^{2}$ extracted using Gaussian fits.

Figure 5 shows AMPT calculations of $R_{\Psi_{2}}(\Delta S^{\prime\prime})$
with $\rho_{00}=$ 0.18, 1/3 and 0.43, respectively, in Au+Au collisions at 200
GeV with the impact parameter of 8 fm. At $\rho_{00}=1/3$, the concave shape,
which suggests a non-CME background, may come from the positive $v_{2}^{\rho}$
and transverse momentum conservation. Figure 5 also presents the $S_{\rm
concavity}/\sigma_{R}^{2}$ values retrieved using Gaussian fits, showing a
$\rho_{00}$ dependence similar to those from the toy model. Thus the AMPT
calculations further confirm the background contribution from globally spin-
aligned $\rho$ mesons to the $R_{\Psi_{2}}(\Delta S)$ correlator.

Compared with Eq. (18), it is more appealing to construct the difference
variable, $\Delta\sigma_{R}^{2}$, as proposed in Ref. [24]:

$\displaystyle\Delta\sigma_{R}^{2}=\sigma^{2}(\Delta S_{\rm
real})-\sigma^{2}(\Delta S_{\rm shuffled})-\sigma^{2}(\Delta S_{\rm
real}^{\perp})$ $\displaystyle+\sigma^{2}(\Delta S_{\rm shuffled}^{\perp}).$
(27)

Note that the CME should give a positive contribution to
$\Delta\sigma_{R}^{2}$. The contribution of $\rho$-decay pions to
$\Delta\sigma_{R}^{2}$ is

$\displaystyle\Delta\sigma_{R}^{2}=\frac{N_{\rho}}{N_{+}N_{-}}\left[\frac{1}{4}(f_{c}+f_{s})(3\rho_{00}-1)+(f_{c}-f_{s})\right].$
(28)

The function form of Eq. (28) is very similar to Eq. (12), and
$\Delta\sigma_{R}^{2}$ also has a linear dependence on $\rho_{00}$.
Consequently, the toy model simulations of $\Delta\sigma_{R}^{2}$ as a
function of $\rho_{00}$ with various $v_{2}^{\rho}$ inputs, as depicted in
Fig.6, are similar to those of $\Delta\gamma_{112}$ in Fig. 2.

Figure 6: Toy model simulations of $\Delta\sigma_{R}^{2}$ as a function of
$\rho_{00}$ with various $v_{2}^{\rho}$ inputs. Linear fits are applied to
guide eyes.

##  4 The signed balance functions

The signed balance functions probe the CME by examining the momentum ordering
between positively and negatively charged particles, based on the following
quantity [20, 24]

$\displaystyle\Delta
B_{y}\equiv\Big{[}\frac{N_{y(+-)}-N_{y(++)}}{N_{+}}-\frac{N_{y(-+)}-N_{y(--)}}{N_{-}}\Big{]}$
$\displaystyle-\Big{[}\frac{N_{y(-+)}-N_{y(++)}}{N_{+}}-\frac{N_{y(+-)}-N_{y(--)}}{N_{-}}\Big{]}$
$\displaystyle=\frac{N_{+}+N_{-}}{N_{+}N_{-}}[N_{y(+-)}-N_{y(-+)}],$ (29)

where $N_{y(\alpha\beta)}$ is the number of pairs in which particle $\alpha$
is ahead of particle $\beta$ along the $y$ axis
($p_{y}^{\alpha}>p_{y}^{\beta}$) in an event. Similarly, $\Delta B_{x}$ can be
constructed along the $x$ axis. The CME will enhance the width of the $\Delta
B_{y}$ distribution via the charge separation along the $y$ axis, and
therefore the final observable is the ratio

$\displaystyle r\equiv\sigma(\Delta B_{y})/\sigma(\Delta B_{x}).$ (30)

$r$ can be calculated in both the laboratory frame ($r_{\rm lab}$) and the
$\rho$ rest frame ($r_{\rm rest}$). The CME will lead to $r_{\rm rest}>r_{\rm
lab}>1$. In this work, we focus on $r_{\rm lab}$, since the extra sensitivity
in $r_{\rm rest}$ is a higher-order effect, and requires substantially more
statistics and computing resources.

Ref. [20] has pointed out that the global spin alignment of $\rho$ mesons will
affect the $r_{\rm lab}$ measurement. Here, we will derive the qualitative
relation. For simplicity, we assume that all the particles have the same
$p_{T}$, and write $\Delta B_{y}$ and $\Delta B_{x}$ as [24]

$\displaystyle\Delta
B_{y}\approx\frac{8M}{\pi^{2}}\left(1+\frac{2}{3}\langle\cos
2\Delta\phi\rangle\right)\left(\langle\sin\Delta\phi_{+}\rangle-\langle\sin\Delta\phi_{-}\rangle\right),$
$\displaystyle\Delta
B_{x}\approx\frac{8M}{\pi^{2}}\left(1-\frac{2}{3}\langle\cos
2\Delta\phi\rangle\right)\left(\langle\cos\Delta\phi_{+}\rangle-\langle\cos\Delta\phi_{-}\rangle\right),$

respectively. Here, the bracket means averaging over all particles of interest
in an event, and we also assume $N_{+}\approx N_{-}=M/2$ and $\langle\cos
2\Delta\phi_{+}\rangle=\langle\cos 2\Delta\phi_{-}\rangle$. After some
derivations (see A for details), we can relate $\sigma(\Delta B_{y})$ and
$\sigma(\Delta B_{x})$ to the core components of the $R_{\Psi_{2}}(\Delta S)$
correlator,

$\displaystyle\sigma^{2}(\Delta
B_{y})\approx\frac{64M^{2}}{\pi^{4}}\left(\frac{4}{9M}+1+\frac{4}{3}v_{2}\right)\sigma^{2}(\Delta
S_{\rm real}),$ (33) $\displaystyle\sigma^{2}(\Delta
B_{x})\approx\frac{64M^{2}}{\pi^{4}}\left(\frac{4}{9M}+1-\frac{4}{3}v_{2}\right)\sigma^{2}(\Delta
S^{\perp}_{\rm real}).$ (34)

According to Eqs. (20) and (22), we immediately see the impact of $\rho_{00}$
on $\sigma(\Delta B_{y})$ and $\sigma(\Delta B_{x})$, as well as their ratio,
$r_{\rm lab}$. However, it is more straightforward to define an observable
based on the difference instead of the ratio,

$\displaystyle\Delta\sigma^{2}(\Delta B)\equiv\sigma^{2}(\Delta
B_{y})-\sigma^{2}(\Delta B_{x})$ $\displaystyle\approx
c_{1}+c_{2}(3\rho_{00}-1),$ (35)

where $c_{1}$ and $c_{2}$ are constant coefficients which depend on the
spectra of $\rho$ mesons, $v_{2}^{\rho}$, and $v_{2}^{\pi}$.

Figure 7 shows the toy model simulations of $\Delta\sigma^{2}(\Delta B)$ as a
function of $\rho_{00}$ with various $v_{2}^{\rho}$ inputs. At a given
$v_{2}^{\rho}$, $\Delta\sigma^{2}(\Delta B)$ exhibits a linear dependence on
$\rho_{00}$. In the case of $v_{2}^{\rho}=0$ and $\rho_{00}=1/3$,
$\Delta\sigma^{2}(\Delta B)$ is zero as expected by Eq. 35). Note that the
global spin alignment effect could give a negative contribution to
$\Delta\sigma^{2}(\Delta B)$, if $\rho_{00}$ is smaller than $1/3$. The
function forms of Eqs. 12), (28) and (35) are very similar to each other, and
therefore these observables should have similar sensitivity to the global spin
alignment of vector mesons. Indeed, Fig. 7 looks very similar to Figs. 2 and
6.

Figure 8 shows AMPT calculations of $\Delta\sigma^{2}(\Delta B)$ as a function
of $\rho_{00}$ in Au+Au collisions at 200 GeV with the impact parameter of 8
fm. $\Delta\sigma^{2}(\Delta B)$ increases linearly with $\rho_{00}$, similar
to the toy model simulations. At $\rho_{00}=1/3$, the non-zero
$\Delta\sigma^{2}(\Delta B)$, which is a non-CME background, may come from the
positive $v_{2}^{\rho}$ and transverse momentum conservation. The slope,
$d\Delta\sigma^{2}(\Delta B)/d\rho_{00}$, could be different between the toy
model and the AMPT model, because of the different $\rho$-meson spectra.

Figure 7: Toy model simulations of $\sigma^{2}(\Delta B)$ as a function of
$\rho_{00}$ for various $v_{2}^{\rho}$ inputs. Linear fits are applied to
guide eyes. Figure 8: AMPT calculations of $\sigma_{\rho}^{2}(\Delta B)$ as a
function of $\rho_{00}$ in Au+Au collisions at 200 GeV with the impact
parameter of 8 fm. A linear fit is applied to guide eyes.

##  5 Summary

The chiral magnetic effect in heavy-ion collisions has aroused extensive
interest, but the experimental search for the CME is hindered by the
incomplete understanding of the background effects. The global spin alignment
of vector mesons can provide a sizeable contribution to the CME-sensitive
observables that involve the decay products of these vector mesons. Since the
spin alignment could arise from non-CME mechanisms, such as the spin-orbit
coupling, it presents a potential background to the CME measurements, in
addition to the commonly admitted flow background.

In this work, we have demonstrated how the globally spin-aligned $\rho$ mesons
affect the CME observables involving pions, the $\Delta\gamma_{112}$
correlator, the $R_{\Psi_{2}}(\Delta S)$ correlator, and the signed balance
functions. For each observable, we first analytically derive its qualitative
dependence on $\rho_{00}$, and then confirm the equations with a toy model as
well as the more realistic AMPT simulations. Qualitative derivations indicate
that the $\rho_{00}$ dependence originates from the anisotropic emission of
the decay products, which imitates elliptic flow in the $\rho$ rest frame. We
find that all these observables are influenced not only by elliptic flow
$v_{2}^{\rho}$, but also by the spin alignment $\rho_{00}$ of $\rho$ mesons.
By constructing new observables using the difference instead of the ratio for
$R_{\Psi_{2}}(\Delta S)$ and $r_{\rm lab}$, we have $\Delta\gamma_{112}$,
$\Delta\sigma_{R}^{2}$, and $\Delta\sigma^{2}(\Delta B)$ all manifest a
similar linear dependence on $\rho_{00}$. Unlike the flow background that is
always positive, the global spin alignment of vector mesons can give a
negative contribution if $\rho_{00}$ is smaller than 1/3, which is likely
according to data. This further warrants the inclusion of the spin alignment
effect in the background estimation, to avoid an over-subtraction of the
background.

We cannot exclude the possibility that the global spin alignment partially
stems from the CME-induced charge separation of $\pi^{+}$ and $\pi^{-}$, some
of whom later form $\rho$ mesons via coalescence. In that case, the CME tends
to give a positive contribution to the $\rho_{00}$ of $\rho$ mesons. Such a
coupling between the CME and the global spin alignment goes beyond the scope
of this work, and calls for more theoretical inputs.

## Acknowledgement

D. Shen and J. Chen are 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 under Contract No. 12147114, No.
12147101 and No. 12025501, by the Strategic Priority Research Program of
Chinese Academy of Sciences under Grant No. XDB34030000, and by the Guangdong
Major Project of Basic and Applied Basic Research No. 2020B0301030008. A.H.
Tang is supported by the US Department of Energy under Grants No. DE-
AC02-98CH10886, DE-FG02-89ER40531. G. Wang is supported by the US Department
of Energy under Grant No. DE-FG02-88ER40424.

##  Appendix A Core components of the Signed Balance Functions

Based on Eq. (4), we write the variance of $\Delta B_{y}$ as

$\displaystyle\sigma^{2}(\Delta
B_{y})\approx\frac{64M^{2}}{\pi^{4}}\sigma^{2}(x_{1}x_{2}),$ (36)

where $x_{1}=1+(2/3)\langle\cos 2\Delta\phi\rangle$ and
$x_{2}=\langle\sin\Delta\phi_{+}\rangle-\langle\sin\Delta\phi_{-}\rangle$.
Because the cosine and sine functions are orthogonal to each other, $x_{1}$
and $x_{2}$ can be treated as independent variables. Therefore, Eq. (36) can
be further expanded as

$\displaystyle\sigma^{2}(\Delta
B_{y})\approx\frac{64M^{2}}{\pi^{4}}[\sigma^{2}(x_{1})\sigma^{2}(x_{2})+\sigma^{2}(x_{1})\langle
x_{2}\rangle^{2}$ $\displaystyle+\sigma^{2}(x_{2})\langle x_{1}\rangle^{2}].$
(37)

We then calculate each term in Eq. (A),

$\displaystyle\sigma^{2}(x_{1})=\frac{8}{9M}(\frac{1}{2}+\frac{v_{4}}{2}-v_{2}^{2}),$
(38) $\displaystyle\sigma^{2}(x_{2})=\sigma^{2}(\Delta S_{\rm real}),$ (39)
$\displaystyle\langle x_{1}\rangle^{2}=(1+\frac{2}{3}v_{2})^{2},$ (40)
$\displaystyle\langle x_{2}\rangle^{2}=0.$ (41)

We ignore higher-order terms like $v_{4}$ and $v_{2}^{2}$, and reach

$\displaystyle\sigma^{2}(\Delta
B_{y})\approx\frac{64M^{2}}{\pi^{4}}\left(\frac{4}{9M}+1+\frac{4}{3}v_{2}\right)\sigma^{2}(\Delta
S_{\rm real}).$ (42)

A similar derivation can be applied to $\sigma^{2}(\Delta B_{x})$ to yield Eq.
(34).

## References

  * Kharzeev [2006] D. Kharzeev,  Parity violation in hot QCD: Why it can happen, and how to look for it,  Phys. Lett. B 633 (2006) 260–264. doi:10.1016/j.physletb.2005.11.075. arXiv:hep-ph/0406125.
  * Kharzeev et al. [1998] D. Kharzeev, R. D. Pisarski, M. H. G. Tytgat,  Possibility of spontaneous parity violation in hot QCD,  Phys. Rev. Lett. 81 (1998) 512–515. doi:10.1103/PhysRevLett.81.512. arXiv:hep-ph/9804221.
  * Kharzeev and Pisarski [2000] D. Kharzeev, R. D. Pisarski,  Pionic measures of parity and CP violation in high-energy nuclear collisions,  Phys. Rev. D 61 (2000) 111901. doi:10.1103/PhysRevD.61.111901. arXiv:hep-ph/9906401.
  * Kharzeev and Liao [2021] D. E. Kharzeev, J. Liao,  Chiral magnetic effect reveals the topology of gauge fields in heavy-ion collisions,  Nature Rev. Phys. 3 (2021) 55–63. doi:10.1038/s42254-020-00254-6. arXiv:2102.06623.
  * Voronyuk et al. [2011] V. Voronyuk, V. D. Toneev, W. Cassing, E. L. Bratkovskaya, V. P. Konchakovski, S. A. Voloshin,  (Electro-)Magnetic field evolution in relativistic heavy-ion collisions,  Phys. Rev. C 83 (2011) 054911. doi:10.1103/PhysRevC.83.054911. arXiv:1103.4239.
  * Deng and Huang [2012] W.-T. Deng, X.-G. Huang,  Event-by-event generation of electromagnetic fields in heavy-ion collisions,  Phys. Rev. C 85 (2012) 044907. doi:10.1103/PhysRevC.85.044907. arXiv:1201.5108.
  * Abelev et al. [2009] B. I. Abelev, et al. (STAR),  Azimuthal Charged-Particle Correlations and Possible Local Strong Parity Violation,  Phys. Rev. Lett. 103 (2009) 251601. doi:10.1103/PhysRevLett.103.251601. arXiv:0909.1739.
  * Adamczyk et al. [2013] L. Adamczyk, et al. (STAR),  Fluctuations of charge separation perpendicular to the event plane and local parity violation in $\sqrt{s_{NN}}=200$ GeV Au+Au collisions at the BNL Relativistic Heavy Ion Collider,  Phys. Rev. C 88 (2013) 064911. doi:10.1103/PhysRevC.88.064911. arXiv:1302.3802.
  * Adamczyk et al. [2014] L. Adamczyk, et al. (STAR),  Beam-energy dependence of charge separation along the magnetic field in Au+Au collisions at RHIC,  Phys. Rev. Lett. 113 (2014) 052302. doi:10.1103/PhysRevLett.113.052302. arXiv:1404.1433.
  * Abdallah et al. [2022] M. Abdallah, et al. (STAR),  Search for the chiral magnetic effect with isobar collisions at $\sqrt{s_{NN}}$=200 GeV by the STAR Collaboration at the BNL Relativistic Heavy Ion Collider,  Phys. Rev. C 105 (2022) 014901. doi:10.1103/PhysRevC.105.014901. arXiv:2109.00131.
  * STA [2022] Search for the Chiral Magnetic Effect in Au+Au collisions at $\sqrt{s_{{}_{\rm{NN}}}}$=27 GeV with the STAR forward Event Plane Detectors (2022). arXiv:2209.03467.
  * Abelev et al. [2013] B. Abelev, et al. (ALICE),  Charge separation relative to the reaction plane in Pb-Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV,  Phys. Rev. Lett. 110 (2013) 012301. doi:10.1103/PhysRevLett.110.012301. arXiv:1207.0900.
  * Khachatryan et al. [2017] V. Khachatryan, et al. (CMS),  Observation of charge-dependent azimuthal correlations in $p$-Pb collisions and its implication for the search for the chiral magnetic effect,  Phys. Rev. Lett. 118 (2017) 122301. doi:10.1103/PhysRevLett.118.122301. arXiv:1610.00263.
  * Sirunyan et al. [2018] A. M. Sirunyan, et al. (CMS),  Constraints on the chiral magnetic effect using charge-dependent azimuthal correlations in $p\mathrm{Pb}$ and PbPb collisions at the CERN Large Hadron Collider,  Phys. Rev. C 97 (2018) 044912. doi:10.1103/PhysRevC.97.044912. arXiv:1708.01602.
  * Acharya et al. [2018] S. Acharya, et al. (ALICE),  Constraining the magnitude of the Chiral Magnetic Effect with Event Shape Engineering in Pb-Pb collisions at $\sqrt{s_{\mathrm{NN}}}$ = 2.76 TeV,  Phys. Lett. B 777 (2018) 151–162. doi:10.1016/j.physletb.2017.12.021. arXiv:1709.04723.
  * Liao et al. [2010] J. Liao, V. Koch, A. Bzdak,  On the Charge Separation Effect in Relativistic Heavy Ion Collisions,  Phys. Rev. C 82 (2010) 054902. doi:10.1103/PhysRevC.82.054902. arXiv:1005.5380.
  * Pratt et al. [2011] S. Pratt, S. Schlichting, S. Gavin,  Effects of Momentum Conservation and Flow on Angular Correlations at RHIC,  Phys. Rev. C 84 (2011) 024909. doi:10.1103/PhysRevC.84.024909. arXiv:1011.6053.
  * Wang and Zhao [2017] F. Wang, J. Zhao,  Challenges in flow background removal in search for the chiral magnetic effect,  Phys. Rev. C 95 (2017) 051901. doi:10.1103/PhysRevC.95.051901. arXiv:1608.06610.
  * Feng et al. [2018] Y. Feng, J. Zhao, F. Wang,  Responses of the chiral-magnetic-effect–sensitive sine observable to resonance backgrounds in heavy-ion collisions,  Phys. Rev. C 98 (2018) 034904. doi:10.1103/PhysRevC.98.034904. arXiv:1803.02860.
  * Tang [2020] A. H. Tang,  Probe chiral magnetic effect with signed balance function,  Chin. Phys. C 44 (2020) 054101. doi:10.1088/1674-1137/44/5/054101. arXiv:1903.04622.
  * Wu et al. [2022] W.-Y. Wu, et al.,  Global constraint on the magnitude of anomalous chiral effects in heavy-ion collisions (2022). arXiv:2211.15446.
  * Voloshin [2004] S. A. Voloshin,  Parity violation in hot QCD: How to detect it,  Phys. Rev. C 70 (2004) 057901. doi:10.1103/PhysRevC.70.057901. arXiv:hep-ph/0406311.
  * Magdy et al. [2018] N. Magdy, S. Shi, J. Liao, N. Ajitanand, R. A. Lacey,  New correlator to detect and characterize the chiral magnetic effect,  Phys. Rev. C 97 (2018) 061901. doi:10.1103/PhysRevC.97.061901. arXiv:1710.01717.
  * Choudhury et al. [2022] S. Choudhury, et al.,  Investigation of experimental observables in search of the chiral magnetic effect in heavy-ion collisions in the STAR experiment *,  Chin. Phys. C 46 (2022) 014101. doi:10.1088/1674-1137/ac2a1f. arXiv:2105.06044.
  * Liang and Wang [2005a] Z.-T. Liang, X.-N. Wang,  Globally polarized quark-gluon plasma in non-central A+A collisions,  Phys. Rev. Lett. 94 (2005a) 102301. doi:10.1103/PhysRevLett.94.102301. arXiv:nucl-th/0410079, [Erratum: Phys.Rev.Lett. 96, 039901 (2006)].
  * Liang and Wang [2005b] Z.-T. Liang, X.-N. Wang,  Spin alignment of vector mesons in non-central A+A collisions,  Phys. Lett. B 629 (2005b) 20–26. doi:10.1016/j.physletb.2005.09.060. arXiv:nucl-th/0411101.
  * Becattini et al. [2013] F. Becattini, L. Csernai, D. J. Wang,  $\Lambda$ polarization in peripheral heavy ion collisions,  Phys. Rev. C 88 (2013) 034905. doi:10.1103/PhysRevC.88.034905. arXiv:1304.4427, [Erratum: Phys.Rev.C 93, 069901 (2016)].
  * Yang et al. [2018] Y.-G. Yang, R.-H. Fang, Q. Wang, X.-N. Wang,  Quark coalescence model for polarized vector mesons and baryons,  Phys. Rev. C 97 (2018) 034917. doi:10.1103/PhysRevC.97.034917. arXiv:1711.06008.
  * Sheng et al. [2020] X.-L. Sheng, L. Oliva, Q. Wang,  What can we learn from the global spin alignment of $\phi$ mesons in heavy-ion collisions?,  Phys. Rev. D 101 (2020) 096005. doi:10.1103/PhysRevD.101.096005. arXiv:1910.13684.
  * Xia et al. [2021] X.-L. Xia, H. Li, X.-G. Huang, H. Zhong Huang,  Local spin alignment of vector mesons in relativistic heavy-ion collisions,  Phys. Lett. B 817 (2021) 136325. doi:10.1016/j.physletb.2021.136325. arXiv:2010.01474.
  * Gao [2021] J.-H. Gao,  Helicity polarization in relativistic heavy ion collisions,  Phys. Rev. D 104 (2021) 076016. doi:10.1103/PhysRevD.104.076016. arXiv:2105.08293.
  * Müller and Yang [2022] B. Müller, D.-L. Yang,  Anomalous spin polarization from turbulent color fields,  Phys. Rev. D 105 (2022) L011901. doi:10.1103/PhysRevD.105.L011901. arXiv:2110.15630.
  * Sheng et al. [2022] X.-L. Sheng, L. Oliva, Q. Wang,  Erratum: What can we learn from the global spin alignment of $\phi$ mesons in heavy-ion collisions? [phys. rev. d 101, 096005 (2020)],  Phys. Rev. D 105 (2022) 099903. URL: https://link.aps.org/doi/10.1103/PhysRevD.105.099903. doi:10.1103/PhysRevD.105.099903.
  * Sheng et al. [2020] X.-L. Sheng, Q. Wang, X.-N. Wang,  Improved quark coalescence model for spin alignment and polarization of hadrons,  Phys. Rev. D 102 (2020) 056013. doi:10.1103/PhysRevD.102.056013. arXiv:2007.05106.
  * Sheng et al. [2022a] X.-L. Sheng, L. Oliva, Z.-T. Liang, Q. Wang, X.-N. Wang,  Spin alignment of vector mesons in heavy-ion collisions (2022a). arXiv:2205.15689.
  * Sheng et al. [2022b] X.-L. Sheng, L. Oliva, Z.-T. Liang, Q. Wang, X.-N. Wang,  Relativistic spin dynamics for vector mesons (2022b). arXiv:2206.05868.
  * Adamczyk et al. [2017] L. Adamczyk, et al. (STAR),  Global $\Lambda$ hyperon polarization in nuclear collisions: evidence for the most vortical fluid,  Nature 548 (2017) 62–65. doi:10.1038/nature23004. arXiv:1701.06657.
  * Adam et al. [2018] J. Adam, et al. (STAR),  Global polarization of $\Lambda$ hyperons in Au+Au collisions at $\sqrt{s_{{}_{NN}}}$ = 200 GeV,  Phys. Rev. C 98 (2018) 014910. doi:10.1103/PhysRevC.98.014910. arXiv:1805.04400.
  * Acharya et al. [2020] S. Acharya, et al. (ALICE),  Global polarization of $\Lambda\bar{\Lambda}$ hyperons in Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 and 5.02 TeV,  Phys. Rev. C 101 (2020) 044611. doi:10.1103/PhysRevC.101.044611. arXiv:1909.01281.
  * Abdallah et al. [2021] M. S. Abdallah, et al. (STAR),  Global $\Lambda$-hyperon polarization in Au+Au collisions at $\sqrt{s_{NN}}$=3 GeV,  Phys. Rev. C 104 (2021) L061901. doi:10.1103/PhysRevC.104.L061901. arXiv:2108.00044.
  * Adam et al. [2021] J. Adam, et al. (STAR),  Global Polarization of $\Xi$ and $\Omega$ Hyperons in Au+Au Collisions at $\sqrt{s_{NN}}$ = 200 GeV,  Phys. Rev. Lett. 126 (2021) 162301. doi:10.1103/PhysRevLett.126.162301. arXiv:2012.13601.
  * Abelev et al. [2008] B. I. Abelev, et al. (STAR),  Spin alignment measurements of the K*0(892) and phi (1020) vector mesons in heavy ion collisions at $\sqrt{s_{NN}}$=200 GeV,  Phys. Rev. C 77 (2008) 061902. doi:10.1103/PhysRevC.77.061902. arXiv:0801.1729.
  * Acharya et al. [2020] S. Acharya, et al. (ALICE),  Evidence of Spin-Orbital Angular Momentum Interactions in Relativistic Heavy-Ion Collisions,  Phys. Rev. Lett. 125 (2020) 012301. doi:10.1103/PhysRevLett.125.012301. arXiv:1910.14408.
  * Abdallah et al. [2022] M. Abdallah, et al. (STAR),  Observation of Global Spin Alignment of $\phi$ and $K^{*0}$ Vector Mesons in Nuclear Collisions (2022). arXiv:2204.02302.
  * ALI [2022] Measurement of the J/$\psi$ polarization with respect to the event plane in Pb-Pb collisions at the LHC (2022). arXiv:2204.10171.
  * Schilling et al. [1970] K. Schilling, P. Seyboth, G. E. Wolf,  On the Analysis of Vector Meson Production by Polarized Photons,  Nucl. Phys. B 15 (1970) 397–412. doi:10.1016/0550-3213(70)90070-2, [Erratum: Nucl.Phys.B 18, 332 (1970)].
  * Abelev et al. [2009] B. I. Abelev, et al. (STAR),  Systematic Measurements of Identified Particle Spectra in $pp,d^{+}$ Au and Au+Au Collisions from STAR,  Phys. Rev. C 79 (2009) 034909. doi:10.1103/PhysRevC.79.034909. arXiv:0808.2041.
  * Adams et al. [2004] J. Adams, et al. (STAR),  Rho0 production and possible modification in Au+Au and p+p collisions at S(NN)**1/2 = 200-GeV,  Phys. Rev. Lett. 92 (2004) 092301. doi:10.1103/PhysRevLett.92.092301. arXiv:nucl-ex/0307023.
  * Sjostrand et al. [2006] T. Sjostrand, S. Mrenna, P. Z. Skands,  PYTHIA 6.4 Physics and Manual,  JHEP 05 (2006) 026\. doi:10.1088/1126-6708/2006/05/026. arXiv:hep-ph/0603175.
  * Lan et al. [2018] S. Lan, Z.-W. Lin, S. Shi, X. Sun,  Effects of finite coverage on global polarization observables in heavy ion collisions,  Phys. Lett. B 780 (2018) 319–324. doi:10.1016/j.physletb.2018.02.076. arXiv:1710.03895.
  * Shen et al. [2021] D. Shen, J. Chen, Z.-W. Lin,  The effect of hadronic scatterings on the measurement of vector meson spin alignments in heavy-ion collisions,  Chin. Phys. C 45 (2021) 054002. doi:10.1088/1674-1137/abe763. arXiv:2102.05266.