Vector meson’s spin alignments in high energy reactions

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. For the measurement of global spin alignment, the spin quantization direction is taken as the direction of global orbital angular momentum, i.e., $\vec{\varepsilon}_{m=0}=\hat{y}$, while other polarization vectors are determined by conditions $\vec{\varepsilon}_{m}^{\,*}\cdot\vec{\varepsilon}_{m^{\prime}}=\delta_{mm^{\prime}}$ and $\vec{\varepsilon}_{m=\pm 1}^{\,*}=-\vec{\varepsilon}_{m=\mp 1}$. Another widely used choice is the direction of meson’s three-momentum in the laboratory frame, $\vec{\varepsilon}_{m=0}=\vec{p}/|\vec{p}|$, corresponding to the helicity polarization. The spin density matrix for different choices of spin quantization directions are related by the following transformation in the spin space,

$$\hat{\rho}_{V}=R\,\hat{\rho}^{\prime}_{V}\,R^{\dagger},$$

(22)

where $\hat{\rho}_{V}$ and $\hat{\rho}^{\prime}_{V}$ are two spin density matrices corresponding to two set of polarization vectors $\vec{\varepsilon}_{m=0,\pm 1}$ and $\vec{\varepsilon}^{\,\prime}_{m=0,\pm 1}$, respectively. The transformation matrix $R$ is a $3\times 3$ matrix in the spin space with elements given by the inner product of polarization vectors, $R_{mm^{\prime}}\equiv\vec{\varepsilon}^{\,*}_{m}\cdot\vec{\varepsilon}^{\,\prime}_{m^{\prime}}$.

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.

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,

$\displaystyle A_{m}(\vec{p};\lambda_{1}\lambda_{2})=$

$\displaystyle\langle\vec{p};\lambda_{1}\lambda_{2}|\hat{U}|j_{A}m_{A}\rangle,$

(23)

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,

	$\displaystyle A_{m}(\vec{p};\lambda_{1}\lambda_{2})=$	$\displaystyle\langle\vec{p};\lambda_{1}\lambda_{2}|M_{A};j_{A}m_{A};\lambda_{1}\lambda_{2}\rangle$
		$\displaystyle\times\langle M_{A};j_{A}m_{A};\lambda_{1}\lambda_{2}|\hat{U}|j_{A}m_{A}\rangle.$		(24)

The space rotation invariance demands that the helicity amplitude

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

(25)

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

	$\displaystyle A_{m}(\vec{p};\lambda_{1}\lambda_{2})$
	$\displaystyle=\langle\vec{p};\lambda_{1}\lambda_{2}|M_{A};j_{A}m_{A};\lambda_{1}\lambda_{2}\rangle H_{A}(\lambda_{1},\lambda_{2}).$		(26)

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

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

(27)

Hence, we obtain the inner product as

	$\displaystyle\langle\vec{p};\lambda_{1}\lambda_{2}|M_{A};j_{A}m_{A};\lambda_{1}\lambda_{2}\rangle$
	$\displaystyle=\sqrt{\frac{2j_{A}+1}{4\pi}}d_{m_{A}\lambda}^{j_{A}*}(\theta)e^{i(m_{A}-\lambda)\varphi},$		(28)

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 (27) 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

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

(29)

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

	$\displaystyle W(\theta,\varphi)=$	$\displaystyle N\sum_{\lambda_{1},\lambda_{2}}\langle\vec{p};\lambda_{1},\lambda_{2}|\hat{\rho}_{12}|\vec{p};\lambda_{1},\lambda_{2}\rangle$
	$\displaystyle=$	$\displaystyle N\sum_{\lambda_{1},\lambda_{2};m_{A},m^{\prime}_{A}}|H_{A}(\lambda_{1},\lambda_{2})|^{2}\rho^{A}_{m_{A}m^{\prime}_{A}}$
		$\displaystyle\times e^{i(m_{A}-m^{\prime}_{A})\varphi}d_{m_{A}\lambda}^{j_{A}*}(\theta)d_{m^{\prime}_{A}\lambda}^{j_{A}}(\theta),$		(30)

where $\rho^{A}_{m_{A}m^{\prime}_{A}}=\langle m_{A}|\hat{\rho}^{A}|m^{\prime}_{A}\rangle$ denotes elements of $\hat{\rho}^{A}$ and $N$ is the normalization constant.

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

$\displaystyle W(\theta,\varphi)=$

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

(31)

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

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

(32)

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. (31) 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

	$\displaystyle W(\theta,\varphi)=$	$\displaystyle\frac{3}{8\pi}\Bigl{\{}(1-\rho^{A}_{00})+(3\rho^{A}_{00}-1)\cos^{2}\theta$
		$\displaystyle-{2}\sin^{2}\theta\bigl{[}\cos 2\varphi{\rm Re}\rho^{A}_{1-1}-\sin 2\varphi{\rm Im}\rho^{A}_{1-1}\bigr{]}$
		$\displaystyle-\sqrt{2}\sin 2\theta\bigl{[}\cos\varphi({\rm Re}\rho^{A}_{10}-{\rm Re}\rho^{A}_{-10})$
		$\displaystyle-\sin\varphi({\rm Im}\rho^{A}_{10}+{\rm Im}\rho^{A}_{-10})\bigr{]}\Bigr{\}},$		(33)

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

$\displaystyle W(\theta)=\frac{3}{4}$

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

(34)

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

	$\displaystyle W(\theta,\varphi)=$	$\displaystyle\frac{3}{8\pi(1+\rho^{A}_{00})}\Bigl{\{}1+\lambda_{\theta}\cos^{2}\theta$
		$\displaystyle+\lambda_{\varphi}\sin^{2}\theta\cos 2\varphi+\lambda_{\theta\varphi}\sin 2\theta\sin 2\varphi$
		$\displaystyle+\lambda^{\perp}_{\varphi}\sin^{2}\theta\sin 2\varphi+\lambda^{\perp}_{\theta\varphi}\sin 2\theta\sin\varphi\Bigr{\}},$		(35)

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

	$\displaystyle\lambda_{\theta}=\frac{1-3\rho^{A}_{00}}{1+\rho^{A}_{00}},$		(36)
	$\displaystyle\lambda_{\varphi}=\frac{2{\rm Re}\rho^{A}_{1-1}}{1+\rho^{A}_{00}},$		(37)
	$\displaystyle\lambda_{\theta\varphi}=\frac{\sqrt{2}{\rm Re}(\rho^{A}_{10}-\rho^{A}_{-10})}{1+\rho^{A}_{00}},$		(38)
	$\displaystyle\lambda_{\varphi}^{\perp}=-\frac{2{\rm Im}\rho^{A}_{1-1}}{1+\rho^{A}_{00}},$		(39)
	$\displaystyle\lambda_{\theta\varphi}^{\perp}=-\frac{\sqrt{2}{\rm Im}(\rho^{A}_{10}+\rho^{A}_{-10})}{1+\rho^{A}_{00}}.$		(40)

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. (31-35) (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. (31) and (33) are involved.

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 as $B$ 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.

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.

New measurements of the vector meson’s spin alignment in heavy-ion collisions are different from conventional studies, where a new polarization axis along the nucleus-nucleus system’s orbital angular momentum is defined Liang:2004ph ; Liang:2004xn , and this is the so-called global spin alignment. In experiment, the quantization axis is determined by the normal of the reaction plane, which can be reconstructed by using the charge particle momentum distribution collected in the detector STAR:2022fan . Particles of interest, for example, the vector-meson $\phi$ and $K^{*0}$ are observed by paring of their decay daughters ($K^{\pm}$, $\pi^{\pm}$) with subtraction of the combinatorial background. Then the polar angle distribution of Eq. (34) is analyzed, and the $\rho_{00}$ is extracted after correction for detection efficiency and acceptance STAR:2022fan .

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.

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 Sec. IV.2.

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

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

(43)

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,

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

(44)

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

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

(45)

which leads to

	$\displaystyle\rho^{V}_{m^{\prime}m}=$	$\displaystyle\sum_{m_{1},m_{2},m_{1}^{\prime},m_{2}^{\prime}}\rho^{q_{1}\bar{q}_{2}}_{m_{1}^{\prime}m_{2}^{\prime},m_{1}m_{2}}$
		$\displaystyle\times\langle j_{V}m^{\prime}|m^{\prime}_{1}m^{\prime}_{2}\rangle\langle m_{1}m_{2}|j_{V}m\rangle,$		(46)

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 ,

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

(47)

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

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

(48)

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

From Eqs. (47) and (48), 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. (47) should be replaced by $\langle P_{q_{1}}P_{\bar{q}_{2}}\rangle$, so we have

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

(49)

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

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

(50)

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. (49) 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 .