Relativistic spin dynamics for vector mesons

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

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

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

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

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

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

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

(1)

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

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

(2)

where the differential operator is defined as

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

(3)

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

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

(4)

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

The free vector field can be quantized as

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

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

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

(6)

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

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

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

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

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

(8)

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

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

(9)

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

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

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

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

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

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

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

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

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

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

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

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

and

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

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

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

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

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

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

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

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

(18)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(25)

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

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

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

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

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

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

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

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

(28)

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

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

(29)

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

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

(30)

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

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

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

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

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