Spin density matrix for neutral $\rho$ mesons in a pion gas in linear response theory

The spin and orbital angular momentum are intrinsically connected and can be converted to each other as demonstrated in the Barnett effect (Barnett:1935, ) and Einstein-de-Haas effect (dehaas:1915, ) in materials. In non-central heavy-ion collisions, a huge orbital angular momentum is generated and partially transferred to the strongly interacting matter in the form of particle’s spin polarization along the reaction plane through spin-orbit couplings (Liang:2004ph, ). This phenomenon is called the global spin polarization as it is with respect to the reaction plane which is fixed for all particles in one event (Liang:2004ph, ; Voloshin:2004ha, ; Liang:2004xn, ; Betz:2007kg, ; Gao:2007bc, ; Becattini:2007sr, ). The global spin polarization of $\Lambda$ and $\overline{\Lambda}$ hyperons was first observed and measured in the STAR experiment (STAR:2017ckg, ; STAR:2018gyt, ), and then measured in experiments of HADES (HADES:2022enx, ) and ALICE (ALICE:2021pzu, ). Since its first observation (STAR:2017ckg, ; STAR:2018gyt, ), there has been tremendous advance in the study of the global spin polarization in theoretical models (Karpenko:2016jyx, ; Li:2017slc, ; Xie:2017upb, ; Sun:2017xhx, ; Baznat:2017jfj, ; Shi:2017wpk, ; Wei:2018zfb, ; Fu:2020oxj, ; Ryu:2021lnx, ; Deng:2021miw, ; Wu:2022mkr, ), see Refs. (Wang:2017jpl, ; Florkowski:2018fap, ; Huang:2020dtn, ; Gao:2020lxh, ; Gao:2020vbh, ; Liu:2020ymh, ; Becattini:2022zvf, ; Hidaka:2022dmn, ; Becattini:2024uha, ) for recent reviews on this topic.

The spin polarization of hyperons can be measured through their parity-breaking weak decays (Bunce:1976yb, ). Vector mesons mostly decay through parity-conserved strong interaction, so there is no way to measure the spin polarization of vector mesons. The spin states of the vector meson can be described by the spin density matrix $\rho_{\lambda_{1}\lambda_{2}}$, where $\lambda_{1}$ and $\lambda_{2}=0,\pm 1$ denote the spin states. The spin density matrix is normalized $\mathrm{tr}\rho\equiv\sum_{\lambda}\rho_{\lambda\lambda}=1$. The $\rho_{00}$ element can be measured in experiments through the polar angle distribution of decay products (Schilling:1969um, ; Liang:2004xn, ; Yang:2017sdk, ; Tang:2018qtu, ). If $\rho_{00}\neq 1/3$, the spin-0 state of the vector meson is not equally populated as $\lambda=\pm 1$ states, which is called the spin alignment. Specifically, $\rho_{00}<1/3$ means the field vector (not the spin vector) is more aligned to the direction perpendicular to the spin quantization direction, while $\rho_{00}>1/3$ means the field vector is more aligned to the spin quantization direction. The spin alignment with respect to the reaction plane is called the global spin alignment and was also predicted in 2004-2005 (Liang:2004xn, ). The global spin alignment of $\phi$ and $K^{0*}$ mesons has recently been measured by the STAR collaboration in Au+Au collisions (STAR:2022fan, ), where a large deviation of $\rho_{00}$ for $\phi$ mesons from 1/3 is observed at lower collision energies, but there is no significant deviation of $\rho_{00}$ for $K^{0*}$ mesons within errors.

A number of sources can contribute to the spin alignment of $\phi$ mesons but not enough to explain such a large effect (Yang:2017sdk, ; Xia:2020tyd, ; Gao:2021rom, ; Muller:2021hpe, ; Kumar:2023ghs, ). It was proposed that a large spin alignment of $\phi$ mesons may come from the local correlation/fluctuation of strong vector fields that are coupled to s and $\bar{\mathrm{s}}$ quarks (Sheng:2019kmk, ). Initially a nonrelativistic quark coalescence model (Yang:2017sdk, ) was used to calculate the spin density matrix of $\phi$ mesons with s and $\bar{\mathrm{s}}$ quarks being polarized by strong vector fields (Sheng:2020ghv, ). Then the nonrelativistic quark coalescence model was generalized to a relativistic one (Sheng:2022ffb, ; Sheng:2022wsy, ). The relativistic model is based on Wigner functions and spin kinetic equations for vector mesons. The model incorporated with strong vector fields can successfully describe the experimental data for the global spin alignment of $\phi$ mesons (Sheng:2022ffb, ; Sheng:2022wsy, ; Sheng:2023urn, ). For recent reviews of the topic, the readers may look at Refs. (Chen:2023hnb, ; Sheng:2023chinphyb, ; Chen:2024bik, ; Chen:2024afy, ).

Recently, the thermal shear contribution to $\phi$ meson’s spin alignment has been studied in the linear response theory (Li:2022vmb, ; Dong:2024nxj, ). In this paper, we will apply the same method to another vector meson, the neutral rho meson or $\rho^{0}$. The spin density matrix of $\rho^{0}$ may serve as a background for the chiral magnetic effect (CME) (Kharzeev:2004ey, ; Kharzeev:2007jp, ; Fukushima:2008xe, ), because the anisotropic angular distribution of its decay daughters (pions) provides a non-vanishing contribution to the $\gamma$ correlator (Voloshin:2004vk, ; STAR:2013ksd, ; STAR:2013zgu, ; Wang:2016iov, ), see Refs. (Tang:2019pbl, ; Shen:2022gtl, ) for quantitative analyses. In Ref. (Yin:2024dnu, ) by some of us, the evolution of $\rho^{0}$ mesons in a pion gas was studied by the spin kinetic or Boltzmann equation for the on-shell spin dependent distribution. The results show that the spin alignment of $\rho^{0}$ mesons is damped rapidly in the pion gas due to $\rho\pi\pi$ interaction and that $\rho_{00}$ and $\mathrm{Re}\rho_{-1,1}$ have a significant contribution to the $\gamma$ correlator for CME. In this paper, we will consider the off-shell contribution as well as the thermal shear contribution to $\rho_{00}$ and $\mathrm{Re}\rho_{-1,1}$ by assuming thermal spin distributions for $\rho^{0}$. We will treat the thermal shear tensor as a perturbation from local equilibrium and apply the Kubo formula (Zubarev_1979, ; Hosoya:1983id, ; Becattini:2019dxo, ) to calculate the linear response of the spin density matrix for $\rho^{0}$ mesons as done for $\phi$ mesons in Refs. (Li:2022vmb, ; Dong:2024nxj, ). The spectral function of $\rho^{0}$ mesons is derived from $\rho\pi\pi$ and $\rho\rho\pi\pi$ interaction in the chiral effective theory (Gale:1990pn, ). Finally, we present numerical results for $\rho_{00}$ and $\mathrm{Re}\rho_{-1,1}$ of $\rho^{0}$ mesons using the simulation results for the thermal shear tensor from the hydrodynamical model (Pang:2018zzo, ; Wu:2021fjf, ; Wu:2022mkr, ).

The paper is organized as follows. In Sec. II, we briefly introduce the Green’s functions for vector mesons and pseudoscalar mesons in closed-time-path (CTP) formalism. In Sec. (III), we define the Wigner function and the matrix valued spin dependent distribution (MVSD) for the $\rho^{0}$ meson which is proportional to its spin density matrix. In Sec. (IV), we use the Dyson-Schwinger equation to derive the spectral function of the $\rho^{0}$ meson. We also derive the explicit form of the Wigner function without nonlocal terms. In Sec. (V), we obtain the nonlocal correction to the Wigner function proportional to the thermal shear tensor through the Kubo formula in the linear response theory. The numerical results are presented in Sec. (VI) and a summary of results and the conclusion are given in the last section.

By convention, the metric tensor is $g^{\mu\nu}=\mathrm{diag}(1,-1,-1,-1)$, four-momenta are denoted as $p^{\mu}=(p_{0},\mathbf{p})$, $p_{\mu}=(p_{0},-\mathbf{p})$, where $\mu,\nu=$0,1,2,3, and Greek letters and Latin letters represent space-time components of four-vectors and space components of three-vectors respectively. For notational clarification, we use $G^{\mu\nu}$ and $p$ for the two-point Green’s function and momentum for the $\rho^{0}$ meson, while $S$ and $k$ for the two-point Green’s function and momentum for the pseudoscalar meson $\pi^{\pm}$.

The closed-time-path (CTP) or Schwinger-Keldysh formalism in quantum field theory is an effective method to describe equilibrium and non-equilibrium physics in many-body systems (Martin:1959jp, ; Keldysh:1964ud, ; Chou:1984es, ; Blaizot:2001nr, ; Wang:2001dm, ; Berges:2004yj, ; Cassing:2008nn, ; Crossley:2015evo, ). The two-point Green’s functions on the CTP for the spin-1 vector meson ($\rho^{0}$) and spin-0 pseudoscalar meson $(\pi^{\pm})$ are defined as

where $T_{C}$ denotes the time-ordered operator on the CTP, and $A^{\mu}$ and $\phi$ are the real vector field and complex scalar field respectively, which can be quantized as

where $p=(E_{p}^{\rho},\mathbf{p})$ and $k=(E_{k}^{\pi},\mathbf{k})$ are on-shell momenta for $\rho$ and $\pi$ with $E_{p}^{\rho}=\sqrt{\mathbf{p}^{2}+m_{\rho}^{2}}$ and $E_{k}^{\pi}=\sqrt{\mathbf{k}^{2}+m_{\pi}^{2}}$, $\lambda$ denotes the spin state, and $\epsilon^{\mu}(\lambda,{\bf p})$ is the spin polarization vector

where $\boldsymbol{\epsilon}_{\lambda}$ is the spin polarization three-vector in the particle’s rest frame given by

We have chosen $+y$ to be the spin quantization direction. The polarization vector satisfies

Note that in the above equation, $p^{\mu}$ is an on-shell momentum, so we put the index “on” to the momentum variable in the projector $\Delta^{\mu\nu}(p_{\mathrm{on}})$ to distinguish it from the off-shell projector $\Delta^{\mu\nu}(p)$ which will be used later. We see in Eq. (2) that we use $G$ and $p$ to denote the Green’s function and momentum for $\rho^{0}$ respectively, while we use $S$ and $k$ to denote the Green’s function and momentum for $\pi^{\pm}$ respectively.

In the CTP formalism, $G_{\mathrm{CTP}}^{\mu\nu}$ in Eq. (1) has four components

depending on whether $x_{1}$ and $x_{2}$ are on the positive or negative time branch. One can verify that all four components satisfy the identity $G_{F}^{\mu\nu}+G_{\overline{F}}^{\mu\nu}=G_{<}^{\mu\nu}+G_{>}^{\mu\nu}$, so only three of them are independent which one can choose the retarded, advanced and correlation Green’s functions $G_{R}^{\mu\nu}$, $G_{A}^{\mu\nu}$ and $G_{C}^{\mu\nu}$ as

The four components of the Green’s function for the complex scalar field can be defined similarly. Other operators can also be defined on the CTP, obeying the relations similar to Eq. (7) as

where $O$ is any operator defined on the CTP.

In order to define the matrix valued spin dependent distribution (MVSD) for the vector meson in phase space, we introduce the Wigner transformation for the two-point Green’s function that defines the Wigner function

For free particles, the Wigner function has the form

where the index “(0)” denotes the leading order of $\hbar$, and $f_{\lambda_{1}\lambda_{2}}^{(0)}(x,{\bf p})$ is the MVSD at leading order defined as

which satisfies $f_{\lambda_{1}\lambda_{2}}^{(0)*}(x,{\bf p})=f_{\lambda_{2}\lambda_{1}}^{(0)}(x,{\bf p})$. Note that the MVSD can be decomposed into the scalar (trace), polarization ($P_{i}$) and tensor ($T_{ij}$) parts as (Sheng:2023chinphyb, ; Becattini:2024uha, )

where $i,j=1,2,3$, $\mathrm{Tr}(f^{(0)})=\sum_{\lambda}f_{\lambda\lambda}^{(0)}$, and $\Sigma_{i}$ and $\Sigma_{ij}$ are traceless matrices defined in Eq. (11) of Ref. (Dong:2023cng, ). The on-shell part of the Wigner function $W^{\mu\nu}(x,\mathbf{p})$ can be obtained by integration of $(p_{0}/\pi)G_{(0)<}^{\mu\nu}(x,p)$ over $p_{0}=\left[0,\infty\right)$. Using the decomposition (12) $W^{\mu\nu}(x,\mathbf{p})$ can be decomposed into the scalar ($\mathcal{S}$), polarization ($G^{[\mu\nu]}$) and tensor ($\mathcal{T}^{\mu\nu}$) parts as

where $\mathcal{S}$, $G^{[\mu\nu]}$ and $\mathcal{T}^{\mu\nu}$ are defined in Eqs. (14) and (15) of Ref. (Dong:2023cng, ).

According to Eq. (10), the MVSD can be inversely obtained from the Green’s function as

We assume that above relations hold at any order

where we have generalized the polarization (field) vector $\epsilon^{\mu}(\lambda,{\bf p})$ to the off-shell four-momentum $p$

One can check that $\epsilon^{\mu}(\lambda,p)$ satisfies

In this paper we use the off-shell field vector, which is different from our previous work in Ref. (Dong:2024nxj, ), because the off-shell effect for the $\rho^{0}$ meson is much more significant than the $\phi$ meson and the expansion in powers of $(p^{0}-E_{p})$ for $\rho^{0}$ does not work well.

In order to calculate the spin density matrix $\rho_{\lambda_{1}\lambda_{2}}$ which is the normalized $f_{\lambda_{1}\lambda_{2}}$, $\rho_{\lambda_{1}\lambda_{2}}\equiv f_{\lambda_{1}\lambda_{2}}/\mathrm{Tr}f$, from the Wigner function, we define the projector

Using Eq. (15) and the above expression for $L_{\mu\nu}$, we obtain

So the deviation of $\rho_{\lambda_{1}\lambda_{2}}$ from its equilibrium value (without polarization) $(1/3)\delta_{\lambda_{1}\lambda_{2}}$ can be written as

Since $\rho_{00}-1/3$ and $\mathrm{Re}\rho_{-1,1}$ are most relevant to the $\gamma$ correlator in search for the CME signal (Yin:2024dnu, ), we will focus on $\delta\rho_{00}$ and $\mathrm{Re}\delta\rho_{-1,1}$ in this paper.

In order to calculate $\rho_{\lambda_{1}\lambda_{2}}$ in Eq. (20), we will evaluate the Green’s function $G_{<}^{\mu\nu}(x,p)$ by expanding $G_{<}^{\mu\nu}(x,p)$ in powers of external sources: the leading order (LO) contribution $G_{<,\mathrm{LO}}^{\mu\nu}(x,p)$ without space-time derivatives and the next-to-leading order (NLO) contribution $G_{<,\mathrm{NLO}}^{\mu\nu}(x,p)$ with space-time derivatives as the linear response correction.

In this section, we will derive the spectral function for the $\rho^{0}$ meson from the Dyson-Schwinger equation for the two-point Green’s function at the leading order. The $\rho\pi\pi$ vertex is taken from the chiral effective Lagrangian.