text:
\section{Introduction}
Recent observations of global and local spin polarization of hadrons in relativistic heavy ion collisions \cite{STAR:2017ckg,STAR:2019erd,ALICE:2019onw} have motivated a number of theoretical studies to understand the origin of spin polarization and spin transport of partons in quark gluons plasmas (QGP). It was originally proposed that the large angular momentum created in peripheral collisions could lead to the spin polarization of partons through the spin-orbit interaction and later inherited by the spin polarization of hadrons \cite{Liang:2004ph,Liang:2004xn}. In global thermal equilibrium, the spin polarization of hadrons is dictated by thermal vorticity \cite{Becattini2013a,Fang:2016vpj}. This theoretical description remarkably matches the measurements of global spin polarization of $\Lambda$ hyperons \cite{Karpenko:2016jyx,Li:2017slc,Xie:2017upb,Wei:2018zfb,Ryu:2021lnx}. Nevertheless, the contribution from solely thermal vorticity \cite{Becattini:2017gcx,Xia:2018tes} disagrees with the later measured local spin polarization \cite{STAR:2019erd}, which implies further corrections beyond the global equilibrium condition need to be considered. Several corrections in \cite{Hidaka:2017auj,Liu:2020dxg,Liu:2021uhn,Fu:2021pok,Becattini:2021suc,Becattini:2021iol,Yi:2021ryh,Florkowski:2021xvy} and out of local equilibrium \cite{Hidaka:2017auj,Hidaka:2018ekt,Yang:2018lew,Shi:2020htn,Fang:2022ttm,Wang:2022yli,Weickgenannt:2022zxs,Weickgenannt:2022qvh,Bhadury:2020puc,Bhadury:2020cop,Buzzegoli:2022kqx,Bhadury:2022ulr} have been recently studied, while these contributions mostly come from gradients of hydrodynamic variables such as the thermal-shear correction \cite{Hidaka:2017auj,Liu:2021uhn,Becattini:2021suc} or the chemical-potential gradient \cite{Hidaka:2017auj,Liu:2020dxg}. See Ref.~\cite{Becattini:2022zvf} for a recent review and more references for the spin polarization in heavy ion collisions.

In addition to the observations of spin polarization of hadrons, there have been further measurements for spin alignment of vector mesons characterized by the deviation of the longitudinal ($00$) component of the spin density matrix $\rho_{00}$ from $1/3$ \cite{ALICE:2019aid,STAR:2022fan,ALICE:2022sli}. As inferred by the spin coalescence model \cite{Liang:2004xn,Yang:2017sdk}, the unexpectedly large deviation observed in experiments implies strong spin polarization of the composite quark and antiquark or more precisely their spin correlation compared with those extracted from the spin polarization of $\Lambda$ hyperons. Moreover, there exist both quantitative and qualitative differences for the spin alignment between distinct flavors and different collision energies. For example, a value $\rho_{00}<1/3$ is observed for both $\phi$ and $K^{*0}$ mesons with small transverse momenta at LHC energies \cite{ALICE:2019aid}, whereas $\rho_{00}>1/3$ for $\phi$ and $\rho_{00}\approx 1/3$ for $K^{*0}$ were found for global spin alignment at RHIC \cite{STAR:2022fan}.
Various theoretical mechanisms \cite{Sheng:2019kmk,Sheng:2020ghv,Xia:2020tyd,Muller:2021hpe,Yang:2021fea,Goncalves:2021ziy,Sheng:2022wsy,Li:2022neh},beyond the contributions from hydrodynamic gradients, have been proposed as one may not expect that the collision energy and flavor dependence could be simultaneously explained by a single effect. At this time the puzzle remains unsolved.

In Refs.~\cite{Muller:2021hpe,Yang:2021fea}, using quantum kinetic theory (QKT) for the coupled vector and axial vecotor current evolution of spin-$1/2$ massless \cite{Son:2012wh,Stephanov:2012ki,Chen:2012ca,Hidaka:2016yjf} and massive fermions \cite{Gao:2019znl,Weickgenannt:2019dks,Hattori:2019ahi,Wang:2019moi,Yang:2020hri,Wang:2020pej,Weickgenannt:2020aaf} in phase space (see also \cite{Hidaka:2022dmn} for a recent review and references), two of us proposed that the turbulent color fields from weakly coupled anisotropic QGP could potentially result in $\rho_{00}<1/3$ for spin alignment. As opposed to most studies focusing on late-time effects upon spin polarization, we also pointed out that the dynamical source term in this framework could capture the early-time effects that results in the spin polarization at freeze-out. In general, such early-time effects will be further affected by collisions in late times, from which the spin polarization could be suppressed by relaxation or enhanced by quantum corrections from gradient terms such as vorticity \cite{Kapusta:2019sad,Li:2019qkf,Yang:2020hri,Weickgenannt:2020aaf,Wang:2020pej,Wang:2021qnt,Fang:2022ttm,Hongo:2022izs,Wang:2022yli}. However, the effect from strong background fields $\sim \mathcal{O}(g^2)$ with $g$ the QCD coupling overwhelms the collisional effect from scattering with hard partons $\sim\mathcal{O}(g^4)$ or $\sim\mathcal{O}(g^4\ln g)$ at weak coupling \cite{Li:2019qkf,Yang:2020hri,Hongo:2022izs} as a systematic analysis of the kinetic equations shows. Nevertheless, the magnitudes of such turbulent color fields, originating from Weibel-type instabilities in expanding QGP \cite{Mrowczynski:1988dz,Mrowczynski:1993qm,Romatschke:2003ms}, remains unknown.

On the other hand, in the early stage of high-energy nuclear collisions before the formation of QGP, the soft gluons with large densities sourced by the hard partons could be described by dynamical color fields encoded in classical Yang-Mills equations and form the highly-dense matter known as glasma \cite{Lappi:2006fp,Lappi:2006hq} in the color glass condensate (CGC) effective theory \cite{McLerran:1993ni,McLerran:1993ka,McLerran:1994vd} (see also \cite{Gelis:2010nm,Albacete:2014fwa} for reviews). It is hence of interest to explore how the strong color fields in the glasma phase could influence the spin polarization and correlation of quarks and antiquarks via the dynamical source term obtained from the QKT \cite{Muller:2021hpe,Yang:2021fea}. Some recent studies have shown a substantial effect from these color fields in glasma on jet quenching of hard probes due to accumulating momentum transfer from the soft gluons even though the glasma phase lasts for only a relatively short period compared with QGP \cite{Carrington:2021dvw,Carrington:2022bnv}. One may analogously anticipate the accumulated angular-momentum transfer from these soft gluons yields the spin polarization and correlation of quarks and antiquarks traveling through the glasma. In this paper, we investigate such effects by studying the dynamical source term from QKT augmented by the color fields analytically solved from linearized (Abelianized) Yang-Mills equations in Ref.~\cite{Guerrero-Rodriguez:2021ask}, from which the spin polarization and correlation of quarks and antiquarks at small momentum and central rapidity are derived in the integral form of the gluon distribution in the glasma. By adopting the Golec-Biernat--W\"usthoff (GBW) dipole distribution \cite{Golec-Biernat:1998zce}, we numerically estimate the non-vanishing spin correlation out of plane and demonstrate its enhancement at weak coupling and large collision energy.

The paper is organized as follows: In Sec.~\ref{sec:general}, we briefly review how dynamical spin polarization of quarks and the related spin correlation associated with spin alignment of vector mesons can be induced by color fields in the framework of QKT. The simplified case at small momentum is further examined. In Sec.~\ref{sec:glasma}, we then review the chromo-electric and -magnetic fields derived from linearized Yang-Mills equations in the glasma. In Sec.~\ref{sec:spin_pol_corr}, the dynamical spin polarization and spin correlation induced by such color fields in the glasma are accordingly investigated. In Sec.~\ref{sec:GBW}, we further analyze these results with the GBW distribution and make a  numerical estimation of the spin correlation. Finally, in Sec.~\ref{sec:conclusions}, we present our conclusions and outlook.  Some details of calculations and derivations are presented in various appendices.

Throughout this paper, we use the signature $\eta^{\mu\nu} = {\rm diag} (1, -1,-1,-1)$ of the Minkowski metric
and the completely antisymmetric tensor $\epsilon^{\mu\nu\rho\lambda}$ with $\epsilon^{0123}= \epsilon^{txyz}= 1$, where $0,1,2,3$ and $t,x,y,z$ will be used as the space-time indices interchangeably unless specified. We introduce the notations $A^{(\mu}B^{\nu)}\equiv A^{\mu}B^{\nu}+A^{\nu}B^{\mu}$
and $A^{[\mu}B^{\nu]}\equiv A^{\mu}B^{\nu}-A^{\nu}B^{\mu}$. We also define the dual field strength of color fields via $\tilde{F}^{a\mu\nu}\equiv\epsilon^{\mu\nu\alpha\beta}F^a_{\alpha\beta}/2$ with $a$ being color indices.

\section{Dynamical spin polarization from quantum kinetic theory}\label{sec:general}

In this section, we briefly summarize the spin polarization and correlation obtained from the framework of QKT and Wigner functions of quarks under background color fields previously derived in Refs.~\cite{Muller:2021hpe,Yang:2021fea}. We also review the connection between spin correlation and spin alignment of vector mesons with an update on the coalescence model. The dynamical spin polarization led by color fields at the small-momentum limit with a more generic form than those in Refs.~\cite{Muller:2021hpe,Yang:2021fea} are presented, which will be later utilized to study the effects from glasma in subsequent sections.

\subsection{From spin correlation to spin alignment by color fields}
As shown in Refs.~\cite{Muller:2021hpe,Yang:2021fea}, the spin-polarization spectrum of a quark is described by the spin Cooper-Frye formula \cite{Becattini2013a,Fang:2016vpj},
\begin{equation}\label{Spin_CooperFrye}
	\mathcal{P}^{\mu}({\bm p})=\frac{\int d\Sigma\cdot p\mathcal{J}_{5}^{s\mu}(\bm p,X)}{2m\int d\Sigma_{\mu}\mathcal{N}^{s\mu}(\bm p,X)},
\end{equation}
where $\mathcal{J}_{5}^{s\mu}(\bm p,X)$ and $\mathcal{N}^{s\mu}(\bm p,X)$ denote the onshell color-singlet axial-charge-current density and number-current density in phase space, respectively. Here $d\Sigma_{\mu}$ is the normal vector of a freeze-out hyper surface and $m$ is the mass of quarks. Also, $\mathcal{J}_{5}^{s\mu}(\bm p,X)$ and $\mathcal{N}^{s\mu}(\bm p,X)$ are associated with the axial-vector and vector components of the color-singlet Wigner functions of quarks, which can be expressed in terms of the effective spin four-vector $\tilde{a}^{s\mu}(p,X)$ and the vector-charge distribution function $f^{s}_{V}(p,X)$, where the former delineates the dynamical spin evolution and the later describes the energy and charge transport of quarks. Their dynamics are governed by an axial kinetic equation and a scalar kinetic equation as the traditional Boltzmann equation, while the collisional effects are neglected due to the suppression compared to relatively strong background fields at weak coupling as our assumption. In this setup, the leading-order $\mathcal{J}_{5}^{s\mu}(\bm p,X)$ and $\mathcal{N}^{s\mu}(\bm p,X)$ in the $\hbar$ expansion are given by \cite{Muller:2021hpe,Yang:2021fea}
\begin{eqnarray}
	\mathcal{N}^{s\mu}(\bm p,X)&=& \big(p^{\mu}f^s_{V}\big)_{p_0=\epsilon_{\bm p}},
	\\\label{eq:J5smu_density}
	\mathcal{J}_5^{s\mu}(\bm p,X)&=&\big(\tilde{a}^{ s\mu}+\hbar\bar{C}_2\mathcal{A}^{\mu}_{Q}\big)_{p_0=\epsilon_{\bm p}},
\end{eqnarray}
where $\epsilon_{\bm p}=\sqrt{|\bm p|^2+m^2}$ is the onshell energy and $\bar{C}_2=1/(2N_c)$ with $N_c$ the number of colors. The $\hbar$ is an expansion parameter characterizing the order of gradient expansion for Wigner functions in phase space. Here the non-dynamical source term, $\mathcal{A}^{\mu}_{Q}$, can be decomposed into
\begin{eqnarray}\label{eq:AQ}
\mathcal{A}^{\mu}_{Q}|_{p_0=\epsilon_{\bm p}}=\mathcal{A}^{\mu}_{Q1}+\mathcal{A}^{\mu}_{Q2}
\end{eqnarray}
with the explicit form of $\mathcal{A}^{\mu}_{Q1}$ and $\mathcal{A}^{\mu}_{Q2}$ as
\begin{equation}\label{AQmu_origin}
	\mathcal{A}^{\mu}_{Q1}=\bigg[\frac{\partial_{p\kappa}}{2}\int^{p,X}_{k,X'}p^{\beta}\big( \tilde{F}^{a\mu\kappa}(X)F^a_{\alpha\beta}(X')\big)\partial_{p}^{\alpha}f^{\rm s}_V(p,X')\bigg]_{p_0=\epsilon_{\bm p}},
\end{equation}
and
\begin{eqnarray}\nonumber
		\mathcal{A}^{\mu}_{Q2}&=&-\frac{\epsilon_{\bm p}}{2}\partial_{p_\perp\kappa}\bigg[\int^{p,X}_{k,X'}\hat{p}^{\beta}\big( \tilde{F}^{a\mu\kappa}(X)F^a_{\alpha\beta}(X')\big)\partial_{p}^{\alpha}f^{\rm s}_V(p,X')\bigg]_{p_0=\epsilon_{\bm p}}
		\\
		&=&\frac{1}{2\epsilon_{\bm p}^2}(p_{\perp\kappa}-\epsilon_{\bm p}^2\partial_{p_{\perp}\kappa})\bigg[\int^{p,X}_{k,X'}p^{\beta}\big( \tilde{F}^{a\mu\kappa}(X)F^a_{\alpha\beta}(X')\big) \partial_{p}^{\alpha}f^{\rm s}_V(p,X')\bigg]_{p_0=\epsilon_{\bm p}},
\end{eqnarray}
where $\hat{p}_{\mu}\equiv p_{\mu}/p_0$,
\begin{eqnarray}
	\int^{p,X}_{k,X'}\equiv \int d^4k\int\frac{d^4X'}{(2\pi)^4}e^{ik\cdot(X'-X)}\big(\pi\delta(p\cdot k)+iPV(1/p\cdot k)\big),
\end{eqnarray}
and $V^{\mu}_{\perp}$ represents the spatial component of an arbitrary four vector $V^{\mu}$ through out this section. The notation $PV(x)$ denotes the principle value of $x$. Note that the integral $\int^{p,X}_{k,X'}$ integrates over $k$ and $X'$ and leads to a function depending on $p$ and $X$. Moreover, the notation $\big( F^a_{\kappa\nu}(X)F^a_{\alpha\lambda}(X')\big)$ implicitly incorporates a gauge link for gauge invariance. In practice, we will neglect the gauge link in perturbative calculations. In addition, $F^a_{\kappa\nu}(X)$ and $F^a_{\alpha\lambda}(X')$ here should be still regarded as field operators and thus $\mathcal{P}^{\mu}({\bm p})$ in Eq.~(\ref{Spin_CooperFrye}) is also an operator. In such a case, one has to further take the ensemble average of the correlation function for color fields, $\langle F^a_{\kappa\nu}(X)F^a_{\alpha\lambda}(X')\rangle$, to evaluate the spin polarization spectrum $\langle\mathcal{P}^{\mu}({\bm p})\rangle$ in connection to the experimental observable.

On the other hand, we have to also include the contribution from $\tilde{a}^{s\mu}$ for dynamical spin polarization. The dynamics of $\tilde{a}^{s\mu}$ is governed by the color-singlet axial kinetic equation,
\begin{eqnarray}\label{AKE_signlet_simplify}
	0=\delta(p^2-m^2)\Big(p\cdot\partial\tilde{a}^{\rm s\mu}(p,X)-\partial_{p}^{\kappa}\mathscr{D}_{\kappa}[\tilde{a}^{s\mu}]
	+\hbar\partial_{p}^{\kappa}\big(\mathscr{A}^{\mu}_{\kappa}[f^{\rm s}_{V}]\big)\Big),
\end{eqnarray}
where
\begin{eqnarray}
	\mathscr{D}_{\kappa}[O]=\bar{C}_2\int^{p,X}_{k,X'}p^{\lambda}p^{\rho}\big(F^a_{\kappa\lambda}(X)
	F^a_{\alpha\rho}(X')\big) \partial_{p}^{\alpha}O(p,X')
\end{eqnarray}
and
\begin{eqnarray}
	\mathscr{A}^{\mu}_{\kappa}[O]=\frac{\bar{C}_2}{2}
	\epsilon^{\mu\nu\rho\sigma}\int^{p,X}_{k,X'}p^{\lambda}p_{\rho}\Big(\partial_{X'\sigma}\big( F^a_{\kappa\lambda}(X)F^a_{\alpha\nu}(X')\big)
	+\partial_{X\sigma}\big( F^a_{\kappa\nu}(X)F^a_{\alpha\lambda}(X')\big)\Big)\partial^{\alpha}_{p}O(p,X').
\end{eqnarray}
At weak coupling, given no initial polarization, $\tilde{a}^{s\mu}$ has to be induced by the dynamical source term $\hbar\partial_{p}^{\kappa}\big(\mathscr{A}^{\mu}_{\kappa}[f^{\rm s}_{V}]\big)$ in Eq.~(\ref{AKE_signlet_simplify}), which yields $\tilde{a}^{s\mu}\sim \mathcal{O}(g^2)$ and the diffusion term $\partial_{p}^{\kappa}\mathscr{D}_{\kappa}[\tilde{a}^{s\mu}]$ is accordingly of $\mathcal{O}(g^4)$. Unless we consider the evolution for sufficiently long time, one may assume $p\cdot\partial\tilde{a}^{\rm s\mu}(p,X)\gg \partial_{p}^{\kappa}\mathscr{D}_{\kappa}[\tilde{a}^{s\mu}]$.
To obtain the solution of $\tilde{a}^{s\mu}$ from the kinetic equation, we may utilize the solution,
\begin{eqnarray}\label{eq:useful_int}\nonumber
	\tilde{a}^{\mu}(p,X)&=&\int^{p,X}_{k,X'}G^{\mu}(p,X')
	\\
	&=&\int\frac{d \delta X_0}{2p_0}\Big(1+{\rm sgn}(\delta X_0)\Big)G^{\mu}(p,X')|_{\delta X_{x,y}=0,\delta X_z=p_z\delta X_0/p_0},
\end{eqnarray}
for the differential equation,
\begin{eqnarray}\label{eq:ex_diffeq}
p\cdot\partial \tilde{a}^{\mu}(p,X)=G^{\mu}(p,X),
\end{eqnarray}
with an arbitrary function $G^{\mu}(p,X)$ independent of $\tilde{a}^{\mu}(p,X)$, where $\delta X=X-X'$.
By neglecting the diffusion term, one finds
\begin{eqnarray}
\tilde{a}^{s\mu}(p,X)=-\frac{\hbar}{2p_0}\int d\delta X_0\Big(1+{\rm sgn}(\delta X_0)\Big)\partial_{p}^{\kappa}\big(\mathscr{A}^{\mu}_{\kappa}[f^{\rm s}_{V}](X')\big)|_{\delta X_{x,y}=0,\delta X_z=p_z\delta X_0/p_0}.
\end{eqnarray}
For notational convenience, we may write
\begin{eqnarray}\nonumber \label{eq:a_sol}
	\tilde{a}^{s\mu}(p,X)&=&-\hbar\int^{p,X}_{k,X'}\partial_{p}^{\kappa}\big(\mathscr{A}^{\mu}_{\kappa}[f^{\rm s}_{V}](X')\big)
	\\\nonumber
	&=&-\frac{\hbar\bar{C}_2}{2}
	\epsilon^{\mu\nu\rho\sigma}\int^{p,X}_{k,X'}\partial_{p}^{\kappa}\int^{p,X'}_{k',X''}p^{\lambda}p_{\rho}\Big(\partial_{X''\sigma}\big( F^a_{\kappa\lambda}(X')F^a_{\alpha\nu}(X'')\big)
	+\partial_{X'\sigma}\big( F^a_{\kappa\nu}(X')F^a_{\alpha\lambda}(X'')\big)\Big)
	\\
	&&\times\partial^{\alpha}_{p}f^{\rm s}_{V}(p,X'').
\end{eqnarray}
Similar to the non-dynamical source term, the field strengths above are also operators and the ensemble average needs to be taken in the end.

Now, given an explicit expression of spin polarization for a quark $\mathcal{P}_{q}^{\mu}({\bm p})$ and for an antiquark $\mathcal{P}_{\bar{q}}^{\mu}({\bm p})$, one could further compute the ensemble average of spin correlation,
\begin{equation}
	\langle\mathcal{P}_q^{\mu}({\bm p})\mathcal{P}_{\bar{q}}^{\mu}({\bm p})\rangle =\frac{\int d\Sigma_X\cdot p\int d\Sigma_Y\cdot p\langle\mathcal{J}_{5}^{s\mu}(\bm p,X)\mathcal{J}_{5}^{s\mu}(\bm p,Y)\rangle}{4m^2\Big(\int d\Sigma_X\cdot\mathcal{N}^{s}(\bm p,X)\Big)^2}.
\end{equation}
Note that $\langle\mathcal{P}_{q}^{\mu}({\bm p})\mathcal{P}_{\bar{q}}^{\mu}({\bm p})\rangle$ needs not be equal to $\langle\mathcal{P}_{q}^{\mu}({\bm p})\rangle\langle\mathcal{P}_{\bar{q}}^{\mu}({\bm p})\rangle$. From the spin-dependent coalescence model, one may evaluate the $00$ component of the spin density matrix for spin-$1$ vector mesons through \cite{Liang:2004xn,Yang:2017sdk}
\begin{eqnarray}\label{eq:rho00_origin}
	\rho_{00}=\frac{1-\langle\mathcal{P}^{i}_{q}\mathcal{P}^{i}_{\bar{q}}\rangle}{3+\langle\mathcal{P}^{i}_{q}\mathcal{P}^{i}_{\bar{q}}\rangle},
\end{eqnarray}
where the superscript $i$ is the assigned spin quantization axis determined by experimental setup, which may be chosen as the $y$ axis along the direction perpendicular to the reaction plane in heavy ion collisions. The deviation of $\rho_{00}$ from $1/3$ thus implies non-vanishing spin correlation in QGP even when the spin polarization of a quark or an antiquark vanishes. However, there exists a caveat for Eq.~(\ref{eq:rho00_origin}), which is in fact derived based on the assumption that the spin of a quark and of an antiquark are fully polarized along the quantization axis. In a more generic case, as shown in Ref.~\cite{Sheng:2022ffb} (see also appendix \ref{app:density_matrix} for a consistent derivation), Eq.~(\ref{eq:rho00_origin}) should be modified as
\begin{eqnarray}\label{eq:rho00_modified}
	\rho_{00}\approx \frac{1+\sum_{j=x,y,z}\langle\mathcal{P}^{j}_{q}\mathcal{P}^{j}_{\bar{q}}\rangle-2\langle\mathcal{P}^{i}_{q}\mathcal{P}^{i}_{\bar{q}}\rangle}{3+\sum_{j=x,y,z}\langle\mathcal{P}^{j}_{q}\mathcal{P}^{j}_{\bar{q}}\rangle},
\end{eqnarray}
where $i$ here again denotes the spin quantization axis yet the spin correlations for all other directions need to be incorporated. The crucial difference is that $\rho_{00}=1/3$ even when $\langle\mathcal{P}^{j}_{q}\mathcal{P}^{j}_{\bar{q}}\rangle\neq 0$ is isotropic (i.e. $\langle\mathcal{P}^{x}_{q}\mathcal{P}^{x}_{\bar{q}}\rangle=\langle\mathcal{P}^{y}_{q}\mathcal{P}^{y}_{\bar{q}}\rangle=\langle\mathcal{P}^{z}_{q}\mathcal{P}^{z}_{\bar{q}}\rangle$).
When evaluating $\rho_{00}$ or $\langle\mathcal{P}^{i}_{q}\mathcal{P}^{i}_{\bar{q}}\rangle$ in the glasma phase, instead of simply taking the ensemble average of two chromo-electromagnetic fields from dynamical source terms, we will consider the ensemble average of four field-strength operators therein.

Notably, when comparing the spin polarization led by the non-dynamical source term in Eq.~(\ref{eq:AQ}) and by the dynamical one in Eq.~(\ref{eq:a_sol}), it is found that the former depends on only the color-field correlator at the late time when spin freezes out. On the contrary, the latter is contributed by integrating the color-field correlator over a whole period before the spin freeze-out. Considering the spin polarization from the strong color fields in the glasma phase in early times, which decay shortly after collisions, the dynamical spin polarization should dominate over the non-dynamical one.
Accordingly, we shall evaluate the spin correlation via
\begin{eqnarray}\nonumber
\langle\mathcal{P}^{i}_{q}\mathcal{P}^{i}_{\bar{q}}\rangle &\sim& \langle \tilde{a}^{si}(p,X)\tilde{a}^{si}(p,Y)\rangle
\\\nonumber
&=&\Big\langle\int^{p,X}_{k,X'}\partial_{p}^{\kappa}\big(\mathscr{A}^{i}_{\kappa}[f^{\rm s}_{V}](X')\big)\int^{p,Y}_{k,Y'}\partial_{p}^{\kappa}\big(\mathscr{A}^{i}_{\kappa}[f^{\rm s}_{V}](Y')\big)\Big\rangle
	\\\nonumber
&=&\frac{\bar{C}^2_2}{4}\epsilon^{i\nu\rho\sigma}\epsilon^{i\nu'\rho'\sigma'}\Big\langle\int^{p,X}_{k,X'}\partial_{p}^{\kappa}\int^{p,X'}_{k',X''}p^{\lambda}p_{\rho}\Big(\partial_{X''\sigma}\big( F^a_{\kappa\lambda}(X')F^a_{\alpha\nu}(X'')\big)
+\partial_{X'\sigma}\big( F^a_{\kappa\nu}(X')F^a_{\alpha\lambda}(X'')\big)\Big)
\\\nonumber
&&\times\partial^{\alpha}_{p}f^{\rm s}_{V}(p,X'')
\int^{p,Y}_{\bar{k},Y'}\partial_{p}^{\kappa'}\int^{p,Y'}_{\bar{k}',Y''}p^{\lambda'}p_{\rho'}\Big(\partial_{Y''\sigma'}\big( F^a_{\kappa'\lambda'}(Y')F^a_{\alpha'\nu'}(Y'')\big)
+\partial_{Y'\sigma'}\big( F^a_{\kappa'\nu'}(Y')F^a_{\alpha'\lambda'}(Y'')\big)\Big)
\\
&&\times\partial^{\alpha'}_{p}f^{\rm s}_{V}(p,Y'')\Big\rangle
\end{eqnarray}
from Eq.~(\ref{eq:a_sol}).
Despite the generic $\rho_{00}$ depends on $\langle\mathcal{P}^{j}_{q}\mathcal{P}^{j}_{\bar{q}}\rangle$ for $j=x,y,z$, we will only focus on $\langle\mathcal{P}^{y}_{q}\mathcal{P}^{y}_{\bar{q}}\rangle$ in this paper for simplicity.

\subsection{Simplification in the small-momentum limit}
For practical applications, one has to convert the field strengths into chromo-electric and -magnetic fields, which are explicitly given by
\begin{eqnarray}
	F^{a}_{\mu\nu}=-\epsilon_{\mu\nu\alpha\beta}B^{a\alpha}\bar{n}^{\beta}+E^a_{[\mu}\bar{n}_{\nu]},\quad
	\tilde{F}^{a\mu\nu}=B^{a[\mu}\bar{n}^{\nu]}+\epsilon^{\mu\nu\alpha\beta}E^a_{\alpha}\bar{n}_{\beta},
\end{eqnarray}
where $\bar{n}^{\mu}=(1,\bm 0)$ denotes the temporal direction.
We will make further approximations to simplify the dynamical source term. Nevertheless, unlike the derivation in Refs.~\cite{Muller:2021hpe,Yang:2021fea}, we will not impose the hierarchy between chromo-electric and -magnetic fields and keep the operator form of color-field correlations for generality. We first assume $f^{s}_V=f_V(p_0)$ as a function only depending on $p_0$ for simplification.
Considering
\begin{align}
	\mathscr{A}^{\mu}_{\kappa}[f_V(p_0)]=-\frac{\bar{C}_2}{2}
	\epsilon^{\mu\nu\rho\sigma}\int^{p,X}_{k,X'}p^{\lambda}p_{\rho}\Big(\partial_{X'\sigma}\left( F^a_{\kappa\lambda}(X)E^a_{\nu}(X')\right)
	+\partial_{X\sigma}\left( F^a_{\kappa\nu}(X)E^a_{\lambda}(X')\right)\Big)\partial_{p0}f_V(p_0), \label{AKE_source_term_cont_}
\end{align}
where we omit the $X$ dependence of $\mathscr{A}^{\mu}_{\kappa}[f_V(p_0)]$ for notational convenience,
and using the relation
\begin{eqnarray}\nonumber\label{eq:DkintG}
	\partial_{p\kappa}\int^{p,X}_{k,X'}G(X,\delta X)&=&-\frac{\delta^{0}_{\kappa}}{p_0}\int^{p,X}_{k,X'}G(X,\delta X)
	+\int^{p,X}_{k,X'}\partial_{p\kappa}G(X,\delta X)
	\\
	&&+\int^{p,X}_{k,X'}\delta X_0\frac{\partial G(X,\delta X)}{\partial \delta X^{\mu}_{\perp}}\partial_{p\kappa}\left(\frac{p_{\perp}^{\mu}}{p_0}\right),
\end{eqnarray}
we derive
\begin{eqnarray}\nonumber
	\partial_{p}^{\kappa}\mathscr{A}^{\mu}_{\kappa}[f_V(p_0)]&=&-\frac{1}{p_0}\int^{p,X}_{k,X'}\hat{\mathscr{A}}^{\mu}_{0}[f_V(p_0)]+\int^{p,X}_{k,X'}\partial_{p\kappa}\hat{\mathscr{A}}^{\mu}_{\kappa}[f_V(p_0)]
	+\int^{p,X}_{k,X'}(X_0-X'_0)\frac{\partial \hat{\mathscr{A}}^{\mu}_{\kappa}[f_V(p_0)]}{\partial  X'^{\nu}_{\perp}}
	\\
	&&\times\frac{1}{p_0}(\eta^{\nu\kappa}-\hat{p}^{\nu}\bar{n}^{\kappa}).
\end{eqnarray}
Here we introduce
\begin{eqnarray}
	\hat{\mathscr{A}}^{\mu}_{\kappa}[f_V(p_0)]=-\frac{\bar{C}_2}{2}
	\epsilon^{\mu\nu\rho\sigma}p^{\lambda}p_{\rho}\Big(\partial_{X'\sigma}\left( F^a_{\kappa\lambda}(X)E^a_{\nu}(X')\right)
	+\partial_{X\sigma}\left( F^a_{\kappa\nu}(X)E^a_{\lambda}(X')\right)\Big)\partial_{p0}f_V(p_0)
\end{eqnarray}
and hence
\begin{eqnarray}\nonumber
	\partial^{\kappa}_{p}\hat{\mathscr{A}}^{\mu}_{\kappa}[f_V(p_0)]&=&-\frac{\bar{C}_2}{2}
	\epsilon^{\mu\nu\rho\sigma}\big(\eta^{\kappa\lambda} p_{\rho}+p^{\lambda}\delta^{\kappa}_{\rho}+p^{\lambda}p_{\rho}\delta^{\kappa}_0\partial_{p0}\big)\Big(\partial_{X'\sigma}\left( F^a_{\kappa\lambda}(X)E^a_{\nu}(X')\right)
	\\
	&&+\partial_{X\sigma}\left( F^a_{\kappa\nu}(X)E^a_{\lambda}(X')\right)\Big)\partial_{p0}f_V(p_0).
\end{eqnarray}

To proceed, we postulate a semi-thermal distribution $f_{V}(p_0)=1/(e^{p_0/\Lambda}+1)$ as the distribution function for quarks or antiquarks created in early times of the glasma phase with $\Lambda$ an effective energy scale that should eventually evolve to temperature of the QGP after thermalization. For simplicity, we will also ignore the time dependence of $\Lambda$. By considering low-energy particles with $p_0\ll \Lambda$ such that $f_{V}(p_0)$ is less suppressed, one finds $|\partial_{p_0}f_V(p_0)|\gg |p^{\mu}||\partial^2_{p_0}f_V(p_0)|$. Considering the most relevant regime that substantial spin alignment is observed in experiments, we will further focus on the small-momentum limit such that $\hat{p}^{\mu}_{\perp}\equiv p^{\mu}_{\perp}/p_0\ll 1$ for $p_0=\epsilon_{\bm p}\equiv\sqrt{\bm p^2+m^2}$ being onshell. Consequently, we can approximate
\begin{eqnarray}\label{eq:hatA_small_p}
	\hat{\mathscr{A}}^{\mu}_{\kappa}[f_V(p_0)]\approx \frac{\bar{C}_2}{2}
	\epsilon^{\mu\nu\rho\sigma}p_0^2\bar{n}_{\rho}\partial_{X'\sigma}\left( E^a_{\kappa}(X)E^a_{\nu}(X')\right)
	\partial_{p0}f_V(p_0),
\end{eqnarray}
and
\begin{eqnarray}
	\partial^{\kappa}_{p}\hat{\mathscr{A}}^{\mu}_{\kappa}[f_V(p_0)]\approx-\frac{\bar{C}_2}{2}
	p_0\Big(\epsilon^{\mu\nu\rho\sigma}\partial_{X'\sigma}\left( E^a_{\rho}(X)E^a_{\nu}(X')\right)
	+\partial_{X\sigma}\left( B^{a[\mu}(X)E^{a\sigma]}(X')\right)\Big)\partial_{p0}f_V(p_0),
\end{eqnarray}
by dropping the higher-order terms of $O(|\hat{p}^{\mu}_{\perp}|)$. From Eq.~(\ref{eq:hatA_small_p}), one also obtains $\hat{\mathscr{A}}^{\mu}_{0}[f_V(p_0)]/p_0\approx 0$ and
\begin{eqnarray}
	\frac{\partial \hat{\mathscr{A}}^{\mu}_{\kappa}[f_V(p_0)]}{p_0\partial  X'^{\kappa}_{\perp}}
	\approx \frac{\bar{C}_2}{2}
	\epsilon^{\mu\nu\rho\sigma}p_0\bar{n}_{\rho}\partial^{\kappa}_{X'}\partial_{X'\sigma}\left( E^a_{\kappa}(X)E^a_{\nu}(X')\right)
	\partial_{p0}f_V(p_0).
\end{eqnarray}
It turns out that
\begin{eqnarray}\nonumber
	\partial_{p}^{\kappa}\mathscr{A}^{\mu}_{\kappa}[f_V(p_0)]&\approx&-
	\frac{\bar{C}_2}{2}
	p_0\big(\partial_{p0}f_V(p_0)\big)\int^{p,X}_{k,X'}\Big[\epsilon^{\mu\nu\rho\sigma}\partial_{X'\sigma}\left( E^a_{\rho}(X)E^a_{\nu}(X')\right)
	+\partial_{X\sigma}\left( B^{a[\mu}(X)E^{a\sigma]}(X')\right)
	\\
	&&+(X_0'-X_0)\epsilon^{\mu\nu\rho\sigma}\bar{n}_{\rho}\partial^{\kappa}_{X'}\partial_{X'\sigma}\left( E^a_{\kappa}(X)E^a_{\nu}(X')\right)
	\Big]
\end{eqnarray}
as the leading-order contribution in the small-momentum limit. One thus arrives at
\begin{eqnarray}\nonumber
	\tilde{a}^{s\mu}(p,X)&=&\frac{\bar{C}_2}{2}
	p_0\big(\partial_{p0}f_V(p_0)\big)\int^{p,X}_{k,X'}\int^{p,X'}_{k',X''}\Big[\epsilon^{\mu\nu\rho\sigma}\partial_{X''\sigma}\left( E^a_{\rho}(X')E^a_{\nu}(X'')\right)
	+\partial_{X'\sigma}\left( B^{a[\mu}(X')E^{a\sigma]}(X'')\right)
	\\
	&&+(X_0''-X_0')\epsilon^{\mu\nu\rho\sigma}\bar{n}_{\rho}\partial^{\kappa}_{X''}\partial_{X''\sigma}\left( E^a_{\kappa}(X')E^a_{\nu}(X'')\right)
	\Big].
\end{eqnarray}
When choosing $\mu=x,y,z$ (or equivalently  $\mu=1,2,3$ ), each component of $\tilde{a}^{s\mu}(p,X)$ explicitly reads
\begin{eqnarray}\nonumber
	\tilde{a}^{sx}(p,X)
	&=&-\Bigg(\frac{\bar{C}_2}{2}
	p_0\big(\partial_{p0}f_V(p_0)\big)\Bigg)\int^{p,X}_{k,X'}\int^{p,X'}_{k',X''}\Big[\partial_{X''0}\left(E_{[3}^a(X') E_{2]}^a (X'')\right)
	-\partial^{\nu}_{X'}\left( B^{a}_{[1}(X')E^{a}_{\nu]}(X'')\right) \\\label{eq:exp_asx_origin}
	&&
	+(X_0''-X_0')\partial_{X''}^{\nu}\partial_{X''[2}\left(E_{\nu}^a(X') E_{3]}^a (X'')\right)\Big],
\end{eqnarray}
\begin{eqnarray}\nonumber
	\tilde{a}^{sy}(p,X)
	&=&-\Bigg(\frac{\bar{C}_2}{2}
	p_0\big(\partial_{p0}f_V(p_0)\big)\Bigg)\int^{p,X}_{k,X'}\int^{p,X'}_{k',X''}\Big[\partial_{X''0}\left(E_{[1}^a(X') E_{3]}^a (X'')\right)
	-\partial^{\nu}_{X'}\left( B^{a}_{[2}(X')E^{a}_{\nu]}(X'')\right) \\\label{eq:exp_asy_origin}
	&&
	+(X_0''-X_0')\partial_{X''}^{\nu}\partial_{X''[3}\left( E_{\nu}^a(X') E_{1]}^a (X'')\right)\Big],
\end{eqnarray}
and
\begin{eqnarray}\nonumber
	\tilde{a}^{sz}(p,X)
	&=&-\Bigg(\frac{\bar{C}_2}{2}
	p_0\big(\partial_{p0}f_V(p_0)\big)\Bigg)\int^{p,X}_{k,X'}\int^{p,X'}_{k',X''}\Big[\partial_{X''0}\left( E_{[2}^a(X') E_{1]}^a (X'')\right)
	-\partial^{\nu}_{X'}\left( B^{a}_{[3}(X')E^{a}_{\nu]}(X'')\right) \\	\label{eq:exp_asz_origin}
	&&
	+(X_0''-X_0')\partial_{X''}^{\nu}\partial_{X''[1}\left( E_{\nu}^a(X') E_{2]}^a (X'')\right)\Big],
\end{eqnarray}
which are the building blocks of the follow-up computations.

\section{Color fields in the glasma}\label{sec:glasma}
In the infinite-momentum frame for the parton content of a nucleus described by the CGC framework \cite{McLerran:1993ni,McLerran:1993ka,McLerran:1994vd}, the dynamics of the soft modes (small-$x$) of gluons with large occupation numbers follows the classical Yang-Mills equation,
\begin{eqnarray}\label{eq:YM_eq}
[D_{\mu},F^{\mu\nu}]=J^{\nu},
\end{eqnarray}
where $D_{\mu}$ is the covariant derivative augmented by the non-Abelian gauge field and $F^{\mu\nu}$ is the gluonic field strength. Here we omit the color indices for brevity. In addition, $J^{\mu}$ serves as a source term coming from valence quarks as the hard modes, for which the explicit expressions in term of color-source densities for collisions of two nuclei and further analyses can be found in Ref.~\cite{Kovner:1995ts}. Generically, Eq.~(\ref{eq:YM_eq}) may only be solved numerically with prescribed initial conditions \cite{Krasnitz:1998ns,Krasnitz:2001qu,Schenke:2012hg,Schenke:2012wb,Mantysaari:2017cni,Jia:2020sbn}, while analytic approximations can be obtained in either the weak-field limit of the dilute-dense system \cite{Kovner:1995ts,Kovchegov:1997ke,Dumitru:2001ux,McLerran:2016snu,Lappi:2017skr} or the early-time limit via the perturbative expansion in an infinitesimal proper time \cite{Fries:2006pv,Fujii:2008km,Chen:2015wia,Carrington:2020ssh}. However, for our purpose to study the memory effect of dynamical spin polarization, the inclusion of late-time dynamics for color fields is indispensable. As a result, we will instead adopt the linearized (Abelianized) approximation in Ref.~\cite{Guerrero-Rodriguez:2021ask}, from which the late-time dynamics can be captured by sacrificing gauge invariance and presuming the nonlinear corrections upon quantities of our interest are small.
According to Ref.~\cite{Guerrero-Rodriguez:2021ask}, based on the definitions $E^{\eta}\equiv F_{\tau\eta}/\tau$, $E^i_T\equiv F_{\tau i}$, $B^{\eta}\equiv-\epsilon^{ij}F_{ij}/2$, and $B^i\equiv -\epsilon^{ij}F_{j\tau}/\tau$ in Milne coordinates with $\tau$ the proper time and $\eta$ spacetime rapidity, the chromo-electric and -magnetic fields in the glasma are found to be
\begin{align}\label{eq:EM_CGC_E3}
	E^{\eta}(\tau,x_{\perp})&=-ig\delta^{ij}\!\!\int \!\frac{d^2q_{\perp}}{(2\pi)^2}\!\!\int \!d^2u_{\perp}[\alpha^{i}_{1}(u_{\perp}),\alpha^{j}_{2}(u_{\perp})]
	\times\!J_0(q\tau)e^{iq_{\perp}(x-u)_{\perp}},
\\
	B^{\eta}(\tau,x_{\perp})&=-ig\epsilon^{ij}\!\!\int \!\frac{d^2q_{\perp}}{(2\pi)^2}\!\!\int \!d^2u_{\perp}[\alpha^{i}_{1}(u_{\perp}),\alpha^{j}_{2}(u_{\perp})]
	\times\!J_0(q\tau)e^{iq_{\perp}(x-u)_{\perp}},
\\
E^{i}_{T}(\tau,x_{\perp})&=-g\epsilon^{ij}\epsilon^{kl}\int \!\frac{d^2q_{\perp}}{(2\pi)^2}\frac{q^j}{q}\!\!\int \!d^2u_{\perp}[\alpha^{k}_{1}(u_{\perp}),\alpha^{l}_{2}(u_{\perp})]
\times\!J_1(q\tau)e^{iq_{\perp}(x-u)_{\perp}},
\\\label{eq:EM_CGC_Bi}
B^{i}_{T}(\tau,x_{\perp})&=-g\epsilon^{ij}\delta^{kl}\int \!\frac{d^2q_{\perp}}{(2\pi)^2}\frac{q^j}{q}\!\!\int \!d^2u_{\perp}[\alpha^{k}_{1}(u_{\perp}),\alpha^{l}_{2}(u_{\perp})]
\times\!J_1(q\tau)e^{iq_{\perp}(x-u)_{\perp}},
\end{align}
which yield
\begin{align}
	E^{c\eta}(\tau,x_{\perp})&=gf^{abc}\delta^{ij}\!\!\int \!\frac{d^2q_{\perp}}{(2\pi)^2}\!\!\int \!d^2u_{\perp}\alpha^{i,a}_{1}(u_{\perp})\alpha^{j,b}_{2}(u_{\perp})
	\times\!J_0(q\tau)e^{iq_{\perp}(x-u)_{\perp}},
	\\
	B^{c\eta}(\tau,x_{\perp})&=gf^{abc}\epsilon^{ij}\!\!\int \!\frac{d^2q_{\perp}}{(2\pi)^2}\!\!\int \!d^2u_{\perp}\alpha^{i,a}_{1}(u_{\perp})\alpha^{j,b}_{2}(u_{\perp})
	\times\!J_0(q\tau)e^{iq_{\perp}(x-u)_{\perp}},
	\\
	E^{ci}_T(\tau,x_{\perp})&=-igf^{abc}\epsilon^{ij}\epsilon^{kl}\int \!\frac{d^2q_{\perp}}{(2\pi)^2}\frac{q^j}{q}\!\!\int \!d^2u_{\perp}\alpha^{k,a}_{1}(u_{\perp})\alpha^{l,b}_{2}(u_{\perp})
	\times\!J_1(q\tau)e^{iq_{\perp}(x-u)_{\perp}},
	\\
	B^{ci}_T(\tau,x_{\perp})&=-igf^{abc}\epsilon^{ij}\delta^{kl}\int \!\frac{d^2q_{\perp}}{(2\pi)^2}\frac{q^j}{q}\!\!\int \!d^2u_{\perp}\alpha^{k,a}_{1}(u_{\perp})\alpha^{l,b}_{2}(u_{\perp})
	\times\!J_1(q\tau)e^{iq_{\perp}(x-u)_{\perp}},
\end{align}
where $f^{abc}$ represents the structure constant, $J_{\alpha}(x)$ denotes the Bessel function of the first kind, and $\alpha^{i,a}_{1,2}$ correspond to the transverse gauge fields sourced by the color-charge densities from nuclei $1,2$. Note that the indices $i,j,k,l$ above are along the transverse direction with respect to the light-cone orientations. Also, the subscript $\perp$ is used to denotes the component perpendicular to the light cones and $q_{\perp}x_{\perp}\equiv \sum_{i}q^i_{\perp}x^i_{\perp}$. That is, $i=x,y$. We will hereafter adopt this convention for an arbitrary vector with the subscript $\perp$ (not to confuse it with the convention used in the previous section).

From the transformation of coordinates, $\tau=\sqrt{t^2-z^2}$ and $\eta=\frac{1}{2}\ln\left(\frac{t+z}{t-z}\right)$, one finds
\begin{eqnarray}
E^{z}\equiv F^a_{0z}=F_{\tau\eta}/\tau\equiv E^{\eta},\quad B^{z}\equiv-\frac{\epsilon^{0zjk}}{2}F_{jk}= -\epsilon^{ij}F^a_{ij}/2\equiv B^{\eta}
\end{eqnarray}
for the longitudinal fields, where $\epsilon^{xy}=-\epsilon^{yx}=1$. For the transverse fields, it is defined that $E^i_T\equiv F_{\tau i}$ and $B^i_{T}\equiv -\epsilon^{ij}F_{j\eta}/\tau$. One hence derives
\begin{eqnarray}
E^y\equiv F_{0y}=\frac{t}{\tau}F_{\tau y}-\frac{z}{\tau^2}F_{\eta y}=\cosh\eta E^y_T-\sinh \eta B^x_T,\quad
E^x=\cosh\eta E^x_T+\sinh \eta B^y_T,
\end{eqnarray}
and
\begin{eqnarray}
B^y=\epsilon^{xy}F_{xz}=\frac{-\epsilon^{yx}}{\tau^2}\big(tF_{x\eta}-zF_{x\tau}\big)=\cosh\eta B_T^y+\sinh\eta E_T^x,
\quad B^x=\cosh\eta B_T^x-\sinh\eta E_T^y.
\end{eqnarray}
Here we have used
\begin{eqnarray}
\partial_{t}\tau=\frac{t}{\tau},\quad \partial_{z}\tau=-\frac{z}{\tau},\quad \partial_{t}\eta=-\frac{z}{\tau^2},\quad \partial_{z}\eta=\frac{t}{\tau^2},
\end{eqnarray}
and $t=\tau\cosh\eta$ and $z=\tau\sinh\eta$.

From  Ref.~\cite{Guerrero-Rodriguez:2021ask}, it turns out that the following correlators vanish,
\begin{eqnarray}
	\langle E_T^{ai}(X')B_T^{aj}(X'')\rangle&=&0,\quad \langle E^{a\eta}(X')B^{a\eta}(X'')\rangle=0,    \label{eq:pocorre}\\
	\langle E_T^{ai}(X')E^{a\eta}(X'')\rangle&=&0, \quad  \langle  B_T^{ai}(X')B^{a\eta}(X'')\rangle=0,\label{eq:pocorre1}
\end{eqnarray}
based on the correlation of gluon fields from the same nucleus,
\begin{eqnarray}
\langle \alpha_{1,2}^{i,a}(u_{\perp})\alpha_{1,2}^{j,b}(v_{\perp})\rangle=\frac{\delta^{ab}}{2}\Big[\delta^{ij}G_{1,2}(u_{\perp},v_{\perp})+\Big(\delta^{ij}-2\frac{(u-v)^i_{\perp}(u-v)^j_{\perp}}{|u_{\perp}-v_{\perp}|^2}\Big)h_{1,2}(u_{\perp},v_{\perp})\Big],
\end{eqnarray}
and $\langle \alpha_{1}^{i,a}(u_{\perp})\alpha_{2}^{j,b}(v_{\perp})\rangle=\langle \alpha_{2}^{i,a}(u_{\perp})\alpha_{1}^{j,b}(v_{\perp})\rangle=0$. Here $G_{1,2}$ and $h_{1,2}$ correspond to the unpolarized and linearly polarized gluon distribution functions of nuclei $1$ and $2$, respectively.
The non-vanishing correlators can be written as
\begin{eqnarray}
	\langle E_T^{ai}(X')E_T^{aj}(X'')\rangle&=&-\bar{N}_c\epsilon^{in}\epsilon^{jm}\int^{X'}_{\perp;q,u}\int^{X''}_{\perp;l,v}\Omega_{-}(u_{\perp},v_{\perp})
	\frac{q^n l^m}{ql}
	\times\!J_1(qX'_0)J_1(lX''_0),    \label{eq:eiejcorr_sim}\\
	%
	%
	\langle B_T^{ai}(X')B_T^{aj}(X'')\rangle&=&-\bar{N}_c\epsilon^{in}\epsilon^{jm}\int^{X'}_{\perp;q,u}\int^{X''}_{\perp;l,v}
	\Omega_{+}(u_{\perp},v_{\perp})
	\frac{q^n l^m}{ql}
	\times\!J_1(qX'_0)J_1(lX''_0),    \label{eq:bibjcorr_sim}\\
	%
	%
	\langle E_T^{ai}(X')B^{a\eta}(X'')\rangle&=&-i\bar{N}_c\epsilon^{in}\int^{X'}_{\perp;q,u}\int^{X''}_{\perp;l,v}
	\Omega_{-}(u_{\perp},v_{\perp})
	\frac{q^n}{q}
	\times\!J_1(qX'_0)J_0(lX''_0),
	\label{eq:eib3corr_sim}\\
	%
	%
	\langle B_T^{ai}(X')E^{a\eta}(X'')\rangle&=&-i\bar{N}_c\epsilon^{in}\int^{X'}_{\perp;q,u}\int^{X''}_{\perp;l,v}
	\Omega_{+}(u_{\perp},v_{\perp})
	\frac{q^n}{q}
	\times\!J_1(qX'_0)J_0(lX''_0),
	\label{eq:bie3corr_sim}\\
	%
	%
	\langle E^{a\eta}(X')E^{a\eta}(X'')\rangle&=&\bar{N}_c\int^{X'}_{\perp;q,u}\int^{X''}_{\perp;l,v}
	\Omega_{+}(u_{\perp},v_{\perp})
	\times\!J_0(qX'_0)J_0(lX''_0),
	\label{eq:e3e3corr_sim}\\
	%
	%
	\langle B^{a\eta}(X')B^{a\eta}(X'')\rangle&=&\bar{N}_c\int^{X'}_{\perp;q,u}\int^{X''}_{\perp;l,v}
	\Omega_{-}(u_{\perp},v_{\perp})
	\times\!J_0(qX'_0)J_0(lX''_0),   \label{eq:b3b3corr}
\end{eqnarray}
where
\begin{eqnarray}
	\bar{N}_c\equiv \frac{1}{2}g^2N_c(N_c^2-1),
\end{eqnarray}
\begin{eqnarray}
	\Omega_{\mp}(u_{\perp},v_{\perp})=\left[G_1(u_{\perp},v_{\perp})G_2(u_{\perp},v_{\perp})\mp h_1(u_{\perp},v_{\perp})h_2(u_{\perp},v_{\perp})\right],
\end{eqnarray}
and
\begin{eqnarray}\nonumber
	\int^{X'}_{\perp;q,u}\equiv\int \!\frac{d^2q_{\perp}}{(2\pi)^2}\int \!d^2u_{\perp}e^{iq_{\perp}(X'-u)_{\perp}}.
\end{eqnarray}
Note that we also have the useful relation such as
\begin{eqnarray}\label{eq:rel_DE_DB}
	\partial_{Xi}E_T^{ai}(X)=\partial_{Xi}B_T^{ai}(X)=0
\end{eqnarray}
from Eqs.~(\ref{eq:EM_CGC_E3})-(\ref{eq:EM_CGC_Bi}). The color-field correlators above will then be employed to evaluate the spin polarization and correlation.

\section{Dynamical spin polarization and correlation from glasma}\label{sec:spin_pol_corr}
This section is dedicated to the derivation of spin polarization and correlation coming from effective spin four vectors in the compact integral form of color-field correlators from glasma or more precisely gluon distribution functions of nuclei introduced in the previous section. The derivation is rather technical and hence only critical steps are presented, while more detailed computations can be found in appendices. In summary, $\langle\tilde{a}^{sx}(p,X)\rangle=\langle\tilde{a}^{sy}(p,X)\rangle=0$ and  $\langle\tilde{a}^{sz}(p,X)\rangle$ is shown in Eq.~(\ref{eq:asz}) with non-vanishing components in Eq.~(\ref{eq:Ab_final}) and Eq.~(\ref{eq:Ac_final}) for spin polarization. For the out-of-plane spin correlation, the primary result is shown in Eq.~(\ref{eq:ayays}) with the explicit expressions in Eqs.~(\ref{eq:I1_final}), (\ref{eq:I2_final}), and (\ref{eq:I3_final}).

\subsection{Spin polarization}\label{sec:spin_pol}
For simplicity and practical reasons, we will focus on the mid-rapidity region for $\eta\rightarrow 0$ such that $E^{x,y}\approx E_T^{x,y}$ and $B^{x,y}\approx B_T^{x,y}$, which allows us to conveniently compute the integrals of color-field correlators in Minkowski coordinates.
We may now first evaluate $\langle \mathcal{P}^y_q\rangle \sim \langle \tilde{a}^{sy}(p,X)\rangle$. Given
Eq.~(\ref{eq:pocorre}), one obtains
\begin{eqnarray}\nonumber\label{eq:exp_asy}
	\langle\tilde{a}^{sy}(p,X)\rangle
	&=&-\Bigg(\frac{\bar{C}_2}{2}
	p_0\big(\partial_{p0}f_V(p_0)\big)\Bigg)\int^{p,X}_{k,X'}\int^{p,X'}_{k',X''}\Big[\partial_{X''0}\left\langle E_{[1}^a(X') E_{3]}^a (X'')\right\rangle\\
	&& -\partial_{X'3}\left\langle B^{a}_{[2}(X')E^{a}_{3]}(X'')\right\rangle
	+(X_0''-X_0')\partial^k_{X''}\partial_{X''[3}\left\langle E_k^a(X') E_{1]}^a (X'')\right\rangle\Big],
\end{eqnarray}
where we have assigned $(x,y,z)=(x^1,x^2,x^3)$. Since $\partial_{X'3}\approx \tau^{-1}\partial_{\eta'}$ and $\partial_{X'0}\approx \partial_{\tau'}$ when $\eta\rightarrow 0$ and the color fields are rapidity-independent, Eq.~(\ref{eq:exp_asy}) reduces to
\begin{eqnarray}\nonumber\label{eq:exp_asy2}
	\langle\tilde{a}^{sy}(p,X)\rangle
	&\approx&-\Bigg(\frac{\bar{C}_2}{2}
	p_0\big(\partial_{p0}f_V(p_0)\big)\Bigg)\int^{p,X}_{k,X'}\int^{p,X'}_{k',X''}\Big[\partial_{X''0}\left\langle E_{[1}^a(X') E_{3]}^a (X'')\right\rangle
	\\
	&&+(X_0''-X_0')\Big(\partial^2_{X''1}\left\langle E_1^a(X') E_3^a (X'')\right\rangle
	+\partial_{X''2}\partial_{X''1}\left\langle E_2^a(X') E_3^a (X'')\right\rangle\Big)\Big].
\end{eqnarray}
According to Eq.~(\ref{eq:pocorre1}), one consequently concludes $\langle\tilde{a}^{sy}(p,X)\rangle=0$ from Eq.~(\ref{eq:exp_asy2}). Similarly, it is found $\langle\tilde{a}^{sx}(p,X)\rangle=0$ by symmetry, while this is not the case for $\langle\tilde{a}^{sz}(p,X)\rangle$.

The dynamical spin polarization now may be contributed by
\begin{eqnarray}\label{eq:asz}
\langle\tilde{a}^{sz}(p,X)\rangle\approx -\Bigg(\frac{\bar{C}_2}{2}
p_0\big(\partial_{p0}f_V(p_0)\big)\Bigg)\big(\mathcal{A}_{a}+\mathcal{A}_{b}+\mathcal{A}_{c}\big),
\end{eqnarray}
where
\begin{eqnarray}
\mathcal{A}_{a}=\int^{p,X}_{k,X'}\int^{p,X'}_{k',X''}\partial_{X''0}\langle E_{[2}^a(X') E_{1]}^a (X'')\rangle,
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{b}=-\int^{p,X}_{k,X'}\int^{p,X'}_{k',X''}\big(\partial^{1}_{X'}\langle B^{a}_{[3}(X')E^{a}_{1]}(X'')\rangle
+\partial^{2}_{X'}\langle B^{a}_{[3}(X')E^{a}_{2]}(X'')\rangle\big),
\end{eqnarray}
and
\begin{eqnarray}
\mathcal{A}_{c}=\int^{p,X}_{k,X'}\int^{p,X'}_{k',X''}(X_0''-X_0')\big(\partial_{X''}^{1}\partial_{X''[1}\langle E_{1}^a(X') E_{2]}^a (X'')\rangle
+\partial_{X''}^{2}\partial_{X''[1}\langle E_{2}^a(X') E_{2]}^a (X'')\rangle\big).
\end{eqnarray}
By using Eq.~(\ref{eq:rel_DE_DB}), $\mathcal{A}_{b}$ reduces to
\begin{eqnarray}
\mathcal{A}_{b}=-\int^{p,X}_{k,X'}\int^{p,X'}_{k',X''}\big(\partial^{1}_{X'}\langle B^{a}_{3}(X')E^{a}_{1}(X'')\rangle
+\partial^{2}_{X'}\langle B^{a}_{3}(X')E^{a}_{2}(X'')\rangle\big).
\end{eqnarray}

From Eq.~(\ref{eq:useful_int}), one finds
\begin{eqnarray}\nonumber
\mathcal{A}_{a}&=&\frac{\bar{N}_c}{4p_0^2}\int^{X_0}_{X'_0}\int^{X'_0}_{X''_0}\int^{X}_{\perp;q,u}\int^{X}_{\perp;l,v}\Omega_{-}(u_{\perp},v_{\perp})\frac{q^x l^y}{ql}
\partial_{X''0}\Big[\big(J_1(qX'_0)J_1(lX''_0)-J_1(qX''_0)J_1(lX'_0)\big)
\\
&&\times\Theta(X'_0)\Theta(X''_0)\Big],
\end{eqnarray}
where we have further used
\begin{eqnarray}
	\int^{X_0}_{X'_0}\equiv \int^{\infty}_{-\infty} d X'_0\Big(1+{\rm sgn}(X_0-X_0')\Big)=2\int^{\infty}_{-\infty} d X'_0\Theta(X_0-X_0').
\end{eqnarray}
Here we have implicitly multiplied the color-field correlators with the unit-step functions for $X_0'$ and $X_0''$, $\Theta(X'_0)$ and $\Theta(X''_0)$, because of setting $X_0'$=$X_0''=0$ as an initial time for the presence of glamsa.
It turns out that $\mathcal{A}_{a}=0$ due to the structure $\langle E_{[2}^a(X') E_{1]}^a (X'')\rangle$ since
\begin{eqnarray}\nonumber
\int^{X'_0}_{X''_0}\partial_{X''0}J_1(qX'_0)J_1(lX''_0)\Theta(X'_0)\Theta(X''_0)&=&\int^{X''_0}_{X'_0}\partial_{X''0}J_1(qX''_0)J_1(lX'_0)\Theta(X'_0)\Theta(X''_0)
\\
&=&2J_1(qX'_0)J_1(lX'_0)\Theta(X'_0)
.
\end{eqnarray}
On the other hand, it is found
\begin{eqnarray}\nonumber
	\mathcal{A}_{b}
	&=&\frac{i\bar{N}_c}{4p_0^2}\int^{X_0}_{X'_0}\int^{X'_0}_{X''_0}\bigg[\partial^1_{X'}\int^{X'}_{\perp;q,u}\int^{X''}_{\perp;l,v}\Omega_{-}(u_{\perp},v_{\perp})\frac{l^y}{l}J_0(qX'_0)J_1(lX''_0)
	\\
	&&-\partial^2_{X'}\int^{X'}_{\perp;q,u}\int^{X''}_{\perp;l,v}\Omega_{-}(u_{\perp},v_{\perp})\frac{l^x}{l}J_0(qX'_0)J_1(lX''_0)\bigg]_{X''_{\perp}=X'_{\perp}=X_{\perp}}\Theta(X'_0)\Theta(X''_0).
\end{eqnarray}
To handle the spatial derivative terms, we introduce the trick,
\begin{eqnarray}\nonumber
	\partial_{X'_{\perp}j}\int^{X'}_{\perp;q,u}G(u_{\perp})&=&\int \!\frac{d^2q_{\perp}}{(2\pi)^2}\int \!d^2u_{\perp}(-iq_{\perp j})e^{iq_{\perp}(X'-u)_{\perp}}G(u_{\perp})
	\\
	&=&-\int \!\frac{d^2q_{\perp}}{(2\pi)^2}\int \!d^2u_{\perp}\big(\partial_{u_{\perp}j}e^{iq_{\perp}(X'-u)_{\perp}}\big)G(u_{\perp}),
\end{eqnarray}
for an arbitrary function $G(u_{\perp})$. When assuming $G(u_{\perp})|_{u^i_{\perp}\rightarrow \pm \infty}\rightarrow 0$, we further obtain the useful relation,
\begin{eqnarray}\label{eq:derivative_trick}
	\partial_{X'_{\perp}j}\int^{X'}_{\perp;q,u}G(u_{\perp})=\int^{X'}_{\perp;q,u}\partial_{u_{\perp}j}G(u_{\perp}).
\end{eqnarray}
Assuming
\begin{eqnarray}
	\Omega_{\pm}(u_{\perp},v_{\perp})|_{u^i_{\perp}\rightarrow \pm \infty}\rightarrow 0,\quad \Omega_{\pm}(u_{\perp},v_{\perp})|_{v^i_{\perp}\rightarrow \pm \infty}\rightarrow 0,
\end{eqnarray}
which are physical boundary conditions, we accordingly derive
\begin{eqnarray}
\mathcal{A}_{b}=\frac{i\bar{N}_c}{4p_0^2}\int^{X_0}_{X'_0}\int^{X'_0}_{X''_0}\int^{X}_{\perp;q,u}\int^{X}_{\perp;l,v}\frac{J_0(qX'_0)J_1(lX''_0)}{l}l^{[y}\partial^{x]}_{u}\Omega_{-}(u_{\perp},v_{\perp})\Theta(X'_0)\Theta(X''_0)
\end{eqnarray}
and similarly
\begin{eqnarray}\nonumber
\mathcal{A}_{c}&=&\frac{\bar{N}_c}{4p_0^2}\int^{X_0}_{X'_0}\int^{X'_0}_{X''_0}(X_0''-X'_0)\int^{X}_{\perp;q,u}\int^{X}_{\perp;l,v}\frac{J_1(qX'_0)J_1(lX''_0)}{ql}\Theta(X'_0)\Theta(X''_0)
\\
&&\times \big[q^{y}l^{x}\partial^{x}_{v}\partial_{vx}-q^{x}l^{y}\partial^{y}_{v}\partial_{vy}
+(q^yl^y-q^xl^x)\partial^{x}_{v}\partial_{vy}
\big]\Omega_{-}(u_{\perp},v_{\perp}).
\end{eqnarray}
Implementing the decomposition in Eq.~(\ref{eq:decomp_qi}) and integrals in Eq.~(\ref{eq:thetaint_1}) to cope with the angular parts of momentum integration, one finds
\begin{eqnarray}\nonumber
\mathcal{A}_{b}&=&\frac{-\bar{N}_c}{4p_0^2}\int^{X_0}_{X'_0}\int^{X'_0}_{X''_0}\int\frac{dqdl ql}{(2\pi)^2}\int d^2u_{\perp}d^2v_{\perp}J_0(qX'_0)J_1(lX''_0)\Theta(X'_0)\Theta(X''_0)
\\
&&\times J_0(q|\bar{u}_{\perp}|)J_1(l|\bar{v}_{\perp}|)\frac{\bar{v}_{\perp}^{[y}\partial^{x]}_{u}}{|\bar{v}_{\perp}|}\Omega_{-}(u_{\perp},v_{\perp})
\end{eqnarray}
and
\begin{eqnarray}\nonumber
\mathcal{A}_{c}&=&-\frac{\bar{N}_c}{4p_0^2}\int^{X_0}_{X'_0}\int^{X'_0}_{X''_0}(X_0''-X'_0)\int\frac{dqdl ql}{(2\pi)^2}\int d^2u_{\perp}d^2v_{\perp}\frac{J_1(qX'_0)J_1(lX''_0)}{|\bar{u}_{\perp}||\bar{v}_{\perp}|}\Theta(X'_0)\Theta(X''_0)
\\
&&\times J_1(q|\bar{u}_{\perp}|)J_1(l|\bar{v}_{\perp}|) \big[\bar{u}_{\perp}^{y}\bar{v}_{\perp}^{x}\partial^{x}_{v}\partial_{vx}-\bar{u}_{\perp}^{x}\bar{v}_{\perp}^{y}\partial^{y}_{v}\partial_{vy}
+(\bar{u}_{\perp}^y\bar{v}_{\perp}^y-\bar{u}_{\perp}^x\bar{v}_{\perp}^x)\partial^{x}_{v}\partial_{vy}
\big]\Omega_{-}(u_{\perp},v_{\perp}),
\end{eqnarray}
where $\bar{u}_{\perp}=X_{\perp}-u_{\perp}$ and $\bar{v}_{\perp}=X_{\perp}-v_{\perp}$. Further applying the orthogonal condition for Bessel functions,
\begin{eqnarray}\label{eq:Bessel_rel}
	\int^{\infty}_{0}dr rJ_{\nu}(kr)J_{\nu}(sr)=\frac{\delta(k-s)}{s},
\end{eqnarray}
$\mathcal{A}_{b}$ and $\mathcal{A}_{c}$ become
\begin{eqnarray}\label{eq:Ab_final}
\mathcal{A}_{b}=\frac{-g^2N_c(N_c^2-1)}{2(2\pi)^2p_0^2}\int^{\perp}_{\bar{u},\bar{v}}\frac{\Theta(X_0-|\bar{u}_{\perp}|)\Theta(|\bar{u}_{\perp}|-|\bar{v}_{\perp}|)}{|\bar{u}_{\perp}||\bar{v}_{\perp}|}
\frac{\bar{v}_{\perp}^{[y}\partial^{x]}_{u}}{|\bar{v}_{\perp}|}\Omega_{-}(u_{\perp},v_{\perp})
\end{eqnarray}
and
\begin{eqnarray}\nonumber\label{eq:Ac_final}
\mathcal{A}_{c}&=&\frac{-g^2N_c(N_c^2-1)}{2(2\pi)^2p_0^2}\int^{\perp}_{\bar{u},\bar{v}}\frac{\Theta(X_0-|\bar{u}_{\perp}|)\Theta(|\bar{u}_{\perp}|-|\bar{v}_{\perp}|)}{|\bar{u}_{\perp}|^2|\bar{v}_{\perp}|^2}(|\bar{v}_{\perp}|-|\bar{u}_{\perp}|)
\\
&&\times  \big[\bar{u}_{\perp}^{y}\bar{v}_{\perp}^{x}\partial^{x}_{v}\partial_{vx}-\bar{u}_{\perp}^{x}\bar{v}_{\perp}^{y}\partial^{y}_{v}\partial_{vy}
+(\bar{u}_{\perp}^y\bar{v}_{\perp}^y-\bar{u}_{\perp}^x\bar{v}_{\perp}^x)\partial^{x}_{v}\partial_{vy}
\big]\Omega_{-}(u_{\perp},v_{\perp}),
\end{eqnarray}
where we have implicitly taken $\Theta(|\bar{u}_{\perp}|)\Theta(|\bar{v}_{\perp}|)=1$ and introduced
\begin{eqnarray}
\int^{\perp}_{u,v}\equiv\int d^2u_{\perp}d^2v_{\perp}.
\end{eqnarray}
Given an explicit expression of $\Omega_{-}(u_{\perp},v_{\perp})$, Eqs.~(\ref{eq:Ab_final}) and (\ref{eq:Ac_final}) can be evaluated numerically.

\subsection{Spin correlation}\label{sec:spin_corr}
We may now evaluate the spin correlation $\langle\mathcal{P}^{y}_{q}\mathcal{P}^{y}_{\bar{q}}\rangle$ dynamically generated by $\langle\tilde{a}^{sy}(p,X)\tilde{a}^{sy}(p,Y)\rangle$. By symmetry, $\langle\tilde{a}^{sy}(p,X)\tilde{a}^{sy}(p,Y)\rangle=\langle\tilde{a}^{sy}(p,Y)\tilde{a}^{sy}(p,X)\rangle$ because we will in the end integrate over spatial $X$ and $Y$ on the freeze-out hyper-surface with $X_0=Y_0$, the integrand in the multiple integrals involved is also invariant under $(X'\leftrightarrow Y',X''\leftrightarrow Y'')$. The explicit form of $\langle\tilde{a}^{sy}(p,X)\tilde{a}^{sy}(p,Y)\rangle$ can be written as
\begin{eqnarray}\nonumber\label{eq:ayay}
	\langle\tilde{a}^{sy}(p,X)\tilde{a}^{sy}(p,Y)\rangle
	&\approx&\Bigg(\frac{\bar{C}_2}{2}
	p_0\big(\partial_{p0}f_V(p_0)\big)\Bigg)^2\int^{p,X}_{k,X'}\int^{p,X'}_{k',X''}\int^{p,Y}_{\bar{k},Y'}\int^{p,Y'}_{\bar{k}',Y''}\nonumber
	\\&&\Big[\partial_{X''0}\partial_{Y''0}\left\langle E_{[1}^a(X') E_{3]}^a (X'')E_{[1}^a(Y') E_{3]}^a (Y'')\right\rangle \nonumber
	\\&& -2\partial_{X''0}\partial_{Y'1}\left\langle E_{[1}^a(X') E_{3]}^a (X'')B^{a[2}(Y') E^{a1]} (Y'')\right\rangle
	\nonumber\\&&
	+2(Y_0''-Y_0')\partial_{X''0}\partial^2_{Y''1}\left\langle E_{[1}^a(X') E_{3]}^a (X'')E_1^a(Y') E_3^a (Y'')\right\rangle
	\nonumber\\&&+2(Y_0''-Y_0')\partial_{X''0}\partial_{Y''2}\partial_{Y''1}\left\langle E_{[1}^a(X') E_{3]}^a (X'')E_2^a(Y') E_3^a (Y'')\right\rangle
	\nonumber\\&&
	+\partial_{X'1}\partial_{Y'1}\left\langle B^{a[2}(X')E^{a1]}(X'')B^{a[2}(Y')E^{a1]}(Y'')\right\rangle	\nonumber\\&&
	-2(Y_0''-Y_0')\partial_{X'1}\partial^2_{Y''1}\left\langle B^{a[2}(X')E^{a1]}(X'')E_1^a(Y') E_3^a (Y'')\right\rangle
	\nonumber\\&&-2(Y_0''-Y_0')\partial_{X'1}\partial_{Y''2}\partial_{Y''1}\left\langle B^{a[2}(X')E^{a1]}(X'')E_2^a(Y') E_3^a (Y'')\right\rangle
	\nonumber\\&&
	+(X_0''-X_0')(Y_0''-Y_0')\Big(\partial^2_{X''1}\partial^2_{Y''1}\left\langle E_1^a(X') E_3^a (X'')E_{1}^a(Y') E_{3}^a (Y'')\right\rangle\nonumber\\
	&&+2\partial^2_{X''1}\partial_{Y''2}\partial_{Y''1}\left\langle E_1^a(X') E_3^a (X'')E_2^a(Y') E_3^a (Y'')\right\rangle\nonumber\\
	&&+\partial_{X''2}\partial_{X''1}\partial_{Y''2}\partial_{Y''1}\left\langle E_2^a(X') E_3^a (X'')E_2^a(Y') E_3^a (Y'')\right\rangle\Big)\Big].
\end{eqnarray}
Nonetheless, one has to derive a more concise form for the multi-dimensional integral with the input of color-field correlators. Such an analytic expression for each term above could be derived in light of the procedure shown in appendix~\ref{app:spin_corr}, where we take $\langle E_{1}^a(X') E_{3}^a (X'')E_{1}^a(Y') E_{3}^a (Y'')\rangle$ as an example.
It turns out that all the terms associated with $\partial_{X0''} E_{[1}^a(X') E_{3]}^a (X'')$ in Eq.~(\ref{eq:ayay}) vanish since such terms involve the integral
\begin{eqnarray}
	\int^{X'_0}_{X_0''}\partial_{X0''}J_1(qX_0'')J_0(lX'_0)\Theta(X_0')\Theta(X_0'')-
	\int^{X'_0}_{X_0''}\partial_{X0''}J_1(qX_0')J_0(lX''_0)\Theta(X_0')\Theta(X_0'')=0,
\end{eqnarray}
which is similar to the reason causing $\mathcal{A}_{a}=0$.
One may refer to appendix~\ref{app:vanishing_terms} for an explicit calculation showing that e.g. the contribution from the second line of Eq.~(\ref{eq:ayay}) vanishes.
Eventually, Eq.~(\ref{eq:ayay}) reduces to
\begin{eqnarray}\label{eq:ayays}
	\langle\tilde{a}^{sy}(p,X)\tilde{a}^{sy}(p,Y)\rangle
	&\approx&\Bigg(\frac{\bar{C}_2}{2}
	p_0\big(\partial_{p0}f_V(p_0)\big)\Bigg)^2\big(\mathcal{I}_1+\mathcal{I}_2+\mathcal{I}_3\big),
\end{eqnarray}
where
\begin{eqnarray}
\mathcal{I}_1=\int^{p,X}_{k,X'}\int^{p,X'}_{k',X''}\int^{p,Y}_{\bar{k},Y'}\int^{p,Y'}_{\bar{k}',Y''}\partial_{X'1}\partial_{Y'1}\left\langle B^{a[2}(X')E^{a1]}(X'')B^{a[2}(Y')E^{a1]}(Y'')\right\rangle,
\end{eqnarray}
\begin{eqnarray}\nonumber
\mathcal{I}_{2}&=& -2\int^{p,X}_{k,X'}\int^{p,X'}_{k',X''}\int^{p,Y}_{\bar{k},Y'}\int^{p,Y'}_{\bar{k}',Y''}(Y_0''-Y_0')\Big(\partial_{X'1}\partial^2_{Y''1}\left\langle B^{a[2}(X')E^{a1]}(X'')E_1^a(Y') E_3^a (Y'')\right\rangle
\\
&&+\partial_{X'1}\partial_{Y''2}\partial_{Y''1}\left\langle B^{a[2}(X')E^{a1]}(X'')E_2^a(Y') E_3^a (Y'')\right\rangle\Big),
\end{eqnarray}
and
\begin{eqnarray}
\mathcal{I}_{3}&=& \int^{p,X}_{k,X'}\int^{p,X'}_{k',X''}\int^{p,Y}_{\bar{k},Y'}\int^{p,Y'}_{\bar{k}',Y''}(X_0''-X_0')(Y_0''-Y_0')\Big(\partial^2_{X''1}\partial^2_{Y''1}\left\langle E_1^a(X') E_3^a (X'')E_{1}^a(Y') E_{3}^a (Y'')\right\rangle\nonumber\\
&&+2\partial^2_{X''1}\partial_{Y''2}\partial_{Y''1}\left\langle E_1^a(X') E_3^a (X'')E_2^a(Y') E_3^a (Y'')\right\rangle\nonumber\\
&&+\partial_{X''2}\partial_{X''1}\partial_{Y''2}\partial_{Y''1}\left\langle E_2^a(X') E_3^a (X'')E_2^a(Y') E_3^a (Y'')\right\rangle\Big).
\end{eqnarray}
It is found that the non-vanishing spin correlation stems from the transverse spatial derivatives upon color-field correlators.

We will compute $\mathcal{I}_1$ as an example, while the other terms in Eq.~(\ref{eq:ayays}) can be derived in a similar manner. By symmetry, the integrand of $\mathcal{I}_1$ can be written as
\begin{eqnarray}\nonumber
&&\partial_{X'1}\partial_{Y'1}\left\langle B^{a[2}(X')E^{a1]}(X'')B^{a[2}(Y')E^{a1]}(Y'')\right\rangle
\\\nonumber
&&= \partial_{X'1}\partial_{Y'1}\big(\left\langle B^{a2}(X')B^{a2}(Y')\right\rangle\left\langle E^{a1}(X'')E^{1}(Y'')\right\rangle
+\left\langle B^{a1}(X')B^{a1}(Y')\right\rangle\left\langle E^{a2}(X'')E^{2}(Y'')\right\rangle
\\
&&\quad-2\left\langle B^{a2}(X')B^{a1}(Y')\right\rangle\left\langle E^{a1}(X'')E^{2}(Y'')\right\rangle\big),
\end{eqnarray}
where we have applied the Wick-theorem like decomposition to decompose a four-field correlator in terms of the products of two-field correlators, e.g.
\begin{eqnarray}\nonumber
&&\left\langle B^{a2}(X')E^{a1}(X'')B^{a2}(Y')E^{a1}(Y'')\right\rangle
\\\nonumber
&&=\left\langle B^{a2}(X')E^{a1}(X'')\right\rangle\left\langle B^{a2}(Y')E^{a1}(Y'')\right\rangle
+\left\langle B^{a2}(Y')E^{a1}(X'')\right\rangle\left\langle B^{a2}(X')E^{a1}(Y'')\right\rangle
\\
&&\quad+\left\langle B^{a2}(X')E^{a1}(Y'')\right\rangle\left\langle B^{a2}(Y')E^{a1}(X'')\right\rangle.
\end{eqnarray}
We then make the further decomposition, $\mathcal{I}_{1}=\mathcal{I}_{1a}+\mathcal{I}_{1b}-2\mathcal{I}_{1c}$, where
\begin{eqnarray}
\mathcal{I}_{1a}\equiv \int^{p,X}_{k,X'}\int^{p,X'}_{k',X''}\int^{p,Y}_{\bar{k},Y'}\int^{p,Y'}_{\bar{k}',Y''}\partial_{X'1}\partial_{Y'1}\left\langle B^{a2}(X')B^{a2}(Y')\right\rangle\left\langle E^{a1}(X'')E^{1}(Y'')\right\rangle,
\end{eqnarray}
\begin{eqnarray}
	\mathcal{I}_{1b}\equiv \int^{p,X}_{k,X'}\int^{p,X'}_{k',X''}\int^{p,Y}_{\bar{k},Y'}\int^{p,Y'}_{\bar{k}',Y''}\partial_{X'1}\partial_{Y'1}\left\langle B^{a1}(X')B^{a1}(Y')\right\rangle\left\langle E^{a2}(X'')E^{2}(Y'')\right\rangle,
\end{eqnarray}
and
\begin{eqnarray}
	\mathcal{I}_{1c}\equiv \int^{p,X}_{k,X'}\int^{p,X'}_{k',X''}\int^{p,Y}_{\bar{k},Y'}\int^{p,Y'}_{\bar{k}',Y''}\partial_{X'1}\partial_{Y'1}\left\langle B^{a2}(X')B^{a1}(Y')\right\rangle\left\langle E^{a1}(X'')E^{2}(Y'')\right\rangle.
\end{eqnarray}

To evaluate $\mathcal{I}_1$, we shall apply the same tricks in Eq.~(\ref{eq:useful_int}) and Eq.~(\ref{eq:derivative_trick}), which yield
\begin{eqnarray}\nonumber
	\mathcal{I}_{1a}(p,X,Y)&\approx& \frac{g^4N_c^2(N_c^2-1)^2}{64p_0^4}
	\int^{X}_{\perp;q,u}\int^{Y}_{\perp;l,v}\int^{X}_{\perp;q',u'}\int^{Y}_{\perp;l',v'}\left(\frac{q^xl^x}{q l}\right)\left(\frac{q'^yl'^y}{q' l'}\right)
	\\
	&&\times\partial_{ux}\partial_{vx}\Omega_{+}(u_{\perp},v_{\perp})\Omega_{-}(u'_{\perp},v'_{\perp})\mathcal{Y}_{1a}(X_0,Y_0,q,l,q',l'),
\end{eqnarray}
\begin{eqnarray}\nonumber
	\mathcal{I}_{1b}(p,X,Y)&\approx& \frac{g^4N_c^2(N_c^2-1)^2}{64p_0^4}
	\int^{X}_{\perp;q,u}\int^{Y}_{\perp;l,v}\int^{X}_{\perp;q',u'}\int^{Y}_{\perp;l',v'}\left(\frac{q^yl^y}{q l}\right)\left(\frac{q'^xl'^x}{q' l'}\right)
	\\
	&&\times\partial_{ux}\partial_{vx}\Omega_{+}(u_{\perp},v_{\perp})\Omega_{-}(u'_{\perp},v'_{\perp})\mathcal{Y}_{1a}(X_0,Y_0,q,l,q',l'),
\end{eqnarray}
and
\begin{eqnarray}\nonumber
	\mathcal{I}_{1c}(p,X,Y)&\approx& \frac{g^4N_c^2(N_c^2-1)^2}{64p_0^4}
	\int^{X}_{\perp;q,u}\int^{Y}_{\perp;l,v}\int^{X}_{\perp;q',u'}\int^{Y}_{\perp;l',v'}\left(\frac{q^xl^y}{q l}\right)\left(\frac{q'^yl'^x}{q' l'}\right)
	\\
	&&\times\partial_{ux}\partial_{vx}\Omega_{+}(u_{\perp},v_{\perp})\Omega_{-}(u'_{\perp},v'_{\perp})\mathcal{Y}_{1a}(X_0,Y_0,q,l,q',l'),
\end{eqnarray}
where we introduced several shorthand notations,
\begin{eqnarray}
	\int^{X'}_{\perp;q,u}\equiv\int \!\frac{d^2q_{\perp}}{(2\pi)^2}\int \!d^2u_{\perp}e^{iq_{\perp}(X'-u)_{\perp}},
\end{eqnarray}
\begin{eqnarray}\nonumber
	&&\mathcal{Y}_{1a}(X_0,Y_0,q,l,q',l')
	\\
	&&\equiv\int^{X_0}_{X'_0}\int^{Y_0}_{Y'_0}\int^{X'_0}_{X''_0}\int^{Y'_0}_{Y''_0}
	J_1(qX'_0)J_1(lY'_0)\Theta(X'_0)\Theta(Y'_0)
	J_1(q'X''_0)J_1(l'Y''_0)\Theta(X''_0)\Theta(Y''_0),
\end{eqnarray}
and
\begin{eqnarray}
	s_{\perp}=X_{\perp}-u_{\perp},\quad
	s'_{\perp}=X_{\perp}-u'_{\perp} \quad
	t_{\perp}=Y_{\perp}-v_{\perp} \quad
	t'_{\perp}=Y_{\perp}-v'_{\perp}.
\end{eqnarray}
One hence obtains
\begin{eqnarray}\nonumber
	\mathcal{I}_{1}
	&\approx&\frac{g^4N_c^2(N_c^2-1)^2}{64p_0^4}
	\int^{X}_{\perp;q,u}\int^{Y}_{\perp;l,v}\int^{X}_{\perp;q',u'}\int^{Y}_{\perp;l',v'}\partial_{ux}\partial_{vx}\Omega_{+}(u_{\perp},v_{\perp})\Omega_{-}(u'_{\perp},v'_{\perp})\mathcal{Y}_{1a}(X_0,Y_0,q,l,q',l')
	\\
	&&\quad\times\frac{1}{qlq'l'}\big[q^xl^xq^{\prime y}l^{\prime y}+q^yl^yq^{\prime x}l^{\prime x}-2q^xl^yq^{\prime y}l^{\prime x}\big].
\end{eqnarray}
Utilizing Eq.~(\ref{eq:decomp_qi}) and Eq.~(\ref{eq:thetaint_1}) again to integrate over the angular parts of momentum integrals, $\mathcal{I}_{1}$ becomes
\begin{eqnarray}\nonumber
	\mathcal{I}_{1}&\approx& \frac{g^4N_c^2(N_c^2-1)^2}{4p_0^4}
\int \!dq\int \!dl\int \!\frac{dq'}{(2\pi)^2}\int \!\frac{dl'}{(2\pi)^2}\int^{\perp}_{u,v,u',v'}qlq'l'
	\\\nonumber
	&&\times \big[\hat{s}_{\perp}^{x}\hat{t}_{\perp}^{x}\hat{s}_{\perp}^{\prime y}\hat{t}_{\perp}^{\prime y}+\hat{s}_{\perp}^{y}\hat{t}_{\perp}^{y}\hat{s}_{\perp}^{\prime x}\hat{t}_{\perp}^{\prime x}-2\hat{s}_{\perp}^{x}\hat{t}_{\perp}^{y}\hat{s}_{\perp}^{\prime y}\hat{t}_{\perp}^{\prime x}\big]J_1(q|s_{\perp}|)J_1(l|t_{\perp}|)J_1(q'|s'_{\perp}|)J_1(l'|t'_{\perp}|)
	\\
	&&\times \partial_{ux}\partial_{vx}\Omega_{+}(u_{\perp},v_{\perp})\Omega_{-}(u'_{\perp},v'_{\perp})J_1(qX'_0)J_1(lY'_0)
	J_1(q'X''_0)J_1(l'Y''_0),
\end{eqnarray}
where $\hat{s}^i_{\perp}=s^i_{\perp}/|s_{\perp}|$ and
\begin{eqnarray}
	\int^{\perp}_{u,v,u'v'}\equiv \int \!d^2u_{\perp}\!\!\int \!d^2v_{\perp}
	\int \!d^2u'_{\perp}\!\!\int \!d^2v'_{\perp}.
\end{eqnarray}
Further using Eq.~(\ref{eq:Bessel_rel}), we arrive at
\begin{eqnarray}\nonumber\label{eq:I1_final}
	\mathcal{I}_{1}&\approx& \frac{g^4N_c^2(N_c^2-1)^2}{4(2\pi)^4p_0^4}
	\int^{\perp}_{s,t,s',t'}\frac{\Theta(X_0-|s_{\perp}|)\Theta(Y_0-|t_{\perp}|)\Theta(|s_{\perp}|-|s'_{\perp}|)\Theta(|t_{\perp}|-|t'_{\perp}|)}{|s_{\perp}||t_{\perp}||s'_{\perp}||t'_{\perp}|}
	\\
	&&\times \big[\hat{s}_{\perp}^{x}\hat{t}_{\perp}^{x}\hat{s}_{\perp}^{\prime y}\hat{t}_{\perp}^{\prime y}+\hat{s}_{\perp}^{y}\hat{t}_{\perp}^{y}\hat{s}_{\perp}^{\prime x}\hat{t}_{\perp}^{\prime x}-2\hat{s}_{\perp}^{x}\hat{t}_{\perp}^{y}\hat{s}_{\perp}^{\prime y}\hat{t}_{\perp}^{\prime x}\big]
    \partial_{ux}\partial_{vx}\Omega_{+}(u_{\perp},v_{\perp})\Omega_{-}(u'_{\perp},v'_{\perp}),
\end{eqnarray}
which could be evaluated numerically with given $\Omega_{\pm}$. Following the same procedure, one can also derive similar expressions for the remaining terms in Eq.~(\ref{eq:ayays}). As shown in appendix.~\ref{app:remaining_terms}, it is found
\begin{eqnarray}\nonumber\label{eq:I2_final}
\mathcal{I}_{2}
&\approx& \frac{g^4N_c^2(N_c^2-1)^2}{2(2\pi)^4p_0^4}
\int^{\perp}_{s,t,s',t'}\frac{\Theta(X_0-|s_{\perp}|)\Theta(Y_0-|t_{\perp}|)\Theta(|s_{\perp}|-|s'_{\perp}|)\Theta(|t'_{\perp}|-|t_{\perp}|)}{|s_{\perp}||t_{\perp}||s'_{\perp}||t'_{\perp}|}
\\
&&\times\big(|t_{\perp}|-|t'_{\perp}|\big)\hat{s}_{\perp}^{[x} \hat{s}'^{ y]}_{\perp}\big(\hat{t}'^y_{\perp}\partial_{vx}-\hat{t}'^x_{\perp}\partial_{vy}\big)\partial_{ux}\partial_{vx}
\Omega_{-}(v'_{\perp},u'_{\perp})\Omega_{+}(u_{\perp},v_{\perp})
\end{eqnarray}
and
\begin{eqnarray}\nonumber\label{eq:I3_final}
\mathcal{I}_{3}
	&\approx& -\frac{g^4N_c^2(N_c^2-1)^2}{4(2\pi)^4p_0^4}
\int^{\perp}_{s,t,s',t'}\frac{\Theta(X_0-|s_{\perp}|)\Theta(Y_0-|t_{\perp}|)\Theta(|s_{\perp}|-|s'_{\perp}|)\Theta(|t_{\perp}|-|t'_{\perp}|)}{|s_{\perp}||t_{\perp}||s'_{\perp}||t'_{\perp}|}
\\\nonumber
&&\times\big(|s'_{\perp}|-|s_{\perp}|\big)\big(|t'_{\perp}|-|t_{\perp}|\big)
\big[\hat{s}^y_{\perp}\hat{t}^y_{\perp}\partial^2_{u'x}\partial^2_{v'x}-2\hat{s}^y_{\perp}\hat{t}^x_{\perp}\partial^2_{u'x}\partial_{v'y}\partial_{v'x}
\\
&&+\hat{s}^x_{\perp}\hat{t}^x_{\perp}\partial_{u'x}\partial_{u'y}\partial_{v'x}\partial_{v'y}\big]
\Omega_{-}(u_{\perp},v_{\perp})\Omega_{+}(u'_{\perp},v'_{\perp}).
\end{eqnarray}
The major difference is that the terms $(Y_0''-Y_0')$ and $(X_0''-X_0')$ give rise to $\big(|t'_{\perp}|-|t_{\perp}|\big)$ and $\big(|s'_{\perp}|-|s_{\perp}|\big)$ in the remaing integrands after integrating over $Y'_0$, $Y''_0$, $X'_0$, and $X''_0$.

\section{Analysis with the GBW distribution}\label{sec:GBW}

Following Ref.~\cite{Guerrero-Rodriguez:2021ask}, we adopt the GBW distribution such that $h_{1,2}=0$ and
\begin{align}
	\Omega_{\pm}(u_{\perp},v_{\perp})=\Omega(u_{\perp},v_{\perp})=\frac{Q_s^4}{g^4N_c^2}\left(\frac{1-e^{-Q_s^2|u_{\perp}-v_{\perp}|^2/4}}{Q_s^2|u_{\perp}-v_{\perp}|^2/4}\right)^2=\frac{Q_s^4}{g^4N_c^2}\left(\frac{1-e^{-Q_s^2|s_{\perp}-t_{\perp}-r_{\perp}|^2/4}}{Q_s^2|s_{\perp}-t_{\perp}-r_{\perp}|^2/4}\right)^2,
\end{align}
where $r_{\perp}\equiv X_{\perp}-Y_{\perp}$. One finds $\mathcal{A}_{b}=0$ and $\mathcal{A}_{c}=0$ after conducting the numerical calculations with the GBW distribution and hence $\langle\tilde{a}^{sz}(p,X)\rangle$=0. It turns out that the spin polarization from glasma vanishes in all directions. We may analytically show $\mathcal{A}_{b}=0$ as an example in the following. From the GBW distribution, it is found
\begin{eqnarray}
	\frac{\bar{v}^{[y}\partial^{x]}_{u}}{|\bar{v}_{\perp}|}\Omega(u_{\perp},v_{\perp})=\mathcal{F}(|\bar{u}_{\perp}|,|\bar{v}_{\perp}|,\cos(\theta_{\bar{v}}-\theta_{\bar{u}}))\sin(\theta_{\bar{v}}-\theta_{\bar{u}})
\end{eqnarray}
where
$\theta_{\bar{u}}$ and $\theta_{\bar{v}}$ are the polar angles of $\bar{u}_{\perp}$ and $\bar{v}_{\perp}$ and the explicit form of $\mathcal{F}$ is unimportant here. By making the change of coordinates,
\begin{eqnarray}\nonumber
	\int d^2\bar{u}_{\perp}\int d^2\bar{v}_{\perp}&=&\int^{\infty}_0 d|\bar{u}_{\perp}||\bar{u}_{\perp}|\int^{\infty}_0 d|\bar{v}_{\perp}||\bar{v}_{\perp}|\int^{2\pi}_0 d\theta_{\bar{u}}\int^{2\pi}_0 d\theta_{\bar{v}}
	\\
	&=&\int^{\infty}_0 d|\bar{u}_{\perp}||\bar{u}_{\perp}|\int^{\infty}_0 d|\bar{v}_{\perp}||\bar{v}_{\perp}|\int^{2\pi}_0 d\bar{\theta}_{\bar{u},\bar{v}}\int^{2\pi}_{-2\pi}d\Theta_{\bar{u},\bar{v}},
\end{eqnarray}
where $\bar{\theta}_{\bar{u},\bar{v}}=(\theta_{\bar{u}}+\theta_{\bar{v}})/2$ and $\Theta_{\bar{u},\bar{v}}=\theta_{\bar{u}}-\theta_{\bar{v}}$. It is then clear to see
\begin{eqnarray}
	\int^{2\pi}_{-2\pi}d\Theta_{\bar{u},\bar{v}}\mathcal{F}(|\bar{u}_{\perp}|,|\bar{v}_{\perp}|,\cos\Theta_{\bar{u},\bar{v}})\sin\Theta_{\bar{u},\bar{v}}=0,
\end{eqnarray}
since $\mathcal{F}(|\bar{u}_{\perp}|,|\bar{v}_{\perp}|,\cos\Theta_{\bar{u},\bar{v}})\sin\Theta_{\bar{u},\bar{v}}$ is an odd function under $\Theta_{\bar{u},\bar{v}}\rightarrow -\Theta_{\bar{u},\bar{v}}$.

For spin correlation, it turns out that $\mathcal{I}=\mathcal{I}_{1}+\mathcal{I}_{2}+\mathcal{I}_{3}$ only depends on $p_0$, $X_0=Y_0$, and $r_{\perp}$, which is easier to be computed in the polar coordinate.
Accordingly, $\mathcal{I}$ can be factorized as
\begin{eqnarray}\label{eq:I_estimate}
	\mathcal{I}(p_0,Q_sX_0,Q_s|r_{\perp}|,\theta_{r})=\frac{g^4N_c^2(N_c^2-1)^2}{4(2\pi)^4p_0^4}\frac{Q_s^8}{g^8N_c^4}\frac{\hat{\mathcal{I}}(Q_sX_0,Q_s|r_{\perp}|,\theta_{r})}{Q_s^2},
\end{eqnarray}
where $\theta_{r}\equiv \cos^{-1}(r^x_{\perp}/|r_{\perp}|)$ and $\hat{\mathcal{I}}(Q_sX_0,Q_s|r_{\perp}|,\theta_r)$ is the eight-dimensional integral as an dimensionless quantity to be evaluated.
Here the first prefactor associated with $1/p_0^4$ comes from the simplified integral form in Eqs.~(\ref{eq:I1_final}), (\ref{eq:I2_final}), and (\ref{eq:I3_final}), while $Q_s^8/(g^8N_c^4)$ stems from the square of GBW distribution and the extra $1/Q_s^2$ is introduced to make $\hat{\mathcal{I}}(Q_sX_0,Q_s|r_{\perp}|,\theta_r)$ dimensionless.

In addition, we will consider isochronous freeze-out in proper time $\tau=\text{const}$, where $\Sigma_x^{\mu}=(\tau\cosh\eta,x^1,x^2,\tau\sinh\eta)$ \cite{Romatschke:2009im}. In our case, we focus on the small-rapidity region such that $-\eta_{\rm m}\leq\eta\leq \eta_{\rm m}$ with $\eta_{\rm m}\ll 1$. We will also approximate that the space-time rapidity is equal to the
momentum rapidity. Then the normal vector $d\Sigma^{\mu}$ gives
\begin{eqnarray}
	d\Sigma_X\cdot p\approx X_0\sqrt{m^2+|p^2_{\perp}|}d^2X_{\perp}d\eta,
\end{eqnarray}
which yields
\begin{equation}\label{eq:spin_corr}
	\langle\mathcal{P}_q^{i}({\bf p})\mathcal{P}_{\bar{q}}^{i}({\bf p})\rangle \approx \frac{\int d^2X_{\perp}\int d^2Y_{\perp}\langle\tilde{a}^{i}(\bm p,X)\tilde{a}^{i}(\bm p,Y)\rangle}{4m^2\Big(\int d^2X_{\perp}f_{V}(p_0)\Big)^2}\Bigg|_{p_0=\epsilon_{\bm p}}.
\end{equation}
Note that $\langle\mathcal{P}_q^{i}({\bf p})\mathcal{P}_{\bar{q}}^{i}({\bf p})\rangle$ depends on $X_0$ as the freeze-out time. For computational convenience, one may further make the change of coordinates,
\begin{eqnarray}
	\int d^2X_{\perp}\int d^2Y_{\perp}=\int d^2r_{\perp}\int d^2R_{\perp},\quad R_{\perp}\equiv (X_{\perp}+Y_{\perp})/2.
\end{eqnarray}

Nevertheless, it is important to note that $\langle\tilde{a}^{i}(\bm p,X)\tilde{a}^{i}(\bm p,Y)\rangle$ will stop evolving much earlier than the freeze-out time close to chemical equilibrium in the QGP phase. Furthermore, we have neglected the time dependence of $f_{V}(p_0)$ as the quark distribution function in early times, which should eventually reach thermal equilibrium in the QGP phase. For practical purposes, we should consider different freeze-out hypersurfaces in the numerator and denominator of Eq.~(\ref{eq:spin_corr}), where the freeze-out time in the numerator is chosen to be at thermalization time $X_0=X^{\rm th}_0$ that roughly characterizes the end of the glasma phase, at which the spin no longer evolves, while the one in the numerator is at chemical equilibrium $X_0=X^{\rm eq}_0$. Consequently, from Eq.~(\ref{eq:spin_corr}), the out-of-plane spin correlation may be approximated as
\begin{equation}\label{eq:spin_corr_2}
	\langle\mathcal{P}_q^{y}({\bf p})\mathcal{P}_{\bar{q}}^{y}({\bf p})\rangle \approx \frac{\bar{C}_2^2p_0^2(\partial_{p0}f_{V})^2\int^{\rm gl} d^2r_{\perp}\int^{\rm gl} d^2R_{\perp}\mathcal{I}(p_0,Q_sX_0,Q_s|r_{\perp}|,\theta_r)}{16m^2{\rm A}_{\rm T}^2f_{\rm eq}(p_0)^2}\Bigg|_{p_0=\epsilon_{\bm p}},
\end{equation}
where $\int^{\rm gl} d^2r_{\perp}\int^{\rm gl} d^2R_{\perp}$ corresponds to the integrals over the transverse plane of glasma (around the transverse size of collided nuclei) and $\rm A_{\rm T}$ denotes the transverse area of the QGP (around chemical freeze-out). Here $f_{\rm eq}(p_0)=1/(e^{p_0/T}+1)$ corresponds to the thermal distribution function with $T$ being the freeze-out temperature and $m$ may be approximated as the constituent quark mass. We may further make an order-of-magnitude estimation of the spin correlation based on Eq.~(\ref{eq:I_estimate}). Although the exact form of $\hat{\mathcal{I}}(Q_sX_0,Q_s|r_{\perp}|,\theta_r)$ can only be obtained from sophisticated multi-dimensional integral, it is physically expected that the dominant contribution should be around $|r_{\perp}|\sim 0$ with short-range correlation as will be also verified numerically. We accordingly estimate
\begin{eqnarray}\label{eq:Gaussian_approx}
	\int^{\rm gl} d^2r_{\perp}\mathcal{I}(p_0,Q_sX_0,Q_s|r_{\perp}|,\theta_r)\sim \pi Q_s^{-2}\mathcal{I}(p_0,Q_sX_0,0,0)
\end{eqnarray}
by postulating $\mathcal{I}(p_0,Q_sX_0,Q_s|r_{\perp}|,\theta_r)\sim \mathcal{I}(p_0,Q_sX_0,0,0)e^{-|r_{\perp}|^2Q_s^2}$ as a Gaussian form with the correlation length of $\mathcal{O}(Q_s^{-1})$ and without angular dependence. Taking $\Lambda\sim Q_s\gg m\gg |\bm p|$ such that $\partial_{p0}f_{V}\approx -1/(4Q_s)$ and $\int^{\rm gl} d^2R_{\perp}\approx\rm A_{\rm N}$ with $\rm A_{\rm N}$ being the transverse area of nuclei, from Eq.~(\ref{eq:I_estimate}), Eq.~(\ref{eq:spin_corr_2}) can be further approximated as
\begin{equation}\label{eq:spin_corr_final}
	\langle\mathcal{P}_q^{y}({\bf p})\mathcal{P}_{\bar{q}}^{y}({\bf p})\rangle \sim \frac{(N_c^2-1)^2(e^{m/T}+1)^2}{N_c^4(16\pi)^4g^4}\frac{\pi Q_s^2\rm A_{\rm N}}{m^4{\rm A}_{\rm T}^2}\hat{\mathcal{I}}(Q_sX^{\rm th}_0,0,0)
\end{equation}
at small momentum. Despite the actual value of $\hat{\mathcal{I}}(Q_sX^{\rm th}_0,0,0)$, the correlation could be enhanced by the factor $Q_s^2/g^4$ at weaker coupling at higher collision energy. Conversely, this effect will be suppressed in low-energy collisions, where the glasma stage does not exist.

To evaluate $\hat{\mathcal{I}}(Q_sX_0,Q_s|r_{\perp}|,\theta_r)$ numerically\footnote{In principle, we may directly evaluate $\int^{\rm gl} d^2r_{\perp}\mathcal{I}(p_0,Q_sX_0,Q_s|r_{\perp}|)$ numerically. However, to get a rough estimate, we instead adopt the approximation in Eq.~(\ref{eq:Gaussian_approx}) for computational efficiency.}, we utilized the local adaptive method built into $\it{Mathematica}$. As shown in Fig.~\ref{figI123}, it is found that $\hat{\mathcal{I}}_3$ dominates over $\hat{\mathcal{I}}_1$ and $\hat{\mathcal{I}}_2$ particularly at late times $Q_sX_0 >2$ and we may approximate $\hat{\mathcal{I}}\approx \hat{\mathcal{I}}_3$, where $\hat{\mathcal{I}}_{1,2,3}$ correspond to the dimensionless quantities coming from $\mathcal{I}_{1,2,3}$ with the factorization in Eq.~(\ref{eq:I_estimate}). The rapid increase of $\hat{\mathcal{I}}_3(Q_sX_0)$ is predominantly led by the factor $\big(|s'_{\perp}|-|s_{\perp}|\big)\big(|t'_{\perp}|-|t_{\perp}|\big)$ originating from $(X_0''-X_0')(Y_0''-Y_0')$ in the integrand, which can be observed in Fig.~\ref{figI3growth} when comparing with the result by removing this factor. In addition, we have confirmed $\hat{\mathcal{I}}_3(Q_sX_0,Q_s|r_{\perp}|,\pi/2)$ is maximized at $Q_s|r_{\perp}|=0$ as shown in Fig.~\ref{figI3r} with also very mild $\theta_r$ dependence illustrated in Fig.~\ref{figI3theta} at small $Q_s|r_{\perp}|$. One could further verify the same scenario for $\hat{\mathcal{I}}_1(Q_sX_0,Q_s|r_{\perp}|,\theta_r)$ and $\hat{\mathcal{I}}_2(Q_sX_0,Q_s|r_{\perp}|,\theta_r)$. Some qualitative features of $\hat{\mathcal{I}}_{1,2,3}$ are further analyzed in appendix~\ref{app:alternative_I}.

For convenience, we adopt the approximation in Eq.~(\ref{eq:spin_corr_final}) for conducting numerical estimation. We take the values $Q_s=2$ GeV, $\alpha_s=g^2/(4\pi)\approx 1/3$, $N_c=3$, $A_{\rm N}\sim A_{\rm T}\approx 100$ fm$^{2}$, $m\approx 500$ MeV as the constituent quark mass for strange quarks, and $T\approx 150$ MeV as the freeze-out temperature. By setting $X^{\rm th}_0 \approx 0.5$ fm and correspondingly $Q_sX^{\rm th}_0\approx 5$, which yields $\hat{\mathcal{I}} \approx 5000$ (see Fig.~\ref{figI3growth}), we obtain the result $\langle\mathcal{P}_q^{y}({\bf p})\mathcal{P}_{\bar{q}}^{y}({\bf p})\rangle\approx 0.002$. When choosing a larger saturation scale such as $Q_s=3$ GeV and $\alpha_s\approx 1/4$ without changing the other parameters, for which now $Q_sX^{\rm th}_0\approx 7.5$ at $X^{\rm th}_0 \approx 0.5$ fm, we obtain $\hat{\mathcal{I}} \approx 15000$ and $\langle\mathcal{P}_q^{y}({\bf p})\mathcal{P}_{\bar{q}}^{y}({\bf p})\rangle\approx 0.03$. Notably, except for the magnification from the prefactor $Q_s^2/g^4$, the numerical value of $\hat{\mathcal{I}}(Q_sX^{\rm th}_0)$ also increases with larger $Q_s$ at a fixed thermalization time.

\begin{figure}[t]
	\begin{minipage}{7cm}
		\begin{center}
		{\includegraphics[width=1.1\hsize,height=6cm,clip]{figI123}}

			\caption{Numerical results for $\hat{\mathcal{I}}_{1,2,3}(Q_sX_0)$ at $Q_s|r_\perp| = 0.05$ and $\theta_r=\pi/2$. The same behavior is found for other parameter values.
	}
	\label{figI123}
		\end{center}
	\end{minipage}
	\hspace {1cm}
	\begin{minipage}{7cm}
		\begin{center}
           {\includegraphics[width=1.1\hsize,height=6cm,clip]{figI3growth}}

			\caption{The origin of the rapid growth for $\hat{I}_3$. The solid line shows the full result; the dashed line shows the result without the secular growth factors.}
			\label{figI3growth}
		\end{center}
	\end{minipage}
\end{figure}

\begin{figure}[t]
	\begin{minipage}{7cm}
		\begin{center}
            {\includegraphics[width=1.1\hsize,height=6cm,clip]{figI3r}}
			\caption{The $|r_{\perp}|$ dependence of $\hat{I}_3$.}\label{figI3r}
		\end{center}
	\end{minipage}
	\hspace {1cm}
	\begin{minipage}{7cm}
		\begin{center}
   {\includegraphics[width=1.1\hsize,height=6cm,clip]{figI3theta}}
			\caption{The $\theta_{r}$ dependence of $\hat{I}_3$ at small $Q_s|r_{\perp}|$.}
			\label{figI3theta}
		\end{center}
	\end{minipage}
\end{figure}

We stress that this estimate is based on rather crude approximations and thus has large uncertainties coming from multiple corrections. Most importantly, we have neglected the spin relaxation in QGP after thermalization and also the distinction between the dressed quark mass in glasma, which should be encoded in $\langle \tilde{a}^{sy}(p,X)\tilde{a}^{sy}(p,Y)\rangle$, and the constituent quark mass for on-shell $f_{\rm eq}(p_0)$. In this scenario, it might be also debatable whether the explicit $1/m^2$ factor in Eq.~(\ref{eq:spin_corr}) for normalization should be set as the constituent quark mass. Both, the spin relaxation time in the strongly coupled QGP and the dressed quark mass in the glasma, are unknown. Fortuitously, the mass-dependent factor $(e^{m/T}+1)^2/m^4$ changes by less than a factor 3 over the range $1 < m/T <4$.
Despite these uncertainties, we emphasize that these estimated values for spin correlation are substantially larger than the expected vorticity contribution\footnote{From the naive extraction of vorticity contribution to global spin polarization of $\Lambda$ hyperons such that $\mathcal{P}^y_{q}\sim\mathcal{P}^y_{\Lambda}\sim \omega/T\lesssim 10^{-3}$ at high-energy nuclear collisions above 200 GeV \cite{STAR:2017ckg,Becattini:2016gvu} ($\mathcal{P}^y_{\Lambda}\sim 10^{-4}$ at $2.76$ TeV \cite{ALICE:2019onw}), where $\omega$ is the global vorticity, the spin correlation will be approximately $\lesssim 10^{-6}$.}.

\section{Conclusions and outlook}\label{sec:conclusions}
In this paper, we derived the dynamical spin polarization and out-of-plane correlation of massive quarks at small momentum and central rapidity from color fields in the glasma phase in terms of the gluon density. Our formalism is based on quantum kinetic theory (QKT) in the Wigner function formalism. For the GBW distribution, the spin polarization is found to vanish, whereas the out-of-plane spin correlation is non-vanishing and enhanced by $Q_s^2/g^4$ at weak coupling. We have also numerically estimated the magnitude of out-of-plane spin correlation, which is found to be around $0.002\sim 0.03$ at $Q_s=2\sim 3$ GeV and by orders of magnitude larger than the correlation from vorticity for high-energy nuclear collisions.

As already mentioned in Sec.~\ref{sec:general}, the updated spin coalescence model incorporates both the in-plane and out-of-plane correlations. By symmetry of the color fields from glasma, we may expect $\langle\mathcal{P}_q^{x}({\bf p})\mathcal{P}_{\bar{q}}^{x}({\bf p})\rangle\sim \langle\mathcal{P}_q^{y}({\bf p})\mathcal{P}_{\bar{q}}^{y}({\bf p})\rangle$ at small momenta, while $\langle\mathcal{P}_q^{z}({\bf p})\mathcal{P}_{\bar{q}}^{z}({\bf p})\rangle$ has to be evaluated separately, which remains to be explored for making an estimation of the impact on spin alignment of vector mesons. In addition, it is also important to obtain the momentum dependence for spin correlation and the corresponding spin alignment. Furthermore, the present study only considers the spin correlation between a quark and an antiquark with the same flavor, which is applicable to the case of $\phi$ mesons. For the $K^{*0}$ vector meson, comprising a strange quark and an anti-down quark, further generalization of our formalism for spin correlation is needed.

Here we used the GBW distribution to evaluate the spin polarization and correlation. Alternatively, we may also apply the gluon distribution \cite{Guerrero-Rodriguez:2021ask} based on the dipole amplitude from the McLerran-Venugopalan model \cite{McLerran:1993ni,McLerran:1993ka,McLerran:1994vd}, which contains a non-vanishing linearly polarized gluon distribution function. It is intriguing to calculate the dynamical spin polarization and correlation with both the unpolarized and linearly polarized gluon distribution functions in such a model using our framework. It is also worth noting that the QKT derived from the $\hbar$ expansion applied to the weak-field limit due to the gradient expansion in phase space. This approximation is similar to the linearization of the Yang-Mills equations by neglecting higher-order effects.  Future studies will be required to construct a self-consistent theory incorporating non-perturbative effects of strong color fields on spin transport. Fortuitously, the nonlinear corrections are more prominent at early times of order $X_0\leq Q_{s}^{-1}$, while the spin correlation is predominantly sensitive to late times. Thus, these higher-order effects may be suppressed.

\acknowledgments
This work was supported by the National Science and Technology Council (Taiwan) under Grant No. MOST 110-2112-M-001-070-MY3 and by the U.S. Department of Energy under Grant DE-FG02-05ER41367.

redpajama_set_name: RedPajamaArXiv


A covey of flushing quail on this graphic pocket T-shirt will speak to any sportsman.
Upland artist's graphic rendition of a covey of quail fly across the back of this substantial long-sleeved pocket T-shirt. Pure soft washed, garment dyed cotton with a double-needle collar and twill-taped neck and shoulders. Left chest pocket and rib cuffs. In white. Pure cotton. Washable. Imported.


When we have a tremendous amount of money for buying the real estate property, then we always stuck in complications while planning. It would be quite complicated for the people in order to choose the right property for the family members. Therefore, you should try The Antares, and this development is available on the Matter Road. Its developer FSKH development is a very talented worker those use their brain for making this fantastic impair for the people. The Antares Macpherson Estate is widespread in 201,115 Sqft of an area. Instead of this, there are about 13 districts that you find in this project.
The development of the Antares is just near to the MRT Station from where you can easily take the train daily in order to travel the city or out of state. Due to this, people are able to travel and save their traveling expense. In addition to this, all you need is the clubhouse that is also available just near to the project. Even you can check out the tennis court and 50m Swimming pool, where you will enjoy your holidays. Not only this, but there is also a sun deck which is really mesmerizing.
Development is perfectly available just near to many shopping malls which are available just along the vicinity. There is also a favorite place called Paya Lebar Quarter, City Plaza and the most mind-blowing one is The Poiz Centre, which is available just near to this development. Nevertheless, if you have issues or question about the development, then you can quickly ask questions for the sellers by visiting their official site. Due to this, shopping centers and many other shops which are available near to the building. Nonetheless, reviews will help you to understand the facts about the development so you should read them correctly.


The St. Tammany Parish Sheriff's Office is looking for a man accused of shooting another man in the buttocks during an altercation in Lacombe.
The St. Tammany Parish Sheriff's Office is looking for a man accused of shooting another man in the buttocks Tuesday during an altercation in Lacombe. Deputies met with the victim at Lacombe Heart Hospital just after 5 p.m. Tuesday, the sheriff's office said in a news release.
The victim told deputies he was outside a friend's house on Chestnut Street in Lacombe when three men walked up. He told deputies he knows two of the men.
The victim said because of an ongoing feud, he became engaged in a physical altercation with one of the men, who he identified as Oliver "Olly" Cousin, the sheriff's office said. Once the two were separated, the victim said Cousin pulled out a pistol and shot him as he attempted to flee, according to the sheriff's office.
A passing motorist picked up the victim and drove him to the hospital, according to the sheriff's office.
A second-degree battery warrant has been issued for Cousin.


text:
"What's the law on what ya can and can't say on a billboard?"
After her daughter is raped and murdered, and several months go by with no arrests, a Missouri mother, Mildred Hayes (Frances McDormand), rents three billboards displaying a message calling out the local police chief (Woody Harrelson) on his failed efforts.
Martin McDonagh is truly a master of dialogue. His previous films, In Bruges and Seven Psychopaths, have become cult classics, with audiences falling in love with their brilliant wordplay. Three Billboards Outside Ebbing, Missouri not only follows suit but eclipses the writer/directors previous work, presenting a more dramatic and heart wrenching tale of redemption set in the American mid-west.
Despite the best efforts of the marketing department at Fox Searchlight to convince us otherwise, this is not comedy. While there are some sharp comedic moments in the film, it's primarily a drama, and one that is grounded by incredible performances from its two leads. The depth McDonagh has given to his characters is truly outstanding, presenting them as fully realized, three-dimensional people who have conflicting emotions and sometimes-dubious motivations.
To call Frances McDormand's portrayal of Mildred Hayes powerful would be an understatement. She commands the film with every second she's on screen, creating a strong female protagonist that welcomely sits side-by-side with iconic characters like Karen Silkwood or her own Marge Gunderson. What makes her so interesting is that at her core Mildred is a conflicted character. While her intentions make sense, she is presented as not necessarily a good person. That's not to say she's bad though, she's incredibly likeable and the audience connect with her immediately, she just has a tendency to fly off the handle, which often has repercussions for herself and those around her. McDormand exudes confidence as the grieving mother, relishing the moments of rage against some of the more unforgiving townsfolk and flooring the audience with quiet reserve in her more reflective moments.
The fiery rage of Mildred is contrast brilliantly with the Harrelson's good-natured Chief Willoughby. Willoughby has a crucial role in the piece, acting as both a martyr and a father figure throughout. His calmness only helps add to the conflicting nature of Mildred's actions, while also working to mediate and control those that are enraged by them.
However, it's Sam Rockwell's performance as Jason Dixon, an unabashed racist police officer, which has the more powerful arc of the film. Rockwell does what he does best, imbedding small moments of comedic flair that quickly give way to an intensity that is profound. Dixon is the type of character that could have easily been played as a loathsome person, but instead Rockwell presents a man who is just ignorant and conflicted. There's a decent person buried deep inside Dixon and this is something Willoughby sees in him, which offers hope for a truly troubled man. It's possibly the finest performance Rockwell has given in his long career and will no doubt see him nominated for (if not taking home) the supporting actor awards at this year's Oscars.
At its core Three Billboards is a film about redemption and growth, a common theme in a lot of McDonagh's work. There's a minimalist nature to the staging of the film, with a lot of the one-on-one dialogue scenes playing out as if they're being presented on stage. However, McDonagh punctuates this style with sumptuous vistas of the rural countryside and the odd flourish of action based cinematography (one scene in particular playing out in a captivating single shot). Carter Burwell's wistful western infused score also helps to set a conflicting tone, giving the audience a sense of underlying dread as the film progresses.
While it may not have the comedic charm of In Bruges or Seven Psychopaths, Three Billboards Outside Ebbing, Missouri is unlike most films you're likely to see this year. It's filled with complex and layered characters all dealing with the aftermath of some horrific acts. McDonagh has outdone himself here, writing dialogue for his characters that is honest, hilarious and heartbreaking, which in turn prompts his cast to deliver career best performances across the board.
Three Billboards, while confronting at times, stands as not only one of the most original and unique films of the year, but will likely stand up as one of the better American films we're likely to see this decade.
Reviewed by Chris Swan.

redpajama_set_name: RedPajamaCommonCrawl


text:
When 6,000 eyes are watching you
Photo by Maria Montero Silva.
Maria Montero Silva, Editor-in-Chief
Around 3,000 attendees showed support for the UC football team in a tight match that the Pioneers lost in overtime.
Co-written by Kaitlyn Tambasco, Managing Editor
The highlight of the Homecoming weekend, the UC football home game brought approximately 3,000 people to the Charles A. Gaetano Stadium last Saturday, Oct. 19.
The Pioneers faced Morrisville in a very close game that kept the audience tied to their seats during the whole experience. Eventually, the UC team fell by a 54-51 score after three hours of intense football.
For head coach Blaise Faggiano, seeing his team play during Homecoming gave him extra motivation, especially when seeing former players and alumni coming back.
"Homecoming at Utica College has been pretty unique for me," Faggiano said. "It's definitely one of the best Homecoming atmospheres I've been around in my time as a college coach."
Having the opportunity of playing during Homecoming weekend is something that the Pioneers embrace although Faggiano admitted that it can also be a distraction.
"They need the ability to stay focused," he said. "In a game like that, it's all about who makes fewer mistakes."
While he regrets losing only by a score, Faggiano admitted that it was an "exciting" and "crazy" game.
"Unfortunately, we lost in overtime but it was a great effort by both teams. We have tough kids who played their hearts out."
Faggiano also stated that one of the secrets for his team's popularity is the fact that they are not just good students and football players but they also do a lot of community service and many of the players are involved in different campus organizations.
After weeks of hard work, seeing his team lose is a "heartbreak," because the players care so much, Faggiano said.
"They are really emotionally and physically invested," he said. "Whether you win or lose, you don't have much time to either celebrate or be upset before the next game because we only have a week so your whole week of work goes into one game."
Like any coach, Faggiano said he enjoys winning because "when you lose, you die a little inside." However, he sees defeat as an opportunity of teaching his guys about life, dealing with adversity and to keep moving forward.
Running Back Jamarious Morgan said he felt relaxed and his ability to focus led to his great performance on the field scoring two rushing touchdowns.
Morgan said it was a very good game and added that even though the team had a few mistakes, he is still "very proud of his teammates."
Having 3,000 people watching the game did not have an effect on Morgan, a junior, who said he felt "no pressure at all" and that it was great to see his friends and family supporting him.
Morgan also agreed with his coach saying that he and the team does not really think about the game after losing and that it is easy for him to move forward and pay attention to the next game.
"It was a tough loss but we all pick each other up," said Morgan, who has been playing football since he was 5 years old.
Losing during Homecoming does not really change Morgan's philosophy to let the past behind and to regard every game as an opportunity to give his 100 percent.
Linebacker and senior Antonio Scala described the Homecoming game as an "awesome atmosphere." He said the weather turned out, alongside a huge turnout for the tailgate. Scala said that as a player, he tries to focus more on the game rather than the fans, even though he said the game received a huge turnout as well.
"We try to make sure we do everything we can to win, so we try to keep our mind focused," Scala said. "Having a home field advantage with the crowd on our side is a blessing."
Like his teammate Jamarious Morgan, Scala has been playing football from a very young age and has been a Pioneer since his freshman year. He described the football game as being "electric," as the game gave him energy, motivation and good vibes.
"We have a buddy system where we have to go to other games as a team, where we show respect to all of the other teams but I do think football is a fun sport to watch," Scala said.
Scala explained that before the game, the team gets "pumped up" in the locker room, where they listen to different types of music. He noted that the locker room's favorite is "Pop Smoke."
"After the (Homecoming) game, we didn't finish the way we wanted to so it was kind of upsetting, especially because we played well in the game too," Scala said. "Something that we always say is 'short-term memory' so it happened and there's nothing you can do to change it and the next thing you can do is learn from it and get ready for the next game and do your best."
Scala said the team always thinks about the results of the game but focusing on it does not help anybody, but learning from it, is the trick.
Scala said that offense played great but defensively they could have done better.
"We're three and three and there's still a lot of football to be played and we could still have one of the best seasons here at Utica College,
Students, faculty, staff, alumni, families and friends attended the Homecoming football game to show support to the Pioneers.
Associate Director of Admissions and UC alum Joanne Pluff was there cheering for UC's team. She said that it was great to see a good turnout at the football game, along with the weather holding out.
"Homecoming is always the same, it's always the same excitement, it's awesome to see this many people come back, people you don't see everyday, so I love it, it's my favorite day on campus," Pluff said. "I love the football tailgate. Everybody is always there and they may not make it to the game but you get to see so many alums from years past."
Sophomore and Occupational Therapy major Brianna Guenter, was also at the game.
Guetner said she was excited to be at the game because she is on the Volleyball team, so going to other sporting events is rare, due to the team's schedule.
"Watching football comes from high-school," Guetner said. "It brings back the school spirit and it's so exciting to watch everyone."
Nick Metzger attended the game, and said he was friends with a lot of the members of the football team, such as football player Antonio Scala.
"I'm a big UC fan, I've been following the team for a few years," Metzger said.
He attended high-school with some of the members of the football team, so for him, "it's kind of a big deal."
Metzger said his favorite activity was the tailgate, where everyone was having fun and having a good time.
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El Comité Paralímpico Maltés es el comité paralímpico nacional que representa a Malta. Esta organización es la responsable de las actividades deportivas paralímpicas en el país. Es miembro del Comité Paralímpico Internacional y del Comité Paralímpico Europeo.

Referencias

Malta

redpajama_set_name: RedPajamaWikipedia


Description: This is an Army Corp of Engineers Lake. Full, Partial and dry camping is available. The lake designated as a is a trophy Bass fishing lake.
This is the type of campground we like. It is at the end of the road, so you have to want to stay here. The campground has large sites, with full and partial hookups. The dry camping spots offer more of a scenic view. Our favorite spot is Number 9, no hookups but is right(above the lake about 150 feet) on the lake. Good fishing for Bass, a boat ramp is available. The campground has a nice playground for children. Best time of the year is March and April, as all the wild flowers are in bloom. Summer can be HOT!
We are back after spending summer at 6700 feet. We forgot just how quiet it is at this campground. A mtn lion had it's cubs just over the hill from our favorite camp site this spring. The road into the campground has been reworked and is better, but it is not interstate smooth!!
Maybe we should tell all to stay away, it is getting harder to get an electric site. Good thing we can rough it w/o electricity and water.
I added the Bobcat picture. We had seen him 3 times in the last year, but always w/o the camera...will we ever learn? I received this photo from one of the Rangers who had a camera handy.
The Corp is adding 21 new electric sites to the campground. These new sites will be very well spaced out, unlike the present electric sites that are mostly all in a row. These electric sites will extend the camping season in comfort. A few of our favorite new electric sites will be; 39, 41, 48, and 51. All of these sites will accommodate a 40 footer plus.
Cons: Bumpy road from Chowchilla to the lake.
Nice place. Sites are ample in size and fairly level. Full hook-up. Great grassy area with a jungle gym type for the kids and area for adult lawn games. Would be too warm for me in the summer months, but for the spring and fall great. Lots of different birds.


Contact Chloe Monger directly; View Chloe's Full Profile. People Also Viewed. Amar Saxena. Consultant Neurosurgeon at NHS - University Hospitals Coventry and Warwickshire NHS Trust. Lisa Gilbert. Client Operations Manager at Aktrion Automotive. Meghana Pandit.
William Tierney (24 Oct Dec 2017) Share obituary on Facebook Chloe Monger lit a candle Comments. Paul and Bernie Collins lit a candle Comments. robert and valerie o'brien wrote. thing of you all sorry we could'nt be there with you all.love from all of us in canada.
Chloe Monger was born on month day 1777, at birth place, Connecticut, to Miles Monger and Sarah Monger. Miles was born on May , in Guilford, New Haven, Ct. Sarah was born on October , in North Bristol, CT.


The light is suspended from the ceiling.
The train was suspended because of the heavy rain.
Beijing is located in the North of China.
Are you clear about the capacity of the cinema?
We have ability to finish the work in two days.