Shear induced polarization: Collisional contributions

It has been suggested that orbital angular momentum carried by participants in off-central heavy ion collisions (HIC) can result in spin polarization of final state particles [1, 2]. Realistic model calculations have indicated that significant vorticity is present in quark-gluon plasma (QGP) produced in HIC [3, 4, 5]. Theoretical predictions of final particle spin polarization have been made based on a spin-orbit coupling picture [6, 7, 8]. Such a picture is indeed consistent with early experimental measurement of Lambda hyperon global polarization [9]. However, recent measurement of Lambda hyperon local polarization [10] shows an overall sign difference from theoretical predictions [11, 12, 13]. Different explanations have been proposed to understand the puzzle [14, 15], yet no consensus has been reached.

Recently it has been realized that shear can also contribute to spin polarization [16, 17]. In particular, it has been found based on a free theory analysis that spin responds to thermal vorticity and thermal shear in the same way. Phenomenological implementations have shown the right trend toward the measured local polarization results [18, 19, 20, 21, 22]. However, as we shall show in this paper, the contribution discussed so far is still incomplete. Vorticity and shear differ in one important aspect: the former does not change the particle distribution while the latter necessarily does. The redistribution of particles by shear flow leads to an extra contribution to spin polarization. The extra contribution can be consistently described in the framework of quantum kinetic theory (QKT), see [23] for a review and references therein. Rapid development of QKT has been made to include collisional term systematically via self-energy over the past few years[24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37]. The QKT is formulated using the Wigner function, whose axial component can be related to spin polarization. The axial component of Wigner function for fermion in a collisional QKT is given by [28, 37, 38]¹¹1The definitions of Wigner function in [28] and [37] differ by a sign. We use the latter definition.

$\displaystyle{\cal A}^{\mu}=-2{\pi}\hbar\left[a^{\mu}f_{A}+\frac{{\epsilon}^{{\mu}{\nu}{\rho}{\sigma}}P_{\rho}u_{\sigma}{\cal D}_{\nu}f}{2(P\cdot u+m)}\right]{\delta}(P^{2}-m^{2}),$

(1)

with $P$ and $u$ being momentum of particle and flow velocity. $a^{\mu}f_{A}$ is a dynamical contribution [38, 39, 40, 41, 42]. ${\cal D}_{\nu}$ is a covariant derivative acting on the distribution function $f$ defined as ${\cal D}_{\nu}={\partial}_{\nu}-{\Sigma}_{\nu}^{>}-{\Sigma}_{\nu}^{<}\frac{1-f}{f}$. The partial derivative term is what has been considered so far, the extra contribution comes from self-energies ${\Sigma}^{>/<}$. Naively one may expect the self-energy term to be suppressed by powers of coupling in a weakly coupled system described by the QKT. In fact this is not true. In a simple relaxation time approximation, the self-energy contribution can be estimated as $\frac{{\delta}f}{{\tau}_{R}}$. The appearance of ${\delta}f$ follows from the fact that the self-energy contribution in the covariant derivative vanishes in equilibrium by detailed balance. The combination $\frac{{\delta}f}{{\tau}_{R}}$ can be further related to ${\partial}f_{0}$ by kinetic equation with $f_{0}$ being local equilibrium distribution. Consequently the self-energy contribution is at the same order as the derivative one, with the dependence on coupling completely canceled between $\frac{1}{{\tau}_{R}}$ and ${\delta}f$.

A second question we attempt to address is the gauge dependence of spin polarization. Since theoretical calculation is usually done in the QGP phase while experiments measure particle after freezeout. The gauge dependence is only present in the partonic level calculations. On general ground, we expect that it is a gauge invariant spin polarization that is passed through freezeout. However, (1) is expressed in terms of self-energy, which is in general gauge dependent. It is necessary to include gauge link contribution to restore gauge invariance. Since collisions are mediated by off-shell particles, it is essential to consider quantum gauge field fluctuations in the gauge link. The quantum gauge field fluctuation also feels the flow via interaction with on-shell fermions. It turns out that there is a similar contribution associated with the gauge link, which is also at the same order as the derivative one. As a conceptual development, we generalize the definition of gauge link to the Schwinger-Keldysh contour, in which the collisional QKT is naturally derived. We also adapt the straight path widely used for background gauge field to the Schwinger-Keldysh contour to allow for consistent treatment of quantum gauge field fluctuations.

The aim of the paper is to evaluate the two contributions mentioned above. We illustrate the calculations by using a massive probe fermion in a massless QED plasma. While the method we use is applicable to arbitrary hydrodynamic flow, we consider the plasma with shear flow only for simplicity. The paper is organized as follows: in Section II, we briefly review the classical limit of QKT, which is the Boltzmann equation widely used in early studies of transport coefficients. By solving the Boltzmann equation we determines the particle redistribution in the presence of shear flow. The information of particle redistribution will be used to calculate the self-energy contribution and the gauge link contribution in Section III and IV respectively. Analytic results can be obtained at the leading logarithmic order. The results will be discussed and compared with the derivative contribution in Section V. Finally we summarize and provide outlook in Section VI.

We consider a QED plasma with $N_{f}$ flavor of massless fermions in a shear flow. The shear flow relaxes on the hydrodynamic scale, which is much slower than the relaxation of plasma constituents, thus we can take a steady shear flow. The presence of shear flow leads to redistribution of fermions and photons, which gives rise to off-equilibrium contribution to energy-momentum tensor responsible for shear viscosity. The kinetic equation addressing this problem has been written down long ago [43, 44, 45]. The kinetic equation is simply the Boltzmann equation with collision term given by elastic and inelastic scatterings. For simplicity we keep to the leading-logarithmic (LL) order, for which the inelastic scatterings are irrelevant. The resulting Boltzmann equations for fermion and photon read respectively

	$\displaystyle\left({\partial}_{t}+{\hat{p}}\cdot\nabla_{x}\right)f_{p}=$	$\displaystyle-\frac{1}{2}\int_{p^{\prime},k^{\prime},k}(2{\pi})^{4}{\delta}^{4}(P+K-P^{\prime}-K^{\prime})\frac{1}{16p_{0}k_{0}p_{0}^{\prime}k_{0}^{\prime}}\times$
		$\displaystyle\bigg{[}\;\;|{\cal M}|_{\text{Coul},f}^{2}\left(f_{p}f_{k}(1-f_{p^{\prime}})(1-f_{k^{\prime}})-f_{p^{\prime}}f_{k^{\prime}}(1-f_{p})(1-f_{k})\right)$
		$\displaystyle+|{\cal M}|_{\text{Comp},f}^{2}\left(f_{p}\tilde{f}_{k}(1+\tilde{f}_{p^{\prime}})(1-f_{k^{\prime}})-\tilde{f}_{p^{\prime}}f_{k^{\prime}}(1-f_{p})(1+\tilde{f}_{k})\right)$
		$\displaystyle+|{\cal M}|_{\text{anni},f}^{2}\left(f_{p}f_{k}(1+\tilde{f}_{p^{\prime}})(1+\tilde{f}_{k^{\prime}})-\tilde{f}_{p^{\prime}}\tilde{f}_{k^{\prime}}(1-f_{p})(1-f_{k})\right)\bigg{]},$		(2a)
	$\displaystyle\left({\partial}_{t}+{\hat{p}}\cdot\nabla_{x}\right)\tilde{f}_{p}=$	$\displaystyle-\frac{1}{2}\int_{p^{\prime},k^{\prime},k}(2{\pi})^{4}{\delta}^{4}(P+K-P^{\prime}-K^{\prime})\frac{1}{16p_{0}k_{0}p_{0}^{\prime}k_{0}^{\prime}}\times$
		$\displaystyle\bigg{[}\;\;|{\cal M}|_{\text{Comp},{\gamma}}^{2}\left(\tilde{f}_{p}f_{k}(1-f_{p^{\prime}})(1+\tilde{f}_{k^{\prime}})-f_{p^{\prime}}\tilde{f}_{k^{\prime}}(1+\tilde{f}_{p})(1-f_{k})\right)$
		$\displaystyle+2N_{f}|{\cal M}|_{\text{anni},{\gamma}}^{2}\left(\tilde{f}_{p}\tilde{f}_{k}(1-\tilde{f}_{p^{\prime}})(1-\tilde{f}_{k^{\prime}})-f_{p^{\prime}}f_{k^{\prime}}(1+\tilde{f}_{p})(1+\tilde{f}_{k})\right)\bigg{]}.$		(2b)

We have used $f_{p}$ and $\tilde{f}_{p}$ to denote distribution functions for fermions and photon carrying momentum $p$ respectively. $|{\cal M}|^{2}$ is partially summed amplitude square with the subscripts “Coul”, “Comp” and “anni” indicate Coulomb, Compton and annihilation processes respectively. The subscripts $f$ and ${\gamma}$ distinguish the fermionic and photonic amplitude squares, whose explicit expressions we shall present shortly. The overall factor $\frac{1}{2}$ on the RHS coming from spin average and $\int_{p}\equiv\int\frac{d^{3}p}{(2{\pi})^{3}}$. When there is imbalance between electron and position, there should be a separate equation for position. We restrict ourselves to neutral plasma, in which the positron distribution is identical to that of anti-fermion.

Now we can work out the redistribution of particles in the presence of thermal shear, given by solution to the Boltzmann equation. We solve (2) by noting that $f_{p}$ and $\tilde{f}_{p}$ on the LHS are the local equilibrium distributions and deviations of equilibrium distributions appear only on the RHS. We can parametrize the local equilibrium distribution by thermal velocity ${\beta}_{\mu}={\beta}u_{\mu}$ as $f^{(0)}_{p}=\frac{1}{e^{P\cdot{\beta}}+1}$ and ${\tilde{f}}^{(0)}_{p}=\frac{1}{e^{P\cdot{\beta}}-1}$ and the thermal shear is given by

$\displaystyle S_{ij}=\frac{1}{2}\left({\partial}_{i}{\beta}_{j}+{\partial}_{j}{\beta}_{i}\right)-\frac{1}{3}{\delta}_{ij}{\partial}\cdot{\beta}.$

(3)

When only thermal shear is present, we can evaluate the LHS as

		$\displaystyle{\hat{p}}_{i}\nabla_{i}f^{(0)}_{p}=-f^{(0)}_{p}(1-f^{(0)}_{p}){\partial}_{i}{\beta}_{j}\frac{p_{i}p_{j}}{E_{p}}=-f^{(0)}_{p}(1-f^{(0)}_{p})S_{ij}I^{p}_{ij}p,$
		$\displaystyle{\hat{p}}_{i}\nabla_{i}{\tilde{f}}^{(0)}_{p}=-{\tilde{f}}^{(0)}_{p}(1+{\tilde{f}}^{(0)}_{p}){\partial}_{i}{\beta}_{j}\frac{p_{i}p_{j}}{E_{p}}=-{\tilde{f}}^{(0)}_{p}(1+{\tilde{f}}^{(0)}_{p})S_{ij}I^{p}_{ij}p,$		(4)

with $I^{p}_{ij}={\hat{p}}_{i}{\hat{p}}_{j}-\frac{1}{3}{\delta}_{ij}$ being a symmetric traceless tensor defined with 3-momentum $p$. We have also replaced $P_{i}P_{j}$ by its traceless part by traceless property of $S_{ij}$. Following the method in [44], we parametrize the deviation of distributions by

$\displaystyle f^{(1)}_{p}=f^{(0)}_{p}(1-f^{(0)}_{p}){\hat{f}_{p}},\quad{\tilde{f}}^{(1)}_{p}={\tilde{f}}^{(0)}_{p}(1+{\tilde{f}}^{(0)}_{p}){\hat{\tilde{f}}_{p}},$

(5)

with the superscripts $(0)$ and $(1)$ counting the order of gradient. To linear order in gradient, the parametrization adopts simple relations for the collision term

		$\displaystyle f_{p}f_{k}(1-f_{p^{\prime}})(1-f_{k^{\prime}})-(p,k\leftrightarrow p^{\prime},k^{\prime})=f^{(0)}_{p}f^{(0)}_{k}(1-f^{(0)}_{p^{\prime}})(1-f^{(0)}_{k^{\prime}})({\hat{f}_{p}}+{\hat{f}_{k}}-{\hat{f}_{p^{\prime}}}-{\hat{f}_{k^{\prime}}}),$
		$\displaystyle f_{p}\tilde{f}_{k}(1+\tilde{f}_{p^{\prime}})(1-f_{k^{\prime}})-(p,k\leftrightarrow p^{\prime},k^{\prime})=f^{(0)}_{p}{\tilde{f}}^{(0)}_{k}(1+{\tilde{f}}^{(0)}_{p^{\prime}})(1-f^{(0)}_{k^{\prime}})({\hat{f}_{p}}+{\hat{\tilde{f}}_{k}}-{\hat{\tilde{f}}_{p^{\prime}}}-{\hat{f}_{k^{\prime}}}),$
		$\displaystyle f_{p}f_{k}(1+\tilde{f}_{p^{\prime}})(1+f_{k^{\prime}})-(p,k\leftrightarrow p^{\prime},k^{\prime})=f^{(0)}_{p}f^{(0)}_{k}(1+{\tilde{f}}^{(0)}_{p^{\prime}})(1+{\tilde{f}}^{(0)}_{k^{\prime}})({\hat{f}_{p}}+{\hat{f}_{k}}-{\hat{\tilde{f}}_{p^{\prime}}}-{\hat{\tilde{f}}_{k^{\prime}}}).$		(6)

By rotational symmetry, we expect

$\displaystyle{\hat{f}}_{p}=S_{ij}I^{p}_{ij}{\chi}(p),\quad{\hat{\tilde{f}}}_{p}=S_{ij}I^{p}_{ij}{\gamma}(p).$

(7)

Using (2) and (7), we obtain a linearized Boltzmann equation from (2):

		$\displaystyle-f_{p}(1-f_{p})S_{ij}I^{p}_{ij}p=-\frac{1}{2}\int_{p^{\prime},k^{\prime},k}(2{\pi}){\delta}^{4}(P+K-P^{\prime}-K^{\prime})\frac{1}{16p_{0}k_{0}p_{0}^{\prime}k_{0}^{\prime}}S_{ij}\times$
		$\displaystyle\qquad\qquad\qquad\bigg{[}|{\cal M}|_{\text{Coul},f}^{2}\left(I^{p}_{ij}{\chi}_{p}+I^{k}_{ij}{\chi}_{k}-I^{p^{\prime}}_{ij}{\chi}_{p^{\prime}}-I^{k^{\prime}}_{ij}{\chi}_{k^{\prime}}\right)f_{p}f_{k}(1-f_{p^{\prime}})(1-f_{k^{\prime}})$
		$\displaystyle\qquad\qquad\qquad+|{\cal M}|_{\text{Comp},f}^{2}\left(I^{p}_{ij}{\chi}_{p}+I^{k}_{ij}{\gamma}_{k}-I^{p^{\prime}}_{ij}{\gamma}_{p^{\prime}}-I^{k^{\prime}}_{ij}{\chi}_{k^{\prime}}\right)f_{p}\tilde{f}_{k}(1+\tilde{f}_{p^{\prime}})(1-f_{k^{\prime}})$
		$\displaystyle\qquad\qquad\qquad{\color[rgb]{0,0,0}+}|{\cal M}|_{\text{anni},f}^{2}\left(I^{p}_{ij}{\chi}_{p}+I^{k}_{ij}{\chi}_{k}-I^{p^{\prime}}_{ij}{\gamma}_{p^{\prime}}-I^{k^{\prime}}_{ij}{\gamma}_{k^{\prime}}\right)f_{p}f_{k}(1+\tilde{f}_{p^{\prime}})(1+\tilde{f}_{k^{\prime}})\bigg{]},$
		$\displaystyle-\tilde{f}_{p}(1+\tilde{f}_{p})S_{ij}I^{p}_{ij}p=-\frac{1}{2}\int_{p^{\prime},k^{\prime},k}(2{\pi}){\delta}^{4}(P+K-P^{\prime}-K^{\prime})\frac{1}{16p_{0}k_{0}p_{0}^{\prime}k_{0}^{\prime}}S_{ij}\times$
		$\displaystyle\qquad\qquad\qquad\bigg{[}|{\cal M}|_{\text{Comp},{\gamma}}^{2}\left(I^{p}_{ij}{\gamma}_{p}+I^{k}_{ij}{\chi}_{k}-I^{p^{\prime}}_{ij}{\chi}_{p^{\prime}}-I^{k^{\prime}}_{ij}{\gamma}_{k^{\prime}}\right)\tilde{f}_{p}f_{k}(1-f_{p^{\prime}})(1+\tilde{f}_{k^{\prime}})$
		$\displaystyle\qquad\qquad\qquad{\color[rgb]{0,0,0}+}|{\cal M}|_{\text{anni},{\gamma}}^{2}\left(I^{p}_{ij}{\gamma}_{p}+I^{k}_{ij}{\gamma}_{k}-I^{p^{\prime}}_{ij}{\chi}_{p^{\prime}}-I^{k^{\prime}}_{ij}{\chi}_{k^{\prime}}\right)\tilde{f}_{p}\tilde{f}_{k}(1-f_{p^{\prime}})(1-f_{k^{\prime}})\bigg{]},$		(8)

where we have used short-hand notations ${\chi}_{p}={\chi}(p)$ and ${\gamma}_{p}={\gamma}(p)$. $S_{ij}$ is arbitrary, thus we can equate its coefficient on two sides. The resulting tensor equations can be converted to scalar ones by contracting with $I^{p}_{ij}$. The flavor dependence in the amplitude squares can be expressed in terms of elementary amplitude squares as

		$\displaystyle|{\cal M}|_{\text{Coul},f}^{2}=2N_{f}|{\cal M}|_{\text{Coul}}^{2}$
		$\displaystyle|{\cal M}|_{\text{Comp},f}^{2}=|{\cal M}|_{\text{Comp}}^{2},\quad|{\cal M}|_{\text{Comp},{\gamma}}^{2}=2N_{f}|{\cal M}|_{\text{Comp}}^{2}$
		$\displaystyle|{\cal M}|_{\text{anni},f}^{2}=\frac{1}{2}|{\cal M}|_{\text{anni}}^{2},\quad|{\cal M}|_{\text{anni},{\gamma}}^{2}=N_{f}|{\cal M}|_{\text{anni}}^{2},$		(9)

with

	$\displaystyle|{\cal M}|_{Coul}^{2}=8e^{4}\frac{s^{2}+u^{2}}{t^{2}}$
	$\displaystyle|{\cal M}|_{Comp}^{2}=8e^{4}\frac{s}{-t}$
	$\displaystyle|{\cal M}|_{anni}^{2}=8e^{4}\left(\frac{u}{t}+\frac{t}{u}\right).$

The factor $2N_{f}$ in Coulomb case comes from scattering with $N_{f}$ fermions and anti-fermions. For scattering between identical fermions, the symmetry factor $\frac{1}{2}$ in the final state is compensated by an identical $u$-channel contribution to the LL accuracy. Similarly the factor $2N_{f}$ in the Compton case comes from scattering of photon with $N_{f}$ fermions and anti-fermions. The factor $N_{f}$ in photon pair annihilation corresponds to $N_{f}$ possible final states and $\frac{1}{2}$ in fermion pair annihilation is a final state symmetry factor.

The phase space integrations are performed in appendix A. The results turn the linearized Boltzmann equations (2) into

	$\displaystyle f^{(0)}_{p}(1-f^{(0)}_{p})\frac{2p}{3}=$	$\displaystyle e^{4}\ln e^{-1}\frac{1}{(2{\pi})^{4}}\Big{[}8N_{f}\frac{{\pi}^{3}\cosh^{-2}\frac{{\beta}p}{2}\left(6{\chi}_{p}+p((-2+{\beta}p\tanh\frac{{\beta}p}{2}){\chi}_{p}^{\prime}-p{\chi}_{p}^{\prime\prime})\right)}{72p^{2}{\beta}^{3}}$
		$\displaystyle\qquad\qquad\qquad+2\frac{{\chi}_{p}-{\gamma}_{p}}{p}\frac{{\pi}^{2}}{8{\beta}^{2}}\frac{4{\pi}}{3}f^{(0)}_{p}(1+{\tilde{f}}^{(0)}_{p})\Big{]}$
	$\displaystyle{\tilde{f}}^{(0)}_{p}(1+{\tilde{f}}^{(0)}_{p})\frac{2p}{3}=$	$\displaystyle e^{4}\ln e^{-1}\frac{1}{(2{\pi})^{4}}4N_{f}\frac{{\gamma}_{p}-{\chi}_{p}}{p}\frac{{\pi}^{2}}{8{\beta}^{2}}\frac{4{\pi}}{3}{\tilde{f}}^{(0)}_{p}(1-f^{(0)}_{p}).$		(10)

The second equation of (2) is algebraic. It is solved by

$\displaystyle\frac{{\gamma}_{p}-{\chi}_{p}}{(2{\pi})^{3}}=\frac{1}{e^{4}\ln e^{-1}}\frac{2{\beta}^{2}}{{\pi}^{2}N_{f}}p^{2}\frac{1+{\tilde{f}}^{(0)}_{p}}{1-f^{(0)}_{p}}.$

(11)

The first equation is differential and need to be solved numerically. In the limit ${\beta}p\gg 1$, the differential terms are subleading, reducing it to an algebraic equation. Combining with (11), we find the following asymptotic solution

$\displaystyle\frac{{\chi}(p\to\infty)}{(2{\pi})^{3}}=\frac{1}{e^{4}\ln e^{-1}}\frac{3(1+2N_{f}){\beta}^{2}p^{2}}{4{\pi}^{2}N_{f}^{2}}.$

(12)

We have combined ${\chi}_{p}$ and ${\gamma}_{p}$ with $\frac{1}{(2{\pi})^{3}}$ in (11) and (12). It is convenient as the same factor will appear in phase space integration measure.

The numerical solution is obtained with the boundary condition (12) and ${\chi}(p=0)=0$²²2A series analysis of the differential equation in (2) around $p=0$ indicate ${\chi}(p)\sim p^{2}$.. In fact, it has been pointed out in [44] that the ansatz ${\chi}_{p},{\gamma}_{p}\sim p^{2}$ gives very good approximation to the numerical solution. Fig. 1 compares (12) with numerical solution, confirming this point. As a further check, we calculate shear viscosity for plasma at constant temperature. In this case $T_{ij}=\eta TS_{ij}$. Expressing $T_{ij}$ using kinetic theory, we obtain

$\displaystyle\eta=\frac{1}{15}\int_{p}p\left[4N_{f}f_{p}(1-f_{p}){\chi}_{p}+2\tilde{f}_{p}(1+\tilde{f}_{p}){\gamma}_{p}\right].$

(13)

Integrations with numerical solution reproduces the corresponding entries in Table I of [45]. Integrations with approximate solution (11) and (12) gives results with an error of about $1\%$ for $N_{f}=1$ and about $3\%$ for $N_{f}=2$. We will simply use the approximate solution in the analysis below.

In the previous section, we have determined the redistribution of constituents in plasma with thermal shear. Now we introduce a massive probe fermion to the plasma and study its polarization in the shear flow. To this end, we need to calculate self-energy correction to axial component of its Wigner function (1). In general both Coulomb and Compton scatterings contribute to the self-energy³³3For probe fermion, pair annihilation is irrelevant.. Following [26], we take the heavy probe limit $m\gg eT$ so that the Coulomb scattering dominates in the self-energy. The Coulomb contribution to the self-energy diagram is depicted in Fig. 2.

We evaluate the self-energy as⁴⁴4${\Sigma}^{>}(x,y)$ is defined by $-e^{2}\langle{\not{A}}(x){\psi}(x)\bar{{\psi}}(y){\not{A}}(y)\rangle$.

	$\displaystyle{\Sigma}^{>}(P)=$	$\displaystyle{\color[rgb]{0,0,0}+}e^{4}N_{f}\int_{P^{\prime},K^{\prime},K}(2{\pi})^{4}{\delta}^{4}(P+K-P^{\prime}-K^{\prime}){\gamma}^{\mu}S^{>}(P^{\prime}){\gamma}^{\nu}D_{{\mu}{\beta}}^{22}(-Q)D_{{\alpha}{\nu}}^{11}(-Q)$
		$\displaystyle\times\text{tr}[{\gamma}^{\alpha}\underline{S}^{<}(K){\gamma}^{\beta}\underline{S}^{>}(K^{\prime})],$		(14)

with $\int_{P}=\int\frac{d^{4}P}{(2{\pi})^{4}}$ and $Q=P^{\prime}-P$. ${\Sigma}^{<}$ can be obtained by the replacement $>\leftrightarrow<$, $11\leftrightarrow 22$. The propagators in (3) are given by

	$\displaystyle S^{>}(P)=2{\pi}{\epsilon}(p_{0})({\not{P}}+m)(1-f_{p}){\delta}(P^{2}-m^{2}),$
	$\displaystyle\underline{S}^{>}(K)=2{\pi}{\epsilon}(k_{0}){\not{K}}(1-f_{k}){\delta}(K^{2}),$
	$\displaystyle D_{{\mu}{\beta}}^{22}(-Q)=\frac{ig_{{\mu}{\beta}}}{Q^{2}},\quad D_{{\alpha}{\nu}}^{11}(-Q)=\frac{-ig_{{\alpha}{\nu}}}{Q^{2}}.$		(15)

We have indicated propagators of medium fermions by an underline. $\underline{S}^{<}$ can be obtained by the replacement $1-f_{k}\to-f_{k}$. Feynman gauge is used for photon propagators. The component of self-energy contributing to polarization is ${\Sigma}^{>{\lambda}}=\frac{1}{4}\text{tr}\left[{\Sigma}^{>}(P){\gamma}^{\lambda}\right]$. The traces involved in this component are evaluated as

	$\displaystyle\text{tr}\left[{\gamma}^{\mu}S^{>}(P^{\prime}){\gamma}^{\nu}{\gamma}^{\lambda}\right]=$	$\displaystyle 4\left(P^{\prime}{}^{\mu}g^{{\nu}{\lambda}}+P^{\prime}{}^{\nu}g^{{\mu}{\lambda}}-P^{\prime}{}^{\lambda}g^{{\mu}{\nu}}\right)2{\pi}{\epsilon}(p_{0}^{\prime}){\delta}(P^{\prime}{}^{2}-m^{2})(1-f_{p^{\prime}}),$
	$\displaystyle\text{tr}\left[{\gamma}^{\alpha}S^{<}(K){\gamma}^{\beta}S^{>}(K^{\prime})\right]=$	$\displaystyle 4\left(K^{\alpha}K^{\prime}{}^{\beta}+K^{\beta}K^{\prime}{}^{\alpha}-K\cdot K^{\prime}g^{{\alpha}{\beta}}\right)(2{\pi})^{2}{\epsilon}(k_{0}){\epsilon}(k_{0}^{\prime})\times$
		$\displaystyle{\delta}(K^{2}){\delta}(K^{\prime}{}^{2})(-f_{k})(1-f_{k}^{\prime}).$		(16)

Note that the LL contribution arises from the regime $q\ll P,K$, we may replace ${\epsilon}(p_{0}^{\prime})\simeq{\epsilon}(p_{0})=1$ for probe fermion and ${\epsilon}(k_{0}){\epsilon}(k_{0}^{\prime})\simeq{\epsilon}(k_{0})^{2}=1$. Below we assume an equilibrium distribution for probe fermion for illustration purpose. Relaxation of this assumption only involves unnecessary complication. It can be important for realistic modeling of phenomenology, which will be studied elsewhere. The medium fermions is off-equilibrium, with the distribution determined in the previous section. The combination needed for polarization is $-f_{p}{\Sigma}_{k}^{>}(P)-(1-f_{p}){\Sigma}_{k}^{<}(P)$. Using (3) and (3), we obtain

		$\displaystyle-f_{p}{\Sigma}_{k}^{>}(P)-(1-f_{p}){\Sigma}_{k}^{<}(P)$
	$\displaystyle=$	$\displaystyle{\color[rgb]{0,0,0}-}16e^{4}N_{f}\int d^{3}kd^{3}q\frac{1}{(2{\pi})^{5}}{\delta}(p_{0}+k_{0}-p_{0}^{\prime}-k_{0}^{\prime})\frac{1}{8p_{0}^{\prime}k_{0}k_{0}^{\prime}}\left[2k_{k}P\cdot K-q_{k}P\cdot K\right]\frac{1}{\left(Q^{2}\right)^{2}}$
		$\displaystyle\times\left(f_{p}(1-f_{p^{\prime}})f_{k}(1-f_{k^{\prime}})-f_{p^{\prime}}(1-f_{p})f_{k^{\prime}}(1-f_{k})\right)$
	$\displaystyle=$	$\displaystyle{\color[rgb]{0,0,0}-}16e^{4}N_{f}\int d^{3}kd^{4}q\frac{1}{(2{\pi})^{5}}{\delta}(p_{0}-p_{0}^{\prime}+q_{0}){\delta}(k_{0}-k_{0}^{\prime}-q_{0})\frac{1}{8p_{0}^{\prime}k_{0}k_{0}^{\prime}}\left[k_{k}P\cdot K^{\prime}+k_{k}^{\prime}P\cdot K\right]$
		$\displaystyle\frac{1}{\left(Q^{2}\right)^{2}}S_{ij}\left(I^{k}_{ij}{\chi}_{k}-I^{k^{\prime}}_{ij}{\chi}_{k^{\prime}}\right)f^{(0)}_{p}f^{(0)}_{k}(1-f^{(0)}_{p^{\prime}})(1-f^{(0)}_{k^{\prime}})$
	$\displaystyle\equiv$	$\displaystyle S_{ij}R_{ijk}.$		(17)

We have inserted a factor of $2$ corresponding to fermions and anti-fermion in the loop and kept term up to $O(q^{2})$ in the square bracket. In the second equality, we have used the assumption that only the distribution of medium fermions is off-equilibrium. $R_{ijk}$ involves complicated tensor integrals of ${{\vec{k}}}$ and ${{\vec{k}}}^{\prime}$. They are evaluated by first converting to tensor integrals of ${{\vec{q}}}$ by rotational symmetry and ${\delta}(k_{0}-k_{0}^{\prime}-q_{0})$, which correlates ${\vec{k}}$ and ${\vec{q}}$. The resulting tensor integrals of ${\vec{q}}$ are further performed with rotational symmetry and ${\delta}(p_{0}-p_{0}^{\prime}+q_{0})$. Details of the evaluation can be found in appendix B. In the end, we find the following component relevant for spin polarization

$\displaystyle{\cal A}^{i}=2{\pi}\frac{{\epsilon}^{ijk}p_{j}R_{mnk}S_{mn}}{2(p_{0}+m)}{\delta}(P^{2}-m^{2})\simeq{\color[rgb]{0,0,0}-}\frac{1}{p_{0}+m}(I_{2}+I_{3})\frac{{\epsilon}^{iml}p_{n}p_{l}S_{mn}}{p^{5}}{\delta}(P^{2}-m^{2})C_{f},$

(18)

with

	$\displaystyle I_{2}=$	$\displaystyle\frac{{\pi}^{2}\cosh^{-2}\frac{{\beta}p_{0}}{2}\left((15p^{4}-87p^{2}p_{0}^{2}+72p_{0}^{4})\ln(\frac{p_{0}-p}{p_{0}+p})+\frac{8p^{5}}{p_{0}}-126p^{3}p_{0}+144pp_{0}^{3}\right)}{72{\beta}}$
		$\displaystyle+\frac{3\cosh^{-2}\frac{{\beta}p_{0}}{2}\left((12p^{2}p_{0}-12p_{0}^{3})\ln\frac{p_{0}-p}{p_{0}+p}+28p^{3}-\frac{28p^{5}}{3p_{0}^{2}}-24pp_{0}^{2}\right)\zeta(3)}{8{\beta}^{2}},$
	$\displaystyle I_{3}=$	$\displaystyle-\frac{\left((p^{4}-9p^{2}p_{0}^{2}+8p_{0}^{4})\ln\frac{p_{0}-p}{p_{0}+p}-\frac{38p_{0}p^{3}}{3}+16p_{0}^{3}p\right)\left({\pi}^{2}-9\tanh\frac{{\beta}p_{0}}{2}\zeta(3)\right)}{4{\beta}\left(1+\cosh({\beta}p_{0})\right)}.$		(19)

and $C_{f}=\frac{3N_{f}(1+2N_{f})}{4{\pi}^{2}N_{f}^{2}}$. We reiterate that the self-energy correction scales as ${\partial}f^{(0)}$, with the dependence on coupling cancels as follows: $e^{4}$ from vertices and $\ln e^{-1}$ from LL enhancement combine to give $\frac{1}{{\tau}_{R}}\sim e^{4}\ln e^{-1}$, which is canceled by a counterpart in $f^{(1)}\sim\frac{{\partial}f^{(0)}}{e^{4}\ln e^{-1}}$.

Before closing this section, we wish to comment on the gauge dependence of (3). We illustrate this with a comparison of Feynman gauge and Coulomb gauge. Let us rewrite (3) as

$\displaystyle{\Sigma}^{>}=e^{2}\int_{Q}{\gamma}^{\mu}S^{>}(P^{\prime}){\gamma}^{\nu}D_{{\mu}{\beta}}^{22}(-Q)D_{{\alpha}{\nu}}^{11}(-Q){\Pi}^{<{\alpha}{\beta}}(Q),$

(20)

with ${\Pi}^{<{\alpha}{\beta}}(Q)$ being the off-equilibrium photon self-energy. In the presence of shear flow, the self-energy can be decomposed into four independent tensor structures as

$\displaystyle{\Pi}^{<{\alpha}{\beta}}(Q)=P_{T}^{{\alpha}{\beta}}{\Pi}_{T}^{<}+P_{L}^{{\alpha}{\beta}}{\Pi}_{L}^{<}+P_{TT}^{{\alpha}{\beta}}{\Pi}_{TT}^{<}+P_{LT}^{{\alpha}{\beta}}{\Pi}_{LT}^{<}.$

(21)

Here $P_{T/L}$ are transverse and longitudinal projectors defined by

$\displaystyle P^{{\alpha}{\beta}}_{T}=P^{{\alpha}{\beta}}-\frac{P^{{\alpha}{\mu}}P^{{\beta}{\nu}}Q_{\mu}Q_{\nu}}{q^{2}},\quad P^{{\alpha}{\beta}}_{L}=P^{{\alpha}{\beta}}-P^{{\alpha}{\beta}}_{T},$

(22)

with $P^{{\alpha}{\beta}}=u^{\alpha}u^{\beta}-g^{{\alpha}{\beta}}$. $P_{TT}^{{\alpha}{\beta}}$ and $P_{LT}^{{\alpha}{\beta}}$ are emergent projectors owning to the shear flow, which are constructed as⁵⁵5The obvious structure constructed by sandwiching $S_{{\rho}{\sigma}}$ with two $P_{L}$ is not independent.

$\displaystyle P_{TT}^{{\alpha}{\beta}}=P_{T}^{{\alpha}{\rho}}S_{{\rho}{\sigma}}P_{T}^{{\sigma}{\beta}},\quad P_{LT}^{{\alpha}{\beta}}=P_{L}^{{\alpha}{\rho}}S_{{\rho}{\sigma}}P_{T}^{{\sigma}{\beta}}+(L\leftrightarrow T).$

(23)

Note that photon self-energy is gauge invariant but propagator is not. Now we illustrate gauge dependence is generically present by using Feynman and Coulomb gauges.

For LL accuracy, we can simply use bare photon propagators in (20). For spacelike momentum $Q$ relevant for our case, we have a simple relation $D^{11}_{{\alpha}{\beta}}=-D^{22}_{{\alpha}{\beta}}=-iD^{R}_{{\alpha}{\beta}}$. The retarded propagator in Feynman and Coulomb gauges have the following representations

	$\displaystyle\text{Feynman}:$	$\displaystyle\;D_{{\alpha}{\beta}}^{R}=P_{{\alpha}{\beta}}^{T}\frac{-1}{Q^{2}}+\left(\frac{Q^{2}}{q^{2}}u^{\alpha}u^{\beta}-\frac{q_{0}(u^{\alpha}Q^{\beta}+u^{\beta}Q^{\alpha})}{q^{2}}+\frac{Q^{\alpha}Q^{\beta}}{q^{2}}\right)\frac{-1}{Q^{2}}$
	$\displaystyle\text{Coulomb}:$	$\displaystyle\;D_{{\alpha}{\beta}}^{R}=P_{{\alpha}{\beta}}^{T}\frac{-1}{Q^{2}}+\left(\frac{Q^{2}}{q^{2}}u^{\alpha}u^{\beta}\right)\frac{-1}{Q^{2}}.$		(24)

Using (21) and (3), we easily seen contribution to ${\Sigma}^{>}$ from ${\Pi}_{T}^{<}$ and ${\Pi}_{TT}^{<}$ are identical in two gauges. For contribution from ${\Pi}_{L}^{<}$ and ${\Pi}_{LT}^{<}$, we use Ward identity ${\Pi}^{{\alpha}{\beta}}Q_{\alpha}=0$ and transverse conditions $P_{T/L}^{{\alpha}{\beta}}Q_{a}=0$, $P_{T}^{a\\ b}u_{\alpha}=0$ to find the following structures, which are present only in Feynman gauge

$\displaystyle{\Pi}_{L}^{<}P^{{\alpha}{\beta}}_{L}u_{\alpha}u_{\beta}Q_{\mu}Q_{\nu},\quad{\Pi}_{L}^{<}P^{{\alpha}{\beta}}_{L}u_{\alpha}u_{\beta}u_{\mu}Q_{\nu}+({\mu}\leftrightarrow{\nu}),\quad{\Pi}_{LT}^{<}P^{{\alpha}{\beta}}_{L}u_{\alpha}Q_{\mu}P^{T}_{{\beta}{\nu}}+({\mu}\leftrightarrow{\nu}).$

(25)

We have also confirmed the gauge dependence of self-energy contribution by explicit calculations.

The gauge dependence we found in the previous section should not be a surprise. The reason is the underlying quantum kinetic theory is derived using a gauge fixed propagators. For Wigner function of the probe fermion, its gauge dependence can be removed by inserting a gauge link. If the gauge field in the link is external, i.e. a classical background, the gauge link simply becomes a complex phase. However, when we consider self-energy of fermions arising from exchanging quantum gauge fields, we need to worry about ordering of quantum field operators from expanding the gauge link and interaction vertex. A systematic treatment of the ordering is still not available at present. We will follow a different approach. Since we have already obtained the axial component of Wigner function without gauge link, we will find correction from expanding the gauge link that contributing at the same order.

When fluctuations of quantum gauge fields appear both in the interaction vertices and in the gauge link, it is natural to order them on the Schwinger-Keldysh contour. The latter is also the base of collisional kinetic theory in the recent development of quantum kinetic theory. However we immediately find the well-known straight path for the gauge link becomes inadequate for the Wigner function joining points on forward and backward contours. To find a proper generalization in Schwingwer-Keldysh contour, let us take a close look at the gauge transformation of the bare Wigner function $S^{<}(x,y)$:

$\displaystyle S^{<}(x,y)\to e^{-ie{\alpha}_{2}(y)}S^{<}(x,y)e^{ie{\alpha}_{1}(x)},$

(26)

with ${\alpha}_{1,2}$ being gauge parameters on contour $1$ and $2$ respectively. If there is only classical background field, the gauge fields on contour $1$ and $2$ are the same, we may take ${\alpha}_{1}={\alpha}_{2}$. In this case, placing the straight path on either contour is equivalent. This is no longer true when quantum fluctuations are present. We propose to use double gauge links

$\displaystyle\bar{S}^{<}(x,y)={\psi}_{1}(x)\bar{{\psi}}_{2}(y)U_{2}(y,\infty)U_{1}(\infty,x),$

(27)

with $U_{i}(y,x)=\exp\left(-ie\int_{y}^{x}dw\cdot A_{i}(w)\right)$ and the $i=1,2$ identifying the forward and backward contours respectively. Assuming quantum fluctuations vanishes at past and future infinities, we easily arrive at the gauge invariance of (27). We have not specified the paths for the gauge links appearing in (27). A natural choice would be to take the straight line joining $x$ and $y$ and extending to future infinite. This is illustrated in Fig. 3. When there is only classical background gauge field, $A_{1}=A_{2}$ so that the two gauge links in (27) cancel partially, leaving a phase from the straight path between $x$ and $y$.

Now we are ready to evaluate possible corrections associated with the gauge link. Note that we need a correction of $O({\partial}f_{0})$. Such a contribution can arise from the diagram in Fig. 4. We shall evaluate its contribution to axial component of Wigner function below. Note that the diagram in Fig.4 contains one quantum fluctuation of gauge field from the link and the other from the interaction vertex. Both fluctuations can occur on either contour $1$ or $2$, and they need to be contour ordered. Enumerating all possible insertions of the two gauge fields along the Schwinger-Keldysh contour, we obtain

		$\displaystyle{\color[rgb]{0,0,0}-}e^{2}S_{11}(x,z){\gamma}^{\lambda}S^{<}(z,y)\left(\int_{\infty}^{x}dw^{\mu}D_{{\lambda}{\mu}}^{11}(z,w)+\int_{y}^{\infty}dw^{\mu}D_{{\lambda}{\mu}}^{<}(z,w)\right)$
		$\displaystyle{\color[rgb]{0,0,0}+}e^{2}S^{<}(x,z){\gamma}^{\lambda}S_{22}(z,y)\left(\int_{\infty}^{x}dw^{\mu}D_{{\lambda}{\mu}}^{>}(z,w)+\int_{y}^{\infty}dw^{\mu}D_{{\lambda}{\mu}}^{22}(z,w)\right),$		(28)

where the two lines corresponding to the vertex coordinate $z$ taking values on contour $1$ and $2$ respectively and the two terms in either bracket corresponding to link coordinate $w$ taking values on contour $1$ and $2$ respectively. The relative sign comes from sign difference of vertices on contour $1$ and $2$. $D_{{\lambda}{\mu}}^{>/<}$ stands for resummed photon propagators in medium with shear flow.

Using $S_{11}=-iS_{R}+S^{<}$ and $S_{22}=S^{<}+iS_{A}$ and the representation

	$\displaystyle S_{R}=ReS_{R}+\frac{i}{2}\left(S^{>}-S^{<}\right)\simeq\frac{i}{2}\left(S^{>}-S^{<}\right),$
	$\displaystyle S_{A}=ReS_{R}-\frac{i}{2}\left(S^{>}-S^{<}\right)\simeq-\frac{i}{2}\left(S^{>}-S^{<}\right),$		(29)

we obtain $S_{11}\simeq S_{22}\simeq\frac{1}{2}\left(S^{>}+S^{<}\right)$ with $ReS_{R}$ ignored in the quasi-particle approximation. Similar expressions can be obtained for $D_{R}$. Plugging the resulting expressions into (4), we have

		$\displaystyle{\color[rgb]{0,0,0}-}\frac{e^{2}}{2}\big{[}S^{>}(x,z){\gamma}^{\lambda}S^{<}(z,y)\int_{y}^{x}dw^{\mu}D_{{\lambda}{\mu}}^{<}(z,w)-S^{<}(x,z){\gamma}^{\lambda}S^{>}(z,y)\int_{y}^{x}dw^{\mu}D_{{\lambda}{\mu}}^{>}(z,w)\big{]}$
		$\displaystyle{\color[rgb]{0,0,0}-}\frac{e^{2}}{2}S^{<}(x,z){\gamma}^{\lambda}S^{<}(z,y)\int_{y}^{x}dw^{\mu}\left(D_{{\lambda}{\mu}}^{<}(z,w)-D_{{\lambda}{\mu}}^{>}(z,w)\right)$
		$\displaystyle{\color[rgb]{0,0,0}-}e^{2}S_{11}(x,z){\gamma}^{\lambda}S^{<}(z,y)\int_{\infty}^{x}dw^{\mu}\frac{1}{2}\left(D_{{\lambda}{\mu}}^{>}(z,w)-D_{{\lambda}{\mu}}^{<}(z,w)\right)$
		$\displaystyle{\color[rgb]{0,0,0}-}e^{2}S^{<}(x,z){\gamma}^{\lambda}S_{22}(z,y)\int_{y}^{\infty}dw^{\mu}\frac{1}{2}\left(D_{{\lambda}{\mu}}^{>}(z,w)-D_{{\lambda}{\mu}}^{<}(z,w)\right).$		(30)

The first line is very similar to what we have considered in self-energy correction. The other lines are all proportional to the photon spectral density ${\rho}_{{\lambda}{\mu}}(z,w)=D_{{\lambda}{\mu}}^{>}(z,w)-D_{{\lambda}{\mu}}^{<}(z,w)$, which is medium independent, thus the other lines are subleading compared to the first one. Below we keep only the first line.