Relaxation time for the alignment between quark spin and
angular velocity in a rotating QCD medium

Relativistic heavy-ion collisions are the best tool to explore, in a controlled manner, the properties of strongly interacting matter under extreme conditions. The study of the different observables emerging from these reactions has produced a wealth of results revealing an ever more complete picture of these properties for temperatures and densities close to or above the deconfinement transition. However, some other phenomena still miss a clearer understanding and pose a challenge for the evolving standard model of heavy-ion reactions. One of these observables is the relatively large degree of polarization of $\Lambda$ and $\overline{\Lambda}$ hyperons measured in semicentral collision for energies 2.5 GeV $\lesssim\sqrt{s_{NN}}\lesssim 27$ GeV, which shows an increasing trend as the energy and centrality of the collision decreases. The raising trend is different for $\Lambda$s than $\overline{\Lambda}$s STAR:2017ckg; STAR:2021beb; STAR:2023nvo; HADES:2022enx. For semicentral collisions, the matter density profile in the transverse plane induces the development of a global angular momentum, quantified in terms of the thermal vorticity Becattini2008; Becattini:2016gvu. Such angular momentum could be transferred to spin degrees of freedom and be responsible for the observed global polarization Becattini:2022zvf. This expectation is supported by the relation between rotation and spin, nowadays referred to as the Barnett effect, whereby a spinning ferromagnet experiences a change of its magnetization 1915PhRv....6..239B and the closely related Einstein–de Haas effect, based on the observation that a change in the magnetic moment of a free body causes this body to rotate 1915KNAB...18..696E. As a consequence, significant efforts have been devoted to quantify how this vorticity may be responsible for the magnitude of the observed polarization, assuming that the medium rotation is transferred to the spin polarization regardless of the microscopic mechanisms responsible for the effect Karpenko:2021wdm; DelZanna:2013eua; Karpenko:2013wva; Karpenko:2016jyx; Ivanov:2019ern; Wei:2018zfb; Vitiuk:2019rfv; Xie:2019jun; Ivanov:2020udj. However, the transferring of rotational motion to spin can only happen provided the medium induced reactions occur fast enough so that the alignment of the spin and angular velocity takes place on average within the lifetime of the medium. In the recent literature, this question has been addressed using different approaches Montenegro:2018bcf; Kapusta:2019ktm; Kapusta:2019sad; Montenegro:2020paq; Kapusta:2020dco; Kapusta:2020npk; Torrieri:2022ogj.

In a couple of recent works, we have explored whether this relaxation time for the alignment is short enough so that the observed polarization of hyperons can be attributed to the transferring of rotation to spin degrees of freedom Ayala:2020ndx; Ayala:2019iin. This is achieved by computing the interaction rate for the spin of a strange quark to align with the thermal vorticity, assuming an effective spin-vorticity coupling in a thermal QCD medium. The findings have been used to compute the $\Lambda$ and $\overline{\Lambda}$ polarization in the context of a core-corona model Ayala:2023xyn; Ayala:2022yyx; Ayala:2021xrn; Ayala:2020soy. The calculation resorts to computing the imaginary part of the self-energy of a vacuum quark whose propagator does not experience the effects of the rotational motion. To improve the description, also in a recent work, we have computed the propagation of a spin one-half fermion immersed in a rigid, cylindrical rotating environment Ayala:2021osy. For these purposes, we have followed the method introduced in Ref. Iablokov:2020upc which requires knowledge of the explicit solutions of the Dirac equation. These have been previously studied in different contexts by imposing different boundary conditions Chodos:1974je; Chernodub:2016kxh; Ambrus:2015lfr; Chen:2015hfc; Ebihara:2016fwa; Ambrus:2014uqa; Yamamoto:2013zwa; Gaspar:2023nqk.

In this work we use the propagator found in Ref. Ayala:2021osy to compute the imaginary part of the self-energy of a quark immersed in a rotating QCD medium at finite temperature ($T$) and baryo-chemical potential ($\mu_{B}$). We show that for values of $T$ and $\mu_{B}$ where the chiral symmetry restoration/deconfinement transition is thought to take place, the relaxation time for quarks turns out to be small enough, compared to the medium life-time, for the inferred, commonly accepted values of the medium angular velocity, after a semicentral heavy-ion collision. However, this is not the case for the antiquarks except for collision energies $\sqrt{s_{NN}}\gtrsim 50$ GeV. The work is organized as follows: In Sec. II we briefly revisit the derivation of the fermion propagator in a rotating environment. In Sec. III we use this propagator to compute the interaction rate for a quark spin to align with the vorticity in a QCD rotating medium at finite temperature and baryo-chemical potential. In Sec. LABEL:IV we compute the relaxation time for values of $T$ and $\mu_{B}$ close to the chiral symmetry restoration/deconfinement transition and show that for quarks this relaxation time is within the putative life-time of the system produced in the reaction, although this is not the case for antiquarks except for large collision energies. We finally summarize and conclude in Sec. LABEL:concl.

The physics within a relativistic rotating frame is most easily described in terms of a metric tensor resembling that of a curved space-time. We consider that the interaction region can be thought of as a rigid cylinder rotating around the $\hat{z}$-axis with constant angular velocity $\Omega$ which is produced in semicentral collisions. We can thus write the metric tensor as

$$g_{\mu\nu}=\begin{pmatrix}1-(x^{2}+y^{2})\Omega^{2}&y\Omega&-x\Omega&0\\ y\Omega&-1&0&0\\ -x\Omega&0&-1&0\\ 0&0&0&-1\\ \end{pmatrix}.$$

(1)

A fermion with mass $m$ within the cylinder is described by the Dirac equation

$$\left[i\gamma^{\mu}\left(\partial_{\mu}+\Gamma_{\mu}\right)-m\right]\Psi=0,$$

(2)

where $\Gamma_{\mu}$ is the affine connection. In this context, the $\gamma^{\mu}$-matrices in Eq. (2) correspond to the Dirac matrices in the rotating frame, which satisfy the usual anti-commutation relations

$$\{\gamma^{\mu},\gamma^{\nu}\}=2g^{\mu\nu}.$$

(3)

The relation between the gamma matrices in the rotating frame and the usual gamma matrices are

		$\displaystyle\gamma^{t}=\gamma^{0},\;\;\;\;\;\;\;\gamma^{x}=\gamma^{1}+y\Omega\gamma^{0},$		(4)
		$\displaystyle\gamma^{z}=\gamma^{3},\;\;\;\;\;\;\;\gamma^{y}=\gamma^{2}-x\Omega\gamma^{0}.$

In this notation, $\mu=\{t,x,y,z\}$ refers to the rotating frame while $\mu=\{0,1,2,3\}$ refers to the local rest frame. Therefore, Eq. (2) can be written as

	$\displaystyle\Big{[}i\gamma^{0}$	$\displaystyle\left(\partial_{t}-x\Omega\partial_{y}+y\Omega\partial_{x}-\frac{i}{2}\Omega\sigma^{12}\right)$		(5)
		$\displaystyle+i\gamma^{1}\partial_{x}+i\gamma^{2}\partial_{y}+i\gamma^{3}\partial_{z}-m\Big{]}\Psi=0.$

In the Dirac representation,

$$\sigma^{12}=\begin{pmatrix}\sigma^{3}&0\\ 0&\sigma^{3}\end{pmatrix},$$

(6)

where $\sigma^{3}=\mbox{diag}(1,-1)$ is the Pauli matrix associated with the third component of the spin. Therefore, we can rewrite Eq. (5) as

$\displaystyle\left[i\gamma^{0}\left(\partial_{t}+\Omega\hat{J}_{z}\right)+i\vec{\gamma}\cdot\vec{\nabla}-m\right]\Psi=0,$

(7)

where

$$\hat{J}_{z}\equiv\hat{L}_{z}+\hat{S}_{z}=-i(x\partial_{y}-y\partial_{x})+\frac{1}{2}\sigma^{12}.$$

(8)

This expression defines the total angular momentum in the $\hat{z}$ direction. The term $\hat{L}_{z}$ represents the orbital angular momentum, whereas $\hat{S}_{z}$ is the spin. On the other hand, the term $-i\vec{\nabla}$ is the usual momentum operator. We can find solutions to Eq. (7) in the form

$$\Psi(x)=\left[i\gamma^{0}\left(\partial_{t}+\Omega\hat{J}_{z}\right)+i\vec{\gamma}\cdot\vec{\nabla}+m\right]\phi(x),$$

(9)

and then, the function $\phi(x)$ satisfies a Klein-Gordon like equation

$$\left[\left(i\partial_{t}+\Omega\hat{J}_{z}\right)^{2}+\partial^{2}_{x}+\partial^{2}_{y}+\partial^{2}_{z}-m^{2}\right]\phi(x)=0.$$

(10)

Notice that the spin operator $\hat{S}_{z}$ when applied to $\phi(x)$ produces eigenvalues $s=\pm$1/2. Consequently, conservation of the total angular momentum expressed in terms of the eigenvalues $j=s+l$ imposes solutions with $l$ for $s=$1/2 and $l+1$ for $s=-$1/2. With these considerations, the solution of Eq. (10) can be written in cylindrical coordinates $(t,x,y,z)\to(t,\rho\sin\varphi,\rho\cos\varphi,z)$ as

$$\phi(x)=\begin{pmatrix}J_{l}(k_{\perp}\rho)\\ J_{l+1}(k_{\perp}\rho)e^{i\varphi}\\ J_{l}(k_{\perp}\rho)\\ J_{l+1}(k_{\perp}\rho)e^{i\varphi}\end{pmatrix}e^{-Et+ik_{z}z+il\varphi},$$

(11)

where $J_{l}$ are Bessel functions of the first kind,

$\displaystyle k_{\perp}^{2}=\tilde{E}^{2}-k_{z}^{2}-m^{2},$

(12)

is the transverse momentum squared and we have defined $\tilde{E}\equiv E+j\>\Omega$, representing the fermion energy observed from the inertial frame. Hence, the solution of Eq. (9) is

	$\displaystyle\Psi(x)=$	$\displaystyle\begin{pmatrix}\left[E+j\Omega+m-k_{z}+ik_{\perp}\right]J_{l}(k_{\perp}\rho)\\ \left[E+j\Omega+m+k_{z}-ik_{\perp}\right]J_{l+1}(k_{\perp}\rho)e^{i\varphi}\\ \left[-E-j\Omega+m-k_{z}+ik_{\perp}\right]J_{l}(k_{\perp}\rho)\\ \left[-E-j\Omega+m+k_{z}-ik_{\perp}\right]J_{l+1}(k_{\perp}\rho)e^{i\varphi}\end{pmatrix}$		(13)
		$\displaystyle\times e^{-(E+j\Omega)t+ik_{z}z+il\varphi}.$

Before writing the expression for the fermion propagator in the rotating environment, it is important to highlight some features of the solution. First, causality requires that $\Omega R<1$, where $R$ is the radius of the cylinder. Therefore, the solution is valid as long as $\Omega<1/R$. Second, we can simplify the solution assuming that the fermion is totally dragged by the vortical motion such that the angular position is determined by the product of the angular velocity and the time, specifically $\varphi+\Omega t=0$. This is a reasonable approximation when considering that during the early stages of a peripheral heavy-ion collision, particle interactions have not yet produced the development of a radial expansion. With this approximation, the propagator is translational invariant and can be simply Fourier transformed. With these features in mind, we write the fermion propagator $S(x,x^{\prime})$ as

$$S(x,x^{\prime})=\left[i\gamma^{0}\left(\partial_{t}+\Omega\hat{J}_{z}\right)+i\vec{\gamma}\cdot\vec{\nabla}+m\right]G(x,x^{\prime}),$$

(14)

where

$$G(x,x^{\prime})=(-i)\int_{-\infty}^{0}d\tau\sum_{\lambda}\exp{\left[-i\tau\lambda\right]}\phi_{\lambda}(x)\phi_{\lambda}^{\dagger}(x).$$

(15)

In this last expression, $\lambda$ and $\phi_{\lambda}(x)$ represent the eigenvalues and eigenvectors of Eq. (10). Taking $E,k_{\perp},k_{z},l$ as independent quantum numbers, the closure relation is written as

$$\sum_{\lambda}\phi_{\lambda}(x)\phi_{\lambda}^{\dagger}(x)=\sum_{l=\infty}^{\infty}\int\frac{dEdk_{z}dk_{\perp}k_{\perp}}{(2\pi)^{3}}\phi(x)\phi^{\dagger}(x)\\ =\begin{pmatrix}1&0&1&0\\ 0&1&0&1\\ 1&0&1&0\\ 0&1&0&1\end{pmatrix}\delta^{4}(x-x^{\prime}).$$

(16)

Hence, Eq. (14) becomes

$$S(x,x^{\prime})=\sum_{l=\infty}^{\infty}\int\frac{dEdk_{z}dk_{\perp}k_{\perp}}{(2\pi)^{3}}\Phi(\rho,\rho^{\prime})\frac{e^{-i(E-(l+1/2)\Omega)(t-t^{\prime})}e^{ik_{z}(z-z^{\prime})}e^{il(\varphi-\varphi^{\prime})}}{E^{2}-k_{z}^{2}-m^{2}-k_{\perp}^{2}+i\epsilon},$$

(17)

where

$$\Phi(\rho,\rho^{\prime})=\begin{pmatrix}\mathcal{A}\mathcal{J}_{l,l}&\mathcal{A}\mathcal{J}_{l,l+1}e^{-i\varphi^{\prime}}&\mathcal{A}\mathcal{J}_{l,l}&\mathcal{A}\mathcal{J}_{l,l}\\ \mathcal{B}\mathcal{J}_{l,l+1}e^{i\varphi}&\mathcal{B}\mathcal{J}_{l+1,l+1}e^{i(\varphi-\varphi^{\prime})}&\mathcal{B}\mathcal{J}_{l+1,l}e^{i\varphi}&\mathcal{B}\mathcal{J}_{l+1,l+1}e^{i(\varphi-\varphi^{\prime})}\\ \mathcal{C}\mathcal{J}_{l,l}&\mathcal{C}\mathcal{J}_{l,l+1}e^{-i\varphi^{\prime}}&\mathcal{C}\mathcal{J}_{l,l}&\mathcal{C}\mathcal{J}_{l,l}\\ \mathcal{D}\mathcal{J}_{l,l+1}e^{i\varphi}&\mathcal{D}\mathcal{J}_{l+1,l+1}e^{i(\varphi-\varphi^{\prime})}&\mathcal{D}\mathcal{J}_{l+1,l}e^{i\varphi}&\mathcal{D}\mathcal{J}_{l+1,l+1}e^{i(\varphi-\varphi^{\prime})}\end{pmatrix},$$

(18)

and we have defined

	$\displaystyle\mathcal{A}$	$\displaystyle\equiv\left[E+m-k_{z}+ik_{\perp}\right],$		(19)
	$\displaystyle\mathcal{B}$	$\displaystyle\equiv\left[E+m+k_{z}-ik_{\perp}\right],$
	$\displaystyle\mathcal{C}$	$\displaystyle\equiv\left[-E+m-k_{z}k+ik_{\perp}\right],$
	$\displaystyle\mathcal{D}$	$\displaystyle\equiv\left[-E+m+k_{z}k-ik_{\perp}\right],$

with

$$\mathcal{J}_{l,l^{\prime}}\equiv J_{l}(k_{\perp}\rho)J_{l^{\prime}}(k_{\perp}\rho^{\prime}).$$

(20)

Carrying out the integration and the summation, and taking the Fourier transform, we obtain

$$S(p)=\begin{pmatrix}\frac{p_{0}+\Omega/2-p_{z}+m+ip_{\perp}}{\left(p_{0}+\Omega/2\right)^{2}-p^{2}-m^{2}+i\epsilon}&0&\frac{p_{0}+\Omega/2-p_{z}+m+ip_{\perp}}{\left(p_{0}+\Omega/2\right)^{2}-p^{2}-m^{2}+i\epsilon}&0\\ 0&\frac{p_{0}+\Omega/2+p_{z}+m-ip_{\perp}}{\left(p_{0}-\Omega/2\right)^{2}-p^{2}-m^{2}+i\epsilon}&0&\frac{p_{0}+\Omega/2+p_{z}+m-ip_{\perp}}{\left(p_{0}-\Omega/2\right)^{2}-p^{2}-m^{2}+i\epsilon}\\ \frac{-(p_{0}+\Omega/2)-p_{z}+m+ip_{\perp}}{\left(p_{0}+\Omega/2\right)^{2}-p^{2}-m^{2}+i\epsilon}&0&\frac{-(p_{0}+\Omega/2)-p_{z}+m+ip_{\perp}}{\left(p_{0}+\Omega/2\right)^{2}-p^{2}-m^{2}+i\epsilon}&0\\ 0&\frac{-(p_{0}+\Omega/2)+p_{z}+m-ip_{\perp}}{\left(p_{0}-\Omega/2\right)^{2}-p^{2}-m^{2}+i\epsilon}&0&\frac{-(p_{0}+\Omega/2)+p_{z}+m-ip_{\perp}}{\left(p_{0}-\Omega/2\right)^{2}-p^{2}-m^{2}+i\epsilon}\end{pmatrix}.$$

(21)

We can write Eq. (21) in terms of the Dirac-gamma matrices as

$$S(P)=\frac{\left(p_{0}+\Omega/2-p_{z}+ip_{\perp}\right)\left(\gamma_{0}+\gamma_{3}\right)+m\left(1+\gamma_{5}\right)}{(p_{0}+\Omega/2)^{2}-p^{2}-m^{2}+i\epsilon}\mathcal{O}^{+}+\frac{(p_{0}-\Omega/2+p_{z}-ip_{\perp})(\gamma_{0}-\gamma_{3})+m(1+\gamma_{5})}{(p_{0}-\Omega/2)^{2}-p^{2}-m^{2}+i\epsilon}\mathcal{O}^{-},$$

(22)

where

$$\mathcal{O}^{\pm}=\frac{1}{2}\left[1\pm i\gamma^{1}\gamma^{2}\right]$$

(23)

is the spin projection operator. Notice that the derivation of the fermion propagator is performed in vacuum. To use this propagator including a finite temperature and chemical potential, recall that in equilibrium it is sufficiently general to make the replacement $p_{0}\to i\tilde{\omega}_{n}+\mu$ where $\omega_{n}=(2n+1)\pi T$ are Matsubara frequencies for fermions. Equation (22) represents our approximation for the fermion propagator in a cylindrical rigidly rotating environment. We now use this propagator to compute the relaxation time for the fermion spin to align with the angular velocity in the rotating medium.

In a QCD plasma in thermal equilibrium at temperature $T$ and baryon chemical potential $\mu_{B}$, the interaction rates $\Gamma^{\pm}$ for a quark with spin components $s=\pm 1/2$ in the direction of $\vec{\Omega}$ and four-momentum $P=(p_{0},\vec{p})$ to align its spin in the direction of the angular momentum vector can be expressed in terms of the total interaction rate, which in turn is given by the probability (per unit time) for a transition between the same quantum quark state $u^{\pm}$, represented by properly normalized spinors with a definite spin projection ($\pm$) along the direction of the angular velocity. This transition is mediated by the imaginary part of the self-energy, Im$\Sigma$. In symbols,

$$\Gamma^{\pm}(p_{0})\sim\bar{u}_{a}^{\pm}\text{Im}\Sigma_{ab}u_{b}^{\pm}.$$

(24)

Shuffling the indexes around, we can also write

	$\displaystyle\Gamma^{\pm}(p_{0})$	$\displaystyle\sim$	$\displaystyle u_{b}^{\pm}\bar{u}_{a}^{\pm}\text{Im}\Sigma_{ab}$		(25)
		$\displaystyle=$	$\displaystyle\text{Tr}[\mathcal{O}^{\pm}\text{Im}\Sigma_{ab}],$

where we used that the spin projection operators $\mathcal{O}^{\pm}$ are given by

$\displaystyle\mathcal{O}^{\pm}\equiv u^{\pm}\bar{u}^{\pm}.$

(26)

To extract the creation rate for e for spin-aligned quark states from the total interaction rate, as discussed in Ref. LeBellac, we multiply the total interaction rate by the fermion distribution function $\tilde{f}$, for a grand-canonical ensemble in the presence of a conserved charge to which a quark chemical potential $\mu=1/3\mu_{B}$ is associated, namely,

$\displaystyle\Gamma^{\pm}(p_{0})=\tilde{f}(p_{0}-\mu\mp\Omega/2)\text{Tr}\left[O^{\pm}\;\text{Im}\Sigma\right].$

(27)

In previous analyses Ayala:2020ndx; Ayala:2019iin the interaction has been modeled using an effective vertex coupling the thermal vorticity and the quark spin. To improve the description, hereby we consider the case where the fermion is subject to the effect of a rotation within a rigid cylinder. The one-loop contribution to $\Sigma$, depicted in Fig. 1, is given by

$$\Sigma(P)=T\sum_{n}\int\frac{d^{3}k}{(2\pi)^{3}}\gamma^{\mu}T_{a}S(P-K)\gamma^{\nu}T_{b}\;^{*}G_{\mu\nu}^{ab},$$

(28)

where $S$ is the quark propagator in a rotating environment obtained in Eq. (22), ${}^{*}G_{\mu\nu}^{ab}$ is the effective gluon propagator in the thermal medium and $T_{a}$ are the $SU(3)$ group generators. The four-momenta are $P=(i\tilde{\omega},\vec{p})$ for the fermion and $K=(i\omega_{n},\vec{k})$ for the gluon, with $\omega_{n}$ being the gluon Matsubara frequencies. Also, ${}^{*}G_{\mu\nu}^{ab}=^{*}\!\!G_{\mu\nu}\delta^{ab}$ and in the hard thermal loop (HTL) approximation ${}^{*}G_{\mu\nu}$ is given by

$${}^{*}G_{\mu\nu}(K)=\Delta_{L}(K)P_{L\;\mu\nu}+\Delta_{T}(K)P_{T\;\mu\nu}$$

(29)

where $P_{L,T\;\mu\nu}$ are the polarization tensors for three-dimensional longitudinal and transverse gluons LeBellac. The gluon propagator functions for longitudinal and transverse modes, $\Delta_{L,T}(K)$, are given by

$$\Delta_{L}^{-1}(K)=K^{2}+2m_{T}^{2}\frac{K^{2}}{k^{2}}\left[1-\frac{i\omega_{n}}{k}Q_{0}\left(\frac{i\omega_{n}}{k}\right)\right],$$

(30)

	$\displaystyle\Delta_{T}^{-1}(K)$	$\displaystyle=$	$\displaystyle-K^{2}-m_{T}^{2}\left(\frac{i\omega_{n}}{k}\right)$		(31)
		$\displaystyle\times$	$\displaystyle\left\{\left[1-\left(\frac{i\omega_{n}}{k}\right)^{2}\right]Q_{0}\left(\frac{i\omega_{n}}{k}\right)+\left(\frac{i\omega_{n}}{k}\right)\right\},$

where

$$Q_{0}(x)=\frac{1}{2}\ln\left(\frac{x+1}{x-1}\right),$$

(32)

and $m_{T}^{2}$ is the gluon thermal mass squared given by

$$m_{T}^{2}=\frac{1}{6}g^{2}\mathcal{C}_{A}T^{2}+\frac{1}{12}g^{2}\mathcal{C}_{F}(T^{2}+\frac{3}{\pi^{2}}\mu^{2}),$$

(33)

where $\mathcal{C}_{A}$ and $\mathcal{C}_{F}$ are the Casimir factors for the adjoint and fundamental representations of $SU(3)$.

It is convenient to first look at the sum over Matsubara frequencies for the products of the propagator functions for longitudinal and transverse gluons, $\Delta_{i}$ with $i=L,T$, and the Matsubara propagator for the quark in a rotating environment $\tilde{\Delta}$, wich is described in Ref. LeBellac, can be obtained as the inverse of the denominator of each of the components of Eq. 22 with the replacement $p_{0}\to i\tilde{\omega}_{n}+\mu$.

$$S_{i}(i\omega)=T\sum_{n}\Delta_{i}(i\omega_{n})\tilde{\Delta}(i(\omega-\omega_{n})).$$

(34)

The sum can be performed introducing the spectral densities $\rho_{i}$ and $\rho_{F}$ for the gluon and fermion, respectively. The imaginary part of $S_{i}$ can be written as

	$\displaystyle\text{Im}(S_{i})\!\!\!$	$\displaystyle=$	$\displaystyle\!\!\!\pi(e^{\beta(p_{0}-\mu-\Omega/2)}+1)\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{dk_{0}}{2\pi}\frac{dp^{\prime}_{0}}{2\pi}f(k_{0})$		(35)
		$\displaystyle\times$	$\displaystyle\!\!\tilde{f}(p^{\prime}_{0}-\mu\mp\Omega/2)\delta(p_{0}-k_{0}-p^{\prime}_{0})$
		$\displaystyle\times$	$\displaystyle\rho_{i}(k_{0},k)\rho_{F}(p^{\prime}_{0},p-k),$

where $f(k_{0})$ is the Bose-Einstein distribution. The spectral densities $\rho_{i}$ are obtained from the imaginary part of $\Delta_{i}(i\omega_{n})$ after the analytic continuation $i\omega_{n}\rightarrow k_{0}+i\epsilon$ and contain the discontinuities of the gluon propagator across the real $k_{0}$ axis. Their support depends on the ratio $x=k_{0}/k$. For $|x|>1$, $\rho_{i}$ have support on the (timelike) quasi-particle poles. For $|x|<1$, their support coincides with the branch cut of $Q_{0}(x)$ and corresponds to Landau damping. On the other hand, the fermion spectral density is

$$\rho_{F}(p_{0}^{\prime},p)=-2\pi\delta\left((p_{0}^{\prime}\pm\Omega/2)^{2}-p^{2}-m^{2}\right),\\$$

(36)

We now concentrate on the trace factors required for the computation of Eq. (27). The term proportional to the fermion momentum and angular velocity

$\displaystyle P_{L,T\;\mu\nu}\text{Tr}\left[\gamma^{\mu}\left(\gamma^{0}\pm\gamma^{3}\right)\left(1\pm i\gamma^{1}\gamma^{2}\right)\gamma^{\nu}\right]=0,$

(37)

vanishes identically, whereas the terms proportional to the fermion mass are given by

	$\displaystyle P_{L\;\mu\nu}\text{Tr}\left[\gamma^{\mu}\left(1+\gamma^{5}\right)\left(1\pm i\gamma^{1}\gamma^{2}\right)\gamma^{\nu}\right]$	$\displaystyle=$	$\displaystyle-4$
	$\displaystyle P_{T\;\mu\nu}\text{Tr}\left[\gamma^{\mu}\left(1+\gamma^{5}\right)\left(1\pm i\gamma^{1}\gamma^{2}\right)\gamma^{\nu}\right]$	$\displaystyle=$	$\displaystyle-8.$		(38)

The delta functions in Eqs. (35) and (36) restrict the integration over gluon energies to the spacelike region, $|x|<1$. Therefore, the parts of the gluon spectral densities that contribute to the interaction rate are given by

$$\rho_{L}(k_{0},k)=\frac{2\pi m_{T}^{2}x\theta(1-x^{2})}{\left[k^{2}+2m_{T}^{2}\left(1-\frac{x}{2}\ln\left|\frac{x+1}{x-1}\right|\right)\right]^{2}+\pi^{2}m_{T}^{4}x^{2}},$$

(39)

$$\rho_{T}(k_{0},k)=\frac{2\pi m_{T}^{2}x(1-x^{2})\theta(1-x^{2})}{2\left[\left(k^{2}(x^{2}-1)-m_{T}^{2}\left[x^{2}+\frac{x(1-x^{2})}{2}\ln\left|\frac{x+1}{x-1}\right|\right]\right)^{2}+\frac{\pi^{2}}{4}m_{T}^{4}x^{2}(1-x^{2})^{2}\right]}.$$

(40)

With all these ingredients we write the interaction rate as {IEEEeqnarray}rCl Γ^±(p_0) & = g2m CFπ2 ∫d3k(2π)3 ∫_-∞ ^∞dk02π ∫_-∞ ^∞dp_0 ^′
× f(k_0)(4ρ_L(k_0) + 8ρ_T(k_0) )
×~f(p_0 ^′- μ∓Ω/ 2) δ(p_0 - k_0 - p_0 ^′)
× δ((p_0 ^′±Ω/2)^2 - E^2), with

$$E^{2}=|\vec{p}-\vec{k}|^{2}-m^{2}.$$

(41)

Notice that

	$\displaystyle\delta\left(\left(p_{0}^{\prime}\pm\Omega/2\right)^{2}-E^{2}\right)=$
	$\displaystyle\frac{1}{2E}\Big{[}\delta(p_{0}^{\prime}\pm\Omega/2-E)+\delta(p_{0}^{\prime}\pm\Omega/2+E)\Big{]}.$		(42)

Therefore, we can integrate Eq. (1) over $p_{0}^{\prime}$ to obtain {IEEEeqnarray}rCl Γ^±(p_0) & = g2m CFπ2 ∫d3k(2π)3 ∫_-∞ ^∞12Edk02π f(k_0)
× (4ρ_L(k_0) + 8ρ_T(k_0) )
× [~f(E - μ∓Ω)δ(p_0 - k_0 - E ±Ω/2)
+ ~f(-E - μ∓Ω)δ(p_0 - k_0 + E ±Ω/2)]. Notice that for the considered angular velocities appropriate to the early stages of the collision (10 MeV $\lesssim\Omega\lesssim 14$ MeV), and for a strange quark mass $m\sim 100$ MeV, the combination $E\pm\Omega/2$ can always be safely regarded as being positive. The kinematical constraint imposed by the first of the delta functions in Eq. (III) corresponds to the rate to produce rotating and thermalized quarks originated by the dispersion of vacuum nonrotating quarks as a result of dispersion with medium quarks. This is depicted in Fig. 2.

We then single out this contribution from the total rate which can then be written as {IEEEeqnarray}rCl Γ^±(p_0) & = g2m CFπ2 ∫k2dk d(cosθ)dϕ(2π)3 ∫_-∞ ^∞12Edk02π
× (4ρ_L(k_0) + 8ρ_T(k_0) )δ(p_0 - k_0 - E ±Ω/2)
× f(k_0) ~f(E -μ∓Ω).

The kinematical restrictions for the $k_{0}$ integration translate into integration regions $\mathcal{R^{\pm}}$. After integrating over the angle $\theta$ between $\vec{p}$ and $\vec{k}$, and over the azimuthal angle $\phi$, and finally using that $E^{2}=p^{2}+\ m^{2}=|\vec{p}-\vec{k}|^{2}-m^{2}=p^{2}+k^{2}-2pk\cos{\theta}+m^{2}$, we obtain {IEEEeqnarray}rCl Γ^±(p_0) & = g2m CFπ2∫_0 ^∞dk k2(2π)3∫_R^±dk_0 f(k0)2pk
×