0.0.0.1 Introduction.

Relativistic heavy-ion collisions provides a unique environment for studying strongly interacting matter, known as the quark-gluon plasma (QGP), under extremely high temperature [3]–[5]. After a short pre-equilibrium evolution, the system is assumed to achieve local thermodynamic equilibrium (LTE) and its evolution is thereafter effectively described by relativistic hydrodynamics [6]–[10]. Physical quantities are then characterized by slowly varying macroscopic variables, such as flow velocity, temperature, and chemical potential. The Zubarev’s approach and its reformulation [11]–[15] offers a rigorous framework to obtain physical observables in a relativistic fluid based on quantum statistical field theory at local equilibrium, which is especially useful for spin [1], [15]–[23], an inherently quantum observable [24]–[34].

If the QGP achieves LTE on some initial hypersurface \(\Sigma_0\), the density operator is obtained by maximizing the entropy \(S=-\text{tr}(\widehat\rho\,\text{log}\widehat\rho)\) with constrained energy-momentum and charge densities. With the Belinfante stress-energy operator \(\widehat{T}^{\mu\nu}\) [35], and neglecting the charge density for the sake of simplicity (this is a good approximation in very high energy nuclear collisions), the density operator takes the following form [11]–[13]: \[\label{LE-density-operator} \widehat{\rho}=\frac{1}{Z}\exp\left[-\int_{\Sigma_0} \!\! {\rm d}\Sigma_{\mu}(y)\; \widehat{T}^{\mu\nu}(y)\beta_{\nu}(y)\right]\,,\tag{1}\] where \(\beta\) is the four-temperature vector. By means of the Gauss’ theorem, the exponent in the operator 1 can be transformed into the sum of a 3D integral at the decoupling hypersurface \(\Sigma_{\rm D}\), that is when the fluid breaks up, ¹ and a 4D integral over the region encompassed by \(\Sigma_{\rm D}\) and \(\Sigma_0\), which is essentially the region of existence of the QGP: \[\label{gauss} \int_{\Sigma_{\rm D}} \!\! {\rm d}\Sigma_{\mu}(y) \; \widehat{T}^{\mu\nu}(y)\beta_{\nu}(y) + \int_\Omega {\rm d}^4 y \; \widehat{T}^{\mu\nu}(y) \partial_\mu \beta_\nu(y)\;.\tag{2}\] While the first term is the LTE integral at the decoupling, the second term accounts for the dissipative terms of physical observables [12], [15]. In this work, we will confine ourselves to the calculation of observables, notably spin polarization, at LTE at the decoupling, so that we will neglect the second integral in the eq. 2 .

In previous spin polarization calculations with the LTE density operator, significant geometric approximations were employed, explicitly or not [1], [2]. However, according to analytical models [36]–[38] and numerical simulations [39]–[42], the decoupling hypersurface in the central region of the QGP has roughly constant proper time \(\tau=\sqrt{t^2-z^2}\) and the geometric approximations in ref. [1] appear not fully appropriate. In this work we propose a new approach for evaluating quantities at LTE, which applies to hypersurfaces with arbitrary geometry; in essence, this method relies on an exchange of the order of integrations for the leading order correction of the Wigner function, which sheds light on the nature of the correction itself. Although our derivation is focused on the spin-1/2 case the method can be readily extended to particles with arbitrary spin.

0.0.0.2 Local thermodynamic equilibrium.

For an arbitrary local operator \(\widehat{O}(x)\), its mean value at LTE at the decoupling is obtained by neglecting the 4D integral in equation 2 and it is thus given by: \[O_{\rm LE}(x)\! \equiv\! \frac{1}{Z}{\rm Tr}\! \left( \exp\left[-\int_{\Sigma_{\rm D}} \!\! {\rm d}\Sigma_{\mu}(y) \widehat{T}^{\mu\nu}(y)\beta_{\nu}(y)\right]\widehat{O}(x)\! \right).\] Since we are interested in the value of the local operator in \(x\), we can replace \(\beta_\nu(y)\) in the above equation with: \[\beta_\nu(y) = \beta_\nu(x) + \left( \beta_\nu(y)-\beta_\nu(x) \right) \equiv \beta_\nu(x) + \Delta \beta_\nu(y,x)\] and rewrite the above equation as: \[\begin{align} \label{lteop} O_{\rm LE}(x) = \frac{1}{Z}& {\rm Tr}\left( \exp\left[- \beta(x) \cdot \widehat P \right.\right. \nonumber \\ & \left.\left. - \int_{\Sigma_{\rm D}} \!\! {\rm d}\Sigma_{\mu}(y)\; \widehat{T}^{\mu\nu}(y) \Delta\beta_{\nu}(y)\right]\, \widehat{O}(x)\right), \end{align}\tag{3}\] where we have taken out \(\beta_\nu(x)\) from the integral and used the identity of the four-momentum operator \(\widehat{P}^\nu = \int_{\Sigma_{\rm D}} \!\! {\rm d}\Sigma_{\mu}(y)\; \widehat{T}^{\mu\nu}(y)\).

In the hydrodynamic limit, with a clear separation between the microscopic scale, associated to the correlation lengths of the operators, and the scale over which the \(\beta\) field significantly varies, the first term in the exponent of 3 dominates over the second which is expected to provide a correction. We can then use linear response theory to obtain a good approximation of the mean value of the operator \(\widehat{O}(x)\) at LTE: \[O_{\rm LE}(x)=O_{\rm GE}(x)+\Delta O_{\rm LE}(x)\,,\] where \(O_{\rm GE}(x)\) is the global equilibrium mean value calculated with a four-temperature equal to \(\beta(x)\) and \(\Delta O_{\rm LE}(x)\) is the leading correction in the linear response theory, that is: \[\begin{align} \label{deltaO} \Delta O_{\rm LE}(x)&=& -\int_{\Sigma_{\rm D}} \!\! {\rm d}\Sigma_{\mu}(y) \; \int_{0}^{1} {\rm d}z \; \Delta\beta_{\nu}(y,x) \nonumber\\ &&\!\!\!\! \times\left\langle \widehat{O}(x),\,{\rm e}^{- z \beta(x) \cdot {\widehat P}} \widehat{T}^{\mu\nu}(y) {\rm e}^{z\beta(x)\cdot {\widehat P}} \right\rangle _{c,\text{GE}}, \end{align}\tag{4}\] where \(\left\langle\cdots\right\rangle_{c,\text{GE}}\) denotes the connected mean value at GE, i.e., with subtraction of the product of the mean values of the two operators separated by the comma.

0.0.0.3 Wigner function.

The covariant Wigner function \(W(x,p)\) is a quantum generalization of the classical phase space distribution [46]–[49]; all physical observables can be expressed with suitable integrals of the Wigner function. For free spin-1/2 particles, it is defined as the mean value of the Wigner operator: \[\label{Wigner-operator} \widehat{W}_{ab}^+(x,p)\equiv\theta(p^0)\!\int\frac{{\rm d}^{4}y}{(2\pi)^{4}}{\rm e}^{-ip\cdot y} \overline{\Psi}_{b}\left(x+\frac{y}{2}\right)\Psi_a\left(x-\frac{y}{2}\right),\tag{5}\] where \(\Psi\) denotes the Dirac field and \(a\), \(b\) are spinor indices. The step function \(\theta(p^0)\) ensures that only particle contributions are included (for antiparticles the procedure is alike). By using the plane wave expansion of the field, one can obtain the following integral expression: \[\begin{align} \label{Wigner-operator2} \widehat{W}^+(x,p) &=& \frac{1}{2(2\pi)^3}\sum_{r,s} \int {\rm d}^{4} q \; {\rm e}^{iq\cdot x} \theta(p_+^0)\theta(p_-^0) \delta(p \cdot q) \nonumber \\ && \times \delta \left( p^2 + \frac{q^2}{4} - m^2 \right) \widehat a^{\dagger}_r (p_+) \widehat a_s(p_-) u_s (p_-) \bar u_r(p_+)\,, \end{align}\tag{6}\] where \(p_\pm\equiv p\pm q/2\) and \(\widehat{a}_s(p_-)\) is the annihilation operator for a particle with momentum \(p_-\) (note that \(p_\pm\) is on-shell) and spin component \(s\) and the \(u\)’s are the familiar spinors solution of the Dirac equation. Plugging the equation 6 into the equation 4 with \(\widehat{O}(x)=\widehat{W}^+(x,p)\) we obtain: \[\begin{align} \label{Wigner-off-GTE} &\Delta W_{\rm LE}^+(x,p) =\frac{1}{2(2\pi)^3}\sum_{r,s} \int_{\Sigma_{\rm D}} \!\! {\rm d}\Sigma_{\mu}(y) \int {\rm d}^{4}q\; {\rm e}^{iq\cdot (x-y)}\nonumber\\ &\times \frac{1 -{\rm e}^{\beta(x)\cdot q}}{\beta(x) \cdot q} \delta(p\cdot q) \delta \left( p^2 + \frac{q^2}{4} - m^2 \right) \theta(p_+^0)\theta(p_-^0) \nonumber \\ & \times u_s (p_-)\bar u_r(p_+) \left\langle \widehat a^{\dagger}_r (p_+) \widehat a_s(p_-), {\widehat T}^{\mu\nu}(0) \right\rangle_{c,{\rm GE}} \Delta \beta_\nu(y,x)\,. \end{align}\tag{7}\] In ref. [1], at this stage, the integration over \(\Sigma_{\rm D}\) was carried out first, introducing a non-trivial geometric assumption, followed by the integration in \(q\). In fact, it is more convenient to carry out the integration in \(q\) first because it does not require any special assumption other than the usual hydrodynamic limit; eventually, this order of integrations yields a nice expression, as we shall see later.

The formula 7 can be rewritten in a way which makes it apparent the effect of the hydrodynamic limit: \[\label{deltaWig} \Delta W_{\rm LE}^+(x,p) =\frac{1}{(2\pi)^3} \int {\rm d}^4 q\; \delta (p \cdot q) G^{\mu\nu}(q) F_{\mu\nu}(q)\,,\tag{8}\] where: \[\begin{align} \label{Gfunct} && G^{\mu\nu}(q) = \frac{1}{2} \delta \left( p^2 + \frac{q^2}{4} - m^2 \right) \frac{1 - {\rm e}^{\beta(x)\cdot q}}{\beta(x) \cdot q} \theta(p_+^0)\theta(p_-^0) \nonumber \\ && \times \sum_{r,s} u_s (p_-) \bar u_r(p_+)\left\langle \widehat a^{\dagger}_r (p_+) \widehat a_s(p_-), {\widehat T}^{\mu\nu}(0) \right\rangle_{c,{\rm GE}} \end{align}\tag{9}\] and \[F_{\mu\nu}(q) = \int_{\Sigma_{\rm D}} \!\! {\rm d}\Sigma_{\mu}(y) \; {\rm e}^{-iq\cdot(y-x)} \Delta \beta_\nu(y,x)\,.\] In the hydrodynamic limit, \(\Delta\beta_{\nu}\) is a slowly varying function in space and time implying that the function \(F_{\mu\nu}\), which is a Fourier transform in the variable \(q\) integrated in the variable \(y\), is a peaked function about \(q^{\mu}=0\). This allows us to take a small-\(q\) expansion of the function \(G^{\mu\nu}(q)\) in the integral 8 . Later, in equation 16 , it will become clear that this expansion in \(q\) generates an expansion of the Wigner function in the gradients of the field \(\beta\). We can rewrite the leading order correction to the Wigner function as: \[\begin{align} \label{eq:W-LTE-In} \Delta W_{\rm LE}^+(x,p) &&= \frac{1}{(2\pi)^3} \sum_{N=0}^{\infty}\frac{1}{N!}\int_{\Sigma_{\rm D}} \!\! {\rm d}\Sigma_{\mu}(y) \, I_{n}^{\nu_{1}\nu_{2}\cdots\nu_{N}}(y-x)\nonumber \\ && \times\left. \Delta\beta_\nu(y,x) \left[\partial_{\nu_{1}}^{q}\partial_{\nu_{2}}^{q}\cdots\partial_{\nu_{}}^{q} G^{\mu\nu}(q)\right]\right|_{q=0}\,, \end{align}\tag{10}\] where integrals over \(q\) are encoded in the rank-\(N\) function \[\begin{align} \label{eq:I-n-final} && I_{N}^{\nu_{1}\nu_{2}\cdots\nu_{n}}(y-x)\equiv\int {\rm d}^{4}q\,\delta(p\cdot q) {\rm e}^{-iq\cdot(y-x)}q^{\nu_{1}}\cdots q^{\nu_{N}} \nonumber \\ & &=(2\pi)^{3}\frac{(-i)^{N}}{|p^{0}|} \partial_x^{\nu_1}\cdots \partial_x^{\nu_N} \delta^{3}\left({\boldsymbol{y}}-{\boldsymbol{x}}-\frac{\boldsymbol{p}}{p^{0}}(y^{0}-x^{0})\right). \end{align}\tag{11}\] Plugging the eq. 11 into the 10 and repeatedly using the Leibniz rule we get: \[\begin{align} \label{eq:delta-W-2} && \Delta W_{\text{LE}}^+(x,p) = \frac{1}{|p^0|} \sum_{N=0}^{\infty} \frac{(-i)^N}{N!} \left.\left[\partial^{q}_{\nu_{1}}\cdots\partial^{q}_{\nu_{N}}G^{\mu\nu}(q)\right]\right|_{q=0} \nonumber \\ &&\times\sum_{M=0}^N \frac{N!(-1)^M}{M!(N-M)!}\partial^{\nu_{M+1}}_{x}\cdots\partial^{\nu_{N}}_{x} \int_{\Sigma_{\rm D}}\!\! {\rm d}\Sigma_{\mu}(y) \nonumber \\ && \times\left[\partial^{\nu_{1}}_{x}\cdots\partial^{\nu_{M}}_{x}\Delta\beta_{\nu}(y,x)\right] \delta^3 \left({\boldsymbol{y}}-{\boldsymbol{x}}-\frac{\boldsymbol{p}}{p^0}(y^0-x^0)\right). \end{align}\tag{12}\] In general, \(\Sigma_{\rm D}\) can be a topological complex hypersurface, with both space-like and time-like parts. Notably, for a given spatial position \({\boldsymbol{x}}\) there might be multiple \(x^0\) times belonging to \(\Sigma_{\rm D}\) (see fig. 1). Nevertheless, the hypersurface can be split into single-valued branches \(x^0=f_k({\boldsymbol{x}})\) and the integration in 12 can be decomposed into the sum of integrals over those branches. For any arbitrary function \(\Theta(x,y)\) the integration in \(y\) in the eq. 12 yields (assuming that \(p\cdot \sigma \ne 0\)): \[\begin{align} \label{deltas} && \int_{\Sigma_{\rm D}} \!\! {\rm d}\Sigma_{\mu}(y) \; \delta^{3}\left({\boldsymbol{y}}-{\boldsymbol{x}}-\frac{\boldsymbol{p}}{p^{0}} (y^{0}-x^{0})\right) \Theta(y,x) \nonumber \\ && = \sum_{k} s_k \int_{\Sigma_k} \!\! {\rm d}^3 {\rm y} \; \sigma_{\mu}(y)\; \delta^{3}\left({\boldsymbol{y}}-{\boldsymbol{x}}-\frac{\boldsymbol{p}}{p^{0}}(y^{0}-x^{0})\right) \Theta(y,x) \nonumber \\ && = \sum_{k,i} s_k \sigma_{\mu}(\bar y_{k,i})\; \frac{|p^0|}{|p\cdot\sigma(\bar y_{k,i})|} \Theta(\bar y_{k,i},x)\,, \end{align}\tag{13}\] where the vector \(\sigma\) is the vector perpendicular to the hypersurface: \[\label{eq:normal-vector} \sigma_{\mu}(x)=\left(1,-\partial f_k/\partial{\boldsymbol{x}}\right)\,,\tag{14}\] and \(\bar{\boldsymbol{y}}_{k,i}\) is the \(i\)-th solution of the equation \({\boldsymbol{y}} = {\boldsymbol{x}}-({\boldsymbol{p}}/p^0) (f_k({\boldsymbol{y}})-x^{0}({\boldsymbol{x}}))\); \(s_k\) in eq. 13 is a sign, which is +1 if \(\sigma_\mu\) has the same direction of the outward pointing vector \({\rm d}\Sigma_\mu\) and \(-1\) otherwise. For any given \({\boldsymbol{x}}\) and \(x^0\) lying on \(\Sigma_{\rm D}\), the solutions are found by intersecting the world-line of a particle off the mass-shell (being \(p^2 \ne m^2\)) moving with velocity \({\boldsymbol{p}}/p^0\): \[\label{worldline} {\boldsymbol{y}} = {\boldsymbol{x}}-\frac{\boldsymbol{p}}{p^{0}}(y^0-x^0)\tag{15}\] with all branches of the hypersurface \(\Sigma_{\rm D}\). This is depicted in fig. 1; there is at least one solution which is \(y=x\), but there might be more if the topology of the hypersurface makes it possible. We can then replace the sum over \(k,i\) in the equation 13 with the sum over the \(p\)-dependent intersections \(\bar y(x,p)\) for a given \(x\).

Plugging the eq. 13 into the eq. 10 , a compact form of the linear correction in \(\Delta\beta\) to the Wigner function can be achieved after some elaboration: \[\begin{align} \label{eq:wignerfinal} \Delta W_{\text{LE}}^+(x,p) &=&\sum_{N=0}^{\infty}\sum_{\bar{y}(x,p)}\frac{1}{N!} D^N_y(\bar{y}) \nonumber \\ && \times \left[ G^{\mu\nu}(q) \frac{n_{\mu}(y)}{|p\cdot n(y)|} \Delta\beta_{\nu}(y,x) \right] \Bigg|_{q=0,y=\bar y(x,p)} \end{align}\tag{16}\] where \(n_\mu= {\rm d}\Sigma_\mu /| {\rm d}\Sigma|\) is the outward pointing unit normal vector to \(\Sigma_{\rm D}\) and the sum over \(\bar y(x,p)\) runs over all the intersections of the particle world-line with the hypersurface for a chosen space-time point \(x\), including the trivial solution \(y=x\). In the above equation, the operator \(D_y\) is defined as: \[\label{eq:operator-D-q-y} D_y(\bar{y}) \equiv-i\Delta^{\nu\rho}(\bar{y}) \partial_{\rho}^{y}\partial_{\nu}^{q}\,,\tag{17}\] where \(\Delta^{\nu\rho}\) is the operator: \[\label{operatorDelta} \Delta^{\nu\rho}(\bar{y}) = g^{\nu\rho} - \frac{n^\nu(\bar{y}) p^\rho}{p \cdot n(\bar{y})}\,.\tag{18}\] Note that the derivatives in 17 do not act on the operator in 18 which is evaluated over the intersections. The formula 16 includes linear terms in the gradients of \(\Delta\beta\) to all orders as well as gradients of the normal vector \(n\) to the hypersurface. The latter are obviously absent if the hypersurface is a hyperplane, as it is tacitly assumed in many calculations in thermal field theory.

It is somewhat surprising that the Wigner function at the point \(x\) receives contributions from the gradients of the four-temperature field \(\beta\) evaluated at points \(\bar y(x,p)\) which can be relatively distant from \(x\). These points lie on the world-line 15 but this by no means implies that there is an actual particle travelling from \(\bar y(x,p)\) to \(x\); its possible physical origin is discussed towards the end of the paper. The mathematical origin of this long-distance contribution can be understood rewriting the equation 7 as: \[\begin{align} \label{deltawig2} \Delta W_{\rm LE}^+(x,p) &=&\frac{1}{(2\pi)^3} \int_{\Sigma_{\rm D}} \!\! {\rm d}\Sigma_{\mu}(y) \Delta \beta_\nu(y,x) \nonumber \\ &&\times \left[ \int {\rm d}^{4}q\; \delta(p\cdot q) {\rm e}^{iq\cdot (x-y)} G^{\mu\nu}(q) \right]\,. \end{align}\tag{19}\] In order for \(y=x\) to be the only dominant contribution of the integral 19 , the function of \(y-x\) between square brackets should be peaked around zero, but because of the presence of \(\delta(p\cdot q)\), this function is constant through the whole worldline 15 , so that points other than \(x\) may give a large contribution to the integral.

A notable feature of eq. 16 is that, due to the presence of \(\Delta^{\nu\rho}\), the derivative in the direction perpendicular to \(\Sigma_{\rm D}\) does not contribute, As a result, the Wigner function depends only on values of the thermodynamic fields on \(\Sigma_{\rm D}\) and their space-time derivatives in tangential directions. This result is a built-in feature of this new method of deriving the gradient expansion; in the old method [1], the Taylor expansion of the \(\beta_\nu(y)\) field in the exponent of 3 did not remove the gradient perpendicular to \(\Sigma_{\rm D}\).

The speed of convergence of the series 16 apparently depends on the ratio between two lengths; first, the correlation length \(l_c\) associated to the function \(G^{\mu\nu}\), that is \(\left|\partial_q G^{\mu\nu}(q)\right|/\left|G^{\mu\nu}(q)\right|\), which depends on the microscopic scales of the quantum field and the inverse temperature. Second, the characteristic variation lengths of the thermo-hydrodynamic field \(\beta\), \(L_H = | \beta |/ \partial \beta|\), and that of the normal vector \(\sigma\), \(L_G = | n |/ \partial n|\) which is a purely geometric quantity. So, the series will rapidly converge if: \[\label{converge-requirement} l_{c}/L_{H}\ll1,\;\;\;l_{c}/L_{G}\ll1\,.\tag{20}\] Such conditions are believed to be generally satisfied in high-energy heavy-ion collisions, yet the second one has not been carefully considered thus far.

0.0.0.4 Spin polarization.

With our formula 16 one can in principle calculate gradient corrections at LTE to all measurable quantities, notably particle spectra and spin polarization. We focus on the spin polarization of spin-\(1/2\) fermions, which is one of the most studied cases and also plays a crucial role for studying vector meson’s spin alignment [32]–[34], hyperon’s spin correlation [50]–[52], and hypernuclei’s spin polarization [53], [54].

Since \(\widehat{T}^{\mu\nu}\) in the equation 3 is evaluated on the decoupling hypersurface, we can take it as being the stress-energy tensor operator of free fermions; in this case, the function \(G^{\mu\nu}(q)\) in eq. (9 ) can be computed explicitly, \[\begin{align} \label{Gmunu-spin-1472} G^{\mu\nu}(q)&=& \frac{n_{F}(x,p_+)-n_{F}(x,p_-)}{(2\pi)^3(\beta\cdot q)} \delta\left(p^{2}+\frac{q^{2}}{4}-m^2\right) \nonumber \\ &&\times\theta(p_{+}^{0})\theta(p_-^{0}) (\cancel{p}_- + m)(\gamma^{\mu} p^\nu+\gamma^\nu p^\mu)(\cancel{p}_+ + m). \end{align}\tag{21}\] With the previous function, one can readily calculate the mean spin polarization using the formula [17]: \[\label{spin-polarization-Wigner} S^{\mu}(p)=\frac{\int_{\Sigma_{\rm D}} \!\! {\rm d}\Sigma\cdot p\,\text{tr}\left[\gamma^{\mu}\gamma^{5}W^+(x,p)\right]}{2\int_{\Sigma_{\rm D}} \!\! {\rm d}\Sigma\cdot p\,\text{tr}\left[W^+(x,p)\right]}\,,\tag{22}\] where the leading contribution to the denominator arises from \(W^+_\text{GE}\) and the numerator depends on \(\Delta W^+_{\rm LE}\). At the leading order the spin polarization reads: \[\begin{align} \tag{23} S^{\mu}(p) &\simeq& -\frac{1}{8mN_p}\int_{\Sigma_{\rm D}}\!\! {\rm d}\Sigma(x) \cdot p\, n_{F}(1-n_{F}) \epsilon^{\mu\nu\rho\lambda}p_{\nu} \nonumber \\ &&\times\sum_{\bar{y}(x,p)}\text{sgn}[p\cdot n(\bar{y})] \left[\varpi_{\rho\lambda}(\bar{y})+\frac{2 n_{\rho}(\bar{y})\xi_{\lambda\alpha}(\bar{y}) p^{\alpha}}{p\cdot n(\bar{y})}\right], \tag{24} \end{align}\] where \(N_p=\int_{\Sigma_{\rm D}}\!\! {\rm d}\Sigma\cdot p\,\theta(p^{0})n_{F}(x,p)\) is the total particle number. The two terms in Eq. (24 ) can be identified as contributions of the thermal vorticity [16], [29], [55] \(\varpi^{\mu\nu} \equiv(\partial^\nu\beta^\mu-\partial^\mu\beta^\nu)/2\), and the thermal shear tensor [1], [2], [18], [56] \(\xi^{\mu\nu}\equiv(\partial^\nu\beta^\mu+\partial^\mu\beta^\nu)/2\), respectively, while the curvature of \(\Sigma_{\rm D}\) does not contribute at this order.

The equation 23 improves upon the formulae known in the literature in a threefold manner. First, it gives a new expression for the shear-induced polarization. In contrast to previous derivations, the factor \(\hat{t}^{\mu}/p\cdot\hat{t}\) in refs. [1], [18] or \(u^{\mu}/p\cdot u\) in refs. [2], [56] (where \(\hat{t}^{\mu}\) is the unit time vector in the QGP frame and \(u^{\mu}\) is the fluid velocity), obtained as the result of significant geometric approximations, is replaced by \(n^{\mu}/|p\cdot n|\). Remarkably, in the eq. 23 , the vorticity and the shear induced terms both receive contributions from the derivative of the \(\beta\) field along \(n_\mu\), but in the combination these contributions cancel out. Hence, when the decoupling hypersurface is isothermal and the direction of the normal vector \(n_\mu\) coincides with \(\partial_{\mu} T\), derivatives of the temperature do not contribute. Therefore, in this upgraded formalism, the absence of temperature gradients at all orders of the expansion [18], [57], [58] arises naturally. Secondly, the thermal vorticity contribution acquires a mofidication by the sign function \(\text{sgn}(p\cdot n)\), which is apparently necessary when \(\Sigma_{\rm D}\) is not entirely space-like and future oriented.

Finally, the formula 23 features additional contributions from the non-trivial solution of eq. 15 with \(\bar y(x,p)\neq x\). Apparently, they can be interpreted as the residual effect of a quantum interference between particles emitted from different points of the hypersurface, the interference yielding a non-vanishing result only for the classical path from \(\bar y(x,p)\) to \(x\). However, these additional contributions are characterized by a different sign of \({\rm d}\Sigma \cdot p\) and \(p \cdot n(\bar y)\) in the formula 23 and are geometrically related to particles moving inward across the hypersurface and traversing the fluid region [59]–[62]. Classically, this appears un-physical and it may be, in our formalism, a spurious effect generated by not having used interacting fields in the Wigner operator when dealing with in-plasma region in the formula 5 . It is possible that such additional terms are thus to be discarded and a usual way to do this for the classical Cooper-Frye formula is to introduce a cutoff \(\theta(p\cdot n)\) [60]–[63]. Likewise, introducing \(\theta(p\cdot n(x))\theta(p\cdot n(\bar y))\) in eq. 23 will remove the additional intersections in eq. 23 .

0.0.0.5 Conclusion.

In conclusion, we have developed a new method to calculate the Wigner function at LTE on the decoupling hypersurface in heavy-ion collisions. By inverting the momentum and the hypersurface integrals, we have been able to remove specific assumptions on the geometry of the hypersurface and to include in the gradient expansion those of the normal vector to the hypersurface. The space-.time derivatives in the normal direction of the hypersurface are naturally excluded, which has a remarkable implication in the vanishing contributions from the temperature gradient if the hypersurface is isothermal. We have derived an upgraded expression for the spin polarization of spin 1/2 fermions, which merits verifications in numerical simulations.

0.0.0.6 Acknowledgments.

The authors thank G.-L. Ma and Q. Wang for insightful discussions. This work is supported in part by the Italian Ministry of University and Research, project PRIN2022 “Advanced probes of the Quark Gluon Plasma”, Next Generation EU, Mission 4 Component 1.

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1 On the Wigner function at local thermodynamic equilibrium↩︎

In this section we show how to determine the expression 6 and 7 . The free Dirac field is expressed as: \[\label{eq:quantized-psi} \Psi(x)=\frac{1}{\sqrt{(2\pi)^{3}}}\sum_s\int\frac{{\rm d}^{3}{\boldsymbol{\rm p}}}{2\varepsilon} \left(u_{s}(p) \widehat{a}_{s}(p){\rm e}^{-ip\cdot x} + v_s(p) \widehat{b}_s v_s(p) {\rm e}^{ip \cdot x} \right)\,,\tag{25}\] where \(\varepsilon =\sqrt{{\boldsymbol{p}}^{2}+m^{2}}\). The creation and annihilation operators obey the covariant anti-commutation relation: \[\left\{ \widehat{a}_{s}(p),\widehat{a}_{s^{\prime}}^{\dagger}(p^{\prime})\right\} = 2\varepsilon_{{\boldsymbol{p}}}\delta^{3}({\boldsymbol{p}}-{\boldsymbol{p}}^{\prime})\delta_{ss^{\prime}}\] and likewise for antiparticles. Replacing the integral measure in eq. 25 with: \[\int\frac{{\rm d}^3 {\rm p}}{2\varepsilon}\rightarrow \int {\rm d}^4 p\; \delta(p^{2}-m^{2})\theta(p^{0})\] and substituting the Dirac field into 5 , the Wigner operator for the particle part turns out to be: \[\begin{align} \label{eq:Wigner-operator} \widehat{W}^{+}(x,p) & = & \frac{1}{(2\pi)^3} \sum_{r,s} \int {\rm d}^4 p_1 \, {\rm d}^4 p_2 \; \delta(p_1^{2}-m^{2})\theta(p_1^{0}) \delta(p_2^2-m^{2})\theta(p_2^{0}) \, \delta\left( p-\frac{p_1+p_2}{2}\right) \nonumber \\ && \times {\rm e}^{i(p_1-p_2) \cdot x} \widehat{a}_{r}^{\dagger}(p_1)\widehat{a}_{s}(p_2) u_{s}(p_2)\bar{u}_{r}(p_1)\,. \end{align}\tag{26}\] One can now change the integration variables: \[\label{varchange} P \equiv \frac{p_1+p_2}{2}\,, \qquad q \equiv p_1-p_2 \qquad\implies\qquad p_1 = P + q/2\,, \qquad p_2 = P - q/2\tag{27}\] so that: \[{\rm d}^4 p_1 \; {\rm d}^4 p_2 = {\rm d}^4 P \; {\rm d}^4 q\] and \[\delta(p_1^{2}-m^{2})\delta(p_2^{2}-m^{2})=\frac{1}{2}\delta\left(P^{2}+\frac{q^{2}}{4}-m^{2}\right) \delta(P\cdot q)\,.\] Integrating in \({\rm d}^4 P\) is now straightforward because of the \(\delta^4 (p - P)\) in the equation 26 and, after replacing all variables according to the 27 one readily gets the 6 .

Now we focus on the leading correction to the Wigner function in the linear response theory, which can be derived by replacing \(\widehat{O}\) in eq. (4 ) by the Wigner operator \(\widehat{W}^+(x,p)\): \[\Delta W_{\text{LE}}^{+}(x,p) = -\int_{\Sigma_{\rm D}}\!\! {\rm d}\Sigma_{\mu}(y)\int_{0}^{1} {\rm d}z \; \Delta\beta_{\nu}(y,x) \left\langle \widehat{W}^{+}(x,p),e^{-z\beta(x)\cdot\widehat{P}} \widehat{T}^{\mu\nu}(y) {\rm e}^{z\beta(x)\cdot\widehat{P}}\right\rangle _{c,\text{GE}}\,.\] With the help of the Wigner operator in eq. 6 , \(\Delta W_{\text{LE}}^{+}\) can be written in terms of creation and annihilation operators as: \[\begin{align} \label{eq:delta-W-LE} \Delta W_{\text{LE}}^{+}(x,p) & = & -\frac{1}{2(2\pi)^{3}}\sum_{r,s}\int {\rm d}^{4}q \; \theta(p_{+}^{0})\theta(p_{-}^{0})e^{iq\cdot x}\delta(p\cdot q)\int_{\Sigma_{\rm D}}\!\! {\rm d}\Sigma_{\mu}(y) \int_{0}^{1}{\rm d}z \; \Delta\beta_{\nu}(y,x) \, \delta\left(p^{2}+\frac{q^{2}}{4}-m^{2}\right) \nonumber \\ && \times \left\langle \widehat{a}_{r}^{\dagger}(p_{+})\widehat{a}_{s}(p_{-}), {\rm e}^{-z\beta(x)\cdot\widehat{P}} \widehat{T}^{\mu\nu}(y) {\rm e}^{z\beta(x)\cdot\widehat{P}}\right\rangle_{c,\text{GE}}u_{s}(p_{-})\bar{u}_{r}(p_{+})\,. \end{align}\tag{28}\] where \(p_\pm = p \pm q/2\). Since the four-momentum operator \(\widehat{P}^{\mu}\) is the generator of spacetime translation, we have: \[{\widehat T}^{\mu\nu}(y)={\rm e}^{i{\widehat P}\cdot y}{\widehat T}^{\mu\nu}(0){\rm e}^{-i{\widehat P}\cdot y}\] and taking into account that the operator \(\beta(x)\cdot{\widehat P}\) commutes with the generator of translations as well as with the density operator \({\rm e}^{-\beta(x)\cdot {\widehat P}}/{\rm Tr}({\rm e}^{-\beta(x)\cdot{\widehat P}})\), we obtain: \[\label{connected1} \left\langle \widehat a^{\dagger}(p_+) \widehat a(p_-), {\rm e}^{-z\beta(x)\cdot\widehat{P}} {\widehat T}^{\mu\nu}(y) {\rm e}^{z\beta(x)\cdot\widehat{P}} \right\rangle_{c,{\rm GE}} =\langle {\rm e}^{-C \cdot {\widehat P}} \widehat a^{\dagger}(p_+) \widehat a(p_-) {\rm e}^{C \cdot {\widehat P}}, {\widehat T}^{\mu\nu}(0) \rangle_{c,{\rm GE}},\tag{29}\] where \(C = -z\beta(x) + iy\). Since \(\widehat a^{\dagger}(p_\pm)\),\(\widehat a(p_\pm)\) are the creation/ annihilation operators of physical states, they transform under translation as: \[\begin{align} {\rm e}^{i{\widehat P}\cdot y}\widehat a^{\dagger}(p){\rm e}^{-i{\widehat P}\cdot y}&={\rm e}^{ip\cdot y}\widehat a^{\dagger}(p),\\ {\rm e}^{i{\widehat P}\cdot y}\widehat a(p'){\rm e}^{-i{\widehat P}\cdot y}&={\rm e}^{-ip'\cdot y}\widehat a(p'), \end{align}\] so that: \[{\rm e}^{-C \cdot {\widehat P}} \widehat a^{\dagger}(k_+) \widehat a(k_-) {\rm e}^{C \cdot {\widehat P}} ={\rm e}^{- C \cdot q }\widehat a^{\dagger}(k_+)\widehat a(k_-) ={\rm e}^{-iy\cdot q }{\rm e}^{z\beta(x)\cdot q}\widehat a^{\dagger}(k_+)\widehat a(k_-),\] thus the 29 becomes: \[\label{connected2} \left\langle \widehat a^{\dagger}(p_+) \widehat a(p_-), {\rm e}^{-z\beta(x)\cdot\widehat{P}} {\widehat T}^{\mu\nu}(y) {\rm e}^{z\beta(x)\cdot\widehat{P}} \right\rangle_{c,{\rm GE}} ={\rm e}^{-iy\cdot q }{\rm e}^{z\beta(x)\cdot q} \langle \widehat a^{\dagger}(p_+) \widehat a(p_-) , {\widehat T}^{\mu\nu}(0) \rangle_{c,{\rm GE}}\,.\tag{30}\] Plugging the 30 and completing the integral over \(z\) leads to the eq. 7 in the main text.

2 Compact form of Wigner function↩︎

We show how to obtain the compact form of the Wigner function correction in eq. 16 starting from the eq. (8 ). As discussed in the main text, the function \(F_{\mu\nu}(q)\) is a peaked function about \(q^\mu=0\), allowing us to perform a small-\(q\) expansion for \(G^{\mu\nu}(q)\), \[G^{\mu\nu}(q)=\sum_{N=0}^{\infty}\frac{1}{N!}\left.\left[\partial^{q}_{\nu_{1}} \partial^q_{\nu_2}\cdots\partial^{q}_{\alpha_{N}}G^{\mu\nu}(q)\right]\right|_{q^{\mu}=0}q^{\nu_1}q^{\nu_2}\cdots q^{\nu_N}\,.\] Substituting this expansion into eq. 8 , the eq. 10 is obtained. The integrals over \(q\) are then encoded in \(I_n^{\nu_1\nu_2\cdots\nu_N}\), which is given by eq. 11 . Plugging the eq. 11 into the eq. 10 , we obtain: \[\begin{align} \Delta W_{\text{LE}}^{+}(x,p) & = & \frac{1}{|p^{0}|}\sum_{N=0}^{\infty}\frac{(-i)^{N}}{N!}\left. \left[\partial^{q}_{\nu_{1}}\partial^{q}_{\nu_{2}}\cdots\partial^{q}_{\nu_{N}} G^{\mu\nu}(q)\right]\right|_{q=0}\nonumber \\ & & \times\int_{\Sigma_{\rm D}}\!\! {\rm d}\Sigma_{\mu}(y)\, \Delta\beta_{\nu}(y,x) \left[ \partial^{\nu_{1}}_{x}\partial^{\nu_{2}}_{x}\cdots\partial^{\nu_{N}}_{x}\delta^{3} \left({\boldsymbol{y}}-{\boldsymbol{x}}-\frac{\boldsymbol{p}}{p^{0}}(y^{0}-x^{0})\right)\right] \,.\label{eq:delta-W-1} \end{align}\tag{31}\] The presence of the partial derivatives of the \(\delta\)-function makes the integral difficult to evaluate. It is convenient to convert the above equation to the form in 12 through the following steps. First, we split the partial derivative into the sum of two different operators \[\partial_x = \partial^{(1)}_{x} + \partial^{(2)}_{x}\,,\] acting only on the \(\delta\)-function and \(\Delta\beta_{\nu}(y,x)\), respectively. Thereby, the partial derivatives in eq. 31 can be recast as \(\left[\partial^{q}\cdot\partial^{(1)}_{x}\right]^{N}\) and taken out of the integral. The latter can be expressed as follows, by means of the Binomial theorem: \[\left[\partial^{q}\cdot\partial^{(1)}_x\right]^{N} = \left[\partial^{q}\cdot(\partial^{(1)}_x+ \partial^{(2)}_x)-\partial^{q}\cdot\partial_{(2)}^{x}\right]^{N} = \sum_{M=0}^{N}\frac{N!(-1)^{M}}{M!(N-M)!}\left[\partial^{q}\cdot \partial_{x} \right]^{N-M} \left[\partial^{q}\cdot\partial^{(2)}_{x}\right]^{M}\,,\] The derivative \(\partial_x^{(2)}\) only acts on the \(\Delta\beta\) function and can be moved into the integral, yielding the expression in the eq. 12 .

In the eq. 12 , the space-time point \(y\) is located on \(\Sigma_{\rm D}\). As described in the main text, \(\Sigma_{\rm D}\) can be split into several branches, where in each the decoupling time \(y^0\) is a single-valued function of \({\boldsymbol{y}}\), i.e., \(y^{0}=f_{k}({\boldsymbol{y}})\). Then the equation 13 follows because, on each branch: \[\delta^{3}\left({\boldsymbol{y}}-{\boldsymbol{x}}- \frac{\boldsymbol{p}}{p^0}(y^{0}-x^{0}) \right)= \sum_{i}\frac{|p^{0}|}{|p\cdot\sigma(\bar{y}_{k,i})|}\delta^{3}({\boldsymbol{y}}-\bar{{\boldsymbol{y}}}_{k,i})\,,\] where \(\bar{y}_{k,i}^{\mu}(x)\) denotes the \(i\)-th crossing point of the worldline 15 with the \(k\)-th branch of \(\Sigma_{\rm D}\). The pre-factor on the right hand side is the inverse of the determinant of the matrix: \[\frac{\partial}{\partial y^j}\left[{\boldsymbol{y}}-{\boldsymbol{x}}- \frac{\boldsymbol{p}}{p^0}(f_k({\boldsymbol{y}})-x^{0})\right]^i = \delta^i_j - \frac{p^i}{p^0} \frac{\partial f_k({\boldsymbol{y}})}{\partial y^j} = \delta^i_j - \frac{p^i}{p^0} \sigma^j = \frac{p^0 \delta^i_j - p^i \sigma^j}{p^0}\] Carrying out the integration in \({\rm d}^3 {\rm y}\), and using the relations 13 , the equation 12 is converted to: \[\begin{align} \label{eq:delta-W-3} \Delta W_{\text{LE}}^{+}(x,p) & = & \sum_{N=0}^{\infty}\sum_{k,i}s_k \frac{(-1)^{N}}{N!} \left.\left[\partial^{q}_{\nu_{1}}\ldots\partial^{q}_{\nu_{N}}G^{\mu\nu}(q)\right]\right|_{q=0}\nonumber \\ && \times\sum_{M=0}^{N}\frac{N!(-1)^{M}}{M!(N-M)!}{\rm d}^{\nu_{M+1}}_{x}\ldots{\rm d}^{\nu_{N}}_{x} \left[\left.\frac{\sigma_{\mu}(y)}{|p\cdot\sigma(y)|}\partial^{\nu_{1}}_{x}\ldots\partial^{\nu_{M}}_{x} \Delta\beta_{\nu}(y,x)\right|_{y=\bar{y}_{k,i}(x)}\right]\,, \end{align}\tag{32}\] where \(s_k\) is a sign, which is +1 if \(\sigma_\mu\) has the same direction of the outward pointing vector \({\rm d}\Sigma_\mu\) and \(-1\) otherwise. In eq. 32 we have introduced the total derivative: \[{\rm d}^\mu_x = \frac{{\rm d}}{{\rm d}x_\mu}\] to emphasize the difference between the derivative acting on the function before setting \(y=\bar{y}_{k,i}(x)\) (that is \(\partial_x\)) and the derivative acting on the function after setting \(y=\bar{y}_{k,i}(x)\). In general, using the chain rule for a function \(g(x,\bar{y}_{k,i}(x))\) we have: \[\label{chain-rule} \frac{{\rm d}}{{\rm d}x^\mu} g(x,\bar{y}_{k,i}(x))= \frac{\partial}{\partial x^\mu} g(x,\bar{y}_{k,i}(x))+ \frac{\partial \bar{y}_{k,i}^\nu(x)}{\partial x^\mu} \frac{\partial g(x,y)}{\partial y^\nu}\Bigg|_{y=\bar{y}_{k,i}(x)}\tag{33}\] Now, the derivative of the function \(y^\nu = \bar y_{k,i}^\nu(x)\) is obtained by taking into account that: \[\bar{{\boldsymbol{y}}}_{k,i}-\frac{{\boldsymbol{p}}}{p^{0}}\bar{y}_{k,i}^{0}={\boldsymbol{x}}-\frac{{\boldsymbol{p}}}{p^{0}}x^{0}\,,\] with \(\bar{y}_{k,i}^{0} =f_{k}(\bar{{\boldsymbol{y}}}_{k,i})\). Taking partial derivatives with respect to \(x^\mu\) of the above equation: \[\frac{\partial\bar{y}_{k,i}^{j}}{\partial x^{\mu}}-\frac{p^{j}}{p^{0}}\frac{\partial\bar{y}_{k,i}^{0}}{\partial\bar{y}_{k,i}^{l}}\frac{\partial\bar{y}_{k,i}^{l}}{\partial x^{\mu}} = \left[ \delta^j_l + \frac{p^j}{p^0} \sigma_l \right] \frac{\partial\bar{y}_{k,i}^{l}}{\partial x^{\mu}} = \delta_{\mu}^{j}-\frac{p^{j}}{p^{0}}\delta_{\mu}^{0}\] where \(j,l=1,2,3\) and where \(\sigma_{\mu}(\bar{y}_{k,i})\) is the normal vector of \(\Sigma_{\rm D}\) at the spacetime point \(\bar{y}_{k,i}\), \[\sigma_{\mu}(\bar{y}_{k,i})=\left(1,-\frac{\partial f_{k}(\bar{{\boldsymbol{y}}}_{k,i})}{\partial\bar{{\boldsymbol{y}}}_{k,i}}\right)\,.\] The \(3 \times 3\) matrix: \[A^j_l = \left[ \delta^j_l + \frac{p^j}{p^0} \sigma_l \right]\] can be inverted and one obtains: \[\frac{\partial\bar{y}_{k,i}^{j}}{\partial x^{\mu}} = A^{-1j}_{\;\;\; l} \left(\delta_{\mu}^{l}- \frac{p^{l}}{p^{0}}\delta_{\mu}^{0}\right) = \left( \delta^j_l - \frac{p^j \sigma_l}{p \cdot \sigma} \right) \left(\delta_{\mu}^{l}-\frac{p^{l}}{p^{0}}g_{\mu}^{0}\right)\] If \(\mu = 0\) we then have: \[\frac{\partial\bar{y}_{k,i}^{j}}{\partial x^{0}} = -\frac{p^j}{p \cdot \sigma}\] while if \(\mu = m\) spacial index \[\frac{\partial\bar{y}_{k,i}^{j}}{\partial x^{m}} = \delta^j_m - \frac{p^j \sigma_m}{p \cdot \sigma}\] so that, altogether: \[\label{deriv1} \frac{\partial\bar{y}_{k,i}^{j}}{\partial x^{\mu}} = \delta^j_\mu - \frac{p^j \sigma_\mu}{p \cdot \sigma}\,.\tag{34}\] Similarly, one can calculate the derivative of \(y^0\) with respect to \(x\): \[\frac{\partial\bar{y}_{k,i}^{0}}{\partial x^{\mu}} = \frac{\partial\bar{y}_{k,i}^{0}}{\partial\bar{y}_{k,i}^{j}}\frac{\partial\bar{y}_{k,i}^{j}}{\partial x^{\mu}} = -\sigma_j \left( \delta^j_\mu - \frac{p^j \sigma_\mu}{p \cdot \sigma} \right)\] whence we obtain: \[\label{deriv2} \frac{\partial\bar{y}_{k,i}^{0}}{\partial x^{0}} = \sigma_0 - \frac{p^0}{p \cdot \sigma} \qquad\qquad \frac{\partial\bar{y}_{k,i}^{0}}{\partial x^{m}} = - \frac{p^0 \sigma_m}{ p \cdot \sigma}\tag{35}\] The equations 34 and 35 can be written in a compact form as: \[\frac{\partial\bar{y}_{k,i}^{\nu}}{\partial x^{\mu}}=\delta_{\mu}^{\nu}-\frac{p^{\nu} \sigma_{\mu}(\bar{y}_{k,i})}{p\cdot\sigma(\bar{y}_{k,i})} \equiv \Delta_{\mu}^{\;\nu}(\bar{y}_{k,i})\] so that eq. 33 can be rewritten as: \[\frac{{\rm d}}{{\rm d}x^\mu} g(x,\bar{y}_{k,i}(x))= \frac{\partial}{\partial x^\mu} g(x,\bar{y}_{k,i}(x))+ \Delta^{\;\nu}_\mu(\bar{y}_{k,i}) \frac{\partial g(x,y)}{\partial y^\nu}\Bigg|_{y=\bar{y}_{k,i}(x)}\] By using the last result, the eq. 32 can be rewritten as: \[\begin{align} \Delta W_{\text{LE}}^{+}(x,p) & = &\sum_{N=0}^{\infty} \sum_{k,i} s_k \frac{(-1)^{N}}{N!} \left.\left[\partial^{q}_{\nu_{1}}\cdots\partial^{q}_{\nu_{N}}G^{\mu\nu}(q)\right]\right|_{q=0} \sum_{M=0}^{N}\frac{N!(-1)^{M}}{M!(N-M)!}\left[\partial^{\nu_{M+1}}_{x}+ \Delta^{\alpha_{M+1}\rho_{M+1}}(\bar{y}_{k,i})\partial_{\rho_{M+1}}^{y}\right]\nonumber \\ & & \times\left.\cdots\left[\partial^{\nu_{N}}_{x}+\Delta^{\nu_N\rho_{N}}(\bar{y}_{k,i}) \partial_{\rho_{N}}^{y}\right]\partial^{\nu_{1}}_{x}\cdots\partial^{\nu_{M}}_{x} \frac{\sigma_{\mu}(y)}{|p\cdot\sigma(y)|}\Delta\beta_{\nu}(y,x)\right|_{y=\bar{y}_{k,i}}\,. \end{align}\] Using again the Binomial theorem, the last expression can be recast as: \[\Delta W_{\text{LE}}^{+}(x,p)=\sum_{N=0}^{\infty}\sum_{k,i} s_k \frac{(-i)^{N}}{N!} \left.\left[\Delta^{\alpha\rho}(\bar{y}_{k,i})\partial_{\rho}^{y}\partial_{\alpha}^{q} \right]^{N} G^{\mu\nu}(p,q)\frac{\sigma_{\mu}(y)}{|p\cdot\sigma(y)|}\Delta\beta_{\nu}(y,x) \right|_{q=0,y=\bar{y}_{k,i}}\;.\] Now, because of its definition (see text below eq. 14 ), the sign \(s_k\) is such that: \[s_k \sigma_\mu = n_\mu \sqrt{|\sigma \cdot \sigma|}\] where \(n_\mu\) is the outward pointing unit vector normal to the hypersurface \(\Sigma_{\rm D}\). Besides: \[\frac{n_\mu \sqrt{|\sigma \cdot \sigma|}}{|p \cdot \sigma|} = \frac{n_\mu}{|p \cdot n|}\] and \[\Delta^{\;\nu}_\mu = \delta_{\mu}^{\nu}-\frac{p^{\nu} \sigma_{\mu}}{p\cdot\sigma} = \delta_{\mu}^{\nu}-\frac{p^{\nu}n_{\mu}}{p\cdot n}\,.\] By taking the above equalities into account and introducing the operator \(D_y\) in eq. 17 , we can express \(\Delta W_{\rm LE}^+\) as shown in eq. 16 . Note the argument of the operator \(\Delta^{\alpha\rho}\) is \(\bar{y}_{k,i}\), such that the derivative with respect to \(y\) does not act on \(\Delta^{\alpha\rho}\).

3 Spin polarization for spin-1/2 fermions↩︎

In this section we will show the details for calculating the spin polarization using eq. 22 . For the denominator, the leading contribution stems from \(W^+_{\rm GE}\), which can be readily calculated by using the well-known formula \[\langle \widehat a^{\dagger}_r(p) a_s(p') \rangle_{\rm GE} = 2 \varepsilon \, \delta^3({\boldsymbol{p}}-{\boldsymbol{p}}') \frac{1}{{\rm e}^{\beta(x) \cdot p}+1} \equiv 2 \varepsilon \, \delta^3({\boldsymbol{p}}-{\boldsymbol{p}}') n_F(x,p)\,,\] Plugging this equation into the definition of the Wigner operator for the particles eq. 6 we get: \[W_{\rm GE}^+(x,p)=\frac{1}{(2\pi)^3}\delta(p^2-m^2)\theta(p^0)(\cancel{p}+m)n_F(x,p)\,,\] whose trace (in the 4-dimensional spinor space) reads \[\label{scalar-component} \text{tr}\left[ W_{\rm GE}^+(x,p) \right] =\frac{4m}{(2\pi)^3}\delta(p^2-m^2) \theta(p^0)n_F(x,p)\,.\tag{36}\] On the other hand, since \(\text{tr}\left[\gamma^\mu\gamma^5 W_{\rm GE}^+(x,p)\right]=0\), the leading contribution to the numerator in eq. 22 depends on \(\Delta W_{\rm LE}^+\). For a free Dirac field, the Belinfante symmetric energy-momentum tensor is given by \[\widehat{T}^{\mu\nu}(x)=\frac{i}{4}\bar{\psi}(x)\left[\gamma^{\mu}\left(\overrightarrow{\partial}^{\nu}-\overleftarrow{\partial}^{\nu}\right)+\gamma^{\nu}\left(\overrightarrow{\partial}^{\mu}- \overleftarrow{\partial}^{\mu}\right)\right]\psi(x)\,,\] Using the quantized field operator in eq. 25 , we can express \(\widehat{T}^{\mu\nu}(0)\) as (with \(p= (p_+ + p_-)/2\)): \[\widehat{T}^{\mu\nu}(0) = \frac{1}{2(2\pi)^{3}} \sum_{r,s} \int {\rm d}^{4}p_{+}{\rm d}^{4}p_{-} \; \delta(p_{+}^{2}-m^{2})\delta(p_{-}^{2}-m^{2})\bar{u}_{s}(p_{-})(\gamma^{\mu}p^{\nu}+\gamma^{\nu}p^{\mu}) u_{r}(p_{+})\widehat{a}_{s}^{\dagger}(p_{-})\widehat{a}_{r}(p_{+}) \theta(p^0_+)\theta(p^0_-)\] for the particle term only; the other terms, like anti-particles and mixed ones, do not contribute to the the connected mean value in eq. 9 : \[\begin{align} \label{connected} \left\langle \widehat{a}_{r}^{\dagger}(p_{+})\widehat{a}_{s}(p_{-}),\widehat{T}^{\mu\nu}(0) \right\rangle _{c,\text{GE}} & = & \frac{1}{2(2\pi)^{3}}\int {\rm d}^{4}p_{+}^{\prime}{\rm d}^{4}p_{-}^{\prime}\, \delta(p_{+}^{\prime2}-m^{2})\delta(p_{-}^{\prime2}-m^{2}) \theta(p^{\prime 0}_+)\theta(p^{\prime 0}_-) \nonumber \\ & & \times\bar{u}_{s^{\prime}}(p_{-}^{\prime})(\gamma^{\mu}p^{\prime\nu}+\gamma^{\nu}p^{\prime\mu}) u_{r^{\prime}}(p_{+}^{\prime})\left\langle \widehat{a}_{r}^{\dagger}(p_{+})\widehat{a}_{s}(p_{-}), \widehat{a}_{s^{\prime}}^{\dagger}(p_{-}^{\prime})\widehat{a}_{r^{\prime}}(p_{+}^{\prime})\right \rangle_{c,\text{GE}}\nonumber \\ & = & \frac{1}{2(2\pi)^3}\bar{u}_{s}(p_{-})(\gamma^{\mu}p^{\nu}+\gamma^{\nu}p^{\mu}) u_{r}(p_{+})n_{F}(x,p_{+})(1-n_{F}(x,p_{-}))\,, \end{align}\tag{37}\] where we have used \[\left\langle \widehat{a}_{r}^{\dagger}(p_{+})\widehat{a}_{s}(p_{-}),\widehat{a}_{s^{\prime}}^{\dagger} (p_{-}^{\prime})\widehat{a}_{r^{\prime}}(p_{+}^{\prime})\right\rangle _{c,\text{GE}} = 4\varepsilon_{+}\varepsilon_{-}\delta^{3}({\boldsymbol{p}}_{-}-{\boldsymbol{p}}_{-}^{\prime}) \delta^{3}({\boldsymbol{p}}_{+}-{\boldsymbol{p}}_{+}^{\prime}) \delta_{rr^{\prime}}\delta_{ss^{\prime}} n_{F}(x,p_{+})(1-n_{F}(x,p_{-}))\] with \(\varepsilon_\pm=\sqrt{m^2+{\boldsymbol{p}}_\pm^2}\) being the on-shell energies. Substituting the equation 37 into 9 and using the relation: \[(1-{\rm e}^{\beta\cdot q})n_{F}(x,p_{+})\left( 1-n_{F}(x,p_{-}) \right)=n_{F}(x,p_{+})-n_{F}(x,p_{-})\,,\] we obtain \(G^{\mu\nu}(q)\) as in the equation 21 .

In order to calculate the spin polarization, we need to evaluate the axial-vector component of the Wigner function \({\cal A}^\mu(x,p) = {\rm tr}(\gamma^\mu \gamma^5 W^+(x,k))\). Making use of the following trace: \[\text{tr}\left[\gamma^{\mu}\gamma^{5} (\cancel{p}_{-}+m)\gamma^{\alpha}(\cancel{p}_++m)\right] =-4i\epsilon^{\mu\alpha\lambda\tau} p_{\lambda}q_{\tau}\,,\] the trace of the function \(\eqref{Gfunct}\) can be readily calculated from the eq. 21 : \[\text{tr}\left[\gamma^{\mu}\gamma^{5}G^{\nu\alpha}(p,q)\right] = -i \frac{n_{F}(x,p_{+})-n_{F}(x,p_{-})}{(2\pi)^{3}(\beta\cdot q)}\delta\left(p^{2}+ \frac{q^{2}}{4}-m^{2}\right)\theta(p_{+}^{0})\theta(p_{-}^{0}) (g_{\rho}^{\nu}p^{\alpha}+g_{\rho}^{\alpha}p^{\nu})\epsilon^{\mu\rho\lambda\tau}p_{\lambda}q_{\tau}\,,\] whence, because of the eq. 16 : \[\begin{align} \label{axialfinal} {\cal A}_{\text{LE}}^\mu(x,p) &=& -i\sum_{N=0}^{\infty}\sum_{\bar{y}(x,p)}\frac{1}{N!} D^N_y \left[ \frac{n_{F}(x,p_{+})-n_{F}(x,p_{-})}{(2\pi)^{3}(\beta\cdot q)}\delta\left(p^{2}+ \frac{q^{2}}{4}-m^{2}\right)\theta(p_{+}^{0})\theta(p_{-}^{0})\right. \nonumber\\ &&\left.\times(g_{\rho}^{\nu}p^{\alpha}+g_{\rho}^{\alpha}p^{\nu})\epsilon^{\mu\rho\lambda\tau}p_{\lambda}q_{\tau}\, \frac{n_{\mu}(y)}{|p\cdot n(y)|} \Delta\beta_{\nu}(y,x) \right] \Bigg|_{q=0,y=\bar y(x,p)}\,. \end{align}\tag{38}\] Since the function to be derived in \(q\) is odd in that variable, only odd terms in the series contribute. The leading order contribution arises from the term of \(n=1\), which turns out to be: \[\begin{align} \label{Amu-all} \mathcal{A}^{\mu}_{\text{LE}}(x,p) & \simeq & \frac{2}{(2\pi)^{3}}n_{F}(x,p)[1-n_{F}(x,p)] \delta(p^{2}-m^{2})\epsilon^{\mu\rho\lambda\tau}p_{\lambda}\nonumber \\ & & \times\frac{1}{2}(g_{\rho}^{\nu}p^{\alpha}+g_{\rho}^{\alpha}p^{\nu})\sum_{\bar{y}(x,p)} \Delta_{\tau\kappa}(\bar{y})\left.\left[\partial_{y}^{\kappa}\frac{n_{\nu}(y)}{|p\cdot n(y)|} \Delta\beta_{\alpha}(y,x)\right]\right|_{y=\bar{y}(x,p)}\,. \end{align}\tag{39}\] The derivative with respect to \(y\) acts on both \(\Delta\beta_{\alpha}(y,x)\) and \(n_{\nu}(y)\), giving rise to two terms. For the former, one obtains a factor which is proportional to: \[\begin{align} && \frac{1}{2}\epsilon^{\mu\rho\lambda\tau}p_{\lambda}\Delta_{\tau\kappa}(\bar{y}) n_{\nu}(\bar{y})(g_{\rho}^{\nu}p^{\alpha}+g_{\rho}^{\alpha}p^{\nu})\left[\left.\partial_{y}^{\kappa} \beta_{\alpha}(y)\right|_{y=\bar{y}(x,p)}\right]\nonumber \\ & = & \frac{1}{2}\epsilon^{\mu\nu\lambda\tau}p_{\lambda}\Delta_{\tau\kappa}(\bar{y}) n_{\nu}(\bar{y})p^{\alpha}\left[\left.\partial_{y}^{\kappa}\beta_{\alpha}(y) \right|_{y=\bar{y}(x,p)}\right]+\left[p\cdot n(\bar{y})\right]\frac{1}{2}\epsilon^{\mu\alpha\lambda\tau} p_{\lambda}\Delta_{\tau\kappa}(\bar{y})\left[\left.\partial_{y}^{\kappa}\beta_{\alpha}(y) \right|_{y=\bar{y}(x,p)}\right]\,. \end{align}\] Plugging the explicit form of \(\Delta_{\tau\kappa}(\bar{y})\) in eq. 18 it becomes: \[\frac{1}{2}\epsilon^{\mu\nu\lambda\kappa}p_{\lambda} n_{\nu}(\bar{y})p^{\alpha} \left[\left.\partial_{\kappa}^{y}\beta_{\alpha}(y)\right|_{y=\bar{y}(x,p)}\right] +[p\cdot n(\bar{y})]\frac{1}{2}\epsilon^{\mu\alpha\lambda\kappa}p_{\lambda} \left[\left.\partial_{\kappa}^{y}\beta_{\alpha}(y)\right|_{y=\bar{y}(x,p)}\right]- \frac{1}{2}\epsilon^{\mu\alpha\lambda\tau}p_{\lambda}p^{\kappa} n_{\tau}(\bar{y}) \left[\left.\partial_{\kappa}^{y}\beta_{\alpha}(y)\right|_{y=\bar{y}(x,p)}\right],\] whence, renaming repeated indices: \[\label{Amu-LE-1} [p\cdot n(x)]\frac{1}{2}\epsilon^{\mu\nu\rho\lambda}p_{\nu}\left[\left.\partial_{\rho}^{y} \beta_{\lambda}(y)\right|_{y=\bar{y}(x,p)}\right] -\frac{1}{2}\epsilon^{\mu\nu\rho\lambda} p_{\nu} n_{\rho}(x)p^{\alpha}\left[\left.\partial_{\lambda}^{y}\beta_{\alpha}(y)+ \partial_{\alpha}^{y}\beta_{\lambda}(y)\right|_{y=\bar{y}(x,p)}\right]\tag{40}\] By using the definitions of thermal vorticity and thermal shear: \[\begin{align} \varpi_{\rho\lambda} & = & -\frac{1}{2}\left(\partial_{\rho} \beta_{\lambda}- \partial_{\lambda}\beta_{\rho} \right) \nonumber \\ \xi_{\rho\lambda} & = & \frac{1}{2}\left(\partial_{\rho} \beta_{\lambda} + \partial_{\lambda}\beta_{\rho} \right) \end{align}\] in the eq. 40 and reinstating the missing factors, we obtain the first part of \(\mathcal{A}_{\text{LE}}^{\mu}\): \[\label{eq:axial-vector} \mathcal{A}_{\text{LE}-1}^\mu (x,p) = -\frac{\delta(p^{2}-m^{2})}{(2\pi)^{3}}n_{F}(x,p)(1-n_{F}(x,p))\sum_{\bar{y}(x,p)} \text{sgn}[p\cdot n(\bar{y})] \epsilon^{\mu\nu\rho\lambda}p_{\nu}\left[\varpi_{\rho\lambda}(\bar{y})+ \frac{2}{p\cdot n(\bar{y})} n_{\rho}(\bar{y})\xi_{\lambda\alpha}(\bar{y})p^{\alpha}\right]\,.\tag{41}\] On the other hand, the contribution from the derivative of the normal vector \(n\) is proportional to \[\label{eq:Amu-LE-2} \epsilon^{\mu\rho\lambda\tau}p_{\lambda}(g_{\rho}^{\nu}p^{\alpha}+g_{\rho}^{\alpha}p^{\nu}) \Delta_{\tau\kappa}(\bar{y})\Delta\beta_{\alpha}(\bar{y},x)\left.\left[\partial_{y}^{\kappa} \frac{ n_{\nu}(y)}{|p\cdot n(y)|}\right]\right|_{y=\bar{y}}\,,\tag{42}\] which in turn gives rise to two terms. The first, obtained by contracting with \(p^\nu\), is proportional to: \[p^{\nu}\left.\left[\partial_{y}^{\kappa}\frac{ n_{\nu}(y)}{|p\cdot n(y)|}\right] \right|_{y=\bar{y}}=\left.\left[\partial_{y}^{\kappa}\text{sgn}[p\cdot n(y)]\right] \right|_{y=\bar{y}}\propto\delta( p\cdot n(\bar y))\] which vanishes unless \(p\cdot n(\bar{y})=0\). In our derivation, in order to ensure the that the \(\delta\) function can be solved in terms of the \({\boldsymbol{y}}\) variable, we had to assume that \(p\cdot n(\bar{y})=0\) so we neglect the above term. Therefore, we can rewrite the above term, without changing it, as: \[\label{eq:Amu-LE-2-step-1} \epsilon^{\mu\rho\lambda\tau}p_{\lambda} (g_{\rho}^{\nu}p^{\alpha}-g_{\rho}^{\alpha}p^{\nu})\Delta_{\tau\kappa}(\bar{y}) \Delta\beta_{\alpha}(\bar{y},x)\left.\left[\partial_{y}^{\kappa} \frac{ n_{\nu}(y)}{|p\cdot n(y)|}\right]\right|_{y=\bar{y}}\,,\tag{43}\] that is by changing the sign of the second term in the bracket. Using the Schouten identity: \[\label{Schouten-identity} \epsilon^{\mu\nu\lambda\tau}p^{\alpha}+\epsilon^{\nu\lambda\tau\alpha}p^{\mu}+ \epsilon^{\lambda\tau\alpha\mu}p^{\nu}+\epsilon^{\tau\alpha\mu\nu}p^{\lambda}+ \epsilon^{\alpha\mu\nu\lambda}p^{\tau}=0\,,\tag{44}\] we have: \[\epsilon^{\mu\rho\lambda\tau}p_{\lambda}(g_{\rho}^{\nu}p^{\alpha}-g_{\rho}^{\alpha}p^{\nu}) = (\epsilon^{\mu\nu\lambda\tau}p^{\alpha}+\epsilon^{\lambda\tau\alpha\mu}p^{\nu})p_{\lambda} = -(\epsilon^{\nu\lambda\tau\alpha}p^{\mu}+\epsilon^{\tau\alpha\mu\nu}p^{\lambda} +\epsilon^{\alpha\mu\nu\lambda}p^{\tau})p_{\lambda}\,,\] Plugging this relation into eq. 43 and taking into account that \(p^{\tau}\Delta_{\tau\kappa}(\bar{y})=0\), the expression in 43 can be rewritten as: \[\begin{align} \label{eq:Amu-LE-2-step-2} && -(\epsilon^{\nu\lambda\tau\alpha}p^{\mu}+\epsilon^{\tau\alpha\mu\nu}p^{\lambda})p_{\lambda} \Delta_{\tau\kappa}(\bar{y})\Delta\beta_{\alpha}(\bar{y},x)\left.\left[\partial_{y}^{\kappa} \frac{ n_{\nu}(y)}{|p\cdot n(y)|}\right]\right|_{y=\bar{y}}\nonumber \\ & = & -\epsilon^{\nu\lambda\tau\alpha}(p^{\mu}p_{\lambda}-g_{\lambda}^{\mu}p^{2}) \Delta_{\tau\kappa}(\bar{y})\Delta\beta_{\alpha}(\bar{y},x)\left.\left[\partial_{y}^{\kappa} \frac{ n_{\nu}(y)}{|p\cdot n(y)|}\right]\right|_{y=\bar{y}}\,. \end{align}\tag{45}\] With the help of the definition of \(\Delta_{\tau\kappa}(\bar{y})\), one can prove that: \[\begin{align} \label{eq:schouten-2} \epsilon^{\nu\lambda\tau\alpha}\Delta_{\tau}^{\;\kappa}(\bar{y}) & = & \epsilon^{\nu\lambda\tau\alpha}\left[g_{\tau}^{\kappa}-\frac{p^{\kappa} n_{\tau}(\bar{y})}{p\cdot n(\bar{y})}\right] = \left(p^{\tau}\epsilon^{\nu\lambda\kappa\alpha}-p^{\kappa}\epsilon^{\nu\lambda\tau\alpha}\right)\frac{ n_{\tau}(\bar{y})}{p\cdot n(\bar{y})}\nonumber \\ & = & -\left(p^{\nu}\epsilon^{\lambda\kappa\alpha\tau}+p^{\lambda}\epsilon^{\kappa\alpha\tau\nu} +p^{\alpha}\epsilon^{\tau\nu\lambda\kappa}\right)\frac{ n_{\tau}(\bar{y})}{p\cdot n(\bar{y})}\,, \end{align}\tag{46}\] where we have again used the Schouten identity (44 ) in the last step. Substituting 46 into eq. 45 and noting that (see also above discussion): \[\begin{align} &&p^{\lambda}(p^{\mu}p_{\lambda}-g_{\lambda}^{\mu}p^{2})=0\,, \\ && p^{\nu}\left.\left[\partial_{y}^{\kappa}\frac{ n_{\nu}(y)}{|p\cdot n(y)|}\right] \right|_{y=\bar{y}}=\left.\left[\partial_{y}^{\kappa}\text{sgn}[p\cdot n(y)]\right] \right|_{y=\bar{y}} = 0\,, \end{align}\] the expression in eq. 45 gets its final form: \[\label{finalform} (p^{\mu}p_{\lambda}-g_{\lambda}^{\mu}p^{2})\epsilon^{\lambda\tau\nu\kappa} \frac{ n_{\tau}(\bar{y})}{p\cdot n(\bar{y})}\left.\left[\partial_{\kappa}^{y} \frac{ n_{\nu}(y)}{|p\cdot n(y)|}\right]\right|_{y=\bar{y}} p^{\alpha}\Delta\beta_{\alpha}(\bar{y},x)\,.\tag{47}\] Defining the new vector: \[\label{defOmega} \Omega^{\lambda}(\bar{y})\equiv\epsilon^{\lambda\tau\nu\kappa}\frac{ n_{\tau}(\bar{y})}{p\cdot n(\bar{y})}\left.\left[\partial_{\kappa}^{y}\frac{ n_{\nu}(y)}{|p\cdot n(y)|}\right]\right|_{y=\bar{y}}=\frac{\text{sgn}[p\cdot n(\bar{y})]}{[p\cdot n(\bar{y})]^{2}}\epsilon^{\lambda\tau\nu\kappa} n_{\tau}(\bar{y})\left. \left[\partial_{\kappa}^{y} n_{\nu}(y)\right]\right|_{y=\bar{y}}\,,\tag{48}\] which depends on the curvature of the decoupling hypersurface at point \(\bar{y}\), the equation 47 can be rewritten as: \[(p^{\mu}p_{\lambda}-g_{\lambda}^{\mu}p^{2}) p^{\alpha}\Delta\beta_{\alpha}(\bar{y},x) \Omega^{\lambda}(\bar{y})\,.\] However, according to the definition of the normal unit vector we have: \[n_\mu = A \sigma_\mu\,,\] with \(A\) suitable scalar and \(\sigma\) like in eq. 14 . Hence: \[\label{dersigma} \partial_{\kappa} n_{\nu} = A \, \partial_\kappa \sigma_\nu + (\partial_\kappa A) n_\nu\tag{49}\] Since: \[\partial_{\kappa}^{y} \sigma_{\nu}(y)= \begin{cases} 0 & \kappa=0\text{ or }\nu=0\\ -\frac{\partial f_{k}({\boldsymbol{y}})}{\partial y^{i}\partial y^{j}} & \kappa=i,\;\nu=j,\;i,j=1,2,3 \end{cases}\] it turns out that the gradient of \(\sigma\) in eq. 49 is symmetric under the exchange of its indices. Therefore, the contraction with \(\epsilon^{\lambda\tau\nu\kappa}\) in equation 48 of both the terms in equation 49 vanishes and so \(\Omega^{\lambda}=0\). In conclusion, the expression in 47 vanishes, demonstrating that the curvature of \(\Sigma_{\rm D}\) does not contribute to the spin polarization at the leading order.

Finally, plugging eqs. 36 and 41 into eq. 22 , the formula 23 is obtained.

4 Isothermal decoupling hypersurface↩︎

We show that the temperature gradient does not contribute to the spin polarization vector in eq. 24 if the hypersurface is \(T={\rm const}\). Thermal vorticity and thermal shear tensors can be split as follows: \[\begin{align} \varpi^{\mu\nu} & = & -\frac{1}{2T}(\partial^{\mu}u^{\nu}-\partial^{\nu}u^{\mu})+ \frac{1}{2T^{2}}(u^{\nu}\partial^{\mu}T-u^{\mu}\partial^{\nu}T)\,, \nonumber \\ \xi^{\mu\nu} & = & \frac{1}{2T}(\partial^{\mu}u^{\nu}+\partial^{\nu}u^{\mu})- \frac{1}{2T^{2}}(u^{\nu}\partial^{\mu}T+u^{\mu}\partial^{\nu}T)\,. \end{align}\] Substituting them into eq. 24 , the vorticity-induced polarization and shear-induced polarization read: \[\begin{align} S_{\text{vorticity}}^{\mu}(p) & = & -\frac{1}{8mN_p}\int_{\Sigma_{\rm D}} \!\! {\rm d}\Sigma(x) \cdot p\, n_{F}(x,p)(1-n_{F}(x,p))\epsilon^{\mu\nu\rho\lambda}p_{\nu} \sum_{\bar{y}(x,p)} \text{sgn}[p\cdot n]\frac{1}{T}\left( \partial_{\lambda} u_{\rho}+ \frac{1}{T} u_{\lambda} \partial_{\rho} T \right)\Bigg|_{\bar{y}(x,p)}\,,\nonumber \\ S_{\text{shear}}^{\mu}(p) & = & -\frac{1}{8mN_p}\int_{\Sigma_{\rm D}} \!\! {\rm d}\Sigma(x) \cdot p\, n_{F}(x,p)(1-n_{F}(x,p)) \epsilon^{\mu\nu\rho\lambda}p_{\nu}\nonumber \\ && \times\sum_{\bar{y}(x,p)}\text{sgn}[p\cdot n] \frac{n_{\rho} p^{\alpha}}{T p\cdot n} \left[ \left( \partial_{\lambda} u_{\alpha} +\partial_{\alpha} u_{\lambda}\right) - \frac{1}{T} \left( u_{\lambda}\partial_{\alpha} T + u_{\alpha}\partial_{\lambda} T \right) \right] \Bigg|_{\bar{y}(x,p)}\,, \end{align}\] where the \(|_{\bar y(x,p)}\) implies that all functions and derivatives in the integrand are evaluated at the crossing points \(\bar y(x,p)\) of the worldline 15 and \(\Sigma_{\rm D}\), depending on both \(x\) and \(p\), as described in the text. We can put together the total contribution from the temperature gradient so as to obtain: \[\begin{align} \label{eq:T-gradient-polarization} S_{\text{T-grad}}^{\mu}(p) & = & -\frac{1}{8mN_p}\int_{\Sigma_{\rm D}} \!\! {\rm d}\Sigma(x) \cdot p\, n_{F}(x,p)[1-n_{F}(x,p)]\epsilon^{\mu\nu\rho\lambda}p_{\nu}\nonumber \\ && \times\sum_{\bar{y}(x,p)}\text{sgn}[p\cdot n] \frac{1}{T^{2}} \left\{ u_{\lambda}\left[\partial_{\rho}T-n_{\rho}\frac{p\cdot\partial T}{p\cdot n}\right] -\frac{p\cdot u}{p\cdot n}n_{\rho}\partial_{\lambda}T]\right\}\Bigg|_{\bar{y}(x,p)}\,. \end{align}\tag{50}\] We can now separate \(\partial_{\mu} T(y)\) into the derivative in the direction of \(n_{\mu}\) and those in the directions perpendicular to \(n\), that is tangent to the hypersurface: \[\partial_{\mu} T =n_{\mu} \frac{n \cdot\partial T}{n \cdot n} +\partial_{\mu}^{\perp}T \,.\] If the decoupling hypersurface is iso-thermal, that is \(T=\text{const}\), for any point on \(\Sigma_{\rm D}\) the projection \(\partial_{\mu}^{\perp} T\) vanishes and so: \[\partial_{\mu}T \propto n_{\mu} \,,\] indicating that \(\partial_{\mu}^{y}T(y)\) align in the same direction as \(n_{\mu}(y)\). We therefore have, in the equation 50 : \[\partial_\rho T - n_{\rho} \frac{p\cdot\partial T}{p\cdot n} = \partial_\rho T - \partial_{\rho} T = 0\,, \qquad\qquad \epsilon^{\mu\nu\rho\lambda}n_{\rho} \partial_{\lambda} T \propto \epsilon^{\mu\nu\rho\lambda}n_{\rho}n_{\lambda} =0\,,\] leading to \(S_{\text{T-grad}}^{\mu}(p)=0\). We can then conclude that, for an iso-thermal \(\Sigma_{\rm D}\), the temperature gradient contribute to the vorticity-induced and the shear-induced spin polarizations, respectively, but these two parts eventually cancel with each other and consequently the total spin polarization does not depend on the temperature gradient. If, on the other hand, \(\Sigma_{\rm D}\) is not iso-thermal, \(\partial_{\mu} T\) is not proportional to \(n_{\mu}\) and it could provide a nonvanishing contribution to the spin polarization.

References↩︎