Hybrid Framework of Fermi–Dirac Spin Hydrodynamics

Abstract

The paper outlines the hybrid framework of spin hydrodynamics, combining classical kinetic theory with the Israel–Stewart method of introducing dissipation. The local equilibrium expressions for the baryon current, the energy–momentum tensor, and the spin tensor of particles with spin 1/2 following the Fermi–Dirac statistics are obtained and compared with the earlier derived versions where the Boltzmann approximation was used. The expressions in the two cases are found to have the same form, but the coefficients are shown to be governed by different functions. The relative differences between the tensor coefficients in the Fermi–Dirac and Boltzmann cases are found to grow exponentially with the baryon chemical potential. In the proposed formalism, nonequilibrium processes are studied including mathematically possible dissipative corrections. Standard conservation laws are applied, and the condition of positive entropy production is shown to allow for the transfer between the spin and orbital parts of angular momentum.

1. Introduction

Experimental results indicating a nonzero spin polarization of particles produced in heavy-ion collisions [1,2,3,4] have generated a growing interest in observables related to spin degrees of freedom [5], motivating at the same time attempts at expanding the theoretical framework of relativistic hydrodynamics [6,7] to include a description of spin, thus giving rise to the field of spin hydrodynamics. This challenge of developing a new theoretical formalism has been pursued along several paths based on different assumptions. (a) The final particle spin polarization is determined on the basis of gradients of hydrodynamic fields on the freezeout hypersurface; the observables that are considered to be of particular importance in this case are the vorticity ${\varpi }_{\mu \nu }=-\frac{1}{2}\left({\partial }_{\mu }{\beta }_{\nu }-{\partial }_{\nu }{\beta }_{\mu }\right)$ and the thermal shear ${\xi }_{\mu \nu }=-\frac{1}{2}\left({\partial }_{\mu }{\beta }_{\nu }+{\partial }_{\nu }{\beta }_{\mu }\right)$, where the Greek letter indices, taking values 0 for time, and 1, 2, and 3 for space components, enumerate the components of a tensor in a basis of the tangent (upper index) and cotangent (lower index) spaces at a point of the 4-dimensional spacetime manifold, ${\partial }_{\mu }\equiv \partial /\partial x_{\mu }$ is the partial derivative, and ${\beta }^{\mu }=u^{\mu }/T$ is the ratio of the flow 4-vector $u^{\mu }$ and the temperature T (see Refs. [8,9,10,11,12]). (b) The spin hydrodynamics equations are derived from quantum or classical kinetic theory of particles with spin $\frac{1}{2}$ [13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28]. (c) Following the method of Israel and Stewart [29], the laws of conservation and entropy production are applied to general mathematically allowed forms of the energy–momentum and spin tensors [30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45]. Lastly, (d) Lagrangian effective field theory techniques are used in the context of a spin-polarizable medium [46,47,48].

Reexamination of the assumptions and establishment of relations between formalisms is considered to be helpful and essential. Any clarification and synthesis thus achieved may facilitate further progress and, as it seems, eventually lead to removing the present discrepancies. Hence, a recent proposal introduced a hybrid framework that combines the use of kinetic theory for the perfect-fluid description and the Israel–Stewart method for the inclusion of dissipation [49,50,51]. The attractive features of this approach include the following: the definition of the local thermodynamic equilibrium of particles with spin (conservation of the spin part of the total angular momentum); consistent and independent power counting in the expansion in the spin polarization tensor ${\omega }^{\mu \nu }$ and in the gradients (the latter appear in dissipative terms, while the second-order expansion in the former is already necessary on the perfect-fluid level to obtain nontrivial thermodynamic relations with spin); and the addition of dissipative effects without resorting to the complicated formalism of nonlocal collisions [19,20,21,22,23].

The current study focuses on a single species of spin-$\frac{1}{2}$ particles. An extension to the hydrodynamics of a medium composed of multiple species of spin-$\frac{1}{2}$ and spin-0 particles can be performed in a straightforward way by introducing a sum over all the particle species, with spin-$\frac{1}{2}$ particles described within the presented framework and spin-0 particles described according to standard relativistic fluid dynamics [52]. The inclusion of spin-1 particles (e.g., gluons) and particles with higher spin is the subject of investigations, as their theoretical description within a spin hydrodynamics framework analogous to the present one is not yet complete.

The study is conducted within the de Groot–van Leeuwen–van Weert (GLW) pseudo-gauge [53], which is particularly appropriate for the description of spin dynamics in hydrodynamic models of quark–gluon plasma (QGP) in heavy-ion collisions. The advantage of the GLW pseudo-gauge is that the spin tensor is conserved in the case of no nonlocal collisions [54], or, in the framework considered here, in the case of no dissipative corrections. In some pseudo-gauge choices, in particular the Belinfante pseudo-gauge, which leads to a symmetric energy–momentum tensor, the spin tensor is zero, making such choices unsuitable for the approach used; therefore, one needs a pseudo-gauge in which the spin tensor is not identically zero. Consequences of different pseudo-gauge choices for formulations of relativistic spin hydrodynamics and their results, such as the production of entropy, are a complex topic, inviting further investigation; see, e.g., Refs. [27,31,55,56].

This study takes as a starting point a perfect-fluid description based on the conservation of baryon number, energy, linear momentum, entropy, and, moreover, the spin part of the total angular momentum [13,57]. Polarization measurements suggest that the regime of low energies is of special interest, and in this regime, the assumption of separate conservation of spin is justified by the domination of the s-wave scattering [58]; conceptually, lower momentum leads to lower angular momentum. The assumption taken may also be justified by the relatively long spin relaxation times [59] (compare, for example, the systems described in Refs. [60,61]). The process of spin–orbit exchange is introduced in the considered approach by dissipative terms through the nonsymmetric part of the energy–momentum tensor.

This paper summarizes the hydrodynamic tensors appearing in the hybrid framework, expands the consideration of the more realistic case of the Fermi–Dirac statistics, and compares the results with the previously derived Boltzmann approximation.

2. Results

2.1. Tensors of Perfect Spin Hydrodynamics

Here, the prescription from Ref [50] is followed to derive the tensors of perfect spin hydrodynamics via the kinetic theory. The focus is in particular on the case of the Fermi–Dirac statistics. In the kinetic theory with classical treatment of spin, one introduces the internal angular momentum [62]

$$s^{\alpha \beta }=\frac{1}{m}{ϵ}^{\alpha \beta \gamma \delta }{p}_{\gamma }{s}_{\delta },$$

(1)

where the spin 4-vector $s^{\mu }$ is orthogonal to the 4-momentum, $s·p=0$, $s^{\alpha }={ϵ}^{\alpha \beta \gamma \delta }{p}_{\beta }{s}_{\gamma \delta }/\left(2m\right)$, with m denoting the particle mass and ${ϵ}^{\alpha \beta \gamma \delta }$ the Levi-Civita symbol. Throughout this study, the mostly negative metric $g^{\mu \nu }=\mathrm{diag}(1,-1,-1,-1)$ is used; in the particle rest frame, $p^{\mu }=(m,0,0,0)$, $s^{\alpha }=(0,{\mathbf{s}}_{*})$, $|{\mathbf{s}}_{*}|=\mathbf{ᵴ}$, and ${\mathbf{ᵴ}}^2=\frac{1}{2}\left(1+\frac{1}{2}\right)=\frac{3}{4}$.

The local equilibrium distribution functions for particles (+) and for antiparticles (−) have the Fermi–Dirac form

$$f_{\mathrm{eq}}^{\pm }(x,p,s)={\left[exp\left(\mp \xi \left(x\right)+p·\beta \left(x\right)-{\displaystyle \frac{1}{2}}\omega \left(x\right):s\right)+1\right]}^{-1}=\frac{1}{e^{y^{\pm }}+1},$$

(2)

with

$$y^{\pm }=\mp \xi \left(x\right)+p·\beta \left(x\right)-{\displaystyle \frac{1}{2}}\omega \left(x\right):s$$

(3)

and $\zeta =\mu /T$, where $\mu$ is the baryon chemical potential. Henceforward, the colon denotes contraction over both indices, $\omega :s\equiv {\omega }_{\mu \nu }{s}^{\mu \nu }$. The spin polarization tensor is parametrized using arbitrary 4-vectors $k^{\mu }$ and ${\omega }^{\mu }$ orthogonal to $u^{\mu }$ as follows:

$${\omega }_{\alpha \beta }=k_{\alpha }{u}_{\beta }-k_{\beta }{u}_{\alpha }+t_{\alpha \beta },$$

(4)

with $t_{\alpha \beta }={ϵ}_{\alpha \beta \gamma \delta }{u}^{\gamma }{\omega }^{\delta }$. The set of 4-vectors orthogonal to a given nonzero 4-vector can be parametrized with three parameters; thus, two such vectors have the same number of degrees of freedom as a rank-two antisymmetric tensor. The parametrization (4) introduced in Ref [13] ensures that ${\omega }_{\alpha \beta }$ is antisymmetric. Hereafter, the use is also made of the vector $t^{\mu }=t^{\mu \nu }{k}_{\nu }={ϵ}^{\mu \nu \alpha \beta }{k}_{\nu }{u}_{\alpha }{\omega }_{\beta }$.

In local equilibrium, the baryon current, the energy–momentum tensor, and the spin tensor can be written as

$$N_{\mathrm{eq}}^{\mu }=\int \mathrm{d}P\mathrm{d}S{p}^{\mu }\left[f_{\mathrm{eq}}^{+}(x,p,s)-f_{\mathrm{eq}}^{-}(x,p,s)\right],$$

(5)

$$T_{\mathrm{eq}}^{\mu \nu }=\int \mathrm{d}P\mathrm{d}S{p}^{\mu }{p}^{\nu }\left[f_{\mathrm{eq}}^{+}(x,p,s)+f_{\mathrm{eq}}^{-}(x,p,s)\right],$$

(6)

and

$$S_{\mathrm{eq}}^{\lambda ,\mu \nu }=\int \mathrm{d}P\mathrm{d}S{p}^{\lambda }{s}^{\mu \nu }\left[f_{\mathrm{eq}}^{+}(x,p,s)+f_{\mathrm{eq}}^{-}(x,p,s)\right],$$

(7)

respectively, with the integration measures

$$\mathrm{d}P=\frac{{\mathrm{d}}^{3}P}{{\left(2\pi \right)}^3{E}_p}\mathrm{and}\mathrm{d}S=\frac{m}{\pi \mathfrak{s}}\mathrm{d}s\delta (s·s+{\mathfrak{s}}^2)\delta (p·s),$$

(8)

where $E_p$ is the particle energy, and $\delta (·)$ represents the Dirac delta function.

For small enough values of ${\omega }^{\mu \nu }$, the Fermi–Dirac distribution function (2) can be expanded around $y_s\equiv -{\displaystyle \frac{1}{2}}\omega :s=0$:

$$f_{\mathrm{eq}}^{\pm }(x,p,s)=\frac{1}{e^{y_0^{\pm }+y_s}+1}=\frac{1}{e^{y_0^{\pm }}+1}-\frac{e^{y_0^{\pm }}}{{(e^{y_0^{\pm }}+1)}^2}{y}_s+\frac{e^{y_0^{\pm }}{(e^{y_0^{\pm }}-1)}^2}{2{(e^{y_0^{\pm }}+1)}^3}{y}_s^2+\dots ,$$

(9)

where $y_0^{\pm }\equiv \mp \xi \left(x\right)+p·\beta \left(x\right)$. Thus,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill f_{\mathrm{eq}}^{\pm }& =f_0^{\pm }-f_0^{\pm }(1-f_0^{\pm })y_s+{\displaystyle \frac{1}{2}}{f}_0^{\pm }(1-f_0^{\pm })(1-2f_0^{\pm })y_s^2+\dots \hfill \\ & \equiv f_0^{\pm }+{\displaystyle \frac{1}{2}}{f}_1^{\pm }\omega :s+{\displaystyle \frac{1}{8}}{f}_2^{\pm }{(\omega :s)}^2+\dots \hfill \end{array}\end{array}$$

(10)

Expansion up to the quadratic order in ${\omega }^{\mu \nu }$ in the case of $N_{\mathrm{eq}}^{\mu }$ (5) and $T_{\mathrm{eq}}^{\mu \nu }$ (6) (the linear terms vanish), and up to the linear order in ${\omega }^{\mu \nu }$ in the case of $S_{\mathrm{eq}}^{\lambda ,\mu \nu }$ (7) yields

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill N_{\mathrm{eq}}^{\mu }& =\int \mathrm{d}P\mathrm{d}S{p}^{\mu }\left[f_{\mathrm{eq}}^{+}(x,p,s)-f_{\mathrm{eq}}^{-}(x,p,s)\right]=2(Z_0^{+\mu }-Z_0^{-\mu })\hfill \\ & +{\mathfrak{s}}^2\frac{\omega :\omega }{6}(Z_2^{+\mu }-Z_2^{-\mu })+\frac{{\mathfrak{s}}^2}{3m^2}(Z_2^{+\mu \alpha \beta }-Z_2^{-\mu \alpha \beta }){\omega }_{\phantom{\gamma }\alpha }^{\gamma }{\omega }_{\beta \gamma }+\dots ,\hfill \end{array}\end{array}$$

(11)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill T_{\mathrm{eq}}^{\mu \nu }& =\int \mathrm{d}P\mathrm{d}S{p}^{\mu }{p}^{\nu }\left[f_{\mathrm{eq}}^{+}(x,p,s)+f_{\mathrm{eq}}^{-}(x,p,s)\right]=2\left(Z_0^{+\mu \nu }+Z_0^{-\mu \nu }\right)\hfill \\ \hfill & +{\mathfrak{s}}^2\frac{\omega :\omega }{6}\left(Z_2^{+\mu \nu }+Z_2^{-\mu \nu }\right)+\frac{{\mathfrak{s}}^2}{3m^2}\left(Z_2^{+\mu \nu \alpha \beta }+Z_2^{-\mu \nu \alpha \beta }\right){\omega }_{\phantom{\gamma }\alpha }^{\gamma }{\omega }_{\beta \gamma }+\dots ,\hfill \end{array}\end{array}$$

(12)

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill S_{\mathrm{eq}}^{\lambda ,\mu \nu }& =\int \mathrm{d}P\mathrm{d}S{p}^{\lambda }{s}^{\mu \nu }\left[f_{\mathrm{eq}}^{+}(x,p,s)+f_{\mathrm{eq}}^{-}(x,p,s)\right]=\frac{2{\mathfrak{s}}^2}{3}{\omega }^{\mu \nu }(Z_1^{+\lambda }+Z_1^{-\lambda })\hfill \\ & +\frac{2{\mathfrak{s}}^2}{3m^2}\left[(Z_1^{+\lambda \alpha \mu }+Z_1^{-\lambda \alpha \mu }){\omega }_{\alpha }^{\nu }-(Z_1^{+\lambda \alpha \nu }+Z_1^{-\lambda \alpha \nu }){\omega }_{\alpha }^{\mu }\right]+\dots \hfill \end{array}\end{array}$$

(13)

with the tensors $Z_n$ defined as

$$Z_n^{\pm \alpha \beta \dots }\equiv \int \mathrm{d}P{p}^{\alpha }{p}^{\beta }\dots f_n^{\pm }.$$

(14)

In the Boltzmann approximation, the local distribution function is considerably simpler, $f_{\mathrm{eq}}^{\pm }=exp\left(y^{\pm }\right)$ (compare Equations (2) and (3)), and the tensors $Z_n$ simplify to the form

$$Z_n^{\pm \alpha \beta \dots }\to e^{\pm \xi }{Z}^{\alpha \beta \dots }\equiv e^{\pm \xi }\int \mathrm{d}P{p}^{\alpha }{p}^{\beta }\cdots e^{-u·p/T},n=0,1,2,\dots$$

(15)

The tensors $Z_0$, $Z_1$, and $Z_2$ with up to four indices were expressed in terms of the function

$${\mathcal{J}}_{mn}(\xi ,z)={\int }_0^{\infty }\frac{{\sinh }^{m}y{\cosh }^{n}y}{\exp (-\xi +z\cosh y)+1}\mathrm{d}y,m,n=0,1,2,\dots$$

(16)

and its derivatives in Appendix A of Ref. [50], although subsequently only the Boltzmann approximation was considered. Here, the Fermi–Dirac case is addressed. Let us denote the first and the second derivative of the function ${\mathcal{J}}_{mn}$ with respect to $\xi$ with ${\dot{\mathcal{J}}}_{mn}$ and ${\ddot{\mathcal{J}}}_{mn}$, respectively:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{\mathcal{J}}}_{mn}(\xi ,z)& \equiv \frac{\partial }{\partial \xi }{\mathcal{J}}_{mn}(\xi ,z)={\int }_0^{\infty }\frac{{\sinh }^{m}y{\cosh }^{n}y}{2+2\cosh (\xi -z\cosh y)}\mathrm{d}y,\hfill \\ \hfill {\ddot{\mathcal{J}}}_{mn}(\xi ,z)& \equiv \frac{{\partial }^2}{\partial {\xi }^2}{\mathcal{J}}_{mn}(\xi ,z)={\int }_0^{\infty }\frac{{\sinh }^{m}y{\cosh }^{n}y\sinh (\xi -z\cosh y)}{2{\left(1+\cosh (\xi -z\cosh y)\right)}^2}\mathrm{d}y.\hfill \end{array}\end{array}$$

(17)

For brevity, let us introduce the following notation using the plus and minus superscript:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathcal{J}}_{mn}^{+}\equiv {\mathcal{J}}_{mn}(\xi ,z),{\mathcal{J}}_{mn}^{-}\equiv {\mathcal{J}}_{mn}(-\xi ,z),\\ \hfill {\mathcal{J}}_{mn}^{\pm }\equiv {\mathcal{J}}_{mn}^{+}+{\mathcal{J}}_{mn}^{-},{\mathcal{J}}_{mn}^{\mp }\equiv {\mathcal{J}}_{mn}^{+}-{\mathcal{J}}_{mn}^{-}\end{array}\end{array}$$

(18)

and the same for $\dot{\mathcal{J}}$ and $\ddot{\mathcal{J}}$.

This study derives the Fermi–Dirac tensors by gathering the expressions derived in Appendix A of Ref. [50], inserting those into Equations (11)–(13), and performing contractions and further simplifications, while noticing the benefit of hyperbolic identities in expressing the functions ${\mathcal{J}}_{mn}$, ${\dot{\mathcal{J}}}_{mn}$, and ${\ddot{\mathcal{J}}}_{mn}$ by functions with different indices, as needed to convert the expressions into their most compact forms. Finally, the three tensors (5)–(7) read

$$\begin{array}{cc}\hfill N_{\mathrm{eq}}^{\mu }& =(n_0+n_2^{\omega }+n_2^k)u^{\mu }+n_t{t}^{\mu },\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill T_{\mathrm{eq}}^{\mu \nu }& =({\epsilon }_0+{\epsilon }_2^{\omega }+{\epsilon }_2^k)u^{\mu }{u}^{\nu }-(P_0+P_2^{\omega }+P_2^k){\Delta }^{\mu \nu }\hfill \\ \hfill & +P_t(t^{\mu }{u}^{\nu }+t^{\nu }{u}^{\mu })+P_{k\omega }(k^{\mu }{k}^{\nu }+{\omega }^{\mu }{\omega }^{\nu }),\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill S_{\mathrm{eq}}^{\lambda ,\mu \nu }& =A_1{u}^{\lambda }{\omega }^{\mu \nu }+\frac{A_2}{2}{u}^{\lambda }(u^{\mu }{k}^{\nu }-u^{\nu }{k}^{\mu })+\frac{A_3}{2}{t}^{\lambda \mu \nu }\hfill \\ \hfill & =u^{\lambda }\left[A_3\left(k^{\mu }{u}^{\nu }-k^{\nu }{u}^{\mu }\right)+A_1{t}^{\mu \nu }\right]+{\displaystyle \frac{A_3}{2}}\left(t^{\lambda \mu }{u}^{\nu }-t^{\lambda \nu }{u}^{\mu }+{\Delta }^{\lambda \mu }{k}^{\nu }-{\Delta }^{\lambda \nu }{k}^{\mu }\right),\hfill \end{array}$$

(21)

where ${\Delta }^{\mu \nu }\equiv g^{\mu \nu }-u^{\mu }{u}^{\nu }$ and ns, ($\epsilon \mathrm{s}$, Ps), and $A_n$s constitute the sets of coefficients for each of the three tensors.

The forms of the tensors are the same in the Fermi–Dirac and the Boltzmann cases, but the functional dependences on m, T, and $\mu$ of the coefficients differ, as, in the Boltzman case, the much simpler integrals in Equation (15) can be expressed in terms of Bessel functions of the second kind $K_n\left(z\right)$. Let us note that compared with the standard perfect-fluid expressions,

$$N_{\mathrm{eq}}^{\mu }=n{u}^{\mu },$$

(22)

$$T_{\mathrm{eq}}^{\mu \nu }=\epsilon u^{\mu }{u}^{\nu }-P{\Delta }^{\mu \nu },$$

(23)

and the “phenomenological” form of the spin tensor [63]

$$S_{\mathrm{eq}}^{\lambda ,\mu \nu }=u^{\lambda }{S}^{\mu \nu },$$

(24)

there are additional terms in the framework considered here. Although seemingly their mathematical structure is typical of dissipative corrections, in this framework, the additional terms appear already in the description of the perfect fluid. The entropy current, defined here as

$$S_{\mathrm{eq}}^{\mu }=T_{\mathrm{eq}}^{\mu \alpha }{\beta }_{\alpha }-\frac{1}{2}{\omega }_{\alpha \beta }{S}_{\mathrm{eq}}^{\mu ,\alpha \beta }-\xi N_{\mathrm{eq}}^{\mu }+{\mathcal{N}}^{\mu },$$

(25)

where ${\mathcal{N}}^{\mu }$ is a spin-hydrodynamical analog of $P{\beta }^{\mu }$, remains conserved. The complete results for the Fermi–Dirac and Boltzmann tensor coefficients obtained are displayed in

Table 1. The baryon current (19).

Coefficient	Fermi–Dirac Case	Boltzmann Case
$n_0$	$\frac{m^3}{{\pi }^2}{\mathcal{J}}_{21}^{\mp }$	$\frac{2\sinh \xi }{{\pi }^2}{z}^2{T}^3{K}_2\left(z\right)$
$n_2^{\omega }$	$-{\omega }^2\frac{{\mathfrak{s}}^2{m}^3}{18{\pi }^2}({\ddot{\mathcal{J}}}_{21}^{\mp }+2{\ddot{\mathcal{J}}}_{23}^{\mp })$	$-{\omega }^2\frac{{\mathfrak{s}}^2\sinh \xi }{3{\pi }^2}z{T}^3\left[z{K}_2\left(z\right)+2K_3\left(z\right)\right]$
$n_2^k$	$-k^2\frac{{\mathfrak{s}}^2{m}^3}{9{\pi }^2}{\ddot{\mathcal{J}}}_{41}^{\mp }$	$-k^2\frac{2{\mathfrak{s}}^2\sinh \xi }{3{\pi }^2}z{T}^3{K}_3\left(z\right)$
$n_t$	$-\frac{{\mathfrak{s}}^2{m}^3}{9{\pi }^2}{\ddot{\mathcal{J}}}_{41}^{\mp }$	$-{\displaystyle \frac{2{\mathfrak{s}}^2\sinh \xi }{3{\pi }^2}}z{T}^3{K}_3\left(z\right)$

,

Table 2. The energy–momentum tensor (20).

Coefficient	Fermi–Dirac Case	Boltzmann Case
${\epsilon }_0$	$\frac{m^4}{{\pi }^2}{\mathcal{J}}_{22}^{\pm }$	$\frac{2 \cosh \xi }{{\pi }^2}{z}^2{T}^4[z{K}_3\left(z\right)-K_2\left(z\right)]$
${\epsilon }_2^{\omega }$	$-{\omega }^2\frac{{\mathfrak{s}}^2{m}^4}{18{\pi }^2}({\ddot{\mathcal{J}}}_{22}^{\pm }+2{\ddot{\mathcal{J}}}_{24}^{\pm })$	$-{\omega }^2\frac{{\mathfrak{s}}^2\cosh \xi }{3{\pi }^2}z{T}^4[z{K}_2\left(z\right)+(z^2+10)K_3\left(z\right)]$
${\epsilon }_2^k$	$-k^2\frac{{\mathfrak{s}}^2{m}^4}{9{\pi }^2}{\ddot{\mathcal{J}}}_{42}^{\pm }$	$-k^2\frac{2 {\mathfrak{s}}^2\cosh \xi }{3{\pi }^2}z{T}^4[z{K}_2\left(z\right)+5K_3\left(z\right)]$
$P_0$	$\frac{m^4}{3{\pi }^2}{\mathcal{J}}_{40}^{\pm }$	$\frac{2 \cosh \xi }{{\pi }^2}{z}^2{T}^4{K}_2\left(z\right)$
$P_2^{\omega }$	$-{\omega }^2\frac{{\mathfrak{s}}^2{m}^4}{90{\pi }^2}({\ddot{\mathcal{J}}}_{40}^{\pm }+4{\ddot{\mathcal{J}}}_{42}^{\pm })$	$-{\omega }^2\frac{{\mathfrak{s}}^2\cosh \xi }{3{\pi }^2}z{T}^4[z{K}_2\left(z\right)+4K_3\left(z\right)]$
$P_2^k$	$-k^2\frac{2{\mathfrak{s}}^2{m}^4}{45{\pi }^2}{\ddot{\mathcal{J}}}_{60}^{\pm }$	$-k^2\frac{4 {\mathfrak{s}}^2\cosh \xi }{3{\pi }^2}z{T}^4{K}_3\left(z\right)$
$P_{k\omega }$	$-\frac{{\mathfrak{s}}^2{m}^4}{45{\pi }^2}{\ddot{\mathcal{J}}}_{60}^{\pm }$	$-\frac{2 {\mathfrak{s}}^2\cosh \xi }{3{\pi }^2}z{T}^4{K}_3\left(z\right)$
$P_t$	$-\frac{{\mathfrak{s}}^2{m}^4}{9{\pi }^2}{\ddot{\mathcal{J}}}_{42}^{\pm }$	$-\frac{2 {\mathfrak{s}}^2\cosh \xi }{3{\pi }^2}z{T}^4[z{K}_2\left(z\right)+5K_3\left(z\right)]$

and

Table 3. The spin tensor (21).

Coefficient	Fermi–Dirac Case	Boltzmann Case
$A_1$	$\frac{{\mathfrak{s}}^2{m}^3}{9{\pi }^2}({\dot{\mathcal{J}}}_{21}^{\pm }+2{\dot{\mathcal{J}}}_{23}^{\pm })$	$\frac{2 {\mathfrak{s}}^2\cosh \xi }{3{\pi }^2}z{T}^3\left[z{K}_2\left(z\right)+2K_3\left(z\right)\right]$
$A_2$	$\frac{2{\mathfrak{s}}^2{m}^3}{3{\pi }^2}(2{\dot{\mathcal{J}}}_{23}^{\pm }-{\dot{\mathcal{J}}}_{21}^{\pm })$	$\frac{4 {\mathfrak{s}}^2\cosh \xi }{3{\pi }^2}{z}^2{T}^3{K}_4\left(z\right)$
$A_3$	$-\frac{2{\mathfrak{s}}^2{m}^3}{9{\pi }^2}{\dot{\mathcal{J}}}_{41}^{\pm }$	$-\frac{4 {\mathfrak{s}}^2\cosh \xi }{3{\pi }^2}z{T}^3{K}_3\left(z\right)$

for the three tensors. Note that the transition from the first line to the second line in Equation (21) used the observation that $A_2=2A_1-4A_3$, which can be verified using hyperbolic identities in the Fermi–Dirac case and Bessel function identities in the Boltzmann case.

Figure 1 compares the Fermi–Dirac and Boltzmann cases. As can be readily seen, for constant m and T, the relative difference increases exponentially with $\mu$ for most of $\mu$-range. In the low-$\mu$ regime, where the Fermi–Dirac distribution can be satisfactorily approximated by the Boltzmann distribution, the difference between the coefficient values in the two cases is negligible, but in the high-$\mu$ regime the difference becomes significant. The relative differences between the two cases are largest for $n_2^{\omega }$ and ${ϵ}_2^{\omega }$, reaching the order of 1 for $\xi =\mu /T\approx 10$ (for the sample values of $m=1000$ MeV and $T=100$ MeV chosen).

2.2. Tensors of Dissipative Spin Hydrodynamics

Following the Israel–Stewart method, let us write down general nonequilibrium expressions as a sum of the equilibrium terms and dissipative corrections:

$$N^{\mu }=N_{\mathrm{eq}}^{\mu }+\delta N^{\mu },T^{\mu \nu }=T_{\mathrm{eq}}^{\mu \nu }+\delta T^{\mu \nu },S^{\mu ,\alpha \beta }=S_{\mathrm{eq}}^{\mu ,\alpha \beta }+\delta S^{\mu ,\alpha \beta }.$$

(26)

It is through dissipative terms that the spin–orbit interaction is introduced; in general, $T^{\mu \nu }$ contains now nonsymmetric parts, and the spin tensor is no longer seperately conserved:

$${\partial }_{\mu }{J}^{\mu ,\alpha \beta }=0,J^{\mu ,\alpha \beta }=x^{\alpha }{T}^{\mu \beta }-x^{\beta }{T}^{\mu \alpha }+S^{\mu ,\alpha \beta },$$

(27)

$${\partial }_{\mu }{S}^{\mu ,\alpha \beta }=T^{\beta \alpha }-T^{\alpha \beta },{\partial }_{\mu }{N}^{\mu }=0,{\partial }_{\mu }{T}^{\mu \nu }=0.$$

(28)

From Equations (25), (26), and (28), one finds

$${\partial }_{\mu }{S}^{\mu }=\delta N^{\mu }{\partial }_{\mu }\xi +\delta T_s^{\mu \nu }{\partial }_{\mu }{\beta }_{\nu }+\delta T_a^{\mu \nu }({\partial }_{\mu }{\beta }_{\nu }-{\omega }_{\nu \mu })-\frac{1}{2}\delta S^{\mu ,\alpha \beta }{\partial }_{\mu }{\omega }_{\alpha \beta }$$

(29)

(compare, for example, with Refs. [30,55]). To find the form of the deviations $\delta N^{\mu }$, $\delta T^{\mu \nu }$, and $\delta S^{\mu ,\alpha \beta }$, let us use a decomposition of general tensors of the given symmetry via projections along $u^{\mu }$ and separation of those tensors into the symmetric (labeled by the subscript s) and antisymmetric (a) parts, obtaining, through the Landau matching conditions (see [50] for details):

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \delta N^{\mu }& =V^{\mu },\hfill \\ \hfill \delta T_s^{\mu \nu }& =-\mathsf{\Pi }{\Delta }^{\mu \nu }+W^{\mu }{u}^{\nu }+W^{\nu }{u}^{\mu }+{\pi }^{\mu \nu },\hfill \\ \hfill \delta T_a^{\mu \nu }& =d_a^{\mu }{u}^{\nu }-d_a^{\nu }{u}^{\mu }+e_a^{\mu \nu },\hfill \\ \hfill \delta S^{\lambda ,\mu \nu }& ={\mathsf{\Sigma }}^{\lambda \mu }{u}^{\nu }-{\mathsf{\Sigma }}^{\lambda \nu }{u}^{\mu }+{\varphi }^{\lambda \mu \nu },\hfill \end{array}\end{array}$$

(30)

with

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill V^{\mu }& =b^{\mu }-n_t{t}^{\mu },\mathsf{\Pi }=e-(P_0+P_2^{\omega }+P_2^k)+(1/3)P_{k\omega }(k^2+{\omega }^2),\hfill \\ \hfill W^{\mu }& =d_s^{\mu }-P_t{t}^{\mu },{\pi }^{\mu \nu }=e_s^{\langle \mu \nu \rangle }-P_{k\omega }(k^{\langle \mu }{k}^{\nu \rangle }+{\omega }^{\langle \mu }{\omega }^{\nu \rangle }),\hfill \\ \hfill {\mathsf{\Sigma }}^{\lambda \mu }& =i^{\lambda \mu }-2t^{\lambda \mu },{\varphi }^{\lambda \mu \nu }=j^{\lambda \mu \nu }-2({\Delta }^{\lambda \mu }{k}^{\nu }-{\Delta }^{\lambda \nu }{k}^{\mu }).\hfill \end{array}\end{array}$$

(31)

Equation (30) has a form that was analyzed in Ref. [37], and the coefficients $b^{\mu }$, $d_s^{\mu }$, $d_a^{\mu }$e, $e_s^{\langle \mu \nu \rangle }$, $e_a^{\mu \nu }$, $i^{\lambda \mu }$, and $j^{\lambda \mu \nu }$ can be expressed in terms of gradients of hydrodynamic variables multiplied by kinetic coefficients, including the shear and bulk viscosities $\eta$ and $\zeta$, respectively, the thermal conductivity $\kappa$, the coefficients ${\lambda }_a$ and $\gamma$ from Ref. [30], and the coefficients ${\chi }_1,{\chi }_2,{\chi }_3,{\chi }_4$ introduced in Ref. [37]. To the first order in gradients, one finds

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill b^{\mu }& =\lambda {\nabla }^{\mu }\xi +n_t{t}^{\mu },d_s^{\mu }=-\kappa (u^{\nu }{\partial }_{\nu }{u}^{\mu }-\beta {\nabla }^{\mu }T)+P_t{t}^{\mu },\hfill \\ \hfill d_a^{\mu }& ={\lambda }_a{\beta }^{-1}(\beta u^{\nu }{\partial }_{\nu }{u}^{\mu }+{\beta }^2{\nabla }^{\mu }T-2k^{\mu }),\hfill \\ \hfill e& =P_0+P_2^{\omega }+P_2^k-\zeta \theta -(1/3)P_{k\omega }(k^2+{\omega }^2),\hfill \\ \hfill e_s^{\langle \mu \nu \rangle }& =2\eta {\sigma }^{\mu \nu }+P_{k\omega }(k^{\langle \mu }{k}^{\nu \rangle }+{\omega }^{\langle \mu }{\omega }^{\nu \rangle }),e_a^{\mu \nu }=\gamma \beta {\nabla }^{[\mu }{u}^{\nu ]},\hfill \\ \hfill i^{\lambda \mu }& =-{\chi }_1{\Delta }^{\lambda \mu }{u}^{\beta }{\nabla }^{\alpha }{\omega }_{\alpha \beta }-{\chi }_2{u}_{\nu }{\nabla }^{\langle \lambda }{\omega }^{\mu \rangle \nu }-{\chi }_3{u}_{\nu }{\Delta }_{\rho }^{[\mu }{\nabla }^{\lambda ]}{\omega }^{\rho \nu }+\frac{A^3}{2}{t}^{\lambda \mu },\hfill \\ \hfill j^{\lambda \mu \nu }& =\frac{{\chi }^4}{2}{\nabla }^{\lambda }{\omega }^{\langle \mu \rangle \langle \nu \rangle }+\frac{A^3}{2}({\Delta }^{\lambda \mu }{k}^{\nu }-{\Delta }^{\lambda \nu }{k}^{\mu }).\hfill \end{array}\end{array}$$

(32)

Angular brackets around a pair of indices denote the orthogonal, symmetric, and traceless part of the tensor, defined via a contraction with the projector ${\Delta }_{\alpha \beta }^{\mu \nu }$: $B^{\langle \alpha \beta \rangle }\equiv {\Delta }_{\alpha \beta }^{\mu \nu }{B}^{\alpha \beta }\equiv \frac{1}{2}\left({\Delta }_{\alpha }^{\mu }{\Delta }_{\beta }^{\nu }+{\Delta }_{\alpha }^{\nu }{\Delta }_{\beta }^{\mu }-\frac{2}{3}{\Delta }^{\mu \nu }{\Delta }_{\alpha \beta }\right)B^{\alpha \beta }$. Squared brackets denote the antisymmetric part: $B^{\left[\mu \nu \right]}\equiv {\displaystyle \frac{1}{2}}(B^{\mu \nu }-B^{\nu \mu })$. A ngular brackets around a single index denote an orthogonal projection: $C^{\alpha \beta \langle \gamma \rangle \delta ...}\equiv {\Delta }_{\rho }^{\gamma }{C}^{\alpha \beta \rho \delta ...}$. Differential operators take precedence over the orthogonal projection. The nabla symbol denotes the transverse gradient: ${\nabla }^{\mu }\equiv {\Delta }^{\mu \nu }{\partial }_{\nu }$.

Importantly, the spin polarization tensor ${\omega }_{\mu \nu }$, which in natural units is a dimensionless quantity, is not considered to be a gradient term. Already, the perfect-fluid case is based on an expansion in ${\omega }_{\mu \nu }$ to the second order. Gradient terms are only introduced when considering dissipation, and the order in gradients is counted separately to the order in ${\omega }_{\mu \nu }$. Therefore, in the expansion (32), which is linear in gradients, terms such as ${\nabla }^{\alpha }{\omega }_{\alpha \beta }$ survive and are not neglected as higher-order contributions.

In the future, the Israel–Stewart method may be used to construct a second-order theory, similar to the derivation performed in Ref. [37]; the main difference, compared to the study in Ref. [37], is that the order in gradients is to be counted independently of the order in ${\omega }_{\mu \nu }$, as stressed just above.

3. Discussion

The hybrid framework of spin hydrodynamics combines the perfect-fluid results of kinetic theory for particles with spin $\frac{1}{2}$ with the Israel–Stewart approach for including nonequilibrium processes. The framework involves a two-fold expansion: in the spin polarization tensor ${\omega }_{\mu \nu }$ (already in local equilibrium) and in gradients of hydrodynamic variables (when considering close-to-equilibrium dynamics). If the spin polarization is nonvanishing, the perfect-fluid description contains seemingly dissipative, transverse terms. However, genuine dissipative terms appear only at the level of the dissipative fluid. Those terms are determined by the condition of positive entropy production.

The current study focuses on summarizing the derivation of the tensors appearing in the hybrid framework, with the emphasis on Fermi–Dirac statistics. Thus, it complements the outline given in Ref. [51], which discusses in more detail the motivation behind the framework, as well as the modification of the ordinary thermodynamic relations of relativistic hydrodynamics that makes them consistent with the observation that the spin tensor (21) derived from kinetic theory, in contrast with the “phenomenological” spin tensor (24), contains parts orthogonal to the flow vector $u^{\mu }$.

It was shown that the difference between the tensor coefficients derived for the Fermi–Dirac statistics and their Boltzmann approximation is significant in the regime of high baryon chemical potential.

The coefficients derived herein may be employed in lieu of the Boltzmann approximation in future computer simulations, thereby enhancing their realism. Although the functions ${\mathcal{J}}_{mn}$, ${\dot{\mathcal{J}}}_{mn}$, and ${\ddot{\mathcal{J}}}_{mn}$, unlike Bessel functions, are not a part of standard mathematical packages, their suitable tabulation and interpolation for use in practical codes presents only little technical complication.

The first numerical test of the hybrid framework was performed in a relatively simple boost-invariant perfect-fluid case in Ref. [64]. That study can be extended to include the Fermi–Dirac statistics and dissipative terms. The framework is also quite straightforward to implement in numerical simulations of spin hydrodynamics in more realistic expansion models, similar to those presented in Refs. [65,66].