#  Spin waves in spin hydrodynamics

Victor E. Ambrus<EMAIL_ADDRESS>Institut für Theoretische Physik,
Johann Wolfgang Goethe-Universität, Max-von-Laue-Strasse 1, D-60438 Frankfurt
am Main, Germany  Department of Physics, West University of Timisoara,
Bd. Vasile Pârvan 4, Timisoara 300223, Romania  Radoslaw Ryblewski
<EMAIL_ADDRESS>Rajeev Singh<EMAIL_ADDRESS>Institute
of Nuclear Physics Polish Academy of Sciences, PL 31-342 Kraków, Poland

###### Abstract

The propagation properties of spin degrees of freedom are analyzed in the
framework of relativistic hydrodynamics with spin based on the de Groot–van
Leeuwen–van Weert definitions of the energy-momentum and spin tensors. We
derive the analytical expression for the spin wave velocity for arbitrary
statistics and show that it goes to half the speed of light in the
ultrarelativistic limit. We find that only the transverse degrees of freedom
propagate, analogously to electromagnetic waves. Finally, we consider the
effect of dissipative corrections and calculate the damping coefficients for
the case of Maxwell-Jüttner statistics.

heavy-ion collisions, non-boost-invariant dynamics,spin polarization,vorticity

††preprint:

##  I Introduction

Recent spin polarization measurements of $\Lambda(\bar{\Lambda})$ hyperons [1,
2, 3, 4, 5, 6, 7, 8] have sparked a huge interest in the heavy-ion physics
community. In this context, many theoretical studies have been performed
referring to spin-orbit coupling [9, 10, 11, 12]. Fundamentally, the
polarization of particles with spin can be induced through the spin-orbit
coupling implied by the Dirac equation [13, 14]. Starting with the works by
Vilenkin in the 1980s [15], it is now understood that a gas of Dirac particles
in rigid motion develops a flow of chirality along the vorticity direction
[16]. Due to its close relation with the axial anomaly, the flow of chirality
due to either background vorticity or electromagnetic fields is understood as
“anomalous transport” [16]. Attempts to incorporate such effects dynamically
have lead to the development of the so-called hydrodynamics with triangle
anomalies [17]. While the persistent polarization of massless particles can be
modeled via an axial chemical potential, such an approach is not justified for
massive particles, where the conservation of the axial current is explicitly
broken (alternatively the helical chemical potential may be used, as discussed
in Refs. [18, 19, 20]).

Various models based on the thermodynamic equilibrium of spin degrees of
freedom [21, 22, 23, 24] have shown good agreement with experimental data of
spin polarization, for recent reviews and papers on this topic see, e.g.,
Refs. [25, 26, 27, 28, 29, 30, 31, 32]. Nevertheless, the differential
measurements of polarization [4, 8] lack a clear explanation. This led to the
idea of including spin degrees of freedom in standard hydrodynamics, first
proposed in Refs. [33, 34] based on the definitions of the energy-momentum and
spin tensors introduced by de Groot, van Leeuwen, and van Weert (GLW) [35].
For recent studies on this formalism see Refs. [36, 37, 38, 39, 40, 41].

In this work we consider the propagation properties of linear perturbations
[42, 43, 44, 45, 46] in the framework of the perfect-fluid spin hydrodynamics
[26, 36], for other similar studies using the effective action approach, see
Refs. [47, 48, 49]. At the level of the spin conservation equation, we find
that the spin degrees of freedom are decoupled from the background fluid, and
therefore their wave spectrum can be analyzed separately from the fluid
degrees of freedom. Conversely, the fluid degrees of freedom are also
decoupled from the spin ones leading to the well-known sound waves [42, 43,
44, 45]. In this study, we consider a linearized expression for the spin
tensor in which quadratic or higher order terms are neglected. For this
reason, our results are strictly valid only for the case of propagation
through an unpolarized background. In this case, we obtain a general analytic
expression for the spin wave velocity, which we apply to the case of both
Maxwell-Jüttner (MJ) and Fermi-Dirac (FD) statistics. In addition, we also
derive the relativistic and nonrelativistic limits of the spin wave velocity
$c_{\rm spin}$. In both cases, $c_{\rm spin}=c/2$ in the ultrarelativistic
limit, with $c$ being the speed of light. The spin degrees of freedom can be
split into an electric part, $C_{\bm{\kappa}}$, and a magnetic part,
$C_{\bm{\omega}}$, in analogy with the electromagnetic degrees of freedom. We
find that the degrees of freedom corresponding to the longitudinal direction
(which is parallel to the wave vector $\mathbf{k}$) do not propagate, while
the four transverse ones support the usual linear or circular polarization, in
perfect analogy to the case of electromagnetic waves [50]. Finally, using the
dissipative spin tensor derived in Refs. [38, 51] for the case of the ideal MJ
gas, we discuss the effect of dissipative corrections leading to exponential
damping of both the transverse and the longitudinal components.

The paper is organized as follows. We begin with a brief review of the
formalism of spin hydrodynamics in Sec. II. Then, in Sec. III, we study the
propagation of perturbations in the spin polarization components and present
the spin wave solutions. Subsequently, in Sec. IV, we analyze the effects of
dissipation on the spin wave propagation. Finally, we conclude in Sec. V.
Technical details about the spin tensor for arbitrary statistics, the ideal
gas, and the FD gas can be found in Appendices A, B, and C, respectively.

In this work, we use the convention of the Minkowski metric
$g_{\mu\nu}=\hbox{diag}(+1,-1,-1,-1)$, while the dot product of two four-
vectors $a^{\alpha}$ and $b^{\alpha}$ reads $a\cdot
b=a^{\alpha}b_{\alpha}=g_{\alpha\beta}a^{\alpha}b^{\beta}=a^{0}b^{0}-{\bm{a}}\cdot{\bm{b}}$,
where boldface indicates three-vectors. For the Levi-Civita tensor
$\epsilon^{\alpha\beta\gamma\delta}$ we use the convention
$\epsilon^{txyz}=+1$. We denote the antisymmetrization by a pair of square
brackets as $M_{[\mu\nu]}=\frac{1}{2}\left(M_{\mu\nu}-M_{\nu\mu}\right)$.
Moreover, we assume natural (Planck) units, i.e. $c=\hbar=k_{B}~{}=1$ (unless
stated explicitly).

##  II Perfect-fluid spin hydrodynamics

In this section, we briefly review the hydrodynamic framework based on the GLW
definitions of energy-momentum and spin tensors for the case of
spin-$\frac{1}{2}$ particles with mass $m$ [36, 26]. In this framework, the
spin effects are assumed to be small so that the conservation laws for charge,
energy, and momentum are independent of the spin tensor. The spin effects
arise only from the conservation of angular momentum [36, 26]. The
conservation laws of baryon current and energy-momentum tensor are defined,
respectively, as [33, 36, 26]

$\displaystyle\partial_{\alpha}N^{\alpha}(x)=0\,,\quad\partial_{\beta}T^{\alpha\beta}(x)=0\,,$
(1)

where the baryon current, $N^{\alpha}$, and the energy-momentum tensor,
$T^{\alpha\beta}$, are of the form [33]

$\displaystyle N^{\alpha}={\cal N}U^{\alpha}\,,\quad T^{\alpha\beta}={\cal
E}U^{\alpha}U^{\beta}-{\cal P}~{}\Delta^{\alpha\beta}\,,$ (2)

with ${\cal N}$, ${\cal E}$, and ${\cal P}$ being the baryon charge density,
energy density, and pressure respectively. The fluid four-velocity is denoted
by $U^{\mu}$ and $\Delta^{\alpha\beta}=g^{\alpha\beta}-U^{\alpha}U^{\beta}$ is
the projector onto the hypersurface orthogonal to $U^{\mu}$.

Due to the symmetric nature of the energy-momentum tensor (2), the
conservation of total angular momentum dictates the separate conservation of
spin [36]

$\displaystyle\partial_{\alpha}S^{\alpha,\beta\gamma}(x)$ $\displaystyle=$
$\displaystyle 0.$ (3)

Violations of the above conservation equation can be induced through quantum
effects such as nonlocal collisions [52, 27, 53, 54, 55], leading most likely
to a relaxation of the spin polarization tensor $\omega^{\mu\nu}$ (6) towards
the local thermal vorticity. Since the exact form of this relaxation equation
is not known yet, we do not consider such effects in this analysis. To the
leading order in $\omega^{\beta\gamma}$, the spin tensor can be decomposed as
[36, 26, 41]

$\displaystyle S^{\alpha,\beta\gamma}$ $\displaystyle=$ $\displaystyle
S^{\alpha,\beta\gamma}_{\rm ph}+S^{\alpha,\beta\gamma}_{\Delta},$ (4a) where
the phenomenological $S^{\alpha,\beta\gamma}_{\rm ph}$ and the auxiliary
$S^{\alpha,\beta\gamma}_{\Delta}$ contributions are given by [33, 41]
$\displaystyle S^{\alpha,\beta\gamma}_{\rm ph}$ $\displaystyle=$
$\displaystyle(\mathcal{A}_{1}+\mathcal{A}_{3})U^{\alpha}\omega^{\beta\gamma},$
(4b) $\displaystyle S^{\alpha,\beta\gamma}_{\Delta}$ $\displaystyle=$
$\displaystyle(2\mathcal{A}_{1}-\mathcal{A}_{3})\,U^{\alpha}U^{\delta}U^{[\beta}\omega^{\gamma]}_{\hphantom{\alpha}\delta}+\mathcal{A}_{3}\Big{(}\Delta^{\alpha\delta}U^{[\beta}\omega^{\gamma]}_{\hphantom{\alpha}\delta}$
(4c)
$\displaystyle+U^{\alpha}\Delta^{\delta[\beta}\omega^{\gamma]}_{\hphantom{\alpha}\delta}+U^{\delta}\Delta^{\alpha[\beta}\omega^{\gamma]}_{\hphantom{\alpha}\delta}\Big{)}.$

The thermodynamic coefficients that appear above can be expressed as follows
(see Appendix A for details)

$\displaystyle\mathcal{A}_{1}$ $\displaystyle=$
$\displaystyle\frac{\mathbf{\mathfrak{s}}^{2}}{9}\left[\left(\frac{\partial\mathcal{N}}{\partial\xi}\right)_{\beta}-\frac{2}{m^{2}}\left(\frac{\partial\mathcal{E}}{\partial\beta}\right)_{\xi}\right],$
$\displaystyle\mathcal{A}_{3}$ $\displaystyle=$
$\displaystyle\frac{2\mathbf{\mathfrak{s}}^{2}}{9}\left[\left(\frac{\partial\mathcal{N}}{\partial\xi}\right)_{\beta}+\frac{1}{m^{2}}\left(\frac{\partial\mathcal{E}}{\partial\beta}\right)_{\xi}\right],$
(5)

where we used general expressions for $\mathcal{A}_{1}$ and $\mathcal{A}_{3}$
which are independent of the underlying statistics of the kinetic
model.111See, e.g., Refs. [36, 51, 26, 41] for the corresponding expressions
for the MJ statistics of an ideal gas, which we summarize in Eq. 65. The case
of the FD statistics is discussed in Appendix C. In the above formula,
$\xi=\mu/T$ is the ratio of the chemical potential to the temperature, $\beta$
is the inverse of the temperature, while $\mathbf{\mathfrak{s}}^{2}=s(s+1)$ is
the magnitude of spin angular momentum, which is equal to $3/4$ for
spin-$\frac{1}{2}$ particles [26]. For future convenience, we also introduce
$z=m/T$ representing the ratio of particle mass $m$ and the temperature.

The (antisymmetric) spin polarization tensor $\omega_{\mu\nu}$ can be
decomposed as [33]

$\displaystyle\omega_{\mu\nu}$ $\displaystyle=$
$\displaystyle\kappa_{\mu}U_{\nu}-\kappa_{\nu}U_{\mu}+\epsilon_{\mu\nu\alpha\beta}U^{\alpha}\omega^{\beta}.$
(6)

where $\kappa_{\mu}$ and $\omega_{\mu}$ together form six independent
components [26, 36]. These four-vectors are orthogonal to $U^{\mu}$ by
construction, $\kappa_{\mu}~{}U^{\mu}=\omega_{\mu}~{}U^{\mu}=0$, such that
[26, 36]

$\displaystyle\kappa_{\mu}=\omega_{\mu\alpha}U^{\alpha},\quad\omega_{\mu}=\frac{1}{2}\epsilon_{\mu\alpha\beta\gamma}\omega^{\alpha\beta}U^{\gamma}.$
(7)

In the fluid rest frame, $\kappa^{\mu}$ and $\omega^{\mu}$ reduce to

$\displaystyle\kappa^{\mu}=\left(0,C_{\bm{\kappa}}\right),\quad\omega^{\mu}=\left(0,C_{\bm{\omega}}\right)\,,$
(8)

where $C_{\bm{\kappa}}=(C_{\kappa X},C_{\kappa Y},C_{\kappa Z})$ and
$C_{\bm{\omega}}=(C_{\omega X},C_{\omega Y},C_{\omega Z})$ are the spin
polarization components [26, 36].

##  III Wave analysis

###  III.1 Dispersion relation for the spin modes

Let us now consider the propagation of infinitesimal excitations in a fluid
with spin degrees of freedom. Since the conservation equations (1)
corresponding to the background fluid are independent of polarization [36,
26], their solutions will give the well-known spectrum of sound waves [42, 43,
44, 45], which propagate with the sound speed satisfying

$\displaystyle
c_{s}^{2}=\left(\frac{\partial\mathcal{P}}{\partial\mathcal{E}}\right)_{\mathcal{N}}+\frac{\mathcal{N}}{\mathcal{E}+\mathcal{P}}\left(\frac{\partial\mathcal{P}}{\partial\mathcal{N}}\right)_{\mathcal{E}}.$
(9)

Focusing now on the excitations propagating at the level of the spin tensor
(4a), the background fluid can be regarded as quiescent, i.e.,
$U^{\mu}=g^{t\mu}$. Treating $\omega^{\mu\nu}$ as a small quantity, which
amounts to assuming that the background fluid is unpolarized, Eqs. (4b) and
(4c) reduce to

$\displaystyle S^{\alpha,\mu\nu}_{\rm ph}$ $\displaystyle=$
$\displaystyle(\mathcal{A}_{1}+\mathcal{A}_{3})g^{t\alpha}\omega^{\mu\nu},$
$\displaystyle S^{\alpha,\mu\nu}_{\Delta}$ $\displaystyle=$ $\displaystyle
2(\mathcal{A}_{1}-2\mathcal{A}_{3})g^{t\alpha}g^{t[\mu}\omega^{\nu]t}$ (10)
$\displaystyle+\mathcal{A}_{3}(g^{t[\mu}\omega^{\nu]\alpha}+g^{\alpha[\mu}\omega^{\nu]t}-g^{t\alpha}\omega^{\mu\nu}).$

Considering that the system is homogeneous with respect to the $x$ and $y$
directions, the divergence of Eq. (10) yields

$\displaystyle\partial_{\alpha}S^{\alpha,\mu\nu}_{\rm ph}$ $\displaystyle=$
$\displaystyle(\mathcal{A}_{1}+\mathcal{A}_{3})\partial_{t}\omega^{\mu\nu},$
$\displaystyle\partial_{\alpha}S^{\alpha,\mu\nu}_{\Delta}$ $\displaystyle=$
$\displaystyle(2\mathcal{A}_{1}-3\mathcal{A}_{3})g^{t[\mu}\partial_{t}\omega^{\nu]t}$
(11)
$\displaystyle+\mathcal{A}_{3}(\partial^{[\mu}\omega^{\nu]t}-\partial_{t}\omega^{\mu\nu}+g^{t[\mu}\partial_{z}\omega^{\nu]z}).$

For the cases $\mu=0,\nu=i$ and $\mu=i,\nu=j$, we find, respectively,

$\displaystyle\partial_{\alpha}S^{\alpha,ti}$ $\displaystyle=$
$\displaystyle\mathcal{A}_{3}\left(\partial_{t}\omega^{ti}+\frac{1}{2}\partial_{z}\omega^{iz}\right),$
$\displaystyle\partial_{\alpha}S^{\alpha,ij}$ $\displaystyle=$
$\displaystyle\mathcal{A}_{1}\partial_{t}\omega^{ij}+\mathcal{A}_{3}\partial^{[i}\omega^{j]t}.$
(12)

Figure 1: Top, middle: the speed of the spin wave $c_{\rm spin}$ as a function
of $z=m/T$ corresponding to the (top) MJ and (middle) FD statistics, together
with the asymptotic forms for small and large $z$ given in Eqs. (19) and (20),
respectively. Bottom: comparison between $c_{\rm spin}$ obtained for the MJ
and FD statistics, for various values of $\xi=\mu/T$ (the MJ result is
independent of $\xi$). The MJ curves are obtained using Eq. (17), while the FD
curves are obtained using Eqs. (18) and (80) when $z>|\xi|$ and $z<|\xi|$,
respectively.

Taking into account Eq. (8), the components of the spin polarization tensor
$\omega^{\mu\nu}$ can be written in terms of the spin polarization components
$C_{\kappa i}$ and $C_{\omega k}$ as

$\displaystyle\omega^{ti}=-C_{\kappa
i},\qquad\omega^{ij}=-\epsilon^{tijk}C_{\omega k}.$ (13)

Demanding that $\partial_{\alpha}S^{\alpha,\mu\nu}=0$, we obtain

$\displaystyle\partial_{t}C_{\kappa
i}-\frac{1}{2}\epsilon^{tijz}\partial_{z}C_{\omega
j}=0,\quad\partial_{t}C_{\omega
i}-\frac{\mathcal{A}_{3}}{2\mathcal{A}_{1}}\epsilon^{tijz}\partial_{z}C_{\kappa
j}=0.~{}~{}~{}~{}~{}~{}~{}$ (14)

Due to the presence of the Levi-Civita symbol, $\partial_{t}C_{\kappa
Z}=\partial_{t}C_{\omega Z}=0$, such that the longitudinal components do not
propagate. Thus, the polarization degrees of freedom propagate only as
transverse waves, similar to the electromagnetic waves [50]. Their equation
can be obtained by setting $i=x,y$ in Eq. (14), leading to

$\displaystyle\left(\frac{\partial^{2}}{\partial t^{2}}-c_{\rm
spin}^{2}\frac{\partial^{2}}{\partial z^{2}}\right)\mathcal{C}=0,$ (15)

where $\mathcal{C}\in\\{C_{\kappa X},C_{\kappa Y},C_{\omega X},C_{\omega
Y}\\}$ and the speed of the spin wave satisfies

$\displaystyle c_{\rm
spin}^{2}=-\frac{1}{4}\frac{\mathcal{A}_{3}}{\mathcal{A}_{1}}=\frac{1}{4}\frac{(\partial\mathcal{E}/\partial
T)_{\xi}-z^{2}(\partial\mathcal{N}/\partial\xi)_{T}}{(\partial\mathcal{E}/\partial
T)_{\xi}+\frac{z^{2}}{2}(\partial\mathcal{N}/\partial\xi)_{T}}.$ (16)

In the ultrarelativistic limit $z\rightarrow 0$, we can observe that $c_{\rm
spin}$ takes the value $1/2$ irrespective of statistics.

The expression for $c_{\rm spin}^{2}$ can be written explicitly for the
(ideal) MJ gas [56]

$\displaystyle c_{\rm
spin}^{2}=\frac{1}{4}\frac{K_{3}(z)}{K_{3}(z)+\frac{z}{2}K_{2}(z)},$ (17)

hence being independent of $\xi=\mu/T$. In the case of the FD gas [57, 58],
$c_{\rm spin}$ becomes an even function of $\xi$:222Equation (18) is valid
only when $|\xi|<z$. At higher values of $|\xi|$, the formal series with
respect to $\ell$ diverge and the integral representation in Eq. (80) must be
employed.

$\displaystyle c_{\rm
spin}^{2}=\frac{\frac{1}{4}\sum_{\ell=1}^{\infty}\frac{(-1)^{\ell+1}}{\ell}\cosh(\ell\xi)K_{3}(\ell
z)}{\sum_{\ell=1}^{\infty}\frac{(-1)^{\ell+1}}{\ell}\cosh(\ell\xi)[K_{3}(\ell
z)+\frac{\ell z}{2}K_{2}(\ell z)]}.~{}~{}~{}$ (18)

For small values of $z$, we find

MJ: $\displaystyle c_{\rm spin}=$
$\displaystyle\frac{1}{2}\left[1-\frac{z^{2}}{16}+O(z^{4})\right],$ FD:
$\displaystyle c_{\rm spin}=$
$\displaystyle\frac{1}{2}\left[1-\frac{15z^{2}}{4\pi^{2}}\frac{1+\frac{3\xi^{2}}{\pi^{2}}}{7+30\frac{\xi^{2}}{\pi^{2}}+15\frac{\xi^{4}}{\pi^{4}}}+O(z^{4})\right].$
(19)

In the nonrelativistic limit, when $z\gg 1$, we get

$\displaystyle c_{\rm spin}\simeq\frac{1}{\sqrt{2z}}\,.$ (20)

The details of these calculations are provided in Appendices B and C. The
above limits are validated by comparison with the exact expressions in Eqs.
(17) and (18) in Figs. 1. It can be observed that $c_{\rm spin}$ is a
monotonically decreasing function of $z$, such that

$\displaystyle 0<c_{\rm spin}\leq\frac{1}{2}\,,$ (21)

where the lower limit is reached for a cold gas of massive particles (the
nonrelativistic limit), while the upper limit is achieved at high temperatures
or for massless particles.

###  III.2 Linear and circular polarization of spin waves

We now construct explicit expressions for the spin wave. Taking as before a
wave propagating along the $z$ direction, Eqs. (14) reduce to

$\displaystyle\partial_{t}C_{\kappa X}-\frac{1}{2}\partial_{z}C_{\omega Y}$
$\displaystyle=$ $\displaystyle 0,\quad\frac{1}{2}\partial_{t}C_{\omega
Y}-c_{\rm spin}^{2}\partial_{z}C_{\kappa X}=0,$
$\displaystyle\partial_{t}C_{\kappa Y}+\frac{1}{2}\partial_{z}C_{\omega X}$
$\displaystyle=$ $\displaystyle 0,\quad\frac{1}{2}\partial_{t}C_{\omega
X}+c_{\rm spin}^{2}\partial_{z}C_{\kappa Y}=0.$ (22)

The linearly polarized solutions for the three-vectors $C_{\bm{\kappa}}$ and
$C_{\bm{\omega}}$ are

$\displaystyle C_{\bm{\kappa}}$ $\displaystyle=$ $\displaystyle C_{0}{\rm
Re}[e^{-ik(c_{\rm spin}t-z)}](\bm{e}_{1}\cos\theta+\bm{e}_{2}\sin\theta),$
$\displaystyle C_{\bm{\omega}}$ $\displaystyle=$ $\displaystyle 2c_{\rm
spin}C_{0}{\rm Re}[e^{-ik(c_{\rm
spin}t-z)}](\bm{e}_{1}\sin\theta-\bm{e}_{2}\cos\theta),$ (23)

where $C_{0}$ is the real amplitude of the wave and $\theta$ is the
inclination angle with respect to the $x$ axis. It can be observed that

$\displaystyle C_{\bm{\omega}}=2c_{\rm spin}\hat{\bm{n}}\times
C_{\bm{\kappa}}\,,$ (24)

where $\hat{\bm{n}}=\bm{e}_{3}$ is the direction vector of the wave. The above
equation is analogous to the relation $\bm{H}=c\hat{\bm{n}}\times\bm{D}$ from
electromagnetism [50] where $c$ is the speed of light.

Right- and left-handed (R / L) circularly polarized waves can be constructed
in the standard fashion,

$\displaystyle C_{{\bm{\kappa}};R/L}$ $\displaystyle=$
$\displaystyle\frac{C_{0}}{\sqrt{2}}{\rm Re}[e^{-ik(c_{\rm
spin}t-z)}(\bm{e}_{1}\cos\theta\pm i\bm{e}_{2}\sin\theta)],$ (25)
$\displaystyle C_{{\bm{\omega}};R/L}$ $\displaystyle=$
$\displaystyle\frac{2c_{\rm spin}C_{0}}{\sqrt{2}}{\rm Re}[e^{-ik(c_{\rm
spin}t-z)}(\bm{e}_{1}\sin\theta\mp i\bm{e}_{2}\cos\theta)],$

where again Eq. (24) holds.

##  IV Dissipative effects

Figure 2: The $z$ dependence of the damping coefficients for the longitudinal
($\nu_{\omega,||}$, $\nu_{\kappa,||}$, starting at $0.2$) and transverse
($\nu_{\kappa,\perp}$, $\nu_{\perp}$ and $\nu_{\omega,\perp}$, starting at
$0.15$) modes calculated for $\mathbf{\mathfrak{s}}^{2}=3/4$ based on Eqs.
(34), (36), and (39).

In this section we consider the effects of dissipation on the propagation of
spin modes. In performing this analysis, we rely on the analysis of
dissipative effects presented in Ref. [51]. Since Ref. [51] employs the MJ
statistics of the ideal gas, we restrict the discussion in this section to
this particular case.

In the context of the relaxation time approximation, the dissipative
corrections to $T^{\mu\nu}$ and $N^{\mu}$ turn out to be independent of the
spin tensor. The correction to the spin term due to dissipation can be written
as follows

$\delta
S^{\lambda,\mu\nu}=\tau_{R}(B^{\lambda,\mu\nu}_{\Pi}\theta+B^{\kappa\lambda,\mu\nu}_{n}\nabla_{\kappa}\xi+B^{\kappa\delta\lambda,\mu\nu}_{\pi}\sigma_{\kappa\delta}\\\
+B^{\eta\beta\gamma\lambda,\mu\nu}_{\Sigma}\nabla_{\eta}\omega_{\beta\gamma}),$
(26)

where $\tau_{R}$ is the relaxation time and
$\nabla_{\mu}=\Delta_{\mu}{}^{\nu}\partial_{\nu}=\partial_{\mu}-U_{\mu}U^{\nu}\partial_{\nu}.$
As in the previous sections, we consider small perturbations on top of a
quiescent, unpolarized background state in thermal equilibrium. In order to
investigate the effect of dissipation on the propagation of small
perturbations, and in particular to assert the stability of the theory of spin
hydrodynamics, we will regard the perturbations amplitudes (including the
magnitude of $\delta S^{\lambda,\mu\nu}\sim\omega^{\mu\nu}$) as infinitesimal,
while allowing the gradients, proportional to the wave number $k$, to be
arbitrarily large, thereby retaining higher order terms with respect to $k$
and $\tau_{R}$. Within this framework, we can test if any instabilities emerge
in the $k\rightarrow\infty$ limit (small wavelengths). We note that, due to
the truncation procedure leading to Eq. (26), we cannot expect the physics of
the $\tau_{R}\gg 1$ and/or $k\gg 1$ regimes to be correctly recovered.

Coming back to Eq. (26), the coefficients $B^{\lambda,\mu\nu}_{\Pi}$,
$B^{\kappa\lambda,\mu\nu}_{n}$, and $B^{\kappa\delta\lambda,\mu\nu}_{\pi}$ are
proportional to the spin polarization tensor $\omega^{\mu\nu}$ [51], which we
assume to be of first order with respect to the perturbation amplitude (the
background state is assumed to be unpolarized). These terms are multiplied by
$\theta=\partial_{\mu}u^{\mu}$, $\nabla_{\kappa}\xi$, and
$\sigma_{\kappa\delta}=\frac{1}{2}(\nabla_{\kappa}u_{\delta}+\nabla_{\delta}u_{\kappa})-\frac{1}{3}\theta\Delta_{\kappa\delta}$,
respectively, which are already of first order with respect to the gradients
of the background state. Since they are of second order with respect to the
perturbation amplitude, the first three terms appearing in Eq. (26) can be
safely neglected and we focus only on the last term given by [51]

$B^{\eta\beta\gamma\lambda,\mu\nu}_{\Sigma}=B^{(1)}_{\Sigma}\Delta^{\lambda\eta}g^{\beta[\mu}g^{\nu]\gamma}+B^{(2)}_{\Sigma}\Delta^{\lambda\eta}u^{\gamma}u^{[\mu}\Delta^{\nu]\beta}\\\
+B^{(3)}_{\Sigma}(\Delta^{\lambda\eta}\Delta^{\gamma[\mu}g^{\nu]\beta}+\Delta^{\lambda\gamma}\Delta^{\eta[\mu}g^{\nu]\beta}+\Delta^{\gamma\eta}\Delta^{\lambda[\mu}g^{\nu]\beta})\\\
+B^{(4)}_{\Sigma}\Delta^{\gamma\eta}\Delta^{\lambda[\mu}\Delta^{\nu]\beta}+B^{(5)}_{\Sigma}u^{\gamma}\Delta^{\lambda\beta}u^{[\mu}\Delta^{\nu]\eta}.$
(27)

The quantities $B^{(i)}_{\Sigma}$ are [51]

$\displaystyle B^{(1)}_{\Sigma}=$
$\displaystyle-\frac{4\mathbf{\mathfrak{s}}^{2}}{3}\cosh\xi I^{(1)}_{21},$
$\displaystyle B^{(2)}_{\Sigma}=$
$\displaystyle-\frac{8\mathbf{\mathfrak{s}}^{2}}{3m^{2}}\cosh\xi\left(I^{(1)}_{41}+\frac{I^{(1)}_{41}I^{(0)}_{31}}{m^{2}I^{(0)}_{10}-2I^{(0)}_{31}}\right),$
$\displaystyle B^{(3)}_{\Sigma}=$
$\displaystyle-\frac{8\mathbf{\mathfrak{s}}^{2}}{3m^{2}}\cosh\xi
I^{(1)}_{42},$ $\displaystyle B^{(4)}_{\Sigma}=$
$\displaystyle-\frac{8\mathbf{\mathfrak{s}}^{2}}{3m^{2}}\cosh\xi\frac{I^{(1)}_{41}I^{(0)}_{31}}{m^{2}I^{(0)}_{10}-(I^{(0)}_{30}+I^{(0)}_{31})},$
$\displaystyle B^{(5)}_{\Sigma}=$
$\displaystyle\frac{8\mathbf{\mathfrak{s}}^{2}}{3m^{2}}\cosh\xi\frac{I^{(1)}_{41}I^{(0)}_{31}}{m^{2}I^{(0)}_{10}-2I^{(0)}_{31}},$
(28)

where $I^{(r)}_{nq}$ are thermodynamic integrals of the form

$\displaystyle I^{(r)}_{nq}=\frac{1}{(2q+1)!!}\int\mathrm{dP}(u\cdot
p)^{n-2q-r}(\Delta_{\alpha\beta}p^{\alpha}p^{\beta})^{q}e^{-\beta u\cdot p},$

and $\mathrm{dP}=d^{3}p/(2\pi)^{3}p^{0}$ defines the invariant integration
measure. Since $A^{\eta}\nabla_{\eta}\omega_{\beta\gamma}$ reduces to
$A^{z}\partial_{z}\omega_{\beta\gamma}$, the index $\eta$ can be safely set to
$z$. Performing the splitting

$\displaystyle\partial_{\lambda}\delta
S^{\lambda,\mu\nu}=\tau_{R}\sum_{i}B_{\Sigma}^{(i)}\mathfrak{T}^{(i)\mu\nu},$
(30)

we find

$\displaystyle\mathfrak{T}^{(1)\mu\nu}=$
$\displaystyle-\partial_{z}^{2}\omega^{\mu\nu},$
$\displaystyle\mathfrak{T}^{(2)\mu\nu}=$
$\displaystyle-g^{t[\mu}\partial_{z}^{2}\omega^{\nu]t},$
$\displaystyle\mathfrak{T}^{(3)\mu\nu}=$
$\displaystyle\partial_{z}^{2}\omega^{\mu\nu}+g^{t[\mu}\partial^{2}_{z}\omega^{\nu]t}+2g^{z[\mu}\partial^{2}_{z}\omega^{\nu]z},$
$\displaystyle\ \mathfrak{T}^{(4)\mu\nu}=$ $\displaystyle
g^{z[\mu}\partial^{2}_{z}\omega^{\nu]z}-g^{z[\mu}g^{\nu]t}\partial^{2}_{z}\omega^{tz},$
$\displaystyle\mathfrak{T}^{(5)\mu\nu}=$ $\displaystyle
g^{z[\mu}g^{\nu]t}\partial^{2}_{z}\omega^{tz}.$ (31)

Grouping all terms together gives

$\frac{1}{\tau_{R}}\partial_{\lambda}\delta
S^{\lambda,\mu\nu}=-\left(B^{(1)}_{\Sigma}-B^{(3)}_{\Sigma}\right)\partial_{z}^{2}\omega^{\mu\nu}\\\
-\left(B^{(2)}_{\Sigma}-B^{(3)}_{\Sigma}\right)g^{t[\mu}\partial^{2}_{z}\omega^{\nu]t}+2B^{(3)}_{\Sigma}g^{z[\mu}\partial^{2}_{z}\omega^{\nu]z}\\\
-\left(B^{(4)}_{\Sigma}-B^{(5)}_{\Sigma}\right)g^{z[\mu}g^{\nu]t}\partial^{2}_{z}\omega^{tz}+B^{(4)}_{\Sigma}g^{z[\mu}\partial^{2}_{z}\omega^{\nu]z}.$
(32)

Noting that $\omega^{ti}=-C_{\kappa i}$ and
$\omega^{ij}=-\epsilon^{0ijk}C_{\omega k}$, we have:

$\displaystyle\partial_{\lambda}\delta S^{\lambda,tx}=$
$\displaystyle\nu_{\kappa,\perp}\mathcal{A}_{3}\partial^{2}_{z}C_{\kappa X},$
$\displaystyle\partial_{\lambda}\delta S^{\lambda,yz}=$
$\displaystyle\nu_{\omega,\perp}\mathcal{A}_{1}\partial^{2}_{z}C_{\omega X},$
$\displaystyle\partial_{\lambda}\delta S^{\lambda,ty}=$
$\displaystyle\nu_{\kappa,\perp}\mathcal{A}_{3}\partial^{2}_{z}C_{\kappa Y},$
$\displaystyle\partial_{\lambda}\delta S^{\lambda,zx}=$
$\displaystyle\nu_{\omega,\perp}\mathcal{A}_{1}\partial^{2}_{z}C_{\omega Y},$
$\displaystyle\partial_{\lambda}\delta S^{\lambda,tz}=$
$\displaystyle\nu_{\kappa,||}\mathcal{A}_{3}\partial^{2}_{z}C_{\kappa Z},$
$\displaystyle\partial_{\lambda}\delta S^{\lambda,xy}=$
$\displaystyle\nu_{\omega,||}\mathcal{A}_{1}\partial^{2}_{z}C_{\omega Z}.$
(33)

In Eq. (33) we identified the longitudinal ($\nu_{\kappa,||}$,
$\nu_{\omega,||}$) and transverse ($\nu_{\kappa,\perp}$, $\nu_{\omega,\perp}$)
kinematic viscosities,

$\displaystyle\nu_{\kappa,||}=$
$\displaystyle\frac{\tau_{R}}{\mathcal{A}_{3}}\left(B^{(1)}_{\Sigma}-\frac{1}{2}B^{(2)}_{\Sigma}-\frac{3}{2}B^{(3)}_{\Sigma}-\frac{1}{2}B^{(5)}_{\Sigma}\right)=\frac{\tau_{R}}{\mathcal{A}_{3}}B^{(3)}_{\Sigma},$
$\displaystyle\nu_{\omega,||}=$
$\displaystyle\frac{\tau_{R}}{\mathcal{A}_{1}}\left(B^{(1)}_{\Sigma}-B^{(3)}_{\Sigma}\right),$
$\displaystyle\nu_{\kappa,\perp}=$
$\displaystyle\frac{\tau_{R}}{\mathcal{A}_{3}}\left(B^{(1)}_{\Sigma}-\frac{1}{2}B^{(2)}_{\Sigma}-\frac{1}{2}B^{(3)}_{\Sigma}\right),$
$\displaystyle\nu_{\omega,\perp}=$
$\displaystyle\frac{\tau_{R}}{\mathcal{A}_{1}}\left(B^{(1)}_{\Sigma}-2B^{(3)}_{\Sigma}-\frac{1}{2}B^{(4)}_{\Sigma}\right).$
(34)

The expression for $\nu_{\kappa,||}$ follows after applying the recurrence
relation
$I^{(r)}_{nq}=\frac{1}{2q+1}\left(m^{2}I^{(r)}_{n-2,q-1}-I^{(r)}_{n,q-1}\right)$
[51] to $I^{(1)}_{42}$.

Combining the above results with Eq. (12), one can find that $C_{\kappa Z}$
and $C_{\omega Z}$ exhibit exponential decay,

$\displaystyle\partial_{t}C_{\kappa
Z}-\nu_{\kappa,||}\partial^{2}_{z}C_{\kappa Z}=0,\quad\partial_{t}C_{\omega
Z}-\nu_{\omega,||}\partial^{2}_{z}C_{\omega Z}=0.~{}~{}~{}~{}$ (35)

Setting now $C_{\kappa/\omega;Z}\\!\\!\sim\\!\\!e^{-i\omega
t+ikz}\widetilde{C}_{\kappa/\omega;Z}$, where
$\widetilde{C}_{\kappa/\omega;Z}$ is a constant, we find
$\omega=-ik^{2}\nu_{\kappa/\omega,||}$ with

$\displaystyle\nu_{\kappa,||}$ $\displaystyle=$
$\displaystyle\frac{4\mathbf{\mathfrak{s}}^{2}\tau_{R}}{45G(z)}[-5z+G(z)(3+z^{2})-z^{2}{\rm
Gi}(z)]$ $\displaystyle\simeq$
$\displaystyle\frac{4\mathbf{\mathfrak{s}}^{2}\tau_{R}}{15}\left[1-\frac{z^{2}}{12}+O(z^{4})\right],$
$\displaystyle\nu_{\omega,||}$ $\displaystyle=$
$\displaystyle\frac{4\mathbf{\mathfrak{s}}^{2}\tau_{R}}{15(2G(z)+z)}[5z+G(z)(2-z^{2})+z^{2}{\rm
Gi}(z)]$ (36) $\displaystyle\simeq$
$\displaystyle\frac{4\mathbf{\mathfrak{s}}^{2}\tau_{R}}{15}\left[1-\frac{z^{4}}{16}+O(z^{5})\right],$

where $G(z)=K_{3}(z)/K_{2}(z)$ and ${\rm Gi}(z)={\rm Ki}_{1}(z)/K_{2}(z)$. The
above expressions are represented as functions of $z$ in Fig. 2.

Performing now the Fourier decomposition
$\mathcal{C}=\widetilde{\mathcal{C}}e^{-i\omega t+ikz}$ of the transverse
modes, we find

$\displaystyle\begin{pmatrix}\omega+ik^{2}\nu_{\kappa,\perp}&-k/2\\\
\frac{k\mathcal{A}_{3}}{2\mathcal{A}_{1}}&\omega+ik^{2}\nu_{\omega,\perp}\end{pmatrix}\begin{pmatrix}\widetilde{C}_{\kappa
X}\\\ \widetilde{C}_{\omega Y}\end{pmatrix}$ $\displaystyle=$ $\displaystyle
0,$ $\displaystyle\begin{pmatrix}\omega+ik^{2}\nu_{\kappa,\perp}&k/2\\\
-\frac{k\mathcal{A}_{3}}{2\mathcal{A}_{1}}&\omega+ik^{2}\nu_{\omega,\perp}\end{pmatrix}\begin{pmatrix}\widetilde{C}_{\kappa
Y}\\\ \widetilde{C}_{\omega X}\end{pmatrix}$ $\displaystyle=$ $\displaystyle
0.$ (37)

The dispersion relation implied by the above system is

$\displaystyle\omega_{\pm}=$ $\displaystyle-ik^{2}\nu_{\perp}\pm kc_{\rm
spin}\,,\qquad\nu_{\perp}=\frac{\nu_{\kappa,\perp}+\nu_{\omega,\perp}}{2}\,,$
$\displaystyle c_{\rm spin}^{2}=$
$\displaystyle-\frac{\mathcal{A}_{3}}{4\mathcal{A}_{1}}-\frac{k^{2}}{4}\left(\nu_{\kappa,\perp}-\nu_{\omega,\perp}\right)^{2}.$
(38)

The damping coefficient $\nu_{\perp}$ is just the average of the damping
coefficients found separately for the $\kappa$ and $\omega$ sectors,

$\displaystyle\nu_{\perp}=$
$\displaystyle\frac{2\mathbf{\mathfrak{s}}^{2}\tau_{R}[3G(z)+2z]}{45G(z)[2G(z)+z]}[-5z+G(z)(3+z^{2})-z^{2}{\rm
Gi}(z)]$ $\displaystyle\simeq$
$\displaystyle\frac{\mathbf{\mathfrak{s}}^{2}\tau_{R}}{5}\left[1-\frac{z^{2}}{24}+O(z^{4})\right].$
(39)

The above expression is represented as a function of $z$ in Fig. 2.

The speed of the spin wave receives a dissipative correction of negative sign,
which can be estimated by writing $c_{\rm spin}^{2}=c^{2}_{\rm
spin;0}(1-\delta c^{2}_{\rm spin})$, where $c_{\rm
spin;0}^{2}=-\mathcal{A}_{3}/4\mathcal{A}_{1}>0$ and

$\delta c^{2}_{\rm
spin}=\frac{k^{2}(\nu_{\kappa,\perp}-\nu_{\omega,\perp})^{2}}{-\mathcal{A}_{3}/\mathcal{A}_{1}}\simeq\frac{k^{2}\mathbf{\mathfrak{s}}^{4}\tau_{R}^{2}}{8100}z^{4}[1+O(z^{6})].$
(40)

The above correction is heavily suppressed at small values of $z$. At finite
$z$, the wave number can become large enough to render $c_{\rm spin}^{2}$
negative. This happens when $k$ exceeds the threshold value given by

$\displaystyle k_{\rm th}=\frac{2c_{\rm
spin;0}}{|\nu_{\kappa,\perp}-\nu_{\omega,\perp}|}.$ (41)

When $k>k_{\rm th}$, $c_{\rm spin}$ becomes imaginary and the wave no longer
propagates. This is reminiscent of similar effects occurring in first-order
hydrodynamics for spinless systems. One example is the case of sound modes in
ultrarelativistic fluids, where $\tau_{R}k_{\rm
th}=\frac{5\eta}{4\mathcal{P}}k_{\rm th}=15/2$ [45]. Considering now the
regime when $k\gg k_{\rm th}$, Eq. (38) shows that the modes remain stable
provided

$\displaystyle\nu_{\perp}-\frac{1}{2}|\nu_{\kappa,\perp}-\nu_{\omega,\perp}|={\rm
min}(\nu_{\kappa,\perp},\nu_{\omega,\perp})>0.$ (42)

The above inequality holds true within the framework studied here. We show
this in the regime of small $z$, when

$\displaystyle\nu_{\kappa,\perp}$ $\displaystyle\simeq$
$\displaystyle\frac{\mathfrak{s}^{2}\tau_{R}}{5}\left[1-\frac{z^{2}}{72}+O(z^{4})\right],$
$\displaystyle\nu_{\omega,\perp}$ $\displaystyle\simeq$
$\displaystyle\frac{\mathfrak{s}^{2}\tau_{R}}{5}\left[1-\frac{5z^{2}}{72}+O(z^{4})\right],$
(43)

while $\tau_{R}k_{\rm th}\simeq 18/(5z^{2}\mathfrak{s}^{2})$. Figure 2
confirms that both $\nu_{\kappa,\perp}$ and $\nu_{\omega,\perp}$ remain
positive at large $z$, thus the theory is stable under linear perturbations.

Let us now consider the impact of dissipation on the propagation of the spin
waves in the context of heavy-ion collisions. For simplicity, let us focus on
the $z\ll 1$ case, when the shear viscosity, $\eta$, can be related to the
relaxation time via $\eta=\frac{4}{5}\tau_{R}\mathcal{P}$ [42, 45]. Assuming
that the ratio $\eta/\mathcal{S}$ is constant, where
$\mathcal{S}=(\mathcal{E}+\mathcal{P}-\mu\mathcal{N})/T\simeq 4\mathcal{P}/T$
is the entropy density (we considered also $|\xi|\ll 1$), we have

$\displaystyle\tau_{R}\simeq\frac{5}{4\pi^{2}T}\times(4\pi\eta/\mathcal{S}).$
(44)

Setting now $\mathfrak{s}^{2}=3/4$, the damping time $t_{\rm
damp;\perp}=1/k^{2}\nu_{\perp}$ can be estimated as

$\displaystyle t_{\rm damp;\perp}\simeq$
$\displaystyle\frac{4\lambda^{2}T/3}{4\pi\eta/\mathcal{S}}$ $\displaystyle=$
$\displaystyle\left(\frac{\lambda}{1\ {\rm fm}}\right)^{2}\left(\frac{T}{600\
{\rm MeV}}\right)\times\frac{4\ {\rm fm}/c}{4\pi\eta/\mathcal{S}},$ (45)

where $\lambda=2\pi/k$ is the wavelength. Thus the lifetime of spin waves is
of the same order of magnitude as the lifetime of the QGP fireball.

##  V Conclusions

In this work we have studied the wave spectrum of the theory of spin
hydrodynamics based on the GLW pseudogauge. As an antisymmetric tensor of rank
two, the spin chemical potential $\omega^{\mu\nu}$ has six independent degrees
of freedom, which can be divided into three electric and three magnetic ones.
Our analysis has revealed the transverse nature of the spin wave. In the
limiting case of the ideal fluid, the longitudinal magnetic and electric
components do not propagate, while the transverse ones oscillate, leading to
the linearly or circularly polarized waves known from the theory of
electromagnetism.

The speed of the spin wave, $c_{\rm spin}$, generally depends on the
parameters of the medium (temperature $T$, chemical potential $\mu$) and on
the properties of the particles (particle mass $m$ or the statistics obeyed by
the particles). A generic feature of the modes is that in the
ultrarelativistic limit ($z=m/T\ll 1$), $c_{\rm spin}\simeq 1/2$, a property
that is independent of the statistics. In the case of the ideal MJ gas,
$c_{\rm spin}$ becomes independent of $\xi=\mu/T$. In the case of FD
statistics, we found that the chemical potential enhances $c_{\rm spin}$ and
maintains the ultrarelativistic threshold for small values of $z/\xi=m/\mu$.
At the other end of the spectrum, when $z\gg 1$, we find the leading-order
behavior $c_{\rm spin}\sim 1/\sqrt{2z}$, again independent of the statistics.

Finally, we have studied the effects of dissipation on the spin waves. At the
level of first-order spin hydrodynamics, the transverse components are all
damped via the same coefficient $\nu_{\perp}$. The longitudinal components
$C_{\kappa Z}$ and $C_{\omega Z}$ decay exponentially with different
coefficients, $\nu_{\kappa,||}$ and $\nu_{\omega,||}$. The speed $c_{\rm
spin}$ receives a viscous correction which becomes dominant at large wave
numbers $k$. Above the threshold $\tau_{R}k_{\rm th}\simeq
18/(5z^{2}\mathfrak{s}^{2})$, $c_{\rm spin}$ becomes imaginary and the wave no
longer propagates.

The approach considered in this paper, based on the spin polarization tensor
$\omega^{\alpha\beta}$, does not account for anomalous transport phenomena.
The addition of vortical terms in $N^{\alpha}$ and $T^{\alpha\beta}$ modifies
the wave spectrum corresponding to the fluid sector, giving rise to a rich
spectrum of excitations, such as the chiral magnetic wave, chiral vortical
wave, chiral heat wave, or helical vortical wave [59, 60, 19] . An
investigation of the interplay between anomalous transport effects and the
dynamics of the spin polarization tensor represents an intriguing avenue for
future research.

###### Acknowledgements.

We thank D. H. Rischke, M. Shokri, and P. Aasha for fruitful discussions and
A. Palermo for the critical reading of the manuscript. V.E.A. gratefully
acknowledges the support of the Alexander von Humboldt Foundation through a
Research Fellowship for postdoctoral researchers, as well as the kind
hospitality of the Institute of Nuclear Physics Polish Academy of Sciences,
Krakow where this work was initiated. R.S. acknowledges the support of Polish
NAWA PROM Program No.: PROM PPI/PRO/2019/1/00016/U/001 and the hospitality of
the Institute for Theoretical Physics, Goethe University Germany where this
work was finalized. This research was also supported in part by the Polish
National Science Centre Grants No. 2016/23/B/ST2/00717 and No.
2018/30/E/ST2/00432.

##  Appendix A Spin tensor decomposition for general statistics

We start with the equilibrium phase-space distribution function which is
constructed after the identification of the collisional invariants for MJ
statistics [26, 38]

$f^{\pm}_{\rm eq}=\exp\left[-\beta p\cdot
U\pm\xi\right]\exp\left[\frac{1}{2}\omega_{\mu\nu}s^{\mu\nu}\right],$ (46)

where $s^{\alpha\beta}=\frac{1}{m}\epsilon^{\alpha\beta\mu\nu}p_{\mu}s_{\nu}$
is the internal angular momentum and $s^{\mu}$ is the spin four-vector [61,
13].

The spin tensor (4a) is derived through the moments of (46) as [26]

$\displaystyle S^{\lambda,\mu\nu}$ $\displaystyle=$
$\displaystyle\int\mathrm{dP}~{}\mathrm{dS}\,\,p^{\lambda}\,s^{\mu\nu}\left[f^{+}_{\rm
eq}+f^{-}_{\rm eq}\right]$ (47) $\displaystyle=$ $\displaystyle
2\cosh\xi\int\mathrm{dP}\,p^{\lambda}\exp\left(-\beta p\cdot U\right)$
$\displaystyle\hskip
28.45274pt\times\int\mathrm{dS}\,s^{\mu\nu}\,\exp\left(\frac{1}{2}\omega_{\alpha\beta}s^{\alpha\beta}\right),$

where $\mathrm{dS}=\frac{m}{\pi\mathbf{\mathfrak{s}}}d^{4}s\,\delta(s\cdot
s+\mathbf{\mathfrak{s}}^{2})\delta(p\cdot s)$ is the invariant spin measure
[26]. To the leading order in $\omega_{\alpha\beta}$ the second integral is
expressed as

$\displaystyle\int\mathrm{dS}\,s^{\mu\nu}\,\exp\left(\frac{1}{2}\omega_{\alpha\beta}s^{\alpha\beta}\right)\simeq\int\mathrm{dS}\,s^{\mu\nu}\,\left(1+\frac{1}{2}\omega_{\alpha\beta}s^{\alpha\beta}\right)$
$\displaystyle\hskip
42.67912pt=\frac{2\mathfrak{s}^{2}}{3m^{2}}\left(m^{2}\omega^{\mu\nu}+2p^{\alpha}p^{[\mu}\omega^{\nu]}_{\hphantom{\alpha}\alpha}\right),$
(48)

where the integral in the spin space was performed using the following
relations [26]:

$\displaystyle\int\mathrm{dS}$ $\displaystyle=$ $\displaystyle
2,\qquad\int\mathrm{dS}\,s^{\mu}=0,$
$\displaystyle\int\mathrm{dS}\,s^{\mu}s^{\nu}$ $\displaystyle=$
$\displaystyle\frac{2\mathfrak{s}^{2}}{3m^{2}}(p^{\mu}p^{\nu}-m^{2}g^{\mu\nu}),$
(49)

leading to $\int\mathrm{dS}\,s^{\mu\nu}=0$ and

$\displaystyle\int\mathrm{dS}\,s^{\mu\nu}s^{\alpha\beta}=\frac{4\mathfrak{s}^{2}}{3m^{2}}(m^{2}g^{\mu[\alpha}g^{\beta]\nu}+2p^{[\alpha}g^{\beta][\mu}p^{\nu]}).$
(50)

Using (48) in (47) we have [26]

$\displaystyle S^{\lambda,\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{4{{\mathfrak{s}}}^{2}}{3m^{2}}\cosh\xi\\!\\!\int\\!\\!\mathrm{dP}\,p^{\lambda}\,e^{-p\cdot\beta}\left(m^{2}\omega^{\mu\nu}\\!+\\!2p^{\alpha}p^{[\mu}\omega^{\nu]}_{\hphantom{\alpha}\alpha}\right),~{}~{}~{}~{}~{}~{}~{}$
(51)

which is the spin tensor for the MJ statistics (4a).

Now we extend the distribution function (46) to general statistics, where
$f_{\rm eq}^{\sigma}\equiv f_{\rm eq}^{\sigma}(y_{\sigma})$, and

$\displaystyle y_{\sigma}$ $\displaystyle=$ $\displaystyle y_{\sigma;0}+y_{\rm
spin},\quad y_{\sigma;0}=\beta p\cdot U-\sigma\xi,$ (52) $\displaystyle\qquad
y_{\rm spin}=-\frac{1}{2}\omega^{\mu\nu}s_{\mu\nu}.$

We consider $y_{\rm spin}\ll y_{\sigma;0}$, such that

$\displaystyle f_{\rm eq}^{\sigma}(y_{\sigma})=f_{\rm
eq}^{\sigma}(y_{\sigma;0})+f^{\sigma\prime}_{\rm eq}(y_{\sigma;0})y_{\rm
spin}+\dots,$ (53)

where $\sigma=+1$ and $-1$ for particles and antiparticles, respectively. The
derivative of the distribution function is evaluated at vanishing spin
chemical potential, such that

$\displaystyle f^{\prime\sigma}_{\rm
eq}(y_{\sigma;0})=-\sigma\left(\frac{\partial f^{\sigma}_{\rm
eq}}{\partial\xi}\right)_{\beta}=\frac{1}{p\cdot U}\left(\frac{\partial
f^{\sigma}_{\rm eq}}{\partial\beta}\right)_{\xi}.$ (54)

Therefore, Eq. (47) can be written for the general statistics as

$\displaystyle S^{\lambda,\mu\nu}_{\rm eq}$ $\displaystyle=$
$\displaystyle-\frac{2{{\mathfrak{s}}}^{2}}{3m^{2}}\sum_{\sigma=\pm}\,\,\\!\\!\int\\!\\!\mathrm{dP}\,p^{\lambda}f^{\prime\sigma}_{\rm
eq}\left(m^{2}\omega^{\mu\nu}\\!+\\!2p^{\alpha}p^{[\mu}\omega^{\nu]}_{\hphantom{\alpha}\alpha}\right).$
(55)

The integral of $p^{\lambda}f^{\sigma\prime}_{\rm eq}$ can be written in terms
of the number density via

$\displaystyle
2\sum_{\sigma=\pm}\int\mathrm{dP}\,p^{\lambda}f^{\sigma\prime}_{\rm
eq}=a_{1}U^{\lambda},\quad
a_{1}=-\left(\frac{\partial\mathcal{N}}{\partial\xi}\right)_{\beta},$ (56)

where the factor of $2$ accounts for the spin degeneracy ($\int\mathrm{dS}=2$)
and we used the perfect fluid form $N^{\alpha}=\mathcal{N}U^{\alpha}$ for the
charge current. The integral involving
$p^{\lambda}p^{\mu}p^{\alpha}f^{\prime\sigma}_{\rm eq}$ allows to perform the
tensor decomposition

$2\sum_{\sigma=\pm}\int\mathrm{dP}f^{\sigma\prime}_{\rm
eq}\,p^{\lambda}p^{\alpha}p^{\mu}\\\
=a_{2}\,U^{\lambda}U^{\alpha}U^{\mu}+b_{2}(U^{\lambda}\Delta^{\alpha\mu}+U^{\alpha}\Delta^{\lambda\mu}+U^{\mu}\Delta^{\lambda\alpha}),$
(57)

where the coefficients $a_{2}$ and $b_{2}$ can be obtained by contracting the
above expression with $U_{\lambda}U_{\alpha}U_{\mu}$ and
$U_{\lambda}g_{\alpha\mu}$, respectively:

$\displaystyle
a_{2}=\left(\frac{\partial\mathcal{E}}{\partial\mathcal{\beta}}\right)_{\xi},\qquad
a_{2}+3b_{2}=-m^{2}\left(\frac{\partial\mathcal{N}}{\partial\xi}\right)_{\beta}.$
(58)

Substituting the above results in Eq. (55) and comparing with Eq. (4) shows
that $a_{1}=-\frac{3}{\mathfrak{s}^{2}}(\mathcal{A}_{1}+\mathcal{A}_{3})$,
$a_{2}=\frac{3m^{2}}{2\mathfrak{s}^{2}}(\mathcal{A}_{3}-2\mathcal{A}_{1})$,
and $b_{2}=-\frac{3m^{2}}{2\mathfrak{s}^{2}}\mathcal{A}_{3}$, where the
coefficients $\mathcal{A}_{1}$ and $\mathcal{A}_{3}$ are given in Eq. (5) and
are compatible with Eq. (58).

##  Appendix B Ideal gas

The ideal gas is modeled using the MJ distribution,

$\displaystyle f^{\sigma}_{\rm eq}=e^{-\beta U\cdot p+\sigma\xi},$ (59)

where $\beta$ is the inverse of the temperature and $\xi=\mu\beta$. The charge
current $N^{\mu}$ and energy-momentum tensor $T^{\mu\nu}$ can be obtained as

$\displaystyle N^{\mu}$ $\displaystyle=$ $\displaystyle
2\sum_{\sigma}\sigma\int\mathrm{dP}\,p^{\mu}f^{\sigma}_{\rm eq},$
$\displaystyle T^{\mu\nu}$ $\displaystyle=$ $\displaystyle
2\sum_{\sigma}\int\mathrm{dP}\,p^{\mu}p^{\nu}f^{\sigma}_{\rm eq},$ (60)

where the factor of 2 accounts for the spin degeneracy. The integrals yield
the perfect fluid form in Eq. (2), where ${\cal N}$, ${\cal E}$, and ${\cal
P}$ are given by [62, 33, 62]

$\displaystyle{\cal N}$ $\displaystyle=$ $\displaystyle 4\sinh(\xi){\cal
N}_{(0)},\quad\begin{pmatrix}{\cal E}\\\ {\cal
P}\end{pmatrix}=4\cosh(\xi)\begin{pmatrix}{\cal E}_{(0)}\\\ {\cal
P}_{(0)}\end{pmatrix}.$ (61)

The number density ${\cal N}_{(0)}$, pressure $\mathcal{P}_{(0)}$, and energy
density ${\cal E}_{(0)}$ for the spinless and neutral classical massive
particles read [33, 62]

$\displaystyle{\cal N}_{(0)}$ $\displaystyle=$
$\displaystyle\frac{T^{3}}{2\pi^{2}}z^{2}\,K_{2}\left(z\right),\quad{\cal
P}_{(0)}=T{\cal N}_{(0)},$ $\displaystyle{\cal E}_{(0)}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi^{2}}T^{4}\,z^{2}\,\left[zK_{1}\left(z\right)+3K_{2}\left(z\right)\right].$
(62)

In the above equations $K_{n}(z)$ are the modified Bessel functions of the
second kind [63]

$\displaystyle
K_{n}(z)=\frac{z^{n}}{(2n+1)!!}\int_{1}^{\infty}dx(x^{2}-1)^{n-\frac{1}{2}}e^{-xz}.$
(63)

The derivatives of $\mathcal{E}$ with respect to $\beta$ and of $\mathcal{N}$
with respect to $\xi$ are

$\displaystyle\left(\frac{\partial\mathcal{E}}{\partial\beta}\right)_{\xi}$
$\displaystyle=$
$\displaystyle-\frac{2m^{3}T^{2}}{\pi^{2}}\cosh\xi\,[zK_{2}(z)+3K_{3}(z)],$
$\displaystyle\left(\frac{\partial\mathcal{N}}{\partial\xi}\right)_{\beta}$
$\displaystyle=$ $\displaystyle\frac{2m^{2}T}{\pi^{2}}\cosh\xi\,K_{2}(z).$
(64)

Taking into account that $(\partial\mathcal{E}/\partial
T)_{\xi}=-\frac{1}{T^{2}}(\partial\mathcal{E}/\partial\beta)_{\xi}$, the
functions $\mathcal{A}_{1}$ and $\mathcal{A}_{3}$ introduced in Eq. (5) can be
readily computed:

$\displaystyle\mathcal{A}_{1}$ $\displaystyle=$
$\displaystyle\frac{4\mathfrak{s}^{2}mT^{2}}{3\pi^{2}}\cosh\xi\left[K_{3}(z)+\frac{z}{2}K_{2}(z)\right],$
$\displaystyle\mathcal{A}_{3}$ $\displaystyle=$
$\displaystyle-\frac{4\mathfrak{s}^{2}mT^{2}}{3\pi^{2}}\cosh\xi\,K_{3}(z),$
(65)

in agreement with the results reported in Ref. [41] for the ideal gas case.
Substituting the above results in Eq. (16) gives Eq. (17).

We now discuss the asymptotic properties of the spin velocity $c_{\rm spin}$
in Eq. (17) in the nonrelativistic and ultrarelativistic limits. At large
values of their argument, the modified Bessel functions admit the following
asymptotic expansion [63]:

$\displaystyle K_{\nu}(z)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\pi}{2z}}e^{-z}\sum_{k=0}^{\infty}\frac{a_{k}(\nu)}{z^{k}},$
$\displaystyle a_{k}(\nu)$ $\displaystyle=$
$\displaystyle\frac{\left(\frac{1}{2}-\nu\right)_{k}\left(\frac{1}{2}+\nu\right)_{k}}{(-2)^{k}k!}.$
(66)

In particular,

$\displaystyle K_{2}(z)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\pi}{2z}}e^{-z}\left(1+\frac{15}{8z}+\frac{105}{128z^{2}}+\dots\right),$
$\displaystyle K_{3}(z)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\pi}{2z}}e^{-z}\left(1+\frac{35}{8z}+\frac{945}{128z^{2}}+\dots\right).$
(67)

From here, we obtain

$\displaystyle c_{\rm spin}(z\gg 1)\simeq\frac{1}{\sqrt{2z}}.$ (68)

For small values of their argument, the modified Bessel functions of the
second kind $K_{n}(z)$ of integer order $n$ admit the series representation
[63]

$K_{n}(z)=\frac{1}{2}\left(\frac{z}{2}\right)^{-n}\sum_{k=0}^{n-1}\frac{(n-k-1)!}{k!}\left(-\frac{z^{2}}{4}\right)^{k}\\\
+(-1)^{n+1}\ln\left(\frac{z}{2}\right)I_{n}(z)\\\
+\frac{(-1)^{n}}{2}\left(\frac{z}{2}\right)^{n}\sum_{k=0}^{\infty}[\psi(k+1)+\psi(n+k+1)]\frac{(z^{2}/4)^{k}}{k!(n+k)!},$
(69)

where $\psi(z)=\Gamma^{\prime}(z)/\Gamma(z)$ is the digamma function. The
modified Bessel functions of the first kind $I_{n}(z)$ have the series
representation

$\displaystyle
I_{n}(z)=\left(\frac{z}{2}\right)^{n}\sum_{k=0}^{\infty}\frac{(z^{2}/4)^{k}}{k!(n+k)!}.$
(70)

Thus, the leading order contributions to $K_{2}(z)$ and $K_{3}(z)$ are given
by the terms in the sum appearing on the first line of Eq. (69),

$\displaystyle K_{2}(z)=$ $\displaystyle\frac{2}{z^{2}}-\frac{1}{2}+O(z^{2}),$
$\displaystyle K_{3}(z)=$ $\displaystyle\frac{8}{z^{3}}-\frac{1}{z}+O(z).$
(71)

Substituting the above into Eq. (17) gives

$\displaystyle c_{\rm spin}^{2}(z\ll 1)\simeq$
$\displaystyle\frac{1}{4}\left[1-\frac{z^{2}}{8}+O(z^{4})\right].$ (72)

##  Appendix C Fermi-Dirac gas

The FD distribution is

$\displaystyle f^{\sigma}_{\rm eq}=\frac{1}{e^{\beta p\cdot
U-\sigma\xi}+1}\,.$ (73)

The charge density, energy density, and pressure can be computed as [64]

$\displaystyle\begin{pmatrix}\mathcal{N}\\\ \mathcal{E}\\\
\mathcal{P}\end{pmatrix}$ $\displaystyle=$
$\displaystyle\frac{1}{\pi^{2}}\sum_{\sigma}\int_{m}^{\infty}dE\,p\begin{pmatrix}\sigma
E\\\ E^{2}\\\ \frac{1}{3}p^{2}\end{pmatrix}\frac{1}{e^{\beta E-\sigma\xi}+1}.$
(74)

In the case when $\xi<z=\beta m$, the Fermi-Dirac factor $[e^{\beta
E-\sigma\xi}+1]^{-1}$ can be expanded as

$\displaystyle\frac{1}{e^{\beta
E-\sigma\xi}+1}=\sum_{\ell=1}^{\infty}(-1)^{\ell+1}e^{-\ell\beta
E+\ell\sigma\xi}.$ (75)

This allows $\mathcal{N}$ and $\mathcal{E}$ to be computed as

$\displaystyle\mathcal{N}=$
$\displaystyle\frac{2m^{2}T}{\pi^{2}}\sum_{\ell=1}^{\infty}\frac{(-1)^{\ell+1}}{\ell}\sinh(\ell\xi)K_{2}(\ell
z),$ (76) $\displaystyle\mathcal{E}=$
$\displaystyle\frac{2m^{2}T^{2}}{\pi^{2}}\sum_{\ell=1}^{\infty}\frac{(-1)^{\ell+1}}{\ell^{2}}\cosh(\ell\xi)[\ell
zK_{1}(\ell z)+3K_{2}(\ell z)],$

which has as its $\ell=1$ contribution the result for the MJ statistics given
in Eqs. (61) and (62).

The derivatives of $\mathcal{E}$ and $\mathcal{N}$ with respect to $\beta$ and
$\xi$, respectively, can be calculated as

$\displaystyle\left(\frac{\partial\mathcal{E}}{\partial\beta}\right)_{\xi}$
$\displaystyle=$
$\displaystyle-\frac{1}{\pi^{2}\beta}\sum_{\sigma}\int_{m}^{\infty}dE\,E^{2}\frac{3p+\frac{E^{2}}{p}}{e^{\beta
E-\sigma\xi}+1},$
$\displaystyle\left(\frac{\partial\mathcal{N}}{\partial\xi}\right)_{\beta}$
$\displaystyle=$
$\displaystyle\frac{1}{\pi^{2}\beta}\sum_{\sigma}\int_{m}^{\infty}dE\frac{p+\frac{E^{2}}{p}}{e^{\beta
E-\sigma\xi}+1}.$ (77)

In the case when $|\xi|<z$, the above integrals can be computed using the
method introduced in Eq. (75). This allows the functions $\mathcal{A}_{1}$ and
$\mathcal{A}_{3}$ introduced in Eq. (5) to be expressed as

$\displaystyle\mathcal{A}_{1}$ $\displaystyle=$
$\displaystyle\frac{4\mathfrak{s}^{2}mT^{2}}{3\pi^{2}}\sum_{\ell=1}^{\infty}\frac{(-1)^{\ell+1}}{\ell}\cosh(\ell\xi)\left[K_{3}(\ell
z)+\frac{\ell z}{2}K_{2}(\ell z)\right],$ $\displaystyle\mathcal{A}_{3}$
$\displaystyle=$
$\displaystyle-\frac{4\mathfrak{s}^{2}mT^{2}}{3\pi^{2}}\sum_{\ell=1}^{\infty}\frac{(-1)^{\ell+1}}{\ell}\cosh(\ell\xi)K_{3}(\ell
z),$ (78)

where again the $\ell=1$ term coincides with the expressions obtained in Eq.
(65) for the MJ statistics. The above result is useful to derive the
nonrelativistic and ultrarelativistic limits of $c_{\rm spin}$. In the latter
case, the modified Bessel functions can be approximated by their large $z$
expansion, given in Eq. (67). In this case, the terms with $\ell>1$ are
penalized by the exponential function, $K_{n}(\ell z)\sim e^{-\ell
z}/\sqrt{\ell z}$, such that the $\ell=1$ term already provides a good
approximation. For this reason, the value of $c_{\rm spin}$ corresponding to
FD particles converges to the MJ one, given in Eq. (68).

Figure 3: Comparison between the numerical results for $c_{\rm spin}$
corresponding to FD statistics and the limits corresponding to the degenerate
Fermi gas derived in Eq. (82) (thick gray lines, only when $|\xi|=10$, $20$,
and $50$) and to the small $z$ limit (19) (dotted black lines).

In the relativistic limit, $z$ can be assumed to be small and the modified
Bessel functions can be replaced by their asymptotic expansions in Eq. (71).
Denoting
$S_{n}=\sum_{\ell=1}^{\infty}\frac{(-1)^{\ell+1}}{\ell^{n}}\cosh(\ell\xi)$,the
functions $\mathcal{A}_{1}$ and $\mathcal{A}_{3}$ converge to

$\displaystyle\mathcal{A}_{1}$ $\displaystyle=$
$\displaystyle\frac{32\mathfrak{s}^{2}T^{4}}{3\pi^{2}m^{2}}[S_{4}+O(z^{4})],$
$\displaystyle\mathcal{A}_{3}$ $\displaystyle=$
$\displaystyle-\frac{32\mathfrak{s}^{2}T^{4}}{3\pi^{2}m^{2}}\left[S_{4}-\frac{z^{2}}{8}S_{2}+O(z^{4})\right].$
(79)

Taking into account that
$S_{4}=\frac{1}{720}(7\pi^{4}+30\pi^{2}\xi^{2}+15\xi^{4})$and
$S_{2}=\frac{1}{12}(\pi^{2}+3\xi^{2})$, we arrive at

$\displaystyle c_{\rm spin}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left[1-\frac{z^{2}}{8}\frac{S_{2}}{S_{4}}+O(z^{4})\right]$
(80) $\displaystyle=$
$\displaystyle\frac{1}{4}\left[1-\frac{15z^{2}}{2\pi^{2}}\frac{1+\frac{3\xi^{2}}{\pi^{2}}}{7+30\frac{\xi^{2}}{\pi^{2}}+15\frac{\xi^{4}}{\pi^{4}}}+O(z^{4})\right].$

Before ending this section, we discuss another interesting limit relevant for
the FD statistics. In the degenerate case ($T\rightarrow 0$ and $\mu>m$), we
have

$\displaystyle\mathcal{N}=\frac{p_{F}^{3}}{3\pi^{2}},\,\,\,\,\mathcal{E}=\frac{1}{8\pi^{2}}\left[p_{F}\,\mu(p_{F}^{2}+\mu^{2})+m^{4}\ln\frac{m}{p_{F}+\mu}\right],$

where $p_{F}=\sqrt{\mu^{2}-m^{2}}$. Since in the degenerate limit,
$\mathcal{E}\equiv\mathcal{E}(\mu)$, we have
$\left(\partial\mathcal{E}/\partial\beta\right)_{\xi}=-\frac{\mu}{\beta}(\partial\mathcal{E}/\partial\mu)$,
such that

$\displaystyle\left(\frac{\partial\mathcal{E}}{\partial\beta}\right)_{\xi}=-\frac{\mu^{3}\sqrt{\mu^{2}-m^{2}}}{\pi^{2}\beta},\,\,\,\,\left(\frac{\partial\mathcal{N}}{\partial\xi}\right)_{\beta}=\frac{\mu\sqrt{\mu^{2}-m^{2}}}{\pi^{2}\beta}.~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(81)

This leads to the approximate formula

$\displaystyle c_{\rm
spin}^{2}=\frac{1}{4}\frac{\xi^{2}-z^{2}}{\xi^{2}+z^{2}/2},$ (82)

which is valid when $\xi\gg z$, see Fig. 3 for the comparison between $c_{\rm
spin}$ and the degenerate limit for the FD gas. We can attempt to link $c_{\rm
spin}$ to the sound speed for a degenerate gas. Taking into account the
expression for the pressure,

$\displaystyle\mathcal{P}=\frac{1}{24\pi^{2}}\left[p_{F}\mu(5p_{F}^{2}-3\mu^{2})-3m^{4}\ln\frac{m}{p_{F}+\mu}\right],$
(83)

it can be shown that

$\displaystyle
c_{s}^{2}=\frac{d\mathcal{P}}{d\mathcal{E}}=\frac{1}{3}-\frac{z^{2}}{3\xi^{2}}\,.$
(84)

Comparing the above expression with Eq. (82), the following relation can be
established between the spin wave velocity and the sound velocity in the
degenerate limit for the FD gas

$\displaystyle c_{\rm spin}^{2}=\frac{c_{s}^{2}/2}{1-c_{s}^{2}}.$ (85)

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