aainstitutetext: Department of Physics, University of Illinois, Chicago, IL
60607, USAbbinstitutetext: RIKEN iTHEMS, RIKEN, Wako 351-0198,
Japanccinstitutetext: Physics Department and Center for Field Theory and
Particle Physics, Fudan University, Shanghai 200433, Chinaddinstitutetext: Key
Laboratory of Nuclear Physics and Ion-beam Application (MOE), Fudan
University, Shanghai 200433, Chinaeeinstitutetext: Department of Physics and
Astronomy, University of Alabama, Tuscaloosa, AL 35487, USA

#  Relativistic spin hydrodynamics with torsion
and linear response theory for spin relaxation

Masaru Hongo c,d Xu-Guang Huang e Matthias Kaminski a Mikhail Stephanov a Ho-
Ung Yee<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Using the second law of local thermodynamics and the first-order Palatini
formalism, we formulate relativistic spin hydrodynamics for quantum field
theories with Dirac fermions, such as QED and QCD, in a torsionful curved
background. We work in a regime where spin density, which is assumed to relax
much slower than other non-hydrodynamic modes, is treated as an independent
degree of freedom in an extended hydrodynamic description. Spin hydrodynamics
in our approach contains only three non-hydrodynamic modes corresponding to a
spin vector, whose relaxation time is controlled by a new transport
coefficient: the rotational viscosity. We study linear response theory and
observe an interesting mode mixing phenomenon between the transverse shear and
the spin density modes. We propose several field-theoretical ways to compute
the spin relaxation time and the rotational viscosity, via the Green-Kubo
formula based on retarded correlation functions.

##  1 Introduction

Spin in the non-relativistic limit, where the spin-orbit coupling is
suppressed, becomes an approximately conserved quantity (good quantum number).
This is the case for the heavy quarks in QCD (e.g. charm quark); the resulting
heavy-quark symmetry shows up as the degeneracy in the spectrum of particular
hadrons Isgur:1989vq ; Isgur:1990yhj ; Isgur:1991wq ; Neubert:1993mb ;
Manohar-Wise2000 . Another familiar realization is a certain class of
condensed matter systems (e.g., magnetic materials), which has motivated
recent development of the hydrodynamic theory describing transport phenomena
of spin, leading to a fruitful research area known as spintronics (see Ref.
maekawa2017spin  for a review). On the other hand, spin in relativistic
systems is not a conserved quantity even approximately – It is only a part of
the total angular momentum, and is not conserved due to the spin-orbit
coupling inherent in relativistic dynamics, whose corresponding transport
theory has not been formulated until recently.

However, the recent experimental observation of spin polarization of hadrons
in relativistic heavy-ion collisions STAR:2017ckg ; STAR:2019erd ;
ALICE:2019aid ; STAR:2020xbm  strongly motivates the development of the theory
describing spin transport in relativistic plasma, in particular, the quark-
gluon plasma (QGP). This motivation has led to several theoretical studies of
relativistic hydrodynamics with spin polarization, based on the second law of
thermodynamics Hattori:2019lfp ; Fukushima:2020ucl ; Li:2020eon ; She:2021lhe
; Gallegos:2021bzp , equilibrium partition functions Gallegos:2021bzp ,
quantum kinetic theory of relativistic fermions Florkowski:2017ruc ;
Gao:2019znl ; Hattori:2019ahi ; Li:2019qkf ; Yang:2020hri ;
Weickgenannt:2020aaf ; Liu:2020flb ; Bhadury:2020puc ; Shi:2020htn ;
Peng:2021ago , holographic approach for strongly-coupled plasma
Hashimoto:2013bna ; Garbiso:2020puw ; Gallegos:2020otk , effective Lagrangian
approach Montenegro:2017rbu ; Montenegro:2017lvf ; Montenegro:2018bcf , and
quantum statistical density operators Becattini:2007nd ; Becattini:2009wh ;
Becattini:2012pp ; Becattini:2018duy ; Hu:2021lnx  (see also Refs.
Florkowski:2018fap ; Becattini:2020sww ; Speranza:2020ilk  and references
therein for a review). These works have shed light on different aspects of
relativistic hydrodynamics including spin degrees of freedom, often referred
to as (relativistic) spin hydrodynamics.

Spin angular momentum in the hydrodynamic regime of long time and distance
scales is in general not a conserved quantity, due to exchanges with orbital
angular momentum via spin-orbit coupling. However, the total angular momentum
is conserved. As a result, an equilibrium state is characterized by a
corresponding thermodynamic parameter Landau:Statistical  – the thermodynamic
conjugate to the total angular momentum, which we refer to as the angular
momentum “chemical” potential. In the absence of torsion, this chemical
potential is equal to the thermal fluid vorticity Becattini:2007nd ;
Becattini:2009wh .111In a globally rotating equilibrium, the temperature in
the local rest frame of a fluid is not a constant in space, and the thermal
vorticity is indeed different from the usual fluid vorticity. In a globally
rotating (torsion-free) equilibrium, the angular momentum chemical potential,
i.e. the thermal vorticity, is constant in space. Since there is only one
angular momentum chemical potential for the conserved total angular momentum,
both spin and orbital angular momenta in equilibrium must be determined
uniquely by that angular momentum chemical potential. That is, the spin
polarization in equilibrium is fixed by the thermal vorticity, and their
relation is one of the equilibrium thermodynamic properties.

On the other hand, since the system has a finite microscopic correlation
length, the state of a local fluid element and its time evolution can only
experience the local environment, and hence should be determined by the local
thermal vorticity as well as other local thermodynamic parameters such as
temperature and chemical potentials of conserved charges. This means that the
fluid element in off-equilibrium relaxes to “local” equilibrium, with the
thermal vorticity as one of the parameters characterizing local thermodynamic
equilibrium.222 We note that the term “local equilibrium” has been used in
literature in different contexts. Here, we use it to specify the local state
where the strict hydrodynamic description can apply. In Refs. Becattini:2012pp
; Becattini:2018duy ; Speranza:2020ilk , however, the same term was used to
refer to the state in which local entropy density is maximized with fixed
local densities of conserved quantities, where the angular momentum “chemical”
potential is not necessarily equal to the local thermal vorticity. In other
words, spin polarization in the strict regime of hydrodynamics based on local
equilibrium is not an independent variable, but is slaved to the conventional
hydrodynamic variable of fluid velocity. Any deviation of spin polarization
from its local equilibrium value is a non-hydrodynamic mode, which relaxes to
zero with a finite relaxation rate that is determined by microscopic dynamics
of the theory Li:2019qkf ; Kapusta:2019sad ; Yang:2020hri ;
Weickgenannt:2020aaf ; Weickgenannt:2021cuo , similar to any other non-
hydrodynamic modes. In a theory where the relaxation rate of spin polarization
is much slower than other non-hydrodynamic modes, one can have two distinct
hydrodynamic descriptions of spin polarization, depending on the time scale of
interest compared to the relaxation time of spin polarization.

When the relevant time scale is much longer than the spin relaxation time, the
spin is in local equilibrium with the thermal vorticity, and is not an
independent degree of freedom. The system is described in a strict sense by
hydrodynamics Landau:Fluid . The true novelty of spin hydrodynamics in this
regime is two-fold Li:2020eon : 1) the spin, or equivalently the fluid
vorticity, affects the local thermodynamic laws used in hydrodynamics, as a
second order gradient correction to the first law of thermodynamics, 2) the
energy-momentum tensor has an anti-symmetric part which is proportional to the
rate of change of the spin tensor, i.e., the fluid vorticity. One can
formulate spin hydrodynamics in this regime based on the gradient expansion of
thermal vorticity. In leading order it gives the ideal limit of spin
hydrodynamics with no production of entropy Li:2020eon , which is in accord
with the existence of globally rotating equilibrium with a constant, but non-
zero thermal vorticity. Interestingly, this ideal spin hydrodynamics has been
shown to be equivalent, by a pseudo-gauge transformation Becattini:2012pp ;
Becattini:2018duy ; Florkowski:2018fap ; Speranza:2020ilk , to conventional
hydrodynamics based on the symmetric energy-momentum tensor with no spin
degrees of freedom, with certain non-dissipative second order transport
coefficients involving fluid vorticity Li:2020eon . The equivalence provides
an important conceptual bridge between the two different formulations in the
strict regime of hydrodynamics.

On the other hand, in the regime where the time scale of interest is
comparable to or shorter than the spin relaxation time (but is still longer
than relaxation time of other non-hydrodynamic modes), one can include spin
polarization as an additional independent dynamical mode Hattori:2019lfp  in
an extended hydrodynamic framework generally called Hydro+Stephanov:2017ghc .
The essential feature of this extended framework is that spin polarization as
a non-hydrodynamic mode relaxes to its local equilibrium value, due to spin-
orbit coupling inherent in the microscopic relativistic theory Hattori:2019lfp
; Weickgenannt:2020aaf . A new transport coefficient appears in this regime,
dictated by the second law of thermodynamics Hattori:2019lfp : the rotational
viscosity, which determines the characteristic time scale for spin relaxation.
In the present paper, we focus on this regime of spin hydrodynamics.

We address several important theoretical issues, founded upon the microscopic
definition of the spin tensor in a background with torsion. First, the
definition of spin current has been unclear in some of the previous works.
Although the canonical spin current of Dirac fermions based on quantum field
theory is totally anti-symmetric with respect to its three Lorentz indices,
that was often not assumed in previous works.333One can perform a pseudo-gauge
transformation to bring the spin current into a desired form, which however
changes the definition of the spin current. Therefore, it is desirable to work
with a microscopic definition of the spin-current operator from the underlying
theory, possessing the appropriate anti-symmetric property of the spin-
current. Second, the regime of applicability of relativistic spin
hydrodynamics is rather subtle. A naive application of derivative expansion is
problematic, and the proper expansion scheme needs to be clarified. Last but
not least is the issue of how rapidly the spin density in a relativistic
plasma relaxes to its equilibrium – one of the most crucial information
relevant to the ongoing heavy-ion experiments. To answer this question, we
need theoretical methods to evaluate the spin relaxation rate in the
microscopic theory. One approach is the quantum kinetic theory with collisions
Li:2019qkf ; Kapusta:2019sad ; Yang:2020hri ; Weickgenannt:2020aaf ;
Weickgenannt:2021cuo . In this work, we follow an alternative path that does
not rely on weak-coupling approximation and leads to the Green-Kubo formula
based on retarded correlation functions of spin observables, which is similar
to those for other transport coefficients Green ; Kubo ; Nakano , but also
with important differences due to non-conservation of spin density.444 Ref.
Hu:2021lnx  has recently derived a Green-Kubo formula based on the statistical
operator approach, which has some overlap with our results.  Our technique
involves torsion which makes spin connection an independent source for the
spin current and facilitates derivation of Green-Kubo formulae.

Our approach is based on the combination of entropy-current analysis and
linear response theory. Our starting point is the Ward-Takahashi identity for
the local Lorentz symmetry in the underlying quantum field theory of Dirac
fermions, such as QED and QCD. The first-order formalism in the background
geometry with torsion naturally leads to a totally anti-symmetric spin
current, which is not conserved due to spin-orbit coupling. As a result of the
total anti-symmetric nature of the spin current, our relativistic spin
hydrodynamics contains three non-hydrodynamic modes, corresponding to a spin
vector that is the generator of spatial rotations.

We then perform the entropy-current analysis in a general background with
torsion, which introduces a new transport coefficient, the rotational
viscosity $\eta_{s}$. The constitutive relations we find in a general
torsionful background allow us to derive the Green-Kubo formula for the
rotational viscosity, as well as other possible ways to compute it from the
underlying quantum field theory.

Furthermore, the constitutive relations tell us that the local equilibrium
value of the angular momentum chemical potential receives additional
contributions coming from the background torsion. We study the linear response
theory of our extended hydrodynamics with spin modes, and obtain the
dispersion relations of dynamical modes. The relaxation rate of the spin modes
is shown to be given by $\Gamma_{s}=2\eta_{s}/\chi_{s}$, where $\chi_{s}$ is
the spin susceptibility (see (56) for the definition).

We consider the frequency scale $\omega\gtrsim\Gamma_{s}$, where these spin
modes are treated as independent dynamical variables in our relativistic spin
hydrodynamics. We also assume that $\omega\ll\Gamma$, where $\Gamma$ is the
relaxation rate of other non-hydrodynamic modes. This window of scale, where
our extended hydrodynamics with additional spin modes is justified, is
possible only when $\Gamma_{s}$ is parametrically smaller than $\Gamma$ (see
Figure 1). For example, in a weakly coupled quark-gluon plasma, when the mass
$M$ of fermions is much larger than the temperature $T$, $\Gamma_{s}$ is
expected to be smaller than $\Gamma$ by additional powers of $T/M$.

One of the interesting features we find in our linear response theory is the
mixing between the transverse shear mode and the spin mode, that can be seen
in Figure 1. This mode mixing arises because the transverse shear gradient
$\partial_{z}u_{x}$ is a linear combination of shear tensor and fluid
vorticity, the latter of which couples to spin modes.

Figure 1:  A schematic picture of frequency scales as well as sound, shear,
and spin density modes. The spin hydrodynamics discussed in this work extends
the regime of validity of hydrodynamics to $\omega\gtrsim\Gamma_{s}$, where
$\Gamma_{s}$ is the relaxation rate for spin density. For
$\omega\ll\Gamma_{s}$, the spin is no longer an independent degree of freedom,
and though spin hydrodynamics still applies, we also have pure hydrodynamics
as an alternative and simpler description. The green and red lines show a mode
mixing phenomenon between transverse shear and spin modes at the scale
$\omega\gtrsim\Gamma_{s}$.

The paper is organized as follows: In section 2, we review the definition of
energy-momentum tensor and spin current in quantum field theory of Dirac
fermions, together with the associated Ward-Takahashi identity of local
Lorentz symmetry. In section 3, we derive the constitutive relations of
relativistic spin hydrodynamics in a torsionful curved background, where spin
density is treated as an independent dynamical variable. In section 4, we
develop the linear response theory of our spin hydrodynamics, which gives the
dispersion relation of dynamical modes, as well as the derivation of the
Green-Kubo formula for the spin relaxation rate. Section 5 is devoted to
summary and discussion. In Appendix A, the relation between the currents in
the vierbein-spin connection formalism and the metric-affine connection
formalism is presented.

##  2 Spin current in the first-order formalism

In this section, we review the definition of the energy-momentum tensor and
the spin current from the viewpoint of the first-order (or Palatini) formalism
for background spacetime Palatini1919 . We consider quantum field theory (QFT)
in a torsionful (Einstein-Cartan) background geometry Hehl:1976kj  and
introduce currents and the Ward-Takahashi identities Ward:1950xp ;
Takahashi:1957xn  associated with diffeomorphism, local Lorentz invariance and
flavor symmetry.

###  2.1 QFT in a torsionful background

When the system enjoys a continuous global symmetry, it brings about a
conserved current thanks to Noether’s theorem Noether1971 . Nevertheless, the
naive definition of the Noether current often contains conceptual problems, an
example of which is the gauge non-invariance of the canonical energy-momentum
tensor for gauge theories. Although these problems may be resolved in an ad
hoc way by using, e.g., the Belinfante improvement Belinfante1939 ;
Belinfante1940 ; Rosenfeld1940 , it is not clear whether the improved one is
the best definition of current operators (see, e.g., Refs. Leader:2013jra ;
Wakamatsu:2014zza  and references therein for reviews on the definition of
spin currents in gauge theories).

In order to avoid potential problems in the Noether current, one can instead
utilize the definition of the gauge current in the following systematic way:
we introduce the background gauge field that couples to the global symmetry of
the system, and by taking the variation of the gauged action with respect to
the background gauge field, one immediately gets a well-behaved current
operator. A famous example is the symmetric and gauge-invariant energy-
momentum tensor in general relativity, which is canonically defined by the
variation of the matter action with respect to the metric.

In the same manner, we can define the spin current operator, although it is
not a conserved current due to its possible exchanges with orbital angular
momentum. The key concept here is the spin connection, which enables us to
promote the Lorentz invariance to a local one. Let us consider QFTs in a
background of torsionful curved spacetime, whose action reads as

$\mathcal{S}\big{[}\varphi;j\big{]}=\int\mathrm{d}^{4}xe\mathcal{L}(\varphi,\partial_{\mu}\varphi;j),$
(1)

where we introduce a set of dynamical fields $\varphi$ and background fields
$j$, respectively. As an explicit example, we will consider QCD in this paper,
but the whole discussion also applies to QED or any other theories including
Dirac fermions. We consider the following QCD Lagrangian density

$\mathcal{L}_{\mathrm{QCD}}\equiv-\frac{1}{2}{\bar{q}}\Big{(}\gamma^{\hat{a}}e_{\hat{a}}^{~{}\mu}\overrightarrow{D}_{\mu}-\overleftarrow{D}_{\mu}e_{\hat{a}}^{~{}\mu}\gamma^{\hat{a}}\Big{)}q-{\bar{q}}Mq-\frac{1}{2}\mathop{\mathrm{tr}}\left(g^{\mu\nu}g^{\alpha\beta}G_{\mu\alpha}G_{\nu\beta}\right),$
(2)

where $q=(u,d,\cdots)^{t}$ denotes the quark fields with
${\bar{q}}\equiv\mathrm{i}q^{\dagger}\gamma^{0}$,
$M=\operatorname{diag}(m_{u},m_{d},\cdots)$ is their mass matrix, and
$G_{\mu\nu}\equiv\partial_{\mu}a_{\nu}-\partial_{\nu}a_{\mu}-\mathrm{i}g[a_{\mu},a_{\nu}]$
is the field strength tensor for the gluon field $a_{\mu}\equiv
a_{\mu}^{a}t_{a}$, with $\mathrm{su}(3)_{\mathrm{c}}$ generators $t_{a}$, and
the QCD coupling constant $g$. The dynamical fields are quarks and gluons:
$\varphi=\\{q,a_{\mu}\\}$, and corresponding to global symmetries of QCD, a
set of background fields is given by
$j=\\{e_{\mu}^{~{}\hat{a}},\omega_{\mu}^{~{}\hat{a}\hat{b}},A_{\mu}\\}$, where
$e_{\mu}^{~{}\hat{a}}~{}(e_{\hat{a}}^{~{}\mu})$ and
$\omega_{\mu}^{~{}\hat{a}\hat{b}}=-\omega_{\mu}^{~{}\hat{b}\hat{a}}$ denote
the (inverse) vierbein and spin connection with $e\equiv\det
e_{\mu}^{~{}\hat{a}}$, respectively, and $A_{\mu}$ is a flavour gauge field,
coupled to, e.g., the $\text{U}(1)$ baryon symmetry (generalization to non-
Abelian flavour symmetry is straightforward). We also introduced the covariant
derivative of the quark field as

$\begin{split}\overrightarrow{D}_{\mu}q&=\partial_{\mu}q-\mathrm{i}ga_{\mu}q-\frac{\mathrm{i}}{2}\omega_{\mu}^{~{}\hat{a}\hat{b}}\Sigma_{\hat{a}\hat{b}}q-\frac{\mathrm{i}}{3}A_{\mu}q,\\\
\bar{q}\overleftarrow{D}_{\mu}&=\partial_{\mu}\bar{q}+\mathrm{i}g\bar{q}a_{\mu}+\frac{\mathrm{i}}{2}\omega_{\mu}^{~{}\hat{a}\hat{b}}\bar{q}\Sigma_{\hat{a}\hat{b}}+\frac{\mathrm{i}}{3}A_{\mu}q,\end{split}$
(3)

where $\gamma^{\hat{a}}$ are the Dirac gamma matrices, and
$\Sigma_{\hat{a}\hat{b}}\equiv\mathrm{i}[\gamma_{\hat{a}},\gamma_{\hat{b}}]/4$
are the generators of the Lorentz group in the spinor representation. In this
paper, we use Greek (hatted Latin) letters for coordinate (local Lorentz)
indices. The Minkowski metric convention is chosen to be the mostly plus one:
$\eta_{\hat{a}\hat{b}}=\operatorname{diag}(-1,+1,+1,+1)$.

It is worth emphasizing that the fundamental variables describing the
background geometry are the vierbein and the spin connection (or contorsion
which will be introduced shortly). We consider a torsionful background, where
the vierbein and the spin connection are independent background fields. One
can then define the space-time metric $g_{\mu\nu}$ and the affine (curved
space-time) connection $\Gamma^{\rho}_{~{}\mu\nu}$ through

$g_{\mu\nu}=e_{\mu}^{~{}\hat{a}}e_{\nu}^{~{}\hat{b}}\eta_{\hat{a}\hat{b}}\quad\mathrm{and}\quad
D_{\mu}e_{\nu}^{~{}\hat{a}}=\partial_{\mu}e_{\nu}^{~{}\hat{a}}-\Gamma^{\rho}_{~{}\mu\nu}e_{\rho}^{~{}\hat{a}}+\omega_{\mu~{}\hat{b}}^{~{}\hat{a}}e_{\nu}^{~{}\hat{b}}=0,$
(4)

where the second equation, or the so-called tetrad postulate, defines the
affine connection from the given vierbein and spin connection. Throughout the
paper, we use $\nabla_{\mu}$ as the covariant derivative including only the
affine connection, and $D_{\mu}$ as the one including both the affine and spin
connections. The anti-symmetric part of the tetrad postulate gives the first
Cartan equation, which defines the torsion
$T^{\rho}_{~{}\mu\nu}\equiv\Gamma^{\rho}_{~{}\mu\nu}-\Gamma^{\rho}_{~{}\nu\mu}$
as follows:

$T^{\hat{a}}=\mathrm{d}e^{\hat{a}}+\omega^{\hat{a}}_{~{}\hat{b}}e^{\hat{b}}~{}\Leftrightarrow~{}T^{\hat{a}}_{~{}\mu\nu}=\partial_{\mu}e_{\nu}^{~{}\hat{a}}-\partial_{\nu}e_{\mu}^{~{}\hat{a}}+\omega_{\mu~{}\hat{b}}^{~{}\hat{a}}e_{\nu}^{~{}\hat{b}}-\omega_{\nu~{}\hat{b}}^{~{}\hat{a}}e_{\mu}^{~{}\hat{b}}.$
(5)

The introduction of the torsion is reasonable because it promotes the spin
connection to an independent background field. This becomes manifest by
solving the tetrad postulate for the spin connection as

$\begin{split}\omega_{\mu}^{~{}\hat{a}\hat{b}}&={\mathring{\omega}}_{\mu}^{~{}\hat{a}\hat{b}}(e)+K_{\mu}^{~{}\hat{a}\hat{b}},\end{split}$
(6)

where we introduced the torsion-free spin connection
${\mathring{\omega}}_{\mu}^{~{}\hat{a}\hat{b}}(e)$ and the contorsion
$K_{\mu}^{~{}\hat{a}\hat{b}}$ as follows Hehl:1976kj :

$\displaystyle{\mathring{\omega}}_{\mu}^{~{}\hat{a}\hat{b}}$
$\displaystyle\equiv\frac{1}{2}e^{\hat{a}\nu}e^{\hat{b}\rho}(-C_{\mu\nu\rho}+C_{\nu\rho\mu}-C_{\rho\nu\mu}),$
(7) $\displaystyle K_{\mu}^{~{}\hat{a}\hat{b}}$
$\displaystyle\equiv\frac{1}{2}e^{\hat{a}\nu}e^{\hat{b}\rho}(T_{\mu\nu\rho}-T_{\nu\rho\mu}+T_{\rho\nu\mu}),$
(8)

where $C_{\mu\nu\rho}$ is the Ricci rotation coefficient given by

$C_{\mu\nu\rho}\equiv
e_{\mu}^{~{}\hat{c}}(\partial_{\nu}e_{\rho\hat{c}}-\partial_{\rho}e_{\nu\hat{c}}).$
(9)

Note that the torsion-free part
${\mathring{\omega}}_{\mu}^{~{}\hat{a}\hat{b}}(e)$ is fixed by the vierbein.
Thus, introducing the spin connection as an independent field is equivalent to
putting the system into a background with torsion. One observes that the quark
couples to the background torsion through the spin connection.

It is worthwhile to emphasize that the gluon (or photon in the case of QED) is
assumed to have no coupling to the torsion, in contrast to the quark field.
This is a consequence of imposing ${\rm SU}(3)_{\mathrm{c}}$ gauge invariance
of the action Hehl:1976kj . Note that we define the gluon field strength
tensor with a simple partial derivative, not with the covariant derivative. At
first glance, this choice does not seem compatible with diffeomorphism
invariance, but it indeed is. To see this, notice that both covariant
derivatives with or without torsion are consistent with diffeomorphism
invariance, since their difference is proportional to the torsion tensor.
However, one sees that the use of the covariant derivative with torsion for
the field strength tensor spoils ${\rm SU}(3)_{\mathrm{c}}$ gauge invariance
due to the last term appearing in

$\nabla_{\mu}a_{\nu}-\nabla_{\nu}a_{\mu}-\mathrm{i}g[a_{\mu},a_{\nu}]=\partial_{\mu}a_{\nu}-\partial_{\nu}a_{\mu}-\mathrm{i}g[a_{\mu},a_{\nu}]-T^{\rho}_{~{}\mu\nu}a_{\rho}.$
(10)

It should also be pointed out that we have the symmetrized derivative in the
fermion kinetic term in Eq. (2). This symmetrized form ensures that the
Lagrangian is real-valued, and as a consequence, the resulting spin current is
a Hermitian operator as it should be. These choices are crucial for the proper
definition of the spin current, or equivalently, for our decomposition of the
total angular momentum into spin and orbital angular momenta.

###  2.2 The Ward-Takahashi identities

The systems enjoying the Poincaré invariance and some flavour symmetries are
equipped with conserved charges, given by the energy, momentum, total angular
momentum, and flavour charges. Since we would like to discuss the dynamics of
spin polarization, we need to identify the proper definition of the spin
current, in order to decompose the total angular momentum into the orbital and
the spin parts. The decomposition of the angular momentum tensor in quantum
gauge theories is still a controversial subject (see, e.g., Refs.
Leader:2013jra ; Wakamatsu:2014zza  and references therein for a review).555
In our view, any decomposition should be acceptable as long as it respects
${\rm SU}(3)_{\mathrm{c}}$ gauge invariance. The real issue is what component
we measure in a given experiment.  Based on the background vierbein and spin
connection, we here introduce one way to define a gauge-invariant spin
current, which eventually is shown to involve only the fermion sector.

First of all, note that the introduction of background vierbein and spin
connection promotes the Poincaré symmetry to local symmetries - diffeomorphism
and local Lorentz invariance. As a consequence of these local symmetries, one
can derive in the usual manner the covariant (non-)conservation laws as the
Ward-Takahashi identities. The spatial components of the associated spin
current, i.e. $\Sigma^{ijk}$, turn out to be completely determined by the
axial charge density $J_{5}^{0}$. Also, it gives us a useful prescription to
define the spin current through the variation of the QCD action (1)-(2) with
respect to the spin connection. One can regard this procedure as a familiar
way to define the gauge current. We thus define the “canonical” energy-
momentum tensor $\Theta^{\mu}_{~{}\hat{a}}$ and the spin current
$\Sigma^{\mu}_{~{}\hat{a}\hat{b}}$, as well as the flavour current $J^{\mu}$,
by taking the variation with respect to $e_{\mu}^{~{}\hat{a}}$ and
$\omega_{\mu}^{~{}\hat{a}\hat{b}}$, as well as $A_{\mu}$, respectively:

$\Theta^{\mu}_{~{}\hat{a}}(x)\equiv\frac{1}{e(x)}\left.\frac{\delta\mathcal{S}_{\mathrm{QCD}}}{\delta
e_{\mu}^{~{}\hat{a}}(x)}\right|_{\omega,\,A},\quad\Sigma^{\mu}_{~{}\hat{a}\hat{b}}(x)\equiv-\frac{2}{e(x)}\left.\frac{\delta\mathcal{S}_{\mathrm{QCD}}}{\delta\omega_{\mu}^{~{}\hat{a}\hat{b}}(x)}\right|_{e,\,A},\quad
J^{\mu}(x)\equiv\frac{1}{e(x)}\left.\frac{\delta\mathcal{S}_{\mathrm{QCD}}}{\delta
A_{\mu}(x)}\right|_{e,\,\omega},$ (11)

where subscripts $\omega$ and $e$ indicate that they are fixed when we compute
the functional derivative.

After a straightforward calculation, we find that the QCD action (1) gives us
the following expression for our currents

$\begin{split}\Theta^{\mu}_{~{}\hat{a}}&=\frac{1}{2}\bar{q}\big{(}\gamma^{\mu}\overrightarrow{D}_{\hat{a}}-\overleftarrow{D}_{\hat{a}}\gamma^{\mu}\big{)}q+2\mathop{\mathrm{tr}}\left(G^{\mu\rho}G_{\hat{a}\rho}\right)+\mathcal{L}_{\mathrm{QCD}}e^{~{}\mu}_{\hat{a}},\\\
\Sigma^{\mu}_{~{}\hat{a}\hat{b}}&=-\frac{\mathrm{i}}{2}{\bar{q}}e^{\mu}_{~{}\hat{c}}\\{\gamma^{\hat{c}},\Sigma_{\hat{a}\hat{b}}\\}q,\\\
J^{\mu}&=\frac{\mathrm{i}}{3}{\bar{q}}e^{\mu}_{~{}\hat{a}}\gamma^{\hat{a}}q,\end{split}$
(12)

where $\\{A,B\\}\equiv AB+BA$ is the anti-commutator. One finds that only the
local (or on-site) quark field contributes to the spin current, so that our
spin current is manifestly ${\rm SU}(3)_{\mathrm{c}}$ gauge-invariant.
Moreover, thanks to the anti-commutator, $\Sigma^{\mu}_{~{}\hat{a}\hat{b}}$
are Hermitian gauge-invariant operators, as all physical observables should
be. In other words, the spin current would not be a Hermitian operator, if the
fermion kinetic term in the Lagrangian was not symmetrized. The absence of
gluon contributions to the spin current results is resulted from the absence
of the torsion in the gluon field strength tensor due to gauge invariance. In
that sense, our requirement of ${\rm SU}(3)_{\mathrm{c}}$ gauge invariance in
torsionful geometry fixes our choice of the decomposition of the angular
momentum.

There is another useful expression for the spin current, thanks to a gamma
matrix identity
$\\{\gamma^{\mu},\Sigma^{\nu\rho}\\}=\varepsilon^{\mu\nu\rho\sigma}\gamma_{\sigma}\gamma_{5}$
with
$\gamma_{5}\equiv-\mathrm{i}\gamma^{\hat{0}}\gamma^{\hat{1}}\gamma^{\hat{2}}\gamma^{\hat{3}}$,
where the normalized totally anti-symmetric tensor is defined as
$\varepsilon^{\mu\nu\rho\sigma}\equiv\epsilon^{\mu\nu\rho\sigma}/e$ with
$\epsilon^{0123}=+1$. This formula tells us that the spin current is totally
anti-symmetric with respect to its three indices. One can then easily show
that it is equal to the Hodge dual of the axial $\text{U}(1)_{A}$ current
$J_{5}^{\hat{a}}$ as

$\Sigma^{\mu}_{~{}\hat{a}\hat{b}}=-\frac{1}{2}\varepsilon^{\mu}_{~{}\hat{a}\hat{b}\hat{c}}J_{5}^{\hat{c}}\quad\mathrm{with}\quad
J_{5}^{\hat{a}}\equiv\mathrm{i}{\bar{q}}\gamma^{\hat{a}}\gamma_{5}q.$ (13)

Therefore, the number of independent components of the spin current operator
is only four, which are in one-to-one correspondence to the $\text{U}(1)_{A}$
current $J_{5}^{\mu}$. This means that some components of the spin density,
e.g. $\Sigma^{00i}$, are identically zero, and the other components such as
$\Sigma^{0ij}$ are equivalent to the spatial components of the axial current
$J_{5}^{i}$. The remaining component of $\Sigma^{ijk}$ is completely
determined by the axial charge density $J_{5}^{0}$.

Let us now derive the Ward-Takahashi identities that follow from local
symmetries. The vital point is that the QCD action (1) remains invariant, i.e.
$\delta_{\chi}\mathcal{S}_{\mathrm{QCD}}=0$, under the following infinitesimal
transformations:

$\begin{cases}\delta_{\chi}e_{\mu}^{~{}\hat{a}}=\xi^{\nu}\partial_{\nu}e_{\mu}^{~{}\hat{a}}+e_{\nu}^{~{}\hat{a}}\partial_{\mu}\xi^{\nu}-\alpha^{\hat{a}}_{~{}\hat{b}}e_{\mu}^{~{}\hat{b}},\\\
\delta_{\chi}\omega_{\mu~{}b}^{~{}\hat{a}}=\xi^{\nu}\partial_{\nu}\omega_{\mu~{}\hat{b}}^{~{}\hat{a}}+\omega_{\nu~{}\hat{b}}^{~{}\hat{a}}\partial_{\mu}\xi^{\nu}+\partial_{\mu}\alpha^{\hat{a}}_{~{}\hat{b}}-\alpha^{\hat{a}}_{~{}\hat{c}}\omega_{\mu~{}\hat{b}}^{~{}\hat{c}}+\alpha^{\hat{c}}_{~{}\hat{b}}\omega_{\mu~{}\hat{c}}^{~{}\hat{a}},\vspace{3pt}\\\
\delta_{\chi}A_{\mu}=\xi^{\nu}\partial_{\nu}A_{\mu}+A_{\nu}\partial_{\mu}\xi^{\nu}+\partial_{\mu}\theta,\\\
\delta_{\chi}q=\xi^{\nu}\partial_{\nu}q+\dfrac{\mathrm{i}}{2}\alpha^{\hat{a}\hat{b}}\Sigma_{\hat{a}\hat{b}}q+\dfrac{\mathrm{i}}{3}\theta
q,\vspace{3pt}\\\
\delta_{\chi}\bar{q}=\xi^{\nu}\partial_{\nu}{\bar{q}}-\dfrac{\mathrm{i}}{2}\alpha^{\hat{a}\hat{b}}\bar{q}\Sigma_{\hat{a}\hat{b}}-\dfrac{\mathrm{i}}{3}\theta
q,\\\
\delta_{\chi}a_{\mu}=\xi^{\nu}\partial_{\nu}a_{\mu}+a_{\nu}\partial_{\mu}\xi^{\nu}.\end{cases}$
(14)

We introduced a set of local infinitesimal parameters,
$\chi\equiv\\{\xi^{\mu},\alpha^{\hat{a}}_{~{}\hat{b}},\theta\\}$, which
generates the general coordinate and local Lorentz transformations – four-
vector $\xi^{\mu}$ for diffeomorphism, anti-symmetric tensor
$\alpha_{\hat{a}\hat{b}}=-\alpha_{\hat{b}\hat{a}}$ for local Lorentz symmetry,
and $\theta$ for the $\text{U}(1)$ symmetry. On the other hand, a direct
computation of the induced variation of the QCD action leads to another
expression for $\delta_{\chi}\mathcal{S}_{\mathrm{QCD}}$ with the arbitrary
parameters $\chi$. For $\delta_{\chi}\mathcal{S}_{\mathrm{QCD}}=0$ to hold for
arbitrary $\chi$, their coefficients need to vanish, which gives us the Ward-
Takahashi identities for the currents. Since the computation is a little
complicated, we first consider the identities associated with the local
Lorentz and $\text{U}(1)$ symmetries, and then move on to that for
diffeomorphism.

The variation of the action under the local Lorentz transformation is given by

$\displaystyle\delta_{\alpha}\mathcal{S}_{\mathrm{QCD}}$
$\displaystyle=\int\mathrm{d}^{4}x\left[\frac{\delta\mathcal{S}_{\mathrm{QCD}}}{\delta
e_{\mu}^{~{}\hat{a}}}\delta_{\alpha}e_{\mu}^{~{}\hat{a}}+\frac{\delta\mathcal{S}_{\mathrm{QCD}}}{\delta\omega_{\mu~{}\hat{b}}^{~{}\hat{a}}}\delta_{\alpha}\omega_{\mu~{}\hat{b}}^{~{}\hat{a}}+\frac{\delta\mathcal{S}_{\mathrm{QCD}}}{\delta
q}\delta_{\alpha}q+\frac{\delta\mathcal{S}_{\mathrm{QCD}}}{\delta\bar{q}}\delta_{\alpha}\bar{q}+\frac{\delta\mathcal{S}_{\mathrm{QCD}}}{\delta
a_{\mu}}\delta_{\alpha}a_{\mu}\right]$
$\displaystyle=\frac{1}{2}\int\mathrm{d}^{4}xe\alpha^{\hat{a}\hat{b}}\bigg{[}(\Theta_{\hat{a}\hat{b}}-\Theta_{\hat{b}\hat{a}})+(D_{\mu}-\mathcal{G}_{\mu})\Sigma^{\mu}_{~{}\hat{a}\hat{b}}\bigg{]},$
(15)

where we used the equation of motion for dynamical variables
($\delta\mathcal{S}_{\mathrm{QCD}}/\delta
q=\delta\mathcal{S}_{\mathrm{QCD}}/\delta\bar{q}=\delta\mathcal{S}_{\mathrm{QCD}}/\delta
a_{\mu}=0$). We here introduced a torsional contribution
$\mathcal{G}_{\mu}\equiv T^{\nu}_{~{}\nu\mu}$, which results from the
integration by parts (all surface terms are neglected) in the presence of the
torsion as

$\partial_{\mu}e=e\Gamma^{\nu}_{~{}\mu\nu}=e(\Gamma^{\nu}_{~{}\nu\mu}-T^{\nu}_{~{}\nu\mu})=e(\Gamma^{\nu}_{~{}\nu\mu}-\mathcal{G}_{\mu}).$
(16)

The local Lorentz invariance of the action tells us
$\delta_{\alpha}\mathcal{S}_{\mathrm{QCD}}=0$ holds for arbitrary position-
dependent $\alpha^{\hat{a}\hat{b}}$. As a result, we obtain the following
identity:

$(D_{\mu}-\mathcal{G}_{\mu})\Sigma^{\mu}_{~{}\hat{a}\hat{b}}=-(\Theta_{\hat{a}\hat{b}}-\Theta_{\hat{b}\hat{a}}).$
(17)

Therefore, based on Eq. (17), the covariant divergence of our spin current is
tied to the anti-symmetric part of our energy-momentum tensor. We regard this
Ward-Takahashi identity as the equation of motion for the spin density.
However, we also note that not all the components of Eq. (17) give equations
of motion for the spin density. This is because our spin current is totally
anti-symmetric, and thus, we only have three spin components as dynamical
variables. As a result, the remaining three equations are shown to be
constraints for a part of the energy-momentum tensor
$\Theta^{\mu}_{~{}\hat{a}}$. We will use these constraints when we derive the
constitutive relations with the help of the second law of local
thermodynamics. A similar computation leads to the following identity attached
to the $\text{U}(1)$:

$(\nabla_{\mu}-\mathcal{G}_{\mu})J^{\mu}=0,$ (18)

which gives a conservation law for the flavour current.

Being equipped with the above identities, we further evaluate the variation of
the action induced by the general coordinate transformation as

$\begin{split}\delta_{\xi}\mathcal{S}_{\mathrm{QCD}}&=-\int\mathrm{d}^{4}xe\xi^{\nu}\left[(\nabla_{\mu}-\mathcal{G}_{\mu})\Theta^{\mu}_{~{}\nu}+\Theta^{\mu}_{~{}\rho}T^{\rho}_{~{}\mu\nu}-\frac{1}{2}\Sigma^{\mu~{}\hat{b}}_{~{}\hat{a}}\mathcal{R}^{\hat{a}}_{~{}\hat{b}\mu\nu}-F_{\nu\mu}J^{\mu}\right],\end{split}$
(19)

where we again used the equation of motion and performed the integration by
parts. Here, we introduced the field strength tensor
$\mathcal{R}^{a}_{~{}b\mu\nu}$ attached to the spin connection, and
$F_{\mu\nu}$ as

$\begin{split}\mathcal{R}^{\hat{a}}_{~{}\hat{b}\mu\nu}&\equiv\partial_{\mu}\omega_{\nu~{}\hat{b}}^{~{}\hat{a}}-\partial_{\nu}\omega_{\mu~{}\hat{b}}^{~{}\hat{a}}+\omega_{\mu~{}\hat{c}}^{~{}\hat{a}}\omega_{\nu~{}\hat{b}}^{~{}\hat{c}}-\omega_{\nu~{}\hat{c}}^{~{}\hat{a}}\omega_{\mu~{}\hat{b}}^{~{}\hat{c}},\\\
F_{\mu\nu}&\equiv\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}.\end{split}$ (20)

For $\delta_{\xi}\mathcal{S}_{\mathrm{QCD}}=0$ to hold for arbitrary
$\xi^{\nu}$, we obtain the identity

$(\nabla_{\mu}-\mathcal{G}_{\mu})\Theta^{\mu}_{~{}\nu}=-\Theta^{\mu}_{~{}\rho}T^{\rho}_{~{}\mu\nu}+\frac{1}{2}\Sigma^{\mu~{}\hat{b}}_{~{}\hat{a}}\mathcal{R}^{\hat{a}}_{~{}\hat{b}\mu\nu}+F_{\nu\mu}J^{\mu}.$
(21)

Therefore, our energy-momentum tensor has source terms coming from the
spacetime torsion, curvature, and external $\text{U}(1)$ gauge field.

In summary, we have obtained the following Ward-Takahashi identities:

$\begin{split}(D_{\mu}-\mathcal{G}_{\mu})\Theta^{\mu}_{~{}\hat{a}}&=-\Theta^{\mu}_{~{}\hat{b}}T^{\hat{b}}_{~{}\mu\hat{a}}+\frac{1}{2}\Sigma^{\mu~{}\hat{c}}_{~{}\hat{b}}\mathcal{R}^{\hat{b}}_{~{}\hat{c}\mu\hat{a}}+F_{\hat{a}\mu}J^{\mu},\\\
(D_{\mu}-\mathcal{G}_{\mu})\Sigma^{\mu}_{~{}\hat{a}\hat{b}}&=-(\Theta_{\hat{a}\hat{b}}-\Theta_{\hat{b}\hat{a}}),\\\
(\nabla_{\mu}-\mathcal{G}_{\mu})J^{\mu}&=0,\end{split}$ (22)

which result from diffeomorphism, local Lorentz invariance and $\text{U}(1)$
symmetry of the QCD action (1)-(2). When we are interested in QCD in the flat
spacetime with vanishing $\text{U}(1)$ gauge field, we turn off the background
fields $e_{\mu}^{~{}\hat{a}}$, $\omega_{\mu}^{~{}\hat{a}\hat{b}}$, and
$A_{\mu}$, which reduces Eqs. (22) to

$\partial_{\mu}\Theta^{\mu}_{~{}\nu}=0,\quad\partial_{\mu}\Sigma^{\mu}_{~{}\hat{a}\hat{b}}=-(\Theta_{\hat{a}\hat{b}}-\Theta_{\hat{b}\hat{a}}),\quad\mathrm{and}\quad\partial_{\mu}J^{\mu}=0.$
(23)

These equations give the conservation laws for the canonical-like energy-
momentum density, total angular momentum, and flavour charge (the first two
were used in Ref. Hattori:2019lfp ). We note that the spin current
$\Sigma^{\mu}_{~{}\hat{a}\hat{b}}$ is totally anti-symmetric with respect to
its indices owing to the gamma matrix structure of Eq. (12) [recall Eq. (13)],
which is not assumed in Ref. Hattori:2019lfp . As a result of the anti-
symmetric property, the number of independent degrees of freedom in the spin
current is not six but three in the present work.

One remark here is that we defined the energy-momentum tensor
$\Theta^{\mu}_{~{}\hat{a}}$ by considering the variation with the fixed spin
connection $\omega_{\mu}^{~{}\hat{a}\hat{b}}$ instead of the fixed contorsion
$K_{\mu}^{~{}\hat{a}\hat{b}}$. As a consequence, one finds that
$\Theta^{\mu}_{~{}\hat{a}}$ corresponds to the canonical-like energy-momentum
tensor. If we introduce the energy-momentum tensor $T^{\mu}_{~{}\hat{a}}$ with
the fixed contorsion $K_{\mu}^{~{}\hat{a}\hat{b}}$, we have

$\begin{split}T^{\mu}_{~{}\hat{a}}&\equiv\frac{1}{e(x)}\left.\frac{\delta\mathcal{S}_{\mathrm{QCD}}}{\delta
e_{\mu}^{~{}\hat{a}}(x)}\right|_{K}=\Theta^{\mu}_{~{}\hat{a}}+\frac{1}{2}(D_{\rho}-\mathcal{G}_{\rho})\Sigma^{\mu~{}\rho}_{~{}\hat{a}},\end{split}$
(24)

where we used the totally anti-symmetric property of the spin current to
obtain the rightmost side. We will see that this energy-momentum tensor
becomes useful when performing the linear-mode analysis in Sec. 4.

##  3 Derivation of spin hydrodynamics

In this section, we provide a derivation of hydrodynamics with spin
polarization Hattori:2019lfp ; Fukushima:2020ucl . Based on the Ward-Takahashi
identities and the second law of local thermodynamics Landau:Fluid , we
specify a first-order constitutive relation of hydrodynamics with the totally
anti-symmetric spin current. In addition to two (shear and bulk) viscosities
and a flavour charge conductivity, we find one kinetic coefficient, called
rotational viscosity, appearing in the anti-symmetric part of the energy-
momentum tensor.

###  3.1 Setup

Let us first introduce the dynamical variables. We employ the Landau-Lifshitz
frame with respect to the symmetric part of the energy-momentum tensor in
defining the fluid four-velocity $u^{\mu}$. Therefore, we introduce $u^{\mu}$
and the associated energy density $\epsilon$ in the rest frame by

$\Theta^{\mu}_{~{}\hat{a}}\big{|}_{(s)}u^{\hat{a}}=-\epsilon
u^{\mu}\quad\mathrm{with}\quad\eta_{\hat{a}\hat{b}}u^{\hat{a}}u^{\hat{b}}=g_{\mu\nu}u^{\mu}u^{\nu}=-1,$
(25)

where we denote the symmetric part of the energy-momentum tensor as
$\Theta^{\mu}_{~{}\hat{a}}\big{|}_{(s)}\equiv(\Theta^{\mu}_{~{}\hat{a}}+\Theta^{~{}\mu}_{\hat{a}})/2$.
Owing to the normalization $u_{\mu}u^{\mu}=-1$, we can parameterize the four-
velocity as $u^{\mu}=\gamma(\bm{v})(1,\bm{v})^{t}$ with
$\gamma(\bm{v})=1/\sqrt{1-\bm{v}^{2}}$. We next introduce the spin density
$\sigma_{\hat{a}\hat{b}}=-\sigma_{\hat{b}\hat{a}}$ and the flavour charge
density $n$ by

$\sigma_{\hat{a}\hat{b}}=-u_{\mu}\Sigma^{\mu}_{~{}\hat{a}\hat{b}}\quad\mathrm{and}\quad
n=-u_{\mu}J^{\mu}.$ (26)

As given in Eq. (12), we microscopically know the spin current for QFTs with
Dirac fermions is totally anti-symmetric so that the spin density for such
QFTs satisfies $\sigma_{\hat{a}\hat{b}}u^{\hat{a}}=0$. This means that we only
have three independent components of spin density attached to the spatial
rotation symmetry, while the other three associated with the boost symmetry
are absent. Thus, our dynamical variables are three spin components in
addition to four energy-momentum densities and one flavour charge density. As
for the spin density, we also use the dual variable $\sigma^{\hat{a}}$ defined
by

$\sigma^{\hat{a}}\equiv\frac{1}{2}\varepsilon^{\hat{a}\hat{b}\hat{c}\hat{d}}u_{\hat{b}}\sigma_{\hat{c}\hat{d}}~{}\Leftrightarrow~{}\sigma_{\hat{a}\hat{b}}=-\varepsilon_{\hat{a}\hat{b}\hat{c}\hat{d}}u^{\hat{c}}\sigma^{\hat{d}}\quad\mathrm{satisfying}\quad\sigma^{\hat{a}}u_{\hat{a}}=0=\sigma_{\hat{a}\hat{b}}u^{\hat{a}}.$
(27)

In short, we identify eight dynamical variables as

$\epsilon,~{}n,~{}u^{\hat{a}},~{}\sigma^{\hat{a}}~{}(\mathrm{or}~{}\sigma_{\hat{a}\hat{b}}),\quad\mathrm{satisfying}\quad\eta_{\hat{a}\hat{b}}u^{\hat{a}}u^{\hat{b}}=-1,\quad\sigma^{\hat{a}}u_{\hat{a}}=0=\sigma_{\hat{a}\hat{b}}u^{\hat{a}}.$
(28)

We then introduce thermodynamic variables conjugate to the energy, spin, and
flavour charge densities. For that purpose, we introduce a generalized entropy
density $s(\epsilon,n,\sigma_{\hat{a}\hat{b}})$, or equivalently
$s(\epsilon,n,\sigma^{\hat{a}})$, and introduce the inverse temperature
$\beta$, flavour chemical potential $\mu$, and spin chemical potential
$\mu^{\hat{a}\hat{b}}$ (or its dual $\mu_{\hat{a}}$) by

$\beta\equiv\frac{\partial
s}{\partial\epsilon},\quad\beta\mu\equiv-\frac{\partial s}{\partial
n},\quad\beta\mu^{\hat{a}\hat{b}}\equiv-2\frac{\partial
s}{\partial\sigma_{\hat{a}\hat{b}}}\quad\mathrm{and}\quad\left(\mathrm{or}\quad\beta\mu_{\hat{a}}\equiv-\frac{\partial
s}{\partial\sigma^{\hat{a}}}\right).$ (29)

In other words, we require the generalized first law of local thermodynamics
in the rest frame including spin density:

$T\mathrm{d}s=\mathrm{d}\epsilon-\mu\mathrm{d}n-\frac{1}{2}\mu^{\hat{a}\hat{b}}\mathrm{d}\sigma_{\hat{a}\hat{b}}(=\mathrm{d}\epsilon-\mu\mathrm{d}n-\mu_{\hat{a}}\mathrm{d}\sigma^{\hat{a}}).$
(30)

Equation (27) forces two spin chemical potentials to be related with each
other by

$\mu_{\hat{a}}=\frac{1}{2}\varepsilon_{\hat{a}\hat{b}\hat{c}\hat{d}}u^{\hat{b}}\mu^{\hat{c}\hat{d}}~{}\Leftrightarrow~{}\mu^{\hat{a}\hat{b}}=-\varepsilon^{\hat{a}\hat{b}\hat{c}\hat{d}}u_{\hat{c}}\mu_{\hat{d}}.$
(31)

Note that the spin chemical potential satisfies
$\mu_{\hat{a}}u^{\hat{a}}=0=\mu^{\hat{a}\hat{b}}u_{\hat{b}}$.

Our purpose is to express $\Theta^{\mu}_{~{}\hat{a}}$,
$\Sigma^{\mu}_{~{}\hat{a}\hat{b}}$, and $J^{\mu}$ by using the eight dynamical
variables listed in Eq. (28). We then rewrite these currents by performing the
tensor decomposition as follows:

$\begin{split}\Theta^{\mu}_{~{}\hat{a}}&=\epsilon
u^{\mu}u_{\hat{a}}+p\Delta^{\mu}_{\hat{a}}+u^{\mu}\delta q_{\hat{a}}-\delta
q^{\mu}u_{\hat{a}}+\delta\Theta^{\mu}_{~{}\hat{a}},\\\
\Sigma^{\mu}_{~{}\hat{a}\hat{b}}&=\varepsilon^{\mu}_{~{}\hat{a}\hat{b}\hat{c}}(\sigma^{\hat{c}}+\delta\sigma
u^{\hat{c}}),\\\ J^{\mu}&=nu^{\mu}+\delta J^{\mu},\end{split}$ (32)

where we defined a pressure $p$ with a projection matrix
$\Delta^{\mu}_{\hat{a}}$:

$\Delta^{\mu}_{\hat{a}}\equiv
e^{~{}\mu}_{\hat{a}}+u^{\mu}u_{\hat{a}}\quad\mathrm{satisfying}\quad\Delta^{\mu}_{\hat{a}}u^{\hat{a}}=0=\Delta^{\mu}_{\hat{a}}u_{\mu}.$
(33)

Note that the Landau-Lifshitz definition of the fluid velocity restricts
$\delta\Theta^{\mu}_{~{}\hat{a}}$ and $\delta q_{\hat{a}}$ to satisfy
$\delta\Theta^{\mu}_{~{}\hat{a}}u^{\hat{a}}=0=u_{\mu}\delta\Theta^{\mu}_{~{}\hat{a}}$
and $\delta q_{\hat{a}}u^{\hat{a}}=0$, and the definition of $n$ leads to
$u_{\mu}\delta J^{\mu}=0$. Equation (32) just gives one possible
parameterization of $\Theta^{\mu}_{~{}\hat{a}}$,
$\Sigma^{\mu}_{~{}\hat{a}\hat{b}}$ and $J^{\mu}$, and thus,
$\delta\Theta^{\mu}_{~{}\hat{a}}$, $\delta q_{\hat{a}}$, $\delta\sigma$, and
$\delta J^{\mu}$ could contain nondissipative terms. We will, however, show
that such a nondissipative term is absent in the constitutive relation for
$\delta\Theta^{\mu}_{~{}\hat{a}}$, $\delta q_{\hat{a}}$ and $\delta\sigma$ in
the first-order spin hydrodynamics.

###  3.2 Derivation of constitutive relations

To derive hydrodynamic equations with spin polarization, we need to express
$\delta\Theta^{\mu}_{~{}\hat{a}}$, $\delta q_{\hat{a}}$, $\delta\sigma$ and
$\delta J^{\mu}$ using our dynamical variables and background fields. For that
purpose, we require a generalized second law of local thermodynamics, which
require the existence of the entropy current

$\exists\,s^{\mu}(\beta,\mu,u^{\mu},\mu^{\hat{a}\hat{b}})\quad\mathrm{satisfying}\quad(\nabla_{\mu}-\mathcal{G}_{\mu})s^{\mu}\geq
0\quad\mathrm{for}\quad\forall\,\beta,~{}\mu,~{}u^{\mu},~{}\mathrm{and}~{}\mu^{\hat{a}\hat{b}}.$
(34)

Let us specify the condition following from the local second law (34). By
parametrizing the entropy current as $s^{\mu}=su^{\mu}+\delta s^{\mu}$, we can
express its divergence as

$(\nabla_{\mu}-\mathcal{G}_{\mu})s^{\mu}=u^{\mu}\nabla_{\mu}s+s(\nabla_{\mu}u^{\mu}-u^{\mu}\mathcal{G}_{\mu})+(\nabla_{\mu}-\mathcal{G}_{\mu})\delta
s^{\mu}.$ (35)

The vital point here is that we can write the material derivative
$u^{\mu}\nabla_{\mu}$ of the entropy density
$s(\epsilon,n,\sigma_{\hat{a}\hat{b}})$ as

$u^{\mu}\nabla_{\mu}s=u^{\mu}\left(\frac{\partial
s}{\partial\epsilon}\nabla_{\mu}\epsilon+\frac{\partial
s}{\partial\sigma_{\hat{a}\hat{b}}}D_{\mu}\sigma_{\hat{a}\hat{b}}+\frac{\partial
s}{\partial n}\nabla_{\mu}n\right)=\beta
u^{\mu}\left(\nabla_{\mu}\epsilon-\frac{1}{2}\mu^{\hat{a}\hat{b}}D_{\mu}\sigma_{\hat{a}\hat{b}}-\mu\nabla_{\mu}n\right),$
(36)

where we used the definition of the thermodynamic variables $\beta$, $\mu$,
and $\mu^{\hat{a}\hat{b}}$. Noting that the left-hand side of this equation is
a scalar, we put the covariant derivative $D_{\mu}\sigma_{\hat{a}\hat{b}}$ to
make the right-hand sides to be scalar too. We can rewrite the rightmost side
of the equation (36) by using the Ward-Takahashi identities with the assumed
constitutive relation (32). To do so, we take contractions of Eqs. (22) with
$\beta^{\hat{a}}=\beta u^{\hat{a}}$, $\frac{1}{2}\beta\mu^{\hat{a}\hat{b}}$
and $\beta\mu$, which results in

$\displaystyle\beta u^{\mu}\nabla_{\mu}\epsilon=$
$\displaystyle-\beta(\epsilon+p)(\nabla_{\mu}u^{\mu}-u^{\mu}\mathcal{G}_{\mu})-\delta\Theta^{\mu}_{~{}\hat{a}}(D_{\mu}\beta^{\hat{a}}-T^{\hat{a}}_{~{}\mu\hat{b}}\beta^{\hat{b}})+(\nabla_{\mu}-\mathcal{G}_{\mu})(\beta\delta
q^{\mu})$ $\displaystyle-\delta
q^{\hat{a}}\big{[}\nabla_{\hat{a}}\beta+u^{\mu}(D_{\mu}\beta_{\hat{a}}-\beta_{\hat{b}}T^{\hat{b}}_{~{}\mu\hat{a}})\big{]}$
$\displaystyle-\frac{1}{2}\varepsilon^{\mu}_{~{}\hat{a}\hat{b}\hat{c}}(\sigma^{\hat{c}}+\delta\sigma
u^{\hat{c}})\mathcal{R}^{\hat{a}\hat{b}}_{~{}~{}\mu\hat{d}}\beta^{\hat{d}}-\beta
u^{\nu}F_{\nu\mu}\delta J^{\mu},$ (37) $\displaystyle-\frac{1}{2}\beta
u^{\mu}\mu^{\hat{a}\hat{b}}D_{\mu}\sigma_{\hat{a}\hat{b}}=$
$\displaystyle\frac{1}{2}\beta\mu^{\hat{a}\hat{b}}\sigma_{\hat{a}\hat{b}}(\nabla_{\mu}u^{\mu}-u^{\mu}\mathcal{G}_{\mu})+\frac{1}{2}\beta\mu^{\hat{a}\hat{b}}\left(\delta\Theta_{\hat{a}\hat{b}}-\delta\Theta_{\hat{b}\hat{a}}\right)$
$\displaystyle+\mu_{\hat{a}\hat{b}}\sigma^{\hat{b}\mu}D_{\mu}\beta^{\hat{a}}+(\nabla_{\mu}-\mathcal{G}_{\mu})(\beta\delta\sigma\mu^{\mu})-\frac{1}{2}\delta\sigma
u^{\hat{c}}\varepsilon^{\mu}_{~{}\hat{a}\hat{b}\hat{c}}D_{\mu}(\beta\mu^{\hat{a}\hat{b}}),$
(38) $\displaystyle-\beta\mu u^{\mu}\nabla_{\mu}n=$ $\displaystyle\beta\mu
n(\nabla_{\mu}u^{\mu}-u^{\mu}\mathcal{G}_{\mu})-\delta
J^{\mu}\nabla_{\mu}(\beta\mu)+(\nabla_{\mu}-\mathcal{G}_{\mu})(\beta\mu\delta
J^{\mu}),$ (39)

where we used $\delta\Theta^{\mu}_{~{}\hat{a}}u^{\hat{a}}=0=\delta
q_{\hat{a}}u^{\hat{a}}$ to obtain the first equation.

At this point, $\delta q^{\hat{a}}$ seems to enjoy the same status as
$\delta\Theta^{\mu}_{~{}\hat{a}}$, $\delta\sigma$ and $\delta J^{\mu}$, whose
constitutive relation should be determined independently of other parts.
However, this is not the case: the constitutive relation for $\delta
q_{\hat{a}}$ is determined by that for $\Sigma^{\mu}_{~{}\hat{a}\hat{b}}$. To
show this, recall that the Ward-Takahashi identity for the spin current
contains three constraints rather than six dynamical equations due to the
totally anti-symmetric property of our spin current. This becomes manifest by
contracting Eq. (17) with $u_{\hat{b}}$ and noticing that the resulting
identity

$\delta
q^{\hat{a}}=-\frac{1}{2}\varepsilon^{\mu\hat{a}}_{~{}~{}\hat{b}\hat{c}}u^{\hat{b}}\big{[}(D_{\mu}-\mathcal{G}_{\mu})\sigma^{\hat{c}}+\delta\sigma
D_{\mu}u^{\hat{c}}\big{]},$ (40)

does not contain the time derivative in the right-hand side. As a consequence,
we only need to derive the constitutive relation for
$\delta\Theta^{\mu}_{~{}\hat{a}}$, $\delta\sigma$ and $\delta J^{\mu}$, the
second of which will determine that for $\delta q_{\hat{a}}$ in accordance
with the constraint (40). It is thus useful to rewrite (37) by substituting
the constraint (40) as

$\begin{split}\beta
u^{\mu}\nabla_{\mu}\epsilon=&-\beta(\epsilon+p)(\nabla_{\mu}u^{\mu}-u^{\mu}\mathcal{G}_{\mu})-\delta\Theta^{\mu}_{~{}\hat{a}}(D_{\mu}\beta^{\hat{a}}-T^{\hat{a}}_{~{}\mu\hat{b}}\beta^{\hat{b}})+(\nabla_{\mu}-\mathcal{G}_{\mu})(\beta\delta
q^{\mu})\\\
&+\frac{1}{2}\varepsilon^{\mu}_{~{}\hat{a}\hat{b}\hat{c}}u^{\hat{b}}\big{[}(D_{\mu}-\mathcal{G}_{\mu})\sigma^{\hat{c}}+\delta\sigma
D_{\mu}u^{\hat{c}}\big{]}\big{[}\nabla^{\hat{a}}\beta+u^{\nu}(D_{\nu}\beta^{\hat{a}}-\beta_{\hat{d}}T^{\hat{d}~{}\hat{a}}_{~{}\nu})\big{]}\\\
&-\frac{1}{2}\varepsilon^{\mu}_{~{}\hat{a}\hat{b}\hat{c}}(\sigma^{\hat{c}}+\delta\sigma
u^{\hat{c}})\mathcal{R}^{\hat{a}\hat{b}}_{~{}~{}\mu\hat{d}}\beta^{\hat{d}}-\beta
u^{\nu}F_{\nu\mu}\delta J^{\mu}.\end{split}$ (41)

The result so far is exact without assuming the derivative expansion.

Let us then apply the derivative expansion to derive relativistic spin
hydrodynamic equations, which capture leading dissipative dynamics of charge
densities. For that purpose, we first need to specify our power counting
scheme for the dynamical variables and background fields. In this paper, we
employ the power counting scheme defined by

$O(\partial^{0})=\\{\beta,~{}u^{\mu},~{}\mu,~{}e_{\mu}^{~{}\hat{a}}\\}\quad\mathrm{and}\quad
O(\partial^{1})=\\{\mu^{\hat{a}\hat{b}},~{}\sigma_{\hat{a}\hat{b}},~{}\omega_{\mu}^{~{}\hat{a}\hat{b}}\\},$
(42)

and derive the constitutive relations up to $O(\partial^{1})$. This allows us
to keep the restricted number of terms appearing in the constitutive
relations. To demonstrate this, using Eqs. (37)-(39), we rewrite the
divergence of the entropy current as

$\begin{split}(\nabla_{\mu}-\mathcal{G}_{\mu})s^{\mu}=&\big{[}s-\beta(\epsilon+p-\mu
n)\big{]}(\nabla_{\mu}u^{\mu}-u^{\mu}\mathcal{G}_{\mu})+(\nabla_{\mu}-\mathcal{G}_{\mu})(\delta
s^{\mu}+\beta\mu\delta J^{\mu})\\\
&-\delta\Theta^{\mu}_{~{}\hat{a}}\big{|}_{(s)}(D_{\mu}\beta^{\hat{a}}-T^{\hat{a}}_{~{}\mu\hat{b}}\beta^{\hat{b}})-\delta\Theta^{\mu}_{~{}\hat{a}}\big{|}_{(a)}(D_{\mu}\beta^{\hat{a}}-T^{\hat{a}}_{~{}\mu\hat{b}}\beta^{\hat{b}}-\beta\mu_{\mu}^{~{}\hat{a}})\\\
&-\delta
J^{\mu}\big{[}(\nabla_{\mu}(\beta\mu)-\beta^{\nu}F_{\mu\nu}\big{]}+\Delta
S,\end{split}$ (43)

where we defined the symmetric and anti-symmetric parts of the energy-momentum
tensor

$\delta\Theta^{\mu}_{~{}\hat{a}}\big{|}_{(s)}\equiv\frac{1}{2}(\delta\Theta^{\mu}_{~{}\hat{a}}+\delta\Theta_{\hat{a}}^{~{}\mu})\quad\mathrm{and}\quad\delta\Theta^{\mu}_{~{}\hat{a}}\big{|}_{(a)}\equiv\frac{1}{2}(\delta\Theta^{\mu}_{~{}\hat{a}}-\delta\Theta_{\hat{a}}^{~{}\mu}).$
(44)

Here, we also introduced a collection of the higher-order terms $\Delta
S=O(\partial^{3})$, whose explicit form is given by

$\displaystyle\Delta S\equiv$
$\displaystyle\frac{1}{2}\beta\mu^{\hat{a}\hat{b}}\sigma_{\hat{a}\hat{b}}(\nabla_{\mu}u^{\mu}-u^{\mu}\mathcal{G}_{\mu})+\sigma^{\mu\hat{b}}\mu_{\hat{b}\hat{a}}D_{\mu}\beta^{\hat{a}}$
$\displaystyle+\frac{1}{2}\varepsilon^{\mu}_{~{}\hat{a}\hat{b}\hat{c}}\left(u^{\hat{b}}\big{[}\nabla^{\hat{a}}\beta+u^{\nu}(D_{\nu}\beta^{\hat{a}}-\beta_{\hat{d}}T^{\hat{d}~{}\hat{a}}_{~{}\nu})\big{]}(D_{\mu}-\mathcal{G}_{\mu})\sigma^{\hat{c}}+\sigma^{\hat{c}}\mathcal{R}^{\hat{b}}_{~{}\hat{a}\mu\hat{d}}\beta^{\hat{d}}\right)$
$\displaystyle-\frac{1}{2}\delta\sigma\varepsilon^{\mu}_{~{}\hat{a}\hat{b}\hat{c}}u^{\hat{c}}\left(D_{\mu}(\beta\mu^{\hat{a}\hat{b}})+\mathcal{R}^{\hat{a}\hat{b}}_{~{}~{}\mu\hat{d}}\beta^{\hat{d}}+D_{\mu}u^{\hat{b}}\big{[}\nabla^{\hat{a}}\beta+u^{\nu}(D_{\nu}\beta^{\hat{a}}-\beta_{\hat{d}}T^{\hat{d}~{}\hat{a}}_{~{}\mu})\big{]}\right)$
$\displaystyle+(\nabla_{\mu}-\mathcal{G}_{\mu})\left(\beta\delta
q^{\mu}+\beta\delta\sigma\mu^{\mu}\right).$ (45)

We shall explain the reason why these terms are counted as $O(\partial^{3})$.
First of all, our power counting scheme (42) immediately tells us the terms in
the first and second lines are the third order in derivative. The first term
in the first line, if considered, leads to a second-order correction of the
Gibbs-Duhem relation as $\epsilon+p=Ts+\mu
n+\frac{1}{2}\mu^{\hat{a}\hat{b}}\sigma_{\hat{a}\hat{b}}$ (see also the
following discussion). From the third line, one finds the form of
$\delta\sigma$ compatible with the second law as

$\delta\sigma=-T\kappa_{s}\varepsilon^{\mu}_{~{}\hat{a}\hat{b}\hat{c}}u^{\hat{c}}\left(D_{\mu}(\beta\mu^{\hat{a}\hat{b}})+\mathcal{R}^{\hat{a}\hat{b}}_{~{}~{}\mu\hat{d}}\beta^{\hat{d}}+D_{\mu}u^{\hat{b}}\big{[}\nabla^{\hat{a}}\beta+u^{\nu}(D_{\nu}\beta^{\hat{a}}-\beta_{\hat{d}}T^{\hat{d}~{}\hat{a}}_{~{}\mu})\big{]}\right),$
(46)

with a positive coefficient $\kappa_{s}>0$. Equation (46) represents a
diffusive transport of the spin current and its response to the curvature
tensor; namely, $\kappa_{s}$ is identified as a spin conductivity. Besides,
this closes the constitutive relation for $\delta q_{\hat{a}}$ thanks to the
constraint (40). Our power counting scheme tells us
$\delta\sigma=O(\partial^{2})$ and $\delta q_{\hat{a}}=O(\partial^{2})$, which
proves that all terms in the last two lines of Eq. (45) are $O(\partial^{3})$.
Note that the last term, if considered, leads to a second-order correction of
the entropy current as $\delta s^{\mu}=-\beta\delta
q^{\mu}-\beta\delta\sigma\mu^{\mu}=O(\partial^{2})$. As a result, we find
$\delta\sigma=0$ and $\delta q_{\hat{a}}=0$ in the first-order spin
hydrodynamics.

Going back to Eq. (43), one finds the entropy current $s^{\mu}$ and the
constitutive relations for $\delta\Theta^{\mu}_{~{}\hat{a}}$ and $\delta
J^{\mu}$ compatible with the generalized second law of local thermodynamics
(34) in the following way. Firstly, the first line of Eq. (43) vanishes by
identifying

$\displaystyle\epsilon+p=Ts+\mu n+O(\partial^{2})\quad\mathrm{and}\quad\delta
s^{\mu}=\beta\mu\delta J^{\mu}+O(\partial^{2}),$ (47)

where the first equation is the usual Gibbs-Duhem relation up to
$O(\partial)$. We note that both equations have second-order derivative
corrections present in $\Delta S$ as we discussed above. The second line and
the first term in the third line of Eq. (43) will always be positive by
identifying the following constitutive relations:

$\begin{split}\delta\Theta^{\mu}_{~{}\hat{a}}\big{|}_{(s)}&=-T\eta^{\mu~{}\nu}_{~{}\hat{a}~{}\hat{b}}(D_{\nu}\beta^{\hat{b}}-T^{\hat{b}}_{~{}\nu\hat{c}}\beta^{\hat{c}})=-\eta^{\mu~{}\nu}_{~{}\hat{a}~{}\hat{b}}(D_{\nu}u^{\hat{b}}-T^{\hat{b}}_{~{}\nu\hat{c}}u^{\hat{c}}),\\\
\delta\Theta^{\mu}_{~{}\hat{a}}\big{|}_{(a)}&=-T(\eta_{s})^{\mu~{}\nu}_{~{}\hat{a}~{}\hat{b}}(D_{\nu}\beta^{\hat{b}}-T^{\hat{b}}_{~{}\nu\hat{c}}\beta^{\hat{c}}-\beta\mu_{\nu}^{~{}\hat{b}})=-(\eta_{s})^{\mu~{}\nu}_{~{}\hat{a}~{}\hat{b}}(D_{\nu}u^{\hat{b}}-T^{\hat{b}}_{~{}\nu\hat{c}}u^{\hat{c}}-\mu_{\nu}^{~{}\hat{b}}),\\\
\delta
J^{\mu}&=-T\kappa_{n}\Delta^{\mu\nu}\big{[}\nabla_{\nu}(\beta\mu)-\beta^{\rho}F_{\nu\rho}\big{]},\end{split}$
(48)

where one can decompose the viscous tensors
$\eta^{\mu~{}\nu}_{~{}\hat{a}~{}\hat{b}}$ and
$(\eta_{s})^{\mu~{}\nu}_{~{}\hat{a}~{}\hat{b}}$ as

$\begin{split}\eta^{\mu~{}\nu}_{~{}\hat{a}~{}\hat{b}}&=2\eta\left(\frac{1}{2}(\Delta^{\mu\nu}\Delta_{\hat{a}\hat{b}}+\Delta^{\mu}_{\hat{b}}\Delta^{\nu}_{\hat{a}})-\frac{1}{3}\Delta^{\mu}_{\hat{a}}\Delta^{\nu}_{\hat{b}}\right)+\zeta\Delta^{\mu}_{\hat{a}}\Delta^{\nu}_{\hat{b}},\\\
(\eta_{s})^{\mu~{}\nu}_{~{}\hat{a}~{}\hat{b}}&=\frac{1}{2}\eta_{s}(\Delta^{\mu\nu}\Delta_{\hat{a}\hat{b}}-\Delta^{\mu}_{\hat{b}}\Delta^{\nu}_{\hat{a}}).\end{split}$
(49)

Here, all four kinetic coefficients $\eta$, $\zeta$, $\eta_{s}$, and
$\kappa_{n}$ are assumed to take positive values. They are shear viscosity,
bulk viscosity, rotational viscosity, and flavour conductivity. The positivity
of these kinetic coefficients makes the entropy production rate a positive
semidefinite quadratic form within the second-order derivative to satisfy the
second law of local thermodynamics.

One can also rewrite the constitutive relation by decomposing the spin
connection to the torsion-free and torsion parts as

$\begin{split}D_{\mu}u^{\hat{a}}-T^{\hat{a}}_{~{}\mu\hat{b}}u^{\hat{b}}&={\mathring{D}}_{\mu}u^{\hat{a}}-u^{\hat{b}}K_{\hat{b}\mu}^{~{}~{}\hat{a}},\\\
\end{split}$ (50)

where the circled derivative contains the torsion-free spin connection. From
this decomposition, one sees that the symmetric part of the energy-momentum
tensor has no contribution from the torsion. This is because the contorsion
$K_{\hat{b}\mu}^{~{}~{}\hat{a}}$ is anti-symmetric under the exchange of
$(\mu,\hat{a})$ indices so that it disappears from the constitutive relation.
Then, we obtain another expression for the constitutive relation as follows:

$\begin{split}\delta\Theta^{\mu}_{~{}\hat{a}}\big{|}_{(s)}&=-\eta^{\mu~{}\nu}_{~{}\hat{a}~{}\hat{b}}{\mathring{D}}_{\nu}u^{\hat{b}},\\\
\delta\Theta^{\mu}_{~{}\hat{a}}\big{|}_{(a)}&=-(\eta_{s})^{\mu~{}\nu}_{~{}\hat{a}~{}\hat{b}}({\mathring{D}}_{\nu}u^{\hat{b}}-u^{\hat{c}}K_{\hat{c}\nu}^{~{}~{}\hat{b}}-\mu_{\nu}^{~{}\hat{b}}).\end{split}$
(51)

Equations (48) or (51) give the constitutive relations for the first-order
spin hydrodynamics. In addition to two viscosities $\eta$ and $\zeta$, we have
another viscosity known as the rotational viscosity $\eta_{s}$.

In summary, we find the constitutive relations for the first-order spin
hydrodynamics in the torsionful curved background as follows:

$\begin{split}\Theta^{\mu}_{~{}\hat{a}}&=\epsilon
u^{\mu}u_{\hat{a}}+p\Delta^{\mu}_{\hat{a}}-\eta^{\mu~{}\nu}_{~{}\hat{a}~{}\hat{b}}(D_{\nu}u^{\hat{b}}-T^{\hat{b}}_{~{}\nu\hat{c}}u^{\hat{c}})-(\eta_{s})^{\mu~{}\nu}_{~{}\hat{a}~{}\hat{b}}(D_{\nu}u^{\hat{b}}-T^{\hat{b}}_{~{}\nu\hat{c}}u^{\hat{c}}-\mu_{\nu}^{~{}\hat{b}})\\\
&=\epsilon
u^{\mu}u_{\hat{a}}+p\Delta^{\mu}_{\hat{a}}-\eta^{\mu~{}\nu}_{~{}\hat{a}~{}\hat{b}}{\mathring{D}}_{\nu}u^{\hat{b}}-(\eta_{s})^{\mu~{}\nu}_{~{}\hat{a}~{}\hat{b}}({\mathring{D}}_{\nu}u^{\hat{b}}-u^{\hat{c}}K_{\hat{c}\nu}^{~{}~{}\hat{b}}-\mu_{\nu}^{~{}\hat{b}}),\\\
\Sigma^{\mu}_{~{}\hat{a}\hat{b}}&=\varepsilon^{\mu}_{~{}\hat{a}\hat{b}\hat{c}}\sigma^{\hat{c}},\\\
J^{\mu}&=nu^{\mu}-T\kappa_{n}\Delta^{\mu\nu}\big{[}\nabla_{\nu}(\beta\mu)-\beta^{\rho}F_{\nu\rho}\big{]},\end{split}$
(52)

where the rank-four viscous tensors are decomposed as Eq. (49). Thus, the
first-order spin hydrodynamics with one flavour current is equipped with three
viscosities $\eta$, $\zeta$, and $\eta_{s}$ and one conductivity $\kappa_{n}$.
Once we know their values (and also the equation of state), the Ward-Takahashi
identities form a closed set of equations for the energy-momentum, spin, and
flavor charge densities. This completes the phenomenological derivation of
hydrodynamics with spin polarization.

A few remarks are in order here. The present analysis uses the same strategy
as Ref. Hattori:2019lfp , but we have some modifications that originate from
the total anti-symmetry of the spin current. This difference eliminates three
dynamical spin densities attached to the boost, and, as a result, one of the
transport coefficients, the boost heat conductivity, found in Ref.
Hattori:2019lfp , disappears. Also, the power counting scheme is a little
different from that used in Ref. Gallegos:2021bzp , which makes it difficult
to give a direct comparison with their constitutive relation (see the
discussion at the end of Sec. 4.1).666 In principle, there can be second order
derivative corrections in $\Theta_{\hat{a}\hat{b}}$, which contribute to the
right-hand side of the spin-current conservation equation (17). Those would
have to be included if derivatives were our only expansion quantity, see for
example Gallegos:2021bzp . Recall, however, that we here consider the spin
relaxation rate $\Gamma_{s}$, which is counted on the same order as the
frequency because we are considering the spin hydrodynamic regime (recall
Figure 1).  Lastly, one could try to use the novel derivative expansion scheme
introduced in Ref. Li:2020eon , treating fluid vorticity and temperature
gradients differently from other dissipative first order gradients such as
shear and bulk tensors. This allows to rigorously extend the entropy-current
analysis to second or higher order terms that involve vorticity and
temperature gradients, producing new constraints for second or higher order
transport coefficients Li:2020eon . The application of this scheme to the
present analysis may, for example, justify (46) more rigorously.

##  4 Linear response theory for spin hydrodynamics

In this section, we present the linear response theory of hydrodynamics with
spin polarization. The purpose of this section is twofold: the first one is to
specify the spectrum (or dispersion relation) of the modes described by spin
hydrodynamics Kadanoff-Martin1963 . We will find that the rotational viscosity
controls the (gapped) relaxation rate of the spin density. The second one is
to find the field-theoretical expression for the rotational viscosity.
Applying the linear response theory with the background (mechanical)
perturbation Green ; Kubo ; Nakano ; Luttinger:1964zz , we provide the Green-
Kubo formula for the rotational viscosity. Since the flavour currents (e.g.
the baryon current) do not affect the dynamics of the spin density, we neglect
them in the following discussion.

###  4.1 Linear-mode analysis and non-hydrodynamic spin mode

Let us consider a spinful relativistic fluid in flat space-time, where the
Ward-Takahashi identities and the constitutive relations are simplified as

$\begin{split}&\partial_{\mu}\Theta^{\mu}_{~{}\hat{a}}=0,\\\
&\partial_{\mu}\Sigma^{\mu}_{~{}\hat{a}\hat{b}}=-(\Theta_{\hat{a}\hat{b}}-\Theta_{\hat{b}\hat{a}}),\\\
&\Theta^{\mu}_{~{}\hat{a}}=(\epsilon+p)u^{\mu}u_{\hat{a}}+p\delta^{\mu}_{\hat{a}}-\eta^{\mu~{}\nu}_{~{}\hat{a}~{}\hat{b}}\partial_{\nu}u^{\hat{b}}-(\eta_{s})^{\mu~{}\nu}_{~{}\hat{a}~{}\hat{b}}(\partial_{\nu}u^{\hat{b}}-\mu_{\nu}^{~{}\hat{b}}),\\\
&\Sigma^{\mu}_{~{}\hat{a}\hat{b}}=\varepsilon^{\mu}_{~{}\hat{a}\hat{b}\hat{c}}\sigma^{\hat{c}}.\end{split}$
(53)

It is not necessary to distinguish the curved and local Lorentz indices in
flat space-time, so we remove hats over the indices in the following analysis.
We perform the linear-mode analysis of the spin hydrodynamic equations (53)
around the global equilibrium without background flow and spin density, that
is, we consider small perturbations given by

$\epsilon(x)=\epsilon_{0}+\delta\epsilon(x),\quad v^{i}(x)=0+\delta
v^{i}(x),\quad\sigma^{\hat{a}}(x)=0+\delta\sigma^{\hat{a}}(x),$ (54)

and keep only $O(\delta)$-terms in the following analysis. In this expansion,
the four-velocity and spin densities are given by $u^{\mu}=(1,\delta
v^{i})^{t}+O(\delta^{2})$ and
$\delta\sigma^{a}=(0,\delta\sigma^{i})^{t}+O(\delta^{2})$, and the
fluctuations of all other quantities (such as the pressure) are expressed by
those in Eq. (54). Introducing
$\delta\pi_{i}\equiv\Theta^{0}_{~{}i}=(\epsilon_{0}+p_{0})\delta v_{i}$, one
finds that the linearized equations of motion become

$\begin{split}0&=\partial_{0}\delta\epsilon+\partial_{i}\delta\pi^{i},\\\
0&=\partial_{0}\delta\pi_{i}+c_{s}^{2}\partial_{i}\delta\epsilon-\gamma_{\parallel}\partial_{i}\partial^{j}\delta\pi_{j}-(\gamma_{\perp}+\gamma_{s})(\delta_{i}^{j}\bm{\nabla}^{2}-\partial_{i}\partial^{j})\delta\pi_{j}+\frac{1}{2}\Gamma_{s}\varepsilon_{0ijk}\partial^{j}\delta\sigma^{k},\\\
0&=\partial_{0}\delta\sigma_{i}+\Gamma_{s}\delta\sigma_{i}+2\gamma_{s}\varepsilon_{0ijk}\partial^{j}\delta\pi^{k},\\\
\end{split}$ (55)

where we introduced a set of static/kinetic coefficients as

$\begin{split}&c_{s}^{2}\equiv\frac{\partial
p}{\partial\epsilon},\quad\gamma_{\parallel}\equiv\frac{1}{\epsilon_{0}+p_{0}}\left(\zeta+\frac{4}{3}\eta\right),\quad\gamma_{\perp}\equiv\frac{\eta}{\epsilon_{0}+p_{0}},\\\
&\chi_{s}\delta_{ij}\equiv\frac{\partial\sigma_{i}}{\partial\mu^{j}},\quad\gamma_{s}\equiv\frac{\eta_{s}}{2(\epsilon_{0}+p_{0})},\quad\Gamma_{s}\equiv\frac{2\eta_{s}}{\chi_{s}}.\end{split}$
(56)

The $\chi_{s}$ is the spin susceptibility, and $c_{s}$ is the speed of sound.
We work in the Fourier space with
$\delta\tilde{\mathcal{O}}(k)\equiv\displaystyle{\int}\mathrm{d}^{4}x\,\,\mathrm{e}^{\mathrm{i}\omega
t-\mathrm{i}\bm{k}\cdot\bm{x}}\delta\mathcal{O}(x)$, and we choose the
momentum direction to be along the $z$-direction, $\bm{k}=(0,0,k)$. Then, the
linearized equations of motion become the block diagonal form as

$\left(\begin{array}[]{c|c}A_{\parallel}^{3\times 3}&O\\\ \hline\cr
O&A_{\perp}^{4\times
4}\end{array}\right)\delta\vec{c}=0\quad\mathrm{with}\quad\delta\vec{c}\equiv\begin{pmatrix}\delta\widetilde{\epsilon}&\delta\widetilde{\pi}_{z}&\delta\widetilde{\sigma}_{z}&\delta\widetilde{\pi}_{x}&\delta\widetilde{\sigma}_{y}&\delta\widetilde{\pi}_{y}&\delta\widetilde{\sigma}_{x}\end{pmatrix}^{t},$
(57)

where we introduce the parallel and perpendicular matrices as

$\begin{split}A_{\parallel}^{3\times
3}&\equiv\begin{pmatrix}-\mathrm{i}\omega&\mathrm{i}|\bm{k}|&0\\\
\mathrm{i}c_{s}^{2}|\bm{k}|&-\mathrm{i}\omega+\gamma_{\parallel}\bm{k}^{2}&0\\\
0&0&-\mathrm{i}\omega+\Gamma_{s}\end{pmatrix},\\\ A_{\perp}^{4\times
4}&\equiv\begin{pmatrix}-\mathrm{i}\omega+(\gamma_{\perp}+\gamma_{s})\bm{k}^{2}&\frac{\mathrm{i}}{2}\Gamma_{s}|\bm{k}|&0&0\\\
-2\mathrm{i}\gamma_{s}|\bm{k}|&-\mathrm{i}\omega+\Gamma_{s}&0&0\\\
0&0&-\mathrm{i}\omega+(\gamma_{\perp}+\gamma_{s})\bm{k}^{2}&\frac{\mathrm{i}}{2}\Gamma_{s}|\bm{k}|\\\
0&0&-2\mathrm{i}\gamma_{s}|\bm{k}|&-\mathrm{i}\omega+\Gamma_{s}\end{pmatrix}.\end{split}$
(58)

Solving the characteristic equations, $\det A_{\parallel}^{3\times 3}=0$ and
$\det A_{\perp}^{4\times 4}=0$, we find the dispersion relations for the modes
in our spin hydrodynamics.

The longitudinal modes, described by $A^{3\times 3}_{\parallel}$, decompose
into the decoupled sound and spin modes, whose dispersion relations read

$\begin{cases}\bullet~{}\mathrm{One~{}pair~{}of~{}sound~{}modes}:~{}\omega_{\mathrm{sound}}(\bm{k})=\pm
c_{s}|\bm{k}|-\dfrac{\mathrm{i}}{2}\gamma_{\parallel}\bm{k}^{2}+O(\bm{k}^{3}),\\\
\bullet~{}\mathrm{One~{}longitudinal~{}spin~{}mode}:~{}\omega_{\mathrm{spin,\parallel}}(\bm{k})=-\mathrm{i}\Gamma_{s}.\end{cases}$
(59)

It is clear that we have the relaxational longitudinal spin mode in addition
to the gapless sound mode (see Figure 2(a)). In contrast, the transverse
momentum (shear mode) and spin modes are coupled together, which gives rise to
two pairs of the following dispersion relations (see Figure 2(b)):

$\begin{split}\omega_{\mathrm{shear}}(\bm{k})&=-\dfrac{\mathrm{i}\Gamma_{s}+\mathrm{i}(\gamma_{\perp}+\gamma_{s})\bm{k}^{2}-\mathrm{i}\sqrt{\Gamma_{s}^{2}-2\Gamma_{s}(\gamma_{\perp}-\gamma_{s})\bm{k}^{2}+(\gamma_{\perp}+\gamma_{s})^{2}\bm{k}^{4}}}{2},\\\
\omega_{\mathrm{spin,\perp}}(\bm{k})&=-\dfrac{\mathrm{i}\Gamma_{s}+\mathrm{i}(\gamma_{\perp}+\gamma_{s})\bm{k}^{2}+\mathrm{i}\sqrt{\Gamma_{s}^{2}-2\Gamma_{s}(\gamma_{\perp}-\gamma_{s})\bm{k}^{2}+(\gamma_{\perp}+\gamma_{s})^{2}\bm{k}^{4}}}{2}.\end{split}$
(60)

Here, we call each branch as the shear and the transverse spin mode, because
of their low-momentum behaviors

$\begin{cases}\bullet~{}\mathrm{Two~{}shear~{}modes}:~{}\omega_{\mathrm{shear}}(\bm{k})=-\mathrm{i}\gamma_{\perp}\bm{k}^{2}+O(\bm{k}^{4}),\\\
\bullet~{}\mathrm{Two~{}transverse~{}spin~{}modes}:~{}\omega_{\mathrm{spin,\perp}}(\bm{k})=-\mathrm{i}\Gamma_{s}-\mathrm{i}\gamma_{s}\bm{k}^{2}+O(\bm{k}^{4}).\end{cases}$
(61)

From the low-momentum behaviors in (59) and (61), we identify three non-
hydrodynamic (gapped) spin modes, in addition to the hydrodynamic (gapless)
sound and shear modes. All spin modes show the relaxation behavior with the
characteristic time scale of $\Gamma_{s}^{-1}$. This non-hydrodynamic behavior
is originated from the fact that the spin angular momentum is not a conserved
quantity. The relaxation of spin modes persists until the source term in the
spin equation of motion vanishes, namely when the spin potential $\mu^{ab}$
coincides with the fluid vorticity when it reaches the local equilibrium. In
this strict hydrodynamic regime of $\omega\ll\Gamma_{s}$, we thus find only
the gapless sound and shear modes, as expected in the conventional
relativistic hydrodynamics.

On the other hand, in the regime of our spin hydrodynamics
($\omega\sim\Gamma_{s}$), three components of spin density start to behave as
independent dynamical modes. In contrast to the decoupled longitudinal spin
mode, the transverse spin modes feature an interesting coupled dynamics with
the transverse shear mode. In fact, in high momentum region, the shear and
transverse spin branches in the dispersion relations behave as

$\omega_{\mathrm{shear}}(\bm{k})\sim-\frac{\mathrm{i}\gamma_{\perp}\Gamma_{s}}{\gamma_{\perp}+\gamma_{s}}\quad\mathrm{and}\quad\omega_{\mathrm{spin,\perp}}(\bm{k})\sim-\mathrm{i}(\gamma_{\perp}+\gamma_{s})\bm{k}^{2}.$
(62)

It is notable that they interchange their momentum dependence between the low
and the high momentum regimes: the shear branch approaches to a constant
relaxation rate independent of momentum, while the spin branch shows a
quadratic dependence on momentum (see Figure 2(b)). At the cross-over momentum
scale, we see the level repulsion phenomenon between the two branches. We
should also point out that the eigenvector of each branch is a linear
combination of $\delta\pi_{x}$ and $\delta\sigma_{y}$ (or $\delta\pi_{y}$ and
$\delta\sigma_{x}$), due to the off-diagonal mixing terms in the matrix
$A_{\perp}^{4\times 4}$ in Eq. (58). This mixing is a result of the fact that
the transverse velocity gradient $\partial_{z}\delta v_{x}$ is a linear
combination of shear tensor and vorticity, and the former couples to momentum
fluctuation while the latter couples to the spin density. In particular, the
strict hydrodynamic shear mode in low momentum is described by the energy-
momentum tensor $T^{\mu}_{~{}\hat{a}}$ defined in Eq. (24), which is a linear
combination of our energy-momentum tensor $\Theta^{\mu}_{~{}\hat{a}}$ and the
spin density. In fact, the flat space-time limit of Eq. (24) gives us

$\delta\widetilde{T}^{0}_{~{}i}=\delta\widetilde{\pi}_{i}+\frac{\mathrm{i}}{2}\varepsilon_{0ijk}k^{k}\delta\sigma^{j},$
(63)

and we find that the transverse fluctuation, e.g.,
$\delta\widetilde{T}^{0}_{~{}x}$, is one of the eigenvectors in low momentum
regime satisfying

$0=\big{[}-\mathrm{i}\omega+\gamma_{\perp}k^{2}\big{]}\delta\widetilde{T}^{0}_{~{}x}+O(\bm{k}^{3}).$
(64)

This shows that $T^{0}_{~{}\hat{a}}$ describes the gapless hydrodynamic modes
in the strict hydrodynamic regime. However, we emphasize that we need
$\Theta^{\mu}_{~{}\hat{a}}$ to describe the dynamics of spin polarization,
because its anti-symmetric part appears in the equation of motion for spin
polarization.

Figure 2: Dispersion relations for (a) longitudinal modes and (b) transverse
modes, with the parameters
$(c_{s},\gamma_{\parallel},\gamma_{\perp},\Gamma_{s},\gamma_{s})=(1/\sqrt{3},0.7,0.5,0.1,0.05)$.

The zero-momentum relaxation rate of the spin modes is controlled by
$\Gamma_{s}=2\eta_{s}/\chi_{s}$, and thus, the rotational viscosity $\eta_{s}$
is essentially the spin relaxation rate. Note that the finite-momentum part of
the spin relaxation rate is different between the longitudinal and transverse
components: the former does not depend on momentum, while the latter is given
also by the rotational viscosity $\eta_{s}$ as
$\gamma_{s}\bm{k}^{2}=\frac{\eta_{s}}{2(\epsilon_{0}+p_{0})}\bm{k}^{2}$. If we
consider the next-leading-order correction of the spin current, the relaxation
rate for the longitudinal spin density also shows a momentum dependence. While
the finite-momentum dependence of the spin relaxation is an interesting
subject, we focus on the zero-momentum behavior in the remainder of this
paper. We also note that the parameter set used in Figure 2 (and also in
Figure 1) is chosen to have a well-defined spin hydrodynamic regime. This is
manifest in our choice of small $\Gamma_{s}$, compared to the scale where
hydrodynamic gradient expansion breaks down, i.e., $\omega_{c}\sim
c_{s}^{2}/\gamma_{\parallel}$. This is a required assumption that ensures the
existence of the spin hydrodynamic regime (or the spin hydro$+$ regime in
Figure 1), in which only spin modes appear as an additional non-hydrodynamic
mode in the time scale much larger than that of other non-hydrodynamic modes.
This may be justified when, e.g., there exist sufficiently heavy fermions in
the system.

Let us translate the above results into field-theoretic language. For that
purpose, we introduce the retarded Green’s function for arbitrary operators
$\hat{A}$ and $\hat{B}$ as

$\widetilde{G}_{\mathrm{R}}^{A,B}(\omega,\bm{k})\equiv\int\mathrm{d}^{4}x\,\,\mathrm{e}^{\mathrm{i}\omega
t-\mathrm{i}\bm{k}\cdot\bm{x}}\,\mathrm{i}\theta(t)\big{\langle}[\hat{A}(t,\bm{x}),\hat{B}(0,\bm{0})]\big{\rangle}_{\mathrm{eq}},$
(65)

where $\langle\hat{\mathcal{O}}\rangle_{\mathrm{eq}}$ denotes the expectation
value in homogeneous thermal equilibrium. Then, the dispersion relation (59)
and (61) implies that the retarded Green’s function for the spin density
$G_{\mathrm{R}}^{\sigma^{i}\sigma^{j}}(\omega,\bm{k}=0)$ has a pole at the
location $\omega_{\mathrm{spin}}(\bm{k}=0)=-\mathrm{i}\Gamma_{s}$. Besides,
taking into account that the zero-frequency limit of the retarded Green’s
function gives the spin susceptibility

$\lim_{\bm{k}\to
0}\widetilde{G}_{\mathrm{R}}^{\sigma_{i}\sigma_{j}}(\omega=0,\bm{k})=\chi_{s}\delta^{ij},$
(66)

we identify the low frequency-momentum behavior of the retarded Green’s
function for spin densities as

$\widetilde{G}_{\mathrm{R}}^{\sigma^{i}\sigma^{j}}(\omega,\bm{k})=\frac{\mathrm{i}\chi_{s}\Gamma_{s}+\cdots}{\omega+\mathrm{i}\Gamma_{s}+O(\bm{k}^{2})}\delta^{ij},$
(67)

where the ellipsis stands for higher-order corrections with respect to
frequency and momentum, which is beyond our first-order spin hydrodynamics. As
a result, we can directly extract the value of the spin relaxation rate
$\Gamma_{s}$, if we identify the location of the pole in the retarded spin-
spin correlator. This give one direct way to evaluate the spin relaxation rate
from the correlation function in quantum field theory.

Before closing this section, we comment on our power counting scheme. Noting
that the equation of motion for the spin density is given by Eq. (17), one
might worry whether our constitutive relation is consistent with (naive)
derivative expansion scheme. This is a natural concern, because the left-hand
side of Eq. (17) is of second order in derivatives, while the right-hand side
is of first order. However, the crucial point is that we need to distinguish
spatial derivatives from time derivatives in the spin hydrodynamic regime we
are working on, since our frequency scale is implicitly assumed to be
$\omega\gtrsim\Gamma_{s}$, and therefore, time derivative should not be
counted equally to spatial derivative. Then, there is no discrepancy in the
number of derivatives between both sides of Eq. (17): they are both
$O(\Gamma_{s}\partial)$, where $\partial$ denotes a spatial derivative. This
point will also be important when we check the consistency between the linear-
mode analysis and the Green-Kubo formula in the next section.

###  4.2 Green-Kubo formula in the first-order formalism

As shown in the previous section, the rotational viscosity $\eta_{s}$ gives
the relaxation rate of spin polarization. Our primary question is how to
evaluate it from an underlying QFT such as QCD. We formulate in this section
the linear-response theory based on the first-order formalism, which gives the
Green-Kubo formula to evaluate the spin relaxation rate from the field-theory
Green’s functions.

First of all, we note that the spin density is a non-hydrodynamic mode with a
characteristic relaxation rate $\Gamma_{s}$, and therefore, it disappears in
the strict hydrodynamic regime $\omega\ll\Gamma_{s}$ in our setup, where the
local equilibrium state does not support vorticity. Thus, when writing the
equations of spin hydrodynamics, we implicitly assume that we are working in
the frequency scale given by $\omega\gtrsim\Gamma_{s}$ (recall Figure 1). As a
result, the small frequency limit which we will use below is not the strict
$\omega\to 0$ limit, but should be regarded as a constrained limit
$\Gamma_{s}\ll\omega\ll\Gamma$, where $\Gamma$ represents the relaxation scale
of other non-hydrodynamic modes. This is in sharp contrast to the usual
$\omega\to 0$ limit used in the Green-Kubo formulae for other transport
coefficients for conserved charges.

We then introduce a mechanical perturbation Luttinger:1964zz , and identify
the rotational viscosity from the linear response of the system to the
background perturbation. The basic strategy is to compute the perturbation of
currents in first order of the background torsion, and to match the result
with the constitutive relation in our spin hydrodynamics, (48) or (51). From
the first equation of (51), one sees that the bulk and shear viscosities are
captured by the torsion-free spin connection, which results in the usual
Green-Kubo formula for them. Thus, we focus on the rotational viscosity in the
following discussion.

The vital point for the subsequent discussion is to identify Eqs. (48) or (51)
as a non-equilibrium expectation value of the quantum operator
$\langle\delta\hat{\Theta}^{\mu}_{~{}\hat{a}}\rangle$ in a torsionful curved
background. We here use the vierbein and contorsion as independent sources,
while it is also possible to use the vierbein and spin connection, instead.
Similarly to the previous section, we assume that the system is initially in
homogeneous thermal equilibrium with $\beta=\mathrm{const.}$,
$u^{\hat{a}}=(1,\bm{0})^{t}$, and $\mu_{\mu}^{~{}\hat{a}}=0$. On the time
scale of $\omega\gg\Gamma_{s}$, the spin chemical potential
$\mu_{\mu}^{~{}\hat{a}}$ remains zero even after introducing background
perturbations, as we explained in the previous paragraph. This simplifies the
constitutive relation for the anti-symmetric part of the energy-momentum
tensor as

$\begin{split}\langle\delta\hat{\Theta}^{\mu}_{~{}\hat{a}}\big{|}_{(a)}\rangle_{e,K}&=-\frac{1}{2}\eta_{s}(\Delta^{\mu\nu}\Delta_{\hat{a}\hat{b}}-\Delta^{\mu}_{\hat{b}}\Delta_{\hat{a}}^{\nu})\big{[}{\mathring{\omega}}_{\nu~{}\hat{0}}^{~{}\hat{b}}(e)-K_{\hat{0}\nu}^{~{}~{}\hat{b}}\big{]}.\end{split}$
(68)

This equation serves as a basis to obtain the Green-Kubo formula for the
rotational viscosity $\eta_{s}$. Below, we present two ways to compute the
rotational viscosity, whose equivalence will be shown by the Ward-Takahashi
identity.

Firstly, we turn off the background vierbein by putting
$e_{\mu}^{~{}\hat{a}}=\delta_{\mu}^{\hat{a}}$, while keeping the non-vanishing
contorsion, so that one has only the second term in Eq. (68). On the other
hand, we can also compute the same
$\langle\delta\hat{\Theta}^{\mu}_{~{}\hat{a}}\rangle_{e.K}$ in a different
manner based on QFT. To describe this in the underlying quantum theory, we
start from the initial density operator at past infinity $t=-\infty$ given by
the equilibrium Gibbs distribution

$\hat{\rho}_{\mathrm{eq}}=\frac{1}{Z(\beta)}\mathrm{e}^{-\beta\hat{H}}\quad\mathrm{with}\quad
Z(\beta)\equiv\mathop{\mathrm{Tr}}\mathrm{e}^{-\beta\hat{H}},$ (69)

where $\hat{H}$ denotes the Hamiltonian of, e.g., QCD in the absence of
background fields. Then, noting that Eq. (68) indicates that the contorsion
$K_{\hat{0}\nu}^{~{}~{}\hat{b}}$ plays a role of a relevant background field,
we perturb the system by adiabatically turning on the small contorsion and
compute the induced response of the anti-symmetric part of
$\langle\hat{\Theta}^{\mu}_{~{}\hat{a}}\rangle$. Since the contorsion couples
to the spin current in the microscopic field theory, one can describe the
situation by considering the time-evolution generated by a time-dependent
Hamiltonian

$\hat{H}_{\mathrm{tot}}(t)=\hat{H}+\hat{H}_{\mathrm{ext}}(t)\quad\mathrm{with}\quad\hat{H}_{\mathrm{ext}}(t)\equiv\frac{1}{2}\int\mathrm{d}^{3}xK_{\hat{0}\nu}^{~{}~{}\hat{b}}(t,\bm{x})\hat{\Sigma}^{\hat{0}\nu}_{~{}~{}\hat{b}}(t,\bm{x}).$
(70)

We shall then evaluate the first-order perturbation induced by
$\hat{H}_{\mathrm{ext}}(t)$. A usual computation easily gives the energy-
momentum tensor $\langle\hat{\Theta}^{\mu}_{~{}\hat{a}}(x)\rangle_{K}$ as

$\displaystyle\langle\hat{\Theta}^{\mu}_{~{}\hat{a}}(x)|_{(a)}\rangle_{K}$
$\displaystyle=\langle\hat{\Theta}^{\mu}_{~{}\hat{a}}(x)|_{(a)}\rangle_{\mathrm{eq}}-\frac{\mathrm{i}}{2}\int_{-\infty}^{\infty}\mathrm{d}t^{\prime}\mathrm{d}^{3}x^{\prime}\theta(t-t^{\prime})\big{\langle}[\hat{\Theta}^{\mu}_{~{}\hat{a}}(x)|_{(a)},\hat{\Sigma}^{\hat{0}\nu}_{~{}~{}\hat{b}}(x^{\prime})]\big{\rangle}_{\mathrm{eq}}K_{\hat{0}\nu}^{~{}~{}\hat{b}}(x^{\prime})$
$\displaystyle=\langle\hat{\Theta}^{\mu}_{~{}\hat{a}}(x)|_{(a)}\rangle_{\mathrm{eq}}-\frac{1}{2}\int_{-\infty}^{\infty}\mathrm{d}t^{\prime}\mathrm{d}^{3}x^{\prime}G_{\mathrm{R}}^{\hat{\Theta}^{\mu}_{~{}\hat{a}}|_{(a)},\Sigma^{\hat{0}\nu}_{~{}~{}\hat{b}}}(x-x^{\prime})K_{\hat{0}\nu}^{~{}~{}\hat{b}}(x^{\prime}),$
(71)

where we use the definition of the retarded Green’s function (65) in the
second line. The first term with the subscript “eq” on the right-hand side
represents the equilibrium expectation value in the absence of the background
contorsion. Thus, after applying the Markov approximation, which essentially
means a small frequency limit, we find the linear response result for
$\langle\delta\hat{\Theta}^{\mu}_{~{}\hat{a}}(x)|_{(a)}\rangle=\langle\hat{\Theta}^{\mu}_{~{}\hat{a}}(x)|_{(a)}\rangle_{K}-\langle\hat{\Theta}^{\mu}_{~{}\hat{a}}(x)|_{(a)}\rangle_{\mathrm{eq}}$
as

$\begin{split}\langle\delta\hat{\Theta}^{\mu}_{~{}\hat{a}}(x)|_{(a)}\rangle\simeq-\frac{1}{2}\lim_{\Gamma_{s}\ll\omega\ll\Gamma}\lim_{\bm{k}\to
0}\widetilde{G}_{\mathrm{R}}^{\Theta^{\mu}_{~{}\hat{a}}|_{(a)},\Sigma^{\hat{0}\nu}_{~{}~{}\hat{b}}}(\omega,\bm{k})K_{\hat{0}\nu}^{~{}~{}\hat{b}}(x)\,.\end{split}$
(72)

Note that we use the constrained limit,
$\displaystyle{\lim_{\Gamma_{s}\ll\omega\ll\Gamma}}$, that was discussed
previously, since in the strict $\omega\to 0$ limit, both sides are expected
to vanish due to relaxation dynamics of spin density. By comparing this result
with Eq. (68), we find the rotational viscosity given by

$\eta_{s}=-\lim_{\Gamma_{s}\ll\omega\ll\Gamma}\lim_{\bm{k}\to
0}\widetilde{G}_{\mathrm{R}}^{\Theta^{x}_{~{}\hat{y}}|_{(a)},\Sigma^{\hat{0}x}_{~{}~{}\hat{y}}}(\omega,\bm{k}).$
(73)

This gives one Green-Kubo formula for the rotational viscosity.

As we mentioned previously, there is another way to compute the rotational
viscosity. This is because Eq. (68) also involves the vierbein through the
torsion-free spin connection
${\mathring{\omega}}_{\nu~{}\hat{0}}^{~{}\hat{b}}(e)$. Thus, considering the
linear response induced by the anti-symmetric part of the vierbein instead of
the contorsion, we can also compute the rotational viscosity. However, we need
to be careful at this point because the operator coupled to the vierbein is
not $\hat{\Theta}^{\mu}_{~{}\hat{a}}$ but $\hat{T}^{\mu}_{~{}\hat{a}}$ defined
in Eq. (24) when we treat the vierbein and contorsion as independent sources.
This second way to compute the rotational viscosity suggests that we can use
the torsion-free background to compute the rotational viscosity, but the
identification of $\Theta^{\mu}_{~{}\hat{a}}$ will be a little complicated
(see also Appendix A).

We can, however, derive the same result while avoiding such a direct
computation with the vierbein perturbation. To do so, we simply need to use
the Ward-Takahashi identity for the spin current. In flat space-time, we find
the following identity in the Fourier space:

$-\mathrm{i}\omega\widetilde{\Sigma}^{\hat{0}x}_{~{}~{}\hat{y}}(\omega,\bm{k})+\mathrm{i}k\widetilde{\Sigma}^{zx}_{~{}~{}\hat{y}}(\omega,\bm{k})=-2\widetilde{\Theta}^{x}_{~{}\hat{y}}\big{|}_{(a)}(\omega,\bm{k}).$
(74)

Substituting this relation into Eq. (73), we obtain another Green-Kubo formula
for the rotational viscosity:

$\eta_{s}=2\lim_{\Gamma_{s}\ll\omega\ll\Gamma}\lim_{\bm{k}\to
0}\frac{1}{\omega}\mathop{\mathrm{Im}}\widetilde{G}_{\mathrm{R}}^{\Theta^{x}_{~{}\hat{y}}|_{(a)},\Theta^{x}_{~{}\hat{y}}|_{(a)}}(\omega,\bm{k}).$
(75)

This looks more familiar because it takes a similar form to the Green-Kubo
formula for the shear and bulk viscosities.

We can show that the constrained limit
$\displaystyle{\lim_{\Gamma_{s}\ll\omega\ll\Gamma}}$ is indeed necessary in
these Green-Kubo formulas. To see this, using the Ward-Takahashi identity
(74), we can replace $\Theta^{x}_{~{}\hat{y}}$ in the first Green-Kubo formula
(73) with the spin density, which leads to another expression for the
rotational viscosity:

$\eta_{s}=\frac{1}{2}\lim_{\Gamma_{s}\ll\omega\ll\Gamma}\lim_{\bm{k}\to
0}\omega\mathop{\mathrm{Im}}\widetilde{G}_{\mathrm{R}}^{\Sigma^{\hat{0}x}_{~{}~{}\hat{y}},\Sigma^{\hat{0}x}_{~{}~{}\hat{y}}}(\omega,\bm{k}).$
(76)

This equation is consistent with the result of the linear-mode analysis given
in Eq. (67) (recall $\sigma^{z}=\Sigma^{\hat{0}x}_{~{}~{}\hat{y}}$ so that
$\widetilde{G}_{\mathrm{R}}^{\sigma^{z}\sigma^{z}}(\omega,\bm{k})=\widetilde{G}_{\mathrm{R}}^{\Sigma^{\hat{0}x}_{~{}~{}\hat{y}},\Sigma^{\hat{0}x}_{~{}~{}\hat{y}}}(\omega,\bm{k})$).
On the other hand, Eq. (67) tells us that the strict $\omega\to 0$ limit of
$\omega\mathop{\mathrm{Im}}\widetilde{G}_{\mathrm{R}}^{\Sigma^{\hat{0}x}_{~{}~{}\hat{y}},\Sigma^{\hat{0}x}_{~{}~{}\hat{y}}}(\omega,\bm{k})$
does not give the rotational viscosity, because it simply vanishes.

While the above Green-Kubo formula utilizes the constrained limit, the strict
$\omega\to 0$ limit can also provide yet another formula for the rotational
viscosity. Taking the imaginary part of Eq. (67), it is straightforward to
obtain

$\frac{\chi_{s}^{2}}{2\eta_{s}}=\lim_{\omega\to
0}\frac{1}{\omega}\mathop{\mathrm{Im}}\widetilde{G}_{\mathrm{R}}^{\Sigma^{\hat{0}x}_{~{}~{}\hat{y}},\Sigma^{\hat{0}x}_{~{}~{}\hat{y}}}(\omega,\bm{k}=0).$
(77)

We emphasize that $\displaystyle{\lim_{\omega\to 0}}$ in this equation is the
usual zero-frequency limit. Interestingly, the rotational viscosity $\eta_{s}$
appears in the denominator of the equation, in contrast to the other Green-
Kubo formula that we derive in this section. This can be understood from the
requirement $\omega\ll\Gamma_{s}$ in this limit, which probes infrared
singularities that should be regularized by a finite $\Gamma_{s}$. The Eq.
(77) may provide another practical way to evaluate the rotational viscosity in
quantum field theory.

##  5 Summary and Outlook

In this paper, we addressed several theoretical issues in relativistic
hydrodynamics with spin polarization. First, we clarified the definition of
the energy-momentum tensor and spin current for quantum field theories with
Dirac fermions (such as QED and QCD) based on the first-order formalism in a
background geometry with torsion. The resulting spin current is totally anti-
symmetric with respect to its three indices, and we performed the consistent
entropy current analysis to derive the first-order constitutive relations in
the torsionful background. The total anti-symmetry of the spin current leads
to three dynamical degrees of freedom arising from spin polarization, and we
find one kinetic coefficient known as the rotational viscosity, which controls
the relaxation rate for the spin density. Second, we presented a number of
ways to evaluate the spin relaxation rate, or equivalently the rotational
viscosity, from the retarded correlation functions of spin-related operators;
the several Green-Kubo formulae for the rotational viscosity are derived with
the help of mechanical perturbations described by the time-dependent
Hamiltonian. We also point out a subtlety of the low-frequency limit in the
obtained Green-Kubo formula, which originates from the non-conservation of
spin angular momentum that is inherent in relativistic spin hydrodynamics.

Although we presented a rigorous framework for the first-order relativistic
spin hydrodynamics, we must also point out the underlying assumptions for its
validity, which have not been clearly explained in the previous literature. As
was already pointed out in the old paper by Martin, Parodi, and Pershan
Martin1972 , we cannot model-independently assume that the relaxational spin
mode is in the near hydrodynamic regime. In other words, we cannot always find
the regime for the spin hydrodynamics we discuss, which is defined by
$\Gamma_{s}\ll\omega\ll\Gamma$, while we use this assumption to derive the
Green-Kubo formula for the rotational viscosity. This is in sharp contrast to
the strict hydrodynamic modes whose gapless property is protected by
conservation laws, so that the presence of the hydrodynamic regime is always
guaranteed in the $\bm{k}\to 0$ limit. There is a certain similarity between
spin hydrodynamics and Müller-Israel-Stewart theory of second-order
relativistic hydrodynamics Muller:1967zza ; Israel:1976tn ; Israel:1976-2 ;
Israel:1979wp  in that both introduce additional degrees of freedom. Neither
of these two theories could be a universal effective description unless the
additional degrees of freedom are parametrically slow. However, when an
approximate spin symmetry emerges by considering, e.g., a large mass limit for
fermions (e.g. charm quarks in QCD, and protons and/or electrons in QED),
$\Gamma_{s}^{\mathrm{heavy}}$ for heavy-fermion spin is expected to be
parametrically smaller than $\Gamma$ for other non-hydrodynamic modes, which
guarantees the existence of the spin hydrodynamic regime.

As an outlook let us point out a few research directions along the lines of
the present paper. The most important one is to evaluate the rotational
viscosity (or the spin relaxation rate) for QCD or gauge theories. The Green-
Kubo formula for the rotational viscosity derived in this paper enables us to
compute the spin relaxation rate in the underlying quantum field theory. For
that purpose, similarly to the computation of the shear viscosity, we may rely
on thermal perturbation theory for the weakly-coupled regime of QCD
Arnold:2000dr ; Arnold:2003zc , or the AdS/CFT correspondence for the
strongly-coupled limit of QCD-like theories Policastro:2001yc ;
Policastro:2002se . The results would have direct implications for the
theoretical study of spin polarization in the QGP created in relativistic
heavy-ion collision experiments.

Our analysis in this paper employs the entropy-current analysis to derive the
constitutive relations. Another interesting direction is to explore the non-
equilibrium statistical operator method Nakajima ; Mori1 ; McLennan ;
McLennan1 ; Zubarev1979 ; Zubarev1 ; Zubarev2 ; Kawasaki-Gunton ; Huang:2011dc
; Sasa2014 ; Becattini:2014yxa ; Hayata:2015lga ; Becattini:2019dxo ;
Hongo:2020qpv . This will provide a complementary starting point to discuss
spin transport with the help of thermal perturbation theory, which describes
the relaxation process based on density operators (see, e.g., Ref. Hu:2021lnx
for a recent attempt to derive the Green-Kubo formula with the statistical
operator method). Besides, it will also be interesting to investigate possible
anomalous transport phenomena in the spin transport, with the help of the
path-integral formulation of the local thermodynamic functional Hongo:2016mqm
. In fact, non-dissipative anomalous currents such as the chiral magnetic and
vortical effects Fukushima:2008xe ; Erdmenger:2008rm ; Banerjee:2008th ;
Son:2009tf ; Landsteiner:2011cp ; Landsteiner:2016led ; Kharzeev:2020jxw  are
captured by the local thermodynamic functional, or equivalently called the
hydrostatic partition function Banerjee:2012iz ; Jensen:2012jh ; Jensen:2012jy
; Jensen:2013kka ; Golkar:2015oxw ; Chowdhury:2016cmh ; Glorioso:2017lcn ;
Manes:2018llx ; Hongo:2019rbd ; Manes:2019fyw . Guided by the result obtained
in Ref. Hongo:2016mqm , an extra amount of spin density, which corresponds to
the deviation of the spin potential from its equilibrium value of thermal
vorticity, would lead to the presence of thermal torsion emerging in the
imaginary-time formalism. We speculate that this geometric description has a
promising potential to provide a solid basis for systematic investigation of
possible anomalous transport phenomena associated with the spin current. We
leave all these interesting questions to future work.

###### Acknowledgements.

We thank Casey Cartwright, Markus Garbiso, Yoshimasa Hidaka, Marco Knipfer,
Yu-Chen Liu, and Yuya Tanizaki for useful discussions. This work is supported
by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics
under Award Number DE-FG0201ER41195, and within the framework of the Beam
Energy Scan Theory (BEST) Topical Collaboration, and also partially by RIKEN
iTHEMS Program (in particular iTHEMS Non-Equilibrium Working group). X. G. H
is supported by NSFC under Grant No. 12075061 and Shanghai NSF under Grant No.
20ZR1404100. This work is supported, in part, by the U.S. Department of Energy
grant DE-SC0012447.

##  Appendix A Currents in metric-affine connection formulation

The vierbein-spin connection formulation used in the main text is a useful
tool to define the spin current and the canonical energy-momentum tensor.
Since our energy-momentum tensor has a possible anti-symmetric component, one
may be interested in how it appears in the formulation relying on the metric.
We here provide a way to define the currents used in this paper with a help of
the first-order metric-connection formulation (see also, e.g., Appendix G of
Ref. Jensen:2013kka ).

Let us now use the metric $g_{\mu\nu}$ and affine connection
$\Gamma^{\mu}_{~{}\nu\rho}$, instead of $e_{\mu}^{~{}\hat{a}}$ and
$\omega_{\mu~{}b}^{~{}\hat{a}}$, as independent backgrounds, which also allows
a possible contribution from the non-vanishing torsion. While QCD enjoys
diffeomorphism invariance as $\delta_{\xi}\mathcal{S}_{\mathrm{QCD}}=0$ even
if we use these backgrounds, its variation with respect to the metric and
affine connection leads to a different set of currents. We then introduce the
following currents as an alternative to Eq. (11):

$t^{\mu\nu}(x)\equiv\frac{2}{\sqrt{-g(x)}}\frac{\delta\mathcal{S}_{\mathrm{QCD}}}{\delta
g_{\mu\nu}(x)},\quad
s^{\mu\nu}_{~{}~{}\rho}(x)\equiv\frac{1}{\sqrt{-g(x)}}\frac{\delta\mathcal{S}_{\mathrm{QCD}}}{\delta\Gamma^{\rho}_{~{}\mu\nu}(x)}.$
(78)

By noting Eqs. (4), one can find their relations to
$\Theta^{\mu}_{~{}\hat{a}}$ and $\Sigma^{\mu}_{~{}\hat{a}\hat{b}}$ in Eq.
(11). In fact, Eqs. (4) allows us to express variations of the metric and
affine connection as

$\begin{split}\delta
g_{\mu\nu}&=\eta_{\hat{a}\hat{b}}\big{(}e_{\nu}^{~{}\hat{b}}\delta
e_{\mu}^{~{}\hat{a}}+e_{\mu}^{~{}\hat{a}}\delta
e_{\nu}^{~{}\hat{b}}\big{)},\\\
\delta\Gamma^{\rho}_{~{}\mu\nu}&=e_{\hat{a}}^{~{}\rho}\partial_{\mu}\delta
e_{\nu}^{~{}\hat{a}}-\Gamma^{\lambda}_{~{}\mu\nu}e_{\hat{a}}^{~{}\rho}\delta
e_{\lambda}^{~{}\hat{a}}+\omega_{\mu~{}\hat{b}}^{~{}\hat{a}}e_{\hat{a}}^{~{}\rho}\delta
e_{\nu}^{~{}\hat{b}}+e_{\nu}^{~{}\hat{b}}e_{\hat{a}}^{~{}\rho}\delta\omega_{\mu~{}\hat{b}}^{~{}\hat{a}}.\end{split}$
(79)

Thus, the variation of the QCD action induced by diffeomorphism is expressed
as follows:

$\displaystyle\delta_{\xi}\mathcal{S}_{\mathrm{QCD}}$
$\displaystyle=\int\mathrm{d}^{4}x\left[\frac{\delta\mathcal{S}_{\mathrm{QCD}}}{\delta
g_{\mu\nu}}\delta_{\xi}g_{\mu\nu}+\frac{\delta\mathcal{S}_{\mathrm{QCD}}}{\delta\Gamma^{\mu}_{~{}\nu\rho}}\delta_{\xi}\Gamma^{\mu}_{~{}\nu\rho}+\frac{\delta\mathcal{S}_{\mathrm{QCD}}}{\delta
q}\delta_{\xi}q+\frac{\delta\mathcal{S}_{\mathrm{QCD}}}{\delta\bar{q}}\delta_{\xi}\bar{q}+\frac{\delta\mathcal{S}_{\mathrm{QCD}}}{\delta
A_{\mu}}\delta_{\xi}A_{\mu}\right]$
$\displaystyle=\int\mathrm{d}^{4}x\sqrt{-g}\bigg{[}\Big{(}t^{\mu\nu}e_{\nu\hat{a}}-(D_{\lambda}-\mathcal{G}_{\lambda})s^{\lambda\mu}_{~{}~{}\hat{a}}\Big{)}\delta_{\xi}e_{\mu}^{~{}\hat{a}}-\frac{1}{2}(s^{\mu}_{~{}\hat{a}\hat{b}}-s^{\mu}_{~{}\hat{b}\hat{a}})\delta_{\xi}\omega_{\mu}^{~{}\hat{a}\hat{b}}\bigg{]},$
(80)

from which we find the relation between the two definitions of currents, (11)
and (78), as

$\Theta^{\mu\nu}=t^{\mu\nu}-(\nabla_{\lambda}-\mathcal{G}_{\lambda})s^{\lambda\mu\nu},\quad\Sigma^{\mu\nu\rho}=s^{\mu\nu\rho}-s^{\mu\rho\nu}.$
(81)

This relation allows us to investigate dynamics of $\Theta^{\mu\nu}$ and
$\Sigma^{\mu\nu\rho}$, using $t^{\mu\nu}$ and $s^{\lambda\mu\nu}$ that are
defined by the variation with respect to the metric and affine connection.

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