# Shear viscosity of rotating, hot, and dense spin-half fermionic systems from
quantum field theory

Sarthak Satapathy<EMAIL_ADDRESS>Department of Physics, Indian
Institute of Technology Roorkee, Roorkee - 247667, India  Rajeev Singh
<EMAIL_ADDRESS>Center for Nuclear Theory, Department of Physics
and Astronomy, Stony Brook University, Stony Brook, New York, 11794-3800, USA
Pushpa Panday<EMAIL_ADDRESS>Department of Physics, Indian
Institute of Technology Roorkee, Roorkee - 247667, India  Salman Ahamad Khan
<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology Roorkee, Roorkee - 247667, India  Debarshi Dey<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology Roorkee, Roorkee -
247667, India

###### Abstract

In this study, we calculate the shear viscosity for rotating fermions with
spin-half under conditions of high temperature and density. We employ the Kubo
formalism, rooted in finite-temperature quantum field theory, to compute the
field correlation functions essential for this evaluation. The one-loop
diagram pertinent to shear viscosity is analyzed within the context of curved
space, utilizing tetrad formalism as an effective approach in cylindrical
coordinates. Our findings focus on extremely high angular velocities, ranging
from 0.1 to 1 GeV, which align with experimental expectations. Furthermore, we
explore the interrelationship between the chemical potential and angular
velocity within the scope of this study.

##  I Introduction

The study of Quark Gluon Plasma (QGP), formed during heavy-ion collisions, has
been a focal point of scientific inquiry [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
12], especially when exposed to high electromagnetic fields [13, 14, 15] or
considerable angular velocities [12, 16, 17]. Initial formation phases of off-
center collisions lead to high angular velocities, documented by the STAR
collaboration, with an estimated angular momentum on the order of $10^{3\sim
5}\hbar$, adding a layer of complexity to the exploration of these extreme
states of matter [12].

Significant advances have been made in understanding the hot and dense state
of nuclear matter under extreme conditions via experiments at RHIC and LHC.
These experiments have shown that the QGP evolves quickly from an initial
glasma state to an equilibrated phase, ultimately recombining into hadrons [5,
6, 8, 7]. Theoretical frameworks, particularly the theory of relativistic
hydrodynamics, have corroborated that the QGP behaves as the smallest fluid
droplets ever observed, exhibiting almost perfect fluidity [18, 19, 20, 21,
22]. This has ushered in new theoretical avenues, including developments in
off-equilibrium hydrodynamics [23].

In addition, the STAR collaboration’s measurements show that emitted
particles, such as $\Lambda$ hyperons, exhibit spin polarization, making the
QGP the most vortical system ever observed in heavy-ion collisions [24, 25,
26, 27, 28, 29, 30]. This spin alignment aligns with their angular momentum,
sparking further interest in the underlying dynamics. Numerous studies have
explored these dynamics using different approaches [31, 32, 33, 34, 35, 36,
37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 16, 52, 53, 54,
55, 56, 57, 58, 59, 46, 49, 60, 61, 43, 44, 45, 62, 63, 64, 65, 66, 67, 68,
47, 69, 50, 70, 71, 72, 73, 74, 75, 76, 66, 77, 78, 79, 48, 49, 80, 81, 82,
83]. Recent reviews have delved into both local and global spin polarization,
and spin hydrodynamics from holographic and quantum field theoretical
perspectives have been covered, further complementing the field of study.

Recent trends also encompass the investigation of spin polarization in nuclear
matter and its interplay with electromagnetic fields. Newly acquired
measurements have not only enriched our understanding of the QGP but have also
spurred a slew of theoretical advances. Thus, the study of QGP, especially
under extreme conditions, remains an exciting and rapidly evolving field of
research.

Theoretical studies have anticipated many interesting phenomena to occur in
this condition, such as chiral vortical effect [32, 84, 85, 86], chiral
vortical wave [87], splitting of masses under rotation [88], magnetic chiral
density wave [89], vortical effects in AdS space [90] etc. These are anomalous
processes which depend on the experimental signatures and can be observed in
heavy-ion collision experiments. Under the influence of large angular
velocities, the QCD medium offers the scope to investigate anomalous
properties in greater detail as they will be properly exhibited, thus allowing
an easier extraction of the experimental signatures [91]. The study of QCD
phase diagram [12, 92, 93] under rotation is another interesting topic of
research. The effect of rotation has been studied on the confining and
deconfining phases of QGP in Ref. [92], where it has been reported that the
deconfining transition is observed at a particular distance from the rotation
axis. There has been a reporting of a mixed inhomogeneous phase, which
contains spatially separated confinement and deconfinement regions, thus two
deconfining temperatures are predicted to be present. Quantities such as the
isothermal moment of inertia for a rigidly rotating QGP has been calculated in
the context of lattice SU(3) gauge theory [94]. It has been found that the
moment of inertia assumes negative values below the supervortical temperature
[94], which is 1.5 times the critical temperature. This observation indicates
that a rigidly rotating system is thermodynamically unstable beyond a certain
temperature.

One of the areas of the study of rotating QCD matter is transport coefficients
[65], which serve as inputs in hydrodynamic simulations. Various approaches
such as kinetic theory [42, 41], based on the relativistic Boltzmann equation,
and field theoretic based formulations [80, 95, 96, 97, 98], which depend on
the calculation of correlation functions, provide a scope for calculating the
transport coefficients. Recently, Kubo formulas for various transport
coefficients appearing in first-order spin hydrodynamics, calculated from
Zubarev’s non-equilibrium statistical operator formalism [95, 96], have been
reported [65], where two new transport coefficients viz. rotational viscosity
and boost heat conductivity are found to be present in addition to the shear
viscosity, bulk viscosity, and thermal conductivity. A rigorous treatment of
the Kubo formulas has been done in Ref. [80] by studying the QCD action and
its coupling with the background torsion. It is particularly important in
understanding the role of rotational effects of the medium which induce a
background torsion, that couples with the fermionic fields. The coupling of
gauge fields with torsion does not occur owing to the SU(3)${}_{\text{c}}$
gauge invariance of the action [99, 80]. The effect of rotation thus
translates into a field theory with background torsion, reminiscent of the
situation of background magnetic field [100, 101, 102, 103]. In such a
situation, the propagator loses its property of translational symmetry between
two points in spacetime. For magnetic fields, this appears as the Schwinger
phase factor [100, 101, 102, 103] which is a exponential function depicting a
phase shift. A similar factor appears in the propagator of fermions in a
rotating environment [104], but the translational symmetry is recovered in the
limit of large angular velocities, allowing a easier computation of the
correlation function of fields [104].

In the present study, we delve into the intricacies of how finite rotation
affects transport coefficients in a medium, specifically focusing on shear
viscosity in a fermionic system under the influence of high angular velocity.
Using the Kubo formalism, we integrate this impact into our calculations by
utilizing the momentum space propagator for the fermion, obtained through the
Fock-Schwinger approach. The role of angular velocity becomes evident in two
key aspects: it influences the distribution function via Matsubara frequency
summation and also appears in the numerator of the expression for shear
viscosity through trace evaluations. Preliminary results indicate that both
temperature and angular velocity contribute to an increase in shear viscosity,
while a rise in chemical potential leads to its decrease. Intriguingly, the
study reveals that the chemical potential and angular velocity, despite being
functionally equivalent, have opposing effects on shear dissipation in
fermionic systems.

The layout of this paper is organized in the following manner: Section II
delves into the dynamics of fermions within a rotating setting. Subsequently,
in section III, we present calculations pertaining to the spectral function
and the shear viscosity of fermions. Section IV is dedicated to a discussion
of our study’s findings and we finally summarize in sec. V.

##  II Fermions in a rotating medium

In this section, we provide a concise introduction to the behavior of fermions
in a rotating setting. To investigate this complex system, our approach will
be based in curved spacetime. The metric tensor, akin to that of a curved
spacetime, serves as a useful tool for characterizing the geometric properties
of the region generated following non-central collisions, which rotates at an
angular velocity $\Omega$ about the $z$-axis. For this specific system, the
metric tensor $g^{\mu\nu}$ is represented by

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

The Dirac equation describing a massive fermion with mass $m$ in cylindrical
coordinates [104, 105] is expressed as

$\displaystyle\left(i\widetilde{\gamma}^{\mu}\widetilde{D}_{\mu}-m\right)\psi=0\,,$
(2)

where $\widetilde{\gamma}^{\mu}$ are the gamma matrices and
$\widetilde{D}_{\mu}$ is the covariant derivative

$\widetilde{D}_{\mu}=\partial_{\mu}+\Gamma_{\mu}\,,$ (3)

where $\Gamma_{\mu}=(1/8)\,\omega_{\mu ab}\big{[}\gamma^{a},\gamma^{b}\big{]}$
is the affine connection with $\omega_{\mu ab}$ being the spin connection. To
compute $\omega_{\mu ab}$ one has to use the definition of vierbein (also
known as tetrad) and metric tensor $g_{\mu\nu}$ which are given as

$\displaystyle e_{a}^{~{}\mu}$ $\displaystyle=$
$\displaystyle\delta_{a}^{~{}\mu}-\delta_{a}^{~{}0}\delta_{i}^{~{}\mu}v_{i}\,,$
$\displaystyle e^{a}_{~{}\mu}$ $\displaystyle=$
$\displaystyle\delta^{a}_{~{}\mu}+\delta^{a}_{~{}i}\delta^{0}_{~{}\mu}v_{i}\,,$
$\displaystyle g_{\mu\nu}$ $\displaystyle=$
$\displaystyle\eta_{ab}e^{a}_{~{}\mu}e^{b}_{~{}\nu}\,,$ (4)

with $(i=1,2,3)$. Thus,

$\displaystyle\omega_{\mu
ab}=g_{\alpha\beta}e_{a}^{~{}\alpha}\left[\partial_{\mu}e_{b}^{~{}\beta}+\Gamma^{\beta}_{~{}\mu\nu}e_{b}^{~{}\nu}\right],$
(5)

where $\Gamma^{\beta}_{~{}\mu\nu}$ is the Christoffel symbol

$\displaystyle\Gamma^{\beta}_{~{}\mu\nu}=\frac{g^{\beta\alpha}}{2}\left[\partial_{\nu}g_{\alpha\mu}+\partial_{\mu}g_{\alpha\nu}-\partial_{\alpha}g_{\mu\nu}\right].$
(6)

In Eq. (4), $\eta_{ab}=\text{diag}(1,-1,-1,-1)$ is the Minkowski spacetime
metric tensor. In curved spacetime the energy-momentum tensor is given as
[105, 106, 107]

$\displaystyle
T^{\mu\nu}=\frac{i}{4}\left(\overline{\psi}\widetilde{\gamma}^{\mu}\widetilde{D}^{\nu}\psi+\overline{\psi}\widetilde{\gamma}^{\nu}\widetilde{D}^{\mu}\psi\right)+\rm{h.c},$
(7)

where $\psi$ and $\overline{\psi}$ are the Dirac field operator and its
conjugate, respectively, and h.c denotes the Hermitian conjugate. In a
uniformly rotating frame, $\widetilde{D}^{\mu}$ is

$\displaystyle\widetilde{D}_{\mu}=\left(\partial_{t}-i\frac{\Omega\Sigma_{3}}{2},-\partial_{x},-\partial_{y},-\partial_{z}\right),$
(8)

where $\Sigma_{3}=(i/2)\left[\gamma^{1},\gamma^{2}\right]$. The spacetime
dependent gamma matrices $\widetilde{\gamma}^{\mu}$ [105, 104] in tetrad
system present in Eq. (7) are defined as

$\displaystyle\widetilde{\gamma}^{0}=\gamma^{0}\,,\quad\quad\quad~{}\quad\widetilde{\gamma}_{0}=\gamma^{0}-x\Omega\gamma^{2}+y\Omega\gamma^{1}\,,$
$\displaystyle\widetilde{\gamma}^{1}=\gamma^{1}+y\Omega\gamma^{0}\,,\quad\widetilde{\gamma}_{1}=-\gamma^{1}\,,$
$\displaystyle\widetilde{\gamma}^{2}=\gamma^{2}-x\Omega\gamma^{0}\,,\quad\widetilde{\gamma}_{2}=-\gamma^{2}\,,$
$\displaystyle\widetilde{\gamma}^{3}=\gamma^{3}\,,\quad\quad\quad~{}~{}\quad\widetilde{\gamma}_{3}=-\gamma^{3}\,,$
(9)

where $\gamma^{\mu}$ are the gamma matrices in Minkowski space. In off-central
heavy-ion collisions the direction of rotation is perpendicular to the
reaction $(x-y)$ plane. Following the above mentioned details, the Lagrangian
[108] with finite chemical potential $\mu$ for a medium rotating with constant
angular velocity $\Omega$ is expressed as

$\displaystyle\mathcal{L}=\overline{\psi}\left[i\gamma^{\mu}\partial_{\mu}+\gamma^{0}\left(\Omega
J_{z}+\mu\right)-m\right]\psi\,,$ (10)

with $J_{z}$ being the third component of the total angular momentum given by
$\vec{J}=\vec{x}\times\vec{p}+\vec{S}$ where
$\vec{S}=(1/2)\begin{pmatrix}\vec{\sigma}&0\\\ 0&\vec{\sigma}\end{pmatrix}$.
The fermion propagator [104] in the momentum space derived from the above
Lagrangian by the help of Fock-Schwinger method [103] is given by

$\displaystyle S(p)$ $\displaystyle=$
$\displaystyle\frac{\big{(}p_{0}+\frac{\Omega}{2}-p_{z}+ip_{\perp}\big{)}\big{(}\gamma_{0}+\gamma_{3}\big{)}+m\big{(}1+\gamma_{5}\big{)}}{\big{(}p_{0}+\frac{\Omega}{2}\big{)}^{2}-\vec{p}^{2}-m^{2}+i\epsilon}\mathcal{O}^{+}+\frac{\big{(}p_{0}-\frac{\Omega}{2}+p_{z}-ip_{\perp}\big{)}\big{(}\gamma_{0}-\gamma_{3}\big{)}+m\big{(}1+\gamma_{5}\big{)}}{\big{(}p_{0}-\frac{\Omega}{2}\big{)}^{2}-\vec{p}^{2}-m^{2}+i\epsilon}\mathcal{O}^{-}\,,$
(11)

where $p_{0}$ is the temporal component, $p_{z}$ is the $z$ component which is
parallel to the axis of rotation and $p_{\perp}=\sqrt{p_{x}^{2}+p_{y}^{2}}$ is
the transverse component of the four-momentum and
$\mathcal{O}^{\pm}\equiv(1/2)\left[1\pm i\gamma^{1}\gamma^{2}\right]$. We have
employed Eq. (11) to perform calculations at finite temperature using
Imaginary Time Formalism (ITF).

Figure 1: One loop diagram representing the photon polarization tensor with
$q$ being the momentum of the photon, and $k$ and $p$ being the momentum of
the anti-fermion and fermion, respectively.

##  III Spectral function and Shear viscosity of fermionic systems from
correlation functions

In this section we will illustrate the computation of the shear viscosity of a
rotating, hot and dense fermionic system by the help of Kubo formalism which
proceeds via calculation of correlation functions for a given quantum field
theory at finite temperature. The Kubo formula for shear viscosity is given by

$\displaystyle\eta=-\frac{1}{10}\displaystyle{\lim_{q_{0}\to
0}}\frac{\rho_{\eta}(q_{0})}{q_{0}}\,,$ (12)

where $\rho_{\eta}(q)$ is the spectral function of shear viscosity calculated
from the two-point correlation function of the shear stress tensor
$\pi^{\mu\nu}$ and $q_{0}$ is the temporal component of the four-momentum. The
expression of $\rho_{\eta}(q)$ is given by

$\displaystyle\rho_{\eta}(q)=\text{Im}\left[\Pi_{\eta}(q)\right]\,,$ (13)

where

$\displaystyle\Pi_{\eta}(q)=i\int d^{4}r~{}e^{iq\cdot
r}\big{\langle}\pi^{\mu\nu}(0)\pi_{\mu\nu}(r)\big{\rangle}_{R}\,.$ (14)

The expression
$\big{\langle}\mathcal{O}_{1}(x)\mathcal{O}_{2}(y)\big{\rangle}_{R}$ is the
retarded thermal average of $\mathcal{O}_{1}(x)$ and $\mathcal{O}_{2}(y)$,
which are some composite expression of fields, that are separated at the
spacetime points $x$ and $y$, with the subscript “$R$” denoting the retarded
two-point function. Here $\pi^{\mu\nu}$ is the the viscous stress tensor
calculated from the relation

$\displaystyle\pi^{\mu\nu}=\Delta^{\mu\nu}_{~{}~{}~{}\alpha\beta}\,T^{\alpha\beta}\,,$
(15)

where $\Delta_{\mu\nu\rho\sigma}$ is a rank-4 trace-less projector orthogonal
to $u^{\mu}$ and $\Delta^{\mu\nu}$ given by

$\displaystyle\Delta_{\mu\nu\rho\sigma}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\big{(}\Delta_{\mu\rho}\Delta_{\nu\sigma}+\Delta_{\mu\sigma}\Delta_{\nu\rho}\big{)}-\frac{\Delta_{\mu\nu}\Delta_{\rho\sigma}}{3}\,,$
$\displaystyle\Delta_{\mu\nu}$ $\displaystyle=$ $\displaystyle
g_{\mu\nu}-u_{\mu}u_{\nu}\,,$ (16)

where

$\displaystyle g_{\mu\nu}=\text{diag}\left(1,-1,-1,-1\right),\quad
u^{\mu}=\left(1,\Omega y,-\Omega x,0\right),$ (17)

and $T^{\mu\nu}$ is the energy-momentum tensor that can be determined from the
Lagrangian of the quantum field theory under study. Here $u^{\mu}$ is the
fluid 4-velocity in local rest frame, rotating around the $z$-axis and the
expression of $T^{\mu\nu}$ for Dirac field theory [105] is given by Eq. (7).
Equation (13) is calculated in cylindrical coordinates where the spacetime
coordinate at any point is given by $r=(t,\rho,\phi,z)$. To proceed further
with the calculations we have to perform Wick’s contraction of fields as

$\displaystyle\wick{\c{\psi}(0)\c{\ovpsi}(r)}=S(0,r)=\int\frac{d^{4}p}{(2\pi)^{4}}e^{-ip\cdot
r}\left(-iS(p)\right)\,,$ (18)

where $S(0,r)$ and $S(p)$ is the fermion propagator in coordinate and momentum
space, respectively. In a rotating system, the translational invariance of the
coordinate space propagator is lost [104], due to the orientation of the
rotation axis along a preferred direction. On making use of Eq. (18), the two-
point correlation function of energy-momentum tensors at two points in
spacetime $x$ and $y$ is given by

$\displaystyle\langle T^{\mu\nu}(x)T^{\alpha\beta}(y)\rangle$ $\displaystyle=$
$\displaystyle-\frac{1}{16}\Big{\langle}\left[\overline{\psi}\widetilde{\gamma}^{\\{\mu}(\widetilde{D}^{\nu\\}}\psi)-(\widetilde{D}^{\\{\mu}\overline{\psi})\widetilde{\gamma}^{\nu\\}}\psi\right]_{x}\left[\overline{\psi}\widetilde{\gamma}^{\\{\alpha}(\widetilde{D}^{\beta\\}}\psi)-(\widetilde{D}^{\\{\alpha}\overline{\psi})\widetilde{\gamma}^{\beta\\}}\psi\right]_{y}\Big{\rangle}\,,$
(19) $\displaystyle=$
$\displaystyle-\frac{1}{16}\Big{[}\big{[}\wick{\c{2}\overline{\psi}\widetilde{\gamma}^{\\{\mu}(\widetilde{D}^{\nu\\}}\c{1}\psi)\big{]}_{x}\big{[}\c{1}\overline{\psi}\widetilde{\gamma}^{\\{\alpha}(\widetilde{D}^{\beta\\}}\c{2}\psi)\big{]}_{y}}-\wick{\big{[}\c{2}\overline{\psi}\widetilde{\gamma}^{\\{\mu}(\widetilde{D}^{\nu\\}}\c{1}\psi)\big{]}_{x}\big{[}(\widetilde{D}^{\\{\alpha}\c{1}\overline{\psi})\widetilde{\gamma}^{\beta\\}}\c{2}\psi\Big{]}_{y}}$
$\displaystyle-\wick{\big{[}(\widetilde{D}^{\\{\mu}\c{2}\overline{\psi})\widetilde{\gamma}^{\nu\\}}\c{1}\psi\big{]}_{x}\big{[}\c{1}\overline{\psi}\widetilde{\gamma}^{\\{\alpha}(\widetilde{D}^{\beta\\}}\c{2}\psi)\big{]}_{y}}+\wick{\big{[}(\widetilde{D}^{\\{\mu}\c{2}\overline{\psi})\widetilde{\gamma}^{\nu\\}}\c{1}\psi\big{]}_{x}\big{[}(\widetilde{D}^{\\{\alpha}\c{1}\overline{\psi})\widetilde{\gamma}^{\beta\\}}\c{2}\psi\big{]}_{y}}\Big{]}\,,$

where $A^{\\{\alpha\beta\\}}=A^{\alpha\beta}+A^{\beta\alpha}$. In Eq. (19) we
have considered the general spacetime points $x$ and $y$, which can be
specified for cylindrical coordinates at the points $r$ and $r^{\prime}$,
where $r=(t,\rho,\phi,z)$ in cylindrical coordinates where $t$ is the time,
$\rho$ is the radius of the cylinder, $\phi$ is the azimuthal angle and $z$ is
the height of the cylinder. On substituting Eq. (18) in Eq. (19), the general
expression for the two-point function of energy-momentum tensors in
cylindrical coordinates is given by

$\displaystyle\big{\langle}T^{\mu\nu}(0)T^{\alpha\beta}(r)\big{\rangle}$
$\displaystyle=$
$\displaystyle-\frac{1}{16}\Big{[}\operatorname{Tr}\big{\\{}\widetilde{\gamma}^{\\{\mu}\widetilde{D}^{\nu\\}}S(0,r)\widetilde{\gamma}^{\\{\alpha}\widetilde{D}^{\beta\\}}S(r,0)\big{\\}}-\operatorname{Tr}\big{\\{}\widetilde{\gamma}^{\\{\mu}\widetilde{D}^{\nu\\}}\widetilde{D}^{\\{\alpha}S(0,r)\widetilde{\gamma}^{\beta\\}}S(r,0)\big{\\}}$
(20)
$\displaystyle-\operatorname{Tr}\big{\\{}\widetilde{\gamma}^{\\{\mu}S(0,r)\widetilde{\gamma}^{\\{\alpha}\widetilde{D}^{\beta\\}}\widetilde{D}^{\nu\\}}S(r,0)\big{\\}}+\operatorname{Tr}\big{\\{}\widetilde{\gamma}^{\\{\mu}\widetilde{D}^{\\{\alpha}S(0,r)\widetilde{\gamma}^{\beta\\}}\widetilde{D}^{\nu\\}}S(r,0)\big{\\}}\Big{]}.$

As shown in Ref. [104], the translational invariance of the propagator under
rotation is lost due to the preference of the rotation axis along a particular
direction. A similar situation occurs in the presence of a background magnetic
field [100, 101, 102], where a phase factor shift, known as the Schwinger
phase factor breaks the translational invariance of the propagator. However
for our case, we have considered very large $\Omega$ of the order $0.1$ GeV,
which leads to the complete disappearance of the phase factor [104]. This
leads to the expression of the propagator depend only on the relative
coordinates i.e.,

$S(r,r^{\prime})\xrightarrow{\text{Large~{}}\Omega}S(r-r^{\prime}).$

A situation of this type can be anticipated in the early stages of off-central
heavy-ion collisions when $\Omega$ is very large of the order of $0.1$ GeV.
Thus, $\Pi_{\eta}(q)$ calculated by following Eq. (20) and utilizing the
translational invariance property of the propagator, is given by

$\displaystyle\Pi_{\eta}(q)$ $\displaystyle=$ $\displaystyle i\int
d^{4}r~{}e^{iq\cdot
r}\big{\langle}\Delta^{\mu\nu\alpha\beta}T_{\alpha\beta}\Delta_{\mu\nu\rho\lambda}T^{\rho\lambda}\big{\rangle}_{R}$
(21) $\displaystyle=$ $\displaystyle\frac{i}{40}\int
d^{4}r\int\int\frac{d^{4}k}{(2\pi)^{4}}\frac{d^{4}p}{(2\pi)^{4}}e^{i(q-p-k)\cdot
r}\Bigg{[}(k_{z}-p_{z})^{2}\Big{\\{}\operatorname{Tr}\big{\\{}\widetilde{\gamma}_{1}S(p)\widetilde{\gamma}^{1}S(k)+\widetilde{\gamma}_{3}S(p)\widetilde{\gamma}^{3}S(k)\big{\\}}\Big{\\}}$
$\displaystyle+\frac{1}{3}\,\Bigg{\\{}-2\operatorname{Tr}\Big{\\{}\widetilde{\gamma}^{0}\Big{(}p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(p)\widetilde{\gamma}^{0}\Big{(}k_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(k)\Big{\\}}+\operatorname{Tr}\Big{\\{}\widetilde{\gamma}^{0}\Big{(}p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}^{2}S(p)\widetilde{\gamma}_{0}S(k)\Big{\\}}$
$\displaystyle+\operatorname{Tr}\Big{\\{}\widetilde{\gamma}^{0}S(p)\widetilde{\gamma}_{0}\Big{(}k_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}^{2}S(k)\Big{\\}}+\operatorname{Tr}\Big{\\{}\widetilde{\gamma}^{0}\Big{(}p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(p)\widetilde{\gamma}^{3}k_{z}S(k)\Big{\\}}$
$\displaystyle+\operatorname{Tr}\Big{\\{}\widetilde{\gamma}^{0}p_{z}\Big{(}p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(p)\widetilde{\gamma}_{3}S(k)\Big{\\}}-\operatorname{Tr}\Big{\\{}\widetilde{\gamma}_{0}S(p)\widetilde{\gamma}^{3}k_{z}\Big{(}k_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(k)\Big{\\}}$
$\displaystyle+\operatorname{Tr}\Big{\\{}\widetilde{\gamma}_{0}p_{z}S(p)\widetilde{\gamma}_{3}\Big{(}k_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(k)\Big{\\}}+\operatorname{Tr}\Big{\\{}\widetilde{\gamma}^{3}p_{z}S(p)\widetilde{\gamma}^{0}\Big{(}k_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(k)\Big{\\}}$
$\displaystyle-\operatorname{Tr}\Big{\\{}\widetilde{\gamma}^{3}p_{z}\Big{(}p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(p)\widetilde{\gamma}_{0}S(k)\Big{\\}}-\operatorname{Tr}\Big{\\{}\widetilde{\gamma}_{3}S(p)\widetilde{\gamma}_{0}k_{z}\Big{(}k_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(k)\Big{\\}}$
$\displaystyle-\operatorname{Tr}\Big{\\{}\widetilde{\gamma}_{3}\Big{(}p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(p)\widetilde{\gamma}_{0}k_{z}S(k)\Big{\\}}-(p_{z}+k_{z})^{2}\operatorname{Tr}\Big{\\{}\widetilde{\gamma}^{3}S(p)\widetilde{\gamma}^{3}S(k)\Big{\\}}$
$\displaystyle+\operatorname{Tr}\Big{\\{}\widetilde{\gamma}^{0}\Big{(}p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(p)\widetilde{\gamma}^{0}\Big{(}k_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(k)\Big{\\}}-\operatorname{Tr}\Big{\\{}\widetilde{\gamma}^{0}\Big{(}p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}^{2}S(p)\widetilde{\gamma}^{0}S(k)\Big{\\}}$
$\displaystyle-\operatorname{Tr}\Big{\\{}\widetilde{\gamma}^{0}S(p)\widetilde{\gamma}^{0}\Big{(}k_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}^{2}S(k)\Big{\\}}-\operatorname{Tr}\Big{\\{}\widetilde{\gamma}^{3}p_{z}S(p)\widetilde{\gamma}^{0}\Big{(}k_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(k)\Big{\\}}$
$\displaystyle+\operatorname{Tr}\Big{\\{}\widetilde{\gamma}^{3}p_{z}\Big{(}p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(p)\widetilde{\gamma}_{0}S(k)\Big{\\}}-\operatorname{Tr}\Big{\\{}\widetilde{\gamma}_{3}S(p)\widetilde{\gamma}^{0}k_{z}\Big{(}k_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(k)\Big{\\}}$
$\displaystyle+\operatorname{Tr}\Big{\\{}\widetilde{\gamma}_{3}\Big{(}p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(p)\widetilde{\gamma}_{0}k_{z}S(k)\Big{\\}}-\operatorname{Tr}\Big{\\{}\widetilde{\gamma}_{0}\Big{(}p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}^{2}S(p)\widetilde{\gamma}_{0}S(k)\Big{\\}}$
$\displaystyle-\operatorname{Tr}\Big{\\{}\widetilde{\gamma}_{0}S(p)\widetilde{\gamma}^{0}\Big{(}k_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}^{2}S(k)\Big{\\}}+\operatorname{Tr}\Big{\\{}\widetilde{\gamma}_{0}\Big{(}p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(p)\widetilde{\gamma}_{0}\Big{(}k_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(k)\Big{\\}}$
$\displaystyle+\operatorname{Tr}\Big{\\{}\widetilde{\gamma}_{0}\Big{(}p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(p)\widetilde{\gamma}^{3}k_{z}S(k)\Big{\\}}+\operatorname{Tr}\Big{\\{}\widetilde{\gamma}_{0}\Big{(}p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}p_{z}S(p)\widetilde{\gamma}_{3}S(k)\Big{\\}}$
$\displaystyle-\operatorname{Tr}\Big{\\{}\widetilde{\gamma}_{0}S(p)\widetilde{\gamma}^{3}k_{z}\Big{(}k_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(k)\Big{\\}}-\operatorname{Tr}\Big{\\{}\widetilde{\gamma}_{0}p_{z}S(p)\widetilde{\gamma}_{3}\Big{(}k_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(k)\Big{\\}}$
$\displaystyle-\operatorname{Tr}\Big{\\{}\widetilde{\gamma}_{0}\Big{(}p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(p)\widetilde{\gamma}^{0}\Big{(}k_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(k)\Big{\\}}+\operatorname{Tr}\Big{\\{}\widetilde{\gamma}_{0}\Big{(}p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}^{2}S(p)\widetilde{\gamma}^{0}S(k)\Big{\\}}$
$\displaystyle-\operatorname{Tr}\Big{\\{}\widetilde{\gamma}_{0}S(p)\widetilde{\gamma}^{0}\Big{(}k_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}^{2}S(k)\Big{\\}}\Bigg{\\}}\Bigg{]}\,.$

On performing the $d^{4}r$ and $d^{4}k$ integration we get the delta function
$(2\pi)^{4}\delta(q-p-k)$ on account of momentum conservation at the vertex,
which then fixes the relation between $p$, $k$, and $q$ i.e.,
$k_{0}=q_{0}-p_{0}$, $k_{z}=q_{z}-p_{z}$, and $k_{\perp}=q_{\perp}-p_{\perp}$
in accordance with the one-loop diagram representing photon polarization in
Fig. 1. Following Eq. (13), spectral function is calculated, by using Eq. (21)
and is given by

$\displaystyle\rho_{\eta}(q)$ $\displaystyle=$ $\displaystyle\text{Im~{}}i\int
d^{4}r~{}e^{iq\cdot
r}\big{\langle}\Delta^{\mu\nu\alpha\beta}T_{\alpha\beta}\Delta_{\mu\nu\rho\lambda}T^{\rho\lambda}\big{\rangle}_{R}$
(22) $\displaystyle=$ $\displaystyle\text{Im}\frac{i}{40}\int
d^{4}r\int\frac{d^{4}p}{(2\pi)^{4}}\Bigg{[}(q_{z}-2p_{z})^{2}\Big{\\{}\operatorname{Tr}\big{\\{}\widetilde{\gamma}_{1}S(p)\widetilde{\gamma}^{1}S(q-p)+\widetilde{\gamma}_{3}S(p)\widetilde{\gamma}^{3}S(q-p)\big{\\}}\Big{\\}}$
$\displaystyle+\frac{1}{3}\Bigg{\\{}-2\operatorname{Tr}\Big{\\{}\widetilde{\gamma}^{0}\Big{(}p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(p)\widetilde{\gamma}^{0}\Big{(}q_{0}-p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(q-p)\Big{\\}}+\operatorname{Tr}\Big{\\{}\widetilde{\gamma}^{0}\Big{(}p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}^{2}S(p)\widetilde{\gamma}_{0}S(q-p)\Big{\\}}$
$\displaystyle+\operatorname{Tr}\Big{\\{}\widetilde{\gamma}^{0}S(p)\widetilde{\gamma}_{0}\Big{(}q_{0}-p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}^{2}S(q-p)\Big{\\}}+\operatorname{Tr}\Big{\\{}\widetilde{\gamma}^{0}\Big{(}p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(p)\widetilde{\gamma}^{3}k_{z}S(q-p)\Big{\\}}$
$\displaystyle+\operatorname{Tr}\Big{\\{}\widetilde{\gamma}^{0}p_{z}\Big{(}p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(p)\widetilde{\gamma}_{3}S(q-p)\Big{\\}}-\operatorname{Tr}\Big{\\{}\widetilde{\gamma}_{0}S(p)\widetilde{\gamma}^{3}(q_{z}-p_{z})\Big{(}q_{0}-p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(q-p)\Big{\\}}$
$\displaystyle+\operatorname{Tr}\Big{\\{}\widetilde{\gamma}_{0}p_{z}S(p)\widetilde{\gamma}_{3}\Big{(}q_{0}-p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(q-p)\Big{\\}}+\operatorname{Tr}\Big{\\{}\widetilde{\gamma}^{3}p_{z}S(p)\widetilde{\gamma}^{0}\Big{(}q_{0}-p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(q-p)\Big{\\}}$
$\displaystyle-\operatorname{Tr}\Big{\\{}\widetilde{\gamma}^{3}p_{z}\Big{(}p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(p)\widetilde{\gamma}_{0}S(q-p)\Big{\\}}-\operatorname{Tr}\Big{\\{}\widetilde{\gamma}_{3}S(p)\widetilde{\gamma}_{0}(q_{z}-p_{z})\Big{(}q_{0}-p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(q-p)\Big{\\}}$
$\displaystyle-\operatorname{Tr}\Big{\\{}\widetilde{\gamma}_{3}\Big{(}p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(p)\widetilde{\gamma}_{0}(q_{z}-p_{z})S(q-p)\Big{\\}}-q_{z}^{2}\operatorname{Tr}\Big{\\{}\widetilde{\gamma}^{3}S(p)\widetilde{\gamma}^{3}S(q-z)\Big{\\}}$
$\displaystyle+\operatorname{Tr}\Big{\\{}\widetilde{\gamma}^{0}\Big{(}p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(p)\widetilde{\gamma}^{0}\Big{(}q_{0}-p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(q-p)\Big{\\}}-\operatorname{Tr}\Big{\\{}\widetilde{\gamma}^{0}\Big{(}p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}^{2}S(p)\widetilde{\gamma}^{0}S(q-p)\Big{\\}}$
$\displaystyle-\operatorname{Tr}\Big{\\{}\widetilde{\gamma}^{0}S(p)\widetilde{\gamma}^{0}\Big{(}q_{0}-p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}^{2}S(q-p)\Big{\\}}-\operatorname{Tr}\Big{\\{}\widetilde{\gamma}^{3}p_{z}S(p)\widetilde{\gamma}^{0}\Big{(}q_{0}-p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(q-p)\Big{\\}}$
$\displaystyle+\operatorname{Tr}\Big{\\{}\widetilde{\gamma}^{3}p_{z}\Big{(}p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(p)\widetilde{\gamma}_{0}S(q-p)\Big{\\}}-\operatorname{Tr}\Big{\\{}\widetilde{\gamma}_{3}S(p)\widetilde{\gamma}^{0}(q_{z}-p_{z})\Big{(}q_{0}-p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(q-p)\Big{\\}}$
$\displaystyle+\operatorname{Tr}\Big{\\{}\widetilde{\gamma}_{3}\Big{(}p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(p)\widetilde{\gamma}_{0}(q_{z}-p_{z})S(q-p)\Big{\\}}-\operatorname{Tr}\Big{\\{}\widetilde{\gamma}_{0}\Big{(}p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}^{2}S(p)\widetilde{\gamma}_{0}S(q-p)\Big{\\}}$
$\displaystyle-\operatorname{Tr}\Big{\\{}\widetilde{\gamma}_{0}S(p)\widetilde{\gamma}^{0}\Big{(}q_{0}-p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}^{2}S(q-p)\Big{\\}}+\operatorname{Tr}\Big{\\{}\widetilde{\gamma}_{0}\Big{(}p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(p)\widetilde{\gamma}_{0}\Big{(}q_{0}-p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(q-p)\Big{\\}}$
$\displaystyle+\operatorname{Tr}\Big{\\{}\widetilde{\gamma}_{0}\Big{(}p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(p)\widetilde{\gamma}^{3}(q_{z}-p_{z})S(q-p)\Big{\\}}+\operatorname{Tr}\Big{\\{}\widetilde{\gamma}_{0}\Big{(}p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}p_{z}S(p)\widetilde{\gamma}_{3}S(q-p)\Big{\\}}$
$\displaystyle-\operatorname{Tr}\Big{\\{}\widetilde{\gamma}_{0}S(p)\widetilde{\gamma}^{3}(q_{z}-p_{z})\Big{(}q_{0}-p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(q-p)\Big{\\}}-\operatorname{Tr}\Big{\\{}\widetilde{\gamma}_{0}p_{z}S(p)\widetilde{\gamma}_{3}\Big{(}q_{0}-p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(q-p)\Big{\\}}$
$\displaystyle-\operatorname{Tr}\Big{\\{}\widetilde{\gamma}_{0}\Big{(}p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(p)\widetilde{\gamma}^{0}\Big{(}q_{0}-p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}S(q-p)\Big{\\}}+\operatorname{Tr}\Big{\\{}\widetilde{\gamma}_{0}\Big{(}p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}^{2}S(p)\widetilde{\gamma}^{0}S(q-p)\Big{\\}}$
$\displaystyle-\operatorname{Tr}\Big{\\{}\widetilde{\gamma}_{0}S(p)\widetilde{\gamma}^{0}\Big{(}q_{0}-p_{0}+\frac{\Omega\Sigma_{3}}{2}\Big{)}^{2}S(q-p)\Big{\\}}\Bigg{\\}}\Bigg{]}\,.$

In order to perform the integration in Eq. (22) we will perform the Matsubara
summation [109] at finite angular velocity and chemical potential. Here we
provide the details of the Matsubara frequency summation used for calculating
the spectral function of energy-momentum tensors. At finite temperature

$\displaystyle p_{0}$ $\displaystyle\to$ $\displaystyle
i\widetilde{\omega}_{N}=(2N+1)\pi T\,,\quad q_{0}\to i\nu_{N}=2\pi NT\,,$
$\displaystyle\int\frac{d^{4}p}{(2\pi)^{4}}$ $\displaystyle\equiv$
$\displaystyle\sum\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{\\{P\\}}=iT\sum_{N=-\infty}^{+\infty}\int\frac{d^{3}p}{(2\pi)^{3}},$
(23)

where $N\in\mathbb{Z}$, and $p_{0}$ and $q_{0}$ are the temporal components of
the four-momentum of the fermion and boson, respectively, in the photon
polarization. In one-loop calculations of spectral functions, the fermion
propagator under rotation, Eq. (11), leads to fermionic Matsubara frequency
summations of the type

$\displaystyle(a)$ $\displaystyle
T\sum_{\\{p_{N}\\}}\frac{1}{\big{[}i\widetilde{\omega}_{N}+\frac{\Omega}{2}+\mu\big{]}^{2}-\vec{p}^{2}-m^{2}}\frac{1}{\big{[}i\nu_{N}-i\widetilde{\omega}_{N}+\frac{\Omega}{2}-\mu\big{]}^{2}-\vec{k}^{2}-m^{2}}\,,$
(24) $\displaystyle(b)$ $\displaystyle
T\sum_{\\{p_{N}\\}}\frac{1}{\big{[}i\widetilde{\omega}_{N}+\frac{\Omega}{2}+\mu\big{]}^{2}-\vec{p}^{2}-m^{2}}\frac{1}{\big{[}i\nu_{N}-i\widetilde{\omega}_{N}-\frac{\Omega}{2}-\mu\big{]}^{2}-\vec{k}^{2}-m^{2}}\,,$
(25) $\displaystyle(c)$ $\displaystyle
T\sum_{\\{p_{N}\\}}\frac{1}{\big{[}i\widetilde{\omega}_{N}-\frac{\Omega}{2}+\mu\big{]}^{2}-\vec{p}^{2}-m^{2}}\frac{1}{\big{[}i\nu_{N}-i\widetilde{\omega}_{N}+\frac{\Omega}{2}-\mu\big{]}^{2}-\vec{k}^{2}-m^{2}}\,,$
(26) $\displaystyle(d)$ $\displaystyle
T\sum_{\\{p_{N}\\}}\frac{1}{\big{[}i\widetilde{\omega}_{N}-\frac{\Omega}{2}+\mu\big{]}^{2}-\vec{p}^{2}-m^{2}}\frac{1}{\big{[}i\nu_{N}-i\widetilde{\omega}_{N}-\frac{\Omega}{2}-\mu\big{]}^{2}-\vec{k}^{2}-m^{2}}\,.$
(27)

Following the Saclay method [109] of the evaluation of Matsubara frequency
sums, we obtain the results for the above frequency summations, Eqs.
(24)-(27), as follows

$\displaystyle(a)\quad
T\sum_{\\{p_{N}\\}}\frac{1}{\big{[}i\widetilde{\omega}_{N}+\frac{\Omega}{2}+\mu\big{]}^{2}-\vec{p}^{2}-m^{2}}\frac{1}{\big{[}i\nu_{N}-i\widetilde{\omega}_{N}+\frac{\Omega}{2}-\mu\big{]}^{2}-\vec{k}^{2}-m^{2}}$
$\displaystyle=\frac{1}{4E_{p}E_{k}}\Bigg{\\{}\frac{n_{F}(E_{p}+\mu+\Omega/2)+n_{F}(E_{k}-\mu+\Omega/2)-1}{q_{0}-E_{p}-E_{k}}+\frac{n_{F}(E_{k}+\mu+\Omega/2)-n_{F}(E_{p}+\mu+\Omega/2)}{q_{0}+E_{k}-E_{p}}$
$\displaystyle+\frac{n_{F}(E_{p}-\mu-\Omega/2)-n_{F}(E_{k}-\mu-\Omega/2)}{q_{0}+E_{p}-E_{k}}+\frac{1-n_{F}(E_{p}-\mu-\Omega/2)-n_{F}(E_{k}+\mu-\Omega/2)}{q_{0}+E_{k}+E_{p}}\Bigg{\\}}\,,$
(28) $\displaystyle(b)\quad
T\sum_{\\{p_{N}\\}}\frac{1}{\big{[}i\widetilde{\omega}_{N}+\frac{\Omega}{2}+\mu\big{]}^{2}-\vec{p}^{2}-m^{2}}\frac{1}{\big{[}i\nu_{N}-i\widetilde{\omega}_{N}-\frac{\Omega}{2}-\mu\big{]}^{2}-\vec{k}^{2}-m^{2}}$
$\displaystyle=\frac{1}{4E_{p}E_{k}}\Bigg{\\{}\frac{n_{F}(E_{p}+\mu+\Omega/2)+n_{F}(E_{k}-\mu-\Omega/2)-1}{q_{0}-E_{p}-E_{k}}+\frac{n_{F}(E_{k}+\mu+\Omega/2)-n_{F}(E_{p}+\mu+\Omega/2)}{q_{0}+E_{k}-E_{p}}$
$\displaystyle+\frac{n_{F}(E_{p}-\mu-\Omega/2)-n_{F}(E_{k}-\mu-\Omega/2)}{q_{0}+E_{p}-E_{k}}+\frac{1-n_{F}(E_{p}-\mu-\Omega/2)-n_{F}(E_{k}+\mu+\Omega/2)}{q_{0}+E_{k}+E_{p}}\Bigg{\\}}\,,$
(29) $\displaystyle(c)\quad
T\sum_{\\{p_{N}\\}}\frac{1}{\big{[}i\widetilde{\omega}_{N}-\frac{\Omega}{2}+\mu\big{]}^{2}-\vec{p}^{2}-m^{2}}\frac{1}{\big{[}i\nu_{N}-i\widetilde{\omega}_{N}+\frac{\Omega}{2}-\mu\big{]}^{2}-\vec{k}^{2}-m^{2}}$
$\displaystyle=\frac{1}{4E_{p}E_{k}}\Bigg{\\{}\frac{n_{F}(E_{p}+\mu-\Omega/2)+n_{F}(E_{k}-\mu+\Omega/2)-1}{q_{0}-E_{p}-E_{k}}+\frac{n_{F}(E_{k}+\mu-\Omega/2)-n_{F}(E_{k}+\mu-\Omega/2)}{q_{0}+E_{k}-E_{p}}$
$\displaystyle+\frac{n_{F}(E_{p}-\mu+\Omega/2)-n_{F}(E_{k}-\mu+\Omega/2)}{q_{0}+E_{p}-E_{k}}+\frac{1-n_{F}(E_{p}-\mu+\Omega/2)-n_{F}(E_{k}+\mu-\Omega/2)}{q_{0}+E_{k}+E_{p}}\Bigg{\\}}\,,$
(30) $\displaystyle(d)\quad
T\sum_{\\{p_{N}\\}}\frac{1}{\big{[}i\widetilde{\omega}_{N}-\frac{\Omega}{2}+\mu\big{]}^{2}-\vec{p}^{2}-m^{2}}\frac{1}{\big{[}i\nu_{N}-i\widetilde{\omega}_{N}-\frac{\Omega}{2}-\mu\big{]}^{2}-\vec{k}^{2}-m^{2}}$
$\displaystyle=\frac{1}{4E_{p}E_{k}}\Bigg{\\{}\frac{n_{F}(E_{p}+\mu-\Omega/2)+n_{F}(E_{k}-\mu-\Omega/2)-1}{q_{0}-E_{p}-E_{k}}+\frac{n_{F}(E_{k}+\mu-\Omega/2)-n_{F}(E_{k}+\mu-\Omega/2)}{q_{0}+E_{k}-E_{p}}$
$\displaystyle+\frac{n_{F}(E_{p}-\mu+\Omega/2)-n_{F}(E_{k}-\mu+\Omega/2)}{q_{0}+E_{p}-E_{k}}+\frac{1-n_{F}(E_{p}-\mu+\Omega/2)-n_{F}(E_{k}+\mu-\Omega/2)}{q_{0}+E_{k}+E_{p}}\Bigg{\\}}.$
(31)

In Eqs. (24)-(27) we have four combinations of $q_{0}$, $E_{k}$ and $E_{p}$
appearing in the denominator as $q_{0}\pm E_{k}\pm E_{p}$ and $q_{0}\mp
E_{k}\pm E_{p}$ which can be considered to be the limiting values of energy
from the upper half plane. By setting $\nu_{N}\to-i(q_{0}+i0^{+})$ via the
analytic continuation of discrete Matsubara frequencies to continuous
energies, we can take the imaginary part as required in Eq. (22) by using the
relation

$\displaystyle\frac{1}{\Delta\pm
i0^{+}}=\mathbb{P}\left(\frac{1}{\Delta}\right)\mp i\pi\delta(\Delta).$ (32)

where $\mathbb{P}$ is the principal part of the function. On employing Eq.
(32), $\rho_{\eta}(q)$ is given by

$\displaystyle\rho_{\eta}(q)$ $\displaystyle=$
$\displaystyle-\pi\int\frac{d^{3}p}{(2\pi)^{3}}\Big{(}\frac{\mathcal{C}(p_{\perp},p_{z},p_{0},q_{\perp},q_{z},q_{0})}{4E_{p}E_{q-p}}\Big{)}\Bigg{[}\Big{\\{}n_{F}(E_{q-p}+\bar{\mu})-n_{F}(E_{p}+\bar{\mu})\Big{\\}}\Big{\\{}\delta(q_{0}+E_{q-p}-E_{p})\Big{\\}}$
(33)
$\displaystyle+\,\Big{\\{}n_{F}(E_{p}-\bar{\mu})-n_{F}(E_{q-p}-\bar{\mu})\Big{\\}}\Big{\\{}\delta(q_{0}+E_{p}-E_{q-p})\Big{\\}}$
$\displaystyle+\,\Big{\\{}n_{F}(E_{q-p}+\bar{\mu})+n_{F}(E_{p}-\bar{\mu})-1\Big{\\}}\Big{\\{}\delta(q_{0}-E_{q-p}-E_{p})\Big{\\}}$
$\displaystyle+\,\Big{\\{}n_{F}(E_{q-p}-\bar{\mu})+n_{F}(E_{p}+\bar{\mu})-1\Big{\\}}\Big{\\{}\delta(q_{0}+E_{q-p}+E_{p})\Big{\\}}\Bigg{]}$
$\displaystyle-\,\pi\int\frac{d^{3}p}{(2\pi)^{3}}\Big{(}\frac{\mathcal{D}(p_{\perp},p_{z},p_{0},q_{\perp},q_{z},q_{0})}{4E_{p}E_{q-p}}\Big{)}\Bigg{[}\Big{\\{}n_{F}(E_{q-p}+\widetilde{\mu})-n_{F}(E_{p}+\widetilde{\mu})\Big{\\}}\Big{\\{}\delta(q_{0}+E_{q-p}-E_{p})\Big{\\}}$
$\displaystyle+\,\Big{\\{}n_{F}(E_{p}-\widetilde{\mu})-n_{F}(E_{q-p}-\widetilde{\mu})\Big{\\}}\Big{\\{}\delta(q_{0}+E_{p}-E_{q-p})\Big{\\}}$
$\displaystyle+\,\Big{\\{}n_{F}(E_{q-p}+\widetilde{\mu})+n_{F}(E_{p}-\widetilde{\mu})-1\Big{\\}}\Big{\\{}\delta(q_{0}-E_{q-p}-E_{p})\Big{\\}}$
$\displaystyle+\,\Big{\\{}n_{F}(E_{q-p}-\widetilde{\mu})+n_{F}(E_{p}+\widetilde{\mu})-1\Big{\\}}\Big{\\{}\delta(q_{0}+E_{q-p}+E_{p})\Big{\\}}\Bigg{]}\,,$

where
$\bar{\mu}=\mu+\frac{\Omega}{2}\text{~{}~{}and~{}~{}}\widetilde{\mu}=\mu-\frac{\Omega}{2}$
are the combinations of chemical potential and angular velocity and the
functions $\mathcal{C}(p_{\perp},p_{z},p_{0},q_{\perp},q_{z},q_{0})$ and
$\mathcal{D}(p_{\perp},p_{z},p_{0},q_{\perp},q_{z},q_{0})$ are expressed as

$\displaystyle\mathcal{C}(p_{\perp},p_{z},p_{0},q_{\perp},q_{z},q_{0})=-\frac{1}{480}\Big{\\{}2(p_{z}-q_{z})\big{(}q_{0}-p_{0}+8p_{z}-2p_{0}+\Omega\big{)}+8(q_{z}-p_{z})^{2}+3(q_{0}-p_{0})\Omega$
$\displaystyle+\,p_{z}\big{(}8p_{z}-2p_{0}+\Omega\big{)}\big{]}\big{[}(q_{z}-p_{z})\big{(}4p_{z}+4p_{0}-2\Omega\big{)}-4(q_{\perp}-p_{\perp})p_{\perp}+\big{(}2q_{0}-2p_{0}-\Omega\big{)}\big{(}2p_{z}+2p_{0}-\Omega\big{)}\big{]}$
$\displaystyle+\frac{1}{10}\big{[}p_{z}\big{(}q_{z}-2p_{z}\big{)}\big{]}\big{[}(q_{z}-p_{z})\big{(}4p_{z}-4p_{0}-2\Omega\big{)}-4(q_{\perp}-p_{\perp})p_{\perp}+\big{(}2q_{0}-2p_{0}-\Omega\big{)}\big{(}2p_{z}+2p_{0}+\Omega\big{)}\Big{\\}}\,,$
(34)
$\displaystyle\mathcal{D}(p_{\perp},p_{z},p_{0},q_{\perp},q_{z},q_{0})=-\frac{1}{480}\Big{\\{}2(p_{z}-q_{z})\big{(}q_{0}-p_{0}+8p_{z}+2p_{0}+\Omega\big{)}+8(q_{z}-p_{z})^{2}+3(q_{0}-p_{0})\Omega$
$\displaystyle+\,p_{z}\big{(}8p_{z}+2p_{0}+\Omega\big{)}\big{]}\big{[}-k_{z}\big{(}-4p_{z}+4p_{0}+2\Omega\big{)}-4(q_{\perp}-p_{\perp})p_{\perp}+\big{(}2q_{0}-2p_{0}+\Omega\big{)}\big{(}-2p_{z}+2p_{0}+\Omega\big{)}\big{]}$
$\displaystyle+\frac{1}{10}\big{[}p_{z}\big{(}q_{z}-2p_{z}\big{)}\big{]}\big{[}(q_{z}-p_{z})\big{(}4p_{z}+4p_{0}+2\Omega\big{)}-4(q_{\perp}-p_{\perp})p_{\perp}+\big{(}2q_{0}-2p_{0}+\Omega\big{)}\big{(}-2p_{z}+2p_{0}+\Omega\big{)}\Big{\\}}.$
(35)

Here it is worth noting that $\rho_{\eta}(q)$ calculated in Eq. (33), from an
analytic continuation and the cut of an Euclidean correlator represents real
scatterings of on-shell particles which are kinematically allowed and whose
distribution function is given by the Fermi-Dirac distribution function. In a
more elegant way, the imaginary parts correspond to the real scatterings of
particles whose information is carried by the delta functions appearing in Eq.
(33). The delta functions $\delta(q_{0}\pm E_{p}\mp E_{k})$ are known as
Landau cuts and $\delta(q_{0}\pm E_{p}\pm E_{k})$ are known as unitary cuts
[110]. The Landau cuts appear only at finite temperature whereas the unitary
cuts are already present due to vacuum contributions. The unitary cuts do not
contribute to the calculation of transport coefficients [110] and are thus
ignored in our calculations. From Eq. (12) the shear viscosity, calculated
from Eq. (33) is given by

$\displaystyle\eta$ $\displaystyle=$
$\displaystyle\displaystyle{\lim_{q_{0}\to
0,\,\vec{q}=\vec{0}}}\text{Im}\Bigg{[}-\pi\int\frac{d^{3}p}{(2\pi)^{3}}\frac{\mathcal{C}(p_{\perp},p_{z},p_{0},q_{\perp},q_{z},q_{0})}{4E_{q-p}E_{p}}$
(36) $\displaystyle\displaystyle{\lim_{\Gamma\to
0}}\Bigg{\\{}\frac{\big{(}n_{F}(E_{q-p}+\bar{\mu})-n_{F}(E_{p}+\bar{\mu})\big{)}/q_{0}}{q_{0}+E_{q-p}-E_{p}+i\Gamma}+\frac{\big{(}n_{F}(E_{q-p}-\bar{\mu})-n_{F}(E_{p}-\bar{\mu})\big{)}/q_{0}}{q_{0}+E_{p}-E_{q-p}+i\Gamma}\Bigg{\\}}\Bigg{]}$
$\displaystyle+$ $\displaystyle\displaystyle{\lim_{q_{0}\to
0,\,\vec{q}=\vec{0}}}\text{Im}\Bigg{[}-\pi\int\frac{d^{3}p}{(2\pi)^{3}}\frac{\mathcal{D}(p_{\perp},p_{z},p_{0},q_{\perp},q_{z},q_{0})}{4E_{q-p}E_{p}}$
$\displaystyle\displaystyle{\lim_{\Gamma\to
0}}\Bigg{\\{}\frac{\big{(}n_{F}(E_{q-p}+\widetilde{\mu})-n_{F}(E_{p}+\widetilde{\mu})\big{)}/q_{0}}{q_{0}+E_{q-p}-E_{p}+i\Gamma}+\frac{\big{(}n_{F}(E_{q-p}-\widetilde{\mu})-n_{F}(E_{p}-\widetilde{\mu})\big{)}/q_{0}}{q_{0}+E_{p}-E_{q-p}+i\Gamma}\Bigg{\\}}\Bigg{]}\,,$

where we have substituted $\delta(x)=-(1/\pi)\displaystyle{\lim_{\Gamma\to
0}}\frac{1}{x+i\Gamma}$, with parameter $\Gamma$ being the thermal width of
the medium arising from interactions in the medium at finite temperature,
where the temperature of the system being very high acts effective enough to
generate interactions (i.e., a finite $\Gamma$) and $\Gamma$ is basically the
inverse of the relaxation time $\tau$ i.e., $\Gamma=1/\tau$ [110]. It should
be noted that in the present work $\Gamma$ has been introduced through the
Breit-Wigner form and similar study can be performed by introducing half-width
$i\Gamma/2$ [111] in the momentum space propagators. This can be done by
considering the expression of the inverse of the retarded momentum space
propagator $G^{-1}_{\text{R}}(p)$ which includes the complex valued self-
energy $\Sigma_{\text{R}}(p)$, given by
$G_{\text{R}}^{-1}(p)=p^{2}-m_{0}^{2}-\text{Re}\Sigma_{\text{R}}(p)-i\text{Im}\Sigma_{\text{R}}(p)\sim(p_{0}+i\gamma(p))^{2}-E_{p}^{2}$,
where
$\gamma=-\frac{1}{2p_{0}}\text{Im}\Sigma_{\text{R}}(p)=\frac{\Gamma}{2}$. This
shows that the information of $\Gamma$ is already present in the propagator of
the theory. For our case, we have employed a particular representation of the
$\delta(x)$ function to inoculate this information in our calculations. One
can calculate $\Gamma$ by considering an interacting Lagrangian and employing
the techniques of thermal field theory [109, 111]. We see that in the
$\vec{q}=\vec{0},\,q_{0}\to 0$ limit, $\eta$ takes an indeterminate $0/0$
form, which is unphysical. Therefore we apply the L’Hospital rule to treat
this divergence. By turning off the interaction i.e., $\Gamma\to 0$, which
happens for a free theory, the shear viscosity diverges. Therefore one has to
keep $\Gamma$ finite for obtaining a non-divergent shear viscosity. Thus the
final expression of $\eta(T,\mu,\Omega)$ is given by

$\displaystyle\eta(T,\mu,\Omega)=\frac{1}{240}\int\frac{d^{3}p}{(2\pi)^{3}}\left(\frac{1}{4E_{p}^{2}\Gamma
T}\right)$ (37)
$\displaystyle\Bigg{\\{}\Big{\\{}160p_{z}^{4}+4p_{\perp}^{2}\Omega^{2}+\big{(}\Omega-
E_{p}-3p_{z}\big{)}\Omega^{3}+8p_{\perp}^{2}p_{z}\big{(}\Omega-4E_{p}\big{)}+32p_{z}^{3}\big{(}\Omega+5E_{p}\big{)}-8p_{z}^{2}\Omega\big{(}2E_{p}+3\Omega\big{)}\Big{\\}}\mathcal{N}_{\pm\mu,\mp\Omega/2}$
$\displaystyle+\Big{\\{}160p_{z}^{4}+4p_{\perp}^{2}\Omega^{2}+\big{(}\Omega-
E_{p}-3p_{z}\big{)}\Omega^{3}+8p_{\perp}^{2}p_{z}\big{(}\Omega-4E_{p}\big{)}+32p_{z}^{3}\big{(}\Omega-5E_{p}\big{)}-8p_{z}^{2}\Omega\big{(}2E_{p}-3\Omega\big{)}\Big{\\}}\mathcal{N}_{\pm\mu,\pm\Omega/2}\Bigg{\\}}\,,$

where

$\displaystyle\mathcal{N}_{\pm\mu,\pm\Omega/2}=\sum_{s=\pm
1}n_{F}(E_{p}+s\mu+s\Omega/2)\big{\\{}1-n_{F}(E_{p}+s\mu+s\Omega/2)\big{\\}}$
$\displaystyle\mathcal{N}_{\pm\mu,\mp\Omega/2}=\sum_{\lambda=\pm
1}n_{F}(E_{p}+\lambda\mu-\lambda\Omega/2)\big{\\{}1-n_{F}(E_{p}+\lambda\mu-\lambda\Omega/2)\big{\\}},$
(38)

and $n_{F}(x)$ is the Fermi-Dirac distribution function. Equation (III) gives
the formula for the shear viscosity of a rotating, hot and dense fermionic
system. The results offer an immediate applicability in hadronic medium
composed of spin-$1/2$ particles. But one has to be careful with the magnitude
of $\Omega$ for this case, which should not drop below the order of $0.1$ GeV
for applying the result of this work. In Eq. (III) the effect of $\Omega$ is
present in the Fermi-Dirac distribution function and in the numerator of the
expression. From this we assert that $\Omega$ acts as an effective chemical
potential in the medium.

##  IV Results

We aim, to discuss in this section the numerical results on the $T$, $\mu$ and
$\Omega$ dependence of $\eta$ which has been calculated for a system of
spin-$1/2$ fermions, subjected to a very high angular frequency. The quantum
field theoretical study of $\eta$ helps us to study the role of $\mu$ and its
effective counterpart $\Omega$ from a first principle approach, where we see
that both of them enter into the distribution function through the Matsubara
frequency summation, but additionally $\Omega$ also enters through the
momentum space propagator, which is evident from Eq. (11). Therefore it plays
the role of an effective chemical potential, instead of an exact one. We have
considered the temperature range of $0.15-0.6$ GeV, which is roughly the
regime in which deconfined QCD matter exists, with $T=0.6$ GeV being the upper
bound of achievable temperature. In this deconfined state, $\mu$ has been
taken to be 0.05, 0.2 and 0.4 GeV, respectively. Finally since the work
focuses the regime of large angular velocity, $\Omega$ is taken to be of the
order of $0.1-1$ GeV.

It is important to mention in the beginning that $\Omega$ cannot be taken to
be very large for two physical reasons. The first issue is that a large
$\Omega$ would violate causality, due to which a global equilibrium of the
system would cease to exist. The explanation goes as follows. For thermal
equilibrium to be maintained under rotation [80], we should first note that
the local temperature $T$ should vary in such a way that the quantity
$T/\gamma$, where $\gamma=1/\left(\sqrt{1-\Omega^{2}r^{2}}\right)$ is the time
dilation factor, $r$ being the radius of the cylinder, should remain constant.
One can therefore deduce that $\Omega<1/r$, should be followed for a valid
Lorentz transformation, which would otherwise make $T$ imaginary.
Additionally, it is worth mentioning that for $\Omega=1/r,\,T\to\infty$,
leading to an unphysical situation. Thus, $\Omega$ should increase in way that
it remains comparable to other energy scales associated with the system such
as $T$ and $\mu$, maintaining thermal equilibrium in the process. The other
reason can be attributed to the appearance of rotational instabilities in a
rotating QGP as reported in Ref. [94], where the moment of inertia has been
found to be negative below $T_{s}=1.5\,T_{c}$, where $T_{s}$ is the super-
vortical temperature and $T_{c}$ is the deconfining transition temperature of
a non-rotating plasma, showing a coincidence with the results obtained for
spinning Kerr and Myers-Perry black holes [112]. The negative moment of
inertia is a function of the velocity of the system $v=\Omega r$, where $r$,
which is the radius of the cylinder as mentioned previously, acts as the
system size. The functional dependence of the system on $\Omega$, up to any
order in $\Omega$, is through the quantity $\Omega\,r$. For a QCD system under
rotation, various energy scales associated with the system are
$T,\,\mu,\,\Omega$, and $\Lambda_{\text{QCD}}$, which are inter-dependent on
each other. Therefore caution needs to be exercised while dealing with
thermodynamic and transport quantities, so that negative values coming from
rotational instabilities do not affect the physical quantities of the system.
In our case, from Eq. (38) we see that $\eta(T,\mu,\Omega)$ has a complicated
dependence on $\Omega$, which needs to be handled carefully, as there is no
exact relation that fixes the upper and lower bounds of $\Omega$. One of the
attempts of this work is to show the interdependence of $\mu$ and $\Omega$.

### Chemical potential $\mu$ vs Effective chemical potential $\Omega$

Figure 2: Behaviour of $\eta$ with $T$ for $\Omega$ = 0.1, 0.5, 1.0 GeV at
$\mu$ = 0.05 GeV. Figure 3: Same as fig. 2 but for $\mu$ = 0.2 GeV. Figure 4:
Same as fig. 2 but for $\mu$ = 0.4 GeV.

The viscous description of QGP is mainly done through the shear viscosity,
although both shear and bulk viscosities are important. Even then, one has to
resort to the quantities such as specific shear $(\eta/s)$ and bulk
$(\zeta/s)$ viscosity, where $s$ is the entropy density, for studying
phenomenological properties of QGP. To start with, our study of $\eta$
provides a generic framework for studying fermionic systems, which can be
extended to the case of quarks and spin-$1/2$ baryons.

In Figs. 2–4 we have shown the variation of $\eta$ with $T$ for various values
of $\mu$. Figure 2 shows the behaviour of $\eta$ with temperature ranging from
$0.15-0.6$ GeV keeping $\mu=0.05$ GeV. We compute and plot for three different
values of $\Omega=0.1,\,0.5,\,1$ GeV corresponding to different collision
energies, respectively. It is found that shear viscosity increases with
temperature which is a common feature of relativistic fluids. The trend of the
plots shows that $\Omega$ increases $\eta$ with a converging nature in the
regime of low $T$, which is expected. But a similar behaviour may not be
obvious for smaller values of $\Omega$ which lie below the order of $0.1$ GeV,
as the propagator that has been employed for our calculations is valid in the
regime of large $\Omega$. The increasing nature of $\eta$ with $\Omega$ may be
deduced mathematically from Eq. (III), where the expression shows that there
is a $\mathcal{O}(\Omega^{4})$ dependence of $\eta$ on $\Omega$. Further, at
finite rotation, the shear induced flow of the fluid coming from velocity
gradients is enhanced due to the fact that the velocity $v$ and its gradient
$\vec{\nabla}v$ given by

$v=\Omega r\,,\quad\vec{\nabla}v\sim\mathcal{O}(\Omega)\,,$

induce an additional velocity gradient of $\mathcal{O}(\Omega)$ in the system,
which contributes to the shear viscosity. A similar trend of increasing nature
is followed by $\eta$ with $T$ in Fig. 3 and 4. But there are subtle
differences to be noted and the discussions are as follows. In Fig. 3 the
curves have been plotted for $\mu$ = 0.2 GeV with the same values of $\Omega$
as in Fig. 2. But the curve with $\Omega$ = 0.1 GeV is found to take a middle
position with respect to the $\Omega$ = 0.5 GeV and 1.0 GeV taking lower and
higher values relative to it. The trend completely reverses in Fig. 4 as
compared to Fig. 2, when the curves with lower value of $\Omega$ take a higher
value as compared to the curves with a larger value $\Omega$. From Figs. 3 and
4 we can clearly see an interplay of $\mu$ and its effective counterpart
$\Omega$. Both $\mu$ and $\Omega$ are present in the distribution functions,
but additionally $\Omega$ is present in the numerator of the expression of
$\eta$. The build up of chemical potential in the medium clearly suppresses
the effect of $\Omega$ as it increases. As discussed previously, the issue of
causality also comes into play, thus restricting the value of $\Omega$ to
certain values which are physically allowed. A larger $\Omega$ implies that
the fermions escape the rotating cylinder, violating causality. Further
increase in $\mu$ implies that the number of fermions violating causality are
also increasing, leading to unphysical behaviour. In the end we remark that
the plots show that the temperature and angular velocity complement each
other, which is understood as we can expect larger angular velocities at
collider experiments if the temperature gets sufficiently high above the
present attainable value. But a contrasting effect is seen as the chemical
potential increases.

##  V Summary

In a medium undergoing finite rotation, transport coefficients are influenced
and exhibit a dependency on the medium’s angular velocity. Utilizing the Kubo
formalism, this study quantifies the shear viscosity in a generalized
fermionic system subjected to high angular velocity. The impact of angular
velocity is integrated into the calculations through the momentum space
propagator of the fermion, obtained via the Fock-Schwinger approach. This
influence manifests in the distribution function via Matsubara frequency
summation and also appears in the numerator of the shear viscosity expression
through trace evaluations. Our findings indicate that shear viscosity rises
with increasing temperature and angular velocity, while it diminishes with
higher chemical potential. Notably, the chemical potential and its functional
equivalent, the angular velocity, exert contrasting effects on the dissipation
via shear viscosity in fermionic systems. Research to generalize these
findings for arbitrary strengths of angular velocity is ongoing, and those
outcomes will be reported in future work.

## Acknowledgements

We thank Maxim N. Chernodub and Amaresh Jaiswal for their encouragement and
the useful discussions on this topic and also acknowledge the cooperation
received from Abhishek Tiwari and Sumit. S.S. is supported by Institute Post
Doctoral Scheme of IIT Roorkee under the grant IITR/Estt-(A)-Rect-
Cell-E-5001(130)18490. R.S. acknowledges the support of Polish NAWA Bekker
program No. BPN/BEK/2021/1/00342 and Polish NCN Grant No. 2018/30/E/ST2/00432,
and thank Francesco Becattini, Derek Teaney, and Kirill Tuchin for the
fruitful discussions.

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