# Distilling circularly polarized photon emission from magnetized chiral
plasmas

Xinyang Wang<EMAIL_ADDRESS>Center for Fundamental Physics, School of
Mechanics and Physics, Anhui University of Science and Technology, Huainan,
Anhui 232001, People’s Republic of China  Igor A. Shovkovy
<EMAIL_ADDRESS>College of Integrative Sciences and Arts, Arizona State
University, Mesa, Arizona 85212, USA  Department of Physics, Arizona State
University, Tempe, Arizona 85287, USA

###### Abstract

We investigate the emission of circularly polarized photons from a magnetized
quark-gluon plasma with nonzero quark-number and chiral charge chemical
potentials. These chemical potentials qualitatively influence the differential
emission rates of circularly polarized photons. A nonzero net electric charge
density, induced by quark-number chemical potentials, enhances the overall
emission of one circular polarization over the other, while a nonzero chiral
charge density introduces a spatial asymmetry in the emission with respect to
reflection in the transverse plane. The signs of the electrical and chiral
charge densities determine which circular polarization dominates overall and
whether the emission preferentially aligns with or opposes the magnetic field.
Based on these findings, we propose that polarized photon emission is a
promising observable for characterizing the quark-gluon plasma produced in
heavy-ion collisions.

##  I Introduction

Over the past two decades, the study of exotic states of matter in extreme
environments has become a central focus of modern nuclear physics. This
interest is largely driven by successful experimental programs at the
relativistic heavy-ion colliders in Brookhaven and CERN. The ultrarelativistic
collisions produce quark-gluon plasma (QGP) at temperatures well above the
quark deconfinement transition. By analyzing particle spectra and
correlations, one uncovers valuable information about the fundamental
properties of QGP.

High temperature and energy density are not the only extreme characteristics
of the QGP produced by heavy-ion collisions. The system is, without
exaggeration, rich in extremes. Since heavy ions carry electric charges, they
induce extreme magnetic fields [1, 2, 3, 4, 5]. Furthermore, large values of
angular momentum in non-central collisions can produce high vorticity in the
resulting QGP fluid [6, 7]. Many theoretical aspects of QGP associated with
strong magnetic fields and high vorticity are routinely tested in experimental
studies [8, 9, 10, 11]. (For reviews, see Refs. [12, 13, 14, 15, 16].)

One aspect we would like to discuss in more detail is the effect of chiral
charge density on the electromagnetic probes of QGP. A nonzero chiral charge
arises from an imbalance between left-handed and right-handed quarks in the
plasma. At first glance, such a state may seem too exotic as it would break
local parity symmetry. However, in a hot plasma, local bubbles of chiral
plasma can be produced by topological configurations of gauge fields [17].
This idea is not new, and it is taken seriously enough to motivate the
challenging search for its observable signatures via the dipole and quadrupole
chiral magnetic effects (CME) in multiple experiments. Unfortunately, the
evidence supporting CME remains inconclusive.

This study was motivated by the proposition that there might be unexplored,
better observables for detecting chiral plasma effects. Electromagnetic
probes, such as photon and dilepton emissions, play a crucial role in
revealing the properties of QGP in strong magnetic fields. These probes
provide valuable information about temperature, magnetic fields, flow
characteristics, and possibly much more. Naturally, one might ask how a
nonzero chiral charge affects electromagnetic probes. Some attempts to address
this question have been made before, see Ref. [18, 19, 20]. Here, we delve
into more detail to illuminate the role of strong magnetic fields and propose
new observables that may resolve the issue. While the task remains
challenging, we believe it is not impossible.

This paper is organized as follows. In Sec. II, we briefly present the
formalism used for studying polarized photon emission from the plasma.
Numerical results for the polarized photon emission are given in Sec. III. In
Sec. IV, we discuss our main findings and summarize the results. Several
technical derivations and auxiliary results are provided in the appendices at
the end of the paper.

##  II Formalism

To investigate the impact of a nonzero chiral charge density on photon
emission from a strongly magnetized plasma, we will utilize the general
framework developed in Refs. [21, 22]. Specifically, we will utilize a
relationship between the emission rate and the imaginary part of the retarded
photon polarization tensor [23]

${\cal R}_{\rm diff}\equiv
k^{0}\frac{d^{3}R}{dk_{x}dk_{y}dk_{z}}=-\frac{1}{(2\pi)^{3}}\frac{\mbox{Im}\left[g^{\mu\nu}\Pi_{\mu\nu}(k)\right]}{\exp\left(\frac{k_{0}}{T}\right)-1}.$
(1)

Assuming that the background magnetic field is sufficiently strong, the
polarization tensor $\Pi_{\mu\nu}(k)$ is determined by the simplest one-loop
diagram at the leading order of coupling [22]. For QED plasma, this
approximation should hold true for magnetic field strengths $|eB|\gtrsim\alpha
T^{2}$, where $\alpha=1/137$ denotes the fine structure constant. In the
context of QGP, the requirement is $|eB|\gtrsim\alpha_{s}T^{2}$, where
$\alpha_{s}$ is the strong coupling constant. Additionally, in the case of
QCD, one should require that $T\gg\Lambda_{\rm QCD}$.

To separate total emission into polarized components, we use the following
completeness relation:

$g^{\mu\nu}=\epsilon_{0}^{\mu}\epsilon_{0}^{\nu*}-\epsilon_{+}^{\mu}\epsilon_{+}^{\nu*}-\epsilon_{-}^{\mu}\epsilon_{-}^{\nu*}-\epsilon_{\parallel}^{\mu}\epsilon_{\parallel}^{\nu*}.$
(2)

where $\epsilon_{r}^{\mu}$ are photon polarization vectors. Here, we will use
the same basis for $\epsilon_{r}^{\mu}$ as in Refs. [19, 20], i.e.,
$\epsilon_{0}^{\mu}=(1,0,0,0)$, $\epsilon_{\parallel}^{\mu}=(0,0,0,1)$,
$\epsilon_{\pm}^{\mu}=(0,1,\pm i,0)/\sqrt{2}$. (A slightly different
decomposition was used in the study of dilepton production in Ref. [24].)

This study focuses on the emission of circularly polarized photons,
represented by the polarization vectors $\epsilon_{\pm}^{\mu}$. Before
proceeding further, it is important to clarify the exact physical meaning of
these photon states. For emission along (or opposite to) the direction of the
magnetic field, these states correspond to the left-handed and right-handed
circular polarizations, or equivalently, photons with positive and negative
helicities, respectively. For photons emitted in other directions in the upper
and lower hemispheres, they serve only as proxies for representing photons
with given circular polarizations. For our qualitative discussion, we ignore
the subtle difference between true circular polarization and its proxy defined
by $\epsilon_{\pm}^{\mu}$. This is acceptable since, from symmetry
considerations, both characterize closely related photon emission observables.

Note that the projection operators for circularly polarized photons take the
following explicit form:

$\epsilon_{\pm}^{\mu}\epsilon_{\pm}^{\nu*}=(-g_{\perp}^{\mu\nu}\pm
i\epsilon_{\perp}^{\mu\nu})/2,$ (3)

where $g_{\perp}^{\mu\nu}=g^{\mu\nu}-\delta^{\mu}_{0}\delta^{\nu}_{0}$ and
$\epsilon_{\perp}^{\mu\nu}=\delta^{\mu}_{1}\delta^{\nu}_{2}-\delta^{\mu}_{2}\delta^{\nu}_{1}$.
For completeness, let us note that the projectors for the other two states are
$\epsilon_{0}^{\mu}\epsilon_{0}^{\nu*}=\mbox{diag}(1,0,0,0)$ and
$\epsilon_{\parallel}^{\mu}\epsilon_{\parallel}^{\nu*}=\mbox{diag}(0,0,0,1)$.
When these projectors are contracted with $\Pi_{\mu\nu}(k)$, as in Eq. (1),
the photon polarization tensor naturally separates into the following four
parts:

$g^{\mu\nu}\Pi_{\mu\nu}(k)\to\Pi^{(0)}(k)+\Pi^{(+)}(k)+\Pi^{(-)}(k)+\Pi^{(\parallel)}(k).\
$ (4)

Then, the emission rate in Eq. (1) similarly splits into four parts. It is
natural to interpret the contributions proportional to
$\mbox{Im}\left[\Pi^{(+)}(k)\right]$ and $\mbox{Im}\left[\Pi^{(-)}(k)\right]$,

$\displaystyle{\cal R}^{(+)}_{\rm diff}$ $\displaystyle\equiv$ $\displaystyle
k^{0}\frac{d^{3}R}{dk_{x}dk_{y}dk_{z}}=-\frac{1}{(2\pi)^{3}}\frac{\mbox{Im}\left[\Pi^{(+)}(k)\right]}{\exp\left(\frac{k_{0}}{T}\right)-1},$
(5) $\displaystyle{\cal R}^{(-)}_{\rm diff}$ $\displaystyle\equiv$
$\displaystyle
k^{0}\frac{d^{3}R}{dk_{x}dk_{y}dk_{z}}=-\frac{1}{(2\pi)^{3}}\frac{\mbox{Im}\left[\Pi^{(-)}(k)\right]}{\exp\left(\frac{k_{0}}{T}\right)-1},$
(6)

as the emission rates for photons with the right-handed and the left-handed
circular polarizations, respectively. It is important to emphasize, however,
that such an interpretation is not entirely correct.

Recall that photons have only two physical degrees of freedom. Nevertheless,
in the conventional quantum field theoretical description of a gauge theory
like QED, it is convenient to introduce four states. Two of these states are
unphysical. While they ensure the gauge invariance of the theory, they should
not affect physical observables such as the photon emission rate. Thus, if all
physical contributions were included in ${\cal R}^{(+)}_{\rm diff}$ and ${\cal
R}^{(-)}_{\rm diff}$, the remaining parts would cancel out. However, this is
not the case, partly due to the use of approximate rather than exact
definitions of the polarization projectors in Eq. (3).

Interestingly, the extra contribution ${\cal R}^{\rm(extra)}_{\rm diff}$ to
the rate, coming from
$\Pi^{\rm(extra)}(k)\equiv\Pi^{(0)}(k)+\Pi^{(\parallel)}(k)$, is negative.
This may seem problematic since the rates are expected to be positive
definite. However, as one can verify, the total rate ${\cal
R}^{\rm(extra)}_{\rm diff}$ is always positive. This implies that the extra
contribution ${\cal R}^{\rm(extra)}_{\rm diff}$ should be distributed between
the two physical circular polarizations, modifying the definitions of rates in
Eqs. (5) and (6). For simplicity, we will use the unmodified definitions
above, which should suffice to approximately extract parity-odd contributions
(albeit somewhat overestimated) to the rates for circularly polarized photons.

To calculate $\Pi_{\mu\nu}(k)$ in the QGP, we will assume that the plasma is
made of the lightest up and down quarks. Their flavor-dependent quark charges
are $e_{f}=eq_{f}$, where $f=u,d$, $q_{u}=2/3$, $q_{d}=-1/3$, and $e$ is the
absolute value of the electron charge. When investigating the role of the
chiral chemical potential $\mu_{5}$, it is convenient to use the chiral limit
for quarks by setting their masses to zero. Noting that typical temperatures
of a deconfined QGP are of the order of $200~{}\mbox{MeV}$ or higher and the
current quark masses are only about $5~{}\mbox{MeV}$, the chiral limit is a
very good approximation for the purposes of our study.

Without loss of generality, we will take the background magnetic field
$\mathbf{B}$ to be along the $+z$ direction. In the Dirac equation for the
quarks, it will be described by a vector potential in the Landau gauge, i.e.,
$\mathbf{A}=(-By,0,0)$. The corresponding quark Green’s functions
$G_{f}(u,u^{\prime})$ take the form of a product of the so-called Schwinger
phase $e^{i\Phi(u,u^{\prime})}$ and a translation invariant function
$\bar{G}_{f}(u-u^{\prime})$, where $u=(t,x,y,z)$. Note, however, that the
Schwinger phase plays no role in the calculation of the polarization tensor
$\Pi_{\mu\nu}(k)$ in the leading one-loop order approximation [22].

In a mixed coordinate-momentum space representation, the Fourier transform of
the translation invariant part of Green’s function can be conveniently given
as a sum over the Landau levels [15]

$\bar{G}_{f}(p_{0},p_{z};\mathbf{r}_{\perp})=i\frac{e^{-\mathbf{r}_{\perp}^{2}/(4\ell_{f}^{2})}}{2\pi\ell_{f}^{2}}\sum_{n=0}^{\infty}\frac{\tilde{D}_{n}^{f}(p_{0},p_{z};\mathbf{r}_{\perp})}{\mathcal{M}-2n|e_{f}B|}.$
(7)

where the function in the numerator has the following explicit form:

$\tilde{D}_{n}^{f}(p_{0},p_{z};\mathbf{r}_{\perp})=\left[(p_{0}+\mu)\gamma^{0}-p_{z}\gamma^{3}-\mu_{5}\gamma^{0}\gamma^{5}\right]\left[\mathcal{P}_{+}^{f}L_{n}\left(\frac{\mathbf{r}_{\perp}^{2}}{2\ell_{f}^{2}}\right)+\mathcal{P}_{-}^{f}L_{n-1}\left(\frac{\mathbf{r}_{\perp}^{2}}{2\ell_{f}^{2}}\right)\right]-\frac{i}{\ell_{f}^{2}}(\mathbf{r}_{\perp}\cdot\bm{\gamma}_{\perp})L_{n-1}^{1}\left(\frac{\mathbf{r}_{\perp}^{2}}{2\ell_{f}^{2}}\right).$
(8)

Here, $L_{n}^{\alpha}(z)$ are the generalized Laguerre polynomials [25],
$\mathcal{P}_{\pm}^{f}\equiv\frac{1}{2}\left(1\pm
is_{\perp}\gamma^{1}\gamma^{2}\right)$ are the spin projectors,
$\mathbf{r}_{\perp}=(x,y)$ is the transverse position vector, and
$\ell_{f}=1/\sqrt{|e_{f}B|}$ is the magnetic length. Here, we included a
nonzero quark number and chiral chemical potentials, denoted by $\mu$ and
$\mu_{5}$, respectively. By definition, $s_{\perp}=\,\mbox{sign}(e_{f}B)$ and
$L_{-1}^{\alpha}(z)\equiv 0$.

The function in the denominator of Eq. (7) is a Dirac matrix. In the chiral
limit, the corresponding fraction takes the following form:

$\frac{1}{\mathcal{M}-2n|e_{f}B|}=\frac{\mathcal{P}_{5}^{(-)}}{(p_{0}+\mu+\mu_{5})^{2}-2n|e_{f}B|-p_{z}^{2}}+\frac{\mathcal{P}_{5}^{(+)}}{(p_{0}+\mu-\mu_{5})^{2}-2n|e_{f}B|-p_{z}^{2}},$
(9)

where $\mathcal{P}_{5}^{(\pm)}=\frac{1}{2}(1\pm\gamma^{5})$ are the chirality
projectors. Note that we included both the quark-number and chiral-charge
chemical potentials, $\mu$ and $\mu_{5}$, respectively. Both play important
role in polarized photon emission. For the sake of simplicity, we assume that
$\mu$ and $\mu_{5}$ are same for both flavors. (The generalization to the case
of flavor-dependent chemical potentials is straightforward and will be briefly
discussed later.)

By using the above Green’s functions in the Landau-level representation, we
derive the following formal expression for the polarization tensor [22]:

$\Pi^{\mu\nu}(i\Omega_{m};\mathbf{k})=4\pi\sum_{f=u,d}N_{c}\alpha_{f}T\sum_{k=-\infty}^{\infty}\int\frac{dp_{z}}{2\pi}\int
d^{2}\mathbf{r}_{\perp}e^{-i\mathbf{r}_{\perp}\cdot\mathbf{k}_{\perp}}\,\mbox{tr}\left[\gamma^{\mu}\bar{G}_{f}(i\omega_{k},p_{z};\mathbf{r}_{\perp})\gamma^{\nu}\bar{G}_{f}(i\omega_{k}-i\Omega_{m},p_{z}-k_{z};-\mathbf{r}_{\perp})\right],$
(10)

where we used the Matsubara finite-temperature formalism. By definition, the
fermionic and bosonic Matsubara frequencies are $\omega_{k}=(2k+1)\pi T$ and
$\Omega_{m}=2n\pi T$, respectively. Regarding other notations,
$\alpha_{f}=q_{f}^{2}\alpha$ is the flavor-dependent coupling constant, and
$N_{c}=3$ is the number of quark colors.

By substituting Green’s function (7) into the definition for
$\Pi^{\mu\nu}(i\Omega_{m};\mathbf{k})$, calculating the Dirac traces,
integrating over $r_{\perp}$ and performing the Matsubara summation, we derive
the following expression for the components of the polarization tensor:

$\Pi^{(i)}(i\Omega_{m};\mathbf{k})=-\sum_{f=u,d}\frac{N_{c}\alpha_{f}}{\pi\ell_{f}^{4}}\sum_{n,n^{\prime}=0}^{\infty}\sum_{s,\eta,\lambda=\pm
1}\int\frac{dp_{z}}{2\pi}\frac{[n_{F}(E_{n,p_{z},f}+\eta\mu+\eta
s\mu_{5})-n_{F}(\lambda E_{n^{\prime},p_{z}-k_{z},f}+\eta\mu+\eta
s\mu_{5})]\mathcal{F}^{(i)}_{s,f}(\xi,i\Omega_{m})}{4\lambda
E_{n,p_{z},f}E_{n^{\prime},p_{z}-k_{z},f}\left[E_{n,p_{z},f}-\lambda
E_{n^{\prime},p_{z}-k_{z},f}+i\eta\Omega_{m}\right]},$ (11)

where $i$ labels the photon polarizations (i.e., $i=0,+,-,\parallel$),
$E_{n,p_{z},f}=\sqrt{p_{z}^{2}+2n|e_{f}B|}$ are Landau-level energies, and
$n_{F}(E)=1/\left[\exp(E/T)+1\right]$ is the Fermi-Dirac distribution
function. The corresponding technical details of the derivation are given in
Appendices A, B, C, and D. The explicit forms of the polarization-dependent
functions $\mathcal{F}^{(i)}_{s,f}(\xi,i\Omega_{m})$ are presented in Eqs.
(40) – (43) in Appendix E.

After replacing $i\Omega_{m}\rightarrow k_{0}+i\epsilon$, it is
straightforward to extract the absorptive part of the individual polarization
component functions,

$\displaystyle\mbox{Im}\Pi^{(i)}(k_{0};\mathbf{k})$ $\displaystyle=$
$\displaystyle\sum_{f=u,d}\frac{N_{c}\alpha_{f}}{4\pi\ell_{f}^{4}}\sum_{s,s^{\prime}=\pm
1}\sum_{n>n^{\prime}}^{\infty}\frac{g(n,n^{\prime})\,\theta\left[(k_{-}^{f})^{2}+k_{z}^{2}-k_{0}^{2}\right]}{\sqrt{\left[(k_{-}^{f})^{2}+k_{z}^{2}-k_{0}^{2}\right]\left[(k_{+}^{f})^{2}+k_{z}^{2}-k_{0}^{2}\right]}}\left.\mathcal{F}^{(i)}_{s,f}(\xi,k_{0})\right|_{p_{z}=p_{z,f}^{(s^{\prime})}}$
(12) $\displaystyle+$
$\displaystyle\sum_{f=u,d}\frac{N_{c}\alpha_{f}}{4\pi\ell_{f}^{4}}\sum_{s,s^{\prime}=\pm
1}\sum_{n<n^{\prime}}^{\infty}\frac{g(n,n^{\prime})\,\theta\left[(k_{-}^{f})^{2}+k_{z}^{2}-k_{0}^{2}\right]}{\sqrt{\left[(k_{-}^{f})^{2}+k_{z}^{2}-k_{0}^{2}\right]\left[(k_{+}^{f})^{2}+k_{z}^{2}-k_{0}^{2}\right]}}\left.\mathcal{F}^{(i)}_{s,f}(\xi,k_{0})\right|_{p_{z}=p_{z,f}^{(s^{\prime})}}$
$\displaystyle-$
$\displaystyle\sum_{f=u,d}\frac{N_{c}\alpha_{f}}{4\pi\ell_{f}^{4}}\sum_{s,s^{\prime}=\pm
1}\sum_{n,n^{\prime}=0}^{\infty}\frac{g(n,n^{\prime})\,\theta\left[k_{0}^{2}-k_{z}^{2}-(k_{+}^{f})^{2}\right]}{\sqrt{\left[k_{0}^{2}-k_{z}^{2}-(k_{-}^{f})^{2}\right]\left[k_{0}^{2}-k_{z}^{2}-(k_{+}^{f})^{2}\right]}}\left.\mathcal{F}^{(i)}_{s,f}(\xi,k_{0})\right|_{p_{z}=p_{z,f}^{(s^{\prime})}},$

where $\theta(x)$ is the Heaviside step function. In Eq. (12), we also
introduced two transverse momentum thresholds,

$k_{\pm}^{f}=\left|\sqrt{2n|e_{f}B|}\pm\sqrt{2n^{\prime}|e_{f}B|}\right|.$
(13)

and the following function:

$g(n,n^{\prime})=1-\sum_{s_{1}=\pm
1}n_{F}\left(\left[\frac{k_{0}}{2}+s_{1}\frac{k_{0}(n-n^{\prime})|e_{f}B|}{k_{0}^{2}-k_{z}^{2}}+s_{1}s^{\prime}\frac{|k_{z}|}{2}\sqrt{\left(1-\frac{(k_{-}^{f})^{2}}{k_{0}^{2}-k_{z}^{2}}\right)\left(1-\frac{(k_{+}^{f})^{2}}{k_{0}^{2}-k_{z}^{2}}\right)}\right]-s_{1}\mu-
s_{1}s\mu_{5}\right).\\\ $ (14)

Note that the photon energy $k_{0}$ in the final expression satisfies the
mass-shell condition, i.e., $k_{0}=\sqrt{k_{\perp}^{2}+k_{z}^{2}}$.

As expected, the expression in Eq. (12) closely resembles the result in Ref.
[22]. It should be noted, however, that $g(n,n^{\prime})$ depends on both
fermion-number and chiral chemical potentials. The most important difference
appears in the definition of functions
$\mathcal{F}^{(i)}_{s,f}(\xi,\Omega)|_{p_{z}=p_{z,f}^{(s^{\prime})}}$, which
define individual polarization components labeled by $i=0,+,-,\parallel$.
Their explicit expressions are given in Eqs. (49) – (52) in Appendix E.

##  III Numerical results

Here, we present the main numerical results for the emission rates of
circularly polarized photons. Considering potential applications to heavy-ion
phenomenology, we will rewrite the differential rate using new variables:

${\cal R}^{(\pm)}_{\rm
diff}(k_{T},\phi,y)=\frac{d^{3}R^{(\pm)}}{k_{T}dk_{T}d\phi dy},$ (15)

where we introduced the transverse momentum $k_{T}$, the azimuthal angle
$\phi$, and the rapidity $y$, replacing the Cartesian components of the
momentum, $k_{x}$, $k_{y}$, and $k_{z}$. By definition, $k_{y}=k_{T}\cos\phi$,
$k_{z}=k_{T}\sin\phi$, and $y=\frac{1}{2}\ln\frac{k_{0}+k_{x}}{k_{0}-k_{x}}$.
For simplicity, below we will be presenting the results for $y=0$ (i.e.,
$k_{x}=0$).

In calculating the imaginary part of the polarization tensor, defined by Eq.
(12), we must sum over all Landau levels. For the numerical calculation, we
will truncate the sum at a finite number of terms, specifically $n_{\rm
max}=500$. This large number of Landau levels ensures high accuracy of the
results across a wide range of model parameters.

For this study, we need to calculate the differential emission rate over the
entire range of azimuthal angles, from $\phi_{\rm min}=-\pi/2$ to $\phi_{\rm
max}=\pi/2$. To achieve high angular resolution, we use a relatively small
step size, $\delta\phi=\pi/600$.

Our goal is to identify qualitative features of polarized photon emission in
the presence of nonzero chemical potentials $\mu$ and $\mu_{5}$. Therefore, we
limit our numerical analysis to a few representative values of the model
parameters. Specifically, we consider two different values of the magnetic
field strength, $|eB|=m_{\pi}^{2}$ and $|eB|=5m_{\pi}^{2}$, and two different
values of temperature, $T=200~{}\mbox{MeV}$ and $T=350~{}\mbox{MeV}$.

###  III.1 Nonzero quark-number chemical potential $\mu$

Let us begin with the case of a nonzero quark-number chemical potential $\mu$.
For our analysis, we will use a representative value of
$\mu=200~{}\mbox{MeV}$. It is important to note that using the same chemical
potential for both quark flavors implies that the plasma is positively charged
overall, due to the up quark’s charge being twice as large in magnitude
compared to the down quark’s charge. This analysis can be easily generalized
to cases with different chemical potentials for the up and down quarks. As we
argue below, it is the sign of the charge carriers in the plasma that
determines the dominance of one circular polarization over the other in photon
emission.

Typical numerical results for the emission rates of left-handed and right-
handed circularly polarized photons are presented in Fig. 1. The data is shown
for $T=350~{}\mbox{MeV}$ and two different values of the magnetic field,
$|eB|=m_{\pi}^{2}$ and $|eB|=5m_{\pi}^{2}$, in the left and right panels,
respectively. We verified that the qualitative features at
$T=200~{}\mbox{MeV}$ are similar, though the rates are overall lower.

Figure 1: The angular dependence of left-handed (top) and right-handed
(bottom) circularly polarized photon emission for two magnetic fields,
$|eB|=m_{\pi}^{2}$ (left) and $|eB|=5m_{\pi}^{2}$ (right). The plasma
temperature is $T=350~{}\mbox{MeV}$.

Each panel in Fig. 1 shows the differential rates for four different fixed
values of the transverse momentum,
$k_{T}=0.1m_{\pi},~{}0.25m_{\pi},~{}0.5m_{\pi},~{}0.9m_{\pi}$. Note that the
transverse momentum also coincides with the photon energy since we set the
rapidity to zero, $y=0$. The two top and two bottom panels display the results
for left-handed and right-handed circular polarizations, respectively. As seen
from the figure, the rates for both polarizations are symmetric under changing
$\phi\to-\phi$, which is equivalent to reflection in the transverse plane.

We observe, however, that the emission rates for left-handed and right-handed
circular polarizations differ significantly, with the latter being a few times
larger than the former. To quantify this effect, it is instructive to
calculate the degree of circular polarization, defined by

$\displaystyle{\cal P}_{\rm circ}(k_{T},\phi,y)=\frac{{\cal R}^{(+)}_{\rm
diff}(k_{T},\phi,y)-{\cal R}^{(-)}_{\rm diff}(k_{T},\phi,y)}{{\cal
R}^{(+)}_{\rm diff}(k_{T},\phi,y)+{\cal R}^{(-)}_{\rm diff}(k_{T},\phi,y)}.$
(16)

The corresponding numerical results are presented in Fig. 2, which confirm a
substantial degree of circular polarization in the photon emission. This sign
of ${\cal P}_{\rm circ}(k_{T},\phi,y)$ is determined primarily by a nonzero
electrical charge density in the plasma. Indeed, by separating the partial
contributions from the up and down quarks, we find that the up quarks emit
more left-handed photons (${\cal P}^{\rm(u)}_{\rm circ}<0$), while the down
quarks emit more right-handed photons (${\cal P}^{\rm(d)}_{\rm circ}>0$).
Since the up quarks have twice the electric charge of the down quarks, their
emission rate is higher overall. Consequently, the net circular polarization
from QGP is negative, ${\cal P}_{\rm circ}<0$.

Figure 2: The degree of circular polarization as a function of the azimuthal
angle for two magnetic fields, $|eB|=m_{\pi}^{2}$ (left) and
$|eB|=5m_{\pi}^{2}$ (right), and two temperatures, $T=200~{}\mbox{MeV}$ (top)
and $T=350~{}\mbox{MeV}$ (bottom).

The effect of a nonzero chemical potential, resulting in a predominant
emission of photons with one circular polarization over the other may resemble
the underlying physics of helicons (whistlers). Recall that helicons are low-
frequency electromagnetic excitations in strongly magnetized plasmas, driven
by the Lorentz force [26]. Their circular polarization is determined by the
sign of the electric charge carriers (e.g., electrons) with higher mobility,
moving in an approximately static background of opposite charges (e.g.,
positive ions).

For the model parameters explored, the degree of circular polarization ranges
from approximately $0.12$ to $0.5$. It tends to be larger at stronger magnetic
fields and lower transverse momenta (or, equivalently, lower photon energies).
Comparing the top and bottom panels in Fig. 2, we also observe that the effect
tends to diminish as the temperature rises.

###  III.2 Nonzero chiral chemical potential $\mu_{5}$

Let us now consider the case of a magnetized quark-gluon plasma (QGP) with a
nonzero chiral chemical potential. Specifically, we choose
$\mu_{5}=100~{}\mbox{MeV}$ as a representative value, although the actual
value is not crucial for identifying the qualitative effects of $\mu_{5}$.

Fig. 3 presents sample numerical results for the emission rates of left-handed
and right-handed circularly polarized photons. The data is shown for a
temperature of $T=200~{}\mbox{MeV}$ and for two different magnetic field
strengths: $|eB|=m_{\pi}^{2}$ (left panel) and $|eB|=5m_{\pi}^{2}$ (right
panel). Although we do not show the rates at $T=350~{}\mbox{MeV}$, we have
verified that they exhibit a similar qualitative behavior, albeit with
generally higher values.

Figure 3: The angular dependence of right-handed (solid lines) and left-handed
(dashed lines) circularly polarized photon emission rates for two magnetic
fields, $|eB|=m_{\pi}^{2}$ (left) and $|eB|=5m_{\pi}^{2}$ (right), and fixed
temperature, $T=200~{}\mbox{MeV}$.

The emission rates in Fig. 3 possess an interesting property, namely they are
asymmetric with respect to reflection in the transverse plane (i.e.,
$\phi\to-\phi$). To visualize this effect, we construct the following two
observables:

$\displaystyle{\cal P}^{(+)}_{\rm asym}(k_{T},\phi,y)$ $\displaystyle=$
$\displaystyle\frac{{\cal R}^{(+)}_{\rm diff}(k_{T},\phi,y)-{\cal
R}^{(+)}_{\rm diff}(k_{T},-\phi,y)}{{\cal R}^{(+)}_{\rm
diff}(k_{T},\phi,y)+{\cal R}^{(+)}_{\rm diff}(k_{T},-\phi,y)},$ (17)
$\displaystyle{\cal P}^{(-)}_{\rm asym}(k_{T},\phi,y)$ $\displaystyle=$
$\displaystyle\frac{{\cal R}^{(-)}_{\rm diff}(k_{T},\phi,y)-{\cal
R}^{(-)}_{\rm diff}(k_{T},-\phi,y)}{{\cal R}^{(-)}_{\rm
diff}(k_{T},\phi,y)+{\cal R}^{(-)}_{\rm diff}(k_{T},-\phi,y)},$ (18)

to measure the degree of emission asymmetry for the left-handed and the right-
handed circularly polarized photons, respectively. It is sufficient to define
them only for positive azimuthal angles in the range $0\leq\phi\leq\pi/2$. The
corresponding numerical results for the emission asymmetry ${\cal
P}^{(+)}_{\rm asym}(k_{T},\phi,y)$ are presented in Fig. 4 for two magnetic
fields, $|eB|=m_{\pi}^{2}$ (left) and $|eB|=5m_{\pi}^{2}$ (right), and two
temperatures, $T=200~{}\mbox{MeV}$ (top) and $T=350~{}\mbox{MeV}$ (bottom). We
do not show any numerical data for ${\cal P}^{(-)}_{\rm asym}(k_{T},\phi,y)$
because it is not truly independent. Indeed, ${\cal P}^{(-)}_{\rm
asym}(k_{T},\phi,y)=-{\cal P}^{(+)}_{\rm asym}(k_{T},\phi,y)$. As we will see
later, this relation will not remain valid at nonzero $\mu$.

Figure 4: The emission asymmetry for the right-handed circularly polarized
photons as a function of the azimuthal angular for two magnetic fields,
$|eB|=m_{\pi}^{2}$ (left) and $|eB|=5m_{\pi}^{2}$ (right), and two
temperatures, $T=200~{}\mbox{MeV}$ (top) and $T=350~{}\mbox{MeV}$ (bottom).

Fig 4 shows that for a positive $\mu_{5}$, the emission asymmetry is positive
for photons with the right-handed circular polarization. Since ${\cal
P}^{(-)}_{\rm asym}(k_{T},\phi,y)=-{\cal P}^{(+)}_{\rm asym}(k_{T},\phi,y)$,
the asymmetry is negative for photons with left-handed circular polarization.
This implies that the emission rate for right-handed polarized photons is
higher in the direction along the magnetic field, while for left-handed
polarized photons, it is higher in the direction opposite to the magnetic
field. Additionally, we find that the degree of asymmetry is most pronounced
at $\phi\simeq\pi/2$ and that its magnitude increases with increasing
transverse momentum. We also observe that the asymmetry grows with increasing
magnetic field strength but diminishes with rising temperature.

###  III.3 Nonzero quark-number and chiral chemical potentials $\mu$ and
$\mu_{5}$

After considering the effects of $\mu$ and $\mu_{5}$ separately in the
previous two subsections, let us now study their combined effect on polarized
photon emission. As one might expect, the emission is characterized by a
combined set of the same qualitative features, namely (i) the overall degree
of circular polarization, indicating that the emission of one circular
polarization dominates over the other and (ii) the degree of asymmetry for
each circular polarization with respect $\phi\to-\phi$, measuring how
asymmetric is the emission rate with respect to reflection in the transverse
plane.

To quantify the effects in the case of nonzero $\mu$ and $\mu_{5}$, we use the
same set of observables that we introduced earlier, namely ${\cal P}_{\rm
circ}(k_{T},\phi,y)$ in Eq. (16), ${\cal P}^{(+)}_{\rm asym}(k_{T},\phi,y)$ in
Eq. (17), and ${\cal P}^{(-)}_{\rm asym}(k_{T},\phi,y)$ in Eq. (18).

The numerical results for the overall degree of circular polarization in
photon emission are presented in Fig. 5. The magnitude of the effect is
comparable to that discussed in Sec. III.1, and the qualitative dependence on
the strength of the magnetic field, temperature, and transverse momentum is
similar. However, there is a significant difference too. The angular
dependence is not symmetric under the reflection $\phi\to-\phi$. The degree of
circular polarization is larger in magnitude in the direction opposite to the
magnetic field. Of course, this is expected since the dominant left-handed
emission is asymmetric.

Figure 5: The degree of circular polarization as a function of the azimuthal
angle for two magnetic fields, $|eB|=m_{\pi}^{2}$ (left) and
$|eB|=5m_{\pi}^{2}$ (right), and two temperatures, $T=200~{}\mbox{MeV}$ (top)
and $T=350~{}\mbox{MeV}$ (bottom).

The degrees of asymmetry for the emission of photons of both circular
polarizations are shown in Fig. 6 for two magnetic fields, $|eB|=m_{\pi}^{2}$
(left panels) and $|eB|=5m_{\pi}^{2}$ (right panels), and two temperatures,
$T=200~{}\mbox{MeV}$ (top panels) and $T=350~{}\mbox{MeV}$ (bottom panels).
For both temperatures, the degree of asymmetry for the right-handed emission
is larger than for the left-handed emission. The largest asymmetry is seen
around $\phi\simeq\pi/2$, with the magnitudes lying in the range from about
$0.12$ (right-handed) to $0.3$ (left-handed) at $T=200~{}\mbox{MeV}$ and from
about $0.08$ (right-handed) to $0.13$ (left-handed) at $T=350~{}\mbox{MeV}$.

Figure 6: The emission asymmetry for the right-handed (solid lines) and left-
handed (dashed lines) circularly polarized photons as a function of the
azimuthal angular for two magnetic fields, $|eB|=m_{\pi}^{2}$ (left) and
$|eB|=5m_{\pi}^{2}$ (right), and two temperatures, $T=200~{}\mbox{MeV}$ (top)
and $T=350~{}\mbox{MeV}$ (bottom).

##  IV Discussion and Summary

In this study, we considered the effect of nonzero quark-number (electric
charge) and chiral chemical potentials on the polarization of photon emission
from a strongly magnetized QGP. In particular, we showed that the composition
of circularly polarized photon emission changes in qualitative ways when the
charge densities are nonzero.

A nonzero quark-number (electrical charge) density produces an overall
dominance of one circular polarization over the other. For example, when the
QGP has a positive quark-number (electrical charge) chemical potential
($\mu>0$), the emission rate of left-handed circular polarization photons
dominates over the rate of right-handed circular polarization photons. In
effect, such polarized emission can be viewed as a photon equivalent of the
Hall effect in a plasma with a nonzero charge density.

It should be mention that the circularly polarized photon emission rates are
symmetric with respect to the reflection in the transverse (reaction) plane
when the electrical charge density is nonzero ($\mu\neq 0$), assuming the
chiral charge density vanishes ($\mu_{5}=0$). However, this changes at nonzero
chiral charge density ($\mu_{5}\neq 0$). When $\mu_{5}>0$, we show that the
emission rate of right-handed (left-handed) circularly polarized photons is
higher in the upper (lower) hemisphere. (Here the two hemispheres are defined
by the magnetic field direction.) The roles of the two circular polarizations
interchange when the sign of $\mu_{5}$ changes.

As expected, the combined effect of nonzero electrical and chiral charge
densities in the QPG is given by a superposition of their individual effects.
Generally, a nonzero electrical charge leads to an overall dominance of the
photon emission with one circular polarization over the other. The additional
chiral charge density produces spatial asymmetries (with respect to reflection
in the transverse plane) in photon emission for each circular polarization.
Moreover, while the rate for one circular polarization is higher in the upper
hemisphere, the rate for the other circular polarization is higher in the
lower hemisphere.

Our findings suggest that the composition of circularly polarized photon
emission and its asymmetry with respect to reflection in the transverse plane
can be used as unambiguous observable signatures of nonzero electrical and
chiral charge densities in a strongly magnetized plasma.

###### Acknowledgements.

We would like to thank Kirill Tuchin for valuable comments regarding the early
draft of the paper. The work of X. W. was supported by Anhui University of
Science and Technology under Grant No. YJ20240001. The work of I. A. S. was
supported in part by the U.S. National Science Foundation under Grant No.
PHY-2209470.

##  Appendix A Matsubara sums

In the calculation of the polarization tensor in the main text, we perform
Matsubara sums by using the following general result:

$\displaystyle
T\sum_{k=-\infty}^{\infty}\frac{(i\omega_{k}+\mu+s\mu_{5})(i\omega_{k}-i\Omega_{m}+\mu+s\mu_{5})X+(i\omega_{k}+\mu+s\mu_{5})Y_{1}+(i\omega_{k}-i\Omega_{m}+\mu+s\mu_{5})Y_{2}+Z}{\left[(i\omega_{k}+\mu+s\mu_{5})^{2}-a^{2}\right]\left[(i\omega_{k}-i\Omega_{m}+\mu+s\mu_{5})^{2}-b^{2}\right]}$
$\displaystyle=\sum_{\lambda=\pm 1}\sum_{\eta=\pm
1}\frac{\left[n_{F}(a+\eta\mu+\eta s\mu_{5})-n_{F}(\lambda b+\eta\mu+\eta
s\mu_{5})\right]}{4\lambda ab\left(a-\lambda b+\eta
i\Omega_{m}\right)}\left[\lambda abX-\eta\left(aY_{1}+\lambda
bY_{2}\right)+Z\right],$ (19)

where coefficient $X$, $Y_{1}$, $Y_{2}$, and $Z$ are arbitrary functions of
momenta. In the calculation of the polarization tensor, parameters $a$ and $b$
are replaced with the Landau level energies,
$E_{n,p_{z},f}=\sqrt{p_{z}^{2}+2n|e_{f}B|}$ and
$E_{n^{\prime},p_{z}-k_{z},f}=\sqrt{(p_{z}-k_{z})^{2}+2n^{\prime}|e_{f}B|}$.

Note that the expression in the parenthesis of the final result in Eq. (19)
can be formally obtained from the numerator of the original expression by
making the following replacements:
$(i\omega_{k}+s\mu_{5})(i\omega_{k}-i\Omega_{m}+s\mu_{5})\to\lambda ab$,
$(i\omega_{k}+s\mu_{5})\to-\eta a$, and
$(i\omega_{k}-i\Omega_{m}+s\mu_{5})\to-\eta\lambda b$.

##  Appendix B Dirac traces

In the derivation of the photon polarization function, one has to calculate
four different types of Dirac traces. For completeness, here we present the
corresponding results:

$\displaystyle T_{1,f}^{\mu\nu}(s_{1},s_{2})$ $\displaystyle=$
$\displaystyle\,\mbox{tr}\left\\{\gamma^{\mu}\left(\bar{p}_{\parallel}\gamma_{\parallel}-\mu_{5}\gamma^{0}\gamma^{5}\right)\left(\mathcal{P}_{+}^{f}L_{n}+\mathcal{P}_{-}^{f}L_{n-1}\right)\mathcal{P}_{5}^{s_{1}}\gamma^{\nu}\left((\bar{p}_{\parallel}-k_{\parallel})\gamma_{\parallel}-\mu_{5}\gamma^{0}\gamma^{5}\right)\left(\mathcal{P}_{+}^{f}L_{n^{\prime}}+\mathcal{P}_{-}^{f}L_{n^{\prime}-1}\right)\mathcal{P}_{5}^{s_{2}}\right\\},$
$\displaystyle T_{2,f}^{\mu\nu}(s_{1},s_{2})$ $\displaystyle=$
$\displaystyle\frac{i}{\ell_{f}^{2}}\,\mbox{tr}\left\\{\gamma^{\mu}\left(\bar{p}_{\parallel}\gamma_{\parallel}-\mu_{5}\gamma^{0}\gamma^{5}\right)\left(\mathcal{P}_{+}^{f}L_{n}+\mathcal{P}_{-}^{f}L_{n-1}\right)\mathcal{P}_{5}^{s_{1}}\gamma^{\nu}(\mathbf{r}_{\perp}\cdot\bm{\gamma}_{\perp})L_{n^{\prime}-1}^{1}\mathcal{P}_{5}^{s_{2}}\right\\},$
(21) $\displaystyle T_{3,f}^{\mu\nu}(s_{1},s_{2})$ $\displaystyle=$
$\displaystyle-\frac{i}{\ell_{f}^{2}}\,\mbox{tr}\left\\{\gamma^{\mu}(\mathbf{r}_{\perp}\cdot\bm{\gamma}_{\perp})L_{n-1}^{1}\mathcal{P}_{5}^{s_{1}}\gamma^{\nu}\left[(\bar{p}_{\parallel}-k_{\parallel})\gamma_{\parallel}-\mu_{5}\gamma^{0}\gamma^{5}\right]\left(\mathcal{P}_{+}^{f}L_{n^{\prime}}+\mathcal{P}_{-}^{f}L_{n^{\prime}-1}\right)\mathcal{P}_{5}^{s_{2}}\right\\},$
(22) $\displaystyle T_{4,f}^{\mu\nu}(s_{1},s_{2})$ $\displaystyle=$
$\displaystyle\frac{1}{\ell_{f}^{4}}\,\mbox{tr}\left\\{\gamma^{\mu}(\mathbf{r}_{\perp}\cdot\bm{\gamma}_{\perp})L_{n-1}^{1}\mathcal{P}_{5}^{s_{1}}\gamma^{\nu}(\mathbf{r}_{\perp}\cdot\bm{\gamma}_{\perp})L_{n^{\prime}-1}^{1}\mathcal{P}_{5}^{s_{2}}\right\\},$
(23)

where we introduced the shorthand notation
$\bar{p}_{\parallel}\gamma_{\parallel}=\bar{p}_{0}\gamma^{0}-p^{3}\gamma^{3}$
and $\bar{p}_{0}=p_{0}+\mu$. Also, for brevity of presentation, we omitted the
explicit dependence of all functions on their arguments, i.e.,
$L_{n}^{\alpha}\equiv L_{n}^{\alpha}\left(\zeta\right)$, where
$\zeta=r_{\perp}^{2}/(2\ell_{f}^{2})$.

##  Appendix C Polarization projections of the traces

Here we separate the results of traces in Eqs. (B) – (23) into individual
polarization contributions. By using the definition in Eq. (2), we introduce
the following polarization projections:

$\displaystyle\mathcal{T}_{0}^{f}(s_{1},s_{2})$ $\displaystyle=$
$\displaystyle\epsilon_{0}^{\mu}\epsilon_{0}^{\nu*}\sum_{i=1}^{4}T_{i,f}^{\mu\nu}(s_{1},s_{2}),$
(24) $\displaystyle\mathcal{T}_{-}^{f}(s_{1},s_{2})$ $\displaystyle=$
$\displaystyle-\epsilon_{-}^{\mu}\epsilon_{-}^{\nu*}\sum_{i=1}^{4}T_{i,f}^{\mu\nu}(s_{1},s_{2}),$
(25) $\displaystyle\mathcal{T}_{+}^{f}(s_{1},s_{2})$ $\displaystyle=$
$\displaystyle-\epsilon_{+}^{\mu}\epsilon_{+}^{\nu*}\sum_{i=1}^{4}T_{i,f}^{\mu\nu}(s_{1},s_{2}),$
(26) $\displaystyle\mathcal{T}_{\parallel}^{f}(s_{1},s_{2})$ $\displaystyle=$
$\displaystyle\epsilon_{\parallel}^{\mu}\epsilon_{\parallel}^{\nu*}\sum_{i=1}^{4}T_{i,f}^{\mu\nu}(s_{1},s_{2}).$
(27)

For each function, the result is nonzero only when $s_{1}=s_{2}$. The
corresponding nonvanishing results read

$\displaystyle\mathcal{T}_{0}^{f}(s,s)$ $\displaystyle=$
$\displaystyle\left[(\bar{p}_{0}-\mu_{5}-p_{z}s_{\perp}s)(\bar{p}_{0}-k_{0}-s\mu_{5})+(p_{z}-k_{z})\left(p_{z}-s_{\perp}s\bar{p}_{0}+s_{\perp}\mu_{5}\right)\right]L_{n}L_{n^{\prime}}$
(28) $\displaystyle+$
$\displaystyle\left[(\bar{p}_{0}-s\mu_{5}+p_{z}s_{\perp}s)(\bar{p}_{0}-k_{0}-s\mu_{5})+(p_{z}-k_{z})(p_{z}+s_{\perp}s\bar{p}_{0}-s_{\perp}s\mu_{5})\right]L_{n-1}L_{n^{\prime}-1}$
$\displaystyle+$
$\displaystyle\frac{2r_{\perp}^{2}}{\ell_{f}^{4}}L_{n-1}^{1}L_{n^{\prime}-1}^{1},$
$\displaystyle\mathcal{T}_{-}^{f}(s,s)$ $\displaystyle=$
$\displaystyle(s\bar{p}_{0}+p_{z}-\mu_{5})(sk_{0}-s\bar{p}_{0}-k_{z}+p_{z}+\mu_{5})\left[L_{n-1}L_{n^{\prime}}(1+s_{\perp})+L_{n}L_{n^{\prime}-1}(1-s_{\perp})\right],$
(29) $\displaystyle\mathcal{T}_{+}^{f}(s,s)$ $\displaystyle=$
$\displaystyle(s\bar{p}_{0}-p_{z}-\mu_{5})(sk_{0}-s\bar{p}_{0}+k_{z}-p_{z}+\mu_{5})\left[L_{n}L_{n^{\prime}-1}(1+s_{\perp})+L_{n-1}L_{n^{\prime}}(1-s_{\perp})\right],$
(30) $\displaystyle\mathcal{T}_{\parallel}^{f}(s,s)$ $\displaystyle=$
$\displaystyle-\mathcal{T}_{0}^{f}(s,s)+\frac{4L_{n-1}^{1}L_{n^{\prime}-1}^{1}r_{\perp}^{2}}{\ell_{f}^{4}}.$
(31)

##  Appendix D Integration over $r_{\perp}$

We have the following results for the integrals:

$\displaystyle\mathcal{J}^{f}_{0}(s,s)$ $\displaystyle=$ $\displaystyle\int
d^{2}\mathbf{r}_{\perp}e^{-i\mathbf{r}_{\perp}\cdot\mathbf{k}_{\perp}}e^{-\mathbf{r}_{\perp}^{2}/(2\ell_{f}^{2})}\mathcal{T}_{0}^{f}(s,s)$
(32) $\displaystyle=$ $\displaystyle
2\pi\ell_{f}^{2}\left[(\bar{p}_{0}-\mu_{5}-p_{z}s_{\perp}s)(\bar{p}_{0}-k_{0}-s\mu_{5})+(p_{z}-k_{z})\left(p_{z}-s_{\perp}s\bar{p}_{0}+s_{\perp}\mu_{5}\right)\right]\,\mathcal{I}_{0,f}^{n,n^{\prime}}$
$\displaystyle+$ $\displaystyle
2\pi\ell_{f}^{2}\left[(\bar{p}_{0}-s\mu_{5}+p_{z}s_{\perp}s)(\bar{p}_{0}-k_{0}-s\mu_{5})+(p_{z}-k_{z})(p_{z}+s_{\perp}s\bar{p}_{0}-s_{\perp}s\mu_{5})\right]\,\mathcal{I}_{0,f}^{n-1,n^{\prime}-1}(\xi)$
$\displaystyle+$ $\displaystyle
8\pi\,\mathcal{I}_{2,f}^{n-1,n^{\prime}-1}(\xi),$
$\displaystyle\mathcal{J}^{f}_{-}(s,s)$ $\displaystyle=$ $\displaystyle\int
d^{2}\mathbf{r}_{\perp}e^{-i\mathbf{r}_{\perp}\cdot\mathbf{k}_{\perp}}e^{-\mathbf{r}_{\perp}^{2}/(2\ell_{f}^{2})}\mathcal{T}_{+}^{f}(s,s)$
(33) $\displaystyle=$ $\displaystyle
2\pi\ell_{f}^{2}(s\bar{p}_{0}+p_{z}-\mu_{5})(sk_{0}-s\bar{p}_{0}-k_{z}+p_{z}+\mu_{5})\left[(1+s_{\perp})\mathcal{I}_{0,f}^{n-1,n^{\prime}}(\xi)+(1-s_{\perp})\mathcal{I}_{0,f}^{n,n^{\prime}-1}(\xi)\right],$
$\displaystyle\mathcal{J}^{f}_{+}(s,s)$ $\displaystyle=$ $\displaystyle\int
d^{2}\mathbf{r}_{\perp}e^{-i\mathbf{r}_{\perp}\cdot\mathbf{k}_{\perp}}e^{-\mathbf{r}_{\perp}^{2}/(2\ell_{f}^{2})}\mathcal{T}_{-}^{f}(s,s)$
(34) $\displaystyle=$ $\displaystyle
2\pi\ell_{f}^{2}(s\bar{p}_{0}-p_{z}-\mu_{5})(sk_{0}-s\bar{p}_{0}+k_{z}-p_{z}+\mu_{5})\left[(1+s_{\perp})\mathcal{I}_{0,f}^{n,n^{\prime}-1}(\xi)+(1-s_{\perp})\mathcal{I}_{0,f}^{n-1,n^{\prime}}(\xi)\right],$
$\displaystyle\mathcal{J}^{f}_{\parallel}(s,s)$ $\displaystyle=$
$\displaystyle\int
d^{2}\mathbf{r}_{\perp}e^{-i\mathbf{r}_{\perp}\cdot\mathbf{k}_{\perp}}e^{-\mathbf{r}_{\perp}^{2}/(2\ell_{f}^{2})}\mathcal{T}_{\parallel}^{f}(s,s)=8\pi\,\mathcal{I}_{2,f}^{n-1,n^{\prime}-1}(\xi)-\mathcal{J}^{f}_{0}(s,s).$
(35)

where $\xi=k_{\perp}^{2}\ell_{f}^{2}/2$ and functions

$\displaystyle\mathcal{I}_{0,f}^{n,n^{\prime}}(\xi)$ $\displaystyle=$
$\displaystyle\frac{(n^{\prime})!}{n!}e^{-\xi}\xi^{n-n^{\prime}}\left(L_{n^{\prime}}^{n-n^{\prime}}\left(\xi\right)\right)^{2}=\frac{n!}{(n^{\prime})!}e^{-\xi}\xi^{n^{\prime}-n}\left(L_{n}^{n^{\prime}-n}\left(\xi\right)\right)^{2},$
(36) $\displaystyle\mathcal{I}_{2,f}^{n,n^{\prime}}(\xi)$ $\displaystyle=$
$\displaystyle\frac{n+n^{\prime}+2}{2}\left[\mathcal{I}_{0,f}^{n,n^{\prime}}(\xi)+\mathcal{I}_{0,f}^{n+1,n^{\prime}+1}(\xi)\right]-\frac{\xi}{2}\left[\mathcal{I}_{0,f}^{n+1,n^{\prime}}(\xi)+\mathcal{I}_{0,f}^{n,n^{\prime}+1}(\xi)\right]$
(37)

are the same as those introduced in Ref. [22].

##  Appendix E Polarization projections of the imaginary part of
$\Pi^{\mu\nu}(\Omega;\mathbf{k})$

The polarization projections of the imaginary part of
$\Pi^{\mu\nu}(i\Omega_{m};\mathbf{k})$ are given by the following formal
expressions:

$\Pi^{(i)}(i\Omega_{m};\mathbf{k})=-\sum_{f=u,d}\frac{N_{c}\alpha_{f}T}{\pi\ell_{f}^{4}}\sum_{n,n^{\prime}=0}^{\infty}\sum_{k=-\infty}^{\infty}\sum_{s=\pm
1}\int\frac{dp_{z}}{2\pi}\frac{\mathcal{J}_{(i)}^{f}(-s,-s)}{[(i\omega_{k}+\mu+s\mu_{5})^{2}-E_{n,p_{z},f}^{2}][(i\omega_{k}+\mu-i\Omega_{m}+s\mu_{5})^{2}-E_{n^{\prime},p_{z}-k_{z},f}^{2}]}.$
(38)

By using Matsubara sum in Appendix A, we derive

$\Pi^{(i)}(i\Omega_{m};\mathbf{k})=-\sum_{f=u,d}\frac{N_{c}\alpha_{f}}{\pi\ell_{f}^{4}}\sum_{n,n^{\prime}=0}^{\infty}\sum_{s,\eta,\lambda=\pm
1}\int\frac{dp_{z}}{2\pi}\frac{[n_{F}(E_{n,p_{z},f}+\eta\mu+\eta
s\mu_{5})-n_{F}(\lambda E_{n^{\prime},p_{z}-k_{z},f}+\eta\mu+\eta
s\mu_{5})]\mathcal{F}^{(i)}_{s,f}(\xi,i\Omega_{m})}{4\lambda
E_{n,p_{z},f}E_{n^{\prime},p_{z}-k_{z},f}\left[E_{n,p_{z},f}-\lambda
E_{n^{\prime},p_{z}-k_{z},f}+i\eta\Omega_{m}\right]},$ (39)

where

$\displaystyle\mathcal{F}^{(0)}_{s,f}(\xi,i\Omega_{m})$ $\displaystyle=$
$\displaystyle 2\pi\ell_{f}^{2}\left(p_{z}-s_{\perp}s\eta
E_{n,p_{z},f}\right)\left[p_{z}-k_{z}-s_{\perp}s\eta\lambda
E_{n^{\prime},p_{z}-k_{z},f}\right]\mathcal{I}_{0,f}^{n,n^{\prime}}(\xi)$ (40)
$\displaystyle+$ $\displaystyle 2\pi\ell_{f}^{2}\left(p_{z}+s_{\perp}s\eta
E_{n,p_{z},f}\right)\left[p_{z}-k_{z}+s_{\perp}s\eta\lambda
E_{n^{\prime},p_{z}-k_{z},f}\right]\mathcal{I}_{0,f}^{n-1,n^{\prime}-1}(\xi)+4\pi\mathcal{I}_{2,f}^{n-1,n^{\prime}-1}(\xi),$
$\displaystyle\mathcal{F}^{(-)}_{s,f}(\xi,i\Omega_{m})$ $\displaystyle=$
$\displaystyle 2\pi\ell_{f}^{2}\left[p_{z}(p_{z}-k_{z})-k_{z}s\eta
E_{n,p_{z},f}-p_{z}si\Omega_{m}-\lambda
E_{n,p_{z},f}E_{n^{\prime},p_{z}-k_{z},f}\right]$ (41) $\displaystyle\times$
$\displaystyle\left[(1+s_{\perp})\mathcal{I}_{0,f}^{n-1,n^{\prime}}(\xi)+(1-s_{\perp})\mathcal{I}_{0,f}^{n,n^{\prime}-1}(\xi)\right],$
$\displaystyle\mathcal{F}^{(+)}_{s,f}(\xi,i\Omega_{m})$ $\displaystyle=$
$\displaystyle 2\pi\ell_{f}^{2}\left[p_{z}(p_{z}-k_{z})+k_{z}s\eta
E_{n,p_{z},f}+p_{z}si\Omega_{m}-\lambda
E_{n,p_{z},f}E_{n^{\prime},p_{z}-k_{z},f}\right]$ (42) $\displaystyle\times$
$\displaystyle\left[(1+s_{\perp})\mathcal{I}_{0,f}^{n,n^{\prime}-1}(\xi)+(1-s_{\perp})\mathcal{I}_{0,f}^{n-1,n^{\prime}}(\xi)\right],$
$\displaystyle\mathcal{F}^{(\parallel)}_{s,f}(\xi,i\Omega_{m})$
$\displaystyle=$
$\displaystyle-\mathcal{F}^{(0)}_{s,f}(\xi,i\Omega_{m})+8\pi\mathcal{I}_{2,f}^{n-1,n^{\prime}-1}(\xi).$
(43)

After replacing $i\Omega_{m}\to\Omega+i\epsilon$ and using the Sokhotski
formula, we extract the following imaginary part of the polarization
functions:

$\displaystyle\mbox{Im}\Pi^{(i)}(\Omega;\mathbf{k})$ $\displaystyle=$
$\displaystyle\sum_{f=u,d}\frac{N_{c}\alpha_{f}}{\ell_{f}^{4}}\sum_{n,n^{\prime}=0}^{\infty}\sum_{s,\eta,\lambda=\pm
1}\int\frac{dp_{z}}{2\pi}\frac{[n_{F}(E_{n,p_{z},f}+\eta\mu+\eta
s\mu_{5})-n_{F}(\lambda E_{n^{\prime},p_{z}-k_{z},f}+\eta\mu+\eta
s\mu_{5})]}{4\lambda E_{n,p_{z},f}E_{n^{\prime},p_{z}-k_{z},f}}$ (44)
$\displaystyle\times$
$\displaystyle\mathcal{F}_{s,f}^{(i)}(\xi,\Omega)\,\delta\left(E_{n,p_{z},f}-\lambda
E_{n^{\prime},p_{z}-k_{z},f}+\eta\Omega\right).$

The solutions of the energy conservation equation $E_{n,p_{z},f}-\lambda
E_{n^{\prime},p_{z}-k_{z},f}+\eta\Omega=0$ are given by the following explicit
expressions [21, 22, 27]:

$\displaystyle p_{z,f}^{(\pm)}$ $\displaystyle=$
$\displaystyle\frac{k_{z}}{2}\left[1+\frac{2(n-n^{\prime})|e_{f}B|}{\Omega^{2}-k_{z}^{2}}\pm\frac{\Omega}{|k_{z}|}\sqrt{\left(1-\frac{(k_{-}^{f})^{2}}{\Omega^{2}-k_{z}^{2}}\right)\left(1-\frac{(k_{+}^{f})^{2}}{\Omega^{2}-k_{z}^{2}}\right)}\right].$
(45)

The corresponding fermions energies, satisfying the energy conservation
relation, are

$\left.E_{n,p_{z},f}\right|_{p_{z}=p_{z,f}^{(\pm)}}=-\frac{\eta\Omega}{2}\left[1+\frac{2(n-n^{\prime})|e_{f}B|}{\Omega^{2}-k_{z}^{2}}\pm\frac{|k_{z}|}{\Omega}\sqrt{\left(1-\frac{(k_{-}^{f})^{2}}{\Omega^{2}-k_{z}^{2}}\right)\left(1-\frac{(k_{+}^{f})^{2}}{\Omega^{2}-k_{z}^{2}}\right)}\right],$
(46a)
$\left.E_{n^{\prime},p_{z}-k_{z},f}\right|_{p_{z}=p_{z,f}^{(\pm)}}=\frac{\lambda\eta\Omega}{2}\left[1-\frac{2(n-n^{\prime})|e_{f}B|}{\Omega^{2}-k_{z}^{2}}\mp\frac{|k_{z}|}{\Omega}\sqrt{\left(1-\frac{(k_{-}^{f})^{2}}{\Omega^{2}-k_{z}^{2}}\right)\left(1-\frac{(k_{+}^{f})^{2}}{\Omega^{2}-k_{z}^{2}}\right)}\right].$
(46b)

By making use of these solutions, we can easily perform the integration over
the longitudinal momentum $p_{z}$ in Eq. (44). The result reads

$\displaystyle\mbox{Im}\Pi^{(i)}(\Omega;\mathbf{k})$ $\displaystyle=$
$\displaystyle\sum_{f=u,d}\frac{N_{c}\alpha_{f}}{4\pi\ell_{f}^{4}}\sum_{n,n^{\prime}=0}^{\infty}\sum_{s,s^{\prime},\eta,\lambda=\pm
1}\Theta_{\lambda,\eta}^{n,n^{\prime}}(\Omega,k_{z})\mathcal{F}_{s,f}^{(i)}(\xi,\Omega)$
(47) $\displaystyle\times$
$\displaystyle\frac{n_{F}(E_{n,p_{z},f}+\eta\mu+\eta s\mu_{5})-n_{F}(\lambda
E_{n^{\prime},p_{z}-k_{z},f}+\eta\mu+\eta
s\mu_{5})}{\eta\lambda\sqrt{\left(\Omega^{2}-k_{z}^{2}-(k_{-}^{f})^{2}\right)\left(\Omega^{2}-k_{z}^{2}-(k_{+}^{f})^{2}\right)}}\Bigg{|}_{p_{z}=p_{z,f}^{(s^{\prime})}},$

where the threshold function
$\Theta_{\lambda,\eta}^{n,n^{\prime}}(\Omega,k_{z})$ is defined as follows:

$\Theta_{\lambda,\eta}^{n,n^{\prime}}(\Omega,k_{z})=\left\\{\begin{array}[]{lll}\theta\left((k_{-}^{f})^{2}+k_{z}^{2}-\Omega^{2}\right)&\mbox{for}&\lambda=1,\eta=-1,n>n^{\prime},\\\
\theta\left((k_{-}^{f})^{2}+k_{z}^{2}-\Omega^{2}\right)&\mbox{for}&\lambda=1,\eta=1,n<n^{\prime},\\\
\theta\left(\Omega^{2}-k_{z}^{2}-(k_{+}^{f})^{2}\right)&\mbox{for}&\lambda=-1,\eta=-1,\end{array}\right.$
(48)

and $\Theta_{\lambda,\eta}^{n,n^{\prime}}(\Omega,k_{z})=0$ otherwise. By
definition, $\theta(x)$ is the Heaviside step function.

The explicit expressions for functions in Eqs. (40) – (43) on the solutions to
the energy-momentum relation read

$\displaystyle\left.\mathcal{F}_{s,f}^{(0)}(\xi,\Omega)\right|_{p_{z}=p_{z,f}^{(s^{\prime})}}$
$\displaystyle=$
$\displaystyle\mathcal{A}_{s,s^{\prime},f}(\xi,\Omega)+\mathcal{B}_{f}(\xi,\Omega),$
(49)
$\displaystyle\left.\mathcal{F}_{s,f}^{(-)}(\xi,\Omega)\right|_{p_{z}=p_{z,f}^{(s^{\prime})}}$
$\displaystyle=$
$\displaystyle\pi\ell_{f}^{2}\left(\Omega^{2}-k_{z}^{2}\right)\left[1-\frac{2(n+n^{\prime})}{\left(\Omega^{2}-k_{z}^{2}\right)\ell_{f}^{2}}-ss^{\prime}\frac{k_{z}}{|k_{z}|}\sqrt{\left(1-\frac{(k_{-}^{f})^{2}}{\Omega^{2}-k_{z}^{2}}\right)\left(1-\frac{(k_{+}^{f})^{2}}{\Omega^{2}-k_{z}^{2}}\right)}\right]$
(50)
$\displaystyle\times\left[(1+s_{\perp})\mathcal{I}_{0,f}^{n-1,n^{\prime}}(\xi)+(1-s_{\perp})\mathcal{I}_{0,f}^{n,n^{\prime}-1}(\xi)\right],$
$\displaystyle\left.\mathcal{F}_{s,f}^{(+)}(\xi,\Omega)\right|_{p_{z}=p_{z,f}^{(s^{\prime})}}$
$\displaystyle=$
$\displaystyle\pi\ell_{f}^{2}\left(\Omega^{2}-k_{z}^{2}\right)\left[1-\frac{2(n+n^{\prime})}{\left(\Omega^{2}-k_{z}^{2}\right)\ell_{f}^{2}}+ss^{\prime}\frac{k_{z}}{|k_{z}|}\sqrt{\left(1-\frac{(k_{-}^{f})^{2}}{\Omega^{2}-k_{z}^{2}}\right)\left(1-\frac{(k_{+}^{f})^{2}}{\Omega^{2}-k_{z}^{2}}\right)}\right]$
(51)
$\displaystyle\times\left[(1-s_{\perp})\mathcal{I}_{0,f}^{n,n^{\prime}-1}(\xi)+(1-s_{\perp})\mathcal{I}_{0,f}^{n-1,n^{\prime}}(\xi)\right],$
$\displaystyle\left.\mathcal{F}_{s,f}^{(\parallel)}(\xi,\Omega)\right|_{p_{z}=p_{z,f}^{(s^{\prime})}}$
$\displaystyle=$
$\displaystyle-\mathcal{A}_{s,s^{\prime},f}(\xi,\Omega)+\mathcal{B}_{f}(\xi,\Omega).$
(52)

Here, we used the following two auxiliary functions:

$\displaystyle\mathcal{A}_{s,s^{\prime},f}(\xi,\Omega)$ $\displaystyle=$
$\displaystyle
2\pi\frac{\Omega^{2}+k_{z}^{2}}{\Omega^{2}-k_{z}^{2}}\Bigg{\\{}\frac{2(n-n^{\prime})^{2}}{\left(\Omega^{2}-k_{z}^{2}\right)\ell_{f}^{2}}-(n+n^{\prime})+\frac{2s^{\prime}\Omega|k_{z}|(n-n^{\prime})}{\Omega^{2}+k_{z}^{2}}$
(53) $\displaystyle\times$
$\displaystyle\sqrt{\left(1-\frac{(k_{-}^{f})^{2}}{\Omega^{2}-k_{z}^{2}}\right)\left(1-\frac{(k_{+}^{f})^{2}}{\Omega^{2}-k_{z}^{2}}\right)}\Bigg{\\}}\left[\mathcal{I}_{0,f}^{n,n^{\prime}}(\xi)+\mathcal{I}_{0,f}^{n-1,n^{\prime}-1}(\xi)\right]$
$\displaystyle+$ $\displaystyle\frac{4\pi s_{\perp}s\Omega
k_{z}}{\Omega^{2}-k_{z}^{2}}\Bigg{\\{}\frac{2(n-n^{\prime})^{2}}{\left(\Omega^{2}-k_{z}^{2}\right)\ell_{f}^{2}}-(n+n^{\prime})+\frac{s^{\prime}(n-n^{\prime})(\Omega^{2}+k_{z}^{2})}{2\Omega|k_{z}|}$
$\displaystyle\times$
$\displaystyle\sqrt{\left(1-\frac{(k_{-}^{f})^{2}}{\Omega^{2}-k_{z}^{2}}\right)\left(1-\frac{(k_{+}^{f})^{2}}{\Omega^{2}-k_{z}^{2}}\right)}\Bigg{\\}}\left[\mathcal{I}_{0,f}^{n,n^{\prime}}(\xi)-\mathcal{I}_{0,f}^{n-1,n^{\prime}-1}(\xi)\right],$
$\displaystyle\mathcal{B}_{f}(\xi,\Omega)$ $\displaystyle=$ $\displaystyle
4\pi\left\\{(n+n^{\prime})\left[\mathcal{I}_{0,f}^{n-1,n^{\prime}-1}(\xi)+\mathcal{I}_{0,f}^{n,n^{\prime}}(\xi)\right]-\xi\left[\mathcal{I}_{0,f}^{n,n^{\prime}-1}(\xi)+\mathcal{I}_{0,f}^{n-1,n^{\prime}}(\xi)\right]\right\\}.$
(54)

Note that $\mathcal{B}_{f}(\xi,\Omega)$ is independent of $s$ and
$s^{\prime}$.

## References

  * Skokov  _et al._ [2009] V. Skokov, A. Y. Illarionov, and V. Toneev, Estimate of the magnetic field strength in heavy-ion collisions, Int. J. Mod. Phys. A24, 5925 (2009), arXiv:0907.1396 .
  * Voronyuk  _et al._ [2011] V. Voronyuk, V. Toneev, W. Cassing, E. Bratkovskaya, V. Konchakovski, _et al._ , (Electro-)magnetic field evolution in relativistic heavy-ion collisions, Phys. Rev. C 83, 054911 (2011), arXiv:1103.4239 .
  * Deng and Huang [2012] W.-T. Deng and X.-G. Huang, Event-by-event generation of electromagnetic fields in heavy-ion collisions, Phys. Rev. C 85, 044907 (2012), arXiv:1201.5108 .
  * Bloczynski  _et al._ [2013] J. Bloczynski, X.-G. Huang, X. Zhang, and J. Liao, Azimuthally fluctuating magnetic field and its impacts on observables in heavy-ion collisions, Phys. Lett. B718, 1529 (2013), arXiv:1209.6594 .
  * Guo  _et al._ [2020] X. Guo, J. Liao, and E. Wang, Spin Hydrodynamic Generation in the Charged Subatomic Swirl, Sci. Rep. 10, 2196 (2020), arXiv:1904.04704 [hep-ph] .
  * Jiang  _et al._ [2016] Y. Jiang, Z.-W. Lin, and J. Liao, Rotating quark-gluon plasma in relativistic heavy ion collisions, Phys. Rev. C 94, 044910 (2016), [Erratum: Phys.Rev.C 95, 049904 (2017)], arXiv:1602.06580 [hep-ph] .
  * Deng and Huang [2016] W.-T. Deng and X.-G. Huang, Vorticity in Heavy-Ion Collisions, Phys. Rev. C 93, 064907 (2016), arXiv:1603.06117 [nucl-th] .
  * Sirunyan  _et al._ [2018] A. M. Sirunyan _et al._ (CMS), Constraints on the chiral magnetic effect using charge-dependent azimuthal correlations in $p\mathrm{Pb}$ and PbPb collisions at the CERN Large Hadron Collider, Phys. Rev. C 97, 044912 (2018), arXiv:1708.01602 [nucl-ex] .
  * Acharya  _et al._ [2020] S. Acharya _et al._ (ALICE), Constraining the Chiral Magnetic Effect with charge-dependent azimuthal correlations in Pb-Pb collisions at $\sqrt{s_{\mathrm{NN}}}$ = 2.76 and 5.02 TeV, J. High Energy Phys. 09, 160, arXiv:2005.14640 [nucl-ex] .
  * Abdallah  _et al._ [2022] M. S. Abdallah _et al._ (STAR), Search for the Chiral Magnetic Effect via Charge-Dependent Azimuthal Correlations Relative to Spectator and Participant Planes in Au+Au Collisions at $\sqrt{s_{NN}}$ = 200 GeV, Phys. Rev. Lett. 128, 092301 (2022), arXiv:2106.09243 [nucl-ex] .
  * Aboona  _et al._ [2023] B. Aboona _et al._ (STAR), Search for the Chiral Magnetic Effect in Au+Au collisions at $\sqrt{s_{{}_{\rm{NN}}}}=27$ GeV with the STAR forward Event Plane Detectors, Phys. Lett. B 839, 137779 (2023), arXiv:2209.03467 [nucl-ex] .
  * Tuchin [2013] K. Tuchin, Particle production in strong electromagnetic fields in relativistic heavy-ion collisions, Adv. High Energy Phys. 2013, 490495 (2013), arXiv:1301.0099 [hep-ph] .
  * Kharzeev  _et al._ [2016] D. E. Kharzeev, J. Liao, S. A. Voloshin, and G. Wang, Chiral magnetic and vortical effects in high-energy nuclear collisions—A status report, Prog. Part. Nucl. Phys. 88, 1 (2016), arXiv:1511.04050 .
  * Huang [2016] X.-G. Huang, Electromagnetic fields and anomalous transports in heavy-ion collisions — A pedagogical review, Rept. Prog. Phys. 79, 076302 (2016), arXiv:1509.04073 .
  * Miransky and Shovkovy [2015] V. A. Miransky and I. A. Shovkovy, Quantum field theory in a magnetic field: From quantum chromodynamics to graphene and Dirac semimetals, Phys. Rept. 576, 1 (2015), arXiv:1503.00732 .
  * Kharzeev  _et al._ [2024] D. E. Kharzeev, J. Liao, and P. Tribedy, Chiral Magnetic Effect in Heavy Ion Collisions: The Present and Future, (2024), arXiv:2405.05427 [nucl-th] .
  * Abdulhamid  _et al._ [2023] M. I. Abdulhamid _et al._ (STAR), Event-by-event correlations between $\Lambda$ ($\Lambda$¯) hyperon global polarization and handedness with charged hadron azimuthal separation in Au+Au collisions at $\sqrt{s_{{}_{\rm{NN}}}}=27$ GeV from STAR, Phys. Rev. C 108, 014909 (2023), arXiv:2304.10037 [nucl-ex] .
  * Yee [2013] H.-U. Yee, Flows and polarization of early photons with magnetic field at strong coupling, Phys. Rev. D 88, 026001 (2013), arXiv:1303.3571 [nucl-th] .
  * Mamo and Yee [2013] K. A. Mamo and H.-U. Yee, Spin polarized photons and dileptons from axially charged plasma, Phys. Rev. D 88, 114029 (2013), arXiv:1307.8099 [nucl-th] .
  * Mamo and Yee [2016] K. A. Mamo and H.-U. Yee, Spin polarized photons from an axially charged plasma at weak coupling: Complete leading order, Phys. Rev. D 93, 065053 (2016), arXiv:1512.01316 [hep-ph] .
  * Wang  _et al._ [2020] X. Wang, I. A. Shovkovy, L. Yu, and M. Huang, Ellipticity of photon emission from strongly magnetized hot QCD plasma, Phys. Rev. D 102, 076010 (2020), arXiv:2006.16254 .
  * Wang and Shovkovy [2021a] X. Wang and I. Shovkovy, Photon polarization tensor in a magnetized plasma: Absorptive part, Phys. Rev. D 104, 056017 (2021a), arXiv:2103.01967 [nucl-th] .
  * Kapusta and Gale [2011] J. I. Kapusta and C. Gale,  _Finite-Temperature Field Theory: Principles and Applications_, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2011).
  * Hattori  _et al._ [2021] K. Hattori, H. Taya, and S. Yoshida, Di-lepton production from a single photon in strong magnetic fields: vacuum dichroism, J. High Energy Phys. 01, 093, arXiv:2010.13492 [hep-ph] .
  * Gradshteyn and Ryzhik [1980] I. S. Gradshteyn and I. M. Ryzhik,  _Table of integrals, series and products_ (Academic Press, Orlando, 1980).
  * Maxfield [1969] B. W. Maxfield, Helicon waves in solids, Am. J. Phys. 37, 241 (1969).
  * Wang and Shovkovy [2021b] X. Wang and I. Shovkovy, Polarization tensor of magnetized quark-gluon plasma at nonzero baryon density, Eur. Phys. J. C 81, 901 (2021b), arXiv:2106.09029 [nucl-th] .


# Quantum physics-informed neural networks
for simulating computational fluid dynamics in complex shapes

Alexandr Sedykh  Terra Quantum AG, 9000 St. Gallen, Switzerland  Maninadh
Podapaka  Evonik Industries AG, 45128 Essen, Germany  Asel Sagingalieva  Terra
Quantum AG, 9000 St. Gallen, Switzerland  Nikita Smertyak  Terra Quantum AG,
9000 St. Gallen, Switzerland  Karan Pinto  Terra Quantum AG, 9000 St. Gallen,
Switzerland  Markus Pflitsch  Terra Quantum AG, 9000 St. Gallen, Switzerland
Alexey Melnikov  Terra Quantum AG, 9000 St. Gallen, Switzerland

###### Abstract

Finding the distribution of the velocities and pressures of a fluid (by
solving the Navier-Stokes equations) is a principal task in the chemical,
energy, and pharmaceutical industries, as well as in mechanical engineering
and the design of pipeline systems. With existing solvers, such as OpenFOAM
and Ansys, simulations of fluid dynamics in intricate geometries are
computationally expensive and require re-simulation whenever the geometric
parameters or the initial and boundary conditions are altered. Physics-
informed neural networks (PINNs) are a promising tool for simulating fluid
flows in complex geometries, as they can adapt to changes in the geometry and
mesh definitions, allowing for generalization across different shapes. We
present a hybrid quantum physics-informed neural network that simulates
laminar fluid flows in 3D Y-shaped mixers. Our approach combines the
expressive power of a quantum model with the flexibility of a PINN, resulting
in a 21% higher accuracy compared to a purely classical neural network. Our
findings highlight the potential of machine learning approaches, and in
particular quantum PINNs, for complex shape optimization tasks in
computational fluid dynamics. By improving the accuracy of fluid simulations
in complex geometries, our research using quantum PINNs contributes to the
development of more efficient and reliable fluid dynamics solvers.

##  I Introduction

Computational fluid dynamics (CFD) solvers are primarily used to find the
distribution of the velocity vector, $\bm{v}$, and pressure, $p$, of a fluid
(or several fluids) given the initial conditions (e.g. initial velocity
profile) and the geometrical domain in which the fluid flows [1, 2]. To do
this, it is necessary to solve a system of differential equations [3] called
the Navier-Stokes (NS) equations that govern the fluid flow [4]. A well-
established approach is to use numerical CFD solvers from several vendors, as
well as publicly accessible alternatives, such as OpenFOAM [5] or Ansys [6].
These solvers discretize a given fluid volume into several small parts known
as cells [7], where it is easier to get an approximate solution and then join
the solutions of all the cells to get a complete distribution of the pressure
and velocity over the entire geometrical domain.

While this is a rather crude explanation of how CFD solvers actually work, the
idea of discretizing a large domain into smaller pieces accurately captures
one of their main working principles [8]. The runtime of the computation and
the accuracy of the solution both sensitively depend on the fineness of
discretization, with finer grids taking longer but giving more accurate
solutions. Furthermore, any changes to the geometrical parameters necessitate
the creation of a new mesh and a new simulation. This process consumes both
time and resources since one has to remesh and rerun the simulation every time
a geometrical parameter is altered [9].

We propose a workflow employing physics-informed neural networks (PINNs) [10]
to escape the need to completely restart the simulations whenever a
geometrical property is changed. A PINN is a new promising tool for solving
all kinds of parameterized partial differential equations (PDEs) [9], as it
does not require many prior assumptions, linearization or local time-stepping.
One defines an architecture of the neural network (number of neurons, layers,
etc.) and then embeds physical laws and boundary conditions into it via
constructing an appropriate loss function, so the prior task immediately
travels to the domain of optimization problems.

For the classical solver, in the case of a parameterized geometric domain
problem, getting accurate predictions for new modified shapes requires a
complete restart of the program, even if the geometry has changed slightly. In
the case of a PINN, to overcome this difficulty, one can use the transfer
learning method [11] (Sec. III.1.2), which allows a model previously trained
on some geometry to be trained on a slightly modified geometry without the
need for a complete reset.

Also, using a trained PINN, it is easy to obtain a solution for other
parameters of the PDE equation (e.g. kinematic viscosity in the NS equation
[4], thermal conductivity in the heat equation, etc.) with no additional
training or restart of the neural network, but in the case of traditional
solvers, restarts cannot be avoided.

One of the features which makes PINNs appealing is that they suffer less from
the curse of dimensionality. Finite discretization of a $d$-dimensional cube
with $N$ points along each axis would require $N^{d}$ points for a traditional
solver. In other words, the complexity of the problem grows exponentially as
the sampling size $d$ increases. Using a neural network, however, one can
define a $\mathbb{R}^{d}\rightarrow\mathbb{R}$ mapping (in case of just one
target feature) with some weight parameters. Research on the topic suggests
that the amount of weights/complexity of a problem in such neural networks
grows polynomially with the input dimension $d$ [12, 13]. This theoretical
foundation alone allows PINNs to be a competitive alternative to solvers.

It is worth noting that although a PINN does not require $N^{d}$ points for
inference, it does require many points for training. There exist a variety of
sampling methods, such as latin hypercube sampling [14], Sobol sequences [15],
etc., which can be used to improve a PINN’s convergence on training [16].
However, a simple static grid of points is used in this work for the purpose
of simplicity.

Classical machine learning can benefit substantially from quantum
technologies. In [17], quantum computing is used in a similar problem setting.
The performance of the current classical models is constrained by the high
computing requirements. Quantum computing models can improve the learning
process of existing classical models [18, 19, 20, 21, 22, 23], allowing for
better target function prediction accuracy with fewer iterations [24]. In many
industries, including the pharmaceutical [25, 26], space [27], automotive
[28], and financial [29, 30, 31, 32, 33] sector quantum technologies can
provide unique advantages over classical computing. Many traditionally
important machine learning domains are also getting potential benefits from
utilizing quantum technologies, e.g., in image processing [34, 35, 36, 37] and
natural language processing [38, 39, 40, 41].

Recent developments in automatic differentiation enable us to compute the
exact derivatives of any order of a PINN, so there is no need to use finite
differences or any other approximate differentiation techniques. It therefore
seems that we do not require a discretized mesh over the computational domain.
However, we still need a collection of points from the problem’s domain to
train and evaluate a PINN. For a PINN to provide an accurate solution for a
fluid dynamics problem, it is important to have a high expressivity (ability
to learn solutions for a large variety of, possibly complex, problems).
Fortunately, expressivity is a known strength of quantum computers [42].
Furthermore, quantum circuits are differentiable, which means that their
derivatives can be calculated analytically, which is essential on noisy
intermediate-scale quantum devices.

In this article, we propose a quantum PINN to solve the NS equations with a
steady flow in 3D Y-shape mixer. The general principles and loss function of
PINN workflow are described in Sec. II. The problem description, including the
geometrical details, is presented in Sec. III while in Sec. III.1 and Sec.
III.2, we describe classical and quantum PINNs in detail. Sec. III.1.1
explains the intricacies of PINN’s training process and simulation results. A
transfer learning approach, applied to PINNs, is presented in Sec. III.1.2.
Conclusions and further plans are described in Sec. IV.

##  II Physics-informed neural networks for solving partial differential
equations

Physics-informed neural networks (PINNs) were originally introduced in [10].
The main idea is to use a neural network - usually a feedforward neural
network like a multilayer perceptron - as a trial function for a PDE’s
solution. Let us consider an abstract PDE:

$\mathcal{D}[f(\bm{r},t);\lambda]=0,$ (1)

where $\Omega\subset\mathbb{R}^{d}$ is the computational domain,
$\bm{r}\in\Omega$ is a coordinate vector, $t\in\mathbb{R}$ is time,
$\mathcal{D}$ is a nonlinear differential operator with $\lambda$ standing in
for the physical parameters of the fluid and $f(\bm{r},t)$ is a solution
function.

Let us consider a neural network $u(\bm{r},t)$ that takes coordinates,
$\bm{r}$, and time, $t$, as input and yields some real value (e.g. the
pressure of a liquid at this coordinate at this particular moment of time).

We can evaluate $u(\bm{r},t)$ at any point in the computational domain via a
forward pass and compute its derivatives (of any order)
$\partial_{t}^{n}u(\bm{r},t)$, $\partial_{\bm{r}}^{n}u(\bm{r},t)$ through
backpropagation [43]. Therefore, we could just substitute
$f(\bm{r},t)=u(\bm{r},t)$ and try to learn the correct solution for the PDE
via common machine learning gradient optimization methods [44] (e.g., gradient
descent).

This approach is inspired firstly by the ability to calculate the exact
derivatives of a neural network via autodifferentiation [45] and secondly by
neural networks being universal function approximators [46].

The loss, $\mathcal{L}$, that the PINN tries to minimize is defined as

$\mathcal{L}=\mathcal{L}_{\text{PDE}}+\mathcal{L}_{\text{BC}},$ (2)

where $\mathcal{L}_{\text{BC}}$ is the boundary condition loss and
$\mathcal{L}_{\text{PDE}}$ is the partial differential equation loss.

The boundary condition loss is responsible for satisfying the boundary
conditions of the problem (e.g., a fixed pressure on the outlet of a pipe).
For any field value, $u$, let us consider a Dirichlet (fixed-type) boundary
condition [47]

$u(\bm{r},t)|_{\bm{r}\in B}=u_{0}(\bm{r},t),$ (3)

where $u_{0}(\bm{r},t)$ is a boundary condition and $B\subset\mathbb{R}^{d}$
is the region where a boundary condition is applied.

If $u(\bm{r},t)$ is a neural network function (see Sec. II), the boundary
condition loss is calculated in a mean-squared error (MSE) manner:

$\mathcal{L}_{\text{BC}}=\langle(u(\bm{r},t)-u_{0}(\bm{r},t))^{2}\rangle_{B},$
(4)

where $\langle\cdot\rangle_{B}$ denotes averaging over all the data points
$\bm{r}\in B$ that have this boundary condition.

[width=1]figures/Fig_1.pdf

Figure 1:  Scheme for the implementation of the hybrid PINN’s training process
and its architecture. PINN takes the $(x,y,z)$ coordinates of the points,
sampled in the geometrical domain, and yields $(\bm{v},p)$ as output, where
$\bm{v}$ is the velocity vector and $p$ is the pressure. The neural network
itself begins with a classical part, which is a multilayer perceptron with $l$
layers of $n$ neurons. Then there is a quantum layer (a variational quantum
circuit). “Filter” divides input points into 4 groups, where each group has
its own constraint (a boundary condition or a PDE). The error of each
constraint is calculated and added to the total loss, which is minimized via
gradient descent. The training process is described in Sec. III.1.1.

The PDE loss is responsible for solving the governing PDE. If we have an
abstract PDE (1) and a neural network function $u(\bm{r},t)$, substituting
$f(\bm{r},t)=u(\bm{r},t)$ and calculating the mean-squared error of the PDE
gives:

$\mathcal{L}_{\text{PDE}}=\langle(\mathcal{D}[u(\bm{r},t);\lambda])^{2}\rangle_{\Omega},$
(5)

where $\langle\cdot\rangle_{\Omega}$ means averaging over all the data points
in the domain of the PDE.

##  III Simulations

In this work, we consider the steady (i.e., time-independent) flow of an
incompressible fluid in 3D without any external forces.

The NS equations (6) and the continuity equation (7) describe this scenario as
follows:

$-(\bm{v}\cdot\nabla)\bm{v}+\nu\Delta\bm{v}-\frac{1}{\rho}\nabla p=0,$ (6)
$\nabla\cdot\bm{v}=0,$ (7)

where $\bm{v}(\bm{r})$ is the velocity vector, $p(\bm{r})$ is the pressure,
$\nu$ is the kinematic viscosity and $\rho$ is the fluid density. The PDE
parameters $\nu$ and $\rho$ were previously referred to as $\lambda$. For each
of the 4 PDEs (3 projections of vector equation 6 and 1 scalar equation 7),
the $\mathcal{L}_{\text{PDE}}$ is calculated separately and then summed up.

Our task is to simulate the flow of liquid in a Y-shaped mixer consisting of
three tubes (Fig. 2). The mixing fluids are identical and have parameters
$\rho=1.0\ \text{kg}/\text{m}^{3}$ and $\nu=1.0\ \text{m}^{2}/\text{s}$.

The imposed boundary conditions are as follows:

  * •

no-slip BC on walls: $\bm{v}(\bm{r})|_{\text{walls}}=0$,

  * •

fixed velocity profile on inlets:
$\bm{v}(\bm{r})|_{\text{inlets}}=\bm{v}_{0}(\bm{r})$,

  * •

fixed pressure on outlets: $p(\bm{r})|_{\text{outlets}}=p_{0}$,

where, in this work, $\bm{v}_{0}(\bm{r})$ are parabolic velocity profiles on
each inlet and $p_{0}(\bm{r})=0$.

###  III.1 Classical PINN

[width=1]figures/Fig_2.pdf

Figure 2: 1) Classical model. (a) Loss curve. The classical PINN managed to
learn solution near the entry point well. However, it vanishes to zero further
down the mixer. 2) Transfer learned models. (a) Loss curve. The transfer-
learned models trained better with each iteration, surpassing the original
model. (b, c, d) Distributions of the fluid velocity for the last model with
$\alpha=35^{\circ}$.

In this section, we provide details on the PINN’s architecture, training
process, and simulation results. The PINN is a neural network whose
architecture is a multilayer perceptron with several fully connected layers.
As shown in Fig. 1, the first layer consists of $3$ neurons (since the problem
is 3D), then there are $l=5$ hidden layers with $n=64$ neurons, and the
penultimate layer has $16$ neurons. For the classical PINN, the “quantum
layer” box is replaced with one fully-connected layer $16\rightarrow 4$. There
is no quantum layer in the classical case, so the output goes straight into
the filter. Between adjacent layers, there is a sigmoid linear unit (SiLU)
activation function [48]. The PINN takes the $(x,y,z)$ coordinates as inputs
and yields $(\bm{v},p)$ as its output, where $\bm{v}$ is the velocity vector
with three components and $p$ is the pressure.

####  III.1.1 Training

The geometry is a collection of points organized in a .csv file. It is split
into four groups: fluid domain, walls, inlets, and outlets. The fluid domain
is the domain in which the NS equations are solved, i.e. where the fluid
flows. The other three groups have boundary conditions described in Sec. II.

While untrained, PINN produces some random distribution of velocities and
pressures. These values and their gradients are substituted into the
corresponding NS equations and boundary conditions. With every iteration, the
weights of the neural network are updated to minimize the error in the
governing equations, and our solution becomes more and more accurate.

The PINN is trained via full-batch gradient descent with the Adam optimizer
[49] for $1000$ epochs, and then with L-BFGS optimizer [50] for $100$ epochs.
All points from the geometrical domain are processed at once, so one step of
the optimizer equals one epoch. The training iteration is simple: the point
cloud is passed through the PINN, the MSE loss is calculated (getting it
requires taking gradients of ($\bm{v},p$) at each point of the geometry), the
gradient of the loss with respect to the weights is taken and the parameters
are updated.

Classical PINN training (1100 total epochs) took $5$ minutes on single NVIDIA
A100 GPU. To visualize the training outcomes of the neural network, we used
ParaView [51] and the simulation results are shown in Fig. 2 (1) for loss and
velocity distribution.

After the training, the PINN manages to learn a non-trivial downward flow at
the beginning of both pipes. On the edges of these pipes, the velocities
become zero, as they should due to no-slip wall boundary conditions. However,
further down the mixer, the solution degenerates to zero, so it does not even
reach the mixing point. This fact should at least violate the continuity
equation because there is an inward flow without matching outward flow, but
the model still treats this situation as an optimum and refuses to propagate
the solution further. Possible reasons for this could be a mismatch between
the scales of the different loss terms (continuity, NS and BC) in the total
loss, the limited expressivity of the underlying neural network, the points
sampling strategy or the choice of the optimizer. Still, we stick to our
simple workflow and use this incomplete solution as a reference to further
explore the properties of the PINN without overcomplicating the problem.

####  III.1.2 Transfer Learning

Transfer learning is a powerful method of using the knowledge and experience
of one model that has been pretrained on one problem to solve another problem
[11]. It is extremely useful because it means that a second model does not
have to be trained from scratch. This is especially helpful for fluid modeling
where the crucial issue of selecting the most appropriate geometrical
hyperparameters would otherwise lead to the simulations being rerun many
times.

For transfer learning, we used a model from the previous section as a base,
which had $\alpha_{0}=30^{\circ}$, where $\alpha$ is the angle between the
right pipe and the $x$ axis (see Fig. 2). Then, for each
$\alpha=\\{31^{\circ},32^{\circ},33^{\circ},34^{\circ},35^{\circ}\\}$, we
tried to obtain the solution, each time using the previously trained model as
a initializer. For example, to transfer learn from $31^{\circ}$ to
$32^{\circ}$, we used the $31^{\circ}$ model as a base and so on. Each
iteration is trained for $100$ epochs with L-BFGS. Fig. 2 shows that the PINN
adapts well to changes in the value of $\alpha$. That is, our hypothesis was
correct: with PINNs, one does not need to completely rerun the simulation on a
parameter change, transfer learning from the base model will suffice.

###  III.2 Hybrid Quantum PINN

[width=1]figures/Fig_3.pdf

Figure 3: Hybrid quantum model. (a) Loss curve. (b, c, d) Distribution of the
fluid velocities for the hybrid PINN at ($\alpha=30^{\circ}$). Similarly to
the classical model, a non-trivial solution near the entry point is present.
However, there is an asymmetry between the left and the right pipe, which
should not be there. This could be an effect of the data encoding strategy.

Quantum machine learning may help to improve the performance of the classical
PINN. We propose a quantum PINN, which is a hybrid quantum-classical neural
network. In the Noisy Intermediate-Scale Quantum (NISQ) era, when quantum
computers do not have access to a quantum memory or a large number of qubits,
the algorithms that are known to be superior to classical ones cannot be
implemented. Heurisitc quantum algorithms, however, have less stringent
requirements for their implementation. Here, we use a variational quantum
circuit (VQC) approach, which is one of the most promising NISQ algorithms
today. It combines the best ideas of classical and quantum machine learning
[52, 53, 54, 25, 55].

At the moment, using only VQCs is not enough to solve large problems,
especially for industrial tasks, so several more classical neural layers are
used with a VQC (hence the name, “hybrid”). Also, the use of a large number of
qubits and layer repetitions leads to the barren plateau problem [56, 57].

As an overview of the workflow, preliminary data processing takes place on a
classical computer. This data is then encoded into the parameters of the
quantum gates of an encoding layer of the VQC, followed by a variational
layer. As the algorithm progresses, the gate parameters in the variational
layer of the VQC are varied and finally, in the measurement layer of the VQC
the qubits are measured, producing a set of classical bits as the output.

A quantum algorithm can be represented as a kind of black box into which
classical information enters, classical information comes out. The goal is to
choose such variational parameters so that the measurement result is as
accurate as possible in terms of the prediction function. In this way, finding
the optimal gate parameters is like finding the optimal weights in a classical
neural network. Hence, we refer to this as training the VQC.

####  III.2.1 Variational Quantum Circuit

Quantum gates are the basic building blocks for any quantum circuit, including
those used for machine learning. Quantum gates come in single-qubit (e.g.
rotation gate $R_{y}(\theta)$, gate that plays a central role in quantum
machine learning) and multiple-qubit gates (e.g. CNOT) and modulate the state
of qubits to perform computations. The $R_{y}(\theta)$ gate rotates the qubit
around the y-axis of the Bloch sphere by an angle $\theta$, while the two-
qubit CNOT gate changes the state of one qubit based on the current state of
another qubit. Gates can be fixed, which means they perform fixed
calculations, such as the Hadamard gate, or they can be variable, such as the
rotation gate that depends on the rotation angle and may perform computations
with tunable parameters.

To get the results, the qubits are measured (i.e., projected onto some basis)
and the expected value is calculated. If we use the $\sigma_{z}$ Pauli matrix
observable, the expected value of that measurement is expressed as:
$\bra{\psi}\sigma_{z}\ket{\psi}$, where $\psi$ is the wave function which
describes the current state of our quantum system. An introduction to quantum
circuits, including logic gates and measurements, can be found in standard
quantum computing textbooks, such as [58].

####  III.2.2 Training

The hybrid PINN consists of the classical PINN with weights preinitialized
from the previous stage (as described in Sec. III.1), a VQC and a fully-
connected layer at the end. The VQC’s design is inspired by [59].

The training process of the hybrid PINN does not differ from that of the
classical PINN except in the following ways. Firstly, all calculations are
done on the classical simulator of quantum hardware, the QMware server [60],
which has recently been shown to be quite good for running hybrid algorithms
[61].

Secondly, how does one backpropagate through the quantum circuit layer? The
answer is to use the “adjoint differentiation” method, introduced in [62],
which helps to efficiently compute derivatives of a VQC on a classical
simulator.

This time the model was trained for $100$ epochs using mini-batch gradient
descent with the Adam optimizer (Fig. 3). The mini-batch strategy was employed
due to the very low speed of training of quantum circuits, as they train on a
CPU. We will then compare this model with a purely classical one, with exactly
the same architecture from Sec. III.1, but this time trained only with mini-
batch Adam. All learning hyperparameters (learning rate, scheduler parameters,
batch size) are shared between the quantum and classical model. Comparing the
two, Fig. 4 shows that the quantum model outperforms the classical one in
terms of the loss value by $21\%$.

We tried to do inference of our model on real quantum processing units (QPUs)
Lucy [63] and Rigetti’s Aspen-M-3 [64]. That means, we have chosen some points
from the geometrical domain and predicted velocity and pressure fields for
them. On comparison, results from both QPUs greatly differ from the simulator
ones. Most likely, these errors are caused by noise and decoherence, which are
hard to mitigate. Also, particular QPUs work better with circuits that take
into account their particular topology and gate implementations. We did not
consider such limitations, as the primary tests of our quantum circuit were
conducted on a simulator.

[width=0.97]figures/Fig_4.pdf

Figure 4:  Losses for the classical and hybrid quantum PINNs. Hybrid quantum
PINN outperforms the classical PINN in terms of loss value due to higher
expressivity.

##  IV Discussion

In this work we examine a recently introduced [10] way of solving
computational fluid simulation tasks, PINNs. This approach is based on
substituting the solution function of the problem with a neural network of an
arbitrary architecture (a multi-layer perceptron is used in this work) and
optimizing trainable parameters via gradient descent (Adam, L-BFGS). In this
problem setting, the loss function becomes a combination of the residual error
of corresponding PDE and boundary conditions. We try to use this approach to
solve a laminar mixing problem in a 3D Y-shaped geometrical domain and achieve
plausible solution in the upper part of the mixer. The solution, however,
vanishes to zero as it goes lower through the mixer. The reason behind such
behaviour could be insufficient expressivity of the model, static point
sampling or even the choice of the optimizers.

We then explore a transfer learning technique with application to a classical
PINN, which shows that the PINN is actually capable of generalizing an
existing solution to a weakly changed geometrical domain with little
additional training. This enables PINNs to be very helpful in shape
optimization tasks because traditional solvers lack this feature and demand a
full restart on any parameter change. Also, with each step of the angle
$\alpha$, we see that the PINN achieves an even lower loss than before. It may
be that this step-by-step learning helps the PINN to mitigate overfitting to a
particular geometrical domain of the problem.

Afterwards, we introduce a new PINN framework – a quantum PINN. We build it on
top of a classical fully-conntected PINN by adding a quantum circuit with
trainable gates. We explore the potential of this model to solve the NS
equations and show its superiority to purely classical network in terms of the
loss value on 21%, which could mean that the chosen quantum circuit increases
the PINN’s expressiveness.

The future plan includes exploring better architectures for the quantum PINN,
investigating their impact on expressiveness, generalizability and
optimization landscape, as well as trying out data-driven approaches. Entirely
different networks, such as neural operators [65, 66] and graph neural
networks [67, 68], could also be considered in a quantum setting and enhanced
with quantum circuits.

## References

  * [1] Mohd Hafiz Zawawi, A Saleha, A Salwa, NH Hassan, Nazirul Mubin Zahari, Mohd Zakwan Ramli, and Zakaria Che Muda.  A review: Fundamentals of computational fluid dynamics (CFD).  In AIP conference proceedings. AIP Publishing LLC, 2018.
  * [2] John David Anderson and John Wendt.  Computational fluid dynamics, volume 206.  Springer, 1995.
  * [3] George Finlay Simmons.  Differential equations with applications and historical notes.  In Differential Equations With Applications and Historical Notes. McGraw-Hill, 1972.
  * [4] Aaron Jon Katz.  Meshless methods for computational fluid dynamics.  Stanford University, 2009.
  * [5] OpenFOAM.  https://www.openfoam.com/, 2022.
  * [6] Ansys.  https://www.ansys.com/, 2022.
  * [7] Tomislav Marić, Douglas B. Kotheb, and Dieter Bothe.  Unstructured un-split geometrical volume-of-fluid methods - A review.  Journal of Computational Physics, 420, 2020.
  * [8] Tomislav Marić, Holger Marschall, and Dieter Bothe.  vofoam - a geometrical volume of fluid algorithm on arbitrary unstructured meshes with local dynamic adaptive mesh refinement using OpenFOAM.  arXiv preprint arXiv: 1305.3417, 2013.
  * [9] Shengze Cai, Zhiping Mao, Zhicheng Wang, Minglang Yin, and George Em Karniadakis.  Physics-informed neural networks (PINNs) for fluid mechanics: A review.  Acta Mechanica Sinica, pages 1–12, 2022.
  * [10] Maziar Raissi, Paris Perdikaris, and George E Karniadakis.  Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations.  Journal of Computational Physics, 378:686–707, 2019.
  * [11] Behnam Neyshabur, Hanie Sedghi, and Chiyuan Zhang.  What is being transferred in transfer learning?  arXiv preprint arXiv:2008.11687, 2020.
  * [12] Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse, and Tuan Anh Nguyen.  A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations.  SN partial differential equations and applications, 1:1–34, 2020\.
  * [13] Philipp Grohs, Fabian Hornung, Arnulf Jentzen, and Philippe Von Wurstemberger.  A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations.  arXiv preprint arXiv:1809.02362, 2018.
  * [14] Michael Stein.  Large sample properties of simulations using latin hypercube sampling.  Technometrics, 29(2):143–151, 1987.
  * [15] Ilya M Sobol’, Danil Asotsky, Alexander Kreinin, and Sergei Kucherenko.  Construction and comparison of high-dimensional Sobol’generators.  Wilmott, 2011(56):64–79, 2011.
  * [16] Kirill Zubov, Zoe McCarthy, Yingbo Ma, Francesco Calisto, Valerio Pagliarino, Simone Azeglio, Luca Bottero, Emmanuel Luján, Valentin Sulzer, Ashutosh Bharambe, et al.  Neuralpde: Automating physics-informed neural networks (PINNs) with error approximations.  arXiv preprint arXiv:2107.09443, 2021.
  * [17] Frank Gaitan.  Finding flows of a Navier–Stokes fluid through quantum computing.  npj Quantum Information, 6(1):1–6, 2020.
  * [18] Vedran Dunjko and Hans J Briegel.  Machine learning & artificial intelligence in the quantum domain: a review of recent progress.  Reports on Progress in Physics, 81(7):074001, 2018.
  * [19] Alexey Melnikov, Mohammad Kordzanganeh, Alexander Alodjants, and Ray-Kuang Lee.  Quantum machine learning: from physics to software engineering.  Advances in Physics: X, 8(1):2165452, 2023.
  * [20] Hartmut Neven, Vasil S. Denchev, Geordie Rose, and William G. Macready.  QBoost: Large scale classifier training with adiabatic quantum optimization.  In Steven C. H. Hoi and Wray Buntine, editors, Proc. Asian Conf. Mach. Learn., volume 25 of Proceedings of Machine Learning Research, pages 333–348. PMLR, 2012.
  * [21] Patrick Rebentrost, Masoud Mohseni, and Seth Lloyd.  Quantum support vector machine for big data classification.  Physical Review Letters, 113:130503, Sep 2014.
  * [22] Valeria Saggio, Beate E Asenbeck, Arne Hamann, Teodor Strömberg, Peter Schiansky, Vedran Dunjko, Nicolai Friis, Nicholas C Harris, Michael Hochberg, Dirk Englund, et al.  Experimental quantum speed-up in reinforcement learning agents.  Nature, 591(7849):229–233, 2021.
  * [23] Mohammad Kordzanganeh, Daria Kosichkina, and Alexey Melnikov.  Parallel hybrid networks: an interplay between quantum and classical neural networks.  arXiv preprint arXiv:2303.03227, 2023.
  * [24] Michael Perelshtein, Asel Sagingalieva, Karan Pinto, Vishal Shete, Alexey Pakhomchik, Artem Melnikov, Florian Neukart, Georg Gesek, Alexey Melnikov, and Valerii Vinokur.  Practical application-specific advantage through hybrid quantum computing.  arXiv preprint arXiv:2205.04858, 2022.
  * [25] Asel Sagingalieva, Mohammad Kordzanganeh, Nurbolat Kenbayev, Daria Kosichkina, Tatiana Tomashuk, and Alexey Melnikov.  Hybrid quantum neural network for drug response prediction.  arXiv preprint arXiv:2211.05777, 2022.
  * [26] A. I. Gircha, A. S. Boev, K. Avchaciov, P. O. Fedichev, and A. K. Fedorov.  Training a discrete variational autoencoder for generative chemistry and drug design on a quantum annealer.  arXiv:2108.11644, 2021.
  * [27] Serge Rainjonneau, Igor Tokarev, Sergei Iudin, Saaketh Rayaprolu, Karan Pinto, Daria Lemtiuzhnikova, Miras Koblan, Egor Barashov, Mohammad Kordzanganeh, Markus Pflitsch, et al.  Quantum algorithms applied to satellite mission planning for Earth observation.  arXiv preprint arXiv:2302.07181, 2023.
  * [28] Asel Sagingalieva, Andrii Kurkin, Artem Melnikov, Daniil Kuhmistrov, Michael Perelshtein, Alexey Melnikov, Andrea Skolik, and David Von Dollen.  Hyperparameter optimization of hybrid quantum neural networks for car classification.  arXiv preprint arXiv:2205.04878, 2022.
  * [29] Javier Alcazar, Vicente Leyton-Ortega, and Alejandro Perdomo-Ortiz.  Classical versus quantum models in machine learning: insights from a finance application.  Machine Learning: Science and Technology, 1(3):035003, 2020.
  * [30] Brian Coyle, Maxwell Henderson, Justin Chan Jin Le, Niraj Kumar, Marco Paini, and Elham Kashefi.  Quantum versus classical generative modelling in finance.  Quantum Science and Technology, 6(2):024013, 2021.
  * [31] Marco Pistoia, Syed Farhan Ahmad, Akshay Ajagekar, Alexander Buts, Shouvanik Chakrabarti, Dylan Herman, Shaohan Hu, Andrew Jena, Pierre Minssen, Pradeep Niroula, Arthur Rattew, Yue Sun, and Romina Yalovetzky.  Quantum Machine Learning for Finance.  arXiv preprint arXiv:2109.04298, 2021.
  * [32] Dimitrios Emmanoulopoulos and Sofija Dimoska.  Quantum Machine Learning in Finance: Time Series Forecasting.  arXiv preprint arXiv:2202.00599, 2022.
  * [33] El Amine Cherrat, Snehal Raj, Iordanis Kerenidis, Abhishek Shekhar, Ben Wood, Jon Dee, Shouvanik Chakrabarti, Richard Chen, Dylan Herman, Shaohan Hu, Pierre Minssen, Ruslan Shaydulin, Yue Sun, Romina Yalovetzky, and Marco Pistoia.  Quantum Deep Hedging.  arXiv preprint arXiv:2303.16585, 2023.
  * [34] Arsenii Senokosov, Alexander Sedykh, Asel Sagingalieva, and Alexey Melnikov.  Quantum machine learning for image classification.  arXiv preprint arXiv:2304.09224, 2023.
  * [35] Wei Li, Peng-Cheng Chu, Guang-Zhe Liu, Yan-Bing Tian, Tian-Hui Qiu, and Shu-Mei Wang.  An Image Classification Algorithm Based on Hybrid Quantum Classical Convolutional Neural Network.  Quantum Engineering, 2022:1–9, 2022.
  * [36] A Naumov, Ar Melnikov, V Abronin, F Oxanichenko, K Izmailov, M Pflitsch, A Melnikov, and M Perelshtein.  Tetra-AML: Automatic machine learning via tensor networks.  arXiv preprint arXiv:2303.16214, 2023.
  * [37] Farina Riaz, Shahab Abdulla, Hajime Suzuki, Srinjoy Ganguly, Ravinesh C. Deo, and Susan Hopkins.  Accurate Image Multi-Class Classification Neural Network Model with Quantum Entanglement Approach.  Sensors, 23(5):2753, 2023.
  * [38] Zhenhou Hong, Jianzong Wang, Xiaoyang Qu, Chendong Zhao, Wei Tao, and Jing Xiao.  QSpeech: Low-Qubit Quantum Speech Application Toolkit.  arXiv preprint arXiv:2205.13221, 2022.
  * [39] Robin Lorenz, Anna Pearson, Konstantinos Meichanetzidis, Dimitri Kartsaklis, and Bob Coecke.  QNLP in Practice: Running Compositional Models of Meaning on a Quantum Computer.  arXiv preprint arXiv:2102.12846, 2021.
  * [40] Bob Coecke, Giovanni de Felice, Konstantinos Meichanetzidis, and Alexis Toumi.  Foundations for Near-Term Quantum Natural Language Processing.  arXiv preprint arXiv:2012.03755, 2020.
  * [41] Konstantinos Meichanetzidis, Alexis Toumi, Giovanni de Felice, and Bob Coecke.  Grammar-Aware Question-Answering on Quantum Computers.  arXiv preprint arXiv:2012.03756, 2020.
  * [42] Oleksandr Kyriienko, Annie E Paine, and Vincent E Elfving.  Solving nonlinear differential equations with differentiable quantum circuits.  Physical Review A, 103(5):052416, 2021.
  * [43] David E Rumelhart, Geoffrey E Hinton, and Ronald J Williams.  Learning representations by back-propagating errors.  Nature, 323(6088):533–536, 1986.
  * [44] Sebastian Ruder.  An overview of gradient descent optimization algorithms.  arXiv preprint arXiv: 1609.04747, 2016.
  * [45] Atilim Gunes Baydin, Barak A. Pearlmutter, Alexey Andreyevich Radul, and Jeffrey Mark Siskind.  Automatic differentiation in machine learning: a survey.  Journal of Machine Learning Research, 18(153):1–43, 2018.
  * [46] Kurt Hornik, Maxwell B. Stinchcombe, and Halbert L. White.  Multilayer feedforward networks are universal approximators.  Neural Networks, 2:359–366, 1989.
  * [47] C Greenshields and H Weller.  Notes on computational fluid dynamics: General principles.  CFD Direct Ltd.: Reading, UK, 2022.
  * [48] Stefan Elfwing, Eiji Uchibe, and Kenji Doya.  Sigmoid-weighted linear units for neural network function approximation in reinforcement learning.  Neural Networks, 107:3–11, 2018.
  * [49] Diederik P Kingma and Jimmy Ba.  Adam: A method for stochastic optimization.  arXiv preprint arXiv:1412.6980, 2014.
  * [50] Dong C Liu and Jorge Nocedal.  On the limited memory BFGS method for large scale optimization.  Mathematical programming, 45(1-3):503–528, 1989.
  * [51] ParaView.  https://www.paraview.org/, 2022.
  * [52] Andrea Mari, Thomas R. Bromley, Josh Izaac, Maria Schuld, and Nathan Killoran.  Transfer learning in hybrid classical-quantum neural networks.  Quantum, 4:340, 2020.
  * [53] Chen Zhao and Xiao-Shan Gao.  QDNN: DNN with quantum neural network layers.  arXiv preprint arXiv:1912.12660, 2019.
  * [54] Tong Dou, Kaiwei Wang, Zhenwei Zhou, Shilu Yan, and Wei Cui.  An unsupervised feature learning for quantum-classical convolutional network with applications to fault detection.  In 2021 40th Chinese Control Conference (CCC), pages 6351–6355. IEEE, 2021.
  * [55] Mohammad Kordzanganeh, Pavel Sekatski, Leonid Fedichkin, and Alexey Melnikov.  An exponentially-growing family of universal quantum circuits.  arXiv preprint arXiv:2212.00736, 2022.
  * [56] Jarrod R. McClean, Sergio Boixo, Vadim N. Smelyanskiy, Ryan Babbush, and Hartmut Neven.  Barren plateaus in quantum neural network training landscapes.  Nature Communications, 9(1):4812, 2018.
  * [57] Marco Cerezo, Akira Sone, Tyler Volkoff, Lukasz Cincio, and Patrick J Coles.  Cost function dependent barren plateaus in shallow parametrized quantum circuits.  Nature Communications, 12(1):1–12, 2021.
  * [58] Michael A Nielsen and Isaac Chuang.  Quantum computation and quantum information, 2002.
  * [59] Maria Schuld, Ryan Sweke, and Johannes Jakob Meyer.  Effect of data encoding on the expressive power of variational quantum-machine-learning models.  Physical Review A, 103(3):032430, 2021.
  * [60] QMware.  https://qm-ware.com/, 2022.
  * [61] Mohammad Kordzanganeh, Markus Buchberger, Maxim Povolotskii, Wilhelm Fischer, Andrii Kurkin, Wilfrid Somogyi, Asel Sagingalieva, Markus Pflitsch, and Alexey Melnikov.  Benchmarking simulated and physical quantum processing units using quantum and hybrid algorithms.  arXiv preprint arXiv:2211.15631, 2022.
  * [62] Tyson Jones and Julien Gacon.  Efficient calculation of gradients in classical simulations of variational quantum algorithms.  arXiv preprint arXiv:2009.02823, 2020.
  * [63] Lucy.  https://oxfordquantumcircuits.com/, 2022.
  * [64] Rigetti.  https://qcs.rigetti.com/, 2022.
  * [65] Nikola Kovachki, Zongyi Li, Burigede Liu, Kamyar Azizzadenesheli, Kaushik Bhattacharya, et al.  Neural operator: Learning maps between function spaces.  arXiv preprint arXiv:2108.08481, 2021.
  * [66] Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, et al.  Fourier neural operator for parametric partial differential equations.  arXiv preprint arXiv:2010.08895, 2020.
  * [67] Tobias Pfaff, Meire Fortunato, Alvaro Sanchez-Gonzalez, and Peter W. Battaglia.  Learning mesh-based simulation with graph networks.  In International Conference on Learning Representations, 2021.
  * [68] Alvaro Sanchez-Gonzalez, Jonathan Godwin, Tobias Pfaff, Rex Ying, Jure Leskovec, and Peter Battaglia.  Learning to simulate complex physics with graph networks.  In International Conference on Machine Learning, pages 8459–8468. PMLR, 2020.