# Electromagnetic bremsstrahlung and energy loss in chiral medium

Jeremy Hansen  Kirill Tuchin  Department of Physics and Astronomy, Iowa State
University, Ames, Iowa, 50011, USA

###### Abstract

We study electromagnetic radiation by a fast particle carrying electric charge
in chiral medium. The medium is homogeneous and isotropic and supports the
chiral magnetic current which renders the fermion and photon states unstable.
The instability manifests as the chirality-dependent resonances in the
bremsstrahlung cross section, which enhance the energy loss in the chiral
medium. We compute the corresponding cross sections in the single scattering
approximation and derive the energy loss in the high energy approximation.

##  I Introduction

This paper is the third in the series of papers dedicated to the problem of
energy loss by a fast particle in the chiral medium due to the electromagnetic
interactions. The pivotal feature of chiral media is the peculiar response to
the external electromagnetic field caused by the chiral anomaly [1, 2]. In
homogeneous media, it is given by the chiral magnetic current
${\bm{j}}=b_{0}{\bm{B}}$ [3, 4, 5, 6, 7], where the chiral magnetic
conductivity $b_{0}$ is assumed to be a constant. It was discussed in the
context of Weyl and Dirac semimetals [8, 9, 10], the quark-gluon plasma [11,
12] and the axion phenomenology [13]. In our first paper [14] we computed the
collisional energy loss and found that at high energies it is dominated by the
chiral Cherenkov radiation. In the second paper [15] we considered the part of
the radiative energy loss that is driven by the resonance at
${\bm{q}}^{2}=b_{0}^{2}$ in the spatial part of the photon propagator
$D_{ij}(q)$. The corresponding contribution to the bremsstrahlung cross
section is proportional to the magnetic moment of the scatterer because
$D_{ij}$ couple only to the spatial part of the source current whose leading
multipole term is the magnetic moment $\textgoth{M}$. It will be further
referred to as the “magnetic channel”. The emergence of the resonance is a
signature of the chiral magnetic instability [16, 17, 18]. Another
manifestation of this instability is the appearance of the chiral run-away
modes in the modified photon dispersion relation. At high energies the
anomalous term in the photon dispersion relation is small and can be neglected
compared to the main contribution stemming from the pole in $D_{ij}$. This is
to say that the resonance in the magnetic bremsstrahlung cross section is
weakly dependent on the radiated photon kinematics.

In this article we consider the photon bremsstrahlung in the chiral medium due
to the Coulomb part $D_{00}(q)$ of the photon propagator. We dub it the
“electric channel”. Although $D_{00}(q)$ itself has no conspicuous dependence
on $b_{0}$, the resonance does emerge in the photon propagator due to the
anomaly in the photon dispersion relation. It causes the momentum transfer
${\bm{q}}^{2}$ to vanish at finite photon emission angles $\theta$. Thus,
unlike the magnetic channel considered in [15], the electric channel is driven
exclusively by the anomaly in the photon dispersion relation. Along with the
photon propagator, the fermion one also becomes resonant, indicating
instability of fermions in the chiral medium [17, 18]. The main goal of this
paper is to compute the photon bremsstrahlung cross section in the chiral
medium due to the Coulomb term in the photon propagator.

The paper is structured as follows. In Sec. II we derive the general
expression for the scattering cross section. In Sec. III we observe that the
cross section is divergent due to the resonances in the photon and fermion
propagators. The divergences are regulated by the finite width of the unstable
modes which is proportional to the fermion relaxation rate $\tau^{-1}$.
Significantly, the same parameter regulates the divergences in the fermion and
photon propagators in the electric channel (but not in the magnetic one
discussed in [15]). The final expression for the scattering cross section is
quite bulky and is given in Appendix A. The high energy limit, relevant in
most applications, is developed in Sec. IV where the resonant structures
become very clear. Particularly simple expressions for the cross section are
derived at low and high temperatures. Our main results are Eqs. (26), (IV.1)
and (34),(IV.2) for the bremsstrahlung cross section in various limits. The
results for the corresponding energy loss are given in Sec. V. The summary and
outlook are presented in Sec. VI.

##  II scattering cross section

Following the original Bethe and Heitler calculation [19], we consider
scattering of a charged fermion off a heavy nucleus of mass $M$, electric
charge $eZ$ and magnetic moment $\textgoth{M}$. The differential cross section
for photon radiation is given by the familiar expression

$\displaystyle
d\sigma=\frac{1}{2}\sum_{ss^{\prime}\lambda}\left|\mathcal{M}\right|^{2}\frac{1}{8(2\pi)^{5}}\frac{\omega|{\bm{p}}^{\prime}|}{|{\bm{p}}|}d\Omega_{\bm{k}}d\Omega^{\prime}d\omega\,,$
(1)

where $p=(E,{\bm{p}})$, $p^{\prime}=(E^{\prime},{\bm{p}}^{\prime})$ and
$k=(\omega,{\bm{k}})$ are the incoming fermion, outgoing fermion, and radiated
photon momenta respectively. The matrix element $\mathcal{M}$ is represented
by two Feynman diagrams shown in Fig. 1.

Figure 1: The two diagrams corresponding to the matrix element for the
scattering cross-section. Double lines indicate photons in chiral medium.

The corresponding analytical expression reads:

$\displaystyle\mathcal{M}=e^{2}\overline{u}(p^{\prime})\left(\not{e}_{k\lambda}^{*}\frac{\not{p}^{\prime}+\not{k}+m}{2p^{\prime}\cdot
k+k^{2}}\not{A}({\bm{q}})-\not{A}(\bm{q})\frac{\not{p}-\not{k}+m}{2p\cdot
k-k^{2}}\not{e}_{k\lambda}^{*}\right)u(p)\,,$ (2)

where $A({\bm{q}})$ indicates the external field, $q=p^{\prime}-p+k$ is the
momentum transfer, $m$ is the mass of the fermion, and $e_{k\lambda}$ is the
photon polarization vector. Whereas in vacuum the photon four-momentum is
light-like, in chiral media it is not. Moreover, in the Lorentz gauge, photon
is forced to be in one of the two circularly polarized states. This eliminates
the residual gauge invariance that is reflected in the Ward identity [15]. The
photon dispersion relation is

$\displaystyle\omega^{2}={\bm{k}}^{2}+k^{2}={\bm{k}}^{2}-\lambda
b_{0}|{\bm{k}}|\,,$ (3)

where $\lambda=\pm 1$ is right or left photon polarization.

The electromagnetic potential induced by the the electric current $J^{\mu}$,
associated with the nucleus, in a chiral medium is
$A_{\mu}=-iD_{\mu\nu}J^{\nu}$. The photon propagator in chiral medium in the
Lorentz/Landau gauge take form [15]:

$\displaystyle
D_{\mu\nu}(q)=-i\frac{q^{2}g_{\mu\nu}+i\epsilon_{\mu\nu\rho\sigma}b^{\rho}q^{\sigma}+b_{\mu}b_{\nu}}{q^{4}+b^{2}q^{2}-(b\cdot
q)^{2}}+i\frac{\left[q^{2}-(b\cdot q)^{2}/q^{2}\right]q_{\mu}q_{\nu}+b\cdot
q(b_{\mu}q_{\nu}+b_{\nu}q_{\mu})}{q^{2}\left[q^{4}+b^{2}q^{2}-(b\cdot
q)^{2}\right]}\,,$ (4)

where $b_{\mu}=(b_{0},{\bm{0}})$. In the static limit $q^{0}=0$ the components
of the photon propagator read [20]

$\displaystyle D_{00}({\bm{q}})=\frac{i}{{\bm{q}}^{2}}\,,$ (5a) $\displaystyle
D_{0i}({\bm{q}})=D_{0i}({\bm{q}})=0\,,$ (5b) $\displaystyle
D_{ij}({\bm{q}})=-\frac{i\delta_{ij}}{{\bm{q}}^{2}-b_{0}^{2}}-\frac{\epsilon_{ijk}q^{k}}{b_{0}({\bm{q}}^{2}-b_{0}^{2})}+\frac{\epsilon_{ijk}q^{k}}{b_{0}{\bm{q}}^{2}}+\frac{iq_{i}q_{j}}{{\bm{q}}^{2}({\bm{q}}^{2}-b_{0}^{2})}\,.$
(5c)

The gauge-dependent terms proportional to $q_{\mu}$ and $q_{\nu}$ vanish when
substituted into the scattering amplitude. The spatial components (5c) couple
to the nucleus magnetic moment and have a resonance at
${\bm{q}}^{2}=b_{0}^{2}$. We analysed this magnetic channel in our previous
paper [15]. In this paper we are interested in the monopole component of the
external field which is determined by the nuclear electric charge. Convolution
of the current
$J^{\nu}({\bm{x}})=eZ\delta\indices{{}^{\nu}_{0}}\delta({\bm{x}})$ with the
$D_{00}$ component of the photon propagator gives rise to the Coulomb
potential:

$\displaystyle
A^{0}({\bm{q}})=eZ/{\bm{q}}^{2}\,,\qquad{\bm{A}}({\bm{q}})=0\,.$ (6)

Plugging (6) into (2) and averaging over the fermion spin directions, we can
obtain a fairly compact expression for the differential cross section of the
electric channel:

$\displaystyle
d\sigma=\frac{Z^{2}}{2}\sum_{\lambda}\frac{d\Omega_{\bm{k}}d\Omega^{\prime}d\omega}{4(2\pi)^{5}}\frac{|{\bm{p}}^{\prime}|}{|{\bm{p}}|}\frac{e^{6}}{\omega{\bm{q}}^{4}}\Bigg{\\{}4\left|\frac{Ep^{\prime}\cdot
e_{\lambda}}{\kappa^{\prime}+\frac{k^{2}}{2\omega}}-\frac{E^{\prime}p\cdot
e_{\lambda}}{\kappa-\frac{k^{2}}{2\omega}}\right|^{2}-{\bm{q}}^{2}\left|\frac{p^{\prime}\cdot
e_{\lambda}}{\kappa^{\prime}+\frac{k^{2}}{2\omega}}-\frac{p\cdot
e_{\lambda}}{\kappa-\frac{k^{2}}{2\omega}}\right|^{2}$
$\displaystyle+\frac{\omega^{2}({\bm{q}}^{2}-(\kappa-\kappa^{\prime}-\frac{k^{2}}{\omega})^{2})}{(\kappa-\frac{k^{2}}{2\omega})(\kappa^{\prime}+\frac{k^{2}}{2\omega})}$
$\displaystyle+\frac{k^{2}}{4}\Bigg{[}{\bm{q}}^{2}\left(\frac{1}{\kappa-\frac{k^{2}}{2\omega}}-\frac{1}{\kappa^{\prime}+\frac{k^{2}}{2\omega}}\right)^{2}+\frac{4|p\cdot
e_{\lambda}-p^{\prime}\cdot
e_{\lambda}|^{2}}{(\kappa-\frac{k^{2}}{2\omega})(\kappa^{\prime}+\frac{k^{2}}{2\omega})}-4\left(\frac{E^{\prime}}{\kappa-\frac{k^{2}}{2\omega}}-\frac{E}{\kappa^{\prime}+\frac{k^{2}}{2\omega}}\right)^{2}\Bigg{]}\Bigg{\\}}\,,$
(7)

where $\kappa=p\cdot k/\omega$ and $\kappa^{\prime}=p^{\prime}\cdot k/\omega$.
In (II) the sum over $\lambda$ is left explicit in order to isolate individual
photon polarizations. Eq. (II) reduces to the Bether-Heitler formula in the
limit $|{\bm{k}}|\rightarrow\omega$, or, equivalently, $k^{2}\rightarrow 0$.

The photon plane waves can only have a circular polarization which satisfies
the transversality condition $k\cdot e_{k}={\bm{k}}\cdot{\bm{e}}_{\lambda}=0$
in the Lorentz gauge. We seek computing the cross section for each photon
polarization. To perform the angular integrals in (II) it is advantageous to
eliminate the photon polarization vectors in favor of particle momenta. To
this end, we introduce a Cartesian reference frame with the $z$-axis pointing
in the direction of the photon momentum ${\bm{k}}$:

$\displaystyle{\bm{k}}=(0,0,|{\bm{k}}|)\,,\quad{\bm{e}}_{\lambda}=\frac{1}{\sqrt{2}}(1,i\lambda,0)\,,\quad{\bm{p}}=(p_{\perp},0,p_{\parallel})\,,\quad{\bm{p}}^{\prime}=(p^{\prime}_{\perp}\cos{\phi},p^{\prime}_{\perp}\sin{\phi},p^{\prime}_{\parallel})\,,$
(8)

where $\phi$ is the azimuthal angle. We derive the following identities:

$\displaystyle|{\bm{p}}\cdot{\bm{e}}_{\lambda}|^{2}$
$\displaystyle=\frac{|{\bm{p}}\times{\bm{k}}|^{2}}{2{\bm{k}}^{2}}\,,$ (9a)
$\displaystyle|{\bm{p}}^{\prime}\cdot{\bm{e}}_{\lambda}|^{2}$
$\displaystyle=\frac{|{\bm{p}}^{\prime}\times{\bm{k}}|^{2}(\cos^{2}\phi+\lambda^{2}\sin^{2}\phi)}{2{\bm{k}}^{2}}=\frac{|{\bm{p}}^{\prime}\times{\bm{k}}|^{2}}{2{\bm{k}}^{2}}\,,$
(9b)
$\displaystyle({\bm{p}}\cdot{\bm{e}}_{\lambda})({\bm{p}}^{\prime}\cdot{\bm{e}}_{\lambda}^{*})$
$\displaystyle=\frac{|{\bm{p}}\times{\bm{k}}||{\bm{p}}^{\prime}\times{\bm{k}}|(\cos\phi-i\lambda\sin\phi)}{2{\bm{k}}^{2}}\,,$
(9c) $\displaystyle{\bm{q}}^{2}={\bm{p}}^{2}+{\bm{p}}^{\prime
2}+{\bm{k}}^{2}+2{\bm{k}}\cdot{\bm{p}}^{\prime}-2{\bm{k}}\cdot{\bm{p}}-2\frac{{\bm{k}}\cdot{\bm{p}}\,{\bm{k}}\cdot{\bm{p}}^{\prime}+|{\bm{k}}\times{\bm{p}}||{\bm{k}}\times{\bm{p}}^{\prime}|\cos{\phi}}{{\bm{k}}^{2}}\,.$
(9d)

We observe that the dependence on the azimuthal angle $\phi$ appears in (II)
only by the way of (9c) and (9d). Therefore, all terms proportional to
$\sin\phi$ in (II) vanish after integration over the directions of the
outgoing fermion. This conclusion holds even after we regulate the divergence
in the fermion propagator using (13) in the next section. Dropping the
imaginary part of
$({\bm{p}}\cdot{\bm{e}}_{\lambda})({\bm{p}}^{\prime}\cdot{\bm{e}}_{\lambda}^{*})$,
observing that

$\displaystyle p\cdot
p^{\prime}=m^{2}+\omega(\kappa^{\prime}-\kappa)+\frac{k^{2}-q^{2}}{2}\,,$ (10)

and using (8), the non vanishing terms proportional to
${\bm{e}}_{\lambda}^{i}{\bm{e}}_{\lambda}^{*j}$ can be written in terms of
$\kappa,\kappa^{\prime}$ and ${\bm{q}}^{2}$:

$\displaystyle|{\bm{e}}_{\lambda}\cdot\bm{p}|^{2}=$
$\displaystyle\frac{|{\bm{k}}\times{\bm{p}}|^{2}}{2{\bm{k}}^{2}}=\frac{{\bm{p}}^{2}}{2}-\frac{({\bm{p}}\cdot{\bm{k}})^{2}}{2{\bm{k}}^{2}}=\frac{\omega^{2}E\kappa}{{\bm{k}}^{2}}-\frac{k^{2}}{2{\bm{k}}^{2}}E^{2}-\frac{m^{2}}{2}-\frac{\omega^{2}}{2{\bm{k}}^{2}}\kappa^{2}\,,$
(11a) $\displaystyle|{\bm{e}}_{\lambda}\cdot\bm{p^{\prime}}|^{2}=$
$\displaystyle\frac{\omega^{2}E^{\prime}\kappa^{\prime}}{{\bm{k}}^{2}}-\frac{k^{2}}{2{\bm{k}}^{2}}E^{\prime
2}-\frac{m^{2}}{2}-\frac{\omega^{2}}{2{\bm{k}}^{2}}\kappa^{\prime 2}\,,$ (11b)
$\displaystyle\mathrm{Re}[({\bm{e}}_{\lambda}\cdot{\bm{p}})({\bm{e}}_{\lambda}^{*}\cdot{\bm{p}}^{\prime})]=$
$\displaystyle\frac{{\bm{p}}\cdot{\bm{p}}^{\prime}}{2}-\frac{({\bm{p}}\cdot{\bm{k}})({\bm{p}}^{\prime}\cdot{\bm{k}})}{2{\bm{k}}^{2}}$
$\displaystyle=$
$\displaystyle\frac{\omega^{2}(\kappa^{\prime}E^{\prime}+\kappa
E-\kappa\kappa^{\prime})-\omega
k^{2}(\kappa-\kappa^{\prime})-k^{2}EE^{\prime}}{2{\bm{k}}^{2}}-\frac{m^{2}}{2}-\frac{k^{2}+{\bm{q}}^{2}}{4}\,.$
(11c)

Eqs. (11) can then be used to rewrite (II) without an explicit dependence on
the photon polarization vectors.

##  III Regulation of the Resonances

An examination of the fermion and photon propagators used in the calculation
of the differential cross section reveals divergences in three kinematic
regions: ${\bm{q}}^{2}=0$, $2\omega\kappa=k^{2}$ and
$2\omega\kappa^{\prime}=-k^{2}$. The first one is the familiar infrared
Coulomb pole ${\bm{q}}^{2}=0$ which is regulated in the usual way

$\displaystyle\frac{1}{{\bm{q}}^{2}}\to\frac{1}{{\bm{q}}^{2}+\mu^{2}}\,,$ (12)

where $\mu$ is the Debye mass of the medium. If the scattering particle mass
$m$ is much larger than $\mu$, the minimum momentum transfer is of the order
$m$. However, the Debye mass scales with the temperature and becomes much
larger than $m$ in a sufficiently hot medium. Therefore, in the following
analysis it will be convenient to consider two limiting cases depending on the
relative magnitude of $m$ and $\mu$. In the next section we will revisit the
behavior of the photon propagator at small momentum transfers and show that
due to the anomaly, ${\bm{q}}^{2}$ can vanish at finite $m$. We will revise
the regularization procedure accordingly.

The other two divergences occur in the fermion propagator and reflect its
instability in chiral matter with respect to spontaneous photon emission [16,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 18, 36]. With the
account of the finite width the fermion propagators in (II) are modified as
follows:

$\displaystyle\frac{1}{2\omega\kappa-k^{2}}\to\frac{1}{2\omega\kappa-k^{2}+iE/\tau}\,,\qquad\frac{1}{2\omega\kappa^{\prime}+k^{2}}\to\frac{1}{2\omega\kappa^{\prime}+k^{2}-iE^{\prime}/\tau}\,,$
(13)

where $\tau$ is the relaxation time as measured in the medium rest frame ***In
the fermion rest frame the relaxation rate is $(E/m)\tau^{-1}$.. A number of
inelastic processes contribute to the relaxation of the chiral state of the
fermion. Among them is the spontaneous photon emission, also known in the
literature as the vacuum or chiral Cherenkov radiation. Its rate at the
leading order in the perturbation theory is:

$\displaystyle
W=\frac{e^{2}}{8\pi}\int\frac{d\omega}{\omega}\left[b_{0}\left(\frac{\omega^{2}}{2E^{2}}-\frac{\omega}{E}+1\right)-\frac{m^{2}\omega}{E^{2}}\right]\Theta\left(\omega^{*}-\omega\right)\Theta(\lambda
b_{0})\,,$ (14)

where the threshold photon energy is

$\displaystyle\omega^{*}=\frac{\lambda b_{0}E^{2}}{\lambda b_{0}E+m^{2}}\,$
(15)

and $\Theta$ is the step-function [18]. In particular, when $m^{2}\gg\lambda
b_{0}E$, the fermion decay rate is $W\approx\alpha b_{0}/2$. One can take this
as the low bound of the total relaxation rate: $\tau^{-1}>\alpha b_{0}/2$.

We now have all necessary ingredients to complete our calculation of the
differential cross section. Eq. (II) may be integrated over directions term by
term while making substitutions (13),(12) in order to obtain the frequency
dependence of the differential cross section. The result is given by (A), (48)
in Appendix A. We use it for the numerical calculation presented in Fig. 2.

##  IV high energy limit

The exact expression for the cross section (A) is rather complicated. In
applications one is usually interested in the high energy limit
$E,E^{\prime}\gg m,\mu$ where the expression for the differential cross
section and squared matrix element significantly simplify. To derive the high
energy limit, it is convenient to expand the squared amplitude before the
integration over the fermion directions:

$\displaystyle\frac{1}{2}\sum_{s,s^{\prime}}|\mathcal{M}|^{2}$
$\displaystyle\approx\frac{2Z^{2}e^{6}}{\omega^{2}({\bm{q}}^{2}+\mu^{2})^{2}}\operatorname{\mathrm{Re}}\Bigg{[}4\left|\frac{Ep^{\prime}\cdot
e_{\lambda}}{\kappa^{\prime}+\frac{k^{2}}{2\omega}+i\frac{E^{\prime}}{\tau\omega}}-\frac{E^{\prime}p\cdot
e_{\lambda}}{\kappa-\frac{k^{2}}{2\omega}+i\frac{E}{\tau\omega}}\right|^{2}$
$\displaystyle+\frac{\omega^{2}({\bm{q}}^{2}-(\kappa-\kappa^{\prime}-\frac{k^{2}}{\omega})^{2})}{(\kappa-\frac{k^{2}}{2\omega}+i\frac{E}{\tau\omega})(\kappa^{\prime}+\frac{k^{2}}{2\omega}-i\frac{E^{\prime}}{\tau\omega})}$
$\displaystyle+k^{2}\Bigg{(}\frac{{\bm{q}}^{2}}{4}\left|\frac{1}{\kappa-\frac{k^{2}}{2\omega}+i\frac{E}{\tau\omega}}-\frac{1}{\kappa^{\prime}+\frac{k^{2}}{2\omega}+i\frac{E^{\prime}}{\tau\omega}}\right|^{2}+\frac{|p\cdot
e_{\lambda}-p^{\prime}\cdot
e_{\lambda}|^{2}}{(\kappa-\frac{k^{2}}{2\omega}+i\frac{E}{\tau\omega})(\kappa^{\prime}+\frac{k^{2}}{2\omega}-i\frac{E^{\prime}}{\tau\omega})}$
$\displaystyle-\left|\frac{E^{\prime}}{\kappa-\frac{k^{2}}{2\omega}+i\frac{E}{\tau\omega}}-\frac{E}{\kappa^{\prime}+\frac{k^{2}}{2\omega}+i\frac{E^{\prime}}{\tau\omega}}\right|^{2}\Bigg{)}\Bigg{]}\,,$
(16)

Additionally, we are interested in photon energies $\omega\gg b_{0}$ which
allows us to write the dispersion relation (3) as

$\displaystyle|{\bm{k}}|\approx\omega+\frac{\lambda b_{0}}{2}\,,$ (17)

and treat $k^{2}/\omega^{2}\approx-\lambda b_{0}/\omega$ as a small parameter.
In this limit Eq. (11) may be rewritten as

$\displaystyle|{\bm{e}}_{\lambda}\cdot\bm{p}|^{2}\approx\left(1-\lambda\frac{b_{0}}{\omega}\right)^{2}E\kappa+\left(\frac{\lambda
b_{0}}{2\omega}-\frac{b_{0}^{2}}{2\omega^{2}}\right)E^{2}-\frac{m^{2}+(1-\lambda\frac{b_{0}}{\omega})^{2}\kappa^{2}}{2}\,,$
(18a)
$\displaystyle|{\bm{e}}_{\lambda}\cdot\bm{p^{\prime}}|^{2}\approx\left(1-\lambda\frac{b_{0}}{\omega}\right)^{2}E^{\prime}\kappa^{\prime}+\left(\frac{\lambda
b_{0}}{2\omega}-\frac{b_{0}^{2}}{2\omega^{2}}\right)E^{\prime
2}-\frac{m^{2}+(1-\lambda\frac{b_{0}}{\omega})^{2}\kappa^{\prime 2}}{2}\,,$
(18b)
$\displaystyle\mathrm{Re}[({\bm{e}}_{\lambda}\cdot{\bm{p}})({\bm{e}}_{\lambda}^{*}\cdot{\bm{p}}^{\prime})]=\frac{(1-\lambda\frac{b_{0}}{\omega})^{2}(\kappa^{\prime}E^{\prime}+\kappa
E-\kappa\kappa^{\prime})-m^{2}}{2}+\left(\frac{\lambda
b_{0}}{2\omega}-\frac{b_{0}^{2}}{2\omega^{2}}\right)EE^{\prime}+\frac{\lambda
b_{0}\omega-{\bm{q}}^{2}}{4}\,.$ (18c)

Employing Eqs. (18) to get rid of the photon polarization vectors in favor of
the momenta in (IV) we derive

$\displaystyle\frac{1}{2}\sum_{s,s^{\prime}}|\mathcal{M}|^{2}\approx\frac{2Z^{2}e^{6}}{\omega^{2}({\bm{q}}^{2}+\mu^{2})^{2}}$
$\displaystyle\operatorname{\mathrm{Re}}\Bigg{\\{}\frac{({\bm{q}}^{2}+\mu^{2})(E^{2}+E^{\prime
2}+4\frac{\lambda
b_{0}}{\omega}EE^{\prime})+4b_{0}^{2}EE^{\prime}}{(\kappa^{\prime}+\frac{\lambda
b_{0}}{2}+i\frac{E^{\prime}}{\tau\omega})(\kappa-\frac{\lambda
b_{0}}{2}-i\frac{E}{\tau\omega})}$
$\displaystyle-2\omega^{2}\left(\frac{\kappa^{\prime}+\frac{\lambda
b_{0}}{2}+i\frac{E^{\prime}}{\tau\omega}}{\kappa-\frac{\lambda
b_{0}}{2}-i\frac{E}{\tau\omega}}+\frac{\kappa-\frac{\lambda
b_{0}}{2}-i\frac{E}{\tau\omega}}{\kappa^{\prime}+\frac{\lambda
b_{0}}{2}+i\frac{E^{\prime}}{\tau\omega}}\right)$
$\displaystyle-4\left[m^{2}-\frac{\lambda
b_{0}}{\omega}(EE^{\prime}-m^{2}+\omega^{2})\right]\left(\frac{E^{\prime}}{\kappa-\frac{\lambda
b_{0}}{2}-i\frac{E}{\tau\omega}}-\frac{E}{\kappa^{\prime}+\frac{\lambda
b_{0}}{2}+i\frac{E^{\prime}}{\tau\omega}}\right)^{2}$
$\displaystyle+\frac{\lambda
4b_{0}}{\omega}\left(\frac{(E+E^{\prime})E^{\prime}}{\kappa-\frac{\lambda
b_{0}}{2}+i\frac{E}{\tau\omega}}-\frac{(E+E^{\prime})E}{\kappa^{\prime}+\frac{\lambda
b_{0}}{2}+i\frac{E^{\prime}}{\tau\omega}}\right)\Bigg{\\}}\,.$ (19)

The largest contributions come from the resonances at $2\omega\kappa-k^{2}=0$
and $2\omega\kappa^{\prime}+k^{2}=0$. In the high energy approximation these
equations have a solution only for $m\omega-b_{0}EE^{\prime}\leq 0$. This
inequality can be equivalently expressed as the requirements
$\omega\leq\omega^{*}$ and $b_{0}\lambda>0$ which is consistent with (14) and
(15). Evidently, only one of the two photon polarizations (the one with
$b_{0}\lambda>0$) is resonant. As such, the differing polarization cases must
be treated separately. Consider, for example the pole at

$\displaystyle k^{2}=2\omega\kappa=2p\cdot k\approx\omega
E\left(\frac{m^{2}}{E^{2}}+\frac{k^{2}}{\omega^{2}}+\theta^{2}\right)\,,$ (20)

where $\theta$ is the angle between ${\bm{k}}$ and ${\bm{p}}$. The denominator
of the corresponding fermion propagator is

$\displaystyle\frac{1}{2p\cdot k-k^{2}+iE/\tau}=\frac{1}{\omega
E\left(\frac{m^{2}}{E\omega}\,\frac{\omega-\omega^{*}}{E-\omega^{*}}+\theta^{2}+\frac{i}{\omega\tau}\right)}\,,$
(21)

where we used (15) to eliminate $\lambda b_{0}$ in favor of $\omega^{*}$:

$\displaystyle k^{2}\approx-\lambda
b_{0}\omega=-\frac{\omega\omega^{*}m^{2}}{E(E-\omega^{*})}\,,\quad(\lambda
b_{0}>0)\,.$ (22)

Note that (22) makes sense only when $\lambda b_{0}>0$, for otherwise
$\omega^{*}$ is negative, indicating that there is no instability when
$\lambda b_{0}<0$. Eq. (21) implies that above the threshold, viz. when
$\omega>\omega^{*}$, photon is radiated mostly at angles
$\theta\lesssim\theta_{0}=\sqrt{\frac{m^{2}}{E\omega}\,\frac{\omega-\omega^{*}}{E-\omega^{*}}}$.
However, at and below the threshold, the angular distribution diverges at
$\theta=\sqrt{\frac{m^{2}}{E\omega}\,\frac{\omega^{*}-\omega}{E-\omega^{*}}}$
which is regulated by the cutoff introduced in the previous section.

Let us now examine the photon propagator in the resonant case $\lambda
b_{0}>0$. Writing
${\bm{q}}^{2}=[({\bm{k}}\times{\bm{q}})^{2}+({\bm{k}}\cdot{\bm{q}})^{2}]/{\bm{k}}^{2}$
and expanding at small photon emission angles we obtain

$\displaystyle{\bm{q}}^{2}=-(k+p^{\prime}-p)^{2}\approx\theta^{2}E^{2}+\theta^{\prime
2}E^{\prime
2}-2EE^{\prime}\theta\theta^{\prime}\cos\phi+\frac{1}{4}\left[\frac{m^{2}(\omega-\omega^{*})}{E^{\prime}(E-\omega^{*})}-E\theta^{2}+E^{\prime}\theta^{\prime
2}\right]^{2}\,.$ (23)

In the non-anomalous case $\omega^{*}=0$, and the momentum transfer is bounded
from below by $\frac{m^{4}\omega^{2}}{4E^{2}E^{\prime 2}}$. In contrast, in
the presence of the anomaly the momentum transfer is allowed to vanish. We can
find the corresponding kinematic region by first observing that the sum of the
first three terms and the last term in the r.h.s. of (23) are non-negative and
therefore have to vanish independently. The sum of the first three terms
vanishes only when $\phi=0$ and $E\theta=E^{\prime}\theta^{\prime}$ in which
case the momentum transfer reads

$\displaystyle{\bm{q}}^{2}|_{\phi=0,E\theta=E^{\prime}\theta^{\prime}}\approx\frac{1}{4}\left[\frac{m^{2}(\omega-\omega^{*})}{E^{\prime}(E-\omega^{*})}+\frac{\omega
E}{E^{\prime}}\theta^{2}\right]^{2}=\frac{1}{4}\frac{\omega^{2}E^{2}}{E^{\prime
2}}\left[\frac{m^{2}(\omega-\omega^{*})}{\omega
E(E-\omega^{*})}+\theta^{2}\right]^{2}\,.$ (24)

This imbues the photon propagator with the same resonant behavior as the
fermion propagator, as can be seen by comparing with (21). Apparently, we need
to regulate the divergence at ${\bm{q}}^{2}=0$ by replacing
$\theta^{2}\to\theta^{2}+\frac{i}{\omega\tau}$ in (24). This is tantamount to
the replacement ${\bm{q}}^{2}\rightarrow{\bm{q}}^{2}+\frac{E^{2}}{4E^{\prime
2}\tau^{2}}$. Along with the Debye mass $\mu$ introduced in (12) it provides
the regulator of the photon propagator at small momentum transfers:

$\displaystyle{\bm{q}}^{2}\rightarrow{\bm{q}}^{2}+\frac{E^{2}}{4E^{\prime
2}\tau^{2}}+\mu^{2}\,.$ (25)

The physics of bremsstrahlung in chiral medium is most transparent in two
limiting cases: (i) low temperature $m\gg\mu$ and (ii) high temperature
$\mu\gg m$. In the anomaly-free medium $b_{0}=0$, the first case reduces to
the scattering off a single nucleus. In this case the momentum transfer
${\bm{q}}^{2}$ never vanishes. On the contrary, in the presence of anomaly,
negative $k^{2}$ can drive the momentum transfer towards zero for one of the
photon polarizations, as we explained.

###  IV.1 Low temperature $\mu\ll m$

We first consider the low temperature/heavy fermion regime. We also assume,
for the sake of simplicity, that $\mu^{2}\ll 1/\tau^{2}$ so that $\tau$
regulates both the fermion and the photon propagators. In the anomaly-free
medium, the typical momentum transfer is of the order $m$ and therefore the
scattering cross section is insensitive to the cutoffs $\tau$ and $\mu$. This
conclusion is upended in the anomalous medium due to the resonances discussed
in Sec. III.

The bremsstrahlung cross section is obtained by substituting (IV) into (1) and
integrating over the fermion and photon directions. The angular integrals are
performed in Appendix B. The results are essentially different for positive
and negative values of the parameter $b_{0}\lambda$. For $b_{0}\lambda<0$
there are no resonances and the angular integrals simplify greatly. In this
case, $\tau^{-1}$ and $\mu$ may be neglected given that all integrals are
convergent. In this case the cross section reads:

$\displaystyle\frac{d\sigma(b_{0}\lambda<0)}{d\omega}\approx\frac{Z^{2}e^{6}E^{\prime}}{4(2\pi)^{3}(m^{2}\omega-\lambda
b_{0}EE^{\prime})E}\left(\frac{E}{E^{\prime}}+\frac{E^{\prime}}{E}-\frac{2}{3}\right)\left[\ln\frac{4E^{2}E^{\prime
2}}{\omega(m^{2}\omega-\lambda b_{0}EE^{\prime})}-1\right]\,.$ (26)

In the anomaly-free medium $b_{0}=0$ (26) reduces to the well-known Bethe-
Heitler expression for the bremsstrahlung cross section on a heavy nucleus
[19, 37]:

$\displaystyle\frac{d\sigma_{\text{BH}}}{d\omega}\approx\frac{Z^{2}e^{6}E^{\prime}}{4(2\pi)^{3}m^{2}\omega
E}\left(\frac{E}{E^{\prime}}+\frac{E^{\prime}}{E}-\frac{2}{3}\right)\left(\ln\frac{2EE^{\prime}}{m\omega}-\frac{1}{2}\right)\,.$
(27)

The effect of the anomaly on this photon polarization is most significant in
the infrared region $\omega\ll b_{0}EE^{\prime}/m^{2}$ where the cross section
scales as $\log(1/\omega)$. In contrast, without the anomaly it scales as
$(1/\omega)\log(1/\omega)$. Thus the anomaly tends to suppress emission of
photons with $b_{0}\lambda<0$ polarization.

The situation is essentially different for the $b_{0}\lambda>0$ polarziation
since the corresponding cross section diverges in the limit $\tau^{-1}\to 0$,
i.e. at small photon emission angles when $\omega\leq\omega^{*}$. As we
explained in the previous section, this divergence occurs concurrently in the
fermion and the photon propagators and is regulated by shifting
$\theta^{2}\to\theta^{2}+\frac{i}{\omega\tau}$ and similarly for
$\theta^{\prime}$. Keeping only the essential terms and setting $\mu=0$ we
obtain:

$\displaystyle\frac{d\sigma(b_{0}\lambda>0)}{d\omega}\approx\frac{Z^{2}e^{6}E^{\prime}}{4(2\pi)^{3}Em^{2}\omega}\left\\{\frac{\left(\frac{E}{E^{\prime}}+\frac{E^{\prime}}{E}-\frac{2}{3}\right)\left(\ln\frac{4E^{2}E^{\prime
2}}{m^{2}\omega^{2}\sqrt{\frac{E^{2}(\omega^{*}-\omega)^{2}}{\omega^{2}(E-\omega^{*})^{2}}+\frac{4E^{4}E^{\prime
2}}{m^{4}\omega^{4}\tau^{2}}}}-1\right)}{\sqrt{\frac{E^{2}(\omega^{*}-\omega)^{2}}{\omega^{2}(E-\omega^{*})^{2}}+\frac{4E^{4}}{m^{4}\omega^{2}\tau^{2}}}}\right.$
$\displaystyle+\frac{2m^{4}(\omega^{*}-\omega)\tau^{3}}{E^{3}E^{\prime
4}(E-\omega^{*})}\left[E^{\prime
6}\arctan\frac{m^{2}(\omega^{*}-\omega)\tau}{E^{\prime}(E-\omega^{*})}+E^{6}\arctan\frac{m^{2}(\omega^{*}-\omega)\tau}{E(E-\omega^{*})}\right]\Theta(\omega^{*}-\omega)\Bigg{\\}}\,,$
(28)

where $\Theta$ is the step function and we replaced $b_{0}\lambda$ in favor of
$\omega^{*}$ using (22).

The are two kinds of terms in (IV.1). (i) The first line of (IV.1) reduces to
the anomaly-free result (27) in the limit $b_{0}\to 0$ and $E/\tau\to 0$. We
neglected all terms proportional to $\tau^{-1}$ with exception of those
appearing under the radicals where their role is to regulate the divergence at
the threshold $\omega=\omega^{*}$. (ii) The second and the third lines of
(IV.1) represent the most singular resonance contributions, viz. the terms
that are most divergent at small $\tau^{-1}$. In the limit $b_{0}\to 0$ the
step function can only be satisfied when $\omega\to 0$, hence the anomalous
contribution vanishes.

We stress that the chiral resonance in the fermion propagator contributes only
to one of the photon polarizations, namely $b_{0}\lambda>0$. The other
polarization $b_{0}\lambda<0$ is suppressed. This is in contrast to the lack
of a distinction between handedness in the absence of the anomaly. A
remarkable feature of the anomalous contribution, dominated by the second and
the third lines of (IV.1), is that at small $\omega$ its spectrum scales as
$1/\omega$, which is the same form as the soft photon spectrum without the
chiral anomaly (27).

|
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Figure 2: The bremsstrahlung spectrum at $\mu\ll m$. The dots: the exact
leading order formula (A), solid line: the high energy approximation given by
the sum of equations (26) and (IV.1), dotted line: the Bethe-Heitler
expression (27), dashed line: the magnetic moment contribution (30). Left
panel: $b_{0}=0.1m$, $\mu=10^{-3}b_{0}$, $\tau^{-1}=0.1b_{0}$. Right panel:
$b_{0}=10^{-2}m$, $\mu=10^{-3}b_{0}$, $\tau^{-1}=0.1b_{0}$. Both panels:
$E=10m$, $Z=33$, $\textgoth{M}=\mu_{N}$, $\Gamma=0.1b_{0}$. The vertical line
indicates $\omega^{*}/E$.

The bremsstrahlung photon spectra are plotted in Fig. 2. Inspection of Fig. 2
reveals two significant features. Firstly, at $\omega<\omega^{*}$ the cross
section is enhanced as compared to the Bethe-Heitler formula. To better
understand the parametric dependence of the cross section, consider the soft
photon region $\omega\ll\omega^{*}\ll E$, in particular
$\omega^{*}\approx\lambda b_{0}E^{2}/m^{2}$. Eq. (IV.1) can be written as

$\displaystyle\frac{d\sigma(b_{0}\lambda>0)}{d\omega}\approx\frac{3\pi}{2}\,\frac{m^{2}\tau^{3}b_{0}}{\ln\frac{2E^{2}}{m\omega}}\,\frac{d\sigma_{\text{BH}}}{d\omega}.$
(29)

Since $m\tau\gg b_{0}\tau\gg 1$ we indeed observe that the anomalous cross
section is strongly enhanced in the soft photon region. Secondly, when
focusing on the production of the right-handed photons, we see a sharp cut-off
at $\omega=\omega^{*}$. The anomalous contribution vanishes at higher photon
energies. A similar behavior was discussed in our paper [15] for the case of
magnetic moment contribution.

To complete our discussion, we cite here the magnetic moment contribution to
the bremsstrahlung in the semi-soft photon limit $b_{0}\ll\omega\ll E$
computed in [15]:

$\displaystyle\frac{d\sigma_{M}}{d\omega}\approx\frac{2e^{4}\textgoth{M}^{2}}{3(2\pi)^{3}\omega}\left[\frac{3b_{0}^{2}}{m^{2}}\ln(\frac{4E^{4}}{m^{2}\omega^{2}})+\ln^{2}\frac{4E^{2}}{m^{2}}+\frac{2b_{0}^{4}\pi}{m^{2}\Gamma^{2}}\Theta(\omega_{0}-\omega)\right]\,,$
(30)

where $\textgoth{M}$ is the magnetic moment. In Fig. 2 one can see that it
gives a minor correction to the total cross section, even though it exhibits
qualitative features similar to (26) and (IV.1). Overall the anomaly in the
magnetic channel may amplify photon production over a wider range of
frequencies but produces less radiation when compared to the electric channel.

###  IV.2 High temperatures $\mu\gg m$

We now consider the differential cross-section in the opposite limit where
$\mu\gg m$ and $\mu^{2}\gg 1/\tau^{2}$, so that $\tau^{-1}$ is neglected in
the photon propagator. The process of computing the differential cross-section
and the necessary integrals is similar to the one outlined in Appendix B,
except that $m$ may be taken to zero in the high-temperature limit. It is
well-known that $\mu$ not only regulates the Coulomb pole, but also sets the
smallest photon emission angle $\theta_{\text{min}}=\mu/E\ll 1$ [38, 39]. In
the previous subsection both roles were played by $m$. In the high energy
limit, one can take the integrals $I_{j,n,l}$, by expanding the integrands at
small photon emission angles $\theta$ and $\theta^{\prime}$:

$\displaystyle I_{j,n,l}\approx$
$\displaystyle\int\frac{d\Omega^{\prime}d\Omega_{k}}{\left[(E^{2}\theta^{2}-2EE\theta\theta^{\prime}\cos\phi+E^{\prime
2}\theta^{\prime 2})+\frac{1}{4}(E\theta^{2}-E^{\prime}\theta^{\prime
2}+\lambda b_{0})^{2}\right]^{j}}$
$\displaystyle\times\frac{1}{\left[(E\omega\theta^{2}-\lambda
b_{0}E^{\prime})^{n}-(-i\frac{E}{\tau})^{n}\right]\left[(E^{\prime}\omega\theta^{\prime
2}-\lambda b_{0}E)^{l}-(i\frac{E^{\prime}}{\tau})^{l}\right]}\,,$ (31)

where $\phi$ is the azimuthal angle which may be integrated over directly. As
an example consider $I_{1,1,1}$. Integrating of $\phi$ yields:

$\displaystyle I_{1,1,1}\approx$
$\displaystyle\int_{\frac{\mu^{2}}{E^{2}}}^{\infty}\int_{\frac{\mu^{2}}{E^{\prime
2}}}^{\infty}\frac{\pi^{2}d\theta^{\prime
2}d\theta^{2}}{\sqrt{((E^{2}\theta^{2}+E^{\prime 2}\theta^{\prime
2})+\frac{1}{4}(E\theta^{2}-E^{\prime}\theta^{\prime 2}+\lambda
b_{0})^{2})^{2}-4E^{2}E^{\prime 2}\theta^{2}\theta^{\prime 2}}}$
$\displaystyle\times\frac{1}{((E\omega\theta^{2}-\lambda
b_{0}E^{\prime})+i\frac{E}{\tau})((E^{\prime}\omega\theta^{\prime 2}-\lambda
b_{0}E)-i\frac{E^{\prime}}{\tau})}\,,$ (32)

where we replaced the upper integration limits by infinities thanks to the
fast convergence of the integrals. Taking the remaining integrals we find that
$I_{1,1,1}$ gives the following contribution to the differential cross-section

$\displaystyle I_{1,1,1}\approx$
$\displaystyle\frac{4\pi^{2}\ln\frac{4E^{2}E^{\prime
2}}{\sqrt{(\mu^{2}\omega^{2}-\lambda b_{0}\omega
EE^{\prime})^{2}+4\frac{E^{4}E^{\prime 2}}{\tau^{2}}}}}{\omega
EE^{\prime}\sqrt{(\mu^{2}\omega-\lambda
b_{0}EE^{\prime})^{2}+4\frac{E^{3}E^{\prime}}{\tau^{2}}}}\,.$ (33)

The other terms may be computed in a similar fashion. In those integrals where
the integral over $\theta$ (or $\theta^{\prime}$) is not convergent at large
angles, one needs to first integrate over the region $\theta<\theta^{\prime}$
(or $\theta^{\prime}<\theta$); the remaining integral then converges well. The
result of the calculation is different for the two photon polarziations and
reads

$\displaystyle\frac{d\sigma(b_{0}\lambda<0)}{d\omega}\approx\frac{Z^{2}e^{6}E^{\prime}}{4(2\pi)^{3}(\mu^{2}\omega-\lambda
b_{0}EE^{\prime})E}\left(\frac{E}{E^{\prime}}+\frac{E^{\prime}}{E}\right)\left[\ln\frac{4E^{2}E^{\prime
2}}{\mu^{2}\omega^{2}-\lambda b_{0}\omega EE^{\prime}}-1\right]\,,$ (34)
$\displaystyle\frac{d\sigma(b_{0}\lambda>0)}{d\omega}\approx\frac{Z^{2}E^{\prime}e^{6}}{4(2\pi)^{3}E\mu^{2}\omega}\Bigg{\\{}\frac{\mu^{2}\omega(\frac{E}{E^{\prime}}+\frac{E^{\prime}}{E})\big{[}\ln\frac{4E^{2}E^{\prime
2}}{\sqrt{(\mu^{2}\omega^{2}-\lambda b_{0}\omega
EE^{\prime})^{2}+4\frac{E^{4}E^{\prime
2}}{\tau^{2}}}}-1\big{]}}{\sqrt{(\mu^{2}\omega-\lambda
b_{0}EE^{\prime})^{2}+\frac{E^{3}E^{\prime}}{\tau^{2}}}}$
$\displaystyle+\frac{\lambda b_{0}\mu\tau}{E}\arctan\left(\frac{(\lambda
b_{0}EE^{\prime}-\mu^{2}\omega)\tau}{EE^{\prime}}\right)\Theta(\omega^{*}-\omega)\bigg{\\}}\,,$
(35)

where $\omega^{*}$ is given by (15) and $E^{\prime},E\gg\mu$.

Consider the soft photon limit in a medium such that $\mu\gg m\gg\tau^{-1}$.
Soft photons region corresponds to $\omega\ll E$ and $\omega\ll\omega^{*}$,
the latter condition implying $\omega\ll b_{0}E^{2}/m^{2}$. Then (IV.2)
reduces to

$\displaystyle\frac{d\sigma(b_{0}\lambda>0)}{d\omega}\approx\frac{\pi}{2}\frac{\mu}{E}\frac{b_{0}\tau}{\ln\frac{2E^{2}}{\mu\omega}}\frac{d\sigma_{\text{BH}}}{d\omega}\,,$
(36)

where the Bethe-Heitler cross section in this case is obtained by setting
$b_{0}=0$ in (34). One observes only a modest enhancement, if any, as compared
to (29).

|
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Figure 3:  The bremsstrahlung spectrum at $\mu\gg m$. Solid line: the high
energy approximation given by the sum of equations (34) and (IV.2), dotted
line: the Bethe-Heitler limit $b_{0}=0$, dashed line: the magnetic moment
contribution (30). Left panel: $b_{0}=10^{-1}m$, right panel:
$b_{0}=10^{-2}m$, both panels: $E=10^{2}m$, $\mu=10m$,
$\tau^{-1}=\Gamma=0.1b_{0}$, $Z=33$. The vertical line indicates
$\omega^{*}/E$.

##  V Energy loss

A fast particle moving across a medium losses energy. There are two main
mechanisms for energy loss at sufficiently high energies. The collisional
energy loss occurs due to momentum transfer in the course of elastic
collisions. The radiative energy loss is caused by the bremsstrahlung. In
chiral media there is an additional contribution to the energy loss driven by
the chiral anomaly. The results of the previous section allow us to compute
the amount of energy lost in a scattering off a single source of electric
charge $eZ$.

Throughout this paper we neglected the quantum interference effects assuming
that the photon formation time is much shorter than the mean-free-path $\ell$.
In this approximation the energy loss per unit length is:

$\displaystyle-\frac{dE}{dz}=n\int_{0}^{E}\omega\frac{d\sigma^{eZ\to
eZ\gamma}}{d\omega}d\omega\,,$ (37)

where $n$ is number of scatterers per unit volume which can be expressed in
terms of the elastic scattering cross section $n=1/\ell\sigma^{eZ\to eZ}$. At
high energies,

$\displaystyle\sigma^{eZ\to
eZ}=\frac{\alpha(eZ)^{2}}{\max\\{\mu^{2},m^{2}\\}}+\mathcal{O}(\textgoth{M})\,,$
(38)

where $\alpha=e^{2}/4\pi$. The omitted term in (38) is proportional to the
small magnetic moment $\textgoth{M}$ of the source. Actually, this term is
affected by the anomaly and can become essential at very high energies [40].
However, in the present calculation it can be neglected.

As in the previous section we will be concerned with the two limits.

###  V.1 Low temperatures $\mu\ll m$

Substituting (26) or (IV.1) into (37) and integrating over $\omega$ gives the
energy loss for two photon polarizations. For brevity we will record these
expressions in the limit $m^{2}\gg b_{0}E$:

$\displaystyle-\frac{dE(b_{0}\lambda<0)}{dz}\approx\frac{e^{2}E}{16\pi^{2}\ell}\left[\ln\frac{2E}{m}-\frac{1}{3}-\frac{2b_{0}E}{9m^{2}}\left(\pi+2\ln\frac{E}{b_{0}}\right)\right]\,,$
(39) $\displaystyle-\frac{dE(b_{0}\lambda>0)}{dz}\approx$
$\displaystyle\frac{e^{2}E}{16\pi^{2}\ell}\Bigg{\\{}\ln\frac{2E}{m}-\frac{1}{3}+\frac{2b_{0}E}{9m^{2}}\left(\pi+2\ln\frac{E}{b_{0}}\right)+2\tau(E-\omega^{*})\arctan\frac{2m^{2}\omega^{*}\tau}{E(E-\omega^{*})}\Bigg{\\}}\,.$
(40)

Eq. (39) agrees with Eq. (93.24) in [37] when $b_{0}=0$. We observe that the
anomaly slightly reduces the amount of energy lost due to the bremsstrahlung
of $b_{0}\lambda<0$ photons. This is consistent with our discussion in the
previous section that the bremsstrahlung cross section is reduced in this
case.

As far as the $b_{0}\lambda>0$ polarization is concerned, we consider again
the soft photon region $\omega\ll\omega^{*}\ll E$ were $\lambda b_{0}\ll m$ so
that the last term on the r.h.s. of (40) is dominant. This allows us to cast
the energy loss (40) in a convenient form:

$\displaystyle-\frac{dE(b_{0}\lambda>0)}{dz}\approx\frac{\pi\tau
E}{\ln\frac{2E}{m}}\left(-\frac{dE}{dz}\right)_{\text{BH}}\,.$ (41)

Thus, the energy loss due to the chiral anomaly is driven by the
$b_{0}\lambda>0$ photon polarization and is enhanced by the large factor $\tau
E$ over the Bethe-Heitler result.

###  V.2 High temperature $\mu\gg m$

The rate of energy loss in the high-temperature limit is computed similarly.
Plugging (34) and (IV.2) into (37) we derive

$\displaystyle-\frac{dE(b_{0}\lambda<0)}{dz}$
$\displaystyle\approx\frac{e^{2}E}{16\pi^{2}\ell}\left(\ln\frac{2E}{\mu}-\frac{6b_{0}E}{7\mu^{2}}\ln\frac{E}{b_{0}}\ln\frac{4E^{2}}{\mu^{2}}\right)\,,$
(42) $\displaystyle-\frac{dE(b_{0}\lambda>0)}{dz}$
$\displaystyle\approx\frac{e^{2}E}{16\pi^{2}\ell}\left(\ln\frac{2E}{\mu}+\frac{6b_{0}E}{7\mu^{2}}\ln\frac{E}{b_{0}}\ln\frac{4E^{2}}{\mu^{2}}+\frac{2b_{0}\mu\tau}{3E}\right)\,.$
(43)

Eq. (42) agrees with [39] when $b_{0}=0$. As in the low temperature case, the
energy loss is mostly driven by $b_{0}\lambda>0$ polarization. However, the
overall magnitude of the lost energy sensitively depends on the actual values
of the parameters.

##  VI Summary

In this paper we computed and analysed the spectrum of electromagnetic
radiation emitted by a fast particle traveling in the chiral medium supporting
the chiral magnetic current with constant chiral conductivity $b_{0}$. We
observed in Fig. 2 that at low temperatures $\mu\ll m$ the anomalous
contribution to bremsstrahlung, stemming from one of the photon polarizations,
is several orders of magnitude larger than the non-anomalous one. We also
computed the corresponding enhancement of the energy loss. If confirmed by
experimental observation, it can serve as an effective tool to study the
chiral anomaly in the materials. It can also be used to search for the new
forms of the chiral matter. In particular, a cosmic ray moving through the
chiral domain generated by an axion field would radiate and lose energy in a
peculiar way described in this paper. In the opposite limit of very high
temperatures $\mu\gg m$, the effect of the anomaly is much smaller as seen in
Fig. 3.

In arriving at our conclusions we made a number of assumptions. First of all,
we treated the chiral conductivity $b_{0}$ as a constant. However, it does
evolve in time albeit slowly. Its temporal evolution is characterized by the
relaxation time $\tau$, which we assumed to be the large parameter in our
calculation. The spatial extent of the domain can be neglected as long as it
is larger than photon formation time $t_{f}$ which is a very good
approximation considering that the formation time must be shorter than the
mean free path $\ell$ in the Bethe-Heitler limit. Also, for the sake of
simplicity we neglected the chiral displacement ${\bm{b}}$. The electric
current ${\bm{j}}={\bm{b}}\times\bm{\mathcal{E}}$ is induced at finite
${\bm{b}}$ and is responsible for the anomalous Hall effect. At $b_{0}=0$ it
induces the radiative instability of the chiral matter which is similar in
many ways to the homogeneous and isotropic case $b_{0}\neq 0$, ${\bm{b}}=0$,
especially at high energies [41, 42]. In fact, one can obtain the rate of the
Cherenkov radiation by simply replacing $b_{0}\to|{\bm{b}}|\cos\beta$, where
$\beta$ is the angle between ${\bm{b}}$ and ${\bm{k}}$. The relationship
between the two cases is more delicate for bremmstrahlung. The photon
propagator takes now the following form, in place of (5):

$\displaystyle
D_{00}=\frac{i{\bm{q}}^{2}}{{\bm{q}}^{4}+({\bm{b}}\times{\bm{q}})^{2}}\,,$
(44) $\displaystyle
D_{0i}=\frac{({\bm{b}}\times{\bm{q}})_{i}}{{\bm{q}}^{4}+({\bm{b}}\times{\bm{q}})^{2}}\,,$
(45) $\displaystyle
D_{ij}=-\frac{i}{{\bm{q}}^{4}+({\bm{b}}\times{\bm{q}})^{2}}\left\\{{\bm{q}}^{2}\delta_{ij}+b_{i}b_{j}-\frac{[{\bm{q}}^{4}-({\bm{b}}\cdot{\bm{q}})^{2}]q_{i}q_{j}}{{\bm{q}}^{4}}-\frac{{\bm{b}}\cdot{\bm{q}}}{{\bm{q}}^{2}}(b_{i}q_{j}+b_{j}q_{i})\right\\}\,.$
(46)

The denominator of the $D_{00}$ component that drives the electric channel is
merely shifted $\mu^{2}\to\mu^{2}+({\bm{b}}\times\hat{{\bm{q}}})^{2}$. The
corresponding energy loss is anisotropic as it depends on the angle between
the incident fermion momentum and the displacement vector. Otherwise, we
believe that the qualitative similarity will persist. The magnetic channel is
quite different, but its contribution to the overall energy loss is not
significant. Clearly, due to possible applications to Weyl and Dirac semi-
metals, the case of finite ${\bm{b}}$ deserves a dedicated study.

Another critical assumption we made is that photon formation time $t_{f}$ is
much shorter than the mean-free path $\ell$. As argued in [39], the condition
$t_{f}\ll\ell$ translates into the requirement that $m,\mu\ll
E\ll\mu\sqrt{\omega\ell}$. Since $t_{f}$ is proportional to $\omega$, this
approximation breaks down at high energies where one must take account of the
multiple scatterings of the projectile in the medium. The resulting LPM
quantum interference effect [43, 44] is an important feature of the
bremsstrahlung spectrum and must certainly be taken into account at higher
energies.

Throughout this paper, we have treated the chiral anomaly as it pertains to
QED. The non-abelian version of the chiral anomaly has a similar effect in QCD
generating contributions to gluon and photon bremsstrahlung. Calculation of
bremsstrahlung and energy loss in QCD in the presence of the chiral magnetic
current is relevant to the quark-gluon plasma phenomenology. We plan to
address this elsewhere.

###### Acknowledgements.

This work was supported in part by the U.S. Department of Energy Grants No.
DE-FG02-87ER40371 and No. DE-SC0023692.

##  Appendix A The bremsstrahlung cross section at the leading order

The bremsstrahlung cross section corresponding to Fig. 1 including the cutoffs
reads:

$\displaystyle d\sigma=\sum_{\lambda}$ $\displaystyle\frac{Z^{2}\omega
e^{6}d\omega}{16(2\pi)^{5}}\frac{|{\bm{p}}^{\prime}|}{|{\bm{p}}|}$
$\displaystyle\times\operatorname{\mathrm{Re}}\Bigg{\\{}\left(2E^{2}+2E^{\prime
2}+k^{2}\right)I_{1,1,1}-\frac{\omega^{2}+{\bm{k}}^{2}}{4{\bm{k}}^{2}}(I_{2,1,-1}+I_{2,-1,1})+(I_{1,0,1}-I_{1,1,0})$
$\displaystyle+\frac{2m^{2}(\omega^{2}+{\bm{k}}^{2})+4EE^{\prime}k^{2}+k^{4}}{4{\bm{k}}^{2}}(I_{1,2,0}-2I_{1,1,1}+I_{1,0,2})$
$\displaystyle-\frac{4}{{\bm{k}}^{2}}\big{[}m^{2}{\bm{k}}^{2}(E^{\prime
2}I_{2,2,0}-2EE^{\prime}I_{2,1,1}+E^{2}I_{2,0,2})+\frac{{\bm{k}}^{2}k^{2}}{2}\left(E^{2}I_{2,0,2}+E^{\prime
2}I_{2,2,0}\right)$ $\displaystyle+E^{\prime 2}E^{\prime
2}k^{2}(I_{2,2,0}-2I_{2,1,1}+I_{2,0,2})-EE^{\prime}k^{2}(I_{2,0,1}+I_{2,1,0})\big{]}$
$\displaystyle+4\pi^{2}\frac{\text{arctanh\,}\left(\frac{2|{\bm{k}}||{\bm{p}}|}{2\omega
E-k^{2}+i\frac{E}{\tau}}\right)\text{arctanh\,}\left(\frac{2|{\bm{k}}||{\bm{p}}^{\prime}|}{2\omega
E^{\prime}+k^{2}-i\frac{E^{\prime}}{\tau}}\right)}{{\bm{k}}^{2}|{\bm{p}}||{\bm{p}}^{\prime}|}$
$\displaystyle-k^{2}(4\pi)^{2}\frac{\text{arctanh\,}\left(\frac{2\omega
E+2|{\bm{p}}||{\bm{k}}|-k^{2}+\mu^{2}}{\mu|{\bm{p}}^{\prime}|}\right)-\text{arctanh\,}\left(\frac{2\omega
E+2|{\bm{p}}||{\bm{k}}|-k^{2}+\mu^{2}}{\mu|{\bm{p}}^{\prime}|}\right)}{{\bm{k}}^{3}|{\bm{p}}||{\bm{p}}^{\prime}|\mu}\Bigg{\\}}\,,$
(47)

where the angular integrals $I_{j,n,l}$ are defined as

$\displaystyle
I_{j,n,l}=\int\frac{d\Omega^{\prime}d\Omega_{k}}{({\bm{q}}^{2}+\mu^{2})^{j}\left[(2\omega\kappa-k^{2})^{n}-(-i\frac{E}{\tau})^{n}\right]\left[(2\omega\kappa^{\prime}+k^{2})^{l}-(i\frac{E^{\prime}}{\tau})^{l}\right]}\,.$
(48)

where $j,n,l$ are integers, $k^{2}=-\lambda b_{0}|{\bm{k}}|$. When any of
these integers is negative the corresponding cutoffs can be set to zero.

##  Appendix B Integrals $I_{j,n,l}$ in the ultrarelativistic Heavy fermion
limit

An analysis of (IV) reveals several dominant contributions to the differential
ultrarelativistic cross-section in terms of $I_{j,n,l}$. We consider these
contributions under two different regimes. In the first, we consider the case
of heavy fermions relative to the Debye mass such that $\mu\ll m$, $\mu^{2}\ll
E/\tau$, and in the second we consider the case of high temperature which
takes on the opposite limit $m\ll\mu$. We focus on the first case in this
appendix, however in both cases we take the high energy limit $\mu,m\ll
E,E^{\prime}$ and $\omega\gg b_{0}$. Additionally, the dominance of small
emission angles allows us to neglect the contribution due to large angles.
Letting $\theta$ and $\theta^{\prime}$ be the angles between ${\bm{k}}$ and
${\bm{p}}$, and ${\bm{k}}$ and ${\bm{p}}^{\prime}$ leads to the following
approximation

$\displaystyle
I_{j,n,l}\approx\int\frac{d\Omega^{\prime}d\Omega_{k}}{\left[\left(E^{2}\theta^{2}-2E^{\prime}E\theta\theta^{\prime}\cos\phi+E^{\prime
2}\theta^{\prime
2}\right)+\frac{1}{4}\left(\frac{m^{2}\omega}{EE^{\prime}}-E\theta^{2}+E^{\prime}\theta^{\prime
2}-\lambda b_{0}+\frac{2i}{\omega\tau}\right)^{2}\right]^{j}}$
$\displaystyle\times\frac{1}{\left[\left(E\omega\theta^{2}+\frac{m^{2}\omega-\lambda
b_{0}E^{\prime}E}{E}\right)^{n}-\left(-i\frac{2E}{\tau}\right)^{n}\right]\left[\left(E^{\prime}\omega\theta^{\prime
2}+\frac{m^{2}\omega-\lambda
b_{0}E^{\prime}E}{E^{\prime}}\right)^{l}-\left(i\frac{2E^{\prime}}{\tau}\right)^{l}\right]}\,,$
(49)

where $\phi$ is the azimuthal angle ranging from $0$ to $2\pi$. Integrating up
to emission angles for the two possible cases $j=1,2$

$\displaystyle I_{1,n,l}\approx\int\frac{\pi^{2}d\theta^{\prime
2}d\theta^{2}}{\sqrt{\left[E^{2}\theta^{2}+E^{\prime 2}\theta^{\prime
2}+\frac{1}{4}\left(\frac{m^{2}\omega}{EE^{\prime}}-E\theta^{2}+E^{\prime}\theta^{\prime
2}-\lambda b_{0}+\frac{2i}{\omega\tau}\right)^{2}\right]^{2}-4E^{\prime
2}E^{2}\theta^{2}\theta^{\prime 2}}}$
$\displaystyle\times\frac{1}{\left[\left(E\omega\theta^{2}+\frac{m^{2}\omega-\lambda
b_{0}E^{\prime}E}{E}\right)^{n}-\left(-i\frac{2E}{\tau}\right)^{n}\right]\left[\left(E^{\prime}\omega\theta^{\prime
2}+\frac{m^{2}\omega-\lambda
b_{0}E^{\prime}E}{E^{\prime}}\right)^{l}-\left(i\frac{2E^{\prime}}{\tau}\right)^{l}\right]}\,,$
(50) $\displaystyle I_{2,n,l}\approx\int\frac{\left[E^{2}\theta^{2}+E^{\prime
2}\theta^{\prime
2}+\frac{1}{4}\left(\frac{m^{2}\omega}{EE^{\prime}}-E\theta^{2}+E^{\prime}\theta^{\prime
2}-\lambda b_{0}\right)^{2}+\mu^{2}\right]\pi^{2}d\theta^{\prime
2}d\theta^{2}}{\left\\{\left[E^{2}\theta^{2}+E^{\prime 2}\theta^{\prime
2}+\frac{1}{4}\left(\frac{m^{2}\omega}{EE^{\prime}}-E\theta^{2}+E^{\prime}\theta^{\prime
2}-\lambda b_{0}+\frac{2i}{\omega\tau}\right)^{2}\right]^{2}-4E^{\prime
2}E^{2}\theta^{2}\theta^{\prime 2}\right\\}^{\frac{3}{2}}}$
$\displaystyle\times\frac{1}{\left[\left(E\omega\theta^{2}+\frac{m^{2}\omega-\lambda
b_{0}E^{\prime}E}{E}\right)^{n}-\left(-i\frac{2E}{\tau}\right)^{n}\right]\left[\left(E^{\prime}\omega\theta^{\prime
2}+\frac{m^{2}\omega-\lambda
b_{0}E^{\prime}E}{E^{\prime}}\right)^{l}-\left(i\frac{2E^{\prime}}{\tau}\right)^{l}\right]}\,$
(51)

These integrals may then be taken for the relevant factors of $n$ and $l$,
noting the dominance of the regions $\omega
E^{2}\theta^{2}<\sqrt{(m^{2}\omega-\lambda
b_{0}E^{\prime}E)^{2}+\frac{E^{4}}{\tau^{2}}}$, and $\omega E^{\prime
2}\theta^{\prime 2}<\sqrt{(m^{2}\omega-\lambda
b_{0}E^{\prime}E)^{2}+\frac{E^{\prime 4}}{\tau^{2}}}$, acting as effective
bounds for the integration. It is convenient to replace the integrals over
emission angles with the difference
$\Delta=|E\theta-E^{\prime}\theta^{\prime}|$ given that the integrands are
largest at $\Delta=0$. The results for various relevant integrals for the
differential cross-section are given. For instances

$\displaystyle I_{1,1,1}\approx$
$\displaystyle\frac{4\pi^{2}\ln\frac{4E^{2}E^{\prime
2}}{\sqrt{(m^{2}\omega^{2}-\lambda b_{0}\omega
EE^{\prime})^{2}+\frac{16E^{4}E^{\prime 2}}{\tau^{2}}}}}{\omega
EE^{\prime}\sqrt{(m^{2}\omega-\lambda
b_{0}EE^{\prime})^{2}+\frac{16E^{3}E^{\prime}}{\tau}}}\,$ (52)

and

$\displaystyle I_{1,-1,1}+I_{1,1,-1}\approx$
$\displaystyle\frac{4\pi^{2}}{\omega\sqrt{(m^{2}\omega-\lambda
b_{0}EE^{\prime})^{2}+\frac{16E^{3}E^{\prime}}{\tau^{2}}}}\left(\frac{E^{\prime}}{E}+\frac{E}{E^{\prime}}\right)\,.$
(53)

Other integrals are more easily taken together, such as

$\displaystyle E^{\prime 2}I_{2,2,0}$
$\displaystyle-2EE^{\prime}I_{2,1,1}+E^{2}I_{2,0,2}\approx\frac{8\pi^{2}\ln\frac{4E^{2}E^{\prime
2}}{\sqrt{(m^{2}\omega^{2}-\lambda b_{0}\omega
EE^{\prime})^{2}+\frac{16E^{4}E^{\prime
2}}{\tau^{2}}}}}{3\left[(m^{2}\omega-\lambda
b_{0}EE^{\prime})^{2}+\frac{16E^{3}E^{\prime}}{\tau^{2}}\right]}$
$\displaystyle\times\frac{\pi^{2}\tau^{3}}{4E^{4}E^{\prime
4}\omega}\left[E^{\prime
6}\arctan\frac{m^{2}(\omega^{*}-\omega)\tau}{E^{\prime}(E-\omega^{*})}+E^{6}\arctan\frac{m^{2}(\omega^{*}-\omega)\tau}{2E(E-\omega^{*})}\right]\Theta(\omega^{*}-\omega)\,,$
(54)

where $\omega^{*}$ is given by (15). The remaining integrals may be computed
similarly, or related using the symmetry $p\rightarrow-p^{\prime}$ such that

$\displaystyle
2\omega\kappa-k^{2}\rightarrow-(2\omega\kappa^{\prime}+k^{2})\,,\qquad{\bm{q}}^{2}=-(p^{\prime}-p+k)^{2}\rightarrow-(-p+p^{\prime}+k)^{2}={\bm{q}}^{2}\,.$
(55)

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# Machine Learning-based Reconfigurable Intelligent Surface-aided MIMO Systems

Nhan Thanh Nguyen1, Ly V. Nguyen2, Thien Huynh-The3, Duy H. N. Nguyen4, A. Lee
Swindlehurst5,
and Markku Juntti1 Email: {nhan.nguyen<EMAIL_ADDRESS>{vnguyen6,
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>The research has
been supported in part by Academy of Finland under 6Genesis Flagship (grant
318927) and EERA Project (grant 332362). Corresponding author: N. T. Nguyen
(email: nhan.nguyen@oulu.fi). 1Centre for Wireless Communications, University
of Oulu, P.O.Box 4500, FI-90014, Finland  2Computational Science Research
Center, San Diego State University, CA, USA  3ICT Convergence Research Center,
Kumoh National Institute of Technology, Gyeongsangbuk-do 39177, Korea
4Department of Electrical and Computer Engineering, San Diego State
University, CA, USA  5Department of Electrical Engineering and Computer
Science, University of California, Irvine, CA, USA.

###### Abstract

Reconfigurable intelligent surface (RIS) technology has recently emerged as a
spectral- and cost-efficient approach for wireless communications systems.
However, existing hand-engineered schemes for passive beamforming design and
optimization of RIS, such as the alternating optimization (AO) approaches,
require a high computational complexity, especially for multiple-input-
multiple-output (MIMO) systems. To overcome this challenge, we propose a low-
complexity unsupervised learning scheme, referred to as learning-phase-shift
neural network (LPSNet), to efficiently find the solution to the spectral
efficiency maximization problem in RIS-aided MIMO systems. In particular, the
proposed LPSNet has an optimized input structure and requires a small number
of layers and nodes to produce efficient phase shifts for the RIS. Simulation
results for a $16\times 2$ MIMO system assisted by an RIS with $40$ elements
show that the LPSNet achieves $97.25\%$ of the SE provided by the AO
counterpart with more than a $95\%$ reduction in complexity.

###### Index Terms:

Reconfigurable intelligent surface, RIS, IRS, MIMO, machine learning, passive
beamforming.

##  I Introduction

Reconfigurable intelligent surface (RIS) technology has recently been shown to
be a promising solution for substantially enhancing the performance of
wireless communications systems [1]. An RIS is often realized by a planar
array comprising a large number of reconfigurable passive reflecting elements
that can be configured to reflect incoming signals by a predetermined phase
shift. Theoretical analyses have shown that an RIS of $N$ elements can achieve
a total beamforming gain of $N^{2}$ [2] and an increase in the received signal
power that increases quadratically with $N$ [3, 4].

An important research direction for RIS-assisted communications systems is how
to optimize the RIS reflecting coefficients to maximize the spectral
efficiency (SE), e.g., [5, 6, 7, 8, 9, 10, 11, 12, 13]. Yang et al. [6]
jointly optimize the transmit power allocation and the passive reflecting
coefficients for an RIS-assisted single-input-single-output (SISO) system. By
contrast, the work in [7, 8, 9, 10] considers multiple-input-single-output
(MISO) systems assisted by RISs with continuous [7, 8] or coarsely quantized
phase shifts [9, 10]. Efficient alternating optimization (AO) methods are
developed in [11, 12, 13] for the SE maximization problem of RIS-aided
multiple-input-multiple-output (MIMO) systems.

Machine learning (ML) has recently attracted much attention for wireless
communication systems [14, 15, 16], and particularly for RIS, e.g., [17, 18,
19, 20, 21, 22]. The work in [17, 18] considers SISO systems where the RIS
contains some active elements. While a deep neural network (DNN) is used in
[17], the work in [18] exploits advances in deep reinforcement learning (DRL).
Their solutions can approach the upper bound achieved with perfect channel
state information (CSI). However, the introduction of active elements at the
RIS causes additional hardware costs and power consumption. DRL is also
exploited to optimize the RIS in assisting a system with a single-antenna user
in [19], and for MIMO channels without a direct base station (BS) – mobile
station (MS) link [20]. Gao et al. [21] propose a DNN-based passive
beamforming (PBF) design, but only for a single-antenna user. Ma et al. [22]
exploit federated learning (FL) to enhance the PBF performance and user
privacy, but again for a simplified system with a single-antenna BS and no
direct link to the MS.

Unlike the aforementioned studies on ML-based PBF, we consider a more general
RIS-aided MIMO system in this paper. Furthermore, both the direct BS-MS and
the reflected BS-RIS-MS links are taken into consideration, which imposes
significantly more difficulty in the optimization of the ML models. In
particular, the ML model needs to be optimized so that the signals through the
different links add constructively at the MS. This implies another challenge,
in that a large amount of information must be extracted by the ML model from
the extremely high-dimension input data. Furthermore, unless an exhaustive
search is performed, it is challenging to develop supervised learning-based ML
models due to the unavailability of data labels. Fortunately, these challenges
can be overcome by the findings in our paper. Specifically, by formulating the
SE maximization problem and studying the AO solution, we discover an
informative input structure that enables a DNN, even with a single hidden
layer and an unsupervised learning strategy, to generate efficient phase
shifts for PBF. The proposed DNN-based PBF scheme, referred to as a learning-
phase-shift neural network (LPSNet), is numerically shown to perform very
close to the AO method in [11] with a substantial complexity reduction.

##  II System Model and Problem Formulation

###  II-A System Model

We consider downlink transmission between a BS and MS that are equipped with
$N_{t}$ and $N_{r}$ antennas, respectively. The communication is assisted by
an RIS with $N$ passive reflecting elements. Let
$\textbf{{H}}_{d}\in\mathbb{C}^{N_{r}\times N_{t}}$,
$\textbf{{H}}_{t}\in\mathbb{C}^{N\times N_{t}}$, and
$\textbf{{H}}_{r}\in\mathbb{C}^{N_{r}\times N}$ denote the BS-MS, BS-RIS, and
RIS-MS channels, respectively, and let
$\boldsymbol{\Phi}=\mathrm{diag}\\{\alpha_{1},\alpha_{2},\ldots,\alpha_{N}\\}$
denote the diagonal reflecting matrix of the RIS. The RIS coefficients are
assumed to have unit-modulus, i.e., $\alpha_{n}=e^{j\theta_{n}},\forall n$,
and, thus, it only introduces phase shifts $\theta_{n}\in[0,2\pi),\forall n$
to the impinging signals. Let $\textbf{{x}}\in\mathbb{C}^{N_{t}\times 1}$ be
the transmitted signal vector. We focus on the design of the PBF and assume a
uniform power allocation, i.e.,
$\mathbb{E}\left\\{\textbf{{x}}\textbf{{x}}^{H}\right\\}=P_{\mathrm{BS}}\textbf{{I}}_{N_{t}}$,
where $P_{\mathrm{BS}}$ is the transmit power at the BS. The MS receives the
signals through the direct channel and via the reflection by the RIS.
Therefore, the received signal vector at the MS can be expressed as

y
$\displaystyle=\textbf{{H}}_{d}\textbf{{x}}+\textbf{{H}}_{r}\boldsymbol{\Phi}\textbf{{H}}_{t}\textbf{{x}}+\textbf{{n}}=\textbf{{H}}\textbf{{x}}+\textbf{{n}},$
(1)

where
$\textbf{{H}}\triangleq\textbf{{H}}_{d}+\textbf{{H}}_{r}\boldsymbol{\Phi}\textbf{{H}}_{t}$
is the combined effective channel, and
$\textbf{{n}}\sim\mathcal{CN}(0,\sigma^{2}\textbf{{I}}_{N_{r}})$ is complex
additive white Gaussian noise (AWGN) at the MS.

###  II-B Problem Formulation

Based on (1), the SE of the RIS-aided MIMO system can be expressed as [11]

$\displaystyle\mathrm{SE}\left(\\{\alpha_{n}\\}\right)=\log_{2}\mathrm{det}(\textbf{{I}}_{N_{r}}+\rho\textbf{{H}}\textbf{{H}}^{H}),$
(2)

where $\\{\alpha_{n}\\}=\\{\alpha_{1},\alpha_{2},\ldots,\alpha_{N}\\}$ is the
set of the phase shifts at the RIS that needs to be optimized, and
$\rho=\frac{P_{\mathrm{BS}}}{\sigma^{2}}$. The PBF design maximizing the SE
can be formulated as

$\displaystyle(\mathrm{P0})\quad\underset{\\{\alpha_{n}\\}}{\textrm{maximize}}\quad$
$\displaystyle\mathrm{SE}\left(\\{\alpha_{n}\\}\right)$ (3a) subject to
$\displaystyle\left|\alpha_{n}\right|=1,\forall n.$ (3b)

The objective function $\mathrm{SE}\left(\\{\alpha_{n}\\}\right)$, given in
(2), is nonconvex with respect to $\\{\alpha_{n}\\}$, and the feasible set for
$(\mathrm{P0})$ is nonconvex due to the unit-modulus constraint (3b).
Therefore, $(\mathrm{P0})$ is intractable and is difficult to find an optimal
solution. Efficient solutions to $(\mathrm{P0})$ can be solved by the AO [11,
13, 12] or projected gradient descent (PGD) [23] methods. However, these
algorithms require an iterative update of $\\{\alpha_{n}\\}$. In particular,
to obtain the phase shifts in each iteration, computationally expensive
mathematical operations are performed, and as a result these algorithms have
high complexity and latency. In the next section, we propose the low-
complexity LPSNet to find an efficient solution to $(\mathrm{P0})$.

##  III DNN-based PBF and Proposed LPSNet

###  III-A DNN-based PBF

Instead of employing a high-complexity algorithm (e.g., AO or PGD), a DNN can
be modeled and trained to solve $(\mathrm{P0})$, i.e., to predict
$\\{\alpha_{n}\\}$. While the phase shifts $\\{\alpha_{n}\\}$ are complex
numbers and cannot be directly generated by a DNN, since
$\left|\alpha_{n}\right|=1,\forall n$, the coefficients $\\{\alpha_{n}\\}$ are
completely specified only by their real-valued phase shifts
$\\{\theta_{n}\\}$. Hence, $(\mathrm{P0})$ is equivalent to

$\displaystyle(\mathrm{P})\quad\underset{\\{\theta_{n}\\}}{\textrm{maximize}}\quad$
$\displaystyle\mathrm{SE}\left(\\{\theta_{n}\\}\right),$ (4)

whose solution is found by a DNN in this work.

Let $\bar{\textbf{{h}}}$ be a vector containing the CSI parameters (i.e., the
information in $\textbf{{H}}_{d}$, $\textbf{{H}}_{t}$, and
$\textbf{{H}}_{r}$). The phase shifts learned by a DNN can be mathematically
modeled as

$\displaystyle\\{\hat{\theta}_{n}\\}=f_{\mathrm{NN}}\left(\bar{\textbf{{h}}}\right),$
(5)

where $f_{\mathrm{NN}}(\cdot)$ represents the non-linear mapping from
$\bar{\textbf{{h}}}$ to $\\{\hat{\theta}_{n}\\}$ performed by the DNN. The
efficiency and structure of $f_{\mathrm{NN}}$ significantly depend on the
input. Therefore, properly designing $\bar{\textbf{{h}}}$ is one of the first
and most important tasks in modeling $f_{\mathrm{NN}}$. Let
$\bar{\textbf{{h}}}_{0}$ be the input vector constructed from the original CSI
(i.e., entries of $\textbf{{H}}_{d}$, $\textbf{{H}}_{t}$, and
$\textbf{{H}}_{r}$), i.e.,

$\displaystyle\bar{\textbf{{h}}}_{0}\triangleq\mathrm{vec}\left(\textbf{{H}}_{d},\textbf{{H}}_{t},\textbf{{H}}_{r}\right),$
(6)

where $\mathrm{vec}(\cdot)$ is a vectorization. Obviously,
$\bar{\textbf{{h}}}_{0}$ contains the required CSI for a conventional hand-
engineered algorithm to solve $(\mathrm{P0})$. However,
$\bar{\textbf{{h}}}_{0}$ does not possess any structure or meaningful patterns
that the DNN can exploit. Furthermore, if a very deep DNN is used, the
resulting computational complexity may be as high as that of the conventional
schemes, making it computationally inefficient. Our experimental results
obtained by training DNNs with the input $\bar{\textbf{{h}}}_{0}$ show that
the DNNs cannot escape from local optima even after extensive fine tuning.
This motivates us to select more informative features as input of the proposed
DNN.

###  III-B Proposed Efficient Input Structure

To design a better-performing input structure for $f_{\mathrm{NN}}$, we first
extract the role of each individual phase shift $\theta_{n}$ in
$\mathrm{SE}\left(\\{\theta_{n}\\}\right)$. Let $\textbf{{r}}_{n}$ be the
$n$th column of $\textbf{{H}}_{r}$ and $\textbf{{t}}_{n}^{H}$ be the $n$th row
of $\textbf{{H}}_{t}$, i.e.,
$\textbf{{H}}_{r}=[\textbf{{r}}_{1},\ldots,\textbf{{r}}_{N}]$ and
$\textbf{{H}}_{t}=[\textbf{{t}}_{1},\ldots,\textbf{{t}}_{N}]^{H}$.
Furthermore, for ease of notation, define
$\bar{\textbf{{H}}}_{(0)}\triangleq\textbf{{H}}_{d}$ and $\theta_{0}\triangleq
0$. Since $\boldsymbol{\Phi}$ is diagonal, one has

$\displaystyle\textbf{{H}}=\textbf{{H}}_{d}+\sum_{n=1}^{N}e^{j\theta_{n}}\textbf{{r}}_{n}\textbf{{t}}_{n}^{H}=\sum_{i=0}^{N}e^{j\theta_{i}}\bar{\textbf{{H}}}_{(i)},$
(7)

where
$\bar{\textbf{{H}}}_{(n)}\triangleq\textbf{{r}}_{n}\textbf{{t}}_{n}^{H}$,
$n=1,\ldots,N$. Consequently, $\mathrm{SE}\left(\\{\theta_{n}\\}\right)$ can
be recast in an explicit form

$\displaystyle\mathrm{SE}(\theta_{n})=\log_{2}\left|\textbf{{A}}_{n}+e^{j\theta_{n}}\textbf{{B}}_{n}+e^{-j\theta_{n}}\textbf{{B}}_{n}^{H}\right|,$
(8)

where

$\displaystyle\textbf{{A}}_{n}$
$\displaystyle\triangleq\textbf{{I}}_{N_{r}}+\rho\left(\sum_{i=0,i\neq
n}^{N}e^{j\theta_{i}}\bar{\textbf{{H}}}_{(i)}\right)\left(\sum_{i=0,i\neq
n}^{N}e^{j\theta_{i}}\bar{\textbf{{H}}}_{(i)}\right)^{H}$ $\displaystyle\hskip
142.26378pt+\rho\bar{\textbf{{H}}}_{(n)}\bar{\textbf{{H}}}_{(n)}^{H},$ (9)
$\displaystyle\textbf{{B}}_{n}$
$\displaystyle\triangleq\rho\bar{\textbf{{H}}}_{(n)}\left(\sum_{i=0,i\neq
n}^{N}e^{j\theta_{i}}\bar{\textbf{{H}}}_{(i)}\right)^{H},$ (10)

$n=1,\ldots,N$. It is observed that neither $\textbf{{A}}_{n}$ or
$\textbf{{B}}_{n}$ involves $\theta_{n}$. This means that if all variables
$\\{\theta_{i}\\}_{i=0,i\neq n}^{N}$ are fixed, $\textbf{{A}}_{n}$ and
$\textbf{{B}}_{n}$ are determined, and $\theta_{n}$ can be found by [11]

$\displaystyle\theta_{n}=\mathrm{arg}\\{\lambda(\textbf{{A}}_{n}^{-1}\textbf{{B}}_{n})\\},$
(11)

where $\lambda(\textbf{{A}}_{n}^{-1}\textbf{{B}}_{n})$ denotes the sole non-
zero eigenvalue of $\textbf{{A}}_{n}^{-1}\textbf{{B}}_{n}$ [11]. This implies
that the information necessary to obtain $\theta_{n}$ is contained in
$\textbf{{A}}_{n}^{-1}\textbf{{B}}_{n}$, which can be computed if the values
$\\{\bar{\textbf{{H}}}_{(i)}\\}$ are available, as observed in (9) and (10).
We summarize this important result in the following remark.

###### Remark 1

In a DNN predicting the phase shifts $\\{\theta_{n}\\}$ of the RIS, the input
vector should contain the coefficients of $\\{\bar{\textbf{{H}}}_{(i)}\\}$.
More specifically, it can be constructed as

$\displaystyle\bar{\textbf{{h}}}=\mathrm{vec}\left(\left\\{\mathfrak{R}{\left(\bar{\textbf{{H}}}_{(i)}\right)}\right\\},\left\\{\mathfrak{I}{\left(\bar{\textbf{{H}}}_{(i)}\right)}\right\\}\right)\in\mathbb{R}^{2N_{t}N_{r}(N+1)\times
1},$ (12)

where $\mathfrak{R}{\left(\textbf{{A}}\right)}$ and
$\mathfrak{I}{\left(\textbf{{A}}\right)}$ represent the real and imagine parts
of the entries of A, respectively. There are $N+1$ matrices
$\bar{\textbf{{H}}}_{(i)}\in\mathbb{C}^{N_{r}\times N_{t}}$. Thus,
$\bar{\textbf{{h}}}$ consists of $2N_{t}N_{r}(N+1)$ real elements.

Some interesting observations can be made from (7) and (12), as noted in the
following remark.

###### Remark 2

The efficient input structure of Remark 1 is constructed from the pairwise
products of the $N$ columns of $\textbf{{H}}_{r}$ and the $N$ rows of
$\textbf{{H}}_{t}$. This is of interest because there are various ways to
extract/combine the information from the entries of $\textbf{{H}}_{t}$ and
$\textbf{{H}}_{r}$. Based on Remark 1, it is reasonable to expect that the raw
CSI $\bar{\textbf{{h}}}_{0}$ is not readily exploited by the DNN to learn our
regression model successfully. Furthermore, $\textbf{{H}}_{d}$, i.e.,
$\\{\bar{\textbf{{H}}}_{(0)}\\}$ in (12), contributes to $\bar{\textbf{{h}}}$
with its original entries, and no manipulation is required.

In addition, we note that although additional mathematical operations are
required to obtain $\\{\bar{\textbf{{H}}}_{(i)}\\}$, they are just low-
complexity multiplications of a column and a row vector and matrix additions.
Therefore, generating the proposed input structure requires low computational
complexity, but it has a significant impact on the learning ability and
structure of the DNN employed for PBF, as will be shown in the next section.

###  III-C Proposed LPSNet

Here we propose a learning-phase-shift neural network, referred to as LPSNet,
for solving the PBF problem $(\mathrm{P})$ in (4). We stress that the
informative input structure in (12) plays a deciding role in the efficiency of
the LPSNet. Specifically, $\bar{\textbf{{h}}}$ in (12) already contains the
meaningful information necessary to obtain $\\{\theta_{n}\\}$. The network
structure, training strategy, and computational complexity of LPSNet are
presented next.

####  III-C1 Network Structure

LPSNet has a fully-connected DNN architecture, and like any other DNN model,
it is optimized by fine-tuning the number of hidden layers and nodes, and the
activation function. However, thanks to the handcrafted feature selection, the
LPSNet can predict $\\{\theta_{n}\\}$ efficiently with only a small number of
layers. Through fine-tuning, we have found that only a single or a few hidden
layers with $N$ nodes in each is sufficient to ensure satisfactory
performance. As a result, the sizes of the input, hidden, and output layers
are $2N_{t}N_{r}(N+1)$ (i.e., the size of $\bar{\textbf{{h}}}$), $N$, and $N$,
respectively. Furthermore, we have found that Sigmoid activation functions at
both the hidden and output layers can provide better training of the LPSNet
than other activation functions. With such a shallow network, the complexity
of LPSNet is very low. Moreover, we have found that to optimize the LPSNet,
fine-tuning can be done by slightly increasing the number of hidden layers and
nodes, e.g., for large MIMO systems. For example, to predict the phase shifts
of an RIS equipped with $N=40$ elements assisting an $8\times 2$ MIMO system,
a single hidden layer with $N$ nodes can guarantee an efficient training,
while for a $16\times 2$ MIMO system, two hidden layers, each deployed with
$N$ nodes, is sufficient (see Table I). For other systems, it is recommended
to begin fine-tuning with such a proposed simple network to avoid overfitting.

####  III-C2 Training Strategy

For offline training, we propose employing an unsupervised training strategy
[21]. We note that, on the other hand, if supervised training is used, labels
($\\{\theta_{n}\\}$) have to be obtained using a conventional high-complexity
method such as AO [11] or PGD [23]. To avoid this high computational load,
unsupervised training is adopted for training the LPSNet so that it can
maximize the achievable SE without the labels $\\{\theta_{n}\\}$. To this end,
we set the loss function to

$\displaystyle\mathrm{Loss}(\\{\hat{\theta}_{n}\\})=-\log_{2}\mathrm{det}(\textbf{{I}}_{N_{r}}+\rho_{\mathrm{train}}\textbf{{H}}_{\mathrm{train}}\textbf{{H}}_{\mathrm{train}}^{H}),$
(13)

where $\rho_{\mathrm{train}}$ is randomly picked from the range
$[-\rho_{0},\rho_{0}]$ dB, and
$\textbf{{H}}_{\mathrm{train}}=\textbf{{H}}_{d,\mathrm{train}}+\textbf{{H}}_{r,\mathrm{train}}\hat{\boldsymbol{\Phi}}\textbf{{H}}_{t,\mathrm{train}}$.
Here, $\textbf{{H}}_{d,\mathrm{train}}$, $\textbf{{H}}_{t,\mathrm{train}}$,
and $\textbf{{H}}_{r,\mathrm{train}}$ are obtained by scaling
$\textbf{{H}}_{d}$, $\textbf{{H}}_{t}$, and $\textbf{{H}}_{r}$ by their
corresponding large-scale fading coefficients so that their entries have zero-
mean and unit-variance. This pre-processing is also applied to obtain the
input data in $\bar{\textbf{{h}}}$. In other words, $\bar{\textbf{{h}}}$ is
obtained from $\textbf{{H}}_{d,\mathrm{train}}$,
$\textbf{{H}}_{t,\mathrm{train}}$, and $\textbf{{H}}_{r,\mathrm{train}}$
rather than $\textbf{{H}}_{d}$, $\textbf{{H}}_{t}$, and $\textbf{{H}}_{r}$.
Furthermore,
$\hat{\boldsymbol{\Phi}}=\mathrm{diag}\\{e^{j\hat{\theta}_{1}},\ldots,e^{j\hat{\theta}_{N}}\\}$,
where $\\{\hat{\theta}_{n}\\}$ are the outputs of the LPSNet, as given in (5).
During training, the weights and biases of the LPSNet are optimized such that
$\mathrm{Loss}$ is minimized, or equivalently, SE is maximized.

####  III-C3 Online LPSNet-based PBF

Once LPSNet has been trained, it is ready to be used online for PBF. At first,
$\bar{\textbf{{h}}}$ is constructed from the CSI based on Remark 1. Then, the
LPSNet takes $\bar{\textbf{{h}}}$ as the input for output
$\\{\hat{\theta}_{n}\\}$, which are then transformed to the reflecting
coefficients $\\{\hat{\alpha}_{n}\\}$, with
$\hat{\alpha}_{n}=\cos{\hat{\theta}}+j\sin{\hat{\theta}},\forall n$. At this
stage, the PBF matrix is obtained as
$\boldsymbol{\Phi}=\mathrm{diag}\\{\hat{\alpha}_{1},\ldots,\hat{\alpha}_{N}\\}$.

####  III-C4 Computational Complexity Analysis

Let $L$ denote the number of hidden layers of LPSNet. LPSNet requires a
complexity of $\mathcal{O}(N_{r}N_{t}N)$ for constructing the input
$\bar{\textbf{{h}}}$, $\mathcal{O}(N_{r}N_{t}N^{2})$ for the first hidden
layer, and $\mathcal{O}(N^{2})$ for subsequent hidden layers and the output
layer. Therefore, the total complexity of the proposed LPSNet is
$\mathcal{O}(\max(N_{r}N_{t}N^{2},LN^{2}))$. When $L<N_{r}N_{t}$, the
complexity of LPSNet is $\mathcal{O(}N_{r}N_{t}N^{2})$. This should be the
case since we found that LPSNet requires a small $L$ to produce efficient
phase shift values; for example in Table I, it is seen that only $1$ or $2$
hidden layers are required.

For comparison, the computational complexity of the AO method in [11] is
$\mathcal{O}\big{(}N_{r}N_{t}(N+M)K+((3N_{r}^{3}+2N_{r}^{2}N_{t}+N_{t}^{2})N+N_{r}N_{t}M)I\big{)}$
where $M=\min(N_{r},N_{t})$, $K$ is the number of random sets
$\\{\alpha_{n}\\}$ initializing the AO method, and $I$ is the number of
iterations. Since $I$ is polynomial over $N_{r}$, $N_{t}$, and $N$, the total
complexity of the AO method is at least quadratic in $N_{r}$, cubic in
$N_{t}$, and quadratic in $N$. The complexity of LPSNet is only linear in
$N_{r}$ and $N_{t}$, and quadratic in $N$. Therefore, LPSNet is less
computationally demanding than the AO method. The computational complexity for
each iteration of the PGM method in [23] is
$\mathcal{O}(2NN_{t}N_{r}+2N_{t}^{2}N_{r}+\frac{3}{2}N_{t}N_{r}^{2}+N_{r}^{3}+N_{r}N+N_{t}N+3N+\frac{3}{2}N_{t}^{3})$.
Similarly, when the number of iterations is large, the PGM method is also more
computationally expensive than LPSNet.

##  IV Simulation Results

In this section, we numerically investigate the SE performance and
computational complexity of the proposed LPSNet. We assume that the BS, MS,
and RIS are deployed in a two-dimensional coordinate system at $(0,0)$,
$(x_{\mathrm{MS}},y_{\mathrm{MS}})$, and $(x_{\mathrm{RIS}},0)$, respectively.
The path loss for link distance $d$ is given by
$\beta(d)=\beta_{0}(d/1\mathrm{m})^{\epsilon}$, where $\beta_{0}$ is the path
loss at the reference distance of $1$ m, and $\epsilon$ is the path loss
exponent [2, 11]. Following [11, 2, 12], we assume a Rician fading channel
model for the small-scale fading of all the involved channels, including
$\textbf{{H}}_{d}$, $\textbf{{H}}_{t}$, and $\textbf{{H}}_{r}$. For more
details on the the small-scale fading, please refer to [11, 2, 12]. We
consider two systems, namely, $8\times 2$ MIMO and $16\times 2$ MIMO, both of
which are aided by a RIS with $N=40$ reflecting elements [11, 2]. The
bandwidth is set to $10$ MHz, and the noise power is $-170$ dBm/Hz [11]. Let
$\epsilon_{(\cdot)}$ and $\kappa_{(\cdot)}$ respectively denote the path loss
exponent and Rician factor of the corresponding channel
$\textbf{{H}}_{(\cdot)}$. We set
$\\{\epsilon_{d},\epsilon_{t},\epsilon_{r}\\}=\\{3.5,2,2.8\\}$ [2], and due to
the large distance and random scattering on the BS-MS channel, we set
$\kappa_{d}=0$ [2]. By contrast, to show that the proposed LPSNet can
generalize to perform well for various physical deployments of the devices, in
the simulations the RIS is randomly located on the $x-$axis, at a maximum
distance of $100$m from the BS. Then, the MS is randomly placed $2$m away from
the RIS. This setting implies that the MS is near the RIS, as widely assumed
in the literature [2, 11, 13]. Consequently, $\kappa_{t}$ and $\kappa_{r}$ are
both randomly generated. This procedure applies to the generation of both the
training and testing data sets.

Table I: LPSNets employed for $8\times 2$ and $16\times 2$ MIMO systems. MIMO system |  No. of hidden layers |  No. of nodes in hidden layers | Batch size
---|---|---|---
$8\times 2$ MIMO | $1$ | $N$ |  $10$ samples
$16\times 2$ MIMO | $2$ | $N\times N$ |  $20$ samples

LPSNet is implemented and trained using Python with the Tensorflow and Keras
libraries and an NVIDIA Tesla V100 GPU processor. For the training phase, we
employ the Adam optimizer (with a decaying learning rate of $0.99$ and a
starting learning rate of $0.001$) and the loss function (13) with
$\rho_{0}=30$ dB. We train and test the performance of the LPSNets, summarized
in Table I, for two MIMO systems, namely $8\times 2$ MIMO and $16\times 2$
MIMO. For the former, LPSNet has only one hidden layer with $N$ nodes, whereas
for the latter, two hidden layers, each with $N$ nodes, are employed. The
batch sizes for training these LPSNets are $10$ and $20$ samples,
respectively, selected from data sets of $40,$$000$ samples.

###  IV-A The Significance of Input Structure

Fig. 1: Loss of LPSNet when using the proposed input $\bar{\textbf{{h}}}$ and
$\bar{\textbf{{h}}}_{0}$ for $8\times 2$ MIMO and $16\times 2$ MIMO systems.

In Fig. 1, we justify the efficiency of the proposed input structure
$\bar{\textbf{{h}}}$ in (12) by comparing the resultant loss to that offered
by $\bar{\textbf{{h}}}_{0}$ in (6). It is observed for both the $8\times 2$
MIMO (Fig. 1(a)) and $16\times 2$ MIMO (Fig. 1(b)) systems that, as the epoch
number increases, the loss value of the LPSNet with $\bar{\textbf{{h}}}$
quickly decreases until reaching convergence, with almost no or acceptable
overfitting/underfitting. In contrast, when $\bar{\textbf{{h}}}_{0}$ is used,
LPSNet cannot escape from the local optima, providing almost constant loss
values (Figs. 1(c)). This is not overcome even when we increase the learning
rate to $0.002$ and batch size to $100$ samples as in Fig. 1(d). The results
in Fig. 1 demonstrate that the input structure plays a key role in the
learning ability of LPSNet, and the proposed informative input structure
$\bar{\textbf{{h}}}$ enables LPSNet to be trained well for PBF even with
simple architectures.

###  IV-B SE Performance and Complexity of the Proposed LPSNet

Fig. 2: SE and complexity of the proposed LPSNet compared to other
conventional schemes for (a) $8\times 2$ and (b) $16\times 2$ MIMO systems.

In Fig. 2, the SE performance and computational complexity of the proposed
LPSNet are shown for $8\times 2$ and $16\times 2$ MIMO systems. For
comparison, we also consider the cases where $\\{\theta_{n}\\}$ are either
randomly generated, obtained based on the AO method [11], or when RISs are not
employed. For both considered MIMO systems we see that the proposed LPSNet
performs far better than when using random phases for the RIS or without RIS.
It performs similarly to the AO method but requires much lower complexity. For
example, in the $16\times 2$ MIMO system, LPSNet achieves $97.25\%$ of the SE
provided by the AO approach with more than a $95\%$ reduction in complexity.

Fig. 3: SE of the proposed LPSNet versus the positions of the MS for (a)
$8\times 2$ and (b) $16\times 2$ MIMO systems with $P_{\mathrm{BS}}=40$ dBm.

In Fig. 3, we show the SE of the considered schemes when fixing
$x_{\mathrm{RIS}}=100$ m, $x_{\mathrm{MS}}\in[80,120]$ m, and
$P_{\mathrm{BS}}=40$ dBm. Clearly, the proposed LPSNet attains good
performance over the entire considered range of $x_{\mathrm{MS}}$, which
demonstrates its ability to perform well regardless of the physical distances
between devices. This is because the training data (i.e., the channel
coefficients) have been pre-processed to remove the effect of the large-scale
fading. Furthermore, LPSNet provides the highest SE at
$x_{\mathrm{MS}}=x_{\mathrm{RIS}}=100$ m, i.e., when the RIS and MS are
closest to each other, in agreement with the findings of [2, 24].

##  V Conclusion

In this paper, we have considered the application of DNN to passive
beamforming in RIS-assisted MIMO systems. Specifically, we formulated the SE
maximization problem, and showed how it can be solved efficiently by the
proposed unsupervised learning-based LPSNet. With the optimized input
structure, LPSNet requires only a small number of layers and nodes to output
reliable phase shifts for PBF. It achieves almost the same performance as the
AO method with considerably less computational complexity. Furthermore, the
proposed unsupervised learning scheme does not require any computational load
for data labeling.

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