# Vavilov-Cherenkov emission with a twist:
a study of the final entangled state

A.D. Chaikovskaia<EMAIL_ADDRESS>School of Physics and Engineering, ITMO
University, 197101 St. Petersburg, Russia  D.V. Karlovets  School of Physics
and Engineering, ITMO University, 197101 St. Petersburg, Russia  V.G. Serbo
Novosibirsk State University, 630090 Novosibirsk, Russia
Sobolev Institute of Mathematics, 630090 Novosibirsk, Russia  School of
Physics and Engineering, ITMO University, 197101 St. Petersburg, Russia

###### Abstract

The Vavilov-Cherenkov radiation by a plane-wave electron or a twisted electron
is considered within quantum electrodynamics. A twisted electron has a
definite projection of the total angular momentum (TAM) $m=\pm 1/2,\,\pm
3/2,\ldots$ on the direction of motion. An exact analytical expression in the
coordinate and momentum representations is found for the evolved wave function
of the final bipartite system, which has arisen as a result of the radiation
process itself and does not depend on the properties of a detector. It is
shown that this evolved wave function is an entangled state of an electron and
a photon, and both particles can be twisted. It provides an interesting
possibility: if we use a detector sensitive to the twisted electron with the
definite TAM projection, then the final photon also is automatically projected
onto the twisted state. Approximations of soft photons as well as of ultra-
relativistic electrons are considered. Besides, we point out a close
similarity between the discussed problem and the problem of the evolved wave
function for the emission of the virtual photons in the deep inelastic
$ep$-scattering as well as for the emission of the equivalent photons in the
Weizsäcker-Williams method.

##  I Introduction

The Vavilov–Cherenkov (VC) radiation was discovered in 1934 discovery  and
very soon explained by Frank and Tamm Tamm-Frank  in the framework of
classical electrodynamics. A few years later, Ginzburg VLG1940  and Sokolov
sokolov1940  gave the quantum derivation of this phenomenon and found quantum
corrections to the classical Frank–Tamm result.

Historical and contemporary reviews on the subject can be found in
Bolotovsky-2009 ; VLG ; Afanasyev . A renewed motivation for the study of VC
radiation within quantum electrodynamics (QED) is brought by recent
theoretical and experimental advances with the so-called twisted particles,
i.e. those carrying nonzero orbital angular momentum (OAM). The reviews
Blioch-Ivanov-2017 ; Knyazev-Serbo-2018 ; Ivanov:2022jzh  give an ample
discussion on the properties of twisted electrons and photons and their
experimental status. A QED description of VC radiation emitted by a twisted
electron in the transparent medium has recently been considered in
Kaminer-2016 ; ISZ-2016 . There, the analytical expressions have been obtained
for the spectral and spectral-angular distributions as well as for the
polarization properties of the emitted radiation.

In any quantum process the detection scheme plays a crucial role in the
measurement of the final quantum state characteristics. In the previous
studies of VC radiation the most common method is to perform calculations for
the final states being detected in the form of plane waves both for the
electron and the photon as in Refs. VLG1940 ; sokolov1940 ; Kaminer-2016 ;
ISZ-2016 . Another approach KBGSS-2022 ; KBGSS-2023 , based on the so-called
generalized measurements, focuses on the entanglement between a pair of the
final particles and implies in practice the usage of a certain post-selection
protocol for the final electron only when the momentum of the electron is
measured with a large uncertainty. In this approach, the evolved state of the
emitted photon appears.

In the present work we aim to give a complementary approach that allows one to
perform a quantum tomography of the scattered radiation without the need to
introduce detector states. In particular, we develop an exact analytical
expression for the final system wave function in the coordinate and momentum
representations that is irrespective of the detector properties. We use the
well–established $S$–matrix theory of the evolved state that provides the wave
function of the outgoing photon as it results from the scattering process
itself. The method was also used recently in Ref. KSS-2021  for the
description of the resonance scattering of twisted laser photons on the ultra-
relativistic partially striped ions in the project of the Gamma Factory at the
Large Hadron Collider.

As a part of motivation for this study a few words on the somewhat natural
relation between twisted photons and VC radiation are in order. The VC
radiation has a very specific angular distribution of final photons of energy
$\omega$: they are concentrated in a narrow cone near the polar angle

$\theta_{\gamma}\approx\arccos\left(\frac{{\displaystyle 1}}{{\displaystyle
vn(\omega)}}\right),$ (1)

where $v$ is the velocity of the emitting electron moving along the $z$-axis.
Just the same feature is possessed by the wave function of a twisted photon,
for which an opening (or conical) angle is equal to

$\theta_{\gamma}=\arccos\left(\frac{{\displaystyle k_{z}}}{{\displaystyle|{\bf
k}|}}\right)=\arccos\left(\frac{{\displaystyle 1}}{{\displaystyle
vn(\omega)}}\right),$ (2)

meaning that it is determined by the magnitude of the longitudinal momentum of
the photon $k_{z}=\omega/v$ and its energy ${\omega=|{\bf k}|/n(\omega)}$.
Second, VC radiation is linearly polarized in the scattering plane. As we show
below, our result for the case of the low energy photons reproduces this
observation.

The paper is organized as follows. We start with some general formalism:
introducing the notion of the evolved wave function of the final system, the
basic notations and the standard expression for the amplitude of the VC
process in Section II. Then, in Section III we perform an analysis for the
case of an ordinary plane-wave initial electron and derive the evolved wave
function of the quantum system in the coordinate and momentum representations
that corresponds to the entangled state of an emitted photon and an outgoing
electron. It is remarkable that a symmetrical and relatively simple wave
function could be obtained. In Section IV we generalize the calculation for
the case of the initial twisted electron represented as a proper superposition
of the plane waves comprising a Bessel wave state. We discuss the impact of
the achieved result and its possible implications in Section V. In conclusion
we also point out a close similarity between the discussed problem and the
problem of the evolved wave functions for the emission of a virtual photon in
the deep inelastic $ep$-scattering and for the emission an equivalent photon
in the Weizsäcker-Williams method (see Refs. BLP  and BGMS-1975 ). Through the
paper we use the relativistic system of units $\hbar=1$, $c=1$ and
$\alpha=e^{2}/(\hbar c)\approx 1/137$, where $e$ is the electron charge.

##  II General formalism

###  II.1 Definition of evolved state

Let us start with emission of a photon by a charged particle (for
definiteness, an electron),

$e\to e^{\prime}+\gamma,$ (3)

in the lowest order of the perturbation theory in QED. It can generate either
VC radiation or transition radiation in medium, synchrotron radiation,
undulator radiation in a given electromagnetic field, and so on. A quantum
state of the final system as it evolves from the reaction irrespectively of
the measurement protocol can be called pre-selected or evolved because no
measurements have been done yet. An initial state $|\text{in}\rangle$ of the
electron and an evolved two-particle state of the final system of the electron
and the photon are connected via an evolution operator $\hat{S}$ BLP

$\hat{S}=\text{T}\exp\left\\{-ie\int
d^{4}x\,\hat{j}_{\mu}\hat{A}^{\mu}\right\\}$ (4)

as follows:

$|e^{\prime},\gamma\rangle^{(\text{ev})}=\hat{S}\,|\text{in}\rangle=\sum\limits_{f}|f\rangle\langle
f|\hat{S}|\text{in}\rangle,$ (5)

where we have expanded the unitary operator over a complete set of two-
particle states (with no virtual particles on the tree-level)

$\displaystyle\hat{1}=\sum\limits_{f}|f\rangle\langle
f|=\sum\limits_{f_{e}}\sum\limits_{f_{\gamma}}|f_{e},f_{\gamma}\rangle\langle
f_{e},f_{\gamma}|.$ (6)

In particular, this can be the plane-wave states with the definite four-
momenta $p^{\prime}=(E^{\prime},{\bf p}^{\prime}),\ k=(\omega,{\bf k})$ and
the helicities $\lambda^{\prime}=\pm 1/2,\lambda_{\gamma}=\pm 1$ for
$e^{\prime}$ and $\gamma$, respectively, so that

$\hat{1}=\sum\limits_{\lambda^{\prime}\lambda_{\gamma}}\int\frac{d^{3}p^{\prime}}{(2\pi)^{3}}\frac{d^{3}k}{(2\pi)^{3}}|{\bf
p}^{\prime}\lambda^{\prime},{\bf k}\lambda_{\gamma}\rangle\langle{\bf
p}^{\prime}\lambda^{\prime},{\bf k}\lambda_{\gamma}|.$ (7)

In this case, the average

$S^{(1)}_{fi}=\langle f_{e},f_{\gamma}|\hat{S}|\text{in}\rangle=\langle{\bf
p}^{\prime}\lambda^{\prime},{\bf k}\lambda_{\gamma}|\hat{S}|\text{in}\rangle$
(8)

is a customary first-order matrix element with two final plane-wave states
whose polarization properties are described by bispinor $u_{{\bf
p}^{\prime}\lambda^{\prime}}$ and four-vector $e_{{\bf
k}\lambda_{\gamma}}=(0,{\bf e}_{{\bf k}\lambda_{\gamma}})$.

Let us first assume that no final particle is detected and discuss how one can
define “the wave function” of the entangled evolved state. The field operators
in the Heisenberg representation of the photon are expanded into series of the
creation and annihilation operators of the plane-wave states

$\displaystyle\hat{\bf A}(x)=\sum\limits_{\lambda_{\gamma}=\pm
1}\int\frac{d^{3}k}{(2\pi)^{3}}\left({\bf A}_{{\bf
k}\lambda_{\gamma}}(x)\,\hat{c}_{{\bf k}\lambda_{\gamma}}+\text{h.c.}\right),$
(9) $\displaystyle{\bf A}_{{\bf k}\lambda_{\gamma}}(x)=\frac{\sqrt{4\pi}\,{\bf
e}_{{\bf k}\lambda_{\gamma}}}{\sqrt{2\omega}\,n}\,e^{-ikx},$ (10)

where $x=(t,{\bf r})$ and $n=n(\omega)$ is the refraction index of medium.
Analogously, for the electron we have

$\displaystyle\hat{\psi}(x)=\sum\limits_{\lambda^{\prime}=\pm
1/2}\int\frac{d^{3}p^{\prime}}{(2\pi)^{3}}\psi_{{\bf
p}^{\prime}\lambda^{\prime}}(x)\hat{a}_{{\bf p}^{\prime}\lambda^{\prime}},$
(11) $\displaystyle\psi_{{\bf
p}^{\prime}\lambda^{\prime}}(x)=\frac{{\displaystyle u_{{\bf
p}^{\prime}\lambda^{\prime}}}}{{\displaystyle\sqrt{2E^{\prime}}}}\,e^{-ip^{\prime}x},$
(12)

where we have omitted the positron part. Now, following the standard
interpretation BLP , the two-particle evolved state in four-dimensional space-
time looks as follows (recall Eq. (5)):

$\displaystyle\langle 0|\hat{\psi}(x_{e})\hat{{\bf
A}}(x_{\gamma})|e^{\prime},\gamma\rangle^{(\text{ev})}$ (13)
$\displaystyle=\int\frac{d^{3}p^{\prime}}{(2\pi)^{3}}\frac{d^{3}k}{(2\pi)^{3}}\,\sum\limits_{\lambda^{\prime}\lambda_{\gamma}}\frac{{\displaystyle
u_{{\bf
p}^{\prime}\lambda^{\prime}}}}{{\displaystyle\sqrt{2E^{\prime}}}}\,e^{-ip^{\prime}x_{e}}\,$
$\displaystyle\times\frac{{\displaystyle\sqrt{4\pi}\,{\bf e}_{{\bf
k}\lambda_{\gamma}}}}{{\displaystyle\sqrt{2\omega}n}}\,e^{-ikx_{\gamma}}S_{fi}^{(1)},$

with $S_{fi}^{(1)}$ constructed following the Feynman rules, see Eq. (17)
further. So the combination

$\displaystyle\sum\limits_{\lambda^{\prime}\lambda_{\gamma}}u_{{\bf
p}^{\prime}\lambda^{\prime}}\,{\bf e}_{{\bf k}\lambda_{\gamma}}\,S_{fi}^{(1)}$
(14)

can be called the wave function of the two-particle evolved state in the
momentum representation. Clearly, it bears both the spinor and vectorial
indices. Importantly, the space-time amplitude of the two-particle evolved
state (13) is a functional of the customary plane-wave matrix element
$S_{fi}^{(1)}$. The case when only one of the final particles is detected is
considered in Refs. KBGSS-2022 ; KBGSS-2023  .

Figure 1: Feynman diagram for the Vavilov–Cherenkov process. The initial
electron has a four-momentum $p=(E,{\bf p})$ and helicity $\lambda$, the final
electron is characterised by $p^{\prime}=(E^{\prime},{\bf p}^{\prime})$ and
$\lambda^{\prime}$, and the radiated photon by $k=(\omega,{\bf k})$ and
$\lambda_{\gamma}$.

###  II.2 Matrix element of the process

We proceed with details of the VC radiation by an electron of mass $m_{e}$ in
a transparent nonmagnetic homogeneous medium with the refraction index
$n=n(\omega)$ within the lowest order of QED. This process looks like a decay
of an electron (see Fig. 1)

$e(p,\lambda)\to e(p^{\prime},\lambda^{\prime})+\gamma(k,\lambda_{\gamma}).$
(15)

where for definiteness we have indicated momenta and helicities of the
particles. In the plane-wave amplitude below, the initial state is described
by an electron plane wave $u_{{\bf p}\lambda}\,e^{-ipx}$ with momentum ${\bf
p}$, energy $E=\sqrt{{\bf p}^{2}+m_{e}^{2}}$, and helicity $\lambda=\pm 1/2$.
The final state is described by a plane wave $u_{{\bf
p}^{\prime}\lambda^{\prime}}\,e^{-ip^{\prime}x}$ and a photon plane wave ${\bf
e}_{{\bf k}\lambda_{\gamma}}\,e^{-ikx}$, where the photon frequency is
$\omega=|{\bf k}|/n$ and the helicity is $\lambda_{\gamma}=\pm 1$.

However, we bear in mind that we intend to use the amplitude calculated here
for the process in which initial electron is not exclusively described by a
plane wave, and its movement is not necessarily a propagation along a specific
axis. Thus, in what follows we keep the spherical angles $\theta,\,\varphi$
and $\theta^{\prime},\,\varphi^{\prime}$ of the initial and final electrons
and the spherical angles $\theta_{\gamma},\,\varphi_{\gamma}$ of the final
photon. An apparent dependance of the spinors $u\equiv u_{{\bf p}\lambda}$ and
$u^{\prime}\equiv u_{{\bf p}^{\prime}\lambda^{\prime}}$ as well as the
polarization vector ${\bf e}\equiv{\bf e}_{{\bf k}\lambda_{\gamma}}$ on the
corresponding spherical angles can be given in terms of an expansion over the
eigenstates of the spin operator $\hat{s}_{z}$, $\hat{s}^{\prime}_{z}$ and
$\hat{s}^{\gamma}_{z}$. See the particularities in Appendix A. That
representation is convenient when we discuss both the plane wave and twisted
states.

In the process discussed, the plane-wave matrix element reads

$\displaystyle S^{(1)}_{fi}=i(2\pi)^{4}\,N\,\delta(p^{\prime}+k-p)\,M_{fi},$
(16) $\displaystyle
M_{fi}=M_{fi}(p,p^{\prime},k)=\sqrt{4\pi\alpha}\,\;\bar{u}^{\prime}\,\gamma_{\mu}u\,(e^{\mu})^{*},$
(17)

where the normalization factor is $N=\sqrt{4\pi/(2E2E^{\prime}2\omega n^{2})}$
(here we set the factor representing the phase space volume to be equal to
unity for brevity). We express the standard amplitude $M_{fi}$ using Eq. (A5)
from Ref. ISZ-2016 ):

$\displaystyle M_{fi}=\sum_{\sigma=\pm 1/2}\,\sum_{\sigma_{\gamma}=\pm
1,\,0}e^{i\sigma(\varphi^{\prime}-\varphi)-i\sigma_{\gamma}(\varphi^{\prime}-\varphi_{\gamma})}$
(18) $\displaystyle\times
M^{\lambda\lambda^{\prime}\lambda_{\gamma}}_{\sigma\sigma_{\gamma}}(E,\omega,\theta,\theta^{\prime},\theta_{\gamma}),$

where the dependence on the azimuthal angles
$\varphi,\varphi^{\prime},\varphi_{\gamma}$ is factorized and the coefficients

$\displaystyle
M^{\lambda\lambda^{\prime}\lambda_{\gamma}}_{\sigma\sigma_{\gamma}}$
$\displaystyle=$
$\displaystyle-\sqrt{4\pi\alpha}\,2\sigma\,(2\lambda\,E_{\lambda\lambda^{\prime}}\,d^{(1/2)}_{\sigma\lambda}(\theta)$
$\displaystyle\times$ $\displaystyle
d^{(1/2)}_{\sigma-\sigma_{\gamma},\lambda^{\prime}}(\theta^{\prime})\,d^{(1)}_{\sigma_{\gamma},\,\lambda_{\gamma}}(\theta_{\gamma})\,\left(\delta_{0,\,\sigma_{\gamma}}-\sqrt{2}\,\delta_{2\sigma,\sigma_{\gamma}}\right)$

do not depend on these angles. Here

$\displaystyle E_{\lambda\lambda^{\prime}}$ $\displaystyle=$
$\displaystyle\sqrt{(E-m_{e})(E^{\prime}+m_{e})}$ $\displaystyle+$
$\displaystyle 2\lambda 2\lambda^{\prime}\sqrt{(E^{\prime}-m_{e})(E+m_{e})}$

and $d_{MM^{\prime}}^{\;\;\;\,(J)}\left(\theta\right)$ are the small Wigner
matrices Varshalovich :

$\displaystyle d^{\,(1/2)}_{\sigma\lambda}(\theta)$ $\displaystyle=$
$\displaystyle\delta_{\sigma\lambda}\cos(\theta/2)-2\sigma\delta_{\sigma,-\lambda}\sin(\theta/2),$
(21) $\displaystyle d^{\;(1)}_{0,\lambda_{\gamma}}(\theta_{\gamma})$
$\displaystyle=$
$\displaystyle\frac{{\displaystyle\lambda_{\gamma}}}{{\displaystyle\sqrt{2}}}\,\sin\theta_{\gamma},\;\;d^{\;(1)}_{2\sigma,\,\lambda_{\gamma}}(\theta_{\gamma})=\frac{1}{2}\,\left(1+2\sigma\lambda_{\gamma}\cos\theta_{\gamma}\right).$

It is useful to point out the following feature of the discussed process: the
angle $\theta_{kp}$ between vectors ${\bf k}$ and ${{\bf p}}$ is completely
determined by the energies of the photon and the initial electron. Indeed, if
we rewrite the equality $E^{\prime}=E-\omega$ in the form

$(E^{\prime})^{2}={\bf p}^{2}+{\bf k}^{2}-2|{\bf p}||{\bf
k}|\cos\theta_{kp}+m_{e}^{2}=(E-\omega)^{2},$ (22)

we get (cf. (1))

$\cos\theta_{kp}=\frac{{\displaystyle 1}}{{\displaystyle
vn}}+\frac{{\displaystyle\omega}}{{\displaystyle 2E}}\,\frac{{\displaystyle
n^{2}-1}}{{\displaystyle vn}}.$ (23)

##  III Plane-wave initial electron

###  III.1 Evolved state

Figure 2: The geometry of the VC process with the initial electron moving
along the $z$-direction.

Let us suppose that the incoming electron has a definite momentum, directed
along the $z$-axis, so that $\theta=\varphi=0$, $p=(E,0,0,|{\bf p}|)$ as
depicted in the left panel of Fig. 2. Therefore, we immediately obtain from
(23) the important equality

$\theta_{\gamma}=\theta_{kp}=\theta_{0},$ (24)

which means that the VC radiation with a defined energy $\omega$ is
concentrated on the cone with the opening angle $\theta_{0}$.

In that case we deal with an electron that can be described standardly as a
plane wave with $\hat{j}_{z}=\hat{s}_{z}$ (see Appendix A). For the matrix
element, this results in the reduction
$d^{\,(1/2)}_{\sigma\lambda}(\theta=0)=\delta_{\sigma\lambda}$ and, hence, we
can lift both sums in Eq. (18)

$M_{fi}=M^{\lambda\lambda^{\prime}\lambda_{\gamma}}_{\lambda,0}\,e^{i\lambda\varphi^{\prime}}+M^{\lambda\lambda^{\prime}\lambda_{\gamma}}_{\lambda,2\lambda}\,e^{-i\lambda\varphi^{\prime}+i2\lambda\varphi_{\gamma}},$
(25)

where the new factorized amplitudes are

$\displaystyle
M^{\lambda\lambda^{\prime}\lambda_{\gamma}}_{\lambda,0}=-\sqrt{4\pi\alpha}E_{\lambda\lambda^{\prime}}\,d^{(1/2)}_{\lambda,\lambda^{\prime}}(\theta^{\prime})\,d^{(1)}_{0,\,\lambda_{\gamma}}(\theta_{\gamma}),$
$\displaystyle
M^{\lambda\lambda^{\prime}\lambda_{\gamma}}_{\lambda,2\lambda}=+\sqrt{8\pi\alpha}E_{\lambda\lambda^{\prime}}\,d^{(1/2)}_{-\lambda,\lambda^{\prime}}(\theta^{\prime})\,d^{(1)}_{2\lambda,\,\lambda_{\gamma}}(\theta_{\gamma}).$

The evolved state in the space-time representation, given by Eq. (13), defines
the probability amplitude to detect the photon in a region of space-time
centered in the point $x_{\gamma}=(t_{\gamma},{\bm{r}}_{\gamma})$ while the
electron is jointly detected in a region centered in the point
$x_{e}=(t_{e},{\bm{r}}_{e})$ (both the points may or may not coincide). First,
we integrate over the three-momentum of the final electron and then integrate
over the polar angle of the photon $\theta_{\gamma}$ rendering it to the value
defined by Eq. (23). This leads to the replacement

$\displaystyle d\Gamma=$
$\displaystyle(2\pi)^{4}\delta(p^{\prime}+k-p)\,\frac{{\displaystyle
d^{3}p^{\prime}}}{{\displaystyle(2\pi)^{3}}}\frac{{\displaystyle
d^{3}k}}{{\displaystyle(2\pi)^{3}}}$ $\displaystyle\rightarrow\
\frac{{\displaystyle(E-\omega)\omega n(\omega)}}{{\displaystyle
vE}}\,\frac{{\displaystyle d(\omega n(\omega))}}{{\displaystyle
2\pi}}\,\frac{{\displaystyle d\varphi_{\gamma}}}{{\displaystyle 2\pi}}.$

When the dispersion is weak, $|\frac{\omega}{n}\frac{dn}{d\omega}|\ll 1$, we
can have $d(\omega n(\omega))\approx n(\omega)d\omega$ and this approximation
is used in some formulas below. The evolved state looks as follows:

$\displaystyle\displaystyle\Psi^{\rm(pw)}_{{\bf p}\lambda}(x_{e},x_{\gamma})$
$\displaystyle\equiv\langle 0|\hat{\psi}(x_{e})\hat{{\bf
A}}(x_{\gamma})|e^{\prime},\gamma\rangle^{(\text{ev})}$ (27)
$\displaystyle\displaystyle=\frac{i}{v(2E)^{3/2}}\,\int\frac{d(\omega
n)}{n}\,e^{-i\omega t_{\gamma}-i(E-\omega)t_{e}}$ (28)
$\displaystyle\displaystyle\times\sum_{\lambda^{\prime}\lambda_{\gamma}}\int\frac{{\displaystyle
d\varphi_{\gamma}}}{{\displaystyle 2\pi}}\,u_{{\bf
p}^{\prime}\lambda^{\prime}}\,e^{i({\bf p}-{\bf
k})\cdot{\bm{r}}_{e}}\,{\bm{e}}_{{\bf k}\lambda_{\gamma}}\,e^{i{\bf
k}\cdot{\bm{r}}_{\gamma}}M_{fi}.$

Here we have used the following four-vectors

$\displaystyle p^{\prime}=(E^{\prime},{\bf p}^{\prime})=(E-\omega,{\bf p}-{\bf
k}),$ (29) $\displaystyle
k=(\omega,k_{\perp}\cos\varphi_{\gamma},k_{\perp}\sin\varphi_{\gamma},\omega
n\cos\theta_{kp}),$

where $k_{\perp}=\omega n\sin\theta_{kp}$ and
$\theta_{kp}=\theta_{kp}(E,\omega)$ is defined by Eq. (23). Moreover, in the
used reference frame we have ${\bf p}^{\prime}_{\perp}=-{\bf k}_{\perp}$ and,
therefore, only two points give contribution to the integral over
$\varphi_{\gamma}$:

$\displaystyle\varphi^{\prime}=\varphi_{\gamma}+\pi\;\;\mbox{when}\;\;\varphi_{\gamma}\subset(0;\pi)$
(30) or
$\displaystyle\varphi^{\prime}=\varphi_{\gamma}-\pi\;\;\mbox{when}\;\;\varphi_{\gamma}\subset(\pi;2\pi),$

the first option is presented in the right panel of Fig. 2, and the second is
easy to imagine with ${\bf p}^{\prime}$ and ${\bf k}$ swapping places
($\varphi_{\gamma}$ becoming greater than $180^{\circ}$ with
$\varphi^{\prime}$ remaining in $(0;2\pi)$ domain). Therefore

$\displaystyle M_{fi}|_{{\bf p}^{\prime}_{\perp}=-{\bf
k}_{\perp}}=e^{i\lambda(\varphi_{\gamma}\pm\pi)}\left(M^{\lambda\lambda^{\prime}\lambda_{\gamma}}_{\lambda,0}-M^{\lambda\lambda^{\prime}\lambda_{\gamma}}_{\lambda,2\lambda}\right).$
(31)

Now, let us expand the momentum states of the final particles in Eq. (27) over
the complete sets of the twisted states Knyazev-Serbo-2018 :

$\displaystyle\displaystyle u_{{\bf
p}^{\prime}\lambda^{\prime}}\,e^{-ip^{\prime}x_{e}}=\sum\limits_{m^{\prime}=-\infty}^{+\infty}i^{m^{\prime}}\,e^{-im^{\prime}\varphi^{\prime}}\psi_{p^{\prime}_{\perp}p^{\prime}_{z}m^{\prime}\lambda^{\prime}}(x_{e}),$
(32) $\displaystyle\displaystyle{\bm{e}}_{{\bf
k}\lambda_{\gamma}}\,e^{-ikx_{\gamma}}=\sum\limits_{m_{\gamma}=-\infty}^{+\infty}i^{m_{\gamma}}\,e^{-im_{\gamma}\varphi_{\gamma}}{\bm{A}}_{k_{\perp}k_{z}m_{\gamma}\lambda_{\gamma}}(x_{\gamma})$
(33)

where $m^{\prime}$ is a half-integer, $m_{\gamma}$ is an integer, and the
functions ${\bf A}_{k_{\perp}k_{z}m_{\gamma}\lambda_{\gamma}}(x_{\gamma})$ and
$\psi_{p^{\prime}_{\perp}p^{\prime}_{z}m^{\prime}\lambda^{\prime}}(x_{e})$ are
defined in Appendix A. After this, the azimuthal integral in Eq. (27) yields
$\delta_{\lambda,m^{\prime}+m_{\gamma}}$ revealing the conservation of the
angular momentum projection. The final result can be presented in the
following way:

$\displaystyle\displaystyle\Psi^{\rm(pw)}_{{\bf
p}\lambda}(x_{e},x_{\gamma})=\frac{i^{\lambda+1}}{v(2E)^{3/2}}\,\sum_{\lambda^{\prime}\lambda_{\gamma}}\sum\limits_{m^{\prime},m_{\gamma}=-\infty}^{+\infty}\int\frac{d(\omega
n)}{n}\,$
$\displaystyle(-1)^{m_{\gamma}}\left(M^{\lambda\lambda^{\prime}\lambda_{\gamma}}_{\lambda,0}-M^{\lambda\lambda^{\prime}\lambda_{\gamma}}_{\lambda,2\lambda}\right)\,\delta_{\lambda,m^{\prime}+m_{\gamma}}$
$\displaystyle\times{\bm{A}}_{k_{\perp}k_{z}m_{\gamma}\lambda_{\gamma}}(x_{\gamma})\,\psi_{p^{\prime}_{\perp}p^{\prime}_{z}m^{\prime}\lambda^{\prime}}(x_{e})$
(34)

where $E^{\prime}=E-\omega,\,p^{\prime}_{\perp}=k_{\perp}=\omega
n\sin\theta_{kp},\,p^{\prime}_{z}=p-k_{z}$.The integration is performed over
the region defined by the energy conservation law $0<\omega<E-m_{e}$ and the
condition $0<\cos\theta_{kp}<1$.

Let us take a closer look at Eq. (34): it contains the function ${\bf
A}_{k_{\perp}k_{z}m_{\gamma}\lambda}(x_{\gamma})$, which describes a twisted
photon with the corresponding total angular momentum (TAM)
$\hat{j}^{\gamma}_{z}$ equal to $m_{\gamma}$ and certain values of
$k_{\perp}$, $k_{z}$ and $\lambda_{\gamma}$; there is also the function
$\psi_{p^{\prime}_{\perp}p^{\prime}_{z},\lambda-
m_{\gamma},\lambda^{\prime}}(x_{e})$, which describes a twisted electron with
$\hat{j}^{\prime}_{z}$ equal $\lambda-m_{\gamma}$, and certain values of
$p^{\prime}_{\perp}$, $p^{\prime}_{z}$ and $\lambda^{\prime}$. Therefore, the
evolved wave function of the final system (34) as a whole describes an
entangled state of a twisted electron and twisted photon. If we define the TAM
operator of the evolved state as a sum of TAM operators,
$\hat{J}_{z}=\hat{j}^{\prime}_{z}+\hat{j}^{\gamma}_{z}$, the function
$\Psi^{\rm(pw)}_{{\bf p}\lambda}(x_{e},x_{\gamma})$ is the eigenstate of
$\hat{J}_{z}$ with the eigenvalue $\lambda$. This result is in accordance with
the TAM conservation law
$\hat{j}^{\prime}_{z}+\hat{j}^{\gamma}_{z}=\hat{j}_{z}$, since the initial
state in the considered case is an eigenfunction of the TAM operator
$\hat{j}_{z}=\hat{s}_{z}$ having the eigenvalue $\lambda$.

Alternatively, we can obtain the evolved wave function in the momentum
representation in the form

$\displaystyle\Phi^{(\rm pw)}_{{\bf p}\lambda}({\bf p}^{\prime},{\bf k})$
$\displaystyle=$
$\displaystyle\sum\limits_{\lambda^{\prime}\lambda_{\gamma}}u_{{\bf
p}^{\prime}\lambda^{\prime}}\,{\bf e}_{{\bf k}\lambda_{\gamma}}\,S_{fi}^{(1)}$
$\displaystyle=$ $\displaystyle
i(2\pi)^{4}\delta(p^{\prime}+k-p)Ne^{i\lambda(\varphi_{\gamma}\pm\pi)}$
$\displaystyle\sum\limits_{\lambda^{\prime}\lambda_{\gamma}}u_{{\bf
p}^{\prime}\lambda^{\prime}}\,{\bf e}_{{\bf
k}\lambda_{\gamma}}\,\left(M^{\lambda\lambda^{\prime}\lambda_{\gamma}}_{\lambda,0}-M^{\lambda\lambda^{\prime}\lambda_{\gamma}}_{\lambda,2\lambda}\right).$

The explicit expression for this helicity sum is given separately in Appendix
B. Certainly, the evolved wave-function (III.1) also is the eigenfunction of
$\hat{J}_{z}$ with the same eigenvalue $\lambda$.

###  III.2  Soft photon approximation

In the commonly used detectors of the Cherenkov radiation the photon energy
$\omega$ is about several eV and the conditions $\omega\ll m_{e}$ and
$\omega\ll E-m_{e}$ are satisfied. In this soft photon approximation several
simplifications to the calculation of the evolved wave function can be made.
As $E^{\prime}\approx E$, we get $\delta_{\lambda^{\prime}\lambda}$ from Eq.
(II.2). Eventually, with ${\bf p}^{\prime}\approx{\bf p}$ and
$\theta^{\prime}$ set to zero, only one term remains from the Wigner function
multiplication and the expression for the scattering amplitude is

$M_{fi}=-\lambda_{\gamma}\sqrt{8\pi\alpha}\,e^{i\lambda(\varphi_{\gamma}\pm\pi)}\,vE\sin\theta_{0}\delta_{\lambda^{\prime}\lambda},$
(36)

where

$\sin\theta_{0}=\frac{{\displaystyle k_{\perp}}}{{\displaystyle|{\bf
k}|}}=\sqrt{1-\frac{{\displaystyle 1}}{{\displaystyle v^{2}n^{2}}}}.$ (37)

The summation over $\lambda^{\prime}$ and $\lambda_{\gamma}$ in (III.1) is
performed as detailed in Appendix B, and the resulting evolved wave function
is greatly simplified when written in terms of the longitudinal photon
polarization ${\bf e}_{\parallel}$ (see Eq. (A) in Appendix A)

$\displaystyle\Phi^{\rm(pw)}_{{\bf p}\lambda}({\bf p}^{\prime},\,{\bf k})$
$\displaystyle=$ $\displaystyle
i4\sqrt{\pi\alpha}(2\pi)^{4}\delta(p^{\prime}+k-p)\,$ $\displaystyle\times
NvE\,\sin\theta_{0}\,u_{{\bf p}^{\prime}\lambda}\,{\bf
e}_{\parallel}\,e^{i\lambda(\varphi_{\gamma}\pm\pi)}$

in the momentum representation and

$\displaystyle\Psi^{\rm(pw)}_{{\bf p}\lambda}(x_{e},x_{\gamma})$
$\displaystyle=$ $\displaystyle i\,\sqrt{\frac{{\displaystyle
2\pi\alpha}}{{\displaystyle E}}}\,u_{{\bf p}\lambda}\,e^{-ipx_{e}}$
$\displaystyle\times\int\sin\theta_{0}\,{\bf
A}_{k_{\perp},k_{z},0,\parallel}(x_{\gamma})\,d\omega$

in the coordinate representation, where the photon wave function with the
longitudinal polarization is given in Eq. (99) and the region of integration
over $\omega$ is determined by the condition $vn(\omega)>1$. The obtained
result corresponds to the linear polarization in the plane of the vectors
${\bf p}$ and ${\bf k}$. This fact is well known from the first experiments of
Cherenkov.

This result also shows that the value $m_{\gamma}=0$ is consistent with the
conservation law for the $z$-projection of the system TAM,
$\lambda=\lambda^{\prime}+m_{\gamma}$, since in the approximations used, the
helicity state and momentum of the final electron almost coincide with those
of the initial electron, i. e. $\lambda^{\prime}=\lambda,\;\;p^{\prime}=p$.

###  III.3 Approximation of ultra-relativistic electrons

The amplitude of the process could also be considered in the approximation of
ultra-relativistic electrons, implying that $E\gg m_{e}$ and $E^{\prime}\gg
m_{e}$:

$\displaystyle
M^{\lambda\lambda^{\prime}\lambda_{\gamma}}_{\sigma\sigma_{\gamma}}$
$\displaystyle=$
$\displaystyle-\sqrt{4\pi\alpha}\,8\sigma\lambda\delta_{\lambda\lambda^{\prime}}\sqrt{EE^{\prime}}\,d^{\,(1/2)}_{\sigma\lambda}(\theta)\,$
$\displaystyle
d^{\,(1/2)}_{\sigma-\sigma_{\gamma},\lambda}(\theta^{\prime})\,d^{\;(1)}_{\sigma_{\gamma},\,\lambda_{\gamma}}(\theta_{\gamma})\,\left(\delta_{0,\,\sigma_{\gamma}}-\sqrt{2}\,\delta_{2\sigma,\sigma_{\gamma}}\right).$

In this limit, helicities of electrons are conserved and summation over
$\lambda^{\prime}$ becomes trivial due to the appearance of
$\delta_{\lambda\lambda^{\prime}}$. Indeed, using relations (31), we obtain

$\displaystyle M_{fi}$ $\displaystyle=$ $\displaystyle-\sqrt{8\pi\alpha
EE^{\prime}}\,\delta_{\lambda\lambda^{\prime}}\,e^{i\lambda(\varphi_{\gamma}\pm\pi)}$
$\displaystyle\left[\lambda_{\gamma}\sin(\theta_{\gamma}-\mbox{$\frac{1}{2}$}\,\theta^{\prime})-2\lambda\sin(\mbox{$\frac{1}{2}$}\,\theta^{\prime})\right].$

Substituting this expression into (III.1) and performing summation over
$\lambda_{\gamma}$ with the help of (100), we obtain the result in terms of
the twisted photon states with the linear polarization for the evolved wave
function in the momentum representation (see Appendix B))

$\displaystyle\Phi^{\rm(pw)}_{{\bf p}\lambda}({\bf p}^{\prime},\,{\bf k})$
$\displaystyle=$ $\displaystyle-4i\,(2\pi)^{4}\delta(p^{\prime}+k-p)\,$
$\displaystyle\times$ $\displaystyle\,N\,\sqrt{\pi\alpha EE^{\prime}}u_{{\bf
p}^{\prime}\lambda}e^{i\lambda(\varphi_{\gamma}\pm\pi)}$ $\displaystyle\times$
$\displaystyle\left[{\bf
e}_{\parallel}\,\sin(\theta_{\gamma}-\mbox{$\frac{1}{2}$}\,\theta^{\prime})-i\,2\lambda\,{\bf
e}_{\perp}\,\sin(\mbox{$\frac{1}{2}$}\,\theta^{\prime})\right]$

and in the coordinate representation

$\displaystyle\displaystyle\Psi^{\rm(pw)}_{{\bf
p}\lambda}(x_{e},x_{\gamma})=\frac{{\displaystyle
i^{\lambda+1}}}{{\displaystyle v}}\sqrt{\frac{{\displaystyle
2\pi\alpha}}{{\displaystyle E}}}\sum_{m_{\gamma}}\int
d\omega\sqrt{1-\omega/E}\,(-1)^{m_{\gamma}}$
$\displaystyle\times\psi_{p^{\prime}_{\perp},p^{\prime}_{z},\lambda-
m_{\gamma},\lambda}(x_{e})\bigg{[}{\bf
A}_{k_{\perp}k_{z}m_{\gamma},\parallel}(x_{\gamma})\sin(\theta_{\gamma}-\mbox{$\frac{1}{2}$}\theta^{\prime})$
$\displaystyle-i\,2\lambda{\bf
A}_{k_{\perp}k_{z}m_{\gamma},\perp}(x_{\gamma})\sin(\mbox{$\frac{1}{2}$}\theta^{\prime})\bigg{]}\,.$
(43)

It is seen from this expression that the linear polarization in the plane of
the vectors ${\bf p}$ and ${\bf k}$ is dominating in the region where
$\theta^{\prime}\ll\theta_{\gamma}$. Moreover, this expression coincides with
the aforementioned soft photons regime when $\omega\ll E$ is set.

##  IV Twisted initial electron

Figure 3: The geometry in the process with initial twisted electron having
non-zero $p_{\bot}$. The lengths and directions of the vectors are
illustrative but not exactly natural for the process.

If the initial electron is in the twisted state, then its plane-wave function
$u\,e^{-ipx}$ is replaced by a superposition of the plane waves

$u_{{\bf p}\lambda}\,e^{-ipx}\to\int\frac{{\displaystyle
d\varphi}}{{\displaystyle 2\pi}}\,i^{-m}\,e^{im\varphi}\,u_{{\bf
p}\lambda}\,e^{-ipx}.$ (44)

This is the stationary state (the Bessel wave) corresponding to a twisted
electron with the longitudinal momentum $p_{z}$, the absolute value of
transverse momentum $p_{\perp}$, the energy
$E=\sqrt{p^{2}_{\perp}+p_{z}^{2}+m_{e}^{2}}$, the projection of an electron
TAM onto the $z$-axis equal to a half-integer number $m$ and the helicity
$\lambda=\pm 1/2$. In this case, the evolved wave function of the final state
has the form

$\displaystyle\Phi^{\rm(tw)}_{p_{\perp}p_{z}m\lambda}({\bf p}^{\prime},{\bf
k})$ $\displaystyle=$ $\displaystyle\int\frac{{\displaystyle
d\varphi}}{{\displaystyle 2\pi}}\,i^{-m}\,e^{im\varphi}\,\Phi^{\rm(pw)}_{{\bf
p}\lambda}({\bf p}^{\prime},{\bf k})$ (45)

in the momentum representation and the similar expression in the coordinate
representation. Note, that the function $\Phi^{\rm(pw)}_{{\bf p}\lambda}({\bf
p}^{\prime},{\bf k})$ in the wave function of Eq. (45) depends now on $M_{fi}$
in the form given by Eq. (18) and bears all three components of the vector
${\bf p}=(p_{\perp}\cos\varphi,\,p_{\perp}\sin\varphi,\,p_{z})$ (as for
example in Fig. 3), while we took the frame with $z$-directed ${\bf
p}=(0,\,0,\,|{\bf p}|)$ when deriving $\Phi^{\rm(pw)}_{{\bf p}\lambda}({\bf
p}^{\prime},{\bf k})$ in Eq. (III.1).

####  IV.0.1 Integration over the azimuth of initial particle

To get the final state in either representation the integration over $\varphi$
can be performed first. We must take into account that $\cos\theta_{pk}$
generally depends on $\varphi$ and also that the plane-wave integrand has some
dependence on $\varphi$ and $\varphi^{\prime}$ – let us call it
$f(\varphi,\varphi^{\prime})$. The following integral arises

$I=\int_{0}^{2\pi}\frac{{\displaystyle d\varphi}}{{\displaystyle
2\pi}}\delta\left(\cos\theta_{kp}-\cos\theta_{0}\right)f(\varphi,\varphi^{\prime}),$
(46)

where $\theta_{0}=\arccos\left[\frac{{\displaystyle 1}}{{\displaystyle
vn}}+\frac{{\displaystyle\omega}}{{\displaystyle 2E}}\,\frac{{\displaystyle
n^{2}-1}}{{\displaystyle vn}}\right]$ is the value determined by the
conservation law (recall Eq. (23)). The integral is of the same type as the
one studied in Ref. ISZ-2016 . To calculate it we follow the same route with
appropriate extension treating the appearance of $\varphi^{\prime}$.

It is straightforward to get the following equations for the angles between
vectors $({\bf k},{\bf p})$ or $({\bf k},{\bf p}^{\prime})$ if the definitions
$\cos\theta_{kp}=\frac{{\displaystyle{\bf k}\cdot{\bf p}}}{{\displaystyle|{\bf
k}|\,|{\bf p}|}}$ and $\cos\theta_{kp^{\prime}}=\frac{{\displaystyle{\bf
k}\cdot{\bf p}^{\prime}}}{{\displaystyle|{\bf k}|\,|{\bf p}^{\prime}|}}$ are
written in spherical coordinates:

$\displaystyle\cos\theta_{kp}=\cos\theta_{\gamma}\cos\theta+\sin\theta_{\gamma}\sin\theta\cos\delta,$
$\displaystyle\cos\theta_{kp^{\prime}}=\cos\theta_{\gamma}\cos\theta^{\prime}+\sin\theta_{\gamma}\sin\theta^{\prime}\cos\delta^{\prime},$
(47) $\displaystyle\delta=\pm(\varphi-\varphi_{\gamma}),\
\delta^{\prime}=\pm(\varphi^{\prime}-\varphi_{\gamma}).$

Further, since $\cos\theta_{kp}$ is fixated by the delta function, $\varphi$
could be expressed other way round with $\delta$, while for $\varphi^{\prime}$
there could be only one option as is evident from the geometrical construction
akin to that on the right panel of Fig. 3. In the end, only two points give
nonzero contribution to the integral (46):
$\varphi=\varphi_{\gamma}+\delta,\;\varphi^{\prime}=\varphi_{\gamma}+\delta^{\prime}$
and
$\varphi=\varphi_{\gamma}-\delta,\;\varphi^{\prime}=\varphi_{\gamma}-\delta^{\prime}$
(see Fig. 3), where

$\displaystyle\delta$ $\displaystyle=$
$\displaystyle\arccos\left(\frac{{\displaystyle\cos\theta_{0}-\cos\theta_{\gamma}\cos\theta}}{{\displaystyle\sin\theta_{\gamma}\sin\theta}}\right),$
(48) $\displaystyle\delta^{\prime}$ $\displaystyle=$
$\displaystyle\arccos\left(\frac{{\displaystyle\cos\theta_{kp^{\prime}}-\cos\theta_{\gamma}\cos\theta^{\prime}}}{{\displaystyle\sin\theta_{\gamma}\sin\theta^{\prime}}}\right).$
(49)

The final expression is

$I=\frac{{\displaystyle 1}}{{\displaystyle
2}}\left[f(\varphi_{\gamma}+\delta,\varphi_{\gamma}+\delta^{\prime})+f(\varphi_{\gamma}-\delta,\,\varphi_{\gamma}-\delta^{\prime})\right]F(\theta,\theta_{\gamma},\theta_{0})$
(50)

where the function $F(\theta,\theta_{\gamma},\theta_{0})$ reads

$\displaystyle F(\theta,\theta_{\gamma},\theta_{0})=\frac{{\displaystyle
1}}{{\displaystyle\pi\,\sin\theta_{\gamma}\sin\theta|\sin\delta|}}$ (51)
$\displaystyle=\frac{{\displaystyle
1}}{{\displaystyle\pi}}\;\left\\{\left[\cos\theta_{\gamma}-\cos(\theta+\theta_{0})\right]\left[\cos(\theta-\theta_{0})-\cos\theta_{\gamma}\right]\right\\}^{-1/2}$

####  IV.0.2 Momentum representation

To calculate the evolved state of Eq. (45), we collect the azimuthal
dependencies into

$f(\varphi,\varphi^{\prime})=i^{-m}e^{im\varphi-i\sigma^{\prime}\varphi^{\prime}+i\sigma(\varphi^{\prime}-\varphi)-i\sigma_{\gamma}(\varphi^{\prime}-\varphi_{\gamma})}.$
(52)

and the resulting expression for (46) is

$\displaystyle I$ $\displaystyle=$ $\displaystyle
i^{-m}e^{i(m-\sigma^{\prime})\varphi_{\gamma}}$ $\displaystyle\times$
$\displaystyle
F(\theta,\theta_{\gamma},\theta_{0})\,\cos\left[(m-\sigma)\delta+(\sigma-\sigma^{\prime}-\sigma_{\gamma})\delta^{\prime}\right],$

Then, the evolved wave function in the momentum representation is

$\displaystyle\Phi^{(\rm tw)}_{p_{\perp}p_{z}m\lambda}({\bf p}^{\prime},{\bf
k})=i^{1-m}(2\pi)^{4}\delta({\bf p}^{\prime}+{\bf k}-{\bf p})$ (54)
$\displaystyle\times\frac{{\displaystyle(E-\omega)N}}{{\displaystyle vE\omega
n}}F(\theta,\theta_{\gamma},\theta_{0})$
$\displaystyle\,\sum\limits_{\lambda^{\prime}\lambda_{\gamma}}\sum_{\sigma\sigma^{\prime}\sigma_{\gamma}}e^{i(m-\sigma^{\prime})\varphi_{\gamma}}\,d^{(1/2)}_{\sigma^{\prime}\lambda^{\prime}}(\theta^{\prime})M^{\lambda\lambda^{\prime}\lambda_{\gamma}}_{\sigma\sigma_{\gamma}}(E,\omega,\theta,\theta^{\prime},\theta_{\gamma})$
$\displaystyle\times\cos\left[(m-\sigma)\delta+(\sigma-\sigma^{\prime}-\sigma_{\gamma})\delta^{\prime}\right]{\bf
e}_{{\bf
k}\lambda_{\gamma}}U^{(\sigma^{\prime})}(E^{\prime},\lambda^{\prime}),$

where we give a full expression in terms of bispinors $U^{(\sigma)}$ defined
in Eq. (76) and
$M^{\lambda\lambda^{\prime}\lambda_{\gamma}}_{\sigma\sigma_{\gamma}}$ is the
helicity amplitude from Eq. (II.2).

####  IV.0.3 Two-particle wave function in coordinate representation

Let us write down explicitly the definition of the evolved state in the
coordinate representation with additional integration over $\varphi$
signifying the vortex character of the incoming electron:

$\displaystyle\displaystyle\Psi^{\rm(tw)}_{p_{\perp}p_{z}m\lambda}(x_{e},x_{\gamma})\equiv\langle
0|\hat{\psi}(x_{e})\hat{{\bf
A}}(x_{\gamma})|e^{\prime},\gamma\rangle^{(\text{ev})}$ (55)
$\displaystyle\displaystyle=\int\frac{d^{3}p^{\prime}}{(2\pi)^{3}}\frac{d^{3}k}{(2\pi)^{3}}\,\sum\limits_{\lambda^{\prime}\lambda_{\gamma}}\frac{{\displaystyle
u_{{\bf
p}^{\prime}\lambda^{\prime}}}}{{\displaystyle\sqrt{2E^{\prime}}}}\,e^{-ip^{\prime}x_{e}}\,\frac{{\displaystyle\sqrt{4\pi}\,{\bf
e}_{{\bf
k}\lambda_{\gamma}}}}{{\displaystyle\sqrt{2\omega}n}}\,e^{-ikx_{\gamma}}\;$
$\displaystyle\times\int\frac{{\displaystyle d\varphi}}{{\displaystyle
2\pi}}i^{-m}e^{im\varphi}S_{fi}^{(1)}$

Using again the series expansions (32), (33) we collect all azimuthal
exponents into

$f(\varphi,\varphi^{\prime})=i^{-m+m^{\prime}+m_{\gamma}}e^{im\varphi-
im^{\prime}\varphi^{\prime}-im_{\gamma}\varphi_{\gamma}+i\sigma(\varphi^{\prime}-\varphi)-i\sigma_{\gamma}(\varphi^{\prime}-\varphi_{\gamma})}.$
(56)

The integral (46) results in

$\displaystyle\displaystyle I$ $\displaystyle=$ $\displaystyle
i^{-m+m^{\prime}+m_{\gamma}}e^{i(m-m^{\prime}-m_{\gamma})\varphi_{\gamma}}$
$\displaystyle\times$
$\displaystyle\cos[(m-\sigma)\delta+(\sigma-m^{\prime}-\sigma_{\gamma})\delta^{\prime}]F(\theta,\theta_{\gamma},\theta_{0}).$

The integration in the phase space could be performed with elimination of the
$d^{3}p^{\prime}$ integral, and we are left with

$d\Gamma\rightarrow\frac{{\displaystyle(E-\omega)\omega n^{2}}}{{\displaystyle
2\pi vE}}\,\delta\left(\cos\theta_{kp}-\cos\theta_{0}\right)\frac{d(\omega
n)}{n}\sin\theta_{\gamma}d\theta_{\gamma}\frac{{\displaystyle
d\varphi_{\gamma}}}{{\displaystyle 2\pi}}$ (58)

The dependence of the wave function on $\varphi_{\gamma}$ is trivial and
implies appearance of the Kronecker delta
$\delta_{m-m^{\prime}-m_{\gamma},0}$. Finally, the evolved wave-function can
be expressed in two equivalent ways

$\displaystyle\displaystyle\Psi^{\rm(tw)}_{p_{\perp}p_{z}m\lambda}(x_{e},x_{\gamma})=\frac{i}{v(2E)^{3/2}}\,\sum\limits_{m^{\prime}m_{\gamma}=-\infty}^{+\infty}\sum_{\lambda^{\prime}\lambda_{\gamma}}\sum_{\sigma\sigma_{\gamma}}\delta_{m,m^{\prime}+m_{\gamma}}$
$\displaystyle\displaystyle\int\frac{d(\omega
n)}{n}d\theta_{\gamma}\sin\theta_{\gamma}\,\
F(\theta,\theta_{\gamma},\theta_{0})M^{\lambda\lambda^{\prime}\lambda_{\gamma}}_{\sigma\sigma_{\gamma}}(E,\omega,\theta,\theta^{\prime},\theta_{\gamma})$
$\displaystyle\cos[(m-\sigma)(\delta-\delta^{\prime})+(m_{\gamma}-\sigma_{\gamma})\delta^{\prime}]\,$
(59)
$\displaystyle\times{\bm{A}}_{k_{\perp}k_{z}m_{\gamma}\lambda_{\gamma}}(x_{\gamma})\,\psi_{p^{\prime}_{\perp}p^{\prime}_{z}m^{\prime}\lambda^{\prime}}(x_{e})$
(60)

The further integration is performed over the region determined by the energy
conservation law $0<\omega<E-m_{e}$ and the condition (61) discussed below.

It is seen from Eq. (60) as well as from (54) that the evolved wave function
of the final system is an entangled state of two particles of different type -
an electron and a photon, either of which can carry OAM. Note, also, that the
evolved functions for the twisted initial electron has the same structure as
the wave function (III.1) for the plane-wave initial electron except two
important points:

(i) $\Phi^{(\rm pw)}_{{\bf p}\lambda}({\bf p}^{\prime},{\bf k})$ and
$\Psi^{\rm(pw)}_{{\bf p}\lambda}(x_{e},x_{\gamma})$ being the eigenfunctions
of $\hat{J}_{z}$ with the eigenvalue $\lambda$ are replaced with $\Phi^{(\rm
tw)}_{p_{\perp}p_{z}m\lambda}({\bf p}^{\prime},{\bf k})$ and $\Psi^{(\rm
tw)}_{p_{\perp}p_{z}m\lambda}(x_{e},x_{\gamma})$, respectively, being the
eigenfunctions of $\hat{J}_{z}$ with the eigenvalue $m$;

(ii) an additional function $F(\theta,\theta_{\gamma},\theta_{0})$ appears.
This new important function is the same as in Ref. ISZ-2016  (see there the
graphs for it) and it is non-zero only when the three theta angles satisfy the
“triangle inequality”

$\left|\theta-\theta_{0}\right|<\theta_{\gamma}<\theta+\theta_{0}\,.$ (61)

Thus, the situation here differs in comparison with the standard VC radiation
of the plane wave electron. Namely, the final photon is emitted not along the
surface of the cone defined by the angle $\theta_{0}$, but in the region
between the two cones given by Eq. (61). It was proven in Ref. ISZ-2016  that,
though $F(\theta,\theta_{\gamma},\theta_{0})$ diverges at the borders of the
interval (61), this singularity can be integrated

$\int_{\left|\theta-\theta_{0}\right|}^{\theta+\theta_{0}}F(\theta,\theta_{\gamma},\theta_{0})\,\sin\theta_{\gamma}\,d\theta_{\gamma}=1.$
(62)

###  IV.1 Soft photons approximation

Once again, in the soft photon approximation, the conditions $\omega\ll m_{e}$
and $\omega\ll E-m_{e}$ are satisfied and the momenta and helicities of the
initial and final electrons coincide. In this case we can simplify the
corresponding wave-functions with the following relations

$\lambda^{\prime}=\lambda,\;\;\theta^{\prime}=\theta,\;\;\delta^{\prime}=\delta$
(63)

and perform the summation over $\sigma$ and $\sigma_{\gamma}$. Let us consider
in detail this limit for the evolved wave-function in the coordinate
representation. First, we need to study separately the coefficient in
expression (60)

$\displaystyle C_{\lambda^{\prime}\lambda_{\gamma}m_{\gamma}}$
$\displaystyle=$
$\displaystyle\sum_{\sigma\sigma_{\gamma}}M^{\lambda\lambda^{\prime}\lambda_{\gamma}}_{\sigma\sigma_{\gamma}}(E,\omega,\theta,\theta^{\prime},\theta_{\gamma})$
$\displaystyle\times$
$\displaystyle\,\cos[(m-\sigma)(\delta-\delta^{\prime})+(m_{\gamma}-\sigma_{\gamma})\delta^{\prime}]$
$\displaystyle=$
$\displaystyle-\sqrt{8\pi\alpha}\,vE\,\delta_{\lambda\lambda^{\prime}}\,(A+B),$
$\displaystyle A$ $\displaystyle=$ $\displaystyle
2\lambda\sqrt{2}\cos(m_{\gamma}\delta)\sum_{\sigma}2\sigma\left[d^{(1/2)}_{\sigma\lambda}(\theta)\right]^{2}d^{\;\;(1)}_{0,\lambda_{\gamma}}(\theta_{\gamma}),$
$\displaystyle B$ $\displaystyle=$ $\displaystyle-4\lambda\sum_{\sigma}2\sigma
d^{(1/2)}_{\sigma\lambda}(\theta)d^{(1/2)}_{-\sigma,\lambda}(\theta)d^{\;\;(1)}_{2\sigma,\lambda_{\gamma}}(\theta_{\gamma})$
$\displaystyle\times$ $\displaystyle\cos[(m_{\gamma}-2\sigma)\delta]$

Using definitions (21) for the Wigner matrices and the relation

$\cos[(m_{\gamma}-2\sigma)\delta]=\cos(m_{\gamma}\delta)\cos\delta+2\sigma\sin(m_{\gamma}\delta)\sin\delta,$
(65)

we find

$\displaystyle C_{\lambda^{\prime}\lambda_{\gamma}m_{\gamma}}$
$\displaystyle=$
$\displaystyle-\sqrt{8\pi\alpha}\,vE\,\delta_{\lambda\lambda^{\prime}}\,(\lambda_{\gamma}G_{1}+G_{2}),$
(66) $\displaystyle G_{1}$ $\displaystyle=$
$\displaystyle\left[\cos\theta\sin\theta_{\gamma}-\sin\theta\cos\theta_{\gamma}\cos\delta\right]\,\cos(m_{\gamma}\delta),\;\;$
$\displaystyle G_{2}$ $\displaystyle=$
$\displaystyle-\sin\theta\sin\delta\sin(m_{\gamma}\delta).$

It is useful to note that Eq. (60) can be presented in another form by
introducing the linear photon polarizations with the help of Eqs. (100):

$\displaystyle\Psi^{\rm(tw)}_{p_{\perp}p_{z}m\lambda}(x_{e},x_{\gamma})=i\sqrt{\frac{{\displaystyle
2\pi\alpha}}{{\displaystyle
E}}}\int\,d\omega\sin\theta_{\gamma}d\theta_{\gamma}F(\theta,\theta_{\gamma},\theta_{0})\,$
$\displaystyle\sum_{m_{\gamma}}\left[G_{1}{\bf
A}_{k_{\perp}k_{z}m_{\gamma},\parallel}(x_{\gamma})-iG_{2}{\bf
A}_{k_{\perp}k_{z}m_{\gamma},\perp}(x_{\gamma})\right]$
$\displaystyle\times\psi_{p_{\perp}p_{z},m-m_{\gamma},\lambda}(x_{e})$ (67)

In the paraxial approximation we have $\theta\ll 1$, $|G_{2}|\ll|G_{1}|$ and
the linear polarization in the scattering plane becomes dominant.

Finally we note that at $\theta\to 0$ and $m\to\lambda$, the result (67)
coincides with Eq. (III.2) up to factor $i^{-\lambda}$ as it should be. To
prove that, we must take into account that in the considered limit
$\theta_{0}\pm\theta\approx\theta_{0}$, therefore, we can replace
$\theta_{\gamma}\to\theta_{0}$ everywhere except the function
$F(\theta,\theta_{\gamma},\theta_{0})$ and perform the integration over
$\theta_{\gamma}$ in accordance with Eq. (62). Using further Eq. (85), we
obtain that

$\displaystyle\psi_{p^{\prime}_{\perp}p^{\prime}_{z},m-m_{\gamma},\lambda^{\prime}}(x_{e})|_{\theta^{\prime}\to\,0}\to\delta_{m-m_{\gamma},\lambda}\,i^{-\lambda}\,U^{(\lambda)}(E,\lambda)$
$\displaystyle\times e^{-i(Et_{e}-|{\bf p}|z_{e})}$ (68)

and that

$C_{\lambda^{\prime}\lambda_{\gamma}m_{\gamma}}\to-\delta_{\lambda^{\prime}\lambda}\sqrt{2\pi\alpha}\,2vE\lambda_{\gamma}\,\sin\theta_{0}.$
(69)

As a result, we find

$\Psi^{\rm(tw)}_{p_{\perp}p_{z}m\lambda}(x_{e},x_{\gamma})\to
i^{-\lambda}\Psi^{\rm(pw)}_{{\bf p}\lambda}(x_{e},x_{\gamma}),$ (70)

where the expression $\Psi^{\rm(pw)}_{{\bf p}\lambda}(x_{e},x_{\gamma})$ is
given by Eq. (III.2).

##  V Discussion

Through the paper we focused on the process of VC radiation. However, as a
means to provide an unburdened version of the mathematics that was used, we
also refer the reader to Appendix C.

In previous sections we obtained the evolved wave function given by Eq. (34)
for the process with initial electron having a definite momentum (being
described by a plane wave), and by Eq. (60) when the initial electron is
described by the Bessel wave. Our results show that the evolved two-particle
state represents an entangled superposition of a photon and an electron and
that this superposition can be expanded either into pairs of the plane-wave
states or of the twisted states. The second choice is convenient when one of
the particles is registered with a detector sensitive to the TAM projection
(and insensitive to the momentum azimuthal angle). The evolved state of the
second particle in this case will automatically be projected onto the twisted
state with the definite TAM and only one term in the sums (34) or (60) will
survive. Alternatively, when one measures the momentum azimuthal angle of one
of the particles with a large error within the generalized measurement scheme
(see KBGSS-2023 ), the resultant state represents a wave packet – that is, a
superposition of several TAM states. The second particle will, therefore, also
be in a quantum state with a finite TAM dispersion (a bandwidth).

The question in what processes twisted photons could be emitted is both
theoretically and experimentally challenging. Let us briefly recall the known
basic methods for the production of twisted photons. In paper Sasaki-
McNulty-2008  it was shown that when an electron is moving in a helical
undulator, it emits the photons of the second harmonic in the twisted state. A
little later it was shown in paper Afanasev-Michailichenko-2009  that higher
harmonics have the same property. Then in 2011, it was shown in papers
jentschura-serbo-2011-1 , jentschura-serbo-2011-2  and ivanov-serbo-2011  that
in the Compton back scattering of twisted laser photons by relativistic
electrons, the final high-energy photon is also twisted in the main region of
the emitted angles. The twisted photons with the energy 99 eV have been
produced by utilizing a helical undulator at the synchrotron light source
BESSY II bahrdt2013 . In 2015–2017, the experiments carried out at BNL on the
nonlinear Compton effect Brookhaven-experiment-2015  were interpreted in
Taira-I-2017  as an observation of the second harmonic which is twisted. In
this case, the circularly polarized laser light plays role of the spiral
undulator in the BESSY II experiments. A more general study Taira-II-2017
shows that twisted photons are emitted at the harmonics higher than the first
one during the spiral motion of an electron.

In this respect, the VC emission, specifically in the case of an initial
plane-wave electron, is a new example of the generation of twisted photons.
Recently, it was argued in Refs. KBGSS-2022 ; KBGSS-2023  that the use of
generalized measurements is indeed an effective method for obtaining twisted
photons. Our results follow up these works and give an interesting perspective
for non-conventional measurement schemes.

The method used here can also be applied to the problem of characterizing the
final wave function in the process of radiation of an equivalent photon (EP)
by high-energy charged particle. To clarify this point, let us consider the
process of inelastic $ep$-scattering

$e(p)+p(P)\to e(p^{\prime})+X(P^{\prime}),$ (71)

where $X$ is the final hadronic state with the total momentum $P^{\prime}$.
This process may be seen as two steps reaction:

(i) an emission of the virtual photon (or EP) $\gamma^{*}$ with momentum
$q=p-p^{\prime}$ and virtuality $q^{2}<0$ by the initial electron;

(ii) an absorption of this EP by proton with the production of the final
hadronic state $X$.

The first step of the discussed process corresponds to the virtual process

$e(p)\to e(p^{\prime})+\gamma^{*}(q),$ (72)

which has a close similarity with the VC radiation. This is a problem we would
like to address in a future work.

##  VI Conclusions

In this paper we explored the application of the evolved state formalism in
the processes of VC emission by a plane wave and Bessel wave (i.e., carrying
OAM) electron. This way we obtained the quantum mechanically complete
description of the final entangled state of the photon and the electron
independent of the detection scheme or the properties of a detector. We
emphasize that the final system is an entangled state of two particles of
different type, however, the obtained form of the wave-function is such that
both particles contribute symmetrically.

In addition, we considered the limiting cases of low-energy radiation and
high-energy initial electron confirming that the evolved state reproduces
correctly the expected behaviour. We also verify the results of paper ISZ-2016
that the photonic part of the final state for the case of the twisted initial
electron has the angular distribution and polarization which differ
considerable from the ordinary VC radiation.

###### Acknowledgements.

We are thankful to I. Ivanov for useful discussion. The studies in Sec. II are
supported by the Russian Science Foundation (Project No. 21-42-04412;
https://rscf.ru/en/project/21-42-04412/). The studies in Sec. III are
supported by the Government of the Russian Federation through the ITMO
Fellowship and Professorship Program. The studies in Sec.IV are supported by
the Russian Science Foundation (Project No. 23-62-10026;
https://rscf.ru/en/project/23-62-10026/).

##  Appendix A Properties of twisted electrons and photons

In this Appendix we collect some useful formulae related to properties of
twisted electrons and photons (for more detail see reviews Blioch-Ivanov-2017
and Knyazev-Serbo-2018 , respectively).

### Electrons

The bispinor $u_{{\bf p}^{\prime}\lambda^{\prime}}$ has the following evident
dependence on the spherical angles $\theta^{\prime}$ and $\varphi^{\prime}$
(see Eq. (A1) from ISZ-2016 ):

$u_{{\bf p}^{\prime}\lambda^{\prime}}=\sum_{\sigma^{\prime}=\pm
1/2}d^{(1/2)}_{\sigma^{\prime}\lambda^{\prime}}(\theta^{\prime})\,U^{(\sigma^{\prime})}(E^{\prime},\lambda^{\prime})\,e^{-i\sigma^{\prime}\varphi^{\prime}},$
(73)

where the basis bispinors $U^{(\sigma^{\prime})}(E^{\prime},\lambda^{\prime})$
are expressed as follows:

$\displaystyle
U^{(\sigma^{\prime})}(E^{\prime},\lambda^{\prime})=\left(\begin{array}[]{c}\sqrt{E^{\prime}+m_{e}}\,w^{(\sigma^{\prime})}\\\\[5.69054pt]
2\lambda^{\prime}\sqrt{E^{\prime}-m_{e}}\,w^{(\sigma^{\prime})}\end{array}\right),$
(76) $\displaystyle w^{(+1/2)}=\left(\begin{tabular}[]{c}1\\\ 0\\\
\end{tabular}\right),\;\;w^{(-1/2)}=\left(\begin{tabular}[]{c}0\\\ 1\\\
\end{tabular}\right)\,.$ (81)

They do not depend on the direction of ${\bf p}^{\prime}$ and are eigenstates
of the spin projection operator $\hat{s}^{\prime}_{z}$ with eigenvalues
$\sigma^{\prime}=\pm 1/2$. Note also that the expression $u_{{\bf
p}^{\prime}\lambda^{\prime}}\,e^{im^{\prime}\varphi^{\prime}}$ is the
eigenfunction in the momentum representation of the TAM operator
$\hat{j}^{\prime}_{z}$ with the eigenvalue $m^{\prime}$ :

$\hat{j}^{\prime}_{z}\,u_{{\bf
p}^{\prime}\lambda^{\prime}}\,e^{im^{\prime}\varphi^{\prime}}=m^{\prime}\,u_{{\bf
p}^{\prime}\lambda^{\prime}}\,e^{im^{\prime}\varphi^{\prime}}.$ (82)

The function

$\displaystyle\psi_{p^{\prime}_{\perp}p^{\prime}_{z}m^{\prime}\lambda^{\prime}}(x)$
$\displaystyle=$ $\displaystyle\int_{0}^{2\pi}i^{-m^{\prime}}\,u_{{\bf
p}^{\prime}\lambda^{\prime}}\,e^{im^{\prime}\varphi^{\prime}}\,e^{-ip^{\prime}x}\frac{{\displaystyle
d\varphi^{\prime}}}{{\displaystyle 2\pi}}$ $\displaystyle=$ $\displaystyle
e^{-i(E^{\prime}t-p^{\prime}_{z}z)}\sum_{\sigma^{\prime}=\pm
1/2}i^{-\sigma^{\prime}}d^{\;(1/2)}_{\sigma^{\prime}\,\lambda^{\prime}}(\theta^{\prime})$
$\displaystyle\times$ $\displaystyle
J_{m^{\prime}-\sigma^{\prime}}(p^{\prime}_{\perp}r_{\perp})\,e^{i(m^{\prime}-\sigma^{\prime})\varphi_{r}}\,U^{(\sigma^{\prime})}(E^{\prime},\lambda^{\prime})$

corresponds to the twisted electron with longitudinal momentum
$p^{\prime}_{z}$, transverse momentum modulus $p^{\prime}_{\perp}$, energy
$E^{\prime}=\sqrt{p^{\prime 2}_{\perp}+p^{\prime 2}_{z}+m^{2}_{e}}$,
projection of the electron TAM onto the $z$-axis equal $m^{\prime}$ and
helicity $\lambda^{\prime}=\pm 1/2$. In the paraxial approximation the above
sum is dominated by the term with $\sigma^{\prime}=\lambda^{\prime}$:

$\displaystyle\psi_{p^{\prime}_{\perp}p^{\prime}_{z}m^{\prime}\lambda^{\prime}}(x)\approx
i^{-\lambda^{\prime}}\cos(\theta^{\prime}/2)\,J_{m^{\prime}-\lambda^{\prime}}(p^{\prime}_{\perp}r_{\perp})$
$\displaystyle
e^{i(m^{\prime}-\lambda^{\prime})\varphi_{r}}\,U^{(\lambda^{\prime})}(E^{\prime},\lambda^{\prime})\,e^{-i(E^{\prime}t-p^{\prime}_{z}z)}.$
(84)

Finally, in the limit $\theta^{\prime}\to 0$ , we obtain

$\psi_{p^{\prime}_{\perp}p^{\prime}_{z}m^{\prime}\lambda^{\prime}}(x)|_{\theta^{\prime}\to\,0}\to\delta_{m^{\prime}\lambda^{\prime}}\,i^{-\lambda^{\prime}}\,U^{(\lambda^{\prime})}(E^{\prime},\lambda^{\prime})\,e^{-i(E^{\prime}t-p^{\prime}_{z}z)}\,.$
(85)

i. e. in this limit and for $m^{\prime}=\lambda^{\prime}$, the wave function
of a twisted electron coincides with a plane wave along the $z$ axis up to the
phase factor $i^{-\lambda^{\prime}}$.

Similarly, the function

$\psi_{p_{\perp}p_{z}m\lambda}(x)=\int_{0}^{2\pi}i^{-m}\,u_{{\bf
p}\lambda}\,e^{im\varphi}\,e^{-ipx}\frac{{\displaystyle
d\varphi}}{{\displaystyle 2\pi}}$ (86)

corresponds to the initial twisted electron with longitudinal momentum
$p_{z}$, transverse momentum modulus $p_{\perp}$, energy
$E=\sqrt{p^{2}_{\perp}+p^{2}_{z}+m^{2}_{e}}$, projection of the electron TAM
onto the $z$-axis equal $m$ and helicity $\lambda=\pm 1/2$. In the limit ${\bf
p}_{\perp}=\theta=0$, momentum ${\bf p}=(0,0,|{\bf p}|)$, the $z$-projection
of the orbital angular momentum disappears because in this limit
$\hat{l}_{z}\propto\hat{\bf l}\cdot{\bf p}=0$, and the state helicity
coincides with the $z$-projection of TAM. Indeed, if $\theta\to 0$, we have

$\psi_{p_{\perp}p_{z}m\lambda}(x)|_{\theta\to\,0}\to\delta_{m\lambda}\,i^{-\lambda}\,U^{(\lambda)}(E,\lambda)\,e^{-i(Et-
p_{z}z)}\,.$ (87)

i. e. in this limit the wave function of the initial electron coincides (up to
the phase factor $i^{-\lambda}$) with a plane wave along the $z$ axis.
Moreover, this plane wave has the same eigenvalue $\lambda$ both for
$\hat{j}_{z}$ and helicity operators, since in this limit
$\hat{j}_{z}=\hat{s}_{z}$.

### Photons

The evident dependence of the photon polarization vector on spherical angles
$\theta_{\gamma}$ and $\varphi_{\gamma}$ reads (see detail in Ref. MSSF-2013
):

${\bf e}_{{\bf k}\lambda_{\gamma}}=\sum_{\sigma_{\gamma}=0,\pm
1}d^{\;\;(1)}_{\sigma_{\gamma}\lambda_{\gamma}}(\theta_{\gamma})\,\bm{\chi}_{\sigma_{\gamma}}\,e^{-i\sigma_{\gamma}\varphi_{\gamma}},$
(88)

where the basis vectors

${\bm{\chi}}_{0}=\left(0,\,0,\,1\right),\;\;{\bm{\chi}}_{\pm
1}=\mp\frac{{\displaystyle 1}}{{\displaystyle\sqrt{2}}}\left(1,\,\pm
i,\,0\right)$ (89)

represent the eigenstates of the photon spin $z$-projection operator
$\hat{s}^{\gamma}_{z}$ with the eigenvalues $\sigma_{\gamma}=0,\,\pm 1$.

The function

$\displaystyle\displaystyle{\bf A}_{k_{\perp}k_{z}m\lambda_{\gamma}}(x)$
$\displaystyle=$ $\displaystyle\int_{0}^{2\pi}i^{-m}\,{\bf e}_{{\bf
k}\lambda_{\gamma}}\,e^{im\varphi_{\gamma}}\,e^{-ikx}\frac{{\displaystyle
d\varphi_{\gamma}}}{{\displaystyle 2\pi}}$ $\displaystyle=$ $\displaystyle
e^{-i(\omega t-k_{z}z)}\sum_{\sigma_{\gamma}=0;\pm
1}i^{-\sigma_{\gamma}}d^{\;(1)}_{\sigma_{\gamma}\,\lambda_{\gamma}}(\theta_{\gamma})$
$\displaystyle\times$ $\displaystyle
J_{m-\sigma_{\gamma}}(k_{\perp}r_{\perp})\,e^{i(m-\sigma_{\gamma})\varphi_{r}}\,\bm{\chi}_{\sigma_{\gamma}},$
(91)

where $J_{n}(x)$ is the Bessel function of the first kind, corresponds to the
twisted photon with longitudinal momentum $k_{z}$, transverse momentum modulus
$k_{\perp}$, energy $\omega=\sqrt{k^{2}_{\perp}+k^{2}_{z}}\,/n$, projection of
the photon TAM onto the $z$-axis equals to the integer number $m$ and helicity
$\lambda_{\gamma}=\pm 1$ Knyazev-Serbo-2018 . This function is normalized by
the condition

$\displaystyle\int{\bf
A}^{*}_{k^{\prime}_{\perp}k^{\prime}_{z}m^{\prime}\lambda^{\prime}_{\gamma}}(x)\,{\bf
A}_{k_{\perp}k_{z}m\lambda_{\gamma}}(x)\,d^{3}r$
$\displaystyle=\frac{{\displaystyle 4\pi^{2}}}{{\displaystyle
k_{\perp}}}\,\delta(k^{\prime}_{\perp}-k_{\perp})\delta_{m^{\prime}m}\,\delta(k^{\prime}_{z}-k_{z})\,\delta_{\lambda^{\prime}_{\gamma}\lambda_{\gamma}}.$
(92)

Note also that the expression ${\bf e}_{{\bf
k}\lambda_{\gamma}}\,e^{im_{\gamma}\varphi_{\gamma}}$ is the eigenfunction in
the momentum representation of the TAM operator $\hat{j}^{\gamma}_{z}$ with
the eigenvalue $m_{\gamma}$ :

$\hat{j}^{\gamma}_{z}\,{\bf e}_{{\bf
k}\lambda_{\gamma}}\,e^{im_{\gamma}\varphi_{\gamma}}=m_{\gamma}\,{\bf e}_{{\bf
k}\lambda_{\gamma}}\,e^{im_{\gamma}\varphi_{\gamma}}$ (93)

For small values of the conical angle $\theta_{\gamma}$ (which corresponds to
the so-called paraxial approximation) the sum (91) is dominated by the term
with $\sigma_{\gamma}=\lambda_{\gamma}$:

$\displaystyle{\bf A}_{k_{\perp}k_{z}m\lambda_{\gamma}}(x)\approx
i^{-\lambda_{\gamma}}\,\cos^{2}(\theta_{\gamma}/2)\,J_{m-\lambda_{\gamma}}(k_{\perp}r_{\perp})$
$\displaystyle\times\,{\bf
e}^{i(m-\lambda_{\gamma})\varphi_{r}}\,\bm{\chi}_{\lambda_{\gamma}}\,{\bf
e}^{-i(\omega t-k_{z}z)}.$ (94)

Therefore in this approximation, the $z$-projection $m$ of the photon TAM is
unambiguously made up of the spin angular momentum projection, approximately
equal to $\lambda_{\gamma}$, and the projection of the OAM, approximately
equal to $m-\lambda_{\gamma}$. Finally, in the limit $\theta_{\gamma}\to 0$
(in this case, $k_{\perp}\to 0$, $k_{z}\to k=\omega n$,
$d^{\;1}_{\sigma\lambda_{\gamma}}(\theta_{\gamma})\to\,\delta_{\sigma\lambda_{\gamma}}$
and $J_{m-\sigma}(k_{\perp}r_{\perp})\to\delta_{m\,\sigma}$) we obtain

${\bf
A}_{k_{\perp}k_{z}m\lambda_{\gamma}}(x)|_{\theta_{\gamma}\to\,0}\to\delta_{m\lambda_{\gamma}}\,i^{-\lambda_{\gamma}}\,\bm{\chi}_{\lambda_{\gamma}}\,{\bf
e}^{-i(\omega t-k_{z}z)},$ (95)

i. e. in this limit and for $m=\lambda_{\gamma}$, the wave function of a
twisted photon coincides with a plane wave along the $z$ axis up to the phase
factor $i^{-\lambda_{\gamma}}$.

In the momentum representation, the wave function of a twisted photon has a
particularly simple form

$\displaystyle\tilde{\bf A}_{k_{\perp}k_{z}m\lambda_{\gamma}}({\bf
K})=\int{\bf A}_{k_{\perp}k_{z}m\lambda_{\gamma}}(x)\,e^{-i{\bf K}{\bf
r}}d^{3}r$ (96) $\displaystyle=\frac{{\displaystyle 4\pi^{2}}}{{\displaystyle
k_{\perp}}}\delta(K_{\perp}-k_{\perp})\,\delta(K_{z}-k_{z})\,i^{-m}{\bf
e}_{{\bf K}\lambda_{\gamma}}\,e^{-im\varphi_{\gamma}-i\omega t}$

and it corresponds to plane waves concentrated on the cone with the polar
angle $\theta_{\gamma}=\arctan(k_{\perp}/k_{z})$.

### Photon polarizations

To analyze the polarization of the photon, it is also convenient to introduce
the following linear combinations of vectors ${\bf e}_{{\bf
k}\lambda_{\gamma}}$ with helicities $\lambda_{\gamma}=\pm 1$:

$\displaystyle{\bf e}_{\parallel}$ $\displaystyle=$
$\displaystyle\frac{{\displaystyle-1}}{{\displaystyle\sqrt{2}}}\left({\bf
e}_{{\bf k},1}-{\bf e}_{{\bf k},-1}\right)$ $\displaystyle=$
$\displaystyle\left(\cos\theta_{\gamma}\cos\varphi_{\gamma},\,\cos\theta_{\gamma}\sin\varphi_{\gamma},\,-\sin\theta_{\gamma}\right),$
$\displaystyle{\bf e}_{\perp}$ $\displaystyle=$
$\displaystyle\frac{{\displaystyle i}}{{\displaystyle\sqrt{2}}}\left({\bf
e}_{{\bf k},1}+{\bf e}_{{\bf k},-1}\right)$ $\displaystyle=$
$\displaystyle\left(-\sin\varphi_{\gamma},\,\cos\varphi_{\gamma},\,0\right).$

It is easy to verify that these vectors are mutually orthogonal and orthogonal
to the vector ${\bf k}$. The vector ${\bf e}_{\parallel}$ defines the
longitudinal linear polarization (lying in the scattering plane given by the
$z$-axis and the vector ${\bf k}$) and the vector ${\bf e}_{\perp}$ – the
linear polarization which is orthogonal to the scattering plane. A twisted
photon state with either a longitudinal $l=\parallel$ or an orthogonal
$l=\perp$ polarization looks similar to Eq. (A):

${\bf A}_{k_{\perp}k_{z}ml}(x)=\int_{0}^{2\pi}i^{-m}{\bf
e}_{l}\,e^{im\varphi_{\gamma}}\,e^{-ikx}\frac{{\displaystyle
d\varphi_{\gamma}}}{{\displaystyle 2\pi}}.$ (99)

The following relations are exploited in the paper:

$\sum_{\lambda_{\gamma}}\lambda_{\gamma}{\bf e}_{{\bf
k}\lambda_{\gamma}}=-\sqrt{2}\,{\bf
e}_{\parallel},\;\;\sum_{\lambda_{\gamma}}{\bf e}_{{\bf
k}\lambda_{\gamma}}=-i\sqrt{2}\,{\bf e}_{\perp},$ (100)

and could be checked straightforwardly.

##  Appendix B Summation over $\lambda^{\prime}$ and $\lambda_{\gamma}$ in Eq.
(III.1)

Here we calculate the sum

${\bf S}=\sum_{\lambda^{\prime}\lambda_{\gamma}}u_{{\bf
p}^{\prime}\lambda^{\prime}}{\bf e}_{{\bf
k}\lambda_{\gamma}}\left(M^{\lambda\lambda^{\prime}\lambda_{\gamma}}_{\lambda,0}-M^{\lambda\lambda^{\prime}\lambda_{\gamma}}_{\lambda,2\lambda}\right),$
(101)

the expression in parentheses as in Eq. (31). First, using Eq. (100), we find
the partial sums

$\displaystyle\sum_{\lambda_{\gamma}}{\bf e}_{{\bf
k}\lambda_{\gamma}}d^{\;(1)}_{0\lambda_{\gamma}}(\theta_{\gamma})$
$\displaystyle=$ $\displaystyle-{\bf e}_{\parallel}\sin\theta_{\gamma},$ (102)
$\displaystyle\sum_{\lambda_{\gamma}}{\bf e}_{{\bf
k}\lambda_{\gamma}}d^{\;(1)}_{2\lambda,\lambda_{\gamma}}(\theta_{\gamma})$
$\displaystyle=$ $\displaystyle-{\bf
e}_{\parallel}2\lambda\cos\theta_{\gamma}-i{\bf e}_{\perp}.$ (103)

Using these expression we obtain

$\displaystyle{\bf S}$ $\displaystyle=$
$\displaystyle\sqrt{4\pi\alpha}E_{\lambda\lambda}u_{{\bf
p}^{\prime}\lambda}\left[{\bf
e}_{\parallel}\sin(\theta_{\gamma}+\mbox{$\frac{1}{2}$}\,\theta^{\prime})+i{\bf
e}_{\perp}2\lambda\sin(\mbox{$\frac{1}{2}$}\,\theta^{\prime})\right]$
$\displaystyle+$ $\displaystyle\sqrt{4\pi\alpha}E_{\lambda,-\lambda}u_{{\bf
p}^{\prime},-\lambda}\bigg{[}{\bf
e}_{\parallel}2\lambda\cos(\theta_{\gamma}+\mbox{$\frac{1}{2}$}\,\theta^{\prime})$
$\displaystyle+$ $\displaystyle i{\bf
e}_{\perp}\cos(\mbox{$\frac{1}{2}$}\,\theta^{\prime})\bigg{]},$

with $E_{\lambda\lambda^{\prime}}$ defined in Eq. (II.2). In the limiting
cases we have

${\bf S}=4\sqrt{\pi\alpha}vEu_{{\bf p}^{\prime}\lambda}{\bf
e}_{\parallel}\sin\theta_{\gamma}$ (105)

for the soft-photon approximation and

${\bf S}=4\sqrt{\pi\alpha EE^{\prime}}u_{{\bf p}^{\prime}\lambda}\left[{\bf
e}_{\parallel}\sin(\theta_{\gamma}+\mbox{$\frac{1}{2}$}\,\theta^{\prime})+i{\bf
e}_{\perp}2\lambda\sin(\mbox{$\frac{1}{2}$}\,\theta^{\prime})\right]$ (106)

for the ultra-relativistic electron approximation.

##  Appendix C Three scalar fields

In this appendix we would like to use the developed formalism and in a
simplified toy model of $1\rightarrow 2$ process with three scalar particles.
For definiteness, consider a process of a scalar particle with mass $M$ and
momentum $p$ decaying into two scalar particles with smaller masses
$\mu^{\prime}$ and $\mu^{\prime\prime}$ ($\mu^{\prime}+\mu^{\prime\prime}\leq
M$) and momenta $p^{\prime}$ and $p^{\prime\prime}$. The tree level matrix
element is given by

$S^{(1)}_{fi}=-i\lambda(2\pi)^{4}\,N\,\delta(p^{\prime}+p^{\prime\prime}-p),\
$ (107)

$N=\frac{1}{\sqrt{2E}\sqrt{2E^{\prime}}\sqrt{2E^{\prime\prime}}}$, and should
be inserted into the definition of the evolved wave function in coordinate
representation

$\displaystyle\langle
0|\hat{\phi}(x^{\prime\prime})\hat{\phi}(x^{\prime})|\phi,\phi\rangle^{(\text{ev})}$
(108)
$\displaystyle=\int\frac{d^{3}p^{\prime}}{(2\pi)^{3}}\frac{d^{3}p^{\prime\prime}}{(2\pi)^{3}}\,\frac{{\displaystyle
1}}{{\displaystyle\sqrt{2E^{\prime}}}}\,e^{-ip^{\prime}x^{\prime}}\,\frac{{\displaystyle
1}}{{\displaystyle\sqrt{2E^{\prime\prime}}}}\,e^{-ip^{\prime\prime}x^{\prime\prime}}\;S_{fi}^{(1)}.$

Using delta function to eliminate the integral over $p^{\prime\prime}$ leads
to reduction

$\displaystyle\int d\Gamma$ $\displaystyle=$
$\displaystyle(2\pi)^{4}\int\delta(p^{\prime}+p^{\prime\prime}-p)\,\frac{{\displaystyle
d^{3}p^{\prime}}}{{\displaystyle(2\pi)^{3}}}\frac{{\displaystyle
d^{3}p^{\prime\prime}}}{{\displaystyle(2\pi)^{3}}}$ $\displaystyle\rightarrow$
$\displaystyle\
\int\delta(E-E^{\prime}-E^{\prime\prime})|p^{\prime}|^{2}\sin\theta^{\prime}\frac{d|p^{\prime}|d\theta^{\prime}d\varphi^{\prime}}{(2\pi)^{2}},$

and writing down the momenta conservation law ${\bf p}^{\prime\prime}={\bf
p}-{\bf p}^{\prime}$ transforms the energy conservation
$\delta(E-E^{\prime}-E^{\prime\prime})$ into

$\delta(E-E^{\prime}-E^{\prime\prime})=\frac{E^{\prime\prime}}{|p^{\prime}||p|}\delta(\cos\theta_{pp^{\prime}}-\cos\theta_{0}),$

where $\cos\theta_{0}=\frac{1}{vv^{\prime}}\left(1-\frac{M^{2}+\mu^{\prime
2}-\mu^{\prime\prime 2}}{2EE^{\prime}}\right)$, $|p|=vE$,
$|p^{\prime}|=v^{\prime}E^{\prime}$. In case the initial particle moves along
the $z$ direction, $\cos\theta_{pp^{\prime}}=\cos\theta^{\prime}$ and the
integral over $\theta^{\prime}$ can be lifted imposing henceforth
$\theta^{\prime}=\theta_{0}$.

After substituting series expansions in terms of cylindrical waves

$\displaystyle
e^{-ip^{\prime}x^{\prime}}=\sum\limits_{m^{\prime}=-\infty}^{+\infty}i^{m^{\prime}}\,e^{-im^{\prime}\varphi^{\prime}}\phi_{p^{\prime}_{\perp}p^{\prime}_{z}m^{\prime}}(x^{\prime}),$
(110) $\displaystyle
e^{-ip^{\prime\prime}x^{\prime\prime}}=\sum\limits_{m^{\prime\prime}=-\infty}^{+\infty}i^{m^{\prime\prime}}\,\
e^{-im^{\prime\prime}\varphi^{\prime\prime}}\phi_{p^{\prime\prime}_{\perp}p^{\prime\prime}_{z}m^{\prime\prime}}(x^{\prime\prime}),$
(111)
$\displaystyle\phi_{p_{\perp}p_{z}m}(x)=e^{-iEt}J_{m}(p_{\perp}r_{\perp})e^{i(m\phi_{r}+k_{z}z)}$
(112)

into the evolved wave function, and considering that momenta conservation
allows only for $\varphi^{\prime\prime}=\varphi^{\prime}\pm\pi$ we can readily
evaluate the remaining azimuthal integral in Eq. (108). Finally, we obtain a
simple expression

$\displaystyle\langle
0|\hat{\phi}(x^{\prime\prime})\hat{\phi}(x^{\prime})|\phi,\phi\rangle^{(\text{ev})}=\frac{i\lambda}{4\pi
v}\frac{1}{(2E)^{3/2}}$ (113)
$\displaystyle\times\int\frac{p^{\prime}dp^{\prime}}{E^{\prime}}\,\sum_{m}(-1)^{m}\phi_{p^{\prime}_{\perp}p^{\prime}_{z}m}(x^{\prime})\phi_{p^{\prime}_{\perp},p_{z}-p^{\prime}_{z},-m}(x^{\prime\prime}).$

This could be generalized for the case of initial twisted particle with OAM
$m$ and manifest the conservation law $m=m^{\prime}+m^{\prime\prime}$.

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# Optical Variability of "Light-weight" Supermassive Black Holes at a Few
Percent Level from ZTF Forced-Photometry Light Curves

Mariia Demianenko,1,2,3 Kirill Grishin,2 Victoria Toptun,2,5 Igor
Chilingarian,4,2 Ivan Katkov,6,7,2 Vladimir Goradzhanov,2,5 and Ivan Kuzmin2,5

###### Abstract

Large time-domain surveys provide a unique opportunity to detect and explore
variability of millions of sources on timescales from days to years. Broadband
photometric variability can be used as the key selection criteria for weak
type-I active galactic nuclei (AGN), when other “direct” confirmation criteria
like X-ray or radio emission are unavailable. However, to detect variability
of rather weak AGN powered by intermediate-mass black holes, typical
sensitivity provided by existing light curve databases is insufficient. Here
we present an algorithm for post-processing of light curves for sources with
stochastic variability, retrieved from the The Zwicky Transient Facility (ZTF)
Forced Photometry service. Using our approach, we can filter out spurious data
points related to data reduction artefacts and also eliminate long-term trends
related to imperfect photometric calibration. We can now confidently detect
the broad-band variability at the 1-3 % level which can potentially be used as
a substitute for expensive X-ray follow-up observations.

1Moscow Institute of Physics and Technology (National Research University),
Moscow, Russia<EMAIL_ADDRESS>
2Sternberg Astronomical Institute, M.V. Lomonosov Moscow State University,
Moscow, Russia

3HSE University, Moscow, Russia

4Center for Astrophysics – Harvard and Smithsonian, Cambridge, USA

5Department of Physics, M.V. Lomonosov Moscow State University

6New York University Abu Dhabi, UAE

7Center for Astro, Particle, and Planetary Physics, NYU Abu Dhabi, UAE

##  1 Introduction and Motivation

An important property of active galactic nuclei (AGN) is flux variability on
all timescales at all wavelengths. The shortest timescales are defined solely
by the black hole mass $M_{BH}$ and no variability is expected at timescales
$\lesssim GM_{BH}/c^{3}$; intermediate timescales (hours to weeks) are
presumably driven by as yet unidentified processes within the disk and corona,
and bracketed by the dust torus size around the AGN, which is proportional to
X-ray luminosity $L_{X}^{0.5}$, which is also related to $M_{BH}$ via the
Eddington limit; the longer timescales (years) reflect the matter distribution
in the nuclear region of a galaxy, e.g. the size and mass of the nuclear star
cluster; finally, the AGN duty cycle depends on the availability of the gas
supply that feeds the BH and can become totally unrelated to the BH itself.
Because of the mass dependence of shorter timescales, intermediate-mass black
holes (IMBHs; $M_{BH}<10^{5}M_{\odot}$) work as a “time-squeezing machine”:
the shortest timescales change $\propto M_{BH}$, so 1 year for a $7\times
10^{4}M_{\odot}$ IMBH corresponds to a few centuries for a typical
(2–5)$\times 10^{7}M_{\odot}$ super-massive black hole (SMBH). At the same
time, the variability properties of IMBHs and “light-weight” SMBHs
($M_{BH}<10^{6}M_{\odot}$) remain completely unexplored.

Optical variability analysis can select and confirm many more “light-weight”
SMBH and IMBH candidates, because X-ray follow-up observations cannot be
performed for large samples of objects with high enough time cadence. ZTF Data
Release (Masci et al. 2019) aperture light curves cannot be used for
unambiguous AGN confirmation because the systematic uncertainties reaching
15-20$\%$ make the detection of variability of weak AGN powered by “light-
weight” SMBHs unfeasible. Instead, we build more reliable light curves from
difference images using the ZTF Forced Photometry service to detect optical
variability at a few percent level from a large sample weak AGN selected from
archival X-ray observations and H$\alpha$ broad line width. Still, to tackle
the systematic errors related to zero-point calibration issues, we need
additional post-processing if the desired level of uncertainty is a few
percent.

Chilingarian et al. (2018) identified a sample of IMBHs ($3\times
10^{4}<M_{BH}<2\times 10^{5}M_{\odot}$) by applying the NBursts full spectral
fitting (Chilingarian et al. 2007a, b) to SDSS and SDSS/eBOSS optical spectra
in the RCSED catalog (Chilingarian et al. 2017) with the virial $M_{BH}$
calibration (Greene & Ho 2004; Bon et al. 2019). For our analysis we use a
part of this list complemented with new sample of “light-weight” SMBHs
selected in a similar fashion, for which we found X-Ray counterparts in
archival data from XMM-Newton, Chandra, and Swift observatories. To illustrate
how our algorithm works, we use 2 objects: (i) J163159.59+243740.2 is a
candidate binary “light-weight” SMBH with a distinct variability seen even in
the ZTF Data Release light curves; (ii) J152442.58+292701.7 is an X-ray
confirmed “light-weight” SMBH with clearly detected broad H$\alpha$ lines in
its optical spectra.

##  2 Methods

###  2.1 Filtering spurious outliers

We use the guidelines for the Forced Photometry service to filter by the
following parameters: signal-to-noise ratio, distance to the nearest reference
source, pixel quality indicators to remove observations with uncorrected
cosmic ray hits and hot pixels in scientific images, sharpness values, and the
nearestrefchi parameter of the reference source to avoid spikes or hard edges
in extended reference sources (Masci et al. 2019). According to the available
information, a large fraction of ZTF data was acquired in non-photometric
conditions (e.g. scattered clouds, variable transparency). Therefore, outliers
in light curves might be artefacts originating from data analysis rather than
real values reflecting physical variability. Hence, a formally computed
statistical criterion of variability becomes unreliable unless spurious
measurements have been removed.

Figure 1.: _Top row:_ J163159.59+243740.3 (left) and J152442.58+292701.7
(right) after filtering out the outliers shown by black star markers, color
correction, zero point correction. _Bottom row:_ J163159.59+243740.3 (left)
and J152442.58+292701.7 (right) after the reference star correction.

For each ZTF photometric band we used the following criteria: (i) a
photometric zero point (ZP) of a given image should not be higher than a
ZP${}_{\rm mean}-$0.2$\times$_(airmass)_ for the $g$ band
(0.15$\times$_(airmass)_ for the $r$ band); (ii) a ZP RMS should be $<0.06$
mag; (iii) there should be at least 80 color calibrators in the $g$ band (or
120 in the $r$ band).

###  2.2 Color and zero-point corrections

We perform a color ZP correction using the $g-r$ source color as described in
the ZTF documentation. Then we apply a ZP correction by minimizing the
$\chi^{2}$ statistics between data points originating from different 64 CCD
quadrants in the ZTF detector.

We rank the quadrants by the number of points in them, and at the first step
we calculate the $\chi^{2}$ statistics for the quadrant with the largest
number of points. Next, we add measurements from the second most common
quadrant and derive the ZP shift for this quadrant, which minimizes the
combined $\chi^{2}$ statistics of the dataset. Then we repeat the algorithm
iteratively for all quadrants, data points from which constitute the light
curve. In other words, we derive such values of ZP shifts for individual
quadrants so that the variability in a combined dataset is the lowest. Such a
technique can reduce a derived measure of real physical variability if a
source was placed in different CCD quadrants at different variability phases
(which is unlikely especially for stochastically variable sources), but this
is perhaps the simplest approach to eliminate ZP deviations among CCD
quadrants, which can introduce a spurious source variability.

###  2.3 Reference star correction

The final step of light curves post-processing removes global trends
originating from long-term ZP changes by using a “median reference star”. We
select up-to 50 bright nearby stars around the source of interest, which are
marked as non-variable in the GAIA catalog and retrieve their light curves
from the ZTF Forced Photometry Service. Each light curve is corrected using
the filtering/correction as described above.

Then we adjust the reference star light curves by applying the difference
between the median magnitude of the source of interest (candidate AGN) and the
median magnitude of each star and keep only those data points, which originate
from the same ZTF images used for the construction of the AGN light curve.
Next, we calculate the difference between the magnitude of the source of
interest and the median value of the magnitude of all reference stars measured
at the same image (after applying the median magnitude difference as described
above).

This simple correction allows us to correct flux measurements of an AGN
candidate by the median reference star also removing potential variable stars,
which are not flagged in GAIA. Finally, we derive a fully corrected light
curve of an AGN candidate, which can then be analyzed statistically.

The light curves before and after median reference star correction are shown
in Fig .1 (top and bottom rows). The object J163159.59+243740.2 (left panels)
exhibits strong variability both on small timescales related to the size of
the accretion disk and, accordingly, the mass of the black hole and on large
timescales: here the median reference star correction does not change the
variability assessment. However, for the object J152442.58+292701.7 the
apparent variability seen after filtering spurious data points becomes less
significant when we apply the median reference star correction. We can see
that for a typical SDSS-selected AGN candidate ($g\sim 17-17.5$ mag) we can
probe the variability at the 1–3% level.

##  3 Implications

By filtering the outliers originating from data processing imperfections,
correcting ZP differences and then correcting for systematic uncertainties
using reference stars, one can reach the sensitivity of 1–3% in the detection
of stochastic variability of weak type-I AGN candidates using publicly
available data from modern time-domain surveys such as ZTF. Currently, the
bottleneck that limits the scalability of our analysis is that the ZTF Forced-
Photometry Service can process no more than 100 objects at a time and it takes
hours to weeks to generate light curves from difference images. In the future,
when a full database with difference image light curves becomes available, we
will be able to process tens of thousands of objects and complete the first
census of weak AGN including those powered by IMBHs and “light-weight” SMBHs.

#### Acknowledgments

MD acknowledges the ADASS 2021 organizing committee for providing financial
aid. The authors are supported by the RSF grant 17-72-20119.

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