# Observability of the superkick effect within a quantum-field-theoretical
approach

Igor P. Ivanov<EMAIL_ADDRESS>Bei Liu<EMAIL_ADDRESS>Pengming Zhang<EMAIL_ADDRESS>School of Physics and Astronomy, Sun
Yat-sen University, 519082 Zhuhai, China

###### Abstract

An atom placed in an optical vortex close to the axis may, upon absorbing a
photon, acquire a transverse momentum much larger than the transverse momentum
of any plane-wave component of the vortex lightfield. This surprising
phenomenon dubbed superkick has been clarified previously in terms of the atom
wave packet evolution in the field of an optical vortex treated classically.
Here, we study this effect within the quantum field theoretical (QFT)
framework. We consider collision of a Bessel twisted wave with a compact
Gaussian beam focused to a small focal spot $\sigma$ located at distance $b$
from the twisted beam axis. Through a qualitative discussion supported by
exact analytical and numerical calculations, we recover the superkick
phenomenon for $\sigma\ll b$ and explore its limits when $\sigma$ becomes
comparable to $b$. On the way to the final result within the QFT treatment, we
encountered and resolved apparent paradoxes related to subtle issues of the
formalism. These results open a way to a detailed QFT exploration of other
superkick-related effects recently suggested to exist in high-energy
collisions.

##  I Introduction

Physics of structured light—that is, a nonplane-wave light field with phase
singularities and other topological features—is a fascinating and actively
studied topic of modern optics rubinsztein2016roadmap ; babiker2019atoms ;
forbes2021structured . A prototypical example of structured light is an
optical vortex: the electromagnetic field configuration near a singularity
line whose phase depends on the azimuthal variable $\varphi$ as
$\exp(im\varphi)$, with an integer winding number $m$. Such a light field is
readily available in the focal region of the Laguerre-Gaussian modes of a
laser beam known since early 1990’s Allen:1992zz  and nowadays routinely used
in numerous directions of fundamental research and in applications
andrews2012angular ; Paggett:2017 ; Knyazev:2019 .

Optical vortex is characterized with a swirling Poynting vector density, which
rotates around the phase singularity axis. As a result, light with optical
vortex carries nonzero orbital angular momentum (OAM) proportional to $m$. If
an atom is placed in the light field of an optical vortex, also called
“twisted light”, it experiences light-induced torque proportional to the
optical OAM babiker1994 . Also, selection rules governing its internal
transitions are modified, and these modifications depend on its distance to
the axis babiker2002orbital ; Afanasev:2014 ; schmiegelow2016transfer ;
afanasev2018experimental ; solyanik2019excitation ; schulz2019modification .
This observation is intuitively clear: since each photon of the optical vortex
carries a nonzero OAM, it can enhance multipole transitions.

What is more surprising—and even counter-intuitive—is the prediction of
barnett2013superkick  that an atom placed in the vicinity of the optical
vortex axis can acquire a much larger transverse momentum than any of the
photons of the twisted light beam can deliver. This unexpectedly large
momentum transfer was called in barnett2013superkick  a “superkick”.

Semiclassical arguments behind the superkick phenomenon were outlined in
barnett2013superkick  and investigated with specific examples in
Afanasev:2020nur . Indeed, if we follow a circular path with radius $b$ around
the phase singularity axis, we accumulate a phase change of $2\pi m$ along the
circumference $2\pi b$. The resulting phase gradient can be interpreted as the
local momentum Berry2013fivemomenta . If we choose unit vectors ${\bf e}_{x}$
and ${\bf e}_{y}$ on the transverse plane and consider the point ${\bf
b}=b{\bf e}_{x}$, then the local phase gradient on this plane produces the
local momentum

${\bf p}_{\perp}=\frac{m}{b}\,{\bf e}_{y}\,,$ (1)

which is orthogonal to ${\bf b}$. If $b$ is sufficiently small, this local
momentum can be arbitrarily large and can be transferred to an atom placed
sufficiently close to the axis. However, if we interpret the light-atom
interaction in terms of photons, each photon can carry much smaller transverse
momentum. We arrive at a paradox: photon absorption induces much larger
momentum transfer than it actually carries.

The semiclassical arguments outlined above blend in uncontrollable way quantum
and classical concepts. They rely on pointlike classical atoms, which are,
nevertheless, supposed to absorb photons, quantum entities. In
barnett2013superkick , Barnett and Berry made one step forward and represented
the atom as a compact wave packet of spatial extent $\sigma$. The smallness of
$\sigma$ implies that, in momentum space, the wavefunction of this wave packet
extends up to the large values of $\approx 1/\sigma$. In the initial wave
packet, all these plane-wave components balance each other leading to
$\langle{\bf k}_{\perp}\rangle=0$. However, when distorted by the interaction,
this set of plane-wave components can easily produce a nonzero, large average
transverse momentum. This picture offers the qualitative resolution to the
paradox: the photon does not supply large transverse momentum to the atom, but
only rearranges and reweights the plane-wave components inside the wave
packets which produce the superkick. In Berry2013fivemomenta , Berry also
interpreted this effect in terms of “superweak” values which arise in
experiments with postselection and can exceed the limits of the true spectrum
of the operator.

This explanation, although enlightening, is still only partially quantum, as
the atom wave packet is supposed to evolve in the classical external
lightfield of the optical vortex. Although it may be sufficient for this
particular problem, one may want to explore similar effects in high-energy
physics collisions, and for that task, a full quantum field theoretic approach
is needed.

The suggestion that superkick related effects may indeed occur and be
observable in nuclear and high-energy physics realm was put forth in the very
recently papers Afanasev:2020nur ; Afanasev:2021fda . A significant shift of
the energy threshold as well as other phenomena were predicted in vicinity of
the twisted photon axis. However the analysis in these publications followed
the formalism used for light-atom interaction in optical vortices.

The purpose of the present paper is to address the superkick effect—and other
puzzling observations which arise on the way—in the quantum field theoretic
(QFT) framework. We will reformulate the process as a QFT scattering problem,
in which a twisted (Bessel) photon collides with a compact Gaussian wave
packet representing the counterpropagating particle offset by an impact
parameter $b$. We will compute the expectation value of the transverse
momentum of the final state and explore its dependence on the parameters of
the collision process. Through exact analytical results, qualitative
discussions, and numerical calculations, we will clarify all the features of
this process which seem paradoxical and, eventually, place limits on the
superkick observability in typical high-energy collisions.

Recovering the superkick phenomenon and resolving the subtle technical issues
which arise when treating this problem within the QFT formalism with
monochromatic beams provide, eventually, useful insights of how to treat such
problems within the full QFT framework. Thus, we view this work as a first
step towards a detailed systematic study, and the experience gained here will
be instrumental in adapting the formalism and exploring additional effects.

The structure of this paper is the following. In the next Section, we will
describe in more details the semiclassical roots of the superkick phenomenon.
We will also describe how we are going to treat this problem in the QFT
framework, highlighting yet another seemingly paradoxical observation. Then in
Section III we will outline the necessary QFT formalism and resolve the second
paradox just mentioned. The following Section IV analyzes the kinematics of
the Bessel-Gaussian collision. These results will finally lead us in Section V
to the superkick phenomenon and its limits. With these results, we give our
comments on previous works in Section VI and finally draw conclusions.
Appendices contain further technical calculations used in the main text.
Vectors are denoted with bold symbols and relativistic units $\hbar=c=1$ are
used throughout the paper.

##  II The semiclassical origin of the superkick

Consider a Bessel photon, that is, a cylindrical monochromatic solution to
Maxwell’s equations with a definite energy $\omega$, a definite longitudinal
momentum $p_{z}$, and a definite modulus of the transverse momentum $|{\bf
p}_{\perp}|=\varkappa$. Within the paraxial approximation, one can treat spin
and OAM separately Allen:1992zz . Since we focus on the kinematic features of
the process, from now on we will suppress the polarization vector of the
photon and discuss only its coordinate or momentum-space wavefunction. The
Bessel photon in the coordinate space can be represented as

$\psi_{p_{z},\varkappa,m}(\rho,z)\propto
e^{ip_{z}z}e^{im\varphi}J_{|m|}(\varkappa\rho)\,.$ (2)

Here, $\rho=|{\bf r}_{\perp}|$, with the point $\rho=0$ corresponding to the
phase singularity. The transverse intensity profile has the shape of
concentric rings, with the first ring having radius $\rho=m/\varkappa$, see
Fig. 1.

Figure 1: An atom placed at distance $b\ll m/\varkappa$ from the Bessel photon
axis can acquire the transverse momentum $p_{\perp}\gg\varkappa$.

Consider now a circle of radius $b$ circumscribed in the transverse plane
around the point of phase singularity. The accumulated phase change of $2\pi
m$ along the circumference $2\pi b$ produces the phase gradient $m/b$, which
leads to the local transverse momentum value (1). At the first intensity ring,
$b=m/\varkappa$, and we get $p_{\perp}(b)=\varkappa$, which coincides with the
transverse momentum carried by every plane wave component of the Bessel
photon. However, for smaller $b$, one gets the local transverse momentum
arbitrarily larger than $\varkappa$. Of course, at $b\ll m/\varkappa$, the
intensity of the light field is suppressed, so that the probability of
absorbing a photon is very small. But when the atom does absorb the photon, it
is predicted to acquire a superkick, a recoil transverse momentum much larger
than $\varkappa$. Thus, we confirm the apparent paradox: a superposition of
plane waves with fixed $|{\bf p}_{\perp}|=\varkappa$ is able to deliver a
transverse momentum (much) larger than $\varkappa$.

One may argue that the local momentum density defined via the local phase
gradient is not the same thing as the momentum of the twisted photon. Since
the twisted state is not a plane wave, it is not a transverse momentum
eigenstate. One could then think of the expectation value of the transverse
momentum operator calculated for the Bessel photon state. However, due to the
azimuthal symmetry of the Bessel twisted state, this expectation value is
zero: $\langle{\bf p}_{\perp}\rangle=0$.

Thus, trying to make sense of the process, we encounter another puzzling
observation: an initial state (the photon and the atom) with a zero average
transverse momentum goes into a final state (an excited atom or a scattered
system) with a nonzero—and potentially large—transverse momentum. How does it
agree with momentum conservation which must hold at the fundamental level?

Notice that this question would not arise if we considered a wave packet
evolving in an external field, as in barnett2013superkick . It is only in the
empty space collision setting that we can appeal to the momentum conservation
law and present its nonconservation as a paradox.

To clarify all these observations and resolve the paradoxes, we will consider
the process in the full QFT setting. We will analyze free-space scattering of
a monochromatic Bessel twisted state with parameters $\varkappa$ and $m$ with
a monochromatic Gaussian beam of transverse spatial extent $\sigma$ localized
at the impact parameter $b$ from the axis. Replacing a semiclassical pointlike
particle with a wave packet of size $\sigma$ will allows us to check how the
average transverse momentum of the final system changes as we vary $\sigma$
and $b$ with respect to $m/\varkappa$. Also, by varying the ratio $\sigma/b$,
we will be able to see how the superkick phenomenon which exists for
$\sigma\ll b$ gradually disappears for $\sigma\sim b$.

##  III Scattering of nonplane-wave states

###  III.1 General considerations

Computation of non-planewave states scattering within the QFT framework does
not represent, by itself, a conceptual novelty. In fact, many QFT textbooks,
when deriving the planewave scattering amplitude, first regularize the
intermediate expressions with localized wavefunctions for the initial and
final states and then take the limit of plane waves, see e.g. Peskin . Since
the scattering amplitude is a linear functional of these initial
wavefunctions, the construction of QFT itself is not altered, so that one can
re-use the planewave scattering matrix amplitude by inserting it between the
appropriate wavefunctions.

However, the kinematical distributions change significantly, especially when
dealing with vortex states of non-zero OAM $m$. Indeed, some of the energy and
momentum delta functions are integrated out when performing convolution with
the initial wavefunctions and do not appear anymore in the final particles
phase space. As a result, one observes novel distributions in the final state
kinematics, which are unattainable for planewave collisions. The superkick
phenomenon studied here also belongs to the family of kinematical
peculiarities driven by the non-planewave nature of the colliding states. This
is why we focus in this work on universal kinematical features, not on
specific particles which collide.

It should be kept in mind that the way these kinematical features are derived
differs significantly within a semiclassical treatment, in quantum mechanical
scattering on a fixed scattering center, which can absorb any momentum
transfer, and in the QFT description where all initial states are represented
as wavepackets and the total energy-momentum conservation law is imposed.
Therefore, even if a phenomenon is demonstrated within a semiclassical
approach, it is desirable to rederive it within the full QFT. This derivation
will shed more light on possible limitations of the effect and on possible
subtle issues. Also, it can be a step towards a systematic study of other
particle production processes which require the QFT framework.

With these reservations in mind, we begin with a brief reminder of the general
QFT approach to calculating collision of free propagating quantum particles.
To illustrate the main kinematic effects, we consider elastic scattering of
spinless particles. The inclusion of the polarization degrees of freedom for
fermions or vector bosons may lead to interesting additional effects but they
are not essential in our discussion of the superkick phenomenon and the
apparent kinematic paradoxes outlined above.

First, we consider the textbook case, in which two plane wave states with
three-momenta ${\bf k}_{1}$, ${\bf k}_{2}$ and energies $E_{1}$, $E_{2}$
scatter to two final-state particles with three-momenta ${\bf
k}^{\prime}_{1}$, ${\bf k}^{\prime}_{2}$ and energies $E^{\prime}_{1}$,
$E^{\prime}_{2}$. With a slight abuse of notation, we denote the total initial
momentum as ${\bf P}_{0}={\bf k}_{1}+{\bf k}_{2}$, the total energy as
$E_{0}=E_{1}+E_{2}$, and the total final momentum as ${\bf P}={\bf
k}^{\prime}_{1}+{\bf k}^{\prime}_{2}$. The plane wave $S$-matrix element has
the form

$S_{PW}(k_{1},k_{2};k^{\prime}_{1},k^{\prime}_{2})=i(2\pi)^{4}\delta(E_{0}-E^{\prime}_{1}-E^{\prime}_{2})\delta^{(3)}({\bf
P}_{0}-{\bf P}){{\cal
M}\over\sqrt{16E_{1}E_{2}E^{\prime}_{1}E^{\prime}_{2}}}\,.$ (3)

Here, ${\cal M}$ is the plane-wave invariant amplitude calculated according to
the standard Feynman rules. It can depend on kinematic invariants involved; in
the simplest case of the pointlike quartic interaction, it is just a constant.
Squaring this amplitude, regularizing the squares of delta-functions in the
usual way, one gets the differential cross section

$d\sigma_{PW}\propto\delta(E_{0}-E^{\prime}_{1}-E^{\prime}_{2})\delta^{(3)}({\bf
P}_{0}-{\bf P})|{\cal M}|^{2}\,\frac{d^{3}{\bf
k}^{\prime}_{1}}{2E^{\prime}_{1}}\frac{d^{3}{\bf
k}^{\prime}_{2}}{2E^{\prime}_{2}}\,.$ (4)

Clearly, for fixed initial ${\bf k}_{1}$ and ${\bf k}_{2}$, the total final
state momentum ${\bf P}$ is also fixed. The plane wave cross section can only
display a non-trivial distribution in ${\bf k}_{1}^{\prime}$ or ${\bf
k}_{2}^{\prime}$ but not in ${\bf P}$.

Let us now assume that the two initial particles are prepared in nonplane-wave
states; for the final particles we still use the plane wave basis. Scattering
theory of arbitrarily shaped, partially coherent beams was developed in the
paraxial approximation in Kotkin:1992bj . The most general formalism, capable
of going beyond the paraxial approximation, was presented in Karlovets:2016jrd
; Karlovets:2020odl . The formalism was applied to twisted particle scattering
in Jentschura:2010ap ; Jentschura:2011ih ; Ivanov:2011kk ; Karlovets:2012eu ;
Ivanov:2016oue ; Karlovets:2016jrd ; Karlovets:2020odl . For the present
analysis, it suffices to stick to the paraxial approximation and consider both
initial particles as monochromatic states with energies $E_{1}$ and $E_{2}$,
as before, but with non-trivial momentum space wavefunctions $\phi_{1}({\bf
k}_{1})$ and $\phi_{2}({\bf k}_{2})$. The exact normalization condition for
these wavefunctions is a subtle issue Jentschura:2011ih ; Ivanov:2011kk ;
Ivanov:2011bv ; Karlovets:2012eu . However, it is inessential to our
discussion because we will be interested in the average values of operators,
not the absolute magnitude of the cross section. With these reservations, we
can write the $S$ matrix element for scattering of the initial nonplane-wave
state $i$ to the final plane wave state as

$S(i;k^{\prime}_{1},k^{\prime}_{2})=N\int d^{3}k_{1}d^{3}k_{2}\,\phi_{1}({\bf
k}_{1})\phi_{2}({\bf k}_{2})\cdot
S_{PW}(k_{1},k_{2};k^{\prime}_{1},k^{\prime}_{2})\,,$ (5)

where the (inessential) factor $N$ takes care of all the normalization
conditions.

We remark in passing that, up to normalization, this very quantity
$S(i;k^{\prime}_{1},k^{\prime}_{2})$ can be viewed as the momentum space
wavefunction of the final two-particle system $\Psi_{f}({\bf
k}^{\prime}_{1},{\bf k}^{\prime}_{2})$. It is the same object as the evolved
wavefunction discussed very recently in Karlovets:2021gcm , that is, the
wavefunction emerging from the scattering process itself and freely expanding
in vacuum before hitting the detector. In general, $\Psi_{f}({\bf
k}^{\prime}_{1},{\bf k}^{\prime}_{2})$ can depend on ${\bf k}^{\prime}_{1}$
and ${\bf k}^{\prime}_{2}$ individually, but in the simplest case of pointlike
interaction it depends only on the total momentum ${\bf P}$ and total energy
$E_{1}^{\prime}+E_{2}^{\prime}$.

The plane-wave $S$-matrix element (3) contains the four-dimensional kinematic
delta-function. Since we deal with monochromatic initial states, the energy
delta function remains unchanged and can be taken out of the integral (5).
Then, squaring this expression, regularizing the energy delta-function squared
as before, we can write the cross section in the following generic way:

$d\sigma\propto|{\cal I}|^{2}d\Gamma^{\prime}\,,\quad\mbox{where}\quad
d\Gamma^{\prime}=\delta(E_{0}-E^{\prime}_{1}-E^{\prime}_{2})\,\frac{d^{3}{\bf
k}^{\prime}_{1}}{2E^{\prime}_{1}}\frac{d^{3}{\bf
k}^{\prime}_{2}}{2E^{\prime}_{2}}\,,$ (6)

while the function ${\cal I}$ includes the integration over all plane wave
components of the two initial states:

${\cal I}=\int d^{3}{\bf k}_{1}d^{3}{\bf k}_{2}\,\phi_{1}({\bf
k}_{1})\phi_{2}({\bf k}_{2})\,\delta^{(3)}({\bf k}_{1}+{\bf k}_{2}-{\bf
P})\cdot{\cal M}\,.$ (7)

The proportionality symbol in (6) refers to the normalization coefficients
whose accurate analysis is inessential.

Let us examine the key features of the general nonplane-wave cross section
$d\sigma$ in (6). Unlike the plane wave case where the fixed initial momenta
forced the total final ${\bf P}$ to be fixed, here we have a distribution in
${\bf P}$. This is illustrated by changing the final phase space from
$d^{3}{\bf k}^{\prime}_{1}d^{3}{\bf k}^{\prime}_{2}$ to $d^{3}{\bf
k}^{\prime}_{1}d^{3}{\bf P}$ and observing the absence of $\delta^{(3)}({\bf
k}_{1}+{\bf k}_{2}-{\bf P})$ in (6). This distribution is a new dimension in
the final state kinematics which was absent in the pure plane-wave scattering
and which can now be explored.

Further simplifications are possible in the realistic situation when
$\phi_{1}({\bf k}_{1})$ and $\phi_{2}({\bf k}_{2})$ are compact wavefunctions
localized around the average momenta ${\bf p}_{1}=\langle{\bf k}_{1}\rangle$
and ${\bf p}_{2}=\langle{\bf k}_{2}\rangle$. We assume that these two average
momenta are antiparallel to each other, thus defining the common axis $z$. We
write them as ${\bf p}_{1}=(0,0,p_{1z})$, ${\bf p}_{2}=(0,0,p_{2z})$, with the
convention that $p_{1z}>0$, $p_{2z}=-|p_{2z}|<0$. We denote their sum as
$P_{z0}=p_{1z}+p_{2z}$. Although the individual final particle momenta ${\bf
k}^{\prime}_{1}$ and ${\bf k}^{\prime}_{2}$ can be large, the total final
transverse momentum ${\bf P}_{\perp}$ as well as the deviation of the final
longitudinal momentum from $P_{z0}$

$\Delta P_{z}=P_{z}-P_{z0}=P_{z}-p_{1z}+|p_{2z}|$ (8)

are expected to be small.

We use the integration over $dk_{1z}$ to eliminate the energy delta function.
As we show in Appendix A, for the compact initial momentum-space
wavefunctions, the vast majority of the final phase space $d\Gamma^{\prime}$
can be presented in the factorized form separating the total momentum and the
relative motion:

$d\Gamma^{\prime}=\sum_{k^{*}_{1z}}\frac{d^{2}{\bf
k}^{\prime}_{1\perp}}{2\sqrt{\lambda}}\cdot d^{2}{\bf P}_{\perp}\,d\Delta
P_{z}\,.$ (9)

where the function $\lambda$ is depends on ${\bf k}_{1\perp}^{\prime 2}$ and
$P_{0z}$ but not on ${\bf P}_{\perp}$ or $d\Delta P_{z}$. The summation here
refers to the two possible values of $k^{\prime}_{1z}$ available for a given
${\bf k}^{\prime}_{1\perp}$ and ${\bf P}$.

Using this approximate factorization of the final phase space and recalling
that ${\cal I}$ also depends on ${\bf P}$ but not on ${\bf
k}^{\prime}_{1\perp}$, we can compute the average transverse momentum of the
final state as

$\langle{\bf P}_{\perp}\rangle=\frac{\langle\Psi_{f}|{\bf
P}_{\perp}|\Psi_{f}\rangle}{\langle\Psi_{f}|\Psi_{f}\rangle}=\frac{\int|{\cal
I}|^{2}{\bf P}_{\perp}d\Gamma^{\prime}}{\int|{\cal
I}|^{2}d\Gamma^{\prime}}\approx\frac{\int|{\cal I}|^{2}{\bf
P}_{\perp}d^{2}{\bf P}_{\perp}\,d\Delta P_{z}}{\int|{\cal I}|^{2}d^{2}{\bf
P}_{\perp}\,d\Delta P_{z}}\,,$ (10)

###  III.2 The momentum non-conservation paradox and its resolution

Before we proceed with the computation of $\langle{\bf P}_{\perp}\rangle$, let
us address the apparent non-conservation of the transverse momentum mentioned
in Section II. Indeed, with the above convention for axis $z$, we see that the
average transverse momentum of the initial state is zero ${\bf
p}_{1\perp}+{\bf p}_{2\perp}=0$. However, we are computing the final
transverse momentum (10) and expect it to be nonzero. How can it happen, if
momentum is conserved?

Figure 2: Schematic illustration of the origin of a nonzero transverse
momentum in the scattered part of the final wavefunction.

The answer is that collision of two incoming waves leads not only to the
scattered state we have considered but also to a nonscattered initial state
which is only partially depleted by the interaction, see Fig. 2. The full
outgoing wavefunction is a superposition of these components, which even
belong to different Fock states if the collision is inelastic. The $S$-matrix
element presented above describes only the scattered component $\Psi_{f}$ of
the total final wavefunction.

Depending on the wavefunctions and on the fundamental interactions, it may
happen that the intensity of the scattering process exhibits certain bias in
the transverse momentum space. As a result, the scattered component of the
outgoing wavefunction may get a nonzero $\langle{\bf P}_{\perp}\rangle$, while
the depleted nonscattered component acquires a nonzero compensating average
momentum. The transverse momentum of the entire final state, which includes
both scattered and nonscattered components, is zero. Thus, scattering does not
produce transverse momentum out of nothing; it just rearranges the individual
plane-wave components.

In the usual plane wave scattering, we never run into this paradox because the
momentum space wavefunction of the initial state is assumed to be infinitely
narrow: $\phi({\bf k})\propto\delta^{(3)}({\bf k}-{\bf p})$. There is simply
no room to induce momentum bias within this infinitely narrow momentum
distributions. Such effects can arise only with non-trivially shaped initial
states $\varphi({\bf k}_{i})$. Even an exact Bessel beams would suffice.
Although in such a state the values of $k_{z}$ and $|{\bf k}_{\perp}|$ are
fixed, the azimuthal angle distribution of $\varphi({\bf k}_{i})$, when
disturbed, can produce a nonzero overall transverse momentum.

##  IV Scattering of Bessel twisted state and Gaussian beam

###  IV.1 The setting

Now we focus on scattering of a Bessel twisted state with a Gaussian beam
whose center is shifted from the twisted state beam axis by ${\bf b}_{\perp}$.
The Bessel twisted state is defined by a fixed longitudinal momentum
$k_{1z}=p_{1z}>0$, a fixed modulus of the transverse momentum $|{\bf
k}_{1\perp}|=\varkappa$, and the OAM $m$, see Jentschura:2010ap ;
Jentschura:2011ih ; Ivanov:2011kk ; Karlovets:2012eu  for more details. The
momentum space wavefunction $\phi_{1}({\bf k}_{1})$, up to normalization, is
then given by

$\phi_{1}({\bf k}_{1})=\delta(k_{1z}-p_{1z})\delta(|{\bf
k}_{1\perp}|-\varkappa)e^{im\varphi_{1}}\,.$ (11)

To construct the second initial particle wave packet, we begin with the plane-
wave state with momentum ${\bf p}_{2}=(0,0,p_{2z})$, where $p_{2z}<0$
satisfies the energy-momentum relation $E_{2}^{2}=p_{2z}^{2}+M^{2}$. We
construct the Gaussian beam by smearing the momentum distribution around ${\bf
p}_{2}$ and keeping the beam monochromatic:

$\phi_{2}({\bf k}_{2})=\exp\left[-\frac{({\bf k}_{2}-{\bf
p}_{2})^{2}\sigma^{2}}{2}\right]\delta({\bf k}_{2}^{2}+M^{2}-E_{2}^{2})\cdot
e^{-i{\bf k}_{2\perp}{\bf b}_{\perp}}\,.$ (12)

Here, the parameter $\sigma$ indicates the transverse spatial extent of the
wavefunction in the coordinate space, while the phase factor with ${\bf
b}_{\perp}$ describes a shift of the beam with respect to the twisted particle
axis. In the calculations below, we take the paraxial approximation for the
Gaussian state:

$\sigma|p_{2z}|\gg 1\,.$ (13)

To make it clear, we do not claim that the expression (12) is the best choice
for description of realistic wave packets. It is explicitly non-Lorentz-
invariant, which by itself does not represent any problem. Indeed, nonplane-
wave states can be produced in an experimental setting which, by definition,
resides in a certain reference frame. Second, one could certainly consider
other monochromatic transversely localized beams as well as 3D localized wave
packets which are unavoidably non-monochromatic. One could also choose
different longitudinal and transverse sizes $\sigma_{z}\not=\sigma_{\perp}$ or
consider a non-Gaussian longitudinal profile. As long as we stay within the
paraxial approximation, these modifications will differ by minor details of
the longitudinal momentum distribution of the cross section. All the
transverse space phenomena, in particular the phenomenon of superkick, are
expected to be insensitive to these details.

Thus, we deal with a four length scale problem: we have the wavelength of the
Gaussian state $\lambda=1/|p_{2z}|$, its transverse wave packet size $\sigma$,
its distance to the twisted photon axis $b$, and the radius of the first
Bessel beam ring in the transverse plane $m/\varkappa$. The kinematic
configuration appropriate for observation of the superkick phenomenon implies
the following hierarchy of these length scales:

$\frac{1}{|p_{2z}|}\ll\sigma\ll b\ll\frac{m}{\varkappa}\,.$ (14)

When testing the limits of this configuration, we will also explore values
$\sigma\sim b$ or even $\sigma\gg b$. Notice that, by taking
$\sigma\to\infty$, we can recover the plane wave limit for the Gaussian state.
In this limit, the results should not depend on $b$, and the cross section
should be represented as the azimuthal averaged value of the plane wave cross
section, as was derived in Ivanov:2011kk .

###  IV.2 Towards the superkick: an initial attempt

With all these considerations, we substitute the initial state wavefunctions
into the integral ${\cal I}$ of (7):

$\displaystyle{\cal I}$ $\displaystyle=$ $\displaystyle\int d^{3}{\bf
k}_{1}d^{3}{\bf k}_{2}\,\delta(k_{1z}-p_{1z})\delta(|{\bf
k}_{1\perp}|-\varkappa)e^{im\varphi_{1}}\cdot\,e^{-\frac{({\bf k}_{2}-{\bf
p}_{2})^{2}\sigma^{2}}{2}}\delta({\bf k}_{2}^{2}+M^{2}-E_{2}^{2})\,e^{-i{\bf
k}_{2\perp}{\bf b}_{\perp}}\delta^{(3)}({\bf k}_{1}+{\bf k}_{2}-{\bf
P})\cdot{\cal M}\,.\quad$ (15)

It contains six integrations and six delta function and, therefore, can be
calculated exactly, see details in Appendix B. As we calculated in that
Appendix, delta functions place constraints on the possible values of $|{\bf
P}_{\perp}|$ and $\Delta P_{z}$, which carve a crescent-like region on the
$(\Delta P_{z},|{\bf P}_{\perp}|)$ plane. In the small-$\Delta P_{z}$ region,
where the cross section is expected to be concentrated, the borders of this
region can be represented by the two inequalities

$(|{\bf P}_{\perp}|-\varkappa)^{2}<2|p_{2z}|\Delta P_{z}<(|{\bf
P}_{\perp}|+\varkappa)^{2}\,.$ (16)

The expression $|{\cal I}|^{2}$, which we compute exactly in Appendix B,
exhibits strong intensity oscillations inside this region and diverges near
the boundaries. Apart from these small-scale ripples, it is concentrated at
small values of ${\bf P}_{\perp}$ and $\Delta P_{z}$ and exhibits exponential
attenuation at their large values. The typical ranges of $\Delta P_{z}$ and
$|{\bf P}_{\perp}|$ are

$\Delta P_{z}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}\frac{1}{\sigma^{2}|p_{2z}|}\,,\quad|{\bf
P}_{\perp}|\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}\frac{1}{\sigma}\,,\quad\Delta
P_{z}\ll|{\bf P}_{\perp}|\,.$ (17)

We conclude that the total momentum of the final state is predominantly
transverse, and it is driven by the transverse momenta of the initial compact
Gaussian state, not by the transverse momentum of the Bessel twisted state.
For the purpose of illustration, we show in Fig. 3 the behavior of $|{\cal
I}|^{2}$ on the $(|{\bf P}_{\perp}|,\Delta P_{z})$-plane for $|p_{2z}|=10$,
$\varkappa=1$, $1/\sigma=2$ (all expressed in arbitrary but the same unit of
momentum), $b=\sigma/5$, and $m=2$.

Figure 3: The $(|{\bf P}_{\perp}|,\Delta P_{z})$-distribution of the function
$|{\cal I}|^{2}$, whose value is shown in arbitrary units and encoded in color
shades. The values of the parameters are (in units of an arbitrary momentum
scale): $|p_{2z}|=10$, $\varkappa=1$, $1/\sigma=2$, $b=\sigma/5$, $m=2$.

With these observations, we confirm the explanation of the origin of the
superkick phenomenon given in barnett2013superkick . The initial compact wave
packet already has large transverse momentum components of the order of
$1/\sigma\gg\varkappa$. However, these large transverse momentum components
are neatly balanced in the initial Gaussian state, so that its averaged
transverse momentum is zero. The role of the twisted photon, which does not
possess any significant transverse momentum, is just to disturb this balance
in a particular, momentum-biased way. The outgoing scattered state still has
large transverse momentum components of the order of $1/\sigma$, but they can
now lead to an overall nonzero average transverse momentum, which can, of
course, be greater than $\varkappa$.

Are these qualitative considerations confirmed by analytical results? In
Appendix B we show that, if we proceed with the calculation of the average
transverse momentum (10) using the exact expression for $|{\cal I}|^{2}$, we
run into a non-integrable divergence at the boundary. This divergence is
driven by the fact that the exact Bessel state is not normalizable and
requires a regularization scheme. Following Jentschura:2010ap ;
Jentschura:2011ih ; Ivanov:2011kk ; Karlovets:2012eu , one can normalize the
Bessel state to a large but finite cylindrical volume and render the integrals
finite. However in this case the average final transverse momentum momentum
becomes parametrically suppressed and, at small $b$, it is proportional to $b$
itself, not to $1/b$, as was expected from the pointlike probe semiclassical
considerations. Thus, this computation does not confirm the superkick
phenomenon.

Figure 4: Transverse spreading of the Gaussian state along $z$ axis. The gray
solid line corresponds to the axis of the Bessel state.

The origin of this new discrepancy becomes evident if we consider the
scattering process in coordinate space. The transverse distribution of the
Bessel twisted state is, by construction, $z$-invariant. In contrast, the
transverse distribution of the Gaussian state (12) evolves along the $z$ axis:

$|\psi({\bf r}_{\perp},z)|^{2}\propto\exp\left[-\frac{{\bf
r}_{\perp}^{2}}{\sigma^{2}(z)}\right]\,,\quad\sigma^{2}(z)=\sigma^{2}\left(1+\frac{z^{2}}{L_{\sigma}^{2}}\right)\,,\quad\mbox{where}\quad
L_{\sigma}=\sigma^{2}|p_{2z}|\,.$ (18)

The Gaussian beam stays well focused only over the longitudinal focal length
$|z|\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}L_{\sigma}$ and broadens at larger values of $|z|$, Fig. 4. It
is clear that the superkick phenomenon is expected only when $\sigma(z)$ stays
much smaller than $b$, that is, over a limited $z$ interval. Beyond this $z$
interval, the Gaussian state is unable to locally probe the very strong phase
gradient of the twisted state. When computing $\langle{\bf P}_{\perp}\rangle$
of the final state, we effectively performed the integration over all values
of $z$ up to the size of the normalization volume. The Gaussian state was
spreading in the transverse plane at large $|z|$, but its interaction with the
Bessel twisted state never dies out because the Bessel state is not localized
in the transverse plane. Therefore, the resulting integral is dominated by
very large $z$ values where no superkick can be expected.

We conclude that the absence of superkick which is detailed in Appendix B is
an artifact of the inappropriate setting we used so far. That is, by
considering an exact monochromatic Bessel vs. Gaussian beam collision in the
entire coordinate space, we inadvertently dilute the effect which is expected
to exist only within a short $z$ range.

###  IV.3 Towards the superkick: limiting the $z$-range

In order to remove the artifact and to recover the superkick phenomenon within
the QFT framework, we need to modify the collision setting to make sure that
scattering takes place only where the Gaussian state is well focused. Several
approaches can be used. One is to pass from monochromatic beams to non-
monochromatic, longitudinally localized wave packets. This method is
especially appropriate for interaction of atoms just released from a trap with
a short twisted laser pulse. Another approach is to replace the Bessel twisted
state with Laguerre-Gaussian beams which are normalized in the transverse
plane Karlovets:2020odl . In this way, the twisted state itself will be
subject to defocusing, and the interaction will be saturated by the
overlapping focal spots of the two beams.

These schemes will make the calculations cumbersome and, perhaps, hinder the
essence of the effect. To keep computations clear, we adopt here yet another
prescription. We keep the same exact monochromatic Bessel and Gaussian beams
but introduce an auxiliary regulating function $f(z)$ which switches on the
scattering within the focal length, with $f(z)=1$ at $z=0$, and switches off
for $|z|\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}\ell$, well before the Gaussian state
begins to significantly defocus. For concreteness, we take the following
regulating function:

$f(z)=\exp(-|z|/\ell)\quad\mbox{with}\quad\sigma\ll\ell\ll L_{\sigma}\,.$ (19)

The motivation for the inequality $\sigma\ll\ell$ will become clear below,
while the last condition guarantees that the integration is limited to the
focal length of the beam, that is, the $z$-range where the Gaussian beam stays
focused to its waist size $\sigma(z)\approx\sigma$. The final result—that is,
the value of the superkick and its dependence on the transverse
parameters—should not depend on $\ell$.

The regulating function modifies the longitudinal momentum conservation in
(15). Instead of

$\delta(k_{1z}+k_{2z}-P_{z})=\frac{1}{2\pi}\int
dz\,e^{-i(k_{1z}+k_{2z}-P_{z})z}$

we now have $g(k_{1z}+k_{2z}-P_{z})$ defined as

$g(k_{1z}+k_{2z}-P_{z})=\frac{1}{2\pi}\int
dz\,e^{-i(k_{1z}+k_{2z}-P_{z})z}e^{-|z|/\ell}=\frac{1}{\pi}\frac{\ell}{1+(k_{1z}+k_{2z}-P_{z})^{2}\ell^{2}}\,.$
(20)

As a result, the expression for ${\cal I}$ has one less delta function than in
(15). The integration over $k_{2z}$ is now compensated not by
$\delta(k_{1z}+k_{2z}-P_{z})$ but by $\delta({\bf k}_{2}^{2}-p_{2z}^{2})$,
which leaves the following $\Delta P_{z}$-depending functions:

${\cal I}\propto
g(\tilde{P}_{z})\,e^{-\tilde{P}_{z}^{2}\sigma^{2}/2}\,,\quad\tilde{P}_{z}=\Delta
P_{z}+\sqrt{p_{2z}^{2}-{\bf k}_{2\perp}^{2}}-|p_{2z}|\approx\Delta
P_{z}-\frac{{\bf k}_{2\perp}^{2}}{2|p_{2z}|}\,.$ (21)

The two functions—the regulating and the exponential functions—have different
ranges. The regulating function $g(\tilde{P}_{z})$ stays flat for
$|\tilde{P}_{z}|\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}1/\ell$ and then decreases. Within
this range, the argument of the exponential function stays below
$\sigma^{2}/\ell^{2}\ll 1$, where we used the first relation of (19).
Therefore, the exponential function can be safely replaced by 1 in front of
$g(\tilde{P}_{z})$.

Figure 5: Schematical distribution of $|{\cal I}|^{2}$ on the $(|{\bf
P}_{\perp}|,\Delta P_{z})$ plane upon restricting the $z$ integration to a
distance within the focal length. Small scale oscillations and feature seen in
Fig. 3 are smoothed out upon smearing.

The resulting $\Delta P_{z}$ distribution becomes much wider than before. In
(17), we had $\Delta P_{z}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}1/(\sigma^{2}|p_{2z}|)=1/L_{\sigma}$.
Now, the $\Delta P_{z}$ distribution is governed by $g(\tilde{P}_{z})$ and
extends up to $1/\ell\gg 1/L_{\sigma}$, where we used the second inequality of
(19). This effectively leads to a dramatic smearing of the entire picture of
Fig. 3 which is illustrated in Fig. 5.

The end result is that ${\cal I}$ displays an approximately factorized
dependence on $\Delta P_{z}$ and ${\bf P}_{\perp}$. The $\Delta P_{z}$
distribution comes mostly from $g(\tilde{P}_{z})\approx g(\Delta P_{z})$.
Therefore, when calculating the average final momentum (10), one can factor
both the numerator and the denominator into the transverse and longitudinal
integrals and cancel the longitudinal ones. Thus, the expression for the
average final transverse momentum can be simplified to

$\langle{\bf P}_{\perp}\rangle\approx\frac{\int|{\cal J}|^{2}{\bf
P}_{\perp}d^{2}{\bf P}_{\perp}}{\int|{\cal J}|^{2}\,d^{2}{\bf P}_{\perp}}\,,$
(22)

where

${\cal J}=\frac{1}{2\pi}\int d\varphi_{1}\exp\left[im\varphi_{1}-i({\bf
P}_{\perp}-{\bf k}_{1\perp}){\bf b}_{\perp}-\frac{({\bf P}_{\perp}-{\bf
k}_{1\perp})^{2}\sigma^{2}}{2}\right]\,.$ (23)

This is the expression that will display the superkick phenomenon in the
appropriate kinematic setting and also will demonstrate its limits. We explore
it in detail in the next Section.

Notice that, when defining ${\cal J}$, we assumed that the invariant amplitude
${\cal M}$ depends on ${\bf P}_{\perp}$ very weakly and, therefore, can be
factored out and canceled in (22). This is a very natural assumption for all
non-resonant processes because the range of ${\bf P}_{\perp}$ is very small
compared to the individual final particles’ momenta ${\bf k}^{\prime}_{1}$ and
${\bf k}^{\prime}_{2}$. This assumption may be violated in the sharply
resonant processes such as resonant photon scattering on atoms or ions.
Analysis of these subtleties is delegated to a future work.

##  V Superkick and its limits

###  V.1 Recovering the superkick

Using integrals 3.937 from integrals , we can formally express the integral
${\cal J}$ in (23) as

${\cal J}=e^{im\varphi_{P}}e^{-i{\bf P}_{\perp}{\bf
b}_{\perp}-\frac{(P_{\perp}^{2}+\varkappa^{2})\sigma^{2}}{2}}\left(\frac{P_{\perp}^{2}\sigma^{4}+b^{2}e^{2i(\varphi_{b}-\varphi_{P})}}{b^{2}+P_{\perp}^{2}\sigma^{4}-2bP_{\perp}\sigma^{2}\sin(\varphi_{b}-\varphi_{P})}\right)^{m/2}I_{m}\left(\varkappa\sqrt{P_{\perp}^{2}\sigma^{4}-b^{2}-2ibP_{\perp}\sigma^{2}\cos\varphi_{bP}}\right)\,.$
(24)

Here, $\varphi_{P}$ and $\varphi_{b}$ are the azimuthal angles of ${\bf
P}_{\perp}$ and ${\bf b}$, respectively. Notice that in the limit $b\to 0$ all
functions apart from the very first phase factor $e^{im\varphi_{P}}$ become
real. This illustrates the OAM conservation for head-on, zero impact parameter
collision. However, this expression itself is not very enlightening, so
instead of inserting it in the exact expression (22), we find it instructive
to perform the ${\bf P}_{\perp}$ integration first in the numerator and
denominator of this expression. The denominator becomes

$\displaystyle\int|{\cal J}|^{2}d^{2}{\bf P}_{\perp}=\int d^{2}{\bf
P}_{\perp}\frac{1}{(2\pi)^{2}}\int
d\varphi_{1}d\varphi^{\prime}_{1}e^{im(\varphi_{1}-\varphi^{\prime}_{1})+i({\bf
k}_{1\perp}-{\bf k}^{\prime}_{1\perp}){\bf b}_{\perp}}\exp\left\\{-\left[({\bf
P}_{\perp}-{\bf k}_{1\perp})^{2}+({\bf P}_{\perp}-{\bf
k}^{\prime}_{1\perp})^{2}\right]\frac{\sigma^{2}}{2}\right\\}$
$\displaystyle=\frac{1}{(2\pi)^{2}}\int
d\varphi_{1}d\varphi^{\prime}_{1}e^{im(\varphi_{1}-\varphi^{\prime}_{1})+i({\bf
k}_{1\perp}-{\bf k}^{\prime}_{1\perp}){\bf b}_{\perp}}\int d^{2}{\bf
P}_{\perp}\exp\left[-\left({\bf P}_{\perp}-\frac{{\bf k}_{1\perp}+{\bf
k}^{\prime}_{1\perp}}{2}\right)^{2}\sigma^{2}-\frac{({\bf k}_{1\perp}-{\bf
k}^{\prime}_{1\perp})^{2}\sigma^{2}}{4}\right]$
$\displaystyle=\frac{\pi}{\sigma^{2}}\cdot\frac{1}{(2\pi)^{2}}\int
d\varphi_{1}d\varphi^{\prime}_{1}e^{im(\varphi_{1}-\varphi^{\prime}_{1})+i({\bf
k}_{1\perp}-{\bf k}^{\prime}_{1\perp}){\bf b}_{\perp}}\exp\left[-\frac{({\bf
k}_{1\perp}-{\bf k}^{\prime}_{1\perp})^{2}\sigma^{2}}{4}\right]\,.$ (25)

To recover the superkick regime, we consider $\sigma$ to be much smaller than
$b$ and $1/\varkappa$. Then, the last factor here can be omitted as it always
stays very close to 1. As a result, the integrations with respect to
$\varphi_{1}$ and $\varphi^{\prime}_{1}$ decouple:

$\int|{\cal J}|^{2}d^{2}{\bf
P}_{\perp}=\frac{\pi}{\sigma^{2}}\cdot\frac{1}{2\pi}\int
d\varphi_{1}e^{im\varphi_{1}+i{\bf k}_{1\perp}{\bf
b}_{\perp}}\cdot\frac{1}{2\pi}\int
d\varphi^{\prime}_{1}e^{-im\varphi^{\prime}_{1}-i{\bf k}^{\prime}_{1\perp}{\bf
b}_{\perp}}=\frac{\pi}{\sigma^{2}}\cdot\left[J_{m}(\varkappa b)\right]^{2}\,.$
(26)

For the numerator of (22), we have an additional ${\bf P}_{\perp}$ inserted.
Performing the ${\bf P}_{\perp}$ integral as before, we get

$\int d^{2}{\bf P}_{\perp}\,{\bf P}_{\perp}\,\exp\left[-\left({\bf
P}_{\perp}-\frac{{\bf k}_{1\perp}+{\bf
k}^{\prime}_{1\perp}}{2}\right)^{2}\sigma^{2}\right]=\frac{\pi}{\sigma^{2}}\cdot\frac{{\bf
k}_{1\perp}+{\bf k}^{\prime}_{1\perp}}{2}\,.$

Instead of Eq. (26), we now have

$\int|{\cal J}|^{2}{\bf P}_{\perp}d^{2}{\bf
P}_{\perp}=\frac{\pi}{\sigma^{2}}\cdot\int
d\varphi_{1}d\varphi^{\prime}_{1}\frac{{\bf k}_{1\perp}+{\bf
k}^{\prime}_{1\perp}}{2}e^{im(\varphi_{1}-\varphi^{\prime}_{1})+i({\bf
k}_{1\perp}-{\bf k}^{\prime}_{1\perp}){\bf b}_{\perp}}\,.$ (27)

Let us define the unit vectors ${\bf e}_{x}$ and ${\bf e}_{y}$ along axes $x$
and $y$, with the axis $x$ chosen along the impact parameter vector ${\bf
b}_{\perp}$ as in Fig. 1. Then ${\bf
k}_{1\perp}=\cos(\varphi_{1}-\varphi_{b}){\bf
e}_{x}+\sin(\varphi_{1}-\varphi_{b}){\bf e}_{y}$. Substituting this expression
into (27) and using the integrals

$\displaystyle\frac{1}{2\pi}\int d\varphi\,e^{im\varphi+i\varkappa
b\cos\varphi}\cos\varphi$ $\displaystyle=$
$\displaystyle\frac{i^{m+1}}{2}[J_{m+1}(\varkappa b)-J_{m-1}(\varkappa b)]\,,$
$\displaystyle\frac{1}{2\pi}\int d\varphi\,e^{im\varphi+i\varkappa
b\cos\varphi}\sin\varphi$ $\displaystyle=$
$\displaystyle\frac{i^{m}}{2}[J_{m+1}(\varkappa b)+J_{m-1}(\varkappa
b)]=i^{m}\frac{m}{\varkappa b}J_{m}(\varkappa b)\,,$ (28)

we obtain that only the ${\bf e}_{y}$ component survives:

$\int|{\cal J}|^{2}{\bf P}_{\perp}d^{2}{\bf
P}_{\perp}=\frac{\pi}{\sigma^{2}}\cdot\frac{m}{b}[J_{m}(\varkappa
b)]^{2}\,{\bf e}_{y}\,.$ (29)

As a result, the transverse momentum of the final state, within the
approximation $\sigma\ll b,1/\varkappa$, is given by

$\langle{\bf P}_{\perp}\rangle=\frac{m}{b}\,{\bf e}_{y}\,.$ (30)

This expression is valid for any value of $\varkappa b$. Thus, we recovered
the superkick phenomenon in the full quantum field theoretic computation of
the Bessel twisted state collision with compact a Gaussian wave packet
provided its size $\sigma\ll b,1/\varkappa$.

###  V.2 Exploring the limits of the superkick phenomenon

Figure 6: Transverse momentum of the final state expressed in units of
$\varkappa$ as a function of $\varkappa b$ up to $\varkappa b=1$ (the upper
plot) and up to $\varkappa b=5$ (the lower plot) for various localization
parameters $\sigma$. The value of $m=5$. In each plot, the uppermost black
line shows the superkick phenomenon in the limit $\sigma\to 0$.

In order to explore how the superkick phenomenon emerges and disappears for
finite $\sigma$, we evaluated the transverse momentum $\langle{\bf
P}_{\perp}\rangle$ numerically using the exact expression for the integral
${\cal J}$. The results are shown in Fig. 6 for $m=5$ and several values of
$\sigma$, from very small, $\sigma\varkappa=0.04$ to moderately small
$\sigma\varkappa=0.5$. The uppermost black curve shows the expected superkick
behavior $m/b=\varkappa\cdot(m/\varkappa b)$.

We see that for small $\sigma$, with $b\gg\sigma$, the curves nicely approach
the black superkick regime. However, starting from $\sigma\approx
0.5/\varkappa$ (the lowest curve in Fig. 6), the superkick behavior does not
resemble the semiclassical Eq. (1): the $b$ dependence is rather flat. At
small $b$, it still produces a mild superkick, as $\langle{\bf
p}_{\perp}\rangle$ is larger than $\varkappa$, but the curve does not display
the pronounced $1/b$ enhancement characteristic to the semiclassical
expectation.

At small values of $\sigma$, when the $1/b$ dependence is clear, we can
observe that the maximal value of $\langle{\bf P}_{\perp}\rangle$ which is
attained at $b\approx\sigma$, lies significantly below the semiclassical
curve. This factor is about 3 for the example shown, $m=5$, and slowly grows
for larger $m$, reaching about 10 for $m=50$. One can also notice that, in
order for $\langle{\bf P}_{\perp}\rangle$ to be close (say, within 20%) to the
semiclassical superkick expectation, the value of $\sigma$ must be smaller
than $b/m$, meaning that the compact Gaussian wave packet must contain
sufficiently high momenta for a given $m$ and $b$.

##  VI Relation with previous works

Our qualitative and quantitative observations are in general agreement with
the findings of barnett2013superkick . Thus, although the approach of
barnett2013superkick  is not fully quantum, we confirm that it is adequate for
the problem of a localized atom in the vicinity of an optical vortex. These
results highlight the necessity of treating the localized particle as a
compact wave packet rather than a classical particle which, by definition, is
pointlike and stays pointlike. Without (transverse) wave packets and their
localization scale $\sigma$, neither can the superkick phenomenon be properly
understood nor its observability limits be detected.

We showed that, in order for the superkick to be close to the semiclassical
prediction, $\sigma$ must be smaller than $b/m$, which in turn must be smaller
than $\varkappa$. However, the smaller $\sigma$, the faster is spreading of
the Gaussian wave packet and the shorter is the focal length $L_{\sigma}$.
This leads to a new requirement for the scattering process occurring in free
space: not only must it involve a particle localized to a fraction of
transverse ring radius of the twisted beam $m/\varkappa$ and placed with high
accuracy close to the beam axis, but also the interaction itself must be
longitudinally localized to within the focal length $L_{\sigma}$, or
alternatively, occur over a very short time.

Satisfying these conditions may be feasible in experiments with trapped atoms
or ions placed in the vicinity of an optical vortex. Indeed, trapped atom or
ion wavefunction can be localized to submicron (in fact sub-100 nm) sizes and
manipulated with comparable accuracy. For example, the recent paper
Drechsler:2021  reports optical superresolution sensing of an individual
trapped 40Ca+ ion, whose wavefunction size was measured to be about 40 nm, in
agreement with a less direct sideband spectroscopic evaluation. At the same
time, the transverse size of the first Bessel ring $1/\varkappa$, which is
larger than the optical wavelength, can reach microns. When released from the
trap, the wave packet begins to spread, but the spreading time may be large
compared to the interaction with a short twisted light pulse. For example, a
heavy atom with $m=100$ GeV localized in a trap to a region of $\Delta x=10$
nm size will begin to spread when released and double its extent on the
timescale $\tau\approx m(\Delta x)^{2}/\hbar\sim 100$ ns. Thus, nanosecond-
long light pulses effectively probe a frozen wavepacket.

Observing superkick and related effects in nuclear and particle physics
processes such as deuteron photodisintegration or $\Delta$ baryon
photoproduction by twisted gamma rays as suggested in Afanasev:2020nur ;
Afanasev:2021fda  is very challenging. For that, one would need not only to
produce twisted gamma beam but also to focus it to sub-picometer or even
femtometer focal spots, localize the target proton or deuteron wavefunction to
a femtometer-scale size and manipulate the beam and the target with comparable
precision. Besides, a free evolving proton wavefunction of the initial
localization size of the order of femtometer will rapidly expand doubling its
size over typical nuclear times scale of $\sim 10^{-22}$ s. Thus, one needs
either to arrange for the hard twisted photon to interact with the proton
during this short time, or to find a way to keep the proton localized.
Achieving this level of control is challenging and requires dedicated
experimental efforts.

##  VII Conclusions and outlook

In summary, we rederived within the QFT framework the superkick phenomenon
predicted in barnett2013superkick . Since in our calculations we treated both
initial particles—the vortex state and the compact Gaussian state—as freely
propagating wavepackets, we included the full energy-momentum conservation
law, which led to non-trivial effects when inserted between the initial and
final states. On the way to the final result, we encountered several apparent
paradoxes, which we resolved with qualitative insights and quantitative
calculations. We believe that this part of our work, by itself, is a valuable
result as it clarifies several aspects of vortex states scattering which may
at first seem puzzling.

Concerning the superkick phenomenon, we stressed that its observability relies
on a hierarchy of several length scales, see Eqs. (14) and (19), and we
explored the roles of these scales. Our QFT results confirm the interpretation
and dependences of barnett2013superkick . In particular, they highlight the
critical role of the wave packet localization parameter $\sigma$. In the
semiclassical language of pointlike particles, without taking into account the
transverse localization and evolution of the wave packets, the superkick
cannot be properly understood.

The requirements on $\sigma$ and $b$ needed for observation of the superkick
and other related phenomena are readily met for atoms or ions interacting with
optical vortex, as it was already noted in previous works on the subject.
However we remarked that observing similar effects in the nuclear and high-
energy physics domains as recently proposed and discussed in Afanasev:2020nur
; Afanasev:2021fda  place severe requirements on target localization and
alignment, which are not yet achievable with the present-day technology.

With our results and discussion, we believe that the effect of superkick is
not as surprising as it may originally appear. Still, there remains room for
improvement. Here, we used exact monochromatic Bessel beams and, running into
an artifact, devised a somewhat artificial scheme for limiting the
longitudinal range of interaction. It would be interesting to see whether more
realistic schemes such as Laguerre-Gaussian vs Gaussian beam collision
Karlovets:2020odl  which are more computationally heavy but free from
singularities lead to the same results. Once a more realistic treatment is
constructed, one can turn to specific processes and investigate, in
particular, their energy behavior and the energy threshold shift proposed in
Afanasev:2020nur ; Afanasev:2021fda .

## Acknowledgments

We thank Pengcheng Zhao and Liping Zou for many stimulating discussions and
Dmitry Karlovets for valuable comments. This work was supported by the
National Natural Science Foundation of China (Grant No. 11975320) and the
Fundamental Research Funds for the Central Universities, Sun Yat-sen
University.

##  Appendix A The two-particle phase space

The expression (6) for a generic $2\to 2$ scattering cross section for
monochromatic but nonplane-wave initial states involves the two-particle phase
space without the three-momentum constraints:

$d\Gamma^{\prime}=\delta(E_{0}-E_{1}^{\prime}-E_{2}^{\prime})\,\frac{d^{3}{\bf
k}^{\prime}_{1}}{2E^{\prime}_{1}}\frac{d^{3}{\bf
k}^{\prime}_{2}}{2E^{\prime}_{2}}\,,$ (31)

where $E_{0}=E_{1}+E_{2}$ is fixed. The energy delta function can be used to
integrate one of the variables leading to a five-dimensional phase space
distribution. In this section, we find a convenient way to perform
it.111Technically, this expression is similar to the three-particle phase
space in a usual plane-wave collision, but we will not rely on this analogy.

In plane-wave scattering processes, when analyzing final particle
distributions, one is free to select a convenient reference frame where the
analysis simplifies. In our case, however, we cannot perform any boost since
it will spoil the monochromaticity of the colliding beams. We are constrained
to work in a generic frame; the only simplification is related to the natural
choice of axis $z$ and the transverse momenta orthogonal to it. Switching from
$d^{3}{\bf k}^{\prime}_{1}d^{3}{\bf k}^{\prime}_{2}$ to $d^{2}{\bf
k}^{\prime}_{1\perp}dk^{\prime}_{1z}\,d^{3}{\bf P}$, where ${\bf P}={\bf
k}^{\prime}_{1}+{\bf k}^{\prime}_{2}$, we perform the $dk^{\prime}_{1z}$
integration to remove the delta-function while keeping ${\bf
k}^{\prime}_{1\perp}$, ${\bf P}_{\perp}$, and $P_{z}$ as independent
variables.

If $m_{1}$ and $m_{2}$ are the masses of the two final particles, we can
define their “transverse masses” as

$m_{1\perp}^{2}=m_{1}^{2}+{\bf k}_{1\perp}^{\prime 2}=E_{1}^{\prime
2}-k_{1z}^{\prime 2}\,,\quad m_{2\perp}^{2}=m_{2}^{2}+({\bf P}_{\perp}-{\bf
k}^{\prime}_{1\perp})^{2}=E_{2}^{\prime 2}-(P_{z}-k^{\prime}_{1z})^{2}\,.$
(32)

In a similar way, we define the invariant mass of the final two-particle
system defined as $M_{inv}^{2}=E_{0}^{2}-{\bf P}^{2}$ and denote its
“transverse mass”

$M_{\perp}^{2}=M_{inv}^{2}+{\bf P}_{\perp}^{2}=E_{0}^{2}-P_{z}^{2}\,.$ (33)

The energy delta-function can be now expressed as a function of
$k^{\prime}_{1z}$:

$\delta(E_{0}-E_{1}^{\prime}-E_{2}^{\prime})=\delta(f(k^{\prime}_{1z}))\,,\quad
f(k^{\prime}_{1z})=E_{0}-\sqrt{m_{1\perp}^{2}+k_{1z}^{\prime
2}}-\sqrt{m_{2\perp}^{2}+(P_{z}-k^{\prime}_{1z})^{2}}$

and integrated out as

$\int
dk^{\prime}_{1z}\frac{\delta(E_{0}-E_{1}^{\prime}-E_{2}^{\prime})}{E_{1}^{\prime}E_{2}^{\prime}}=\int
dk^{\prime}_{1z}\frac{\delta(f(k^{\prime}_{1z}))}{E_{1}^{\prime}E_{2}^{\prime}}=\sum_{k^{\prime}_{1z}=k^{*}_{1z}}\frac{1}{|df/dk^{\prime}_{1z}|E_{1}^{\prime}E_{2}^{\prime}}\,,$
(34)

where $k^{*}_{1z}$ are the solutions of the equation $f(k^{\prime}_{1z})=0$.
The solutions of this equations can be expressed as

$\displaystyle k^{*}_{1z}$ $\displaystyle=$
$\displaystyle\frac{1}{2}P_{z}\left(1+\frac{m_{1\perp}^{2}-m_{2\perp}^{2}}{M_{\perp}^{2}}\right)\pm\frac{E_{0}}{2M_{\perp}^{2}}\sqrt{\lambda(M_{\perp}^{2},m_{1\perp}^{2},m_{2\perp}^{2})}\,,$
$\displaystyle P_{z}-k^{*}_{1z}$ $\displaystyle=$
$\displaystyle\frac{1}{2}P_{z}\left(1-\frac{m_{1\perp}^{2}-m_{2\perp}^{2}}{M_{\perp}^{2}}\right)\mp\frac{E_{0}}{2M_{\perp}^{2}}\sqrt{\lambda(M_{\perp}^{2},m_{1\perp}^{2},m_{2\perp}^{2})}\,,$
(35)

where

$\lambda(M_{\perp}^{2},m_{1\perp}^{2},m_{2\perp}^{2})=M_{\perp}^{4}+m_{1\perp}^{4}+m_{2\perp}^{4}-2M_{\perp}^{2}m_{1\perp}^{2}-2M_{\perp}^{2}m_{2\perp}^{4}-2m_{1\perp}^{2}m_{2\perp}^{2}\,.$
(36)

This solution exists only if the transverse masses satisfy the following
inequality:

$M_{\perp}\geq m_{1\perp}+m_{2\perp}\,,$ (37)

which places a constraint on the possible values of ${\bf
k}^{\prime}_{1\perp}$, ${\bf P}_{\perp}$, and $P_{z}$. The value of
$k^{\prime}_{1z}=k^{*}_{1z}$ corresponds to the following energies of the
final particles:

$\displaystyle E^{\prime}_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}E_{0}\left(1+\frac{m_{1\perp}^{2}-m_{2\perp}^{2}}{M_{\perp}^{2}}\right)\pm\frac{P_{z}}{2M_{\perp}^{2}}\sqrt{\lambda(M_{\perp}^{2},m_{1\perp}^{2},m_{2\perp}^{2})}\,,$
$\displaystyle E^{\prime}_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}E_{0}\left(1-\frac{m_{1\perp}^{2}-m_{2\perp}^{2}}{M_{\perp}^{2}}\right)\mp\frac{P_{z}}{2M_{\perp}^{2}}\sqrt{\lambda(M_{\perp}^{2},m_{1\perp}^{2},m_{2\perp}^{2})}\,,$

The derivative $|df/dk^{\prime}_{1z}|$ entering (34) can be expressed as
$|k^{*}_{1z}/E^{\prime}_{1}-(P_{z}-k^{*}_{1z})/E^{\prime}_{2}|$ so that

$\left|\frac{df}{dk^{\prime}_{1z}}\right|E^{\prime}_{1}E^{\prime}_{2}=|k^{*}_{1z}E_{0}-P_{z}E^{\prime}_{1}|=\frac{1}{2}\sqrt{\lambda(M_{\perp}^{2},m_{1\perp}^{2},m_{2\perp}^{2})}\,.$
(39)

This result allows us to finally represent the final particle phase space as

$d\Gamma^{\prime}=\sum_{k^{*}_{1z}}\frac{d^{2}{\bf
k}^{\prime}_{1\perp}\,d^{2}{\bf
P}_{\perp}\,dP_{z}}{2\sqrt{\lambda(M_{\perp}^{2},m_{1\perp}^{2},m_{2\perp}^{2})}}\,.$
(40)

Here, we keep the summation over the two solutions for $k^{*}_{1z}$ because
the values of the invariant amplitude ${\cal M}$, and consequently ${\cal I}$,
can differ for the two final state momenta.

Let us understand the behavior of the $1/\sqrt{\lambda}$ factor within the
available phase space. In our kinematic situation, we have in mind a high-
energy collision with large $M_{inv}^{2}$ and small ${\bf P}_{\perp}^{2}$.
Thus, when studying ${\bf P}_{\perp}$ distribution, we can safely assume that
$M_{\perp}^{2}$ stays almost constant and approximately equal to
$M_{inv}^{2}\approx E_{0}^{2}-P_{z0}^{2}$, where $P_{z0}=p_{1z}+p_{2z}$. The
values of $m_{1\perp}^{2}$ and $m_{2\perp}^{2}$ depend on the final momenta,
which are large, $|{\bf k}^{\prime}_{1\perp}|\gg|{\bf P}_{\perp}|$. In
particular, in the vast majority of the phase space we have
$m_{2\perp}^{2}=m_{2}^{2}+({\bf P}_{\perp}-{\bf
k}^{\prime}_{1\perp})^{2}\approx m_{2}^{2}+{\bf k}_{1\perp}^{\prime 2}$, so
that $\lambda$ shows almost no dependence on ${\bf P}_{\perp}$ or $P_{z}$ but
is a function of ${\bf k}^{\prime}_{1\perp}$ only. We conclude that the final
phase space can be approximately factorized:

$d\Gamma^{\prime}=\mbox{relative motion}\times d^{2}{\bf
P}_{\perp}\,dP_{z}\,.$ (41)

This factorized form will allows us to evaluate the total transverse momentum
of the final state for a generic value of ${\bf k}^{\prime}_{1\perp}$.

##  Appendix B Absence of the superkick due to non-decoupling of the Bessel
and Gaussian states

Here, we continue calculating the integral (15) for the exact Bessel and
Gaussian state scattering:

$\displaystyle{\cal I}$ $\displaystyle=$ $\displaystyle\int d^{3}{\bf
k}_{1}d^{3}{\bf k}_{2}\,\delta(k_{1z}-p_{1z})\delta(|{\bf
k}_{1\perp}|-\varkappa)e^{im\varphi_{1}}\cdot\,e^{-\frac{({\bf k}_{2}-{\bf
p}_{2})^{2}\sigma^{2}}{2}}\delta({\bf k}_{2}^{2}+M^{2}-E_{2}^{2})\,e^{-i{\bf
k}_{2\perp}{\bf b}_{\perp}}\delta^{(3)}({\bf k}_{1}+{\bf k}_{2}-{\bf
P})\cdot{\cal M}\,.$

Five out of six delta functions can be taken care of by the integrations
leading to

$\displaystyle{\cal I}$ $\displaystyle=$ $\displaystyle\varkappa\int
d\varphi_{1}\exp\left[im\varphi_{1}-i({\bf P}_{\perp}-{\bf k}_{1\perp}){\bf
b}_{\perp}-\frac{({\bf P}_{\perp}-{\bf
k}_{1\perp})^{2}\sigma^{2}}{2}-\frac{(P_{z}-p_{1z}-p_{2z})^{2}\sigma^{2}}{2}\right]$
(42) $\displaystyle\qquad\times\ \delta\left[({\bf P}_{\perp}-{\bf
k}_{1\perp})^{2}+(P_{z}-p_{1z})^{2}-p_{2z}^{2}\right]\cdot{\cal M}\,.$

Here, the only remaining variable is the azimuthal angle of ${\bf
k}_{1\perp}$, with $|{\bf k}_{1\perp}|=\varkappa$ already fixed. Also, in the
last delta-function we replaced $E_{2}^{2}-M^{2}$ with $p_{2z}^{2}$. To
perform the remaining integration, we denote the azimuthal angles of ${\bf
P}_{\perp}$ and ${\bf b}$ as $\varphi_{P}$ and $\varphi_{b}$, respectively,
and their difference as $\varphi_{bP}=\varphi_{b}-\varphi_{P}$. Then we
introduce the shifted variable $\tilde{\varphi}_{1}=\varphi_{1}-\varphi_{P}$
and rewrite (42) as

${\cal I}=\varkappa\,e^{im\varphi_{P}}e^{-i{\bf P}_{\perp}{\bf
b}_{\perp}}e^{-\sigma^{2}|p_{2z}|(P_{z}-p_{1z}-p_{2z})}\int
d\tilde{\varphi}_{1}e^{im\tilde{\varphi}_{1}+i\varkappa
b\cos(\tilde{\varphi}_{1}-\varphi_{bP})}\delta(a-c\cos\tilde{\varphi}_{1})\cdot{\cal
M}\,,$ (43)

where

$a={\bf P}_{\perp}^{2}+\varkappa^{2}+(P_{z}-p_{1z})^{2}-p_{2z}^{2}\,,\quad
c=2|{\bf P}_{\perp}|\varkappa\,.$ (44)

The integral is nonzero only when $|a|<c$, which places constraints on
possible values of ${\bf P}_{\perp}$ and $P_{z}$. The delta function fixes the
angle $\tilde{\varphi}_{1}=\pm\varphi_{*}$, where $\cos\varphi_{*}=a/c$. The
value of the invariant amplitude ${\cal M}$ can, in principle, be different at
these two values of $\tilde{\varphi}_{1}$, which may lead to interesting
effects similar to those predicted in double-twisted electron scattering
Ivanov:2016oue . However since we are interested here in kinematic effects, we
assume that ${\cal M}$ takes the same value at these two angles
$\tilde{\varphi}_{1}$. This allows us to simplify the integral as

${\cal I}=e^{im\varphi_{P}}e^{-ib\cos\varphi_{bP}(|{\bf P}_{\perp}|-a/2|{\bf
P}_{\perp}|)}e^{-\sigma^{2}|p_{2z}|(P_{z}-p_{1z}-p_{2z})}\cdot\frac{\cal
M}{|{\bf P}_{\perp}|\sin\varphi_{*}}\cos(m\varphi_{*}+\varkappa
b\sin\varphi_{*}\sin\varphi_{bP})\,.$ (45)

This expression represents the outgoing wavefunction of the center of mass
motion of the final state $\Psi_{f}({\bf P})$ in momentum representation. Its
modulus squared

$|{\cal I}|^{2}=e^{-2\sigma^{2}|p_{2z}|\Delta P_{z}}\cdot\frac{|{\cal
M}|^{2}}{{\bf
P}_{\perp}^{2}\sin^{2}\varphi_{*}}\cos^{2}(m\varphi_{*}+\varkappa
b\sin\varphi_{*}\sin\varphi_{bP})$ (46)

defines the differential cross section (6).

Figure 7: The crescent-like region on the $(|{\bf P}_{\perp}|,\Delta P_{z})$
plane defined by the kinematical constraints in Eq. (48). The inset shows the
zoomed region near the origin. The plot corresponds to
$\varkappa=0.1|p_{2z}|$.

Let us understand the kinematic constraints on $P_{z}$ and ${\bf P}_{\perp}$
which arise from the restriction $|a|<c$ in (44). To this end, we use $\Delta
P_{z}=P_{z}-p_{1z}+|p_{2z}|$ defined in Eq. (8), which shows how the value of
$P_{z}$ differs from the paraxial limit value, and rewrite

$(P_{z}-p_{1z})^{2}-p_{2z}^{2}=\Delta P_{z}(2p_{2z}+\Delta P_{z})\,.$ (47)

The two conditions $a<c$ and $-a<c$ take the following form:

$(|{\bf P}_{\perp}|-\varkappa)^{2}<2|p_{2z}|\Delta P_{z}-(\Delta
P_{z})^{2}<(|{\bf P}_{\perp}|+\varkappa)^{2}\,,$ (48)

from which one immediately sees that $\Delta P_{z}\geq 0$. These two
conditions cut out a crescent-like region on the $(\Delta P_{z},|{\bf
P}_{\perp}|)$ plane bounded by two arcs as shown in Fig. 7 (this plot
corresponds to $\varkappa=0.1|p_{2z}|$). It is inside this region that the
function $|{\cal I}|^{2}$ of (46) is defined. The maximal value of $\Delta
P_{z}$ is $2|p_{2z}|$, the maximal value of $|{\bf P}_{\perp}|$ is
$|p_{2z}|+\varkappa$. In the region of small $\Delta P_{z}$ and $|{\bf
P}_{\perp}|$, the arcs do not reach the origin but meet at the $|{\bf
P}_{\perp}|=0$ axis at the value $\Delta P_{z}\approx\varkappa^{2}/2|p_{2z}|$,
shown in the inset of Fig. 7. The larger arc touches the $\Delta P_{z}=0$ axis
at $|{\bf P}_{\perp}|=\varkappa$.

The cross section is not distributed homogeneously over this crescent-like
region. The first exponential factor in $|{\cal I}|^{2}$ in Eq. (46) together
ith the strong inequality $\sigma|p_{2z}|\gg 1$ make it clear that most of the
cross section comes from the origin of small $\Delta P_{z}$. In this region,
we can neglect $(\Delta P_{z})^{2}$ when compared to $2|p_{2z}|\Delta P_{z}$
and replace the exact inequalities of Eq. (48) with the approximation Eq. (16)
given in the main text. The typical ranges of $\Delta P_{z}$ and $|{\bf
P}_{\perp}|$ are expected to be

$\Delta P_{z}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}\frac{1}{\sigma^{2}|p_{2z}|}\,,\quad|{\bf
P}_{\perp}|\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}\frac{1}{\sigma}\,,\quad\Delta
P_{z}\ll|{\bf P}_{\perp}|\,.$ (49)

The last relation confirms the expectation that the total momentum of the
final state is predominantly transverse, and that, for a well-localized
Gaussian beam with $\sigma\ll 1/\varkappa$, it is mostly driven by the
transverse momenta of the initial compact Gaussian state, not by the
transverse momentum of the Bessel twisted state.

Inside this small-$\Delta P_{z}$, small-$|{\bf P}_{\perp}|$ region, the cross
section exhibits strong variations. As we pick up a $|{\bf P}_{\perp}|$ value
and scan the allowed region horizontally by varying $\Delta P_{z}$, the value
of $\varphi_{*}$ varies from 0 to $\pi$. So, at the boundaries, we expect to
observe endpoint divergences $\propto 1/\sin^{2}\varphi_{*}$, while inside
this region we expect to see the oscillations of the cosine squared.

Next, we need to compute the average transverse momentum of the final state
via (10). Let us focus on the integration over $\Delta P_{z}$ for any given
${\bf P}_{\perp}$. Since

$d\Delta P_{z}\approx\sin\varphi_{*}d\varphi_{*}\,\frac{|{\bf
P}_{\perp}|\varkappa}{|p_{2z}|}\,,$

we see that the integration over $\Delta P_{z}$ diverges at the boundaries:

$\int_{0}^{\pi}\frac{d\varphi_{*}}{\sin\varphi_{*}}\cos^{2}(m\varphi_{*}+\varkappa
b\sin\varphi_{*}\sin\varphi_{bP})\,.$ (50)

is undefined. This divergence is omnipresent in all scattering processes with
exact Bessel twisted states Jentschura:2010ap ; Jentschura:2011ih ;
Ivanov:2011kk ; Karlovets:2012eu  and is driven by the fact that such states
are not normalizable. To make the integrals finite, one must define a
regularization scheme. One possibility is to normalize a Bessel twisted state
in a large but finite cylindrical volume of length $L$ and radius $R$
Jentschura:2010ap . Another option is to assume that the twisted state is not
an exact Bessel beam with fixes $\varkappa$ but is represented by some
distribution over a small range of $\varkappa$ Ivanov:2011bv . Whatever
prescription for regularizing the divergent $\int
d\varphi_{*}/\sin\varphi_{*}$ integration, we obtain a finite result. Even if
it is enhanced by a large logarithm, it will be the same in the numerator and
denominator of (10) and the potentially large factors cancel.

We conclude from this discussion that, via a suitable regularization
procedure, it is possible to make the $P_{z}$ integrals finite, with the
largest contributions coming from the regions near the boundaries, that is,
where $\varphi_{*}$ is close to 0 or $\pi$.

Next, we switch to the transverse momentum integration $d^{2}{\bf
P}_{\perp}=d{\bf P}_{\perp}^{2}d\varphi_{P}/2$. We remind the reader that our
goal is to verify the phenomenon of superkick. The key expression is

$\int d\varphi_{P}{\bf P}_{\perp}\cos^{2}(m\varphi_{*}+\varkappa
b\sin\varphi_{*}\sin\varphi_{bP})$ (51)

which we need to analyze for $\varkappa b\ll 1$. Let us select axis $x$ along
${\bf b}_{\perp}$ as in Fig. 1. Then $\varphi_{bP}=\varphi_{P}$ and ${\bf
P}_{\perp}=|{\bf P}_{\perp}|(\cos\varphi_{P},\,\sin\varphi_{P})$. We see that
only $P_{y}$ gives a nonvanishing result, which, at $\varkappa b\ll 1$, can be
written as

$\frac{1}{2}|{\bf P}_{\perp}|\int
d\varphi_{P}\sin\varphi_{P}\left[-\sin(2m\varphi_{*})\cdot 2\varkappa
b\sin\varphi_{*}\sin\varphi_{P}\right]=-\pi|{\bf P}_{\perp}|\varkappa
b\cdot\sin(2m\varphi_{*})\sin\varphi_{*}\,.$ (52)

This result does not confirm the superkick phenomenon. We see that the
transverse momentum transfer is proportional to $b$, not $1/b$, as expected
from the semiclassical analysis of the pointlike probe recoil. Moreover, its
value is additionally suppressed by the two sine factors computed for
$\varphi_{*}$ near $0$ or $\pi$.

Thus, we seem to run into yet another paradox: the exact QFT treatment of the
Gaussian vs. Bessel beam scattering fails to reproduce the superkick
phenomenon obtained with semiclassical considerations.

In the main text, we show that this conclusion is an artifact of the
inappropriate setting we just used and demonstrate how the problem can be
cured.

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