#  Searches for Large Extra Dimensions at the Tevatron

Vyacheslav Krutelyov1 for the CDF and the DØ Collaborations
1- University of California at Santa Barbara - Dept of Physics
Santa Barbara CA 93106-9530 - U.S.A.
Work supported by the U.S. Department of Energy under contract No. DE-
AC02-07CH11359.

###### Abstract

The presence of extra dimensions can be probed in high energy collisions via
the production or exchange of gravitons. The former corresponds to signatures
with missing energy while the latter corresponds to modifications of the final
state spectra. Here I review results of analyses performed by the CDF and DØ
Collaborations on $p\overline{p}$ collisions at
$\sqrt{s}=1.96~{}\mathrm{Te\kern-1.00006ptV}$ in signatures sensitive to large
extra dimensions. These include analyses of
$\gamma+\mbox{${E\\!\\!\\!\\!/_{T}}$}$ and
$\mathrm{jet}+\mbox{${E\\!\\!\\!\\!/_{T}}$}$ as signatures of graviton
production as well as analyses of dilepton and diboson final states sensitive
to graviton exchange.

##  1 Introduction

The standard model of particle physics (SM) does not account for gravitational
interactions. A number of theoretical models beyond the SM naturally include
gravity and require the existence of extra dimensions where only gravity
propagates freely. In particular, a model proposed in Ref. [1] assumes there
are large extra dimensions (LED) compactified on a sub-millimeter scale $R$.
In this model gravity can become strong at the $\mathrm{Te\kern-1.00006ptV}$
scale, which effectively solves the hierarchy problem present in the SM. For
$n$ extra dimensions the four-dimensional Planck mass $M_{\mathrm{Pl}}$ is
related to the $4+n$-dimensional fundamental scale $M_{D}$ by
$M_{\mathrm{Pl}}^{2}=8\pi M_{D}^{2}(M_{D}R)^{n}$. For $M_{D}$ in the
$\mathrm{Te\kern-1.00006ptV}$ range $R$ is of the order of 0.1 $\mathrm{mm}$
(10 $\mathrm{fm}$) for $n=2$ ($6$). To solve the hierarchy problem $M_{D}$
should not be too large, which rules out $n=1$. Tests of the Newton’s law
constrain $R$ to be below $37~{}\mathrm{\mu m}$ for $n=2$ [2]. Other
constraints come from astrophysics (up to $n=3$) and from high energy
colliders [2].

The cross sections of the LED processes depend on $M_{D}$ and can be at levels
detectable at the Tevatron [3]. In $p\overline{p}$ collisions the graviton can
be produced in $q\overline{q}\to gG$, $qg\to qG$, or $gg\to gG$ corresponding
to a $\mathrm{jet}+\mbox{${E\\!\\!\\!\\!/_{T}}$}$ final state, and
$q\overline{q}\to\gamma G$ corresponding to a
$\gamma+\mbox{${E\\!\\!\\!\\!/_{T}}$}$ detector signature [4]. Graviton
exchange can be studied in a range of $2\to 2$ processes with the best
sensitivity in final states with two leptons, photons, or $Z$ bosons.

In this review [5] I first describe searches for graviton production in the
$\gamma+\mbox{${E\\!\\!\\!\\!/_{T}}$}$ and
$\mathrm{jet}+\mbox{${E\\!\\!\\!\\!/_{T}}$}$ signatures reported by CDF and in
the $\gamma+\mbox{${E\\!\\!\\!\\!/_{T}}$}$ signature reported by DØ. Then I
briefly describe searches sensitive to graviton exchange in dimuon, dielectron
and diphoton final states reported by DØ and in $ZZ$ final states by both CDF
and DØ.

##  2 CDF and DØ detectors

A detailed description of the CDF and the DØ detectors can be found in Ref.
[6, 7]. The vertexing and the tracking detectors surrounding the interaction
region are used to reconstruct charged particle trajectories. Further out are
the calorimeters with electromagnetic and hadronic longitudinal segmentation
used to identify and measure photons, electrons, and jets. Outside the
calorimeter are the muon detectors used to identify muons from the collisions
and cosmic rays.

In analyses of $\gamma+\mbox{${E\\!\\!\\!\\!/_{T}}$}$ events one of the major
backgrounds is from cosmic ray muons misidentified as photons. The detector
features used to reject this background are: photon pointing available at DØ
allowing the reconstruction of direction of the photon from the longitudinal
segmentation of the electromagnetic calorimeter [8]; photon timing available
at CDF from the EMTiming system [9] providing the measurement of the time of
the energy deposit and allowing to suppress cosmic-ray background by a factor
of 20 or more.

##  3 Searches for graviton production

The production of gravitons in hadron collisions is probed in events where the
final-state hadronic jet or photon recoils against the graviton, which is not
detected and results in ${E\\!\\!\\!\\!/_{T}}$.

Background Events $Z\to\nu\nu$ $388\pm 30$ $W\to\ell\nu$ $362\pm 17$ Multi-jet
$23\pm 20$ $\gamma+$jet $17\pm 5$ Non-collision $10\pm 10$ Total predicted
$808\pm 62$ Data observed $809$

Table 1: The numbers of expected background and observed events in the
$\mathrm{jet}+\mbox{${E\\!\\!\\!\\!/_{T}}$}$ sample used by CDF.

Searches in both the $\mathrm{jet}+\mbox{${E\\!\\!\\!\\!/_{T}}$}$ and the
$\gamma+\mbox{${E\\!\\!\\!\\!/_{T}}$}$ require exclusive signatures: only a
jet or a photon is in the event with a veto on the presence of other objects.
These searches have similar backgrounds. The dominant and only irreducible
background is production of $Z\to\nu\nu$ in association with a photon or a
jet.

A search for LED in $\mathrm{jet}+\mbox{${E\\!\\!\\!\\!/_{T}}$}$ events has
been performed by CDF using 1.1 $\mathrm{fb}^{-1}$ of data, updating the
analysis in Ref. [10]. Events with high ${E\\!\\!\\!\\!/_{T}}$, a high-$E_{T}$
jet and no second jet with $E_{T}>60~{}\mathrm{Ge\kern-1.00006ptV}$ are
selected. All major backgrounds are estimated in a data-driven way. The
contributions from $Z\to\nu\nu$ and from $W\to\ell\nu$, where the lepton is
lost, are estimated using measured $Z\to\ell\ell$ and $W\to\ell\nu$ events.
After an a priori optimization for the best sensitivity to LED, events with
the leading jet $E_{T}>150~{}\mathrm{Ge\kern-1.00006ptV}$ and
$\mbox{${E\\!\\!\\!\\!/_{T}}$}>120~{}\mathrm{Ge\kern-1.00006ptV}$ are
selected. The summary of predicted and observed events is given in Table 1.
The constraints on LED are given at the end of this section.

Background CDF DØ $Z\gamma\to\nu\nu\gamma$ $24.8\pm 2.8$ $12.1\pm 1.3$
$W\to(e\to\gamma)\nu$ $2.6\pm 0.4$ $3.8\pm 0.3$ $W\to(\mu/\tau\to\gamma)\nu$
$1.0\pm 0.2$ – $W\gamma\to\ell\nu\gamma$ $5.0\pm 1.4$ $1.5\pm 0.2$
jet$\to\gamma$ – $2.2\pm 1.5$ $\gamma\gamma\to\gamma$ $2.3\pm 0.6$ - Non-
collision $9.8\pm 1.3$ $2.8\pm 1.4$ Total predicted $46.3\pm 3.0$ $22.4\pm
2.5$ Observed data $40$ $29$

Table 2: The numbers of expected background and observed events in the
$\gamma+\mbox{${E\\!\\!\\!\\!/_{T}}$}$ samples analyzed by CDF and DØ. The
contribution from jet$\to\gamma$ in CDF is included in $(W/\gamma)\gamma$ by
virtue of the background estimation method.

Results for the $\gamma+\mbox{${E\\!\\!\\!\\!/_{T}}$}$ signature in the
Tevatron Run II have been recently reported by CDF in 2 $\mathrm{fb}^{-1}$ of
data and by DØ in 1.05 $\mathrm{fb}^{-1}$ of data [8]. Both analyses use
events with high ${E\\!\\!\\!\\!/_{T}}$ and a high-$E_{T}$ photon in the
central detector region with $|\eta|<1.1$ [4]. No jets with
$E_{T}>15~{}\mathrm{Ge\kern-1.00006ptV}$ are allowed in the event in order to
suppress QCD backgrounds. Also, no high-$p_{T}$ track is allowed, suppressing
contributions from leptonic $W$ and $Z$ decays: tracks with
$p_{T}>10~{}\mathrm{Ge\kern-1.00006ptV}/c$ (isolated with
$p_{T}>6.5~{}\mathrm{Ge\kern-1.00006ptV}/c$) are vetoed by CDF (DØ). The
contribution from cosmic rays is suppressed based on presence of hits in the
muon detectors as well as using the photon pointing or the photon timing
mentioned in Section 2.

The background contributions for both analyses are summarized in Table 2. The
number of events from $Z\to\nu\nu$ is estimated from simulation. The
contribution from $W\to e\nu$, where the electron is identified as the photon,
is extracted from the number of events with electrons passing all other
requirements applied to the photon signal sample, scaled by the
misidentification rate. Contributions from cosmic rays and jets misidentified
as photons are estimated at DØ using the photon pointing method. The remaining
backgrounds in the analysis by DØ are estimated using simulation. The
contribution from cosmics at CDF is estimated using events with the photon
time significantly different from the collision time. The remaining
backgrounds at CDF are from processes where an object is lost. Except for
$W\to\tau\nu$, which is estimated from simulation, each of these backgrounds
is given by the number of events in data with this object identified, scaled
by the simulated rate for it to be lost. This approach implicitly includes
contributions from a jet identified as a photon.

Source LEP DØ CDF $n$ $\gamma+\mbox{${E\\!\\!\\!\\!/_{T}}$}$
$\gamma+\mbox{${E\\!\\!\\!\\!/_{T}}$}$
$\mathrm{jet}+\mbox{${E\\!\\!\\!\\!/_{T}}$}$
$\gamma+\mbox{${E\\!\\!\\!\\!/_{T}}$}$ combined 2 1600 921 1310 1080 1400 3
1200 877 1080 1000 1150 4 940 848 980 970 1040 5 770 821 910 930 980 6 660 810
880 900 940

Table 3: Lower limits on $M_{D}$ in $\mathrm{Ge\kern-1.00006ptV}/{c^{2}}$ at
$95\%$ C.L. for $n$ from 2 to 6 observed in the
$\mathrm{jet}+\mbox{${E\\!\\!\\!\\!/_{T}}$}$ and
$\gamma+\mbox{${E\\!\\!\\!\\!/_{T}}$}$ signatures and the combination of the
two at CDF, and in the $\gamma+\mbox{${E\\!\\!\\!\\!/_{T}}$}$ signature at DØ,
along with the constraints from the LEP experiments [11]. Constraints provided
by DØ for $n$ above 6 can be found in [8].

After an a priori optimization for the best sensitivity, events with a photon
with $E_{T}>90~{}\mathrm{Ge\kern-1.00006ptV}$ and
$\mbox{${E\\!\\!\\!\\!/_{T}}$}>50(70)~{}\mathrm{Ge\kern-1.00006ptV}$ are
selected in the CDF (DØ) signal sample. Event counts observed in the data are
consistent with expectations from the backgrounds. The constraints on the LED
model are summarized in Table 3. Since the sensitivity in the
$\gamma+\mbox{${E\\!\\!\\!\\!/_{T}}$}$ is similar to that in the
$\mathrm{jet}+\mbox{${E\\!\\!\\!\\!/_{T}}$}$ mode, the constraints can be
improved in a combination. The result of this combination using the CDF data
is shown in Table 3. Depending on the number of extra dimensions, the
sensitivity to LED at the Tevatron is comparable to or better than that from
the LEP experiments [11].

##  4 Searches for virtual graviton exchange

In hadron collisions the processes most sensitive to virtual graviton exchange
in LED are those with two leptons (electrons or muons), two photons, or two
$Z$ bosons in the final state, where it is possible to reconstruct the full
kinematics of the final state with high precision. The value of the graviton
exchange amplitude depends on the cutoff energy $\Lambda$, presumably of order
$M_{D}$ [3]. The cross section is proportional to $\Lambda^{-8}$.

A search for LED in the dimuon final state has been reported by DØ using 200
$\mathrm{pb}^{-1}$ of data [12]. The sample is comprised of events with two
opposite charge muons with $p_{T}$ above $15~{}\mathrm{Ge\kern-1.00006ptV}/c$
and both muons passing additional quality and isolation requirements. The
dilepton invariant mass and the scattering angle in the dilepton center of
mass are analyzed simultaneously to improve sensitivity to the exchange of a
graviton (spin 2 particle). A similar analysis using a di-em final state where
no discrimination is made between electrons and photons has been performed at
DØ with 240 $\mathrm{pb}^{-1}$ of data [13]. In this case events were selected
if they had two em objects with $E_{T}>25~{}\mathrm{Ge\kern-1.00006ptV}$
passing quality and isolation requirements. No excess over the expected
backgrounds is observed. The constraints on the parameter $\Lambda$ are in the
range of 1 to 2 $\mathrm{Te\kern-1.00006ptV}$ depending on the theoretical
parameters.

Both CDF and DØ have recently reported on searches for $p\overline{p}\to ZZ$
production in a four-lepton final state [14, 15]. The observed number of
events is consistent with the SM expectations. Although no constraint has been
provided on LED by either experiment, based on Ref. [16] the constraint on
$\Lambda$ is expected to be in the range of 1.5 to 2.5
$\mathrm{Te\kern-1.00006ptV}$.

##  5 Summary

Searches for LED in signatures with direct graviton production and those with
virtual graviton exchange are explored by the CDF and the DØ experiments. No
evidence of LED has been seen. Additional $p\overline{p}$ collision data
currently produced at the Tevatron is expected to give more insight about the
possible presence of large extra spacial dimensions.

## Acknowledgements

I would like to acknowledge the funding institutions supporting the CDF and DØ
Collaborations. The full list of agencies can be found in, e.g, [6, 7].

## References

  * [1] N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. B 429, 263 (1998) [arXiv:hep-ph/9803315].
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  * [4] The polar coordinate system is used with an origin at the center of a detector and the $z$-axis ($\theta=0$) along the proton beam. The pseudorapidity $\eta$ is defined as $\eta=-\ln[\tan(\theta/2)]$. The transverse momentum $p_{T}$ is defined as $p_{T}=p\cdot\sin{\theta}$, where $p$ is the particle’s momentum. The transverse energy is defined as $E_{T}=E\cdot\sin{\theta}$, where $E$ is the energy measured by the calorimeter. The missing $E_{T}$ (${E\\!\\!\\!\\!/_{T}}$) is defined as $\vec{\mbox{${E\\!\\!\\!\\!/_{T}}$}}=-\sum_{i}E_{T}^{i}\hat{n}_{i}$, where $i$ is the index of a calorimeter tower (a segmentation unit in $\eta$-$\phi$) and $\hat{n}_{i}$ is a unit vector perpendicular to the beam axis pointing to the tower from the origin.
  * [5] Slides:
`http://indico.cern.ch/contributionDisplay.py?contribId=104&sessionId=15&confId=24657`

  * [6] A. Abulencia et al. [CDF Collaboration], J. Phys. G 34, 2457 (2007) [arXiv:hep-ex/0508029].
  * [7] V. M. Abazov et al. [D0 Collaboration], Nucl. Instrum. Meth. A 565, 463 (2006) [arXiv:physics/0507191].
  * [8] V. M. Abazov et al. [D0 Collaboration], arXiv:0803.2137 [hep-ex], accepted by Phys. Rev. Lett.
  * [9] M. Goncharov et al., Nucl. Instrum. Meth. A 565, 543 (2006) [arXiv:physics/0512171].
  * [10] A. Abulencia et al. [CDF Collaboration], Phys. Rev. Lett. 97, 171802 (2006) [arXiv:hep-ex/0605101].
  * [11] J. Abdallah et al. [DELPHI Collaboration], Eur. Phys. J. C 38, 395 (2005) [arXiv:hep-ex/0406019]; P. Achard et al. [L3 Collaboration], Phys. Lett. B 587, 16 (2004) [arXiv:hep-ex/0402002].
  * [12] V. M. Abazov et al. [D0 Collaboration], Phys. Rev. Lett. 95, 161602 (2005) [arXiv:hep-ex/0506063].
  * [13] V. M. Abazov et al. [D0 Collaboration], DØ note 4336 (2004), unpublished.
  * [14] T. Aaltonen et al. [CDF Collaboration], arXiv:0801.4806 [hep-ex].
  * [15] V. M. Abazov et al. [D0 Collaboration], Phys. Rev. Lett. 100, 131801 (2008) [arXiv:0712.0599 [hep-ex]].
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# Resonant photoproduction of ultrarelativistic electron-positron pairs on a
nucleus in strong monochromatic light field

Sergei P. Roshchupkin<EMAIL_ADDRESS>Peter the Great St.Petersburg
Polytechnic University, Saint-Petersburg, Russia  Nikita R. Larin
<EMAIL_ADDRESS>Peter the Great St.Petersburg Polytechnic University,
Saint-Petersburg, Russia  Victor V. Dubov<EMAIL_ADDRESS>Peter the Great
St.Petersburg Polytechnic University, Saint-Petersburg, Russia

###### Abstract

For complete development of quantum electrodynamics in the presence of a
strong external field, the proper understanding of resonant processes and all
their peculiarities is essential. We present our attempt to analytically
investigate the resonant case of laser-assisted electron-positron pair
photoproduction on a nucleus. Due to the presence of external field, the
intermediate virtual particle may become real, herewith the second order
process in the fine structure constant effectively reduces into the two
successive first order processes. All inherent kinematics features were
discussed in details and the resonant differential cross section was obtained.
We established that the resonant energies of produced particles ambiguously
depend on the positron (channel A) or electron (channel B) outgoing angle, and
the certain minimal amount of absorbed wave photons are required for resonance
to happen. Furthermore, the resonant cross section significantly exceeds the
corresponding one in the absence of the external field within the particular
kinematic regions and consequently, the considered process can be used qua a
marker for probing theoretical predictions of quantum electrodynamics with
strong background field.

###### pacs:

25.75.Dw, 42.62.$-$b, 42.50.Hz

##  I Introduction

Nowadays, the nonlinear phenomena of quantum electrodynamics (QED) within the
strong electromagnetic fields attract enormous interest Di Piazza  _et al._
(2012); Gonoskov  _et al._ (2021); Blackburn (2020); Salamin  _et al._ (2006);
Ehlotzky  _et al._ (2009); Cajiao Vélez  _et al._ (2019); Ritus and Nikishov
(1979); Roshchupkin and Voroshilo (2008); Roshchupkin and Lebed’ (2013) due to
the development of the contemporary high-intensity laser radiation facilities
Danson  _et al._ (2019); Yoon  _et al._ (2021); Turcu  _et al._ (2019);
Papadopoulos  _et al._ (2016); Bromage  _et al._ (2019); Rossbach  _et al._
(2019); Heinemann, Beate  _et al._ (2020) and high-energy particles sources
Gonoskov  _et al._ (2017); Magnusson  _et al._ (2019); Zhu  _et al._ (2018).
Amongst of such phenomena, the resonant behavior of the second order processes
in the fine structure constant Oleinik (1967, 1968); Krajewska (2011); Bos
_et al._ (1979); Roshchupkin  _et al._ (2012); Roshchupkin (1996). The feature
of these processes is that the intermediate virtual particle can possibly
become a real one, and by virtue of it the initial process of the second order
effectively splits into the two successive first order processes. Wherein, the
resonant differential cross section may significantly surpass the
corresponding non-resonant one within the certain kinematic region. Therefore,
it makes resonant processes potential candidates to become markers for probing
the predictions of QED in the presence of a strong external field.

The conversion of electromagnetic radiation into the matter is one of the most
intriguing phenomena since the dawn of quantum field theory. There are diverse
scenarios of the electron-positron pair productions in nature Ruffini  _et
al._ (2010); Fradkin and Gitman (1981), amid them the famous Bethe-Haitler
(BH) Bethe and Heitler (1934) and Breit-Wheeler (BW) Breit and Wheeler (1934)
processes. In turn, the former may be modified by the presence of strong
external field, and one refers to it as the laser-assisted BH process. In the
present paper, we are concerned with the resonant case of this process.
Notwithstanding the large number of fruitful investigations devoted to the
laser-assisted PPP on a nucleus Lötstedt  _et al._ (2009); Di Piazza  _et al._
(2010); Augustin and Müller (2014); Müller  _et al._ (2003); Krajewska and
Kamiński (2010); Hafizi  _et al._ (2019), the complete description of this
problem, especially the resonant situation, is hitherto far away from the
completeness. We want explicitly to highlight previous works, where attention
was paid to the resonant laser-assisted BH process for the case of weak
monochromatic Larin  _et al._ (2019, 2020a) and pulsed Roshchupkin  _et al._
(2021a); Larin  _et al._ (2020b) plane wave field. Within the current
research, our analytical investigation is extended to the case of strong
external field.

The inherent feature about the processes within external electromagnetic wave
field is that there are two characteristic parameters that govern their
behavior. The first is a classical relativistic invariant parameter, which
defines the interactions of fermions with background plane wave field:

$\displaystyle\eta=\frac{{eF\mathchar
22\relax\mkern-10.0mu\lambda}}{{m{c^{2}}}},$ (1)

which numerically equals to the ratio of the field work at a wavelength to the
electron rest energy ($e$ and $m$ are the charge and the electron mass, $F$
and $\;\mathchar
22\relax\mkern-10.0mu\lambda={c\mathord{\left/{\vphantom{c\omega}}\right.\kern-1.2pt}\omega}$
are the field strength and wavelength, $\omega$ is a wave frequency). The
second is a quantum multiphoton parameter appears when particles interact with
the Coulomb center within the plane electromagnetic wave Bunkin and Fedorov
(1965):

$\displaystyle\gamma=\eta\frac{{mvc}}{{\hbar\omega}}$ (2)

Herein $v$ is the particle velocity, $c$ is the speed of light. However, this
parameter (2) plays an essential role only for the case, when particles are
scattered on a large angle by the Coulomb potential. Otherwise, when the
scattering angle is small, this parameter does not appear Lebed’ (2016). Thus,
the main parameter that determines the multiphoton processes is the classical
relativistic parameter (1). Henceforth, we will employ the relativistic system
of units $\hbar=c=1$.

##  II The amplitude of the process

In order to deduce analytical expressions for the resonant differential cross
section we adhere to the model of infinitely spatially and temporally extended
electromagnetic wave with circular polarization, which propagates along the
$z$ axes. Let us choose the corresponding four-potential in the following
form:

$\displaystyle
A\left(\phi\right)=\frac{F}{\omega}\left({{e_{x}}\cos\phi+\delta{e_{y}}\sin\phi}\right),\quad\phi=kx=\omega\left({t-z}\right),$
(3)

where $k=\left({\omega,{\bf{k}}}\right)$ is the wave vector, $\delta=\pm 1$ is
the ellipticity parameter of the wave and
${e_{x,y}}=\left({0,{{\bf{e}}_{x,y}}}\right)$ are the polarization four-
vectors of the wave, particularly
$e_{x,y}^{2}=-1,{\rm{}}\left({{e_{x,y}}k}\right)={k^{2}}=0$. We treat
interaction with the Coulomb potential of the nucleus within the first Born
approximation, therefore, we restrict ourselves with the condition
$Z\alpha/v\ll 1$ ($Z$ is the nuclear charge, $\alpha$ is the fine structure
constant).

The considered process is of the second order in the fine structure constant,
consequently it is described by two Feynman diagrams (see. Fig 1), which
differ from each other by the intermediate state.

Figure 1: Feynman diagrams of the PPP process on the nucleus in the external
electromagnetic field. Double incoming and outgoing lines correspond to the
dressed electron and positron functions in the initial and final states. The
inner lines stand for the intermediate dressed state, initial gamma quantum
and pseudo photon of recoil are depicted by wavy and dashed lines accordingly.

The amplitude of such process can be represented as the sum over total number
absorbed (emitted) photons of external wave:

$\displaystyle S=\sum\limits_{l=-\infty}^{\infty}{{S_{l}},}$ (4)
$\displaystyle{S_{l}}=\frac{{8Z{e^{3}}{\pi^{5/2}}}}{{\sqrt{2{{\tilde{E}}_{+}}{{\tilde{E}}_{-}}{\omega_{i}}}}}\exp\left({i\psi}\right)\left[{{{\bar{u}}_{{p_{-}},\lambda}}{B_{l}}{v_{{p_{+}},\lambda^{\prime}}}}\right]\frac{{\delta\left({{q^{0}}}\right)}}{{{{\bf{q}}^{2}}}},$
(5)
$\displaystyle{B_{l}}=\sum\limits_{r=-\infty}^{+\infty}{\left[{{M_{r-l}}\left({{{\tilde{p}}_{-}},{{\tilde{q}}_{-}}}\right)\frac{{{{\hat{\tilde{q}}}_{-}}-\frac{{{\eta^{2}}{m^{2}}}}{{2\left({k{q_{-}}}\right)}}\hat{k}+m}}{{\tilde{q}_{-}^{2}-m_{*}^{2}}}{F_{-r}}\left({{{\tilde{q}}_{-}},{{\tilde{p}}_{+}}}\right)+{M_{r-l}}\left({{{\tilde{p}}_{+}},{{\tilde{q}}_{+}}}\right)\frac{{{{\hat{\tilde{q}}}_{+}}-\frac{{{\eta^{2}}{m^{2}}}}{{2\left({k{q_{+}}}\right)}}\hat{k}+m}}{{\tilde{q}_{+}^{2}-m_{*}^{2}}}{F_{-r}}\left({{{\tilde{q}}_{+}},{{\tilde{p}}_{-}}}\right)}\right]}$
(6)

Hereinafter, all notations with hat imply the contraction of the corresponding
vector with Dirac gamma matrices
${\tilde{\gamma}^{\mu}}=\left({{{\tilde{\gamma}}^{0}},{\bf{\tilde{\gamma}}}}\right),{\rm{}}\mu=0,1,2,3$
(e.g.
$\hat{k}={k_{\mu}}{\tilde{\gamma}^{\mu}}={k_{0}}{\tilde{\gamma}^{0}}-{\bf{k\tilde{\gamma}}}$).
In the expression (5) ${\bar{u}_{{p_{-}},\lambda}}$ and
${v_{{p_{+}},\lambda^{\prime}}}$ are free Dirac bispinors for electron in the
final and positron in the initial state, respectively, and $\psi$ is an
insignificant phase that does not depend either on the summation index or
momenta of particles. Here we introduced the notations for the electron and
positron four-quasimomenta
${\tilde{p}_{\pm}}=\left({{{\tilde{E}}_{\pm}},{{{\bf{\tilde{p}}}}_{\pm}}}\right)$,
as well four-quasimomenta of intermediate states
${\tilde{q}_{\pm}}=\left({{{\tilde{E}}_{\pm}},{{{\bf{\tilde{q}}}}_{\pm}}}\right)$:

$\displaystyle{\tilde{q}_{-}}=-{\tilde{p}_{+}}+{k_{i}}+rk,\quad{\tilde{q}_{+}}=-{\tilde{p}_{-}}+{k_{i}}+rk,$
(7) $\displaystyle q={\tilde{p}_{+}}+{\tilde{p}_{-}}-{k_{i}}-lk.$ (8)
$\displaystyle{\tilde{p}_{\pm}}={p_{\pm}}+{\eta^{2}}\frac{{{m^{2}}}}{{2\left({k{p_{\pm}}}\right)}}k,\quad{\tilde{q}_{\pm}}={q_{\pm}}+{\eta^{2}}\frac{{{m^{2}}}}{{2\left({k{q_{\pm}}}\right)}}k,$
(9)
$\displaystyle\tilde{p}_{\pm}^{2}=m_{*}^{2},\quad{m_{*}}=m\sqrt{1+{\eta^{2}}}.$
(10)

Herein ${k_{i}}={\omega_{i}}\left({1,{{\bf{n}}_{i}}}\right)$ is the four-
momentum of the initial gamma quantum and ${m_{*}}$ is an effective mass of
fermion within the external electromagnetic field (3). The amplitudes
${M_{r-l}}$ and ${F_{-r}}$ in the relation (6) have the following expressions:

$\displaystyle{M_{l-r}}\left({{{\tilde{p}}_{2}},{{\tilde{p}}_{1}}}\right)={a^{0}}{L_{r-l}}\left({{{\tilde{p}}_{2}},{{\tilde{p}}_{1}}}\right)+b_{-}^{0}{L_{r-l-1}}+b_{+}^{0}{L_{r-l+1}},\quad$
(11)
$\displaystyle{F_{-r}}\left({{{\tilde{p}}_{2}},{{\tilde{p}}_{1}}}\right)=\left({a\varepsilon}\right){L_{-r}}\left({{{\tilde{p}}_{2}},{{\tilde{p}}_{1}}}\right)+\left({{b_{-}}\varepsilon}\right){L_{-r-1}}+\left({{b_{+}}\varepsilon}\right){L_{-r+1}},$
(12)

where we denoted by parentheses the dot product of the initial gamma quantum
polarization four-vector ${\varepsilon^{\mu}}$ with matrices
${a^{\mu}},{\rm{}}b_{\pm}^{\mu}$ that are defined in the following way:

$\displaystyle{a^{\mu}}={\tilde{\gamma}^{\mu}}+{\eta^{2}}\frac{{{m^{2}}}}{{2\left({k{{\tilde{p}}_{1}}}\right)\left({k{{\tilde{p}}_{2}}}\right)}}{k^{\mu}}\hat{k},$
(13) $\displaystyle b_{\pm}^{\mu}=\frac{1}{4}\eta
m\left[{\frac{{{{\hat{\varepsilon}}_{\pm}}\hat{k}{\gamma^{\mu}}}}{{\left({k{{\tilde{p}}_{2}}}\right)}}+\frac{{{\gamma^{\mu}}\hat{k}{{\hat{\varepsilon}}_{\pm}}}}{{\left({k{{\tilde{p}}_{1}}}\right)}}}\right],{\rm{}}{\hat{\varepsilon}_{\pm}}={\hat{e}_{x}}\pm
i\delta{\hat{e}_{y}},$ (14)

Special functions
${L_{r-l}}\left({{{\tilde{p}}_{2}},{{\tilde{p}}_{1}}}\right)$,
${L_{-r}}\left({{{\tilde{p}}_{2}},{{\tilde{p}}_{1}}}\right)$ and their
arguments are given by the expressions Roshchupkin  _et al._ (2000):

$\displaystyle{L_{n}}\left({{{\tilde{p}}_{2}},{{\tilde{p}}_{1}}}\right)=\exp\left({-in{\chi_{{{\tilde{p}}_{2}}{{\tilde{p}}_{1}}}}}\right){J_{n}}\left({{\gamma_{{{\tilde{p}}_{2}}{{\tilde{p}}_{1}}}}}\right)$
(15)
$\displaystyle\tan{\chi_{{{\tilde{p}}_{2}}{{\tilde{p}}_{1}}}}=\delta\frac{{\left({{e_{y}}{Q_{{{\tilde{p}}_{2}}{{\tilde{p}}_{1}}}}}\right)}}{{\left({{e_{x}}{Q_{{{\tilde{p}}_{2}}{{\tilde{p}}_{1}}}}}\right)}},\quad{Q_{{{\tilde{p}}_{2}}{{\tilde{p}}_{1}}}}=\frac{{{{\tilde{p}}_{2}}}}{{\left({k{{\tilde{p}}_{2}}}\right)}}+\frac{{{{\tilde{p}}_{1}}}}{{\left({k{{\tilde{p}}_{1}}}\right)}},$
(16) $\displaystyle{\gamma_{{{\tilde{p}}_{2}}{{\tilde{p}}_{1}}}}=\eta
m\sqrt{-Q_{{{\tilde{p}}_{2}}{{\tilde{p}}_{1}}}^{2}}.$ (17)

We note, that $\left({k{{\tilde{p}}_{1,2}}}\right)=\left({k{p_{1,2}}}\right)$
and thus, to obtain the appropriate expressions for the channel A (i.e. for
the first term in (6)) we need to replace
${\tilde{p}_{1}}\to-{\tilde{p}_{+}},{\rm{}}{\tilde{p}_{2}}\to{\tilde{q}_{-}}$
for ${F_{-r}}\left({{{\tilde{p}}_{2}},{{\tilde{p}}_{1}}}\right)$ and
${\tilde{p}_{1}}\to{\tilde{q}_{-}},{\rm{}}{\tilde{p}_{2}}\to{\tilde{p}_{-}}$
for ${M_{l-r}}\left({{{\tilde{p}}_{2}},{{\tilde{p}}_{1}}}\right)$ in the
relations (13)-(17). For the channel B (i.e. for the second term in (6)) one
has to act in similar way and make the replacement
${\tilde{p}_{1}}\to{\tilde{p}_{-}},{\rm{}}{\tilde{p}_{2}}\to-{\tilde{q}_{+}}$
for ${F_{-r}}\left({{{\tilde{p}}_{2}},{{\tilde{p}}_{1}}}\right)$ and
${\tilde{p}_{1}}\to-{\tilde{q}_{+}},{\rm{}}{\tilde{p}_{2}}\to-{\tilde{p}_{+}}$
for ${M_{l-r}}\left({{{\tilde{p}}_{2}},{{\tilde{p}}_{1}}}\right)$. It is
important to emphasize, that obtained amplitude (5)-(17) is valid for the
arbitrary intensities and frequencies of the plane monochromatic wave with
circular polarization.

##  III POLES OF THE AMPLITUDE IN A STRONG FIELD

In the presence of the external electromagnetic field (3) the intermediate
particle momentum may satisfy its dispersion relation:

$\displaystyle\tilde{q}_{-}^{2}=m_{*}^{2},$ (18)
$\displaystyle\tilde{q}_{+}^{2}=m_{*}^{2}.$ (19)

Such behavior is caused by the quasi-discrete energy spectrum of fermion
propagating within the plane electromagnetic wave. Due to that fact, one may
interpret it as the reduction of the second order process (see. Fig.1) into
the two successive second order processes in fine structure constant (see
Fig.2).

Figure 2: Resonant photoproduction of the electron-positron pair in the field
of a nucleus and plane electromagnetic wave.

Within the external field of a plane monochromatic wave there is four-
quasimomentum conservation law, which can be written for both channels in
every vertex (see Fig.2) in the following way:

$\displaystyle{k_{i}}+rk={\tilde{p}_{+}}+{\tilde{q}_{-}},$ (20) $\displaystyle
q={\tilde{q}_{-}}-{\tilde{p}_{-}}+\left({l-r}\right)k$ (21)

and

$\displaystyle{k_{i}}+rk={\tilde{p}_{-}}+{\tilde{q}_{+}},$ (22) $\displaystyle
q={\tilde{q}_{+}}-{\tilde{p}_{+}}+\left({l-r}\right)k.$ (23)

Insofar as
$\tilde{p}_{\pm}^{2}=\tilde{q}_{\pm}^{2}=m_{*}^{2},{\rm{and}}\;{k^{2}}=k_{i}^{2}=0,$
the equalities (20) and (22) are fulfilled only for the $r\geq 1$. This fact
in conjunction with the form of amplitude (5), (6) (see also Fig.2) allows us
to conclude that ${F_{-r}}$ (12) represents the amplitude of the laser-
stimulated Breit-Wheeler process Ritus and Nikishov (1979) with absorption of
$r$ wave photons. In turn, ${M_{r-l}}$ (11) is nothing but laser-assisted Mott
scattering of electron (channel A) or positron (channel B) on a nucleus with
the absorption (emission) of $\left|{r-l}\right|$ wave photons Cajiao Vélez
_et al._ (2019); Ehlotzky  _et al._ (1998). Hence, that verified that in the
absence of interference, the initial second order process in the fine
structure constant effectively reduces into two successive first order
processes, as was mentioned above. One also may check that simultaneous
fulfillment of the resonant conditions (18), (19) and four-quasimomentum law
conservation (20), (22) is impossible unless the initial gamma quantum and
external plane wave propagate in one direction.

The thorough examination of the resonant conditions (18), (19) and
conservation laws (20)-(23) shows us that for resonance to occur, one of the
possibilities is to require the ultrarelativistic energies of produced
particles and thus the sufficient energy of the initial gamma quantum.
Moreover, the resonant kinematics region is confined with the configuration,
where all produced particles have to propagate within the narrow cone with
initial gamma quantum direction. Additionally, we demand the directions of
initial gamma quantum and external wave propagation do not coincidence,
otherwise resonances are merely impossible:

$\displaystyle{\omega_{i}}\gg m$ (24)
$\displaystyle{\theta_{i\pm}}=\left({{{\bf{k}}_{i}},{{\bf{p}}_{\pm}}}\right)\ll
1,\quad{{\bar{\theta}}_{\pm}}=\left({{{\bf{p}}_{-}},{{\bf{p}}_{+}}}\right)\ll
1,$ $\displaystyle{\theta_{i}}=\left({{{\bf{k}}_{i}},{\bf{k}}}\right)\sim
1,\quad{\theta_{\pm}}=\left({{\bf{k}},{{\bf{p}}_{\pm}}}\right)\sim 1.\quad$
(25)

In a matter of fact, the condition (24) has to be rewritten for the case of
strong field, when the classical parameter becomes not small
$\eta\mathbin{\lower
1.29167pt\hbox{$\buildrel>\over{\smash{\scriptstyle\sim}\vphantom{{}_{x}}}$}}1$.
We face the necessity to replace the particle mass with the effective mass
Roshchupkin  _et al._ (2021b). Hence, the condition (24) takes form:

$\displaystyle\frac{{{\omega_{i}}}}{{{m_{*}}}}=\frac{{{\omega_{i}}}}{{m\sqrt{1+{\eta^{2}}}}}\sim\left\\{{\begin{array}[]{*{20}{c}}{{{{\omega_{i}}}\mathord{\left/{\vphantom{{{\omega_{i}}}{m\gg
1,\quad{\rm{if}}\quad\eta\ll{\rm{1}}}}}\right.\kern-1.2pt}{m\gg
1,\quad{\rm{if}}\quad\eta\ll{\rm{1}}}}}\\\
{{{{\omega_{i}}}\mathord{\left/{\vphantom{{{\omega_{i}}}{\left({\eta
m}\right)\gg 1,\quad{\rm{if}}\quad\eta\mathbin{\lower
1.29167pt\hbox{$\buildrel>\over{\smash{\scriptstyle\sim}\vphantom{{}_{x}}}$}}{\rm{1}}}}}\right.\kern-1.2pt}{\left({\eta
m}\right)\gg 1,\quad{\rm{if}}\quad\eta\mathbin{\lower
1.29167pt\hbox{$\buildrel>\over{\smash{\scriptstyle\sim}\vphantom{{}_{x}}}$}}{\rm{1}}}}}\end{array}}\right.$
(28)

From the second string in the condition (28) we obtain the restriction on the
maximum intensity of the external field:

$\displaystyle\eta\ll{\eta_{\max}}=\frac{{{\omega_{i}}}}{m}.$ (29)

By the similar reasoning, we formulate the new ultrarelativistic condition for
produced particles:

$\displaystyle\frac{{{{\tilde{E}}_{\pm}}}}{{{m_{*}}}}\approx\frac{{{E_{\pm}}}}{{m\sqrt{1+{\eta^{2}}}}}\sim\left\\{{\begin{array}[]{*{20}{c}}{{{{E_{\pm}}}\mathord{\left/{\vphantom{{{E_{\pm}}}{m\gg
1,\quad{\rm{if}}\quad\eta\ll{\rm{1}}}}}\right.\kern-1.2pt}{m\gg
1,\quad{\rm{if}}\quad\eta\ll{\rm{1}}}}}\\\
{{{{E_{\pm}}}\mathord{\left/{\vphantom{{{E_{\pm}}}{\left({\eta m}\right)\gg
1,\quad{\rm{if}}\quad\eta\mathbin{\lower
1.29167pt\hbox{$\buildrel>\over{\smash{\scriptstyle\sim}\vphantom{{}_{x}}}$}}{\rm{1}}}}}\right.\kern-1.2pt}{\left({\eta
m}\right)\gg 1,\quad{\rm{if}}\quad\eta\mathbin{\lower
1.29167pt\hbox{$\buildrel>\over{\smash{\scriptstyle\sim}\vphantom{{}_{x}}}$}}{\rm{1}}}}}\end{array}}\right.$
(32)

Deliberately, throughout our research, we consider the initial gamma quantum
energy ${\omega_{i}}\mathbin{\lower
1.29167pt\hbox{$\buildrel<\over{\smash{\scriptstyle\sim}\vphantom{{}_{x}}}$}}100\;{\rm{GeV}}$.
This value leads us to the estimation of the classical invariant parameter
$\eta<<{\eta_{\max}}\sim{10^{5}}$, which corresponds to
$F<<{F_{\max}}\sim{10^{15}}\;{{\rm{V}}\mathord{\left/{\vphantom{{\rm{V}}{{\rm{cm}}}}}\right.\kern-1.2pt}{{\rm{cm}}}}$
$\left({I<<{I_{\max}}\sim{{10}^{28}}\;{{\rm{W}}\mathord{\left/{\vphantom{{\rm{W}}{{\rm{c}}{{\rm{m}}^{\rm{2}}}}}}\right.\kern-1.2pt}{{\rm{c}}{{\rm{m}}^{\rm{2}}}}}}\right)$
for the optical frequency range. Therefore, all further results are valid for
sufficiently large intensity, however, they are still not applicable to the
fields of the critical Schwinger limit ${F_{cr}}\approx
1.3\cdot{10^{16}}\;{{\rm{V}}\mathord{\left/{\vphantom{{\rm{V}}{{\rm{cm}}}}}\right.\kern-1.2pt}{{\rm{cm}}}}$.

With use of relations (18) and (20) we can derive the expression for the
resonant positron energy in channel A:

$\displaystyle{x_{\eta+(r)}}=\frac{{r\pm\sqrt{r\left({r-{r_{\eta}}}\right)-r_{\eta}^{2}\delta_{\eta+}^{2}}}}{{2\left({r+r_{\eta}\delta_{\eta+}^{2}}\right)}}.$
(33)

Analogously, relations (19) and (22) help us to deduce expression for the
resonant electron energy for channel B:

$\displaystyle{x_{\eta-(r)}}=\frac{{r\pm\sqrt{r\left({r-{r_{\eta}}}\right)-r_{\eta}^{2}\delta_{\eta-}^{2}}}}{{2\left({r+{r_{\eta}}\delta_{\eta-}^{2}}\right)}}.$
(34)

Here we introduced notations:

$\displaystyle{x_{\eta\pm(r)}}=\frac{{{E_{\eta\pm(r)}}}}{{{\omega_{i}}}}{\rm{,}}\;{r_{\eta}}{\rm{=}}\frac{{m_{*}^{2}}}{{{\omega_{i}}\omega{{\sin}^{2}}\left({{{{\theta_{i}}}\mathord{\left/{\vphantom{{{\theta_{i}}}2}}\right.\kern-1.2pt}2}}\right)}}{\rm{,}}\;{\delta_{\eta\pm}}=\frac{{{\omega_{i}}{\theta_{i\pm}}}}{{2{m_{*}}}}.$
(35)

In formulae (33) and (34) $r$ is a number of resonance (namely, it is the
number of wave photons absorbed within the laser-stimulated BW process),
${r_{\eta}}$ is a characteristic parameter that determines the minimal amount
of wave photons that are required for the laser-stimulated BW process to
happen: $r\geq r_{min}$, where

$\displaystyle{r_{\min}}=\left\lceil{{r_{\eta}}}\right\rceil.$ (36)

Throughout this paper, we will use for assessments the certain set of
parameters: ${\omega_{i}}=50\;{\rm{GeV}}$, $\omega=1\;{\rm{eV}}$,
${\theta_{i}}=\pi$. For such set, it follows:

$\displaystyle{r_{\eta}}\approx 5.2\left({1+{\eta^{2}}}\right).$ (37)

One can see that in strong fields $\left({\eta>>1}\right)$, the resonant
process involves large number of absorbed wave photons
$\left({{r_{\eta}}\approx{\eta^{2}}\gg 1}\right)$. By the definition of
${r_{\eta}}$ it follows, as well, that the number of absorbed photons
increases proportional to intensity $\left({{r_{\eta}}\sim{\eta^{2}}\sim
I\;\left({{\rm{Wc}}{{\rm{m}}^{{\rm{-2}}}}}\right)}\right)$. We underline, that
all obtained expressions (33)-(35) are in complete agreement with the weak
field limit $\eta\ll 1$, particularly the parameter ${r_{\eta}}$ reduces to
the threshold energy for initial gamma quantum Larin  _et al._ (2019).

Another peculiarity of the resonant behavior is that resonant energy of
produced particles ambiguously depends on corresponding outgoing angles
(positron outgoing angle for channel A (33) and electron outgoing angle for
channel B (34)) (see. Fig.3 and Larin  _et al._ (2019); Roshchupkin  _et al._
(2021a)). Henceforth, we will refer to the expressions with «+» sign in
numerators of (33) and (34) as high-energy solutions and to expressions with
«-» sign as low-energy. Also, the outgoing angle of particle (positron for
channel A and electron for channel B) enclosed in the interval, which is
defined by the following inequality:

$\displaystyle
0\leq\delta_{\eta\pm}^{2}\leq\delta_{\eta\pm\max}^{2}=\frac{r}{{{r_{\eta}}}}\left({\frac{r}{{{r_{\eta}}}}-1}\right).$
(38)

Figure 3: The resonant positron (channel A) and electron (channel B) energies
as functions of absorbed wave photons and corresponding outgoing angle,
plotted for the parameters (37). Fig.3a and Fig.3c represent high-energy
solution, meantime Fig.3b and Fig.3d correspond to the low-energy solutions
(33), (34).

Figure 4: The resonant positron (channel A) and electron (channel B) energies
as functions of the corresponding outgoing angle, plotted for the different
values of absorbed wave photons and different intensities for certain values
of the parameters (37). Solid lines represent the high-energy solution,
meantime dotted lines stand for the low-energy solutions (33), (34).

Inasmuch there are no intersections between energy’s value with different
within the frame of particular channel (see. Fig.4), we can distinguish one
process with different number of absorbed photons from another and thus, they
do not interfere.

##  IV THE RESONANT DIFFERENTIAL CROSS SECTION OF THE PPP IN THE
ULTRARELATIVISTIC ENERGY LIMIT

As long as we confine ourselves by the condition (29) we may neglect the
second and the third term in fermion scattering amplitude (11)
($\left|{b_{\pm}^{0}}\right|\mathbin{\lower
1.29167pt\hbox{$\buildrel<\over{\smash{\scriptstyle\sim}\vphantom{{}_{x}}}$}}\eta{m\mathord{\left/{\vphantom{m{{\omega_{i}}}}}\right.\kern-1.2pt}{{\omega_{i}}}}\ll
1$, see Eqs. (14) and (28)). As a result, the expression for ${M_{r-l}}$
essentially simplifies:

$\displaystyle
M_{r-l}=\exp\left[{-i\left({r-l}\right){\chi_{{{\tilde{p}}_{-}}{{\tilde{q}}_{-}}}}}\right]{J_{r-l}}\left[{\gamma\left({{{\tilde{p}}_{-}},{{\tilde{q}}_{-}}}\right)}\right]{\gamma^{0}}$
(39)

For the conciseness, we represent our derivations for the channel A. To obtain
the corresponding relations for channel B, one must substitute
${\tilde{q}_{-}}\to{\tilde{q}_{+}},\;{\tilde{p}_{-}}\to-{\tilde{p}_{+}}$.
Also, we introduce the subscript «+» for further expressions to signify that
within the channel A all deduced quantities are defined by the positron
outgoing angle, in contrast to the channel B, where we use the subscript «-»
to accentuate the similar role of the electron outgoing angle. The influence
of interference between different channels is left out of our consideration.
Nevertheless, we emphasize that there is indeed interference in resonance
case, and its impact requires further investigation.

We perform the standard procedure Berestetskii  _et al._ (2012) to derive
resonant differential cross section for the unpolarized particles from the
amplitude (4)-(6), (12), (39):

$\displaystyle d\sigma_{+(l,r)}=\frac{2}{{{\pi^{2}}}}d{{\rm
M}_{+\left({l-r}\right)}}\frac{{{m^{2}}{E_{-}}}}{{{{\left|{\tilde{q}_{-}^{2}-m_{*}^{2}}\right|}^{2}}}}d{{\rm{P}}_{+\left(r\right)}}.$
(40)

Herein $d{{\rm M}_{+\left({l-r}\right)}}$ represents differential cross
section of the intermediate electron scattering on the nucleus with emission
(absorption) of $\left|{l-r}\right|$ wave photons (lasser-assisted Mott
scattering) Roshchupkin and Voroshilo (2008):

$\displaystyle d{{\rm
M}_{+\left({l-r}\right)}}={Z^{2}}r_{e}^{2}\frac{{{m^{2}}}}{{{{\bf{q}}^{4}}}}J_{l-r}^{2}\left({{\gamma_{{{\tilde{p}}_{-}},{{\tilde{q}}_{-}}}}}\right)\delta\left[{\tilde{q}_{-}^{0}-{{\tilde{E}}_{-}}+\left({l-r}\right)\omega}\right]{d^{3}}{\tilde{p}_{-}},$
(41)

where transferred to nucleus momentum ${\bf{q}}$ and argument of the Bessel
function have the following forms:

$\displaystyle{\bf{q}}={{\bf{\tilde{q}}}_{-}}-{{\bf{\tilde{p}}}_{-}}+\left({l-r}\right){\bf{k}},$
(42) $\displaystyle{\gamma_{{{\tilde{p}}_{-}},{{\tilde{q}}_{-}}}}=\eta
m\sqrt{-Q_{{{\tilde{p}}_{-}},{{\tilde{q}}_{-}}}^{2}},\quad
Q_{{{\tilde{p}}_{-}},{{\tilde{q}}_{-}}}=\frac{{{{\tilde{p}}_{-}}}}{{\left({k{{\tilde{p}}_{-}}}\right)}}-\frac{{{{\tilde{q}}_{-}}}}{{\left({k{{\tilde{q}}_{-}}}\right)}}.\
$ (43)

Function $d{{\rm{P}}_{+\left(r\right)}}$ determines the differential
probability (per unit of time) of the lasser-stimulated BW process with the
absorption of wave photons Ritus and Nikishov (1979):

$\displaystyle
d{{\rm{P}}_{+\left(r\right)}}=\frac{\alpha}{{{\omega_{i}}{E_{+}}}}P\left({{u_{\eta+\left(r\right)}},{v_{\eta\left(r\right)}}}\right){d^{3}}{\tilde{p}_{+}},$
(44)

here

$\displaystyle
P\left({{u_{\eta+\left(r\right)}},{v_{\eta\left(r\right)}}}\right)=J_{r}^{2}\left({{\gamma_{{{\tilde{q}}_{-}}{{\tilde{p}}_{+}}}}}\right)+{\eta^{2}}\left({2{u_{\eta+\left(r\right)}}-1}\right)\left[{\left({\frac{{{r^{2}}}}{{\gamma_{{{\tilde{q}}_{-}}{{\tilde{p}}_{+}}}^{2}}}-1}\right)J_{r}^{2}+J_{r}^{\prime
2}}\right],$ (45)
$\displaystyle{\gamma_{\eta+\left(r\right)}}=2r\frac{\eta}{{\sqrt{1+{\eta^{2}}}}}\sqrt{\frac{{{u_{\eta+\left(r\right)}}}}{{{v_{\eta\left(r\right)}}}}\left({1-\frac{{{u_{\eta+\left(r\right)}}}}{{{v_{\eta\left(r\right)}}}}}\right)},$
(46)
$\displaystyle{u_{\eta+(r)}}=\frac{{{{\left({k{k_{i}}}\right)}^{2}}}}{{4\left({k{q_{-}}}\right)\left({k{p_{+}}}\right)}}\approx\frac{1}{{4{x_{\eta+\left(r\right)}}\left({1-{x_{\eta+\left(r\right)}}}\right)}},$
(47)
$\displaystyle{v_{\eta(r)}}=r\frac{{\left({k{k_{i}}}\right)}}{{2m_{*}^{2}}}=\frac{r}{{{r_{\eta}}}}.$
(48)

Due to the condition (32) it is permissible to put
${d^{3}}{\tilde{p}_{-}}\approx{d^{3}}{p_{-}}\approx
E_{-}^{2}d{E_{-}}d{\Omega_{-}}$ in the cross section (41) and easily carry out
the integration with respect to the electron energy (or the positron energy
for channel B).

The appearance of the resonant infinity is caused by the idealized spatial and
temporal dependence of the external filed, which allows analytical
investigation. Complete treatment of such kind divergences involves cumbersome
calculations of all radiative corrections to fermion propagator Oleinik
(1967). Other ways to elaborate this issue are to consider the finite size and
duration of the external field Roshchupkin  _et al._ (2021a); Mackenroth and
Di Piazza (2013) or to engage the Sokhotski-Plemelj theoremSeipt and Kämpfer
(2012); Ilderton (2011). The last method, in fact, also implies the finite
duration of external laser pulse. In turn, we achieve the elimination of the
resonant infinity in channels A and B by means of the renowned Breit-Wigner
procedure Breit and Wigner (1936):

$\displaystyle{m_{*}}\to{\mu_{*}}={m_{*}}-i{\Gamma_{\eta+\left(r\right)}},\quad{\Gamma_{\eta+\left(r\right)}}=\frac{{\tilde{q}_{-}^{0}}}{{2{m_{*}}}}W\left({{r_{\eta}}}\right).$
(49)

Here ${W_{\eta+\left(r\right)}}$ is the total probability (per unit of time)
of laser-stimulated BW process with absorption of wave photons Ritus and
Nikishov (1979):

$\displaystyle
W\left({{r_{\eta}}}\right)=\frac{{\alpha{m^{2}}}}{{8\pi{\omega_{i}}}}{\rm{P}}\left({{r_{\eta}}}\right),$
(50)
$\displaystyle{\rm{P}}\left({{r_{\eta}}}\right)=\sum\limits_{n={r_{\min}}}^{\infty}{{{\rm{P}}_{n}}\left({{r_{\eta}}}\right),}$
(51)
$\displaystyle{{\rm{P}}_{n}}\left({{r_{\eta}}}\right)=\int\limits_{1}^{{n\mathord{\left/{\vphantom{n{{r_{\eta}}}}}\right.\kern-1.2pt}{{r_{\eta}}}}}{\frac{{du}}{{u\sqrt{u\left({u-1}\right)}}}P\left({u,\frac{n}{{{r_{\eta}}}}}\right)},$
(52)

where function
$P\left({u,{n\mathord{\left/{\vphantom{n{{r_{\eta}}}}}\right.\kern-1.2pt}{{r_{\eta}}}}}\right)$
is derived from the expressions (45)-(48) by substitution:
${u_{\eta+\left(r\right)}}\to u,\;r\to n$. Given the relations (49)-(52) the
resonant denominator for channel A can be represented as follows:

$\displaystyle{\left|{\tilde{q}_{-}^{2}-\mu_{*}^{2}}\right|^{2}}=16m_{*}^{4}x_{\eta+\left(r\right)}^{2}\left[{{{\left({\delta_{\eta+}^{2}-\delta_{\eta+\left(r\right)}^{2}}\right)}^{2}}+\Upsilon_{\eta+\left(r\right)}^{2}}\right].\quad$
(53)

Here $\Upsilon_{\eta+\left(r\right)}$ is the angular resonant width:

$\displaystyle{\Upsilon_{\eta+\left(r\right)}}=\frac{\alpha}{{32\pi\left({1+{\eta^{2}}}\right)}}\frac{{\left({1-{x_{\eta+\left(r\right)}}}\right)}}{{{x_{\eta+\left(r\right)}}}}{\rm{P}}\left({{r_{\eta}}}\right).$
(54)

In the Eq.(53) parameter $\delta_{\eta+\left(r\right)}^{2}$ is related to the
positron resonant energy by the formula (33), meanwhile parameter
$\delta_{\eta+}^{2}$ varies independently. We also note that the function
${\rm{P}}\left({{r_{\eta}}}\right)$ has the most general form (51) and valid
for any values of the classical relativistic parameter $\eta$. However, for
the case $\eta\gg 1$ it is more convenient to work with another expression,
that was deduced by Nikishov and Ritus Ritus and Nikishov (1979):

$\displaystyle{\rm{P}}\left(\eta\right)=\frac{3}{{16}}\sqrt{\frac{3}{2}}\frac{{\left({1+{\eta^{2}}}\right)\eta}}{{{r_{\eta}}}}\exp\left[{-\frac{4}{3}\frac{{{r_{\eta}}}}{{\left({1+{\eta^{2}}}\right)\eta}}\left({1-\frac{1}{{15{\eta^{2}}}}}\right)}\right],$
(55)

In contrast to (51) and (52) in formula (55) were performed summation over all
processes with different $r$ and integration with respect to $u$. The
corresponding calculations may be analytically carried out only for the case
$\eta\gg 1$. We emphasize that angular width
${\Upsilon_{\eta+\left(r\right)}}$ increases with the wave intensity growth.
Noteworthy that due to the field strength limitation, which we consider
throughout this paper $F\mathbin{\lower
1.29167pt\hbox{$\buildrel<\over{\smash{\scriptstyle\sim}\vphantom{{}_{x}}}$}}{10^{14}}\;{{\rm{V}}\mathord{\left/{\vphantom{{\rm{V}}{{\rm{cm}}}}}\right.\kern-1.2pt}{{\rm{cm}}}}$
(see Eq. (29)), the angular radiation width ${\Upsilon_{\eta+\left(r\right)}}$
is significantly greater than the radiative corrections Hartin (2018).

Further calculations with use of relations (28)-(34) and (53) lead us to the
following expressions for cross sections for channel A and B:

$\displaystyle\frac{{d{\sigma_{\eta\pm\left({l,r}\right)}}}}{{d{x_{\eta\pm\left(r\right)}}d\delta_{\eta\pm}^{2}}}=\frac{{\left({{Z^{2}}\alpha
r_{e}^{2}}\right)}}{{\pi{{\left({1+{\eta^{2}}}\right)}^{2}}}}\frac{{J_{l-r}^{2}\left({{\alpha_{\eta\pm\left(r\right)}}}\right)}}{{g_{\pm}^{4}}}\frac{{{{\left({1-{x_{\eta\pm\left(r\right)}}}\right)}^{3}}}}{{\left[{{{\left({\delta_{\eta\pm}^{2}-\delta_{\eta\pm\left(r\right)}^{2}}\right)}^{2}}+\Upsilon_{\eta+\left(r\right)}^{2}}\right]{x_{\eta\pm\left(r\right)}}}}P\left({{u_{\eta\pm\left(r\right)}},\frac{r}{{{r_{\eta}}}}}\right)d\delta_{\eta\mp}^{2}d\varphi,$
(56)

where

$\displaystyle g_{\pm}^{2}={g_{\eta
0}}+{\left({\frac{{{m_{*}}}}{{2{\omega_{i}}}}}\right)^{2}}{g_{\eta\pm(r,l)}}.$
(57)

Herein $\phi$ is the angle between planes
$\left({{{\bf{k}}_{i}},{{\bf{p}}_{+}}}\right)$ and
$\left({{{\bf{k}}_{i}},{{\bf{p}}_{-}}}\right)$. Similarly to the channel A,
the parameter $\delta_{\eta-\left(r\right)}^{2}$ is expressed via the resonant
energy of electron for channel B (34) and $\delta_{\eta-}^{2}$ varies
independently. The relativistic invariant parameter
${u_{\eta-\left(r\right)}}$ and the resonant width for channel B obey to
relations:

$\displaystyle{u_{\eta-(r)}}=\frac{{{{\left({k{k_{i}}}\right)}^{2}}}}{{4\left({k{q_{+}}}\right)\left({k{p_{-}}}\right)}}\approx\frac{1}{{4{x_{\eta-\left(r\right)}}\left({1-{x_{\eta-\left(r\right)}}}\right)}},$
(58)
$\displaystyle{\Upsilon_{\eta-\left(r\right)}}=\frac{\alpha}{{32\pi\left({1+{\eta^{2}}}\right)}}\frac{{1-{x_{\eta-\left(r\right)}}}}{{{x_{\eta-\left(r\right)}}}}{\rm{P}}\left({{r_{\eta}}}\right).$
(59)

The function
$P\left({{u_{\eta-\left(r\right)}},{r\mathord{\left/{\vphantom{r{{r_{\eta}}}}}\right.\kern-1.2pt}{{r_{\eta}}}}}\right)$
has the likewise to channel A form (45) except the substitution
${u_{\eta+\left(r\right)}}\to{u_{\eta-\left(r\right)}}$. The impact of
transferred to nucleus momentum contains in the functions $g_{+}^{2}$ and
$g_{-}^{2}$ (57), where we took into account the influence of corrections
proportional to
${{m_{*}^{2}}\mathord{\left/{\vphantom{{m_{*}^{2}}{\omega_{i}^{2}}}}\right.\kern-1.2pt}{\omega_{i}^{2}}}$:

$\displaystyle{g_{\eta
0}}=\tilde{\delta}_{\eta+}^{2}+\tilde{\delta}_{\eta-}^{2}+2\tilde{\delta}_{\eta+}\tilde{\delta}_{\eta-}\cos\varphi,\;\;{\rm{}}\tilde{\delta}_{\eta\pm}=2{x_{\eta\pm(r)}}\delta_{\eta\pm},\quad$
(60)
$\displaystyle{g_{\eta\pm(r,l)}}=g_{\eta\pm(r,l)}^{(0)}+\frac{1}{{1+{\eta^{2}}}}g_{\eta\pm(r,l)}^{(1)}+\frac{1}{{{{\left({1+{\eta^{2}}}\right)}^{2}}}}g_{\eta\pm(r,l)}^{(2)},\quad$
(61) $\displaystyle
g_{\eta\pm(r,l)}^{(0)}=\frac{{\tilde{\delta}_{\eta\pm}^{2}}}{3}\frac{{x_{\eta\pm(r)}^{2}{{\left({1-{x_{\eta\pm(r)}}}\right)}^{2}}-{{\left({1-{x_{\eta\pm(r)}}}\right)}^{3}}-x_{\eta\pm(r)}^{3}}}{{x_{\eta\pm(r)}^{3}{{\left({1-{x_{\eta\pm(r)}}}\right)}^{3}}}}-\frac{{4{\beta_{\eta\pm(l,r)}}\tilde{\delta}_{\eta\pm}^{2}}}{{x_{\eta\pm(r)}\left({1-{x_{\eta\pm(r)}}}\right)}}\qquad$
(62) $\displaystyle
g_{\eta\pm(r,l)}^{(1)}=2\tilde{\delta}_{\eta\pm}^{2}\frac{{x_{\eta\pm(r)}^{3}\left({2-{x_{\eta\pm(r)}}}\right)-{{\left({1-{x_{\eta\pm(r)}}}\right)}^{4}}}}{{x_{\eta\pm(r)}^{3}{{\left({1-{x_{\eta\pm(r)}}}\right)}^{3}}}}+\frac{{4{\beta_{\eta\pm(l,r)}}}}{{x_{\eta\pm(r)}\left({1-{x_{\eta\pm(r)}}}\right)}},\qquad$
(63) $\displaystyle
g_{\eta\pm(r,l)}^{(2)}=\frac{{{{\left({1-{x_{\eta\pm(r)}}}\right)}^{3}}+x_{\eta\pm(r)}^{3}-{{\left({1-2{x_{\eta\pm(r)}}}\right)}^{2}}}}{{x_{\eta\pm(r)}^{3}{{\left({1-{x_{\eta\pm(r)}}}\right)}^{3}}}},$
(64)
$\displaystyle{\beta_{\eta\pm(l,r)}}=\frac{l}{{{r_{\eta}}}}-\frac{1}{4}\frac{{{\eta^{2}}}}{{1+{\eta^{2}}}}\frac{1}{{{x_{\eta\pm(r)}}\left({1-{x_{\eta\pm(r)}}}\right)}}.$
(65)

The arguments of the Bessel functions that define the processes of emission or
absorption of $\left|{l-r}\right|$ wave photons during the scattering of
intermediate fermion on the nucleus for channel A and B have the form:

$\displaystyle{\alpha_{\eta\pm\left(r\right)}}\approx
2{r_{\eta}}\frac{\eta}{{\sqrt{1+{\eta^{2}}}}}\sqrt{g_{\eta 0}^{2}}$ (66)

As we have already mentioned, the corrections of the order of were introduced
in transmitted momentum (57). These corrections are of the great importance
for the certain kinematic regions. Namely, they make the dominant contribution
to the differential cross section under the conditions:

$\displaystyle\left|{\varphi-\pi}\right|\mathbin{\lower
1.29167pt\hbox{$\buildrel<\over{\smash{\scriptstyle\sim}\vphantom{{}_{x}}}$}}\frac{{{m_{*}}}}{{{\omega_{i}}}}\ll
1,\quad\left|{{{\tilde{\delta}}_{\eta+}}-{{\tilde{\delta}}_{\eta-}}}\right|\mathbin{\lower
1.29167pt\hbox{$\buildrel<\over{\smash{\scriptstyle\sim}\vphantom{{}_{x}}}$}}\frac{{{m_{*}}}}{{{\omega_{i}}}}\ll
1.$ (67)

Under such conditions, the function $g_{\eta 0}$ tends to zero and
consequently, there is a sharp maximum in the corresponding differential cross
section. This notorious behavior of differential cross section in
ultrarelativistic limit is due to the long-range Coulomb potential
Berestetskii  _et al._ (2012). We perform the saddle point method to integrate
resonant cross sections (56) (for the channel B analogically) within the
vicinity of maxima points (67):

$\displaystyle\frac{{d{\sigma_{\eta+\left({l,r}\right)}}}}{{d{x_{\eta+\left(r\right)}}d\delta_{\eta+}^{2}}}=\frac{{\left({{Z^{2}}\alpha
r_{e}^{2}}\right)}}{{4\pi{{\left({1+{\eta^{2}}}\right)}^{2}}}}\frac{{\left({1-{x_{\eta+\left(r\right)}}}\right)P\left({{u_{\eta+\left(r\right)}},{r\mathord{\left/{\vphantom{r{{r_{\eta}}}}}\right.\kern-1.2pt}{{r_{\eta}}}}}\right)}}{{\left[{{{\left({\delta_{\eta+}^{2}-\delta_{\eta+\left(r\right)}^{2}}\right)}^{2}}+\Upsilon_{\eta+\left(r\right)}^{2}}\right]{x_{\eta+\left(r\right)}}}}{C_{\eta+\left({l,r}\right)}},$
(68)

where

$\displaystyle{C_{\eta+\left({l,r}\right)}}=\int\limits_{0}^{2\pi}{d\varphi}\int\limits_{0}^{\infty}{J_{l-r}^{2}\left({{\alpha_{\eta+\left(r\right)}}}\right){{\left[{g_{\eta
0}^{2}+{{\left({\frac{{{m_{*}}}}{{2{\omega_{i}}}}}\right)}^{2}}{g_{\eta+(r,l)}}}\right]}^{-2}}}d\tilde{\delta}_{\eta-}^{2}.$
(69)

The integrand in (69) has an abrupt maximum within the interval (67) herewith,
the function
$J_{l-r}^{2}\left({{\alpha_{\eta+\left(r\right)}}}\right)\mathbin{\lower
1.29167pt\hbox{$\buildrel<\over{\smash{\scriptstyle\sim}\vphantom{{}_{x}}}$}}1$
and we are allowed to take it out of the integral

$\displaystyle{C_{\eta+\left({l,r}\right)}}\approx
J_{l-r}^{2}\left({{\alpha_{\eta+\left(r\right)}}}\right)\int\limits_{0}^{2\pi}{d\varphi}\int\limits_{0}^{\infty}{d\tilde{\delta}_{\eta-}^{2}}\exp\left[{{f_{\eta}}\left({\varphi,{{\tilde{\delta}}_{\eta-}}}\right)}\right],$
(70)

here

$\displaystyle{f_{\eta}}\left({\varphi,{{\tilde{\delta}}_{\eta-}}}\right)=-2\ln\left({g_{\eta
0}^{2}+{\kappa_{\eta+}}}\right),\quad{\kappa_{\eta+}}={\left({\frac{{{m_{*}}}}{{2{\omega_{i}}}}}\right)^{2}}{g_{\eta+(r,l)}}.$
(71)

We employ the Taylor expansion of the ${f_{\eta}}$ in the vicinity of the
point $\varphi=\pi,\;\;{\tilde{\delta}_{\eta-}}={\tilde{\delta}_{\eta+}}$

$\displaystyle{f_{\eta}}\left({\varphi,{{\tilde{\delta}}_{\eta-}}}\right)\approx-2\ln\left({{\kappa_{\eta+}}}\right)-\frac{{2\tilde{\delta}_{\eta+}^{2}}}{{{\kappa_{\eta+}}}}{\left({\varphi-\pi}\right)^{2}}-\frac{1}{{2{\kappa_{\eta+}}\tilde{\delta}_{\eta+}^{2}}}{\left({\tilde{\delta}_{\eta-}^{2}-\tilde{\delta}_{\eta+}^{2}}\right)^{2}}.$
(72)

Eventually, we perform integration (70) with use of the expansion (72)

$\displaystyle{C_{\eta+\left({l,r}\right)}}\approx\frac{\pi}{{{g_{\eta+\left({l,r}\right)}}}}\left({\frac{{2\omega_{i}^{2}}}{{m_{*}^{2}}}}\right)J_{l-r}^{2}\left(0\right)=\frac{{4\pi}}{{{g_{\eta+\left(r\right)}}}}\left({\frac{{\omega_{i}^{2}}}{{m_{*}^{2}}}}\right).$
(73)

In so doing, we take into account that the argument of Bessel function (66) is
virtually zero and thus, the Bessel function itself is not zero only for the
case $l=r$. In other words, the most probable situation is the scattering of
ultrarelativistic fermion on the nucleus without absorption or emission of
wave photonsLebed’ (2016); Larin  _et al._ (2019). Ultimately, the
differential cross section for both channels may be written in the following
way:

$\displaystyle\frac{{d{\sigma_{\eta\pm\left({l,r}\right)}}}}{{d{x_{\eta\pm\left(r\right)}}d\delta_{\eta\pm}^{2}}}={\left({\frac{{{\omega_{i}}}}{{{m_{*}}}}}\right)^{2}}\frac{{\left({{Z^{2}}\alpha
r_{e}^{2}}\right)}}{{{{\left({1+{\eta^{2}}}\right)}^{2}}}}\frac{{\left({1-{x_{\eta\pm\left(r\right)}}}\right)}}{{{g_{\eta\pm\left(r\right)}}\left[{{{\left({\delta_{\eta\pm}^{2}-\delta_{\eta\pm\left(r\right)}^{2}}\right)}^{2}}+\Upsilon_{\eta\pm\left(r\right)}^{2}}\right]{x_{\eta+\left(r\right)}}}}P\left({{u_{\eta\pm\left(r\right)}},\frac{r}{{{r_{\eta}}}}}\right),$
(74)

Herein the functions ${g_{\eta+\left(r\right)}}$ and
${g_{\eta-\left(r\right)}}$ are defined by the expressions (61)-(65) but
instead of the index $l$ one should write $r$. When the following conditions
are met

$\displaystyle{\left({\delta_{\eta+}^{2}-\delta_{\eta+\left(r\right)}^{2}}\right)^{2}}\ll\Upsilon_{\eta+\left(r\right)}^{2},\>{\left({\delta_{\eta-}^{2}-\delta_{\eta-\left(r\right)}^{2}}\right)^{2}}\ll\Upsilon_{\eta-\left(r\right)}^{2}\quad$
(75)

we obtain the maximum resonant differential cross section for channels A and
B:

$\displaystyle
R_{\eta\pm\left(r\right)}^{\max}=\frac{{d\sigma_{\eta\pm\left(r\right)}^{\max}}}{{d{x_{\eta\pm\left(r\right)}}d\delta_{\eta\pm}^{2}}}=\left({{Z^{2}}\alpha
r_{e}^{2}}\right){F_{\eta\pm\left(r\right)}}.$ (76)

The functions ${F_{\eta+\left(r\right)}}$ and ${F_{\eta-\left(r\right)}}$
determine the spectral-angular distribution at fixed intensity of the resonant
PPP differential cross section for channels A and B, correspondingly:

$\displaystyle{F_{\eta\pm\left(r\right)}}=D\frac{{{x_{\eta\pm\left(r\right)}}}}{{\left({1-{x_{\eta\pm\left(r\right)}}}\right){g_{\eta\pm\left(r\right)}}{{\rm{P}}^{2}}\left({{r_{\eta}}}\right)}}P\left({{u_{\eta\pm\left(r\right)}},\frac{r}{{{r_{\eta}}}}}\right),$
(77) $\displaystyle
D={\left({\frac{{32\pi{\omega_{i}}}}{{\alpha{m_{*}}}}}\right)^{2}}.$ (78)

Figure 5: Dependence of the resonant cross section (77) (in units
${Z^{2}}\alpha r_{e}^{2}$) on the corresponding outgoing angle for the certain
parameters (37) and different intensity values

Figure 6: Temperature maps of the resonant cross section with respect to the
corresponding outgoing angle and amount of absorbed wave photons for different
intensities and parameters (37)

From Fig.6a-Fig.6d we can conclude, that only for particular values of the
outgoing angles and absorbed wave photons resonant differential cross sections
have significant magnitudes. In plane ($\delta_{\pm}^{2},r$) these values form
a region, which shape depends on the intensity. More detailed behavior of this
region is presented in Fig.5a-Fig.5d, where we can see that with increase of
intensity, the number of maxima grows, and they become more distinguishable.
Herewith, the global maximum exists for every value of intensity, and
therefore it allows us to determine the most probable energies of produced
particles and corresponding outgoing angles. We stress, that the magnitude of
the resonant differential cross section decreases with increase of intensity.
Such dependence is explained by the behavior of the resonant angular width
(54), (59), which increases with intensity. It is noteworthy, that we
represented in the Fig.5 and Fig.6 the corresponding expressions with "high-
energy" solutions (33), (34) therein. The reason is, that expressions with
"low-energy" solutions utterly suppressed in comparison to the "high-energy"
ones and don’t make any impact into the total resonant cross section.

##  V Conclusion

We have considered the resonant photoproduction of electron-positron pair on a
nucleus within the strong external field. The thorough examination of four-
quasimomentum conservation laws along with resonant conditions allowed us to
formulate one of the possible resonant conditions. Due to these conditions, we
have to demand ultrarelativistic energies of the produced particles and
consequently sufficient energy of the initial gamma quantum. Besides, the
propagation of the produced particles has to be enclosed within the narrow
cone in the initial gamma quantum direction. We established, that there is
minimal amount of absorbed wave photons in resonance. This amount is
completely determined by the experimental set up and increases proportional to
the external field intensity. The energies of produced particles were derived
as the functions of outgoing angle and number of absorbed photons. The
corresponding dependencies are ambiguous. In addition, there is maximum
outgoing angle for every value of absorbed photons.

We obtained the resonant differential cross section of laser-assisted BH
process with simultaneous registration of the particle energies and the
corresponding outgoing angle (positron for channel A and electron for channel
B) for vast intensity range (from $10^{18}$ to
$10^{24}\;{\rm{W/c}}{{\rm{m}}^{2}}$). Also, we verified that under the
resonant conditions, it factorizes into the product of differential
probability of BW process and differential cross section of Mott scattering.
Herewith, the most probable is the situation, when intermediate particle
scatters on nucleus without absorption or emission of wave photons. The
obtained angular distribution has distinguishable maxima for each value of the
absorbed wave photons. The number of maxima varies from one to four, depending
on the number of absorbed photons and intensity. Moreover, there are global
maxima of the resonant cross section with respect to the outgoing angle and
number of absorbed wave photons for certain intensity value. This fact gives
opportunity to determine the most probable energies of the particles and their
outgoing angles, and thus to define a resonant process with high accuracy.
Noteworthy, that with increase of the intensity the resonant differential
cross section decreases due to the growth of the resonant width. There are
also kinematic regions, where resonant differential cross section is totally
suppressed.

The above analysis was carried out with considering the model of plane
monochromatic electromagnetic wave, which allowed us to provide analytical
investigation, but led us to notorious divergence of the resonant cross
section. We employed the phenomenological procedure to eliminate this
divergence, whereas, we emphasized that there are other more elegant and, in a
matter of fact, more rigorous ways to deal with this problem. However, by
virtue of inconsistency of the mentioned methods with the model of plane
monochromatic wave, we can’t perform them here. The detailed analysis
involving proper elaboration of resonant infinity and influence of
interference between different channels will be a subject of future research.

##  VI Acknowledgment

The work was supported by the State Program for the Fundamental Research
(theme code FSEG-2020-0024)

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