# Elastic scattering of a muon neutrino by an electron in the presence of a
circularly polarized laser field

S. El Asri  Sultan Moulay Slimane University, Polydisciplinary Faculty,
Laboratory of Research in Physics $\&$ Engineering Sciences, Team of Modern
and Applied Physics, Beni Mellal, 23000, Morocco.  S. Mouslih  Faculty of
Sciences and Techniques, Laboratory of Materials Physics (LMP), Beni Mellal,
23000, Morocco.  Sultan Moulay Slimane University, Polydisciplinary Faculty,
Laboratory of Research in Physics $\&$ Engineering Sciences, Team of Modern
and Applied Physics, Beni Mellal, 23000, Morocco.  M. Jakha  Sultan Moulay
Slimane University, Polydisciplinary Faculty, Laboratory of Research in
Physics $\&$ Engineering Sciences, Team of Modern and Applied Physics, Beni
Mellal, 23000, Morocco.  B. Manaut  Sultan Moulay Slimane University,
Polydisciplinary Faculty, Laboratory of Research in Physics $\&$ Engineering
Sciences, Team of Modern and Applied Physics, Beni Mellal, 23000, Morocco.  Y.
Attaourti  High Energy Physics and Astrophysics Laboratory, FSSM, Cadi Ayyad
University, Marrakesh, Morocco.  S. Taj<EMAIL_ADDRESS>Sultan Moulay Slimane
University, Polydisciplinary Faculty, Laboratory of Research in Physics $\&$
Engineering Sciences, Team of Modern and Applied Physics, Beni Mellal, 23000,
Morocco.  R. Benbrik  Polydisciplinary Faculty, Laboratoire de Physique
Fondamentale et Appliquée, Sidi Bouzid, B.P. 4162 Safi, Morocco.

(September 3, 2024)

###### Abstract

In view of the great contribution of neutrino-electron scattering to the deep
understanding of electroweak interactions, we focus in this paper on the study
of elastic scattering of a muon neutrino by an electron
$(e^{-}\nu_{\mu}\rightarrow e^{-}\nu_{\mu})$ in the presence of a circularly
polarized electromagnetic field. We perform our theoretical calculation within
the framework of Fermi theory using the exact wave functions of charged
particles in an electromagnetic field. The expression of the differential
cross section (DCS) for this process is obtained analytically in the absence
and presence of the laser field. The effect of the field strength and
frequency on the exchange of photons as well as on the DCS is presented and
analyzed. Massive neutrino effects are also included and discussed. This
study, added to the previous ones, will significantly enrich our knowledge in
fundamental physics.

##  I Introduction

In particle physics, the scattering experiments are the most effective
research tools that allow us to study the interactions between particles and
probe the structure of matter. It is through them that we gather a lot of
information about the physical world. From the Rutherford’s gold-foil
experiment [1] revealing the atomic nucleus to the discovery of the Higgs
boson at the Large Hadron Collider (LHC) [2], the observation and
interpretation of quantum scattering processes have been pivotal to the
advancement of particle physics. Laser-assisted or -induced scattering
processes are a type of scattering processes that has received considerable
attention since the invention of the laser in the $1960$s until the recent
development of high-power optical lasers [3, 4, 5]. G. Mourou and D.
Strickland were jointly awarded the 2018 Nobel Prize in Physics for inventing
the use of chirped pulse amplification (CPA) as a means to generate high-
intensity and ultra-short optical pulses [6, 7]. Meanwhile, scientists have
been enthusiastic and focused on the study of laser-assisted quantum
processes, whether in quantum electrodynamics (QED) [8, 9, 10, 11],
electroweak theory [12, 13, 14, 15, 16] or atomic physics [17, 18, 19, 20],
and generally on the study of the laser-matter interactions [21, 22]. In
parallel with the production of the high-power femtosecond lasers,
experimental testing of these studies became feasible [23, 24, 25]. In our
turn, we have decided to study, through this research paper, one of the most
important scattering processes that have played an important role in the
development of modern physics, namely the elastic scattering process of a muon
neutrino by an electron $(e^{-}\nu_{\mu}\rightarrow e^{-}\nu_{\mu})$ in the
presence of a circularly polarized electromagnetic field. Nine years ago, the
same process has been studied, within the framework of Fermi’s theory, in the
presence of a linearly polarized laser field and the phenomena of multi photon
absorption and emission are induced [26]; the authors found that the
distributions of a multiphoton energy transfer (MPET) spectrum are largely
affected by the laser even at moderate intensities, while the differential
cross section (DCS) can be notably changed by several orders of magnitude only
in superstrong fields with ultrarelativistic electrons. In general, the study
of (anti)muon neutrino scattering on electrons has offered important results
for understanding the electroweak sector of the Standard Model [27]. In
particular, the first observation of a few
$\bar{\nu}_{\mu}e\rightarrow\bar{\nu}_{\mu}e$ scattering events by the
Gargamelle bubble chamber in 1973 at CERN [28] provided the first empirical
proof of the existence of weak neutral-current (NC) interactions. An extensive
theory and a collection of new results for total and differential cross
sections of the elastic neutrino-electron scattering have been reported in
Refs. [29, 30]. The purpose of this paper is mainly to reveal the effect of a
strong electromagnetic field on the scattering process
$e^{-}\nu_{\mu}\rightarrow e^{-}\nu_{\mu}$ and in particular on its calculated
DCS. The theoretical calculations are performed within the Fermi theory using
the method of exact solutions for electron states in the presence of a
circularly polarized electromagnetic wave field. Although the electromagnetic
field does not couple to the standard model neutrino, it affects neutrino
physics by altering the behavior of the charged particles with which the
neutrino may interact [31, 32]. Moreover, a short review of other processes
involving neutrinos affected by the presence of a magnetic field can be found
in Ref. [33]. In this paper, our plan is as follows. In order to proceed in a
pedagogical way and to give the reader the necessary material, we present
first, in Sec. II, the detailed theoretical calculation of the DCS of the
scattering process in the absence of any external field. Then in Sec. III, the
explicit expression of the laser-assisted DCS is derived. The theoretical
changes required when considering the nonzero neutrino mass are summarized in
Sec. IV. The results obtained, whether in the absence or in the presence of
the laser, are presented and discussed in Sec. V. Finally, Sec. VI summarizes
the results of this work and draws conclusions. The relativistic system of
units, where $\hbar=c=1$, and the metric tensor $g^{\mu\nu}=diag(1,-1,-1,-1)$
will be used throughout this paper.

##  II Laser-free scattering process

In this section, we calculate analytically the general formula of the
unpolarized DCS for the scattering of a muon neutrino by an electron in the
absence of a laser field. This scattering process can be schematized as

$e^{-}(p_{i})+\nu_{\mu}(k_{i})\longrightarrow e^{-}(p_{f})+\nu_{\mu}(k_{f}),$
(1)

where the arguments are our labels for the associated four-momentum for each
particle, the indices $i$ and $f$ stand, respectively, for the initial and
final states. This process is a weak interaction process; it can be described
by the lowest Feynman diagrams. Therefore, in the first Born approximation,
the transition matrix element can be written as

$S_{fi}=\dfrac{-iG}{\sqrt{2}}\int_{-\infty}^{\infty}d^{4}x\bigg{[}\overline{\psi}^{\textsl{f}}_{\nu_{\mu}}(x)\gamma^{\mu}(1-\gamma_{5})\psi^{\textsl{i}}_{\nu_{\mu}}(x)\bigg{]}\bigg{[}\overline{\psi}^{\textsl{f}}_{e^{-}}(x)\gamma_{\mu}(g_{V}-g_{A}\gamma_{5})\psi^{\textsl{i}}_{e^{-}}(x)\bigg{]},$
(2)

where $\textit{G}=(1.166\leavevmode\nobreak\ 37\pm 0.000\leavevmode\nobreak\
02)\times 10^{-11}\leavevmode\nobreak\ \text{MeV}^{-2}$ is the coupling
constant of Fermi measured from muon decay [34].
$g_{V}=1/2-2\sin^{2}(\theta_{W})$ and $g_{A}=-1/2$ are, respectively, the
vector and axial-vector coupling constants and $\theta_{W}$ is the weak-mixing
angle. Here, for $g_{V}$ and $g_{A}$, we choose the corrected (loop-level)
values from the tree-level ones as follows: $g_{V}=0.043\pm 0.063$ and
$g_{A}=-0.545\pm 0.056$ [35]. The Dirac wave functions, normalized to the
volume $V$, that describe the incoming and outgoing muon neutrinos have the
following form:

$\begin{split}&\psi^{\textsl{i}}_{\nu_{\mu}}(x)=\dfrac{1}{\sqrt{2E_{i}V}}u_{\nu_{\mu}}(k_{i},t_{i})e^{-ik_{i}.x},\\\
&\psi^{\textsl{f}}_{\nu_{\mu}}(x)=\dfrac{1}{\sqrt{2E_{f}V}}u_{\nu_{\mu}}(k_{f},t_{f})e^{-ik_{f}.x},\end{split}$
(3)

where $u_{\nu_{\mu}}(k_{i,f},t_{i,f})$ represents the Dirac bispinor with
four-momentum $k_{i,f}$ and spin $t_{i,f}$ satisfying

$\sum_{t_{i,f}}\bar{u}_{\nu_{\mu}}(k_{i,f},t_{i,f})u_{\nu_{\mu}}(k_{i,f},t_{i,f})=\not{k}_{i,f}.$
(4)

In the first stage, we will consider the muon neutrinos as massless particles.
The wave functions $\psi_{e^{-}}^{\textsl{i}}(x)$ and
$\psi_{e^{-}}^{\textsl{f}}(x)$ are, respectively, the initial and final states
of the electron

$\begin{split}&\psi^{\textsl{i}}_{e^{-}}(x)=\dfrac{1}{\sqrt{2p_{i}^{0}V}}u_{e^{-}}(p_{i},s_{i})e^{-ip_{i}.x},\\\
&\psi^{\textsl{f}}_{e^{-}}(x)=\dfrac{1}{\sqrt{2p_{f}^{0}V}}u_{e^{-}}(p_{f},s_{f})e^{-ip_{f}.x},\end{split}$
(5)

where $p_{i}^{0}$ and $p_{f}^{0}$ are, respectively, the total energies of the
incident and outgoing electrons. $u_{e^{-}}(p_{i,f},s_{i,f})$ is the free
Dirac spinor satisfying
$\sum_{s_{i,f}}\bar{u}_{e^{-}}(p_{i,f},s_{i,f})u_{e^{-}}(p_{i,f},s_{i,f})=\not{p}_{i,f}+m$,
where $m$ is the rest mass of the electron. After introducing the different
wave functions describing the particles involved in the process (1), we
substitute them into the S-matrix element (2). After some manipulations, we
get

$S_{fi}=\dfrac{-iG}{\sqrt{32E_{i}E_{f}p_{i}^{0}p_{f}^{0}V^{4}}}(2\pi)^{4}\delta^{4}(p_{f}+k_{f}-p_{i}-k_{i})M_{fi},$
(6)

where

$M_{fi}=\big{[}\bar{u}_{\nu_{\mu}}(k_{f},t_{f})\gamma^{\mu}(1-\gamma^{5})u_{\nu_{\mu}}(k_{i},t_{i})\big{]}\big{[}\bar{u}_{e^{-}}(p_{f},s_{f})\gamma_{\mu}(g_{V}-g_{A}\gamma_{5})u_{e^{-}}(p_{i},s_{i})\big{]}.$
(7)

What interests us in the evaluation of the DCS is the square of the S-matrix
element $|S_{fi}|^{2}$ multiplied by the density of final states and divided
by the flux of the incoming particles $|J_{inc}|$. In the case of unpolarized
DCS, we must sum over the final spin states and average over the initial spin
states. We point out here, by the way, that electrons can be in two spin
states, while neutrinos exist in only one state of negative helicity [35].
This yields

$\begin{split}d\overline{\sigma}(e^{-}\nu_{\mu}\rightarrow
e^{-}\nu_{\mu})&=\leavevmode\nobreak\
V\int\dfrac{d^{3}p_{f}}{(2\pi)^{3}}\leavevmode\nobreak\
V\int\dfrac{d^{3}k_{f}}{(2\pi)^{3}}\leavevmode\nobreak\
\dfrac{1}{2}\sum_{t_{i,f},s_{i,f}}\dfrac{|S_{fi}(e^{-}\nu_{\mu}\rightarrow
e^{-}\nu_{\mu})|^{2}}{T|J_{inc}|},\\\
&=\dfrac{G^{2}}{64p_{i}^{0}p_{f}^{0}E_{i}E_{f}V}\dfrac{\delta^{4}(p_{f}+k_{f}-p_{i}-k_{i})}{(2\pi)^{2}|J_{inc}|}\int
d^{3}k_{f}\int d^{3}p_{f}\sum_{t_{i,f},s_{i,f}}|M_{fi}|^{2},\end{split}$ (8)

where we have used
$[(2\pi)^{4}\delta^{4}(p_{f}+k_{f}-p_{i}-k_{i})]^{2}=VT(2\pi)^{4}\delta^{4}(p_{f}+k_{f}-p_{i}-k_{i})$
and $J_{inc}$ is the incoming neutrinos current in the laboratory system given
by $J_{inc}=(k_{i}.p_{i})/(E_{i}p_{i}^{0}V)$. With the help of the following
relations $d^{3}p_{f}=|\textbf{p}_{f}|^{2}d|\textbf{p}_{f}|d\Omega$ and
$\delta^{4}(p_{f}+k_{f}-p_{i}-k_{i})=\delta^{0}(p_{f}^{0}+E_{f}-p_{i}^{0}-E_{i})\delta^{3}(\textbf{p}_{f}+\textbf{k}_{f}-\textbf{p}_{i}-\textbf{k}_{i})$,
the DCS becomes

$\dfrac{d\overline{\sigma}}{d\Omega}(e^{-}\nu_{\mu}\rightarrow
e^{-}\nu_{\mu})=\dfrac{G^{2}}{64(2\pi)^{2}(k_{i}.p_{i})}\int\dfrac{|\textbf{p}_{f}|^{2}d|\textbf{p}_{f}|}{E_{f}p_{f}^{0}}\delta^{0}(p_{f}^{0}+E_{f}-p_{i}^{0}-E_{i})\sum_{t_{i,f},s_{i,f}}|M_{fi}|^{2}\bigg{|}_{\textbf{p}_{f}+\textbf{k}_{f}-\textbf{p}_{i}-\textbf{k}_{i}=0.}$
(9)

The remaining integral over $d|\textbf{p}_{f}|$ can be solved by using the
following formula [35]

$\displaystyle\int
dxf(x)\delta(g(x))=\dfrac{f(x)}{|g^{\prime}(x)|}\bigg{|}_{g(x)=0.}$ (10)

Finally, we get

$\begin{split}\dfrac{d\overline{\sigma}}{d\Omega}(e^{-}\nu_{\mu}\rightarrow
e^{-}\nu_{\mu})=&\dfrac{G^{2}}{256\pi^{2}p_{f}^{0}E_{f}}\dfrac{|\textbf{p}_{f}|^{2}}{(k_{i}.p_{i})|g^{\prime}(|\textbf{p}_{f}|)|}\\\
&\times\text{Tr}\big{[}(\not{p}_{f}+m)\gamma_{\mu}(g_{V}-g_{A}\gamma_{5})(\not{p}_{i}+m)\gamma_{\nu}(g_{V}-g_{A}\gamma_{5})\big{]}\\\
&\times\text{Tr}\big{[}\not{k}_{f}\gamma^{\mu}(1-\gamma^{5})\not{k}_{i}\gamma^{\nu}(1-\gamma^{5})\big{]},\end{split}$
(11)

where

$g^{\prime}(|\textbf{p}_{f}|)=\dfrac{|\textbf{p}_{f}|}{\sqrt{|\textbf{p}_{f}|^{2}+m^{2}}}+\dfrac{|\textbf{p}_{f}|+E_{i}\cos(\theta_{f})-|\textbf{p}_{i}|F(\phi_{i},\phi_{f},\theta_{i},\theta_{f})}{E_{f}},$
(12)

and

$\begin{split}F(\phi_{i},\phi_{f},\theta_{i},\theta_{f})=&\cos(\phi_{i})\sin(\theta_{i})\cos(\phi_{f})\sin(\theta_{f})+\sin(\theta_{i})\sin(\phi_{i})\sin(\theta_{f})\sin(\phi_{f})\\\
&+\cos(\theta_{i})\cos(\theta_{f}).\end{split}$ (13)

The product of the two traces in Eq. (11) is evaluated and gives the following
result:

$\begin{split}\sum_{t_{i,f},s_{i,f}}|M_{fi}|^{2}=&64\big{[}g_{A}^{2}\big{(}(k_{f}.p_{f})(k_{i}.p_{i})+(k_{f}.p_{i})(k_{i}.p_{f})+(k_{i}.k_{f})m^{2}\big{)}+2g_{A}g_{V}\big{(}(k_{f}.p_{f})(k_{i}.p_{i})\\\
&-(k_{f}.p_{i})(k_{i}.p_{f})\big{)}+g_{V}^{2}\big{(}(k_{f}.p_{f})(k_{i}.p_{i})+(k_{f}.p_{i})(k_{i}.p_{f})-(k_{i}.k_{f})m^{2}\big{)}\big{]}.\end{split}$
(14)

The reader can refer to the textbook in Ref. [35] for another independent,
pedagogical, and more detailed calculation of this scattering process in the
absence of a laser field.

##  III Laser-assisted scattering process

Now, we consider the process (1) in the presence of a laser field. For reasons
of mathematical simplicity, the laser field is considered as a monochromatic
plane wave of circular polarization, whose classical four-potential can be
expressed in a unified notation such as

$A^{\mu}(x)=|\textbf{a}|\big{[}\eta_{1}^{\mu}\cos(k.x)+\eta_{2}^{\mu}\sin(k.x)\big{]},$
(15)

where $|\textbf{a}|=\mathcal{E}_{0}/\omega$ denotes the magnitude of the four-
potential with $\mathcal{E}_{0}$ is the electric field strength and $\omega$
is the laser frequency. $k=(\omega,0,0,\omega)$ is the wave 4-vector chosen to
be directed along the $z$-axis. The polarization 4-vectors $\eta_{1}^{\mu}$
and $\eta_{2}^{\mu}$ satisfy the normalization and orthogonality conditions,
$\eta_{1}^{2}=\eta_{2}^{2}=-1$ and $(\eta_{1}.\eta_{2})=0$. The Lorentz gauge
condition, $k_{\mu}.A^{\mu}=0$, applied to the four-potential implies
$(k.\eta_{1})=0$ and $(k.\eta_{2})=0$. Now, in the presence of a laser field,
the electron obeys the following Dirac equation [36]:

$\big{[}(p_{i,f}-eA)^{2}-m^{2}-\dfrac{ie}{2}F_{\mu\nu}\sigma^{\mu\nu}\big{]}\psi^{\textsl{i,f}}_{e^{-}}(x)=0,$
(16)

where $e=-|e|<0$ is the charge of the electron,
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the
electromagnetic field tensor and
$\sigma^{\mu\nu}=\tfrac{1}{2}[\gamma^{\mu},\gamma^{\nu}]$. The general
solution of this equation is the normalized relativistic Dirac-Volkov wave
functions [37], which describe the incident and outgoing electrons in a laser
field

$\psi^{\textsl{i,f}}_{e^{-}}(x)=\bigg{[}1+\dfrac{e\not{k}\not{A}}{2(k.p_{i,f})}\bigg{]}\frac{u(p_{i,f},s_{i,f})}{\sqrt{2Q_{i,f}V}}\times
e^{iS(q_{i,f},x)},$ (17)

where

$S(q_{i,f},x)=-q_{i,f}.x-\dfrac{e|\textbf{a}|(\eta_{1}.p_{i,f})}{(k.p_{i,f})}\sin(k.x)+\dfrac{e|\textbf{a}|(\eta_{2}.p_{i,f})}{(k.p_{i,f})}\cos(k.x).$
(18)

$q_{i,f}=(Q_{i,f},\textbf{q}_{i,f})$ is the Volkov momentum of the electron in
the presence of the laser field, which is given by

$q_{i,f}=p_{i,f}+\frac{e^{2}|\textbf{a}|^{2}}{2(k.p_{i,f})}k.$ (19)

Squaring this four-momentum shows that the mass of the dressed electron
(effective mass) has a dependence on the strength of the EM field

$m_{*}^{2}=m^{2}+e^{2}|\textbf{a}|^{2}.$ (20)

Note that in the absence of the EM field $(|\textbf{a}|\rightarrow 0)$, the
Volkov wave function, Eq. (17), reduces to the free-field wave function given
in Eq. (5), and the mass $m_{*}$ and four-momentum $q_{i,f}$ of the dressed
electron reduce also in that case to the electron mass $m$ and four-momentum
$p_{i,f}$, respectively. Since the incoming and outgoing muon neutrinos are
electrically neutral and thus do not interact with the laser field, their wave
functions are unchanged. We will therefore describe them using the same
previous unaffected plane waves given in Eq. (3). Thus, the transition matrix
element (2) becomes

$\begin{split}S_{fi}=&\dfrac{-iG}{\sqrt{32E_{i}E_{f}Q_{i}Q_{f}V^{4}}}\int
d^{4}x\leavevmode\nobreak\ e^{i(k_{f}-k_{i}).x}\leavevmode\nobreak\
e^{i(S(q_{i},x)-S(q_{f},x))}\Big{[}\bar{u}(p_{f},s_{f})\Big{(}1+C(p_{f})\not{A}\not{k}\Big{)}\\\
&\times\gamma_{\mu}(g_{V}-g_{A}\gamma_{5})\Big{(}1+C(p_{i})\not{k}\not{A}\Big{)}u(p_{i},s_{i})\Big{]}\Big{[}\bar{u}_{\nu_{\mu}}(k_{f},t_{f})\gamma^{\mu}(1-\gamma^{5})u_{\nu_{\mu}}(k_{i},t_{i})\Big{]},\end{split}$
(21)

where $C(p_{i})=e/(2(k.p_{i}))$ and $C(p_{f})=e/(2(k.p_{f}))$. We can recast
the exponential term in the appropriate form,
$e^{i(S(q,x)-S(q_{1},x))}=e^{i(q_{f}-q_{i}).x}\leavevmode\nobreak\
e^{-iz\sin(k.x-\varphi)}$, where

$z=e|\textbf{a}|\sqrt{\bigg{(}\dfrac{\eta_{1}.p_{i}}{k.p_{i}}-\dfrac{\eta_{1}.p_{f}}{k.p_{f}}\bigg{)}^{2}+\bigg{(}\dfrac{\eta_{2}.p_{i}}{k.p_{i}}-\dfrac{\eta_{2}.p_{f}}{k.p_{f}}\bigg{)}^{2}},$
(22)

and

$\varphi=\arctan[\frac{(\eta_{2}.p_{i})(k.p_{f})-(\eta_{2}.p_{f})(k.p_{i})}{(\eta_{1}.p_{i})(k.p_{f})-(\eta_{1}.p_{f})(k.p_{i})}\bigg{]}.$
(23)

Therefore, the transition matrix element becomes

$\begin{split}S_{fi}=&\dfrac{-iG}{\sqrt{32E_{i}E_{f}Q_{i}Q_{f}V^{4}}}\int
d^{4}x\leavevmode\nobreak\
e^{i(k_{f}+q_{f}-k_{i}-q_{i}).x}\leavevmode\nobreak\
e^{-iz\sin(k.x-\varphi)}\Big{[}\bar{u}(p_{f},s_{f})\big{(}\Delta_{0\mu}\\\
&+\Delta_{1\mu}\cos(k.x)+\Delta_{2\mu}\sin(k.x)\big{)}u(p_{i},s_{i})\Big{]}\Big{[}\bar{u}_{\nu_{\mu}}(k_{f},t_{f})\gamma^{\mu}(1-\gamma^{5})u_{\nu_{\mu}}(k_{i},t_{i})\Big{]},\end{split}$
(24)

where

$\begin{split}&\Delta_{0\mu}=\gamma_{\mu}\leavevmode\nobreak\
(g_{V}-g_{A}\gamma_{5})+2\leavevmode\nobreak\ C(p_{i})\leavevmode\nobreak\
C(p_{f})\leavevmode\nobreak\ |\textbf{a}|^{2}\leavevmode\nobreak\
k_{\mu}\leavevmode\nobreak\ \not{k}\leavevmode\nobreak\
(g_{V}-g_{A}\gamma_{5}),\\\
&\Delta_{1\mu}=C(p_{i})|\textbf{a}|\gamma_{\mu}\leavevmode\nobreak\
(g_{V}-g_{A}\gamma_{5})\not{k}\leavevmode\nobreak\
\not{\eta}_{1}+C(p_{f})|\textbf{a}|\leavevmode\nobreak\
\not{\eta}_{1}\leavevmode\nobreak\ \not{k}\leavevmode\nobreak\
\gamma_{\mu}\leavevmode\nobreak\ (g_{V}-g_{A}\gamma_{5}),\\\
&\Delta_{2\mu}=C(p_{i})|\textbf{a}|\gamma_{\mu}\leavevmode\nobreak\
(g_{V}-g_{A}\gamma_{5})\not{k}\leavevmode\nobreak\
\not{\eta}_{2}+C(p_{f})|\textbf{a}|\leavevmode\nobreak\
\not{\eta}_{2}\leavevmode\nobreak\ \not{k}\leavevmode\nobreak\
\gamma_{\mu}\leavevmode\nobreak\ (g_{V}-g_{A}\gamma_{5}).\end{split}$ (25)

The linear combination of the three different quantities in Eq. (24) can be
transformed by the well-known Jacobi-Anger identity involving ordinary Bessel
functions $J_{n}(z)$

$\begin{split}\begin{Bmatrix}1\\\ \cos(k.x)\\\ \sin(k.x)\end{Bmatrix}\times
e^{-iz\sin(k.x-\varphi)}&=\sum_{n=-\infty}^{+\infty}e^{-in(k.x)}\begin{Bmatrix}J_{n}(z)e^{in\varphi}\\\
\frac{1}{2}\big{\\{}J_{n+1}(z)e^{i(n+1)\varphi}+J_{n-1}(z)e^{i(n-1)\varphi}\big{\\}}\\\
\frac{1}{2i}\big{\\{}J_{n+1}(z)e^{i(n+1)\varphi}-J_{n-1}(z)e^{i(n-1)\varphi}\big{\\}}\end{Bmatrix},\\\
&=\sum_{n=-\infty}^{+\infty}e^{-in(k.x)}\begin{Bmatrix}b_{n}(z)\\\
b_{1n}(z)\\\ b_{2n}(z)\end{Bmatrix},\end{split}$ (26)

where $z$ is the argument of the Bessel functions defined in Eq. (22) and $n$,
their order, is interpreted as the number of exchanged photons. After the
integration, the S-matrix element $S_{fi}$ becomes

$\begin{split}S_{fi}=\dfrac{-iG}{\sqrt{32E_{i}E_{f}Q_{i}Q_{f}V^{4}}}\sum_{n=-\infty}^{+\infty}(2\pi)^{4}\delta^{4}(q_{f}+k_{f}-q_{i}-k_{i}-nk)\leavevmode\nobreak\
M^{n}_{fi}.\end{split}$ (27)

The quantity $M^{n}_{fi}$ is defined by

$M^{n}_{fi}=\big{[}\bar{u}(p_{f},s_{f})\Gamma_{\mu}^{n}u(p_{i},s_{i})\big{]}\big{[}\bar{u}_{\nu_{\mu}}(k_{f},t_{f})\gamma^{\mu}(1-\gamma^{5})u_{\nu_{\mu}}(k_{i},t_{i})\big{]},$
(28)

where

$\Gamma_{\mu}^{n}=\Delta_{0\mu}b_{n}(z)+\Delta_{1\mu}b_{1n}(z)+\Delta_{2\mu}b_{2n}(z).$
(29)

To evaluate the DCS in the presence of a laser field, we follow the same steps
as in the absence of the laser field in the previous section. This yields

$\begin{split}d\overline{\sigma}=&\sum_{n,l=-\infty}^{+\infty}\dfrac{G^{2}}{32E_{i}E_{f}Q_{i}Q_{f}V^{4}T|J_{inc}|}V\int\dfrac{d^{3}q_{f}}{(2\pi)^{3}}\leavevmode\nobreak\
V\int\dfrac{d^{3}k_{f}}{(2\pi)^{3}}\leavevmode\nobreak\
\dfrac{1}{2}\sum_{t_{i,f},s_{i,f}}(2\pi)^{8}\delta^{4}(q_{f}+k_{f}-q_{i}-k_{i}-nk)\\\
&\times\delta^{4}(q_{f}+k_{f}-q_{i}-k_{i}-lk)M^{l*}_{fi}M^{n}_{fi}.\\\
\end{split}$ (30)

From the inspection, one can see that both 4-dimensional $\delta$ functions
imply that for there to be any contribution to the summation, either there is
no incoming photon energy ($E_{\gamma}=0$) or, more appropriately, that $l=n$.
We can therefore replace $M^{l*}_{fi}M^{n}_{fi}$ by the square of the norm of
the scattering amplitude $|M^{n}_{fi}|^{2}$ and exclude the sum over $l$. We
get for the unpolarized DCS

$\begin{split}d\overline{\sigma}=&\sum_{n=-\infty}^{+\infty}\dfrac{G^{2}}{64E_{i}E_{f}Q_{i}Q_{f}V^{2}T|J_{inc}|}\int\dfrac{d^{3}q_{f}}{(2\pi)^{3}}\leavevmode\nobreak\
\int\dfrac{d^{3}k_{f}}{(2\pi)^{3}}[(2\pi)^{4}\delta^{4}(q_{f}+k_{f}-q_{i}-k_{i}-nk)]^{2}\sum_{t_{i,f},s_{i,f}}|M^{n}_{fi}|^{2},\\\
\end{split}$ (31)

which is simplified after some manipulation into the following form:

$\dfrac{d\overline{\sigma}}{d\Omega}=\sum_{n=-\infty}^{+\infty}\dfrac{G^{2}}{64Q_{f}Q_{i}E_{f}E_{i}}\dfrac{|\textbf{q}_{f}|^{2}d|\textbf{q}_{f}|}{(2\pi)^{2}|J_{inc}|V}\delta^{0}(Q_{f}+E_{f}-Q_{i}-E_{i}-n\omega)\sum_{t_{i,f},s_{i,f}}|M^{n}_{fi}|^{2}\bigg{|}_{\textbf{q}_{f}+\textbf{k}_{f}-\textbf{q}_{i}-\textbf{k}_{i}-n\textbf{k}=0.}$
(32)

Using the formula (10) to perform the remaining integral over
$d|\textbf{q}_{f}|$ and replacing the incoming current
$|J_{inc}|=(k_{i}.q_{i})/(Q_{i}E_{i}V)$, we obtain

$\begin{split}\Big{(}\dfrac{d\overline{\sigma}}{d\Omega}\Big{)}^{\text{with
laser}}=\sum_{n=-\infty}^{+\infty}\dfrac{d\overline{\sigma}^{n}}{d\Omega}=&\sum_{n=-\infty}^{+\infty}\dfrac{G^{2}}{256\pi^{2}Q_{f}E_{f}}\dfrac{|\textbf{q}_{f}|^{2}}{(k_{i}.q_{i})|g^{\prime}(|\textbf{q}_{f}|)|}\\\
&\times\text{Tr}\big{[}(\not{p}_{f}+m)\Gamma_{\mu}^{n}(\not{p}_{i}+m)\overline{\Gamma}_{\nu}^{n}\big{]}\\\
&\times\text{Tr}\big{[}\not{k}_{f}\gamma^{\mu}(1-\gamma^{5})\not{k}_{i}\gamma^{\nu}(1-\gamma^{5})\big{]},\end{split}$
(33)

where

$g^{\prime}(|\textbf{q}_{f}|)=\dfrac{|\textbf{q}_{f}|}{\sqrt{|\textbf{q}_{f}|^{2}+m_{*}^{2}}}+\dfrac{|\textbf{q}_{f}|+E_{i}\cos(\theta_{f})-n\omega\cos(\theta_{f})-|\textbf{q}_{i}|F(\phi_{i},\phi_{f},\theta_{i},\theta_{f})}{E_{f}},$
(34)

and

$\begin{split}\overline{\Gamma}_{\nu}^{n}&=\gamma^{0}\Gamma_{\nu}^{n+}\gamma^{0},\\\
&=\overline{\Delta}_{0\nu}b_{n}^{*}(z)+\overline{\Delta}_{1\nu}b_{1n}^{*}(z)+\overline{\Delta}_{2\nu}b_{2n}^{*}(z),\end{split}$
(35)

where

$\begin{split}&\overline{\Delta}_{0\nu}=\gamma_{\nu}\leavevmode\nobreak\
(g_{V}-g_{A}\gamma_{5})+2\leavevmode\nobreak\ C(p_{i})\leavevmode\nobreak\
C(p_{f})\leavevmode\nobreak\ |\textbf{a}|^{2}\leavevmode\nobreak\
k_{\nu}\leavevmode\nobreak\ \not{k}\leavevmode\nobreak\
(g_{V}-g_{A}\gamma_{5}),\\\
&\overline{\Delta}_{1\nu}=C(p_{i})|\textbf{a}|\leavevmode\nobreak\
\not{\eta}_{1}\leavevmode\nobreak\ \not{k}\gamma_{\nu}\leavevmode\nobreak\
(g_{V}-g_{A}\gamma_{5})+C(p_{f})|\textbf{a}|\leavevmode\nobreak\
\gamma_{\nu}\leavevmode\nobreak\
(g_{V}-g_{A}\gamma_{5})\not{k}\leavevmode\nobreak\ \not{\eta}_{1},\\\
&\overline{\Delta}_{2\nu}=C(p_{i})|\textbf{a}|\leavevmode\nobreak\
\not{\eta}_{2}\leavevmode\nobreak\ \not{k}\gamma_{\nu}\leavevmode\nobreak\
(g_{V}-g_{A}\gamma_{5})+C(p_{f})|\textbf{a}|\leavevmode\nobreak\
\gamma_{\nu}\leavevmode\nobreak\
(g_{V}-g_{A}\gamma_{5})\not{k}\leavevmode\nobreak\ \not{\eta}_{2}.\end{split}$
(36)

The evaluation of the traces is commonly performed with the help of FEYNCALC
[38, 39, 40]. The result we obtained is attached in the Appendix. In order to
distinguish between them and to avoid any confusion, it would be very
appropriate to consider the summed differential cross section (SDCS),
$(d\overline{\sigma}/d\Omega)^{\text{with laser}}$, as the sum of discrete
individual differential cross sections (IDCS),
$d\overline{\sigma}^{n}/d\Omega$, for each photon exchange process.
Note that in the absence of an EM field (i.e., when we take the limit
$|\textbf{a}|\rightarrow 0$ and without the exchange of any photons $(n=0)$),
the argument of the Bessel functions vanishes $(z=0)$. Thus, all the terms
that contribute to $\Gamma_{\mu}^{n}$ given in Eq. (29) will be null except
for the term $\Delta_{0\mu}$, since $b_{0}(0)=1$ and $b_{10}(0)=b_{20}(0)=0$.
The expression of $\Delta_{0\mu}$ in Eq. (25) reduces, for
$|\textbf{a}|\rightarrow 0$, to $\gamma_{\mu}\leavevmode\nobreak\
(g_{V}-g_{A}\gamma_{5})$, and then the quantity $M^{n=0}_{fi}$ (in Eq. (28))
becomes that obtained in the absence of the laser field (Eq. (7)). The same
thing applies to the S-matrix element $S_{fi}$ and all other quantities that
compose the DCS in Eq. (33).

##  IV Nonzero neutrino mass effect

Throughout the calculation performed in the two previous sections, we have
disregarded the mass of the muon neutrinos $(m_{\nu_{\mu}}=0)$. In this
section, and in order to reveal the effect induced by the nonzero mass of
neutrinos, we will show what changes will theoretically occur in some of the
relations when considering the mass of neutrinos. But first, we will try to
give some brief words about the discovery that neutrinos have a nonnegligible
mass. Within the standard model of elementary particle physics, the masses of
the neutrinos are assumed to be exactly zero. It is only recently that it has
been confirmed that neutrinos have small but nonzero masses by discovering the
phenomenon of neutrino oscillations [41]. Neutrino oscillation is a quantum
mechanical phenomenon in which a neutrino produced in a specific lepton flavor
can later be measured to have a different lepton flavor after traveling some
distances. Neutrino oscillation has been discovered through studies of
neutrinos produced by cosmic-ray interactions in the atmosphere [42, 43].
Since we are interested in the present research on the muon neutrino
$\nu_{\mu}$, its discovery in 1962 [44] by Leon Lederman, Melvin Schwartz and
Jack Steinberger was rewarded with the 1988 Nobel Prize in Physics [45]. An
evidence of its oscillations to $\nu_{\tau}$ in the KEK to Kamioka (K2K)
accelerator-based experiment has been reported in Ref. [46]. An upper limit of
0.17 MeV (at the $90\%$ confidence level) for the muon-neutrino mass
$(m_{\nu_{\mu}}<0.17\leavevmode\nobreak\ \text{MeV})$ has been derived in [47]
by analyzing the decay at rest of positively charged pions. To be more
precise, it is very important to point out here that in spite of all the
above, the mass of the muon neutrino remains a not well defined principle, and
in contrast we introduce what is called the effective muon neutrino mass,
which is an incoherent sum of neutrino masses and lepton mixing matrix
elements [48]. For illustration purposes and in order to make the effect
visible we will take some liberties in the determination of the muon neutrino
mass. On this basis, when presenting the results for the nonzero neutrino
mass, we choose the muon neutrino mass as
$m_{\nu_{\mu}}=0.15\leavevmode\nobreak\ \text{MeV}$. Now, let us summarize
some of the changes that will occur in the theoretical calculation. We start
with the relation, Eq. (4), that Dirac bispinor of neutrino verifies, which
will be the following:
$\sum_{t_{i,f}}\bar{u}_{\nu_{\mu}}(k_{i,f},t_{i,f})u_{\nu_{\mu}}(k_{i,f},t_{i,f})=\not{k}_{i,f}+m_{\nu_{\mu}}.$
This change will be manifested in the calculation of the trace related to the
neutrino current. Note also that the relation $E_{i,f}=|\textbf{k}_{i,f}|$
will not be valid anymore, we retain instead the original relativistic energy-
momentum relation, $E_{i,f}=\sqrt{|\textbf{k}_{i,f}|^{2}+m_{\nu_{\mu}}^{2}}$.
The other thing that will change is the incoming neutrinos flux $J_{inc}$,
which will remain in its original formula as follows:
$J_{inc}=\sqrt{(k_{i}.p_{i})^{2}-m_{\nu_{\mu}}^{2}m^{2}}/(E_{i}p_{i}^{0}V)$.
With all of this, the entire dependence on the mass of neutrino is fully taken
into account. The results of this effect on the DCS will be presented and
discussed in the next section.

##  V Results and discussion

This section will be devoted to the presentation and discussion of numerical
results related to the DCS of the electron and muon neutrino elastic
scattering process (1) in the absence and presence of the electromagnetic
field. The DCS is derived with respect to the solid angle of the electron due
to the fact that neutrinos are somewhat difficult to capture experimentally.
In order to obtain the total cross section (TCS), the integral over that solid
angle must be computed using numerical methods. For the geometry, we consider
the momentum of the incoming muon neutrino $\textbf{k}_{i}$, with kinetic
energy $E_{\nu}^{\text{kin}}$, along the opposite direction of the $z$-axis.
We set both the initial and final electrons in a general geometry with
spherical coordinates $\theta_{i}$, $\theta_{f}$, $\phi_{i}$ and $\phi_{f}$.
The angles $\phi_{i}$ and $\phi_{f}$ were chosen to be zero
($\phi_{i}=\phi_{f}=0^{\circ}$) in all results presented in this section. The
momentum of the outgoing muon neutrino $\textbf{k}_{f}$ can be deduced from
the previous ones by using the momentum conservation relationship,
$\textbf{q}_{f}+\textbf{k}_{f}-\textbf{q}_{i}-\textbf{k}_{i}-n\textbf{k}=0$.
Considering the effects of relativity and spin, we choose the kinetic energy
of the incoming electron as (unless otherwise stated)
$E_{e}^{\text{kin}}=10^{6}\leavevmode\nobreak\ \text{eV}$, and that of muon
neutrinos varies in the range between $0.1$ and $10^{15}\leavevmode\nobreak\
\text{eV}$ depending on their sources [49].

###  V.1 Without laser field

In addition to pedagogical purposes, the main goal of performing calculations
in the absence of a laser field is simply to provide a means that enables us
to ensure the consistency of our calculations when a laser is added. This is
done by taking the limit of the field strength $\mathcal{E}_{0}$ and the
number of photons exchanged $n$ as they all tend to zero and then checking if
the dressed DCS gives, under these conditions, its corresponding one in the
absence of the laser. By the way, this check has been carried out successfully
and the result is shown in Fig. 1.

Figure 1: The two DCSs with and without laser, drawn as a function of the
scattering angle $\theta_{f}$ for an electric field strength of
$\mathcal{E}_{0}=0\leavevmode\nobreak\ \text{V/cm}$ and without any exchange
of photons ($n=0$). The kinetic energy of the incoming muon neutrino is
$E_{\nu}^{\text{kin}}=0.5\times 10^{6}$ eV. The initial angle $\theta_{i}$ is
(a) $\theta_{i}=1^{\circ}$ and (b) $\theta_{i}=45^{\circ}$.

Due to their perfect overlap, it is hardly possible to distinguish between the
two curves that represent the changes in the two DCSs (with and without laser)
as a function of the final angle $\theta_{f}$ of the electron. This represents
a consistency check of our calculations. The difference between Figs. 1blue(a)
and 1blue(b) is only in the geometry, where in Fig. 1blue(a) the initial angle
is $\theta_{i}=1^{\circ}$ and in Fig. 1blue(b) it is $\theta_{i}=45^{\circ}$.
We note here that the choice $\theta_{i}=1^{\circ}$ was intentionally made
only to avoid the divergence that occurs in the DCS in the case of
$\theta_{i}=\theta_{f}=0^{\circ}$. In Fig. 1blue(a), the DCS exhibits a
symmetry in its curve with respect to the angle $\theta_{f}=0^{\circ}$, where
it has a maximum value, in contrast to Fig. 1blue(b) in which the DCS is
asymmetrical and its peak is shifted to large final angles ($\theta_{f}\simeq
50^{\circ}$). This clearly shows the effect of geometry on the angular
distribution of the DCS. Another thing worth paying attention to when
examining these figures is to get information about the final angle
$\theta_{f}$ at which the electron will go out after scattering from the
initial angle $\theta_{i}$ at which it comes in. On the basis of these two
figures, it appears that the peak is located approximately around the final
angle that is equal to the initial one ($\theta_{f}\sim\theta_{i}$). This fact
is valid for any geometry, provided that $\phi_{i}=\phi_{f}=0^{\circ}$. This
means that an electron incoming at a small angle $\theta_{i}$ has a high
probability to be scattered at a small final angle $\theta_{f}$ with a value
approximately equal to the initial angle. As long as $\theta_{i}$ increases,
the final angle $\theta_{f}$ also increases.
For further comparison, we give here a value for the total cross section
obtained theoretically at the kinetic energy of the electron
$E_{e}^{\text{kin}}=0.05\times 10^{6}$ eV:

$\frac{1}{E_{\nu}^{\text{kin}}}\overline{\sigma}(e^{-}\nu_{\mu}\rightarrow
e^{-}\nu_{\mu})=1.40\times 10^{-42}\leavevmode\nobreak\
\text{cm}^{2}/\text{GeV},$ (37)

where $E_{\nu}^{\text{kin}}=0.5\times 10^{6}$ eV, and the error from the
numerical integration performed using Simpson’s rule is estimated to be less
than $10^{-18}$. This theoretical value given in Eq. (37) is to be compared
with that obtained experimentally [35, 50, 51]

$\frac{1}{E_{\nu}^{\text{kin}}}\overline{\sigma}(e^{-}\nu_{\mu}\rightarrow
e^{-}\nu_{\mu})=(1.45\pm 0.26)\times 10^{-42}\leavevmode\nobreak\
\text{cm}^{2}/\text{GeV}.$ (38)

Table 1: Numerical values of the laser-free DCS (Eq. (11)) taking into account the mass of muon neutrino for different neutrino kinetic energies. The initial and final angles are $\theta_{i}=1^{\circ}$ and $\theta_{f}=0^{\circ}$. | Laser-free DCS ($10^{-36}\leavevmode\nobreak\ \text{eV}^{-2}$)
---|---
$E_{\nu}^{\text{kin}}$ (eV) | $m_{\nu_{\mu}}=0.15\leavevmode\nobreak\ \text{MeV}$ | $m_{\nu_{\mu}}=0$
$0.1$ | $4.95$ | $4.58$
$0.5$ | $4.95$ | $4.58$
$1$ | $4.95$ | $4.58$
$5$ | $4.94$ | $4.58$
$10$ | $4.93$ | $4.58$
$50$ | $4.91$ | $4.58$
$10^{2}$ | $4.89$ | $4.57$
$5\times 10^{2}$ | $4.83$ | $4.53$
$10^{3}$ | $4.78$ | $4.47$
$5\times 10^{4}$ | $4.31$ | $4.15$
$10^{5}$ | $4.24$ | $4.15$
$5\times 10^{5}$ | $4.17$ | $4.16$
$10^{6}$ | $4.16$ | $4.16$
$5\times 10^{6}$ | $4.16$ | $4.16$
$10^{7}$ | $4.16$ | $4.16$
$5\times 10^{7}$ | $4.16$ | $4.16$
$10^{8}$ | $4.16$ | $4.16$
$5\times 10^{8}$ | $4.16$ | $4.16$
$10^{9}$ | $4.16$ | $4.16$
$10^{10}$ | $4.16$ | $4.16$
$10^{14}$ | $2.85\times 10^{12}$ | $2.85\times 10^{12}$
$10^{15}$ | $1.65\times 10^{18}$ | $1.65\times 10^{18}$

Instead of neglecting the neutrinos mass, let us now see what happens to the
DCS if we consider their mass. Some of the changes that will occur in the
theoretical calculation were summarized previously. In what follows, we will
discuss the results of the nonzero neutrinos mass effect on the DCS in the
absence of the laser and discover under what conditions this effect
disappears. In Table 1, we list the values of the laser-free DCS as a function
of the kinetic energy of the neutrino. The results for massless neutrinos are
also included for comparison. The kinetic energy of the neutrino has been
intentionally varied from its very low values to very high ones in order to
make the effect of a nonzero neutrino mass more noticeable. From this table,
one can clearly see the slight difference that exists between the DCS values
at lower neutrino kinetic energies. The DCS when considering neutrino mass
develops a small correction to its equivalent when neglecting neutrino mass,
and thus it always remains the largest. With the increase of the neutrino
kinetic energy, from value $E_{\nu}^{\text{kin}}=10^{6}\leavevmode\nobreak\
\text{eV}$, we see that the difference between the values disappears gradually
until it completely vanishes at very high kinetic energies. This result could
be used as a justification for choosing the value of the neutrino kinetic
energy in the case of zero neutrino mass to $10^{6}\leavevmode\nobreak\
\text{eV}$ order of magnitude and beyond. Hereafter, we take the kinetic
energy of the neutrino exactly as $E_{\nu}^{\text{kin}}=0.5\times
10^{6}\leavevmode\nobreak\ \text{eV}$.

###  V.2 With laser field

After a thorough discussion of the results obtained in the absence of a laser
field, let us now turn to see what happens when we embed our scattering
process (1) in a circularly polarized electromagnetic field. The latter can be
supplied experimentally by applying a laser instrument [52]. The direction of
the field wave vector k is along the $z$-axis, whereas the polarization
vectors $\boldsymbol{\eta_{1}}$ and $\boldsymbol{\eta_{2}}$ perpendicular to k
are along the $x$\- and $y$-axes, respectively. In the absence of a laser, our
DCS depends on the total energies of the incoming electron and neutrino, as
well as on the spherical angles. In the presence of the laser, three other
variables are added to these ones; two of them, the field strength
$\mathcal{E}_{0}$ and frequency $\omega$, characterize the laser and the third
is the number of photons exchanged $n$ between the laser and the scattering
system, which emerged due to the introduction of ordinary Bessel functions in
the theoretical calculation. Therefore, to cover the laser-free results, it is
sufficient, if the calculation is accurate, to cancel the number of photons
$n$ and the field strength $\mathcal{E}_{0}$. Similarly, the effect of the
laser on the DCS will be highlighted by studying its variations as a function
of those parameters related to the electromagnetic field. Let us start by
presenting the results related to how the laser affects the photons exchange
phenomenon.

Figure 2: The behavior of the IDCS, $d\overline{\sigma}^{n}/d\Omega$, versus
the number of photons $n$. The different parameters are (a) $\hbar\omega=1.17$
eV, $\theta_{i}=1^{\circ}$ and $\theta_{f}=0^{\circ}$, (b)
$\mathcal{E}_{0}=10^{7}$ V/cm, $\theta_{i}=1^{\circ}$ and
$\theta_{f}=0^{\circ}$, (c) $\mathcal{E}_{0}=10^{7}$ V/cm, $\hbar\omega=1.17$
eV, $\theta_{i}=1^{\circ}$ and $\theta_{f}=0^{\circ}$ and (d)
$\mathcal{E}_{0}=10^{7}$ V/cm and $\hbar\omega=1.17$ eV.

Figure 2 depicts the effect of field strength $\mathcal{E}_{0}$ (Fig.
2blue(a)), frequency $\hbar\omega$ (Fig. 2blue(b)), electron kinetic energy
(Fig. 2blue(c)) and geometry (Fig. 2blue(d)) on the photon exchange process.
These graphs, which represent the changes in the IDCS in terms of photon
number $n$, exhibit symmetrical envelopes with respect to the $n=0$-axis,
which indicates that the photon emission processes $(n>0)$ are exactly equal
to the photon absorption processes $(n<0)$. We did not join the points
together because the number of photons $n$ is a discrete relative integer. The
oscillations of these envelopes and their abrupt drop on the sides of each
figure, as well as the positions of the peaks, are due to the well-known
properties of Bessel functions [53]. In Fig. 2blue(a), we show the IDCS
changes in terms of the number of photons $n$ for two different field
strengths, $\mathcal{E}_{0}=10^{6}$ V/cm and $\mathcal{E}_{0}=10^{7}$ V/cm, at
a specific laser frequency $\hbar\omega=1.17$ eV. It seems clear to us that
the photon exchange enhances with the increase of the field strength. In the
case of $\mathcal{E}_{0}=10^{6}$ V/cm, only a few photons are exchanged and
the cutoff number is about $n=\pm 7$ compared to the case of
$\mathcal{E}_{0}=10^{7}$ V/cm where the cutoff number is $n=\pm 45$. A large
number of exchanged photons indicates that the scattering process interacts
strongly with the laser, which makes the effect of the latter on it important
and prominent. In Fig. 2blue(b), the field strength is fixed at $10^{7}$ V/cm
and the multiphoton processes are displayed for two different frequencies,
$\hbar\omega=1.17$ eV and $\hbar\omega=2$ eV. It appears that at lower
frequencies, the exchange of photons is important compared to higher
frequencies. Based on the two Figs. 2blue(a) and 2blue(b), we conclude that
the photon exchange process between the laser and the scattering process is
related to the properties of the applied laser. Figure 2blue(c) shows the
dependence of the multiphoton phenomenon on the kinetic energy of the
electron. We note that when the kinetic energy of the incoming electron
increases, its interaction with the laser field enhances, which causes an
exchange of a large number of photons between them. Figure 2blue(d)
illustrates how the chosen geometry can also affect the multiphoton processes.
The number of exchanged photons in geometry $\theta_{i}=1^{\circ}$ and
$\theta_{f}=0^{\circ}$ is greater than that exchanged in geometry
$\theta_{i}=45^{\circ}$ and $\theta_{f}=50^{\circ}$. The difference between
these two geometries is that the first means, according to well-known
spherical coordinates, that the electron is incoming almost in the same
direction as the field vector k, which is the $z$-axis direction, while in the
second geometry the electron comes at an angle of $\theta_{i}=45^{\circ}$ to
the $z$-axis. This indicates that the electron interacts with the laser if
they are incoming together in the same direction more than if they are in two
different directions.

Figure 3: The variations of the IDCS, $d\overline{\sigma}^{n}/d\Omega$, versus
the number of photons $n$ for very low electron kinetic energies
$E_{e}^{\text{kin}}=4\times 10^{2}$ and $10^{3}$ eV. The laser parameters are
$\hbar\omega=1.17$ eV and $\mathcal{E}_{0}=10^{9}$ V/cm. The geometry
parameters are $\theta_{i}=1^{\circ}$ and $\theta_{f}=0^{\circ}$.

In Fig. 3, we have included the result obtained in the case of very low
electron kinetic energy in order to consider the situation where the electron
is (at least approximately) at rest. This case is more natural and closer to
realistic experimental conditions. From Fig. 3, it appears that in the case
where the electron is almost at rest, it interacts little with the laser
field, since it has exchanged very few photons, even at high field strengths
($10^{9}$ V/cm), compared to the case in which it has more kinetic energy.
Therefore, the laser will not have a significant effect on an electron that is
almost at rest. The effect of the laser at moderate intensities will be clear
and significant only when the electron moves with a certain kinetic energy
that is not neglected. For example at $\mathcal{E}_{0}=10^{7}$ V/cm and for
the kinetic energy $E_{e}^{\text{kin}}=10^{3}$ eV, we found that the electron
has emitted and absorbed only one photon ($n=\pm 1$).

Figure 4: The variations of the SDCS as a function of the final angle $\theta_{f}$ for various numbers of photons exchanged. The laser field strength and frequency are, respectively, $\mathcal{E}_{0}=10^{7}$ V/cm and $\hbar\omega=1.17$ eV. The initial angle is (a) $\theta_{i}=1^{\circ}$ and (b) $\theta_{i}=45^{\circ}$. The notation $n=\pm N$ means that we have summed over the range of values $-N\leq n\leq+N$. Table 2: Numerical values of the SDCS summed over a fixed number of photons $-20\leq n\leq+20$ in terms of field strength for three different frequencies. The initial and final angles are $\theta_{i}=1^{\circ}$ and $\theta_{f}=0^{\circ}$, respectively. | SDCS ($10^{-36}\leavevmode\nobreak\ \text{eV}^{-2}$)
---|---
$\mathcal{E}_{0}\leavevmode\nobreak\ (\text{V/cm})$ | $\hbar\omega=0.117\leavevmode\nobreak\ \text{eV}$ | $\hbar\omega=1.17\leavevmode\nobreak\ \text{eV}$ | $\hbar\omega=2\leavevmode\nobreak\ \text{eV}$
$10$ | $4.16$ | $4.16$ | $4.16$
$10^{2}$ | $4.16$ | $4.16$ | $4.16$
$10^{3}$ | $4.16$ | $4.16$ | $4.16$
$10^{4}$ | $4.16$ | $4.16$ | $4.16$
$10^{5}$ | $1.43$ | $4.16$ | $4.16$
$10^{6}$ | $1.32\times 10^{-1}$ | $4.16$ | $4.16$
$10^{7}$ | $1.33\times 10^{-2}$ | $1.43$ | $4.16$
$10^{8}$ | $1.35\times 10^{-3}$ | $1.32\times 10^{-1}$ | $4.05\times 10^{-1}$
$10^{9}$ | $1.49\times 10^{-4}$ | $1.37\times 10^{-2}$ | $3.94\times 10^{-2}$
$10^{10}$ | $2.29\times 10^{-5}$ | $1.50\times 10^{-3}$ | $4.06\times 10^{-3}$
$10^{11}$ | $3.54\times 10^{-6}$ | $2.36\times 10^{-4}$ | $1.54\times 10^{-3}$

So far, we have discussed the variations of the IDCS in terms of the number of
photons exchanged and reported the effect of different parameters on the
photon exchange process. Let us now show the results of the laser-assisted DCS
summed over a given number of photons. Figure 4 presents the changes of the
SDCS summed over a different numbers of exchanged photons as a function of the
final scattering angle $\theta_{f}$. The variations of the laser-free DCS are
also included for comparison. From this figure, we see that the laser-free DCS
curve is always the highest. It is also noted that with the increase of the
number of photons over which we summed, the SDCS tends towards the laser-free
one so that they overlap perfectly when reaching the cutoff number. This can
be translated mathematically as follows:

$\sum_{n=-\text{cutoff}}^{+\text{cutoff}}\dfrac{d\overline{\sigma}^{n}}{d\Omega}=\bigg{(}\dfrac{d\overline{\sigma}}{d\Omega}\bigg{)}^{laser-
free}.$ (39)

The cutoff number differs from one geometry to another, being $n=\pm 60$ in
geometry $\theta_{i}=1^{\circ}$ and $n=\pm 30$ in geometry
$\theta_{i}=45^{\circ}$ (see Fig. 2blue(d)). This convergence reached in these
two figures is known by the so-called sum rule (Eq. (39)) [54], which has been
previously confirmed to be fulfilled in several processes [12, 55, 56, 57,
58]. At first, it was demonstrated in the atomic processes that occur in the
presence of the laser in the nonrelativistic regime [59, 60, 61, 62], and then
it was applied to other relativistic scattering and weak decay processes. This
sum rule is fulfilled only when summing over a specified number of photons,
called the cutoff number, and above $(|n|\geq|\text{cutoff}|)$. The cutoff
number is defined as a fixed number of photons from which the IDCS
$d\overline{\sigma}^{n}/d\Omega$ suddenly falls off
$(d\overline{\sigma}^{n}/d\Omega=0)$. It can be explained by the well-known
properties of the Bessel function, which decreases considerably when its order
$n$ is approximately equal to its argument $z$ [see Fig. (5)]. Our main
objective in drawing the envelopes shown in Fig. (2) was to determine the
cutoff number, at each specific field strength and frequency. In the context
of laser-matter interaction, the only special functions included in the
theoretical calculation are the Bessel functions. Consequently, their
mathematical properties inevitably play an important role in the behavior of
the calculated quantities, for example the SDCS here. The sum rule is thus,
like the cutoff number, a reflection of the properties of Bessel functions.
Let us give an example to illustrate this. If we look at the SDCS expression
in Eq. (33) and the expression for the two traces given in the Appendix, we
find that each term in the sum corresponds to a process in which a net number
$n$ of photons are emitted (or absorbed for negative $n$). Let us take, for
example, the first three terms of Eq. (43) in the Appendix, which are
multiplied by the square of the ordinary Bessel functions $J_{n}^{2}(z)$,
$J_{n+1}^{2}(z)$ and $J_{n-1}^{2}(z)$. In Fig. (5), we give the changes of
these functions in terms of order $n$ and for different arguments $z$. It is
clear that the shape of the envelopes in this figure is the same as the one
shown in Fig. (2). It is these functions that give this shape to the envelopes
that present the variations of the IDCS with respect to the number of photons
exchanged $n$. The sum of these functions over order $n$ leads to two
different cases. The first is the summation over an order $n$ less than the
cutoff number $(|n|<|\text{cutoff}|)$ for each Bessel function, which gives

$\sum_{-80}^{80}J_{n}^{2}(100)=0.588359;\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\sum_{-100}^{100}J_{n+1}^{2}(150)=0.467531;\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\sum_{-150}^{150}J_{n-1}^{2}(200)=0.541836.$ (40)

The second is the summation over an order $n$ greater than or equal to the
cutoff number $(|n|\geq|\text{cutoff}|)$

$\sum_{-120}^{120}J_{n}^{2}(100)=1;\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \sum_{-180}^{180}J_{n+1}^{2}(150)=1;\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\sum_{-220}^{220}J_{n-1}^{2}(200)=1.$ (41)

If we focus as an example on $J_{n}^{2}(100)$, according to Fig. (5) its
cutoff number is approximately equal to $100$ (or $-100$). The relationship
$\sum_{-120}^{120}J_{n}^{2}(100)=1$ is in fact a kind of sum rule that brings
us, together with the other functions, to the last one that we have included
in Eq. (39); this means that summing the square of function $J_{n}(z)$,
$J_{n}^{2}(z)$, over its cutoff number at a specific argument $z$ is exactly
equal to its square when the order $n$ and the argument $z$ are zero (this
indicates in our case the absence of a laser),

$\sum_{-\text{cutoff}}^{+\text{cutoff}}J_{n}^{2}(z)=J_{0}^{2}(0)=1.$ (42)

One can connect this last equation with the one given in Eq. (39). All this is
just to show how much the Bessel functions play a significant role in the
results obtained.

Figure 5: The Bessel functions $J_{n}^{2}(z)$, $J_{n+1}^{2}(z)$ and
$J_{n-1}^{2}(z)$ as a function of the order $n$, with different arguments $z$.
The cutoff at $n\approx z$ is clearly visible.

Table 2 contains numerical values of the SDCS summed over a fixed number of
photons $-20\leq n\leq+20$ in terms of field strength and for three different
frequencies. It is evident from this table that the SDCS is not affected by
the laser at low field strengths as it takes a constant value, but as soon as
the laser becomes strong it begins to decrease. As for the frequency
dependence, it is found that the effect of the laser diminishes with
increasing frequency and that the low-frequency laser affects the SDCS faster
than that of high frequency. As if the electron, at high frequencies, is not
able to follow the electric field oscillations, and therefore the influence of
the laser field on it decreases. The coupling of the electron to the laser
field, and thus the effect of the laser, is theoretically determined by the
argument of the ordinary Bessel functions,
$z\propto\mathcal{E}_{0}/\omega^{2}$, defined in Eq. (22). Thus, one would
expect an increased effect of the laser on the SDCS, which implies an enhanced
exchange of photons, for stronger fields or lower frequencies (see Figs.
2blue(a) and 2blue(b)).

Table 3: Numerical values of the SDCS summed over a fixed number of photons $-20\leq n\leq+20$ taking into account the mass of muon neutrino for different field strengths. The laser frequency is $\hbar\omega=1.17$ eV. The initial and final angles are $\theta_{i}=1^{\circ}$ and $\theta_{f}=0^{\circ}$. The neutrino kinetic energy is $E^{\text{kin}}_{\nu}=0.5\times 10^{6}\text{eV}$. | SDCS ($10^{-36}\leavevmode\nobreak\ \text{eV}^{-2}$)
---|---
$\mathcal{E}_{0}\leavevmode\nobreak\ (\text{V/cm})$ | $m_{\nu_{\mu}}=0.15\leavevmode\nobreak\ \text{MeV}$ | $m_{\nu_{\mu}}=0$
$10$ | $4.17$ | $4.16$
$10^{2}$ | $4.17$ | $4.16$
$10^{3}$ | $4.17$ | $4.16$
$10^{4}$ | $4.17$ | $4.16$
$10^{5}$ | $4.17$ | $4.16$
$10^{6}$ | $4.17$ | $4.16$
$10^{7}$ | $1.43$ | $1.43$
$10^{8}$ | $1.32\times 10^{-1}$ | $1.32\times 10^{-1}$
$10^{9}$ | $1.37\times 10^{-2}$ | $1.37\times 10^{-2}$
$10^{10}$ | $1.50\times 10^{-3}$ | $1.50\times 10^{-3}$

The effect of the nonzero neutrino mass on the SDCS is displayed by the values
listed in Table 3, where we give numerical values for the SDCS, considering
the mass of the muon neutrino, summed over a number of photons $-20\leq
n\leq+20$ for different values of field strength. From this table, it is clear
that the effect of the neutrino mass is noticeable only in the range of very
weak field strengths. The same result was found by A. V. Kurilin when he
studied the leptonic decays of the $W$ boson
($W^{\pm}\rightarrow\ell^{\pm}\bar{\nu}_{\ell}$) in the presence of a strong
electromagnetic field [63]. He found, in the case of $m_{\nu}\neq 0$, that the
vacuum decay width correction develops a nontrivial oscillatory term, which
can be served as an indication of the existence of massive neutrinos. Note
that oscillations in the probabilities of quantum processes in an external
field arise for a number of allowed reactions in vacuum if the participating
particles have nonzero rest mass. For instance, it is found that there are
oscillations that occur in the cross section for the process $\gamma
e\rightarrow W\nu_{e}$ if the relevant calculations take into account the
effects of a nonzero neutrino mass [64]. In order to be more practical and
accurate, we would like to point out here that the results presented in both
Tables 2 and 3 are more illustrative than realistic because fixing the number
of photons and truncating the sum over it at a specific number, for example
here $-20\leq n\leq+20$, does not give a realistic physical effect as long as
we do not have any idea how many photons have actually been exchanged.

##  VI Conclusion

Our goal in this paper was the theoretical investigation of the elastic
scattering of a muon neutrino by an elctron in the presence of a circularly
polarized monochromatic electromagnetic background. The DCS for this process
has been analytically expressed in terms of the kinematical and laser
parameters in both in the absence and presence of the laser. The results
obtained show that the laser field has a considerable influence, via its
strength and frequency, on the DCS as well as on the photon exchange process
between the laser and the scattering system. Summing up the results, it can be
concluded that the phenomenon of absorption and emission of multiple photons
depends on the properties of the implemented laser as well as on the
kinematical and geometrical conditions. Regarding the effect of the laser on
the DCS, it was found that it increases with the field strength and gradually
decreases at high frequencies. A sum rule, saying that the laser-assisted DCS
summed over all the exchanged photon numbers gives the laser-free DCS, is
found to be successfully fulfilled. The results, which took into account the
neutrino mass, proved that the effect of a nonzero neutrino mass is noticeable
only in the case of very low field strengths and neutrino kinetic energies.
However, this approach based on Fermi theory, although adopted here and in a
previous work [26] for the same process, may not be practical in all
situations, especially at high energies, due to the well-known limitations of
Fermi theory. Therefore, the next stage of our research will be a study of
this process within the framework of electroweak theory, in which the relevant
scattering process is mediated only by the exchange of the neutral $Z$ boson.
*

##  Appendix A Explicit expression of $\sum_{t_{i,f},s_{i,f}}|M^{n}_{fi}|^{2}$

The evaluation of the two traces shown in Eq. (33) gives the following result:

$\begin{split}\sum_{t_{i,f},s_{i,f}}|M^{n}_{fi}|^{2}=&AJ_{n}^{2}(z)+BJ_{n+1}^{2}(z)+CJ_{n-1}^{2}(z)+DJ_{n}(z)J_{n+1}(z)+EJ_{n}(z)J_{n-1}(z)\\\
&+FJ_{n-1}(z)J_{n+1}(z),\end{split}$ (43)

where the coefficients $A$, $B$, $C$, $D$, $E$ and $F$ are explicitly
expressed, in terms of different 4-vector products, by:

$\begin{split}A=&\dfrac{32}{(k.p_{f})(k.p_{i})}\Big{[}|\textbf{a}|^{4}e^{4}(g_{A}^{2}+g_{V}^{2})(k.k_{f})(k.k_{i})+2(k.p_{f})(k.p_{i})(2g_{A}\leavevmode\nobreak\
g_{V}(-(k_{f}.p_{i})(k_{i}.p_{f})+(k_{f}.p_{f})\\\
&\times(k_{i}.p_{i}))+g_{V}^{2}((k_{f}.p_{i})(k_{i}.p_{f})+(k_{f}.p_{f})(k_{i}.p_{i})-(k_{i}.k_{f})m^{2})+g_{A}^{2}((k_{f}.p_{i})(k_{i}.p_{f})+(k_{f}.p_{f})\\\
&\times(k_{i}.p_{i})+(k_{i}.k_{f})m^{2}))+|\textbf{a}|^{2}e^{2}\Big{(}2g_{A}\leavevmode\nobreak\
g_{V}(-(k_{i}.p_{i})(k.k_{f})(k.p_{f})+(k_{f}.p_{i})(k.k_{i})(k.p_{f})+(k_{i}.p_{f})\\\
&\times(k.k_{f})(k.p_{i})-(k_{f}.p_{f})(k.k_{i})(k.p_{i}))+g_{A}^{2}((k_{i}.p_{i})(k.k_{f})(k.p_{f})+(k_{f}.p_{i})(k.k_{i})(k.p_{f})+(k_{i}.p_{f})\\\
&\times(k.k_{f})(k.p_{i})+(k_{f}.p_{f})(k.k_{i})(k.p_{i})-2(k_{i}.k_{f})(k.p_{f})(k.p_{i})-2(k.k_{f})(k.k_{i})m^{2}-2(k.k_{f})\\\
&\times(k.k_{i})(p_{i}.p_{f}))+g_{V}^{2}((k_{i}.p_{i})(k.k_{f})(k.p_{f})+(k_{f}.p_{i})(k.k_{i})(k.p_{f})+(k_{i}.p_{f})(k.k_{f})(k.p_{i})+(k_{f}.p_{f})\\\
&\times(k.k_{i})(k.p_{i})-2(k_{i}.k_{f})(k.p_{f})(k.p_{i})+2(k.k_{f})(k.k_{i})m^{2}-2(k.k_{f})(k.k_{i})(p_{i}.p_{f}))\Big{)}\Big{]},\end{split}$
(44) $\begin{split}B=&\dfrac{4e^{2}\leavevmode\nobreak\
|\textbf{a}|^{2}}{(k.p_{f})^{2}(k.p_{i})^{2}}\big{[}4(k.p_{f})(k.p_{i})\leavevmode\nobreak\
\big{(}-g_{A}^{2}(k_{i}.p_{f})(k.k_{f})(k.p_{f})+2\leavevmode\nobreak\
g_{A}\leavevmode\nobreak\ g_{V}\leavevmode\nobreak\
(k_{i}.p_{f})(k.k_{f})(k.p_{f})\\\ &-g_{V}^{2}\leavevmode\nobreak\
(k_{i}.p_{f})(k.k_{f})(k.p_{f})+g_{A}^{2}\leavevmode\nobreak\
(k_{i}.p_{i})(k.k_{f})(k.p_{f})-2\leavevmode\nobreak\
g_{A}\leavevmode\nobreak\ g_{V}\leavevmode\nobreak\
(k_{i}.p_{i})(k.k_{f})(k.p_{f})\\\ &+g_{V}^{2}\leavevmode\nobreak\
(k_{i}.p_{i})(k.k_{f})(k.p_{f})-(\eta_{1}.k_{f})(\eta_{1}.p_{i})\leavevmode\nobreak\
g_{A}^{2}\leavevmode\nobreak\
(k.k_{i})(k.p_{f})-(\eta_{2}.k_{f})(\eta_{2}.p_{i})\leavevmode\nobreak\
g_{A}^{2}\leavevmode\nobreak\ (k.k_{i})(k.p_{f})\\\
&-2(\eta_{1}.k_{f})(\eta_{1}.p_{i})\leavevmode\nobreak\
g_{A}\leavevmode\nobreak\ g_{V}\leavevmode\nobreak\
(k.k_{i})(k.p_{f})-2(\eta_{2}.k_{f})(\eta_{2}.p_{i})\leavevmode\nobreak\
g_{A}\leavevmode\nobreak\ g_{V}\leavevmode\nobreak\
(k.k_{i})(k.p_{f})-(\eta_{1}.k_{f})(\eta_{1}.p_{i})\leavevmode\nobreak\
g_{V}^{2}\\\
&\times(k.k_{i})(k.p_{f})-(\eta_{2}.k_{f})(\eta_{2}.p_{i})g_{V}^{2}\leavevmode\nobreak\
(k.k_{i})(k.p_{f})-g_{A}^{2}\leavevmode\nobreak\
(k_{f}.p_{f})(k.k_{i})(k.p_{f})-2\leavevmode\nobreak\
g_{A}\leavevmode\nobreak\ g_{V}\\\
&\times(k_{f}.p_{f})(k.k_{i})(k.p_{f})-g_{V}^{2}(k_{f}.p_{f})(k.k_{i})(k.p_{f})+g_{A}^{2}(k_{f}.p_{i})\times(k.k_{i})(k.p_{f})\\\
&+2g_{A}g_{V}(k_{f}.p_{i})(k.k_{i})(k.p_{f})+g_{V}^{2}(k_{f}.p_{i})(k.k_{i})(k.p_{f})+g_{A}^{2}(k_{i}.p_{f})(k.k_{f})(k.p_{i})\\\
&+2g_{A}g_{V}(k_{i}.p_{f})(k.k_{f})(k.p_{i})+g_{V}^{2}(k_{i}.p_{f})(k.k_{f})(k.p_{i})-g_{A}^{2}(k_{i}.p_{i})(k.k_{f})(k.p_{i})\\\
&-2g_{A}g_{V}(k_{i}.p_{i})(k.k_{f})(k.p_{i})-g_{V}^{2}(k_{i}.p_{i})(k.k_{f})(k.p_{i})+g_{A}^{2}(k_{f}.p_{f})(k.k_{i})(k.p_{i})\\\
&-2g_{A}g_{V}(k_{f}.p_{f})(k.k_{i})(k.p_{i})+g_{V}^{2}(k_{f}.p_{f})(k.k_{i})(k.p_{i})-g_{A}^{2}(k_{f}.p_{i})(k.k_{i})(k.p_{i})\\\
&+2g_{A}g_{V}(k_{f}.p_{i})(k.k_{i})(k.p_{i})-g_{V}^{2}(k_{f}.p_{i})(k.k_{i})(k.p_{i})-2g_{A}^{2}(k_{i}.k_{f})(k.p_{f})(k.p_{i})\\\
&-2g_{V}^{2}(k_{i}.k_{f})(k.p_{f})(k.p_{i})+(\eta_{1}.p_{f})(k.k_{i})(2(\eta_{1}.p_{i})(g_{A}^{2}+g_{V}^{2})(k.k_{f})-(\eta_{1}.k_{f})(g_{A}-g_{V})^{2}\\\
&\times(k.p_{i}))+(\eta_{2}.p_{f})(k.k_{i})(2(\eta_{2}.p_{i})(g_{A}^{2}+g_{V}^{2})(k.k_{f})-(\eta_{2}.k_{f})(g_{A}-g_{V})^{2}(k.p_{i}))-2g_{A}^{2}\\\
&\times(k.k_{f})(k.k_{i})m^{2}+2g_{V}^{2}(k.k_{f})(k.k_{i})m^{2}-2(g_{A}^{2}+g_{V}^{2})(k.k_{f})(k.k_{i})(p_{i}.p_{f})\big{)}-2g_{A}g_{V}\\\
&\times\big{(}(k_{i}.p_{i})(k.p_{f})((k.p_{f})^{2}-2(k.p_{f})(k.p_{i})+2(k.p_{i})^{2})-((k.p_{f})-(k.p_{i}))((k_{i}.p_{f})(k.p_{f})(k.p_{i})\\\
&+(k.k_{i})((k.p_{f})+(k.p_{i}))(p_{i}.p_{f}))\big{)}\epsilon(\eta_{1},\eta_{2},k,k_{f})-2\leavevmode\nobreak\
g_{A}g_{V}\big{(}(k_{f}.p_{i})(k.p_{f})((k.p_{f})^{2}-2(k.p_{f})\\\
&\times(k.p_{i})+2(k.p_{i})^{2})-((k.p_{f})-(k.p_{i}))((k_{f}.p_{f})(k.p_{f})(k.p_{i})+(k.k_{f})((k.p_{f})+(k.p_{i}))(p_{i}.p_{f})\big{)}\\\
&\times\epsilon(\eta_{1},\eta_{2},k,k_{i})+2g_{A}g_{V}\epsilon(\eta_{1},\eta_{2},k,p_{f})\big{(}2(k_{i}.k_{f})(k.p_{f})^{2}(k.p_{i})-(k_{i}.p_{i})(k.k_{f})(k.p_{i})^{2}\\\
&-(k_{f}.p_{i})(k.k_{i})(k.p_{i})^{2}+(k_{i}.k_{f})(k.p_{f})(k.p_{i})^{2}\big{)}+2\leavevmode\nobreak\
g_{A}g_{V}\epsilon(\eta_{1},\eta_{2},k,p_{i})\big{(}(k_{i}.p_{f})(k.k_{f})(k.p_{f})^{2}\\\
&+(k_{f}.p_{f})(k.k_{i})(k.p_{f})^{2}+2(k_{i}.k_{f})(k.p_{f})^{3}-4(k_{i}.k_{f})(k.p_{f})^{2}(k.p_{i})\big{)}+\epsilon(\eta_{1},\eta_{2},k_{f},k_{i})\\\
&\times\big{(}4g_{A}^{2}(k.p_{f})^{3}(k.p_{i})+4g_{V}^{2}(k.p_{f})^{3}(k.p_{i})+4g_{A}^{2}(k.p_{f})(k.p_{i})^{3}+4g_{V}^{2}(k.p_{f})(k.p_{i})^{3}\big{)}-2g_{A}g_{V}\\\
&\times\epsilon(\eta_{1},\eta_{2},k_{f},p_{f})\big{(}(k.k_{i})(k.p_{f})^{2}(k.p_{i})+(k.k_{i})(k.p_{f})(k.p_{i})^{2}+(k.k_{i})(k.p_{i})^{3}\big{)}+4g_{A}g_{V}(k.k_{i})\\\
&\times(k.p_{f})^{2}(k.p_{i})\epsilon(\eta_{1},\eta_{2},k_{f},p_{i})-2\leavevmode\nobreak\
g_{A}g_{V}\epsilon(\eta_{1},\eta_{2},k_{i},p_{f})\big{(}(k.k_{f})(k.p_{f})^{2}(k.p_{i})+(k.k_{f})\\\
&\times(k.p_{f})(k.p_{i})^{2}+(k.k_{f})\big{)}+4g_{A}g_{V}(k.k_{f})(k.p_{f})^{2}(k.p_{i})\epsilon(\eta_{1},\eta_{2},k_{i},p_{i})-4\leavevmode\nobreak\
\epsilon(\eta_{1},k,k_{f},k_{i})\\\
&\times\big{(}(\eta_{2}.p_{f})g_{A}^{2}(k.p_{f})^{2}(k.p_{i})+(\eta_{2}.p_{f})g_{V}^{2}(k.p_{f})^{2}(k.p_{i})+(\eta_{2}.p_{i})g_{A}^{2}(k.p_{f})(k.p_{i})^{2}+(\eta_{2}.p_{i})\\\
&\times g_{V}^{2}(k.p_{f})(k.p_{i})^{2}\big{)}-g_{A}g_{V}\leavevmode\nobreak\
\epsilon(\eta_{1},k,k_{f},p_{f})\big{(}(\eta_{2}.p_{i})(k.k_{i})(k.p_{f})^{2}-4(\eta_{2}.p_{i})(k.k_{i})(k.p_{f})(k.p_{i})\\\
&-(\eta_{2}.p_{i})(k.k_{i})(k.p_{i})^{2}\big{)}-g_{A}g_{V}\leavevmode\nobreak\
\epsilon(\eta_{1},k,k_{f},p_{i})\big{(}(\eta_{2}.p_{f})(k.k_{i})(k.p_{f})^{2}+4(\eta_{2}.p_{f})(k.k_{i})(k.p_{f})\\\
&\times(k.p_{i})-(\eta_{2}.p_{f})(k.k_{i})(k.p_{i})^{2}\big{)}-g_{A}g_{V}\leavevmode\nobreak\
\leavevmode\nobreak\
\epsilon(\eta_{1},k,k_{i},p_{f})\big{(}(\eta_{2}.p_{i})(k.k_{f})(k.p_{f})^{2}-4(\eta_{2}.p_{i})(k.k_{f})\\\
&\times(k.p_{f})(k.p_{i})-2(\eta_{2}.k_{f})(k.p_{f})^{2}(k.p_{i})-(\eta_{2}.p_{i})(k.k_{f})(k.p_{i})^{2}-2(\eta_{2}.k_{f})(k.p_{f})(k.p_{i})^{2}\big{)}\\\
&-g_{A}g_{V}\leavevmode\nobreak\
\epsilon(\eta_{1},k,k_{i},p_{i})\big{(}(\eta_{2}.p_{f})(k.k_{f})(k.p_{f})^{2}-2(\eta_{2}.k_{f})(k.p_{f})^{3}+4(\eta_{2}.p_{f})(k.k_{f})(k.p_{f})(k.p_{i})\\\
&-(\eta_{2}.p_{f})(k.k_{f})(k.p_{i})^{2}\big{)}-g_{A}\leavevmode\nobreak\
g_{V}\leavevmode\nobreak\ \leavevmode\nobreak\
\epsilon(\eta_{1},k,p_{f},p_{i})\big{(}(\eta_{2}.k_{f})(k.k_{i})(k.p_{f})^{2}+(\eta_{2}.k_{f})(k.k_{i})(k.p_{i})^{2}\big{)}\\\
&+\epsilon(\eta_{2},k,k_{f},k_{i})\big{(}4g_{A}^{2}(\eta_{1}.p_{f})(k.p_{f})^{2}(k.p_{i})+4g_{V}^{2}(\eta_{1}.p_{f})(k.p_{f})^{2}(k.p_{i})+4g_{A}^{2}(\eta_{1}.p_{i})(k.p_{f})\\\
&\times(k.p_{i})^{2}+4g_{V}^{2}(\eta_{1}.p_{i})(k.p_{f})(k.p_{i})^{2}\big{)}+g_{A}\leavevmode\nobreak\
g_{V}\leavevmode\nobreak\ \leavevmode\nobreak\
\epsilon(\eta_{2},k,k_{f},p_{f})\big{(}(\eta_{1}.p_{i})(k.k_{i})(k.p_{f})^{2}\\\
&-4(\eta_{1}.p_{i})(k.k_{i})(k.p_{f})(k.p_{i})-(\eta_{1}.p_{i})(k.k_{i})(k.p_{i})^{2}\big{)}+g_{A}\leavevmode\nobreak\
g_{V}\leavevmode\nobreak\ \leavevmode\nobreak\
\epsilon(\eta_{2},k,k_{f},p_{i})\big{(}(\eta_{1}.p_{f})\\\
&\times(k.k_{i})(k.p_{f})^{2}+4(\eta_{1}.p_{f})(k.k_{i})(k.p_{f})(k.p_{i})-(\eta_{1}.p_{f})(k.k_{i})(k.p_{i})^{2}\big{)}+g_{A}\leavevmode\nobreak\
g_{V}\leavevmode\nobreak\ \leavevmode\nobreak\ \\\
&\times\epsilon(\eta_{2},k,k_{i},p_{f})\big{(}(\eta_{1}.p_{i})(k.k_{f})(k.p_{f})^{2}-4(\eta_{1}.p_{i})(k.k_{f})(k.p_{f})(k.p_{i})-2(\eta_{1}.k_{f})(k.p_{f})^{2}(k.p_{i})\\\
&-(\eta_{1}.p_{i})(k.k_{f})(k.p_{i})^{2}+2(\eta_{1}.k_{f})(k.p_{f})(k.p_{i})^{2}\big{)}+g_{A}\leavevmode\nobreak\
g_{V}\leavevmode\nobreak\
\epsilon(\eta_{2},k,k_{i},p_{i})\big{(}(\eta_{1}.p_{f})(k.k_{f})\\\
&\times(k.p_{f})^{2}-2(\eta_{1}.k_{f})(k.p_{f})^{3}+4(\eta_{1}.p_{f})(k.k_{f})(k.p_{f})(k.p_{i})-(\eta_{1}.p_{f})(k.k_{f})(k.p_{i})^{2}\big{)}\\\
&+(\eta_{1}.k_{f})g_{A}g_{V}(k.k_{i})((k.p_{f})^{2}+(k.p_{i})^{2})\epsilon(\eta_{2},k,p_{f},p_{i})\big{]},\end{split}$
(45) $\begin{split}C=&\dfrac{4e^{2}\leavevmode\nobreak\
|\textbf{a}|^{2}}{(k.p_{f})^{2}(k.p_{i})^{2}}\big{[}4(k.p_{f})(k.p_{i})\big{(}-g_{A}^{2}(k_{i}.p_{f})(k.k_{f})(k.p_{f})+2\leavevmode\nobreak\
g_{A}\leavevmode\nobreak\
g_{V}(k_{i}.p_{f})(k.k_{f})(k.p_{f})-g_{V}^{2}(k_{i}.p_{f})\\\
&\times(k.k_{f})(k.p_{f})+g_{A}^{2}(k_{i}.p_{i})(k.k_{f})(k.p_{f})-2g_{A}\leavevmode\nobreak\
g_{V}(k_{i}.p_{i})(k.k_{f})(k.p_{f})+g_{V}^{2}(k_{i}.p_{i})(k.k_{f})(k.p_{f})\\\
&-(\eta_{1}.k_{f})(\eta_{1}.p_{i})g_{A}^{2}(k.k_{i})(k.p_{f})-(\eta_{2}.k_{f})(\eta_{2}.p_{i})g_{A}^{2}(k.k_{i})(k.p_{f})-2(\eta_{1}.k_{f})(\eta_{1}.p_{i})g_{A}g_{V}(k.k_{i})(k.p_{f})\\\
&-2(\eta_{2}.k_{f})(\eta_{2}.p_{i})g_{A}g_{V}(k.k_{i})(k.p_{f})-(\eta_{1}.k_{f})(\eta_{1}.p_{i})g_{V}^{2}(k.k_{i})(k.p_{f})-(\eta_{2}.k_{f})(\eta_{2}.p_{i})g_{V}^{2}(k.k_{i})(k.p_{f})\\\
&-g_{A}^{2}(k_{f}.p_{f})(k.k_{i})(k.p_{f})-2g_{A}g_{V}(k_{f}.p_{f})(k.k_{i})(k.p_{f})-g_{V}^{2}(k_{f}.p_{f})(k.k_{i})(k.p_{f})+g_{A}^{2}(k_{f}.p_{i})(k.k_{i})\\\
&\times(k.p_{f})+2g_{A}g_{V}(k_{f}.p_{i})(k.k_{i})(k.p_{f})+g_{V}^{2}(k_{f}.p_{i})(k.k_{i})(k.p_{f})+g_{A}^{2}(k_{i}.p_{f})(k.k_{f})(k.p_{i})+2g_{A}g_{V}\\\
&\times(k_{i}.p_{f})(k.k_{f})(k.p_{i})+g_{V}^{2}(k_{i}.p_{f})(k.k_{f})(k.p_{i})-g_{A}^{2}(k_{i}.p_{i})(k.k_{f})(k.p_{i})-2g_{A}g_{V}(k_{i}.p_{i})(k.k_{f})\\\
&\times(k.p_{i})-g_{V}^{2}(k_{i}.p_{i})(k.k_{f})(k.p_{i})+g_{A}^{2}(k_{f}.p_{f})(k.k_{i})(k.p_{i})-2g_{A}g_{V}(k_{f}.p_{f})(k.k_{i})(k.p_{i})+g_{V}^{2}(k_{f}.p_{f})\\\
&\times(k.k_{i})(k.p_{i})-g_{A}^{2}(k_{f}.p_{i})(k.k_{i})(k.p_{i})+2g_{A}g_{V}(k_{f}.p_{i})(k.k_{i})(k.p_{i})-g_{V}^{2}(k_{f}.p_{i})(k.k_{i})(k.p_{i})\\\
&-2g_{A}^{2}(k_{i}.k_{f})(k.p_{f})(k.p_{i})-2g_{V}^{2}(k_{i}.k_{f})(k.p_{f})(k.p_{i})+(\eta_{1}.p_{f})(k.k_{i})(2(\eta_{1}.p_{i})(g_{A}^{2}+g_{V}^{2})(k.k_{f})\\\
&-(\eta_{1}.k_{f})(g_{A}-g_{V})^{2}(k.p_{i}))+(\eta_{2}.p_{f})(k.k_{i})(2(\eta_{2}.p_{i})(g_{A}^{2}+g_{V}^{2})(k.k_{f})-(\eta_{2}.k_{f})(g_{A}-g_{V})^{2}(k.p_{i}))\\\
&-2g_{A}^{2}(k.k_{f})(k.k_{i})m^{2}+2g_{V}^{2}(k.k_{f})(k.k_{i})m^{2}-2(g_{A}^{2}+g_{V}^{2})(k.k_{f})(k.k_{i})(p_{i}.p_{f})\big{)}+2\leavevmode\nobreak\
g_{A}\leavevmode\nobreak\ g_{V}\big{(}(k_{i}.p_{i})\\\
&\times(k.p_{f})((k.p_{f})^{2}-2(k.p_{f})(k.p_{i})+2(k.p_{i})^{2})-((k.p_{f})-(k.p_{i}))((k_{i}.p_{f})(k.p_{f})(k.p_{i})+(k.k_{i})\\\
&\times((k.p_{f})+(k.p_{i}))(p_{i}.p_{f}))\big{)}\epsilon(\eta_{1},\eta_{2},k,k_{f})+2\leavevmode\nobreak\
g_{A}\leavevmode\nobreak\
g_{V}\big{(}(k_{f}.p_{i})(k.p_{f})((k.p_{f})^{2}-2(k.p_{f})(k.p_{i})\\\
&+2(k.p_{i})^{2})-((k.p_{f})-(k.p_{i}))((k_{f}.p_{f})(k.p_{f})(k.p_{i})+(k.k_{f})((k.p_{f})+(k.p_{i}))(p_{i}.p_{f}))\big{)}\\\
&\times\leavevmode\nobreak\
\epsilon(\eta_{1},\eta_{2},k,k_{i})+2\leavevmode\nobreak\
g_{A}\leavevmode\nobreak\ g_{V}\leavevmode\nobreak\
\big{(}-2\leavevmode\nobreak\
(k_{i}.k_{f})(k.p_{f})^{2}(k.p_{i})+(k_{i}.p_{i})(k.k_{f})(k.p_{i})^{2}+(k_{f}.p_{i})(k.k_{i})\\\
&\times(k.p_{i})^{2}-2\leavevmode\nobreak\
(k_{i}.k_{f})(k.p_{f})(k.p_{i})^{2}\big{)}\epsilon(\eta_{1},\eta_{2},k,p_{f})+2\leavevmode\nobreak\
g_{A}\leavevmode\nobreak\
g_{V}\big{(}-(k_{i}.p_{f})(k.k_{f})((k.p_{f})^{2})\\\
&-(k_{f}.p_{f})(k.k_{i})(k.p_{f})^{2}-2(k_{i}.k_{f})(k.p_{f})^{3}+4\leavevmode\nobreak\
(k_{i}.k_{f})(k.p_{f})^{2}(k.p_{i})\big{)}\leavevmode\nobreak\
\epsilon(\eta_{1},\eta_{2},k,p_{i})\\\ &+4\leavevmode\nobreak\
\big{(}-g_{A}^{2}(k.p_{f})^{3}(k.p_{i})-g_{V}^{2}(k.p_{f})^{3}(k.p_{i})-g_{A}^{2}(k.p_{f})(k.p_{i})^{3}-g_{V}^{2}(k.p_{f})(k.p_{i})^{3}\big{)}\\\
&\times\epsilon(\eta_{1},\eta_{2},k_{f},k_{i})+2\leavevmode\nobreak\
g_{A}\leavevmode\nobreak\ g_{V}\leavevmode\nobreak\
\big{(}(k.k_{i})(k.p_{f})^{2}(k.p_{i})+(k.k_{i})(k.p_{f})(k.p_{i})^{2}+(k.k_{i})(k.p_{i})^{3}\big{)}\\\
&\times\epsilon(\eta_{1},\eta_{2},k_{f},p_{f})-4\leavevmode\nobreak\
\leavevmode\nobreak\ g_{A}\leavevmode\nobreak\
g_{V}(k.k_{i})(k.p_{f})^{2}(k.p_{i})\leavevmode\nobreak\
\epsilon(\eta_{1},\eta_{2},k_{f},p_{i})+2\leavevmode\nobreak\
g_{A}\leavevmode\nobreak\ g_{V}\big{(}(k.k_{f})(k.p_{f})^{2}(k.p_{i})\\\
&+(k.k_{f})(k.p_{f})(k.p_{i})^{2}+(k.k_{f})(k.p_{i})^{3}\big{)}\epsilon(\eta_{1},\eta_{2},k_{i},p_{f})-4\leavevmode\nobreak\
\leavevmode\nobreak\ g_{A}\leavevmode\nobreak\
g_{V}(k.k_{f})(k.p_{f})^{2}(k.p_{i})\leavevmode\nobreak\
\epsilon(\eta_{1},\eta_{2},k_{i},p_{i})\\\ &+4\leavevmode\nobreak\
\big{(}g_{A}^{2}(\eta_{2}.p_{f})(k.p_{f})^{2}(k.p_{i})+g_{V}^{2}(\eta_{2}.p_{f})(k.p_{f})^{2}(k.p_{i})+g_{A}^{2}(\eta_{2}.p_{i})(k.p_{f})(k.p_{i})^{2}+g_{V}^{2}(\eta_{2}.p_{i})(k.p_{f})\\\
&\times(k.p_{i})^{2}\big{)}\leavevmode\nobreak\
\epsilon(\eta_{1},k,k_{f},k_{i})+\leavevmode\nobreak\
g_{A}\leavevmode\nobreak\ g_{V}\leavevmode\nobreak\
(\eta_{2}.p_{i})\leavevmode\nobreak\
\big{(}(k.k_{i})(k.p_{f})^{2}-4\leavevmode\nobreak\
(k.k_{i})(k.p_{f})(k.p_{i})-(k.k_{i})(k.p_{i})^{2}\big{)}\\\
&\times\epsilon(\eta_{1},k,k_{f},p_{f})+\leavevmode\nobreak\
g_{A}\leavevmode\nobreak\ g_{V}\leavevmode\nobreak\
(\eta_{2}.p_{f})\leavevmode\nobreak\
\big{(}(k.k_{i})(k.p_{f})^{2}+4\leavevmode\nobreak\
(k.k_{i})(k.p_{f})(k.p_{i})-(k.k_{i})(k.p_{i})^{2}\big{)}\\\
&\times\epsilon(\eta_{1},k,k_{f},p_{i})+\leavevmode\nobreak\
g_{A}\leavevmode\nobreak\
g_{V}\big{(}(\eta_{2}.p_{i})(k.k_{f})(k.p_{f})^{2}-4(\eta_{2}.p_{i})(k.k_{f})(k.p_{f})(k.p_{i})-2(\eta_{2}.k_{f})(k.p_{f})^{2}(k.p_{i})\\\
&-(\eta_{2}.p_{i})(k.k_{f})(k.p_{i})^{2}+2(\eta_{2}.k_{f})(k.p_{f})(k.p_{i})^{2}\big{)}\leavevmode\nobreak\
\epsilon(\eta_{1},k,k_{i},p_{f})+\leavevmode\nobreak\
g_{A}\leavevmode\nobreak\ g_{V}\leavevmode\nobreak\
\big{(}(\eta_{2}.p_{f})(k.k_{f})(k.p_{f})^{2}\\\
&-2(\eta_{2}.k_{f})(k.p_{f})^{3}+4(\eta_{2}.p_{f})(k.k_{f})(k.p_{f})(k.p_{i})-(\eta_{2}.p_{f})(k.k_{f})(k.p_{i})^{2}\big{)}\leavevmode\nobreak\
\epsilon(\eta_{1},k,k_{i},p_{i})\\\ &+\leavevmode\nobreak\
g_{A}\leavevmode\nobreak\ g_{V}\leavevmode\nobreak\
(\eta_{2}.k_{f})\big{(}(k.k_{i})(k.p_{f})^{2}+(k.k_{i})(k.p_{i})^{2}\big{)}\leavevmode\nobreak\
\epsilon(\eta_{1},k,p_{f},p_{i})+4\big{(}-\leavevmode\nobreak\
g_{A}^{2}\leavevmode\nobreak\ (\eta_{1}.p_{f})(k.p_{f})^{2}(k.p_{i})\\\
&-g_{V}^{2}\leavevmode\nobreak\
(\eta_{1}.p_{f})(k.p_{f})^{2}(k.p_{i})-g_{A}^{2}(\eta_{1}.p_{i})(k.p_{f})(k.p_{i})^{2}-g_{V}^{2}\leavevmode\nobreak\
(\eta_{1}.p_{i})(k.p_{f})(k.p_{i})^{2}\big{)}\leavevmode\nobreak\
\epsilon(\eta_{2},k,k_{f},k_{i})\\\ &+\leavevmode\nobreak\
g_{A}\leavevmode\nobreak\ g_{V}\leavevmode\nobreak\
(\eta_{1}.p_{i})\big{(}-(k.k_{i})(k.p_{f})^{2}+4\leavevmode\nobreak\
(k.k_{i})(k.p_{f})(k.p_{i})+(k.k_{i})(k.p_{i})^{2}\big{)}\leavevmode\nobreak\
\epsilon(\eta_{2},k,k_{f},p_{f})\\\ &+\leavevmode\nobreak\
g_{A}\leavevmode\nobreak\ g_{V}\leavevmode\nobreak\
(\eta_{1}.p_{f})\big{(}-(k.k_{i})(k.p_{f})^{2}-4\leavevmode\nobreak\
(k.k_{i})(k.p_{f})(k.p_{i})+(k.k_{i})(k.p_{i})^{2}\big{)}\leavevmode\nobreak\
\epsilon(\eta_{2},k,k_{f},p_{i})\\\ &+\leavevmode\nobreak\
g_{A}\leavevmode\nobreak\
g_{V}\big{(}-(\eta_{1}.p_{i})(k.k_{f})(k.p_{f})^{2}+4(\eta_{1}.p_{i})(k.k_{f})(k.p_{f})(k.p_{i})+2(\eta_{1}.k_{f})(k.p_{f})^{2}(k.p_{i})\\\
&+(\eta_{1}.p_{i})(k.k_{f})(k.p_{i})^{2}-2(\eta_{1}.k_{f})(k.p_{f})(k.p_{i})^{2}\big{)}\leavevmode\nobreak\
\epsilon(\eta_{2},k,k_{i},p_{f})+g_{A}\leavevmode\nobreak\
g_{V}\leavevmode\nobreak\ \big{(}-(\eta_{1}.p_{f})(k.k_{f})\\\
&\times(k.p_{f})^{2}+2(\eta_{1}.k_{f})(k.p_{f})^{3}-4(\eta_{1}.p_{f})(k.k_{f})(k.p_{f})(k.p_{i})+(\eta_{1}.p_{f})(k.k_{f})(k.p_{i})^{2}\big{)}\leavevmode\nobreak\
\epsilon(\eta_{2},k,k_{i},p_{i})\\\ &-\leavevmode\nobreak\
g_{A}\leavevmode\nobreak\
g_{V}(\eta_{1}.k_{f})(k.k_{i})((k.p_{f})^{2}+(k.p_{i})^{2})\leavevmode\nobreak\
\epsilon(\eta_{2},k,p_{f},p_{i})\big{]},\end{split}$ (46)
$\begin{split}D=&\dfrac{16e}{(k.p_{f})^{2}(k.p_{i})^{2}}\Big{[}(k.p_{f})(k.p_{i})\Big{(}2|\textbf{a}|(\eta_{1}.p_{i})((g_{A}-g_{V})^{2}(k_{i}.p_{f})(k.k_{f})+(g_{A}+g_{V})^{2}(k_{f}.p_{f})(k.k_{i}))\\\
&\times(k.p_{f})+2(|\textbf{a}|(\eta_{1}.p_{f})(g_{A}+g_{V})^{2}(k_{i}.p_{i})(k.k_{f})+|\textbf{a}|(\eta_{1}.p_{f})(g_{A}-g_{V})^{2}(k_{f}.p_{i})(k.k_{i})-|\textbf{a}|\\\
&\times(\eta_{1}.k_{f})((g_{A}-g_{V})^{2}(k_{i}.p_{f})+(g_{A}+g_{V})^{2}(k_{i}.p_{i}))(k.p_{f}))(k.p_{i})+|\textbf{a}|^{2}e^{2}(k.k_{i})(2|\textbf{a}|(\eta_{1}.p_{f})\\\
&\times(g_{A}^{2}+g_{V}^{2})(k.k_{f})+2|\textbf{a}|(\eta_{1}.p_{i})(g_{A}^{2}+g_{V}^{2})(k.k_{f})-|\textbf{a}|(\eta_{1}.k_{f})(g_{A}+g_{V})^{2}(k.p_{f})-|\textbf{a}|(\eta_{1}.k_{f})\\\
&\times(g_{A}-g_{V})^{2}(k.p_{i}))\Big{)}+|\textbf{a}|(k.p_{f})(k.p_{i})\big{(}(k.p_{f})+(k.p_{i})\big{)}\Big{(}|\textbf{a}|^{2}e^{2}(g_{A}^{2}+g_{V}^{2})+2g_{V}^{2}(m^{2}-(p_{i}.p_{f}))\\\
&-2g_{A}^{2}(m^{2}+(p_{i}.p_{f}))\Big{)}\epsilon(\eta_{2},k,k_{f},k_{i})+g_{A}g_{V}|\textbf{a}|(k.p_{i})\Big{(}2(k_{i}.p_{i})(k.p_{f})((k.p_{f})-2(k.p_{i}))+|\textbf{a}|^{2}e^{2}\\\
&\times(k.k_{i})((k.p_{f})-(k.p_{i}))\Big{)}\epsilon(\eta_{2},k,k_{f},p_{f})+g_{A}g_{V}|\textbf{a}|\Big{(}|\textbf{a}|^{2}e^{2}(k.k_{i})(k.p_{f})^{2}-|\textbf{a}|^{2}e^{2}(k.k_{i})(k.p_{f})\\\
&\times(k.p_{i})+2(k_{i}.p_{f})(k.p_{f})^{2}(k.p_{i})\Big{)}\epsilon(\eta_{2},k,k_{f},p_{i})+g_{A}g_{V}|\textbf{a}|\Big{(}|\textbf{a}|^{2}e^{2}(k.k_{f})(k.p_{f})(k.p_{i})+2(k_{f}.p_{i})\\\
&\times(k.p_{f})^{2}(k.p_{i})-|\textbf{a}|^{2}e^{2}(k.k_{f})(k.p_{i})^{2}-4(k_{f}.p_{i})(k.p_{f})(kp_{i})^{2}\Big{)}\epsilon(\eta_{2},k,k_{i},p_{f})+g_{A}g_{V}|\textbf{a}|\Big{(}|\textbf{a}|^{2}e^{2}(k.k_{f})\\\
&\times(k.p_{f})^{2}-|\textbf{a}|^{2}e^{2}(k.k_{f})(k.p_{f})(k.p_{i})+2(k_{f}.p_{f})(k.p_{f})^{2}(k.p_{i})\Big{)}\epsilon(\eta_{2},k,k_{i},p_{i})-2g_{A}g_{V}|\textbf{a}|\Big{(}|\textbf{a}|^{2}e^{2}\\\
&\times(k.k_{f})(k.k_{i})(k.p_{f})+|\textbf{a}|^{2}e^{2}(k.k_{f})(k.k_{i})(k.p_{i})-2(k_{i}.k_{f})(k.p_{f})^{2}(k.p_{i})\Big{)}\epsilon(\eta_{2},k,p_{f},p_{i})+|\textbf{a}|\Big{(}2g_{A}^{2}\\\
&\times(k.p_{f})(k.p_{i})^{3}+2g_{V}^{2}(k.p_{f})(k.p_{i})^{3}\Big{)}\epsilon(\eta_{2},k_{f},k_{i},p_{f})+|\textbf{a}|\Big{(}2g_{A}^{2}(k.p_{f})^{3}(k.p_{i})+2g_{V}^{2}(k.p_{f})^{3}(k.p_{i})\Big{)}\\\
&\times\epsilon(\eta_{2},k_{f},k_{i},p_{i})-2g_{A}g_{V}|\textbf{a}|(k.k_{i})(k.p_{f})^{2}(k.p_{i})\epsilon(\eta_{2},k_{f},p_{f},p_{i})-2|\textbf{a}|g_{A}g_{V}(k.k_{f})(k.p_{f})^{2}(k.p_{i})\\\
&\times\epsilon(\eta_{2},k_{i},p_{f},p_{i})-2g_{A}^{2}\Big{(}|\textbf{a}|(\eta_{2}.p_{i})(k.p_{f})(k.p_{i})^{2}+|\textbf{a}|(\eta_{2}.p_{i})(k.p_{f})(k.p_{i})^{2}\Big{)}\epsilon(k,k_{f},k_{i},p_{f})-2g_{A}^{2}\\\
&\times\Big{(}|\textbf{a}|(\eta_{2}.p_{f})(k.p_{f})^{2}(k.p_{i})-|\textbf{a}|(\eta_{2}.p_{f})(k.p_{f})^{2}(k.p_{i})\Big{)}\epsilon(k,k_{f},k_{i},p_{i})+2|\textbf{a}|(\eta_{2}.k_{f})g_{A}g_{V}\\\
&\times(k.p_{f})^{2}(k.p_{i})\epsilon(k,k_{i},p_{f},p_{i})\Big{]},\end{split}$
(47)
$\begin{split}E=&\dfrac{-16e}{(k.p_{f})^{2}(k.p_{i})^{2}}\Big{[}(k.p_{f})(k.p_{i})\Big{(}-2|\textbf{a}|(\eta_{1}.p_{i})((g_{A}-g_{V})^{2}(k_{i}.p_{f})(k.k_{f})+(g_{A}+g_{V})^{2}(k_{f}.p_{f})\\\
&\times(k.k_{i}))(k.p_{f})+2(-|\textbf{a}|(\eta_{1}.p_{f})(g_{A}+g_{V})^{2}(k_{i}.p_{i})(k.k_{f})-|\textbf{a}|(\eta_{1}.p_{f})(g_{A}-g_{V})^{2}(k_{f}.p_{i})(k.k_{i})\\\
&+|\textbf{a}|(\eta_{1}.k_{f})((g_{A}-g_{V})^{2}(k_{i}.p_{f})+(g_{A}+g_{V})^{2}(k_{i}.p_{i}))(k.p_{f}))(k.p_{i})+|\textbf{a}|^{2}e^{2}(k.k_{i})(-2|\textbf{a}|(\eta_{1}.p_{f})\\\
&\times(g_{A}^{2}+g_{V}^{2})(k.k_{f})-2|\textbf{a}|(\eta_{1}.p_{i})(g_{A}^{2}+g_{V}^{2})(k.k_{f})+|\textbf{a}|(\eta_{1}.k_{f})(g_{A}+g_{V})^{2}(k.p_{f})+|\textbf{a}|(\eta_{1}.k_{f})\\\
&\times(g_{A}-g_{V})^{2}(k.p_{i}))\Big{)}+(k.p_{f})(k.p_{i})((k.p_{f})+|\textbf{a}|(k.p_{i}))\Big{(}|\textbf{a}|^{2}e^{2}(g_{A}^{2}+g_{V}^{2})+2g_{V}^{2}(m^{2}-(p_{i}.p_{f}))\\\
&-2g_{A}^{2}(m^{2}+(p_{i}.p_{f}))\Big{)}\epsilon(\eta_{2},k,k_{f},k_{i})+g_{A}g_{V}|\textbf{a}|\Big{(}2(k_{i}.p_{i})(k.p_{f})((k.p_{f})-2(k.p_{i}))+|\textbf{a}|^{2}e^{2}(k.k_{i})\\\
&\times((k.p_{f})-(k.p_{i}))\Big{)}(k.p_{i})\epsilon(\eta_{2},k,k_{f},p_{f})+g_{A}g_{V}|\textbf{a}|\Big{(}|\textbf{a}|^{2}e^{2}(k.k_{i})(k.p_{f})^{2}-|\textbf{a}|^{2}e^{2}(k.k_{i})(k.p_{f})\\\
&\times(k.p_{i})+2(k_{i}.p_{f})(k.p_{f})^{2}(k.p_{i})\Big{)}\epsilon(\eta_{2},k,k_{f},p_{i})+g_{A}g_{V}|\textbf{a}|\Big{(}|\textbf{a}|^{2}e^{2}(k.k_{f})(k.p_{f})(k.p_{i})+2(k_{f}.p_{i})\\\
\
&\times(k.p_{f})^{2}(k.p_{i})-|\textbf{a}|^{2}e^{2}(k.k_{f})(k.p_{i})^{2}-4(k_{f}.p_{i})(k.p_{f})(k.p_{i})^{2}\Big{)}\epsilon(\eta_{2},k,k_{i},p_{f})+g_{A}g_{V}|\textbf{a}|\Big{(}|\textbf{a}|^{2}e^{2}\\\
&\times(k.k_{f})(k.p_{f})^{2}-|\textbf{a}|^{2}e^{2}(k.k_{f})(k.p_{f})(k.p_{i})+2(k_{f}.p_{f})(k.p_{f})^{2}(k.p_{i})\Big{)}\epsilon(\eta_{2},k,k_{i},p_{i})+2g_{A}g_{V}|\textbf{a}|\\\
&\times\Big{(}|\textbf{a}|^{2}e^{2}(k.k_{f})(k.k_{i})(k.p_{f})-|\textbf{a}|^{2}e^{2}(k.k_{f})(k.k_{i})(k.p_{i})+2(k_{i}.k_{f})(k.p_{f})^{2}(k.p_{i})\Big{)}\epsilon(\eta_{2},k,p_{f},p_{i})\\\
&+2(k.p_{f})(k.p_{i})^{3}|\textbf{a}|\big{(}g_{A}^{2}+g_{V}^{2}\big{)}\epsilon(\eta_{2},k_{f},k_{i},p_{f})+2(k.p_{f})^{3}(k.p_{i})|\textbf{a}|\big{(}g_{A}^{2}+g_{V}^{2}\big{)}\epsilon(\eta_{2},k_{f},k_{i},p_{i})\\\
&-2g_{A}g_{V}|\textbf{a}|(k.k_{i})(k.p_{f})^{2}(k.p_{i})\epsilon(\eta_{2},k_{f},p_{f},p_{i})-2g_{A}g_{V}|\textbf{a}|(k.k_{f})(k.p_{f})^{2}(k.p_{i})\epsilon(\eta_{2},k_{i},p_{f},p_{i})\\\
&-2|\textbf{a}|(\eta_{2}.p_{i})(k.p_{f})(k.p_{i})^{2}\big{(}g_{A}^{2}+g_{V}^{2}\Big{)}\epsilon(k,k_{f},k_{i},p_{f})-2|\textbf{a}|(\eta_{2}.p_{f})(k.p_{f})^{2}(k.p_{i})\big{(}g_{A}^{2}+g_{V}^{2}\big{)}\\\
&\times\epsilon(k,k_{f},k_{i},p_{i})+2|\textbf{a}|(\eta_{2}.k_{f})g_{A}g_{V}(k.p_{f})^{2}(k.p_{i})\epsilon(k,k_{i},p_{f},p_{i})\Big{]},\end{split}$
(48)
$\begin{split}F=&\dfrac{32e^{2}(k.k_{i})}{(k.p_{f})(k.p_{i})}\Big{[}2|\textbf{a}|^{2}(\eta_{1}.p_{f})(\eta_{1}.p_{i})(g_{A}^{2}+g_{V}^{2})(k.k_{f})-2|\textbf{a}|^{2}(\eta_{2}.p_{f})(\eta_{2}.p_{i})(g_{A}^{2}+g_{V}^{2})(k.k_{f})\\\
&-|\textbf{a}|^{2}((\eta_{1}.k_{f})(\eta_{1}.p_{i})-(\eta_{2}.k_{f})(\eta_{2}.p_{i}))(g_{A}+g_{V})^{2}(k.p_{f})-|\textbf{a}|^{2}(\eta_{1}.k_{f})(\eta_{1}.p_{f})(g_{A}-g_{V})^{2}(k.p_{i})\\\
&+|\textbf{a}|^{2}(\eta_{2}.k_{f})(\eta_{2}.p_{f})(g_{A}-g_{V})^{2}(k.p_{i})\Big{]},\end{split}$
(49)

where, for all 4-vectors $a$, $b$, $c$ and $d$, we have

$\begin{split}\epsilon(a,b,c,d)=\epsilon_{\mu\nu\rho\sigma}a^{\mu}b^{\nu}c^{\rho}d^{\sigma}.\end{split}$
(50)

We notice that, in the two coefficients $A$ and $F$, there is no appearance of
the antisymmetric tensors. This obviously means that they have been completely
contracted. However, the other coefficients $B$, $C$, $D$ and $E$ contained
different noncontracted tensors. To calculate analytically these tensors, we
recall here that we used the Grozin convention

$\begin{split}\epsilon_{0123}=1,\end{split}$ (51)

which means that $\epsilon_{\mu\nu\rho\sigma}=1\leavevmode\nobreak\ (-1)$ for
an even (odd) permutation of the Lorentz indices and
$\epsilon_{\mu\mu\rho\sigma}=0$ otherwise. For instance, we give below how we
have calculated one of these tensors, for example in the coefficient $D$

$\begin{split}\epsilon(\eta_{2},k,k_{f},k_{i})=&\epsilon_{\mu\nu\rho\sigma}\eta_{2}^{\mu}k^{\nu}k_{f}^{\rho}k_{i}^{\sigma},\\\
=&\omega\big{(}\epsilon_{2013}k_{f}^{1}k_{i}^{3}+\epsilon_{2031}k_{f}^{3}k_{i}^{1}\big{)}+\omega\big{(}\epsilon_{2301}k_{f}^{0}k_{i}^{1}+\epsilon_{2310}k_{f}^{1}k_{i}^{0}\big{)},\\\
=&-2\omega
E_{i}\big{(}|\textbf{q}_{i}|\cos(\phi_{i})\sin(\theta_{i})-|\textbf{q}_{f}|\cos(\phi_{f})\sin(\theta_{f})\big{)}.\end{split}$
(52)

One can verify that in the case where $|\textbf{a}|\rightarrow 0$ and $n=0$,
which implies $z=0$, all the terms that contribute to Eq. (43) vanish except
for the term multiplied by the coefficient $A$, since $J_{n}(0)=\delta_{0n}$
where $\delta_{0n}$ is the Kronecker’s delta satisfying $\delta_{0n}=1$ if
$n=0$ and $\delta_{0n}=0$ otherwise. Looking at the expression of the
coefficient $A$ in Eq. (44), we find that in this case it simplifies to match
the expression obtained in the absence of the laser field given in Eq. (14).

## References

  * [1] E. Rutherford, The scattering of $\alpha$ and $\beta$ particles by matter and the structure of the atom, Phil. Mag. 21, 669 (1911).
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