# Relativistic elastic scattering of a muon neutrino by an electron in an
elliptically polarized laser field

R. Chahri  Polydisciplinary Faculty, Laboratory of Research in Physics and
Engineering Sciences, Team of Modern and Applied Physics, Sultan Moulay
Slimane University, Beni Mellal, 23000, Morocco.  S. El Asri  Polydisciplinary
Faculty, Laboratory of Research in Physics and Engineering Sciences, Team of
Modern and Applied Physics, Sultan Moulay Slimane University, Beni Mellal,
23000, Morocco.  S. Mouslih  Faculté des Sciences et Techniques, Laboratoire
de l’Ingénierie en Chimie et Physique de la Matière (LICPM), Beni Mellal,
23000, Morocco.  Polydisciplinary Faculty, Laboratory of Research in Physics
and Engineering Sciences, Team of Modern and Applied Physics, Sultan Moulay
Slimane University, Beni Mellal, 23000, Morocco.  M. Jakha  Polydisciplinary
Faculty, Laboratory of Research in Physics and Engineering Sciences, Team of
Modern and Applied Physics, Sultan Moulay Slimane University, Beni Mellal,
23000, Morocco.  B. Manaut  Polydisciplinary Faculty, Laboratory of Research
in Physics and Engineering Sciences, Team of Modern and Applied Physics,
Sultan Moulay Slimane University, Beni Mellal, 23000, Morocco.  S. Taj
<EMAIL_ADDRESS>Polydisciplinary Faculty, Laboratory of Research in Physics and
Engineering Sciences, Team of Modern and Applied Physics, Sultan Moulay
Slimane University, Beni Mellal, 23000, Morocco.

(August 28, 2024)

###### Abstract

Within the framework of electroweak theory, we investigate the elastic
scattering process $e^{-}+\nu_{\mu}\rightarrow e^{-}+\nu_{\mu}$ in the
presence of an intense elliptically polarized laser field. We derive an
analytical expression for the spin-unpolarized differential cross section
using the first Born approximation and the Dirac-Volkov states to describe the
incident and scattered electrons. Our results generalize those found for the
linearly polarized field by Bai et al. [Phys. Rev. A 85, 013402 (2012)] and
for the circularly polarized field by El Asri et al. [Phys. Rev. D 104, 113001
(2021)]. We find that the differential cross section is significantly enhanced
for linear polarization and reduced for circular and elliptical polarizations.

##  I Introduction

Due to the rapid and advanced progress in the field of laser technology since
the early 1960s, lasers have become widely used in industrial, medical,
commercial, scientific and military domains. Lasers can provide us with
sources having extreme properties in terms of energy, pulse width and
wavelength, helping researchers to understand the fundamental concepts of
radiation-matter interaction. Development of lasers with shorter wavelengths,
shorter pulses and higher intensities continues unabated. The achievement of a
maximum laser intensity of $10^{22}$ W/cm2 [1] should lead to a better
understanding of the behavior of various scattering processes [2]. Therefore,
lasers are currently indispensable tools for investigating physical processes,
in particular, laser-matter interaction. The early studies of laser-assisted
scattering in the nonrelativistic regime and at moderate field strengths are
well established and documented in the literature. A comprehensive overview of
this can be found in the books of Mittleman [3], Faisal [4], Delone [5],
Fedorov [6] and in some recent reviews [7, 8, 9]. With the advent of very
powerful laser sources, it has become important to consider laser-assisted
processes in the relativistic regime. Therefore, in a laser field of
relativistic intensity, many processes have been studied such as Mott
scattering in an elliptically and linearly polarized laser field [10, 11],
laser-assisted bremsstrahlung for circular and linear polarization [12].
Electron-proton elastic scattering has been investigated in the presence of a
linearly and circularly polarized laser field in [15, 13, 14]. In addition, in
[16, 17], the authors studied new phenomena in laser-assisted scattering of an
electron (positron) by a muon with different polarizations. There are also
some papers that studied decay processes in the presence of laser field [18,
19, 20, 21, 22]. In this paper, using the first Born approximation, we give
complete analytical and numerical results of the scattering process
$e^{-}+\nu_{\mu}\rightarrow e^{-}+\nu_{\mu}$, assisted by an elliptically
polarized laser field. These results can generate those found in previous
works for a circularly polarized laser field by El Asri et al. [23] and for a
linearly polarized laser field by Bai et al. [24]. In addition, this process
has been recently studied in the framework of electroweak theory for a
circularly polarized laser field by [25]. Overall, the goal of this work is to
generalize the previous research, with a more detailed calculation in the
presence of an elliptically polarized laser field. We have also compared the
differential cross section (DCS) in the absence and presence of a laser field
at different polarizations, and examined its dependence on the laser
parameters. For the elliptical polarization of the laser field, our current
calculations lead to a new form of ordinary and generalized Bessel functions
[28, 27, 26]. The paper is organized as follows. In Sec. II, we establish the
detailed analytical calculation of the S-matrix element, in the first Born
approximation, of the process $e^{-}+\nu_{\mu}\rightarrow e^{-}+\nu_{\mu}$, as
well as the expression for the DCS in the presence of an external elliptically
polarized laser field. Then, in Sec. III, we present numerical results for the
laser-assisted DCS and discuss its dependence on the relevant parameters. Sec.
IV is devoted to conclusions. Throughout this work, we use the natural units
($\hbar=c=1$) and the Minkowski metric tensor
$g^{\mu\nu}=\text{diag}(1,-1,-1,-1)$.

##  II Theory

We consider the scattering process of a muon neutrino by an electron
schematized as follows:

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

where the labels $(p_{i,f},k_{i,f})$ and $(s_{i,f},t_{i,f})$ are the
associated four-momenta and spin respectively, with $i$ and $f$ stand for the
initial and final states. In the framework of electroweak theory, this
scattering process is mediated by the exchange of only the neutral $Z$ boson.
The corresponding lowest order Feynman diagram is given by Fig. 1.

Figure 1: The tree-level Feynman diagram at the $t$-channel of the process
$e^{-}+\nu_{\mu}\rightarrow e^{-}+\nu_{\mu}$ in the framework of electroweak
theory. The time flows from left to right.

We treat this process in the presence of an elliptically polarized laser field
described by the following classical four-potential [10]

$\begin{split}A^{\mu}=a_{1}^{\mu}\cos(\phi)+a_{2}^{\mu}\sin(\phi)\tan(\eta/2),\end{split}$
(2)

where $\phi=k^{\mu}x_{\mu}=(k.x)$, and $\eta$ is the degree of ellipticity of
the external electromagnetic (EM) field. The linear polarization is obtained
for $\eta=0^{\circ}$, while the circular polarization is obtained for
$\eta=90^{\circ}$. We choose the wave four-vector as
$k^{\mu}=(\omega,\textbf{k})=\omega(1,0,0,1)$. The polarization four-vectors
$a_{1}^{\mu}=(0,\textbf{a}_{1})=|\textbf{a}|(0,1,0,0)$ and
$a_{2}^{\mu}=(0,\textbf{a}_{2})=|\textbf{a}|(0,0,1,0)$ are along the $x$ and
$y$-axis, respectively. $a_{1}^{\mu}$ and $a_{2}^{\mu}$ satisfy the
normalization
$a_{1}^{2}=a_{2}^{2}=a^{2}=-|\textbf{a}|^{2}=-(\mathcal{E}_{0}/\omega)^{2}$
and the orthogonality conditions $(a_{1}.a_{2})=0$, with $\mathcal{E}_{0}$ is
the laser field strength and $\omega$ is the frequency. Moreover, the four-
potential $A^{\mu}$ satisfies the Lorenz gauge condition $k_{\mu}A^{\mu}=0$,
which implies that $k_{\mu}a_{1}^{\mu}=k_{\mu}a_{2}^{\mu}=0$, forcing the wave
vector k to be along the $z$-axis. The incoming and outgoing muon neutrinos,
which do not interact with the laser field, are treated as non-mass particles,
described by Dirac wave functions, normalized to the volume $V$ given by the
following formula [29]

$\begin{split}\begin{matrix}\psi^{i,f}_{\nu_{\mu}}(x)=\dfrac{1}{\sqrt{2k^{0}_{i,f}V}}u_{\nu_{\mu}}(k_{i,f},t_{i,f})\leavevmode\nobreak\
e^{-ik_{i,f}.x}.\end{matrix}\end{split}$ (3)

In the presence of a laser field, the electron obeys the following Dirac-
Volkov equation [30]

$\begin{split}\big{\\{}(\widehat{p_{i,f}}-eA)^{2}-m^{2}-\dfrac{1}{2}ieF_{\mu\nu}\sigma^{\mu\nu}\big{\\}}\psi^{i,f}_{e^{-}}(x)=0,\end{split}$
(4)

where $e=-|e|<0$ is the electric charge of electron. Dressed by an
elliptically polarized laser field, the incoming and outgoing electrons can be
considered as Dirac-Volkov states normalized to volume $V$ [30]

$\begin{split}\psi^{i,f}_{e^{-}}(x)=\Big{[}1+\dfrac{e}{2(k.p_{i,f})}\Big{(}\not{k}\not{a}_{1}\cos(\phi)+\not{k}\not{a}_{2}\sin(\phi)\tan(\eta/2)\Big{)}\Big{]}\dfrac{u_{e}(p_{i,f},s_{i,f})}{\sqrt{2Q_{i,f}V}}e^{iS(q_{i,f},x)},\end{split}$
(5)

where

$\begin{split}S(q_{i,f},x)=&-(q_{i,f}.x)-\dfrac{e(p_{i,f}.a_{1})}{(k.p_{i,f})}\sin(\phi)+\dfrac{e^{2}a^{2}}{8(k.p_{i,f})}(1-\tan^{2}(\eta/2))\sin(2\phi)+\dfrac{e(p_{i,f}.a_{2})}{(k.p_{i,f})}\tan(\eta/2)\cos(\phi),\end{split}$
(6)

where $u_{\nu_{\mu}}(k_{i,f},t_{i,f})$ and $u_{e}(p_{i,f},s_{i,f})$ represent
the Dirac bispinors satisfying
$\sum_{t_{i,f}}\bar{u}_{\nu_{\mu}}(k_{i,f},t_{i,f})u_{\nu_{\mu}}(k_{i,f},t_{i,f})=\not{k}_{i,f}$
and
$\sum_{s_{i,f}}\bar{u}_{e}(p_{i,f},s_{i,f})u_{e}(p_{i,f},s_{i,f})=\not{p}_{i,f}+m_{e}$.
$q_{i,f}=(Q_{i,f},\textbf{q}_{i,f})$ is the Volkov momentum of the electron in
the presence of a laser field. That is

$q_{i,f}^{\mu}=p_{i,f}^{\mu}-\dfrac{e^{2}a^{2}}{4(k.p_{i,f})}(1+\tan^{2}(\eta/2))k^{\mu}.$
(7)

The square of this four-momentum shows that the mass of the dressed electron
(effective mass) is proportional to the strength of the EM field as follows :

$q_{i,f}^{2}=m_{*}^{2}=m_{e}^{2}-\dfrac{e^{2}a^{2}}{2}(1+\tan^{2}(\eta/2)),$
(8)

where $m_{e}$ is the mass of electron, and the quantity $m_{*}$ represents the
effective mass of the electron in the elliptically polarized EM field.
In the first Born approximation, the transition matrix element can be
expressed using Feynman rules as follows:

$\begin{split}S_{fi}(e^{-}\nu_{\mu}\rightarrow e^{-}\nu_{\mu})=&-i\int
d^{4}x\int
d^{4}y\Big{[}\bar{\psi}^{f}_{\nu_{\mu}}(x)\dfrac{-ig}{4\cos(\theta_{W})}\gamma^{\mu}(1-\gamma^{5})\psi^{i}_{\nu_{\mu}}(x)\Big{]}D_{\mu\nu}(x-y)\\\
&\times\Big{[}\bar{\psi}^{f}_{e}(y)\dfrac{-ig}{2\cos(\theta_{W})}\gamma^{\nu}(g_{V}-g_{A}\gamma^{5})\psi^{i}_{e}(y)\Big{]},\end{split}$
(9)

with $g_{V}=1/2-2\sin^{2}(\theta_{W})$ and $g_{A}=-1/2$ are the coupling
constants [31], where $\theta_{W}$ is the Weinberg angle. Here, we choose
$g_{V}=0,043$ and $g_{A}=-0,545$ [29]. $D_{\mu\nu}(x-y)$ is the Feynman
propagator for the coupling between the $Z$-boson and the fermions given by
[32]

$\begin{split}D_{\mu\nu}(x-y)=-i\int\dfrac{d^{4}q}{(2\pi)^{4}}\dfrac{e^{-iq(x-y)}}{q^{2}-M_{Z}^{2}+iM_{Z}\Gamma_{Z}}\leavevmode\nobreak\
\Big{[}g_{\mu\nu}-\dfrac{q_{\mu}q_{\nu}}{M_{Z}^{2}}\Big{]},\end{split}$ (10)

where $M_{Z}=(91.1876\pm 0.0021)\leavevmode\nobreak\ \text{GeV}$ is the rest
mass of the $Z$-boson and $\Gamma_{Z}=(2.4952\pm 0.0023)\leavevmode\nobreak\
\text{GeV}$ is its total decay rate [33]. After inserting Eqs. (3), (5) and
(10) into Eq. (9) and after some algebraic manipulations, we find

$\begin{split}S_{fi}(e^{-}\nu_{\mu}\rightarrow
e^{-}\nu_{\mu})=&\dfrac{g^{2}}{8\cos^{2}(\theta_{W})\leavevmode\nobreak\
(q^{2}-M_{Z}^{2}+iM_{Z}\Gamma_{Z})}\int
d^{4}xd^{4}y\dfrac{d^{4}q}{(2\pi)^{4}}\dfrac{e^{-iq(x-y)}e^{i(k_{f}-k_{i}).x}}{\sqrt{16k^{0}_{i}k^{0}_{f}Q_{i}Q_{f}V^{4}}}\\\
&\times
e^{i(S(q_{i},y)-S(q_{f},y))}\Big{[}\bar{u}_{\mu_{\mu}}(k_{f},t_{i})\gamma^{\mu}(1-\gamma_{5})u_{\nu_{\mu}}(k_{i},t_{f})\Big{]}\\\
&\times\Big{[}\bar{u}_{e}(p_{f},s_{f})\Big{(}1+\dfrac{e\not{A}\not{k}}{2(k.p_{f})}\Big{)}\gamma_{\mu}(g_{V}-g_{A}\gamma^{5})\Big{(}1+\dfrac{e\not{k}\not{A}}{2(k.p_{i})}\Big{)}\leavevmode\nobreak\
u_{e}(p_{i},s_{i})\Big{]}.\end{split}$ (11)

We expand the term $e^{i(S(q_{i},y)-S(q_{f},y))}$ in Eq. (11) under the
following transformation

$\begin{split}e^{i(S(q_{i},y)-S(q_{f},y))}=e^{i(q_{f}-q_{i}).y}\leavevmode\nobreak\
e^{i\big{(}\xi_{1}\sin(\phi)+\xi_{2}\sin(2\phi)\big{)}}\leavevmode\nobreak\
e^{i\xi_{3}\cos(\phi)},\end{split}$ (12)

where

$\xi_{1}=e\Big{(}\dfrac{p_{f}.a_{1}}{k.p_{f}}-\dfrac{p_{i}.a_{1}}{k.p_{i}}\Big{)},\leavevmode\nobreak\
\leavevmode\nobreak\
\xi_{2}=\dfrac{-e^{2}a^{2}}{8}\Big{(}\dfrac{1}{k.p_{f}}-\dfrac{1}{k.p_{i}}\Big{)}\Big{(}1-\tan^{2}(\eta/2)\Big{)},\leavevmode\nobreak\
\leavevmode\nobreak\
\xi_{3}=-e\Big{(}\dfrac{p_{f}.a_{2}}{k.p_{f}}-\dfrac{p_{i}.a_{2}}{k.p_{i}}\Big{)}\tan(\eta/2).$
(13)

By introducing Eq. (12) and using
$g^{2}/(8\cos^{2}(\theta_{W}))=G_{F}M^{2}_{Z}/\sqrt{2}$, where
$\textit{G}_{F}=(1.16637\pm 0.00002)\times 10^{-11}\leavevmode\nobreak\
\text{MeV}^{-2}$ is the Fermi coupling constant, the expression of S-matrix
becomes

$\begin{split}S_{fi}(e^{-}\nu_{\mu}\rightarrow
e^{-}\nu_{\mu})=&\dfrac{G_{F}M^{2}_{Z}}{\sqrt{2}\leavevmode\nobreak\
(q^{2}-M_{Z}^{2}+iM_{Z}\Gamma_{Z})}\int
d^{4}xd^{4}y\dfrac{d^{4}q}{(2\pi)^{4}}\dfrac{e^{i(k_{f}-k_{i}-q).x}\leavevmode\nobreak\
e^{i(q_{f}-q_{i}+q).y}}{\sqrt{16k^{0}_{i}k^{0}_{f}Q_{i}Q_{f}V^{4}}}\\\ &\times
e^{i\big{(}\xi_{1}\sin(\phi)+\xi_{2}\sin(2\phi)\big{)}}e^{i\xi_{3}\cos(\phi)}\leavevmode\nobreak\
\Big{[}\bar{u}_{\mu_{\mu}}(k_{f},t_{i})\gamma^{\mu}(1-\gamma_{5})\leavevmode\nobreak\
u_{\nu_{\mu}}(k_{i},t_{f})\Big{]}\\\
&\times\Big{[}\bar{u}_{e}(p_{f},s_{f})\Big{(}\chi_{0\mu}+\chi_{1\mu}\cos(\phi)+\chi_{2\mu}\sin(\phi)+\chi_{3\mu}\cos^{2}(\phi)\Big{)}u_{e}(p_{i},s_{i})\Big{]},\end{split}$
(14)

where the quantities $\chi_{0\mu}$, $\chi_{1\mu}$, $\chi_{2\mu}$ and
$\chi_{3\mu}$ are expressed as follows :

$\begin{split}&\chi_{0\mu}=\gamma_{\mu}(g_{V}-g_{A}\gamma_{5})-2C(p_{i})C(p_{f})a^{2}k_{\mu}\not{k}(g_{V}-g_{A}\gamma_{5})\tan^{2}(\eta/2),\\\
&\chi_{1\mu}=C(p_{f})\not{a}_{1}\not{k}\gamma_{\mu}(g_{V}-g_{A}\gamma_{5})+C(p_{i})\gamma_{\mu}(g_{V}-g_{A}\gamma_{5})\not{k}\not{a}_{1},\\\
&\chi_{2\mu}=\big{[}C(p_{f})\not{a}_{2}\not{k}\gamma_{\mu}(g_{V}-g_{A}\gamma_{5})+C(p_{i})\gamma_{\mu}(g_{V}-g_{A}\gamma_{5})\not{k}\not{a}_{2}\big{]}\tan(\eta/2),\\\
&\chi_{3\mu}=-2C(p_{i})C(p_{f})a^{2}k_{\mu}\not{k}(g_{V}-g_{A}\gamma_{5})\big{(}1-\tan^{2}(\eta/2)\big{)},\end{split}$
(15)

where $C(p_{i,f})=e/[2(k.p_{i,f})]$. Now, we use a transformation, known as
the ordinary and generalized Jacobi-Anger identity, involving ordinary
$J_{s}(\xi_{3})$ and generalized $B_{s}(\xi_{1},\xi_{2})$ Bessel functions
[26]:

$\begin{split}e^{i(\xi_{1}\sin(\phi)+\xi_{2}\sin(2\phi))}&=\sum_{s=-\infty}^{+\infty}B_{s}(\xi_{1},\xi_{2})\leavevmode\nobreak\
e^{is\phi},\\\
e^{i\xi_{3}\cos(\phi)}&=\sum_{s=-\infty}^{+\infty}i^{s}j_{s}(\xi_{3})\leavevmode\nobreak\
e^{is\phi},\\\ \end{split}$ (16)

where $s$, the order of Bessel functions, is commonly interpreted as the
number of photons exchanged between the two particles involved in our
scattering process and the laser field. Explicitly, we apply the following
transformation

$\begin{split}\begin{Bmatrix}1\\\ \cos(\phi)\\\ \sin(\phi)\\\
\cos^{2}(\phi)\end{Bmatrix}e^{i(\xi_{1}\sin(\phi)+\xi_{2}\sin(2\phi))}\leavevmode\nobreak\
e^{i\xi_{3}\cos(\phi)}=\sum_{s=-\infty}^{+\infty}\begin{Bmatrix}B_{s}(\xi_{3})\\\
B_{1s}(\xi_{3})\\\ B_{2s}(\xi_{3})\\\
B_{3s}(\xi_{3})\end{Bmatrix}B_{s}(\xi_{1},\xi_{2})\leavevmode\nobreak\
e^{i2s\phi},\end{split}$ (17)

where the coefficients $B_{s}(\xi_{3})$, $B_{1s}(\xi_{3})$, $B_{2s}(\xi_{3})$,
$B_{3s}(\xi_{3})$ and $B_{s}(\xi_{1},\xi_{2})$ are expressed in terms of
ordinary Bessel functions as follows [34, 26]:

$\begin{split}B_{s}(\xi_{3})&=i^{s}j_{s}(\xi_{3}),\\\
B_{1s}(\xi_{3})&=\big{(}i^{s-1}j_{s-1}(\xi_{3})+i^{s+1}j_{s+1}(\xi_{3})\big{)}/2,\\\
B_{2s}(\xi_{3})&=\big{(}i^{s-1}j_{s-1}(\xi_{3})-i^{s+1}j_{s+1}(\xi_{3})\big{)}/2i,\\\
B_{3s}(\xi_{3})&=\big{(}2i^{s}j_{s}(\xi_{3})+i^{s-2}j_{s-2}(\xi_{3})+i^{s+2}j_{s+2}(\xi_{3})\big{)}/4,\\\
B_{s}(\xi_{1},\xi_{2})&=\sum_{\lambda=-\infty}^{+\infty}j_{s-2\lambda}(\xi_{1})j_{\lambda}(\xi_{2}).\end{split}$
(18)

After integration over space-time and $d^{4}q$, and after some algebraic
manipulations, we can decompose the transition matrix element into a series of
terms in the form of ordinary and generalized Bessel functions

$\begin{split}S_{fi}(e^{-}\nu_{\mu}\rightarrow
e^{-}\nu_{\mu})=\sum_{s=-\infty}^{+\infty}\dfrac{G_{F}M_{Z}^{2}}{\sqrt{2}(q^{2}-M_{Z}^{2}+iM_{Z}\Gamma_{Z})}\dfrac{(2\pi)^{4}\leavevmode\nobreak\
\delta^{4}(q_{f}-q_{i}+k_{f}-k_{i}+2sk)}{\sqrt{16k^{0}_{i}k^{0}_{f}Q_{i}Q_{f}V^{4}}}\leavevmode\nobreak\
M^{(s)}_{fi},\end{split}$ (19)

where $q=q_{i}-q_{f}-2sk$ is the relativistic four-momentum transfer in the
presence of the EM field. The quantity $M^{(s)}_{fi}$ in Eq. (19) is defined
by

$\begin{split}M^{(s)}_{fi}=\big{[}\bar{u}_{\nu_{\mu}}(k_{f},t_{i})\gamma^{\mu}(1-\gamma_{5})u_{\nu_{\mu}}(k_{i},t_{f})\big{]}\big{[}\bar{u}_{e}(p_{f},s_{f})\Gamma_{\mu}^{(s)}u_{e}(p_{i},s_{i})\big{]},\end{split}$
(20)

where

$\begin{split}\Gamma_{\mu}^{(s)}=\big{[}\chi_{0\mu}B_{s}(\xi_{3})+\chi_{1\mu}B_{1s}(\xi_{3})+\chi_{2\mu}B_{2s}(\xi_{3})+\chi_{3\mu}B_{3s}(\xi_{3})\big{]}B_{s}(\xi_{1},\xi_{2}).\end{split}$
(21)

The DCS can be obtained by summing over the final spin states and averaging
over the initial ones. Note that electrons can be in two spin states, while
neutrinos are only in one negative helicity state [29]. Therefore, we divide
the square of the matrix element $|S_{fi}|^{2}$ by the incident particle flux
$|J_{inc}|$ and the observation time interval $T$, and multiply it by the
density of the final states. The unpolarized DCS can then be written as
follows:

$d\bar{\sigma}(e^{-}\nu_{\mu}\rightarrow e^{-}\nu_{\mu})=\leavevmode\nobreak\
\int V\dfrac{d^{3}q_{f}}{(2\pi)^{3}}\leavevmode\nobreak\ \int
V\dfrac{d^{3}k_{f}}{(2\pi)^{3}}\leavevmode\nobreak\
\dfrac{1}{2}\sum_{t_{i,f},s_{i,f}}\dfrac{|S_{fi}(e^{-}\nu_{\mu}\rightarrow
e^{-}\nu_{\mu})|^{2}}{T|J_{inc}|},$ (22)

where $|J_{inc}|=(k_{i}.q_{i})/(k^{0}_{i}Q_{i}V)$ denotes the current of the
incoming particles in the laboratory system. Applying the following relations:
$d^{3}q_{f}=|\textbf{q}_{f}|^{2}d|\textbf{q}_{f}|d\Omega$ and
$\delta^{4}(q_{f}-q_{i}+k_{f}-k_{i}+2sk)=\delta^{0}(Q_{f}+k^{0}_{f}-Q_{i}-k^{0}_{i}+2s\omega)\delta^{3}(\textbf{q}_{f}+\textbf{k}_{f}-\textbf{q}_{i}-\textbf{k}_{i}+2s\textbf{k})$
and taking simplifications, we obtain

$\begin{split}\dfrac{d\bar{\sigma}}{d\Omega}(e^{-}\nu_{\mu}\rightarrow
e^{-}\nu_{\mu})=&\sum_{s=-\infty}^{+\infty}\dfrac{\textit{G}_{F}^{2}\leavevmode\nobreak\
M_{Z}^{4}}{64(2\pi)^{2}((q^{2}-M_{Z}^{2})^{2}+M_{Z}^{2}\Gamma_{Z}^{2})(k_{i}.q_{i})}\int\dfrac{|\textbf{q}_{f}|^{2}d|\textbf{q}_{f}|}{k^{0}_{f}Q_{f}}\delta^{0}(Q_{f}+k^{0}_{f}-Q_{i}-k^{0}_{i}+2s\omega)\\\
&\times\sum_{s^{\prime}_{i,f},s_{i,f}}|M_{fi}^{(s)}|^{2}\bigg{|}_{\textbf{q}_{f}+\textbf{k}_{f}-\textbf{q}_{i}-\textbf{k}_{i}+2s\textbf{k}=0}.\end{split}$
(23)

Using the following relation [32]

$\begin{split}\int
dxf(x)\delta(g(x))=\dfrac{f(x)}{|g^{\prime}(x)|}\Big{|}_{g(x)=0},\end{split}$
(24)

we can perform the remaining integral over $d|\textbf{q}_{f}|$. Therefore, the
summed differential cross section (SDCS) can be decomposed into a series of
discrete individual differential cross section (IDCS) for different numbers of
photons exchanged. This yields

$\begin{split}\Big{(}\dfrac{d\bar{\sigma}}{d\Omega}\Big{)}^{\text{with
laser}}(e^{-}\nu_{\mu}\rightarrow
e^{-}\nu_{\mu})=\sum_{s=-\infty}^{+\infty}\dfrac{d\sigma^{(s)}}{d\Omega},\end{split}$
(25)

where the IDCS can be written as

$\begin{split}\dfrac{d\sigma^{(s)}}{d\Omega}=\dfrac{\textit{G}_{F}^{2}\leavevmode\nobreak\
M_{Z}^{4}}{64(2\pi)^{2}((q^{2}-M_{Z}^{2})^{2}+M_{Z}^{2}\Gamma_{Z}^{2})(k_{i}.q_{i})k^{0}_{f}Q_{f}}\dfrac{|\textbf{q}_{f}|^{2}}{\Big{|}g^{\prime}(|\textbf{q}_{f}|)\Big{|}}\times\sum_{s_{i},s_{f}}\sum_{t_{i},t_{f}}|M_{fi}^{(s)}|^{2}\bigg{|}_{\textbf{q}_{f}+\textbf{k}_{f}-\textbf{q}_{i}-\textbf{k}_{i}+2s\textbf{k}=0},\end{split}$
(26)

where

$\begin{split}g^{\prime}(|\textbf{q}_{f}|)=\dfrac{|\textbf{q}_{f}|}{Q_{f}}+\dfrac{|\textbf{q}_{f}|+k^{0}_{i}\cos(\theta_{f})-|\textbf{q}_{i}|F(\theta_{i},\theta_{f},\varphi_{i},\varphi_{f})+2s\omega\cos(\theta_{f})}{k^{0}_{f}},\end{split}$
(27)

with

$\begin{split}F(\theta_{i},\theta_{f},\varphi_{i},\varphi_{f})=\sin(\theta_{i})\cos(\varphi_{i})\sin(\theta_{f})\cos(\varphi_{f})+\sin(\theta_{i})\sin(\varphi_{i})\sin(\theta_{f})\sin(\varphi_{f})+\cos(\theta_{i})\cos(\theta_{f}).\end{split}$
(28)

The sum over spin can be converted to trace calculation as follows:

$\begin{split}\sum_{s_{i},s_{f}}\sum_{t_{i},t_{f}}|M_{fi}^{(s)}|^{2}=\text{Tr}\big{[}\not{k}_{f}\gamma^{\mu}(1-\gamma_{5})\not{k}_{i}\gamma^{\nu}(1-\gamma_{5})\big{]}\text{Tr}\big{[}(\not{p}_{f}+m_{e})\Gamma_{\mu}^{(s)}(\not{p}_{i}+m_{e})\bar{\Gamma}_{\nu}^{(s)}\big{]},\end{split}$
(29)

where

$\begin{split}&\bar{\Gamma}_{\nu}^{(s)}=\gamma^{0}\Gamma_{\nu}^{(s)\dagger}\gamma^{0},\\\
&\bar{\Gamma}_{\nu}^{(s)}=\big{[}\bar{\chi}_{0\nu}B^{*}_{s}(\xi_{3})+\bar{\chi}_{1\nu}B^{*}_{1s}(\xi_{3})+\bar{\chi}_{2\nu}B^{*}_{2s}(\xi_{3})+\bar{\chi}_{3\nu}B^{*}_{3s}(\xi_{3})\big{]}\leavevmode\nobreak\
B^{*}_{s}(\xi_{1},\xi_{2}),\end{split}$ (30)

and

$\begin{split}&\bar{\chi}_{0\nu}=\gamma_{\nu}(g_{V}-g_{A}\gamma_{5})-2C(p_{i})C(p_{f})a^{2}k_{\nu}\not{k}(g_{V}-g_{A}\gamma_{5})\tan^{2}(\eta/2),\\\
&\bar{\chi}_{1\nu}=C(p_{f})\gamma_{\nu}(g_{V}-g_{A}\gamma_{5})\not{k}\not{a}_{1}+C(p_{i})\not{a}_{1}\not{k}\gamma_{\nu}(g_{V}-g_{A}\gamma_{5}),\\\
&\bar{\chi}_{2\nu}=\big{[}C(p_{f})\gamma_{\nu}(g_{V}-g_{A}\gamma_{5})\not{k}\not{a}_{2}+C(p_{i})\not{a}_{2}\not{k}\gamma_{\nu}(g_{V}-g_{A}\gamma_{5})\big{]}\tan(\eta/2),\\\
&\bar{\chi}_{3\nu}=-2C(p_{i})C(p_{f})a^{2}k_{\nu}\not{k}(g_{V}-g_{A}\gamma_{5})\big{(}1-\tan^{2}(\eta/2)\big{)}.\end{split}$
(31)

The FeynCalc program [35] is used to compute the traces in Eq.(29). The result
obtained is written in terms of the coefficients $\Lambda_{1}$ to
$\Lambda_{16}$ as follows:

$\begin{split}\sum_{s_{i},s_{f}}\sum_{t_{i},t_{f}}|M_{fi}^{(s)}|^{2}=&\big{[}\Lambda_{1}|B_{s}(\xi_{3})|^{2}+\Lambda_{2}|B_{1s}(\xi_{3})|^{2}+\Lambda_{3}|B_{2s}(\xi_{3})|^{2}+\Lambda_{4}\leavevmode\nobreak\
|B_{3s}(\xi_{3})|^{2}+\Lambda_{5}B_{s}(\xi_{3})B^{*}_{1s}(\xi_{3})\\\
&+\Lambda_{6}B_{1s}(\xi_{3})B^{*}_{s}(\xi_{3})+\Lambda_{7}B_{s}(\xi_{3})B^{*}_{2s}(\xi_{3})+\Lambda_{8}\leavevmode\nobreak\
B_{2s}(\xi_{3})B^{*}_{s}(\xi_{3})+\Lambda_{9}B_{s}(\xi_{3})B^{*}_{3s}(\xi_{3})\\\
&+\Lambda_{10}\leavevmode\nobreak\
B_{3s}(\xi_{3})B^{*}_{s}(\xi_{3})+\Lambda_{11}\leavevmode\nobreak\
B_{1s}(\xi_{3})B^{*}_{2s}(\xi_{3})+\Lambda_{12}\leavevmode\nobreak\
B_{2s}(\xi_{3})B^{*}_{1s}(\xi_{3})+\Lambda_{13}B_{1s}(\xi_{3})B^{*}_{3s}(\xi_{3})\\\
&+\Lambda_{14}\leavevmode\nobreak\
B_{3s}(\xi_{3})B^{*}_{1s}(\xi_{3})+\Lambda_{15}\leavevmode\nobreak\
B_{2s}(\xi_{3})B^{*}_{3s}(\xi_{3})+\Lambda_{16}B_{3s}(\xi_{3})B^{*}_{2s}(\xi_{3})\big{]}|B_{s}(\xi_{1},\xi_{2})|^{2}.\end{split}$
(32)

We give the explicit expression of first four coefficients in the Appendix.

##  III Numericla results and discussion

In this section, we will present the numerical results obtained and discuss
their physical interpretation. We should focus on experimentally measurable
quantities, in particular the behavior of the DCS assisted by an external
elliptically polarized EM field. We set the incident electron momentum
$\textbf{p}_{i}$ and outgoing $\textbf{p}_{f}$ in a general geometry with
spherical coordinates $\theta_{i}$, $\theta_{f}$, $\varphi_{i}$ and
$\varphi_{f}$. Except for Figs. 3, 4 and 5, we have chosen
$\theta_{i}=\varphi_{i}=\varphi_{f}=1^{\circ}$ in all the results obtained.
For the incident muon neutrino, the momentum $\textbf{k}_{i}$ remains in the
direction opposite to the $z$ axis. Except for Fig. 3, we fix the kinetic
energy of the incident electron at $E_{e}^{\text{kin}}=10^{-3}$ GeV, and the
initial energy of the muon neutrino is chosen as the mass of an electron at
rest, $E_{\nu}^{\text{kin}}=0.5\times 10^{-3}$ GeV. The direction of the field
wave vector k is along the $z$-axis, while the polarization vectors
$\textbf{a}_{1}$ and $\textbf{a}_{2}$ perpendicular to k are along the $x$ and
$y$-axes, respectively, as illustrated in Fig. 2. Every
$d\sigma^{(s)}/d\Omega$ given by Eq. (26), considering the four-momentum
conservation, can be interpreted as the IDCS that describes the scattering
process for each number of photons $s$ ($s>0$ for absorption and $s<0$ for
emission). Summing over a number of exchanged photons $s$, we obtain the SDCS
given by Eq. (25).

Figure 2: Geometry of the coordinate system for the scattering process. The
incident electron $e^{-}(p_{i})$ moves in a spherical coordinate geometry
($\theta_{i}$, $\varphi_{i}$), while the muon neutrino $\nu_{\mu}(k_{i})$
remains in the direction opposite to the $z$-axis. The outgoing muon neutrino
and final electron move in space ($xyz$).

We will show how the DCS, in the presence and absence of the laser field,
varies as a function of the final scattering angle $\theta_{f}$, the kinetic
energy of the incident electron $E^{kin}_{e}$ and various incident angles of
the electron $\theta_{i}$. Then, we make comparisons of the IDCS and SDCS with
other research papers. Afterward, we show how the IDCS depends on the number
of photons exchanged $s$, on the geometry, on the parameters characterizing
the EM field ($\mathcal{E}_{0}$, $\omega$) and on the kinetic energy of the
incident electron. Before finishing, we illustrate the variation of the SDCS
as a function of the final scattering angle $\theta_{f}$ and the kinetic
energy of the incident electron $E^{kin}_{e}$ at different polarizations,
frequencies, and electric field strengths $\mathcal{E}_{0}$. Finally, we show
how it evolves as a function of the electric field strength at different
polarizations. We start our discussion with something we are used to do in
such processes occurring in an external EM field. That is we make sure that
the DCS in the presence of the laser field with different polarizations is
exactly equal to that in the absence of the laser field when the laser
parameters tend to zero. In Fig. 3, we illustrate the comparison between the
laser-assisted DCS of the electron-muon neutrino scattering in the framework
of electroweak theory, given by Eq.(26), and the corresponding one in the
absence of the laser field for a geometry $\theta_{i}=1^{\circ}$,
$\varphi_{i}=\varphi_{f}=0^{\circ}$ in Fig. 3(a), and
$\theta_{i}=\varphi_{i}=\varphi_{f}=1^{\circ}$, $\theta_{f}=2^{\circ}$ in Fig.
3(b).

Figure 3: The two DCSs with and without laser, illustrated as a function of
the final scattering angle $\theta_{f}$ and the kinetic energy of the incident
electron for an electric field strength $\mathcal{E}_{0}=0\leavevmode\nobreak\
\text{V/cm}$ and without any exchange of photons $(s=0)$. The remaining
parameters are chosen as follows: (a) $E_{\nu}^{\text{kin}}=0.5\times 10^{-3}$
GeV, $E_{e}^{\text{kin}}=10^{-3}$ GeV,
$\theta_{i}=1^{\circ},\leavevmode\nobreak\ \varphi_{i}=\varphi_{f}=0^{\circ}$;
(b) $E_{\nu}^{\text{kin}}=10$ GeV,
$\theta_{i}=\varphi_{i}=\varphi_{f}=1^{\circ}$, $\theta_{f}=2^{\circ}$.

The comparison allows us to verify our results by taking the limit of the
electric field strength $\mathcal{E}_{0}=0\leavevmode\nobreak\ \text{V/cm}$
and of the number of photons exchanged $(s=0)$, where they all tend to zero.
Regarding the laser field strength and frequency, these are parameters that
characterize the external EM field, while the number of photons exchanged $s$
appeared due to the introduction of ordinary and generalized Bessel functions
in our theoretical calculation. We note that the four graphs shown in Figs.
3(a) and 3(b) are so identical as they are indistinguishable for all final
scattering angles $\theta_{f}$ and kinetic energies $E^{kin}_{e}$ of the
incident electron. This proves the consistency and validity of our theoretical
calculations. In the next step, we will try to see the effect of the geometry
in Fig. (4), which displays the variations of the DCS in the absence of the
laser field as a function of the final scattering angle $\theta_{f}$. We
observe that the graphs of the DCS are distinct, which clearly shows that the
geometry influences the angular distribution of the DCS. We also observe that
as the incidence angle $\theta_{i}$ increases, the pic of the DCS increases
with a shift towards large final scattering angles.

Figure 4: The variations of the IDCS as a function of scattering angle
$\theta_{f}$ for differents angles of the incident electron. The laser field
strength and the number of photons are, respectively $\mathcal{E}_{0}=0$ V/cm
and $s=0$. The kinetic energy of the incident muon neutrino and the incoming
electron are, respectively, $E_{\nu}^{\text{kin}}=0.5\times 10^{-3}$ GeV and
$E_{e}^{\text{kin}}=10^{-3}$ GeV.

Another important remark is that the electron has a high probability to be
scattered under a final angle $\theta_{f}$ with a value approximately
approaching the incidence angle $(\theta_{f}\approx\theta_{i})$. For example,
if the incidence angle is $\theta_{i}=15^{\circ}$, the peak is located around
the final scattering angle $\theta_{f}\approx 15^{\circ}$. In the same
context, and to highlight the correctness and accuracy of our calculations in
the presence of an EM field with elliptical polarization, this calculation
which is general allows us to find the results previously obtained in two
research papers that deal with the same scattering process in the presence of
a laser field with circular [23] or linear [24] polarization. We begin with
Fig. 5(a) which presents the variations of the IDCS as a function of the
number of photons exchanged $s$ with a degree of ellipticity $\eta=90$
(circular polarization) for two different field strengths $\mathcal{E}_{0}$
and for a frequency $\hbar\omega=1.17\leavevmode\nobreak\ \text{eV}$. This is
the same envelope obtained in a previous paper by El Asri et al. (see Fig.
2(a) in [23]). Thus, the theoretical formalism adopted here is general and can
lead to all the results obtained in [23]. For linear polarization, we display,
in Fig. 5(b), the IDCS as a function of the number of photons exchanged $s$
with a degree of ellipticity $\eta=0$ for the laser field strength
$\mathcal{E}_{0}=5.18\times 10^{7}\leavevmode\nobreak\ \text{V/cm}$ and
frequency $\hbar\omega=1.17\leavevmode\nobreak\ \text{eV}$.

Figure 5: The IDCS as a function of the number of photons exchanged. The free
parameters are (a) $E_{\nu}^{\text{kin}}=0.5\times 10^{-3}$ GeV,
$E_{e}^{\text{kin}}=10^{-3}$ GeV, $\hbar\omega=1.17\leavevmode\nobreak\
\text{eV}$, $\theta_{i}=1^{\circ}$,
$\theta_{f}=\varphi_{i}=\varphi_{f}=0^{\circ}$ (to be compared with [23]); and
(b) $E_{\nu}^{\text{kin}}=0.5\times 10^{-3}$ GeV, $E_{e}^{\text{kin}}=10^{-3}$
GeV, $\mathcal{E}_{0}=5.18\times 10^{7}\leavevmode\nobreak\ \text{V/cm}$,
$\hbar\omega=1.17\leavevmode\nobreak\ \text{eV}$, $\theta_{i}=45^{\circ}$,
$\varphi_{i}=\varphi_{f}=0^{\circ}$, $\theta_{f}=1^{\circ}$ (to be compared
with [24]).

We observe, that a large number of photons are exchanged between the laser
field and the scattering process, and the cutoff number is about $s=\pm 1400$.
Comparing, this figure with the one obtained by Bai et al. (see Fig. 2(a) in
[24]), we get the same result. Another comparison concerns Fig. 6, which
displays the dependence of the SDCS given in Eq. (25) on the final scattering
angle $\theta_{f}$ for different polarizations of the EM field. In Fig. 6(a),
we show the variation of the SDCS as a function of the final scattering angle
$\theta_{f}$ for the degree of ellipticity $\eta=90$.

Figure 6: Dependence of SDCS on the scattering angle $\theta_{f}$ for various
polarizations of the EM field, taking the laser parameters such as:
$\mathcal{E}_{0}=10^{8}\leavevmode\nobreak\ \text{V/cm}$,
$\hbar\omega=1.17\text{eV}$. The parameters of the geometry are taken as
$\theta_{i}=\varphi_{i}=\varphi_{f}=1^{\circ}$ and the remaining parameters
are selected as shown in Fig. 5.

We obtain a symmetrical graph, which presents a peak in the vicinity of
$\theta_{f}=0$. Furthermore, the SDCS remains lower than the DCS without a
laser. In Fig. 6(b), we represent the same graphs, but with a degree of
ellipticity $\eta=0$. In this case, we see an enhancement of the SDCS compared
to the DCS without laser, which is consistent with previous research in the
case of linear polarization [16, 17, 12]. After a detailed discussion and a
comparison of the results obtained with the previous work, let us see what
happens if we introduce an external EM field with elliptical polarization
having a degree of ellipticity $\eta=120^{\circ}$.

Figure 7: The behavior of the IDCS, as a function of photon number $s$. The
various parameters are (a) $\theta_{i}=1^{\circ}$, $\theta_{f}=2^{\circ}$ and
$\hbar\omega=1.17$ eV, (b) $\mathcal{E}_{0}=5.18\times 10^{8}$ V/cm,
$\theta_{i}=1^{\circ}$ and $\theta_{f}=2^{\circ}$, (c)
$\mathcal{E}_{0}=5.18\times 10^{8}$ V/cm, $\hbar\omega=1.17$ eV,
$\theta_{i}=1^{\circ}$ and $\theta_{f}=2^{\circ}$ and (d)
$\mathcal{E}_{0}=5.18\times 10^{8}$ V/cm and $\hbar\omega=1.17$ eV.

In Fig. 7, we display the variations of the IDCS (the multi-photon energy
transfer phenomenon) as a function of the number of photons exchanged $s$, at
different field strengths and frequencies, and at different kinetic energies
and geometry of the incident electron. From Fig. 7(a), we can observe that the
electron exchanges a large number of photons with the high-intensity EM field
($\mathcal{E}_{0}=10^{9}$ V/cm), where the cutoff number is approximately
$s=\pm 125$, compared to the low-intensity EM field
($\mathcal{E}_{0}=5.18\times 10^{8}$ V/cm), where the cutoff number is
approximately equal to $s=\pm 65$. This implies that the influence of the
laser field on the scattering process is more significant at higher field
strengths, i.e. the electron interacts powerfully with the strong EM field.
Additionally, the order of magnitude of the IDCS decreases with the increase
of the field strength. In Fig. 7(b), we visualize the variations of the IDCS
as a function of the number of photons exchanged $s$ at different frequencies
of the laser field. We see that at low frequencies ($\hbar\omega=1.17$ eV),
the number of exchanged photons is large and the cutoff number is about $s=\pm
65$, compared to high frequencies ($\hbar\omega=2$ eV) where the cutoff number
is about $s=\pm 25$. Moreover, the order of magnitude of the IDCS increases
with the increase of the frequency. Consequently, the multi-photon energy
transfer phenomenon between the laser and the scattering process is related to
the properties of the applied laser field. Figure 7(c) describes the
variations of the IDCS as a function of the number of photons exchanged $s$ at
different kinetic energies of the incident electron. The multi-photon energy
transfer phenomenon increases as the kinetic energy of the incident electron
increases, leading to an exchange of a large number of photons, while the
order of magnitude of the IDCS decreases. In Fig. 7(d), we illustrate the
influence of the chosen geometry on the multi-photon process. We observe that
the number of photons exchanged in the geometry
($\theta_{i}=1^{\circ},\leavevmode\nobreak\ \theta_{f}=2^{\circ}$ ) is more
important than the one exchanged in the geometry
($\theta_{i}=15^{\circ},\leavevmode\nobreak\ \theta_{f}=20^{\circ}$). While
the first geometry means that the incident electron is almost in the same
direction as the k field vector, the second one indicates that the electron
arrives at an angle of $\theta_{i}=15^{\circ}$ with respect to the $z$-axis.
Physically, this states that the electron interacts with the laser if they
arrive together in the same direction more than if they are in two different
directions. We also see that the oscillations of these envelopes fall sharply
to the sides, which can also be best explained by the well-known behavior and
properties of ordinary and generalized Bessel functions (GBF)[28]. Let us now
see what happens if we sum over all the possible number of photons exchanged.
In Fig. 8, we have plotted the variations of the SDCS as a function of the
final scattering angle $\theta_{f}$, for different known frequencies and
electric field strengths.

Figure 8: Variation of the laser-assisted SDCS (with the degree of ellipticity
$\eta=120^{\circ}$) as a function of the scattering angle $\theta_{f}$ for
various electric field strengths $\mathcal{E}_{0}$ and laser frequencies
$\hbar\omega$. The kinetic energy of the incident muon neutrino and the
incoming electron are, respectively, $E_{\nu}^{\text{kin}}=0.5\times 10^{-3}$
GeV et $E_{e}^{\text{kin}}=10^{-3}$ GeV. The other parameters are taken as
follows: (a) $\mathcal{E}_{0}=10^{8}\leavevmode\nobreak\ \text{V/cm}$,
$\theta_{i}=\varphi_{i}=\varphi_{f}=1^{\circ}$, (b)
$\hbar\omega=1.17\leavevmode\nobreak\ \text{eV}$,
$\theta_{i}=\varphi_{i}=\varphi_{f}=1^{\circ}$.

Figure 8(a) shows the dependence of the SDCS assisted by elliptically
polarized laser field with a degree of ellipticity $\eta=120^{\circ}$ as a
function of the final scattering angle $\theta_{f}$, for different known laser
frequencies $\hbar\omega$ which are the CO2 laser ($\hbar\omega=0.117$ eV),
the Nd:YAG laser ($\hbar\omega=1.17$ eV) and the He-Ne laser ($\hbar\omega=2$
eV). For $0^{\circ}\leqslant\theta_{f}\leqslant 180^{\circ}$, the order of
magnitude of the SDCS increases with increasing laser field frequency, which
confirms the result of Fig. 7(b). Since the laser field strength is also a
crucial parameter, we performed the same analysis for different values of the
laser field strength in Fig. 8(b). We observe that the order of magnitude of
SDCS decreases with increasing laser field strength, which is consistent with
the result obtained in Fig. 7(a).

Let us now turn to the discussion of the laser field polarization effect on
the SDCS. Figure 9 shows the variations of the SDCS as a function of the final
scattering angle $\theta_{f}$ and the kinetic energy of the incident electron
$E_{e}^{\text{kin}}$ at different polarizations of the strong EM field. The
parameters of the laser field are chosen as follows:
$\hbar\omega=1.17\leavevmode\nobreak\ \text{eV}$ and
$\mathcal{E}_{0}=10^{10}\leavevmode\nobreak\ \text{V/cm}$. Our results show
that the effect of the laser field polarization is clearly highlighted, since
the three SDCSs with laser and without laser are now obviously
distinguishable.

Figure 9: Variations of the SDCS as a function of the scattering angle
$\theta_{f}$ and the kinetic energy of the incident electron for different
polarizations of the strong EM field. The laser field parameters are selected
as follows: $\hbar\omega=1.17\leavevmode\nobreak\ \text{eV}$ and
$\mathcal{E}_{0}=10^{10}\leavevmode\nobreak\ \text{V/cm}$. The other
parameters are taken as: (a) $E_{\nu}^{\text{kin}}=0.5\times
10^{-3}\leavevmode\nobreak\ \text{GeV}$,
$E_{e}^{\text{kin}}=10^{-3}\leavevmode\nobreak\ \text{GeV}$,
$\theta_{i}=\varphi_{i}=\varphi_{f}=1^{\circ}$; (b)
$E_{\nu}^{\text{kin}}=10\leavevmode\nobreak\ \text{GeV}$,
$\theta_{i}=\varphi_{i}=\varphi_{f}=1^{\circ}$, $\theta_{f}=2^{\circ}$.

We also see that the SDCS in the case of linear polarization
($\eta=0^{\circ}$) is enhanced compared to that without laser. Furthermore, we
observe that there is a reduction of the SDCS in the case of an elliptical
($\eta=120^{\circ}$) and circular ($\eta=90^{\circ}$) polarizations. We note
here that the SDCS for elliptical polarization is lower or higher than the
SDCS for circular polarization depending on the value of the degree of
ellipticity $\eta$. The same thing can be said about the result presented in
Fig. 9(b) as a function of the kinetic energy of the incident electron
$E_{e}^{\text{kin}}$.
Finally, to understand most clearly the effect of polarization, we display the
dependence of the SDCS on the laser field strength at different polarizations
in Fig. 10. The laser field strength appears in the equations for determining
the behavior of the SDCS through the arguments $\xi_{1}$, $\xi_{2}$ and
$\xi_{3}$ (see Eq. (13)) of the ordinary and generalized Bessel functions and
the coefficients $\chi_{0\mu}$, $\chi_{1\mu}$, $\chi_{2\mu}$ and $\chi_{3\mu}$
given by Eq. (15).

Figure 10: Variations of SDCS as a function of the laser field strength
$\mathcal{E}_{0}$ for different polarizations. The kinetic energy of the
incident muon neutrino and the incoming electron are respectively
$E_{\nu}^{\text{kin}}=0.5\times 10^{-3}$ and $E_{e}^{\text{kin}}=10$ GeV. The
frequency of the laser field is $\hbar\omega=0.117$ eV. The other parameters
are chosen as follows: $\theta_{i}=\varphi_{i}=\varphi_{f}=1^{\circ}$ and
$\theta_{f}=2^{\circ}$.

In Fig. 10(a), we can see that the SDCS for linear polarization
($\eta=0^{\circ}$) remains constant and equal to the DCS without laser in the
field strength range between $5.18\times 10^{9}$ and $6.21\times 10^{10}$
V/cm. Outside this interval, we find that the SDCS increases progressively
with the laser field strengh, which is consistent with the results obtained
previously in [24, 16, 13, 11]. Although the ordinary and generalized Bessel
functions in the SDCS vary with the laser field strength, the electron state
is more perturbed, and thus the cross section is strongly changed.
Consequently, the SDCS increases because the energy of the electron increases
after the absorption of laser photons. On the other hand, in Figs. 10(b) and
10(c), which illustrate the variations of the SDCS in the case of circular
($\eta=90^{\circ}$) and elliptical ($\eta=60^{\circ}$) polarizations, it is
shown that the SDCS decreases and remains lower than the DCS without laser,
which is consistent with the results found in [10, 12]. Therefore, the effect
of electron dressing induces very significant changes in the DCS, and a large
enhancement is found in the linear polarization case.

##  IV Conclusion

Laser-assisted electron-muon neutrino scattering is investigated, for
elliptical polarization, in the first Born approximation and in the framework
of electroweak theory. We have extended the study of this scattering process
for a general polarization that leads to all particular results obtained
before in linear and circular polarizations. The numerical results show that
the SDCS is significantly modified by the polarization type depending on the
degree of ellipticity and the laser field parameters. Moreover, the SDCS in
the case of linear polarization is enhanced compared to that without laser,
while it is reduced in the case of elliptical and circular polarizations. We
have shown that the scattering geometry, as well as the laser field parameters
and the kinetic energy of the incident electron influence the multi-photon
process. We hope that the present work will serve as a stimulus for
experimenters to perform such scattering experiments on electrons dressed in
linear and elliptical polarization. *

##  Appendix A Explicit expression of the first four coefficients in Eq. (32)

The expressions for the coefficients $\Lambda_{1}$ to $\Lambda_{16}$ have been
computed using the FeynCalc package. In order to limit the length of our
article, we give here the first four coefficients $\Lambda_{1}$ to
$\Lambda_{4}$:

$\begin{split}\Lambda_{1}=&64a^{2}g_{A}^{2}(k_{f}.k_{i})\tan^{2}(\eta/2)e^{2}+(64a^{2}g_{A}g_{V}(k.k_{i})(k_{f}.p_{f})\tan^{2}(\eta/2)e^{2})/(k.p_{f})\\\
&-(32a^{2}g_{V}^{2}(k.k_{i})(k_{f}.p_{f})\tan^{2}(\eta/2)e^{2})/(k.p_{f})-(32a^{2}g_{A}^{2}(k.k_{i})(k_{f}.p_{f})\tan^{2}(\eta/2)e^{2})/(k.p_{f})\\\
&-(32a^{2}g_{A}^{2}(k.k_{i})(k_{f}.p_{i})\tan^{2}(\eta/2)e^{2})/(k.p_{i})-(64a^{2}g_{A}g_{V}(k.k_{i})(k_{f}.p_{i})\tan^{2}(\eta/2)e^{2})/(k.p_{i})\\\
&-(32a^{2}g_{V}^{2}(k.k_{i})(k_{f}.p_{i})\tan^{2}(\eta/2)e^{2})/(k.p_{i})-(32a^{2}g_{A}^{2}(k.k_{f})(k_{i}.p_{f})\tan^{2}(\eta/2)e^{2})/(k.p_{f})\\\
&-(32a^{2}g_{V}^{2}(k.k_{f})(k_{i}.p_{f})\tan^{2}(\eta/2)e^{2})/(k.p_{f})+(64a^{2}g_{A}g_{V}(k.k_{f})(k_{i}.p_{i})\tan^{2}(\eta/2)e^{2})/(k.p_{i})\\\
&-(32a^{2}g_{V}^{2}(k.k_{f})(k_{i}.p_{i})\tan^{2}(\eta/2)e^{2})/(k.p_{i})-(32a^{2}g_{A}^{2}(k.k_{f})(k_{i}.p_{i})\tan^{2}(\eta/2)e^{2})/(k.p_{i})\\\
&-(64a^{2}g_{A}g_{V}(k.k_{f})(k_{i}.p_{f})\tan^{2}(\eta/2)e^{2})/(k.p_{f})+(64a^{2}g_{A}^{2}m^{2}(k.k_{f})(k.k_{i})\tan^{2}(\eta/2)e^{2})/((k.p_{f})(k.p_{i}))\\\
&-(64a^{2}g_{V}^{2}m^{2}(k.k_{f})(k.k_{i})\tan^{2}(\eta/2)e^{2})/((k.p_{f})(k.p_{i}))+(64a^{2}g_{A}^{2}(k.k_{f})(k.k_{i})(p_{f}.p_{i})\tan^{2}(\eta/2)\\\
&\times
e^{2})/((k.p_{f})(k.p_{i}))+64gA^{2}m^{2}(k_{f}.k_{i})-64g_{V}^{2}m^{2}(k_{f}.k_{i})+64g_{A}^{2}(k_{f}.p_{i})(k_{i}.p_{f})+64g_{V}^{2}(k_{f}.p_{i})(k_{i}.p_{f})\\\
&-128g_{A}g_{V}(k_{f}.p_{i})(k_{i}.p_{f})+64g_{A}^{2}(k_{f}.p_{f})(k_{i}.p_{i})+64g_{V}^{2}(k_{f}.p_{f})(k_{i}.p_{i})+128g_{A}g_{V}(k_{f}.p_{f})(k_{i}.p_{i})\\\
&+64a^{2}g_{V}^{2}(k_{f}.k_{i})\tan^{2}(\eta/2)e^{2}+(64a^{2}g_{V}^{2}(k.k_{f})(k.k_{i})(p_{f}.p_{i})\tan^{2}(\eta/2)e^{2})/((k.p_{f})(k.p_{i}))\\\
&+(32a^{4}g_{A}^{2}(k.k_{f})(k.k_{i})\tan^{4}(\eta/2)e^{4})/((k.p_{f})(k.p_{i}))+(32a^{4}g_{V}^{2}(k.k_{f})(k.k_{i})\tan^{4}(\eta/2)e^{4})/((k.p_{f})(k.p_{i})),\end{split}$
(33)
$\begin{split}\Lambda_{2}=&(32g_{V}^{2}a^{2}(k.k_{f})(k_{i}.p_{f})e^{2})/(k.p_{f})+(32g_{A}^{2}a^{2}(k.k_{i})(k_{f}.p_{f})e^{2})/(k.p_{f})+(32g_{A}^{2}a^{2}(k.k_{f})(k_{i}.p_{f})e^{2})/(k.p_{f})\\\
&+(32g_{V}^{2}a^{2}(k.k_{i})(k_{f}.p_{f})e^{2})/(k.p_{f})-(64g_{A}g_{V}a^{2}(k.k_{i})(k_{f}.p_{f})e^{2})/(k.p_{f})-(32g_{A}^{2}a^{2}(k.k_{i})(k_{f}.p_{f})e^{2})/(k.p_{i})\\\
&-(32g_{V}^{2}a^{2}(k.k_{i})(k_{f}.p_{f})e^{2})/(k.p_{i})-(64g_{A}g_{V}a^{2}(k.k_{i})(k_{f}.p_{f})e^{2})/(k.p_{i})-(32g_{A}^{2}a^{2}(k.k_{i})(k_{f}.p_{i})e^{2})/(k.p_{f})\\\
&-(32g_{V}^{2}a^{2}(k.k_{i})(k_{f}.p_{i})e^{2})/(k.p_{f})+(64g_{A}g_{V}a^{2}(k.k_{i})(k_{f}.p_{i})e^{2})/(k.p_{f})-64gA^{2}a^{2}(k_{f}.k_{i})e^{2}\\\
&-64g_{V}^{2}a^{2}(k_{f}.k_{i})e^{2}+128g_{A}^{2}(a_{1}.k_{f})(a_{1}.k_{i})e^{2}+128g_{V}^{2}(a_{1}.k_{f})(a_{1}.k_{i})e^{2}+(32g_{A}^{2}a^{2}(k.k_{f})(k_{i}.p_{i})e^{2})/(k.p_{i})\\\
&+(32g_{V}^{2}a^{2}(k.k_{f})(k_{i}.p_{i})e^{2})/(k.p_{i})-(64g_{A}g_{V}a^{2}(k.k_{f})(k_{i}.p_{i})e^{2})/(k.p_{i})+(32g_{A}^{2}a^{2}(k.k_{i})(k_{f}.p_{i})e^{2})/(k.p_{i})\\\
&+(32g_{V}^{2}a^{2}(k.k_{i})(k_{f}.p_{i})e^{2})/(k.p_{i})+(64g_{A}g_{V}a^{2}(k.k_{i})(k_{f}.p_{i})e^{2})/(k.p_{i})+(64g_{A}g_{V}a^{2}(k.k_{f})(k_{i}.p_{f})e^{2})/(k.p_{f})\\\
&-(32g_{A}^{2}a^{2}(k.k_{f})(k_{i}.p_{f})e^{2})/(k.p_{i})-(32g_{V}^{2}a^{2}(k.k_{f})(k_{i}.p_{f})e^{2})/(k.p_{i})+(64g_{A}g_{V}a^{2}(k.k_{f})(k_{i}.p_{f})e^{2})/(k.p_{i})\\\
&-(32g_{A}^{2}a^{2}(k.k_{f})(k_{i}.p_{i})e^{2})/(k.p_{f})-(32g_{V}^{2}a^{2}(k.k_{f})(k_{i}.p_{i})e^{2})/(k.p_{f})-(64g_{A}^{2}a^{2}(k.k_{f})(k.k_{i})\\\
&\times(p_{f}.p_{i})e^{2})/((k.p_{f})(k.p_{i}))-(64g_{V}^{2}a^{2}(k.k_{f})(k.k_{i})(p_{f}.p_{i})e^{2})/((k.p_{f})(k.p_{i}))-(64g_{A}^{2}(a_{1}.k_{i})(a_{1}.p_{f})\\\
&\times(k.k_{f})e^{2})/(k.p_{f})-(64g_{V}^{2}(a_{1}.k_{i})(a_{1}.p_{f})(k.k_{f})e^{2})/(k.p_{f})-(128g_{A}g_{V}(a_{1}.k_{i})(a_{1}.p_{f})(k.k_{f})e^{2})/(k.p_{f})\\\
&-(64g_{A}^{2}(a_{1}.k_{f})(a_{1}.p_{f})(k.k_{i})e^{2})/(k.p_{f})-(64g_{V}^{2}(a_{1}.k_{f})(a_{1}.p_{f})(k.k_{i})e^{2})/(k.p_{f})+(128g_{A}g_{V}(a_{1}.k_{f})\\\
&\times(a_{1}.p_{f})(k.k_{i})e^{2})/(k.p_{f})-(64g_{A}^{2}(a_{1}.k_{i})(a_{1}.p_{i})(k.k_{f})e^{2})/(k.p_{i})-(64g_{V}^{2}(a_{1}.k_{i})(a_{1}.p_{i})(k.k_{f})e^{2})/(k.p_{i})\\\
&+(128g_{A}g_{V}(a_{1}.k_{i})(a_{1}.p_{i})(k.k_{f})e^{2})/(k.p_{i})-(64g_{A}^{2}(a_{1}.k_{f})(a_{1}.p_{i})(k.k_{i})e^{2})/(k.p_{i})-(64g_{V}^{2}(a_{1}.k_{f})(a_{1}.p_{i})\\\
&\times(k.k_{i})e^{2})/(k.p_{i})-(64g_{A}^{2}m^{2}a^{2}(k.k_{f})(k.k_{i})e^{2})/((k.p_{f})(k.p_{i}))+(64g_{V}^{2}m^{2}a^{2}(k.k_{f})(k.k_{i})e^{2})/((k.p_{f})(k.p_{i}))\\\
&+(128g_{A}^{2}(a_{1}.p_{f})(a_{1}.p_{i})(k.k_{f})(k.k_{i})e^{2})/((k.p_{f})(k.p_{i}))+(128g_{V}^{2}(a_{1}.p_{f})(a_{1}.p_{i})(k.k_{f})(k.k_{i})\\\
&\times
e^{2})/((k.p_{f})(k.p_{i}))-(128g_{A}g_{V}(a_{1}.k_{f})(a_{1}.p_{i})(k.k_{i})e^{2})/(k.p_{i})-(64g_{A}g_{V}a^{2}(k.k_{f})(k_{i}.p_{i})e^{2})/(k.p_{f}),\end{split}$
(34)
$\begin{split}\Lambda_{3}=&(32g_{A}^{2}a^{2}(k.k_{f})(k_{i}.p_{f})\tan^{2}(\eta/2)e^{2})/(k.p_{f})+(32g_{V}^{2}a^{2}(k.k_{f})(k_{i}.p_{f})\tan^{2}(\eta/2)e^{2})/(k.p_{f})\\\
&+(32g_{A}^{2}a^{2}(k.k_{i})(k_{f}.p_{i})\tan^{2}(\eta/2)e^{2})/(k.p_{i})+(32g_{V}^{2}a^{2}(k.k_{i})(k_{f}.p_{i})\tan^{2}(\eta/2)e^{2})/(k.p_{i})\\\
&+(64g_{A}g_{V}a^{2}(k.k_{i})(k_{f}.p_{i})\tan^{2}(\eta/2)e^{2})/(k.p_{i})-64g_{A}^{2}a^{2}(k_{f}.k_{i})\tan^{2}(\eta/2)e^{2}-64g_{V}^{2}a^{2}\\\
&\times(k_{f}.k_{i})\tan^{2}(\eta/2)e^{2}+(64g_{A}g_{V}a^{2}(k.k_{f})(k_{i}.p_{f})\tan^{2}(\eta/2)e^{2})/(k.p_{f})-(32g_{A}^{2}a^{2}(k.k_{f})(k_{i}.p_{f})\\\
&\times\tan^{2}(\eta/2)e^{2})/(k.p_{i})-(32g_{V}^{2}a^{2}(k.k_{f})(k_{i}.p_{f})\tan^{2}(\eta/2)e^{2})/(k.p_{i})+(32g_{A}^{2}a^{2}(k.k_{f})(k_{i}.p_{i})\\\
&\times\tan^{2}(\eta/2)e^{2})/(k.p_{i})+(32g_{A}^{2}a^{2}(k.k_{i})(k_{f}.p_{f})\tan^{2}(\eta/2)e^{2})/(k.p_{f})+(32g_{V}^{2}a^{2}(k.k_{i})(k_{f}.p_{f})\\\
&\times\tan^{2}(\eta/2)e^{2})/(k.p_{f})-(64g_{A}g_{V}a^{2}(k.k_{i})(k_{f}.p_{f})\tan^{2}(\eta/2)e^{2})/(k.p_{f})+128g_{A}^{2}(a_{2}.k_{f})(a_{2}.k_{i})\\\
&\times\tan^{2}(\eta/2)e^{2}+128g_{V}^{2}(a_{2}.k_{f})(a_{2}.k_{i})\tan^{2}(\eta/2)e^{2}-(32g_{A}^{2}a^{2}(k.k_{i})(k_{f}.p_{f})\tan^{2}(\eta/2)e^{2})/(k.p_{i})\\\
&-(32g_{V}^{2}a^{2}(k.k_{i})(k_{f}.p_{f})\tan^{2}(\eta/2)e^{2})/(k.p_{i})-(64g_{A}g_{V}a^{2}(k.k_{i})(k_{f}.p_{f})\tan^{2}(\eta/2)e^{2})/(k.p_{i})\\\
&-(32g_{A}^{2}a^{2}(k.k_{i})(k_{f}.p_{i})\tan^{2}(\eta/2)e^{2})/(k.p_{f})-(32g_{V}^{2}a^{2}(k.k_{i})(k_{f}.p_{i})\tan^{2}(\eta/2)e^{2})/(k.p_{f})\\\
&+(64g_{A}g_{V}a^{2}(k.k_{i})(k_{f}.p_{i})\tan^{2}(\eta/2)e^{2})/(k.p_{f})+(32g_{V}^{2}a^{2}(k.k_{f})(k_{i}.p_{i})\tan^{2}(\eta/2)e^{2})/(k.p_{i})\\\
&-(64g_{A}g_{V}a^{2}(k.k_{f})(k_{i}.p_{i})\tan^{2}(\eta/2)e^{2})/(k.p_{i})+(64g_{A}g_{V}a^{2}(k.k_{f})(k_{i}.p_{f})\tan^{2}(\eta/2)e^{2})/(k.p_{i})\\\
&-(32g_{A}^{2}a^{2}(k.k_{f})(k_{i}.p_{i})\tan^{2}(\eta/2)e^{2})/(k.p_{f})-(32g_{V}^{2}a^{2}(k.k_{f})(k_{i}.p_{i})\tan^{2}(\eta/2)e^{2})/(k.p_{f})\\\
&-(64g_{A}^{2}a^{2}(k.k_{f})(k.k_{i})(p_{f}.p_{i})\tan^{2}(\eta/2)e^{2})/((k.p_{f})(k.p_{i}))-(64g_{V}^{2}a^{2}(k.k_{f})(k.k_{i})(p_{f}.p_{i})\\\
&\times\tan^{2}(\eta/2)e^{2})/((k.p_{f})(k.p_{i}))-(64g_{A}^{2}(a_{2}.k_{i})(a_{2}.p_{f})(k.k_{f})\tan^{2}(\eta/2)e^{2})/(k.p_{f})-(64gV^{2}(a_{2}.k_{i})\\\
&\times(a_{2}.p_{f})(k.k_{f})\tan^{2}(\eta/2)e^{2})/(k.p_{f})-(128g_{A}g_{V}(a_{2}.k_{i})(a_{2}.p_{f})(k.k_{f})\tan^{2}(\eta/2)e^{2})/(k.p_{f})\\\
&-(64g_{A}^{2}(a_{2}.k_{f})(a_{2}.p_{f})(k.k_{i})\tan^{2}(\eta/2)e^{2})/(k.p_{f})-(64g_{V}^{2}(a_{2}.k_{f})(a_{2}.p_{f})(k.k_{i})\tan^{2}(\eta/2)e^{2})/(k.p_{f})\\\
&+(128g_{A}g_{V}(a_{2}.k_{f})(a_{2}.p_{f})(k.k_{i})\tan^{2}(\eta/2)e^{2})/(k.p_{f})-(64g_{A}^{2}(a_{2}.k_{i})(a_{2}.p_{i})(k.k_{f})\tan^{2}(\eta/2)e^{2})/(k.p_{i})\\\
&-(64g_{V}^{2}(a_{2}.k_{i})(a_{2}.p_{i})(k.k_{f})\tan^{2}(\eta/2)e^{2})/(k.p_{i})+(128g_{A}g_{V}(a_{2}.k_{i})(a_{2}.p_{i})(k.k_{f})\tan^{2}(\eta/2)e^{2})/(k.p_{i})\\\
&-(64g_{A}^{2}(a_{2}.k_{f})(a_{2}.p_{i})(k.k_{i})\tan^{2}(\eta/2)e^{2})/(k.p_{i})-(64g_{A}^{2}m^{2}a^{2}(k.k_{f})(k.k_{i})\tan^{2}(\eta/2)e^{2})/((k.p_{f})(k.p_{i}))\\\
&+(64gV^{2}m^{2}a^{2}(k.k_{f})(k.k_{i})\tan^{2}(\eta/2)e^{2})/((k.p_{f})(k.p_{i}))+(128g_{A}^{2}(a_{2}.p_{f})(a_{2}.p_{i})(k.k_{f})(k.k_{i})\\\
&\times\tan^{2}(\eta/2)e^{2})/((k.p_{f})(k.p_{i}))+(128g_{V}^{2}(a_{2}.p_{f})(a_{2}.p_{i})(k.k_{f})(k.k_{i})\tan^{2}(\eta/2)e^{2})/((k.p_{f})(k.p_{i}))\\\
&-(64gV^{2}(a_{2}.k_{f})(a_{2}.p_{i})(k.k_{i})\tan^{2}(\eta/2)e^{2})/(k.p_{i})-(128g_{A}g_{V}(a_{2}.k_{f})(a_{2}.p_{i})(k.k_{i})\tan^{2}(\eta/2)e^{2})/(k.p_{i})\\\
&-(64g_{A}g_{V}a^{2}(k.k_{f})(k_{i}.p_{i})\tan^{2}(\eta/2)e^{2})/(k.p_{f}),\end{split}$
(35)
$\begin{split}\Lambda_{4}=&-((32a^{4}g_{A}^{2}(k.k_{f})(k.k_{i})\tan^{4}(\eta/2)e^{4})/((k.p_{f})(k.p_{i})))-(32a^{4}g_{V}^{2}(k.k_{f})(k.k_{i})\tan^{4}(\eta/2)e^{4})/((k.p_{f})(k.p_{i}))\\\
&+(64a^{4}g_{A}^{2}(k.k_{f})(k.k_{i})\tan^{2}(\eta/2)e^{4})/((k.p_{f})(k.p_{i}))-(32a^{4}g_{A}^{2}(k.k_{f})(k.k_{i})e^{4})/((k.p_{f})(k.p_{i}))\\\
&-(32a^{4}g_{V}^{2}(k.k_{f})(k.k_{i})e^{4})/((k.p_{f})(k.p_{i}))+(64a^{4}g_{V}^{2}(k.k_{f})(k.k_{i})\tan^{2}(\eta/2)e^{4})/((k.p_{f})(k.p_{i})).\end{split}$
(36)

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