#  __
Effect of electron spin polarization in laser-assisted electron-proton
scattering

I. Dahiri1, M. Baouahi1, M. Jakha1, S. Mouslih1,2 B. Manaut1, S. Taj1,
1 Polydisciplinary Faculty, Laboratory of Research in Physics and Engineering
Sciences,
Team of Modern and Applied Physics, Sultan Moulay Slimane University, Beni
Mellal, 23000, Morocco.
2Faculty of Sciences and Techniques, Laboratory of Materials Physics (LMP),
Beni Mellal, 23000, Morocco. Corresponding author, E-mail<EMAIL_ADDRESS>
###### Abstract

This work aims to study theoretically the electron-proton scattering for
initially spin-polarized electrons in the presence of a circularly polarized
electromagnetic field. Using the first Born approximation and the Dirac-Volkov
states for dressed electrons, we derive an analytic expression for the
differential cross-section with the help of the spin-helicity formalism.
Meanwhile, a well-known concept of spin-flip and spin non-flip differential
cross sections is applied. The influence of the electromagnetic field on the
degree of polarization of the scattered electron is investigated, and the
effect of electron spin polarization is examined. We found that the initial
spin of the electron can be affected depending on its kinetic energy during
the scattering process. This work is a continuation of a recent paper in which
we studied the unpolarized electron-proton scattering in the presence of a
laser field (Dahiri, et al., Laser Phys. Lett. 18 (2021) 096001).

Keywords: spin effects, QED calculation, laser-assisted processes

##  1\. Introduction

In view of the new developments in laser technology, high intensity and short
duration laser pulses are generated nowadays in the laboratory [1, 2, 3]. This
has greatly inspired researchers to study various ultrafast processes in the
presence of strong laser fields, such as atomic processes [4, 5, 6, 7, 8, 9,
10] and those that occur in the frameworks of quantum electrodynamics (QED)
[11, 12, 13, 14, 15] and electroweak theory [17, 16, 18, 19, 20, 21], as well
as recently the processes of Higgs boson production beyond the Standard Model
[22, 23, 24]. New experimental perspectives for the study of these processes
in the presence of laser field are proposed by future high field laboratories,
such as the Extreme Light Infrastructure (ELI) [25] and the Exawatt Center for
Extreme Light Studies (XCELS) [26]. The majority of theoretical studies
investigating scattering processes in QED treat the involved particles as
unpolarized and therefore their spins are averaged over. At high energies, the
scattering of spin-polarized particles is an important aspect of physics [27],
owing much to the development of new detectors and sources of such particles.
Polarized beams and targets are used to study various effects of spin physics.
As a definition, an ensemble of particles (e.g., electrons) is said to be
polarized if the electron spin has a preferential orientation so that there
exists a direction for which the two possible spin states are not equally
populated. In early experiments with free electrons, the direction of their
spins was rarely considered. Whenever the spin direction plays a role, one has
to average over all spin polarizations in order to describe the experiments
properly. It has long been possible to produce electron beams in which the
spin has a preferential orientation [28, 29, 30, 31]. There are many reasons
for the interest in polarized electrons. The most important one is that in
physical experiments one seeks to define as accurately as possible the initial
and/or final states of the considered systems [32]. Several theoretical
studies have considered the effects of spin polarization in various processes.
Deep inelastic scattering of polarized leptons and protons revealed
interesting details on the spin-structure of the proton’s constituents [33,
34], and the parity non-conservation effects have been investigated by using
polarized beams [35, 36]. The polarization effects on Compton scattering have
been studied by several works [37, 38, 39]. Mott scattering of polarized
electrons has been studied in a strong laser field of circular [8] and linear
[9] polarization. The unpolarized scattering of electrons by protons is one of
the basic processes in QED that has been widely studied, as it is considered
the simpler way to access more detailed information on the internal structure
of proton. The effect of a circularly polarized laser field on this process
has been the focus of a recent paper for unpolarized electrons [11]. As an
extension of this paper, the present work aims to study the effect of electron
spin polarization on the electron-proton scattering process in the presence of
a laser field using the concept of a spin-polarized relativistic Dirac
particle. Polarization effects in elastic proton-electron scattering have been
investigated in [40] using polarized beams for both electron and proton. The
same approach adopted here has been used to study laser-assisted processes in
relativistic atomic physics [8, 9, 6]. The organization of this paper is as
follows: in section 2., we develop a theoretical analytical calculation of the
differential cross section (DCS) of electron-proton (e-p) scattering in the
absence and in the presence of the laser field for both unpolarized and
polarized electrons. Then, in section 3., we will discuss the different
results obtained and we end with a conclusion in section 4.. We mention that
this work is developed in the laboratory system of proton and for a space-time
metric of signature $(+,-,-,-)$. We use the atomic unit system
$\hbar=e=m_{e}=1$ during the theoretical calculation.

##  2\. Overview of the theory

In the theoretical treatment, we will present an analytical calculation of the
free DCS in the first Born approximation. Then, we introduce the concept of
spin polarization, and at the end the expression of the DCS in the presence of
a monochromatic laser field is derived based on the approach of describing the
electrons by the Volkov states [41] and the spin projection formalism.

###  2.1. Laser-free unpolarized differential cross section

The elastic scattering of a proton by electron impact, shown in Fig. 1, is
studied by considering the proton as a structureless Dirac particle.

Figure 1: Electron-proton scattering in the laboratory system, with $A^{\mu}$
is the four-potential produced by the proton.

In Fig. 1, the $p_{n}$ (for $n=\\{1,3\\}$) represent the 4-momentum of the
incident ($p_{1}$) and scattered ($p_{3}$) electrons, while $p_{m}$ (for
$m=\\{2,4\\}$) represent the 4-momentum of the proton in its initial ($p_{2}$)
and final ($p_{4}$) states. In the absence of the laser field, the particles
studied are described by Dirac wave functions, normalized by the quantum
volume $V$, as follows:

$\displaystyle\left\\{\begin{array}[]{c}\psi_{p_{n}}(x)=\frac{1}{\sqrt{2E_{n}V}}u(p_{n},s_{n})e^{-ip_{n}.x},\\\
\\\
~{}~{}\psi_{p_{m}}(y)=\frac{1}{\sqrt{2E_{m}V}}u(p_{m},s_{m})e^{-ip_{m}.y},\end{array}\right.$
(4)

$E_{n,m}$ and $s_{n,m}$ represent the energy and the spin of the particles
respectively. At the leading order, the matrix element $S_{fi}$ representing
our scattering process is:

$S_{fi}=-i\int d^{4}x~{}\overline{\psi}_{p_{3}}(x)\not{A}(x)\psi_{p_{1}}(x),$
(5)

with $\not{A}(x)=A_{\mu}(x)\gamma^{\mu}$, where $A_{\mu}$ is the
electromagnetic 4-potential of the scattering given by:

$\displaystyle A_{\mu}(x)$ $\displaystyle=$ $\displaystyle-\int
d^{4}y~{}D_{F}(x-y)\bar{\psi}_{p_{4}}(y)\gamma_{\mu}\psi_{p_{2}}(y),$ (6)
$\displaystyle=$ $\displaystyle\int
d^{4}y~{}\frac{d^{4}q}{(2\pi)^{4}}\frac{e^{-iq(x-y)}}{q^{2}+i\epsilon}\big{[}\bar{\psi}_{p_{4}}(y)\gamma_{\mu}\psi_{p_{2}}(y)\big{]},$

such as the element $D_{F}(x-y)=-\int
d^{4}qe^{-iq(x-y)}/\big{[}(2\pi)^{4}(q^{2}+i\epsilon)\big{]}$ is the Feynman
electromagnetic propagator. By replacing Eqs. (4) and (6) in the expression of
$S_{fi}$ (5), the matrix element becomes:

$\displaystyle S_{fi}$ $\displaystyle=$
$\displaystyle\frac{-i}{\sqrt{16E_{1}E_{2}E_{3}E_{4}V^{4}}}\int
d^{4}x~{}d^{4}y~{}\frac{d^{4}q}{(2\pi)^{4}}\frac{e^{i(p_{3}-p_{1}-q).x}e^{i(p_{4}-p_{2}+q).y}}{q^{2}+i\epsilon}\mathcal{M}_{fi},$
(7) $\displaystyle=$
$\displaystyle\frac{-i}{\sqrt{16E_{1}E_{2}E_{3}E_{4}V^{4}}}\frac{(2\pi)^{4}\delta^{4}(p_{4}-p_{2}+p_{3}-p_{1})}{(p_{3}-p_{1})^{2}+i\epsilon}\mathcal{M}_{fi},$

where the element $\mathcal{M}_{fi}$ represents the scattering amplitude such
that:

$\displaystyle\mathcal{M}_{fi}=\big{[}\bar{u}(p_{3},s_{3})\gamma^{\mu}u(p_{1},s_{1})\big{]}\big{[}\bar{u}(p_{4},s_{4})\gamma_{\mu}u(p_{2},s_{2})\big{]}.$
(8)

To express the unpolarized DCS, the squared matrix element $|S_{fi}|^{2}$ is
weighted by the flux of the incident electrons, the phase space of the
particles in the final state and by unit of time $T$. Thus, we get:

$\displaystyle
d\bar{\sigma}=\frac{Vd^{3}\vec{p}_{3}}{(2\pi)^{3}}\frac{Vd^{3}\vec{p}_{4}}{(2\pi)^{3}}\frac{1}{|J_{\text{inc.}}|T}\frac{(2\pi)^{4}VT\delta^{4}(p_{4}-p_{2}+p_{3}-p_{1})}{16E_{1}E_{2}E_{3}E_{4}V^{4}q^{4}}|\overline{\mathcal{M}_{fi}}|^{2},$
(9)

where we average over the initial spins and sum over the final ones as
follows:

$|\overline{\mathcal{M}_{fi}}|^{2}=\frac{1}{4}\sum_{s_{n,m}}|\mathcal{M}_{fi}|^{2}.$
(10)

The incident particle flux is $|J_{\text{inc.}}|=|\vec{p}_{1}|c^{2}/(E_{1}V)$.
To integrate with respect to $d^{3}\vec{p}_{4}$, we will use the following
formula:

$\displaystyle\frac{d^{3}\vec{p}_{4}}{E_{4}}=2\int_{-\infty}^{+\infty}d^{4}p_{4}\delta(p_{4}^{2}-M^{2}c^{2})\Theta(p_{4}^{0});\quad\,\,\Theta(p_{4}^{0})=\left\\{\begin{array}[]{ccc}&1&~{}~{}\text{for}~{}~{}p_{4}^{0}>0\\\
&0&~{}~{}\text{for}~{}~{}p_{4}^{0}<0,\end{array}\right.$ (13)

where $M$ is the rest mass of the proton. Using
$d^{3}\vec{p}_{3}=|\vec{p}_{3}|^{2}d|\vec{p}_{3}|d\Omega_{f}$ and
$E_{3}dE_{3}=c^{2}|\vec{p}_{3}|d|\vec{p}_{3}|$, we find that the unpolarized
DCS is expressed as :

$\displaystyle\frac{d\overline{\sigma}}{d\Omega_{f}}=\frac{1}{8(2\pi)^{2}c^{6}Mq^{4}}\frac{|\vec{p}_{3}|}{|\vec{p}_{1}|}\int\delta(p_{4}^{2}-M^{2}c^{2})dE_{3}|\overline{\mathcal{M}_{fi}}|^{2}.$
(14)

The integration over $dE_{3}$ is performed using the following property [42]:

$\displaystyle\int
dx~{}f(x)\delta(g(x))=\frac{f(x)}{|g^{\prime}(x)|}\bigg{|}_{g(x)=0}.$ (15)

In our case:
$g^{\prime}(E_{3})=M+\frac{E_{1}}{c^{2}}-\frac{E_{3}}{c^{2}}\frac{|\vec{p}_{1}|}{|\vec{p}_{3}|}F(\theta_{i},\theta_{f},\varphi_{i},\varphi_{f})$,
where

$\displaystyle
F(\theta_{i},\theta_{f},\varphi_{i},\varphi_{f})=\sin(\theta_{i})\sin(\theta_{f})\big{[}\cos(\varphi_{i})\cos(\varphi_{f})+\sin(\varphi_{i})\sin(\varphi_{f})\big{]}+\cos(\theta_{i})\cos(\theta_{f}).$
(16)

Finally, we give the expression of the unpolarized DCS as follows:

$\displaystyle\bigg{(}\frac{d\overline{\sigma}}{d\Omega_{f}}\bigg{)}^{\text{laser-
free}}_{\text{unpolarized}}=\frac{1}{16(2\pi)^{2}Mc^{6}q^{4}}\frac{|\vec{p}_{3}|}{|\vec{p}_{1}|}\frac{|\overline{\mathcal{M}_{fi}}|^{2}}{|g^{\prime}(E_{3})|},$
(17)

where

$\displaystyle|\overline{\mathcal{M}_{fi}}|^{2}=\frac{1}{4}\text{Tr}\big{[}(c\not{p}_{3}+c^{2})\gamma^{\mu}(c\not{p}_{1}+c^{2})\gamma^{\nu}\big{]}\text{Tr}\big{[}(c\not{p}_{4}+Mc^{2})\gamma_{\mu}(c\not{p}_{2}+Mc^{2})\gamma_{\nu}\big{]}.$
(18)

###  2.2. Laser-free differential cross section for polarized electrons

Until date, all calculations have been done assuming that the electron’s spin
has not been detected. Now, we introduce the calculation of the DCS for
polarised electrons. As a first step, we define the spin 4-vector
$S_{\lambda}^{\mu}$ as:

$\displaystyle
S_{\lambda}^{\mu}=\frac{\lambda}{c}\Big{(}|\vec{p}|,\frac{E}{c}\hat{p}\Big{)},~{}~{}~{}~{}\text{where}~{}~{}~{}\,\,\hat{p}=\frac{\vec{p}}{|\vec{p}|},$
(19)

which verifies the normalization relation $S^{\mu}S_{\mu}=-1$, and the
orthogonality relation with the 4-momentum $p$ such that:
$(p.S)=p^{\mu}.S_{\mu}=0$. The parameter $\lambda$ defines the spin
orientation (spin Up for $\lambda=+1$ and Down for $\lambda=-1$).
Subsequently, the spin projection operator is introduced:

$\displaystyle\hat{\Sigma}(S)=\frac{1}{2}(1+\gamma_{5}\not{S}),~{}~{}~{}\quad\text{with}~{}~{}~{}\gamma_{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}=-i\gamma_{0}\gamma_{1}\gamma_{2}\gamma_{3}.$
(20)

The spin projection operator $\hat{\Sigma}(S)$ verifies the following
property:

$\displaystyle\hat{\Sigma}(S)u(p,S^{\prime})=\delta_{SS^{\prime}}u(p,S^{\prime}).$
(21)

If we consider the spin polarization of electrons, the only quantity that
changes in the expression of the laser-free unpolarized DCS (17) is
$|\overline{\mathcal{M}_{fi}}|^{2}$ because it is the only part that depends
on the spin. It becomes

$|\overline{\mathcal{M}_{fi}}|^{2}=\frac{1}{2}\sum_{s_{2},s_{4}}|\mathcal{M}_{fi}|^{2}.$
(22)

In Eq. (22), one has to be careful since there are no summations on electron
spin (either initial or final) polarizations. The factor $\tfrac{1}{2}$ is due
to the average over the initial polarizations of the unpolarized proton.
Introducing the two spin projection operators for the initial and final
electrons, we find

$\begin{split}|\overline{\mathcal{M}_{fi}}|^{2}=&\text{Tr}\Big{[}\big{(}c\not{p}_{3}+c^{2}\big{)}\gamma^{\mu}\Big{(}\frac{1+\lambda_{i}\gamma_{5}\not{S}_{i}}{2}\Big{)}\big{(}c\not{p}_{1}+c^{2}\big{)}\gamma^{\nu}\Big{(}\frac{1+\lambda_{f}\gamma_{5}\not{S}_{f}}{2}\Big{)}\Big{]}\\\
&\times\dfrac{1}{2}\text{Tr}\big{[}(c\not{p}_{4}+Mc^{2})\gamma_{\mu}(c\not{p}_{2}+Mc^{2})\gamma_{\nu}\big{]},\end{split}$
(23)

where $\lambda_{i}$ and $\lambda_{f}$ are the helicity states respectively of
initial and final electrons, and $S_{i}$ and $S_{f}$ are their 4-vectors spin.
Now, the DCS for polarized electrons depends, in addition to other parameters,
on $\lambda_{i}$ and $\lambda_{f}$. Therefore, if a flip of the electron spin
occurs during the process, which means that:

$\lambda_{f}=-\lambda_{i}=\pm
1,~{}~{}~{}\Longrightarrow~{}~{}~{}\lambda_{i}\lambda_{f}=-1,$ (24)

the obtained DCS is called spin-flip DCS. In the opposite case where no spin
flip occurs ($\lambda_{f}=\lambda_{i}=\pm
1,~{}\Rightarrow~{}\lambda_{i}\lambda_{f}=1$), it is called spin non-flip DCS.
One of the most important consistency tests to check is that the sum of the
spin-flip and the spin non-flip DCSs must give the unpolarized DCS given in
Eq. (17)

$\bigg{(}\frac{d\overline{\sigma}}{d\Omega_{f}}\bigg{)}_{\text{unpolarized}}^{\text{laser-
free}}=\bigg{(}\frac{d\overline{\sigma}}{d\Omega_{f}}\bigg{)}_{\text{flip}}+\bigg{(}\frac{d\overline{\sigma}}{d\Omega_{f}}\bigg{)}_{\text{non-
flip}}.$ (25)

Another important quantity to be introduced here is the degree of polarization
(DP), which is defined as:

$\displaystyle\text{DP}=\frac{\big{(}d\overline{\sigma}/d\Omega_{f}\big{)}_{\text{non-
flip}}-\big{(}d\overline{\sigma}/d\Omega_{f}\big{)}_{\text{flip}}}{\big{(}d\overline{\sigma}/d\Omega_{f}\big{)}_{\text{non-
flip}}+\big{(}d\overline{\sigma}/d\Omega_{f}\big{)}_{\text{flip}}}.$ (26)

###  2.3. Laser-assisted differential cross section for polarized electrons

The monochromatic laser field of circular polarization is described by the
following classical 4-potential:

$A_{\text{laser}}^{\mu}(\phi)=a^{\mu}_{1}\cos(\phi)+a^{\mu}_{2}\sin(\phi),$
(27)

where $a_{1}^{\mu}=(0,|\vec{a}|,0,0)$ and $a_{2}^{\mu}=(0,0,|\vec{a}|,0)$ are
the orthogonal polarization 4-vectors satisfying the following relation:
$a_{1}^{2}=a_{2}^{2}=a^{2}=-|\vec{a}|^{2}=(c\varepsilon_{0}/\omega)^{2}$,
where $\varepsilon_{0}$ is the electric field strength and $\omega$ is the
electromagnetic field frequency. $\phi=k.x$ is the phase of the
electromagnetic field of wave 4-vector $k=(\omega/c,\vec{k})$ (with
$k^{2}=0$). To describe an electron dressed by an electromagnetic field, we
use the solution of the following Dirac equation in the presence of the laser
field:

$\displaystyle\Big{[}\Big{(}\hat{p}-\frac{A}{c}\Big{)}^{2}-c^{2}-\frac{i}{2c}F_{\mu\nu}\sigma^{\mu\nu}\Big{]}\psi(x)=0,$
(28)

where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ represents the
electromagnetic tensor and
$\sigma^{\mu\nu}=\tfrac{1}{2}[\gamma^{\mu},\gamma^{\nu}]$. The general
solution of Eq. (28) is an exact Dirac-Volkov plane wave function expressed as
follows [41]:

$\displaystyle\psi_{p_{n}}(x)=\bigg{(}1+\frac{\not{k}\not{A}}{2c(k.p_{n})}\bigg{)}\frac{u(p_{n},s_{n})}{\sqrt{2Q_{n}V}}e^{iS(q_{n},x)},$
(29)

where $u(p_{n},s_{n})$ is the free Dirac bispinor of the electron, which
verifies the following relation:

$\displaystyle\bar{u}(p_{n},s_{n})u(p_{n},s_{n})=2c^{2}.$ (30)

The dressed electron is of 4-momentum $q_{n}=(Q_{n}/c,\vec{q}_{n})$ given by:

$\displaystyle
q_{n}=p_{n}-\frac{a^{2}}{2c^{2}(k.p_{n})}k,\quad\text{with}~{}~{}\,q_{n}^{2}={m^{*}}^{2}c^{2},$
(31)

where $m^{*}=\sqrt{1-a^{2}/c^{4}}$ is the effective mass of the electron in
the presence of the laser field. The classical action $S(q_{n},x)$ in Eq. (29)
has the following expression:

$\displaystyle
S(q_{n},x)=-q_{n}.x-\frac{a_{1}.p_{n}}{c(k.p_{n})}\sin(\phi)+\frac{a_{2}.p_{n}}{c(k.p_{n})}\cos(\phi).$
(32)

In the first Born approximation, using Feynman’s rules and after a simple
treatment, we express $S_{fi}$ in the presence of an electromagnetic field by:

$\displaystyle S_{fi}$ $\displaystyle=$
$\displaystyle\frac{-i}{\sqrt{16Q_{1}Q_{3}E_{2}E_{4}V^{4}}}\int
d^{4}x~{}d^{4}y~{}\frac{d^{4}q}{(2\pi)^{4}}\frac{e^{i(q_{3}-q_{1}-q).x}e^{i(p_{4}-p_{2}+q).y}e^{-iz\sin(\phi-\phi_{0})}}{q^{2}+i\varepsilon}$
(33) $\displaystyle\times$
$\displaystyle\bigg{[}\overline{u}(p_{3},s_{3})\bigg{(}C_{\mu}^{0}+C_{\mu}^{1}\cos(\phi)+C_{\mu}^{2}\sin(\phi)\bigg{)}u(p_{1},s_{1})\bigg{]}\bigg{[}\overline{u}(p_{4},s_{4})\gamma^{\mu}u(p_{2},s_{2})\bigg{]},$

where

$\displaystyle
z=\sqrt{\alpha_{1}^{2}+\alpha_{2}^{2}}~{}~{}~{}\text{and}~{}~{}~{}\phi_{0}=\arctan(\alpha_{2}/\alpha_{1}),$
(34)

with

$\displaystyle\alpha_{1}=\frac{a_{1}.p_{1}}{c(k.p_{1})}-\frac{a_{1}.p_{3}}{c(k.p_{3})}\quad\text{and}\quad\alpha_{2}=\frac{a_{2}.p_{1}}{c(k.p_{1})}-\frac{a_{2}.p_{3}}{c(k.p_{3})}.$
(35)

The terms $C^{0}_{\mu}$, $C^{1}_{\mu}$ and $C^{2}_{\mu}$ are given by:

$\displaystyle\left\\{\begin{array}[]{ccc}C^{0}_{\mu}&=&\gamma_{\mu}-2k_{\mu}a^{2}\not{k}C(p_{1})C(p_{3}),\\\
C^{1}_{\mu}&=&C(p_{1})\gamma_{\mu}\not{k}\not{a}_{1}+C(p_{3})\not{a}_{1}\not{k}\gamma_{\mu},\\\
C^{2}_{\mu}&=&C(p_{1})\gamma_{\mu}\not{k}\not{a}_{2}+C(p_{3})\not{a}_{2}\not{k}\gamma_{\mu}.\end{array}\right.\qquad\text{where}\quad
C(p_{i})=\frac{1}{2c(k.p_{i})}.$ (39)

Then, by integrating with respect to $d^{4}x$, $d^{4}y$ and $d^{4}q$, we
obtain:

$\displaystyle S_{fi}$ $\displaystyle=$
$\displaystyle\frac{-i}{\sqrt{16Q_{1}Q_{3}E_{2}E_{4}V^{4}}}\sum_{\ell=-\infty}^{+\infty}\frac{(2\pi)^{4}\delta^{4}(p_{4}-p_{2}+q_{3}-q_{1}-\ell
k)}{q^{2}+i\varepsilon}$ (40) $\displaystyle\times$
$\displaystyle\bigg{[}\overline{u}(p_{3},s_{3})\bigg{(}C_{\mu}^{0}B_{\ell}(z)+C_{\mu}^{1}B_{1\ell}(z)+C_{\mu}^{2}B_{2\ell}(z)\bigg{)}u(p_{1},s_{1})\bigg{]}\bigg{[}\overline{u}(p_{4},s_{4})\gamma^{\mu}u(p_{2},s_{2})\bigg{]},$

where $\ell$ is the number of photons exchanged and $q=q_{3}-q_{1}-\ell k$ is
the 4-momentum transfer. The functions $B_{\ell}$, $B_{1\ell}$ and $B_{2\ell}$
are obtained using Fourier transforms into Bessel functions [43], such that:

$\displaystyle\Big{(}1,\cos(\phi),\sin(\phi)\Big{)}e^{-iz\sin(\phi-\phi_{0})}=\sum_{\ell=-\infty}^{+\infty}\Big{(}B_{\ell}(z),B_{1\ell}(z),B_{2\ell}(z)\Big{)},$
(41)

with

$\displaystyle\begin{split}\begin{bmatrix}B_{\ell}(z)\\\ B_{1\ell}(z)\\\
B_{2\ell}(z)\end{bmatrix}=\begin{bmatrix}J_{\ell}(z)e^{i\ell\phi_{0}}\\\
\big{(}J_{\ell+1}(z)e^{i(\ell+1)\phi_{0}}+J_{\ell-1}(z)e^{i(\ell-1)\phi_{0}}\big{)}/2\\\
\big{(}J_{\ell+1}(z)e^{i(\ell+1)\phi_{0}}-J_{\ell-1}(z)e^{i(\ell-1)\phi_{0}}\big{)}/2i\end{bmatrix}.\end{split}$
(42)

As in the laser-free case of the DCS (9), we give the DCS in the presence of
an external laser field by the following formula:

$\displaystyle
d\bar{\sigma}=\sum_{\ell=-\infty}^{+\infty}\frac{Vd^{3}\vec{q}_{3}}{(2\pi)^{3}}\frac{Vd^{3}\vec{p}_{4}}{(2\pi)^{3}}\frac{1}{|J_{\text{inc.}}|T}\frac{(2\pi)^{4}VT\delta^{4}(p_{4}-p_{2}+q_{3}-q_{1}-\ell
k)}{16Q_{1}E_{2}Q_{3}E_{4}V^{4}q^{4}}|\overline{\mathcal{M}_{fi}^{\ell}}|^{2},$
(43)

where $|J_{\text{inc.}}|=[|\vec{q}_{1}|c^{2}]/[VQ_{1}]$ is the flux of
incident electrons dressed by the laser field. The associated scattering
amplitude $|\overline{\mathcal{M}_{fi}^{\ell}}|$, in case of well defined spin
electrons, will take the following form:

$\begin{split}|\overline{\mathcal{M}_{fi}^{\ell}}|^{2}=&\text{Tr}\big{[}(c\not{p}_{3}+c^{2})\Gamma_{\mu}^{\ell}(\frac{1+\lambda_{1}\gamma_{5}\not{S}_{1}}{2})(c\not{p}_{1}+c^{2})\Gamma_{\nu}^{\ell\dagger}(\frac{1+\lambda_{3}\gamma_{5}\not{S}_{3}}{2})\big{]}\\\
&\times\frac{1}{2}\text{Tr}\big{[}(c\not{p}_{4}+Mc^{2})\gamma^{\mu}(c\not{p}_{2}+Mc^{2})\gamma^{\nu}\big{]},\end{split}$
(44)

with $\Gamma_{\nu}^{\ell\dagger}$ is the conjugate of $\Gamma_{\mu}^{\ell}$
defined by:

$\displaystyle\Gamma_{\mu}^{\ell}=C_{\mu}^{0}B_{\ell}(z)+C_{\mu}^{1}B_{1\ell}(z)+C_{\mu}^{2}B_{2\ell}(z).$
(45)

Finally, the DCS expression in the presence of an electromagnetic wave and for
well-defined spin states of the electron is given by:

$\displaystyle\bigg{(}\frac{d\overline{\sigma}}{d\Omega_{f}}\bigg{)}^{\text{laser-
assisted}}_{\text{$e$-polarized}}(\lambda_{1},\lambda_{3})=\sum_{\ell=-\infty}^{+\infty}\frac{d\overline{\sigma}^{\ell}}{d\Omega_{f}}(\lambda_{1},\lambda_{3})=\frac{1}{16(2\pi)^{2}Mc^{6}q^{4}}\sum_{\ell=-\infty}^{+\infty}\frac{|\vec{q}_{3}|}{|\vec{q}_{1}|}\frac{|\overline{\mathcal{M}^{\ell}_{fi}}|^{2}}{|g^{\prime}(Q_{3})|},$
(46)

where the function $g^{\prime}(Q_{3})$ is obtained by using the property in
Eq. (15):

$\displaystyle
g^{\prime}(Q_{3})=\bigg{[}\frac{Q_{1}}{c^{2}}+M+\frac{\ell\omega}{c^{2}}\bigg{]}-\frac{Q_{3}}{c^{2}|\vec{q_{3}}|}\bigg{[}|\vec{q}_{1}|F(\theta_{i},\theta_{f},\varphi_{i},\varphi_{f})+\frac{\ell\omega}{c^{2}}\cos(\theta_{f})\bigg{]}.$
(47)

For the spin-flip and spin non-flip laser-assisted DCSs, we have:

$\begin{split}\bigg{(}\frac{d\overline{\sigma}}{d\Omega_{f}}\bigg{)}_{\text{flip}}&=\bigg{(}\frac{d\overline{\sigma}}{d\Omega_{f}}\bigg{)}^{\text{laser-
assisted}}_{\text{$e$-polarized}}(\lambda_{1}=-\lambda_{3}=\pm 1),\\\
\bigg{(}\frac{d\overline{\sigma}}{d\Omega_{f}}\bigg{)}_{\text{non-
flip}}&=\bigg{(}\frac{d\overline{\sigma}}{d\Omega_{f}}\bigg{)}^{\text{laser-
assisted}}_{\text{$e$-polarized}}(\lambda_{1}=\lambda_{3}=\pm 1).\end{split}$
(48)

The sum of the two (spin-flip and non-flip DCSs) should again yield the
unpolarized laser-assisted DCS that was previously calculated in a recent work
[11].
The calculation of the traces appearing in Eqs. (18), (23) and (44) is
performed with the help of the symbolic-algebra program Feyncalc [44]. We
mention here that the result obtained is too long and cumbersome to be
included here due to the complexity of the calculations. For example, the
result of the trace calculation in the presence of the laser (given in eq.
(44)) yielded about 63 different antisymmetric tensors,
$\epsilon(a,b,c,d)=\epsilon_{\mu\nu\rho\sigma}a^{\mu}b^{\nu}c^{\rho}d^{\sigma}$
for all four-vectors $a,b,c$ and $d$, which appear when four matrices $\gamma$
meet the fifth matrix $\gamma_{5}$ inside the trace
$(\text{Tr}[\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}\gamma^{5}]=-4i\epsilon^{\mu\nu\rho\sigma})$.
These resulting tensors were calculated analytically and replaced.

##  3\. Results and discussion

After this detailed theoretical treatment of elastic e-p scattering in the
presence of a laser field and for electrons of well-defined spin polarization,
we will now present the different numerical results of this work. The geometry
that has been found to give good results is one with spherical angles of
$\theta_{i}=\varphi_{i}=15^{\circ}$ and $\varphi_{f}=105^{\circ}$. The final
angle $\theta_{f}$ is varied from $-180^{\circ}$ to $180^{\circ}$ to reveal
the angular distribution of the DCS. The incident elecron kinetic energy is
(unless otherwise stated) $T_{i}=2.7$ keV. The direction of the
electromagnetic field is that represented by the wave vector $\vec{k}$; it is
chosen along the axis ($Oz$). We start with the first result in which the
element $d\overline{\sigma}^{\ell}/d\Omega_{f}(\lambda_{1},\lambda_{3})$ of
Eq. (46) represents the DCS of each number of photons exchanged.

Figure 2: Differential cross section
$d\overline{\sigma}^{\ell}/d\Omega_{f}(\lambda_{1},\lambda_{3})$ for spin-flip
($\lambda_{1}=-\lambda_{3}=1$) and spin non-flip ($\lambda_{1}=\lambda_{3}=1$)
as a function of the number of photons exchanged $\ell$. The scattering final
angle is $\theta_{f}=0^{\circ}$. The field strength $\varepsilon_{0}$ and
frequency $\hbar\omega$ are indicated above each figure.

Fig. 2 gives information about the number of possible photons to be exchanged
during the scattering process, treating the two cases of polarization of spins
initially defined in spin Up ($\lambda_{1}=1$). For a laser field frequency
$\hbar\omega=1.17$ eV and strength $\varepsilon_{0}=5\times 10^{6}$ V/cm, the
DCS, $d\overline{\sigma}^{\ell}/d\Omega_{f}(\lambda_{1},\lambda_{3})$, in the
two Figs. 2A and 2B for spin non-flip and spin-flip varies in magnitude for
the same number of photons exchanged $-3\leq\ell\leq 3$. This result is also
seen by comparing Fig. 2C with Fig. 2D when the strength of the same laser
field increases to $\varepsilon_{0}=10^{7}$ V/cm, but now the number of
photons exchanged $\ell$ varies from $-5$ to $5$. We notice that by increasing
the field strength the number of possible exchanged photons becomes larger.
This number of photons exchanged $\ell$ corresponds to non-zero values of DCS,
while the DCS becomes zero from some values higher than $\ell$ in absolute
value (the number of exchanged photons $\ell$ depends strongly on the laser
field strength). This value from which the DCS cancels is called the cutoff of
the multiphoton scattering process. For a value $|\ell|$ greater than the
cutoff, the summed laser-assisted DCS (46) becomes equal to its corresponding
one in the absence of the field. Then, we say that the laser assisted process
verifies the sum rule formulated as follows:

$\frac{d\overline{\sigma}_{free}}{d\Omega_{f}}(\lambda_{1},\lambda_{3})=\sum_{\ell=-\text{cutoff}}^{+\text{cutoff}}\frac{d\overline{\sigma}^{\ell}}{d\Omega_{f}}(\lambda_{1},\lambda_{3}).$
(49)

Figure 3: Variation of the summed laser-assisted DCS (46) for polarized
electrons as a function of the scattering angle $\theta_{f}$ for different
summations over $\ell$. The notation $\ell=\pm i$ means that we have summed
over $-i\leq\ell\leq+i$.

The variation of the photon number $\ell$ given in the final expression of DCS
(eq. (46)), which was introduced by Bessel functions of the first order, was a
subject of observation of the absorption and free multiphoton emission in
elastic electron-atom collisions in the presence of a cw-CO2 laser [45] and in
the case of pulsed CO2 laser [46]. In similar studies, the experimental data
of these laser-assisted elastic electron scattering processes are in
reasonable agreement with the predictions of the Kroll-Watson theory [47]. The
results in Fig. 2 will be considered in the discussion of the effect of the
circularly polarized laser field on the DCS in Fig. 3 by varying the
scattering angle $\theta_{f}$. In Figs. 3A and 3B, we have plotted the summed
laser-assisted DCS (46) respectively for spin non-flip and spin-flip of
electrons, summing over different numbers of photons exchanged $\ell$ from
$-i$ to $i$ (with $i=\\{1,2,3,5\\}$). For these two figures, we can see that
the applied Nd:YAG laser field, of frequency $\hbar\omega=1.17$ eV, has a
decreasing effect on the DCS. This diminution becomes more important in the
case of a single photon exchange ($\ell=\pm 1$). While by increasing the
number of photons, the DCS increases and becomes closer to that without field.
We notice, in these two figures, that when the multiphoton process exchanges
$5$ photons ($\ell=\pm 5$), the DCS in the presence of the laser field becomes
closer to the one without the laser field, which confirms the result of the
sum rule (49) already found in Fig. 2A and 2B. On the other hand and according
to the results obtained so far, we note that the spin non-flip DCS
($\lambda_{1}=\lambda_{3}=1$) is higher than the spin-flip one
($\lambda_{1}=-\lambda_{3}=1$). This remark is clearly observed in Fig. 3C by
comparing the blue curve with the red one for a laser field strength
$\varepsilon_{0}=5\times 10^{6}$ V/cm. We moved on to Fig. 3D, where we raised
the strength to $\varepsilon_{0}=10^{7}$ V/cm, we still found the difference
between these two DCSs (flip and non-flip) for $\ell=\pm 1$. Figs. 3C and 3D
show that the sum of these two DCSs always coincides with the unpolarized DCS.
To give a closer picture of the effect of the laser field, for another type of
laser, on each spin polarization, we will introduce numerical results in table
1 by increasing the field strength up to the value $\varepsilon_{0}=10^{8}$
V/cm.

$\varepsilon_{0}\,\text{[V/cm]}$ non-flip DCS($\lambda_{1}=\lambda_{3}=1$)
[a.u.] flip DCS($\lambda_{1}=-\lambda_{3}=1$) [a.u.] $\hbar\omega=0.117$ eV
$\hbar\omega=1.17$ eV $\hbar\omega=0.117$ eV $\hbar\omega=1.17$ eV $10$
$2.575\,55$ $2.575\,55$ $4.417$$\times 10^{-2}$ $4.417$$\times 10^{-2}$
$10^{2}$ $2.575\,55$ $2.575\,55$ $4.417$$\times 10^{-2}$ $4.417$$\times
10^{-2}$ $10^{3}$ $2.575\,55$ $2.575\,55$ $4.417$$\times 10^{-2}$
$4.417$$\times 10^{-2}$ $10^{4}$ $2.573\,02$ $2.575\,55$ $4.412$$\times
10^{-2}$ $4.417$$\times 10^{-2}$ $10^{5}$ $4.726$$\times 10^{-1}$ $2.575\,55$
$8.105$$\times 10^{-3}$ $4.417$$\times 10^{-2}$ $10^{6}$ $5.722$$\times
10^{-2}$ $2.573\,02$ $9.808$$\times 10^{-4}$ $4.412$$\times 10^{-2}$ $10^{7}$
$7.697$$\times 10^{-3}$ $4.726$$\times 10^{-1}$ $1.319$$\times 10^{-4}$
$8.101$$\times 10^{-3}$ $10^{8}$ $7.373$$\times 10^{-3}$ $5.644$$\times
10^{-2}$ $1.141$$\times 10^{-5}$ $9.598$$\times 10^{-4}$

Table 1: Numerical values of spin-flip and non-flip DCSs as a function of
$\varepsilon_{0}$ for different frequencies and for summation over the $\ell$
number from $-1$ to $1$. The scattering angle $\theta_{f}=0^{\circ}$.

In table 1, we give a numerical vision of the effect of the electromagnetic
field strength on the e-p scattering process. We see that the effect of a
Nd:YAG laser field ($\hbar\omega=1.17$ eV) begins from a strength
$\varepsilon_{0}=10^{6}$ V/cm and is manifested by a decrease in both spin-
flip and non-flip DCSs. When the laser field is pulsed CO2
($\hbar\omega=0.117$ eV), its effect on DCSs starts when the strength
$\varepsilon_{0}=10^{4}$ V/cm. For low strengths ranging from $10$ to $10^{4}$
V/cm and for $\ell=\pm 1$, we have noted that the DCS for both polarization
cases takes constant values that correspond to those in the absence of the
field, which verifies the sum rule for $\ell=\pm 1$. In terms of spin, the
non-flip DCS in the case where $\hbar\omega=0.117$ eV and the strength ranges
from $10^{5}$ to $10^{8}$ V/cm has decreased slightly compared to the spin-
flip DCS. In Fig. 4, we plot the curves representing the values given in table
1, with the addition of another laser type (He:Ne) of frequency 2 eV. We
notice that the frequency of the laser field can change the variation of the
DCS with respect to $\varepsilon_{0}$ for both cases of spin polarization. We
also observe that for the CO2 laser, the effect of the DCS is very important.
By comparing between the DCSs in the spin-flip and non-flip cases, both
figures show that they have the same decrease by varying the strength
$\varepsilon_{0}$. So, if there is a difference between these two
polarizations, it will be small. From Fig. 4, it appears that the laser field
effect diminishes at higher frequencies.

Figure 4: Variation of (A) spin non-flip and (B) spin-flip DCSs as a function
of the laser field strength for different laser frequencies.

Figure 5: Variation of the laser-assisted degree of polarization (DP) as a
function of the field strength for different frequencies and for electrons
initially in (A) spin Up and (B) spin Down.

Finally, we present, in Fig. 5, the effect of the laser field on the degree of
polarization (DP) defined in Eq. (26). Fig. 5A depicts the variation of the
degree of polarization as a function of the strength $\varepsilon_{0}$ for
electrons initially polarized in spin Up. We found that the degree of
polarization slightly increased when we use a CO2 laser. This increasing
effect starts to appear from $\varepsilon_{0}=10^{7}$ V/cm, where
$\text{DP}=96,62\,\%$, up to $\varepsilon_{0}=10^{8}$ V/cm where
$\text{DP}=96,95\,\%$. For the other frequencies, the degree of polarization
has increased less than before. In Fig. 5B, where the electron is initially in
spin Down, the degree of polarization increases this time from $96.63\,\%$ to
$97.45\,\%$ respectively for $\varepsilon_{0}=10^{7}$ and $10^{8}$ V/cm in the
case of a CO2 pulsed laser, while it remains almost constant for other lasers.
At the end, we conclude that the elastic e-p scattering process of initially
well-defined electron spin assisted by laser field of frequency
$\hbar\omega=0.117$ eV is more probable for the scattering of non-flip
electrons than for flip ones. Also, we conclude, as in [9], that the degree of
polarization is weak depending on the number of photons exchanged $\ell$.

Figure 6: Degree of polarization without laser as a function of scattering
angle $\theta_{f}$ for different incident electron kinetic energies. The
electron is considered to be initially in spin Up ($\uparrow$).

To see the effect of the incident electron kinetic energy on the degree of
polarization, we include in Fig. (6) its changes in terms of the final angle
$\theta_{f}$ with respect to various kinetic energies. At low-energy
scattering, the degree of polarization is strongly changed with the scattering
angle $\theta_{f}$. On the other hand, at very high energy, when the
scattering becomes relativistic, the degree of polarization becomes less
dependent on the scattering angle $\theta_{f}$ and approaches the constant
value $\text{DP}=1$, which implies the complete dominance of spin non-flip
DCS. The physical meaning of this result is that at very high energy, the
probability that the incident electron changes its spin is zero. These results
are coherent with those obtained in previous articles [6, 42] which follow the
same methodology.

##  4\. Conclusion

In this study, we have treated the process of laser-assisted electron-proton
scattering considering the effect of electron spin polarization. By applying
the concept of polarized electrons, the expressions of the DCS and the degree
of polarization are derived and their behavior in the presence of the laser
field is studied. We found that the effect of high laser field strengths is
important in each well defined polarization case (spin-flip and non-flip) for
an exchanged number pf photons $\ell=\pm 1$. The comparison of the different
spin polarizations, made by introducing the degree of polarization, shows that
the laser field can modify both spin-flip and non-flip DCSs with a continuous
dominance of the non-flip DCS. It is concluded that at high kinetic energy of
the incident electron, the electron’s probability for changing its spin is
zero. A consistency check that is carried out successfully is that the sum of
the two spin-flip and spin non-flip DCSs always gives the unpolarized DCS.
This approach, developed to describe the electrons by Dirac-Volkov functions,
is based on the fact that the involved proton is unpolarized and not dressed
by the laser field. In order to provide an in-depth and comprehensive study,
we are planning to also consider, in addition to the electron, the effect of
proton dressing and its spin polarization in a future work.

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