# Analysis of $e^{+}e^{-}\rightarrow H^{\pm}W^{\mp}$ in the presence of a
circularly polarized laser field

J. Ou aali,1 M. Ouhammou,1 M. Ouali,1 L. Rahili,2 S. Taj,1 and B. Manaut1, 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  2 EPTHE, Faculty of Sciences, Ibn
Zohr University, B.P 8106, Agadir, Morocco<EMAIL_ADDRESS>
###### Abstract

In this study, we have investigated the production of a charged Higgs boson in
assiciation with a charged weak $W$-boson ($H^{\pm}W^{\mp}$) at tree level via
$e^{+}e^{-}$ annihilation in the presence of a laser field with circular
polarization. At first step and by using analytical techniques, we have
derived the expression of the differential cross section involving both $Z$
and $\gamma$ diagrams. Next, we have shown that the cross section of this
process inside the laser field is proportional to the form factor $F_{HWV}$,
and it depends on the mass of the charged Higgs boson and the energy of the
center of mass. Then, we have analyzed the dependence of the total laser-
assisted cross section on the number of photons exchanged for two different
cases of form factor. Finally, we have found that the maximum number of
photons that can be transferred depends only on the laser parameters.

Keywords: Laser-assisted processes, electron-positron annihilation, Cross
section, Standard model and beyond.

##  1 Introduction

Since 1960s, the laser technology was and remains the focus of scientists as
it receives a great attention among them. Its importance lies not only in
studing the physical phenomena associated with laser, but also in assisting
our understanding of fundamental particle physics and atomic physics. The
first careful studies of laser-matter interaction in non relativistic atomic
physics was hypothesized by Francken and Joachain [1]. After, for a well
comprehension of the behavior of particles and their properties, some
phyisicists have been elaborated laser-atom interaction in relativistic
regime. They first studied laser field effects on relativistic atomic physics
scattering processes [2, 3]. Then, they studied some laser-assisted weak decay
processes [4, 5]. Due to the significant advancements achieved by laser
technology lately[6], it becomes possible to encourage the laser-assisted
production of heavy elementary particles such as neutral higgs pair
production[7], charged higgs pair production[8] and Higgsstrahlung
production[9]. This is being achieved by relying on high-intensity laser
sources which provide the incident beam $e^{+}e^{-}$ with high kinetic
energies. In ref [11], it is found that the laser field has a great impact on
the pion’s behavior and its properties such as the decay rate and the
particle’s lifetime. Moreover, it is shown that the laser field decreases the
total cross section of heavy elementary particles’ production (see [9, 10] for
more details).

The incredible discovery of the last absent particle of the Standard Model
(SM) with a mass around $125$ GeV at the Large Hadron Collider (LHC) at
CERN[12, 13] opens the door to a deeper understanding of the molecular
structure of matter and also phenomena related to high energy processes and
particles’ behaviors. Despite the fact that the observed properties of the new
neutral Higgs boson are in a great agreement with the prediction of the SM,
there are some questions that SM does not solve such as the existence of dark
matter and the origin of neutrino mass. These unexplained problems in the
standard model motivate physicists to continue their researches and
experiments for new physics, and to gives information about the extended Higgs
sector (Charged Higgs $H^{+}$, $H^{-}$)[14, 15] and analyse the coverage of
resonant di-Higgs production in the four final state [16, 17]. In addition, it
is well known that the insertion of the laser field in particle physics has a
great impact on the particles properties and its behavior. Therefore, the
study of laser-assisted high energy processes may lead to significant results
such as determining the properties of the charged Higgs boson.

The pair production of charged Higgs boson as well as neutral higgs boson from
$e^{-}e^{+}$ annihilation are analyzed within many extensions BSM, whether in
Inert Higgs Doublet Model (IHDM)[18], Two Higgs Doublet Model (2HDM)[19], or
in the SM which is extended with Triplet (HTM)[20]. Moreover, the process
which acts as a source of charged Higgs boson $H^{\pm}$ associated with a
$W^{\mp}$ boson at $e^{+}e^{-}$ collider is studied [21], and the signal
process proceeds through $s$-channel via virtual $\gamma$ and $Z$-boson
exchange. The authors of this work (ref[21]) showed how much the cross section
can be ameliorated by quark- and Higgs-loop effects. Furthermore, the coupling
$H^{\pm}W^{\mp}Z$ is the most related to the production of charged Higgs, and,
in ref[22], the authors studied the possibility of measuring it by using the
recoil method at ILC. In this respect, we have investigated the production of
a charged Higgs bosons in association with a charged weak gauge boson inside a
circularly polarized electromagnetic field to analyze theoretically and
analytically its effect on the production cross section.

This work is outlined as follows: In section 2, we start our discussion by
providing the theoretical calculation of the total cross section of charged
Higgs production in association with W-boson from $e^{+}e^{-}$ annihilation in
the presence of a circularly polarized laser field. Section 3 is devoted to
analyze and discuss the obtained results. The conclusion will be presented in
section 4. It should be pointed out that, in this paper, we have used natural
units $\hbar=c=1$, and the metric tensor $g^{\mu\nu}$ is chosen such as
$g^{\mu\nu}=(1,-1,-1,-1)$.

##  2 Outline of the theory

We consider the process which acts as a source of the charged Higgs boson
production in association with a $W$ boson at $e^{-}e^{+}$ colliders such as:

$e^{-}(p_{1},s_{1})+e^{+}(p_{2},s_{2})\rightarrow
W^{\mp}(k_{1},\lambda)+H^{\pm}(k_{2})$ (1)

where $p_{1}$ and $p_{2}$ are the free four-momentum of incident electron and
positron. $k_{1}$ and $k_{2}$ indicate the free four-momentum of $W$ boson and
the charged Higgs, respectively. Obviously, the charged Higgs boson associated
with a $W$ boson are produced via the s-channel process mediated by virtual
$\gamma$ and $Z$-boson exchange. These processes are described by the Feynman
diagrams which are depicted in figure 1.

Figure 1: Leading-order Feynman diagrams for the electron-positron
annihilation process $e^{-}e^{+}\rightarrow H^{\pm}W^{\mp}$.

The effective Lagrangian needed in our study [23, 24] is given by the
following expression:

$\eqalign{\hskip
71.13188pt\mathcal{L}_{eff}&=gM_{W}f_{HWV}H^{\pm}W^{\mp}_{\mu}V^{\mu}+g_{HWV}H^{\pm}F^{\mu\nu}_{V}F_{W\mu\nu}\cr&+(ih_{HWV}\epsilon_{\mu\nu\rho\sigma}F^{\mu\nu}_{V}F^{\rho\sigma}_{W}+h.c),}$
(2)

Where, $V=(Z,\gamma)$. $F^{\mu\nu}_{V}$ and $F^{\mu\nu}_{W}$ stand for the
field strength tensors for the weak gauge bosons ($Z$ and $W$). Let us remind
that the antisymmetric tensors $\epsilon_{\mu\nu\rho\sigma}$ is defined in
such a way to satisfy $\epsilon_{0123}=1$. It should be noted that $f_{HWV}$
is the coefficient of the dimension three operator, while $g_{HWV}$ and
$h_{HWV}$ are those of the dimension five operators. Therefore, only the form
factor $f_{HWV}$ may appear at the tree level. Besides all mentioned before,
the electron and positron are embedded in a circularly polarized laser field
which is chosen to be along the $z$-axis. Its polarization vector is given as
$k=\omega(1,0,0,1$) where $k^{2}=0$, and it is described by the following
classical four potential:

$A^{\mu}(\phi)=\chi_{1}^{\mu}\cos\phi+\chi_{2}^{\mu}\sin\phi.$ (3)

Here, $\phi=k.x$ stands for the phase of laser field, and $\chi_{1,2}^{\mu}$
are the polarization four vectors chosen as $\chi_{1}^{\mu}=(0,\chi,0,0)$ and
$\chi_{2}^{\mu}$ = $(0,0,\chi,0)$, with $\chi$ denotes the amplitude of the
vector potential. We assume that $\chi_{1,2}^{\mu}$ are taken in such a manner
that the following equations $\chi_{1}^{2}=\chi_{2}^{2}$ = $\chi^{2}$ =
$-|\boldsymbol{\chi}|^{2}$ = $-(\varepsilon_{0}/\omega)^{2}$ are checked, with
$\varepsilon_{0}$ is the electric field’s amplitude and $\omega$ its
frequency. The application of Lorentz gauge condition $\partial_{\mu}A^{\mu}$
= $0$ implies that $(k.\chi_{1})=(k.\chi_{2})=0$. The scattering-matrix
element at tree level [26] for the process represented by figure 1 can be
expressed as follows:

$\eqalign{S_{fi}({e}^{+}{e}^{-}\rightarrow H^{\pm}W^{\mp})=&\int d^{4}x\int
d^{4}y\Big{\\{}\bar{\psi}_{p_{2},s_{2}}(x)\Big{(}\frac{-ie}{2C_{W}S_{W}}\gamma^{\mu}(g_{v}^{e}-g_{a}^{e}\gamma^{5})\Big{)}\psi_{p_{1},s_{1}}(x)\cr&\times
D_{\mu\nu}(x-y)\varphi^{*\nu}_{k_{1},\lambda}(y)(igM_{W}f_{HWZ})\varphi^{*}_{k_{2}}(y)+\bar{\psi}_{p_{2},s_{2}}(x)(-ie\gamma^{\mu})\psi_{p_{1},s_{1}}(x)\cr&\times
G_{\mu\nu}(x-y)\varphi^{*\nu}_{k_{1},\lambda}(y)(igM_{W}f_{HW\gamma})\varphi^{*}_{k_{2}}(y)}$
(4)

In the equation above, the short notations $C_{W}$ and $S_{W}$ stands
respectively for $\cos_{\theta_{W}}$ and $\sin_{\theta_{W}}$, where
$\theta_{W}$ is the Weinberg angle. $x$ denotes the space time coordinates of
the electron and positron, while $y$ is the space-time coordinates of the
Higgs bosons and $W$ boson. $g_{v}^{e}=-1+4\sin^{2}_{\theta_{W}}$ and
$g_{a}^{e}=1$ are the vector and axial vector coupling constants. $g$ is the
electroweak coupling which is defined by $g^{2}$ =
$e^{2}/\sin^{2}_{\theta_{W}}=8G_{F}M^{2}_{Z}\cos^{2}_{\theta_{W}}/\sqrt{2}$.
In the Feynman gauge, the $Z^{*}$-boson propagator $D_{\mu\nu}(x-y)$ and the
$\gamma^{*}$-boson propagator $G_{\mu\nu}(x-y)$ are given by [26]:

$D_{\mu\nu}(x-y)=\int\frac{d^{4}q}{(2\pi)^{4}}\frac{-ig_{\mu\nu}}{q^{2}-M_{Z}^{2}}e^{-iq(x-y)}$
(5)
$G_{\mu\nu}(x-y)=\int\frac{d^{4}q}{(2\pi)^{4}}\frac{-ig_{\mu\nu}}{q^{2}}e^{-iq(x-y)}$
(6)

Here, $q$ is the four-momentum of the off-shell $V$ ($V=Z$ or $\gamma$) boson
propagators. The quantity $\psi_{p_{1},s_{1}}(x)$ is the Dirac-Volkov state
[26] of the electron, while $\psi_{p_{2},s_{2}}(x)$ is the Dirac-Volkov state
of the positron in the presence of the laser field. Their expressions can be
written as:

$\cases{\psi_{p_{1},s_{1}}(x)=\Big{[}1-\frac{e\not{k}\not{A}}{2(k.p_{1})}\Big{]}\frac{u(p_{1},s_{1})}{\sqrt{2Q_{1}V}}\exp^{iS(q_{1},s_{1})}&\\\
\psi_{p_{2},s_{2}}(x)=\Big{[}1+\frac{e\not{k}\not{A}}{2(k.p_{2})}\Big{]}\frac{v(p_{2},s_{2})}{\sqrt{2Q_{2}V}}\exp^{iS(q_{2},s_{2})}&\\\
}$ (7)

with, $u(p_{1},s_{1})$ and $v(p_{2},s_{2})$ are the Dirac spionrs of electron
and positron, respectively. $s_{i}$($i$ = $1,2$) are their spins. $p_{1}$ =
($E_{1}$,$|p_{1}|$,0,0) and $p_{2}$ = ($E_{2}$,$-|p_{1}|$,0,0) are referring
to their corresponding free four momenta in the centre of mass frame. $Q_{1}$
and $Q_{2}$ denote the effective energy acquired by the electron and positron
inside the electromagnetic field. We define the arguments of the exponential
terms in equation 7 as:

$\cases{S(q_{1},s_{1})=-q_{1}x+\frac{e(\chi_{1}.p_{1})}{k.p_{1}}\sin\phi-\frac{e(\chi_{2}.p_{1})}{k.p_{1}}\cos\phi&\\\
S(q_{2},s_{2})=+q_{2}x+\frac{e(\chi_{1}.p_{2})}{k.p_{2}}\sin\phi-\frac{e(\chi_{2}.p_{2})}{k.p_{2}}\cos\phi&\\\
}$ (8)

where, $q_{i}$, $i$ = (1,2) stands respectively for the effective momentum of
electron and positron inside the laser field. They are related to their
corresponding free four momentum such that:

$q_{i}=p_{i}+\frac{e^{2}a^{2}}{2(k.p_{i})}k.$ (9)

In this case, the square of the effective momentum reads as,
$q_{i}^{2}=m_{e}^{*2}$ = ($m_{e}^{2}+e^{2}a^{2}$), with $m_{e}^{*}$ is the
effective mass of incident particles, $m_{e}$ is the electron mass and $e$
denotes its charge. Moreover, in our study, we consider that there is no
interaction between the produced particles and the laser field. Therefore,
they are described by Klein-Gordon states such that:

$\varphi^{\mu}_{k_{1},\lambda}(y)=\frac{\epsilon^{\mu}(k_{1},\lambda)}{\sqrt{2E_{W^{\mp}}V}}e^{-ik_{1}y}\hskip
2.84544pt,\hskip 14.22636pt\\\ \
\varphi_{k_{2}}(y)=\frac{1}{\sqrt{2E_{H^{\pm}}V}}e^{-ik_{2}y},$ (10)

with, $k_{1}$ = ($E_{W^{\mp}}$,$|k_{1}|\cos\theta$,$|k|\sin\theta$,0) and
$k_{2}$ = ($E_{H^{\pm}}$,$-|k_{2}|\cos\theta$,$-|k_{2}|\sin\theta$,0) are
respectively the free four-momentum of $W$-boson and the charged Higgs-boson.
$\epsilon^{\mu}(k_{1},\lambda)$ stands for $W$-boson polarization vector where
$\lambda$ denotes its polarization such that
$\sum_{\lambda=1}^{3}\epsilon^{\mu}(k_{1},\lambda)\epsilon^{*\nu}(k_{1},\lambda)=-g^{\mu\nu}+k_{1}^{\mu}k_{1}^{\nu}/M_{W}^{2}$.
Then, to determine the expression of the scattering matrix element, we will
insert the equations given by 5, 6, 7, 8 and 10 into equation 4, and we get:

$S_{fi}^{n}({e}^{+}{e}^{-}\rightarrow
H^{\pm}W^{\mp})=\frac{(2\pi)^{4}\delta^{4}(k_{1}+k_{2}-q_{1}-q_{2}-nk)}{4V^{2}\sqrt{Q_{1}Q_{2}E_{H^{\pm}}E_{W^{\mp}}}}\big{(}M_{Z}^{n}+M_{\gamma}^{n}\big{)}$
(11)

where $n$ indicates the number of exchanged photons between the laser field
and the colliding physical system. The two parts which constitute the total
scattering amplitude $M_{Z}^{n}$ and $M_{\gamma}^{n}$ are coming from the
contribution of the free-photon exchange and the $Z$-boson contribution,
respectively. The first part can be expressed in terms of Bessel functions as
follows:

$\eqalign{M_{Z}^{n}&=\frac{e}{2C_{W}S_{W}}gM_{W}F_{HWZ}\frac{\epsilon^{*\mu}(k_{1},\lambda)}{(q_{1}+q_{2}+nk)^{2}-M_{Z}^{2}}\Bigg{\\{}\bar{v}(p_{2},s_{2})\cr&\times\Bigg{[}\kappa^{0}_{\mu}\,J_{n}(z)e^{-in\phi_{0}}(z)+\frac{1}{2}\,\,\kappa^{1}_{\mu}\Big{(}J_{n+1}(z)e^{-i(n+1)\phi_{0}}+J_{n-1}(z)e^{-i(n-1)\phi_{0}}\Big{)}\cr&+\frac{1}{2\,i}\,\kappa^{2}_{\mu}\Big{(}J_{n+1}(z)e^{-i(n+1)\phi_{0}}-J_{n-1}(z)e^{-i(n-1)\phi_{0}}\Big{)}\Bigg{]}u(p_{1},s_{1})\Bigg{\\}},}$
(12)

where, the quantities $\kappa^{0}_{\mu}$, $\kappa^{1}_{\mu}$ and
$\kappa^{2}_{\mu}$ can be written as follows:

$\cases{\kappa^{0}_{\mu}=\gamma_{\mu}(g_{v}^{e}-g_{a}^{e}\gamma^{5})+2a_{p_{1}}a_{p_{2}}\chi^{2}k_{\mu}\not{k}(g_{v}^{e}-g_{a}^{e}\gamma^{5})&\\\
\kappa^{1}_{\mu}=a_{p_{1}}\gamma_{\mu}(g_{v}^{e}-g_{a}^{e}\gamma^{5})\not{k}\not{\chi}_{1}-a_{p_{2}}\not{\chi}_{1}\not{k}\gamma_{\mu}(g_{v}^{e}-g_{a}^{e}\gamma^{5})&\\\
\kappa^{2}_{\mu}=a_{p_{1}}\gamma_{\mu}(g_{v}^{e}-g_{a}^{e}\gamma^{5})\not{k}\not{\chi}_{2}-a_{p_{2}}\not{\chi}_{2}\not{k}\gamma_{\mu}(g_{v}^{e}-g_{a}^{e}\gamma^{5})&\\\
}$ (13)

with $a_{p_{i}}=e/2(kp_{i})$($i=1,2$). Similarly, one gets $M_{\gamma}^{n}$
as:

$\eqalign{M_{\gamma}^{n}&=\frac{e}{(q_{1}+q_{2}+nk)^{2}}gM_{W}F_{HW\gamma}\epsilon^{*\mu}(k_{1},\lambda)\Bigg{\\{}\bar{v}(p_{2},s_{2})\Bigg{[}\xi^{0}_{\mu}\,J_{n}(z)e^{-in\phi_{0}}(z)\cr&+\frac{1}{2}\,\,\xi^{1}_{\mu}\Big{(}J_{n+1}(z)e^{-i(n+1)\phi_{0}}+J_{n-1}(z)e^{-i(n-1)\phi_{0}}\Big{)}+\frac{1}{2\,i}\,\xi^{2}_{\mu}\cr&\times\Big{(}J_{n+1}(z)e^{-i(n+1)\phi_{0}}-J_{n-1}(z)e^{-i(n-1)\phi_{0}}\Big{)}\Bigg{]}u(p_{1},s_{1})\Bigg{\\}}.}$

In equation 2, the factors $\xi^{0}_{\mu}$, $\xi^{1}_{\mu}$ and
$\xi^{2}_{\mu}$ are related to Dirac matrices as follows:

$\cases{\xi^{0}_{\mu}=\gamma_{\mu}+2a_{p_{1}}a_{p_{2}}\chi^{2}k_{\mu}\not{k}&\\\
\xi^{1}_{\mu}=a_{p_{1}}\gamma_{\mu}\not{k}\not{\chi}_{1}-a_{p_{2}}\not{\chi}_{1}\not{k}\gamma_{\mu}&\\\
\xi^{2}_{\mu}=a_{p_{1}}\gamma_{\mu}\not{k}\not{\chi}_{2}-a_{p_{2}}\not{\chi}_{2}\not{k}\gamma_{\mu}&\\\
}$ (14)

The two form factors $F_{HWZ}$ and $F_{HW\gamma}$ in equations 12 and 2 are
successively related to the coefficients $f_{HWZ}$ and $f_{HW\gamma}$ that
appear in the effective Lagrangian given by the equation 2. Moreover, the
reason of $U(1)_{em}$ gauge invariance imposes that the form factor
$F_{HW\gamma}$ should be vanished in any extended model beyond the SM at the
tree level. So, the $Z$-boson couplings has a dominant contribution. However,
due to the violation of the Custiodal Symmetry in the kinetic term, all the
form factors containing $F_{HWZ}$ are zero at the tree level in some extended
Higgs models with only two doublet scalar fields such as two-Higgs-doublet
model (2HDM) and beyond. The simplest models in which the form factor
$F_{HWZ}$ appears at the tree level are given in [22, 25]. The argument of the
Bessel function $z$ and the phase $\phi_{0}$ are determined by,
$z=\sqrt{\alpha_{1}^{2}+\alpha_{2}^{2}}$ and
$\phi_{0}=\arctan(\alpha_{2}/\alpha_{1})$, where:

$\alpha_{1}=\frac{e(\chi_{1}.p_{1})}{(k.p_{1})}-\frac{e(\chi_{1}.p_{2})}{(k.p_{2})}\hskip
2.84544pt,\hskip 14.22636pt\\\ \
\alpha_{2}=\frac{e(\chi_{2}.p_{1})}{(k.p_{1})}-\frac{e(\chi_{2}.p_{2})}{(k.p_{2})},$
(15)

In general, the differential cross section in the center of mass frame is
calculated by dividing the squared matrix element of equation 11 by $VT$, and
by the current of incident particles defined by
$|J_{inc}|=(\sqrt{(q_{1}q_{2})^{2}-m_{e}^{*^{4}}}/{Q_{1}Q_{2}V})$, then by the
density of the particles $\rho=V^{-1}$. We get the expression of the partial
differential cross section as follows:

$d\sigma_{n}=\frac{|S_{fi}^{n}|^{2}}{VT}\frac{1}{|J_{inc}|}\frac{1}{\varrho}V\int\frac{d^{3}k_{2}}{(2\pi)^{3}}V\int\frac{d^{3}k_{1}}{(2\pi)^{3}}.$
(16)

After some simplification and by summing over the final states and averaging
over the initial ones, the differential cross section becomes:

$\eqalign{d\bar{\sigma_{n}}({e}^{+}{e}^{-}\rightarrow
H^{\pm}W^{\mp})&=\frac{1}{16\sqrt{(q_{1}q_{2})^{2}-m_{e}^{*^{4}}}}\big{|}\overline{M_{Z}^{n}+M_{\gamma}^{n}}\big{|}^{2}\int\frac{|\mathbf{k}_{2}|^{2}d|\mathbf{k}_{2}|d\Omega}{(2\pi)^{2}E_{H^{\pm}}}\int\frac{d^{3}k_{1}}{E_{W^{\mp}}}\cr&\times\delta^{4}(k_{1}+k_{2}-q_{1}-q_{2}-nk).}$
(17)

The remaining integral over $d^{3}k_{1}$ can be estimated by using the well
known formula [26] given by:

$\int
d\mathbf{y}f(\mathbf{y})\delta(g(\mathbf{y}))=\frac{f(\mathbf{y})}{|g^{{}^{\prime}}(\mathbf{y})|_{g(\mathbf{y})=0}}.$
(18)

Finally, we find the final expression of the differential cross section as
follows:

$\eqalign{\frac{d\sigma_{n}}{d\Omega}({e}^{+}{e}^{-}\rightarrow
H^{\pm}W^{\mp})&=\frac{1}{16\sqrt{(q_{-}q_{+})^{2}-m_{e}^{*^{4}}}}\big{|}\overline{M_{Z}^{n}+M_{\gamma}^{n}}\big{|}^{2}\frac{2|\mathbf{k}_{2}|^{2}}{(2\pi)^{2}E_{H^{\pm}}}\frac{1}{|g^{{}^{\prime}}(|\mathbf{k}_{2}|)|_{g(|\mathbf{k}_{2}|)=0}},}$
(19)

where the function $g^{{}^{\prime}}(|\mathbf{k}_{2}|)$ is given by:

$\big{|}g^{{}^{\prime}}(|\textbf{k}_{2}|)\big{|}=-4\Big{[}\frac{(\sqrt{s}+n\omega)|\textbf{k}_{2}|}{2\sqrt{|\textbf{k}_{2}|^{2}+M_{H^{\pm}}^{2}}}-\frac{|\textbf{k}_{2}|}{\sqrt{|\textbf{k}_{2}|^{2}+M_{H^{\pm}}^{2}}}\frac{e^{2}a^{2}}{\sqrt{s}}\Big{]}.$
(20)

By using FeynCalc tools [27], we have derived the expression of the quantity
$\big{|}\overline{M_{Z}^{n}+M_{\gamma}^{n}}\big{|}^{2}$ given in equation 19,
and it can be written as:

$\eqalign{\big{|}\overline{M_{Z}^{n}+M_{\gamma}^{n}}\big{|}^{2}&=\frac{1}{4}\sum_{n=-\infty}^{+\infty}\sum_{s}\big{|}M_{Z}^{n}+M_{\gamma}^{n}\big{|}^{2}=\frac{1}{4}\sum_{n=-\infty}^{+\infty}\Bigg{\\{}(2gM_{W})^{2}(F_{HW\gamma})^{2}\frac{e^{4}}{(q_{1}+q_{2}+nk)^{4}}\cr&\times
Tr\Bigg{[}(\not{p}_{1}-m_{e})\Big{[}\xi^{0}_{\mu}\,J_{n}(z)e^{-in\phi_{0}}(z)+\xi^{1}_{\mu}\,\,\frac{1}{2}\Big{(}J_{n+1}(z)e^{-i(n+1)\phi_{0}}+J_{n-1}(z)e^{-i(n-1)\phi_{0}}\Big{)}\cr&+\xi^{2}_{\mu}\,\frac{1}{2\,i}\Big{(}J_{n+1}(z)e^{-i(n+1)\phi_{0}}-J_{n-1}(z)e^{-i(n-1)\phi_{0}}\Big{)}\Big{]}(\not{p}_{2}+m_{e})\Big{[}\xi^{0}_{\nu}\,J^{*}_{n}(z)e^{+in\phi_{0}}(z)\cr&+\xi^{1}_{\nu}\,\,\frac{1}{2}\Big{(}J^{*}_{n+1}(z)e^{+i(n+1)\phi_{0}}+J^{*}_{n-1}(z)e^{+i(n-1)\phi_{0}}\Big{)}-\xi^{2}_{\nu}\,\frac{1}{2\,i}\Big{(}J^{*}_{n+1}(z)e^{+i(n+1)\phi_{0}}\cr&-J^{*}_{n-1}(z)e^{+i(n-1)\phi_{0}}\Big{)}\Big{]}\Bigg{]}\Big{(}-g^{\mu\nu}+\frac{(k_{1}^{\mu}.k_{1}^{\nu})}{M_{W}^{2}}\Big{)}+(2gM_{W})^{2}(F_{HWZ})^{2}\left(\frac{e}{2C_{W}S_{W}}\right)^{2}\cr&\times\left(\frac{1}{(q_{1}+q_{2}+nk)^{2}-M_{Z}^{2}}\right)^{2}Tr\Bigg{[}(\not{p}_{1}-m_{e})\Big{[}\kappa^{0}_{\mu}\,J_{n}(z)e^{-in\phi_{0}}(z)+\kappa^{1}_{\mu}\,\,\frac{1}{2}\Big{(}J_{n+1}(z)e^{-i(n+1)\phi_{0}}\cr&+J_{n-1}(z)e^{-i(n-1)\phi_{0}}\Big{)}+\kappa^{2}_{\mu}\,\frac{1}{2\,i}\Big{(}J_{n+1}(z)e^{-i(n+1)\phi_{0}}-J_{n-1}(z)e^{-i(n-1)\phi_{0}}\Big{)}\Big{]}\cr&\times(\not{p}_{2}+m_{e})\Big{[}\kappa^{0}_{\nu}\,J^{*}_{n}(z)e^{+in\phi_{0}}(z)+\kappa^{1}_{\nu}\,\,\frac{1}{2}\Big{(}J^{*}_{n+1}(z)e^{+i(n+1)\phi_{0}}+J^{*}_{n-1}(z)e^{+i(n-1)\phi_{0}}\Big{)}\cr&-\kappa^{2}_{\nu}\,\frac{1}{2\,i}\Big{(}J^{*}_{n+1}(z)e^{+i(n+1)\phi_{0}}-J^{*}_{n-1}(z)e^{+i(n-1)\phi_{0}}\Big{)}\Big{]}\Bigg{]}\Big{(}-g^{\mu\nu}+\frac{(k_{1}^{\mu}.k_{1}^{\nu})}{M_{W}^{2}}\Big{)}\cr&+(2gM_{W})^{2}(F_{HWZ}F_{HW\gamma})\frac{e^{2}}{(q_{1}+q_{2}+nk)^{2}}\left(\frac{e}{2C_{W}S_{W}}\right)\frac{1}{(q_{1}+q_{2}+nk)^{2}-M_{Z}^{2}}\cr&\times
Tr\Bigg{[}(\not{p}_{1}-m_{e})\Big{[}\xi^{0}_{\mu}\,J_{n}(z)e^{-in\phi_{0}}(z)+\xi^{1}_{\mu}\,\,\frac{1}{2}\Big{(}J_{n+1}(z)e^{-i(n+1)\phi_{0}}+J_{n-1}(z)e^{-i(n-1)\phi_{0}}\Big{)}\cr&+\xi^{2}_{\mu}\,\frac{1}{2\,i}\Big{(}J_{n+1}(z)e^{-i(n+1)\phi_{0}}-J_{n-1}(z)e^{-i(n-1)\phi_{0}}\Big{)}\Big{]}(\not{p}_{2}+m_{e})\Big{[}\kappa^{0}_{\nu}\,J^{*}_{n}(z)e^{+in\phi_{0}}(z)\cr&+\kappa^{1}_{\nu}\,\,\frac{1}{2}\Big{(}J^{*}_{n+1}(z)e^{+i(n+1)\phi_{0}}+J^{*}_{n-1}(z)e^{+i(n-1)\phi_{0}}\Big{)}-\kappa^{2}_{\nu}\,\frac{1}{2\,i}\Big{(}J^{*}_{n+1}(z)e^{+i(n+1)\phi_{0}}\cr&-J^{*}_{n-1}(z)e^{+i(n-1)\phi_{0}}\Big{)}\Big{]}\Bigg{]}\Big{(}-g^{\mu\nu}+\frac{(k_{1}^{\mu}.k_{1}^{\nu})}{M_{W}^{2}}\Big{)}+(2gM_{W})^{2}(F_{HW\gamma}F_{HWZ})\frac{e^{2}}{(q_{1}+q_{2}+nk)^{2}}\cr&\times\left(\frac{e}{2C_{W}S_{W}}\right)\frac{1}{(q_{1}+q_{2}+nk)^{2}-M_{Z}^{2}}Tr\Bigg{[}(\not{p}_{1}-m_{e})\Big{[}\kappa^{0}_{\mu}\,J_{n}(z)e^{-in\phi_{0}}(z)\cr&+\kappa^{1}_{\mu}\,\,\frac{1}{2}\Big{(}J_{n+1}(z)e^{-i(n+1)\phi_{0}}+J_{n-1}(z)e^{-i(n-1)\phi_{0}}\Big{)}+\kappa^{2}_{\mu}\,\frac{1}{2\,i}\Big{(}J_{n+1}(z)e^{-i(n+1)\phi_{0}}\cr&-J_{n-1}(z)e^{-i(n-1)\phi_{0}}\Big{)}\Big{]}(\not{p}_{2}+m_{e})\Big{[}\xi^{0}_{\nu}\,J^{*}_{n}(z)e^{+in\phi_{0}}(z)+\xi^{1}_{\nu}\,\,\frac{1}{2}\Big{(}J^{*}_{n+1}(z)e^{+i(n+1)\phi_{0}}\cr&-J^{*}_{n-1}(z)e^{+i(n-1)\phi_{0}}\Big{)}-\xi^{2}_{\nu}\,\frac{1}{2\,i}\Big{(}J^{*}_{n+1}(z)e^{+i(n+1)\phi_{0}}-J^{*}_{n-1}(z)e^{+i(n-1)\phi_{0}}\Big{)}\Big{]}\Bigg{]}\Big{(}-g^{\mu\nu}+\frac{(k_{1}^{\mu}.k_{1}^{\nu})}{M_{W}^{2}}\Big{)}\Bigg{\\}}.}$

##  3 Results and discussion

In this section, we have performed a systematic numerical scan to investigate
the scenario which can lead to significant cross section for the production
process of the charged Higgs boson associated with a $W$-boson from
$e^{+}e^{-}$ annihilation. As we have shown in equation 19, in the centre of
mass frame, this cross section depends on the mass of charged Higgs boson
$M_{H^{\pm}}$, the centre of mass energy $\sqrt{s}$ and the laser field
parameters such as the number of exchanged photons $n$, the laser field
strength $\varepsilon_{0}$ and its frequency $\omega$. Moreover, we can get
the total cross section by numerically integrating the equation 19 over the
solid angle defined by $d\Omega=\sin(\theta)d\theta d\phi$, where $\theta$ is
the angle between the momentum of the outgoing Higgs-bosons and the beam axis.
The SM parameters used in our analysis such as $m_{e}=0.511\,MeV$,
$M_{Z}=91.1875\,GeV$, $M_{W}=80.379\,GeV$ and $\theta_{W}=28.75^{\circ}$ are
taken from PDG [28]. As it is previously mentioned, we assume that
$g_{HWV}=h_{HWV}=0$. Consequenctly, the cross section is proportional to
$F_{HWV}$, $V$ = ($Z$ or $\gamma$). Throughout this work, we have considered
two sample benchmark points, namely $F_{HW\gamma}$ = $1$ and $F_{HW\gamma}=0$,
while $F_{HWZ}=1$[22].

Table 1: Laser-assisted total cross section as a function of the number of exchanged photons for different laser field strengths and frequencies by taking $F_{HW\gamma}=1$. The centre of mass energy and the charged Higgs mass are chosen as: $\sqrt{s}=300\,GeV$ ; $M_{H^{\pm}}=200\,GeV$. |  |  $\sigma$[fb] |  |  $\sigma$[fb] |  |  $\sigma$[fb] |
---|---|---|---|---|---|---|---
$\varepsilon_{0}(\,V.cm^{-1})$ |  | He:Ne Laser |  | Nd:YAG laser |  |  $CO_{2}$ Laser |
| n | $\omega=2\,eV$ | n | $\omega=1.17\,eV$ | n | $\omega=0.117\,eV$ |
| $\pm 7$ | $0$ | $\pm 20$ | $0$ | $\pm 1200$ | $0$ |
| $\pm 6$ | $0$ | $\pm 15$ | $0$ | $\pm 1100$ | $0$ |
| $\pm 5$ | $1.44717$ | $\pm 12$ | $1.24094$ | $\pm 900$ | $0.00167942$ |
$10^{5}$ | $\pm 4$ | $9.36089$ | $\pm 9$ | $20.568$ | $\pm 600$ | $0.165453$ |
| $\pm 3$ | $33.5847$ | $\pm 6$ | $0.893134$ | $\pm 300$ | $0.104116$ |
| $\pm 2$ | $47.3638$ | $\pm 3$ | $2.63859$ | $\pm 100$ | $0.0191935$ |
| $0$ | $32.4356$ | $0$ | $13.9913$ | $0$ | $0.134855$ |
| $\pm 40$ | $0$ | $\pm 120$ | $0$ | $\pm 6000$ | $0$ |
| $\pm 35$ | $0$ | $\pm 110$ | $0$ | $\pm 5100$ | $0$ |
| $\pm 32$ | $9.62478$ | $\pm 80$ | $1.98117$ | $\pm 4000$ | $0.00227526$ |
$10^{6}$ | $\pm 24$ | $0.782794$ | $\pm 60$ | $1.6538$ | $\pm 3000$ | $0.0141595$ |
| $\pm 16$ | $4.39571$ | $\pm 40$ | $1.02198$ | $\pm 2000$ | $0.00252853$ |
| $\pm 8$ | $3.06533$ | $\pm 20$ | $1.30663$ | $\pm 1000$ | $0.000287841$ |
| $0$ | $3.5543$ | $0$ | $0.565689$ | $0$ | $0.0117641$ |
Table 2: Laser-assisted total cross section as a function of the number of exchanged photons for different laser field strengths and frequencies by taking $F_{HW\gamma}=0$. The centre of mass energy and the charged Higgs mass are chosen as: $\sqrt{s}=300\,GeV$ ; $M_{H^{\pm}}=200\,GeV$. |  |  $\sigma$[fb] |  |  $\sigma$[fb] |  |  $\sigma$[fb] |
---|---|---|---|---|---|---|---
$\varepsilon_{0}(\,V.cm^{-1})$ |  | He:Ne Laser |  | Nd:YAG laser |  |  $CO_{2}$ Laser |
| n | $\omega=2\,eV$ | n | $\omega=1.17\,eV$ | n | $\omega=0.117\,eV$ |
| $\pm 7$ | $0$ | $\pm 20$ | $0$ | $\pm 1200$ | $0$ |
| $\pm 6$ | $0$ | $\pm 15$ | $0$ | $\pm 1100$ | $0$ |
| $\pm 5$ | $1.19267$ | $\pm 12$ | $1.02271$ | $\pm 900$ | $0.00138408$ |
$10^{5}$ | $\pm 4$ | $7.71467$ | $\pm 9$ | $16.9509$ | $\pm 600$ | $0.136356$ |
| $\pm 3$ | $27.6784$ | $\pm 6$ | $0.736066$ | $\pm 300$ | $0.0858061$ |
| $\pm 2$ | $39.0343$ | $\pm 3$ | $2.17457$ | $\pm 100$ | $0.0158181$ |
| $0$ | $26.7314$ | $0$ | $11.5307$ | $0$ | $0.111139$ |
| $\pm 40$ | $0$ | $\pm 120$ | $0$ | $\pm 6000$ | $0$ |
| $\pm 35$ | $0$ | $\pm 110$ | $0$ | $\pm 5100$ | $0$ |
| $\pm 32$ | $7.93215$ | $\pm 80$ | $1.63275$ | $\pm 4000$ | $0.00187531$ |
$10^{6}$ | $\pm 24$ | $0.645131$ | $\pm 60$ | $1.36296$ | $\pm 3000$ | $0.0116694$ |
| $\pm 16$ | $3.62268$ | $\pm 40$ | $0.842252$ | $\pm 2000$ | $0.00208386$ |
| $\pm 8$ | $2.52625$ | $\pm 20$ | $1.07684$ | $\pm 1000$ | $0.000237221$ |
| $0$ | $2.92923$ | $0$ | $0.46206$ | $0$ | $0.0096528$ |

In Tables 1 and 2, we present the variation of the partial total cross section
of the process $e^{+}e^{-}\rightarrow H^{\pm}W^{\mp}$ as a function of the
number of exchanged photons inside the electromagnetic field. This variation
is presented for different laser parameters such as the laser field strength
and its frequency. Table 1 illustrates the case where $F_{HW\gamma}=1$, while
Table 2 represents the results obtained for $F_{HW\gamma}=0$. As we can see
from these tables, the number of photons that can be exchanged varies by
changing the laser parameters, and, as a consequence, the partial total cross
section changes. Moreover, when we increase the number of exchanged photons,
this partial total cross section progressively decreases until it becomes
null. This result allows us to define the ”cutoff” as the greatest value of
photons’ number that can be exchanged and for which the partial total cross
section becomes null. We note that the cross section takes the same values for
the absorption of photons ($+n$) as well as for the emission of photons
($-n$). We remark also that the value of the ”cutoff” increases as long as the
intensity of the laser field is increased or by decreasing its frequency.
Furthermore, by comparing the ”cutoffs” given in tables 1 and 2 for a
specified values of $\varepsilon_{0}$ and $\omega$, we can see that they do
not change. For instance, for $\varepsilon_{0}=10^{6}\,V.cm^{-1}$ and
$\omega=2\,eV$, the cutoffs are equal to $\pm 35$ for both $F_{HW\gamma}=1$
and $F_{HW\gamma}=0$. Therefore, the cutoff does not depend on the choice of
$F_{HW\gamma}$, and it depends only on the laser field parameters. To discuss
these results more deeply, we have plotted in figure 6 the partial total cross
section as a function of the laser photons’ number for both $F_{HW\gamma}$ =
$1$ (green curve) and $F_{HW\gamma}=0$ (blue curve). The laser parameters are
chosen as $\varepsilon_{0}=10^{6}\,V.cm^{-1}$ and $\omega=1.17\,eV$. As we can
see from this figure, the maximum number of photons transferred between the
laser beam and the colliding system, which cancels the partial total cross
section in both cases $F_{HW\gamma}=1$ and $F_{HW\gamma}=0$, is equal to $\pm
110$. Therefore, theses results are in good agreement with those given in
tables 1 and 2. In figure 6, the charged Higgs mass is chosen as
$M_{H^{\pm}}=200$ GeV which is consistent with the IHDM constraints [29].

Figure 2: Laser-assisted partial total cross section versus the number of
exchanged photons in the case $F_{HWZ}=1$. The green color shows the case
$F_{HW\gamma}=1$, and the blue one shows the case $F_{HW\gamma}=0$. The laser
field strength and its frequency are chosen as
$\varepsilon_{0}=10^{5}\,V.cm^{-1}$ and $\omega=1.17\,eV$. The centre of mass
energy and the charged Higgs mass are taken as $\sqrt{s}=300\,GeV$ and
$M_{H^{\pm}}=200\,GeV$.

Now, we will discuss the dependence of the laser-assisted total cross section
of the process $e^{+}e^{-}\rightarrow H^{\pm}W^{\mp}$ on the centre of mass
energy for different transferred photons number and for different laser field
strengths and frequencies in the case $F_{HWZ}=1$ and $F_{HW\gamma}=0$. The
phenomenology of these scenarios is illustrated in figure 6. Let’s start our
discussion with the laser-free total cross section. As it is shown in this
figure, the cross section increases severely for low centre of mass energies
region $\sqrt{s}<326$ GeV until it reaches its maximum at $\sqrt{s}$ $\simeq$
$326$ GeV. Then, it decreases gradually as long as $\sqrt{s}$ increases. The
existence of the peak at $\sqrt{s}$ $\simeq$ $326$ GeV is a consequence of
competition among the phase-space enlargement and $s$-channel suppression as
much as $\sqrt{s}$ increases[30, 31]. By applying the circularly polarized
laser field, we notice that the behavior of the production cross section as a
function of $\sqrt{s}$ for different values of the laser’s parameters and
different exchanged photons number are quite similar. However, compared to its
corresponding laser-free cross section, we can see that the laser-assisted
cross section decreases by several orders of magnitude. The maximum value of
the cross section can surpass $66fb$ for $n=\pm 4$, and it decreases to about
$60fb$ and $50fb$ for $n=\pm 3$ and $n=\pm 2$, respectively. As shown in
figure 6(a) in which $\varepsilon_{0}=10^{5}\,V.cm^{-1}$ and $\omega=2\,eV$,
the maximal value is reached around $68fb$ for the region of high $n$ ($n=\pm
6$). In this region, the total cross section will be equal to its
corresponding laser-free total cross section in all centre of mass energies.
This means that the effect of laser field on the incident particles is
strongly suppressed. As a result no photon will be exchanged between the laser
field and the incident particles. In figures 6(b), 6(c) and 6(d), the total
cross section increases as long as the number $n$ raises until $n=\pm$ cutoff
similarly to figure 6(a). This summation over $\pm$ cutoff number is called
sum-rule [32], and it leads to a cross section which is equal to the laser-
free cross section in all centre of mass energies. Moreover, the total cross
section strongly depends on the values of the laser field amplitude and its
frequency. By comparing figures 6(a) and 6(b), we remark that, for the same
laser field amplitude which is $\varepsilon_{0}=10^{5}\,V.cm^{-1}$ and for the
same number of exchanged photons such as $n=\pm 3$, the corresponding cross
section is about $61fb$ for $\omega$ = $2eV$ and $14fb$ for $\omega=1.17\,eV$.
Another important point to be mentioned, here, is that the peak does not
change for all laser field’s parameters. This result push us to ask an
important question which is: How can these free-parameters change the behavior
and properties of the cross section of the studied process?

Figure 3: Dependence of the laser-assisted total cross section of the process
${e}^{+}{e}^{-}\rightarrow H^{\pm}W^{\mp}$ on $\sqrt{s}$ in the case where
$F_{HWZ}=1$ and $F_{HW\gamma}=0$ for different exchanged photons number and by
taking the charged Higgs mass as $M_{H^{\pm}}=200\,GeV$. The laser field
strength and its frequency are chosen as: (a):
$\varepsilon_{0}=10^{5}\,V.cm^{-1}$ , $\,\omega=2\,eV$ ; (b):
$\varepsilon_{0}=10^{5}\,V.cm^{-1}$ , $\,\omega=1.17\,eV$ ; (c):
$\varepsilon_{0}=10^{6}\,V.cm^{-1}$ , $\,\omega=2\,eV$ ; (d):
$\varepsilon_{0}=10^{6}\,V.cm^{-1}$ , $\,\omega=1.17\,eV$.

To answer this question, let’s begin by analyzing the dependence of the total
cross section on the mass of the charged Higgs. Figure 4 shows the variation
of the total cross section as a function of $M_{H^{\pm}}$ for three known
laser frequencies (left panel) and for different laser field strengths (right
panel) by summing over $n$ from $-4$ to $+4$. According to this figure, we
find that the total cross section of ${e}^{+}{e}^{-}\rightarrow
H^{\pm}W^{\mp}$ reaches up to $85-86\,fb$. This maximal value of the cross
section is reached in the range of intermediate masses of $H^{\pm}$ (
$M_{H^{\pm}}\sim 150-176\,GeV$) for $\omega=2\,eV$ and
$\varepsilon_{0}=10^{5}\,V.cm^{-1}$. Therefore, the cross section decreases by
increasing the mass of the charged Higgs. This effect is due to the fact that
the cross-section is inversely proportional to the mass $M_{H^{\pm}}$. Another
important observation which should be discussed here is that the order of
magnitude of the cross section is dropping rapidly as long as the laser
frequency decreases (left panel). For instance, in the low masses region
($M_{H^{\pm}}\,<\,160\,GeV$), the total cross section reaches up to $85\,fb$
for $\omega=2\,eV$, while it remains under $24\,fb$ for $\omega=1.17\,eV$.
Moreover, for $\omega=0.117\,eV$, the total cross section’s order is very
close to zero. This is due to the presence of a strong correlation between the
number of photons needed to fulfill the well-known sum rule and the frequency
$\omega$. In general, this number increases so far as the frequency of the
laser field decreases. In the right panel of figure 4, we can see that the
order of magnitude of the cross section decreases by increasing the laser
field strength, specifically, as long as
$\varepsilon_{0}\,\geq\,10^{5}\,V.cm^{-1}$. However, we find that the laser-
assisted cross section is equal to its corresponding laser-free cross section
for $\varepsilon_{0}\,\leq\,10^{4}\,V.cm^{-1}$. Therefore, the electromagnetic
field does not affect the integrated laser-assisted cross section at low
strengths. To confirm these results, we have made the same analysis for the
same laser field strengths and frequencies. In figure 5, we have plotted the
variation of laser-assisted total cross section versus the centre of mass
energy for different strengths of the electromagnetic field (left panel) and
different frequencies (right panel). As this figure shows, the peak does not
change for all laser field’s parameters. Consequently, it confirms the
validity of our previous results.

Figure 4: Laser-assisted total cross section of the process
${e}^{+}{e}^{-}\rightarrow H^{\pm}W^{\mp}$ as a function of the charged Higgs-
boson for different laser field strengths and frequencies in the case where
$F_{HWZ}=1$ and $F_{HW\gamma}=0$. The $\sqrt{s}$ and $n$ are taken as
$300\,GeV$ and $\pm 4$, respectively. (left panel)
$\varepsilon_{0}=10^{5}\,V.cm^{-1}$; (right panel) $\omega=2\,eV$.

Figure 5: Laser-assisted total cross section of the process
${e}^{+}{e}^{-}\rightarrow H^{\pm}W^{\mp}$ as a function of the centre of mass
energy for different laser field strengths and frequencies in the case where
$F_{HWZ}=1$ and $F_{HW\gamma}=0$. The charged Higgs mass and the number of
exchanged photons are chosen as $M_{H^{\pm}}=200\,GeV$ and $n=\pm 4$,
respectively. (left panel) $\varepsilon_{0}=10^{5}\,V.cm^{-1}$; (right panel)
$\omega=2\,eV$.

Figure 6: Laser-assisted total cross section of the process
${e}^{+}{e}^{-}\rightarrow H^{\pm}W^{\mp}$ for some typical values of
$M_{H^{\pm}}$ (left panel) and $\sqrt{s}$ (right panel) in the case where
$F_{HWZ}=1$ and $F_{HW\gamma}=0$. The number of exchanged photons, the laser
field strength and its frequency are chosen such that: $n=\pm 4$,
$\varepsilon_{0}=10^{5}\,V.cm^{-1}$ and $\omega=2\,eV$, respectively.

As a last result, we illustrate in figure 6 the dependence of the laser-
assisted total cross section on the centre of mass energy and the charged
Higgs mass for different frequencies and for some typical values of
$M_{H^{\pm}}$ (left panel) and centre of mass energies (right panel). The
laser field parameters are taken such as $\varepsilon_{0}=10^{5}\,V.cm^{-1}$
and $\omega=2\,eV$. As we can see from the left panel of figure 6, at low
values of $\sqrt{s}$ ($\sqrt{s}<430\,GeV$), the $M_{H^{\pm}}=180\,GeV$
scenario dominates, followed by $M_{H^{\pm}}=220\,GeV$ and then
$M_{H^{\pm}}=260\,GeV$. In addition, there is a peak at $\sqrt{s}\simeq
290\,GeV$ for $M_{H^{\pm}}=180\,GeV$ while it is located at $\sqrt{s}\simeq
335\,GeV$, $\sqrt{s}\simeq 385\,GeV$ and $\sqrt{s}\simeq 420\,GeV$ for
$M_{H^{\pm}}=220\,GeV$, $M_{H^{\pm}}=260\,GeV$ and $M_{H^{\pm}}=280\,GeV$,
respectively. Whereas, at large values of $\sqrt{s}$ ($\sqrt{s}>430\,GeV$), we
can see that the curves which represent the $\sqrt{s}$ dependence of the cross
section are very close to each other for all values of $M_{H^{\pm}}$. In
figure 6 (right panel), we can see that the order of magnitude of the cross
section depends on the centre of mass energy. For example, its maximum values
are $\sigma\simeq 90fb$, $\sigma\simeq 78fb$, $\sigma\simeq 57fb$ and
$\sigma\simeq 43fb$ for $\sqrt{s}=250\,GeV$, $\sqrt{s}=300\,GeV$,
$\sqrt{s}=350\,GeV$ and $\sqrt{s}=400\,GeV$, respectively. By comparing these
plots, we can deduce that the cross section drops rapidly for
$\sqrt{s}=250\,GeV$, while it decreases slowly for high centre of mass
energies (threshold effect).

##  4 Conclusion

In summary, we have investigated the process which acts as a source of charged
Higgs boson associated with W-boson at tree level in the presence of a
circularly polarized electromagnetic field. By theoretically analyzing this
process, we have presented the dependence of the partial total cross section
on the number of exchanged photons, and we have shown that the maximum number
of photons that can be exchanged depends only on the laser field parameters.
Then, we have found that there is a strong correlation between the order of
magnitude and the laser field parameters such as the number of exchanged
photons, the laser field strength and its frequency. More precisely, this
order of magnitude of the total laser-assisted cross section decreases as long
as the laser field strength increases or by decreasing its frequency. However,
radiations with low intensities have no effect on the cross section.
Furthermore, we have shown that, for a given laser field strength and
frequency, the laser-assisted total cross section falls down rapidly for
$\sqrt{s}=250\,GeV$ as compared to high values of centre of mass energy.

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#  On wide-angle photo- and electroproduction of pions to twist-3 accuracy

K. Passek-Kumerički111Presented at HADRON2021. Division of Theoretical
Physics, Rudjer Bošković Institute, HR-10002 Zagreb, Croatia

###### Abstract

The wide-angle photo- and electroproduction of pions is investigated within
the handbag mechanism in which the $\gamma^{(*)}N\to\pi N^{\prime}$ amplitudes
factorize into subprocess amplitudes,$\gamma^{(*)}q\to\pi q^{\prime}$, and
form factors representing $1/x$-moments of generalized parton distributions
(GPDs). The subprocess is calculated to twist-3 accuracy taking into account
both the 2- and 3-body Fock components of the pion. The cross-sections are
compared to experiment and spin-effects are discussed.

###### pacs:

13.60.-r, 13.60.Le, 12.38.Bx

day month yearday month year exclusive processes, meson production,
generalized parton distributions, distribution amplitudes, higher twist

##  1 Introduction

The hard exclusive processes are successfully described by the handbag
mechanism in which only one quark from the incoming nucleon and one from the
outgoing nucleon participate in the hard subprocess while all other partons
are spectators. In particular, it was applied to the Compton scattering
$\gamma^{(*)}N\rightarrow\gamma N$, the simplest probe of the nucleon
structure, as well as, to the meson electroproduction
$\gamma^{(*)}N\rightarrow MN^{\prime}$. The prerequisite of this approach is
the existence of at least one large scale in the process enabling the use of
the perturbative expansion in the strong coupling constant, as well as, the
twist expansion. Two kinematical regions were extensively investigated in the
past. The deeply virtual (DV) region is characterized by the large virtuality
$Q^{2}$ of the incoming photon and a small momentum transfer $(-t)$ from the
incoming to the outgoing nucleon. In the wide-angle (WA) region $(-t)$ is
large, as well as $(-u)$ and $s$, while $Q^{2}$ is smaller than $(-t)$
($Q^{2}=0$ in the case of photoproduction). The all order proofs of
factorization exist for DV Compton scattering (DVCS) [1] and DV meson
production (DVMP) [2]. The process amplitudes factorize in hard perturbatively
calculable subprocess amplitudes and generalized parton distributions (GPDs)
which encapsulate the soft hadron-parton transitions, i.e., and the hadron
structure. In contrast, for WA processes the general factorization proofs are
still missing but it was shown that the factorization holds to next-to-leading
order in the strong coupling for WA Compton scaterring (WACS) [3, 4] and to
leading-order for WA meson production (WAMP) [5]. In these cases it was argued
that in the symmetric frame where skewness is zero the process amplitudes can
be represented as a product of subprocess amplitudes and form factors that
represent $1/x$ moments of GPDs at zero-skewness.

Both DVCS and WACS were widely investigated in the last decades and the
handbag factorization achieved a good description of the experimental data.
But for the DV pion production the experimental data indicate the importance
of the transversely polarized virtual photons [6, 7, 8, 9] and the need to go
beyond the leading twist-2 calculation since only the longitudinally polarized
photons contribute to the latter. The twist-3 calculation with transversity
GPDs was proposed and the calculation including twist-3 2-body pion Fock
component achieved already the successful agreement with the data [10].
Similarly, the experimental data for photoproduction [12, 13, 11] show that
the the twist-2 contributions for WA pion production [5] are not sufficient.
In contrast to DVMP, it was found that the twist-3 contribution to pion
photoproduction vanishes in often used Windzura-Wilczek approximation, i.e.,
when just twist-3 2-body pion Fock components are considered [14]. Both 2- and
3-body twist-3 Fock components of $\pi_{0}$ were considered in [15] and the
results were successfully fitted to CLAS data [11]. We report here on
extension of this work to the twist-3 prediction for $P=\pi^{\pm},\pi^{0}$ WA
photo- and electroproduction [16].

##  2 Application of handbag mechanism

The helicity amplitudes ${\cal M}^{P}_{\nu,\kappa^{\prime},\mu,\kappa}$ for
$\gamma^{(*)}(\mu)N(\kappa)\rightarrow P(\nu)N^{\prime}(\kappa^{\prime})$
process, i.e, the (electro)production of the pseudoscalar meson $P$, in the WA
angle region can be expressed as

$\displaystyle{\cal M}^{P}_{0+,\mu+}$ $\displaystyle=$
$\displaystyle\frac{e_{0}}{2}\,\sum_{\lambda}\Big{[}{\cal
H}^{P}_{0\lambda,\mu\lambda}\,\left(R_{V}^{P}(t)+2\lambda\,R_{A}^{P}(t)\right)$
$\displaystyle-2\lambda\,\frac{\sqrt{-t}}{2m}\,{\cal
H}^{P}_{0-\lambda,\mu\lambda}\,\bar{S}^{P}_{T}(t)\Big{]}$ $\displaystyle{\cal
M}^{P}_{0-,\mu+}$ $\displaystyle=$
$\displaystyle\frac{e_{0}}{2}\,\sum_{\lambda}\Big{[}\frac{\sqrt{-t}}{2m}\,{\cal
H}^{P}_{0\lambda,\mu\lambda}\,R_{T}^{P}(t)$ (1)
$\displaystyle-2\lambda\,\frac{t}{2m^{2}}\,{\cal
H}^{P}_{0-\lambda,\mu\lambda}\,S^{P}_{S}(t)\Big{]}$ $\displaystyle+e_{0}{\cal
H}^{P}_{0-,\mu+}\,S^{P}_{T}(t)\,,$

where $m$ is a nucleon mass.

The soft form factors, $R_{i}^{P}$ and $S_{i}^{P}$ ($F_{i}^{P}$), represent
specific flavor combinations of $1/x$-moments of zero-skewness GPDs
($K_{i}^{a}$)

$F_{i}^{a}(t)=\int_{0}^{1}\frac{dx}{x}\,K_{i}^{a}(x,t)\,,$ (2)

where $x$ is the average momentum fraction of an active quark and $a$ is its
flavour. The form factors $R_{V}^{P}$, $R_{A}^{P}$ and $R_{T}^{P}$ are related
to the helicity non-flip GPDs $H$, $\widetilde{H}$ and $E$, respectively. The
$S$-type form factors, $S^{P}_{T}$, $\bar{S}_{T}^{P}$ and $S^{P}_{S}$ are
related to the helicity-flip or transversity GPDs $H_{T}$, $\bar{E}_{T}$ and
$\widetilde{H}_{T}$, respectively222The GPDs $\tilde{E}$ and $\tilde{E}_{T}$
and their associated form factors decouple at zero skewness. .

The amplitudes ${\cal H}_{\mu,\lambda,\nu,\lambda^{\prime}}$ correspond to the
subprocesses $\gamma^{(*)}(\mu)q(\lambda)\rightarrow
P(\nu)q^{\prime}(\lambda^{\prime})$. and they are calculated using handbag
diagrams as the ones depicted on Fig. 1. The meson $P$ with momentum
$q^{\prime}$ is replaced by an appropriate 2- or 3-body Fock state. The 2-body
projector $\pi\rightarrow q\bar{q}$ [17] given by

$\displaystyle{\cal P}_{2}^{P}$ $\displaystyle\sim$ $\displaystyle
f_{\pi}\,\Big{\\{}\gamma_{5}\,q/^{\prime}\phi_{\pi}(\tau,\mu_{F})$ (3)
$\displaystyle+\mu_{\pi}(\mu_{F})\Big{[}\gamma_{5}\,\phi_{\pi
p}(\tau,\mu_{F})$
$\displaystyle-\frac{i}{6}\,\gamma_{5}\,\sigma_{\mu\nu}\,\frac{q^{\prime}{}^{\mu}n^{\nu}}{q^{\prime}\cdot
n}\,\phi^{\prime}_{\pi\sigma}(\tau,\mu_{F})$
$\displaystyle+\frac{i}{6}\,\gamma_{5}\,\sigma_{\mu\nu}\,q^{\prime}{}^{\mu}\phi_{\pi\sigma}(\tau,\mu_{F})\frac{\partial}{\partial
k_{\perp\nu}}\Big{]}\Big{\\}}_{k_{\perp}\to 0}\,,$

contributes to the subprocess amplitudes corresponding to the diagrams
depicted on Fig. 1a. The quark (antiquark) longitudinal momentum fraction is
denoted by $\tau$ ($\bar{\tau}$), and $k_{T}$ represents its intrinsic
transverse momentum. The first term in (3) corresponds to the twist-2 part,
while the twist-3 part is proportional to the chiral condensate
$\mu_{\pi}=m_{\pi}^{2}/(m_{u}+m_{d})\cong 2$ GeV (at the factorization scale
$\mu_{F}=2$ GeV). This parameter is large and although the twist-3 cross
section for pion electroproduction is suppressed by $\mu_{\pi}^{2}/Q^{2}$ as
compared to twist-2 cross section, for the range of $Q^{2}$ accessible in
current experiments the suppression factor is of order unity333 Twist-3
effects can also be generated by twist-3 GPDs. However, these are expected to
be small and therefore neglected.. The 3-body $\pi\rightarrow q\bar{q}g$
projector [15]

${\cal P}_{3}^{P}\sim
f_{3\pi}(\mu_{F})\,\frac{i}{g}\,\gamma_{5}\,\sigma_{\mu\nu}q^{\prime}{}^{\mu}g_{\perp}^{\nu\rho}\,\frac{\phi_{3\pi}(\tau_{a},\tau_{b},\tau_{g},\mu_{F})}{\tau_{g}}$
(4)

contributes to the subprocesses amplitudes corresponding to Fig. 1b. The pion
constituents are taken to be collinear to the pion and $\tau_{i}$ denote their
longitudinal momentum fractions. The helicity non-flip amplitudes ${\cal
H}_{0\lambda,\mu,\lambda^{\prime}}$ are generated by twist-2, while the
helicity flip ones ${\cal H}_{0-\lambda,\mu,\lambda^{\prime}}$ are of twist-3
origin.

Figure 1: Generic diagrams corresponding to the 2- and 3-body subprocess
amplitudes.

Similarly to GPDs, the pion distribution amplitudes (DAs) $\phi$ encode the
long distance effects and thus the pion structure. In addition to twist-2 DA
$\phi_{\pi}$ there are two 2-body twist-3 DAs, $\phi_{\pi p}$ and
$\phi_{\pi\sigma}$, and 3-body twist-3 DA $\phi_{3\pi}$. Twist-3 DAs are
connected by equations of motion (EOMs)

$\displaystyle\tau\phi_{\pi
p}(\tau)+\frac{\tau}{6}\,\phi^{\prime}_{\pi\sigma}(\tau)-\frac{1}{3}\,\phi_{\pi\sigma}(\tau)$
$\displaystyle=$ $\displaystyle\phi^{EOM}_{\pi 2}(\bar{\tau})$
$\displaystyle\bar{\tau}\phi_{\pi
p}(\tau)-\frac{\bar{\tau}}{6}\phi^{\prime}_{\pi\sigma}(\tau)-\frac{1}{3}\phi_{\pi\sigma}(\tau)$
$\displaystyle=$ $\displaystyle\phi^{EOM}_{\pi 2}(\tau)$ (5)

where

$\phi^{EOM}_{\pi 2}(\tau)=\displaystyle
2\frac{f_{3\pi}}{f_{\pi}\mu_{\pi}}\int_{0}^{\bar{\tau}}\frac{d\tau_{g}}{\tau_{g}}\,\phi_{3\pi}(\tau,\bar{\tau}-\tau_{g},\tau_{g})\,.$

Using EOMs and DA symmetry properties the twist-3 subprocess amplitudes can be
expressed in terms of just two twist-3 DAs and 2- and 3-body contributions can
be combined. Moreover, EOMs lead to the inhomogeneous linear first order
differential equation which from known 3-body DA $\phi_{3\pi}$ [18] fixes
$\phi_{\pi p}$ (and $\phi_{\pi\sigma}$). We note that in the derivation of
$q\bar{q}g$ projector and EOMs the same gauge has to be used for the
constituent gluon, and we have consistently used the light-cone gauge.

For meson electroproduction both transverse and longitudinal photons
contribute to twist-2 subprocess amplitudes. As expected in photoproduction
limit ($Q^{2}=0$) the longitudinal contribution vanishes, while in DVMP limit
($t\to 0$) only the longitudinal photons contribute. The twist-3 contributions
both for transverse and longitudinal photons have general structure

$\displaystyle{\cal H}^{P,tw3}$ $\displaystyle=$ $\displaystyle{\cal
H}^{P,tw3,q\bar{q}}+{\cal H}^{P,tw3,q\bar{q}g}$ $\displaystyle=$
$\displaystyle\big{(}{\cal H}^{P,\phi_{\pi p}}+\underbrace{{\cal
H}^{P,\phi_{\pi 2}^{EOM}}\big{)}+\big{(}{\cal H}^{P,q\bar{q}g,C_{F}}}+{\cal
H}^{P,q\bar{q}g,C_{G}}\big{)}$ $\displaystyle=$ $\displaystyle{\cal
H}^{P,\phi_{\pi p}}\;\;+\quad\qquad{\cal
H}^{P,\phi_{3\pi},C_{F}}\qquad\quad+{\cal H}^{P,\phi_{3\pi},C_{G}}\,.$

The twist-3 2-body contribution ${\cal H}^{P,tw3,q\bar{q}}$ is proportional to
$C_{F}$ colour factor, while twist-3 3-body contribution has $C_{F}$ and
$C_{G}$ proportional parts. The $C_{G}$ part is separately gauge invariant,
while for $C_{F}$ contributions only the sum of 2- and 3-body parts is gauge
invariant (with respect to the choice of photon or virtual gluon gauge). We
have used EOMs to obtain that sum, as well as, the complete twist-3
contribution expressed through just two twist-3 DAs, $\phi_{3\pi}$ and
$\phi_{\pi p}$. The twist-3 subprocess amplitude for longitudinal photons
vanishes both for $Q\to 0$ and $t\to 0$, i.e, for photoproduction and DVMP.
One finds that for photoproduction ${\cal H}^{P,\phi_{\pi p}}=0$ [15]. For
DVMP ${\cal H}^{P,\phi_{\pi 2}^{EOM}}=0$, and although for $t\neq 0$ one finds
no end-point singularities, in this limit one has to deal with end-point
singularities in ${\cal H}^{P,\phi_{\pi p}}$ [10].

##  3 Numerical results

For details on the form of the pion DAs, GPDs and parameters used to obtain
the numerical results as well as the explicit expressions derived for the
subprocess amplitudes we refer to [16]. The form factors related to both
helicity non-flip and transversity GPDs and only two independent pion DAs
(twist-2 pion DA $\phi_{\pi}$ and twist-3 3-body pion DA $\phi_{3\pi}$) remain
as soft physics input. The latter are accompanied by decay constants $f_{\pi}$
and $f_{3\pi}$.

We comment here on few selected results. In [15] the cross-section for
$\pi^{0}$ photoproduction has been fitted to [11] data. The results are
displayed in Fig. 3.

Figure 2: Photoproduction. Figure 3:  The cross section for $\pi^{0}$
photoproduction. The solid (dotted) curve represents the full (twist 2)
result. The dashed curve is obtained with the amplitudes taken at the fixed
renormalization and factorization scale $\mu_{R}=\mu_{F}=1\,$ GeV. Data taken
CLAS [11] (open circles) and from SLAC [12] (the latter provided at $s=10.3$
GeV2).

As it was already noted in [5], one can see from Fig. 3 that twist-2
prediction lies well beyond the data. But by including the twist-3
contributions one obtains reasonable agreement with the experiment. Twist-3 is
more important in backward hemisphere ($\theta$ is c.m.s. scattering angle).
The twist-2 and twist-3 cross sections scale as $s^{-7}$ and $s^{-8}$,
respectively. Our results have effective $s^{-9}$ scaling which is a bit too
strong. As an attempt to modify this the fixed renormalization and
factorization scales were tested. In [16] such an analysis was extended to
$\pi^{+}$ and $\pi^{-}$ for which only few experimental data are available
[12, 13], The similar behaviour of photoproduction cross-sections was found.

For pion electroproduction there are four partial cross sections:
$d\sigma_{L}/dt$, $d\sigma_{T}/dt$, $d\sigma_{LT}/dt$, and $d\sigma_{TT}/dt$.
In [16] the theoretical predictions were given and importance of the
measurement was stressed. In the context of obtaining new information about
transverse GPDs we note that both for $\sigma_{L}$ and $\sigma_{LT}$ there is
no twist-2 and twist-3 interference. Given that the form factor $R_{A}^{P}$
contributing to twist-2 part is not unknown at large $(-t)$, one could thus
obtain the information on the $S_{T}$ form factor and gain knowledge on
$H_{T}$ GPD. Furthermore, $\sigma_{TT}$ that is suppressed for DVMP could be
used in WA limit to obtain the information on $S_{S}$ and thus on completely
unknown transversity GPD $\tilde{H}_{T}$.

Figure 4: Results for the helicity correlation parameters $A_{LL}$ and
$K_{LL}$ for $\pi^{+}$ and $\pi^{-}$ photoproduction.

As often the spin-dependent observables offer additional insight which is less
sensitive to particular parameters. For meson photoproduction the most
interesting are the correlations of the helicities of the photon and that of
either the incoming or the outgoing nucleon, i.e., $A_{LL}$ and $K_{LL}$,
respectively. It can be shown that

$\displaystyle A_{LL}^{P,tw2}$ $\displaystyle=$ $\displaystyle
K_{LL}^{P,tw2}\,,$ $\displaystyle A_{LL}^{P,tw3}$ $\displaystyle=$
$\displaystyle-K_{LL}^{P,tw3}$ (7)

and so the measurement of $A_{LL}$ and $K_{LL}$ offers characteristic
signature for dominance of twist-2 or twist-3. This is similar to the role
that the comparison of $\sigma_{T}$ and $\sigma_{L}$ has in DVMP. From Fig. 4
it is clear that our numerical results suggest the dominance of twist-3 for
large $\theta$, while twist-2 increases in the forward direction.

##  4 Conclusions

In this work we have summarized the recent application of the handbag
factorization to WA photo- and electroproduction of pions [16]. In contrast to
WACS, but like DVMP, the leading twist-2 analysis which involves helicity non-
flip GPDs fails for WA photoproduction by order of magnitude. We have obtained
the twist-3 prediction for WAMP which includes both 2- and 3-body twist-3 Fock
components of the pions. The $\pi^{0}$ photoproduction was fitted to the data
[15] Interesting helicity correlations show that twist-3 dominates for
$\pi^{0}$ channel, and mostly for $\pi^{\pm}$, while in latter case twist-2
matters in forward hemisphere. Different combinations of form factors along
with available data should allow to extract the form factors and to learn
about large $-t$ behaviour of transversity GPDs important for parton
tomography. The application to other pseudoscalar mesons [19] is
straightforward.

Acknowledgements This publication is supported by the Croatian Science
Foundation project IP-2019-04-9709, by the EU Horizon 2020 research and
innovation programme, STRONG-2020 project, under grant agreement No 824093 and
by Deutsche Forschungsgemeinschaft (DFG) through the Research Unit FOR 2926,
”Next Generation pQCD for Hadron Structure: Preparing for the EIC”, project
number 40824754.

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