#  __
Higgs-strahlung boson production in the presence of a circularly polarized
laser field

M. Ouhammou,1 M. Ouali,1 S. Taj,1 and B. Manaut1,
1 Sultan Moulay Slimane University, Polydisciplinary Faculty,
Research Team in Theoretical Physics and Materials (RTTPM), Beni Mellal,
23000, Morocco.
Corresponding author, E-mail<EMAIL_ADDRESS>
###### Abstract

In the framework of the electroweak standard model, we investigated, in the
center of mass frame, the Higgs boson production in the presence of an intense
laser field via ${e}^{+}{e}^{-}$ annihilation $({e}^{+}{e}^{-}\rightarrow
ZH)$. By comparing our results with those obtained by Djouadi [1] for laser-
free process, we show that the circularly polarized laser field affects
significantly the $s$-channel Higgs boson production. We find that for a given
number of exchanged photons, laser field strength and frequency, the total
cross section decreases by several order of magnitude. These effects of laser
field on cross section are found to be consistent with what was found for muon
pair production via QED process in the presence of a circularly polarized
laser field [2].

Keywords: Standard model, Electroweak interaction, Laser assisted processes,
Cross section.

##  1\. Introduction

The effect of the laser on the behavior of physical systems is relevant in the
study of phenomena related to the electroweak standard model of particle
physics, where the electromagnetic field could have an important impact on
particles properties and its interactions. It is well known that the
electromagnetic field affects the physical properties of particles, then it
would be interesting to investigate how the transition probability is modified
in the presence of an ultraintense laser fields. This could eventually play an
important role especially on high energy production and decay processes. These
effect on particle physics interaction has been a subject of several works.
In[3], the authors showed that the insertion of an external field has a big
impact on $Z$ boson decay as it enhances its mode decay and affects its life
time. The same results was obtained for muon decay [4] and pion decay [5]. The
muon pair production via electron-positron annihilation in QED was studied in
the presence of both circularly [2] and linearly [6] polarized laser field. It
is found that the insertion of a linearly polarized laser field enhances the
total cross section of the muon pair production while the circularly polarized
laser field reduces the total cross section by several order of magnitude. In
this respect, we have investigated the electroweak process
${e}^{+}{e}^{-}\rightarrow ZH$ in the presence of a circularly polarized laser
field to show how the cross section of the Higgs production is affected.

Higgs boson, which play a key role in the symmetry breaking mechanism by
generating particle masses in the standard model, was discovered by ATLAS and
CMS experiment in 2012 [7, 8] with a mass around 125 GeV. The noble prize in
physics was awarded to Peter Higgs and Francois Englert in 2013 for their
works in identifying and discovering it. Many of Higgs boson properties have
been measured by ATLAS and CMS collaboration [9, 10, 11, 12, 13, 14, 15, 16,
17] using data collected during run 1 and run 2 of the Large Hadron Collider
(LHC); moreover, all these properties are found to be consistent with the
prediction of the standard model of particle physics. As a result, the
standard model get the full recognition as a very successful model even though
there are hints of existence of new physics beyond the standard model. The
latter is a quantum field theory that unifies three of the four fundamental
interactions known today, which are the weak nuclear force, the
electromagnetic force and the strong fore. It gives an almost complete
description of the elementary particle physics. The Large Electron-Positron
Collider (LEP), which operated from 1989 to 2000, was the first particle
accelerator to have significant reach into the potential mass range of the
Higgs boson. It makes significant headway in the search, determinating that
the mass should be larger than 114 GeV. The future Circular Electron Positron
collider (CEPC) is mainly designed to operate at center of mass energies
around $240-250\,GeV$ during 10 years in three different modes which are
${e}^{+}{e}^{-}\rightarrow{Z}{H}$, ${e}^{+}{e}^{-}\rightarrow{Z}$ and
${e}^{+}{e}^{-}\rightarrow{W}^{+}{W}^{-}$. The first 7 years will dedicated to
the production of more than one million Higgs bosons via the Higgs strahlung
process. This fact highly motivate us to introduce the circularly polarized
laser field in Higgs strahlung process ${e}^{+}{e}^{-}\rightarrow{Z}{H}$ in
order to show its effect on the Higgs production cross section.

The aim of this paper is to study the Higgs boson production via the Higgs
strahlung process in the framework of electroweak standard model in the
presence of a circularly polarized laser field. The Higgs boson is produced
when an electron and positron annihilate to create a Higgs boson and a
supermassive $Z$ boson. This process is the main mechanism by which a Higgs
boson can be produced in the future Circular Electron Positron collider
(CEPC). Even-though the circularly polarized laser field obviously decreases
the cross section, the results found may help in particle physics experiments,
especially at CEPC where the the first run will dedicated to the Higgs
strahlung production. We hope that we study this process in the presence of a
linearly polarized field in a future work to show its effect on Higgs
production cross section.

The remainder of this paper is presented in three sections: In section 2, we
begin by presenting the analytical calculation of the total cross section of
the process ${e}^{+}{e}^{-}\rightarrow{Z}{H}$ in the presence of a circularly
polarized laser field. Then, the data obtained are presented and discussed in
section 3. Finally, a brief conclusion is given in section 4. Some Bessel
functions multiplying coefficients are listed in the appendix. In this work
the natural units are adopted ($\hbar=c=1$). The choice made for Livi-Civita
tensor is that $\epsilon^{0123}=1$ and the metric $g^{\mu\nu}$ is chosen such
that $g^{\mu\nu}=(1,-1,-1,-1)$.

##  2\. Outline of the theory

The cross section is one of the most important measured quantities in
production scattering processes. In this research paper’s part, we present the
analytical calculation of the total cross section of Higgs strahlung process
in the presence of a circularly polarized laser field. We find that this
quantity depends on the energy of the incident particles, i.e., the center of
mass energy and the laser’s parameters such as the exchanged photons number,
the laser field strength and the laser frequency.

### A-Description of the laser field and wave functions

Figure 1: Feynman diagram for $s$-channel Higgs production in the standard
model during the $e^{+}e^{-}$ collision (annihilation) in the lowest order.

$p_{1}$, $p_{2}$, $p_{3}$ and $p_{4}$ are the free four vector momenta of the
electron, positron, $Z$-boson and Higgs boson, respectively. In the present
work, we consider an electromagnetic, circularly polarized and monochromatic
wave field propagating along the $z$-axis. It is represented by the classical
four-vector potential $A^{\mu}$ such that:

$A^{\mu}(\phi)=a_{1}^{\mu}\cos\phi+a_{2}^{\mu}\sin\phi\hskip 28.45274pt;\hskip
28.45274pt\phi=(k.x)$ (1)

Where $k^{\mu}$ is taken to be parallel to the $z$-axis
$(k=(\omega,0,0,\omega))$ and $(k^{2}=0)$, $\phi$ is the phase of the laser
field and $\omega$ its frequency. The Lorentz gauge is satisfied
$\partial_{\mu}A^{\mu}=0$, which implies that
$(k_{\mu}A_{\mu}=0;\,a_{1}.k=0;\,a_{2}.k=0)$. $a_{1,2}^{\mu}$ are the
polarization four vectors chosen as $a_{1}^{\mu}=(0,a,0,0)$ and
$a_{2}^{\mu}=(0,0,a,0)$ where $a$ is the amplitude of the four-vector
potential. They are orthogonals, which implies that $(a_{1}.a_{2})=0$ and
$a_{1}^{2}=a_{2}^{2}=a^{2}=-|\mathbf{a}|^{2}=-\big{(}\frac{\epsilon_{0}^{2}}{\omega}\big{)}$
with $\epsilon_{0}$ is the laser’s field strength.
The incoming electron and positron are electrically charged particle and its
antiparticle, respectively, with mass $m_{e}$. They are described by Dirac-
Volkov states [18] which satisfies the Dirac equation in an external field

$\Big{[}(p_{i}-eA)^{2}-m_{e}^{2}-\dfrac{ie}{2}F_{\mu\nu}\sigma^{\mu\nu}\Big{]}\psi_{p_{i},s_{i}}(x)=0$
(2)

Here, the indices $p_{i}(i=1,2)$ and $s_{i}(i=1,2)$ refer to the particle
momentum and its spin outside the electromagnetic field. Dressed by such a
laser field, the incoming electron and positron can be viewed as Dirac-Volkov
states [18] described by:

$\begin{cases}\psi_{p_{1},s_{1}}(x)=\Big{[}1-\dfrac{e\not{k}\not{A}}{2(k.p_{1})}\Big{]}\frac{u(p_{1},s_{1})}{\sqrt{2Q_{1}V}}\exp^{iS(q_{1},s_{1})}&\\\
\psi_{p_{2},s_{2}}(x)=\Big{[}1+\dfrac{e\not{k}\not{A}}{2(k.p_{2})}\Big{]}\frac{v(p_{2},s_{2})}{\sqrt{2Q_{2}V}}\exp^{iS(q_{2},s_{2})}\end{cases}$
(3)

where the quantities $S(q_{i},s_{i})(i=1,2)$ are defined as:

$\begin{cases}S(q_{1},s_{1})=-q_{1}x+\frac{e(a_{1}.p_{1})}{k.p_{1}}\sin\phi-\frac{e(a_{2}.p_{1})}{k.p_{1}}\cos\phi&\\\
S(q_{2},s_{2})=+q_{2}x+\frac{e(a_{1}.p_{2})}{k.p_{2}}\sin\phi-\frac{e(a_{2}.p_{2})}{k.p_{2}}\cos\phi\end{cases}$
(4)

$u(p_{1},s_{1})$ and $v(p_{2},s_{2})$ represent the bispinor of the electron
and positron, respectively, with a momentum $p_{i}(i=1,2)$ and spin
$s_{i}(i=1,2)$. They satisfy the following equation:
$\sum_{s}u(p_{1},s_{1})\bar{u}(p_{1},s_{1})=(\not{p}_{1}-m_{e})$ and
$\sum_{s}v(p_{2},s_{2})\bar{v}(p_{2},s_{2})=(\not{p}_{2}+m_{e})$
The four vector $q_{i}=(Q_{i},q_{i})(i=1,2)$ is the effective four vector
momentum of the electron and the positron inside laser field with:

$q_{i}=p_{i}+\dfrac{e^{2}a^{2}}{2(k.p_{i})}k$ (5)

Its zero component $q_{i}^{0}=Q_{i}$ is the effective energy appearing in the
normalization factor of the Volkov wave function. The effective mass can be
obtained as:

$m_{e}^{*^{2}}=q_{i}^{2}=(m_{e}^{2}+e^{2}a^{2})$ (6)

The scattered Higgs boson $H$ and $Z^{0}$-boson are massive particles with
spin $0$ and $1$, respectively. They are electrically neutral which means that
they will never interact with the laser field and they are described by the
following equations:

$Z^{\nu}(y)=\dfrac{\epsilon^{\nu}(p_{3},\lambda)}{\sqrt{2Q_{Z}V}}e^{-ip_{3}y}\hskip
14.22636pt;\hskip 14.22636pt\\\ \
\phi(y)=\dfrac{1}{\sqrt{2Q_{H}V}}e^{-ip_{4}y}$ (7)

Where $p_{3}$ and $p_{4}$ are the four vector momenta of the $Z^{0}$-boson and
Higgs boson, respectively. $Q_{Z}=p_{3}^{0}$ and $Q_{H}=p_{4}^{0}$.
$\epsilon^{\nu}(p_{3},\lambda)$ is the $Z^{0}$-boson polarisation vector and
$(\lambda)$ denotes its polarization.

### B- Scattering amplitude and cross section

The lowest scattering-matix element [19] for the laser assisted Higgs
strahlung process can be written as:

$S_{fi}({e}^{+}{e}^{-}\rightarrow
ZH)=\frac{gM_{Z}^{2}}{2\cos_{\theta_{W}}v}\int d^{4}x\int
d^{4}y\phi^{*}(y)Z^{*\nu}(y)D_{\mu\nu}(x-y)\bar{\psi}_{p_{2},s_{2}}(x)\Big{[}\gamma^{\nu}(g_{v}^{e}-g_{a}^{e}\gamma^{5})\Big{]}\psi_{p_{1},s_{1}}(x)$
(8)

where $M_{Z}$ is the scattered $Z$-boson mass,
$v=(\sqrt{2}G_{F})^{-\frac{1}{2}}$ is the vacuum expectation value,
$\theta_{W}$ is the Weinberg angle, $g_{v}^{e}=-1+4\sin^{2}\theta_{w}$ and
$g_{a}^{e}=1$ are the vector and axial vector coupling constants,
respectively. $g$ is the electroweak coupling constant which is related to the
electroweak mixing angle by
$g^{2}=\frac{e^{2}}{\sin^{2}\theta_{w}}=\frac{8G_{F}M_{Z}^{2}\cos^{2}_{\theta_{w}}}{\sqrt{2}}$
where $e=1eV$ and $G_{F}=1.1663787\times 10^{-5}GeV^{-2}$ are the electron’s
charge and the Fermi coupling constant, respectively.

$D_{\mu\nu}(x-y)$ is the $Z^{*}$-boson propagator [19]. It is given by:

$D_{\mu\nu}(x-y)=\int\dfrac{d^{4}q}{(2\pi)^{4}}\frac{e^{-iq(x-y)}}{q^{2}-M_{Z}^{2}}\Bigg{[}-ig_{\mu\nu}+i\dfrac{q_{\mu}q_{\nu}}{M_{Z}^{2}}\Bigg{]}$
(9)

where $q$ is the four vector momentum of the $Z^{*}$-boson propagator.

By inserting the equations (3), (7) and (9) into the scattering-matrix element
given by equation (8), it becomes:

$S_{fi}({e}^{+}{e}^{-}\rightarrow
HZ)=\frac{gM_{Z}^{2}}{2\cos_{\theta_{W}}v}\dfrac{1}{\sqrt{16V^{4}Q_{Z}Q_{H}Q_{1}Q_{2}}}\Bigg{[}\dfrac{1}{(q_{1}+q_{2})^{2}-M_{Z}^{2}}\Bigg{]}M_{fi}^{n}\,(2\pi)^{4}\,\\\
\delta^{4}\big{(}p_{3}+p_{4}-q_{1}-q_{2}-nk\big{)}$ (10)

Where $(n)$ is the number of photons that are exchanged with the charged
incident particles, with $(n)$ might be either positive or negative. $n<0$
refers to the emission of $n$ photons and $n>0$ refers to the absorption.
$M_{fi}^{n}$ is the amplitude of the Higgs strahlung process which can be
decomposed in terms of Bessel function series and it is given by:

$M_{fi}^{n}=\epsilon^{*\nu}(p_{3},\lambda)\bar{v}(p_{2},s_{2})\Lambda^{n}_{\mu}u(p_{1},s_{1})$
(11)

with $\Lambda^{n}_{\mu}$ is derived by using the generating function for the
Bessel functions [20]

$\sum_{n=-\infty}^{+\infty}J_{n}(z)e^{-in\phi}=e^{-iz\sin\phi}$ (12)

We find that:

$\Lambda_{\mu}^{n}=C^{0}_{\mu}\,\xi_{0n}(z)+C^{1}_{\mu}\,\xi_{1n}(z)+C^{2}_{\mu}\,\xi_{2n}(z)$
(13)

Where the quantities $C^{0}_{\mu}$, $C^{1}_{\mu}$ and $C^{2}_{\mu}$ are
explicitly expressed by:

$\begin{cases}C^{0}_{\mu}=\gamma_{\mu}(g_{v}^{e}-g_{a}^{e}\gamma^{5})+2c_{p_{1}}c_{p_{2}}a^{2}k_{\mu}\not{k}(g_{v}^{e}-g_{a}^{e}\gamma^{5})&\\\
C^{1}_{\mu}=c_{p_{1}}\gamma_{\mu}(g_{v}^{e}-g_{a}^{e}\gamma^{5})\not{k}\not{a}_{1}-c_{p_{2}}\not{a}_{1}\not{k}\gamma_{\mu}(g_{v}^{e}-g_{a}^{e}\gamma^{5})&\\\
C^{2}_{\mu}=c_{p_{1}}\gamma_{\mu}(g_{v}^{e}-g_{a}^{e}\gamma^{5})\not{k}\not{a}_{2}-c_{p_{2}}\not{a}_{2}\not{k}\gamma_{\mu}(g_{v}^{e}-g_{a}^{e}\gamma^{5})\end{cases}$
(14)

With $c_{p_{1}}=\dfrac{e}{2(k.p_{1})}$ and $c_{p_{2}}=\dfrac{e}{2(k.p_{2})}$
The quantities $\xi_{0n}(z)$, $\xi_{1n}(z)$ and $\xi_{2n}(z)$ are expressed in
terms of Bessel functions by:

$\left.\begin{cases}\xi_{0n}(z)\\\ \xi_{1n}(z)\\\
\xi_{2n}(z)\end{cases}\right\\}=\left.\begin{cases}J_{n}(z)e^{-in\phi_{0}}\\\
\frac{1}{2}\Big{(}J_{n+1}(z)e^{-i(n+1)\phi_{0}}+J_{n-1}(z)e^{-i(n-1)\phi_{0}}\Big{)}\\\
\frac{1}{2\,i}\Big{(}J_{n+1}(z)e^{-i(n+1)\phi_{0}}-J_{n-1}(z)e^{-i(n-1)\phi_{0}}\Big{)}\end{cases}\right\\}$
(15)

The argument of the Bessel function is expressed by:
$z=\sqrt{\alpha_{1}^{2}+\alpha_{2}^{2}}$ and
$\phi_{0}=\arctan(\frac{\alpha_{2}}{\alpha_{1}})$, where:

$\alpha_{1}=\dfrac{e(a_{1}.p_{1})}{(k.p_{1})}-\dfrac{e(a_{1}.p_{2})}{(k.p_{2})}$
$\alpha_{2}=\dfrac{e(a_{2}.p_{1})}{(k.p_{1})}-\dfrac{e(a_{2}.p_{2})}{(k.p_{2})}$

In the center of mass frame, the differential cross section is obtained by
dividing the square of the S-matrix element by $VT$ to obtain the transition
probability per volume, by
$|J_{inc}|=(\sqrt{(q_{1}q_{2})^{2}-m_{e}^{*^{4}}}/{Q_{1}Q_{2}V})$, and by the
particle density $\rho=V^{-1}$ and finally one has to average over the initial
spins and sum over the final ones. We obtain:

$\dfrac{d\sigma_{n}}{d\Omega}=\dfrac{g^{2}M_{Z}^{4}}{64\cos_{\theta_{W}}^{2}v^{2}}\Bigg{(}\dfrac{1}{(q_{1}+q_{2})^{2}-M_{Z}^{2}}\Bigg{)}^{2}\dfrac{1}{\sqrt{(q_{1}q_{2})^{2}-m_{e}^{*^{4}}}}\big{|}\overline{M_{fi}^{n}}\big{|}^{2}\int\dfrac{2|\mathbf{p}_{4}|^{2}d|\mathbf{p}_{4}|}{(2\pi)^{4}Q_{H}}\int\dfrac{d^{3}p_{3}}{Q_{Z}}\\\
\times\delta^{4}(p_{3}+p_{4}-q_{1}-q_{2}-nk)$ (16)

By integrating over $d^{3}p_{3}$, the differential cross section becomes:

$\dfrac{d\sigma_{n}}{d\Omega}=\dfrac{g^{2}M_{Z}^{4}}{64\cos_{\theta_{W}}^{2}v^{2}}\Bigg{(}\dfrac{1}{(q_{1}+q_{2})^{2}-M_{Z}^{2}}\Bigg{)}^{2}\dfrac{1}{\sqrt{(q_{1}q_{2})^{2}-m_{e}^{*^{4}}}}\big{|}\overline{M_{fi}^{n}}\big{|}^{2}\int\dfrac{2|\mathbf{p}_{4}|^{2}d|\mathbf{p}_{4}|}{(2\pi)^{2}Q_{H}}\delta(Q_{Z}+Q_{H}-Q_{1}-Q_{2}-n\omega)$
(17)

With the argument of the delta function in the formula (16) determines the law
of the momentum conservation:

$\mathbf{p_{3}}+\mathbf{p_{4}}-\mathbf{q_{1}}-\mathbf{q_{2}}-\mathbf{nk}=0$
(18)

The remaining integral in the differential cross section expression can be
performed by using the well known formula [19] given by:

$\int
d\mathbf{x}f(\mathbf{x})\delta(g(\mathbf{x}))=\dfrac{f(\mathbf{x})}{|g^{{}^{\prime}}(\mathbf{x})|_{g(\mathbf{x})=0}}$
(19)

thus, we get:

$\dfrac{d\sigma_{n}}{d\Omega}=\dfrac{g^{2}M_{Z}^{4}}{64\cos_{\theta_{W}}^{2}v^{2}}\Bigg{(}\dfrac{1}{(q_{1}+q_{2})^{2}-M_{Z}^{2}}\Bigg{)}^{2}\dfrac{1}{\sqrt{(q_{1}q_{2})^{2}-m_{e}^{*^{4}}}}\big{|}\overline{M_{fi}^{n}}\big{|}^{2}\dfrac{2|\mathbf{p_{4}}|^{2}}{(2\pi)^{2}Q_{H}}\dfrac{1}{\big{|}g^{{}^{\prime}}(|\mathbf{p}_{4}|)\big{|}_{g(|\mathbf{p}_{4}|)=0}}$
(20)

with

$g^{{}^{\prime}}(|\mathbf{p}_{4}|)=\dfrac{-4e^{2}a^{2}}{\sqrt{s}}\dfrac{|\mathbf{p}_{4}|}{\sqrt{|\mathbf{p}_{4}|^{2}+M_{H}^{2}}}-\dfrac{2|\mathbf{p}_{4}|(\sqrt{s}+n\omega)}{\sqrt{|\mathbf{p}_{4}|^{2}+M_{H}^{2}}}$
(21)

where $(\sqrt{s})$ is the center of mass energy.
The equation (20) indicates that the total differential cross section can be
decomposed into series of partial discrete differential cross section for
different number of exchanged photons.
The total cross section is obtained by performing a numerical integration over
the solid angle $d\Omega=\sin\theta d\theta d\phi$. The term
$\big{|}\overline{M_{fi}^{n}}\big{|}^{2}$ appearing in the cross section can
be evaluated as follows:

$\big{|}\overline{M_{fi}^{n}}\big{|}^{2}=\dfrac{1}{4}\sum_{n=-\infty}^{+\infty}\sum_{\lambda}\sum_{s}\big{|}M_{fi}^{n}\big{|}^{2}=\frac{1}{4}\sum_{n=-\infty}^{+\infty}\big{(}-g^{\mu\nu}+\dfrac{p_{3}^{\mu}p_{3}^{\nu}}{M_{Z}^{2}}\big{)}Tr\big{[}(\not{p}_{1}-m_{e})\Lambda^{n}_{\mu}(\not{p}_{2}+m_{e})\bar{\Lambda}^{n}_{\nu}\big{]}$
(22)

Where,
$\bar{\Lambda}_{\nu}^{n}=\bar{C^{0}_{\nu}}\,\xi_{0n}^{*}(z)+\bar{C^{1}_{\nu}}\,\xi_{1n}^{*}(z)+\bar{C^{2}_{\nu}}\,\xi_{2n}^{*}(z)$
and :

$\begin{cases}\bar{C^{0}_{\nu}}=\gamma_{\nu}(g_{v}^{e}-g_{a}^{e}\gamma^{5})+2c_{p_{1}}c_{p_{2}}a^{2}k_{\nu}\not{k}(g_{v}^{e}-g_{a}^{e}\gamma^{5})&\\\
\bar{C^{1}_{\nu}}=c_{p_{1}}\not{a}_{1}\not{k}\gamma_{\nu}(g_{v}^{e}-g_{a}^{e}\gamma^{5})-c_{p_{2}}\gamma_{\nu}(g_{v}^{e}-g_{a}^{e}\gamma^{5})\not{k}\not{a}_{1}&\\\
\bar{C^{2}_{\nu}}=c_{p_{1}}\not{a}_{2}\not{k}\gamma_{\nu}(g_{v}^{e}-g_{a}^{e}\gamma^{5})-c_{p_{2}}\gamma_{\nu}(g_{v}^{e}-g_{a}^{e}\gamma^{5})\not{k}\not{a}_{2}\end{cases}$
(23)

Because of its hardness, the trace calculation is performed by using FeynCalc
program. The result obtained is too long to be presented in this paper, but,
in general, it has the following form:

$\big{|}\overline{M_{fi}^{n}}\big{|}^{2}=\frac{1}{4}\sum_{n=-\infty}^{+\infty}\Big{[}AJ_{n}^{2}(z)+BJ_{n+1}^{2}(z)+CJ_{n-1}^{2}(z)+DJ_{n}(z)J_{n+1}(z)+EJ_{n}(z)J_{n-1}(z)+FJ_{n-1}(z)J_{n+1}(z)\Big{]}$
(24)

The coefficients A, B, C, D, E and F are calculated using FeynCalc tools. We
give here the expression of the first coefficient multiplied by
$J_{n}^{2}(z)$, the others are given in the appendix.

$\displaystyle A$ $\displaystyle=$
$\displaystyle\dfrac{2}{((k.p_{1})(k.p_{2})M_{Z}^{2})}\Big{[}(a^{4}e^{4}(g_{a}^{e^{2}}+g_{v}^{e^{2}})(k.p_{3})^{2}+2(k.p_{1})(k.p_{2})(M_{Z}^{2}(g_{a}^{e^{2}}(-3m_{e}^{2}+(p_{1}.p_{2}))$
$\displaystyle+$ $\displaystyle
g_{v}^{e^{2}}(3m_{e}^{2}+(p_{1}.p_{2})))+2(g_{a}^{e^{2}}+g_{v}^{e^{2}})(p_{1}.p_{3})(p_{2}.p_{3}))+2a^{2}e^{2}(g_{a}^{e^{2}}(-2(k.p_{1})(k.p_{2})M_{Z}^{2}$
$\displaystyle+$
$\displaystyle(k.p_{3})((k.p_{3})m_{e}^{2}-(k.p_{3})(p_{1}.p_{2})+(k.p_{2})(p_{1}.p_{3})+(k.p_{1})(p_{2}.p_{3})))+g_{v}^{e^{2}}(-2(k.p_{1})(k.p_{2})$
$\displaystyle\times$ $\displaystyle
M_{Z}^{2}(k.p_{3})(-(k.p_{3})(m_{e}^{2}+(p_{1}.p_{2}))+(k.p_{2})(p_{1}.p_{3})+(k.p_{1})(p_{2}.p_{3})))))\Big{]}$

We have checked the validity of the sum-rule [21] for multiphotonic Higgs
strahlung process, we find after calculation that:

$\sum_{n=-\infty}^{+\infty}\dfrac{d\sigma_{n}}{d\Omega}=\Big{(}\dfrac{d\sigma}{d\Omega}\Big{)}_{laser-
free}$ (26)

and

$\sum_{n=-\infty}^{+\infty}\sigma_{n}=\big{(}\sigma\big{)}_{laser-free}$ (27)

##  3\. Results and Discussion

In this section, we discuss the numerical results obtained for one of the most
fundamental process in the standard model in the presence of a circularly
polarized laser field. In the center of mass frame, we show the dependence of
the total cross section on the photon number transfer, the number of exchanged
photons, the laser field strength and its frequency. The numerical data and
curves presented in this section are obtained by using Mathematica. All curves
are plotted by taking the center of mass energy as $\sqrt{s}=250\,GeV$ except
in Fig.2 and Fig.6, eventhough we can use various other energy point such as
$240GeV$ to satisfy the CEPC concerns.

Figure 2: Comparaison between DJOUADI’s total cross section [1] and the total
cross section in the presence of the laser field for the Higgs strahlung
process $e^{+}e^{-}\rightarrow ZH$ as a function of the center of mass energy
$\sqrt{s}$. The strength of the laser field and the number of exchanged
photons are $\epsilon_{0}=0\,\,V.cm^{-1}$ and $n=0$, respectively.

Figure 2 shows the total cross section for the Higgs strahlung production in
the presence of a circularly polarized laser field in comparison with that
obtained by DJOUADI in [1]. The laser’s parameters are chosen as follows: the
laser strength ($\epsilon_{0}=0\,V.cm^{-1}$) and the number of exchanged
photons is ($n=0$). The aim of this comparison is to check that our results
converge to those obtained in the absence of the laser field by taking
$\epsilon_{0}=0\,\,V.cm^{-1}$ and $n=0$. According to the Fig.2, the two
curves are in full agreement for every center of mass energy $\sqrt{s}$. This
comparison makes as more comfortable not only about our analytical calculation
but also about the results that will be discussed bellow.

Figure 3: The total cross section of the Higgs strahlung process
$e^{+}e^{-}\rightarrow ZH$ in natural units as a function of the photons
energy transfer number for the center of mass energy of $(\sqrt{s}=250GeV)$.
The laser field strength and the laser frequency are chosen as
$\epsilon_{0}=10^{7}V.cm^{-1}$ and $\omega=1.17\,eV$, respectively. Figure 4:
The total cross section of the Higgs strahlung process $e^{+}e^{-}\rightarrow
ZH$ in natural units as a function of the photons energy transfer number for
the center of mass energy of $(\sqrt{s}=250GeV)$. The laser field strength and
the laser frequency are chosen as $\epsilon_{0}=10^{8}V.cm^{-1}$ and
$\omega=2\,eV$ , respectively. Figure 5: The total cross section of the Higgs
strahlung process $e^{+}e^{-}\rightarrow ZH$ in natural units as a function of
the photons energy transfer number for the center of mass energy of
$(\sqrt{s}=250GeV)$. The laser field strength and the laser frequency are
chosen as $\epsilon_{0}=10^{7}V.cm^{-1}$ and $\omega=2\,eV$, respectively.

Figures 3, 4 and 5 show the total cross section versus the net photon number
transferred between the colliding system ($e^{+}e^{-}$) and the circularly
polarized laser field. The strength and frequency of the laser field in Fig.3,
Fig.4 and Fig.5 are ($\epsilon_{0}=10^{7}\,\,V.cm^{-1}$ ; $\omega=1.17eV$),
($\epsilon_{0}=10^{8}\,\,V.cm^{-1}$ ; $\omega=2eV$) and
($\epsilon_{0}=10^{7}\,\,V.cm^{-1}$ ; $\omega=2eV$), respectively. The
magnitude of the total cross section varies in a range of a few orders for
different photon number $n$. A large number of photons are exchanged between
the laser field and the colliding system in all curves and the cutoff number
is about $n=\pm 1100$ and $n=\pm 3700$ and $n=\pm 400$ in Fig.3, Fig.4 and
Fig.5, respectively. The process of photon absorption and emission exhibits a
symmetric envelope scenario as shown in Fig.3, Fig.4 and Fig.5. Comparison
between Fig.3 and Fig.5 shows that, for the same laser intensity, the cutoff
number decreases as far as the laser frequency increases. Contrarily, for the
same value of laser frequency (Fig.4 and Fig.5), the required number of
exchanged photons to reach the sum-rule increases as far as the laser field
strength increases. This symmetry is due to one of the Bessel function
properties which is $J_{-n}(z)=(-1)^{n}J_{n}(z)$. The case of net zero-
exchanged photons $n=0$ does not correspond to the field-free case, for this
situation, the laser field modifies the angular range of the final-state
particles momenta and hence the total cross section. Having understood the
circularly polarized laser field effect on the Higgs strahlung cross section,
we now proceed to explore how the total cross section is influenced when the
laser field is embedded in the $e^{+}e^{-}$ collision. In order to get a
distinct picture of the pure effects of circularly polarized laser field, we
mainly discuss the dependence of the total cross section as a function of the
center of mass energy $\sqrt{s}$ for different number of exchanged photons.

Figure 6: The total cross section of the Higgs strahlung process
$e^{+}e^{-}\rightarrow ZH$ as a function of the center of mass energy
$\sqrt{s}$ for different number of exchanged photons.

Figure 6 shows the total cross section of the Higgs strahlung production in
the absence of an external field and the case of multiphoton process described
by equation (27) where the electron and the positron exchange a number of
photons with a circularly polarized laser field. The laser field strength is
chosen as $\epsilon_{0}=10^{7}\,V.cm^{-1}$ and its frequency is
$\omega=1.17\,eV$. In contrast to the linearly polarized laser field which
amplifies the cross section [6], the circularly polarized laser field
decreases the cross section by several order of magnitude by decreasing the
transition probability. For a number of exchanged photons $(n=\pm 300)$, the
total cross section reaches its maximum at $0.04\,pb$ which is much smaller
than the laser-free cross section. As far as the number of exchanged photons
increases, the total cross section converges to the laser-free cross section.
To reach this convergency, the physical system has to exchange, at least, a
number of photons of $(n=\pm 1100)$. This results is in full agreement with
the Fig.3 which determines the cutoff number as $(n=\pm 1100)$. So, the
multiphoton process cross-section can easily performed by summing over all
possible processes of absorption and emission of the photons. At electron
positron collider (CEPC for example), the Higgs bosons are produced through
Higgs strahlung and vector fusion process. In the Higgs strahlung process,
which is the leading production process at the center of mass energy of
$240\,GeV$-$250\,GeV$, the Higgs boson is produced in association with a
$Z$-boson. Above the threshold of $ZH$ $(\sqrt{s}=215\,GeV)$, the cross
section of Higgs strahlung increases rapidly and reaches its maximum at
$250\,GeV$, then it decreases asymptotically as $(1/s)$ like any typical
$s$-channel. Therefore, the vector boson fusion will become the dominant
contribution to the Higgs production far beyond the $ZH$ threshold
$(\sqrt{s}\geq 500\,GeV)$ [22, 23].
The quantity that measures the ability of a particle collider to produce the
required number of events (useful interactions) is called the luminosity and
it is defined as the proportionality factor between the number of events per
second $dR/dt$ and the total cross section such as: $dR/dt=L\times\sigma$
where $L$ and $\sigma$ are the luminosity of the collider and the total cross
section, respectively. The proposed high energy electron positron collider
CEPC anticipates integrated luminosity of $3\times 10^{34}cm^{-2}s^{-1}$,
$32\times 10^{34}cm^{-2}s^{-1}$ and $10\times 10^{34}cm^{-2}s^{-1}$ for center
of mass energy $240\,GeV$, $91\,GeV$ and $160\,GeV$, respectively. As we see
from Fig.6, the cross section diminishes as a function of the photon exchanged
number which means that the number of events (Higgs production) rises as far
as the number of exchanged photon between the laser field and the colliding
system increases. Thus, the expected number of events per second is
$0.729\times 10^{-2}$, $0.471\times 10^{-2}$ and $0.14\times 10^{-2}$ for the
number of exchanged photons $n=\pm 1100$ (which corresponds to laser-free
process), $n=\pm 900$ and $n=\pm 300$, respectively.

Figure 7: Higgs strahlung process $e^{+}e^{-}\rightarrow ZH$ cross section as
a function of the laser field strength for different exchanged photons number.
The center of mass energy and the laser frequency are taken as
$\sqrt{s}=250GeV$ and $\,\omega=0.117eV$, respectively Figure 8: Higgs
strahlung process $e^{+}e^{-}\rightarrow ZH$ cross section as a function of
the laser field strength for different exchanged photons number. The center of
mass energy and the laser frequency are taken as $\sqrt{s}=250GeV$ and
$\omega=1,17eV$, respectively. Figure 9: Higgs strahlung process
$e^{+}e^{-}\rightarrow ZH$ cross section as a function of the laser field
strength for different exchanged photons number. The center of mass energy and
the laser frequency are taken as $\sqrt{s}=250GeV$ $\omega=2eV$, respectively.

Figures 7, 8 and 9 displays the behavior of the total cross section of the
Higgs strahlung production in the center of mass frame as a function of the
circularly polarized laser field strength for different numbers of exchanged
photons and for different laser frequencies. In the range of small
intensities, the laser field has no effect at all on the total cross section,
regardless of the laser frequency and the number of exchanged photons. In
general, this range depends on the laser field frequency. In Fig.7, the laser
frequency is $\omega=0.117\,eV$ and the effect of laser begins at
$\epsilon_{0}=10^{3}\,V.cm^{-1}$ whereas in Fig.8 and Fig.9 the effect of the
laser field begins at $\epsilon_{0}=10^{5}\,V.cm^{-1}$ and
$\epsilon_{0}=10^{6}\,V.cm^{-1}$, respectively, for the number of exchanged
photons $n=\pm\,50$. The total cross section diminishes considerably as far as
the laser field strength increases, moreover, this decrease depends on the
number of exchanged photons and it is due to Bessel functions. For example,
the first term $(\sum_{n=-50}^{50}J_{n}^{2}(z)=1)$ in equation (24) is equal
to $1$ for the laser field strengths between $10$ and $10^{5}\,V.cm^{-1}$,
then it begins to decrease and it is equal to $0.313439$ for
$\epsilon_{0}=10^{6}\,V.cm^{-1}$. It is obvious that if we sum over the cutoff
number of each laser strength and frequency, the curves will not show any
dependence of the total cross section on the laser field.
At electron positron collider, one of the most used method to measure the
Higgs boson is through tagging the Higgs decay final states. This method
allows the number of Higgs events to be counted in several final states. Thus,
it allows us to measure the product of Higgs boson cross section and its decay
branching ratio to a given final states $\sigma(e^{+}e^{-}\rightarrow
ZH)\times BR(H\rightarrow X\bar{X})$, where $X$ is the possible finale state
and $\sigma(ZH)$ is the cross section of the Higgs strahlung process. Another
method is through the recoil mass to the associated $Z$ boson, especially if
the $Z$ boson decays to pair of electrons or muons. This recoil mass allow us
to better determine the Higgs cross section $\sigma(e^{+}e^{-}\rightarrow ZH)$
and the Higgs coupling $g(HZZ)$ in a model-independent manner [24]. Thus, we
have: $\sigma(e^{+}e^{-}\rightarrow ZH)\propto g_{HZZ}^{2}$,
$\sigma(e^{+}e^{-}\rightarrow ZH)\times BR(H\rightarrow X\bar{X})\propto
g_{HZZ}^{2}g_{HXX}^{2}/\Gamma_{H}$ and $\sigma(e^{+}e^{-}\rightarrow
H\nu_{e}\bar{\nu_{e}})\times BR(H\rightarrow X\bar{X})\propto
g_{HZZ}^{2}g_{HWW}^{2}/\Gamma_{H}$. With the recoil method, we can detect the
$125\,GeV$ Higgs boson without using its decay informations. In other words,
the clean environment and low cross section for background processes at
$e^{+}e^{-}$ collider, allow the Higgs strahlung events to be tagged based
only on the measurement of the $4$-vector momentum of the $Z$ boson regardless
of the Higgs final states in the Higgs decay process. So, we assume a common
modification to the Higgs coupling $HZZ$ and $HWW$ for that the circularly
polarized laser field reduces the Higgs strahlung cross section.

Figure 10: Variation of the total cross section of the Higgs strahlung process
$e^{+}e^{-}\rightarrow ZH$ as a function of the laser field strength
$(\epsilon_{0})$ for a Nd:YAG laser $(\omega=1.17eV)$ and a CO2
laser$(\omega=0.117eV)$. The center of mass energy and the number of exchanged
photons are taken as $(-100\leq n\leq 100)$ and $(\sqrt{s}=250GeV)$,
respectively.

Figure 10 illustrates the effect of laser frequency on Higgs strahlung
production where the colliding electron and positron exchanges a number of
photons $(-100\leq n\leq 100)$. We consider two different laser frequencies
which are the Nd:YAG laser $(\omega=1.17eV)$ and the CO2 laser
$(\omega=0.117eV)$. Regardless of the laser frequency, each laser field has a
precise range of laser field strengths in which no effect on the total cross
section is observed. These ranges are $0\leq\epsilon_{0}\leq 10^{3}\,\,V/cm$
and $0\leq\epsilon_{0}\leq 10^{5}\,\,V/cm$ for the CO2 laser and the Nd:YAG
laser, respectively. According to Fig.10, as far as the laser frequency
increases, the effect of the laser field on the total cross section
diminishes. Apparently, the total cross section is reduced by the same order
of magnitude by which the field strength increases.

Figure 11: dependence of the Higgs strahlung process’s total cross section as
a function of the laser field frequency for $(\epsilon_{0}=10^{7}V/cm)$. The
number of exchanged photons and the center of mass energy are taken as
$(-100\leq n\leq 100)$ and $(\sqrt{s}=250GeV)$, respectively.

Figure 11 shows the variation of the Higgs starhlung multiphon process cross
section as a function of the laser frequency. The center of mass energy and
the laser field strength are chosen as $(\sqrt{s}=250GeV)$ and
$\epsilon_{0}=10^{7}V/cm$, respectively. We have chosen a small number of
exchanged photons ($(-100\leq n\leq 100)$) to avoid intensive computing
calculations that take a long time. According to Fig.11, it is clear that at
low frequencies, the laser beam does not affects the total cross section. In
addition, as the laser frequency overcomes the threshold $(\omega=0.2\,eV)$,
the total cross section rises exponentially as much as the laser frequency
increases.
The Higgs width measurement is an essential ingredient to determine its
partial width and coupling constant. The natural width of a $125\,GeV$ Higgs
boson, as predicted in the standard model, is only $4.2\,MeV$. As the best
precision achieved with current LHC data is $10\,MeV$, the Higgs boson width
is very small to be measured experimentally at LHC. The introduction of the
laser field in the future $e^{+}e^{-}$ high luminosity collider could have a
significant impact on the measurements.

##  4\. Conclusion

In the present paper, we investigated, in the framework of the electroweak
interaction, the laser-assisted Higgs stahlung process. We showed how the
total cross section of the Higgs strahlung production is reduced by the
circularly polarized laser field by several order of magnitude. The reduction
on the cross section depends on the centre of mass energy of the colliding
particles and on the laser’s parameters such as the electric field strength
$(\epsilon_{0})$ and its frequency $(\omega)$. The dependence of the total
cross section of the multiphoton Higgs strahlung process on the number of
exchanged photons is presented in details. Moreover, The variation of the
total cross section as a function of the electric field amplitude is shown for
the Nd:YAG laser $(\omega=1.17eV)$ and the CO2 laser $(\omega=0.117eV)$.

##  5\. Appendix

In this appendix, we give the remaining coefficients of the equation (24):

$\displaystyle B$ $\displaystyle=$
$\displaystyle-\dfrac{e^{2}}{2((k.p_{1})(k.p_{2})M_{Z}^{2})}\Big{[}(2(k.p_{1})(k.p_{2})(2(a_{1}.p_{3})^{2}(g_{a}^{2}+g_{v}^{2})(k.p_{1})(k.p_{2})+2(a_{2}.p_{3})^{2}(g_{a}^{2}+g_{v}^{2})$
$\displaystyle\times$
$\displaystyle(k.p_{1})(k.p_{2})-2(a_{1}.p_{3})(g_{a}^{2}+g_{v}^{2})((a_{1}.p_{2})(k.p_{1})+(a_{1}.p_{1})(k.p_{2}))(k.p_{3})+2(a_{1}.p_{1})(a_{1}.p_{2})g_{a}^{2}$
$\displaystyle\times$
$\displaystyle(k.p_{3})^{2}+2(a_{1}.p_{1})(a_{1}.p_{2})g_{v}^{2}(k.p_{3})^{2}+2a^{2}g_{a}^{2}(k.p_{3})^{2}m_{e}^{2}-2a^{2}g_{v}^{2}(k.p_{3})^{2}m_{e}^{2}+a^{2}g_{a}^{2}(k.p_{1})^{2}M_{Z}^{2}$
$\displaystyle+$ $\displaystyle
a^{2}g_{v}^{2}(k.p_{1})^{2}M_{Z}^{2}-2a^{2}g_{a}^{2}(k.p_{1})(k.p_{2})M_{Z}^{2}-2a^{2}g_{v}^{2}(k.p_{1})(k.p_{2})M_{Z}^{2}+a^{2}g_{a}^{2}(k.p_{2})^{2}M_{Z}^{2}+a^{2}g_{v}^{2}$
$\displaystyle\times$
$\displaystyle(k.p_{2})^{2}M_{Z}^{2}-2a^{2}g_{a}^{2}(k.p_{3})^{2}(p_{1}.p_{2})-2a^{2}g_{v}^{2}(k.p_{3})^{2}(p_{1}.p_{2})+2a^{2}g_{a}^{2}(k.p_{1})(k.p_{3})(p_{1}.p_{3})+2a^{2}$
$\displaystyle\times$ $\displaystyle
g_{v}^{2}(k.p_{1})(k.p_{3})(p_{1}.p_{3})+2a^{2}g_{a}^{2}(k.p_{2})(k.p_{3})(p_{1}.p_{3})+2a^{2}g_{v}^{2}(k.p_{2})(k.p_{3})(p_{1}.p_{3})+2a^{2}(g_{a}^{2}+g_{v}^{2})$
$\displaystyle\times$
$\displaystyle((k.p_{1})+(k.p_{2}))(k.p_{3})(p_{2}.p_{3}))-g_{a}g_{v}(k.p_{2})(-2(3(k.p_{1})-(k.p_{2}))((k.p_{1})+(k.p_{2}))M_{Z}^{2}$
$\displaystyle+$
$\displaystyle((k.p_{1})+3(k.p_{2}))(k.p_{3})(p_{2}.p_{3}))\epsilon(a_{1},a_{2},k,p_{1})+g_{a}g_{v}(k.p_{1})(2((k.p_{1})-3(k.p_{2}))((k.p_{1})$
$\displaystyle+$
$\displaystyle(k.p_{2}))M_{Z}^{2}+(3(k.p_{1})+(k.p_{2}))(k.p_{3})(p_{1}.p_{3}))\epsilon(a_{1},a_{2},k,p_{2})+g_{a}g_{v}(((k.p_{1})-(k.p_{2}))((k.p_{1})$
$\displaystyle\times$
$\displaystyle(k.p_{2}))(k.p_{3})(p_{1}.p_{2})\epsilon(a_{1},a_{2},k,p_{3})+3((k.p_{1})-(k.p_{2}))^{2}(k.p_{3})^{2}\epsilon(a_{1},a_{2},p_{1},p_{2})+(k.p_{1})^{2}$
$\displaystyle\times$
$\displaystyle(k.p_{2})(k.p_{3})\epsilon(a_{1},a_{2},p_{1},p_{3})+5(k.p_{1})(k.p_{2})^{2}(k.p_{3})\epsilon(a_{1},a_{2},p_{1},p_{3})-2(k.p_{2})^{3}(k.p_{3})$
$\displaystyle\times$
$\displaystyle\epsilon(a_{1},a_{2},p_{1},p_{3})2(k.p_{1})^{3}(k.p_{3})\epsilon(a_{1},a_{2},p_{2},p_{3})-5(k.p_{1})^{2}(k.p_{2})(k.p_{3})\epsilon(a_{1},a_{2},p_{2},p_{3})-(k.p_{1})$
$\displaystyle\times$
$\displaystyle(k.p_{2})^{2}(k.p_{3})\epsilon(a_{1},a_{2},p_{2},p_{3})+4(a_{2}.p_{3})(k.p_{1})(k.p_{2})(k.p_{3})\epsilon(a_{1},k,p_{1},p_{2})-4(a_{2}.p_{3})(k.p_{1})^{2}$
$\displaystyle\times$
$\displaystyle(k.p_{2})\epsilon(a_{1},k,p_{1},p_{3})4(a_{2}.p_{3})(k.p_{1})(k.p_{2})^{2}\epsilon(a_{1},k,p_{1},p_{3})+4(a_{2}.p_{3})(k.p_{1})^{2}(k.p_{2})\epsilon(a_{1},k,p_{2},p_{3})$
$\displaystyle+$ $\displaystyle
4(a_{2}.p_{3})(k.p_{1})(k.p_{2})^{2}\epsilon(a_{1},k,p_{2},p_{3})-4(a_{1}.p_{3})(k.p_{1})(k.p_{2})(k.p_{3})\epsilon(a_{2},k,p_{1},p_{2})+4(a_{1}.p_{3})$
$\displaystyle\times$
$\displaystyle(k.p_{1})^{2}(k.p_{2})\epsilon(a_{2},k,p_{1},p_{3})4(a_{1}.p_{3})(k.p_{1})(k.p_{2})^{2}\epsilon(a_{2},k,p_{1},p_{3})-(a_{1}.p_{2})(k.p_{1})(k.p_{2})$
$\displaystyle\times$
$\displaystyle(k.p_{3})\epsilon(a_{2},k,p_{1},p_{3})+(a_{1}.p_{2})(k.p_{2})^{2}(k.p_{3})\epsilon(a_{2},k,p_{1},p_{3})-(k.p_{1})(4(a_{1}.p_{3})(k.p_{2})((k.p_{1})$
$\displaystyle+$
$\displaystyle(k.p_{2}))+(a_{1}.p_{1})((k.p_{1})(k.p_{2}))(k.p_{3}))\epsilon(a_{2},k,p_{2},p_{3})))\Big{]}$
$\displaystyle C$ $\displaystyle=$
$\displaystyle-\dfrac{e^{2}}{2((k.p_{1})(k.p_{2})M_{Z}^{2})}\Big{[}(2(k.p_{1})(k.p_{2})(2(a_{1}.p_{3})^{2}(g_{a}^{2}+g_{v}^{2})(k.p_{1})(k.p_{2})+2(a_{2}.p_{3})^{2}(g_{a}^{2}+g_{v}^{2})$
(29) $\displaystyle\times$
$\displaystyle(k.p_{1})(k.p_{2})-2(a_{1}.p_{3})(g_{a}^{2}+g_{v}^{2})((a_{1}.p_{2})(k.p_{1})+(a_{1}.p_{1})(k.p_{2}))(k.p_{3})+2(a_{1}.p_{1})(a_{1}.p_{2})g_{a}^{2}$
$\displaystyle\times$
$\displaystyle(k.p_{3})^{2}+2(a_{1}.p_{1})(a_{1}.p_{2})g_{v}^{2}(k.p_{3})^{2}+2a^{2}g_{a}^{2}(k.p_{3})^{2}m_{e}^{2}-2a^{2}g_{v}^{2}(k.p_{3})^{2}m_{e}^{2}+a^{2}g_{a}^{2}(k.p_{1})^{2}M_{Z}^{2}a^{2}$
$\displaystyle\times$ $\displaystyle
g_{v}^{2}(k.p_{1})^{2}M_{Z}^{2}-2a^{2}g_{a}^{2}(k.p_{1})(k.p_{2})M_{Z}^{2}-2a^{2}g_{v}^{2}(k.p_{1})(k.p_{2})M_{Z}^{2}+a^{2}g_{a}^{2}(k.p_{2})^{2}M_{Z}^{2}+a^{2}g_{v}^{2}$
$\displaystyle\times$
$\displaystyle(k.p_{2})^{2}M_{Z}^{2}-2a^{2}g_{a}^{2}(k.p_{3})^{2}(p_{1}.p_{2})-2a^{2}g_{v}^{2}(k.p_{3})^{2}(p_{1}.p_{2})+2a^{2}g_{a}^{2}(k.p_{1})(k.p_{3})(p_{1}.p_{3})+2a^{2}$
$\displaystyle\times$ $\displaystyle
g_{v}^{2}(k.p_{1})(k.p_{3})(p_{1}.p_{3})+2a^{2}g_{a}^{2}(k.p_{2})(k.p_{3})(p_{1}.p_{3})+2a^{2}g_{v}^{2}(k.p_{2})(k.p_{3})(p_{1}.p_{3})+2a^{2}(g_{a}^{2}+g_{v}^{2})$
$\displaystyle\times$
$\displaystyle((k.p_{1})(k.p_{2}))(k.p_{3})(p_{2}.p_{3}))+g_{a}g_{v}(k.p_{2})(-2(3(k.p_{1})-(k.p_{2}))((k.p_{1})+(k.p_{2}))M_{Z}^{2}$
$\displaystyle+$
$\displaystyle((k.p_{1})+3(k.p_{2}))(k.p_{3})(p_{2}.p_{3}))\epsilon(a_{1},a_{2},k,p_{1})-g_{a}g_{v}(k.p_{1})(2((k.p_{1})-3(k.p_{2}))((k.p_{1})$
$\displaystyle+$
$\displaystyle(k.p_{2}))M_{Z}^{2}+(3(k.p_{1})(k.p_{2}))(k.p_{3})(p_{1}.p_{3}))\epsilon(a_{1},a_{2},k,p_{2})+g_{a}g_{v}((-(k.p_{1})^{2}+(k.p_{2})^{2})(k.p_{3})$
$\displaystyle\times$
$\displaystyle(p_{1}.p_{2})\epsilon(a_{1},a_{2},k,p_{3})3((k.p_{1})-(k.p_{2}))^{2}(k.p_{3})^{2}\epsilon(a_{1},a_{2},p_{1},p_{2})-(k.p_{1})^{2}(k.p_{2})(k.p_{3})$
$\displaystyle\times$
$\displaystyle\epsilon(a_{1},a_{2},p_{1},p_{3})-5(k.p_{1})(k.p_{2})^{2}(k.p_{3})\epsilon(a_{1},a_{2},p_{1},p_{3})+2(k.p_{2})^{3}(k.p_{3})\epsilon(a_{1},a_{2},p_{1},p_{3})-2$
$\displaystyle\times$
$\displaystyle(k.p_{1})^{3}(k.p_{3})\epsilon(a_{1},a_{2},p_{2},p_{3})+5(k.p_{1})^{2}(k.p_{2})(k.p_{3})\epsilon(a_{1},a_{2},p_{2},p_{3})+(k.p_{1})(k.p_{2})^{2}(k.p_{3})$
$\displaystyle\times$
$\displaystyle\epsilon(a_{1},a_{2},p_{2},p_{3})-4(a_{2}.p_{3})(k.p_{1})(k.p_{2})(k.p_{3})\epsilon(a_{1},k,p_{1},p_{2})+4(a_{2}.p_{3})(k.p_{1})^{2}(k.p_{2})$
$\displaystyle\times$
$\displaystyle\epsilon(a_{1},k,p_{1},p_{3})+4(a_{2}.p_{3})(k.p_{1})(k.p_{2})^{2}\epsilon(a_{1},k,p_{1},p_{3})4(a_{2}.p_{3})(k.p_{1})^{2}(k.p_{2})\epsilon(a_{1},k,p_{2},p_{3})$
$\displaystyle-$ $\displaystyle
4(a_{2}.p_{3})(k.p_{1})(k.p_{2})^{2}\epsilon(a_{1},k,p_{2},p_{3})+4(a_{1}.p_{3})(k.p_{1})(k.p_{2})(k.p_{3})\epsilon(a_{2},k,p_{1},p_{2})-4(a_{1}.p_{3})$
$\displaystyle\times$
$\displaystyle(k.p_{1})^{2}(k.p_{2})\epsilon(a_{2},k,p_{1},p_{3})-4(a_{1}.p_{3})(k.p_{1})(k.p_{2})^{2}\epsilon(a_{2},k,p_{1},p_{3})+(a_{1}.p_{2})(k.p_{1})(k.p_{2})(k.p_{3})$
$\displaystyle\times$
$\displaystyle\epsilon(a_{2},k,p_{1},p_{3})-(a_{1}.p_{2})(k.p_{2})^{2}(k.p_{3})\epsilon(a_{2},k,p_{1},p_{3})(k.p_{1})(4(a_{1}.p_{3})(k.p_{2})((k.p_{1})+(k.p_{2}))$
$\displaystyle+$
$\displaystyle(a_{1}.p_{1})((k.p_{1})-(k.p_{2}))(k.p_{3}))\epsilon(a_{2},k,p_{2},p_{3})))\Big{]}$
$\displaystyle D$ $\displaystyle=$
$\displaystyle\dfrac{2\,e}{((k.p_{1})(k.p_{2})M_{Z}^{2})}\Big{[}(g_{a}^{2}+g_{v}^{2})(a^{2}e^{2}(k.p_{3})((a_{1}.p_{3})((k.p_{1})-(k.p_{2}))+(-(a_{1}.p_{1})+(a_{1}.p_{2}))$
(30) $\displaystyle\times$
$\displaystyle(k.p_{3}))+(a_{1}.p_{2})(k.p_{1})(-((k.p_{1})+(k.p_{2}))M_{Z}^{2}-2(k.p_{3})(p_{1}.p_{3}))+(k.p_{2})((a_{1}.p_{1})((k.p_{1})$
$\displaystyle+$
$\displaystyle(k.p_{2}))M_{Z}^{2}+2(a_{1}.p_{3})(k.p_{1})((p_{1}.p_{3})-(p_{2}.p_{3}))+2(a_{1}.p_{1})(k.p_{3})(p_{2}.p_{3})))\Big{]}$
$\displaystyle E$ $\displaystyle=$
$\displaystyle\dfrac{2\,e}{((k.p_{1})(k.p_{2})M_{Z}^{2})}\Big{[}((g_{a}^{2}+g_{v}^{2})(k.p_{1})(k.p_{2})(a^{2}e^{2}(k.p_{3})((a_{1}.p_{3})((k.p_{1})-(k.p_{2}))+(-(a_{1}.p_{1})$
(31) $\displaystyle+$
$\displaystyle(a_{1}.p_{2}))(k.p_{3}))+(a_{1}.p_{2})(k.p_{1})(-((k.p_{1})+(k.p_{2}))M_{Z}^{2}-2(k.p_{3})(p_{1}.p_{3}))+(k.p_{2})((a_{1}.p_{1})$
$\displaystyle\times$
$\displaystyle((k.p_{1})+(k.p_{2}))M_{Z}^{2}+2(a_{1}.p_{3})(k.p_{1})((p_{1}.p_{3})-(p_{2}.p_{3}))+2(a_{1}.p_{1})(k.p_{3})(p_{2}.p_{3})))+g_{a}g_{v}$
$\displaystyle\times$
$\displaystyle(((k.p_{1})-(k.p_{2}))(a^{2}e^{2}(k.p_{3})^{2}-4(k.p_{1})(k.p_{2})M_{Z}^{2})\epsilon(a_{2},k,p_{1},p_{2})+(k.p_{2})((k.p_{1})+(k.p_{2}))$
$\displaystyle\times$
$\displaystyle(a^{2}e^{2}(k.p_{3})+2(k.p_{1})(p_{2}.p_{3}))\epsilon(a_{2},k,p_{1},p_{3})+(k.p_{1})(((k.p_{1})+(k.p_{2}))(a^{2}e^{2}(k.p_{3})+2(k.p_{2})$
$\displaystyle\times$
$\displaystyle(p_{1}.p_{3}))\epsilon(a_{2},k,p_{2},p_{3})+2(k.p_{2})(-(k.p_{1})+(k.p_{2}))(-(k.p_{3})\epsilon(a_{2},p_{1},p_{2},p_{3})$
$\displaystyle+$
$\displaystyle(a_{2}.p_{3})\epsilon(k,p_{1},p_{2},p_{3})))))\Big{]}$
$\displaystyle F$ $\displaystyle=$
$\displaystyle-\dfrac{4\,e^{2}}{2((k.p_{1})(k.p_{2})M_{Z}^{2})}\Big{[}(g_{a}^{2}+g_{v}^{2})(((a_{1}.p_{3})-(a_{2}.p_{3}))((a_{1}.p_{3})+(a_{2}.p_{3}))(k.p_{1})$
(32) $\displaystyle\times$
$\displaystyle(k.p_{2})-(a_{1}.p_{3})((a_{1}.p_{2})(k.p_{1})+(a_{1}.p_{1})(k.p_{2}))(k.p_{3})+(a_{1}.p_{1})(a_{1}.p_{2})(k.p_{3})^{2}))\Big{]}$

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