#  __
Laser-assisted kaon decay and CPT symmetry violation

M Baouahi1, M Ouali1, M Jakha1, S Mouslih2, Y Attaourti3, B Manaut1, S Taj1,
1 USMS, Polydisciplinary Faculty, ERPTM, Beni Mellal, 23000, Morocco.
2 FST-BM, Laboratory of Materials Physics (LMP), 23000, Morocco.
3 High Energy Physics and Astrophysics Laboratory, FSSM, UCAM, Morocco.
Corresponding author, E-mail<EMAIL_ADDRESS>
###### Abstract

In this paper, we have investigated the charged kaons decay at the lowest
order in the presence of a circularly polarized laser field. To be more
precis, we have examined the leptonic decay of both positive (matter) and
negative (antimatter) kaon which weakly decay via the exchange of $W$ boson.
Indeed, we have derived the expression of the leptonic decay width, the
leptonic branching ratio, the leptonic ratio and the charged kaon lifetime by
using the decay matrix approach. In addition, by using numerical computation,
we have presented and discussed how the laser field influences these physical
quantities. Moreover, we have analyzed the effect of the laser field on the
parameter associated to the CPT symmetry. Then, we have concluded that, in the
presence of an electromagnetic field and based on this CPT symmetry parameter,
it is possible to control the dominance of matter over anti-matter or vice-
virsa by applying an external field to either violate or conserve the CPT
symmetry.

Keywords: QED and Weak processes, Laser-assisted, CPT symmetry.

##  1\. Introduction

The great progress in laser technology since its introduction in 1960,
especially by increasing its intensities and shortening its pulse duration,
have received a great scientific interest[1, 3, 4, 5, 6, 2, 7]. This interest
is due to many previously unknown phenomena that are caused by the application
of the laser radiation, allowing us to understand not only the atomic and
molecular structure of matter but also phenomena related to high energy
processes and particles’ behaviors[5, 6]. Moreover, laser field can pave a new
ways to high energy physics. Therefore, the study of various aspects of the
influence of the electromagnetic radiation on high energy scattering and decay
processes is one of the most important issues of modern physics.

The study of quantum effects in laser-matter interactions may allow us to
obtain new information about the nature of the interactions of the particles
involved in the scattering and decay processes. Moreover, the dependence of
these processes on laser field’s parameters that can be easily controlled in
experiments may lead to the detection of new effects that do not exist in the
case of laser-free interactions between particles. In some processes, the
electromagnetic field is a determining element and without it they do not
occur (see for example [1, 2]), while in other processes that occur in the
absence of the laser field, they are modified by its presence [3, 4, 5, 6].

The precise study of rare processes where kaons play a predominant role since
their discovery in 1947 [8] gives us today the light to understand the
structure of the weak interaction. In particle physics, CP symmetry violation
has been one of the most sensitive issues since its discovery in 1964 in
neutral kaon decays [9] and later in B mesons[10, 11]. Therefore, it is
expected that the weak interaction violating CP symmetry also violates the
invariance of time inversion T which was found in $B$ mesons when exchanging
initial and final states in transitions which can be connected by a T symmetry
transformation [12]. In this context, an elegant explanation of this effect
was proposed by Kobayashi and Maskawa, as a CP violation phase in the Cabibbo-
Kobayashi-Maskawa (CKM) three-generation quark mixing matrix[13] in the
Standard Model (SM). So with this phenomena related to the CP symmetry
characterizing this interaction [9], the question arises as to whether there
exists an effect capable of controlling unique properties of an interaction
(weak interaction) by another interaction (electromagnetic interaction), and
what are the tangible variations of the measurable quantities characterizing
the interacting particles. For that, the process of the charged kaons decay
into two leptonic bodies, which is recognized in the Standard Model by the
exchange of a bosonic propagator $W$ to give two leptons, is developed in this
work. The laser-assisted decay process is the preferred choice for charged
particles not only because of the fact that the electromagnetic field
interacts only with charged particles but also for that this decay process may
give us the possibility to extend the particles’ lifetime. We note that at
high laser field’s intensites where the Schwinger limit is respected
$\epsilon_{0}=(m^{2}c^{3})/(e\hbar)=1.32\times 10^{16}V/cm$ [14, 15], the two
leptonic branching ratios are affected in such a way that they will be of the
same order.

CP violation allows the two partial decay widths of the particle and
antiparticle to be different even if CPT symmetry is valid. In addition, CPT
symmetry also imposes equality of lifetimes for CP-conjugate states. In the
case where the produced particles interact only via weak interaction or if the
final state interactions does not involve transitions between different decay
channels, CPT invariance is sufficient to ensure equality of the partial
rates. Therefore, the comparison of leptonic decay widths of the charged kaon
is a test of CPT invariance [16, 17, 18]. For this reason, we will use the
asymmetry parameter defined in the theoretical part.

In this work, the laser-assisted charged kaon decay process is studied without
taking into account the radiative corrections and by considering the kaon as a
structureless particle. Moreover, the external field, which is considered to
be monochromatic and circularly polarized, will influence all charged
particles of the initial and final states and exchanges multiphotons [19].
This effect of the laser field is examined through the theoretical measurement
of measurable quantities such as that associated with the CPT symmetry, the
partial decay width, the leptonic ratio, the leptonic branching ratio and the
lifetime of the charged kaons. We consider, for the kaon decay process without
laser field, that matter and antimatter have the same mass, and also their
found decay widths are equal. This equality ensures the conservation of CPT
symmetry at the lowest order.

The remainder of this research paper is organized as follows: The section (2.)
is about the theoretical analysis of some quantities which characterize the
two body leptonic charged kaons decay process at the lowest order in the
absence and in the presence of a circularly polarized electromagnetic field.
The obtained results and data are presented and analyzed in section (3.). In
section (4.), we have given a short conclusion. Some calculation’s details of
the charged kaon decay width are given in the appendix. We mention that we
have used natural units such that $\hbar=c=1$, and the signature metric is
taken as $(+---)$.

##  2\. Outline of theory

This paper’s part is devoted to the analytical calculation of the charged kaon
partial decay width as it is one of the most important measured quantities in
decay processes. It is well known that the charged kaon may decays via several
channels [20]. However, these decays modes are different in terms of their
decay rates. We will focus on the leptonic decay modes at the lowest order
which are $K^{+}\rightarrow l^{+}+\nu_{l}$ and $K^{-}\rightarrow
l^{-}+\overline{\nu}_{l}$. Indeed, we begin by giving the expression of the
laser-free decay width then we will move on to derive its expression in the
presence of an electromagnetic field.

###  2.1. Laser-free kaon decay

In general, the leptonic tow-body decay width of a charged mesons $P^{c}$,
such as $\pi^{\pm}$, $K^{\pm}$, $Ds^{\pm}$ and $B^{\pm}$ , in the absence of
the electromagnetic field is given by:

$\displaystyle\Gamma$ $\displaystyle=$
$\displaystyle\dfrac{G_{F}^{2}F_{P^{c}}^{2}}{4\pi}m_{1}m_{l}^{2}(1-\dfrac{m_{l}^{2}}{m_{1}^{2}})^{2},$
(1)

where $G_{F}$ and $F_{P^{c}}$ are successively the Fermi constant and the
charged meson’s structure constant. $m_{1}$ is the mass of the charged mesons
$P^{c}$. For the charged kaon decay, we have $m_{1}=m_{K^{\pm}}=493.677\,MeV$.
We notice that the mass of neutrinos is taken to be zero. $m_{l}$ is the mass
of the produced charged lepton such that:
$m_{l}=m_{\mu^{\pm}}=105,6583745\,MeV$ for the muon and the antimuon, or
$m_{l}=m_{e^{\pm}}=0.5109989\,MeV$ for the electron and the positron.

###  2.2. Laser-assisted kaon decay

The leptonic decay of the charged kaon in the presence of an external
electromagnetic field can be described as follows:

$\displaystyle K^{\pm}(q_{1})\rightarrow l^{\pm}(q_{2})+\nu_{l_{i}}(P_{3}),$
(2)

where $l$ can be either an electron or a muon, and $\nu_{l_{i}}$ represents
$\nu_{l}$ for $K^{+}$ decay and $\bar{\nu_{l}}$ for $K^{-}$ decay. We consider
that the charged kaon is initially represented by the 4-vector momentum
$P_{1}=(E_{1},\vec{0})$, but when it is dressed by the laser field, its
4-vector momentum becomes the effective 4-vector momentum
$q_{1}=P_{1}-(e^{2}a^{2})/(2k.P_{1})k$, where $k=(\omega,\vec{k})$, $e$ and
$a$ are respectively the electromagnetic wave 4-vector, the charge of the kaon
and the polarization of the laser field.
$q_{2}=P_{2}-(e^{2}a^{2})/(2k.P_{2})k$ is the effective 4-vector momentum of
the lepton $l^{\pm}$, and $P_{3}=(E_{3},\vec{P_{3}})$ is the 4-vector momentum
of $\nu_{l_{i}}$. The classical four-vector potential which represents the
circularly polarized monochromatic laser field is given by:

$\displaystyle A^{\mu}(\phi)=a_{1}^{\mu}\cos\phi+a_{2}^{\mu}\sin\phi,$ (3)

with $\phi=(k.x)$ is the phase of the laser field. The 4-vectors $a_{1}$ and
$a_{2}$ are orthogonal, and chosen such that $a_{1}^{\mu}=|a|(0,1,0,0)$ and
$a_{2}^{\mu}=|a|(0,0,1,0)$. Thus, they verify the following conditions:
$a_{1}.a_{2}=a_{2}.a_{1}=0$ and
$a_{1}^{2}=a_{2}^{2}=a^{2}=-|a|^{2}=-(\epsilon_{0}/\omega)^{2}$, with
$\epsilon_{0}$ is the amplitude of the laser’s electric field. The produced
charged lepton inside the laser field is described by the analytical solution
of Dirac equation (Dirac-Volkov wave function)[21] such that:

$\displaystyle\psi_{l^{\pm}}(x)=\left[1+\dfrac{e\not{k}\not{A}}{2k.P_{2}}\right]\dfrac{DS(P_{2},s_{2})}{\sqrt{2Q_{2}V}}e^{\mp
iS(q_{2},x)},$ (4)

$V$ is the quantization volume. In general, $DS(P_{i},s_{i})$ represents the
Dirac bispinor which can be either $u(P_{i},s_{i})$ for lepton $l^{-}$ or
$v(P_{i},s_{i})$ for antilepton $l^{+}$. It verifies the following relations:

$\displaystyle\sum_{s_{i}}u(P_{i},s_{i})\bar{u}(P_{i},s_{i})=\not{P_{i}}+m_{l}\qquad\text{and}\qquad\sum_{s_{i}}v(P_{i},s_{i})\bar{v}(P_{i},s_{i})=\not{P_{i}}-m_{l},$
(5)

where $s_{i}$ and $m_{l}$ denotes successively the particle’s spin and its
mass. The classical action of the produced charged lepton in the plane
electromagnetic wave (phase of the Dirac-Volkov state) is given by the
following expression:

$\displaystyle
S(q_{2},x)=q_{2}.x-e\dfrac{a_{1}.P_{2}}{k.P_{2}}\sin\phi+e\dfrac{a_{2}.P_{2}}{k.P_{2}}\cos\phi,$
(6)

The square of the effective four-vector momentum $q_{2}=(Q_{2},\vec{q_{2}})$,
where its zero component $Q_{2}$ is the charged lepton’s effective energy, is
defined such that:

$\displaystyle{q_{2}}^{2}={P_{2}}^{2}-e^{2}a^{2}=m_{l}^{2}-e^{2}a^{2}={m_{l}^{*}}^{2},$
(7)

where $m_{l}^{*}$ is the charged lepton’s effective mass. Inside the laser
field, the charged kaon is described by the Klein-Gordon equation of a charged
scalar coupled to a classical electromagnetic field[21, 22], which is given by
the following expression.

$\displaystyle\left[(i\partial-eA)^{2}-m_{K}\right]\psi_{K}(x)=0,$ (8)

where $m_{K}$ is the charged kaon mass. From the equation (8), we can derive
the wave function of the charged kaon as follows:

$\displaystyle\psi_{K^{\pm}}(x)$ $\displaystyle=$
$\displaystyle\dfrac{1}{\sqrt{2Q_{1}V}}e^{\pm iS(q_{1},x)},$ (9)

where the classical action $S(q_{1},x)$ is expressed as follows:

$\displaystyle
S(q_{1},x)=q_{1}.x-e\dfrac{a_{1}.P_{1}}{k.P_{1}}\sin\phi+e\dfrac{a_{2}.P_{1}}{k.P_{1}}\cos\phi.$
(10)

The square of the effective four-vector momentum $q_{1}=(Q_{1},\vec{q_{1}})$
is given by:

$\displaystyle{q_{1}}^{2}={P_{1}}^{2}-e^{2}a^{2}=m_{K}^{2}-e^{2}a^{2}={m_{K}^{*}}^{2}.$
(11)

We mention that the neutrino / anti neutrino is considered as massless
particle. It is a neutral particles which means that it doesn’t interact with
the electromagnetic field. Indeed, it is described by a free wave function as
follows:

$\displaystyle\psi_{\nu_{l_{i}}}(x)$ $\displaystyle=$
$\displaystyle\dfrac{DS(P_{3},s_{3})}{\sqrt{2E_{3}V}}e^{\mp iP_{3}.x},$ (12)

where $E_{3}$ represents the neutrino’s energy. In the first Born
approximation and by following Feynman’s rules, the matrix elements $S_{fi}$
of the leptonic decay modes of the charged kaon ($S_{fi}^{+}$ for the matter
decay and $S_{fi}^{-}$ for the antimatter decay) can be expressed as follows
[23]:

$\displaystyle S_{fi}^{\pm}=\dfrac{-iG_{F}}{\sqrt{2}}\int
d^{4}x\mathcal{J}_{\mu}^{K^{\pm}\dagger}(x)\mathcal{J}^{l^{\pm}\mu}(x),$ (13)

where $G_{F}=(1.16637\pm 0.00002)\times 10^{-11}\,MeV^{-2}$ is the Fermi
coupling constant. The laser-assisted leptonic currents for matter decay
($\mathcal{J}^{l^{+}}$) and antimatter decay ($\mathcal{J}^{l^{-}}$) are given
by:

$\displaystyle\mathcal{J}^{l^{+}\mu}(x)=\bar{\psi}_{l^{+}}(x)\gamma^{\mu}(1-\gamma^{5})\psi_{\nu_{l}}(x)\qquad\text{and}\qquad\mathcal{J}^{l^{-}\mu}(x)=\bar{\psi}_{l^{-}}(x)\gamma^{\mu}(1-\gamma^{5})\psi_{\bar{\nu_{l}}}(x).$
(14)

The associated laser-assisted hadronic currents are expressed as follows :

$\displaystyle\mathcal{J}_{\mu}^{K^{\pm}}(x)=i\sqrt{2}F_{K}P_{1\mu}\dfrac{1}{\sqrt{2Q_{1}V}}e^{\pm
iS(q_{1},x)},$ (15)

where the kaon decay constant $F_{K}$ is obtained from the leptonic decay
modes of the charged kaon in the absence of the laser field [23] such that:
$F_{K}=24.9414\,MeV$ for the muonic channel and $F_{K}=24.5501\,MeV$ for the
electronic channel. After calculation and by substituting the equations (15)
and (14) into equation (13), the decay matrix element of the two leptonic
decays become as follows:

$\displaystyle S_{fi}^{\pm}$ $\displaystyle=$
$\displaystyle\dfrac{-G_{F}F_{K}}{\sqrt{8Q_{1}Q_{2}E_{3}V^{3}}}\int
d^{4}x\bar{DS}(P_{2},s_{2})(1+c(P_{2})\not{a_{1}}\not{k}\cos\phi+c(P_{2})\not{a_{2}}\not{k}\sin\phi)$
(16) $\displaystyle\times$
$\displaystyle\not{P_{1}}(1-\gamma^{5})DS(P_{3},s_{3})e^{i(\mp S(q_{1},x)\pm
S(q_{2},x)\mp P_{3}.x)},$

with $c(P_{2})=\dfrac{e}{2k.P_{2}}$. The exponential term in equation (16) can
be evaluated as follows:

$\displaystyle\pm\left[-S(q_{1},x)+S(q_{2},x)-P_{3}.x\right]=\pm\left[(q_{1}-q_{2}-P_{3}).x+z\sin(\phi-\phi_{0})\right],$
(17)

where $z$ and $\phi_{0}$ are defined as:

$\displaystyle
z=\sqrt{\alpha_{1}^{2}+\alpha_{2}^{2}},\qquad\phi_{0}=\arctan(\dfrac{\alpha_{2}}{\alpha_{1}}),$
(18)

with:

$\displaystyle\alpha_{1}=e(\dfrac{a_{1}.P_{1}}{k.P_{1}}-\dfrac{a_{1}.P_{2}}{k.P_{2}}),\qquad\alpha_{2}=e(\dfrac{a_{2}.P_{1}}{k.P_{1}}-\dfrac{a_{2}.P_{2}}{k.P_{2}}).$
(19)

By using the generating function of Bessel functions,
$e^{iz\sin\theta}=\sum_{n=-\infty}^{n=+\infty}J_{n}(z)e^{in\theta}$, we can
express the charged kaon decay matrix element $S_{fi}^{\pm}$ in terms of
$Bn^{\pm}(z)$, $B1n^{\pm}(z)$ and $B2n^{\pm}(z)$ which are defined as follows:

$\displaystyle\boldsymbol{B}n^{\pm}(z)$ $\displaystyle=$ $\displaystyle
J_{n}(z)e^{\mp in\phi_{0}},$ $\displaystyle\boldsymbol{B}1n^{\pm}(z)$
$\displaystyle=$ $\displaystyle\frac{J_{n+1}(z)e^{\mp
i\phi_{0}(n+1)}+J_{n-1}(z)e^{\mp i\phi_{0}(n-1)}}{2},$ (20)
$\displaystyle\boldsymbol{B}2n^{\pm}(z)$ $\displaystyle=$
$\displaystyle\frac{J_{n+1}(z)e^{\mp i\phi_{0}(n+1)}-J_{n-1}(z)e^{\mp
i\phi_{0}(n-1)}}{2i},$

where $n$ is the number of exchanged photons between the electromagnetic field
and the particles involved in the decay process. By using some of the Bessel
function properties, we get the following transformations:

$\displaystyle e^{\pm iz\sin(\phi-\phi_{0})}$ $\displaystyle=$
$\displaystyle\sum_{n=-\infty}^{n=+\infty}\boldsymbol{B}n^{\pm}(z),$ (21)
$\displaystyle\cos\phi e^{\pm iz\sin(\phi-\phi_{0})}$ $\displaystyle=$
$\displaystyle\sum_{n=-\infty}^{n=+\infty}\boldsymbol{B}1n^{\pm}(z),$ (22)
$\displaystyle\sin\phi e^{\pm iz\sin(\phi-\phi_{0})}$ $\displaystyle=$
$\displaystyle\sum_{n=-\infty}^{n=+\infty}\boldsymbol{B}2n^{\pm}(z).$ (23)

By inserting these transformations in equation (16), the decay matrix element
S${}_{fi}^{\pm}$ becomes as follows:

$\displaystyle{S_{fi}}^{\pm}=\dfrac{-G_{F}F_{K}}{\sqrt{8Q_{1}Q_{2}E_{3}V^{3}}}\sum_{n=-\infty}^{n=+\infty}(2\pi)^{4}\delta^{4}(\pm
q_{1}\mp q_{2}\mp P_{3}\pm nk){{{\mathcal{M}}_{fi}}^{n}}^{\pm},$ (24)

where ${{{\mathcal{M}}_{fi}}^{n}}^{\pm}$ is the decay amplitude which is given
by:

$\displaystyle{{\mathcal{M}}_{fi}}^{n\pm}$ $\displaystyle=$
$\displaystyle\bar{DS}(P_{2},s_{2})\left[\boldsymbol{B}n^{\pm}(z)+c(P_{2})\not{a_{1}}\not{k}\boldsymbol{B}1n^{\pm}(z)+c(P_{2})\not{a_{2}}\not{k}\boldsymbol{B}2n^{\pm}(z)\right]$
(25) $\displaystyle\times$
$\displaystyle\not{P_{1}}(1-\gamma^{5})DS(P_{3},s_{3}).$

To express the partial decay width of matter ($\Gamma^{+}$) and antimatter
($\Gamma^{-}$), the decay matrix element is weighted by the phase space and
time unit $Ts$uch that:

$\displaystyle\Gamma^{\pm}$ $\displaystyle=$
$\displaystyle\dfrac{1}{T}\int\dfrac{Vd^{3}\vec{q}_{2}}{(2\pi)^{3}}\int\dfrac{Vd^{3}\vec{P}_{3}}{(2\pi)^{3}}|\bar{S}_{fi}^{\pm}|^{2}=\sum_{n=-\infty}^{n=+\infty}\Gamma^{n\pm},$
(26)

where $\Gamma^{n\pm}$ has the following expression:

$\displaystyle\Gamma^{n\pm}=\dfrac{G_{F}^{2}F_{K}^{2}}{32\pi^{2}Q_{1}}\int
d\Omega_{q_{2}}\int_{0}^{+\infty}\dfrac{\vec{q}_{2}^{2}d|\vec{q}_{2}|}{Q_{2}E_{3}}\delta(\pm
Q_{1}\mp Q_{2}\mp E_{3}\pm
nw)\displaystyle{\sum_{s_{2},s_{3}}}|\mathcal{M}_{fi}^{n\pm}|^{2},$ (27)

with $\Omega_{q_{2}}$ is the solid angle associated to the dressed charged
lepton $l^{\pm}$. The evaluation of both the integral over $|\vec{q}_{2}|$ and
the term $\displaystyle{\sum_{s_{2},s_{3}}}|\mathcal{M}_{fi}^{n\pm}|^{2}$ in
equation (27) are given in the appendix.
Let’s define the total decay width $\Gamma^{\pm}_{T}$ of the kaon decay at the
lowest order such that:

$\displaystyle\Gamma_{T}^{\pm}=\Gamma^{\pm}\left[K^{\pm}\rightarrow
e^{\pm}+\nu_{e_{i}}\right]+\Gamma^{\pm}\left[K^{\pm}\rightarrow\mu^{\pm}+\nu_{\mu_{i}}\right]+\Gamma_{O.C.}^{\pm},$
(28)

where $\Gamma_{O.C.}^{\pm}$ is the summation of the non leptonic decay widths
of the charged kaons. Thus, the leptonic decay branching ratio $BR_{l}^{\pm}$
can be expressed as follows:

$\displaystyle BR_{l}^{\pm}\left[K^{\pm}\rightarrow
l^{\pm}+\nu_{l_{i}}\right]=\dfrac{\Gamma^{\pm}\left[K^{\pm}\rightarrow
l^{\pm}+\nu_{l_{i}}\right]}{\Gamma_{T}^{\pm}}.$ (29)

Eventhough the charged kaon has fifty disintegration modes [20] and for the
fact that we only have taken into account its leptonic decay modes, we shall
define the lifetime $\tau(K^{\pm})$ which is associated to each decay process
such that:

$\displaystyle\tau(K^{\pm})=\dfrac{1}{\Gamma^{\pm}_{T}}.$ (30)

We have also defined the leptonic ratio $R_{e/\mu}$, which is given
experimentally by $R_{e/\mu}=(2.488\pm 0.009)\times 10^{-5}$, in the presence
of a circularly polarized laser field as follows:

$\displaystyle R_{e/\mu}=\dfrac{\Gamma^{\pm}\left[K^{\pm}\rightarrow
e^{\pm}+\nu_{e_{i}}\right]}{\Gamma^{\pm}\left[K^{\pm}\rightarrow\mu^{\pm}+\nu_{\mu_{i}}\right]}.$
(31)

To discuss the effect of the laser field on the matter-antimatter decay, we
have defined the decay factor $\vartriangle_{CPT}(l)$, which is experimentally
equal to $(-0.27\pm 0.21)\%$ without laser field in the muonic channel[20], by
the following relation:

$\displaystyle\triangle_{CPT}(l)=\dfrac{\Gamma^{+}\left[K^{+}\rightarrow
l^{+}+\nu_{l}\right]-\Gamma^{-}\left[K^{-}\rightarrow
l^{-}+\bar{\nu_{l}}\right]}{\Gamma^{+}\left[K^{+}\rightarrow
l^{+}+\nu_{l}\right]+\Gamma^{-}\left[K^{-}\rightarrow
l^{-}+\bar{\nu_{l}}\right]}.$ (32)

##  3\. Results and discussion

After dealing with the theoretical calculation of the laser-assisted two-body
leptonic decay of the charged mesons $K^{\pm}$, the results and discussion
part will be devoted to study the variation of different quantities and
observables already illustrated in the previous section in order to understand
the behavior of the charged kaon inside the laser field. We have used the
FormCalc-9.3.1 [24, 25, 26] package to evaluate the spinorial part of the
decay width. The different numerical results are obtained in the natural units
system by using unit conversions such as: The amplitude of the electric field
$\epsilon_{0}\,[eV^{2}]=43290.844\times\epsilon_{0}\,[V/cm]$, the time
$t\,[s]=t/6.582122\times 10^{-16}\,[eV^{-1}]$ and the electric charge
$|e|=8.5424546\times 10^{-2}$. Throughout this work, we have chosen spherical
coordinates such that the spherical angle $\varphi$ is chosen as equal to
zero, and the wave vector $\vec{k}$ is taken as along the $z$ axis. We begin
our discussion by analyzing the variation of the differential partial decay
width as a function of the number of emitted or absorbed photons.

Figure 1: Envelope of the differential partial decay width of the charged kaon
$\Gamma^{s+}(K^{+}\rightarrow\mu^{+}+\nu_{\mu})$(27) as a function of the
photons number $n$ for different values of $\hbar\omega$ and $\epsilon_{0}$,
by choosing the spherical coordinates such as $\theta=90^{\circ}$ and
$\varphi=0^{\circ}$.

Figure 1 illustrates the dependence of the differential partial decay width of
the charged kaon on the photons number $n$ for different laser field
amplitudes and frequencies. This figure aims to perform two comparisons.
Firstly, by comparing the two figures 1(a) and 1(b), for the same laser field
strength $\epsilon_{0}=10^{6}\,V/cm$, the laser-assisted differential
multiphoton decay rate d$\Gamma^{n}$/d$\theta$ depends on the laser frequency
as it moves from the order of $10^{-11}\,eV$ for $\hbar\omega=0.117\,eV$ to
$10^{-9}\,eV$ for the case where $\hbar\omega=2\,eV$. In addition, the number
of photons that can be exchanged between the particles involved in the decay
process and the laser field decreases by increasing the laser frequency. For
instance, this number is approximately $n=1330$ in the case of
$\hbar\omega=0.117\,eV$, while $n=9$ for $\hbar\omega=2\,eV$. Secondly, in the
two figures 1(c) and 1(d), where the laser frequency is fixed at
$\hbar\omega=1.17\,eV$, the differential partial decay width presents
symmetric cutoffs with respect to $n=0$. Moreover, the partial decay width
decreases by decreasing the laser field strength. Consequently, the cutoff
number increases by either decreasing the laser frequency or increasing the
laser field strength. This first result is in full agreement with that found
in other studies[5] as it gives us information about the number of photons $n$
that can be exchanged when the laser-assisted decay width is different from
its corresponding laser-free decay width. We will now focus our attention on
discussing the variation of the laser-assisted differential decay width
d$\Gamma^{\pm}$/d$\theta$ as a function of the angle $\theta$ for the two
leptonic modes of the charged kaon decay.

Figure 2: Dependence of the laser-assisted differential decay width of the
charged kaon on the angle $\theta$ for different numbers of exchanged photons
$n$. (a) represents the case of $K^{+}\rightarrow e^{+}+\nu_{e}$ decay. (b)
represents the case of $K^{-}\rightarrow\mu^{-}+\bar{\nu}_{\mu}$ decay.

Figure 2 displays the effect of the number of exchanged photons $n$ on the
differential decay width by varying the angle $\theta$ and taking the laser
parameters as $\hbar\omega=2\,eV$ and $\epsilon_{0}=10^{6}\,V/cm$. These two
figures show that the circularly polarized laser field reduces the
differential decay width by several orders of magnitudes. In addition, the
differential decay width in the presence of the laser field converges to the
laser-free differential decay width as much as we increase the number of
exchanged photons $n$. Furthermore, when the number of exchanged photons
reaches the cutoff which is the first value of $n$ for which
d$\Gamma^{n}$/d$\theta$ will be zero, the differential decay width of the
charged kaon decay in the presence of the laser field will be equal to its
corresponding laser-free differential decay width. This convegency is called
the sum-rule shown by Bunkin and Fedorov, and also by Kroll and Watson[27]. It
remains valid for any value of $n\geq$ cutoff. This result confirms that
already obtained in figure 1(b). An other important point to mention here is
that the order of magnitude of the exchanged photons number is different for
the two studied processes. This is interpreted by the fact that the cutoff
number increases as much as the mass of the produced charged lepton $l$
decreases. Let’s move now to discuss the effect of the laser field on the two
leptonic decay channels of the antimatter $K^{-}$ (27) for different laser
field strengths and frequencies.

| $\Gamma(K^{-}\rightarrow e^{-}+\bar{\nu}_{e})\>\ [eV]$ | $\Gamma(K^{-}\rightarrow\mu^{-}+\bar{\nu}_{\mu})\>\ [eV]$
---|---|---
$\epsilon_{0}\>\ [\text{V/cm}]$ |  $\hbar\omega=0.117\,eV$ |  $\hbar\omega=1.17\,eV$ |  $\hbar\omega=0.117\,eV$ |  $\hbar\omega=1.17\,eV$
10 |  $8.41108\times 10^{-13}$ |  $8.41108\times 10^{-13}$ |  $3.37932\times 10^{-8}$ |  $3.37932\times 10^{-8}$
$10^{2}$ |  $8.41108\times 10^{-13}$ |  $8.41108\times 10^{-13}$ |  $3.37932\times 10^{-8}$ |  $3.37932\times 10^{-8}$
$10^{3}$ |  $8.38928\times 10^{-13}$ |  $8.41108\times 10^{-13}$ |  $3.37932\times 10^{-8}$ |  $3.37932\times 10^{-8}$
$10^{4}$ |  $6.89024\times 10^{-13}$ |  $8.41108\times 10^{-13}$ |  $3.03290\times 10^{-8}$ |  $3.37932\times 10^{-8}$
$10^{5}$ |  $1.18529\times 10^{-13}$ |  $8.38928\times 10^{-13}$ |  $5.24955\times 10^{-9}$ |  $3.37932\times 10^{-8}$
$10^{6}$ |  $1.17859\times 10^{-14}$ |  $6.89024\times 10^{-14}$ |  $5.19151\times 10^{-10}$ |  $3.03290\times 10^{-8}$
$10^{7}$ |  $1.17459\times 10^{-15}$ |  $1.18529\times 10^{-13}$ |  $5.18861\times 10^{-11}$ |  $5.24955\times 10^{-9}$
$10^{8}$ |  $1.18045\times 10^{-16}$ |  $1.17861\times 10^{-14}$ |  $5.15933\times 10^{-12}$ |  $5.19151\times 10^{-10}$
$10^{9}$ |  $1.33016\times 10^{-17}$ |  $1.17586\times 10^{-15}$ |  $5.27825\times 10^{-13}$ |  $5.18861\times 10^{-11}$
$10^{10}$ |  $1.42073\times 10^{-17}$ |  $1.30758\times 10^{-16}$ |  $5.29958\times 10^{-14}$ |  $5.15869\times 10^{-12}$
$10^{11}$ |  $1.30649\times 10^{-16}$ |  $1.43220\times 10^{-16}$ |  $5.38011\times 10^{-15}$ |  $5.26224\times 10^{-13}$
$10^{12}$ |  $1.22483\times 10^{-15}$ |  $1.30750\times 10^{-15}$ |  $1.72866\times 10^{-15}$ |  $5.39979\times 10^{-14}$
Table 1: Laser-assisted partial decay width $\Gamma$ of the two processes
$K^{-}\rightarrow\mu^{-}+\bar{\nu}_{\mu}$ and $K^{-}\rightarrow
e^{-}+\bar{\nu}_{e}$ for different values of $\hbar\omega$ and $\epsilon_{0}$
by taking $n$ as ranging from $-20$ to $20$.

Table 1 shows the effect of the circularly polarized laser field on the
partial decay width of the two processes $K^{-}\rightarrow
e^{-}+\bar{\nu}_{e}$ and $K^{-}\rightarrow\mu^{-}+\bar{\nu}_{\mu}$. We mention
that, in the absence of an external electromagnetic field, the partial decay
width of $K^{-}$ in the electronic channel is $\Gamma^{-}=8.41108\times
10^{-13}\,eV$ while $\Gamma^{-}=3.37932\times 10^{-8}\,eV$ in the muonic
channel [20]. This table indicates clearly that, for low intensities, the
electromagnetic field doesn’t affect the partial decay width regardless of the
laser field frequency. For instance, for the process $K^{-}\rightarrow
e^{-}+\bar{\nu}_{e}$, the effect of the laser field begins to appear at
$\epsilon_{0}\simeq 10^{3}\,V/cm$ for $\hbar\omega=0.117\,eV$. However, for
$\hbar\omega=1.17\,eV$, this effect appears at $\epsilon_{0}\simeq
10^{5}\,V/cm$. This result is explained by the fact that the number of
exchanged photons $n=\pm 20$ verifies the well known sum-rule for
$\epsilon_{0}\lesssim 10^{2}\,V/cm$ and $\hbar\omega=0.117\,eV$ as in the
first column, or $\epsilon_{0}\lesssim 10^{5}\,V/cm$ and
$\hbar\omega=1.17\,eV$ as in the second column. To explain more clearly, we
can deduce from figure 2(b) that, for $\hbar\omega=2\,eV$,
$\epsilon_{0}=10^{6}\,V/cm$ and $n\geq 20$, the partial decay width in the
muonic mode is equal to that obtained in the absence of the laser field. This
is explained by the fact that there will be no photons to be exchanged for
$n\geq 20$ (over limit of cutoff). Table 1 also shows that at high laser field
strengths, the partial decay width of $K^{-}$ in the ($e^{-},\bar{\nu}_{e}$)
channel approaches that in ($\mu^{-},\bar{\nu}_{\mu}$) channel. Therefore, to
understand clearly this behavior, it is recommended to evaluate the effect of
the circularly polarized laser field on the branching ratio. However, the fact
that the charged kaon can decay via fifty modes, it is difficult to take into
account all these modes to discuss the branching ratio and average lifetime.
For this reason, we will discuss the laser-assisted leptonic branching ratio
$BR_{l}$ given by the equation (29) and the charged kaon lifetime represented
by the equation (30).

Figure 3: Laser-assisted leptonic branching ratio as a function of the laser
field strength $\epsilon_{0}$. (a) and (b) are successively the leptonic
branching ratios of $K^{+}\rightarrow e^{+}+\nu_{e}$ and
$K^{+}\rightarrow\mu^{+}+\nu_{\mu}$ modes for different values of $n$ and for
$\hbar\omega=1.17\,eV$. (c) and (d) are the leptonic branching ratios of
$K^{+}\rightarrow e^{+}+\nu_{e}$ and $K^{+}\rightarrow\mu^{+}+\nu_{\mu}$
modes, respectively, for different values of $\hbar\omega$, and by summing
over $n$ from $-60$ to $60$.

Figure 3 illustrates the dependence of the leptonic branching ratio $BR_{l}$
of the positive kaon decay on the laser field strength for different number of
exchanged photons and laser field frequencies. As we can see from figures 3(a)
and 3(b), for $\hbar\omega=1.17eV$, the variation of the $BR_{l}$ as a
function of the laser field strength $\epsilon_{0}$ for different values of
exchanged photons $n$ ($-j\leqslant n\leqslant j$, with
$j=\\{5,20,60,100,150\\}$) indicates that, for low intensities, the laser
field doesn’t influence the leptonic branching ratio $BR_{l}$. Moreover, it
starts to decrease in both channels when the laser field amplitude overcomes
the threshold value $\epsilon_{0}=10^{5}V/cm$. However, as long as the laser
field overcomes $\epsilon_{0}=10^{10}\,V/cm$, the branching ratio begins to
increase in the electronic channel($e^{+},\nu_{e}$). It is well known that, in
the absence of an external field, the experimental branching ratios of
$K^{+}\rightarrow\mu^{+}+\nu_{\mu}$ is $63.56(11)\%$ while the experimental
branching ratio of $K^{+}\rightarrow e^{+}+\nu_{e}$ is equal to
$1.582(7)\times 10^{-5}$ [20]. Thus, we can deduce from Fig. 3 that the
circularly polarized laser field reduces the branching ratio of
$K^{+}\rightarrow\mu^{+}+\nu_{\mu}$ and $K^{+}\rightarrow e^{+}+\nu_{e}$,
however, the $BR(e^{+},\nu_{e})$ begins to increase at high laser field
amplitudes. Figures 3(c) and 3(d) illustrate the dependence of the $BR_{l}$ as
a function of the laser field strength by summing over the number of exchanged
photons $n$ from -60 to 60 and for different known laser field frequencies
which are $\hbar\omega=0.117\,eV$, $\hbar\omega=1.17\,eV$ and
$\hbar\omega=2\,eV$. It is obvious from these figures that, at high laser
field strengths and regardless of the laser frequency, the leptonic branching
ratio of the positive kaon decay always increases via the ($e^{+},\nu_{e}$)
channel and keeps decreasing in the ($\mu^{+},\nu_{\mu}$) channel. In
addition, when the laser field strength reaches a certain value, the two
leptonic branching ratios become comparable. For instance, for
$\hbar\omega=0.117\,eV$ and $\epsilon_{0}=10^{12}\,V/cm$, the leptonic
branching ratios of the positive kaon decay modes are
BR($e^{+},\nu_{e})=1.86508\times 10^{-7}$ and
BR($\mu^{+},\nu_{\mu})=2.63299\times 10^{-7}$. Furthermore, the variation of
the leptonic branching ratio, regardless of the leptonic decay mode, is
important as much as the laser frequency decreases. It is recognized that
lifetime of an unstable particle is linked to its decay width, and since the
leptonic decay width is affected by the laser field, it is worthy and
important to study the effect of the external electromagnetic field on the
lifetime $\tau$ of the charged kaon as a function of the laser parameters.

Figure 4: Lifetime of the positive kaon as a fucntion of the laser field
strength. (a) the liftime of the positive kaon for different values of $n$ and
for $\hbar\omega=1.17\,eV$. (b) the liftime of the positive kaon for different
known laser frequencies and by taking the number of exchanged photons as
$-60\leq n\leq+60$.

Figure 4 displays the variation of the positive kaon lifetime $\tau$ given by
equation (30) on the laser field strength $\epsilon_{0}$ for different numbers
of exchanged photons $n$ and laser frequencies $\hbar\omega$. According to
figure 4(a), for the laser frequency $\hbar\omega=1.17\,eV$ and regardless of
the number of exchanged photons, the laser field does not affect the positive
kaon lifetime unless its strength overcomes a threshold
$\epsilon_{0}=10^{5}\,V/cm$. Whereas, for the number of exchanged photons
$-60\leq n\leq+60$ as in figures 4(b), the laser field effect on the lifetime
appears at different threshold values of the laser field strength which
depends on the laser frequency. Furthermore, this threshold value decreases as
much as the laser frequency decreases. Moreover, as $\epsilon_{0}$ overcomes
this threshold value, the positive kaon lifetime increases as far as the laser
field strength increases; it moves from the scale of $1.238\times 10^{-8}\,s$
to $3.397\times 10^{-8}\,s$. Therefore, the lifetime of the charged kaon is
modified inside the electromagnetic field, despite the fact that we have only
dressed its leptonic decay modes. To give a complete picture of the laser
field effect on the leptonic decay of the charged kaon, we move to discuss the
behavior of the leptonic ratio $R_{e/\mu}$ which is given by equation (31)
inside the circularly polarized laser field.

Figure 5: Leptonic ratio as a function of the laser field strength
$\epsilon_{0}$. (a): $R_{e/\mu}$ for different values of $n$ and
$\hbar\omega=1.17\,eV$. (b): $R_{e/\mu}$ for different values of $\hbar\omega$
and by summing over $n$ from $-60$ to $60$.

Figure 5 displays the variation of the leptonic ratio $R_{e/\mu}$ as a
function of the laser field strength for different number of exchanged photons
by summing over $n$ from $-i$ to $i$ such that $i=\\{5,20,40,80,150\\}$ as in
figure 5(a), and for different laser frequencies $\hbar\omega$ as in figure
5(b). We mention that, in the absence of an external field, the quantity
$R_{e/\mu}$ is experimentally equal to $2.488(9)\times 10^{-5}$. Thus, we
notice that the effect of the laser field on the experimental factor
$R_{e/\mu}$ becomes important when the intensity $\epsilon_{0}$ is large, and
that confirms the significant decrease of the charged kaon decay width in the
muonic channel in comparison with that found in the electronic mode. For
example, for $\hbar\omega=0.117\,eV$ and $\epsilon_{0}=10^{12}\,V/cm$, we can
deduce from figure 5(a) that $\Gamma(K^{+}\rightarrow
e^{+}+\nu_{e})=0.708347\times\Gamma(K^{+}\rightarrow\mu^{+}+\nu_{\mu})$. From
this result, we can confirm that, in the presence of the laser field and by
dressing all the decay modes of the charged kaon, the branching ratio in the
electronic channel will be of the same importance as the branching ratio in
the muonic channel. The last important point to be discussed in this research
paper is the variation of the quantity associated to CPT symmetry (32). It is
an experimental quantity which represents one of the most challenging issues
that are not explained by the standard model.

| $\Delta_{CPT}(e,\nu_{e})$ | $\Delta_{CPT}(\mu,\nu_{\mu})$
---|---|---
$\epsilon_{0}\>\ [\text{V/cm}]$ |  $\hbar\omega=0.117\,eV$ |  $\hbar\omega=1.17\,eV$ |  $\hbar\omega=0.117\,eV$ |  $\hbar\omega=1.17\,eV$
10 |  $-1.24871\times 10^{-11}$ |  $5.53546\times 10^{-12}$ |  $9.79109\times 10^{-17}$ |  $9.79109\times 10^{-17}$
$10^{2}$ |  $-4.72363\times 10^{-12}$ |  $-2.51076\times 10^{-12}$ |  $-3.91644\times 10^{-16}$ |  $-2.93733\times 10^{-16}$
$10^{3}$ |  $-1.08599\times 10^{-11}$ |  $2.42769\times 10^{-11}$ |  $-4.89555\times 10^{-16}$ |  $1.95822\times 10^{-16}$
$10^{4}$ |  $5.80142\times 10^{-11}$ |  $-3.06077\times 10^{-12}$ |  $8.32847\times 10^{-11}$ |  $0$
$10^{5}$ |  $1.59613\times 10^{-10}$ |  $1.27065\times 10^{-11}$ |  $2.56883\times 10^{-10}$ |  $-1.95822\times 10^{-16}$
$10^{6}$ |  $9.85136\times 10^{-10}$ |  $-5.24162\times 10^{-11}$ |  $1.11873\times 10^{-9}$ |  $8.32833\times 10^{-10}$
$10^{7}$ |  $8.25115\times 10^{-9}$ |  $-3.49834\times 10^{-10}$ |  $1.57303\times 10^{-9}$ |  $2.56878\times 10^{-9}$
$10^{8}$ |  $1.17815\times 10^{-7}$ |  $7.86274\times 10^{-9}$ |  $3.28896\times 10^{-8}$ |  $1.11873\times 10^{-8}$
$10^{9}$ |  $-8.78207\times 10^{-7}$ |  $8.16060\times 10^{-8}$ |  $1.09299\times 10^{-6}$ |  $1.57062\times 10^{-8}$
$10^{10}$ |  $-1.36747\times 10^{-8}$ |  $5.59991\times 10^{-7}$ |  $2.25762\times 10^{-6}$ |  $3.35539\times 10^{-7}$
$10^{11}$ |  $-6.10246\times 10^{-5}$ |  $7.12923\times 10^{-7}$ |  $1.82039\times 10^{-4}$ |  $-4.19000\times 10^{-6}$
$10^{12}$ |  $7.56812\times 10^{-4}$ |  $-8.71563\times 10^{-5}$ |  $5.54973\times 10^{-4}$ |  $-2.13747\times 10^{-4}$
Table 2: $\Delta_{CPT}$ as a function of the laser field strength
$\epsilon_{0}$ for different laser field frequencies $\hbar\omega$.

Table 2 represents the variation of the parameter associated to CPT symmetry,
which is given by equation (32), as a function of the laser field strength
$\epsilon_{0}$ for different values of $\hbar\omega$ and by summing over the
exchanged photons number $n$ form $-20$ to $+20$. As we can see from the table
2, it is clear that the circularly polarized laser field affects both matter
and antimatter decay of the charged kaon. Additionally, it is recognized that
CPT symmetry is preserved ($\vartriangle_{CPT}=0\%$) or broken
($\vartriangle_{CPT}\neq 0\%$) in both cases where the disintegrated particles
$K^{\pm}$ have the same or different masses. According to PDG [20], the
experimental value of the CPT associated parameter in the muonic mode is of
the order of $(-0.27\pm 0.21)\%$. However, in the absence of the laser field,
its theoretical value is equal to zero (equation (1)) while it goes from an
order of $10^{-17}$ to $10^{-4}$ when we increase the electric field strength
form $\epsilon_{0}=10\,V/cm$ to $\epsilon_{0}=10^{12}\,V/cm$. Moreover, for
the positive values of $\vartriangle_{CPT}$, we see that the matter $K^{+}$
becomes dominant at high precision width value $\Gamma$ in comparison to the
antimatter $K^{-}$ while its negative values indiquate the dominance of the
antimatter over the matter. CPT symmetry is also affected by the external
field in the electronic channel, and the dominance of matter over antimatter
or vice-versa depends of the laser field parameters. The results obtained
about CPT conservation and CPT symmetry breaking are due to the fact that
phases associated to the Bessel functions which are expressed as a function of
$\phi_{0}$ in equation (2.2.) depends on the type of disintegration (matter or
antimatter). Furthermore, the elements associated to the lepton mass $m_{l}$
which is obtained from the Dirac bispinors part are not equal to zero as in
the case of the laser-free disintegration. All these results of CPT breaking
symmetry can represent, in cosmology, a probable reason for the asymmetry
observed in the universe between matter and antimatter [28].

##  4\. Conclusion

In this study, we have investigated the laser-assisted leptonic decay of the
charged kaon at the lowest order. We have found that the circularly polarized
laser field reduces the differential decay width by several orders of
magnitudes. In addition, at high laser field intensities, the partial decay
width of the negative kaon ($k^{-}$) in the ($e^{-}$,$\nu_{e}$) channel
approaches that in $(\mu^{-},\nu_{\mu})$. Moreover, the branching ratio of the
two processes $K^{+}\rightarrow e^{+}+\nu_{e}$ and
$K^{+}\rightarrow\mu^{+}+\nu_{\mu}$ decreases in the presence of an external
field. However, the branching ratio of $K^{+}\rightarrow e^{+}+\nu_{e}$ begins
to increases when the laser field strength reaches
$\epsilon_{0}=10^{10}\,V/cm$. Since lifetime is inversely proportional to the
total decay width, which is affected by the laser field despite the fact that
we have only dressed the leptonic modes, we assume that lifetime of the
charged kaon extends. Furthermore, the variation of the leptonic ratio
$R_{e/\mu}$ as a function of the laser field strength indicates that the two
leptonic decay widths of the the charged kaon will be of the same importance
at high laser field intensities. By theoretically measuring the parameter
associated to CPT symmetry inside the electromagnetic field, we have come out
with a surprising result which indicates that we can control the charged kaon
decay by applying an external field to conserve or violate the CPT symmetry.
Consequently, by violating this symmetry, we have the possibility to make the
antimatter dominant over the matter or vice versa. Additionally, the laser
field also enables us to conserve this symmetry in order to produce the same
amount of matter and antimatter.

## Appendix

The integral over $|\vec{q}_{2}|$ in the equation (27) can be evaluated by
using the following mathematical formula:

$\displaystyle\int
F(x)\delta(G(x))dx=\dfrac{F(x_{0})}{|G^{\prime}(x_{0})|}\qquad\text{where}\quad
G(x_{0})=0,$ (33)

and so, our next integral becomes:

$\displaystyle\int_{0}^{+\infty}\dfrac{\vec{q}_{2}^{2}d|\vec{q}_{2}|}{Q_{2}E_{3}}\delta(Q_{1}-Q_{2}-E_{3}+nw)\displaystyle{\sum_{s_{2},s_{3}}}|\mathcal{M}_{fi}^{n\pm}|^{2}=\dfrac{\displaystyle{\vec{\textbf{q}}_{2}^{2}\sum_{s_{2},s_{3}}}|\mathcal{M}_{fi}^{n\pm}|^{2}}{Q_{2}E_{3}|G^{\prime}(|\vec{\textbf{q}}_{2}|)|}$
(34)

where $Q_{2}=\sqrt{\vec{\textbf{q}}_{2}^{2}+m_{l}^{*2}}$ and
$E_{3}=\sqrt{\vec{P}_{3}^{2}}=|\vec{q}_{1}-\vec{q}_{2}+n\vec{k}|$. The
absolute value of $G^{\prime}(|\vec{\textbf{q}}_{2}|)$ is deduced from the
first derivation with respect to $|\vec{\textbf{q}}_{2}|$. Thus ,it becomes as
follows:

$\displaystyle G^{\prime}(|\vec{\textbf{q}}_{2}|)$ $\displaystyle=$
$\displaystyle-\dfrac{|\vec{\textbf{q}}_{2}|}{\sqrt{|\vec{\textbf{q}}_{2}|^{2}+\omega^{2}-e^{2}a^{2}}}-\left[|\vec{\textbf{q}}_{2}|-\omega\cos\theta(n-\dfrac{e^{2}a^{2}}{2k.P_{1}})\right]$
(35) $\displaystyle\times$
$\displaystyle\dfrac{1}{\sqrt{|\vec{\textbf{q}}_{2}|^{2}-2|\vec{\textbf{q}}_{2}|\omega\cos\theta(n-\dfrac{e^{2}a^{2}}{2k.P_{1}})+\omega^{2}(n-\dfrac{e^{2}a^{2}}{2k.P_{1}})^{2}}},$

where the four-vector $\vec{\textbf{q}}_{2}$ has the following components
$\vec{\textbf{q}}_{2}\equiv|\vec{\textbf{q}}_{2}|(\sin\theta\cos\varphi,\sin\sin,\cos\theta)$
is in spherical coordinates. Its norm $|\vec{\textbf{q}}_{2}|$ is given by:

$\displaystyle|\vec{\textbf{q}}_{2}|$ $\displaystyle=$
$\displaystyle(\omega(e^{2}a^{2}-2nk.P_{1})\times(e^{4}a^{4}\omega^{2}-4e^{2}a^{2}n\omega^{2}k.P_{1}-4k.P_{1}$
(36) $\displaystyle\times$
$\displaystyle(m^{*2}_{l}+(E_{1}-\dfrac{e^{2}a^{2}}{2k.P_{1}}w)^{2}+2(E_{1}-\dfrac{e^{2}a^{2}}{2k.P_{1}}w)\omega))\cos\theta$
$\displaystyle+$ $\displaystyle
2k.P_{1}\sqrt{((E_{1}-\dfrac{e^{2}a^{2}}{2k.P_{1}}w)+n\omega)^{2}}(a^{8}e^{8}\omega^{4}-8a^{6}e^{6}k.P_{1}n\omega^{4}$
$\displaystyle+$ $\displaystyle
32a^{2}e^{2}(k.P_{1})^{3}(E_{1}-\dfrac{e^{2}a^{2}}{2k.P_{1}}w)n\omega^{2}(E_{1}-\dfrac{e^{2}a^{2}}{2k.P_{1}}w+2n\omega)$
$\displaystyle-$ $\displaystyle
8a^{4}e^{4}(k.P_{1})^{2}\omega^{2}((E_{1}-\dfrac{e^{2}a^{2}}{2k.P_{1}}w)^{2}+2(E_{1}-\dfrac{e^{2}a^{2}}{2k.P_{1}}w)n\omega$
$\displaystyle-$ $\displaystyle
2s^{2}\omega^{2})+16(k.P_{1})^{4}(m^{*4}_{l}-2m^{*2}_{l}(E_{1}-\dfrac{e^{2}a^{2}}{2k.P_{1}}w+n\omega)^{2}$
$\displaystyle+$
$\displaystyle(E_{1}-\dfrac{e^{2}a^{2}}{2k.P_{1}}w)^{2}(E_{1}-\dfrac{e^{2}a^{2}}{2k.P_{1}}w+2n\omega)^{2})$
$\displaystyle+$ $\displaystyle
8(k.P_{1})^{2}m^{*2}_{l}(a^{2}e^{2}-2nk.P_{1})^{2}\omega^{2}\cos
2\theta)^{1/2})$ $\displaystyle\times$
$\displaystyle\dfrac{1}{(k.P_{1})^{3}(E_{1}-\dfrac{e^{2}a^{2}}{2k.P_{1}}w+n\omega)^{2}-4k.P_{1}(a^{2}e^{2}-2nk.P_{1})^{2}\omega^{2}\cos^{2}\theta}.$

To give the expression of the quantity
$\sum_{s_{2},s_{3}}|{{\mathcal{M}}_{fi}}^{n\pm}|^{2}$ in equation (27), we
have used the FeynCalc-9.3.0 package. Thus, we have:

$\displaystyle\sum_{s_{2},s_{3}}$
$\displaystyle|{{\mathcal{M}}_{fi}}^{n\pm}|^{2}=\mp
4em_{K}^{2}(a_{1}.P_{3})\boldsymbol{B}1n^{\pm}(z)\boldsymbol{B}n^{\pm}(z)^{*}\mp
4em_{K}^{2}(a_{2}.P_{3})\boldsymbol{B}2n^{\pm}(z)\boldsymbol{B}n^{\pm}(z)^{*}$
(37) $\displaystyle\mp$ $\displaystyle
4em_{K}^{2}(a_{1}.P_{3})\boldsymbol{B}1n^{\pm}(z)^{*}\boldsymbol{B}n^{\pm}(z)\mp
4em_{K}^{2}(a_{2}.P_{3})\boldsymbol{B}2n^{\pm}(z)^{*}\boldsymbol{B}n^{\pm}(z)$
$\displaystyle+$
$\displaystyle\varepsilon[a_{1},a_{2},k,P_{3}]\left[\dfrac{4ie^{2}m_{K}^{2}\boldsymbol{B}2n^{\pm}(z)\boldsymbol{B}1n^{\pm}(z)^{*}-4ie^{2}m_{K}^{2}\boldsymbol{B}1n^{\pm}(z)\boldsymbol{B}2n^{\pm}(z)^{*}}{k.P_{2}}\right]$
$\displaystyle\pm$
$\displaystyle\varepsilon[a_{1},k,P_{2},P_{3}]\left[\dfrac{4iem_{K}^{2}\boldsymbol{B}1n^{\pm}(z)\boldsymbol{B}n^{\pm}(z)^{*}-4iem_{K}^{2}\boldsymbol{B}1n^{\pm}(z)^{*}\boldsymbol{B}n^{\pm}(z)}{k.P_{2}}\right]$
$\displaystyle\pm$
$\displaystyle\varepsilon[a_{2},k,P_{2},P_{3}]\left[\dfrac{4iem_{K}^{2}\boldsymbol{B}2n^{\pm}(z)\boldsymbol{B}n^{\pm}(z)^{*}-4iem_{K}^{2}\boldsymbol{B}2n^{\pm}(z)^{*}\boldsymbol{B}n^{\pm}(z)}{k.P_{2}}\right]$
$\displaystyle+$
$\displaystyle\varepsilon[a_{1},a_{2},k,P_{1}]\left[\dfrac{8ie^{2}(P_{1}.P_{3})\boldsymbol{B}1n^{\pm}(z)\boldsymbol{B}2n^{\pm}(z)^{*}-8ie^{2}(P_{1}.P_{3})\boldsymbol{B}2n^{\pm}(z)\boldsymbol{B}1n^{\pm}(z)^{*}}{k.P_{2}}\right]$
$\displaystyle\pm$
$\displaystyle\varepsilon[a_{1},k,P_{1},P_{2}]\left[\dfrac{8ie(P_{1}.P_{3})\boldsymbol{B}1n^{\pm}(z)\boldsymbol{B}n^{\pm}(z)^{*}-8ie(P_{1}.P_{3})\boldsymbol{B}n^{\pm}(z)\boldsymbol{B}1n^{\pm}(z)^{*}}{k.P_{2}}\right]$
$\displaystyle\pm$
$\displaystyle\varepsilon[a_{2},k,P_{1},P_{2}]\left[\dfrac{8ie^{2}(P_{1}.P_{3})\boldsymbol{B}2n^{\pm}(z)\boldsymbol{B}n^{\pm}(z)^{*}-8ie(P_{1}.P_{3})\boldsymbol{B}n^{\pm}(z)\boldsymbol{B}2n^{\pm}(z)^{*}}{k.P_{2}}\right]$
$\displaystyle+$
$\displaystyle\left[\dfrac{4e^{2}m_{K}^{2}a^{2}(k.P_{3})\boldsymbol{B}1n^{\pm}(z)\boldsymbol{B}1n^{\pm}(z)^{*}+4e^{2}m_{K}^{2}a^{2}(k.P_{3})\boldsymbol{B}2n^{\pm}(z)\boldsymbol{B}2n^{\pm}(z)^{*}}{k.P_{2}}\right]$
$\displaystyle\pm$
$\displaystyle\left[\dfrac{4em_{K}^{2}(a_{1}.P_{2})(k.P_{3})\boldsymbol{B}1n^{\pm}(z)\boldsymbol{B}n^{\pm}(z)^{*}+4em_{K}^{2}(a_{2}.P_{2})(k.P_{3})\boldsymbol{B}2n^{\pm}(z)\boldsymbol{B}n^{\pm}(z)^{*}}{k.P_{2}}\right]$
$\displaystyle\pm$
$\displaystyle\left[\dfrac{4em_{K}^{2}(a_{1}.P_{2})(k.P_{3})\boldsymbol{B}1n^{\pm}(z)^{*}\boldsymbol{B}n^{\pm}(z)+4em_{K}^{2}(a_{2}.P_{2})(k.P_{3})\boldsymbol{B}2n^{\pm}(z)^{*}\boldsymbol{B}n^{\pm}(z)}{k.P_{2}}\right]$
$\displaystyle-$
$\displaystyle\left[\dfrac{8e^{2}a^{2}(P_{1}.P_{3})(k.P_{1})\boldsymbol{B}1n^{\pm}(z)\boldsymbol{B}1n^{\pm}(z)^{*}+8e^{2}a^{2}(P_{1}.P_{3})(k.P_{1})\boldsymbol{B}2n^{\pm}(z)\boldsymbol{B}2n^{\pm}(z)^{*}}{k.P_{2}}\right]$
$\displaystyle\mp$
$\displaystyle\left[\dfrac{8e(a_{1}.P_{2})(P_{1}.P_{3})(k.P_{1})\boldsymbol{B}1n^{\pm}(z)\boldsymbol{B}n^{\pm}(z)^{*}+8e(a_{2}.P_{2})(P_{1}.P_{3})(k.P_{1})\boldsymbol{B}2n^{\pm}(z)\boldsymbol{B}n^{\pm}(z)^{*}}{k.P_{2}}\right]$
$\displaystyle\mp$
$\displaystyle\left[\dfrac{8e(a_{1}.P_{2})(P_{1}.P_{3})(k.P_{1})\boldsymbol{B}1n^{\pm}(z)^{*}\boldsymbol{B}n^{\pm}(z)+8e(a_{2}.P_{2})(P_{1}.P_{3})(k.P_{1})\boldsymbol{B}2n^{\pm}(z)^{*}\boldsymbol{B}n^{\pm}(z)}{k.P_{2}}\right]$
$\displaystyle+$
$\displaystyle\left[16(P_{1}.P_{2})(P_{1}.P_{3})\boldsymbol{B}n^{\pm}(z)\boldsymbol{B}n^{\pm}(z)^{*}\right]-\left[8m_{K}^{2}(P_{2}.P_{3})\boldsymbol{B}n^{\pm}(z)\boldsymbol{B}n^{\pm}(z)^{*}\right],$

We have used Grozin’s convention, $\varepsilon_{0123}=1$, to calculate the
terms $\varepsilon[A,B,C,D]$ that appear during the evaluation of the quantity
$\sum_{s_{2},s_{3}}|{{\mathcal{M}}_{fi}}^{n\pm}|^{2}$. These terms are
calculated by using the following calculation techniques:

$\displaystyle\varepsilon[a_{1},a_{2},k,P_{3}]$ $\displaystyle=$
$\displaystyle\varepsilon_{1203}a_{1}^{1}a_{2}^{2}k^{0}P_{3}^{3}+\varepsilon_{1230}a_{1}^{1}a_{2}^{2}k^{3}P_{3}^{0},$
$\displaystyle=$
$\displaystyle\dfrac{\epsilon_{0}^{2}}{\omega}\\{\left[\omega(n-\dfrac{e^{2}a^{2}}{2k.P_{1}})-|\vec{\textbf{q}}_{2}|\cos\theta\right]$
$\displaystyle-$
$\displaystyle\sqrt{|\vec{\textbf{q}}_{2}|^{2}-2|\vec{\textbf{q}}_{2}|\omega\cos\theta(n-\dfrac{e^{2}a^{2}}{2k.P_{1}})+\omega^{2}(n-\dfrac{e^{2}a^{2}}{2k.P_{1}})^{2}}\;\
\\},$ (39) $\displaystyle\varepsilon[a_{1},k,P_{2},P{3}]$ $\displaystyle=$
$\displaystyle\varepsilon_{1023}a_{1}^{1}k^{0}P_{2}^{2}P_{3}^{3}+\varepsilon_{1032}a_{1}^{1}k^{0}P_{2}^{3}P_{3}^{2}+\varepsilon_{1302}a_{1}^{1}k^{3}P_{2}^{0}P_{3}^{2}+\varepsilon_{1320}a_{1}^{1}k^{3}P_{2}^{2}P_{3}^{0}$
(40) $\displaystyle=$
$\displaystyle\epsilon_{0}\dfrac{|\vec{\textbf{q}}_{2}|}{2k.P_{1}}\\{2k.P_{1}\sqrt{-a^{2}e^{2}+m_{l}^{2}+|\vec{\textbf{q}}_{2}|^{2}}+a^{2}e^{2}\omega-2n\omega
k.P_{1}+2k.P_{1}$ $\displaystyle\times$
$\displaystyle\sqrt{|\vec{\textbf{q}}_{2}|^{2}-2|\vec{\textbf{q}}_{2}|\omega\cos\theta(n-\dfrac{e^{2}a^{2}}{2k.P_{1}})+\omega^{2}(n-\dfrac{e^{2}a^{2}}{2k.P_{1}})^{2}}\;\
\\}\sin\theta\sin\phi,$ $\displaystyle\varepsilon[a_{2},k,P_{2},P{3}]$
$\displaystyle=$
$\displaystyle\varepsilon_{2013}a_{2}^{2}k^{0}P_{2}^{1}P_{3}^{3}+\varepsilon_{2031}a_{2}^{2}k^{0}P_{2}^{3}P_{3}^{1}+\varepsilon_{2301}a_{2}^{2}k^{3}P_{2}^{0}P_{3}^{1}+\varepsilon_{2310}a_{2}^{2}k^{3}P_{2}^{1}P_{3}^{0}$
(41) $\displaystyle=$
$\displaystyle-\epsilon_{0}\dfrac{|\vec{\textbf{q}}_{2}|}{2k.P_{1}}\\{2k.P_{1}\sqrt{-a^{2}e^{2}+m_{l}^{2}+|\vec{\textbf{q}}_{2}|^{2}}+a^{2}e^{2}\omega-2n\omega
k.P_{1}+2k.P_{1}$ $\displaystyle\times$
$\displaystyle\sqrt{|\vec{\textbf{q}}_{2}|^{2}-2|\vec{\textbf{q}}_{2}|\omega\cos\theta(n-\dfrac{e^{2}a^{2}}{2k.P_{1}})+\omega^{2}(n-\dfrac{e^{2}a^{2}}{2k.P_{1}})^{2}}\;\
\\}\sin\theta\cos\phi,$ $\displaystyle\varepsilon[a_{1},a_{2},k,P{1}]$
$\displaystyle=$
$\displaystyle\varepsilon_{1203}a_{1}^{1}a_{2}^{2}k^{3}P_{1}^{0}=\dfrac{\epsilon_{0}^{2}E_{1}}{\omega},$
(42) $\displaystyle\varepsilon[a_{1},k,P{1},P_{2}]$ $\displaystyle=$
$\displaystyle\varepsilon_{1302}a_{1}^{1}k^{3}P_{1}^{0}P_{2}^{2}=\epsilon_{0}E_{1}|\vec{\textbf{q}}_{2}|\sin\theta\sin\phi,$
(43) $\displaystyle\varepsilon[a_{2},k,P{1},P_{2}]$ $\displaystyle=$
$\displaystyle\varepsilon_{2301}a_{2}^{2}k^{3}P_{1}^{0}P_{2}^{1}=\epsilon_{0}E_{1}|\vec{\textbf{q}}_{2}|\sin\theta\cos\phi.$
(44)

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