Laser-assisted decay of charged mesons into leptonic modes

Abstract

The two-body leptonic decays of both light and heavy-charged mesons $X_{l2}^{\pm }$ are investigated in the first Born approximation in the presence of a monochromatic and circularly polarized laser field. As a result of using Volkov functions, the analytic study of these processes has brought a change in the parameters associated with each decay process. The decay width, lifetime, and branching ratio of each decay process are affected by increasing the classical parameter $\xi$ of the meson. A comparison between the matter decay $X^{+}_{l2}$ and that of the anti-matter $X^{-}_{l2}$ in the presence of a circularly polarized laser wave, indicates that increasing laser field strength (increasing the classical parameter $\xi$ ) can induce a slight dominance of matter over antimatter or vise versa.

1 Introduction

The principle of stimulated or induced emission, which was described by Albert Einstein in 1917, is one of the most scientific successes that led to the invention of lasers in 1960 [1]. In the 1950 s, Charles Townes, Nicolay Basov, and Aleksandr Prokhorov contributed to experiments related to these emission phenomena in masers and lasers, which produce coherent and concentrated beams of micro-waves and light, respectively [2]. In 1966, a Nobel Prize was won by Alfred Kastler for his discovery and development of optical pumping which allows, in particular, the population inversion in lasers. Plenty of applications in several domains are based on lasers. In 1997, the real interest in lasers as a potential tool for piegeage and atoms cooling was a subject of a Nobel prize by Steven Chu, Claude Cohen-Tannoudji, and William D. Phillips. In chemistry, the use of lasers is also very important. In fact, femtosecond lasers are used in spectroscopy by Ahmed Zewail, and they allowed the observation of atoms’ movement inside molecules in a chemical reaction. Nowadays, the physics field of extreme light is a domain that largely benefits from the experimental development of ultra-short lasers by LUXE in DESY [3] and ultra-intense lasers by "Extreme Light Infrastructure" (ELI) [4].

In addition to practical research, extreme light physics is developed also in quantum electrodynamics (QED) in the regime of high field via high power laser sources [5,6,7,8,9]. In particle physics, the possibility of using an intense laser to influence processes beyond the atomic physics energy scale has allowed us to push down fundamental light-electron interactions in the extreme limit. This is done in order to illuminate the quantum vacuum and to induce the production of particles such as electrons, muons, pions, and their corresponding antiparticles [9,10,11]. In QED, the strong field scale corresponds to the characteristic separation of the electron from the positron, and it is known as the Schwinger field limit: $E_{cr}=m_{e}^{2}c^{3}/(e\hbar )\simeq 1.3\times 10^{16}\,V/cm$, where $m_{e}$ and e are the mass and the electron charge, c is the speed of light and $\hbar$ is the reduced Planck constant [12,13,14].

In the majority of studies on decay and scattering processes in the presence of a laser field, the dynamic of participating particles is studied. For instance, in atomic physics, the scattering processes by an atomic target, named Laser-Assisted Elastic Electron Scattering (LAES ), are studied in the framework of laser-assisted processes (See [15,16,17] for more details). In 1976, Andrick and Langhans made the first observation of energy transfer in LAES by using a laser field with continuum waves [18]. One year later, Weingartshofer et al., have signaled the multiphotonic phenomena in electron-argon collision in the presence of a CO$_{2}$ pulsed laser [19]. Moreover, there have been discussions about the role of using ultra-intense laser fields in radioactive processes such as $\alpha$ decay, especially on the modification of decay rates of the nuclear processes [20,21,22,23,24]. In both non-relativistic and relativistic regimes, the electron-proton, proton-nucleus, and nucleus-nucleus elastic scatterings and also electron-atom (positron-atom) inelastic scattering have been studied [25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41]. In particle physics, studying the behavior of a charged particle inside a laser field and the impact of this laser field on the properties of the charged particle was a subject of several researches [42,43,44]. In addition, the circular polarization may decrease the branching ratio of a particle decay and increases its lifetime. In one of these theoretical studies [45], which is about the charged kaons’ decay, we have found that the decay width, the branching ratio, and the ratio of the electronic channel over the muonic one are modified in the presence of a laser field with circular polarization. The dominance of $K^{+}_{l2}$ decay over $K^{-}_{l2}$ decay mode or vice-versa is also studied in the presence of a laser field. This matter–antimatter dominance is introduced by the asymmetric parameter $\Delta _{Asy.}$ which is related to $\Delta _{CPT}$ symmetry.

The CPT symmetry has strong theoretical evidences and supports. In fact, the CPT theorem is a fundamental principle in theoretical physics which states that charge, parity, and time symmetries are conserved in all physical interactions. Indeed, if we reverse the charge, parity, and time in a physical interaction, we obtain the same result as in the original interaction [46,47,48,49,50,51,52]. This result signifies that laws of physics are the same for particles and antiparticles. They do not depend on their direction in space, and they are the same in all directions of displacement. In the case of matter decay, the CPT theorem states that if we change the particles by their corresponding antiparticles and by simultaneously reversing the charge, parity, and time, then the result remains the same as for the original interaction. Therefore, the laws of physics that govern the decay of matter will govern also the antimatter decay though the particles are replaced by their corresponding antiparticles. Consequently, the matter and anti-matter decays have to be produced in the same manner except that particles of matter are replaced by antiparticles of anti-matter [53]. The difference between CP and CPT violations is due to the difference in symmetries violated in each case. The CPT symmetry is a fundamental property of physics, and it is considered exact in the majority of actual theories. This signifies that if a theory conforms to the CPT symmetry, then it must be consistent with the laws of physics. However, the CP symmetry is often violated in interactions of subatomic particles. This leads to a difference between matter and antimatter since the violation of CPT symmetry implies a difference between the nature and its reversed mirror image. Further, the violation of CP and CPT symmetries are different; the violation of CPT symmetry implies a simultaneous violation of both CP and time reversion (T) symmetries while the CP symmetry violation implies only a violation of symmetry between particles and their corresponding antiparticles. It is important to note that elementary particles theory predicts that CPT symmetry must always be conserved. In addition, an experiment showing a CPT violation may be a hint for new physics beyond the actual theory.

In this work, we have studied the decay of a charged meson $X^{\pm }_{l2}$ in the presence of a circularly polarized and monochromatic pulsed laser field in the first Born approximation. The charged particle $X^{\pm }$ is considered either a light charge meson ($\pi ^{\pm }$ or $K^{\pm }$), which will decay via a leptonic channel to an electron ( or a muon ) and its anti-neutrino, or a heavy charged meson ($D^{\pm }$ or $Ds^{\pm }$) which decays via the two channels $X^{\pm }_{\mu 2}$ and $X^{\pm }_{\tau 2}$. We introduce the effect of the laser field, in natural units $\hbar =c=1$, on the decay width of matter and antimatter decay, the branching ratio and lifetime via the introduction of Volkov formalism [55] to describe charged bosons and fermions involved in the decay.

2 Overview of the theory

The decay width, which is an observable, enables us to determine the different products of a particle’s decay. To analytically study the decay of charged mesons in the presence of an electromagnetic field, we will be based on Dirac-Volkov formalism [55] to describe the charged particles inside a laser field. The quantity that represents a decay process is the decay width, which is a measurable quantity used to test the branching ratio and lifetime, and it is used also to investigate the asymmetry between a particle and its corresponding antiparticle. In this section, we perform an analytic calculation of the charged mesons’ decay width in the presence of a pulsed laser wave with circular polarization. The meson is considered as a structureless particle with a charge $e=-|e|$. The calculations of the decay width $\Gamma$ are done for the two-body decay $X^{\pm }_{l2}$ with the same procedure for all different charged mesons. The meson $X^{\pm }$ in the initial state and the leptons in the final state are dressed with the same electromagnetic field. The expression of the decay width of the two-body charged meson decay is given by:

$$\Gamma = \frac{1}{T}\int {\int {\frac{{Vd{{\vec{q}}}_{2} }}{{(2\pi )^{3} }}} } \frac{{Vd{{\vec{p}}}_{3} }}{{(2\pi )^{3} }}|S_{{fi}} |^{2} ,$$

(1)

where T represents the time of observation, V is the quantization volume of each particle. $\vec{q}_{2}$ and $\vec{p}_{3}$ are the four-momentum of the produced charged lepton and that of its corresponding anti-neutrino, respectively. $S_{fi}$ is the transition matrix element from the initial state i to the final state f, and it is defined by the following expression [56]:

$$\begin{aligned} S_{fi}=\frac{-iG_{F}}{\sqrt{2}}\int d^{4}x \bigg (J^{X^{+}}_{\mu }(x)\bigg )^{\dag }.J^{l^+_{\mu }}(x), \end{aligned}$$

(2)

with $G_{F}$ is the Fermi constant. Thus, the transition matrix element $S_{fi}$, in the first Born approximation, is a multiplication of two currents; the meson current $J^{X^{+}}_{\mu }$ and the leptonic current $J^{l^{+}_{\mu }}(x)$ such that:

$$\begin{aligned} J^{X^{+}}_{\mu }(x)= & {} \frac{i\sqrt{2}}{\sqrt{2Q_{1}V}}F_{X^{+}}p_{1\mu }e^{iS(q_{1},x)},\nonumber \\ J^{l^{+}_{\mu }}(x)= & {} \overline{\psi }_{\nu _{l}}(x)\gamma ^{\mu }(1-\gamma ^{5})\psi _{l^{+}}(x), \end{aligned}$$

(3)

with $F_{X^{+}}$ denotes the form factor which is expressed as the normalization of the experimental value of the decay width with its theoretical expression in the absence of a laser field [56]. The four-vectors $p_{i}=(E_{i},\vec{p}_{i})$ ($i=\{1,\, 2 \,\text {and}\, 3\}$) are the laser-free four-momenta of the charged meson, the produced lepton, and its corresponding anti-neutrino, respectively. In the presence of a laser field, the four-momenta $p_{1}$ and $p_{2}$ will be $q_{1} = (Q_{1} ,{{\vec{q}}}_{1} )$ and $q_{2} = (Q_{2} ,{{\vec{q}}}_{2} )$. $S(q_{1},x)$ can be obtained by solving the Klein-Gordon equation (Dirac equation for $S(q_{2},x)$) in the presence of an external field such that:

$$\begin{aligned} S(q_{i},x)=(p_{i}.x)-\int _{0}^{(k.x)}\dfrac{(-e)}{(k.p_{i})}\bigg [(A.p_{i})+\dfrac{(-e)A^{2}}{2}\bigg ]d\phi , \end{aligned}$$

(4)

where $A=A(\phi )$ is the four-vector potential of the electromagnetic field with the wave vector $k = (\omega ,{{\vec{k}}})$. In the leptonic current (equation (3)), $\psi _{l^{+}}(x)$ and $\psi _{\nu _{l}}(x)$ denote successively the wave functions of the produced lepton and its anti-neutrino such that:

(5)

where u(p, s) and v(p, s) are free Dirac bispinors associated successively to matter and antimatter. They satisfy the following equations:

(6)

For a laser wave with circular polarization, the four-vector potential $A(\phi )$ can be expressed by a summation of two orthogonal polarizations $a_{1}^{\mu }=|a|(0,1,0,0)$ and $a_{2}^{\mu }=|a|(0,0,1,0)$ which verify the following condition: $a_{1}^{2}=a_{2}^{2}=a^{2}=-|a|^{2}=-(\varepsilon _{0}/\omega )^{2}$, with $\varepsilon _{0}$ is the strength of the laser field. Thus,

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

(7)

where $\phi =(k.x)$ is the phase of the laser field. The electromagnetic four-potential $A^\mu$ can be used to calculate the classical parameter $\xi$ in the case of a charged particle with a mass m [61]. The parameter $\xi$ comes from the Lorentz equation which describes the force applied by an electromagnetic field on a charged particle. By using this equation, the ratio between the Lorentz force work and the particle’s energy leads to a dimensionless equation in which $\xi$ is used as a parameter to describe the electromagnetic field strength. For a particle at rest, the $\xi$ parameter can be expressed as follows:

$$\begin{aligned} \xi ^2 = \dfrac{e^2 a^2}{m^2}. \end{aligned}$$

(8)

This expression shows that the quantity $\xi$, which describes the movement of a charged particle inside an electric or magnetic field, can be expressed in terms of four-potential $A^\mu$ that describes this field in space-time.

The classical parameter is important for the study of laser-matter interaction as it enables us to characterize the effect of laser field on charged particles and to determine the operating regime of a laser source such as high and low intensities regimes. To calculate the $\xi _{l}$ parameter for charged leptons within the framework of charged meson decay, it is essential to determine the four-momentum of leptons in final states versus the initial state of the physical system. Indeed, the four momenta of particles produced during the decay are measured with respect to this physical system. Once the final four momenta of leptons are determined, we can use this information to calculate the $\xi _{l}$ parameter. In our case, the $\xi$ parameter for the interaction of a circularly pulsed laser field with charged lepton l is expressed as follows;

$$\xi _{l}^{2} = \frac{{e^{2} |a|^{2} }}{{E_{2}^{2} (Q_{2} - |{{\vec{q}}}_{2} |\cos \theta _{2} )^{2} }}|\langle p_{{2_{z} }}^{2} \rangle + \langle p_{{2_{t} }}^{2} \rangle - 2\langle p_{{2_{t} }} p_{{2_{z} }} \rangle |,$$

(9)

where $\langle \rangle$ denotes the average over the phase angle $\phi =(k\cdot x)$. Once the leptonic final four-momenta are determined, we can use them to calculate the $\xi _{l}$ parameter for charged leptons. The quantities $p_{2_z}$ and $p_{2_t}$ represent the spatial component along the z-axis and the temporal component, respectively. A charged particle dressed by a laser field will no longer have the four-momentum p. It will have a new effective four-momentum q. For a charged meson $X^+$ and the charged lepton $l^+$, we have:

$$\begin{aligned} {\left\{ \begin{array}{ll} q_{1}^{\mu }=(m_{X},0,0,\textbf{k})=p_{1}^{\mu }-\dfrac{e^{2}a^{2}}{2(k.p_{1})}k^{\mu } \quad \Longrightarrow \quad {q_{1}}^{2}=m_{X}^{2}-e^{2}a^{2}= {m_{X}^{*}}^{2}=m_{X}^{2}(1+\xi _{X}^{2}),\\ \\ q_{2}^{\mu }=\bigg (Q_{2},\textbf{q}_2{\left\{ \begin{array}{ll} \sin (\theta )\cos (\varphi )\\ \sin (\theta )\sin (\varphi )\\ \cos (\theta )\end{array}\right. }\bigg ) =p_{2}^{\mu }-\dfrac{e^{2}a^{2}}{2(k.p_{2})}k^{\mu } \quad \Longrightarrow \quad {q_{2}}^{2}= {m_{l}^{*}}^{2}, \end{array}\right. } \end{aligned}$$

(10)

with $m_{X}^{*}$ is the effective mass of the charged meson $X^+$ inside the laser field. For circular polarization, the function $S(q_{i},x)$ in equation (4) takes the following form:

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

(11)

After an analytic development of the transition matrix element $S_{fi}$ (equation (2)), the obtained argument of the exponential term can be expressed as:

$$\begin{aligned} (p_{3}.x)+S(q_{2},x)-S(q_{1},x)= & {} ((-q_{1}+q_{2}+p_{3}).x)+e\left[ \dfrac{(a_{1}.p_{1})}{(k.p_{1})}-\dfrac{(a_{1}.p_{2})}{(k.p_{2})}\right] \sin (\phi )\nonumber \\{} & {} -e\left[ \dfrac{(a_{2}.p_{1})}{(k.p_{1})}-\dfrac{(a_{2}.p_{2})}{(k.p_{2})}\right] (\cos (\phi )-1). \end{aligned}$$

(12)

Then, we make the following variables change:

${\left\{ \begin{array}{ll} \alpha _{1}=e\left[ \dfrac{(a_{1}.p_{1})}{(k.p_{1})}-\dfrac{(a_{1}.p_{2})}{(k.p_{2})}\right] \\ \alpha _{2}=e\left[ \dfrac{(a_{2}.p_{1})}{(k.p_{1})}-\dfrac{(a_{2}.p_{2})}{(k.p_{2})}\right] , \end{array}\right. }$ $\qquad \text {and:}\qquad {\left\{ \begin{array}{ll} z=\sqrt{\alpha _{1}^{2}+\alpha _{2}^{2}} \\ \phi _{0}=\arctan \left( \dfrac{\alpha _{2}}{\alpha _{1}}\right) \\ e=-|e|. \end{array}\right. }$

The expression of the $S_{fi}$ element will be as follows:

(13)

By applying the Jacobi-Anger identity [58, 59], we can develop the term $e^{iz\sin (\phi -\phi _{0})}$, in the case of laser wave with circular polarization, by an infinite harmonic summation over the number $\ell$ as follows:

$$\begin{aligned} \begin{aligned} \begin{bmatrix} 1\\ \cos (\phi )\\ \sin (\phi ) \end{bmatrix}e^{iz\sin (\phi -\phi _{0})}=\sum _{\ell =-\infty }^{+\infty }\begin{bmatrix}J_{\ell }(z)e^{-i\ell \phi _{0}}\\ \dfrac{J_{\ell -1}(z)e^{-i(\ell -1)\phi _{0}}+J_{\ell +1}(z)e^{-i(\ell +1)\phi _{0}}}{2}\\ \dfrac{J_{\ell -1}(z)e^{-i(\ell -1)\phi _{0}}- J_{\ell +1}(z)e^{-i(\ell +1)\phi _{0}}}{2i} \end{bmatrix}e^{i\ell \phi }=\sum _{\ell =-\infty }^{+\infty }\begin{bmatrix} B_{\ell }(z)\,\\ B_{1\ell }(z)\\ B_{2\ell }(z) \end{bmatrix}e^{i\ell \phi }. \end{aligned} \end{aligned}$$

(14)

For the case of the matter decay ($X^-$), we get:

$$\begin{aligned} \begin{bmatrix} 1\\ \cos (\phi )\\ \sin (\phi ) \end{bmatrix}e^{-iz\sin (\phi -\phi _{0})}= & {} \sum _{\ell =-\infty }^{+\infty }\begin{bmatrix}J_{\ell }(z)e^{i\ell \phi _{0}}\\ \dfrac{J_{\ell +1}(z)e^{i(\ell +1)\phi _{0}}+J_{\ell -1}(z)e^{i(\ell -1)\phi _{0}}}{2}\\ \dfrac{J_{\ell +1}(z)e^{i(\ell +1)\phi _{0}}- J_{\ell -1}(z)e^{i(\ell -1)\phi _{0}}}{2i} \end{bmatrix}e^{-i\ell \phi }\nonumber \\= & {} \sum _{\ell =-\infty }^{+\infty }\begin{bmatrix} C_{\ell }(z)\,\\ C_{1\ell }(z)\\ C_{2\ell }(z) \end{bmatrix}e^{-i\ell \phi }. \end{aligned}$$

(15)

The expression of the decay width, which is expressed as a function of the transition matrix element $|S_{fi}|^{2}$ (see equation (1)), is integrated over $d\vec{p}_{3}$ of the phase space as follows:

$$\Gamma _{{X_{{l2}}^{ + } }} = \sum\limits_{{\ell = - \infty }}^{{ + \infty }} {\frac{{G_{F}^{2} F_{{X^{ + } }}^{2} }}{{32\pi ^{2} Q_{1} }}} \int {\frac{{|{\vec{{q}}}_{2} |^{2} d|{\vec{{q}}}_{2} |d\Omega _{2} }}{{E_{3} Q_{2} }}} \delta ^{0} (Q_{1} - Q_{2} - E_{3} + \ell \omega )|{\mathcal{M}}_{{fi}}^{{ + \ell }} |^{2} ,$$

(16)

where $d\Omega _{2}$ represents the solid angle associated with charged lepton $l^+$. The function $\delta ^{0}$ ensures the conservation of energy. The decay system is no longer constituted of the three particles $X^{+}$, $l^+$ and $\nu _{l}$. Indeed, the function $\delta ^{0}$ introduces a new conservation form of energy in which the dressed decaying physical system interacts with photons having the four-momentum k. The element $\mathcal {M}_{fi}^{\pm \ell }$, which represents the decay amplitude ($+\ell$ for $X^{+}_{l2}$ decay and $-\ell$ for $X^{-}_{l2}$ decay), and the leptonic neutrino energy $E_{3}$ are expressed as follows:

(17)

The final expression of the charged meson decay width is obtained by performing a numerical integral over the solid angle $\Omega _{2}$.

$$\Gamma _{{X_{{l2}}^{ + } }} = \sum\limits_{{\ell = - \infty }}^{{ + \infty }} {\frac{{G_{F}^{2} F_{{X^{ + } }}^{2} }}{{32\pi ^{2} Q_{1} }}} \int f (|{{\vec{q}}}_{2} |)\delta (g(|{{\vec{q}}}_{2} |))d|{{\vec{q}}}_{2} |d\Omega _{2} = \left. {\sum\limits_{{\ell = - \infty }}^{{ + \infty }} {\frac{{G_{F}^{2} F_{{X^{ + } }}^{2} }}{{32\pi ^{2} Q_{1} }}} \smallint _{{\Omega _{2} }} \frac{{f(|{{\vec{q}}}_{2} |)}}{{g^{\prime}\left( {|{\vec{q}}_{2} |} \right)}}} \right|_{{_{{g(|{\vec{q}}_{2} |) = 0}} .}} d\Omega _{2}$$

(18)

The term $|\vec{q}_{2}|$ is determined by resolving the equation $g(|\vec{q}_{2}|)=0$. The functions f, g and $g'$ are defined by the following expressions:

$$\left\{ {\begin{array}{*{20}l} {f(|\vec{q}_{2} |) = \frac{{|\vec{q}_{2} |^{2} }}{{E_{3} Q_{2} }}|{\mathcal{M}}_{{fi}}^{{ + \ell }} |^{2} , } \hfill \\ {g(|\vec{q}_{2} |) = Q_{1} - \sqrt {\vec{q}_{2}^{2} + m_{l}^{{*2}} } - E_{3} + \ell \omega , } \hfill \\ {g^{\prime } (|\vec{q}_{2} |) = - \frac{{|\vec{q}_{2} |}}{{\sqrt {|\vec{q}_{2} |^{2} + m_{l}^{{*2}} } }} - \frac{{\left[ {|\vec{q}_{2} | - \omega \left( {\ell - \frac{{e^{2} a^{2} }}{{2(k.p_{1} )}}} \right)\cos \theta } \right]}}{{\sqrt {\omega ^{2} \left( {\ell - \frac{{e^{2} a^{2} }}{{2(k.p_{1} )}}} \right)^{2} + |\vec{q}_{2} |^{2} - 2|\vec{q}_{2} |\omega \left( {\ell - \frac{{e^{2} a^{2} }}{{2(k.p_{1} )}}} \right)\cos \theta } }}. } \hfill \\ \end{array} } \right.$$

(19)

To describe the effect of the laser field on the $X^{-}_{l2}$ decay, we calculate the decay width of $\Gamma _{X^{-}_{l2}}$ by following the same procedure as that of $X^{+}_{l2}$. In this case, we inverse the sign of the four-momentum $p_{i}$ and the sign of electric charges, and the free Dirac bispinors will also change. In this study, we are interested in the asymmetric parameter $\Delta _{Asy.}$ which is a quantity that expresses the dominance of matter over antimatter or vice versa. This parameter is expressed as a function of the decay width $\Gamma _{X^{+}_{l2}}$ and $\Gamma _{X^{-}_{l2}}$. For this parameter, we assume that the effect of the laser field on the matter and antimatter decay of the charged meson is the same for the other channels (semi-leptonic and hadronic channels). Therefore:

$$\begin{aligned} \Delta _{Asy.}(X^{\pm }_{l2})= & {} \dfrac{\Gamma _{X^{+}_{l2}}-\Gamma _{X^{-}_{l2}}}{\Gamma _{X^{+}_{l2}}+\Gamma _{X^{-}_{l2}}}. \end{aligned}$$

(20)

To analyze the effect of the laser field with circular polarization on different channels, we study the branching ratio and the lifetime in each leptonic channel. We associate with other channels their values outside the laser field. Thus, the total decay width $\Gamma _{X^{\pm }}^{T}$ is given by:

$$\begin{aligned} \Gamma _{X^{\pm }}^{T}= & {} \Gamma _{X^{\pm }_{l2}}+\Gamma _{\text {Laser free}}, \end{aligned}$$

(21)

with $\Gamma _{\text {Laser free}}$ denotes the decay width in the absence of the laser field.

3 Results and Discussion

To numerically evaluate all the presented theoretical quantities, we use a geometry in which the meson, initially not dressed by the laser field, is at rest. The leptonic neutrino and its anti-neutrino are considered as massless particles. The decay process is studied in a spherical geometry where the spherical angle $\varphi$ associated with charged lepton is fixed at zero degrees. The element $|\mathcal {M}_{fi}^{+\ell }|^{2}$ in equation (16) is developed by using FeynCalc$-$9.3.0 [60] package. The development of the decay width $X^{\pm }_{l2}$ for matter and antimatter is discussed for light mesons such as $\pi ^{\pm }$ and $K^{\pm }$ which may decay into leptons l ($l=\{e\,\text {or}\,\mu \}$). It is also discussed for the decay of heavy mesons $D^{\pm }$ and $Ds^{\pm }$ into leptons $l=\{e,\,\mu \,\text {or}\,\tau \}$. We begin our discussion by introducing the effect of the laser field on the $\xi$ parameter in $\pi ^+_{e2}$ decay as a function of $\theta$ of the positron in its final state and also as a function of the laser strength $\varepsilon _0$.

Fig. 1

Variation of the parameters $\xi _{e^{+}}$ and $\xi _{\mu ^{+}}$ in $\pi ^{+}_{e2}$, $D^{+}_{\mu 2}$ and $K^{+}_{e2}$ decays as a function of the laser field strength $\varepsilon _0$ and the angle $\theta$ of the charged lepton. Figure 1A and C): Variation of $\xi _{e^{+}}$ and $\xi _{\mu ^{+}}$, respectively, as a function of $\theta$ for different values of $\hbar \omega$ and for $\varepsilon _0=5\times 10^{10}$ V/cm. Figure 1B and D): Variation of $\xi _{e^{+}}$ and $\xi _{\mu ^{+}}$, respectively, as a function of $\varepsilon _0$ for different values $\hbar \omega$ with $\theta =37.7^\circ$. Figure 1E and F) illustrate a comparison between two frequencies of the laser field ($\hbar \omega =0.117$ eV and $\hbar \omega =2$ eV) in $K^{+}_{e2}$ decay through the dependence of $\xi _{e^{+}}$ on $\theta$ and $\varepsilon _0$

We begin by examining the effect of the laser field on the parameters $\xi _{e^{+}}$ and $\xi _{\mu ^{+}}$ for the multiphotonic processes $\pi ^{+}_{e2}$ and $D^{+}_{\mu 2}$, respectively, as a function of the laser strength $\varepsilon _0$ and the angle $\theta$ of the charged leptons and for different values of $\hbar \omega$. Figure 1A) illustrates the variation of $\xi _{e^{+}}$ as a function of $\theta$ for different frequencies; $\hbar \omega =0.117$ eV, $\hbar \omega =1.17$ eV and $\hbar \omega =2$ eV. In this figure, we can distinguish two distinct regions. For large values of the classical parameter, the positron may undergo nonlinear processes such as high harmonics generation. In addition, this classical parameter can be used to determine the operating regime of the laser sources such as the regimes of high and low intensities. If $\xi _{e^{+}}<1$, thus the effect of the laser field is low, and the interaction can be considered as a perturbation of the positron movement (Similarly for the meson X when $\xi _X<1$). However, the effect of the laser field is strong if $\xi _{e^{+}}>1$. In this figure and for discrete values of $\theta$, the laser field strength $\varepsilon _0=5\times 10^{10}$ V/cm corresponds to important values of $\xi _{e^{+}}$. The variation of $\xi _{e^{+}}$ as a function of $\theta$ shows also sinusoidal perturbations when $\xi _{e^{+}}<1$ and for different values of $\hbar \omega$.

In Fig. 1B), we plot the variation of $\xi _{e^{+}}$ as a function of $\varepsilon _0$ for different values of $\hbar \omega$ and by fixing $\theta$ at $37.7^\circ$. We remark that for this value of $\theta$ and for the values of $\varepsilon _0$ inferior to $2\times 10^{10}$ V/cm, the $\xi _{e^{+}}$ parameter undergoes a linear increase with respect to the increase of the laser strength. However, when $\varepsilon _0$ overcomes this limit, the interaction of the laser field with the charged particle presents a decrease followed by an increase of the $\xi _{e^{+}}$ parameter. After this increasing process, the value of $\xi _\pi$ will be equal to 1. This is represented in the hatched region that corresponds to the red curve for $\hbar \omega =0.117$ eV. This observation suggests that the effect of the laser field becomes more significant in this region where nonlinear phenomena and strong coupling effects are induced. This transition in the behavior of $\xi _{e^{+}}$ is important as it indicates a change in the nature of the interaction between the charged particles ($\pi ^+,e^+$) and the laser field with a laser strength overcoming a certain threshold. This observation illustrates the importance of taking into consideration the effect of the laser field strength in the analysis of interaction processes between charged particles and electromagnetic fields. We move to Fig. 1C) in which we have plotted the variation of $\xi _{\mu ^{+}}$ as a function of $\theta$ of the muon for the same value of $\hbar \omega$ and for $\varepsilon _0=5\times 10^{10}$ V/cm. We observe that the maximal value of the $\xi _{\mu ^{+}}$ parameter is less than that associated with the positron in Fig. 1A). In addition, the minimal value of $\xi _{\mu ^{+}}$ is less than that associated to the positron. This leads to the conclusion that for the decay of charged particles heavier than ($\pi ^+,e^+$), the laser field required to obtain $\xi$ value greater than 1 must be more intense than that required in the case of light-charged particles decay.

In Fig. 1D), we have plotted the variation of $\xi _{\mu ^{+}}$ for $D^{+}_{\mu 2}$ as a function of $\varepsilon _0$. The variation of $\xi _{\mu ^{+}}$ are similar to those observed in Fig. 1B) except the laser strength, at which $\xi _{\mu ^{+}}$ and $\xi _{D^{+}}$ are greater than 1, is different. In this case and for $\hbar \omega =0.117$ eV, this laser strength is superior to $10^{13}$ V/cm, while it is equal to $8\times 10^{11}$ V/cm for the $\pi ^{+}_{e2}$ decay. To sum up, high laser field intensities are required as far as the charged particle’s mass increases. This means that intense interaction with the laser field is required to influence massive particles as compared to light ones. These observations illustrate the importance of considering the charged particle’s mass in the study of the laser field effect on the multi-photonic decay processes. In Fig. 1E) and F), we present a comparison between two laser sources with $\hbar \omega =0.117$ and $\hbar \omega =2$ eV in $K^{+}_{e2}$ decay. For this reason, we plot the variations of $\xi _{e^{+}}$ for each decay by varying $\theta$ of the positron and the laser field strength $\varepsilon _0$. We observe that the positron-laser interaction is more encouraged at low angles $\theta$ of the produced positron. In addition, if the laser is intense enough (about $7\times 10^{11}$ V/cm for $K^{+}_{e2}$ decay with a laser frequency $\hbar \omega =0.117$ eV and $1.2\times 10^{12}$ V/cm for $K^{+}_{e2}$ decay with $\hbar \omega =2$ eV), it can conduct the produced positron with small angles $\theta$. This is due to the fact that the work performed by the laser field on the positron in the range of small angles is more important as compared to that performed at big angles $\theta$. Moreover, when the laser frequency increases, high laser field strengths are required to obtain these effects. In the hatched region the parameter $\xi _{e^{+}}$ is less than 1.

To study the effect of the variation of the $\xi$ parameter on the number of summation $\ell$, we plotted in Fig. 2 the differential partial decay width as a function of $\theta$ for different values of $\ell$. In this regard, we can obtain important information about the dynamic of the physical system and also the effect of the laser on the final states of particles.

Fig. 2

A and B Variation of $\xi$, $\theta$ and $\ell$, respectively, in $\pi ^{+}_{\mu 2}$ and $D^{+}_{\mu 2}$ decays for $\hbar \omega =1.117$ eV. Left panel: Variation of $d\Gamma /d\theta$ as a function of $\theta$ for different summations over $\ell$. Right panel: Variation of $d\Gamma ^{\ell }/d\theta$ as a function of $\ell$ for different values of $\xi$ of meson and for $\theta =90^\circ$

In this figure, we show the effect of $\xi$ variations for mesons and charged leptons on muliphotonic decay processes such as $\pi ^{+}_{\mu 2}$ and $D^{+}_{\mu 2}$ decays. First, for a laser field with circular polarization, the decay width is analytically described as an infinite and symmetric summation with respect to $\ell =0$. However, for low values of $\xi$, this summation over $\ell$ becomes finite. To find this finite value, we plot in Fig. 2A and B) the variation of $d\Gamma /d\theta$ as a function of $\theta$ (left panel) and $d\Gamma ^{\ell }/d\theta$ as a function of the number of exchanged laser photons (right panel) [61, 62]. We begin by the left panel of Fig. 2A), in which we have plotted the differential decay width in the absence and presence of a laser field by varying the angle $\theta$. For the decay process $\pi ^{+}_{\mu 2}$, which is assisted by a laser field with $\hbar \omega =1.17$ eV and $\varepsilon _{0}=10^{5}$ V/cm (corresponds to $\xi _{\pi ^{+}}=1.20839\times 10^{-8}$), the number of exchanged photons $\ell$ takes different values such as $\ell =\{1, 2, 5, 10 \text { and } 20\}$. Moreover, it is well shown in the small figure in the center that near $\theta =90^{\circ }$, the curve $d\Gamma /d\theta$ in the absence of the laser field is the same as those in the presence of the laser field for $\ell$ equal to 2, 5, 10, and 20. This result implies that the multiphotonic process is stopped when the first finite value $\ell =2$ is reached, and this value is named as the "cut-off" of the multiphotonic process. The right panel of Fig. 2A) shows clearly the possible number of photons that may be exchanged for $\theta =90^{\circ }$. In this figure, we have plotted $d\Gamma ^{\ell }/d\theta$ as a function of $\ell$ for values of the classical parameter $\xi _{\pi ^+}$ equal to $1.20839\times 10^{-8},\, 6.04195\times 10^{-8}\text { and } 1.20839\times 10^{-7}$ and for the frequency $\hbar \omega =1.17$ eV. We remark that the quantity $d\Gamma ^{\ell }/d\theta$ is not null unless the number of laser photons overcomes certain limits which are symmetric with respect to $\ell =0$. This symmetry is due to the properties of Bessel functions that appear in the decay amplitude $|\mathcal {M}_{fi}^{\pm \ell }|^{2}$ (see equation (17)). In addition, the value of the "cut-off" changes by varying the electric strength of the laser field. As we have seen, for $\xi _{\pi ^{+}}=1.20839\times 10^{-8}$, the quantity $d\Gamma ^{\ell }/d\theta$ will be equal to zero from $\ell =2$. However, for $\xi _{\pi ^{+}}$ equal to $6.04195\times 10^{-8}$ and $1.20839\times 10^{-7}$ (correspond to $\varepsilon _{0}$ equal to $5\times 10^{5}$ V/cm and $10^{6}$ V/cm, respectively), $d\Gamma ^{\ell }/d\theta$ will be equal to zero from $\ell =4$ and $\ell =5$, respectively. We conclude that the possible number of photons to be exchanged will be great as long as the value of $\xi _{\pi ^{+}}$ increases. We move on to Fig. 2B) in which we have studied the $D^{+}_{\mu 2}$ decay in the absence and presence of a laser field where the latter’s parameters are the same as in Fig. 2A). In the left panel of figure 2B) and for low values of $\theta$, the curves are different unless for $\ell =20$ for which the laser-assisted differential decay width is equal to its corresponding laser-free decay width. For the cases where $\ell =\{5, 10 \text { and } 20\}$, the curve of $d\Gamma /d\theta$ coincides with that in the absence of the laser field for $\theta =90^{\circ }$, and this result is obviously indicated in the right figure. We remark, in the right panel, that $d\Gamma ^{\ell }/d\theta$ is equal to zero for $\varepsilon _{0}=10^{5}$ V/cm and $\ell =4$ which is the new "cut-off" for the $D^{+}_{\mu 2}$ decay process. To sum up the results presented in the two Fig. 2A) and B), the charged meson particles $\pi ^{+}$ and $D^{+}$ inside a laser field behave differently by exchanging different numbers of photons $\ell$. This result is interpreted by the mass effect, for that $D^{+}$ is much heavier and unstable than $\pi ^{+}$. Therefore, the latter meson is more affected by the laser field.

In the third figure, we will discuss the effect of the laser strength and its frequency on the decay width when the number of laser photons transferred, $\ell$, varies from $-10$ to 10.

Fig. 3

Variation of decay width for different decay channels in the presence of a laser fields CO$_{2}$ and He : Ne for $\varepsilon _0=6\times 10^{6}$ V/cm and $\varepsilon _0=4\times 10^{13}$ V/cm

In Fig. 3, we have plotted two histograms in which we vary the decay width for two laser sources; the CO$_{2}$ laser with frequency $\hbar \omega =0.117$ eV and He : Ne laser with frequency $\hbar \omega =2$ eV. In this figure, we have chosen two laser strengths $\varepsilon _{0}=6\times 10^{6}$ V/cm and $\varepsilon _{0}=4\times 10^{13}$ V/cm ( they correspond to $\xi _{X}\ll 1$ and $\xi _{X}>1$, receptively) and the summation over $\ell$ is from $-10$ to 10. $\Gamma _{W.L}$ denotes the decay width of each decay system in the absence of the laser field. Figure 3A) represents the decay widths of light mesons via the two electronic channels ($K^{+}_{e2}$ and $\pi ^{+}_{e2}$) and two muonic channels ($K^{+}_{\mu 2}$ and $\pi ^{+}_{\mu 2}$), while Fig. 3B) represents that of the heavy charged meson via the muonic channels ($D^{+}_{\mu 2}$ and $Ds^{+}_{\mu 2}$) and the tauic channels ($D^{+}_{\tau 2}$ and $Ds^{+}_{\tau 2}$). We observe that the laser-assisted decay width is less than its corresponding decay width in the absence of the laser field, $\Gamma _{W.L}$, for all decay channels. This is due to the properties of the circular polarization of the electromagnetic field which is characterized by minimizing measurable quantities such as the decay widths and cross sections [25, 26][42,43,44,45]. By comparing the obtained results, we remark that for all decay channels except $\pi ^{+}_{e2}$ and $K^{+}_{e2}$ for $\hbar \omega =0.117$ eV, the decay width $\Gamma$ at $\varepsilon _{0}=10^{6}$ V/cm is always greater than that of $\varepsilon _{0}=4\times 10^{13}$ V/cm. This is due to the fact that the number of exchanged photons is as large as the laser field strength increases. However, in our case, we have truncated the number of exchanged photons as ranging from $\ell =-10$ to $\ell =+10$. For two laser strengths $\varepsilon _{0}^{1}$ and $\varepsilon _{0}^{2}$, if $\varepsilon _{0}^{1}$ is less then $\varepsilon _{0}^{2}$, we have $\Gamma ^{\ell }(\varepsilon _{0}^{1})$ is higher than $\Gamma ^{\ell }(\varepsilon _{0}^{2})$. Thus, the number of exchanged photons $\ell$ is small in the first case, and the maximum of $\Gamma ^{\ell }(\varepsilon _{0}^{1})$ is higher than that corresponds to $\varepsilon _{0}^{2}$. This property is illustrated in the right panel of Fig. 3. For $\pi ^{+}_{e2}$ and $K^{+}_{e2}$ decays at $\hbar \omega =0.117$, the obtained results are interpreted by the fact that the value $\ell =10$ corresponds to a band of $\ell$ values at which the decay width for $\varepsilon _{0}=4\times 10^{13}$ V/cm will be slightly superior to that corresponds to $\varepsilon _{0}=6\times 10^{6}$ V/cm. This result is illustrated for $D^{+}_{\mu 2}$ decay for $\ell =8$ in the right panel of Fig. 3B) in which we compare $\Gamma ^{\ell }$ for $\xi _{D^{+}}=4.51035\times 10^{-9}$ to that corresponds to $\xi _{D^{+}}=9.0207\times 10^{-9}$. Another remark from the histograms presented in the Fig. 3A and B), is that inside an He : Ne laser with $\varepsilon _{0}=6\times 10^{6}$ V/cm, the decay widths of $\pi ^{+}_{\mu 2}$, $D^{+}_{\tau 2}$ and $Ds^{+}_{\tau 2}$ are maximal and they are equal to their corresponding free-decay widths $\Gamma _{W.L}$. This implies that the summation over $\ell =\pm 5$ is great or equal to the laser "Cut-off". For the other channels where $\Gamma <\Gamma _{W.L}$, the number $\ell$ is less than the laser "Cut-off". From Fig. 3, we can not see if the laser field affects the two decay channels of each meson decay in the same manner. To know the channel which is the most sensitive to the laser field, we calculate the branching ratio in each channel. Then, we discuss the effect of the laser field on the meson $D^{+}$.

Fig. 4

Variation of the branching ratio of $\pi ^{+}_{e2}$ channel and the leptonic ratio of decay widths $\Gamma _{\pi ^{+}_{e2}}/\Gamma _{\pi _{\mu 2}^+}$ as a function of $\xi _{\pi ^+}$. Figures 4C and D): Variation of the branching ratio of $D^{+}_{\tau 2}$ channel and leptonic decay width $\Gamma _{D^{+}_{\mu 2}}/\Gamma _{D^{+}_{\tau 2}}$ as a function of $\xi _{D^{+}}$. Fig. 4E and F): Variation of the lifetime of pion and meson D as a function of $\xi _{\pi ^+}$ and $\xi _{D^{+}}$, respectively. We adopt the following values in all figure: $\ell =\pm 10$, $\hbar \omega =0.117$ eV and $\hbar \omega =2$ eV

In Fig. 4A and B), we study the effect of the laser field on both the branching ratio of $\pi ^{+}_{e2}$ channel and the leptonic ratio between the decay widths of $\pi ^{+}_{e2}$ and $\pi ^+_{\mu 2}$. This is done by varying the $\xi _{\pi ^+}$ parameter from $10^{-10}$ to 10 for $\hbar \omega =0.117$ eV and $\hbar \omega =2$ eV, and by considering the two leptonic channels. We observe that this figure presents four distinct regions. In the first region which corresponds to low laser field ($\xi _{\pi ^+}$ between $10^{-10}$ and $10^{-9}$ for $\hbar \omega =0.117$ eV, and $\xi _{\pi ^+}$ between $10^{-10}$ and $10^{-8}$ for $\hbar \omega =2$ eV), the branching ratio remains the same as in the absence of the laser field. This is due to the fact that we have performed a summation over $\ell$ from $-10$ to 10, and this overcomes the value associated to the corresponding cut-off. In the second region, the branching ratio diminishes progressively as it passes from $0.0123\%$ to $0.0033\%$ when $\xi _{\pi ^+}$ reaches $10^{-7}$. In this region, the channel $\pi _{\mu 2}^+$ becomes dominant as compared to $\pi ^{+}_{e2}$ channel in the absence of the laser field. In the third region where $\xi _{\pi ^+}$ is between $10^{-7}$ and $10^{-3}$, the branching ratio and the leptonic ratio remains constant. In this range ($\xi _{\pi ^+}<1$), the $\xi _{e^{+}}$ parameter may reach values superior to 10, while the value of $\xi _{\mu ^{+}}$ remains under 1. After third region in which $\xi _{\pi ^+}$ varies from $10^{-3}$ to 10, the leptonic branching ratio increases from $0.001\%$ to $50\%$ when $\xi _{\pi ^+}$ reaches 10. In this case, the laser field induces an effect of equalization between the probabilities of muon and electron production in the final state. These results illustrate the influence of the laser field strength on the branching ratios and leptonic ratios. In addition, whenever $\xi _{\pi ^+}$ is less than $10^{-3}$, the decay processes encourage muon production. However, as $\xi _{\pi ^+}$ increases, the branching ratio and leptonic ratio change, and this leads to a significant increase in the electron production probability as compared to that of the muon.

To study the effect of the laser field variation and the masses of dressed particles on the branching ratio and the ratio between leptonic partial decay widths, we have plotted in Fig. 4C and D) the variation of these two quantities for the heavy meson $D^{+}_{\tau 2}$. Similarly, we remark that there exist four regions with a slight difference as compared to that of the pion. In the case of branching ratio of $D^{+}_{\tau 2}$, the decrease of this quantity with an order of $10^{-7}$ in the second region needs an increase of $\xi _{D^{+}}$ from $10^{-9}$ to $10^{-1}$ for $\hbar \omega =0.117$ eV and from $10^{-8}$ to $10^{-1}$ for $\hbar \omega =2$ eV. Beyond the second region, the branching ratio remains the same, and it undergoes a slight increase whenever $\xi _{D^{+}}$ overcomes 1. In addition, to obtain a similar effect to that observed in $\pi ^{+}_{e2}$ decay, it is necessary to reach the values of $\xi _{D^{+}}$ superior to 10. By comparing the decay modes of $D^{+}_{\mu 2}$ and $D^{+}_{\tau 2}$, we remark that the tauon production will be dominant as compared to the light leptonic production of a muon when $\xi _{D^{+}}$ is included in the regime of intense laser field ($\xi _{D^{+}}$ superior to 1). This result illustrates the influence of dressed particles mass on the branching ratio and also on the final states of particles in the context of their interaction with an intense laser field. These branching ratio variations indicate the importance of the laser field interaction on the probability of each decay channel. The laser field modifies the dynamic of the physical system and encourages some channels as compared to the others. These results provide essential information about how the laser field influences the decay processes and the particles final states.

In Fig. 4E and F), we examine the impact of the laser field on the lifetime of pion and meson D, respectively, by dressing the leptonic channels with a laser field with circular polarization. We plotted the lifetime of each particle as a function of the parameter $\xi$ for $\hbar \omega =0.117$ eV and $\hbar \omega =2$ eV by summing over $\ell$ from $-10$ to 10. In Fig. 4E), we observe that the laser field with frequency $\hbar \omega =0.117$ can increase the lifetime of the pion to $2.6033\times 10^{-8}$ s for the values of $\xi _{\pi ^+}$ between $10^{-10}$ and $10^{-8}$ and to $1.6775\times 10^{-1}$ s for $\xi _{\pi ^+}=1$. Moreover, for $\hbar \omega =2$ eV, the lifetime increases until it reaches $9.7751\times 10^{-3}$ s. This lifetime increase is due to the fact that when the laser strength $\varepsilon _0$ increases (see Fig. 3A), a remarkable decrease of the decay width in the muonic channel, which is dominant in terms of branching ratio in the absence of the laser field, is observed. However, in Fig. 4F), we observe that the lifetime of heavy meson increases slightly because the leptonic channel doesn’t contribute strongly in the absence of the laser field. For the meson D, the lifetime reaches its maximum from $\xi _{D^{+}}=10^{-6}$ for $\hbar \omega =0.117$ eV and $\xi _{D^{+}}=10^{-5}$ for $\hbar \omega =2$ eV, and it remains constant ($1.0416\times 10^{-12}$ s) until $\xi _{D^{+}}=10$. These results reveal the significant effect of the laser field on the pion and D meson lifetime. They indicate that the laser field induces an increase in the lifetime and a decrease in the leptonic decay width. However, the effect of the laser effect on the meson D is less strong due to the low contribution of leptonic channels.

In the previous figures, we have examined the effect of the laser field on different quantities as a function of different parameters in order to study the behavior of decayed particles in the presence of an electromagnetic field with circular polarization. In the following Table 1, we present some results about the $\Delta _{Asy.}$ parameter, which is given by equation (20), for the channels $\pi ^\pm _{ e2}$, $\pi ^\pm _{\mu 2}$, $D^\pm _{\mu 2}$ and $D^\pm _{\tau 2}$. The He : Ne laser field is considered, and the summation over the number of transferred photons is performed for $\ell =\pm 1$ and $\ell =\pm 10$.

Table 1 $\Delta _{Asy.}$ of $\pi ^{\pm }_{e2}$, $\pi ^{\pm }_{\mu 2}$, $D^{\pm }_{\mu 2}$ and $D^{\pm }_{\tau 2}$ as a unction of $\xi _X$ for $\hbar \omega =2$ eV and for different summations over $\ell$

The difference between the decay width of matter and antimatter $\Delta _{Asy.}$ is a phenomenon known as CP symmetry violation. In the standard model, this phenomenon is related to an explicit violation of CP symmetry known as the CKM phase. In fact, matter and antimatter are two opposed forms of particles that are characterized by identical properties and opposed electric charges. However, in our universe, matter is dominant as compared to antimatter. The CP symmetry violation, which is a mechanism by which the interaction between matter and antimatter may affect the symmetry between particles’ decay processes, can help in explaining this asymmetry. In this table, we introduce a new source of symmetry violation between matter and antimatter, and it is induced by a laser field with circular polarization in leptonic decay of charged mesons by varying the value of $\xi _X$ with $\hbar \omega =2$ eV. For low values of $\xi _{\pi }$ and $\xi _D$, $\Delta _{Asy.}$ is almost null when the summation over $\ell$ is performed from $-1$ to 1 and from $-10$ to 10 for all decay channels. These results indicate that the symmetry between matter and antimatter interactions is conserved. In addition, for these values, the laser field doesn’t affect the symmetry of particles during the decay process. Whenever the value of $\xi$ increases, the laser strength increases, and the value of $\Delta _{Asy.}$ will no longer be null and it takes negative or positive values of the order 0 to $10^{-6}$ for $\ell =\pm 1$ or $\ell =\pm 10$. These variations indicate a breaking of the interaction symmetry between the matter and the antimatter. For the values of $\xi _{\pi }$ varying from $10^{-10}$ to 50, the parameter $\Delta _{Asy.}$ varies from the order $10^{-7}$ to $10^{-6}$ for different summations over $\ell$. They can be negative or positive, and they depend on the value of $\xi _{\pi }$ as for the meson D decay. Negative values of the asymmetric parameter indicate that the antimatter channel is dominant as compared to the matter channel. In the case where the asymmetric parameter is positive, the matter decay channel is dominant. Similar results about the conservation and violation of CP symmetry are obtained for the decay of other mesons such as $K^{\pm }_{e2}$, $K^{\pm }_{\mu 2}$, $Ds^{\pm }_{\mu 2}$ and $Ds^{\pm }_{\tau 2}$. The origin of the symmetry violation in this study is introduced by the multiphotonic phenomena during the decay process. In fact, this violation appears in the element $\mathcal {M}_{fi}^{\pm \ell }$ given by equation (17). This element is expressed for the decay of $X^+_{l2}$ with an electric charge $e'=-e$, while $e'=e$ for $X^-_{l2}$ decay. In addition, the phase $e^{\pm i\ell \phi _0}$ (see equations (14) and (15)) which couples the ordinary Bessel functions in the cases of matter and antimatter is different. To sum up, the CP symmetry violation can explain the dominance of matter over antimatter in our universe. Indeed, if this symmetry is violated, the decay processes may slightly produce more particles than antiparticles (or vice-versa). This leads to an asymmetry mater-antimatter. However, the violation of CP symmetry is not enough to explain entirely this asymmetry, and other processes have to be taken into consideration.

4 Conclusion

In this study, we have deepened the understanding of two body leptonic decay of charged heavy and light mesons ($X^{\pm }$) in the presence of a laser pulse with circular polarization. We have adopted the Klein-Gordon-Volkov formalism to describe the charged mesons with spin zero, and the Dirac-Volkov for charged fermions. We have also adopted the approximation of plane wave and the natural units system. The decay process is characterized by a symmetric exchange of photons with emission and/or absorption of one or several photons as a function of the parameter $\xi$, laser field strength, and its frequency. We have examined the multiphotonic effect of the laser field on different quantities such as the decay width, the branching ratio, the ratio between leptonic channels, the lifetime, and the asymmetry parameter associated with decay widths of matter–antimatter. We have noted that the $\xi$ parameter of the meson and the frequency of the laser field play a crucial role in reducing the decay width of the meson and increasing its lifetime. In addition, these two parameters of the laser field may induce a slight dominance of matter over antimatter or vice-versa, and this depends on the used value of $\xi$. We have also observed how the increasing laser field strength leads to the increase of $\xi _X$ and $\xi _l$ values. Moreover, our results demonstrate that, whatever the value of the laser field strength, the circular polarization of the pulsed laser field can not increase the decay width of the meson beyond its laser-free value. This result shows the importance of taking into consideration the effect of the laser field in studying the decay of charged mesons and their properties. Overall, these results allow us to comprehend the mechanism of charged mesons’ decay in the presence of a laser field, and they reveal the importance of the parameter $\xi$ and the laser field strength in the modification of the properties of decayed charged particles. These results also contribute to enriching our knowledge about fundamental interactions and decay processes in particle physics.