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Analysis of the laser geometry effect on the Higgs-strahlung process

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Abstract

In this paper, we have investigated the process of Higgs-strahlung production \(e^{+}e^{-}\rightarrow Z H\), at the leading order inside an intense electromagnetic wave propagating in different directions. The differential cross section is analytically calculated in the centre of mass frame by using the scattering matrix approach. Then, the integrated cross section is computed for three different directions of the wave four-vector. We have found that in the case where the wave four-vector is along the \(e^{+}e^{-}\) direction, the laser field significantly affects the total cross section, and this effect begins from low intensities as compared to the case where the wave four-vector is perpendicular to the incoming \(e^{+}e^{-}\) beam.

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The authors received no financial support for the research, authorship, and/or publication of this article.

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Authors and Affiliations

  1. Research Laboratory in Physics and Engineering Sciences, Team of Modern and Applied Physics FPBM, USMS, Beni Mellal, Morocco

    M. Ouhammou, M. Ouali, S. Taj, Y. Mekaoui & B. Manaut

Authors
  1. M. Ouhammou
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  2. M. Ouali
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  3. S. Taj
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  4. Y. Mekaoui
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  5. B. Manaut
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Contributions

M.O and M.O developed the theoretical formalism, performed the analytic calculations, and performed the numerical simulations. Both M.O and S.T contributed to data analysis and wrote the final version of the manuscript. B.M supervised the project.

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Correspondence to M. Ouali.

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Appendix

Appendix

The coefficients \(A_{i}(i=1,...,6)\) of Eq. (24) are expressed as follows:

$$\begin{aligned} A_{1}= & {} \dfrac{2}{\Big (k.p_{-}\, k.p_{+}\, M_{Z}^{2}\Big )}\bigg [\Big (a^{4} e^{4} (g_{a}^{e}{}^{2} \nonumber \\{} & {} + g_{v}^{e}{}^{2}) (k.k_{1})^2 +2 k.p_{-}\, k.p_{+}\, \Big (M_{Z}^{2} \Big (g_{a}^{e}{}^{2} \Big (-3 m_{e}^{2} + p_{-}.p_{+}\Big ) \nonumber \\ {}{} & {} + g_{v}^{e}{}^{2} \Big (3 m_{e}^{2}+p_{-}.p_{+}\Big )\Big )\nonumber \\{} & {} +2 (g_{a}^{e}{}^{2} + g_{v}^{e}{}^{2}) p_{-}.k_{1}\, p_{+}.k_{1}\Big )+2 a^{2} e^2 \Big (g_{a}^{e}{}^{2} \Big (-2 \,k.p_{-}\, k.p_{+}\, M_{Z}^{2} \nonumber \\{} & {} + k.k_{1}\, \Big (k.k_{1}\, m_{e}^{2} - \,k.k_{1}\, p_{-}.p_{+}\, +\, k.p_{+}\, p_{-}.k_{1}\, + \,k.p_{-}\, p_{+}.k_{1}\Big )\Big )+g_{v}^{e}{}^{2} \Big (-2 \,k.p_{-}\, k.p_{+}\nonumber \\\times & {} M_{Z}^{2}\,k.k_{1}\, \Big (-k.k_{1}\, \Big (m_{e}^{2} + p_{-}.p_{+}\Big ) \nonumber \\{} & {} + k.p_{+}\, p_{-}.k_{1}\,+\, k.p_{-}\, p_{+}.k_{1}\Big )\Big )\Big )\Big )\bigg ]. \end{aligned}$$
(30)
$$\begin{aligned} A_{2}= & {} -\dfrac{e^2}{2\Big (k.p_{-}\, k.p_{+}\, M_{Z}^{2}\Big )} \bigg [\Big (2 k.p_{-}\, k.p_{+}\, \Big (2 (a_{1}.k_{1})^2 (g_{a}^{e}{}^{2} + g_{v}^{e}{}^{2}) k.p_{-}\, k.p_{+}\,\nonumber \\{} & {} +2 (a_{2}.k_{1})^2 (g_{a}^{e}{}^{2} + g_{v}^{e}{}^{2})\nonumber \\\times & {} k.p_{-}\, k.p_{+}-2 a_{1}.k_{1}\, (g_{a}^{e}{}^{2} + g_{v}^{e}{}^{2}) \Big (a_{1}.p_{+}\, k.p_{-}\, + \,a_{1}.p_{-}\, k.p_{+}\Big ) \,k.k_{1}\,+2 a_{1}.p_{-} \, a_{1}.p_{+}\, g_{a}^{e}{}^{2} \nonumber \\\times & {} (k.k_{1})^2 + 2 a_{1}.p_{-}\, a_{1}.p_{+}\, g_{v}^{e}{}^{2} (k.k_{1})^2+2 a^{2} g_{a}^{e}{}^{2} (k.k_{1})^2 m_{e}^{2} \nonumber \\{} & {} - 2 a^{2} g_{v}^{e}{}^{2} (k.k_{1})^2 m_{e}^{2} + a^{2} g_{a}^{e}{}^{2} (k.p_{-})^2 M_{Z}^{2} \nonumber \\{} & {} + a^{2} g_{v}^{e}{}^{2} (k.p_{-})^2 M_{Z}^{2}-2 a^{2} g_{a}^{e}{}^{2} \,k.p_{-}\,\nonumber \\{} & {} k.p_{+}\, M_{Z}^{2} - 2 a^{2} g_{v}^{e}{}^{2} \,k.p_{-}\, k.p_{+}\,\nonumber \\{} & {} M_{Z}^{2}+a^{2} g_{a}^{e}{}^{2} (k.p_{+})^2 M_{Z}^{2} + a^{2} g_{v}^{e}{}^{2} \nonumber \\\times & {} (k.p_{+})^2 M_{Z}^{2} -2 a^{2} g_{a}^{e}{}^{2} (k.k_{1})^2 \,p_{-}.p_{+}\, \nonumber \\{} & {} - 2 a^{2} g_{v}^{e}{}^{2} (k.k_{1})^2 \,p_{-}.p_{+}\,\nonumber \\{} & {} +2 a^{2} g_{a}^{e}{}^{2} \,k.p_{-}\, k.k_{1}\, p_{-}.k_{1}\, + 2 a^{2}\nonumber \\\times & {} g_{v}^{e}{}^{2}\, k.p_{-}\, k.k_{1}\, p_{-}.k_{1}\, +2 a^{2} g_{a}^{e}{}^{2} \,k.p_{+}\, k.k_{1}\,p_{-}.k_{1}\,\nonumber \\{} & {} + 2 a^{2} g_{v}^{e}{}^{2} \,k.p_{+}\, k.k_{1}\, p_{-}.k_{1}\, \nonumber \\{} & {} +2 a^{2} (g_{a}^{e}{}^{2} + g_{v}^{e}{}^{2}) \nonumber \\ {}\times & {} \Big (\Big (k.p_{-}\, + \,k.p_{+}\Big ) \,k.k_{1}\, p_{+}.k_{1}\Big )-g_{a} g_{v} \,k.p_{+}\, \Big (-2 \Big (3 \,k.p_{-}\, \nonumber \\{} & {} - \,k.p_{+}\Big ) \Big (k.p_{-}\, + \,k.p_{+}\Big ) M_{Z}^{2} \nonumber \\{} & {} + \Big (\,k.p_{-}\,+3 \,k.p_{+}\Big ) \,k.k_{1}\, p_{+}.k_{1}\Big )\nonumber \\{} & {} \epsilon (a_{1},a_{2},k,p_{-})+g_{a} g_{v} (k.p_{-})\nonumber \\{} & {} \Big (2 \Big (k.p_{-}\, - 3 \,k.p_{+}\Big ) \Big (k.p_{-} \nonumber \\{} & {} + \,k.p_{+}\Big ) M_{Z}^{2} + \Big (3 \,k.p_{-}\, + \,k.p_{+}\Big ) \,k.k_{1}\, p_{-}.k_{1}\Big ) \epsilon (a_{1},a_{2},k,p_{+})\nonumber \\{} & {} +g_{a} g_{v} \Big (\Big (k.p_{-}\, - \,k.p_{+}\Big ) \Big (k.p_{-} \nonumber \\\times & {} k.p_{+}\Big ) \,k.k_{1}\, p_{-}.p_{+}\, \epsilon (a_{1},a_{2},k,k_{1})+3 \nonumber \\{} & {} \Big (k.p_{-}\, - \,k.p_{+}\Big )^2 (k.k_{1})^2 \epsilon (a_{1},a_{2},p_{-},p_{+})+(k.p_{-})^2\nonumber \\\times & {} k.p_{+}\, k.k_{1} \epsilon (a_{1},a_{2},p_{-},k_{1})+5 \,k.p_{-}\, (k.p_{+})^2 \,k.k_{1}\,\nonumber \\{} & {} \epsilon (a_{1},a_{2},p_{-},k_{1})-2 (k.p_{+})^3 \,k.k_{1} \nonumber \\\times & {} \epsilon (a_{1},a_{2},p_{-},k_{1}) 2 (k.p_{-})^3 \,k.k_{1}\, \epsilon (a_{1},a_{2},p_{+},k_{1})-5 (k.p_{-})^2 \,k.p_{+}\,k.k_{1}\,\nonumber \\{} & {} \epsilon (a_{1},a_{2},p_{+},k_{1})-\,k.p_{-}\nonumber \\ {}\times & {} (k.p_{+})^2 \,k.k_{1}\, \epsilon (a_{1},a_{2},p_{+},k_{1})+4 \,a_{2}.k_{1}\, k.p_{-}\,\nonumber \\{} & {} k.p_{+}\, k.k_{1}\,\epsilon (a_{1},k,p_{-},p_{+})-4 \,a_{2}.k_{1}\, (k.p_{-})^2\nonumber \\\times & {} \,k.p_{+}\, \epsilon (a_{1},k,p_{-},k_{1}) 4 \,a_{2}.k_{1}\,k.p_{-}\,(k.p_{+})^2 \epsilon (a_{1},k,p_{-},k_{1})\nonumber \\{} & {} +4 \,a_{2}.k_{1}\, (k.p_{-})^2 \,k.p_{+}\, \epsilon (a_{1},k,p_{+},k_{1})\nonumber \\{} & {} + 4 \,a_{2}.k_{1}\, k.p_{-}\, (k.p_{+})^2\nonumber \\{} & {} \epsilon (a_{1},k,p_{+},k_{1})-4 \,a_{1}.k_{1}\, k.p_{-}\, k.p_{+}\, k.k_{1}\epsilon (a_{2},k,p_{-},p_{+})\nonumber \\{} & {} +4 \,a_{1}.k_{1}\nonumber \\\times & {} (k.p_{-})^2 \,k.p_{+}\,\nonumber \\{} & {} \epsilon (a_{2},k,p_{-},k_{1}) 4 \,a_{1}.k_{1}\, k.p_{-}\, (k.p_{+})^2\nonumber \\{} & {} \epsilon (a_{2},k,p_{-},k_{1})-\,a_{1}.p_{+}\, k.p_{-}\, k.p_{+}\nonumber \\\times & {} k.k_{1}\, \epsilon (a_{2},k,p_{-},k_{1})+\, a_{1}.p_{+}\, (k.p_{+})^2 \,k.k_{1}\,\nonumber \\{} & {} \epsilon (a_{2},k,p_{-},k_{1})-\,k.p_{-}\, \Big (4 \,a_{1}.k_{1}\, k.p_{+}\, \nonumber \\{} & {} \Big (k.p_{-} \nonumber \\ {}{} & {} + k.p_{+}\Big ) + \, a_{1}.p_{-}\, \Big (k.p_{-}\, k.p_{+}\Big ) \,k.k_{1}\Big )\nonumber \\{} & {} \epsilon (a_{2},k,p_{+},k_{1})\Big )\Big )\bigg ] \end{aligned}$$
(31)
$$\begin{aligned} A_{3}= & {} -\dfrac{e^2}{2\Big (k.p_{-}\, k.p_{+}\, M_{Z}^{2}\Big )}\nonumber \\{} & {} \bigg [\Big (2 \,k.p_{-}\, k.p_{+}\, \Big (2 (a_{1}.k_{1})^2 (g_{a}^{e}{}^{2}\nonumber \\{} & {} + g_{v}^{e}{}^{2}) \,k.p_{-}\, k.p_{+}\, \nonumber \\{} & {} + 2 (a_{2}.k_{1})^2 (g_{a}^{e}{}^{2} + g_{v}^{e}{}^{2})\nonumber \\\times & {} k.p_{-}\, k.p_{+}\,-2 \,a_{1}.k_{1}\, (g_{a}^{e}{}^{2} + g_{v}^{e}{}^{2}) \Big (a_{1}.p_{+}\,k.p_{-}\,\nonumber \\{} & {} + \,a_{1}.p_{-}\, k.p_{+}\Big ) \,k.k_{1}\, +2 \,a_{1}.p_{-}\, a_{1}.p_{+}\, g_{a}^{e}{}^{2} \nonumber \\\times & {} (k.k_{1})^2 + 2\, a_{1}.p_{-}\, a_{1}.p_{+}\, g_{v}^{e}{}^{2} (k.k_{1})^2\nonumber \\{} & {} +2 a^{2} g_{a}^{e}{}^{2} (k.k_{1})^2 m_{e}^{2} \nonumber \\{} & {} - 2 a^{2} g_{v}^{e}{}^{2} (k.k_{1})^2 m_{e}^{2}+a^{2} g_{a}^{e}{}^{2} (k.p_{-})^2 M_{Z}^{2} a^{2} \nonumber \\\times & {} g_{v}^{e}{}^{2} (k.p_{-})^2 M_{Z}^{2}-2 a^{2} g_{a}^{e}{}^{2} \,k.p_{-}\, k.p_{+} M_{Z}^{2}\nonumber \\{} & {} - 2 a^{2} g_{v}^{e}{}^{2} \,k.p_{-}\, k.p_{+}\, M_{Z}^{2}\nonumber \\{} & {} +a^{2} g_{a}^{e}{}^{2} (k.p_{+})^2 M_{Z}^{2} + a^{2} g_{v}^{e}{}^{2}\nonumber \\\times & {} (k.p_{+})^2 M_{Z}^{2}-2 a^{2} g_{a}^{e}{}^{2} (k.k_{1})^2 \, p_{-}.p_{+}\, - 2 a^{2} g_{v}^{e}{}^{2} (k.k_{1})^2 \, p_{-}.p_{+}\, \nonumber \\{} & {} +2 a^{2} g_{a}^{e}{}^{2} \, k.p_{-}\, k.k_{1}\, p_{-}.k_{1}\, + 2 a^{2}\nonumber \\\times & {} g_{v}^{e}{}^{2} \, k.p_{-}\, k.k_{1}\, p_{-}.k_{1}\, + 2 a^{2} g_{a}^{e}{}^{2}\, k.p_{+}\, k.k_{1}\, p_{-}.k_{1}\,\nonumber \\{} & {} + 2 a^{2} g_{v}^{e}{}^{2} \,k.p_{+}\, k.k_{1}\, p_{-}.k_{1}\, +2 a^{2} (g_{a}^{e}{}^{2} + g_{v}^{e}{}^{2})\nonumber \\ {}\times & {} \Big (k.p_{-}\, k.p_{+}\Big ) \, k.k_{1}\, p_{+}.k_{1}\Big )+g_{a} g_{v}\,k.p_{+}\, \Big (-2 \Big (3 \, k.p_{-}\, - \,k.p_{+}\Big ) \Big (k.p_{-}\, + \,k.p_{+}\Big ) M_{Z}^{2} \nonumber \\{} & {} + \Big (k.p_{-}\, + 3 k.p_{+}\Big ) \,k.k_{1}\, p_{+}.k_{1}\Big ) \epsilon (a_{1},a_{2},k,p_{-})- g_{a} g_{v} \,k.p_{-} \Big (2 \Big (k.p_{-}\, - 3 \,k.p_{+}\Big ) \Big (k.p_{-}\nonumber \\ {}{} & {} + k.p_{+}\Big ) M_{Z}^{2} \nonumber \\{} & {} + \Big (3 \,k.p_{-}\, k.p_{+}\Big ) \,k.k_{1}\, p_{-}.k_{1}\Big ) \epsilon (a_{1},a_{2},k,p_{+})+ g_{a} g_{v} \Big (\Big (-(k.p_{-})^2 + (k.p_{+})^2\Big ) \,k.k_{1}\nonumber \\\times & {} p_{-}.p_{+}\,\epsilon (a_{1},a_{2},k,k_{1}) 3 \Big (k.p_{-}\, - k.p_{+}\Big )^2 (k.k_{1})^2 \nonumber \\{} & {} \epsilon (a_{1},a_{2},p_{-},p_{+})\nonumber \\{} & {} -(k.p_{-})^2 \,k.p_{+}\, k.k_{1}\nonumber \\\times & {} \epsilon (a_{1},a_{2},p_{-},k_{1})- 5 \,k.p_{-}\, (k.p_{+})^2 \,k.k_{1}\, \epsilon (a_{1},a_{2},p_{-},k_{1})\nonumber \\{} & {} +2 (k.p_{+})^3 \,k.k_{1}\, \epsilon (a_{1},a_{2},p_{-},k_{1})- 2 \nonumber \\ {}\times & {} (k.p_{-})^3 \,k.k_{1}\,\nonumber \\{} & {} \epsilon (a_{1},a_{2},p_{+},k_{1})+5 (k.p_{-})^2 \,k.p_{+}\, k.k_{1}\, \epsilon (a_{1},a_{2},p_{+},k_{1})+\,k.p_{-}\, (k.p_{+})^2 \,k.k_{1} \nonumber \\ {}\times & {} \epsilon (a_{1},a_{2},p_{+},k_{1})- 4 \,a_{2}.k_{1}\, k.p_{-}\, k.p_{+}\, k.k_{1}\, \epsilon (a_{1},k,p_{-},p_{+})+4 \,a_{2}.k_{1}\, (k.p_{-})^2 \,k.p_{+}\, \nonumber \\\times & {} \epsilon (a_{1},k,p_{-},k_{1})+4 \,a_{2}.k_{1}\, k.p_{-}\, (k.p_{+})^2 \epsilon (a_{1},k,p_{-},k_{1})4 \,a_{2}.k_{1}\, (k.p_{-})^2 \,k.p_{+}\,\epsilon (a_{1},k,p_{+},k_{1})\nonumber \\ {}{} & {} - 4 \,a_{2}.k_{1}\, k.p_{-}\, (k.p_{+})^2 \epsilon (a_{1},k,p_{+},k_{1})\nonumber \\{} & {} +4 \,a_{1}.k_{1}\, k.p_{-}\, k.p_{+}\, k.k_{1}\, \epsilon (a_{2},k,p_{-},p_{+})-4 \,a_{1}.k_{1}\nonumber \\\times & {} (k.p_{-})^2 \,k.p_{+} \epsilon (a_{2},k,p_{-},k_{1})-4 \,a_{1}.k_{1}\, k.p_{-}\, (k.p_{+})^2 \epsilon (a_{2},k,p_{-},k_{1})+\,a_{1}.p_{+}\, k.p_{-}\, k.p_{+}\, k.k_{1} \nonumber \\\times & {} \epsilon (a_{2},k,p_{-},k_{1})-\, a_{1}.p_{+}\, \nonumber \\{} & {} (k.p_{+})^2 \,k.k_{1}\, \epsilon (a_{2},k,p_{-},k_{1})\,k.p_{-}\,\nonumber \\{} & {} \Big (4 \,a_{1}.k_{1}\, k.p_{+}\, \Big (k.p_{-}\, + \,k.p_{+}\Big ) \nonumber \\ {}{} & {} + a_{1}.p_{-}\, \nonumber \\{} & {} \Big (k.p_{-}\, - \,k.p_{+}\Big ) \,k.k_{1}\Big ) \epsilon (a_{2},k,p_{+},k_{1})\Big )\Big )\bigg ] \end{aligned}$$
(32)
$$\begin{aligned} A_{4}= & {} \dfrac{2\, e}{\Big (k.p_{-}\, k.p_{+}\, M_{Z}^{2}\Big )}\bigg [(g_{a}^{e}{}^{2}\nonumber \\{} & {} +g_{v}^{e}{}^{2}) \Big (a^{2} e^2 \,k.k_{1}\, \Big (a_{1}.k_{1}\,\nonumber \\{} & {} \Big (k.p_{-}\, - \,k.p_{+}\Big ) + \Big (-a_{1}.p_{-}\,+\,a_{1}.p_{+}\Big ) \nonumber \\\times & {} k.k_{1}\Big )+\,a_{1}.p_{+} \,k.p_{-} \Big (-\Big (k.p_{-}\, + \,k.p_{+}\Big ) M_{Z}^{2} - 2 \,k.k_{1}\, p_{-}.k_{1}\Big )\nonumber \\{} & {} +\,k.p_{+}\, \Big (a_{1}.p_{-}\, \Big (k.p_{-} \nonumber \\ {}{} & {} + k.p_{+}\Big ) M_{Z}^{2}\nonumber \\{} & {} + 2\, a_{1}.k_{1}\, k.p_{-} \Big (p_{-}.k_{1}\,-\,p_{+}.k_{1}\Big )\nonumber \\{} & {} +2 \,a_{1}.p_{-}\, k.k_{1}\, p_{+}.k_{1}\Big )\Big )\bigg ] \end{aligned}$$
(33)
$$\begin{aligned} A_{5}= & {} \dfrac{2\, e}{\Big (k.p_{-}\, k.p_{+}\, M_{Z}^{2}\Big )}\bigg [\Big ((g_{a}^{e}{}^{2}+g_{v}^{e}{}^{2}) \,k.p_{-}\,k.p_{+}\,\nonumber \\{} & {} \Big (a^{2} e^2 \,k.k_{1} \Big (a_{1}.k_{1}\, \Big (k.p_{-}\, - \,k.p_{+}\Big ) + \Big (-a_{1}.p_{-}\nonumber \\ {}{} & {} + a_{1}.p_{+}\Big ) \,k.k_{1}\Big ) \nonumber \\{} & {} + \,a_{1}.p_{+}\, k.p_{-} \Big (-\Big (k.p_{-}\, + \,k.p_{+}\Big ) M_{Z}^{2} - 2 \,k.k_{1}\, p_{-}.k_{1}\Big )+\,k.p_{+} \Big (a_{1}.p_{-} \nonumber \\ {}\times & {} \Big (k.p_{-}\, + \,k.p_{+}\Big ) M_{Z}^{2} + 2 \,a_{1}.k_{1}\, k.p_{-} \nonumber \\{} & {} \Big (p_{-}.k_{1}\, - \, p_{+}.k_{1}\Big )+2 \,a_{1}.p_{-}\, k.k_{1}\, p_{+}.k_{1}\Big )\Big )+g_{a}^{e} g_{v}^{e}\nonumber \\\times & {} \Big (\Big (k.p_{-}\, - \,k.p_{+}\Big ) \Big (a^{2} e^2 (k.k_{1})^2 - 4 \,k.p_{-}\,k.p_{+}\, M_{Z}^{2}\Big )\nonumber \\{} & {} \epsilon (a_{2},k,p_{-},p_{+})+\,k.p_{+} \Big (k.p_{-}\, + \,k.p_{+}\Big ) \nonumber \\\times & {} \Big (a^{2} e^2 \,k.k_{1}\, + 2 \,k.p_{-}\, p_{+}.k_{1}\Big ) \epsilon (a_{2},k,p_{-},k_{1}) + \,k.p_{-} \Big (\Big (k.p_{-}\, \nonumber \\{} & {} + \,k.p_{+}\Big ) \Big (a^{2} e^2 \,k.k_{1}\, + 2 \,k.p_{+} \nonumber \\ {}\times & {} p_{-}.k_{1}\Big ) \epsilon (a_{2},k,p_{+},k_{1})+2 \, k.p_{+} \Big (-k.p_{-}\, +\, k.p_{+}\Big )\nonumber \\{} & {} \Big (-k.k_{1}\epsilon (a_{2},p_{-},p_{+},k_{1})\nonumber \\{} & {} + a_{2}.k_{1} \epsilon (k,p_{-},p_{+},k_{1})\Big )\Big )\Big )\Big )\bigg ] \end{aligned}$$
(34)
$$\begin{aligned} A_{6}= & {} - \dfrac{4\, e^2}{2\Big (k.p_{-}\,k.p_{+}\, M_{Z}^{2}\Big )}\nonumber \\{} & {} \bigg [(g_{a}^{e}{}^{2} + g_{v}^{e}{}^{2}) \Big (\Big (a_{1}.k_{1}\, - \,a_{2}.k_{1}\Big ) \Big (a_{1}.k_{1}\, + \,a_{2}.k_{1}\Big ) k.p_{-} \nonumber \\\times & {} k.p_{+}\, -\, a_{1}.k_{1} \Big (a_{1}.p_{+}\, k.p_{-}\, \nonumber \\{} & {} + \,a_{1}.p_{-}\, k.p_{+}\Big ) k.k_{1}\, + \,a_{1}.p_{-}\,a_{1}.p_{+}(k.k_{1})^2\Big )\Big )\bigg ] \end{aligned}$$
(35)

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Ouhammou, M., Ouali, M., Taj, S. et al. Analysis of the laser geometry effect on the Higgs-strahlung process. Indian J Phys (2024). https://doi.org/10.1007/s12648-024-03160-0

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  • DOI: https://doi.org/10.1007/s12648-024-03160-0

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