\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,amsfonts}
\begin{document}
In this work, the conversion of linear polarization of a laser beam to circular one through its forward scattering by a TeV order charged lepton beam in the presence of Lorentz violation correction is explored. We calculate the ratio of circular polarization to linear one (Faraday conversion phase Ã¯Â¿Â½?Ã¯Â¿Â½?FC ) of the laser beam interacting with either electron or the muon beam in the framework of the quantum Boltzmann equation. Regarding the experimentally available sensitivity to the Faraday conversion Ã¯Â¿Â½?Ã¯Â¿Â½?FCÃ¯Â¿Â½??10Ã¯Â¿Â½??3Ã¯Â¿Â½??10Ã¯Â¿Â½??2 , we show that the scattering of a linearly polarized laser beam with energy k0Ã¯Â¿Â½?Ã¯Â¿Â½0.1 eV and an electron/muon beam with flux ÃÂµÃÂ¯e,ÃÂ¼Ã¯Â¿Â½?Ã¯Â¿Â½1010/1012 TeV cm Ã¯Â¿Â½??2 s Ã¯Â¿Â½??1 places an upper bound on the combination of lepton sector Lorentz violation coefficients cÃÂ¼ÃÂ½ components (cTT+1.4 c(TZ)+0.25(cXX+cYY+2 cZZ) . The obtained bound on the combination for the electron beam is at the 4.35Ã¯Â¿Â½?10Ã¯Â¿Â½??15 level and for the muon beam at the 3.9Ã¯Â¿Â½?10Ã¯Â¿Â½??13 level. It should be mentioned that the laser and charged lepton beams considered here to reach the experimentally measurable Ã¯Â¿Â½?Ã¯Â¿Â½?FC are currently available or will be accessible in the near future. This study provides a valuable supplementary to other theoretical and experimental frameworks for measuring and constraining Lorentz violation coefficients.
\end{document}