Probing Lorentz violation effects via a laser beam interacting with a high-energy charged lepton beam

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 $\Delta\phi_{\rm{FC}}$) 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 $\Delta\phi_{\rm{FC}}\simeq 10^{-3}-10^{-2}$, we show that the scattering of a linearly polarized laser beam with energy $k_0\sim 0.1$ eV and an electron/muon beam with flux $\bar{\epsilon}_{e,\mu}\sim 10^{10}/10^{12}$ TeV cm$^{-2}$ s$^{-1}$ places an upper bound on the combination of lepton sector Lorentz violation coefficients $c_{\mu\nu}$ components $(c_{TT}+1.4~c_{(TZ)}+0.25(c_{XX}+c_{YY}+2~c_{ZZ}))$. The obtained bound on the combination for the electron beam is at the $4.35\times 10^{-15}$ level and for the muon beam at the $3.9\times 10^{-13}$ level. It should be mentioned that the laser and charged lepton beams considered here to reach the experimentally measurable $\Delta\phi_{\rm{FC}}$ 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.