Gravitational Lorentz-violating $e^-+e^+\to\ell^-+\ell^+$ scattering

We investigate the gravitational $e^-+e^+\to\ell^-+\ell^+$ scattering process within the framework of gravitoelectromagnetism, a weak-field approximation of gravity analogous to Maxwell's theory of electromagnetism. This process involves the interaction between a fermion and an antifermion mediated by graviton exchange. We consider the nonminimal gravitational sector of the standard model extension and calculate the corrections to the scattering cross section arising from Lorentz violation. The analysis is carried out in two scenarios: (i) at zero temperature and (ii) at finite temperature. To incorporate thermal effects, we employ the thermo field dynamics formalism, which allows for a consistent treatment of quantum fields at finite temperature. The results provide insights into how Lorentz-violating and thermal corrections influence gravitational interactions, particularly relevant in high-energy or astrophysical environments.