Nonrelativistic effective potential of the bumblebee model

Abstract

In this paper, we explicitly obtain the nonrelativistic Breit potential in the bumblebee model arising in the weak gravity limit of the metric-affine bumblebee gravity, coupled to the spinor matter. In this theory, in the lower (second) order in the small coupling constant $ξ$ (and the second order in the LV vector $β_μ$) it demonstrates the $1/r$ asymptotics, which naturally corresponds to the massless character of the theory, while higher orders in $ξ$ yield anisotropic modifications of the Coulomb potential due to the Lorentz symmetry breaking. For the lower-order modification of the effective potential, we calculate LV corrections to energy levels of the hydrogen atom.