Nonrelativistic effective potential of the bumblebee model

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 $\xi$ (and the second order in the LV vector ${\beta }_{\mu }$) it demonstrates the $1\mathrm{/}r$ asymptotics, which naturally corresponds to the massless character of the theory, while higher orders in $\xi$ 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.