Nonrelativistic effective potential of the bumblebee model

TL;DR

This work derives the nonrelativistic Breit potential in the weak-gravity limit of metric-affine bumblebee gravity coupled to spinor matter. The leading nonrelativistic potential retains the $1/r$ Coulomb form, while higher orders in the LV coupling $\xi$ introduce anisotropic corrections. The authors compute the LV-induced shift of hydrogenic energy levels arising at $O(\xi^2)$ and express it as a modification of the Coulomb strength: $V_{eff} = -\frac{1}{4\pi r}\left(1-\left(\frac{m\beta\xi}{2}\right)^2\right)$. They estimate the magnitude of these shifts under current bounds, finding them exceedingly small, and discuss the potential for higher-order corrections and future work.

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.