A spacetime-varying fine structure constant α(x µ) could generate quantum corrections in some of the coefficients of the Lorentz-violating standard model extension (SME) associated with electrically charged fermions. The quantum corrections depend on ∂µα, the spacetime gradient of the fine structure constant. Lorentz-violating operators involving fermions arise from the one-loop corrections to the quantum electrodynamics (QED) vertex function and fermion self-energy. Both g λµν and c µν terms are generated, at O(∂µα) and O[(∂µα) 2 ], respectively. The g λµν terms so generated are different in the vertex and self-energy, which represents a radiatively induced violation of gauge invariance.
Ferrero, A., & Altschul, B. (2012). Quantum effects of a spacetime varying alpha on the propagation of electrically charged fermions. ArXiv:1210.3826 [Hep-Ph, Physics:Hep-Th]. http://arxiv.org/abs/1210.3826