Quantum Effects of a Spacetime Varying alpha on the Propagation of   Electrically Charged Fermions
  A spacetime-varying fine structure constant alpha(x^mu) 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 d_mu alpha, 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^(lambda mu nu) and c^(mu nu) terms are generated, at O(d_mu alpha) and O[(d_mu alpha)^2], respectively. The g^(lambda mu nu) terms so generated are different in the vertex and self-energy, which represents a radiatively induced violation of gauge invariance. 
Alejandro Ferrero, Brett Altschul
hep-ph hep-th