Radiatively induced Lorentz and gauge symmetry violation in electrodynamics with varying $\alpha$

A time-varying fine structure constant $\alpha (t)$ could give rise to Lorentz- and $CPT$-violating changes to the vacuum polarization, which would affect photon propagation. Such changes to the effective action can violate gauge invariance, but they are otherwise permitted. However, in the minimal theory of varying $\alpha$, no such terms are generated at lowest order. At second order, vacuum polarization can generate an instability—a Lorentz-violating analogue of a negative photon mass squared $-m_{\gamma }^2\propto \alpha (\frac{\dot{\alpha }}{2\alpha }{)}^2\log ({\Lambda }^2)$, where $\Lambda$ is the cutoff for the low-energy effective theory.