  We examine the renormalization group flow in the vicinity of the free-field
fixed point for effective field theories in the presence of a constant,
nondynamical vector potential background. The interaction with this vector
potential represents the simplest possible form of Lorentz violation. We search
for any normal modes of the flow involving nonpolynomial interactions. For
scalar fields, the inclusion of the vector potential modifies the known modes
only through a change in the field strength renormalization. For fermionic
theories, where an infinite number of particle species are required in order
for nonpolynomial interactions to be possible, we find no evidence for any
analogous relevant modes. These results are consistent with the idea that the
vector potential interaction, which may be eliminated from the action by a
gauge transformation, should have no physical effects.
