Gauge propagator and physical consistency of the $CPT$-even part of the standard model extension

In this work, we explicitly evaluate the gauge propagator of the Maxwell theory supplemented by the $CPT$-even term of the standard model extension. First, we specialize our evaluation for the parity-odd sector of the tensor $W_{\mu \nu \rho \sigma }$, using a parametrization that retains only the three nonbirefringent coefficients. From the poles of the propagator, it is shown that physical modes of this electrodynamics are stable, noncausal and unitary. In the sequel, we carry out the parity-even gauge propagator using a parametrization that allows to work with only the isotropic nonbirefringent element. In this case, we show that the physical modes of the parity-even sector of the tensor $W$ are causal, stable and unitary for a limited range of the isotropic coefficient.