extended electrodynamics

Could one allow a term \(\propto F_{\mu\nu}\tilde{F}^{\mu\nu}\) in the Maxwell-Lagrange-density \(\mathcal{L}\)? if no, what principle would it violate? (\(\tilde{F}^{\mu\nu}\) is dual to \(F^{\mu\nu}\).) Is the term Lorentz-invariant and if yes, could it be expressed in the fields \(\vec{E}\) and \(\vec{B}\)?

The term \(F_{\mu\nu}\tilde{F}^{\mu\nu}\) Lorentz-invariant as a contraction of two Lorentz-tensors, and is \(\propto \vec{E}\cdot\vec{B}\) expressed in terms of the fields. There one can see that the invariant is in fact pseudoscalar instead of scalar, as a product between a polar and an axial vector, and is therefore not admissible in the Lagrange-density. Multiplying the term with a pseudoscalar field \(\theta\) would give a valid term with parity-positive properties, which is the entry point to axion electrodynamics.