My professor has given me the following action stating that $a(x)$ is an axionic field and told us in class that for this action to be Lorentz invariant the field must be a pseudoscalar.
$$ S = -\int d^4x \frac{1}{2} \partial^\mu a \partial_\mu a - \frac{1}{4} F_{\mu \nu} F^{\mu \nu} - \frac{1}{f}[aF_{\mu \nu} \left.^* F^{\mu \nu} - 2 \partial_\mu (a A_\nu \left.^*F^{\mu \nu}) \right. \right.]$$
This is the first time that I encounter such fields in my life. I tried googling around to find some information but I didn't even understand what such a field is.
I tried proving that $S$ is Lorentz invariant iff $a$ is a pseudoscalar using the definition of a Lorentz transformation
$$\delta x^b = \delta \omega^b_a x^a$$
But the computations seem to be too difficult to me. Is there any shortcut or alternative way to prove this?