Don't forget that the polarization tensors depend on the gauge choice via reference vectors (call them $q$, $q'$) Now you have to check what happen when you chance the reference vectors from $q, q'$ to some new vectors $r,r'$. The change of the vectors will lead to the new polarization vectors aquire a term proportional to its momentum $p$.
$$\epsilon(p,r)^\mu \sim \epsilon(p,q)^\mu+p^\mu$$

The contraction of the last term with $M_{\mu\nu}$ vanishes, i.e. you have shown gauge invariance.
Do you also have to show that $M_{\mu\nu}$ contracted into one of its momenta vanishes?

This post imported from StackExchange Physics at 2014-04-15 16:46 (UCT), posted by SE-user A friendly helper