# How to verify gauge invariance of an amplitude

I have calculated a tree level amplitude for Compton scattering (${e\left(p\right)+\gamma\left(k\right)\to e\left(p\prime\right)+\gamma\left(k\prime\right)}$):

$${ i\mathcal{M}=M_{\mu\nu}\epsilon^{*\mu}\left(k\prime\right)\epsilon^{\nu}\left(k\right)\textrm{.} }$$

How should I go about trying to verify that it is gauge invariant?

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?