I have to work on vacuum polarization and gauge contributions for a given problem.
I have to compute and show that their sum is gauge invariant, which according to the exercise, is equivalent to showing that
$$ p_\mu M^{\mu,\nu;a,b} = 0 $$ or equivalently, $$ M^{\mu,\nu;a,b} (p) = \delta^{ab}(g^{\mu \nu} p ^2 - p^\mu p ^\nu)\Pi(p^2) $$ Why does $M$ having this form imply that it is gauge invariant? I have read that the fact that this holds is due to current conservation and also the Ward Identity. I am looking for insights on this and how to see gauge invariance from it.
