I am considering the following scalar QED lagrangian: $$L = −\frac{1}{4}F_{\mu\nu}^2 + |D_{\mu\varphi}|^2 − m^2|\varphi|^2− \frac{1}{2\xi}(\partial_\mu A^\mu)^2.$$
Where I want to show that the correlation function in the path integral formulation is independent on the choice of $\xi$ by using the identity: $$1 =\frac{\int{D\pi e^{−i\int{d^4x\frac{1}{2\xi}(\square\pi−\partial_\mu A^\mu)^2}}}}{\int{D\pi e^{−i\int{d^4x\frac{1}{2\xi}(\square\pi)^2}}}}.$$
I think the answer lies in the given identity where using a redefinition of the integration variable $π$ should prove this. I've tried showing this by using $$\pi' = \pi - \frac{\partial_\mu A^\mu}{\square}.$$ however this does not seem to give me the correct answer.
I think it has to do with the fact that the transformation of $\pi$ only adds boundary terms to the exponential, which vanish at spatial infinity, which would also hold for the action.
Is this a correct way of interpreting this problem or am I missing something?
