 Solution

Let $\hat{C}|\Omega ⟩=|\Omega ⟩$. Since charge conjugation sends both $A_{\mu }$ and either Noether current to its negative and commutes with time ordering,For odd $n$ the correlator equals its negative and hence vanishes. This operator argument is exact and does not use a perturbation expansion. It is Furry's theorem: scattering amplitudes with an odd number of external photons and no charged external particles vanish to every order, whereas even-photon scattering is allowed.