Suppose we have a lagrangian $\mathcal{L}$ made by different fields, i.e. \begin{equation} \mathcal{L}= \mathcal{L_0} + g\phi\partial_\mu\phi A^\mu, \end{equation} where $\mathcal{L_0}$ is the free lagrangian for the fields $A^\mu$ and $\phi$.
Why is the 2-point correlation function $$\langle\Omega|T(\phi A^\mu)|\Omega\rangle = 0~?$$ In my lecture notes, my professor just assume this. Is it true in general for all types of contractions between different fields? Why? I get the intuitive explanation: if in the free lagrangian there are no terms like $\phi A^\mu$ we cannot turn a $\phi$ particle into an $A^\mu$ particle, but how can I see this more rigorously?