In an exercise is given the following Lagrangian $$\mathcal{L} = -\frac{1}{4}F^{\mu\nu}F_{\mu\nu} + \frac{1}{2}MA^\mu A_\mu + \bar\psi(i\gamma^\mu\partial_\mu - m)\psi + \mathcal{L}_{int}$$ with the following interaction term $$\mathcal{L}_{int} = k\,\bar\psi(c_V+ic_A\gamma_5)\psi\,\partial_\mu A^\mu$$ Assuming that $\mathcal H_{int} = -\mathcal L_{int}$, it's asked at first, to evaluate the expected value: $$\langle0|\partial_\mu A^\mu|a_{(\lambda)}(k)\rangle $$ where $|a_{(\lambda)}(k)\rangle$ is a state with one vector particle of momentum $k$ and polarization $\lambda$.
Then it's asked to derive the Feynman rules for this theory.
How shall I deal with this kind of derivative interaction term, since I know that it has to vanish $\partial_\mu A^\mu= 0$ on the free field solution (also related to $k^\mu \epsilon_\mu^\lambda = 0$ for $\lambda = 1,2,3$)?


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