I was looking at the theory with interaction Lagrangian $L_\text{int}=\phi^3 \cdot \partial_{\mu}\phi$.
I was computing the following self-loop diagram \begin{equation} \langle \phi_x \phi_z^3 \cdot \partial_{\mu}\phi_z \phi_y \rangle =3\cdot D_{xz}\partial_{\mu}D_{zz}D_{zy}. \end{equation} But since in euclidean signature \begin{equation} \partial_{\mu}D_{zz}=\partial_{\mu}\int_p e^{-i(x-x)}\frac{1}{p^2+m^2} =0 \end{equation} the diagram should vanish.
So is it true to generally conclude that Feynman graphs vanish when the derivative acts upon a self-loop (i.e. propagator evaluated at same spacetime point)?
(Hence also every diagram with a sub-diagram like this?)
