Path-integral perturbation by source derivatives ($Z[J]\propto e^{i{S}_{\mathrm{int}}[(1/i)\delta /\delta J]}{Z}_0[J]$)

An interaction depending on the field can be represented by substituting $(1/i)\delta /\delta J$ into its action and acting on the free generating functional. Expanding that exponential produces Feynman diagrams, contractions and vertex weights. Normalize by the resulting value at $J=0$ to cancel vacuum diagrams.