= Derivative expansion of a four-point vertex
The small-external-momentum expansion of a <four-point one-particle-irreducible correlation function> generates a local quartic term followed by four-field terms with derivatives, for example $\phi^2(\partial\phi)^2$ and higher-derivative operators, modulo <integration by parts>. This is an analytic expansion when there is an appropriate gap and regulator smoothness. A sharp projector on every internal momentum can instead generate nonanalytic boundary artifacts; then the local expansion is a smooth-regulator or fixed-loop-cutoff convention, not an automatic theorem about the entire sharp-cutoff vertex.
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