Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 304 / 3 / ii

The box has four external fields at all momenta. Its constant term gives ${\varphi }^4$; the momentum-dependent terms describe a derivative expansion of a four-point vertex. With an analytic local-expansion prescription, quadratic momentum terms give operators such as ${\varphi }^2{\partial }_{\mu }\varphi {\partial }_{\mu }\varphi$ or ${\varphi }^3{\partial }^2\varphi$, equivalent up to integration by parts, followed by four-field operators with more derivatives. The kernel also retains its full momentum dependence as a nonlocal four-field interaction. ThereforeThe latter come from different diagrams. For the strictly sharp projector on every propagator in the preceding part, moving shell boundaries can produce nonanalytic momentum dependence. The usual derivative expansion is consequently a smooth-regulator or fixed-loop-cutoff local expansion; the zero-momentum quartic coefficient calculated next does not depend on that distinction.