Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 303 / 2 / b / ii / Solution

The first correction to the four-point coupling containing ${\lambda }_0$ is of order ${\lambda }_0^2$: two cubic vertices are joined by one fast internal propagator, leaving four slow external legs. The three exchange channels pair the external legs in the $s$, $t$, and $u$ ways. In a sharp momentum-shell scheme this contribution is retained when the exchanged momentum lies in the eliminated shell; its derivative expansion supplies the induced local quartic interaction.