Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 303 / 2 / c / ii

Writeand, for the three-line topology,and define the four-line integralwith every propagator momentum restricted to the fast shell.

The first cumulant containsSince the quadratic free energy is $\frac{1}{2}{\mu }^2({\varphi }^{-}{)}^2$, this gives

The mixed term in the second cumulant isWick contraction with the printed normalization $V_6={\lambda }_0\int {\varphi }^6$ and $V_4=g_0\int {\varphi }^4$ gives, at zero external momentum,The coefficients respectively combine the two placements of both external legs in the two-line topology, the three-line topology with one external leg on each vertex, and the four-line topology. Couplings normalized as $g_0{\varphi }^4/4!$ and ${\lambda }_0{\varphi }^6/6!$ absorb the corresponding factorials, which is why formulas in that convention have much smaller numerical coefficients.