Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 304 / 2 / a / Solution

For the Euclidean quartic scalar field theory, expansion of $e^{-S}$ gives the momentum-space rules

At one loop, the four-point one-particle-irreducible correlation function receives the three bubble diagrams in the $s$, $t$, and $u$ channels. If $P$ is the momentum through one channel, its contribution isUsing a Feynman parameter, shifting the loop momentum, and writing ${\Delta }_x=m^2+x(1-x)P^2$ gives, up to terms caused by shifting the boundary of a hard cutoff,Thus every channel has the logarithmic ultraviolet divergenceThe complete one-loop vertex isLet $I_{m,\mathrm{os}}$ denote the bracket evaluated at the chosen on-shell kinematic point. The on-shell renormalization scheme requires $V_a^{(4)}{|}_{\mathrm{os}}=-{\lambda }_{\mathrm{phys}}$, hence the perturbative solution isThe cutoff dependence of ${\lambda }_a$ is the coupling counterterm needed to hold the measured coupling fixed.