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

Split $\varphi ={\varphi }_{<}+{\varphi }_{>}$ and write $I_n={\int }_{\Lambda /b<|k|<\Lambda }\frac{d^{d}k}{(2\pi {)}^d}{G}_0(k{)}^n$. The first cumulant of $g_0{\varphi }^4$ contains $6g_0{\varphi }_{<}^2⟨{\varphi }_{>}^2⟩=6g_0{I}_1{\varphi }_{<}^2$. The connected second cumulant of the two terms $3{\lambda }_0{\varphi }_{<}{\varphi }_{>}^2$ uses Wick theorem to give $⟨{\varphi }_{>}^2(x){\varphi }_{>}^2(y){⟩}_c=2G_{>}(x-y{)}^2$. Keeping its leading local term, the quadratic coefficient after integrating out the shell is thereforeThe quartic tadpole raises the mass parameter, while the pair of cubic vertices gives the negative bubble contribution. A cubic interaction also generates a linear tadpole $3{\lambda }_0{I}_1{\varphi }_{<}$, which may be removed by shifting the background but is not part of ${\mu }^{\prime 2}$.