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

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 ${\mu }^{\prime 2}$.

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.

If ${\lambda }_0=0$, the action has the exact discrete symmetry $\varphi \mapsto -\varphi$. Integrating out modes and rescaling preserve this symmetry, so no odd operator such as ${\varphi }^3$ can be generated from the even quartic coupling. Thus $g_0$ corrects $\lambda$ at no order in perturbation theory: the hypersurface $\lambda =0$ is invariant under the renormalization-group flow.