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

A Wilsonian renormalization group step first splits the field into slow and fast Fourier modes and integrates out the shell $\Lambda /b<|k|<\Lambda$. It then rescales coordinates, or momenta, to restore the cutoff to $\Lambda$, and finally rescales the field to normalize the kinetic term. The resulting local effective free energy has the same allowed operators with changed coefficients. Repeating the operation therefore composes maps on the set of couplings and defines a renormalization-group flow.

For couplings $g_i(\ell )$ under coarse-graining scale $b=e^{\ell }$, their beta function is ${\beta }_i(g)=d{g}_i/d\ell$. At a renormalization-group fixed point $g_{\ast }$ all beta functions vanish. Linearization givesThe eigenvalues $y_a$ of the stability matrix of a renormalization-group fixed point are the scaling dimensions of the associated coupling directions: $\delta g_a(\ell )=e^{y_a\ell }\delta g_a(0)$.

A coupling direction is relevant, irrelevant, or marginal according as its renormalization-group eigenvalue is positive, negative, or zero. The critical surface is the stable manifold of a critical fixed point: initial couplings on it flow toward that fixed point, while relevant perturbations take the theory away from criticality. Microscopic models with different irrelevant couplings lose those distinctions under coarse-graining and approach the same fixed point, which explains their shared long-distance behavior and universality class.

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.