Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 303 / 2 / a

A momentum-shell renormalization group step has three parts. First split the Fourier transform of the field into slow modes ${\varphi }_{<}$ with $|k|<\Lambda /\zeta$ and fast modes ${\varphi }_{>}$ with $\Lambda /\zeta <|k|<\Lambda$, then perform the functional integral over ${\varphi }_{>}$. Second rescale momenta by $k^{\prime }=\zeta k$, equivalently coordinates by $x^{\prime }=x/\zeta$, to restore the cutoff from $\Lambda /\zeta$ to $\Lambda$. Third rescale the field so that the coefficient of $(\nabla \varphi {)}^2/2$ again has its chosen normalization. The effective free energy contains every operator allowed by the symmetries, with transformed coefficients. Repeating the step composes these coefficient maps and produces a renormalization-group flow.