Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 303 / 2 / a

A momentum-shell renormalization group transformation consists of three steps. First split $\varphi ={\varphi }_{-}+{\varphi }_{+}$ into slow modes with $|k|<\Lambda /\zeta$ and fast modes with $\Lambda /\zeta <|k|<\Lambda$, and integrate out ${\varphi }_{+}$ to obtain a Wilsonian effective action for ${\varphi }_{-}$. Next rescale momenta by $k^{\prime }=\zeta k$, equivalently coordinates by $x^{\prime }=x/\zeta$, to restore the cutoff to $\Lambda$. Finally rescale the field to restore the chosen normalization of the gradient term. Repeating these operations produces a renormalization-group flow of every permitted mass and coupling.