Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 303 / 2 / a

A momentum-shell renormalization group step has three parts:

1. Split the field into slow modes ${\varphi }^{-}$ with $|k|<\Lambda /\zeta$ and fast modes ${\varphi }^{+}$ with $\Lambda /\zeta <|k|<\Lambda$, then integrate out ${\varphi }^{+}$ to obtain a Wilsonian effective action for ${\varphi }^{-}$.
2. Rescale momenta by $k^{\prime }=\zeta k$, equivalently coordinates by $x^{\prime }=x/\zeta$, so the reduced cutoff returns from $\Lambda /\zeta$ to $\Lambda$.
3. Rescale the field, at the Gaussian fixed point by ${\varphi }^{\prime }(x^{\prime })={\zeta }^{(d-2)/2}{\varphi }^{-}(x)$, so the coefficient of $(\nabla \varphi {)}^2/2$ retains its chosen normalization.

The resulting functional has the original cutoff but changed coefficients. Iterating the operation gives a renormalization-group flow on masses and interaction couplings.