Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 303 / 2 / a / Solution

A momentum-shell renormalization group step has three parts:

1. Split $\varphi ={\varphi }_{<}+{\varphi }_{>}$ into slow modes $|k|<\Lambda /\zeta$ and fast modes $\Lambda /\zeta <|k|<\Lambda$, then integrate out ${\varphi }_{>}$ using a cumulant expansion to obtain a Wilsonian effective action for ${\varphi }_{<}$.
2. Rescale momenta by $k^{\prime }=\zeta k$, equivalently coordinates by $x^{\prime }=x/\zeta$, restoring the cutoff from $\Lambda /\zeta$ to $\Lambda$.
3. Rescale the field by ${\varphi }^{\prime }(x^{\prime })={\zeta }^{(d-2)/2}{\varphi }_{<}(x)$ at the Gaussian fixed point, restoring the normalized kinetic term.

The resulting functional has the original form and cutoff but new masses and couplings. Iteration produces a renormalization-group flow.