Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 303 / 3 / c / Solution

For small $\mathbf{\pi }$, expand $(1-{\mathbf{\pi }}^2{)}^{-1}=1+O({\mathbf{\pi }}^2)$ and retain the quartic interaction $(\mathbf{\pi }\cdot {\partial }_i\mathbf{\pi }{)}^2/(2g_0)$. Split $\mathbf{\pi }={\mathbf{\pi }}^{-}+{\mathbf{\pi }}^{+}$ and contract the fast fields in the shell $\Lambda /\zeta <|q|<\Lambda$. To first order inthe contraction adds $I_d$ to the inverse coefficient of the slow-field kinetic term. The prescribed wave-function renormalizationthen multiplies that coefficient by $A^2$. HenceFinally the coordinate rescaling contributes ${\zeta }^{d-2}$, and thus