Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 304 / 2 / a / ii / Solution

When $h_0=0$, the cubic contribution is absent andAlthough terms odd in ${\varphi }^{+}$ occur for a fixed low field, the simultaneous transformation $(\varphi ,{\varphi }^{+})\mapsto (-\varphi ,-{\varphi }^{+})$ shows that integrating out the shell preserves the original ${\mathbb{Z}}_2$ discrete symmetry of the effective action.