Past exam of the quantum field theory course of the University of Oxford / 2017 / 4 / c / Solution

Write ${\mathcal{D}}_0=\mu {\partial }_{\mu }+{\beta }_0(g_0){\partial }_{g_0}$. Allow a general bare coefficient ${\gamma }_0$ temporarily, so the starting equation is $({\mathcal{D}}_0-n{\gamma }_0/2){\hat{\Gamma }}^{(n)}=0$. Substitute ${\hat{\Gamma }}^{(n)}=Z_{\varphi }(g_0{)}^{-n/2}{\Gamma }^{(n)}(p,g(g_0),\mu )$ and apply the product and chain rules:Multiplying by $Z_{\varphi }^{n/2}$ identifiesThus the renormalization-group beta function transforms by the chain rule under a coupling reparametrization, while the field coefficient also records the scale dependence of the finite or divergent field normalization. For the actual bare parametrization here, substitute ${\beta }_0=-ϵg_0$ and ${\gamma }_0=0$.