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

The dimensionless bare parameter is $g_0={\lambda }_0{\mu }^{-ϵ}$, so direct differentiation at fixed ${\lambda }_0$ givesThe dimensionful bare vertex itself is scale independent. Reexpressing it as ${\hat{\Gamma }}^{(n)}(p,g_0,\mu )={\Gamma }_0^{(n)}(p,g_0{\mu }^{ϵ})$ introduces only the combination $g_0{\mu }^{ϵ}$. HenceThere is no multiplicative field rescaling in this bare parametrization, soThis does not assert that the interacting renormalized field has zero anomalous dimension. It says that the unrescaled bare field has no additional normalization factor under this change of the arbitrary scale.