Past exam of the quantum field theory course of the University of Oxford / 2017 / 3 / f / Solution

To first order, invert $m_0^2=m^2[1+g/(8{\pi }^2ϵ)]$ and use $g={\lambda }_0{\mu }^{-ϵ}+O({\lambda }_0^2)$:The remainder is a perturbative statement at fixed regulator. When differentiating, hold the bare mass $m_0^2$ fixed as well as ${\lambda }_0$; otherwise the mass running would not be defined by a fixed bare theory. Since $\mu {\partial }_{\mu }{\mu }^{-ϵ}=-ϵ{\mu }^{-ϵ}$,Thus, with the source's extra factor of one half,The factor $-ϵ$ from the running dimensionless bare coupling cancels the counterterm pole. Differentiating as though $g$ were independent of $\mu$ would miss this finite result. Some conventions call $\mu d\log m^2/d\mu$ itself the mass anomalous dimension; that convention is twice the one defined here.