Source renormalization of a scalar generating functional ($h_0={\mu }^{-ϵ/2}{Z}^{-1/2}h$)

For $d=4-ϵ$, write the bare field as ${\varphi }_0=Z^{1/2}\varphi$ and the renormalized source coupling as ${\mu }^{-ϵ/2}h\varphi$. In bare variables this is $h_0{\varphi }_0$, with the displayed source. At fixed bare parameters, the source running is $\mu d\log h/d\mu =ϵ/2+\frac{1}{2}\hat{\beta }{\partial }_{\lambda }\log Z$. Normalizing the generating functional by its zero-source value cancels source-independent vacuum diagrams and the constant measure factor from field rescaling. It does not remove source-dependent ultraviolet divergences or infrared divergences; the former require the ordinary field, mass and interaction counterterms.