Wilsonian rescaling of a scalar coupling (${\kappa }^{\prime }=(\kappa +\Delta \kappa )b^{m+n(d-2)/2-d}/(1+\Delta Z{)}^{n/2}$)

After shell integration and $x^{\prime }=bx$, restore the kinetic term by ${\varphi }^{\prime }=b^{-(d-2)/2}\sqrt{1+\Delta Z}\varphi$. An operator with $n$ scalar fields and $m$ derivatives scales by $b^{m+n(d-2)/2-d}$. In four dimensions this gives the displayed factor with $m+n-4$; the general exponent is $m+n(d-2)/2-d$. The coupling shift and wavefunction renormalization arise before the rescaling.