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After integrating out the shell, let a chosen operator's coefficient be $\kappa +\Delta \kappa$ and write the two-derivative quadratic term as $\frac{1}{2}(1+\Delta Z)(\partial \varphi {)}^2$. Here $\Delta \kappa$ is the shell-induced change before rescaling, and $\Delta Z$ is the shell correction to the wavefunction renormalization. Work in an operator basis in which that quadratic term has this form; the other generated operators remain in the Wilsonian effective action.

Under $x^{\prime }=bx$, the volume element changes by $d^{4}x=b^{-4}{d}^4{x}^{\prime }$ and each derivative by ${\partial }_x=b{\partial }_{x^{\prime }}$. Canonical field normalization is restored by definingIndeed, the volume factor, two derivatives and two field factors cancel in the kinetic term. A term with $n$ fields and $m$ derivatives consequently obtains the factor $b^{-4}{b}^m{b}^n(1+\Delta Z{)}^{-n/2}$. HenceThis is Wilsonian rescaling of a scalar coupling. The exponent is minus the coupling's engineering mass dimension, $[\kappa ]=4-n-m$. The rescaling also returns the low-momentum cutoff to $\Lambda$, because $p^{\prime }=p/b$.