Wilsonian effective action

A Wilsonian effective action is obtained by integrating out field modes above a chosen momentum scale while retaining lower-momentum modes as backgrounds. Its local couplings change with the scale according to a renormalization-group flow.

The Polchinski equation changes an interaction action with its regulated Gaussian covariance so that the normalized functional integral stays unchanged. In operator notation, ${\dot{e}}^{-S_1}=-\dot{C}:{\delta }^2{e}^{-S_1}/2$. The resulting action flow has a product of first functional derivatives, joining two vertices, and a second derivative, closing two legs into a loop. It generates a Wilsonian effective action, which is not the same functional as the one-particle-irreducible correlation functions encoded by the quantum effective action.

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.

The cumulant expansion of the shell integral keeps precisely connected Feynman diagrams built from shell-restricted scalar propagators. The external low fields are backgrounds, not propagators. Connected graphs with bridges can contribute, so the expansion is not limited to one-particle-irreducible Feynman diagrams.

If a bridge is the only line carrying momentum away from a low external field, tadpole pairs do not change its momentum and the shell-restricted scalar propagator has zero support there. A bridge attached to a product such as ${\varphi }_{<}^3$ can survive because the sum of three low momenta can enter the shell. This matters before a local derivative expansion.

Split Fourier transform modes into a low-momentum ball and its disjoint shell. The quadratic scalar action has no cross term between these supports. Integrating the shell at fixed low field defines the lower-cutoff Wilsonian effective action.