Finite-regulator Wilsonian action

Take a finite-regulator model datum with retained coordinates $\mathrm{\ell }$, integrated coordinates $h$, a normalized Borel probability measure ${\nu }_H$ on the finite-dimensional $h$-space, and a real Borel ultraviolet action $S_{\mathrm{U}\mathrm{V}}(\mathrm{\ell },h)$. For each retained configuration $\mathrm{\ell }$ define

On the domain of $\mathrm{\ell }$ for which $0<Z_H(\mathrm{\ell })<\mathrm{\infty }$, the finite-regulator Wilsonian action at scale ${\mu }_W$ is

Thus “integrating out” the high sector means this specified finite integral over $h$ while $\mathrm{\ell }$ is held fixed. The measure, action, regulator, and low/high split are inputs; the definition does not assert that every interacting model has a finite or well-defined integral. Replacing ${\nu }_H$ by a positive constant multiple changes $S_W$ only by an additive constant independent of $\mathrm{\ell }$, which is fixed by the chosen normalization convention.

For comparison, fix the following finite Euclidean convention for a one-particle-irreducible (1PI) effective action. Separately supply a Borel probability $\rho$ on the retained space ${\mathbb{R}}^X$, and pair a source $j\in ({\mathbb{R}}^X{)}^{\ast }$ with $\mathrm{\ell }\in {\mathbb{R}}^X$ by $j\cdot \mathrm{\ell }={\sum }_{x\in X}{j}_x{\mathrm{\ell }}_x$, with source units chosen to make the pairing dimensionless. On a nonempty open set $U\subset ({\mathbb{R}}^X{)}^{\ast }$ require $0<\int e^{j\cdot \mathrm{\ell }}d\rho (\mathrm{\ell })<\mathrm{\infty }$ and define the connected source-generating function $W(j)=\log \int e^{j\cdot \mathrm{\ell }}d\rho (\mathrm{\ell })$. Require $W$ to be $C^2$ on $U$ and its gradient $\mathrm{\nabla }W:U\to V$ to be a $C^1$ diffeomorphism onto an open set $V\subset {\mathbb{R}}^X$ (a bijection with continuously differentiable map and inverse). For $\phi \in V$ set $j(\phi )=(\mathrm{\nabla }W{)}^{-1}(\phi )$ and define the Legendre transform $\mathrm{\Gamma }(\phi )=j(\phi )\cdot \phi -W(j(\phi ))$. This defines the finite Euclidean 1PI action on $V$ under these additional hypotheses; no graph expansion or existence of these additional data is asserted. It is a different construction from the high-coordinate integral above, which needs none of $\rho ,U,V,W,\mathrm{\Gamma }$.

Finite-regulator EFT model and scale data