Gradient Flows from an Approximation to the Exact Renormalization Group

TL;DR

This paper develops a formalism to analyze fixed points and critical phenomena in scalar field theories across dimensions 2<d<4, revealing gradient flows with monotonic c-functions and matching epsilon-expansion results near critical dimensions.

Contribution

It introduces a novel approach to study renormalization group flows using gradient flows and a positive-definite metric, connecting to known epsilon-expansion results.

Findings

Identification of fixed points in scalar theories across dimensions

Existence of a gradient flow with a decreasing c-function

Results match epsilon-expansion near critical dimensions

Abstract

Through appropriate projections of an exact renormalization group equation, we study fixed points, critical exponents and nontrivial renormalization group flows in scalar field theories in $2<d<4$. The standard upper critical dimensions $d_k={2k\over k-1}$, $k=2,3,4,\ldots$ appear naturally encoded in our formalism, and for dimensions smaller but very close to $d_k$ our results match the $\ee$-expansion. Within the coupling constant subspace of mass and quartic couplings and for any $d$, we find a gradient flow with two fixed points determined by a positive-definite metric and a $c$-function which is monotonically decreasing along the flow.