C Function Representation of the Local Potential Approximation

TL;DR

This paper introduces a function within the Local Potential Approximation of the exact renormalization group that monotonically decreases along flows and can serve as a basis for a variational approximation method.

Contribution

It constructs a new function c of coupling constants that tracks degrees of freedom and proposes a variational approximation approach within the framework.

Findings

The function c decreases monotonically along renormalization group flows.

The function c is stationary at fixed points, indicating stable theories.

A promising variational approximation method is proposed based on restrictions to sub-manifolds.

Abstract

Within the Local Potential Approximation to Wilson's, or Polchinski's, exact renormalization group, and for general spacetime dimension, we construct a function, c, of the coupling constants; it has the property that (for unitary theories) it decreases monotonically along flows, and is stationary only at fixed points ---where it `counts degrees of freedom', i.e. is extensive, counting one for each Gaussian scalar. Furthermore, by choosing restrictions to some sub-manifold of coupling constant space, we arrive at a very promising variational approximation method.