Smoothing of Singular Legendre Transforms in Renormalization Group Flows

TL;DR

This paper demonstrates that the renormalization group flow smooths out logarithmic singularities in certain symmetric potentials, enabling well-defined initial conditions and efficient numerical analysis.

Contribution

It shows that the RG flow for these potentials smooths singularities, allowing for well-defined initial conditions and improved numerical methods.

Findings

Logarithmic singularities are rapidly smoothed in the RG flow.

The RG equation for the local potential has well-defined initial conditions.

An efficient numerical method for studying these flows is proposed.

Abstract

We consider O(N)-symmetric potentials with a logarithmic singularity in the second field derivative. This class includes BCS and Gross Neveu potentials. Formally, the exact renormalization group equation for the Legendre transform of these potentials seems to have ill-defined initial conditions. We show that the renormalization group equation for the local potential has well-defined initial conditions and that the logarithmic singularity is smoothed rapidly in the flow. Our analysis also provides an efficient method for numerical studies.