Gradient Flow of O(N) nonlinear sigma model at large N

TL;DR

This paper analyzes the gradient flow in the large N limit of the 2D O(N) nonlinear sigma model, deriving explicit solutions and demonstrating finiteness of the two-point function at finite flow time, with applications to defining a non-perturbative running coupling.

Contribution

It provides a novel large N analysis of the gradient flow in the O(N) sigma model, including explicit solutions and applications to non-perturbative coupling definitions.

Findings

Explicit solution for the n=1 case of the flow equations.

Demonstration that the two-point function at finite flow time is finite.

Application to defining a non-perturbative running coupling.

Abstract

We study the gradient flow equation for the O(N) nonlinear sigma model in two dimensions at large N. We parameterize solution of the field at flow time t in powers of bare fields by introducing the coefficient function X_n for the n-th power term (n=1,3,...). Reducing the flow equation by keeping only the contributions at leading order in large N, we obtain a set of equations for X_n's, which can be solved iteratively starting from n=1. For n=1 case, we find an explicit form of the exact solution. Using this solution, we show that the two point function at finite flow time t is finite. As an application, we obtain the non-perturbative running coupling defined from the energy density. We also discuss the solution for n=3 case.