Flow Equation of N=1 Supersymmetric O(N) Nonlinear Sigma Model in Two Dimensions

TL;DR

This paper derives a supersymmetric flow equation for the N=1 O(N) nonlinear sigma model in two dimensions, proposes a simple form, and analyzes its large N behavior, showing finiteness of the two-point function.

Contribution

It introduces a specific flow equation respecting supersymmetry and O(N) symmetry, and provides the leading order large N solution demonstrating finiteness.

Findings

Flow equation respects supersymmetry and O(N) symmetry.

Leading order large N solution shows scalar dominance in the flow.

Two-point function remains finite at leading order.

Abstract

We study the flow equation for the $\mathcal{N}=1$ supersymmetric $O(N)$ nonlinear sigma model in two dimensions, which cannot be given by the gradient of the action, as evident from dimensional analysis. Imposing the condition on the flow equation that it respects both the supersymmetry and the $O(N)$ symmetry, we show that the flow equation has a specific form, which however contains an undetermined function of the supersymmetric derivatives $D$ and $\bar D$. Taking the most simple choice, we propose a flow equation for this model. As an application of the flow equation, we give the solution of the equation at the leading order in the large $N$ expansion. The result shows that the flow of the superfield in the model is dominated by the scalar term, since the supersymmetry is unbroken in the original model. It is also shown that the two point function of the superfield is finite at the leading order of the large $N$ expansion.