Renormalizability of the gradient flow in the 2D $O(N)$ non-linear sigma model

TL;DR

This paper proves that the gradient flow in the 2D O(N) non-linear sigma model results in UV finite composite operators without wave function renormalization, facilitating the construction of a conserved energy-momentum tensor in lattice simulations.

Contribution

It demonstrates the UV finiteness of the gradient flow in the 2D O(N) model and introduces a method to construct the energy-momentum tensor on the lattice.

Findings

Gradient flow yields UV finite operators without wave function renormalization.

Constructed a lattice energy-momentum tensor that restores continuum properties.

Proved UV finiteness using a (2+1)-dimensional field theoretical representation.

Abstract

It is known that the gauge field and its composite operators evolved by the Yang--Mills gradient flow are ultraviolet (UV) finite without any multiplicative wave function renormalization. In this paper, we prove that the gradient flow in the 2D $O(N)$ non-linear sigma model possesses a similar property: The flowed $N$-vector field and its composite operators are UV finite without multiplicative wave function renormalization. Our proof in all orders of perturbation theory uses a $(2+1)$-dimensional field theoretical representation of the gradient flow, which possesses local gauge invariance without gauge field. As application of the UV finiteness of the gradient flow, we construct the energy--momentum tensor in the lattice formulation of the $O(N)$ non-linear sigma model that automatically restores the correct normalization and the conservation law in the continuum limit.