Walking in the 3-dimensional large $N$ scalar model

TL;DR

This paper investigates the three-dimensional large N scalar model using lattice methods, demonstrating a 'walking' beta function behavior under certain conditions, and analyzing lattice artifacts and finite size effects.

Contribution

It provides the first lattice study of the 3D large N scalar model showing walking behavior and examines its robustness through physical observables.

Findings

The beta function exhibits walking behavior when the mass-to-coupling ratio is small.

Lattice artifacts can be significant at realistic correlation lengths.

Finite size effects are manageable at sizes around three correlation lengths.

Abstract

The solvability of the three-dimensional O($N$) scalar field theory in the large $N$ limit makes it an ideal toy model exhibiting "walking" behavior, expected in some SU($N$) gauge theories with a large number of fermion flavors. We study the model using lattice regularization and show that when the ratio of the particle mass to an effective 4-point coupling (with dimension mass) is small, the beta function associated to the running 4-point coupling is "walking". We also study lattice artifacts and finite size effects, and find that while the former can be sizable at realistic correlation length, the latter are under control already at lattice sizes a few ($\sim$3) correlation lengths. We show the robustness of the walking phenomenon by showing that it can also be observed by studying physical observables such as the scattering phase shifts and the mass gap in finite volume.