Improved actions, the perfect action, and scaling by perturbation theory in Wilsons renormalization group: the two dimensional $O(N)$-invariant non linear $\sigma$-model in the hierarchical approximation

TL;DR

This paper introduces a perturbation theory-based method combined with scaling ideas to determine improved lattice actions, tested on the two-dimensional hierarchical $O(N)$ sigma model, enhancing the accuracy of lattice field theory simulations.

Contribution

It presents a novel approach integrating perturbation theory with renormalization group scaling to improve lattice actions, specifically applied to the 2D hierarchical $O(N)$ sigma model.

Findings

Successful determination of improved actions using the proposed method.

Enhanced scaling behavior observed in the hierarchical $O(N)$ model.

Potential for broader application in lattice field theories.

Abstract

We propose a method using perturbation theory in the running coupling constant and the idea of scaling to determine improved actions for lattice field theories combining Wilson's renormalization group with Symanzik's improvement program . The method is based on the analysis of a single renormalization group transformation. We test it on the hierarchical $O(N)$ invariant $\sigma$ model in two dimensions.