Non-Perturbative Aspects of Scalar Field theory

TL;DR

This paper investigates the non-perturbative properties of scalar field theories using hierarchical approximations, focusing on cut-off dependence, triviality bounds, and the behavior of critical exponents.

Contribution

It introduces a two-parameter family of recursion formulas that interpolate between different hierarchical models, providing insights into the dependence of critical exponents on model parameters.

Findings

Critical exponent gamma varies continuously with parameter zeta.

Oscillatory corrections to scaling laws are observed in hierarchical approximations.

Zeta-independence is proposed as a criterion for improved recursion formulas.

Abstract

Using the hierarchical approximation, we discuss the cut-off dependence of the renormalized quantities of a scalar field theory. The naturalness problem and questions related to triviality bounds are briefly discussed. We discuss unphysical features associated with the hierarchical approximation such as the recently observed oscillatory corrections to the scaling laws. We mention a two-parameter family of recursion formulas which allows one to continuously extrapolate between Wilson's approximate recursion formula and the recursion formula of Dyson's hierarchical model. The parameters are the dimension D and 2^zeta, the number of sites integrated in one RG transformation. We show numerically that at fixed D, the critical exponent gamma depends continuously on zeta. We suggest the requirement of zeta -independence as a guide for constructing improved recursion formulas.