Short-range Interaction and Nonrelativistic Phi**4 Theory in Various Dimensions

TL;DR

This paper uses effective field theory to analyze short-range interactions in phi**4 theory across various dimensions, revealing how relativistic effects influence renormalization and challenging the triviality claim in 4D.

Contribution

It provides a systematic nonrelativistic effective theory matching for phi**4 in different dimensions, including exact S-wave amplitude expressions and insights into renormalization group flows.

Findings

Relativistic effects qualitatively alter RG flow in 3D.

Exact Lorentz-invariant S-wave amplitude derived.

Challenged the zero-range interaction assumption in 4D phi**4 theory.

Abstract

We employ the effective field theory method to systematically study the short-range interaction in two-body sector in 2, 3 and 4 spacetime dimensions, respectively. The phi**4 theory is taken as a specific example and matched onto the nonrelativistic effective theory to one loop level. An exact, Lorentz-invariant expression for the S-wave amplitude is presented, from which the nonperturbative information can be easily extracted. We pay particular attention to the renormalization group analysis in the 3 dimensions, and show that relativistic effects qualitatively change the renormalization group flow of higher-dimensional operators. There is one ancient claim that triviality of the 4-dimensional phi**4 theory can be substantiated in the nonrelativistic limit. We illustrate that this assertion arises from treating the interaction between two nonrelativistic particles as literally zero-range, which is incompatible with the Uncertainty Principle. The S-wave effective range in this theory is identified to be approximately 16/3pi times the Compton wavelength.