Is \phi^4 theory trivial ?

TL;DR

This paper clarifies the concept of triviality in four-dimensional ^4 theory, showing that recent results do not support the existence of true triviality, which involves the impossibility of constructing a non-trivial continuum limit.

Contribution

The paper distinguishes between two definitions of triviality and reviews recent evidence, concluding that true triviality in ^4 theory is not supported by current results.

Findings

Wilson's triviality is confirmed by existing data.

Recent results indicate the absence of true triviality.

The distinction clarifies longstanding ambiguities in the literature.

Abstract

The four-dimensional \phi^4 theory is usually considered to be trivial in the continuum limit. In fact, two definitions of triviality were mixed in the literature. The first one, introduced by Wilson, is equivalent to positiveness of the Gell-Mann -- Low function \beta(g) for g\ne 0; it is confirmed by all available information and can be considered as firmly established. The second definition, introduced by mathematical community, corresponds to the true triviality, i.e. principal impossibility to construct continuous theory with finite interaction at large distances: it needs not only positiveness of \beta(g) but also its sufficiently quick growth at infinity. Indications of true triviality are not numerous and allow different interpretation. According to the recent results, such triviality is surely absent.