Triviality from the Exact Renormalization Group

TL;DR

This paper uses the exact renormalization group to demonstrate the non-existence of physically acceptable non-trivial fixed points with positive anomalous dimension in certain theories and shows that fixed points with negative anomalous dimension are non-unitary.

Contribution

It provides a rigorous analysis showing the absence of physically acceptable fixed points with positive anomalous dimension in specific quantum field theories using the exact renormalization group.

Findings

No non-trivial fixed points with positive anomalous dimension in O(N) scalar theories in ≥4D.

No physically acceptable fixed points with positive anomalous dimension in Abelian gauge theories.

Fixed points with negative anomalous dimension are non-unitary.

Abstract

Using the exact renormalization group, it is shown that no physically acceptable non-trivial fixed points, with positive anomalous dimension, exist for (i) O(N) scalar field theory in four or more dimensions, (ii) non-compact, pure Abelian gauge theory in any dimension. It is then shown, for both theories in any dimension, that otherwise physically acceptable non-trivial fixed points with negative anomalous dimension are non-unitary. In addition, a very simple demonstration is given, directly from the exact renormalization group, that should a critical fixed point exist for either theory in any dimension, then the n-point correlation functions exhibit the expected behaviour.