An Extension of Pohlmeyer's Theorem

TL;DR

This paper proves that in scalar field theory, the only fixed point with zero anomalous dimension is Gaussian, using the Exact Renormalization Group and heat equation methods, extending previous results to general dimensions.

Contribution

It extends Pohlmeyer's theorem to non-integer dimensions and non-gauge theories using a heat equation approach within the Exact Renormalization Group framework.

Findings

Only Gaussian fixed point has zero anomalous dimension.

Method applies to scalar theories in arbitrary dimensions.

Extension to non-gauge theories with matter content.

Abstract

Applying the Exact Renormalization Group to scalar field theory in Euclidean space of general (not necessarily integer) dimension, it is proven that the only fixed-point with vanishing anomalous dimension is the Gaussian one. The proof requires positivity of the two-point connected correlation function together with a technical assumption concerning solutions of the flow equation. The method, in which the representation of the flow equation as a heat equation plays a central role, extends directly to non-gauge theories with arbitrary matter content (though non-linear sigma models are beyond the scope of the current method).