Unitarity violation at the Wilson-Fisher fixed point in 4-epsilon dimensions

TL;DR

This paper shows that scalar field theories continued to non-integer dimensions violate unitarity, leading to negative norm states and complex anomalous dimensions at the Wilson-Fisher fixed point, even at leading order.

Contribution

It demonstrates that non-integer dimensional scalar theories are inherently non-unitary and reveals the emergence of complex anomalous dimensions at the Wilson-Fisher fixed point.

Findings

Correlation functions are incompatible with unitarity in non-integer dimensions.

Negative norm states arise from the operator product expansion.

Complex anomalous dimensions appear at leading order in epsilon expansion.

Abstract

We consider the continuation of free and interacting scalar field theory to non-integer spacetime dimension d. We find that the correlation functions in these theories are necessarily incompatible with unitarity (or with reflection positivity in Euclidean signature). In particular, the theories contain negative norm states unless d is a positive integer. These negative norm states can be obtained via the OPE from simple positive norm operators, and are therefore an integral part of the theory. At the Wilson-Fisher fixed point the non-unitarity leads to the existence of complex anomalous dimensions. We demonstrate that they appear already at leading order in the epsilon expansion.