Interacting scale but non-conformal field theories

TL;DR

This paper investigates the existence of interacting scale invariant but non-conformal field theories, showing their presence in certain gauge-fixed models but not in elastic theories coupled with fermions.

Contribution

It demonstrates the absence of non-trivial fixed points in elastic theories with fermions and identifies examples in gauge-fixed Banks-Zaks theories.

Findings

Elastic theories with fermions lack interacting scale invariant fixed points.

Interacting scale invariant but non-conformal theories exist in gauge-fixed Banks-Zaks models.

Unphysical fixed points with infinite compression coefficients are found in elastic theories.

Abstract

There is a dilemma in constructing interacting scale invariant but not conformal invariant Euclidean field theories. On one hand, scale invariance without conformal invariance seems more generic by requiring only a smaller symmetry. On the other hand, the existence of a non-conserved current with exact scaling dimension $d-1$ in $d$ dimensions seems to require extra fine-tuning. To understand the competition better, we explore some examples without the reflection positivity. We show that a theory of elasticity (a.k.a Riva-Cardy theory) coupled with massless fermions in $d=4-\epsilon$ dimensions does not possess an interacting scale invariant fixed point except for unstable (and unphysical) one with an infinite coefficient of compression. We do, however, find interacting scale invariant but non-conformal field theories in gauge fixed versions of the Banks-Zaks fixed points in $d=4$ dimensions.