Interacting scale invariant but nonconformal field theories

There is a dilemma in constructing interacting scale invariant Euclidean field theories that are not conformal invariant. 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 nonconserved 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 (also known as Riva-Cardy theory) coupled with massless fermions in $d=4-\epsilon$ dimensions does not possess an interacting scale invariant fixed point except for an unstable (and unphysical) one with an infinite coefficient of compression. We do, however, find interacting scale invariant but nonconformal field theories in gauge fixed versions of the Banks-Zaks fixed points in $d=4$ dimensions.

Figure 1

Renormalization group flow of the gauge coupling $g$ and the gauge fixing parameter $\alpha$ in $SU(3)$ gauge theories with $N_f=16$ fermions.