Interacting scale but non-conformal field theories

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.