  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.
