Scale without Conformal Invariance in relativistic Quantum Field Theory

In 2-dim it is known that a unitary, well defined quantum field theory, if scale 
invariant must also be invariant under conformal transformations. Whether this is 
also true in dimensions higher than two has been an open question for decades. 
We have discovered renomalization group flows in 4-epsilon dimensions 
corresponding to scale but not conformal invariant theories. The flows 
correspond to limit cycles or ergodic behavior, neither of which had been 
reported in relativistic quantum field theories either. There seems to be a deep 
connection between scale but not conformal invariance and this type of 
renormalization group behavior. We will present these results and list some of 
open questions, including the possibility of such behavior in integral dimensions.