I understand that asymptotically free theories must be based on UV fixed points rather than IR ones, because the RG flow goes into rather than out of an IR fixed point, so an asymptotically free theory based on an IR fixed point is trivial at low energies. But at higher energies the coupling will be nonzero/nontrivial, so it seems you can have an asymptotically free theory based on an IR fixed point, but at low energies there's no interaction, so what's the problem with this: I see the statement in books that you can't have an asymptotically free theory with an IR fixed point. Do all UV fixed points imply asymptotic freedom, and all IR fixed points imply triviality? I'm reading Kerson Huang's QFT book and he says for QED perturbation theory shows there's no UV fixed point and hence theory is not asymptotically free. That's fine as I get that asymptotically free theories must have UV fixed points, so QED can't be asymptotically free. But he then says perturbation theory suggests there's an IR fixed point, which leads to triviality. Does this mean all IR fixed points have to be at the origin of coupling constant space?