Price of Asymptotic Freedom

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Abstract
A renormalizable field theory is said to be asymptotically free if the origin of coupling-constant space is an ultraviolet-stable fixed point in the sense of Wilson. Asymptotically free theories are of great interest because they have almost-canonical light-cone singularities, and thus predict phenomena very close to Bjorken scaling. All known examples of asymptotically free theories involve non-Abelian gauge fields. We show that this is not coincidence: No renormalizable field theory without non-Abelian gauge fields can be asymptotically free.