# Asymptotic freedom requires perturbative renormalizability?

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1. Feb 4, 2016

### metroplex021

I have read many times that a theory (such as gravity) that contains couplings with negative mass dimensions cannot be asymptotically free. Does anyone have a reference that proves that that's the case? The argument is usually just that the coupling grows with energy, as seen in the Callen-Symantek equation. But if that's the argument, then it's hard to see how such a theory could be asymptotically safe either, and plenty of people think that gravity might be asymptotically safe. Thanks so much!

2. Feb 9, 2016