Asymptotic freedom requires perturbative renormalizability?

[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!

 

[AndyW]

Hello,

Thank you for bringing up this interesting topic. I can understand your curiosity about the asymptotic behavior of theories with negative mass dimensions. To answer your question, I would like to provide some background information on asymptotic freedom and asymptotic safety.

Asymptotic freedom is a property of quantum field theories where the coupling constant decreases at high energies, allowing for a perturbative approach to calculations. This behavior was first observed in quantum chromodynamics (QCD), the theory of strong interactions, by David Gross, Frank Wilczek, and David Politzer in the 1970s. This has been confirmed by numerous experiments and is now considered a fundamental property of QCD.

On the other hand, asymptotic safety is a concept in quantum gravity where the theory is well-defined and non-perturbatively renormalizable at all energy scales. This means that the theory can be consistently defined even at the Planck scale, where the effects of gravity are expected to be significant. The idea of asymptotic safety was first proposed by Steven Weinberg in the 1970s and has gained attention in recent years as a possible solution to the problems with quantizing gravity.

Now, coming to your question, the argument that theories with negative mass dimensions cannot be asymptotically free is based on the Callen-Symanzik equation, which relates the beta function (the rate of change of the coupling constant with energy) to the anomalous dimensions (the scaling behavior of fields and operators). In theories with negative mass dimensions, the anomalous dimensions are also negative, which leads to a positive beta function, indicating that the coupling constant will grow with energy. This is in contrast to asymptotically free theories, where the beta function is negative and the coupling constant decreases with energy.

However, this argument does not necessarily rule out the possibility of asymptotic safety in theories with negative mass dimensions. Asymptotic safety is based on the existence of a non-trivial fixed point, where the beta function vanishes, and the theory is scale-invariant. This fixed point can exist even in the presence of negative anomalous dimensions, as has been shown in some studies of quantum gravity. Therefore, it is possible for a theory with negative mass dimensions to be both asymptotically free and asymptotically safe.

In conclusion, while the argument based on the Callen-Symanzik equation suggests that theories with negative mass dimensions cannot be asymptotically free,