Electroweak asymptotic freedom

[star apple]

First, the coupling constant of the strong force can be computed.. does it mean it should also (ought to be) for the electroweak?

Also U(1) is not asymptotically free but electroweak SU(2)xU(1) is asymptotically free.. why is SU(2)xU(1) asymptotically free?

 

[king vitamin]

why is SU(2)xU(1) asymptotically free?

I don't believe that it is. In (3+1) dimensions, the U(1) coupling should grow in the UV.

 

[star apple]

I don't believe that it is. In (3+1) dimensions, the U(1) coupling should grow in the UV.

I queried this as a result of reading Peter Woit book "Not Even Wrong".. he quoted:

"Why SU(3)xSU(2)xU(1)? A truly fundamental theory should explain where this precise set of symmetry groups is coming from. In addition, whereas QCD (the SU(3) part of this) has the beautiful property of having no free parameters, introducing the two other groups SU(2) and U(1)) introduces two free parameters and one would like some explanation of why they have the values they do. One of these is the fine structure constant a, and the question of where this number comes from goes back to the earliest days of QED. Another related concern is that the U(1) part of the gauge theory is not asymptotically free, and as a result it may not be completely mathematical consistent."

I thought he meant that SU(2)xU(1) was asymptotically free.. if not.. why did he mentioned that "Another related concern is that the U(1) part of the gauge theory is not asymptotically free, and as a result it may not be completely mathematical consistent"? This statement implied SU(2)xU(1) was asymptotically free.

 

[vanhees71]

The Glashow-Salam-Weinberg theory is not asymptotically free. There's a U(1) in the gauge group and also it's Higgsed. I'm not certain anymore, what's the precise statement but if I remember right, one can show that for a wide range of Higgsed local gauge symmetries the resulting models are not asymptotically free, even if there are no U(1)'s are in the gauge group:

Higgs phenomena in asymptotically free gauge theories
T. P. Cheng, E. Eichten, and Ling-Fong Li
https://doi.org/10.1103/PhysRevD.9.2259