Electroweak asymptotic freedom

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SUMMARY

The discussion centers on the concept of electroweak asymptotic freedom, particularly the behavior of the SU(2)xU(1) gauge group in contrast to the U(1) component. Participants debate whether SU(2)xU(1) is asymptotically free, with references to Peter Woit's book "Not Even Wrong" highlighting the inconsistency of the U(1) coupling in (3+1) dimensions. The Glashow-Salam-Weinberg theory is identified as not being asymptotically free due to the presence of the U(1) gauge group and its Higgs mechanism. The conversation emphasizes the need for a deeper understanding of the parameters and symmetries in fundamental theories.

PREREQUISITES
  • Understanding of gauge theories, specifically SU(2) and U(1) groups.
  • Familiarity with asymptotic freedom in quantum field theory.
  • Knowledge of the Glashow-Salam-Weinberg theory and its implications.
  • Basic concepts of the Higgs mechanism and its role in particle physics.
NEXT STEPS
  • Research the implications of asymptotic freedom in gauge theories.
  • Study the Glashow-Salam-Weinberg theory in detail.
  • Examine the role of the Higgs mechanism in electroweak symmetry breaking.
  • Explore Peter Woit's arguments in "Not Even Wrong" regarding fundamental theories and symmetry groups.
USEFUL FOR

The discussion is beneficial for theoretical physicists, particle physicists, and researchers interested in quantum field theory, particularly those exploring the nuances of electroweak interactions and gauge symmetries.

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?
 
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star apple said:
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.
 
king vitamin said:
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.
 
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
 
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