Why we must demonstrate the electroweak theory to be renormalizable?

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SUMMARY

The discussion centers on the renormalizability of the electroweak theory, specifically the Weinberg-Salam model, which is confirmed to be renormalizable due to its dimensionless coupling constant, g~e. The necessity for Gerard 't Hooft to demonstrate this renormalizability in 1971 is highlighted, alongside the challenge of addressing the anomaly problem, which is crucial for ensuring the theory's stability. Dimensional regularization, a technique introduced by 't Hooft, plays a significant role in managing divergences and preserving the physical anomaly structure. The relationship between anomalies and renormalization, particularly through the Ward-Taylor identity and BPHZ theorem, is also explored.

PREREQUISITES
  • Understanding of electroweak theory and the Weinberg-Salam model
  • Familiarity with quantum field theory and renormalization concepts
  • Knowledge of dimensional regularization techniques
  • Awareness of anomalies in quantum field theories
NEXT STEPS
  • Research "Gerard 't Hooft's dimensional regularization" for insights on regularization methods
  • Study "Ward-Taylor identities" to understand their role in quantum field theory
  • Explore "BPHZ theorem" and its implications for renormalization
  • Investigate the "anomaly cancellation in chiral theories" for a deeper understanding of stability in electroweak theory
USEFUL FOR

This discussion is beneficial for theoretical physicists, quantum field theorists, and advanced students studying electroweak interactions and renormalization techniques.

ndung200790
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In general speaking,if the coupling constant is (mass) dimensionless then the quantum field theory is renormalizable.In electroweak theory the coupling constant g~e,so the coupling constant is dimensionless,then the electroweak theory(Weinberg-Salam theory) would be renormalizable.So I do not understand why in 1971(I have heard that) t' Hooft must demonstrate the Weinberg-Salam to be renormalizable.I also can not find in any textbook the t'Hooft's demonstration,where can I find it?
Please forgive me if my question is not good question(I have to self-study the subject)
 
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I have known that in electroweak theory we have to solve the anomaly problem.But I do not understand why the anomaly problem relates with the renormalization problem(divergent problem)
 
In general proving that a theory is renormalizable is not an easy task. First you must make sure that any divergent integrals can be absorbed by a finite set of counter terms, you must properly fix the gauge while at the same time making sure that any non-physical degrees of freedom do not appear in physical calculations.

I do not quite remember what Thooft did but I do know one of his contributions was dimensional regularization which brought leaps and bounds on our abilities to do regularization in a Lorentz covariant way.
 
Thank jarod very much!
 
The dimensional analysis gives you a hint on the structure of counter terms; it doesn't say anything regarding quantization anomalies which could arise in loop calculations. So one major step was to provide a regularization method which preserves the "physical anomaly structure" (triangle anomalies in chiral theories) and to study their cancellation w/o introducing "unphysical anomalies" which would arise e.g. in the unmodified Pauli–Villars regularization approach.
 
We do not need it to be renormalizable but we need it to be stable. That y we made it renormalizable.
 
So,is there any relation between anomalies and renormalization characteristic?
 
It seems that there is a close relation between Ward(Taylor) Identity and BPHZ theorem?
 

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