Why we must demonstrate the electroweak theory to be renormalizable?

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Discussion Overview

The discussion centers around the renormalizability of the electroweak theory, specifically the Weinberg-Salam model. Participants explore the relationship between coupling constants, anomalies, and the requirements for a theory to be considered renormalizable. The scope includes theoretical aspects and technical challenges associated with proving renormalizability.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant states that if the coupling constant is dimensionless, the theory is renormalizable, suggesting that since the coupling constant g in electroweak theory is dimensionless, it should be renormalizable.
  • Another participant raises a question about the connection between the anomaly problem and renormalization, indicating a lack of understanding of how these concepts relate.
  • A participant notes that proving a theory's renormalizability involves ensuring divergent integrals can be absorbed by counter terms and managing gauge fixing without introducing non-physical degrees of freedom.
  • One contribution mentions t'Hooft's work on dimensional regularization as a significant advancement in regularization techniques, although the specifics of his demonstration are not recalled.
  • Another participant emphasizes that dimensional analysis aids in understanding counter terms but does not address quantization anomalies that may arise in loop calculations.
  • One participant argues that stability, rather than renormalizability alone, is crucial for the theory, suggesting that renormalizability was pursued to achieve stability.
  • A question is posed regarding the relationship between anomalies and renormalization characteristics, indicating ongoing uncertainty in this area.
  • Another participant suggests a potential connection between Ward (Taylor) identities and the BPHZ theorem, hinting at deeper theoretical links that may exist.

Areas of Agreement / Disagreement

Participants express various viewpoints on the necessity of renormalizability versus stability, and there is no consensus on the relationship between anomalies and renormalization. The discussion remains unresolved with competing perspectives on these issues.

Contextual Notes

Some participants note the complexity of proving renormalizability and the specific challenges posed by anomalies, but the discussion does not resolve these complexities or provide definitive answers.

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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