SU(2)L, SU(2)R, other symmetric groups and SSB

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SUMMARY

The discussion focuses on the differences between the SU(2)L and SU(2)R groups in the context of electroweak interactions and their implications for Lagrangian invariance. It is established that a Lagrangian can be invariant under the combined transformations of SU(2)L x SU(2)R, but upon spontaneous symmetry breaking, the resulting Lagrangian becomes invariant under SU(2)L+R. This transition signifies the emergence of a Higgs field and the loss of certain symmetries, raising questions about which symmetries persist post-symmetry breaking.

PREREQUISITES
  • Understanding of group theory, specifically SU(2) groups
  • Familiarity with Lagrangian mechanics in quantum field theory
  • Knowledge of the Higgs mechanism and spontaneous symmetry breaking
  • Basic comprehension of electroweak interactions
NEXT STEPS
  • Study the implications of SU(2)L x SU(2)R invariance in quantum field theories
  • Research the Higgs mechanism and its role in electroweak symmetry breaking
  • Examine the concept of spontaneous symmetry breaking in detail
  • Explore the differences between SU(2)L+R and other symmetry groups in particle physics
USEFUL FOR

The discussion is beneficial for theoretical physicists, graduate students in particle physics, and anyone interested in the mathematical foundations of the Higgs mechanism and symmetry in quantum field theories.

Anne-Sylvie
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Hello everyone,

When we speak about the SU(2)L group (in electroweak interactions for example), about what group do we talk ? What is the difference with the SU(2) group ? And with the SU(2)R ? Why is the label so important ?

I ask this because I see that a Lagrangien can be invariant under SU(2)L x SU(2)R. I think that means that if we apply any transformation of the SU(2)L and then SU(2)R on the lagrangien, it will give the same lagrangien as before. Is it right ?
Then, when we consider the Higgs mechanism. There is an non zero expectation value, and then via the spontaneous symmetry breaking mechanism, a higgs field appear, etc, and we have another lagrangien. This lagrangien is no longer invariant under SU(2)L x SU(2)R transformations, but well under SU(2)L+R. What does that mean ? What is SU(2)L+R ? How can we know, when we spontaneously break a symmetry, which symmetries will survive ?

I'm a bit confused with all of that. I hope I'm relatively clear in my message... Please excuse me for my English but I'm non native and I'm still learning this usefull langage. :-)
 
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See chapters 83, 88, and 89 in the book by Srednicki (draft version freely downloadable at his web page).
 
Thanks for your advice. :-)
 

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