One family of fermions SU(2)LXU(1)

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SUMMARY

The discussion focuses on the implications of having a single family of fermions—comprising a neutrino, a charged lepton, an up-type quark, and a down quark—on the Lagrangian and Feynman rules of the SU(2)L×U(1) electroweak theory. It is established that most interactions in the Standard Model (SM), including those mediated by the Z boson, photon, and gluons, do not mix between different generations of fermions. The Lagrangian remains a sum of identical terms for each generation, differing only by mass. However, interactions mediated by the W boson do involve mixing between generations, characterized by coefficients from the CKM matrix, highlighting the significance of multiple generations in electroweak interactions.

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  • Understanding of SU(2)L×U(1) electroweak theory
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  • Knowledge of the CKM matrix and its role in particle interactions
  • Basic concepts of Lagrangian mechanics in quantum field theory
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Euphemia
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Hello, all
If now I only have one family of fermions (a neutrino, a charged lepton, an up-type quark and a down quark), what is going to change of the Lagragian and also the Feynman rules of SU(2)LXU(1)
electroweak theory?
Euphemia
 
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Almost all the interactions in the SM, including those mediated by the Z, photon and gluons ( strong, electromagnetic and weak neutral current) don't mix between the different generations ( families) of fermions. For these interactions, the lagrangain is simply a sum of an identical lagrangian for each generation ( up to different masses, of course) with the same feynman rules for each generation.
Only the interaction mediated by the W boson (charged neutral current) mixes between the generations with coefficents given by the CKM matrix. This is where the fact that we have three generations and not one come to play.
 

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