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It is said in textbooks that if the $SU(2)_f$ or $SU(3)_f$ flavor symmetry were exact for sstrong isospin, then all members of the multiplets would be exactly equally massive. By looking at quark masses then one can conclude that the symmetry is not exact. But how does the argument work with the weak force ? It is said to be an exact symmetry, yet electron and neutrino have different masses even though they are in the same doublet. What is the difference ?

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What is in the same electroweak doublet are just left-handed components of electron and neutrino. Note that the mass of the particle can be thought of as the strength of the coupling between left-handed and right-handed components of particle's field. You have such coupling for electron (in a gauge-invariant way, via Higgs), and not for neutrino (which lacks right-handed component altogether in a pure Standard model).

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