What are the other components of the weak isospin?
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Weak isospin, or SU(2)LSU(2)L, like conventional spin, is characterized by 3 generators, which do not commute, of course. However, as in plain spin settings, choosing a reference direction, such as the third, to moor the charges of the particles in a weak multiplet,
Q=T3+12YW,Q=T3+12YW,
we usually utilize the other two generators together through their quadratic Casimir,
T23+T22+T21≡T(T+1).T32+T22+T12≡T(T+1).
The eigenvalue T, then, shared by all particles in a given multiplet (so T=1/2 for the electron and neutrino, T=1 for the original pre-SSB gauge-boson triplet, etc), are necessary to account for the gauge invariance of the Lagrangian.
As a result, the couplings of the various fields, even after SSB, reflect the weak isospin invariance underlying them. Thus, e.g., the coupling
esinθW2–√W+μ(νL¯¯¯¯¯γμeL+u¯γμdL)esinθW2Wμ+(νL¯γμeL+u¯γμdL)
"knows about the other two generators" in ensuring the two lepton isodoublets have composed into an isotriplet to match the W isotriplet into an isosinglet. And likewise for the quarks.
So it is not enough that T3T3 adds up to 0, across the reaction, here, e.g., Wdecay: the residual symmetries of the multiplet composition structure are also severely constrained, and you cannot have the W decaying to particle groupings conserving T3T3 but not T.
Q=T3+12YW,Q=T3+12YW,
we usually utilize the other two generators together through their quadratic Casimir,
T23+T22+T21≡T(T+1).T32+T22+T12≡T(T+1).
The eigenvalue T, then, shared by all particles in a given multiplet (so T=1/2 for the electron and neutrino, T=1 for the original pre-SSB gauge-boson triplet, etc), are necessary to account for the gauge invariance of the Lagrangian.
As a result, the couplings of the various fields, even after SSB, reflect the weak isospin invariance underlying them. Thus, e.g., the coupling
esinθW2–√W+μ(νL¯¯¯¯¯γμeL+u¯γμdL)esinθW2Wμ+(νL¯γμeL+u¯γμdL)
"knows about the other two generators" in ensuring the two lepton isodoublets have composed into an isotriplet to match the W isotriplet into an isosinglet. And likewise for the quarks.
So it is not enough that T3T3 adds up to 0, across the reaction, here, e.g., Wdecay: the residual symmetries of the multiplet composition structure are also severely constrained, and you cannot have the W decaying to particle groupings conserving T3T3 but not T.
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