
#1
Jan1214, 06:15 PM

P: 3

In demonstrating that Higgs mechanism gives mass to gauge boson fields, we used the fact that hypercharge Y=1/2, which is due to "Higgs particle is a complex doublet of the weak isospin SU(2) symmetry". But why?
In other words, can you show the details about why the Higgs field has charge +1/2 under the weak hypercharge U(1) symmetry? 



#2
Jan1214, 10:55 PM

P: 514

That's because in the unbroken Standard Model, everything must be massless except perhaps the Higgs particle. This is because lefthanded and righthanded parts have gaugemultiplet mismatches, and the Higgs particle is necessary for bridging this gap.
The Standard Model's charged elementary fermions have mass terms that look like this: (mass) . (lefthanded part of EF field) . (righthanded part of EF field)^{+} + Hermitian conjugate (+ = HC) In the unbroken SM, the EF fields break down into these gauge multiplets: Lefthanded quark, I = 1/2, Y = 1/6 Righthanded up quark, I = 0, Y = 2/3 Righthanded down quark, I = 0, Y = 1/3 Lefthanded lepton, I = 1/2, Y = 1/2 Righthanded neutrino (if it exists), I = 0, Y = 0 Righthanded electron, I = 0, Y = 1 I = weak isospin, Y = weak hypercharge Hermitian conjugate, same I,  Y I'm ignoring generations here for simplicity. The muon and the tau are essentially additional flavors of electron, etc. Electric charge Q = I3 + Y I3 = I to I in integer steps, like angular momentum That makes bare Dirac masses impossible in the Standard Model, or at least so it seems. A lefthanded part and a righthanded part, when combined, have I = 1/2 and Y = + 1. That means that there must be some additional field with I = 1/2 and Y = 1 or 1 to cancel that out and make a proper interaction term. That field is the Higgs particle, with I = 1/2, Y = 1. We get Higgscoupling terms (Higgs) . (coupling) . (lefthanded quark) . (righthanded up quark)^{+} (Higgs)^{+} . (coupling) . (lefthanded quark) . (righthanded down quark)^{+} (Higgs) . (coupling) . (lefthanded lepton) . (righthanded neutrino)^{+} (Higgs)^{+} . (coupling) . (lefthanded lepton) . (righthanded electron)^{+} Their (I,Y) sets: (1/2,1/2) . (1/2,1/6) . (0,2/3) (1/2,1/2) . (1/2,1/6) . (0,1/3) (1/2,1/2) . (1/2,1/2) . (0,0) (1/2,1/2) . (1/2,1/2) . (0,1) If the Higgs particle has a nonzero vacuum field value, then that field value can combine with the coupling to make a Dirac mass. 


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