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?
That's because in the unbroken Standard Model, everything must be massless except perhaps the Higgs particle. This is because left-handed and right-handed parts have gauge-multiplet 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) . (left-handed part of EF field) . (right-handed part of EF field)^{+} + Hermitian conjugate (+ = HC) In the unbroken SM, the EF fields break down into these gauge multiplets: Left-handed quark, I = 1/2, Y = 1/6 Right-handed up quark, I = 0, Y = 2/3 Right-handed down quark, I = 0, Y = -1/3 Left-handed lepton, I = 1/2, Y = -1/2 Right-handed neutrino (if it exists), I = 0, Y = 0 Right-handed 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 left-handed part and a right-handed 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 Higgs-coupling terms (Higgs) . (coupling) . (left-handed quark) . (right-handed up quark)^{+} (Higgs)^{+} . (coupling) . (left-handed quark) . (right-handed down quark)^{+} (Higgs) . (coupling) . (left-handed lepton) . (right-handed neutrino)^{+} (Higgs)^{+} . (coupling) . (left-handed lepton) . (right-handed 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.
It seems like you are confusing "Higgs field" with "Higgs boson". You add a complex doublet field (4 degrees of freedom), and are left with but a single Higgs boson.