Why is Higgs particle a doublet

  1. 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. jcsd
  3. 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.
     
  4. Vanadium 50

    Vanadium 50 17,571
    Staff Emeritus
    Science Advisor
    Education Advisor

    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.
     
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