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I How does SU(5) explain proton-electron charge equivalence?

  1. Apr 19, 2016 #1

    In my notes it says that SU(5) operates on a five component vector. Fine. But the example given is (d d d e v)

    ie. electron, neutrino and 3 down quarks (one for each colour) it says this explains why the proton charge is the equal and opposite to the electrons but why?

    A proton is uud not ddd.

  2. jcsd
  3. Apr 20, 2016 #2


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    Q(up) = Q(down) + 1. See e.g. chapter VII of Zee's QFT.
  4. May 30, 2016 #3
    Strictly speaking, the SU(5) GUT breaks into the unbroken Standard Model, and its multiplets have weak-isospin and weak-hypercharge degrees of freedom. Standard-Model breaking then makes electric charge out of them.

    Q = I3 + Y
    (electric charge) = (projected weak isospin) + (weak hypercharge)

    Here are the Standard Model's unbroken elementary-fermion multiplets with (QCD, weak isospin, weak hypercharge) values. A c means charge conjugate:
    • L (left-handed electron, neutrino): (1, 2, -1/2) -- Lc: (1, 2, 1/2)
    • E (right-handed electron): (1,1,-1) -- Ec: (1,1,1)
    • N (right-handed neutrino): (1,1,0) -- Nc: (1,1,0)
    • Q (left-handed up, down quarks): (3,2,1/6) -- Qc: (3*,2,-1/6)
    • U (right-handed up quark): (3,1,2/3) -- Uc: (3*,1,-2/3)
    • D (right-handed down quark: (3,1,-1/3) -- Dc: (3*,1,1/3)

    Here is where they are in the Georgi-Glashow SU(5) model (multiplet number, chirality, indices in asymmetric tensor,):
    • 1L (0): Nc
    • 5R (1): D + Lc
    • 10L (2): Q + Uc + Ec
    • 10R (3): Qc + U + E
    • 5L (4): Dc + L
    • 1R (5): N
    In all of these multiplets, the weak-hypercharge values add up to 0. That's because the weak hypercharge is derived from a SU(5) operator, and SU(N) operators are traceless (diagonal components adding to 0). One can show that these are the only WHC values that are consistent with SU(5), to within a multiplicative factor.

    This is what makes the up quark charged +2/3, the down quark charged -1/3, the neutrino charged 0, and the electron charge -1. Since quarks are confined as multiples of 3, antiquarks being negative quarks, that means that hadrons all have integer charges, just like the electron.
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