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

[cletus]

Hi,

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.

Thanks.

 

[haushofer]

Q(up) = Q(down) + 1. See e.g. chapter VII of Zee's QFT.

 

[lpetrich]

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.