Posted: Jul 21, 2011 12:32 am
The unbroken (Minimal Supersymmetric) Standard Model seems as bad as the broken one. I'll now consider a series of GUT symmetries.
First, the GeorgiGlashow one: SU(5)
The elementary fermions go:
Right: 1 = N
Left: 5 = D* + L
Right: 10 = Q* + U + E
Left: 10* = Q + U* + E*
Right: 5* = D + L*
Left 1 = N*
So we have a nice alternating binomial sequence of degeneracies and chiralities (left vs. right handed):
1R, 5L, 10R, 10*L, 5*R, 1L
The Higgses go:
Left: 5 = Hd + (3*,1,1/3)
Left: 5* = Hu + (3,1,1/3)
Right: 5 = Hu* + (3*,1,1/3)
Right: 5* = Hd* + (3,1,1/3)
The (3,1,1/3) and (3*,1,1/3) look like a down quark, and they form the "Higgs triplet". This particle can cause proton and boundneutron decay, and it has to have a GUTscale mass to keep proton decay from being observed.
Note that lefthanded and righthanded sets have both 5 and 5* in them.
The gauge particles go:
24 = g + W + B + (3,2,5/6) + (3*,2,5/6)
Only one multiplet, but it contains the familiar ones and some extra ones, which can also cause proton decay.
SU(5) unification has the interesting consequence that the EFHiggs coupling is the same for electrons and down quarks; at GUT energies, m(tau) = m(bottom).

The next one up: SO(10), which breaks into SU(5) * U(1)
The second symmetry is related to (baryon number)  (lepton number) or BL
Elementary fermions:
Left: 16 = (1,5) + (5,3) + (10*,1)
Right: 16* = (1,5) + (5*,3) + (10,1)
One multiplet for all the elementary fermions in each generation
Higgses:
Left, Right: 10 = (5,2) + (5*,2)
One multiplet for all the Higgses
Gauge:
45 = (24,0) + (1,0) + (10,4) + (10*,4)
The SU(5) gauge multiplet + some extra ones
SO(10) predicts complete mass unification for each generation of EF's  and forbids crossgeneration decay. That sort of decay must therefore be the result of SO(10) breaking.

Now, E6. It breaks into SO(10) * U(1)
Elementary fermions:
Left: 27 = (16,1) + (10,2) + (1,4)
Right: 27* = (16*,1) + (10,2) + (1,4)
SO(10) EF"s + possible SO(10) Higgs + extra
The Higgses can be included in one of the EF multiplets, which may explain why they couple to one generation much more than to the others. The others must be kept at GUT energies by symmetry breaking.
Gauge:
78 = (45,0) + (1,0) + (16,3) + (16*,3)
The SO(10) gauge multiplet + some extra ones

Finally, E8, breaking into E6 * SU(3) in the E8 * E8 heterotic superstring
Gauge:
248 = (78,1) + (1,8) + (27,3) + (27*,3*)
The first one is the E6 gauge multiplet, while the third and fourth are E6 EF's and Higgses. The different spins are the result of "compactification" from 10 to 4 spacetime dimensions. E8 gets broken into E6 * SU(3), and the compactification interacts with the SU(3) parts.
So one multiplet can contain all the Standard Model.
First, the GeorgiGlashow one: SU(5)
The elementary fermions go:
Right: 1 = N
Left: 5 = D* + L
Right: 10 = Q* + U + E
Left: 10* = Q + U* + E*
Right: 5* = D + L*
Left 1 = N*
So we have a nice alternating binomial sequence of degeneracies and chiralities (left vs. right handed):
1R, 5L, 10R, 10*L, 5*R, 1L
The Higgses go:
Left: 5 = Hd + (3*,1,1/3)
Left: 5* = Hu + (3,1,1/3)
Right: 5 = Hu* + (3*,1,1/3)
Right: 5* = Hd* + (3,1,1/3)
The (3,1,1/3) and (3*,1,1/3) look like a down quark, and they form the "Higgs triplet". This particle can cause proton and boundneutron decay, and it has to have a GUTscale mass to keep proton decay from being observed.
Note that lefthanded and righthanded sets have both 5 and 5* in them.
The gauge particles go:
24 = g + W + B + (3,2,5/6) + (3*,2,5/6)
Only one multiplet, but it contains the familiar ones and some extra ones, which can also cause proton decay.
SU(5) unification has the interesting consequence that the EFHiggs coupling is the same for electrons and down quarks; at GUT energies, m(tau) = m(bottom).

The next one up: SO(10), which breaks into SU(5) * U(1)
The second symmetry is related to (baryon number)  (lepton number) or BL
Elementary fermions:
Left: 16 = (1,5) + (5,3) + (10*,1)
Right: 16* = (1,5) + (5*,3) + (10,1)
One multiplet for all the elementary fermions in each generation
Higgses:
Left, Right: 10 = (5,2) + (5*,2)
One multiplet for all the Higgses
Gauge:
45 = (24,0) + (1,0) + (10,4) + (10*,4)
The SU(5) gauge multiplet + some extra ones
SO(10) predicts complete mass unification for each generation of EF's  and forbids crossgeneration decay. That sort of decay must therefore be the result of SO(10) breaking.

Now, E6. It breaks into SO(10) * U(1)
Elementary fermions:
Left: 27 = (16,1) + (10,2) + (1,4)
Right: 27* = (16*,1) + (10,2) + (1,4)
SO(10) EF"s + possible SO(10) Higgs + extra
The Higgses can be included in one of the EF multiplets, which may explain why they couple to one generation much more than to the others. The others must be kept at GUT energies by symmetry breaking.
Gauge:
78 = (45,0) + (1,0) + (16,3) + (16*,3)
The SO(10) gauge multiplet + some extra ones

Finally, E8, breaking into E6 * SU(3) in the E8 * E8 heterotic superstring
Gauge:
248 = (78,1) + (1,8) + (27,3) + (27*,3*)
The first one is the E6 gauge multiplet, while the third and fourth are E6 EF's and Higgses. The different spins are the result of "compactification" from 10 to 4 spacetime dimensions. E8 gets broken into E6 * SU(3), and the compactification interacts with the SU(3) parts.
So one multiplet can contain all the Standard Model.