What are the proposed gauge-symmetry groups for Grand Unified Theories?

[lpetrich]

I first thought of posting on cataloguing various Grand Unified Theory proposals, but that would be an enormous task, so I decided on something simpler: cataloguing proposed GUT gauge-symmetry groups.

The unbroken Standard-Model symmetry is SU(3)_C * SU(2)_L * U(1)_Y
QCD:
SU(3)_C -- color
Electroweak:
SU(2)_L -- weak isospin
U(1)_Y -- weak hypercharge

I'll consider gauge particles, Higgs particles, and elementary fermions (EF's), complete with right-handed neutrinos (RHN).

The GUT's:

SU(5) -- Georgi-Glashow
Gauge: 1, Higgs: 2, EF's: 3 (one of them is RHN)

SU(4)*SU(2)*SU(2) or SO(6)*SO(4) -- Pati-Salam
Gauge: 3, Higgs: 1, EF's: 2 (includes RHN)

SO(10)
Gauge: 1, Higgs: 1, EF's: 1 (includes RHN)

SO(10) can break into Georgi-Glashow or Pati-Salam

SU(3)*SU(3)*SU(3) -- trinification
Gauge: 3, EF's: 3 (includes RHN; one of them also can contain the Higgs)

E(6)
Gauge: 1, EF's: 1 (can also contain the Higgs)

E(6) can break into SO(10) or trinification

E(8)
Everything in the 248 fundamental / adjoint multiplet, including all three generations of EF's.

E(8) can break into E(6)*SU(3), SO(10)*SU(4), or SU(5)*SU(5)

SU(6)
Gauge: 1, EF's: 3 (includes RHN; can also contain the Higgs)

SU(6) can break into SU(5)

ETA: E(6) can break into SU(6)

Any others that anyone has proposed?

 

[arivero]

A break of Pati Salam is important too: U(1)xSU(3)xSU(2)xSU(2) where the first U(1) is not a 4th colour but just B-L, baryon minus lepton number.

There is also some SU(10+2k) groups proposed with the goal of family unification. SO(14)? 16? 18?

 

[lpetrich]

A break of Pati Salam is important too: U(1)xSU(3)xSU(2)xSU(2) where the first U(1) is not a 4th colour but just B-L, baryon minus lepton number.

Also, one of the SU(2)'s becomes U(1), and the U(1)'s mix.

Here are all the symmetry breakings in the symmetry groups that I've listed:

E(8) ->
- E(6) * SU(3)
- SO(10) * SU(4)/SO(6)
- SU(5) * SU(5)
E(6) ->
- SO(10) * U(1)
- SU(6) * U(1)
- SU(3) * SU(3) * SU(3)
SO(10) ->
- SU(5) * U(1)
- SU(4) * SU(2) * SU(2) / SO(6) * SO(4)
SU(6) ->
- SU(5) * U(1)
Georgi-Glashow: SU(5) -> SM
Pati-Salam: SU(4) * SU(2) * SU(2) ->
- SU(3) * U(1) * SU(2) * U(1) -> SM
Trinification: SU(3) * SU(3) * SU(3) ->
- SU(3) * SU(2) * U(1) * U(1) * U(1) -> SM

 

[lpetrich]

There is also some SU(10+2k) groups proposed with the goal of family unification. SO(14)? 16? 18?

I think that you mean horizontal / cross-generation symmetry and trying to unify a horizontal symmetry group with a gauge one.

E(8) appears in the heterotic superstring, and it gets broken to E(6) * SU(3) or SO(10) * SU(4), where the first group contains the SM gauge groups and the second group becomes a horizontal-symmetry group.

I'll see what I can come up with for SO(10), where the elementary fermions and their antiparticles are in conjugate spinor representations with dimension 16. A higher SO must have a spinor rep that contains these spinor reps, and SO(14) and SO(15) are the first candidates that come to my mind.

SO(14) / D(7) -> SO(10) * SO(4) / D(2) / SU(2)*SU(2):
EF's:
64 = (16,2,1) + (16*,1,2)
64* = (16,1,2) + (16*,2,1)
64 and 64* are complex conjugates, 2 is pseudoreal
Gauge: 91 = (45,1,1) + (1,3,1) + (3,1,1) + (10,2,2) - adjoints, then vector-vector
Higgs: 14 -> (10,1,1) + (1,2,2) - SO(10) Higgs + vector

SO(15) / B(7) -> SO(10) * SO(5) / B(2):
EF's:
128 = (16,4) + (16*,4)
128 is real and 4 is pseudoreal
Gauge: 105 = (45,1) + (1,10) + (10,5) - adjoints, then vector-vector
Higgs: 15 -> (10,1) + (1,5) - SO(10) Higgs + vector

SO(16) / D(8) -> SO(10) * SO(6) / D(3) / SU(4) * A(3):
EF's:
128 = (16,4) + (16*,4*)
128' = (16,4*) + (16*,4)
128, 128' are real and 4, 4* are complex conjugates
Gauge: 120 = (45,1) + (1,15) + (10,6) - adjoints, then vector-vector
Higgs: 16 -> (10,1) + (1,6) - SO(10) Higgs + vector

So it's possible to get a horizontal symmetry by extending the SO, though with 2 or 4 generations. It does not multiply the Higgs particles, however.

For SU(5), we need SU(15) / A(14), and the EF's are in SU(5) reps 5 and 10.
The fundamental rep is easy:
15 -> (5,3)
However, when one gets the antisymmetrized product, things become more difficult. We want that to make the 10 of SU(5).
105 -> (10,6) + (15,3*)
6 and not 3 generations for the 10, and a *symmetrized* product in SU(5).

 

[lpetrich]

A further connection. The group SO(16) is a subgroup of E(8), while SU(15) is not.

E(8) -> E(6) * SU(3)
248 -> (78,1) + (27,3) + (27*,3*) + (1,8)
Three EF generations and Higgs sets

E(8) -> SO(10) * SU(4)
248 -> (45,1) + (16,4) + (16*,4*) + (10,6) + (1,15)
Four EF generations, but 6 Higgs sets

E(8) -> SU(5) * SU(5)
248 -> (24,1) + (5,10) + (5*,10*) + (10,5*) + (10*,5) + (1,24)
Five EF generations, 5 Higgs sets

E(8) -> SO(16) / D(8)
248 -> 120 + 128 -- adjoint + *one* of the spinors

 

[lpetrich]

For reference, here's the content of the (Minimal Supersymmetric) Standard Model, as

(QCD multiplicity, weak-isospin multiplicity, weak hypercharge) with chirality L or R

Gauge particles: gluon (QCD: SU(3)), W (WIS: SU(2)), B (WHC: U(1))
g (8,1,0) ... W (1,3,0) ... B (1,1,0)

Higgs particles, up Higgs, down Higgs (the MSSM needs 2 Higgs doublets). I'll be listing the chirality of the Higgsinos
Hu (1,2,1/2)L ... Hd (1,2,-1/2)L ... Hu* (1,2,-1/2)R ... Hd* (1,2,1/2)R

Quarks: doublet Q, singlets U and D, up-like and down-like
Q (3,2,1/6)L ... U (3,1,2/3)R ... D (3,1,-1/3)R ... Q* (3*,2,-1/6)R ... U* (3*,1,-2/3)L ... D* (3*,1,1/3)L

Leptons: doublet L, singlets N and E, neutrinos and electron-like
L (1,2,-1/2)L ... N (1,1,0)R ... E (1,1,-1)R ... L* (1,2,1/2)R ... N* (1,1,0)L ... E* (1,1,1)LHiggs terms with coupling-constant matrices yu,yd,yn,ye:
yu.Hu.Q.U* ... yd.Hd.Q.D* ... yn.Hu.L.N* ... ye.Hd.L.E* ... yu*.Hu*.Q*.U ... yd*.Hd*.Q*.D ... yn*.Hu*.L*.N ... ye*.Hd*.L*.E

MSSM Higgs self-interaction with mass mhh:
mhh.Hu.Hd ... mhh*.Hu*.Hd*

Right-handed neutrino seesaw term with mass matrix mnr:
mnr.N*.N* ... mnr*.N.N

All these terms have chiralities LL/RR or LLL/RRR, as expected for Wess-Zumino multiplets

 

[lpetrich]

Georgi-Glashow SU(5) -> Standard Model:
ETA: Hypercharge = (-5/6)*(U(1) factor)

Gauge:
24 -> g + W + B + (3,2,-5/6) + (3*,2,5/6)
Adds a leptoquark with charges -4/3 and -1/3

Higgs:
5L -> Hu + (3,1,-1/3) ... 5*L -> Hd + (3*,1,1/3) ... 5R -> Hd* + (3,1,-1/3) ... 5*R -> Hu* + (3*,1,1/3)
Adds a down-quark-like Higgs triplet, producing the doublet-triplet problem

Elementary fermions:
1L -> N* ... 5R -> D + L* ... 10L -> Q + U* + E* ... 10*R -> Q* + U + E ... 5*L -> D* + L ... 1R -> N
Note the interesting alternation of left-handed and right-handed multipletsHiggs interaction terms:
yu.H5L.F10L.F10L ... yn.H5L.F1L.F5*L ... yde.H5*L.F5*L.F10L ... yu*.H5*R.F10*R.F10*R ... yn*.H5*R.F1*R.F5R ... yde*.H5R.F5R.F10*R ...
where yd = ye = yde -- mass unification for the tau lepton and the bottom quark

Higgs self-interaction:
mhh.H5L.H5*L + mhh*.H5*R.H5R

Right-handed neutrino seesaw term:
mnr.F1L.F1L ... mnr*.F1R.F1R
Still possible in Georgi-Glashow

 

[lpetrich]

SO(10) -> SU(5) * U(1)
Baryon - lepton number (B - L) = - (4/5)*(U(1) factor) + (4/5)*(weak hypercharge)

Gauge:
45 -> (24,0) + (10,1) + (10*,-1) + (1,0)
SU(5) gauge multiplet with additional leptoquarks and a Z_B-L particle.

Higgs:
10L -> 5L + 5*L ... 10R -> 5R + 5*R
One multiplet

Elementary fermions:
16L -> (10,-1/4)L + (5*,3/4)L + (1,-5/4)L ... 16*R -> (10*,1/4)R + (5,-3/4)R + (1,5/4)R
One multiplet

There is one Higgs-EF-interaction term, y.H.F.F, meaning complete mass unification and thus no cross-generation decays. So SO(10) breaking must break mass unification.

Higgs self-interaction, mhh.H.H, also exists, but there is no right-handed-neutrino seesaw term, because such a term breaks B - L.

 

[lpetrich]

One more.
E(6) -> SO(10) * U(1)
The U(1) I will call EFH, because it's involved in distinguishing elementary fermions from Higgs particles.

Gauge:
78 -> (45,0) + (16,1) + (16*,-1) + (1,0)

Elementary fermions and Higgs:
27L -> (16,-1/3)L + (10,2/3)L + (1,-4/3)L ... 27*R -> (16*,1/3)R + (10,-2/3)R + (1,4/3)R

A SO(10) scalar shows up here, which I will call S. There is one Higgs-like interaction term, y.X.X.X, which breaks down into EF's (F), Higgses (H), and those scalars as
y.H.F.F ... y.S.H.H ... y.S.S.S

One of these scalars could appear in accelerator-accessible energies in the Next-to-Minimal Supersymmetric Standard Model (NMSSM).

This model predicts 3 Higgs-doublet pairs and 3 SO(10) scalars, but only one of them seems to be present at low energies. The others must be forced to higher masses by some sort of horizontal symmetry breaking.

 

[lpetrich]

Seems like I found most of the more commonly-discussed GUT gauge-symmetry algebras.

I should also note that
E(6) -> SU(6) * SU(2)
is another way to get from E(6) to SU(6), a superset of SU(6) * U(1)GUT symmetries must be broken to produce the Standard Model, but some proposed GUT Higgs fields are rather large. This is unlike the Standard-Model situation, where the Higgs multiplets are about the size of the others.

First, the highest-weight vectors for the Standard-Model groups.
Standard Model:
SU(2) / SO(3): dimension 2j+1, homework vector (2j), j = 3D angular momentum
SU(3); 3: 10 ... 3*: 01 ... 8: 11

SU(5)
5: 1000 ... 10: 0100 ... 10*: 0010 ... 5*: 0001 ... 24: 1001
Proposed GUT Higgs: 24, ...

SO(10)
10: 10000 ... 45: 01000 ... 16: 00010 ... 16*: 00001
54: 20000 ... 120: 00100 ... 126: 00020 ... 126*: 00002 ... 144: 10010 ... 144*: 10001 ... 210: 00011
Proposed GUT Higgs: 10, 16, 16*, 45, 54, 126, 126*, 144, 144*, 210

E(6)
27: 100000 ... 27* 000010 ... 78: 000001
351: 200000 ... 351*: 000020 ... 351': 010000 ... 351'*: 000100 ... 650: 100010
(short Dynkin-diagram branch is root 6)
Proposed GUT Higgs: 351, 351*, 351', 351'*, 650However, the HE heterotic superstring has only fundamental/adjoint reps of its two E(8) gauge fields, and these break down only to 27, 27*, and 78 of E(6) and 10, 16, 16*, and 45 of SO(10), which strongly limits possible GUT Higgs mechanisms. The more usual symmetry breaking I've seen proposed is from compactification of 10 space-time dimensions into 4 large ones and 6 small ones.