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

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Discussion Overview

The discussion revolves around various proposed gauge-symmetry groups for Grand Unified Theories (GUTs), exploring their structures, symmetry breakings, and implications for elementary particles. Participants examine different GUT models, including their gauge particles, Higgs particles, and elementary fermions, while considering the relationships and transitions between these groups.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Exploratory

Main Points Raised

  • Some participants outline the unbroken Standard-Model symmetry as SU(3)C * SU(2)L * U(1)Y and propose various GUTs such as SU(5), SO(10), and E(8), detailing their gauge structures and particle content.
  • One participant mentions the Pati-Salam model, suggesting a breaking into U(1)xSU(3)xSU(2)xSU(2) with B-L as a significant component.
  • Another participant introduces the idea of SU(10+2k) groups aimed at family unification, mentioning SO(14), SO(16), and SO(18) as potential candidates.
  • There are discussions about the breaking patterns of E(8) into various combinations of other groups, including E(6) and SO(10), with specific symmetry breakings detailed.
  • Some participants explore the implications of higher SO groups, such as SO(14) and SO(15), in relation to elementary fermion representations and gauge structures.
  • One participant highlights the relationship between E(8) and SO(16), noting that SO(16) is a subgroup of E(8) while SU(15) is not.
  • Details about the Minimal Supersymmetric Standard Model (MSSM) are provided, including gauge particles, Higgs particles, and their interactions.
  • Participants discuss the implications of the Georgi-Glashow SU(5) model and its breaking to the Standard Model, including the introduction of leptoquarks and the structure of Higgs and fermion multiplets.
  • SO(10) is presented as breaking into SU(5) * U(1), with discussions on the implications for baryon-lepton number and mass unification.
  • There is mention of E(6) breaking into SO(10) * U(1), with a focus on distinguishing elementary fermions from Higgs particles.

Areas of Agreement / Disagreement

Participants express a range of views on the proposed gauge-symmetry groups and their implications, with no consensus reached on the superiority or correctness of any specific model. Multiple competing views remain regarding the structure and implications of these GUTs.

Contextual Notes

Some discussions involve complex representations and breaking patterns that may depend on specific assumptions or definitions, which are not fully resolved within the thread.

lpetrich
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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?
 
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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?
 
arivero said:
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
 
arivero said:
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).
 
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
 
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
 
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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
 
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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 ZB-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.
 
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.
 
  • #10
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.
 

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