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GUT Proposed Gauge Symmetries

  1. Sep 8, 2011 #1
    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?
     
    Last edited: Sep 8, 2011
  2. jcsd
  3. Sep 8, 2011 #2

    arivero

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    Gold Member

    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?
     
  4. Sep 9, 2011 #3
    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
     
  5. Sep 9, 2011 #4
    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).
     
  6. Sep 9, 2011 #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
     
  7. Sep 9, 2011 #6
    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)L


    Higgs 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
     
    Last edited: Sep 9, 2011
  8. Sep 9, 2011 #7
    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 multiplets


    Higgs 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
     
    Last edited: Sep 9, 2011
  9. Sep 9, 2011 #8
    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.
     
  10. Sep 9, 2011 #9
    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.
     
  11. Sep 21, 2011 #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, hw 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'*, 650


    However, 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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