Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

A Exceptional group and grand unified theory

  1. Apr 13, 2016 #1
    Dear All
    I have a project about exceptional group as a candidate group for grand unified theories. Can any one suggest me any paper or reference to use.
    Thank you
     
  2. jcsd
  3. Apr 13, 2016 #2

    phyzguy

    User Avatar
    Science Advisor

  4. Apr 14, 2016 #3

    haushofer

    User Avatar
    Science Advisor

    Georgi's book on group theory treats, afaik, unification and exceptional groups.
     
  5. Apr 14, 2016 #4

    arivero

    User Avatar
    Gold Member

    Slansky report
     
  6. May 30, 2016 #5
    The smallest exceptional algebra that's suitable for GUT's is E6. The others are either supersets of it (E7, E8) or unsuitable (G2, F4). The algebra E6 can break down into the Standard-Model one by several routes:

    E6 -> SO(10) * U(1)
    SO(10) -> SU(5) * U(1) -- Georgi-Glashow
    SU(5) -> SM: SU(3) * SU(2) * U(1)

    SO(10) -> SO(6) * SO(4) ~ SU(4) * SU(2) * SU(2) -- Pati-Salam
    SU(4) * SU(2) * SU(2) -> SM * U(1)

    E6 -> SU(3) * SU(3) * SU(3)
    SU(3) * SU(3) * SU(3) -> SM * SU(2) * U(1)

    E6 -> SU(6) * SU(2)
    SU(6) -> SU(5) * U(1)
    SU(5) -> SM * SU(2) * U(1)

    Turning to supersets of E6, the most-discussed one is E8, and that can come out of the HE heterotic superstring. Thus getting the Standard Model from string theory:
    E8 -> E6 * SU(3)
     
  7. May 30, 2016 #6

    arivero

    User Avatar
    Gold Member

    How does the question of having complex representations work from E8 down to E6?
     
  8. May 30, 2016 #7
    E8 has a fundamental representation, its 248 one. That representation is also its adjoint one, something unique to this algebra. Not surprisingly, it is a real rep.

    This rep of E8 breaks down into E6*SU(3) as follows:
    248 -> (78,1) + (1,8) + (27,3) + (27*,3*)

    I've also seen E8 -> SO(10)*SU(4):
    248 -> (45,1) + (1,15) + (16,4) + (16*,4*) + (10,6)

    I don't know if anyone has proposed E8 -> SU(5)*SU(5), however. But here goes:
    248 -> (24,1) + (1,24) + (5,10) + (5*,10*) + (10,5*) + (10*,5)

    So both conjugates of the Standard Model's complex reps fit inside of E8's fundamental rep.
     
  9. May 30, 2016 #8

    arivero

    User Avatar
    Gold Member

    Never seen it; but if the (24,1) -and (1,24)- are leptons, I think it is worthwhile an effort to try to extract 36 quarks from the (5,10) and claim the other 14 states as predictions for the LHC 750 GeV thing :-)
     
  10. May 30, 2016 #9
    The (24,1) and (1,24) are SU(5)*SU(5) gauge multiplets. The 5, 5*, 10, and 10* are elementary fermions and Higgs particles.
     
  11. May 30, 2016 #10

    arivero

    User Avatar
    Gold Member

    I am not sure why; I would expect all of them to be particles, the gauge multiplets being the matrices that act over them. It would be nice to have some paper describing this; I understand from your first comment that there is none?
     
  12. May 31, 2016 #11
    I don't know of any paper describing (superstring E8) -> SU(5) * SU(5).

    I have been able to find several on (superstring E8) -> E6 * SU(3) and some on (superstring E8) -> SO(10) * SU(4). Shall I link to some of them?
     
  13. May 31, 2016 #12

    arivero

    User Avatar
    Gold Member

    E8 to E6 is the traditional. They look interesting, but well known :-)

    Also surely are aware of the half-joke E6 -> E5 -> E4 to describe the standard model GUT hierarchy
     
  14. May 31, 2016 #13
    Yes, and one can go even further to get the Standard Model as E3 * U(1).

    Here are the appropriate Dynkin diagrams, in ASCII form:
    • E8: 1 - 2 - 3 ( - 8) - 4 - 5 - 6 - 7
    • E7: 1 - 2 - 3 ( - 7) - 4 - 5 - 6
    • E6: 1 - 2 - 3 ( - 6) - 4 - 5
    • E5: 1 - 2 - 3 ( - 5) - 4 ... D5 = SO(10)
    • E4: 1 - 2 - 3 ( - 4) ... A4 = SU(5)
    • E3: 1 - 2 (3) ... A2 * A1 = SU(3) * SU(2)
    At this point, we get two possibilities for E2:
    • 1 - 2 ... A2 = SU(3)
    • 1 (2) ... A1 * A1 = SU(2) * SU(2)
    Standard-Model electroweak symmetry breaking is the first possibility, with the SU(3) being the QCD algebra.

    These root removals can be handled as demoting the roots to U(1) factors. Some of these resulting U(1)'s have various meanings.

    In electroweak symmetry breaking, the SU(2) weak-isospin root gets demoted to a U(1) projected-weak-isospin (component 3) factor. A mixture of it and the weak-hypercharge U(1) gives the electromagnetic U(1).

    In SU(5) -> SU(3) * SU(2) * U(1) the U(1) factor is for weak hypercharge.

    In SO(10) -> SU(5) * U(1) the U(1) factor is a mixture of weak hypercharge and B - L (baryon number - lepton number).

    I can't think of any simple interpretation of the U(1) in E6 -> SO(10) * U(1).
     
  15. Jun 1, 2016 #14
    Burt Ovrut and others have done work on (superstring E8) -> SO(10) * SO(6) (SO(6) ~ SU(4))

    Now for how E6 breaks down into SO(10) * U(1). Here are E6's smallest nonscalar irreps:
    27 -> (10, -2) + (16, 1) + (1, 4)
    27* -> (10,2) + (16*,-1) + (1,-4)
    78 -> (45,0) + (1,0) + (10,-3) + (10*,3)
    where I use * for the conjugate rep.

    From A121737 - OEIS, "Dimensions of the irreducible representations of the simple Lie algebra of type E6 over the complex numbers, listed in increasing order." -- 1, 27, 78, 351, 650, 1728, 2430, 2925, 3003, 5824, 7371, 7722, 17550, 19305, 34398, 34749, 43758, 46332, 51975, 54054, 61425, 70070, 78975, 85293, 100386, 105600, 112320, 146432, 252252, 314496, 359424, 371800, 386100, 393822, 412776, 442442

    Likewise, from A121732 - OEIS, "Dimensions of the irreducible representations of the simple Lie algebra of type E8 over the complex numbers, listed in increasing order." -- 1, 248, 3875, 27000, 30380, 147250, 779247, 1763125, 2450240, 4096000, 4881384, 6696000, 26411008, 70680000, 76271625, 79143000, 146325270, 203205000, 281545875, 301694976, 344452500, 820260000, 1094951000, 2172667860

    E6 has the nice feature that a symmetrized product of three 27's gives a scalar. A 27 contains three SO(10) irreps: the scalar, the vector, and one of the two spinors. Its conjugate 27* is similar, but with the other spinor, the conjugate of the first one. The vector can be identified with the (Minimal-Symmetric-)Standard-Model Higgs particles, and the spinor with its elementary fermions. So I'll call the three SO(10) parts S, H, and F.

    The three-27 product contain these possible products of them, and only these possible ones: H.F.F and S.H.H.

    The H.F.F has a simple interpretation. It's what the (MS)SM EF-Higgs interactions become in SO(10). However, the S.H.H term resembles what the MSSM Higgs-mass "μ" term becomes in SO(10), with the S field instead of the inserted-by-hand mass value μ. The S as a "Higgs singlet" is a part of an extension of the MSSM, the Next-to-MSSM or NMSSM.

    Symmetry breaking from E6 to SO(10) will be rather complicated, it must be noted. Three of the 27's contain the EF multiplets, while some mixture of them or else some additional one contains the Higgs particles.
     
  16. Jun 1, 2016 #15

    arivero

    User Avatar
    Gold Member

    I hope the OP will post some link to the work when finished :-)
     
  17. Jun 3, 2016 #16
    Here is (weak hypercharge) + (B-L) for the elementary fermions and Higgs particles. I'll assume that MSSM, and I'll make all the fermionic parts left-handed. In the MSSM, both the EF's and the Higgses are Wess-Zumino multiplets, unifying them by that much.

    SU(3), SU(2), WHC U(1), B-L U(1), combination (right-handed parts are here their antiparticles, which are left-handed), SU(5) and SO(10) multiplets
    Left-handed quarks (6): 3, 2, 1/6, 1/3, 1, (10, 16)
    Right-handed up (3): 3*, 1, -2/3, -1/3, 1, (10, 16)
    Right-handed down (3): 3*, 1, 1/3, -1/3, -3, (5*, 16)
    Left-handed leptons (2): 1, 2, -1/2, -1, -3, (5*, 16)
    Right-handed neutrino (1): 1, 1, 0, 1, 5, (1, 16)
    Right-handed electron (1): 1, 1, 1, 1, 1, (10, 16)
    Up Higgs (2): 1, 2, 1/2, 0, -2, (5, 10)
    Down Higgs (2): 1, 2, -1/2, 0, 2, (5*, 10)
    Combination = -4*(WHC) + 5*(B-L)
     
  18. Jun 3, 2016 #17
    phyzguy's Garrett Lisi paper with its page title: [0711.0770v1] An Exceptionally Simple Theory of Everything. Its abstract:
    It has E8 multiplet 248 which breaks down into G2 and F4 ones as
    248 -> (14,1) + (1,52) + (7,26)
    Adjoint -> adjoint + (fundamental, fundamental)

    Strictly speaking, he uses a noncompact analytic continuation of E8, one that breaks down into G2 * (NCAC of F4).

    He proposes G2 -> SU(3)/A2 of QCD:
    7 -> 1 + 3 + 3*, (fundamentals, scalar)
    14 -> 8 + 3 + 3*, (adjoint, fundamental)

    He also proposes F4 -> SO(8)/D4, though that would first go through SO(9)/A4:
    26 -> 1 + 9 + 16, (fundamental, scalar, vector, spinor)
    52 -> 36 + 16 (adjoint, spinor)
    9 -> 1 + 8 (scalar, vectors)
    16 -> 8 + 8 (spinors)
    36 -> 28 + 8 (adjoint, vector)
    Combined:
    26 -> 1 + 1 + 8 + 8 + 8
    52 -> 28 + 8 + 8 + 8

    Strictly swpeaking, it's (NCAC of F4) -> SO(7,1)
    It breaks down further to SO(3,1) * SO(4) and the second one is equivalent to SU(2) * SU(2)
    Thus getting the Lorentz group (space-time), weak isospin, and a SU(2) that breaks down to the U(1) of weak hypercharge.

    SO(8) -> SO(4) * SO(4) ~ SU(2) * SU(2) * SU(2) * SU(2)
    giving us
    8 -> (4,1) + (1,4) -> (2,2,1,1) + (1,1,2,2)
    8 -> (2,2) + (2',2') -> (2,1,2,1) + (1,2,1,2)
    8 -> (2,2') + (2',2) -> (2,1,1,2) + (1,2,2,1)
    28 -> (6,1) + (1,6) + (4,4) -> (3,1,1,1) + (1,3,1,1) + (1,1,3,1) + (1,1,1,3) + (2,2,2,2,2)

    This model requires some complicated symmetry breaking to keep many of its particles from being observed at low energies. By comparison, SU(5), SO(10), and E6 look much simpler -- and they also can be derived from E8.
     
  19. Jun 7, 2016 #18
    I've tried searching the literature on GUT's, but much of it describes specific models. The overviews that I've found, like at the Particle Data Group, mainly discuss SU(5) and SO(10), with only a little mention of E6 and E8.

    I had worked out E8 -> SU(5)*SU(5) with some software I'd written: SemisimpleLieAlgebras.zip It's in Mathematica, Python, and C++, and it's feature-parallel except for the graphics parts in Mma. I do the algebras and also their representations, expressed as {root, weight, multiplicity} sets. The software does products of reps, powers of reps with various symmetries, subalgebras, and reps in them, complete with values of U(1) factors as appropriate.
     
  20. Jun 8, 2016 #19

    arivero

    User Avatar
    Gold Member

    I see the OP is still around (shereen1 was last seen: Yesterday at 1:33 PM) so I hope will appreciate the work!!

    My own intestest in SU(5)xSU(5) is related to flavour. Time ago I looked to a way to reconstuct the three families from SU(5) flavour, the symmetry that transforms the five light quark, generalization of the SU(2) of isospin. In that idea, the families appear from 5 x 5 giving the lepton charges in the 24, 5 x 5 giving the antiquark charges in the 15, and 5 x 5 similarly giving the quark charges. So I think that a similar assignment could also be searched in the product SU(5)xSU(5), but I did not tried. Note that the 24 cointains all the +1, 0 and -1 charges, while the 15 of the quarks contains six of +2/3, six of -1/3 and, well, three of +4/3. This extra content was a sort of failure for the idea.

    Part of the atractive of E8 into SM, at least for the amateurs, is that it seems to invite ways to introduce a gneration structure. Different ways, it seems, depending of the author.My attempt was not connected to E8 but only because I had not thought in its SU(5)xSU(5) substructure :-D
     
  21. Jun 8, 2016 #20
    One gets SU(5)*SU(5) from E8 from what I like to call extension splitting. For an algebra's Dynkin diagram, add an extra root at a suitable place. For some algebras, at least, it's to a root that has a nonzero highest weight in the algebra's adjoint rep. For E8, this extra root is added to the end of its diagram's long branch. The resulting diagram will not be a legitimate diagram for an algebra, but that's not a problem.

    The next step is to remove a root, and one gets for E8: SU(2)*E7, SU(3)*E6, SU(4)*SO(10), SU(5)*SU(5), SU(6)*SU(3)*SU(2), SU(8)*SU(2), SO(16)
    Not all of them are maximal. For instance, E7 -> SU(8) and SO(16) -> SO(6)*SO(10) ~ SU(4)*SO(10)

    I'll now consider the exceptional algebra between E6 and E8: E7.

    E7: fundamental 56, adjoint 133
    E6: fundamental 27 with conjugate 27*, adjoint 78
    SO(12): vector 12, adjoint 66, spinors 32, 32'
    SO(10): vector 10, adjoint 45, spinors 16, 16*

    One can get SO(10) from E7:
    E7 -> SO(12)*SU(2) 56 -> (32,1) + (12,2)
    133 -> (66,1) + (1,3) + (32',2)
    SO(12) -> SO(10)*U(1)
    12 -> (10,0) + (1,1) + (1,-1)
    66 -> (45,0) + (1,0) + (10,1) + (10,-1)
    32 -> (16,1/2) + (16*,-1/2)
    32' -> (16,-1/2) + (16*,1/2)
    Combined:
    56 -> (16,1,1/2) + (16*,1,-1/2) + (10,2,0) + (1,2,1) + (1,2,-1)
    133 -> (45,1,0) + (1,3,0) + (1,1,0) + (10,1,1) + (10,1,-1) + (16,2,-1/2) + (16*,2,1/2)

    One can also get E6 from E7:
    E7 -> E6*U(1)
    56 -> (27,1/2) + (27*,-1/2) + (1,3/2) + (1,-3/2)
    133 -> (78,0) + (1,0) + (27,-1/2) + (27*,1/2)

    One won't be able to get multiple elementary-fermion generations from a single E7 fundamental rep, as one can do with E8.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Exceptional group and grand unified theory
Loading...