# Exceptional group and grand unified theory

• A
• shereen1

#### shereen1

Dear All
I have a project about exceptional group as a candidate group for grand unified theories. Can anyone suggest me any paper or reference to use.
Thank you

Georgi's book on group theory treats, afaik, unification and exceptional groups.

Demystifier
Dear All
I have a project about exceptional group as a candidate group for grand unified theories. Can anyone suggest me any paper or reference to use.
Thank you
Slansky report

Demystifier
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)

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)

How does the question of having complex representations work from E8 down to E6?

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.

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)

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 :-)

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.

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.

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?

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?

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?

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

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).

arivero
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.

I hope the OP will post some link to the work when finished :-)

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)

phyzguy's Garrett Lisi paper with its page title: [0711.0770v1] An Exceptionally Simple Theory of Everything. Its abstract:
All fields of the standard model and gravity are unified as an E8 principal bundle connection. A non-compact real form of the E8 Lie algebra has G2 and F4 subalgebras which break down to strong su(3), electroweak su(2) x u(1), gravitational so(3,1), the frame-Higgs, and three generations of fermions related by triality. The interactions and dynamics of these 1-form and Grassmann valued parts of an E8 superconnection are described by the curvature and action over a four dimensional base manifold.

It has E8 multiplet 248 which breaks down into G2 and F4 ones as
248 -> (14,1) + (1,52) + (7,26)

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.

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.

arivero
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.
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

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.

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.

A potential problem with supersets of the Standard Model is quantum-mechanical anomalies. Certain diagrams cannot be evaluated consistently.

[0802.0634] Lectures on Anomalies, PHY 522 Topics in Particle Physics and Cosmology: Anomalies in Quantum Field Theory, like this one on the Standard Model: Lecture 7: Fri Oct 9

In particular, for the Standard Model and its supersets, it's the "triangle anomaly", a fermion loop with three gauge particles coming out of it. This is for 4 space-time dimensions. For D dimensions, it only exists for D being even, and it has D/2+1 gauge particles. Thus, for 10 dimensions, the anomaly is a hexagon anomaly. In the diagram, even numbers of gauge particles can be replaced with gravitons, giving a gauge-gravitational anomaly or even a pure gravitational anomaly.

The anomaly consists of two parts multipled together: an interaction part and a loop-integral part. It's the loop integral that causes the trouble, so the only way to get rid of anomalies is to make the interaction part cancel. That implies constraints on possible quantum field theories.

Let us see how to evaluate the interaction part. For triangle anomalies, with gauge particles a, b, and c, the interactions with the fermions can be given by matrices Ta, Tb, and Tc. Since we want to sum over all the fermions, we get

Tr(Ta.Tb.Tc + Ta.Tc.Tb)left - Tr(Ta.Tb.Tc + Ta.Tc.Tb)right

If (say) b and c are gravitons, then Tb = Tc = I * (some constant), and we get

Tr(Ta)left - Tr(Ta)right

The interaction parts are proportional to the "Casimir invariants" of the gauge algebra:

Tr(C(T,p))left - Tr(C(T,p))right
for p = 1,3 (4D) and p = 0, 2, 4, 6 (10D)

Every algebra has an infinite number of them, but only a finite number of linearly independent ones, and all of them can be constructed from the smallest linearly-independent ones.

Thus, U(1) has the smallest independent one C(1) = (U(1)-factor value). All the others are C(p) = C(1)p.

Let's now look at the sizes of the smallest independent ones:
U(1): 1
SU(n) = A(n+1): 2, 3, ..., n
SO(2n) = D(n): 2, 4, ..., 2n-2, n
SO(2n+1) = B(n): 2, 4, ..., 2n
Sp(2n) = C(n): 2, 4, ..., 2n
G2: 2, 6
F4: 2, 6, 8, 12
E6: 2, 5, 6, 8, 9, 12
E7: 2, 6, 8, 10, 12, 14, 18
E8: 2, 8, 12, 14, 18, 20, 24, 30

For all but U(1), the smallest one is 2.

So for the Standard-Model anomalies, the contributors are: C(QCD,3), C(QCD,2)*C(WHC,1), C(WIS,2)*C(WHC,1), C(WHC,1)3, C(WHC,1) for QCD = SU(3) = quantum chromodynamics, WIS = SU(2) = weak isospin, and WIH = U(1) = weak hypercharge.

In the Standard Model's elementary-fermion multiplets, the triangle anomaly cancels. The Minimal Supersymmetric Standard Model includes a pair of Higgs doublets, the up Higgs and the down Higgs, and the triangle anomaly cancels for them also. One has to evaluate it for the MSSM's
higgsinos, because supersymmetry gives the Higgs particles fermionic counterparts.

Looking at GUT's, we start at Georgi-Glashow SU(5). C(GUT,3) is nonvanishing here, but the interaction part does vanish for both the extrapolated elementary-fermion content and the extrapolated higgsino content.

But for SO(10), E6, E7, and E8, the interaction part automatically vanishes, because C(GUT,3) = 0 for all of them. This counts in favor of such theories.

But in 10D, the hexagon anomaly contains C(GUT,2), C(GUT,4), and C(GUT,6), and these are in general nonzero. Thus, in heterotic superstrings, their cancellation only for SO(32) and E8*E8 is a nontrivial result.

arivero, shereen1 and haushofer