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\(E_6\) grand unification, F-theory prevail in the \(750\GeV\) phenomenology today

Models of numerous kinds have been proposed in about 130 model builder papers so far that try to explain the diphoton resonance of invariant mass around \(750\GeV\) seen at the LHC – which may turn out to be a manifestation of new physics or a fluke.

Tommaso Dorigo is a cynic but I found his recent comment amusing, even if it is wrong. He claims that the \(750\GeV\) bump can be neither the Loch Ness Monster nor Mickey Mouse, because of some differences in the shape. Otherwise, the bump can be anything.

The bump may actually be due to Loch Ness and Mickey Mouse assuming that both of them cooperate and Loch Ness is properly twisted by the Donald Duck. Dorigo has overlooked that model.

More seriously, in the real world, the bumps may be real and many of the explanations are viable if not highly intriguing. I've discussed sgoldstinos, radions, D3-branes and closed strings, sbinos and NMSSM, and a few other generic options.

But today, we have a clear new winner, the \(E_6\) grand unified theories.

On the hep-ph arXiv, we were shown the first dose of new papers submitted in 2016. There are 84 entries on the hep-ph arXiv today. 33 of them are new papers primarily classified as hep-ph articles. I believe that 11 of them are dedicated to the new diphoton resonance: 2, 9, 12, 17, 24, 28-33.

Previously, the total number of hep-ph papers about the resonance was 118 on December 29th. I believe that 11 were added on December 31st but only 2 on January 1st. So as of now, the number of papers should be 118+11+2+11=142 but I hope you understand that it will be increasingly difficult to be any certain about the total number (the papers can't be found by any easy reliable "reference"). See this page with a graph and hyperlinks claiming that the current number is 146 (thanks, Djack).

At the end of the year, I could mention a fun Chilean paper that claims to explain the patterns in quark and lepton masses, on top of the diphoton resonance, from a scalar with a \(\ZZ_{14}\times\ZZ_2\) group and a new quark of charge \(+8/3\); and Ian Low's and Joe Lykken's paper imposing model-independent bounds on the branching ratios.

But among the 11 new papers today, there is one rather specific kind of models that dominate. 4 out of 11 papers (36%) argue that the diphoton resonance may be obtained from the grand unified theories whose gauge group is \(E_6\), the only simple exceptional compact Lie group that has complex representations and is therefore directly viable as the gauge group of grand unified field theories. It is not quite a new category – I've reminded you of some \(E_6\) model building e.g. in the context of the paper [25] here, marketed as an extension of NMSSM. (Maybe I was the first one to talk about \(E_6\) in the context of the diphoton resonance and the new papers have copied me – and I am not upset at all if that is the case.)

But the "synergy" with which the today's authors write about \(E_6\) is kind of remarkable.

The \(E_6\) Dynkin diagram is left-right-symmetric, much like the \(SO(2N)\) and \(SU({}^\geq 3)\) diagrams (\(SO(8)\) has the higher \(S_3\) triality symmetry). This \(\ZZ_2\) symmetry is what exchanges the representations with their inequivalent complex conjugate friends. The existence of this left-right symmetry and therefore these complex representations (reps inequivalent to their complex conjugates) is ironically needed to explain the left-right-asymmetric character of the electroweak force which is why \(E_6\) is the only exceptional group that may be directly used as a grand unified gauge group.

We are talking about four papers

Palti: Vector-Like Exotics in F-Theory and \(750\GeV\) Diphotons

Ko+2: Diphoton Excess at \(750\GeV\) in leptophobic \(U(1)'\) model inspired by \(E_6\) GUT

Chao: The Diphoton Excess from an Exceptional Supersymmetric Standard Model

Karozas+3: Diphoton excess from \(E_6\) in F-theory GUTs

Bonus from end of January 2016:
King+Nevzorov: \(750\GeV\) Diphoton Resonance from Singlets in an Exceptional Supersymmetric Standard Model
Three papers among the four (except for Ko+2) talk about the stringy origin of these effective field theories. The first and the last one explicitly talk about F-theory, Vafa's geometric generalized nonperturbative description of type IIB string vacua, while the third paper by Chao envisions a heterotic string model as the origin (there exist heterotic-F-theory dualities so a model may often have both descriptions at the same moment).

As I have previously mentioned, relatively to the 16-dimensional representation for fermions in \(SO(10)\) grand unified model building (the 16 components include the right-handed neutrino, unlike the 5+10-dimensional representation in \(SU(5)\) models), the fundamental representation of \(E_6\) is 27-dimensional and decomposes as \[

{\bf 27} = {\bf 16}\oplus {\bf 10}\oplus {\bf 1}

\] under the \(SO(10)\) subgroup. The extra 10-dimensional representation behaves as \({\bf 5}\oplus \bar{\bf 5}\) under the \(SU(5)\) subgroup. Because we get both the representation and its complex conjugate, it means that the fermions will transform as full uniform Dirac spinors under the \(SU(5)\) group. So the gauge couplings will be non-chiral i.e. vector-like, as people call it. In the Dirac spinor notation, you won't need any \(\gamma_5\) to write the Lagrangian for those fields.

The chiral interactions are the cornerstone of the electroweak theory, as you know (only the left-handed fermions and right-handed antifermions interact weakly) but to explain the diphoton resonance, we need the couplings to the pairs of gluons (initial state) and pairs of photons (final state) and non-chiral interactions are good for that.

However, as most of the 4 papers notice, the new vector-like quarks interact chirally with a new \(U(1)'\) gauge group that arises out of the \(E_6\). At least the first two papers among the four papers above explain that this is great because this non-chiral interaction of the new exotic quarks guarantees that they have masses linked to the Higgs that breaks this \(U(1)'\) symmetry – which they put close to \(1\TeV\). This scale used to be expected to be near the GUT scale instead. The first paper claims that by reducing this scale, we get an extra bonus: the proton decay slows down!

Note that in the original "normal" Georgi-Glashow \(SU(5)\) grand unified models, we need fermions in \({\bf 5}\oplus \bar{\bf 10}\) under \(SU(5)\) and the known leptons and quarks are organized so that the five-dimensional representation contains a lepton doublet and a right-handed down-quark which is an electroweak singlet. The identity \(3+2=5\) is helpful here. In the "flipped" \(SU(5)\) models (Barr 1982; Nanopoulos et al. 1984; Antoniadis, Ellis, Hagelin for the supersymmetric, string-inspired version), the assignments of the charges are switched to another possible solution and the 5-dimensional representation contains a lepton doublet plus the right-handed up-quark instead. See a 2013 TRF text about the basic GUT embeddings.

Off-topic: 21 days are left to the global destruction.

At any rate, the today's papers assume the original Georgi-Glashow embedding. Because the new exotic vector-like fermions coming from the \(E_6\) models transform as a 5-dimensional representation of \(SU(5)\) plus its complex conjugate, the only color-charged states they contain are those resembling the down-quark.

So these \(E_6\) models have new Dirac spinors that look like the down-quark – the same electric charge, three colors, but no chiral (left-right-asymmetric) electroweak interactions. It's this new exotic quark that runs in the loop (triangle) that produces the intermediate scalar – and that allows it to decay, too. The intermediate scalar is the new Higgs that breaks the \(U(1)'\) symmetry in these models; and it has an even CP-even "real part" which is said to be the \(750\GeV\) new particle allegedly observed by the LHC.

This model makes a lot of sense. The \(E_6\) grand unification has predicted the extra states. There were no experimental signs of them – which was one reason why people preferred to assume that all of them were heavy. But aside from the "absence of experimental evidence" which is not the "evidence of absence", there has never been any good reason to be convinced that the "rest of the 27-dimensional representation" must be GUT-scale heavy.

In 2016, the LHC may very well start to discover the new exotic down-quarks, three generations of them, which combine to the 27-dimensional representation along with the known quarks and leptons. That would be truly remarkable.

Supersymmetry and grand unification are two of the beautiful principles in the beyond-the-standard-model model building. It's somewhat hard to say which of them is more radical – the first and only non-bosonic symmetry; or a complete unification of all non-gravitational gauge groups? Well, SUSY would still win. But one reason why it was considered "hotter" was the assumption that SUSY must emerge before grand unification because SUSY has some good reasons (an explanation of the hierarchy problem) to appear near the Higgs mass scale; while the rest of the GUT model building is likely to have masses near the GUT scale.

But this assumption may have been wrong and the GUT multiplets may show up in front of our eyes very soon. Imagine that. Imagine that the LHC will rather soon discover three generations of fermions that complete the three generations transforming in the 27-dimensional representation (probably just 25 of 27 because the two singlets at two levels are too weakly interacting; but we should also see 3 generations of the new vector-like lepton doublet). The LHC folks may deserve a Nobel prize but so can the theorists.

Who should get a Nobel prize for the \(E_6\) model building? The more general idea, the grand unification initiated by the \(SU(5)\) group, was started by Georgi and Glashow. They're great physicists and entertaining men. But to some extent, they have abandoned their spiritual child like an Islamic father who (fortunately non-lethally) stabs his offspring because he's upset that it's a girl. I have a problem to consider Georgi and especially Glashow as flagships of grand unification today. The physicists celebrated for something shouldn't be "accidental encounterers" of something but folks who were convinced that the discovery is real and important and who kept on pushing it for years, perhaps despite some opposition. Georgi and Glashow (who basically abandoned GUT model building after the first simple-minded prediction of the proton decay was falsified) are not examples of that.

Moreover, it's the \(E_6\) group that is needed. The possibility to replace \(SU(5)\) by \(SO(10)\) or \(E_6\) was somewhat obvious. I admit that I don't know who wrote the first papers on grand unification involving these larger groups. But I think it's true that the exceptional groups are sufficiently exotic for most non-stringy particle physicists so that a huge portion of the \(E_6\) grand unified literature is actually stringy or at least string-inspired.

\(E_6\) and the beauty of string theory

As I just hinted, I want to end up with a section emphasizing how terribly close \(E_6\) – which may get experimentally discovered – would be to some key aspects of the beauty hiding in string theory model building. The largest compact exceptional simple Lie group is \(E_8\). It can't be used as a starting point (gauge group) in field theory because it has no complex representations. But it does work as the starting point in string theory because strings have new, intrinsically stringy ways to break the gauge symmetry.

And indeed, there exists a beautiful geometric description of this \(E_8\) starting point, the \(E_8\times E_8\) heterotic string, the main "detailed candidate for a stringy TOE" of the 1980s (and for some, including your humble correspondent, even today). The \(E_8\) gauge group may also be shown (and Hořava+Witten did it) to arise on the boundaries of the spacetime in the 11-dimensional M-theory on spaces with boundaries. A "board" of the 11-dimensional spacetime has two boundaries (upper and lower; the vertical dimension is a line interval) which is why we get two \(E_8\) factors in the dual heterotic string at each point of the 10 "horizontal" spacetime dimensions.

Also, F-theory as well as M-theory may be shown to admit codimension-4 singularities with the \(E_8\) gauge supermultiplet living on them. Such M/F-theoretical vacua with singularities may be shown dual (equivalent) to the heterotic string, at least in some simple (and some hard) cases. It's very interesting to know which singularities produce the \(E_6,E_7,E_8\) gauge group. The answer is given by the ADE classification. They arise from \(\CC^2/\Gamma\) where \(\Gamma\) is a finite subgroup of \(SU(2)\), basically a subgroup of \(SO(3)\) translated to its action on the 2-dimensional spinor. If \(\Gamma\) is the symmetry group of the tetrahedron, you get an \(E_6\); if you pick a cube or an octahedron instead, you obtain \(E_7\); for a dodecahedron or the dual icosahedron, one ends up with the \(E_8\). Many of these facts are well-known and rigorous mathematical and physical facts. Some of them are intriguing speculations. For example, we may get an \(E_6\) from the orbifold by the tetrahedron isometry group which resembles an \(A_4\) or an \(S_4\). It's very tempting to identify this group with one of the most realistic choices of the "family groups" that produce realistic spectra of fermion masses!

The fundamental representation of \(E_8\) is 248-dimensional and coincides with the adjoint representation (the only simple Lie group where it's true). The only smaller representation is the trivially transforming 1-dimensional singlet! Under the maximum \(E_6\times SU(3)\) subgroup, \({\bf 248}\) decomposes as\[

({\bf 78},{\bf 1}) \oplus (1,{\bf 8}) \oplus ({\bf 27},{\bf 3}) \oplus (\bar{\bf 27},\bar{\bf 3})

\] It's very intriguing. Aside from the adjoint representation of the subgroup, we get 3 copies of the 27-dimensional representation of \(E_6\) – it almost looks like the \(E_8\) has the right content for three families of the right \(E_6\) type. Some orbifolds link this number 3 to the number of "complex hidden dimensions" and in this sense, they naturally explain why the number of generations is three.

It's also interesting to see how the \(E_8\) representation \({\bf 248}\) decomposes under another maximal subgroup, \(G_2\times F_4\).\[

({\bf 14}, {\bf 1}) \oplus
({\bf 1}, {\bf 52}) \oplus
({\bf 7}, {\bf 26})

\] Here, \(G_2\) is the automorphism group of the octonions (the pure imaginary octonions form a 7-dimensional real space); \(F_4\) ends up being a centralizer of a \(G_2\) inside \(E_8\) and it's also the automorphism group of the "Hermitian" \(3\times 3\) octonion matrices equipped with the "anticommutator" binary operation. There is also an interesting relationship between \(F_4\) and \(E_6\): \(F_4\) may be obtained as a \(\ZZ_2\) orbifold from \(E_6\). Well, if you "divide" the \(E_6\) Dynkin diagram by a \(\ZZ_2\) reflecting the left and right sides, you get the \(F_4\) Dynkin diagram.

All these facts have some beautiful geometric explanations in string theory. The character of and relationships between the exceptional gauge group may look like some artificially added mathematics in the context of quantum field theories. But in the context of string theory, they're unavoidable and "obvious" geometric properties of the possible shapes of extra dimensions. String theory loves to produce the right equations whose solutions cover the boring, systematic cases as well as the exceptional ones. The fact that we have a rather clever gauge group – such as the Standard Model gauge group or, ideally in 2016, the \(E_6\) group – may be attributed to the extra dimensions. The extra dimensions are what "produces" the complexity and structure of these groups and the representations in which the matter fields transform.

If the LHC happened to discover the \(750\GeV\) particle as well as some new particles predicted by the \(E_6\) models, that would be spectacular. It wouldn't bring the grand unification scale closer to our experiments. But it would bring some scales ("aspects of the GUT scale") that were assumed to coincide with the GUT scale closer.

That would obviously change the world when it comes to the "testability" of different claims. If some new particles implied by the \(E_6\)-related structures of extra dimensions in string theory become observable, we may compare lots of the natural predictions of the stringy models with the reality. We would start to be seeing some new aspects of the stringy physics.

I am really intrigued by many clever ideas that could explain the excess – sgoldstinos, sbinos, perhaps even radions etc. But if I were to pick the most "conservative" or "obvious" or "natural" or "economic" (according to my previous evaluation of the explanations of the observed complexity of the Standard Model) model, I would probably end up picking the (supersymmetric, but in this case it's not the most essential thing) models built out of the \(E_6\) GUT, i.e. the \(E_6 SSM\). Go, e-six-es-es-em, go. (ESSM: King, Moretti, Nevzorov 2005-6. Slides.)

If you want an old TRF blog post that said that \(E_6SSM\) models predict some new exotics at the \({\rm TeV}\) scale and things should get interesting approximately in 2015, try e.g. this one from 2013 and the pre-rumor, November 2015 text about the LHC's going to prove \(E_6\) SUSY GUT.. ;-)

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