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Leptoquarks may arrive: LHC to prove \(E_6\) SUSY GUT?

The most conservative stringy scenario to explain all the anomalies

The LHC has glimpsed numerous small anomalies. Some of them may be easily related to leptoquarks.



For our purposes, we define a leptoquark as a new elementary spinless particle that is capable of decaying to a lepton and a quark. So it is not a bound state of a lepton and a quark, it is a genuinely new elementary particle, but it carries the same quantum numbers as such a bound state would carry. We want the decay to be allowed by statistics (and by all other possible constraints) – so the new particle has to be a boson.




In Summer 2014, the CMS has observed a 2.5-sigma excess in the search of leptoquarks, suggesting that a leptoquark of the mass around \(650\GeV\) might exist.



But there exist more well-known anomalies that have been used as evidence in favor of the existence of leptoquarks.




In particular, it has been known for years that the observed muon's magnetic moment or, equivalently, its \((g-2)\) is slightly different than the Standard Model predicts. The expected relative precision is poorer than for the electron counterpart of this quantity – the most accurately verified prediction of science – and the prediction is off, anyway.

But as this blog has discussed in detail, the LHCb Collaboration – a smaller brother of ATLAS and CMS – has revealed numerous experimental results that could be resolved if leptoquarks exist, too.

In August, I described the apparent violation of the lepton universality observed when the LHCb measured the B-mesons' decay rates. (See also a June 2014 blog post.) That paper had D-mesons in the final state.

The LHCb has also claimed some anomalies in decays that involve K-mesons in the final state; see also a July 2013 blog post. And the CMS has possibly seen (with the fuzzy vision that a 2.5-sigma deviation represents) a flavor-violating decay of the Higgs, \(h\to\mu^\pm\tau^\mp\), which seemed to take place in 1% of the Higgs boson's decays. (See Jester's summary of these anomalies, too.)

As some phenomenology papers such as the fresh Bauer-Neubert paper point out, all or almost all these anomalies may be easily explained if you add a leptoquark to the Standard Model. If such a new particle is capable of decaying to the right-handed (singlet) up-quark and a right-handed (singlet) charged lepton, it can explain the correction to the muon magnetic moment as well the LHCb anomalies both with K-mesons and D-mesons.

There are obviously tons of papers about leptoquarks and (recently) their application as explanations of the possibly emerging LHC anomalies. See e.g. Hewett-Rizzo 1997, Reuter-Wiesler 2010, Freytsis et al. June 2015, Crivelin et al. July 2015, and Baek-Nishiwaki September 2015.

Now, what would be the bigger message?

If the experiments proved the existence of leptoquarks that have no other reasons to exist and no ancestry or relatives or purpose, we would have to accept this fact. But should we actually expect something like that to take place in the future? Well, I mostly don't. I think that in isolation, they're artificial purpose-less new particles. Occam's razor is a reason to favor a simpler model without such arbitrary additions.

However, they don't have to be arbitrary and optional. In grand unified theories, they (or at least some of their variations – depending on the choice of GUT groups and representations) may become unavoidable. They may become mandatory because they may be relatives of the fermions we know.

Leptoquarks are scalars so they can't be related to fermions by an old-fashioned, "bosonic" symmetry. But supersymmetry may change it. Leptoquarks may be superpartners of new fermions that are members of the same multiplets with the fermions we know so well.

\(SO(10)\) and \(E_6\) multiplets

Because of the excesses that look like a new gauge boson of mass \(2\TeV\) and perhaps also other signals at higher masses, I have repeatedly discussed the left-right-symmetric models (extending the Standard Model) whose gauge group is ideally\[

SU(3)_c \times SU(2)_L \times SU(2)_R \times U(1)_{B-L}.

\] The minimum GUT gauge group is \(SU(5)\) but it needs two different representations, basically \({\bf 5}\oplus\bar{\bf 10}\), to account for the known 15 two-spinors in one generation of fermions (three colors times two from doublet/flavor times two from Dirac-is-left-and-right, which is 12, plus 3 leptonic from a Weyl/Majorana neutrino and a Dirac charged lepton).

If you want to unify these reps as well, you surely prefer the \(SO(10)\) gauge group. And you actually do need at least \(SO(10)\) if you want the \(U(1)_{B-L}\) generator to be a part of the gauge group. The \(SU(5)\) GUTs simply don't contain the left-right-symmetric (or non-really-unified Pati-Salam) models.

With \(SO(10)\), things are nicer. All the leptons of one generation transform as \({\bf 16}\) which is a chiral spinor of \(SO(10)\) that decomposes under \(SU(5)\) as\[

{\bf 5}\oplus\bar{\bf 10} \oplus{\bf 1}

\] Note that \(5+10+1=16\) – but just to be sure (beginners listen now), the claims about the representations say much more than that the dimensions work. ;-) Aside from the \(5+10\) two-spinors from \(SU(5)\) GUT that are exactly enough for the Standard Model fermions, there is an extra singlet that is capable of completing the neutrino to a Dirac particle (the right-handed neutrino is what is added).

If you feel more familiar with the orthogonal groups than the unitary groups, \(SO(10)\) must be simpler for you. You may obviously embed \(SO(6)\times SO(4)\) group to it – by dividing the 10-dimensional vector to two segregated pieces \(6+4\) – and \(SO(6)\sim SU(4)\) [thanks Bill Z. for the fix] while \(SO(4)\sim SU(2)\times SU(2)\). It's enough to see that you have all the required groups for the left-right-symmetric extension of the Standard Model.



Xindl X recorded a "relative song" for the 100th anniversary of general relativity although it mostly says that life in Czechia is relatively OK unless you compare it with the civilized countries. ;-)

The left-right-symmetric models predict the new gauge bosons. Those from \(SU(2)_R\) which is new may have been seen by the LHC – the \(2\TeV\) and \(3\TeV\) gauge bosons – and (all?) the other gauge bosons of the GUT group almost certainly have to be much heavier, close to the GUT scale. However, the matter spectrum – away from the gauge multiplets – may always be partly or completely accessible.

This is particularly important if we try to increase the gauge group \(SO(10)\) to a larger and potentially cooler gauge group, \(E_6\). It's one of the five compact exceptional Lie groups, the others are \(E_7\),\(E_8\),\(F_4\),\(G_2\). The group \(E_6\) is the only one among these five that has complex representations (which are not equivalent to their complex conjugates). This is needed for the chiral and CP-violating spectrum of the Standard Model.



The \(E_6\) Dynkin diagram is the only exceptional group Dynkin diagram with an exact left-right symmetry - the only other simple compact Lie groups with this property are \(SU(\geq 3)\) – which is equivalent to its having complex (not real, not pseudoreal) representations because the reflection of the diagram indeed nontrivially acts on the set of irreps as well and a reflection, generating \(\ZZ_2\), has to be a mirroring operation and the complex conjugation of these reps is the only possibility.

The group is 78-dimensional but the smallest nontrivial, fundamental representations are \({\bf 27}\) and \(\bar{\bf 27}\). Those extend and replace the 16-dimensional spinor of \(SO(10)\). The decomposition under the \(SO(10)\) subgroup is obviously\[

{\bf 27} = {\bf 16}_{+1} + {\bf 10}_{-2} + {\bf 1}_{+4}

\] up to some possible bars that partially depend on your conventions (but one must be careful about certain correlations between bars and nonbars at different places). I've included some subscripts that actually express, in a certain simple normalization, the charge of the representations under a \(U(1)\): the maximum subgroup we may embed to \(E_6\) is actually \(SO(10)\times U(1)\). Note that the trace of the new \(U(1)\) vanishes over the 27-dimensional representation because\[

16\times 1 - 10\times 2 + 1\times 4 = 0,

\] as expected from traces of generators in non-Abelian simple groups. So it is interesting. Aside from the well-known 16 fermionic two-spinors, we also have an extra representation \({\bf 10}\) of \(SO(10)\). It decomposes as \({\bf 5}\oplus \bar{\bf 5}\) under the \(SU(5)\) subgroup which is why you shouldn't confuse it with the representation \({\bf 10}\) of \(SU(5)\) which is an antisymmetric tensor, \(5\times 4/2\times 1 = 10\).

There is a difference between the new 10 components of the matter fields and the previous 16 that we already had in \(SO(10)\). In SUSY GUT, particles have superpartners and because SUSY is broken, one of these two partners is expected to be lighter or the "normal one". It's the one that is R-parity-even. Evil tongues say that 1/2 of the particles predicted by SUSY have already been discovered – it's the particles with the even R-parity. But for all known particle species, the R-parity may be defined or written as\[

P_R = (-1)^{B-3L +2J}

\] and we may extend this definition to all new particle species as well if we assume that \(B,L,P_R\) remain well-defined and conserved. So if \(B-3L\) is even, the R-parity-even (\(P_R=+1\)) particle is a boson (\(j\in\ZZ\)) which is the case for the gauge bosons and the Higgs. But if \(B-3L\) is odd, which is the case for leptons and quarks because exactly one of the terms in \(B-3L\) is nonzero and \(\pm 1\), then the normal R-parity-even parts of the supermultiplet are the fermions with \(j\in \ZZ+1/2\). It works.

Now, the funny thing is that \(B-3L\) of the new 10 components of the 27-dimensional multiplet is even, so the "normal", R-parity-even parts of the supermultiplet that we can see are bosons. SUSY predicts new bosonic as well as fermionic particles from the 10-dimensional representation of \(SO(10)\) but it's the bosons that may be lighter because their R-parity is even. All of the things have to work in this way, obviously, because we postulated that the leptoquarks must be allowed to decay to known R-parity-even fermions and no LSP (lightest R-parity-odd) particle is ever found among the decay products.

A funny thing is that it is possible that it will turn out to be harder or hard to find any R-parity-odd particles at the LHC. But even if that were the case, the discovery of the R-parity-even leptoquarks would be significant evidence in favor of SUSY because those particles could be relatives of the leptons and quarks via a combination of SUSY and a new gauge symmetry. Some consistency checks could work.

If you think about "all possible spectra" left by string/M-theory to field theory to deal with, I do think that the SUSY \(E_6\) GUT is the single most likely possibility offered by string/M-theory. Recall that one of the two ten-dimensional heterotic string vacua, conventionally known as the \(E_8\times E_8\) heterotic string theory (the outdated terminology involving "many string theories" is from the 1980s but people don't want to rename things all the time), is the only class of models in string theory that has a sufficient gauge group for particle physics that is also "bounded from above" and bounded from above by a unique and preferred choice of the group.

To get down to \(\NNN=1\) supersymmetry, we need to compactify heterotic string theory on a Calabi-Yau manifold. This 6-real-dimensional manifold has to have the holonomy of \(SU(3)\), a subgroup of the generic (unoriented) potato manifold's holonomy \(O(6)\), and the simplest bundle to choose is one that identifies the gauge connection with the gravitational connection on the manifold.

This forces us to embed an \(SU(3)\) to \(E_8\). And the unbroken group we are left with is unavoidably an \(E_6\) because \(E_6\times SU(3)\) is a maximum subgroup of \(E_8\). I still believe that these are the most beautiful relationships between groups that may be related to the gauge groups in particle physics. Note that \(E_8\) is the largest exceptional group but it is no good for grand unified theory model building. The only viable exceptional group is \(E_6\) because of its complex representations.

And string theory contains an explanation why the "best" group \(E_8\) gets broken exactly to the "viable" \(E_6\): it's because the latter is the centralizer of \(SU(3)\) within \(E_8\) and we pick \(SU(3)\) because we basically have "three complex extra dimensions" predicted by the \(D=10\) string theory. All these things make so much sense. Many other structures of a similar degree of beauty (and maybe sometimes prettier) have been found in string theory during the following 3 decades after 1985 when heterotic string theory was born and proven to be at least semi-realistic.

But I believe that this path from \(E_8\) to \(E_6\) GUTs etc. remains the most persuasive scenario offered by string theory and telling us "what happens with the gauge groups" before the extra dimensions and stringy stuff may be forgotten.

We may be looking at early hints of all this new wonderful physics that is waiting to be found in coming years. We've been used to "nothing beyond the Standard Model" all the time and we tend to be very humble and shy. It's normal that phenomenologists are only willing to add one or two new particle species (components).

But if and when the LHC starts to see physics beyond the Standard Model, I am sure that the atmosphere will change or should change. Model builders should immediately become generous again. Many models with lots of new component fields are way more natural and justifiable than models where you humbly add one or two unmotivated extra components to your field content. If new physics is found at the LHC, we will have to think big again because the new experimental findings may tell us much more about the super-fundamental stringy architecture of Nature than we may believe at this moment.

By the way, I've encountered lots of models in my life, about \(10^{272,000}\) LOL. But if I had to choose one, I would still choose a \(T_6/\ZZ_3\) orbifold compactification of heterotic string theory; it's even simpler than the generically curved low-Hodge-number manifolds, it seems to me. I believe that it may actually explain three generations most sensibly, along with something like the \(S_4\) family symmetry. I am spending some time with the phenomenology of this model now.

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