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Grand unified groups: a beauty contest

A few days ago, I discussed the LHCb B-meson anomalies that could be signs of either Z'-bosons or leptoquarks. I wanted to know whether I would be able to find out that such new particles are natural and pretty enough to be almost guaranteed to be exploited by Nature.


The Standard Model has the gauge group \(SU(3)_c\times SU(2)_L\times U(1)_Y\). Symmetry breaking is a mundane thing and it's common sense that this group may be a subgroup of a larger one that is broken to the Standard Model group, either by Higgs-like field-theoretical methods or some fancy, stringy ones.

Under the Standard Model group, we have 15 two-component left-handed spinors of "matter", i.e. leptons and quarks. A sixteenth one, the left-handed antineutrino, might exist but there is no experimental proof that it's really needed.

The minimum gauge group that has a single factor and incorporates the Standard Model is \(SU(5)\). That was "the" canonical grand unified group that was tested in the first proton-decay experiments etc. The fifteen spinors decompose as \({\bf 10}\oplus\overline {\bf 5}\) under \(SU(5)\). It's similar for the flipped SU(5) which needs \(SU(5)\times U(1)\) and is arguably more natural within the superstring models.

While the gauge group is unified into a single factor, the leptons and quarks are divided into two irreducible representations. You may say that it's an aesthetic defect just like the numerous factors of the Standard Model gauge group. The minimum gauge group that removes this aesthetic defect is \(SO(10)\). Inside that, \(SU(5)\) is embedded in the normal way picturing a complex number \(x+iy\) as a real \(2\times 2\) matrix. \[

\pmatrix{x&+y\\ -y&x}

\] The spinor \({\bf 16}\) of that \(SO(10)\) automatically incorporates the 15 left-handed Standard Model spinors as well as the left-handed antineutrino.

But you may extend the Standard Model in a different direction. The hypercharge \(U(1)_Y\) may be taken to be a part of the new \(SU(2)_R\) group which is actually analogous to the electroweak \(SU(2)_L\) group. The starting point \(SU(3)_c\times SU(2)_L\times SU(2)_R\) is the group defining the left-right-symmetric models (where the two \(SU(2)\) groups are broken at different scales, however). Enhance \(SU(3)\) to \(SU(4)\) and you get a Pati-Salam model.

The largest simple grand unified group incorporating \(SO(10)\) is \(E_6\). Its fundamental representation is 27-dimensional. There is really no smaller nontrivial representation of that exceptional Lie group. Under the \(SO(10)\) group, the 27-dimensional rep decomposes into the spinor \({\bf 16}\) discussed above, the known quarks and leptons, plus an \({\bf 10}\) and a single \({\bf 1}\).

The new 10-dimensional representation mainly contains the leptoquarks. They are elementary particles that may decay into a lepton plus a quark. Most typically, the first scalar leptoquark is \(\tilde D\) which may decay to the electron \(e^-\) and the up-quark \(u\). So it has the same charge (and the same color representation) as the down quark \(d\) but is a boson, not a fermion, and has the same \(L=+1\) as the electron, on top of the \(B=+1/3\) of the up-quark (or down-quark).

While a fermion with this charge, \(B\), and \(L\) does exist in the 27-dimensional representation of \(E_6\), there is a good reason why the scalar superpartner could be lighter. Why? Because it has the even R-parity just like leptons and quarks. It's because\[

\text{R-parity} = (-1)^{3B+L +2J}

\] This power of minus one ends up being positive for fermions in the Standard Model. They have \(2J=1\), one minus sign, and the other minus sign comes either from \(3B\) from quarks or from \(L\) for leptons. Well, the leptoquarks have minus signs both from \(3B\) and \(L\) so they cancel. To have an even R-parity, you want an even \(2J\), not an odd one, so you want the bosonic superpartners, not the fermions.

String theory may bring some extra signs from the breaking of gauge groups by the Wilson lines and other things.

But if you believe that some \(E_6\) and probably \(E_8\) group is a relevant starting point, e.g. because we live in a heterotic string theory world, then it is natural to think that the rest of the 27-dimensional representation is somewhere out there. And assuming that the superpartners of the known quarks and leptons are heavy, it's very likely that the rule is reversed for the 10 states in the 27-dimensional representation: the light ones should be bosons.

In fact, I feel – but can't prove at this moment – that there may exist a swampland-like consistency condition that says that such new light bosonic states must exist, in order to cancel some supertrace. This claim is something I really can't prove at this moment and the reason why I believe it could be true is as aesthetic in character as you can get. But when the dust settles, it's clear that some valid aesthetic criteria are the most solid anchors that make physicists certain that some statements must be fundamentally true.

The CMS and ATLAS detectors have looked for leptoquarks and other new matter and they have found nothing so far. The leptoquarks, if they exist at all, should be a few \({\rm TeV}\) in mass. But I think that there is a sentiment that a proper theoretical physicist has – which is that he or she doesn't really give a damn about this higher mass and experimental inaccessibility. It matters whether those particles exist or not, the direct experimental search isn't really the only way how physics may become certain about the answer, and whatever the mass is, the "amount of the beauty" that Nature stores in these constructions is pretty much exactly the same.

Needless to say, there may exist leptoquarks with other quantum numbers. They may be spin-1 bosons and not spin-0 scalars; and they may transform in the diquark, symmetric-tensor \({\bf 6}\) representation of the colorful \(SU(3)\). Additional choices probably exist. It's possible that one may incorporate them in some stringy vacua in a comparably aesthetically pleasing way.

Beauty probably isn't enough to uniquely determined the spectrum of matter that will be discovered first, after the Standard Model era. On the other hand, the ugliness may very well a sufficient tool to eliminate some possibilities. The Standard Model is OK at the level of non-gravitational field theory (except for the Landau pole etc.) but it's possible that within the more demanding quantum gravity conditions, it's not a consistent truncation of the full theory.

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