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Fermion masses from a Δ(54) heterotic orbifold

Stephen King, the King of Horror, didn't receive the Nobel prize in literature for his 200 short stories yesterday – the award went to Austria and Poland instead. Similarly, neither Trump nor the aggressive Swedish teenager got the Nobel prize in peace today – instead, it actually went to a guy (prime minister) who established peace in Ethiopia. ;-) Clearly, the committee in Oslo needed and still needs to recover some credibility after it was overspent in recent years.

But Stephen King (I guess it is a different one) is the most famous author name of an exciting 4-author hep-ph preprint today

Flavon alignments from orbifolding: \(SU(5) \times SU(3)\) model with \({\mathbb T}^6/\Delta(54)\)
The masses of quarks and leptons are free and arbitrary parameters in the Standard Model and one of the most obvious collections of data that expects to be explained by a deeper theory – some SUSY/GUT or ideally string theory.



There are three generations of quarks and leptons and they may naturally transform as a 3-component representation of a group such as \(A_4,A_5,S_3,S_4,D_4,D_6,Q_4,Q_6,T',\Delta(27)\), or others. Well, these people take a more straightforward option – they are a triplet under a new \(SU(3)_f\) that may perhaps be found as a part of the heterotic string theory's gauge group.



Clearly, there aren't any massless gluons of this \(SU(3)_f\) so the group must be broken in some way. It's normally assumed to be broken by Higgs-like fields called "flavons". Their vevs have some values which break the flavor symmetry and impose a constraint on the allowed mass matrices. We ideally want a special Ansatz for the mass matrices of neutrinos – the so-called tribimaximal-reactor lepton mixing which is compatible with the state-of-the-art experimental data on neutrino oscillations.

To make the story short, these people succeed in finding a rather new type of an orbifold of heterotic string theory where the six compactified dimensions have the shape of the\[

{\mathbb T}^6/\Delta(54)

\] orbifold which produces realistic mass matrices for the fermions.

It's more or less the first time when this orbifold is studied in the flavor literature. Note that the six-torus is a product of the three two-tori and the group \(\Delta(54)\) is a 54-element semidirect product of \(\ZZ_3\times \ZZ_3\) and an \(S_3\), the permutation group of three elements. The most skillful readers will be capable of finding a calculator to check that \(3\times 3 \times 6 = 54\), good. All these 54 elements of the group act as some permutation of the 3 complex coordinates of the six-torus composed with the multiplication of each of these three complex coordinates by a power of the sixth root of unity. So the matrices are \(3\times 3\) permutation matrices except that \(1\) may be replaced with \(\exp(2\pi ik /6)\) for some \(k\in\ZZ_6\), here \(k\) may be different for each of the three nonzero matrix elements.

Well, it's not just a new orbifold that they properly study for the first time. This heterotic orbifold is special because for the first time, both fermions and the flavons exist at fixed points of the orbifold group within the 6 compact dimensions.

They realize that some previous papers that claimed to have constructed "heterotic orbifold models of the flavor groups" have actually ignored the stringy conditions arising from orbifolds – so these papers were at most string-inspired or, equivalently, "sloppy string theorists' papers". Their ambition is to be careful string theorists here.

Note that a stringy orbifold starts with a group \(G\) that the original space is divided by (i.e. the original space is reduced to a quotient by that group). For every element (or conjugacy class) of \(G\), one needs to add a twisted sector – which increases the number of spacetime fields. On the other hand, one also requires all the first-quantized states of the strings to be invariant or singlets under \(G\), which reduces the number of spacetime fields or stringy vibrations by the same factor (the number of elements in the group \(G\)).

Whether the flavons live in the twisted sectors or not, their vevs must obey the "GSO-like projection" from the second part of the previous sentence. And they find out that this condition is rather nontrivial. And the combination of this condition with the experimental constraints – which they reduce to the tribimaximal-reactor lepton mixing – is even more stringent.

All the four qualitatively different orbifolds they study are capable of producing the proper GUT breaking. But except for their \(\Delta(54)\) orbifold, all the other choices fail in some other conditions. \(T^2/\ZZ_2\) and \(T^6/\ZZ_2\times \ZZ_2\) fail in the patterns required from the boundary conditions by the string theory consistency; \(T^6/\ZZ_2\times \ZZ_2\) and \(T^6/S_4\) fail in the SM fermion localization which is bad because the GSO-like condition would probably remove the flavor triplets out of the spectrum.

They really select the only winner which is realistic – and requires the 54-element group to act on the simple six-torus in heterotic string theory. At some level, this is a type of research that could have been possible shortly after the 1985 discovery of heterotic string theory (similar heterotic orbifolds of the six-torus were studied right away). But there has never been a sufficient number of stringy brains to study the interrelations between the heterotic orbifolds and patterns in the fermion mass matrices, among other things, so many of these interesting things and promising orbifolds are found now, 34 years after the discovery of the heterotic string!

If such a heterotic vacuum were right, I suspect that some intensely non-landscape solution would exist for the cosmological constant problem and other puzzles.

Another new paper on neutrino mass matrices uses \(A_4\) and talks about the breaking of the \(\mu\)-\(\tau\) symmetry.

Another remotely related stringy orbifold phenomenology paper uses type IIA string theory on \(T^6/\ZZ_2\times \ZZ_2\) with D6-branes and achieves the first Pati-Salam model with the wrapping number 5.

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