**The most similar previous blog post was one about the \(\Delta(27)\) group**

The first hep-ph paper today is dedicated to heterotic string phenomenology.

Delta(54) flavor phenomenology and stringswas written by Mexicans, Ms Brenda Carballo-Perez, Eduardo Peinado, Saul Ramos-Sanchez, but that can't prevent it from being more interesting than many papers from the U.S. The first hep-ph papers often look more interesting than the rest. I believe that also in this case, the authors struggled to get the #1 spot because they're more excited about their work than the authors of the remaining papers today.

*Michal Tučný, "Everyone is already in Mexico". Buenos días, I am also going. One of his top 20 best country music songs.*

The Standard Model of particle physics is usually formulated as a gauge theory based on the \(SU(3)\times SU(2)\times U(1)\) gauge group. The particles carry the color and the electroweak charges. The gauge group is continuous which implies that there are gauge bosons in the spectrum.

However, the Standard Model also requires 3 generations of fermions – quarks and leptons. Because of this repetitive structure, it's natural to imagine that they transform as "triplets" under another, family group as well. However, there are apparently no \(SU(3)_{\rm flavor}\) gauge bosons, at least not available at the LHC yet. For this and other reasons, it's more sensible to assume that the 3 generations of fermions are "triplets" under a discrete, and not continuous, family symmetry.

The family groups that have been tried and that admit three-dimensional representations have been \(\ZZ_3\), \(S_3\), and \(\Delta(27)\). However, there exists one larger discrete group with a three-dimensional representation, the \(\Delta(54)\) group. With some definitions, it's the maximal one – the maximal "exceptional" discrete symmetry with three-dimensional representations, and the previous three cases are all subgroups of \(\Delta(54)\).

Surely many people share the gut feeling that the maximum symmetries of a similar exceptional type are (just like \(E_8\) among the simple Lie groups) the "most beautiful ones" and therefore most promising ones, at least from a certain aesthetic viewpoint. The larger symmetry with 54 elements probably makes the models more constrained and therefore more predictive.

These Mexican heterotic string theorists point out that the heterotic orbifolds of tori haven't led to viable models with this \(\Delta(54)\) symmetry yet. But that's because people were focusing on orbifolds\[

T^6 / \ZZ_3, \quad T^6 / \ZZ_3 \times \ZZ_2.

\] However, they propose different orbifolds instead:\[

T^6 / \ZZ_3 \times \ZZ_3

\] There are various discrete groups here and I need to emphasize that the group \(G\) in the description of the orbifold, \(T^6 / G\), is

*not*the same group as the group produced as the family symmetry by the resulting heterotic string model. The group \(G\) is being "gauged" on the world sheet. It means that all the one-string states have to be invariant under \(G\) i.e. there are no charged states or non-singlets under \(G\); and closed string sectors with almost periodic boundary conditions up to the action of \(G\) have to be added, the twisted sectors.

The family group produced by orbifolds is in some way a "dual" group. The \(\ZZ_3\times \ZZ_3\) orbifold produces three fixed points not identified with each other. Dynamics at each of them is the same so there is actually an \(S_3\) symmetry (the full permutation group with 6 elements) exchanging them. A semidirect product of this group with a \(\ZZ_3\times \ZZ_3\) group (different from \(G\)) has to be considered because an additional global symmetry results from the action on localization charges \(m,q\) of the twisted sectors etc. The semidirect product with 54 elements is what we call \(\Delta(54)\).

All the fields and terms in the low-energy Lagrangian should simply be symmetric under this \(\Delta(54)\) and that leads to constraints on the Yukawa couplings and other things. It seems that their models – they found some 700 heterotic vacua using some software – have several great phenomenological properties such as

- qualitatively realistic quark and lepton masses in general, including the recently proven nonzero \(\theta_{13}\) neutrino mixing angle
- the right relationship between the Cabibbo angle and the strange-to-down quark mass ratio (the Gatto-Sartori-Tonin relation)
- a cool equation relating ratios of both (down-type) quark and (charged) lepton masses: \[

\frac{m_s-m_d}{m_b} = \frac{m_\mu - m_e}{m_\tau}

\] Well, this relationship isn't obeyed by the observed masses but corrections could make it true - the normal, not inverted, hierarchy of neutrino masses, as preferred by latest experiments
- PMNS neutrino matrix compatible with the known data

But it's amazing how fine questions about the spectrum and parameters of the Standard Model are already being "almost completely explained" by a theory based on a completely different and much more concise starting point – the hybrid of the only bosonic \(D=26\) string theory and the only \(D=10\) superstring with the maximum family group produced at low energies.

When you combine these conditions, you rather naturally obtain three generations with realistic constraints on the quarks' and leptons' masses and mixing angles. Whoever isn't intrigued by those hints is a cold-blooded animal.

*I added the label/category "music" to this blog post because of the extensive discussion of (mostly Czech) country music in the comment section.*

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