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Fermion masses from the Δ(27) group

Ivo de Medeiros Varzielas of Basel, Switzerland and Daniel Pidt of Dortmund, Germany released an interesting paper about the family symmetries

Geometrical CP violation with a complete fermion sector
They continue in the authors' three-weeks-old research of quark masses and Varzielas' 2012 research and other developments and argue that the \(\Delta(27)\) family symmetry seems fully appropriate to obtain not only quark masses but also lepton masses and the CP violation.

These models have (not just one but) several Higgs doublets – e.g. three Higgs doublets or a multiple of three – and a discrete symmetry is required to be respected by the scalar potential. This condition implies a relationship between the vacuum expectation values and leads to realistic patterns for quark and lepton masses. In some cases with three or more generations, the CP violation is made inevitable, too.

It's interesting because the same multi-Higgs paradigm using the \(\Delta(27)\) symmetry was exploited in Standard-Model-like braneworld models in type IIB string theory written down by Berenstein, Jejjala, and Leigh in their 2000 and 2001 papers. They had considered type IIB string theory on the \(\RR^4\times \CC^3/\Delta(27)\) orbifold.

At that time, their braneworlds looked particularly intriguing because they reduced to the almost pure supersymmetric Standard Model and inevitably predicted SUSY breaking approximately at the \(3\TeV\) scale and the stringy spectrum (!!!) at the \(10\TeV\) scale. David Berenstein believes that the model has been ruled out years ago but I forgot the exact reasons.

Nevertheless, the maths of such models seems irresistibly attractive. For decades, I tended to think that it's no coincidence that the number of extra dimensions in string theory compactifications is six, a multiple of the number of generations of fermions (three). There seem to be various constructions that promote this observation to much more than just numerology.

Symmetries are important in physics. Steven Weinberg just recorded a monologue on that very theme yesterday. Well, I have seen this "dancing Weinberg" video some time ago ;-) but if it were new, it would be a funnier coincidence. Hat tip: Phil Gibbs

Moreover, the \(\Delta(27)\) group is extremely simple and natural for the \(T^6\) compactifications: yes, it may act not just on \(\CC^3\) but also on the simply compactified sibling of it, the six-torus. How does it act? It's simple. Consider the group \(\Delta(3n^2)\) – where \(3n^2\) is the number of elements – generated by the following three transformations of three complex variables \(z_1,z_2,z_3\):\[

e_1:\quad & (z_1,z_2,z_3)\to (\omega_n z_1,\omega_n^{-1} z_2,z_3),\\
e_2:\quad & (z_1,z_2,z_3)\to (z_1,\omega_n z_2,\omega_n^{-1} z_3),\\
e_3:\quad & (z_1,z_2,z_3)\to (z_3,z_1,z_2).

\] Here, \(\omega_n\) is an/the \(n\)-th root of unity. The first two generators only change the phases of the three complex variables while the third generator cyclically permutes them. Note that for \(n=3\), the group has \(27\) elements and preserves a honeycomb-cubed-like hexagonal lattice that makes it compatible with a toroidal compactification. (I can't resist thinking that there are other copies of the fixed points in the six-torus and the matter living there may look like dark matter to us although it may follow the same laws of particle physics as the matter we know. This possibility could even imply that the ratio of the dark and visible matter density in the Universe is a simple integer such as five.)

These finite groups may be easily seen to be subgroups of \(SU(3)\); see e.g. this paper on discrete subgroups of \(SU(3)\). Recall that the finite subgroups of \(SU(2)\) are classified by the ADE classification.

These groups \(\Delta(3n^2)\) may also be fully specified by the following short exact sequence:\[

0\to \ZZ_n\times \ZZ_n \to \Delta(3n^2)\to \ZZ_3\to 0.

\] The adjective "short" means that the sequence only has five elements if we also count the trivial one-element groups \(0\) at both sides. The term "exact sequence" means that in the sequence, each arrow describes a homomorphism of groups and the image (the set of possible results) of one homomorphism coincides with the kernel (the subset of the group that is mapped to the identity) of the following homomorphism.

This concept beloved by mathematicians may sound convoluted but they use it often and in this case, it's trivial to see how it works. The \(\ZZ_n\times \ZZ_n\) group is generated by the mutually commuting generators \(e_1,e_2\) above – but not \(e_3\). The first map starting in the trivial group \(0\) has image composed of the identity element of \(\ZZ_n\times\ZZ_n\) – because that's where the identity (i.e. only) element of the trivial group is mapped.

This image, the identity element of \(\ZZ_n\times\ZZ_n\), must coincide with the kernel of the following map. And indeed, the following map is simple because \(\ZZ_n\times \ZZ_n\) is mapped to a larger group by a (technically) "simple" map. The image of that map is composed of all the elements of \(\Delta(3n^2)\) that don't need the generator \(e_3\) to be written down. And indeed, this image coincides with the kernel of the following map that classifies the elements of \(\Delta(3n^2)\) by the exponent we need above \(e_3\) which is either \(0\) or \(1\) or \(2\), thus producing a \(\ZZ_3\) group. The image of that map in \(\ZZ_3\) is "everything" because the whole \(\ZZ_3\) may appear as a result and indeed, it's the kernel of the last map going to \(0\) because each element of \(\ZZ_3\) has to be mapped to the identity element of the trivial group (because this group has no other element).

There are many short exact sequences and if you just replace the 3 non-trivial groups above by something else, the story is pretty much "isomorphic" (in the colloquial sense) to the story above.

But that was just a segment of the text dedicated to some group theory. The dynamics of D-branes in the braneworlds on the orbifolds by the non-Abelian groups \(\Delta(3n^2)\) are described by the Douglas-Moore quiver theories – gauge theories with numerous simple factors (nodes in the quiver/moose diagram) and lots of added "bifundamental" matter (depicted as arrows in between those nodes). Berenstein et al. are among those who have played with this exciting technical tools in string theory for quite some time.

The anthropic fanatics may argue that there exists an overwhelming majority of \(10^{500}\) stringy compactifications that don't respect such a structure but I don't care about these majority arguments. The probability that the right compactification of string/M-theory uses the \(\Delta(27)\) group in a rather fundamental way and explains the three generations and the mass matrices of the leptons and quarks in these generations in this way seems very high to me – perhaps 10 percent if not higher – because this construction is mathematically natural and seems to explain certain things.

It can't be excluded that experimental hints of this scenario could arrive in a few years. A possible discovery of "several or many new Higgs bosons" could be a straightforward method for Nature to strengthen this sort of reasoning among the intelligent humans.

Stay tuned.

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