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Beyond the Standard Models
What is new with Koide sum rules?
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[QUOTE="mitchell porter, post: 3702460, member: 103130"] I have been thinking about this for weeks now and I have lots of ideas, but nothing decisive, so it's time to talk. First, let's consider what a "standard" approach to a discovery like this is. If someone guesses a pattern in the masses and mixing angles, the answer usually involves some combination of multiple Higgses, "flavons" whose VEVs contribute to the Yukawas, and family symmetries (usually discrete). Let us suppose provisionally that the quantity appearing in the Koide relation is a VEV (and that the corresponding Yukawa is the square of this VEV). It seems that some of these VEVs are negative, thus the "minus sqrt mass" term appearing in s-c-b (and in Brannen's neutrino triplet). For the quarks we then have a set of six quantities, which to a first approximation satisfy the Koide relation in four sets of three (dus, usc, scb, cbt). The Koide relation only holds well for scb and cbt, but there is some evidence that the actual values for dus and usc are highly perturbed away from a "primordial" set of mass values which includes m_u = 0. Another aspect of this perturbation is that the primordial Koide phase for the scb triplet is 45 degrees, but the real value is 2/3 radians. In his "yukawaon" papers, Koide obtains VEV relations from supersymmetric vacuum conditions. So that is one way to get a set of four chained Koide triplets - construct a superpotential which implies Koide VEV relations for the four sets of three. (It would also be good to do this without supersymmetry.) However, it's clear (from the relation between e-mu-tau and s-c-b) that the important parametrization of the Koide relation is the one (due to Carl Brannen?) featuring a mass scale and a phase. These parameters don't stand forth in Koide's constructions. Since Brannen uses circulant matrices, perhaps we should therefore be interested in models like Stephen Adler's multi-Higgs models with Z_3 symmetry, where there are three or six Higgs doublets, and where there are circulant (or "retrocirculant") mass matrices. Another thing we can learn from Koide is the importance of the Sumino mechanism. The running of the masses ought to spoil the original Koide relation for the charged leptons, but it remains exact. Sumino suggested that the bosons of a gauged family symmetry could cancel the electromagnetic radiative corrections which would otherwise spoil the relation. Koide's latest yukawaon models are SU(5)-compliant supersymmetric models in which the Koide relation for the charged leptons comes from SUSY vacuum conditions, and in which the Sumino family symmetry exists and is gauged. So one way forward is to follow his lead: look for a basic explanation of these extended Koide relations - perhaps using Adler-Brannen circulant mass matrices, perhaps using a version of Georgi-Jarlskog to explain the factor of 3 difference between s-c-b and e-mu-tau - and then use the Sumino mechanism to protect the relations (though it's not yet clear whether the quark mass relations are exact enough to need protection). However, this still leaves one more clue unused - the appearance of QCD mass scales in the Brannen parametrization of the Koide formula. This leads me to think in terms of holographic QCD and Alejandro's own "sBootstrap". The basic paradigm of holographic QCD is that you have a stack of color branes and a stack of flavor branes that intersect. A gluon is a string between color branes, a quark is a string between a flavor brane and a color brane, a meson is a string between flavor branes, and a baryon is a brane instanton connected to multiple flavor branes by strings. The sBootstrap is a combinatorial construction in which all the SM fermions are made from pairings of the five "light" quarks ("light" here means lighter than the top quark). Leptons are made from mesonlike pairings, quarks from diquarklike pairings. Since holographic QCD contains fermionic meson strings ("mesinos"), an hQCD implementation of the sBootstrap would say that the leptons are mesinos. The situation for the quarks is less satisfactory; but one might imagine that there is some mixing between quark strings and fermionic "diquarkinos". Top-down holographic QCD constructions (Sakai-Sugimoto is the best known) so far don't resemble QCD exactly. For one example, they are usually studied in the large-N limit, N being the number of colors, whereas reality involves N=3. But also, the spectrum has extra stuff not seen in reality. The fermionic mesons already mentioned are one of these trouble spots. However, if we expect the leptons to come from the mesino sector, then the trouble becomes a virtue. We might look for a hQCD model that contains the whole standard model via the sBootstrap. (Or we might look for a more conventional string model which nonetheless realizes the leptons in this fashion.) How is this relevant to explaining the extended Koide relations? The point is that it offers an avenue whereby QCD mass scales may show up in lepton mass formulae, since the leptons would just be the mesinos of some SQCD-like theory. [/QUOTE]
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What is new with Koide sum rules?
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