This thread is getting old, so I'll type in the latest Koide fits. These are supposed to be practical applications of triality to quarks interacting to form mesons. As such, they might give a clue on how to fit the quarks better in with the leptons and quarks.
The first observation is that the (handed) quarks and anti quarks are 2/3 or 1/3 of the way between leptons and anti leptons in quantum numbers, and they come in 3 colors, so one ends up with a (1,3,3,1) multiplet with the 1's two types of leptons, say left handed positron and right handed electron, while the 3's two types of quarks, say right handed up quark and left handed anti-down quark. The implication is that the quarks and leptons could be built from three preons each of eight types, charged or neutral, left handed or right handed, and preon or antipreon (with positive charge). Then the electron is composed of three preons with charge +1/3 each, the up quark is made from two +1/3 preons and a neutral preon, etc.
Koide's formula for the masses of the charged leptons reads as follows (ignoring an overall mass scale factor of 25.054 sqrt(MeV)):
\sqrt{m_{en}} = \sqrt{0.5} + \cos(2/9 + 0\pi/12 + 2n\pi/3)
A similar formula fits the neutrino oscillation mass differenes (with a mass scale of 0.1414 sqrt(eV) also not included):
\sqrt{m_{\nu n}} = \sqrt{0.5} + \cos(2/9 + 1\pi/12 + 2n\pi/3)
The formula for the charged leptons is quite old and famous. I found the neutrino mass formula a couple years ago and it's now in the literature in various places, for example, Mod. Phys. Lett. A, Vol. 22, No. 4 (2007) 283-288:
http://www.worldscinet.com/mpla/22/2204/S0217732307022621.html
If the quarks are to be composed of a mixture of preons, which form should they follow?
Thinking of the above formulas as resonance conditions for the preons, perhaps a meson made from two quarks could resonate either way. The simplest place to test this is on the mesons that are most carefully and exactly studied, the b-bbar (Upsilon) and c-cbar (J/psi) mesons.
In the particle data group information on the b-bbar and c-cbar mesons:
http://pdg.lbl.gov/2007/listings/contents_listings.html
there are six of each type given:
The Upsilon b-bbar mesons are:
Name, quarks, I^G(J^PC) mass(error) koide_type
\Upsilon(1S) b/b 0^-(1^{--}) 9460.30(26) 1
\Upsilon(2S) b/b 0^-(1^{--}) 10023.26(31) 1
\Upsilon(3S) b/b 0^-(1^{--}) 10355.20(50) 0
\Upsilon(4S) b/b 0^-(1^{--}) 10579.40(120) 1
\Upsilon(10860) b/b 0^-(1^{--}) 10865.00(800) 0
\Upsilon(11020) b/b 0^-(1^{--}) 11019.00(800) 0
The Koide_type is 0 or 1 according as that mass is part of a triplet of states that follow the Koide formula with 0 or 1 copies of pi/12 in the angle. The resulting equations for the Upsilon masses are (leaving off the factor of 25.054 again):
\sqrt{m_{\Upsilon 1 n}} = 3.994433 - 0.128815\cos(2/9 + 1\pi/12 + 2n\pi/3)
\sqrt{m_{\Upsilon 0 n}} = 4.137251 - 0.077550\cos(2/9 + 0\pi/12 + 2n\pi/3)
The measured and calculated masses are as follows (MeV):
9460.3(3) ~= 9451.8
10023.2(3) ~= 10041.0
10355.2(5) ~= 10355.1
10579.4(12) ~= 10569.1
10865.0(80) ~= 10864.3
11019.0(80) ~= 11019.5
which is considerably more accurate than random chance would suggest. To put this into perspective, the mass difference between the charged and neutral pions is about 5 MeV.
Fitting the Koide formula to six masses like this is similar to how one would fit a spin-1/2 splitting to six masses. In that case one looks for how one can put the six masses into three pairs of masses with the same difference between the masses. If that were the case for the Upsilons, you can be sure that there would be papers showing why a quark interaction causes this kind of splitting. After you divided the six particles up into three pairs, you need only four degrees of freedom to describe the particles, say the three averages of the pairs, and the split amount. Similarly, with the above Koide fit, you end up removing two degrees of freedom from the six masses. The new four degrees of freedom are 3.994433, -0.128815, 4.137251, and -0.077550.
It would be easy to suppose that this is random chance, but the c-cbar mesons also come in exactly six masses, and these also are very closely fit by four Koide parameters. In this case the mass formulas are:
\sqrt{m_{\Psi 1 n}} = 2.442070 - 0.249554\cos(2/9 + 1\pi/12 + 2n\pi/3)
\sqrt{m_{\Psi 0 n}} = 2.510507 - 0.089433\cos(2/9 + 0\pi/12 + 2n\pi/3)
and the mass fits are unnaturally accurate:
J\psi(1S) c/c 0^-(1^{--}) 3096.916(11) 1 ~= 3096.9
\psi(2S) c/c 0^-(1^{--}) 3686.093(34) 0 ~= 3686.1
\psi(3770) c/c 0^-(1^{--}) 3771.1(2.4) 1 ~= 3773.8
\psi(4040) c/c 0^-(1^{--}) 4039(1) 0 ~= 4040.4
\psi(4160) c/c 0^-(1^{--}) 4153(3) 0 ~= 4149.8
\psi(4415) c/c 0^-(1^{--}) 4421(4) 1 ~= 4418.4
I've not yet figured out how to derive this from the assumption that the quarks are composites made from the same things that make up the leptons. I think it has to be doen with perturbation theory. My instinct is that the quarks are acting as a body that has two possible resonances, the type 0 (electron - muon - tau) and the type 1 (neutrinos). Either of these two resonances shows up in threes just like the generations of particles do. But only one resonance can be excited at a time. The result is six resonances that satisfy a Koide relationship.
Among the 8 numbers that give the four Koide fits here, some of them are rather close to rational numbers, or square roots, or what have you. For example, the first of the 8 Koide fit numbers, 3.994433, is very close to 4. But I don't see an overall pattern to the numbers.
My suspicion is that if this can be put into a perturbation expansion, we will see how to derive the 8 Koide fits from simpler assumptions. But I haven't figured out how to do this yet. This may be the reader's opportunity to score a quick paper. Like I mention above, my suspicion is that one should model this as a system that has two available, but mutually exclusive, resonances. And I'm working on this, but I haven't yet got anything worth writing up.
It may or may not help to read the incomplete paper I'm writing that is driving the search for these kinds of coincidences:
http://www.brannenworks.com/qbs.pdf
