The charged lepton masses obey to high precision the so-called Koide relation. We propose a generalization of this relation to quarks. It includes up and down quarks of the three generations and is numerically reasonably close to the Koide limit.
The paper suggests that it may make sense to include all six leptons in the Koide formula (which has no impact on the relationship within the range of experimental accuracy because neutrino masses are so small), and in turn to include all six quark masses in the generalization of the Koide formula to quarks. In this generalization, the arccosine of the Koide formula for leptons implies an angle theta lepton, of pi/3 and the arccosine of the Koide forumla for quarks implies an angle theta quark of pi/4.
Between Koide's original formula, developing analysis of the idea of quark-lepton complementarity and the most recent paper, Arivero notes that:
Werner Rodejohann and He Zhang, from the MPI in Heidelberg, proposed that the quark sector did not need to match triplets following weak isospin, and then empirically found that it was possible to build triplets choosing either the massive or the massless quarks. This was preprint http://arxiv.org/abs/1101.5525 and it is already published in Physics Letters B. . . .
Then myself, answering to a question here in PF, checked that there was also a Koide triplet for the quarks of intermediate mass. I have not tried to find a link between this and the whole six quarks generalisation, but I found other interesting thing: that the mass constant AND the phase for the intermediate quarks is three times the one of the charged leptons. This seems to be a reflect of the limit when the mass of electron is zero, jointly with an orthogonality between the triplets of quarks and leptons in this limit: it implies a phase of 15 degrees for leptons and 45 degrees for quarks, so that 45+120+15=280. If besides orthogonality of Koide-Foot vectors we ask for equality of the masses (charm equal to tau, strange equal muon), the mass constant needs to be three too.
with the premises
1. Top, Bottom, Charm have a Koide sum rule
2. Strange, Charm, Bottom have a Koide sum rule
3. Electron, Muon, Tau have a Koide sum rule
4. phase and mass of S-C-B are three times the phase and mass of e-mu-tau
All of this continues to add to the appearance of some method to the madness of the many constants of the Standard Model, although it isn't quite yet clear precisely why this arises and precisely how the reliationship should be stated.