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.
Thanks for your dispatch! Let me add that the note is now available both in arxiv:1111.7232 and vixra:1111.0062. Vixra allows to add comments in the abstract page.
Most probably I will not pursue publication in the short term; the first try proved that there are some issues basically of scholarly style. A lot of work is already from other authors, it must be reviewed because the topic is not well known anymore, and then the letter must evolve into an article; a time consuming task that I can not afford by now.
"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."
No, they're actually much closer than that. In each case you have a three-dimensional angle of pi/4 and then a six-dimensional generalization of pi/3.
In each case, there are three heavy particles and three light particles. For the leptons, the heavy particles are e-mu-tau, and the light particles are the neutrinos. For the quarks, the heavy particles are c-b-t, and the light particles are d-u-s. e-mu-tau and c-b-t are conventional Koide triples, which satisfy Foot's d=3 geometric relation.
(By the way, while talking about angles, let us take care to note the difference between these angles - the angle between the mass vector and the "democratic vector" (1,1,1...) - and the "phase" parameter, which has to do with the directions of the components of the mass vector, in the plane perpendicular to the democratic vector.)
When we go from three dimensions to six dimensions (i.e. from three masses to six masses), we are no longer considering the angle between (M1,M2,M3,0,0,0) and (1,1,1,0,0,0), but the angle between (M1,M2,M3,0,0,0) and (1,1,1,1,1,1). Evidently (I haven't bothered to do the calculation) the angle between the mass vector and the democratic vector goes from pi/4 towards pi/3 with the jump in dimensions. Then, introducing the three lighter masses (M1,M2,M3,m1,m2,m3) makes that angle almost exactly pi/3.
I should correct my parenthetical remark about the meaning of the phase parameter - I mean the delta0 in equation 3 of arxiv:1111.7232. The way I described it is wrong, its geometric interpretation is a little more complex. But the main point is that it's a very important angle in discussions about the Koide relation, and it's a different angle to the one which Foot introduced.
She also: http://arxiv.org/abs/1205.4068, a more recent Koide formula generalization paper focused on estimating neutrino masses.
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