## Wednesday, November 16, 2011

### Quarks and Neutrinos

Recall that some time ago we were thinking about quark lepton scale complementarity, since the Koide charged lepton scale of $313.8$ MeV happens to equal the dynamical quark mass. Today, Alejandro adds another quark lepton coincidence for the proton mass, using strange quarks. Note that our Koide fit for the quarks resembles the neutrinos in having one minus sign out of the six eigenvalue signs, which is for the down quark, the way I parameterised it. Alejandro, please provide references for your comments!

1. I was waiting to upload the note to some repository online, but ok, you can check it on my old website. It is only two pages long and all the meat is in page 2.

2. A thing I do not understand yet is the phase. It is very puzzling to have an integer phase, and I approve the cautious approach of Carl in the published articles, when he prefers to use the experimental number. Of course, using Lenz-Motl mnemonic, we can write an equation for the lepton phase:

2*pi^5*(1+sqrt(2)*cos(2*pi/3+phase))^2=1

But it is still sort of random. Another clue comes from Dave observation that in some cases the phase and the mass constant are multiplied by the same factor. It happens to go from the U to the D sector, and now it also happens from lepton to quarks, but it is not a general rule.

Best guess is that we are seeing some sort of instanton tunnel correction.

3. Hi, Alejandro. Thanks! So that's a Koide eigenvalue of

$2 \pi^5 (1 + \sqrt{2} \cos (2 \pi/3 + \delta)) = 1$

with some integer phase. Integer phases are not unexpected in constructive M theory, but yes, we do need to understand them.

4. Oops, I forgot the squaring ...

5. Anyway, Lenz is more a mnemonic than a equation, IMO the important thing is the QCD scale, the point of getting the 313 MeV from the lepton side and the 939 MeV from the quark side.

Besides, the negative sign in the square root of the strange quark has other consecuences, particularly it allows for almost-orthogonality of some tuples of square roots. Particularly, [e,mu,tau] and [bottom,charm,strange] are almost orthogonal when the sign of sqr(s) is negative, while the positive sign fails badly. But I have not searched for all the possible combinations of quarks orthogonal to the leptons, so I can not say how significative it is.

6. Well, fitted triplets can only work in one way. Even the charged triplets (although the masses are only approximate) are significant, because the $r$ value follows from the lepton $r = \sqrt{2}$, as pointed out by Dave.