Kea & Arivero,
I had the most useful plane flight from Seattle.
The most controversial thing in the neutrino mass predictions is the supposition that the masses scale like
\mu_1/\mu_0 = 3^{12} / 3^1 = 3^{11}
And this was supposed to be associated with the assumption that the neutrino mixing matrix is a 12th root of unity (after you correct it by making it translate from mass eigenstates (\nu_1,\nu_2,\nu_3) to mass eigenstates (e,\mu,\tau)which you do by multiplying it by the matrix of eigenvectors of a circulant matrix). I hadn't even included the neutrino mixing angle stuff in the slides.
Now the justification for all this was that the mass conversion L->R->L is correct for the charged leptons, but is off by an angle of 2 pi/12 for the neutrinos. However, if that angle were taken literally, you'd end up with a complex mass, not a mass that was taken to the 12th power.
On the plane ride over, I realized that the two circulant matrices that represent the charged leptons and the neutrinos were simply:
<br />
\begin{array}{rcl}<br />
M_1 &=& \left(\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right),\\<br />
M_0 &=& \left(\begin{array}{ccc}0&i&0\\0&0&i\\i&0&0\end{array}\right),<br />
\end{array}
where the charged lepton transformation acts like L_e -> R_e -> M_1L_e and the neutrino transformation acts like L_\nu -> R_\nu ->M_0L_\nu
The point is that the M_0 matrix can act like a true 12th (or is it 6th) root of unity in a way that does not overuse (burn up) any degrees of freedom the way that trying to make it be a complex 12th root of unity does.
Now the place where all this gets tied back into the theory is this: I now assume that the mass interaction is a change in the sign of \gamma_0. But \gamma_0 is not one of the commuting roots of unity that defines the primitive idempotents associated with the elementary particles. So what does nature do? The elementary particles are mass eigenstates and therefore have to be eigenstates of parity.
Nature does the same thing as happens when you want to make a spin-1/2 in the +x direction spinor out of the two +/-z spinors (1,0) and (0,1). She makes a complex linear combination. The result is that one of the linear combinations ends up as M_1, the other as M_0. But instead of the complex linear combination being built out of nice simple complex numbers, instead nature has to use those nasty 3x3 matrices. The result is that the mass interaction is far less probable with the neutrinos than the charged leptons. Neutrinos have to go through the 12 step program in order to achieve sobriety.
Also, I realized that I didn't explain how one got the Koide formula from the 3x3 solutions to the idempotent equation. One does this by adding together the right handed and left handed matrices. When you do this, all the vectors cancel and all that is left is the scalars, which have the form of the Koide mass formula 1+\sqrt{2}\cos(\delta+2n\pi/3). The only problem is that the angle delta is wrong.
The reason that all the non scalar parts canceled is because the potential energy was assumed to be order Planck energy for every part of a multivector except the scalar part (which corresponds to order of magnitude of muon mass, I guess). The potential energy is a sum of squares of coefficients, and so the mass is the square of the scalar coefficient, which explains why Koide's formula works for square roots of masses instead of masses.
It is possible to derive that the observed particles are what you get when you demand that all but the scalar parts sum to zero but it isn't easy. It is easy, however, to show that this happens with the observed leptons. And the quarks show that of the three (non scalar) roots of unity that define the ``fermion cube'', one of those roots has a potential energy that, while being very high compared to the scalar energy, its low enough that we can sort of see it a little.
As a last addition, I thought I need to put in a slide showing how all this came about, particularly the PhysicsForums contribution, which I sem to recall was Alejandro's doing.
None of this has taken place yet. I got in 7 hours ago, and since I was in my relatively inexpensive hotel by 3PM local time, I engaged a very helpful cab driver to drive me all over Honolulu looking for what we eventually figured out was the Punchbowl Memorial. My father lost a brother, missing in action, in the Pacific in 1944. He had always imagined that some day his children would play with his big brother's children but that was not to be. Instead, as an inadequate substitute, he named his first born son (me) after his brother. In the US, not returning from wars gets your name listed on a memorial, in this case at the Punchbowl Memorial, so of course I went to visit it. I had been thinking that I would do this on the day I flew out, but this worked out much better. It was a beautiful day to be sad. The sky was 40% covered with those sharp edged, very white clouds that make the blue look more intense. Like this:
http://en.wikipedia.org/wiki/Cumulus_cloud
Carl
Somehow I modded -1 out.
