I've had some interesting results in rewriting the Koide equation as a sort of "field energy" equation. The idea is to treat the square in the mass as coming from the energy of a field.
Field energies are quadratic, for instance, E&M field gives mass as m = (E^2 + B^2)/c^2, where I've left off some units. So begin with electromagnetism as a toy example.
Then the thing to notice is that E and B end up quantized at different amplitudes. Magnetic monopoles are much heavier than electrons, so assume that when you quantize E and B, the contribution of B dominates, giving you m = B^2/c^2.
From there, you assume that the angle I've called "delta" is 2/9exactly, and that the reason this doesn't exactly fit the electron, muon and tau masses is cause "E" does contribute slightly.
That converts the Koide formula from being a two parameter fit, with mu and delta, (the mass scale and the angle), to being a two parameter fit with a B scale and an E scale. To write the masses we have
m_n = |B|^2(1 + \sqrt{2}\cos(2/9+2n\pi/3))^2 + |E|^2(\sqrt{2}\sin(2/9+2n\pi/3))^2
where B and E are constants. The contribution to B is split into two parts, 1 + \sqrt{2}\cos(2/9+2n\pi/3), so we write B_v = B, B_s = \sqrt{2}\cos(2/9+2n\pi/3), and E_s=\sqrt{2}\sin(2/9+2n\pi/3). That is, the "v" field is a valence field that is shared between the electron, muon, and tau, and the "s" field is a sea field that distinguishes the three generations.
Then the mass equation is m = (B_v + B_s)^2 + E_s^2.
What's more interesting is that if you write down the vectors (B_v,B_s,E_s) for the electron, muon, and tau, you get the tribimaximal mixing matrix (after scaling the B stuff and E stuff so that each vector has length 1).
Another way of saying this is that the vectors (B_v,B_s,E_s) are orthogonal. Making them orthonormal defines the tribimaximal neutrino mixing matrix. (Except that when you see it in the literature, it is usually has two columns reversed so you should put the three contributions in the order (B_s,B_v,E_s) instead.)
Using the best PDG numbers for the electron and muon masses to predict the tau mass, the equations for the charged lepton masses are (ignore the precision, I haven't had time to compute the ranges and fix everything up yet):
\begin{array}{rcl}<br />
m_n &=& 313.8561002547\;\textrm{MeV}\;(1 + \sqrt{2}\cos(2/9 + 2n\pi/3))^2\\<br />
&&+4.6929703\;\textrm{eV}\; (\sqrt{2}\sin(2/9+2n\pi/3))^2<br />
\end{array}
And the three vectors (which ignore the phase angle 2/9 because it is presumably canceled in the neutrinos) are:
\begin{array}{ccc}<br />
(1,& \sqrt{2},& 0)\\<br />
(1,& -\sqrt{2}/2,& +\sqrt{3/2})\\<br />
(1,& -\sqrt{2}/2,& -\sqrt{3/2})<br />
\end{array}
In the above, note that the angle 2/9 has been removed as it is presumably canceled in the neutrinos, which also use a similar angle. And the scaling to B and E has been removed because in computing phases, one needs to normalize by particle number rather than energy.
After dividing by the lengths of the vectors, sqrt(3), and turning the three vectors into a matrix, one has:
\left(\begin{array}{ccc}<br />
\sqrt{1/3},& \sqrt{2/3},& 0\\<br />
\sqrt{1/3},& -\sqrt{1/6},& +\sqrt{1/2}\\<br />
\sqrt{1/3},& -\sqrt{1/6},& -\sqrt{1/2}<br />
\end{array}\right)
Carl
Koide paper giving Tribimaximal mixing matrix, see eqn (3.2):
http://arxiv.org/abs/hep-ph/0605074
