This post was originally going to be a comment to Kea’s blog, but she has developed a slightly unappreciative attitude towards my comments, and I can’t put subscripts or superscripts in comments, so I’ll put my idea here instead.
“This might give another explanation of the square root.” – Carl Brannen
Kinetic energy can be expressed as
E = (1/2)p2/m
p2/(Em) = 2
Kinetic energy, E = (1/2)mv2, gives velocity v = (2E/m)1/2, hence:
p = mv = m(2E/m)1/2
Hence, we have a square-root of mass appear, which may be useful in the theory for the Koide formula! So could the Koide formula be an averaging of the momenta of the three leptons in the situation that they are in thermal equilibrium, with identical kinetic energy, only differing masses? Simply use Eq. 1 to sum the momenta given by Eq. 2 for three particles. Let’s assume we can justify writing Eq. 1 for 3 particles as the following (note that we add up momenta before squaring the sum):
(p1 + p2 + p3)2/[E(m1 + m2 + m3)] = 2
Inserting Eq. 2, pn = (2Emn)1/2, gives:
[(2Em1)1/2 + (2Em2)1/2 + (2Em3)1/2]2/[E(m1 + m2 + m3)] = 2
which will simplify to:
(m11/2 + m21/2 + m31/2)2/(m1 + m2 + m3) = 1
which is close to Koide’s formula, which differs in having the dimensionless numerical factor on the right hand side 3/2 instead of 1. Possibly this may arise from relativistic effects and/or angular momentum (spin) of the leptons. Maybe we should be thinking of all leptons having identical angular momentum (spin) energy and just differing masses; which might be the deeper physical meaning of the Koide formula. In any case, it seems to me to be an average of the momenta of the generations of leptons in thermal or energy equilibrium. I’m not claiming to yet have a complete and rigorous theoretical derivation of Koide’s formula. Maybe by putting out this idea, someone else with more time available will be motivated to investigate further along these or related lines.
Carl has a PDF paper about Koide’s formula here. Yoshio Koide in 1982 proposed that if you square the sum of the square roots of the 3 lepton masses, and then divide that result into the simple sum of the 3 lepton masses, you get the ratio 3/2. This is treated by some as a chance result of random numerology, but Carl and others treat it as a more interesting type of coincidence. After all, Balmer’s formula for the curious distance spacings between discrete atomic lines in light spectra was originally entirely empirical, with no theoretical justification. It was decades before Bohr came up with the explanation in his atomic theory of quantum leaps. Although the Koide formula only predicts one mass after requiring two masses as inputs, it has nevertheless been vindicated ever more precisely as more accurate mass data for the leptons has become available over the last 27 years. In particular, Carl has also found it possible to apply it to the 3 neutrino masses if the lightest neutrino mass square root is given a negative sign instead of the positive sign: