The Koide formula revisited

Important update: The Left-Right symmetric extension of the standard model yields the Koide ratio to be exactly 2/3

The Koide formula is the following empirical observation about the experimentally measured masses of the three charged leptons (electron, muon, tau-lepton): if the sum of their masses is divided by the square of the sum of the square-roots of their masses, the result is the number

0.666661(7) ~ 2/3

Is there a theory in which this number is shown to be exactly 2/3? Yes.

The first important assumption, justifiably, is: the unification of gravity with the standard model is required at all energies, not just at the Planck scale. And the exact description of fermions is on a spinor spacetime, not on Minkowski spacetime. Such a description, unifying gravity with the standard model, has been set up on an octonionic space.

A quantum system such as an electron obeys such L-R unification at all energies, lives in a spinor spacetime, the neutrino is a Dirac neutrino, and the Koide ratio comes out to be exactly 2/3.

However, and this is crucial, when we make a measurement on the electron, L-R symmetry is broken, because we make this measurement not in a spinor spacetime but in the emergent Minkowski spacetime: classical gravity is present and RH sector is hidden. The neutrino appears as if Majorana, and we find the Koide ratio to be 0.669163, departing a little both from 2/3 and from the measured value quoted above.

So the big picture is this: before we make a measurement on the charged leptons to measure their masses, the Koide ratio is exactly 2/3. After the measurement is made, the theoretical prediction for the resulting value is 0.669163, whereas the measured value is smaller. [The uncertainty in the mass of the tau-lepton is such that by demanding the Koide ratio to be 2/3 one can predict the mass of the tau-lepton.]

The above is an important development as we now know when the Koide ratio is exactly 2/3 [it is when the electron is not being observed]. And we understand why the measured value of this ratio is not exactly 2/3. In principle, we could demand the measured value to be equal to the value in Minkowski spacetime, and thereby fix the mass of the tau-lepton.

We also realise that a quantum measurement breaks Left-Right symmetry !!

Reference: eqn. 63 in https://arxiv.org/abs/2108.05787 [hep-ph]