Koide Lepton Relation Explained by Lepton Radius (Compton Wavelength)/4

The intriguing Koide relation between lepton masses,


Introduction
People have wondered about the nature of fundamental particles for centuries. Our best model of fundamental particles at present is the Standard Model of particle physics that identifies the fundamental particles in our universe as twelve spin 1/2 point particles. Those  servations [1] of neutrino oscillations show that neutrinos must have non-zero mass.
The quantum mechanical uncertainty principle tells us we cannot discern any structure within a particle of mass m that has dimensions smaller than the Compton wavelength 2 c l mc = . That does not mean there can be no structure with dimensions less than the Compton wavelength, but only that we can't directly observe it. We can make progress by assuming [2] fundamental fermions in the Standard Model have radius (Compton wavelength)/4, even though we can't directly observe objects of that size. That assumption avoids the problem of the infinite densities of point particles and it has consequences relevant to the Koide relation [3] between lepton masses. The mass and pressure distribution inside fermions with radius (Compton wavelength)/4, involving their volume 3 l , their surface area 2 l , and their diameter 2 l , results in cubic equations [2] for fermion Compton wavelengths l , allowing at most three particles in each charge state. Describing the mass and pressure distribution with surface and linear elements requires minimum surface shell thicknesses and axial core radii on the order of the Planck length 3 P G l c = .
The total particle mass is then the sum of the mass equivalents of pressure, m/2, in the volume, the mass equivalent of surface pressure , and the core mass Ll . So   The background presented above regarding fermions with radius (Compton wavelength)/4 provides some insights into the otherwise somewhat puzzling Koide relation between lepton masses. First, it explains why only three lepton masses can be involved. Second, as shown in the following, the Koide parameter is a consequence of the fact that the electron Compton wavelength is more than 200 times the muon and tau Compton wavelengths.
The following analysis is based on Particle Data Group 2020 data [4]   l l , and 1 θ related to the three roots of the cubic equation [2] in l that describe the axial, surface, and volume energy distribution within leptons. Then, as seen below, the Koide parameter 1 2 3 Q ≈ because the electron Compton wavelength is more than 200 times the muon and tau Compton wavelengths, and a tau mass equivalent of 1776.98 MeV, within experimental [4] error bars, would result in 1 2 3 Q = . Written in terms of Compton wavelengths, Koide's parameter is

Koide Q and Lepton Wavelengths
Solutions to the cubic equation [2] in l for leptons with radius 4 l , specifying the axial, surface, and volume energy distributions

Concluding Remarks
None of the results and conclusions in the preceding discussion, except those related to neutrino mass estimates, are affected by the following remarks. However, we can go further in considering the nature of Standard Model fermions by identifying them with Godel solutions to Einstein's equations rotating with angular velocity ω , with an average matter density ρ equal to the average fermion mass density, pressure  nation of ideas from general relativity and quantum mechanics may provide further insight into the nature of elementary particles, but it is certainly not the long sought unification of general relativity and quantum mechanics. Any such unification must face the fundamental difficulty that general relativity can be seen as a theory of space, while quantum mechanics is a theory describing events within space.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.