In its original form, it is not fully empirical but a set of guesses for a model for masses of quarks and leptons, as well as CKM angles.
From this model it survives the observation about the masses of the three charged leptons; later authors have extended the relation to neutrinos, quarks, and other families of particles.
[a] No matter what masses are chosen to stand in place of the electron, muon, and tau, the ratio Q is constrained to 1 / 3 ≤ Q < 1.
The experimentally determined value, 2 / 3 , lies at the center of the mathematically allowed range.
But note that removing the requirement of positive roots, it is possible to fit an extra tuple in the quark sector (the one with strange, charm and bottom).
Not only is the result peculiar, in that three ostensibly arbitrary numbers give a simple fraction, but also in that in the case of electron, muon, and tau, Q is exactly halfway between the two extremes of all possible combinations: 1 / 3 (if the three masses were equal) and 1 (if one mass dwarfs the other two).
Q is a dimensionless quantity, so the relation holds regardless of which unit is used to express the magnitudes of the masses.
Robert Foot also interpreted the Koide formula as a geometrical relation, in which the value
[4] While the original formula arose in the context of preon models, other ways have been found to derive it (both by Sumino and by Koide – see references below).
Similar matches have been found for triplets of quarks depending on running masses.
[8] The Koide relation exhibits permutation symmetry among the three charged lepton masses
The Koide relation is scale invariant; that is, multiplying each mass by a common constant
Carl Brannen has proposed[4] the lepton masses are given by the squares of the eigenvalues of a circulant matrix with real eigenvalues, corresponding to the relation which can be fit to experimental data with η2 = 0.500003(23) (corresponding to the Koide relation) and phase δ = 0.2222220(19), which is almost exactly 2/9 .
[4] This kind of relation has also been proposed for the quark families, with phases equal to low-energy values 2/27 = 2/9 × 1/3 and 4/27 = 2/9 × 2/3, hinting at a relation with the charge of the particle family (1/3 and 2/3 for quarks vs. 3/3 = 1 for the leptons, where 1/3 × 2/3 × 3/3 ≈ δ ).
Quark masses depend on the energy scale used to measure them, which makes an analysis more complicated.
[13] Taking the heaviest three quarks, charm (1.275±0.03 GeV/c2), bottom (4.180±0.04 GeV/c2) and top (173.0±0.40 GeV/c2), regardless of their uncertainties, one arrives at the value cited by F. G. Cao (2012):[14] This was noticed by Rodejohann and Zhang in the preprint of their 2011 article,[15] but the observation was removed in the published version,[5] so the first published mention is in 2012 from Cao.
[14] The relation is published as part of the analysis of Rivero,[16] who notes (footnote 3 in the reference) that an increase of the value for charm mass makes both equations, heavy and middle, exact.
The masses of the lightest quarks, up (2.2±0.4 MeV/c2), down (4.7±0.3 MeV/c2), and strange (95.0±4.0 MeV/c2), without using their experimental uncertainties, yield a value also cited by Cao in the same article.
[14] An older article, H. Harari, et al.,[17] calculates theoretical values for up, down and strange quarks, coincidentally matching the later Koide formula, albeit with a massless up-quark.
In quantum field theory, quantities like coupling constant and mass "run" with the energy scale.
[18] That is, their value depends on the energy scale at which the observation occurs, in a way described by a renormalization group equation (RGE).
[19] One usually expects relationships between such quantities to be simple at high energies (where some symmetry is unbroken) but not at low energies, where the RG flow will have produced complicated deviations from the high-energy relation.
The Koide relation is exact (within experimental error) for the pole masses, which are low-energy quantities defined at different energy scales.
[20] However, the Japanese physicist Yukinari Sumino has proposed mechanisms to explain origins of the charged lepton spectrum as well as the Koide formula, e.g., by constructing an effective field theory with a new gauge symmetry that causes the pole masses to exactly satisfy the relation.
[22][23] François Goffinet's doctoral thesis gives a discussion on pole masses and how the Koide formula can be reformulated to avoid using square roots for the masses.
[24] A cubic equation usually arises in symmetry breaking when solving for the Higgs vacuum, and is a natural object when considering three generations of particles.
Taking the ratio of these symmetric polynomials, but squaring the first so we divide out the unknown parameter
For the relativistic case, Goffinet's dissertation presented a similar method to build a polynomial with only even powers of
Minimising does not give the mass scale, which would have to be given by additional terms of the potential, so the Koide formula might indicate existence of additional scalar particles beyond the Standard Model's Higgs boson.
which when expanded out the determinant in terms of traces would simplify using the Koide relations.