The Koide formula is where the masses of the electron, muon, and tau are measured respectively as = , = , and = ; the digits in parentheses are the uncertainties in the last figures. This gives = . Therefore the experimentally determined value,, lies at the center of the mathematically allowed range. Clearly, no matter what masses are chosen to stand in place of the electron, muon, and tau, The upper boundfollows from the fact that the square roots are necessarily positive. The lower bound follows from the Cauchy–Bunyakovsky–Schwarz inequality. It can also be shown by interpreting the value as the squared cosine of the angle between the vector and the vector . The mystery is in the physical value. 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, is exactly halfway between the two extremes of all possible combinations: and 1. The value of = corresponds to = . While the original formula arose in the context of preonmodels, other ways have been found to derive it. As a whole, however, understanding remains incomplete. Similar matches have been found for triplets of quarks depending on running masses. With alternating quarks, chaining Koide equations for consecutive triplets, it is possible to reach a result of 173.263947 GeV for the mass of the top quark.
Similar formulae
There are similar empirical formulae which relate other masses. Quark masses depend on the energy scale used to measure them, which makes an analysis more complicated. Taking the heaviest three quarks, charm, bottom and top, and without using their uncertainties gives the value cited by F. G. Cao, This was noticed by Rodejohann and Zhang in the first version of their 2011 article but the observation was removed in the published version, so the first published mention is in 2012 from Cao. Similarly, the masses of the lightest quarks, up, down, and strange, without using their experimental uncertainties yield, a value also cited by Cao in the same paper.
Running of particle masses
In quantum field theory, quantities like coupling constant and mass "run" with the energy scale. That is, their value depends on the energy scale at which the observation occurs, in a way described by a renormalization group equation. One usually expects relationships between such quantities to be simple at high energies but not at low energies, where the RG flow will have produced complicated deviations from the high-energy relation. The Koide relation is exact for the pole masses, which are low-energy quantities defined at different energy scales. For this reason, many physicists regard the relation as "numerology". 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 in which a new gauge symmetry causes the pole masses to exactly satisfy the relation. François Goffinet's doctoral thesis gives a discussion on pole masses and how the Koide formula can be reformulated without taking the square roots of masses.
