of the eigenvalues of a circulant matrix with real eigenvalues, corresponding to the relation for = 0, 1, 2, ... which can be fit to experimental data with = 0.500003(23) (corresponding to the Koide relation) and phase = 0.2222220(19), which is almost exactly . However, the experimental data are in conflict with simultaneous equality of η = and = . This kind of relation has also been proposed for the quark families, with phases equal to low-energy values = × and = × , hinting at a relation with the charge of the particle family and for quarks vs. = 1 for the leptons, where 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 , regardless of their uncertainties, one arrives at the value cited by F. G. Cao (2012): 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 article. Note that an older article, H. Harari, et al., calculates theoretical values for up, down and strange quarks, coincidentally matching the later Koide formula, albeit with a massless up-quark. That should be, with modern values, 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 (RGE). 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. 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