Koide's Formula and Running Fermion Masses

One of the criticisms that Lubos Motl has directed towards Koide's formula (which in its original form shows that the sum of the three charged lepton masses divided by the square root of the sum of their squares is exactly two-thirds, midway between the minimum 1/3rd value of that ratio and the maximal 1 value of that ratio) is that in its naiive version it, one uses a not particularly fundamental definition of mass (the real valued rest mass of the energy level of that particle).  Motl would personally favor using a complex valued "pole mass" in any fundamental formulation of mass relations in the Standard Model, but hasn't tested that approach to see if it would produce a different result.

One way to see if this criticism has any substance to it is to apply Koide's formula to lepton and quark triples that have been proposed to be Koide triples and to see if the formula is substantially distorted by the running of quark and lepton masses at different energy levels. 

Koide's himself wrote one of the seminal papers calculating these running masses at various energy levels in 1998, and it was updated to reflect more recent experimental measurements and more energy levels in 2008.

While the masses of quarks and leptons respectively due "run" with the energy level of the interaction in which they are measured, this has far less of an impact on the relative masses of particles within a particular Koide triple than it does on the absolute values.  So, it is not obvious that the formula fails to fit the facts by anything more than the inherent uncertainty in the experiments measurements available to check it against, when differently defined particle masses are involved.

Given the rather large uncertainties with which we know the quark masses, I suspect that the running of the mass values for quarks won't have much of an impact on the validity of Koide's formula for these triples.  The quoted uncertainty in the strange quark mass, for example, is on the order of 20-30%.

The charged lepton masses provide a sterner test as they are known with much more precision.  But, they do seem to run at rates closely (if not exactly) to proportionate to the energy scale involved. 

A Few Calculations

For example, the running electron, muon and tau masses are each not quite 2% lower at the energy scale of a top quark than they are at the energy scale of a charm quark (which is about 138 times smaller than a top quark, give or take).  Similarly, the masses of the charged leptons between the Fermi scale (the Z boson mass energy level of 91.2 GeV) and the GUT scale (2*10^16 GeV) each declines, to three significant digits anyway, by a factor of 3.60% over fourteen orders of magnitude of running mass distortion.  And, since Koide's formula is true independent of the units of mass used, any proportionate adjustment of all of the charged lepton masses should be immaterial to its accuracy.

There are more elegant ways to show this, I am sure, from direct comparison of the running mass renormalization equations themselves, but suffice it to say that it is safe to assume that Koide's formula should continue to hold true to a quite high degree of precision in any consistently applied redefinition of the charged lepton masses.

Put another way, Koide's formula is an empirical one that seems to hold to a high degree of precision but has not really deep theoretical basis, so it is hard to know what definitions of mass to use in a deeper theoretical framework that reproduces the Standard Model with fewer fundamental constants and some deeper relationships between the constants that are now known. 

But, any definitions that produce the same relative masses of the charged leptons as the most commonly used version of his formula should produce the same result, any there are many alternative defintions of those masses in which this is the case.

A Footnote on the Precision of What We Know About The SM Constants.

We have some experimental estimates for each of the Standard Model constants except the CP violating phase of the PMNS matrix (i.e. a term that would make neutrino and antineutrino oscillation rates asymmetrical in carefully constucted circumstances).  Some of these constants, like the proton and neutron mass, the charged lepton masses and the electromagnetic couple constants (including the running of that constant at sub-TeV levels) are measured with extreme precision.  But, some of those constants like the absolute neutrino masses and oscillation constants and the strange quark mass are constants for which we have only very approximate values.  The values of the CKM mixing matrix entries are known with considerable precision, while the values of equivalent matrix for leptons are know only at a one or two significant digit level of accuracy for the most part and a great deal of published work that tries to make calculations involving neutrino oscillations use theoretically suggested approximations instead of experimentally measured values.

Also, since the masses of quarks other than the top quark cannot be measured directly, since the other five quarks are never observed outside of composite multiquark hadrons bound by gluons and many of the measurements are only available in high energy physics interactions, all of these constants are highly theory dependent.

Theorists have confidence that the QCD Lagrangian and numerical approximations of it derived from it and selective measurement of QCD constants at high energies are correct and can be extrapolated to lower energies.  And, generally speaking the agreement between theory and experiment has been excellent.  But, imprecision in both the experimental values and the numerical calculations of theoretical predictions from first principles means that few QCD measurements, particularly in lower energy systems, are accurate to much more than two significant digits, which leaves considerable room for introducing tweaks to the Standard Model QCD formulas without doing any injustice to current experimental knowledge.  Indeed, even gross simplifications of the Standard Model formulation, such as the assumption that up, or both up and down quarks are massless, that they are fewer than six or even as few as four quarks, are routinely used to improve computational ease in numerical QCD calculations that aren't particularly sensitive to these constants.

Similarly, the "beta functions" that govern how the electromagnetic, weak and strong force coupling constants run at different energy levels are all extrapolated from fairly thin data, which is particularly thin at the highest energy levels, without any direct experimental confirmation that they are correctly formulated for energy levels many orders of magnitudes higher than any experiment has every measured (a span of about thirteen orders of magnitude between experiment and the GUT scale).