A. C. Kleppe has uploaded a preprint entitled "Quark mass matrices inspired by a numerical relation" that explores how Koide's rule for charged lepton masses can be extended to quarks. This conference paper presentation is Kleppe's first paper on arXiv and it isn't clear that the author has a university affiliation.
The abstract, after stating Koide's 1981 charged lepton mass rule (which still holds to high precision as the inputs have become more accurate over the last 44 years) states that:
Inspired by this relation, we introduce tentative mass matrices, using numerical values, and find matrices that display an underlying democratic texture.
I have discussed other attempts to make this extension and my own thoughts on it, in several previous posts at this blog. The paper does not, however, meaningfully engage with (or even mention) most of the prior literature in this area.
The statement in the paper that:
It should be noted that for the square roots of the running charged lepton masses at MZ around 91 GeV, the results no longer give the exact Koide formula.
is particularly concerning when it comes to understanding, because Koide's rule is a rule about the pole masses of particles and not about the running mass of those particles at a consistent energy scale. And, Koide's rule is, in fact, exquisitely confirmed when applied to pole masses.
This distinction matters because the proper definition of mass to use for light quarks when extending Koide's rule is not self-evident.
The conclusion, which I have screenshotted rather than cut and paste from to preserve the integrity of the notation, states:
