A possible explanation for the discrepancy is that one of the two formulae is exact, for deep physical reasons, and the other is approximate, and depends on exactly how the underlying theory is gauged. My attempt to explain in detail how this might happen can be found in all the usual places.

]]>The tell-tale phrase is “some are more approximate than others”. As far as I am aware there are only two exact mass formulae: the Coleman-Glashow relation, and mine. All the rest are more or less approximate. And that includes the Koide formula.

]]>A4 or S4 appear in Koide formula in a simple way. Consider the GUT limit where each family has a given mass, and it is composed of a tuple ([e,nu],[u-type,d-type]). We have three mass levels and any selection of three quarks choosing one of each family will meet Koide formula if electrÃ³n, muon and tau still meet it. So you have A4 lurking here. For instance put each generation quarks in opposite faces of a cube, and each vertex is a Koide formula for three quarks. Now, it happens that this symmetry is broken and some formulae remain more approximate than others: (tbc) (bcs) (csu) (sud), and perhaps (bcd).

For the argument to work, of course, you need the unbroken unified tuples to be not the expected ones, but [ub] [cd] [ts], This is not different to the argument used by Harari et al in the first paper having a Koide mass tuple.

]]>Ha, ha! Try me! Have a look at my arXiv papers and see how radical you think I am.

]]>Well, if you are more radical that suggesting that QCD mass is related to yukawa-electroweak mass, then you have a really radical proposal :-D

]]>Yes, that seems like a reasonable interpretation. My own interpretations are rather more radical, which I suppose is why few serious physicists pay any attention to them.

]]>Ah yes we have here another peculiarity of Koide formula, that builds upon a peculiarity of electroweak theory: that the mass scale of electroweak particles happens to be in the same range that QCD mass scale. And then one half of the average of electron, tau and muon is about the same that one third of the average of neutron and proton. One could interpret this guessing that both sets of masses are generated from “current quark mass, 313 MeV”, somehow. And one interpretation of your formula is that the “current mass” should be calculated considering the contribution of isospin breaking, this is, the mass difference between proton and neutron.

]]>In fact, Marni came back to me at our next meeting a week or two later, and said that my formula wasn’t quite correct. I didn’t understand how she came to this conclusion, and I didn’t pursue it at the time, but it seems likely that she was using the Koide value of the tau mass, rather than the experimental value, in which case her conclusion was indeed logically correct.

]]>What I find remarkable about Marni’s thesis was that she mentioned the utility of associahedra for QCD. This was vindicated, as Melia’s 2013 work on quark color decomposition used associahedra. Also, Nima Arkani-Hamed et al. by 2017 showed that associahedra give tree amplitudes of biadjoint scalar theory, which is known to double copy to Yang-Mills theory. Therefore, it seems that Marni’s thesis had some good moments.

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