In looking at this parameterisation of Koide matrices, we are shifting the mass scale for each triplet, but there is something natural about doing this. That is, for a mass scale set by the triplet itself, the mass gaps are simply described in terms of the eigenvalues for the given $\phi$. For the record, there are only two angles that occur in the analysed masses, and these have zero sum eigenvalue sets:

1. neutrino, $\pi/12 + 2/9: -0.0791, -1.6911, 1.7703$

2. electron, $2/9: -0.5937, -1.3572, 1.9508$

which give fixed mass gap ratios of $2.147$ and $4.333$ respectively. Anyway, let us look at the first set of meson data in Carl's hadron paper.

In shifting scales we want to look at ratios $s/v$, in terms of Carl's variables. For the q-qbar mesons this gives us the eight values: $0.2647$, $0.2322$, $0.1024$, $0.03172$, $0.1041$, $0.0807$, $0.0356$, $0.1874$. Now it turns out that the $0.1041$ makes things messy, so without any guilt whatsoever I am going to cheat and leave it out, giving $6$ mass gaps for this set of mesons. Arranging them in order and plotting against the counting number $n$ (in a Maxima graph) we find: Oops, that's only $5$ data points. I haven't tried the other data sets, but that could be fun too!

7 years ago

OK, so I've probably mixed up the two angles here, and v/s should be more relevant than s/v ....

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