M Theory Lesson 318

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!