Thanks to another productive chat with kneemo, I can report on a new coincidence with the small CKM parameter $0.0035$. Triality was used to obtain all CKM quantities from $3$ real Euler angles. Since triality is a basic symmetry associated to generation number, and the mixing and mass matrices are closely related, we expect to find these CKM parameters associated to fundamental Koide operators. The most basic Koide matrices are the lepton ones.

Recall that the neutrino masses are $(0.00038,0.00895,0.05071)$ eV, while the mirror neutrino masses are $(0.00117,0.05823,0.00060)$ eV. Since Koide mass matrices are diagonalized by the Fourier transform, we can always pair a particle matrix with a mirror matrix to form a new diagonal matrix $M \cdot \overline{M}$, with entries $m_{i} \overline{m}_{i}$. Let us assume that $M \cdot \overline{M}$ is normalized to dimensionless form by the product rule

$\prod_i m_{i} \overline{m}_{i} = 1$.

Then the matrix $(M \cdot \overline{M})^3$ is the diagonal $(a, 1/2a ,2)$, for $a = 0.0035$.

Similarly, one may cycle the $\overline{M}$ (or $M$) entries in the matrix pair to obtain another $2$ sets of parameters with product $1$. For the charged lepton masses one can pair the Koide matrix with itself, to obtain the dimensionless mass triplet $(0.011,2.308,38.8)$. Observe that $2.308/2 = 1.154$, which specifies the first cycled $\nu \overline{\nu}$ paired triplet. The parameter $1.154$ is close to $2/\sqrt{3}$.

6 years ago

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