As can be seen in the period $\omega$ graph below, which searches for solutions to the new mass relation, there is a second solution for the new $\theta$ at the fixed Koide parameter value of $r = \sqrt{2}$. That is, the mass relation for neutrinos and antineutrinos also holds at $\theta = 0.8857$. Using the phase $\pm \pi/12$ there is a corresponding set of six new masses associated to this $\theta$, given by

$m_{i}: 0.00166, 0.0250, 0.0333$ eV

$\overline{m}_{i}: 0.00084, 0.0130, 0.0461$ eV

These all lie within the mass range of the neutrinos and antineutrinos but have smaller $\Delta m^2$ values. The mass triplet sums are still $0.06$ eV, as for the usual Koide rule. Now however, the Koide rule holds when (i) the lightest antiparticle mass (at $0.00084$ eV) occurs with a negative eigenvalue and (ii) the lightest particle mass ($0.00166$ eV) also occurs with a negative eigenvalue. This is like exchanging a basic $SO(2,4)$ metric light cone (for a six dimensional $\nu$ $\overline{\nu}$ square root eigenvalue space) for an $SO(3,3)$ metric. So we do not expect these particles (assuming they exist) to participate in the weak interaction with ordinary matter, like the neutrino sector, but if localisable then their detection would in principle be similar to that of the neutrino sector. This brings the total count of neutral CPT violating chiral Koide particles to $24$, and no more are possible.

7 years ago

We are looking for a $y$ value of zero in the graph. The negative cusp that gives the only two solutions (at $\theta = 0.886$ and $0.223$) occurs at the phase $\theta = \pi /6$, the twelfth root of unity.

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