When Graham Dungworth and I wrote about mirror neutrinos, I was rather fixated on the appearance of the last braid in a natural set of $B_3$ ribbons. Let us look once more at the Z boson, which comes in two forms, from the Fourier transform $F$ of the mass splitting circulants.

A pair of Koide phases, $\phi$ and $\overline{\phi}$, also result in a conjugate pair of diagonals, each a rotation of the cubed roots in the plane.

We use minus signs to denote a mirror pair of circulants, giving a triplet of sixth roots on the $Z$ boson side. Altogether there are $12$ neutrinos, including generation, and $12$ $Z$ bosons, including color. Recall that the mirror neutrino $- \pi/12$ phase results in the CMB temperature. For the charged leptons, this mirror phase component essentially vanishes, or equals $2 m \pi$. Thus we associate the interval $[- \phi, + \phi]$ with neutral (or partially neutral) mixing.

The CMB suggests not one, but a vast collection of particles behind the mirror. Infinity meets small ordinals, as in the contrast between the analysis of the CMB and the quark gluon plasma. Across the mirror there are three mirror neutrino temperatures: the past, the present and the future.

When the phase $\phi$ is tripled, it splits $[- 3\phi, + 3\phi]$ into six segments, shrinking the six segments formed by the basic sixth roots of unity. The phase shrinkage of $2 \pi$ to $2 \phi$ is reminiscent of the conical singularities in topological gravity, where one glues the two edges of a phase interval to create a cone, defining a point mass at the vertex.

6 years ago

The matrix product of any phase pair gives the photon identity matrix, which fixes the remaining Koide parameter. Recall also that the tripling of $\pi /12$ gives the basic $\pi /4$ phase.

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