From Neutrinos to Pions

If neutrinos can be superluminal, what happens with the pions? Neutral pions are built from quarks on the sides of the tetractys. Recall that the honeycomb dual to the tetractys becomes a triangle diagram for the Fano plane, when the up and down quarks are merged, as in CKM mixing.

The honeycomb is a rule on a set of three $3 \times 3$ matrices. Thus the triangle diagram looks like a rule for three $2 \times 2$ matrices, just like the triality of mixing. Moreover, the color kinematic duality of Bern et al suggests a special role for the gluon preons in perturbative gravity. These are obtained (as non braid objects) from the Fourier transform of up and down quarks, just as the photon arises as the transform of the (left handed) neutrino.

If gluon preons are essentially superluminal, what does this say about the pions? A natural hypothesis is that all the local aspects of mixing, for quarks and neutrinos, are governed by underlying superluminal phenomena. Pions might travel at close to but slightly above $c$. However, for now we will stick to the hypothesis that the small rest masses of neutrinos are responsible for their superluminal behavour, in line with the surface of last scattering as a honeycomb boundary that separates the neutrinos from all other localised states. Nonetheless, pion triangles are also a crucial piece of the mixing puzzle.