Theory Update 74

The fun representation for braids in $B_3$ now has two forms for the neutrino braid. First, using $\omega$ and $\overline{\omega}$ for a positive or negative crossing, we obtain the matrix shown. But the mirror particle representation on the left results in a dual diagonal matrix $(1, \overline{\omega}^2, \omega^2)$. This forces $\overline{\omega} = \omega^2$, which says that $\omega$ is a cubed root of unity.

As we know, the cubed root limits the allowed number of twists in a $B_2$ section of a braid. The cubed roots thus also label the charges $\{ 0, \pm 1 \}$ of particle ribbon diagrams. We have now described the $Z$ boson diagonal in $B_3$, rather than in $B_6$. Instead of three separated ribbons, we use the strand groupings $(1,2)$, $(2,3)$ and $(3,1)$ as holders for a twist.