Koide Review

In an ancient discussion thread a post by Hans de Vries caught my eye. Let us be more explicit about the connection between tribimaximal mixing for neutrinos and the Koide matrix itself. Recall that the three Koide eigenvalues are given by for $n = 0,1,2$. We could rewrite the $n = 0$ eigenvalue in the form so that it looks very much like one row of the (square root) tribimaximal matrix. Moreover, the remaining entries of the tribimaximal matrix are expressed as and so with basic trigonometric rules we can write down the other members of the Koide triplet in terms of cosines and sines for the angles $\theta$ and $\omega$, using the entries of the tribimaximal matrix. This holds for all Koide triplets using the factor $r = \sqrt{2}$, since $\theta$ is arbitrary. In contrast, the CKM quark mixing matrix rows will not look like a basic Koide triplet, because there is no zero element to produce the simple expression for the eigenvalues. We can however use the CKM entries to define sets of triplets in an analogous manner.