Monday, April 26, 2010

M Theory Lesson 317

The Koide mass matrices are written in terms of two parameters, which we will call $n$ (placed on the diagonal) and $\phi$ (the phase angle). The eigenvalues of such a matrix may be expressed as $\lambda + n$ for an eigenvalue $\lambda$ of since adding $n \mathbf{1}$ to the Koide matrix leaves the eigenvectors intact. Note that the eigenvalue set for $K$ always sums to zero, so these three $\lambda$ describe the mass differences within a triplet. For example, at $\phi = 2/9$ the eigenvalues of $K$ are $\{ -0.59367, -1.35715, 1.9508 \}$. At a cubed root, the eigenvalues are $\{ -1, -1, 2 \}$, like the entries of the idempotent $S$. All possible zero mass triplets are thus described by the $U(1)$ of the variable $\phi$. As $\phi$ varies, a given eigenvalue varies between $-2$ and $+2$.