Generalized scaling ansatz and minimal seesaw mechanism

Abstract

Generalized scaling in flavor neutrino masses ${M}_{ij}$ ($i$, $j=e$, $\ensuremath{\mu}$, $\ensuremath{\tau}$) expressed in terms of ${\ensuremath{\theta}}_{\mathrm{SC}}$ and the atmospheric neutrino mixing angle ${\ensuremath{\theta}}_{23}$ is defined by ${M}_{i\ensuremath{\tau}}/{M}_{i\ensuremath{\mu}}=\ensuremath{-}{\ensuremath{\kappa}}_{i}{t}_{23}$ ($i=e$, $\ensuremath{\mu}$, $\ensuremath{\tau}$) with ${\ensuremath{\kappa}}_{e}=1$, ${\ensuremath{\kappa}}_{\ensuremath{\mu}}=B/A$ and ${\ensuremath{\kappa}}_{\ensuremath{\tau}}=1/B$, where ${t}_{23}=\mathrm{tan}{\ensuremath{\theta}}_{23}$, $A={cos}^{2}{\ensuremath{\theta}}_{\mathrm{SC}}+{sin}^{2}{\ensuremath{\theta}}_{\mathrm{SC}}{t}_{23}^{4}$ and $B={cos}^{2}{\ensuremath{\theta}}_{\mathrm{SC}}\ensuremath{-}{sin}^{2}{\ensuremath{\theta}}_{\mathrm{SC}}{t}_{23}^{2}$. The generalized scaling ansatz predicts the vanishing reactor neutrino mixing angle ${\ensuremath{\theta}}_{13}=0$. It is shown that the minimal seesaw mechanism naturally implements our scaling ansatz. There are textures satisfying the generalized scaling ansatz that yield vanishing baryon asymmetry of the Universe (BAU). Focusing on these textures, we discuss effects of ${\ensuremath{\theta}}_{13}\ensuremath{\ne}0$ to evaluate a $CP$-violating Dirac phase $\ensuremath{\delta}$ and BAU and find that BAU is approximately controlled by the factor ${sin}^{2}{\ensuremath{\theta}}_{13}\mathrm{sin}(2\ensuremath{\delta}\ensuremath{-}\ensuremath{\phi})$, where $\ensuremath{\phi}$ stands for the $CP$-violating Majorana phase whose magnitude turns out to be at most 0.1.