Quark and lepton masses and mixing in the landscape

Abstract

Even if quark and lepton masses are not uniquely predicted by the fundamental theory, as may be the case in the string theory landscape, nevertheless their pattern may reveal features of the underlying theory. We use statistical techniques to show that the observed masses appear to be representative of a scale-invariant distribution, $\ensuremath{\rho}(m)\ensuremath{\sim}1/m$. If we extend this distribution to include all the Yukawa couplings, we show that the resulting Cabibbo-Kobayashi-Maskawa matrix elements typically show a hierarchical pattern similar to observations. The Jarlskog invariant measuring the amount of $CP$ violation is also well reproduced in magnitude. We also apply this framework to neutrinos using the seesaw mechanism. The neutrino results are ambiguous, with the observed pattern being statistically allowed even though the framework does not provide a natural explanation for the observed two large mixing angles. Our framework highly favors a normal hierarchy of neutrino masses. We also are able to make statistical predictions in the neutrino sector when we specialize to situations consistent with the known mass differences and two large mixing angles. Within our framework, we show that with 95% confidence the presently unmeasured Maki-Nakagawa-Sakata mixing angle $\mathrm{sin}{\ensuremath{\theta}}_{13}$ is larger than 0.04 and typically of order 0.1. The leptonic Jarlskog invariant is found to be typically of order ${10}^{\ensuremath{-}2}$ and the magnitude of the effective Majorana mass ${m}_{ee}$ is typically of order 0.001 eV.