Anarchy and hierarchy: An approach to study models of fermion masses and mixings

We advocate a new approach to study models of fermion masses and mixings, namely, the anarchy proposed by Hall, Murayama, and Weiner. In this approach, we scan the $O\left(1\right)\mathrm{}$ coefficients randomly. We argue that this is the correct approach when the fundamental theory is sufficiently complicated. Assuming that there is no physical distinction among three generations of neutrinos, the probability distributions in Maki-Nakagawa-Sakata mixing angles can be predicted independent of the choice of the measure. This is because the mixing angles are distributed according to the Haar measure of the Lie groups whose elements diagonalize the mass matrices. The near-maximal mixings, as observed in the atmospheric neutrino data and as required in the large mixing angle solution to the solar neutrino problem, are highly probable. A small hierarchy between $\Delta m^2$ for the atmospheric and the solar neutrinos is obtained very easily; the complex seesaw case gives a hierarchy of a factor of 20 as the most probable one, even though this conclusion is more measure dependent. $U_{e3\mathrm{}}$ has to be just below the current limit from the CHOOZ experiment. The $\mathrm{CP}$-violating parameter $\sin \delta$ is preferred to be maximal. We present a simple $\mathrm{SU}\left(5\right)\mathrm{}$-like extension of anarchy to the charged lepton and quark sectors that works well phenomenologically.