Probability Density Functions for CP-Violating Rephasing Invariants

The implications of the anarchy principle on CP violation in the lepton sector are investigated. A systematic method is introduced to compute the probability density functions for the CP-violating rephasing invariants of the PMNS matrix from the Haar measure relevant to the anarchy principle. Contrary to the CKM matrix which is hierarchical, it is shown that the Haar measure, and hence the anarchy principle, are very likely to lead to the observed PMNS matrix. Predictions on the CP-violating Dirac rephasing invariant $\mathrm{\mid }{j}_D\mathrm{\mid }$ and Majorana rephasing invariant $\mathrm{\mid }{j}_1\mathrm{\mid }$ are also obtained. They correspond to $⟨\mathrm{\mid }{j}_D\mathrm{\mid }{⟩}_{\text{Haar}}=\pi \mathrm{/}105\approx 0.030$ and $⟨\mathrm{\mid }{j}_1\mathrm{\mid }{⟩}_{\text{Haar}}=1\mathrm{/}(6\pi )\approx 0.053$ respectively, in agreement with the experimental hint from T2K of $\mathrm{\mid }{j}_D^{\text{exp}}\mathrm{\mid }\approx 0.032\pm 0.005$ (or $\approx 0.033\pm 0.003$) for the normal (or inverted) hierarchy.