Unified Formulation of the Dirac Phase of Quark and Lepton Mixing from a Mixing-Equilibrium Condition

The Dirac CP phase of quarks and leptons plays a very important role in flavour mixing. Both the CKM and PMNS matrices contain a Dirac phase component. The cosine and sine of the phase enter the real and imaginary parts of a Bargmann invariant of the mixing matrix. We have developed a unified formulation of the Dirac phase of quark and lepton mixing in a phenomenological model in which the mixing angles follow from a mixing-equilibrium condition. This condition gives the real and imaginary parts of the Bargmann invariant from the mixing-weight and its variance at equilibrium in each sector. The quark phase is found to be 65.4^∘, in agreement with the measured value of 65.7^∘ ± 1.5^∘. For leptons, the phase is 285.0^∘ in inverted ordering and 195.0^∘ in normal ordering, both within 1σ of the current global fit. The two orderings differ because the splitting of the pair ν₁, ν₂, compared with its gap to the third state, is small in inverted ordering and sizeable in normal ordering. We find strong CP violation in inverted ordering, |JCP| 0.032, and weak CP violation in normal ordering, |JCP| 0.009, with sinδCP < 0 in both. The value for normal ordering differs from the predictions of flavour-symmetry models by more than 60^∘. DUNE and Hyper-Kamiokande will test it.