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 $65.4^\circ$, in agreement with the measured value of $65.7^\circ \pm 1.5^\circ$. For leptons, the phase is $285.0^\circ$ in inverted ordering and $195.0^\circ$ in normal ordering, both within $1\sigma$ of the current global fit. The two orderings differ because the splitting of the pair $\nu_1$, $\nu_2$, 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, $|J_{\rm CP}| \simeq 0.032$, and weak CP violation in normal ordering, $|J_{\rm CP}| \simeq 0.009$, with $\sin\delta_{\rm CP} < 0$ in both. The value for normal ordering differs from the predictions of flavour-symmetry models by more than $60^\circ$. DUNE and Hyper-Kamiokande will test it.