Rephasing invariant structure of Dirac CP phase and basis independent reduction of unitarity constraints for mixing matrices

Abstract

In this paper, we explore rephasing invariant structures of the Dirac CP phase $\delta$ under an approximation $U^{e}_{13} = 0$, where the 1-3 element of the diagonalization of charged leptons $U^{e}$ is neglected. With the further simplified condition $U^{e}_{12} = 0$, the Dirac phase reduces to a compact form $\delta = \delta^{\nu} + \arg [ (U_{33}^e / U_{23}^{e}) - ( U_{33}^{\nu} / U_{23}^{\nu}) ] - \arg [ ( U_{33}^e / U_{23}^{e}) + (U_{23}^{\nu *} / U_{33}^{\nu *} ) ] $, and the CP phase for finite $U_{12}^{e}$ can be understood as a generalization of this compact form. These results encompass almost all perturbative calculations of the CP phases in quark and lepton mixing matrices with hierarchical masses of charged fermions, and are independent of any specific parametrization. As a second result of this work, we derive a basis independent reduction of the unitarity constraints for an arbitrary unitary matrix $V$ by eliminating the elements $V_{21}, V_{22}, V_{31}, V_{32}$ using the inversion formula. Applying the explicit rephasing transformation to this reduction yields a rephasing invariant representation of the PDG parametrization, which allows the translation of theoretical results expressed in the PDG parametrization directly into rephasing invariants.