Perturbative determination of quark CP phase and rephasing invariants of hierarchical mass matrices using their inverses

Abstract

In this paper, we derive approximate expressions for the CP phase $δ$ in the CKM matrix by a perturbative singular value decomposition for hierarchical mass matrices of quarks $m_{u,d}$. The diagonalization is achieved through a seesaw-like procedure in which the heavier generations are successively integrated out, naturally leading to mixing matrices expressed in terms of the mass matrices and their inverses $m_{u,d}^{-1}$. As a result, in the basis where the up-type quark mass matrix is diagonal, $δ$ is reduced to a fourth-order invariant $δ\simeq \arg [ - m^{-1}_{d12} m_{d23}^{} / m^{-1}_{d11} m_{d13}^{} ]$. Furthermore, in the Kobayashi--Maskawa parametrization, the CP phase is likewise expressed by an invariant $δ_{\rm KM} \simeq \arg [ m^{-1}_{d13} m_{d33}^{} / m^{-1}_{d11} m_{d13}^{} ] \simeq π/2$ in the same basis. Since the next-to-leading-order corrections are suppressed at second order in the perturbation, the accuracy of these expressions is typically of order $O(λ^{2}) \sim 4 \%$.