Perturbative determination of CP phase in CKM matrix using rephasing invariants of hierarchical mass matrices and 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 quark mass matrices $m_{q}$. 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 matrix and its inverse $m_{q}^{-1}$. As a result, in the basis where the up-type quark mass matrix is diagonal, $δ$ is reduced to the fourth-order invariant $δ\simeq \arg [ - m^{-1}_{d12} m_{d23}^{} / m^{-1}_{d11} m_{d13}^{} ]$, constructed from the down-type mass matrix and its inverse. Furthermore, in the Kobayashi--Maskawa parametrization, the CP phase is likewise expressed in the same basis as the invariant $δ_{\rm KM} \simeq \arg [ m^{-1}_{d13} m_{d33}^{} / m^{-1}_{d11} m_{d13}^{} ] \simeq π/2$.