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 $\delta$ 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, $\delta$ is reduced to a fourth-order invariant $\delta \simeq \arg [-m_{d12}^{-1}{m}_{d23}^{}/m_{d11}^{-1}{m}_{d13}^{}]$. Furthermore, in the Kobayashi--Maskawa parametrization, the CP phase is likewise expressed by an invariant ${\delta }_{\mathrm{KM}}\simeq \arg [m_{d13}^{-1}{m}_{d33}^{}/m_{d11}^{-1}{m}_{d13}^{}]\simeq \pi /2$ 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 $O({\lambda }^2)\sim 4\%$.

arXiv comment9 pages, title is changed, some discussion is added