On the CP-violating phase ${\delta }_{\mathrm{CP}}$ in fermion mixing matrices

Abstract

The recent established large ${\theta }_{13}$ in neutrino mixing provides an optimistic possibility for the investigation of the CP violation, therefore it is necessary to study the CP-violating phase ${\delta }_{\mathrm{CP}}$ in detail. Based on the maximal CP violation hypothesis in the original Kobayashi-Maskawa (KM) scheme of neutrino mixing matrix, i.e., ${\delta }_{\mathrm{KM}}=90^{\circ }$, we calculate ${\delta }_{\mathrm{CK}}$ for both quarks and leptons in the Chau-Keung (CK) scheme of the standard parametrization and find that ${\delta }_{\mathrm{CK}}^{\mathrm{quark}}=(68.62_{-0.81}^{+0.89}{)}^{\circ }$ and ${\delta }_{\mathrm{CK}}^{\mathrm{lepton}}=(85.39_{-1.82}^{+4.76}{)}^{\circ }$, provided with three mixing angles to be given. We also examine the sensitivity of $|V_{ij}|$ and $|U_{ij}|$ to ${\delta }_{\mathrm{CK}}$ and ${\delta }_{\mathrm{KM}}$. As a convention-independent investigation, we discuss the $\Phi$ matrix, which has elements correspond to angles of the unitarity triangles. We demonstrate the $\Phi$ matrices for both quark and lepton sectors and discuss the implications as well as the variations of the $\Phi$ matrix elements with ${\delta }_{\mathrm{CP}}$.

Comments: 10 pages, 6 figures, 3 tables. Numerical results improved