On the CP-violating phase $δ_{\rm CP}$ in fermion mixing matrices

Abstract

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