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

TL;DR

This paper investigates the CP-violating phase in fermion mixing matrices, deriving values for quarks and leptons based on a maximal CP violation hypothesis, and explores the implications for unitarity triangles.

Contribution

It introduces a calculation of the CP-violating phase in both quark and lepton sectors under a maximal CP violation hypothesis, and analyzes the $ ext{ extPhi}$ matrix's sensitivity to $ ext{ extdelta}_{ ext{CP}}$.

Findings

Calculated $ ext{ extdelta}_{ ext{CP}}$ for quarks and leptons.

Demonstrated the $ ext{ extPhi}$ matrix variations with $ ext{ extdelta}_{ ext{CP}}$.

Discussed the implications for unitarity triangles.

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 $\delta_{\rm CP}$ in detail. Based on the maximal CP violation hypothesis in the original Kobayashi-Maskawa (KM) scheme of neutrino mixing matrix, i.e., $\delta_{\rm KM}=90^\circ$, we calculate $\delta_{\rm CK}$ for both quarks and leptons in the Chau-Keung (CK) scheme of the standard parametrization and find that $\delta^{\mathrm{quark}}_{\mathrm{CK}}=(68.62^{+0.89}_{-0.81})^\circ$ and $\delta^{\mathrm{lepton}}_{\mathrm{CK}}=(85.39^{+4.76}_{-1.82})^\circ$, provided with three mixing angles to be given. We also examine the sensitivity of $|V_{ij}|$ and $|U_{ij}|$ to $\delta_{\rm CK}$ and $\delta_{\rm KM}$. 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 $\delta_{\rm CP}$.