Direct Evaluation of CP Phase of CKM matrix, General Perturbative Expansion and Relations with Unitarity Triangles

TL;DR

This paper derives an analytic expression for the CP phase in the CKM matrix using a perturbative approach, relates it to unitarity triangles, and compares it with experimental data to explore CP phase hypotheses.

Contribution

It introduces a perturbative formula for the CKM CP phase and explores its relation to unitarity triangles and experimental observations.

Findings

Derived an analytic relation for the CKM CP phase.

Found the uncertainty of the relation is about 4%.

Suggested some CP phases may be maximal.

Abstract

In this letter, using a rephasing invariant formula $\delta = \arg [ { V_{ud} V_{us} V_{c b} V_{tb} / V_{ub} \det V_{\rm CKM} }]$, we evaluate the CP phase $\delta$ of the CKM matrix $V_{\rm CKM}$ perturbatively for small quark mixing angles $s_{ij}^{u,d}$ with associated phases $\rho_{ij}^{u,d}$. Consequently, we derived a relation $\delta \simeq \arg [\Delta s_{12} \Delta s_{23} / ( \Delta s_{13} - s^u_{12} e^{-i \rho^u_{12}} \Delta s_{23} )]$ with $\Delta s_{ij} \equiv s^d_{ij} e^{-i \rho^d_{ij}} - s^u_{ij} e^{-i \rho^u_{ij}}$. Such a result represents the analytic behavior of the CKM phase. The uncertainty in the relation is of order $O(\lambda^{2}) \sim 4\%$, which is comparable to the current experimental precision. Comparisons with experimental data suggest that the hypothesis of some CP phases being maximal. We also discussed relationships between the phase $\delta$ and unitarity triangles. As a result, several relations between the angles $\alpha, \beta, \gamma$ and $\delta$ are identified through other invariants $V_{il} V_{jm} V_{kn} / \det V_{\rm CKM}$.