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

TL;DR

This paper presents a direct, rephasing-invariant method to determine the CKM CP phase $\delta$ by using $\delta = \arg \left[ { V_{ud} V_{us} V_{cb} V_{tb} \over V_{ub} \det V_{\rm CKM}} \right]$. It develops a perturbative expansion in small quark-mixing angles with associated phases, yielding a compact analytic formula $\delta \simeq \arg \left[ { \Delta s_{12} \Delta s_{23} \over \Delta s_{13} - s^{u}_{12} e^{-i \rho^{u}_{12}} \Delta s_{23} } \right]$, where $\Delta s_{ij} = s^{d}_{ij} e^{-i \rho^{d}_{ij}} - s^{u}_{ij} e^{-i \rho^{u}_{ij}}$. The approach provides $O(\lambda^{2})$ (\approx 4\%) uncertainty and connects $\delta$ to unitarity-triangle angles, with further implications for leptonic CP phase and grand unified considerations. The results align with current data, suggesting near-maximal CP phases in some sectors, and offer a framework that improves analytic transparency and error control over the conventional Jarlskog-invariant approach.

Abstract

In this letter, using a rephasing invariant formula $δ= \arg [ { V_{ud} V_{us} V_{c b} V_{tb} / V_{ub} \det V_{\rm CKM} }]$, we evaluate the CP phase $δ$ of the CKM matrix $V_{\rm CKM}$ perturbatively for small quark mixing angles $s_{ij}^{u,d}$ with associated phases $ρ_{ij}^{u,d}$. Consequently, we derived a relation $δ\simeq \arg [Δs_{12} Δs_{23} / ( Δs_{13} - s^u_{12} e^{-i ρ^u_{12}} Δs_{23} )]$ with $Δs_{ij} \equiv s^d_{ij} e^{-i ρ^d_{ij}} - s^u_{ij} e^{-i ρ^u_{ij}}$. Such a result represents the analytic behavior of the CKM phase. The uncertainty in the relation is of order $O(λ^{2}) \sim 4\%$, 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 $δ$ and unitarity triangles. As a result, several relations between the angles $α, β, γ$ and $δ$ are identified through other invariants $V_{il} V_{jm} V_{kn} / \det V_{\rm CKM}$.