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

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 $V_{il} V_{jm} V_{kn} / \det V_{\rm CKM}$.

💡 Research Summary

The paper introduces a novel, rephasing‑invariant expression for the CP‑violating phase of the Cabibbo–Kobayashi–Maskawa matrix:
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