Rephasing Invariant Formula for CP Phase in Kobayashi-Maskawa Parametrization and Exact Sum Rule with Unitarity Triangle $δ_{\rm PDG} + δ_{\rm KM} = π- α+ γ$

TL;DR

This work derives a rephasing-invariant expression for the Kobayashi–Maskawa CP phase $δ_{ m KM}$ in the KM parametrization, showing it can be written as $δ_{ m KM} = ext{arg} ig[ - { V_{ud} \, ext{det} V_{ m CKM} / (V_{us} V_{ub} V_{cd} V_{td}) } ig]$, and analyzes its perturbative expansion in small quark mixing angles. A key finding is that $δ_{ m KM}$ is nearly maximal ($ imes rac{π}{2}$) when the phase difference between 1–2 mixings, $e^{i( ho^d_{12}- ho^u_{12})}$, is close to $-i$ for small $s_{13}^{u,d}$; this is connected to the ratio $|V_{ub}/V_{td}| e^{i δ_{ m KM}}$ being governed by $V_{ub}^*/V_{td}$ at leading order. Furthermore, combining this with a PDG-phase expression yields an exact sum rule $δ_{ m PDG} + δ_{ m KM} = π - α + γ$, tying the CP phases to unitarity-triangle angles $α, β, γ$. The results provide a robust, parametrization-agnostic view of CP violation and offer a framework for exploring CP phases across different CKM parametrizations, with numerical consistency against UTfit values.

Abstract

In this letter, we obtain a rephasing invariant formula for the CP phase in the Kobayashi--Maskawa parameterization $δ_{\rm KM} = \arg [ - { V_{ud} \det V_{\rm CKM} / V_{us} V_{ub} V_{cd} V_{td}} ]$. General perturbative expansion of the formula and observed value $δ_{\rm KM} \simeq π/2$ reveal that the phase difference of the 1-2 mixings $e^{i (ρ_{12}^{d} - ρ_{12}^{u})}$ is close to maximal for sufficiently small 1-3 quark mixings $s_{13}^{u,d}$. Moreover, combining this result with another formula for the CP phase $δ_{\rm PDG}$ in the PDG parameterization, we derived an exact sum rule $δ_{\rm PDG} + δ_{\rm KM} = π- α+ γ$ which relating the phases and the angles $α, β, γ$ of the unitarity triangle.