Rephasing Invariant Formula for CP Phase in Kobayashi-Maskawa Parametrization and Exact Sum Rule with Unitarity Triangle $\delta_{\rm PDG} + \delta_{\rm KM} = \pi - \alpha + \gamma$

Abstract

In this letter, we obtain a rephasing invariant formula for the CP phase in the Kobayashi--Maskawa parameterization $\delta_{\rm KM} = \arg [ - { V_{ud} \det V_{\rm CKM} / V_{us} V_{ub} V_{cd} V_{td}} ]$. General perturbative expansion of the formula and observed value $\delta_{\rm KM} \simeq \pi/2$ reveal that the phase difference of the 1-2 mixings $e^{i (\rho_{12}^{d} - \rho_{12}^{u})}$ 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 $\delta_{\rm PDG}$ in the PDG parameterization, we derived an exact sum rule $\delta_{\rm PDG} + \delta_{\rm KM} = \pi - \alpha + \gamma$ which relating the phases and the angles $\alpha, \beta, \gamma$ of the unitarity triangle.