Rephasing Invariant Formula for CP Phase in Kobayashi-Maskawa Parametrization and Exact Sum Rule with Unitarity Triangle ${\delta }_{\mathrm{P}\mathrm{D}\mathrm{G}}+{\delta }_{\mathrm{K}\mathrm{M}}=\pi -\alpha +\gamma$

In this letter, we obtain a rephasing invariant formula for the CP phase in the Kobayashi–Maskawa parameterization ${\delta }_{\mathrm{K}\mathrm{M}}=\arg [-V_{ud}\det V_{\mathrm{C}\mathrm{K}\mathrm{M}}\mathrm{/}{V}_{us}{V}_{ub}{V}_{cd}{V}_{td}]$. General perturbative expansion of the formula and observed value ${\delta }_{\mathrm{K}\mathrm{M}}\simeq \pi \mathrm{/}2$ 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 ${\delta }_{\mathrm{P}\mathrm{D}\mathrm{G}}$ in the PDG parameterization, we derived an exact sum rule ${\delta }_{\mathrm{P}\mathrm{D}\mathrm{G}}+{\delta }_{\mathrm{K}\mathrm{M}}=\pi -\alpha +\gamma$ which relating the phases and the angles $\alpha ,\beta ,\gamma$ of the unitarity triangle.