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

Abstract

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

Comments: 6 pages, published version in Prog Theor Exp Phys (2025)