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

In this letter, using a rephasing invariant formula $\delta =\arg [V_{ud}{V}_{us}{V}_{cb}{V}_{tb}\mathrm{/}{V}_{ub}\det V_{\mathrm{C}\mathrm{K}\mathrm{M}}]$, we evaluate the CP phase $\delta$ of the CKM matrix $V_{\mathrm{C}\mathrm{K}\mathrm{M}}$ perturbatively for small quark mixing angles $s_{ij}^{u,d}$ with associated phases ${\rho }_{ij}^{u,d}$. Consequently, we derived a relation $\delta \simeq \arg [\mathrm{\Delta }{s}_{12}\mathrm{\Delta }{s}_{23}\mathrm{/}(\mathrm{\Delta }{s}_{13}-s_{12}^u{e}^{-i{\rho }_{12}^u}\mathrm{\Delta }{s}_{23})]$ with $\mathrm{\Delta }{s}_{ij}\equiv s_{ij}^d{e}^{-i{\rho }_{ij}^d}-s_{ij}^u{e}^{-i{\rho }_{ij}^u}$. Such a result represents the analytic behavior of the CKM phase. The uncertainty in the relation is of order $O({\lambda }^2)\sim 4\mathrm{\%}$, 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 $\delta$ and unitarity triangles. As a result, several relations between the angles $\alpha ,\beta ,\gamma$ and $\delta$ are identified through other invariants $V_{il}{V}_{jm}{V}_{kn}\mathrm{/}\det V_{\mathrm{C}\mathrm{K}\mathrm{M}}$.