Unified description of sum rules and duality between CP phases and unitarity triangles through third-order rephasing invariants

In this letter, we demonstrate that products of third-order rephasing invariants $V_{\alpha i} V_{\beta j} V_{\gamma k} / \det V$ of flavor mixing matrix $V$ reproduce all the nine angles of unitarity triangles and all the CP phases in the nine parameterizations of $V$. The sum rules relating the CP phases and angles are also decomposed into terms of these rephasing invariants. Furthermore, through ninth-order invariants, these fourth- and fifth-order invariants become equivalent, which can be regarded as a certain duality. For the phase matrix $\Delta$ and the angle matrix $\Phi$, $\Delta \pm \Phi$ are expressed in terms of even-permutations $X$ and odd-permutations $\Psi$ of third-order invariant. As a result, these are represented by the two concise matrix equations $\Phi = \Psi - {\rm X}$ and $\Delta = \Pi' - \Psi - {\rm X}$.