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 $V_{αi} V_{βj} V_{γk} / \det V$ 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 $Δ$ and the angle matrix $Φ$, $Δ\pm Φ$ are expressed in terms of even-permutations $X$ and odd-permutations $Ψ$ of third-order invariant. As a result, these are represented by the two concise matrix equations $Φ= Ψ- {\rm X}$ and $Δ= Π' - Ψ- {\rm X}$.