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

In this paper, we demonstrate that products of third-order rephasing invariants V α i V β j V γ 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 Δ and the angle matrix Φ , Δ ± Φ are expressed in terms of even-permutations X and odd-permutations Ψ of third-order invariants. As a result, these are represented by the two concise matrix equations Φ = Ψ − X and Δ = Π ′ − Ψ − X .