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

TL;DR

This work addresses the apparent scatter of CP-violating phases across multiple parameterizations of the flavor-mixing matrix by developing a unified description based on rephasing invariants. It introduces six third-order rephasing invariants with determinant and demonstrates that they, along with ninth-order invariants, reproduce all nine CP angles and all nine CP phases, enabling a decomposition of sum rules and a duality between phases and unitarity-triangle angles. The key contributions are the concise matrix relations Φ = Ψ − X and Δ = Π′ − Ψ − X, where X and Ψ encode even and odd permutations of third-order invariants, and the expression of CP-violating observables through higher-order invariants. The approach complements Jarlskog invariants, provides a deeper algebraic structure for CP violation, and extends naturally to the lepton sector (MNS matrix), aiding precision tests and new physics searches.

Abstract

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 $Δ= Π' - Ψ- {\rm X}$.