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

2025/08/31 by Yang, Masaki J. S.
#FOS: Physical sciences #High Energy Physics - Phenomenology (hep-ph)

paper · doi:10.48550/arxiv.2509.00702

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± 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.

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