Unified description of sum rules and duality between CP phases and unitarity triangles through third-order rephasing invariants
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918171024293888 |
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| author | Yang, Masaki J. S. |
| author_facet | Yang, Masaki J. S. |
| contents | 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 $Δ= Π' - Ψ- {\rm X}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_00702 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Unified description of sum rules and duality between CP phases and unitarity triangles through third-order rephasing invariants Yang, Masaki J. S. High Energy Physics - Phenomenology 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 $Δ= Π' - Ψ- {\rm X}$. |
| title | Unified description of sum rules and duality between CP phases and unitarity triangles through third-order rephasing invariants |
| topic | High Energy Physics - Phenomenology |
| url | https://arxiv.org/abs/2509.00702 |