Geometric representation of CP phases $δ_{\rm PDG}, δ_{\rm KM}$ in flavor mixing matrix and its sum rule by alternative unitarity triangle and quadrangle
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911305859858432 |
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| author | Yang, Masaki J. S. |
| author_facet | Yang, Masaki J. S. |
| contents | In this letter, we present a geometric representation of the CP phases $δ_{\rm PDG}$ and $δ_{\rm KM}$ in the PDG and Kobayashi--Maskawa parameterizations of the flavor mixing matrix in the complex plane. The sum rule with the unitarity triangle $δ_{\rm PDG} + δ_{\rm KM} = π- α+ γ$ is expressed as a quadrangle, which is a combination of a unitarity triangle and an alternative triangle. Through the unitarity quadrangle, the CP phases are also identified with specific geometric angles. Furthermore, a new set of inverse unitarity triangles is defined from the inversion formula of a unitary matrix $U^{\dagger} = U^{-1}$. These novel triangles contain standard angles of the form $\arg [U_{αi } U_{βj} U_{αj}^{*} U_{βi}^{*}]$ and new angles $\arg [U_{αi } U_{βj} U_{γk} / \det U]$, which directly determine nontrivial arguments of the mixing matrix elements. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_11596 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometric representation of CP phases $δ_{\rm PDG}, δ_{\rm KM}$ in flavor mixing matrix and its sum rule by alternative unitarity triangle and quadrangle Yang, Masaki J. S. High Energy Physics - Phenomenology In this letter, we present a geometric representation of the CP phases $δ_{\rm PDG}$ and $δ_{\rm KM}$ in the PDG and Kobayashi--Maskawa parameterizations of the flavor mixing matrix in the complex plane. The sum rule with the unitarity triangle $δ_{\rm PDG} + δ_{\rm KM} = π- α+ γ$ is expressed as a quadrangle, which is a combination of a unitarity triangle and an alternative triangle. Through the unitarity quadrangle, the CP phases are also identified with specific geometric angles. Furthermore, a new set of inverse unitarity triangles is defined from the inversion formula of a unitary matrix $U^{\dagger} = U^{-1}$. These novel triangles contain standard angles of the form $\arg [U_{αi } U_{βj} U_{αj}^{*} U_{βi}^{*}]$ and new angles $\arg [U_{αi } U_{βj} U_{γk} / \det U]$, which directly determine nontrivial arguments of the mixing matrix elements. |
| title | Geometric representation of CP phases $δ_{\rm PDG}, δ_{\rm KM}$ in flavor mixing matrix and its sum rule by alternative unitarity triangle and quadrangle |
| topic | High Energy Physics - Phenomenology |
| url | https://arxiv.org/abs/2509.11596 |