Rephasing invariant formulae for CP phases in general parameterizations of flavor mixing matrix and exact sum rules with unitarity triangles
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911210633428992 |
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| author | Yang, Masaki J. S. |
| author_facet | Yang, Masaki J. S. |
| contents | In this letter, we present rephasing invariant formulae $δ^{(αi)} = \arg [ { V_{α1} V_{α2} V_{α3} V_{1i} V_{2i} V_{3i} / V_{αi }^{3} \det V } ] $ for CP phases $δ^{(αi)}$ associated with nine Euler-angle-like parameterizations of a flavor mixing matrix. Here, $α$ and $i$ denote the row and column carrying the trivial phases in a given parameterization. Furthermore, we show that the phases $δ^{(αi)}$ and the nine angles $Φ_{αi}$ of unitarity triangles satisfy compact sum rules $ δ^{(α, i+2)} - δ^{(α, i+1)} = Φ_{α-2, i} - Φ_{α-1, i}$ and $ δ^{(α-2, i)} - δ^{(α-1, i)} = Φ_{α, i+2} - Φ_{α, i+1}$ where all indices are taken cyclically modulo three. These twelve relations are natural generalizations of the previous result $δ_{\mathrm{PDG}}+δ_{\mathrm{KM}}=π-α+γ$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_17866 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rephasing invariant formulae for CP phases in general parameterizations of flavor mixing matrix and exact sum rules with unitarity triangles Yang, Masaki J. S. High Energy Physics - Phenomenology In this letter, we present rephasing invariant formulae $δ^{(αi)} = \arg [ { V_{α1} V_{α2} V_{α3} V_{1i} V_{2i} V_{3i} / V_{αi }^{3} \det V } ] $ for CP phases $δ^{(αi)}$ associated with nine Euler-angle-like parameterizations of a flavor mixing matrix. Here, $α$ and $i$ denote the row and column carrying the trivial phases in a given parameterization. Furthermore, we show that the phases $δ^{(αi)}$ and the nine angles $Φ_{αi}$ of unitarity triangles satisfy compact sum rules $ δ^{(α, i+2)} - δ^{(α, i+1)} = Φ_{α-2, i} - Φ_{α-1, i}$ and $ δ^{(α-2, i)} - δ^{(α-1, i)} = Φ_{α, i+2} - Φ_{α, i+1}$ where all indices are taken cyclically modulo three. These twelve relations are natural generalizations of the previous result $δ_{\mathrm{PDG}}+δ_{\mathrm{KM}}=π-α+γ$. |
| title | Rephasing invariant formulae for CP phases in general parameterizations of flavor mixing matrix and exact sum rules with unitarity triangles |
| topic | High Energy Physics - Phenomenology |
| url | https://arxiv.org/abs/2508.17866 |