{
  "abstract": "In this paper, we derive approximate expressions for the CP phase $δ$ in the CKM matrix by a perturbative singular value decomposition for hierarchical mass matrices of quarks $m_{u,d}$. The diagonalization is achieved through a seesaw-like procedure in which the heavier generations are successively integrated out, naturally leading to mixing matrices expressed in terms of the mass matrices and their inverses $m_{u,d}^{-1}$. As a result, in the basis where the up-type quark mass matrix is diagonal, $δ$ is reduced to a fourth-order invariant $δ\\simeq \\arg [ - m^{-1}_{d12} m_{d23}^{} / m^{-1}_{d11} m_{d13}^{} ]$. Furthermore, in the Kobayashi--Maskawa parametrization, the CP phase is likewise expressed by an invariant $δ_{\\rm KM} \\simeq \\arg [ m^{-1}_{d13} m_{d33}^{} / m^{-1}_{d11} m_{d13}^{} ] \\simeq π/2$ in the same basis. Since the next-to-leading-order corrections are suppressed at second order in the perturbation, the accuracy of these expressions is typically of order $O(λ^{2}) \\sim 4 \\%$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.06985",
  "author_total": 1,
  "authors": [
    {
      "name": "Masaki J. S. Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ff4m14xz2x9s3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.06985",
  "fulltext_url": null,
  "id": "ff4m14xz2x9s3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.06985",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "hep-ph"
    ],
    [
      "openalex",
      "3106|Particle physics theoretical and experimental studies"
    ]
  ],
  "openalex": "W7167846740",
  "pdf_url": "https://arxiv.org/pdf/2607.06985",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ff4m14xz2x9s3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:41Z",
      "native_id": "2607.06985",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.06985"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:hep-ph",
      "subfield": "high-energy"
    }
  ],
  "title": "Perturbative determination of quark CP phase and rephasing invariants of hierarchical mass matrices using their inverses",
  "title_text": "Perturbative determination of quark CP phase and rephasing invariants of hierarchical mass matrices using their inverses",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:41Z",
  "url": "https://research.draconitic.com/paper/ff4m14xz2x9s3",
  "venue": "arXiv",
  "versions": []
}