pith. sign in

arxiv: 2602.14513 · v2 · pith:OQVMNW62new · submitted 2026-02-16 · ✦ hep-ph

Rephasing invariant structure of CP phase for simplified mixing matrices in Fritzsch--Xing parametrization

Masaki J. S. Yang This is my paper

Pith reviewed 2026-05-15 22:18 UTC · model grok-4.3

classification ✦ hep-ph
keywords Fritzsch-Xing parametrizationrephasing transformationCP phasemixing matrixneutrino mixingquark mixinghierarchical fermionsPMNS matrix
0
0 comments X

The pith

An explicit rephasing transformation converts any unitary mixing matrix to the Fritzsch-Xing parametrization, and under the approximations U13^e=0 and U23^e=0 the FX phase simplifies to the sum of the neutrino-intrinsic phase and the 1-2 re

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs an explicit rephasing transformation that puts an arbitrary unitary mixing matrix into the Fritzsch-Xing form by making the third row and third column real. It then examines the structure of the CP-violating phase δ_FX when the charged-lepton mixing matrix has vanishing 1-3 element. Adding the further condition that the 2-3 element also vanishes reduces δ_FX to a simple sum of the intrinsic neutrino phase and the difference of two phases from the first two generations. This compact result applies to perturbative treatments of both the CKM quark mixing matrix and the MNS neutrino mixing matrix whenever the charged fermions are strongly hierarchical.

Core claim

We construct an explicit rephasing transformation that converts an arbitrary unitary mixing matrix into the Fritzsch-Xing parametrization obtained by trivializing arguments of the matrix elements in the third row and third column. Under the approximation U13^e = 0 the FX phase δ_FX has a rephasing-invariant structure. With the additional approximation U23^e = 0, δ_FX reduces to the sum of the neutrino-intrinsic FX phase δ^ν_FX and the contribution from the relative phase ρ'1 − ρ'2 between the lighter generations. For finite U23^e the expression generalizes this compact form, covering almost all perturbative calculations of CP phases for the CKM and MNS matrices with hierarchical charged

What carries the argument

The rephasing transformation that trivializes phases in the third row and column of the mixing matrix to reach the Fritzsch-Xing parametrization, together with the approximations U13^e = 0 and U23^e = 0 that allow the FX phase to be written as δ^ν_FX + (ρ'1 - ρ'2)

Load-bearing premise

The approximations that the 1-3 and 2-3 elements of the charged-lepton diagonalization matrix U^e are zero

What would settle it

A direct calculation of δ_FX for a specific unitary matrix with U13^e and U23^e set exactly to zero that yields a value different from δ^ν_FX + (ρ'1 − ρ'2) would falsify the claimed simplification

read the original abstract

In this paper, we construct an explicit rephasing transformation that converts an arbitrary unitary mixing matrix into the Fritzsch--Xing (FX) parametrization, which is obtained by trivializing arguments of the matrix elements in the third row and third column. We further analyze rephasing invariant structure of the FX phase $\delta_{\rm FX}$ under an approximation $U_{13}^{e} = 0$, where the 1-3 element of the diagonalization matrix of charged leptons $U^{e}$ is neglected. With an additional approximation $U_{23}^{e} = 0$, the FX phase becomes highly simplified, reducing to a sum of the neutrino-intrinsic FX phase $\delta^{\nu}_{\rm FX}$ and the contribution from the relative phase $\rho'_{1}- \rho'_{2}$ between the lighter 1-2 generations. The phase $\delta_{\rm FX}$ for finite $U_{23}^{e}$ is understood as a generalization of the compact expression. This result covers almost all perturbative calculations of CP phases for the CKM and MNS matrices with hierarchical charged fermions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript constructs an explicit rephasing transformation that converts an arbitrary 3×3 unitary mixing matrix into the Fritzsch-Xing (FX) parametrization by rendering the third row and third column real. It then examines the rephasing-invariant structure of the FX phase δ_FX under the approximation U_{13}^e = 0 and shows that the further approximation U_{23}^e = 0 reduces δ_FX to the sum of the neutrino-intrinsic FX phase δ^ν_FX plus the relative phase contribution (ρ'_1 − ρ'_2). The result is presented as applicable to perturbative calculations of CP phases in both the CKM and MNS matrices for hierarchical charged fermions.

Significance. If the derivation is correct, the work supplies a concrete rephasing-invariant framework for the CP phase in the FX parametrization under controlled approximations. The explicit mapping from a general unitary matrix and the isolation of charged-lepton phase contributions provide a transparent tool that can streamline phenomenological analyses of CP violation in both quark and lepton sectors when fermion masses are hierarchical. The approach builds directly on standard unitary rephasing freedoms without introducing new parameters.

minor comments (2)
  1. [Section 3] In the paragraph following Eq. (12), the primed phases ρ'_i are introduced without an explicit statement of their relation to the original rephasing parameters; adding one sentence linking them to the general freedom in the unitary matrix would improve readability.
  2. [Conclusion] The claim that the result 'covers almost all perturbative calculations' (abstract and concluding paragraph) would benefit from a short footnote or sentence listing the two or three most common hierarchies to which the approximations apply.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive assessment. The referee's summary correctly captures the construction of the rephasing transformation to the Fritzsch-Xing parametrization and the analysis of the rephasing-invariant structure of δ_FX under the stated approximations. We are pleased that the work is viewed as providing a transparent tool for phenomenological analyses of CP violation in hierarchical fermion sectors.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central derivation is an explicit construction of a rephasing transformation that maps any 3x3 unitary matrix to the FX form by rendering the third row and column real. This follows directly from the standard rephasing freedom of unitary matrices and does not define the target phase in terms of itself. The subsequent reduction of δ_FX under the stated approximations U_{13}^e=0 and U_{23}^e=0 is obtained by isolating charged-lepton phases in the product U = U^e† U^ν and collecting relative phases; the resulting compact expression is a direct algebraic consequence rather than a fit or self-referential definition. No load-bearing self-citation, uniqueness theorem imported from prior work, or renaming of known results is used to force the outcome. The derivation remains self-contained and independent of the authors' earlier results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard mathematical property that any unitary matrix can be rephased by left and right multiplication by diagonal phase matrices, together with the physical assumption that charged-lepton mixing is nearly diagonal because of the strong mass hierarchy.

axioms (2)
  • standard math Any 3×3 unitary matrix can be transformed into the Fritzsch-Xing form by a suitable choice of rephasing phases that set the arguments of the third row and third column to zero.
    This is the defining property of the FX parametrization invoked in the first sentence of the abstract.
  • domain assumption The charged-lepton diagonalization matrix U^e is nearly diagonal, so that its off-diagonal elements U13^e and U23^e can be set to zero in a perturbative treatment.
    Explicitly stated as the two approximations used to obtain the simplified phase expression.

pith-pipeline@v0.9.0 · 5502 in / 1575 out tokens · 56113 ms · 2026-05-15T22:18:27.345544+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

56 extracted references · 56 canonical work pages · 26 internal anchors

  1. [1]

    Before proceeding to the general situation, let us consider another simplified scenario, in whichU e 23 = 0 is imposed by sacrificing the conditionU ν 13 = 0. The mixing matrix is then given by U=   U e∗ 11 − U e 12U e 33 detU e 0 U e∗ 12 U e 11U e 33 detU e 0 0 0U e∗ 33     U ν 11 U ν 12 U ν 13 U ν 21 U ν 22 U ν 23 U ν 31 U ν 32 U ν 33   =  ...

    work page

  2. [2]

    If|U e 23|is sufficiently small, we can perturbatively expand the expression ofδ FX

    These arguments are functions ofδ ν FX after the rephasing (7). If|U e 23|is sufficiently small, we can perturbatively expand the expression ofδ FX. By separating terms that does not involveU e 23, δFX = arg " U e∗ 33 U ν 31 (U e∗ 33 U ν 32)∗ U e∗ 11 U ν 13 − U e 12U e 33 detU e U ν 23 (U e∗ 12 U ν 13 + U e 11U e 33 detU e U ν 23)∗ detU ν∗ detU e U e∗ 33 ...

    work page

  3. [3]

    Jarlskog, Phys

    C. Jarlskog, Phys. Rev. Lett.55, 1039 (1985)

    work page 1985

  4. [4]

    Wu, Phys

    D.-d. Wu, Phys. Rev. D33, 860 (1986)

    work page 1986

  5. [5]

    Bernabeu, G

    J. Bernabeu, G. C. Branco, and M. Gronau, Phys. Lett. B169, 243 (1986)

    work page 1986

  6. [6]

    Gronau, A

    M. Gronau, A. Kfir, and R. Loewy, Phys. Rev. Lett.56, 1538 (1986)

    work page 1986

  7. [7]

    G. C. Branco and L. Lavoura, Phys. Lett. B208, 123 (1988)

    work page 1988

  8. [8]

    J. D. Bjorken and I. Dunietz, Phys. Rev. D36, 2109 (1987)

    work page 1987

  9. [9]

    J. F. Nieves and P. B. Pal, Phys. Rev. D36, 315 (1987)

    work page 1987

  10. [10]

    F. J. Botella and J. P. Silva, Phys. Rev. D51, 3870 (1995), arXiv:hep-ph/9411288

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  11. [11]

    Rephasing Invariant Parametrization of Flavor Mixing Matrices

    T.-K. Kuo and T.-H. Lee, Phys. Rev. D71, 093011 (2005), arXiv:hep-ph/0504062

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  12. [12]

    E. E. Jenkins and A. V. Manohar, Nucl. Phys. B792, 187 (2008), arXiv:0706.4313

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  13. [13]

    S. H. Chiu and T. K. Kuo, Phys. Lett. B760, 544 (2016), arXiv:1510.07368

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  14. [14]

    P. B. Denton and R. Pestes, JHEP05, 139 (2021), arXiv:2006.09384

    work page arXiv 2021

  15. [15]

    Chau and W.-Y

    L.-L. Chau and W.-Y. Keung, Phys. Rev. Lett.53, 1802 (1984)

    work page 1984

  16. [16]

    Kobayashi and T

    M. Kobayashi and T. Maskawa, Prog. Theor. Phys.49, 652 (1973)

    work page 1973

  17. [17]

    Maximal CP Violation Hypothesis and Phase Convention of the CKM Matrix

    Y. Koide, Phys. Lett.B607, 123 (2005), arXiv:hep-ph/0411280

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  18. [18]

    Maximal CP Violation Hypothesis and a Lepton Mixing Matrix

    Y. Koide and H. Nishiura, Phys. Rev. D79, 093005 (2009), arXiv:0811.2839

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  19. [19]

    P. H. Frampton and X.-G. He, Phys. Lett. B688, 67 (2010), arXiv:1003.0310

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  20. [20]

    On Leptonic Unitary Triangles and Boomerangs

    A. Dueck, S. Petcov, and W. Rodejohann, Phys. Rev. D82, 013005 (2010), arXiv:1006.0227

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  21. [21]

    P. H. Frampton and X.-G. He, Phys. Rev. D82, 017301 (2010), arXiv:1004.3679

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  22. [22]

    Unitarity boomerangs of quark and lepton mixing matrices

    S.-W. Li and B.-Q. Ma, Phys. Lett. B691, 37 (2010), arXiv:1003.5854

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  23. [23]

    A new relation between quark and lepton mixing matrices

    N. Qin and B. Q. Ma, Phys. Lett. B702, 143 (2011), arXiv:1106.3284

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  24. [24]

    The Kobayashi-Maskawa Parametrization of Lepton Flavor Mixing and Its Application to Neutrino Oscillations in Matter

    Y.-L. Zhou, Phys. Rev. D84, 113012 (2011), arXiv:1110.5023

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  25. [25]

    Parametrization of fermion mixing matrices in Kobayashi-Maskawa form

    N. Qin and B.-Q. Ma, Phys. Rev. D83, 033006 (2011), arXiv:1101.4729

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  26. [26]

    A new simple form of quark mixing matrix

    N. Qin and B.-Q. Ma, Phys. Lett. B695, 194 (2011), arXiv:1011.6412

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  27. [27]

    A prediction of neutrino mixing matrix with CP violating phase

    X. Zhang and B.-Q. Ma, Phys. Lett. B713, 202 (2012), arXiv:1203.2906

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  28. [28]

    The $\alpha$, $\beta$ and $\gamma$ parameterizations of CP violating CKM phase

    G.-N. Li, H.-H. Lin, D. Xu, and X.-G. He, Int. J. Mod. Phys. A28, 1350014 (2013), arXiv:1204.1230

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  29. [29]

    On the CP-violating phase $\delta_{\rm CP}$ in fermion mixing matrices

    X. Zhang and B.-Q. Ma, The Universe1, 16 (2013), arXiv:1204.6604

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  30. [30]

    Flavor Symmetries and the Description of Flavor Mixing

    H. Fritzsch and Z.-Z. Xing, Phys. Lett.B413, 396 (1997), arXiv:hep-ph/9707215

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  31. [31]

    Shin, Phys

    M. Shin, Phys. Lett. B160, 411 (1985)

    work page 1985

  32. [32]

    Shin, Phys

    M. Shin, Phys. Lett. B154, 205 (1985)

    work page 1985

  33. [33]

    Lehmann, C

    H. Lehmann, C. Newton, and T. T. Wu, Phys. Lett. B384, 249 (1996)

    work page 1996

  34. [34]

    New Class of Quark Mass Matrix and Calculability of Flavor Mixing Matrix

    K. Kang and S. K. Kang, Phys. Rev.D56, 1511 (1997), arXiv:hep-ph/9704253

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  35. [35]

    The breaking of the flavour permutational symmetry: Mass textures and the CKM matrix

    A. Mondragon and E. Rodriguez-Jauregui, Phys. Rev.D59, 093009 (1999), arXiv:hep-ph/9807214

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  36. [36]

    Maximal Neutrino Mixing and Maximal CP Violation

    H. Fritzsch and Z.-z. Xing, Phys. Rev. D61, 073016 (2000), arXiv:hep-ph/9909304

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  37. [37]

    Four-Zero Texture of Hermitian Quark Mass Matrices and Current Experimental Tests

    H. Fritzsch and Z.-z. Xing, Phys. Lett. B555, 63 (2003), arXiv:hep-ph/0212195

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  38. [38]

    Complete Parameter Space of Quark Mass Matrices with Four Texture Zeros

    Z.-z. Xing and H. Zhang, J. Phys. G30, 129 (2004), arXiv:hep-ph/0309112

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  39. [39]

    Right Unitarity Triangles, Stable CP-violating Phases and Approximate Quark-Lepton Complementarity

    Z.-z. Xing, Phys. Lett. B679, 111 (2009), arXiv:0904.3172. 7

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  40. [40]

    Universal Mass Texture, CP violation and Quark-Lepton Complementarity

    J. Barranco, F. Gonzalez Canales, and A. Mondragon, Phys. Rev.D82, 073010 (2010), arXiv:1004.3781

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  41. [41]

    On the four-zero texture of quark mass matrices and its stability

    Z.-z. Xing and Z.-h. Zhao, Nucl. Phys. B897, 302 (2015), arXiv:1501.06346

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  42. [42]

    M. J. S. Yang, Phys. Lett. B806, 135483 (2020), arXiv:2002.09152

    work page arXiv 2020

  43. [43]

    M. J. S. Yang, Chin. Phys. C45, 043103 (2021), arXiv:2003.11701

    work page arXiv 2021

  44. [44]

    Fritzsch, Z.-z

    H. Fritzsch, Z.-z. Xing, and D. Zhang, Nucl. Phys. B974, 115634 (2022), arXiv:2111.06727

    work page arXiv 2022

  45. [45]

    Awasthi, M

    N. Awasthi, M. Kumar, M. Randhawa, and M. Gupta, Eur. Phys. J. C82, 7 (2022), arXiv:2207.10381

    work page arXiv 2022

  46. [46]

    M. J. S. Yang, Phys. Lett. B868, 139784 (2025), arXiv:2507.04720

    work page arXiv 2025

  47. [47]

    M. J. S. Yang, Nucl. Phys. B1020, 117187 (2025), arXiv:2509.00702

    work page arXiv 2025

  48. [48]

    M. J. S. Yang, (2025), arXiv:2508.02058

    work page arXiv 2025

  49. [49]

    M. J. S. Yang, PTEP2026, 023B01 (2026), arXiv:2508.10249

    work page arXiv 2026

  50. [50]

    M. J. S. Yang, Phys. Lett. B872, 140075 (2026), arXiv:2509.11596

    work page arXiv 2026

  51. [51]

    M. J. S. Yang, Chin. Phys.50, 011002 (2026), arXiv:2508.17866

    work page arXiv 2026

  52. [52]

    M. J. S. Yang, (2025), arXiv:2512.14074

    work page arXiv 2025

  53. [53]

    M. J. S. Yang, (2025), arXiv:2512.20889

    work page arXiv 2025

  54. [54]

    M. J. S. Yang, (2026), arXiv:2601.09389, will be published in PTEP

    work page arXiv 2026

  55. [55]

    M. J. S. Yang, Phys. Lett. B860, 139169 (2025), arXiv:2410.08686

    work page arXiv 2025

  56. [56]

    M. J. S. Yang, Nucl. Phys. B1018, 117092 (2025), arXiv:2504.00365

    work page arXiv 2025