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

Masaki J. S. Yang

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

T0 review · 0 major / 2 minor · reviewed 2026-05-15 · grok-4.3

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Pith's one-line read 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

desk verdict This paper offers a practical rephasing map and simplified CP phase expression for the Fritzsch-Xing parametrization under common approximations. read the letter →

arxiv 2602.14513 v2 pith:OQVMNW62 submitted 2026-02-16 hep-ph

Masaki J. S. Yang This is my paper · ORCID

classification hep-ph

keywords Fritzsch-XingparametrizationrephasingtransformationCPphasemixingmatrixneutrinoquarkhierarchicalfermionsPMNS

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

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.

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)

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

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Extended reading notes

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

Load-bearing premise

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

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The main point is that this work provides an explicit rephasing transformation to bring any unitary mixing matrix into the Fritzsch-Xing parametrization and then derives a compact expression for the CP phase δ_FX when the charged lepton mixing is approximated by setting its 1-3 and 2-3 elements to zero. The construction begins with the general rephasing freedom of unitary matrices to make the third row and column real. The authors then separate the phases coming from the charged lepton sector U^e and the neutrino sector. Under U13^e = 0 the phase relation simplifies, and with the extra U23^e = 0 it reduces to δ_FX equal to the intrinsic neutrino FX phase plus the difference in relative phases ρ'_1 minus ρ'_2 from the first two generations. They also give the form for nonzero U23^e as a generalization. This applies to both quark and lepton mixing in the hierarchical limit. What stands out is that the steps are direct and use only standard properties of unitary matrices, without circular definitions or extra assumptions beyond the stated approximations. The result is useful for anyone doing perturbative expansions of mixing angles and phases in flavor models. The approximations are the main limitation, as they are good for strong mass hierarchies but may require higher-order terms when those elements are small but nonzero. The paper handles this by providing the generalized expression, so the central claim holds up. No problems with consistency or unitarity show up. This paper is for phenomenologists in hep-ph who work with the FX parametrization for CKM or MNS matrices. It offers a time-saving formula for CP phase calculations in data analysis and model building. A reader who needs quick analytical expressions will find it practical. It deserves peer review because the derivation is straightforward and the result is new in its compact form. I would send it to a journal for refereeing.

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)

[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.
[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 · score 0.0 of 10

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.

Assumptions & free parameters 0 free parameters · 2 assumptions · 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.

assumptions (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.

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Cite this review

Pith. "Pith review of Rephasing invariant structure of CP phase for simplified mixing matrices in Fritzsch--Xing parametrization." pith.science (2026). https://pith.science/paper/OQVMNW62

@misc{pith2026260214513,
  author       = {Pith},
  title        = {Pith review of: Rephasing invariant structure of CP phase for simplified mixing matrices in Fritzsch--Xing parametrization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OQVMNW62}},
  note         = {Machine review of arXiv:2602.14513}
}

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.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Perturbative determination of CP phase in CKM matrix using rephasing invariants of hierarchical mass matrices and their inverses
hep-ph 2026-07 conditional novelty 6.0 of 10

The CKM CP phase is expressed as a fourth-order rephasing invariant constructed from the down-quark mass matrix and its inverse via perturbative singular value decomposition.
Rephasing invariant CP phases and sum rules in TM$_{1,2}$ mixing
hep-ph 2026-05 unverdicted novelty 5.0 of 10

TM1,2 mixing phases φ1,2 equal specific rephasing-invariant phase combinations of the PMNS matrix and satisfy exact sum rules with the Dirac phase δ.

Reference graph

Works this paper leans on

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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   =  ...

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Pith tools

Reviewed May 15, 2026 · model on record in the stance chip above.

[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