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arxiv: 2606.00980 · v1 · pith:MHB6KZ7Cnew · submitted 2026-05-31 · ✦ hep-ph

Rephasing invariant CP phases and sum rules in TM_(1,2) mixing

Masaki J. S. Yang This is my paper

Pith reviewed 2026-06-28 17:12 UTC · model grok-4.3

classification ✦ hep-ph
keywords neutrino mixingCP phasesrephasing invariantsTM1,2 mixingsum rulesPMNS matrixDirac phase
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The pith

The CP phases in TM1,2 neutrino mixing are rephasing invariants that equal the Dirac phase minus arguments of specific PDG matrix elements.

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

The paper shows that the two CP phases appearing in the TM1,2 mixing texture are identical to particular rephasing-invariant combinations built from the elements of the full neutrino mixing matrix. It further derives explicit relations that tie each of these phases to the standard Dirac CP phase through the arguments of two elements taken from the PDG parametrization. The resulting identities are presented as concrete instances of broader sum rules that link physical observables. These steps remove rephasing ambiguity from the TM1,2 phases and place them inside a general framework of phase relations.

Core claim

The CP phases φ1,2 appearing in the TM1,2 mixing are directly identified with rephasing invariants φ1 = -arg[U_e2 U_e3 U_μ1 U_τ1 / U_e1 det U], φ2 = -arg[U_e1 U_e3 U_μ2 U_τ2 / U_e2 det U]. Furthermore, φi = δ - arg[U_μi^0 U_τi^0] among φ1,2, the Dirac CP phase δ and matrix elements in the PDG parametrization U0. These relations of CP phases are interpreted as specific elements of general sum rules among physical quantities.

What carries the argument

The rephasing-invariant combinations of PMNS matrix elements that define φ1 and φ2 in the TM1,2 texture.

If this is right

  • The TM1,2 phases become measurable through rephasing-invariant observables built from the mixing matrix.
  • Explicit links appear between the TM1,2 phases and the Dirac phase δ.
  • The relations constitute concrete cases of general sum rules connecting physical quantities.
  • The identification holds inside any model that enforces the TM1,2 texture.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same rephasing construction could be applied to other trimaximal variants to obtain analogous phase relations.
  • If the sum-rule interpretation generalizes, it may reduce the number of independent CP parameters needed in global fits.
  • The relations supply a direct test of the TM1,2 assumption once all matrix elements are measured to sufficient precision.

Load-bearing premise

The neutrino mixing matrix follows the TM1,2 texture and the PDG parametrization is taken as the reference form.

What would settle it

An experimental extraction of the full set of PMNS matrix elements whose computed values violate the equality φi = δ - arg[U_μi^0 U_τi^0] would falsify the claimed identification.

read the original abstract

We show that the CP phases $\phi_{1,2}$ appearing in the TM$_{1,2}$ mixing are directly identified with rephasing invariants $\phi _{1} = - \arg \left[ { U_{e2} U_{e3} U_{\mu 1} U_{\tau 1} / U_{e 1} \det U } \right]$, $\phi_{2} = - \arg \left[ { U_{e1} U_{e3} U_{\mu 2} U_{\tau 2} / U_{e 2} \det U} \right]$. Furthermore, we demonstrate relations $\phi_{i} = \delta - \arg [ U_{\mu i}^{0} U_{\tau i}^{0} ]$ among $\phi_{1,2}$, the Dirac CP phase $\delta$ and matrix elements in the PDG parametrization $U^{0}$. These relations of CP phases are interpreted as specific elements of general sum rules among physical quantities.

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 claims that the CP phases φ_{1,2} appearing in the TM_{1,2} mixing parametrization of the PMNS matrix U are rephasing invariants, with explicit expressions φ_1 = -arg[U_{e2} U_{e3} U_{μ1} U_{τ1} / U_{e1} det U] and φ_2 = -arg[U_{e1} U_{e3} U_{μ2} U_{τ2} / U_{e2} det U]. It further derives the relations φ_i = δ - arg[U_{μi}^0 U_{τi}^0] linking these phases to the Dirac phase δ in the PDG reference form U^0, and interprets the relations as elements of general sum rules among physical quantities.

Significance. If the algebraic identifications hold, the result supplies rephasing-invariant definitions of the TM_{1,2} phases and direct links to the standard Dirac phase, which can serve as model-independent sum rules connecting CP-violating observables in neutrino oscillations. The derivation is parameter-free and relies only on substitution of the assumed texture into standard rephasing transformations, which is a methodological strength.

minor comments (2)
  1. [Abstract] Abstract and § on relations: the phrase 'specific elements of general sum rules among physical quantities' is stated without an explicit example of a sum rule or reference to prior literature on sum rules; adding one concrete illustration would clarify the interpretation.
  2. Notation: the factor det U appearing in the argument expressions should be explicitly identified as the determinant of the full 3×3 mixing matrix (or clarified if it denotes something else) to avoid ambiguity for readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of our manuscript. We appreciate the recognition that the algebraic identifications, if they hold, provide rephasing-invariant definitions and direct links to the Dirac phase as model-independent sum rules. The referee's summary accurately captures the content of the paper.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central results are algebraic identities obtained by substituting the assumed TM1,2 texture into standard rephasing transformations and the PDG parametrization. These equalities (φ1,2 equal to the listed arg expressions, and φi = δ − arg[U0μi U0τi]) hold identically by construction from the definitions of the matrix elements and phase conventions; no fitted parameters, external uniqueness theorems, or self-citation chains are invoked to force the outcomes. The derivations are self-contained reparametrization steps with no reduction to inputs by definition beyond the explicit premise of the TM1,2 form.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the work relies on standard properties of unitary matrices and rephasing invariance without introducing new fitted parameters or entities.

axioms (1)
  • standard math The neutrino mixing matrix U is unitary, so det U is well-defined and phases can be rephased.
    Invoked implicitly when defining the arg expressions involving det U and rephasing invariants.

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