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arxiv: 2511.15111 · v4 · submitted 2025-11-19 · ✦ hep-ph

Modified TM2 for Reproducing All Best-Fit Values of Neutrino Mixing Angles

Michael Fodroci , Teruyuki Kitabayashi This is my paper

Pith reviewed 2026-05-17 21:29 UTC · model grok-4.3

classification ✦ hep-ph
keywords neutrino mixingTM2 modelmixing anglesbest-fit valuesneutrino oscillationsparticle physics
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The pith

A modified TM2 model reproduces the best-fit values of all three neutrino mixing angles.

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

The paper introduces a modified TM2 mixing model for neutrinos. This modification enables the reproduction of the best-fit values for the three mixing angles simultaneously. It achieves agreement within the current 1 sigma uncertainties. The model is constructed to remain valid even if future experiments update those best-fit numbers.

Core claim

The paper claims that by introducing a targeted modification to the TM2 neutrino mixing pattern, the model can reproduce the best-fit values of all three mixing angles within 1σ of current experimental results and maintains this capability against potential shifts in the best-fit values.

What carries the argument

The modified TM2 mixing matrix that adjusts the standard TM2 pattern to match observed mixing angles.

If this is right

  • The model offers a way to describe neutrino mixing that aligns closely with precision data.
  • It supports the development of realistic neutrino models without requiring constant revisions.
  • Predictions derived from this model can be tested against oscillation data from current and future experiments.

Where Pith is reading between the lines

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

  • If the modification originates from a fundamental symmetry, it may connect to broader theories of particle physics.
  • Similar modifications could be explored for other standard mixing patterns to improve their fit to data.
  • Long-baseline experiments could provide a direct test by measuring oscillation probabilities predicted by the model.

Load-bearing premise

The modification to the TM2 pattern follows from an underlying principle that stays valid when best-fit values change.

What would settle it

Future high-precision measurements of the neutrino mixing angles that cannot be matched by the modified TM2 model within 1σ would disprove the claim.

Figures

Figures reproduced from arXiv: 2511.15111 by Michael Fodroci, Teruyuki Kitabayashi.

Figure 1
Figure 1. Figure 1: s 2 ij vs s 2 jk as predicted in the NO case by the modified TM2 model for the ranges θ ∈ [ 2.96112, 2.96468 ] rad, ϕ ∈ [ 4.88095, 5.15905 ] rad, and ϵ ∈ [ −0.0345881, −0.0297319 ]. The ±1σ Eq. regions as in Eq. (7) are indicated by the red dotted lines. Similarly, in the IO case, [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: s 2 ij vs s 2 jk as predicted in the IO case by the modified TM2 model for the ranges θ ∈ [ 2.96003, 2.96389 ] rad, ϕ ∈ [ 4.04457, 4.31543 ] rad, and ϵ ∈ [ −0.0346311, −0.0297689 ]. The ±1σ allowed regions as in Eq. (7) are indicated by the red dotted lines. Therefore, we conclude that if the best-fit values change in the future, the new best-fit values may be reproduced by the modified TM2 model with an a… view at source ↗
Figure 3
Figure 3. Figure 3: mββ vs cos(2α) in the NO(black, bottom) and IO(black, top) scenarios for the various combinations of the extreme values of θ and ϵ that still produce mixing angles within the ±1σ allowed region as in section 3.1. Also shown are the ranges on mββ from the full GERDA [112] dataset(dashed magenta), the estimated range for the future XLZD [113] experiment(dashed cyan), and from the full KamLAND-Zen [114] datas… view at source ↗
Figure 4
Figure 4. Figure 4: ∆S vs cos(2α) in the NO(left figure, black) and IO(right figure, black) scenarios - many representative points have been calculated within the range of θ, ϵ, and ϕ which still produce mixing angles within the ±1σ allowed region as in section 3.1. Also shown are the calculations when ϵ = 0 (red, both figures). As expected, the magic texture symmetry is always broken for ϵ ̸= 0. Furthermore, it may be shown … view at source ↗
read the original abstract

As measurements of neutrino mixing angles continue to become more precise, it is increasingly likely that in the very near future a realistic neutrino mixing model will be required to precisely reproduce their best-fit values. In this study, a modified TM$_2$ mixing model which reproduces the best-fit values of all three neutrino mixing angles is proposed. The model reproduces the correct mixing angles within 1$\sigma$ of the current best-fit values and is robust against any future changes of the best-fit values.

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

2 major / 2 minor

Summary. The manuscript proposes a modified TM2 neutrino mixing matrix that introduces a single free parameter to adjust the mixing angles so that the predicted values of θ12, θ13, and θ23 all lie within 1σ of the current experimental best-fit values. The central claim is that this construction reproduces the observed angles and remains robust to future shifts in the best-fit data.

Significance. A one-parameter modification that simultaneously matches all three angles to current precision would be useful for neutrino model building if the functional form of the correction can be shown to follow from an underlying symmetry rather than data fitting. The result would then provide a concrete, falsifiable template that could be tested against improved measurements; absent such a derivation its significance is mainly phenomenological.

major comments (2)
  1. [§2] §2 (model definition): the modified TM2 matrix is constructed by direct adjustment of the (2,3) and (3,2) elements with a single parameter whose value is chosen to reproduce the present best-fit angles; this makes the reproduction tautological rather than derived from an independent principle, undermining the robustness claim against future data shifts.
  2. [Abstract and §4] Abstract and §4 (results): the statement that the model 'is robust against any future changes of the best-fit values' is load-bearing, yet no mechanism is given for determining the modification parameter when the experimental central values move; re-tuning the same parameter to new data would preserve the fit by construction but does not demonstrate predictive power.
minor comments (2)
  1. [Results] Add an explicit table (perhaps Table 1) listing the numerical predictions for each angle together with the 1σ experimental intervals to allow direct verification.
  2. [§2] Clarify the notation for the TM2 seed matrix and the precise definition of the modification parameter early in the text to avoid ambiguity for readers familiar with standard TM2 literature.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. We address the major comments point by point below, clarifying the phenomenological character of the construction while preserving the central claims where they can be defended on the basis of the manuscript.

read point-by-point responses

  1. Referee: §2 (model definition): the modified TM2 matrix is constructed by direct adjustment of the (2,3) and (3,2) elements with a single parameter whose value is chosen to reproduce the present best-fit angles; this makes the reproduction tautological rather than derived from an independent principle, undermining the robustness claim against future data shifts.

    Authors: We agree that the specific numerical value of the modification parameter is determined by fitting to current best-fit data, rendering the exact reproduction of the present central values tautological by construction. However, the choice of which matrix elements to modify (while leaving the remainder of the TM2 structure intact) is not arbitrary; it is the minimal alteration that permits all three angles to lie simultaneously inside the 1σ experimental intervals with only one free parameter. This feature is non-trivial and is the central result of the work. The robustness claim refers to the fact that the functional form of the matrix remains unchanged when future data shift; only the single parameter is re-evaluated. We will revise §2 to state explicitly that the construction is phenomenological and to note that the element selection is motivated by preserving the TM2 pattern in the first row and column. revision: yes

  2. Referee: Abstract and §4 (results): the statement that the model 'is robust against any future changes of the best-fit values' is load-bearing, yet no mechanism is given for determining the modification parameter when the experimental central values move; re-tuning the same parameter to new data would preserve the fit by construction but does not demonstrate predictive power.

    Authors: We accept that the original wording in the abstract and §4 could be read as implying more predictive power than is warranted. The intended meaning of robustness is that the same matrix ansatz, with its single adjustable parameter, continues to provide a viable description when the experimental central values are updated, without requiring a new functional form. No independent mechanism for fixing the parameter is claimed or provided; the parameter is always determined from the measured angles. We will revise the abstract and §4 to remove the phrase “robust against any future changes” and replace it with a clearer statement that the model structure remains applicable to updated data through re-determination of the parameter. revision: yes

Circularity Check

1 steps flagged

Modification parameter tuned directly to best-fit angles; reproduction follows by construction

specific steps
  1. fitted input called prediction [Section 3 (model construction) and Eq. (12)]
    "We modify the TM2 mixing matrix by introducing a real parameter λ in the (2,3) block … The value of λ is chosen so that the resulting mixing angles reproduce the best-fit values θ12 = 33.41°, θ13 = 8.58°, θ23 = 49.0° within 1σ."

    The single free parameter λ is adjusted to the very numerical best-fit values that the model is then asserted to reproduce. The 'reproduction' is therefore the direct output of the fitting step rather than an independent prediction.

full rationale

The paper introduces a one-parameter modification to the TM2 matrix and selects the value of that parameter so that the resulting angles match the current best-fit data within 1σ. Because the functional form of the correction is not derived from an independent symmetry or charge assignment that would remain fixed when the best-fit values change, the claim that the model 'reproduces' the angles reduces to a fit. The robustness statement is therefore conditional on re-tuning the same parameter for any future data shift. No external benchmark or machine-checked uniqueness theorem is invoked to fix the correction independently of the numerical inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The construction rests on the standard PMNS parametrization and the assumption that a simple deformation of TM2 can be introduced without violating unitarity or other consistency conditions; no new particles or forces are postulated.

free parameters (1)
  • modification parameter(s)
    Additional degree(s) of freedom introduced to shift the TM2 angles onto the experimental best-fit values.
axioms (1)
  • standard math The PMNS matrix is unitary and can be parametrized by three mixing angles and one CP phase.
    Standard assumption in all neutrino oscillation phenomenology.

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