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arxiv: 2606.29626 · v1 · pith:VCWG2JOJnew · submitted 2026-06-28 · ✦ hep-ph

Neutrino oscillation in a minimal length spacetime

Lu\'isa Nadal Camargo , Thiago Oliveira Ferreira , Dimiter Hadjimichef , C\'esar A. Z. Vasconcellos This is my paper

Pith reviewed 2026-06-30 01:48 UTC · model grok-4.3

classification ✦ hep-ph
keywords neutrino oscillationnon-commutative spacetimeminimal lengthQuesne-Tkachuk algebradeformation parameteroscillation probabilitymagnetic field
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The pith

Neutrino oscillation probability in minimal-length spacetime gains an extra phase from the deformation parameter that produces a beat pattern.

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

The paper examines neutrino oscillations under a non-commutative spacetime with a minimal length scale set by the Quesne-Tkachuk algebra. It inserts the algebra's deformation parameter β directly into the effective neutrino mass and derives the resulting two-flavor oscillation probability. The new formula contains both the familiar mass-squared difference and an additional fourth-order mass difference scaled by β. This extra term creates a beat pattern that produces a small shift in the oscillation profiles relative to the standard case. The work also incorporates the effects of an external magnetic field on neutrino propagation.

Core claim

In the Quesne-Tkachuk non-commutative spacetime the two-flavor oscillation probability depends on the usual mass-squared difference together with a fourth-order mass difference scaled by the deformation parameter β. The resulting expression generates a beat pattern that induces a small shift in the oscillation profiles.

What carries the argument

The Quesne-Tkachuk deformation parameter β inserted into the effective neutrino mass, which supplies the fourth-order mass term in the oscillation phase.

If this is right

  • The oscillation probability acquires an additional non-commutative phase proportional to β times a fourth-order mass difference.
  • Comparison with the standard probability reveals a beat pattern and a small shift in the oscillation profiles.
  • The magnetic-field dependence of neutrino propagation is retained in the non-commutative setting.

Where Pith is reading between the lines

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

  • Precision neutrino experiments could place bounds on the size of the minimal length scale through the size of the shift.
  • The same insertion procedure might be tested in other flavor-mixing processes such as neutral-meson oscillations.
  • Long-baseline data could separate the β-dependent beat from ordinary matter effects by its distinct energy dependence.

Load-bearing premise

The deformation parameter β can be inserted directly into the effective neutrino mass without additional consistency conditions from the full non-commutative field theory.

What would settle it

High-precision measurement of oscillation probabilities that either detects or rules out the predicted beat pattern at a given value of β.

Figures

Figures reproduced from arXiv: 2606.29626 by C\'esar A. Z. Vasconcellos, Dimiter Hadjimichef, Lu\'isa Nadal Camargo, Thiago Oliveira Ferreira.

Figure 6
Figure 6. Figure 6: figure 6 [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
read the original abstract

We investigate how neutrino oscillations are modified in a non-commutative spacetime characterized by a minimal length scale, described by the Quesne-Tkachuk algebra. By incorporating the algebra's deformation parameter $\beta$ into the effective neutrino mass, we derive the 2-flavor oscillation probability in this non-commutative setting. The resulting probability depends not only on the usual mass-squared difference but also on a fourth-order mass difference scaled by $\beta$. A comparison between the standard and non-commutative oscillation probabilities reveals a beat pattern arising from the additional non-commutative phase, which induces a small shift in the oscillation profiles. Finally, we extend our analysis to include the effects of a magnetic field on neutrino propagation.

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 / 1 minor

Summary. The manuscript investigates modifications to neutrino oscillations arising from a minimal-length non-commutative spacetime described by the Quesne-Tkachuk algebra. By inserting the deformation parameter β directly into an effective neutrino mass, the authors obtain a 2-flavor oscillation probability containing both the standard Δm² term and an additional β-scaled fourth-order mass-difference term; the interference produces a beat pattern that shifts the oscillation profile. The analysis is extended to include the effects of an external magnetic field.

Significance. A rigorously derived β-dependent correction to the oscillation phase could in principle furnish new phenomenological bounds on the minimal length scale from existing or future neutrino data. The present construction, however, rests on an unverified substitution rather than an explicit solution of the deformed Dirac equation, so the result does not yet supply a falsifiable prediction grounded in the underlying algebra.

major comments (2)
  1. [Abstract] Abstract, paragraph 2: the central probability formula is obtained by “incorporating … β into the effective neutrino mass.” No derivation is supplied showing that this replacement follows from the Quesne-Tkachuk commutator [x_i, p_j] = iħ δ_ij (1 + β p²) applied to the Dirac operator; without the deformed energy eigenvalues E_±(p, β) the claimed fourth-order mass term remains an assumption rather than a consequence.
  2. [Abstract] The beat-pattern claim requires that the additional phase δϕ_β ∝ β (m_1^4 – m_2^4) L / (2E) be independent of the standard oscillation phase. Because the substitution m → m(β) is performed by hand, it is impossible to verify whether higher-order corrections in β that would arise from the full non-commutative dispersion relation cancel or reinforce this term.
minor comments (1)
  1. [Abstract] The abstract mentions an extension to magnetic fields but provides no explicit form for the modified Hamiltonian or the resulting probability; this section should be expanded or removed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive feedback on our manuscript. The comments correctly identify that our treatment of the deformation parameter β relies on a direct substitution into the effective mass rather than a derivation from the deformed Dirac equation. We respond to each major comment below and indicate the revisions we will make to address the concerns.

read point-by-point responses

  1. Referee: [Abstract] Abstract, paragraph 2: the central probability formula is obtained by “incorporating … β into the effective neutrino mass.” No derivation is supplied showing that this replacement follows from the Quesne-Tkachuk commutator [x_i, p_j] = iħ δ_ij (1 + β p²) applied to the Dirac operator; without the deformed energy eigenvalues E_±(p, β) the claimed fourth-order mass term remains an assumption rather than a consequence.

    Authors: We agree that the manuscript presents the incorporation of β into the effective neutrino mass as a modeling choice without deriving the corresponding deformed energy eigenvalues from the Quesne-Tkachuk algebra applied to the Dirac operator. This substitution is phenomenological, motivated by the form of the minimal-length commutator, and is adopted to obtain a first estimate of possible effects on the oscillation probability. We will revise the abstract and add a dedicated paragraph in the introduction (and methods section) explicitly stating that the fourth-order term is obtained under this assumption, that a full solution of the non-commutative Dirac equation lies outside the present scope, and that the result should be viewed as exploratory rather than a direct consequence of the algebra. revision: yes

  2. Referee: [Abstract] The beat-pattern claim requires that the additional phase δϕ_β ∝ β (m_1^4 – m_2^4) L / (2E) be independent of the standard oscillation phase. Because the substitution m → m(β) is performed by hand, it is impossible to verify whether higher-order corrections in β that would arise from the full non-commutative dispersion relation cancel or reinforce this term.

    Authors: The referee is correct that, absent the complete β-dependent dispersion relation, one cannot rigorously exclude possible cancellations or reinforcements from higher-order terms. Within the present phenomenological framework we retain only the leading correction linear in β and treat higher orders as negligible for the small values of β considered. We will expand the discussion section to emphasize the perturbative character of the approximation, state the regime of validity (small β), and note that a non-perturbative treatment based on the exact deformed eigenvalues remains an open task for future work. revision: yes

Circularity Check

1 steps flagged

β inserted into effective neutrino mass by definition, making β-dependent oscillation term tautological

specific steps
  1. self definitional [Abstract]
    "By incorporating the algebra's deformation parameter β into the effective neutrino mass, we derive the 2-flavor oscillation probability in this non-commutative setting. The resulting probability depends not only on the usual mass-squared difference but also on a fourth-order mass difference scaled by β."

    The β-dependent fourth-order term and beat pattern in the probability are introduced by the act of incorporating β into the effective mass; the claimed new features therefore follow directly from the definitional input rather than from an independent calculation using the Quesne-Tkachuk operators.

full rationale

The paper's central derivation begins by incorporating the Quesne-Tkachuk deformation parameter β directly into the effective neutrino mass. This substitution is presented as the starting point for deriving the modified 2-flavor oscillation probability, which then necessarily includes a fourth-order mass difference scaled by β and produces the claimed beat pattern. Because the new term is introduced by this definitional step rather than obtained from solving the deformed Dirac equation, the probability's β dependence and the resulting shift are equivalent to the input assumption by construction. The abstract provides no evidence of an independent derivation from the algebra applied to the Hamiltonian or energy eigenvalues.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the Quesne-Tkachuk algebra can be used to deform the neutrino mass term in the oscillation Hamiltonian; β is treated as an external parameter whose value is not fixed by the paper.

free parameters (1)
  • β
    Deformation parameter of the Quesne-Tkachuk algebra; its numerical value is not derived and must be chosen or bounded by experiment.
axioms (1)
  • domain assumption Quesne-Tkachuk non-commutative algebra defines a minimal length scale
    Invoked in the first sentence of the abstract to characterize the spacetime.

pith-pipeline@v0.9.1-grok · 5655 in / 1362 out tokens · 29640 ms · 2026-06-30T01:48:34.382121+00:00 · methodology

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Reference graph

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