Integral representation of the neutrino mass-squared differences
Source: arXiv:2607.19340 · Published 2026-07-21 · By I. Alikhanov
TL;DR
This paper addresses the challenging problem of determining the absolute scale of neutrino masses, which remains unresolved despite extensive neutrino oscillation data measuring mass-squared differences. The author proposes a novel integral representation of the neutrino mass-squared differences (Δm²_ij) as a tool to express these parameters through simple definite integrals. This mathematical perspective offers complementary insights and practical utility for bounding the individual neutrino masses. Using recent high-precision data from the JUNO experiment combined with cosmological constraints on the sum of neutrino masses, the paper derives competitive upper limits on the lightest neutrino mass m1 (<0.0023 eV at 95% confidence level) and practical bounds on m2 and m3. Furthermore, the integral framework naturally recovers known neutrino mass relations of the Gatto-Sartori-Tonin type without extra model assumptions. These results validate the integral approach as a valuable method for refining neutrino mass scale estimates given oscillation parameters and cosmological limits.
Key findings
- Integral representation links neutrino mass-squared difference Δm²_21 to a definite integral with rapid convergence properties (Eq. 2.2).
- Analytical bound on lightest neutrino mass: m1 < sqrt(61 * Δm²_21) / 30, yielding m1 < 0.0023 eV at 95% C.L. using JUNO data (Eq. 3.7).
- Bounds on m2 and m3: 0.0085 eV ≤ m2 < 0.0091 eV and 0.0497 eV ≤ m3 < 0.0508 eV (95% C.L.) from oscillation and cosmological data (Eqs. 3.8, 3.9).
- Total neutrino mass constrained to 0.0582 eV ≤ Σm_ν < 0.0622 eV (95% C.L.) consistent with normal ordering (Eq. 3.10).
- Allowed ranges for effective electron neutrino mass m_ν_e (0.0085–0.0096 eV) and effective Majorana mass m_ββ (0–0.0057 eV) derived analytically (Eqs. 3.11, 3.12).
- Integral approach implies massless neutrino state ν1 as a viable spectrum candidate, consistent with current limits and theory (Sec. 3).
- Gatto-Sartori-Tonin type neutrino mass relations emerge from the integral representation and the mean value theorem without model assumptions (Sec. 4).
- The trapezoidal rule with 12 subintervals provides precise integral approximations with errors smaller than 3.8×10⁻¹¹ in relevant mass regimes (Eq. 3.2).
Methodology — deep read
Threat model & assumptions: The paper assumes the standard three-neutrino paradigm with normal mass ordering (m1 < m2 < m3), incorporating precise measured neutrino oscillation parameters (mass-squared differences Δm²_21 and Δm²_31) and stringent cosmological upper bounds on the total neutrino mass sum. It does not consider adversarial scenarios but focuses on mathematical and phenomenological constraints. Uncertainties from experimental measurements and cosmological analyses are folded into confidence limits.
Data: The work uses experimental inputs from the recent JUNO collaboration measuring Δm²_21 = (7.50 ± 0.12)×10⁻⁵ eV², and the NuFIT 6.1 global fit providing Δm²_31 = (2.529 ± 0.021)×10⁻³ eV². Cosmological mass bounds taken include Σm_ν < 0.0642 eV (95% C.L.) from DESI and other cosmological data. Mixing angles and phases are from NuFIT 6.1 for computing effective neutrino masses.
Integral representation and algebraic relations: The core methodology introduces a smooth, 2π-periodic function F(θ) = 1/(m₂² - m₁² sin θ), relating the neutrino mass-squared difference Δm²_21 to the reciprocal of the squared average of F(θ) over [0,2π]. Specifically, the integral equation
(1/2π) ∫₀^{2π} dθ / (m₂² - m₁² sin θ) = 1/√(Δm²_21)
is used as a defining relation connecting masses and oscillation parameters. Similar relations hold for Δm²_31.
The trapezoidal numerical integration rule with 12 uniform intervals approximates the integral to an algebraic form (Eq. 3.1), controlling approximation error via analytical estimates (Eq. 3.2). This yields inequalities constraining individual masses in terms of measured Δm² values.
Training and hyperparameters: Not applicable since this is a theoretical/mathematical analysis rather than a machine learning model.
Evaluation protocol: The approach evaluates bounds for neutrino masses using global fits and oscillation data as inputs, comparing derived analytical inequalities with current experimental confidence intervals. Competitive upper bounds are validated against existing literature results and cosmological constraints. Error bounds from numerical integration are carefully estimated and included conservatively.
Reproducibility: The paper provides explicit formulas and numerical procedures that can be applied by others given publicly available experimental oscillation and cosmological data. Code is not released, but all steps (integral approximation, trapezoidal partitions, inequality derivations) are described in sufficient detail to reproduce.
Concrete example end-to-end: Using the trapezoidal rule with 12 partitions, the integral representation of Δm²_21 is approximated via Eq. (3.1), bounding m₂ in terms of Δm²_21 from JUNO. Propagating inequalities yields m₁ < 0.0023 eV at 95% confidence, an experimentally competitive bound. Corresponding effective neutrino masses m_ν_e and m_ββ ranges are then derived using mixing angles from NuFIT, demonstrating the practical utility of the integral approach in constraining the absolute neutrino mass scale from current data.
Technical innovations
- Proposed a novel integral representation of neutrino mass-squared differences Δm²_ij as reciprocals of integrals over smooth periodic functions, linking oscillation parameters directly to neutrino masses in an analytically tractable form.
- Derived rigorous analytical bounds for individual neutrino masses (m1, m2, m3) based on integral approximations and current oscillation plus cosmological data, improving or validating existing upper limits.
- Showed that classical neutrino mass relations of Gatto-Sartori-Tonin type arise naturally from the integral representation via the first mean value theorem, eliminating reliance on model assumptions.
- Identified the configuration with a massless neutrino state ν1 as a mathematically justified physical candidate emerging from the integral formalism and consistency with data.
- Demonstrated that standard numerical methods (trapezoidal rule with a modest number of subdivisions) yield highly precise integral approximations enabling sharp mass bounds with quantified, controllable errors.
Datasets
- JUNO experiment neutrino oscillation data — Δm²_21 = (7.50 ± 0.12)×10⁻⁵ eV² — public
- NuFIT 6.1 global neutrino oscillation fit — Δm²_31 = (2.529 ± 0.021)×10⁻³ eV² plus mixing angles — public
- Cosmological neutrino mass bounds (e.g., DESI) — Σm_ν < 0.0642 eV (95% C.L.) — public
Baselines vs proposed
- Current cosmological bound on sum of neutrino masses Σm_ν < 0.0642 eV (95% C.L.) vs integral approach constrained total mass range 0.0582 eV ≤ Σm_ν < 0.0622 eV (95% C.L.).
- KATRIN effective electron neutrino mass limit m_ν_e < 0.45 eV (90% C.L.) vs integral method derived effective mass range 0.0085 eV < m_ν_e < 0.0096 eV (95% C.L.).
- Previous theoretical predictions (e.g., Fritzsch & Xing 2006) m1 ≃ √0.2 Δm²_21 vs integral bound m1 < 0.0023 eV (95% C.L.) consistent and competitive.
- Trapezoidal numerical integration error < 3.8 × 10⁻¹¹ in estimating integral for Δm²_21, ensuring precise analytical relations (Eq. 3.2).
Limitations
- Analysis assumes normal neutrino mass ordering; although extension to inverted ordering is stated as straightforward, it is not explicitly developed or evaluated.
- No assessment of possible impacts from unknown CP-violating phases and Majorana phases beyond current approximations in effective mass calculations.
- Integral approximations use the trapezoidal rule with fixed partitioning; more sophisticated numerical and functional choices might yield tighter bounds but are unexplored.
- Results depend on cosmological bounds that are model-dependent and have some variation across analyses; relaxing these bounds could weaken mass constraints significantly.
- No explicit treatment or testing under distribution shifts such as new oscillation data or future revised cosmological parameters.
- No public release of computational tools or code accompanying the integral and numerical analyses, which may challenge direct reproducibility despite detailed formulas.
Open questions / follow-ons
- Can the integral representation and associated analytical methods be extended explicitly and validated for inverted neutrino mass ordering?
- How would including complex CP-violating and Majorana phases quantitatively impact the derived analytical bounds on effective masses and mass relations?
- Could alternative integral kernels or numerical integration schemes produce significantly improved or tighter neutrino mass constraints?
- Is it possible to leverage the integral representation framework to directly constrain or measure the unknown geometric parameter (angle θ′) that determines the full neutrino mass spectrum?
Why it matters for bot defense
While not directly related to bot-defense or CAPTCHA systems, this paper provides an interesting example of leveraging integral representations to transform physical parameter constraints into analytically tractable bounds. For bot-defense engineers, this work illustrates the value of reframing hard-to-estimate parameters into integral or functional forms that allow rigorous bounding and error control. Similarly, the use of mathematical theorems (e.g., the mean value theorem) to uncover hidden parameter relations without model assumptions can inspire methods that detect or estimate behavioral patterns from noisy or incomplete data. Although focused on neutrino physics, the approach exemplifies rigorous, mathematically grounded reasoning that can inform robust feature extraction or model interpretability techniques in security contexts.
Cite
@article{arxiv2607_19340,
title={ Integral representation of the neutrino mass-squared differences },
author={ I. Alikhanov },
journal={arXiv preprint arXiv:2607.19340},
year={ 2026 },
url={https://arxiv.org/abs/2607.19340}
}