Another relation among the neutrino mass-squared differences?

Problem Statement
Determining the absolute masses of neutrinos remains a fundamental unsolved problem in particle physics. While oscillation experiments have precisely measured the neutrino mass-squared differences ($\Delta m^2_{21}$, $\Delta m^2_{31}$, and $\Delta m^2_{32}$), these observables are insufficient to determine the absolute mass scale or the mass ordering (normal vs. inverted) with high confidence. The number of unknowns (absolute masses $m_1, m_2, m_3$) exceeds the number of available equations derived from oscillation data. Although direct measurement experiments (e.g., KATRIN) and cosmological observations provide upper bounds on the effective electron neutrino mass ($m_{\nu e}$) and the sum of neutrino masses ($\sum m_i$), respectively, they do not yet pinpoint the specific mass values. The paper posits that discovering additional algebraic relations among the mass-squared differences could facilitate the determination of absolute masses and offer insights into the underlying mass-generation mechanism.

Methodology
The author proposes a phenomenological approach inspired by recent global fits of neutrino oscillation parameters. The methodology involves:

1. Empirical Observation: Analyzing the ratio of square roots of the mass-squared differences, defined as $\lambda(\Delta m^2_{21}, \Delta m^2_{31}) = \frac{\sqrt{\Delta m^2_{31}} + \sqrt{\Delta m^2_{21}}}{\sqrt{\Delta m^2_{31}} - \sqrt{\Delta m^2_{21}}}$.
2. Hypothesis Formulation: Noting that substituting best-fit values from recent analyses (e.g., NuFIT-6.1, JUNO data) yields a value extremely close to $\sqrt{2}$ (approximately 1.414), the paper hypothesizes that this ratio is exactly equal to $\sqrt{2}$.
3. Algebraic Derivation: Assuming the exact relation $\frac{\sqrt{\Delta m^2_{31}} + \sqrt{\Delta m^2_{21}}}{\sqrt{\Delta m^2_{31}} - \sqrt{\Delta m^2_{21}}} = \sqrt{2}$, the author derives consequences for the mass spectrum, specifically under the assumption of normal mass ordering ($m_1 < m_2 < m_3$).
4. Numerical Validation: The derived relations are tested against current best-fit values from various global analyses (NuFIT, Capozzi et al., de Salas et al.) to assess consistency within experimental uncertainties.

Key Contributions and Results

• Proposed Algebraic Relation: The paper introduces a simple algebraic relation between $\Delta m^2_{21}$ and $\Delta m^2_{31}$ that approximates $\sqrt{2}$ with high precision (within 0.27% relative error across various datasets).
• Implication of Massless $\nu_1$: The proposed relation implies that the lightest neutrino mass eigenstate is effectively massless ($m_1 \simeq 0$). Under this assumption, the entire mass-squared difference spectrum can be reconstructed from just two masses ($m_2$ and $m_3$).
• Structural Similarity to Koide Formula: The derived relation mirrors the structural form of the Koide formula for charged lepton masses. Specifically, the paper derives $\frac{\Delta m^2_{21} + \Delta m^2_{31}}{(\sqrt{\Delta m^2_{21}} + \sqrt{\Delta m^2_{31}})^2} = \frac{3}{4}$, which is analogous to the Koide sum rule equaling $2/3$.
• Predicted Mass Values: Assuming $m_1 = 0$ and using current experimental inputs (JUNO for $\Delta m^2_{21}$ and NuFIT-6.1 for $\Delta m^2_{31}$), the paper calculates:
  • $m_2 \approx 0.00866$ eV
  • $m_3 \approx 0.05029$ eV
  • $\sum m_i \approx 0.05895$ eV
  • Effective electron neutrino mass $m_{\nu e} \approx 0.00892$ eV
  • Effective Majorana mass $m_{\beta\beta}$ is constrained to the range $0.00141 \text{ eV} \leq m_{\beta\beta} \leq 0.00383$ eV (dependent on unknown Majorana phases).
• Consistency: These results are consistent with current 95% CL constraints from oscillation data, the KATRIN experiment, and cosmological bounds (e.g., DESI).

Significance and Claims
The paper claims that the proposed relation provides a useful algebraic framework for manipulating neutrino oscillation parameters and deriving physically meaningful interpretations. The primary significance lies in the suggestion that the lightest neutrino is massless, a scenario that reduces the number of free parameters in the neutrino sector. The author notes that while measuring an absolute zero mass is experimentally indistinguishable from a very small non-zero mass, the derived results hold for a vanishing $\nu_1$ mass within the order of experimental uncertainty.

The paper frames the emergence of the specific number $\sqrt{2}$ as potentially indicative of underlying quantum physics, such as state normalization or specific mixing angles, though it does not provide a rigorous quantum theoretical derivation. The author emphasizes that the relation serves as a hypothesis suitable for experimental verification. Future high-precision data, particularly from the JUNO detector and upcoming projects like DUNE and Hyper-Kamiokande, are expected to test the validity of this relation and the assumption of a massless lightest neutrino. The paper concludes that while the current precision of oscillation experiments supports the relation, further refinements are necessary to confirm it definitively.