Integral representation of the neutrino mass-squared differences

Technical Summary: Integral Representation of Neutrino Mass-Squared Differences

Problem Statement
Determining the absolute neutrino mass scale remains a primary challenge in particle physics. While oscillation experiments have precisely measured the differences between the squares of neutrino masses ($\Delta m^2_{ij}$), these measurements alone are insufficient to determine the individual masses ($m_1, m_2, m_3$) or their ordering (normal vs. inverted). The number of unknown mass parameters exceeds the number of available equations derived from oscillation data. Although cosmological bounds on the sum of neutrino masses ($\Sigma m_\nu$) and direct kinematic measurements (e.g., KATRIN) provide constraints, a rigorous, model-independent analytical framework to relate mass-squared differences directly to absolute masses is needed to further constrain theoretical models and reduce the parameter space.

Methodology
The paper proposes a novel mathematical approach by representing neutrino mass-squared differences as Riemann integrals. The core formalism begins with the function:
$$F(\theta) = \frac{1}{m_2 - m_1 \sin \theta}$$
where $m_1$ and $m_2$ are neutrino mass eigenstates. The author demonstrates that the solar mass splitting, $\Delta m^2_{21}$, is related to the reciprocal of the period average of this function via Cauchy's residue theorem:
$$\frac{1}{2\pi} \int_0^{2\pi} \frac{d\theta}{m_2 - m_1 \sin \theta} = \frac{1}{\sqrt{\Delta m^2_{21}}}$$
This integral representation is treated as an equation for the neutrino masses. To extract practical bounds, the author applies the trapezoidal rule to approximate the integral over 12 uniform subintervals. The error term ($r$) of this approximation is rigorously estimated, noting that for smooth periodic functions, the error decays exponentially, though the presence of a peak near $\theta = \pi/2$ requires careful handling.

The methodology leverages the rich mathematical properties of the Riemann integral to transform the transcendental relationship between masses and oscillation parameters into an algebraic inequality. The analysis assumes normal mass ordering ($m_1 < m_2 < m_3$) and incorporates stringent cosmological bounds on the sum of neutrino masses ($\Sigma m_\nu < 0.0642$ eV) to constrain the error term and derive strict inequalities.

Key Contributions and Results

1. Analytical Mass Bounds: By treating the integral approximation error conservatively, the paper derives a strict inequality linking $m_2$ to $\sqrt{\Delta m^2_{21}}$:
   $$\sqrt{\Delta m^2_{21}} \le m_2 < \frac{31}{30}\sqrt{\Delta m^2_{21}}$$
   From this, analytical upper bounds for the lightest ($m_1$) and heaviest ($m_3$) neutrino masses are derived:
   $$m_1 < \frac{1}{30}\sqrt{61\Delta m^2_{21}}$$
   $$\sqrt{\Delta m^2_{31}} \le m_3 < \sqrt{\Delta m^2_{31} + \frac{61\Delta m^2_{21}}{30^2}}$$

2. Numerical Constraints: Utilizing recent data from the JUNO experiment ($\Delta m^2_{21} = (7.50 \pm 0.12) \times 10^{-5} \text{ eV}^2$) and NuFIT-6.1 global analysis ($\Delta m^2_{31}$), the paper calculates 95% confidence level (C.L.) limits:

   • $m_1 < 0.0023$ eV
   • $0.0085 \text{ eV} \le m_2 < 0.0091$ eV
   • $0.0497 \text{ eV} \le m_3 < 0.0508$ eV
   • Total mass sum: $0.0582 \text{ eV} \le \Sigma m_\nu < 0.0622$ eV

3. Effective Mass Ranges: The derived mass ranges are translated into allowed intervals for the effective electron neutrino mass ($m_{\nu e}$) and the effective Majorana mass ($m_{\beta\beta}$):

   • $0.0085 \text{ eV} < m_{\nu e} < 0.0096$ eV
   • $0 \le m_{\beta\beta} < 0.0057$ eV

4. Massless Neutrino Candidate: The analysis highlights that in the limit of vanishing approximation error, the solution converges to $m_3 = \sqrt{\Delta m^2_{31}}$, implying $m_1 = 0$. This suggests that a massless $\nu_1$ state is a viable candidate for the physical spectrum, consistent with cosmological trends pushing $\Sigma m_\nu$ toward the normal ordering floor.

5. Emergence of Mass Relations: The paper demonstrates that Gatto–Sartori–Tonin type mass relations emerge directly from the integral representation. By applying the first mean value theorem for integrals, the author shows that the neutrino mass spectrum can be resolved by determining a specific geometric angle $\theta'$, reducing the problem of finding masses to evaluating a single parameter.

Significance and Claims
The paper claims that this integral representation provides a "complementary perspective" on oscillation parameters, offering a unique structural insight distinct from standard algebraic or numerical approaches. The primary significance lies in the ability to derive competitive, analytical upper limits for the lightest neutrino mass ($m_1$) without relying on complex global fits, solely by combining integral calculus with current experimental data and cosmological bounds.

The author asserts that the framework is robust and generalizable; while the primary examples focus on $\Delta m^2_{21}$, the method applies equally to $\Delta m^2_{31}$. The approach is presented as a tool to facilitate the determination of absolute neutrino masses by establishing precise relations among them, potentially guiding future theoretical models and aiding in the interpretation of upcoming high-precision data from experiments like JUNO, DUNE, and Hyper-Kamiokande. The paper concludes that the extensive mathematical properties of the Riemann integral make this framework promising for uncovering novel relations among neutrino masses.