⚛️ phenomenology

A Catalog of Lepton Mass Textures from Koide and Alikhanov Relations

This paper systematically constructs and evaluates 400 viable lepton mass-matrix texture pairs that satisfy the empirical Koide and Alikhanov relations while remaining consistent with experimental mixing data and cosmological mass bounds, thereby predicting specific ranges for the absolute neutrino mass scale and effective Majorana mass.

Original authors: Shaikh Saad

Published 2026-10-21

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CC BY 4.0

Original authors: Shaikh Saad

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The universe is built from a small family of fundamental particles, but among them, the leptons hold a peculiar secret. This group includes the electron, which orbits atoms and powers our technology, and its heavier cousins, the muon and the tau, which appear briefly in high-energy collisions. Alongside them are the ghostly neutrinos, particles so light and elusive that they pass through entire planets without stopping. For decades, physicists have struggled to understand why these particles have the specific masses they do. The Standard Model, our best theory of how the universe works, treats these masses as random numbers that must be measured and plugged in by hand, offering no explanation for their values or the patterns they form. While the charged leptons follow a steep hierarchy where each is much heavier than the last, the neutrinos tell a different story with much smaller mass differences and a tendency to mix and change identities as they travel. Finding a hidden rule that connects these masses could unlock a deeper layer of reality, suggesting that the apparent randomness is actually the result of a precise, underlying mathematical structure.

Two such patterns have recently captured the attention of researchers. The first is a relationship discovered years ago involving the three charged leptons, where a specific combination of their masses and their square roots fits a simple fraction with remarkable precision. The second is a newer observation concerning the neutrinos, linking the differences between their mass-squared values in a way that suggests a clean, geometric connection. These are not just numerical coincidences; they are hints that the mass matrices—the mathematical grids that determine how heavy each particle is—might be constructed from simple, elegant building blocks. A new study by Shaikh Saad at the Jožef Stefan Institute takes these hints seriously, treating them as a blueprint to reconstruct the possible shapes of these mass matrices. The goal was not to guess the answer, but to systematically build every possible version of these matrices that could satisfy these two strict rules and then see if the resulting predictions match the real world.

The researcher began by constructing a catalog of possible structures for the charged leptons. Instead of guessing a single shape, the team explored three distinct categories of mathematical families. These categories were defined by how many fixed directions, or axes, remained unchanged as the shape of the matrix varied. In one category, three directions stayed fixed; in another, only one remained fixed while the others rotated; and in the third, no direction stayed fixed at all. For each of these categories, the team generated specific mathematical forms that perfectly reproduced the known masses of the electron, muon, and tau. This process narrowed down the infinite possibilities of particle physics to a manageable set of concrete candidates that obey the first rule.

Next, the focus shifted to the neutrinos, where the challenge was to find simple, integer-based matrices that satisfied the second rule regarding their mass differences. The researchers performed an exhaustive computer search, testing every possible combination of whole numbers between negative five and five for the entries in the neutrino mass matrix. This brute-force approach, while computationally heavy, ensured that no simple solution was missed. The search identified twelve distinct families of neutrino matrices that worked. Six of these families predicted that the lightest neutrino has zero mass, while the other six predicted that the lightest neutrino has a small, but non-zero, mass. These twelve families represent the complete set of simple, integer-based solutions that fit the observed data for neutrino mass differences.

The final and most complex step was to combine the charged lepton candidates with the neutrino candidates. Since the way these two groups of particles mix to form the observable universe depends on how their internal axes are oriented relative to one another, the team had to test every possible alignment. They rotated the charged lepton structures and paired them with the neutrino families, checking each combination against the measured angles of lepton mixing. This massive combinatorial effort resulted in a list of 400 viable structural pairs. Each of these pairs successfully reproduces the known masses of the charged particles, fits the neutrino mass differences, and predicts the correct mixing angles observed in experiments. The study confirms that a surprisingly rich variety of simple mathematical textures can accommodate all current experimental data, suggesting that the underlying structure of lepton masses is far more flexible than previously thought.

The study also made specific predictions about the absolute masses of the neutrinos, which are currently unknown. For the families where the lightest neutrino has zero mass, the model predicts a specific range for the effective mass relevant to neutrinoless double-beta decay, a rare process that scientists are hunting for in underground laboratories. For the families where the lightest neutrino has a small mass, the model predicts that this mass falls between 16.77 and 17.22 milli-electronvolts. In this scenario, the effective mass for double-beta decay is predicted to be between 4.07 and 18.67 milli-electronvolts. These numbers provide clear targets for future experiments. If a future detector measures a value outside these ranges, it would rule out these specific simple textures. Conversely, if a measurement lands within these windows, it would strongly support the idea that the lepton masses are governed by the simple integer relations and geometric constraints explored in this work.

All the viable solutions found in the study also respect the strict limits set by cosmology and laboratory experiments regarding the total sum of neutrino masses. The models predict a total mass well below the current cosmological upper bound of 0.11 electronvolts, ensuring they are consistent with our understanding of the universe's evolution. However, the study notes that if future cosmological data tightens this bound significantly, as some recent analyses suggest, the branch of solutions where the lightest neutrino has a small mass might be excluded. This leaves the massless neutrino scenario as a robust possibility that survives even stricter constraints. The research demonstrates that by starting with simple empirical relations and systematically exploring the mathematical space, it is possible to map out a landscape of viable theories that connect the microscopic world of particles to the macroscopic structure of the cosmos.

The findings do not prove that the universe is built exactly this way, but they show that such a construction is entirely possible and consistent with every measurement we have today. The existence of 400 distinct families that work suggests that the true theory of lepton masses might be one of these simple structures, waiting to be identified by more precise measurements. Future experiments, particularly those measuring the neutrino mass differences with higher precision, will serve as the ultimate test. As detectors like JUNO gather more data, they will either confirm the tightness of these relations or force physicists to look for more complex explanations. For now, the catalog of textures stands as a comprehensive map of the possibilities, offering a clear path forward for understanding the hidden order behind the lepton masses.

Technical Summary: A Catalog of Lepton Mass Textures from Koide and Alikhanov Relations

Problem Statement
The hierarchical structures of charged-fermion masses and quark mixing angles, as well as the distinct pattern of large lepton mixing angles and neutrino mass-squared differences, remain unexplained within the Standard Model (SM). While flavor symmetries are often invoked, no underlying symmetry has been experimentally established. However, two empirical relations have attracted attention: the Koide relation for charged-lepton masses [1–10] and the recently proposed Alikhanov relation for neutrino mass-squared differences [11–13]. Both exhibit remarkable numerical agreement with experimental data. The central problem addressed is whether specific, structured mass-matrix textures can simultaneously satisfy these empirical relations and reproduce the observed lepton mixing parameters (PMNS matrix) and mass spectra.

Methodology
The author employs a systematic, algebraic approach to construct and enumerate lepton mass textures, avoiding statistical fitting in favor of exact algebraic matching and exhaustive enumeration within bounded integer spaces.

Charged-Lepton Sector:
- The charged-lepton mass matrix is modeled as $M_e = m^* K^2$ , where $K$ is a positive Hermitian matrix.
- To satisfy the Koide relation exactly, $K$ is constrained such that $\text{tr}(K^2)/(\text{tr} K)^2 = 2/3$ .
- The search focuses on one-parameter families of such matrices, $K(\theta) = I + \rho(\cos\theta T + \sin\theta S)$ $K (θ) = I + ρ (cos θ T + sin θ S)$ , classified by the number of mutually orthogonal eigenvector directions common to the generators $T$ $T$ and $S$ $S$ (denoted as $\zeta$ $ζ$ ). Three explicit families are investigated:
  - $K_1$ ( $\zeta=3$ ): Three fixed eigenvector directions (Fourier vectors).
  - $K_2$ ( $\zeta=1$ ): One fixed direction; the other two rotate.
  - $K_3$ ( $\zeta=0$ ): No fixed eigenvector directions.
- Parameters are fixed algebraically to reproduce the central values of the electron, muon, and tau masses.
Neutrino Sector:
- The neutrino mass matrix is modeled as $M_\nu = \mu_\nu A$ , where $A$ is a real symmetric integer matrix with entries in $[-5, 5]$ .
- The Alikhanov relation, $\frac{\sqrt{\Delta m^2_{31}} + \sqrt{\Delta m^2_{21}}}{\sqrt{\Delta m^2_{31}} - \sqrt{\Delta m^2_{21}}} = \sqrt{2}$ , is imposed exactly on the mass-squared differences.
- An exhaustive enumeration of all $11^6$ possible integer matrices is performed. Matrices are filtered by rank (rank-2 for $m_1=0$ and rank-3 for $m_1>0$ ) and the exact Alikhanov spectral condition.
- Redundancies are removed via signed flavor permutations and overall sign/scaling, resulting in 12 distinct texture classes (6 for $m_1=0$ and 6 for $m_1>0$ ).
Combination and Mixing:
- Charged and neutrino textures are combined by applying relative flavor orientations (rotations) and phase alignments.
- The PMNS matrix is constructed as $U_{\text{PMNS}} = U_e^\dagger \Phi W_\nu$ , where $\Phi$ represents relative alignment phases.
- A numerical search scans bounded integer orientation frames ( $B=2$ for free phases, $B=5$ for identity alignment) to find pairs where the resulting mixing angles ( $\sin^2\theta_{12}, \sin^2\theta_{23}, \sin^2\theta_{13}$ ) fall within the experimental $3\sigma$ intervals.
- Predictions for neutrino mass sums ( $\Sigma m_\nu$ ), effective electron-neutrino mass ( $m_\beta$ ), and effective Majorana mass ( $m_{\beta\beta}$ ) are calculated and checked against cosmological and laboratory bounds.

Key Contributions and Results

Systematic Catalog: The study produces a comprehensive catalog of 402 distinct physical prediction families (reducible to 400 representative families) that satisfy the Koide and Alikhanov relations and current mixing data. This significantly extends previous work [14] which presented only two explicit constructions.
Neutrino Mass Predictions:
- The analysis identifies two distinct branches: a massless lightest neutrino scenario ( $m_1=0$ ) and a full-rank scenario.
- For the full-rank branch, the bounded integer prescription predicts a specific range for the lightest neutrino mass: $m_1 \simeq 16.77\text{--}17.22$ meV.
- Effective Majorana masses ( $m_{\beta\beta}$ ) are predicted to be in the range $1.29\text{--}4.03$ meV for $m_1=0$ and $4.07\text{--}18.67$ meV for $m_1>0$ .
Consistency with Constraints: All retained solutions satisfy the adopted cosmological bound ( $\Sigma m_\nu < 0.11$ eV) and the KATRIN bound ( $m_\beta < 0.45$ eV). They also satisfy the KamLAND-Zen limit on $m_{\beta\beta}$ (up to 122 meV). The author notes that a stricter cosmological bound ( $\Sigma m_\nu < 0.072$ eV from DESI) would exclude the full-rank ( $m_1>0$ ) branch.
Structural Diversity: The results demonstrate that a rich landscape of textures exists across all three eigenvector categories ( $\zeta=3, 1, 0$ ) and both neutrino mass branches, capable of accommodating the empirical relations without fine-tuning of continuous parameters beyond the initial algebraic matching.

Significance and Claims
The paper claims to demonstrate that the Koide and Alikhanov relations are not merely numerical coincidences but can be embedded within a broad class of structured mass matrices that are phenomenologically viable. By restricting the search to specific algebraic forms (Hermitian square-roots for charged leptons and bounded integer matrices for neutrinos), the author shows that the observed lepton mixing and mass hierarchies can be reproduced by a discrete set of structural pairs.

The significance lies in the identification of specific, testable predictions:

The existence of a "full-rank" branch with a narrow prediction for $m_1$ ( $\sim 17$ meV).
Specific, low-range predictions for $m_{\beta\beta}$ that fall within the sensitivity reach of future high-sensitivity neutrinoless double-beta decay experiments.
The validation that these empirical relations can coexist with the observed mixing angles, suggesting a potential underlying structure in lepton flavor physics.

The author remains modest, acknowledging that their classification is restricted to the specific search domain (bounded integers, specific generator forms) and does not claim to be the unique solution to lepton flavor. They emphasize that future improvements in precision, particularly from JUNO regarding $\Delta m^2$ measurements and absolute mass experiments, will be necessary to further test the Alikhanov relation and distinguish between the identified spectral branches.

Technical Summary: A Catalog of Lepton Mass Textures from Koide and Alikhanov Relations

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