⚛️ phenomenology

Simple Modular $S_3$ Models for Lepton Masses and Mixing

This paper presents a systematic classification and Bayesian analysis of economical modular $S_3$ lepton-flavor models without right-handed neutrinos or extra flavons, identifying 18 viable normal-ordering and 19 inverted-ordering scenarios that successfully fit current neutrino data while offering distinct, testable predictions for absolute neutrino mass scales and CP-violating phases.

Original authors: V. V. Vien, Mayengbam Kishan Singh

Published 2026-09-14

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: V. V. Vien, Mayengbam Kishan Singh

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 set of fundamental particles, but the rules governing how they interact remain one of physics' deepest mysteries. While scientists have mapped out the basic ingredients of matter, the Standard Model of particle physics cannot explain why these particles have the specific masses they do, nor why they mix together in the strange patterns observed in experiments. A particularly puzzling case involves neutrinos, ghostly particles that pass through everything almost unnoticed. For decades, physicists assumed these particles were massless, but experiments have confirmed they possess a tiny, non-zero mass and that they oscillate, or change their identity, as they travel. This discovery proves the Standard Model is incomplete, yet the origin of these masses and the specific way they mix remains unknown. To solve this, researchers often propose new symmetries—hidden rules that dictate how particles behave—but these solutions frequently require adding many new, unseen particles to the theory, making the models complicated and difficult to test.

A team of physicists has now explored a different path, using a mathematical framework called modular symmetry to explain the behavior of neutrinos without inventing a host of new particles. Instead of relying on a large number of extra fields, they utilized a specific, simple symmetry group known as S3, which describes how objects can be rearranged. By applying this symmetry to the known particles, they systematically tested every possible way the math could work to generate the observed neutrino masses and mixing patterns. Their goal was to find the simplest, most economical versions of these theories that still matched the precise data collected from neutrino experiments around the world. They did not just look for one solution; they mapped out the entire landscape of possibilities, checking which configurations could survive the rigorous constraints of current measurements.

The researchers began by organizing the known families of leptons—particles like electrons and neutrinos—according to the rules of this S3 symmetry. They assigned the first family of particles to one category and the second and third families to another, then calculated how these groups could interact to produce mass. They examined twenty-seven distinct mathematical structures, or "textures," that could arise from this setup. For each structure, they ran extensive computer simulations to see if the resulting predictions matched the real-world data, specifically looking at the angles at which neutrinos mix and the differences in their masses. They tested these models against two competing scenarios for how the neutrino masses are ordered: a "normal" order where the lightest is first, and an "inverted" order where the heaviest is first.

The analysis revealed that nature is far more selective than a simple guess might suggest. Out of the twenty-seven possibilities, eighteen were found to be viable for the normal mass ordering, while nineteen worked for the inverted ordering. Remarkably, fourteen of these models were flexible enough to accommodate both scenarios, suggesting that the underlying symmetry is robust. However, four models were ruled out entirely because they could not produce results consistent with any known data, and others were only compatible with one specific ordering. The study identified a particular model, labeled MIII5, as the most statistically favored option, performing well in both the normal and inverted scenarios. This model serves as a prime example of how a simple, elegant mathematical rule can reproduce the complex behavior of the subatomic world without needing to clutter the theory with unnecessary new particles.

Beyond simply finding models that fit the data, the researchers discovered that these theories make sharp, testable predictions about the absolute mass of neutrinos, a value that has not yet been directly measured. The models predict that if the neutrino masses follow the normal ordering, the total mass of the three neutrinos would be relatively low, falling between 59.2 and 112.5 units of mass. If the masses follow the inverted ordering, the total would be significantly higher, ranging from 99.5 to 120.0 units. This difference is crucial because it means that future experiments designed to measure the sum of neutrino masses could definitively tell us which ordering is correct. The predictions also extend to the effective mass of the neutrino as it appears in beta decay and neutrinoless double-beta decay, two processes that are currently being hunted by sensitive detectors. The models suggest that if the inverted ordering is correct, these decay signals should be stronger and easier to detect than if the normal ordering is true.

The study also shed light on the mysterious "phases" that govern how neutrinos change identity, specifically the Dirac CP-violating phase, which determines whether neutrinos and antineutrinos behave differently. The models predict a wide variety of values for this phase, with some scenarios favoring values close to zero and others pointing toward maximum violation. While current experimental data is not yet precise enough to rule out these specific predictions, the range of possibilities narrows the field for future experiments. The researchers found that the models generally predict a higher absolute mass scale for the inverted ordering, making those scenarios more likely to be constrained by cosmological observations of the universe's expansion and more accessible to neutrinoless double-beta decay searches. In contrast, the normal ordering models predict a lower mass scale that sits comfortably within current cosmological limits but might be harder to detect in the near future.

Ultimately, this work demonstrates that a highly economical framework, relying on a simple symmetry and no extra particles, can successfully explain the intricate patterns of lepton flavor. The researchers did not find a single "winner" that explains everything perfectly, but rather a diverse set of viable candidates that are all consistent with current data. The most significant outcome is the clear distinction these models draw between the two possible mass orderings. By predicting specific ranges for the total neutrino mass and the strength of neutrinoless double-beta decay signals, these theories provide a concrete roadmap for the next generation of experiments. As detectors become more sensitive and cosmological data improves, scientists will be able to test these specific predictions, potentially confirming whether the universe follows the normal or inverted path and validating the power of modular symmetry in explaining the fundamental nature of matter.

Technical Summary: Simple Modular S3 Models for Lepton Masses and Mixing

Problem Statement
The Standard Model (SM) successfully describes fundamental particles but fails to account for non-zero neutrino masses and the observed lepton mixing pattern. While non-Abelian discrete flavor symmetries offer a framework to explain these structures, traditional implementations often require extended scalar sectors, additional Abelian symmetries, and numerous flavon fields. These extensions introduce complications in scalar potential minimization and a proliferation of free parameters, reducing predictive power. Modular symmetry has emerged as an economical alternative where Yukawa couplings are determined by modular forms, eliminating the need for auxiliary flavon fields. However, previous applications of the smallest non-Abelian finite modular group, $\Gamma_2 \simeq S_3$ , to lepton flavor have generally relied on enlarged gauge, fermion, or scalar sectors (e.g., introducing right-handed neutrinos or additional symmetries). Furthermore, earlier economical constructions (e.g., Ref. [74]) were guided by minimality rather than an exhaustive classification, potentially missing viable realizations, particularly for the inverted ordering (IO) of neutrino masses. The central problem addressed is whether a systematic, exhaustive exploration of modular $S_3$ models with minimal field content—without enlarging the continuous gauge symmetry, introducing right-handed neutrinos, or adding extra flavon fields—can accommodate both normal (NO) and inverted (IO) neutrino mass orderings.

Methodology
The authors perform a systematic classification and phenomenological analysis of modular $S_3$ lepton-flavor models under strict economy constraints:

Field Content and Assignments: The model utilizes only the SM field content plus the modulus $\tau$ . The first family of left-handed lepton doublets ( $L_1$ ) and right-handed charged leptons ( $l^c_1$ ) are assigned to $S_3$ singlets ($1$), while the second and third families ( $L_\alpha, l^c_\alpha$ ) form an $S_3$ doublet ($2$). The Higgs doublets $H_u$ and $H_d$ are singlets under $S_3$ .
Modular Weight Constraints: The analysis restricts modular weights to $k \leq 6$ . The structure of $S_3$ modular forms (singlets $1, 1'$ and doublet $2$) dictates specific constraints on the sum of modular weights for invariant Yukawa couplings.
Model Classification: By solving the weight constraints for the neutrino sector (Weinberg operator) and the charged-lepton sector, the authors identify four distinct classes of models (Type I–IV) based on the modular weight assignments ( $k_l, k_L, k_r, k_R$ ). This leads to a complete classification of 27 inequivalent models (MI1–MIV7).
Phenomenological Analysis:
- Mass Matrices: Explicit forms for charged-lepton ( $M_l$ ) and neutrino ( $M_L$ ) mass matrices are derived for each model, containing complex coefficients and modular forms $Y(\tau)$ .
- Parameter Scanning: A Bayesian analysis is performed using the Multinest algorithm. Free parameters (Yukawa couplings and the modulus $\tau$ ) are scanned to fit five observables: charged-lepton mass ratios ( $m_e/m_\mu, m_\mu/m_\tau$ ), two mixing angles ( $\sin^2\theta_{12}, \sin^2\theta_{13}$ ), and one mass-squared difference ( $\Delta m^2_{21}$ ).
- Ordering Tests: The analysis is conducted separately for NO and IO. Models are ranked by Bayesian evidence ( $\ln Z$ ) and $\chi^2$ values.
- Predictions: For viable models, predictions are generated for the Dirac CP phase ( $\delta_{CP}$ ), Majorana phases ( $\alpha_{21}, \alpha_{31}$ ), the sum of neutrino masses ( $\sum m_i$ ), the effective electron-neutrino mass ( $m_\beta$ ), and the effective Majorana mass ( $m_{\beta\beta}$ ).

Key Contributions

Exhaustive Classification: The paper provides the first complete classification of all inequivalent modular $S_3$ lepton models with minimal field content (no right-handed neutrinos or extra flavons) and modular weights up to six.
Viability of Inverted Ordering: Contrary to previous minimal studies that found no viable IO solutions, this work identifies 19 viable models for IO, demonstrating that IO is not excluded within the broader class of economical modular $S_3$ frameworks.
Model Counting: Out of 27 total textures, 18 are viable for NO, 19 for IO, and 14 accommodate both orderings. Four models are exclusive to NO, five to IO, and four are excluded by both.
Parameter Correlation Analysis: The study identifies strong parameter degeneracies (e.g., between $\beta_l$ and $\gamma_l$ in the charged-lepton sector) and notes that IO models exhibit systematically more multi-parameter degeneracy than NO models.

Results

Best-Fit Models: The model MIII5 emerges as the highest-evidence realization for both NO and IO, serving as a benchmark.
Absolute Mass Observables: The most significant distinction between NO and IO lies in observables probing the absolute mass scale:
- Sum of Masses ( $\sum m_i$ ): NO predictions range from $\approx 59$ to $112$ meV, while IO predictions range from $\approx 99$ to $120$ meV.
- Effective Electron Mass ( $m_\beta$ ): The NO range ( $8.9–28.5$ meV) and IO range ( $48.9–51.4$ meV) are completely disjoint at the best-fit points.
- Effective Majorana Mass ( $m_{\beta\beta}$ ): NO models predict values from $0.5$ to $25.6$ meV (with some in the "funnel" region $<1$ meV), whereas IO models predict $15.9–50.3$ meV.
Cosmological Constraints: All best-fit solutions satisfy the Planck bound ( $\sum m_i < 120$ meV). However, the more stringent DESI-based constraint ( $\sum m_i < 64$ meV) would retain only a subset of NO models and exclude all IO models.
CP Violation: The models predict a diverse range of $\delta_{CP}$ values. For NO, 17 of 18 viable models fall within the joint T2K–NOvA $3\sigma$ interval. For IO, the predictions are more varied, with some models lying within recent global analysis intervals and others outside, reflecting the current experimental tension in determining $\delta_{CP}$ for IO.
Modulus Space: For the benchmark model MIII5, viable points in the fundamental domain of $\tau$ are concentrated in the interval $1.7 \lesssim \text{Im}(\tau) \lesssim 1.9$ , showing that phenomenological constraints restrict the imaginary part of the modulus more strongly than the real part.

Significance
The paper demonstrates that the modular $S_3$ framework, even with an extremely economical field content and without auxiliary symmetries, accommodates a diverse set of phenomenologically viable lepton-flavor realizations. The study refutes the notion that IO is inherently incompatible with minimal modular $S_3$ models. Crucially, the authors conclude that while oscillation parameters and mixing angles show substantial overlap between NO and IO realizations, observables related to the absolute neutrino-mass scale ( $\sum m_i, m_\beta, m_{\beta\beta}$ ) provide the clearest separation between the two orderings. Consequently, future improvements in cosmological mass bounds, beta-decay endpoint measurements (e.g., Project 8), and neutrinoless double-beta decay searches (e.g., CUPID, LEGEND-1000) will serve as the primary discriminators for testing these specific modular $S_3$ scenarios.

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