⚛️ phenomenology

Modular TM $_1$ mixing in light of precision measurement in JUNO

This paper investigates modular $S_4$ symmetry models predicting the TM $_1$ lepton mixing pattern, utilizing JUNO's precision measurements to constrain their parameter spaces and demonstrate how distinct model-building strategies lead to different, distinguishable predictions for the lightest neutrino mass and the effective Majorana mass in neutrinoless double beta decay.

Original authors: Wen-Hao Jiang, Ruiwen Ouyang, Ye-Ling Zhou

Published 2026-10-01

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Wen-Hao Jiang, Ruiwen Ouyang, Ye-Ling Zhou

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

Neutrinos are the most abundant massive particles in the universe, yet they remain among the most elusive. These ghostly particles zip through everything, rarely interacting with the matter around them, and they possess a strange ability to change their identity as they travel. A neutrino born as one type can transform into another, a phenomenon known as oscillation. For decades, physicists have measured the angles that describe how these transformations happen, slowly piecing together a map of the neutrino world. Recently, a massive experiment in China called JUNO began taking data with unprecedented precision, measuring the specific rates at which neutrinos change flavor. This new data is so sharp that it acts as a strict filter, capable of confirming or rejecting the theoretical blueprints scientists have built to explain why neutrinos behave the way they do.

The central mystery driving this research is the pattern of these flavor changes. For a long time, theorists proposed a simple, elegant arrangement called "trimaximal mixing," which suggested that the first column of the mathematical table describing neutrino behavior was fixed and unchanging. This idea, known as TM1 mixing, was a popular candidate because it fit the data reasonably well. However, with JUNO's new, high-precision measurements of the mixing angles, the question is no longer just whether TM1 is possible, but exactly how nature constructs it. The researchers in this study set out to test three different theoretical models that could produce this specific pattern, using the new JUNO data as a rigorous test to see which, if any, could survive.

The scientists focused on a framework called modular symmetry, a mathematical structure that suggests the properties of particles are not random but arise from the geometry of an underlying field. In this view, the universe has a hidden shape, and the way particles interact depends on how they sit within that shape. To explain the TM1 pattern, the team explored three distinct ways this symmetry could break down in the real world. In the first approach, they imagined a scenario where the symmetry breaks in the charged lepton sector (the family of particles that includes electrons) using a specific alignment of fields, while the neutrino sector relies on a fixed point in the modular field itself. In the second approach, they reversed the roles: the modular field sets the pattern for the charged leptons, while the neutrinos require a specific arrangement of auxiliary fields to achieve the correct symmetry. The third approach was the most complex, utilizing two separate modular fields, one for charged leptons and one for neutrinos, which are then linked together by a connecting field to create the desired mixing pattern.

When the researchers ran their calculations against the new JUNO data, the results were decisive. The first model, despite its mathematical elegance, proved to be too rigid. It created a relationship between the neutrino masses and the mixing angles that simply could not match the numbers JUNO observed. The equations forced a contradiction, meaning this specific way of building the universe is ruled out. The other two models, however, fared much better. They possessed enough flexibility to accommodate the new measurements while still preserving the core TM1 structure. These surviving models predict specific relationships between the mass of the lightest neutrino and the effective mass that would be seen in a rare process called neutrinoless double beta decay.

The study also looked at the broader implications of these surviving models by comparing them with cosmological data, which measures the total mass of all neutrinos in the universe. The analysis showed that while the models can still work, they place significant pressure on the idea that neutrinos have an "inverted" mass ordering, where the two heavier neutrinos are nearly equal and the lightest is much lighter. In fact, the data strongly favors the "normal" ordering, where the masses increase in a step-like fashion. For the model that survived the best, the predictions align comfortably with current limits on the total mass of neutrinos, whereas the other surviving model struggles to stay within the tightest cosmological bounds.

Ultimately, this work does not claim to have solved the mystery of neutrino mixing, but it has cleared the field of several dead ends. By applying the new, precise measurements from JUNO, the researchers have demonstrated that only specific ways of arranging the modular symmetry can explain the observed world. The study narrows the search for the correct theory, suggesting that if the TM1 pattern is indeed the key, it is likely realized through a mechanism similar to the second model they tested, where the modular field dictates the charged lepton behavior and auxiliary fields guide the neutrinos. This leaves physicists with a clearer, more constrained path forward, knowing exactly which mathematical structures are still viable candidates for describing the fundamental nature of the universe.

Technical Summary: Modular TM1 Mixing in Light of Precision Measurement in JUNO

Problem Statement
Leptonic flavor mixing remains an unresolved mystery in particle physics. While three mixing angles and two mass-squared differences have been measured, the precise determination of neutrino mass ordering and the Dirac CP phase $\delta$ are ongoing goals. Recent high-precision data from the JUNO experiment, specifically on $\theta_{12}$ and $\Delta m^2_{21}$ , provide stringent constraints on theoretical models. The paper investigates whether the Trimaximal TM1 mixing pattern—a specific ansatz preserving the first column of the Tri-Bimaximal (TBM) mixing matrix—is consistent with these new precision measurements. Furthermore, it seeks to distinguish between different model-building strategies within the framework of modular $S_4$ symmetry that can realize this pattern.

Methodology
The authors employ a modular symmetry approach, where flavor textures arise from modular forms of a modular field $\tau$ , rather than solely from the vacuum expectation values (VEVs) of scalar flavon fields. The study focuses on the finite modular group $\Gamma_4 \simeq S_4$ .

Symmetry Analysis: The paper establishes that TM1 mixing arises when the residual symmetry in the charged lepton sector is $Z_3^T$ and the residual symmetry in the neutrino sector is $Z_2^{SU}$ . This holds true regardless of whether the symmetry breaking is achieved via traditional flavon alignment or modular field fixed points.
Model Construction: Three distinct models are constructed to realize these residual symmetries using modular $S_4$ $S_{4}$ :
- Model A: The charged lepton sector breaks $S_4 \to Z_3^T$ via a flavon alignment, while the neutrino sector preserves $Z_2^{SU}$ via the modular field $\tau_\nu$ stabilizing at the fixed point $\tau_{SU}$ .
- Model B: The charged lepton sector preserves $Z_3^T$ via the modular field $\tau_l$ at the fixed point $\tau_T$ , while the neutrino sector breaks to $Z_2^{SU}$ via flavon alignments.
- Model C: Both sectors utilize distinct modular fields ( $\tau_l$ and $\tau_\nu$ ). A bi-triplet flavon $\Phi$ breaks the product symmetry $S_4^l \times S_4^\nu$ to a diagonal $S_4$ , with $\tau_l$ and $\tau_\nu$ fixed at $\tau_T$ and $\tau_{SU}$ respectively.
Parameterization and Fitting: The authors derive the Majorana mass matrices for each model and re-parameterize them to fit five observables: three mixing angles ( $\theta_{12}, \theta_{13}, \theta_{23}$ ) and two mass-squared differences ( $\Delta m^2_{21}, \Delta m^2_{31}$ ). They perform numerical scans using the latest JUNO and NuFIT 6.0 data to constrain the parameter spaces, specifically examining the lightest neutrino mass ( $m_{\text{lightest}}$ ) and the effective Majorana mass ( $m_{ee}$ ) relevant for neutrinoless double beta decay.

Key Contributions and Results

Validation of TM1 with JUNO Data: The paper confirms that the TM1 mixing pattern is marginally consistent with JUNO's first data release ( $\sin^2 \theta_{12} = 0.3092 \pm 0.0087$ ), particularly when small Renormalization Group (RG) running effects are included. In contrast, the TM2 pattern is shown to be inconsistent with JUNO data even at the $3\sigma$ level.
Exclusion of Model A: Numerical analysis reveals that Model A is over-constrained. The correlation between $\theta_{13}$ and the neutrino mass eigenvalues derived from its specific mass matrix structure cannot simultaneously satisfy the experimental values for $\theta_{13}$ , $\Delta m^2_{31}$ , and $\Delta m^2_{32}$ . Consequently, Model A is excluded unless additional free parameters (via higher modular weights) are introduced, which would reduce its predictivity.
Viability of Models B and C: Models B and C successfully fit the oscillation data. The authors derive analytical sum rules connecting mixing angles and the CP phase $\delta$ for these models.
Predictions for $m_{ee}$ and Cosmology:
- The paper presents correlations between the lightest neutrino mass ( $m_{\text{lightest}}$ ) and the effective Majorana mass ( $m_{ee}$ ) for Models B and C.
- Cosmological Constraints: When compared with cosmological upper bounds on the sum of neutrino masses ( $\sum m_\nu < 0.112$ eV from CMB + DESI + NuFit-6.0), the Inverted Ordering (IO) scenario is strongly disfavored for both models, and almost entirely excluded for Model C. The Normal Ordering (NO) scenario remains allowed, particularly for Model B.
- Model B Superiority: Among the three constructions, Model B is identified as providing the best fit to the data, maintaining compatibility with both oscillation data and cosmological bounds in the NO scenario.

Significance
The paper demonstrates that modular $S_4$ symmetry offers a viable framework for realizing the TM1 mixing pattern, which is currently the most consistent simple mixing ansatz with high-precision JUNO data. By systematically comparing three different realization strategies (varying the use of flavons vs. modular fields in different sectors), the study highlights that while the mixing pattern itself is robustly predicted by the residual symmetries $Z_3^T$ and $Z_2^{SU}$ , the specific model-building choices significantly impact the phenomenological predictions for neutrino masses and the effective Majorana mass. The exclusion of the minimal Model A and the identification of Model B as the most robust candidate provide a refined landscape for future theoretical developments and experimental tests, particularly regarding the neutrino mass ordering and the potential observation of neutrinoless double beta decay.

Technical Summary: Modular TM1 Mixing in Light of Precision Measurement in JUNO

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