⚛️ phenomenology

A $\Gamma_{3}$ modular symmetric approach for two-zero textures in left-right symmetric model

This paper proposes a predictive left-right symmetric model utilizing $A_4$ modular symmetry without flavon fields to systematically construct and analyze all seven two-zero neutrino mass textures, identifying specific classes that successfully accommodate current neutrino data, reproduce the baryon asymmetry, and yield testable predictions for neutrinoless double beta decay.

Original authors: Ankita Kakoti, Happy Borgohain

Published 2026-08-11

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Ankita Kakoti, Happy Borgohain

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

Imagine the universe as a giant, cosmic orchestra. For decades, physicists have been trying to write the sheet music for this orchestra using a rulebook called the Standard Model. It's a brilliant book that explains how most of the particles in the universe play their parts, from the electrons buzzing in your phone to the protons in your body. But there's a problem: the music sheet has a few missing notes. The biggest mystery is the neutrino, a ghostly particle that zips through everything without bumping into much. We know they exist, and we know they have a tiny, non-zero mass, but the Standard Model's book says they should be massless. It's like the sheet music says the drummer is silent, but we can clearly hear a faint tap.

To fix this, scientists are looking for a "Beyond Standard Model" theory—a new, expanded rulebook. One of the most popular ideas is that neutrinos get their mass through a "seesaw" mechanism, where heavy, invisible partners pull the light ones down, making them heavy in a way that looks tiny to us. But this theory has a lot of free knobs to turn, making it hard to predict exactly what the music should sound like. To tighten the screws, physicists use "flavor symmetries," which are like strict conductors telling the particles how to arrange themselves. Recently, a new kind of conductor called "modular symmetry" has become a hot topic. It's a mathematical rule that forces the particles to fit together in specific patterns without needing extra, messy ingredients. This paper dives deep into using this new conductor to see if it can explain the specific, weird patterns we see in neutrino data.

The Paper's Mission: Tuning the Cosmic Orchestra

This paper is a detective story set in the world of particle physics. The authors, Ankita Kakoti and Happy Borgohain, are investigating a specific theory called the Left-Right Symmetric Model (LRSM). Think of this model as a grand, symmetrical building where the "left" side (our familiar world) and the "right" side (a hidden, heavier world) are mirror images. In this building, neutrinos get their mass from two sources at once: a "Type-I" seesaw (involving heavy right-handed partners) and a "Type-II" seesaw (involving heavy triplets of particles).

The big question the authors ask is: Can we use the new "modular symmetry" conductor to force the neutrino mass matrix (the mathematical map of how heavy the neutrinos are) to have exactly two zeros? In math land, a "zero" in a matrix is like a missing instrument in the orchestra—it means a specific connection between two particles simply doesn't exist. There are 15 possible ways to arrange two zeros in a 3x3 grid, but only seven of those arrangements actually match the real-world data we've collected from experiments like Super-Kamiokande and T2K. The authors wanted to see if their specific version of the Left-Right model, guided by modular symmetry, could naturally produce all seven of these allowed "two-zero" patterns without needing to invent any extra, messy particles (called flavons) to make it work.

The Investigation: Weighing the Particles

To solve this, the authors treated the "modular weight" of the particles like different weights on a scale. In this theory, every particle has a weight, and for the math to work, the weights in any interaction must add up to zero. The team tried three specific weight settings: 4, 8, and 10. They systematically swapped the charges and weights of the particles (like swapping the seats of the violinists and cellists) to see which combinations would naturally result in those seven allowed two-zero patterns.

Here is what they found:

The Weight 4 Experiment: When they set the weight to 4, they successfully generated two of the allowed patterns, called Class B1 and Class B2.
- Class B2 looked promising at first, but when they checked the "Dirac CP phase" (a number that tells us if neutrinos and anti-neutrinos behave differently, which is crucial for understanding why the universe has more matter than antimatter), it failed. Specifically, for the "inverted hierarchy" (a specific ordering of neutrino masses), no data points satisfied the strict 3-sigma range (a high standard of certainty). It also struggled with the solar mixing angle.
- Class B1 was better at matching the mixing angles, but it also failed to produce any viable data points for the CP phase. It seemed to force the atmospheric mixing angle into a very narrow range, particularly for the inverted hierarchy.
The Weight 8 Experiment: Cranking the weight up to 8 allowed them to find Class B4, Class B3, Class A2, and Class A1.
- Class B4 managed to fit the mixing angles but, like the others, couldn't find a happy spot for the CP phase. It did, however, successfully narrow down the total sum of neutrino masses to a range between 0.05 eV and 0.12 eV.
- Class A2 was a bust; it couldn't satisfy the constraints for the solar mixing angle.
- Class A1 was a mixed bag. It satisfied the bounds for all three oscillation parameters, but the results for the CP phase only worked for the "normal hierarchy" (the other ordering of masses). Interestingly, while the best-fit value for the solar mixing angle in the normal hierarchy fell outside the 3-sigma range, the inverted hierarchy showed satisfactory results. The authors note that recent data from the JUNO experiment puts a stringent bound on the solar mixing angle, making the inverted hierarchy results for Class A1 favorable, whereas the normal hierarchy best-fit is currently outside the 3σ range.
The Weight 10 Experiment: Finally, at weight 10, they uncovered Class C. This one was a star. All three neutrino oscillation parameters satisfied the 3-sigma bounds, and the CP phase results were favorable for the normal hierarchy.

The Aftermath: Baryon Asymmetry and Double Beta Decay

The authors didn't just stop at the mass patterns; they checked if these patterns could explain two other huge cosmic mysteries: why the universe is made of matter instead of antimatter (baryon asymmetry) and a rare process called neutrinoless double beta decay (0νββ).

Baryon Asymmetry: To explain why we exist, the universe needed a process called "resonant leptogenesis." The authors found that this mathematical process only worked for Classes B1, B2, and C. For the other classes, the math simply didn't add up to produce the observed amount of matter.
Neutrinoless Double Beta Decay: This is a hypothetical process where a nucleus decays by emitting two electrons but no neutrinos. The authors calculated the "effective mass" for this process for all seven classes. They found that all seven classes produced results that were consistent with current experimental limits (specifically data from KamLAND-Zen), meaning none of them were ruled out by this test.

The Verdict

In the end, this paper suggests that the Left-Right Symmetric Model, when guided by modular symmetry, is a very strong candidate for explaining the universe's neutrino secrets. It successfully generated all seven allowed two-zero texture classes without needing extra, messy particles. However, it also ruled out several specific scenarios. For instance, Class B2 and Class A2 were effectively ruled out because they couldn't match the data for the solar mixing angle or the CP phase. Class B1 and Class B4 were also constrained, failing to produce viable CP phase data.

The study highlights that Class C and Class A1 (specifically for the inverted hierarchy) are the most promising, as they satisfy the most experimental bounds. Class C stands out for satisfying all bounds for the normal hierarchy, while Class A1 shows strong potential when the inverted hierarchy is considered, despite the normal hierarchy best-fit falling outside the 3σ range. The authors conclude that this framework makes specific, testable predictions for future experiments like DUNE and Hyper-Kamiokande. If future experiments measure the neutrino mass sum or the CP phase and find values that match the predictions for Class C or the inverted hierarchy of Class A1, it would be a massive win for this theory. If they find values that match the ruled-out classes, this specific version of the theory would need to be rewritten. For now, the modular symmetry approach remains a compelling, predictive, and elegant way to tune the cosmic orchestra.

Technical Summary: A $\Gamma_3$ Modular Symmetric Approach for Two-Zero Textures in Left-Right Symmetric Model

Problem Statement
The Standard Model (SM) fails to explain the origin of neutrino masses, flavor mixing, the baryon asymmetry of the universe (BAU), and the nature of dark matter. While the Left-Right Symmetric Model (LRSM) provides a mechanism for neutrino mass generation via the seesaw mechanism and offers a pathway for leptogenesis, it lacks a predictive framework for the flavor structure of the Yukawa sector. Traditional discrete flavor symmetries often require the introduction of numerous "flavon" fields to constrain mass matrices, increasing the number of free parameters. This work addresses the need for a predictive framework that simultaneously explains neutrino mass textures, mixing parameters, and cosmological observables without relying on extra flavon fields.

Methodology
The authors investigate the realization of all seven phenomenologically viable two-zero neutrino mass textures within a generic LRSM augmented by $\Gamma_3$ modular symmetry (isomorphic to the non-abelian discrete group $A_4$ ).

Model Framework: The model utilizes the gauge group $SU(3)_C \otimes SU(2)_R \otimes SU(2)_L \otimes U(1)_{B-L}$ . The scalar sector includes a Higgs bidoublet $\phi$ and two scalar triplets $\Delta_L$ and $\Delta_R$ . Spontaneous symmetry breaking occurs in two steps, generating light neutrino masses as a sum of Type-I and Type-II seesaw contributions ( $M_\nu = M_\nu^I + M_\nu^{II}$ ).
Modular Symmetry Implementation: Instead of introducing flavon fields, Yukawa couplings are promoted to modular forms $Y(\tau)$ dependent on the modulus $\tau$ . The authors assign specific modular weights ( $k$ ) and $A_4$ charges to the lepton doublets ( $L_L$ ), right-handed leptons ( $L_R^c$ ), and scalar fields.
Texture Construction: The study systematically constructs mass matrices by varying the modular weights of the Yukawa forms. The authors focus on modular weights $k_Y = 4, 8,$ and $10$. By assigning these weights to the particle content, they derive specific structures for the Dirac mass matrix ( $M_D$ ), the right-handed Majorana mass matrix ( $M_R$ ), and the left-handed Majorana mass matrix ( $M_L$ ).
Phenomenological Analysis: The resulting light neutrino mass matrices are diagonalized to extract oscillation parameters (mixing angles $\theta_{12}, \theta_{23}, \theta_{13}$ $θ_{12}, θ_{23}, θ_{13}$ and CP phase $\delta_{CP}$ $δ_{C P}$ ). The viability of each texture is tested against:
- Current neutrino oscillation data (NuFIT).
- Cosmological constraints on the sum of neutrino masses ( $\Sigma m_\nu$ ) from Planck.
- Resonant Leptogenesis (RL) requirements for generating the observed Baryon Asymmetry of the Universe (BAU).
- Effective Majorana mass ( $m_{\beta\beta}$ ) predictions for neutrinoless double beta decay ( $0\nu\beta\beta$ ), including new physics contributions from heavy right-handed neutrinos and scalar triplets in the LRSM context.

Key Contributions

Flavon-Free Construction: The paper demonstrates that all seven allowed two-zero textures (Classes $A_1, A_2, B_1, B_2, B_3, B_4,$ and $C$ ) can be realized in the LRSM using only modular symmetry and specific weight assignments, eliminating the need for additional flavon fields.
Systematic Classification: The authors explicitly derive the superpotentials and mass matrices for each texture class using modular weights $k=4, 8,$ and $10$.
Comprehensive Viability Check: The study provides a detailed phenomenological analysis of each texture class against current experimental bounds, identifying which classes are compatible with normal and inverted mass orderings.

Results
The analysis yields the following specific findings for the seven texture classes:

Class $B_2$ ( $k=4$ ): Compatible with oscillation data for normal ordering (NO) but fails for inverted ordering (IO) regarding the Dirac CP phase ( $\delta_{CP}$ ) and solar mixing angle ( $\sin^2\theta_{12}$ ).
Class $B_1$ ( $k=4$ ): Satisfies bounds for mixing angles but fails to produce viable data points for $\delta_{CP}$ in either ordering. It constrains the atmospheric mixing angle, particularly in the IO regime.
Class $B_4$ ( $k=8$ ): While oscillation parameters fall within NuFit ranges, no satisfactory data points are found for $\delta_{CP}$ . It successfully constrains the sum of neutrino masses between 0.05 eV and 0.12 eV.
Class $B_3$ ( $k=8$ ): Similar to $B_4$ , it fails to satisfy constraints for $\delta_{CP}$ .
Class $A_2$ ( $k=8$ ): Fails to satisfy the current $3\sigma$ constraints on $\sin^2\theta_{12}$ .
Class $A_1$ ( $k=8$ ): All three oscillation parameters satisfy NuFit bounds. However, results for $\delta_{CP}$ are favorable only for NO. For $\sin^2\theta_{12}$ , the model best-fit for NO lies outside the $3\sigma$ range, suggesting the results are more favorable for IO.
Class $C$ ( $k=10$ ): All three oscillation parameters satisfy $3\sigma$ bounds. Results for $\delta_{CP}$ are favorable for NO.

Phenomenological Implications:

Leptogenesis: Resonant leptogenesis calculations indicate that only Classes $B_1, B_2,$ and $C$ yield baryon asymmetry parameters ( $\eta_B$ ) consistent with the Planck 2018 bound.
Neutrinoless Double Beta Decay: All seven texture classes yield effective Majorana mass values ( $m_{\beta\beta}$ ) that are consistent with current experimental limits (KamLAND-Zen) and future sensitivity projections.

Significance
The paper claims that the modular symmetric LRSM framework provides a simple and predictive mechanism for understanding neutrino flavor. By successfully generating all seven allowed two-zero textures without flavon fields, the model reduces the number of free parameters while maintaining consistency with neutrino oscillation data, cosmological bounds, and baryogenesis requirements. The framework produces correlated predictions for the sum of neutrino masses, the absolute mass scale, the CP-violating phase, and the effective Majorana mass, making the model directly testable through future neutrino oscillation experiments (e.g., DUNE, Hyper-Kamiokande), cosmological surveys, and $0\nu\beta\beta$ searches. The authors suggest that embedding this study into Grand Unified or supersymmetric frameworks could further enhance its phenomenological consequences.

Technical Summary: A Γ3\Gamma_3Γ3​ Modular Symmetric Approach for Two-Zero Textures in Left-Right Symmetric Model

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Technical Summary: A $\Gamma_3$ Modular Symmetric Approach for Two-Zero Textures in Left-Right Symmetric Model