⚛️ phenomenology

Neutrino Phenomenology and Leptogenesis from Non-Invertible Fusion Symmetry

This paper proposes a minimal type-I seesaw model utilizing non-invertible fusion symmetry to predict an inverted neutrino mass ordering with a vanishing lightest mass, a nearly maximal Dirac CP phase, and a specific effective Majorana mass, while simultaneously explaining the observed baryon asymmetry through thermal leptogenesis.

Original authors: Pulakesh Borah, Mrinal Kumar Das

Published 2026-10-27

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Pulakesh Borah, Mrinal Kumar Das

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 filled with particles that rarely interact with anything, slipping through solid matter as if it were not there. Among these ghostly travelers are neutrinos, tiny particles that come in three distinct "flavors" and have long puzzled scientists. For decades, the prevailing view was that these particles were massless, but experiments have since confirmed they possess a tiny, non-zero mass and can change their identity as they travel. This discovery opened a new chapter in physics, yet it left a deeper mystery unsolved: why do these particles have the specific masses they do, and why do they mix in the particular patterns we observe? Furthermore, the origin of the matter that makes up our world remains a profound question. The Big Bang should have created equal amounts of matter and antimatter, which would have annihilated each other, leaving behind a universe of pure light. Instead, we exist. Something tipped the balance, creating a surplus of matter, and many physicists believe the answer lies in the behavior of these elusive neutrinos.

A team of researchers at Tezpur University in India has proposed a new way to solve these puzzles by introducing a fresh kind of mathematical rule to govern how particles interact. Rather than relying on the traditional symmetry groups that have guided particle physics for years, they utilized a concept known as a non-invertible fusion symmetry. In standard physics, combining two specific properties usually results in a single, predictable outcome. In this new framework, combining two properties can split into a sum of several different possibilities. The researchers used this unique rule to construct a model that explains the masses and mixing of neutrinos without needing to invent a host of new, unseen fields to force the numbers to work. Their approach is remarkably minimal, requiring only two heavy, unseen neutrino partners and a second type of Higgs particle, a field responsible for giving other particles mass, to make the theory function.

By applying this new symmetry, the team found that the structure of the neutrino world is not random but follows a strict, predictive pattern. Their calculations indicate that the three known neutrinos do not all have different masses; instead, the lightest one has zero mass. This specific arrangement, known as inverted ordering, is the only version of the model that fits with current experimental data. The model also predicts a very specific value for the sum of all three neutrino masses, placing it between 0.098 and 0.100 electron volts. This prediction is sharp enough to be tested by future experiments; if a more sensitive measurement finds a value lower than this range, the model would be ruled out. Additionally, the theory points to a nearly maximum level of a property called CP violation, a subtle difference in how matter and antimatter behave, which is essential for explaining why our universe is made of matter.

The researchers did not stop at explaining the particles themselves; they traced the consequences of their model back to the earliest moments of the cosmos. They simulated the thermal history of the early universe to see if the heavy neutrino partners in their model could generate the observed imbalance between matter and antimatter. The results were successful. The simulations showed that the decay of these heavy particles, driven by the specific rules of their new symmetry, naturally produced the exact amount of matter surplus we see today. This connection is significant because it links the tiny, unmeasured properties of neutrinos to the grand scale of the universe's existence. The model also makes a clear prediction for a rare nuclear process called neutrinoless double beta decay, suggesting that the effective mass of the neutrino in this process should fall within a specific range that upcoming experiments will be able to probe.

What makes this work stand out is its elegance and its reliance on a single, unifying principle. Previous attempts to explain the variety of particle masses often required adding many invisible fields and complex mechanisms to align the vacuum of space, a process that felt somewhat arbitrary. This new model achieves the same predictive power without those extra layers, deriving the necessary patterns directly from the fusion rules of the symmetry itself. The authors have demonstrated that a minimal framework, grounded in this non-invertible symmetry, can simultaneously account for the mass of neutrinos, their mixing patterns, the existence of matter in the universe, and the potential for future experimental discovery. It offers a coherent story where the smallest particles and the largest cosmic structures are connected by a single, underlying mathematical logic.

Technical Summary: Neutrino Phenomenology and Leptogenesis from Non-Invertible Fusion Symmetry

Problem Statement
The origin of fermion flavor hierarchies and mixing patterns remains a primary unresolved issue in the Standard Model (SM). While discrete flavor symmetries (e.g., $A_4$ , $S_4$ ) and modular symmetries have been employed to explain neutrino mass textures, they typically require extended flavon sectors and complex vacuum-alignment mechanisms. Furthermore, fundamental parameters such as the absolute neutrino mass scale, mass ordering, and CP-violating phases remain undetermined. Concurrently, the dynamical origin of the baryon asymmetry of the Universe (BAU) via leptogenesis often introduces new free parameters (heavy right-handed neutrino masses and Yukawa couplings) that mirror the flavor puzzle. This work addresses the challenge of constructing a predictive neutrino model that simultaneously explains flavor textures and generates the BAU without relying on extended flavon sectors or vacuum alignment, utilizing the emerging concept of non-invertible fusion symmetries.

Methodology
The authors construct a minimal Type-I seesaw model extended by a non-invertible $\mathbb{Z}_5^{\text{IN}}$ fusion symmetry. The framework includes:

Symmetry Structure: The model utilizes the $\mathbb{Z}_5^{\text{IN}}$ algebra, obtained by gauging the $\mathbb{Z}_2$ automorphism ( $g_k \to g_{-k}$ ) of $\mathbb{Z}_5$ . This results in three fusion sectors: $[0]$ , $[1]$ , and $[2]$ . Unlike ordinary groups, the fusion of two sectors decomposes into a sum of sectors (e.g., $[1] \times [1] = [0] + [2]$ ), generating selection rules that dictate allowed interactions without external flavon fields.
Field Content: The model extends the SM with two right-handed Majorana neutrinos ( $N_1, N_2$ $N_{1}, N_{2}$ ), a second $SU(2)_L$ $S U (2)_{L}$ Higgs doublet ( $H_\nu$ $H_{ν}$ ), and an auxiliary $\mathbb{Z}_2^\nu$ $Z_{2}^{ν}$ parity.
- Left-handed lepton doublets ( $L_e, L_\mu, L_\tau$ ) and right-handed charged leptons are assigned to sectors $[0]$ , $[1]$ , and $[2]$ respectively.
- $N_1$ and $N_2$ are assigned to $[1]$ and $[2]$ .
- Higgs doublets $H_\ell$ and $H_\nu$ are assigned to $[0]$ and $[1]$ .
Lagrangian Construction: The fusion rules strictly determine the texture of the Yukawa matrices. The charged-lepton mass matrix becomes diagonal by construction. The Dirac neutrino mass matrix ( $M_D$ ) and the heavy Majorana mass matrix ( $M_R$ ) acquire specific zero textures dictated by the fusion product containing the identity sector $[0]$ .
Numerical Analysis: The light neutrino mass matrix ( $M_\nu = -M_D M_R^{-1} M_D^T$ ) is analyzed against NuFIT 6.1 oscillation data (3 $\sigma$ ranges). The model is tested for both Normal Ordering (NO) and Inverted Ordering (IO).
Leptogenesis: The thermal leptogenesis mechanism is evaluated using the Boltzmann equations for unflavored leptogenesis. The CP asymmetry ( $\epsilon_1$ ) is calculated for the decay of the lightest heavy neutrino ( $N_1$ ), and the final baryon-to-photon ratio ( $\eta_B$ ) is computed using the ULYSSES package.

Key Contributions and Results

Texture Generation without Flavons: The study demonstrates that the $\mathbb{Z}_5^{\text{IN}}$ fusion symmetry alone fixes the neutrino mass textures. The resulting light neutrino mass matrix has rank 2, implying one massless neutrino ( $m_3 = 0$ for IO).
Exclusion of Normal Ordering: The specific texture derived from the fusion rules is incompatible with Normal Ordering. For NO, the texture requires a mass ratio $m_2/m_3$ in the range of $0.25\text{--}0.46$ , whereas oscillation data dictates $m_2/m_3 \simeq 0.17\text{--}0.18$ . This discrepancy leads to the exclusive selection of Inverted Ordering (IO) with $m_3 = 0$ .
Predictive Neutrino Parameters:
- Mass Sum: The model predicts a sum of neutrino masses $\sum m_\nu \simeq 0.098\text{--}0.100$ eV. This satisfies the Planck 2018 bound ( $<0.12$ eV) but lies in tension with stricter cosmological bounds ( $\Lambda$ CDM) while remaining consistent with $w_0w_a$ CDM scenarios.
- CP Violation: The model predicts a nearly maximal Dirac CP phase, $\delta_{CP} \approx [268^\circ, 271^\circ]$ . The Jarlskog invariant is constrained to $J_{CP} \in [-0.0349, -0.0322]$ .
- Neutrinoless Double Beta Decay ( $0\nu\beta\beta$ ): The effective Majorana mass is predicted to be $m_{\beta\beta} \approx 47\text{--}48$ meV. This falls within current KamLAND-Zen limits but is well above the projected sensitivity of future experiments like nEXO ( $5\text{--}20$ meV), offering a decisive test.
Successful Leptogenesis: The framework successfully reproduces the observed baryon asymmetry of the Universe ( $\eta_B \approx 6.1 \times 10^{-10}$ ) via thermal leptogenesis. The authors identify three benchmark points (BP1, BP2, BP3) with heavy neutrino masses in the range of $10^{13}\text{--}10^{15}$ GeV that satisfy all oscillation constraints and yield the correct BAU.

Significance
The authors claim this work establishes the first connection between predictive neutrino phenomenology and leptogenesis within a framework based on non-invertible fusion symmetry. The significance lies in demonstrating that non-invertible symmetries can serve as a viable organizing principle for flavor structures, replacing the need for extended flavon sectors and vacuum alignment. By constraining the model to minimal field content (two right-handed neutrinos and a second Higgs doublet), the paper provides a testable scenario where neutrino mass ordering, CP phases, and the baryon asymmetry are linked through a single symmetry structure. The prediction of a specific mass sum and a near-maximal CP phase offers sharp, falsifiable targets for future cosmological and oscillation experiments.

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