⚛️ phenomenology

Non-Invertible Froggatt-Nielsen Symmetry

This paper proposes a non-invertible Froggatt-Nielsen symmetry framework, realized through topological defects and instanton effects in a hidden gauge sector, to naturally generate the hierarchical Dirac neutrino Yukawa couplings and quark/charged-lepton masses while forbidding Majorana masses.

Original authors: Hajime Otsuka, Ryo Yokokura

Published 2026-10-16

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Hajime Otsuka, Ryo Yokokura

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 a vast array of particles, from the heavy protons that make up our bodies to the ghostly neutrinos that stream through the Earth by the trillions every second. One of the most persistent mysteries in physics is why these particles have the specific masses they do. Why is an electron so light while a top quark is nearly as heavy as a gold atom? Why do neutrinos have such incredibly tiny masses compared to everything else? For decades, physicists have used a framework called flavor symmetry to organize these differences, treating them like a set of rules that dictate how particles interact. However, a complete explanation for the extreme smallness of neutrino masses has remained elusive, often requiring separate, unconnected theories for different types of particles.

A new study by Hajime Otsuka and Ryo Yokokura offers a unified solution to this puzzle by combining two distinct ideas: a mechanism for generating particle mass hierarchies and a novel type of symmetry that exists only in the quantum realm. The researchers propose that the tiny masses of neutrinos are not accidental, but are the result of a specific, hidden process involving invisible particles and a unique kind of symmetry that cannot be reversed. By constructing a detailed model, they show how the same set of rules that explains the masses of quarks and charged leptons can also generate the minuscule masses of neutrinos, all without needing to invent new, unrelated mechanisms.

The core of this work relies on a concept known as the Froggatt-Nielsen mechanism, which suggests that the differences in particle masses arise from a hidden symmetry that is slightly broken. Imagine a set of rules that allows certain interactions to happen easily while making others very difficult, much like a filter that lets small particles pass through but blocks larger ones. In this model, a field called a flavon acquires a value that breaks the symmetry, creating a hierarchy where some particles are naturally heavy and others are light. The researchers applied this idea to the Standard Model of particle physics, assigning specific charges to quarks and leptons that successfully reproduce the observed patterns of their masses. However, this standard approach alone could not explain why neutrinos are so much lighter than even the lightest charged particles.

To solve this, the authors introduced a hidden sector of physics that operates alongside the known particles. This sector contains a hidden gauge force, similar to the electromagnetic force but acting on a different set of particles. Crucially, this hidden force is broken in a specific way that allows for the existence of magnetic monopoles, which are hypothetical particles carrying a single magnetic charge. In the quantum world, the presence of these monopoles can break a symmetry that would otherwise be perfect. The researchers discovered that by carefully designing the charges of the particles in this hidden sector, they could create a "non-invertible" symmetry. This is a type of symmetry that, unlike a mirror reflection which can be undone, cannot be reversed once it acts. It is a one-way street in the quantum landscape.

This non-invertible symmetry plays a pivotal role in the model. In the absence of certain heavy particles, this symmetry would strictly forbid neutrinos from having mass. However, the model includes a process where a hidden instanton—a fleeting, quantum tunneling event—temporarily violates this symmetry. This violation is extremely rare and suppressed by a factor that depends on the strength of the hidden force. Because this suppression is so strong, it naturally generates the tiny neutrino masses observed in experiments. The researchers calculated that the mass of the neutrino is determined by this exponential suppression factor, combined with the flavor charges that dictate the hierarchy. This means the extreme smallness of the neutrino mass is not a coincidence but a direct consequence of the hidden sector's dynamics.

The model also addresses the structure of the neutrino masses themselves. The researchers found that the interactions in their model naturally lead to a pattern where two neutrinos have mass while the third remains massless at the leading level. This results in a "normal ordering" of masses, where the lightest neutrino is significantly lighter than the other two, a pattern favored by current experimental data. The specific values of the masses and the way they mix together are determined by the same charge assignments that explain the masses of quarks and charged leptons. This unification is a significant achievement, as it suggests that the entire spectrum of particle masses, from the heaviest quarks to the lightest neutrinos, can be explained by a single, coherent set of principles.

Furthermore, the model includes a discrete gauge symmetry that acts as a strict guardian for the nature of neutrinos. This symmetry ensures that neutrinos remain Dirac particles, meaning they are distinct from their antiparticles, rather than Majorana particles, which would be their own antiparticles. This distinction is crucial because it forbids certain types of mass terms that would otherwise ruin the delicate balance of the model. The researchers verified that their proposed particle content and charge assignments are consistent with all known quantum mechanical rules, including the cancellation of potential anomalies that could make the theory inconsistent. They also showed that the model is stable against small perturbations, meaning the delicate mechanism generating the neutrino masses is robust.

The paper concludes by demonstrating that this framework is not just a theoretical possibility but a viable explanation for real-world data. By plugging in the known values for quark and lepton masses, the model successfully reproduces the observed hierarchies and mixing angles without requiring fine-tuned parameters. The authors provide a numerical example showing that with reasonable choices for the hidden sector's energy scales, the predicted neutrino masses match the values measured by experiments like Super-Kamiokande. This work suggests that the mystery of neutrino mass might be solved by looking not just at the particles we see, but at the hidden, non-invertible symmetries that govern the quantum world. It offers a compelling vision where the vast differences in particle masses are all part of a single, elegant story written in the language of symmetry and hidden forces.

Technical Summary: Non-Invertible Froggatt-Nielsen Symmetry and Dirac Neutrino Masses

Problem Statement
The origin of fermion mass hierarchies and the smallness of Dirac neutrino masses remain open questions in flavor physics. While the Froggatt-Nielsen (FN) mechanism successfully explains hierarchical structures in quark and charged-lepton Yukawa couplings via powers of a small parameter $\epsilon = \langle \phi \rangle / \Lambda_{FN}$ , it typically requires ad-hoc assumptions to generate the extreme suppression ( $\sim 10^{-13}$ ) observed in Dirac neutrino Yukawa couplings. Furthermore, standard Abelian flavor symmetries are often anomalous, raising questions about their consistency at the quantum level. Recent developments suggest that anomalous chiral symmetries can survive as non-invertible defects when dressed by topological quantum field theories (TQFTs), provided specific higher-form symmetries exist. This paper investigates whether such non-invertible completions of anomalous flavor symmetries can naturally generate the required exponential suppression for Dirac neutrino masses while simultaneously reproducing the leading quark and lepton hierarchies within a single framework.

Methodology
The authors construct a concrete extension of the Standard Model (SM) incorporating a hidden gauge sector and non-invertible symmetry defects. The methodology proceeds through the following steps:

Non-Invertible Symmetry Construction: The authors review the construction of non-invertible chiral defects in four dimensions. An anomalous chiral current, when combined with a 3D TQFT (specifically an Abelian $A_{n,p}$ theory) coupled to the bulk magnetic one-form symmetry, forms a gauge-invariant, topological defect. This defect acts as a non-invertible symmetry operator.
Model Building (Hidden Sector): They introduce a hidden $SU(2)_H$ gauge group that breaks to an intermediate $U(1)_H$ via an adjoint scalar $\Phi$ , and subsequently to the trivial group via a fundamental scalar $\Omega$ . The theory contains hidden Weyl doublets $N^c_D$ (identified with right-handed neutrinos) and $\eta$ .
Anomaly Matching and Defect Formation: The SM gauge sectors preserve a $\mathbb{Z}^{\text{FN}}_4$ subgroup of the global $U(1)_{\text{FN}}$ symmetry. The hidden $U(1)_H$ sector possesses a mixed anomaly with $U(1)_{\text{FN}}$ such that the order-four FN transformation acquires a rational anomaly phase. This allows the $\mathbb{Z}^{\text{FN}}_4$ transformation to be completed into a non-invertible defect $D^{\text{FN}}_4$ in the limit where magnetic monopoles are decoupled.
Instanton-Induced Breaking: The non-invertible nature of the defect relies on the exactness of the magnetic one-form symmetry. Introducing dynamical monopoles (finite mass) breaks this symmetry, rendering the defect non-topological. In the ultraviolet $SU(2)_H$ theory, a unit instanton generates a fermion mixing term $\mu_H (\eta N^c_D)$ that violates the FN charge by one unit. The coefficient of this term is exponentially suppressed by the instanton action: $\mu_H \propto \exp(-8\pi^2/g^2_H)$ .
Messenger Mechanism: Two singlet messenger fields ( $X_1, X_2$ ) couple the hidden sector to the SM leptons. Integrating out the heavy hidden fermions and messengers generates an effective neutrino Yukawa matrix. The structure of this matrix is determined by the FN charges (powers of $\epsilon$ ) and the common nonperturbative suppression factor $\zeta_H$ from the instanton.
Discrete Gauge Symmetry: An independent discrete gauge symmetry $\mathbb{Z}^D_8$ is imposed to forbid Majorana mass terms and the Weinberg operator, ensuring the neutrinos remain Dirac particles.

Key Contributions and Results

Non-Invertible FN Completion: The paper explicitly constructs a non-invertible completion of an anomalous $\mathbb{Z}^{\text{FN}}_4$ subgroup. The defect is formed by dressing the anomalous wall with a TQFT ( $A_{2,1}$ ) to cancel the rational anomaly phase arising from the hidden $U(1)_H$ sector.
Exponential Suppression via Instantons: The smallness of Dirac neutrino masses is attributed to the nonperturbative factor $\zeta_H \sim \exp(-8\pi^2/g^2_H(v_H))$ generated by $SU(2)_H$ instantons. This factor provides the common exponential suppression for all neutrino Yukawa couplings, distinct from the flavor hierarchy generated by FN charges.
Neutrino Mass Hierarchy and Texture:
- The resulting neutrino Yukawa matrix has a rank of at most two at tree level, leading to one massless neutrino state in the minimal setup.
- The ratio of the two non-zero masses is predicted to be $m_2/m_3 \sim \epsilon$ . With $\epsilon \approx 0.2$ , this favors a normal ordering of neutrino masses.
- The mixing angles are determined by the messenger couplings and FN charges. The model naturally reproduces the observed hierarchy where $\sin \theta_{12}, \sin \theta_{13} \sim \epsilon$ and $\sin \theta_{23} \sim \mathcal{O}(1)$ .
Unified Flavor Structure: The same FN charge assignment that generates the leading quark and charged-lepton hierarchies (e.g., $y_u:y_c:y_t \sim \epsilon^7:\epsilon^4:1$ ) also dictates the neutrino texture. The neutrino mass scale is separated from the flavor hierarchy, with the former set by the hidden sector dynamics and the latter by the FN charges.
Stability and Protection: The authors demonstrate that the charge-one mixing $\mu_H$ is protected from large perturbative corrections by the $\mathbb{Z}^{\text{FN}}_4$ selection rule and the discrete gauge symmetry $\mathbb{Z}^D_8$ . A nonminimal $\mathbb{Z}^9$ extension is discussed to protect the chiral selection rule at the renormalizable level against independent UV contributions.
Discrete Gauge Consistency: The model includes a singlet fermion $N^c_S$ specifically to cancel the mixed gravitational anomaly of the $\mathbb{Z}^D_8$ gauge symmetry, ensuring the consistency of the discrete gauge theory.

Significance and Claims
The authors claim that this work provides a unified framework where:

Non-invertible symmetries offer a rigorous mechanism to handle anomalous flavor symmetries, transforming them into topological defects that are broken only by specific nonperturbative effects (monopoles/instantons).
Dirac neutrino masses can be naturally small due to hidden instanton effects, without invoking a Majorana seesaw mechanism or fine-tuning of parameters.
Flavor hierarchies in both the quark/charged-lepton sectors and the neutrino sector arise from a single set of FN charges, with the neutrino mass scale decoupled from the flavor structure.
The model is phenomenologically viable, reproducing the observed neutrino mass ratios and mixing patterns (specifically normal ordering with $m_2/m_3 \sim \epsilon$ ) and the quark/lepton mass textures, while forbidding dangerous Majorana operators via a discrete gauge symmetry.

The paper emphasizes that the exponential suppression is a dynamical consequence of the hidden sector's instanton physics, while the flavor structure is a consequence of the Froggatt-Nielsen mechanism. The work extends previous applications of non-invertible symmetries to flavor physics and provides a concrete UV-complete realization involving hidden gauge dynamics.

Technical Summary: Non-Invertible Froggatt-Nielsen Symmetry and Dirac Neutrino Masses

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