⚛️ phenomenology

Two-loop rainbow neutrino masses in a non-invertible symmetry

This paper proposes a two-loop rainbow neutrino mass model based on a gauged $\mathbb{Z}_2$ subgroup of a $\mathbb{Z}_6$ non-invertible symmetry, which simultaneously explains neutrino oscillations, provides a stable vector-like fermion dark matter candidate, and predicts a sum of neutrino masses in the normal hierarchy that exceeds that of the inverted hierarchy.

Original authors: Hiroshi Okada, Yoshihiro Shigekami

Published 2026-01-23

📖 5 min read🧠 Deep dive

CC0 1.0

Original authors: Hiroshi Okada, Yoshihiro Shigekami

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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 Big Picture: Why Do Neutrinos Have Mass?

Imagine the Standard Model of physics as a giant, mostly complete puzzle. For a long time, one piece was missing: neutrinos. We knew they existed, but we thought they were weightless (massless). Experiments later proved they have a tiny bit of weight, but the puzzle didn't explain how they got it.

Usually, scientists try to solve this by adding new "heavy" particles that interact with neutrinos. But in this paper, the authors (Hiroshi Okada and Yoshihiro Shigekami) propose a different, more complex recipe. They suggest that neutrinos get their mass not from a single interaction, but from a two-step, two-loop process involving a hidden "rulebook" called a non-invertible symmetry.

The Ingredients: A New Particle Family

To make this work, the authors added a new "family" of particles to the universe. Think of these like a secret society of particles that don't play by the usual rules:

Vector-like Fermions: These are like "twin" particles (one neutral, one charged) that are heavy.
Heavy Right-Handed Neutrinos: Extra, very heavy versions of the neutrinos we know.
Inert Bosons: Invisible particles that don't interact with light but help pass messages between the others.

The Secret Rulebook (Non-Invertible Symmetry):
Imagine a game where you can swap players around, but there's a special rule: you can't always undo the swap. This is a "non-invertible symmetry."

The Good News: This rulebook has a leftover "Z2 symmetry" (like a simple "Yes/No" switch) that acts as a security guard. It ensures that the lightest, neutral member of this new family cannot decay or disappear. Because it's stable and invisible, it becomes a perfect candidate for Dark Matter—the invisible stuff holding galaxies together.

The Mechanism: The "Rainbow" Mass

The paper is titled "Two-loop rainbow neutrino masses." Here is what that means:

The Loop: In physics, particles can interact by creating a temporary "loop" of other particles. Usually, a neutrino gets mass from a simple loop. Here, the authors say the neutrino has to go through two loops (a more complex path).
The Rainbow: Imagine a neutrino trying to cross a river. Instead of a single bridge, it has to cross a series of bridges that look like a rainbow (different colors representing different particles). The neutrino hops from one particle to another in a complex chain before finally getting its tiny bit of mass.
Why Two Loops? This complexity is necessary because the "security guard" (the symmetry) forbids the neutrino from getting mass easily. It forces the process to be very rare and slow, which explains why neutrinos are so incredibly light compared to other particles.

The Dark Matter Connection

The authors focus on the lightest neutral particle from their new family as the Dark Matter candidate.

The "Co-Annihilation" Party: Usually, Dark Matter particles just bump into each other and disappear (annihilate). But here, the authors suggest that the Dark Matter particle is so similar in weight to its two "cousins" (the other two families of neutral fermions) that they hang out together.
The Analogy: Imagine a group of friends at a party. If they are all wearing the same outfit (similar mass), they can swap places easily. When they try to leave the party (annihilate), they do it in groups. This "co-annihilation" helps explain exactly how much Dark Matter is left over in the universe today, matching what astronomers observe.

The Results: A Surprising Twist

The authors ran the numbers to see if their model fits with real-world data (like how neutrinos mix, how fast the universe is expanding, and experiments looking for rare particle decays).

They found two possible scenarios for how the neutrinos are ordered (Normal Hierarchy vs. Inverted Hierarchy). Here is the big surprise:

In most models: If the neutrinos are ordered one way (Normal), the total mass is usually lighter. If they are ordered the other way (Inverted), the total mass is heavier.
In this model: It's the opposite. Because of how the Dark Matter interacts in their specific setup, the "Normal" ordering results in a heavier total mass than the "Inverted" ordering.

What This Means for the Future

The paper doesn't claim to have solved everything yet, but it offers a specific "fingerprint" to look for:

Neutrino Mass Sum: If future experiments measure the total weight of neutrinos and find it is heavy (around 140–150 meV for the Normal case), it might support this model.
Double Beta Decay: They predict a specific signal for a rare event called "neutrinoless double beta decay." Future experiments like LEGEND-1000 or nEXO might be able to see this.
Muon Decay: They predict a very specific, tiny chance of a muon turning into an electron and a photon. Future experiments (like MEG II) might catch this rare event.

In Summary:
The authors built a complex machine where a secret, unbreakable rule creates a stable Dark Matter particle and forces neutrinos to take a long, winding "rainbow" path to get their tiny mass. The result is a unique prediction that flips the usual expectations about neutrino weights, offering a new way to test our understanding of the universe.

Technical Summary: Two-loop rainbow neutrino masses in a non-invertible symmetry

Problem and Motivation
The nature of neutrino masses and mixing remains an open question beyond the Standard Model (SM). While radiative seesaw scenarios offer a promising framework to explain tiny neutrino masses while correlating them with other phenomenology—such as lepton flavor violations (LFVs), the muon anomalous magnetic moment ( $g-2$ ), and dark matter (DM)—there is no unique solution yet. Recent theoretical developments have introduced non-invertible symmetries, which have been applied to various sectors including mass matrix textures and DM stability. This paper addresses the construction of a neutrino mass model that incorporates a $Z_2$ gauging of a $Z_6$ non-invertible symmetry, aiming to generate neutrino masses at the two-loop level while providing a stable dark matter candidate.

Methodology and Model Setup
The authors propose a model extending the SM with the following new fields, all charged under the $Z_6$ non-invertible symmetry ( $F_R(6)$ ):

Three families of isospin doublet vector-like fermions ( $L' \equiv [n', e']^T$ ).
Heavy right-handed neutrinos ( $N_R$ ).
An inert isospin doublet boson ( $\eta$ ) and an inert singlet boson ( $S^0$ ).

The SM fermions and the Higgs doublet are singlets under $F_R(6)$ . The model utilizes the remnant $Z_2$ symmetry of the non-invertible symmetry to stabilize the neutral components of the new fields, making them DM candidates. The non-invertible symmetry is dynamically broken at the one-loop level, but the authors argue this violation does not invalidate their scenario.

Key Contributions and Mechanisms

Two-Loop Neutrino Mass Generation: The dominant contribution to the active neutrino mass matrix arises from "rainbow" type diagrams at the two-loop level. The mass matrix consists of four distinct diagram topologies (labeled I–IV). Under the assumed mass hierarchy where the vector-like fermion mass $M'$ is the lightest new scale ( $10^2 \text{ GeV} \lesssim M' \lesssim 10^4 \text{ GeV}$ ) and other new particles are heavier, Diagram (I) is found to be the dominant contribution (contributing at least 50% to the total mass).
Dark Matter Candidate and Co-annihilation: The lightest neutral component of the vector-like fermions ( $n'$ ), denoted as $\psi_{R1}$ or $X_R$ , is identified as the primary DM candidate. Crucially, the model generates small Majorana mass terms ( $\delta m_{R,L}$ ) for these fermions at the one-loop level. These terms induce a small mass splitting between the three families of neutral fermions. This splitting allows for a co-annihilation system among the nearly degenerate neutral fermions ( $\psi_{R1}, \psi_{R2}, \psi_{R3}$ ) to explain the observed relic density, a mechanism necessary because annihilation via Yukawa couplings alone is insufficient.
Inverted Hierarchy vs. Normal Hierarchy Sum Rule: A distinct feature of this model is the relationship between the sum of neutrino masses ( $\sum m_\nu$ ) and the mass hierarchy. Due to the specific DM phenomenology and co-annihilation requirements, the model predicts that the sum of neutrino masses in the Normal Hierarchy (NH) case is larger than in the Inverted Hierarchy (IH) case. This is contrary to typical active neutrino models where the IH sum is generally larger.

Numerical Results and Constraints
The authors perform a numerical analysis scanning parameter space subject to:

Neutrino oscillation data (mixing angles, mass squared differences).
Constraints on the sum of neutrino masses from cosmology (CMB and extended models).
Limits on neutrinoless double beta decay ( $m_{ee}$ ).
LFV bounds (specifically $\mu \to e\gamma$ ).
Muon $g-2$ ( $\Delta a_\mu$ ).
The observed DM relic density ( $\Omega h^2$ ).

Findings:

Normal Hierarchy (NH): The allowed region for the sum of neutrino masses is localized around $141\text{--}151$ meV, with the effective Majorana mass $m_{ee}$ in the range $13.5\text{--}40$ meV. These values satisfy the upper bounds from extended cosmological models but exceed the strictest bounds from minimal standard cosmological models ( $\sum m_\nu \leq 120$ meV). The predicted DM mass is approximately $3250\text{--}3700$ GeV. The branching ratio for $\mu \to e\gamma$ is close to current experimental limits, suggesting potential testability by future experiments like MEG II.
Inverted Hierarchy (IH): The allowed region for the sum of neutrino masses is localized around $103\text{--}104$ meV, which is consistent with minimal standard cosmological bounds. The $m_{ee}$ range is $18\text{--}49$ meV. The predicted DM mass is slightly lower, around $2800\text{--}3350$ GeV.
Other Observables: The model predicts contributions to $\Delta a_\mu$ that are positive but small ( $\sim 10^{-9}$ ), well within current experimental errors. LFV processes involving tau leptons ( $\tau \to e\gamma, \tau \to \mu\gamma$ ) are predicted to be below current bounds, though $\tau \to e\gamma$ could be probed by future experiments like Belle II.

Significance and Claims
The paper claims that the proposed model successfully generates neutrino masses at the two-loop level via a non-invertible symmetry framework while providing a viable DM candidate stabilized by a remnant symmetry. The authors highlight that the model's prediction—where the sum of neutrino masses is larger in the NH case than the IH case—is a unique signature distinguishing it from typical active neutrino models. Furthermore, the model suggests that future precision measurements of $\mu \to e\gamma$ and neutrinoless double beta decay ( $m_{ee}$ ) have the potential to test the allowed parameter space of this scenario. The authors conclude that the interplay between the non-invertible symmetry, the two-loop mass generation, and the co-annihilation DM mechanism creates a testable phenomenological framework.