Radiative Dirac Neutrino Masses from Modular $S_3$ Symmetry in an Axion Model

This paper proposes a unified KSVZ-type axion model incorporating modular $S_3$ symmetry and a global $U(1)_{\rm PQ}$ symmetry to simultaneously explain the origin of radiative Dirac neutrino masses, the strong CP problem, and dark matter, while predicting a massless neutrino and testable signatures for charged lepton flavor violation and axion-photon coupling.

The Gist

Imagine the universe as a giant, complex machine. For a long time, physicists have been trying to fix four specific, stubborn glitches in this machine using the "Standard Model" (the current rulebook of physics). These glitches are:

1. The Ghostly Neutrinos: We know these tiny particles exist and have mass, but the rulebook says they should be weightless.
2. The Flavor Mystery: Why do neutrinos and electrons mix and switch identities in such specific, weird patterns?
3. The Strong CP Problem: A mathematical glitch in the "strong force" (which holds atoms together) that suggests the universe should behave differently than it actually does.
4. The Dark Matter Mystery: We know 85% of the universe is made of invisible stuff that holds galaxies together, but we have no idea what it is.

Usually, scientists try to fix these problems one by one with different tools. This paper proposes a single, unified toolkit that fixes all four problems at once. They call this a "Modular $S_3$ Symmetry" model, but let's break it down into everyday concepts.

The Master Key: The Axion

Think of the Axion as a magical, invisible thread that runs through the whole machine.

• Fixing the Strong Force: The axion acts like a self-correcting dial. If the "strong force" tries to get out of balance (the Strong CP problem), the axion automatically turns itself to zero out the error, keeping the universe stable.
• The Dark Matter Candidate: Because this axion is light, invisible, and stable, it is a perfect candidate for the missing "Dark Matter" that holds the universe together.

The Neutrino Puzzle: A One-Loop Detour

In the standard rulebook, neutrinos get their mass directly, like a straight line from A to B. But in this model, the Peccei-Quinn (PQ) symmetry (a set of rules governing the axion) acts like a bouncer at a club. It says, "No direct mass for neutrinos!"

So, how do neutrinos get mass? They have to take a scenic route.

• The paper introduces "exotic" particles (colored fermions and scalars) that don't exist in our everyday world.
• Neutrinos borrow mass by interacting with these exotic particles in a one-loop process. Imagine a neutrino trying to cross a river. Instead of a bridge (tree-level), it has to take a ferry that stops at two exotic islands (the loop) before reaching the other side.
• Because of the specific rules of this model, this detour only allows two of the three neutrinos to get mass. The third one remains massless. This is a unique prediction: the universe has a "massless neutrino."

The Flavor Pattern: The Modular $S_3$ Symmetry

Why do the particles mix in the specific ways we see? The authors use a concept called Modular $S_3$ Symmetry.

• Think of this as a geometric dance floor. The particles aren't just random; they are arranged in a specific pattern (like a triangle or a specific dance step).
• This symmetry dictates exactly how the neutrinos and electrons mix. It's like a recipe that ensures the ingredients (particles) combine in the right proportions to create the flavor patterns we observe in experiments.

The Results: What the Math Says

The authors ran the numbers (a "numerical analysis") to see if their machine actually works with real-world data.

• Two Scenarios: They tested two possibilities: "Normal Hierarchy" (lightest neutrino is very light) and "Inverted Hierarchy" (two heavy, one light).
• The Mass Sum: Because one neutrino is massless, the total weight of all three neutrinos is tightly constrained.
  • In the "Normal" case, the total mass is predicted to be around 58 meV.
  • In the "Inverted" case, it's around 100 meV.
  • These numbers fit perfectly with recent data from space telescopes (like DESI and CMB) that measure the universe's expansion.
• The CP Phase: They also predicted the "Dirac CP phase" (a measure of how much neutrinos violate symmetry), finding values that match current experimental hints.

Can We Test This?

The best part of this paper is that it's not just theory; it's testable.

• The Axion Hunt: The model predicts a specific strength for how the axion interacts with light (photons). This prediction falls right in the "sweet spot" where upcoming experiments like IAXO, ADMX, and MADMAX are looking.
• No Conflict: The model respects all current safety limits. It doesn't break any known rules about how stars cool down or how the universe evolved.

Summary

This paper builds a single, elegant house to solve four different problems.

1. It uses a magic thread (Axion) to fix the strong force and provide Dark Matter.
2. It uses a detour (One-loop) to give neutrinos mass while keeping one massless.
3. It uses a geometric dance (Modular $S_3$) to explain why particles mix the way they do.
4. It predicts specific numbers that upcoming experiments can check in the near future.

It's a "unified theory" that suggests the universe's deepest secrets are all connected by the same underlying symmetry.

Technical Summary

Technical Summary: Radiative Dirac Neutrino Masses from Modular $S_3$ Symmetry in an Axion Model

Problem Statement
The Standard Model (SM) fails to explain several fundamental phenomena: the origin of non-zero neutrino masses and their flavor mixing structure, the absence of CP violation in Quantum Chromodynamics (the strong CP problem), and the nature of dark matter (DM). While various extensions address these issues individually, unified frameworks that simultaneously resolve neutrino mass generation, the strong CP problem, and DM phenomenology within a single theoretical structure remain a significant goal in beyond-Standard-Model (BSM) physics. Furthermore, the specific nature of neutrinos (Dirac vs. Majorana) and the predictive power of flavor models require robust symmetry mechanisms.

Methodology and Model Framework
The authors propose a unified framework based on a KSVZ-type axion model augmented by a global $U(1)_{PQ}$ symmetry and a modular $S_3$ symmetry. The model introduces the following new fields:

• Vector-like Fermions ($\Psi_{L,R}$): Two generations of color-triplet, $SU(2)_L$-singlet fermions with hypercharge $Y=0$.
• Scalars: A complex singlet $\sigma$ (responsible for $U(1)_{PQ}$ breaking), an $SU(2)_L$ doublet $\eta$ ($Y=1/2$), and an $SU(2)_L$ singlet $\chi$ ($Y=0$), all carrying color charges.
• Symmetry Assignments:
  • $U(1)_{PQ}$: Assigned to forbid tree-level neutrino masses. The spontaneous breaking of this symmetry generates a residual $Z_3$ symmetry, which enforces the Dirac nature of neutrinos by forbidding Majorana mass terms.
  • Modular $S_3$: Lepton doublets ($L_{2,3}$), right-handed charged leptons ($e_{R2,3}$), and right-handed neutrinos ($\nu_{R2,3}$) transform as doublets, while $\Psi_{L2}$ and $\Psi_{R2}$ are singlets. Modular weights are assigned such that Yukawa couplings become modular forms, providing a predictive structure for mixing angles.

Mechanism of Neutrino Mass Generation
Neutrino masses are generated radiatively at the one-loop level. The effective operator $L \tilde{H} \nu_R \sigma^*$ is UV-completed by the exchange of the exotic colored fermions ($\Psi$) and scalars ($\eta, \chi$).

• The colored scalars $\eta$ and $\chi$ mix via a term in the scalar potential, leading to mass eigenstates $\zeta_1$ and $\zeta_2$.
• The one-loop diagram involves the exchange of these mixed scalars and the vector-like fermions.
• Due to the minimal realization involving only two generations of vector-like fermions, the resulting neutrino mass matrix is of rank two. This predicts one massless neutrino eigenstate, consistent with current oscillation data constraints.

Numerical Analysis and Results
The authors perform a $\chi^2$ analysis to fit the model to neutrino oscillation data (mass squared differences $\Delta m^2_{sol}, \Delta m^2_{atm}$ and mixing angles $s^2_{12}, s^2_{13}, s^2_{23}$) for both Normal Hierarchy (NH) and Inverted Hierarchy (IH) scenarios.

• Normal Hierarchy (NH):

  • The best-fit point yields $\chi^2_{min} = 2.48$.
  • The modular parameter $\tau$ is constrained such that $\text{Im}[\tau] \approx 1.7$, while $\text{Re}[\tau]$ is allowed over a wide range.
  • The sum of neutrino masses is predicted to be $\sum m_\nu \approx 58.8$ meV.
  • The effective electron antineutrino mass is constrained to $m_{\nu e} \approx 4.88$ meV.
  • No parameter points were found within the $1\sigma$ region for the mixing angles in this specific scan, though the model remains consistent with data up to $5\sigma$.

• Inverted Hierarchy (IH):

  • The best-fit point yields $\chi^2_{min} = 1.88$.
  • Both $\text{Re}[\tau]$ and $\text{Im}[\tau]$ are confined to narrow intervals ($\text{Re}[\tau] \in [-0.156, 0.147]$, $\text{Im}[\tau] \in [1.63, 1.74]$).
  • The sum of neutrino masses is predicted to be $\sum m_\nu \approx 99.1$ meV.
  • The effective electron antineutrino mass is localized near the lower bound of global analyses, $m_{\nu e} \approx 48.9$ meV.

Flavor Violation and $g-2$
The model induces charged lepton flavor violation (cLFV) processes ($\ell_\alpha \to \ell_\beta \gamma$) and contributions to the lepton anomalous magnetic moments ($g-2$) via the same loop diagrams.

• cLFV: The branching ratios for $\mu \to e\gamma$ are constrained to be below current experimental limits ($< 3.10 \times 10^{-13}$). For NH, $\tau \to e\gamma$ is predicted to be potentially verifiable ($< 1.54 \times 10^{-8}$), while for IH, $\tau \to \mu\gamma$ is within reach ($< 4.15 \times 10^{-8}$).
• $g-2$: The contributions to electron and muon $g-2$ are found to be negligible ($|\Delta a_\mu| \lesssim 10^{-13}$), consistent with current experimental precision but not explaining the existing muon $g-2$ anomaly.

Axion and Dark Matter
The spontaneous breaking of $U(1)_{PQ}$ by the vacuum expectation value of $\sigma$ generates the axion, solving the strong CP problem.

• Color Anomaly: The model yields a color anomaly factor $N=1$, consistent with KSVZ models.
• Dark Matter: The axion is a viable cold DM candidate produced via the misalignment mechanism.
  • In a post-inflationary scenario (including cosmic string contributions), the model predicts an axion decay constant $f_a \simeq (4.5 - 7.0) \times 10^{10}$ GeV and an axion mass $m_a \simeq (81 - 127)$ $\mu$eV.
  • This mass range corresponds to an axion-photon coupling $|g_{a\gamma\gamma}| \simeq (3 - 5) \times 10^{-14}$ GeV$^{-1}$.
• Constraints: The predicted coupling and mass lie well within the sensitivity of upcoming experiments (IAXO, ADMX, MADMAX, CAPP) and are consistent with existing astrophysical bounds (stellar cooling, SN 1987A, Bullet Cluster).

Significance and Claims
The paper claims to present a unified framework that simultaneously addresses the origin of neutrino masses, the strong CP problem, and dark matter. Key contributions include:

1. Radiative Dirac Masses: A mechanism where the residual $Z_3$ symmetry from $U(1)_{PQ}$ breaking naturally ensures Dirac neutrinos without tree-level masses.
2. Predictive Flavor Structure: The use of modular $S_3$ symmetry provides a predictive setup for lepton mixing, linking the flavor structure to the modular parameter $\tau$.
3. Minimal Realization: The model achieves these goals with a minimal particle content (two generations of vector-like fermions), resulting in a rank-two neutrino mass matrix and a massless neutrino.
4. Experimental Accessibility: The predicted axion parameters fall within the reach of next-generation axion searches, offering a concrete target for experimental verification. The model remains consistent with all current constraints on cLFV, lepton $g-2$, and cosmological bounds.