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Modular $S_4$ Inverse Seesaw at the LZ High-Recoil Frontier

This paper proposes a modular $S_4$ inverse seesaw model that unifies lepton flavor physics with high-recoil inelastic dark matter, demonstrating that a common complex modulus can simultaneously satisfy neutrino oscillation data and predict a TeV-scale pseudo-Dirac fermion with a $\sim371.5$ keV mass splitting that aligns with extended LUX-ZEPLIN nuclear-recoil analyses.

Original authors: Waqas Ahmed, Vasileios Basiouris, George K. Leontaris

Published 2026-10-14

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CC BY 4.0

Original authors: Waqas Ahmed, Vasileios Basiouris, George K. Leontaris

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

Deep in the silence of the universe, invisible particles known as dark matter are thought to drift through everything, including our planet. For decades, scientists have tried to catch these particles using massive detectors buried underground, waiting for a dark matter particle to bump into an atom in the detector and create a tiny flash of light. Usually, scientists expect these bumps to happen at very low energies, like a gentle tap. However, the LUX-ZEPLIN experiment, a giant tank of liquid xenon deep in a South Dakota mine, recently reported something unusual. It found a single event where the energy was much higher than expected, around 248 thousand electron volts. This high-energy bump suggests that the dark matter particle might not just be tapping the atom, but perhaps kicking it hard enough to change its internal state, a process that requires a specific amount of energy to get started. This idea, called endothermic scattering, changes the rules of the game: it means the dark matter must be moving very fast to have enough energy to cause this reaction, and it would leave a very specific signature that looks different from the gentle taps scientists usually search for.

A team of physicists has now built a new theoretical model to explain this strange signal, connecting it to the fundamental rules that govern how particles like electrons and neutrinos behave. They propose that the dark matter particle is a heavy, invisible partner to the known particles, weighing in at about 1.8 trillion electron volts, which is nearly two thousand times heavier than a proton. Crucially, this particle has a twin with a slightly different mass, separated by a tiny gap of about 371 thousand electron volts. This small difference is the key to the high-energy bump seen in the detector. When the dark matter hits a xenon atom, it must use part of its speed to bridge this gap, which explains why the event happened at such a high energy and why the detector did not see the usual low-energy signals. The researchers did not just guess these numbers; they built a complex mathematical structure based on a concept called modular symmetry, which acts like a set of geometric rules that dictate how particles interact. By using these rules, they showed that the same hidden patterns that determine the masses of neutrinos also naturally create the specific heavy dark matter particle and its tiny mass gap needed to explain the LUX-ZEPLIN event.

The scientists started by looking at the known particles, specifically the charged leptons and neutrinos, to see if their model could reproduce the real-world data we already have. They ran a massive computer simulation, testing millions of different combinations of parameters within their modular framework. They found over 180,000 solutions that perfectly matched the known properties of neutrinos, such as how they mix and oscillate. Within these successful solutions, they searched for the specific conditions that would create the heavy dark matter particle. They discovered that the same mathematical "modulus," a complex number that acts as a master key for the model's geometry, controls both the neutrino sector and the dark matter sector. This means the model is not just a random guess; it is a unified theory where the rules for light particles and heavy dark matter are written in the same language. The simulation showed that when the model fits the neutrino data, it naturally produces a dark matter candidate with a mass between 300 GeV and 2 TeV and a mass splitting between 300 and 400 keV, which is exactly the range needed to explain the high-energy event.

To test if this specific combination of mass and energy gap could actually produce the signal seen in the detector, the researchers analyzed the physics of the collision. They calculated how the speed of the dark matter particle, the mass of the xenon atom, and the energy gap interact. They found that for a particle with a mass of about 1.8 TeV and a gap of 371 keV, the collision is only possible if the dark matter is moving at the very highest speeds allowed by the galaxy's rotation. This creates a sharp cutoff: if the gap were any larger, the dark matter would not be fast enough to trigger the reaction, and if it were smaller, the signal would appear at lower energies. The model predicts that the single event seen by LUX-ZEPLIN sits right at this theoretical limit, the absolute maximum energy possible for this type of interaction in our galaxy. The researchers also checked if this heavy particle would be caught by the Sun and produce neutrinos that we could detect. They noted that while some other theories are ruled out by solar neutrino data, their specific model involves a different type of particle that interacts much more weakly with the Sun, meaning it could survive these constraints and still be a valid explanation for the underground signal.

The work represents a significant step in connecting the invisible world of dark matter with the known world of particle physics. Instead of treating the dark matter signal as an isolated mystery, the researchers embedded it into a broader framework that also explains the masses of neutrinos. They showed that the heavy particle and its tiny mass difference are not arbitrary numbers inserted to fit the data, but are the natural result of the same geometric rules that govern the rest of the particle zoo. The model suggests that the universe is organized by a deep, underlying symmetry that links the lightest particles we know to the heaviest ones we are trying to find. While this is currently a theoretical construction supported by computer simulations and mathematical consistency, it offers a clear path forward. If future data from the LUX-ZEPLIN experiment or other detectors confirms the presence of more high-energy events with this specific signature, it would provide strong evidence that this modular framework is the correct way to understand the dark sector. For now, the model stands as a precise, testable hypothesis that turns a single, puzzling event into a potential window into the fundamental architecture of the universe.

Technical Summary: Modular S4 Inverse Seesaw at the LZ High-Recoil Frontier

Problem Statement
The LUX–ZEPLIN (LZ) collaboration recently reported a nuclear-recoil event at approximately 248 keV with a local significance of 3.4σ, extending their analysis to energies near 270 keV. While not definitive evidence for dark matter (DM), this event suggests a qualitatively new regime where endothermic scattering (inelastic DM) could suppress low-energy recoils and concentrate the spectrum near the high-velocity tail of the Galactic halo. The central challenge is to embed this TeV-scale pseudo-Dirac DM state and its specific few-hundred-keV mass splitting ( $\delta$ ) into a predictive flavor framework that also accounts for lepton mixing and neutrino masses, without treating the splitting as an arbitrary low-energy parameter.

Methodology
The authors construct a model based on modular $S_4$ symmetry that unifies the lepton sector and a distinct dark sector through a common complex modulus $\tau$ .

Lepton and Neutrino Sector:
- The model employs a minimal 2NR + 2S inverse seesaw mechanism. This involves three active neutrinos ( $\nu_L$ ), two right-handed neutrinos ( $N_R$ ), and two sterile singlets ( $S$ ).
- Yukawa couplings are promoted to holomorphic modular forms of weight $k$ dependent on $\tau$ .
- The inverse seesaw generates a rank-2 light neutrino mass matrix ( $m_\nu$ ), predicting one effectively massless neutrino at leading order.
- The heavy neutral fermions form two quasi-Dirac pairs at the TeV scale, with small Majorana splittings controlled by a lepton-number-violating parameter $\mu_S$ .
- A global scan is performed over the fundamental domain of $\tau$ and dimensionless coupling ratios to fit charged-lepton masses and neutrino oscillation data ( $\sin^2\theta_{12}, \sin^2\theta_{23}, \sin^2\theta_{13}, \delta_{CP}, R_\nu$ ). Solutions are retained if $\chi^2_{\ell+\nu} < 30$ .
Dark Sector:
- A separate pair of Standard-Model singlet fermions ( $\chi_1, \chi_2$ ) constitutes the DM candidate.
- The DM mass ( $M_\chi$ ) and the inelastic splitting ( $\delta \equiv m_{\chi_2} - m_{\chi_1}$ ) are generated by higher-dimensional operators involving a symmetry-breaking field $\Phi$ and a cutoff scale $\Lambda$ .
- Crucially, the same modular forms $Y^{(4)}_1(\tau)$ and $Y^{(6)}_1(\tau)$ that determine the lepton textures also enter the operators generating the DM Dirac mass and Majorana splitting.
- The dark sector scan is conditional: it is performed only for values of $\tau$ already accepted by the lepton-sector fit. This "common-modulus" requirement filters the DM parameter space based on flavor constraints.
Phenomenological Analysis:
- The model is tested against the LZ high-recoil data using an exploratory two-bin Poisson likelihood.
- The analysis incorporates endothermic kinematics, the Helm nuclear form factor, Standard Halo Model velocities, and Gaussian energy smearing.
- The likelihood profiles over an unknown signal normalization to identify the best-fit $(m_\chi, \delta)$ region.

Key Results

Flavor Fit: The scan yields 181,447 accepted solutions in the lepton sector, distributed across several disconnected modular regions. The best-fit point achieves $\chi^2_{\ell+\nu} \approx 4.87 \times 10^{-7}$ at $\tau \approx -0.068 + 1.424i$ . The model successfully reproduces oscillation parameters and predicts a negligible lightest neutrino mass.
Inverse Seesaw Scales: The physical scan allows for heavy quasi-Dirac fermions at the TeV scale ( $M_0 \sim \text{TeV}$ ) while maintaining small active neutrino masses via the inverse-seesaw suppression. The non-unitarity parameter $R_\eta$ remains within current experimental bounds for the majority of the viable parameter space.
Dark Sector Correlations: Of $2 \times 10^6$ sampled dark-sector points (restricted to flavor-accepted $\tau$ ), 905 points fall within the LZ-oriented region ( $300 \text{ GeV} \le m_\chi \le 2 \text{ TeV}$ and $300 \text{ keV} \le \delta \le 400 \text{ keV}$ ).
Likelihood Best Fit: The likelihood analysis identifies a best grid point at $(m_\chi, \delta) \simeq (1.847 \text{ TeV}, 371.5 \text{ keV})$ . This point lies approximately 0.19 keV below the terrestrial kinematic ceiling for xenon scattering, explaining the likelihood boundary geometry.
UV Structure: The small splitting $\delta$ is not a free parameter but arises from the hierarchy of scales: $\delta/m_\chi \propto (v_\Phi/\Lambda) \times (\text{modular functions})$ . The accepted points form a correlated band in the $(v_\Phi, \Lambda)$ plane, pointing toward an intermediate symmetry-breaking scale ( $v_\Phi \sim 10^9$ GeV) and a cutoff near the grand unification regime.

Significance and Claims
The paper claims to provide a direct link between modular lepton flavor, TeV-scale inverse-seesaw dynamics, and high-recoil inelastic dark matter. The primary significance lies in the predictive bridge created by the common modulus $\tau$ : the dark sector is not scanned independently but is filtered by the flavor structure required to explain neutrino data.

Key distinctions and claims include:

Separation of Sectors: Unlike previous models that might identify the sterile neutrino with the DM particle, this construction keeps the 2NR+2S inverse seesaw (for neutrino masses) and the pseudo-Dirac singlet (for DM) physically distinct. Their connection is purely through the shared modular background.
Naturalness of Splitting: The few-hundred-keV splitting required for the LZ event is derived from the ratio of symmetry-breaking and cutoff scales modulated by calculable modular functions, rather than being inserted as a tiny bare parameter.
Kinematic Interpretation: The model naturally populates the region near the kinematic ceiling of endothermic scattering, providing a consistent explanation for why the LZ event appears at the high-energy edge of the spectrum.
Solar Neutrino Constraints: The authors note that their DM candidate is a Standard-Model singlet coupled via a vector portal, distinct from the Higgsino interpretations often constrained by solar neutrino limits. While solar constraints are powerful discriminators for Higgsinos, the modular singlet model requires a dedicated capture calculation, which is not performed here but is presented as a complementary probe.

The construction demonstrates that the LZ high-recoil regime can be embedded consistently into a broader theoretical framework linking flavor physics, neutrino mass generation, and ultraviolet physics, without requiring independent tuning of the dark sector parameters.

Technical Summary: Modular S4 Inverse Seesaw at the LZ High-Recoil Frontier

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