⚛️ phenomenology

Constraints on Neutron-Electron Quantum Dark Forces

This paper extends previous constraints on quantum dark forces by deriving new limits on the mass and couplings of dark scalar particles exchanged bilinearly between neutrons and electrons.

Original authors: V. Kurmangaliyeva, B. Massak, O. Agyl-Mussapar, S. Amangeldinova, W. M. Snow

Published 2026-09-10

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CC BY 4.0

Original authors: V. Kurmangaliyeva, B. Massak, O. Agyl-Mussapar, S. Amangeldinova, W. M. Snow

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 things we cannot see. Astronomers know that ordinary matter—the stars, the planets, and the people reading this—makes up only a tiny fraction of the cosmos. The rest is hidden in a "dark sector," a realm of particles and forces that do not interact with light and therefore remain invisible to our telescopes. While we cannot see this dark matter directly, physicists suspect it might still whisper to the ordinary world through very faint, new kinds of forces. For decades, scientists have looked for these whispers by studying how particles bump into one another. They have focused mostly on how heavy particles, like protons and neutrons, might feel a tug from dark matter. However, a new study shifts the focus to a much lighter partner: the electron. By asking whether neutrons and electrons might be feeling a hidden pull from dark matter, researchers are testing a different kind of interaction, one that behaves in a unique way depending on the distance between the particles.

A team of physicists from Kazakhstan and the United States has taken a fresh look at this possibility. They investigated a specific theory where dark matter particles do not just push or pull on ordinary matter in a simple, direct line. Instead, they proposed that these dark particles interact through a more complex process involving quantum fluctuations, which are temporary, jittery changes in energy that happen even in empty space. In this scenario, the dark matter acts like a mediator that exchanges energy between a neutron and an electron in a way that creates a force with a very distinct signature. Unlike the familiar forces of gravity or magnetism, which fade away at a steady rate as objects move apart, this new type of force would change its strength in a more complicated pattern as the distance between the neutron and electron shifts. The researchers wanted to know if this specific kind of "quantum dark force" exists, and if it does, how strong it could be.

To find the answer, the team did not build a new machine or launch a new satellite. Instead, they acted as detectives of data, turning their attention to a vast collection of measurements already taken by other scientists over many years. These measurements come from experiments where beams of slow-moving neutrons are fired at various atoms, such as neon, argon, and lead. In these collisions, physicists have carefully measured how the neutrons scatter, or bounce off, the electrons inside those atoms. These scattering patterns are incredibly precise, and they have been used to map out the standard interactions between neutrons and electrons that we already understand. The new study asked a simple but profound question: if we add the effect of this mysterious dark force to our existing models, does it fit the data better, or does it break the picture?

The researchers built a mathematical model that included the known forces between neutrons and electrons, and then they added the potential influence of the dark force. They tested three different versions of this dark force, each corresponding to a different type of dark particle that might be doing the mediating. They then compared their combined model against the real-world data from the neutron scattering experiments. The goal was to see if the data required the presence of the dark force to make sense, or if the standard model alone was sufficient. If the dark force were strong enough to be detected, the data would show a clear mismatch with the standard predictions, revealing a gap that only the new force could fill.

The result of this careful analysis was a definitive silence. The data from the neutron scattering experiments matched the predictions of the standard model perfectly, with no need to invoke any new dark forces. The researchers found no evidence that neutrons and electrons are feeling a hidden tug from dark matter. This does not mean that dark matter does not exist, nor does it prove that these specific quantum forces are impossible. Rather, it places strict limits on how strong such a force could be. The study effectively rules out any version of this dark force that is strong enough to be noticed with current technology. It tells us that if this force exists, it is incredibly weak, far weaker than the team had hoped to find.

The team translated these findings into concrete boundaries for the properties of the hypothetical dark particles. They calculated that for the force to remain invisible to these experiments, the particles carrying it must be either extremely light or interact with ordinary matter so feebly that they are essentially undetectable at the scales they tested. They provided specific numbers for the range of distances where this force could not be hiding, covering scales from the size of an atom down to the tiny distances inside an atomic nucleus. By narrowing the search area, they have helped other scientists know where not to look, allowing the global effort to hunt for dark matter to focus its energy on other possibilities.

This work is part of a larger, ongoing effort to understand the invisible universe. Just as a map is made more useful by marking the areas where treasure is not found, this study helps define the boundaries of the unknown. The researchers used existing data to test a sophisticated theory, and in doing so, they confirmed that the known laws of physics hold up even under the scrutiny of these subtle interactions. While they did not discover a new force, they successfully constrained the possibilities for what might be hiding in the shadows. Their work ensures that future searches for dark matter can proceed with a clearer understanding of the landscape, knowing that this particular path has been walked and found empty.

Technical Summary: Constraints on Neutron-Electron Quantum Dark Forces

Problem Statement
Recent theoretical work by Brax, Fichet, and Pignol introduced the concept of "quantum dark forces," arising from the exchange of light (sub-GeV) dark sector particles that couple bilinearly to Standard Model (SM) particles via quantum fluctuations. Unlike standard single-boson exchange, these bilinear interactions generate potentials with distinct distance dependencies (power-law falloff at short distances and Yukawa-type exponential falloff at long distances). While previous analyses constrained these forces for nucleon-nucleon interactions, the specific case of neutron-electron interactions via bilinear scalar exchange remained unexplored. This paper addresses the gap by deriving new constraints on neutron-electron quantum dark forces, leveraging the unique distance scales probed by neutron scattering data, which lie between nuclear and atomic scales.

Methodology
The authors extend the effective field theory (EFT) framework established in Refs. [10, 11] to the neutron-electron system. The methodology proceeds in three main stages:

Theoretical Formulation: The authors define three effective operators ( $O_a, O_b, O_c$ ) describing the coupling of fermion bilinears ( $\bar{n}n$ and $\bar{e}e$ ) to dark scalar fields. Unlike the nucleon-nucleon case where the loop involves identical fermions, the neutron-electron loop involves a product of distinct Wilson coefficients ( $c_n c_e$ ). The authors derive the coordinate-space potentials ( $V(r)$ ) and the corresponding momentum-space scattering amplitudes ( $l_{NP}(Q)$ ) for these operators. The potentials exhibit characteristic behaviors: $1/r^3$ , $1/r^5$ , and $1/r^7$ at short distances ( $mr \ll 1$ ), and exponential suppression modulated by $r^{-5/2}$ , $r^{-7/2}$ , and $r^{-9/2}$ at long distances ( $mr \gg 1$ ).
Data Integration: The analysis utilizes the Particle Data Group (PDG) average for the neutron-electron scattering length ( $b_{ne}$ ) and the neutron mean-square charge radius ( $r_n^2$ ). These values are derived from thermal neutron scattering experiments on noble gases (Ne, Ar, Kr, Xe) and heavy nuclei (Pb, Bi). The standard model contribution to the scattering amplitude includes terms for the nuclear strong interaction, the neutron charge radius (proportional to $b_{ne}$ ), and neutron polarizability.
Statistical Analysis: A least-squares fit is performed to constrain the new physics contribution. The total scattering length is modeled as a sum of the SM "contact" term, a non-contact term dependent on momentum transfer $Q$ derived from the dark force potentials, and a coherent nucleon-sector companion interaction. The fit minimizes $\chi^2$ over the effective coupling scale ( $1/\Lambda_{ne}$ ) and the nucleon coupling parameter ( $\beta_i$ ), treating the sign of the neutron-electron coupling ( $\alpha_i$ ) as a free parameter.

Key Contributions

Extension of EFT to Neutron-Electron Systems: The paper explicitly applies the Brax-Fichet-Pignol formalism to the neutron-electron sector, deriving the specific loop factors and scattering amplitudes required for this mixed-fermion interaction.
Derivation of Potentials and Amplitudes: The authors provide explicit expressions for the potentials and scattering amplitudes for three distinct operator structures ( $O_a, O_b, O_c$ ), detailing their behavior in both the short- and long-distance limits.
New Experimental Constraints: By analyzing existing neutron scattering data, the paper sets 95% confidence level (C.L.) limits on the effective inverse scale ( $1/\Lambda_{ne}$ ) for dark scalar exchange between neutrons and electrons across a range of mediator masses ( $m$ ) from 1 eV to 10 keV.

Results
The analysis yields no significant deviation from the Standard Model, with no data point exceeding a $1.5\sigma$ deviation. Consequently, the authors establish exclusion limits on the coupling strength:

For the operator $O_a$ (dimension-5 coupling), the limit on $1/\Lambda_{ne}$ ranges from approximately $0.478 \, \text{GeV}^{-1}$ at $m=1$ eV to $5.96 \, \text{GeV}^{-1}$ at $m=10$ keV.
For higher-dimensional operators $O_b$ and $O_c$ , the constraints are weaker (limits on $1/\Lambda_{ne}$ are larger), particularly at the few-keV momentum transfers relevant to the neutron measurements. This is attributed to the additional powers of momentum in the non-contact pieces of these operators, which suppress their sensitivity in the current experimental regime.
The results are presented in Table I and visualized in exclusion plots (Figs. 1–3), showing the allowed parameter space for the effective scale versus the mediator mass.

Significance and Claims
The authors position this work as a complementary analysis to existing constraints on exotic single-boson exchange between nucleons. They claim that by utilizing neutron-electron scattering data, they probe a distinct regime of distance scales that bridges the gap between nuclear and atomic physics.

The paper modestly frames its contributions as a necessary extension of previous work, noting that while the theoretical framework was established for nucleons, the application to electron-nucleon systems required explicit calculation and data analysis. The authors suggest that their results provide a baseline for future constraints. They note that extending these constraints to shorter distance scales would require further theoretical work to calculate the sensitivity of other observables, such as Compton scattering, transition form factor ratios, and charge radii of protons and deuterons. Additionally, they acknowledge that future analyses of isotope shift spectroscopy (King plots) could offer improved constraints, provided higher-order Standard Model corrections are carefully accounted for. The paper does not claim to have discovered new forces but rather to have rigorously constrained the parameter space for such forces using existing high-precision data.

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