⚛️ phenomenology

Ultra-relativistic freeze-out of fermion dark matter through a heavy scalar mediator

This paper investigates the production of Majorana fermion dark matter via a heavy scalar mediator during reheating, identifying three distinct regimes (freeze-in, WIMP-like freeze-out, and ultra-relativistic freeze-out) and demonstrating that accurate relic abundance predictions in the ultra-relativistic freeze-out scenario require a consistent treatment of continued production and entropy dilution, which are significantly influenced by scalar cubic interactions.

Original authors: Jin-Lei Yang, Ke-Shuo Cai, Yin-Jie Zhang, Hai-Bin Zhang, Tai-Fu Feng

Published 2026-10-20

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CC BY 4.0

Original authors: Jin-Lei Yang, Ke-Shuo Cai, Yin-Jie Zhang, Hai-Bin Zhang, Tai-Fu Feng

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 invisible matter that we cannot see, touch, or smell, yet we know it is there because of the way it pulls on stars and bends light. This substance, known as dark matter, makes up about a quarter of everything in existence, but scientists have never been able to identify what particle creates it. For decades, the leading idea was that dark matter consists of heavy, slow-moving particles that once floated freely in the hot early universe before cooling down and freezing into their current state. However, as experiments have ruled out many of these heavy candidates, physicists have begun to look at other possibilities, including particles that are much lighter and move at incredible speeds, or those that never quite settled into a calm state with the rest of the universe.

A team of researchers has now explored a specific scenario where dark matter is created through a heavy, invisible particle that acts as a bridge between the known world and the hidden one. They focused on a period just after the Big Bang when the universe was still heating up and expanding rapidly, a time known as reheating. In their model, the dark matter particles are fermions, a type of matter that includes electrons and protons, and they interact with the rest of the universe only through a heavy scalar particle, which is a type of force carrier. The researchers used detailed computer simulations to track how these particles would behave as the universe cooled, calculating exactly how many would be produced and how fast they would move.

The study reveals that the history of dark matter is not limited to just one story. Depending on the mass of the heavy bridge particle and the temperature of the early universe, the dark matter could have been created in three distinct ways. In some cases, the particles were so weakly connected to the rest of the universe that they were slowly produced over time, never reaching a state of balance. In other cases, they behaved like the traditional heavy particles, reaching a perfect balance with the surrounding heat before freezing out as the universe cooled. Most notably, the researchers found a third, more complex path called ultra-relativistic freeze-out. In this scenario, the dark matter particles reached a state of balance with the hot soup of the early universe, but then fell out of sync while still moving at nearly the speed of light, long before they slowed down to a crawl.

What makes this third path particularly interesting is that the production of dark matter did not stop the moment the particles fell out of balance. Even after they stopped interacting frequently enough to stay in equilibrium, the universe continued to churn out new dark matter particles. At the same time, the rapid expansion of the universe during reheating was injecting new energy and stretching space, which effectively diluted the amount of dark matter present. The researchers found that to understand how much dark matter exists today, one must carefully balance these two competing effects: the continued creation of new particles and the dilution caused by the expanding universe. If scientists only looked at the moment the particles stopped interacting, they would significantly underestimate the final amount of dark matter.

The team also examined how the internal structure of the heavy bridge particle affects these outcomes. They discovered that the strength of the connection between the heavy bridge and the lighter particles in the universe plays a much larger role than the strength of the bridge's own internal interactions. By adjusting these connections, the boundaries between the different creation stories shift, changing which scenario is most likely to produce the exact amount of dark matter we observe today. Their calculations show that the ultra-relativistic freeze-out scenario is a viable and robust explanation for the dark matter we see, working across a wide range of masses and temperatures. This suggests that the dark matter in our universe could be much lighter and faster than previously thought, and that its history is written in the complex interplay of creation and dilution during the universe's earliest, hottest moments.

Technical Summary: Ultra-relativistic freeze-out of fermion dark matter through a heavy scalar mediator

Problem Statement
The Standard Model (SM) lacks a viable particle candidate for dark matter (DM), which constitutes approximately one-quarter of the cosmic energy density. While thermal freeze-out (WIMP) and freeze-in (FIMP) mechanisms are well-studied, the parameter space for WIMPs is increasingly constrained by direct detection experiments, and FIMPs are difficult to probe due to their feeble couplings. An intermediate mechanism, ultra-relativistic freeze-out (UFO), has been proposed where DM reaches chemical equilibrium but decouples at a temperature significantly higher than its mass ( $T_{FO} \gg M_\chi$ ). Previous investigations into UFO have focused on heavy vector mediators. This paper addresses whether the UFO mechanism persists in models mediated by heavy scalars, a scenario motivated by the prevalence of additional scalar fields in SM extensions. The authors aim to determine if scalar-mediated interactions can support UFO and to clarify the connection between WIMP-like freeze-out, freeze-in, and UFO in this context.

Methodology
The authors extend the SM with a real gauge-singlet scalar $\phi$ and a Majorana fermion $\chi$ . The model includes a scalar potential with cubic and quartic interactions, leading to mixing between the SM Higgs doublet and the singlet. The DM production is analyzed during the reheating epoch and the subsequent radiation-dominated era.

Production Rates: The authors calculate the total DM particle-number production rate per unit volume, $R_\chi(T)$ . This includes contributions from:
- Scattering processes: $h_1 h_1 \to \chi\chi$ , $h_2 h_2 \to \chi\chi$ , $h_1 h_2 \to \chi\chi$ , and annihilation of SM particles ( $W^+W^-, ZZ, t\bar{t}$ ) via scalar exchange.
- Decays: $h_1 \to \chi\chi$ and $h_2 \to \chi\chi$ .
- The calculation retains full mass dependence, includes fermion exchange in $t$ and $u$ channels, and accounts for scalar mixing.
- To avoid double counting, the real-intermediate-state (RIS) contribution from resonant scattering is subtracted from the decay terms.
Cosmological Evolution: The evolution of the DM abundance is governed by the Boltzmann equation. The authors model perturbative reheating where the inflaton condensate behaves like matter, leading to a temperature scaling $T \propto a^{-3/8}$ during reheating and $T \propto a^{-1}$ during radiation domination. The Hubble expansion rate $H(T)$ is approximated piecewise to match continuously at the reheating temperature $T_{RH}$ .
Regime Classification: The production regimes are identified by solving the Boltzmann equation numerically:
- Freeze-in (FI): DM never reaches chemical equilibrium ( $\Gamma_{int}/H < 1$ ).
- WIMP-like Freeze-out (FO): DM reaches equilibrium and decouples in the non-relativistic regime ( $T_{FO} \lesssim M_\chi$ ).
- Ultra-relativistic Freeze-out (UFO): DM reaches equilibrium but decouples while relativistic ( $T_{FO} > M_\chi$ ). The authors define specific operational criteria based on the ratio of interaction rate to expansion rate and the deviation from equilibrium density to distinguish these regimes.

Key Contributions and Results

Identification of Three Regimes: Depending on the mediator mass ( $M_{h_2}$ ), Yukawa coupling ( $y_\chi$ ), and reheating temperature ( $T_{RH}$ ), the model realizes three distinct production regimes (FI, WIMP-like FO, and UFO), all capable of reproducing the observed relic abundance $\Omega_{DM}h^2 \approx 0.12$ .
UFO Dynamics in Scalar Models: The study confirms that UFO is a viable mechanism for scalar-mediated fermion DM. A key finding is that in the UFO regime, significant DM production persists after chemical decoupling. Simultaneously, entropy injection during reheating dilutes the DM yield. Consequently, an accurate determination of the final relic abundance requires a consistent treatment of both continued production and entropy dilution; approximating the final abundance solely by the equilibrium value at decoupling leads to underestimation.
Impact of Scalar Cubic Interactions: The authors examine the role of scalar cubic couplings, specifically the mixing-induced coupling $\kappa_{H\phi}$ $κ_{H ϕ}$ and the heavy-scalar self-coupling $\kappa$ $κ$ .
- The coupling $\kappa_{H\phi}$ significantly modifies the boundaries between production regimes and reshapes the relic-density contours. Increasing $\kappa_{H\phi}$ enhances interactions with the SM bath (via $W^+W^-, ZZ, t\bar{t}$ and $h_1$ decays), allowing chemical equilibrium to be sustained to lower temperatures, thereby shifting regions from UFO to WIMP-like FO.
- In contrast, the heavy-scalar self-coupling $\kappa$ has a modest effect on the production regimes and relic contours within the explored parameter space.
Parameter Space Scans: Numerical scans in the $(M_\chi, T_{RH})$ plane demonstrate that UFO can account for the relic abundance over extended parameter regions, including for large Yukawa couplings ( $y_\chi = 1$ ). The results highlight that the transition between regimes is sensitive to the interplay between the DM mass, mediator mass, and reheating history.

Significance
The paper establishes that the ultra-relativistic freeze-out mechanism is robust across different mediator types, persisting in heavy scalar-mediated scenarios just as it does in vector-mediated ones. The work provides a rigorous framework for calculating DM abundance in the presence of simultaneous entropy dilution and post-decoupling production, which is critical for accurately predicting relic densities in reheating scenarios. Furthermore, it clarifies that scalar cubic interactions, particularly those linking the heavy mediator to the light scalar sector, play a decisive role in determining the thermal history and the viability of the UFO mechanism, offering a distinct phenomenological signature compared to vector portals.

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