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Inelastic Kinetic Decoupling in WIMP Leptogenesis: A Minimal $Z_4$ Model

This paper investigates a minimal $Z_4$ -symmetric WIMP leptogenesis model where inelastic scattering drives early kinetic decoupling, significantly altering dark matter relic abundance through temperature evolution while leaving the final $B-L$ asymmetry largely unchanged due to the timing of asymmetry production relative to chemical decoupling.

Original authors: An Liu, Hong-Hao Zhang

Published 2026-10-30

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CC BY 4.0

Original authors: An Liu, Hong-Hao Zhang

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 holds galaxies together, yet we have never seen a single particle of it. This dark matter makes up about 27 percent of the cosmos, while the ordinary matter that forms stars, planets, and people accounts for only a tiny fraction. At the same time, the universe contains a profound mystery regarding why there is more matter than antimatter. In the early moments after the Big Bang, physics suggests that matter and antimatter should have been created in equal amounts, only to annihilate each other completely, leaving behind a void of pure energy. Instead, a small surplus of matter survived, eventually forming everything we see today. For decades, physicists have searched for a theory that could explain both the existence of dark matter and the origin of this cosmic imbalance, hoping that a single mechanism might solve both puzzles at once.

A recent study by researchers at Sun Yat-Sen University explores a specific scenario where these two mysteries are linked. They propose a model involving a new, invisible particle that acts as dark matter. In their theory, this particle does not simply sit still; it interacts with other particles in a way that creates an imbalance between matter and antimatter. The researchers focused on a process called "leptogenesis," where an initial difference in the number of leptons (a family of particles that includes electrons) is eventually converted into the excess of matter we observe. Their work investigates how the dark matter particles behave as the universe cools down, specifically looking at a moment when they stop exchanging energy with the hot soup of particles that filled the early cosmos.

The team built a mathematical framework based on a simple set of rules, adding a few new particles to the known laws of physics. In their model, the dark matter particle is a type of fermion, a fundamental building block of matter, which is kept stable by a specific symmetry that prevents it from decaying. To generate the necessary matter-antimatter imbalance, these dark matter particles annihilate with each other, producing a heavy partner particle and a standard lepton. This process is designed to favor the creation of matter over antimatter through subtle quantum effects. The researchers then simulated the history of the universe, tracking how the number of dark matter particles changed and how the temperature of this dark matter evolved relative to the temperature of the rest of the universe.

A key discovery in their work concerns the timing of when dark matter stops moving in sync with the rest of the cosmic soup. Usually, physicists assume that dark matter stays in thermal equilibrium, meaning it shares the same temperature as the surrounding particles, until it freezes out and stops interacting. However, this study shows that in their specific model, the dark matter can lose this connection much earlier than expected. This happens because the dark matter particles try to convert into a heavier partner particle, but as the universe cools, there is not enough energy left to make this conversion happen frequently. The process becomes inefficient, and the dark matter effectively decouples from the rest of the universe while it is still relatively hot.

This early separation has a significant impact on how much dark matter remains today. When the dark matter particles stop sharing energy with the rest of the universe, their internal temperature drops at a different rate than the surrounding particles. This change in temperature alters how often they collide and annihilate with each other. The researchers found that this effect can change the final amount of dark matter by a noticeable margin, sometimes increasing it and sometimes decreasing it, depending on the specific properties of the particles involved. In some scenarios, the annihilation rate becomes so sensitive to the speed of the particles that the early temperature shift leads to a much larger or smaller leftover population than standard calculations would predict.

Despite these changes in the amount of dark matter, the study found that the generation of the matter-antimatter imbalance is surprisingly robust. The process that creates the excess of matter happens mostly before the dark matter fully separates from the rest of the universe. By the time the dark matter temperature shifts significantly, the crucial work of creating the asymmetry is already nearly finished. As a result, the final amount of matter in the universe remains largely unchanged, even when the amount of dark matter varies. This suggests that the two phenomena, while linked in the model, are not equally sensitive to the timing of the dark matter's cooling.

The researchers also checked if their model could be tested by looking for signals from space. If dark matter particles are annihilating today, they should produce high-energy photons that could be detected by telescopes. The team calculated the expected signal and compared it with data from the Fermi-LAT telescope, which observes dwarf galaxies for such emissions. They found that the surviving versions of their model, which produce the correct amount of dark matter and matter, are consistent with current observations. The predicted signals are generally weak enough to avoid being ruled out, though they remain within reach of future, more sensitive instruments.

Ultimately, this work demonstrates that the history of dark matter is more complex than previously assumed. The moment when dark matter stops exchanging energy with the rest of the universe is a critical turning point that can reshape our understanding of how much of it exists. By treating the interactions of these particles with full mathematical precision, the researchers showed that early decoupling is a natural outcome in certain models. While the exact nature of dark matter remains unknown, this study provides a clearer picture of how it might have behaved in the early universe and how its behavior is tied to the very existence of the matter that makes up our world.

Technical Summary: Inelastic Kinetic Decoupling in WIMP Leptogenesis: A Minimal Z4 Model

Problem Statement
The paper addresses two fundamental gaps in the Standard Model (SM): the nature of dark matter (DM) and the origin of the cosmic baryon asymmetry. While WIMP leptogenesis offers a mechanism where DM annihilations generate a lepton asymmetry (later converted to baryon asymmetry via sphalerons), most existing studies assume DM remains in kinetic equilibrium with the thermal bath until chemical decoupling ( $T_\chi = T$ ). This assumption neglects the possibility of early kinetic decoupling (eKD), where momentum exchange becomes inefficient before chemical freeze-out. In scenarios involving resonant annihilation or specific interaction channels, eKD can significantly alter the DM relic abundance. Furthermore, the impact of eKD on the generation of the lepton asymmetry in WIMP leptogenesis models has received limited attention. The authors investigate this interplay within a minimal $Z_4$ -symmetric extension of the SM, specifically focusing on a regime where kinetic equilibrium is maintained primarily by inelastic conversion processes rather than elastic scattering.

Methodology
The authors construct a model extending the SM with a singlet Dirac fermion $\chi$ (DM), a vectorlike fermion doublet $\Psi$ , and two pseudoscalar mediators $\phi_{1,2}$ . The model is stabilized by a $Z_4$ symmetry. Key features include:

Kinetic Maintenance: Kinetic equilibrium is driven mainly by the inelastic conversion $\chi L \leftrightarrow \bar{\chi} \Psi$ , where $L$ is the SM lepton doublet. The mass gap $\Delta m = m_\Psi - m_L$ leads to Boltzmann suppression of this process as the Universe cools, naturally inducing eKD.
Collision Operators: Unlike studies relying on the Fokker-Planck approximation (valid only for small momentum transfers), the authors utilize the full inelastic collision operator. They derive an effective momentum transfer rate ( $\gamma_{kin}$ ) by linearizing the inelastic conversion contribution around equilibrium.
Coupled Evolution: The authors solve a system of coupled Boltzmann equations for:
1. The DM number density ( $Y_\chi$ ).
2. The DM kinetic temperature ( $T_\chi$ , parameterized by $y = m_\chi T_\chi s^{-2/3}$ ).
3. The lepton asymmetry ( $Y_{\Delta L_a}$ ).
Spectator Relations: The visible sector $B-L$ asymmetry is reconstructed from the lepton doublet asymmetry using spectator coefficients that depend on the decoupling time of the hidden sector fermion $n_R$ .
Numerical Scan: A parameter scan is performed over $m_\chi$ , the mass ratio $r = m_\Psi/m_\chi$ , the resonance parameter $\delta$ , and couplings. The analysis compares results from the coupled Boltzmann equations (cBE, allowing $T_\chi \neq T$ ) against the standard treatment (nBE, assuming $T_\chi = T$ ). Constraints include the observed DM relic density ( $\Omega_{DM}h^2 \simeq 0.12$ ), the baryon asymmetry, and indirect detection limits from Fermi-LAT, H.E.S.S., and CTA.

Key Contributions

Formalism for Inelastic Decoupling: The paper provides a rigorous derivation of the effective momentum transfer rate for kinetic decoupling driven by inelastic conversion, avoiding the limitations of the Fokker-Planck expansion when large mass gaps exist.
Interplay of Timescales: The study systematically categorizes the impact of eKD based on the relative ordering of three critical timescales:
- $x_{kd}$ : Kinetic decoupling ( $m_\chi/T$ ).
- $x_{cd}$ : Chemical decoupling.
- $x_{lep}$ : The end of the main asymmetry production stage.
Resonance and Heating Effects: The authors demonstrate that near resonance, the annihilation cross-section's velocity dependence can lead to "annihilation-induced heating." Even after kinetic decoupling, preferential annihilation of low-velocity DM can temporarily raise $T_\chi$ , modifying the expected suppression or enhancement of the relic abundance.
Indirect Detection Constraints: The paper evaluates the indirect detection signals for this specific cascade scenario ( $\chi\chi \to L\bar{\Psi} \to \text{SM} + n_R$ ) using Pythia-generated spectra and Fermi-LAT dwarf spheroidal likelihoods.

Results

Relic Abundance: Early kinetic decoupling can significantly modify the DM relic abundance, particularly when $x_{kd} \lesssim x_{cd}$ and the annihilation cross-section has strong velocity dependence (e.g., near resonance). The ratio of cBE to nBE relic densities can deviate significantly from unity (e.g., factors of $\sim 0.5$ to $\sim 1.3$ in benchmark points). The direction and magnitude of the shift depend on the coefficient $C$ in the velocity expansion $\langle \sigma v \rangle \approx \sigma_0(1 + Cv^2)$ and the subsequent thermal evolution.
Lepton Asymmetry: The final $B-L$ asymmetry is generally robust against eKD effects in viable parameter regions. For most allowed points, asymmetry production concludes before chemical decoupling ( $x_{lep} < x_{cd}$ ), meaning the asymmetry is largely fixed before significant kinetic departures occur. Consequently, the cBE and nBE predictions for the asymmetry remain close to unity. Significant deviations in the asymmetry ratio ( $>1$ ) are observed primarily in excluded parameter regions where $x_{lep} > x_{cd}$ , allowing the source term to remain sensitive to the non-thermal DM temperature evolution.
Indirect Detection: The viable parameter space, consistent with the observed relic density and baryon asymmetry, satisfies current Fermi-LAT constraints. Near resonance, the present-day annihilation rate can be enhanced due to lower DM velocities, but the allowed points generally remain below or near the thermal cross-section limit ( $\sim 2 \times 10^{-26} \text{ cm}^3 \text{ s}^{-1}$ ) or within projected CTA sensitivities.
Dark Radiation: The contribution of the hidden sector to dark radiation ( $\Delta N_{\text{eff}}$ ) is calculated to be $\sim 0.06$ , well within current cosmological bounds.

Significance and Claims
The paper claims that neglecting early kinetic decoupling in WIMP leptogenesis models with inelastic conversion channels can lead to inaccurate predictions for the DM relic abundance, potentially by factors of order unity. However, it emphasizes that the generated baryon asymmetry is less sensitive to these kinetic effects, provided the asymmetry production finishes before chemical decoupling. The study highlights that the "standard" assumption of kinetic equilibrium is not merely a simplification but can be a source of significant error in resonant or inelastic scenarios. The work establishes a framework for treating coupled number density and temperature evolution in models where inelastic processes dominate momentum exchange, offering a more complete picture of the thermal history of WIMP leptogenesis. The authors note that while their scan identifies viable regions, the largest deviations in asymmetry occur in regions that are currently excluded by relic density or other constraints, suggesting that for viable models, the standard treatment of asymmetry generation remains a reasonable approximation.

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