⚛️ phenomenology

Annihilation feedback on elastic scattering in self-interacting dark matter

This paper investigates how dark matter annihilation, enhanced by long-range forces, creates a feedback loop that significantly modifies elastic scattering amplitudes—suppressing low-velocity cross-section growth near resonances and generating non-zero elastic floors near scattering zeros—by enforcing unitarity constraints that couple elastic and inelastic processes.

Original authors: Kalliopi Petraki, Thomas Pignard

Published 2026-10-21

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CC BY 4.0

Original authors: Kalliopi Petraki, Thomas Pignard

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

Dark matter is the invisible scaffolding of the universe, a substance that does not emit light but exerts a gravitational pull strong enough to hold galaxies together. For decades, the standard model of cosmology has treated this substance as "cold" and "collisionless," meaning the particles that make it up simply pass through one another like ghosts, interacting only through gravity. However, observations of galaxy clusters and smaller dwarf galaxies have revealed a puzzle. In some places, the centers of these galaxies are less dense than predicted, while in others, they are far too dense. These discrepancies suggest that dark matter might not be a ghost at all, but rather a substance that bumps into itself, exchanging energy and reshaping the galaxies it inhabits. To explain this, physicists have proposed that dark matter particles interact with each other through a new, invisible force, carried by light particles called mediators.

When these mediators are very light, the force they carry becomes long-range, similar to how gravity reaches across vast distances. This long-range force can cause dark matter particles to scatter off one another with surprising intensity, especially when they move slowly. In fact, under certain conditions, the probability of them colliding can grow so large that it threatens to break the fundamental laws of physics that govern how particles behave. Specifically, a principle called unitarity, which ensures that the total probability of all possible outcomes adds up to one, can be violated if the calculations ignore what happens when particles do not just bounce off each other, but disappear into something else. This paper investigates exactly that scenario: what happens when dark matter particles, instead of just scattering, annihilate each other, and how that annihilation changes the way they scatter in the first place.

The researchers, Kalliopi Petraki and Thomas Pignard, focused on a specific type of dark matter model where particles interact via a long-range force and can also annihilate into other particles. In previous calculations, physicists often treated the scattering of particles and their annihilation as separate events. They would calculate how likely particles are to bounce off one another, and then separately calculate how likely they are to vanish. However, the laws of quantum mechanics dictate that these two processes are deeply linked. If particles can disappear, the calculation of how they bounce must account for the possibility of that disappearance. The authors set out to see how this "feedback" from annihilation alters the scattering behavior, particularly in the extreme environments of dwarf galaxies where dark matter moves slowly.

They found that the effect of this feedback is dramatic and depends entirely on the speed of the particles and the specific strength of the force between them. In regions where the force causes a resonant enhancement—a situation where the scattering probability spikes to its maximum allowed limit—the inclusion of annihilation acts as a brake. Without this feedback, calculations predicted that the scattering rate would grow infinitely large as the particles slowed down, eventually exceeding the physical limits of the universe. The new calculations show that the annihilation process steps in to cap this growth. As the particles slow down and the scattering rate tries to explode, the chance of them annihilating increases, which effectively drains energy from the system and prevents the scattering rate from becoming unphysical. The result is a stable, finite scattering rate that respects the laws of physics, rather than the runaway growth predicted by simpler models.

Conversely, the researchers discovered that annihilation can also create scattering where there was none before. In certain specific conditions, the baseline calculation predicts that dark matter particles should pass right through each other without interacting at all, a phenomenon known as a scattering zero. However, because the particles can still annihilate, the feedback mechanism forces a small amount of scattering to appear, filling in this "zero" with a non-zero value. This ensures that the fundamental rules of probability are never broken, even when the primary force seems to vanish. This effect is particularly important because it means that even in scenarios where dark matter was thought to be effectively invisible to itself, it might still interact just enough to influence the structure of galaxies.

The study also revealed that these feedback effects make the predictions for dark matter behavior much more robust. In the old models, the scattering rate was incredibly sensitive to tiny changes in the mass of the dark matter particles or the strength of the force; a minute shift in these values could change the predicted interaction rate by orders of magnitude. By including the annihilation feedback, the researchers found that the results become much more stable. The wild swings in predicted behavior are smoothed out, suggesting that the universe is less finicky about the exact values of these parameters than previously thought. This stability is crucial for comparing theoretical models with real astronomical observations, as it reduces the uncertainty in what we should expect to see in the sky.

These findings have direct implications for understanding the structure of the universe, particularly the dense cores of dwarf galaxies and the compact substructures seen in gravitational lensing. If dark matter is indeed self-interacting, the mechanisms described here suggest that its behavior is self-regulating. The same forces that might cause it to clump together are also limited by the very processes that allow it to vanish. This interplay means that models relying on extreme resonances to explain the density of dark matter cores must be revised to include these annihilation effects. The authors conclude that while the qualitative picture of self-interacting dark matter remains valid, the quantitative details are significantly altered by this feedback, offering a more consistent and physically sound framework for interpreting the cosmos.

Technical Summary: Annihilation Feedback on Elastic Scattering in Self-Interacting Dark Matter

Problem Statement
Self-interacting dark matter (SIDM) models motivated by galactic structure observations often rely on light force mediators to generate velocity-dependent scattering cross sections. While light mediators induce nonperturbative scattering (Sommerfeld enhancement) and resonant behavior, they also render inelastic processes, such as dark matter annihilation, significant. A critical theoretical gap exists in treating elastic and inelastic scattering independently. Unitarity, via the optical theorem, imposes coupled constraints: inelastic processes introduce absorptive contributions to the two-particle self-energy kernel, which in turn modifies the elastic amplitude. Conversely, the resulting dynamics of the incoming state alters inelastic amplitudes. Previous calculations that resum only the purely elastic interaction kernel often violate global unitarity bounds when inelastic channels become large, particularly near Sommerfeld resonances where annihilation cross sections can grow rapidly at low velocities. This paper addresses the feedback of dark matter annihilation on elastic scattering in the presence of a long-range force, specifically investigating how unitarity restoration modifies cross sections near resonances and elastic-scattering zeros.

Methodology
The authors employ a nonrelativistic effective field theory framework where dark matter particles ( $X$ ) interact via a Yukawa potential mediated by a light scalar ( $\varphi$ ). The methodology proceeds through the following steps:

Unitarity Framework: The authors utilize the optical theorem to derive coupled constraints on partial-wave elastic ( $\sigma^{\text{elas}}_\ell$ ) and inelastic ( $\sigma^{\text{inel}}_\ell$ ) cross sections. They formulate the problem using a non-Hermitian effective potential in the Schrödinger equation, where the anti-Hermitian part accounts for probability flux loss into inelastic channels (annihilation).
Baseline vs. Regulated Calculation:
- Unregulated (Baseline): Solves the Schrödinger equation with a Hermitian potential (Yukawa exchange) to obtain baseline phase shifts and cross sections. This approach neglects the feedback of annihilation on the incoming state.
- Regulated (Unitarized): Solves the Schrödinger equation with the full non-Hermitian kernel, including the absorptive (and dispersive) contributions from annihilation.
Renormalization: Since annihilation is a contact-like interaction in the nonrelativistic limit, its iteration leads to ultraviolet (UV) divergences. The authors apply a renormalization procedure (following Refs. [30, 35]) to handle these divergences. They introduce a subtraction scale and renormalized contact coefficients (absorptive and dispersive couplings) to define a finite, unitary theory.
Analytic and Numerical Implementation: The authors derive analytic expressions for the regulated cross sections in terms of the unregulated amplitudes and a projected Green's function ( $W_\ell$ ). They implement a numerical framework for the Yukawa potential to compute Sommerfeld factors and phase shifts, applying the renormalization prescription to specific benchmark scenarios.

Key Contributions and Results

Unitarity Restoration Mechanism: The paper demonstrates that when inelastic cross sections approach their unitarity limits, the feedback from annihilation becomes non-negligible. This feedback is encoded in the non-Hermitian kernel, which modifies the elastic amplitude. The resulting "regulated" cross sections satisfy the coupled unitarity constraints by construction, whereas the "unregulated" baseline calculations can violate them.
Suppression near Sommerfeld Resonances: Near $s$ -wave Sommerfeld resonances, the baseline (unregulated) elastic cross section grows as $v_{\text{rel}}^{-2}$ , and the annihilation cross section grows even faster, eventually exceeding unitarity bounds. The inclusion of annihilation feedback cuts off this growth. As the velocity decreases, the regulated elastic cross section saturates to a constant value (or scales as $v_{\text{rel}}^0$ ) rather than diverging, and the annihilation cross section is similarly suppressed. This effect is most pronounced at low velocities where the Sommerfeld enhancement is strongest.
Regeneration of Elastic Scattering at Zeros: Near zeros of the baseline elastic amplitude (antiresonances), the unregulated elastic cross section vanishes. However, unitarity requires a non-zero elastic cross section if inelastic channels are open. The authors find that annihilation feedback generates a non-zero "elastic floor" in these regions. Specifically, while the baseline elastic cross section scales as $k^6$ (vanishing rapidly), the regulated cross section scales as $k^2$ , approaching a finite constant at zero velocity.
Reduction of Parameter Sensitivity: The study reveals that the extreme sensitivity of baseline cross sections to small variations in model parameters (e.g., the mediator mass or coupling) near resonances and antiresonances is significantly tamed by unitarization. Once feedback is included, cross sections for nearby parameter values converge to similar regulated results, reducing the fine-tuning required to match phenomenological benchmarks.
Phenomenological Impact on SIDM: Applying these results to dark matter halos, the authors show that annihilation feedback modifies the velocity dependence of the viscosity cross section ( $\sigma_V$ ). Near resonances, it prevents the cross section from growing indefinitely at low velocities, potentially altering the viability of resonant SIDM models intended to explain core collapse in subhalos. Near antiresonances, it lifts the suppression, generating a finite cross section where the baseline model predicts none.

Significance
The paper claims that the feedback of inelastic processes on elastic scattering is a necessary component of any consistent calculation of SIDM cross sections in the presence of light mediators. The authors emphasize that:

Consistency: Treating elastic and inelastic channels independently violates unitarity in regimes where inelasticity is enhanced by long-range forces.
Reliability of Resonant Models: The mechanism of using Sommerfeld resonances to generate large low-velocity cross sections must be reassessed when inelastic channels are open, as the feedback can suppress the very enhancement sought.
Robustness: Unitarization reduces the extreme sensitivity of theoretical predictions to small parameter variations, making resonant SIDM models more robust against fine-tuning.
Generality: While the specific calculations focus on a scalar dark sector with annihilation into scalars, the qualitative features (suppression at resonances, regeneration at zeros, reduced sensitivity) are expected to persist across a broad class of SIDM models with symmetric dark matter populations.

The authors note that this work focuses specifically on annihilation feedback and that a companion paper addresses the qualitatively different feedback from radiative bound-state formation (BSF). They conclude that a complete treatment of SIDM phenomenology may eventually require the joint inclusion of multiple inelastic processes (e.g., both annihilation and BSF) in the self-energy kernel.

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