⚛️ high-energy theory

Unitarity violation and restoration in radiative bound-state formation

This paper demonstrates that the severe partial-wave unitarity violations found in non-relativistic radiative bound-state formation calculations are resolved by properly resumming the inelastic contributions to the incoming state's self-energy.

Original authors: Marcos M. Flores, Kalliopi Petraki

Published 2026-06-24

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CC BY 4.0

Original authors: Marcos M. Flores, Kalliopi Petraki

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 Big Picture: A Traffic Jam in the Quantum World

Imagine you are watching two cars (particles) driving toward each other on a highway. In the world of quantum physics, sometimes these cars don’t just bounce off each other; they crash together and stick, forming a single, larger vehicle (a "bound state"). To make this happen, they have to shed some energy, usually by shooting out a spark or a piece of debris (radiating a particle). This process is called Radiative Bound-State Formation.

Physicists are very interested in this because it might explain Dark Matter—the invisible stuff that makes up most of the universe’s mass. If Dark Matter particles stick together like this, it changes how we calculate how much of it exists today.

The Problem: The "Speed Limit" Violation

In physics, there is a strict rule called Unitarity. Think of it as a cosmic speed limit or a conservation law for probability. It says that the total chance of something happening (the cars bouncing, sticking, or disappearing) must add up to 100%. You can’t have a 150% chance of an event occurring.

The paper points out a major glitch in current calculations. When physicists calculate the likelihood of these Dark Matter cars sticking together at very slow speeds, the math says the probability shoots up way past 100%. It suggests that at low speeds, the cars are more than certain to stick together, which is physically impossible.

Why does this happen?
Imagine the two cars are approaching each other. In standard calculations, we look at all the possible "parking spots" (energy levels) they could end up in. There are infinite parking spots available. The math shows that as the cars slow down, the number of available spots that attract them grows so fast that the total probability breaks the "speed limit."

The Solution: Fixing the Roadmap

The authors argue that the old calculations were incomplete. They were looking at the cars in isolation, ignoring how the act of sticking together actually changes the road the cars are driving on.

Here is the analogy:

The Old Way: Imagine calculating how likely a car is to crash by only looking at the driver’s skill, ignoring the fact that the road itself might be slippery or curvy.
The New Way: The authors say, "Wait, the act of potentially crashing changes the road."

In quantum terms, the possibility of the particles sticking together creates an "absorptive potential." This is like a fog or a drag on the road that slows the particles down or changes their path before they even stick. The old math ignored this fog.

The authors use a mathematical technique called resummation. Instead of calculating the crash probability once, they calculate how the "fog" (the potential) changes the cars' movement, which changes the crash probability, which changes the fog, and so on, until the numbers settle down.

The Key Discovery: The "Hidden Poles"

The most technical part of the paper involves looking at the math in the "complex momentum plane." Don’t worry about the jargon; think of it as looking at the hidden structure of the equation.

The authors found that the equations have specific "singularities" (think of them as mathematical black holes or sharp spikes) that were previously ignored. These spikes represent the infinite number of excited states (parking spots) the particles could fall into.

By properly accounting for these spikes, the authors show that:

The "fog" (absorptive potential) gets stronger as the particles slow down.
This fog effectively "absorbs" some of the probability that was causing the violation.
The final probability stops growing infinitely and settles at a safe, physical number (specifically, it caps out at about 25% of the maximum possible limit for that specific type of interaction).

Why This Matters

The paper concludes that if you want to accurately predict Dark Matter behavior—like how much of it is left over from the Big Bang (freeze-out) or what signals we might detect from it today (indirect detection)—you must use this corrected math.

If you use the old, broken math, you might think Dark Matter freezes out differently than it actually does, or you might predict signals that are too strong or too weak. By fixing the "unitarity violation," the authors provide a reliable tool for physicists to make accurate predictions about the invisible universe.

Summary in a Nutshell

The Issue: Old calculations said Dark Matter particles sticking together at slow speeds had an impossible probability (>100%).
The Cause: The calculations ignored how the possibility of sticking together changes the particles' motion beforehand.
The Fix: The authors added a mathematical correction that accounts for this feedback loop.
The Result: The probability is tamed and stays within physical limits, allowing for accurate predictions about Dark Matter.

Technical Summary: Unitarity Violation and Restoration in Radiative Bound-State Formation

Problem Statement
The paper addresses a severe violation of partial-wave unitarity in calculations of radiative bound-state formation (BSF) within the non-relativistic regime. This issue is particularly relevant for dark matter (DM) phenomenology, where long-range interactions and bound states play a critical role in freeze-out and indirect detection. Standard calculations for BSF processes, particularly those involving monopole transitions where the emitted boson carries a conserved charge (e.g., charged scalars or non-Abelian gauge bosons), yield cross-sections that grow unphysically at low velocities. This growth leads to an apparent violation of the unitarity bound on inelastic cross-sections, even for arbitrarily small couplings. The authors identify that this violation stems from the neglect of the absorptive contributions to the self-energy of the incoming scattering state generated by the BSF processes themselves.

Methodology
The authors employ a two-pronged approach: first, quantifying the severity of the unitarity violation in existing "unregulated" calculations, and second, applying a formal unitarization scheme to restore consistency.

Quantification of Violation: The authors derive Kramers-like formulae for individual partial waves ( $\ell$ ) by summing BSF cross-sections over all accessible bound levels ( $n$ ). They analyze the scaling of these cross-sections with velocity for monopole transitions ( $\Delta \ell = 0$ ) in theories with Coulomb-like potentials. They distinguish between cases where the scattering potential coupling $\alpha_S$ is zero and where it is non-zero, defining the ratio $\lambda = \alpha_S/\alpha_B$ .
Unitarization Framework: The authors apply the formalism developed in Refs. [2, 15], which relates regulated (unitarized) quantities to unregulated ones via the resummation of inelastic contributions to the incoming-state self-energy. This involves:
- Constructing an anti-Hermitian potential from irreducible inelastic amplitudes.
- Solving the Schrödinger equation with this potential to obtain regulated wavefunctions.
- Evaluating the regularization matrix $N_\ell(k)$ , which encodes the interplay between absorptive (on-shell) and dispersive (off-shell) effects.
- Crucially, the authors account for the specific analytic structure of the BSF amplitudes in the complex momentum plane, identifying singularities (poles) associated with off-shell bound states that must be included in the principal value integrals defining the dispersive corrections.

Key Contributions

Derivation of Kramers-like Formulae: The paper provides analytical approximations for the sum of BSF cross-sections over bound levels for individual partial waves. These formulae explicitly demonstrate that for $\lambda \le 1$ , the inclusive cross-section grows monotonically at low velocities, violating unitarity.
Role of Singularities: The authors highlight that the unitarization of BSF is not merely a matter of resumming the on-shell absorptive part (as in simple contact interactions) but requires careful treatment of the non-analytic structure of the amplitudes. The singularities in the complex momentum plane, arising from the bound-state spectrum, generate dispersive mixing terms that are essential for taming the low-velocity growth.
General Applicability: While focusing on monopole transitions with charged scalar emission as a prototype, the methodology is shown to be generalizable to other radiative transitions, including dipole transitions in Abelian and non-Abelian theories.

Results

Unitarity Violation: For $\alpha_S = 0$ (and generally $\lambda < 1$ ), the unregulated inclusive BSF cross-section $\sigma_{BSF}^\ell$ scales as $\zeta_B \ln(\zeta_B)$ or faster at low velocities (where $\zeta_B \propto 1/v_{rel}$ ). This leads to $\sigma_{BSF}^\ell > \sigma_U^\ell/4$ (the unitarity limit) for sufficiently low velocities, regardless of how small the coupling $\alpha_{rad}$ is.
Restoration of Unitarity: By including the dispersive corrections from the off-shell singularities (encoded in the matrix $W_\ell$ ), the regulated cross-sections saturate at the unitarity ceiling. Specifically, the regulated inclusive cross-section $\sigma_{BSF, reg}^\ell / \sigma_U^\ell$ approaches $\sim 1/4$ at low velocities for perturbative couplings.
Importance of Off-Diagonal Terms: Numerical evaluations show that the off-diagonal elements of the regularization matrix $W_\ell$ , which represent mixing between different off-shell bound levels, are crucial. Neglecting these terms (keeping only diagonal elements) fails to fully restore unitarity, resulting in residual growth.
Dependence on $\lambda$ : For $\lambda > 1$ , the contributions from highly excited bound states are exponentially suppressed, and the unregulated cross-sections do not exhibit severe unitarity violation; thus, the unitarization corrections are minimal.

Significance
The paper claims that proper unitarization is essential for quantitatively reliable phenomenological predictions in scenarios where BSF is significant. This includes:

DM Freeze-out: Ensuring that freeze-out occurs reliably and yielding correct relic density predictions.
Indirect Detection: Providing accurate rates for high-energy signals from metastable bound state decays and low-energy emission from BSF.
Theoretical Consistency: Resolving the apparent paradox of unitarity violation in weakly-coupled theories by demonstrating that the resummation of self-energy contributions, including their non-analytic structure, restores consistency with the optical theorem.

The authors emphasize that their results apply to a variety of theories, including WIMP models, colored coannihilation scenarios, and hidden-sector models, and provide a robust framework for future phenomenological studies.