Perturbative Unitarity Violation in Radiative Capture Transitions to Dark Matter Bound States

TL;DR

This paper analyzes the formation of dark matter bound states via radiative capture, revealing conditions under which perturbative unitarity is violated due to strong overlaps of wave functions in Coulomb potentials.

Contribution

It provides closed-form cross sections for bound-state formation with various multipole transitions and explains unitarity violations in terms of wave function overlaps in Coulomb potentials.

Findings

Bound-state formation cross sections derived for different quantum numbers.

Unitarity violation occurs at specific velocities for large quantum numbers.

Overlap of wave functions causes effective strong interactions leading to unitarity issues.

Abstract

We investigate the formation of bound states of non-relativistic dark matter particles subject to long-range interactions through radiative capture. The initial scattering and final bound states are described by Coulomb potentials with different strengths, as relevant for non-abelian gauge interactions or theories featuring charged scalars. For bound states with generic quantum numbers $n$ and $\ell$, we provide closed-form expressions for the bound-state formation (BSF) cross sections of monopole, dipole and quadrupole transitions, and of arbitrary multipole order when $\ell=n-1$. This allows us to investigate in detail a strong enhancement of BSF that occurs for initial states in a repulsive potential. For $\ell=n-1\gg 1$, we show that the BSF cross section for each single bound state violates the perturbative unitarity bound in the vicinity of a certain critical initial velocity, and provide an interpretation in terms of a smooth matching of classical trajectories. When summing the BSF cross section over all possible bound states in the final state, this leads to a unitarity violation below a certain velocity, but within the validity range of the weakly coupled non-relativistic description. We identify an effectively strong interaction as the origin of this unitarity violation, which is caused by an "anomalously" large overlap of scattering and bound-state wave functions in Coulomb potentials of different strength.