# Universal physics of bound states of a few charged particles

## TL;DR

This paper explores the universal properties of shallow bound states of a few charged particles, revealing unique scaling laws influenced by Coulomb interactions, and emphasizing the importance of Coulomb-modified scattering parameters.

## Contribution

It introduces universal scaling laws for charged particle bound states that depend on Coulomb-modified scattering length and effective range, extending understanding beyond neutral systems.

## Key findings

- Bound states obey unique Coulomb-influenced scaling laws.
- Both Coulomb-modified scattering length and effective range are essential for accurate descriptions.
- States with large spatial extent exhibit universality independent of short-range potential shape.

## Abstract

We study few-body bound states of charged particles subject to attractive zero-range/short-range plus repulsive Coulomb interparticle forces. The characteristic length scales of the system at zero energy are set by the Coulomb length scale $D$ and the Coulomb-modified effective range $r_{\mathrm{eff}}$. We study shallow bound states of charged particles with $D\gg r_{\mathrm{eff}}$ and show that these systems obey universal scaling laws different from neutral particles. An accurate description of these states requires both the Coulomb-modified scattering length and the effective range unless the Coulomb interaction is very weak ($D\to \infty$). Our findings are relevant for bound states whose spatial extent is significantly larger than the range of the attractive potential. These states enjoy universality -- their character is independent of the shape of the short-range potential.