Universal physics of bound states of a few charged particles

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 Dand the Coulomb-modified effective range r(eff). We study shallow bound states of charged particles with D >> r(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 -> infinity). 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. (C) 2019 The Authors. Published by Elsevier B.V.