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arXiv:1904.00913 · Physics Letters B

Universal physics of bound states of a few charged particles

Christiane H. Schmickler, H.‐W. Hammer, Artem G. Volosniev

Published Oct 10, 2019• 18 citations
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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 reff. We study shallow bound states of charged particles with D≫reff 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→∞). 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.

DOI
10.1016/j.physletb.2019.135016

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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 reff. We study shallow bound states of charged particles with D≫reff 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→∞). 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.
We study charged particles in three dimensions interacting via a short-range potential in addition to the Coulomb potential. When the Bohr radius and the scattering length are much larger than the potential range, low-energy physics of the system becomes independent from details of the short-range potential. We develop the zero-range theory to describe such universal physics in terms of the Bohr radius and the scattering length by generalizing the Bethe-Peierls boundary condition, which is then applied to two charged particles to reveal their bound states and resonances. Infinite resonances are found for a repulsive Coulomb potential, one of which turns into a bound state with increasing inverse scattering length, whereas infinite bound states exist for an attractive Coulomb potential with no resonances at any scattering length. The zero-range theory is also applied to three equally charged particles at infinite scattering length under the variational Born-Oppenheimer approximation. We find that the effective potential between two heavy particles induced by a light particle is an inverse-square attraction at distances shorter than the Bohr radius, leading to infinite deep bound states, whereas shallow ones successively turn into resonances with increasing Coulomb repulsion.
We study the structure of bound states appearing in systems governed by the Coulomb and short-range interactions. We analyze the binding energies and wave functions of the bound states generated by the Coulomb plus short-range potential. We demonstrate that Coulomb-induced shifts of the binding energy are closely correlated with the spatial distribution of the wave function. Furthermore, we show that the asymptotic behavior of wave functions of weakly bound states is qualitatively altered by Coulomb repulsion, leading to a modification of the near-threshold mass scaling that is otherwise universal for short-range interactions.
Very weakly bound systems may manifest intriguing "universal" properties, independent of the specific interaction which keeps the system bound. An interesting example is given by relations between the size of the system and the separation energy, or scaling laws. So far, scaling laws have been investigated for short-range and long-range (repulsive) potentials. We report here on scaling laws for weakly bound two-body systems valid for a larger class of potentials, i.e. short-range potentials having a repulsive core and long-range attractive potentials. We emphasize analogies and differences between the short- and the long-range case. In particular, we show that the emergence of halos is a threshold phenomenon which can arise when the system is bound not only by short-range interactions but also by long-range ones, and this for any value of the orbital angular momentum ℓ\ellℓ. These results enlarge the image of halo systems we are accustomed to.
Universal physics manifests in low-energy phenomena over a large array of different systems from nuclear to atomic physics. Here, two interesting aspects of universality are studied using the Gaussian Expansion Method (GEM). This is a variational method that uses products of Gaussians as basis functions. Gaussian potentials are used as a short-range potential. The first aspect is the behaviour close to the dimer + atom breakup threshold of Efimov states and associated universal tetramers in ultra-cold mixtures of alkali atoms. These can be treated as bosons here. It is shown that trimer and tetramer vanish into the threshold at almost the same point. The predictions of effective Efimov states in the vicinity of the dimer + atom breakup threshold are addressed, but the results are inconclusive. The second aspect is the interplay between universal states of up to four bosons and the Coulomb interaction. This is interesting because it opens up nuclear physics to the investigation. First, the effect of the Coulomb interaction on universal states is studied in natural units. This introduces a scale for the Coulomb potential's strength relative to the strength of the short-range Gaussian potential. A generalised Efimov plot of the binding energies of states of charged bosons versus the Coulomb-modified scattering length is shown. This plot illustrates the impact of different relative strengths of the Coulomb potential on universal states. To complement this analysis, the structure is also calculated via root mean square (rms) radius calculations and contour plots. The results are then applied to the excited state of 17 F, which has a proton halo. The binding energy of this state can be reproduced after fixing the effective range of the potential to the physical value. The N α system proved more problematic. It was found that the ground state of 12 C was too deep to be described within the framework used and the highest excited state of 16 O below the 4α breakup threshold proved difficult to describe. I discuss possible reasons for this. In the last part I studied the zero-range limit in the presence of the Coulomb interaction. I show that my results for the dimer and trimer can be rescaled to coincide with the zero-range result. An extrapolation towards the zero-range limit for the tetramer ground state is also presented.
We investigate the internal structure of near-threshold states in a system with a repulsive Coulomb interaction combined with a short-range potential, using the compositeness. We construct a model in which the eigenmomentum is expressed in terms of three observables: the Coulomb scattering length, the Coulomb effective range, and the Bohr radius. In the presence of the Coulomb interaction, a bound state directly goes into a resonance as parameters are varied, bypassing a virtual state, in contrast to the case with only the short-range interaction. We show that the compositeness of near-threshold states can be expressed solely in terms of these observables. When the magnitude of the Coulomb effective range is much smaller than that of the Bohr radius, both shallow bound states and near-threshold resonances exhibit common structures with large compositeness, reflecting the remnant of the low-energy universality.
We consider the nonrelativistic four-boson system in two dimensions interacting via a short-range attractive potential. For a weakly attractive potential with one shallow two-body bound state with binding energy B-2, the binding energies B-N of shallow N-body bound states are universal and thus do not depend on the details of the interaction potential. We compute the four-body binding energies in an effective quantum mechanics approach. There are exactly two bound states: the ground state with B-4((0)) = 197.33(1)B-2 and one excited state with B-4((1)) = 25.5 (1)B-2. We compare our results to recent predictions for N-body bound states with large N much greater than 1.