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Physical Review C

Linear relation between the scattering length and the size of a loosely bound two-body system: One-dimensional model analysis

F. Masafumi Toyama, Yukihisa Nogami

Published Dec 1, 1988• 8 citations
Publisher

Abstract

Relativistic and nonrelativistic models in one dimension which simulate the deuteron and the $^{3}\mathrm{S}_{1}$ nucleon-nucleon (NN) scattering are constructed and the relation between the scattering length ${a}_{t}$ and the root-mean-square radius ${r}_{m}$ of the simulated deuteron is examined for a variety of interactions. The linear relation between ${a}_{t}$ and ${r}_{m}$, which was found by Klarsfeld et al. for realistic NN potentials, holds for a wide class of potentials, relativistic as well as nonrelativistic. Within the limitation of our one-dimensional models, this suggests that the discrepancy between theory and experiment which was pointed out by Klarsfeld et al. regarding ${a}_{t}$ and ${r}_{m}$ cannot be resolved by relativistic effects.

DOI
10.1103/physrevc.38.2881

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Continuum structures of three short-range interacting particles in a deformed external one-body field are investigated. We use the equivalent ddd method employing noninteger dimension, ddd, in a spherical calculation with a dimension-dependent angular momentum barrier. We focus on dimensions close to the critical dimension, d=dEd={d}_{E}d=dE​, between 2 and 3, defined by zero two-body energies, where the Efimov effect can occur. We design for this dimension region a schematic, long-distance realistic, square-well based, three-body spherical model, which is used to derive analytic expressions for the wave functions, scattering lengths, phase shifts, and elastic-scattering cross sections. The procedure and the results are universal, valid for all short-range potentials, and for large scattering lengths. We discuss the properties and validity of the derived expressions by means of the simplest system of three identical bosons. The derived expressions are particularly useful for very small energies, where full numerical calculations are often not feasible. For energies where the numerical calculations can be performed, a good agreement with the analytic results is found. These model results may be tested by scattering experiments for three particles in an equivalent external deformed oscillator potential. The cross sections all vanish in the zero-energy limit for d<3d<3d<3 with definite ddd-dependent power of energy.