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

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.