Universal nature and finite-range corrections in elastic atom-dimer scattering below the dimer breakup threshold

We investigate universal behavior in elastic atom-dimer scattering below the dimer breakup threshold calculating the atom-dimer effective-range function $ak\cot \delta$. Using the He-He system as a reference, we solve the Schrödinger equation for a family of potentials having different values of the two-body scattering length $a$ and we compare our results to the universal zero-range form deduced by Efimov, $ak\cot \delta =c_1\left(ka\right)+c_2\left(ka\right)\cot [s_0\ln \left({\kappa }_{*}a\right)+\varphi \left(ka\right)]$, for selected values of the three-body parameter ${\kappa }_{*}$. Using the parametrization of the universal functions $c_1,c_2,\varphi$ given in the literature, a good agreement with the universal formula is obtained after introducing a particular type of finite-range corrections. Furthermore, we show that the same parametrization describes a very different system: nucleon-deuteron scattering below the deuteron breakup threshold. Our analysis confirms the universal character of the process, and relates the pole energy in the effective-range function of nucleon-deuteron scattering to the three-body parameter ${\kappa }_{*}$.