Scale-(in)dependence in quantum 4-body scattering

Abstract

We investigate the multi-channel 4-body scattering system using regularized 2- and 3-body contact interactions. The analysis determines the sensitivity of bound-state energies, scattering phase shifts and cross-sections on the cutoff parameter (\(\lambda \)), and on the energy gaps between scattering thresholds. The latter dependency is obtained with a 2-body scale fixed to an unnaturally large value and a floating 3-body parameter. Specifically, we calculate the binding energies of the shallow 3- and 4-body states, dimer-dimer and trimer-atom scattering lengths, and the trimer-atom to dimer-dimer reaction rates. Employing a potential renormalized by a large 2-body scattering length and a 3-body scale, we find all calculated observables to remain practically constant over the range \(6\,\text {fm}^{-2}<\lambda <10\,\text {fm}^{-2}\). Divergences in scattering lengths emerge for critical 3-body parameters at which thresholds are degenerate. For those critical points where the dimer-dimer and trimer-atom thresholds overlap, we predict an enhancement of the inelastic over the elastic scattering event. Such an inversion between elastic- and rearrangement-collision probabilities indicates a strong sensitivity of the 4-body reaction dynamics on the 3-body parameter at finite 2-body scale. As this enhancement is found here for all considered cutoffs but not in earlier studies which employ different renormalization schemes, we conclude that the sole cutoff variation does not necessarily imply non-perturbative changes in reaction rates. We conjecture the enhancement to follow from specific combinations of the finite 2-body range and the 3-body parameter.