Momentum Space Calculation of Heteronuclear Three-Body Parameter using Pairwise Separable Potentials

In the 1970s, Efimov predicted the existence of a geometric series of trimer states when pairwise interactions are resonant, i.e., the two-body scattering length is much larger than the range of the interaction [1]. The three-body parameter, commonly determined by the trimer ground state energy, is an essential quantity that uniquely defines the spectrum of three interacting particles. Over the past few decades, numerous studies have demonstrated the universality of this parameter among various ultracold three-body systems, independent of particles structure or the details of two-body interactions at short separations. This work aims to study the three-body parameter in the heteronuclear case with two identical particles. Pairwise van der Waals interactions, described by separable s-wave potentials, are utilized. The insertion of these separable potentials into the three-body Schrödinger’s equation in momentum space leads to a system of coupled homogeneous integral equations. Specializing to states with total angular momentum L = 0, these equations are solved numerically for trimer energies at different heteronuclear scattering lengths by searching for roots of the corresponding determinantal equation. The separable potential form factor is selected, as in [2], to give the zero energy two-body wavefunction of the van der Waals interaction exactly. This choice is shown to reproduce the low-energy features of the original potential (e.g. scattering length, dimer bound state).



[1] V. Efimov, Energy levels of three resonantly interacting particles, Nuclear Physics A 210, 157 (1973)

[2] P. Naidon, S. Endo, and M. Ueda, Physical origin of the universal three-body parameter in atomic Efimov physics, Phys. Rev. A 90, 022106 (2014).