Exact diagrammatic approach for dimer-dimer scattering and bound states of three and four resonantly interacting particles

Abstract

We present a diagrammatic approach for the dimer-dimer scattering problem in two or three spatial dimensions, within the resonance approximation where these dimers are in a weakly bound resonant state. This approach is first applied to the calculation of the dimer-dimer scattering length ${a}_{B}$ in three spatial dimensions, for dimers made of two fermions in a spin-singlet state, with corresponding scattering length ${a}_{F}$, and the already known result ${a}_{B}=0.60$ ${a}_{F}$ is recovered exactly. Then we make use of our approach to obtain results in two spatial dimensions for fermions as well as for bosons. Specifically, we calculate bound-state energies for three $bbb$ and four $bbbb$ resonantly interacting bosons in two dimensions. We consider also the case of a resonant interaction between fermions and bosons, and we obtain the exact bound-state energies of two bosons plus one fermion $bbf$, two bosons plus two fermions $b{f}_{\ensuremath{\uparrow}}b{f}_{\ensuremath{\downarrow}}$, and three bosons plus one fermion $bbbf$.