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

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 aB in three spatial dimensions, for dimers made of two fermions in a spin-singlet state, with corresponding scattering length aF, and the already known result aB=0.60 aF 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 bf_↑bf_↓, and three bosons plus one fermion $bbbf$.