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 $a_B$ 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 $b{f}_{\uparrow }b{f}_{\downarrow }$, and three bosons plus one fermion $bbbf$.

Figure 1

The graphic representation of the two-particle vertex $T_2\left(P\right)$ [the four external propagators do not belong to $T_2\left(P\right)$].

Figure 2

The graphic representation of the simplest dimer-fermion scattering process ${\Delta }_3$ (the two external fermion propagators and the two external dimer propagators do not belong to ${\Delta }_3$).

Figure 3

The diagrammatic representation of the equation for the full dimer-fermion scattering vertex $T_3$.

Figure 4

The graphic representation of the dimer-dimer scattering processes contributing to $T_4$.

Figure 5

Diagrammatic representation of the relation between the full dimer-dimer scattering matrix $T_4$ and the vertex $\Phi$.

Figure 6

The diagrammatic representation of the integral equation for the function $\Phi$ introduced by dimer-dimer scattering.

Figure 7

(Color online) Eigenvalues $\lambda$ found for $\mid E_4\mid =2.4$, for both the $bbbf$ and $b{f}_{\uparrow }b{f}_{\downarrow }$ cases. When one of the eigenvalues is equal to 1, $E_4$ is the energy of a possible eigenstate of the complex. The number $n$ appearing on the $x$ axis is just here to number the first few eigenvalues which are displayed by decreasing order.

Figure 8

(Color online) Eigenvalues $\lambda$ found for $\mid E_4\mid =16.5$ and $\mid E_4\mid =22$, for the $bbbb$ case. When one of the eigenvalues is equal to 1, $E_4$ is the energy of a possible eigenstate of the complex. The number $n$ appearing on the $x$ axis is just here to number the first few eigenvalues which are displayed by decreasing order.

Figure 9

The diagrammatic representation of the equation for the full dimer-dimer scattering vertex $T_4(p_1,p_2;P)$.

Figure 10

The diagrammatic representation of the sum of all irreducible diagrams ${\Delta }_4(p_1,p_2;P)$.

Figure 11

The graphic representation of the equation on the full vertex $\Gamma (q_1,q_2;p_2,P)$.