# 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{sub B} in three spatial dimensions, for dimers made of two fermions in a spin-singlet state, with corresponding scattering length a{sub F}, and the already known result a{sub B}=0.60 a{sub 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 bf{sub {up_arrow}}bf{sub {down_arrow}}, and three bosons plus one fermion bbbf.

- Authors:

- P.L. Kapitza Institute for Physical Problems, Kosygin street 2, Moscow, Russia, 119334 (Russian Federation)
- (Russian Federation)
- (France)

- Publication Date:

- OSTI Identifier:
- 20786930

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. A; Journal Volume: 73; Journal Issue: 3; Other Information: DOI: 10.1103/PhysRevA.73.032724; (c) 2006 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 74 ATOMIC AND MOLECULAR PHYSICS; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; APPROXIMATIONS; BOSE-EINSTEIN CONDENSATION; BOSONS; BOUND STATE; DIMERS; FERMIONS; INTERACTING BOSON MODEL; RESONANCE; SCATTERING; SCATTERING LENGTHS; SPIN

### Citation Formats

```
Brodsky, I. V., Kagan, M. Yu., Klaptsov, A. V., Combescot, R., Leyronas, X., Russian Research Centre 'Kurchatov Institute', Kurchatov square 1, Moscow, Russia, 123182, and Laboratoire de Physique Statistique, Ecole Normale Superieure, 24 rue Lhomond, 75231 Paris Cedex 05.
```*Exact diagrammatic approach for dimer-dimer scattering and bound states of three and four resonantly interacting particles*. United States: N. p., 2006.
Web. doi:10.1103/PHYSREVA.73.0.

```
Brodsky, I. V., Kagan, M. Yu., Klaptsov, A. V., Combescot, R., Leyronas, X., Russian Research Centre 'Kurchatov Institute', Kurchatov square 1, Moscow, Russia, 123182, & Laboratoire de Physique Statistique, Ecole Normale Superieure, 24 rue Lhomond, 75231 Paris Cedex 05.
```*Exact diagrammatic approach for dimer-dimer scattering and bound states of three and four resonantly interacting particles*. United States. doi:10.1103/PHYSREVA.73.0.

```
Brodsky, I. V., Kagan, M. Yu., Klaptsov, A. V., Combescot, R., Leyronas, X., Russian Research Centre 'Kurchatov Institute', Kurchatov square 1, Moscow, Russia, 123182, and Laboratoire de Physique Statistique, Ecole Normale Superieure, 24 rue Lhomond, 75231 Paris Cedex 05. Wed .
"Exact diagrammatic approach for dimer-dimer scattering and bound states of three and four resonantly interacting particles". United States.
doi:10.1103/PHYSREVA.73.0.
```

```
@article{osti_20786930,
```

title = {Exact diagrammatic approach for dimer-dimer scattering and bound states of three and four resonantly interacting particles},

author = {Brodsky, I. V. and Kagan, M. Yu. and Klaptsov, A. V. and Combescot, R. and Leyronas, X. and Russian Research Centre 'Kurchatov Institute', Kurchatov square 1, Moscow, Russia, 123182 and Laboratoire de Physique Statistique, Ecole Normale Superieure, 24 rue Lhomond, 75231 Paris Cedex 05},

abstractNote = {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{sub B} in three spatial dimensions, for dimers made of two fermions in a spin-singlet state, with corresponding scattering length a{sub F}, and the already known result a{sub B}=0.60 a{sub 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 bf{sub {up_arrow}}bf{sub {down_arrow}}, and three bosons plus one fermion bbbf.},

doi = {10.1103/PHYSREVA.73.0},

journal = {Physical Review. A},

number = 3,

volume = 73,

place = {United States},

year = {Wed Mar 15 00:00:00 EST 2006},

month = {Wed Mar 15 00:00:00 EST 2006}

}