Dimer-dimer scattering length for fermions with different masses: Analytical study for large mass ratio

Abstract

We study the dimer-dimer scattering length ${a}_{4}$ for a two-component Fermi mixture in which the different fermions have different masses ${m}_{\ensuremath{\uparrow}}$ and ${m}_{\ensuremath{\downarrow}}$. This is made in the framework of the exact field-theoretic method. In the large mass ratio domain the equations are simplified enough to lead to an analytical solution. In particular we link ${a}_{4}$ to the fermion-dimer scattering length ${a}_{3}$ for the same fermions and obtain the very simple relation ${a}_{4}={a}_{3}/2$. The result ${a}_{4}\ensuremath{\simeq}{a}_{3}/2$ is actually valid whatever the mass ratio with quite good precision. As a result we find an analytical expression providing ${a}_{4}$ with fairly good precision for any mass. To dominant orders for large mass ratio it agrees with the literature. We show that in this large mass ratio domain, the dominant processes are the repeated dimer-dimer Born scatterings, considered earlier by Pieri and Strinati [Phys Rev. B 61, 15370 (2000)]. We conclude that their approximation of retaining only these processes is a fairly good one whatever the mass ratio.