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

We study the dimer-dimer scattering length $a_4$ for a two-component Fermi mixture in which the different fermions have different masses $mus$ and $mds$. This is made in the framework of the exact field theoretical 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 $a_4 simeq a_3/2$ is actually valid whatever the mass ratio with quite good precision. As a result we find an analytical expression providing $a_4$ with a fairly good precision for any masses. 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. We conclude that their approximation, of retaining only these processes, is a fairly good one whatever the mass ratio.