Atom-dimer scattering length for fermions with different masses: analytical study of limiting cases

TL;DR

This paper analytically derives the atom-dimer scattering length for fermions with different masses in the universal regime, focusing on limiting cases where one fermion's mass is very large or very small.

Contribution

It generalizes the Skorniakov and Ter-Martirosian solution to fermions with unequal masses and provides explicit expressions in limiting mass cases.

Findings

Explicit formulas for scattering length in limiting mass cases

Analytical solutions for the integral equation functions

Extension of previous equal-mass results to mass-imbalanced systems

Abstract

We consider the problem of obtaining the scattering length for a fermion colliding with a dimer, formed from a fermion identical to the incident one and another different fermion. This is done in the universal regime where the range of interactions is short enough so that the scattering length $a$ for non identical fermions is the only relevant quantity. This is the generalization to fermions with different masses of the problem solved long ago by Skorniakov and Ter-Martirosian for particles with equal masses. We solve this problem analytically in the two limiting cases where the mass of the solitary fermion is very large or very small compared to the mass of the two other identical fermions. This is done both for the value of the scattering length and for the function entering the Skorniakov-Ter-Martirosian integral equation, for which simple explicit expressions are obtained.