Atom-dimer scattering in heteronuclear mixture with finite intra-species scattering length

We study the three-body problem of two ultracold identical bosonic atoms (denoted by $B$) and one extra atom (denoted by $X$), where the scattering length $a_{BX}$ between each bosonic atom and atom $X$ is resonantly large and positive. We calculate the scattering length $a_{{\rm ad}}$ between one bosonic atom and the shallow dimer formed by the other bosonic atom and atom $X$, and investigate the effect induced by the interaction between the two bosonic atoms. We find that even if this interaction is weak (i.e., the corresponding scattering length $a_{BB}$ is of the same order of the van der Waals length $r_{{\rm vdW}}$ or even smaller), it can still induce a significant effect for the atom--dimer scattering length $a_{{\rm ad}}$. Explicitly, an atom--dimer scattering resonance can always occur when the value of $a_{BB}$ varies in the region with $|a_{BB}|\lesssim r_{{\rm vdW}}$. As a result, both the sign and the absolute value of $a_{{\rm ad}}$, as well as the behavior of the $a_{{\rm ad}}$-$a_{BX}$ function, depends sensitively on the exact value of $a_{BB}$. Our results show that, for a good quantitative theory, the intra-species interaction is required to be taken into account for this heteronuclear system, even if this interaction is weak.