Boson-induceds-wave pairing in dilute boson-fermion mixtures

We show that in dilute boson-fermion mixtures with fermions in two internal states, even when the bare fermion-fermion interaction is repulsive, the exchange of density fluctuations of the Bose condensate may lead to an effective fermion-fermion attraction, and thus to a Cooper instability in the s-wave channel. We give a simple condition to know when this is going to happen, and an analytic expression for the associated $T_c$ in the most important limit where the phonon branch of the Bogoliubov excitation spectrum of the bosons is important. We find a $T_c$ of the same order as for a pure Fermi gas with bare attraction.