  We study the instabilities towards (exotic) superconductivity of mixtures of
spin-$1/2$ fermions coupled to scalar bosons on a two-dimensional square
lattice with the Dynamical-Cluster-Approximation (DCA) using a numerically
exact continuous-time Monte-Carlo solver. The Bogoliubov bosons provide an
effective phononic bath for the fermions and induce a non-local retarded
interaction between the fermions, which can lead to (exotic) superconductivity.
Because of the sign problem the biggest clusters we can study are limited to $2
\times 2$ in size, but this nevertheless allows us to study the pairing
instablilities, and their possible divergence, in the $s$- and $d$ -wave
channels as well as the competition with antiferromagnetic fluctuations. At
fermionic half-filling we find that $d$-wave is stable when the mediated
interaction by the bosons is of the same order as the bare fermionic repulsion.
Its critical temperature can be made as high as the maximum one for $s$-wave,
which opens perspectives for its detection in a cold atom experiment.
