  We propose an approach allowing the computation of currents and their
correlations in interacting multiterminal mesoscopic systems involving quantum
dots coupled to normal and/or superconducting leads. The formalism relies on
the expression of branching currents and noise crossed correlations in terms of
one- and two-particle Green's functions for the dots electrons, which are then
evaluated self-consistently within a conserving approximation. We then apply
this to the Cooper-pair beam-splitter setup recently proposed [L. Hofstetter et
al. Nature (London) 461 960 (2009); Phys. Rev. Lett. 107 136801 (2011); L. G.
Herrmann et al. Phys. Rev. Lett. 104 026801 (2010)], which we model as a double
quantum dot with weak interactions, connected to a superconducting lead and two
normal ones. Our method not only enables us to take into account a local
repulsive interaction on the dots, but also to study its competition with the
direct tunneling between dots. Our results suggest that even a weak Coulomb
repulsion tends to favor positive current cross correlations in the
antisymmetric regime (where the dots have opposite energies with respect to the
superconducting chemical potential).
