  We consider a double quantum dot coupled to two normal leads and one
superconducting lead, modeling the Cooper pair beam splitter studied in two
recent experiments. Starting from a microscopic Hamiltonian we derive a general
expression for the branching current and the noise crossed correlations in
terms of single and two-particle Green's function of the dot electrons. We then
study numerically how these quantities depend on the energy configuration of
the dots and the presence of direct tunneling between them, isolating the
various processes which come into play. In absence of direct tunneling, the
antisymmetric case (the two levels have opposite energies with respect to the
superconducting chemical potential) optimizes the Crossed Andreev Reflection
(CAR) process while the symmetric case (the two levels have the same energies)
favors the Elastic Cotunneling (EC) process. Switching on the direct tunneling
tends to suppress the CAR process, leading to negative noise crossed
correlations over the whole voltage range for large enough direct tunneling.
