  Adiabatic quantum pumping in noninteracting, phase coherent quantum dots is
elegantly described by Brouwer's formula. Interactions within the dot, while
suppressing phase coherence, make Brouwer's formalism inapplicable. In this
paper, we discuss the nature of the physical processes forcing a description of
pumping beyond Brouwer's formula, and develop, using a controlled adiabatic
expansion, a useful formalism to study the effect of interactions within a
generic perturbative scheme. The pumped current consists of a first
contribution, analogous to Brouwer's formula and accounting for the remanent
coherence, and of interaction corrections describing inelastic scattering. We
apply the formalism to study the effect of interaction with a bosonic bath on a
resonant level pump and discuss the robustness of the quantization of the
pumped charge in turnstile cycles.
