Renormalization Group Approach to Zero Temperature Bose Condensation

We study the problem of Bose condensation at zero temperature and weak coupling for a three dimensional system of bosons, interacting with a repulsive short range potential, in the Bogoliubov approximation. We prove that the properties of the model can be explained in terms of an anomalous asymptotically free renormalization group ow and we show that the two-point correlation function has the typical superruid behaviour at long wavelengths, as generally expected. The proof is, for the moment, only at the level of perturbation theory in the running coupling constants. We also obtain an expression for the sound speed, whose leading term (when the coupling goes to zero) coincides with the sound speed in the exactly soluble Bogoliubov model.