Transition temperature of the weakly interacting Bose gas: perturbative solution of the crossover equations in the canonical ensemble

We compute the shift of the critical temperature Tcwith respect to the ideal case for a weakly interacting uniform Bose gas. We work in the framework of the canonical ensemble, extending the criterion of condensation provided by the canonical particle counting statistics for the zero-momentum state of the uniform ideal gas. The perturbative solution of the crossover equation to lowest order in powers of the scattering length yields (Tc-Tc0)/Tc0= -0.93a?1/3, where Tc0is the transition temperature of the corresponding ideal Bose gas, ais the scattering length, and ? is the particle number density. This result is at variance with the standard grand canonical prediction of a null shift of the critical temperature in the lowest perturbative order. The non-equivalence of statistical ensembles for the ideal Bose gas is thus confirmed (at the lowest perturbative level) also in the presence of interactions.