Condensate density and superfluid mass density of a dilute Bose-Einstein condensate near the condensation transition

We derive via diagrammatic perturbation theory the scaling behavior of the condensate and superfluid mass density of a dilute Bose gas just below the condensation temperature, T(c). Sufficiently below T(c) particle excitations are described by mean field (Bogoliubov). Near T(c), however, mean field fails, and the system undergoes a second order phase transition, rather than first order as predicted by Bogoliubov theory. Both condensation and superfluidity occur at the same T(c), and have similar scaling functions below T(c), but different finite size scaling at T(c) to leading order in the system size. A self-consistent two-loop calculation yields the condensate fraction critical exponent, 2beta approximately 0.66.