Condensate density and superfluid mass density of a dilute Bose gas near the condensation transition

We derive, through analysis of the structure of diagrammatic perturbation theory, the scaling behavior of the condensate and superfluid mass density of a dilute Bose gas just below the condensation transition. Sufficiently below the critical temperature, $T_c$, the system is governed by the mean field (Bogoliubov) description of the particle excitations. Close to $T_c$, however, mean field breaks down and the system undergoes a second order phase transition, rather than the first order transition predicted in Bogoliubov theory. Both condensation and superfluidity occur at the same critical temperature, $T_c$ and have similar scaling functions below $T_c$, but different finite size scaling at $T_c$ to leading order in the system size. Through a simple self-consistent two loop calculation we derive the critical exponent for the condensate fraction, $2\beta\simeq 0.66$.