The transition temperature of the dilute interacting Bose gas for N internal states

We calculate explicitly the variation delta T-c of the Bose-Einstein condensation temperature T-c induced by weak repulsive two-bod interactions to leading order in the interaction strength. As shown earlier by general arguments, delta T-c/T-c is linear in the dimensionless product an(1/3) to leading order, where n is the density and a the scattering length. This result is non-perturbative, and a direct perturbative calculation of the amplitude is impossible due to infrared divergences familiar from the study of the superfluid helium lambda transition. Therefore, we introduce here another standard expansion scheme, generalizing the initial model which depends on one complex field to one depending on N real fields, and calculating the temperature shift at leading order for large N. The result is explicit and finite.. The reliability of the result depends on the relevance of the large-N expansion to the situation N = 2, which can in principle be checked by systematic higher-order calculations. The large-M result agrees remarkably well with recent numerical simulations.