Exact 1/N and Optimized Perturbative Evaluation of mu_c for Homogeneous   Interacting Bose Gases
  In the framework of the O(N) three-dimensional effective scalar field model for homogeneous dilute weakly interacting Bose gases we use the 1/N expansion to evaluate, within the large N limit, the parameter r_c which is directly related to the critical chemical potential mu_c. This quantity enters the order-a^2 n^{2/3} coefficient contributing to the critical temperature shift Delta T_c where a represents the s-wave scattering length and n represents the density. Compared to the recent precise numerical lattice simulation results, our calculation suggests that the large N approximation performs rather well even for the physical case N=2. We then calculate the same quantity but using different forms of the optimized perturbative (variational) method, showing that these produce excellent results both for the finite N and large-N cases. 
Jean-Loic Kneur and Marcus B. Pinto