Self-consistent equation for an interacting Bose gas

We consider interacting Bose gas in thermal equilibrium assuming a positive and bounded pair potential V(r) such that 0< integral dr V(r)=a< infinity. Expressing the partition function by the Feynman-Kac functional integral yields a classicallike polymer representation of the quantum gas. With the Mayer graph summation techniques, we demonstrate the existence of a self-consistent relation rho(mu)=F(mu-(a)rho(mu)) between the density rho and the chemical potential mu, valid in the range of convergence of Mayer series. The function F is equal to the sum of all rooted multiply connected graphs. Using Kac's scaling V(gamma)(r)=gamma3V(gamma(r)), we prove that in the mean-field limit gamma-->0, only the tree diagrams contribute and function F reduces to the free gas density. We also investigate how to extend the validity of the self-consistent relation beyond the convergence radius of the Mayer series (vicinity of Bose-Einstein condensation), and study the dominant corrections to the mean field. At the lowest order, the form of function F is shown to depend on a single polymer partition function for which we derive the lower and the upper bounds and on the resummation of ring diagrams which can be analytically performed.