  We consider interacting Bose gas in thermal equilibrium assuming a positive
and bounded pair potential $V(r)$ such that $0<\int d\br V(r) = a<\infty$.
Expressing the partition function by the Feynman-Kac functional integral yields
a classical-like polymer representation of the quantum gas. With 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 $V_{\gamma}(\br)=\gamma^{3}V(\gamma r)$ we prove that in
the mean-field limit $\gamma\to 0$ only 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 Mayer series (vicinity of
Bose-Einstein condensation) and study dominant corrections to mean field. At
lowest order, the form of function $F$ is shown to depend on single polymer
partition function for which we derive lower and upper bounds and on the
resummation of ring diagrams which can be analytically performed.
