  We rewrite the Lagrangian for a dilute Bose gas in terms of auxiliary fields
related to the normal and anomalous condensate densities. We derive the loop
expansion of the effective action in the composite-field propagators. The
lowest-order auxiliary field (LOAF) theory is a conserving mean-field
approximation consistent with the Goldstone theorem without some of the
difficulties plaguing approximations such as the Hartree and Popov
approximations. LOAF predicts a second-order phase transition. We give a set of
Feynman rules for improving results to any order in the loop expansion in terms
of composite-field propagators. We compare results of the LOAF approximation
with those derived using the Popov approximation. LOAF allows us to explore the
critical regime for all values of the coupling constant and we determine
various parameters in the unitarity limit.
