Many-body t-matrix theory eliminated the infrared divergence on the Bose-Einstein-condensed systems.

• We study the many-body t-matrix approximation for the weakly interacting bose gas. • We solved the coupled equation for the t-matrix and the condensate density. • This results lead to the correct order of the phase transition. We have proved in the many-body t-matrix approximation for the weakly interacting Bose gas the infrared divergences are exactly removed. We also show that this t-matrix becomes very temperature-dependent and, as the temperature approaches to T c it vanishes at the transition point. This leads to a smooth change of condensate density and compressibility at T c , implying the second order phase transition occurs at T c. [ABSTRACT FROM AUTHOR]