Evidence for the breakdown of momentum independent many-body t-matrix approximation in the normal phase of Bosons

We revisit the momentum independent many-body t-matrix approach for boson systems developed by Shi and Griffin~\cite{sg} and Bijlsma and Stoof~\cite{bs}. Despite its popularity, simplicity, and expected advantage of being its applicability to both normal and superfluid phases, we find that the theory breaks down in the normal phase of bosons. We conjecture that this failure is due to neglecting of momentum dependence on the t-matrix.