Convergent Resummed Linear $\delta$ Expansion in the Critical $O(N)$ $({\varphi }_i^2{)}_{3d}^2$ Model

The linear $\delta$ expansion (LDE) is applied to the critical $O(N)$ ${\varphi }^4$ three-dimensional field theory which has been widely used to study the temperature of condensation of dilute weakly interacting homogeneous Bose gases. We study the higher order convergence of the LDE as it is usually applied to this problem. We show how to improve both the large $N$ and finite $N=2$ LDE results with an efficient resummation technique which accelerates convergence. In the large $N$ limit, it reproduces the known exact result within numerical integration accuracy. In the finite $N=2$ case, our improved results support the recent numerical Monte Carlo estimates for the transition temperature.