Asymptotically improved convergence of optimized perturbation theory in the Bose-Einstein condensation problem

Abstract

We investigate the convergence properties of optimized perturbation theory, or linear $\ensuremath{\delta}$ expansion (LDE), within the context of finite temperature phase transitions. Our results prove the reliability of these methods, recently employed in the determination of the critical temperature ${T}_{c}$ for a system of a weakly interacting homogeneous dilute Bose gas. We carry out explicit LDE optimized calculations and also the infrared analysis of the relevant quantities involved in the determination of ${T}_{c}$ in the large-$N$ limit, when the relevant effective static action describing the system is extended to $\mathrm{O}(N)$ symmetry. Then, using an efficient resummation method, we show how the LDE can already exactly reproduce the known large-$N$ result for ${T}_{c}$ at the first nontrivial order. Next, we consider the finite $N=2$ case where, using similar resummation techniques, we improve the analytical results for the nonperturbative terms involved in the expression for the critical temperature, allowing comparison with recent Monte Carlo estimates of them. To illustrate the method, we have considered a simple geometric series showing how the procedure as a whole works consistently in a general case.