Transition Temperature for Weakly Interacting Homogeneous Bose Gases

We apply the non-perturbative optimized linear $\delta$ expansion method to the O(N) scalar field model in three-dimensions to determine the transition temperature of a dilute homogeneous Bose gas. Our results show that the shift of the transition temperature, $\mathrm{\Delta }{T}_c\mathrm{/}{T}_c$, of the interacting model, compared with the ideal gas transition temperature, really behaves as $\gamma a n^{1/3}$where$a$isthes-wavescatteringlengthand$n$ is the number density. For $N=2$ our calculations yield the value $\gamma =3.059$.