Transition temperature for weakly interacting homogeneous Bose gases

We apply the nonperturbative optimized linear $\ensuremath{\delta}$ 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 $\ensuremath{\Delta}{T}_{c}{/T}_{c}$ of the interacting model, compared with the ideal-gas transition temperature, really behaves as $\ensuremath{\gamma}{\mathrm{an}}^{1/3}$ where a is the s-wave scattering length and n is the number density. For $N=2$ our calculations yield the value $\ensuremath{\gamma}=3.059.$

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