Transition temperature for weakly interacting homogeneous Bose gases

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