Interaction effects on atomic laboratory trapped Bose-Einstein condensates

We discuss the effect of inter-atoms interactions on the condensation temperature $T_c$ of an atomic laboratory trapped Bose-Einstein condensate. We show that, in the mean-field Hartree-Fock and semiclassical approximations, interactions produce a shift $\Delta T_{c}/T_{c}^{0} \approx b_1 (a/\lambda_{T_c}) + b_2 (a/\lambda_{T_c})^2 + \psi[a/\lambda_{T_c}]$ with $a$ the s-wave scattering length, $\lambda_T$ the thermal wavelength and $\psi[a/\lambda_{T_c}]$ a non-analytic function such that $\psi[0] = \psi'[0] = \psi"[0] = 0$ and $|\psi"'[0]| = \infty$. Therefore, with no more assumptions than Hartree-Fock and semiclassical approximations, interaction effecs are perturbative to second order in $a/\lambda_{T_c}$ and the expected non-perturbativity of physical quantities at critical temperature appears only to third order. We compare this finding with different results by other authors, which are based on more than the Hartree-Fock and semiclassical approximations. Moreover, we obtain an analytical estimation for $b_2 \simeq 18.8$ which improves a previous numerical result. We also discuss how the discrepancy between $b_2$ and the empirical value of $b_2 = 46 \pm 5$ may be explained with no need to resort to beyond-mean field effects.