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) approximate to b(1)(a/lambda(Tc)) + b(2)(a/lambda(Tc))(2) + psi [a/lambda(Tc)] with a the s-wave scattering length, lambda(T) the thermal wavelength and psi [a/lambda(Tc)] a non-analytic function such that psi[0] = psi' [0] = psi '' [0] = 0 and vertical bar psi''' [0] vertical bar = infinity. Therefore, with no more assumptions than Hartree-Fock and semiclassical approximations, interaction effecs are perturbative to second order in a/lambda(Tc) 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) similar or equal to 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 +/- 5 may be explained with no need to resort to beyond-mean field effects.