Origin of nonlinear contribution to the shift of the critical temperature in atomic Bose-Einstein condensates

We discuss a possible origin of the experimentally observed nonlinear contribution to the shift $\Delta T_{c}=T_c-T_{c}^{0}$ of the critical temperature $T_{c}$ in an atomic Bose-Einstein condensate (BEC) with respect to the critical temperature $T_{c}^{0}$ of an ideal gas. We found that accounting for a nonlinear (quadratic) Zeeman effect (with applied magnetic field closely matching a Feshbach resonance field $B_0$) in the mean-field approximation results in a rather significant renormalization of the field-free nonlinear contribution $b_{2}$, namely $\Delta T_{c}/T_{c}^{0}\simeq b_{2}^{\ast }(a/\lambda _{T})^{2}$ (where $a$ is the s-wave scattering length, $\lambda _{T}$ is the thermal wavelength at $T_{c}^{0}$) with $b_{2}^{\ast }=\gamma ^{2}b_{2}$ and $\gamma =\gamma (B_0)$. In particular, we predict $b_{2}^{\ast }\simeq 42.3$ for the $B_{0}\simeq 403G$ resonance observed in the $\ ^{39}K$ BEC.