Exact 1/N and optimized perturbative evaluation of μ c for homogeneous interacting Bose gases

In the framework of the $\mathrm{O}(N)$ three-dimensional effective scalar field model for homogeneous dilute weakly interacting Bose gases we use the $1∕N$ expansion, within the large-$N$ limit, to evaluate the parameter ${r}_{c}$ which is directly related to the critical chemical potential ${\ensuremath{\mu}}_{c}$. This quantity enters the order-${a}^{2}{n}^{2∕3}$ coefficient contributing to the critical temperature shift $\ensuremath{\Delta}{T}_{c}$ where $a$ represents the $s$-wave scattering length and $n$ represents the density. Compared to the recent precise numerical lattice simulation results, our calculation suggests that the large-$N$ approximation performs rather well even for the physical case $N=2$. We then calculate the same quantity but using different forms of the optimized perturbative (variational) method, showing that these produce excellent results for both the finite-$N$ and large-$N$ cases.