$T_{\mathrm{c}}$ for dilute Bose gases: Beyond leading order in $1/N$

Baym, Blaizot, and Zinn-Justin have recently used the large-$N$ approximation to calculate the effect of interactions on the transition temperature of dilute Bose gases. We extend their calculation to next-to-leading order in $1/N$ and find a relatively small correction of $-26\%$ to the leading order result. This suggests that the large-$N$ approximation works surprisingly well in this application.