Four-point renormalized coupling constant in O(N) models

The renormalized zero-momentum four-point coupling $g_r$ of $O(N)$-invariant scalar field theories in $d$ dimensions is studied by applying the $1\mathrm{/}N$ expansion and strong coupling analysis. The $O(1\mathrm{/}N)$ correction to the $\beta$-function and to the fixed point value $g_r^{\ast }$ are explictly computed. Strong coupling series for lattice non-linear $\sigma$ models are analyzed near criticality in $d=2$ and $d=3$ for several values of $N$ and the corresponding values of $g_r^{\ast }$ are extracted. Large-$N$ and strong coupling results are compared with each other, finding a good general agreement. For small $N$ the strong coupling analysis in 2-d gives the best determination of $g_r^{\ast }$ to date (or comparable for $N=2,3$ with the available Monte Carlo estimates), and in 3-d it is consistent with available ${\varphi }^4$ field theory results.