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/N expansion and strong coupling analysis. The O(1/N) correction to the $\beta$-function and to the fixed point value $g_r^*$ 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 $g_r^*$ 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 $g^*_r$ to date (or comparable for N=2,3 with the available Monte Carlo estimates), and in 3-d it is consistent with available $\phi^4$ field theory results.