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We extend our earlier work on the massive $O(N)$ nonlinear sigma model toother observables. We derive expressions at leading order in the large $N$expansion at all orders in the loop expansion for the decay constant, vacuumexpectation value, meson-meson scattering and the scalar and vector formfactors. This is done using cactus diagram resummation using a generalized gapequation and other recursion relations. For general $N$ we derive theexpressions for the $n$-th loop order leading logarithms$\left(M^2/F^2\log(\mu^2/M^2)\right)^n$, up to five-loops for the decayconstant and vacuum expectation value (VEV) and up to four-loops formeson-meson scattering, the scalar and vector form factors. We also quote ourearlier result for the mass. The large $N$ results do not give a goodapproximation for the case $N=3$. We use our results to study the convergenceof the perturbative series and compare with elastic unitarity.

The Massive O(N) Non-linear Sigma Model at High Orders