The massive O(N)O(N) non-linear sigma model at high orders

We extend our earlier work on the massive O(N)O(N) non-linear sigma model to other observables. We derive expressions at leading order in the large N expansion at all orders in the loop expansion for the decay constant, vacuum expectation value, meson–meson scattering and the scalar and vector form factors. This is done using cactus diagram resummation using a generalized gap equation and other recursion relations. For general N we derive the expressions for the n  -th-loop-order leading logarithms (M2/F2log(μ2/M2))n, up to five loops for the decay constant and vacuum expectation value (VEV) and up to four loops for meson–meson scattering, the scalar and vector form factors. We also quote our earlier result for the mass. The large N   results do not give a good approximation for the case N=3N=3. We use our results to study the convergence of the perturbative series and compare with elastic unitarity.