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

Abstract

We extend our earlier work on the massive $O(N)$ nonlinear 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 ${\left(M^2/F^2\log ({\mu }^2/M^2)\right)}^n$, 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 $N=3$. We use our results to study the convergence of the perturbative series and compare with elastic unitarity.

39 pages

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• Hilbert series and higher-order Lagrangians for the $O(N)$ model
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• Chiral Perturbation Theory Reflections on Effective Theories of the Standard Model
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• Leading Logarithms of the Two Point Function in Massless O(N) and SU(N) Models to any Order from Analyticity and Unitarity
• ChPT calculations of pion formfactors
• Quasi-Renormalizable Quantum Field Theories
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