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

TL;DR

This paper extends the analysis of the massive O(N) nonlinear sigma model to higher orders, deriving explicit expressions for various observables at large N and examining the convergence of the perturbative series.

Contribution

It provides all-order leading logarithm expressions for key observables in the O(N) model using cactus diagram resummation, up to five loops for some quantities.

Findings

Derived explicit leading logarithm expressions up to five loops for decay constant and VEV.

Provided results for meson-meson scattering and form factors up to four loops.

Found that large N results are not accurate for N=3 and analyzed series convergence.

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.