Testing the Bethe ansatz with large N renormalons

TL;DR

This paper tests the Bethe ansatz predictions for the ground state energy of integrable theories by analyzing large N renormalons in the non-linear sigma model and its supersymmetric extension, providing analytical results at next-to-leading order.

Contribution

It offers a non-trivial analytical test of Bethe ansatz predictions in asymptotically free theories through detailed diagram calculations and renormalon analysis at large N.

Findings

Confirmed the Bethe ansatz predictions for the energy expansion.

Located the position of renormalons in the theories.

Derived the beta function at next-to-leading order in 1/N.

Abstract

The ground state energy of integrable asymptotically free theories can be conjecturally computed by using the Bethe ansatz, once the theory has been coupled to an external potential through a conserved charge. This leads to a precise prediction for the perturbative expansion of the energy. We provide a non-trivial test of this prediction in the non-linear sigma model and its supersymmetric extension, by calculating analytically the associated Feynman diagrams at next-to-leading order in the $1/N$ expansion, and at all loops. By investigating the large order behaviour of the diagrams, we locate the position of the renormalons of the theory and we obtain an analytic expression for the large $N$ trans-series associated to each. As a spin-off of our calculation, we provide a direct derivation of the beta function of these theories, at next-to-leading order in the $1/N$ expansion.