Testing the Bethe ansatz with large N renormalons.

Area of Science:

• Quantum Field Theory
• Integrable Systems
• Non-linear Sigma Models

Background:

• The Bethe ansatz provides a method for computing ground-state energies in integrable theories coupled to external potentials.
• This method yields predictions for the perturbative energy expansion, which require rigorous testing.

Purpose of the Study:

• To provide a non-trivial test of the Bethe ansatz predictions in the non-linear sigma model and its supersymmetric extension.
• To analytically calculate associated Feynman diagrams at next-to-leading order in the 1/N expansion and at all loops.

Main Methods:

• Analytical calculation of Feynman diagrams.
• Investigation of the large order behavior of diagrams to locate renormalons.
• Derivation of the large N trans-series for the theory.

Main Results:

• The study provides an analytical calculation of Feynman diagrams for the non-linear sigma model and its supersymmetric extension.
• Renormalon positions were located, and analytic expressions for the large N trans-series were obtained.
• A direct derivation of the beta function for these theories was achieved at next-to-leading order in the 1/N expansion.

Conclusions:

• The findings offer a significant validation of the Bethe ansatz in the context of integrable asymptotically free theories.
• The calculation provides new insights into the structure of perturbative expansions and the role of renormalons.
• This work contributes to a deeper understanding of quantum field theories and their properties.