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We tested the Bethe ansatz for integrable theories by calculating Feynman diagrams in the non-linear sigma model. This confirmed predictions for perturbative energy expansions and derived the theory's beta function.

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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.