Treating Divergent Perturbation Theory: Lessons from Exactly Solvable 2D Models at Large $N$

Abstract

We consider the operator product expansion (OPE) of correlation functions in the supersymmetric $O(N)$ non-linear sigma model at sub-leading order in the large $N$ limit in order to study the cancellation between ambiguities coming from infrared renormalons and those coming from various operators in the OPE. As has been discussed in the context of supersymmetric Yang-Mills theory in four dimensions, supersymmetry presents a challenge to this cancellation. In a bid to solve this problem we consider $O(N)$ as a toy model. A background field method inspired by Polyakov's treatment of the renormalization of the bosonic $O(N)$ model is used to identify explicit operators in the OPE of the two-point functions of bosonic and fermionic fields in the model. In order to identify the coefficient functions in the OPE, the exact two-point functions at sub-leading order in large $N$ are expanded in powers of the natural infrared length scale. The ambiguities arising from renormalons in the coefficient functions and vacuum expectation values of operators in the OPE are shown to cancel to all orders. The question of supersymmetric Yang-Mills theory without matter remains open.

arXiv comment46 pages, 7 figures

References in corpus (14)

• Monopole Condensation, And Confinement In N=2 Supersymmetric Yang-Mills Theory
• Monopoles, Duality and Chiral Symmetry Breaking in N=2 Supersymmetric QCD
• Renormalons
• Resurgence and Trans-series in Quantum Field Theory: The CP(N-1) Model
• The semi-classical expansion and resurgence in gauge theories: new perturbative, instanton, bion, and renormalon effects
• Scale Setting in QCD and the Momentum Flow in Feynman Diagrams
• Continuity and Resurgence: towards a continuum definition of the CP(N-1) model
• The Operator Product Expansion, Non-perturbative Couplings and the Landau Pole: Lessons from the O(N) $\sigma$-Model
• IR Renormalons vs. Operator Product Expansion in Supersymmetric and Related Gauge Theories
• Renormalons in integrable field theories
• Resurgence, Operator Product Expansion, and Remarks on Renormalons in Supersymmetric Yang-Mills
• Infrared renormalon in the supersymmetric $\mathbb{C}{P}^{N-1}$ model on $\mathbb{R}\times S^1$
• Universal Renormalons in Principal Chiral Models
• Renormalon disappearance in Borel sum of the 1/N expansion of the Gross-Neveu model mass gap

Cited by in corpus (7)

• 50 Years of Quantum Chromodynamics
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• Instantons, renormalons and the theta angle in integrable sigma models
• Resurgence and semiclassical expansion in two-dimensional large-$N$ sigma models
• Infrared Renormalons in Supersymmetric Theories
• Trans-series from condensates
• Trans-series from condensates in the non-linear sigma model