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<p>Instantons and renormalons play important roles at the interface between perturbative and non-perturbative quantum field theory. They are both associated with branch points in the Borel transform of asymptotic series, and as such can be detected in perturbation theory. However, while instantons are associated with non-perturbative saddle points of the path integral, renormalons have mostly been understood in terms of Feynman diagrams and operator product expansions. We suggest a non-perturbative path integral explanation of how both instantons and renormalons produce singularities in the Borel plane using representative finite-dimensional integrals. In particular, we build evidence that renormalons can be understood as saddle points of the 1-loop effective action, enabled by a crucial contribution from the quantum scale anomaly. These results are illustrated in simple toy models and indicate a possible route toward studying renormalons within realistic asymptotically-free field theories.</p>

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Renormalons as saddle points

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年4月17日

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  • Arindam Bhattacharya,
  • Jordan Cotler,
  • Aurélien Dersy,
  • Matthew D. Schwartz

摘要

Instantons and renormalons play important roles at the interface between perturbative and non-perturbative quantum field theory. They are both associated with branch points in the Borel transform of asymptotic series, and as such can be detected in perturbation theory. However, while instantons are associated with non-perturbative saddle points of the path integral, renormalons have mostly been understood in terms of Feynman diagrams and operator product expansions. We suggest a non-perturbative path integral explanation of how both instantons and renormalons produce singularities in the Borel plane using representative finite-dimensional integrals. In particular, we build evidence that renormalons can be understood as saddle points of the 1-loop effective action, enabled by a crucial contribution from the quantum scale anomaly. These results are illustrated in simple toy models and indicate a possible route toward studying renormalons within realistic asymptotically-free field theories.