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The absence of IR renormalons in gauge theories on $\mathbb R^3\times \mathbb S^1$ and what it means for resurgence

Mohamed M. Anber, Tin Sulejmanpasic

TL;DR

The paper addresses whether IR renormalons persist for gauge theories on $\mathbb R^3\times \mathbb S^1$ by performing exact one-loop vacuum-polarization calculations in QCD with adjoint matter. It demonstrates that IR logarithms cancel between vacuum and finite-volume (holonomy) contributions, eliminating IR renormalon ambiguities in this setting and clarifying the role of neutral bions as diagrammatic proliferations rather than renormalon effects. The authors develop a perturbative strategy using the background-field method in a center-symmetric holonomy, compute the polarization tensor for both fermionic and non-abelian sectors, and obtain a fully resummed gluon propagator that remains well-behaved in the IR. These results suggest a clean semi-classical regime on small circles and have implications for resurgence, with exact one-loop results applicable to thermal QCD and broader non-perturbative questions. The work thus provides a tangible link between perturbative control on $\mathbb R^3\times \mathbb S^1$ and non-perturbative semi-classical dynamics, while clarifying the nature of singularities in the Borel plane in this setting.

Abstract

We analyze the renormalon diagram of gauge theories on $\mathbb R^3\times \mathbb S^1$. In particular, we perform exact one loop calculations for the vacuum polarization in QCD with adjoint matter and observe that all infrared logarithms, as functions of the external momentum, cancel between the vacuum part and finite volume part, which eliminates the IR renormalon problem. We argue that the singularities in the Borel plane, arising from the topological neutral bions, are not associated with renormalons, but with the proliferation of the Feynman diagrams. As a byproduct, we obtain, for the first time, an exact one-loop result of the vacuum polarization which can be adapted to the case of thermal compactification of QCD.

The absence of IR renormalons in gauge theories on $\mathbb R^3\times \mathbb S^1$ and what it means for resurgence

TL;DR

The paper addresses whether IR renormalons persist for gauge theories on by performing exact one-loop vacuum-polarization calculations in QCD with adjoint matter. It demonstrates that IR logarithms cancel between vacuum and finite-volume (holonomy) contributions, eliminating IR renormalon ambiguities in this setting and clarifying the role of neutral bions as diagrammatic proliferations rather than renormalon effects. The authors develop a perturbative strategy using the background-field method in a center-symmetric holonomy, compute the polarization tensor for both fermionic and non-abelian sectors, and obtain a fully resummed gluon propagator that remains well-behaved in the IR. These results suggest a clean semi-classical regime on small circles and have implications for resurgence, with exact one-loop results applicable to thermal QCD and broader non-perturbative questions. The work thus provides a tangible link between perturbative control on and non-perturbative semi-classical dynamics, while clarifying the nature of singularities in the Borel plane in this setting.

Abstract

We analyze the renormalon diagram of gauge theories on . In particular, we perform exact one loop calculations for the vacuum polarization in QCD with adjoint matter and observe that all infrared logarithms, as functions of the external momentum, cancel between the vacuum part and finite volume part, which eliminates the IR renormalon problem. We argue that the singularities in the Borel plane, arising from the topological neutral bions, are not associated with renormalons, but with the proliferation of the Feynman diagrams. As a byproduct, we obtain, for the first time, an exact one-loop result of the vacuum polarization which can be adapted to the case of thermal compactification of QCD.

Paper Structure

This paper contains 20 sections, 111 equations, 3 figures.

Table of Contents

  1. Introduction
  2. IR renormalons: the sickness and cure
  3. IR renormalons on ${\mathbb R}^4$
  4. Overview of the theory on ${\mathbb R}^3\times {\mathbb S}^1$ and the absence of the renormalon problem
  5. Strategy and the calculation method
  6. Perturbative dynamics of QCD(adj) on $\mathbb R^3 \times \mathbb S^1$
  7. The background field method on $\mathbb R^3 \times \mathbb S^1$ in the presence of non-trivial holonomy
  8. The general form of the gluon propagator
  9. Calculating the polarization tensor on $\mathbb R^3 \times \mathbb S^1$
  10. The one-loop fermion correction
  11. The non-abelian part of QCD one-loop correction
  12. The static limit, resummation and absence of IR renormalons
  13. Conclusion
  14. Lie algebra and propagators on $\mathbb R^3 \times \mathbb S^1$ in the presence of holonomy
  15. Elements of the Lie algebra
  16. ...and 5 more sections

Figures (3)

  • Figure 1: Left: the vacuum polarization with all corrections. Right: Particular contribution to the vacuum polarization often referred to as the renormalon diagram.
  • Figure 2: Fermion contribution to the vacuum polarization.
  • Figure 3: Diagrams contributing to the non-ableian part of the vacuum polarization \ref{['the nonabelian polarization']}. The dashed lines are the ghosts, and that the second line of diagrams does not contribute on ${\mathbb R}^4$ in a normalization that doesn't violate gauge invariance.