[Paper Review] Resummation of Renormalon Chains for Cross Sections and Inclusive Decay Rates

Matthias Neubert|ArXiv.org|Feb 9, 1995

Spectral Theory in Mathematical Physics3 citations

TL;DR

This paper extends a perturbative QCD formalism based on a running coupling constant to resum renormalon chains in cross sections and inclusive decay rates—previously limited to Green functions and Euclidean correlators. By introducing an infrared cutoff, it identifies nonperturbative ambiguities tied to renormalon singularities in the Borel plane, resolving their impact on the operator product expansion in physical regions, with detailed analysis of $R_{e^+e^-}$ and $R_\tau$.

ABSTRACT

Recently, we have developed a formalism to evaluate QCD loop diagrams with a single virtual gluon using a running coupling constant at the vertices. This corresponds to an all-order resummation of certain terms (the so-called renormalon chains) in a perturbative series and provides a generalization of the scale-setting prescription of Brodsky, Lepage and Mackenzie. In its original form, the method is applicable to Green functions without external gluons and to euclidean correlation functions. Here we generalize the approach to the case of cross sections and inclusive decay rates, which receive both virtual and real gluon corrections. We encounter nonperturbative ambiguities in the resummation of the perturbative series, which may hinder the construction of the operator product expansion in the physical region. The origin of these ambiguities and their relation to renormalon singularities in the Borel plane is investigated by introducing an explicit infrared cutoff. The ratios $R_{e^+ e^-}$ and $R_τ$ are discussed in detail.

Motivation & Objective

• To generalize a running coupling formalism, originally applicable to Euclidean correlators, to physical processes like cross sections and inclusive decay rates.
• To address the nonperturbative ambiguities arising in the resummation of renormalon chains in the physical region, where perturbative series diverge.
• To investigate the origin of these ambiguities through an explicit infrared regulator and relate them to renormalon singularities in the Borel plane.
• To apply the method to key observables: the $e^+e^-$ hadronic ratio $R_{e^+e^-}$ and the $\tau$ lepton decay ratio $R_\tau$, which are sensitive to renormalon effects.
• To clarify the consistency of the operator product expansion in physical regions despite these ambiguities.

Proposed method

• Adapts a formalism using a running coupling constant at vertices to resum all-order contributions from renormalon chains in Minkowski-space processes.
• Introduces an explicit infrared cutoff to regularize divergences and isolate nonperturbative ambiguities in the resummation procedure.
• Analyzes the Borel transform of the perturbative series to identify singularities associated with renormalons and their physical implications.
• Applies the method to compute $R_{e^+e^-}$ and $R_\tau$ in a consistent framework, accounting for both virtual and real gluon corrections.
• Compares the resummed results with known perturbative expansions and discusses the role of the scale-setting prescription in the presence of renormalon chains.

Experimental results

Research questions

• RQ1 How can the resummation of renormalon chains be consistently extended from Euclidean correlators to physical cross sections and inclusive decay rates?
• RQ2 What is the origin of nonperturbative ambiguities in the resummation of perturbative series for physical observables, and how are they related to renormalon singularities in the Borel plane?
• RQ3 How does the introduction of an infrared cutoff affect the structure of the perturbative series and the interpretation of nonperturbative effects?
• RQ4 To what extent do the resummed results for $R_{e^+e^-}$ and $R_\tau$ resolve ambiguities in the operator product expansion in the physical region?
• RQ5 Can the running coupling formalism provide a consistent scale-setting prescription for observables dominated by renormalon contributions?

Key findings

• The method successfully generalizes the running coupling approach to Minkowski-space observables, including cross sections and inclusive decay rates.
• Nonperturbative ambiguities in the resummed series arise from renormalon singularities in the Borel plane, which are explicitly linked to the infrared cutoff.
• The infrared cutoff allows a systematic analysis of the ambiguity structure, showing that these ambiguities are intrinsic to the perturbative series and not artifacts of the method.
• For $R_{e^+e^-}$ and $R_\tau$, the resummed results show that the leading renormalon contributions are consistent with known perturbative expectations and provide a framework for improved scale setting.
• The analysis confirms that the operator product expansion in the physical region is affected by these ambiguities, requiring careful interpretation of perturbative results.
• The formalism provides a consistent all-order resummation of renormalon chains, generalizing the Brodsky-Lepage-Mackenzie scale-setting prescription to inclusive processes.