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Renormalons and Power Corrections

M. Beneke, V. M. Braun

TL;DR

The paper surveys infrared renormalons as a bridge between perturbative QCD and nonperturbative power corrections, articulating how factorial growth in perturbation theory maps to concrete $1/Q^p$ corrections via the Borel plane and the OPE framework. It presents the Large-$\beta_0$ limit as a tractable model to study renormalon structures and applies these ideas to a broad set of observables, including deep inelastic scattering, hadronic event shapes, and heavy-quark processes. A key theme is the interplay between perturbative ambiguities and higher-twist parameters, leading to practical renormalon-based models (e.g., shape functions, Milan factor) and mass definitions (PS mass) that improve convergence and phenomenological accuracy. The review also distinguishes IR from UV renormalons, discusses their respective implications, and highlights remaining formal and computational challenges in universal quantification of power corrections and their lattice connections.

Abstract

Even for short-distance dominated observables the QCD perturbation expansion is never complete. The divergence of the expansion through infrared renormalons provides formal evidence of this fact. In this article we review how this apparent failure can be turned into a useful tool to investigate power corrections to hard processes in QCD.

Renormalons and Power Corrections

TL;DR

The paper surveys infrared renormalons as a bridge between perturbative QCD and nonperturbative power corrections, articulating how factorial growth in perturbation theory maps to concrete corrections via the Borel plane and the OPE framework. It presents the Large- limit as a tractable model to study renormalon structures and applies these ideas to a broad set of observables, including deep inelastic scattering, hadronic event shapes, and heavy-quark processes. A key theme is the interplay between perturbative ambiguities and higher-twist parameters, leading to practical renormalon-based models (e.g., shape functions, Milan factor) and mass definitions (PS mass) that improve convergence and phenomenological accuracy. The review also distinguishes IR from UV renormalons, discusses their respective implications, and highlights remaining formal and computational challenges in universal quantification of power corrections and their lattice connections.

Abstract

Even for short-distance dominated observables the QCD perturbation expansion is never complete. The divergence of the expansion through infrared renormalons provides formal evidence of this fact. In this article we review how this apparent failure can be turned into a useful tool to investigate power corrections to hard processes in QCD.

Paper Structure

This paper contains 22 sections, 96 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Infrared Renormalons - Basic Concepts
  3. The Borel Plane
  4. IR Renormalons and the Operator Product Expansion
  5. The Large-$\beta_0$ Limit
  6. IR Renormalons and Power Corrections in Quantities without an Operator Product Expansion
  7. The Landau Pole
  8. Applications
  9. Deep Inelastic Scattering
  10. The GLS Sum Rule
  11. The $x$-Dependence of Power Corrections to Structure Functions
  12. Hadronic Event Shape Variables
  13. Mean Values of Event Shape Variables
  14. Event Shape Distributions
  15. The Energy Flow Correlation Function
  16. ...and 7 more sections

Figures (9)

  • Figure 1: Singularities in the Borel plane of $\Pi(Q^2)$. The singular points are shown, but not the cuts attached to each of them.
  • Figure 2: The set of "bubble" diagrams consists of all diagrams with any number of fermion loops inserted into a single gluon line.
  • Figure 3: Relative twist-4 contribution $D(x)$ (called $C_{p,d}(x)$ here) defined by Eq. (\ref{['par']}) to the proton (deuteron) structure function $F_2$ in the "renormalon model" MSSM97 (dashed line) compared with proton (filled circles) and deuteron (empty circles) data. VM92 The solid curve shows the unrescaled estimate of the renormalon ambiguity.
  • Figure 4: Twist-4 correction to $x F_3$ as extracted from the (revised) CCFR data. The three plots show the effect of including leading order (LO), next-to-leading order (NLO) and next-to-next-to-leading order (NNLO) QCD corrections in the twist-2 term. The data points KKPS97 are overlaid with the shape obtained from the "renormalon model" for the $1/Q^2$ power correction.
  • Figure 5: Energy dependence of $\langle 1-T\rangle$ (upper panel) and the heavy jet mass $\langle M_H^2/Q^2\rangle$ (lower panel) plotted as function of $1/Q$. Jad98 Dotted line: second-order perturbation theory with scale $\mu=Q$. Solid line: second-order perturbation theory with power correction added according to Eq. (\ref{['ksmu']}) and with $\mu=Q$, $\mu_I=2\,$GeV. For $\bar{\alpha}_0(2\,\hbox{GeV})$ the fit values 0.543 for thrust and 0.457 for the heavy jet mass Jad98 are taken. (Note that this reference uses $c_{1-T}=c_{M_H^2/s}=1$.) The dashed line shows second order perturbation theory at the very low scale $0.07 Q$ with no power correction added. For both observables $\alpha_s(M_Z)$ has been fixed to 0.12.
  • ...and 4 more figures