Renormalons

TL;DR

This work provides a comprehensive survey of renormalons, linking the factorial growth of perturbative coefficients to both ultraviolet and infrared momentum regions. It develops a diagrammatic, 1/N_f framework to locate and characterize renormalon poles in the Borel plane, and connects these to operator product expansions and power corrections. The paper then surveys a wide range of phenomenological applications, including event shapes, DIS, Drell-Yan, and heavy-quark physics, illustrating how renormalon ideas shape our understanding of non-perturbative effects and guide modeling of power corrections. A key message is that infrared renormalons encode universal scaling patterns for power corrections but leave absolute magnitudes model-dependent, motivating both disciplined use of effective mass definitions and renormalon-inspired modeling in data analyses. Overall, renormalons illuminate the boundary between perturbative QCD and non-perturbative dynamics, underscoring the need for careful factorization, scheme choices, and non-perturbative input to achieve precise predictions in high-energy processes.

Abstract

A certain pattern of divergence of perturbative expansions in quantum field theories, related to their small and large momentum behaviour, is known as renormalons. We review formal and phenomenological aspects of renormalon divergence. We first summarize what is known about ultraviolet and infrared renormalons from an analysis of Feynman diagrams. Because infrared renormalons probe large distances, they are closely connected with non-perturbative power corrections in asymptotically free theories such as QCD. We discuss this aspect of the renormalon phenomenon in various contexts, and in particular the successes and failures of renormalon-inspired models of power corrections to hard processes in QCD.

Figures (23)

• Figure 1: The simplest set of 'bubble' diagrams for the Adler function consists of all diagrams with any number of fermion loops inserted into a single gluon line.
• Figure 2: The integrand of (\ref{['basint']}) for $n=0$ and $n=2$ as function of $\hat{k}^2$. The vertical scale is arbitrary.
• Figure 3: Singularities in the Borel plane of $\Pi(Q^2)$, the current-current correlation function in QCD. Shown are the singular points, but not the cuts attached to each of them. Recall that $\beta_0 < 0$ according to (\ref{['beta0']}).
• Figure 4: Pair creation of quarks by an external current: (a) Leading order in the flavour expansion; (b) representatives at next-to-leading order. Chains are displayed as dashed lines.
• Figure 5: Leading order contribution in the flavour expansion to the two point function of vector currents $j_V$ (left) and its reduced diagrams with operator insertions (right). The momentum $p$ is the loop momentum for the fermion loop.