Resummation of Renormalon Chains for Cross Sections and Inclusive Decay Rates

TL;DR

This paper extends renormalon-chain resummation to time-like observables such as cross sections and inclusive decay rates by employing a running coupling at gluon vertices, generalizing the BLM scale-setting framework. It exposes infrared ambiguities tied to Landau-pole regularization, and compares linear and non-linear (contour) resummations for $R_{e^+e^-}$ and $R_ au$, highlighting that in the physical region these ambiguities can exceed standard IR renormalon effects and may challenge the applicability of the OPE. Through Euclidean-regulated analyses and various cutoff schemes, the work demonstrates sizable nonperturbative contributions (e.g., $1/m^2$ or $1/m^4$ terms) that depend on the resummation prescription. The numerical results illustrate modest but meaningful shifts from resummation, underline the non-uniqueness of resummed outcomes in the physical region, and have potential implications for precise determinations of $ar{ ext{MS}}$-scale parameters and CKM elements from hadronic tau decays and heavy-quark decays.

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.

Figures (8)

• Figure 1: The distribution function $\tau\,\widehat{w}_D(\tau)$ as a function of $\ln\tau$. The long arrow indicates the average value of $\ln\tau$, which determines the BLM scale. The short arrows show the point $\tau=\lambda^2/Q^2$ for $\lambda=1$ GeV and $Q^2=m_\tau^2$ (right) and $(20~\hbox{GeV})^2$ (left).
• Figure 2: Short-distance contribution to $D(m_\tau^2)$ as a function of the factorization scale. The dotted line is used in the region below the Landau pole, where the short-distance contribution is no longer well defined.
• Figure 3: Effective coupling constants $F_1(a)$ (dash-dotted line) and $F_2(a)$ (solid line) as a function of $1/a$. Both functions coincide for $a>0$. The thin line shows the bare coupling constant $a$.
• Figure 4: The distribution functions $\tau\,W_{e^+ e^-}(\tau)$ (solid line) and $\tau\,W_\tau(\tau)$ (dash-dotted line) as a function of $\ln\tau$. The arrows indicate the average values of $\ln\tau$, which determine the BLM scales.
• Figure 5: Effective coupling constants $G_1(a)$ (dash-dotted line), $G_2(a)$ (dotted line) and $G_3(a)$ (solid line) as a function of (a) the inverse coupling constant $1/a$, and (b) the coupling constant $a$. For small positive values of $a$, the three functions have identical Taylor expansions.