Resummation of $(β_0 α_s)^n$ Corrections in QCD: Techniques and Applications to the $τ$ Hadronic Width and the Heavy Quark Pole Mass

TL;DR

Ball, Beneke, and Braun develop a dispersion- and Borel-based resummation framework to exactly sum an infinite class of (β0 α_s)^n corrections in QCD, extending the Brodsky-Lepage-Mackenzie approach to all orders via Naive Nonabelianization. The method converts multi-loop vacuum-polarization insertions into a one-dimensional integral over a finite-gluon-mass parameter, yielding an effective charge that clarifies scale-setting and renormalization effects. Applying the technique to hadronic τ decays and to heavy-quark pole masses, they find that resummation lowers α_s(m_τ) by about 10% and increases the bottom pole–MS mass difference by roughly 30%, with finite-mass effects and two-loop running explored. The work provides practical, numerically tractable tools for incorporating large higher-order perturbative corrections, clarifies renormalon and analyticity issues, and has significant implications for precision determinations of α_s and heavy-quark parameters.

Abstract

We propose to resum exactly any number of one-loop vacuum polarization insertions into the scale of the coupling of lowest order radiative corrections. This makes maximal use of the information contained in one-loop perturbative corrections combined with the one-loop running of the effective coupling and provides a natural extension of the familiar BLM scale-fixing prescription to all orders in the perturbation theory. It is suggested that the remaining radiative corrections should be reduced after resummation. In this paper we implement this resummation by a dispersion technique and indicate a possible generalization to incorporate two-loop evolution. We investigate in some detail higher order perturbative corrections to the $τ$ decay width and the pole mass of a heavy quark. We find that these corrections tend to reduce $α_s(m_τ)$ determined from $τ$ decays by approximately 10\% and increase the difference between the bottom pole and $\MS$-renormalized mass by 30\%.

Figures (7)

• Figure 1: A typical diagram with multiple fermion loop insertions into the lowest order correction to a generic physical quantity.
• Figure 2: One-loop running coupling in the V-scheme, $C=0$, (broken line) and effective coupling $-\Phi(\lambda^2)$ (solid line) as functions of $\lambda^2/\Lambda_V^2$ .
• Figure 3: Perturbative corrections to the $\tau$ hadronic width. I: After resummation of one-loop running effects; II: Exact order $\alpha_s^3$-approximation; III: Exact order $\alpha_s^3$-approximation including the resummation of running coupling effects along a circle in the complex plane (taken from Pich); IV: Exact order $\alpha_s^2$-approximation for comparison. The shaded bar gives the experimental value with experimental errors only.
• Figure 4: Ratio $d_n(m_i)/d_n(0)$ for different values of internal quark masses $m_i$ as a function of the number of fermion loop insertions. The value of $m_i^2/m^2$ is indicated to the right of each curve.
• Figure 5: QED-like diagrams incorporating evolution of the coupling to leading (a) and next-to-leading order (b). A circle with letter $m$ denotes a chain of $m$ fermion loops. At order $\alpha^{k+1}$, the relevant diagrams are specified by $n=k$ and $n_1+n_2+n_3=k-2$. At NLO, the diagrams, where the $n_2$-chain forms a self-energy-type insertion, are not depicted.