Renormalon resummation and exponentiation of soft and collinear gluon radiation in the thrust distribution

TL;DR

The paper introduces Dressed Gluon Exponentiation (DGE) to resum both Sudakov logarithms and infrared renormalons in the thrust distribution for e+e− annihilation, by exponentiating a single dressed-gluon (SDG) cross-section with a dispersive running coupling. It develops a scheme-invariant Borel formulation, analyzes the renormalon structure, and shows how power corrections can be described by a shape-function, with even moments suppressed. Exponentiation is implemented via Laplace transforms, yielding a resummed cross-section that is matched to fixed-order NLO results (log-R matching) and integrated with non-perturbative corrections through a shift or a shape-function. Phenomenological fits to data across a wide range of energies yield $\alpha_s^{\overline{MS}}(M_Z) \approx 0.109$–$0.111$ with ~5% theoretical uncertainty and demonstrate a consistent perturbative-nonperturbative description, highlighting the importance of sub-leading logs and the predictive power of the renormalon-based power corrections. The work establishes a general framework linking Sudakov resummation with renormalon physics and provides a path for applying DGE to other observables with soft and collinear radiation.

Abstract

The thrust distribution in e+e- annihilation is calculated exploiting its exponentiation property in the two-jet region t = 1-T << 1. We present a general method (DGE) to calculate a large class of logarithmically enhanced terms, using the dispersive approach in renormalon calculus. Dressed Gluon Exponentiation is based on the fact that the exponentiation kernel is associated primarily with a single gluon emission, and therefore the exponent is naturally represented as an integral over the running coupling. Fixing the definition of Lambda is enough to guarantee consistency with the exact exponent to next-to-leading logarithmic accuracy. Renormalization scale dependence is avoided by keeping all the logs. Sub-leading logs, that are usually neglected, are factorially enhanced and are therefore important. Renormalization-group invariance as well as infrared renormalon divergence are recovered in the sum of all the logs. The logarithmically enhanced cross-section is evaluated by Borel summation. Renormalon ambiguity is then used to study power corrections in the peak region Qt \gsim Lambda, where the hierarchy between the renormalon closest to the origin (~1/Qt) and others (~1/(Qt)^n) starts to break down. The exponentiated power-corrections can be described by a shape-function, as advocated by Korchemsky and Sterman. Our calculation suggests that the even central moments of the shape-function are suppressed. Good fits are obtained yielding alpha_s^{MSbar} (M_Z) = 0.110 \pm 0.001, with a theoretical uncertainty of ~5%.

Figures (12)

• Figure 1: The functions $h_j(t)$, $j=1$ through $4$, in different approximations normalised to the leading logarithm (LL). The complete result (calculated numerically) is the full line and the sum of the logarithmically enhanced terms $\left.h_j(t)\right\vert_{\rm log}$ (calculated analytically) is dashed, the NLL approximation is dotted. For the first moment, the $\beta_0$ dependent part of the NLO result is also shown for comparison.
• Figure 2: $f_k(\xi)$ for $k=1$ (LL) through $k=8$ (${\rm N}^7{\rm LL}$) as a function of $\xi\equiv {\bar{A}}(Q^2)\ln\nu$. The absolute value $\left\vert f_k(\xi)\right\vert$ is plotted on a logarithmic scale.
• Figure 3: The relative significance (in absolute value) of terms ${{\bar{A}(Q^2)}^{k-2}}\,\, f_{k}\!\left({\bar{A}}(Q^2)\ln\nu\right)$ in (\ref{['log_expansion']}) for $k=2$ (NLL) through $k=7$ (${\rm N}^6{\rm LL}$) as a function of $1/\nu$ ($1/\nu\sim t$ in the small $t$ region). In both plots, the terms are normalized by the LL function ($1$ on the vertical axis).
• Figure 4: DGE, the standard NLL and the LO and NLO results as a function of $t$ in the two-jet region, at $Q={\rm M_Z}$. Fixed-order and NLL results are in the $\overline{\rm MS}$ scheme. As an example of scale dependence the NLL result is shown at two different renormalization points $\mu_R=Q$ and $\mu_R=Q/2$. For the LO and NLO results $\mu_R=Q$. We assume $\alpha_s^{\hbox{$\overline{\hbox{\tiny MS}}\,$}}({\rm M_Z})=0.110$. $\aleph$ data ALEPH91 is shown for orientation.
• Figure 5: The convergence of increasing approximants to the perturbative cross-section, according to the technique of section 3.3, eq. (\ref{['R_res_sum_I']}). The upper plot shows the differential cross-section for $p_{\rm max}=3,5$ and $8$ and the lower plot shows the relative error of the differential cross-section with respect to the $p_{\rm max}=8$ calculation, for $p_{\rm max}=5,6$ and $7$. We fix $\alpha_s^{\hbox{$\overline{\hbox{\tiny MS}}\,$}}({\rm M_Z})=0.110$ and use two-loop running coupling; log-R matching is applied.