Perturbative and non-perturbative aspects of moments of the thrust distribution in e+e- annihilation

TL;DR

This paper investigates perturbative and non-perturbative aspects of moments of the thrust distribution $igra t^m\big\bra$ in $e^+e^-$ annihilation using the single dressed gluon (SDG) framework with a dispersive running coupling. It derives the thrust distribution within SDG, analyzes the first few moments to extract characteristic gluon virtualities and renormalon structure, and quantifies how running-coupling effects and power corrections evolve with $m$. The study finds that while the leading $1/Q$ power correction dominates for the average thrust, higher moments exhibit increasingly suppressed infrared corrections (roughly $1/Q^3$ or $1/Q^5$), though potential three-jet configurations could reintroduce $\alpha_s(Q^2)/Q$ terms. The results illuminate the interplay between perturbative resummation, renormalon ambiguities, and non-perturbative universality, with implications for precision determinations of $\alpha_s$ and for combining SDG with Sudakov/shape-function approaches.

Abstract

Resummation and power-corrections play a crucial role in the phenomenology of event-shape variables like the thrust T. Previous investigations showed that the perturbative contribution to the average thrust is dominated by gluons of small invariant mass, of the order of 10% of Q, where Q is the center-of-mass energy. The effect of soft gluons is also important, leading to a non-perturbative 1/Q correction. These conclusions are based on renormalon analysis in the single dressed gluon (SDG) approximation. Here we analyze higher moments of the thrust distribution using a similar technique. We find that the characteristic gluon invariant mass contributing to <(1-T)^m> increases with m. Yet, for m=2 this scale is quite low, around 27% of Q, and therefore renormalon resummation is still very important. On the other hand, the power-correction to <(1-T)^2> from a single soft gluon emission is found to be highly suppressed: it scales as 1/Q^3. In practice, <(1-T)^2> and higher moments depend also on soft gluon emission from configurations of three hard partons, which may lead to alpha_s(Q^2)/Q power-corrections. This issue is yet to be investigated.

Figures (3)

• Figure 1: Phase-space for the emission of a "massive" gluon with $\mu^2\,=\,\epsilon\,Q^2\,=\,0.1\,Q^2$ in the plane of the quark and anti-quark energy fractions ($x_{1,2}$). The outer continuous lines represent phase-space limits, the dashed lines separate the phase-space according to the identification of the thrust axis $\vec{n}_T$ and internal continuous lines are constant thrust contours. These contours are drawn with separation of $\Delta T=0.03$, stating at the lowest possible value of $1-T=\epsilon=0.1$ where two of the three partons are roughly collinear, up to the highest possible value of $1-T= \left(2 \sqrt{1+3\epsilon}-1\right)/3\simeq 0.427$ where the momentum splits equally between the three partons.
• Figure 2: The derivative of the characteristic functions $\dot{{\cal F}}(\epsilon)$ for ${\large<}1-T{\large>}$ (upper plot) and ${\large<}(1-T)^2{\large>}$ (lower plot) as a function of $\log_{10}(\epsilon)$, where $\mu^2=\epsilon Q^2$ is the gluon virtuality. The dashed lines in the two plots represent the ${\cal O}(\epsilon^{1/2})$ and ${\cal O}(\epsilon^{3/2})$ approximations to $\dot{{\cal F}}(\epsilon)$, respectively.
• Figure 3: The derivative of the characteristic functions $\dot{{\cal F}}(\epsilon)$ for ${\large<}(1-T)^3{\large>}$ (upper plot) and ${\large<}(1-T)^4{\large>}$ (lower plot) as a function of $\log_{10}(\epsilon)$, where $\mu^2=\epsilon Q^2$ is the gluon virtuality.