On Thrust Resummation Ambiguities in $e^+e^-$ Annihilation into Hadrons

TL;DR

This paper investigates how different legitimate resummation methods in QCD predictions for $e^+e^-$ annihilation into hadrons, especially for the thrust variable, can lead to significant differences in results, highlighting the need for more conservative error estimates.

Contribution

It compares conjugate-space and direct-space resummation approaches for thrust, revealing their differences and the impact of approximations on predictions.

Findings

Different resummation formalisms can produce non-negligible numerical differences.

The inverse Laplace transform introduces a convergent tower of subleading terms.

Systematic uncertainties from formalism choices often exceed standard theoretical errors.

Abstract

In $e^+e^-$ shape-variable studies, and in particular for the case of thrust, fixed-order QCD predictions are typically supplemented with the resummation of contributions enhanced near the two-jet limit. In this work we examine whether different, yet legitimate, resummation prescriptions can induce significant differences in the resulting predictions. This can occur because formally equivalent prescriptions may differ by terms that, although subleading, are characterised by asymptotic expansions and may therefore lead to slow convergence. We first compare two alternative formulations of resummation: the conjugate-space (or Laplace-space) approach, in which resummation is performed in a variable conjugate to thrust, such that the observable factorises exactly in the soft-collinear limit; and the direct-space formulation, where resummation is instead carried out directly in the thrust variable. We show that, at double-logarithmic level, the inverse Laplace transform generates a convergent tower of subleading terms. Starting from leading-logarithmic accuracy, the expansion becomes asymptotic due to the presence of the Landau pole, leading to a mild log-factorial growth of the coefficients. When including the highest available logarithmic order in the resummation, matched to fixed-order results, we still find non-negligible differences between predictions obtained in the two spaces. We then consider a formulation of the resummation that avoids certain approximations commonly used in the derivation of conjugate-space resummation. We observe that this also has a non-negligible numerical impact. In general, we find that the systematics stemming from the adoption of different formalisms typically exceeds the quoted theoretical uncertainties, suggesting the need for more conservative theory-error estimates when using the thrust distribution in determinations of the strong coupling.