$C$-parameter hadronisation in the symmetric 3-jet limit and impact on $\alpha_s$ fits

TL;DR

This paper analytically computes the leading non-perturbative correction to the $C$ parameter in the symmetric 3-jet limit and assesses its impact on the extraction of the strong coupling constant, revealing potential explanations for discrepancies in previous $eta_s$ fits.

Contribution

It provides the first analytical calculation of hadronisation corrections for the $C$ parameter in the symmetric 3-jet limit and evaluates their influence on $eta_s$ determinations.

Findings

The correction differs by more than a factor of two from the 2-jet limit result.

Interpolating hadronisation corrections can increase the extracted $eta_s$ by up to 4%.

This work offers a new perspective on the $eta_s$ discrepancy in event-shape analyses.

Abstract

Hadronisation corrections are crucial in extractions of the strong coupling constant ($\alpha_s$) from event-shape distributions at lepton colliders. Although their dynamics cannot be understood rigorously using perturbative methods, their dominant effect on physical observables can be estimated in singular configurations sensitive to the emission of soft radiation. The differential distributions of some event-shape variables, notably the $C$ parameter, feature two such singular points. We analytically compute the leading non-perturbative correction in the symmetric three-jet limit for the $C$ parameter, and find that it differs by more than a factor of two from the known result in the two-jet limit. We estimate the impact of this result on strong coupling extractions, considering a range of functions to interpolate the hadronisation correction in the region between the 2 and 3-jet limits. Fitting data from ALEPH and JADE, we find that most interpolation choices increase the extracted $\alpha_s$, with effects of up to $4\%$ relative to standard fits. This brings a new perspective on the long-standing discrepancy between certain event-shape $\alpha_s$ fits and the world average.