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Hadronisation corrections are crucial in extractions of the strong couplingconstant ($\alpha_s$) from event-shape distributions at lepton colliders.Although their dynamics cannot be understood rigorously using perturbativemethods, their dominant effect on physical observables can be estimated insingular configurations sensitive to the emission of soft radiation. Thedifferential distributions of some event-shape variables, notably the $C$parameter, feature two such singular points. We analytically compute theleading 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 theknown result in the two-jet limit. We estimate the impact of this result onstrong coupling extractions, considering a range of functions to interpolatethe hadronisation correction in the region between the 2 and 3-jet limits.Fitting data from ALEPH and JADE, we find that most interpolation choicesincrease the extracted $\alpha_s$, with effects of up to $4\%$ relative tostandard fits. This brings a new perspective on the long-standing discrepancybetween certain event-shape $\alpha_s$ fits and the world average.

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