Semi-numerical resummation of event shapes

TL;DR

The paper tackles the challenge of resumming single-logarithmic contributions in event-shape distributions by introducing a semi-numerical framework that relates a complex observable to a simple reference via a probability P(v|v_s) and a function F(R') encoding multi-emission effects. It systematically constructs a simple observable with an analytically tractable resummation, then numerically maps the complex observable's resummation to this reference, ensuring NNLL contamination is controlled and NL accuracy is preserved. The method reproduces known analytical results for thrust and broadening, extends to three-jet thrust minor, and yields new fully NLL resummed predictions for thrust major, oblateness, and the Durham y3 distribution, including careful treatment of subleading terms and matching to fixed order. This approach offers a general, automation-friendly route to resummed distributions for a wide class of global event shapes, providing both validation against established results and new predictions that were previously intractable analytically.

Abstract

For many event-shape observables, the most difficult part of a resummation in the Born limit is the analytical treatment of the observable's dependence on multiple emissions, which is required at single logarithmic accuracy. We present a general numerical method, suitable for a large class of event shapes, which allows the resummation specifically of these single logarithms. It is applied to the case of the thrust major and the oblateness, which have so far defied analytical resummation and to the two-jet rate in the Durham algorithm, for which only a subset of the single logs had up to now been calculated.