arXiv2025
We introduce a class of collider observables, named Lund-Tree Shapes (LTS), defined from declustering trees originating from the Lund jet plane representation of the QCD radiation pattern in multi-jet scattering processes. At the differential level, they are continuous global variables akin classical event shapes and $n\to n+1$ jet-resolution parameters, which probe the geometry and hierarchical structure of the radiation in an event. At the integrated, cumulative level, they naturally define $n$ jet rates, providing a jet-multiplicity-based characterisation of multi-jet final states. Their definition applies to scattering processes with any number of resolved jets in the final state, as well as to groomed jets. They are thus usable as resolution variables in the context of higher-order calculations via phase-space slicing, matching fixed-order calculations to parton showers, and testing the logarithmic accuracy of shower algorithms. From a theoretical viewpoint, such observables feature a simple all-order structure and are free of non-global logarithmic corrections. As an initial application, we derive next-to-next-to-leading-logarithmic accurate predictions for processes with two QCD legs at $e e$, $pp$ and $e p$ colliders, and matched predictions to next-to-next-to-leading order for the LHC, discussing aspects of collider phenomenology.