Soft Functions for Generic Jet Algorithms and Observables at Hadron Colliders

We introduce a method to compute one-loop soft functions for exclusive \(N\)-jet processes at hadron colliders, allowing for different definitions of the algorithm that determines the jet regions and of the measurements in those regions. In particular, we generalize the \(N\)-jettiness hemisphere decomposition of [Jouttenus 2011] in a manner that separates the dependence on the jet boundary from the observables measured inside the jet and beam regions. Results are given for several factorizable jet definitions, including anti-\(k_T\), XCone, and other geometric partitionings. We calculate explicitly the soft functions for angularity measurements, including jet mass and jet broadening, in \(pp \to L + 1\) jet and explore the differences for various jet vetoes and algorithms. This includes a consistent treatment of rapidity divergences when applicable. We also compute analytic results for these soft functions in an expansion for a small jet radius \(R\). We find that the small-\(R\) results, including corrections up to \(\mathcal{O}(R^2)\), accurately capture the full behavior over a large range of \(R\).