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

Soft Functions for Generic Jet Algorithms and Observables at Hadron Colliders