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 T. T. Joutennus et al. (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→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 O(R²), accurately capture the full behavior over a large range of R.