Soft function for exclusive N-jet production at hadron colliders

The N-jettiness event shape divides phase space into N + 2 regions, each containing one jet or beam. Using a geometric measure, these regions correspond to jets with circular boundaries. We give a factorization theorem for the cross section fully differential in the mass of each jet, and compute the corresponding soft function at next-to-leading order (NLO). The ultraviolet divergences are analytically extracted by exploiting hemispheres for interactions between each pair of hard partons, leaving only convergent integrals that are sensitive to the precise boundaries. This hemisphere decomposition can also be applied to other N-jet soft functions, including other observables. For N-jettiness, the final result for the soft function involves stable one-dimensional numerical integrals, and all ingredients are now available to extend NLO cross sections to resummed predictions at next-to-next-to-leading logarithmic order.