  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 (transverse) 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 method can be
applied in general to 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.
