$N$-Jettiness Soft Functions Made Simple

We present a new method to compute the soft function for the $N$-Jettiness variable for arbitrary $N$ at high perturbative orders in QCD. It is based on the observation that the most singular part of the soft function, the dipole contribution, can be represented by a sum of an analytically calculable inclusive soft function and a remainder. The latter is absent at NLO, is immediately finite at NNLO and can be made finite with the help of simple NLO-like infrared subtractions at N$^3$LO. As a byproduct of this approach, we derive a very simple formula for the tripole contribution to the $N$-Jettiness NNLO soft function, which results in a fast numerical evaluation. We apply this method to compute the $N$-Jettiness soft function at NNLO, and report numerical results for up to five jets for the hadron-collider soft function. We finally outline the prospects for applications at N$^3$LO.