Next-to-next-to-leading order N-jettiness soft function for tW production

We calculate the $N$-jettiness soft function for $tW$ production up to next-to-next-to-leading order in QCD, which is an important ingredient of the $N$-jettiness subtraction method for predicting the differential cross sections of massive coloured particle productions. The divergent parts of the results have been checked using the renormalization group equations controlled by the soft anomalous dimension.