Next-to-next-to-leading order threshold soft function for $tW$ production

We compute the two-loop soft function for the associated production of a top quark and a $W$ boson near the threshold, where the invariant mass of the $tW$ system approaches the collider energy. We employ the reverse unitarity technique and integration-by-parts identities to reduce soft loop integrals to a minimal basis of master integrals. These master integrals are analytically evaluated using the method of differential equations, yielding results expressed in terms of multiple polylogarithms. Additionally, we analyze the asymptotic behavior of the soft function in the low and high energy limits. Our results provide a vital component for threshold resummation at the next-to-next-to-next-to-leading logarithmic level in this process.