Renormalization of the Next-to-Leading-Power Soft Function for the Drell-Yan Off-diagonal Channel

We renormalize the soft function entering the factorization and resummation of the $qg$ parton-scattering channel of the Drell-Yan process near the kinematic threshold $\widehat{s}\to Q^2$ at next-to-leading power in the expansion around $z\equiv Q^2\mathrm{/}\widehat{s}=1$, and solve its renormalization-group equation.