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We present a factorization theorem of the partonic Drell-Yan off-diagonalprocesses $g\bar{q}\,(qg) \to \gamma^* + X$ in the kinematic threshold regime$z=Q^2/\hat{s} \to 1$ at general subleading powers in the $(1-z)$ expansion.Focusing on the first order of the expansion (next-to-leading power accuracywith respect to the leading power $q \bar{q}$ channel), we validate the barefactorization formula up to $\mathcal{O}(\alpha^2_s)$. This is achieved bycarrying out an explicit calculation of the generalized soft function in$d$-dimensions using the reduction to master integrals and the differentialequations method. The collinear function is a universal object which we computefrom an operator matching equation at one-loop level. Next, we integrate thesoft and collinear functions over the convolution variables and remove theremaining initial state collinear singularities through PDF renormalization.The resulting expression agrees with the known cross section in the literature.

Threshold factorization of the Drell-Yan quark-gluon channel and two-loop soft function at next-to-leading power