Relating Transverse Momentum Dependent and Collinear Factorization Theorems in a Generalized Formalism

We construct an improved implementation for combining transverse-momentum-dependent (TMD) factorization and collinear factorization. TMD factorization is suitable for low transverse momentum physics, while collinear factorization is suitable for high transverse momenta and for a cross section integrated over transverse momentum. The result is a modified version of the standard $W+Y$ prescription traditionally used in the Collins-Soper-Sterman (CSS) formalism and related approaches. We further argue that questions regarding the shape and $Q$-dependence of the cross sections at lower $Q$ are largely governed by the matching to the $Y$-term.