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We extend the improved Collins-Soper-Sterman (iCSS) $W+Y$ constructionrecently presented in~\cite{Collins:2016hqq} to the case of polarizedobservables, where we focus in particular on the Sivers effect insemi-inclusive deep-inelastic scattering. We further show how one recovers theexpected leading-order collinear twist-3 result from a (weighted)$q_T$-integral of the differential cross section. We are also able todemonstrate the validity of the well-known relation between the (TMD) Siversfunction and the (collinear twist-3) Qiu-Sterman function within the iCSSframework. This relation allows for their interpretation as functions yieldingthe average transverse momentum of unpolarized quarks in a transverselypolarized spin-$\frac{1}{2}$ target. We further outline how this study can begeneralized to other polarized quantities.

Connections between collinear and transverse-momentum-dependent polarized observables within the Collins-Soper-Sterman formalism