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

TL;DR

This paper extends the Collins-Soper-Sterman formalism to polarized observables, specifically analyzing the Sivers effect, and demonstrates the relation between TMD and collinear twist-3 functions within this framework.

Contribution

It generalizes the iCSS $W+Y$ formalism to polarized observables and confirms the TMD-collinear relation for the Sivers function.

Findings

Recovered leading-order collinear twist-3 results from weighted $q_T$ integrals.

Validated the relation between the Sivers function and Qiu-Sterman function.

Outlined potential generalizations to other polarized quantities.

Abstract

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