Light-cone divergence in twist-3 correlation functions

It is argued that the definition of the twist-3 transverse momentum dependent correlation functions must be modified if they contain light-like Wilson lines. In the framework of a simple spectator model of the nucleon we show explicitly the presence of a light-cone divergence for a specific twist-3 time-reversal odd parton density. This divergence emerges for all eight twist-3 T-odd correlators and appears also in the case of a quark target in perturbative QCD. The divergence can be removed by using non-light-like Wilson lines. Based on our results we argue that currently there exists no established factorization formula for transverse momentum dependent twist-3 observables in semi-inclusive DIS and related processes. Partonic correlation functions (distributions and fragmentation functions) that depend on both the longitudinal and transverse parton momentum (pT -dependent) have received considerable attention over the past years. The most prominent examples are the (naive) time-reversal odd (T-odd) Sivers parton density [1, 2] and the Collins fragmentation function [3], where the status of the mere existence of T-odd parton distributions was clarified only relatively recently [4, 5]. A great deal of theoretical and experimental effort has been devoted to determining these functions by characterizing and measuring single spin asymmetries in semi-inclusive deep-inelastic scattering (DIS) [6, 7, 8] and azimuthal asymmetries in e+e−-annihilation [9] (see also [10, 11]). The first experimental studies on single spin asymmetries in semi-inclusive DIS dealt with the observables AUL (unpolarized lepton beam and longitudinally polarized target) [12, 13, 14, 15] and ALU (longitudinally polarized lepton beam and unpolarized target) [16]. By contrast they are suppressed like 1/Q (with Q denoting the virtuality of the exchanged photon) with respect to the leading twist asymmetry AUT (transversely polarized target). Consequently, a parton model description of these asymmetries invokes (pT -dependent) twist-3 correlation functions [17, 18, 19] (see, e.g., Refs. [20, 21, 22, 23, 24, 25, 26, 27, 28] for related theoretical work on twist-3 single spin asymmetries). A complete list of pT -dependent twist-3 correlation functions was obtained only recently [29, 30, 31, 32, 33]. It is non-trivial to establish QCD-factorization for transverse momentum dependent hard scattering processes in terms of pT -dependent partonic correlators. While the twist-2 case was studied in some detail in the literature (see, e.g., Refs. [34, 35, 36, 37, 38]), for twist-3 observables only a tree-level formalism is available up to now [17, 18, 19, 39]. To some extent the difficulty is intimately related to the task of finding a suitable Wilson line that ensures color gauge invariance of the pT -dependent correlation functions (see in particular the discussion in [40]). In this context