Abstract
We derive new Lorentz invariance and equation of motion relations between twist-three generalized parton distributions (GPDs) and moments in the parton transverse momentum, ${k}_{T}$, of twist-two generalized transverse momentum-dependent distributions (GTMDs), as a function of the parton longitudinal momentum fraction $x$. Although GTMDs in principle define the observables for partonic orbital motion, experiments that can unambiguously detect them appear remote at present. The relations presented here provide a solution to this impasse in that, e.g., the orbital angular momentum density is connected to directly measurable twist-three GPDs. Out of 16 possible equation of motion relations that can be written in the $T$-even sector, we focus on three helicity configurations that can be detected analyzing specific spin asymmetries: two correspond to longitudinal proton polarization and are associated with quark orbital angular momentum and spin-orbit correlations; the third, obtained for transverse proton polarization, is a generalization of the relation obeyed by the ${g}_{2}$ structure function. We also exhibit an additional relation connecting the off-forward extension of the Sivers function to an off-forward Qiu-Sterman term.
Highlights
A fundamental way of characterizing the internal structure of the proton is through sum rules that express how global properties of the proton are composed from corresponding quark and gluon quantities
We presented the derivation of a set of relations connecting k2T-moments of generalized transverse momentum-dependent distributions (GTMDs) and twist-two as well as twist-three generalized parton distributions (GPDs), known as Lorentz invariance relations and equation of motion relations
Within our general scheme of constructing Lorentz invariance relations (LIR), we focus on ones involving the k2T-moments of the GTMDs F14 and G11, which describe the x-density distributions of the quark orbital angular momentum (OAM), Lz, and longitudinal spin-orbit interaction, LzSz
Summary
A fundamental way of characterizing the internal structure of the proton is through sum rules that express how global properties of the proton are composed from corresponding quark and gluon quantities. The three GTMDs referenced in these relations are G11, which was observed to provide information on the longitudinal part of the quark spin-orbit interaction, or the projection of quark OAM along the quark spin [8]; G12, which corresponds to a transverse proton spin configuration and generalizes the TMD g1T leading to the original Wandzura-Wilczek relation [23,24]; and, the naive T-odd part of F12 which corresponds to the off-forward generalization of the Sivers function, f⊥1T [25], which we relate to a generalized Qiu-Sterman term represented by MF12 in Eq (9).
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