Lorentz invariance and equation of motion relations for twist-three GTMDs

A number of sum rules between twist-two Generalized Transverse Momentum Distributions (GTMDs) and twist-three Generalized Parton Distributions (GPDs) have been proven through the method of QCD equations of motion relations (EoMs) and Lorentz invariance relations (LIRs). Among these are the sum rules that relate the quark transverse momentum squared moment of the GTMDs F₁₄, which measures longitudinal quark orbital angular momentum, and F₁₂, which measures transverse quark orbital angular momentum, to combinations of twist-two and twist-three GPDs that can be, in principle, probed in experiment. Motivated by the success of such results on the lattice, we derive new EoMs and LIRs to relate twist-three GTMDs and twist-four GPDs. We use such sum rules to inform the phenomenology of the longitudinal structure function F_L of deep inelastic scattering, thus providing a direct connection between average quark transverse momentum and QCD correlation functions.