Lorentz-invariance relations connecting twist-3 parton distributions with transverse momentum dependent twist-2 distributions have been proposed previously. Naively, these relations can be extracted from a covariant decomposition of the quark-quark correlator. It is argued that the derivation of the Lorentz-invariance relations fails if the path-ordered exponential, which ensures gauge invariance, is taken into account in the correlator. Our model-independent analysis is supplemented by an explicit calculation of the corresponding parton distributions in perturbative QCD with a quark target, and in a simple model of Brodsky-Hwang-Schmidt.

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