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We study the semi-classical interpretation of the $x^3$ and $x^4$ moments oftwist-3 parton distribution functions (PDFs). While no semi-classicalinterpretation for the higher moments of $g_T(x)$ and $e(x)$ was find, the$x^3$ moment of the chirally odd spin-dependent twist-3 PDF $h_L^3(x)$ can berelated to the longitudinal gradient of the transverse force on transverselypolarized quarks in longitudinally polarized nucleons in a DIS experiment. Wediscuss how this result relates to the torque acting on a quark in the sameexperiment. This has further implications for comparisons between thaJaffe-Manohar and the Ji decompositions of the nucleon spin.

Transverse Force on Transversely Polarized Quarks in Longitudinally Polarized Nucleons