Color Lorentz force on a transversely polarized quark in the nucleon

Abstract

The $x^2$ moment of the genuine twist-3 component of the unpolarized scalar parton distribution function $e(x)$ is related to the average transverse color Lorentz force experienced by a transversely polarized quark in an unpolarized nucleon. This higher-twist quantity probes quark--gluon correlations and provides unique insight into the nonperturbative dynamics of quantum chromodynamics. For the first time, we study the corresponding off-forward matrix element and derive its Lorentz-covariant decomposition in terms of generalized form factors. We calculate these form factors from light-front wave functions obtained by solving the light-front Hamiltonian eigenvalue problem within the Basis Light-Front Quantization framework. Their Fourier transforms provide a spatial picture of the transverse color forces in impact-parameter space. In particular, $\mathscr{F}_2$, whose forward limit is directly related to the twist-3 matrix element $e_2$, can be interpreted as the impact-parameter-space representation of the Boer--Mulders force, providing an intuitive spatial picture of the final-state interaction mechanism underlying the Boer--Mulders effect.