Insights into the higher-twist distribution e(x) at CLAS

This preprint has been superseded by (A. Courtoy, A. S. Miramontes, H. Avakian, M. Mirazita, and S. Pisano, Phys. Rev. D 106, 014027 (2022), 2203.14975). The latter should be consulted and referred to for the extraction of the $e(x)$ PDF. We present the extraction of the twist-3 PDF, $e(x)$, through the analysis of the preliminary data for the $\sin\phi$-moment of the beam-spin asymmetry for di-hadron Semi-Inclusive DIS at CLAS at 6 GeV. Pion-pair production off unpolarized target in the DIS regime provide an access to the higher-twist Parton Distribution Functions $e(x)$ and to Di-hadron Fragmentation Functions. The latter have been extracted from the semi-inclusive production of two hadron pairs in back-to-back jets in $e^+e^-$ annihilation at Belle. The $e(x)$ PDF offers important insights into the physics of the largely-unexplored quark-gluon correlations, and its $x$-integral is related to the marginally-known scalar-charge of the nucleon, and to the pion-nucleon $\sigma$-term, a fundamental property of the nucleon.