The connection between Nucleon Energy Correlators and Fracture Functions

We establish a sum rule that connects fracture functions to nucleon energy-energy correlators~(NEECs) in a one-to-one correspondence. Using this sum rule, we study the energy pattern in the target fragmentation region of deep inelastic scatterings. Through investigations up to twist-3, we express all eighteen energy-pattern structure functions in terms of associated NEECs, elucidating various azimuthal and spin asymmetries critical for nucleon tomography. Additionally, we investigate the perturbative matching of the twist-2 quark NEECs. We demonstrate that the Sivers-type and worm-gear-type quark NEECs match onto twist-3 multi-parton distributions. Our work provides a framework for examining energy-weighted observables through hadron production processes in the target fragmentation region, offering new insights into nucleon tomography.