This paper presents the first calculation of single inclusive hadron and jet production at large transverse momentum in lepton–hadron collisions using a joint Quantum Chromodynamics (QCD) and Quantum Electrodynamics (QED) factorization approach. In plain terms, the authors study collisions where a lepton (for example an electron) hits a proton or nucleus and a single high‑momentum hadron or a jet is observed. They treat the strong and electromagnetic radiation produced in the collision together, rather than handling QED effects as a separate correction.

The authors factorize the scattering cross section into short‑distance “hard” parts that can be calculated with perturbation theory, convolved with three kinds of universal, non‑perturbative functions: lepton distribution functions (LDFs), parton distribution functions (PDFs) of the hadron, and fragmentation functions (FFs) that describe how a parton turns into the observed hadron. They show that the standard evolution equations that describe how these distributions change with the scale — known as DGLAP‑type evolution equations (Dokshitzer‑Gribov‑Lipatov‑Altarelli‑Parisi) — must include both QCD and QED splitting kernels in this joint framework. For practical predictions they derive a default set of LDFs at an input scale equal to the charm quark mass and evolve them to higher scales. For hadron PDFs and FFs they use existing sets such as JAM20 and MAP1.0, and for jets they use jet functions instead of FFs.

Why this matters: when a large momentum is exchanged between the lepton and the hadron, both QCD and QED radiation can change the observed particle distributions and the effective momentum of the probing particle. Treating QED radiation on the same footing as QCD radiation avoids a class of singular behaviors that can appear when the exchanged photon becomes nearly real (the “pinch singularity”). The joint factorization makes the short‑distance parts infrared safe (insensitive to low‑energy radiation) and defines universal LDFs that can in principle be extracted from data. The authors present calculations aimed at the energies of Jefferson Lab and the future Electron‑Ion Collider (EIC), and they discuss how single high‑pT hadrons or jets can probe nuclear modifications of PDFs, such as the EMC effect at high momentum fraction.