Researchers report the first next-to-next-to-next-to-leading order (N3LO) calculation for unpolarized semi-inclusive deep-inelastic scattering (SIDIS). SIDIS is a process in which an electron scatters from a proton and a specific hadron (a bound state of quarks, like a pion) is observed in the final state. The new work introduces a two-dimensional transverse-momentum subtraction method that makes this high-precision prediction possible.

To obtain the result the team developed a subtraction scheme designed to handle the tricky parts of quantum chromodynamics (QCD) calculations. In QCD, contributions from very soft radiation (low-energy particles) and collinear radiation (particles emitted nearly along the same direction) create mathematical singularities, or infinities, that must be cancelled before one can get finite predictions. The authors used insights from factorization and soft-collinear effective theory (SCET), an approximation method that separates effects at different energy scales, to guide a subtraction that removes these singularities in two transverse-momentum directions. They applied the method to the SIDIS process e− + p → e− + h + X, where h is the identified hadron, and produced a fully differential framework that can include arbitrary selection cuts.

In plain terms, “subtraction” here means isolating the infinite pieces that occur in intermediate steps and subtracting them analytically so the remaining numerical parts are well behaved. The two-dimensional transverse-momentum subtraction is a generalization of existing qT-subtraction methods (qT stands for transverse momentum) that were used for simpler, more inclusive processes. By working differentially, the calculation can describe detailed final-state configurations rather than only overall rates.

The authors report that the N3LO corrections are generally moderate, but they can become significant in threshold regions where the observed hadron carries a large fraction of the available energy. The higher-order result shows good perturbative convergence and smaller dependence on the arbitrary theory scales that are usually varied to estimate uncertainty. These features matter because future measurements, notably at the planned Electron‑Ion Collider (EIC), aim for high precision. A reliable N3LO theory prediction helps extract information about how quarks and gluons form hadrons and supports precise “nucleon tomography,” the mapping of how partons are distributed inside protons.