Two-Dimensional Transverse-Momentum Subtraction and Semi-Inclusive Deep-Inelastic Scattering at N$^3$LO in QCD

Abstract

Identified hadron production is essential for the study of nucleon structure and QCD hadronization at high energies. We present the first calculation of unpolarized semi-inclusive deep-inelastic scattering (SIDIS) at next-to-next-to-next-to-leading order (N$^3$LO) in perturbative QCD. Our calculation is based on a novel method of two-dimensional transverse-momentum subtraction motivated by QCD factorization of soft and collinear singularities. The N$^3$LO corrections are moderate in general but can be significant in threshold regions, and exhibit excellent perturbative convergence and reduced scale variations. The fully differential framework allows for arbitrary selection cuts and directly enables precision nucleon tomography at the upcoming Electron-Ion Collider, establishing the theory foundation needed to match the anticipated experimental accuracy. Generalization of the method to calculations of polarized SIDIS is also feasible.

Figures (5)

• Figure 1: Left: schematic definition of the kinematic variables used for the two-dimensional $q_T$-subtraction method in the Breit frame. The incoming virtual photon $\gamma^*$ scatters off the proton, producing an identified hadron and a recoiling leading jet ${\cal J}_k$. The blue lines along the proton direction denote beam collinear radiation. The green curved lines indicate the soft radiations. The slicing variables are chosen as the beam polar angle $\delta \theta$ and the azimuthal decorrelation $\delta\phi$. Right: partition of the phase space into different regions according to $\delta \theta > \Delta$ and $\delta \phi > \lambda \Delta$, with the two slicing parameters $\Delta,\,\lambda \ll 1$.
• Figure 2: Numerical stability of corrections to the total cross section with respect to the slicing parameter $\Delta$. The panels display the $\mathcal{O}(\alpha_s^2)$ (red) and $\mathcal{O}(\alpha_s^3)$ (blue) contributions for six representative partonic channels. Error bars indicate Monte Carlo statistical uncertainties. The solid cyan lines represent a fit to the asymptotic limit $\Delta \to 0$. The dashed purple lines denote the reference values of $\mathcal{O}(\alpha_s^2)$.
• Figure 3: Differential cross section (multiplied by bin center) as functions of the hadron momentum fraction $z$ for charged pion production at the EIC at various orders in QCD, including NLO, NNLO and N$^3$LO. In the lower panel all predictions including scale variations are normalized to the central NNLO results. The diamond-shaped scatters represent approximate N$^3$LO predictions.
• Figure 4: Differential cross section (multiplied by bin center) as functions of the Bjorken variable $x$ for charged pion production at the EIC at various orders in QCD, including NLO, NNLO and N$^3$LO. In the lower panel all predictions including scale variations are normalized to the central NNLO results. The diamond-shaped scatters represent approximate N$^3$LO predictions.
• Figure 5: Differential multiplicity as functions of the hadron momentum fraction $z$ for charged pion production at COMPASS experiment (proton target with $\sqrt s=17.3\,{\rm GeV}$) at various orders in QCD, including NLO, NNLO and N$^3$LO. In the lower panel all predictions including scale variations are normalized to the central NNLO results. The error bars represent the COMPASS $\pi^+$ data with full uncertainties.