The third-order QCD corrections to deep-inelastic scattering by photon exchange

TL;DR

This work delivers the full three-loop (N3LO) coefficient functions for the electromagnetic structure functions F2 and FL in massless perturbative QCD, completing NNLO predictions for FL and driving F2 toward N3LO accuracy in not-too-small x. Using the operator-product expansion in Mellin-N space, the authors derive analytic N-dependence through a sophisticated reduction of three-loop four-point integrals to master integrals and reconstruct the complete x-dependence via harmonic sums and harmonic polylogarithms. The results pass stringent checks against known small- and large-x limits and threshold-resummation predictions, and are provided in compact parametrizations suitable for fast phenomenology. Numerical studies show improved perturbative stability, enabling sub-1% truncation uncertainties in DIS scaling violations and highlighting the differing convergence behavior between F2 and FL, with practical implications for precise determinations of alpha_s. The work also supplies practical code and parametrizations to facilitate integration into global PDF fits and future higher-order resummations.

Abstract

We compute the full three-loop coefficient functions for the structure functions F_2 and F_L in massless perturbative QCD. The results for F_L complete the next-to-next-to-leading order description of unpolarized electromagnetic deep-inelastic scattering. The third-order coefficient functions for F_2 form, at not too small values of the Bjorken variable x, the dominant part of the next-to-next-to-next-to-leading order corrections, thus facilitating improved determinations of the strong coupling alpha_s from scaling violations. The three-loop corrections to F_L are larger than those for F_2. Especially for the latter quantity the expansion in powers of alpha_s is very stable, for photon virtualities Q^2 >> 1 GeV^2, over the full x-range accessible to fixed-target and collider measurements.

Figures (20)

• Figure 1: Representative three-loop diagrams of the flavour classes (from left to right) $fl_{2}$, $fl_{02}$ and $fl_{11}$ for photon--quark scattering and $fl^{\,g}_{2}$ and $fl^{\,g}_{11}$ for photon--gluon scattering.
• Figure 2: The top-level two-point topologies ${\rm LA}$ (left), ${\rm BE}$ (center) and ${\rm NO}$ (right) at three loops. The arrows indicate the assigned momentum flow. The external momentum is $q$ with $q^2 \neq 0$.
• Figure 3: As Fig. \ref{['fig:bbbtop']}, but for the second-level topologies ${\rm FA}$ (left) and ${\rm BU}$ (right).
• Figure 4: Generic ${\rm BE}_{15}$ (${\rm BU}$-type) and ${\rm BE}_{25}$ subtopologies. The momenta $q$ and $p$ (fat lines) flow from right to left and from top to bottom through the diagram, respectively.
• Figure 5: As Fig. \ref{['fig:basicbe']}, but for the generic ${\rm NO}_{16}$ and ${\rm NO}_{25}$ topologies.