Towards the NNLO evolution of polarised parton distributions

TL;DR

The paper delivers the first calculation of the spin-dependent structure function $g_1$ at three-loop order in massless QCD, using a dispersive forward-Compton amplitude approach and a $D$-dimensional treatment of $\gamma_5$. It extracts the NNLO helicity-difference splitting functions $\Delta P_{qq}^{(2)}$ and $\Delta P_{qg}^{(2)}$, confirms two-loop coefficient functions, and analyzes behavior at small and large $x$ as well as sum-rule constraints. The work extends the unpolarised three-loop program to the polarised sector, validating the methodological framework and providing crucial inputs for precise predictions of polarised DIS observables. It also identifies ongoing challenges, notably the computation of the lower-row splitting functions $\Delta P_{gf}^{(2)}$ and the completion of full $N$- and $x$-space results, planned for future publications. This advances NNLO evolution for polarised parton distributions and enhances the potential impact of future data from facilities like the Electron-Ion Collider.

Abstract

We report on the first calculation of the structure function g_1 in polarised deep-inelastic scattering to the third order in massless perturbative QCD. The calculation follows the dispersive approach already used for the corresponding unpolarised cases of F_2,L, but additionally involves higher tensor integrals and the Dirac matrix gamma_5 in D unequal 4 dimensions. Our results confirm all known two-loop expressions including the coefficient functions of Zijlstra and van Neerven not independently verified before. At three loops we extract the helicity-difference next-to-next-to-leading order (NNLO) quark-quark and gluon-quark splitting functions Delta P_qq and Delta P_qg. The results exhibit interesting features concerning sum rules and the momentum-fraction limits x to 1 and x to 0.

Figures (3)

• Figure 1: The end-2006 world data on the proton structure function $g_1^{}$ with some NLO QCD fits g1p-plot
• Figure 2: DIS in the QCD-improved parton model$\!$
• Figure 3: The LO, NLO, and NNLO approximations for the splitting function $\Delta P_{\rm qg}$ at values of $\alpha_{\rm s}$ and the number ${n^{}_{\! f}}$ of light flavours relevant to present data on the spin structure function $g_1^{}$.