NNLO evolution of deep-inelastic structure functions: the non-singlet case

TL;DR

This paper develops NNLO (three-loop) insights for the non-singlet sector of deep-inelastic scattering, deriving compact $x$-space parametrizations for the two-loop coefficient functions and approximate reconstructions of the three-loop splitting functions. By combining these with the general formalism for evolution and scale dependence, it assesses the size of NNLO corrections, the stability under renormalization-scale variations, and the impact on precise $\alpha_s$ determinations. The results show NNLO effects are modest (~2%) for evolution kernels and significantly reduce theoretical uncertainties, especially in $\alpha_s$, while highlighting residual uncertainties in $P_{ m NS}^{(2)\pm}$ at very small $x$. The work provides practical tools (parametrizations and Mellin-space forms) enabling accurate NNLO analyses of non-singlet DIS data and sets the stage for analogous treatment of the singlet sector.

Abstract

We study the next-to-next-to-leading order (NNLO) evolution of flavour non-singlet quark densities and structure functions in massless perturbative QCD. Present information on the corresponding three-loop splitting functions is used to derive parametrizations of these quantities, including Bjorken-x dependent estimates of their residual uncertainties. Compact expressions are also provided for the exactly known, but rather involved two-loop coefficient functions. The size of the NNLO corrections and their effect on the stability under variations of the renormalization scale are investigated. The residual uncertainty of the three-loop splitting functions does not lead to appreciable effects for x > 10^-2. Inclusion of the NNLO contributions reduces the main theoretical uncertainty of alpha_s determinations from non-singlet scaling violations by more than a factor of two.

Figures (13)

• Figure 1: Approximations for the $N_f$-independent part of $P^{(1)+} _{\rm NS}$, derived from the lowest even-integer moments by means of Eqs. (\ref{['ansatz']}) and (\ref{['pnlo']}), compared to the exact result.
• Figure 2: Approximations for the $N_f$-independent part of $P^{(2)+} _{\rm NS}$, denoted by $P_0^{(2)+}$ in Eq. (\ref{['nf_dep']}), as derived from the five lowest even-integer moments by means of Eqs. (\ref{['ansatz']}), (\ref{['pnnlo0']}) and (\ref{['mo1']}). The full lines represent those functions selected for further consideration.
• Figure 3: The convolution of the approximations 'A' -- 'D' of $P_0^{(2)+}$ selected in Fig. 2 with a shape typical of hadronic non-singlet initial distributions.
• Figure 4: Left: approximations to the $N_f^1$ part of $P^{(2)+} _{\rm NS}$, obtained from the five lowest even-integer moments using Eqs. (\ref{['ansatz']}) and (\ref{['pnlo']}). Right: approximate results for the $N_f^0$ and $N_f^1$ terms of the 3-loop splitting function $P^{(2)-}_{\rm NS}$.
• Figure 5: Left: The perturbative expansion of the QCD $\beta$-function up to order $\alpha_s^5$, for four flavours in the $\overline {\hbox{MS}}$ renormalization scheme. Right: Illustration of the resulting scale dependence of $\alpha_s$, using a variable $N_f$ as detailed in the text. $\mu_r^2$ is given in GeV$^2$.