Non-singlet structure functions beyond the next-to-next-to leading order

TL;DR

The authors extend non-singlet deep-inelastic scattering analyses to N^3LO, constructing approximate three-loop coefficient functions c_{a,3} from existing moments and soft-gluon constraints, and deriving x-space evolution kernels with parametrized convolutions up to N^3LO. They demonstrate that N^3LO corrections are small for x > 0.01 and substantially reduce the renormalization-scale uncertainty in alpha_s, enabling more precise determinations from DIS data. The work also evaluates soft-gluon resummation and alternative extrapolation methods (PMS, ECH, Padé) to estimate higher-order effects, finding these approaches provide reliable guidance for N^4LO and beyond, though current uncertainties in B_2 and D_2 limit resummed predictions at very large x. Overall, the paper delivers practical, accurate tools for NS evolution and highlights remaining theoretical gaps, notably four-loop splitting functions, that constrain ultimate precision in alpha_s extraction.

Abstract

We study the evolution of the flavour non-singlet deep-inelastic structure functions F_{2,NS} and F_3 at the next-to-next-to-next-to-leading order (N^3LO) of massless perturbative QCD. The present information on the corresponding three-loop coefficient functions is used to derive approximate expressions of these quantities which prove completely sufficient for values x > 10^{-2} of the Bjorken variable. The inclusion of the N^3LO corrections reduces the theoretical uncertainty of alpha_s determinations from non-singlet scaling violations arising from the truncation of the perturbation series to less than 1%. We also study the predictions of the soft-gluon resummation, of renormalization-scheme optimizations by the principle of minimal sensitivity (PMS) and the effective charge (ECH) method, and of the Pade' summation for the structure-function evolution kernels. The PMS, ECH and Pade' approaches are found to facilitate a reliable estimate of the corrections beyond N^3LO.

Figures (12)

• Figure 1: Approximations for the two-loop coefficient functions $c_{2,2} ^{}(x,N_f\! =\! 4)$ for $F_{2,\rm NS^{\,\!}}^{\,\rm e.m._{\,\!}}$ obtained from the lowest seven even-integer moments and the two leading soft-gluon terms, compared to the exact result of ref. ZvN. Also shown is the corresponding exact coefficient function for $F_{2,\rm NS}$ in charged-current DIS.
• Figure 2: Approximations for the three-loop coefficient functions for $F_{2,\rm NS^{\,\!}}^{\,\rm e.m._{\,\!}}$ (left) and the charged- current $F_{3^{\,\!}}^{\,\nu+\bar{\nu}_{\,\!}}$ (right) derived from the respective seven lowest moments momsmnew and the soft-gluon terms (\ref{['eq35']}). The full lines show the selected functions (\ref{['eq36']}) and (\ref{['eq37']}).
• Figure 3: The convolution of the approximations (\ref{['eq36']}) and (\ref{['eq37']}) selected from the previous figure with a shape typical of hadronic non-singlet distributions.
• Figure 4: The perturbative expansion of the scale derivative $\dot{F}_{2,\rm NS} \equiv d\ln F_{2,\rm NS\,} / d\ln Q^2$ of the electromagnetic structure function $F_{2,\rm NS}$ at $\mu_r^2 = Q^2 \simeq 30 \hbox{GeV}^2$ for the initial conditions specified in Eqs. (\ref{['eq51']}) and (\ref{['eq52']}). The differences between the predictions at different orders in $\alpha_s$ are shown on a larger scale in the right part.
• Figure 5: As Fig. 4, but for the charged-current combination $F_3 \equiv F_3^{\,\nu + \bar{\nu}}$. In all figures the subscripts $A$ and $B$ at N$^3$LO refer to the approximations discussed below Eq. (\ref{['eq53']}).