Non-Singlet Structure Functions at Three Loops: Fermionic Contributions

TL;DR

This work computes the fermionic ($n_f$) three-loop contributions to the non-singlet structure functions in unpolarized electromagnetic DIS, providing the $n_f$ parts of the three-loop anomalous dimensions and coefficient functions for $F_2$ and $F_L$ in the MS-bar scheme. The authors advance a Mellin-space approach to obtain all even Mellin moments $N$ via recursion/difference equations, express results with harmonic sums, and supply compact $x$-space parametrizations for practical use. The nf results enable the full NNLO evolution of non-singlet distributions and complete threshold resummation at NNLL accuracy, including the crucial finding that the DIS-specific $D_2^{DIS}$ vanishes in the MS-bar scheme. Overall, the paper confirms consistency with known moments and soft-gluon predictions while delivering the missing nf-3-loop ingredients necessary for high-precision DIS phenomenology and related hard-scattering processes.

Abstract

We compute the fermionic (n_f) contributions to the flavour non-singlet structure functions in unpolarized electromagnetic deep-inelastic scattering at third order of massless perturbative QCD. Complete results are presented for the corresponding nf-parts of the three-loop anomalous dimension and the three-loop coefficient functions for the structure functions F_2 and F_L. Our results agree with all partial and approximate results available in the literature. The present calculation also facilitates a complete determination of the threshold-resummation parameters B_2 and D_2^DIS of which only the sum was known so far, thus completing the information required for the next-to-next-to-leading logarithmic resummation. We find that D_2^DIS vanishes in the MSbar scheme.

Figures (4)

• Figure 1: The topologies $\rm BE$ (left) and O4 (right) of propagator-type diagrams, with the line numbering as employed in figs. \ref{['pic:benz']} and \ref{['pic:o4']} below. The external lines carry the momentum $Q$.
• Figure 2: The diagrams of topology $\rm BE$ which contribute to the fermionic part of the non-singlet structure functions $F_2$ and $F_L$. The subtopologies are ${\rm BE}_{1\rm Q}$ (left) and ${\rm BE}_{13}$ (right).
• Figure 3: The diagrams of topology O4, which contribute to the fermionic part of the non-singlet structure functions $F_2$ and $F_L$. These diagrams are examples of the subtopologies O4$_{\rm 1Q}$ and O4$_{18}$.
• Figure 4: The ${n^{}_{\! f}}^{\!\!1}$ and ${n^{}_{\! f}}^{\!\!2}$ parts $P^{(2)}_{+,1}(x)$ and $P^{(2)}_{+,2}(x)$ of the three-loop non-singlet splitting function (\ref{['eq:Pqq2']}), multiplied by $(1\!-\!x)$ for display purposes. Also shown in the left part (dashed curve) is the uncertainty band derived in ref. vanNeerven:2000wp from the lowest six even-integer moments Larin:1994vuLarin:1997wdRetey:2000nq.