Analytic continuation of the Mellin moments of deep inelastic structure functions

TL;DR

The paper develops a systematic method to analytically continue the Mellin moments of deep inelastic structure functions to complex and real n at NNLO. Building on KaKoKo94, it extends the continuation to complex nested sums that appear in NNLO anomalous dimensions and coefficient functions, preserving a form close to the original MVV representations. The authors provide explicit formulas and prescriptions for both nonalternating and alternating sums, including detailed treatment of even/odd n and the associated Ψ and Zeta constructs, and validate the approach with nonsinglet anomalous dimensions and known NNLO results. This framework enables robust n-space representations for DIS, facilitating precise fits, sum-rule evaluations, and improved small-x evolution analyses.

Abstract

We derive the analytic continuation of the Mellin moments of deep inelastic structure functions at the next-to-next-to-leading order accuracy.

Figures (6)

• Figure 1: The diagrams contributing to $T_{\mu\nu}$ for a gluon target. They should be multiplied by a factor of 2 because of the opposite direction of the fermion loop. The diagram (a) should be also doubled because of crossing symmetry.
• Figure 2: The circles are represented the sums $S_{-2}(n)$ and $S_{-3}(n)$. The triangles show results for $\overline S_{-2}^+(n)$ and $\overline S_{-3}^+(n)$.
• Figure 3: As in Fig. \ref{['s-a']} but for the sums $S_{-3, -1, 1, 1}(n)$ and $S_{2, -1, 2, 1}(n)$.
• Figure 4: The circles are represented the sum $S_{2,-1,-1,3}(n)$ and the difference $\overline S_{2,-1,-1,3}^+(n)-S_{2,-1,-1,3}(n)$ (in the parts (a) and (b), respectively).
• Figure 5: As in Fig. \ref{['s-a']} but for the sum $S_{-2, 1, -2, 1, 1}(n)$ and $S_{-3, -1, -1, 2}(n)$.