Single-valued representation of unpolarized and polarized semi-inclusive deep inelastic scattering at next-to-next-to-leading order

TL;DR

This work tackles NNLO SIDIS coefficient functions which previously required region-based case distinctions in the (x,z) plane. The authors rewrite the NNLO terms in terms of single-valued polylogarithms to obtain a unified, case-free representation across all kinematics, achieving a 30–60% reduction in analytic length. This approach clarifies the analytic structure by canceling spurious branch cuts associated with the one-loop box integral and paves the way for analytic Mellin transforms of the cross section. They also provide a numerical library BEAVER and ancillary files, enabling faster evaluations essential for NNLO PDF fits and future multi-loop generalizations.

Abstract

We revisit the recently published analytic results for unpolarized and polarized semi-inclusive deep inelastic scattering (SIDIS) at next-to-next-to-leading order (NNLO) in QCD. These expressions for the hard scattering coefficients contain case distinctions in the kinematic $(x,z)$ plane splitting the analytic result in four regions. By re-expressing the coefficient functions in terms of single-valued polylogarithms we remove these case distinctions and can present a unified result valid in the entire kinematic range of SIDIS. This reduces the length of the overall expressions by 30% to 60%.

Figures (5)

• Figure 1: Comparison of expression sizes before (full bars) and after (horizontal red lines) removing case distinctions. $U_j$ indicates all parts of the expressions from Bonino:2024unpolBonino:2024pol involving case distinctions, $\boxtimes$ the rest (see eq. \ref{['eq: SIDIS regions']}). Expressions are consistently sorted, LeafCount (number of indivisible subexpressions in Mathematica) serves as proxy to measure expression length.
• Figure 2: Sketch of the SIDIS process $e(k)\,p(P)\rightarrow e(k^\prime)\,h(P_h)+X$ in the parton model. Here, $e$ denotes the scattered electron, $\gamma^\ast$ the intermediate virtual photon, $p$ the incoming proton, $h$ the identified hadron, and $X$ the proton remnant (including all unidentified particles in the final state). In SIDIS the electron momentum and the identified hadron's longitudinal momentum are measured. Radiative corrections are not included in the picture.
• Figure 3: The four regions of SIDIS for which there are case distinctions in the NNLO coefficient functions as presented in Goyal:2023unpolBonino:2024unpolBonino:2024polGoyal:2024polGoyal:2024emo. The red and blue boundaries depict the kinematic endpoints for $x$ and $z$ where physical thresholds are present.
• Figure 4: Illustrative one-loop box diagram contributing to the real-virtual correction at NNLO responsible for the spurious branch cuts.
• Figure 5: Branch cut structure of the one loop box integral. Each color coded region boundary corresponds to a physical branch cut. $x_{1,2}$ are ratios of Mandelstam variables defined in Haug:2022. The dashed gray lines indicate the spurious branch cuts at $x_i=1$ induced by the hypergeometric functions in the representation of eq. \ref{['eq: Scalar box integral']} running right through the DIS regions.