Fragmentation Functions at Next-to-Next-to-Leading Order Accuracy

TL;DR

The paper advances the determination of parton-to-pion fragmentation functions by performing a first NNLO analysis of SIA data, using Mellin-space techniques to implement time-like evolution and cross-section calculations. It extends the Pegasus framework to handle NNLO time-like evolution, validates the NNLO coefficient functions and kernels, and fits FFs with a DSS-inspired parameterization. The results show improved agreement with data and a substantial reduction in scale uncertainties at NNLO, particularly at small z, while highlighting residual effects from large-logarithm regions that motivate resummation. The work lays the groundwork for more precise FF determinations and motivates incorporating additional processes (SIDIS, pp) and all-order resummations in future global analyses.

Abstract

We present a first analysis of parton-to-pion fragmentation functions at next-to-next-to-leading order accuracy in QCD based on single-inclusive pion production in electron-positron annihilation. Special emphasis is put on the technical details necessary to perform the QCD scale evolution and cross section calculation in Mellin moment space. We demonstrate how the description of the data and the theoretical uncertainties are improved when next-to-next-to-leading order QCD corrections are included.

Figures (7)

• Figure 1: The dashed line represents the contour $\mathcal{C}_N$ in complex $N$-space to perform the inverse Mellin transformation (\ref{['eq:inverse1']}). The poles of the integrand along the real axis are schematically represented by the crosses.
• Figure 2: The value of the real part of $\mathcal{K}_{12}$ in Eq. (\ref{['eq:mtgeneralsolutionschema']}) in a region of the complex $N$ plane for both the evolution of FFs (upper panel) and PDFs (lower panel). The lines correspond to three different integration contours ${\cal C}_N$ in (\ref{['eq:inverse1']}). $\mathcal{C}_1$ is the default choice in the Pegasus package ref:pegasus; see text.
• Figure 3: Ratios for [data-theory]/theory for our LO (dot-dashed), NLO (dashed), and NNLO (solid lines) fits computed with the scale $\mu=Q$ for the data sets listed in Tab. \ref{['tab:exppiontab']}. The shaded bands illustrate the remaining scale ambiguity at NNLO accuracy in the range $Q/2\le\mu \le2Q$. The points along the zero axis indicate the relative experimental uncertainty.
• Figure 4: Comparison of our LO, NLO, and NNLO FFs $D_{i}^{\pi^{+}}(z,Q^2)$ at $Q^2=10\,\mathrm{GeV}^2$ for $i = u+\bar{u}$, $s+\bar{s}$, $g$, and the flavor singlet combination in (\ref{['eq:singlet']}) for $N_f=4$. Also shown are the optimum NLO FFs from Kretzer ref:kretzer, obtained also solely from SIA data, and the latest global analysis of the DSS group ref:dssnew based on SIA, SIDIS, and $pp$ data. For the latter, we also illustrate their $90\%$ C.L. uncertainty estimates (shaded bands).
• Figure 5: NNLO/NLO (solid) and NLO/LO (dashed lines) $K$-factors for the SIA process for three different c.m.s. energies. All computations are performed with our NLO set of parton-to-pion FFs; see text.