Higher Orders in A(α_s)/[1-x]_+ of Non-Singlet Partonic Splitting Functions

TL;DR

This work develops a streamlined framework to compute higher-order contributions to the singular A(α_s)/[1−x]_{+} term of non-singlet parton splitting functions by exploiting factorization and nonabelian eikonal exponentiation. By expressing the relevant PDFs near x→1 in terms of a color-singlet eikonal cross section and applying Ward identities, the authors isolate the A-coefficients via renormalization of eikonal vertices using LCOPT. The method reproduces the known A^{(1)} and A^{(2)} results and yields the N_f^{n−1} contributions at higher loops; notably, the complete A^{(3)}_f term’s N_f contribution matches independent numerical estimates, validating the approach and suggesting feasibility for A^{(4)} with further work. The technique reduces diagrammatic complexity and provides a complementary route to OPE-based three-loop results, with potential use in NNLL threshold resummations.

Abstract

We develop a simplified method for obtaining higher orders in the perturbative expansion of the singular term A(α_s)/[1-x]_+ of non-singlet partonic splitting functions. Our method is based on the calculation of eikonal diagrams. The key point is the observation that the corresponding cross sections exponentiate in the case of two eikonal lines, and that the exponent is directly related to the functions A(α_s) due to the factorization properties of parton distribution functions. As examples, we rederive the one- and two-loop coefficients A^(1) and A^(2). We go on to derive the known general formula for the contribution to A^(n) proportional to N_f^{n-1}, where N_f denotes the number of flavors. Finally, we determine the previously uncalculated term proportional to N_f of the three-loop coefficient A^(3) to illustrate the method. Our answer agrees with the existing numerical estimate. The exact knowledge of the coefficients A^(n) is important for the resummations of large logarithmic corrections due to soft radiation. Although only the singular part of the splitting functions is calculable within our method, higher-order computations are much less complex than within conventional methods, and even the calculation of A^(4) may be possible.

Figures (15)

• Figure 1: Feynman rules for eikonal lines in the fundamental representation with velocities $\beta^\mu$, represented by the double lines. The vertical line represents the cut separating the amplitude and its complex conjugate. For an eikonal line in the adjoint representation one has to replace $T^a_{ij}$ with $i f_{ija}$.
• Figure 2: a) Graphical representation of a parton-in-parton distribution function, Eq. (\ref{['pdfdef']}), and b) its factorized form for arbitrary momentum fraction $x$, drawn as a reduced diagram. c) The reduced diagram for the PDF in the limit $x \rightarrow 1$. Here we have suppressed the labels $L$ and $R$ for purely virtual contributions to the left and to the right of the cut, respectively, compared to the notation in the text. The cut represents the final state.
• Figure 3: a) Ward identity for a scalar polarized gluon. b) Identity for a single longitudinally polarized gluon attaching to an eikonal line. c) Resulting identity after iterative application of Figs. a) and b). Repeated gauge-group indices are summed over.
• Figure 4: Parton distribution function for $x \rightarrow 1$ with the jet functions factorized from the hard part, graphical representation of Eq. (\ref{['factform1']}).
• Figure 5: Parton distribution function for $x \rightarrow 1$, factorized into hard scatterings, an eikonal cross section, and purely virtual jet-remainders, as derived in Eq. (\ref{['finalform']}). The virtual jet functions are normalized by their eikonal analogs, as in Eq. (\ref{['jetvirdef']}).