Two-Loop Spacelike Splitting Amplitudes in Full-Color QCD

The study of QCD scattering amplitudes in the collinear regime provides crucial insight into the factorization properties of hadronic cross sections. In this paper, we present the first complete results for two-loop spacelike splitting amplitudes in full-color QCD, in all partonic channels and helicity configurations. We confirm the universality of a class of contributions already found in N=4 super Yang–Mills (sYM) theory, and identify previously unknown sources of collinear factorization-violating (CFV) effects. Consistent with recent observations in N=4 sYM, all CFV contributions cancel in color-summed squared amplitudes, implying the universality of single-parton collinear factorization for jet cross sections at third order in QCD.

💡 Research Summary

The paper presents the first complete calculation of two‑loop spacelike splitting amplitudes in full‑color Quantum Chromodynamics (QCD), covering every partonic channel and helicity configuration. The authors begin by reviewing the role of collinear factorization in QCD cross‑section predictions and the potential breakdown caused by Glauber gluon exchanges, which generate color‑coherence‑violating (CFV) phases suppressed by 1/Nc. They emphasize that such CFV effects could, in principle, jeopardize the universality of parton distribution functions (PDFs) at next‑to‑next‑to‑next‑to‑leading order (N³LO).

To obtain the two‑loop spacelike splitting amplitudes, the authors exploit the recently computed full‑color five‑point two‑loop amplitudes. They analyze the collinear limit p₂‖p₃ by scaling the invariant s₂₃∼δ², where δ→0. Two complementary techniques are employed: (i) an analytic‑continuation method using twistor variables and a contour encircling the branch point τ=0, which captures the discontinuity between spacelike and timelike regimes; and (ii) a differential‑equation approach that expands the pentagon functions in powers of δ, solves one‑dimensional DEs for the δ‑dependence, and reconstructs the full functional form using generalized polylogarithms (GPLs) up to weight four. Both methods agree, providing a robust cross‑check.

The resulting splitting amplitude is expressed in an exponential form:
Sp = √(4π αs) Split · exp