Evolution Kernels from Splitting Amplitudes

TL;DR

This paper presents a gauge-invariant, amplitude-based method to derive Altarelli–Parisi evolution kernels by exploiting collinear factorization of splitting amplitudes. It re-derives the LO gluon kernel and then constructs the NLO kernel by carefully analyzing three- and four-parton final states with phase-space slicing, isolating collinear singularities, and canceling soft divergences. The resulting P^(0) and P^(1) reproduce known time-like kernels and illustrate a modular framework wherein radiative corrections to splitting amplitudes—plus their integration—generate the evolution kernels. The approach provides a transparent decomposition into physically interpretable pieces and outlines a clear path toward NNLO using higher-order splitting amplitudes.

Abstract

We recalculate the next-to-leading order Altarelli-Parisi kernel using a method which relates it to the splitting amplitudes describing the collinear factorization properties of scattering amplitudes. The method breaks up the calculation of the kernel into individual pieces which have an independent physical interpretation.

Figures (2)

• Figure 1: A schematic depiction of the collinear factorization of tree-level amplitudes, with the amplitudes labeled clockwise.
• Figure 2: A schematic depiction of the collinear factorization of one-loop amplitudes.