Two-Loop g -> gg Splitting Amplitudes in QCD

TL;DR

This work computes the two-loop g→gg splitting amplitudes in QCD, N=1, and N=4 super-Yang-Mills theories to characterize universal collinear behavior of massless gauge-theory amplitudes and to inform NNLO evolution kernels.A unitarity-based sewing method is employed to construct two-loop integrands, which are reduced to 13 master integrals via IBP/Lorentz identities and then solved with differential equations in the splitting momentum fraction z.The resulting poles match Catani's infrared formula in color space, providing an inductive proof-like consistency and enabling the extraction of finite-remainder relations for the Catani subtraction scheme; in N=4 the two-loop amplitude exhibits a remarkable iterative structure.These results yield detailed, theory-dependent expressions for the splitting amplitudes, clarify the role of color factors, and provide essential building blocks for NNLO parton evolution and for cross-checking higher-point two-loop amplitudes.

Abstract

Splitting amplitudes are universal functions governing the collinear behavior of scattering amplitudes for massless particles. We compute the two-loop g -> gg splitting amplitudes in QCD, N=1, and N=4 super-Yang-Mills theories, which describe the limits of two-loop n-point amplitudes where two gluon momenta become parallel. They also represent an ingredient in a direct x-space computation of DGLAP evolution kernels at next-to-next-to-leading order. To obtain the splitting amplitudes, we use the unitarity sewing method. In contrast to the usual light-cone gauge treatment, our calculation does not rely on the principal-value or Mandelstam-Leibbrandt prescriptions, even though the loop integrals contain some of the denominators typically encountered in light-cone gauge. We reduce the integrals to a set of 13 master integrals using integration-by-parts and Lorentz invariance identities. The master integrals are computed with the aid of differential equations in the splitting momentum fraction z. The epsilon-poles of the splitting amplitudes are consistent with a formula due to Catani for the infrared singularities of two-loop scattering amplitudes. This consistency essentially provides an inductive proof of Catani's formula, as well as an ansatz for previously-unknown 1/epsilon pole terms having non-trivial color structure. Finite terms in the splitting amplitudes determine the collinear behavior of finite remainders in this formula.