Abstract
We compute in conventional dimensional regularisation the tree-level splitting amplitudes for a gluon parent which splits into four collinear partons. This is part of the universal infrared behaviour of the QCD scattering amplitudes at next-to-next-to-next-to-leading order (N3LO) in the strong coupling constant. Combined with our earlier results for a quark parent, this completes the set of tree-level splitting amplitudes required at this order. We also study iterated collinear limits where a subset of the four collinear partons become themselves collinear.
Highlights
The two-loop currents for the emission of two collinear or one soft parton were obtained in refs. [61,62,63,64,65,66], and the one-loop current for the emission of three collinear partons is given in refs. [67, 68]
Combined with our previous results for a quark parent [73], this completes the set of tree-level splitting amplitudes describing all collinear singularities for the emission of up to four collinear partons at N3LO
We have considered the strongly-ordered limit when a subset of the four collinear partons become collinear to each other, and we have derived the corresponding factorisation formulæ
Summary
We examine the behaviour of tree-level QCD amplitudes in the limit where a given number of massless partons become collinear. Pm become simultaneously collinear to some light-like direction P In this limit, the leading behaviour is described by the amplitude for the production of a massless particle of momentum P from a scattering of the particles that do not take part in the collinear limit, multiplied by a universal factor, termed the splitting amplitude, which depends only on the m partons in the collinear set. The dependence of the splitting amplitude on the transverse momenta k⊥i and momentum fractions zi of the particles in the collinear set is understood. Splitting amplitudes for the squared matrix element have been computed at tree level for the emission of up to three collinear partons in refs. The goal of this paper is to compute the tree-level splitting amplitudes for the squared matrix element for the emission of up to four partons out of a parent gluon, completing the set of splitting amplitudes for the emission of up to four collinear partons in QCD.
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