Multi-Gluon Collinear Limits from MHV diagrams

TL;DR

The paper addresses the infrared structure of tree-level multi-gluon amplitudes in QCD when many gluons become collinear, proposing a method based on MHV rules to derive timelike splitting functions for arbitrary numbers of positive or negative helicity gluons.By exploiting the CSW MHV vertices connected by scalar propagators and classifying collinear limits by ΔM, the authors obtain compact, helicity-dependent splitting functions and show that only MHV-diagrams with on-shell internal propagators contribute to the collinear limit.Explicit results are provided for up to six gluons, with ΔM = 0, 1, 2 cases treated generally and higher-n specifics demonstrated up to n=6, all of which agree with existing literature where applicable while delivering new compact expressions.The findings hold potential for improving higher-order perturbative predictions and jet observables at the LHC, and can be extended to quark-gluon collinear limits through the MHV framework.

Abstract

We consider the multi-collinear limit of multi-gluon QCD amplitudes at tree level. We use the MHV rules for constructing colour ordered tree amplitudes and the general collinear factorization formula to derive timelike splitting functions that are valid for specific numbers of negative helicity gluons and an arbitrary number of positive helicity gluons (or vice versa). As an example we present new results describing the collinear limits of up to six gluons.

Figures (6)

• Figure 1: Factorisation of an $N$-point colour ordered amplitude with gluons $p_1,\ldots,p_n$ collinear into splitting function for $P \to 1, \ldots, n$ multiplied by an $(N-n+1)$-point amplitude.
• Figure 2: MHV diagrams contributing to $\mathrm{Split}^{(n)}_+(m_1)$. Negative helicity gluons are indicated by solid lines, while arbitrary numbers of positive helicity gluons emitted from each vertex are shown as dotted arcs.
• Figure 3: MHV diagrams contributing to $\mathrm{Split}^{(n)}_-(m_1,m_2)$.
• Figure 4: MHV diagrams contributing to $\mathrm{Split}^{(n)}_+(m_1,m_2)$.
• Figure 5: MHV topologies contributing to $\mathrm{Split}^{(n)}_-(m_1,m_2,m_3)$. The negative helicity gluons $m_1$, $m_2$ and $m_3$ are distributed in a cyclic way around each diagram. The remaining leg is the negative helicity gluon that remains after the collinear limit is taken.