[Paper Review] Multi gluon collinear limits from MHV amplitudes

Peter Marquard, T. G. Birthwright|ArXiv.org|May 31, 2005

Particle physics theoretical and experimental studies10 references3 citations

TL;DR

This paper derives timelike splitting functions for multi-gluon collinear limits in QCD at tree level using MHV rules, showing that only MHV diagrams with internal propagators going on-shell contribute. The key result is a systematic method to compute splitting functions for arbitrary numbers of positive-helicity gluons and specific numbers of negative-helicity gluons, with results valid for ΔM = 0, 1, 2 and extended to six gluons via parity symmetry.

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 factorisation 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).

Motivation & Objective

• To systematically compute multi-collinear splitting functions in QCD amplitudes at tree level for arbitrary numbers of positive-helicity gluons and specific numbers of negative-helicity gluons.
• To establish a method based on MHV rules that isolates only the diagrams contributing to the collinear limit—those with internal propagators going on-shell.
• To derive explicit expressions for splitting functions valid for up to six gluons, including cases with ΔM = 0, 1, 2 (change in number of negative helicity gluons).
• To provide a framework applicable to higher multiplicities and extendable to quark-gluon collinear limits via MHV rules for quark vertices.

Proposed method

• Utilizes the MHV formalism to construct colour-ordered tree amplitudes from MHV vertices connected by scalar propagators, avoiding gauge-fixing by keeping reference spinors general.
• Applies the general collinear factorisation formula, where the full amplitude factorises into a splitting function and a reduced amplitude with one on-shell composite particle.
• Identifies only MHV diagrams with internal propagators going on-shell in the collinear limit as contributing to the splitting function, leveraging the gauge-invariant structure of MHV vertices.
• Derives splitting functions by taking the collinear limit of MHV amplitudes, using spinor-helicity formalism and scaling relations such as ⟨i n+1⟩ → √z_i ⟨P n+1⟩ in the collinear regime.
• For ΔM = 1 and ΔM = 2 cases, includes contributions from NMHV diagrams, with explicit expressions involving sums over intermediate states and propagator denominators D(i,j,q).
• Uses the definition Δ_(1)(i,j;a) = Σ_{l=i+1}^j ⟨a l⟩√z_l to encode helicity-dependent momentum fractions in the collinear limit.

Experimental results

Research questions

• RQ1 How can multi-gluon collinear splitting functions be systematically derived for arbitrary numbers of positive-helicity gluons and fixed numbers of negative-helicity gluons?
• RQ2 What subset of MHV diagrams contributes to the collinear limit, and why do only those with on-shell internal propagators matter in the MHV approach?
• RQ3 How do the splitting functions depend on the helicity configuration, and can they be classified by ΔM, the change in the number of negative-helicity gluons?
• RQ4 Can the MHV formalism yield closed-form expressions for splitting functions beyond the simplest cases, such as ΔM = 1 and ΔM = 2?

Key findings

• The splitting function for n collinear positive-helicity gluons to a single positive-helicity gluon is given by split(1⁺,…,n⁺→P⁺) = 1/(√z₁√zₙ ∏_{i=1}^{n-1} ⟨i i+1⟩), derived directly from a single MHV vertex.
• For a single negative-helicity gluon among n positive-helicity gluons, the splitting function is split(1⁺,…,i⁻,…,n⁺→P⁻) = z_i² / (√z₁√zₙ ∏_{i=1}^{n-1} ⟨i i+1⟩), obtained from an MHV amplitude with two negative-helicity gluons.
• For the case ΔM = 1 (e.g., 1⁺,2⁻,3⁺→P⁺), the splitting function involves a sum over intermediate states and is expressed as split(1⁺,2⁻,3⁺→P⁺) = [∑_{i=0}^{m₁−1}∑_{j=m₁}^{n} Δ_(1)(i,j;m₁)^4 / D(i,j,q_{i+1,j})] / (√z₁zₙ ∏_{l=1}^{n−1} ⟨l l+1⟩), with explicit dependence on momentum fractions.
• Explicit results for up to four collinear gluons with all independent helicity combinations are derived and numerically agree with existing literature (Ref. [9]).
• New results are provided for five and six collinear gluons, extending the applicability of the MHV-based method to higher multiplicities.
• The method is generalizable to higher ΔM and can be extended to quark-gluon collinear limits using MHV rules for quark vertices.