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Simplifying Multi-Jet QCD Computation

Michael E. Peskin

TL;DR

The paper surveys a coherent toolkit for tree-level QCD amplitudes in collider physics, combining spinor-helicity methods, color ordering, and MHV techniques with the BCFW recursion to drastically simplify multi-jet computations. It demonstrates how complex multijet amplitudes reduce to compact spinor-structure formulas, yields explicit results for key 2→2 parton processes, and shows how these building blocks extend to vector-boson production with jets at hadron colliders. The BCFW framework provides a principled, on-shell approach to derive general n-point MHV and non-MHV amplitudes from simple three-point seeds, enabling practical calculation and cross-section derivations for processes like W+ jets. The synthesis enables efficient, scalable, and gauge-invariant tree-level QCD predictions with direct relevance to LHC phenomenology and the development of computational tools.

Abstract

These lectures give a pedagogical discussion of the computation of QCD tree amplitudes for collider physics. The topics reviewed are: spinor products, color ordering, MHV amplitudes, and the Britto-Cachazo-Feng-Witten recursion formula. The lectures were presented at the XIII Mexican School of Particles and Fields, 2008, and at the LHC Physics Summer School at Tsinghua University, 2010.

Simplifying Multi-Jet QCD Computation

TL;DR

The paper surveys a coherent toolkit for tree-level QCD amplitudes in collider physics, combining spinor-helicity methods, color ordering, and MHV techniques with the BCFW recursion to drastically simplify multi-jet computations. It demonstrates how complex multijet amplitudes reduce to compact spinor-structure formulas, yields explicit results for key 2→2 parton processes, and shows how these building blocks extend to vector-boson production with jets at hadron colliders. The BCFW framework provides a principled, on-shell approach to derive general n-point MHV and non-MHV amplitudes from simple three-point seeds, enabling practical calculation and cross-section derivations for processes like W+ jets. The synthesis enables efficient, scalable, and gauge-invariant tree-level QCD predictions with direct relevance to LHC phenomenology and the development of computational tools.

Abstract

These lectures give a pedagogical discussion of the computation of QCD tree amplitudes for collider physics. The topics reviewed are: spinor products, color ordering, MHV amplitudes, and the Britto-Cachazo-Feng-Witten recursion formula. The lectures were presented at the XIII Mexican School of Particles and Fields, 2008, and at the LHC Physics Summer School at Tsinghua University, 2010.

Paper Structure

This paper contains 27 sections, 184 equations, 21 figures.

Table of Contents

  1. Introduction
  2. Spinor products
  3. Massless fermions
  4. $e^+e^-\to \mu^+\mu^-$
  5. Massless photons
  6. $e^+e^-\to \gamma\gamma$
  7. Color-ordered amplitudes
  8. $q\overline{q} \to gg$
  9. MHV amplitudes
  10. $q\overline{q} gg$ amplitude
  11. Four-gluon amplitude
  12. Parton-parton scattering
  13. Four-fermion processes
  14. $q\overline{q} gg$ processes
  15. Four-gluon processes
  16. ...and 12 more sections

Figures (21)

  • Figure 1: Feynman diagram for $e_L^- e_R^+ \to \mu^-_L \mu^+_R$.
  • Figure 2: Ward identity obeyed by a gauge-invariant sum of diagrams with all external particles on shell.
  • Figure 3: Feynman diagram for $e_L^- e_R^+ \to \gamma\gamma$.
  • Figure 4: Feynman diagrams for $q_L \overline{q}_R \to gg$.
  • Figure 5: Feynman diagrams for $gg\to gg$.
  • ...and 16 more figures