All tree-level amplitudes in massless QCD

Abstract

We derive compact analytical formulae for all tree-level color-ordered gauge theory amplitudes involving any number of external gluons and up to three massless quark-anti-quark pairs. A general formula is presented based on the combinatorics of paths along a rooted tree and associated determinants. Explicit expressions are displayed for the next-to-maximally helicity violating (NMHV) and next-to-next-to-maximally helicity violating (NNMHV) gauge theory amplitudes. Our results are obtained by projecting the previously-found expressions for the super-amplitudes of the maximally supersymmetric Yang-Mills theory (N=4 SYM) onto the relevant components yielding all gluon-gluino tree amplitudes in N=4 SYM. We show how these results carry over to the corresponding QCD amplitudes, including massless quarks of different flavors as well as a single electroweak vector boson. The public Mathematica package GGT is described, which encodes the results of this work and yields analytical formulae for all N=4 SYM gluon-gluino trees. These in turn yield all QCD trees with up to four external arbitrary-flavored massless quark-anti-quark-pairs.

40 pages, Mathematica package GGT.m and example notebook is included in submission, v2: QCD four fermion line translations provided; GGT version 1.1 update with a numerical evaluation function; comments on computer speed optimizations, v3: Minor changes, version to be published in JHEP, v4: published version in JHEP

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• On the Double Copy for Spinning Matter
• Helicity amplitudes for QCD with massive quarks
• Positroids, Plabic Graphs, and Scattering Amplitudes in Mathematica
• Efficient Tree-Amplitudes in N=4: Automatic BCFW Recursion in Mathematica
• Connected formulas for amplitudes in standard model
• Getting more flavour out of one-flavour QCD
• Prescriptive Unitarity for Non-Planar Six-Particle Amplitudes at Two Loops
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• Manifestly Dual-Conformal Loop Integration
• Two-loop $\mathcal{N}=2$ SQCD amplitudes with external matter from iterated cuts
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• What can we learn about QCD and collider physics from N=4 super Yang-Mills?
• The $L_{\infty }$ structure of gauge theories with matter
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• Color ordering in QCD
• The colourful antenna subtraction method
• Comparing efficient computation methods for massless QCD tree amplitudes: Closed Analytic Formulae versus Berends-Giele Recursion
• The Coulomb-branch S-matrix from massless amplitudes
• New relations for graviton-matter amplitudes
• Double-Soft Limits of Gluons and Gravitons
• All rational one-loop Einstein-Yang-Mills amplitudes at four points
• Recursion relations for multi-gluon off-shell amplitudes on the light-front and Wilson lines
• All-multiplicity amplitudes with four massive quarks and identical-helicity gluons
• Complete Form Factors in Yang-Mills from Unitarity and Spinor Helicity in Six Dimensions
• Recursion relations and scattering amplitudes in the light-front formalism
• A Monte Carlo Approach to the 4D Scattering Equations
• Recursion relations for scattering amplitudes with massive particles II: massive vector bosons
• Evaluating EYM amplitudes in four dimensions by refined graphic expansion
• SpinorsExtras - Mathematica implementation of massive spinor-helicity formalism
• A new Wilson line-based action for gluodynamics
• Leading multi-soft limits from scattering equations
• All one-loop NMHV gluon amplitudes in N=1 SYM
• Scattering amplitudes in gauge theories with and without supersymmetry
• Perturbative Methods for Superconformal Quantum Field Theories in String - Gauge Theory Dualities
• High multiplicity processes at NLO with BlackHat and Sherpa