Kinematic numerators from the worldsheet: cubic trees from labelled trees

Abstract

In this note we revisit the problem of explicitly computing tree-level scattering amplitudes in various theories in any dimension from worldsheet formulas. The latter are known to produce cubic-tree expansion of tree amplitudes with kinematic numerators automatically satisfying Jacobi-identities, once any half-integrand on the worldsheet is reduced to logarithmic functions. We review a natural class of worldsheet functions called "Cayley functions", which are in one-to-one correspondence with labelled trees, and natural expansions of known half-integrands onto them with coefficients that are particularly compact building blocks of kinematic numerators. We present a general formula expressing kinematic numerators of all cubic trees as linear combinations of coefficients of labelled trees, which satisfy Jacobi identities by construction and include the usual combinations in terms of master numerators as a special case. Our results provide an efficient algorithm, which is implemented in a Mathematica package, for computing all tree amplitudes in theories including non-linear sigma model, special Galileon, Yang-Mills-scalar, Einstein-Yang-Mills and Dirac-Born-Infeld.

An important typo corrected. References updated

References in corpus (13)

• New Relations for Gauge-Theory Amplitudes
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• The Five-Loop Four-Point Integrand of N=8 Supergravity as a Generalized Double Copy
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• One-loop Correlators and BCJ Numerators from Forward Limits
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• Local BCJ numerators for ten-dimensional SYM at one loop
• Planar binary trees in scattering amplitudes
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Cited by in corpus (12)

• Covariant Color-Kinematics Duality
• Kinematic Hopf Algebra for BCJ Numerators in Heavy-Mass Effective Field Theory and Yang-Mills Theory
• Off-Shell Color-Kinematics Duality for Chern-Simons
• Off-shell extended graphic rule and the expansion of Berends-Giele currents in Yang-Mills theory
• Universal expansions of scattering amplitudes for gravitons, gluons and Goldstone particles
• One-loop diagrams with quadratic propagators from the worldsheet
• One-loop Bern-Carrasco-Johansson Numerators on Quadratic Propagators from the Worldsheet
• Covariant color-kinematics duality, Hopf algebras and permutohedra
• Massive Color-Kinematics Duality and Double-Copy for Kaluza-Klein Scattering Amplitudes
• Next-to-MHV Yang-Mills kinematic algebra
• New relations for tree-level form factors and scattering amplitudes
• Form factors from string amplitudes