Kinematic numerators from the worldsheet: cubic trees from labelled trees

TL;DR

This paper presents a new method to compute tree-level scattering amplitudes using worldsheet formulas, linking cubic tree expansions with labelled trees, and provides an efficient algorithm implemented in Mathematica for various theories.

Contribution

It introduces a general formula for kinematic numerators from labelled trees, ensuring Jacobi identities and enabling efficient amplitude computations across multiple theories.

Findings

Derived a formula expressing kinematic numerators as linear combinations of labelled tree coefficients.

Implemented an algorithm in Mathematica for computing amplitudes in diverse theories.

Validated the approach with applications to theories like Yang-Mills, DBI, and non-linear sigma models.

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.