Kinematic numerators from the worldsheet: cubic trees from labelled trees

2021-03-30

Authors:  Linghui Hou, Song He, Jintian Tian and Yong Zhang

Preprint number:  UUITP-14/21

Abstract: In this note we revisit the problem of explicitly computing tree-level scatter-ing amplitudes in various theories in any dimension from worldsheet formulas. The latterare known to produce cubic-tree expansion of tree amplitudes with kinematic numeratorsautomatically satisfying Jacobi-identities, once any half-integrand on the worldsheet is re-duced 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 natu-ral expansions of known half-integrands onto them with coefficients that are particularlycompact building blocks of kinematic numerators. We present a general formula expressingthe kinematic numerator of any cubic tree as a linear combination of these coefficients oflabelled trees, including the usual combination in terms of master numerators as a specialcase. Our results provide an efficient algorithm, which is implemented in aMathemat-icapackage, for computing tree amplitudes in non-linear sigma model, special Galileon,Yang-Mills-scalar, Einstein-Yang-Mills, Dirac-Born-Infeld and so on.