Kinematic numerators from the worldsheet: cubic trees from labelled trees

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 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 the kinematic numerator of any cubic tree as a linear combination of these coefficients of labelled trees, including the usual combination in terms of master numerators as a special case. Our results provide an efficient algorithm, which is implemented in a Mathematica package, for computing tree amplitudes in non-linear sigma models, special Galileon, Yang-Mills-scalar, Einstein-Yang-Mills, Dirac-Born-Infeld and so on.