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 model, special Galileon,Yang-Mills-scalar, Einstein-Yang-Mills, Dirac-Born-Infeld and so on.