Virtuous trees at five- and six-point levels for Yang-Mills theory and gravity

We present a particularly nice D-dimensional graph-based representation of the full color-dressed five-point tree-level gluon amplitude. It possesses the following virtues: (1)it satisfies the color-kinematic correspondence, and thus trivially generates the associated five-point graviton amplitude, (2)all external-state information is encoded in color-ordered partial amplitudes, and (3)one function determines the kinematic contribution of all graphs in the Yang-Mills amplitude, so the associated gravity amplitude is manifestly permutation symmetric. The third virtue, while shared among all known loop-level correspondence-satisfying representations, is novel for tree-level representations sharing the first two virtues. This new D-dimensional representation makes contact with the recently found multiloop five-point representations, suggesting all-loop, all-multiplicity ramifications through unitarity. Additionally we present a slightly less virtuous representation of the six-point maximally helicity-violating (MHV) and MHV̄ amplitudes that holds only in four dimensions.