Virtuous Trees at Five and Six Points for Yang-Mills and Gravity

TL;DR

This work constructs two distinct D-dimensional, tree-level, color-kinematic (BCJ) representations for the five-point Yang–Mills amplitude that are amplitude-encoded and symmetric, enabling straightforward gravity double-copy formulations and linking to recent five-point loop results. It shows how a single master function can govern the kinematic content across all graphs, and unpacks a second representation grounded in loop-level BCJ structures with a β-function, providing a gauge-parameterized bridge between them. Additionally, it provides a four-dimensional, six-point MHV/MHVbar tree representation and discusses the broader implications for all-loop, all-multiplicity unitarity via these virtuous tree-level constructions. The results illuminate constructive pathways for moving between representations and hint at an underlying, dimensionally robust kinematic algebra yet to be fully understood.

Abstract

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 MHV and MHVbar amplitudes which holds only in four dimensions.

Figures (4)

• Figure 1: Illustration of a generic $m$-point half-ladder diagram.
• Figure 2: The four point cubic diagram. It appears with three distinct labelings of external legs $(a,b,c,d)$, corresponding to the "$s$"-channel diagram: $(1,2,3,4)$, the "$t$"-channel diagram: $(2,3,4,1)$, and the "$u$"-channel diagram: $(3,1,4,2)$.
• Figure 3: The five point half-ladder diagram. All contributions in cubic-graph representations of five-loops involve this topology or are related by antisymmetry around vertices.
• Figure 4: Illustration of the kinematic Jacobi relation associated with the indicated edge, as given in eq. (\ref{['sixJacRule']}), which expresses the numerator of the trimerous topology on the left in terms of the difference between the two half-ladders on the right.