New Relations for Gauge-Theory Amplitudes

TL;DR

The paper introduces a kinematic numerator identity for n-point gauge-theory tree amplitudes that parallels the color Jacobi identity, enabling new relations among color-ordered partial amplitudes and reducing the independent basis from (n-1)! to (n-3)!. By leveraging the unitarity method, these relations extend to higher loops, as illustrated in a two-loop QCD example, and illuminate the structure of gravity amplitudes via KLT-like diagram-by-diagram squaring. The authors also generalize the construction to higher points, conjecturing all-n relations and providing an all-n framework for gravity-gauge relations. These insights promise to streamline multi-loop computations and deepen understanding of gauge/gravity dualities and their UV behavior.

Abstract

We present an identity satisfied by the kinematic factors of diagrams describing the tree amplitudes of massless gauge theories. This identity is a kinematic analog of the Jacobi identity for color factors. Using this we find new relations between color-ordered partial amplitudes. We discuss applications to multi-loop calculations via the unitarity method. In particular, we illustrate the relations between different contributions to a two-loop four-point QCD amplitude. We also use this identity to reorganize gravity tree amplitudes diagram by diagram, offering new insight into the structure of the KLT relations between gauge and gravity tree amplitudes. This can be used to obtain novel relations similar to the KLT ones. We expect this to be helpful in higher-loop studies of the ultraviolet properties of gravity theories.

Figures (7)

• Figure 1: The "parent" diagrams for the two-loop four-point identical-helicity amplitudes of QCD and ${{\cal N}=4}$ super-Yang-Mills theory. All other diagrams appearing in the amplitude are obtained by collapsing propagators. These parent diagrams also determine the color factors appearing in eq. (\ref{['FColor']}), by dressing them with $\widetilde{f}^{abc}$s in a clockwise ordering.
• Figure 2: The Jacobi identity relating the color factors of the $u, s, t$ channel "color diagrams". The color factors are given by dressing each vertex with an $\widetilde{f}^{abc}$ following a clockwise ordering.
• Figure 3: The unitarity method constructs multiloop amplitudes from lower-loop amplitudes. The blobs represent amplitudes, the white holes loops and the dotted lines indicate cuts which replace propagators with on-shell delta functions. Generalized cuts which decompose loop amplitude solely in terms of tree amplitudes are particularly useful in carrying out multiloop calculations.
• Figure 4: The Jacobi identity at five points. These diagrams can be interpreted as relations for color factors, where each color factor is obtained by dressing the diagrams with $\widetilde{f}^{abc}$ at each vertex in a clockwise ordering. Alternatively it can be interpreted as relations between the kinematic numerator factors of corresponding diagrams, where the diagrams are nontrivially rearranged compared to Feynman diagrams.
• Figure 5: Near-maximal cuts with specified color order given by the uncut blob. All visible lines are cut, thus on-shell. The blobs are tree amplitudes with implied sums over all helicity and particle types entering and leaving the blobs.