Color-Kinematics Duality for Pure Yang-Mills and Gravity at One and Two Loops

TL;DR

This work extends color-kinematics (BCJ) duality to nonsupersymmetric pure Yang–Mills by constructing a manifestly dual one-loop four-point integrand in D dimensions and a duality-satisfying two-loop identical-helicity amplitude. Via the double-copy prescription, gravity integrands for a graviton–dilaton–antisymmetric tensor theory are obtained, allowing explicit UV-divergence analyses across dimensions. The authors provide detailed one-loop divergences in D=4,6,8 and a two-loop R^3 divergence in D=4, illustrating that the double-copy structure alone does not guarantee ultraviolet finiteness without additional cancellations such as supersymmetry. Overall, the paper expands evidence for BCJ duality in non-supersymmetric settings and clarifies the UV behavior of gravity theories constructed through double-copy.

Abstract

We provide evidence in favor of the conjectured duality between color and kinematics for the case of nonsupersymmetric pure Yang-Mills amplitudes by constructing a form of the one-loop four-point amplitude of this theory that makes the duality manifest. Our construction is valid in any dimension. We also describe a duality-satisfying representation for the two-loop four-point amplitude with identical four-dimensional external helicities. We use these results to obtain corresponding gravity integrands for a theory containing a graviton, dilaton, and antisymmetric tensor, simply by replacing color factors with specified diagram numerators. Using this, we give explicit forms of ultraviolet divergences at one loop in four, six, and eight dimensions, and at two loops in four dimensions.

Figures (10)

• Figure 1: The basic Jacobi relation for either color or numerator factors. These three diagrams can be embedded in a larger diagram, including loops.
• Figure 2: The Jacobi relations determining either color or kinematic numerators of the four-point diagrams containing either a triangle or internal bubble.
• Figure 3: The color or kinematic Jacobi relations involving a bubble on an external leg or a tadpole. These diagrams have vanishing contribution to the integrated amplitude.
• Figure 4: The seven diagrams for the color-ordered amplitude with ordering $(1,2,3,4)$.
• Figure 5: The (a) $s$-channel and (b) $t$-channel unitarity cuts used to determine the amplitude. The exposed intermediate legs are on shell.