Color-kinematics duality for pure Yang-Mills and gravity at one and two loops

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.

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.

Figure 6

The diagrams needed to describe an integrand for the identical-helicity amplitude where the duality between color and kinematics is manifest. When integrated diagrams (d)–(k) vanish. The (a) planar double-box, (b) nonplanar double-box, and (c) double-triangle integrals are the only nonvanishing ones under integration.

Figure 7

The bow-tie integral appearing in the identical-helicity pure Yang-Mills amplitude.

Figure 8

The nontrivial duality relations (a)–(m) satisfied by the numerators of the identical-helicity two-loop amplitude. The shaded (red) leg marks the central leg of the applied Jacobi identity.

Figure 9

Sample duality relations (a)–(f) involving graphs with vanishing numerators. In each relation, the leftmost diagram has a vanishing numerator. The shaded (red) leg marks the central leg of the applied dual-Jacobi identity.

Figure 10

The $R^3$ operator diagrams that contribute to the all-plus-helicity four-graviton amplitude. The solid dot represents vertices generated by the $R^3$ operator.