Gravity Amplitudes as Generalized Double Copies

Abstract

Whenever the integrand of a gauge-theory loop amplitude can be arranged into a form where the BCJ duality between color and kinematics is manifest, a corresponding gravity integrand can be obtained simply via the double-copy procedure. However, finding such gauge-theory representations can be challenging, especially at high loop orders. Here we show that we can instead start from generic gauge-theory integrands, where the duality is not manifest, and apply a modified double-copy procedure to obtain gravity integrands that include contact terms generated by violations of dual Jacobi identities. We illustrate this with three-, four- and five-loop examples in N=8 supergravity.

Figures (3)

• Figure 1: Sample N$^k$-maximal cuts at three, four and five loops. Exposed lines are all on shell.
• Figure 2: Diagrams (a)-(i) define the three-loop four-point amplitude of ${{\cal N}=4}$ sYM theory; diagrams (j)-(m) are the additional contact terms needed for ${{\cal N}=8}$ supergravity.
• Figure 3: The three diagrams whose kinematic numerators contribute to $J_{\{1,1\},1}$. The thick shaded (red) cross marks the off-shell legs participating in the dual Jacobi relation. The shaded (red) dot indicates the off-shell leg of the second amplitude factor.