Double-Copy Constructions and Unitarity Cuts

TL;DR

The paper addresses the challenge of constructing multiloop gravity amplitudes by leveraging the color-kinematics (BCJ) duality and the double-copy method. It introduces a strategy that relaxes the requirement of manifest BCJ duality in the full integrand, enforcing it instead on a spanning set of generalized unitarity cuts to preserve correct gravity cuts while keeping gauge-theory ansätze simple. The two-loop, four-point examples in both identical-helicity and general D-dimensional polarization cases demonstrate that this cut-based duality suffices for reliable double-copy gravity amplitudes, with explicit analysis of ultraviolet divergences and evanescent effects. The results suggest a practical pathway to higher-loop gravity calculations and offer insights into the UV behavior of gravity theories beyond standard symmetry expectations, pointing to future explorations of evanescent operators and duality transformations.

Abstract

The duality between color and kinematics enables the construction of multiloop gravity integrands directly from corresponding gauge-theory integrands. This has led to new nontrivial insights into the structure of gravity theories, including the discovery of enhanced ultraviolet cancellations. To continue to gain deeper understandings and probe these new properties, it is crucial to further improve techniques for constructing multiloop gravity integrands. In this paper, we show by example how one can alleviate difficulties encountered at the multiloop level by relaxing the color-kinematics duality conditions to hold manifestly only on unitarity cuts instead of globally on loop integrands. As an example, we use a minimal ansatz to construct an integrand for the two-loop four-point nonsupersymmetric pure Yang-Mills amplitude in $D$ dimensions that is compatible with these relaxed color-kinematics duality constraints. We then immediately obtain a corresponding gravity integrand through the double-copy procedure. Comments on ultraviolet divergences are also included.

Figures (11)

• Figure 1: The basic Jacobi relation for either color or numerator factors. These three diagrams can be embedded in a larger diagram at tree level or loop level. The propagator around which the Jacobi relation is performed is shaded (red).
• Figure 2: The Jacobi relations determining the triangle and bubble numerators in terms of box numerators. The shaded (red) propagator indicates the line around which the Jacobi identities are applied.
• Figure 3: Sample color- or kinematic-numerator Jacobi relations for the two-loop four-point amplitudes. The shaded (red) propagator indicates the line around which the Jacobi identities are applied.
• Figure 4: Two-particle cut evaluated in all three channels determines one-loop four-point amplitudes.
• Figure 5: The one-loop Jacobi relations with cut conditions imposed. The shaded (red) internal lines indicate the leg around which the Jacobi identities are applied. Internal legs intersected by the dashed lines are put on shell.