Pure Gravities via Color-Kinematics Duality for Fundamental Matter

TL;DR

This work extends color-kinematics duality to gauge theories with fundamental matter and develops a ghost-based double-copy prescription to construct pure gravity amplitudes, including Einstein gravity and pure ${ m N}<4$ supergravities. By promoting fundamental double copies to opposite-statistics ghosts and counting matter loops with a sign factor, the authors cancel unwanted dilaton-axion states and enable tunable non-self-interacting matter in gravity. They provide a formal framework, explicit one-loop numerators for multiple matter representations, and two-loop unitarity-cut checks supporting the approach, with scalars found to be problematic at two loops. The results yield a coherent, extensible method for obtaining a wide class of pure supergravity and gravity-matter amplitudes, with potential applications to multiloop ultraviolet structure and dimensional generalizations.

Abstract

We give a prescription for the computation of loop-level scattering amplitudes in pure Einstein gravity, and four-dimensional pure supergravities, using the color-kinematics duality. Amplitudes are constructed using double copies of pure (super-)Yang-Mills parts and additional contributions from double copies of fundamental matter, which are treated as ghosts. The opposite-statistics states cancel the unwanted dilaton and axion in the bosonic theory, as well as the extra matter supermultiplets in supergravities. As a spinoff, we obtain a prescription for obtaining amplitudes in supergravities with arbitrary non-self-interacting matter. As a prerequisite, we extend the color-kinematics duality from the adjoint to the fundamental representation of the gauge group. We explain the numerator relations that the fundamental kinematic Lie algebra should satisfy. We give nontrivial evidence supporting our construction using explicit tree and loop amplitudes, as well as more general arguments.

Figures (15)

• Figure 1: The kinematic algebra in the vector case (a) and in the fundamental matter case (b). Curly lines represent gluons or vector supermultiplets, solid lines represents fermions, scalars or supersymmetric matter. Alternatively, these diagrams describe standard Lie-algebra relations for the color factors of the gauge group.
• Figure 2: A typical one-loop graph with external adjoint and internal fundamental particles.
• Figure 3: An additional two-term kinematic identity that can be imposed on the numerators for indistinguishable fermions in spacetime dimensions $D=3,4,6,10$, or for indistinguishable complex scalars in any dimension. However, this relation is not mandatory for the color-kinematics duality. In fact, the corresponding color factors will in general not satisfy this relation.
• Figure 4: A double copy \ref{['dilatonaxionexample']} of diagrams that obeys the color-kinematics duality in the fundamental representation. It gives a four-scalar amplitude with gravitational interactions.
• Figure 5: An equality between two different double copies: with two distinct fermion lines and with a single fermion line. Note that the propagators are implicitly included in this identity, and the $t$- and $u$-channel poles on the second line are spurious.