The Entire Four-Graviton EFT from the Duality Between Color and Kinematics

TL;DR

The paper extends the BCJ color-kinematics duality to the full four-point color algebra, including $d^{abc}$ structures, by introducing a universal $N$-copy that assembles gauge-theory numerators across the antisymmetric, mixed-symmetry, symmetric, and permutation-invariant sectors. By classifying parity-even four-graviton operators in any dimension and constructing their gravity amplitudes via double copies (primarily from $ff$, $dd$, and $df$ sectors) and a necessary triple-copy for the Lovelock $R^3$ term in $D>6$, the authors show that the entire four-point gravitational EFT can be factorized into gauge-theory building blocks governed by color-kinematics duality. The work identifies a finite, constructible basis of gravitational polynomials $ ext{G}$ and demonstrates that most higher-derivative gravity amplitudes arise from $(gggg) ext{ or }(ggss) ext{ constructions}$, with the $R^3$ Lovelock term demanding a genuine triple-copy. These results reveal a deeper organizing principle for gravity, extending the double-copy paradigm beyond string-theoretic embeddings and suggesting new directions for higher-multiplicity amplitudes and potential UV completions.

Abstract

The Bern-Carrasco-Johansson (BCJ) double-copy construction reveals a fundamental structural connection between gauge and gravity theories. At its core, the BCJ double copy is directly due to a duality between the algebraic relations of a color root and those of a kinematic root. We generalize this principle beyond the conventional Lie algebra structure of tree-level Yang-Mills theory. By demanding color-kinematics duality for the complete basis of four-point color structures -- including those involving the symmetric $d^{abc}$ constants -- we define the universal double copy. We systematically classify the bases of all such parity-even generalized gauge-theory numerators and, independently, the space of all parity-even four-graviton higher-derivative operators. We demonstrate that our universal double-copy construction precisely spans the entire tower of parity-even four-graviton amplitudes in any dimension, except for the Lovelock $R^3$ contribution in $D >6$ which we can express in terms of a particularly simple universal triple-copy involving gauge theories coupled to scalars. Explicit machine-readable expressions for the complete basis of gauge-theory numerators and fundamental gravitational building blocks are provided in the ancillary files. This establishes that all possible four-point gravitational interactions can be factorized into products of gauge-theory building blocks governed by this universal notion of color-kinematics duality.