Gravitational Effective Field Theory Islands, Low-Spin Dominance, and the Four-Graviton Amplitude

TL;DR

The paper develops a comprehensive framework linking heavy-mass-induced corrections to gravity with perturbative unitarity, crossing, and dispersion relations in four dimensions. It computes one-loop four-graviton amplitudes with massive matter up to spin 2 via generalized unitarity and the BCJ double copy, then matches their large-mass expansion to a low-energy EFT containing R^3 and R^4 operators, extracting Wilson coefficients. Through dispersive sum rules and the principle of low-spin dominance, the authors derive two-sided bounds on EFT coefficients, revealing small, string-like theory islands within the broader EFThedron; these islands align for both string theories and one-loop field-theory data. The results suggest that physically reasonable gravitational theories occupy a much more constrained region of EFT space than naive 2-to-2 scattering bounds would imply, with LSD providing a unifying explanation for the observed clustering. Overall, the work connects explicit amplitude computations to geometric bounds on gravitational EFTs, offering a quantitative pathway to test UV completions and guiding future exploration of gravity’s perturbative landscape.

Abstract

We analyze constraints from perturbative unitarity and crossing on the leading contributions of higher-dimension operators to the four-graviton amplitude in four spacetime dimensions, including constraints that follow from distinct helicity configurations. We focus on the leading-order effect due to exchange by massive degrees of freedom which makes the amplitudes of interest infrared finite. In particular, we place a bound on the coefficient of the $R^3$ operator that corrects the graviton three-point amplitude in terms of the $R^4$ coefficient. To test the constraints we obtain nontrivial effective field-theory data by computing and taking the large-mass expansion of the one-loop minimally-coupled four-graviton amplitude with massive particles up to spin 2 circulating in the loop. Remarkably, we observe that the leading EFT coefficients obtained from both string and one-loop field-theory amplitudes lie in small islands. The shape and location of the islands can be derived from the dispersive representation for the Wilson coefficients using crossing and assuming that the lowest-spin spectral densities are the largest. Our analysis suggests that the Wilson coefficients of weakly-coupled gravitational physical theories are much more constrained than indicated by bounds arising from dispersive considerations of $2 \to 2$ scattering. The one-loop four-graviton amplitudes used to obtain the EFT data are computed using modern amplitude methods, including generalized unitarity, supersymmetric decompositions and the double copy.

Figures (23)

• Figure 1: The (a) $s$-, (b) $t$- and (c) $u$-channel two-particle cuts of a one-loop four-point amplitude. The exposed lines are all on shell and the blobs represent tree-level amplitudes.
• Figure 2: Example of a color or numerator relation for the one-loop four-point amplitudes. Here the diagram represent either color or kinematic numerators. We use these relations on the generalized-unitarity cuts, as indicated by the dashed lines. The relations are effectively tree-level ones except that the state and color sums on the cut legs are carried out.
• Figure 3: The tree-level four-point color or numerator Jacobi identity. This can be used to set the numerator of one of the diagrams to zero.
• Figure 4: The (a) $s$-, (b) $t$- and (c) $u$-channel two-particle cuts of a one-loop four-point amplitude with two negative- and two positive-helicity external gluons or gravitons (double-minus configuration). The internal lines represent massive spinning particles. The exposed lines are all on shell and the blobs represent tree-level amplitudes.
• Figure 5: Generic box diagram whose color factor and denominator are given by Eqs. (\ref{['colorGen']}) and (\ref{['denGen']}) respectively. The external momenta are taken incoming while the direction of the loop momentum is indicated by the arrow.