Energy correlators in four-dimensional gravity

TL;DR

This work develops infrared-finite observables for four-dimensional gravity by focusing on energy correlators measured at null infinity. The authors compute the one- and two-point energy-energy correlators at one loop in both ${\mathcal{N}}=8$ supergravity and pure gravity, demonstrating explicit cancellation of infrared divergences and verifying energy–momentum Ward identities, including the role of contact terms. In the back-to-back limit, they derive an all-order soft-graviton result that exponentiates and is governed by a lightlike cusp anomalous dimension, with a universal soft dynamics description and an explicit resummation formula, while a beam-averaged EEC exhibits analytic and dispersion-relations properties and remains well-behaved under stringy corrections. The analysis is extended to tree-level string theory corrections, yielding a controlled expansion in $\alpha'$ and highlighting how stringy effects enter the EEC beyond SG and pure gravity; dispersion relations further illuminate positivity structures and multipole expansions of the EEC. Overall, the paper provides a coherent IR-safe framework for gravitational collider observables, reveals universal soft-graviton dynamics in endpoint regions, and suggests bootstrap-like constraints for gravity in flat space. The results offer a bridge between gravitational amplitudes, celestial holography, and IR/UV consistency conditions with potential phenomenological and foundational implications.

Abstract

We investigate energy correlators in four-dimensional gravitational theories, which provide a simple class of infrared-finite observables. We compute the one- and two-point energy correlators at one loop in $\mathcal{N}=8$ supergravity and in pure Einstein gravity, with particular emphasis on the contact terms arising from the interplay between virtual corrections and real emissions. We explicitly demonstrate the cancellation of infrared divergences and verify the Ward identities associated with energy-momentum conservation. In the back-to-back limit, we derive an all-order expression for the energy-energy correlator, showing that it is governed by universal soft-graviton dynamics. We further introduce a particularly simple beam-averaged energy-energy correlator and compute it in different gravitational theories, including tree-level string theory. The resulting correlators exhibit analyticity and polynomial boundedness, allowing for the formulation of dispersion relations, which we explore. Finally, we discuss additional singularities of the gravitational energy correlators, absent in QCD, that originate from the long-range nature of the gravitational interactions.

Figures (9)

• Figure 1: We consider an initial state of two gravitons colliding in the center-of-mass frame with total energy $2E$. We then compute, perturbatively in $(\kappa E)$, where $\kappa^2 = 32 \pi G_N$, the angular distribution of the radiation in the final state, as measured by calorimeters (blue) placed at null infinity and labeled by points on the celestial sphere. This figure is a slight modification of Figure 4 in Kologlu:2019mfz, adapted to the initial state studied here.
• Figure 2: Geometry of the two-point energy correlator. In orange, we plot the direction of the beam. In blue, the location of the calorimeters at $\vec{n}_i$ and $\vec{n}_2$, and the angle between them, $\vec{n}_1 \cdot \vec{n}_2 = \cos \theta$, $0\le \theta \le \pi$. The angle between the planes spanned by $(\vec{n}_1, \vec{n}_2)$ and $(\vec{n}, \vec{n}_1)$ is denoted by $0\le \beta \le \pi$. We also introduce the variables $y_i = (1 - \vec{n} \cdot \vec{n}_i)/ 2$ and $z=(1 - \vec{n}_1 \cdot \vec{n}_2)/2$.
• Figure 3: The leading contribution to the beam-averaged EEC in ${\cal N}=8$ supergravity, see \ref{['eq:introform']}. Its overall shape is reminiscent of the familiar gauge-theory result. In gravity, however, the singular behavior as $z\to 0$ is tied to the forward-scattering singularity (as opposed to the gauge-theory collinear divergences). In the back-to-back limit $z\to 1$, the EEC is controlled by soft radiation.
• Figure 4: Kinematics of the two-to-two scattering in the center-of-mass frame. In a two-particle final state, the only way to obtain a nontrivial correlator between the particles is to place the detectors in the collinear or back-to-back configurations, see \ref{['conterm']}.
• Figure 5: Comparison of energy correlators in maximal supergravity and in pure gravity, for two-graviton pure initial states of polarizations $(+,+)$ and $(+,-)$. The angle variables are defined in (\ref{['ys']}) and $y \equiv y_1$. Left: the NLO correction to the EC. The functions are symmetric under $y \to 1 - y$ and exhibit different asymptotic behavior near the endpoints. Note that the presence of negative values does not contradict the positivity of the EC, see (\ref{['eq:ECpertexp']}). Right: the NLO correction to the beam-averaged EEC, see (\ref{['eq:avEECdef']}). All three functions take positive values for $0<z<1$, grow as $O(1/z)$ at small $z$ and display a universal behavior for $z\to 1$, see (\ref{['EECbar']}) and (\ref{['b2b-gr']}).