Energy correlators in four-dimensional gravity

Technical Summary: Energy Correlators in Four-Dimensional Gravity

Problem and Motivation
Standard scattering theory in four-dimensional quantum gravity faces a fundamental obstruction: the presence of long-range gravitational forces renders the S-matrix between plane-wave states ill-defined due to infrared (IR) divergences. While asymptotic symmetries (BMS) and dressed S-matrix formalisms address these issues theoretically, practical collider-like observables remain challenging to define. This paper investigates energy correlators as a class of IR-finite observables in four-dimensional gravitational theories. Unlike inclusive cross-sections which diverge due to the infinite total cross-section of plane-wave scattering, energy correlators measure the energy flux detected on the celestial sphere, weighted by the energies of the detected particles. The authors aim to compute these correlators perturbatively, demonstrating their IR finiteness, verifying conservation laws, and exploring their analytic structure and asymptotic behaviors.

Methodology
The study focuses on the scattering of two gravitons in the center-of-mass frame ($p_1 + p_2 \to q_1 + q_2 + X$), where $X$ represents unobserved radiation. The analysis is conducted in two primary theories:

1. $\mathcal{N}=8$ Supergravity (SG): Treated as a maximally supersymmetric theory and the low-energy limit of Type II string theory compactified on $T^6$.
2. Pure Einstein Gravity: Without matter.

The methodology involves a perturbative expansion in the gravitational coupling $\kappa$ (where $\kappa^2 = 32\pi G_N$). The authors compute the one-point ($\langle E(\vec{n}_1) \rangle$) and two-point ($\langle E(\vec{n}_1)E(\vec{n}_2) \rangle$) energy correlators at the first non-trivial order (Next-to-Leading Order, NLO).

• Virtual Corrections: Derived from the one-loop four-point amplitude squared.
• Real Corrections: Derived from the tree-level five-point amplitude squared, integrated over the phase space of undetected soft radiation.
• Regularization: Dimensional regularization ($d = 4 - 2\epsilon$) is employed to handle IR divergences, which are shown to cancel between virtual and real contributions.
• Kinematics: The correlators are expressed in terms of angular variables $y_i$ (angle between beam and detector) and $z$ (angle between detectors). The analysis specifically examines the collinear limit ($z \to 0$) and the back-to-back limit ($z \to 1$).
• Stringy Corrections: For $\mathcal{N}=8$ SG, leading stringy corrections are computed using the Kawai-Lewellen-Tye (KLT) relations to express closed-string amplitudes in terms of open-string disk amplitudes, expanded in powers of the Regge slope $\alpha'$.

Key Contributions and Results

1. Infrared Finiteness and Contact Terms:
   The authors explicitly demonstrate that while virtual and real contributions to the energy correlators are individually IR divergent, their sum is IR finite. A crucial aspect of this calculation is the treatment of contact terms localized at $z=0$ (collinear) and $z=1$ (back-to-back). These terms arise from the interplay between virtual corrections and real emissions. The paper shows that these contact terms are essential for the consistency of the perturbative expansion and are themselves IR finite.

2. Ward Identities (Sum Rules):
   The study formulates and explicitly verifies sum rules (Ward identities) associated with energy and momentum conservation. The authors show that the validity of these identities, specifically:
   $$\int d\Omega_{\vec{n}_2} \text{EEC} = 2E \cdot \text{EC}, \quad \int d\Omega_{\vec{n}_2} \vec{n}_2 \cdot \text{EEC} = 0$$
   crucially depends on the inclusion of the calculated contact terms. Without these terms, the conservation laws would be violated at the NLO level.

3. Explicit NLO Results:

   • One-Point Correlator: The NLO correction to the one-point energy correlator in $\mathcal{N}=8$ SG is computed, exhibiting uniform transcendental weight two and symmetry under the exchange of incoming particles.
   • Two-Point Correlator (EEC): The NLO EEC is decomposed into a regular part and contact terms. The regular part is non-zero for $0 < z < 1$. The contact terms at $z=0$ and $z=1$ are computed explicitly, showing the cancellation of IR poles.
   • Beam-Averaged EEC: A simplified observable, averaged over the beam direction, is introduced. The NLO result for this observable in $\mathcal{N}=8$ SG is presented as a function of the angle between detectors, exhibiting positivity, analyticity, and polynomial boundedness.

4. Stringy Corrections:
   The leading stringy corrections to the beam-averaged EEC are computed for $0 < z < 1$. The expansion in $\alpha'$ reveals that the leading terms involve products of odd zeta values ($\zeta_{2n+1}$). In the collinear and back-to-back limits, the stringy corrections are found to be softer than the supergravity contributions, with the leading asymptotics remaining governed by the supergravity result.

5. Back-to-Back Asymptotics and Soft Graviton Resummation:
   In the back-to-back limit ($z \to 1$), the behavior of the EEC is governed by soft graviton radiation. Using the eikonal approximation and the properties of soft theorems, the authors derive an all-order expression for the EEC:
   $$\langle E(\vec{n}_1)E(\vec{n}_2) \rangle \sim \frac{C(y_1, \beta)}{(1-z)^{1 - B_{gr}(y_1, E)/2}}$$
   where $B_{gr}$ is the gravitational Bremsstrahlung function. This result demonstrates the exponentiation of logarithmically enhanced terms generated by soft gravitons. The function governing the singularity is identified as the lightlike gravitational cusp anomalous dimension.

6. Analyticity and Dispersion Relations:
   The beam-averaged EEC is shown to be an analytic function in the complex $z$-plane with branch points at $z=0$ and $z=1$. It satisfies polynomial boundedness, allowing for the formulation of dispersion relations. The authors derive these relations and use them to establish positivity constraints on the Taylor coefficients of the correlator and its multipole expansion.

7. Pure Gravity vs. Supergravity:
   While pure gravity results share the same qualitative features (IR finiteness, universal back-to-back behavior), the analytic expressions are more complex, lacking uniform transcendental weight and depending on the initial-state helicity configuration.

Significance and Claims
The paper claims to substantially extend previous studies on gravitational energy correlators (specifically Refs. [19, 21]) by:

• Shifting the physical setup from massive scalar scattering to two-griton scattering in four dimensions.
• Computing both one- and two-point correlators at NLO in two distinct gravitational theories ($\mathcal{N}=8$ SG and pure gravity).
• Explicitly calculating contact terms at $z=0$ and $z=1$, which are shown to be necessary for IR finiteness and the satisfaction of Ward identities.
• Introducing and computing the beam-averaged EEC, a new observable not previously analyzed in this context.
• Establishing the analytic structure (analyticity, polynomial boundedness) of the correlators and deriving dispersion relations and positivity constraints.
• Deriving the all-order back-to-back asymptotics driven by soft-graviton dynamics, including the exponentiation of logarithmic terms.
• Computing the leading stringy corrections and discussing the expected behavior in the high-energy limit where black hole formation may dominate.

The authors emphasize that the contact terms are not minor details but essential components for the consistency of the perturbative expansion and the definition of IR-finite observables in gravity. The work provides a concrete framework for studying quantum gravitational effects using collider-style observables, bridging the gap between S-matrix bootstrap methods and standard perturbative calculations.