Gravitational Regge bounds

TL;DR

This work derives Regge growth bounds for gravitational 2→2 scattering by extending standard nonperturbative QFT assumptions to gravity and incorporating known large-impact-parameter physics. By combining unitarity, analyticity, subexponentiality, crossing, and gravitational clustering, the authors show that gravitational amplitudes admit dispersion relations with two subtractions (2SDR) for fixed t<0, and that a smeared class of amplitudes can reduce to one subtraction in black-hole–dominated regimes. They develop a detailed partial-wave framework, including smeared partial waves, and analyze pointwise and smeared Regge bounds using Born and Eikonal models, with black hole formation tightening bounds further; they also establish local growth bounds akin to a gravitational CRG bound and discuss implications for AdS/CFT and potential extensions to d=4. The results provide a nonperturbative constraint on the high-energy behavior of gravity and connect dispersion-relation techniques to semi-classical gravity, offering tools for constraining EFTs coupled to gravity. Overall, the paper furnishes a coherent dispersive framework for gravitational scattering with concrete bounds on both global and local growth, highlighting where 2SDR are mandatory and where BH physics can reduce subtractions.

Abstract

We review the basic assumptions and spell out the detailed arguments that lead to the bound on the Regge growth of gravitational scattering amplitudes. The minimal extra ingredient compared to the gapped case - in addition to unitarity, analyticity, subexponentiality, and crossing - is the assumption that scattering at large impact parameters is controlled by known semi-classical physics. We bound the Regge growth of amplitudes both with the fixed transferred momentum and smeared over it. Our basic conclusion is that gravitational scattering amplitudes admit dispersion relations with two subtractions. For a sub-class of smeared amplitudes, black hole formation reduces the number of subtractions to one. Finally, using dispersion relations with two subtractions we derive bounds on the local growth of relativistic scattering amplitudes. Schematically, the local bound states that the amplitude cannot grow faster than $s^2$.

Figures (9)

• Figure 1: Different energy scales in the impact parameter space $b$. The scales are arranged as they appear in the large $s$ limit in $d \geq 7$. The Born approximation to the gravitational interaction is valid up to impact parameters $b \gtrsim s^{{1 \over d-4}}$. For impact parameters $s^{{1 \over d-2}} \lesssim b \lesssim s^{{1 \over d-4}}$ gravitational interaction is controlled by an eikonalized Born result. Moreover in the Regge limit, the leading contribution to the scattering amplitude comes from impact parameters $\sim s^{{1 \over d-3}}$, see e.g. Muzinich:1987in. For $b \lesssim s^{{1 \over d-2}}$ tidal effects become important. Similarly, for $b \lesssim s^{{2 \over 3d-10}}$ emission of gravitational waves becomes relevant. Finally, for impact parameters less than the corresponding Schwarzschild radius $b \lesssim s^{{1 \over 2(d-3)}}$ scattering is dominated by the black hole formation. For $d \geq 7$ the leading inelastic effect is due to tidal excitations; for $d=6$ tidal exciatations and gravity wave emission scale with $s$ in the same way; finally, for $d\leq 5$ the leading inelastic effect at large energies is due to the emission of gravity waves.
• Figure 2: The upper half-plane in the complex $s$ variable. In this paper we mostly concern ourselves with bounding the scattering amplitude $T(s,t)$ along the real axes (regions (a) and (b)). We then combine this with analyticity in the upper half-plane together with the subexponentiality assumption in region (c) to derive the Regge bounds on the amplitude everywhere in the upper half-plane.
• Figure 3: Every elastic amplitude in a gravitational theory is singular at $t=0$ due to universal, long-range nature of gravity.
• Figure 4: The classical picture of scattering at fixed impact parameter $b$.
• Figure 5: Regions of validity of the bound \ref{['eq:localboundgravity']} for the functionals \ref{['eq:funclocal4']} and \ref{['eq:funclocal5']} in the $s$-complex plane, where $s_0=2m^2 + x$. For the detailed plots and explanations of the underlying procedure, see \ref{['app:smearedboundonscattering']}. The units are fixed by choosing $M_{\text{gap}}=1$ and we chose to smear up to $q_0=1$.