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A tale of two exponentiations in ${\cal N}=8$ supergravity

Paolo Di Vecchia, Andrés Luna, Stephen G. Naculich, Rodolfo Russo, Gabriele Veneziano, Chris D. White

TL;DR

This paper studies the four-graviton scattering amplitude in ${\cal N}=8$ supergravity, focusing on the Regge limit to uncover the structure of the finite remainder ${\cal F}_4$ after IR exponentiation. By working in impact-parameter space, the authors connect the eikonal phase to a convolution in momentum space, enabling an all-orders, ${\cal O}(\epsilon^0)$ resummation of the Regge-limit remainder and revealing a compact dependence on odd zeta values. The results not only corroborate the three-loop findings but also provide predictions for higher-loop terms and a consistent transcendental structure, offering nontrivial consistency checks for future gravity calculations. The work also outlines potential extensions to subleading eikonal effects and non-maximal supersymmetry scenarios.

Abstract

The structure of scattering amplitudes in supergravity theories continues to be of interest. Recently, the amplitude for $2\rightarrow 2$ scattering in ${\cal N}=8$ supergravity was presented at three-loop order for the first time. The result can be written in terms of an exponentiated one-loop contribution, modulo a remainder function which is free of infrared singularities, but contains leading terms in the high energy Regge limit. We explain the origin of these terms from a well-known, unitarity-restoring exponentiation of the high-energy gravitational $S$-matrix in impact-parameter space. Furthermore, we predict the existence of similar terms in the remainder function at all higher loop orders. Our results provide a non-trivial cross-check of the recent three-loop calculation, and a necessary consistency constraint for any future calculation at higher loops.

A tale of two exponentiations in ${\cal N}=8$ supergravity

TL;DR

This paper studies the four-graviton scattering amplitude in supergravity, focusing on the Regge limit to uncover the structure of the finite remainder after IR exponentiation. By working in impact-parameter space, the authors connect the eikonal phase to a convolution in momentum space, enabling an all-orders, resummation of the Regge-limit remainder and revealing a compact dependence on odd zeta values. The results not only corroborate the three-loop findings but also provide predictions for higher-loop terms and a consistent transcendental structure, offering nontrivial consistency checks for future gravity calculations. The work also outlines potential extensions to subleading eikonal effects and non-maximal supersymmetry scenarios.

Abstract

The structure of scattering amplitudes in supergravity theories continues to be of interest. Recently, the amplitude for scattering in supergravity was presented at three-loop order for the first time. The result can be written in terms of an exponentiated one-loop contribution, modulo a remainder function which is free of infrared singularities, but contains leading terms in the high energy Regge limit. We explain the origin of these terms from a well-known, unitarity-restoring exponentiation of the high-energy gravitational -matrix in impact-parameter space. Furthermore, we predict the existence of similar terms in the remainder function at all higher loop orders. Our results provide a non-trivial cross-check of the recent three-loop calculation, and a necessary consistency constraint for any future calculation at higher loops.

Paper Structure

This paper contains 7 sections, 31 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Review of previous results
  3. The remainder function up to three-loop order
  4. Impact-parameter exponentiation and the eikonal phase
  5. The three-loop remainder function in the Regge limit
  6. The remainder function to all orders in the Regge limit
  7. Discussion

Figures (2)

  • Figure 1: Labelling of 4-momenta for the four-graviton scattering process.
  • Figure 2: A representative (crossed) ladder graph, where all particles are gravitons. The sum of all such diagrams in the leading Regge limit builds up the exponentiated amplitude of eq. (\ref{['Meik']}).