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The two-loop five-particle amplitude in $\mathcal{N}=8$ supergravity

D. Chicherin, T. Gehrmann, J. M. Henn, P. Wasser, Y. Zhang, S. Zoia

TL;DR

This work computes the first symbol-level two-loop five-point amplitude in ${\mathcal N}=8$ supergravity, expressing the IR-finite hard function as a uniform-weight weight-4 symbol built from pentagon-function letters and a small set of rational prefactors. By derived integration-by-parts reductions, uniform-transcendental-weight bases, and canonical differential equations, the authors reveal a remarkably compact representation: a single seed function summed over all permutations, with the full result exhibiting correct soft, collinear, and multi-Regge limits. The analysis confirms exponentiation of soft divergences and provides detailed symbol-level structure that will aid future explorations of gravity amplitudes, including their Regge behavior and subleading soft theorems. Ancillary files supply the full symbol of the IR-subtracted hard function and seed data for reproducibility and further study.

Abstract

We compute for the first time the two-loop five-particle amplitude in $\mathcal{N}=8$ supergravity. Starting from the known integrand, we perform an integration-by-parts reduction and express the answer in terms of uniform weight master integrals. The latter are known to evaluate to non-planar pentagon functions, described by a 31-letter symbol alphabet. We express the final result for the amplitude in terms of uniform weight four symbols, multiplied by a small set of rational factors. The amplitude satisfies the expected factorization properties when one external graviton becomes soft, and when two external gravitons become collinear. We verify that the soft divergences of the amplitude exponentiate, and extract the finite remainder function. The latter depends on fewer rational factors, and is independent of one of the symbol letters. By analyzing identities involving rational factors and symbols we find a remarkably compact representation in terms of a single seed function, summed over all permutations of external particles. Finally, we work out the multi-Regge limit, and present explicitly the leading logarithmic terms in the limit. The full symbol of the IR-subtracted hard function is provided as an ancillary file.

The two-loop five-particle amplitude in $\mathcal{N}=8$ supergravity

TL;DR

This work computes the first symbol-level two-loop five-point amplitude in supergravity, expressing the IR-finite hard function as a uniform-weight weight-4 symbol built from pentagon-function letters and a small set of rational prefactors. By derived integration-by-parts reductions, uniform-transcendental-weight bases, and canonical differential equations, the authors reveal a remarkably compact representation: a single seed function summed over all permutations, with the full result exhibiting correct soft, collinear, and multi-Regge limits. The analysis confirms exponentiation of soft divergences and provides detailed symbol-level structure that will aid future explorations of gravity amplitudes, including their Regge behavior and subleading soft theorems. Ancillary files supply the full symbol of the IR-subtracted hard function and seed data for reproducibility and further study.

Abstract

We compute for the first time the two-loop five-particle amplitude in supergravity. Starting from the known integrand, we perform an integration-by-parts reduction and express the answer in terms of uniform weight master integrals. The latter are known to evaluate to non-planar pentagon functions, described by a 31-letter symbol alphabet. We express the final result for the amplitude in terms of uniform weight four symbols, multiplied by a small set of rational factors. The amplitude satisfies the expected factorization properties when one external graviton becomes soft, and when two external gravitons become collinear. We verify that the soft divergences of the amplitude exponentiate, and extract the finite remainder function. The latter depends on fewer rational factors, and is independent of one of the symbol letters. By analyzing identities involving rational factors and symbols we find a remarkably compact representation in terms of a single seed function, summed over all permutations of external particles. Finally, we work out the multi-Regge limit, and present explicitly the leading logarithmic terms in the limit. The full symbol of the IR-subtracted hard function is provided as an ancillary file.

Paper Structure

This paper contains 16 sections, 51 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Kinematics and pentagon functions
  3. Five-graviton scattering amplitudes: tree-level and one-loop cases
  4. Structure of infrared divergences and hard function
  5. Calculation of the two-loop five-graviton amplitude
  6. Expected structure of the result
  7. Two-loop integrand
  8. Uniform transcendental weight integral basis
  9. Integration-by-parts reduction
  10. Test of expected form of the result
  11. Main result for the remainder function at two loops
  12. Limits of the amplitude
  13. Soft limit
  14. Collinear limit
  15. Multi-Regge limit
  16. ...and 1 more sections

Figures (1)

  • Figure 1: The six Feynmal integral topologies used to define the integrand of the two-loop five-point amplitude in $\mathcal{N}=8$ supergravity amplitude Carrasco:2011mn.