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Multi-Regge Limit of the Two-Loop Five-Point Amplitudes in $\mathcal{N} = 4$ Super Yang-Mills and $\mathcal{N} = 8$ Supergravity

Simon Caron-Huot, Dmitry Chicherin, Johannes Henn, Yang Zhang, Simone Zoia

TL;DR

The paper advances from symbol-level to full functional results for the two-loop five-point amplitudes in ${\mathcal{N}}=4$ sYM and ${\mathcal{N}}=8$ supergravity, constructing explicit expressions from pure master integrals organized into three families and validating infrared factorisation through hard functions. It then analyzes the multi-Regge limit, where the pentagon alphabet collapses to a compact set, expressing results in terms of a small basis of single- and multi-valued polylogarithms, and provides both gauge-theory and gravity MRK predictions. A key outcome is the independent BFKL-based confirmation for certain non-planar color structures in ${\mathcal{N}}=4$ sYM, demonstrating consistency between perturbative amplitudes and Regge-theory expectations. The work also highlights subtle analytic features near hypersurfaces like $\epsilon_5=0$ and reveals a richer MRK structure in gravity, including power-suppressed terms and non-uniform transcendental weight.

Abstract

In previous work, the two-loop five-point amplitudes in $\mathcal{N}=4$ super Yang-Mills theory and $\mathcal{N}=8$ supergravity were computed at symbol level. In this paper, we compute the full functional form. The amplitudes are assembled and simplified using the analytic expressions of the two-loop pentagon integrals in the physical scattering region. We provide the explicit functional expressions, and a numerical reference point in the scattering region. We then calculate the multi-Regge limit of both amplitudes. The result is written in terms of an explicit transcendental function basis. For certain non-planar colour structures of the $\mathcal{N}=4$ super Yang-Mills amplitude, we perform an independent calculation based on the BFKL effective theory. We find perfect agreement. We comment on the analytic properties of the amplitudes.

Multi-Regge Limit of the Two-Loop Five-Point Amplitudes in $\mathcal{N} = 4$ Super Yang-Mills and $\mathcal{N} = 8$ Supergravity

TL;DR

The paper advances from symbol-level to full functional results for the two-loop five-point amplitudes in sYM and supergravity, constructing explicit expressions from pure master integrals organized into three families and validating infrared factorisation through hard functions. It then analyzes the multi-Regge limit, where the pentagon alphabet collapses to a compact set, expressing results in terms of a small basis of single- and multi-valued polylogarithms, and provides both gauge-theory and gravity MRK predictions. A key outcome is the independent BFKL-based confirmation for certain non-planar color structures in sYM, demonstrating consistency between perturbative amplitudes and Regge-theory expectations. The work also highlights subtle analytic features near hypersurfaces like and reveals a richer MRK structure in gravity, including power-suppressed terms and non-uniform transcendental weight.

Abstract

In previous work, the two-loop five-point amplitudes in super Yang-Mills theory and supergravity were computed at symbol level. In this paper, we compute the full functional form. The amplitudes are assembled and simplified using the analytic expressions of the two-loop pentagon integrals in the physical scattering region. We provide the explicit functional expressions, and a numerical reference point in the scattering region. We then calculate the multi-Regge limit of both amplitudes. The result is written in terms of an explicit transcendental function basis. For certain non-planar colour structures of the super Yang-Mills amplitude, we perform an independent calculation based on the BFKL effective theory. We find perfect agreement. We comment on the analytic properties of the amplitudes.

Paper Structure

This paper contains 30 sections, 183 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Kinematics and pentagon functions
  3. The two-loop five-point $\mathcal{N}=4$ sYM and $\mathcal{N}=8$ supergravity amplitudes
  4. Expected structure of the two-loop amplitudes
  5. Two-loop integrands
  6. Pure integral bases
  7. Permutation of the external legs and integrated expressions
  8. Infrared factorisation and hard functions
  9. $\mathcal{N}=4$ super Yang-Mills
  10. $\mathcal{N}=8$ supergravity
  11. The two-loop hard functions
  12. Multi-Regge limit and pentagon functions
  13. Multi-Regge kinematics
  14. Multi-Regge limit of the pure integrals
  15. Feynman integrals with non-trivial analytic properties
  16. ...and 15 more sections

Figures (4)

  • Figure 1: Diagrams in the representation of the integrands of the two-loop five-point amplitudes given by Ref. Carrasco:2011mn.
  • Figure 2: Pictorial representation of the multi-Regge kinematics in the $s_{12}$ channel.
  • Figure 3: Examples of Feynman integrals to illustrate the analytic properties near the hypersurface $\epsilon_5=0$. The scalar integrals depicted in this figure are multiplied by a factor of $\epsilon_5$. As a consequence, they have uniform transcendental weight and odd parity. While the integral (a) does not vanish on the hypersurface $\epsilon_5=0$ approached from within the physical $s_{12}$ scattering region, the integral (b) does.
  • Figure 4: Family of reggeon diagrams in which a maximal number of reggeons (horizontal lines) is exchanged in both the $t_1$ and $t_2$ channels. We also include all permutations of the insertions along the vertical (eikonal) lines.