The Imaginary Part of the N = 4 Super-Yang-Mills Two-Loop Six-Point MHV Amplitude in Multi-Regge Kinematics

TL;DR

This work investigates whether the two-loop six-point MHV amplitude in planar ${\cal N}=4$ SYM violates the BDS ansatz in multi-Regge kinematics by examining the imaginary part of the amplitude. Using the amplitude/Wilson-loop correspondence, the authors compute the imaginary part via analytic continuation of the hexagon Wilson loop and compare it to the leading-log predictions of Bartels, Lipatov, and Sabio Vera (BLSV), finding solid agreement. They parameterize a principal MRK slice with variables $(a,b,c)$ to obtain dual conformal cross-ratios $u_1=c^2$, $u_2=a^2$, $u_3=b^2$, and perform a careful continuation of 84 integrals, validating the leading-log structure ${\rm Im}\{R_{6;2}\} = \frac{\pi}{2} \ln(1-u_1) \ln\left(\frac{1-u_1}{u_1 u_2}\right) \ln\left(\frac{1-u_1}{u_1 u_3}\right) + \cdots$. The results support the BLSV picture, show sub-leading terms depend mainly on the ratio $\xi=u_2/u_3$, and address critiques by DDG, reinforcing the reliability of the amplitude/Wilson-loop approach for MRK analyses and the understanding of BDS violation at two loops. Overall, the study provides numerical benchmarks and strengthens the interpretation of remainder-function dynamics in MRK for ${\cal N}=4$ SYM.

Abstract

The precise form of the multi-Regge asymptotics of the two-loop six-point MHV amplitude in N = 4 Super-Yang-Mills theory has been a subject of recent controversy. In this paper we utilize the amplitude/Wilson loop correspondence to obtain precise numerical results for the imaginary part of these asymptotics. The region of phase-space that we consider is interesting because it allowed Bartels, Lipatov, and Sabio Vera to determine that the two-loop six-point MHV amplitude is not fixed by the BDS ansatz. They proceeded by working in the framework of a high energy effective action, thus side-stepping the need for an arduous two-loop calculation. Our numerical results are consistent with the predictions of Bartels, Lipatov, and Sabio Vera for the leading-log asymptotics.