Higher point MHV amplitudes in N=4 Supersymmetric Yang-Mills Theory

TL;DR

This paper computes the even part of the two-loop seven-point planar MHV amplitude in $N=4$ SYM using a unitarity-based approach with a conformal integral basis, finding that it can be expressed with dual conformal integrals having simple rational coefficients and that no hexagon or higher-polygon integrals contribute. The authors also derive all-$n$ cuts to test the structure across arbitrary numbers of external legs and introduce a leg addition rule that relates some $n$-point coefficients to $(n+1)$-point coefficients, facilitating extrapolation to higher points. The results support a highly constrained integral basis and suggest a practical connection to Wilson-loop computations, while leaving open the complete basis problem for two-loop MHV amplitudes and the treatment of odd parts and $ ext{mu}$-terms. Overall, the work advances understanding of higher-point, two-loop MHV amplitudes and provides tools for extending the analysis to broader classes of amplitudes.

Abstract

We compute the even part of the two-loop seven-point planar MHV amplitude in N=4 supersymmetric Yang-Mills theory. We find that the even part is expressed in terms of conformal integrals with simple rational coefficients. We also compute the even part of two all-n cuts. An important feature of the result is that no hexagon (or higher polygon) loops appear among the integrals detected by the cuts we computed. We also present a "leg addition rule," which allows us to express some integral coefficients in the n+1-point MHV amplitude in terms of the integral coefficients of the n-point MHV amplitude.

Figures (6)

• Figure 1: The unitarity cuts used to compute the seven-point MHV amplitude.
• Figure 2: The integral topologies which can be made conformal, but appear with zero coefficient in the result for the seven-point MHV amplitude. Sometimes two or more external massless legs are attached at the same point and their sum is generically a massive momentum. These massive momenta are denoted by thick solid lines and the massless legs are denoted by thin solid lines. The dot marks the position of the dual variable $x_1$, while the rest of the dual variables are ordered clockwise.
• Figure 3: The integral topologies which can be made conformal and appear with nonzero coefficient in the result for the MHV amplitude. The notations for the integrals are the same as those in Fig. \ref{['fig:zero-7pt']}. For the integrals $15$ and $18$ the notation is ambiguous because one can distribute the external legs in several ways to form massive momenta. In those cases we take the convention that the massive leg which appears first when going clockwise starting at $x_1$ is a sum of two massless momenta, while the next massive leg is a sum of three massless momenta. The dual variable of the left (right) loop is $x_p$ ($x_q$).
• Figure 4: The two-loop $n$-points cuts used to constrain the $n$-point MHV amplitude. For the cut (b) we take $r$ such that $2 < r < n-2$.
• Figure 5: Two examples of applying the leg addition rule. The dotted line is an internal line, carrying loop momentum. When computing the coefficients, the sum of momenta marked by arrows before the transformation must be replaced with the sum of momenta after the transformation.