Lifting Heptagon Symbols to Functions

A Symbol of Uniqueness: The Cluster Bootstrap for the 3-Loop MHV Heptagon

Seven-particle scattering amplitudes in planar super-Yang-Mills theory are believed to belong to a special class of generalised polylogarithm functions called heptagon functions. These are functions with physical branch cuts whose symbols may be written in terms of the 42 cluster A-coordinates on Gr(4,7). Motivated by the success of the hexagon bootstrap programme for constructing six-particle amplitudes we initiate the systematic study of the symbols of heptagon functions. We find that there is exactly one such symbol of weight six which satisfies the MHV last-entry condition and is finite in…

七边形符号在五重循环中的应用及所有循环序列

Heptagon Symbols at Five Loops and All-Loop Sequences

本研究通过E6聚类代数和物理约束，发现七边形符号在五重循环中存在唯一解，并预测所有循环序列。

Cluster adjacency and the four-loop NMHV heptagon

We exploit the recently described property of cluster adjacency for scattering amplitudes in planar $\mathcal{N}=4$ super Yang-Mills theory to construct the symbol of the four-loop NMHV heptagon amplitude. We use a manifestly cluster adjacent ansatz and describe how the parameters of this ansatz are determined using simple physical consistency requirements. We then specialise our answer for the amplitude to the multi-Regge limit, finding agreement with previously available results up to the next-to-leading logarithm, and obtaining new predictions up to (next-to)$^3$-leading-logarithmic accurac…

Heptagons from the Steinmann Cluster Bootstrap

We reformulate the heptagon cluster bootstrap to take advantage of the Steinmann relations, which require certain double discontinuities of any amplitude to vanish. These constraints vastly reduce the number of functions needed to bootstrap seven-point amplitudes in planar $\mathcal{N} = 4$ supersymmetric Yang-Mills theory, making higher-loop contributions to these amplitudes more computationally accessible. In particular, dual superconformal symmetry and well-defined collinear limits suffice to determine uniquely the symbols of the three-loop NMHV and four-loop MHV seven-point amplitudes. We …

七点金字形神塔图的第一类Landau奇异性 I

Landau Singularities of the 7-Point Ziggurat I

本文计算了与七点金字形神塔图相关的四圈七点图的第一类Landau奇异性，发现其与平面N=4超杨-米尔斯理论中的七边形符号字母表子集完全一致，其余字母留待后续论文处理。

Cluster adjacency properties of scattering amplitudes

We conjecture a new set of analytic relations for scattering amplitudes in planar N=4 super Yang-Mills theory. They generalise the Steinmann relations and are expressed in terms of the cluster algebras associated to Gr(4,n). In terms of the symbol, they dictate which letters can appear consecutively. We study heptagon amplitudes and integrals in detail and present symbols for previously unknown integrals at two and three loops which support our conjecture.