Abstract
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 $$ \mathcal{A} $$ -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 the 7 ∥ 6 collinear limit. This unique symbol is both dihedral and parity-symmetric, and remarkably its collinear limit is exactly the symbol of the three-loop six-particle MHV amplitude, although none of these properties were assumed a priori. It must therefore be the symbol of the threeloop seven-particle MHV amplitude. The simplicity of its construction suggests that the n-gon bootstrap may be surprisingly powerful for n > 6.
Highlights
The analytic structure of the S-matrices of general quantum field theories are notoriously complicated [2]
In the case of maximally helicity-violating (MHV) amplitudes, the relevant piece which is not fixed by dual conformal symmetry is called the remainder function
The relation to Wilson loops means that the remainder function must obey constraints on its discontinuities [26,27,28,29] and on the power-suppressed corrections [30,31,32,33,34,35,36] in the collinear limit coming from an operator product expansion (OPE) for light-like Wilson loops
Summary
Following the definition of hexagon functions given in [18], we define a heptagon function of weight k to be a polylogarithm function of weight k whose symbol may be written in the alphabet (2.12) and which is free of branch points in the Euclidean region. It can happen that a symbol which satisfies the first-entry condition and is well-defined in a collinear limit can be promoted to a function with physical branch cuts only by adding certain terms of lower weight, which may end up diverging in the collinear limit. An example of this phenomenon has already been seen at three loops in the MHV hexagon case [16, 18].
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