Pdf:R73_HU

Seven-particle 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 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
three-loop seven-particle MHV amplitude. The simplicity of its construction
suggests that the n-gon bootstrap may be surprisingly powerful for n > 6.