Abstract
The amplituhedron provides a beautiful description of perturbative superamplitude integrands in mathcal{N}=4 SYM in terms of purely geometric objects, generalisations of polytopes. On the other hand the Wilson loop in supertwistor space also gives an explicit description of these superamplitudes as a sum of planar Feynman diagrams. Each Feynman diagram can be naturally associated with a geometrical object in the same space as the amplituhedron (although not uniquely). This suggests that these geometric images of the Feynman diagrams give a tessellation of the amplituhedron. This turns out to be the case for NMHV amplitudes. We argue however that beyond NMHV this is not true. Specifically, each Feynman diagram leads to an image with a physical boundary and spurious boundaries. The spurious ones should be “internal”, matching with neighbouring diagrams. We however show that there is no choice of geometric image of the Wilson loop Feynman diagrams which yields a geometric object without leaving unmatched spurious boundaries.
Highlights
Let us emphasise that we make no assumptions about positivity, or convexity or any particular specific form for this geometrical shape
Since each Wilson loop Feynman diagrams (WLDs) contains spurious poles which have a geometrical interpretation as spurious boundaries we ask if it is possible to choose these regions in such a way that all spurious boundaries locally glue together correctly pairwise with those of other diagrams so that the union of regions leaves no remaining unmatched spurious boundaries
If on the other hand we are given a canonical form, written as a sum of terms each containing spurious poles that cancel in the sum, the assigning of a geometrical region to each term can not be done independently for each term: the cancelling of spurious poles should correspond geometrically to a matching of the corresponding spurious boundaries
Summary
We provide a brief description of planar Wilson loops in N = 4 Super Yang Mills in super twistor space and define the WLDs that arise. The WLDs we are discussing here are the Feynman diagrams describing a polygonal holomorphic Wilson-loop in super twistor space with vertices being the super twistors Z1 . In the planar theory this is equivalent, via the Wilson loop/amplitude duality [36,37,38], to n-point superamplitudes. In the planar theory diagrams are only valid if we can draw all the propagators inside the Wilson loop without crossing. The NkMHV Wilson loop is the sum over all such diagrams involving k propagators (see figure 3 for an example of a diagram contributing to 8-point N4MHV). In figure 4 we illustrate these rules with two examples firstly an example diagram contributing to the NMHV six-point amplitude and secondly one contributing to the N2MHV six-point amplitude
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
