TY - JOUR

T1 - A note on polytopes for scattering amplitudes

AU - Arkani-Hamed, N.

AU - Bourjaily, J.

AU - Cachazo, F.

AU - Hodges, A.

AU - Trnka, J.

PY - 2012/5/2

Y1 - 2012/5/2

N2 - In this note we continue the exploration of the polytope picture for scattering amplitudes, where amplitudes are associated with the volumes of polytopes in generalized momentum-twistor spaces. After a quick warm-up example illustrating the essential ideas with the elementary geometry of polygons in CP 2, we interpret the 1-loop MHV integrand as the volume of a polytope in CP 3×CP 3, which can be thought of as the space obtained by taking the geometric dual of the Wilson loop in each CP 3 of the product. We then review the polytope picture for the NMHV tree amplitude and give it a more direct and intrinsic definition as the geometric dual of a canonical \square of the Wilson-Loop polygon, living in a certain extension of momentum-twistor space into CP 4. In both cases, one natural class of triangulations of the polytope produces the BCFW/CSW representations of the amplitudes; another class of triangulations leads to a striking new form, which is both remarkably simple as well as manifestly cyclic and local.

AB - In this note we continue the exploration of the polytope picture for scattering amplitudes, where amplitudes are associated with the volumes of polytopes in generalized momentum-twistor spaces. After a quick warm-up example illustrating the essential ideas with the elementary geometry of polygons in CP 2, we interpret the 1-loop MHV integrand as the volume of a polytope in CP 3×CP 3, which can be thought of as the space obtained by taking the geometric dual of the Wilson loop in each CP 3 of the product. We then review the polytope picture for the NMHV tree amplitude and give it a more direct and intrinsic definition as the geometric dual of a canonical \square of the Wilson-Loop polygon, living in a certain extension of momentum-twistor space into CP 4. In both cases, one natural class of triangulations of the polytope produces the BCFW/CSW representations of the amplitudes; another class of triangulations leads to a striking new form, which is both remarkably simple as well as manifestly cyclic and local.

UR - http://www.scopus.com/inward/record.url?scp=84860302886&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84860302886&partnerID=8YFLogxK

U2 - 10.1007/JHEP04(2012)081

DO - 10.1007/JHEP04(2012)081

M3 - Article

AN - SCOPUS:84860302886

VL - 2012

JO - Journal of High Energy Physics

JF - Journal of High Energy Physics

SN - 1126-6708

IS - 4

M1 - 081

ER -