Scattering Amplitudes and Simple Canonical Forms for Simple Polytopes

Abstract

We provide an efficient recursive formula to compute the canonical forms of arbitrary d-dimensional simple polytopes, which are convex polytopes such that every vertex lies precisely on d facets. For illustration purposes, we explicitly derive recursive formulae for the canonical forms of Stokes polytopes, which play a similar role for a theory with quartic interaction as the Associahedron does in planar bi-adjoint φ 3 theory. As a by-product, our formula also suggests a new way to obtain the full planar amplitude in φ 4 theory by taking suitable limits of the canonical forms of constituent Stokes polytopes.

Double Soft Theorem for Generalised Biadjoint Scalar Amplitudes

“…However, recent interest in this direction came from the twistor formulation of the N = 4 super-Yang-Mills theory [4,5], leading to the BCFW formulation of the scattering amplitudes [6][7][8]. The representation of the N = 4 super-Yang-Mills theory in terms of Grassmannians [9][10][11] as well as polytope realisation of the scattering amplitudes [9,[12][13][14][15][16][17][18][19][20][21][22][23][24], and the Cachazo-He-Yuan(CHY) formulation of the scattering amplitudes [25][26][27][28][29][30] gave further impetus to unraveling their underlying structure.…”

Positive Geometries for all Scalar Theories from Twisted Intersection Theory

“…To be precise, these surfaces are not known to be tiled by any regular polytope, making the analysis somewhat tricky. Some progress has been reported at genus one, for which the reader can consult [13,14,36].…”

“…This can be seen from the formula for intersection numbers, which was first used in the context of scattering amplitudes in [2]. In our case, we are interested in the self intersection number of ϕ (13,36) , for which it is sufficient to note that the intersection number is localized on the vertices of the accordiohedron. Schematically, for a given accordiohedron of dimension n, if the vertices are labelled by V I , the self intersection number of the corresponding form would be given by,…”

“…We will focus on the reference dissection (13,14), which gives rise to a pentagon. The accordiohedron of this reference is labelled by the vertices {(13, 14), (24,14), (24,26), (26,36), (13,36)}. The facets may be read off from the set of vertices; they are {( 13), (36), ( 26), ( 24), ( 14)}.…”

One-loop integrand from generalised scattering equations

“…Moduli space of open string worldsheet is an associahedron, and the scattering equations of CHY act as diffeomorphism between the associahedron in the worldsheet and that described in the kinematic space. This led to a fascinating series of investigations into the connection between scattering amplitudes and positive geometries for various scalar theories [61][62][63][64][65][66][67][68][69][70][71][72] . Stringy deformations of the scattering forms have been considered in [73].…”