Abstract
In recent years, it has been understood that color-ordered scattering amplitudes can be encoded as logarithmic differential forms on positive geometries. In particular, amplitudes in maximally supersymmetric Yang-Mills theory in spinor helicity space are governed by the momentum amplituhedron. Due to the group-theoretic structure underlying color decompositions, color-ordered amplitudes enjoy various identities which relate different orderings. In this paper, we show how the Kleiss-Kuijf relations arise from the geometry of the momentum amplituhedron. We also show how similar relations can be realised for the kinematic associahedron, which is the positive geometry of bi-adjoint scalar cubic theory.
Highlights
Where the sum is over the (n − 1)! non-cyclic permutations of particle labels for which particle 1 has been fixed to the first position
We show how similar relations can be realised for the kinematic associahedron, which is the positive geometry of bi-adjoint scalar cubic theory
As we will show in this paper, the answer is affirmative for tree-level amplitudes in N = 4 supersymmetric Yang-Mills (sYM), as well as for those in bi-adjoint scalar cubic theory, where positive geometries provide a beautiful geometrical realization of the KK relations!
Summary
We give more details on how the color structure of SU(N ) gauge amplitudes is organised; we recall the definition of color-ordered amplitudes and the relations between them. The objects Atnree are called color-ordered or partial amplitudes and carry only kinematic information, since the color dependence has been stripped off. They receive contributions only from planar diagrams in a particular ordering and have singularities only when the sum of adjacent momenta in this ordering go on-shell. {βT } denotes the reverse ordering of the labels {β}, nβ is the number of elements in {β} and {α} {βT } denotes the set of all shuffles of {α} with {βT }, i.e. the set of permutations on {α} ∪ {β} preserving the ordering within {α} and {βT } These relations can be used to put any two legs next to each other, these being 1 and n in (2.3). Both the reflection symmetry relations and the U(1) decoupling identities are particular cases of the KK relations
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