Deep Into the Amplituhedron: Amplitude Singularities at All Loops and Legs

Abstract

In this Letter we compute a canonical set of cuts of the integrand for maximally helicity violating amplitudes in planar N ¼ 4 supersymmetric Yang-Mills theory, where all internal propagators are put on shell. These "deepest cuts" probe the most complicated Feynman diagrams and on-shell processes that can possibly contribute to the amplitude, but are also naturally associated with remarkably simple geometric facets of the amplituhedron. The recent reformulation of the amplituhedron in terms of combinatorial geometry directly in the kinematic (momentum-twistor) space plays a crucial role in understanding this geometry and determining the cut. This provides us with the first nontrivial results on scattering amplitudes in the theory valid for arbitrarily many loops and external particle multiplicities.

Emergent unitarity from the amplituhedron

“…ii + 1i + 2i − 1 (−1) k−1 has k sign flips. Loop level (AB) a ii + 1 > 0 and the sequence This topological definition has already been used to investigate the structure of "deep" cuts to all loop orders in [14,15]. It is well known that the branch cut structure of amplitudes is intimately tied to perturbative unitarity.…”

All-loop cuts from the Amplituhedron

“…The results we derive are valid at all multiplicities and loop orders for the maximally helicity violating (MHV) configurations. These include detailed derivations for the results in [1]. We conclude by indicating how one might move beyond trivial mutual positivity by presenting a series of configuration which re-introduce it bit by bit.…”

The positive geometry for 𝜙p interactions

“…The accordiohedron obtained by starting with a particular p-angulation is also completely determined by the relative configuration of diagonals. 5 The n = 1 accordiohedron AC…”

“…In section(4) we derive a formula for the number of primitves of a given dimension n and provide a classification all primitves up to n ≤ 3, we also provide general prescription for determining the weights. In section (5) we prove factorization for accordiohedra and also discuss the role of factorisation in determining the weights. We finally end with some conclusions and future directions.…”

“…It was further shown that unitarity and locality emerge quite naturally from geometric properties of the amplituhedron. Understanding scattering amplitudes in terms on canonical form on the amplituhedron makes the dual conformal symmetry manifest and has already produced striking results regarding positivity properties of the amplitudes as well as computational simplification for higher loop amplitudes [4,5].…”