The SAGEX review on scattering amplitudes Chapter 7: Positive geometry of scattering amplitudes

Abstract

Scattering amplitudes are both a wonderful playground to discover novel ideas in Quantum Field Theory and simultaneously of immense phenomenological importance to make precision predictions for e.g.~particle collider observables and more recently also for gravitational wave signals. In this review chapter, we give an overview of some of the exciting recent progress on reformulating QFT in terms of mathematical, geometric quantities, such as polytopes, associahedra, Grassmanians, and the amplituhedron. In this novel approach, standard notions of locality and unitarity are derived concepts rather than fundamental ingredients in the construction which might give us a handle on a number of open questions in QFT that have evaded an answer for decades. We first give a basic summary of positive geometry, before discussing the associahedron---one of the simplest physically relevant geometric examples---and its relation to tree-level scattering amplitudes in bi-adjoint $\phi^3$ theory. Our second example is the amplituhedron construction for scattering amplitudes in planar maximally supersymmetric Yang-Mills theory.

Large $N$ analytical functional bootstrap I: 1D CFTs and total positivity

“…Do the AdS analogs of the conformal blocks, the Witten diagrams also satisfy total positivity? The role of total positivity in the 4D amplitude in flat spacetime has been extensively studied recently [41][42][43][44]. Our results on the 1D CFTs suggest that the AdS 2 could provide another interesting and technically tractable laboratory to explore the role of (total) positivity in quantum field theories.…”

“…It has deep connections to quantum field theories, see e.g. [41][42][43][44]. The possible role of total positivity in conformal bootstrap has been proposed in [45], in which the authors focused on the geometrical configuration supported by the SL(2, R) conformal blocks.…”

The SAGEX review on scattering amplitudes*

“…In the remainder of the chapter, we utilise the advantage provided by the underlying model to showcase diverse applications, ranging from loop amplitudes to scattering on curved backgrounds, and interesting connections to celestial holography. [7] In this chapter, we summarise some of the recent developments in rephrasing perturbative scattering amplitudes in an entirely novel geometric way, where traditional notions such as locality and unitarity are secondary concepts derived from fundamental mathematical properties of positive geometries. We discuss these geometric ideas on two concrete examples: first for tree-level scattering amplitudes in bi-adjoint scalar φ 3 theory and their connection to the associahedron polytope with its boundaries of all co-dimensions.…”

Internal boundaries of the loop amplituhedron

“…The amplituhedron is a geometrical object introduced in [1,2] and was discovered to yield a beautiful intrinsic definition of the perturbative expansion of planar amplitudes in N = 4 super Yang-Mills (SYM), allowing for entirely novel expressions for amplitudes to be found (see [3] for a review). In this framework, amplitude integrands are obtained as a differential form, called the canonical form, of the amplituhedron.…”