Nonperturbative Negative Geometries: Amplitudes at Strong Coupling and the Amplituhedron

Abstract

The amplituhedron determines scattering amplitudes in planar $\mathcal{N}=4$ super Yang-Mills by a single "positive geometry" in the space of kinematic and loop variables. We study a closely related definition of the amplituhedron for the simplest case of four-particle scattering, given as a sum over complementary "negative geometries", which provides a natural geometric understanding of the exponentiation of infrared (IR) divergences, as well as a new geometric definition of an IR finite observable $\mathcal{F}(g,z)$ - dually interpreted as the expectation value of the null polygonal Wilson loop with a single Lagrangian insertion - which is directly determined by these negative geometries. This provides a long-sought direct link between canonical forms for positive (negative) geometries, and a completely IR finite post-loop-integration observable depending on a single kinematical variable $z$, from which the cusp anomalous dimension ${\Gamma }_{\mathrm{cusp}}(g)$ can also be straightforwardly obtained. We study an especially simple class of negative geometries at all loop orders, associated with a "tree" structure in the negativity conditions, for which the contributions to $\mathcal{F}(g,z)$ and ${\Gamma }_{\mathrm{cusp}}$ can easily be determined by an interesting non-linear differential equation immediately following from the combinatorics of negative geometries. This lets us compute these "tree" contributions to $\mathcal{F}(g,z)$ and ${\Gamma }_{\mathrm{cusp}}$ for all values of the 't Hooft coupling. The result for ${\Gamma }_{\mathrm{cusp}}$ remarkably shares all main qualitative characteristics of the known exact results obtained using integrability.

arXiv comment38 pages, 45 figures

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• Amplitudes meet Cosmology: A (Scalar) Primer
• The SAGEX Review on Scattering Amplitudes, Chapter 8: Half BPS correlators
• All-Loop Four-Point Aharony-Bergman-Jafferis-Maldacena Amplitudes from Dimensional Reduction of the Amplituhedron
• Two-Loop Five-Point Two-Mass Planar Integrals and Double Lagrangian Insertions in a Wilson Loop
• An Operator Product Expansion for Form Factors III. Finite Coupling and Multi-Particle Contributions
• Integrated negative geometries in ABJM
• The Loop Momentum Amplituhedron
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• Emergent unitarity, all-loop cuts and integrations from the ABJM amplituhedron
• Non-planar BCFW Grassmannian Geometries
• Poles At Infinity in On-shell Diagrams
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• Positive Geometry for Stringy Scalar Amplitudes
• On Solving Dual Conformal Integrals in Coulomb-branch Amplitudes and Their Periods
• The Geometry of BCFW for ABJM Loop Integrands