Nonperturbative Negative Geometries: Amplitudes at Strong Coupling and the Amplituhedron

The amplituhedron determines scattering amplitudes in planar ${\cal N}=4$ 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 ${\cal F}(g,z)$ - 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 $\Gamma_{\rm cusp}(g)$ 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 ${\cal F}(g,z)$ and $\Gamma_{\rm cusp}$ 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 ${\cal F}(g,z)$ and $\Gamma_{\rm cusp}$ for all values of the 't Hooft coupling. The result for $\Gamma_{\rm cusp}$ remarkably shares all main qualitative characteristics of the known exact results obtained using integrability.