Nonperturbative negative geometries: amplitudes at strong coupling and the amplituhedron

A bstract The amplituhedron determines scattering amplitudes in planar $$ \mathcal{N} N = 4 super Yang-Mills by a single “positive geometry” the space of kinematic and loop variables. We study closely related definition for simplest case four-particle scattering, given as sum over complementary “negative geometries”, which provides natural geometric understanding exponentiation infrared (IR) divergences, well new an IR finite observable \mathcal{F} F ( g, z ) — dually interpreted expectation value null polygonal Wilson with Lagrangian insertion is directly determined these negative geometries. This long-sought direct link between canonical forms positive (negative) geometries, completely post-loop-integration depending on kinematical variable , from cusp anomalous dimension ? g can also be straightforwardly obtained. especially simple class geometries at all orders, associated “tree” structure negativity conditions, contributions to easily interesting non-linear differential equation immediately following combinatorics lets us compute values ‘t Hooft coupling. result remarkably shares main qualitative characteristics known exact results obtained using integrability.