Nonperturbative negative geometries: amplitudes at strong coupling and the amplituhedron
Abstract
The amplituhedron determines scattering amplitudes in planar N = 4 super YangMills 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 fourparticle 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 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 longsought direct link between canonical forms for positive (negative) geometries, and a completely IR finite postloopintegration observable depending on a single kinematical variable z, from which the cusp anomalous dimension Γ_{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 F (g, z) and Γ_{cusp} can easily be determined by an interesting nonlinear differential equation immediately following from the combinatorics of negative geometries. This lets us compute these "tree" contributions to F (g, z) and Γ_{cusp} for all values of the `t Hooft coupling. The result for Γ_{cusp} remarkably shares all main qualitative characteristics of the known exact results obtained using integrability.
 Publication:

Journal of High Energy Physics
 Pub Date:
 March 2022
 DOI:
 10.1007/JHEP03(2022)108
 arXiv:
 arXiv:2112.06956
 Bibcode:
 2022JHEP...03..108A
 Keywords:

 Scattering Amplitudes;
 Supersymmetric Gauge Theory;
 AdSCFT Correspondence;
 Resummation;
 High Energy Physics  Theory
 EPrint:
 38 pages, 45 figures