Nonperturbative negative geometries: amplitudes at strong coupling and the amplituhedron
Nima Arkani–Hamed, Johannes M. Henn, Jaroslav Trnka
Abstract
A bstract The amplituhedron determines scattering amplitudes in planar $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 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 $$ \mathcal{F} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> ( 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 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 $$ \mathcal{F} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> ( g, z ) and Γ 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 $$ \mathcal{F} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> ( 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.