An All-Loop Amplituhedron in Two Dimensions

Abstract

We define and study a positive geometry $\Delta^{(L)}$ which serves as a natural generalization of loop amplituhedra to two-dimensional Minkowski space $\mathbb{R}^{1,1}$. The geometry is formulated in the framework of lightcone geometries in dual momentum space, and can equivalently be obtained as a specific boundary of the $L$-loop amplituhedron for $\mathcal{N}=4$ super Yang--Mills. The simplicity of the two-dimensional setting allows us to calculate the canonical form of $\Delta^{(L)}$ at any loop order, which is shown to correspond to massless banana graphs. We integrate the canonical form at all loop orders in dimensional regularization, and find that the full IR divergence structure at $L$-loops is captured by the $L$th power of the one-loop result, a phenomenon analogous to IR exponentiation. Furthermore, these integrated functions can be resummed into a closed-form non-perturbative result given by a Fox--Wright function. In the limit where $L\to\infty$, the geometry gives rise to a path integral over worldlines, suggesting the emergence of a dual description at strong coupling. This construction provides a simple and tractable setting in which to explore the geometry of loop amplitudes, and offers a controlled toy model for investigating loop amplituhedra beyond their standard scope.