Deformed Amplituhedra and Interpolating Anomalous Dimensions on the Coulomb Branch

Abstract

We study planar four-point scattering amplitudes on the Coulomb branch of $\mathcal N=4$ super Yang--Mills theory, focusing on multi-parameter families of massive kinematics. We extend the deformed Amplituhedron construction of \cite{Arkani-Hamed:2023epq,Flieger:2025ekn} to general external kinematics characterized by a six-parameter embedding-space Gram matrix. We show that copositivity constraints on the inverse Gram matrix control the real geometry and restrict the numerator of the two-loop canonical form to the span of the two planar double-box structures expected for the scattering amplitude. We then specialize to a three-parameter massive configuration and evaluate the relevant one- and two-loop integrals using differential equations directly in four dimensions, obtaining analytic expressions in terms of multiple polylogarithms. We investigate multiscale infrared limits of the proposed two-loop amplitude and find exponentiation through two loops. The kinematic family contains the limits governed by the cusp and octagon anomalous dimensions and interpolates between them through the walking anomalous dimension introduced in~\cite{Alday:2026cpk}. Our result reproduces the one-loop interpolation and fixes the two-loop coefficients in their proposed all-loop ansatz. Interestingly, our results suggest that the walking anomalous dimension may be entirely determined through all loop orders by the cusp and octagon anomalous dimensions.