The Two-loop MHV Momentum Amplituhedron from Fibrations of Fibrations

Recently, a new approach to computing the canonical forms of the momentum amplituhedron in dual-momentum space was proposed by the authors. These are relevant for the integrands of scattering amplitudes in planar N=4 super-Yang-Mills. At one-loop the idea was to view the set of all loop momenta, which we refer to as the one-loop fiber geometry, as a fibration over the tree-level kinematic data. This led to the notion of tree-level chambers, subsets of the tree-level kinematic space for which the combinatorial structure of the one-loop fiber remains unchanged, that allowed for a novel representation of the one-loop integrand. The goal of this paper is to extend these ideas to two loops for MHV integrands. Our approach will be to view the geometry accessed by the second loop momentum, similarly referred to as the two-loop fiber geometry, as a fibration over both the one-loop kinematic data and the position of the first loop momentum in the one-loop fiber. This will lead to the notion of one-loop chambers, subsets of the one-loop fibers for which the combinatorial structure of the two-loop fiber remains unchanged. We will characterise the full set of one-loop chambers and their corresponding two-loop fibers and present formulae for their canonical forms. Ultimately, this will result in a new formula for the two-loop MHV integrand written as a fibration of fibration.