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Uplifting Amplitudes in Special Kinematics

We consider scattering amplitudes in planar N = 4 supersymmetric Yang-Mills theory in special kinematics where all external four-dimensional momenta are restricted to a (1+1)-dimensional subspace. The amplitudes are known to satisfy non-trivial factorisation properties arising from multi-collinear limits, which we further study here. We are able to find a general solution to these multi-collinear limits. This results in a simple formula which represents an n-point superamplitude in terms of a linear combination of functions S_m which are constrained to vanish in all appropriate multi-collinear limits. These collinear-vanishing building blocks, S_m, are dual-conformally-invariant functions which depend on the reduced m-point kinematics with 8 \leq m \leq 4l. For MHV amplitudes they can be constructed directly using, for example, the approach in Ref. [1]. This procedure provides a universal uplift of lower-point collinearly vanishing building blocks S_m to all higher-point amplitudes. It works at any loop-level l \geq 1 and for any MHV or N^kMHV amplitude. We compare this with explicit examples involving n-point MHV amplitudes at 2-loops and 10-point MHV amplitudes at 3-loops. Tree-level superamplitudes have different properties and are treated separately from loop-level amplitudes in our approach. To illustrate this we derive an expression for n-point tree-level NMHV amplitudes in special kinematics.