Planar loop integrands from cuts in D dimensions

A bstract We present a direct reconstruction formula for planar loop integrands from D -dimensional generalized unitarity cuts in any colored theory. The reconstruction combinatorics is separated from the theory-dependent tree amplitudes entering the cuts: for the L -loop n -point color-ordered amplitude, the integrand is expressed as a sum over admissible non-scaleless scalar graphs dressed by corresponding cuts in D dimensions; the coefficients are given by the universal Möbius-inversion formula of the refinement poset, or equivalently one minus the Euler characteristics of associated complexes. As an application we write down closed-formulas for loop integrands in pure Yang-Mills theory, where the required cuts are generated by gluing D -dimensional tree amplitudes and summing over internal gluon states. We also use the two-loop five-point case as a validation, comparing with known integrand data and after integration-by-parts reduction, with known integrated helicity amplitudes. The same framework also produces compact cut-organized data for larger examples, which are included in the accompanying supplementary material. We also describe the corresponding simplification in maximally supersymmetric Yang-Mills theory, where the absence of bubble and triangle subgraphs reduces the relevant cut poset substantially.