Planar loop integrands from cuts in D dimensions
Abstract
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, including the two-loop six-point and three-loop four-point cases. 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.
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BibTeX
@misc{zhu2026planar,
title = {{Planar loop integrands from cuts in $D$ dimensions}},
author = {Fan Zhu and Song He and Zhenqi Han},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.28052},
eprint = {2606.28052},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.28052},
}
RIS
TY - UNPB
TI - Planar loop integrands from cuts in D dimensions
AU - Fan Zhu
AU - Song He
AU - Zhenqi Han
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2606.28052
UR - https://arxiv.org/abs/2606.28052
AB - 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, including the two-loop six-point and three-loop four-point cases. 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.
ER -