Planar loop integrands from cuts in D dimensions
- 1Graduate School of China Academy of Engineering Physics
- 2Chinese Academy of Sciences
- 3Hangzhou Institute for Advanced Study and ICTP-AP, UCAS
Abstract
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.
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BibTeX
@article{zhu2026planar,
title = {{Planar loop integrands from cuts in D dimensions}},
author = {Fan Zhu and Song He and Zhenqi Han},
journal = {Journal of High Energy Physics},
year = {2026},
month = oct,
doi = {10.1007/jhep10(2026)221},
url = {https://link.springer.com/article/10.1007/JHEP10(2026)221},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Planar loop integrands from cuts in D dimensions
AU - Fan Zhu
AU - Song He
AU - Zhenqi Han
PY - 2026
DA - 2026/10/21
JO - Journal of High Energy Physics
DO - 10.1007/jhep10(2026)221
UR - https://link.springer.com/article/10.1007/JHEP10(2026)221
AB - 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.
SN - 1029-8479
ER -