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All-loop planar integrands in Yang-Mills theory from recursion

Qu Cao1,2,* and Fan Zhu2,3,4,†

  • *Contact author: qucao@zju.edu.cn
  • †Contact author: zhufan25@gscaep.ac.cn
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Phys. Rev. D 112, 085012 – Published 16 October, 2025

DOI: https://doi.org/10.1103/prxt-x26h

Abstract

In this article, we generalize the recursion methods based on cut equations [N. Arkani-Hamed, H. Frost, and G. Salvatori, arXiv:2412.21027.], originally developed for scalar theories, to gluons in pure Yang-Mills theory. In gauge theories, planar loop integrands are subtle to defined and obtained due to the existence of scaleless integrals. A critical challenge arises when constructing higher-loop integrands via recursion methods: integrands with or without all scaleless terms both are incompatible with the reconstruction formalism in Arkani-Hamed et al. To address this, we introduce a refined integrand by systematically removing specific scaleless contributions, and develop an algorithmic implementation of the recursion to all-loop level. We explicitly demonstrate the framework by three steps and obtain the recursion formula in pure Yang-Mills theory. In the Supplemental Material, we provide the results up to the two-loop five-point integrand and the simplified result in the large-D limit for the three-loop four-point case.

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Phys. Rev. D 112, 085012– Published 16 October, 2025
  • Received 2 April 2025
  • Accepted 13 September 2025
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