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

In this article, we generalize the recursion methods based on cut equations [N. Arkani-Hamed, H. Frost, and G. Salvatori, .], 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 To address this, we introduce a 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-<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi>D</a:mi></a:mrow></a:math> limit for the three-loop four-point case.