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On-shell recursion relations for generic integrands

The quantum effects encapsulated in loop corrections are crucial in quantum field theory for a wide variety of formal and phenomenological applications. In this article we propose and check a definition of the so-called single cut contributions needed to complete on-shell recursion relations for the integrand of scattering amplitudes in generic power-counting renormalisable theories at conjecturally any loop order. Our proposal meshes well with standard dimensional regularisation and applies in particular directly to much of the standard model of particle physics. Apart from a diagrammatic construction, at one loop order we provide a direct cross-check for box coefficients. Interestingly, at one loop our proposal can be related to a specific subset of all double unitarity cuts as well as tree-level poles by iterated recursion. We focus in particular on demonstrating the method in rational-term one-loop examples in pure Yang-Mills theory. For the finite amplitudes we present all-order arguments. First steps toward gravity integrands are taken.