Strongly-ordered infrared counterterms from factorisation

In the context of infrared subtraction algorithms beyond next-to-leading order, it becomes necessary to consider multiple infrared limits of scattering amplitudes, in which several particles become soft or collinear in a strongly-ordered sequence. We study these limits from the point of view of infrared factorisation, and we provide general definitions of strongly-ordered soft and collinear kernels in terms of gauge-invariant operator matrix elements. With these definitions in hand, it is possible to construct local subtraction counterterms for strongly-ordered configurations. Because of their factorised structure, these counterterms cancel infrared poles of real-virtual contributions by construction. We test these ideas at tree level for multiple emissions, and at one loop for single and double emissions, contributing to NNLO and N3LO distributions, respectively.