Factorization and subtraction

We explore the connection between the factorisation of virtual corrections to multi-particle massless gauge theory amplitudes and the problem of subtraction at NNLO and beyond. Taking inspiration from virtual factorisation, we provide a set of definitions for local soft and collinear counterterms, expressed in terms of matrix elements of operators involving fields and Wilson lines, and valid to all orders in perturbation theory. We hope that the connection between factorisation and subtraction will help in the construction of minimal, stable, and efficient subtraction algorithms, taking maximal advantage of existing analytic information.