Catani's generalization of collinear factorization breaking

We consider the most general form of soft and collinear factorization for hard-scattering amplitudes to all orders in perturbative Quantum Chromodynamics. Specifically, we present the generalization of collinear factorization to configurations with several collinear directions, where the most singular behaviour is encoded by generalized collinear splitting amplitudes that manifestly embed the breaking of strict collinear factorization in space-like collinear configurations. We also extend the analysis to the simultaneous soft-collinear factorization with multiple collinear directions where na\"{\i}ve multiplicative factorization does not hold. As an illustrative example of factorization breaking, we present explicit results at the one-loop level in the soft-collinear limit.