Next-to-soft radiation from a different angle

Soft and collinear radiation in collider processes can be described in a universal way, that is independent of the underlying process. Recent years have seen a number of approaches for probing whether radiation beyond the leading soft approximation can also be systematically classified. In this paper, we study a formula that captures the leading next-to-soft QCD radiation affecting processes with both final- and initial-state partons, by shifting the momenta in the nonradiative squared amplitude. We first examine $W+\text{jet}$ production, and show that a previously derived formula of this type indeed holds in the case in which massive color singlet particles are present in the final state. Next, we develop a physical understanding of the momentum shifts, showing precisely how they disrupt the well-known angular ordering property of leading soft radiation.