Rapidity regulators in the semi-inclusive deep inelastic scattering and Drell-Yan processes

We study the semi-inclusive limit of the deep inelastic scattering and Drell-Yan (DY) processes in soft collinear effective theory. In this regime so-called threshold logarithms must be resummed to render perturbation theory well behaved. Part of this resummation occurs via the Dokshitzer, Gribov, Lipatov, Altarelli, Parisi (DGLAP) equation, which at threshold contains a large logarithm that calls into question the convergence of the anomalous dimension. We demonstrate here that the problematic logarithm is related to rapidity divergences, and by introducing a rapidity regulator can be tamed. We show that resumming the rapidity logarithms allows us to reproduce the standard DGLAP running at threshold as long as a set of potentially large non-perturbative logarithms are absorbed into the definition of the parton distribution function (PDF). These terms could, in turn, explain the steep fall-off of the PDF in the endpoint. We then go on to show that the resummation of rapidity divergences does not change the standard threshold resummation in DY, nor do our results depend on the rapidity regulator we choose to use.