On the approaches to threshold resummation of rapidity distributions for the Drell-Yan process

We consider threshold resummation of rapidity distributions, for which various approaches exist in the literature. Recently, a work by Lustermans, Michel, Tackmann suggested that older approaches by Becher, Neubert, Xu (BNX) and Bonvini, Forte, Ridolfi (BFR) were wrong because they miss some leading-power contributions at threshold. In this work, we prove and demonstrate that the BNX and BFR approaches are correct and able to resum threshold logarithms to leading-power accuracy. We then show that the BNX and BFR approaches can provide rather good alternatives to more modern approaches to threshold resummation of rapidity distributions, provided the threshold logarithms are resummed according to the $\psi$-soft definition introduced in the context of Higgs production.