Threshold resummation in SCET vs. perturbative QCD: an analytic comparison

We compare threshold resummation in QCD, as performed using soft-collinear effective theory (SCET) in the Becher–Neubert approach, to the standard perturbative QCD formalism based on factorization and resummation of Mellin moments of partonic cross-sections. We consider various forms of the SCET result, which correspond to different choices of the soft scale μs that characterizes this approach. We derive a master formula that relates the SCET resummation to the QCD result for any choice of μs. We then use it first, to show that if SCET resummation is performed in N-Mellin moment space by suitable choice of μs it is equivalent to the standard perturbative approach. Next, we show that if SCET resummation is performed by choosing for μs a partonic momentum variable, the perturbative result for partonic resummed cross-sections is again reproduced, but, like its standard perturbative counterpart, it is beset by divergent behaviour at the endpoint. Finally, using the master formula we show that when μs is chosen as a hadronic momentum variable the SCET and standard approach are related through a multiplicative (convolutive) factor, which contains the dependence on the Landau pole and associated divergence. This factor depends on the luminosity in a non-universal way; it lowers by one power of log the accuracy of the resummed result, but it is otherwise subleading if one assumes the luminosity not to contain logarithmically-enhanced terms. Therefore, the SCET approach can be turned into a prescription to remove the Landau pole from the perturbative result, but the price to pay for this is the reduction by one logarithmic power of the accuracy at each order and the need to make assumptions on the parton luminosity.