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

TL;DR

This paper compares threshold resummation techniques in SCET and perturbative QCD, deriving a master formula that relates the two and analyzing the implications of different scale choices on the resummation accuracy and divergence issues.

Contribution

It provides a unified framework linking SCET and perturbative QCD resummation methods through a master formula for various scale choices.

Findings

SCET resummation in Mellin space is equivalent to standard QCD.

Choosing a partonic momentum scale reproduces standard results but with endpoint divergences.

Using a hadronic momentum scale relates SCET and QCD results via a factor that addresses the Landau pole.

Abstract

We compare threshold resummation in QCD, as performed using soft-collinear effective theory (SCET), 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 mu_s that characterizes this approach. We derive a master formula that relates the SCET resummation to the QCD result for any choice of mu_s. We then use it first, to show that if SCET resummation is performed in N-Mellin moment space by suitable choice of mu_s it is equivalent to the standard perturbative approach. Next, we show that if SCET resummation is performed by choosing for mu_s 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 mu_s 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.