The scale of soft resummation in SCET vs perturbative QCD

Abstract

We summarize and extend previous results on the comparison of threshold resummation, performed, using softcollinear 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 show that the logarithmic accuracy of this SCET result can be extended by half a logarithmic order, thereby bringing it in full agreement with the standard QCD result if a suitable choice is made for the soft scale µ s which characterizes the SCET result. We provide a master formula relating the two approaches for other scale choices. We then show that with the Becher-Neubert scale choice the Landau pole, which in the perturbative QCD approach is usually removed through power-or exponentially suppressed terms, in the SCET approach is removed by logarithmically subleading terms which break factorization. Such terms may become leading for generic choices of parton distributions, and are always leading when resummation is used far enough from the hadronic threshold.Keywords: QCD, soft-gluon, threshold, resummation, soft-collinear effective theory Soft resummation and scale choicesThreshold resummation [1,2] plays an important role in extending and stabilizing the accuracy of perturbative results, and it may be of some relevance even for hadronic processes which are quite far from threshold [3,4], due to the fact that the underlying partonic process can be rather closer to threshold than the hadronic one [5]. All-order resummed results are known to lead to a divergent series when expanded out in powers of the strong coupling: this is physically due to the fact that resummation is obtained by choosing as a scale of the parton-level process the maximum energy of the radiated partons [6,7], which tends to zero in the threshold limit. The divergence can be tamed by introducing suitable subleading contributions, such as exponentially suppressed terms outside the physical kinematic region [8], or power-suppressed terms [9,10]. * Speaker. In Ref.[11] it was suggested, within the context of a SCET approach to threshold resummation, that the divergence can be tamed by making a hadronic choice of scale. In SCET this is possible because resummed results are characterized by a "soft scale" µ s : the BecherNeubert (BN) scale choice consists of expressing µ s in terms of kinematic variables of the hadronic scattering process. The meaning of this choice is not obvious in the conventional QCD approach, where, because of perturbative factorization, the partonic cross section, which is being resummed, is independent of the hadronic kinematic variables.In Ref.[13] we have clarified this issue by explicitly exhibiting a relation between the µ s dependent resummed SCET result, and the standard (µ s independent) QCD expression. Specializing to the BN scale choice (while taking the Drell-Yan process [12] as an example) we were able to show that in the SCET result, with the BN scale choice, the divergence is removed through arXiv:130...

Three loop soft function forN3LL′gluon fusion Higgs boson production in soft-collinear effective theory

“…I we also show different ways of counting the logarithmic accuracy (unprimed and primed, or, using a notation introduced in Ref. [23], starred and unstarred), depending on the order of the matching functions H ands Higgs . A natural logarithmic counting at the exponent would lead to the unprimed (starred) accuracies; however, it is known [19] that the inclusion of the next order matching functions, leading to the primed (unstarred) accuracy, contains the bulk of the next logarithmic order correction.…”

“…This is a direct consequence of the way SCET is constructed: the functions H ands Higgs are found by subsequent matchings of full QCD onto SCET. Alternatively, this can be understood in terms of the equivalence of SCET and dQCD [16,19,[21][22][23]: for a particular choice of the scales µ S and µ H , SCET and dQCD coincide to all orders in α s . Additionally, the expansion of the SCET N k LL result does not depend on µ S and µ H up to order α k s : so, up to this order, the expansion must coincide with the same expansion in dQCD, independently on the scale choice.…”

“…1 Exploiting the equivalence of SCET and dQCD [16,19,[21][22][23], we extract the three loop expression of the soft function from the N 3 LL dQCD result. We also consider a modification of the form of the soft logarithms that includes important collinear contributions, and discuss the comparison to the dQCD result of Ref.…”

Resummed Higgs cross section at N3LL

“…See ref. [44]. Note that the four-loop contribution to A(α s ) is yet unknown and in our study we take a Padé approximation [45].…”

Phenomenology of QCD threshold resummation for gluino pair production at NNLL

“…A dedicated discussion can be found in ref. [34]. Within the Mellin-space formalism on the other hand, one has the implicit scale choices µ s = m g /N and µ h = µ f , and the two approaches are formally identical up to O(1/N )-terms (see for instance the discussion in ref.…”