Abstract
We compare methods to resum logarithms in event shape distributions as they have been used in perturbative QCD directly and in effective field theory. We demonstrate that they are equivalent. In showing this equivalence, we are able to put standard soft-collinear effective theory (SCET) formulae for cross sections in momentum space into a novel form more directly comparable with standard QCD formulae, and endow the QCD formulae with dependence on separated hard, jet, and soft scales, providing potential ways to improve estimates of theoretical uncertainty. We show how to compute cross sections in momentum space to keep them as accurate as the corresponding expressions in Laplace space. In particular, we point out that that care is required in truncating differential distributions at N$^k$LL accuracy to ensure they match the accuracy of the corresponding cumulant or Laplace transform. We explain how to avoid such mismatches at N$^k$LL accuracy, and observe why they can also be avoided by working to N$^k$LL$'$ accuracy.
Highlights
The theory Quantum Chromodynamics (QCD) is remarkably successful in describing the strong interaction
We summarize these methods in parallel discussions below in the context of “direct” QCD and soft collinear effective theory (SCET) [17,18,19,20,21] techniques to resum logs, applied to jet cross sections [22, 23] and event shape distributions [24]
If we base the definition of NkLL accuracy on this object, we argue that in order to evaluate cross section in momentum space to the equivalent accuracy, some additional care is warranted in evaluating typical formulae for the cumulant R(τa), and even greater care in evaluating the differential distribution σ(τa)
Summary
The theory Quantum Chromodynamics (QCD) is remarkably successful in describing the strong interaction. [16], which are naturally derivable using the methods of effective field theory We summarize these methods in parallel discussions below in the context of “direct” QCD (dQCD) and soft collinear effective theory (SCET) [17,18,19,20,21] techniques to resum logs, applied to jet cross sections [22, 23] and event shape distributions [24]. These distributions look similar but not identical at first, but we will put the two into forms that are precisely and transparently equivalent. We hope that the issues resolved here will help ensure that the most precise QCD predictions will be available at the LHC and future facilities
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