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.
Comment: 65 pages plus Appendices, 8 figures, uses JHEP style. v2: improved clarity, added references, clarified that potential mismatches in accuracy of distribution vs. cumulant/Laplace transform exist at unprimed but not primed orders. v3: added references, footnote in Sec. 3.5 on earlier related results, minor corrections, version published in JHEP