Observables which distinguish boosted topologies from QCD jets are playing an
increasingly important role at the Large Hadron Collider (LHC). These
observables are often used in conjunction with jet grooming algorithms, which
reduce contamination from both theoretical and experimental sources. In this
paper we derive factorization formulae for groomed multi-prong substructure
observables, focusing in particular on the groomed D2β observable, which is
used to identify boosted hadronic decays of electroweak bosons at the LHC. Our
factorization formulae allow systematically improvable calculations of the
perturbative D2β distribution and the resummation of logarithmically enhanced
terms in all regions of phase space using renormalization group evolution. They
include a novel factorization for the production of a soft subjet in the
presence of a grooming algorithm, in which clustering effects enter directly
into the hard matching. We use these factorization formulae to draw robust
conclusions of experimental relevance regarding the universality of the D2β
distribution in both e+eβ and pp collisions. In particular, we show that
the only process dependence is carried by the relative quark vs. gluon jet
fraction in the sample, no non-global logarithms from event-wide correlations
are present in the distribution, hadronization corrections are controlled by
the perturbative mass of the jet, and all global color correlations are
completely removed by grooming, making groomed D2β a theoretically clean QCD
observable even in the LHC environment. We compute all ingredients to one-loop
accuracy, and present numerical results at next-to-leading logarithmic accuracy
for e+eβ collisions, comparing with parton shower Monte Carlo simulations.
Results for pp collisions, as relevant for phenomenology at the LHC, are
presented in a companion paper.Comment: 66 pages, 18 figure