The D-parameter is one of the oldest and most experimentally well-studied hadronic observables for e + e − collisions. Nevertheless, unlike other classic observables like the C-parameter or thrust, the D-parameter has never been resummed throughout its entire singular phase space. Using insights and techniques motivated by modern multi-differential jet substructure calculations, we are able to predict the D-parameter distribution with no additional phase space cuts. Our approach is to measure both the C-and D-parameters on hadronic final states in e + e − collisions. We can tune the value of the C-parameter with respect to the D-parameter to specify simple, physical configurations of final state particles in which to perform calculations. There are three parametric regions that exist: D C 2 ∼ 1, D C 2 1, and D ∼ C 2 1, and we calculate the D-parameter in each region separately. In the first two of these three regions, we present all-orders factorization theorems and explicitly demonstrate resummation to next-to-leading logarithmic accuracy. The region in which D ∼ C 2 1 corresponds to the dijet limit and where the D-parameter loses the property of additivity. In this region we introduce a systematically-improvable procedure exploiting properties of conditional probabilities and resum to approximate next-to-leading logarithmic accuracy. The contributions from these regions can be consistently combined, and the value of the C-parameter integrated over to produce the cross section for the Dparameter. With these results, we match to leading fixed order as proof of principle and compare our resummed and matched prediction to data from LEP.
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