New Insights on an Old Problem: Resummation of the D-parameter

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: $D \ll C^2\sim 1$, $D\ll C^2\ll 1$, and $D\sim C^2\ll1$, 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 $D\sim C^2\ll1$ 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 $D$-parameter. With these results, we match to leading fixed order as proof of principle and compare our resummed and matched prediction to data from LEP.