Joint resummation of two angularities at next-to-next-to-leading logarithmic order

Multivariate analyses are emerging as important tools to understand properties of hadronic jets which play a key role in the LHC experimental program. We take a first step towards precise and differential theory predictions by calculating the cross section for $e^+ e^- \to$ 2 jets differential in the angularities $e_\alpha$ and $e_\beta$. The logarithms of $e_\alpha$ and $e_\beta$ in the cross section are jointly resummed to next-to-next-to-leading logarithmic accuracy, using the SCET+ framework we developed, and are matched to the next-to-leading order cross section. We perform analytic one-loop calculations that serve as input for our numerical analysis, provide controlled theory uncertainties, and compare our results to Pythia. We also obtain predictions for the cross section differential in the ratio $e_\alpha/e_\beta$, which cannot be determined from a fixed-order calculation. The effect of nonperturbative corrections is also investigated. Using Event2, we validate the logarithmic structure of the single angularity cross section predicted by factorization theorems at ${\mathcal O}(\alpha_s^2)$, demonstrating that for specific angularities recoil must be taken into account when using the thrust axis, while it can be ignored if these are measured with respect to the winner-take-all axis.