Probing factorization violation with vector angularities

Factorization underlies all predictions at the Large Hadron Collider, but has only been rigorously proven in a few cases. One of these cases is the Drell-Yan process, $pp \to Z/\gamma + X$, in the limit of small boson transverse momentum. We introduce a one-parameter family of observables, that we call vector angularities, of which the transverse momentum is a special case. This enables the study of factorization violation, with a smooth transition to the limit for which factorization has been established. Like the angularity event shapes, vector angularities are a sum of transverse momenta weighted by rapidity, but crucially this is a vector sum rather than a sum of the magnitude of transverse momenta. We study these observables in Pythia, using the effect of multi-parton interactions (MPI) as a proxy factorization violation, finding a negligible effect in the case where factorization is established but sizable effects away from it. We also present a factorization formula for the cross section, that does not include factorization violating contributions from Glauber gluons, and thus offers a baseline for studying factorization violation experimentally using vector angularities. Our predictions at next-to-leading logarithmic accuracy (NLL$'$) are in good in agreement with Pythia (not including MPI), and can be extended to higher order.