Kinematic regions in the $e^+e^- \to h \, X$ factorized cross section in a $2$-jet topology with thrust

Factorization theorems allow to separate out the universal, non-perturbative content of the hadronic cross section from its perturbative part, which can be computed in perturbative QCD, up to the desired order. In this paper, we derive a rigorous proof of factorization of the $e^+ e^- \to h\,X$ cross section, sensitive to the transverse momentum of the detected hadron with respect to the thrust axis, in a completely general framework, based on the Collins-Soper-Sterman approach. The results are explicitly computed to NLO-NLL accuracy and subsequently generalized to all orders in perturbation theory. This procedure naturally leads to a partition of the $e^+ e^- \to h\,X$ kinematics into three different regions, each associated to a different factorization theorem. In one of these regions, which covers the central and widest range, the factorization theorem has a new structure, which shares the features of both TMD and collinear factorization schemes. In the corresponding cross section, the role of the rapidity cut-off is investigated, as its physical meaning becomes increasingly evident. An algorithm to identify these three kinematic regions, based on ratios of observable quantities, is provided.