Abstract
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− → hX 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. This procedure naturally leads to a partition of the e+e− → hX 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.
Highlights
Modern studies of high energy QCD processes are based on factorization theorems, that play a pivotal role in the study of strong interactions, as they allow to write the cross sections of hadronic processes in a form suitable for phenomenological analyses
Explicit perturbative computations require to extend the definitions of the soft factors and of the Transverse Momentum Dependent (TMD) in order to take into account the dependence on thrust
Within this general approach we were able to recover the same collinear-TMD factorization theorem obtained in ref. [16] by means of an explicit perturbative computations, in a completely natural way
Summary
Modern studies of high energy QCD processes are based on factorization theorems, that play a pivotal role in the study of strong interactions, as they allow to write the cross sections of hadronic processes in a form suitable for phenomenological analyses. We show that, if the size of the transverse momentum of the detected hadron is neither too large to affect significantly the topological configuration of the final state nor too small to be sensitive to the deflection due to soft radiation, the cross section of e+e− → h X factorizes in the convolution of a partonic cross section, fully computable in perturbation theory, and a TMD Fragmentation Function This is a new kind of structure, never encountered before in any known factorization theorem. Instead, when the transverse momentum of the detected hadron is large enough to significantly affect the topology of the final state (and, the measured value of thrust) the factorization theorem does not involve a TMD FF anymore, and it shares many of the features of collinear factorization We will call these two kinematical ranges Region 1 and Region 3, respectively. This approach looks very promising and it might be one of the future keys to fundamental QCD issues
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