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Factorizing the cross section for single hadron production in $e^+e^-$annihilations is a highly non trivial task when the transverse momentum of theoutgoing hadron with respect to the thrust axis is taken into account. We workin a scheme that allows to factorize the $e^+e^- \to HX$ cross section as aconvolution of a calculable hard coefficient and a Transverse MomentumDependent (TMD) fragmentation function. The result, differential in $z_h$,$P_T$ and thrust, will be given to all orders in perturbation theory andexplicitly computed to Next to Leading Order (NLO) and Next to Leading Log(NLL) accuracy. The predictions obtained from our computation, applying thesimplest and most natural ansatz to model the non-perturbative part of the TMD,are in exceptional agreement with the experimental measurements of the BELLECollaboration. The factorization scheme we propose relates the TMD partondensities defined in 1-hadron and 2-hadron processes, restoring the possibilityto perform global phenomenological studies of TMD physics includingexperimental data from semi-inclusive deep inelastic scattering, Drell-Yanprocesses, $e^+e^- \to H_1 H_2 X$ and $e^+e^- \to HX$ annihilations.

Factorization of $e^+e^- \to H \, X$ cross section, differential in $z_h$, $P_T$ and thrust, in the $2$-jet limit