The effects of the final state interaction in slow proton production in semi inclusive deep inelastic scattering processes off nuclei, A(e,ep)X, are investigated in details within the spectator and target fragmentation mechanisms; in the former mechanism, the hard interaction on a nucleon of a correlated pair leads, by recoil, to the emission of the partner nucleon, whereas in the latter mechanism proton is produced when the diquark, which is formed right after the visrtual photon-quark interaction, captures a quark from the vacuum. Unlike previous papers on the subject, particular attention is paid on the effects of the final state interaction of the hadronizing quark with the nuclear medium within an approach based upon an effective time-dependent cross section which combines the soft and hard parts of hadronization dynamics in terms of the string model and perturbative QCD, respectively. It is shown that the final state interaction of the hadronizing quark with the medium plays a relevant role both in deuteron and complex nuclei; nonetheless, kinematical regions where final state interaction effects are minimized can experimentally be selected, which would allow one to investigate the structure functions of nucleons embedded in the nuclear medium; likewise, regions where the interaction of the struck hadronizing quark with the nuclear medium is maximized can be found, which would make it possible to study non perturbative hadronization mechanisms.
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