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Effects of Quark Spin Flip on the Collins Fragmentation Function in a Toy Model

The recent extension of the NJL-jet model to hadronization of transversely polarized quarks allowed the study of the Collins fragmentation function. Both favored and unfavored Collins fragmentation functions were generated, the latter purely by multiple hadron emissions, with 1/2 moments of opposite sign in the region of the light-cone momentum fraction $z$ accessible in current experiments. Hints of such behavior has been seen in the measurements in several experiments. Also, in the transverse momentum dependent (TMD) hadron emission probabilities, modulations of up to fourth order in sine of the polar angle were observed, while the Collins effect describes just the linear modulations. A crucial part of the extended model was the calculation of the quark spin flip probability after each hadron emission in the jet. Here we study the effects of this probability on the resulting unfavored and favored Collins functions by setting it as a constant and use a toy model for the elementary single hadron emission probabilities. The results of the Monte Carlo simulations showed that preferential quark spin flip in the elementary hadron emission is needed to generate the favored and unfavored Collins functions with opposite sign 1/2 moments. For the TMD hadron emission modulations, we showed that the model quark spin flip probabilities are a partial source of the higher rode modulations, while the other source is the Collins modulation of the remnant quark from the hadron emission recoil.