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Experimental data from Belle Collaboration for the transverse polarization of$\Lambda$'s measured in $e^+ e^-$ annihilation processes are used to extractthe polarizing fragmentation function (FF) within a TMD approach. We considerboth associated and inclusive $\Lambda$ production, showing a quite consistentscenario. Good separation in flavor is obtained, leading to four independentFFs. Predictions for SIDIS processes at the EIC, crucial for understandingtheir universality and evolution properties, are also presented.

$\Lambda$ polarizing fragmentation function from Belle $e^+e^-$ data