We demonstrate that spontaneous transverse polarization of Lambda baryon ($Lambda$) production in $e^+e^-$ annihilation can be described using the transverse momentum dependent polarizing fragmentation functions (TMD PFFs). Using a simple Gaussian model, we perform an extraction of the TMD PFFs by fitting the BELLE collaborations recent measurement of the $Lambda$ transverse polarization in back-to-back $Lambda+h$ production in $e^+ e^-$ collisions, $e^{-} + e^{+} rightarrow Lambda^{uparrow}+h+X$. We find that this simple model accurately describes the experimental data for $Lambda$ production associated with pions and kaons, and we are able to determine TMD PFFs for different quark flavors. We use these newly extracted TMD PFFs to make predictions for the transverse polarization of $Lambda$ produced in semi-inclusive deep inelastic scattering at a future electron-ion collider, and find that such a polarization is around $10%$ and should be measurable.
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