Polarizing fragmentation function and the Λ polarization in e + e − processes

The surprising polarization of $\mathrm{\ensuremath{\Lambda}}$'s and other hyperons measured in many unpolarized hadronic processes, $pN\ensuremath{\rightarrow}{\mathrm{\ensuremath{\Lambda}}}^{\ensuremath{\uparrow}}X$, has been a long-standing challenge for QCD phenomenology. One possible explanation was suggested, related to nonperturbative properties of the quark hadronization process, and encoded in the so-called polarizing fragmentation function (PFF). Recent Belle data have shown a nonzero $\mathrm{\ensuremath{\Lambda}}$ polarization also in unpolarized ${e}^{+}{e}^{\ensuremath{-}}$ processes, ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}\mathrm{\ensuremath{\Lambda}}X$ and ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}\mathrm{\ensuremath{\Lambda}}hX$. We consider the single inclusive case and the role of the PFFs. Adopting a simplified kinematics, we show how they can originate a polarization ${P}_{\mathrm{\ensuremath{\Lambda}}}\ensuremath{\ne}0$ and give explicit expressions for it in terms of the PFFs. Although the Belle data do not allow yet, in our approach, an extraction of the PFFs, some clear predictions are given, suggesting crucial measurements, and estimates of ${P}_{\mathrm{\ensuremath{\Lambda}}}$ are computed, in qualitative agreement with the Belle data.