Transverse $\Lambda$ polarization in unpolarized $pp\rightarrow \, {\rm jet}\, \Lambda^\uparrow\, X $

In this Letter, we investigate the spontaneous transverse polarization of $\Lambda$ hyperons produced in unpolarized $pp$ collisions inside a jet, by adopting a TMD approach where transverse momentum effects are included only in the fragmentation process. We will present predictions based on the parametrizations of the $\Lambda$ polarizing fragmentation function as extracted from fits to Belle $e^+e^-$ data. These estimates will be compared against preliminary STAR data. We will then be able to explore the universality properties of the quark polarizing fragmentation function and, for the first time, the role of its gluon counterpart.