The transverse polarization of $\Lambda$ hyperons in $e^+e^-\to\Lambda^\uparrow h X$ processes within TMD factorization

We investigate the transverse polarization of the $\Lambda$ hyperon in the processes $e^+e^-\to\Lambda^\uparrow \pi^\pm X$ and $e^+e^-\to\Lambda^\uparrow K^\pm X$ within the framework of the transverse momentum dependent~(TMD) factorization. The transverse polarization is contributed by the convolution of the transversely polarizing fragmentation function~(PFF) $D_{1T}^\perp$ of the lambda hyperon and the unpolarized fragmentation function $D_1$ of pion/kaon. We adopt the spectator diquark model result for $D_{1T}^{\perp}$ to numerically estimate the transverse polarization in $e^+e^-\to\Lambda^\uparrow h X$ process at the kinematical region of Belle Collaboration. To implement the TMD evolution formalism of the fragmentation functions, we apply two different parametrizations on the nonperturbative Sudakov form factors associated with the fragmentation functions of the $\Lambda$, pion and kaon. It is found that our prediction on the polarization in the $\Lambda \pi^+$ production and $\bar{\Lambda} \pi^-$ is consistent with the recent Belle measurement in size and sign, while the model predictions on the polarizations in $\Lambda\pi^-$ and $\Lambda K^\pm$ productions show strong disagreement with the Belle data. The reason for the discrepancies is discussed and possible approaches to improve the calculation in the future are also discussed.