Matching between TMD and twist-3 factorizations in the transversely polarized hyperon production

We focus on the origin of the transverse polarization of $\Lambda$ hyperon produced in the proton-proton collision which has been a mystery in high energy hadron reactions over the past half century. There are two different perturbative QCD frameworks for the description of the origin with respect to the size of the transverse momentum of the produced hyperon $P_{hT}$, transverse-momentum-dependent(TMD) factorization in small $P_{hT}$ and the collinear twist-3 factorization in high $P_{hT}$. We can consider the intermediate region of $P_{hT}$ where both frameworks are valid in semi-inclusive deep inelastic scattering and it was once shown that the TMD Boer-Mulders effect and the twist-3 distribution effect from the initial state proton give consistent cross section formulas in this region. We will here show that the consistency can be also found between the TMD polarizing fragmentation and the twist-3 fragmentation contributions generated by the polarized hyperon. This result supports the idea that the two frameworks describe a unique origin of the transverse polarization in different kinematic regions.