In this paper we study transverse polarization of $Lambda$ hyperons in single-inclusive leptonic annihilation. We show that when the transverse momentum of the $Lambda$ baryon is measured with respect to the thrust axis, a transverse momentum dependent (TMD) factorization formalism is required and the polarization is generated by the TMD polarizing fragmentation function (TMD PFF), $D_{1T}^perp$. However, when the transverse momentum of the $Lambda$ baryon is measured with respect to the momentum of the initial leptons, a collinear twist-3 formalism is required and the polarization is generated by the intrinsic collinear twist-3 fragmentation function $D_{T}$. Thus while these measurements differ from one another only by a change in the measurement axis, they probe different distribution functions. Recently, Belle measured a significant polarization in single-inclusive $Lambda$ baryon production as a function of the transverse momentum with respect to the thrust axis. However, this data can in principle be re-analyzed to measure the polarization as a function of the transverse momentum of the $Lambda$ baryon with respect to the lepton pair. This observable could be the first significant probe of the function, $D_{T}$. In this paper, we first develop a TMD formalism for $Lambda$ polarization; we then present a recent twist-3 formalism that was established to describe $Lambda$ polarization. Using the TMD formalism, we demonstrate that the $Lambda$ polarization at OPAL and Belle can be described using the twist-2 TMD factorization formalism. Finally, we make a theoretical prediction for this polarization in the collinear twist-3 formalism at Belle.
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