We present the systematic results for three-dimensional fragmentation functions of spin-1 hadrons defined via the quark-quark correlator. There are totally 72 such fragmentation functions, among them 18 are twist-2, 36 are twist-3 and 18 are twist-4. We also present the relationships between the twist-3 parts and those defined via the quark-gluon-quark correlator obtained from the QCD equation of motion. We show that the two particle semi-inclusive hadron production process $e^{+}{e}^{-}\to V\pi X$ at high energies is one of the best places to study the three-dimensional tensor polarization dependent fragmentation functions. We present the general kinematic analysis of this process and show that the cross section should be expressed in terms of 81 independent structure functions. After that we present parton model results for the hadronic tensor, the structure functions, and the azimuthal and spin asymmetries in terms of these gauge invariant fragmentation functions at the leading order perturbative quantum chromodynamics up to twist-3.

We present for the first time the complete twist-4 result for the semi-inclusive deeply inelastic scattering $e^{-}N\to e^{-}qX$ with polarized electron and proton beams at the tree level of perturbative QCD. The calculations have been carried out using the formalism obtained after collinear expansion where the multiple gluon scatterings are taken into account and gauge links are obtained automatically in a systematical way. The results show in particular that there are twist-4 contributions to all the eight twist-2 structure functions for $e^{-}N\to e^{-}hX$ that correspond to the eight twist-2 transverse-momentum-dependent parton distribution functions. Such higher-twist effects could be very significant and therefore have important impacts on extracting these three-dimensional parton distribution functions from the asymmetry data on $e^{-}N\to e^{-}hX$. We suggest also an approximate way to obtain a rough estimation of such higher-twist contributions.

By applying the collinear expansion to the semi-inclusive hadron production process $e^{+}+e^{-}\to h+\overline{q}(\text{jet})+X$ at high energies, we construct a theoretical framework where leading and higher twist contributions at the leading perturbative QCD can be calculated systematically. With this framework, we calculate the contributions up to twist-3 for spin-0, spin-$1/2$, and spin-1 hadrons, respectively. We present the results for the hadronic tensors, the differential cross sections, the azimuthal asymmetries, and the polarizations of the hadrons.

The spin alignment of vector mesons produced in high energy reactions is determined by the spin-dependent fragmentation function $D_{1LL}(z,{\mu }_f)$ that is shown to be independent of the polarization of the fragmenting quark. In this paper, we extract the spin-dependent fragmentation function $D_{1LL}(z,{\mu }_f)$ from data on the spin alignment of $K^{*0}$ in $e^{+}{e}^{-}$ annihilation at the large electron-position collider (LEP) in two different scenarios and apply them to make predictions in $pp$ collisions. We make detailed analysis of contributions from different subprocesses and show that the spin alignment should be quite significant also in high energy $pp$ collisions.

We make a systematic study of $\mathrm{\Lambda }$ hyperon polarizations in unpolarized lepton induced semi-inclusive reactions such as $e^{-}N\to e^{-}\mathrm{\Lambda }X$ and $e^{+}{e}^{-}\to \mathrm{\Lambda }hX$. We present the general form of cross sections in terms of structure functions obtained from a general kinematic analysis. This already shows that the produced hyperons can be polarized in three orthogonal directions, i.e., the longitudinal direction along the hyperon momentum, the normal direction of the production plane, and the transverse direction in the production plane. We present the parton model results at the leading twist and leading order in perturbative QCD using transverse momentum dependent factorization and provide the expressions for these structure functions and polarizations in terms of three-dimensional parton distribution functions and fragmentation functions. We emphasize in particular that by studying the longitudinal polarization and the transverse polarization in the production plane, we can extract the corresponding chiral-odd fragmentation functions $H_{1Lq}^{\perp \mathrm{\Lambda }}$, $H_{1Tq}^{\mathrm{\Lambda }}$, and $H_{1Tq}^{\perp \mathrm{\Lambda }}$. We also present numerical results of rough estimates utilizing available parametrizations of fragmentation functions and approximations. We discuss how to measure these polarizations and point out in particular that they can be carried out in future EIC and/or $e^{+}{e}^{-}$ annihilation experiments such as Belle.

The longitudinal polarization of a hyperon in $e^{+}{e}^{-}$ annihilation at high energies depends on the longitudinal polarization of the quark produced at the $e^{+}{e}^{-}$ annihilation vertex, whereas the spin alignment of vector mesons is independent of it. They exhibit very different energy dependences. We use the longitudinal polarization of the Lambda hyperon and the spin alignment of $K^{*}$ as representative examples to present numerical results of energy dependences and demonstrate such distinct differences. We present the results at the leading twist with perturbative QCD evolutions of fragmentation functions at the leading order.