Useful relations and sum rules for PDFs and multiparton distribution functions of spin-1 hadrons

There are two types of polarizations in spin-1 hadrons, and they are vector and tensor polarizations. The latter is a unique one since it does not exist in the spin-1/2 proton. The vector-polarized PDFs are the same for both the proton and spin-1 hadrons; therefore, we mainly investigate the unique PDFs in tensor-polarized hadrons. By using the operator product expansion, the twist-3 PDF $f_{LT}$ can be expressed by two terms in the same way with $g_T$ of the proton. The first term is determined by the twist-2 PDF $f_{1LL}$ (or $b_1$) which was measured by an experiment, and the second term is expressed by twist-3 quark-gluon distributions. If we neglect the higher-twist effects, $f_{LT}$ is simply given by $f_{1LL}$, and this relation is similar to the Wandzura-Wilczek relation of $g_T$. Furthermore, a new sum rule is also obtained for $f_{2LT}=2/3 f_{LT}- f_{1LL}$, which is analogous to the Burkhardt-Cottingham sum rule, in the tensor-polarized spin-1 hadrons. In future, these interesting relations could be studied at accelerator facilities, such as the Jefferson Laboratory, the Fermilab, the nuclotron-based ion collider facility in Russia, the proposed electron-ion colliders in US and China.

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