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We show transverse-momentum-dependent parton distribution functions (TMDs)for spin-1 hadrons including twist-3 and 4 functions by taking thedecomposition of a quark correlation function in the Lorentz-invariant way withthe conditions of Hermiticity and parity invariance. We found 30 new TMDs inthe tensor-polarized spin-1 hadron at twists 3 and 4 in addition to 10 TMDs attwist 2. Since time-reversal-odd terms of the collinear correlation functionshould vanish after integrals over the partonic transverse momentum, weobtained new sum rules for the time-reversal-odd structure functions, $\bm{\intd^2 k_T g_{LT}  = \int d^2 k_T h_{LL} = \int d^2 k_T h_{3LL}}$$\bm{=0}$, at twists 3 and 4.We also indicated that transverse-momentum-dependent fragmentation functionsexist in tensor-polarized spin-1 hadrons. The TMDs can probe color degrees offreedom, so that they are valuable in providing unique opportunities forcreating interdisciplinary physics fields such as gluon condensate, colorAharonov-Bohm effect, and color entanglement. We also found three new collinearPDFs at twists 3 and 4, and a twist-2 relation and a sum rule were derived inanalogy to the Wandzura-Wilczek relation and the Burkhardt-Cottingham sum ruleon the structure function $\bm{g_2}$.

Transverse-momentum-dependent parton distribution functions for spin-1 hadrons