TMDs for spin-1 hadrons

Transverse-momentum-dependent parton distribution functions (TMDs) were investigated at the twists 3 and 4 for spin-1 hadrons in addition to the twist-2 ones. They were found by studying all the possible decomposition of a quark correlation function in a Lorentz-invariant way with the Hermiticity and parity invariance. The time-reversal invariance was not imposed for the TMDs due to an active role of gauge links; however, they were used for collinear parton distribution functions (PDFs) by integrating the TMDs over the transverse momentum. We found that 30 TMDs exist in the tensor-polarized spin-1 hadron at the twists 3 and 4, whereas there are 10 TMDs at the twist 2. We also showed that there are 3 collinear PDFs at the twists 3 and 4. The corresponding TMD fragmentation functions exist sat the twists 2, 3, and 4 simply by changing function names and variables. Since the time-reversal invariance is valid in the collinear PDFs, the integrals of time-reversal-odd TMDs over the transverse momentum should vanish. It leads to the sum rules $\int d^2 k_T \, h_{1LT} (x, k_T^{\, 2}) = \int d^2 k_T g_{LT} (x, k_T^{\, 2}) = \int d^2 k_T h_{LL} (x, k_T^{\, 2}) = \int d^2 k_T h_{3LL}(x, k_T^{\, 2}) =0$ on the time-reversal odd TMDs at the twists 3 and 4.