Quark spin-orbit correlations in spin-1 targets
The quark spin-orbit correlation probes the alignment of quark helicity with longitudinal kinetic orbital angular momentum inside a hadron. This correlation is defined by a QCD operator: the position moment of the asymmetric parity-odd quark energy-momentum tensor. The matrix element of this rank-two tensor decomposes into symmetric-traceless, antisymmetric, and trace parts. The symmetric-traceless part is matched to moments of axial generalized parton distributions. The QCD equations of motion relate the antisymmetric part to vector and tensor form factors and set the trace to zero. Using these relations, we derive two gauge-invariant sum rules for the spin-orbit correlation in a spin-$1$ hadron. One gives the correlation in an unpolarized target. The other gives its tensor-polarization dependence, which is absent for spin-$0$ and spin-$1/2$ targets. We estimate the unpolarized spin-orbit correlations for the $\rho$ meson and deuteron using existing lattice and phenomenological inputs, respectively.