Spin alignment of vector mesons by second-order hydrodynamic gradients

Starting with the polarization dependent Wigner function of vector mesons, we derive an expression for the 00-component ($\rho_{00}$) of spin density matrix in terms of the second order gradients of the vector meson distribution functions. We further apply a thermal model to analyze the transverse momentum and the azimuthal angle dependence of $\rho_{00}$ for $\phi$ and $K^{*0}$ mesons resulting from distribution gradients in Au-Au collisions with $\sqrt{s_{NN}}=130$ GeV at mid-rapidity. Our results for the transverse momentum dependence indicate that the deviations of $\rho_{00}$ from $1/3$ as the signal for spin alignment are greatly enhanced at large transverse momenta and have a strong centrality dependence while analysis of the azimuthal angle ($\phi_q$) dependence suggest that such deviations have a $\cos(2\phi_q)$ structure with opposite sign for $\phi$ and $K^{*0}$. Our finding may be considered as a baseline for probing spin-alignment mechanisms beyond hydrodynamic gradients.