Tensor spin polarization induced by a curved freeze-out hypersurface

We investigate how the curvature of the freeze-out hypersurface polarizes massive vector bosons in relativistic heavy-ion collisions. Starting from the Proca Lagrangian and using the Wigner function formalism, we perform a systematic gradient expansion to obtain a covariant spin-polarization tensor expressed in terms of hydrodynamic fields and the curvature tensor of the freeze-out hypersurface. Analytic results for Bjorken and Gubser flows show that curvature anisotropy generates a nonzero tensor polarization. For ϕ meson, we estimate the curvature contribution to its spin alignment as δ Θ y y ∼ − 10 − 4 to − 10 − 3 . We also find that the curvature contribution grows as the system size decreases. A rough estimate for central O-O collisions gives a spin alignment of order − 10 − 2 , suggesting that spin-alignment measurements in such small systems may provide a clean probe of this geometric effect.