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Description
We derive a Cooper-Frye-type formula for the spin alignment of neutral vector mesons, such as $\phi$ mesons, at local thermal equilibrium. We describe the local equilibrium state with a grand canonical ensemble specified by temperature, fluid velocity, and spin potential. We develop a set of Feynman rules to evaluate the Wigner function order by order in space-time gradient.
We assume that the vector mesons freeze out on a space-like hypersurface in the Minkowski space-time that is close to a hyperplane. We find that the leading order of the spin alignment is proportional to the curvature of the hypersurface and the hydrodynamic fields at first-order space-time gradient, such as the thermal shear. It is a non-dissipative mechanism that induces the spin alignment proportional to the hydrodynamic fields with the first-order space-time gradient.
