Spin alignment of $K^\ast$ induced by strange-baryon density inhomogeneity

The difference between the spin alignments of $K^\ast$ and those of $\phi$ at the low collision energies is a puzzle raised by the recent experiments. Unlike $\phi$ meson, $K^\ast$, carrying a unit strange charge, should react to strange chemical potential $\mu_S$. In this paper, we shall first convince you that $\mu_S$ is not small in a brayon-rich medium for keeping strange neutrality, and then derive the spin alignment induced by the gradient of $\mu_S$, and hence of baryon chemical potential $\mu_B$, using linear response theory, with the transport coefficients expressed, without any approximation, in terms of the $K^\ast$'s in-medium spectral properties by employing Ward-Takahashi identity. It turns out that such an effect applies mainly to the particles whose longitudinal and transverse modes diverge, and induces only the local spin alignment in a static medium. The magnitudes of these coefficients will be further estimated under the quasi-particle approximation.