Emergent spin polarization from $\rho$ meson condensation in rotating hadronic matter

The behavior of vector mesons in extreme environments provides a unique probe of non-perturbative Quantum Chromodynamics. We investigate the conditions for Bose-Einstein condensation (BEC) of spin-1 $\rho$ mesons in dense rotating hadronic matter, a regime relevant to the peripheral heavy-ion collisions and the interiors of rapidly rotating neutron stars. When the $\rho$ meson chemical potential ($\mu_\rho$) approaches its effective mass ($m_\rho^*$), a phase transition to BEC occurs. We demonstrate that this transition is non-trivially influenced by global rotation, which couples to the spin of the $\rho$ mesons, leading to a macroscopic spin alignment of the condensate along the axis of rotation. This interplay between condensation and rotation results in distinct polarization patterns, which can serve as a possible signature of a BEC in experiments. The results suggest that rapidly rotating neutron stars may harbor an anisotropic, spin-polarized $\rho$-condensed phase, which could impact their equation of state.