Separating the transverse and longitudinal modes of $\phi$, $\rho$, $K^{\ast }$ and $K_1$ mesons through their angular-dependent two-body decay modes

The mass shift a spin-1 particle moving in the nuclear medium will depend on its polarization direction. To study polarization-independent mass shifts in the medium, we explore methods to isolate each polarization direction of spin-1 mesons through the angular-dependent two-body decay modes. Specifically, we study $\phi \to K^{+}{K}^{-}$, $\rho \to \pi \pi$, $K^{\ast }\to K\pi$, $\phi \to e^{+}{e}^{-}$ and $K_1\to \rho K(K^{\ast }\pi )$ decays. While each polarization mode can be isolated through angular dependencies, accomplishing the decomposition for the $K_1$ meson further requires measuring the polarization of the $\rho (K^{\ast })$ meson. Concerning $K^{\ast }$ and $K_1$ mesons, since both particles have vacuum widths smaller than 100 MeV, they are ideal candidates for experimentally measuring chiral partners. The simultaneous observation of mass shifts of these chiral partners would provide valuable insights into the contribution of chiral symmetry breaking to the generation of hadron masses.