Practical Considerations for Measuring Global Spin Density Matrix Elements of Vector Mesons in Heavy-Ion Collisions

The STAR Collaboration has reported a significant $\phi$-meson global spin alignment ($\rho_{00}$) signal in Au+Au collisions at $\sqrt{s_{NN}}\leq62$ GeV by measuring the polar angle distribution of $\phi$-meson daughters with respect to the orbital angular momentum (OAM) direction of the collision system. In this paper, a new method is explored for studying vector-meson global spin alignment in heavy-ion collisions by examining the two dimensional polar and azimuthal angle distribution. This method allows simultaneous extraction of $\rho_{00}$ and off-diagonal spin density matrix elements (SDMEs), providing unique access to local quark-antiquark spin correlations and spin hydrodynamics in quark-gluon plasma (QGP). The new 2D method also removes potential biases from non-zero off-diagonal SDMEs on $\rho_{00}$ with the 1D method. A detailed procedure to correct for detector acceptance and resolution effects is also presented and validated by simulation studies.