Exploring the interplay of helicity and global-frame spin density matrix elements

A significant $\phi$-meson global spin alignment ($\rho_{00}$) signal was measured in Au+Au collisions at $\sqrt{s_{NN}}\le62$ GeV. Conventional physics mechanisms such as $s(\bar{s})$ spin polarization fail to accommodate this $\rho_{00}$ signal, motivating further investigation into its origin. Recent studies from a general framework utilizing the gauge/gravity duality predict that there could be non-zero helicity-frame spin alignment generated by the relative motion of $s\bar{s}$ pairs to the thermal background in heavy-ion collisions, leading to significant global $\rho_{00}$. In this work, we derive the expected relationships between $\phi$-meson helicity and global-frame spin density matrix elements and confirm these analytical results using Monte Carlo simulations. The effects on these relationships from finite elliptic flow and various kinematic selections are also examined. Additionally, we extract $\rho_{00}$ simultaneously with the off-diagonal elements from the 2-dimensional $\theta^{*}$ and $\beta$ angular dimensions, as opposed to the standard 1-dimensional extraction of $\rho_{00}$. These angles describe the direction of a daughter kaon's momentum in the $\phi$-meson's rest frame, where $\theta^{*}$ is the polar angle with respect to the chosen spin-quantization axis, and $\beta$ is the azimuthal angle in the perpendicular plane. This 2-dimensional method allows us to consider possible effects from finite geometric acceptance and detector efficiencies, which could couple to the spin density matrix elements and influence the extraction of these elements within the same frame. The studies presented in this work explore the relationships between helicity and global-frame spin density matrix elements and possible detector effects on the 2-dimensional extraction of spin density matrix elements.