Impact of Tracking Resolutions on $\phi$-Meson Spin Alignment Measurement

Measurements of global spin alignment of vector mesons in relativistic heavy-ion collisions can provide unique insights into spin-orbit interactions and vector meson dynamics in the Quark-Gluon Plasma (QGP) produced in those collisions. The global spin alignment is measured by the $00^{\rm th}$ coefficient of the spin density matrix, $\rho_{00}$, via the polar angle ($\theta^{*}$) of the decay-daughter momentum in the parent rest frame with respect to the direction of the orbital angular momentum of the collision. Such measurements are affected by the angular and momentum resolutions of the reconstructed tracks in the experiment. Such effects are nontrivial because of kinematic complications caused by the boost to the parent rest frame, and could be important given that the global spin alignment signal is weak. In this paper, we investigate the effects of experimental tracking resolutions on measurements of the $\phi$(1020) meson $\rho_{00}$. We study these effects for two methods of $\rho_{00}$ measurements, the conventional method analyzing the $\phi$-meson yield versus $\cos^2 \theta^*$ and the invariant mass ($m_{\rm inv}$) method utilizing $\langle\cos^2\theta^*\rangle$ versus $m_{\rm inv}$. Using typical resolution values from experiments, we find that the effect of track resolution on $\rho_{00}$ is small, well within typical measurement uncertainties.