Holographic spin alignment for vector mesons

We develop a general framework for studying the spin alignment ${\rho }_{00}$ for flavorless vector mesons by using the gauge/gravity duality. Focusing on the dilepton production through vector meson decay, we derive the relation between production rates at each spin channel and meson's spectral function in a thermal medium, which can be evaluated by holographic models for a strongly coupled system. As examples, we study ${\rho }_{00}$ for $J/\psi$ and $\varphi$ mesons, induced by the relative motion to a thermal background, within the soft-wall model. Spectral functions for $J/\psi$ at $T=150\mathrm{MeV}$ have significant resonance peaks, indicating that $c-\overline{c}$ can exist as quasistable $J/\psi$ mesons. However, no significant peaks for the $\varphi$ meson are observed at the same temperature, implying that $\varphi$ mesons are melting down at this temperature and spectral functions are interpreted as probabilities of unstable $s-\overline{s}$ pairs. These pairs are related to free-streaming $\varphi$ mesons by an instantaneous freeze-out assumption, which is performed by sandwiching the in-medium spectral function between the meson's free-streaming propagators. Then only states with invariant masses closing to the meson's vacuum mass survive during the freeze-out stage and contribute to final results. We show that ${\rho }_{00}$ in the helicity frame for $J/\psi$ and $\varphi$ mesons have positive and negative deviations from $1/3$ at $T=150\mathrm{MeV}$, respectively, which consequently leads to different properties for their global spin alignments.