Quark Anomalous Magnetic Moments and Neutral Pseudoscalar Meson Dynamics in Magnetized QCD Matter

We investigate the influence of quark anomalous magnetic moments (AMM) on the mass spectra of neutral pseudoscalar mesons ($\pi$, $K$, $\eta$, $\eta^{'}$) under external magnetic fields, finite temperatures, and quark chemical potentials using the three-flavor Nambu-Jona-Lasinio (NJL) model. By incorporating AMM at the quark level, we reveal that AMM significantly alters the magnetic field dependence of constituent quark masses, inducing first-order phase transitions for light quarks at critical fields, while strange quarks exhibit non-monotonic mass behavior. The inclusion of AMM reshapes the QCD phase diagram, suppressing chiral transition temperatures and shifting critical endpoints (CEP) toward lower $\mu$ and $T$. Notably, crossover transitions observed without AMM are replaced by first-order transitions under strong fields, aligning with lattice QCD predictions for IMC. For mesons, AMM triggers abrupt mass collapses and enhances flavor mixing, accelerating chiral restoration for $K$ and $\eta$ mesons via thresholds tied to strange quark masses. The $\eta'$ meson, as a resonance state, shows suppressed mass growth and instability at strong fields, highlighting limitations in handling non-perturbative decay widths within the NJL framework. These findings underscore AMM critical role in reconciling effective model predictions with LQCD results, particularly in explaining IMC and phase transition dynamics.