Thermal properties of the $\rho$ meson from a machine-learning-improved holographic QCD model

The thermal properties of the $\rho$ meson are revisited within the soft-wall AdS/QCD framework. The dilaton background is calibrated against lattice QCD screening-mass data through a neural-network-assisted inverse procedure. Using this calibrated background, we compute the temporal vector spectral function and the lowest vector quasi-normal modes. The pole mass $m_{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{e}}(T)$ and thermal width $\mathrm{\Gamma }(T)$ are then extracted from the corresponding complex quasi-normal mode frequencies. Adopting the vanishing real part of the quasi-normal mode frequency, ${\omega }_{\mathrm{R}\mathrm{e}}(T_d)=0$, as a criterion for meson dissociation, we estimate $T_d\approx 168\mathrm{M}\mathrm{e}\mathrm{V}$. As the temperature increases, the thermal width grows, and the spectral function evolves from a narrow Breit-Wigner-like resonance peak into a transport peak, confirming the QNM prediction that the vanishing of the pole mass is accompanied by the emergence of a transport peak and making it difficult to identify a quasiparticle peak. A numerical extraction of the dispersion relation shows that the propagator can be described by a finite-temperature simple-pole form with momentum-dependent thermal width, characterized by a diffusion coefficient $D(T)$ and an effective propagation speed $u(T)$, whose temperature dependences are also extracted. This form modifies the usual relation between the screening and pole masses. These results may provide a useful reference for effective studies of the hot hadronic medium.