Holographic vector mesons from spectral functions at finite baryon or isospin density

TL;DR

The paper investigates vector-meson spectral functions in a strongly coupled gauge theory with flavor at finite temperature and finite baryon or isospin density using gauge/gravity duality with D7-brane probes in an AdS-Schwarzschild background. It computes spectral functions from linearized flavor gauge-field fluctuations (q=0) and analyzes both baryon-dense and isospin-dense regimes. Key findings include a nonmonotonic dependence of vector-meson resonance frequencies on the quark mass to temperature ratio, a finite diffusion constant with a first-order density-driven transition, and isospin-induced splitting of resonances, qualitatively mirroring QCD expectations for meson spectra in dense media. The results illuminate how holographic models capture melting-to-bound-state transitions, density effects on transport, and isospin symmetry breaking in vector channels, with potential relevance to heavy-ion phenomenology and lattice studies.

Abstract

We consider gauge/gravity duality with flavor for the finite-temperature field theory dual of the AdS-Schwarzschild black hole background with embedded D7-brane probes. In particular, we investigate spectral functions at finite baryon density in the black hole phase. We determine the resonance frequencies corresponding to meson-mass peaks as function of the quark mass over temperature ratio. We find that these frequencies have a minimum for a finite value of the quark mass. If the quotient of quark mass and temperature is increased further, the peaks move to larger frequencies. At the same time the peaks narrow, in agreement with the formation of nearly stable vector meson states which exactly reproduce the meson mass spectrum found at zero temperature. We also calculate the diffusion coefficient, which has finite value for all quark mass to temperature ratios, and exhibits a first-order phase transition. Finally we consider an isospin chemical potential and find that the spectral functions display a resonance peak splitting, similar to the isospin meson mass splitting observed in effective QCD models.

Figures (12)

• Figure 1: The phase diagram for quarks: The quark chemical potential $\mu_q$ divided by the quark mass is plotted versus the temperature $T$ divided by $\bar{M}= 2 m_q/\sqrt{\lambda}$. Two different regions are displayed: The shaded region with vanishing baryon density and the region above the transition line with finite baryon density, in which we work here. The multivalued region at the lower tip of the transition line is not resolved here. The curves are lines of equal baryon density parametrized by $\tilde{d}=2^{5/2}n_B/(N_f\sqrt{\lambda}T^3)$. The critical density $\tilde{d}^*=0.00315$, at which the first order phase transition between two black hole phases disappears, is shown as short-dashed line close to the transition line. It virtually coincides with the short-dashed line for $\tilde{d}=0.002$.
• Figure 2: The dependence of the scaled quark mass $m=2 m_q/\sqrt{\lambda} T$ on the horizon value $\chi_0=\lim_{\rho\to1}\chi$ of the embedding.
• Figure 3: The three figures of the left column show the embedding function $\chi$ versus the radial coordinate $\rho$, the corresponding background gauge fields $\tilde{A}_0$ and the distance $L=\rho\,\chi$ between the D3 and the D7-branes at $\tilde{d}=10^{-4}/4$. $L$ is plotted versus $r$, given by $\rho^2 = r^2 + L^2$. In the right column, the same three quantities are depicted for $\tilde{d}=0.25$. The five curves in each plot correspond to parametrizations of the quark mass to temperature ratio with $\chi_0=\chi(1)=0,\,0.5,\,0.9,\,0.99$ (all solid) and $0.99998$ (dashed) from bottom up. These correspond to scaled quark masses $m=2 m_q/T\sqrt{\lambda}=0,\,0.8089,\,1.2886,\,1.3030,\,1.5943$ in the left plot and to $m=0,\,0.8342,\,1.8614,\,4.5365,\,36.4028$ on the right. The curves on the left exhibit $\mu\approx 10^{-4}$. Only the upper most curve on the left at $\chi_0=0.99998$ develops a large chemical potential of $\mu=0.107049$. In the right column curves correspond to chemical potential values $\mu=0.1241,\,0.1606,\, 0.5261,\,2.2473,\,25.3810$ from bottom up.
• Figure 4: The diffusion coefficient times temperature is plotted against the mass-scaled temperature for diverse baryon densities parametrized by $\tilde{d}=0.1$ (uppermost line in upper plot, not visible in lower plot), $0.004,$ (long-dashed), $0.00315$ (thin solid), $0.002$ (long-short-dashed), $0.000025$ (short-dashed) and $0$ (thick solid). The finite baryon density lifts the curves at small temperatures. Therefore the diffusion constant never vanishes but is only minimized near the phase transition. The lower plot zooms into the region of the transition. The phase transition vanishes above a critical value $\tilde{d}^*=0.00315$. The position of the transition shifts to smaller $T/\bar{M}$, as $\tilde{d}$ is increased towards its critical value.
• Figure 5: The finite temperature part of the spectral function $\mathfrak{R}-\mathfrak{R}_0$ (in units of $N_f N_c T^2/4$) at finite baryon density $\tilde{d}$. The maximum grows and shifts to smaller frequencies as $\chi_0$ is increased towards $\chi_0=0.7$, but then turns around to approach larger frequency values.