Quasinormal modes of massive charged flavor branes

TL;DR

This work provides a comprehensive holographic study of quasinormal modes for vector and scalar fluctuations on D7-branes in a D3/D7 setup at finite temperature and baryon density. By combining numerical shooting/relaxation methods with Schrödinger-potential analysis, it reveals a tachyonic purely imaginary scalar mode at low T and density, a critical density above which the scalar sector stabilizes, and a density-driven desingularization of the spectrum that connects to SUSY-like normal modes at zero density. At finite momentum, the authors uncover a hydrodynamic-to-collisionless crossover in the vector and scalar channels, as well as spiraling QNM trajectories and attractor frequencies; finite density introduces a horizon barrier that suppresses dissipation and drives resonances toward higher frequencies near the SUSY spectrum. The results illuminate the phase structure and stability of the D3/D7 system, link spectral features to the underlying brane embeddings (Minkowski vs black-hole), and suggest extensions to isospin density and Goldstone modes in phases with spontaneously broken symmetry. Overall, the paper advances understanding of how temperature, density, and brane geometry shape the late-time dynamics of flavored holographic plasmas and provides tools for interpreting spectral functions in strongly coupled gauge theories.

Abstract

We present an analysis and classification of vector and scalar fluctuations in a D3/D7 brane setup at finite termperature and baryon density. The system is dual to an N=2 supersymmetric Yang-Mills theory with SU(N_c) gauge group and N_f hypermultiplets in the fundamental representation in the quenched approximation. We improve significantly over previous results on the quasinormal mode spectrum of D7 branes and stress their novel physical interpretation. Amongst our findings is a new purely imaginary scalar mode that becomes tachyonic at sufficiently low temperature and baryon density. We establish the existence of a critical density above which the scalar mode stays in the stable regime for all temperatures. In the vector sector we study the crossover from the hydrodynamic to the quasiparticle regime and find that it moves to shorter wavelengths for lower temperatures. At zero baryon density the quasinormal modes move toward distinct discrete attractor frequencies that depend on the momentum as we increase the temperature. At finite baryon density, however, the trajectories show a turning behavior such that for low temperature the quasinormal mode spectrum approaches the spectrum of the supersymmetric zero temperature normal modes. We interpret this as resolution of the singular quasinormal mode spectrum that appears at the limiting D7 brane embedding at vanishing baryon density.

Figures (40)

• Figure 1: (a) Sketch of the free energy $F$ of the flavor fields versus the quark mass over temperature ratio $m\propto M_q/T$ close to the first order phase transition. (b) Pressure versus volume of the van-der-Waals gas. The red line marks the phase transition which is obtained by the Maxwell construction.
• Figure 2: Plot of the dimensionless mass parameter $m$ vs the cosine $\chi_0=\cos\Theta_0$ of the embedding angle $\Theta(z)$ at the horizon, i.e. $\Theta_0=\Theta(z=1)$. The mass is not a single valued function of $\chi_0$ on $[0,1]$! It takes a maximum value of $m=1.31$ at $\chi_0=0.962$. The horizontal line indicates the first order phase transition at $\chi_0=0.939$.
• Figure 3: At finite charge density: The dependence of the dimensionless quark mass $m=2 M_q/\sqrt{\lambda} T$ on the horizon value $\chi_0=\lim_{\rho\to1}\chi$ of the embedding as shown in Erdmenger:2007ja.
• Figure 4: Location of the first (left) and second (right) quasinormal modes in the complex frequency plane for the vector fluctuations at vanishing momentum ($k=0$) as a function of the embedding $\chi_0$. Red color indicates small quark mass, or high temperature, while the temperature decreases towards blue colors. The horizontal (black) dash indicates the frequency at the first order phase transition where the angle is $\chi_0=0.939$. The vertical (red) dash indicates the frequecy at which the embeddings become locally unstable at $\chi_0=0.962$. The modes are followed down to embeddings with $\chi_0=0.999875$.
• Figure 5: Shoot and relax: Comparison of the shooting method result (green squares) with the relaxation method results (black circles) for the location of the first transverse vector quasinormal mode at vanishing momentum $k=0$, density $\tilde{d}=0$. Along the curves the temperature is varied. Our two methods agree very well.