On the stability of multi-scale models of dynamical symmetry breaking from holography

We consider two classes of backgrounds of Type IIB supergravity obtained by wrapping D5-branes on a two-cycle inside the conifold. The field theory dual exhibits confinement and, in addition, a region in which the dynamics is walking, at least in the weak sense that the running of the coupling is anomalously slow. We introduce quenched matter in the fundamental, modelled by probe D7-branes which wrap an internal three-dimensional manifold and lie at the equator of the transverse two-sphere. In the space spanned by the remaining internal angle and the radial coordinate the branes admit two embeddings. The first one is U-shaped: the branes merge at some finite value of the radius. The second one is disconnected and extends along the entire radial direction at fixed angular separation. We interpret these two configurations as corresponding to chiral-symmetry breaking and preserving phases, respectively. We present a simple diagnostic tool to examine the classical stability of the embedding, based on the concavity/convexity conditions for the relevant thermodynamic potentials. We use this criterion to show that U-shaped probes that explore the walking region are unstable, hence providing a dynamical origin for the tachyonic mode found in the literature. Whenever this occurs, the disconnected solution becomes favored energetically. We find that in one of the two classes of backgrounds the U-shaped embedding is always unstable, and thus never realised dynamically. Consequently, these models cannot be used to describe chiral-symmetry breaking. In the second category of solutions, our analysis reveals the presence of a first-order phase transition between chiral-symmetry broken and restored phases. Interestingly, this is in the same class that contains a parametrically light scalar in the spectrum of glueballs of the dual field theory.