Dynamical stabilization of runaway potentials and landscape of vacua at finite density

We study a SU(2) gauge theory with a classical complex modulus. Introducing a chemical potential for a conserved modulus hypercharge causes it to become unstable and start condensing. We show that the modulus condensation in turn generates homogeneous but anisotropic non-abelian field strength condensates. The existence of a stable vacuum at the end point of the condensation process depends on a modulus representation under the gauge group. For a modulus in the fundamental representation, the global vacuum of the theory is a state both with the rotational symmetry and the electromagnetic U(1) being spontaneously broken. In other words, the system describes an anisotropic superconducting medium. We further explore the landscape of vacua of this theory and identify metastable vacua with an abnormal number of Nambu-Goldstone bosons. The SO(2) symmetry of these vacua corresponds to locking gauge, flavor, and spin degrees of freedom. There are also metastable SO(3) rotationally invariant vacua. For a modulus in the adjoint representation, we show that the theory does not have stable vacua with homogeneous anisotropic non-abelian field strength condensates, although there are metastable vacua. The reason of that is connected with a larger number of the physical components of the modulus in the case of the adjoint representation as compared to the fundamental one.