  We propose in this article that if the chemical potential exceeds a critical
value in dense hadronic medium, a first-order phase transition to a new state
of matter with Lorentz symmetry spontaneously broken (in addition to the
explicit breaking) takes place. As a consequence, light vector mesons get
excited as ``almost'' Goldstone bosons. Since the light vector mesons
dominantly couple to photons, the presence of these new vector mesons could
lead to an enhancement in the dilepton production from dense medium at an
invariant mass lower than the free-space vector-meson mass. We provide a
low-energy quark model which demonstrates that the above scenario is a generic
case for quark theories with a strong interaction in the vector channel. We
discuss possible relevance of this phase to the phenomenon of the enhanced
dilepton production at low invariant masses in relativistic heavy-ion
collisions.
