Superconductivity of QCD vacuum in strong magnetic field

We show that in a sufficiently strong magnetic field the QCD vacuum may undergo a transition to a new phase where charged $\rho^\pm$ mesons are condensed. In this phase the vacuum behaves as an anisotropic inhomogeneous superconductor which supports superconductivity along the axis of the magnetic field. In the directions transverse to the magnetic field the superconductivity is absent. The magnetic-field-induced anisotropic superconductivity -- which is realized in the cold vacuum, i.e. at zero temperature and density -- is a consequence of a nonminimal coupling of the $\rho$ mesons to the electromagnetic field. The onset of the superconductivity of the charged $\rho^\pm$ mesons should also induce an inhomogeneous superfluidity of the neutral $\rho^0$ mesons. We also argue that due to simple kinematical reasons a strong enough magnetic field makes the lifetime of the $\rho$ mesons longer by closing the main channels of the strong decays of the $\rho$ mesons into charged pions.