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.Comment: 17 pages, 9 figures; v2: one figure and references are added, quality of figures is improved, a comment about a superconducting electroweak phase is added, published versio

The structure of the phase diagram for strong interactions becomes richer in the presence of a magnetic background, which enters as a new control parameter for the thermodynamics. Motivated by the relevance of this physical setting for current and future high-energy heavy ion collision experiments and for the cosmological QCD transitions, we use the linear sigma model coupled to quarks and to Polyakov loops as an effective theory to investigate how the chiral and the deconfining transitions are affected, and present a general picture for the temperature-magnetic field phase diagram. We compute and discuss each contribution to the effective potential for the approximate order parameters, and uncover new phenomena such as the paramagnetically-induced breaking of global Z3 symmetry, and possible splitting of deconfinement and chiral transitions in a strong magnetic field.