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The charged vector $\rho$ mesons in the presence of external magnetic fieldsat finite temperature $T$ and chemical potential $\mu$ have been investigatedin the framework of the Nambu--Jona-Lasinio model. We compute the masses ofcharged $\rho$ mesons numerically as a function of the magnetic field fordifferent values of temperature and chemical potential. The self-energy of the$\rho$ meson contains the quark-loop contribution, i.e. the leading ordercontribution in $1/N_c$ expansion. The charged $\rho$ meson mass decreases withthe magnetic field and drops to zero at a critical magnetic field $eB_c$, whichmeans that the charged vector meson condensation, i.e. the electromagneticsuperconductor can be induced above the critical magnetic field. Surprisingly,it is found that the charged $\rho$ condensation can even survive at hightemperature and density. At zero temperature, the critical magnetic field justincreases slightly with the chemical potential, which indicates that thecharged $\rho$ condensation might occur inside compact stars. At zero density,in the temperature range $0.2-0.5~ {\rm GeV}$, the critical magnetic field forcharged $\rho$ condensation is in the range of $0.2-0.6~ {\rm GeV}^2$, whichindicates that the high temperature electromagnetic superconductor could becreated at LHC.

Survival of charged rho condensation at high temperature and density