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Survival of charged rho condensation at high temperature and density

The charged vector $ρ$ mesons in the presence of external magnetic fields at finite temperature $T$ and chemical potential $μ$ have been investigated in the framework of the Nambu--Jona-Lasinio model. We compute the masses of charged $ρ$ mesons numerically as a function of the magnetic field for different values of temperature and chemical potential. The self-energy of the $ρ$ meson contains the quark-loop contribution, i.e. the leading order contribution in $1/N_c$ expansion. The charged $ρ$ meson mass decreases with the magnetic field and drops to zero at a critical magnetic field $eB_c$, which means that the charged vector meson condensation, i.e. the electromagnetic superconductor can be induced above the critical magnetic field. Surprisingly, it is found that the charged $ρ$ condensation can even survive at high temperature and density. At zero temperature, the critical magnetic field just increases slightly with the chemical potential, which indicates that the charged $ρ$ condensation might occur inside compact stars. At zero density, in the temperature range $0.2-0.5~ {\rm GeV}$, the critical magnetic field for charged $ρ$ condensation is in the range of $0.2-0.6~ {\rm GeV}^2$, which indicates that the high temperature electromagnetic superconductor could be created at LHC.