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Charged and neutral vector meson under magnetic field

The vector meson $ρ$ in the presence of external magnetic field has been investigated in the framework of the Nambu--Jona-Lasinio model, where mesons are constructed by infinite sum of quark-loop chains by using random phase approximation. The $ρ$ meson polarization function is calculated to the leading order of $1/N_c$ expansion. It is found that the constituent quark mass increases with magnetic field, the masses of the neutral vector meson $ρ^{0}$ with spin component $s_z=0,\,\pm1$ and the charged vector meson $ρ^{\pm}$ with $s_z=0$ also increases with magnetic field. However, the mass square of the charged vector meson $ρ^{+}$ ($ρ^{-}$) with $s_z=+1$ ($s_z=-1$) decreases linearly with magnetic field and drops to zero at the critical magnetic field $e B_c \simeq 0.2 {\rm GeV}^2$, which indicates the possible condensation of charged vector meson in the vacuum. This critical magnetic field is much lower than the value $eB_c=0.6 {\rm GeV}^2$ predicted by a point-like vector meson. We also show that if we use lowest Landau level approximation, the mass of the charged vector meson $ρ^{\pm}$ for $s_z=\pm1$ cannot drop to zero at high magnetic fields.