Charged and neutral vector meson under magnetic field

Understand

The vector meson $\rho$ 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 $\rho$ 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 $\rho^{0}$ with spin component $s_z=0,\,\pm1$ and the charged vector meson $\rho^{\pm}$ with $s_z=0$ also increases with magnetic field.
• However, the mass square of the charged vector meson $\rho^{+}$ ($\rho^{-}$) 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.