Charged and neutral vector<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>ρ</mml:mi></mml:math>mesons in a magnetic field

Abstract

The vector meson $\ensuremath{\rho}$ in the presence of an external magnetic field has been investigated in the framework of the Nambu--Jona-Lasinio model, in which mesons are constructed by the infinite sum of quark-loop chains by using random phase approximation. The quark loop of $\ensuremath{\rho}$ meson polarization function is calculated to the leading order of $1/{N}_{c}$ expansion by taking the quark propagator in Ritus form as well the Landau level representation. It is found that the constituent quark mass increases with magnetic field, the masses of the neutral vector meson ${\ensuremath{\rho}}^{0}$ with spin component ${s}_{z}=0,\ifmmode\pm\else\textpm\fi{}1$, and the charged vector meson ${\ensuremath{\rho}}^{\ifmmode\pm\else\textpm\fi{}}$ with ${s}_{z}=0$ also increases with magnetic field. However, the mass square of the charged vector meson ${\ensuremath{\rho}}^{+}$ (${\ensuremath{\rho}}^{\ensuremath{-}}$) with ${s}_{z}=+1$ (${s}_{z}=\ensuremath{-}1$) decreases linearly with magnetic field and drops to zero at the critical magnetic field $e{B}_{c}\ensuremath{\simeq}0.2\text{ }\text{ }{\mathrm{GeV}}^{2}$, which indicates the possible condensation of a charged vector meson in the vacuum. The value of the critical magnetic field is the same by using the Ritus quark propagator as well the Landau level representation of the quark propagator, which is much lower than the value $e{B}_{c}\ensuremath{\simeq}0.6\text{ }\text{ }{\mathrm{GeV}}^{2}$ predicted by a pointlike vector meson.