Properties of Mesons in a Strong Magnetic Field

By extending the $\Phi$-derivable approach in Nambu-Jona-Lasinio model to finite magnetic field we calculate the properties of pion, $\sigma$ and $\rho$ mesons in a magnetic field at finite temperature in not only the quark-antiquark bound state scheme but also the pion-pion scattering resonant state scenario. Our calculation results manifest that the masses of $\pi^{0}$ and $\sigma$ meson can be nearly degenerate at the pseudo-critical temperature which increases with increasing the magnetic field strength, and the $\pi^{\pm}$ mass ascends suddenly at almost the same critical temperature. While the $\rho$ mesons' masses decrease with the temperature but increase with the magnetic field strength. We also check the Gell-Mann-Oakes-Renner relation and find that the relation can be violated obviously with increasing the temperature, and the effect of the magnetic field becomes pronounced around the critical temperature. With different criteria, we analyze the effect of the magnetic field on the chiral phase transition and find that the pseudo-critical temperature of the chiral phase cross, $T_{c}^{\chi}$, is always enhanced by the magnetic field. Moreover our calculations indicate that the $\rho$ mesons will get melted as the chiral symmetry has not yet been restored, but the $\sigma$ meson does not disassociate even at very high temperature. Particularly, it is the first to show that there does not exist vector meson condensate in the QCD vacuum in the pion-pion scattering scheme.