Charged and neutral vector $\rho$ mesons in a magnetic field

The vector meson $\rho$ 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 $\rho$ 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 ${\rho }^0$ with spin component $s_z=0,\pm 1$, 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 $e{B}_c\simeq 0.2{\mathrm{GeV}}^2$, 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 $e{B}_c\simeq 0.6{\mathrm{GeV}}^2$ predicted by a pointlike vector meson.

Figure 1

The full propagator of the $\rho$ meson in the random phase approximation. Thick wavy lines indicate the full propagator $D_{ab}^{\mu \nu }$ of the $\rho$ meson, and thin wavy lines indicate the bare propagator $-2G_V{\delta }_{ab}$.

Figure 2

The $\rho$ meson polarization function ${\mathrm{\Pi }}^{\mu \nu ,ab}$ with one quark-loop contribution, i.e., the leading order contribution in $1/N_c$ expansion.

Figure 3

Quark constitute mass $M$ as a function of $eB$ with different Landau levels included in the numerical calculations.

Figure 4

The mass square of charged ${\rho }^{\pm }$ with spin component $s_z=\pm 1$ as a function of $eB$.

Figure 5

Masses of the charged vector meson ${\rho }^{\pm }$ with $s_z=0$ and neutral vector meson ${\rho }_0$ with $s_z=0,\pm 1$ as functions of magnetic field $eB$.

Figure 6

$M_{{\rho }^{\pm }}^2$ for $s_z=\pm 1$ as a function of $eB$ for different Landau levels, with LL representing the Landau Levels.

Figure 7

$a$ and $b$ as functions of $eB$ with Landau levels $\mathrm{LL}=0$ and 20, respectively. We have taken $M_{{\rho }^{-}}=768\mathrm{MeV}$.

Figure 8

The gap function $1-2G_V(a+b)$ at $M_{{\rho }^{-}}=0$ as a function of $eB$ with Landau levels $\mathrm{LL}=0$ and 20, respectively.

Figure 9

The gap function $1-2G_V(a-b)$ at $M_{{\rho }^{-}}=0$ by using Landau level representation for the quark propagator as a function of $eB$ with Landau levels $\mathrm{LL}=0$ and 30, respectively.

Figure 10

The mass square of charged ${\rho }^{-}$ with spin component $s_z=-1$ as a function of $eB$ by using Landau level representation for the quark propagator.