Charged meson masses under strong magnetic fields: Gauge invariance and Schwinger phases

We study the role of the Schwinger phase (SP) that appears in the propagator of a charged particle in the presence of a static and uniform magnetic field $\overrightarrow{B}$. We first note that this phase cannot be removed by a gauge transformation; far from this, we show that it plays an important role in the restoration of the symmetries of the system. Next, we analyze the effect of SPs in the one-loop corrections to charged pion and rho meson self-energies. To carry out this analysis we consider first a simple form for the meson-quark interactions, and then we study the ${\pi }^{+}$ and ${\rho }^{+}$ propagators within the Nambu-Jona-Lasinio model, performing a numerical analysis of the $B$ dependence of meson lowest energy states. For both ${\pi }^{+}$ and ${\rho }^{+}$ mesons, we compare the numerical results arising from the full calculation—in which SPs are included in the propagators, and meson wave functions correspond to states of definite Landau quantum number—and those obtained within alternative schemes in which SPs are neglected (or somehow eliminated) and meson states are described by plane waves of definite four-momentum.

Figure 1

Ratio $E_{{\pi }^{+}}/m_{\pi ,0}$ as a function of $eB$. Here $E_{{\pi }^{+}}$ stands for the energy of the lowest ${\pi }^{+}$ state (corresponding to the Landau level $k=0$), while $m_{\pi ,0}$ is the charged pion mass at vanishing external magnetic field.

Figure 2

Ratio $E_{{\rho }^{+}}/m_{\rho ,0}$ as a function of $eB$. Here $E_{{\rho }^{+}}$ stands for the energy of the lowest ${\rho }^{+}$ state (corresponding to the Landau level $k=-1$), while $m_{\rho ,0}$ is the charged rho mass at vanishing external magnetic field.