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

TL;DR

This paper investigates the impact of Schwinger phases on charged meson masses in strong magnetic fields, emphasizing gauge invariance and comparing full versus simplified models within the NJL framework.

Contribution

It demonstrates the essential role of Schwinger phases in restoring symmetries and provides a detailed numerical analysis of their effect on charged meson energies in magnetic fields.

Findings

Schwinger phases are crucial for gauge invariance and symmetry restoration.

Including Schwinger phases significantly alters meson energy predictions.

Neglecting Schwinger phases leads to inaccurate meson mass calculations.

Abstract

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 $\vec{B}$. 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 selfenergies. 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 wavefunctions 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.