Gauge independence of pion masses in a magnetic field within the Nambu--Jona-Lasinio model

TL;DR

This paper demonstrates that the masses of charged pions in a magnetic field, calculated within the Nambu--Jona-Lasinio model, are gauge independent by deriving correlation functions in coordinate space using linear response theory.

Contribution

It establishes gauge invariance of charged pion masses in a magnetic field within the NJL model using a coordinate-space correlation function approach.

Findings

Charged pion masses are gauge independent.

Coordinate space correlation functions match momentum-space RPA results.

Method confirms gauge invariance in the NJL model.

Abstract

We investigate the properties of neutral and charged pions in a constant background magnetic field mainly at zero temperature within the Nambu--Jona-Lasinio model. In the previous calculations, the Ritus method, involving Schwinger phases in a fixed gauge, was employed within the momentum-space random phase approximation (RPA)~[Phys. Lett. B $\textbf{782}$, 155-161 (2018)]. However, gauge invariance of the charged pion masses has not yet been examined. In this work, by adopting the linear response theory based on the imaginary-time path integral formalism, we derive the correlation functions for pions in the coordinate space, where the corresponding Schwinger phases show up automatically. At sufficiently large imaginary time $\tau$, the meson correlation function approaches an exponential form $\sim\exp(-E_{\rm G}\tau)$, where $E_{\rm G}$ is the ground-state energy of the one-meson state and hence determined as the meson mass. Furthermore, we show that the mass of the charged pions is gauge independent, i.e., independent of the choice of the vector potential for the magnetic field. Actually, we also find that the momentum-space RPA is equivalent to the imaginary-time method used here.