Mass spectra of charged mesons and the quenching of vector meson condensation via exact phase-space diagonalization

TL;DR

This paper develops an exact phase-space diagonalization method within the NJL model to analyze charged meson spectra in magnetic fields, revealing how magnetic catalysis and temperature affect meson stability and condensation.

Contribution

It introduces a novel algebraic diagonalization approach using phase-space techniques to study meson spectra with asymmetric charges in magnetic fields.

Findings

Zeeman splitting arises dynamically from microscopic thresholds.

Vector meson tachyonic instability is quenched by magnetic catalysis.

Meson masses decrease monotonically with temperature but remain bound.

Abstract

We investigate the dynamics and mass spectra of charged pseudoscalar ($\pi^+$) and vector ($\rho^+$) mesons in a background magnetic field at finite temperature using the two-flavor Nambu-Jona--Lasinio (NJL) model. By employing a quark propagator that isolates the Schwinger phase from its Landau level expansion, we formulate an exact non-commutative phase-space framework utilizing the Wigner-Weyl transform and the Moyal star product. This approach enables the algebraic diagonalization of the Bethe-Salpeter equations for composite states with asymmetric fractional constituent charges. For the pseudoscalar channel, we analytically verify the exact cancellation between the dynamical random phase approximation spatial sum rules and the vacuum gap equation. This identity preserves the generalized Goldstone theorem, causing the $\pi^+$ pole mass to strictly track the kinematic zero-point energy drift at order of $eB$. In the vector channel, our full phase-space evaluation reveals that the Zeeman spin-splitting emerges dynamically from microscopic threshold truncations governed by the chiral Dirac algebra. Notably, we find that the tachyonic instability of the spin-aligned $\rho^+$ state is quenched. The magnetic catalysis of the chiral condensate drives the continuum threshold ($2M$) upwards, overtaking the Zeeman attraction and preventing vector meson condensation within this mean-field framework. Furthermore, finite-temperature evaluations show a monotonic thermal suppression of the meson masses driven by Pauli blocking, yet all modes remain bound without undergoing Mott dissociation prior to chiral symmetry restoration.