A Comparative Study of Mass Extraction Schemes and $\pi^\pm-\rho^\pm$ Mixing

We study the origin of the non-monotonic magnetic-field dependence of the lowest charged pion excitation observed in lattice QCD. In a magnetic field, the charged pion mixes with the longitudinally polarized charged rho meson, which shares the same quantum numbers. Within the SU(2)$_f$ Nambu--Jona-Lasinio model supplemented by a gauge invariant tree-level $\pi-\rho$ mixing operator constrained by the experimental $\rho^\pm\rightarrow\pi^\pm\gamma$ decay width, we compare four mass-extraction schemes: rest-mass reconstruction, local bosonization, direct determinant solving with Landau projection, and near-pole expansion. The rest-mass scheme cannot reproduce the lattice-type turnover, while in the local derivative-expansion scheme the turnover presence but is weak which occurs at large magnetic field. By contrast, the direct determinant and near-pole schemes both retain a robust non-monotonic lowest mode. The former is most faithful to the Landau-level kinematics of the charged excitation, while the latter most clearly shows that residue suppression enhances the effective mixing after canonical normalization. Our results indicate that the lattice behavior is a genuine quasiparticle mixing effect, but one whose robustness depends crucially on how the charged-meson pole structure is extracted in a magnetic field.