Residue-Enhanced Pion-Rho Mixing as the Origin of Nonmonotonic Charged Pion Mass in Magnetic Fields

Abstract

We identify the dynamical origin of the non-monotonic magnetic field dependence of the charged pion mass observed in lattice QCD. Using a near-pole effective action derived from the SU(2) Nambu--Jona-Lasinio model, we show that the lowest Landau level charged pion mixes with the longitudinally polarized charged rho meson, which shares the same quantum numbers in a magnetic background. This mixing, generated by quark-loop polarization and a gauge-invariant tree-level operator matched to the vacuum decay $ρ^\pm\rightarrowπ^\pmγ$, induces strong level repulsion. Crucially, this effect is dynamically amplified by a rapid suppression of the rho-meson wave function renormalization near the pole. As a result, the lower eigenmode exhibits a turnover as the magnetic field increases. The mechanism is analogous to singlet-triplet mixing in positronium and provides a natural explanation for the lattice results. Such effects are expected to be generic for charged mesons in magnetic fields when symmetry allowed mixing and near-pole residue suppression are present.

Figures (4)

• Figure 1: Unmixed lowest-Landau-level (LLL) energies $E_\pi(B)$ and $E_\rho(B)$ obtained from the diagonal Landau-projected kernels \ref{['diagonalkernel']}, together with the corresponding rest masses $m_\phi=\sqrt{E_\phi^2-eB}$. Both unmixed LLL energies increase monotonically with the magnetic field.
• Figure 2: Wave-function renormalizations $Z_\pi(B)$ and $Z_\rho(B)$ extracted from the slopes of the diagonal LLL Landau-projected kernels at the poles. While $Z_\pi(B)$ decreases mildly with increasing magnetic field, $Z_\rho(B)$ is rapidly suppressed in the LLL.
• Figure 3: Upper panel: Loop-induced mixing coupling $g_{\rho\pi}^\text{loop}(B)$ defined in \ref{['gloop']}, evaluated at $q_0 = E_\pi(B)$. Lower panel: Total mixing strength $h(B)$ defined in \ref{['hB']}, including both loop-induced and tree-level contributions.
• Figure 4: Lowest eigenmode of the coupled $\pi^+-\rho^+_{s_z=0}$ system in the lowest Landau level as a function of the magnetic field. The dashed blue curve shows the unmixed pion LLL energy $E_{\pi^+}$ . The red curve corresponds to the lower eigenmode $E_-$ obtained from the near-pole effective kernel \ref{['determinantK']}, while the black curve is obtained by solving $\text{det}\,\mathcal{K}(q_0)=0$ directly using the Landau-level–projected kernel \ref{['microscopickernel']}, without invoking the near-pole expansion. The vertical lines at $eB\simeq 0.3$ GeV$^2$ and $eB \simeq 0.65$ GeV$^2$ indicate the characteristic field strengths where lattice simulations observe the onset of deviation from lowest-Landau-level behavior and the location of the maximal effective mass, respectively.