Spectral function for pions in magnetic field

This study examines the spectral functions of neutral ($π_0$) and charged ($π_{\pm}$) pions under a uniform magnetic field using the SU(2) Nambu-Jona-Lasinio (NJL) model with the Ritus method. The analysis highlights the complex interplay of magnetic field effects, thermal influences, and chiral symmetry on meson properties in extreme QCD environments. For $π_0$, whose properties are governed by the behavior of its constituent quarks, magnetic field-induced Landau levels lead to a multi-peak structure in its spectral function, reflecting stable and resonance solutions that evolve with temperature, showing shifts and critical enhancements near chiral restoration. For $π_{\pm}$, cross terms that come from the asymmetry between the constituent quarks introduce Landau cuts alongside Unitary cuts, indicating damping effects, with decay widths narrowing at higher temperatures, suggesting increased stability.

💡 Research Summary

This paper investigates how a uniform external magnetic field modifies the spectral functions of neutral (π⁰) and charged (π±) pions in hot QCD matter, employing the two‑flavor Nambu–Jona-Lasinio (NJL) model together with the Ritus method. The Ritus approach restores translational invariance broken by the magnetic field by introducing conserved “Ritus momenta” that incorporate Landau‑level quantum numbers. Quark propagators are thus expressed as a sum over Landau levels, each characterized by an effective one‑dimensional mass m_f(n)=√(m_q²+2|q_f B|n).

For the neutral pion, the polarization function Π_{π⁰}(ω) splits into a regular term J_f^{(1)} and a dispersive term J_f^{(2)}(ω). The latter contains principal‑value integrals and Heaviside step functions that generate Unitary cuts at ω=±2 m_f(n). Because each Landau level produces a threshold at ω=2 m_f(n), the inverse propagator becomes discontinuous at these points, leading to multiple solutions of the pole equation. Consequently, the spectral function ξ_{π⁰}(ω)= (1/π) Im U_{π⁰}(ω) exhibits a series of distinct peaks, each associated with a specific Landau level of the constituent quarks. As temperature rises, the Fermi‑Dirac distribution suppresses contributions from higher Landau levels, shifting the peaks to lower energies and narrowing them. Near the chiral restoration temperature, the constituent quark mass m_q drops sharply, causing the Landau‑level thresholds to cluster and producing a pronounced critical enhancement of the spectral strength—a manifestation of a Mott‑type transition where the pion dissolves into its quark constituents.

The charged pion case is more intricate because the up and down quarks carry different electric charges (Q_u=2e/3, Q_d=−e/3). This charge asymmetry generates cross‑terms j_{n,n′}(ω²) in the polarization function Π_{π±}(ω,0,−kB,0). These terms couple Landau levels of the two flavors and give rise to two distinct analytic structures: (i) the usual Unitary cuts corresponding to the creation of a quark‑antiquark pair, and (ii) Landau cuts that appear when ω matches the sum or difference of the energies of quarks occupying different Landau levels. The Landau cuts represent Landau damping—processes where one particle is absorbed from the medium while another is emitted. As a result, the charged‑pion spectral function contains a high‑energy Unitary peak together with additional low‑energy structures generated by the Landau cuts. The latter are strongly temperature dependent: at low T they are visible as small shoulders or secondary peaks, but they fade away as T increases because the thermal occupation factors suppress the relevant phase space. Simultaneously, the overall width of the charged‑pion peak narrows with temperature, indicating enhanced stability in hotter media.

Numerically, the authors employ a gauge‑invariant Pauli‑Villars regularization and fix model parameters (current quark mass m₀, coupling G, cutoff Λ) to reproduce vacuum pion properties. Their calculations reveal:

1. Multi‑peak structure for π⁰ – For moderate magnetic fields (eB ≈ 0.5–1 GeV²) and temperatures up to the chiral crossover, up to three clear peaks appear, located at ω≈2√(m_q²+2|Q_f B|n). The peaks shift downward and become sharper as temperature rises, with a dramatic enhancement near the critical temperature.

2. Complex structure for π± – In addition to the dominant Unitary peak, Landau‑cut induced sub‑peaks appear at lower ω. These sub‑peaks are most pronounced at low T and moderate B, and they disappear at higher T. The decay width extracted from the spectral function decreases with temperature, reflecting reduced damping.

3. Mott‑type transition – Both π⁰ and π± exhibit a rapid change in pole position when the constituent quark mass drops, signaling the transition from a bound meson to an unbound quark‑antiquark continuum.

4. Physical implications – The detailed spectral information feeds directly into transport coefficients (shear viscosity, electrical conductivity) and dilepton production rates in magnetized quark‑gluon plasma. The presence of Landau cuts for charged pions suggests additional channels for energy loss in the medium, which could be relevant for interpreting charge‑dependent flow observables in peripheral heavy‑ion collisions where strong magnetic fields are generated.

In summary, the work provides a comprehensive, model‑consistent description of how Landau quantization, temperature, and chiral dynamics intertwine to shape the spectral properties of neutral and charged pions. It highlights the emergence of multi‑peak structures for π⁰, the coexistence of Unitary and Landau cuts for π±, and the temperature‑driven narrowing of decay widths, thereby offering valuable theoretical insight for both heavy‑ion phenomenology and the study of magnetized dense astrophysical matter.