Spectral function for pions in magnetic field

TL;DR

This paper tackles how a uniform magnetic field modifies pion spectral functions at finite temperature using the two-flavor NJL model with the Ritus method. It derives and analyzes polarization functions for $\\pi_0$ and $\\pi_\\pm$, revealing Landau-level induced multi-peak structure for $\\pi_0$ and cross-term–driven Landau cuts for $\\pi_\\pm$, including temperature-dependent threshold effects near chiral restoration. Key findings include stable and resonance pole solutions for $\\pi_0$ with thresholds at $2 m_f^{(n)}$, and prominent Landau cuts and damping for $\\pi_\\pm$, whose widths tend to shrink at high $T$ under strong magnetic fields. These spectral functions provide inputs for transport coefficients and electromagnetic emissivity in magnetized QCD matter, with potential relevance to heavy-ion collision phenomenology.

Abstract

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.

Figures (6)

• Figure 1: Analytic structure of self-energy $\Pi(\omega,0)$ for $\pi_0$ in complex plane. Here is the Unitary cut at $n$th Landau level of quark with flavor $f$.
• Figure 2: Analytic structure of self-energy $\Pi(\omega,0)$ for $\pi_{\pm}$ in complex plane. Here is the Unitary cut (upper subfigure) and Landau cut (lower subfigure) with $n'$th Landau level of $u$ quark and $n$th Landau level of $d$ quark.
• Figure 3: (upper panels) Real part of inverse propagator $1-2G\text{Re}\Pi(\omega)$ as a function of $\omega$; (middle panels) Imaginary part of inverse propagator $2G\text{Im}\Pi(\omega)$ as a function of $\omega$; (lower panels) spectral function of $\pi_0$ as a function of $\omega$. The vertical line represents the delta function, which corresponds to the pole of propagator. The magnetic field strength is set to be $eB=30m_{\pi}^2$, with temperature $T=0$ (first column), $T=0.18\text{GeV}$ (second column), $T=0.25\text{GeV}$ (third column). The vertical dashed lines represents the mass threshold $2m_f^{(n)}$ with $n=0,1,2...$ being the Landau level for constituent quarks.
• Figure 4: Pole mass of $\pi_0$ as a function of T with magnetic field set to $eB=30m_{\pi}^2$. Two solutions of the pole equation are both shown in the figure (black and blue solid lines). Decay width $\Gamma$ for resonance solution is also shown in the figure by $m_{\pi_0}+\frac{\Gamma}{2}$ and $m_{\pi_0}-\frac{\Gamma}{2}$ (light blue lines). For comparison, the mass sum of the two constituent quark are shown in red dashed line.
• Figure 5: (upper panels) Real part of inverse propagator $1-2G\text{Re}\Pi(\omega)$ for $\pi_{\pm}$ as a function of $\omega$; (middle panels) Imaginary part of inverse propagator $2G\text{Im}\Pi(\omega)$ as a function of $\omega$; (lower panels) spectral function of $\pi_{\pm}$ as a function of $\omega$. The vertical line represents the delta function, which corresponds to the pole of propagator. The magnetic field strength is set to be $eB=30m_{\pi}^2$, with temperature $T=0$ (first column), $T=0.18\text{MeV}$ (second column), $T=0.25\text{GeV}$ (third column). For comparison, the spectral function without considering cross terms is also shown in red color.