[Paper Review] Recent progresses on QCD phases in a strong magnetic field -- views from Nambu--Jona-Lasinio model

Gaoqing Cao|arXiv (Cornell University)|Feb 28, 2021

High-Energy Particle Collisions Research187 references4 citations

TL;DR

This review investigates quantum chromodynamics (QCD) phases under strong magnetic fields using the Nambu–Jona-Lasinio (NJL) model, focusing on chiral symmetry breaking/restoration, neutral and charged superfluidity/superconductivity, and color superconductivity. It identifies that self-consistent NJL schemes resolve key puzzles like inverse magnetic catalysis and phase competition, but cautions that the model fails for vector mesons like the ρ due to lack of confinement, especially near 2m thresholds.

ABSTRACT

In this review, we summarize recent progress on the possible phases of quantum chromodynamics (QCD) in the presence of a strong magnetic field, mainly from the views of the chiral effective Nambu--Jona-Lasinio model. Four kinds of phase transitions are explored in detail: chiral symmetry breaking and restoration, neutral pseudoscalar superfluidity, charged pion superfluidity and charged rho superconductivity. In particular, we revisit the unsolved problems of inverse magnetic catalysis effect and competition between the chiral density wave and solitonic modulation phases. It is shown that useful results can be obtained by adopting self-consistent schemes.

Motivation & Objective

• To systematically analyze QCD phases under strong magnetic fields using the Nambu–Jona-Lasinio (NJL) effective field theory.
• To resolve the long-standing puzzle of inverse magnetic catalysis in chiral symmetry restoration at finite temperature.
• To investigate the competition between inhomogeneous phases such as chiral density wave and solitonic modulation in dense matter.
• To assess the viability of neutral pseudoscalar, charged pion, and charged rho superfluidity/superconductivity in parallel EM fields and rotation.
• To evaluate the limitations of the NJL model in describing physical vector mesons like the ρ under magnetic fields.

Proposed method

• Employing the NJL model with self-consistent mean-field approximations to study phase transitions in strong magnetic fields.
• Using the Ritus formalism and proper-time representation to handle Landau level quantization in external magnetic fields.
• Applying three regularization schemes—three-momentum cutoff, four-momentum cutoff, and Pauli-Villars—to assess model dependence and stability.
• Calculating effective inverse propagators for ρ and π⁰ mesons to probe resonance behavior and identify unphysical features.
• Comparing results across different magnetic field strengths and chemical potentials to map phase diagrams.
• Using Feynman parameter integration to evaluate loop corrections in magnetic fields, particularly for vector meson self-energies.

Experimental results

Research questions

• RQ1 How does the inverse magnetic catalysis effect manifest in the NJL model at finite temperature, and what mechanisms explain its suppression?
• RQ2 What is the role of magnetic fields in stabilizing or inhibiting charged pion and charged rho superconducting phases?
• RQ3 How do inhomogeneous phases such as chiral density wave and solitonic modulation compete in the presence of a magnetic field?
• RQ4 Why do standard NJL models fail to describe physical ρ meson properties in strong magnetic fields, and what regularization schemes are viable?
• RQ5 Can the NJL model consistently describe neutral pseudoscalar superfluidity in multi-flavor systems under parallel electric and magnetic fields?

Key findings

• The four-momentum cutoff regularization scheme is the only one that yields physically consistent signs for the ρ meson propagator near its physical pole, unlike three-momentum and Pauli-Villars schemes.
• Strong magnetic fields induce unphysical dips in the ρ⁺₁ meson inverse propagator near 2m, signaling rapid and discontinuous mass shifts due to unphysical quark pair production.
• The instability near 2m is an artifact of the NJL model’s lack of confinement, not a physical resonance effect, invalidating its use for ρ and ω mesons in strong fields.
• Even with the inclusion of the Polyakov loop (PNJL model), confinement effects remain insufficient to stabilize vector meson spectra in strong magnetic fields.
• The NJL model successfully describes chiral symmetry breaking, inverse magnetic catalysis, and neutral pseudoscalar superfluidity, especially when self-consistent schemes are applied.
• Charged pion and charged rho superconducting phases are suppressed by magnetic fields, with no stable solutions found in parallel EM fields under standard NJL assumptions.