Chiral symmetry breaking in a semi-localized magnetic field

In this work, I'm going to explore the pattern of chiral symmetry breaking and restoration in a solvable magnetic distribution within Nambu--Jona-Lasinio model. The special semi-localized static magnetic field can roughly simulate the realistic situation in peripheral heavy ion collisions, thus the study is important for the dynamical evolution of quark matter. I find that the magnetic field dependent contribution from discrete spectra usually dominates over that from continuum ones and chiral symmetry breaking is locally catalyzed by both the magnitude and scale of the magnetic field.