Chiral-symmetry restoration in the Nambu–Jona-Lasinio model with a constant electromagnetic field

The proper-time Schwinger formalism is implemented in a derivation of the gap equation and total energy of a system of interacting fermions described by the Nambu–Jona-Lasinio model that is minimally coupled to a constant electromagnetic field. Inclusion of a Lagrange multiplier term to vary the scalar density enables the calculation of energy curves as a function of the scalar density that plays the role of an order parameter. A consistent gauge- and Lorentz-invariant regularization of the divergent quantities that occur in this theory is implemented in calculating the total energy and gap relation. Specializing to constant electric fields, we find that a chiral-symmetry-restoration phase transition can occur at a critical value of the electric field. For our choice of parameters, g${\Lambda }^2$/2${\pi }^2$=1.12 and Λ=1041 MeV, one finds the dynamically generated mass $m_{E=0\mathrm{}}^{\mathrm{*}}$=208 MeV and critical field ${\mathrm{eE}}_c$=(270 MeV${)}^2$. By contrast, a constant magnetic field is found to inhibit the phase transition by stabilizing the chirally broken vacuum state.