The finite volume effects of the Nambu–Jona–Lasinio model with the running coupling constant

Abstract With Schwinger’s proper-time formalism of the Nambu–Jona–Lasinio model, we investigate the finite volume effects with the anti-periodic boundary condition in the presence of magnetic fields. The model is solved with a running coupling constant G ( B ), which is properly fitted by the lattice average (Σ u + Σ d )/2 and the difference Σ u − Σ d . For the model in a finite or infinite volume, the magnetic fields can increase the constituent quark mass M while the temperatures can decrease it. M is close to the infinite volume limit when the box length L is appropriately large. For a sufficiently small value of L , M is close to the chiral limit. The finite volume effects behave intensely in the narrow ranges of L where the partial derivative ∂ M /∂ L is greater than zero. These narrow ranges can be reduced by stronger magnetic fields and by higher temperatures. In addition, the chiral limit can be restored by a sufficiently small finite volume and be broken by sufficiently strong magnetic fields. Finally, we discuss the thermal susceptibility and the crossover phase transition depending on the temperature at finite volume in the presence of magnetic fields.