Finite volume effects of the Nambu-Jona-Lasinio model with the running coupling constant

With the Schwinger's proper-time formalism of the Nambu-Jona-Lasinio model, we investigate the finite volume effects in the presence of magnetic fields. Since the coupling constant $G$ can be influenced by strong magnetic fields, the model is solved with a running coupling constant $G(B)$ which is fitted by the lattice average $(\Sigma_u+\Sigma_d)/2$ and difference $\Sigma_u-\Sigma_d$. The investigation mainly focuses on the constituent quark mass and the thermal susceptibility depending on the magnetic fields, the temperatures and the finite sizes. For the model in finite or infinite volume, the magnetic fields can increase the constituent quark mass while the temperatures can decrease it inversely. There is a narrow range of the box length that makes the effects of finite volume perform prominently. The model will behave close to infinite volume limit for larger box length. It is shown that the influence of finite volume can be changed by magnetic fields and temperatures. Finally, we discuss the thermal susceptibility depending on the temperature in finite volume in the presence of magnetic fields.