Convergence of Ginzburg-Landau expansions: superconductivity in the BCS theory and chiral symmetry breaking in the NJL model

We study the convergence of the Ginzburg-Landau (GL) expansion in the context of the Bardeen-Cooper-Schrieffer (BCS) theory for superconductivity and the Nambu–Jona-Lasinio (NJL) model for chiral symmetry breaking at finite temperature T and chemical potential μ. We present derivations of the all-order formulas for the coefficients of the GL expansions in both systems under the mean-field approximation. We show that the convergence radii for the BCS gap Δ and dynamical quark mass M are given by Δconv = πT and $M_{\rm conv} = \sqrt{\mu ^2 + (\pi T)^2}$, respectively. We also discuss the implications of these results and the quantitative reliability of the GL expansion near the first-order chiral phase transition.