Convergence of Ginzburg–Landau Expansions: Superconductivity in the Bardeen–Cooper–Schrieffer Theory and Chiral Symmetry Breaking in the Nambu–Jona-Lasinio 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 Mconv = √μ ² + (π 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.