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 $\mu$. 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 $\mathrm{\Delta }$ and dynamical quark mass $M$ are given by ${\mathrm{\Delta }}_{\text{conv}}=\pi T$ and $M_{\text{conv}}=\sqrt{{\mu }^2+(\pi T{)}^2}$, respectively. We also discuss the implications of these results and the quantitative reliability of the GL expansion near the first-order chiral phase transition.