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We study the convergence of the Ginzburg-Landau (GL) expansion in the contextof the Bardeen-Cooper-Schrieffer (BCS) theory for superconductivity and theNambu-Jona-Lasinio (NJL) model for chiral symmetry breaking at finitetemperature $T$ and chemical potential $\mu$. We present derivations of theall-order formulas for the coefficients of the GL expansions in both systemsunder the mean-field approximation. We show that the convergence radii for theBCS gap $\Delta$ and dynamical quark mass $M$ are given by $\Delta_\text{conv}= \pi T$ and $M_\text{conv} = \sqrt{\mu^2 + (\pi T)^2}$, respectively. We alsodiscuss the implications of these results and the quantitative reliability ofthe GL expansion near the first-order chiral phase transition.

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