Pair-field effective action of a superconductor ($S_{\mathrm{eff}}[\Delta ]=\int |\Delta {|}^2/g-\mathrm{Tr}\ln G^{-1}[\Delta ]$)

A pairing-channel Hubbard–Stratonovich transformation replaces an attractive fermion interaction by a complex Cooper pair field coupled to fermion pairs. A Grassmann Gaussian integral then gives a trace-log effective action, with its trace convention fixed to avoid doubling independent spin-pair blocks. The uniform stationary equation is the BCS gap equation. A small-field, slow-gradient expansion near the transition yields Ginzburg-Landau theory of superconductivity; the normalization of the phenomenological field need not equal that of the microscopic gap.