= Solution
Consider a conventional spin-singlet, weak-coupling, three-dimensional <superconductor>. Set $\hbar=k_B=1$ in the microscopic <action> and restore electromagnetic units below. An attraction $\lambda>0$ in an <energy> shell about the <Fermi surface> gives a Grassmann <action>
$$
S=\int d\tau\,d^3x\left[\sum_{\sigma=\uparrow,\downarrow}
\bar\psi_\sigma(\partial_\tau+\xi(-i\nabla))\psi_\sigma
-\lambda\bar\psi_\uparrow\bar\psi_\downarrow\psi_\downarrow\psi_\uparrow\right],
\qquad\xi_{\mathbf k}=\epsilon_{\mathbf k}-\mu.
$$
The attractive sign permits pairing even when the single-particle <Fermi gas> is otherwise stable. A complex <Hubbard–Stratonovich transformation> introduces a pair field $\Delta$ proportional at the saddle to $\lambda\langle\psi_\downarrow\psi_\uparrow\rangle$. The <action> is then quadratic in <fermions>. Integrating them out gives the <BCS pairing-field effective action>
$$
S_{\rm eff}[\Delta]=\int d\tau\,d^3x\,\frac{|\Delta|^2}{\lambda}
-\operatorname{Tr}\ln\mathcal G^{-1}[\Delta]+\text{constant}.
$$
Here the <trace> uses one independent <Nambu spinor> pairing block; a fully doubled particle-hole <representation> instead carries its corresponding one-half factor. At a uniform saddle the <Bogoliubov--de Gennes Hamiltonian> can be written as
$$
\mathcal H_{\rm BdG}(\mathbf k)=\begin{pmatrix}\xi_{\mathbf k}&\Delta\\
\Delta^*&-\xi_{\mathbf k}\end{pmatrix},\qquad
E_{\mathbf k}=\sqrt{\xi_{\mathbf k}^2+|\Delta|^2}.
$$
The negative branch describes the paired vacuum; excitations are <Bogoliubov quasiparticles> mixing particles and holes. Extremizing the <effective action> gives
$$
\boxed{\frac1\lambda=\int_{\rm shell}\frac{d^3k}{(2\pi)^3}
\frac{\tanh(E_{\mathbf k}/2T)}{2E_{\mathbf k}}.}
$$
The physical cutoff or a renormalized interaction is part of the microscopic model; an unregulated contact integral must not be used indiscriminately. With constant <density of states> $N(0)$ per <spin> and shell half-width $\Omega_c$, the zero-temperature weak-coupling solution is $\Delta_0\simeq2\Omega_c e^{-1/[N(0)\lambda]}$. The exponentially small scale follows from the logarithmic pairing susceptibility of states near the <Fermi surface>. For this ideal weak-coupling model, $2\Delta_0/(k_BT_c)\simeq3.53$. The microscopic theory relates the gap, pairing and thermodynamic transition.<Theory of Superconductivity by John Bardeen and others>[pp.1175–1177 and the energy-gap and finite-temperature developments]{ref}
Close to a continuous transition, for fields varying slowly compared with microscopic scales, expand the <effective action> in powers and <gradients> of $\Delta$. Its static part is the <Ginzburg-Landau theory of superconductivity>:
$$
F=\int d^3x\left[a(T)|\Delta|^2+\frac b2|\Delta|^4+
K\left|\left(-i\nabla-\frac q\hbar\mathbf A\right)\Delta\right|^2
+\frac{|\mathbf B|^2}{2\mu_0}\right],
\qquad q=-2e,\quad b,K>0.
$$
The normalization of $\Delta$ fixes the coefficients; here $a(T)$ changes sign at $T_c$. The uniform minimum has $|\Delta_0|^2=-a/b$ for $a<0$, condensation <free-energy density> $-a^2/(2b)$, and characteristic coherence-length scale $\xi=\sqrt{K/|a|}$. Above the transition the minimum is at zero. The quartic stabilization and <gradient> cost are precisely the ingredients of <Landau theory> for a complex <order parameter>. This controlled near-transition expansion supplies a bridge between microscopic <BCS theory> and macroscopic magnetic response; its coefficients and validity depend on the material and fluctuation regime.<Microscopic Derivation of the Ginzburg-Landau Equations in the Theory of Superconductivity by Lev Gor'kov>[pp.1364–1367, microscopic expansion near the transition]{ref}
Write $\Delta=(|\Delta_0|+\eta)e^{i\phi}$. The radial fluctuation has positive local restoring curvature, whereas the phase has no local potential in a neutral theory with global $U(1)$ <symmetry>. Freezing the amplitude gives
$$
F_\phi=\frac{\rho_s}{2}\int d^3x
\left(\nabla\phi-\frac q\hbar\mathbf A\right)^2,
\qquad\rho_s=2K|\Delta_0|^2.
$$
The neutral case supports a gapless collective phase mode. In the charged case the <electromagnetic field> couples to this phase rigidity. Under $\mathbf A\mapsto\mathbf A+\nabla\chi$ and $\phi\mapsto\phi+q\chi/\hbar$, the displayed combination is invariant. Locally choosing a gauge with constant phase leaves a quadratic term in $\mathbf A$. Varying the <free energy> gives the transverse London current and magnetic screening scale,
$$
\mathbf j=-\frac{q^2\rho_s}{\hbar^2}\mathbf A,
\qquad\boxed{\lambda_L^{-2}=\frac{\mu_0q^2\rho_s}{\hbar^2}.}
$$
Combining this relation with static <Maxwell equations> gives exponential field penetration, the <Meissner effect>. Single-valuedness of the pair field around a loop with negligible current gives flux in units $\Phi_0=2\pi\hbar/|q|=\pi\hbar/e$. These are gauge-invariant physical consequences of the condensate stiffness.
For the dynamics, the neutral phase and density mode is reorganized by the Coulomb interaction and the dynamical <electromagnetic field>. In a bulk three-dimensional system the long-wave longitudinal mode has a plasma gap, rather than remaining the neutral sound mode. This is the <Anderson-Higgs mechanism in a superconductor>: a phase coordinate becomes part of the electromagnetic collective response instead of surviving as an independent massless physical particle. Local <gauge transformations> are redundancies of the description, so a literal breaking of a local redundancy is not an observable. Phase coherence, current response and magnetic screening provide the physical statements.<Plasmons, Gauge Invariance, and Mass by Philip Warren Anderson>[pp.439–442, plasma oscillations and the charged versus neutral matter spectrum]{ref}
A static Landau functional alone cannot determine that full dynamical spectrum. A two-dimensional film, for example, need not have the same long-wave plasma dispersion as a bulk charged system. Nor does a positive radial curvature automatically guarantee a sharp Higgs amplitude <resonance>: coupling to quasiparticles and other modes can damp it. The chosen mean-field phase is also a thermodynamic symmetry-breaking description; an isolated finite system can be formulated with definite particle number. \b[The microscopic pairing saddle, the near-transition Landau expansion and the electromagnetic Anderson-Higgs response describe successive scales of the same superconducting state, with distinct assumptions at each step.]
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