= Solution
Use the reduced attractive <reduced BCS pairing Hamiltonian> with volume V, $\xi_k=\varepsilon_k-\mu$ and attraction $\lambda>0$:
$$
\mathcal H=\sum_{k\sigma}\xi_kc_{k\sigma}^\dagger c_{k\sigma}-\frac\lambda V B^\dagger B,\qquad
B=\sum_kc_{-k\downarrow}c_{k\uparrow}.
$$
A specified pairing shell or an ultraviolet renormalization is part of the model. The trace is grand canonical. Insert <fermionic coherent states> into its time-sliced trace. The resulting <Grassmann variables> are antiperiodic, and with inverse-energy <imaginary time> $0\le\tau\le\beta$ the dimensionless action is
$$
\mathcal S=\int_0^\beta d\tau\left[\sum_{k\sigma}\bar\psi_{k\sigma}(\partial_\tau+\xi_k)\psi_{k\sigma}-\frac\lambda V\bar P P\right],\qquad
P=\sum_k\psi_{-k\downarrow}\psi_{k\uparrow}.
$$
This boundary condition is essential for <Fermi-Dirac statistics>; replacing it by a bosonic periodic trace changes the answer. A complex <Hubbard–Stratonovich transformation> uses the local identity
$$
e^{(\lambda/V)\bar P P}\propto\int d\Delta\,d\Delta^*\,
e^{-V|\Delta|^2/\lambda+\Delta\bar P+\Delta^*P},
$$
with time-slice factors understood. The pairing field renders the fermion action quadratic. For a uniform static field the independent <Nambu spinor> block $(\psi_{k\uparrow},\bar\psi_{-k\downarrow})^T$ has
$$
\mathcal G^{-1}(k,i\omega_n)=
\begin{pmatrix}-i\omega_n+\xi_k&-\Delta\\-\Delta^*&-i\omega_n-\xi_k\end{pmatrix},\qquad
\omega_n=(2n+1)\pi/\beta,\qquad E_k=\sqrt{\xi_k^2+|\Delta|^2}.
$$
Its determinant is $-(\omega_n^2+E_k^2)$. The phase-independent determinant sign cancels under consistent reference normalization. Integrating the fermions gives the <BCS pairing-field effective action>, $\mathcal S_{\rm eff}=\int_0^\beta V|\Delta|^2/\lambda\,d\tau-\operatorname{Tr}\ln\mathcal G^{-1}$, with the normal-ordering/reference term retained. The <fermionic pairing-block determinant normalization> uses one independent pairing block for every k; a fully doubled spin-particle-hole trace instead needs one half. The <subtracted grand potential of the reduced BCS model> is particularly unambiguous:
$$
\Omega(\Delta)-\Omega(0)=\frac{V|\Delta|^2}{\lambda}
-\frac1\beta\sum_{kn}\ln\frac{\omega_n^2+E_k^2}{\omega_n^2+\xi_k^2}.
$$
Performing the <Matsubara frequency> sum yields
$$
\Omega(\Delta)=\frac{V|\Delta|^2}{\lambda}
+\sum_k(\xi_k-E_k)-\frac2\beta\sum_k\ln(1+e^{-\beta E_k}),
$$
up to a common constant. At zero gap this reduces exactly to the two-spin free-gas result. The <Matsubara derivation of the BCS saddle> follows by differentiating with respect to $\Delta^*$ and using $\beta^{-1}\sum_n(\omega_n^2+E^2)^{-1}=\tanh(\beta E/2)/(2E)$ gives
$$
\boxed{1=\frac\lambda V\sum_k\frac{\tanh(\beta E_k/2)}{2E_k},\qquad
N=\sum_k\left[1-\frac{\xi_k}{E_k}\tanh\frac{\beta E_k}2\right].}
$$
The second equation fixes the <chemical potential> at specified number. This stationary action is BCS <mean-field theory>, not an exact integration of pairing-field fluctuations. With constant single-spin <density of states> $\nu_F$ per volume and a symmetric energy cutoff $\Lambda$, weak coupling gives $\Delta(0)\simeq2\Lambda e^{-1/(\lambda\nu_F)}$ and $k_BT_c\simeq(2e^{\gamma_E}/\pi)\Lambda e^{-1/(\lambda\nu_F)}$. The zero-temperature condensation grand-potential density is $-\nu_F\Delta^2/2$ to leading weak-coupling order; using a spin-summed density would change these displayed factors. The <BCS coherence factors> are $u_k^2=(1+\xi_k/E_k)/2$ and $v_k^2=(1-\xi_k/E_k)/2$, and the associated <BCS ground state> is the normalized product of empty and occupied pair states. <Lecture Notes for Quantum Matter by Steven H. Simon>[Chapter11, BCS variational state, coherence factors, gap and thermal occupations]{ref}
Allowing a slowly varying complex pairing field generates phase stiffness and, near the transition, the <Ginzburg-Landau theory of superconductivity>. A neutral paired system has a gapless phase sound mode; in a charged bulk superconductor electromagnetic coupling produces the <Anderson-Higgs mechanism in a superconductor> and the <Meissner effect>. These collective fluctuations are distinct from the gapped fermionic excitations. A microscopic dynamical calculation is needed to fix collective-mode dispersion and damping; the static saddle alone does not determine them. <Physics 215C: Quantum Field Theory, Part 3 by John McGreevy>[§§3.1–3.2, coherent-state construction; §2.3, pair-field integration and charged phase dynamics]{ref}
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