= Solution
Define $\mathbf j_n=\bar\psi_n\boldsymbol\sigma\psi_n$ and $D=\bar\psi_\uparrow\bar\psi_\downarrow\psi_\downarrow\psi_\uparrow$. The <Pauli matrix completeness identity>, or direct evaluation of its three components, gives $\mathbf j^2=-6D$. Consequently the quartic action is $3UD=-U\mathbf j^2/2$.
The <Hubbard–Stratonovich decoupling> follows by completing a three-component square at each site and time slice:
$$
\int d^3M\,\exp\left[-\Delta\tau\left(\frac{\mathbf M^2}{2U}-\mathbf M\cdot\mathbf j\right)\right]
=C\exp\left[\frac{U\Delta\tau}{2}\mathbf j^2\right]=C e^{-3U\Delta\tau D}.
$$
The constant $C$ is absorbed into the normalized auxiliary-field measure, and $U>0$ ensures convergence of the real <Gaussian integral>. Thus the <spin-vector Hubbard–Stratonovich decoupling> yields
$$
\boxed{\mathcal S=\int_0^\beta d\tau\sum_{\mathbf k\sigma}\bar\psi_{\mathbf k\sigma}(\partial_\tau+\epsilon_{\mathbf k}-\mu)\psi_{\mathbf k\sigma}
+\int_0^\beta d\tau\sum_n\left[\frac{\mathbf M_n^2}{2U}-\bar\psi_n\mathbf M_n\cdot\boldsymbol\sigma\psi_n\right].}
$$
Integrating out $\mathbf M$ reconstructs the normal-ordered interaction exactly. The auxiliary field is periodic in imaginary time, while the fermionic fields remain antiperiodic.
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