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Define ${\mathbf{j}}_n={\bar{\psi }}_n\mathbf{\sigma }{\psi }_n$ 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: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 yieldsIntegrating out $\mathbf{M}$ reconstructs the normal-ordered interaction exactly. The auxiliary field is periodic in imaginary time, while the fermionic fields remain antiperiodic.