Spin-vector Hubbard–Stratonovich decoupling (${\mathbf{j}}^2=-6{\bar{\psi }}_{\uparrow }{\bar{\psi }}_{\downarrow }{\psi }_{\downarrow }{\psi }_{\uparrow }$)

For two Grassmann spin components, $\mathbf{j}=\bar{\psi }\mathbf{\sigma }\psi$ satisfies ${\mathbf{j}}^2=-6{\bar{\psi }}_{\uparrow }{\bar{\psi }}_{\downarrow }{\psi }_{\downarrow }{\psi }_{\uparrow }$. A real vector auxiliary field with cost ${\mathbf{M}}^2/(2U)$ and coupling $-\mathbf{M}\cdot \mathbf{j}$ therefore reproduces the repulsive $3U$ contact action. All three spin components are essential to that normalization.