= Spin-vector Hubbard–Stratonovich decoupling
{title2=$\mathbf j^2=-6\bar\psi_\uparrow\bar\psi_\downarrow\psi_\downarrow\psi_\uparrow$}
For two Grassmann spin components, $\mathbf j=\bar\psi\boldsymbol\sigma\psi$ 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.
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