Gaussian order-parameter scaling index (${\beta }_G=(D-2)/4$)

The Gaussian momentum-shell scaling has thermal eigenvalue $2$ and source eigenvalue $(D+2)/2$. The formal homogeneous magnetization index is therefore $[D-(D+2)/2]/2=(D-2)/4$. A free quadratic Hamiltonian alone does not possess a stable negative-mass ordered phase, so this index is not a claim of spontaneous magnetization in an unstabilized Gaussian integral. In $2<D<4$, a quartic interaction is relevant at the Gaussian fixed point and changes the generic scalar critical behavior.