Gaussian critical exponent ($\alpha =2-d/2,\Delta =(d+2)/4$)

For quadratic dispersion $p^2+t$ with no interacting anomalous dimension, the thermal and uniform-source scaling eigenvalues are $y_t=2$ and $y_h=(d+2)/2$. The generic homogeneous singular free-energy density has $2-\alpha =d/2$ and gap exponent $\Delta =(d+2)/4$. These Gaussian-model values below the upper critical dimension are not the exponents of an interacting scalar critical point.