Centered quadratic Gaussian transformation ($G+\alpha (G^2-⟨G^2⟩)$)

If $G$ is a centered Gaussian random field, the transformation $G+\alpha (G^2-⟨G^2⟩)$ retains zero mean and introduces a three-point connected correlation function at first order in $\alpha$. Wick contractions give $2\alpha$ times the sum of products of two covariances. The corresponding local cosmological model uses $\alpha =3f_{\mathrm{NL}}/5$.