Loop correction to the local primordial bispectrum ($B_{\mathrm{loop}}=O(f_{\mathrm{NL}}^3)$)

The exact centered quadratic Gaussian transformation has a connected three-quadratic-vertex contribution $8{\alpha }^3\int d^{3}qP(q)P(|{\mathbf{k}}_1-\mathbf{q}|)P(|{\mathbf{k}}_2+\mathbf{q}|)/(2\pi {)}^3$. It is higher order than the usual tree-level local primordial bispectrum. Infrared or ultraviolet regularization may be needed.