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We analyze the energy density spectrum of \acp{SIGW} using the NANOGrav15-year data set, thereby constraining the primordial non-Gaussian parameter$f_{\mathrm{NL}}$. For the first time, we calculate the seventeen missingtwo-loop diagrams proportional to $f_{\mathrm{NL}}A_{\zeta}^3$ that correspondto the two-point correlation function $\langle h^{\lambda,(3)}_{\mathbf{k}}h^{\lambda',(2)}_{\mathbf{k}'} \rangle$ for local-type primordialnon-Gaussianity. The total energy density spectrum of \acp{SIGW} can besignificantly suppressed by these two-loop diagrams. If \acp{SIGW} dominate the\acp{SGWB} observed in \ac{PTA} experiments, the parameter interval$f_{\mathrm{NL}}\in [-5,-1]$ is notably excluded based on NANOGrav 15-year dataset. After taking into account abundance of \acp{PBH} and the convergence ofthe cosmological perturbation expansion, we find that the only possibleparameter range for $f_{\mathrm{NL}}$ might be $-1\le f_{\mathrm{NL}}< 0$.

New constraints on primordial non-Gaussianity from missing two-loop contributions of scalar induced gravitational waves