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$\psi_\alpha$, $\alpha=1,2,\ldots d$, is a column vector of a $d\times d$ unitary matrix $U$; averaging over the Haar measure gives $$\int d\psi\, \psi_{\alpha} \psi^\ast_\beta \psi_{\alpha'}\psi^\ast_{\beta'}=\frac{1}{d+d^2}\left(\delta_{\alpha\beta}\delta_{\alpha'\beta'}+\delta_{\alpha\beta'}\delta_{\alpha'\beta}\right)$$