})\rangle \langle {\bf S}_{-{\bf q}}(t){\bf S}_{{\bf q}}(t^{\prime })\rangle
,$$

$$\Gamma ({\bf k},{\bf q})=\tau \Gamma _\tau ({\bf k},{\bf q})+\frac J2\Gamma
_J({\bf k},{\bf q}),$$ $$\Gamma _\tau ({\bf k},{\bf q})=4\gamma _{{\bf k}-{\bf q}}\left[ \left(
1+\gamma _{{\bf k}-{\bf q}}\right) /2K_{{\bf k}-{\bf q}}-1\right] ,$$ $$\Gamma _J({\bf k},{\bf q})=4\gamma _{{\bf q}}\left[ \left( \frac 34-C_{{\bf
g }}\right) \frac{4\gamma _{{\bf k}-{\bf q}}}{3K_{{\bf k}-{\bf q}}}-1\right]
,$$ $$\langle {\bf S}_{-{\bf q}}(t){\bf S}_{{\bf q}}(t^{\prime })\rangle =\frac 1N
\sum_{{\bf r},{\bf r}^{\prime }}{\rm e}^{i{\bf q\cdot }({\bf r}^{\prime }-
{\bf r})}\left\langle {\bf S}_{{\bf r}}(t){\bf S}_{{\bf r}^{\prime
}}(t^{\prime })\right\rangle$$

Using the spectral representation for Green’s functions we obtain the following intermediate result for the self-energy

$$\Sigma ({\bf k},\omega )=N^{-1}\sum_{{\bf q}}\frac{K_{{\bf k}-{\bf q}}}{K_{
{\bf k}}}\Gamma ^2({\bf k},{\bf q})\int_{-\infty }^\infty \frac{d\omega _1}
\pi \times$$ $$\label{fplak}\int_{-\infty }^\infty \frac{d\omega _2}\pi \frac{{\rm e}
^{\beta (\omega _1+\omega _2)}+1}{\left( {\rm e}^{\beta \omega _1}+1\right)
\left( {\rm e}^{\beta \omega _2}-1\right) }\frac{{\rm Im}[G({\bf k}-{\bf q}
,\omega _1+\imath 0^{+})]{\rm Im}[D({\bf q},\omega _2+\imath 0^{+})]}{\omega
-\left( \omega _1+\omega _2\right) +\imath 0^{+}}.$$

Here the spin excitation Green’s function [@spin; @spinb] $$\label{gfspin}D({\bf q},\omega )=\langle S_{-{\bf q}}^z|S_{{\bf q}}^z\rangle
=-\frac{8JC_{{\bf g}}}3\frac{1-\gamma _{{\bf q}}}{\omega ^2-\omega _{{\bf q}
}^2};$$ $$\omega _{{\bf q}}^2=-32J\alpha _1C_{{\bf g}}(1-\gamma _{{\bf q}})(2\Delta
+1+\gamma _{{\bf q}})$$ we neglect the influence of doped holes on copper spin dynamics and take the spin 