character of polaron site operator (\[singl\]) - four hole site operators surround the copper spin operator. It schematically has the form $$\left\langle Z_{{\bf r}_1}(t)\left( \sum_{{\bf a}_1}c_{{\bf r}_2+{\bf a}
_1}(t)Z_{{\bf r}_2}(t)\right) \left( \sum_{{\bf a}_2}Z_{{\bf r}_3}(t^{\prime
})c_{{\bf r}_3+{\bf a}_2}^{\dagger }(t^{\prime })\right) Z_{{\bf r}
_4}(t^{\prime })\right\rangle \simeq$$ $$\label{decu}\left\langle \left( \sum_{{\bf a}_1}c_{{\bf r}_2+{\bf a}_1}(t)Z_{
{\bf r}_2}(t)\right) \left( \sum_{{\bf a}_2}Z_{{\bf r}_3}(t^{\prime })c_{
{\bf r}_3+{\bf a}_2}^{\dagger }(t^{\prime })\right) \right\rangle
\left\langle Z_{{\bf r}_1}(t)Z_{{\bf r}_4}(t^{\prime })\right\rangle .$$ Let us mention that we also tested the more complex decoupling procedure, and it did not qualitatively alter the results given by approximation (\[decu\]). On the next step we project polaron operators in (\[decu\]) onto $
{\cal B}_{{\bf k}\sigma }$ : $$\label{proj}c_i(t)Z_j(t)\simeq \xi {\cal B}_{{\bf k}\sigma }(t),\xi
=\left\langle \{c_i(t)Z_j(t),{\cal B}_{{\bf k},\sigma }^{\dagger
}\}\right\rangle .$$

Taking into account, that we shall calculate only the irreducible part of Green’s function (\[Rirr\]), the averages $\left\langle 
Z_{{\bf r}_1}(t)Z_{ {\bf r}_4}(t^{\prime })\right\rangle $ are transformed to corresponding spin-spin correlation functions $\left\langle S_{{\bf r}_1}^\alpha (t)S_{ {\bf r}_4}^\alpha 
(t^{\prime })\right\rangle $. Collecting all terms, we have

$$\label{decup}\langle {\cal R}_{{\bf k},\sigma }(t){\cal R}_{{\bf k},\sigma
}^{\dagger }(t^{\prime })\rangle \simeq N^{-1}\sum_{{\bf q}}\frac{K_{{\bf k}
- {\bf q}}}{K_{{\bf k}}}\Gamma ^2({\bf k},{\bf q})\langle {\cal B}_{{\bf k}-
{\bf q},\sigma }(t){\cal B}_{{\bf k}-{\bf q},\sigma }^{\dagger }(t^{\prime
