spectrum parameters calculated in Ref.[@spin] (the vertex correction $\alpha _1=1.7$ , the spin excitations condensation part $
m^2=0.0225$).

As a result we come to the integral equation for the Green’s function that always arises within the framework of SCBA $$\label{gineq}G({\bf k},\omega )=\frac 1{\omega -\Omega _{{\bf k}}-\Sigma (
{\bf k},\omega )},$$

where $$\label{gineq1}\Sigma ({\bf k},\omega )=N^{-1}\sum_{{\bf q}}M^2({\bf k},{\bf q
})\left[ (1+\nu _{{\bf q}})G({\bf k}-{\bf q},\omega -\omega _{{\bf q}})+\nu
_{{\bf q}}G({\bf k}-{\bf q},\omega +\omega _{{\bf q}})\right] ,$$ $\nu _{{\bf q}}=1/\left[ \exp (\beta \omega _{{\bf q}})-1\right] $ is the Bose function, $$\label{vert}M^2({\bf k},{\bf q})=\frac{K_{{\bf k}-{\bf q}}}{K_{{\bf k}}}
\Gamma ^2({\bf k},{\bf q})\frac{\left( -4C_{{\bf g}}\right) \left( 1-\gamma
_{{\bf q}}\right) }{\omega _{{\bf q}}}.$$

$\Gamma ({\bf k},{\bf q})$ corresponds to the bare vertex for the coupling between a spin polaron and a spin wave. It is known [@shrif] that this vertex is substantially renormalized for ${\bf q}$ close to the AFM vector $ {\bf q}_0=(\pi ,\pi )$ . This renormalization is due to the strong interaction of a polaron with the condensation part of spin excitations that must be taken into account from the very beginning. As a result, the renormalized vertex $\tilde \Gamma ({\bf k},{\bf q})$ must be proportional to [@shrif] $\left[ \left( {\bf q}-{\bf q}_0\right) 