Calculations of $\Phi ({\bf k},\omega )$.

In the one-loop approximation expression for $\Phi ({\bf k},\omega )$ has the form (A4). After substitution of $n({\bf k},\omega)$ in the one pole approximation (4.16) one may perform analytically integration over frequencies:

$$\Phi({\bf k},\omega)=\int \frac{d^3 k_1 d^3 k_2}{(2\pi)^3} n({\bf k}_1)n({\bf k}_2)$$ (96)

$$\times \Bigg[ \frac {\vert V({\bf k},{\bf k}_1,{\bf k}_2)\vert^2 ( \gamm... ...ega({\bf k}_2)\right]^2 + \left[ \gamma({\bf k}_1)+ \gamma({\bf k}_2)\right]^2}$$

$$+ \frac {\vert V({\bf k}_2,{\bf k}_1,{\bf k})\vert^2 ( \gamma({\bf ... ..._2)\right]^2 + \left[ \gamma({\bf k}_1)+ \gamma({\bf k}_2)\right]^2} \Bigg] \ .$$

We will analyze this expression in the next section.