One-loop approximation

Let us begin our treatment with the simple one-loop (or direct interaction) approximation for mass operators $\Sigma$ and $\Phi$. This approximation corresponds to taking into account just the second order (in bare vertex $V$ (2.20)) diagrams for the mass operators $\Sigma$ and $\Phi$. Two loop approximation will be considered in Appendix C. We will estimate two-loops diagrams and we will show, that some of them gives the same order contribution to $\gamma_k$ as one-loop diagrams. Therefore, one loop approximation is an uncontrolled approximation, but we believe, that it gives qualitatively correct results. Note that these diagrams include the dressed Green's function in contrast to the approximation of kinetic equation which is nothing but one-loop approximation with the bare Green's function inside. We will see later that this difference is very important in particular case of acoustic turbulence. The KE for waves with linear dispersion law forbids the angular evolution of energy because conservation laws of energy and momentum allow interaction only for waves with parallel wave vectors. In the one-loop approximation with dressed Green's function, the conservation laws $\omega({\bf k})\pm\omega({\bf k}_1) =\omega({\bf k}\pm{\bf k}_1)$ are satisfied with some accuracy [of the order of $\gamma({\bf k})$]. As a result, there exists a cone of allowed angles between ${\bf k}$ and ${\bf k}_1$ in which interactions are allowed. Therefore one has to expect some angle evolution of wave packages within this approximation. Combining (4.11) with (4.15) one has the following expression:

$$n({\bf k},\omega)=\frac{2 \gamma({\bf k})\tilde n({\bf k})} {[\omega-\omega({\bf k})]^2+ \gamma^2(k)} \ .$$ (78)