Approximation of the propagators of virtual particles and the high-energy behavior of Feynman diagrams

A study is made of the high-energy ($s\to\infty$, $t$ fixed) behavior of Feynmaa diagrams in the model $L_{\mathrm {int}}=g\,{:}\psi^2(x)\varphi(x){:}$ when the propagators of the virtual particles are altered as follows:
$$ \frac{1}{(p+\sum_i k_i)^2-m^2+i\varepsilon}\to\frac{1}{\sum_i k_i^2+2p\sum_i k_i+i\varepsilon} $$
($p^2=m^2$; $k_i$ are the momenta of the virtual particles). It is shown that the asymptotic behavior of planar and quasiplanar diagrams of ladder type are not affected by this substitution. In the case of diagrams with crossing lines this assertion is proved in the lowest orders of perturbation theory.