Propagation of waves in a randomly inhomogeneous medium with strongly developed fluctuations. II. Infrared representation and large-distance behavior

In the first part of the present study [1], the problem was treated by the method that employs the renormalization group and the $4-\varepsilon$ expansion, and this was shown to be ineffective at the actual values of the parameters. In this, the second part of the study, the problem of the infrared divergences in the case of massless noise with correlation function $1/k^2$ is studied directly in three-dimensional space by means of an infrared perturbation theory of the type developed by Fradkin [2]. Summation of the infrared divergences leads to an integral representation for the propagator that, first, is completely free of infrared singularities on the mass shell and, second, exactly reproduces when expanded with respect to the coupling constant the series of ordinary perturbation theory. This representation is used to calculate the coordinate asymptotic behavior of the propagator at large distances, and it is shown that instead of the ordinary damping of the type $\exp(-\beta r)$ the damping $\exp(-\beta r\ln (r/r_0))$ is obtained, the parameter $r_0$ also being determined. Moreover, in the momentum representation the singularity of the propagator disappears altogether through the physical mass's becoming infinite on account of the infrared divergences. Such a mechanism is of interest in connection with the quark confinement problem in quantum chromodynamics.