Renormalization group in the theory of developed turbulence. The problem of justifying the Kolmogorov hypotheses for composite operators

Stohastic theory of fully developed turbulence is considered within the framework of the field theoretic renormalization group and short-distance expansion. The problem of verification of the Kolmogorov–Obukhov theory is discussed in connection with correlation functions of composite operators. An explicit expression for the critical dimensionality of a general composite operator is obtained. The Second Kolmogorov hypothesis (indepedence of the correlators on the viscosity) is proved for an arbitrary UV-finite composite operator. It is shown that there exists an infinite number of Galilean invariant scalar operators having negative critical dimensionalities.