Renormalization group in the theory of turbulence: Three-loop approximation as $d\to\infty$

We use the renormalization group method to study the stochastic Navier–Stokes equation with a random force correlator of the form $k^{4-d-2\varepsilon}$ in a $d$-dimensional space in connection with the problem of constructing a $1/d$-expansion and going beyond the framework of the standard $\varepsilon$-expansion in the theory of fully developed hydrodynamic turbulence. We find a sharp decrease in the number of diagrams of the perturbation theory for the Green's function in the large-$d$ limit and develop a technique for calculating the diagrams analytically. We calculate the basic ingredients of the renormalization group approach (renormalization constant, $\beta$-function, fixed-point coordinates, and ultraviolet correction index $\omega$) up to the order $\varepsilon^3$ (three-loop approximation). We use the obtained results to propose hypothetical exact expressions (i.e., not in the form of $\varepsilon$-expansions) for the fixed-point coordinate and the index $\omega$.