Theory of Turbulence for $d \rightarrow \infty$: Four-Loop Approximation of the Renormalization Group

TL;DR

This paper advances the theoretical understanding of turbulence by calculating the four-loop approximation of the renormalization group beta function in high-dimensional spaces, providing detailed insights into fixed points and stability.

Contribution

It presents the first four-loop calculation of the beta function and the stability index in the renormalization group approach to turbulence for large dimensions.

Findings

Calculated the four-loop beta function in high dimensions.

Determined the fixed point position and stability index.

Derived the four-term epsilon expansion of the stability index.

Abstract

Within the framework of the renormalization group approach in the stochastic model of fully developed turbulence, the $\beta$-function has been calculated in the fourth order of perturbation theory for high-dimensional spaces $d \rightarrow \infty$. The position of the fixed point of the renormalization group in the fourth order of the $\varepsilon$-expansion has been determined, and the index $\omega$, which defines the infrared stability of this point, has been calculated. We demonstrate the possibility of significantly reducing the number of Feynman diagrams through mutual cancellation. The results obtained allow us to find the four terms of the $\varepsilon$-expansion of the index $\omega$: $$ \omega = 2\,\varepsilon + \frac{2}{3}\,\varepsilon^2 + \frac{10}{9}\,\varepsilon^3 + \frac{56}{27}\,\varepsilon^4. $$