Hyperlogarithms in the theory of turbulence of infinite dimension

TL;DR

This paper extends the application of hyperlogarithm techniques to a high-dimensional turbulence model, enabling analytical calculations of critical exponents in a novel context of stochastic dynamics.

Contribution

It introduces a new adaptation of hyperlogarithms for critical dynamics in turbulence, including a renormalization scheme and perturbative calculations up to fourth order.

Findings

Analytical renormalization group functions derived up to fourth order.

Explicit epsilon-expansion of the critical exponent aa obtained.

Application of hyperlogarithms to stochastic turbulence models demonstrated.

Abstract

Parametric integration with hyperlogarithms so far has been successfully used in problems of high energy physics (HEP) and critical statics. In this work, for the first time, it is applied to a problem of critical dynamics, namely, a stochastic model of developed turbulence in high-dimensional spaces, which has a propagator that is non-standard with respect to the HEP: $(-i \omega + \nu k^2)^{-1}$. Adaptation of the hyperlogarithm method is carried out by choosing a proper renormalization scheme and considering an effective dimension of the space. Analytical calculation of the renormalization group functions is performed up to the fourth order of the perturbation theory, $\varepsilon$-expansion of the critical exponent $\omega$ responsible for the infrared stability of the fixed point is obtained.