Influence of hydrodynamic fluctuations on the phase transition in the $E$ and $F$ models of critical dynamics

We use the renormalization group method to study the $E$ model of critical dynamics in the presence of velocity fluctuations arising in accordance with the stochastic Navier–Stokes equation. Using the Martin–Siggia–Rose theorem, we obtain a field theory model that allows a perturbative renormalization group analysis. By direct power counting and an analysis of ultraviolet divergences, we show that the model is multiplicatively renormalizable, and we use a two-parameter expansion in $\epsilon$ and $\delta$ to calculate the renormalization constants. Here, $\epsilon$ is the deviation from the critical dimension four, and $\delta$ is the deviation from the Kolmogorov regime. We present the results of the one-loop approximation and part of the fixed-point structure. We briefly discuss the possible effect of velocity fluctuations on the large-scale behavior of the model.