A gauge theory of turbulence: Higgs mechanism for the Gribov parameter, intermittency and the anomalous dimension of the scalar field

Abstract

We formalize the gauge structure of the Navier–Stokes equation for incompressible fluids, interpreting ∇ ⋅ v = 0 as a gauge fixing analogous to the Coulomb gauge in electromagnetism. We construct the Martin–Siggia–Rose action and its full BRST symmetry, introducing BRST doublets for the Gribov parameter γ and the monodromy phase θ . Through a Higgs mechanism, γ acquires a vacuum expectation value γ ₀ , generating a mass scale for the vorticity and the ghosts; we compute the one-loop effective potential and analyze vacuum stability. Intermittency – measured by the exponents ζ _n – is described by fluctuations of the Higgs field σ around the condensate, via a Gribov-inspired log-Poisson hierarchy that satisfies exactly ζ ₃ = 1 (Kolmogorov’s four-fifths law). A one-parameter fit to the DNS exponents of Gotoh et al. favors filamentary vorticity structures, D _f = 1.10 ± 0.15, while a two-parameter fit shows no significant improvement, the ( D _f , Δ) plane being degenerate. The one-loop anomalous dimension in d = 3, Δ _σ ≈ 0.50, supports the identification D _f = 2Δ _σ within the fit uncertainty. The formalism unifies classical hydrodynamics with gauge theory and spontaneous symmetry breaking, opening the study of turbulence and intermittency to quantum field theory methods.