Taming the 3D Wilson-Fisher Fixed Point via Nonlocal Effective Action

Abstract

We present a Renormalization Group (RG) framework based on a nonlocal effective action ansatz to analyze the strong coupling dynamics of the three-dimensional relativistic $ϕ^{4}$ theory. By implementing a Hubbard-Stratonovich transformation, we decouple the quartic interaction into the primary field $ϕ$ and an auxiliary field $φ\sim ϕ^2$, allowing both exponents $Δ_ϕ$ and $Δ_φ$ to act as independent, unconstrained variables rather than fixed scaling dimensions. Within this nonlocal propagator framework, both the field self-energies and vertex corrections are evaluated at the one-loop order. The resulting one-loop logarithmic derivatives determine the renormalization group flows of the couplings and the scaling exponents. For $d = 3$ and $ε= 0.5$, the self-consistent equations yield a physical fixed point at $Δ_ϕ \approx 0.81479$ and $Δ_{ϕ^2} \approx 1.37042$. These exponents result in a kinematic anomalous dimension $η_ϕ \approx 0.37042$, an energy operator dimension $Δ_{ϕ^2} \approx 1.37042$, and a thermal correlation length exponent $ν\approx 0.61366$. For $ε\approx-0.28996$, the self-consistent equations yield another fixed point at $Δ_ϕ\approx0.87284$, $Δ_φ\approx-0.53564$, and $Δ_{ϕ^2}\approx1.25432$, corresponding to $η_ϕ\approx0.25432$ and $ν\approx0.57284$. Although these leading-order results show deviations from high-precision Quantum Monte Carlo (QMC) and conformal bootstrap benchmarks, they characterize the baseline scaling behavior of the Wilson-Fisher universality class.