Taming the 3D Wilson-Fisher Fixed Point via Nonlocal Effective Action
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 $ε\approx-0.198$, the self-consistent equations yield a representative fixed point at $Δ_ϕ\approx0.97714$, $Δ_φ\approx-0.65260$, and $Δ_{ϕ^2}\approx1.04573$, corresponding to $η_ϕ\approx0.04572$ and $ν\approx0.51170$. Relative to the high-precision conformal-bootstrap benchmarks, the deviations are approximately $0.48\%$ for $Δ_ϕ$, $25.97\%$ for $Δ_{ϕ^2}$, and $18.77\%$ for $ν$, demonstrating sub-percent agreement in the fundamental-field sector while revealing substantially larger deviations in the composite and correlation-length sectors within the leading-order truncation.