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 $\phi^{4}$ theory. By implementing a Hubbard-Stratonovich transformation, we decouple the quartic interaction into the primary field $\phi$ and an auxiliary field $\varphi \sim \phi^2$, allowing both exponents $\Delta_{\phi}$ and $\Delta_{\varphi}$ 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 $\epsilon\approx-0.198$, the self-consistent equations yield a representative fixed point at $\Delta_{\phi}\approx0.97714$, $\Delta_{\varphi}\approx-0.65260$, and $\Delta_{\phi^2}\approx1.04573$, corresponding to $\eta_{\phi}\approx0.04572$ and $\nu\approx0.51170$. Relative to the high-precision conformal-bootstrap benchmarks, the deviations are approximately $0.48\%$ for $\Delta_{\phi}$, $25.97\%$ for $\Delta_{\phi^2}$, and $18.77\%$ for $\nu$, 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.