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 ${\varphi }^4$ theory. By implementing a Hubbard-Stratonovich transformation, we decouple the quartic interaction into the primary field $\varphi$ and an auxiliary field $\phi \sim {\varphi }^2$, allowing both exponents ${\Delta }_{\varphi }$ and ${\Delta }_{\phi }$ 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 ${\Delta }_{\varphi }\approx 0.97714$, ${\Delta }_{\phi }\approx -0.65260$, and ${\Delta }_{{\varphi }^2}\approx 1.04573$, corresponding to ${\eta }_{\varphi }\approx 0.04572$ and $\nu \approx 0.51170$. Relative to the high-precision conformal-bootstrap benchmarks, the deviations are approximately $0.48\%$ for ${\Delta }_{\varphi }$, $25.97\%$ for ${\Delta }_{{\varphi }^2}$, 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.

Completely rewritten and fine-tuned