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 $\phi \sim {\phi }^2$, allowing both exponents ${\mathrm{\Delta }}_{\phi }$ and ${\mathrm{\Delta }}_{\phi }$ 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 $\epsilon =0.5$, the self-consistent equations yield a physical fixed point at ${\mathrm{\Delta }}_{\phi }\approx 0.81479$ and ${\mathrm{\Delta }}_{{\phi }^2}\approx 1.37042$. These exponents result in a kinematic anomalous dimension ${\eta }_{\phi }\approx 0.37042$, an energy operator dimension ${\mathrm{\Delta }}_{{\phi }^2}\approx 1.37042$, and a thermal correlation length exponent $\nu \approx 0.61366$. For $\epsilon \approx -0.28996$, the self-consistent equations yield another fixed point at ${\mathrm{\Delta }}_{\phi }\approx 0.87284$, ${\mathrm{\Delta }}_{\phi }\approx -0.53564$, and ${\mathrm{\Delta }}_{{\phi }^2}\approx 1.25432$, corresponding to ${\eta }_{\phi }\approx 0.25432$ and $\nu \approx 0.57284$. 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.