Gaussian infrared limit of positive four-dimensional quartic coupling ($\lambda (\mu {)}^{-1}=\lambda ({\mu }_s{)}^{-1}-3\log (\mu /{\mu }_s)/(16{\pi }^2)$)

For a positive weak quartic scalar field theory coupling in four dimensions, the one-loop renormalization-group beta function is $\beta =a{\lambda }^2$, $a=3/(16{\pi }^2)>0$. Integration gives the displayed inverse-coupling law, so $\lambda$ tends to zero logarithmically as $\mu \to 0$ and grows toward the ultraviolet. The interacting Wilson-Fisher fixed point of $4-ϵ$ dimensions merges with the Gaussian fixed point at $ϵ=0$. This describes the quartic direction in the massless or critically tuned infrared regime, not disappearance of every finite-energy interaction in a massive theory. The ultraviolet Landau pole signals breakdown of that perturbative extrapolation; one loop alone does not prove a nonperturbative triviality theorem.