= Gaussian infrared limit of positive four-dimensional quartic coupling
{c}
{title2=$\lambda(\mu)^{-1}=\lambda(\mu_s)^{-1}-3\log(\mu/\mu_s)/(16\pi^2)$}
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 $\mu\to0$ and grows toward the ultraviolet. The interacting <Wilson-Fisher fixed point> of $4-\epsilon$ dimensions merges with the <Gaussian fixed point> at $\epsilon=0$. 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.
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