= Quartic running coupling below four dimensions
{title2=$\lambda(\mu)=\lambda_*\big/[1+(\lambda_*/\lambda(\mu_0)-1)(\mu/\mu_0)^\epsilon],\quad\lambda_*=16\pi^2\epsilon/3$}
At quadratic order, $\mu\,d\lambda/d\mu=-\epsilon\lambda+3\lambda^2/(16\pi^2)$ for $\epsilon>0$. Solving the linear equation for $1/\lambda$ gives the displayed expression. If $0<\lambda(\mu_0)<\lambda_*$, the ultraviolet limit is zero and the infrared limit is $\lambda_*$. Initial data above $\lambda_*$ reach a <Landau pole> at a finite ultraviolet scale. The interacting value is the <Wilson-Fisher fixed point> along the quartic coupling direction; the full scalar theory also has a <thermal relevant direction at the Wilson-Fisher fixed point>, so a critical infrared limit requires mass tuning.
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