Quartic running coupling below four dimensions ($\lambda (\mu )={\lambda }_{\ast }/[1+({\lambda }_{\ast }/\lambda ({\mu }_0)-1)(\mu /{\mu }_0{)}^{ϵ}],{\lambda }_{\ast }=16{\pi }^2ϵ/3$)

At quadratic order, $\mu d\lambda /d\mu =-ϵ\lambda +3{\lambda }^2/(16{\pi }^2)$ for $ϵ>0$. Solving the linear equation for $1/\lambda$ gives the displayed expression. If $0<\lambda ({\mu }_0)<{\lambda }_{\ast }$, the ultraviolet limit is zero and the infrared limit is ${\lambda }_{\ast }$. Initial data above ${\lambda }_{\ast }$ 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.