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In four dimensions set $b=3/(16{\pi }^2)$ and retain the one-loop beta function $dg/d\log \mu =b{g}^2$. On an interval where $g\ne 0$, differentiate its inverse:Integrating between ${\mu }_s$ and $\mu$ yieldsThe $g_s=0$ solution is identically zero and is also the continuous limit of this expression. Expanding the denominator gives $g(\mu )=g_s+b{g}_s^2\log (\mu /{\mu }_s)+\cdots$, directly checking the perturbative sign.

For positive $g_s$, the running coupling decreases towards the infrared and increases towards the ultraviolet. The one-loop expression has a Landau pole atThis is the scale where the truncated solution diverges, not a controlled perturbative prediction of a physical singularity at arbitrarily strong coupling. The formula is quantitatively reliable while the running coupling remains small enough that omitted higher-loop terms are negligible. It resums the leading logarithms of that approximate renormalization group equation; it is not the exact all-orders solution of the theory.

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.