One-loop quartic scalar beta function ($\beta (\lambda )=3{\lambda }^2/(16{\pi }^2)+O({\lambda }^3)$)

With interaction $\lambda {\varphi }^4/4!$ for one real scalar, the three one-loop four-point bubble channels give the four-dimensional renormalization-group beta function $\beta (\lambda )=3{\lambda }^2/(16{\pi }^2)+O({\lambda }^3)$. Its positive sign for $\lambda >0$ excludes asymptotic freedom and produces a perturbative Landau pole on extrapolation.

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.

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.