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.