Landau pole

A Landau pole is a finite renormalization scale at which a perturbatively running coupling diverges. It signals that the perturbative description cannot be continued to that scale without new physics or a nonperturbative completion.

In the self-coupling-dominated approximation $d{\alpha }_H/d\ln {\mu }^2=b{\alpha }_H^2$, $b>0$, the running coupling reaches a perturbative Landau pole at the displayed scale. With the conventional Higgs potential, $m_H^2=2{\lambda }_H{v}^2$ and one can normalize ${\alpha }_H={\lambda }_H/(4\pi )$. A larger Higgs boson mass therefore decreases the logarithmic interval before the self-coupling becomes strong. Requiring perturbative validity up to a fixed high cutoff gives an upper Higgs-mass bound in this approximation. The full Standard Model beta function also contains gauge and Yukawa couplings; its sign is not universally positive. A perturbative pole is not a theorem excluding every nonperturbative ultraviolet completion.