The QCD Axion and Electroweak Vacuum Stability

The complex field $\Phi$ containing the QCD axion has a natural portal coupling to the Higgs doublet of the form $\lambda_{h \phi}|\Phi|^2 |H|^2$. Here we consider the possibility that $\lambda_{h \phi}$ has a natural magnitude for a dimensionless coupling, $\lambda_{h \phi} \sim 0.1-1$. This is possible if the total mass squared parameter of the Higgs in the vacuum, including quadratic divergent and the $\Phi$ vacuum expectation value contributions, is renormalized to reproduce the observed Higgs boson mass. It is then possible for the axion sector to stabilize the electroweak vacuum. We show the requirement of electroweak vacuum stability implies that the axion decay constant satisfies $f_{a} < 1.3 \times 10^{10}$ GeV.