GeV-scale QCD Axion

In order to solve the strong CP problem, we study the possibility that the Peccei–Quinn symmetry is broken {\it below}/ the QCD scale. We find that a QCD axion can be above GeV, and may be among the observed $\eta$ resonances. It is immune to quantum gravity corrections. The only fermion that has a $U(1)$ Peccei–Quinn charge is the right-handed up quark. Flavor-changing neutral currents are surprisingly small. All accelerator and astrophysical limits can be evaded. The most significant constraint is the mass splitting between ${\pi }^{\pm }$ and ${\pi }^0$. In a UV completed model, LHC can look for a heavy quark pair $\mathcal{D}\bar{\mathcal{D}}$ followed by the decay $\mathcal{D}\to W+u$ or a single production $qu\to q\mathcal{U}$ followed by $\mathcal{U}\to uZ,uh$. There can be an $O(1)$ contribution $h\to u\bar{u}\phi$ in the measurement of $h\to gg$ or permille effects on the hadronic $Z$ width at a Higgs factory.