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 $η$ 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 $π^\pm$ and $π^0$. In a UV completed model, LHC can look for a heavy quark pair ${\cal D} \overline{\cal D}$ followed by the decay ${\cal D} \rightarrow W+u$ or a single production $q u \rightarrow q\, {\cal U}$ followed by ${\cal U} \rightarrow u Z, uh$. There can be an $O(1)$ contribution $h \rightarrow u\bar{u}\varphi$ in the measurement of $h \rightarrow gg$ or permille effects on the hadronic $Z$ width at a Higgs factory.