  A major goal of axion searches is to reach inside the parameter space region
of realistic axion models. Currently, the boundaries of this region depend on
somewhat arbitrary criteria, and it would be desirable to specify them in terms
of precise phenomenological requirements. We consider hadronic axion models and
classify the representations $R_Q$ of the new heavy quarks $Q$. By requiring
that $i)$ the $Q$ are sufficiently short lived to avoid issues with long lived
strongly interacting relics, $ii)$ no Landau poles are induced below the Planck
scale, fifteen cases are selected, which define a phenomenologically preferred
axion window bounded by a maximum (minimum) value of the axion-photon coupling
about twice (four times) larger than commonly assumed. Allowing for more than
one $R_Q$, larger couplings, as well as complete axion-photon decoupling,
become possible.
