  A totally asymptotically free QCD axion model, where all couplings flow to
zero in the infinite energy limit, was recently formulated. A very interesting
feature of this fundamental theory is the ability to predict some low-energy
observables, like the masses of the extra fermions and scalars. Here we find
and investigate a region of the parameter space where the Peccei-Quinn (PQ)
symmetry is broken quantum mechanically through the Coleman-Weinberg mechanism.
This results in an even more predictive framework: the axion sector features
only two independent parameters (the PQ symmetry breaking scale and the QCD
gauge coupling). In particular, we show that the PQ phase transition is
strongly first order and can produce gravitational waves within the reach of
future detectors. The predictivity of the model leads to a rigid dependence of
the phase transition (like its duration and the nucleation temperature) and the
gravitational wave spectrum on the PQ symmetry breaking scale and the QCD gauge
coupling.
