The Minimal UV-induced Effective QCD Axion Theory
  The characteristic axion couplings could be generated via effective couplings
between the Standard Model (SM) fermions to a pseudo-Goldstone from a
high-scale $U(1)$ Peccei-Quinn (PQ) symmetry breaking. Assuming that the
UV-induced effective operators generate necessary couplings before the PQ
symmetry breaking, and any low-scale couplings to the SM are restricted to the
Yukawa sector, three minimal natural scenarios can be formulated, which
provides a connection between the QCD-axions and mediators at the GUT/string
scales. We find that the PQ symmetry breaking scale could be about $10^{15}$
GeV, higher than the classical QCD dark matter axion window but possible if the
anthropic window is considered. We also propose an experiment to probe such
scenarios. If the dark matter axion is discovered, they might suggest that we
live in an atypical Hubble volume.
