  We explore a new framework that furnishes a mechanism to simultaneously
address the electroweak naturalness problem and the axion high quality problem.
The framework is based on a doubly composite dynamics where the second
confinement takes place after the CFT encounters the first confinement and the
theory flows into another conformal fixed point. For a calculable example, we
present a holographic dual description of the 4D model via a warped extra
dimension model with three 3-branes. While the hierarchy problem is taken cared
of by the localization of the Higgs fields on the TeV brane just as in the
original Randall-Sundrum model, the Peccei-Quinn (PQ) symmetry is realized as a
gauge symmetry in the bulk of the extra dimension to solve the axion quality
problem. We introduce a 5D scalar field whose potential at the intermediate
brane drives spontaneous breaking of the PQ symmetry. Then, the PQ breaking
scale is given by the scale of the intermediate brane and is naturally small
compared to the Planck scale. The axion bulk profile is significantly
suppressed around the UV brane, which protects the axion from gravitational
violations of the PQ symmetry on the UV brane. Our model genuinely predicts the
existence of the Kaluza-Klein excitations of the QCD axion at around the TeV
scale and relatively light extra Higgs bosons.
