Scale and quality of Peccei-Quinn symmetry and weak gravity conjectures

The promising solution to the strong CP problem by a Peccei-Quinn (PQ) symmetry may introduce quality and hierarchy problems, which are both relevant to Planck physics. In this paper, we study whether both problems can be explained by introducing a simple hidden gauge group that satisfies a weak gravity conjecture (WGC) or its variant. As a concrete example, we point out that a weakly-coupled hidden $SU(N)$ gauge symmetry, which is broken down to $SO(N)$, can do this job in the context of a Tower/sub-Lattice WGC. Cosmology is discussed.