How is the Peccei-Quinn symmetry protected from gravitational breaking?
In plain words
The axion solution needs a symmetry to be almost exact, but gravity is expected to break all such global symmetries slightly. Even a tiny breaking would spoil the solution.
Precise statement
Planck-suppressed operators $O \sim \Phi^n / M_{\mathrm{Pl}}^{(n-4)}$ shift $\theta_{\mathrm{bar}}$ by $\sim f_a^n / (M_{\mathrm{Pl}}^{(n-4)} m_a^2 f_a^2)$; keeping $\mid\theta_{\mathrm{bar}}\mid < 1e-10$ for $f_a \sim 1e10\,\mathrm{GeV}$ requires n > ~ 10. An answer is a UV mechanism (gauge discrete symmetries, composite or extra-dimensional axions) that provably forbids operators up to the required n with predicted low-energy signatures.
What would settle it
A UV-complete model with the required protection and a distinct prediction, for example heavy axions or modified axion couplings, confirmed by experiment.
Related problems
- Special case of Why is the QCD $\theta$ angle smaller than $1e-10$?