Axion quality problem and nonminimal gravitational coupling in the Palatini formulation

In axion models, the global U(1) Peccei-Quinn (PQ) symmetry is explicitly broken by nonperturbative effects of gravity, such as axionic wormholes. The gravitational violation of the PQ symmetry due to wormholes is large enough to invalidate the PQ mechanism, which is entitled as the axion quality problem. Recently, a novel solution to this quality problem was suggested, where the nonminimal coupling of the axion field to gravity <math display="inline"><mi>ξ</mi></math> is introduced to suppress the wormhole contribution. In this work, we revisit the problem in a different but equally valid formulation of gravity, namely the Palatini formulation, where the Ricci scalar is solely determined by connection. We first find the axionic wormhole solution in the Palatini formulation, taking the full dynamical radial mode as well as the axial mode, then show that the quality problem is still resolved with the nonminimal coupling <math display="inline"><mi>ξ</mi></math>. The requested lower bound of <math display="inline"><mi>ξ</mi></math> in the Palatini formulation turns out to be slightly higher than that in the metric formulation.