  QCD axion models have been proposed as a solution to the strong CP problem.
QCD instanton effects explicitly violate the global U(1) Peccei-Quinn (PQ)
symmetry, and the axion potential is minimized at the CP conserving points.
However, it is expected that the global U(1) PQ symmetry is explicitly violated
by quantum gravity. This gravitational violation may potentially undermine the
PQ solution to the strong CP problem, which is referred to as the axion quality
problem. One source of explicit PQ violation is axionic wormholes, such as the
Giddings-Strominger model. Generally speaking, the extent to which the U(1) PQ
symmetry is violated depends on models. One straightforward extension of the
Einstein gravity involves quadratic scalar curvature, which is motivated in
cosmological contexts, such as inflation. The aim of this work is to
investigate the effects of $R^2$ corrections on axionic wormholes. In this
paper, we compute wormhole solutions and evaluate the size of explicit PQ
violation in the models with the $R^2$ term. Furthermore, We illustrate the
differences between the metric and Palatini formulations.
