Revisiting wormhole-induced global symmetry breaking

It is widely believed that global symmetries cannot exist in a consistent theory of quantum gravity. A prominent mechanism underlying this expectation is provided by gravitational instantons or Euclidean wormholes, whose contributions to the Euclidean path integral generates local symmetry-breaking operators in the effective action with couplings parametrized by the $\alpha$-parameters $\overrightarrow{\alpha}=(\alpha_1^{},\alpha_2^{},\cdots)$. The appearance of these $\alpha$-parameters is often taken as evidence that wormholes explicitly break global symmetries. In this paper, we argue that this conclusion is premature. The fate of the symmetry is determined by the ensemble average over these $\alpha$-parameters. We show that, under broad situations, this average can be sharply dominated by the symmetric point $\overrightarrow{\alpha}=0$, while contributions from other symmetry-breaking critical points $\overrightarrow{\alpha}\neq 0$ are typically suppressed by a doubly exponential factor $\exp(-e^{2S_{\rm ins}})$, where $S_{\rm ins}$ is the instanton action. In particular, standard $\mathrm{U}(1)^{}$ Peccei--Quinn models fall into this class, implying that the conventional axion quality problem does not arise in the wormhole-induced effective theory. Our analysis is formulated for general $\mathrm{U}(1)^{}$ $p$-form global symmetries.