Baby-universe state dependence of the wormhole-induced gravi-axion mass gap

Abstract

Euclidean wormholes can break the axion shift symmetry, and when this symmetry is broken, a periodic potential is created for the axion, and the curvature of this potential at a minimum point can give the axion mass. In this work, we take seriously the assumption that the mere existence of a wormhole does not mean that the axion mass is uniquely determined, and that more information is needed to determine it. We first show that since for a Giddings--Strominger wormhole with symmetry \(O(4)\), the gravitational Pontryagin density is zero, the dynamical Chern--Simons interaction can affect quantum effects, including the determinant phase and fermionic selection rules, without changing the classical wormhole action. We also conclude that in a fixed \(\alpha\) sector, the axion can have a specific mass, but this mass depends on the baby-universe state and is not a universal value. Then, for the mixed case of \(\alpha\) sectors that give a distribution of conditional masses, we show that the lower edge of the averaged spectral measure is determined by the essential infimum of the conditional mass squared over the $\alpha$ distribution. Finally, we conclude that even if almost all individual sectors are massive and the mean mass squared is also non-zero, if the support of the measure accumulates at the symmetric point \(\alpha=0\), the averaged spectral measure will be gapless. To clarify the discussion, we consider a clear example of this situation, which can be a Gaussian baby-universe state, and show that in it the spectral threshold starts at zero and there is no isolated massive pole. Therefore, predicting the mass and cosmology of a gravi-axion from a wormhole without specifying the baby-universe state is not a unique, state-independent prediction.