Scalar Gauss-Bonnet Wormholes and the Imaginary Distance Bound

Abstract

We study $O(4)$ symmetric Euclidean wormholes in four-dimensional gravity with a massless scalar linearly coupled to the Gauss-Bonnet invariant. At nonzero coupling, the Einstein wormhole becomes a regular complex saddle that satisfies the Kontsevich-Segal-Witten criterion until its branch ends at a finite coupling. In asymptotically flat space, the Gauss-Bonnet term modifies the scalar displacement and the on-shell action for finite wormholes, but the size dependent parts of both corrections vanish in the large throat limit. The original imaginary distance threshold therefore remains unchanged. A negative cosmological constant already lowers the Einstein large throat imaginary distance relative to its flat space value. When the Gauss-Bonnet coupling is also present, the scalar is no longer a modulus as it develops a logarithmic running and its trajectory has infinite length. Subtracting the logarithmic asymptotic behavior leaves a finite regulated displacement. Its large throat value is shifted upward from the Einstein AdS result by the Gauss-Bonnet interaction and determines the leading large charge behavior of the renormalized fixed charge action.