Quantum vacuum energy and geometry of extra dimension

TL;DR

This paper investigates how the ultraviolet divergences in the energy-momentum tensor of 5D fields depend on the extra-dimensional geometry, finding that flat or AdS geometries uniquely allow for cancellation of these divergences.

Contribution

It demonstrates that only flat or AdS geometries enable cancellation of UV divergences in 5D energy-momentum tensors, suggesting these geometries are energetically favored.

Findings

UV divergences depend on extra-dimensional geometry.

Cancellation of divergences occurs only in flat or AdS geometries.

Flat or AdS geometries are energetically favored.

Abstract

We discuss the cancellation of the ultraviolet cutoff scale $\Lambda_{\rm cut}$ in the calculation of the expectation value of the five-dimensional (5D) energy-momentum tensor $\langle T_{MN}\rangle$ ($M,N=0,1,\cdots,4$). Since 5D fields feel the background geometry differently depending on their spins, the bosonic and the fermionic contributions to the $\Lambda_{\rm cut}$-dependent part $\langle T_{MN}\rangle^{\rm UV}$ may have different profiles in the extra dimension. In that case, there is no chance for them to be cancelled with each other. We consider arbitrary numbers of scalar and spinor fields with arbitrary bulk masses, calculate $\langle T_{MN}\rangle$ using the 5D propagators, and clarify the dependence of $\langle T_{MN}\rangle^{\rm UV}$ on the extra-dimensional coordinate $y$ for a general background geometry of the extra dimension. We find that if the geometry is not flat nor (a slice of) anti-de Sitter (AdS) space, it is impossible to cancel $\langle T_{MN}\rangle^{\rm UV}$ between the bosonic and the fermionic contributions. This may suggest that the flat (or AdS) space is energetically favored over the other geometries, and thus the dynamics forces the compact space to be flat (or AdS).